{
  "Title": "Special values of the $L$-functions of level one cusp forms",
  "Definition": "For $f_{k,i}(q)=\\sum_{n\\geq1}a_nq^n$, the $i$th normalised Hecke eigenform in $S_k(\\mathrm{SL}_2(\\mathbb{Z}))$ CITE{WikiCusp} ordered by increasing embedded $a_2$, this table gives the analytically continued values $L(f_{k,i},s)$ CITE{WikiL} at the critical integers $1\\leq s\\leq k-1$.",
  "Parameters": {
    "k": {
      "type": "Z",
      "title": "weight",
      "display": "$k$",
      "constraints": "$k$ is even and $\\dim S_k(\\mathrm{SL}_2(\\mathbb{Z}))>0$"
    },
    "i": {
      "type": "Z",
      "title": "position by increasing $a_2$",
      "display": "$i$",
      "constraints": "$1\\leq i\\leq\\dim S_k(\\mathrm{SL}_2(\\mathbb{Z}))$, with positions counted after sorting the eigenforms by increasing $a_2$"
    },
    "s": {
      "type": "Z",
      "title": "argument of $L$",
      "display": "$s$",
      "constraints": "$s$ is an integer with $1\\leq s\\leq k-1$"
    }
  },
  "Comments": {
    "comment-normalisation": "The table uses the arithmetic normalisation with $a_1=1$. In this normalisation the functional equation relates $s$ and $k-s$, the centre is $s=k/2$, and the critical integers are $s=1,\\ldots,k-1$. The analytic normalisation with centre $1/2$ is $L^{\\mathrm{an}}(f,s)=L(f,s+(k-1)/2)$. The $k=12$ rows are values of the Ramanujan tau $L$-function in this normalisation.",
    "comment-ordering": "The index $i$ counts the eigenforms of weight $k$ in increasing order of the embedded $T_2$-eigenvalue $a_2$; this is not necessarily the order Sage or PARI lists them in. Each entry gives its own $a_2$ and links to $\\chi_{k,2}$ in HREF{Hecke_polynomials_of_level_one_cusp_forms}[the Hecke polynomial table], whose roots are the $a_2$ values in that weight.",
    "comment-central-zero": "The sign of the functional equation is $(-1)^{k/2}$, so for $k\\equiv2\\pmod4$ the central value $L(f,k/2)$ vanishes. Those forced zeros are omitted from the entries."
  },
  "Formulas": {
    "formula-completed": "The completed $L$-function is $\\Lambda(f,s)=(2\\pi)^{-s}\\Gamma(s)L(f,s)$, and for level one it satisfies $\\Lambda(f,s)=(-1)^{k/2}\\Lambda(f,k-s)$.",
    "formula-euler-product": "For $\\operatorname{Re}s>(k+1)/2$, $L(f,s)=\\prod_p(1-a_pp^{-s}+p^{k-1-2s})^{-1}$."
  },
  "Programs": {
    "program-pari": {
      "language": "PARI/GP",
      "code": "default(realbitprecision, 330);\nmf = mfinit([1, 12], 0);\nforms = vecsort(concat(lfunmf(mf)), f -> lfunan(f, 2)[2]);\nlfun(forms[1], 6)               \\\\ 0.792122838646030569355944890487..."
    },
    "program-sage": {
      "language": "Sage",
      "code": "from sage.modular.modform.constructor import Newforms\nf = Newforms(1, 12, names='a')[0]\nf.lseries(prec=330)(6)          # 0.792122838646030569355944890487..."
    }
  },
  "Similar tables": [
    {
      "table": "HREF{Hecke_polynomials_of_level_one_cusp_forms}[Hecke polynomials of level one cusp forms]",
      "relation": "stores the characteristic polynomials whose $T_2$ roots order the eigenforms used here"
    },
    {
      "table": "HREF{Q-expansion_of_the_modular_discriminant}[$q$-expansion of the modular discriminant $\\Delta$]",
      "relation": "stores the coefficients of the weight $12$ eigenform whose $L$-values are the $k=12$ rows here"
    },
    {
      "table": "HREF{Values_of_Dirichlet_L-functions_at_positive_integers}[Values of Dirichlet $L$-functions at positive integers]",
      "relation": "stores special values of degree-one $L$-functions"
    },
    {
      "table": "HREF{Special_value_of_L-function_of_elliptic_curves_over_Q_of_rank_1}[Special $L$-value of elliptic curves over $\\mathbb{Q}$ of rank $1$]",
      "relation": "stores leading Taylor coefficients at the central point for weight $2$ modular $L$-functions attached to elliptic curves of rank $1$"
    },
    {
      "table": "HREF{Special_value_of_L-function_of_elliptic_curves_over_Q_of_rank_2}[Special $L$-value of elliptic curves over $\\mathbb{Q}$ of rank $2$]",
      "relation": "stores leading Taylor coefficients at the central point for weight $2$ modular $L$-functions attached to elliptic curves of rank $2$"
    },
    {
      "table": "HREF{Special_value_of_L-function_of_elliptic_curves_over_Q_of_rank_3}[Special $L$-value of elliptic curves over $\\mathbb{Q}$ of rank $3$]",
      "relation": "stores leading Taylor coefficients at the central point for weight $2$ modular $L$-functions attached to elliptic curves of rank $3$"
    },
    {
      "table": "HREF{Special_L-values_of_genus_2_curves_over_Q}[Special $L$-values of genus 2 curves over $\\mathbb{Q}$]",
      "relation": "stores leading Taylor coefficients for Hasse-Weil $L$-functions of genus $2$ curves"
    }
  ],
  "Links": {
    "WikiCusp": {
      "title": "Wikipedia: cusp form",
      "url": "https://en.wikipedia.org/wiki/Cusp_form"
    },
    "WikiL": {
      "title": "Wikipedia: L-function",
      "url": "https://en.wikipedia.org/wiki/L-function"
    },
    "PARILfun": {
      "title": "PARI/GP reference: L-functions",
      "url": "https://pari.math.u-bordeaux.fr/dochtml/html-stable/_L_minusfunctions.html"
    },
    "PARIModular": {
      "title": "PARI/GP reference: modular forms",
      "url": "https://pari.math.u-bordeaux.fr/dochtml/html-stable/Modular_forms.html"
    },
    "SageNewforms": {
      "title": "SageMath reference: Newforms",
      "url": "https://doc.sagemath.org/html/en/reference/modfrm/sage/modular/modform/constructor.html#sage.modular.modform.constructor.Newforms"
    }
  },
  "Keywords": [
    "level one cusp form",
    "modular form L-function",
    "critical L-value",
    "special L-value",
    "Ramanujan tau L-function"
  ],
  "Tags": [
    "number theory",
    "L-function",
    "special values"
  ],
  "Data properties": {
    "type": "R",
    "rigour": "heuristic (agreement-checked)",
    "complete": "no",
    "complete-note": "it holds every even weight $12\\leq k\\leq40$ with $\\dim S_k(\\mathrm{SL}_2(\\mathbb{Z}))>0$, every eigenform in the increasing-$a_2$ ordering, and every critical integer $1\\leq s\\leq k-1$, except the forced central zeros for $k\\equiv2\\pmod4$",
    "rigour details": "The generator evaluates Sage's `Newforms` $L$-series CITE{SageNewforms} at $80$ and $120$ decimal working digits, passing explicit bit precisions to Sage's Dokchitser implementation and creating a fresh $L$-series object for each value, and stores only $30$ digits. It sorts the embeddings of each Hecke orbit by the embedded value of $a_2$. An independent check compares every stored value with PARI/GP's `lfunmf` values CITE{PARILfun} and CITE{PARIModular}, after sorting PARI's embeddings by the same $a_2$ values, and checks the functional equation with PARI's `lfuncheckfeq`. PARI and Sage did not stably agree to $100$ digits in the comparison run; $30$ digits is the precision supported by both implementations."
  },
  "Display properties": {
    "number-header": "$L(f_{k,i},s)$",
    "group parameters": [
      [
        "k"
      ],
      [
        "i"
      ],
      [
        "s"
      ]
    ]
  },
  "Numbers": {
    "12": {
      "1": {
        "1": {
          "number": "0.0374412812685155417387703158412",
          "comment": "$a_2=-24$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#12,2}[$\\chi_{12,2}$]."
        },
        "2": {
          "number": "0.146374542091265989413000913275",
          "comment": "$a_2=-24$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#12,2}[$\\chi_{12,2}$]."
        },
        "3": {
          "number": "0.315241565880993084286937938416",
          "comment": "$a_2=-24$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#12,2}[$\\chi_{12,2}$]."
        },
        "4": {
          "number": "0.501617647510902215168742991995",
          "comment": "$a_2=-24$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#12,2}[$\\chi_{12,2}$]."
        },
        "5": {
          "number": "0.666709188434003643826130221671",
          "comment": "$a_2=-24$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#12,2}[$\\chi_{12,2}$]."
        },
        "6": {
          "number": "0.792122838646030569355944890487",
          "comment": "$a_2=-24$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#12,2}[$\\chi_{12,2}$]."
        },
        "7": {
          "number": "0.877354125388660916453218437561",
          "comment": "$a_2=-24$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#12,2}[$\\chi_{12,2}$]."
        },
        "8": {
          "number": "0.930707030298126094202932748711",
          "comment": "$a_2=-24$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#12,2}[$\\chi_{12,2}$]."
        },
        "9": {
          "number": "0.962126459694425863266316841741",
          "comment": "$a_2=-24$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#12,2}[$\\chi_{12,2}$]."
        },
        "10": {
          "number": "0.979809088251220515857621009051",
          "comment": "$a_2=-24$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#12,2}[$\\chi_{12,2}$]."
        },
        "11": {
          "number": "0.989432913100337599555367833818",
          "comment": "$a_2=-24$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#12,2}[$\\chi_{12,2}$]."
        }
      }
    },
    "16": {
      "1": {
        "1": {
          "number": "0.587014408020281161499499160844",
          "comment": "$a_2=216$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#16,2}[$\\chi_{16,2}$]."
        },
        "2": {
          "number": "1.66545603824784736966882943949",
          "comment": "$a_2=216$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#16,2}[$\\chi_{16,2}$]."
        },
        "3": {
          "number": "2.55866314351437000469843171843",
          "comment": "$a_2=216$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#16,2}[$\\chi_{16,2}$]."
        },
        "4": {
          "number": "2.86834836182577734119868250169",
          "comment": "$a_2=216$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#16,2}[$\\chi_{16,2}$]."
        },
        "5": {
          "number": "2.67834774477184721011770234067",
          "comment": "$a_2=216$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#16,2}[$\\chi_{16,2}$]."
        },
        "6": {
          "number": "2.26475708925865152719116173972",
          "comment": "$a_2=216$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#16,2}[$\\chi_{16,2}$]."
        },
        "7": {
          "number": "1.84620037831065267068059594362",
          "comment": "$a_2=216$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#16,2}[$\\chi_{16,2}$]."
        },
        "8": {
          "number": "1.52056166908472874187639702228",
          "comment": "$a_2=216$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#16,2}[$\\chi_{16,2}$]."
        },
        "9": {
          "number": "1.30151909850483260425922629196",
          "comment": "$a_2=216$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#16,2}[$\\chi_{16,2}$]."
        },
        "10": {
          "number": "1.16723772210316597848017694816",
          "comment": "$a_2=216$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#16,2}[$\\chi_{16,2}$]."
        },
        "11": {
          "number": "1.08991939829013396822496707714",
          "comment": "$a_2=216$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#16,2}[$\\chi_{16,2}$]."
        },
        "12": {
          "number": "1.04728859628972042435818927617",
          "comment": "$a_2=216$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#16,2}[$\\chi_{16,2}$]."
        },
        "13": {
          "number": "1.02448317049494969330903381766",
          "comment": "$a_2=216$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#16,2}[$\\chi_{16,2}$]."
        },
        "14": {
          "number": "1.01253787485567881473868421875",
          "comment": "$a_2=216$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#16,2}[$\\chi_{16,2}$]."
        },
        "15": {
          "number": "1.00637204838518570287442018658",
          "comment": "$a_2=216$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#16,2}[$\\chi_{16,2}$]."
        }
      }
    },
    "18": {
      "1": {
        "1": {
          "number": "-3.53164830549827527971215345253",
          "comment": "$a_2=-528$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#18,2}[$\\chi_{18,2}$]."
        },
        "2": {
          "number": "-8.67835156290327801352117784011",
          "comment": "$a_2=-528$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#18,2}[$\\chi_{18,2}$]."
        },
        "3": {
          "number": "-11.3261252501285280858015428098",
          "comment": "$a_2=-528$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#18,2}[$\\chi_{18,2}$]."
        },
        "4": {
          "number": "-10.4685651619304441313217745933",
          "comment": "$a_2=-528$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#18,2}[$\\chi_{18,2}$]."
        },
        "5": {
          "number": "-7.67749832527182922530061481488",
          "comment": "$a_2=-528$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#18,2}[$\\chi_{18,2}$]."
        },
        "6": {
          "number": "-4.69639076342179298083759011736",
          "comment": "$a_2=-528$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#18,2}[$\\chi_{18,2}$]."
        },
        "7": {
          "number": "-2.38802503366295111895120266634",
          "comment": "$a_2=-528$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#18,2}[$\\chi_{18,2}$]."
        },
        "8": {
          "number": "-0.882886075198155063144990862559",
          "comment": "$a_2=-528$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#18,2}[$\\chi_{18,2}$]."
        },
        "10": {
          "number": "0.484096460746456757698008134138",
          "comment": "$a_2=-528$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#18,2}[$\\chi_{18,2}$]."
        },
        "11": {
          "number": "0.738461421890638540319499470380",
          "comment": "$a_2=-528$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#18,2}[$\\chi_{18,2}$]."
        },
        "12": {
          "number": "0.868698283551820282190440086082",
          "comment": "$a_2=-528$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#18,2}[$\\chi_{18,2}$]."
        },
        "13": {
          "number": "0.934400269170070739866698684155",
          "comment": "$a_2=-528$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#18,2}[$\\chi_{18,2}$]."
        },
        "14": {
          "number": "0.967290178750563974586349168536",
          "comment": "$a_2=-528$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#18,2}[$\\chi_{18,2}$]."
        },
        "15": {
          "number": "0.983697174291201227643873027163",
          "comment": "$a_2=-528$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#18,2}[$\\chi_{18,2}$]."
        },
        "16": {
          "number": "0.991872353800215958357901529375",
          "comment": "$a_2=-528$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#18,2}[$\\chi_{18,2}$]."
        },
        "17": {
          "number": "0.995945900637295114949616210300",
          "comment": "$a_2=-528$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#18,2}[$\\chi_{18,2}$]."
        }
      }
    },
    "20": {
      "1": {
        "1": {
          "number": "27.5105533059560505229898706886",
          "comment": "$a_2=456$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#20,2}[$\\chi_{20,2}$]."
        },
        "2": {
          "number": "60.3948520818062326640058401248",
          "comment": "$a_2=456$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#20,2}[$\\chi_{20,2}$]."
        },
        "3": {
          "number": "70.2652587512273854482770769680",
          "comment": "$a_2=456$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#20,2}[$\\chi_{20,2}$]."
        },
        "4": {
          "number": "58.0327352948612416455748354306",
          "comment": "$a_2=456$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#20,2}[$\\chi_{20,2}$]."
        },
        "5": {
          "number": "38.5272392785997985581164782061",
          "comment": "$a_2=456$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#20,2}[$\\chi_{20,2}$]."
        },
        "6": {
          "number": "22.1551174906640138947829072854",
          "comment": "$a_2=456$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#20,2}[$\\chi_{20,2}$]."
        },
        "7": {
          "number": "11.6999572414120423955797377531",
          "comment": "$a_2=456$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#20,2}[$\\chi_{20,2}$]."
        },
        "8": {
          "number": "6.02375330833359169827427391224",
          "comment": "$a_2=456$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#20,2}[$\\chi_{20,2}$]."
        },
        "9": {
          "number": "3.24926697677742178856549366708",
          "comment": "$a_2=456$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#20,2}[$\\chi_{20,2}$]."
        },
        "10": {
          "number": "1.98173540543352669590857411559",
          "comment": "$a_2=456$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#20,2}[$\\chi_{20,2}$]."
        },
        "11": {
          "number": "1.42528798463630030627238401450",
          "comment": "$a_2=456$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#20,2}[$\\chi_{20,2}$]."
        },
        "12": {
          "number": "1.18539057450068724421910313826",
          "comment": "$a_2=456$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#20,2}[$\\chi_{20,2}$]."
        },
        "13": {
          "number": "1.08207912046047781945323570751",
          "comment": "$a_2=456$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#20,2}[$\\chi_{20,2}$]."
        },
        "14": {
          "number": "1.03708399982660875763951172931",
          "comment": "$a_2=456$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#20,2}[$\\chi_{20,2}$]."
        },
        "15": {
          "number": "1.01711360472606039324881215822",
          "comment": "$a_2=456$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#20,2}[$\\chi_{20,2}$]."
        },
        "16": {
          "number": "1.00805238270336889793084086185",
          "comment": "$a_2=456$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#20,2}[$\\chi_{20,2}$]."
        },
        "17": {
          "number": "1.00385089096121879235163583391",
          "comment": "$a_2=456$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#20,2}[$\\chi_{20,2}$]."
        },
        "18": {
          "number": "1.00186522065141322618834615920",
          "comment": "$a_2=456$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#20,2}[$\\chi_{20,2}$]."
        },
        "19": {
          "number": "1.00091209067413361346543068994",
          "comment": "$a_2=456$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#20,2}[$\\chi_{20,2}$]."
        }
      }
    },
    "22": {
      "1": {
        "1": {
          "number": "-264.522169053328521213659207034",
          "comment": "$a_2=-288$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#22,2}[$\\chi_{22,2}$]."
        },
        "2": {
          "number": "-522.060629388605615488922625113",
          "comment": "$a_2=-288$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#22,2}[$\\chi_{22,2}$]."
        },
        "3": {
          "number": "-542.180180149728177036300822727",
          "comment": "$a_2=-288$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#22,2}[$\\chi_{22,2}$]."
        },
        "4": {
          "number": "-396.065316470856243226116998228",
          "comment": "$a_2=-288$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#22,2}[$\\chi_{22,2}$]."
        },
        "5": {
          "number": "-229.520387532624078980524379428",
          "comment": "$a_2=-288$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#22,2}[$\\chi_{22,2}$]."
        },
        "6": {
          "number": "-112.759845881527281231660347798",
          "comment": "$a_2=-288$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#22,2}[$\\chi_{22,2}$]."
        },
        "7": {
          "number": "-48.9011838194784381190520205769",
          "comment": "$a_2=-288$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#22,2}[$\\chi_{22,2}$]."
        },
        "8": {
          "number": "-19.1014957271868789889584274731",
          "comment": "$a_2=-288$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#22,2}[$\\chi_{22,2}$]."
        },
        "9": {
          "number": "-6.62960630553851819546541396600",
          "comment": "$a_2=-288$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#22,2}[$\\chi_{22,2}$]."
        },
        "10": {
          "number": "-1.78816997842145311173865872490",
          "comment": "$a_2=-288$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#22,2}[$\\chi_{22,2}$]."
        },
        "12": {
          "number": "0.641764737779063165961601946837",
          "comment": "$a_2=-288$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#22,2}[$\\chi_{22,2}$]."
        },
        "13": {
          "number": "0.869742658768751643385358697453",
          "comment": "$a_2=-288$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#22,2}[$\\chi_{22,2}$]."
        },
        "14": {
          "number": "0.951254678748630769083024505736",
          "comment": "$a_2=-288$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#22,2}[$\\chi_{22,2}$]."
        },
        "15": {
          "number": "0.981030396891340738692700844426",
          "comment": "$a_2=-288$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#22,2}[$\\chi_{22,2}$]."
        },
        "16": {
          "number": "0.992281266106880429282459839257",
          "comment": "$a_2=-288$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#22,2}[$\\chi_{22,2}$]."
        },
        "17": {
          "number": "0.996715786166595846512507909923",
          "comment": "$a_2=-288$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#22,2}[$\\chi_{22,2}$]."
        },
        "18": {
          "number": "0.998545146385339963612553842502",
          "comment": "$a_2=-288$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#22,2}[$\\chi_{22,2}$]."
        },
        "19": {
          "number": "0.999333639089277402610849467421",
          "comment": "$a_2=-288$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#22,2}[$\\chi_{22,2}$]."
        },
        "20": {
          "number": "0.999686786622265679987474543538",
          "comment": "$a_2=-288$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#22,2}[$\\chi_{22,2}$]."
        },
        "21": {
          "number": "0.999849941425838259952451625079",
          "comment": "$a_2=-288$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#22,2}[$\\chi_{22,2}$]."
        }
      }
    },
    "24": {
      "1": {
        "1": {
          "number": "3094.59164116119106964614198787",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#24,2}[$\\chi_{24,2}$] in increasing order; $a_2\\approx -4016.351171716245$."
        },
        "2": {
          "number": "5550.55056486031802775310746692",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#24,2}[$\\chi_{24,2}$] in increasing order; $a_2\\approx -4016.351171716245$."
        },
        "3": {
          "number": "5212.44455526652885024630355203",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#24,2}[$\\chi_{24,2}$] in increasing order; $a_2\\approx -4016.351171716245$."
        },
        "4": {
          "number": "3423.34572532746053898465413701",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#24,2}[$\\chi_{24,2}$] in increasing order; $a_2\\approx -4016.351171716245$."
        },
        "5": {
          "number": "1771.87009737453312228514403812",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#24,2}[$\\chi_{24,2}$] in increasing order; $a_2\\approx -4016.351171716245$."
        },
        "6": {
          "number": "771.829857244746155036625891161",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#24,2}[$\\chi_{24,2}$] in increasing order; $a_2\\approx -4016.351171716245$."
        },
        "7": {
          "number": "294.813651074697473949181332188",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#24,2}[$\\chi_{24,2}$] in increasing order; $a_2\\approx -4016.351171716245$."
        },
        "8": {
          "number": "101.452767131960119484515304733",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#24,2}[$\\chi_{24,2}$] in increasing order; $a_2\\approx -4016.351171716245$."
        },
        "9": {
          "number": "32.0735146051052020489858993463",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#24,2}[$\\chi_{24,2}$] in increasing order; $a_2\\approx -4016.351171716245$."
        },
        "10": {
          "number": "9.56111897355254263197699232094",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#24,2}[$\\chi_{24,2}$] in increasing order; $a_2\\approx -4016.351171716245$."
        },
        "11": {
          "number": "2.93585702272107935198862473343",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#24,2}[$\\chi_{24,2}$] in increasing order; $a_2\\approx -4016.351171716245$."
        },
        "12": {
          "number": "1.21946272914692790247256707071",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#24,2}[$\\chi_{24,2}$] in increasing order; $a_2\\approx -4016.351171716245$."
        },
        "13": {
          "number": "0.878052951285365633227365311628",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#24,2}[$\\chi_{24,2}$] in increasing order; $a_2\\approx -4016.351171716245$."
        },
        "14": {
          "number": "0.868382199050543185387399070316",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#24,2}[$\\chi_{24,2}$] in increasing order; $a_2\\approx -4016.351171716245$."
        },
        "15": {
          "number": "0.912720770824074862661695921820",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#24,2}[$\\chi_{24,2}$] in increasing order; $a_2\\approx -4016.351171716245$."
        },
        "16": {
          "number": "0.949803507699992735081880159088",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#24,2}[$\\chi_{24,2}$] in increasing order; $a_2\\approx -4016.351171716245$."
        },
        "17": {
          "number": "0.972879776041970134948205252884",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#24,2}[$\\chi_{24,2}$] in increasing order; $a_2\\approx -4016.351171716245$."
        },
        "18": {
          "number": "0.985808879961763545036572458049",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#24,2}[$\\chi_{24,2}$] in increasing order; $a_2\\approx -4016.351171716245$."
        },
        "19": {
          "number": "0.992705109867542632026282590356",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#24,2}[$\\chi_{24,2}$] in increasing order; $a_2\\approx -4016.351171716245$."
        },
        "20": {
          "number": "0.996288881971066598586984254004",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#24,2}[$\\chi_{24,2}$] in increasing order; $a_2\\approx -4016.351171716245$."
        },
        "21": {
          "number": "0.998123906413627564586599570778",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#24,2}[$\\chi_{24,2}$] in increasing order; $a_2\\approx -4016.351171716245$."
        },
        "22": {
          "number": "0.999055278936330536411853356325",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#24,2}[$\\chi_{24,2}$] in increasing order; $a_2\\approx -4016.351171716245$."
        },
        "23": {
          "number": "0.999525456748817308892894968036",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#24,2}[$\\chi_{24,2}$] in increasing order; $a_2\\approx -4016.351171716245$."
        }
      },
      "2": {
        "1": {
          "number": "3097.94096694344104162714694151",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#24,2}[$\\chi_{24,2}$] in increasing order; $a_2\\approx 5096.351171716245$."
        },
        "2": {
          "number": "5562.54669646931432977690412127",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#24,2}[$\\chi_{24,2}$] in increasing order; $a_2\\approx 5096.351171716245$."
        },
        "3": {
          "number": "5234.92869252478526105089960894",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#24,2}[$\\chi_{24,2}$] in increasing order; $a_2\\approx 5096.351171716245$."
        },
        "4": {
          "number": "3452.80301376661019045584766806",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#24,2}[$\\chi_{24,2}$] in increasing order; $a_2\\approx 5096.351171716245$."
        },
        "5": {
          "number": "1802.27563892996980181447957495",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#24,2}[$\\chi_{24,2}$] in increasing order; $a_2\\approx 5096.351171716245$."
        },
        "6": {
          "number": "798.253761345716351515325931411",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#24,2}[$\\chi_{24,2}$] in increasing order; $a_2\\approx 5096.351171716245$."
        },
        "7": {
          "number": "314.990286980719120327107140233",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#24,2}[$\\chi_{24,2}$] in increasing order; $a_2\\approx 5096.351171716245$."
        },
        "8": {
          "number": "115.400185258993688101681499164",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#24,2}[$\\chi_{24,2}$] in increasing order; $a_2\\approx 5096.351171716245$."
        },
        "9": {
          "number": "40.9976311216145911556408034380",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#24,2}[$\\chi_{24,2}$] in increasing order; $a_2\\approx 5096.351171716245$."
        },
        "10": {
          "number": "14.9373189338486622244818941029",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#24,2}[$\\chi_{24,2}$] in increasing order; $a_2\\approx 5096.351171716245$."
        },
        "11": {
          "number": "6.02655093603133912743910249753",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#24,2}[$\\chi_{24,2}$] in increasing order; $a_2\\approx 5096.351171716245$."
        },
        "12": {
          "number": "2.93320547718128517752123013121",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#24,2}[$\\chi_{24,2}$] in increasing order; $a_2\\approx 5096.351171716245$."
        },
        "13": {
          "number": "1.80241435277709389072957379560",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#24,2}[$\\chi_{24,2}$] in increasing order; $a_2\\approx 5096.351171716245$."
        },
        "14": {
          "number": "1.35667194389854784930694658605",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#24,2}[$\\chi_{24,2}$] in increasing order; $a_2\\approx 5096.351171716245$."
        },
        "15": {
          "number": "1.16667568054188347179160205467",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#24,2}[$\\chi_{24,2}$] in increasing order; $a_2\\approx 5096.351171716245$."
        },
        "16": {
          "number": "1.08037960764198947324097356560",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#24,2}[$\\chi_{24,2}$] in increasing order; $a_2\\approx 5096.351171716245$."
        },
        "17": {
          "number": "1.03946231368897062998202239027",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#24,2}[$\\chi_{24,2}$] in increasing order; $a_2\\approx 5096.351171716245$."
        },
        "18": {
          "number": "1.01955844155424079025280923395",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#24,2}[$\\chi_{24,2}$] in increasing order; $a_2\\approx 5096.351171716245$."
        },
        "19": {
          "number": "1.00974007000096134349809930006",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#24,2}[$\\chi_{24,2}$] in increasing order; $a_2\\approx 5096.351171716245$."
        },
        "20": {
          "number": "1.00486177273924408037269323013",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#24,2}[$\\chi_{24,2}$] in increasing order; $a_2\\approx 5096.351171716245$."
        },
        "21": {
          "number": "1.00242936322465038226416053895",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#24,2}[$\\chi_{24,2}$] in increasing order; $a_2\\approx 5096.351171716245$."
        },
        "22": {
          "number": "1.00121448791402306460021193135",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#24,2}[$\\chi_{24,2}$] in increasing order; $a_2\\approx 5096.351171716245$."
        },
        "23": {
          "number": "1.00060725905765047496744416568",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#24,2}[$\\chi_{24,2}$] in increasing order; $a_2\\approx 5096.351171716245$."
        }
      }
    },
    "26": {
      "1": {
        "1": {
          "number": "-43290.0506485854536729291581862",
          "comment": "$a_2=-48$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#26,2}[$\\chi_{26,2}$]."
        },
        "2": {
          "number": "-71209.1372260170711196825949238",
          "comment": "$a_2=-48$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#26,2}[$\\chi_{26,2}$]."
        },
        "3": {
          "number": "-61113.2820362129149189010910881",
          "comment": "$a_2=-48$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#26,2}[$\\chi_{26,2}$]."
        },
        "4": {
          "number": "-36554.9665982072070745379086995",
          "comment": "$a_2=-48$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#26,2}[$\\chi_{26,2}$]."
        },
        "5": {
          "number": "-17179.6159216917300382860218074",
          "comment": "$a_2=-48$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#26,2}[$\\chi_{26,2}$]."
        },
        "6": {
          "number": "-6781.63645149135998081831629081",
          "comment": "$a_2=-48$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#26,2}[$\\chi_{26,2}$]."
        },
        "7": {
          "number": "-2347.84017876120984880091531064",
          "comment": "$a_2=-48$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#26,2}[$\\chi_{26,2}$]."
        },
        "8": {
          "number": "-734.940365141317694162068166536",
          "comment": "$a_2=-48$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#26,2}[$\\chi_{26,2}$]."
        },
        "9": {
          "number": "-212.630552238851154709329375648",
          "comment": "$a_2=-48$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#26,2}[$\\chi_{26,2}$]."
        },
        "10": {
          "number": "-57.5680211296584869618719414897",
          "comment": "$a_2=-48$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#26,2}[$\\chi_{26,2}$]."
        },
        "11": {
          "number": "-14.4144850024665229280583977581",
          "comment": "$a_2=-48$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#26,2}[$\\chi_{26,2}$]."
        },
        "12": {
          "number": "-2.95155114131575373196401823193",
          "comment": "$a_2=-48$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#26,2}[$\\chi_{26,2}$]."
        },
        "14": {
          "number": "0.746939541906930320436707436761",
          "comment": "$a_2=-48$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#26,2}[$\\chi_{26,2}$]."
        },
        "15": {
          "number": "0.935132788420595656306896663682",
          "comment": "$a_2=-48$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#26,2}[$\\chi_{26,2}$]."
        },
        "16": {
          "number": "0.982933038686974521173351821738",
          "comment": "$a_2=-48$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#26,2}[$\\chi_{26,2}$]."
        },
        "17": {
          "number": "0.995326446335172783649113293060",
          "comment": "$a_2=-48$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#26,2}[$\\chi_{26,2}$]."
        },
        "18": {
          "number": "0.998648848708936819706450988688",
          "comment": "$a_2=-48$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#26,2}[$\\chi_{26,2}$]."
        },
        "19": {
          "number": "0.999581999939780091476656892383",
          "comment": "$a_2=-48$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#26,2}[$\\chi_{26,2}$]."
        },
        "20": {
          "number": "0.999860211422181919415558510165",
          "comment": "$a_2=-48$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#26,2}[$\\chi_{26,2}$]."
        },
        "21": {
          "number": "0.999949272589309915878216666098",
          "comment": "$a_2=-48$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#26,2}[$\\chi_{26,2}$]."
        },
        "22": {
          "number": "0.999980107303485664531876091767",
          "comment": "$a_2=-48$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#26,2}[$\\chi_{26,2}$]."
        },
        "23": {
          "number": "0.999991660416808020217807438902",
          "comment": "$a_2=-48$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#26,2}[$\\chi_{26,2}$]."
        },
        "24": {
          "number": "0.999996314240344731629534198562",
          "comment": "$a_2=-48$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#26,2}[$\\chi_{26,2}$]."
        },
        "25": {
          "number": "0.999998306132242791386415666689",
          "comment": "$a_2=-48$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#26,2}[$\\chi_{26,2}$]."
        }
      }
    },
    "28": {
      "1": {
        "1": {
          "number": "712658.881788275533376013379868",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx -18713.59859471915$."
        },
        "2": {
          "number": "1081949.89593268728171185895563",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx -18713.59859471915$."
        },
        "3": {
          "number": "854032.923764368975686432223185",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx -18713.59859471915$."
        },
        "4": {
          "number": "468011.113951919763865370686447",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx -18713.59859471915$."
        },
        "5": {
          "number": "200601.084436755035725272304841",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx -18713.59859471915$."
        },
        "6": {
          "number": "71829.1228339656870521978114054",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx -18713.59859471915$."
        },
        "7": {
          "number": "22400.6262850437246161843344391",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx -18713.59859471915$."
        },
        "8": {
          "number": "6256.68395684487833587061533773",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx -18713.59859471915$."
        },
        "9": {
          "number": "1593.33311231917738619974904136",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx -18713.59859471915$."
        },
        "10": {
          "number": "372.645788218279228913043256832",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx -18713.59859471915$."
        },
        "11": {
          "number": "79.3213533277205091457818140209",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx -18713.59859471915$."
        },
        "12": {
          "number": "14.7693358393523950012323536592",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx -18713.59859471915$."
        },
        "13": {
          "number": "2.23406916282794111293398076341",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx -18713.59859471915$."
        },
        "14": {
          "number": "0.431363431350265862419908986625",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx -18713.59859471915$."
        },
        "15": {
          "number": "0.484601732786476187884683229981",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx -18713.59859471915$."
        },
        "16": {
          "number": "0.702645948369832596163358271554",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx -18713.59859471915$."
        },
        "17": {
          "number": "0.846472326662659688510286032972",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx -18713.59859471915$."
        },
        "18": {
          "number": "0.923484651170322213264937777315",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx -18713.59859471915$."
        },
        "19": {
          "number": "0.962243017679046165773479118580",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx -18713.59859471915$."
        },
        "20": {
          "number": "0.981383051852985809928808494276",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx -18713.59859471915$."
        },
        "21": {
          "number": "0.990800454738073000780711069871",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx -18713.59859471915$."
        },
        "22": {
          "number": "0.995441611289128705580864864769",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx -18713.59859471915$."
        },
        "23": {
          "number": "0.997735801515358736749125557449",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx -18713.59859471915$."
        },
        "24": {
          "number": "0.998873187359587502267298718121",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx -18713.59859471915$."
        },
        "25": {
          "number": "0.999438425640041746978544927277",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx -18713.59859471915$."
        },
        "26": {
          "number": "0.999719840510740910237620291162",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx -18713.59859471915$."
        },
        "27": {
          "number": "0.999860133672657889327868689723",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx -18713.59859471915$."
        }
      },
      "2": {
        "1": {
          "number": "712814.180877281357878729384702",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx 10433.59859471915$."
        },
        "2": {
          "number": "1082422.27537030389553277997979",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx 10433.59859471915$."
        },
        "3": {
          "number": "854780.671849625591588549413677",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx 10433.59859471915$."
        },
        "4": {
          "number": "468834.023103223532059793532097",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx 10433.59859471915$."
        },
        "5": {
          "number": "201311.003303190259787619608607",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx 10433.59859471915$."
        },
        "6": {
          "number": "72342.5546812063164041151461893",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx 10433.59859471915$."
        },
        "7": {
          "number": "22725.8743677508663323835821785",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx 10433.59859471915$."
        },
        "8": {
          "number": "6442.96173438099577125805709590",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx 10433.59859471915$."
        },
        "9": {
          "number": "1692.19288310164032711717971635",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx 10433.59859471915$."
        },
        "10": {
          "number": "422.255855846480365643776080967",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx 10433.59859471915$."
        },
        "11": {
          "number": "103.268908001142972823304522186",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx 10433.59859471915$."
        },
        "12": {
          "number": "26.0534491165768480200717614337",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx 10433.59859471915$."
        },
        "13": {
          "number": "7.48845805857930646687482250802",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx 10433.59859471915$."
        },
        "14": {
          "number": "2.87274956148603492482699205899",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx 10433.59859471915$."
        },
        "15": {
          "number": "1.62435425521598675125956416843",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx 10433.59859471915$."
        },
        "16": {
          "number": "1.23948366141459563646375391614",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx 10433.59859471915$."
        },
        "17": {
          "number": "1.10202699727629200466800697592",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx 10433.59859471915$."
        },
        "18": {
          "number": "1.04642750319399663646852034431",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx 10433.59859471915$."
        },
        "19": {
          "number": "1.02194624196358612089575795147",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx 10433.59859471915$."
        },
        "20": {
          "number": "1.01060138141409310884472307620",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx 10433.59859471915$."
        },
        "21": {
          "number": "1.00518647877812779724305992589",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx 10433.59859471915$."
        },
        "22": {
          "number": "1.00255699019311118800389501107",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx 10433.59859471915$."
        },
        "23": {
          "number": "1.00126674687989856653303346160",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx 10433.59859471915$."
        },
        "24": {
          "number": "1.00062951720382836519138732316",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx 10433.59859471915$."
        },
        "25": {
          "number": "1.00031348343735732058041757407",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx 10433.59859471915$."
        },
        "26": {
          "number": "1.00015631829756812498014338766",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx 10433.59859471915$."
        },
        "27": {
          "number": "1.00007801823406680357898635952",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\\chi_{28,2}$] in increasing order; $a_2\\approx 10433.59859471915$."
        }
      }
    },
    "30": {
      "1": {
        "1": {
          "number": "-13648674.1351535171899498034030",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx -17433.90502875288$."
        },
        "2": {
          "number": "-19243239.5476901342914033211891",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx -17433.90502875288$."
        },
        "3": {
          "number": "-14067480.1604543375738196575800",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx -17433.90502875288$."
        },
        "4": {
          "number": "-7119116.30332032255023837643033",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx -17433.90502875288$."
        },
        "5": {
          "number": "-2809804.67426978600856995284401",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx -17433.90502875288$."
        },
        "6": {
          "number": "-923928.198166939643793279937116",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx -17433.90502875288$."
        },
        "7": {
          "number": "-264055.199114982573163040552543",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx -17433.90502875288$."
        },
        "8": {
          "number": "-67563.1669602734285554415733907",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx -17433.90502875288$."
        },
        "9": {
          "number": "-15819.0491281240804680258714973",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx -17433.90502875288$."
        },
        "10": {
          "number": "-3445.72273488011856547569249481",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx -17433.90502875288$."
        },
        "11": {
          "number": "-706.784601473905623718741274844",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx -17433.90502875288$."
        },
        "12": {
          "number": "-137.456499397790443635595604146",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx -17433.90502875288$."
        },
        "13": {
          "number": "-25.1327212233421429312958717840",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx -17433.90502875288$."
        },
        "14": {
          "number": "-3.90389828425235057000613578200",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx -17433.90502875288$."
        },
        "16": {
          "number": "0.733903460717375145506258009598",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx -17433.90502875288$."
        },
        "17": {
          "number": "0.896760267244227069929395397224",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx -17433.90502875288$."
        },
        "18": {
          "number": "0.949143094934468681980376162483",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx -17433.90502875288$."
        },
        "19": {
          "number": "0.973078861990040960755277933241",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx -17433.90502875288$."
        },
        "20": {
          "number": "0.985706053348774748837247374186",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx -17433.90502875288$."
        },
        "21": {
          "number": "0.992509566911188240371353268076",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx -17433.90502875288$."
        },
        "22": {
          "number": "0.996127157279203968836963281124",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx -17433.90502875288$."
        },
        "23": {
          "number": "0.998018336646528673263561438573",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx -17433.90502875288$."
        },
        "24": {
          "number": "0.998993517222687784759411613342",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx -17433.90502875288$."
        },
        "25": {
          "number": "0.999491417752265669894724451899",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx -17433.90502875288$."
        },
        "26": {
          "number": "0.999743901035165160155927261812",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx -17433.90502875288$."
        },
        "27": {
          "number": "0.999871341637316451807396922058",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx -17433.90502875288$."
        },
        "28": {
          "number": "0.999935466406847951967859774166",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx -17433.90502875288$."
        },
        "29": {
          "number": "0.999967664724680349278170637037",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx -17433.90502875288$."
        }
      },
      "2": {
        "1": {
          "number": "-13649775.4959396985241604772671",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx 26073.90502875288$."
        },
        "2": {
          "number": "-19246338.5776575839682568455359",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx 26073.90502875288$."
        },
        "3": {
          "number": "-14071996.8301533740409748878449",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx 26073.90502875288$."
        },
        "4": {
          "number": "-7123666.25964116884871399708414",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx 26073.90502875288$."
        },
        "5": {
          "number": "-2813371.10168773523158668260045",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx 26073.90502875288$."
        },
        "6": {
          "number": "-926249.242855275965965804166792",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx 26073.90502875288$."
        },
        "7": {
          "number": "-265361.429805920015939456645472",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx 26073.90502875288$."
        },
        "8": {
          "number": "-68216.3406170292118122160126387",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx 26073.90502875288$."
        },
        "9": {
          "number": "-16114.5495956341623157429738714",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx 26073.90502875288$."
        },
        "10": {
          "number": "-3567.94863775350291653123843891",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx 26073.90502875288$."
        },
        "11": {
          "number": "-753.096418346525068070784139760",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx 26073.90502875288$."
        },
        "12": {
          "number": "-153.344464243010996757996815839",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx 26073.90502875288$."
        },
        "13": {
          "number": "-29.8588117212250531787629085042",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx 26073.90502875288$."
        },
        "14": {
          "number": "-4.94092624507179308489413980744",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx 26073.90502875288$."
        },
        "16": {
          "number": "0.928856903120352151174386134972",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx 26073.90502875288$."
        },
        "17": {
          "number": "1.06539183484247260369069568956",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx 26073.90502875288$."
        },
        "18": {
          "number": "1.05885018184173991766373793110",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx 26073.90502875288$."
        },
        "19": {
          "number": "1.03683951829908124935457265082",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx 26073.90502875288$."
        },
        "20": {
          "number": "1.02067079706385662400191809621",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx 26073.90502875288$."
        },
        "21": {
          "number": "1.01104968513542848307319541477",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx 26073.90502875288$."
        },
        "22": {
          "number": "1.00575731594681607296636586017",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx 26073.90502875288$."
        },
        "23": {
          "number": "1.00295534294602729751660425214",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx 26073.90502875288$."
        },
        "24": {
          "number": "1.00150313712760293338545800176",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx 26073.90502875288$."
        },
        "25": {
          "number": "1.00076005169999835181731121076",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx 26073.90502875288$."
        },
        "26": {
          "number": "1.00038285549073725849554015120",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx 26073.90502875288$."
        },
        "27": {
          "number": "1.00019237202372521078460489859",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx 26073.90502875288$."
        },
        "28": {
          "number": "1.00009650114157808009033945775",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx 26073.90502875288$."
        },
        "29": {
          "number": "1.00004835572532051507216403884",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\\chi_{30,2}$] in increasing order; $a_2\\approx 26073.90502875288$."
        }
      }
    },
    "32": {
      "1": {
        "1": {
          "number": "300786039.242670904207546996408",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx -31347.87172677239$."
        },
        "2": {
          "number": "395812766.614205733306654326713",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx -31347.87172677239$."
        },
        "3": {
          "number": "269406954.943264467385961447262",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx -31347.87172677239$."
        },
        "4": {
          "number": "126608749.038358610474967499030",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx -31347.87172677239$."
        },
        "5": {
          "number": "46275214.7584110629779680923572",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx -31347.87172677239$."
        },
        "6": {
          "number": "14049528.5934921704612474346494",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx -31347.87172677239$."
        },
        "7": {
          "number": "3695918.21374967533473480460635",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx -31347.87172677239$."
        },
        "8": {
          "number": "867664.304440952928175025300591",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx -31347.87172677239$."
        },
        "9": {
          "number": "185795.471669042122710090487015",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx -31347.87172677239$."
        },
        "10": {
          "number": "36894.5888947114200516974403836",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx -31347.87172677239$."
        },
        "11": {
          "number": "6877.33835870824722918178910465",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx -31347.87172677239$."
        },
        "12": {
          "number": "1212.21171658985293163440904933",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx -31347.87172677239$."
        },
        "13": {
          "number": "202.080779746584921309627007928",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx -31347.87172677239$."
        },
        "14": {
          "number": "31.6015203251730726663318655500",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx -31347.87172677239$."
        },
        "15": {
          "number": "4.76343056131498813693083802581",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx -31347.87172677239$."
        },
        "16": {
          "number": "1.08092278453390571792665826076",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx -31347.87172677239$."
        },
        "17": {
          "number": "0.783552920537299347323141685779",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx -31347.87172677239$."
        },
        "18": {
          "number": "0.862262008450325528721883536176",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx -31347.87172677239$."
        },
        "19": {
          "number": "0.930251037265139176090246938381",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx -31347.87172677239$."
        },
        "20": {
          "number": "0.966225558105669060452335200885",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx -31347.87172677239$."
        },
        "21": {
          "number": "0.983688266975766759523060345709",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx -31347.87172677239$."
        },
        "22": {
          "number": "0.992065203396026855152061092911",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx -31347.87172677239$."
        },
        "23": {
          "number": "0.996109800143053912689661389703",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx -31347.87172677239$."
        },
        "24": {
          "number": "0.998080872536339557698435030242",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx -31347.87172677239$."
        },
        "25": {
          "number": "0.999049044895060718013598165868",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx -31347.87172677239$."
        },
        "26": {
          "number": "0.999527360834064295967272590556",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx -31347.87172677239$."
        },
        "27": {
          "number": "0.999764615388416796753494360270",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx -31347.87172677239$."
        },
        "28": {
          "number": "0.999882615950534280751809286309",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx -31347.87172677239$."
        },
        "29": {
          "number": "0.999941409764115061992797659948",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx -31347.87172677239$."
        },
        "30": {
          "number": "0.999970738549615024748293023574",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx -31347.87172677239$."
        },
        "31": {
          "number": "0.999985380427767001674714672613",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx -31347.87172677239$."
        }
      },
      "2": {
        "1": {
          "number": "300800439.723932550204017405982",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx 71307.87172677239$."
        },
        "2": {
          "number": "395850696.249917954832775565874",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx 71307.87172677239$."
        },
        "3": {
          "number": "269458649.325432791196126832940",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx 71307.87172677239$."
        },
        "4": {
          "number": "126657424.961163729413455090864",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx 71307.87172677239$."
        },
        "5": {
          "number": "46310895.3436347346191809010475",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx 71307.87172677239$."
        },
        "6": {
          "number": "14071286.9745836040373976604252",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx 71307.87172677239$."
        },
        "7": {
          "number": "3707441.70255734613559668381237",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx 71307.87172677239$."
        },
        "8": {
          "number": "873130.879703255762793024541867",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx 71307.87172677239$."
        },
        "9": {
          "number": "188174.849223652141517205390134",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx 71307.87172677239$."
        },
        "10": {
          "number": "37864.1897233076799955922031793",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx 71307.87172677239$."
        },
        "11": {
          "number": "7254.05962504666023244438465895",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx 71307.87172677239$."
        },
        "12": {
          "number": "1354.24297882858552551588546562",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx 71307.87172677239$."
        },
        "13": {
          "number": "254.972165282875452461564070988",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx 71307.87172677239$."
        },
        "14": {
          "number": "51.4067525591174460346887522776",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx 71307.87172677239$."
        },
        "15": {
          "number": "12.3470090478611264732812110876",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx 71307.87172677239$."
        },
        "16": {
          "number": "4.09058356427755893807487146966",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx 71307.87172677239$."
        },
        "17": {
          "number": "2.03100158065100508587946139840",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx 71307.87172677239$."
        },
        "18": {
          "number": "1.40265687389173923190509670566",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx 71307.87172677239$."
        },
        "19": {
          "number": "1.17372924592617906087277009565",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx 71307.87172677239$."
        },
        "20": {
          "number": "1.07943534955293709889839485267",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx 71307.87172677239$."
        },
        "21": {
          "number": "1.03757194555733856062015152192",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx 71307.87172677239$."
        },
        "22": {
          "number": "1.01813697359461517535615319891",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx 71307.87172677239$."
        },
        "23": {
          "number": "1.00886641514069689293370593330",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx 71307.87172677239$."
        },
        "24": {
          "number": "1.00436911578855010753658314002",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx 71307.87172677239$."
        },
        "25": {
          "number": "1.00216397596803012104058720080",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx 71307.87172677239$."
        },
        "26": {
          "number": "1.00107532004733044530157024974",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx 71307.87172677239$."
        },
        "27": {
          "number": "1.00053548564259522533932114742",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx 71307.87172677239$."
        },
        "28": {
          "number": "1.00026703021414447561974567869",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx 71307.87172677239$."
        },
        "29": {
          "number": "1.00013328065843983745868309131",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx 71307.87172677239$."
        },
        "30": {
          "number": "1.00006656296215358372239565067",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx 71307.87172677239$."
        },
        "31": {
          "number": "1.00003325589023536451030278416",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\\chi_{32,2}$] in increasing order; $a_2\\approx 71307.87172677239$."
        }
      }
    },
    "34": {
      "1": {
        "1": {
          "number": "-7558007179.89621713143686268041",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx -171359.4371321172$."
        },
        "2": {
          "number": "-9324131789.60250282393386455777",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx -171359.4371321172$."
        },
        "3": {
          "number": "-5936892000.52271226567115275604",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx -171359.4371321172$."
        },
        "4": {
          "number": "-2604004905.22794430246463469614",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx -171359.4371321172$."
        },
        "5": {
          "number": "-886083129.640517143629914909944",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx -171359.4371321172$."
        },
        "6": {
          "number": "-249786091.330899950671048248717",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx -171359.4371321172$."
        },
        "7": {
          "number": "-60832808.4191034820044227237778",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx -171359.4371321172$."
        },
        "8": {
          "number": "-13178871.4656723328532769188536",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx -171359.4371321172$."
        },
        "9": {
          "number": "-2594889.16138421976778559565402",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx -171359.4371321172$."
        },
        "10": {
          "number": "-471919.171970520078735806088388",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx -171359.4371321172$."
        },
        "11": {
          "number": "-80214.1110042367774056532364679",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx -171359.4371321172$."
        },
        "12": {
          "number": "-12838.1322453361551807417720196",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx -171359.4371321172$."
        },
        "13": {
          "number": "-1938.67716544905555362850963320",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx -171359.4371321172$."
        },
        "14": {
          "number": "-274.692434481821843068661176718",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx -171359.4371321172$."
        },
        "15": {
          "number": "-35.8573787040505620522921850857",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx -171359.4371321172$."
        },
        "16": {
          "number": "-3.98376816340662827896175313200",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx -171359.4371321172$."
        },
        "18": {
          "number": "0.578209055859966612286041287872",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx -171359.4371321172$."
        },
        "19": {
          "number": "0.760966158262213842353077918678",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx -171359.4371321172$."
        },
        "20": {
          "number": "0.865190224147675893176004609502",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx -171359.4371321172$."
        },
        "21": {
          "number": "0.927164488118796105628857119909",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx -171359.4371321172$."
        },
        "22": {
          "number": "0.961861042125030208362743526103",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx -171359.4371321172$."
        },
        "23": {
          "number": "0.980405266273915554003067503484",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx -171359.4371321172$."
        },
        "24": {
          "number": "0.990043741126197379451933171061",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx -171359.4371321172$."
        },
        "25": {
          "number": "0.994973603298556706354295470643",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx -171359.4371321172$."
        },
        "26": {
          "number": "0.997471992872029368397764364645",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx -171359.4371321172$."
        },
        "27": {
          "number": "0.998731398105998262142618680831",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx -171359.4371321172$."
        },
        "28": {
          "number": "0.999364253839427379421997975191",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx -171359.4371321172$."
        },
        "29": {
          "number": "0.999681667616567414349498047808",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx -171359.4371321172$."
        },
        "30": {
          "number": "0.999840686413834890892721730584",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx -171359.4371321172$."
        },
        "31": {
          "number": "0.999920295520002899298270711188",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx -171359.4371321172$."
        },
        "32": {
          "number": "0.999960132228828936605774055839",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx -171359.4371321172$."
        },
        "33": {
          "number": "0.999980061029096836068660248434",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx -171359.4371321172$."
        }
      },
      "2": {
        "1": {
          "number": "-7558201572.26678488335654484080",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx 49679.43713211717$."
        },
        "2": {
          "number": "-9324611311.13557145019151011710",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx 49679.43713211717$."
        },
        "3": {
          "number": "-5937502431.82556450759676131552",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx 49679.43713211717$."
        },
        "4": {
          "number": "-2604540114.75186876696498689742",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx 49679.43713211717$."
        },
        "5": {
          "number": "-886447089.116282447942720187055",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx 49679.43713211717$."
        },
        "6": {
          "number": "-249991059.504333518442376079823",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx 49679.43713211717$."
        },
        "7": {
          "number": "-60932479.6584825694053922309707",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx 49679.43713211717$."
        },
        "8": {
          "number": "-13221954.2492491800655872178839",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx 49679.43713211717$."
        },
        "9": {
          "number": "-2611797.14550364964707031081268",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx 49679.43713211717$."
        },
        "10": {
          "number": "-478039.558399674303139595275983",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx 49679.43713211717$."
        },
        "11": {
          "number": "-82280.7968799051560869770735452",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx 49679.43713211717$."
        },
        "12": {
          "number": "-13493.4701465843985263139990824",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx 49679.43713211717$."
        },
        "13": {
          "number": "-2133.83760978234772873372195675",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx 49679.43713211717$."
        },
        "14": {
          "number": "-328.612478972714356797057053169",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx 49679.43713211717$."
        },
        "15": {
          "number": "-49.0963907556941139795351822452",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx 49679.43713211717$."
        },
        "16": {
          "number": "-6.44243860035663408908441111879",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx 49679.43713211717$."
        },
        "18": {
          "number": "0.935063534762173579271000149957",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx 49679.43713211717$."
        },
        "19": {
          "number": "1.04192479227938206143859546209",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx 49679.43713211717$."
        },
        "20": {
          "number": "1.03502051258328678466473046280",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx 49679.43713211717$."
        },
        "21": {
          "number": "1.02049917875017886113327950922",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx 49679.43713211717$."
        },
        "22": {
          "number": "1.01096039587780505191613855387",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx 49679.43713211717$."
        },
        "23": {
          "number": "1.00566503280217860646446078078",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx 49679.43713211717$."
        },
        "24": {
          "number": "1.00288375830997981586221358003",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx 49679.43713211717$."
        },
        "25": {
          "number": "1.00145673103062902646208085268",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx 49679.43713211717$."
        },
        "26": {
          "number": "1.00073280849685796771221147423",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx 49679.43713211717$."
        },
        "27": {
          "number": "1.00036776504092959902628873533",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx 49679.43713211717$."
        },
        "28": {
          "number": "1.00018430696813786577878325918",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx 49679.43713211717$."
        },
        "29": {
          "number": "1.00009228779824892541142533473",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx 49679.43713211717$."
        },
        "30": {
          "number": "1.00004618689376928476038536519",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx 49679.43713211717$."
        },
        "31": {
          "number": "1.00002310733613311240263984503",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx 49679.43713211717$."
        },
        "32": {
          "number": "1.00001155818745262053307673847",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx 49679.43713211717$."
        },
        "33": {
          "number": "1.00000578057261632246060366396",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\\chi_{34,2}$] in increasing order; $a_2\\approx 49679.43713211717$."
        }
      }
    },
    "36": {
      "1": {
        "1": {
          "number": "214806287815.882255787774658531",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -165109.1678324939$."
        },
        "2": {
          "number": "249416807845.873779885902331521",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -165109.1678324939$."
        },
        "3": {
          "number": "149189179632.246908738734177150",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -165109.1678324939$."
        },
        "4": {
          "number": "61350404062.3157190849531034734",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -165109.1678324939$."
        },
        "5": {
          "number": "19531635903.1595747631793179802",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -165109.1678324939$."
        },
        "6": {
          "number": "5140119475.50050099231956410685",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -165109.1678324939$."
        },
        "7": {
          "number": "1166045330.67783382691572162370",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -165109.1678324939$."
        },
        "8": {
          "number": "234790783.994685790177066589062",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -165109.1678324939$."
        },
        "9": {
          "number": "42885087.6198793028746743451081",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -165109.1678324939$."
        },
        "10": {
          "number": "7225610.57472466062876756376640",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -165109.1678324939$."
        },
        "11": {
          "number": "1137887.14274339761141401489613",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -165109.1678324939$."
        },
        "12": {
          "number": "169171.610256260200607828769788",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -165109.1678324939$."
        },
        "13": {
          "number": "23893.7762971066436982428365261",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -165109.1678324939$."
        },
        "14": {
          "number": "3205.20362606942343984032898859",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -165109.1678324939$."
        },
        "15": {
          "number": "401.905556261750433686096121610",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -165109.1678324939$."
        },
        "16": {
          "number": "44.2533362912200383111955871531",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -165109.1678324939$."
        },
        "17": {
          "number": "3.38045206179478357701005767884",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -165109.1678324939$."
        },
        "18": {
          "number": "0.135344088203735118483757043775",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -165109.1678324939$."
        },
        "19": {
          "number": "0.436127118258318855918682051889",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -165109.1678324939$."
        },
        "20": {
          "number": "0.741430557851698744861374927570",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -165109.1678324939$."
        },
        "21": {
          "number": "0.886108608251160178863340180879",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -165109.1678324939$."
        },
        "22": {
          "number": "0.948923008385626204999493169262",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -165109.1678324939$."
        },
        "23": {
          "number": "0.976458631077660474650533205388",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -165109.1678324939$."
        },
        "24": {
          "number": "0.988888196283694436540405352030",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -165109.1678324939$."
        },
        "25": {
          "number": "0.994660129930264445208325135223",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -165109.1678324939$."
        },
        "26": {
          "number": "0.997400999919056316188613049521",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -165109.1678324939$."
        },
        "27": {
          "number": "0.998723867602195962017659505847",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -165109.1678324939$."
        },
        "28": {
          "number": "0.999369655695858188304708594344",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -165109.1678324939$."
        },
        "29": {
          "number": "0.999687385909345412488942000255",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -165109.1678324939$."
        },
        "30": {
          "number": "0.999844541927326355269586660184",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -165109.1678324939$."
        },
        "31": {
          "number": "0.999922553089595471659626652303",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -165109.1678324939$."
        },
        "32": {
          "number": "0.999961370385018801462595122446",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -165109.1678324939$."
        },
        "33": {
          "number": "0.999980716425314247821953923916",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -165109.1678324939$."
        },
        "34": {
          "number": "0.999990368612452563294074801602",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -165109.1678324939$."
        },
        "35": {
          "number": "0.999995187770182470585467106034",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -165109.1678324939$."
        }
      },
      "2": {
        "1": {
          "number": "214807155615.009161975316272478",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -26808.00761087159$."
        },
        "2": {
          "number": "249418826787.701906074215508286",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -26808.00761087159$."
        },
        "3": {
          "number": "149191601533.907787928037050841",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -26808.00761087159$."
        },
        "4": {
          "number": "61352404141.8307705414391503494",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -26808.00761087159$."
        },
        "5": {
          "number": "19532917219.3783452277545588041",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -26808.00761087159$."
        },
        "6": {
          "number": "5140800055.43007157327387076537",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -26808.00761087159$."
        },
        "7": {
          "number": "1166358318.13305273276084729503",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -26808.00761087159$."
        },
        "8": {
          "number": "234919373.097691246637867330931",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -26808.00761087159$."
        },
        "9": {
          "number": "42933460.9544090457614144048240",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -26808.00761087159$."
        },
        "10": {
          "number": "7242622.83647553203379121726758",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -26808.00761087159$."
        },
        "11": {
          "number": "1143585.53696820982374398004801",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -26808.00761087159$."
        },
        "12": {
          "number": "171021.237306319176360734227070",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -26808.00761087159$."
        },
        "13": {
          "number": "24485.0440497684245069867260221",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -26808.00761087159$."
        },
        "14": {
          "number": "3394.11200472056691717981154724",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -26808.00761087159$."
        },
        "15": {
          "number": "462.995229948869506650743357797",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -26808.00761087159$."
        },
        "16": {
          "number": "64.4491416681199069071543855634",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -26808.00761087159$."
        },
        "17": {
          "number": "10.2557702743097091105256047886",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -26808.00761087159$."
        },
        "18": {
          "number": "2.55805889320511492005262697004",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -26808.00761087159$."
        },
        "19": {
          "number": "1.32314242399854279400090782935",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -26808.00761087159$."
        },
        "20": {
          "number": "1.07979571857812378832872869538",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -26808.00761087159$."
        },
        "21": {
          "number": "1.02079718094199357040403086384",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -26808.00761087159$."
        },
        "22": {
          "number": "1.00485065851084523295729139528",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -26808.00761087159$."
        },
        "23": {
          "number": "1.00062176432145834944215999921",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -26808.00761087159$."
        },
        "24": {
          "number": "0.999700142534957752024398163781",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -26808.00761087159$."
        },
        "25": {
          "number": "0.999641261474101349173616999271",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -26808.00761087159$."
        },
        "26": {
          "number": "0.999749320065253849135290629087",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -26808.00761087159$."
        },
        "27": {
          "number": "0.999850403804676866235644889849",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -26808.00761087159$."
        },
        "28": {
          "number": "0.999916985729049401181444991844",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -26808.00761087159$."
        },
        "29": {
          "number": "0.999955719912062374126695436321",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -26808.00761087159$."
        },
        "30": {
          "number": "0.999976926812769153421742096123",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -26808.00761087159$."
        },
        "31": {
          "number": "0.999988150103134967904358380585",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -26808.00761087159$."
        },
        "32": {
          "number": "0.999993970044031941572848833599",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -26808.00761087159$."
        },
        "33": {
          "number": "0.999996949874441909300422488159",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -26808.00761087159$."
        },
        "34": {
          "number": "0.999998463184748439166766236588",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -26808.00761087159$."
        },
        "35": {
          "number": "0.999999227665660768547861871766",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx -26808.00761087159$."
        }
      },
      "3": {
        "1": {
          "number": "214809393769.108921795686771720",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx 331573.1754433655$."
        },
        "2": {
          "number": "249424021665.613230713341679495",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx 331573.1754433655$."
        },
        "3": {
          "number": "149197811435.180792079626648061",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx 331573.1754433655$."
        },
        "4": {
          "number": "61357505689.6592979690660446218",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx 331573.1754433655$."
        },
        "5": {
          "number": "19536160055.2444622063662392032",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx 331573.1754433655$."
        },
        "6": {
          "number": "5142502681.82819892763098923316",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx 331573.1754433655$."
        },
        "7": {
          "number": "1167128034.07544390282696660989",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx 331573.1754433655$."
        },
        "8": {
          "number": "235227743.944403595144482314492",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx 331573.1754433655$."
        },
        "9": {
          "number": "43045283.5056149298750372939223",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx 331573.1754433655$."
        },
        "10": {
          "number": "7279921.86124651080640119339111",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx 331573.1754433655$."
        },
        "11": {
          "number": "1155175.28549454041497624745797",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx 331573.1754433655$."
        },
        "12": {
          "number": "174410.845697846152549460668477",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx 331573.1754433655$."
        },
        "13": {
          "number": "25426.8435839818658216074566088",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx 331573.1754433655$."
        },
        "14": {
          "number": "3645.32587905872158458637614049",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx 331573.1754433655$."
        },
        "15": {
          "number": "528.387354020086875485444173426",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx 331573.1754433655$."
        },
        "16": {
          "number": "81.5997176025277071481794553226",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx 331573.1754433655$."
        },
        "17": {
          "number": "15.0690444341081817538954780844",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx 331573.1754433655$."
        },
        "18": {
          "number": "4.12659565794223284068826296351",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx 331573.1754433655$."
        },
        "19": {
          "number": "1.94412427800111391040708745800",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx 331573.1754433655$."
        },
        "20": {
          "number": "1.36714040596723641759710471761",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx 331573.1754433655$."
        },
        "21": {
          "number": "1.16497165961875981252438022153",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx 331573.1754433655$."
        },
        "22": {
          "number": "1.07922428752033865851155022703",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx 331573.1754433655$."
        },
        "23": {
          "number": "1.03910995775277440406404129572",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx 331573.1754433655$."
        },
        "24": {
          "number": "1.01951400919572713741965442654",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx 331573.1754433655$."
        },
        "25": {
          "number": "1.00977219655722909560920919535",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx 331573.1754433655$."
        },
        "26": {
          "number": "1.00489796241427741276119340503",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx 331573.1754433655$."
        },
        "27": {
          "number": "1.00245456895913293865530132526",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx 331573.1754433655$."
        },
        "28": {
          "number": "1.00122954349499874609861090887",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx 331573.1754433655$."
        },
        "29": {
          "number": "1.00061562163123007869949313299",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx 331573.1754433655$."
        },
        "30": {
          "number": "1.00030811789095771182010849370",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx 331573.1754433655$."
        },
        "31": {
          "number": "1.00015416716051691795157465410",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx 331573.1754433655$."
        },
        "32": {
          "number": "1.00007712109797638480298125578",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx 331573.1754433655$."
        },
        "33": {
          "number": "1.00003857341268520782465004187",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx 331573.1754433655$."
        },
        "34": {
          "number": "1.00001929108292542304813712448",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx 331573.1754433655$."
        },
        "35": {
          "number": "1.00000964702233791718760805557",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\\chi_{36,2}$] in increasing order; $a_2\\approx 331573.1754433655$."
        }
      }
    },
    "38": {
      "1": {
        "1": {
          "number": "-6855803638765.48265346732331236",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx -480411.7539742225$."
        },
        "2": {
          "number": "-7518203695559.77270105920399357",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx -480411.7539742225$."
        },
        "3": {
          "number": "-4240067290272.67559235237858301",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx -480411.7539742225$."
        },
        "4": {
          "number": "-1641066734943.95687089872261344",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx -480411.7539742225$."
        },
        "5": {
          "number": "-490794746148.802716110995990131",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx -480411.7539742225$."
        },
        "6": {
          "number": "-121091978585.496791757824196680",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx -480411.7539742225$."
        },
        "7": {
          "number": "-25698845045.9826364717300743891",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx -480411.7539742225$."
        },
        "8": {
          "number": "-4830109089.97912956476275898782",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx -480411.7539742225$."
        },
        "9": {
          "number": "-821550995.630570888116146761404",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx -480411.7539742225$."
        },
        "10": {
          "number": "-128589516.253241444943500236241",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx -480411.7539742225$."
        },
        "11": {
          "number": "-18768338.1745898829816432366968",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx -480411.7539742225$."
        },
        "12": {
          "number": "-2581489.69866518794404507840673",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx -480411.7539742225$."
        },
        "13": {
          "number": "-337302.834737685865595652919398",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx -480411.7539742225$."
        },
        "14": {
          "number": "-42081.1052095977742367597893862",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx -480411.7539742225$."
        },
        "15": {
          "number": "-5017.46552637429746714623303197",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx -480411.7539742225$."
        },
        "16": {
          "number": "-568.570084862925980769063751164",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx -480411.7539742225$."
        },
        "17": {
          "number": "-60.1787303190139354883672340727",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx -480411.7539742225$."
        },
        "18": {
          "number": "-5.52604208374430189313540708185",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx -480411.7539742225$."
        },
        "20": {
          "number": "0.637892973921962245747210630296",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx -480411.7539742225$."
        },
        "21": {
          "number": "0.806598612996974334075739058299",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx -480411.7539742225$."
        },
        "22": {
          "number": "0.895403763292890423533660039442",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx -480411.7539742225$."
        },
        "23": {
          "number": "0.945290087350826862959597342412",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx -480411.7539742225$."
        },
        "24": {
          "number": "0.972012184609126463623247130613",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx -480411.7539742225$."
        },
        "25": {
          "number": "0.985847422448306574600031946053",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx -480411.7539742225$."
        },
        "26": {
          "number": "0.992884183641291634985700242313",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx -480411.7539742225$."
        },
        "27": {
          "number": "0.996432123069207446284704808114",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx -480411.7539742225$."
        },
        "28": {
          "number": "0.998213490462835304182027490401",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx -480411.7539742225$."
        },
        "29": {
          "number": "0.999106063304543906172440974883",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx -480411.7539742225$."
        },
        "30": {
          "number": "0.999552845179373760782145064115",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx -480411.7539742225$."
        },
        "31": {
          "number": "0.999776370005870497465346087893",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx -480411.7539742225$."
        },
        "32": {
          "number": "0.999888169735071018439369464277",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx -480411.7539742225$."
        },
        "33": {
          "number": "0.999944080315726848170980095740",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx -480411.7539742225$."
        },
        "34": {
          "number": "0.999972038769336153288141826858",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx -480411.7539742225$."
        },
        "35": {
          "number": "0.999986018952868852076081453242",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx -480411.7539742225$."
        },
        "36": {
          "number": "0.999993009340031273366641801727",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx -480411.7539742225$."
        },
        "37": {
          "number": "0.999996504626383878038513830744",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx -480411.7539742225$."
        }
      },
      "2": {
        "1": {
          "number": "-6855841869151.52642354832296198",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx 286011.7539742225$."
        },
        "2": {
          "number": "-7518287543132.15515412900991978",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx 286011.7539742225$."
        },
        "3": {
          "number": "-4240161864697.75259353671880232",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx 286011.7539742225$."
        },
        "4": {
          "number": "-1641139941479.43712756247782435",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx 286011.7539742225$."
        },
        "5": {
          "number": "-490838533036.917381478866876784",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx 286011.7539742225$."
        },
        "6": {
          "number": "-121113584788.993285517748543321",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx 286011.7539742225$."
        },
        "7": {
          "number": "-25708015607.1024991455586496835",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx 286011.7539742225$."
        },
        "8": {
          "number": "-4833556271.00707782856180938409",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx 286011.7539742225$."
        },
        "9": {
          "number": "-822723688.544415358011290367963",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx 286011.7539742225$."
        },
        "10": {
          "number": "-128956663.921517194479359318015",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx 286011.7539742225$."
        },
        "11": {
          "number": "-18875548.5207130704568980066804",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx 286011.7539742225$."
        },
        "12": {
          "number": "-2611003.71106968590026622161377",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx 286011.7539742225$."
        },
        "13": {
          "number": "-345026.284497809383952638902679",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx 286011.7539742225$."
        },
        "14": {
          "number": "-44012.5444458805566049133378468",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx 286011.7539742225$."
        },
        "15": {
          "number": "-5479.14693401254039299504499257",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx 286011.7539742225$."
        },
        "16": {
          "number": "-672.933370173119126565322155002",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx 286011.7539742225$."
        },
        "17": {
          "number": "-81.6031186724187534597879954444",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx 286011.7539742225$."
        },
        "18": {
          "number": "-8.93516986095146592661049297100",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx 286011.7539742225$."
        },
        "20": {
          "number": "1.03142212613014714145617144364",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx 286011.7539742225$."
        },
        "21": {
          "number": "1.09375791061852665522225861967",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx 286011.7539742225$."
        },
        "22": {
          "number": "1.05975866149138667103673315984",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx 286011.7539742225$."
        },
        "23": {
          "number": "1.03227082610439706329928261311",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx 286011.7539742225$."
        },
        "24": {
          "number": "1.01662561531984887228937930307",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx 286011.7539742225$."
        },
        "25": {
          "number": "1.00842103361984665239007944137",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx 286011.7539742225$."
        },
        "26": {
          "number": "1.00423576723559036052139318477",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx 286011.7539742225$."
        },
        "27": {
          "number": "1.00212404058522623583883153030",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx 286011.7539742225$."
        },
        "28": {
          "number": "1.00106358093788667694837483536",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx 286011.7539742225$."
        },
        "29": {
          "number": "1.00053220070422778369642543966",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx 286011.7539742225$."
        },
        "30": {
          "number": "1.00026621200818568181774891792",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx 286011.7539742225$."
        },
        "31": {
          "number": "1.00013313741276867396776385456",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx 286011.7539742225$."
        },
        "32": {
          "number": "1.00006657781396601575886786044",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx 286011.7539742225$."
        },
        "33": {
          "number": "1.00003329162026765066672583732",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx 286011.7539742225$."
        },
        "34": {
          "number": "1.00001664663748475235394026693",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx 286011.7539742225$."
        },
        "35": {
          "number": "1.00000832357596869558239821054",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx 286011.7539742225$."
        },
        "36": {
          "number": "1.00000416186922945206263542182",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx 286011.7539742225$."
        },
        "37": {
          "number": "1.00000208096060064557866016957",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\\chi_{38,2}$] in increasing order; $a_2\\approx 286011.7539742225$."
        }
      }
    },
    "40": {
      "1": {
        "1": {
          "number": "244165804630765.992122066090066",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx -799151.7631007462$."
        },
        "2": {
          "number": "253664884048054.131632974340404",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx -799151.7631007462$."
        },
        "3": {
          "number": "135327826057959.635892978117441",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx -799151.7631007462$."
        },
        "4": {
          "number": "49467568481583.6449936421561165",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx -799151.7631007462$."
        },
        "5": {
          "number": "13949133204438.2019046656443254",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx -799151.7631007462$."
        },
        "6": {
          "number": "3239276032217.18899635412038778",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx -799151.7631007462$."
        },
        "7": {
          "number": "645836174658.236727076089710702",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx -799151.7631007462$."
        },
        "8": {
          "number": "113813525209.756742777415771620",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx -799151.7631007462$."
        },
        "9": {
          "number": "18114305930.5139328382234753304",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx -799151.7631007462$."
        },
        "10": {
          "number": "2647632812.72141315265900218457",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx -799151.7631007462$."
        },
        "11": {
          "number": "360164788.227863365799708914280",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx -799151.7631007462$."
        },
        "12": {
          "number": "46097597.9180522152435420727213",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx -799151.7631007462$."
        },
        "13": {
          "number": "5600688.67205398381443464267723",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx -799151.7631007462$."
        },
        "14": {
          "number": "650448.941460566125574590426395",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx -799151.7631007462$."
        },
        "15": {
          "number": "72549.6734466386887259494148561",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx -799151.7631007462$."
        },
        "16": {
          "number": "7781.75002053517654791409921506",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx -799151.7631007462$."
        },
        "17": {
          "number": "798.686778637229744322767593026",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx -799151.7631007462$."
        },
        "18": {
          "number": "77.0770283333585180283131946171",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx -799151.7631007462$."
        },
        "19": {
          "number": "6.89797237391346665052088974697",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx -799151.7631007462$."
        },
        "20": {
          "number": "0.948096669386525192729575467606",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx -799151.7631007462$."
        },
        "21": {
          "number": "0.716634300001780645591990367200",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx -799151.7631007462$."
        },
        "22": {
          "number": "0.836313368928377057813584715168",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx -799151.7631007462$."
        },
        "23": {
          "number": "0.914763245646265385623813333912",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx -799151.7631007462$."
        },
        "24": {
          "number": "0.956139819039912277468850801594",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx -799151.7631007462$."
        },
        "25": {
          "number": "0.977545154763494948870286204029",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx -799151.7631007462$."
        },
        "26": {
          "number": "0.988567364674957585584928497663",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx -799151.7631007462$."
        },
        "27": {
          "number": "0.994208624731447422714042269517",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx -799151.7631007462$."
        },
        "28": {
          "number": "0.997077959622007985275937093974",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx -799151.7631007462$."
        },
        "29": {
          "number": "0.998529942615199898036529399486",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx -799151.7631007462$."
        },
        "30": {
          "number": "0.999261913136048067986259836210",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx -799151.7631007462$."
        },
        "31": {
          "number": "0.999629929345166003455259169499",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx -799151.7631007462$."
        },
        "32": {
          "number": "0.999814621036505987585587615356",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx -799151.7631007462$."
        },
        "33": {
          "number": "0.999907195811565571934137271737",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx -799151.7631007462$."
        },
        "34": {
          "number": "0.999953559658861740984603717450",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx -799151.7631007462$."
        },
        "35": {
          "number": "0.999976767083362461569190736808",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx -799151.7631007462$."
        },
        "36": {
          "number": "0.999988379294856407991703164772",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx -799151.7631007462$."
        },
        "37": {
          "number": "0.999994188232511685044376000971",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx -799151.7631007462$."
        },
        "38": {
          "number": "0.999997093644834677007820210531",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx -799151.7631007462$."
        },
        "39": {
          "number": "0.999998546665340246723115887568",
          "comment": "$a_2$ is root 1 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx -799151.7631007462$."
        }
      },
      "2": {
        "1": {
          "number": "244166266549071.844420402987462",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 241487.7444577613$."
        },
        "2": {
          "number": "253665843549559.885572043739760",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 241487.7444577613$."
        },
        "3": {
          "number": "135328849386138.297975531545052",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 241487.7444577613$."
        },
        "4": {
          "number": "49468316129727.4940377134966116",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 241487.7444577613$."
        },
        "5": {
          "number": "13949554447136.7883774498440925",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 241487.7444577613$."
        },
        "6": {
          "number": "3239471390185.59392045199117878",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 241487.7444577613$."
        },
        "7": {
          "number": "645913904557.364426363363277105",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 241487.7444577613$."
        },
        "8": {
          "number": "113840831924.947551563963201505",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 241487.7444577613$."
        },
        "9": {
          "number": "18122955587.1923012298012308023",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 241487.7444577613$."
        },
        "10": {
          "number": "2650142823.46809153877073660014",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 241487.7444577613$."
        },
        "11": {
          "number": "360840203.613835990098193949941",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 241487.7444577613$."
        },
        "12": {
          "number": "46267665.4133729060385918472768",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 241487.7444577613$."
        },
        "13": {
          "number": "5641006.59686436722540941659581",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 241487.7444577613$."
        },
        "14": {
          "number": "659474.221811828050357684264167",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 241487.7444577613$."
        },
        "15": {
          "number": "74454.8596875042206344985604914",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 241487.7444577613$."
        },
        "16": {
          "number": "8158.23815759010185006147215238",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 241487.7444577613$."
        },
        "17": {
          "number": "867.254698897004997506001361049",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 241487.7444577613$."
        },
        "18": {
          "number": "88.3456291883308938140656710677",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 241487.7444577613$."
        },
        "19": {
          "number": "8.61758707799187106135236044627",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 241487.7444577613$."
        },
        "20": {
          "number": "1.30703268324762197909347901279",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 241487.7444577613$."
        },
        "21": {
          "number": "0.895286056333887963666559035187",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 241487.7444577613$."
        },
        "22": {
          "number": "0.958581724986059246475712875740",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 241487.7444577613$."
        },
        "23": {
          "number": "0.993296426564909124838257970222",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 241487.7444577613$."
        },
        "24": {
          "number": "1.00239873230292317086339606486",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 241487.7444577613$."
        },
        "25": {
          "number": "1.00321591922324132122814322079",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 241487.7444577613$."
        },
        "26": {
          "number": "1.00228419476505710840318779246",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 241487.7444577613$."
        },
        "27": {
          "number": "1.00136567825198469657739449288",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 241487.7444577613$."
        },
        "28": {
          "number": "1.00075647127751388929057108408",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 241487.7444577613$."
        },
        "29": {
          "number": "1.00040248126595213279243303635",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 241487.7444577613$."
        },
        "30": {
          "number": "1.00020923412733757871300954332",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 241487.7444577613$."
        },
        "31": {
          "number": "1.00010725680819025327689251465",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 241487.7444577613$."
        },
        "32": {
          "number": "1.00005450160474191316264345277",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 241487.7444577613$."
        },
        "33": {
          "number": "1.00002754008541895728235340315",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 241487.7444577613$."
        },
        "34": {
          "number": "1.00001386600323354703066572430",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 241487.7444577613$."
        },
        "35": {
          "number": "1.00000696486739078732138576149",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 241487.7444577613$."
        },
        "36": {
          "number": "1.00000349302448976874720774183",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 241487.7444577613$."
        },
        "37": {
          "number": "1.00000175003455288804600831137",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 241487.7444577613$."
        },
        "38": {
          "number": "1.00000087618935479282642903542",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 241487.7444577613$."
        },
        "39": {
          "number": "1.00000043848485702524404595448",
          "comment": "$a_2$ is root 2 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 241487.7444577613$."
        }
      },
      "3": {
        "1": {
          "number": "244166651111473.314812726974144",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 1106520.018642985$."
        },
        "2": {
          "number": "253666642983841.660225283972958",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 1106520.018642985$."
        },
        "3": {
          "number": "135329702988943.360329653168487",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 1106520.018642985$."
        },
        "4": {
          "number": "49468940863539.2322345667398578",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 1106520.018642985$."
        },
        "5": {
          "number": "13949907360120.1738518884496128",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 1106520.018642985$."
        },
        "6": {
          "number": "3239635705040.18009752606205138",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 1106520.018642985$."
        },
        "7": {
          "number": "645979671448.282700555158973393",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 1106520.018642985$."
        },
        "8": {
          "number": "113864143054.233770871970884771",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 1106520.018642985$."
        },
        "9": {
          "number": "18130439529.5285146521042404903",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 1106520.018642985$."
        },
        "10": {
          "number": "2652359019.92587493381084205887",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 1106520.018642985$."
        },
        "11": {
          "number": "361455066.956028896824736630075",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 1106520.018642985$."
        },
        "12": {
          "number": "46429791.4560760902212748845076",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 1106520.018642985$."
        },
        "13": {
          "number": "5682207.95413190479025490801294",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 1106520.018642985$."
        },
        "14": {
          "number": "669713.753331657490520762225135",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 1106520.018642985$."
        },
        "15": {
          "number": "76983.1099519956202847859965260",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 1106520.018642985$."
        },
        "16": {
          "number": "8789.27302843841426275665129599",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 1106520.018642985$."
        },
        "17": {
          "number": "1029.43060668759704864649841125",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 1106520.018642985$."
        },
        "18": {
          "number": "132.051404061092507408290862745",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 1106520.018642985$."
        },
        "19": {
          "number": "21.1679018091776797248242353411",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 1106520.018642985$."
        },
        "20": {
          "number": "5.19193807618327152515001462613",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 1106520.018642985$."
        },
        "21": {
          "number": "2.19914544060723604707774718733",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 1106520.018642985$."
        },
        "22": {
          "number": "1.43280503919296008823430786367",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 1106520.018642985$."
        },
        "23": {
          "number": "1.17904203265755038261074447859",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 1106520.018642985$."
        },
        "24": {
          "number": "1.07993367825063236762742150188",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 1106520.018642985$."
        },
        "25": {
          "number": "1.03728194155897105998376368415",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 1106520.018642985$."
        },
        "26": {
          "number": "1.01784647190142113728695739415",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 1106520.018642985$."
        },
        "27": {
          "number": "1.00867955465979526767214333243",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 1106520.018642985$."
        },
        "28": {
          "number": "1.00426321156683332453054176799",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 1106520.018642985$."
        },
        "29": {
          "number": "1.00210714390334366759671963504",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 1106520.018642985$."
        },
        "30": {
          "number": "1.00104566458009877390243653761",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 1106520.018642985$."
        },
        "31": {
          "number": "1.00052025484286350706353343901",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 1106520.018642985$."
        },
        "32": {
          "number": "1.00025928225669237538774322233",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 1106520.018642985$."
        },
        "33": {
          "number": "1.00012936279225345122711807101",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 1106520.018642985$."
        },
        "34": {
          "number": "1.00006458944332142757950662689",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 1106520.018642985$."
        },
        "35": {
          "number": "1.00003226427338364623730952745",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 1106520.018642985$."
        },
        "36": {
          "number": "1.00001612203721602307671019585",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 1106520.018642985$."
        },
        "37": {
          "number": "1.00000805766446896854073599031",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 1106520.018642985$."
        },
        "38": {
          "number": "1.00000402771725087817946445947",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 1106520.018642985$."
        },
        "39": {
          "number": "1.00000201348772507251237863243",
          "comment": "$a_2$ is root 3 of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\\chi_{40,2}$] in increasing order; $a_2\\approx 1106520.018642985$."
        }
      }
    }
  }
}