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{"Title": "Zeros of Hecke-type Faber polynomials of monstrous McKay\u2013Thompson series", "Definition": "The entry $z_{g,n,k}$ is the $k$th distinct zero of the monic Hecke-type Faber polynomial $P_{n,g}(X)$, defined by $P_{n,g}(T_g(\\tau))=q^{-n}+O(q)$ for the normalized McKay\u2013Thompson series $T_g$ CITE{BKM}.", "Parameters": {"g": {"type": "Symbolic", "display": "$g$", "title": "Monster conjugacy class", "constraints": "a Monster conjugacy class in the ATLAS notation"}, "n": {"type": "Z", "display": "$n$", "title": "polynomial degree", "constraints": "$n\\geq1$"}, "k": {"type": "Z", "display": "$k$", "title": "distinct root index", "constraints": "$1\\leq k\\leq d_{g,n}$, where $d_{g,n}$ is the number of distinct roots of $P_{n,g}$"}}, "Tags": ["number theory", "modular function", "groups", "polynomial", "zero", "algebraic"], "Keywords": ["McKay-Thompson", "McKay\u2013Thompson", "Monster", "moonshine", "Hauptmodul", "Faber polynomial", "polynomial zeros", "Hecke-type Faber roots"], "Comments": {"comment-order": "Distinct zeros are ordered by increasing exact algebraic real part, then increasing exact algebraic imaginary part, starting at $k=1$. Every zero is listed once and its entry comment gives its multiplicity. Both members of every nonreal conjugate pair are retained.", "comment-normalization": "For the Monster conjugacy class $g$, write $T_g(\\tau)=q^{-1}+\\sum_{m\\geq1}c_g(m)q^m$, with $q=e^{2\\pi i\\tau}$ CITE{Matsusaka}. Every series has constant coefficient zero. The identity series is $T_{1A}=j-744$, whose positive Fourier coefficients are in HREF{Q-expansion_of_the_j-invariant}[the $j$-invariant expansion]. In particular, the class $13A$ series begins $q^{-1}+12q+28q^2+\\cdots$ CITE{OEIS-13A}. No division by $n$ is made in the polynomial normalization.", "comment-classes": "The primes $p=2,3,5,7,13$ are precisely those for which $X_0(p)$ has genus zero. Their classes $pB$ and $pA$ give the non-Fricke and Fricke Hauptmoduln, respectively CITE{Matsusaka}; the identity class $1A$ supplies the classical case.", "comment-plane": "These are roots of HREF{Hecke-type_Faber_polynomials_of_monstrous_McKay_Thompson_series}[the defining polynomials] in the $X$-plane, not the points $\\tau$ in the upper half-plane at which $P_{n,g}(T_g(\\tau))$ vanishes. The HREF{Values_of_monstrous_McKay_Thompson_series_at_rectangular_CM_points}[rectangular CM values] use the same series normalization and supply arguments for evaluating the polynomials, rather than a list of their zeros.", "comment-trivial": "The degree-one polynomial is $P_{1,g}(X)=X$ for every class, so its single zero is $0$. This common case is omitted from the entries.", "comment-real": "All zeros for the classes $1A$ and $2A$ are real for every degree CITE{BKM}, \u00a74. The other selected classes are not assumed to have only real zeros; the algebraic root isolation decides this separately for each polynomial."}, "Formulas": {"formula-eta": "For $p\\in\\{2,3,5,7,13\\}$, set $e_p=b_p=24/(p-1)$, $C_p=p^{12/(p-1)}$ and $R_p(\\tau)=(\\eta(\\tau)/\\eta(p\\tau))^{e_p}$. Then $T_{pB}=R_p+b_p$ and $T_{pA}=R_p+b_p+C_p/R_p$ CITE{Matsusaka}. Here $\\eta(\\tau)=q^{1/24}\\prod_{m\\geq1}(1-q^m)$; its HREF{Dedekind_eta_values_at_rectangular_CM_points}[rectangular CM values] use this product normalization.", "formula-roots": "If $\\mu_{g,n,k}$ is the multiplicity of $z_{g,n,k}$, then $P_{n,g}(X)=\\prod_{k=1}^{d_{g,n}}(X-z_{g,n,k})^{\\mu_{g,n,k}}$. For $n\\geq2$, $\\sum_{k=1}^{d_{g,n}}\\mu_{g,n,k}z_{g,n,k}=0$ and $\\prod_{k=1}^{d_{g,n}}z_{g,n,k}^{\\mu_{g,n,k}}=(-1)^nP_{n,g}(0)$.", "formula-degree-two": "$P_{2,g}(X)=X^2-2c_g(1)$ CITE{BKM}, \u00a72.2. In particular, $P_{2,7B}=X^2-4$ has the exact roots $-2,2$, and $P_{2,13B}=X^2+2$ has the roots $-i\\sqrt2,i\\sqrt2$."}, "Links": {"OEIS-13A": {"title": "OEIS A034319: zero-constant McKay\u2013Thompson series of class 13A", "url": "https://oeis.org/A034319"}}, "References": {"BKM": {"bib": "Eiichi Bannai, Koji Kojima and Tsuyoshi Miezaki, On the zeros of Hecke-type Faber polynomials, Kyushu Journal of Mathematics 62 (2008), 15\u201361, \u00a7\u00a71\u20135.", "doi": "10.2206/kyushujm.62.15"}, "Matsusaka": {"bib": "Toshiki Matsusaka, The Fourier coefficients of the McKay\u2013Thompson series and the traces of CM values, Research in Number Theory 3 (2017), article 23, \u00a71.", "doi": "10.1007/s40993-017-0090-x"}}, "Similar tables": [{"table": "HREF{Hecke-type_Faber_polynomials_of_monstrous_McKay_Thompson_series}[Hecke-type Faber polynomials of monstrous McKay\u2013Thompson series]", "relation": "These are the exact polynomials $P_{n,g}(X)$ whose distinct zeros are listed here, with the same class labels and monic normalization."}, {"table": "HREF{Values_of_monstrous_McKay_Thompson_series_at_rectangular_CM_points}[McKay\u2013Thompson values at rectangular CM points]", "relation": "These give $T_g(i\\sqrt r)$ for the ten nonidentity classes in the same normalization; evaluating $P_{n,g}$ at them gives CM specializations."}, {"table": "HREF{Q-expansion_of_the_j-invariant}[$q$-expansion of the $j$-invariant]", "relation": "The positive Fourier coefficients determine the identity-class polynomials and their roots."}], "Programs": {"program-sage": {"language": "Sage", "code": "# Save the attached generate.py beside this script.\nfrom generate import enclosed_root\n# The first root is ordered by exact real part, then imaginary part.\nenclosed_root('5B', 10, 1, digits=100)"}}, "Display properties": {"number-header": "$z_{g,n,k}$", "group parameters": [["g"], ["n"], ["k"]]}, "Data properties": {"type": "C", "rigour": "proven", "complete": "no", "complete-note": "it holds every distinct root for $2\\leq n\\leq10$ and $g=1A,2A,2B,3A,3B,5A,5B,7A,7B,13A,13B$. This includes both Fricke and non-Fricke cases at every prime with genus-zero $X_0(p)$, together with the classical identity class. The degree bound is a noncanonical practical reference extent across all eleven classes, balancing their real and complex root configurations against the quadratic growth in total root count. Degree one is stated in CITE{comment-trivial}; other Monster classes and larger degrees are outside this coverage", "rigour details": "The generator constructs the polynomials over the integers using finite eta products and the Faber recurrence, and isolates their roots in Sage's exact algebraic closure. Exact algebraic comparisons determine the root order. Conversion to Arb balls at 429 bits gives bounds containing each root; the nonzero approximate components retain at least 427 bits of relative accuracy, supporting the 100 significant digits recorded. The roots $-2,2$ of $P_{2,7B}$ are stored as exact integers. All 99 selected polynomials are squarefree.\n\nIndependent checks on the written intervals use exact rational sign brackets for 504 real irrational roots, rational evaluation for two integer roots, and sign brackets for the two purely imaginary roots after substituting $X=iY$. For the remaining 86 roots, Taylor bounds at 768 bits certify one root in a disk contained in its recorded complex rectangle by Rouch\u00e9's theorem. The disjoint enclosures account for every root. Known-answer controls and deliberately incorrect enclosures precede the certification. The full generated balls lie inside the intervals denoted by the written decimals.\n\nThe exact polynomials agree with HREF{Hecke-type_Faber_polynomials_of_monstrous_McKay_Thompson_series}[the companion table] for all 99 class-degree pairs. Their Laurent principal parts are checked through the constant term. Four explicit degree-two and degree-three examples from CITE{BKM}, pp. 24 and 26, agree after converting the shifts $744$ and $104$ to the zero-constant convention. The first nine positive coefficients of the identity series agree with HREF{Q-expansion_of_the_j-invariant}[the $j$-invariant expansion], and those of class $13A$ agree with CITE{OEIS-13A}. Conjugation, lexicographic ordering, multiplicities, the root sums and the root products are checked on every class-degree pair. The attached checker repeats these tests on entries read from the database."}, "Numbers": {"1A": {"2": {"1": {"number": "-627.5093624799553538530655884523509300222021812648872799599659204447302767233675356699825512778882923", "comment": "Multiplicity 1."}, "2": {"number": "627.5093624799553538530655884523509300222021812648872799599659204447302767233675356699825512778882923", "comment": "Multiplicity 1."}}, "3": {"1": {"number": "-706.6877904338276167148585200684368064120339892630150905047976094532584801614449653819095505294201838", "comment": "Multiplicity 1."}, "2": {"number": "-111.5176703473043583936952580520189699173226718355932504797774204353097371164020826534985884409105157", "comment": "Multiplicity 1."}, "3": {"number": "818.2054607811319751085537781204557763293566610986083409845750298885682172778470480354081389703306995", "comment": "Multiplicity 1."}}, "4": {"1": {"number": "-727.6662591807221986539414337782814498272458562527464478675038487638895994674385090570254105791722652", "comment": "Multiplicity 1."}, "2": {"number": "-428.5799582910797621399141499877213571977962743699490233305820634829014593936555096777383424651242937", "comment": "Multiplicity 1."}, "3": {"number": "267.1244134876527273646315490120398763483006085729917454208214479790678294181766060539508834207141176", "comment": "Multiplicity 1."}, "4": {"number": "889.1218039841492334292240347539629306767415220497037257772644642677232294429174126808128696235824414", "comment": "Multiplicity 1."}}, "5": {"1": {"number": "-735.4521791327011653615339000065642496688664219630377466834263217767195142054933150567950394336168821", "comment": "Multiplicity 1."}, "2": {"number": "-567.7777245097102110451840339152071968691863436494971949643855694554217682111753895607118469491058788", "comment": "Multiplicity 1."}, "3": {"number": 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