{
  "Title": "Fourier coefficients of monstrous McKay\u2013Thompson series",
  "Definition": "The entry is the signed coefficient $c_g(n)=\\operatorname{Tr}(g\\mid V_n^\\natural)$ in the normalized Monster McKay\u2013Thompson series $T_g(\\tau)=q^{-1}+\\sum_{n\\geq1}c_g(n)q^n$, where $q=e^{2\\pi i\\tau}$ and $V_n^\\natural$ is the degree-$n$ component of the moonshine module in the shifted grading CITE{DGO}.",
  "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": "Fourier exponent",
      "constraints": "$n\\geq1$"
    }
  },
  "Tags": [
    "number theory",
    "modular function",
    "groups",
    "Taylor series"
  ],
  "Links": {
    "CTblLib": {
      "title": "GAP Character Table Library 1.3.11, Monster character table",
      "url": "https://www.math.rwth-aachen.de/~Thomas.Breuer/ctbllib/"
    }
  },
  "References": {
    "DGO": {
      "bib": "John F. R. Duncan, Michael J. Griffin and Ken Ono, Moonshine, Research in the Mathematical Sciences 2 (2015), article 11, \u00a73.",
      "arxiv": "1411.6571"
    },
    "CN": {
      "bib": "John H. Conway and Simon P. Norton, Monstrous Moonshine, Bulletin of the London Mathematical Society 11 (1979), 308\u2013339, \u00a7\u00a76, 8, 10 and Tables 1a, 2, 3, 4.",
      "doi": "10.1112/blms/11.3.308",
      "url": "https://gwern.net/doc/math/1979-conway.pdf"
    },
    "Borcherds": {
      "bib": "Richard E. Borcherds, Monstrous moonshine and monstrous Lie superalgebras, Inventiones Mathematicae 109 (1992), 405\u2013444, \u00a7\u00a78\u20139.",
      "doi": "10.1007/BF01232032",
      "url": "https://math.berkeley.edu/~reb/papers/monster/monster.pdf"
    }
  },
  "Data properties": {
    "type": "Z",
    "rigour": "exact",
    "complete": "no",
    "complete-note": "it holds every $1\\leq n\\leq30$ for all 32 nonidentity Monster classes of order at most 10: $2A,2B,3A,3B,3C,4A,4B,4C,4D,5A,5B,6A,6B,6C,6D,6E,6F,7A,7B,8A,8B,8C,8D,8E,8F,9A,9B,10A,10B,10C,10D,10E$. These noncanonical practical bounds cover a shared initial graded segment across all low-order classes, including prime and composite orders and sparse and dense series. Including the identity internally makes the selected classes closed under powers for verification. The exponent bound extends beyond the ten-term published controls while keeping entries short; the longest integer has 20 characters. A single leading $1A$ reference links to HREF{Q-expansion_of_the_j-invariant}[the $j$-invariant expansion] for every $n\\geq1$; those coefficients are not duplicated here",
    "rigour details": "Computed using exact integer arithmetic and the four coefficient recurrences in CITE{Borcherds}, \u00a79, equation (9.1). Every division by two is checked for exact divisibility. The seeds at exponents 1, 2, 3 and 5 are transcribed from the scanned Table 4 in CITE{CN}; the squared-class data are frozen from GAP CTblLib 1.3.11 CITE{CTblLib}. The identity-class series is computed internally from $j=E_4^3/\\Delta$, using HREF{Q-expansion_of_the_Eisenstein_series_E4}[$E_4$] and HREF{Q-expansion_of_the_modular_discriminant}[$\\Delta$].\n\nAll first-ten coefficients for the 32 stored classes and the identity agree with the scanned published controls. Character sums using the decompositions in CITE{CN}, Table 1a, agree for exponents 1 through 5 in all 33 classes. Independent finite eta products agree with all 240 coefficients of the eight prime-order series. Triangular Laurent cancellation checks the replication relations at multipliers 3 and 5 through exponents 10 and 6, respectively, for every stored class. It also reconstructs all 171 overlapping HREF{Hecke-type_Faber_polynomials_of_monstrous_McKay_Thompson_series}[Faber polynomials] through degree 20. The identity coefficients agree with HREF{Q-expansion_of_the_j-invariant}[the separate $j$ expansion] through exponent 30. Known-answer controls and deliberately incorrect signs and exponent offsets precede the comparisons. The attached checker repeats these tests on values read from the database."
  },
  "Display properties": {
    "number-header": "$c_g(n)$",
    "group parameters": [
      [
        "g"
      ],
      [
        "n"
      ]
    ]
  },
  "Numbers": {
    "1A": {
      "equals": "HREF{Q-expansion_of_the_j-invariant}[$q^n$ coefficients of the $j$-invariant, $n\\geq1$]",
      "comment": "The identity class: $T_{1A}=j-744$. For every $n\\geq1$, its coefficient equals the coefficient of $q^n$ in $j$; this single reference covers the whole positive-exponent family."
    },
    "2A": {
      "1": "4372",
      "2": "96256",
      "3": "1240002",
      "4": "10698752",
      "5": "74428120",
      "6": "431529984",
      "7": "2206741887",
      "8": "10117578752",
      "9": "42616961892",
      "10": "166564106240",
      "11": "611800208702",
      "12": "2125795885056",
      "13": "7040425608760",
      "14": "22327393665024",
      "15": "68134255043715",
      "16": "200740384538624",
      "17": "572950514538856",
      "18": "1588198806411264",
      "19": "4286153037862594",
      "20": "11283613372743680",
      "21": "29030493318777216",
      "22": "73105649849655296",
      "23": "180454128099240060",
      "24": "437155796944945152",
      "25": "1040545340935546700",
      "26": "2436001535686266880",
      "27": "5614282459787463036",
      "28": "12748902531008430080",
      "29": "28546382905646406648",
      "30": "63071424165763399680"
    },
    "2B": {
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    "3A": {
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      "1": "12",
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      "4": "0",
      "5": "232",
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      "7": "639",
      "8": "0",
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      "11": "3774",
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      "13": "8328",
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      "17": "34520",
      "18": "0",
      "19": "66882",
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      "21": "125568",
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      "24": "0",
      "25": "409236",
      "26": "0",
      "27": "716412",
      "28": "0",
      "29": "1231048",
      "30": "0"
    },
    "8C": {
      "1": "0",
      "2": "0",
      "3": "26",
      "4": "0",
      "5": "0",
      "6": "0",
      "7": "79",
      "8": "0",
      "9": "0",
      "10": "0",
      "11": "326",
      "12": "0",
      "13": "0",
      "14": "0",
      "15": "755",
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      "17": "0",
      "18": "0",
      "19": "2106",
      "20": "0",
      "21": "0",
      "22": "0",
      "23": "4460",
      "24": "0",
      "25": "0",
      "26": "0",
      "27": "10284",
      "28": "0",
      "29": "0",
      "30": "0"
    },
    "8D": {
      "1": "-4",
      "2": "0",
      "3": "2",
      "4": "0",
      "5": "8",
      "6": "0",
      "7": "-1",
      "8": "0",
      "9": "-20",
      "10": "0",
      "11": "-2",
      "12": "0",
      "13": "40",
      "14": "0",
      "15": "3",
      "16": "0",
      "17": "-72",
      "18": "0",
      "19": "2",
      "20": "0",
      "21": "128",
      "22": "0",
      "23": "-4",
      "24": "0",
      "25": "-220",
      "26": "0",
      "27": "-4",
      "28": "0",
      "29": "360",
      "30": "0"
    },
    "8E": {
      "1": "4",
      "2": "0",
      "3": "2",
      "4": "0",
      "5": "-8",
      "6": "0",
      "7": "-1",
      "8": "0",
      "9": "20",
      "10": "0",
      "11": "-2",
      "12": "0",
      "13": "-40",
      "14": "0",
      "15": "3",
      "16": "0",
      "17": "72",
      "18": "0",
      "19": "2",
      "20": "0",
      "21": "-128",
      "22": "0",
      "23": "-4",
      "24": "0",
      "25": "220",
      "26": "0",
      "27": "-4",
      "28": "0",
      "29": "-360",
      "30": "0"
    },
    "8F": {
      "1": "0",
      "2": "0",
      "3": "-6",
      "4": "0",
      "5": "0",
      "6": "0",
      "7": "15",
      "8": "0",
      "9": "0",
      "10": "0",
      "11": "-26",
      "12": "0",
      "13": "0",
      "14": "0",
      "15": "51",
      "16": "0",
      "17": "0",
      "18": "0",
      "19": "-102",
      "20": "0",
      "21": "0",
      "22": "0",
      "23": "172",
      "24": "0",
      "25": "0",
      "26": "0",
      "27": "-276",
      "28": "0",
      "29": "0",
      "30": "0"
    },
    "9A": {
      "1": "27",
      "2": "86",
      "3": "243",
      "4": "594",
      "5": "1370",
      "6": "2916",
      "7": "5967",
      "8": "11586",
      "9": "21870",
      "10": "39852",
      "11": "71052",
      "12": "123444",
      "13": "210654",
      "14": "352480",
      "15": "581013",
      "16": "942786",
      "17": "1510254",
      "18": "2388204",
      "19": "3734964",
      "20": "5777788",
      "21": "8852004",
      "22": "13434984",
      "23": "20218395",
      "24": "30177684",
      "25": "44704413",
      "26": "65743348",
      "27": "96033357",
      "28": "139368816",
      "29": "201032186",
      "30": "288281592"
    },
    "9B": {
      "1": "0",
      "2": "5",
      "3": "0",
      "4": "0",
      "5": "-7",
      "6": "0",
      "7": "0",
      "8": "3",
      "9": "0",
      "10": "0",
      "11": "15",
      "12": "0",
      "13": "0",
      "14": "-32",
      "15": "0",
      "16": "0",
      "17": "9",
      "18": "0",
      "19": "0",
      "20": "58",
      "21": "0",
      "22": "0",
      "23": "-96",
      "24": "0",
      "25": "0",
      "26": "22",
      "27": "0",
      "28": "0",
      "29": "149",
      "30": "0"
    },
    "10A": {
      "1": "22",
      "2": "56",
      "3": "177",
      "4": "352",
      "5": "870",
      "6": "1584",
      "7": "3412",
      "8": "5952",
      "9": "11442",
      "10": "19240",
      "11": "34377",
      "12": "56256",
      "13": "95560",
      "14": "151824",
      "15": "247965",
      "16": "385024",
      "17": "609756",
      "18": "927864",
      "19": "1431094",
      "20": "2139680",
      "21": "3228516",
      "22": "4752896",
      "23": "7038610",
      "24": "10215552",
      "25": "14885450",
      "26": "21330480",
      "27": "30643161",
      "28": "43407680",
      "29": "61571148",
      "30": "86305680"
    },
    "10B": {
      "1": "6",
      "2": "-8",
      "3": "17",
      "4": "-32",
      "5": "54",
      "6": "-80",
      "7": "116",
      "8": "-192",
      "9": "290",
      "10": "-408",
      "11": "585",
      "12": "-832",
      "13": "1192",
      "14": "-1648",
      "15": "2237",
      "16": "-3072",
      "17": "4156",
      "18": "-5576",
      "19": "7414",
      "20": "-9824",
      "21": "12964",
      "22": "-16896",
      "23": "22002",
      "24": "-28544",
      "25": "36794",
      "26": "-47184",
      "27": "60185",
      "28": "-76736",
      "29": "97388",
      "30": "-122864"
    },
    "10C": {
      "1": "-3",
      "2": "6",
      "3": "2",
      "4": "2",
      "5": "-5",
      "6": "-16",
      "7": "12",
      "8": "2",
      "9": "17",
      "10": "-10",
      "11": "-48",
      "12": "56",
      "13": "10",
      "14": "24",
      "15": "-35",
      "16": "-126",
      "17": "106",
      "18": "14",
      "19": "94",
      "20": "-70",
      "21": "-284",
      "22": "296",
      "23": "60",
      "24": "152",
      "25": "-175",
      "26": "-620",
      "27": "536",
      "28": "80",
      "29": "398",
      "30": "-320"
    },
    "10D": {
      "1": "21",
      "2": "62",
      "3": "162",
      "4": "378",
      "5": "819",
      "6": "1680",
      "7": "3276",
      "8": "6138",
      "9": "11145",
      "10": "19662",
      "11": "33840",
      "12": "57048",
      "13": "94362",
      "14": "153432",
      "15": "245757",
      "16": "388218",
      "17": "605466",
      "18": "933414",
      "19": "1423614",
      "20": "2149586",
      "21": "3215844",
      "22": "4769544",
      "23": "7016572",
      "24": "10243896",
      "25": "14848809",
      "26": "21378276",
      "27": "30582360",
      "28": "43484304",
      "29": "61473438",
      "30": "86428896"
    },
    "10E": {
      "1": "1",
      "2": "2",
      "3": "2",
      "4": "-2",
      "5": "-1",
      "6": "0",
      "7": "-4",
      "8": "-2",
      "9": "5",
      "10": "2",
      "11": "0",
      "12": "8",
      "13": "2",
      "14": "-8",
      "15": "-3",
      "16": "-2",
      "17": "-14",
      "18": "-6",
      "19": "14",
      "20": "6",
      "21": "4",
      "22": "24",
      "23": "12",
      "24": "-24",
      "25": "-11",
      "26": "-4",
      "27": "-40",
      "28": "-16",
      "29": "38",
      "30": "16"
    }
  },
  "Keywords": [
    "moonshine",
    "Monster",
    "McKay-Thompson",
    "McKay\u2013Thompson",
    "graded traces",
    "head characters",
    "Fourier coefficients",
    "replicable function"
  ],
  "Comments": {
    "comment-grading": "The grading is $V^\\natural=\\bigoplus_{n\\geq-1}V_n^\\natural$, with $\\dim V_{-1}^\\natural=1$ and $V_0^\\natural=0$. The pole coefficient and the zero constant coefficient are not repeated as entries. A grading by conformal weight assigns weight $n+1$ to $V_n^\\natural$ CITE{DGO}.",
    "comment-identity": "The identity class satisfies $T_{1A}=j-744$. Its positive Fourier coefficients equal those in HREF{Q-expansion_of_the_j-invariant}[the $j$-invariant expansion]. The first $1A$ row references that whole family for $n\\geq1$, without duplicating its entries.",
    "comment-characters": "The class labels are Monster conjugacy classes in the ATLAS notation CITE{CTblLib}. These values are traces on graded representations, rather than individual HREF{Degrees_of_the_irreducible_characters_of_the_sporadic_simple_groups}[irreducible character degrees]. Negative traces and structural zeros are retained. Conway and Norton call the coefficient characters $H_n$; their Table 4 row $H_0$ is a Rademacher constant, not the constant coefficient of $T_g$ CITE{CN}, \u00a710.",
    "comment-faber": "The HREF{Hecke-type_Faber_polynomials_of_monstrous_McKay_Thompson_series}[Hecke-type Faber polynomials] $P_{m,g}(X)$ are the unique monic polynomials satisfying $P_{m,g}(T_g(\\tau))=q^{-m}+O(q)$. Their coefficients are determined by the coefficients tabulated here. The class parameter selects the acting group element, while the entries always give the same graded-trace quantity."
  },
  "Formulas": {
    "formula-faber": "$P_{2,g}(X)=X^2-2c_g(1)$ and $P_{3,g}(X)=X^3-3c_g(1)X-3c_g(2)$, where $P_{m,g}$ is the monic HREF{Hecke-type_Faber_polynomials_of_monstrous_McKay_Thompson_series}[Hecke-type Faber polynomial].",
    "formula-replication": "$P_{m,g}(T_g(\\tau))=\\sum_{ad=m,\\ a,d>0}\\sum_{b=0}^{d-1}T_{g^a}((a\\tau+b)/d)$ for $m\\geq1$, where $P_{m,g}$ is the monic Faber polynomial and $g^a$ denotes the conjugacy class of the $a$th power. There is no factor $1/m$ CITE{CN}, \u00a78; CITE{Borcherds}, \u00a7\u00a78\u20139.",
    "formula-eta": "For $p\\in\\{2,3,5,7\\}$, put $e_p=24/(p-1)$ and $R_p(\\tau)=(\\eta(\\tau)/\\eta(p\\tau))^{e_p}$, with $\\eta(\\tau)=q^{1/24}\\prod_{r\\geq1}(1-q^r)$. Then $T_{pB}=R_p+e_p$ and $T_{pA}=R_p+e_p+p^{12/(p-1)}/R_p$ CITE{CN}, \u00a76 and Table 3. HREF{Values_of_monstrous_McKay_Thompson_series_at_rectangular_CM_points}[Rectangular CM evaluations] use the same normalization."
  },
  "Similar tables": [
    {
      "table": "HREF{Q-expansion_of_the_j-invariant}[$q$-expansion of the $j$-invariant]",
      "relation": "Its positive Fourier coefficients give the identity-class traces, referenced by the leading 1A row."
    },
    {
      "table": "HREF{Hecke-type_Faber_polynomials_of_monstrous_McKay_Thompson_series}[Hecke-type Faber polynomials]",
      "relation": "These polynomial transforms of the same series are determined by their Fourier coefficients."
    },
    {
      "table": "HREF{Values_of_monstrous_McKay_Thompson_series_at_rectangular_CM_points}[McKay\u2013Thompson values at rectangular CM points]",
      "relation": "These evaluate the same normalized series at principal rectangular CM arguments."
    },
    {
      "table": "HREF{Degrees_of_the_irreducible_characters_of_the_sporadic_simple_groups}[Degrees of sporadic-group irreducible characters]",
      "relation": "The Monster degrees give dimensions of irreducible constituents of the graded moonshine representations."
    }
  ],
  "Programs": {
    "program-generator": {
      "language": "Sage",
      "code": "# Save generate.py, seeds.json and class-data.json from the attachments\n# in the working directory.\n# The output is an exact integer; the digits argument does not round it.\nfrom generate import McKayThompsonCoefficients\ngenerator = McKayThompsonCoefficients()\ngenerator.value({'g': '2A', 'n': 30}, digits=100)",
      "comment": "Save generate.py, seeds.json and class-data.json from the attachments in the working directory. The output is an exact integer; the digits argument does not round it."
    }
  }
}