Thompson moonshine coefficients

edit · history · discussion · files · short url · Taylor series groups modular form number theory

Numbers

$[g]$

$n$

$c_{[g]}(n)$

248

comment: This is the degree of the 248-dimensional irreducible representation.

54000

comment: The even component is the sum of the two irreducible representations of degree $27000$.

-171990

comment: The odd component is the sum of two irreducible representations of degree $85995$, so its supertrace is negative.

3414528

-8192000

12:

88660992

13:

-183902292

16:

1417878000

17:

-2687827968

20:

16555069440

21:

-29466050250

comment: The sign is negative. The positive sign in [2], Table 6, is a typographical error; Table 1 and the exact modular-form construction give $-29466050250$.

24:

154184576000

25:

-261761531904

28:

1208278536192

29:

-1976452670250

32:

8251985424384

33:

-13096231436288

36:

50337746998000

37:

-77897942407350

40:

279250592947200

41:

-422964412416000

44:

1428218123904000

45:

-2123461679148810

48:

6806527762341888

49:

-9956715872256000

52:

30484812119150592

53:

-43956516990730734

56:

129203628422400000

57:

-183919810117435392

60:

521175528272588800

61:

-733341541502130750

64:

2010450258635670000

65:

-2799317972297318400

68:

7446765885087449088

69:

-10269820018890314750

72:

26577801382888723968

73:

-36332456851182256128

76:

91675725880781952000

77:

-124311097789669982268

80:

306416810521123061760

81:

-412391756829925376000

84:

994703198461400064000

85:

-1329424610499790172550

88:

3142537389457197456384

89:

-4172807127458930688000

92:

9679622929050549915648

93:

-12775168664023696440120

96:

29115811870275859456000

97:

-38208606696510213390336

100:

85648872592091236717296

101:

-111795801453256486135500

104:

246720197270785311360000

105:

-320415440871807624806400

108:

696774883043422048997376

109:

-900590096624482232923500

112:

1931312964721758493704192

113:

-2484983923005320719368192

116:

5259105072740190518784000

117:

-6737827082726372594701446

120:

14081877289865234010777600

121:

-17967933476075186380800000

124:

37107122597336702776320000

125:

-47163874069526662951489350

128:

96301285200294294436872192

129:

-121948744507025693859840000

132:

246314320014796897252573184

133:

-310813946752458554276045706

136:

621315214126779430073088000

137:

-781368020307953777040556032

140:

1546538108306530897548748800

141:

-1938650493359943697729463250

144:

3800832540589574135423638000

145:

-4749752790796806748776038400

148:

9227686751471465432298344448

149:

-11497216500879825543994530000

152:

22141874867721972972166333440

153:

-27508855493107238960917315584

156:

52534077850157300011485696000

157:

-65088601844473235105544957084

160:

123298627244529139827095347200

161:

-152360453017205491203145728000

164:

286377961374920230500851712000

165:

-352976324980717853768269502850

168:

658489009772632697023885219840

169:

-809628513691543572511752192000

172:

1499473596179564334448374334464

173:

-1839265403097735693768426282810

176:

3382637149481776458375527424000

177:

-4139655957873551249376641351680

180:

7561972983667014836735795343360

181:

-9233794829321420740879972655250

184:

16757423941228510912851817728000

185:

-20418312349541793973215205785600

188:

36820882483597155472201372557312

189:

-44771725896647054443407643169250

192:

80243871450810208527473250336768

193:

-97374807126201131490812730605568

196:

173488334178223090757113454646000

197:

-210114836893841697455723751703284

200:

372197831185102195550907858009600

201:

-449922048487188586879996747776000

-8

240

-1536

12:

7168

13:

-84

16:

-26640

17:

20:

86016

21:

310

24:

-250880

25:

28:

675840

29:

-810

32:

-1708032

33:

36:

4096240

37:

1866

40:

-9397248

41:

44:

20751360

45:

-3850

48:

-44326912

49:

52:

91951104

53:

7698

56:

-185825280

57:

60:

366833664

61:

-14910

64:

-708952080

65:

68:

1343913984

69:

27650

72:

-2502887936

73:

76:

4586050560

77:

-49212

80:

-8277491712

81:

84:

14733025280

85:

85626

88:

-25884189696

89:

92:

44925947904

93:

-146232

96:

-77092413440

97:

100:

130880766192

101:

244020

104:

-219967749120

105:

108:

366189006848

109:

-398700

112:

-604138930176

113:

116:

988226334720

117:

641914

120:

-1603429849088

121:

124:

2581595627520

125:

-1019718

128:

-4125993566208

129:

132:

6548115718144

133:

1596534

136:

-10322532648960

137:

140:

16168260642816

141:

-2466770

144:

-25168871450640

145:

148:

38948971204608

149:

3770160

152:

-59932348578816

153:

156:

91717963878400

157:

-5704860

160:

-139625301172224

161:

164:

211482206208000

165:

8545918

168:

-318758623602688

169:

172:

478191990408192

173:

-12681018

176:

-714109051576320

177:

180:

1061730839572480

181:

18661230

184:

-1571861115402240

185:

188:

2317515417477120

189:

-27251170

192:

-3403263903334400

193:

196:

4978357936128240

197:

39492876

200:

-7255116957983232

201:

-54

-128

12:

336

13:

378

16:

-702

17:

20:

21:

-1794

24:

2912

25:

3456

28:

-5616

29:

32:

33:

-11648

36:

17836

37:

20358

40:

-30240

41:

44:

45:

-57330

48:

82656

49:

93312

52:

-133056

53:

56:

57:

-234624

60:

326560

61:

363636

64:

-500526

65:

68:

69:

-841232

72:

1138176

73:

1257984

76:

-1690416

77:

80:

81:

-2730752

84:

3619200

85:

3969810

88:

-5226336

89:

92:

93:

-8190702

96:

10663744

97:

11636352

100:

-15076854

101:

104:

105:

-23024640

108:

29553552

109:

32091228

112:

-41008032

113:

116:

117:

-61300512

120:

77712960

121:

84060288

124:

-106181712

125:

128:

129:

-155793792

132:

195435968

133:

210654000

136:

-263390400

137:

140:

141:

-380288454

144:

472625404

145:

507911040

148:

-629463744

149:

152:

153:

-895942656

156:

1104411648

157:

1183645764

160:

-1455330240

161:

164:

165:

-2045380170

168:

2502819904

169:

2676001536

172:

-3267079920

173:

176:

177:

-4539593344

180:

5518356480

181:

5887359450

184:

-7141912128

185:

188:

189:

-9822016152

192:

11868306240

193:

12637168128

196:

-15241442166

197:

200:

201:

-20765087616

-108

12:

13:

-378

16:

702

17:

864

20:

-1512

21:

-12

24:

-4

25:

-3456

28:

5616

29:

6480

32:

-10044

33:

36:

-2

37:

-20358

40:

30240

41:

34560

44:

-50760

45:

48:

49:

-93312

52:

133056

53:

149148

56:

-209520

57:

-48

60:

-32

61:

-363636

64:

500526

65:

556416

68:

-759888

69:

72:

-36

73:

-1257984

76:

1690416

77:

1862190

80:

-2482920

81:

84:

120

85:

-3969810

88:

5226336

89:

5724000

92:

-7495200

93:

-144

96:

-68

97:

-11636352

100:

15076854

101:

16420590

104:

-21169080

105:

108:

-108

109:

-32091228

112:

41008032

113:

44473536

116:

-56606040

117:

126

120:

264

121:

-84060288

124:

106181712

125:

114706044

128:

-144370620

129:

-336

132:

-184

133:

-210654000

136:

263390400

137:

283616640

140:

-353541456

141:

228

144:

-218

145:

-507911040

148:

629463744

149:

675806220

152:

-835226424

153:

288

156:

624

157:

-1183645764

160:

1455330240

161:

1558483200

164:

-1911565440

165:

-732

168:

-392

169:

-2676001536

172:

3267079920

173:

3490487478

176:

-4252049640

177:

464

180:

-504

181:

-5887359450

184:

7141912128

185:

7614406080

188:

-9218646720

189:

594

192:

1236

193:

-12637168128

196:

15241442166

197:

16218729576

200:

-19525407588

201:

-1488

-4

-54

108

160

12:

-312

13:

16:

17:

-864

20:

1512

21:

1770

24:

-2920

25:

28:

29:

-6480

32:

10044

33:

11680

36:

-17840

37:

40:

41:

-34560

44:

50760

45:

57366

48:

-82584

49:

52:

53:

-149148

56:

209520

57:

234528

60:

-326624

61:

64:

65:

-556416

68:

759888

69:

841300

72:

-1138248

73:

76:

77:

-1862190

80:

2482920

81:

2730880

84:

-3618960

85:

88:

89:

-5724000

92:

7495200

93:

8190414

96:

-10663880

97:

100:

101:

-16420590

104:

21169080

105:

23024832

108:

-29553768

109:

112:

113:

-44473536

116:

56606040

117:

61300764

120:

-77712432

121:

124:

125:

-114706044

128:

144370620

129:

155793120

132:

-195436336

133:

136:

137:

-283616640

140:

353541456

141:

380288910

144:

-472625840

145:

148:

149:

-675806220

152:

835226424

153:

895943232

156:

-1104410400

157:

160:

161:

-1558483200

164:

1911565440

165:

2045378706

168:

-2502820688

169:

172:

173:

-3490487478

176:

4252049640

177:

4539594272

180:

-5518357488

181:

184:

185:

-7614406080

188:

9218646720

189:

9822017340

192:

-11868303768

193:

196:

197:

-16218729576

200:

19525407588

201:

20765084640

12:

13:

-84

16:

17:

20:

21:

310

24:

25:

28:

29:

-810

32:

33:

36:

37:

1866

40:

41:

44:

45:

-3850

48:

49:

52:

53:

7698

56:

57:

60:

61:

-14910

64:

65:

68:

69:

27650

72:

73:

76:

77:

-49212

80:

81:

84:

85:

85626

88:

89:

92:

93:

-146232

96:

97:

100:

101:

244020

104:

105:

108:

109:

-398700

112:

113:

116:

117:

641914

120:

121:

124:

125:

-1019718

128:

129:

132:

133:

1596534

136:

137:

140:

141:

-2466770

144:

145:

148:

149:

3770160

152:

153:

156:

157:

-5704860

160:

161:

164:

165:

8545918

168:

169:

172:

173:

-12681018

176:

177:

180:

181:

18661230

184:

185:

188:

189:

-27251170

192:

193:

196:

197:

39492876

200:

201:

-22

12:

13:

108

16:

17:

20:

21:

-330

24:

25:

28:

29:

790

32:

33:

36:

37:

-1782

40:

41:

44:

45:

3894

48:

49:

52:

53:

-7854

56:

57:

60:

61:

14850

64:

65:

68:

69:

-27390

72:

73:

76:

77:

49348

80:

81:

84:

85:

-86022

88:

89:

92:

93:

145992

96:

97:

100:

101:

-243340

104:

105:

108:

109:

399060

112:

113:

116:

117:

-643014

120:

121:

124:

125:

1019194

128:

129:

132:

133:

-1594890

136:

137:

140:

141:

2467630

144:

145:

148:

149:

-3772560

152:

153:

156:

157:

5703588

160:

161:

164:

165:

-8542338

168:

169:

172:

173:

12682758

176:

177:

180:

181:

-18666450

184:

185:

188:

189:

27248670

192:

193:

196:

197:

-39485492

200:

201:

-2

12:

-8

13:

-42

16:

17:

20:

-60

21:

24:

25:

28:

192

29:

32:

-116

33:

-288

36:

37:

150

40:

-300

41:

44:

45:

440

48:

888

49:

52:

-408

53:

-1234

56:

57:

608

60:

-1200

61:

64:

65:

1600

68:

3088

69:

72:

-1532

73:

-4128

76:

77:

1982

80:

-3740

81:

84:

85:

4950

88:

9384

89:

92:

-4352

93:

-12120

96:

97:

5664

100:

-10704

101:

104:

105:

13600

108:

25376

109:

112:

-11808

113:

-32192

116:

117:

14804

120:

-27400

121:

124:

125:

34400

128:

63692

129:

132:

-28816

133:

-79206

136:

137:

35968

140:

-66200

141:

144:

145:

81600

148:

149448

149:

152:

-67560

153:

-183584

156:

157:

82416

160:

-150300

161:

164:

165:

183400

168:

333840

169:

172:

-148536

173:

-405060

176:

177:

180320

180:

-327140

181:

184:

185:

394400

188:

713312

189:

192:

-316232

193:

-857568

196:

197:

378966

200:

-682900

201:

Definition

The entry $c_{[g]}(n)=\operatorname{str}(g\mid W_n)$ is the signed supertrace on grade $n$ of the Thompson supermodule specified by Harvey and Rayhaun [2], Conjecture 3.1.

Parameters

$[g]$

— Thompson conjugacy class (a Thompson conjugacy class in the notation of [1], Appendix A)

$n$

— Fourier grade ($n\geq0$ and $n\equiv0,1\pmod4$)

Formulas

(1)

$\mathcal T_{[g]}(\tau)=2q^{-3}+\sum_{n\geq0,\ n\equiv0,1\ (\mathrm{mod}\ 4)}c_{[g]}(n)q^n$, with $c_{[g]}(1)=0$ [1], Theorem 1.2 and equation (2.6).

(2)

$\mathcal T_{1A}(\tau)=2f_3(\tau)+248\theta(\tau)$, where $f_3(\tau)=q^{-3}+\sum_{n>0,\ n\equiv0,1\ (\mathrm{mod}\ 4)}A(n,3)q^n$ is the canonical weakly holomorphic weight-$1/2$ plus-space form on $\Gamma_0(4)$ with the theta multiplier and zero constant term [2], equations (1.4) and (1.11).

(3)

For a positive fundamental discriminant $D>1$, $c_{1A}(D)=2A_1(D,3)$, where $A_1(D,3)=t_1(D,3)/\sqrt D$ is the integer normalization of the genus-character twisted trace [2], equations (2.10)–(2.14). The theta correction vanishes because $D$ is not a square.

(4)

Put $\Theta_k(n)=[q^n]\theta(k\tau)$, so $\Theta_k(0)=1$, $\Theta_k(n)=2$ when $n/k$ is a positive integer square, and $\Theta_k(n)=0$ otherwise. Then $c_{1A}(n)\equiv c_{5A}(n)\pmod{25}$, $c_{1A}(n)-c_{3A}(n)+9\Theta_9(n)\equiv0\pmod{27}$, and $c_{1A}(n)-c_{2A}(n)-256\Theta_4(n)\equiv0\pmod{4096}$ [1], Appendix B.

(5)

The Thompson irreducible degrees give $c_{1A}(0)=248$, $c_{1A}(4)=2\cdot27000$, $c_{1A}(5)=-2\cdot85995$, $c_{1A}(8)=2\cdot1707264$, $c_{1A}(9)=-2\cdot4096000$ and $c_{1A}(12)=2\cdot44330496$ [2], Table 2.

Comments

(6)

Write $q=e^{2\pi i\tau}$ with $\operatorname{Im}\tau>0$, $\eta(\tau)=q^{1/24}\prod_{r\geq1}(1-q^r)$, and $\theta(\tau)=\sum_{r\in\mathbb Z}q^{r^2}$. The class label $[g]$ always denotes a Thompson class, as in [1], Appendix A. It does not denote a Monster class with the same spelling.

(7)

For $n\geq0$, $W_n$ is purely even when $n\equiv0\pmod4$ and purely odd when $n\equiv1\pmod4$. Thus $c_{1A}(n)$ is the dimension in the even sector and the negative of the dimension in the odd sector. Nonidentity supertraces retain their own signs. The polar coefficient at $n=-3$ is $2$ for every class and is part of the normalization rather than a stored row.

(8)

The selected classes $1A,2A,3A,3B,3C,4A,4B,5A$ have modular levels $N_{[g]}=4,8,36,12,36,16,32,20$ with the ordinary theta multiplier, respectively [1], Table A.5. This level includes the extra multiplier behavior of $3A$, $3C$ and $4B$; it need not equal four times the element order.

(9)

The Thompson supermodule $W$ was proved to exist by Griffin and Mertens [1], Theorem 1.2.

Programs

(P1)

sage

# Download generate.py and eta-data.json from this table.
from generate import ThompsonCoefficients
generator = ThompsonCoefficients()
# One more grade in a selected class; the result is exact.
print(generator.value({'g': '3A', 'n': '205'}, 100))

References

[1]

Michael J. Griffin and Michael H. Mertens, A proof of the Thompson moonshine conjecture, Research in the Mathematical Sciences 3 (2016), article 36. Theorem 1.2, equation (2.6), Section 5, Tables A.5–A.6 and Appendix B in the arXiv v2 numbering. (arXiv) (doi)

[2]

Jeffrey A. Harvey and Brandon C. Rayhaun, Traces of Singular Moduli and Moonshine for the Thompson Group, Communications in Number Theory and Physics 10 (2016), 23–62. Equations (1.3)–(1.11), (2.10)–(2.14), Conjecture 3.1 and Tables 1, 2, 6. (arXiv) (doi)

Links

[3]

Michael H. Mertens, exact eta bases and coefficients for Thompson moonshine

[4]

Griffin–Mertens supplementary MAGMA data and coefficient lists

Similar tables

Genus-character twisted traces of singular moduli — At positive fundamental-discriminant grades, the identity Thompson coefficient is twice its integer-normalized trace with Hecke order one and negative discriminant magnitude three.

O’Nan moonshine coefficients — These complementary arithmetic moonshine traces have weight 3/2 and grades congruent to 0 or 3 modulo four, with a different group and module.

Degrees of the irreducible characters of the sporadic simple groups — The Thompson degrees give dimensions of the irreducible constituents in the early identity-class supertraces.

Fourier coefficients of monstrous McKay–Thompson series — These are Monster graded traces of weight-zero modular functions, with group-specific class labels and a different module.

Mathieu and umbral moonshine coefficients — These are related mock modular moonshine coefficients; Thompson moonshine uses weakly holomorphic modular forms.

Data properties

Entries are of type: integer

How well the digits are known: exact

Table is complete: no (it holds every $n=4m$ and $n=4m+1$ with $0\leq m\leq50$ for $g=1A,2A,3A,3B,3C,4A,4B,5A$, all Thompson classes of element order at most five. This low-order subfamily includes all prime-order classes through five and both order-four classes before extending any class more deeply. The noncanonical grade bound gives consecutive even/odd pairs through $n=201$, retaining signs and structural zeros and supporting the theta-splitting identities at a common depth. There are 816 entries; the longest integer has 34 characters. Higher-order classes and larger grades remain outside this coverage)

How they were obtained:

Computed using exact integer Euler-product recurrences and rational linear combinations of the eta bases supplied in [3] and [4]. Every division in the recurrence is checked for exact divisibility, every final coefficient is required to be integral, and coefficients outside the plus-space residues are required to vanish. No decimal approximation or numerical rounding produces a stored value.

All 816 entries agree with the author’s supplementary coefficient list. An independent Rankin–Cohen construction reproduces all 102 identity-class entries. The rational identities in [1], Table A.6, hold through the full selected range for all eight theta-split total series and for the separate odd sectors of five classes, using finite Euler products rather than the generator’s recurrence. Seven selected congruences hold at all 102 grades. The first six representation decompositions agree with the Thompson degrees, and all 30 overlapping fundamental-discriminant values agree with the separately computed twisted-trace table.

The attached CM checker also tests all 61 positive fundamental-discriminant grades in the selected range with independent reduced quadratic forms and 1024-bit Arb modular-j evaluations. It requires finite enclosures with real and imaginary radii below $10^{-110}$ before checking the exact predicted trace. Known-answer and deliberately incorrect controls precede the comparisons. The attached scripts repeat the checks on values read from the draft; numerical-evidence.json and cm-evidence.json record the outcomes.