O’Nan moonshine coefficients

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Numbers

$[g]$

$n$

$a_{[g]}(n)$

26752

143376

8288256

26124256

11:

561346944

12:

1417904008

15:

18508941312

16:

41113157868

19:

392037661056

20:

798588584512

23:

6103910176768

24:

11670072146880

27:

75506620523008

28:

137458661980137

31:

779869748441088

32:

1364980209159768

35:

6958069062202112

36:

11789251984427088

39:

54957276277708800

40:

90597759912318912

43:

391379553534032256

44:

630184560026074776

47:

2548919136928993280

48:

4020900518689269520

51:

15351901156733989632

52:

23784039396025000512

55:

86286272197016193024

56:

131548365135291713408

59:

455950044005404355712

60:

685168817792423987199

63:

2279151350414516705280

64:

3380583487390931366412

67:

10833618646086027344256

68:

15879663099954841041024

71:

49186850301388438689792

72:

71318882529234281508768

75:

214122181480228091845504

76:

307386361127918188609176

79:

896715314725961329668096

80:

1275487471219214216175536

83:

3623183263892538892528768

84:

5109773808490232332100736

87:

14160573777366188477761536

88:

19812506128358995766680512

91:

53655196863763363454350080

92:

74515481827213101194106786

95:

197497436989313859069954048

96:

272381167208442891461518512

99:

707484021032208293928217728

100:

969386754177436358922834000

103:

2470502755210381978461665280

104:

3364314697227619299637241664

107:

8421849840803567364954352768

108:

11402497621768797914297024032

111:

28064913606589728167148926976

112:

37789767507870794044275585024

115:

91534062769805230642844627712

116:

122612882375843688678935103168

119:

292514719221884854610147856384

120:

389904494464803238120011057024

123:

916856832773179948153208992512

124:

1216393710669960206241033793629

127:

2821313011422022610644046561280

128:

3726341943157422519483563424792

131:

8530456440096443820118676594560

132:

11218938226748386556083089873024

135:

25363628453244521011766676430848

136:

33221676674296166610904777600896

139:

74214703954003082740924318231680

140:

96829448510435211584478649580848

143:

213848587615854277205697516738560

144:

277972924697658009583192287237276

147:

607209169623780817649897091259264

148:

786465097465619925963802598056512

151:

1699983302433376468962825102821376

152:

2194283857099858710837678776521536

155:

4695342127350075223693900280129024

156:

6040604468116473417987264014096148

159:

12800661932051794426206414580783104

160:

16415908202263704510514550274780336

163:

34462946518054640076811525963856256

164:

44061086825099325960346590151430400

167:

91669830811709969261385431741474816

168:

116855043519104765038237706766897024

171:

241013787944639819903520405472747392

172:

306355909156461097839492540803088024

175:

626578190175063842172958998161344512

176:

794265159391915275742727547491497264

179:

1611361741462989831598650337367072640

180:

2037179481658829018829749668468729536

183:

4100647350752766988008669828582402048

184:

5170977118752282488545331613624485760

187:

10330041411803548132551515110533280512

188:

12993977544724561414023957778832691435

191:

25768119468411726612500031966448021504

192:

32335281962374733197073857358441238544

195:

63668971518992902167181255275688257024

196:

79709101173489479673108758889612533904

199:

155870527597737570437127933381389015040

200:

194697726393040334127065372098283380512

128

-2048

-32

11:

16768

12:

-120

15:

-96256

16:

236

19:

442752

20:

576

23:

-1746944

24:

-1088

27:

6144512

28:

-2071

31:

-19746816

32:

3672

35:

58983168

36:

6224

39:

-165767168

40:

-10304

43:

442368384

44:

-16744

47:

-1128919040

48:

26384

51:

2770551552

52:

41536

55:

-6568343552

56:

-63616

59:

15098840192

60:

-96257

63:

-33757603840

64:

143884

67:

73598976384

68:

211072

71:

-156822906880

72:

-307296

75:

327201915776

76:

-442728

79:

-669595144192

80:

630704

83:

1345953793152

84:

894080

87:

-2660880857088

88:

-1254464

91:

5179536507648

92:

-1747038

95:

-9937238964224

96:

2417840

99:

18808030495872

100:

3317840

103:

-35146143139840

104:

-4528320

107:

64891639809152

108:

-6144480

111:

-118458671319040

112:

8286208

115:

213932305613568

116:

11126464

119:

-382436085608448

120:

-14858368

123:

677073420288768

124:

-19746723

127:

-1187710615306240

128:

26131480

131:

2065242896058240

132:

34412672

135:

-3561153496735744

136:

-45142144

139:

6091580417016960

140:

-58983120

143:

-10340420388218880

144:

76764316

147:

17424252776448896

148:

99574336

151:

-29154616293423104

152:

-128691392

155:

48452771475457536

156:

-165767404

159:

-80002068511188992

160:

212855984

163:

131268706320384384

164:

272421120

167:

-214090904537520128

168:

-347648128

171:

347141028938712960

172:

-442368360

175:

-559722337492023296

176:

561296688

179:

897597276472248192

180:

710371008

183:

-1431895134211446784

184:

-896645248

187:

2272668190894080768

188:

-1128918805

191:

-3589437244777926656

192:

1417957392

195:

5642205753028609536

196:

1776660624

199:

-8828095139885426688

200:

-2221097184

-188

11:

-552

12:

709

15:

1644

16:

19:

20:

-5888

23:

-11456

24:

14280

27:

26734

28:

31:

32:

-69906

35:

-120496

36:

143382

39:

239160

40:

43:

44:

-540018

47:

-860032

48:

1001758

51:

1566564

52:

55:

56:

-3204064

59:

-4850136

60:

5553651

63:

8288268

64:

67:

68:

-15834144

71:

-23077824

72:

26124444

75:

37692154

76:

79:

80:

-68318260

83:

-96763256

84:

108504408

87:

152398980

88:

91:

92:

-265092045

95:

-366846528

96:

408356226

99:

561347496

100:

103:

104:

-943942152

107:

-1281712856

108:

1417903315

111:

1914377424

112:

115:

116:

-3129621888

119:

-4181749632

120:

4602206208

123:

6119995524

124:

127:

128:

-9766546650

131:

-12871824680

132:

14102537448

135:

18508939692

136:

139:

140:

-28927826948

143:

-37671580864

144:

41113157886

147:

53345047546

148:

151:

152:

-81861101112

155:

-105488079616

156:

114729068751

159:

147363528540

160:

163:

164:

-222500889984

167:

-284045710144

168:

307984709856

171:

392037661068

172:

175:

176:

-583402796036

179:

-738546828840

180:

798588590400

183:

1008317189904

184:

187:

188:

-1481019250056

191:

-1860689997440

192:

2006960787214

195:

2515491232968

196:

199:

108

200:

-3651183260676

-32

11:

12:

-120

15:

16:

236

19:

20:

576

23:

24:

-1088

27:

28:

-2071

31:

32:

3672

35:

36:

6224

39:

40:

-10304

43:

44:

-16744

47:

48:

26384

51:

52:

41536

55:

56:

-63616

59:

60:

-96257

63:

64:

143884

67:

68:

211072

71:

72:

-307296

75:

76:

-442728

79:

80:

630704

83:

84:

894080

87:

88:

-1254464

91:

92:

-1747038

95:

96:

2417840

99:

100:

3317840

103:

104:

-4528320

107:

108:

-6144480

111:

112:

8286208

115:

116:

11126464

119:

120:

-14858368

123:

124:

-19746723

127:

128:

26131480

131:

132:

34412672

135:

136:

-45142144

139:

140:

-58983120

143:

144:

76764316

147:

148:

99574336

151:

152:

-128691392

155:

156:

-165767404

159:

160:

212855984

163:

164:

272421120

167:

168:

-347648128

171:

172:

-442368360

175:

176:

561296688

179:

180:

710371008

183:

184:

-896645248

187:

188:

-1128918805

191:

192:

1417957392

195:

196:

1776660624

199:

200:

-2221097184

-32

11:

12:

-120

15:

16:

236

19:

20:

576

23:

24:

-1088

27:

28:

-2071

31:

32:

3672

35:

36:

6224

39:

40:

-10304

43:

44:

-16744

47:

48:

26384

51:

52:

41536

55:

56:

-63616

59:

60:

-96257

63:

64:

143884

67:

68:

211072

71:

72:

-307296

75:

76:

-442728

79:

80:

630704

83:

84:

894080

87:

88:

-1254464

91:

92:

-1747038

95:

96:

2417840

99:

100:

3317840

103:

104:

-4528320

107:

108:

-6144480

111:

112:

8286208

115:

116:

11126464

119:

120:

-14858368

123:

124:

-19746723

127:

128:

26131480

131:

132:

34412672

135:

136:

-45142144

139:

140:

-58983120

143:

144:

76764316

147:

148:

99574336

151:

152:

-128691392

155:

156:

-165767404

159:

160:

212855984

163:

164:

272421120

167:

168:

-347648128

171:

172:

-442368360

175:

176:

561296688

179:

180:

710371008

183:

184:

-896645248

187:

188:

-1128918805

191:

192:

1417957392

195:

196:

1776660624

199:

200:

-2221097184

11:

-56

12:

15:

16:

-132

19:

20:

137

23:

24:

130

27:

28:

31:

-912

32:

35:

862

36:

-1537

39:

800

40:

1412

43:

44:

1276

47:

48:

51:

-6368

52:

55:

5524

56:

-9842

59:

4712

60:

8449

63:

64:

7162

67:

68:

71:

-32208

72:

75:

26754

76:

-46324

79:

22096

80:

38036

83:

84:

30986

87:

88:

91:

-129920

92:

95:

104048

96:

-179988

99:

82728

100:

143375

103:

104:

113664

107:

108:

111:

-455024

112:

115:

356462

116:

-610832

119:

278384

120:

475774

123:

124:

369129

127:

128:

131:

-1426440

132:

135:

1095848

136:

-1872604

139:

838680

140:

1433348

143:

144:

1093776

147:

148:

151:

-4114624

152:

155:

3116524

156:

-5301227

159:

2355104

160:

4001586

163:

164:

3013150

167:

168:

171:

-11079608

172:

175:

8288262

176:

-14065736

179:

6185640

180:

10495161

183:

184:

7814510

187:

188:

191:

-28210496

192:

195:

20894524

196:

-35356221

199:

15449040

200:

26124262

-2

11:

-8

12:

-3

15:

-28

16:

19:

-12

20:

23:

24:

-8

27:

122

28:

31:

32:

-18

35:

-240

36:

39:

-344

40:

-8

43:

-12

44:

47:

640

48:

51:

876

52:

-8

55:

56:

59:

-1528

60:

-29

63:

-2044

64:

67:

-12

68:

-32

71:

3392

72:

-36

75:

4334

76:

79:

80:

-52

83:

-6936

84:

87:

-8724

88:

-8

91:

-24

92:

95:

13504

96:

99:

16776

100:

-10

103:

104:

120

107:

-25336

108:

-117

111:

-30928

112:

115:

-24

116:

-128

119:

45696

120:

-160

123:

55308

124:

127:

128:

-218

131:

-80136

132:

200

135:

-96220

136:

-16

139:

-36

140:

252

143:

137280

144:

238

147:

163310

148:

-8

151:

152:

328

155:

-229632

156:

-337

159:

-271436

160:

163:

-12

164:

-384

167:

376768

168:

-448

171:

442740

172:

175:

176:

-516

179:

-607752

180:

576

183:

-710032

184:

-16

187:

-24

188:

632

191:

964480

192:

750

195:

1121496

196:

-18

199:

200:

828

-2

11:

12:

-25

15:

16:

19:

-50

20:

23:

24:

-28

27:

28:

31:

-52

32:

35:

106

36:

39:

40:

-272

43:

44:

47:

-444

48:

298

51:

52:

-154

55:

492

56:

408

59:

-206

60:

63:

608

64:

67:

68:

-1496

71:

72:

75:

-2154

76:

1545

79:

80:

-698

83:

2158

84:

1878

87:

-1040

88:

91:

2654

92:

95:

96:

-5954

99:

100:

103:

-8144

104:

5880

107:

108:

-2536

111:

8040

112:

6680

115:

-3296

116:

119:

8860

120:

123:

124:

-19777

127:

128:

131:

-26158

132:

18752

135:

136:

-7824

139:

24410

140:

20465

143:

-10348

144:

147:

26754

148:

151:

152:

-57700

155:

156:

159:

-74104

160:

53150

163:

164:

-21864

167:

68256

168:

56464

171:

-27694

172:

175:

71568

176:

179:

180:

-152982

183:

184:

187:

-193052

188:

137954

191:

192:

-56190

195:

172636

196:

143378

199:

-70616

200:

-2

11:

12:

-25

15:

16:

19:

-50

20:

23:

24:

-28

27:

28:

31:

-52

32:

35:

106

36:

39:

40:

-272

43:

44:

47:

-444

48:

298

51:

52:

-154

55:

492

56:

408

59:

-206

60:

63:

608

64:

67:

68:

-1496

71:

72:

75:

-2154

76:

1545

79:

80:

-698

83:

2158

84:

1878

87:

-1040

88:

91:

2654

92:

95:

96:

-5954

99:

100:

103:

-8144

104:

5880

107:

108:

-2536

111:

8040

112:

6680

115:

-3296

116:

119:

8860

120:

123:

124:

-19777

127:

128:

131:

-26158

132:

18752

135:

136:

-7824

139:

24410

140:

20465

143:

-10348

144:

147:

26754

148:

151:

152:

-57700

155:

156:

159:

-74104

160:

53150

163:

164:

-21864

167:

68256

168:

56464

171:

-27694

172:

175:

71568

176:

179:

180:

-152982

183:

184:

187:

-193052

188:

137954

191:

192:

-56190

195:

172636

196:

143378

199:

-70616

200:

Definition

The entry $a_{[g]}(n)=\operatorname{tr}(g\mid W_n)$ is the signed trace on grade $n$ of the virtual O’Nan module $W$ constructed by Duncan, Mertens and Ono [1], Theorems 3.1 and 4.1.

Parameters

$[g]$

— O’Nan conjugacy class (an O’Nan conjugacy class in the notation of [1], Appendix A)

$n$

— Fourier grade ($n>0$ and $n\equiv0,3\pmod4$)

Formulas

(1)

For $N=1,\ldots,7$, put $d_N=\gcd(N,2)$ and $A_N(D)=\frac12\bigl(\operatorname{Tr}_D^{(N)}(J_2^{(N)})-\operatorname{Tr}_D^{(N/d_N)}(J^{(N/d_N)})\bigr)$ [1], Proposition 5.1.

(2)

For every positive admissible grade $D$, the coefficients are $$\begin{aligned}a_{1A}(D)&=A_1(D),\\a_{2A}(D)&=A_2(D)+12H_1(D)-12H_2(D),\\a_{3A}(D)&=A_3(D)+12H_1(D)-12H_3(D),\\a_{4A}(D)=a_{4B}(D)&=A_4(D)+12H_2(D)-12H_4(D),\\a_{5A}(D)&=A_5(D)+6H_1(D)-6H_5(D),\\a_{6A}(D)&=A_6(D)-12H_1(D)+8H_2(D)+\tfrac{21}{2}H_3(D)-\tfrac{13}{2}H_6(D),\\a_{7A}(D)=a_{7B}(D)&=A_7(D)+4H_1(D)-4H_7(D).\end{aligned}$$ [1], Appendix D.

(3)

$a_{1A}(n)=\bigl(t_2(n)-t_1(n)\bigr)/2$, where $t_m(n)$ are the Zagier traces of singular moduli with unnormalized Hecke order $m$ [1], Proposition 5.1. This is a difference of two traces, rather than a scalar rescaling of one trace.

(4)

$a_{1A}(n)\equiv a_{3A}(n)\pmod{3^5}$, $a_{1A}(n)\equiv a_{5A}(n)\pmod{5^3}$, $a_{1A}(n)\equiv a_{7A}(n)\pmod{7^3}$ and $a_{2A}(n)\equiv a_{6A}(n)\pmod{3^2}$ [1], Appendix C.

Comments

(5)

Write $q=e^{2\pi i\tau}$ with $\operatorname{Im}\tau>0$. The series is $F_{[g]}(\tau)=-q^{-4}+2+\sum_{n>0}a_{[g]}(n)q^n$. It has weight $3/2$ and satisfies the Kohnen plus condition $a_{[g]}(n)=0$ for $n\equiv1,2\pmod4$. The universal polar and constant terms are part of this normalization. The entries give the admissible positive grades, including their structural zeros [1], Theorem 3.1.

(6)

The labels $[g]$ denote conjugacy classes of O’Nan in [1], Appendix A. They are group-specific: a label such as $3A$ does not denote a Monster class. For the classes represented here, $N=o(g)$ is the element order, the modular group is $\Gamma_0(4N)$, and the character is trivial. The pairs $4A,4B$ and $7A,7B$ have equal series and retain separate class identities [1], Appendix D.

(7)

Traces on nonidentity classes may be negative. These are traces on the specified virtual graded module; absolute values, rescaled traces and individual character multiplicities are different quantities.

(8)

The first two positive grades satisfy $W_3\cong V_7$ and $W_4\cong V_1\oplus V_{12}\oplus V_{18}$, where $V_j$ has character $\chi_j$ in [1], Appendix A. Their dimensions are $26752$ and $1+58311+85064=143376$. The irreducible character degrees give these constituent dimensions.

(9)

For $D>0$, let $\mathcal Q_{-D}^{(N)}$ contain all positive definite integral forms $Q=[a,b,c]$ with $b^2-4ac=-D$ and $N\mid a$, including imprimitive forms. Set $\tau_Q=(-b+i\sqrt D)/(2a)$ using the positive square root, and $\omega_N(Q)=\#\operatorname{Stab}_{\Gamma_0(N)}(Q)/2$. The weighted trace is $\operatorname{Tr}_D^{(N)}(f)=\sum_{[Q]\in\mathcal Q_{-D}^{(N)}/\Gamma_0(N)}f(\tau_Q)/\omega_N(Q)$, and $H_N(D)=\operatorname{Tr}_D^{(N)}(1)$ [1], equation (5.1).

(10)

For the represented levels, let $J^{(N)}=q^{-1}+c_Nq+O(q^2)$ be the normalized Hauptmodul of $\Gamma_0(N)$ in [1], Table 5.1, with zero constant coefficient and no other poles. Put $J_2^{(N)}=(J^{(N)})^2-2c_N$. At level one $J^{(1)}=j-744$, with the classical $j$ normalization, and $J_2^{(1)}=P_{2,1A}(J^{(1)})$ uses the identity-class Faber polynomial.

Programs

(P1)

sage

# Download generate.py and its listed attachments from this table.
from generate import ONanCoefficients
G = ONanCoefficients()
# The default enumeration is all nine classes and every admissible n <= 200.
print(G.value({'g': '1A', 'n': '3'}, 100))  # 26752

References

[1]

John F. R. Duncan, Michael H. Mertens and Ken Ono, O’Nan Moonshine and Arithmetic, final preprint version, 17 March 2019. Theorems 3.1 and 4.1; §5.1; Appendices A–D. (arXiv) https://arxiv.org/html/1702.03516v4

Links

[2]

GAP Character Table Library 1.3.11, ordinary O’Nan table in data/ctospora.tbl

Similar tables

Zagier_traces_of_singular_moduli — The identity-class coefficient is half the difference between its Hecke-order two and Hecke-order one traces.

Hecke-type_Faber_polynomials_of_monstrous_McKay_Thompson_series — The identity-class degree-two Faber polynomial supplies the modular function whose trace occurs in the identity coefficient.

Degrees_of_the_irreducible_characters_of_the_sporadic_simple_groups — These give the dimensions of irreducible constituents of the O’Nan graded module.

Fourier_coefficients_of_monstrous_McKay_Thompson_series — These are graded traces for the Monster, with weight-zero modular functions and a different module.

Mathieu_and_umbral_moonshine_coefficients — These are coefficients of related mock modular moonshine forms, with different groups, support and normalization.

Genus-character_twisted_traces_of_singular_moduli — These use genus characters to twist CM traces; the O’Nan construction instead uses level-dependent traces and class-number corrections.

Data properties

Entries are of type: integer

How well the digits are known: exact

Table is complete: no (it holds every $3\leq n\leq200$ with $n\equiv0,3\pmod4$ for $g=1A,2A,3A,4A,4B,5A,6A,7A,7B$, all conjugacy classes of element order at most seven. This prefix spans prime and composite orders with genus-zero CM constructions, before extending any single class more deeply. The noncanonical grade bound retains both residue classes, signs, structural zeros and imprimitive CM contributions, while keeping entries readable and allowing a full identity-class comparison with the Zagier traces; the longest integer has 39 characters. Other classes and larger grades remain outside this coverage)

How they were obtained:

CM traces are evaluated in 1024-bit Arb complex ball arithmetic using (1) and the class-number corrections in (2). The integrality theorem [1], Theorem 3.1(2), supplies the integer lattice. Every finite enclosure has both component radii below $10^{-110}$, contains zero in its imaginary component, and isolates exactly one integer in its real component. Each result is stored as that exact integer. Imprimitive forms and their exceptional stabilizers are included.

The 100 identity-class values agree with the independently computed Zagier trace table. The 162 class-grade comparisons through grade 36 agree with the multiplicities in [1], Appendix B, evaluated using [2]. Its character ordering agrees with DMO on the nine selected class columns except for a printed sign error: DMO Table A.1 gives $\chi_{22}(7A)=5$, whereas CTblLib gives $-5$. The value $-5$ is the rational value shared with the Galois conjugate $\chi_{21}$ and is congruent to the degree $169290$ modulo seven; the printed positive sign fails that necessary condition. The attached source controls preserve both readings. This correction concerns the independent check and does not alter the modular-form definition.

All 400 stated prime-power congruences and 200 equal-series comparisons are checked throughout the range. All 700 distinct level-grade coefficients agree at 1280 bits. Known modular values and deliberately wrong, nonfinite and broad enclosures test the checking procedure. The attached checker supports repeating the identities on values read from the draft.