K3 elliptic genus coefficients

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Numbers

$D$

$c(D)$

-1:

comment: The polar coefficient fixes the full K3 normalization.

comment: This is the coefficient of the neutral term in the constant Laurent polynomial.

-128

216

-1026

1616

11:

-5504

12:

8032

15:

-23550

16:

33048

19:

-86400

20:

117280

23:

-283652

24:

376608

27:

-854528

28:

1112832

31:

-2402298

32:

3082192

35:

-6378240

36:

8077560

39:

-16135176

40:

20213280

43:

-39153024

44:

48579872

47:

-91617270

48:

112732000

51:

-207612672

52:

253555488

55:

-457194510

56:

554658112

59:

-981229696

60:

1183223040

63:

-2057384940

64:

2467340568

67:

-4223097216

68:

5039175744

71:

-8501335066

72:

10097415792

75:

-16809361280

76:

19880131104

79:

-32689375200

80:

38508444960

83:

-62598913152

84:

73470825024

87:

-118164347946

88:

138210142752

91:

-220074990336

92:

256580228864

95:

-404745665480

96:

470456020512

99:

-735609545856

100:

852598665720

103:

-1322090071110

104:

1528228784480

107:

-2351206830208

108:

2710884725632

111:

-4139809817514

112:

4761580813056

115:

-7220252148480

116:

8285643864416

119:

-12479893198958

120:

14290135805760

123:

-21386689813248

124:

24437971496448

127:

-36351597240180

128:

41455521023952

131:

-61307075022720

132:

69782230061376

135:

-102624978219180

136:

116599873633344

139:

-170564480936064

140:

193453780310080

143:

-281543352161070

144:

318792567657528

147:

-461681651141504

148:

521925386598432

151:

-752302095809796

152:

849156542446944

155:

-1218424944821760

156:

1373248200195840

159:

-1961824578982590

160:

2207947579426080

163:

-3141004194140544

164:

3530185913881984

167:

-5001617051413896

168:

5613850353989184

171:

-7922596572136320

172:

8880939957524256

175:

-12485812978711080

176:

13978732289498400

179:

-19580826881945472

180:

21895714266564960

183:

-30561801873395262

184:

34135069845582144

187:

-47481647296856832

188:

52973311416240640

191:

-73440104399324484

192:

81844421735622496

195:

-113099357522634240

196:

125908367167764504

199:

-173445116824164168

200:

192889545857599920

203:

-264906204566693376

204:

294309144782101056

207:

-402996050858596374

208:

447290503035764256

211:

-610711619918544000

212:

677195417331506784

215:

-922028237623521460

216:

1021461820017684096

219:

-1386975356066274816

220:

1535173275031614720

223:

-2078986390488296028

224:

2299117820540038976

227:

-3105510461364533120

228:

3431415016405333824

231:

-4623290141346270900

232:

5104243708273037088

235:

-6860295828095650560

236:

7567841761003612000

239:

-10147131761581269008

240:

11184846350047530240

243:

-14961884694971137664

244:

16479300830629700448

247:

-21994003412418901356

248:

24206412405404357248

251:

-32235054508858504832

252:

35451597340368142080

255:

-47107343392565867730

256:

51770764310332350744

259:

-68645757811975054848

260:

75388527631863036800

263:

-99754248578841897262

264:

109477613279961439104

267:

-144566744045052187392

268:

158551808401707446304

271:

-208953869532110092422

272:

229017348502196248128

275:

-301232794335166629760

276:

329944914059714620032

279:

-433159468814233410102

280:

474148161167981912640

283:

-621312632300876867712

284:

679687301588291872512

287:

-889020239253743798934

288:

971960881892594917104

291:

-1269035387801211735552

292:

1386609924705763957056

295:

-1807245877215075802380

296:

1973541848467676739744

299:

-2567806503842077817856

300:

2802494977149756948640

303:

-3640226009877056489160

304:

3970718633102990103072

307:

-5149134516290569262208

308:

5613553080777752609408

311:

-7267720146906460458528

312:

7918977166517865374016

315:

-10236179312918534488320

316:

11147575117830892660224

319:

-14387005807883190666288

320:

15659892075683638787360

323:

-20179592278114968464896

324:

21953846556579259646808

327:

-28247490927367083272772

328:

30715808591180195429184

331:

-39462854645885233315968

332:

42890216565111899528288

335:

-55024156473832661151320

336:

59774302637249142818880

339:

-76575399105341558626560

340:

83146770600714500788800

343:

-106367855751165928660542

344:

115442312994158001359008

347:

-147479165977756049517440

348:

159987920246872512982272

351:

-204109658618002260871134

352:

221322359837668218126624

355:

-281982502839778804200960

356:

305627432011065093724608

359:

-388883244868432062451548

360:

421309226638102151110560

363:

-535386192648397040902784

364:

579780385673899461406272

367:

-735830913095235892500444

368:

796511333940739398227712

371:

-1009633054656404033608704

372:

1092440913901125888860736

375:

-1383041440827514920195810

376:

1495866602125931735140992

379:

-1891490048147435712962688

380:

2044973796814330655151360

383:

-2582741901901608772593718

384:

2791215560159957196385824

387:

-3521085780394270555762560

388:

3803822618241232279715904

391:

-4792930732100603234269926

392:

5175813515045778955720656

395:

-6514254069232214275783680

396:

7031993308422332647899744

399:

-8840503908699658638370662

400:

9539584848079429722037560

403:

-11979748171614674886697728

404:

12922340901586154218314976

407:

-16210112127506884526837116

408:

17479253039166652287441984

411:

-21902874148466126411030784

412:

23609323541545938641099520

415:

-29553017781255288377130540

416:

31844324695327498247452256

419:

-39819597978373507443213696

420:

42892068151659430436257920

423:

-53579009772237474640438740

424:

57693487596438464464940640

427:

-71995199849712716036573952

428:

77497855222907878946163552

431:

-96612101351893014019941872

432:

103961776739922828640048000

435:

-129475184987824241181427200

436:

139279331707191638915503968

439:

-173291115149241950458953558

440:

186352963017163436422846400

443:

-231637219863899430431331200

444:

249017622323640942085737216

447:

-309236010632742140678732244

448:

332334441539490265205914368

451:

-412314557266538416203450624

452:

442975074224936598882996352

455:

-549074435577331512634650350

456:

589724156071673056332810624

459:

-730305609751685659979701248

460:

784135483262808856321596480

463:

-970187483659394903853645198

464:

1041388030360088334930008672

467:

-1287333093560803654087064192

468:

1381401503574424082442690720

471:

-1706148834837798733106788848

472:

1830288618923251075313392224

475:

-2258603261748461926743937920

476:

2422243818334008806728905216

479:

-2986525706705127845326628292

480:

3201997109452704443586381120

483:

-3944590439166697120051570176

484:

4227998949027159342452919864

487:

-5204186999638498814049645552

488:

5576549897411367910194904480

491:

-6858434975561799711806138752

492:

7347150100379514958809333312

495:

-9028675367042261484535995300

496:

9669422258058727105428893184

499:

-11872865319719656146565147776

500:

12712062396752258230408836800

503:

-15596424100631896078975426884

504:

16694401542408360186644449728

507:

-20466233030778864805486760576

508:

21901326022447352098971125760

511:

-26828689896404988998487058458

512:

28702514393441152729561090000

515:

-35132970851589476751433509120

516:

37577217334026718378381284288

519:

-45960974844808909455857786202

520:

49146149007887717309041316160

523:

-60065835959060164774813518720

524:

64212494340740483370487291040

527:

-78421411554072114234225022058

528:

83814591614737474002257301312

531:

-102285818848034812754796481920

532:

109293555688604193946650624576

535:

-133282937625623698877993876760

536:

142380004341503775238073660384

539:

-173506870160966350574083459456

540:

185305189646073879923619333120

543:

-225655711835917387605189699618

544:

240943282162217360085158701632

547:

-293202713777855353185522089088

548:

312993389124690727533288963968

551:

-380615108483917310132868868534

552:

406212210741292211150852529792

555:

-493633642154519049298795292160

556:

526711179718330964870735523168

559:

-639629365970966224757706871434

560:

682335649833604931220546799680

563:

-828058674745190175187354852736

564:

883148402995778342001850154112

567:

-1071043186391576795019735027402

568:

1142045686046475115206752073024

571:

-1384108132304070146660683916160

572:

1475541489053938543960476010240

575:

-1787121856875310325485551284030

576:

1904765238055875454119343436856

579:

-2305490280424802641146713146368

580:

2456730001155429367191938090880

583:

-2971674360594594434088719164962

584:

3165943329001520994151869327360

587:

-3827116440231740591399389628032

588:

4076451759543266968322492457376

591:

-4924683830176766767028732476620

592:

5244433801630517949704193194784

595:

-6331766212797613833842948298240

596:

6741486109224632431783632048352

599:

-8134198928190616005260313388360

600:

8658785112938349079146219284640

603:

-10441228745167885305191875079040

604:

11112353517907979783833699742976

607:

-13391794601827649365294819230498

608:

14249720214540191041601650002528

611:

-17162465867501837933226157600512

612:

18258336284441118954553299918528

615:

-21977468471643117514178704469610

616:

23376202660547205331965204986496

619:

-28121339178783577560373211270016

620:

29905281279932646369108915034240

623:

-35954885856243057885052125363148

624:

38228407052961649388680084750080

627:

-45935303618785449884110539308544

628:

48830599882527305563533694944096

631:

-58641511742292879205006525012422

632:

62325903277525143968315412401792

635:

-74806044780382356241161745708800

636:

79491159967800786467193852761856

639:

-95355166513596122719003331934720

640:

101308489192586406180677103977760

643:

-121459293495405723598380134001792

644:

129018672188456991414749135050496

647:

-154596336234776365644878878686118

648:

164188203164566571487873941375184

651:

-196631215517949154436318292704256

652:

208793449149265915292144355695136

655:

-249915620057282534830468631802930

656:

265326216263766138981918109899136

659:

-317413077995478093605050030954624

660:

336926082764923996686866931815040

663:

-402855666357125678391180262292844

664:

427546180768542260258107891382688

667:

-510940238166970150723433234855424

668:

542160750965314269392543996860672

671:

-647573979390249114150129484085594

672:

687024834573460657200280256697408

675:

-820181506979162794691597015774720

676:

869998998993499797710403369912120

679:

-1038088696132463267255659839193500

680:

1100955135146055456493506356285120

683:

-1313002116397620143437267114451840

684:

1392283259678416299012943333751712

687:

-1659607531476919713037246664379492

688:

1759524082221972211860175179444000

691:

-2096316584919431972801560957693824

692:

2222158076231041928377577112514016

695:

-2646197810073102615727385906577490

696:

2804589192159702380175679331181504

699:

-3338136783999241613755171083390720

700:

3537370506983337287728665129780480

703:

-4208280980834904740890150335020406

704:

4458730424394093492417527321500960

707:

-5301838149329437176954078549698816

708:

5616472029835455929279130752111424

711:

-6675313431021152846946922308410742

712:

7070335484135578396714469046563712

715:

-8399290673748532386023113879918080

716:

8894934671096986322654276338955424

719:

-10561888369311711735824974161064878

720:

11183405634283978392991787369409120

723:

-13273051446177764525723040203582976

724:

14051936795897651945061540598814880

727:

-16669878118389310244340430900141398

728:

17645390956562353565446761216225472

731:

-20923227774273033840804794448730624

732:

22144278359054905234547094243504384

735:

-26245913498738640536571077519053260

736:

27773400782262189992027670255433536

739:

-32902853730891247733968280826066816

740:

34812561312520796491446756563729920

743:

-41223644789704579735296685143552344

744:

43609826300442554724328886174154432

747:

-51618123258439635705260918854410624

748:

54597938948067038429866126067846208

751:

-64595619035625929403641520860185032

752:

68314622750227586747541461546437120

755:

-80788761780178611635844280631636480

756:

85427683464610911626919774214483200

759:

-100982902277582591097371126889999510

760:

106766027532688210125168827380334400

763:

-126152454214195154966472508235367936

764:

133357971624596053402145002437830144

767:

-157505761082124904951193515019818022

768:

166478532867460822001759492774438752

771:

-196540459795935781879274752407959808

772:

207707775317337686243356967439859776

775:

-245111762152637364797708878572415110

776:

259002761185674965079546804485452096

779:

-305516625886832050673043822185858816

780:

322786234545072347062204341765394560

783:

-380597461170366092186327455111371246

784:

402055874259358553077714554897270360

787:

-473869843285986499949142145087029120

788:

500518820396955812178058631467640480

791:

-589679711127940112300173822040733050

792:

622757239333867976840721410438654880

795:

-733396764645625710219049398999037440

796:

774431989669666461145144220879374080

799:

-911652281634260646297992795841504342

800:

962533035728305030920697203133581360

803:

-1132631415343686129418079941114935040

804:

1195687190756324390063602899940811712

807:

-1406432282125691997643835051248477164

808:

1484536134514033835909851017278530656

811:

-1745506891380925953834440348719675008

812:

1842200532813018418182258552610697344

815:

-2165202316052557700050316928080069030

816:

2284849602701600817587757215224793664

819:

-2684424581582055577396386371931922176

820:

2832399753672358502922413978685141120

823:

-3326452723294713940267152626057311806

824:

3509371159033833140001107875682626112

827:

-4119936519188002668030647732880162432

828:

4345937474548950479976657692488051968

831:

-5100118780621044435653617407431371776

832:

5379211668692941179492530173758651936

835:

-6310332060624658369480251746353916160

836:

6654820357549454108978829118726501952

839:

-7803830561650823282880997233356872572

840:

8228830507496486187318256677627630720

843:

-9646031307293674997560379295457077504

844:

10170106316560574299662537673148445792

847:

-11917254789265743507900761183521930266

848:

12563191038702217000444344540777248608

851:

-14716074921068179603733028188748955392

852:

15511829113463548189522638266975839872

855:

-18163411960658771162001732736553010810

856:

19143268979600062761815557258706016864

859:

-22407530997754489410999212300216309376

860:

23613517320657376923727615933951351040

863:

-27630143715346460109124424794159893854

864:

29113752343403303402332370685005847168

867:

-34053853732523945207514570436459649408

868:

35878148392653722834153706140185246848

871:

-41951237490872688122660483774163756270

872:

44193418408479398434789517053763616160

875:

-51655915260382580860354330376624366080

876:

54410446426570649658549019458141624960

879:

-63576042707559998571501316789395930828

880:

66958461914686634745693142242091234560

883:

-78210745357125759312432357484651562112

884:

82362304129437940724074625073761304320

887:

-96170129529123628786976855871266347114

888:

101263441368049701305412997339247305152

891:

-118199637972539130859267696556847808896

892:

124445551198914755356865027638869328896

895:

-145209681298547916717823804443960024210

896:

152865638568275974427919603945560747840

899:

-178311673289630257086733637949703652608

900:

187691875225218007103678723688358283160

903:

-218861836268933633856377707051167895374

904:

230349593562097540685572633596705682560

907:

-268514430438250540863402141778216622208

908:

282577169615773735116798572300312005280

911:

-329286408639887004381206201057841723166

912:

346493894306896685207388533290830497088

915:

-403635917647823243610290042147888716800

916:

424682371870667927385377071066630874784

919:

-494557573616203475174428134936486104652

920:

520288515318728943647704077931631368000

923:

-605698050464754376568136704436720838400

924:

637142849260332498183339457664442619392

927:

-741496257101332890958387492941557775150

928:

779907602844272109283999030716342951456

931:

-907353268109473755440358834334310371968

932:

954255006801203625909039506877962071488

935:

-1109838243654286315927578297594074594700

936:

1167083330800473300197049844075275027744

939:

-1356937864849475833214963889675926817792

940:

1426778549278840705297886682740148363840

943:

-1658358365046314567254687227854609630424

944:

1743531151983067921100591890308271877984

947:

-2025891108677694183035230232315906623360

948:

2129719575395932040736931467203922209472

951:

-2473854921251447837572090651914476718090

952:

2600374089784183177655088668753998780032

955:

-3019631083368750337683380163617626191360

956:

3173737813895757150151705200699983703552

959:

-3684310159969257083234844306372806879834

960:

3871944941308379074075712628755608915200

963:

-4493473753268867215318652993055133965696

964:

4721840364060052212564948663404280752832

967:

-5478138975678686941309996859682489651726

968:

5755969808158768622253822357714738217456

971:

-6675899094963684795847055975034985253504

972:

7013775516710647419221677964802145448416

975:

-8132300596635259974228807056640922986060

976:

8543039626999820130862274147644793582688

979:

-9902505063757491627388595056015204488192

980:

10401625923924523563813709511798689519840

983:

-12053294061926968644967499775468147968088

984:

12659580896032573200889082405616909539520

987:

-14665486959941623682788235955988611529728

988:

15401667309521320454152298206272823003392

991:

-17836855700438425164165835717697334203850

992:

18730418253386507528488587896290874613376

995:

-21685637420607796750914280268290846612480

996:

22769817277445290049622473496033369114048

999:

-26354766061748748140937098747426903101476

1000:

27669731419989340131913817424944554249920

Definition

The integer $c(D)$ is determined by $Z_{K3}(\tau,z)=\sum_{n\geq0,\,\ell\in\mathbb Z}c(4n-\ell^2)q^ny^\ell$, where $Z_{K3}$ is the full K3 elliptic genus [1].

Parameters

$D$

— Fourier discriminant ($D\geq-1$ and $D\equiv0,3\pmod4$)

Formulas

(1)

$\theta_1(\tau,z)=-i\sum_{r\in\mathbb Z}(-1)^r q^{(r+1/2)^2/2}y^{r+1/2}$, $\theta_2(\tau,z)=\sum_{r\in\mathbb Z}q^{(r+1/2)^2/2}y^{r+1/2}$, $\theta_3(\tau,z)=\sum_{r\in\mathbb Z}q^{r^2/2}y^r$, and $\theta_4(\tau,z)=\sum_{r\in\mathbb Z}(-1)^r q^{r^2/2}y^r$. Fractional powers mean $q^a=e^{2\pi ia\tau}$ and $y^a=e^{2\pi iaz}$. Also $\eta(\tau)=q^{1/24}\prod_{m\geq1}(1-q^m)$ [1], equations (2.16) and (4.13)–(4.14), with the odd-theta sign chosen for [2], equation (1.4). The Dedekind eta values at rectangular CM points use this normalization.

(2)

$Z_{K3}(\tau,z)=8\sum_{j=2}^{4}\left(\theta_j(\tau,z)/\theta_j(\tau,0)\right)^2$ [2], equation (1.2).

(3)

Put $a(q)=\sum_{r\in\mathbb Z}q^{r^2}$, $b(q)=2\sum_{r\geq0}q^{r(r+1)}$, and $P(q)=\prod_{m\geq1}(1-q^m)^{-6}$. Then $\sum_{n\geq0}c(4n)q^n=(10a^4b-2qb^5)P$ and $\sum_{n\geq0}c(4n-1)q^n=(2a^5-10qab^4)P$ [1], equation (4.33).

(4)

$Z_{K3}(\tau,0)=24$, the Euler characteristic of K3. Equivalently, $\sum_{\ell\in\mathbb Z}c(4n-\ell^2)$ is $24$ for $n=0$ and $0$ for $n>0$, with $c(D)=0$ for $D<-1$ [2], following equation (1.2).

(5)

$Z_{K3}(\tau,z)=\dfrac{\theta_1(\tau,z)^2}{\eta(\tau)^3}\bigl(24\mu(\tau,z)+H(\tau)\bigr)$, where $\mu(\tau,z)=\dfrac{-iy^{1/2}}{\theta_1(\tau,z)}\sum_{r\in\mathbb Z}\dfrac{(-1)^r q^{r(r+1)/2}y^r}{1-yq^r}$ and $H(\tau)=q^{-1/8}(-2+90q+462q^2+1540q^3+\cdots)$. The formula is initially evaluated away from the poles of $\mu$, and the product extends holomorphically [2], equations (1.3)–(1.9).

Comments

(6)

Here $\operatorname{Im}\tau>0$, $z\in\mathbb C$, $q=e^{2\pi i\tau}$ and $y=e^{2\pi iz}$. The normalization is $Z_{K3}=2\phi_{0,1}=2y^{-1}+20+2y+O(q)$, where $\phi_{0,1}$ is the standard weak Jacobi form of weight zero and index one [1].

(7)

The elliptic transformation law makes the Fourier coefficient depend only on $D=4n-\ell^2$; the residue of $D$ modulo $4$ also determines the parity of $\ell$ [1], equations (4.28)–(4.32). Each discriminant is listed once, with its sign preserved. There is no restriction to fundamental or squarefree discriminants, and $c(D)=0$ for $D<-1$.

(8)

The theta functions use the variables $(\tau,z)$ in (1). Their nome is $e^{\pi i\tau}=q^{1/2}$. The theta-three, theta-four and theta-two companion tables give the denominator constants at rectangular CM points.

(9)

The odd theta function $\theta_1$ is defined with the Fourier sign in (1), compatible with the $\mu$ convention of [2]. It is the negative of $\vartheta_1$ used by Kawai in [1], equations (4.13)–(4.14). This sign difference does not change the squared theta quotient or the component formulas.

(10)

The $\mathcal N=4$ character decomposition introduces the mock modular series $H$ in (5) [2]. Its coefficients are different from the signed Fourier coefficients $c(D)$: the decomposition involves characters and convolution, rather than rescaling one coefficient. Here $H$ has the full Mathieu normalization, which is twice the coefficients $A_n$ used in [2], equation (1.9).

Programs

(P1)

Sage

# Save generate.py from this table's attachments.
from generate import K3EllipticGenus
generator = K3EllipticGenus()
generator.value({'D': '1000'}, digits=100)

References

[1]

Toshiya Kawai, K3 Surfaces, Igusa Cusp Form and String Theory, equations (4.28)–(4.33). (arXiv)

[2]

Tohru Eguchi, Hirosi Ooguri and Yuji Tachikawa, Notes on the K3 Surface and the Mathieu group M24, equations (1.2)–(1.12). (arXiv)

Similar tables

Jacobi theta-two constants at singular nomes — These are $\theta_2(\tau,0)$ at $\tau=i\sqrt r$, the denominator of the theta-two term in (2).

Jacobi theta-three constants at singular nomes — These are $\theta_3(\tau,0)$ at the same CM points, the denominator of the theta-three term in (2).

Jacobi theta-four constants at singular nomes — These are $\theta_4(\tau,0)$ at the same CM points, the denominator of the theta-four term in (2).

Fourier coefficients of monstrous McKay–Thompson series — This is the analogous Monster graded-trace family. Its modular functions and coefficients differ from the K3 Jacobi form and its character decomposition.

Mathieu and umbral moonshine coefficients — The $X=A_1^{24}$, $r=1$ component is the full Mathieu series $H$ in (5): $Z_{K3}=\theta_1^2\eta^{-3}(24\mu+H)$. Its positive coefficients $C_{A_1^{24}}(1,8n-1)$ begin $90,462,1540$; they arise from the character decomposition and convolution, rather than rescaling the signed genus coefficients $c(D)$.

Data properties

Entries are of type: integer

How well the digits are known: exact

Table is complete: no (it holds every $-1\leq D\leq1000$ with $D\equiv0,3\pmod4$, equivalently both $c(4n-1)$ and $c(4n)$ for $0\leq n\leq250$. This complete initial segment includes the polar term and both theta components without repeated elliptic translates. The noncanonical cutoff provides reference coverage of the discrete Fourier grades while keeping exact entries compact and the full two-variable verification inexpensive; the longest written value has 42 characters)

How they were obtained:

Computed with exact integer additions and multiplications from the two component series in (3). The inverse eta product is computed by six integer partition-product passes. No floating-point approximation or rounding is used.

An independently implemented rational long division expands the bivariate theta quotient in (2) through $q^{250}$. It agrees at all 15,813 tested Fourier positions, covering every stored coefficient and 5,200 terms with $D<-1$. The checks include ten printed coefficients in [1], equation (4.31), charge reflection, dependence on $4n-\ell^2$, integrality, and the Euler-characteristic identity at every computed grade. The character relation in (5) yields the same normalized $H$ series at $y=-1$ and $y=-2$ through grade 250 and agrees with the polar term and nine positive coefficients printed in [2], equation (1.12). This checks two elliptic specializations, not a numerical proof of the general identity. Known-answer and reciprocal-series controls precede the comparisons. The attached checker supports repeating these checks on the values read from the database.