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Properties

Label 9.40
Level 9
Weight 40
Dimension 92
Nonzero newspaces 2
Newform subspaces 6
Sturm bound 240
Trace bound 1

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Defining parameters

Level: \( N \) = \( 9 = 3^{2} \)
Weight: \( k \) = \( 40 \)
Nonzero newspaces: \( 2 \)
Newform subspaces: \( 6 \)
Sturm bound: \(240\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{40}(\Gamma_1(9))\).

Total New Old
Modular forms 121 97 24
Cusp forms 113 92 21
Eisenstein series 8 5 3

Trace form

\( 92 q - 499719 q^{2} - 1109442855 q^{3} - 5292678516109 q^{4} - 35207996726805 q^{5} - 2704541067462105 q^{6} + 44263855758037957 q^{7} + 2690341251233858634 q^{8} + 3716697415813560297 q^{9} + O(q^{10}) \) \( 92 q - 499719 q^{2} - 1109442855 q^{3} - 5292678516109 q^{4} - 35207996726805 q^{5} - 2704541067462105 q^{6} + 44263855758037957 q^{7} + 2690341251233858634 q^{8} + 3716697415813560297 q^{9} + 23912710916775389472 q^{10} - 81349025432052698778 q^{11} + 1924907744002747720308 q^{12} + 1456922407060398284935 q^{13} - 83609125061181441202464 q^{14} + 54053354277301697262027 q^{15} - 1309200282365622408373873 q^{16} - 909246546278538485954868 q^{17} - 11380583103887060353587924 q^{18} + 871972909419356247483706 q^{19} - 78234951545836579720340268 q^{20} + 106438049614273922779633581 q^{21} + 429747434106422970658357959 q^{22} - 147003029888841567290032665 q^{23} + 3456624493421697901388913837 q^{24} - 4405772107125231868859802199 q^{25} - 25514023438803778877404645260 q^{26} - 7030426083067081600123650768 q^{27} + 42967367463179907203369933884 q^{28} - 110604786717402391489356818163 q^{29} + 133131003987882111950590605648 q^{30} - 117546767710841962239291358535 q^{31} + 201459255988491202272085506447 q^{32} + 2296585588081322803008062992266 q^{33} - 2654416883104235146360738588491 q^{34} + 4906489083741816324031184603766 q^{35} - 15601427942661888299711963656461 q^{36} + 3454533559240155385685012037388 q^{37} - 38998822017341268778096518279549 q^{38} + 16799765774905277627739172917519 q^{39} + 86115538627326777029726102833896 q^{40} - 98595102650221203378753303403530 q^{41} + 32362487841566583852175402037874 q^{42} - 143680719020520575264035299405608 q^{43} - 462004131785605202382167856569430 q^{44} - 143822436574550390198252409800475 q^{45} - 1139941277713534250422200724926936 q^{46} + 326681759299383898120888770341913 q^{47} - 1329941353234894365440704149228189 q^{48} - 3362446677873729983838862683130467 q^{49} - 1489416056678090604237555059464431 q^{50} + 2210966049132966272465573301142485 q^{51} - 11659761420642631452738372935403086 q^{52} + 16729082266141379360128926252263670 q^{53} - 58050380075276251649654897207399559 q^{54} + 45287510078353421374132460970474174 q^{55} - 112447635900469701193713268336038546 q^{56} - 30551351165036904616931017596051105 q^{57} + 45141659757851533323106878898099332 q^{58} - 142962746974259482176379581033759246 q^{59} + 378435953165807977687739757835869084 q^{60} - 209447174788942432877280401096855495 q^{61} + 498444443863420252602269835486813540 q^{62} - 215993983009031929248334700403862629 q^{63} + 1097615169728527386563763771693249794 q^{64} - 201830323609051873720092266154640185 q^{65} + 371737423640114782790520424881506058 q^{66} - 303660837178410059726336859138296780 q^{67} + 277024966857925344284825576405546919 q^{68} - 847980922313473784846734994332130421 q^{69} + 4855128208748542841166133843479622242 q^{70} - 4931910122457648632084977033020904776 q^{71} + 11976661155837547060662433326102635211 q^{72} - 6206220557854730220597811073454406514 q^{73} + 5896287375164629460673963339275603508 q^{74} - 3061332579416061916349068720542852783 q^{75} - 3841728010087732762204192217751050243 q^{76} - 82520997175638002267408984160908283 q^{77} - 3967146945711784816852173408273763818 q^{78} + 1131353596605102299737205915221427887 q^{79} - 95843524931295520350204812474955741408 q^{80} + 278320919688959197262501585820762621 q^{81} + 24702986855137795400602155279747997542 q^{82} - 146930343993522225414795095875763825745 q^{83} + 58733282081676757408201304099440224618 q^{84} + 149406663016197549480451695575547628746 q^{85} - 335815179751382567686050000046428761919 q^{86} - 120709256585818851789178083739290855873 q^{87} + 478956450217774172556266007688558668171 q^{88} - 120776453925266340003393821130368778738 q^{89} + 28734574253322800173923901981342842444 q^{90} + 787259541541765237709728764347547558782 q^{91} - 1076839269286725350906249442796950458814 q^{92} + 668053595719675547032211330874072214251 q^{93} + 1283312716818201467923337060042295649896 q^{94} - 2174278904705845617269273460130216553868 q^{95} + 2263796708150342129453934644250794617128 q^{96} + 877392818361839986105766102428069264336 q^{97} - 983608957417047470372423276057514425832 q^{98} - 1317172023062139780647742073709981634039 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{40}^{\mathrm{new}}(\Gamma_1(9))\)

We only show spaces with even parity, since no modular forms exist when this condition is not satisfied. Within each space \( S_k^{\mathrm{new}}(N, \chi) \) we list available newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
9.40.a \(\chi_{9}(1, \cdot)\) 9.40.a.a 1 1
9.40.a.b 3
9.40.a.c 3
9.40.a.d 3
9.40.a.e 6
9.40.c \(\chi_{9}(4, \cdot)\) 9.40.c.a 76 2

Decomposition of \(S_{40}^{\mathrm{old}}(\Gamma_1(9))\) into lower level spaces

\( S_{40}^{\mathrm{old}}(\Gamma_1(9)) \cong \) \(S_{40}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 3}\)\(\oplus\)\(S_{40}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 2}\)\(\oplus\)\(S_{40}^{\mathrm{new}}(\Gamma_1(9))\)\(^{\oplus 1}\)