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Properties

Label 90.20
Level 90
Weight 20
Dimension 993
Nonzero newspaces 6
Sturm bound 8640
Trace bound 1

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Defining parameters

Level: \( N \) = \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) = \( 20 \)
Nonzero newspaces: \( 6 \)
Sturm bound: \(8640\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{20}(\Gamma_1(90))\).

Total New Old
Modular forms 4168 993 3175
Cusp forms 4040 993 3047
Eisenstein series 128 0 128

Trace form

\( 993 q - 1536 q^{2} - 92874 q^{3} + 6029312 q^{4} - 3828047 q^{5} - 38722560 q^{6} - 37579572 q^{7} + 402653184 q^{8} - 4871453378 q^{9} + O(q^{10}) \) \( 993 q - 1536 q^{2} - 92874 q^{3} + 6029312 q^{4} - 3828047 q^{5} - 38722560 q^{6} - 37579572 q^{7} + 402653184 q^{8} - 4871453378 q^{9} + 12914187776 q^{10} - 91289044162 q^{11} + 32474398720 q^{12} - 211138542050 q^{13} + 256447367168 q^{14} - 857120555904 q^{15} + 2954937499648 q^{16} + 2166004047546 q^{17} - 1891332069376 q^{18} + 7795778920480 q^{19} - 697997983744 q^{20} - 8953011580212 q^{21} + 26994489347072 q^{22} + 33471274181004 q^{23} - 5436623290368 q^{24} - 116983346557709 q^{25} + 9255556690944 q^{26} + 8592780801576 q^{27} - 125058555052032 q^{28} - 226331181403350 q^{29} + 495726584537088 q^{30} - 1530820666460920 q^{31} - 105553116266496 q^{32} + 1550056516890766 q^{33} - 784743512291328 q^{34} + 968380439125328 q^{35} + 1774895372959744 q^{36} - 284612916003902 q^{37} + 4280401301222400 q^{38} + 5857860067144096 q^{39} + 674011231027200 q^{40} + 2694154497522272 q^{41} + 4470848496590848 q^{42} - 14735492253328670 q^{43} + 21573134668791808 q^{44} - 16676276552836228 q^{45} - 31322443145203712 q^{46} + 17060332153028172 q^{47} - 10633789269082112 q^{48} + 66227472957611247 q^{49} + 34538626797216256 q^{50} - 19111395723705042 q^{51} + 28247809930559488 q^{52} + 30951077131706970 q^{53} - 4755271552521216 q^{54} - 343763970276422508 q^{55} - 97656775406256128 q^{56} + 223793187572006726 q^{57} - 71234242343207936 q^{58} - 615190308079927630 q^{59} - 248324156095987712 q^{60} + 1195216255753860718 q^{61} + 132381145133199360 q^{62} + 447203426898737480 q^{63} - 1639310264362860544 q^{64} + 265221415639234902 q^{65} - 646909158134636544 q^{66} - 806351445336478026 q^{67} + 401171984906452992 q^{68} + 1730602097996139560 q^{69} - 1047630399299745792 q^{70} - 2108719071895228056 q^{71} - 803448106644406272 q^{72} + 833774094061864178 q^{73} + 572851998390846464 q^{74} - 4497977479084629330 q^{75} - 1985777699055468544 q^{76} - 4472906809582486668 q^{77} - 5516453031672371200 q^{78} + 965275763173710880 q^{79} + 362610482344886272 q^{80} + 8869108575582546878 q^{81} - 4346606685518199808 q^{82} - 15304368276261333108 q^{83} - 2526997280600358912 q^{84} + 9335631717290280998 q^{85} + 7249398341673960448 q^{86} + 18209438667445282636 q^{87} + 7076443415398842368 q^{88} - 23896675849071850934 q^{89} - 3230613162574782464 q^{90} + 6454384928569261568 q^{91} + 8774293698905112576 q^{92} + 9842607224963838776 q^{93} - 33642372223767373824 q^{94} - 53072979092206879572 q^{95} + 3262294980076503040 q^{96} + 6336497195075471356 q^{97} - 81520838307495627264 q^{98} + 110654393986940132860 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{20}^{\mathrm{new}}(\Gamma_1(90))\)

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
90.20.a \(\chi_{90}(1, \cdot)\) 90.20.a.a 1 1
90.20.a.b 1
90.20.a.c 1
90.20.a.d 1
90.20.a.e 1
90.20.a.f 2
90.20.a.g 2
90.20.a.h 2
90.20.a.i 2
90.20.a.j 2
90.20.a.k 2
90.20.a.l 2
90.20.a.m 3
90.20.a.n 3
90.20.a.o 4
90.20.a.p 4
90.20.c \(\chi_{90}(19, \cdot)\) 90.20.c.a 8 1
90.20.c.b 10
90.20.c.c 10
90.20.c.d 20
90.20.e \(\chi_{90}(31, \cdot)\) n/a 152 2
90.20.f \(\chi_{90}(17, \cdot)\) 90.20.f.a 36 2
90.20.f.b 40
90.20.i \(\chi_{90}(49, \cdot)\) n/a 228 2
90.20.l \(\chi_{90}(23, \cdot)\) n/a 456 4

"n/a" means that newforms for that character have not been added to the database yet

Decomposition of \(S_{20}^{\mathrm{old}}(\Gamma_1(90))\) into lower level spaces

\( S_{20}^{\mathrm{old}}(\Gamma_1(90)) \cong \) \(S_{20}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 12}\)\(\oplus\)\(S_{20}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 6}\)\(\oplus\)\(S_{20}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 8}\)\(\oplus\)\(S_{20}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 6}\)\(\oplus\)\(S_{20}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{20}^{\mathrm{new}}(\Gamma_1(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{20}^{\mathrm{new}}(\Gamma_1(10))\)\(^{\oplus 3}\)\(\oplus\)\(S_{20}^{\mathrm{new}}(\Gamma_1(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{20}^{\mathrm{new}}(\Gamma_1(18))\)\(^{\oplus 2}\)\(\oplus\)\(S_{20}^{\mathrm{new}}(\Gamma_1(30))\)\(^{\oplus 2}\)\(\oplus\)\(S_{20}^{\mathrm{new}}(\Gamma_1(45))\)\(^{\oplus 2}\)\(\oplus\)\(S_{20}^{\mathrm{new}}(\Gamma_1(90))\)\(^{\oplus 1}\)