Properties

Label 90.3
Level 90
Weight 3
Dimension 102
Nonzero newspaces 5
Newform subspaces 10
Sturm bound 1296
Trace bound 1

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Defining parameters

Level: \( N \) = \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) = \( 3 \)
Nonzero newspaces: \( 5 \)
Newform subspaces: \( 10 \)
Sturm bound: \(1296\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(\Gamma_1(90))\).

Total New Old
Modular forms 496 102 394
Cusp forms 368 102 266
Eisenstein series 128 0 128

Trace form

\( 102 q - 2 q^{2} + 18 q^{5} + 24 q^{6} + 16 q^{7} + 4 q^{8} + 16 q^{9} + O(q^{10}) \) \( 102 q - 2 q^{2} + 18 q^{5} + 24 q^{6} + 16 q^{7} + 4 q^{8} + 16 q^{9} + 42 q^{10} + 44 q^{11} - 8 q^{12} + 26 q^{13} - 72 q^{14} - 42 q^{15} - 8 q^{16} - 130 q^{17} - 16 q^{18} - 16 q^{19} - 40 q^{20} + 36 q^{21} + 8 q^{22} + 8 q^{23} + 60 q^{25} - 12 q^{26} + 48 q^{27} - 56 q^{28} - 36 q^{29} - 72 q^{30} - 212 q^{31} + 8 q^{32} - 380 q^{33} - 40 q^{34} - 484 q^{35} - 152 q^{36} - 46 q^{37} - 40 q^{38} - 356 q^{39} + 36 q^{40} - 52 q^{41} - 32 q^{42} + 200 q^{43} + 134 q^{45} - 136 q^{46} + 168 q^{47} + 16 q^{48} + 156 q^{49} + 242 q^{50} + 720 q^{51} - 28 q^{52} + 562 q^{53} + 408 q^{54} - 436 q^{55} - 32 q^{56} + 368 q^{57} - 56 q^{58} + 108 q^{59} + 148 q^{60} + 412 q^{61} + 472 q^{62} - 100 q^{63} + 96 q^{64} + 408 q^{65} + 528 q^{66} + 496 q^{67} + 116 q^{68} + 164 q^{69} + 372 q^{70} + 40 q^{71} + 192 q^{72} + 354 q^{73} - 144 q^{74} - 1002 q^{75} - 96 q^{76} - 620 q^{77} - 448 q^{78} + 356 q^{79} - 568 q^{81} - 520 q^{82} - 456 q^{83} + 48 q^{84} - 410 q^{85} - 480 q^{86} - 164 q^{87} - 16 q^{88} + 64 q^{90} - 992 q^{91} + 16 q^{92} + 404 q^{93} - 8 q^{94} + 1012 q^{95} - 32 q^{96} - 634 q^{97} + 494 q^{98} + 868 q^{99} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(\Gamma_1(90))\)

We only show spaces with odd 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
90.3.b \(\chi_{90}(89, \cdot)\) 90.3.b.a 4 1
90.3.d \(\chi_{90}(71, \cdot)\) None 0 1
90.3.g \(\chi_{90}(37, \cdot)\) 90.3.g.a 2 2
90.3.g.b 2
90.3.g.c 2
90.3.g.d 4
90.3.h \(\chi_{90}(11, \cdot)\) 90.3.h.a 16 2
90.3.j \(\chi_{90}(29, \cdot)\) 90.3.j.a 8 2
90.3.j.b 16
90.3.k \(\chi_{90}(7, \cdot)\) 90.3.k.a 24 4
90.3.k.b 24

Decomposition of \(S_{3}^{\mathrm{old}}(\Gamma_1(90))\) into lower level spaces

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