Properties

Label 90.6
Level 90
Weight 6
Dimension 259
Nonzero newspaces 6
Newform subspaces 20
Sturm bound 2592
Trace bound 1

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Defining parameters

Level: \( N \) = \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) = \( 6 \)
Nonzero newspaces: \( 6 \)
Newform subspaces: \( 20 \)
Sturm bound: \(2592\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_1(90))\).

Total New Old
Modular forms 1144 259 885
Cusp forms 1016 259 757
Eisenstein series 128 0 128

Trace form

\( 259 q + 12 q^{2} - 18 q^{3} + 80 q^{4} + 127 q^{5} + 168 q^{6} + 44 q^{7} - 192 q^{8} - 1406 q^{9} + O(q^{10}) \) \( 259 q + 12 q^{2} - 18 q^{3} + 80 q^{4} + 127 q^{5} + 168 q^{6} + 44 q^{7} - 192 q^{8} - 1406 q^{9} + 1300 q^{10} + 2966 q^{11} + 640 q^{12} - 2150 q^{13} - 1648 q^{14} - 7008 q^{15} + 2304 q^{16} - 3402 q^{17} - 592 q^{18} + 11248 q^{19} + 5936 q^{20} + 22500 q^{21} + 6136 q^{22} + 18828 q^{23} - 1152 q^{24} - 1911 q^{25} - 31944 q^{26} - 46392 q^{27} - 4864 q^{28} - 8106 q^{29} + 7488 q^{30} + 1224 q^{31} + 3072 q^{32} + 66562 q^{33} + 31344 q^{34} + 33488 q^{35} - 9440 q^{36} + 48694 q^{37} - 20280 q^{38} + 1408 q^{39} - 7872 q^{40} - 59128 q^{41} + 7744 q^{42} - 11990 q^{43} - 13952 q^{44} + 143396 q^{45} - 94240 q^{46} - 72084 q^{47} - 3584 q^{48} + 30909 q^{49} - 58916 q^{50} - 21402 q^{51} + 58144 q^{52} + 36030 q^{53} + 75576 q^{54} + 104868 q^{55} + 30976 q^{56} - 80902 q^{57} - 57112 q^{58} - 317638 q^{59} - 96512 q^{60} - 122126 q^{61} - 156960 q^{62} + 29960 q^{63} - 102400 q^{64} + 307002 q^{65} + 42432 q^{66} + 216542 q^{67} + 131904 q^{68} + 172280 q^{69} + 151440 q^{70} + 290376 q^{71} + 119424 q^{72} - 128274 q^{73} - 319144 q^{74} - 323706 q^{75} - 263904 q^{76} - 984324 q^{77} - 585520 q^{78} - 323456 q^{79} - 50432 q^{80} - 423694 q^{81} - 123464 q^{82} + 241164 q^{83} + 199488 q^{84} + 498130 q^{85} + 935464 q^{86} + 806668 q^{87} + 98176 q^{88} + 620590 q^{89} + 441280 q^{90} + 227072 q^{91} + 301248 q^{92} + 882152 q^{93} + 387504 q^{94} + 640044 q^{95} + 16384 q^{96} + 52708 q^{97} - 1247148 q^{98} - 2187572 q^{99} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_1(90))\)

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 the newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
90.6.a \(\chi_{90}(1, \cdot)\) 90.6.a.a 1 1
90.6.a.b 1
90.6.a.c 1
90.6.a.d 1
90.6.a.e 1
90.6.a.f 1
90.6.a.g 1
90.6.c \(\chi_{90}(19, \cdot)\) 90.6.c.a 2 1
90.6.c.b 2
90.6.c.c 4
90.6.c.d 4
90.6.e \(\chi_{90}(31, \cdot)\) 90.6.e.a 8 2
90.6.e.b 10
90.6.e.c 10
90.6.e.d 12
90.6.f \(\chi_{90}(17, \cdot)\) 90.6.f.a 4 2
90.6.f.b 4
90.6.f.c 12
90.6.i \(\chi_{90}(49, \cdot)\) 90.6.i.a 60 2
90.6.l \(\chi_{90}(23, \cdot)\) 90.6.l.a 120 4

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_1(90))\) into lower level spaces

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