Properties

Label 90.10
Level 90
Weight 10
Dimension 469
Nonzero newspaces 6
Newform subspaces 25
Sturm bound 4320
Trace bound 1

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Defining parameters

Level: \( N \) = \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) = \( 10 \)
Nonzero newspaces: \( 6 \)
Newform subspaces: \( 25 \)
Sturm bound: \(4320\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_1(90))\).

Total New Old
Modular forms 2008 469 1539
Cusp forms 1880 469 1411
Eisenstein series 128 0 128

Trace form

\( 469 q + 48 q^{2} - 150 q^{3} + 2816 q^{4} + 887 q^{5} + 6048 q^{6} - 17916 q^{7} - 12288 q^{8} + 53398 q^{9} + O(q^{10}) \) \( 469 q + 48 q^{2} - 150 q^{3} + 2816 q^{4} + 887 q^{5} + 6048 q^{6} - 17916 q^{7} - 12288 q^{8} + 53398 q^{9} - 129200 q^{10} + 245446 q^{11} + 1024 q^{12} + 108510 q^{13} - 404288 q^{14} + 569622 q^{15} + 1245184 q^{16} - 1118202 q^{17} - 294976 q^{18} + 502512 q^{19} - 1712384 q^{20} + 1998144 q^{21} + 2556768 q^{22} - 9138732 q^{23} - 1941504 q^{24} + 953469 q^{25} + 1797216 q^{26} - 4976952 q^{27} - 3535872 q^{28} - 47539650 q^{29} + 9612288 q^{30} + 49485364 q^{31} + 3145728 q^{32} + 10752010 q^{33} - 28685248 q^{34} - 35227232 q^{35} + 35232256 q^{36} - 70251834 q^{37} + 22647456 q^{38} - 18424436 q^{39} + 12177408 q^{40} + 183412420 q^{41} + 22786048 q^{42} - 113257794 q^{43} - 48240640 q^{44} + 76044626 q^{45} + 26900992 q^{46} + 238733580 q^{47} - 29229056 q^{48} + 268515039 q^{49} - 240303376 q^{50} - 94321902 q^{51} + 63297024 q^{52} - 403327458 q^{53} + 116960736 q^{54} + 123972608 q^{55} + 51200000 q^{56} + 326556254 q^{57} + 64208928 q^{58} + 337376074 q^{59} + 108702208 q^{60} + 251199722 q^{61} - 807442944 q^{62} - 2317909732 q^{63} - 721420288 q^{64} + 679644372 q^{65} + 1762854144 q^{66} + 1905341178 q^{67} + 942163968 q^{68} - 453107212 q^{69} - 196398720 q^{70} - 3555829968 q^{71} + 35364864 q^{72} + 1517633142 q^{73} - 345960224 q^{74} - 1980880716 q^{75} - 251300352 q^{76} + 3012737544 q^{77} + 499949120 q^{78} - 1025221908 q^{79} + 125370368 q^{80} - 970192930 q^{81} - 517441440 q^{82} + 426911124 q^{83} - 546020352 q^{84} - 1516275370 q^{85} - 3281688160 q^{86} + 3169880488 q^{87} + 654532608 q^{88} + 7512103658 q^{89} + 2075704960 q^{90} - 3942916896 q^{91} - 2339515392 q^{92} - 8977947844 q^{93} - 1560873920 q^{94} - 1833266796 q^{95} + 696254464 q^{96} + 11880594996 q^{97} + 6273677904 q^{98} + 4463329456 q^{99} + O(q^{100}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_1(90))\)

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
90.10.a \(\chi_{90}(1, \cdot)\) 90.10.a.a 1 1
90.10.a.b 1
90.10.a.c 1
90.10.a.d 1
90.10.a.e 1
90.10.a.f 1
90.10.a.g 1
90.10.a.h 1
90.10.a.i 1
90.10.a.j 1
90.10.a.k 1
90.10.a.l 2
90.10.a.m 2
90.10.c \(\chi_{90}(19, \cdot)\) 90.10.c.a 4 1
90.10.c.b 4
90.10.c.c 6
90.10.c.d 8
90.10.e \(\chi_{90}(31, \cdot)\) 90.10.e.a 16 2
90.10.e.b 18
90.10.e.c 18
90.10.e.d 20
90.10.f \(\chi_{90}(17, \cdot)\) 90.10.f.a 16 2
90.10.f.b 20
90.10.i \(\chi_{90}(49, \cdot)\) 90.10.i.a 108 2
90.10.l \(\chi_{90}(23, \cdot)\) 90.10.l.a 216 4

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_1(90))\) into lower level spaces

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