## 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

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

## 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 the 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

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