## Defining parameters

 Level: $$N$$ = $$90 = 2 \cdot 3^{2} \cdot 5$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$6$$ Newform subspaces: $$12$$ Sturm bound: $$864$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(90))$$.

Total New Old
Modular forms 280 53 227
Cusp forms 153 53 100
Eisenstein series 127 0 127

## Trace form

 $$53 q + 3 q^{2} + 6 q^{3} + 3 q^{4} + 5 q^{5} - 6 q^{6} + 12 q^{7} - 3 q^{8} - 14 q^{9} + O(q^{10})$$ $$53 q + 3 q^{2} + 6 q^{3} + 3 q^{4} + 5 q^{5} - 6 q^{6} + 12 q^{7} - 3 q^{8} - 14 q^{9} - 11 q^{10} - 26 q^{11} - 8 q^{12} - 14 q^{13} - 20 q^{14} - 24 q^{15} + 3 q^{16} - 18 q^{17} - 4 q^{18} - 8 q^{19} - 11 q^{20} - 36 q^{21} - 10 q^{22} - 36 q^{23} + 6 q^{24} - 17 q^{25} + 6 q^{26} - 24 q^{27} - 6 q^{29} - 8 q^{31} + 3 q^{32} + 10 q^{33} + 4 q^{34} + 16 q^{35} + 10 q^{36} - 26 q^{37} + 42 q^{38} + 64 q^{39} - 3 q^{40} + 76 q^{41} + 64 q^{42} + 10 q^{43} + 32 q^{44} + 80 q^{45} + 60 q^{47} + 10 q^{48} + 3 q^{49} + 47 q^{50} + 30 q^{51} + 10 q^{52} + 30 q^{53} + 30 q^{54} - 20 q^{55} + 20 q^{56} + 26 q^{57} + 34 q^{58} + 10 q^{59} - 8 q^{60} + 14 q^{61} - 40 q^{63} - 3 q^{64} - 42 q^{65} - 48 q^{66} - 18 q^{67} - 36 q^{68} - 64 q^{69} - 24 q^{70} - 72 q^{71} - 6 q^{72} - 58 q^{73} - 50 q^{74} + 6 q^{75} - 46 q^{76} - 12 q^{77} - 28 q^{78} - 32 q^{79} + 5 q^{80} + 2 q^{81} - 58 q^{82} + 36 q^{83} - 12 q^{84} - 30 q^{85} - 70 q^{86} - 20 q^{87} - 10 q^{88} - 46 q^{89} - 56 q^{90} - 64 q^{91} - 36 q^{92} + 32 q^{93} - 28 q^{94} + 12 q^{95} - 8 q^{96} - 44 q^{97} - 27 q^{98} - 20 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(90))$$

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
90.2.a $$\chi_{90}(1, \cdot)$$ 90.2.a.a 1 1
90.2.a.b 1
90.2.a.c 1
90.2.c $$\chi_{90}(19, \cdot)$$ 90.2.c.a 2 1
90.2.e $$\chi_{90}(31, \cdot)$$ 90.2.e.a 2 2
90.2.e.b 2
90.2.e.c 4
90.2.f $$\chi_{90}(17, \cdot)$$ 90.2.f.a 4 2
90.2.i $$\chi_{90}(49, \cdot)$$ 90.2.i.a 4 2
90.2.i.b 8
90.2.l $$\chi_{90}(23, \cdot)$$ 90.2.l.a 8 4
90.2.l.b 16

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(90))$$ into lower level spaces

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