## Defining parameters

 Level: $$N$$ = $$900 = 2^{2} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$8$$ Newform subspaces: $$9$$ Sturm bound: $$43200$$ Trace bound: $$13$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(900))$$.

Total New Old
Modular forms 1224 235 989
Cusp forms 104 50 54
Eisenstein series 1120 185 935

The following table gives the dimensions of subspaces with specified projective image type.

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 50 0 0 0

## Trace form

 $$50 q + q^{2} + 5 q^{4} + q^{5} - 4 q^{6} + q^{8} + 2 q^{9} + O(q^{10})$$ $$50 q + q^{2} + 5 q^{4} + q^{5} - 4 q^{6} + q^{8} + 2 q^{9} + q^{10} + 6 q^{13} - 2 q^{14} + q^{16} + 2 q^{17} - 4 q^{20} - 4 q^{21} + 2 q^{24} + q^{25} + 2 q^{26} + 4 q^{31} - 4 q^{32} - 15 q^{34} + 12 q^{36} + q^{37} - 7 q^{40} - 8 q^{41} - 12 q^{46} - 2 q^{49} + q^{50} - 10 q^{52} - 3 q^{53} + 2 q^{54} + 14 q^{56} - 10 q^{58} - 8 q^{61} - q^{64} - 3 q^{65} + 2 q^{68} - 2 q^{69} - 10 q^{73} + 2 q^{74} + q^{80} + 2 q^{81} + 6 q^{82} - 2 q^{84} - 11 q^{85} - 4 q^{86} + q^{89} - 12 q^{91} - 2 q^{94} - 6 q^{96} - 10 q^{97} + q^{98} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(900))$$

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
900.1.b $$\chi_{900}(449, \cdot)$$ None 0 1
900.1.c $$\chi_{900}(451, \cdot)$$ 900.1.c.a 1 1
900.1.c.b 1
900.1.f $$\chi_{900}(199, \cdot)$$ None 0 1
900.1.g $$\chi_{900}(701, \cdot)$$ None 0 1
900.1.l $$\chi_{900}(757, \cdot)$$ 900.1.l.a 4 2
900.1.m $$\chi_{900}(107, \cdot)$$ 900.1.m.a 4 2
900.1.p $$\chi_{900}(101, \cdot)$$ None 0 2
900.1.q $$\chi_{900}(499, \cdot)$$ None 0 2
900.1.t $$\chi_{900}(151, \cdot)$$ 900.1.t.a 4 2
900.1.u $$\chi_{900}(149, \cdot)$$ None 0 2
900.1.x $$\chi_{900}(91, \cdot)$$ 900.1.x.a 4 4
900.1.y $$\chi_{900}(89, \cdot)$$ None 0 4
900.1.ba $$\chi_{900}(161, \cdot)$$ None 0 4
900.1.bb $$\chi_{900}(19, \cdot)$$ 900.1.bb.a 8 4
900.1.bc $$\chi_{900}(157, \cdot)$$ None 0 4
900.1.bd $$\chi_{900}(407, \cdot)$$ 900.1.bd.a 8 4
900.1.bh $$\chi_{900}(287, \cdot)$$ 900.1.bh.a 16 8
900.1.bi $$\chi_{900}(37, \cdot)$$ None 0 8
900.1.bl $$\chi_{900}(79, \cdot)$$ None 0 8
900.1.bm $$\chi_{900}(41, \cdot)$$ None 0 8
900.1.bo $$\chi_{900}(29, \cdot)$$ None 0 8
900.1.bp $$\chi_{900}(31, \cdot)$$ None 0 8
900.1.bu $$\chi_{900}(23, \cdot)$$ None 0 16
900.1.bv $$\chi_{900}(13, \cdot)$$ None 0 16

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(900))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(900)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(100))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(180))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(225))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(300))$$$$^{\oplus 2}$$