## Defining parameters

 Level: $$N$$ = $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$6$$ Newforms: $$12$$ Sturm bound: $$800$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(75))$$.

Total New Old
Modular forms 256 159 97
Cusp forms 145 119 26
Eisenstein series 111 40 71

## Trace form

 $$119q - q^{2} - 9q^{3} - 25q^{4} - 6q^{5} - 23q^{6} - 28q^{7} - 21q^{8} - 13q^{9} + O(q^{10})$$ $$119q - q^{2} - 9q^{3} - 25q^{4} - 6q^{5} - 23q^{6} - 28q^{7} - 21q^{8} - 13q^{9} - 34q^{10} - 4q^{11} - 19q^{12} - 30q^{13} - 24q^{14} - 14q^{15} - 33q^{16} - 2q^{17} - q^{18} - 8q^{19} + 36q^{20} - 6q^{21} + 16q^{22} + 16q^{23} + 51q^{24} + 14q^{25} - 6q^{26} - 9q^{27} + 44q^{28} + 14q^{29} + 14q^{30} - 28q^{31} + 17q^{32} - 10q^{33} - 20q^{34} - 20q^{35} + 9q^{36} - 28q^{37} + 8q^{38} + 12q^{39} + 2q^{40} - 22q^{41} + 66q^{42} + 8q^{43} + 48q^{44} + 64q^{45} + 12q^{46} + 16q^{47} + 67q^{48} + 47q^{49} + 106q^{50} + 18q^{51} + 98q^{52} + 56q^{53} + 115q^{54} + 4q^{55} + 58q^{57} + 26q^{58} + 28q^{59} + 14q^{60} - 54q^{61} + 4q^{62} - 28q^{63} - 101q^{64} - 58q^{65} - 58q^{66} - 112q^{67} - 122q^{68} - 104q^{69} - 140q^{70} - 56q^{71} - 171q^{72} - 114q^{73} - 134q^{74} - 114q^{75} - 124q^{76} - 96q^{77} - 118q^{78} - 100q^{79} - 154q^{80} - 61q^{81} - 46q^{82} - 28q^{83} - 106q^{84} + 38q^{85} - 4q^{86} - 14q^{87} + 64q^{88} + 72q^{89} - 24q^{90} + 12q^{91} + 32q^{92} + 68q^{93} + 172q^{94} + 56q^{95} + 33q^{96} + 158q^{97} + 95q^{98} - 4q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(75))$$

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
75.2.a $$\chi_{75}(1, \cdot)$$ 75.2.a.a 1 1
75.2.a.b 1
75.2.a.c 1
75.2.b $$\chi_{75}(49, \cdot)$$ 75.2.b.a 2 1
75.2.b.b 2
75.2.e $$\chi_{75}(32, \cdot)$$ 75.2.e.a 4 2
75.2.e.b 4
75.2.g $$\chi_{75}(16, \cdot)$$ 75.2.g.a 4 4
75.2.g.b 8
75.2.g.c 12
75.2.i $$\chi_{75}(4, \cdot)$$ 75.2.i.a 16 4
75.2.l $$\chi_{75}(2, \cdot)$$ 75.2.l.a 64 8

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(75))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(75)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 2}$$