## Defining parameters

 Level: $$N$$ = $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$6$$ Newform subspaces: $$12$$ 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

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

## 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

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