## Defining parameters

 Level: $$N$$ = $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$6$$ Newform subspaces: $$15$$ Sturm bound: $$1200$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(75))$$.

Total New Old
Modular forms 456 298 158
Cusp forms 344 256 88
Eisenstein series 112 42 70

## Trace form

 $$256q + 8q^{2} - 2q^{3} - 4q^{4} + 4q^{5} - 2q^{6} - 4q^{7} - 24q^{8} - 42q^{9} + O(q^{10})$$ $$256q + 8q^{2} - 2q^{3} - 4q^{4} + 4q^{5} - 2q^{6} - 4q^{7} - 24q^{8} - 42q^{9} - 44q^{10} - 32q^{11} - 66q^{12} - 20q^{13} + 6q^{15} - 68q^{16} - 60q^{17} + 34q^{18} - 124q^{19} - 244q^{20} - 78q^{21} - 260q^{22} - 152q^{23} - 164q^{24} - 16q^{25} - 96q^{26} - 98q^{27} + 348q^{28} + 200q^{29} + 274q^{30} + 340q^{31} + 712q^{32} + 282q^{33} + 496q^{34} + 260q^{35} - 66q^{36} - 204q^{37} - 468q^{38} - 282q^{39} - 928q^{40} - 208q^{41} - 614q^{42} - 548q^{43} - 700q^{44} - 406q^{45} - 380q^{46} - 416q^{47} - 348q^{48} - 184q^{49} + 16q^{50} - 180q^{51} + 824q^{52} + 248q^{53} + 608q^{54} + 604q^{55} + 840q^{56} + 758q^{57} + 1544q^{58} + 800q^{59} + 1634q^{60} + 188q^{61} + 1124q^{62} + 1052q^{63} + 992q^{64} + 272q^{65} + 782q^{66} + 284q^{67} + 208q^{68} + 868q^{69} + 100q^{70} + 544q^{71} + 582q^{72} + 124q^{73} - 134q^{75} - 1208q^{76} - 176q^{77} - 880q^{78} - 964q^{79} - 164q^{80} - 234q^{81} - 2196q^{82} - 1248q^{83} - 2362q^{84} - 2192q^{85} - 1112q^{86} - 1732q^{87} - 2804q^{88} - 1500q^{89} - 1494q^{90} - 900q^{91} - 1608q^{92} - 920q^{93} - 1124q^{94} - 784q^{95} - 190q^{96} + 124q^{97} + 96q^{98} - 180q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(75))$$

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
75.3.c $$\chi_{75}(26, \cdot)$$ 75.3.c.a 1 1
75.3.c.b 1
75.3.c.c 2
75.3.c.d 2
75.3.c.e 2
75.3.c.f 2
75.3.d $$\chi_{75}(74, \cdot)$$ 75.3.d.a 2 1
75.3.d.b 4
75.3.d.c 4
75.3.f $$\chi_{75}(7, \cdot)$$ 75.3.f.a 4 2
75.3.f.b 4
75.3.f.c 4
75.3.h $$\chi_{75}(14, \cdot)$$ 75.3.h.a 72 4
75.3.j $$\chi_{75}(11, \cdot)$$ 75.3.j.a 72 4
75.3.k $$\chi_{75}(13, \cdot)$$ 75.3.k.a 80 8

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(75))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(75)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 2}$$