## 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

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

## 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

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