## Defining parameters

 Level: $$N$$ = $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$12$$ Newform subspaces: $$32$$ Sturm bound: $$4800$$ Trace bound: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(175))$$.

Total New Old
Modular forms 1368 1163 205
Cusp forms 1033 959 74
Eisenstein series 335 204 131

## Trace form

 $$959q - 25q^{2} - 28q^{3} - 37q^{4} - 38q^{5} - 60q^{6} - 41q^{7} - 97q^{8} - 55q^{9} + O(q^{10})$$ $$959q - 25q^{2} - 28q^{3} - 37q^{4} - 38q^{5} - 60q^{6} - 41q^{7} - 97q^{8} - 55q^{9} - 58q^{10} - 60q^{11} - 100q^{12} - 58q^{13} - 71q^{14} - 116q^{15} - 85q^{16} - 50q^{17} - 83q^{18} - 36q^{19} - 28q^{20} - 82q^{21} - 80q^{22} - 48q^{23} + 40q^{24} - 6q^{25} - 102q^{26} - 4q^{27} + 25q^{28} - 54q^{29} + 4q^{30} - 32q^{31} + 9q^{32} - 8q^{34} - 32q^{35} - 53q^{36} - 48q^{37} - 40q^{38} + 16q^{39} - 6q^{40} - 62q^{41} + 38q^{42} - 56q^{43} - 32q^{44} + 38q^{45} - 72q^{46} - 56q^{47} + 56q^{48} - 105q^{49} - 46q^{50} - 160q^{51} + 6q^{52} - 88q^{53} + 28q^{54} - 44q^{55} - 79q^{56} - 36q^{57} + 46q^{58} + 92q^{60} + 14q^{61} + 132q^{62} + 61q^{63} + 75q^{64} + 14q^{65} + 48q^{66} + 52q^{67} + 158q^{68} + 52q^{69} + 102q^{70} - 124q^{71} + 147q^{72} + 34q^{73} + 126q^{74} + 28q^{75} + 28q^{76} + 18q^{77} - 8q^{78} + 16q^{79} + 98q^{80} - 91q^{81} + 86q^{82} + 48q^{83} + 90q^{84} + 22q^{85} - 36q^{86} + 4q^{87} + 180q^{88} + 84q^{89} + 170q^{90} - 56q^{91} + 12q^{92} + 28q^{93} + 64q^{94} + 28q^{95} - 4q^{96} + 114q^{97} + 125q^{98} - 68q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(175))$$

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
175.2.a $$\chi_{175}(1, \cdot)$$ 175.2.a.a 1 1
175.2.a.b 1
175.2.a.c 1
175.2.a.d 2
175.2.a.e 2
175.2.a.f 2
175.2.b $$\chi_{175}(99, \cdot)$$ 175.2.b.a 2 1
175.2.b.b 4
175.2.b.c 4
175.2.e $$\chi_{175}(51, \cdot)$$ 175.2.e.a 2 2
175.2.e.b 2
175.2.e.c 4
175.2.e.d 6
175.2.e.e 6
175.2.f $$\chi_{175}(118, \cdot)$$ 175.2.f.a 4 2
175.2.f.b 4
175.2.f.c 4
175.2.f.d 8
175.2.h $$\chi_{175}(36, \cdot)$$ 175.2.h.a 4 4
175.2.h.b 28
175.2.h.c 32
175.2.k $$\chi_{175}(74, \cdot)$$ 175.2.k.a 8 2
175.2.k.b 12
175.2.n $$\chi_{175}(29, \cdot)$$ 175.2.n.a 56 4
175.2.o $$\chi_{175}(68, \cdot)$$ 175.2.o.a 4 4
175.2.o.b 4
175.2.o.c 8
175.2.o.d 24
175.2.q $$\chi_{175}(11, \cdot)$$ 175.2.q.a 144 8
175.2.s $$\chi_{175}(13, \cdot)$$ 175.2.s.a 144 8
175.2.t $$\chi_{175}(4, \cdot)$$ 175.2.t.a 144 8
175.2.x $$\chi_{175}(3, \cdot)$$ 175.2.x.a 288 16

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(175))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(175)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(35))$$$$^{\oplus 2}$$