## Defining parameters

 Level: $$N$$ = $$475 = 5^{2} \cdot 19$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$3$$ Newform subspaces: $$4$$ Sturm bound: $$18000$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(475))$$.

Total New Old
Modular forms 521 360 161
Cusp forms 17 11 6
Eisenstein series 504 349 155

The following table gives the dimensions of subspaces with specified projective image type.

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 11 0 0 0

## Trace form

 $$11 q - q^{4} - 2 q^{5} - 4 q^{6} - 2 q^{7} - q^{9} + O(q^{10})$$ $$11 q - q^{4} - 2 q^{5} - 4 q^{6} - 2 q^{7} - q^{9} - 2 q^{11} - 3 q^{16} + 8 q^{17} - q^{19} - 2 q^{23} - 2 q^{25} + 4 q^{26} - 2 q^{28} + 8 q^{35} + q^{36} + 4 q^{39} - 2 q^{43} - 6 q^{44} - 2 q^{47} - 7 q^{49} - 2 q^{61} - 2 q^{63} + 3 q^{64} - 2 q^{68} - 2 q^{73} + 4 q^{74} - 3 q^{76} - 4 q^{77} - 2 q^{80} - 3 q^{81} + 8 q^{83} - 2 q^{85} - 2 q^{92} - 2 q^{95} + 4 q^{96} - 6 q^{99} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(475))$$

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
475.1.c $$\chi_{475}(151, \cdot)$$ 475.1.c.a 1 1
475.1.c.b 2
475.1.d $$\chi_{475}(474, \cdot)$$ None 0 1
475.1.f $$\chi_{475}(343, \cdot)$$ None 0 2
475.1.i $$\chi_{475}(274, \cdot)$$ None 0 2
475.1.k $$\chi_{475}(126, \cdot)$$ None 0 2
475.1.m $$\chi_{475}(94, \cdot)$$ 475.1.m.a 4 4
475.1.o $$\chi_{475}(56, \cdot)$$ 475.1.o.a 4 4
475.1.q $$\chi_{475}(7, \cdot)$$ None 0 4
475.1.s $$\chi_{475}(51, \cdot)$$ None 0 6
475.1.t $$\chi_{475}(124, \cdot)$$ None 0 6
475.1.w $$\chi_{475}(58, \cdot)$$ None 0 8
475.1.y $$\chi_{475}(31, \cdot)$$ None 0 8
475.1.z $$\chi_{475}(69, \cdot)$$ None 0 8
475.1.ba $$\chi_{475}(43, \cdot)$$ None 0 12
475.1.bd $$\chi_{475}(83, \cdot)$$ None 0 16
475.1.bf $$\chi_{475}(21, \cdot)$$ None 0 24
475.1.bh $$\chi_{475}(14, \cdot)$$ None 0 24
475.1.bj $$\chi_{475}(17, \cdot)$$ None 0 48

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(475))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(475)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(95))$$$$^{\oplus 2}$$