## Defining parameters

 Level: $$N$$ = $$500 = 2^{2} \cdot 5^{3}$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$5$$ Newform subspaces: $$6$$ Sturm bound: $$15000$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(500))$$.

Total New Old
Modular forms 498 168 330
Cusp forms 48 40 8
Eisenstein series 450 128 322

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 40 0 0 0

## Trace form

 $$40 q + q^{2} + q^{4} - 4 q^{6} + q^{8} + q^{9} + O(q^{10})$$ $$40 q + q^{2} + q^{4} - 4 q^{6} + q^{8} + q^{9} + 2 q^{13} + 5 q^{16} + 2 q^{17} - 9 q^{18} - 5 q^{20} - 8 q^{21} - 6 q^{26} + 2 q^{29} - 9 q^{32} - 8 q^{34} + q^{36} - 8 q^{37} - 10 q^{41} - 4 q^{46} - 9 q^{49} + 2 q^{52} - 8 q^{53} - 4 q^{56} + 2 q^{58} - 10 q^{61} + q^{64} - 5 q^{65} + 2 q^{68} + q^{72} + 2 q^{73} + 2 q^{74} - 3 q^{81} + 2 q^{82} - 5 q^{85} - 4 q^{86} - 8 q^{89} - 4 q^{96} + 2 q^{97} + q^{98} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(500))$$

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
500.1.b $$\chi_{500}(251, \cdot)$$ 500.1.b.a 4 1
500.1.d $$\chi_{500}(499, \cdot)$$ 500.1.d.a 2 1
500.1.d.b 2
500.1.f $$\chi_{500}(57, \cdot)$$ None 0 2
500.1.h $$\chi_{500}(99, \cdot)$$ 500.1.h.a 8 4
500.1.j $$\chi_{500}(51, \cdot)$$ 500.1.j.a 4 4
500.1.k $$\chi_{500}(93, \cdot)$$ None 0 8
500.1.n $$\chi_{500}(19, \cdot)$$ None 0 20
500.1.p $$\chi_{500}(11, \cdot)$$ 500.1.p.a 20 20
500.1.q $$\chi_{500}(13, \cdot)$$ None 0 40

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(500))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(500)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(100))$$$$^{\oplus 2}$$