## Defining parameters

 Level: $$N$$ = $$2500 = 2^{2} \cdot 5^{4}$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$7$$ Newform subspaces: $$15$$ Sturm bound: $$375000$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(2500))$$.

Total New Old
Modular forms 3058 984 2074
Cusp forms 308 216 92
Eisenstein series 2750 768 1982

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 216 0 0 0

## Trace form

 $$216 q + 4 q^{6} + O(q^{10})$$ $$216 q + 4 q^{6} - 4 q^{16} + 5 q^{18} + 8 q^{21} - 12 q^{26} + 5 q^{32} + 5 q^{34} + 5 q^{37} + 12 q^{41} + 4 q^{46} + 5 q^{49} + 5 q^{53} + 4 q^{56} + 12 q^{61} + 4 q^{81} + 4 q^{86} + 5 q^{89} + 4 q^{96} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(2500))$$

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
2500.1.b $$\chi_{2500}(1251, \cdot)$$ 2500.1.b.a 2 1
2500.1.b.b 2
2500.1.d $$\chi_{2500}(2499, \cdot)$$ 2500.1.d.a 4 1
2500.1.f $$\chi_{2500}(1693, \cdot)$$ None 0 2
2500.1.h $$\chi_{2500}(499, \cdot)$$ 2500.1.h.a 4 4
2500.1.h.b 4
2500.1.h.c 4
2500.1.h.d 4
2500.1.h.e 8
2500.1.j $$\chi_{2500}(251, \cdot)$$ 2500.1.j.a 4 4
2500.1.j.b 4
2500.1.j.c 8
2500.1.j.d 8
2500.1.k $$\chi_{2500}(57, \cdot)$$ None 0 8
2500.1.n $$\chi_{2500}(99, \cdot)$$ 2500.1.n.a 40 20
2500.1.p $$\chi_{2500}(51, \cdot)$$ 2500.1.p.a 20 20
2500.1.q $$\chi_{2500}(93, \cdot)$$ None 0 40
2500.1.t $$\chi_{2500}(11, \cdot)$$ 2500.1.t.a 100 100
2500.1.v $$\chi_{2500}(19, \cdot)$$ None 0 100
2500.1.w $$\chi_{2500}(13, \cdot)$$ None 0 200

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(2500))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(2500)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(100))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(500))$$$$^{\oplus 2}$$