## Defining parameters

 Level: $$N$$ = $$52 = 2^{2} \cdot 13$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$1$$ Newform subspaces: $$1$$ Sturm bound: $$168$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(52))$$.

Total New Old
Modular forms 32 12 20
Cusp forms 2 2 0
Eisenstein series 30 10 20

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 2 0 0 0

## Trace form

 $$2 q - q^{2} - q^{4} - 2 q^{5} + 2 q^{8} - q^{9} + O(q^{10})$$ $$2 q - q^{2} - q^{4} - 2 q^{5} + 2 q^{8} - q^{9} + q^{10} - q^{13} - q^{16} + q^{17} + 2 q^{18} + q^{20} - q^{26} + q^{29} - q^{32} - 2 q^{34} - q^{36} + q^{37} - 2 q^{40} + q^{41} + q^{45} - q^{49} + 2 q^{52} - 2 q^{53} + q^{58} + q^{61} + 2 q^{64} + q^{65} + q^{68} - q^{72} - 2 q^{73} + q^{74} + q^{80} - q^{81} + q^{82} - q^{85} - 2 q^{89} - 2 q^{90} - 2 q^{97} - q^{98} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(52))$$

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
52.1.b $$\chi_{52}(51, \cdot)$$ None 0 1
52.1.c $$\chi_{52}(27, \cdot)$$ None 0 1
52.1.g $$\chi_{52}(5, \cdot)$$ None 0 2
52.1.i $$\chi_{52}(23, \cdot)$$ None 0 2
52.1.j $$\chi_{52}(3, \cdot)$$ 52.1.j.a 2 2
52.1.k $$\chi_{52}(33, \cdot)$$ None 0 4