## Defining parameters

 Level: $$N$$ = $$676 = 2^{2} \cdot 13^{2}$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$6$$ Newform subspaces: $$7$$ Sturm bound: $$28392$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(676))$$.

Total New Old
Modular forms 620 251 369
Cusp forms 50 46 4
Eisenstein series 570 205 365

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 46 0 0 0

## Trace form

 $$46 q + q^{2} + q^{4} + 2 q^{5} - 5 q^{8} + q^{9} + O(q^{10})$$ $$46 q + q^{2} + q^{4} + 2 q^{5} - 5 q^{8} + q^{9} - 4 q^{10} + q^{16} - 4 q^{17} - 5 q^{18} - 4 q^{20} - 3 q^{25} - 4 q^{29} + q^{32} + 2 q^{34} + q^{36} - 4 q^{37} - 10 q^{40} - 4 q^{41} - 4 q^{45} + q^{49} - 3 q^{50} - 3 q^{52} - 10 q^{53} - 4 q^{58} - 4 q^{61} - 5 q^{64} - 3 q^{65} - 4 q^{68} + q^{72} + 2 q^{73} - 4 q^{74} - 4 q^{80} + q^{81} - 4 q^{82} - 2 q^{85} + 2 q^{89} + 2 q^{90} + 2 q^{97} + q^{98} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(676))$$

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
676.1.b $$\chi_{676}(675, \cdot)$$ 676.1.b.a 2 1
676.1.c $$\chi_{676}(339, \cdot)$$ 676.1.c.a 1 1
676.1.c.b 1
676.1.g $$\chi_{676}(437, \cdot)$$ None 0 2
676.1.i $$\chi_{676}(23, \cdot)$$ 676.1.i.a 4 2
676.1.j $$\chi_{676}(191, \cdot)$$ 676.1.j.a 2 2
676.1.k $$\chi_{676}(89, \cdot)$$ None 0 4
676.1.o $$\chi_{676}(27, \cdot)$$ 676.1.o.a 12 12
676.1.p $$\chi_{676}(51, \cdot)$$ None 0 12
676.1.r $$\chi_{676}(5, \cdot)$$ None 0 24
676.1.t $$\chi_{676}(3, \cdot)$$ 676.1.t.a 24 24
676.1.u $$\chi_{676}(43, \cdot)$$ None 0 24
676.1.x $$\chi_{676}(33, \cdot)$$ None 0 48

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(676))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(676)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(52))$$$$^{\oplus 2}$$