## Defining parameters

 Level: $$N$$ = $$13$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$1$$ Newform subspaces: $$1$$ Sturm bound: $$28$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(13))$$.

Total New Old
Modular forms 13 13 0
Cusp forms 2 2 0
Eisenstein series 11 11 0

## Trace form

 $$2 q - 3 q^{2} - 2 q^{3} + q^{4} + 6 q^{6} - q^{9} + O(q^{10})$$ $$2 q - 3 q^{2} - 2 q^{3} + q^{4} + 6 q^{6} - q^{9} - 3 q^{10} - 4 q^{12} - 5 q^{13} + 6 q^{15} + 5 q^{16} + 3 q^{17} - 6 q^{19} + 3 q^{20} + 6 q^{23} - 6 q^{24} + 4 q^{25} + 3 q^{26} - 8 q^{27} - 3 q^{29} - 6 q^{30} - 9 q^{32} + q^{36} + 15 q^{37} + 12 q^{38} + 14 q^{39} + 6 q^{40} - 9 q^{41} - 8 q^{43} - 3 q^{45} - 18 q^{46} + 10 q^{48} - 7 q^{49} - 6 q^{50} - 12 q^{51} + 2 q^{52} - 6 q^{53} + 12 q^{54} + 9 q^{58} + 12 q^{59} - q^{61} + 6 q^{62} - 2 q^{64} - 9 q^{65} + 6 q^{67} - 3 q^{68} + 12 q^{69} + 6 q^{71} + 3 q^{72} - 15 q^{74} - 4 q^{75} - 6 q^{76} - 24 q^{78} + 8 q^{79} - 15 q^{80} + 11 q^{81} + 9 q^{82} + 9 q^{85} - 6 q^{87} - 12 q^{89} + 6 q^{90} + 12 q^{92} - 12 q^{93} - 6 q^{94} + 6 q^{95} + 12 q^{97} + 21 q^{98} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(13))$$

We only show spaces with even 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
13.2.a $$\chi_{13}(1, \cdot)$$ None 0 1
13.2.b $$\chi_{13}(12, \cdot)$$ None 0 1
13.2.c $$\chi_{13}(3, \cdot)$$ None 0 2
13.2.e $$\chi_{13}(4, \cdot)$$ 13.2.e.a 2 2