## Defining parameters

 Level: $$N$$ = $$13$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$2$$ Newform subspaces: $$2$$ Sturm bound: $$42$$ Trace bound: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(13))$$.

Total New Old
Modular forms 20 20 0
Cusp forms 8 8 0
Eisenstein series 12 12 0

## Trace form

 $$8q - 6q^{2} - 6q^{3} - 6q^{4} - 6q^{5} - 6q^{6} + 4q^{7} + 30q^{8} + 18q^{9} + O(q^{10})$$ $$8q - 6q^{2} - 6q^{3} - 6q^{4} - 6q^{5} - 6q^{6} + 4q^{7} + 30q^{8} + 18q^{9} + 24q^{10} - 18q^{13} - 36q^{14} - 42q^{15} - 86q^{16} - 12q^{17} + 30q^{18} + 10q^{19} + 42q^{20} + 72q^{21} + 84q^{22} + 18q^{23} + 66q^{24} - 36q^{26} - 84q^{27} - 40q^{28} + 42q^{29} - 54q^{30} + 20q^{31} - 108q^{33} - 126q^{34} - 36q^{35} - 54q^{36} - 28q^{37} + 66q^{39} + 156q^{40} + 132q^{41} + 120q^{42} + 180q^{43} - 84q^{44} + 54q^{45} + 102q^{46} - 72q^{47} - 126q^{48} - 72q^{49} - 174q^{50} + 80q^{52} + 48q^{53} + 168q^{54} - 36q^{55} - 84q^{56} - 60q^{57} - 180q^{58} - 108q^{59} - 108q^{60} - 420q^{61} + 6q^{62} - 12q^{63} + 108q^{65} + 192q^{66} + 34q^{67} + 516q^{68} + 72q^{69} + 132q^{70} + 198q^{71} + 24q^{72} + 58q^{73} + 168q^{74} + 48q^{75} + 94q^{76} - 336q^{78} + 24q^{79} - 228q^{80} - 222q^{81} + 24q^{82} - 240q^{83} - 180q^{84} - 48q^{85} - 204q^{86} + 198q^{87} - 204q^{88} + 90q^{89} + 208q^{91} - 612q^{92} + 60q^{93} - 426q^{94} - 78q^{95} + 360q^{96} + 110q^{97} + 210q^{98} + 360q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(13))$$

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
13.3.d $$\chi_{13}(5, \cdot)$$ 13.3.d.a 4 2
13.3.f $$\chi_{13}(2, \cdot)$$ 13.3.f.a 4 4