## 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

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

## 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