## Defining parameters

 Level: $$N$$ = $$15 = 3 \cdot 5$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$1$$ Newform subspaces: $$1$$ Sturm bound: $$32$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(15))$$.

Total New Old
Modular forms 16 9 7
Cusp forms 1 1 0
Eisenstein series 15 8 7

## Trace form

 $$q - q^{2} - q^{3} - q^{4} + q^{5} + q^{6} + 3 q^{8} + q^{9} + O(q^{10})$$ $$q - q^{2} - q^{3} - q^{4} + q^{5} + q^{6} + 3 q^{8} + q^{9} - q^{10} - 4 q^{11} + q^{12} - 2 q^{13} - q^{15} - q^{16} + 2 q^{17} - q^{18} + 4 q^{19} - q^{20} + 4 q^{22} - 3 q^{24} + q^{25} + 2 q^{26} - q^{27} - 2 q^{29} + q^{30} - 5 q^{32} + 4 q^{33} - 2 q^{34} - q^{36} - 10 q^{37} - 4 q^{38} + 2 q^{39} + 3 q^{40} + 10 q^{41} + 4 q^{43} + 4 q^{44} + q^{45} + 8 q^{47} + q^{48} - 7 q^{49} - q^{50} - 2 q^{51} + 2 q^{52} - 10 q^{53} + q^{54} - 4 q^{55} - 4 q^{57} + 2 q^{58} - 4 q^{59} + q^{60} - 2 q^{61} + 7 q^{64} - 2 q^{65} - 4 q^{66} + 12 q^{67} - 2 q^{68} - 8 q^{71} + 3 q^{72} + 10 q^{73} + 10 q^{74} - q^{75} - 4 q^{76} - 2 q^{78} - q^{80} + q^{81} - 10 q^{82} + 12 q^{83} + 2 q^{85} - 4 q^{86} + 2 q^{87} - 12 q^{88} - 6 q^{89} - q^{90} - 8 q^{94} + 4 q^{95} + 5 q^{96} + 2 q^{97} + 7 q^{98} - 4 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$

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
15.2.a $$\chi_{15}(1, \cdot)$$ 15.2.a.a 1 1
15.2.b $$\chi_{15}(4, \cdot)$$ None 0 1
15.2.e $$\chi_{15}(2, \cdot)$$ None 0 2