## Defining parameters

 Level: $$N$$ = $$15\( 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

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

## 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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + 2 T^{2}$$
$3$ $$1 + T$$
$5$ $$1 - T$$
$7$ $$1 + 7 T^{2}$$
$11$ $$1 + 4 T + 11 T^{2}$$
$13$ $$1 + 2 T + 13 T^{2}$$
$17$ $$1 - 2 T + 17 T^{2}$$
$19$ $$1 - 4 T + 19 T^{2}$$
$23$ $$1 + 23 T^{2}$$
$29$ $$1 + 2 T + 29 T^{2}$$
$31$ $$1 + 31 T^{2}$$
$37$ $$1 + 10 T + 37 T^{2}$$
$41$ $$1 - 10 T + 41 T^{2}$$
$43$ $$1 - 4 T + 43 T^{2}$$
$47$ $$1 - 8 T + 47 T^{2}$$
$53$ $$1 + 10 T + 53 T^{2}$$
$59$ $$1 + 4 T + 59 T^{2}$$
$61$ $$1 + 2 T + 61 T^{2}$$
$67$ $$1 - 12 T + 67 T^{2}$$
$71$ $$1 + 8 T + 71 T^{2}$$
$73$ $$1 - 10 T + 73 T^{2}$$
$79$ $$1 + 79 T^{2}$$
$83$ $$1 - 12 T + 83 T^{2}$$
$89$ $$1 + 6 T + 89 T^{2}$$
$97$ $$1 - 2 T + 97 T^{2}$$