## Defining parameters

 Level: $$N$$ = $$15 = 3 \cdot 5$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$3$$ Newforms: $$4$$ Sturm bound: $$64$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(15))$$.

Total New Old
Modular forms 32 22 10
Cusp forms 16 14 2
Eisenstein series 16 8 8

## Trace form

 $$14q + 4q^{2} - 6q^{3} - 24q^{4} + 6q^{5} - 20q^{7} - 36q^{8} - 18q^{9} + O(q^{10})$$ $$14q + 4q^{2} - 6q^{3} - 24q^{4} + 6q^{5} - 20q^{7} - 36q^{8} - 18q^{9} - 96q^{10} - 56q^{11} + 108q^{12} + 164q^{13} + 264q^{14} + 174q^{15} + 304q^{16} + 40q^{17} - 204q^{18} - 256q^{19} - 436q^{20} - 588q^{21} - 520q^{22} - 288q^{23} - 288q^{24} + 86q^{25} + 232q^{26} + 702q^{27} + 696q^{28} + 788q^{29} + 1116q^{30} + 632q^{31} + 116q^{32} - 12q^{33} - 568q^{34} - 520q^{35} - 696q^{36} - 1260q^{37} - 392q^{38} - 624q^{39} - 152q^{40} - 364q^{41} + 288q^{42} + 452q^{43} + 728q^{44} + 126q^{45} + 392q^{46} + 232q^{47} - 780q^{48} + 150q^{49} - 596q^{50} - 444q^{51} + 144q^{52} + 112q^{53} - 216q^{54} - 24q^{55} + 996q^{57} + 56q^{58} + 496q^{59} + 576q^{60} + 1116q^{61} + 312q^{62} + 1392q^{63} + 1224q^{64} + 688q^{65} + 1296q^{66} + 124q^{67} + 152q^{68} + 144q^{69} - 1560q^{70} - 688q^{71} - 2124q^{72} - 2980q^{73} - 3376q^{74} - 3486q^{75} - 3920q^{76} - 1728q^{77} - 960q^{78} - 1520q^{79} + 1004q^{80} + 774q^{81} + 3912q^{82} + 1368q^{83} + 2496q^{84} + 4092q^{85} + 4384q^{86} - 756q^{87} + 2184q^{88} - 876q^{89} + 384q^{90} + 1592q^{91} + 1056q^{92} + 1944q^{93} + 3416q^{94} + 3304q^{95} + 2928q^{96} + 2484q^{97} + 404q^{98} + 1008q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(15))$$

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
15.4.a $$\chi_{15}(1, \cdot)$$ 15.4.a.a 1 1
15.4.a.b 1
15.4.b $$\chi_{15}(4, \cdot)$$ 15.4.b.a 4 1
15.4.e $$\chi_{15}(2, \cdot)$$ 15.4.e.a 8 2

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(15))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(15)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 2}$$