## Defining parameters

 Level: $$N$$ = $$15 = 3 \cdot 5$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$3$$ Newform subspaces: $$4$$ 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

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

## 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

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