## Defining parameters

 Level: $$N$$ = $$16 = 2^{4}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$2$$ Newforms: $$2$$ Sturm bound: $$64$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(16))$$.

Total New Old
Modular forms 31 16 15
Cusp forms 17 11 6
Eisenstein series 14 5 9

## Trace form

 $$11q - 2q^{2} + 2q^{3} + 8q^{4} - 4q^{5} - 32q^{6} - 24q^{7} - 44q^{8} - 11q^{9} + O(q^{10})$$ $$11q - 2q^{2} + 2q^{3} + 8q^{4} - 4q^{5} - 32q^{6} - 24q^{7} - 44q^{8} - 11q^{9} - 68q^{10} + 62q^{11} + 100q^{12} + 20q^{13} + 188q^{14} - 132q^{15} + 280q^{16} + 46q^{17} + 174q^{18} - 70q^{19} - 196q^{20} - 44q^{21} - 588q^{22} + 56q^{23} - 848q^{24} - 121q^{25} - 264q^{26} + 32q^{27} + 280q^{28} - 4q^{29} + 1236q^{30} + 528q^{31} + 968q^{32} + 172q^{33} + 436q^{34} + 524q^{35} - 596q^{36} - 172q^{37} - 1232q^{38} + 88q^{39} - 1336q^{40} - 198q^{41} - 680q^{42} - 890q^{43} + 868q^{44} + 216q^{45} + 1132q^{46} - 1472q^{47} + 1768q^{48} + 327q^{49} + 726q^{50} - 1300q^{51} - 236q^{52} - 620q^{53} - 1376q^{54} - 88q^{55} - 488q^{56} - 176q^{57} + 8q^{58} + 2374q^{59} - 192q^{60} + 1460q^{61} - 80q^{62} + 2892q^{63} + 512q^{64} - 536q^{65} - 428q^{66} + 1754q^{67} - 880q^{68} + 804q^{69} + 160q^{70} - 728q^{71} + 1092q^{72} + 154q^{73} - 452q^{74} - 3438q^{75} - 1228q^{76} - 1324q^{77} - 772q^{78} - 3760q^{79} - 2648q^{80} + 171q^{81} - 704q^{82} - 2798q^{83} + 1960q^{84} - 112q^{85} + 3764q^{86} + 792q^{87} + 1528q^{88} + 714q^{89} + 1896q^{90} + 2804q^{91} + 632q^{92} - 1552q^{93} - 3248q^{94} + 6988q^{95} - 4432q^{96} - 482q^{97} + 314q^{98} + 4474q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(16))$$

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
16.4.a $$\chi_{16}(1, \cdot)$$ 16.4.a.a 1 1
16.4.b $$\chi_{16}(9, \cdot)$$ None 0 1
16.4.e $$\chi_{16}(5, \cdot)$$ 16.4.e.a 10 2

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(16))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(16)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 2}$$