## Defining parameters

 Level: $$N$$ = $$16 = 2^{4}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$2$$ Newform subspaces: $$2$$ 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

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

## 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

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