## Defining parameters

 Level: $$N$$ = $$16 = 2^{4}$$ Weight: $$k$$ = $$5$$ Nonzero newspaces: $$2$$ Newform subspaces: $$2$$ Sturm bound: $$80$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{5}(\Gamma_1(16))$$.

Total New Old
Modular forms 39 20 19
Cusp forms 25 16 9
Eisenstein series 14 4 10

## Trace form

 $$16q - 2q^{2} - 2q^{3} - 8q^{4} + 34q^{5} + 64q^{6} - 4q^{7} - 92q^{8} - 222q^{9} + O(q^{10})$$ $$16q - 2q^{2} - 2q^{3} - 8q^{4} + 34q^{5} + 64q^{6} - 4q^{7} - 92q^{8} - 222q^{9} - 100q^{10} + 94q^{11} - 332q^{12} + 354q^{13} + 44q^{14} - 168q^{16} - 256q^{17} + 1390q^{18} - 706q^{19} + 1900q^{20} + 604q^{21} + 900q^{22} + 1148q^{23} - 1872q^{24} - 602q^{25} - 3416q^{26} - 1664q^{27} - 3784q^{28} - 1982q^{29} - 3740q^{30} + 3208q^{32} + 3452q^{33} + 7508q^{34} + 1340q^{35} + 11468q^{36} - 766q^{37} + 3568q^{38} + 2684q^{39} - 5144q^{40} + 324q^{41} - 17064q^{42} + 1694q^{43} - 14636q^{44} - 2586q^{45} - 5316q^{46} + 6888q^{48} + 3948q^{49} + 20070q^{50} - 3012q^{51} + 20452q^{52} + 706q^{53} + 10784q^{54} - 11780q^{55} - 6952q^{56} - 11136q^{57} - 20456q^{58} - 2786q^{59} - 29920q^{60} - 2526q^{61} - 11472q^{62} + 15808q^{64} + 4388q^{65} + 30148q^{66} + 7998q^{67} + 18032q^{68} + 30364q^{69} + 15296q^{70} + 19964q^{71} - 17708q^{72} - 13372q^{73} - 23780q^{74} + 17570q^{75} - 23996q^{76} - 16420q^{77} - 8052q^{78} + 1384q^{80} - 5008q^{81} + 16016q^{82} - 17282q^{83} + 19624q^{84} + 5412q^{85} - 4796q^{86} - 49284q^{87} + 7288q^{88} + 16452q^{89} - 5416q^{90} - 28036q^{91} - 14632q^{92} - 320q^{93} + 432q^{94} + 6064q^{96} - 3200q^{97} - 12246q^{98} + 49214q^{99} + O(q^{100})$$

## Decomposition of $$S_{5}^{\mathrm{new}}(\Gamma_1(16))$$

We only show spaces with odd 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
16.5.c $$\chi_{16}(15, \cdot)$$ 16.5.c.a 2 1
16.5.d $$\chi_{16}(7, \cdot)$$ None 0 1
16.5.f $$\chi_{16}(3, \cdot)$$ 16.5.f.a 14 2

## Decomposition of $$S_{5}^{\mathrm{old}}(\Gamma_1(16))$$ into lower level spaces

$$S_{5}^{\mathrm{old}}(\Gamma_1(16)) \cong$$ $$S_{5}^{\mathrm{new}}(\Gamma_1(4))$$$$^{\oplus 3}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 2}$$