## Defining parameters

 Level: $$N$$ = $$8 = 2^{3}$$ Weight: $$k$$ = $$5$$ Nonzero newspaces: $$1$$ Newform subspaces: $$2$$ Sturm bound: $$20$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{5}(\Gamma_1(8))$$.

Total New Old
Modular forms 11 5 6
Cusp forms 5 3 2
Eisenstein series 6 2 4

## Trace form

 $$3q + 2q^{2} - 2q^{3} - 12q^{4} - 68q^{6} + 152q^{8} + 25q^{9} + O(q^{10})$$ $$3q + 2q^{2} - 2q^{3} - 12q^{4} - 68q^{6} + 152q^{8} + 25q^{9} + 240q^{10} - 98q^{11} - 392q^{12} - 480q^{14} + 528q^{16} - 122q^{17} + 550q^{18} + 702q^{19} - 480q^{20} - 132q^{22} - 368q^{24} - 45q^{25} - 240q^{26} - 1988q^{27} + 960q^{28} + 1440q^{30} - 928q^{32} + 332q^{33} - 2748q^{34} + 3840q^{35} + 3100q^{36} + 1468q^{38} - 2880q^{40} + 742q^{41} - 2880q^{42} - 7266q^{43} - 8q^{44} + 2400q^{46} - 1952q^{48} - 477q^{49} + 3170q^{50} + 10748q^{51} + 480q^{52} - 392q^{54} + 5760q^{56} - 4468q^{57} + 2640q^{58} - 10274q^{59} - 2880q^{60} - 9600q^{62} + 3648q^{64} + 1920q^{65} + 2888q^{66} + 10878q^{67} - 15512q^{68} - 3840q^{70} + 3400q^{72} + 10278q^{73} + 13680q^{74} - 12770q^{75} + 3192q^{76} - 1440q^{78} + 13440q^{80} - 4433q^{81} - 6972q^{82} + 6718q^{83} + 5760q^{84} - 10244q^{86} - 5232q^{88} - 14618q^{89} - 10800q^{90} - 3840q^{91} - 4800q^{92} + 16320q^{94} - 26048q^{96} + 7494q^{97} + 12482q^{98} - 2950q^{99} + O(q^{100})$$

## Decomposition of $$S_{5}^{\mathrm{new}}(\Gamma_1(8))$$

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
8.5.c $$\chi_{8}(7, \cdot)$$ None 0 1
8.5.d $$\chi_{8}(3, \cdot)$$ 8.5.d.a 1 1
8.5.d.b 2

## Decomposition of $$S_{5}^{\mathrm{old}}(\Gamma_1(8))$$ into lower level spaces

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