## Defining parameters

 Level: $$N$$ = $$8 = 2^{3}$$ Weight: $$k$$ = $$6$$ Nonzero newspaces: $$2$$ Newforms: $$2$$ Sturm bound: $$24$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{6}(\Gamma_1(8))$$.

Total New Old
Modular forms 13 7 6
Cusp forms 7 5 2
Eisenstein series 6 2 4

## Trace form

 $$5q - 2q^{2} + 20q^{3} + 20q^{4} - 74q^{5} - 116q^{6} + 72q^{7} - 248q^{8} - 7q^{9} + O(q^{10})$$ $$5q - 2q^{2} + 20q^{3} + 20q^{4} - 74q^{5} - 116q^{6} + 72q^{7} - 248q^{8} - 7q^{9} + 632q^{10} + 124q^{11} + 1576q^{12} + 478q^{13} - 2384q^{14} - 1896q^{15} - 3312q^{16} - 998q^{17} + 4754q^{18} + 3044q^{19} + 4624q^{20} - 480q^{21} - 5636q^{22} + 2520q^{23} - 7792q^{24} + 3907q^{25} + 5608q^{26} - 1720q^{27} + 5152q^{28} - 3282q^{29} - 2128q^{30} - 18656q^{31} + 5408q^{32} + 128q^{33} - 4772q^{34} + 1776q^{35} - 10164q^{36} + 10326q^{37} + 15980q^{38} + 44664q^{39} + 16032q^{40} - 13454q^{41} - 26144q^{42} - 9188q^{43} - 29112q^{44} - 11618q^{45} + 29200q^{46} - 31056q^{47} + 35616q^{48} - 6403q^{49} - 47498q^{50} - 23960q^{51} - 36560q^{52} + 11686q^{53} + 23288q^{54} + 76296q^{55} + 40768q^{56} + 58848q^{57} + 3784q^{58} + 16876q^{59} + 2592q^{60} - 18482q^{61} + 34496q^{62} - 157208q^{63} - 41920q^{64} - 54892q^{65} + 43224q^{66} - 15532q^{67} + 10344q^{68} + 3680q^{69} - 68928q^{70} + 174728q^{71} - 83272q^{72} + 35090q^{73} + 17464q^{74} + 47020q^{75} + 99944q^{76} - 2976q^{77} - 174064q^{78} - 203312q^{79} - 35520q^{80} - 42867q^{81} + 161132q^{82} + 67364q^{83} + 196672q^{84} + 88652q^{85} - 18500q^{86} + 242232q^{87} - 167216q^{88} - 12638q^{89} + 142280q^{90} - 11472q^{91} - 49056q^{92} - 114560q^{93} - 98784q^{94} - 485000q^{95} - 115648q^{96} - 50710q^{97} - 117042q^{98} + 19468q^{99} + O(q^{100})$$

## Decomposition of $$S_{6}^{\mathrm{new}}(\Gamma_1(8))$$

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
8.6.a $$\chi_{8}(1, \cdot)$$ 8.6.a.a 1 1
8.6.b $$\chi_{8}(5, \cdot)$$ 8.6.b.a 4 1

## Decomposition of $$S_{6}^{\mathrm{old}}(\Gamma_1(8))$$ into lower level spaces

$$S_{6}^{\mathrm{old}}(\Gamma_1(8)) \cong$$ $$S_{6}^{\mathrm{new}}(\Gamma_1(4))$$$$^{\oplus 2}$$