## Defining parameters

 Level: $$N$$ = $$8 = 2^{3}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$2$$ Newforms: $$2$$ Sturm bound: $$16$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(8))$$.

Total New Old
Modular forms 9 5 4
Cusp forms 3 3 0
Eisenstein series 6 2 4

## Trace form

 $$3q - 2q^{2} - 4q^{3} - 12q^{4} - 2q^{5} + 28q^{6} + 8q^{7} + 40q^{8} - 13q^{9} + O(q^{10})$$ $$3q - 2q^{2} - 4q^{3} - 12q^{4} - 2q^{5} + 28q^{6} + 8q^{7} + 40q^{8} - 13q^{9} - 56q^{10} - 44q^{11} - 56q^{12} + 22q^{13} + 16q^{14} + 120q^{15} + 16q^{16} + 22q^{17} + 2q^{18} + 44q^{19} + 112q^{20} - 96q^{21} - 84q^{22} - 360q^{23} - 112q^{24} - 95q^{25} + 280q^{26} + 152q^{27} + 96q^{28} + 198q^{29} - 112q^{30} + 288q^{31} - 352q^{32} + 344q^{33} + 28q^{34} - 48q^{35} + 12q^{36} - 162q^{37} - 196q^{38} - 648q^{39} + 224q^{40} - 338q^{41} - 224q^{42} + 52q^{43} + 168q^{44} + 22q^{45} + 304q^{46} + 1200q^{47} + 672q^{48} - 325q^{49} - 26q^{50} - 200q^{51} - 560q^{52} - 242q^{53} + 728q^{54} - 248q^{55} - 320q^{56} + 216q^{57} - 840q^{58} - 668q^{59} - 672q^{60} + 550q^{61} - 448q^{62} - 248q^{63} + 576q^{64} + 1076q^{65} - 168q^{66} + 188q^{67} + 168q^{68} + 224q^{69} + 448q^{70} + 584q^{71} - 40q^{72} - 434q^{73} + 1288q^{74} + 484q^{75} + 392q^{76} - 1056q^{77} + 560q^{78} - 1584q^{79} - 1344q^{80} - 1821q^{81} + 140q^{82} + 236q^{83} + 448q^{84} - 100q^{85} - 2324q^{86} + 888q^{87} + 336q^{88} + 1246q^{89} + 56q^{90} + 528q^{91} + 1824q^{92} + 640q^{93} - 672q^{94} - 872q^{95} - 448q^{96} + 1510q^{97} + 558q^{98} + 484q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(8))$$

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
8.4.a $$\chi_{8}(1, \cdot)$$ 8.4.a.a 1 1
8.4.b $$\chi_{8}(5, \cdot)$$ 8.4.b.a 2 1