## Defining parameters

 Level: $$N$$ = $$8 = 2^{3}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$2$$ Newform subspaces: $$2$$ 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

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

## 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