## Defining parameters

 Level: $$N$$ = $$64 = 2^{6}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$4$$ Newform subspaces: $$9$$ Sturm bound: $$1024$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(64))$$.

Total New Old
Modular forms 420 227 193
Cusp forms 348 205 143
Eisenstein series 72 22 50

## Trace form

 $$205q - 8q^{2} - 6q^{3} - 8q^{4} - 8q^{5} - 8q^{6} - 8q^{7} - 8q^{8} - 37q^{9} + O(q^{10})$$ $$205q - 8q^{2} - 6q^{3} - 8q^{4} - 8q^{5} - 8q^{6} - 8q^{7} - 8q^{8} - 37q^{9} - 8q^{10} - 26q^{11} - 8q^{12} + 64q^{13} - 8q^{14} + 116q^{15} - 8q^{16} + 90q^{17} - 8q^{18} + 18q^{19} - 8q^{20} - 20q^{21} + 464q^{22} - 8q^{23} + 992q^{24} + 41q^{25} - 48q^{26} - 192q^{27} - 768q^{28} - 408q^{29} - 2328q^{30} - 368q^{31} - 1248q^{32} - 988q^{33} - 1008q^{34} - 484q^{35} - 888q^{36} - 528q^{37} + 432q^{38} - 8q^{39} + 1632q^{40} + 1014q^{41} + 3152q^{42} + 830q^{43} + 992q^{44} + 1476q^{45} - 8q^{46} + 936q^{47} - 8q^{48} + 1049q^{49} - 2864q^{50} + 4468q^{51} - 3320q^{52} + 400q^{53} - 1736q^{54} + 568q^{55} + 384q^{56} - 1088q^{57} + 2368q^{58} - 4466q^{59} + 4888q^{60} - 1088q^{61} + 2984q^{62} - 7684q^{63} + 6040q^{64} - 2296q^{65} + 5528q^{66} - 6030q^{67} + 2056q^{68} - 644q^{69} + 2008q^{70} - 456q^{71} - 656q^{72} + 1014q^{73} - 2640q^{74} + 7362q^{75} - 5960q^{76} + 3116q^{77} - 2000q^{78} + 10072q^{79} + 4256q^{80} + 6805q^{81} + 6952q^{82} + 2554q^{83} + 4136q^{84} + 1648q^{85} - 528q^{86} - 8q^{87} - 3128q^{88} - 3578q^{89} - 9368q^{90} - 3340q^{91} - 12624q^{92} - 8384q^{93} - 8936q^{94} - 6900q^{95} - 12928q^{96} - 9358q^{97} - 12112q^{98} - 4750q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(64))$$

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
64.4.a $$\chi_{64}(1, \cdot)$$ 64.4.a.a 1 1
64.4.a.b 1
64.4.a.c 1
64.4.a.d 1
64.4.a.e 1
64.4.b $$\chi_{64}(33, \cdot)$$ 64.4.b.a 2 1
64.4.b.b 4
64.4.e $$\chi_{64}(17, \cdot)$$ 64.4.e.a 10 2
64.4.g $$\chi_{64}(9, \cdot)$$ None 0 4
64.4.i $$\chi_{64}(5, \cdot)$$ 64.4.i.a 184 8

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(64))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(64)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(32))$$$$^{\oplus 2}$$