## Defining parameters

 Level: $$N$$ = $$64 = 2^{6}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$4$$ Newform subspaces: $$4$$ Sturm bound: $$512$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(64))$$.

Total New Old
Modular forms 164 83 81
Cusp forms 93 61 32
Eisenstein series 71 22 49

## Trace form

 $$61 q - 8 q^{2} - 6 q^{3} - 8 q^{4} - 8 q^{5} - 8 q^{6} - 8 q^{7} - 8 q^{8} - 13 q^{9} + O(q^{10})$$ $$61 q - 8 q^{2} - 6 q^{3} - 8 q^{4} - 8 q^{5} - 8 q^{6} - 8 q^{7} - 8 q^{8} - 13 q^{9} - 8 q^{10} - 10 q^{11} - 8 q^{12} - 16 q^{13} - 8 q^{14} - 12 q^{15} - 8 q^{16} - 22 q^{17} - 8 q^{18} - 14 q^{19} - 8 q^{20} - 4 q^{21} - 8 q^{23} + 32 q^{24} + q^{25} + 32 q^{26} + 32 q^{28} + 8 q^{29} + 72 q^{30} + 16 q^{31} + 32 q^{32} + 20 q^{33} + 32 q^{34} - 4 q^{35} + 72 q^{36} + 32 q^{38} - 8 q^{39} + 32 q^{40} - 10 q^{41} + 32 q^{42} - 18 q^{43} - 12 q^{45} - 8 q^{46} - 24 q^{47} - 8 q^{48} - 23 q^{49} - 32 q^{50} + 20 q^{51} - 56 q^{52} - 32 q^{53} - 72 q^{54} + 56 q^{55} - 64 q^{56} - 16 q^{57} - 80 q^{58} + 62 q^{59} - 104 q^{60} - 16 q^{61} - 40 q^{62} + 60 q^{63} - 104 q^{64} - 24 q^{65} - 88 q^{66} + 82 q^{67} - 56 q^{68} - 20 q^{69} - 104 q^{70} + 56 q^{71} - 80 q^{72} - 10 q^{73} - 64 q^{74} + 50 q^{75} - 72 q^{76} - 4 q^{77} - 32 q^{78} + 24 q^{79} + 32 q^{80} - 11 q^{81} + 72 q^{82} - 6 q^{83} + 104 q^{84} + 96 q^{86} - 8 q^{87} + 72 q^{88} + 38 q^{89} + 136 q^{90} - 12 q^{91} + 144 q^{92} + 32 q^{93} + 88 q^{94} + 12 q^{95} + 128 q^{96} + 34 q^{97} + 128 q^{98} + 18 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(64))$$

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
64.2.a $$\chi_{64}(1, \cdot)$$ 64.2.a.a 1 1
64.2.b $$\chi_{64}(33, \cdot)$$ 64.2.b.a 2 1
64.2.e $$\chi_{64}(17, \cdot)$$ 64.2.e.a 2 2
64.2.g $$\chi_{64}(9, \cdot)$$ None 0 4
64.2.i $$\chi_{64}(5, \cdot)$$ 64.2.i.a 56 8

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(64))$$ into lower level spaces

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