## Defining parameters

 Level: $$N$$ = $$128 = 2^{7}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$5$$ Newform subspaces: $$11$$ Sturm bound: $$2048$$ Trace bound: $$9$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(128))$$.

Total New Old
Modular forms 592 312 280
Cusp forms 433 264 169
Eisenstein series 159 48 111

## Trace form

 $$264 q - 16 q^{2} - 12 q^{3} - 16 q^{4} - 16 q^{5} - 16 q^{6} - 12 q^{7} - 16 q^{8} - 20 q^{9} + O(q^{10})$$ $$264 q - 16 q^{2} - 12 q^{3} - 16 q^{4} - 16 q^{5} - 16 q^{6} - 12 q^{7} - 16 q^{8} - 20 q^{9} - 16 q^{10} - 12 q^{11} - 16 q^{12} - 16 q^{13} - 16 q^{14} - 16 q^{15} - 16 q^{16} - 24 q^{17} - 16 q^{18} - 12 q^{19} - 16 q^{20} - 28 q^{21} - 16 q^{22} - 20 q^{23} - 16 q^{24} - 36 q^{25} - 16 q^{26} - 36 q^{27} - 16 q^{28} - 32 q^{29} - 16 q^{30} - 32 q^{31} - 16 q^{32} - 64 q^{33} - 16 q^{34} - 36 q^{35} - 16 q^{36} - 32 q^{37} - 16 q^{38} - 36 q^{39} - 16 q^{40} - 36 q^{41} - 16 q^{42} - 20 q^{43} - 16 q^{44} - 12 q^{45} - 16 q^{46} - 16 q^{47} - 16 q^{48} + 4 q^{49} + 32 q^{50} - 8 q^{51} + 80 q^{52} + 16 q^{53} + 112 q^{54} + 20 q^{55} + 96 q^{56} + 44 q^{57} + 128 q^{58} + 20 q^{59} + 176 q^{60} + 48 q^{61} + 80 q^{62} + 48 q^{63} + 176 q^{64} + 48 q^{65} + 176 q^{66} + 28 q^{67} + 80 q^{68} + 36 q^{69} + 176 q^{70} + 20 q^{71} + 128 q^{72} + 44 q^{73} + 96 q^{74} + 112 q^{76} - 12 q^{77} + 80 q^{78} - 16 q^{79} + 32 q^{80} - 20 q^{81} - 16 q^{82} - 52 q^{83} - 16 q^{84} - 8 q^{85} - 16 q^{86} - 68 q^{87} - 16 q^{88} - 84 q^{89} - 16 q^{90} - 60 q^{91} - 16 q^{92} - 64 q^{93} - 16 q^{94} - 72 q^{95} - 16 q^{96} - 96 q^{97} - 16 q^{98} - 64 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(128))$$

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
128.2.a $$\chi_{128}(1, \cdot)$$ 128.2.a.a 1 1
128.2.a.b 1
128.2.a.c 1
128.2.a.d 1
128.2.b $$\chi_{128}(65, \cdot)$$ 128.2.b.a 2 1
128.2.b.b 2
128.2.e $$\chi_{128}(33, \cdot)$$ 128.2.e.a 2 2
128.2.e.b 2
128.2.g $$\chi_{128}(17, \cdot)$$ 128.2.g.a 4 4
128.2.g.b 8
128.2.i $$\chi_{128}(9, \cdot)$$ None 0 8
128.2.k $$\chi_{128}(5, \cdot)$$ 128.2.k.a 240 16

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(128))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(128)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(32))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(64))$$$$^{\oplus 2}$$