## Defining parameters

 Level: $$N$$ = $$256 = 2^{8}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$6$$ Newform subspaces: $$16$$ Sturm bound: $$8192$$ Trace bound: $$9$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(256))$$.

Total New Old
Modular forms 2224 1204 1020
Cusp forms 1873 1100 773
Eisenstein series 351 104 247

## Trace form

 $$1100 q - 32 q^{2} - 24 q^{3} - 32 q^{4} - 32 q^{5} - 32 q^{6} - 24 q^{7} - 32 q^{8} - 40 q^{9} + O(q^{10})$$ $$1100 q - 32 q^{2} - 24 q^{3} - 32 q^{4} - 32 q^{5} - 32 q^{6} - 24 q^{7} - 32 q^{8} - 40 q^{9} - 32 q^{10} - 24 q^{11} - 32 q^{12} - 32 q^{13} - 32 q^{14} - 24 q^{15} - 32 q^{16} - 48 q^{17} - 32 q^{18} - 24 q^{19} - 32 q^{20} - 32 q^{21} - 32 q^{22} - 24 q^{23} - 32 q^{24} - 40 q^{25} - 32 q^{26} - 24 q^{27} - 32 q^{28} - 32 q^{29} - 32 q^{30} - 32 q^{31} - 32 q^{32} - 56 q^{33} - 32 q^{34} - 24 q^{35} - 32 q^{36} - 32 q^{37} - 32 q^{38} - 24 q^{39} - 32 q^{40} - 40 q^{41} - 32 q^{42} - 24 q^{43} - 32 q^{44} - 56 q^{45} - 32 q^{46} - 24 q^{47} - 32 q^{48} - 76 q^{49} - 32 q^{50} - 56 q^{51} - 32 q^{52} - 64 q^{53} - 32 q^{54} - 88 q^{55} - 32 q^{56} - 104 q^{57} - 32 q^{58} - 88 q^{59} - 32 q^{60} - 96 q^{61} - 32 q^{62} - 96 q^{63} - 32 q^{64} - 136 q^{65} - 32 q^{66} - 104 q^{67} - 32 q^{68} - 96 q^{69} - 32 q^{70} - 88 q^{71} - 32 q^{72} - 104 q^{73} - 32 q^{74} - 88 q^{75} - 32 q^{76} - 64 q^{77} - 32 q^{78} - 56 q^{79} - 32 q^{80} - 84 q^{81} - 32 q^{82} - 24 q^{83} - 32 q^{84} - 72 q^{85} - 32 q^{86} - 24 q^{87} - 32 q^{88} - 40 q^{89} - 32 q^{90} - 24 q^{91} - 32 q^{92} + 16 q^{93} - 32 q^{94} - 32 q^{95} - 32 q^{96} - 56 q^{97} - 32 q^{98} - 48 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(256))$$

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
256.2.a $$\chi_{256}(1, \cdot)$$ 256.2.a.a 1 1
256.2.a.b 1
256.2.a.c 1
256.2.a.d 1
256.2.a.e 2
256.2.b $$\chi_{256}(129, \cdot)$$ 256.2.b.a 2 1
256.2.b.b 2
256.2.b.c 2
256.2.e $$\chi_{256}(65, \cdot)$$ 256.2.e.a 8 2
256.2.e.b 8
256.2.g $$\chi_{256}(33, \cdot)$$ 256.2.g.a 4 4
256.2.g.b 4
256.2.g.c 8
256.2.g.d 8
256.2.i $$\chi_{256}(17, \cdot)$$ 256.2.i.a 56 8
256.2.k $$\chi_{256}(9, \cdot)$$ None 0 16
256.2.m $$\chi_{256}(5, \cdot)$$ 256.2.m.a 992 32

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(256))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(256)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(32))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(64))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(128))$$$$^{\oplus 2}$$