# Properties

 Label 256.8 Level 256 Weight 8 Dimension 8012 Nonzero newspaces 6 Sturm bound 32768 Trace bound 9

## Defining parameters

 Level: $$N$$ = $$256 = 2^{8}$$ Weight: $$k$$ = $$8$$ Nonzero newspaces: $$6$$ Sturm bound: $$32768$$ Trace bound: $$9$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{8}(\Gamma_1(256))$$.

Total New Old
Modular forms 14512 8116 6396
Cusp forms 14160 8012 6148
Eisenstein series 352 104 248

## Trace form

 $$8012 q - 32 q^{2} - 24 q^{3} - 32 q^{4} - 32 q^{5} - 32 q^{6} - 24 q^{7} - 32 q^{8} - 40 q^{9} + O(q^{10})$$ $$8012 q - 32 q^{2} - 24 q^{3} - 32 q^{4} - 32 q^{5} - 32 q^{6} - 24 q^{7} - 32 q^{8} - 40 q^{9} - 32 q^{10} - 24 q^{11} - 32 q^{12} - 32 q^{13} - 32 q^{14} - 24 q^{15} - 32 q^{16} - 48 q^{17} - 32 q^{18} - 24 q^{19} - 32 q^{20} - 32 q^{21} - 32 q^{22} - 24 q^{23} - 32 q^{24} - 40 q^{25} - 32 q^{26} - 24 q^{27} - 32 q^{28} - 32 q^{29} - 32 q^{30} - 32 q^{31} - 32 q^{32} - 56 q^{33} - 32 q^{34} - 24 q^{35} - 32 q^{36} - 32 q^{37} - 32 q^{38} - 24 q^{39} - 32 q^{40} - 40 q^{41} - 32 q^{42} - 24 q^{43} - 32 q^{44} - 17528 q^{45} - 32 q^{46} - 24 q^{47} - 32 q^{48} - 3294220 q^{49} - 32 q^{50} + 5986632 q^{51} - 32 q^{52} + 3631232 q^{53} - 32 q^{54} - 8382040 q^{55} - 32 q^{56} - 12408808 q^{57} - 32 q^{58} - 3671896 q^{59} - 32 q^{60} + 9119520 q^{61} - 32 q^{62} + 20003744 q^{63} - 32 q^{64} + 11415032 q^{65} - 32 q^{66} - 3881384 q^{67} - 32 q^{68} - 19161824 q^{69} - 32 q^{70} - 24697432 q^{71} - 32 q^{72} - 10136552 q^{73} - 32 q^{74} + 22521000 q^{75} - 32 q^{76} + 23828864 q^{77} - 32 q^{78} + 523720 q^{79} - 32 q^{80} - 19131924 q^{81} - 32 q^{82} - 24 q^{83} - 32 q^{84} - 625032 q^{85} - 32 q^{86} - 24 q^{87} - 32 q^{88} - 40 q^{89} - 32 q^{90} - 24 q^{91} - 32 q^{92} + 34960 q^{93} - 32 q^{94} - 32 q^{95} - 32 q^{96} - 56 q^{97} - 32 q^{98} - 17520 q^{99} + O(q^{100})$$

## Decomposition of $$S_{8}^{\mathrm{new}}(\Gamma_1(256))$$

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 available newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
256.8.a $$\chi_{256}(1, \cdot)$$ 256.8.a.a 1 1
256.8.a.b 1
256.8.a.c 1
256.8.a.d 1
256.8.a.e 2
256.8.a.f 2
256.8.a.g 2
256.8.a.h 2
256.8.a.i 2
256.8.a.j 4
256.8.a.k 4
256.8.a.l 4
256.8.a.m 4
256.8.a.n 4
256.8.a.o 4
256.8.a.p 4
256.8.a.q 6
256.8.a.r 6
256.8.b $$\chi_{256}(129, \cdot)$$ 256.8.b.a 2 1
256.8.b.b 2
256.8.b.c 2
256.8.b.d 2
256.8.b.e 2
256.8.b.f 2
256.8.b.g 2
256.8.b.h 4
256.8.b.i 4
256.8.b.j 4
256.8.b.k 6
256.8.b.l 6
256.8.b.m 8
256.8.b.n 8
256.8.e $$\chi_{256}(65, \cdot)$$ n/a 112 2
256.8.g $$\chi_{256}(33, \cdot)$$ n/a 216 4
256.8.i $$\chi_{256}(17, \cdot)$$ n/a 440 8
256.8.k $$\chi_{256}(9, \cdot)$$ None 0 16
256.8.m $$\chi_{256}(5, \cdot)$$ n/a 7136 32

"n/a" means that newforms for that character have not been added to the database yet

## Decomposition of $$S_{8}^{\mathrm{old}}(\Gamma_1(256))$$ into lower level spaces

$$S_{8}^{\mathrm{old}}(\Gamma_1(256)) \cong$$ $$S_{8}^{\mathrm{new}}(\Gamma_1(1))$$$$^{\oplus 9}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(2))$$$$^{\oplus 8}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(4))$$$$^{\oplus 7}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 6}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 5}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(32))$$$$^{\oplus 4}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(64))$$$$^{\oplus 3}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(128))$$$$^{\oplus 2}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(256))$$$$^{\oplus 1}$$