# Properties

 Label 256.9 Level 256 Weight 9 Dimension 9164 Nonzero newspaces 6 Sturm bound 36864 Trace bound 9

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## Defining parameters

 Level: $$N$$ = $$256 = 2^{8}$$ Weight: $$k$$ = $$9$$ Nonzero newspaces: $$6$$ Sturm bound: $$36864$$ Trace bound: $$9$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{9}(\Gamma_1(256))$$.

Total New Old
Modular forms 16560 9268 7292
Cusp forms 16208 9164 7044
Eisenstein series 352 104 248

## Trace form

 $$9164 q - 32 q^{2} - 24 q^{3} - 32 q^{4} - 32 q^{5} - 32 q^{6} - 24 q^{7} - 32 q^{8} - 40 q^{9} + O(q^{10})$$ $$9164 q - 32 q^{2} - 24 q^{3} - 32 q^{4} - 32 q^{5} - 32 q^{6} - 24 q^{7} - 32 q^{8} - 40 q^{9} - 32 q^{10} - 24 q^{11} - 32 q^{12} - 32 q^{13} - 32 q^{14} - 24 q^{15} - 32 q^{16} - 48 q^{17} - 32 q^{18} - 24 q^{19} - 32 q^{20} - 32 q^{21} - 32 q^{22} - 24 q^{23} - 32 q^{24} - 40 q^{25} - 32 q^{26} - 24 q^{27} - 32 q^{28} - 32 q^{29} - 32 q^{30} - 16 q^{31} - 32 q^{32} - 56 q^{33} - 32 q^{34} - 24 q^{35} - 32 q^{36} - 32 q^{37} - 32 q^{38} - 24 q^{39} - 32 q^{40} - 40 q^{41} - 32 q^{42} - 24 q^{43} - 32 q^{44} + 52456 q^{45} - 32 q^{46} - 24 q^{47} - 32 q^{48} + 23059156 q^{49} - 32 q^{50} - 55448088 q^{51} - 32 q^{52} + 10717408 q^{53} - 32 q^{54} + 92653544 q^{55} - 32 q^{56} + 12400088 q^{57} - 32 q^{58} - 89877528 q^{59} - 32 q^{60} - 97904160 q^{61} - 32 q^{62} - 32 q^{63} - 32 q^{64} + 119766456 q^{65} - 32 q^{66} + 186760936 q^{67} - 32 q^{68} + 34546144 q^{69} - 32 q^{70} - 159664152 q^{71} - 32 q^{72} - 251476520 q^{73} - 32 q^{74} - 51674136 q^{75} - 32 q^{76} + 189928672 q^{77} - 32 q^{78} + 144406504 q^{79} - 32 q^{80} - 172186932 q^{81} - 32 q^{82} - 24 q^{83} - 32 q^{84} - 3125032 q^{85} - 32 q^{86} - 24 q^{87} - 32 q^{88} - 40 q^{89} - 32 q^{90} - 24 q^{91} - 32 q^{92} - 105008 q^{93} - 32 q^{94} - 16 q^{95} - 32 q^{96} - 56 q^{97} - 32 q^{98} - 52512 q^{99} + O(q^{100})$$

## Decomposition of $$S_{9}^{\mathrm{new}}(\Gamma_1(256))$$

We only show spaces with odd 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
256.9.c $$\chi_{256}(255, \cdot)$$ 256.9.c.a 1 1
256.9.c.b 1
256.9.c.c 2
256.9.c.d 2
256.9.c.e 2
256.9.c.f 2
256.9.c.g 2
256.9.c.h 2
256.9.c.i 4
256.9.c.j 4
256.9.c.k 8
256.9.c.l 8
256.9.c.m 12
256.9.c.n 12
256.9.d $$\chi_{256}(127, \cdot)$$ 256.9.d.a 2 1
256.9.d.b 4
256.9.d.c 4
256.9.d.d 4
256.9.d.e 4
256.9.d.f 4
256.9.d.g 4
256.9.d.h 4
256.9.d.i 16
256.9.d.j 16
256.9.f $$\chi_{256}(63, \cdot)$$ n/a 128 2
256.9.h $$\chi_{256}(31, \cdot)$$ n/a 248 4
256.9.j $$\chi_{256}(15, \cdot)$$ n/a 504 8
256.9.l $$\chi_{256}(7, \cdot)$$ None 0 16
256.9.n $$\chi_{256}(3, \cdot)$$ n/a 8160 32

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

## Decomposition of $$S_{9}^{\mathrm{old}}(\Gamma_1(256))$$ into lower level spaces

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