# Properties

 Label 1024.2 Level 1024 Weight 2 Dimension 18192 Nonzero newspaces 8 Sturm bound 131072 Trace bound 9

## Defining parameters

 Level: $$N$$ = $$1024 = 2^{10}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$8$$ Sturm bound: $$131072$$ Trace bound: $$9$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(1024))$$.

Total New Old
Modular forms 33600 18672 14928
Cusp forms 31937 18192 13745
Eisenstein series 1663 480 1183

## Trace form

 $$18192 q - 128 q^{2} - 96 q^{3} - 128 q^{4} - 128 q^{5} - 128 q^{6} - 96 q^{7} - 128 q^{8} - 160 q^{9} + O(q^{10})$$ $$18192 q - 128 q^{2} - 96 q^{3} - 128 q^{4} - 128 q^{5} - 128 q^{6} - 96 q^{7} - 128 q^{8} - 160 q^{9} - 128 q^{10} - 96 q^{11} - 128 q^{12} - 128 q^{13} - 128 q^{14} - 96 q^{15} - 128 q^{16} - 192 q^{17} - 128 q^{18} - 96 q^{19} - 128 q^{20} - 128 q^{21} - 128 q^{22} - 96 q^{23} - 128 q^{24} - 160 q^{25} - 128 q^{26} - 96 q^{27} - 128 q^{28} - 128 q^{29} - 128 q^{30} - 96 q^{31} - 128 q^{32} - 224 q^{33} - 128 q^{34} - 96 q^{35} - 128 q^{36} - 128 q^{37} - 128 q^{38} - 96 q^{39} - 128 q^{40} - 160 q^{41} - 128 q^{42} - 96 q^{43} - 128 q^{44} - 128 q^{45} - 128 q^{46} - 96 q^{47} - 128 q^{48} - 192 q^{49} - 128 q^{50} - 96 q^{51} - 128 q^{52} - 128 q^{53} - 128 q^{54} - 96 q^{55} - 128 q^{56} - 160 q^{57} - 128 q^{58} - 96 q^{59} - 128 q^{60} - 128 q^{61} - 128 q^{62} - 96 q^{63} - 128 q^{64} - 256 q^{65} - 128 q^{66} - 96 q^{67} - 128 q^{68} - 128 q^{69} - 128 q^{70} - 96 q^{71} - 128 q^{72} - 160 q^{73} - 128 q^{74} - 96 q^{75} - 128 q^{76} - 128 q^{77} - 128 q^{78} - 96 q^{79} - 128 q^{80} - 192 q^{81} - 128 q^{82} - 96 q^{83} - 128 q^{84} - 128 q^{85} - 128 q^{86} - 96 q^{87} - 128 q^{88} - 160 q^{89} - 128 q^{90} - 96 q^{91} - 128 q^{92} - 128 q^{93} - 128 q^{94} - 96 q^{95} - 128 q^{96} - 224 q^{97} - 128 q^{98} - 96 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(1024))$$

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
1024.2.a $$\chi_{1024}(1, \cdot)$$ 1024.2.a.a 2 1
1024.2.a.b 2
1024.2.a.c 2
1024.2.a.d 2
1024.2.a.e 2
1024.2.a.f 2
1024.2.a.g 4
1024.2.a.h 4
1024.2.a.i 4
1024.2.a.j 4
1024.2.b $$\chi_{1024}(513, \cdot)$$ 1024.2.b.a 2 1
1024.2.b.b 2
1024.2.b.c 2
1024.2.b.d 2
1024.2.b.e 2
1024.2.b.f 2
1024.2.b.g 8
1024.2.b.h 8
1024.2.e $$\chi_{1024}(257, \cdot)$$ 1024.2.e.a 2 2
1024.2.e.b 2
1024.2.e.c 2
1024.2.e.d 2
1024.2.e.e 2
1024.2.e.f 2
1024.2.e.g 4
1024.2.e.h 4
1024.2.e.i 4
1024.2.e.j 4
1024.2.e.k 4
1024.2.e.l 4
1024.2.e.m 4
1024.2.e.n 4
1024.2.e.o 4
1024.2.e.p 8
1024.2.g $$\chi_{1024}(129, \cdot)$$ 1024.2.g.a 16 4
1024.2.g.b 16
1024.2.g.c 16
1024.2.g.d 16
1024.2.g.e 16
1024.2.g.f 16
1024.2.g.g 16
1024.2.g.h 16
1024.2.i $$\chi_{1024}(65, \cdot)$$ n/a 224 8
1024.2.k $$\chi_{1024}(33, \cdot)$$ n/a 480 16
1024.2.m $$\chi_{1024}(17, \cdot)$$ n/a 992 32
1024.2.o $$\chi_{1024}(9, \cdot)$$ None 0 64
1024.2.q $$\chi_{1024}(5, \cdot)$$ n/a 16256 128

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

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(1024))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(1024)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 7}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(32))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(64))$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(128))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(256))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(512))$$$$^{\oplus 2}$$