## Defining parameters

 Level: $$N$$ = $$640 = 2^{7} \cdot 5$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$18$$ Newform subspaces: $$66$$ Sturm bound: $$49152$$ Trace bound: $$33$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(640))$$.

Total New Old
Modular forms 12928 6480 6448
Cusp forms 11649 6192 5457
Eisenstein series 1279 288 991

## Trace form

 $$6192 q - 32 q^{2} - 24 q^{3} - 32 q^{4} - 48 q^{5} - 96 q^{6} - 24 q^{7} - 32 q^{8} - 40 q^{9} + O(q^{10})$$ $$6192 q - 32 q^{2} - 24 q^{3} - 32 q^{4} - 48 q^{5} - 96 q^{6} - 24 q^{7} - 32 q^{8} - 40 q^{9} - 48 q^{10} - 72 q^{11} - 32 q^{12} - 32 q^{13} - 32 q^{14} - 32 q^{15} - 96 q^{16} - 48 q^{17} - 32 q^{18} - 24 q^{19} - 48 q^{20} - 72 q^{21} - 32 q^{22} - 8 q^{23} - 32 q^{24} - 44 q^{25} - 96 q^{26} + 24 q^{27} - 32 q^{28} - 48 q^{30} - 32 q^{31} - 32 q^{32} - 32 q^{34} - 12 q^{35} - 96 q^{36} - 32 q^{38} + 24 q^{39} - 48 q^{40} - 88 q^{41} - 32 q^{42} - 8 q^{43} - 32 q^{44} - 52 q^{45} - 96 q^{46} - 16 q^{47} - 32 q^{48} - 104 q^{49} - 96 q^{50} - 80 q^{51} - 224 q^{52} - 96 q^{53} - 288 q^{54} - 68 q^{55} - 320 q^{56} - 168 q^{57} - 320 q^{58} - 88 q^{59} - 240 q^{60} - 224 q^{61} - 224 q^{62} - 144 q^{63} - 416 q^{64} - 112 q^{65} - 480 q^{66} - 104 q^{67} - 224 q^{68} - 136 q^{69} - 240 q^{70} - 136 q^{71} - 320 q^{72} - 168 q^{73} - 256 q^{74} - 68 q^{75} - 352 q^{76} - 40 q^{77} - 224 q^{78} - 32 q^{79} - 96 q^{80} - 152 q^{81} - 32 q^{82} + 56 q^{83} - 32 q^{84} - 56 q^{85} - 96 q^{86} + 88 q^{87} - 32 q^{88} + 88 q^{89} - 48 q^{90} + 24 q^{91} - 32 q^{92} + 64 q^{93} - 32 q^{94} + 24 q^{95} - 96 q^{96} + 64 q^{97} - 32 q^{98} + 80 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(640))$$

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
640.2.a $$\chi_{640}(1, \cdot)$$ 640.2.a.a 1 1
640.2.a.b 1
640.2.a.c 1
640.2.a.d 1
640.2.a.e 1
640.2.a.f 1
640.2.a.g 1
640.2.a.h 1
640.2.a.i 2
640.2.a.j 2
640.2.a.k 2
640.2.a.l 2
640.2.c $$\chi_{640}(129, \cdot)$$ 640.2.c.a 6 1
640.2.c.b 6
640.2.c.c 6
640.2.c.d 6
640.2.d $$\chi_{640}(321, \cdot)$$ 640.2.d.a 4 1
640.2.d.b 4
640.2.d.c 4
640.2.d.d 4
640.2.f $$\chi_{640}(449, \cdot)$$ 640.2.f.a 2 1
640.2.f.b 2
640.2.f.c 4
640.2.f.d 4
640.2.f.e 4
640.2.f.f 4
640.2.f.g 4
640.2.j $$\chi_{640}(543, \cdot)$$ 640.2.j.a 2 2
640.2.j.b 2
640.2.j.c 18
640.2.j.d 18
640.2.l $$\chi_{640}(161, \cdot)$$ 640.2.l.a 16 2
640.2.l.b 16
640.2.n $$\chi_{640}(127, \cdot)$$ 640.2.n.a 12 2
640.2.n.b 12
640.2.n.c 12
640.2.n.d 12
640.2.o $$\chi_{640}(63, \cdot)$$ 640.2.o.a 2 2
640.2.o.b 2
640.2.o.c 2
640.2.o.d 2
640.2.o.e 2
640.2.o.f 2
640.2.o.g 2
640.2.o.h 2
640.2.o.i 8
640.2.o.j 12
640.2.o.k 12
640.2.q $$\chi_{640}(289, \cdot)$$ 640.2.q.a 2 2
640.2.q.b 2
640.2.q.c 2
640.2.q.d 2
640.2.q.e 16
640.2.q.f 16
640.2.s $$\chi_{640}(223, \cdot)$$ 640.2.s.a 2 2
640.2.s.b 2
640.2.s.c 18
640.2.s.d 18
640.2.u $$\chi_{640}(47, \cdot)$$ 640.2.u.a 88 4
640.2.x $$\chi_{640}(81, \cdot)$$ 640.2.x.a 64 4
640.2.z $$\chi_{640}(49, \cdot)$$ 640.2.z.a 88 4
640.2.ba $$\chi_{640}(207, \cdot)$$ 640.2.ba.a 88 4
640.2.bd $$\chi_{640}(87, \cdot)$$ None 0 8
640.2.be $$\chi_{640}(41, \cdot)$$ None 0 8
640.2.bf $$\chi_{640}(9, \cdot)$$ None 0 8
640.2.bj $$\chi_{640}(7, \cdot)$$ None 0 8
640.2.bl $$\chi_{640}(3, \cdot)$$ 640.2.bl.a 1504 16
640.2.bm $$\chi_{640}(21, \cdot)$$ 640.2.bm.a 1024 16
640.2.bo $$\chi_{640}(29, \cdot)$$ 640.2.bo.a 1504 16
640.2.br $$\chi_{640}(43, \cdot)$$ 640.2.br.a 1504 16

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(640))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(640)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(32))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(40))$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(64))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(80))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(128))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(160))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(320))$$$$^{\oplus 2}$$