# Properties

 Label 1600.2 Level 1600 Weight 2 Dimension 39365 Nonzero newspaces 28 Sturm bound 307200 Trace bound 12

## Defining parameters

 Level: $$N$$ = $$1600 = 2^{6} \cdot 5^{2}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$28$$ Sturm bound: $$307200$$ Trace bound: $$12$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(1600))$$.

Total New Old
Modular forms 78816 40267 38549
Cusp forms 74785 39365 35420
Eisenstein series 4031 902 3129

## Trace form

 $$39365q - 104q^{2} - 78q^{3} - 104q^{4} - 128q^{5} - 168q^{6} - 80q^{7} - 104q^{8} - 133q^{9} + O(q^{10})$$ $$39365q - 104q^{2} - 78q^{3} - 104q^{4} - 128q^{5} - 168q^{6} - 80q^{7} - 104q^{8} - 133q^{9} - 128q^{10} - 130q^{11} - 104q^{12} - 112q^{13} - 104q^{14} - 96q^{15} - 168q^{16} - 190q^{17} - 104q^{18} - 86q^{19} - 128q^{20} - 164q^{21} - 96q^{22} - 80q^{23} - 64q^{24} - 160q^{25} - 288q^{26} - 72q^{27} - 64q^{28} - 88q^{29} - 128q^{30} - 104q^{31} - 64q^{32} - 52q^{33} - 64q^{34} - 96q^{35} - 88q^{36} - 96q^{37} - 64q^{38} - 64q^{39} - 128q^{40} - 210q^{41} - 64q^{42} - 90q^{43} - 96q^{44} - 128q^{45} - 168q^{46} - 96q^{47} - 104q^{48} - 191q^{49} - 128q^{50} - 220q^{51} - 152q^{52} - 128q^{53} - 168q^{54} - 96q^{55} - 224q^{56} - 88q^{57} - 176q^{58} - 10q^{59} - 128q^{60} - 176q^{61} - 136q^{62} + 100q^{63} - 200q^{64} - 320q^{65} - 248q^{66} + 106q^{67} - 152q^{68} + 76q^{69} - 128q^{70} + 64q^{71} - 176q^{72} - 2q^{73} - 160q^{74} + 16q^{75} - 392q^{76} + 92q^{77} - 128q^{78} + 96q^{79} - 128q^{80} - 35q^{81} - 24q^{82} + 82q^{83} + 8q^{84} - 32q^{85} - 64q^{86} + 144q^{87} - 24q^{88} + 46q^{89} - 128q^{90} - 4q^{91} + 48q^{92} + 128q^{93} - 8q^{94} - 48q^{95} - 32q^{96} + 26q^{97} + 32q^{98} + 90q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(1600))$$

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
1600.2.a $$\chi_{1600}(1, \cdot)$$ 1600.2.a.a 1 1
1600.2.a.b 1
1600.2.a.c 1
1600.2.a.d 1
1600.2.a.e 1
1600.2.a.f 1
1600.2.a.g 1
1600.2.a.h 1
1600.2.a.i 1
1600.2.a.j 1
1600.2.a.k 1
1600.2.a.l 1
1600.2.a.m 1
1600.2.a.n 1
1600.2.a.o 1
1600.2.a.p 1
1600.2.a.q 1
1600.2.a.r 1
1600.2.a.s 1
1600.2.a.t 1
1600.2.a.u 1
1600.2.a.v 1
1600.2.a.w 1
1600.2.a.x 1
1600.2.a.y 1
1600.2.a.z 2
1600.2.a.ba 2
1600.2.a.bb 2
1600.2.a.bc 2
1600.2.a.bd 2
1600.2.c $$\chi_{1600}(449, \cdot)$$ 1600.2.c.a 2 1
1600.2.c.b 2
1600.2.c.c 2
1600.2.c.d 2
1600.2.c.e 2
1600.2.c.f 2
1600.2.c.g 2
1600.2.c.h 2
1600.2.c.i 2
1600.2.c.j 2
1600.2.c.k 2
1600.2.c.l 2
1600.2.c.m 2
1600.2.c.n 4
1600.2.c.o 4
1600.2.d $$\chi_{1600}(801, \cdot)$$ 1600.2.d.a 2 1
1600.2.d.b 4
1600.2.d.c 4
1600.2.d.d 4
1600.2.d.e 4
1600.2.d.f 4
1600.2.d.g 4
1600.2.d.h 4
1600.2.d.i 8
1600.2.f $$\chi_{1600}(1249, \cdot)$$ 1600.2.f.a 2 1
1600.2.f.b 2
1600.2.f.c 4
1600.2.f.d 4
1600.2.f.e 4
1600.2.f.f 4
1600.2.f.g 4
1600.2.f.h 4
1600.2.f.i 4
1600.2.f.j 4
1600.2.j $$\chi_{1600}(143, \cdot)$$ 1600.2.j.a 2 2
1600.2.j.b 8
1600.2.j.c 16
1600.2.j.d 18
1600.2.j.e 24
1600.2.l $$\chi_{1600}(401, \cdot)$$ 1600.2.l.a 2 2
1600.2.l.b 2
1600.2.l.c 2
1600.2.l.d 4
1600.2.l.e 4
1600.2.l.f 12
1600.2.l.g 12
1600.2.l.h 16
1600.2.l.i 16
1600.2.n $$\chi_{1600}(1343, \cdot)$$ 1600.2.n.a 2 2
1600.2.n.b 2
1600.2.n.c 2
1600.2.n.d 2
1600.2.n.e 2
1600.2.n.f 2
1600.2.n.g 2
1600.2.n.h 2
1600.2.n.i 2
1600.2.n.j 2
1600.2.n.k 2
1600.2.n.l 2
1600.2.n.m 2
1600.2.n.n 2
1600.2.n.o 4
1600.2.n.p 4
1600.2.n.q 4
1600.2.n.r 4
1600.2.n.s 4
1600.2.n.t 4
1600.2.n.u 8
1600.2.n.v 8
1600.2.o $$\chi_{1600}(543, \cdot)$$ 1600.2.o.a 2 2
1600.2.o.b 2
1600.2.o.c 2
1600.2.o.d 2
1600.2.o.e 8
1600.2.o.f 8
1600.2.o.g 8
1600.2.o.h 8
1600.2.o.i 8
1600.2.o.j 8
1600.2.o.k 8
1600.2.o.l 8
1600.2.q $$\chi_{1600}(49, \cdot)$$ 1600.2.q.a 2 2
1600.2.q.b 2
1600.2.q.c 4
1600.2.q.d 4
1600.2.q.e 12
1600.2.q.f 12
1600.2.q.g 16
1600.2.q.h 16
1600.2.s $$\chi_{1600}(207, \cdot)$$ 1600.2.s.a 2 2
1600.2.s.b 8
1600.2.s.c 16
1600.2.s.d 18
1600.2.s.e 24
1600.2.u $$\chi_{1600}(321, \cdot)$$ n/a 232 4
1600.2.v $$\chi_{1600}(407, \cdot)$$ None 0 4
1600.2.y $$\chi_{1600}(201, \cdot)$$ None 0 4
1600.2.ba $$\chi_{1600}(249, \cdot)$$ None 0 4
1600.2.bb $$\chi_{1600}(7, \cdot)$$ None 0 4
1600.2.be $$\chi_{1600}(289, \cdot)$$ n/a 240 4
1600.2.bg $$\chi_{1600}(129, \cdot)$$ n/a 232 4
1600.2.bj $$\chi_{1600}(161, \cdot)$$ n/a 240 4
1600.2.bl $$\chi_{1600}(43, \cdot)$$ n/a 1136 8
1600.2.bm $$\chi_{1600}(101, \cdot)$$ n/a 1192 8
1600.2.bn $$\chi_{1600}(149, \cdot)$$ n/a 1136 8
1600.2.br $$\chi_{1600}(107, \cdot)$$ n/a 1136 8
1600.2.bt $$\chi_{1600}(303, \cdot)$$ n/a 464 8
1600.2.bu $$\chi_{1600}(81, \cdot)$$ n/a 464 8
1600.2.bx $$\chi_{1600}(223, \cdot)$$ n/a 480 8
1600.2.by $$\chi_{1600}(63, \cdot)$$ n/a 464 8
1600.2.cb $$\chi_{1600}(209, \cdot)$$ n/a 464 8
1600.2.cc $$\chi_{1600}(47, \cdot)$$ n/a 464 8
1600.2.cf $$\chi_{1600}(87, \cdot)$$ None 0 16
1600.2.cg $$\chi_{1600}(9, \cdot)$$ None 0 16
1600.2.ci $$\chi_{1600}(41, \cdot)$$ None 0 16
1600.2.cl $$\chi_{1600}(23, \cdot)$$ None 0 16
1600.2.cm $$\chi_{1600}(3, \cdot)$$ n/a 7616 32
1600.2.cq $$\chi_{1600}(29, \cdot)$$ n/a 7616 32
1600.2.cr $$\chi_{1600}(21, \cdot)$$ n/a 7616 32
1600.2.cs $$\chi_{1600}(67, \cdot)$$ n/a 7616 32

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

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(1600))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(1600)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 9}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 10}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 7}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(32))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(40))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(50))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(64))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(80))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(100))$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(160))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(200))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(320))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(400))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(800))$$$$^{\oplus 2}$$