# Properties

 Label 2400.2 Level 2400 Weight 2 Dimension 59314 Nonzero newspaces 40 Sturm bound 614400 Trace bound 25

## Defining parameters

 Level: $$N$$ = $$2400 = 2^{5} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$40$$ Sturm bound: $$614400$$ Trace bound: $$25$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(2400))$$.

Total New Old
Modular forms 157184 60134 97050
Cusp forms 150017 59314 90703
Eisenstein series 7167 820 6347

## Trace form

 $$59314 q - 38 q^{3} - 104 q^{4} - 84 q^{6} - 76 q^{7} - 74 q^{9} + O(q^{10})$$ $$59314 q - 38 q^{3} - 104 q^{4} - 84 q^{6} - 76 q^{7} - 74 q^{9} - 128 q^{10} - 68 q^{12} - 116 q^{13} - 32 q^{14} - 48 q^{15} - 208 q^{16} - 16 q^{17} - 60 q^{18} - 100 q^{19} - 100 q^{21} - 128 q^{22} - 24 q^{23} - 40 q^{24} - 192 q^{25} + 40 q^{26} - 74 q^{27} - 64 q^{28} - 28 q^{29} - 64 q^{30} - 236 q^{31} + 40 q^{32} - 192 q^{33} - 80 q^{34} - 48 q^{35} - 32 q^{36} - 244 q^{37} + 40 q^{38} - 112 q^{39} - 128 q^{40} - 120 q^{41} + 8 q^{42} - 204 q^{43} + 8 q^{44} - 80 q^{45} - 168 q^{46} - 72 q^{47} - 138 q^{49} - 136 q^{51} - 56 q^{52} - 44 q^{53} - 136 q^{55} + 56 q^{56} - 8 q^{57} - 32 q^{58} + 64 q^{59} - 228 q^{61} + 48 q^{62} + 32 q^{63} + 160 q^{64} + 64 q^{65} + 196 q^{66} + 68 q^{67} + 440 q^{68} + 76 q^{69} + 160 q^{70} + 168 q^{71} + 368 q^{72} + 60 q^{73} + 672 q^{74} + 8 q^{75} + 184 q^{76} + 320 q^{77} + 492 q^{78} + 164 q^{79} + 320 q^{80} + 178 q^{81} + 456 q^{82} + 160 q^{83} + 424 q^{84} + 64 q^{85} + 448 q^{86} + 164 q^{87} + 512 q^{88} + 256 q^{89} + 176 q^{90} + 496 q^{92} + 152 q^{93} + 256 q^{94} + 48 q^{95} + 32 q^{96} - 140 q^{97} + 112 q^{98} + 196 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(2400))$$

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
2400.2.a $$\chi_{2400}(1, \cdot)$$ 2400.2.a.a 1 1
2400.2.a.b 1
2400.2.a.c 1
2400.2.a.d 1
2400.2.a.e 1
2400.2.a.f 1
2400.2.a.g 1
2400.2.a.h 1
2400.2.a.i 1
2400.2.a.j 1
2400.2.a.k 1
2400.2.a.l 1
2400.2.a.m 1
2400.2.a.n 1
2400.2.a.o 1
2400.2.a.p 1
2400.2.a.q 1
2400.2.a.r 1
2400.2.a.s 1
2400.2.a.t 1
2400.2.a.u 1
2400.2.a.v 1
2400.2.a.w 1
2400.2.a.x 1
2400.2.a.y 1
2400.2.a.z 1
2400.2.a.ba 1
2400.2.a.bb 1
2400.2.a.bc 1
2400.2.a.bd 1
2400.2.a.be 1
2400.2.a.bf 1
2400.2.a.bg 1
2400.2.a.bh 1
2400.2.a.bi 2
2400.2.a.bj 2
2400.2.b $$\chi_{2400}(2351, \cdot)$$ 2400.2.b.a 2 1
2400.2.b.b 4
2400.2.b.c 4
2400.2.b.d 4
2400.2.b.e 8
2400.2.b.f 8
2400.2.b.g 12
2400.2.b.h 12
2400.2.b.i 16
2400.2.d $$\chi_{2400}(49, \cdot)$$ 2400.2.d.a 2 1
2400.2.d.b 2
2400.2.d.c 2
2400.2.d.d 2
2400.2.d.e 6
2400.2.d.f 6
2400.2.d.g 8
2400.2.d.h 8
2400.2.f $$\chi_{2400}(1249, \cdot)$$ 2400.2.f.a 2 1
2400.2.f.b 2
2400.2.f.c 2
2400.2.f.d 2
2400.2.f.e 2
2400.2.f.f 2
2400.2.f.g 2
2400.2.f.h 2
2400.2.f.i 2
2400.2.f.j 2
2400.2.f.k 2
2400.2.f.l 2
2400.2.f.m 2
2400.2.f.n 2
2400.2.f.o 2
2400.2.f.p 2
2400.2.f.q 2
2400.2.f.r 2
2400.2.h $$\chi_{2400}(1151, \cdot)$$ 2400.2.h.a 4 1
2400.2.h.b 4
2400.2.h.c 4
2400.2.h.d 4
2400.2.h.e 4
2400.2.h.f 16
2400.2.h.g 16
2400.2.h.h 24
2400.2.k $$\chi_{2400}(1201, \cdot)$$ 2400.2.k.a 2 1
2400.2.k.b 2
2400.2.k.c 6
2400.2.k.d 8
2400.2.k.e 8
2400.2.k.f 12
2400.2.m $$\chi_{2400}(1199, \cdot)$$ 2400.2.m.a 4 1
2400.2.m.b 8
2400.2.m.c 16
2400.2.m.d 16
2400.2.m.e 24
2400.2.o $$\chi_{2400}(2399, \cdot)$$ 2400.2.o.a 4 1
2400.2.o.b 4
2400.2.o.c 4
2400.2.o.d 4
2400.2.o.e 4
2400.2.o.f 4
2400.2.o.g 4
2400.2.o.h 4
2400.2.o.i 4
2400.2.o.j 4
2400.2.o.k 16
2400.2.o.l 16
2400.2.s $$\chi_{2400}(601, \cdot)$$ None 0 2
2400.2.t $$\chi_{2400}(599, \cdot)$$ None 0 2
2400.2.v $$\chi_{2400}(257, \cdot)$$ n/a 144 2
2400.2.w $$\chi_{2400}(607, \cdot)$$ 2400.2.w.a 4 2
2400.2.w.b 4
2400.2.w.c 4
2400.2.w.d 4
2400.2.w.e 4
2400.2.w.f 4
2400.2.w.g 8
2400.2.w.h 8
2400.2.w.i 8
2400.2.w.j 8
2400.2.w.k 8
2400.2.w.l 8
2400.2.y $$\chi_{2400}(7, \cdot)$$ None 0 2
2400.2.bb $$\chi_{2400}(857, \cdot)$$ None 0 2
2400.2.bc $$\chi_{2400}(1207, \cdot)$$ None 0 2
2400.2.bf $$\chi_{2400}(2057, \cdot)$$ None 0 2
2400.2.bh $$\chi_{2400}(943, \cdot)$$ 2400.2.bh.a 16 2
2400.2.bh.b 24
2400.2.bh.c 32
2400.2.bi $$\chi_{2400}(593, \cdot)$$ n/a 136 2
2400.2.bk $$\chi_{2400}(551, \cdot)$$ None 0 2
2400.2.bl $$\chi_{2400}(649, \cdot)$$ None 0 2
2400.2.bo $$\chi_{2400}(481, \cdot)$$ n/a 240 4
2400.2.bp $$\chi_{2400}(43, \cdot)$$ n/a 576 4
2400.2.bs $$\chi_{2400}(893, \cdot)$$ n/a 1136 4
2400.2.bt $$\chi_{2400}(299, \cdot)$$ n/a 1136 4
2400.2.bw $$\chi_{2400}(301, \cdot)$$ n/a 608 4
2400.2.by $$\chi_{2400}(251, \cdot)$$ n/a 1192 4
2400.2.bz $$\chi_{2400}(349, \cdot)$$ n/a 576 4
2400.2.cc $$\chi_{2400}(293, \cdot)$$ n/a 1136 4
2400.2.cd $$\chi_{2400}(643, \cdot)$$ n/a 576 4
2400.2.cg $$\chi_{2400}(191, \cdot)$$ n/a 480 4
2400.2.ci $$\chi_{2400}(289, \cdot)$$ n/a 240 4
2400.2.ck $$\chi_{2400}(529, \cdot)$$ n/a 240 4
2400.2.cm $$\chi_{2400}(431, \cdot)$$ n/a 464 4
2400.2.co $$\chi_{2400}(479, \cdot)$$ n/a 480 4
2400.2.cq $$\chi_{2400}(239, \cdot)$$ n/a 464 4
2400.2.cs $$\chi_{2400}(241, \cdot)$$ n/a 240 4
2400.2.cu $$\chi_{2400}(119, \cdot)$$ None 0 8
2400.2.cv $$\chi_{2400}(121, \cdot)$$ None 0 8
2400.2.cz $$\chi_{2400}(17, \cdot)$$ n/a 928 8
2400.2.da $$\chi_{2400}(367, \cdot)$$ n/a 480 8
2400.2.dc $$\chi_{2400}(137, \cdot)$$ None 0 8
2400.2.df $$\chi_{2400}(103, \cdot)$$ None 0 8
2400.2.dg $$\chi_{2400}(233, \cdot)$$ None 0 8
2400.2.dj $$\chi_{2400}(487, \cdot)$$ None 0 8
2400.2.dl $$\chi_{2400}(127, \cdot)$$ n/a 480 8
2400.2.dm $$\chi_{2400}(353, \cdot)$$ n/a 960 8
2400.2.dq $$\chi_{2400}(169, \cdot)$$ None 0 8
2400.2.dr $$\chi_{2400}(71, \cdot)$$ None 0 8
2400.2.ds $$\chi_{2400}(173, \cdot)$$ n/a 7616 16
2400.2.dv $$\chi_{2400}(67, \cdot)$$ n/a 3840 16
2400.2.dx $$\chi_{2400}(109, \cdot)$$ n/a 3840 16
2400.2.dy $$\chi_{2400}(11, \cdot)$$ n/a 7616 16
2400.2.ea $$\chi_{2400}(61, \cdot)$$ n/a 3840 16
2400.2.ed $$\chi_{2400}(59, \cdot)$$ n/a 7616 16
2400.2.ef $$\chi_{2400}(163, \cdot)$$ n/a 3840 16
2400.2.eg $$\chi_{2400}(53, \cdot)$$ n/a 7616 16

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

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(2400))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(2400)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 16}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 9}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 10}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(32))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(40))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(48))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(50))$$$$^{\oplus 10}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(60))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(75))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(80))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(96))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(100))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(120))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(150))$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(160))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(200))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(240))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(300))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(400))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(480))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(600))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(800))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(1200))$$$$^{\oplus 2}$$