# Properties

 Label 1200.4 Level 1200 Weight 4 Dimension 44609 Nonzero newspaces 28 Sturm bound 307200 Trace bound 25

## Defining parameters

 Level: $$N$$ = $$1200 = 2^{4} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$28$$ Sturm bound: $$307200$$ Trace bound: $$25$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(1200))$$.

Total New Old
Modular forms 116768 44977 71791
Cusp forms 113632 44609 69023
Eisenstein series 3136 368 2768

## Trace form

 $$44609 q - 23 q^{3} - 72 q^{4} - 12 q^{6} - 12 q^{7} + 84 q^{8} + 47 q^{9} + O(q^{10})$$ $$44609 q - 23 q^{3} - 72 q^{4} - 12 q^{6} - 12 q^{7} + 84 q^{8} + 47 q^{9} - 64 q^{10} - 60 q^{11} - 128 q^{12} - 442 q^{13} - 348 q^{14} - 36 q^{15} - 384 q^{16} + 390 q^{17} - 8 q^{18} + 916 q^{19} + 234 q^{21} + 616 q^{22} + 88 q^{23} + 84 q^{24} - 192 q^{25} - 20 q^{26} - 1067 q^{27} + 1360 q^{28} + 1194 q^{29} + 1600 q^{30} + 276 q^{31} + 1760 q^{32} + 590 q^{33} - 2536 q^{34} - 456 q^{35} - 1400 q^{36} - 4226 q^{37} - 7032 q^{38} - 284 q^{39} - 6304 q^{40} - 4322 q^{41} - 3948 q^{42} - 1364 q^{43} - 4456 q^{44} + 66 q^{45} - 80 q^{46} + 240 q^{47} + 4872 q^{48} + 5009 q^{49} + 5680 q^{50} + 1470 q^{51} + 16256 q^{52} + 5810 q^{53} + 1676 q^{54} + 452 q^{55} + 1344 q^{56} - 1226 q^{57} + 1464 q^{58} - 628 q^{59} - 544 q^{60} - 786 q^{61} - 996 q^{62} + 3242 q^{63} - 17616 q^{64} + 1048 q^{65} - 15260 q^{66} + 13772 q^{67} - 13440 q^{68} - 1254 q^{69} - 1648 q^{70} - 3528 q^{71} + 1180 q^{72} - 4242 q^{73} + 14508 q^{74} - 6112 q^{75} + 14544 q^{76} - 4336 q^{77} + 20016 q^{78} - 25628 q^{79} + 12640 q^{80} - 2189 q^{81} + 24296 q^{82} - 8476 q^{83} + 17616 q^{84} + 15328 q^{85} + 7760 q^{86} + 28 q^{87} + 7344 q^{88} + 814 q^{89} - 3752 q^{90} + 11444 q^{91} - 16880 q^{92} - 3482 q^{93} - 18504 q^{94} + 4152 q^{95} - 13856 q^{96} - 14994 q^{97} - 18488 q^{98} + 12296 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(1200))$$

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 available newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
1200.4.a $$\chi_{1200}(1, \cdot)$$ 1200.4.a.a 1 1
1200.4.a.b 1
1200.4.a.c 1
1200.4.a.d 1
1200.4.a.e 1
1200.4.a.f 1
1200.4.a.g 1
1200.4.a.h 1
1200.4.a.i 1
1200.4.a.j 1
1200.4.a.k 1
1200.4.a.l 1
1200.4.a.m 1
1200.4.a.n 1
1200.4.a.o 1
1200.4.a.p 1
1200.4.a.q 1
1200.4.a.r 1
1200.4.a.s 1
1200.4.a.t 1
1200.4.a.u 1
1200.4.a.v 1
1200.4.a.w 1
1200.4.a.x 1
1200.4.a.y 1
1200.4.a.z 1
1200.4.a.ba 1
1200.4.a.bb 1
1200.4.a.bc 1
1200.4.a.bd 1
1200.4.a.be 1
1200.4.a.bf 1
1200.4.a.bg 1
1200.4.a.bh 1
1200.4.a.bi 1
1200.4.a.bj 1
1200.4.a.bk 1
1200.4.a.bl 2
1200.4.a.bm 2
1200.4.a.bn 2
1200.4.a.bo 2
1200.4.a.bp 2
1200.4.a.bq 2
1200.4.a.br 2
1200.4.a.bs 2
1200.4.a.bt 2
1200.4.a.bu 2
1200.4.b $$\chi_{1200}(551, \cdot)$$ None 0 1
1200.4.d $$\chi_{1200}(649, \cdot)$$ None 0 1
1200.4.f $$\chi_{1200}(49, \cdot)$$ 1200.4.f.a 2 1
1200.4.f.b 2
1200.4.f.c 2
1200.4.f.d 2
1200.4.f.e 2
1200.4.f.f 2
1200.4.f.g 2
1200.4.f.h 2
1200.4.f.i 2
1200.4.f.j 2
1200.4.f.k 2
1200.4.f.l 2
1200.4.f.m 2
1200.4.f.n 2
1200.4.f.o 2
1200.4.f.p 2
1200.4.f.q 2
1200.4.f.r 2
1200.4.f.s 2
1200.4.f.t 2
1200.4.f.u 2
1200.4.f.v 4
1200.4.f.w 4
1200.4.f.x 4
1200.4.h $$\chi_{1200}(1151, \cdot)$$ n/a 114 1
1200.4.k $$\chi_{1200}(601, \cdot)$$ None 0 1
1200.4.m $$\chi_{1200}(599, \cdot)$$ None 0 1
1200.4.o $$\chi_{1200}(1199, \cdot)$$ n/a 108 1
1200.4.s $$\chi_{1200}(301, \cdot)$$ n/a 456 2
1200.4.t $$\chi_{1200}(299, \cdot)$$ n/a 856 2
1200.4.v $$\chi_{1200}(257, \cdot)$$ n/a 212 2
1200.4.w $$\chi_{1200}(607, \cdot)$$ n/a 108 2
1200.4.y $$\chi_{1200}(643, \cdot)$$ n/a 432 2
1200.4.bb $$\chi_{1200}(893, \cdot)$$ n/a 856 2
1200.4.bc $$\chi_{1200}(43, \cdot)$$ n/a 432 2
1200.4.bf $$\chi_{1200}(293, \cdot)$$ n/a 856 2
1200.4.bh $$\chi_{1200}(7, \cdot)$$ None 0 2
1200.4.bi $$\chi_{1200}(857, \cdot)$$ None 0 2
1200.4.bk $$\chi_{1200}(251, \cdot)$$ n/a 900 2
1200.4.bl $$\chi_{1200}(349, \cdot)$$ n/a 432 2
1200.4.bo $$\chi_{1200}(241, \cdot)$$ n/a 360 4
1200.4.bq $$\chi_{1200}(191, \cdot)$$ n/a 720 4
1200.4.bs $$\chi_{1200}(289, \cdot)$$ n/a 360 4
1200.4.bu $$\chi_{1200}(169, \cdot)$$ None 0 4
1200.4.bw $$\chi_{1200}(71, \cdot)$$ None 0 4
1200.4.by $$\chi_{1200}(239, \cdot)$$ n/a 720 4
1200.4.ca $$\chi_{1200}(119, \cdot)$$ None 0 4
1200.4.cc $$\chi_{1200}(121, \cdot)$$ None 0 4
1200.4.ce $$\chi_{1200}(59, \cdot)$$ n/a 5728 8
1200.4.cf $$\chi_{1200}(61, \cdot)$$ n/a 2880 8
1200.4.cj $$\chi_{1200}(137, \cdot)$$ None 0 8
1200.4.ck $$\chi_{1200}(103, \cdot)$$ None 0 8
1200.4.cm $$\chi_{1200}(53, \cdot)$$ n/a 5728 8
1200.4.cp $$\chi_{1200}(67, \cdot)$$ n/a 2880 8
1200.4.cq $$\chi_{1200}(173, \cdot)$$ n/a 5728 8
1200.4.ct $$\chi_{1200}(163, \cdot)$$ n/a 2880 8
1200.4.cv $$\chi_{1200}(127, \cdot)$$ n/a 720 8
1200.4.cw $$\chi_{1200}(17, \cdot)$$ n/a 1424 8
1200.4.da $$\chi_{1200}(109, \cdot)$$ n/a 2880 8
1200.4.db $$\chi_{1200}(11, \cdot)$$ n/a 5728 8

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

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(1200))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(1200)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(1))$$$$^{\oplus 30}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(2))$$$$^{\oplus 24}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(3))$$$$^{\oplus 15}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(4))$$$$^{\oplus 18}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 20}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 16}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 9}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 10}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 10}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(40))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(48))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(50))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(60))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(75))$$$$^{\oplus 5}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(80))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(100))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(120))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(150))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(200))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(240))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(300))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(400))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(600))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(1200))$$$$^{\oplus 1}$$