## Defining parameters

 Level: $$N$$ = $$1200 = 2^{4} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$28$$ Sturm bound: $$153600$$ Trace bound: $$25$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(1200))$$.

Total New Old
Modular forms 39968 15115 24853
Cusp forms 36833 14747 22086
Eisenstein series 3135 368 2767

## Trace form

 $$14747q - 19q^{3} - 56q^{4} - 48q^{6} - 44q^{7} - 12q^{8} - 15q^{9} + O(q^{10})$$ $$14747q - 19q^{3} - 56q^{4} - 48q^{6} - 44q^{7} - 12q^{8} - 15q^{9} - 64q^{10} - 12q^{11} - 24q^{12} - 86q^{13} + 12q^{14} - 36q^{15} - 64q^{16} - 30q^{17} - 16q^{18} - 92q^{19} - 78q^{21} - 32q^{22} - 40q^{23} + 4q^{24} - 32q^{25} + 20q^{26} - 31q^{27} + 80q^{28} + 6q^{29} - 12q^{31} + 160q^{32} - 10q^{33} + 200q^{34} + 24q^{35} + 8q^{36} + 66q^{37} + 216q^{38} + 52q^{39} + 96q^{40} + 122q^{41} + 92q^{42} + 28q^{43} + 184q^{44} - 14q^{45} + 128q^{46} + 48q^{47} + 40q^{48} - 13q^{49} + 80q^{50} + 38q^{51} + 48q^{52} + 46q^{53} - 72q^{54} - 28q^{55} + 38q^{57} - 32q^{58} + 28q^{59} - 64q^{60} - 46q^{61} - 12q^{62} + 202q^{63} - 80q^{64} + 8q^{65} - 116q^{66} + 220q^{67} - 192q^{68} + 90q^{69} - 208q^{70} + 312q^{71} - 228q^{72} + 2q^{73} - 300q^{74} + 128q^{75} - 352q^{76} + 112q^{77} - 312q^{78} + 324q^{79} - 160q^{80} - 51q^{81} - 280q^{82} + 436q^{83} - 336q^{84} + 128q^{85} - 272q^{86} + 268q^{87} - 432q^{88} + 74q^{89} - 152q^{90} + 244q^{91} - 304q^{92} + 110q^{93} - 312q^{94} + 312q^{95} - 224q^{96} + 66q^{97} - 136q^{98} + 56q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(1200))$$

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 the newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
1200.2.a $$\chi_{1200}(1, \cdot)$$ 1200.2.a.a 1 1
1200.2.a.b 1
1200.2.a.c 1
1200.2.a.d 1
1200.2.a.e 1
1200.2.a.f 1
1200.2.a.g 1
1200.2.a.h 1
1200.2.a.i 1
1200.2.a.j 1
1200.2.a.k 1
1200.2.a.l 1
1200.2.a.m 1
1200.2.a.n 1
1200.2.a.o 1
1200.2.a.p 1
1200.2.a.q 1
1200.2.a.r 1
1200.2.a.s 1
1200.2.b $$\chi_{1200}(551, \cdot)$$ None 0 1
1200.2.d $$\chi_{1200}(649, \cdot)$$ None 0 1
1200.2.f $$\chi_{1200}(49, \cdot)$$ 1200.2.f.a 2 1
1200.2.f.b 2
1200.2.f.c 2
1200.2.f.d 2
1200.2.f.e 2
1200.2.f.f 2
1200.2.f.g 2
1200.2.f.h 2
1200.2.f.i 2
1200.2.h $$\chi_{1200}(1151, \cdot)$$ 1200.2.h.a 2 1
1200.2.h.b 2
1200.2.h.c 2
1200.2.h.d 2
1200.2.h.e 2
1200.2.h.f 2
1200.2.h.g 2
1200.2.h.h 2
1200.2.h.i 2
1200.2.h.j 4
1200.2.h.k 4
1200.2.h.l 4
1200.2.h.m 4
1200.2.h.n 4
1200.2.k $$\chi_{1200}(601, \cdot)$$ None 0 1
1200.2.m $$\chi_{1200}(599, \cdot)$$ None 0 1
1200.2.o $$\chi_{1200}(1199, \cdot)$$ 1200.2.o.a 4 1
1200.2.o.b 4
1200.2.o.c 4
1200.2.o.d 4
1200.2.o.e 4
1200.2.o.f 4
1200.2.o.g 4
1200.2.o.h 4
1200.2.o.i 4
1200.2.s $$\chi_{1200}(301, \cdot)$$ n/a 152 2
1200.2.t $$\chi_{1200}(299, \cdot)$$ n/a 280 2
1200.2.v $$\chi_{1200}(257, \cdot)$$ 1200.2.v.a 4 2
1200.2.v.b 4
1200.2.v.c 4
1200.2.v.d 4
1200.2.v.e 4
1200.2.v.f 4
1200.2.v.g 4
1200.2.v.h 4
1200.2.v.i 4
1200.2.v.j 4
1200.2.v.k 4
1200.2.v.l 8
1200.2.v.m 16
1200.2.w $$\chi_{1200}(607, \cdot)$$ 1200.2.w.a 4 2
1200.2.w.b 8
1200.2.w.c 8
1200.2.w.d 8
1200.2.w.e 8
1200.2.y $$\chi_{1200}(643, \cdot)$$ n/a 144 2
1200.2.bb $$\chi_{1200}(893, \cdot)$$ n/a 280 2
1200.2.bc $$\chi_{1200}(43, \cdot)$$ n/a 144 2
1200.2.bf $$\chi_{1200}(293, \cdot)$$ n/a 280 2
1200.2.bh $$\chi_{1200}(7, \cdot)$$ None 0 2
1200.2.bi $$\chi_{1200}(857, \cdot)$$ None 0 2
1200.2.bk $$\chi_{1200}(251, \cdot)$$ n/a 292 2
1200.2.bl $$\chi_{1200}(349, \cdot)$$ n/a 144 2
1200.2.bo $$\chi_{1200}(241, \cdot)$$ n/a 120 4
1200.2.bq $$\chi_{1200}(191, \cdot)$$ n/a 240 4
1200.2.bs $$\chi_{1200}(289, \cdot)$$ n/a 120 4
1200.2.bu $$\chi_{1200}(169, \cdot)$$ None 0 4
1200.2.bw $$\chi_{1200}(71, \cdot)$$ None 0 4
1200.2.by $$\chi_{1200}(239, \cdot)$$ n/a 240 4
1200.2.ca $$\chi_{1200}(119, \cdot)$$ None 0 4
1200.2.cc $$\chi_{1200}(121, \cdot)$$ None 0 4
1200.2.ce $$\chi_{1200}(59, \cdot)$$ n/a 1888 8
1200.2.cf $$\chi_{1200}(61, \cdot)$$ n/a 960 8
1200.2.cj $$\chi_{1200}(137, \cdot)$$ None 0 8
1200.2.ck $$\chi_{1200}(103, \cdot)$$ None 0 8
1200.2.cm $$\chi_{1200}(53, \cdot)$$ n/a 1888 8
1200.2.cp $$\chi_{1200}(67, \cdot)$$ n/a 960 8
1200.2.cq $$\chi_{1200}(173, \cdot)$$ n/a 1888 8
1200.2.ct $$\chi_{1200}(163, \cdot)$$ n/a 960 8
1200.2.cv $$\chi_{1200}(127, \cdot)$$ n/a 240 8
1200.2.cw $$\chi_{1200}(17, \cdot)$$ n/a 464 8
1200.2.da $$\chi_{1200}(109, \cdot)$$ n/a 960 8
1200.2.db $$\chi_{1200}(11, \cdot)$$ n/a 1888 8

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

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(1200))$$ into lower level spaces

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