# Properties

 Label 1200.3 Level 1200 Weight 3 Dimension 29677 Nonzero newspaces 28 Sturm bound 230400 Trace bound 9

## Defining parameters

 Level: $$N$$ = $$1200 = 2^{4} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$28$$ Sturm bound: $$230400$$ Trace bound: $$9$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(1200))$$.

Total New Old
Modular forms 78368 30047 48321
Cusp forms 75232 29677 45555
Eisenstein series 3136 370 2766

## Trace form

 $$29677 q - 19 q^{3} - 40 q^{4} - 36 q^{6} - 26 q^{7} - 12 q^{8} - 13 q^{9} + O(q^{10})$$ $$29677 q - 19 q^{3} - 40 q^{4} - 36 q^{6} - 26 q^{7} - 12 q^{8} - 13 q^{9} - 64 q^{10} + 32 q^{11} - 80 q^{12} + 2 q^{13} - 44 q^{14} + 24 q^{15} - 64 q^{16} + 52 q^{17} - 72 q^{18} - 142 q^{19} - 140 q^{21} + 24 q^{22} - 320 q^{23} - 44 q^{24} - 112 q^{25} - 100 q^{26} - 187 q^{27} - 560 q^{28} - 388 q^{29} - 320 q^{30} - 234 q^{31} - 960 q^{32} - 374 q^{33} - 1096 q^{34} - 96 q^{35} + 72 q^{36} - 414 q^{37} - 56 q^{38} - 124 q^{39} + 96 q^{40} + 148 q^{41} + 676 q^{42} + 170 q^{43} + 1208 q^{44} + 6 q^{45} + 1648 q^{46} + 384 q^{47} + 1096 q^{48} + 1167 q^{49} + 880 q^{50} - 124 q^{51} + 1248 q^{52} + 252 q^{53} - 188 q^{54} + 52 q^{55} - 224 q^{56} - 148 q^{57} - 472 q^{58} - 128 q^{59} + 96 q^{60} - 406 q^{61} - 276 q^{62} - 620 q^{63} + 752 q^{64} - 112 q^{65} + 1124 q^{66} - 2598 q^{67} + 2016 q^{68} - 14 q^{69} + 1232 q^{70} - 2048 q^{71} + 1404 q^{72} + 366 q^{73} + 2108 q^{74} - 392 q^{75} + 1456 q^{76} + 320 q^{77} + 976 q^{78} + 358 q^{79} - 160 q^{80} - 41 q^{81} - 184 q^{82} + 288 q^{83} - 592 q^{84} - 2032 q^{85} - 1264 q^{86} + 1298 q^{87} - 2384 q^{88} - 892 q^{89} - 1352 q^{90} + 3368 q^{91} - 3248 q^{92} + 56 q^{93} - 3432 q^{94} + 1152 q^{95} - 2208 q^{96} - 330 q^{97} - 2264 q^{98} + 1672 q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(1200))$$

We only show spaces with odd 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
1200.3.c $$\chi_{1200}(449, \cdot)$$ 1200.3.c.a 2 1
1200.3.c.b 2
1200.3.c.c 2
1200.3.c.d 4
1200.3.c.e 4
1200.3.c.f 4
1200.3.c.g 4
1200.3.c.h 4
1200.3.c.i 4
1200.3.c.j 4
1200.3.c.k 8
1200.3.c.l 12
1200.3.c.m 16
1200.3.e $$\chi_{1200}(751, \cdot)$$ 1200.3.e.a 2 1
1200.3.e.b 2
1200.3.e.c 2
1200.3.e.d 2
1200.3.e.e 2
1200.3.e.f 2
1200.3.e.g 2
1200.3.e.h 2
1200.3.e.i 2
1200.3.e.j 4
1200.3.e.k 4
1200.3.e.l 4
1200.3.e.m 4
1200.3.e.n 4
1200.3.g $$\chi_{1200}(151, \cdot)$$ None 0 1
1200.3.i $$\chi_{1200}(1049, \cdot)$$ None 0 1
1200.3.j $$\chi_{1200}(799, \cdot)$$ 1200.3.j.a 4 1
1200.3.j.b 4
1200.3.j.c 4
1200.3.j.d 8
1200.3.j.e 8
1200.3.j.f 8
1200.3.l $$\chi_{1200}(401, \cdot)$$ 1200.3.l.a 1 1
1200.3.l.b 1
1200.3.l.c 1
1200.3.l.d 1
1200.3.l.e 1
1200.3.l.f 2
1200.3.l.g 2
1200.3.l.h 2
1200.3.l.i 2
1200.3.l.j 2
1200.3.l.k 2
1200.3.l.l 2
1200.3.l.m 2
1200.3.l.n 2
1200.3.l.o 2
1200.3.l.p 2
1200.3.l.q 2
1200.3.l.r 2
1200.3.l.s 2
1200.3.l.t 4
1200.3.l.u 4
1200.3.l.v 6
1200.3.l.w 6
1200.3.l.x 8
1200.3.l.y 12
1200.3.n $$\chi_{1200}(1001, \cdot)$$ None 0 1
1200.3.p $$\chi_{1200}(199, \cdot)$$ None 0 1
1200.3.q $$\chi_{1200}(499, \cdot)$$ n/a 288 2
1200.3.r $$\chi_{1200}(101, \cdot)$$ n/a 596 2
1200.3.u $$\chi_{1200}(407, \cdot)$$ None 0 2
1200.3.x $$\chi_{1200}(457, \cdot)$$ None 0 2
1200.3.z $$\chi_{1200}(107, \cdot)$$ n/a 568 2
1200.3.ba $$\chi_{1200}(493, \cdot)$$ n/a 288 2
1200.3.bd $$\chi_{1200}(443, \cdot)$$ n/a 568 2
1200.3.be $$\chi_{1200}(157, \cdot)$$ n/a 288 2
1200.3.bg $$\chi_{1200}(193, \cdot)$$ 1200.3.bg.a 4 2
1200.3.bg.b 4
1200.3.bg.c 4
1200.3.bg.d 4
1200.3.bg.e 4
1200.3.bg.f 4
1200.3.bg.g 4
1200.3.bg.h 4
1200.3.bg.i 4
1200.3.bg.j 4
1200.3.bg.k 4
1200.3.bg.l 4
1200.3.bg.m 4
1200.3.bg.n 4
1200.3.bg.o 4
1200.3.bg.p 4
1200.3.bg.q 8
1200.3.bj $$\chi_{1200}(143, \cdot)$$ n/a 144 2
1200.3.bm $$\chi_{1200}(149, \cdot)$$ n/a 568 2
1200.3.bn $$\chi_{1200}(451, \cdot)$$ n/a 304 2
1200.3.bp $$\chi_{1200}(89, \cdot)$$ None 0 4
1200.3.br $$\chi_{1200}(391, \cdot)$$ None 0 4
1200.3.bt $$\chi_{1200}(31, \cdot)$$ n/a 240 4
1200.3.bv $$\chi_{1200}(209, \cdot)$$ n/a 472 4
1200.3.bx $$\chi_{1200}(439, \cdot)$$ None 0 4
1200.3.bz $$\chi_{1200}(41, \cdot)$$ None 0 4
1200.3.cb $$\chi_{1200}(161, \cdot)$$ n/a 472 4
1200.3.cd $$\chi_{1200}(79, \cdot)$$ n/a 240 4
1200.3.cg $$\chi_{1200}(221, \cdot)$$ n/a 3808 8
1200.3.ch $$\chi_{1200}(19, \cdot)$$ n/a 1920 8
1200.3.ci $$\chi_{1200}(47, \cdot)$$ n/a 960 8
1200.3.cl $$\chi_{1200}(97, \cdot)$$ n/a 480 8
1200.3.cn $$\chi_{1200}(133, \cdot)$$ n/a 1920 8
1200.3.co $$\chi_{1200}(203, \cdot)$$ n/a 3808 8
1200.3.cr $$\chi_{1200}(13, \cdot)$$ n/a 1920 8
1200.3.cs $$\chi_{1200}(83, \cdot)$$ n/a 3808 8
1200.3.cu $$\chi_{1200}(73, \cdot)$$ None 0 8
1200.3.cx $$\chi_{1200}(23, \cdot)$$ None 0 8
1200.3.cy $$\chi_{1200}(91, \cdot)$$ n/a 1920 8
1200.3.cz $$\chi_{1200}(29, \cdot)$$ n/a 3808 8

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

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(1200))$$ into lower level spaces

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