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Properties

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

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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

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

Display 100 coefficients 10 coefficients

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 the 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

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