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Properties

Label 780.2
Level 780
Weight 2
Dimension 6256
Nonzero newspaces 40
Newform subspaces 84
Sturm bound 64512
Trace bound 21

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Defining parameters

Level: \( N \) = \( 780 = 2^{2} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 40 \)
Newform subspaces: \( 84 \)
Sturm bound: \(64512\)
Trace bound: \(21\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(780))\).

Total New Old
Modular forms 17088 6528 10560
Cusp forms 15169 6256 8913
Eisenstein series 1919 272 1647

Trace form

\( 6256 q - 4 q^{3} - 8 q^{4} - 4 q^{5} - 20 q^{6} + 24 q^{8} - 8 q^{9} + O(q^{10}) \) \( 6256 q - 4 q^{3} - 8 q^{4} - 4 q^{5} - 20 q^{6} + 24 q^{8} - 8 q^{9} + 4 q^{10} - 8 q^{11} + 8 q^{12} - 72 q^{13} + 16 q^{15} - 72 q^{16} + 28 q^{17} - 8 q^{18} + 16 q^{19} + 8 q^{20} - 4 q^{21} + 80 q^{22} + 48 q^{23} + 36 q^{24} - 20 q^{25} + 104 q^{26} + 44 q^{27} + 48 q^{28} + 36 q^{29} - 42 q^{30} + 80 q^{32} + 36 q^{33} + 24 q^{34} - 8 q^{35} - 36 q^{37} - 32 q^{38} + 56 q^{39} - 152 q^{40} + 140 q^{41} - 64 q^{42} + 88 q^{43} - 120 q^{44} + 16 q^{45} - 264 q^{46} + 48 q^{47} - 68 q^{48} + 292 q^{49} - 76 q^{50} + 120 q^{51} - 200 q^{52} + 184 q^{53} + 124 q^{54} + 172 q^{55} - 88 q^{56} + 148 q^{57} - 104 q^{58} + 136 q^{59} + 20 q^{60} + 156 q^{61} - 8 q^{62} + 28 q^{63} - 80 q^{64} + 190 q^{65} - 112 q^{66} + 120 q^{67} + 32 q^{68} - 76 q^{69} - 80 q^{70} + 48 q^{71} - 192 q^{72} - 80 q^{73} + 22 q^{75} - 136 q^{76} - 96 q^{77} - 256 q^{78} - 96 q^{79} - 8 q^{80} - 200 q^{81} - 136 q^{82} - 72 q^{83} - 376 q^{84} - 262 q^{85} - 224 q^{86} - 208 q^{87} - 352 q^{88} - 184 q^{89} - 288 q^{90} - 128 q^{91} - 256 q^{92} - 244 q^{93} - 424 q^{94} - 48 q^{95} - 416 q^{96} - 152 q^{97} - 264 q^{98} + 20 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(780))\)

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
780.2.a \(\chi_{780}(1, \cdot)\) 780.2.a.a 1 1
780.2.a.b 1
780.2.a.c 1
780.2.a.d 1
780.2.a.e 2
780.2.a.f 2
780.2.c \(\chi_{780}(181, \cdot)\) 780.2.c.a 2 1
780.2.c.b 2
780.2.c.c 4
780.2.d \(\chi_{780}(779, \cdot)\) 780.2.d.a 8 1
780.2.d.b 8
780.2.d.c 16
780.2.d.d 128
780.2.g \(\chi_{780}(131, \cdot)\) 780.2.g.a 8 1
780.2.g.b 8
780.2.g.c 8
780.2.g.d 32
780.2.g.e 40
780.2.h \(\chi_{780}(469, \cdot)\) 780.2.h.a 2 1
780.2.h.b 2
780.2.h.c 4
780.2.h.d 4
780.2.j \(\chi_{780}(649, \cdot)\) 780.2.j.a 12 1
780.2.m \(\chi_{780}(311, \cdot)\) 780.2.m.a 8 1
780.2.m.b 8
780.2.m.c 48
780.2.m.d 48
780.2.n \(\chi_{780}(599, \cdot)\) 780.2.n.a 144 1
780.2.q \(\chi_{780}(61, \cdot)\) 780.2.q.a 2 2
780.2.q.b 2
780.2.q.c 4
780.2.q.d 6
780.2.q.e 6
780.2.r \(\chi_{780}(73, \cdot)\) 780.2.r.a 28 2
780.2.u \(\chi_{780}(47, \cdot)\) 780.2.u.a 8 2
780.2.u.b 312
780.2.v \(\chi_{780}(53, \cdot)\) 780.2.v.a 48 2
780.2.y \(\chi_{780}(547, \cdot)\) 780.2.y.a 72 2
780.2.y.b 72
780.2.z \(\chi_{780}(629, \cdot)\) 780.2.z.a 56 2
780.2.bc \(\chi_{780}(31, \cdot)\) 780.2.bc.a 56 2
780.2.bc.b 56
780.2.be \(\chi_{780}(499, \cdot)\) 780.2.be.a 84 2
780.2.be.b 84
780.2.bf \(\chi_{780}(161, \cdot)\) 780.2.bf.a 40 2
780.2.bh \(\chi_{780}(103, \cdot)\) 780.2.bh.a 168 2
780.2.bk \(\chi_{780}(77, \cdot)\) 780.2.bk.a 8 2
780.2.bk.b 48
780.2.bm \(\chi_{780}(697, \cdot)\) 780.2.bm.a 28 2
780.2.bn \(\chi_{780}(203, \cdot)\) 780.2.bn.a 8 2
780.2.bn.b 312
780.2.br \(\chi_{780}(419, \cdot)\) 780.2.br.a 320 2
780.2.bs \(\chi_{780}(251, \cdot)\) 780.2.bs.a 112 2
780.2.bs.b 112
780.2.bv \(\chi_{780}(49, \cdot)\) 780.2.bv.a 24 2
780.2.bx \(\chi_{780}(289, \cdot)\) 780.2.bx.a 32 2
780.2.by \(\chi_{780}(191, \cdot)\) 780.2.by.a 4 2
780.2.by.b 4
780.2.by.c 4
780.2.by.d 4
780.2.by.e 208
780.2.cb \(\chi_{780}(179, \cdot)\) 780.2.cb.a 4 2
780.2.cb.b 4
780.2.cb.c 4
780.2.cb.d 4
780.2.cb.e 304
780.2.cc \(\chi_{780}(121, \cdot)\) 780.2.cc.a 4 2
780.2.cc.b 8
780.2.cc.c 8
780.2.cf \(\chi_{780}(227, \cdot)\) 780.2.cf.a 640 4
780.2.cg \(\chi_{780}(37, \cdot)\) 780.2.cg.a 56 4
780.2.ci \(\chi_{780}(17, \cdot)\) 780.2.ci.a 112 4
780.2.cl \(\chi_{780}(43, \cdot)\) 780.2.cl.a 336 4
780.2.cn \(\chi_{780}(41, \cdot)\) 780.2.cn.a 8 4
780.2.cn.b 64
780.2.co \(\chi_{780}(19, \cdot)\) 780.2.co.a 168 4
780.2.co.b 168
780.2.cq \(\chi_{780}(271, \cdot)\) 780.2.cq.a 112 4
780.2.cq.b 112
780.2.ct \(\chi_{780}(89, \cdot)\) 780.2.ct.a 112 4
780.2.cu \(\chi_{780}(367, \cdot)\) 780.2.cu.a 336 4
780.2.cx \(\chi_{780}(113, \cdot)\) 780.2.cx.a 112 4
780.2.cy \(\chi_{780}(167, \cdot)\) 780.2.cy.a 640 4
780.2.db \(\chi_{780}(97, \cdot)\) 780.2.db.a 56 4

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(780))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(780)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(13))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(15))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(20))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(26))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(39))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(52))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(60))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(65))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(78))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(130))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(156))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(195))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(260))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(390))\)\(^{\oplus 2}\)