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Properties

Label 1680.2
Level 1680
Weight 2
Dimension 27056
Nonzero newspaces 56
Sturm bound 294912
Trace bound 23

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Defining parameters

Level: \( N \) = \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 56 \)
Sturm bound: \(294912\)
Trace bound: \(23\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(1680))\).

Total New Old
Modular forms 76416 27592 48824
Cusp forms 71041 27056 43985
Eisenstein series 5375 536 4839

Trace form

\( 27056 q - 10 q^{3} - 48 q^{4} - 4 q^{5} - 72 q^{6} - 40 q^{7} - 48 q^{8} - 26 q^{9} + O(q^{10}) \) \( 27056 q - 10 q^{3} - 48 q^{4} - 4 q^{5} - 72 q^{6} - 40 q^{7} - 48 q^{8} - 26 q^{9} - 56 q^{10} - 48 q^{11} - 8 q^{12} - 88 q^{13} + 24 q^{14} - 86 q^{15} - 16 q^{16} - 24 q^{17} + 24 q^{18} - 140 q^{19} + 32 q^{20} - 102 q^{21} - 88 q^{23} + 104 q^{24} - 62 q^{25} + 80 q^{26} + 8 q^{27} + 32 q^{28} - 72 q^{29} + 84 q^{30} - 60 q^{31} + 160 q^{32} - 46 q^{33} + 208 q^{34} + 12 q^{35} - 16 q^{36} - 44 q^{37} + 192 q^{38} + 124 q^{39} + 232 q^{40} + 104 q^{41} + 120 q^{42} + 40 q^{43} + 256 q^{44} + 47 q^{45} + 224 q^{46} + 120 q^{47} + 104 q^{48} + 48 q^{49} + 176 q^{50} + 230 q^{51} + 416 q^{52} + 280 q^{53} + 136 q^{54} + 152 q^{55} + 336 q^{56} + 204 q^{57} + 384 q^{58} + 208 q^{59} + 100 q^{60} + 308 q^{61} + 384 q^{62} + 170 q^{63} + 432 q^{64} + 88 q^{65} + 280 q^{66} + 364 q^{67} + 144 q^{68} + 344 q^{69} - 32 q^{70} + 312 q^{71} + 88 q^{72} + 100 q^{73} - 144 q^{74} + 315 q^{75} - 80 q^{76} + 80 q^{77} - 176 q^{78} + 300 q^{79} - 304 q^{80} + 66 q^{81} - 176 q^{82} + 208 q^{83} - 184 q^{84} - 100 q^{85} - 320 q^{86} + 124 q^{87} - 496 q^{88} - 104 q^{89} - 308 q^{90} + 272 q^{91} - 400 q^{92} + 42 q^{93} - 688 q^{94} + 212 q^{95} - 632 q^{96} - 296 q^{97} - 464 q^{98} + 60 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(1680))\)

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
1680.2.a \(\chi_{1680}(1, \cdot)\) 1680.2.a.a 1 1
1680.2.a.b 1
1680.2.a.c 1
1680.2.a.d 1
1680.2.a.e 1
1680.2.a.f 1
1680.2.a.g 1
1680.2.a.h 1
1680.2.a.i 1
1680.2.a.j 1
1680.2.a.k 1
1680.2.a.l 1
1680.2.a.m 1
1680.2.a.n 1
1680.2.a.o 1
1680.2.a.p 1
1680.2.a.q 1
1680.2.a.r 1
1680.2.a.s 1
1680.2.a.t 1
1680.2.a.u 2
1680.2.a.v 2
1680.2.d \(\chi_{1680}(1231, \cdot)\) 1680.2.d.a 4 1
1680.2.d.b 4
1680.2.d.c 12
1680.2.d.d 12
1680.2.e \(\chi_{1680}(71, \cdot)\) None 0 1
1680.2.f \(\chi_{1680}(881, \cdot)\) 1680.2.f.a 2 1
1680.2.f.b 2
1680.2.f.c 2
1680.2.f.d 2
1680.2.f.e 4
1680.2.f.f 4
1680.2.f.g 4
1680.2.f.h 4
1680.2.f.i 4
1680.2.f.j 4
1680.2.f.k 16
1680.2.f.l 16
1680.2.g \(\chi_{1680}(841, \cdot)\) None 0 1
1680.2.j \(\chi_{1680}(169, \cdot)\) None 0 1
1680.2.k \(\chi_{1680}(209, \cdot)\) 1680.2.k.a 4 1
1680.2.k.b 4
1680.2.k.c 4
1680.2.k.d 8
1680.2.k.e 8
1680.2.k.f 8
1680.2.k.g 8
1680.2.k.h 24
1680.2.k.i 24
1680.2.p \(\chi_{1680}(1079, \cdot)\) None 0 1
1680.2.q \(\chi_{1680}(559, \cdot)\) 1680.2.q.a 8 1
1680.2.q.b 8
1680.2.q.c 32
1680.2.t \(\chi_{1680}(1009, \cdot)\) 1680.2.t.a 2 1
1680.2.t.b 2
1680.2.t.c 2
1680.2.t.d 2
1680.2.t.e 2
1680.2.t.f 2
1680.2.t.g 2
1680.2.t.h 4
1680.2.t.i 6
1680.2.t.j 6
1680.2.t.k 6
1680.2.u \(\chi_{1680}(1049, \cdot)\) None 0 1
1680.2.v \(\chi_{1680}(239, \cdot)\) 1680.2.v.a 12 1
1680.2.v.b 12
1680.2.v.c 24
1680.2.v.d 24
1680.2.w \(\chi_{1680}(1399, \cdot)\) None 0 1
1680.2.z \(\chi_{1680}(391, \cdot)\) None 0 1
1680.2.ba \(\chi_{1680}(911, \cdot)\) 1680.2.ba.a 8 1
1680.2.ba.b 8
1680.2.ba.c 16
1680.2.ba.d 16
1680.2.bf \(\chi_{1680}(41, \cdot)\) None 0 1
1680.2.bg \(\chi_{1680}(961, \cdot)\) 1680.2.bg.a 2 2
1680.2.bg.b 2
1680.2.bg.c 2
1680.2.bg.d 2
1680.2.bg.e 2
1680.2.bg.f 2
1680.2.bg.g 2
1680.2.bg.h 2
1680.2.bg.i 2
1680.2.bg.j 2
1680.2.bg.k 2
1680.2.bg.l 2
1680.2.bg.m 2
1680.2.bg.n 2
1680.2.bg.o 4
1680.2.bg.p 4
1680.2.bg.q 4
1680.2.bg.r 4
1680.2.bg.s 4
1680.2.bg.t 4
1680.2.bg.u 6
1680.2.bg.v 6
1680.2.bj \(\chi_{1680}(937, \cdot)\) None 0 2
1680.2.bk \(\chi_{1680}(113, \cdot)\) n/a 144 2
1680.2.bl \(\chi_{1680}(127, \cdot)\) 1680.2.bl.a 4 2
1680.2.bl.b 8
1680.2.bl.c 12
1680.2.bl.d 24
1680.2.bl.e 24
1680.2.bm \(\chi_{1680}(167, \cdot)\) None 0 2
1680.2.bp \(\chi_{1680}(883, \cdot)\) n/a 288 2
1680.2.bs \(\chi_{1680}(853, \cdot)\) n/a 384 2
1680.2.bu \(\chi_{1680}(197, \cdot)\) n/a 576 2
1680.2.bv \(\chi_{1680}(923, \cdot)\) n/a 752 2
1680.2.bx \(\chi_{1680}(629, \cdot)\) n/a 752 2
1680.2.ca \(\chi_{1680}(491, \cdot)\) n/a 384 2
1680.2.cb \(\chi_{1680}(589, \cdot)\) n/a 288 2
1680.2.ce \(\chi_{1680}(811, \cdot)\) n/a 256 2
1680.2.cg \(\chi_{1680}(421, \cdot)\) n/a 192 2
1680.2.ch \(\chi_{1680}(139, \cdot)\) n/a 384 2
1680.2.ck \(\chi_{1680}(461, \cdot)\) n/a 512 2
1680.2.cl \(\chi_{1680}(659, \cdot)\) n/a 576 2
1680.2.co \(\chi_{1680}(13, \cdot)\) n/a 384 2
1680.2.cp \(\chi_{1680}(43, \cdot)\) n/a 288 2
1680.2.cr \(\chi_{1680}(83, \cdot)\) n/a 752 2
1680.2.cu \(\chi_{1680}(533, \cdot)\) n/a 576 2
1680.2.cx \(\chi_{1680}(967, \cdot)\) None 0 2
1680.2.cy \(\chi_{1680}(1007, \cdot)\) n/a 192 2
1680.2.cz \(\chi_{1680}(97, \cdot)\) 1680.2.cz.a 8 2
1680.2.cz.b 8
1680.2.cz.c 16
1680.2.cz.d 16
1680.2.cz.e 24
1680.2.cz.f 24
1680.2.da \(\chi_{1680}(617, \cdot)\) None 0 2
1680.2.df \(\chi_{1680}(199, \cdot)\) None 0 2
1680.2.dg \(\chi_{1680}(1199, \cdot)\) n/a 192 2
1680.2.dh \(\chi_{1680}(89, \cdot)\) None 0 2
1680.2.di \(\chi_{1680}(289, \cdot)\) 1680.2.di.a 4 2
1680.2.di.b 4
1680.2.di.c 12
1680.2.di.d 16
1680.2.di.e 16
1680.2.di.f 20
1680.2.di.g 24
1680.2.dl \(\chi_{1680}(521, \cdot)\) None 0 2
1680.2.dq \(\chi_{1680}(191, \cdot)\) n/a 128 2
1680.2.dr \(\chi_{1680}(871, \cdot)\) None 0 2
1680.2.du \(\chi_{1680}(121, \cdot)\) None 0 2
1680.2.dv \(\chi_{1680}(1361, \cdot)\) n/a 128 2
1680.2.dw \(\chi_{1680}(1031, \cdot)\) None 0 2
1680.2.dx \(\chi_{1680}(31, \cdot)\) 1680.2.dx.a 4 2
1680.2.dx.b 4
1680.2.dx.c 4
1680.2.dx.d 4
1680.2.dx.e 12
1680.2.dx.f 12
1680.2.dx.g 12
1680.2.dx.h 12
1680.2.ea \(\chi_{1680}(1039, \cdot)\) 1680.2.ea.a 16 2
1680.2.ea.b 16
1680.2.ea.c 32
1680.2.ea.d 32
1680.2.eb \(\chi_{1680}(359, \cdot)\) None 0 2
1680.2.eg \(\chi_{1680}(689, \cdot)\) n/a 184 2
1680.2.eh \(\chi_{1680}(1129, \cdot)\) None 0 2
1680.2.ei \(\chi_{1680}(137, \cdot)\) None 0 4
1680.2.ej \(\chi_{1680}(577, \cdot)\) n/a 192 4
1680.2.eo \(\chi_{1680}(47, \cdot)\) n/a 384 4
1680.2.ep \(\chi_{1680}(247, \cdot)\) None 0 4
1680.2.eq \(\chi_{1680}(227, \cdot)\) n/a 1504 4
1680.2.et \(\chi_{1680}(653, \cdot)\) n/a 1504 4
1680.2.ev \(\chi_{1680}(157, \cdot)\) n/a 768 4
1680.2.ew \(\chi_{1680}(163, \cdot)\) n/a 768 4
1680.2.ey \(\chi_{1680}(101, \cdot)\) n/a 1024 4
1680.2.fb \(\chi_{1680}(179, \cdot)\) n/a 1504 4
1680.2.fc \(\chi_{1680}(541, \cdot)\) n/a 512 4
1680.2.ff \(\chi_{1680}(19, \cdot)\) n/a 768 4
1680.2.fh \(\chi_{1680}(109, \cdot)\) n/a 768 4
1680.2.fi \(\chi_{1680}(451, \cdot)\) n/a 512 4
1680.2.fl \(\chi_{1680}(269, \cdot)\) n/a 1504 4
1680.2.fm \(\chi_{1680}(11, \cdot)\) n/a 1024 4
1680.2.fp \(\chi_{1680}(53, \cdot)\) n/a 1504 4
1680.2.fq \(\chi_{1680}(563, \cdot)\) n/a 1504 4
1680.2.fs \(\chi_{1680}(67, \cdot)\) n/a 768 4
1680.2.fv \(\chi_{1680}(493, \cdot)\) n/a 768 4
1680.2.fw \(\chi_{1680}(647, \cdot)\) None 0 4
1680.2.fx \(\chi_{1680}(1087, \cdot)\) n/a 192 4
1680.2.gc \(\chi_{1680}(737, \cdot)\) n/a 368 4
1680.2.gd \(\chi_{1680}(73, \cdot)\) None 0 4

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

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(1680))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(1680)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(15))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(16))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(20))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(21))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(24))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(28))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(30))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(35))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(40))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(42))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(48))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(56))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(60))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(70))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(80))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(84))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(105))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(112))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(120))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(140))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(168))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(210))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(240))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(280))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(336))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(420))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(560))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(840))\)\(^{\oplus 2}\)