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Properties

Label 684.7
Level 684
Weight 7
Dimension 35556
Nonzero newspaces 32
Sturm bound 181440
Trace bound 9

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Defining parameters

Level: \( N \) = \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) = \( 7 \)
Nonzero newspaces: \( 32 \)
Sturm bound: \(181440\)
Trace bound: \(9\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(\Gamma_1(684))\).

Total New Old
Modular forms 78480 35860 42620
Cusp forms 77040 35556 41484
Eisenstein series 1440 304 1136

Trace form

\( 35556 q - 37 q^{2} - 12 q^{3} - 65 q^{4} + 334 q^{5} + 606 q^{6} + 1216 q^{7} + 305 q^{8} - 1332 q^{9} + O(q^{10}) \) \( 35556 q - 37 q^{2} - 12 q^{3} - 65 q^{4} + 334 q^{5} + 606 q^{6} + 1216 q^{7} + 305 q^{8} - 1332 q^{9} + 767 q^{10} + 4428 q^{11} - 11004 q^{12} + 6678 q^{13} + 5133 q^{14} + 10080 q^{15} + 4039 q^{16} - 1142 q^{17} - 29028 q^{18} + 2516 q^{19} - 7022 q^{20} + 7116 q^{21} + 49719 q^{22} + 20376 q^{23} + 54714 q^{24} + 86006 q^{25} - 255299 q^{26} - 80352 q^{27} - 40764 q^{28} + 89542 q^{29} + 41892 q^{30} - 16148 q^{31} + 147878 q^{32} + 215136 q^{33} - 110008 q^{34} - 225432 q^{35} + 37302 q^{36} - 348084 q^{37} - 68859 q^{38} - 447624 q^{39} + 238634 q^{40} + 160450 q^{41} + 141240 q^{42} - 394004 q^{43} + 376044 q^{44} + 1258308 q^{45} + 18456 q^{46} + 1755036 q^{47} + 628854 q^{48} - 8306 q^{49} - 1104978 q^{50} - 1191204 q^{51} - 975703 q^{52} - 2571446 q^{53} + 539850 q^{54} - 358128 q^{55} + 1968228 q^{56} + 170046 q^{57} + 692114 q^{58} + 2545884 q^{59} + 179292 q^{60} + 2521818 q^{61} - 4641366 q^{62} + 1974624 q^{63} - 3282365 q^{64} + 2566226 q^{65} - 22536 q^{66} - 1577852 q^{67} + 3111544 q^{68} - 2428332 q^{69} + 5810253 q^{70} + 1465200 q^{71} + 1601814 q^{72} - 3970008 q^{73} - 1105811 q^{74} - 5197956 q^{75} - 2237202 q^{76} - 940836 q^{77} + 955728 q^{78} - 438272 q^{79} + 835933 q^{80} - 1603380 q^{81} - 3130786 q^{82} + 4864392 q^{83} - 861696 q^{84} - 1577750 q^{85} + 2789994 q^{86} + 8167896 q^{87} + 5951655 q^{88} + 8786710 q^{89} - 1960026 q^{90} - 1584640 q^{91} - 4895052 q^{92} - 15815244 q^{93} + 17100768 q^{94} + 9391356 q^{95} - 1959456 q^{96} - 14561454 q^{97} - 39530224 q^{98} - 10274940 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{7}^{\mathrm{new}}(\Gamma_1(684))\)

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
684.7.b \(\chi_{684}(683, \cdot)\) n/a 240 1
684.7.e \(\chi_{684}(305, \cdot)\) 684.7.e.a 36 1
684.7.g \(\chi_{684}(343, \cdot)\) n/a 270 1
684.7.h \(\chi_{684}(37, \cdot)\) 684.7.h.a 2 1
684.7.h.b 4
684.7.h.c 8
684.7.h.d 16
684.7.h.e 20
684.7.m \(\chi_{684}(353, \cdot)\) n/a 240 2
684.7.p \(\chi_{684}(407, \cdot)\) n/a 1432 2
684.7.q \(\chi_{684}(163, \cdot)\) n/a 596 2
684.7.s \(\chi_{684}(445, \cdot)\) n/a 240 2
684.7.t \(\chi_{684}(265, \cdot)\) n/a 240 2
684.7.v \(\chi_{684}(115, \cdot)\) n/a 1296 2
684.7.x \(\chi_{684}(7, \cdot)\) n/a 1432 2
684.7.y \(\chi_{684}(145, \cdot)\) 684.7.y.a 20 2
684.7.y.b 20
684.7.y.c 20
684.7.y.d 40
684.7.ba \(\chi_{684}(107, \cdot)\) n/a 480 2
684.7.bc \(\chi_{684}(77, \cdot)\) n/a 216 2
684.7.be \(\chi_{684}(425, \cdot)\) n/a 240 2
684.7.bf \(\chi_{684}(335, \cdot)\) n/a 1432 2
684.7.bh \(\chi_{684}(227, \cdot)\) n/a 1432 2
684.7.bj \(\chi_{684}(125, \cdot)\) 684.7.bj.a 80 2
684.7.bl \(\chi_{684}(373, \cdot)\) n/a 240 2
684.7.bm \(\chi_{684}(463, \cdot)\) n/a 1432 2
684.7.br \(\chi_{684}(155, \cdot)\) n/a 4296 6
684.7.bu \(\chi_{684}(43, \cdot)\) n/a 4296 6
684.7.bw \(\chi_{684}(5, \cdot)\) n/a 720 6
684.7.bx \(\chi_{684}(109, \cdot)\) n/a 300 6
684.7.by \(\chi_{684}(17, \cdot)\) n/a 240 6
684.7.ca \(\chi_{684}(193, \cdot)\) n/a 720 6
684.7.cb \(\chi_{684}(283, \cdot)\) n/a 4296 6
684.7.cd \(\chi_{684}(71, \cdot)\) n/a 1440 6
684.7.cg \(\chi_{684}(55, \cdot)\) n/a 1788 6
684.7.ci \(\chi_{684}(59, \cdot)\) n/a 4296 6
684.7.cj \(\chi_{684}(13, \cdot)\) n/a 720 6
684.7.ck \(\chi_{684}(245, \cdot)\) n/a 720 6

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

Decomposition of \(S_{7}^{\mathrm{old}}(\Gamma_1(684))\) into lower level spaces

\( S_{7}^{\mathrm{old}}(\Gamma_1(684)) \cong \) \(S_{7}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 12}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 6}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(\Gamma_1(9))\)\(^{\oplus 6}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(\Gamma_1(18))\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(\Gamma_1(19))\)\(^{\oplus 9}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(\Gamma_1(36))\)\(^{\oplus 2}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(\Gamma_1(38))\)\(^{\oplus 6}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(\Gamma_1(57))\)\(^{\oplus 6}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(\Gamma_1(76))\)\(^{\oplus 3}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(\Gamma_1(114))\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(\Gamma_1(171))\)\(^{\oplus 3}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(\Gamma_1(228))\)\(^{\oplus 2}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(\Gamma_1(342))\)\(^{\oplus 2}\)