Properties

Label 684.3
Level 684
Weight 3
Dimension 11752
Nonzero newspaces 32
Sturm bound 77760
Trace bound 9

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Defining parameters

Level: \( N \) = \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) = \( 3 \)
Nonzero newspaces: \( 32 \)
Sturm bound: \(77760\)
Trace bound: \(9\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(\Gamma_1(684))\).

Total New Old
Modular forms 26640 12056 14584
Cusp forms 25200 11752 13448
Eisenstein series 1440 304 1136

Trace form

\( 11752q - 29q^{2} - 6q^{3} - 33q^{4} - 76q^{5} - 18q^{6} + 2q^{7} + 49q^{8} - 42q^{9} + O(q^{10}) \) \( 11752q - 29q^{2} - 6q^{3} - 33q^{4} - 76q^{5} - 18q^{6} + 2q^{7} + 49q^{8} - 42q^{9} - 33q^{10} + 72q^{11} + 36q^{12} + 32q^{13} - 27q^{14} + 90q^{15} - 57q^{16} + 59q^{17} - 60q^{18} - 23q^{19} - 142q^{20} - 210q^{21} - 57q^{22} - 243q^{23} - 294q^{24} - 254q^{25} - 355q^{26} - 348q^{28} - 124q^{29} - 348q^{30} - 94q^{31} - 554q^{32} + 84q^{33} - 228q^{34} - 72q^{35} + 54q^{36} + 140q^{37} + 261q^{38} - 114q^{39} + 474q^{40} + 170q^{41} + 672q^{42} + 191q^{43} + 972q^{44} - 672q^{45} + 708q^{46} - 387q^{47} + 630q^{48} - 345q^{49} + 882q^{50} - 396q^{51} + 105q^{52} - 946q^{53} - 150q^{54} - 288q^{55} - 540q^{56} - 237q^{57} - 582q^{58} + 153q^{59} - 1188q^{60} - 196q^{61} - 522q^{62} + 222q^{63} + 195q^{64} - 29q^{65} - 1056q^{66} + 173q^{67} - 328q^{68} + 540q^{69} + 213q^{70} + 1143q^{71} + 342q^{72} - 631q^{73} + 857q^{74} + 594q^{75} - 66q^{76} + 1539q^{77} + 1344q^{78} + 272q^{79} + 1373q^{80} + 582q^{81} - 558q^{82} + 603q^{83} + 1248q^{84} + 2370q^{85} - 162q^{86} + 126q^{87} - 1113q^{88} + 1949q^{89} + 54q^{90} + 1096q^{91} - 1260q^{92} + 1218q^{93} + 72q^{94} + 2061q^{95} + 576q^{96} + 2621q^{97} + 2380q^{98} + 1422q^{99} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(\Gamma_1(684))\)

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 the newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
684.3.b \(\chi_{684}(683, \cdot)\) 684.3.b.a 80 1
684.3.e \(\chi_{684}(305, \cdot)\) 684.3.e.a 12 1
684.3.g \(\chi_{684}(343, \cdot)\) 684.3.g.a 4 1
684.3.g.b 14
684.3.g.c 36
684.3.g.d 36
684.3.h \(\chi_{684}(37, \cdot)\) 684.3.h.a 2 1
684.3.h.b 2
684.3.h.c 2
684.3.h.d 4
684.3.h.e 6
684.3.m \(\chi_{684}(353, \cdot)\) 684.3.m.a 80 2
684.3.p \(\chi_{684}(407, \cdot)\) n/a 472 2
684.3.q \(\chi_{684}(163, \cdot)\) n/a 196 2
684.3.s \(\chi_{684}(445, \cdot)\) 684.3.s.a 80 2
684.3.t \(\chi_{684}(265, \cdot)\) 684.3.t.a 80 2
684.3.v \(\chi_{684}(115, \cdot)\) n/a 432 2
684.3.x \(\chi_{684}(7, \cdot)\) n/a 472 2
684.3.y \(\chi_{684}(145, \cdot)\) 684.3.y.a 2 2
684.3.y.b 2
684.3.y.c 2
684.3.y.d 2
684.3.y.e 2
684.3.y.f 6
684.3.y.g 8
684.3.y.h 8
684.3.ba \(\chi_{684}(107, \cdot)\) n/a 160 2
684.3.bc \(\chi_{684}(77, \cdot)\) 684.3.bc.a 72 2
684.3.be \(\chi_{684}(425, \cdot)\) 684.3.be.a 80 2
684.3.bf \(\chi_{684}(335, \cdot)\) n/a 472 2
684.3.bh \(\chi_{684}(227, \cdot)\) n/a 472 2
684.3.bj \(\chi_{684}(125, \cdot)\) 684.3.bj.a 24 2
684.3.bl \(\chi_{684}(373, \cdot)\) 684.3.bl.a 80 2
684.3.bm \(\chi_{684}(463, \cdot)\) n/a 472 2
684.3.br \(\chi_{684}(155, \cdot)\) n/a 1416 6
684.3.bu \(\chi_{684}(43, \cdot)\) n/a 1416 6
684.3.bw \(\chi_{684}(5, \cdot)\) n/a 240 6
684.3.bx \(\chi_{684}(109, \cdot)\) n/a 102 6
684.3.by \(\chi_{684}(17, \cdot)\) 684.3.by.a 84 6
684.3.ca \(\chi_{684}(193, \cdot)\) n/a 240 6
684.3.cb \(\chi_{684}(283, \cdot)\) n/a 1416 6
684.3.cd \(\chi_{684}(71, \cdot)\) n/a 480 6
684.3.cg \(\chi_{684}(55, \cdot)\) n/a 588 6
684.3.ci \(\chi_{684}(59, \cdot)\) n/a 1416 6
684.3.cj \(\chi_{684}(13, \cdot)\) n/a 240 6
684.3.ck \(\chi_{684}(245, \cdot)\) n/a 240 6

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

Decomposition of \(S_{3}^{\mathrm{old}}(\Gamma_1(684))\) into lower level spaces

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