## 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

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