## Defining parameters

 Level: $$N$$ = $$684 = 2^{2} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ = $$5$$ Nonzero newspaces: $$32$$ Sturm bound: $$129600$$ Trace bound: $$9$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{5}(\Gamma_1(684))$$.

Total New Old
Modular forms 52560 23956 28604
Cusp forms 51120 23652 27468
Eisenstein series 1440 304 1136

## Trace form

 $$23652q - 21q^{2} + 18q^{3} - q^{4} - 12q^{5} - 66q^{6} - 298q^{7} - 399q^{8} + 6q^{9} + O(q^{10})$$ $$23652q - 21q^{2} + 18q^{3} - q^{4} - 12q^{5} - 66q^{6} - 298q^{7} - 399q^{8} + 6q^{9} - 353q^{10} + 36q^{11} + 420q^{12} - 264q^{13} + 669q^{14} - 450q^{15} + 1223q^{16} - 795q^{17} - 180q^{18} + 631q^{19} - 2958q^{20} + 1986q^{21} - 2313q^{22} + 4383q^{23} + 1866q^{24} + 6532q^{25} + 3069q^{26} - 1296q^{27} - 36q^{28} - 4260q^{29} + 3924q^{30} - 4174q^{31} + 1374q^{32} - 12576q^{33} + 856q^{34} + 4752q^{35} + 1974q^{36} + 288q^{37} - 1467q^{38} + 16530q^{39} - 5054q^{40} + 11586q^{41} - 6696q^{42} - 25015q^{43} - 33132q^{44} - 38064q^{45} - 20136q^{46} - 16821q^{47} - 4362q^{48} + 25787q^{49} + 12798q^{50} + 13752q^{51} - 7079q^{52} + 66738q^{53} + 9930q^{54} + 38736q^{55} + 18468q^{56} + 9867q^{57} + 21970q^{58} - 4041q^{59} + 32748q^{60} - 95952q^{61} + 48498q^{62} - 49926q^{63} - 14461q^{64} - 121011q^{65} + 34680q^{66} - 14725q^{67} + 5712q^{68} - 28308q^{69} - 49395q^{70} + 109917q^{71} - 8202q^{72} + 106449q^{73} - 64947q^{74} + 60594q^{75} - 10338q^{76} + 83769q^{77} - 69168q^{78} + 26864q^{79} + 1629q^{80} - 37146q^{81} + 121558q^{82} - 102087q^{83} - 102192q^{84} - 246238q^{85} - 48030q^{86} - 44910q^{87} - 35481q^{88} - 102873q^{89} + 20598q^{90} - 17576q^{91} + 17868q^{92} + 48726q^{93} + 178224q^{94} + 302715q^{95} + 389712q^{96} + 114399q^{97} + 284136q^{98} + 16542q^{99} + O(q^{100})$$

## Decomposition of $$S_{5}^{\mathrm{new}}(\Gamma_1(684))$$

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
684.5.b $$\chi_{684}(683, \cdot)$$ n/a 160 1
684.5.e $$\chi_{684}(305, \cdot)$$ 684.5.e.a 24 1
684.5.g $$\chi_{684}(343, \cdot)$$ n/a 180 1
684.5.h $$\chi_{684}(37, \cdot)$$ 684.5.h.a 2 1
684.5.h.b 2
684.5.h.c 4
684.5.h.d 4
684.5.h.e 8
684.5.h.f 14
684.5.m $$\chi_{684}(353, \cdot)$$ n/a 160 2
684.5.p $$\chi_{684}(407, \cdot)$$ n/a 952 2
684.5.q $$\chi_{684}(163, \cdot)$$ n/a 396 2
684.5.s $$\chi_{684}(445, \cdot)$$ n/a 160 2
684.5.t $$\chi_{684}(265, \cdot)$$ n/a 160 2
684.5.v $$\chi_{684}(115, \cdot)$$ n/a 864 2
684.5.x $$\chi_{684}(7, \cdot)$$ n/a 952 2
684.5.y $$\chi_{684}(145, \cdot)$$ 684.5.y.a 2 2
684.5.y.b 2
684.5.y.c 12
684.5.y.d 14
684.5.y.e 14
684.5.y.f 24
684.5.ba $$\chi_{684}(107, \cdot)$$ n/a 320 2
684.5.bc $$\chi_{684}(77, \cdot)$$ n/a 144 2
684.5.be $$\chi_{684}(425, \cdot)$$ n/a 160 2
684.5.bf $$\chi_{684}(335, \cdot)$$ n/a 952 2
684.5.bh $$\chi_{684}(227, \cdot)$$ n/a 952 2
684.5.bj $$\chi_{684}(125, \cdot)$$ 684.5.bj.a 56 2
684.5.bl $$\chi_{684}(373, \cdot)$$ n/a 160 2
684.5.bm $$\chi_{684}(463, \cdot)$$ n/a 952 2
684.5.br $$\chi_{684}(155, \cdot)$$ n/a 2856 6
684.5.bu $$\chi_{684}(43, \cdot)$$ n/a 2856 6
684.5.bw $$\chi_{684}(5, \cdot)$$ n/a 480 6
684.5.bx $$\chi_{684}(109, \cdot)$$ n/a 198 6
684.5.by $$\chi_{684}(17, \cdot)$$ n/a 156 6
684.5.ca $$\chi_{684}(193, \cdot)$$ n/a 480 6
684.5.cb $$\chi_{684}(283, \cdot)$$ n/a 2856 6
684.5.cd $$\chi_{684}(71, \cdot)$$ n/a 960 6
684.5.cg $$\chi_{684}(55, \cdot)$$ n/a 1188 6
684.5.ci $$\chi_{684}(59, \cdot)$$ n/a 2856 6
684.5.cj $$\chi_{684}(13, \cdot)$$ n/a 480 6
684.5.ck $$\chi_{684}(245, \cdot)$$ n/a 480 6

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

## Decomposition of $$S_{5}^{\mathrm{old}}(\Gamma_1(684))$$ into lower level spaces

$$S_{5}^{\mathrm{old}}(\Gamma_1(684)) \cong$$ $$S_{5}^{\mathrm{new}}(\Gamma_1(4))$$$$^{\oplus 6}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 8}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 6}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 4}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 4}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(\Gamma_1(19))$$$$^{\oplus 9}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(\Gamma_1(36))$$$$^{\oplus 2}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(\Gamma_1(38))$$$$^{\oplus 6}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(\Gamma_1(57))$$$$^{\oplus 6}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(\Gamma_1(76))$$$$^{\oplus 3}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(\Gamma_1(114))$$$$^{\oplus 4}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(\Gamma_1(171))$$$$^{\oplus 3}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(\Gamma_1(228))$$$$^{\oplus 2}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(\Gamma_1(342))$$$$^{\oplus 2}$$