## Defining parameters

 Level: $$N$$ = $$684 = 2^{2} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$32$$ Newform subspaces: $$65$$ Sturm bound: $$51840$$ Trace bound: $$10$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(684))$$.

Total New Old
Modular forms 13680 6105 7575
Cusp forms 12241 5797 6444
Eisenstein series 1439 308 1131

## Trace form

 $$5797q - 21q^{2} - 17q^{4} - 36q^{5} - 30q^{6} + 6q^{7} - 27q^{8} - 48q^{9} + O(q^{10})$$ $$5797q - 21q^{2} - 17q^{4} - 36q^{5} - 30q^{6} + 6q^{7} - 27q^{8} - 48q^{9} - 73q^{10} + 6q^{11} - 48q^{12} - 52q^{13} - 51q^{14} - 18q^{15} - 41q^{16} - 87q^{17} - 72q^{18} - 21q^{19} - 90q^{20} - 66q^{21} - 33q^{22} - 15q^{23} - 42q^{24} - 54q^{25} - 27q^{26} - 30q^{28} - 42q^{29} + 36q^{31} + 84q^{32} - 102q^{33} + 44q^{34} + 48q^{35} + 30q^{36} - 106q^{37} + 63q^{38} + 6q^{39} + 32q^{40} - 78q^{41} + 27q^{43} + 18q^{44} - 48q^{45} - 60q^{46} + 81q^{47} - 78q^{48} - 11q^{49} - 42q^{50} + 90q^{51} - 91q^{52} + 78q^{53} - 114q^{54} + 72q^{55} - 90q^{56} + 12q^{57} - 118q^{58} + 129q^{59} - 48q^{60} + 86q^{61} - 72q^{62} + 78q^{63} - 173q^{64} + 189q^{65} + 12q^{66} + 117q^{67} - 60q^{68} + 60q^{69} - 123q^{70} + 147q^{71} + 6q^{72} + 44q^{73} - 93q^{74} + 24q^{75} - 114q^{76} + 63q^{77} - 12q^{78} + 54q^{79} - 99q^{80} - 48q^{81} - 244q^{82} - 27q^{83} - 96q^{84} + 34q^{85} - 132q^{86} + 18q^{87} - 57q^{88} - 51q^{89} - 138q^{90} - 120q^{92} - 162q^{93} - 54q^{94} - 147q^{95} - 240q^{96} - 235q^{97} - 324q^{98} - 144q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(684))$$

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
684.2.a $$\chi_{684}(1, \cdot)$$ 684.2.a.a 1 1
684.2.a.b 1
684.2.a.c 1
684.2.a.d 2
684.2.a.e 2
684.2.c $$\chi_{684}(647, \cdot)$$ 684.2.c.a 4 1
684.2.c.b 32
684.2.d $$\chi_{684}(341, \cdot)$$ 684.2.d.a 8 1
684.2.f $$\chi_{684}(379, \cdot)$$ 684.2.f.a 4 1
684.2.f.b 8
684.2.f.c 10
684.2.f.d 10
684.2.f.e 16
684.2.i $$\chi_{684}(229, \cdot)$$ 684.2.i.a 2 2
684.2.i.b 2
684.2.i.c 16
684.2.i.d 16
684.2.j $$\chi_{684}(49, \cdot)$$ 684.2.j.a 40 2
684.2.k $$\chi_{684}(505, \cdot)$$ 684.2.k.a 2 2
684.2.k.b 2
684.2.k.c 2
684.2.k.d 2
684.2.k.e 2
684.2.k.f 4
684.2.k.g 4
684.2.l $$\chi_{684}(121, \cdot)$$ 684.2.l.a 40 2
684.2.n $$\chi_{684}(65, \cdot)$$ 684.2.n.a 2 2
684.2.n.b 38
684.2.o $$\chi_{684}(11, \cdot)$$ 684.2.o.a 232 2
684.2.r $$\chi_{684}(487, \cdot)$$ 684.2.r.a 16 2
684.2.r.b 20
684.2.r.c 20
684.2.r.d 40
684.2.u $$\chi_{684}(103, \cdot)$$ 684.2.u.a 232 2
684.2.w $$\chi_{684}(151, \cdot)$$ 684.2.w.a 232 2
684.2.z $$\chi_{684}(467, \cdot)$$ 684.2.z.a 8 2
684.2.z.b 72
684.2.bb $$\chi_{684}(293, \cdot)$$ 684.2.bb.a 2 2
684.2.bb.b 38
684.2.bd $$\chi_{684}(113, \cdot)$$ 684.2.bd.a 40 2
684.2.bg $$\chi_{684}(191, \cdot)$$ 684.2.bg.a 4 2
684.2.bg.b 4
684.2.bg.c 208
684.2.bi $$\chi_{684}(83, \cdot)$$ 684.2.bi.a 232 2
684.2.bk $$\chi_{684}(449, \cdot)$$ 684.2.bk.a 16 2
684.2.bn $$\chi_{684}(31, \cdot)$$ 684.2.bn.a 232 2
684.2.bo $$\chi_{684}(73, \cdot)$$ 684.2.bo.a 6 6
684.2.bo.b 6
684.2.bo.c 12
684.2.bo.d 12
684.2.bo.e 12
684.2.bp $$\chi_{684}(25, \cdot)$$ 684.2.bp.a 120 6
684.2.bq $$\chi_{684}(85, \cdot)$$ 684.2.bq.a 120 6
684.2.bs $$\chi_{684}(23, \cdot)$$ 684.2.bs.a 696 6
684.2.bt $$\chi_{684}(67, \cdot)$$ 684.2.bt.a 696 6
684.2.bv $$\chi_{684}(29, \cdot)$$ 684.2.bv.a 120 6
684.2.bz $$\chi_{684}(53, \cdot)$$ 684.2.bz.a 36 6
684.2.cc $$\chi_{684}(211, \cdot)$$ 684.2.cc.a 696 6
684.2.ce $$\chi_{684}(35, \cdot)$$ 684.2.ce.a 240 6
684.2.cf $$\chi_{684}(91, \cdot)$$ 684.2.cf.a 48 6
684.2.cf.b 60
684.2.cf.c 60
684.2.cf.d 120
684.2.ch $$\chi_{684}(47, \cdot)$$ 684.2.ch.a 696 6
684.2.cl $$\chi_{684}(173, \cdot)$$ 684.2.cl.a 120 6

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(684))$$ into lower level spaces

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