# Properties

 Label 630.2 Level 630 Weight 2 Dimension 2402 Nonzero newspaces 30 Newform subspaces 118 Sturm bound 41472 Trace bound 15

## Defining parameters

 Level: $$N$$ = $$630 = 2 \cdot 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$30$$ Newform subspaces: $$118$$ Sturm bound: $$41472$$ Trace bound: $$15$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(630))$$.

Total New Old
Modular forms 11136 2402 8734
Cusp forms 9601 2402 7199
Eisenstein series 1535 0 1535

## Trace form

 $$2402 q - 6 q^{2} - 12 q^{3} - 10 q^{4} - 16 q^{5} + 12 q^{6} - 28 q^{7} + 6 q^{8} + 28 q^{9} + O(q^{10})$$ $$2402 q - 6 q^{2} - 12 q^{3} - 10 q^{4} - 16 q^{5} + 12 q^{6} - 28 q^{7} + 6 q^{8} + 28 q^{9} + 16 q^{10} + 28 q^{11} + 16 q^{12} + 24 q^{13} + 56 q^{14} + 72 q^{15} + 6 q^{16} + 132 q^{17} + 56 q^{18} + 84 q^{19} + 40 q^{20} + 108 q^{21} + 92 q^{22} + 168 q^{23} + 12 q^{24} + 58 q^{25} + 72 q^{26} + 120 q^{27} + 24 q^{28} + 108 q^{29} + 12 q^{30} + 72 q^{31} - 6 q^{32} + 28 q^{33} + 16 q^{34} + 86 q^{35} - 20 q^{36} + 140 q^{37} - 72 q^{38} - 56 q^{39} - 8 q^{41} - 64 q^{42} + 20 q^{43} - 64 q^{44} - 112 q^{45} - 24 q^{46} - 24 q^{47} - 20 q^{48} + 54 q^{49} - 154 q^{50} - 132 q^{51} - 204 q^{53} - 204 q^{54} + 64 q^{55} - 32 q^{56} - 244 q^{57} - 20 q^{58} - 200 q^{59} - 56 q^{60} + 88 q^{61} - 168 q^{62} - 296 q^{63} + 2 q^{64} - 12 q^{65} - 96 q^{66} + 204 q^{67} - 48 q^{68} - 88 q^{69} + 42 q^{70} + 48 q^{71} + 12 q^{72} + 172 q^{73} - 44 q^{74} - 96 q^{75} + 88 q^{76} - 60 q^{77} - 40 q^{78} + 136 q^{79} - 16 q^{80} + 20 q^{81} + 44 q^{82} - 36 q^{83} + 12 q^{84} + 48 q^{85} + 44 q^{86} + 40 q^{87} - 28 q^{88} - 148 q^{89} - 32 q^{90} - 108 q^{91} - 24 q^{92} - 184 q^{93} - 160 q^{94} - 456 q^{95} - 8 q^{96} - 216 q^{97} - 234 q^{98} - 104 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(630))$$

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
630.2.a $$\chi_{630}(1, \cdot)$$ 630.2.a.a 1 1
630.2.a.b 1
630.2.a.c 1
630.2.a.d 1
630.2.a.e 1
630.2.a.f 1
630.2.a.g 1
630.2.a.h 1
630.2.a.i 1
630.2.a.j 1
630.2.b $$\chi_{630}(251, \cdot)$$ 630.2.b.a 8 1
630.2.b.b 8
630.2.d $$\chi_{630}(629, \cdot)$$ 630.2.d.a 4 1
630.2.d.b 4
630.2.d.c 4
630.2.d.d 4
630.2.g $$\chi_{630}(379, \cdot)$$ 630.2.g.a 2 1
630.2.g.b 2
630.2.g.c 2
630.2.g.d 2
630.2.g.e 2
630.2.g.f 2
630.2.g.g 4
630.2.i $$\chi_{630}(121, \cdot)$$ 630.2.i.a 2 2
630.2.i.b 2
630.2.i.c 2
630.2.i.d 2
630.2.i.e 4
630.2.i.f 12
630.2.i.g 12
630.2.i.h 12
630.2.i.i 16
630.2.j $$\chi_{630}(211, \cdot)$$ 630.2.j.a 2 2
630.2.j.b 2
630.2.j.c 2
630.2.j.d 2
630.2.j.e 2
630.2.j.f 4
630.2.j.g 4
630.2.j.h 4
630.2.j.i 6
630.2.j.j 6
630.2.j.k 6
630.2.j.l 8
630.2.k $$\chi_{630}(361, \cdot)$$ 630.2.k.a 2 2
630.2.k.b 2
630.2.k.c 2
630.2.k.d 2
630.2.k.e 2
630.2.k.f 2
630.2.k.g 2
630.2.k.h 2
630.2.k.i 4
630.2.k.j 4
630.2.l $$\chi_{630}(331, \cdot)$$ 630.2.l.a 2 2
630.2.l.b 2
630.2.l.c 2
630.2.l.d 2
630.2.l.e 4
630.2.l.f 12
630.2.l.g 12
630.2.l.h 12
630.2.l.i 16
630.2.m $$\chi_{630}(197, \cdot)$$ 630.2.m.a 4 2
630.2.m.b 4
630.2.m.c 8
630.2.m.d 8
630.2.p $$\chi_{630}(307, \cdot)$$ 630.2.p.a 8 2
630.2.p.b 8
630.2.p.c 8
630.2.p.d 16
630.2.r $$\chi_{630}(59, \cdot)$$ 630.2.r.a 48 2
630.2.r.b 48
630.2.t $$\chi_{630}(311, \cdot)$$ 630.2.t.a 4 2
630.2.t.b 28
630.2.t.c 32
630.2.u $$\chi_{630}(109, \cdot)$$ 630.2.u.a 4 2
630.2.u.b 4
630.2.u.c 4
630.2.u.d 8
630.2.u.e 8
630.2.u.f 12
630.2.z $$\chi_{630}(169, \cdot)$$ 630.2.z.a 4 2
630.2.z.b 24
630.2.z.c 44
630.2.ba $$\chi_{630}(499, \cdot)$$ 630.2.ba.a 96 2
630.2.be $$\chi_{630}(341, \cdot)$$ 630.2.be.a 8 2
630.2.be.b 8
630.2.bf $$\chi_{630}(209, \cdot)$$ 630.2.bf.a 8 2
630.2.bf.b 8
630.2.bf.c 8
630.2.bf.d 8
630.2.bf.e 32
630.2.bf.f 32
630.2.bi $$\chi_{630}(479, \cdot)$$ 630.2.bi.a 48 2
630.2.bi.b 48
630.2.bk $$\chi_{630}(101, \cdot)$$ 630.2.bk.a 4 2
630.2.bk.b 28
630.2.bk.c 32
630.2.bl $$\chi_{630}(41, \cdot)$$ 630.2.bl.a 32 2
630.2.bl.b 32
630.2.bo $$\chi_{630}(89, \cdot)$$ 630.2.bo.a 16 2
630.2.bo.b 16
630.2.bq $$\chi_{630}(79, \cdot)$$ 630.2.bq.a 96 2
630.2.bt $$\chi_{630}(317, \cdot)$$ 630.2.bt.a 192 4
630.2.bv $$\chi_{630}(73, \cdot)$$ 630.2.bv.a 16 4
630.2.bv.b 16
630.2.bv.c 16
630.2.bv.d 32
630.2.bw $$\chi_{630}(103, \cdot)$$ 630.2.bw.a 192 4
630.2.bz $$\chi_{630}(13, \cdot)$$ 630.2.bz.a 192 4
630.2.ca $$\chi_{630}(113, \cdot)$$ 630.2.ca.a 72 4
630.2.ca.b 72
630.2.cd $$\chi_{630}(23, \cdot)$$ 630.2.cd.a 192 4
630.2.ce $$\chi_{630}(53, \cdot)$$ 630.2.ce.a 16 4
630.2.ce.b 16
630.2.ce.c 32
630.2.cg $$\chi_{630}(157, \cdot)$$ 630.2.cg.a 192 4

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(630))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(630)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(21))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(35))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(42))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(45))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(63))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(70))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(90))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(105))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(126))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(210))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(315))$$$$^{\oplus 2}$$