## Defining parameters

 Level: $$N$$ = $$630 = 2 \cdot 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$30$$ Sturm bound: $$82944$$ Trace bound: $$15$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(630))$$.

Total New Old
Modular forms 31872 7210 24662
Cusp forms 30336 7210 23126
Eisenstein series 1536 0 1536

## Trace form

 $$7210 q + 12 q^{2} + 12 q^{3} - 24 q^{4} + 58 q^{5} - 72 q^{6} + 92 q^{7} - 48 q^{8} - 164 q^{9} + O(q^{10})$$ $$7210 q + 12 q^{2} + 12 q^{3} - 24 q^{4} + 58 q^{5} - 72 q^{6} + 92 q^{7} - 48 q^{8} - 164 q^{9} + 32 q^{10} - 16 q^{11} - 32 q^{12} - 152 q^{13} + 592 q^{14} - 252 q^{15} - 224 q^{16} - 1560 q^{17} - 112 q^{18} - 632 q^{19} + 128 q^{20} - 720 q^{21} - 168 q^{22} - 732 q^{23} - 288 q^{24} + 692 q^{25} + 48 q^{26} + 480 q^{27} + 224 q^{28} - 108 q^{29} + 264 q^{30} + 1380 q^{31} + 192 q^{32} + 916 q^{33} + 832 q^{34} - 2084 q^{35} + 1168 q^{36} + 616 q^{37} + 672 q^{38} - 656 q^{39} - 384 q^{40} - 1648 q^{41} + 800 q^{42} - 3332 q^{43} - 272 q^{44} + 4244 q^{45} - 736 q^{46} + 5676 q^{47} + 448 q^{48} + 7146 q^{49} - 812 q^{50} - 6492 q^{51} + 2176 q^{52} + 864 q^{53} - 2040 q^{54} + 5644 q^{55} - 640 q^{56} - 2356 q^{57} - 696 q^{58} + 5900 q^{59} + 4336 q^{60} - 1372 q^{61} + 5616 q^{62} + 18436 q^{63} + 1152 q^{64} + 5328 q^{65} + 10560 q^{66} + 2800 q^{67} + 2544 q^{68} + 9392 q^{69} - 3132 q^{70} - 11520 q^{71} - 96 q^{72} - 2480 q^{73} - 12424 q^{74} - 21156 q^{75} - 6944 q^{76} - 28980 q^{77} - 21520 q^{78} - 14540 q^{79} - 800 q^{80} - 3532 q^{81} - 10584 q^{82} - 1932 q^{83} + 1872 q^{84} + 1564 q^{85} + 7576 q^{86} - 2936 q^{87} - 2208 q^{88} + 9424 q^{89} - 1376 q^{90} + 5500 q^{91} + 1152 q^{92} + 7304 q^{93} + 7664 q^{94} + 10026 q^{95} + 1792 q^{96} + 10552 q^{97} + 15828 q^{98} + 16720 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(630))$$

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
630.4.a $$\chi_{630}(1, \cdot)$$ 630.4.a.a 1 1
630.4.a.b 1
630.4.a.c 1
630.4.a.d 1
630.4.a.e 1
630.4.a.f 1
630.4.a.g 1
630.4.a.h 1
630.4.a.i 1
630.4.a.j 1
630.4.a.k 1
630.4.a.l 1
630.4.a.m 1
630.4.a.n 1
630.4.a.o 1
630.4.a.p 1
630.4.a.q 1
630.4.a.r 1
630.4.a.s 1
630.4.a.t 1
630.4.a.u 1
630.4.a.v 1
630.4.a.w 1
630.4.a.x 1
630.4.a.y 2
630.4.a.z 2
630.4.a.ba 2
630.4.b $$\chi_{630}(251, \cdot)$$ 630.4.b.a 16 1
630.4.b.b 16
630.4.d $$\chi_{630}(629, \cdot)$$ 630.4.d.a 24 1
630.4.d.b 24
630.4.g $$\chi_{630}(379, \cdot)$$ 630.4.g.a 2 1
630.4.g.b 2
630.4.g.c 2
630.4.g.d 4
630.4.g.e 4
630.4.g.f 6
630.4.g.g 6
630.4.g.h 6
630.4.g.i 6
630.4.g.j 6
630.4.i $$\chi_{630}(121, \cdot)$$ n/a 192 2
630.4.j $$\chi_{630}(211, \cdot)$$ n/a 144 2
630.4.k $$\chi_{630}(361, \cdot)$$ 630.4.k.a 2 2
630.4.k.b 2
630.4.k.c 2
630.4.k.d 2
630.4.k.e 2
630.4.k.f 2
630.4.k.g 2
630.4.k.h 2
630.4.k.i 2
630.4.k.j 2
630.4.k.k 4
630.4.k.l 4
630.4.k.m 4
630.4.k.n 4
630.4.k.o 6
630.4.k.p 6
630.4.k.q 6
630.4.k.r 6
630.4.k.s 10
630.4.k.t 10
630.4.l $$\chi_{630}(331, \cdot)$$ n/a 192 2
630.4.m $$\chi_{630}(197, \cdot)$$ 630.4.m.a 16 2
630.4.m.b 16
630.4.m.c 20
630.4.m.d 20
630.4.p $$\chi_{630}(307, \cdot)$$ n/a 120 2
630.4.r $$\chi_{630}(59, \cdot)$$ n/a 288 2
630.4.t $$\chi_{630}(311, \cdot)$$ n/a 192 2
630.4.u $$\chi_{630}(109, \cdot)$$ n/a 120 2
630.4.z $$\chi_{630}(169, \cdot)$$ n/a 216 2
630.4.ba $$\chi_{630}(499, \cdot)$$ n/a 288 2
630.4.be $$\chi_{630}(341, \cdot)$$ 630.4.be.a 32 2
630.4.be.b 32
630.4.bf $$\chi_{630}(209, \cdot)$$ n/a 288 2
630.4.bi $$\chi_{630}(479, \cdot)$$ n/a 288 2
630.4.bk $$\chi_{630}(101, \cdot)$$ n/a 192 2
630.4.bl $$\chi_{630}(41, \cdot)$$ n/a 192 2
630.4.bo $$\chi_{630}(89, \cdot)$$ 630.4.bo.a 48 2
630.4.bo.b 48
630.4.bq $$\chi_{630}(79, \cdot)$$ n/a 288 2
630.4.bt $$\chi_{630}(317, \cdot)$$ n/a 576 4
630.4.bv $$\chi_{630}(73, \cdot)$$ n/a 240 4
630.4.bw $$\chi_{630}(103, \cdot)$$ n/a 576 4
630.4.bz $$\chi_{630}(13, \cdot)$$ n/a 576 4
630.4.ca $$\chi_{630}(113, \cdot)$$ n/a 432 4
630.4.cd $$\chi_{630}(23, \cdot)$$ n/a 576 4
630.4.ce $$\chi_{630}(53, \cdot)$$ n/a 192 4
630.4.cg $$\chi_{630}(157, \cdot)$$ n/a 576 4

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

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(630))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(630)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(7))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(21))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(35))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(42))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(45))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(63))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(70))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(90))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(105))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(126))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(210))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(315))$$$$^{\oplus 2}$$