# Properties

 Label 315.3 Level 315 Weight 3 Dimension 4670 Nonzero newspaces 30 Newform subspaces 50 Sturm bound 20736 Trace bound 9

## Defining parameters

 Level: $$N$$ = $$315 = 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$30$$ Newform subspaces: $$50$$ Sturm bound: $$20736$$ Trace bound: $$9$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(315))$$.

Total New Old
Modular forms 7296 4938 2358
Cusp forms 6528 4670 1858
Eisenstein series 768 268 500

## Trace form

 $$4670 q - 22 q^{2} - 20 q^{3} - 30 q^{4} - 10 q^{5} - 4 q^{6} - 10 q^{7} + 30 q^{8} - 4 q^{9} + O(q^{10})$$ $$4670 q - 22 q^{2} - 20 q^{3} - 30 q^{4} - 10 q^{5} - 4 q^{6} - 10 q^{7} + 30 q^{8} - 4 q^{9} + 52 q^{10} + 52 q^{11} + 128 q^{12} + 88 q^{13} + 246 q^{14} - 20 q^{15} + 214 q^{16} + 56 q^{17} - 8 q^{18} - 44 q^{19} - 150 q^{20} - 96 q^{21} - 64 q^{22} - 220 q^{23} - 324 q^{24} + 176 q^{25} + 52 q^{26} + 52 q^{27} - 222 q^{28} + 84 q^{29} - 34 q^{30} + 164 q^{31} + 170 q^{32} + 44 q^{33} + 416 q^{34} - 26 q^{35} - 764 q^{36} - 124 q^{37} - 852 q^{38} - 680 q^{39} - 738 q^{40} - 752 q^{41} - 384 q^{42} - 288 q^{43} - 1332 q^{44} - 40 q^{45} - 876 q^{46} - 484 q^{47} - 364 q^{48} - 38 q^{49} + 782 q^{50} - 268 q^{51} + 144 q^{52} - 376 q^{53} - 736 q^{54} + 76 q^{55} - 546 q^{56} + 172 q^{57} + 916 q^{58} + 1008 q^{59} + 590 q^{60} + 976 q^{61} + 1096 q^{62} + 252 q^{63} + 118 q^{64} + 308 q^{65} + 964 q^{66} + 12 q^{67} + 1840 q^{68} + 1320 q^{69} + 150 q^{70} + 1252 q^{71} + 2544 q^{72} + 316 q^{73} + 888 q^{74} - 812 q^{75} - 216 q^{76} + 632 q^{77} - 376 q^{78} + 96 q^{79} - 3094 q^{80} - 1612 q^{81} - 1388 q^{82} - 2488 q^{83} - 1836 q^{84} - 1472 q^{85} - 5228 q^{86} - 2492 q^{87} - 3864 q^{88} - 2940 q^{89} - 2294 q^{90} - 2776 q^{91} - 2884 q^{92} - 444 q^{93} - 3328 q^{94} - 1314 q^{95} - 1184 q^{96} - 2748 q^{97} - 3258 q^{98} + 520 q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(315))$$

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 available newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
315.3.c $$\chi_{315}(71, \cdot)$$ 315.3.c.a 16 1
315.3.e $$\chi_{315}(244, \cdot)$$ 315.3.e.a 1 1
315.3.e.b 1
315.3.e.c 4
315.3.e.d 16
315.3.e.e 16
315.3.f $$\chi_{315}(134, \cdot)$$ 315.3.f.a 24 1
315.3.h $$\chi_{315}(181, \cdot)$$ 315.3.h.a 2 1
315.3.h.b 2
315.3.h.c 12
315.3.h.d 12
315.3.n $$\chi_{315}(62, \cdot)$$ 315.3.n.a 64 2
315.3.o $$\chi_{315}(127, \cdot)$$ 315.3.o.a 12 2
315.3.o.b 24
315.3.o.c 24
315.3.q $$\chi_{315}(229, \cdot)$$ 315.3.q.a 184 2
315.3.s $$\chi_{315}(11, \cdot)$$ 315.3.s.a 128 2
315.3.v $$\chi_{315}(254, \cdot)$$ 315.3.v.a 184 2
315.3.w $$\chi_{315}(136, \cdot)$$ 315.3.w.a 8 2
315.3.w.b 12
315.3.w.c 12
315.3.w.d 20
315.3.x $$\chi_{315}(76, \cdot)$$ 315.3.x.a 128 2
315.3.y $$\chi_{315}(44, \cdot)$$ 315.3.y.a 64 2
315.3.ba $$\chi_{315}(29, \cdot)$$ 315.3.ba.a 144 2
315.3.bc $$\chi_{315}(31, \cdot)$$ 315.3.bc.a 128 2
315.3.bd $$\chi_{315}(191, \cdot)$$ 315.3.bd.a 128 2
315.3.bg $$\chi_{315}(34, \cdot)$$ 315.3.bg.a 4 2
315.3.bg.b 4
315.3.bg.c 176
315.3.bi $$\chi_{315}(19, \cdot)$$ 315.3.bi.a 4 2
315.3.bi.b 4
315.3.bi.c 12
315.3.bi.d 24
315.3.bi.e 32
315.3.bk $$\chi_{315}(176, \cdot)$$ 315.3.bk.a 96 2
315.3.bm $$\chi_{315}(116, \cdot)$$ 315.3.bm.a 40 2
315.3.bn $$\chi_{315}(94, \cdot)$$ 315.3.bn.a 184 2
315.3.bp $$\chi_{315}(166, \cdot)$$ 315.3.bp.a 128 2
315.3.br $$\chi_{315}(74, \cdot)$$ 315.3.br.a 184 2
315.3.bt $$\chi_{315}(58, \cdot)$$ 315.3.bt.a 368 4
315.3.bu $$\chi_{315}(38, \cdot)$$ 315.3.bu.a 368 4
315.3.bw $$\chi_{315}(47, \cdot)$$ 315.3.bw.a 368 4
315.3.by $$\chi_{315}(22, \cdot)$$ 315.3.by.a 288 4
315.3.ca $$\chi_{315}(37, \cdot)$$ 315.3.ca.a 24 4
315.3.ca.b 64
315.3.ca.c 64
315.3.cd $$\chi_{315}(17, \cdot)$$ 315.3.cd.a 128 4
315.3.cf $$\chi_{315}(83, \cdot)$$ 315.3.cf.a 368 4
315.3.ch $$\chi_{315}(67, \cdot)$$ 315.3.ch.a 368 4

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(315))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(315)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(1))$$$$^{\oplus 12}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(3))$$$$^{\oplus 8}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(7))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(21))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(35))$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(45))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(63))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(105))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(315))$$$$^{\oplus 1}$$