# Properties

 Label 315.2 Level 315 Weight 2 Dimension 2268 Nonzero newspaces 30 Newform subspaces 93 Sturm bound 13824 Trace bound 9

## Defining parameters

 Level: $$N$$ = $$315 = 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$30$$ Newform subspaces: $$93$$ Sturm bound: $$13824$$ Trace bound: $$9$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(315))$$.

Total New Old
Modular forms 3840 2540 1300
Cusp forms 3073 2268 805
Eisenstein series 767 272 495

## Trace form

 $$2268 q - 4 q^{2} - 8 q^{3} + 12 q^{4} - 6 q^{5} - 40 q^{6} + 2 q^{7} - 12 q^{8} - 16 q^{9} + O(q^{10})$$ $$2268 q - 4 q^{2} - 8 q^{3} + 12 q^{4} - 6 q^{5} - 40 q^{6} + 2 q^{7} - 12 q^{8} - 16 q^{9} - 36 q^{10} - 28 q^{11} - 64 q^{12} - 42 q^{14} - 92 q^{15} - 64 q^{16} - 64 q^{17} - 104 q^{18} - 36 q^{19} - 122 q^{20} - 108 q^{21} - 88 q^{22} - 72 q^{23} - 168 q^{24} - 30 q^{25} - 172 q^{26} - 92 q^{27} - 98 q^{28} - 104 q^{29} - 142 q^{30} - 60 q^{31} - 212 q^{32} - 88 q^{33} - 148 q^{34} - 88 q^{35} - 200 q^{36} - 120 q^{37} - 160 q^{38} - 32 q^{39} - 110 q^{40} - 116 q^{41} - 120 q^{42} - 16 q^{43} - 80 q^{44} - 28 q^{45} - 208 q^{46} - 4 q^{47} + 80 q^{48} - 52 q^{49} - 16 q^{50} - 40 q^{51} - 104 q^{52} + 44 q^{53} + 44 q^{54} - 108 q^{55} + 30 q^{56} + 16 q^{57} - 112 q^{58} + 20 q^{59} + 26 q^{60} - 152 q^{61} + 84 q^{63} - 204 q^{64} - 160 q^{65} - 164 q^{66} - 156 q^{67} + 4 q^{68} - 96 q^{69} - 300 q^{70} - 296 q^{71} + 36 q^{72} - 228 q^{73} - 188 q^{74} - 32 q^{75} - 364 q^{76} - 168 q^{77} - 40 q^{78} - 180 q^{79} + 34 q^{80} - 160 q^{81} - 204 q^{82} - 72 q^{83} - 96 q^{84} - 132 q^{85} - 100 q^{86} - 44 q^{87} - 72 q^{88} + 12 q^{89} + 202 q^{90} - 100 q^{91} + 288 q^{92} + 36 q^{93} + 80 q^{94} + 218 q^{95} + 16 q^{96} + 64 q^{97} + 268 q^{98} - 8 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(315))$$

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
315.2.a $$\chi_{315}(1, \cdot)$$ 315.2.a.a 1 1
315.2.a.b 1
315.2.a.c 2
315.2.a.d 2
315.2.a.e 2
315.2.a.f 2
315.2.b $$\chi_{315}(251, \cdot)$$ 315.2.b.a 4 1
315.2.b.b 4
315.2.d $$\chi_{315}(64, \cdot)$$ 315.2.d.a 2 1
315.2.d.b 2
315.2.d.c 2
315.2.d.d 2
315.2.d.e 6
315.2.g $$\chi_{315}(314, \cdot)$$ 315.2.g.a 16 1
315.2.i $$\chi_{315}(106, \cdot)$$ 315.2.i.a 2 2
315.2.i.b 2
315.2.i.c 8
315.2.i.d 8
315.2.i.e 12
315.2.i.f 16
315.2.j $$\chi_{315}(46, \cdot)$$ 315.2.j.a 2 2
315.2.j.b 2
315.2.j.c 4
315.2.j.d 4
315.2.j.e 4
315.2.j.f 6
315.2.j.g 6
315.2.k $$\chi_{315}(16, \cdot)$$ 315.2.k.a 4 2
315.2.k.b 24
315.2.k.c 36
315.2.l $$\chi_{315}(121, \cdot)$$ 315.2.l.a 4 2
315.2.l.b 24
315.2.l.c 36
315.2.m $$\chi_{315}(8, \cdot)$$ 315.2.m.a 12 2
315.2.m.b 12
315.2.p $$\chi_{315}(118, \cdot)$$ 315.2.p.a 4 2
315.2.p.b 4
315.2.p.c 4
315.2.p.d 8
315.2.p.e 16
315.2.r $$\chi_{315}(184, \cdot)$$ 315.2.r.a 4 2
315.2.r.b 84
315.2.t $$\chi_{315}(101, \cdot)$$ 315.2.t.a 2 2
315.2.t.b 30
315.2.t.c 32
315.2.u $$\chi_{315}(59, \cdot)$$ 315.2.u.a 88 2
315.2.z $$\chi_{315}(104, \cdot)$$ 315.2.z.a 8 2
315.2.z.b 80
315.2.bb $$\chi_{315}(89, \cdot)$$ 315.2.bb.a 8 2
315.2.bb.b 24
315.2.be $$\chi_{315}(236, \cdot)$$ 315.2.be.a 2 2
315.2.be.b 30
315.2.be.c 32
315.2.bf $$\chi_{315}(109, \cdot)$$ 315.2.bf.a 4 2
315.2.bf.b 16
315.2.bf.c 16
315.2.bh $$\chi_{315}(169, \cdot)$$ 315.2.bh.a 4 2
315.2.bh.b 4
315.2.bh.c 64
315.2.bj $$\chi_{315}(26, \cdot)$$ 315.2.bj.a 12 2
315.2.bj.b 12
315.2.bl $$\chi_{315}(41, \cdot)$$ 315.2.bl.a 2 2
315.2.bl.b 2
315.2.bl.c 2
315.2.bl.d 2
315.2.bl.e 2
315.2.bl.f 2
315.2.bl.g 2
315.2.bl.h 2
315.2.bl.i 24
315.2.bl.j 24
315.2.bo $$\chi_{315}(4, \cdot)$$ 315.2.bo.a 4 2
315.2.bo.b 84
315.2.bq $$\chi_{315}(164, \cdot)$$ 315.2.bq.a 88 2
315.2.bs $$\chi_{315}(52, \cdot)$$ 315.2.bs.a 4 4
315.2.bs.b 4
315.2.bs.c 4
315.2.bs.d 4
315.2.bs.e 160
315.2.bv $$\chi_{315}(23, \cdot)$$ 315.2.bv.a 176 4
315.2.bx $$\chi_{315}(2, \cdot)$$ 315.2.bx.a 176 4
315.2.bz $$\chi_{315}(73, \cdot)$$ 315.2.bz.a 4 4
315.2.bz.b 4
315.2.bz.c 32
315.2.bz.d 32
315.2.cb $$\chi_{315}(13, \cdot)$$ 315.2.cb.a 176 4
315.2.cc $$\chi_{315}(92, \cdot)$$ 315.2.cc.a 144 4
315.2.ce $$\chi_{315}(53, \cdot)$$ 315.2.ce.a 64 4
315.2.cg $$\chi_{315}(157, \cdot)$$ 315.2.cg.a 4 4
315.2.cg.b 4
315.2.cg.c 4
315.2.cg.d 4
315.2.cg.e 160

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(315))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(315)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(21))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(35))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(45))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(63))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(105))$$$$^{\oplus 2}$$