# Properties

 Label 315.4 Level 315 Weight 4 Dimension 7076 Nonzero newspaces 30 Sturm bound 27648 Trace bound 9

## Defining parameters

 Level: $$N$$ = $$315 = 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$30$$ Sturm bound: $$27648$$ Trace bound: $$9$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(315))$$.

Total New Old
Modular forms 10752 7348 3404
Cusp forms 9984 7076 2908
Eisenstein series 768 272 496

## Trace form

 $$7076 q - 14 q^{2} - 20 q^{3} - 78 q^{4} - 51 q^{5} - 4 q^{6} - 42 q^{7} + 162 q^{8} + 44 q^{9} + O(q^{10})$$ $$7076 q - 14 q^{2} - 20 q^{3} - 78 q^{4} - 51 q^{5} - 4 q^{6} - 42 q^{7} + 162 q^{8} + 44 q^{9} - 230 q^{10} - 302 q^{11} - 472 q^{12} - 80 q^{13} - 774 q^{14} + 214 q^{15} + 1022 q^{16} + 1534 q^{17} + 1528 q^{18} + 514 q^{19} - 370 q^{20} + 138 q^{21} - 972 q^{22} - 1158 q^{23} - 396 q^{24} - 133 q^{25} - 2348 q^{26} - 1424 q^{27} - 1726 q^{28} + 620 q^{29} + 1286 q^{30} - 274 q^{31} + 470 q^{32} + 1520 q^{33} - 1164 q^{34} + 1729 q^{35} + 2164 q^{36} + 1130 q^{37} + 5980 q^{38} + 2476 q^{39} - 118 q^{40} + 956 q^{41} + 1020 q^{42} + 1060 q^{43} - 1648 q^{44} - 4498 q^{45} - 596 q^{46} - 7790 q^{47} - 4516 q^{48} - 3742 q^{49} - 4754 q^{50} + 1700 q^{51} - 8224 q^{52} - 1874 q^{53} + 6728 q^{54} - 1826 q^{55} - 8334 q^{56} + 2476 q^{57} + 7436 q^{58} - 1022 q^{59} - 2518 q^{60} + 5802 q^{61} - 936 q^{62} - 8130 q^{63} + 4846 q^{64} + 2686 q^{65} - 14420 q^{66} - 1998 q^{67} - 12964 q^{68} - 8004 q^{69} + 1566 q^{70} + 3536 q^{71} + 11856 q^{72} - 9686 q^{73} + 11756 q^{74} + 14746 q^{75} + 8164 q^{76} + 21480 q^{77} + 34112 q^{78} + 14118 q^{79} + 22274 q^{80} + 11012 q^{81} + 34600 q^{82} + 9324 q^{83} + 10932 q^{84} + 7330 q^{85} + 2320 q^{86} - 4748 q^{87} + 2172 q^{88} - 9330 q^{89} - 23402 q^{90} - 912 q^{91} - 24456 q^{92} - 18228 q^{93} - 6708 q^{94} - 11327 q^{95} - 26864 q^{96} - 18688 q^{97} - 48346 q^{98} - 21956 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(315))$$

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 available newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
315.4.a $$\chi_{315}(1, \cdot)$$ 315.4.a.a 1 1
315.4.a.b 1
315.4.a.c 1
315.4.a.d 1
315.4.a.e 1
315.4.a.f 2
315.4.a.g 2
315.4.a.h 2
315.4.a.i 2
315.4.a.j 2
315.4.a.k 2
315.4.a.l 2
315.4.a.m 2
315.4.a.n 3
315.4.a.o 3
315.4.a.p 3
315.4.b $$\chi_{315}(251, \cdot)$$ 315.4.b.a 16 1
315.4.b.b 16
315.4.d $$\chi_{315}(64, \cdot)$$ 315.4.d.a 6 1
315.4.d.b 10
315.4.d.c 10
315.4.d.d 20
315.4.g $$\chi_{315}(314, \cdot)$$ 315.4.g.a 48 1
315.4.i $$\chi_{315}(106, \cdot)$$ n/a 144 2
315.4.j $$\chi_{315}(46, \cdot)$$ 315.4.j.a 2 2
315.4.j.b 2
315.4.j.c 4
315.4.j.d 4
315.4.j.e 6
315.4.j.f 10
315.4.j.g 10
315.4.j.h 10
315.4.j.i 16
315.4.j.j 16
315.4.k $$\chi_{315}(16, \cdot)$$ n/a 192 2
315.4.l $$\chi_{315}(121, \cdot)$$ n/a 192 2
315.4.m $$\chi_{315}(8, \cdot)$$ 315.4.m.a 36 2
315.4.m.b 36
315.4.p $$\chi_{315}(118, \cdot)$$ n/a 116 2
315.4.r $$\chi_{315}(184, \cdot)$$ n/a 280 2
315.4.t $$\chi_{315}(101, \cdot)$$ n/a 192 2
315.4.u $$\chi_{315}(59, \cdot)$$ n/a 280 2
315.4.z $$\chi_{315}(104, \cdot)$$ n/a 280 2
315.4.bb $$\chi_{315}(89, \cdot)$$ 315.4.bb.a 96 2
315.4.be $$\chi_{315}(236, \cdot)$$ n/a 192 2
315.4.bf $$\chi_{315}(109, \cdot)$$ n/a 116 2
315.4.bh $$\chi_{315}(169, \cdot)$$ n/a 216 2
315.4.bj $$\chi_{315}(26, \cdot)$$ 315.4.bj.a 32 2
315.4.bj.b 32
315.4.bl $$\chi_{315}(41, \cdot)$$ n/a 192 2
315.4.bo $$\chi_{315}(4, \cdot)$$ n/a 280 2
315.4.bq $$\chi_{315}(164, \cdot)$$ n/a 280 2
315.4.bs $$\chi_{315}(52, \cdot)$$ n/a 560 4
315.4.bv $$\chi_{315}(23, \cdot)$$ n/a 560 4
315.4.bx $$\chi_{315}(2, \cdot)$$ n/a 560 4
315.4.bz $$\chi_{315}(73, \cdot)$$ n/a 232 4
315.4.cb $$\chi_{315}(13, \cdot)$$ n/a 560 4
315.4.cc $$\chi_{315}(92, \cdot)$$ n/a 432 4
315.4.ce $$\chi_{315}(53, \cdot)$$ n/a 192 4
315.4.cg $$\chi_{315}(157, \cdot)$$ n/a 560 4

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

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(315))$$ into lower level spaces

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