# Properties

 Label 315.5 Level 315 Weight 5 Dimension 9482 Nonzero newspaces 30 Sturm bound 34560 Trace bound 9

## Defining parameters

 Level: $$N$$ = $$315 = 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ = $$5$$ Nonzero newspaces: $$30$$ Sturm bound: $$34560$$ Trace bound: $$9$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{5}(\Gamma_1(315))$$.

Total New Old
Modular forms 14208 9750 4458
Cusp forms 13440 9482 3958
Eisenstein series 768 268 500

## Trace form

 $$9482 q - 18 q^{2} - 20 q^{3} + 98 q^{4} - 96 q^{5} - 436 q^{6} - 154 q^{7} - 210 q^{8} + 668 q^{9} + O(q^{10})$$ $$9482 q - 18 q^{2} - 20 q^{3} + 98 q^{4} - 96 q^{5} - 436 q^{6} - 154 q^{7} - 210 q^{8} + 668 q^{9} + 92 q^{10} + 1200 q^{11} + 224 q^{12} - 1004 q^{13} - 1218 q^{14} - 1712 q^{15} - 3410 q^{16} - 1320 q^{17} + 5224 q^{18} + 2144 q^{19} + 8010 q^{20} + 3396 q^{21} + 5616 q^{22} - 480 q^{23} - 5076 q^{24} - 7330 q^{25} - 5556 q^{26} - 3332 q^{27} - 9222 q^{28} - 16260 q^{29} + 1094 q^{30} + 9336 q^{31} + 15858 q^{32} + 260 q^{33} + 18560 q^{34} - 888 q^{35} - 21404 q^{36} - 20988 q^{37} - 21540 q^{38} + 11392 q^{39} - 21074 q^{40} + 25992 q^{41} + 35544 q^{42} + 16640 q^{43} + 45396 q^{44} + 3644 q^{45} + 59508 q^{46} + 21216 q^{47} - 55036 q^{48} + 43118 q^{49} - 49854 q^{50} - 47020 q^{51} - 58312 q^{52} - 72984 q^{53} - 49168 q^{54} - 57548 q^{55} + 12774 q^{56} + 77356 q^{57} - 21820 q^{58} + 69348 q^{59} + 116342 q^{60} + 54388 q^{61} + 51144 q^{62} - 16152 q^{63} + 58814 q^{64} - 7164 q^{65} - 116636 q^{66} + 59984 q^{67} - 41040 q^{68} - 73056 q^{69} + 69102 q^{70} - 36252 q^{71} - 228096 q^{72} - 79076 q^{73} - 218592 q^{74} - 80504 q^{75} - 169448 q^{76} - 58560 q^{77} + 109928 q^{78} - 111588 q^{79} + 223794 q^{80} + 184052 q^{81} - 5276 q^{82} + 97848 q^{83} + 153012 q^{84} - 23128 q^{85} + 107844 q^{86} + 38932 q^{87} + 172056 q^{88} + 175968 q^{89} + 15394 q^{90} + 228600 q^{91} + 195228 q^{92} + 63060 q^{93} + 252464 q^{94} + 55848 q^{95} - 215984 q^{96} - 49048 q^{97} - 69558 q^{98} - 202256 q^{99} + O(q^{100})$$

## Decomposition of $$S_{5}^{\mathrm{new}}(\Gamma_1(315))$$

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 the newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
315.5.c $$\chi_{315}(71, \cdot)$$ 315.5.c.a 32 1
315.5.e $$\chi_{315}(244, \cdot)$$ 315.5.e.a 1 1
315.5.e.b 1
315.5.e.c 2
315.5.e.d 2
315.5.e.e 8
315.5.e.f 32
315.5.e.g 32
315.5.f $$\chi_{315}(134, \cdot)$$ 315.5.f.a 48 1
315.5.h $$\chi_{315}(181, \cdot)$$ 315.5.h.a 12 1
315.5.h.b 20
315.5.h.c 20
315.5.n $$\chi_{315}(62, \cdot)$$ n/a 128 2
315.5.o $$\chi_{315}(127, \cdot)$$ n/a 120 2
315.5.q $$\chi_{315}(229, \cdot)$$ n/a 376 2
315.5.s $$\chi_{315}(11, \cdot)$$ n/a 256 2
315.5.v $$\chi_{315}(254, \cdot)$$ n/a 376 2
315.5.w $$\chi_{315}(136, \cdot)$$ n/a 108 2
315.5.x $$\chi_{315}(76, \cdot)$$ n/a 256 2
315.5.y $$\chi_{315}(44, \cdot)$$ n/a 128 2
315.5.ba $$\chi_{315}(29, \cdot)$$ n/a 288 2
315.5.bc $$\chi_{315}(31, \cdot)$$ n/a 256 2
315.5.bd $$\chi_{315}(191, \cdot)$$ n/a 256 2
315.5.bg $$\chi_{315}(34, \cdot)$$ n/a 376 2
315.5.bi $$\chi_{315}(19, \cdot)$$ n/a 156 2
315.5.bk $$\chi_{315}(176, \cdot)$$ n/a 192 2
315.5.bm $$\chi_{315}(116, \cdot)$$ 315.5.bm.a 88 2
315.5.bn $$\chi_{315}(94, \cdot)$$ n/a 376 2
315.5.bp $$\chi_{315}(166, \cdot)$$ n/a 256 2
315.5.br $$\chi_{315}(74, \cdot)$$ n/a 376 2
315.5.bt $$\chi_{315}(58, \cdot)$$ n/a 752 4
315.5.bu $$\chi_{315}(38, \cdot)$$ n/a 752 4
315.5.bw $$\chi_{315}(47, \cdot)$$ n/a 752 4
315.5.by $$\chi_{315}(22, \cdot)$$ n/a 576 4
315.5.ca $$\chi_{315}(37, \cdot)$$ n/a 312 4
315.5.cd $$\chi_{315}(17, \cdot)$$ n/a 256 4
315.5.cf $$\chi_{315}(83, \cdot)$$ n/a 752 4
315.5.ch $$\chi_{315}(67, \cdot)$$ n/a 752 4

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

## Decomposition of $$S_{5}^{\mathrm{old}}(\Gamma_1(315))$$ into lower level spaces

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