# Properties

 Label 507.4 Level 507 Weight 4 Dimension 20838 Nonzero newspaces 12 Sturm bound 75712 Trace bound 1

## Defining parameters

 Level: $$N$$ = $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$12$$ Sturm bound: $$75712$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(507))$$.

Total New Old
Modular forms 28848 21246 7602
Cusp forms 27936 20838 7098
Eisenstein series 912 408 504

## Trace form

 $$20838 q - 66 q^{3} - 132 q^{4} - 66 q^{6} - 276 q^{7} - 288 q^{8} - 66 q^{9} + O(q^{10})$$ $$20838 q - 66 q^{3} - 132 q^{4} - 66 q^{6} - 276 q^{7} - 288 q^{8} - 66 q^{9} + 228 q^{10} + 240 q^{11} + 402 q^{12} + 144 q^{13} + 480 q^{14} + 78 q^{15} - 132 q^{16} - 756 q^{17} - 1398 q^{18} - 1908 q^{19} - 2064 q^{20} - 66 q^{21} + 1428 q^{22} + 912 q^{23} + 2238 q^{24} + 1344 q^{25} + 1500 q^{26} + 1530 q^{27} + 1956 q^{28} + 780 q^{29} + 2046 q^{30} - 372 q^{31} - 2040 q^{32} - 1842 q^{33} - 4644 q^{34} - 3120 q^{35} - 6198 q^{36} - 1416 q^{37} - 1824 q^{39} - 8028 q^{40} - 4956 q^{41} - 5070 q^{42} - 2820 q^{43} - 600 q^{44} + 498 q^{45} + 4908 q^{46} + 2976 q^{47} + 6654 q^{48} + 7788 q^{49} + 8832 q^{50} + 4818 q^{51} + 1500 q^{52} + 3792 q^{53} - 3474 q^{54} + 5484 q^{55} + 5544 q^{56} + 2262 q^{57} + 3972 q^{58} + 1968 q^{59} + 5250 q^{60} - 4656 q^{61} + 1272 q^{62} + 10230 q^{63} - 3756 q^{64} - 2430 q^{65} + 4794 q^{66} - 612 q^{67} + 9360 q^{68} + 3270 q^{69} + 1764 q^{70} + 1056 q^{71} - 6246 q^{72} - 2436 q^{73} - 11544 q^{74} - 15870 q^{75} - 25092 q^{76} - 13248 q^{77} - 14682 q^{78} - 11292 q^{79} - 18624 q^{80} - 11298 q^{81} + 420 q^{82} + 2448 q^{83} - 7278 q^{84} - 96 q^{85} - 12768 q^{86} - 5370 q^{87} + 7620 q^{88} - 6960 q^{89} + 4242 q^{90} - 552 q^{91} + 12720 q^{92} + 15030 q^{93} + 21084 q^{94} + 14160 q^{95} + 34410 q^{96} + 21900 q^{97} + 19032 q^{98} + 22710 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(507))$$

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
507.4.a $$\chi_{507}(1, \cdot)$$ 507.4.a.a 1 1
507.4.a.b 1
507.4.a.c 1
507.4.a.d 1
507.4.a.e 1
507.4.a.f 2
507.4.a.g 2
507.4.a.h 3
507.4.a.i 4
507.4.a.j 4
507.4.a.k 4
507.4.a.l 4
507.4.a.m 4
507.4.a.n 9
507.4.a.o 9
507.4.a.p 9
507.4.a.q 9
507.4.a.r 10
507.4.b $$\chi_{507}(337, \cdot)$$ 507.4.b.a 2 1
507.4.b.b 2
507.4.b.c 2
507.4.b.d 2
507.4.b.e 4
507.4.b.f 4
507.4.b.g 6
507.4.b.h 8
507.4.b.i 10
507.4.b.j 18
507.4.b.k 18
507.4.e $$\chi_{507}(22, \cdot)$$ n/a 156 2
507.4.f $$\chi_{507}(239, \cdot)$$ n/a 288 2
507.4.j $$\chi_{507}(316, \cdot)$$ n/a 152 2
507.4.k $$\chi_{507}(80, \cdot)$$ n/a 576 4
507.4.m $$\chi_{507}(40, \cdot)$$ n/a 1080 12
507.4.p $$\chi_{507}(25, \cdot)$$ n/a 1104 12
507.4.q $$\chi_{507}(16, \cdot)$$ n/a 2160 24
507.4.s $$\chi_{507}(5, \cdot)$$ n/a 4320 24
507.4.t $$\chi_{507}(4, \cdot)$$ n/a 2208 24
507.4.x $$\chi_{507}(2, \cdot)$$ n/a 8640 48

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

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(507))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(507)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(13))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(39))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(169))$$$$^{\oplus 2}$$