# Properties

 Label 784.6 Level 784 Weight 6 Dimension 50564 Nonzero newspaces 16 Sturm bound 225792 Trace bound 3

## Defining parameters

 Level: $$N$$ = $$784 = 2^{4} \cdot 7^{2}$$ Weight: $$k$$ = $$6$$ Nonzero newspaces: $$16$$ Sturm bound: $$225792$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{6}(\Gamma_1(784))$$.

Total New Old
Modular forms 94920 51001 43919
Cusp forms 93240 50564 42676
Eisenstein series 1680 437 1243

## Trace form

 $$50564 q - 62 q^{2} - 55 q^{3} - 84 q^{4} - 97 q^{5} + 52 q^{6} - 54 q^{7} + 136 q^{8} + 43 q^{9} + O(q^{10})$$ $$50564 q - 62 q^{2} - 55 q^{3} - 84 q^{4} - 97 q^{5} + 52 q^{6} - 54 q^{7} + 136 q^{8} + 43 q^{9} - 496 q^{10} - 1315 q^{11} - 56 q^{12} - 11 q^{13} - 72 q^{14} + 3843 q^{15} - 932 q^{16} - 743 q^{17} - 3198 q^{18} + 3421 q^{19} + 2912 q^{20} - 6462 q^{21} + 4312 q^{22} - 7609 q^{23} + 8308 q^{24} + 18495 q^{25} + 7308 q^{26} + 36773 q^{27} - 72 q^{28} + 12275 q^{29} - 30504 q^{30} - 23117 q^{31} - 24052 q^{32} - 60731 q^{33} - 1800 q^{34} - 25236 q^{35} + 6784 q^{36} + 5759 q^{37} + 53188 q^{38} + 48829 q^{39} + 75212 q^{40} + 8331 q^{41} - 93384 q^{42} - 42283 q^{43} - 98264 q^{44} + 131781 q^{45} + 207528 q^{46} + 119647 q^{47} + 416188 q^{48} + 76398 q^{49} + 19758 q^{50} + 3755 q^{51} - 206784 q^{52} - 154313 q^{53} - 717068 q^{54} - 165259 q^{55} - 249468 q^{56} - 444731 q^{57} - 208956 q^{58} - 82427 q^{59} - 62028 q^{60} + 49439 q^{61} + 345772 q^{62} + 120876 q^{63} + 538476 q^{64} + 494755 q^{65} + 1037224 q^{66} + 275177 q^{67} + 288572 q^{68} - 49377 q^{69} - 397272 q^{70} - 466277 q^{71} - 316248 q^{72} - 252747 q^{73} + 147088 q^{74} - 418929 q^{75} - 87528 q^{76} - 52458 q^{77} - 631888 q^{78} + 206663 q^{79} - 554516 q^{80} + 428535 q^{81} - 93276 q^{82} + 836719 q^{83} - 72 q^{84} + 579157 q^{85} + 470408 q^{86} + 451101 q^{87} + 590268 q^{88} + 63813 q^{89} + 466716 q^{90} - 466149 q^{91} + 717572 q^{92} - 1430611 q^{93} + 1102724 q^{94} - 1576625 q^{95} - 1787548 q^{96} - 515814 q^{97} - 967836 q^{98} + 3356338 q^{99} + O(q^{100})$$

## Decomposition of $$S_{6}^{\mathrm{new}}(\Gamma_1(784))$$

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
784.6.a $$\chi_{784}(1, \cdot)$$ 784.6.a.a 1 1
784.6.a.b 1
784.6.a.c 1
784.6.a.d 1
784.6.a.e 1
784.6.a.f 1
784.6.a.g 1
784.6.a.h 1
784.6.a.i 1
784.6.a.j 1
784.6.a.k 1
784.6.a.l 1
784.6.a.m 1
784.6.a.n 1
784.6.a.o 2
784.6.a.p 2
784.6.a.q 2
784.6.a.r 2
784.6.a.s 2
784.6.a.t 2
784.6.a.u 2
784.6.a.v 2
784.6.a.w 2
784.6.a.x 2
784.6.a.y 2
784.6.a.z 2
784.6.a.ba 2
784.6.a.bb 2
784.6.a.bc 2
784.6.a.bd 2
784.6.a.be 4
784.6.a.bf 4
784.6.a.bg 4
784.6.a.bh 4
784.6.a.bi 4
784.6.a.bj 5
784.6.a.bk 5
784.6.a.bl 5
784.6.a.bm 5
784.6.a.bn 6
784.6.a.bo 8
784.6.b $$\chi_{784}(393, \cdot)$$ None 0 1
784.6.e $$\chi_{784}(391, \cdot)$$ None 0 1
784.6.f $$\chi_{784}(783, \cdot)$$ 784.6.f.a 4 1
784.6.f.b 12
784.6.f.c 14
784.6.f.d 14
784.6.f.e 16
784.6.f.f 40
784.6.i $$\chi_{784}(177, \cdot)$$ n/a 196 2
784.6.j $$\chi_{784}(195, \cdot)$$ n/a 792 2
784.6.m $$\chi_{784}(197, \cdot)$$ n/a 810 2
784.6.p $$\chi_{784}(31, \cdot)$$ n/a 200 2
784.6.q $$\chi_{784}(215, \cdot)$$ None 0 2
784.6.t $$\chi_{784}(361, \cdot)$$ None 0 2
784.6.u $$\chi_{784}(113, \cdot)$$ n/a 834 6
784.6.w $$\chi_{784}(19, \cdot)$$ n/a 1584 4
784.6.x $$\chi_{784}(165, \cdot)$$ n/a 1584 4
784.6.bb $$\chi_{784}(111, \cdot)$$ n/a 840 6
784.6.bc $$\chi_{784}(55, \cdot)$$ None 0 6
784.6.bf $$\chi_{784}(57, \cdot)$$ None 0 6
784.6.bg $$\chi_{784}(65, \cdot)$$ n/a 1668 12
784.6.bh $$\chi_{784}(29, \cdot)$$ n/a 6696 12
784.6.bk $$\chi_{784}(27, \cdot)$$ n/a 6696 12
784.6.bl $$\chi_{784}(9, \cdot)$$ None 0 12
784.6.bo $$\chi_{784}(87, \cdot)$$ None 0 12
784.6.bp $$\chi_{784}(47, \cdot)$$ n/a 1680 12
784.6.bt $$\chi_{784}(37, \cdot)$$ n/a 13392 24
784.6.bu $$\chi_{784}(3, \cdot)$$ n/a 13392 24

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

## Decomposition of $$S_{6}^{\mathrm{old}}(\Gamma_1(784))$$ into lower level spaces

$$S_{6}^{\mathrm{old}}(\Gamma_1(784)) \cong$$ $$S_{6}^{\mathrm{new}}(\Gamma_1(1))$$$$^{\oplus 15}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(2))$$$$^{\oplus 12}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(4))$$$$^{\oplus 9}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(7))$$$$^{\oplus 10}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 6}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 8}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 3}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(28))$$$$^{\oplus 6}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(49))$$$$^{\oplus 5}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(56))$$$$^{\oplus 4}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(98))$$$$^{\oplus 4}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(112))$$$$^{\oplus 2}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(196))$$$$^{\oplus 3}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(392))$$$$^{\oplus 2}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(784))$$$$^{\oplus 1}$$