## Defining parameters

 Level: $$N$$ = $$784 = 2^{4} \cdot 7^{2}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$16$$ Sturm bound: $$150528$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(784))$$.

Total New Old
Modular forms 57288 30688 26600
Cusp forms 55608 30251 25357
Eisenstein series 1680 437 1243

## Trace form

 $$30251 q - 62 q^{2} - 43 q^{3} - 52 q^{4} - 79 q^{5} - 92 q^{6} - 54 q^{7} - 152 q^{8} - 26 q^{9} + O(q^{10})$$ $$30251 q - 62 q^{2} - 43 q^{3} - 52 q^{4} - 79 q^{5} - 92 q^{6} - 54 q^{7} - 152 q^{8} - 26 q^{9} - 128 q^{10} + 17 q^{11} + 40 q^{12} - 49 q^{13} - 72 q^{14} - 213 q^{15} + 220 q^{16} - 89 q^{17} + 114 q^{18} + 425 q^{19} - 256 q^{20} + 90 q^{21} - 696 q^{22} - 241 q^{23} - 908 q^{24} - 1144 q^{25} - 324 q^{26} - 1315 q^{27} - 72 q^{28} - 739 q^{29} + 1176 q^{30} - 237 q^{31} + 908 q^{32} + 181 q^{33} + 376 q^{34} + 468 q^{35} - 704 q^{36} + 1265 q^{37} - 1292 q^{38} + 1141 q^{39} - 1396 q^{40} - 207 q^{41} - 2664 q^{42} - 2183 q^{43} - 6584 q^{44} - 4977 q^{45} - 3128 q^{46} - 2129 q^{47} + 1468 q^{48} + 462 q^{49} + 4734 q^{50} + 755 q^{51} + 8992 q^{52} + 4009 q^{53} + 16372 q^{54} + 2477 q^{55} + 4548 q^{56} + 9973 q^{57} + 8180 q^{58} + 4393 q^{59} + 11508 q^{60} + 3785 q^{61} + 2668 q^{62} - 84 q^{63} - 2452 q^{64} - 3281 q^{65} - 13688 q^{66} + 1205 q^{67} - 13060 q^{68} - 5361 q^{69} - 7512 q^{70} + 2227 q^{71} - 11256 q^{72} + 1435 q^{73} - 512 q^{74} - 5421 q^{75} - 1288 q^{76} - 2058 q^{77} - 880 q^{78} - 9097 q^{79} - 2708 q^{80} - 7194 q^{81} - 764 q^{82} - 11813 q^{83} - 72 q^{84} - 8023 q^{85} + 3704 q^{86} - 7083 q^{87} + 1468 q^{88} - 4389 q^{89} + 7020 q^{90} - 6333 q^{91} + 6980 q^{92} - 6499 q^{93} + 24772 q^{94} - 16745 q^{95} + 26468 q^{96} + 4120 q^{97} + 5892 q^{98} + 2134 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(784))$$

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 the newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
784.4.a $$\chi_{784}(1, \cdot)$$ 784.4.a.a 1 1
784.4.a.b 1
784.4.a.c 1
784.4.a.d 1
784.4.a.e 1
784.4.a.f 1
784.4.a.g 1
784.4.a.h 1
784.4.a.i 1
784.4.a.j 1
784.4.a.k 1
784.4.a.l 1
784.4.a.m 1
784.4.a.n 1
784.4.a.o 1
784.4.a.p 1
784.4.a.q 1
784.4.a.r 1
784.4.a.s 1
784.4.a.t 2
784.4.a.u 2
784.4.a.v 2
784.4.a.w 2
784.4.a.x 2
784.4.a.y 2
784.4.a.z 2
784.4.a.ba 2
784.4.a.bb 3
784.4.a.bc 3
784.4.a.bd 3
784.4.a.be 3
784.4.a.bf 4
784.4.a.bg 4
784.4.a.bh 4
784.4.b $$\chi_{784}(393, \cdot)$$ None 0 1
784.4.e $$\chi_{784}(391, \cdot)$$ None 0 1
784.4.f $$\chi_{784}(783, \cdot)$$ 784.4.f.a 2 1
784.4.f.b 2
784.4.f.c 2
784.4.f.d 2
784.4.f.e 4
784.4.f.f 4
784.4.f.g 6
784.4.f.h 6
784.4.f.i 8
784.4.f.j 24
784.4.i $$\chi_{784}(177, \cdot)$$ n/a 116 2
784.4.j $$\chi_{784}(195, \cdot)$$ n/a 472 2
784.4.m $$\chi_{784}(197, \cdot)$$ n/a 482 2
784.4.p $$\chi_{784}(31, \cdot)$$ n/a 120 2
784.4.q $$\chi_{784}(215, \cdot)$$ None 0 2
784.4.t $$\chi_{784}(361, \cdot)$$ None 0 2
784.4.u $$\chi_{784}(113, \cdot)$$ n/a 498 6
784.4.w $$\chi_{784}(19, \cdot)$$ n/a 944 4
784.4.x $$\chi_{784}(165, \cdot)$$ n/a 944 4
784.4.bb $$\chi_{784}(111, \cdot)$$ n/a 504 6
784.4.bc $$\chi_{784}(55, \cdot)$$ None 0 6
784.4.bf $$\chi_{784}(57, \cdot)$$ None 0 6
784.4.bg $$\chi_{784}(65, \cdot)$$ n/a 996 12
784.4.bh $$\chi_{784}(29, \cdot)$$ n/a 4008 12
784.4.bk $$\chi_{784}(27, \cdot)$$ n/a 4008 12
784.4.bl $$\chi_{784}(9, \cdot)$$ None 0 12
784.4.bo $$\chi_{784}(87, \cdot)$$ None 0 12
784.4.bp $$\chi_{784}(47, \cdot)$$ n/a 1008 12
784.4.bt $$\chi_{784}(37, \cdot)$$ n/a 8016 24
784.4.bu $$\chi_{784}(3, \cdot)$$ n/a 8016 24

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

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(784))$$ into lower level spaces

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