## Defining parameters

 Level: $$N$$ = $$784 = 2^{4} \cdot 7^{2}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$16$$ Newform subspaces: $$102$$ Sturm bound: $$75264$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(784))$$.

Total New Old
Modular forms 19656 10375 9281
Cusp forms 17977 9938 8039
Eisenstein series 1679 437 1242

## Trace form

 $$9938 q - 62 q^{2} - 47 q^{3} - 60 q^{4} - 77 q^{5} - 56 q^{6} - 54 q^{7} - 104 q^{8} - 15 q^{9} + O(q^{10})$$ $$9938 q - 62 q^{2} - 47 q^{3} - 60 q^{4} - 77 q^{5} - 56 q^{6} - 54 q^{7} - 104 q^{8} - 15 q^{9} - 60 q^{10} - 43 q^{11} - 64 q^{12} - 71 q^{13} - 72 q^{14} - 77 q^{15} - 68 q^{16} - 139 q^{17} - 58 q^{18} - 27 q^{19} - 56 q^{20} - 78 q^{21} - 108 q^{22} - 9 q^{23} - 60 q^{24} + 33 q^{25} - 56 q^{26} - 11 q^{27} - 72 q^{28} - 105 q^{29} - 64 q^{30} - 13 q^{31} - 52 q^{32} - 91 q^{33} - 56 q^{34} - 36 q^{35} - 112 q^{36} - 45 q^{37} - 72 q^{38} - 51 q^{39} - 68 q^{40} - 9 q^{41} - 144 q^{42} - 83 q^{43} - 160 q^{44} - 151 q^{45} - 168 q^{46} - 65 q^{47} - 292 q^{48} - 210 q^{49} - 366 q^{50} - 101 q^{51} - 280 q^{52} - 181 q^{53} - 396 q^{54} - 123 q^{55} - 156 q^{56} - 219 q^{57} - 240 q^{58} - 99 q^{59} - 396 q^{60} - 189 q^{61} - 260 q^{62} - 84 q^{63} - 276 q^{64} - 197 q^{65} - 296 q^{66} - 79 q^{67} - 180 q^{68} - 129 q^{69} - 120 q^{70} - 21 q^{71} - 248 q^{72} + 33 q^{73} - 60 q^{74} + 39 q^{75} - 48 q^{76} - 42 q^{77} - 112 q^{78} + 39 q^{79} - 52 q^{80} - 11 q^{81} - 60 q^{82} + 55 q^{83} - 72 q^{84} - 35 q^{85} - 60 q^{86} + 45 q^{87} - 52 q^{88} + 81 q^{89} + 80 q^{90} - 117 q^{91} - 108 q^{92} + 13 q^{93} + 68 q^{94} - 129 q^{95} + 164 q^{96} - 106 q^{97} + 12 q^{98} - 374 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(784))$$

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
784.2.a $$\chi_{784}(1, \cdot)$$ 784.2.a.a 1 1
784.2.a.b 1
784.2.a.c 1
784.2.a.d 1
784.2.a.e 1
784.2.a.f 1
784.2.a.g 1
784.2.a.h 1
784.2.a.i 1
784.2.a.j 1
784.2.a.k 2
784.2.a.l 2
784.2.a.m 2
784.2.a.n 2
784.2.b $$\chi_{784}(393, \cdot)$$ None 0 1
784.2.e $$\chi_{784}(391, \cdot)$$ None 0 1
784.2.f $$\chi_{784}(783, \cdot)$$ 784.2.f.a 2 1
784.2.f.b 2
784.2.f.c 4
784.2.f.d 4
784.2.f.e 8
784.2.i $$\chi_{784}(177, \cdot)$$ 784.2.i.a 2 2
784.2.i.b 2
784.2.i.c 2
784.2.i.d 2
784.2.i.e 2
784.2.i.f 2
784.2.i.g 2
784.2.i.h 2
784.2.i.i 2
784.2.i.j 2
784.2.i.k 4
784.2.i.l 4
784.2.i.m 4
784.2.i.n 4
784.2.j $$\chi_{784}(195, \cdot)$$ 784.2.j.a 56 2
784.2.j.b 96
784.2.m $$\chi_{784}(197, \cdot)$$ 784.2.m.a 2 2
784.2.m.b 2
784.2.m.c 2
784.2.m.d 4
784.2.m.e 4
784.2.m.f 4
784.2.m.g 8
784.2.m.h 12
784.2.m.i 20
784.2.m.j 24
784.2.m.k 24
784.2.m.l 48
784.2.p $$\chi_{784}(31, \cdot)$$ 784.2.p.a 2 2
784.2.p.b 2
784.2.p.c 2
784.2.p.d 2
784.2.p.e 2
784.2.p.f 2
784.2.p.g 4
784.2.p.h 8
784.2.p.i 8
784.2.p.j 8
784.2.q $$\chi_{784}(215, \cdot)$$ None 0 2
784.2.t $$\chi_{784}(361, \cdot)$$ None 0 2
784.2.u $$\chi_{784}(113, \cdot)$$ 784.2.u.a 6 6
784.2.u.b 12
784.2.u.c 18
784.2.u.d 18
784.2.u.e 24
784.2.u.f 42
784.2.u.g 42
784.2.w $$\chi_{784}(19, \cdot)$$ 784.2.w.a 4 4
784.2.w.b 4
784.2.w.c 8
784.2.w.d 8
784.2.w.e 32
784.2.w.f 56
784.2.w.g 192
784.2.x $$\chi_{784}(165, \cdot)$$ 784.2.x.a 4 4
784.2.x.b 4
784.2.x.c 4
784.2.x.d 4
784.2.x.e 4
784.2.x.f 4
784.2.x.g 4
784.2.x.h 4
784.2.x.i 8
784.2.x.j 16
784.2.x.k 16
784.2.x.l 24
784.2.x.m 24
784.2.x.n 40
784.2.x.o 48
784.2.x.p 96
784.2.bb $$\chi_{784}(111, \cdot)$$ 784.2.bb.a 48 6
784.2.bb.b 120
784.2.bc $$\chi_{784}(55, \cdot)$$ None 0 6
784.2.bf $$\chi_{784}(57, \cdot)$$ None 0 6
784.2.bg $$\chi_{784}(65, \cdot)$$ 784.2.bg.a 24 12
784.2.bg.b 24
784.2.bg.c 48
784.2.bg.d 60
784.2.bg.e 84
784.2.bg.f 84
784.2.bh $$\chi_{784}(29, \cdot)$$ 784.2.bh.a 1320 12
784.2.bk $$\chi_{784}(27, \cdot)$$ 784.2.bk.a 1320 12
784.2.bl $$\chi_{784}(9, \cdot)$$ None 0 12
784.2.bo $$\chi_{784}(87, \cdot)$$ None 0 12
784.2.bp $$\chi_{784}(47, \cdot)$$ 784.2.bp.a 108 12
784.2.bp.b 108
784.2.bp.c 120
784.2.bt $$\chi_{784}(37, \cdot)$$ 784.2.bt.a 2640 24
784.2.bu $$\chi_{784}(3, \cdot)$$ 784.2.bu.a 2640 24

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(784))$$ into lower level spaces

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