# Properties

 Label 1680.2 Level 1680 Weight 2 Dimension 27056 Nonzero newspaces 56 Sturm bound 294912 Trace bound 23

## Defining parameters

 Level: $$N$$ = $$1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$56$$ Sturm bound: $$294912$$ Trace bound: $$23$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(1680))$$.

Total New Old
Modular forms 76416 27592 48824
Cusp forms 71041 27056 43985
Eisenstein series 5375 536 4839

## Trace form

 $$27056 q - 10 q^{3} - 48 q^{4} - 4 q^{5} - 72 q^{6} - 40 q^{7} - 48 q^{8} - 26 q^{9} + O(q^{10})$$ $$27056 q - 10 q^{3} - 48 q^{4} - 4 q^{5} - 72 q^{6} - 40 q^{7} - 48 q^{8} - 26 q^{9} - 56 q^{10} - 48 q^{11} - 8 q^{12} - 88 q^{13} + 24 q^{14} - 86 q^{15} - 16 q^{16} - 24 q^{17} + 24 q^{18} - 140 q^{19} + 32 q^{20} - 102 q^{21} - 88 q^{23} + 104 q^{24} - 62 q^{25} + 80 q^{26} + 8 q^{27} + 32 q^{28} - 72 q^{29} + 84 q^{30} - 60 q^{31} + 160 q^{32} - 46 q^{33} + 208 q^{34} + 12 q^{35} - 16 q^{36} - 44 q^{37} + 192 q^{38} + 124 q^{39} + 232 q^{40} + 104 q^{41} + 120 q^{42} + 40 q^{43} + 256 q^{44} + 47 q^{45} + 224 q^{46} + 120 q^{47} + 104 q^{48} + 48 q^{49} + 176 q^{50} + 230 q^{51} + 416 q^{52} + 280 q^{53} + 136 q^{54} + 152 q^{55} + 336 q^{56} + 204 q^{57} + 384 q^{58} + 208 q^{59} + 100 q^{60} + 308 q^{61} + 384 q^{62} + 170 q^{63} + 432 q^{64} + 88 q^{65} + 280 q^{66} + 364 q^{67} + 144 q^{68} + 344 q^{69} - 32 q^{70} + 312 q^{71} + 88 q^{72} + 100 q^{73} - 144 q^{74} + 315 q^{75} - 80 q^{76} + 80 q^{77} - 176 q^{78} + 300 q^{79} - 304 q^{80} + 66 q^{81} - 176 q^{82} + 208 q^{83} - 184 q^{84} - 100 q^{85} - 320 q^{86} + 124 q^{87} - 496 q^{88} - 104 q^{89} - 308 q^{90} + 272 q^{91} - 400 q^{92} + 42 q^{93} - 688 q^{94} + 212 q^{95} - 632 q^{96} - 296 q^{97} - 464 q^{98} + 60 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(1680))$$

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
1680.2.a $$\chi_{1680}(1, \cdot)$$ 1680.2.a.a 1 1
1680.2.a.b 1
1680.2.a.c 1
1680.2.a.d 1
1680.2.a.e 1
1680.2.a.f 1
1680.2.a.g 1
1680.2.a.h 1
1680.2.a.i 1
1680.2.a.j 1
1680.2.a.k 1
1680.2.a.l 1
1680.2.a.m 1
1680.2.a.n 1
1680.2.a.o 1
1680.2.a.p 1
1680.2.a.q 1
1680.2.a.r 1
1680.2.a.s 1
1680.2.a.t 1
1680.2.a.u 2
1680.2.a.v 2
1680.2.d $$\chi_{1680}(1231, \cdot)$$ 1680.2.d.a 4 1
1680.2.d.b 4
1680.2.d.c 12
1680.2.d.d 12
1680.2.e $$\chi_{1680}(71, \cdot)$$ None 0 1
1680.2.f $$\chi_{1680}(881, \cdot)$$ 1680.2.f.a 2 1
1680.2.f.b 2
1680.2.f.c 2
1680.2.f.d 2
1680.2.f.e 4
1680.2.f.f 4
1680.2.f.g 4
1680.2.f.h 4
1680.2.f.i 4
1680.2.f.j 4
1680.2.f.k 16
1680.2.f.l 16
1680.2.g $$\chi_{1680}(841, \cdot)$$ None 0 1
1680.2.j $$\chi_{1680}(169, \cdot)$$ None 0 1
1680.2.k $$\chi_{1680}(209, \cdot)$$ 1680.2.k.a 4 1
1680.2.k.b 4
1680.2.k.c 4
1680.2.k.d 8
1680.2.k.e 8
1680.2.k.f 8
1680.2.k.g 8
1680.2.k.h 24
1680.2.k.i 24
1680.2.p $$\chi_{1680}(1079, \cdot)$$ None 0 1
1680.2.q $$\chi_{1680}(559, \cdot)$$ 1680.2.q.a 8 1
1680.2.q.b 8
1680.2.q.c 32
1680.2.t $$\chi_{1680}(1009, \cdot)$$ 1680.2.t.a 2 1
1680.2.t.b 2
1680.2.t.c 2
1680.2.t.d 2
1680.2.t.e 2
1680.2.t.f 2
1680.2.t.g 2
1680.2.t.h 4
1680.2.t.i 6
1680.2.t.j 6
1680.2.t.k 6
1680.2.u $$\chi_{1680}(1049, \cdot)$$ None 0 1
1680.2.v $$\chi_{1680}(239, \cdot)$$ 1680.2.v.a 12 1
1680.2.v.b 12
1680.2.v.c 24
1680.2.v.d 24
1680.2.w $$\chi_{1680}(1399, \cdot)$$ None 0 1
1680.2.z $$\chi_{1680}(391, \cdot)$$ None 0 1
1680.2.ba $$\chi_{1680}(911, \cdot)$$ 1680.2.ba.a 8 1
1680.2.ba.b 8
1680.2.ba.c 16
1680.2.ba.d 16
1680.2.bf $$\chi_{1680}(41, \cdot)$$ None 0 1
1680.2.bg $$\chi_{1680}(961, \cdot)$$ 1680.2.bg.a 2 2
1680.2.bg.b 2
1680.2.bg.c 2
1680.2.bg.d 2
1680.2.bg.e 2
1680.2.bg.f 2
1680.2.bg.g 2
1680.2.bg.h 2
1680.2.bg.i 2
1680.2.bg.j 2
1680.2.bg.k 2
1680.2.bg.l 2
1680.2.bg.m 2
1680.2.bg.n 2
1680.2.bg.o 4
1680.2.bg.p 4
1680.2.bg.q 4
1680.2.bg.r 4
1680.2.bg.s 4
1680.2.bg.t 4
1680.2.bg.u 6
1680.2.bg.v 6
1680.2.bj $$\chi_{1680}(937, \cdot)$$ None 0 2
1680.2.bk $$\chi_{1680}(113, \cdot)$$ n/a 144 2
1680.2.bl $$\chi_{1680}(127, \cdot)$$ 1680.2.bl.a 4 2
1680.2.bl.b 8
1680.2.bl.c 12
1680.2.bl.d 24
1680.2.bl.e 24
1680.2.bm $$\chi_{1680}(167, \cdot)$$ None 0 2
1680.2.bp $$\chi_{1680}(883, \cdot)$$ n/a 288 2
1680.2.bs $$\chi_{1680}(853, \cdot)$$ n/a 384 2
1680.2.bu $$\chi_{1680}(197, \cdot)$$ n/a 576 2
1680.2.bv $$\chi_{1680}(923, \cdot)$$ n/a 752 2
1680.2.bx $$\chi_{1680}(629, \cdot)$$ n/a 752 2
1680.2.ca $$\chi_{1680}(491, \cdot)$$ n/a 384 2
1680.2.cb $$\chi_{1680}(589, \cdot)$$ n/a 288 2
1680.2.ce $$\chi_{1680}(811, \cdot)$$ n/a 256 2
1680.2.cg $$\chi_{1680}(421, \cdot)$$ n/a 192 2
1680.2.ch $$\chi_{1680}(139, \cdot)$$ n/a 384 2
1680.2.ck $$\chi_{1680}(461, \cdot)$$ n/a 512 2
1680.2.cl $$\chi_{1680}(659, \cdot)$$ n/a 576 2
1680.2.co $$\chi_{1680}(13, \cdot)$$ n/a 384 2
1680.2.cp $$\chi_{1680}(43, \cdot)$$ n/a 288 2
1680.2.cr $$\chi_{1680}(83, \cdot)$$ n/a 752 2
1680.2.cu $$\chi_{1680}(533, \cdot)$$ n/a 576 2
1680.2.cx $$\chi_{1680}(967, \cdot)$$ None 0 2
1680.2.cy $$\chi_{1680}(1007, \cdot)$$ n/a 192 2
1680.2.cz $$\chi_{1680}(97, \cdot)$$ 1680.2.cz.a 8 2
1680.2.cz.b 8
1680.2.cz.c 16
1680.2.cz.d 16
1680.2.cz.e 24
1680.2.cz.f 24
1680.2.da $$\chi_{1680}(617, \cdot)$$ None 0 2
1680.2.df $$\chi_{1680}(199, \cdot)$$ None 0 2
1680.2.dg $$\chi_{1680}(1199, \cdot)$$ n/a 192 2
1680.2.dh $$\chi_{1680}(89, \cdot)$$ None 0 2
1680.2.di $$\chi_{1680}(289, \cdot)$$ 1680.2.di.a 4 2
1680.2.di.b 4
1680.2.di.c 12
1680.2.di.d 16
1680.2.di.e 16
1680.2.di.f 20
1680.2.di.g 24
1680.2.dl $$\chi_{1680}(521, \cdot)$$ None 0 2
1680.2.dq $$\chi_{1680}(191, \cdot)$$ n/a 128 2
1680.2.dr $$\chi_{1680}(871, \cdot)$$ None 0 2
1680.2.du $$\chi_{1680}(121, \cdot)$$ None 0 2
1680.2.dv $$\chi_{1680}(1361, \cdot)$$ n/a 128 2
1680.2.dw $$\chi_{1680}(1031, \cdot)$$ None 0 2
1680.2.dx $$\chi_{1680}(31, \cdot)$$ 1680.2.dx.a 4 2
1680.2.dx.b 4
1680.2.dx.c 4
1680.2.dx.d 4
1680.2.dx.e 12
1680.2.dx.f 12
1680.2.dx.g 12
1680.2.dx.h 12
1680.2.ea $$\chi_{1680}(1039, \cdot)$$ 1680.2.ea.a 16 2
1680.2.ea.b 16
1680.2.ea.c 32
1680.2.ea.d 32
1680.2.eb $$\chi_{1680}(359, \cdot)$$ None 0 2
1680.2.eg $$\chi_{1680}(689, \cdot)$$ n/a 184 2
1680.2.eh $$\chi_{1680}(1129, \cdot)$$ None 0 2
1680.2.ei $$\chi_{1680}(137, \cdot)$$ None 0 4
1680.2.ej $$\chi_{1680}(577, \cdot)$$ n/a 192 4
1680.2.eo $$\chi_{1680}(47, \cdot)$$ n/a 384 4
1680.2.ep $$\chi_{1680}(247, \cdot)$$ None 0 4
1680.2.eq $$\chi_{1680}(227, \cdot)$$ n/a 1504 4
1680.2.et $$\chi_{1680}(653, \cdot)$$ n/a 1504 4
1680.2.ev $$\chi_{1680}(157, \cdot)$$ n/a 768 4
1680.2.ew $$\chi_{1680}(163, \cdot)$$ n/a 768 4
1680.2.ey $$\chi_{1680}(101, \cdot)$$ n/a 1024 4
1680.2.fb $$\chi_{1680}(179, \cdot)$$ n/a 1504 4
1680.2.fc $$\chi_{1680}(541, \cdot)$$ n/a 512 4
1680.2.ff $$\chi_{1680}(19, \cdot)$$ n/a 768 4
1680.2.fh $$\chi_{1680}(109, \cdot)$$ n/a 768 4
1680.2.fi $$\chi_{1680}(451, \cdot)$$ n/a 512 4
1680.2.fl $$\chi_{1680}(269, \cdot)$$ n/a 1504 4
1680.2.fm $$\chi_{1680}(11, \cdot)$$ n/a 1024 4
1680.2.fp $$\chi_{1680}(53, \cdot)$$ n/a 1504 4
1680.2.fq $$\chi_{1680}(563, \cdot)$$ n/a 1504 4
1680.2.fs $$\chi_{1680}(67, \cdot)$$ n/a 768 4
1680.2.fv $$\chi_{1680}(493, \cdot)$$ n/a 768 4
1680.2.fw $$\chi_{1680}(647, \cdot)$$ None 0 4
1680.2.fx $$\chi_{1680}(1087, \cdot)$$ n/a 192 4
1680.2.gc $$\chi_{1680}(737, \cdot)$$ n/a 368 4
1680.2.gd $$\chi_{1680}(73, \cdot)$$ None 0 4

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

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(1680))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(1680)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 16}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 10}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(21))$$$$^{\oplus 10}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(28))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(35))$$$$^{\oplus 10}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(40))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(42))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(48))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(56))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(60))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(70))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(80))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(84))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(105))$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(112))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(120))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(140))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(168))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(210))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(240))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(280))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(336))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(420))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(560))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(840))$$$$^{\oplus 2}$$