# Properties

 Label 1680.4 Level 1680 Weight 4 Dimension 81704 Nonzero newspaces 56 Sturm bound 589824 Trace bound 23

## Defining parameters

 Level: $$N$$ = $$1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$56$$ Sturm bound: $$589824$$ Trace bound: $$23$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(1680))$$.

Total New Old
Modular forms 223872 82240 141632
Cusp forms 218496 81704 136792
Eisenstein series 5376 536 4840

## Trace form

 $$81704 q - 26 q^{3} - 112 q^{4} + 4 q^{5} + 72 q^{6} + 24 q^{7} + 336 q^{8} + 222 q^{9} + O(q^{10})$$ $$81704 q - 26 q^{3} - 112 q^{4} + 4 q^{5} + 72 q^{6} + 24 q^{7} + 336 q^{8} + 222 q^{9} + 216 q^{10} - 240 q^{11} - 424 q^{12} - 456 q^{13} - 696 q^{14} + 234 q^{15} - 1296 q^{16} + 312 q^{17} + 56 q^{18} - 44 q^{19} + 160 q^{20} - 678 q^{21} + 2592 q^{22} - 152 q^{23} + 424 q^{24} + 954 q^{25} - 80 q^{26} - 1208 q^{27} - 288 q^{28} + 2568 q^{29} + 1764 q^{30} - 444 q^{31} - 1120 q^{32} + 2162 q^{33} - 3056 q^{34} - 1380 q^{35} - 2960 q^{36} - 3868 q^{37} - 3264 q^{38} + 1516 q^{39} - 7768 q^{40} - 3080 q^{41} + 5240 q^{42} + 2248 q^{43} + 9728 q^{44} + 55 q^{45} + 6016 q^{46} - 264 q^{47} + 7144 q^{48} - 13816 q^{49} + 10576 q^{50} - 2346 q^{51} + 1696 q^{52} + 104 q^{53} - 11016 q^{54} + 856 q^{55} - 15792 q^{56} - 4564 q^{57} - 10432 q^{58} - 6640 q^{59} - 7772 q^{60} - 7644 q^{61} - 9600 q^{62} + 1354 q^{63} - 18256 q^{64} + 6152 q^{65} - 8168 q^{66} + 5324 q^{67} + 6096 q^{68} + 4472 q^{69} + 6288 q^{70} - 2952 q^{71} + 536 q^{72} - 18700 q^{73} + 6288 q^{74} - 8573 q^{75} + 7280 q^{76} + 1840 q^{77} + 10288 q^{78} - 7508 q^{79} + 26704 q^{80} - 8486 q^{81} + 26512 q^{82} - 11632 q^{83} + 8360 q^{84} - 3380 q^{85} + 2624 q^{86} - 6836 q^{87} + 22160 q^{88} - 1144 q^{89} - 820 q^{90} + 23248 q^{91} - 13904 q^{92} + 2778 q^{93} - 41968 q^{94} + 53332 q^{95} - 12152 q^{96} + 12840 q^{97} - 22960 q^{98} + 30972 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(1680))$$

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 available newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
1680.4.a $$\chi_{1680}(1, \cdot)$$ 1680.4.a.a 1 1
1680.4.a.b 1
1680.4.a.c 1
1680.4.a.d 1
1680.4.a.e 1
1680.4.a.f 1
1680.4.a.g 1
1680.4.a.h 1
1680.4.a.i 1
1680.4.a.j 1
1680.4.a.k 1
1680.4.a.l 1
1680.4.a.m 1
1680.4.a.n 1
1680.4.a.o 1
1680.4.a.p 1
1680.4.a.q 1
1680.4.a.r 1
1680.4.a.s 1
1680.4.a.t 1
1680.4.a.u 1
1680.4.a.v 1
1680.4.a.w 1
1680.4.a.x 1
1680.4.a.y 2
1680.4.a.z 2
1680.4.a.ba 2
1680.4.a.bb 2
1680.4.a.bc 2
1680.4.a.bd 2
1680.4.a.be 2
1680.4.a.bf 2
1680.4.a.bg 2
1680.4.a.bh 2
1680.4.a.bi 2
1680.4.a.bj 2
1680.4.a.bk 2
1680.4.a.bl 2
1680.4.a.bm 2
1680.4.a.bn 2
1680.4.a.bo 2
1680.4.a.bp 2
1680.4.a.bq 3
1680.4.a.br 3
1680.4.a.bs 3
1680.4.a.bt 3
1680.4.d $$\chi_{1680}(1231, \cdot)$$ 1680.4.d.a 16 1
1680.4.d.b 16
1680.4.d.c 32
1680.4.d.d 32
1680.4.e $$\chi_{1680}(71, \cdot)$$ None 0 1
1680.4.f $$\chi_{1680}(881, \cdot)$$ n/a 192 1
1680.4.g $$\chi_{1680}(841, \cdot)$$ None 0 1
1680.4.j $$\chi_{1680}(169, \cdot)$$ None 0 1
1680.4.k $$\chi_{1680}(209, \cdot)$$ n/a 284 1
1680.4.p $$\chi_{1680}(1079, \cdot)$$ None 0 1
1680.4.q $$\chi_{1680}(559, \cdot)$$ n/a 144 1
1680.4.t $$\chi_{1680}(1009, \cdot)$$ n/a 108 1
1680.4.u $$\chi_{1680}(1049, \cdot)$$ None 0 1
1680.4.v $$\chi_{1680}(239, \cdot)$$ n/a 216 1
1680.4.w $$\chi_{1680}(1399, \cdot)$$ None 0 1
1680.4.z $$\chi_{1680}(391, \cdot)$$ None 0 1
1680.4.ba $$\chi_{1680}(911, \cdot)$$ n/a 144 1
1680.4.bf $$\chi_{1680}(41, \cdot)$$ None 0 1
1680.4.bg $$\chi_{1680}(961, \cdot)$$ n/a 192 2
1680.4.bj $$\chi_{1680}(937, \cdot)$$ None 0 2
1680.4.bk $$\chi_{1680}(113, \cdot)$$ n/a 432 2
1680.4.bl $$\chi_{1680}(127, \cdot)$$ n/a 216 2
1680.4.bm $$\chi_{1680}(167, \cdot)$$ None 0 2
1680.4.bp $$\chi_{1680}(883, \cdot)$$ n/a 864 2
1680.4.bs $$\chi_{1680}(853, \cdot)$$ n/a 1152 2
1680.4.bu $$\chi_{1680}(197, \cdot)$$ n/a 1728 2
1680.4.bv $$\chi_{1680}(923, \cdot)$$ n/a 2288 2
1680.4.bx $$\chi_{1680}(629, \cdot)$$ n/a 2288 2
1680.4.ca $$\chi_{1680}(491, \cdot)$$ n/a 1152 2
1680.4.cb $$\chi_{1680}(589, \cdot)$$ n/a 864 2
1680.4.ce $$\chi_{1680}(811, \cdot)$$ n/a 768 2
1680.4.cg $$\chi_{1680}(421, \cdot)$$ n/a 576 2
1680.4.ch $$\chi_{1680}(139, \cdot)$$ n/a 1152 2
1680.4.ck $$\chi_{1680}(461, \cdot)$$ n/a 1536 2
1680.4.cl $$\chi_{1680}(659, \cdot)$$ n/a 1728 2
1680.4.co $$\chi_{1680}(13, \cdot)$$ n/a 1152 2
1680.4.cp $$\chi_{1680}(43, \cdot)$$ n/a 864 2
1680.4.cr $$\chi_{1680}(83, \cdot)$$ n/a 2288 2
1680.4.cu $$\chi_{1680}(533, \cdot)$$ n/a 1728 2
1680.4.cx $$\chi_{1680}(967, \cdot)$$ None 0 2
1680.4.cy $$\chi_{1680}(1007, \cdot)$$ n/a 576 2
1680.4.cz $$\chi_{1680}(97, \cdot)$$ n/a 288 2
1680.4.da $$\chi_{1680}(617, \cdot)$$ None 0 2
1680.4.df $$\chi_{1680}(199, \cdot)$$ None 0 2
1680.4.dg $$\chi_{1680}(1199, \cdot)$$ n/a 576 2
1680.4.dh $$\chi_{1680}(89, \cdot)$$ None 0 2
1680.4.di $$\chi_{1680}(289, \cdot)$$ n/a 288 2
1680.4.dl $$\chi_{1680}(521, \cdot)$$ None 0 2
1680.4.dq $$\chi_{1680}(191, \cdot)$$ n/a 384 2
1680.4.dr $$\chi_{1680}(871, \cdot)$$ None 0 2
1680.4.du $$\chi_{1680}(121, \cdot)$$ None 0 2
1680.4.dv $$\chi_{1680}(1361, \cdot)$$ n/a 384 2
1680.4.dw $$\chi_{1680}(1031, \cdot)$$ None 0 2
1680.4.dx $$\chi_{1680}(31, \cdot)$$ n/a 192 2
1680.4.ea $$\chi_{1680}(1039, \cdot)$$ n/a 288 2
1680.4.eb $$\chi_{1680}(359, \cdot)$$ None 0 2
1680.4.eg $$\chi_{1680}(689, \cdot)$$ n/a 568 2
1680.4.eh $$\chi_{1680}(1129, \cdot)$$ None 0 2
1680.4.ei $$\chi_{1680}(137, \cdot)$$ None 0 4
1680.4.ej $$\chi_{1680}(577, \cdot)$$ n/a 576 4
1680.4.eo $$\chi_{1680}(47, \cdot)$$ n/a 1152 4
1680.4.ep $$\chi_{1680}(247, \cdot)$$ None 0 4
1680.4.eq $$\chi_{1680}(227, \cdot)$$ n/a 4576 4
1680.4.et $$\chi_{1680}(653, \cdot)$$ n/a 4576 4
1680.4.ev $$\chi_{1680}(157, \cdot)$$ n/a 2304 4
1680.4.ew $$\chi_{1680}(163, \cdot)$$ n/a 2304 4
1680.4.ey $$\chi_{1680}(101, \cdot)$$ n/a 3072 4
1680.4.fb $$\chi_{1680}(179, \cdot)$$ n/a 4576 4
1680.4.fc $$\chi_{1680}(541, \cdot)$$ n/a 1536 4
1680.4.ff $$\chi_{1680}(19, \cdot)$$ n/a 2304 4
1680.4.fh $$\chi_{1680}(109, \cdot)$$ n/a 2304 4
1680.4.fi $$\chi_{1680}(451, \cdot)$$ n/a 1536 4
1680.4.fl $$\chi_{1680}(269, \cdot)$$ n/a 4576 4
1680.4.fm $$\chi_{1680}(11, \cdot)$$ n/a 3072 4
1680.4.fp $$\chi_{1680}(53, \cdot)$$ n/a 4576 4
1680.4.fq $$\chi_{1680}(563, \cdot)$$ n/a 4576 4
1680.4.fs $$\chi_{1680}(67, \cdot)$$ n/a 2304 4
1680.4.fv $$\chi_{1680}(493, \cdot)$$ n/a 2304 4
1680.4.fw $$\chi_{1680}(647, \cdot)$$ None 0 4
1680.4.fx $$\chi_{1680}(1087, \cdot)$$ n/a 576 4
1680.4.gc $$\chi_{1680}(737, \cdot)$$ n/a 1136 4
1680.4.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 $$S_{4}^{\mathrm{old}}(\Gamma_1(1680))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(1680)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(1))$$$$^{\oplus 40}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(2))$$$$^{\oplus 32}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(3))$$$$^{\oplus 20}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(4))$$$$^{\oplus 24}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 20}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 16}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(7))$$$$^{\oplus 20}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 16}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 16}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 16}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 10}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(21))$$$$^{\oplus 10}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(28))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(35))$$$$^{\oplus 10}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(40))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(42))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(48))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(56))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(60))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(70))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(80))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(84))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(105))$$$$^{\oplus 5}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(112))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(120))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(140))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(168))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(210))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(240))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(280))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(336))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(420))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(560))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(840))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(1680))$$$$^{\oplus 1}$$