# Properties

 Label 688.6 Level 688 Weight 6 Dimension 41351 Nonzero newspaces 16 Sturm bound 177408 Trace bound 2

## Defining parameters

 Level: $$N$$ = $$688 = 2^{4} \cdot 43$$ Weight: $$k$$ = $$6$$ Nonzero newspaces: $$16$$ Sturm bound: $$177408$$ Trace bound: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{6}(\Gamma_1(688))$$.

Total New Old
Modular forms 74508 41719 32789
Cusp forms 73332 41351 31981
Eisenstein series 1176 368 808

## Trace form

 $$41351 q - 80 q^{2} - 43 q^{3} - 36 q^{4} - 61 q^{5} - 308 q^{6} - 287 q^{7} - 572 q^{8} - 137 q^{9} + O(q^{10})$$ $$41351 q - 80 q^{2} - 43 q^{3} - 36 q^{4} - 61 q^{5} - 308 q^{6} - 287 q^{7} - 572 q^{8} - 137 q^{9} + 788 q^{10} + 2477 q^{11} - 92 q^{12} - 221 q^{13} + 116 q^{14} - 7911 q^{15} + 1660 q^{16} + 1027 q^{17} + 6192 q^{18} + 12421 q^{19} - 6028 q^{20} - 2225 q^{21} - 8924 q^{22} - 7903 q^{23} - 16820 q^{24} - 4305 q^{25} - 14820 q^{26} - 3727 q^{27} + 14588 q^{28} - 493 q^{29} + 60804 q^{30} + 20065 q^{31} + 47900 q^{32} + 7819 q^{33} + 3396 q^{34} - 23287 q^{35} - 13868 q^{36} + 1139 q^{37} - 106580 q^{38} + 29089 q^{39} - 150628 q^{40} - 16701 q^{41} - 66884 q^{42} - 5969 q^{43} + 80080 q^{44} + 22299 q^{45} + 184980 q^{46} - 43679 q^{47} + 295900 q^{48} + 79219 q^{49} + 170016 q^{50} - 56743 q^{51} - 183228 q^{52} - 111917 q^{53} - 417428 q^{54} + 39905 q^{55} - 382340 q^{56} - 101717 q^{57} - 213636 q^{58} + 77821 q^{59} + 308652 q^{60} + 202499 q^{61} + 547660 q^{62} + 34137 q^{63} + 567468 q^{64} + 60331 q^{65} + 306628 q^{66} + 162917 q^{67} - 267508 q^{68} - 151569 q^{69} - 824404 q^{70} - 157727 q^{71} - 940572 q^{72} - 125373 q^{73} - 294380 q^{74} - 210459 q^{75} + 174852 q^{76} + 18767 q^{77} + 1263476 q^{78} + 40961 q^{79} + 1108828 q^{80} + 221547 q^{81} + 186348 q^{82} - 185579 q^{83} - 381860 q^{84} + 35722 q^{85} - 470552 q^{86} - 117150 q^{87} - 1180740 q^{88} - 203517 q^{89} - 560388 q^{90} + 512889 q^{91} + 443708 q^{92} - 30217 q^{93} + 921740 q^{94} + 140473 q^{95} + 1194572 q^{96} + 146883 q^{97} + 889208 q^{98} + 525493 q^{99} + O(q^{100})$$

## Decomposition of $$S_{6}^{\mathrm{new}}(\Gamma_1(688))$$

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
688.6.a $$\chi_{688}(1, \cdot)$$ 688.6.a.a 3 1
688.6.a.b 3
688.6.a.c 5
688.6.a.d 6
688.6.a.e 8
688.6.a.f 8
688.6.a.g 9
688.6.a.h 10
688.6.a.i 11
688.6.a.j 13
688.6.a.k 14
688.6.a.l 15
688.6.c $$\chi_{688}(345, \cdot)$$ None 0 1
688.6.e $$\chi_{688}(343, \cdot)$$ None 0 1
688.6.g $$\chi_{688}(687, \cdot)$$ n/a 110 1
688.6.i $$\chi_{688}(49, \cdot)$$ n/a 218 2
688.6.j $$\chi_{688}(171, \cdot)$$ n/a 876 2
688.6.k $$\chi_{688}(173, \cdot)$$ n/a 840 2
688.6.o $$\chi_{688}(351, \cdot)$$ n/a 220 2
688.6.q $$\chi_{688}(7, \cdot)$$ None 0 2
688.6.s $$\chi_{688}(393, \cdot)$$ None 0 2
688.6.u $$\chi_{688}(97, \cdot)$$ n/a 654 6
688.6.x $$\chi_{688}(165, \cdot)$$ n/a 1752 4
688.6.y $$\chi_{688}(123, \cdot)$$ n/a 1752 4
688.6.bb $$\chi_{688}(223, \cdot)$$ n/a 660 6
688.6.bd $$\chi_{688}(39, \cdot)$$ None 0 6
688.6.bf $$\chi_{688}(41, \cdot)$$ None 0 6
688.6.bg $$\chi_{688}(17, \cdot)$$ n/a 1308 12
688.6.bh $$\chi_{688}(21, \cdot)$$ n/a 5256 12
688.6.bi $$\chi_{688}(27, \cdot)$$ n/a 5256 12
688.6.bl $$\chi_{688}(9, \cdot)$$ None 0 12
688.6.bn $$\chi_{688}(55, \cdot)$$ None 0 12
688.6.bp $$\chi_{688}(63, \cdot)$$ n/a 1320 12
688.6.bu $$\chi_{688}(3, \cdot)$$ n/a 10512 24
688.6.bv $$\chi_{688}(13, \cdot)$$ n/a 10512 24

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

## Decomposition of $$S_{6}^{\mathrm{old}}(\Gamma_1(688))$$ into lower level spaces

$$S_{6}^{\mathrm{old}}(\Gamma_1(688)) \cong$$ $$S_{6}^{\mathrm{new}}(\Gamma_1(4))$$$$^{\oplus 6}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 4}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 2}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(43))$$$$^{\oplus 5}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(86))$$$$^{\oplus 4}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(172))$$$$^{\oplus 3}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(344))$$$$^{\oplus 2}$$