# Properties

 Label 700.6 Level 700 Weight 6 Dimension 36892 Nonzero newspaces 24 Sturm bound 172800 Trace bound 7

## Defining parameters

 Level: $$N$$ = $$700 = 2^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$6$$ Nonzero newspaces: $$24$$ Sturm bound: $$172800$$ Trace bound: $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{6}(\Gamma_1(700))$$.

Total New Old
Modular forms 72840 37304 35536
Cusp forms 71160 36892 34268
Eisenstein series 1680 412 1268

## Trace form

 $$36892 q - 27 q^{2} + 7 q^{3} - 19 q^{4} - 26 q^{5} - 760 q^{6} + 56 q^{7} + 1185 q^{8} + 1038 q^{9} + O(q^{10})$$ $$36892 q - 27 q^{2} + 7 q^{3} - 19 q^{4} - 26 q^{5} - 760 q^{6} + 56 q^{7} + 1185 q^{8} + 1038 q^{9} + 1104 q^{10} + 787 q^{11} - 3504 q^{12} - 4860 q^{13} + 1831 q^{14} - 4876 q^{15} + 6737 q^{16} + 11289 q^{17} - 19747 q^{18} - 2295 q^{19} - 2756 q^{20} + 1763 q^{21} + 3394 q^{22} - 20743 q^{23} + 528 q^{24} + 13030 q^{25} + 18388 q^{26} - 10166 q^{27} - 5013 q^{28} + 33204 q^{29} + 17172 q^{30} + 10067 q^{31} + 78813 q^{32} + 14703 q^{33} - 40 q^{34} + 48124 q^{35} - 138111 q^{36} - 58169 q^{37} - 231768 q^{38} + 13182 q^{39} + 92984 q^{40} + 1952 q^{41} + 152830 q^{42} - 13096 q^{43} + 123538 q^{44} + 23150 q^{45} + 19282 q^{46} + 28997 q^{47} - 129236 q^{48} - 76336 q^{49} - 194884 q^{50} - 232507 q^{51} - 181044 q^{52} - 132853 q^{53} - 85296 q^{54} - 4584 q^{55} - 299861 q^{56} + 654950 q^{57} + 321630 q^{58} + 666995 q^{59} + 710372 q^{60} + 739661 q^{61} + 84988 q^{62} - 169804 q^{63} - 568615 q^{64} - 784118 q^{65} + 76728 q^{66} - 706417 q^{67} + 113812 q^{68} - 939122 q^{69} - 480178 q^{70} - 190336 q^{71} + 307997 q^{72} + 548317 q^{73} + 122370 q^{74} + 1109660 q^{75} - 311520 q^{76} + 364819 q^{77} - 749272 q^{78} + 362561 q^{79} - 71784 q^{80} - 921505 q^{81} + 814612 q^{82} - 607140 q^{83} - 683870 q^{84} - 209574 q^{85} - 580658 q^{86} + 191002 q^{87} - 519278 q^{88} + 2593099 q^{89} - 365308 q^{90} - 275412 q^{91} + 2064302 q^{92} - 94493 q^{93} + 1131492 q^{94} - 991976 q^{95} - 965224 q^{96} - 3661352 q^{97} + 125003 q^{98} - 1203700 q^{99} + O(q^{100})$$

## Decomposition of $$S_{6}^{\mathrm{new}}(\Gamma_1(700))$$

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
700.6.a $$\chi_{700}(1, \cdot)$$ 700.6.a.a 1 1
700.6.a.b 1
700.6.a.c 1
700.6.a.d 1
700.6.a.e 1
700.6.a.f 1
700.6.a.g 2
700.6.a.h 2
700.6.a.i 3
700.6.a.j 3
700.6.a.k 4
700.6.a.l 4
700.6.a.m 4
700.6.a.n 4
700.6.a.o 8
700.6.a.p 8
700.6.c $$\chi_{700}(699, \cdot)$$ n/a 356 1
700.6.e $$\chi_{700}(449, \cdot)$$ 700.6.e.a 2 1
700.6.e.b 2
700.6.e.c 2
700.6.e.d 2
700.6.e.e 4
700.6.e.f 4
700.6.e.g 6
700.6.e.h 6
700.6.e.i 8
700.6.e.j 8
700.6.g $$\chi_{700}(251, \cdot)$$ n/a 374 1
700.6.i $$\chi_{700}(401, \cdot)$$ n/a 126 2
700.6.k $$\chi_{700}(43, \cdot)$$ n/a 540 2
700.6.m $$\chi_{700}(293, \cdot)$$ n/a 120 2
700.6.n $$\chi_{700}(141, \cdot)$$ n/a 296 4
700.6.p $$\chi_{700}(451, \cdot)$$ n/a 748 2
700.6.r $$\chi_{700}(149, \cdot)$$ n/a 120 2
700.6.t $$\chi_{700}(199, \cdot)$$ n/a 712 2
700.6.w $$\chi_{700}(111, \cdot)$$ n/a 2384 4
700.6.y $$\chi_{700}(29, \cdot)$$ n/a 304 4
700.6.ba $$\chi_{700}(139, \cdot)$$ n/a 2384 4
700.6.bc $$\chi_{700}(157, \cdot)$$ n/a 240 4
700.6.be $$\chi_{700}(107, \cdot)$$ n/a 1424 4
700.6.bg $$\chi_{700}(81, \cdot)$$ n/a 800 8
700.6.bh $$\chi_{700}(13, \cdot)$$ n/a 800 8
700.6.bj $$\chi_{700}(127, \cdot)$$ n/a 3600 8
700.6.bm $$\chi_{700}(19, \cdot)$$ n/a 4768 8
700.6.bo $$\chi_{700}(9, \cdot)$$ n/a 800 8
700.6.bq $$\chi_{700}(31, \cdot)$$ n/a 4768 8
700.6.bt $$\chi_{700}(23, \cdot)$$ n/a 9536 16
700.6.bv $$\chi_{700}(17, \cdot)$$ n/a 1600 16

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

## Decomposition of $$S_{6}^{\mathrm{old}}(\Gamma_1(700))$$ into lower level spaces

$$S_{6}^{\mathrm{old}}(\Gamma_1(700)) \cong$$ $$S_{6}^{\mathrm{new}}(\Gamma_1(1))$$$$^{\oplus 18}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(2))$$$$^{\oplus 12}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(4))$$$$^{\oplus 6}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 12}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(7))$$$$^{\oplus 9}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 8}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 6}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 4}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 6}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(28))$$$$^{\oplus 3}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(35))$$$$^{\oplus 6}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(50))$$$$^{\oplus 4}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(70))$$$$^{\oplus 4}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(100))$$$$^{\oplus 2}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(140))$$$$^{\oplus 2}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(175))$$$$^{\oplus 3}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(350))$$$$^{\oplus 2}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(700))$$$$^{\oplus 1}$$