# Properties

 Label 690.3 Level 690 Weight 3 Dimension 5784 Nonzero newspaces 12 Sturm bound 76032 Trace bound 4

## Defining parameters

 Level: $$N$$ = $$690 = 2 \cdot 3 \cdot 5 \cdot 23$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$12$$ Sturm bound: $$76032$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(690))$$.

Total New Old
Modular forms 26048 5784 20264
Cusp forms 24640 5784 18856
Eisenstein series 1408 0 1408

## Trace form

 $$5784q - 8q^{2} - 8q^{3} + 16q^{6} + 16q^{7} + 16q^{8} + 80q^{9} + O(q^{10})$$ $$5784q - 8q^{2} - 8q^{3} + 16q^{6} + 16q^{7} + 16q^{8} + 80q^{9} + 24q^{10} + 32q^{11} + 16q^{12} + 56q^{13} - 162q^{15} - 32q^{16} - 528q^{17} - 440q^{18} - 424q^{19} - 104q^{20} - 260q^{21} - 64q^{22} + 192q^{23} + 192q^{25} + 304q^{26} + 604q^{27} + 496q^{28} + 616q^{29} + 352q^{30} + 1144q^{31} + 32q^{32} + 532q^{33} + 320q^{34} - 216q^{35} + 48q^{36} - 1016q^{37} + 128q^{38} - 448q^{39} + 48q^{40} - 416q^{41} + 160q^{42} - 560q^{43} + 120q^{45} - 80q^{46} + 160q^{47} - 32q^{48} + 576q^{49} - 184q^{50} + 224q^{51} - 208q^{52} + 504q^{53} + 136q^{54} - 48q^{55} - 128q^{56} + 2108q^{57} - 128q^{58} + 1232q^{59} + 320q^{60} + 448q^{61} + 160q^{62} + 1708q^{63} + 408q^{65} + 992q^{66} + 688q^{67} + 176q^{68} - 20q^{69} + 96q^{70} - 128q^{71} - 272q^{72} - 24q^{73} - 942q^{75} + 192q^{76} + 256q^{77} - 1592q^{78} - 256q^{79} - 3528q^{81} - 448q^{82} - 1088q^{83} - 1184q^{84} - 176q^{85} - 192q^{86} - 912q^{87} + 128q^{88} - 176q^{89} - 72q^{90} - 416q^{91} + 32q^{92} + 128q^{93} + 640q^{94} + 2732q^{95} + 64q^{96} + 4160q^{97} + 3064q^{98} + 2216q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(690))$$

We only show spaces with odd 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
690.3.b $$\chi_{690}(599, \cdot)$$ 690.3.b.a 88 1
690.3.c $$\chi_{690}(91, \cdot)$$ 690.3.c.a 32 1
690.3.f $$\chi_{690}(229, \cdot)$$ 690.3.f.a 48 1
690.3.g $$\chi_{690}(461, \cdot)$$ 690.3.g.a 56 1
690.3.k $$\chi_{690}(277, \cdot)$$ 690.3.k.a 40 2
690.3.k.b 48
690.3.l $$\chi_{690}(137, \cdot)$$ n/a 192 2
690.3.o $$\chi_{690}(41, \cdot)$$ n/a 640 10
690.3.p $$\chi_{690}(19, \cdot)$$ n/a 480 10
690.3.s $$\chi_{690}(61, \cdot)$$ n/a 320 10
690.3.t $$\chi_{690}(29, \cdot)$$ n/a 960 10
690.3.u $$\chi_{690}(17, \cdot)$$ n/a 1920 20
690.3.v $$\chi_{690}(13, \cdot)$$ n/a 960 20

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

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(690))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(690)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(23))$$$$^{\oplus 8}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(46))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(69))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(115))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(138))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(230))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(345))$$$$^{\oplus 2}$$