## Defining parameters

 Level: $$N$$ = $$690 = 2 \cdot 3 \cdot 5 \cdot 23$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$12$$ Newform subspaces: $$45$$ Sturm bound: $$50688$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(690))$$.

Total New Old
Modular forms 13376 2889 10487
Cusp forms 11969 2889 9080
Eisenstein series 1407 0 1407

## Trace form

 $$2889 q + q^{2} + 5 q^{3} + q^{4} + 9 q^{5} + 5 q^{6} + 8 q^{7} + q^{8} + q^{9} + O(q^{10})$$ $$2889 q + q^{2} + 5 q^{3} + q^{4} + 9 q^{5} + 5 q^{6} + 8 q^{7} + q^{8} + q^{9} - 7 q^{10} - 20 q^{11} - 11 q^{12} - 18 q^{13} - 24 q^{14} + 3 q^{15} + q^{16} + 58 q^{17} + 73 q^{18} + 76 q^{19} + 37 q^{20} + 124 q^{21} + 84 q^{22} + 153 q^{23} + 5 q^{24} + 105 q^{25} + 102 q^{26} + 137 q^{27} + 96 q^{28} + 70 q^{29} + 73 q^{30} + 88 q^{31} + q^{32} + 40 q^{33} - 14 q^{34} + 20 q^{35} + q^{36} + 86 q^{37} - 28 q^{38} + 46 q^{39} - 23 q^{40} + 50 q^{41} - 24 q^{42} + 92 q^{43} - 4 q^{44} + 9 q^{45} - 47 q^{46} + 128 q^{47} - 11 q^{48} + 177 q^{49} - 15 q^{50} + 82 q^{51} - 18 q^{52} + 46 q^{53} - 39 q^{54} + 68 q^{55} - 8 q^{56} - 24 q^{57} - 2 q^{58} + 68 q^{59} - 9 q^{60} - 2 q^{61} - 16 q^{62} - 212 q^{63} + q^{64} + 14 q^{65} - 212 q^{66} - 28 q^{67} - 30 q^{68} - 235 q^{69} + 8 q^{70} - 120 q^{71} - 71 q^{72} - 38 q^{73} - 42 q^{74} - 175 q^{75} - 44 q^{76} - 96 q^{77} - 142 q^{78} - 8 q^{79} + 9 q^{80} - 151 q^{81} - 22 q^{82} - 20 q^{83} - 52 q^{84} + 14 q^{85} - 68 q^{86} + 74 q^{87} - 4 q^{88} - 78 q^{89} - 23 q^{90} + 32 q^{91} - 23 q^{92} - 64 q^{93} - 16 q^{94} - 72 q^{95} - 11 q^{96} - 302 q^{97} - 215 q^{98} + 40 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(690))$$

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
690.2.a $$\chi_{690}(1, \cdot)$$ 690.2.a.a 1 1
690.2.a.b 1
690.2.a.c 1
690.2.a.d 1
690.2.a.e 1
690.2.a.f 1
690.2.a.g 1
690.2.a.h 1
690.2.a.i 1
690.2.a.j 1
690.2.a.k 1
690.2.a.l 2
690.2.d $$\chi_{690}(139, \cdot)$$ 690.2.d.a 4 1
690.2.d.b 4
690.2.d.c 6
690.2.d.d 6
690.2.e $$\chi_{690}(551, \cdot)$$ 690.2.e.a 16 1
690.2.e.b 16
690.2.h $$\chi_{690}(689, \cdot)$$ 690.2.h.a 24 1
690.2.h.b 24
690.2.i $$\chi_{690}(47, \cdot)$$ 690.2.i.a 4 2
690.2.i.b 4
690.2.i.c 8
690.2.i.d 8
690.2.i.e 32
690.2.i.f 32
690.2.j $$\chi_{690}(367, \cdot)$$ 690.2.j.a 24 2
690.2.j.b 24
690.2.m $$\chi_{690}(31, \cdot)$$ 690.2.m.a 10 10
690.2.m.b 10
690.2.m.c 20
690.2.m.d 20
690.2.m.e 20
690.2.m.f 20
690.2.m.g 30
690.2.m.h 30
690.2.n $$\chi_{690}(89, \cdot)$$ 690.2.n.a 240 10
690.2.n.b 240
690.2.q $$\chi_{690}(11, \cdot)$$ 690.2.q.a 160 10
690.2.q.b 160
690.2.r $$\chi_{690}(49, \cdot)$$ 690.2.r.a 120 10
690.2.r.b 120
690.2.w $$\chi_{690}(7, \cdot)$$ 690.2.w.a 240 20
690.2.w.b 240
690.2.x $$\chi_{690}(77, \cdot)$$ 690.2.x.a 960 20

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(690))$$ into lower level spaces

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