## Defining parameters

 Level: $$N$$ = $$1210 = 2 \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$12$$ Sturm bound: $$174240$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(1210))$$.

Total New Old
Modular forms 44840 13089 31751
Cusp forms 42281 13089 29192
Eisenstein series 2559 0 2559

## Trace form

 $$13089q - q^{2} - 4q^{3} - q^{4} - q^{5} + 16q^{6} + 32q^{7} - q^{8} + 67q^{9} + O(q^{10})$$ $$13089q - q^{2} - 4q^{3} - q^{4} - q^{5} + 16q^{6} + 32q^{7} - q^{8} + 67q^{9} + 19q^{10} + 20q^{11} + 36q^{12} + 26q^{13} + 32q^{14} + 56q^{15} - q^{16} + 62q^{17} + 7q^{18} + 40q^{19} - q^{20} + 48q^{21} + 56q^{23} + 16q^{24} + 79q^{25} + 66q^{26} + 140q^{27} + 32q^{28} + 130q^{29} + 56q^{30} + 88q^{31} + 19q^{32} + 110q^{33} + 62q^{34} + 112q^{35} + 7q^{36} + 42q^{37} + 100q^{38} + 144q^{39} + 19q^{40} + 118q^{41} + 8q^{42} - 4q^{43} - 10q^{44} - 153q^{45} - 104q^{46} - 128q^{47} - 4q^{48} - 217q^{49} - 121q^{50} - 132q^{51} - 94q^{52} - 94q^{53} - 200q^{54} - 90q^{55} - 8q^{56} - 60q^{57} - 190q^{58} - 40q^{59} - 124q^{60} - 102q^{61} - 72q^{62} - 24q^{63} - q^{64} - 54q^{65} + 40q^{66} + 52q^{67} - 18q^{68} + 184q^{69} - 8q^{70} + 248q^{71} + 67q^{72} + 166q^{73} + 42q^{74} + 176q^{75} + 20q^{76} + 160q^{77} + 104q^{78} + 120q^{79} + 19q^{80} + 419q^{81} + 98q^{82} + 176q^{83} + 128q^{84} + 202q^{85} + 216q^{86} + 280q^{87} + 20q^{88} + 230q^{89} + 167q^{90} + 208q^{91} + 96q^{92} + 152q^{93} + 192q^{94} + 40q^{95} - 4q^{96} + 42q^{97} + 123q^{98} + 60q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(1210))$$

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
1210.2.a $$\chi_{1210}(1, \cdot)$$ 1210.2.a.a 1 1
1210.2.a.b 1
1210.2.a.c 1
1210.2.a.d 1
1210.2.a.e 1
1210.2.a.f 1
1210.2.a.g 1
1210.2.a.h 1
1210.2.a.i 1
1210.2.a.j 1
1210.2.a.k 1
1210.2.a.l 1
1210.2.a.m 1
1210.2.a.n 2
1210.2.a.o 2
1210.2.a.p 2
1210.2.a.q 2
1210.2.a.r 2
1210.2.a.s 2
1210.2.a.t 2
1210.2.a.u 4
1210.2.a.v 4
1210.2.b $$\chi_{1210}(969, \cdot)$$ 1210.2.b.a 2 1
1210.2.b.b 2
1210.2.b.c 2
1210.2.b.d 2
1210.2.b.e 2
1210.2.b.f 4
1210.2.b.g 4
1210.2.b.h 4
1210.2.b.i 4
1210.2.b.j 4
1210.2.b.k 8
1210.2.b.l 8
1210.2.b.m 8
1210.2.f $$\chi_{1210}(483, \cdot)$$ n/a 108 2
1210.2.g $$\chi_{1210}(81, \cdot)$$ n/a 144 4
1210.2.j $$\chi_{1210}(9, \cdot)$$ n/a 216 4
1210.2.k $$\chi_{1210}(111, \cdot)$$ n/a 440 10
1210.2.l $$\chi_{1210}(233, \cdot)$$ n/a 432 8
1210.2.o $$\chi_{1210}(89, \cdot)$$ n/a 660 10
1210.2.q $$\chi_{1210}(43, \cdot)$$ n/a 1320 20
1210.2.s $$\chi_{1210}(31, \cdot)$$ n/a 1760 40
1210.2.u $$\chi_{1210}(49, \cdot)$$ n/a 2640 40
1210.2.x $$\chi_{1210}(7, \cdot)$$ n/a 5280 80

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

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(1210))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(1210)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(11))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(22))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(55))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(110))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(121))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(242))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(605))$$$$^{\oplus 2}$$