## Defining parameters

 Level: $$N$$ = $$210 = 2 \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$12$$ Newform subspaces: $$32$$ Sturm bound: $$4608$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(210))$$.

Total New Old
Modular forms 1344 249 1095
Cusp forms 961 249 712
Eisenstein series 383 0 383

## Trace form

 $$249q + q^{2} + 9q^{3} + 9q^{4} + 21q^{5} + 17q^{6} + 33q^{7} + q^{8} + 17q^{9} + O(q^{10})$$ $$249q + q^{2} + 9q^{3} + 9q^{4} + 21q^{5} + 17q^{6} + 33q^{7} + q^{8} + 17q^{9} + 5q^{10} + 28q^{11} - 7q^{12} + 14q^{13} - 15q^{14} - 19q^{15} + q^{16} - 6q^{17} - 39q^{18} + 20q^{19} - 7q^{20} - 27q^{21} - 4q^{22} - 7q^{24} - 7q^{25} - 10q^{26} - 63q^{27} - 23q^{28} - 66q^{29} - 31q^{30} - 16q^{31} + q^{32} - 76q^{33} - 62q^{34} - 111q^{35} - 31q^{36} - 122q^{37} - 52q^{38} - 106q^{39} - 11q^{40} - 38q^{41} - 27q^{42} - 20q^{43} - 4q^{44} - 51q^{45} - 24q^{46} - 24q^{47} - 7q^{48} + 25q^{49} + 33q^{50} - 30q^{51} - 34q^{52} + 30q^{53} - 31q^{54} - 32q^{55} + 17q^{56} + 28q^{57} - 26q^{58} - 20q^{59} + q^{60} - 66q^{61} + 32q^{62} - 55q^{63} + 9q^{64} - 46q^{65} - 36q^{66} - 172q^{67} - 6q^{68} - 56q^{69} + q^{70} - 72q^{71} + 41q^{72} - 126q^{73} + 30q^{74} - 111q^{75} - 12q^{76} - 12q^{77} + 38q^{78} - 48q^{79} + 21q^{80} + 49q^{81} + 122q^{82} + 36q^{83} + 45q^{84} + 66q^{85} + 52q^{86} + 110q^{87} + 20q^{88} + 26q^{89} + 85q^{90} + 110q^{91} + 24q^{92} + 160q^{93} + 128q^{94} + 80q^{95} + q^{96} + 74q^{97} + q^{98} + 188q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(210))$$

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
210.2.a $$\chi_{210}(1, \cdot)$$ 210.2.a.a 1 1
210.2.a.b 1
210.2.a.c 1
210.2.a.d 1
210.2.a.e 1
210.2.b $$\chi_{210}(41, \cdot)$$ 210.2.b.a 4 1
210.2.b.b 4
210.2.d $$\chi_{210}(209, \cdot)$$ 210.2.d.a 8 1
210.2.d.b 8
210.2.g $$\chi_{210}(169, \cdot)$$ 210.2.g.a 2 1
210.2.g.b 2
210.2.i $$\chi_{210}(121, \cdot)$$ 210.2.i.a 2 2
210.2.i.b 2
210.2.i.c 2
210.2.i.d 2
210.2.j $$\chi_{210}(113, \cdot)$$ 210.2.j.a 12 2
210.2.j.b 12
210.2.m $$\chi_{210}(13, \cdot)$$ 210.2.m.a 8 2
210.2.m.b 8
210.2.n $$\chi_{210}(79, \cdot)$$ 210.2.n.a 4 2
210.2.n.b 12
210.2.r $$\chi_{210}(101, \cdot)$$ 210.2.r.a 12 2
210.2.r.b 12
210.2.t $$\chi_{210}(59, \cdot)$$ 210.2.t.a 4 2
210.2.t.b 4
210.2.t.c 4
210.2.t.d 4
210.2.t.e 8
210.2.t.f 8
210.2.u $$\chi_{210}(73, \cdot)$$ 210.2.u.a 16 4
210.2.u.b 16
210.2.x $$\chi_{210}(23, \cdot)$$ 210.2.x.a 64 4

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(210))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(210)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(21))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(35))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(42))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(70))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(105))$$$$^{\oplus 2}$$