## Defining parameters

 Level: $$N$$ = $$209 = 11 \cdot 19$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$12$$ Newform subspaces: $$25$$ Sturm bound: $$7200$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(209))$$.

Total New Old
Modular forms 1980 1913 67
Cusp forms 1621 1605 16
Eisenstein series 359 308 51

## Trace form

 $$1605 q - 71 q^{2} - 74 q^{3} - 83 q^{4} - 80 q^{5} - 88 q^{6} - 76 q^{7} - 87 q^{8} - 81 q^{9} + O(q^{10})$$ $$1605 q - 71 q^{2} - 74 q^{3} - 83 q^{4} - 80 q^{5} - 88 q^{6} - 76 q^{7} - 87 q^{8} - 81 q^{9} - 86 q^{10} - 84 q^{11} - 162 q^{12} - 70 q^{13} - 68 q^{14} - 58 q^{15} - 23 q^{16} - 68 q^{17} - 47 q^{18} - 49 q^{19} - 118 q^{20} - 66 q^{21} - 53 q^{22} - 166 q^{23} - 60 q^{24} - 59 q^{25} - 62 q^{26} - 80 q^{27} - 42 q^{28} - 66 q^{29} - 22 q^{30} - 82 q^{31} - 61 q^{32} - 29 q^{33} - 134 q^{34} - 44 q^{35} + 61 q^{36} - 62 q^{37} - 3 q^{38} - 84 q^{39} - 62 q^{41} + 20 q^{42} - 26 q^{43} - 20 q^{44} - 74 q^{45} + 14 q^{46} - 46 q^{47} + 48 q^{48} - 45 q^{49} + 17 q^{50} - 40 q^{51} - 54 q^{52} - 42 q^{53} - 10 q^{54} - 53 q^{55} - 48 q^{56} - 58 q^{57} - 162 q^{58} - 52 q^{59} + 70 q^{60} - 26 q^{61} + 58 q^{62} - 16 q^{63} + 93 q^{64} + 34 q^{65} + 47 q^{66} - 30 q^{67} + 62 q^{68} + 28 q^{69} + 64 q^{70} - 12 q^{71} + 129 q^{72} + 14 q^{73} + 6 q^{74} - 18 q^{75} - 25 q^{76} - 67 q^{77} - 58 q^{78} + 72 q^{79} + 74 q^{80} + 39 q^{81} + 112 q^{82} - 18 q^{83} + 128 q^{84} + 40 q^{85} - 30 q^{86} + 96 q^{87} + 66 q^{88} - 120 q^{89} + 100 q^{90} - 2 q^{92} + 18 q^{93} + 50 q^{94} - 60 q^{95} + 52 q^{96} - 30 q^{97} + 87 q^{98} + 45 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(209))$$

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
209.2.a $$\chi_{209}(1, \cdot)$$ 209.2.a.a 1 1
209.2.a.b 2
209.2.a.c 5
209.2.a.d 7
209.2.d $$\chi_{209}(208, \cdot)$$ 209.2.d.a 2 1
209.2.d.b 8
209.2.d.c 8
209.2.e $$\chi_{209}(45, \cdot)$$ 209.2.e.a 18 2
209.2.e.b 18
209.2.f $$\chi_{209}(20, \cdot)$$ 209.2.f.a 4 4
209.2.f.b 28
209.2.f.c 40
209.2.g $$\chi_{209}(65, \cdot)$$ 209.2.g.a 16 2
209.2.g.b 20
209.2.j $$\chi_{209}(23, \cdot)$$ 209.2.j.a 6 6
209.2.j.b 42
209.2.j.c 48
209.2.k $$\chi_{209}(18, \cdot)$$ 209.2.k.a 8 4
209.2.k.b 8
209.2.k.c 56
209.2.n $$\chi_{209}(26, \cdot)$$ 209.2.n.a 144 8
209.2.p $$\chi_{209}(10, \cdot)$$ 209.2.p.a 108 6
209.2.t $$\chi_{209}(8, \cdot)$$ 209.2.t.a 144 8
209.2.u $$\chi_{209}(4, \cdot)$$ 209.2.u.a 432 24
209.2.w $$\chi_{209}(2, \cdot)$$ 209.2.w.a 432 24

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(209))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(209)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(11))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(19))$$$$^{\oplus 2}$$