## Defining parameters

 Level: $$N$$ = $$209 = 11 \cdot 19$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$2$$ Newform subspaces: $$2$$ Sturm bound: $$3600$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(209))$$.

Total New Old
Modular forms 188 160 28
Cusp forms 8 8 0
Eisenstein series 180 152 28

The following table gives the dimensions of subspaces with specified projective image type.

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 4 4 0 0

## Trace form

 $$8 q - 2 q^{3} - q^{4} - 4 q^{5} + 3 q^{7} - q^{9} + O(q^{10})$$ $$8 q - 2 q^{3} - q^{4} - 4 q^{5} + 3 q^{7} - q^{9} - q^{11} - 2 q^{15} + q^{16} + 3 q^{17} - q^{19} - 2 q^{20} + 2 q^{22} - 4 q^{23} - 3 q^{25} + 4 q^{26} - 4 q^{27} - 2 q^{28} - 2 q^{34} + q^{35} - q^{36} - 2 q^{38} - 2 q^{43} + 4 q^{44} - 2 q^{45} - 4 q^{47} + 2 q^{48} + 6 q^{49} + 2 q^{53} + 3 q^{55} - 4 q^{58} - 2 q^{59} + 3 q^{61} + 3 q^{63} - 5 q^{64} + 2 q^{66} + 2 q^{67} + 3 q^{68} + 4 q^{69} + 2 q^{71} - 2 q^{73} + 4 q^{76} - 2 q^{77} - 2 q^{78} + 5 q^{80} + q^{81} + 2 q^{82} - 2 q^{83} + q^{85} - 2 q^{86} - 4 q^{88} + 2 q^{89} - 2 q^{92} + 3 q^{95} + 2 q^{97} - q^{99} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(209))$$

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
209.1.b $$\chi_{209}(56, \cdot)$$ None 0 1
209.1.c $$\chi_{209}(153, \cdot)$$ None 0 1
209.1.h $$\chi_{209}(87, \cdot)$$ 209.1.h.a 4 2
209.1.i $$\chi_{209}(12, \cdot)$$ None 0 2
209.1.l $$\chi_{209}(39, \cdot)$$ None 0 4
209.1.m $$\chi_{209}(37, \cdot)$$ 209.1.m.a 4 4
209.1.o $$\chi_{209}(34, \cdot)$$ None 0 6
209.1.q $$\chi_{209}(43, \cdot)$$ None 0 6
209.1.r $$\chi_{209}(27, \cdot)$$ None 0 8
209.1.s $$\chi_{209}(7, \cdot)$$ None 0 8
209.1.v $$\chi_{209}(6, \cdot)$$ None 0 24
209.1.x $$\chi_{209}(3, \cdot)$$ None 0 24