# Properties

 Label 209.2.e Level $209$ Weight $2$ Character orbit 209.e Rep. character $\chi_{209}(45,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $36$ Newform subspaces $2$ Sturm bound $40$ Trace bound $2$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$209 = 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 209.e (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$19$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$2$$ Sturm bound: $$40$$ Trace bound: $$2$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(209, [\chi])$$.

Total New Old
Modular forms 44 36 8
Cusp forms 36 36 0
Eisenstein series 8 0 8

## Trace form

 $$36 q - 4 q^{3} - 18 q^{4} + 4 q^{6} - 12 q^{7} - 12 q^{8} - 22 q^{9} + O(q^{10})$$ $$36 q - 4 q^{3} - 18 q^{4} + 4 q^{6} - 12 q^{7} - 12 q^{8} - 22 q^{9} + 10 q^{10} + 24 q^{12} + 10 q^{13} - 2 q^{14} + 6 q^{15} - 18 q^{16} + 8 q^{17} + 4 q^{18} + 10 q^{19} - 40 q^{20} - 2 q^{22} + 10 q^{23} + 14 q^{24} - 22 q^{25} - 24 q^{26} + 20 q^{27} + 6 q^{28} + 40 q^{30} - 16 q^{31} + 22 q^{32} - 4 q^{33} + 4 q^{34} - 6 q^{35} - 12 q^{36} - 8 q^{37} + 16 q^{39} - 12 q^{40} - 12 q^{41} + 2 q^{42} + 16 q^{43} - 12 q^{45} - 8 q^{46} - 16 q^{47} - 34 q^{48} - 8 q^{50} + 10 q^{51} + 28 q^{52} - 20 q^{53} + 16 q^{54} - 44 q^{56} - 50 q^{57} + 8 q^{58} - 6 q^{59} + 26 q^{60} - 12 q^{61} + 6 q^{62} + 26 q^{63} + 96 q^{64} - 16 q^{65} + 10 q^{66} - 16 q^{67} - 4 q^{68} + 48 q^{69} - 32 q^{70} - 20 q^{71} + 38 q^{72} - 8 q^{73} + 42 q^{74} + 100 q^{75} - 92 q^{76} - 6 q^{78} + 30 q^{79} + 54 q^{80} - 74 q^{81} + 38 q^{82} + 20 q^{83} - 104 q^{84} + 20 q^{85} + 22 q^{86} - 104 q^{87} - 12 q^{88} - 14 q^{89} + 26 q^{90} - 48 q^{91} + 42 q^{92} + 12 q^{93} - 112 q^{94} + 12 q^{95} - 40 q^{96} - 28 q^{97} - 40 q^{98} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(209, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
209.2.e.a $18$ $1.669$ $$\mathbb{Q}[x]/(x^{18} - \cdots)$$ None $$-1$$ $$-4$$ $$0$$ $$-6$$ $$q-\beta _{1}q^{2}-\beta _{7}q^{3}+(-1-\beta _{4}-\beta _{9}+\cdots)q^{4}+\cdots$$
209.2.e.b $18$ $1.669$ $$\mathbb{Q}[x]/(x^{18} - \cdots)$$ None $$1$$ $$0$$ $$0$$ $$-6$$ $$q+\beta _{1}q^{2}+(\beta _{4}-\beta _{12})q^{3}+(-1+\beta _{3}+\cdots)q^{4}+\cdots$$