# Properties

 Label 117.2.q Level $117$ Weight $2$ Character orbit 117.q Rep. character $\chi_{117}(10,\cdot)$ Character field $\Q(\zeta_{6})$ Dimension $10$ Newform subspaces $4$ Sturm bound $28$ Trace bound $7$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$117 = 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 117.q (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$13$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$4$$ Sturm bound: $$28$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$2$$, $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(117, [\chi])$$.

Total New Old
Modular forms 36 14 22
Cusp forms 20 10 10
Eisenstein series 16 4 12

## Trace form

 $$10 q + 3 q^{2} + 3 q^{4} - 6 q^{7} + O(q^{10})$$ $$10 q + 3 q^{2} + 3 q^{4} - 6 q^{7} + 7 q^{10} + 6 q^{11} + 7 q^{13} - q^{16} - 3 q^{17} - 18 q^{19} - 15 q^{20} - 20 q^{22} - 12 q^{23} - 3 q^{26} - 48 q^{28} + 9 q^{29} + 9 q^{32} + 6 q^{35} + 9 q^{37} - 12 q^{38} + 26 q^{40} + 21 q^{41} + 2 q^{43} + 42 q^{46} + 19 q^{49} + 6 q^{50} - 6 q^{52} - 18 q^{53} - 8 q^{55} - 21 q^{58} - 6 q^{59} - q^{61} - 6 q^{62} + 82 q^{64} + 3 q^{65} - 12 q^{67} + 3 q^{68} + 12 q^{71} + 15 q^{74} - 42 q^{76} - 12 q^{77} + 44 q^{79} + 39 q^{80} + 19 q^{82} + 39 q^{85} + 20 q^{88} - 66 q^{91} + 12 q^{92} - 26 q^{94} - 18 q^{95} - 6 q^{97} - 21 q^{98} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(117, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
117.2.q.a $2$ $0.934$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$-3$$ $$q-2\zeta_{6}q^{4}+(2-4\zeta_{6})q^{5}+(-2+\zeta_{6})q^{7}+\cdots$$
117.2.q.b $2$ $0.934$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$9$$ $$q-2\zeta_{6}q^{4}+(6-3\zeta_{6})q^{7}+(-4+3\zeta_{6})q^{13}+\cdots$$
117.2.q.c $2$ $0.934$ $$\Q(\sqrt{-3})$$ None $$3$$ $$0$$ $$0$$ $$0$$ $$q+(1+\zeta_{6})q^{2}+\zeta_{6}q^{4}+(-1+2\zeta_{6})q^{5}+\cdots$$
117.2.q.d $4$ $0.934$ $$\Q(\sqrt{-3}, \sqrt{-5})$$ None $$0$$ $$0$$ $$0$$ $$-12$$ $$q+\beta _{1}q^{2}+3\beta _{2}q^{4}-\beta _{3}q^{5}+(-4+2\beta _{2}+\cdots)q^{7}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(117, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(117, [\chi]) \simeq$$ $$S_{2}^{\mathrm{new}}(13, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(39, [\chi])$$$$^{\oplus 2}$$