# Properties

 Label 338.2.e Level $338$ Weight $2$ Character orbit 338.e Rep. character $\chi_{338}(23,\cdot)$ Character field $\Q(\zeta_{6})$ Dimension $28$ Newform subspaces $5$ Sturm bound $91$ Trace bound $3$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$338 = 2 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 338.e (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$13$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$5$$ Sturm bound: $$91$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(338, [\chi])$$.

Total New Old
Modular forms 120 28 92
Cusp forms 64 28 36
Eisenstein series 56 0 56

## Trace form

 $$28 q + 14 q^{4} - 14 q^{9} + O(q^{10})$$ $$28 q + 14 q^{4} - 14 q^{9} - 4 q^{14} - 14 q^{16} - 2 q^{17} + 2 q^{22} - 4 q^{23} - 20 q^{25} - 12 q^{27} + 6 q^{29} - 6 q^{30} + 10 q^{35} + 14 q^{36} + 4 q^{38} + 6 q^{42} - 4 q^{43} + 4 q^{49} + 28 q^{51} - 12 q^{53} + 8 q^{55} - 2 q^{56} + 10 q^{61} - 8 q^{62} - 28 q^{64} - 16 q^{66} + 2 q^{68} - 16 q^{69} + 8 q^{74} - 22 q^{75} + 40 q^{77} - 32 q^{79} + 10 q^{81} - 4 q^{82} + 20 q^{87} - 2 q^{88} + 28 q^{90} - 8 q^{92} + 6 q^{94} - 24 q^{95} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(338, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
338.2.e.a $4$ $2.699$ $$\Q(\zeta_{12})$$ None $$0$$ $$-2$$ $$0$$ $$0$$ $$q+\zeta_{12}q^{2}+(-1+\zeta_{12}^{2})q^{3}+\zeta_{12}^{2}q^{4}+\cdots$$
338.2.e.b $4$ $2.699$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{12}q^{2}+\zeta_{12}^{2}q^{4}+\zeta_{12}^{3}q^{5}+(4\zeta_{12}+\cdots)q^{7}+\cdots$$
338.2.e.c $4$ $2.699$ $$\Q(\zeta_{12})$$ None $$0$$ $$2$$ $$0$$ $$0$$ $$q+\zeta_{12}q^{2}+(1-\zeta_{12}^{2})q^{3}+\zeta_{12}^{2}q^{4}+\cdots$$
338.2.e.d $4$ $2.699$ $$\Q(\zeta_{12})$$ None $$0$$ $$6$$ $$0$$ $$0$$ $$q+\zeta_{12}q^{2}+(3-3\zeta_{12}^{2})q^{3}+\zeta_{12}^{2}q^{4}+\cdots$$
338.2.e.e $12$ $2.699$ 12.0.$$\cdots$$.1 None $$0$$ $$-6$$ $$0$$ $$0$$ $$q+\beta _{6}q^{2}+(\beta _{4}+2\beta _{9})q^{3}+\beta _{7}q^{4}+(2\beta _{2}+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(338, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(338, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(13, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(169, [\chi])$$$$^{\oplus 2}$$