# Properties

 Label 338.2.b Level $338$ Weight $2$ Character orbit 338.b Rep. character $\chi_{338}(337,\cdot)$ Character field $\Q$ Dimension $12$ Newform subspaces $4$ Sturm bound $91$ Trace bound $3$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$338 = 2 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 338.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$13$$ Character field: $$\Q$$ Newform subspaces: $$4$$ 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 60 12 48
Cusp forms 32 12 20
Eisenstein series 28 0 28

## Trace form

 $$12 q + 2 q^{3} - 12 q^{4} + 18 q^{9} + O(q^{10})$$ $$12 q + 2 q^{3} - 12 q^{4} + 18 q^{9} - 6 q^{10} - 2 q^{12} + 4 q^{14} + 12 q^{16} - 4 q^{17} - 2 q^{22} + 16 q^{23} - 2 q^{25} - 16 q^{27} - 6 q^{29} + 12 q^{30} - 28 q^{35} - 18 q^{36} - 10 q^{38} + 6 q^{40} - 12 q^{42} + 6 q^{43} + 2 q^{48} + 20 q^{51} + 6 q^{53} - 8 q^{55} - 4 q^{56} + 6 q^{61} + 8 q^{62} - 12 q^{64} - 8 q^{66} + 4 q^{68} + 4 q^{69} - 2 q^{74} + 14 q^{75} + 20 q^{77} - 36 q^{79} - 12 q^{81} + 4 q^{82} - 20 q^{87} + 2 q^{88} + 26 q^{90} - 16 q^{92} + 60 q^{95} + O(q^{100})$$

## 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.b.a $2$ $2.699$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-6$$ $$0$$ $$0$$ $$q+iq^{2}-3q^{3}-q^{4}-iq^{5}-3iq^{6}+\cdots$$
338.2.b.b $2$ $2.699$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-q^{4}-iq^{5}-4iq^{7}-iq^{8}+\cdots$$
338.2.b.c $2$ $2.699$ $$\Q(\sqrt{-1})$$ None $$0$$ $$2$$ $$0$$ $$0$$ $$q+iq^{2}+q^{3}-q^{4}+3iq^{5}+iq^{6}+\cdots$$
338.2.b.d $6$ $2.699$ 6.0.153664.1 None $$0$$ $$6$$ $$0$$ $$0$$ $$q-\beta _{5}q^{2}+(1+\beta _{2}-\beta _{4})q^{3}-q^{4}-2\beta _{1}q^{5}+\cdots$$

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

$$S_{2}^{\mathrm{old}}(338, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(26, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(169, [\chi])$$$$^{\oplus 2}$$