# Properties

 Label 416.2.e Level $416$ Weight $2$ Character orbit 416.e Rep. character $\chi_{416}(337,\cdot)$ Character field $\Q$ Dimension $12$ Newform subspaces $3$ Sturm bound $112$ Trace bound $5$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$416 = 2^{5} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 416.e (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$104$$ Character field: $$\Q$$ Newform subspaces: $$3$$ Sturm bound: $$112$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$3$$, $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(416, [\chi])$$.

Total New Old
Modular forms 64 16 48
Cusp forms 48 12 36
Eisenstein series 16 4 12

## Trace form

 $$12 q - 12 q^{9} + O(q^{10})$$ $$12 q - 12 q^{9} - 8 q^{17} - 8 q^{23} - 4 q^{25} - 16 q^{39} + 12 q^{49} + 56 q^{55} - 16 q^{79} + 4 q^{81} - 8 q^{87} + 16 q^{95} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(416, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
416.2.e.a $2$ $3.322$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$-2$$ $$0$$ $$q+iq^{3}-q^{5}-3iq^{7}+2q^{9}+2q^{11}+\cdots$$
416.2.e.b $2$ $3.322$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$2$$ $$0$$ $$q+iq^{3}+q^{5}+3iq^{7}+2q^{9}-2q^{11}+\cdots$$
416.2.e.c $8$ $3.322$ 8.0.4521217600.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{2}q^{3}+\beta _{4}q^{5}+\beta _{3}q^{7}+(-3-\beta _{5}+\cdots)q^{9}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(416, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(416, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(104, [\chi])$$$$^{\oplus 3}$$