# Properties

 Label 416.2.w Level $416$ Weight $2$ Character orbit 416.w Rep. character $\chi_{416}(225,\cdot)$ Character field $\Q(\zeta_{6})$ Dimension $28$ Newform subspaces $4$ Sturm bound $112$ Trace bound $9$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$416 = 2^{5} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 416.w (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$13$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$4$$ Sturm bound: $$112$$ Trace bound: $$9$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

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

Total New Old
Modular forms 128 28 100
Cusp forms 96 28 68
Eisenstein series 32 0 32

## Trace form

 $$28 q - 14 q^{9} + O(q^{10})$$ $$28 q - 14 q^{9} + 6 q^{13} - 6 q^{17} - 40 q^{25} + 2 q^{29} + 30 q^{37} + 18 q^{41} - 30 q^{45} + 30 q^{49} - 12 q^{53} - 10 q^{61} + 10 q^{65} + 16 q^{69} + 48 q^{77} + 2 q^{81} - 6 q^{85} - 72 q^{93} - 48 q^{97} + O(q^{100})$$

## 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.w.a $4$ $3.322$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(-\zeta_{12}-\zeta_{12}^{3})q^{3}+(-2+4\zeta_{12}^{2}+\cdots)q^{5}+\cdots$$
416.2.w.b $4$ $3.322$ $$\Q(\zeta_{12})$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+(-1+2\zeta_{12}^{2}+\zeta_{12}^{3})q^{5}+(3-3\zeta_{12}^{2}+\cdots)q^{9}+\cdots$$
416.2.w.c $8$ $3.322$ 8.0.56070144.2 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{6}q^{3}+(-\beta _{4}+\beta _{5})q^{5}+\beta _{3}q^{7}+\cdots$$
416.2.w.d $12$ $3.322$ 12.0.$$\cdots$$.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{9}q^{3}+(-2\beta _{4}-\beta _{5}+\beta _{6})q^{5}-\beta _{3}q^{7}+\cdots$$

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

$$S_{2}^{\mathrm{old}}(416, [\chi]) \simeq$$ $$S_{2}^{\mathrm{new}}(13, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(26, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(52, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(104, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(208, [\chi])$$$$^{\oplus 2}$$