# Properties

 Label 396.2.h Level $396$ Weight $2$ Character orbit 396.h Rep. character $\chi_{396}(307,\cdot)$ Character field $\Q$ Dimension $28$ Newform subspaces $4$ Sturm bound $144$ Trace bound $4$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$396 = 2^{2} \cdot 3^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 396.h (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$44$$ Character field: $$\Q$$ Newform subspaces: $$4$$ Sturm bound: $$144$$ Trace bound: $$4$$ Distinguishing $$T_p$$: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(396, [\chi])$$.

Total New Old
Modular forms 80 32 48
Cusp forms 64 28 36
Eisenstein series 16 4 12

## Trace form

 $$28 q - 4 q^{4} + 4 q^{5} + O(q^{10})$$ $$28 q - 4 q^{4} + 4 q^{5} + 4 q^{14} - 16 q^{16} + 24 q^{20} - 16 q^{22} + 24 q^{25} - 12 q^{26} + 12 q^{37} - 24 q^{44} + 12 q^{49} + 24 q^{53} - 12 q^{56} - 16 q^{58} - 64 q^{64} - 16 q^{70} - 16 q^{77} - 60 q^{80} - 56 q^{86} + 12 q^{89} + 48 q^{92} - 52 q^{97} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(396, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
396.2.h.a $4$ $3.162$ $$\Q(\sqrt{-2}, \sqrt{-11})$$ $$\Q(\sqrt{-33})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{2}q^{2}-2q^{4}-\beta _{1}q^{7}-2\beta _{2}q^{8}+\cdots$$
396.2.h.b $4$ $3.162$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$4$$ $$0$$ $$q+\beta _{1}q^{2}+(-1+\beta _{2})q^{4}+q^{5}+(\beta _{1}+\cdots)q^{7}+\cdots$$
396.2.h.c $8$ $3.162$ 8.0.207360000.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{2}q^{2}+(1+\beta _{5})q^{4}+(-\beta _{4}+\beta _{7})q^{5}+\cdots$$
396.2.h.d $12$ $3.162$ 12.0.$$\cdots$$.2 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+\beta _{2}q^{4}+(\beta _{1}-\beta _{4}-\beta _{5}+\beta _{7}+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(396, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(396, [\chi]) \simeq$$ $$S_{2}^{\mathrm{new}}(44, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(132, [\chi])$$$$^{\oplus 2}$$