# Properties

 Label 304.2.h Level $304$ Weight $2$ Character orbit 304.h Rep. character $\chi_{304}(303,\cdot)$ Character field $\Q$ Dimension $10$ Newform subspaces $3$ Sturm bound $80$ Trace bound $5$

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## Defining parameters

 Level: $$N$$ $$=$$ $$304 = 2^{4} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 304.h (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$76$$ Character field: $$\Q$$ Newform subspaces: $$3$$ Sturm bound: $$80$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$3$$, $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(304, [\chi])$$.

Total New Old
Modular forms 46 10 36
Cusp forms 34 10 24
Eisenstein series 12 0 12

## Trace form

 $$10 q - 2 q^{9} + O(q^{10})$$ $$10 q - 2 q^{9} + 12 q^{17} + 10 q^{25} - 26 q^{49} - 28 q^{57} - 32 q^{61} - 44 q^{73} + 12 q^{77} + 34 q^{81} + 36 q^{85} - 56 q^{93} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(304, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
304.2.h.a $2$ $2.427$ $$\Q(\sqrt{-19})$$ $$\Q(\sqrt{-19})$$ $$0$$ $$0$$ $$-2$$ $$0$$ $$q-q^{5}-\beta q^{7}-3q^{9}-\beta q^{11}+7q^{17}+\cdots$$
304.2.h.b $4$ $2.427$ $$\Q(\sqrt{-3}, \sqrt{7})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{3}-\beta _{2}q^{7}+4q^{9}+2\beta _{2}q^{11}+\cdots$$
304.2.h.c $4$ $2.427$ $$\Q(\sqrt{-3}, \sqrt{-19})$$ $$\Q(\sqrt{-19})$$ $$0$$ $$0$$ $$2$$ $$0$$ $$q+(1-\beta _{2})q^{5}+(2\beta _{1}+\beta _{3})q^{7}-3q^{9}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(304, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(304, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(76, [\chi])$$$$^{\oplus 3}$$