# Properties

 Label 304.5.d Level $304$ Weight $5$ Character orbit 304.d Rep. character $\chi_{304}(191,\cdot)$ Character field $\Q$ Dimension $36$ Newform subspaces $2$ Sturm bound $200$ Trace bound $1$

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## Defining parameters

 Level: $$N$$ $$=$$ $$304 = 2^{4} \cdot 19$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 304.d (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$4$$ Character field: $$\Q$$ Newform subspaces: $$2$$ Sturm bound: $$200$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{5}(304, [\chi])$$.

Total New Old
Modular forms 166 36 130
Cusp forms 154 36 118
Eisenstein series 12 0 12

## Trace form

 $$36q - 72q^{5} - 972q^{9} + O(q^{10})$$ $$36q - 72q^{5} - 972q^{9} - 360q^{13} + 360q^{17} - 1824q^{21} + 6204q^{25} + 5688q^{29} - 4320q^{33} - 4200q^{37} - 1080q^{41} + 7992q^{45} - 12828q^{49} + 2520q^{53} - 6600q^{61} + 7056q^{65} - 41472q^{69} + 12840q^{73} + 30384q^{77} + 33828q^{81} + 15360q^{85} - 15768q^{89} - 44640q^{93} - 4920q^{97} + O(q^{100})$$

## Decomposition of $$S_{5}^{\mathrm{new}}(304, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
304.5.d.a $$12$$ $$31.424$$ $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ None $$0$$ $$0$$ $$-18$$ $$0$$ $$q+\beta _{1}q^{3}+(-1+\beta _{3})q^{5}+\beta _{5}q^{7}+(-28+\cdots)q^{9}+\cdots$$
304.5.d.b $$24$$ $$31.424$$ None $$0$$ $$0$$ $$-54$$ $$0$$

## Decomposition of $$S_{5}^{\mathrm{old}}(304, [\chi])$$ into lower level spaces

$$S_{5}^{\mathrm{old}}(304, [\chi]) \cong$$ $$S_{5}^{\mathrm{new}}(4, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(16, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(76, [\chi])$$$$^{\oplus 3}$$