# Properties

 Label 304.5.e Level $304$ Weight $5$ Character orbit 304.e Rep. character $\chi_{304}(113,\cdot)$ Character field $\Q$ Dimension $39$ Newform subspaces $6$ Sturm bound $200$ Trace bound $5$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$304 = 2^{4} \cdot 19$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 304.e (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$19$$ Character field: $$\Q$$ Newform subspaces: $$6$$ Sturm bound: $$200$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$3$$, $$5$$

## Dimensions

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

Total New Old
Modular forms 166 41 125
Cusp forms 154 39 115
Eisenstein series 12 2 10

## Trace form

 $$39q - 2q^{5} - 30q^{7} - 889q^{9} + O(q^{10})$$ $$39q - 2q^{5} - 30q^{7} - 889q^{9} + 2q^{11} - 50q^{17} + 65q^{19} - 718q^{23} + 4373q^{25} + 4132q^{35} + 3008q^{39} + 2818q^{43} - 1090q^{45} + 2882q^{47} + 9845q^{49} + 7828q^{55} + 1136q^{57} + 7902q^{61} - 7742q^{63} - 274q^{73} - 484q^{77} + 5671q^{81} + 6434q^{83} + 15260q^{85} + 33120q^{87} + 14560q^{93} + 15458q^{95} + 24610q^{99} + O(q^{100})$$

## 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.e.a $$1$$ $$31.424$$ $$\Q$$ $$\Q(\sqrt{-19})$$ $$0$$ $$0$$ $$31$$ $$73$$ $$q+31q^{5}+73q^{7}+3^{4}q^{9}+233q^{11}+\cdots$$
304.5.e.b $$2$$ $$31.424$$ $$\Q(\sqrt{57})$$ $$\Q(\sqrt{-19})$$ $$0$$ $$0$$ $$-31$$ $$-73$$ $$q+(-17-3\beta )q^{5}+(-39-5\beta )q^{7}+\cdots$$
304.5.e.c $$4$$ $$31.424$$ 4.0.12107488.1 None $$0$$ $$0$$ $$-42$$ $$-136$$ $$q-\beta _{2}q^{3}+(-10-\beta _{3})q^{5}+(-31-6\beta _{3})q^{7}+\cdots$$
304.5.e.d $$4$$ $$31.424$$ $$\mathbb{Q}[x]/(x^{4} + \cdots)$$ None $$0$$ $$0$$ $$22$$ $$-24$$ $$q+\beta _{1}q^{3}+(6-\beta _{2})q^{5}+(-7+2\beta _{2}+\cdots)q^{7}+\cdots$$
304.5.e.e $$8$$ $$31.424$$ $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ None $$0$$ $$0$$ $$18$$ $$162$$ $$q-\beta _{1}q^{3}+(2-\beta _{5})q^{5}+(20-\beta _{4})q^{7}+\cdots$$
304.5.e.f $$20$$ $$31.424$$ $$\mathbb{Q}[x]/(x^{20} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$-32$$ $$q+\beta _{1}q^{3}-\beta _{3}q^{5}+(-2+\beta _{5})q^{7}+(-19+\cdots)q^{9}+\cdots$$

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

$$S_{5}^{\mathrm{old}}(304, [\chi]) \cong$$ $$S_{5}^{\mathrm{new}}(19, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(38, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(76, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(152, [\chi])$$$$^{\oplus 2}$$