# Properties

 Label 304.7.e Level $304$ Weight $7$ Character orbit 304.e Rep. character $\chi_{304}(113,\cdot)$ Character field $\Q$ Dimension $59$ Newform subspaces $6$ Sturm bound $280$ Trace bound $5$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$304 = 2^{4} \cdot 19$$ Weight: $$k$$ $$=$$ $$7$$ 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: $$280$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$3$$, $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{7}(304, [\chi])$$.

Total New Old
Modular forms 246 61 185
Cusp forms 234 59 175
Eisenstein series 12 2 10

## Trace form

 $$59q - 2q^{5} + 722q^{7} - 14773q^{9} + O(q^{10})$$ $$59q - 2q^{5} + 722q^{7} - 14773q^{9} + 2q^{11} + 6838q^{17} + 7009q^{19} + 8906q^{23} + 171873q^{25} + 7156q^{35} + 134624q^{39} + 174722q^{43} - 29794q^{45} + 125666q^{47} + 857073q^{49} - 392036q^{55} - 367288q^{57} - 502898q^{61} + 704866q^{63} - 613082q^{73} + 5740q^{77} + 2946235q^{81} + 1027698q^{83} - 309844q^{85} + 904272q^{87} - 1328240q^{93} - 3493070q^{95} - 431246q^{99} + O(q^{100})$$

## Decomposition of $$S_{7}^{\mathrm{new}}(304, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
304.7.e.a $$1$$ $$69.936$$ $$\Q$$ $$\Q(\sqrt{-19})$$ $$0$$ $$0$$ $$-54$$ $$-610$$ $$q-54q^{5}-610q^{7}+3^{6}q^{9}+1062q^{11}+\cdots$$
304.7.e.b $$2$$ $$69.936$$ $$\Q(\sqrt{57})$$ $$\Q(\sqrt{-19})$$ $$0$$ $$0$$ $$54$$ $$610$$ $$q+(3^{3}+7\beta )q^{5}+(305-9\beta )q^{7}+3^{6}q^{9}+\cdots$$
304.7.e.c $$8$$ $$69.936$$ $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ None $$0$$ $$0$$ $$2$$ $$-362$$ $$q+\beta _{1}q^{3}+\beta _{7}q^{5}+(-46-\beta _{2}-2\beta _{3}+\cdots)q^{7}+\cdots$$
304.7.e.d $$8$$ $$69.936$$ $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ None $$0$$ $$0$$ $$108$$ $$140$$ $$q+\beta _{1}q^{3}+(13-\beta _{5})q^{5}+(18-2\beta _{3}+\cdots)q^{7}+\cdots$$
304.7.e.e $$10$$ $$69.936$$ $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ None $$0$$ $$0$$ $$-112$$ $$224$$ $$q-\beta _{5}q^{3}+(-11-\beta _{1})q^{5}+(23-2\beta _{1}+\cdots)q^{7}+\cdots$$
304.7.e.f $$30$$ $$69.936$$ None $$0$$ $$0$$ $$0$$ $$720$$

## Decomposition of $$S_{7}^{\mathrm{old}}(304, [\chi])$$ into lower level spaces

$$S_{7}^{\mathrm{old}}(304, [\chi]) \cong$$ $$S_{7}^{\mathrm{new}}(19, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{7}^{\mathrm{new}}(38, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{7}^{\mathrm{new}}(76, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{7}^{\mathrm{new}}(152, [\chi])$$$$^{\oplus 2}$$