# Properties

 Label 304.2.k Level $304$ Weight $2$ Character orbit 304.k Rep. character $\chi_{304}(77,\cdot)$ Character field $\Q(\zeta_{4})$ Dimension $72$ Newform subspaces $2$ Sturm bound $80$ Trace bound $1$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$304 = 2^{4} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 304.k (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$16$$ Character field: $$\Q(i)$$ Newform subspaces: $$2$$ Sturm bound: $$80$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

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

Total New Old
Modular forms 84 72 12
Cusp forms 76 72 4
Eisenstein series 8 0 8

## Trace form

 $$72q - 4q^{4} - 12q^{6} + O(q^{10})$$ $$72q - 4q^{4} - 12q^{6} - 8q^{11} + 4q^{12} + 12q^{14} + 20q^{16} + 16q^{20} - 4q^{22} + 4q^{24} - 16q^{26} + 24q^{27} - 24q^{28} - 16q^{29} + 24q^{30} - 20q^{32} - 32q^{34} + 52q^{36} - 16q^{37} - 8q^{40} - 20q^{42} - 24q^{43} - 12q^{44} - 40q^{47} + 20q^{48} - 72q^{49} + 24q^{50} - 40q^{51} - 28q^{52} + 16q^{53} - 56q^{54} + 28q^{56} + 48q^{58} + 8q^{59} - 96q^{60} - 12q^{62} + 40q^{63} - 64q^{64} - 16q^{65} - 44q^{66} + 40q^{67} - 40q^{68} + 20q^{70} + 76q^{72} + 72q^{74} + 32q^{75} + 16q^{77} + 60q^{78} - 88q^{80} - 72q^{81} + 4q^{82} + 40q^{83} + 84q^{84} - 36q^{86} - 40q^{88} - 56q^{90} + 16q^{91} + 72q^{92} - 48q^{93} - 24q^{94} + 32q^{95} + 56q^{96} - 80q^{98} - 8q^{99} + O(q^{100})$$

## 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.k.a $$4$$ $$2.427$$ $$\Q(\zeta_{8})$$ None $$0$$ $$-4$$ $$-8$$ $$0$$ $$q+(\zeta_{8}+\zeta_{8}^{3})q^{2}+(-1+\zeta_{8}^{2}+\zeta_{8}^{3})q^{3}+\cdots$$
304.2.k.b $$68$$ $$2.427$$ None $$0$$ $$4$$ $$8$$ $$0$$

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

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