# Properties

 Label 385.2.b Level $385$ Weight $2$ Character orbit 385.b Rep. character $\chi_{385}(309,\cdot)$ Character field $\Q$ Dimension $32$ Newform subspaces $4$ Sturm bound $96$ Trace bound $10$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$385 = 5 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 385.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$4$$ Sturm bound: $$96$$ Trace bound: $$10$$ Distinguishing $$T_p$$: $$2$$, $$13$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(385, [\chi])$$.

Total New Old
Modular forms 52 32 20
Cusp forms 44 32 12
Eisenstein series 8 0 8

## Trace form

 $$32q - 28q^{4} + 6q^{5} - 8q^{6} - 44q^{9} + O(q^{10})$$ $$32q - 28q^{4} + 6q^{5} - 8q^{6} - 44q^{9} + 8q^{10} + 8q^{11} - 8q^{14} + 2q^{15} + 20q^{16} + 16q^{19} + 12q^{20} - 32q^{24} - 6q^{25} - 16q^{26} + 16q^{30} - 4q^{31} + 40q^{34} + 4q^{35} + 84q^{36} - 24q^{40} - 8q^{41} - 20q^{44} - 24q^{45} - 40q^{46} - 32q^{49} + 40q^{50} + 8q^{51} - 72q^{54} - 6q^{55} + 24q^{56} + 20q^{59} + 60q^{60} - 4q^{64} - 24q^{65} + 28q^{69} + 12q^{70} + 20q^{71} - 66q^{75} - 104q^{76} - 8q^{79} - 40q^{80} + 88q^{81} - 28q^{85} + 80q^{86} - 36q^{89} + 16q^{90} + 16q^{91} + 128q^{94} - 44q^{95} - 24q^{96} - 4q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(385, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
385.2.b.a $$2$$ $$3.074$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$2$$ $$0$$ $$q+iq^{2}-2iq^{3}+q^{4}+(1+2i)q^{5}+\cdots$$
385.2.b.b $$2$$ $$3.074$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$2$$ $$0$$ $$q+iq^{2}-2iq^{3}+q^{4}+(1-2i)q^{5}+\cdots$$
385.2.b.c $$12$$ $$3.074$$ $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ None $$0$$ $$0$$ $$6$$ $$0$$ $$q+\beta _{1}q^{2}+(\beta _{2}-\beta _{10})q^{3}+(-1+\beta _{4}+\cdots)q^{4}+\cdots$$
385.2.b.d $$16$$ $$3.074$$ $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ None $$0$$ $$0$$ $$-4$$ $$0$$ $$q+\beta _{1}q^{2}+(\beta _{7}+\beta _{13})q^{3}+(-2+\beta _{2}+\cdots)q^{4}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(385, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(385, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(35, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(55, [\chi])$$$$^{\oplus 2}$$