# Properties

 Label 405.2.e Level $405$ Weight $2$ Character orbit 405.e Rep. character $\chi_{405}(136,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $32$ Newform subspaces $12$ Sturm bound $108$ Trace bound $7$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$405 = 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 405.e (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$9$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$12$$ Sturm bound: $$108$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$2$$, $$7$$, $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(405, [\chi])$$.

Total New Old
Modular forms 132 32 100
Cusp forms 84 32 52
Eisenstein series 48 0 48

## Trace form

 $$32q - 16q^{4} + 10q^{7} + O(q^{10})$$ $$32q - 16q^{4} + 10q^{7} + 10q^{13} - 16q^{16} + 4q^{19} - 12q^{22} - 16q^{25} - 80q^{28} + 40q^{31} + 18q^{34} - 20q^{37} + 30q^{40} + 28q^{43} - 36q^{46} - 18q^{49} + 4q^{52} - 36q^{58} - 14q^{61} + 44q^{64} - 14q^{67} - 12q^{70} + 76q^{73} + 34q^{76} - 2q^{79} + 96q^{82} - 12q^{85} - 36q^{88} - 52q^{91} - 48q^{94} - 26q^{97} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(405, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
405.2.e.a $$2$$ $$3.234$$ $$\Q(\sqrt{-3})$$ None $$-2$$ $$0$$ $$-1$$ $$0$$ $$q+(-2+2\zeta_{6})q^{2}-2\zeta_{6}q^{4}-\zeta_{6}q^{5}+\cdots$$
405.2.e.b $$2$$ $$3.234$$ $$\Q(\sqrt{-3})$$ None $$-2$$ $$0$$ $$-1$$ $$3$$ $$q+(-2+2\zeta_{6})q^{2}-2\zeta_{6}q^{4}-\zeta_{6}q^{5}+\cdots$$
405.2.e.c $$2$$ $$3.234$$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$0$$ $$1$$ $$0$$ $$q+(-1+\zeta_{6})q^{2}+\zeta_{6}q^{4}+\zeta_{6}q^{5}-3q^{8}+\cdots$$
405.2.e.d $$2$$ $$3.234$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-1$$ $$-2$$ $$q+2\zeta_{6}q^{4}-\zeta_{6}q^{5}+(-2+2\zeta_{6})q^{7}+\cdots$$
405.2.e.e $$2$$ $$3.234$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$1$$ $$-2$$ $$q+2\zeta_{6}q^{4}+\zeta_{6}q^{5}+(-2+2\zeta_{6})q^{7}+\cdots$$
405.2.e.f $$2$$ $$3.234$$ $$\Q(\sqrt{-3})$$ None $$1$$ $$0$$ $$-1$$ $$0$$ $$q+(1-\zeta_{6})q^{2}+\zeta_{6}q^{4}-\zeta_{6}q^{5}+3q^{8}+\cdots$$
405.2.e.g $$2$$ $$3.234$$ $$\Q(\sqrt{-3})$$ None $$2$$ $$0$$ $$1$$ $$0$$ $$q+(2-2\zeta_{6})q^{2}-2\zeta_{6}q^{4}+\zeta_{6}q^{5}+2q^{10}+\cdots$$
405.2.e.h $$2$$ $$3.234$$ $$\Q(\sqrt{-3})$$ None $$2$$ $$0$$ $$1$$ $$3$$ $$q+(2-2\zeta_{6})q^{2}-2\zeta_{6}q^{4}+\zeta_{6}q^{5}+(3+\cdots)q^{7}+\cdots$$
405.2.e.i $$4$$ $$3.234$$ $$\Q(\zeta_{12})$$ None $$-2$$ $$0$$ $$2$$ $$6$$ $$q+(-\zeta_{12}+\zeta_{12}^{2})q^{2}+(-2+2\zeta_{12}+\cdots)q^{4}+\cdots$$
405.2.e.j $$4$$ $$3.234$$ $$\Q(\sqrt{-3}, \sqrt{13})$$ None $$-1$$ $$0$$ $$2$$ $$-2$$ $$q-\beta _{1}q^{2}+(-1+\beta _{1}+\beta _{2}+\beta _{3})q^{4}+\cdots$$
405.2.e.k $$4$$ $$3.234$$ $$\Q(\sqrt{-3}, \sqrt{13})$$ None $$1$$ $$0$$ $$-2$$ $$-2$$ $$q+\beta _{1}q^{2}+(-1+\beta _{1}+\beta _{2}+\beta _{3})q^{4}+\cdots$$
405.2.e.l $$4$$ $$3.234$$ $$\Q(\zeta_{12})$$ None $$2$$ $$0$$ $$-2$$ $$6$$ $$q+(\zeta_{12}-\zeta_{12}^{2})q^{2}+(-2+2\zeta_{12}-2\zeta_{12}^{2}+\cdots)q^{4}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(405, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(405, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(45, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(81, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(135, [\chi])$$$$^{\oplus 2}$$