# Properties

 Label 585.2.j Level $585$ Weight $2$ Character orbit 585.j Rep. character $\chi_{585}(406,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $48$ Newform subspaces $9$ Sturm bound $168$ Trace bound $4$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$585 = 3^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 585.j (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$13$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$9$$ Sturm bound: $$168$$ Trace bound: $$4$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(585, [\chi])$$.

Total New Old
Modular forms 184 48 136
Cusp forms 152 48 104
Eisenstein series 32 0 32

## Trace form

 $$48 q - 26 q^{4} - 6 q^{7} + 12 q^{8} + O(q^{10})$$ $$48 q - 26 q^{4} - 6 q^{7} + 12 q^{8} - 2 q^{10} + 8 q^{11} + 8 q^{13} - 4 q^{14} - 26 q^{16} + 6 q^{17} + 8 q^{19} + 4 q^{20} - 20 q^{22} + 18 q^{23} + 48 q^{25} + 38 q^{26} - 22 q^{28} - 12 q^{29} + 8 q^{31} - 22 q^{32} + 4 q^{34} + 10 q^{35} + 10 q^{37} - 104 q^{38} + 12 q^{40} - 20 q^{41} - 14 q^{43} - 12 q^{44} + 26 q^{46} + 24 q^{47} - 40 q^{49} + 2 q^{52} + 24 q^{53} - 8 q^{55} + 24 q^{56} - 32 q^{58} + 28 q^{59} - 4 q^{61} - 8 q^{62} + 2 q^{67} - 38 q^{68} - 8 q^{70} - 28 q^{71} - 54 q^{74} - 30 q^{76} + 36 q^{77} - 64 q^{79} + 32 q^{80} - 4 q^{82} + 16 q^{83} + 6 q^{85} - 12 q^{86} - 26 q^{88} + 32 q^{89} - 32 q^{91} - 84 q^{92} - 20 q^{94} + 16 q^{95} + 18 q^{97} + 60 q^{98} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(585, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
585.2.j.a $2$ $4.671$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$2$$ $$1$$ $$q+2\zeta_{6}q^{4}+q^{5}+\zeta_{6}q^{7}+(6-6\zeta_{6})q^{11}+\cdots$$
585.2.j.b $2$ $4.671$ $$\Q(\sqrt{-3})$$ None $$2$$ $$0$$ $$2$$ $$-5$$ $$q+(2-2\zeta_{6})q^{2}-2\zeta_{6}q^{4}+q^{5}-5\zeta_{6}q^{7}+\cdots$$
585.2.j.c $4$ $4.671$ $$\Q(\zeta_{12})$$ None $$-2$$ $$0$$ $$-4$$ $$0$$ $$q+(-\zeta_{12}+\zeta_{12}^{2})q^{2}+(-2+2\zeta_{12}+\cdots)q^{4}+\cdots$$
585.2.j.d $4$ $4.671$ $$\Q(\sqrt{-3}, \sqrt{13})$$ None $$-1$$ $$0$$ $$4$$ $$2$$ $$q-\beta _{1}q^{2}+(-1+\beta _{1}+\beta _{2}+\beta _{3})q^{4}+\cdots$$
585.2.j.e $4$ $4.671$ $$\Q(\sqrt{-3}, \sqrt{5})$$ None $$1$$ $$0$$ $$-4$$ $$-4$$ $$q+\beta _{1}q^{2}+(\beta _{1}+\beta _{2}-\beta _{3})q^{4}-q^{5}+\cdots$$
585.2.j.f $6$ $4.671$ 6.0.1714608.1 None $$0$$ $$0$$ $$-6$$ $$-3$$ $$q+(\beta _{3}-\beta _{5})q^{2}+(-2\beta _{1}-\beta _{2}+\beta _{4}+\cdots)q^{4}+\cdots$$
585.2.j.g $6$ $4.671$ 6.0.591408.1 None $$0$$ $$0$$ $$6$$ $$5$$ $$q+\beta _{5}q^{2}+(\beta _{1}-\beta _{3}-\beta _{4}+\beta _{5})q^{4}+\cdots$$
585.2.j.h $10$ $4.671$ $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ None $$-2$$ $$0$$ $$10$$ $$-1$$ $$q+(\beta _{1}+\beta _{3})q^{2}+(-1+\beta _{4}-\beta _{8})q^{4}+\cdots$$
585.2.j.i $10$ $4.671$ $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ None $$2$$ $$0$$ $$-10$$ $$-1$$ $$q+(-\beta _{1}-\beta _{3})q^{2}+(-1+\beta _{4}-\beta _{8}+\cdots)q^{4}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(585, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(585, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(39, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(65, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(117, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(195, [\chi])$$$$^{\oplus 2}$$