# Properties

 Label 39.2.j Level $39$ Weight $2$ Character orbit 39.j Rep. character $\chi_{39}(4,\cdot)$ Character field $\Q(\zeta_{6})$ Dimension $2$ Newform subspaces $1$ Sturm bound $9$ Trace bound $0$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$39 = 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 39.j (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$13$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$1$$ Sturm bound: $$9$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(39, [\chi])$$.

Total New Old
Modular forms 14 2 12
Cusp forms 6 2 4
Eisenstein series 8 0 8

## Trace form

 $$2q + q^{3} - 2q^{4} - 3q^{7} - q^{9} + O(q^{10})$$ $$2q + q^{3} - 2q^{4} - 3q^{7} - q^{9} - 6q^{11} - 4q^{12} + 7q^{13} + 6q^{15} - 4q^{16} + 6q^{19} + 12q^{20} + 6q^{23} - 14q^{25} - 2q^{27} + 6q^{28} - 6q^{29} - 6q^{33} - 6q^{35} - 2q^{36} + 2q^{39} - 12q^{41} + q^{43} + 6q^{45} + 4q^{48} - 4q^{49} - 10q^{52} + 24q^{53} + 12q^{55} - 6q^{59} - q^{61} + 3q^{63} + 16q^{64} + 6q^{65} + 15q^{67} - 6q^{69} - 18q^{71} - 7q^{75} - 12q^{76} + 12q^{77} - 22q^{79} - 24q^{80} - q^{81} + 6q^{84} + 6q^{87} + 12q^{89} - 9q^{91} - 24q^{92} + 3q^{93} + 12q^{95} - 9q^{97} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(39, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
39.2.j.a $$2$$ $$0.311$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$1$$ $$0$$ $$-3$$ $$q+(1-\zeta_{6})q^{3}-2\zeta_{6}q^{4}+(-2+4\zeta_{6})q^{5}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(39, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(39, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(13, [\chi])$$$$^{\oplus 2}$$