# Properties

 Label 39.2.f Level $39$ Weight $2$ Character orbit 39.f Rep. character $\chi_{39}(5,\cdot)$ Character field $\Q(\zeta_{4})$ Dimension $4$ 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.f (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$39$$ Character field: $$\Q(i)$$ 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 12 12 0
Cusp forms 4 4 0
Eisenstein series 8 8 0

## Trace form

 $$4 q - 4 q^{3} - 4 q^{6} + 4 q^{7} - 4 q^{9} + O(q^{10})$$ $$4 q - 4 q^{3} - 4 q^{6} + 4 q^{7} - 4 q^{9} - 8 q^{13} + 8 q^{15} + 4 q^{16} + 8 q^{18} + 4 q^{19} - 4 q^{21} - 16 q^{22} + 12 q^{24} + 20 q^{27} + 4 q^{28} - 20 q^{31} - 16 q^{33} + 4 q^{37} + 8 q^{39} - 24 q^{40} - 8 q^{42} - 16 q^{45} + 24 q^{46} - 4 q^{48} + 12 q^{52} - 4 q^{54} + 32 q^{55} - 4 q^{57} + 8 q^{58} - 8 q^{60} + 32 q^{61} - 4 q^{63} + 16 q^{66} - 20 q^{67} + 8 q^{70} - 24 q^{72} + 4 q^{73} - 4 q^{76} - 4 q^{78} - 40 q^{79} - 28 q^{81} - 4 q^{84} + 16 q^{87} - 20 q^{91} + 20 q^{93} - 16 q^{94} + 20 q^{96} + 28 q^{97} + 32 q^{99} + O(q^{100})$$

## 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.f.a $4$ $0.311$ $$\Q(\zeta_{8})$$ None $$0$$ $$-4$$ $$0$$ $$4$$ $$q+\zeta_{8}q^{2}+(-1+\zeta_{8}+\zeta_{8}^{3})q^{3}-\zeta_{8}^{2}q^{4}+\cdots$$