# Properties

 Label 39.2.k Level $39$ Weight $2$ Character orbit 39.k Rep. character $\chi_{39}(2,\cdot)$ Character field $\Q(\zeta_{12})$ Dimension $12$ Newform subspaces $2$ Sturm bound $9$ Trace bound $1$

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## Defining parameters

 Level: $$N$$ $$=$$ $$39 = 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 39.k (of order $$12$$ and degree $$4$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$39$$ Character field: $$\Q(\zeta_{12})$$ Newform subspaces: $$2$$ Sturm bound: $$9$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

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

Total New Old
Modular forms 28 28 0
Cusp forms 12 12 0
Eisenstein series 16 16 0

## Trace form

 $$12q - 2q^{3} - 12q^{4} - 2q^{6} - 14q^{7} - 2q^{9} + O(q^{10})$$ $$12q - 2q^{3} - 12q^{4} - 2q^{6} - 14q^{7} - 2q^{9} + 12q^{10} + 8q^{13} - 14q^{15} + 4q^{16} + 4q^{18} - 2q^{19} + 22q^{21} + 4q^{22} + 18q^{24} + 4q^{27} + 18q^{30} - 6q^{31} + 16q^{33} - 36q^{34} - 36q^{36} - 30q^{37} - 38q^{39} - 24q^{40} - 16q^{42} + 30q^{43} - 20q^{45} - 14q^{48} + 18q^{49} + 76q^{52} + 46q^{54} + 4q^{55} + 28q^{57} + 28q^{58} + 44q^{60} + 28q^{61} + 16q^{63} - 40q^{66} - 8q^{67} - 32q^{70} + 12q^{72} - 62q^{73} - 18q^{75} - 36q^{76} - 80q^{78} + 16q^{79} - 14q^{81} - 24q^{82} - 8q^{84} + 12q^{85} - 34q^{87} - 12q^{88} + 10q^{91} + 10q^{93} + 64q^{94} + 16q^{96} - 18q^{97} + 40q^{99} + O(q^{100})$$

## 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.k.a $$4$$ $$0.311$$ $$\Q(\zeta_{12})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$-10$$ $$q+(-\zeta_{12}-\zeta_{12}^{3})q^{3}+2\zeta_{12}q^{4}+(-2+\cdots)q^{7}+\cdots$$
39.2.k.b $$8$$ $$0.311$$ 8.0.56070144.2 None $$0$$ $$-2$$ $$0$$ $$-4$$ $$q+(-\beta _{1}-\beta _{2}+\beta _{3}-\beta _{4}+2\beta _{5}+\beta _{7})q^{2}+\cdots$$