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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 $0$

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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

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

Display 100 coefficients 10 coefficients

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

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
39.2.j.a 39.j 13.e $2$ $0.311$ \(\Q(\sqrt{-3}) \) None \(0\) \(1\) \(0\) \(-3\) $\mathrm{SU}(2)[C_{6}]$ \(q+(1-\zeta_{6})q^{3}-2\zeta_{6}q^{4}+(-2+4\zeta_{6})q^{5}+\cdots\)

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

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