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 $0$

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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

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

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 \(0\) \(-4\) \(0\) \(4\) \(q+\zeta_{8}q^{2}+(-1+\zeta_{8}+\zeta_{8}^{3})q^{3}-\zeta_{8}^{2}q^{4}+\cdots\)