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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

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

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.f.a 39.f 39.f $4$ $0.311$ \(\Q(\zeta_{8})\) None \(0\) \(-4\) \(0\) \(4\) $\mathrm{SU}(2)[C_{4}]$ \(q+\zeta_{8}q^{2}+(-1+\zeta_{8}+\zeta_{8}^{3})q^{3}-\zeta_{8}^{2}q^{4}+\cdots\)