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Properties

Label 39.2.e
Level $39$
Weight $2$
Character orbit 39.e
Rep. character $\chi_{39}(16,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $6$
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.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q(\zeta_{3})\)
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 14 6 8
Cusp forms 6 6 0
Eisenstein series 8 0 8

Trace form

\( 6 q - 2 q^{2} - q^{3} - 4 q^{4} - 8 q^{5} + q^{7} + 12 q^{8} - 3 q^{9} + O(q^{10}) \) \( 6 q - 2 q^{2} - q^{3} - 4 q^{4} - 8 q^{5} + q^{7} + 12 q^{8} - 3 q^{9} - 6 q^{10} + 6 q^{11} + 12 q^{12} - 5 q^{13} - 16 q^{14} + 2 q^{15} - 2 q^{16} + 6 q^{17} + 4 q^{18} - 2 q^{20} - 10 q^{21} + 4 q^{22} + 2 q^{23} - 12 q^{24} - 2 q^{25} + 30 q^{26} + 2 q^{27} + 18 q^{28} - 8 q^{30} + 10 q^{31} - 14 q^{32} + 2 q^{33} + 4 q^{34} + 6 q^{35} - 4 q^{36} - 12 q^{37} - 40 q^{38} - 6 q^{39} - 4 q^{40} - 10 q^{41} + 12 q^{42} - 11 q^{43} - 16 q^{44} + 4 q^{45} + 4 q^{46} + 12 q^{47} - 4 q^{48} + 4 q^{49} + 28 q^{50} + 16 q^{51} - 30 q^{52} + 4 q^{53} - 8 q^{55} + 28 q^{56} + 24 q^{57} + 26 q^{58} + 14 q^{59} - 17 q^{61} + 4 q^{62} + q^{63} + 28 q^{64} + 4 q^{65} - 3 q^{67} - 18 q^{68} - 10 q^{69} - 8 q^{70} - 34 q^{71} - 6 q^{72} - 2 q^{73} + 2 q^{74} - 7 q^{75} - 4 q^{76} + 4 q^{77} - 28 q^{78} + 22 q^{79} - 22 q^{80} - 3 q^{81} - 18 q^{82} - 48 q^{83} + 14 q^{84} - 14 q^{85} - 2 q^{87} + 12 q^{88} - 4 q^{89} + 12 q^{90} + 31 q^{91} + 32 q^{92} + 3 q^{93} + 28 q^{94} + 20 q^{95} + 8 q^{96} + 15 q^{97} - 22 q^{98} - 12 q^{99} + O(q^{100}) \)

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.e.a 39.e 13.c $2$ $0.311$ \(\Q(\sqrt{-3}) \) None \(-1\) \(1\) \(-2\) \(-2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots\)
39.2.e.b 39.e 13.c $4$ $0.311$ \(\Q(\sqrt{-3}, \sqrt{17})\) None \(-1\) \(-2\) \(-6\) \(3\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{1}q^{2}-\beta _{2}q^{3}+(-2+\beta _{1}+2\beta _{2}+\cdots)q^{4}+\cdots\)