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

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

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.k.a 39.k 39.k $4$ $0.311$ \(\Q(\zeta_{12})\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-10\) $\mathrm{U}(1)[D_{12}]$ \(q+(-\zeta_{12}-\zeta_{12}^{3})q^{3}+2\zeta_{12}q^{4}+(-2+\cdots)q^{7}+\cdots\)
39.2.k.b 39.k 39.k $8$ $0.311$ 8.0.56070144.2 None \(0\) \(-2\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{12}]$ \(q+(-\beta _{1}-\beta _{2}+\beta _{3}-\beta _{4}+2\beta _{5}+\beta _{7})q^{2}+\cdots\)