Properties

Label 39.2.a
Level $39$
Weight $2$
Character orbit 39.a
Rep. character $\chi_{39}(1,\cdot)$
Character field $\Q$
Dimension $3$
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.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(9\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(39))\).

Total New Old
Modular forms 6 3 3
Cusp forms 3 3 0
Eisenstein series 3 0 3

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(3\)\(13\)FrickeDim.
\(+\)\(-\)\(-\)\(1\)
\(-\)\(+\)\(-\)\(2\)
Plus space\(+\)\(0\)
Minus space\(-\)\(3\)

Trace form

\( 3 q - q^{2} + q^{3} + q^{4} + 2 q^{5} - 3 q^{6} - 4 q^{7} - 9 q^{8} + 3 q^{9} + O(q^{10}) \) \( 3 q - q^{2} + q^{3} + q^{4} + 2 q^{5} - 3 q^{6} - 4 q^{7} - 9 q^{8} + 3 q^{9} - 6 q^{10} + 3 q^{12} - q^{13} + 4 q^{14} - 2 q^{15} + 5 q^{16} + 6 q^{17} - q^{18} + 14 q^{20} + 4 q^{21} + 8 q^{22} - 8 q^{23} - 3 q^{24} + 5 q^{25} + 3 q^{26} + q^{27} - 12 q^{28} - 6 q^{29} - 10 q^{30} - 4 q^{31} + 11 q^{32} - 8 q^{33} + 14 q^{34} - 24 q^{35} + q^{36} - 6 q^{37} - 8 q^{38} - 3 q^{39} - 14 q^{40} + 22 q^{41} + 12 q^{42} - 4 q^{43} - 8 q^{44} + 2 q^{45} + 8 q^{46} - 12 q^{47} + 7 q^{48} + 11 q^{49} - 7 q^{50} + 2 q^{51} - 3 q^{52} + 2 q^{53} - 3 q^{54} + 8 q^{55} + 20 q^{56} - 14 q^{58} + 16 q^{59} + 18 q^{60} + 2 q^{61} + 20 q^{62} - 4 q^{63} - 7 q^{64} + 2 q^{65} - 30 q^{68} - 8 q^{69} + 8 q^{70} + 4 q^{71} - 9 q^{72} + 14 q^{73} - 14 q^{74} + 7 q^{75} + 16 q^{76} - 16 q^{77} + q^{78} + 8 q^{79} - 2 q^{80} + 3 q^{81} - 18 q^{82} - 20 q^{84} - 28 q^{85} - 36 q^{86} + 14 q^{87} + 22 q^{89} - 6 q^{90} - 4 q^{91} - 8 q^{92} - 12 q^{93} - 4 q^{94} + 16 q^{95} + q^{96} + 6 q^{97} + 7 q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(39))\) into newform subspaces

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