Properties

Label 39.2.a
Level 39
Weight 2
Character orbit a
Rep. character \(\chi_{39}(1,\cdot)\)
Character field \(\Q\)
Dimension 3
Newforms 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: \(\Q\)
Newforms: \( 2 \)
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

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(39))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 13
39.2.a.a \(1\) \(0.311\) \(\Q\) None \(1\) \(-1\) \(2\) \(-4\) \(+\) \(-\) \(q+q^{2}-q^{3}-q^{4}+2q^{5}-q^{6}-4q^{7}+\cdots\)
39.2.a.b \(2\) \(0.311\) \(\Q(\sqrt{2}) \) None \(-2\) \(2\) \(0\) \(0\) \(-\) \(+\) \(q+(-1+\beta )q^{2}+q^{3}+(1-2\beta )q^{4}-2\beta q^{5}+\cdots\)