Properties

Label 31.2.a
Level 31
Weight 2
Character orbit a
Rep. character \(\chi_{31}(1,\cdot)\)
Character field \(\Q\)
Dimension 2
Newforms 1
Sturm bound 5
Trace bound 0

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Defining parameters

Level: \( N \) = \( 31 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 31.a (trivial)
Character field: \(\Q\)
Newforms: \( 1 \)
Sturm bound: \(5\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(31))\).

Total New Old
Modular forms 3 3 0
Cusp forms 2 2 0
Eisenstein series 1 1 0

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

\(31\)Dim.
\(-\)\(2\)

Trace form

\(2q \) \(\mathstrut +\mathstrut q^{2} \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut q^{4} \) \(\mathstrut +\mathstrut 2q^{5} \) \(\mathstrut -\mathstrut 6q^{6} \) \(\mathstrut -\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 6q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut q^{2} \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut q^{4} \) \(\mathstrut +\mathstrut 2q^{5} \) \(\mathstrut -\mathstrut 6q^{6} \) \(\mathstrut -\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 6q^{9} \) \(\mathstrut +\mathstrut q^{10} \) \(\mathstrut +\mathstrut 4q^{11} \) \(\mathstrut -\mathstrut 4q^{12} \) \(\mathstrut -\mathstrut 2q^{13} \) \(\mathstrut +\mathstrut 3q^{14} \) \(\mathstrut -\mathstrut 2q^{15} \) \(\mathstrut -\mathstrut 3q^{16} \) \(\mathstrut +\mathstrut 6q^{17} \) \(\mathstrut +\mathstrut 13q^{18} \) \(\mathstrut -\mathstrut q^{20} \) \(\mathstrut -\mathstrut 6q^{21} \) \(\mathstrut +\mathstrut 2q^{22} \) \(\mathstrut -\mathstrut 2q^{23} \) \(\mathstrut +\mathstrut 10q^{24} \) \(\mathstrut -\mathstrut 8q^{25} \) \(\mathstrut -\mathstrut 6q^{26} \) \(\mathstrut -\mathstrut 20q^{27} \) \(\mathstrut +\mathstrut 7q^{28} \) \(\mathstrut +\mathstrut 10q^{29} \) \(\mathstrut -\mathstrut 6q^{30} \) \(\mathstrut +\mathstrut 2q^{31} \) \(\mathstrut -\mathstrut 9q^{32} \) \(\mathstrut -\mathstrut 4q^{33} \) \(\mathstrut -\mathstrut 2q^{34} \) \(\mathstrut -\mathstrut 4q^{35} \) \(\mathstrut +\mathstrut 7q^{36} \) \(\mathstrut -\mathstrut 4q^{37} \) \(\mathstrut -\mathstrut 5q^{38} \) \(\mathstrut +\mathstrut 12q^{39} \) \(\mathstrut +\mathstrut 14q^{41} \) \(\mathstrut +\mathstrut 2q^{42} \) \(\mathstrut -\mathstrut 2q^{43} \) \(\mathstrut -\mathstrut 2q^{44} \) \(\mathstrut +\mathstrut 6q^{45} \) \(\mathstrut +\mathstrut 14q^{46} \) \(\mathstrut -\mathstrut 4q^{47} \) \(\mathstrut +\mathstrut 18q^{48} \) \(\mathstrut +\mathstrut 4q^{49} \) \(\mathstrut -\mathstrut 4q^{50} \) \(\mathstrut +\mathstrut 4q^{51} \) \(\mathstrut -\mathstrut 4q^{52} \) \(\mathstrut -\mathstrut 12q^{53} \) \(\mathstrut -\mathstrut 20q^{54} \) \(\mathstrut +\mathstrut 4q^{55} \) \(\mathstrut -\mathstrut 10q^{56} \) \(\mathstrut +\mathstrut 10q^{57} \) \(\mathstrut -\mathstrut 4q^{60} \) \(\mathstrut -\mathstrut 6q^{61} \) \(\mathstrut +\mathstrut q^{62} \) \(\mathstrut +\mathstrut 8q^{63} \) \(\mathstrut +\mathstrut 4q^{64} \) \(\mathstrut -\mathstrut 2q^{65} \) \(\mathstrut -\mathstrut 12q^{66} \) \(\mathstrut +\mathstrut 16q^{67} \) \(\mathstrut -\mathstrut 8q^{68} \) \(\mathstrut -\mathstrut 28q^{69} \) \(\mathstrut +\mathstrut 3q^{70} \) \(\mathstrut +\mathstrut 4q^{71} \) \(\mathstrut -\mathstrut 20q^{72} \) \(\mathstrut +\mathstrut 8q^{73} \) \(\mathstrut -\mathstrut 2q^{74} \) \(\mathstrut +\mathstrut 8q^{75} \) \(\mathstrut -\mathstrut 5q^{76} \) \(\mathstrut -\mathstrut 8q^{77} \) \(\mathstrut +\mathstrut 16q^{78} \) \(\mathstrut -\mathstrut 10q^{79} \) \(\mathstrut -\mathstrut 3q^{80} \) \(\mathstrut +\mathstrut 22q^{81} \) \(\mathstrut +\mathstrut 7q^{82} \) \(\mathstrut -\mathstrut 12q^{83} \) \(\mathstrut +\mathstrut 8q^{84} \) \(\mathstrut +\mathstrut 6q^{85} \) \(\mathstrut +\mathstrut 4q^{86} \) \(\mathstrut +\mathstrut 10q^{89} \) \(\mathstrut +\mathstrut 13q^{90} \) \(\mathstrut -\mathstrut 6q^{91} \) \(\mathstrut +\mathstrut 16q^{92} \) \(\mathstrut -\mathstrut 2q^{93} \) \(\mathstrut +\mathstrut 8q^{94} \) \(\mathstrut +\mathstrut 4q^{96} \) \(\mathstrut -\mathstrut 14q^{97} \) \(\mathstrut -\mathstrut 18q^{98} \) \(\mathstrut +\mathstrut 12q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(31))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 31
31.2.a.a \(2\) \(0.248\) \(\Q(\sqrt{5}) \) None \(1\) \(-2\) \(2\) \(-4\) \(-\) \(q+\beta q^{2}-2\beta q^{3}+(-1+\beta )q^{4}+q^{5}+\cdots\)