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Properties

Label 71.2.a
Level $71$
Weight $2$
Character orbit 71.a
Rep. character $\chi_{71}(1,\cdot)$
Character field $\Q$
Dimension $6$
Newform subspaces $2$
Sturm bound $12$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 71 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 71.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(12\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(71))\).

Total New Old
Modular forms 7 7 0
Cusp forms 6 6 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 and the Fricke involution.

\(71\)Dim
\(-\)\(6\)

Trace form

\( 6 q - q^{2} + 7 q^{4} + 2 q^{5} + 4 q^{7} - 9 q^{8} + 8 q^{9} + O(q^{10}) \) \( 6 q - q^{2} + 7 q^{4} + 2 q^{5} + 4 q^{7} - 9 q^{8} + 8 q^{9} - 2 q^{10} - 2 q^{11} - 16 q^{12} + 6 q^{13} - 16 q^{14} - 16 q^{15} - 3 q^{16} - 22 q^{18} + 12 q^{19} - q^{20} + 2 q^{21} - 4 q^{22} - 4 q^{23} + 11 q^{24} + 12 q^{25} + 4 q^{26} - 12 q^{27} + 22 q^{28} + 6 q^{29} + 7 q^{30} + 6 q^{31} - 9 q^{32} + 18 q^{33} + 18 q^{34} - 20 q^{35} + 31 q^{36} - 6 q^{37} + 15 q^{38} - 8 q^{39} + 10 q^{40} + 12 q^{41} - 6 q^{42} - 4 q^{43} + 2 q^{44} + 2 q^{45} - 14 q^{46} - 6 q^{47} - 5 q^{48} + 30 q^{49} + 17 q^{50} - 32 q^{51} - 10 q^{52} - 18 q^{53} + 56 q^{54} - 18 q^{55} - 6 q^{56} - 12 q^{57} + 18 q^{58} - 26 q^{59} - 39 q^{60} + 24 q^{61} - 24 q^{62} - 14 q^{63} - 23 q^{64} - 22 q^{66} - 24 q^{67} + 34 q^{68} - 40 q^{69} - 32 q^{70} + 6 q^{71} - 55 q^{72} + 30 q^{73} + 23 q^{74} - 4 q^{75} + 16 q^{76} - 32 q^{77} + 74 q^{78} + 4 q^{79} + 12 q^{80} + 6 q^{81} - 32 q^{82} + 4 q^{83} + 32 q^{84} - 10 q^{85} - 24 q^{86} - 4 q^{87} - 16 q^{88} + 14 q^{89} + 3 q^{90} + 32 q^{92} - 12 q^{93} - 4 q^{94} + 50 q^{96} + 26 q^{97} + 31 q^{98} + 36 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(71))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 71
71.2.a.a 71.a 1.a $3$ $0.567$ 3.3.257.1 None \(-1\) \(1\) \(5\) \(2\) $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+\beta _{1}q^{3}+(1+\beta _{2})q^{4}+(2-\beta _{1}+\cdots)q^{5}+\cdots\)
71.2.a.b 71.a 1.a $3$ $0.567$ 3.3.257.1 None \(0\) \(-1\) \(-3\) \(2\) $-$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{2}+(-\beta _{1}+\beta _{2})q^{3}+(1+\beta _{1}+\cdots)q^{4}+\cdots\)