Properties

Label 4011.2.a.e
Level 4011
Weight 2
Character orbit 4011.a
Self dual yes
Analytic conductor 32.028
Analytic rank 0
Dimension 3
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4011 = 3 \cdot 7 \cdot 191 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4011.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.0279962507\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} - q^{3} + ( 1 + \beta_{2} ) q^{4} + ( 1 + \beta_{2} ) q^{5} -\beta_{1} q^{6} - q^{7} + q^{8} + q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} - q^{3} + ( 1 + \beta_{2} ) q^{4} + ( 1 + \beta_{2} ) q^{5} -\beta_{1} q^{6} - q^{7} + q^{8} + q^{9} + ( 1 + 2 \beta_{1} ) q^{10} + ( -2 - 2 \beta_{2} ) q^{11} + ( -1 - \beta_{2} ) q^{12} + ( 3 + \beta_{1} - \beta_{2} ) q^{13} -\beta_{1} q^{14} + ( -1 - \beta_{2} ) q^{15} + ( -2 + \beta_{1} - 2 \beta_{2} ) q^{16} + ( -2 + 3 \beta_{1} + \beta_{2} ) q^{17} + \beta_{1} q^{18} + ( 5 - \beta_{1} + \beta_{2} ) q^{19} + ( 4 + \beta_{1} ) q^{20} + q^{21} + ( -2 - 4 \beta_{1} ) q^{22} + ( -3 + 2 \beta_{1} ) q^{23} - q^{24} + ( -1 + \beta_{1} ) q^{25} + ( 2 + 2 \beta_{1} + \beta_{2} ) q^{26} - q^{27} + ( -1 - \beta_{2} ) q^{28} + 3 q^{29} + ( -1 - 2 \beta_{1} ) q^{30} + ( -3 + 3 \beta_{1} - \beta_{2} ) q^{31} + ( -1 - 4 \beta_{1} + \beta_{2} ) q^{32} + ( 2 + 2 \beta_{2} ) q^{33} + ( 10 - \beta_{1} + 3 \beta_{2} ) q^{34} + ( -1 - \beta_{2} ) q^{35} + ( 1 + \beta_{2} ) q^{36} + ( 2 \beta_{1} + \beta_{2} ) q^{37} + ( -2 + 6 \beta_{1} - \beta_{2} ) q^{38} + ( -3 - \beta_{1} + \beta_{2} ) q^{39} + ( 1 + \beta_{2} ) q^{40} + ( 5 - 3 \beta_{1} + \beta_{2} ) q^{41} + \beta_{1} q^{42} + ( 4 + 2 \beta_{1} ) q^{43} + ( -8 - 2 \beta_{1} ) q^{44} + ( 1 + \beta_{2} ) q^{45} + ( 6 - 3 \beta_{1} + 2 \beta_{2} ) q^{46} + ( -2 - 5 \beta_{1} + \beta_{2} ) q^{47} + ( 2 - \beta_{1} + 2 \beta_{2} ) q^{48} + q^{49} + ( 3 - \beta_{1} + \beta_{2} ) q^{50} + ( 2 - 3 \beta_{1} - \beta_{2} ) q^{51} + ( 1 + \beta_{1} + 4 \beta_{2} ) q^{52} + 5 \beta_{1} q^{53} -\beta_{1} q^{54} + ( -8 - 2 \beta_{1} ) q^{55} - q^{56} + ( -5 + \beta_{1} - \beta_{2} ) q^{57} + 3 \beta_{1} q^{58} + ( 1 - \beta_{1} + 4 \beta_{2} ) q^{59} + ( -4 - \beta_{1} ) q^{60} + ( 8 - \beta_{1} + 2 \beta_{2} ) q^{61} + ( 8 - 4 \beta_{1} + 3 \beta_{2} ) q^{62} - q^{63} + ( -7 - 2 \beta_{1} ) q^{64} + ( 1 + \beta_{1} + 4 \beta_{2} ) q^{65} + ( 2 + 4 \beta_{1} ) q^{66} + ( -6 - \beta_{2} ) q^{67} + ( 4 + 7 \beta_{1} - 3 \beta_{2} ) q^{68} + ( 3 - 2 \beta_{1} ) q^{69} + ( -1 - 2 \beta_{1} ) q^{70} + ( -2 + 4 \beta_{1} - 4 \beta_{2} ) q^{71} + q^{72} + ( 9 + 2 \beta_{1} + \beta_{2} ) q^{73} + ( 7 + \beta_{1} + 2 \beta_{2} ) q^{74} + ( 1 - \beta_{1} ) q^{75} + ( 7 - \beta_{1} + 4 \beta_{2} ) q^{76} + ( 2 + 2 \beta_{2} ) q^{77} + ( -2 - 2 \beta_{1} - \beta_{2} ) q^{78} + ( 5 - \beta_{1} - \beta_{2} ) q^{79} -7 q^{80} + q^{81} + ( -8 + 6 \beta_{1} - 3 \beta_{2} ) q^{82} + ( -6 + 3 \beta_{1} + \beta_{2} ) q^{83} + ( 1 + \beta_{2} ) q^{84} + ( 4 + 7 \beta_{1} - 3 \beta_{2} ) q^{85} + ( 6 + 4 \beta_{1} + 2 \beta_{2} ) q^{86} -3 q^{87} + ( -2 - 2 \beta_{2} ) q^{88} + ( 2 + 4 \beta_{1} - \beta_{2} ) q^{89} + ( 1 + 2 \beta_{1} ) q^{90} + ( -3 - \beta_{1} + \beta_{2} ) q^{91} + ( -1 + 4 \beta_{1} - 3 \beta_{2} ) q^{92} + ( 3 - 3 \beta_{1} + \beta_{2} ) q^{93} + ( -14 - \beta_{1} - 5 \beta_{2} ) q^{94} + ( 7 - \beta_{1} + 4 \beta_{2} ) q^{95} + ( 1 + 4 \beta_{1} - \beta_{2} ) q^{96} + ( -1 - 9 \beta_{1} + 4 \beta_{2} ) q^{97} + \beta_{1} q^{98} + ( -2 - 2 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{3} + 2q^{4} + 2q^{5} - 3q^{7} + 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{3} + 2q^{4} + 2q^{5} - 3q^{7} + 3q^{8} + 3q^{9} + 3q^{10} - 4q^{11} - 2q^{12} + 10q^{13} - 2q^{15} - 4q^{16} - 7q^{17} + 14q^{19} + 12q^{20} + 3q^{21} - 6q^{22} - 9q^{23} - 3q^{24} - 3q^{25} + 5q^{26} - 3q^{27} - 2q^{28} + 9q^{29} - 3q^{30} - 8q^{31} - 4q^{32} + 4q^{33} + 27q^{34} - 2q^{35} + 2q^{36} - q^{37} - 5q^{38} - 10q^{39} + 2q^{40} + 14q^{41} + 12q^{43} - 24q^{44} + 2q^{45} + 16q^{46} - 7q^{47} + 4q^{48} + 3q^{49} + 8q^{50} + 7q^{51} - q^{52} - 24q^{55} - 3q^{56} - 14q^{57} - q^{59} - 12q^{60} + 22q^{61} + 21q^{62} - 3q^{63} - 21q^{64} - q^{65} + 6q^{66} - 17q^{67} + 15q^{68} + 9q^{69} - 3q^{70} - 2q^{71} + 3q^{72} + 26q^{73} + 19q^{74} + 3q^{75} + 17q^{76} + 4q^{77} - 5q^{78} + 16q^{79} - 21q^{80} + 3q^{81} - 21q^{82} - 19q^{83} + 2q^{84} + 15q^{85} + 16q^{86} - 9q^{87} - 4q^{88} + 7q^{89} + 3q^{90} - 10q^{91} + 8q^{93} - 37q^{94} + 17q^{95} + 4q^{96} - 7q^{97} - 4q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - 4 x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.86081
−0.254102
2.11491
−1.86081 −1.00000 1.46260 1.46260 1.86081 −1.00000 1.00000 1.00000 −2.72161
1.2 −0.254102 −1.00000 −1.93543 −1.93543 0.254102 −1.00000 1.00000 1.00000 0.491797
1.3 2.11491 −1.00000 2.47283 2.47283 −2.11491 −1.00000 1.00000 1.00000 5.22982
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4011.2.a.e 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4011.2.a.e 3 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(1\)
\(191\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - 4 T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4011))\).