Properties

Label 4022.2.a.b
Level 4022
Weight 2
Character orbit 4022.a
Self dual yes
Analytic conductor 32.116
Analytic rank 1
Dimension 3
CM no
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 4022 = 2 \cdot 2011 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4022.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.1158316930\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.169.1
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 4 x - 1\):

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

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
−1.37720
−0.273891
2.65109
1.00000 −1.00000 1.00000 −1.37720 −1.00000 −2.65109 1.00000 −2.00000 −1.37720
1.2 1.00000 −1.00000 1.00000 −0.273891 −1.00000 1.37720 1.00000 −2.00000 −0.273891
1.3 1.00000 −1.00000 1.00000 2.65109 −1.00000 0.273891 1.00000 −2.00000 2.65109
\(n\): 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 4022.2.a.b 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4022.2.a.b 3 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(2011\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4022))\).