Properties

Label 4022.2.a.e
Level 4022
Weight 2
Character orbit 4022.a
Self dual yes
Analytic conductor 32.116
Analytic rank 0
Dimension 46
CM no
Inner twists 1

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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: \(0\)
Dimension: \(46\)
Coefficient ring index: multiple of None
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 46q - 46q^{2} + 8q^{3} + 46q^{4} + 14q^{5} - 8q^{6} + 28q^{7} - 46q^{8} + 58q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 46q - 46q^{2} + 8q^{3} + 46q^{4} + 14q^{5} - 8q^{6} + 28q^{7} - 46q^{8} + 58q^{9} - 14q^{10} - 6q^{11} + 8q^{12} + 37q^{13} - 28q^{14} + 9q^{15} + 46q^{16} + 6q^{17} - 58q^{18} + 18q^{19} + 14q^{20} + 19q^{21} + 6q^{22} - 4q^{23} - 8q^{24} + 86q^{25} - 37q^{26} + 32q^{27} + 28q^{28} + 15q^{29} - 9q^{30} + 18q^{31} - 46q^{32} + 37q^{33} - 6q^{34} - 2q^{35} + 58q^{36} + 74q^{37} - 18q^{38} - 3q^{39} - 14q^{40} - 18q^{41} - 19q^{42} + 25q^{43} - 6q^{44} + 94q^{45} + 4q^{46} + 18q^{47} + 8q^{48} + 92q^{49} - 86q^{50} - 10q^{51} + 37q^{52} + 17q^{53} - 32q^{54} + 37q^{55} - 28q^{56} + 43q^{57} - 15q^{58} - 24q^{59} + 9q^{60} + 46q^{61} - 18q^{62} + 80q^{63} + 46q^{64} + 24q^{65} - 37q^{66} + 61q^{67} + 6q^{68} + 59q^{69} + 2q^{70} - 8q^{71} - 58q^{72} + 101q^{73} - 74q^{74} + 34q^{75} + 18q^{76} + 40q^{77} + 3q^{78} + 9q^{79} + 14q^{80} + 58q^{81} + 18q^{82} + 18q^{83} + 19q^{84} + 60q^{85} - 25q^{86} + 20q^{87} + 6q^{88} - 25q^{89} - 94q^{90} + 51q^{91} - 4q^{92} + 63q^{93} - 18q^{94} - 31q^{95} - 8q^{96} + 76q^{97} - 92q^{98} + 21q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −1.00000 −3.17262 1.00000 4.25415 3.17262 4.60982 −1.00000 7.06553 −4.25415
1.2 −1.00000 −3.08058 1.00000 −0.00834233 3.08058 −0.451365 −1.00000 6.48998 0.00834233
1.3 −1.00000 −2.98506 1.00000 2.19729 2.98506 −1.49134 −1.00000 5.91056 −2.19729
1.4 −1.00000 −2.89591 1.00000 −0.0804536 2.89591 2.75347 −1.00000 5.38632 0.0804536
1.5 −1.00000 −2.71162 1.00000 −3.39235 2.71162 −1.76888 −1.00000 4.35289 3.39235
1.6 −1.00000 −2.56761 1.00000 4.10115 2.56761 −1.16645 −1.00000 3.59263 −4.10115
1.7 −1.00000 −2.56517 1.00000 1.93763 2.56517 −4.25391 −1.00000 3.58009 −1.93763
1.8 −1.00000 −2.21023 1.00000 −3.23048 2.21023 2.26977 −1.00000 1.88514 3.23048
1.9 −1.00000 −2.06796 1.00000 2.09562 2.06796 5.11749 −1.00000 1.27644 −2.09562
1.10 −1.00000 −2.03510 1.00000 −3.10204 2.03510 −0.000158603 0 −1.00000 1.14162 3.10204
1.11 −1.00000 −1.80060 1.00000 −1.50418 1.80060 4.15262 −1.00000 0.242152 1.50418
1.12 −1.00000 −1.79779 1.00000 3.03526 1.79779 −2.51987 −1.00000 0.232040 −3.03526
1.13 −1.00000 −1.60420 1.00000 −3.64067 1.60420 4.45692 −1.00000 −0.426530 3.64067
1.14 −1.00000 −1.26909 1.00000 1.05177 1.26909 0.0899900 −1.00000 −1.38940 −1.05177
1.15 −1.00000 −1.02851 1.00000 −1.32303 1.02851 −0.0172071 −1.00000 −1.94217 1.32303
1.16 −1.00000 −0.961528 1.00000 0.960632 0.961528 −1.91758 −1.00000 −2.07546 −0.960632
1.17 −1.00000 −0.923967 1.00000 −1.00517 0.923967 −1.65407 −1.00000 −2.14628 1.00517
1.18 −1.00000 −0.848137 1.00000 2.52415 0.848137 4.28843 −1.00000 −2.28066 −2.52415
1.19 −1.00000 −0.498264 1.00000 0.121384 0.498264 −4.42409 −1.00000 −2.75173 −0.121384
1.20 −1.00000 −0.469086 1.00000 0.405337 0.469086 2.00530 −1.00000 −2.77996 −0.405337
See all 46 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.46
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.e 46
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4022.2.a.e 46 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 \(T_{3}^{46} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4022))\).