Properties

Label 4022.2.a.d
Level 4022
Weight 2
Character orbit 4022.a
Self dual yes
Analytic conductor 32.116
Analytic rank 1
Dimension 37
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: \(1\)
Dimension: \(37\)
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.

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

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 −3.25505 1.00000 −0.797463 3.25505 −4.64387 −1.00000 7.59538 0.797463
1.2 −1.00000 −3.25277 1.00000 −3.41922 3.25277 1.95122 −1.00000 7.58049 3.41922
1.3 −1.00000 −3.07138 1.00000 0.455356 3.07138 0.869394 −1.00000 6.43339 −0.455356
1.4 −1.00000 −2.82634 1.00000 2.27969 2.82634 −0.755653 −1.00000 4.98821 −2.27969
1.5 −1.00000 −2.78933 1.00000 0.171910 2.78933 1.94661 −1.00000 4.78038 −0.171910
1.6 −1.00000 −2.37393 1.00000 1.41638 2.37393 3.16627 −1.00000 2.63555 −1.41638
1.7 −1.00000 −2.35142 1.00000 −3.94455 2.35142 −0.732372 −1.00000 2.52919 3.94455
1.8 −1.00000 −1.96616 1.00000 −1.21238 1.96616 −3.03603 −1.00000 0.865786 1.21238
1.9 −1.00000 −1.94782 1.00000 3.45647 1.94782 −2.39115 −1.00000 0.794011 −3.45647
1.10 −1.00000 −1.92131 1.00000 −0.749362 1.92131 −4.27002 −1.00000 0.691416 0.749362
1.11 −1.00000 −1.46057 1.00000 −0.684995 1.46057 −3.06021 −1.00000 −0.866744 0.684995
1.12 −1.00000 −1.41095 1.00000 −3.55049 1.41095 −3.54920 −1.00000 −1.00921 3.55049
1.13 −1.00000 −1.40012 1.00000 −1.65031 1.40012 2.90659 −1.00000 −1.03967 1.65031
1.14 −1.00000 −1.24109 1.00000 −1.11151 1.24109 1.91739 −1.00000 −1.45969 1.11151
1.15 −1.00000 −1.16293 1.00000 3.05784 1.16293 2.13691 −1.00000 −1.64760 −3.05784
1.16 −1.00000 −1.10960 1.00000 2.36799 1.10960 0.505980 −1.00000 −1.76879 −2.36799
1.17 −1.00000 −0.586559 1.00000 2.99859 0.586559 1.29059 −1.00000 −2.65595 −2.99859
1.18 −1.00000 −0.0611935 1.00000 −3.49914 0.0611935 −5.08120 −1.00000 −2.99626 3.49914
1.19 −1.00000 −0.00489128 1.00000 −4.12652 0.00489128 −1.34401 −1.00000 −2.99998 4.12652
1.20 −1.00000 0.0519256 1.00000 −2.57858 −0.0519256 3.17465 −1.00000 −2.99730 2.57858
See all 37 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.37
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.d 37
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4022.2.a.d 37 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 \(T_{3}^{37} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4022))\).