Properties

Label 4022.2.a.f
Level 4022
Weight 2
Character orbit 4022.a
Self dual yes
Analytic conductor 32.116
Analytic rank 0
Dimension 50
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: \(50\)
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.

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

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 −3.20664 1.00000 −0.511117 −3.20664 0.304730 1.00000 7.28252 −0.511117
1.2 1.00000 −2.97777 1.00000 −2.48115 −2.97777 4.88773 1.00000 5.86711 −2.48115
1.3 1.00000 −2.74563 1.00000 −2.60075 −2.74563 0.636977 1.00000 4.53848 −2.60075
1.4 1.00000 −2.61133 1.00000 −0.740567 −2.61133 −2.94518 1.00000 3.81902 −0.740567
1.5 1.00000 −2.59841 1.00000 4.27842 −2.59841 3.34363 1.00000 3.75173 4.27842
1.6 1.00000 −2.50306 1.00000 0.0732871 −2.50306 −1.25699 1.00000 3.26529 0.0732871
1.7 1.00000 −2.49143 1.00000 2.52331 −2.49143 1.07230 1.00000 3.20721 2.52331
1.8 1.00000 −2.46190 1.00000 2.99591 −2.46190 1.50567 1.00000 3.06093 2.99591
1.9 1.00000 −2.20305 1.00000 2.72503 −2.20305 3.76461 1.00000 1.85345 2.72503
1.10 1.00000 −1.73177 1.00000 0.638996 −1.73177 0.168919 1.00000 −0.000964633 0 0.638996
1.11 1.00000 −1.60405 1.00000 1.46390 −1.60405 4.50208 1.00000 −0.427040 1.46390
1.12 1.00000 −1.54111 1.00000 −4.38749 −1.54111 4.20037 1.00000 −0.624992 −4.38749
1.13 1.00000 −1.49263 1.00000 −2.25527 −1.49263 0.477234 1.00000 −0.772062 −2.25527
1.14 1.00000 −1.45431 1.00000 −2.49956 −1.45431 −3.69872 1.00000 −0.884987 −2.49956
1.15 1.00000 −1.41980 1.00000 2.76587 −1.41980 −4.92374 1.00000 −0.984179 2.76587
1.16 1.00000 −1.05832 1.00000 −2.35340 −1.05832 −2.98821 1.00000 −1.87996 −2.35340
1.17 1.00000 −0.537568 1.00000 2.09045 −0.537568 4.78863 1.00000 −2.71102 2.09045
1.18 1.00000 −0.445408 1.00000 −2.87764 −0.445408 −1.74390 1.00000 −2.80161 −2.87764
1.19 1.00000 −0.424610 1.00000 −2.05502 −0.424610 3.71032 1.00000 −2.81971 −2.05502
1.20 1.00000 −0.367906 1.00000 2.32640 −0.367906 1.76735 1.00000 −2.86465 2.32640
See all 50 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.50
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.f 50
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4022.2.a.f 50 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 \(T_{3}^{50} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4022))\).