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

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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: \(32.115831693\)
Analytic rank: \(0\)
Dimension: \(46\)
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.

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

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 have CM; other inner twists have not been computed.

Atkin-Lehner signs

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

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{46} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4022))\).