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

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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: \(1\)
Dimension: \(37\)
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.

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

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 −3.25505 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 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 \(T_{3}^{37} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4022))\).