Properties

Label 4006.2.a.g
Level 4006
Weight 2
Character orbit 4006.a
Self dual yes
Analytic conductor 31.988
Analytic rank 1
Dimension 40
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4006 = 2 \cdot 2003 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4006.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.9880710497\)
Analytic rank: \(1\)
Dimension: \(40\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 40q - 40q^{2} - q^{3} + 40q^{4} - 23q^{5} + q^{6} + 12q^{7} - 40q^{8} + 29q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 40q - 40q^{2} - q^{3} + 40q^{4} - 23q^{5} + q^{6} + 12q^{7} - 40q^{8} + 29q^{9} + 23q^{10} - 28q^{11} - q^{12} - 12q^{13} - 12q^{14} - 14q^{15} + 40q^{16} - 10q^{17} - 29q^{18} + 3q^{19} - 23q^{20} - 40q^{21} + 28q^{22} - 10q^{23} + q^{24} + 43q^{25} + 12q^{26} - 7q^{27} + 12q^{28} - 37q^{29} + 14q^{30} - 16q^{31} - 40q^{32} - 11q^{33} + 10q^{34} - 24q^{35} + 29q^{36} - 17q^{37} - 3q^{38} - 10q^{39} + 23q^{40} - 58q^{41} + 40q^{42} + 37q^{43} - 28q^{44} - 66q^{45} + 10q^{46} - 34q^{47} - q^{48} + 28q^{49} - 43q^{50} - 7q^{51} - 12q^{52} - 43q^{53} + 7q^{54} + 28q^{55} - 12q^{56} - 25q^{57} + 37q^{58} - 92q^{59} - 14q^{60} - 37q^{61} + 16q^{62} + 14q^{63} + 40q^{64} - 54q^{65} + 11q^{66} - 10q^{67} - 10q^{68} - 49q^{69} + 24q^{70} - 87q^{71} - 29q^{72} + 12q^{73} + 17q^{74} - 31q^{75} + 3q^{76} - 53q^{77} + 10q^{78} + 14q^{79} - 23q^{80} - 4q^{81} + 58q^{82} - 42q^{83} - 40q^{84} - 24q^{85} - 37q^{86} + 24q^{87} + 28q^{88} - 118q^{89} + 66q^{90} - 35q^{91} - 10q^{92} - 22q^{93} + 34q^{94} - 18q^{95} + q^{96} + 2q^{97} - 28q^{98} - 48q^{99} + O(q^{100}) \)

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 −3.34571 1.00000 −3.64453 3.34571 5.09237 −1.00000 8.19376 3.64453
1.2 −1.00000 −3.11880 1.00000 1.57138 3.11880 2.57640 −1.00000 6.72691 −1.57138
1.3 −1.00000 −2.96542 1.00000 0.539090 2.96542 −0.220082 −1.00000 5.79371 −0.539090
1.4 −1.00000 −2.86035 1.00000 −2.90727 2.86035 0.924767 −1.00000 5.18162 2.90727
1.5 −1.00000 −2.39522 1.00000 0.491091 2.39522 3.07111 −1.00000 2.73707 −0.491091
1.6 −1.00000 −2.30611 1.00000 3.88128 2.30611 0.245423 −1.00000 2.31816 −3.88128
1.7 −1.00000 −2.23215 1.00000 −2.75227 2.23215 −2.11016 −1.00000 1.98248 2.75227
1.8 −1.00000 −1.94007 1.00000 −0.275091 1.94007 −2.80268 −1.00000 0.763873 0.275091
1.9 −1.00000 −1.91121 1.00000 −1.39712 1.91121 3.46874 −1.00000 0.652717 1.39712
1.10 −1.00000 −1.84543 1.00000 2.20410 1.84543 −0.412853 −1.00000 0.405623 −2.20410
1.11 −1.00000 −1.82093 1.00000 −3.80048 1.82093 −3.54475 −1.00000 0.315788 3.80048
1.12 −1.00000 −1.69686 1.00000 3.10338 1.69686 1.10094 −1.00000 −0.120661 −3.10338
1.13 −1.00000 −1.01040 1.00000 −1.91784 1.01040 4.92355 −1.00000 −1.97910 1.91784
1.14 −1.00000 −0.956879 1.00000 −0.0953014 0.956879 −4.20577 −1.00000 −2.08438 0.0953014
1.15 −1.00000 −0.941592 1.00000 −0.129713 0.941592 0.350515 −1.00000 −2.11340 0.129713
1.16 −1.00000 −0.934245 1.00000 −4.17073 0.934245 1.30479 −1.00000 −2.12719 4.17073
1.17 −1.00000 −0.779235 1.00000 −0.867411 0.779235 −3.45139 −1.00000 −2.39279 0.867411
1.18 −1.00000 −0.712075 1.00000 −4.09209 0.712075 2.33011 −1.00000 −2.49295 4.09209
1.19 −1.00000 −0.535052 1.00000 2.84024 0.535052 2.46812 −1.00000 −2.71372 −2.84024
1.20 −1.00000 −0.0866100 1.00000 2.47499 0.0866100 0.719708 −1.00000 −2.99250 −2.47499
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.40
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 4006.2.a.g 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4006.2.a.g 40 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(2003\) \(1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{40} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4006))\).