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

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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: \(40\)
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.

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

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 −3.34571 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 have CM; other inner twists have not been computed.

Atkin-Lehner signs

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

Hecke kernels

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