Properties

Label 4006.2.a.f
Level 4006
Weight 2
Character orbit 4006.a
Self dual Yes
Analytic conductor 31.988
Analytic rank 1
Dimension 31
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: \(31\)
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.

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

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 −3.10750 1.00000 −0.905636 −3.10750 3.10017 1.00000 6.65658 −0.905636
1.2 1.00000 −2.96975 1.00000 3.59055 −2.96975 −0.0161745 1.00000 5.81940 3.59055
1.3 1.00000 −2.94307 1.00000 −2.92103 −2.94307 −1.64254 1.00000 5.66165 −2.92103
1.4 1.00000 −2.69988 1.00000 2.56179 −2.69988 −4.70163 1.00000 4.28934 2.56179
1.5 1.00000 −2.68137 1.00000 0.326909 −2.68137 2.00522 1.00000 4.18972 0.326909
1.6 1.00000 −2.64840 1.00000 −3.78126 −2.64840 −4.25052 1.00000 4.01402 −3.78126
1.7 1.00000 −2.59151 1.00000 −4.34351 −2.59151 −1.56317 1.00000 3.71590 −4.34351
1.8 1.00000 −2.43870 1.00000 1.19348 −2.43870 2.99932 1.00000 2.94725 1.19348
1.9 1.00000 −1.77148 1.00000 −2.30505 −1.77148 0.622882 1.00000 0.138156 −2.30505
1.10 1.00000 −1.74675 1.00000 −3.10700 −1.74675 3.74168 1.00000 0.0511396 −3.10700
1.11 1.00000 −1.70491 1.00000 −0.390426 −1.70491 −1.43635 1.00000 −0.0932983 −0.390426
1.12 1.00000 −1.50294 1.00000 1.71703 −1.50294 −3.21564 1.00000 −0.741171 1.71703
1.13 1.00000 −0.876341 1.00000 2.54110 −0.876341 −3.25278 1.00000 −2.23203 2.54110
1.14 1.00000 −0.788172 1.00000 4.07685 −0.788172 −1.81597 1.00000 −2.37879 4.07685
1.15 1.00000 −0.724653 1.00000 −3.73538 −0.724653 −3.63449 1.00000 −2.47488 −3.73538
1.16 1.00000 −0.466489 1.00000 1.06478 −0.466489 2.04789 1.00000 −2.78239 1.06478
1.17 1.00000 −0.453821 1.00000 −0.630094 −0.453821 1.98767 1.00000 −2.79405 −0.630094
1.18 1.00000 0.105946 1.00000 1.78036 0.105946 0.955066 1.00000 −2.98878 1.78036
1.19 1.00000 0.115469 1.00000 −3.86988 0.115469 2.67770 1.00000 −2.98667 −3.86988
1.20 1.00000 0.232858 1.00000 −1.02641 0.232858 −0.709779 1.00000 −2.94578 −1.02641
See all 31 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.31
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 \(T_{3}^{31} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4006))\).