Properties

Label 8006.2.a.a
Level 8006
Weight 2
Character orbit 8006.a
Self dual Yes
Analytic conductor 63.928
Analytic rank 1
Dimension 69
CM No

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Newspace parameters

Level: \( N \) = \( 8006 = 2 \cdot 4003 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8006.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(63.9282318582\)
Analytic rank: \(1\)
Dimension: \(69\)
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.

\(\operatorname{Tr}(f)(q) = \) \(69q \) \(\mathstrut +\mathstrut 69q^{2} \) \(\mathstrut -\mathstrut 15q^{3} \) \(\mathstrut +\mathstrut 69q^{4} \) \(\mathstrut -\mathstrut 9q^{5} \) \(\mathstrut -\mathstrut 15q^{6} \) \(\mathstrut -\mathstrut 29q^{7} \) \(\mathstrut +\mathstrut 69q^{8} \) \(\mathstrut +\mathstrut 40q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(69q \) \(\mathstrut +\mathstrut 69q^{2} \) \(\mathstrut -\mathstrut 15q^{3} \) \(\mathstrut +\mathstrut 69q^{4} \) \(\mathstrut -\mathstrut 9q^{5} \) \(\mathstrut -\mathstrut 15q^{6} \) \(\mathstrut -\mathstrut 29q^{7} \) \(\mathstrut +\mathstrut 69q^{8} \) \(\mathstrut +\mathstrut 40q^{9} \) \(\mathstrut -\mathstrut 9q^{10} \) \(\mathstrut -\mathstrut 48q^{11} \) \(\mathstrut -\mathstrut 15q^{12} \) \(\mathstrut -\mathstrut 30q^{13} \) \(\mathstrut -\mathstrut 29q^{14} \) \(\mathstrut -\mathstrut 51q^{15} \) \(\mathstrut +\mathstrut 69q^{16} \) \(\mathstrut -\mathstrut 37q^{17} \) \(\mathstrut +\mathstrut 40q^{18} \) \(\mathstrut -\mathstrut 72q^{19} \) \(\mathstrut -\mathstrut 9q^{20} \) \(\mathstrut -\mathstrut 38q^{21} \) \(\mathstrut -\mathstrut 48q^{22} \) \(\mathstrut -\mathstrut 75q^{23} \) \(\mathstrut -\mathstrut 15q^{24} \) \(\mathstrut +\mathstrut 18q^{25} \) \(\mathstrut -\mathstrut 30q^{26} \) \(\mathstrut -\mathstrut 48q^{27} \) \(\mathstrut -\mathstrut 29q^{28} \) \(\mathstrut -\mathstrut 27q^{29} \) \(\mathstrut -\mathstrut 51q^{30} \) \(\mathstrut -\mathstrut 61q^{31} \) \(\mathstrut +\mathstrut 69q^{32} \) \(\mathstrut -\mathstrut 29q^{33} \) \(\mathstrut -\mathstrut 37q^{34} \) \(\mathstrut -\mathstrut 64q^{35} \) \(\mathstrut +\mathstrut 40q^{36} \) \(\mathstrut -\mathstrut 42q^{37} \) \(\mathstrut -\mathstrut 72q^{38} \) \(\mathstrut -\mathstrut 68q^{39} \) \(\mathstrut -\mathstrut 9q^{40} \) \(\mathstrut -\mathstrut 49q^{41} \) \(\mathstrut -\mathstrut 38q^{42} \) \(\mathstrut -\mathstrut 95q^{43} \) \(\mathstrut -\mathstrut 48q^{44} \) \(\mathstrut -\mathstrut 20q^{45} \) \(\mathstrut -\mathstrut 75q^{46} \) \(\mathstrut -\mathstrut 62q^{47} \) \(\mathstrut -\mathstrut 15q^{48} \) \(\mathstrut -\mathstrut 4q^{49} \) \(\mathstrut +\mathstrut 18q^{50} \) \(\mathstrut -\mathstrut 76q^{51} \) \(\mathstrut -\mathstrut 30q^{52} \) \(\mathstrut -\mathstrut 28q^{53} \) \(\mathstrut -\mathstrut 48q^{54} \) \(\mathstrut -\mathstrut 76q^{55} \) \(\mathstrut -\mathstrut 29q^{56} \) \(\mathstrut -\mathstrut 44q^{57} \) \(\mathstrut -\mathstrut 27q^{58} \) \(\mathstrut -\mathstrut 68q^{59} \) \(\mathstrut -\mathstrut 51q^{60} \) \(\mathstrut -\mathstrut 62q^{61} \) \(\mathstrut -\mathstrut 61q^{62} \) \(\mathstrut -\mathstrut 91q^{63} \) \(\mathstrut +\mathstrut 69q^{64} \) \(\mathstrut -\mathstrut 79q^{65} \) \(\mathstrut -\mathstrut 29q^{66} \) \(\mathstrut -\mathstrut 116q^{67} \) \(\mathstrut -\mathstrut 37q^{68} \) \(\mathstrut -\mathstrut 23q^{69} \) \(\mathstrut -\mathstrut 64q^{70} \) \(\mathstrut -\mathstrut 89q^{71} \) \(\mathstrut +\mathstrut 40q^{72} \) \(\mathstrut -\mathstrut 60q^{73} \) \(\mathstrut -\mathstrut 42q^{74} \) \(\mathstrut -\mathstrut 47q^{75} \) \(\mathstrut -\mathstrut 72q^{76} \) \(\mathstrut +\mathstrut 5q^{77} \) \(\mathstrut -\mathstrut 68q^{78} \) \(\mathstrut -\mathstrut 170q^{79} \) \(\mathstrut -\mathstrut 9q^{80} \) \(\mathstrut -\mathstrut 3q^{81} \) \(\mathstrut -\mathstrut 49q^{82} \) \(\mathstrut -\mathstrut 82q^{83} \) \(\mathstrut -\mathstrut 38q^{84} \) \(\mathstrut -\mathstrut 81q^{85} \) \(\mathstrut -\mathstrut 95q^{86} \) \(\mathstrut -\mathstrut 51q^{87} \) \(\mathstrut -\mathstrut 48q^{88} \) \(\mathstrut -\mathstrut 78q^{89} \) \(\mathstrut -\mathstrut 20q^{90} \) \(\mathstrut -\mathstrut 85q^{91} \) \(\mathstrut -\mathstrut 75q^{92} \) \(\mathstrut -\mathstrut 21q^{93} \) \(\mathstrut -\mathstrut 62q^{94} \) \(\mathstrut -\mathstrut 70q^{95} \) \(\mathstrut -\mathstrut 15q^{96} \) \(\mathstrut -\mathstrut 60q^{97} \) \(\mathstrut -\mathstrut 4q^{98} \) \(\mathstrut -\mathstrut 148q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −3.36322 1.00000 3.90848 −3.36322 −1.71915 1.00000 8.31124 3.90848
1.2 1.00000 −3.28301 1.00000 −1.18461 −3.28301 0.314789 1.00000 7.77813 −1.18461
1.3 1.00000 −3.15967 1.00000 −0.278823 −3.15967 1.77562 1.00000 6.98353 −0.278823
1.4 1.00000 −3.08249 1.00000 1.32186 −3.08249 1.26521 1.00000 6.50177 1.32186
1.5 1.00000 −2.86585 1.00000 2.47496 −2.86585 −3.03619 1.00000 5.21309 2.47496
1.6 1.00000 −2.78283 1.00000 2.27257 −2.78283 −2.57189 1.00000 4.74415 2.27257
1.7 1.00000 −2.75233 1.00000 −1.92549 −2.75233 3.54771 1.00000 4.57534 −1.92549
1.8 1.00000 −2.73399 1.00000 1.40012 −2.73399 2.82233 1.00000 4.47471 1.40012
1.9 1.00000 −2.72077 1.00000 0.316882 −2.72077 −4.75223 1.00000 4.40260 0.316882
1.10 1.00000 −2.65418 1.00000 −3.04088 −2.65418 −4.15627 1.00000 4.04466 −3.04088
1.11 1.00000 −2.51689 1.00000 −2.29473 −2.51689 0.132144 1.00000 3.33476 −2.29473
1.12 1.00000 −2.38200 1.00000 2.27463 −2.38200 −1.51830 1.00000 2.67393 2.27463
1.13 1.00000 −2.26149 1.00000 −2.88813 −2.26149 2.25047 1.00000 2.11432 −2.88813
1.14 1.00000 −2.16991 1.00000 −2.35323 −2.16991 −2.98482 1.00000 1.70849 −2.35323
1.15 1.00000 −2.10040 1.00000 −0.571471 −2.10040 3.01756 1.00000 1.41167 −0.571471
1.16 1.00000 −2.04988 1.00000 4.28371 −2.04988 3.78593 1.00000 1.20199 4.28371
1.17 1.00000 −1.97198 1.00000 2.38225 −1.97198 2.34177 1.00000 0.888724 2.38225
1.18 1.00000 −1.76933 1.00000 3.70543 −1.76933 −4.46804 1.00000 0.130514 3.70543
1.19 1.00000 −1.67001 1.00000 1.85681 −1.67001 0.410918 1.00000 −0.211058 1.85681
1.20 1.00000 −1.63641 1.00000 −3.89748 −1.63641 0.627952 1.00000 −0.322178 −3.89748
See all 69 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.69
Significant digits:
Format:

Inner twists

This newform does not have CM; other inner twists have not been computed.

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(4003\) \(-1\)