Properties

Label 8007.2.a.c
Level 8007
Weight 2
Character orbit 8007.a
Self dual Yes
Analytic conductor 63.936
Analytic rank 1
Dimension 39
CM No

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Newspace parameters

Level: \( N \) = \( 8007 = 3 \cdot 17 \cdot 157 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8007.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(63.9362168984\)
Analytic rank: \(1\)
Dimension: \(39\)
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.

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

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 −2.68305 −1.00000 5.19875 −0.525183 2.68305 1.96908 −8.58242 1.00000 1.40909
1.2 −2.62708 −1.00000 4.90155 1.61007 2.62708 −0.284525 −7.62261 1.00000 −4.22979
1.3 −2.53786 −1.00000 4.44073 −3.08666 2.53786 −3.86465 −6.19424 1.00000 7.83351
1.4 −2.23101 −1.00000 2.97739 0.320461 2.23101 0.645663 −2.18055 1.00000 −0.714950
1.5 −2.20952 −1.00000 2.88199 −2.30853 2.20952 −2.22002 −1.94877 1.00000 5.10076
1.6 −2.11675 −1.00000 2.48063 2.70234 2.11675 2.58592 −1.01738 1.00000 −5.72017
1.7 −2.06382 −1.00000 2.25934 −2.15851 2.06382 0.293019 −0.535232 1.00000 4.45478
1.8 −1.97468 −1.00000 1.89938 1.34463 1.97468 −4.50545 0.198692 1.00000 −2.65521
1.9 −1.86416 −1.00000 1.47510 −3.19016 1.86416 1.30918 0.978494 1.00000 5.94698
1.10 −1.63095 −1.00000 0.660014 1.38859 1.63095 2.37558 2.18546 1.00000 −2.26473
1.11 −1.60951 −1.00000 0.590533 2.17104 1.60951 3.27452 2.26856 1.00000 −3.49431
1.12 −1.19134 −1.00000 −0.580715 −1.39828 1.19134 −2.85096 3.07450 1.00000 1.66582
1.13 −1.11359 −1.00000 −0.759925 3.93850 1.11359 −0.664710 3.07342 1.00000 −4.38586
1.14 −1.03615 −1.00000 −0.926391 −2.19284 1.03615 0.525649 3.03218 1.00000 2.27211
1.15 −0.833201 −1.00000 −1.30578 −2.56183 0.833201 1.99624 2.75438 1.00000 2.13452
1.16 −0.744255 −1.00000 −1.44608 1.71466 0.744255 −3.69118 2.56477 1.00000 −1.27614
1.17 −0.679888 −1.00000 −1.53775 2.83101 0.679888 −0.718741 2.40527 1.00000 −1.92477
1.18 −0.642908 −1.00000 −1.58667 1.17491 0.642908 −0.0115143 2.30590 1.00000 −0.755361
1.19 −0.554461 −1.00000 −1.69257 −2.60082 0.554461 2.71630 2.04739 1.00000 1.44205
1.20 −0.177236 −1.00000 −1.96859 −2.36208 0.177236 −1.71756 0.703377 1.00000 0.418646
See all 39 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.39
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(17\) \(-1\)
\(157\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{39} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8007))\).