Properties

Label 8041.2.a.g
Level 8041
Weight 2
Character orbit 8041.a
Self dual Yes
Analytic conductor 64.208
Analytic rank 1
Dimension 69
CM No

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Newspace parameters

Level: \( N \) = \( 8041 = 11 \cdot 17 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8041.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.2077082653\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 69q - 11q^{2} - 3q^{3} + 65q^{4} - 6q^{5} - 10q^{6} - 11q^{7} - 33q^{8} + 56q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 69q - 11q^{2} - 3q^{3} + 65q^{4} - 6q^{5} - 10q^{6} - 11q^{7} - 33q^{8} + 56q^{9} - q^{10} - 69q^{11} - 3q^{12} - 28q^{13} - 15q^{14} - 45q^{15} + 53q^{16} + 69q^{17} - 17q^{18} - 32q^{19} - 21q^{20} - 38q^{21} + 11q^{22} - 41q^{23} - 11q^{24} + 67q^{25} - 6q^{26} - 3q^{27} - 21q^{28} - 22q^{29} - 22q^{30} - 27q^{31} - 87q^{32} + 3q^{33} - 11q^{34} - 44q^{35} + 59q^{36} + 24q^{37} - 22q^{38} - 59q^{39} + q^{40} - 43q^{41} - 15q^{42} + 69q^{43} - 65q^{44} - 12q^{45} - 21q^{46} - 99q^{47} + 2q^{48} + 64q^{49} - 78q^{50} - 3q^{51} - 57q^{52} - 50q^{53} + 20q^{54} + 6q^{55} - 59q^{56} - 15q^{57} + 22q^{58} - 82q^{59} - 86q^{60} - 24q^{61} + 15q^{62} - 63q^{63} + 63q^{64} - 23q^{65} + 10q^{66} - 54q^{67} + 65q^{68} + 36q^{69} + 9q^{70} - 128q^{71} - 69q^{72} + 2q^{73} - 58q^{74} - 31q^{75} - 76q^{76} + 11q^{77} - 19q^{78} - 43q^{79} - 19q^{80} + 49q^{81} - 2q^{82} - 62q^{83} - 82q^{84} - 6q^{85} - 11q^{86} - 62q^{87} + 33q^{88} - 49q^{89} - 37q^{90} - 2q^{91} - 96q^{92} - 29q^{93} - 75q^{94} - 133q^{95} - 86q^{96} + 5q^{97} - 72q^{98} - 56q^{99} + O(q^{100}) \)

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 −2.81130 −0.519316 5.90341 −3.87319 1.45995 3.50752 −10.9737 −2.73031 10.8887
1.2 −2.75821 0.843025 5.60771 3.43649 −2.32524 −0.344850 −9.95082 −2.28931 −9.47854
1.3 −2.75592 2.83072 5.59510 −1.76173 −7.80125 2.47832 −9.90779 5.01300 4.85519
1.4 −2.67122 2.62074 5.13539 0.949400 −7.00056 −4.03564 −8.37532 3.86827 −2.53605
1.5 −2.56568 −3.40788 4.58273 3.86072 8.74353 2.70727 −6.62646 8.61362 −9.90537
1.6 −2.53420 −1.82986 4.42217 −1.56039 4.63723 −0.999486 −6.13827 0.348383 3.95435
1.7 −2.47238 −0.879695 4.11267 −2.32585 2.17494 1.92542 −5.22332 −2.22614 5.75038
1.8 −2.45687 −0.886506 4.03623 −0.806805 2.17803 −3.77933 −5.00275 −2.21411 1.98222
1.9 −2.42698 0.265461 3.89025 3.56663 −0.644270 −2.27175 −4.58762 −2.92953 −8.65616
1.10 −2.20984 0.750857 2.88340 0.114086 −1.65928 4.65666 −1.95217 −2.43621 −0.252113
1.11 −2.20191 −3.02493 2.84839 −2.51341 6.66061 0.0943373 −1.86807 6.15021 5.53430
1.12 −2.12129 1.35005 2.49986 −1.60395 −2.86383 −4.53308 −1.06034 −1.17738 3.40244
1.13 −2.10431 −0.209589 2.42810 1.85985 0.441039 3.06120 −0.900860 −2.95607 −3.91370
1.14 −2.02215 2.02595 2.08909 −3.90857 −4.09678 −0.616656 −0.180153 1.10448 7.90371
1.15 −1.87945 −2.61165 1.53233 −1.23543 4.90847 −3.71692 0.878969 3.82074 2.32194
1.16 −1.85886 3.31260 1.45535 0.943187 −6.15766 −1.28029 1.01243 7.97335 −1.75325
1.17 −1.71913 3.18493 0.955421 −4.00741 −5.47533 −0.822562 1.79577 7.14381 6.88927
1.18 −1.69256 −2.67363 0.864767 3.80511 4.52529 −4.13916 1.92145 4.14830 −6.44038
1.19 −1.67732 −1.15146 0.813414 0.864739 1.93138 4.27290 1.99029 −1.67413 −1.45045
1.20 −1.66399 1.84747 0.768874 1.24336 −3.07417 0.0311850 2.04859 0.413132 −2.06894
See all 69 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.69
Significant digits:
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Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(11\) \(1\)
\(17\) \(-1\)
\(43\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{69} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8041))\).