Properties

Label 8039.2.a.a
Level 8039
Weight 2
Character orbit 8039.a
Self dual Yes
Analytic conductor 64.192
Analytic rank 1
Dimension 279
CM No

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Newspace parameters

Level: \( N \) = \( 8039 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8039.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.1917381849\)
Analytic rank: \(1\)
Dimension: \(279\)
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.

\(\operatorname{Tr}(f)(q) = \) \(279q \) \(\mathstrut -\mathstrut 13q^{2} \) \(\mathstrut -\mathstrut 12q^{3} \) \(\mathstrut +\mathstrut 227q^{4} \) \(\mathstrut -\mathstrut 20q^{5} \) \(\mathstrut -\mathstrut 40q^{6} \) \(\mathstrut -\mathstrut 57q^{7} \) \(\mathstrut -\mathstrut 39q^{8} \) \(\mathstrut +\mathstrut 175q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(279q \) \(\mathstrut -\mathstrut 13q^{2} \) \(\mathstrut -\mathstrut 12q^{3} \) \(\mathstrut +\mathstrut 227q^{4} \) \(\mathstrut -\mathstrut 20q^{5} \) \(\mathstrut -\mathstrut 40q^{6} \) \(\mathstrut -\mathstrut 57q^{7} \) \(\mathstrut -\mathstrut 39q^{8} \) \(\mathstrut +\mathstrut 175q^{9} \) \(\mathstrut -\mathstrut 42q^{10} \) \(\mathstrut -\mathstrut 53q^{11} \) \(\mathstrut -\mathstrut 36q^{12} \) \(\mathstrut -\mathstrut 75q^{13} \) \(\mathstrut -\mathstrut 31q^{14} \) \(\mathstrut -\mathstrut 60q^{15} \) \(\mathstrut +\mathstrut 127q^{16} \) \(\mathstrut -\mathstrut 55q^{17} \) \(\mathstrut -\mathstrut 57q^{18} \) \(\mathstrut -\mathstrut 113q^{19} \) \(\mathstrut -\mathstrut 43q^{20} \) \(\mathstrut -\mathstrut 103q^{21} \) \(\mathstrut -\mathstrut 73q^{22} \) \(\mathstrut -\mathstrut 30q^{23} \) \(\mathstrut -\mathstrut 106q^{24} \) \(\mathstrut +\mathstrut 75q^{25} \) \(\mathstrut -\mathstrut 42q^{26} \) \(\mathstrut -\mathstrut 45q^{27} \) \(\mathstrut -\mathstrut 146q^{28} \) \(\mathstrut -\mathstrut 92q^{29} \) \(\mathstrut -\mathstrut 76q^{30} \) \(\mathstrut -\mathstrut 84q^{31} \) \(\mathstrut -\mathstrut 71q^{32} \) \(\mathstrut -\mathstrut 117q^{33} \) \(\mathstrut -\mathstrut 106q^{34} \) \(\mathstrut -\mathstrut 49q^{35} \) \(\mathstrut +\mathstrut 67q^{36} \) \(\mathstrut -\mathstrut 123q^{37} \) \(\mathstrut -\mathstrut 21q^{38} \) \(\mathstrut -\mathstrut 92q^{39} \) \(\mathstrut -\mathstrut 97q^{40} \) \(\mathstrut -\mathstrut 116q^{41} \) \(\mathstrut -\mathstrut 19q^{42} \) \(\mathstrut -\mathstrut 126q^{43} \) \(\mathstrut -\mathstrut 131q^{44} \) \(\mathstrut -\mathstrut 85q^{45} \) \(\mathstrut -\mathstrut 183q^{46} \) \(\mathstrut -\mathstrut 42q^{47} \) \(\mathstrut -\mathstrut 47q^{48} \) \(\mathstrut -\mathstrut 22q^{49} \) \(\mathstrut -\mathstrut 64q^{50} \) \(\mathstrut -\mathstrut 90q^{51} \) \(\mathstrut -\mathstrut 158q^{52} \) \(\mathstrut -\mathstrut 60q^{53} \) \(\mathstrut -\mathstrut 117q^{54} \) \(\mathstrut -\mathstrut 99q^{55} \) \(\mathstrut -\mathstrut 65q^{56} \) \(\mathstrut -\mathstrut 182q^{57} \) \(\mathstrut -\mathstrut 93q^{58} \) \(\mathstrut -\mathstrut 58q^{59} \) \(\mathstrut -\mathstrut 141q^{60} \) \(\mathstrut -\mathstrut 217q^{61} \) \(\mathstrut -\mathstrut 16q^{62} \) \(\mathstrut -\mathstrut 141q^{63} \) \(\mathstrut -\mathstrut 47q^{64} \) \(\mathstrut -\mathstrut 197q^{65} \) \(\mathstrut -\mathstrut 53q^{66} \) \(\mathstrut -\mathstrut 147q^{67} \) \(\mathstrut -\mathstrut 90q^{68} \) \(\mathstrut -\mathstrut 103q^{69} \) \(\mathstrut -\mathstrut 118q^{70} \) \(\mathstrut -\mathstrut 78q^{71} \) \(\mathstrut -\mathstrut 135q^{72} \) \(\mathstrut -\mathstrut 282q^{73} \) \(\mathstrut -\mathstrut 98q^{74} \) \(\mathstrut -\mathstrut 53q^{75} \) \(\mathstrut -\mathstrut 296q^{76} \) \(\mathstrut -\mathstrut 53q^{77} \) \(\mathstrut -\mathstrut 27q^{78} \) \(\mathstrut -\mathstrut 153q^{79} \) \(\mathstrut -\mathstrut 52q^{80} \) \(\mathstrut -\mathstrut 89q^{81} \) \(\mathstrut -\mathstrut 81q^{82} \) \(\mathstrut -\mathstrut 54q^{83} \) \(\mathstrut -\mathstrut 164q^{84} \) \(\mathstrut -\mathstrut 303q^{85} \) \(\mathstrut -\mathstrut 82q^{86} \) \(\mathstrut -\mathstrut 29q^{87} \) \(\mathstrut -\mathstrut 203q^{88} \) \(\mathstrut -\mathstrut 185q^{89} \) \(\mathstrut -\mathstrut 56q^{90} \) \(\mathstrut -\mathstrut 163q^{91} \) \(\mathstrut -\mathstrut 66q^{92} \) \(\mathstrut -\mathstrut 156q^{93} \) \(\mathstrut -\mathstrut 134q^{94} \) \(\mathstrut -\mathstrut 69q^{95} \) \(\mathstrut -\mathstrut 189q^{96} \) \(\mathstrut -\mathstrut 212q^{97} \) \(\mathstrut -\mathstrut 13q^{98} \) \(\mathstrut -\mathstrut 181q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.75826 0.543669 5.60799 4.07368 −1.49958 −1.45954 −9.95176 −2.70442 −11.2363
1.2 −2.72307 −0.226737 5.41513 0.679891 0.617421 −1.82848 −9.29967 −2.94859 −1.85139
1.3 −2.72088 2.51741 5.40317 −2.91261 −6.84957 −4.81699 −9.25961 3.33737 7.92486
1.4 −2.69202 −2.07215 5.24695 2.43355 5.57825 −0.413821 −8.74083 1.29379 −6.55116
1.5 −2.68146 1.09614 5.19020 0.161648 −2.93925 3.85315 −8.55439 −1.79847 −0.433452
1.6 −2.66452 2.42009 5.09965 1.98258 −6.44836 1.42101 −8.25906 2.85681 −5.28261
1.7 −2.65513 −1.36605 5.04972 2.66534 3.62703 −0.858733 −8.09741 −1.13392 −7.07683
1.8 −2.65123 3.28882 5.02904 −0.364681 −8.71943 −0.0824305 −8.03069 7.81633 0.966855
1.9 −2.64944 0.544800 5.01955 −2.56078 −1.44342 −0.780813 −8.00012 −2.70319 6.78465
1.10 −2.64457 −1.20149 4.99374 −2.42138 3.17741 3.07885 −7.91715 −1.55643 6.40351
1.11 −2.62410 1.81984 4.88588 −0.731341 −4.77543 −1.89048 −7.57282 0.311808 1.91911
1.12 −2.61158 −1.07466 4.82033 −1.60450 2.80656 −4.85582 −7.36550 −1.84510 4.19028
1.13 −2.60916 −2.67838 4.80773 −2.56224 6.98832 −1.44552 −7.32584 4.17369 6.68531
1.14 −2.59800 −2.63260 4.74962 0.476137 6.83949 1.24122 −7.14351 3.93057 −1.23700
1.15 −2.55932 −0.558527 4.55013 0.514137 1.42945 2.79738 −6.52662 −2.68805 −1.31584
1.16 −2.55424 2.06857 4.52414 −3.35787 −5.28363 0.600266 −6.44725 1.27899 8.57679
1.17 −2.54048 −0.00862555 4.45405 1.93048 0.0219131 2.12668 −6.23447 −2.99993 −4.90434
1.18 −2.53338 3.09967 4.41800 3.04096 −7.85264 −3.65810 −6.12571 6.60796 −7.70391
1.19 −2.51220 −0.224247 4.31117 3.23846 0.563355 −3.25585 −5.80612 −2.94971 −8.13568
1.20 −2.47449 2.58655 4.12308 0.927265 −6.40039 −0.828816 −5.25355 3.69026 −2.29451
See next 80 embeddings (of 279 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.279
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(8039\) \(1\)