Properties

Label 8039.2.a.b
Level 8039
Weight 2
Character orbit 8039.a
Self dual Yes
Analytic conductor 64.192
Analytic rank 0
Dimension 391
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: \(0\)
Dimension: \(391\)
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.

\(\operatorname{Tr}(f)(q) = \) \(391q \) \(\mathstrut +\mathstrut 14q^{2} \) \(\mathstrut +\mathstrut 12q^{3} \) \(\mathstrut +\mathstrut 446q^{4} \) \(\mathstrut +\mathstrut 22q^{5} \) \(\mathstrut +\mathstrut 40q^{6} \) \(\mathstrut +\mathstrut 63q^{7} \) \(\mathstrut +\mathstrut 36q^{8} \) \(\mathstrut +\mathstrut 501q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(391q \) \(\mathstrut +\mathstrut 14q^{2} \) \(\mathstrut +\mathstrut 12q^{3} \) \(\mathstrut +\mathstrut 446q^{4} \) \(\mathstrut +\mathstrut 22q^{5} \) \(\mathstrut +\mathstrut 40q^{6} \) \(\mathstrut +\mathstrut 63q^{7} \) \(\mathstrut +\mathstrut 36q^{8} \) \(\mathstrut +\mathstrut 501q^{9} \) \(\mathstrut +\mathstrut 40q^{10} \) \(\mathstrut +\mathstrut 57q^{11} \) \(\mathstrut +\mathstrut 20q^{12} \) \(\mathstrut +\mathstrut 83q^{13} \) \(\mathstrut +\mathstrut 21q^{14} \) \(\mathstrut +\mathstrut 60q^{15} \) \(\mathstrut +\mathstrut 548q^{16} \) \(\mathstrut +\mathstrut 59q^{17} \) \(\mathstrut +\mathstrut 54q^{18} \) \(\mathstrut +\mathstrut 131q^{19} \) \(\mathstrut +\mathstrut 35q^{20} \) \(\mathstrut +\mathstrut 121q^{21} \) \(\mathstrut +\mathstrut 89q^{22} \) \(\mathstrut +\mathstrut 34q^{23} \) \(\mathstrut +\mathstrut 110q^{24} \) \(\mathstrut +\mathstrut 609q^{25} \) \(\mathstrut +\mathstrut 54q^{26} \) \(\mathstrut +\mathstrut 27q^{27} \) \(\mathstrut +\mathstrut 182q^{28} \) \(\mathstrut +\mathstrut 102q^{29} \) \(\mathstrut +\mathstrut 92q^{30} \) \(\mathstrut +\mathstrut 88q^{31} \) \(\mathstrut +\mathstrut 76q^{32} \) \(\mathstrut +\mathstrut 131q^{33} \) \(\mathstrut +\mathstrut 128q^{34} \) \(\mathstrut +\mathstrut 31q^{35} \) \(\mathstrut +\mathstrut 654q^{36} \) \(\mathstrut +\mathstrut 135q^{37} \) \(\mathstrut +\mathstrut 23q^{38} \) \(\mathstrut +\mathstrut 96q^{39} \) \(\mathstrut +\mathstrut 113q^{40} \) \(\mathstrut +\mathstrut 128q^{41} \) \(\mathstrut +\mathstrut 45q^{42} \) \(\mathstrut +\mathstrut 140q^{43} \) \(\mathstrut +\mathstrut 151q^{44} \) \(\mathstrut +\mathstrut 77q^{45} \) \(\mathstrut +\mathstrut 245q^{46} \) \(\mathstrut +\mathstrut 22q^{47} \) \(\mathstrut +\mathstrut 25q^{48} \) \(\mathstrut +\mathstrut 712q^{49} \) \(\mathstrut +\mathstrut 53q^{50} \) \(\mathstrut +\mathstrut 102q^{51} \) \(\mathstrut +\mathstrut 174q^{52} \) \(\mathstrut +\mathstrut 54q^{53} \) \(\mathstrut +\mathstrut 131q^{54} \) \(\mathstrut +\mathstrut 101q^{55} \) \(\mathstrut +\mathstrut 43q^{56} \) \(\mathstrut +\mathstrut 226q^{57} \) \(\mathstrut +\mathstrut 109q^{58} \) \(\mathstrut +\mathstrut 40q^{59} \) \(\mathstrut +\mathstrut 123q^{60} \) \(\mathstrut +\mathstrut 249q^{61} \) \(\mathstrut +\mathstrut 28q^{62} \) \(\mathstrut +\mathstrut 139q^{63} \) \(\mathstrut +\mathstrut 730q^{64} \) \(\mathstrut +\mathstrut 227q^{65} \) \(\mathstrut +\mathstrut 55q^{66} \) \(\mathstrut +\mathstrut 169q^{67} \) \(\mathstrut +\mathstrut 48q^{68} \) \(\mathstrut +\mathstrut 89q^{69} \) \(\mathstrut +\mathstrut 98q^{70} \) \(\mathstrut +\mathstrut 66q^{71} \) \(\mathstrut +\mathstrut 120q^{72} \) \(\mathstrut +\mathstrut 324q^{73} \) \(\mathstrut +\mathstrut 60q^{74} \) \(\mathstrut +\mathstrut 19q^{75} \) \(\mathstrut +\mathstrut 356q^{76} \) \(\mathstrut +\mathstrut 83q^{77} \) \(\mathstrut -\mathstrut 11q^{78} \) \(\mathstrut +\mathstrut 195q^{79} \) \(\mathstrut +\mathstrut 26q^{80} \) \(\mathstrut +\mathstrut 807q^{81} \) \(\mathstrut +\mathstrut 49q^{82} \) \(\mathstrut +\mathstrut 74q^{83} \) \(\mathstrut +\mathstrut 252q^{84} \) \(\mathstrut +\mathstrut 373q^{85} \) \(\mathstrut +\mathstrut 100q^{86} \) \(\mathstrut +\mathstrut 43q^{87} \) \(\mathstrut +\mathstrut 211q^{88} \) \(\mathstrut +\mathstrut 207q^{89} \) \(\mathstrut +\mathstrut 10q^{90} \) \(\mathstrut +\mathstrut 189q^{91} \) \(\mathstrut +\mathstrut 30q^{92} \) \(\mathstrut +\mathstrut 172q^{93} \) \(\mathstrut +\mathstrut 130q^{94} \) \(\mathstrut +\mathstrut 43q^{95} \) \(\mathstrut +\mathstrut 203q^{96} \) \(\mathstrut +\mathstrut 254q^{97} \) \(\mathstrut +\mathstrut 26q^{98} \) \(\mathstrut +\mathstrut 273q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.81050 0.273089 5.89891 −1.07280 −0.767516 3.48763 −10.9579 −2.92542 3.01510
1.2 −2.80167 −1.48127 5.84934 2.23536 4.15003 2.13740 −10.7846 −0.805840 −6.26274
1.3 −2.79911 −1.99039 5.83499 −0.815598 5.57131 −3.39799 −10.7345 0.961647 2.28295
1.4 −2.79609 −0.0994905 5.81814 −1.46865 0.278185 0.0832220 −10.6759 −2.99010 4.10649
1.5 −2.78936 −2.33525 5.78055 3.95215 6.51386 5.17788 −10.5453 2.45339 −11.0240
1.6 −2.78914 1.94751 5.77928 −0.505263 −5.43186 −0.334314 −10.5409 0.792785 1.40925
1.7 −2.78421 2.45333 5.75180 2.41719 −6.83056 1.41777 −10.4458 3.01881 −6.72994
1.8 −2.77323 3.30634 5.69082 −2.95004 −9.16925 3.29315 −10.2355 7.93189 8.18116
1.9 −2.73934 −2.46230 5.50400 −1.26420 6.74509 −0.402196 −9.59867 3.06293 3.46308
1.10 −2.73322 2.36794 5.47049 3.42125 −6.47211 −3.92603 −9.48561 2.60715 −9.35102
1.11 −2.72603 −3.12098 5.43123 −3.26463 8.50788 −3.62664 −9.35363 6.74052 8.89948
1.12 −2.71686 −1.39132 5.38132 −4.28068 3.78003 2.13526 −9.18657 −1.06422 11.6300
1.13 −2.70592 0.720832 5.32200 1.39019 −1.95051 −3.88734 −8.98906 −2.48040 −3.76173
1.14 −2.67673 −3.14138 5.16489 −3.87922 8.40863 4.00887 −8.47155 6.86827 10.3836
1.15 −2.65165 1.77560 5.03124 −0.216981 −4.70826 −4.11451 −8.03778 0.152748 0.575357
1.16 −2.64673 −3.15136 5.00518 2.70892 8.34080 −2.33255 −7.95389 6.93109 −7.16979
1.17 −2.62368 −0.0902052 4.88368 −3.95870 0.236669 −3.68990 −7.56585 −2.99186 10.3864
1.18 −2.62255 0.822293 4.87774 −3.51986 −2.15650 −0.523743 −7.54702 −2.32383 9.23098
1.19 −2.61852 1.44668 4.85663 3.17736 −3.78815 3.56525 −7.48013 −0.907124 −8.31997
1.20 −2.61507 3.14612 4.83861 2.05578 −8.22734 4.83721 −7.42319 6.89808 −5.37602
See next 80 embeddings (of 391 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.391
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

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