Properties

Label 8009.2.a.b
Level 8009
Weight 2
Character orbit 8009.a
Self dual Yes
Analytic conductor 63.952
Analytic rank 0
Dimension 361
CM No

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Newspace parameters

Level: \( N \) = \( 8009 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8009.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(63.9521869788\)
Analytic rank: \(0\)
Dimension: \(361\)
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.

\(\operatorname{Tr}(f)(q) = \) \(361q \) \(\mathstrut +\mathstrut 10q^{2} \) \(\mathstrut +\mathstrut 23q^{3} \) \(\mathstrut +\mathstrut 414q^{4} \) \(\mathstrut +\mathstrut 21q^{5} \) \(\mathstrut +\mathstrut 49q^{6} \) \(\mathstrut +\mathstrut 106q^{7} \) \(\mathstrut +\mathstrut 30q^{8} \) \(\mathstrut +\mathstrut 406q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(361q \) \(\mathstrut +\mathstrut 10q^{2} \) \(\mathstrut +\mathstrut 23q^{3} \) \(\mathstrut +\mathstrut 414q^{4} \) \(\mathstrut +\mathstrut 21q^{5} \) \(\mathstrut +\mathstrut 49q^{6} \) \(\mathstrut +\mathstrut 106q^{7} \) \(\mathstrut +\mathstrut 30q^{8} \) \(\mathstrut +\mathstrut 406q^{9} \) \(\mathstrut +\mathstrut 65q^{10} \) \(\mathstrut +\mathstrut 33q^{11} \) \(\mathstrut +\mathstrut 52q^{12} \) \(\mathstrut +\mathstrut 89q^{13} \) \(\mathstrut +\mathstrut 32q^{14} \) \(\mathstrut +\mathstrut 55q^{15} \) \(\mathstrut +\mathstrut 512q^{16} \) \(\mathstrut +\mathstrut 42q^{17} \) \(\mathstrut +\mathstrut 34q^{18} \) \(\mathstrut +\mathstrut 191q^{19} \) \(\mathstrut +\mathstrut 48q^{20} \) \(\mathstrut +\mathstrut 53q^{21} \) \(\mathstrut +\mathstrut 61q^{22} \) \(\mathstrut +\mathstrut 52q^{23} \) \(\mathstrut +\mathstrut 139q^{24} \) \(\mathstrut +\mathstrut 458q^{25} \) \(\mathstrut +\mathstrut 57q^{26} \) \(\mathstrut +\mathstrut 80q^{27} \) \(\mathstrut +\mathstrut 194q^{28} \) \(\mathstrut +\mathstrut 47q^{29} \) \(\mathstrut +\mathstrut 32q^{30} \) \(\mathstrut +\mathstrut 254q^{31} \) \(\mathstrut +\mathstrut 55q^{32} \) \(\mathstrut +\mathstrut 40q^{33} \) \(\mathstrut +\mathstrut 122q^{34} \) \(\mathstrut +\mathstrut 93q^{35} \) \(\mathstrut +\mathstrut 519q^{36} \) \(\mathstrut +\mathstrut 43q^{37} \) \(\mathstrut +\mathstrut 25q^{38} \) \(\mathstrut +\mathstrut 210q^{39} \) \(\mathstrut +\mathstrut 184q^{40} \) \(\mathstrut +\mathstrut 54q^{41} \) \(\mathstrut +\mathstrut 48q^{42} \) \(\mathstrut +\mathstrut 151q^{43} \) \(\mathstrut +\mathstrut 56q^{44} \) \(\mathstrut +\mathstrut 82q^{45} \) \(\mathstrut +\mathstrut 101q^{46} \) \(\mathstrut +\mathstrut 117q^{47} \) \(\mathstrut +\mathstrut 77q^{48} \) \(\mathstrut +\mathstrut 563q^{49} \) \(\mathstrut +\mathstrut 38q^{50} \) \(\mathstrut +\mathstrut 143q^{51} \) \(\mathstrut +\mathstrut 241q^{52} \) \(\mathstrut +\mathstrut 14q^{53} \) \(\mathstrut +\mathstrut 164q^{54} \) \(\mathstrut +\mathstrut 452q^{55} \) \(\mathstrut +\mathstrut 52q^{56} \) \(\mathstrut +\mathstrut 21q^{57} \) \(\mathstrut +\mathstrut 55q^{58} \) \(\mathstrut +\mathstrut 125q^{59} \) \(\mathstrut +\mathstrut 39q^{60} \) \(\mathstrut +\mathstrut 227q^{61} \) \(\mathstrut +\mathstrut 58q^{62} \) \(\mathstrut +\mathstrut 292q^{63} \) \(\mathstrut +\mathstrut 710q^{64} \) \(\mathstrut +\mathstrut 15q^{65} \) \(\mathstrut +\mathstrut 105q^{66} \) \(\mathstrut +\mathstrut 120q^{67} \) \(\mathstrut +\mathstrut 125q^{68} \) \(\mathstrut +\mathstrut 136q^{69} \) \(\mathstrut +\mathstrut 88q^{70} \) \(\mathstrut +\mathstrut 105q^{71} \) \(\mathstrut +\mathstrut 78q^{72} \) \(\mathstrut +\mathstrut 108q^{73} \) \(\mathstrut +\mathstrut 41q^{74} \) \(\mathstrut +\mathstrut 128q^{75} \) \(\mathstrut +\mathstrut 461q^{76} \) \(\mathstrut +\mathstrut 28q^{77} \) \(\mathstrut +\mathstrut 13q^{78} \) \(\mathstrut +\mathstrut 400q^{79} \) \(\mathstrut +\mathstrut 59q^{80} \) \(\mathstrut +\mathstrut 485q^{81} \) \(\mathstrut +\mathstrut 175q^{82} \) \(\mathstrut +\mathstrut 97q^{83} \) \(\mathstrut +\mathstrut 76q^{84} \) \(\mathstrut +\mathstrut 144q^{85} \) \(\mathstrut -\mathstrut 14q^{86} \) \(\mathstrut +\mathstrut 327q^{87} \) \(\mathstrut +\mathstrut 145q^{88} \) \(\mathstrut +\mathstrut 52q^{89} \) \(\mathstrut +\mathstrut 60q^{90} \) \(\mathstrut +\mathstrut 192q^{91} \) \(\mathstrut +\mathstrut 11q^{92} \) \(\mathstrut +\mathstrut 32q^{93} \) \(\mathstrut +\mathstrut 366q^{94} \) \(\mathstrut +\mathstrut 182q^{95} \) \(\mathstrut +\mathstrut 275q^{96} \) \(\mathstrut +\mathstrut 117q^{97} \) \(\mathstrut +\mathstrut 42q^{98} \) \(\mathstrut +\mathstrut 111q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.82633 2.37476 5.98811 1.91635 −6.71185 −2.63274 −11.2717 2.63950 −5.41622
1.2 −2.82341 −3.01045 5.97167 2.99967 8.49976 4.35875 −11.2137 6.06283 −8.46931
1.3 −2.80716 0.197946 5.88014 −1.81269 −0.555666 −3.48648 −10.8922 −2.96082 5.08851
1.4 −2.80354 −2.63483 5.85984 1.64291 7.38685 0.300217 −10.8212 3.94232 −4.60596
1.5 −2.78850 0.327361 5.77575 −3.12369 −0.912847 1.40614 −10.5287 −2.89283 8.71042
1.6 −2.76744 2.44821 5.65873 −3.94684 −6.77527 −1.82730 −10.1253 2.99372 10.9226
1.7 −2.74210 −3.01470 5.51909 −3.17594 8.26659 1.49397 −9.64969 6.08840 8.70874
1.8 −2.74193 −0.421446 5.51817 −1.04239 1.15558 0.747202 −9.64659 −2.82238 2.85815
1.9 −2.71747 −2.51708 5.38462 −3.75245 6.84009 −1.33469 −9.19758 3.33572 10.1972
1.10 −2.70375 −0.429580 5.31028 −1.54181 1.16148 3.87074 −8.95019 −2.81546 4.16866
1.11 −2.70301 1.53742 5.30627 −0.0405993 −4.15566 1.54013 −8.93687 −0.636347 0.109740
1.12 −2.69966 0.961365 5.28818 0.760429 −2.59536 −4.15994 −8.87697 −2.07578 −2.05290
1.13 −2.69652 −0.708536 5.27124 1.66526 1.91059 4.37208 −8.82098 −2.49798 −4.49040
1.14 −2.69232 2.96337 5.24860 −1.91075 −7.97835 3.66239 −8.74628 5.78157 5.14435
1.15 −2.65249 1.26544 5.03571 2.87387 −3.35656 2.78404 −8.05219 −1.39867 −7.62292
1.16 −2.64190 −1.64895 4.97966 −2.14113 4.35636 −4.48395 −7.87198 −0.280971 5.65667
1.17 −2.63785 2.50173 4.95827 3.39410 −6.59921 0.113907 −7.80348 3.25867 −8.95313
1.18 −2.63069 −2.68243 4.92052 0.701022 7.05664 −5.11853 −7.68299 4.19544 −1.84417
1.19 −2.63067 1.36161 4.92043 −3.62909 −3.58196 4.78654 −7.68268 −1.14601 9.54694
1.20 −2.62631 −2.69396 4.89749 2.82018 7.07518 −2.28331 −7.60969 4.25744 −7.40666
See next 80 embeddings (of 361 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.361
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(8009\) \(-1\)