Properties

Label 177.12.a.a.1.19
Level $177$
Weight $12$
Character 177.1
Self dual yes
Analytic conductor $135.997$
Analytic rank $1$
Dimension $26$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(135.996742959\)
Analytic rank: \(1\)
Dimension: \(26\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.19
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+38.1800 q^{2} -243.000 q^{3} -590.285 q^{4} +118.373 q^{5} -9277.75 q^{6} -24998.6 q^{7} -100730. q^{8} +59049.0 q^{9} +O(q^{10})\) \(q+38.1800 q^{2} -243.000 q^{3} -590.285 q^{4} +118.373 q^{5} -9277.75 q^{6} -24998.6 q^{7} -100730. q^{8} +59049.0 q^{9} +4519.50 q^{10} +78307.6 q^{11} +143439. q^{12} +516458. q^{13} -954447. q^{14} -28764.7 q^{15} -2.63696e6 q^{16} +707304. q^{17} +2.25449e6 q^{18} +1.93229e7 q^{19} -69874.0 q^{20} +6.07465e6 q^{21} +2.98979e6 q^{22} -4.56770e6 q^{23} +2.44773e7 q^{24} -4.88141e7 q^{25} +1.97184e7 q^{26} -1.43489e7 q^{27} +1.47563e7 q^{28} +1.21637e8 q^{29} -1.09824e6 q^{30} +5.43729e7 q^{31} +1.05615e8 q^{32} -1.90288e7 q^{33} +2.70049e7 q^{34} -2.95916e6 q^{35} -3.48557e7 q^{36} +3.64765e8 q^{37} +7.37749e8 q^{38} -1.25499e8 q^{39} -1.19237e7 q^{40} -9.48045e7 q^{41} +2.31931e8 q^{42} -1.12923e9 q^{43} -4.62238e7 q^{44} +6.98983e6 q^{45} -1.74395e8 q^{46} -1.05885e9 q^{47} +6.40782e8 q^{48} -1.35240e9 q^{49} -1.86372e9 q^{50} -1.71875e8 q^{51} -3.04857e8 q^{52} +1.10685e9 q^{53} -5.47842e8 q^{54} +9.26954e6 q^{55} +2.51810e9 q^{56} -4.69546e9 q^{57} +4.64409e9 q^{58} +7.14924e8 q^{59} +1.69794e7 q^{60} +9.70538e9 q^{61} +2.07596e9 q^{62} -1.47614e9 q^{63} +9.43290e9 q^{64} +6.11348e7 q^{65} -7.26519e8 q^{66} -1.88824e10 q^{67} -4.17511e8 q^{68} +1.10995e9 q^{69} -1.12981e8 q^{70} -1.71949e9 q^{71} -5.94799e9 q^{72} -6.45985e9 q^{73} +1.39268e10 q^{74} +1.18618e10 q^{75} -1.14060e10 q^{76} -1.95758e9 q^{77} -4.79157e9 q^{78} +3.94200e9 q^{79} -3.12146e8 q^{80} +3.48678e9 q^{81} -3.61964e9 q^{82} +1.19778e10 q^{83} -3.58578e9 q^{84} +8.37260e7 q^{85} -4.31142e10 q^{86} -2.95577e10 q^{87} -7.88791e9 q^{88} +2.13002e10 q^{89} +2.66872e8 q^{90} -1.29107e10 q^{91} +2.69624e9 q^{92} -1.32126e10 q^{93} -4.04270e10 q^{94} +2.28732e9 q^{95} -2.56645e10 q^{96} -4.49781e10 q^{97} -5.16346e10 q^{98} +4.62399e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 26q - 78q^{2} - 6318q^{3} + 23070q^{4} + 3808q^{5} + 18954q^{6} - 98819q^{7} - 117645q^{8} + 1535274q^{9} + O(q^{10}) \) \( 26q - 78q^{2} - 6318q^{3} + 23070q^{4} + 3808q^{5} + 18954q^{6} - 98819q^{7} - 117645q^{8} + 1535274q^{9} - 859751q^{10} + 579094q^{11} - 5606010q^{12} - 2018538q^{13} + 4157413q^{14} - 925344q^{15} + 20190274q^{16} - 13084493q^{17} - 4605822q^{18} + 9917231q^{19} + 10165633q^{20} + 24013017q^{21} - 89820518q^{22} - 63513223q^{23} + 28587735q^{24} + 218986852q^{25} - 77999532q^{26} - 373071582q^{27} - 444601862q^{28} + 81530981q^{29} + 208919493q^{30} - 408861231q^{31} - 26253128q^{32} - 140719842q^{33} - 508910076q^{34} - 75731421q^{35} + 1362260430q^{36} - 802381301q^{37} + 732704675q^{38} + 490504734q^{39} - 646130800q^{40} - 1354472849q^{41} - 1010251359q^{42} + 282952194q^{43} + 1846047996q^{44} + 224858592q^{45} + 9629305849q^{46} - 1196794197q^{47} - 4906236582q^{48} + 10889725683q^{49} - 6236232091q^{50} + 3179531799q^{51} - 1968200812q^{52} - 8276044236q^{53} + 1119214746q^{54} - 6672895076q^{55} + 2579741342q^{56} - 2409887133q^{57} - 9401656060q^{58} + 18588031774q^{59} - 2470248819q^{60} - 21181559029q^{61} - 6117706514q^{62} - 5835163131q^{63} + 42975855037q^{64} + 25680681860q^{65} + 21826385874q^{66} + 26234163394q^{67} + 19707344091q^{68} + 15433713189q^{69} + 129203099090q^{70} + 52088830406q^{71} - 6946819605q^{72} + 20943384867q^{73} + 41969200146q^{74} - 53213805036q^{75} + 223987219368q^{76} + 94604773153q^{77} + 18953886276q^{78} + 68965662774q^{79} + 218947784293q^{80} + 90656394426q^{81} + 11938614923q^{82} + 17947446393q^{83} + 108038252466q^{84} - 52849386709q^{85} + 384986147852q^{86} - 19812028383q^{87} - 49061112607q^{88} + 38570593981q^{89} - 50767436799q^{90} - 226268806999q^{91} - 79559686310q^{92} + 99353279133q^{93} - 16709400108q^{94} - 252795831501q^{95} + 6379510104q^{96} - 186894587836q^{97} - 252443311612q^{98} + 34194921606q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 38.1800 0.843668 0.421834 0.906673i \(-0.361387\pi\)
0.421834 + 0.906673i \(0.361387\pi\)
\(3\) −243.000 −0.577350
\(4\) −590.285 −0.288225
\(5\) 118.373 0.0169402 0.00847011 0.999964i \(-0.497304\pi\)
0.00847011 + 0.999964i \(0.497304\pi\)
\(6\) −9277.75 −0.487092
\(7\) −24998.6 −0.562181 −0.281090 0.959681i \(-0.590696\pi\)
−0.281090 + 0.959681i \(0.590696\pi\)
\(8\) −100730. −1.08683
\(9\) 59049.0 0.333333
\(10\) 4519.50 0.0142919
\(11\) 78307.6 0.146604 0.0733018 0.997310i \(-0.476646\pi\)
0.0733018 + 0.997310i \(0.476646\pi\)
\(12\) 143439. 0.166407
\(13\) 516458. 0.385786 0.192893 0.981220i \(-0.438213\pi\)
0.192893 + 0.981220i \(0.438213\pi\)
\(14\) −954447. −0.474294
\(15\) −28764.7 −0.00978044
\(16\) −2.63696e6 −0.628701
\(17\) 707304. 0.120820 0.0604098 0.998174i \(-0.480759\pi\)
0.0604098 + 0.998174i \(0.480759\pi\)
\(18\) 2.25449e6 0.281223
\(19\) 1.93229e7 1.79031 0.895153 0.445758i \(-0.147066\pi\)
0.895153 + 0.445758i \(0.147066\pi\)
\(20\) −69874.0 −0.00488259
\(21\) 6.07465e6 0.324575
\(22\) 2.98979e6 0.123685
\(23\) −4.56770e6 −0.147977 −0.0739885 0.997259i \(-0.523573\pi\)
−0.0739885 + 0.997259i \(0.523573\pi\)
\(24\) 2.44773e7 0.627484
\(25\) −4.88141e7 −0.999713
\(26\) 1.97184e7 0.325475
\(27\) −1.43489e7 −0.192450
\(28\) 1.47563e7 0.162035
\(29\) 1.21637e8 1.10122 0.550612 0.834761i \(-0.314394\pi\)
0.550612 + 0.834761i \(0.314394\pi\)
\(30\) −1.09824e6 −0.00825144
\(31\) 5.43729e7 0.341109 0.170555 0.985348i \(-0.445444\pi\)
0.170555 + 0.985348i \(0.445444\pi\)
\(32\) 1.05615e8 0.556419
\(33\) −1.90288e7 −0.0846416
\(34\) 2.70049e7 0.101932
\(35\) −2.95916e6 −0.00952346
\(36\) −3.48557e7 −0.0960750
\(37\) 3.64765e8 0.864777 0.432388 0.901687i \(-0.357671\pi\)
0.432388 + 0.901687i \(0.357671\pi\)
\(38\) 7.37749e8 1.51042
\(39\) −1.25499e8 −0.222734
\(40\) −1.19237e7 −0.0184112
\(41\) −9.48045e7 −0.127796 −0.0638980 0.997956i \(-0.520353\pi\)
−0.0638980 + 0.997956i \(0.520353\pi\)
\(42\) 2.31931e8 0.273834
\(43\) −1.12923e9 −1.17141 −0.585703 0.810526i \(-0.699182\pi\)
−0.585703 + 0.810526i \(0.699182\pi\)
\(44\) −4.62238e7 −0.0422548
\(45\) 6.98983e6 0.00564674
\(46\) −1.74395e8 −0.124843
\(47\) −1.05885e9 −0.673437 −0.336719 0.941605i \(-0.609317\pi\)
−0.336719 + 0.941605i \(0.609317\pi\)
\(48\) 6.40782e8 0.362981
\(49\) −1.35240e9 −0.683953
\(50\) −1.86372e9 −0.843426
\(51\) −1.71875e8 −0.0697552
\(52\) −3.04857e8 −0.111193
\(53\) 1.10685e9 0.363555 0.181778 0.983340i \(-0.441815\pi\)
0.181778 + 0.983340i \(0.441815\pi\)
\(54\) −5.47842e8 −0.162364
\(55\) 9.26954e6 0.00248349
\(56\) 2.51810e9 0.610997
\(57\) −4.69546e9 −1.03363
\(58\) 4.64409e9 0.929067
\(59\) 7.14924e8 0.130189
\(60\) 1.69794e7 0.00281897
\(61\) 9.70538e9 1.47129 0.735645 0.677367i \(-0.236879\pi\)
0.735645 + 0.677367i \(0.236879\pi\)
\(62\) 2.07596e9 0.287783
\(63\) −1.47614e9 −0.187394
\(64\) 9.43290e9 1.09813
\(65\) 6.11348e7 0.00653529
\(66\) −7.26519e8 −0.0714094
\(67\) −1.88824e10 −1.70862 −0.854311 0.519762i \(-0.826020\pi\)
−0.854311 + 0.519762i \(0.826020\pi\)
\(68\) −4.17511e8 −0.0348232
\(69\) 1.10995e9 0.0854346
\(70\) −1.12981e8 −0.00803464
\(71\) −1.71949e9 −0.113104 −0.0565520 0.998400i \(-0.518011\pi\)
−0.0565520 + 0.998400i \(0.518011\pi\)
\(72\) −5.94799e9 −0.362278
\(73\) −6.45985e9 −0.364709 −0.182354 0.983233i \(-0.558372\pi\)
−0.182354 + 0.983233i \(0.558372\pi\)
\(74\) 1.39268e10 0.729584
\(75\) 1.18618e10 0.577185
\(76\) −1.14060e10 −0.516011
\(77\) −1.95758e9 −0.0824177
\(78\) −4.79157e9 −0.187913
\(79\) 3.94200e9 0.144134 0.0720671 0.997400i \(-0.477040\pi\)
0.0720671 + 0.997400i \(0.477040\pi\)
\(80\) −3.12146e8 −0.0106503
\(81\) 3.48678e9 0.111111
\(82\) −3.61964e9 −0.107817
\(83\) 1.19778e10 0.333771 0.166885 0.985976i \(-0.446629\pi\)
0.166885 + 0.985976i \(0.446629\pi\)
\(84\) −3.58578e9 −0.0935507
\(85\) 8.37260e7 0.00204671
\(86\) −4.31142e10 −0.988278
\(87\) −2.95577e10 −0.635792
\(88\) −7.88791e9 −0.159334
\(89\) 2.13002e10 0.404332 0.202166 0.979351i \(-0.435202\pi\)
0.202166 + 0.979351i \(0.435202\pi\)
\(90\) 2.66872e8 0.00476397
\(91\) −1.29107e10 −0.216881
\(92\) 2.69624e9 0.0426507
\(93\) −1.32126e10 −0.196940
\(94\) −4.04270e10 −0.568157
\(95\) 2.28732e9 0.0303282
\(96\) −2.56645e10 −0.321248
\(97\) −4.49781e10 −0.531810 −0.265905 0.963999i \(-0.585671\pi\)
−0.265905 + 0.963999i \(0.585671\pi\)
\(98\) −5.16346e10 −0.577029
\(99\) 4.62399e9 0.0488678
\(100\) 2.88142e10 0.288142
\(101\) −6.75570e10 −0.639592 −0.319796 0.947486i \(-0.603614\pi\)
−0.319796 + 0.947486i \(0.603614\pi\)
\(102\) −6.56219e9 −0.0588502
\(103\) −9.45356e9 −0.0803509 −0.0401755 0.999193i \(-0.512792\pi\)
−0.0401755 + 0.999193i \(0.512792\pi\)
\(104\) −5.20227e10 −0.419285
\(105\) 7.19077e8 0.00549837
\(106\) 4.22595e10 0.306720
\(107\) −2.46903e11 −1.70182 −0.850912 0.525308i \(-0.823950\pi\)
−0.850912 + 0.525308i \(0.823950\pi\)
\(108\) 8.46994e9 0.0554689
\(109\) −6.18565e10 −0.385070 −0.192535 0.981290i \(-0.561671\pi\)
−0.192535 + 0.981290i \(0.561671\pi\)
\(110\) 3.53911e8 0.00209524
\(111\) −8.86380e10 −0.499279
\(112\) 6.59204e10 0.353444
\(113\) −2.13710e11 −1.09117 −0.545586 0.838055i \(-0.683693\pi\)
−0.545586 + 0.838055i \(0.683693\pi\)
\(114\) −1.79273e11 −0.872043
\(115\) −5.40694e8 −0.00250676
\(116\) −7.18003e10 −0.317400
\(117\) 3.04963e10 0.128595
\(118\) 2.72958e10 0.109836
\(119\) −1.76816e10 −0.0679224
\(120\) 2.89746e9 0.0106297
\(121\) −2.79180e11 −0.978507
\(122\) 3.70552e11 1.24128
\(123\) 2.30375e10 0.0737831
\(124\) −3.20955e10 −0.0983162
\(125\) −1.15582e10 −0.0338756
\(126\) −5.63591e10 −0.158098
\(127\) 2.58686e11 0.694787 0.347394 0.937719i \(-0.387067\pi\)
0.347394 + 0.937719i \(0.387067\pi\)
\(128\) 1.43848e11 0.370041
\(129\) 2.74404e11 0.676312
\(130\) 2.33413e9 0.00551362
\(131\) −4.89314e10 −0.110814 −0.0554071 0.998464i \(-0.517646\pi\)
−0.0554071 + 0.998464i \(0.517646\pi\)
\(132\) 1.12324e10 0.0243958
\(133\) −4.83045e11 −1.00648
\(134\) −7.20931e11 −1.44151
\(135\) −1.69853e9 −0.00326015
\(136\) −7.12466e10 −0.131311
\(137\) −3.61771e11 −0.640429 −0.320214 0.947345i \(-0.603755\pi\)
−0.320214 + 0.947345i \(0.603755\pi\)
\(138\) 4.23780e10 0.0720784
\(139\) −5.60774e11 −0.916657 −0.458328 0.888783i \(-0.651552\pi\)
−0.458328 + 0.888783i \(0.651552\pi\)
\(140\) 1.74675e9 0.00274490
\(141\) 2.57301e11 0.388809
\(142\) −6.56501e10 −0.0954223
\(143\) 4.04426e10 0.0565576
\(144\) −1.55710e11 −0.209567
\(145\) 1.43985e10 0.0186550
\(146\) −2.46637e11 −0.307693
\(147\) 3.28633e11 0.394880
\(148\) −2.15315e11 −0.249250
\(149\) 1.48332e11 0.165467 0.0827333 0.996572i \(-0.473635\pi\)
0.0827333 + 0.996572i \(0.473635\pi\)
\(150\) 4.52885e11 0.486952
\(151\) 7.02729e11 0.728475 0.364238 0.931306i \(-0.381330\pi\)
0.364238 + 0.931306i \(0.381330\pi\)
\(152\) −1.94639e12 −1.94577
\(153\) 4.17656e10 0.0402732
\(154\) −7.47405e10 −0.0695331
\(155\) 6.43631e9 0.00577846
\(156\) 7.40803e10 0.0641974
\(157\) −6.12975e11 −0.512855 −0.256428 0.966563i \(-0.582546\pi\)
−0.256428 + 0.966563i \(0.582546\pi\)
\(158\) 1.50506e11 0.121601
\(159\) −2.68964e11 −0.209899
\(160\) 1.25020e10 0.00942585
\(161\) 1.14186e11 0.0831899
\(162\) 1.33126e11 0.0937408
\(163\) 6.05132e11 0.411925 0.205962 0.978560i \(-0.433968\pi\)
0.205962 + 0.978560i \(0.433968\pi\)
\(164\) 5.59616e10 0.0368340
\(165\) −2.25250e9 −0.00143385
\(166\) 4.57314e11 0.281592
\(167\) 1.19419e12 0.711429 0.355714 0.934595i \(-0.384238\pi\)
0.355714 + 0.934595i \(0.384238\pi\)
\(168\) −6.11899e11 −0.352759
\(169\) −1.52543e12 −0.851169
\(170\) 3.19666e9 0.00172674
\(171\) 1.14100e12 0.596769
\(172\) 6.66570e11 0.337629
\(173\) −4.78670e11 −0.234846 −0.117423 0.993082i \(-0.537463\pi\)
−0.117423 + 0.993082i \(0.537463\pi\)
\(174\) −1.12851e12 −0.536397
\(175\) 1.22028e12 0.562020
\(176\) −2.06495e11 −0.0921698
\(177\) −1.73727e11 −0.0751646
\(178\) 8.13241e11 0.341122
\(179\) −2.16926e12 −0.882308 −0.441154 0.897431i \(-0.645431\pi\)
−0.441154 + 0.897431i \(0.645431\pi\)
\(180\) −4.12599e9 −0.00162753
\(181\) 1.23590e12 0.472881 0.236441 0.971646i \(-0.424019\pi\)
0.236441 + 0.971646i \(0.424019\pi\)
\(182\) −4.92931e11 −0.182976
\(183\) −2.35841e12 −0.849450
\(184\) 4.60104e11 0.160826
\(185\) 4.31785e10 0.0146495
\(186\) −5.04458e11 −0.166151
\(187\) 5.53873e10 0.0177126
\(188\) 6.25024e11 0.194101
\(189\) 3.58702e11 0.108192
\(190\) 8.73298e10 0.0255869
\(191\) 3.89010e12 1.10733 0.553665 0.832740i \(-0.313229\pi\)
0.553665 + 0.832740i \(0.313229\pi\)
\(192\) −2.29219e12 −0.634008
\(193\) −3.05506e12 −0.821210 −0.410605 0.911813i \(-0.634682\pi\)
−0.410605 + 0.911813i \(0.634682\pi\)
\(194\) −1.71726e12 −0.448671
\(195\) −1.48558e10 −0.00377315
\(196\) 7.98300e11 0.197132
\(197\) −5.41761e12 −1.30090 −0.650450 0.759549i \(-0.725419\pi\)
−0.650450 + 0.759549i \(0.725419\pi\)
\(198\) 1.76544e11 0.0412282
\(199\) 2.53133e12 0.574986 0.287493 0.957783i \(-0.407178\pi\)
0.287493 + 0.957783i \(0.407178\pi\)
\(200\) 4.91704e12 1.08652
\(201\) 4.58842e12 0.986473
\(202\) −2.57933e12 −0.539603
\(203\) −3.04074e12 −0.619087
\(204\) 1.01455e11 0.0201052
\(205\) −1.12223e10 −0.00216489
\(206\) −3.60937e11 −0.0677895
\(207\) −2.69718e11 −0.0493257
\(208\) −1.36188e12 −0.242544
\(209\) 1.51313e12 0.262465
\(210\) 2.74544e10 0.00463880
\(211\) −8.82774e12 −1.45310 −0.726551 0.687112i \(-0.758878\pi\)
−0.726551 + 0.687112i \(0.758878\pi\)
\(212\) −6.53355e11 −0.104786
\(213\) 4.17836e11 0.0653007
\(214\) −9.42675e12 −1.43577
\(215\) −1.33671e11 −0.0198439
\(216\) 1.44536e12 0.209161
\(217\) −1.35925e12 −0.191765
\(218\) −2.36169e12 −0.324871
\(219\) 1.56974e12 0.210565
\(220\) −5.47167e9 −0.000715805 0
\(221\) 3.65293e11 0.0466105
\(222\) −3.38420e12 −0.421226
\(223\) −1.48106e11 −0.0179844 −0.00899218 0.999960i \(-0.502862\pi\)
−0.00899218 + 0.999960i \(0.502862\pi\)
\(224\) −2.64023e12 −0.312808
\(225\) −2.88242e12 −0.333238
\(226\) −8.15946e12 −0.920587
\(227\) −1.46590e13 −1.61422 −0.807111 0.590400i \(-0.798970\pi\)
−0.807111 + 0.590400i \(0.798970\pi\)
\(228\) 2.77166e12 0.297919
\(229\) 1.16723e13 1.22478 0.612392 0.790554i \(-0.290207\pi\)
0.612392 + 0.790554i \(0.290207\pi\)
\(230\) −2.06437e10 −0.00211487
\(231\) 4.75692e11 0.0475839
\(232\) −1.22524e13 −1.19685
\(233\) −7.50544e12 −0.716009 −0.358005 0.933720i \(-0.616543\pi\)
−0.358005 + 0.933720i \(0.616543\pi\)
\(234\) 1.16435e12 0.108492
\(235\) −1.25340e11 −0.0114082
\(236\) −4.22009e11 −0.0375237
\(237\) −9.57905e11 −0.0832159
\(238\) −6.75084e11 −0.0573040
\(239\) 4.48254e12 0.371823 0.185911 0.982567i \(-0.440476\pi\)
0.185911 + 0.982567i \(0.440476\pi\)
\(240\) 7.58515e10 0.00614897
\(241\) 2.28198e13 1.80808 0.904041 0.427445i \(-0.140586\pi\)
0.904041 + 0.427445i \(0.140586\pi\)
\(242\) −1.06591e13 −0.825535
\(243\) −8.47289e11 −0.0641500
\(244\) −5.72894e12 −0.424063
\(245\) −1.60088e11 −0.0115863
\(246\) 8.79572e11 0.0622484
\(247\) 9.97946e12 0.690675
\(248\) −5.47698e12 −0.370729
\(249\) −2.91061e12 −0.192703
\(250\) −4.41294e11 −0.0285797
\(251\) −1.04956e13 −0.664972 −0.332486 0.943108i \(-0.607887\pi\)
−0.332486 + 0.943108i \(0.607887\pi\)
\(252\) 8.71343e11 0.0540115
\(253\) −3.57686e11 −0.0216940
\(254\) 9.87663e12 0.586170
\(255\) −2.03454e10 −0.00118167
\(256\) −1.38264e13 −0.785942
\(257\) −1.84143e13 −1.02452 −0.512262 0.858829i \(-0.671192\pi\)
−0.512262 + 0.858829i \(0.671192\pi\)
\(258\) 1.04768e13 0.570582
\(259\) −9.11861e12 −0.486161
\(260\) −3.60870e10 −0.00188364
\(261\) 7.18252e12 0.367075
\(262\) −1.86820e12 −0.0934904
\(263\) −1.21552e13 −0.595668 −0.297834 0.954618i \(-0.596264\pi\)
−0.297834 + 0.954618i \(0.596264\pi\)
\(264\) 1.91676e12 0.0919913
\(265\) 1.31021e11 0.00615870
\(266\) −1.84427e13 −0.849131
\(267\) −5.17594e12 −0.233441
\(268\) 1.11460e13 0.492467
\(269\) −3.42443e13 −1.48235 −0.741175 0.671312i \(-0.765731\pi\)
−0.741175 + 0.671312i \(0.765731\pi\)
\(270\) −6.48499e10 −0.00275048
\(271\) 8.60560e12 0.357643 0.178822 0.983882i \(-0.442772\pi\)
0.178822 + 0.983882i \(0.442772\pi\)
\(272\) −1.86514e12 −0.0759594
\(273\) 3.13730e12 0.125217
\(274\) −1.38124e13 −0.540309
\(275\) −3.82252e12 −0.146561
\(276\) −6.55187e11 −0.0246244
\(277\) 2.62981e13 0.968915 0.484457 0.874815i \(-0.339017\pi\)
0.484457 + 0.874815i \(0.339017\pi\)
\(278\) −2.14104e13 −0.773354
\(279\) 3.21067e12 0.113703
\(280\) 2.98076e11 0.0103504
\(281\) −9.46162e12 −0.322167 −0.161083 0.986941i \(-0.551499\pi\)
−0.161083 + 0.986941i \(0.551499\pi\)
\(282\) 9.82376e12 0.328026
\(283\) −2.59883e13 −0.851046 −0.425523 0.904948i \(-0.639910\pi\)
−0.425523 + 0.904948i \(0.639910\pi\)
\(284\) 1.01499e12 0.0325994
\(285\) −5.55818e11 −0.0175100
\(286\) 1.54410e12 0.0477158
\(287\) 2.36998e12 0.0718445
\(288\) 6.23647e12 0.185473
\(289\) −3.37716e13 −0.985403
\(290\) 5.49737e11 0.0157386
\(291\) 1.09297e13 0.307041
\(292\) 3.81315e12 0.105118
\(293\) 2.86354e13 0.774697 0.387348 0.921933i \(-0.373391\pi\)
0.387348 + 0.921933i \(0.373391\pi\)
\(294\) 1.25472e13 0.333148
\(295\) 8.46280e10 0.00220543
\(296\) −3.67427e13 −0.939869
\(297\) −1.12363e12 −0.0282139
\(298\) 5.66332e12 0.139599
\(299\) −2.35902e12 −0.0570875
\(300\) −7.00186e12 −0.166359
\(301\) 2.82292e13 0.658542
\(302\) 2.68302e13 0.614591
\(303\) 1.64164e13 0.369269
\(304\) −5.09538e13 −1.12557
\(305\) 1.14886e12 0.0249240
\(306\) 1.59461e12 0.0339772
\(307\) −3.67740e13 −0.769625 −0.384813 0.922995i \(-0.625734\pi\)
−0.384813 + 0.922995i \(0.625734\pi\)
\(308\) 1.15553e12 0.0237548
\(309\) 2.29722e12 0.0463906
\(310\) 2.45738e11 0.00487510
\(311\) 7.76531e13 1.51348 0.756740 0.653716i \(-0.226791\pi\)
0.756740 + 0.653716i \(0.226791\pi\)
\(312\) 1.26415e13 0.242074
\(313\) 8.63382e13 1.62446 0.812230 0.583338i \(-0.198254\pi\)
0.812230 + 0.583338i \(0.198254\pi\)
\(314\) −2.34034e13 −0.432680
\(315\) −1.74736e11 −0.00317449
\(316\) −2.32690e12 −0.0415431
\(317\) 7.03310e13 1.23402 0.617008 0.786956i \(-0.288344\pi\)
0.617008 + 0.786956i \(0.288344\pi\)
\(318\) −1.02691e13 −0.177085
\(319\) 9.52508e12 0.161443
\(320\) 1.11660e12 0.0186026
\(321\) 5.99973e13 0.982549
\(322\) 4.35963e12 0.0701846
\(323\) 1.36672e13 0.216304
\(324\) −2.05820e12 −0.0320250
\(325\) −2.52104e13 −0.385675
\(326\) 2.31039e13 0.347528
\(327\) 1.50311e13 0.222320
\(328\) 9.54964e12 0.138893
\(329\) 2.64698e13 0.378594
\(330\) −8.60004e10 −0.00120969
\(331\) −1.27120e13 −0.175857 −0.0879286 0.996127i \(-0.528025\pi\)
−0.0879286 + 0.996127i \(0.528025\pi\)
\(332\) −7.07033e12 −0.0962011
\(333\) 2.15390e13 0.288259
\(334\) 4.55941e13 0.600209
\(335\) −2.23517e12 −0.0289444
\(336\) −1.60186e13 −0.204061
\(337\) −1.20215e13 −0.150659 −0.0753294 0.997159i \(-0.524001\pi\)
−0.0753294 + 0.997159i \(0.524001\pi\)
\(338\) −5.82410e13 −0.718104
\(339\) 5.19315e13 0.629989
\(340\) −4.94222e10 −0.000589913 0
\(341\) 4.25782e12 0.0500078
\(342\) 4.35633e13 0.503475
\(343\) 8.32384e13 0.946686
\(344\) 1.13748e14 1.27312
\(345\) 1.31389e11 0.00144728
\(346\) −1.82757e13 −0.198132
\(347\) 1.61543e13 0.172376 0.0861879 0.996279i \(-0.472531\pi\)
0.0861879 + 0.996279i \(0.472531\pi\)
\(348\) 1.74475e13 0.183251
\(349\) 1.06002e14 1.09591 0.547954 0.836509i \(-0.315407\pi\)
0.547954 + 0.836509i \(0.315407\pi\)
\(350\) 4.65905e13 0.474158
\(351\) −7.41060e12 −0.0742445
\(352\) 8.27048e12 0.0815729
\(353\) 6.62164e13 0.642991 0.321495 0.946911i \(-0.395815\pi\)
0.321495 + 0.946911i \(0.395815\pi\)
\(354\) −6.63289e12 −0.0634139
\(355\) −2.03542e11 −0.00191601
\(356\) −1.25732e13 −0.116539
\(357\) 4.29663e12 0.0392150
\(358\) −8.28225e13 −0.744375
\(359\) −1.63008e13 −0.144275 −0.0721374 0.997395i \(-0.522982\pi\)
−0.0721374 + 0.997395i \(0.522982\pi\)
\(360\) −7.04084e11 −0.00613706
\(361\) 2.56884e14 2.20520
\(362\) 4.71868e13 0.398955
\(363\) 6.78406e13 0.564942
\(364\) 7.62099e12 0.0625106
\(365\) −7.64673e11 −0.00617825
\(366\) −9.00441e13 −0.716653
\(367\) −6.31664e13 −0.495248 −0.247624 0.968856i \(-0.579650\pi\)
−0.247624 + 0.968856i \(0.579650\pi\)
\(368\) 1.20449e13 0.0930334
\(369\) −5.59811e12 −0.0425987
\(370\) 1.64856e12 0.0123593
\(371\) −2.76696e13 −0.204384
\(372\) 7.79921e12 0.0567629
\(373\) 1.55068e13 0.111205 0.0556024 0.998453i \(-0.482292\pi\)
0.0556024 + 0.998453i \(0.482292\pi\)
\(374\) 2.11469e12 0.0149435
\(375\) 2.80865e12 0.0195581
\(376\) 1.06658e14 0.731914
\(377\) 6.28202e13 0.424837
\(378\) 1.36953e13 0.0912779
\(379\) −8.86462e13 −0.582297 −0.291148 0.956678i \(-0.594037\pi\)
−0.291148 + 0.956678i \(0.594037\pi\)
\(380\) −1.35017e12 −0.00874134
\(381\) −6.28606e13 −0.401136
\(382\) 1.48524e14 0.934218
\(383\) −1.41516e13 −0.0877430 −0.0438715 0.999037i \(-0.513969\pi\)
−0.0438715 + 0.999037i \(0.513969\pi\)
\(384\) −3.49552e13 −0.213643
\(385\) −2.31725e11 −0.00139617
\(386\) −1.16642e14 −0.692828
\(387\) −6.66802e13 −0.390469
\(388\) 2.65499e13 0.153281
\(389\) −1.11304e14 −0.633560 −0.316780 0.948499i \(-0.602602\pi\)
−0.316780 + 0.948499i \(0.602602\pi\)
\(390\) −5.67194e11 −0.00318329
\(391\) −3.23075e12 −0.0178785
\(392\) 1.36227e14 0.743343
\(393\) 1.18903e13 0.0639786
\(394\) −2.06845e14 −1.09753
\(395\) 4.66627e11 0.00244166
\(396\) −2.72947e12 −0.0140849
\(397\) 1.49667e14 0.761691 0.380846 0.924639i \(-0.375633\pi\)
0.380846 + 0.924639i \(0.375633\pi\)
\(398\) 9.66464e13 0.485097
\(399\) 1.17380e14 0.581089
\(400\) 1.28721e14 0.628521
\(401\) −2.70108e14 −1.30090 −0.650450 0.759549i \(-0.725419\pi\)
−0.650450 + 0.759549i \(0.725419\pi\)
\(402\) 1.75186e14 0.832255
\(403\) 2.80813e13 0.131595
\(404\) 3.98779e13 0.184346
\(405\) 4.12742e11 0.00188225
\(406\) −1.16096e14 −0.522303
\(407\) 2.85639e13 0.126779
\(408\) 1.73129e13 0.0758123
\(409\) −1.60681e14 −0.694204 −0.347102 0.937827i \(-0.612834\pi\)
−0.347102 + 0.937827i \(0.612834\pi\)
\(410\) −4.28469e11 −0.00182645
\(411\) 8.79104e13 0.369752
\(412\) 5.58029e12 0.0231591
\(413\) −1.78721e13 −0.0731897
\(414\) −1.02978e13 −0.0416145
\(415\) 1.41785e12 0.00565415
\(416\) 5.45458e13 0.214658
\(417\) 1.36268e14 0.529232
\(418\) 5.77714e13 0.221433
\(419\) 4.24501e14 1.60584 0.802919 0.596088i \(-0.203279\pi\)
0.802919 + 0.596088i \(0.203279\pi\)
\(420\) −4.24460e11 −0.00158477
\(421\) −1.38573e14 −0.510654 −0.255327 0.966855i \(-0.582183\pi\)
−0.255327 + 0.966855i \(0.582183\pi\)
\(422\) −3.37044e14 −1.22594
\(423\) −6.25242e13 −0.224479
\(424\) −1.11493e14 −0.395124
\(425\) −3.45264e13 −0.120785
\(426\) 1.59530e13 0.0550921
\(427\) −2.42621e14 −0.827131
\(428\) 1.45743e14 0.490508
\(429\) −9.82755e12 −0.0326535
\(430\) −5.10357e12 −0.0167416
\(431\) 2.32465e14 0.752893 0.376447 0.926438i \(-0.377146\pi\)
0.376447 + 0.926438i \(0.377146\pi\)
\(432\) 3.78376e13 0.120994
\(433\) −3.58882e13 −0.113310 −0.0566551 0.998394i \(-0.518044\pi\)
−0.0566551 + 0.998394i \(0.518044\pi\)
\(434\) −5.18961e13 −0.161786
\(435\) −3.49884e12 −0.0107704
\(436\) 3.65130e13 0.110987
\(437\) −8.82612e13 −0.264924
\(438\) 5.99328e13 0.177647
\(439\) 3.49088e14 1.02183 0.510916 0.859631i \(-0.329306\pi\)
0.510916 + 0.859631i \(0.329306\pi\)
\(440\) −9.33719e11 −0.00269915
\(441\) −7.98577e13 −0.227984
\(442\) 1.39469e13 0.0393238
\(443\) −1.39270e14 −0.387826 −0.193913 0.981019i \(-0.562118\pi\)
−0.193913 + 0.981019i \(0.562118\pi\)
\(444\) 5.23216e13 0.143905
\(445\) 2.52137e12 0.00684947
\(446\) −5.65468e12 −0.0151728
\(447\) −3.60447e13 −0.0955322
\(448\) −2.35809e14 −0.617350
\(449\) 4.43677e14 1.14739 0.573696 0.819068i \(-0.305509\pi\)
0.573696 + 0.819068i \(0.305509\pi\)
\(450\) −1.10051e14 −0.281142
\(451\) −7.42391e12 −0.0187354
\(452\) 1.26150e14 0.314503
\(453\) −1.70763e14 −0.420585
\(454\) −5.59683e14 −1.36187
\(455\) −1.52828e12 −0.00367402
\(456\) 4.72973e14 1.12339
\(457\) −2.75578e14 −0.646703 −0.323352 0.946279i \(-0.604810\pi\)
−0.323352 + 0.946279i \(0.604810\pi\)
\(458\) 4.45647e14 1.03331
\(459\) −1.01490e13 −0.0232517
\(460\) 3.19163e11 0.000722512 0
\(461\) −6.83076e14 −1.52797 −0.763983 0.645236i \(-0.776759\pi\)
−0.763983 + 0.645236i \(0.776759\pi\)
\(462\) 1.81619e13 0.0401450
\(463\) −5.88016e14 −1.28438 −0.642190 0.766545i \(-0.721974\pi\)
−0.642190 + 0.766545i \(0.721974\pi\)
\(464\) −3.20752e14 −0.692341
\(465\) −1.56402e12 −0.00333620
\(466\) −2.86558e14 −0.604074
\(467\) −7.88305e13 −0.164230 −0.0821148 0.996623i \(-0.526167\pi\)
−0.0821148 + 0.996623i \(0.526167\pi\)
\(468\) −1.80015e13 −0.0370644
\(469\) 4.72033e14 0.960554
\(470\) −4.78548e12 −0.00962470
\(471\) 1.48953e14 0.296097
\(472\) −7.20142e13 −0.141494
\(473\) −8.84277e13 −0.171732
\(474\) −3.65728e13 −0.0702066
\(475\) −9.43230e14 −1.78979
\(476\) 1.04372e13 0.0195769
\(477\) 6.53582e13 0.121185
\(478\) 1.71144e14 0.313695
\(479\) −6.85495e13 −0.124211 −0.0621054 0.998070i \(-0.519781\pi\)
−0.0621054 + 0.998070i \(0.519781\pi\)
\(480\) −3.03799e12 −0.00544202
\(481\) 1.88386e14 0.333619
\(482\) 8.71261e14 1.52542
\(483\) −2.77472e13 −0.0480297
\(484\) 1.64795e14 0.282030
\(485\) −5.32421e12 −0.00900897
\(486\) −3.23495e13 −0.0541213
\(487\) −6.19920e14 −1.02548 −0.512739 0.858544i \(-0.671369\pi\)
−0.512739 + 0.858544i \(0.671369\pi\)
\(488\) −9.77621e14 −1.59905
\(489\) −1.47047e14 −0.237825
\(490\) −6.11216e12 −0.00977499
\(491\) −1.56199e14 −0.247018 −0.123509 0.992343i \(-0.539415\pi\)
−0.123509 + 0.992343i \(0.539415\pi\)
\(492\) −1.35987e13 −0.0212661
\(493\) 8.60341e13 0.133049
\(494\) 3.81016e14 0.582700
\(495\) 5.47357e11 0.000827832 0
\(496\) −1.43380e14 −0.214456
\(497\) 4.29848e13 0.0635850
\(498\) −1.11127e14 −0.162577
\(499\) −5.75189e14 −0.832257 −0.416128 0.909306i \(-0.636613\pi\)
−0.416128 + 0.909306i \(0.636613\pi\)
\(500\) 6.82265e12 0.00976378
\(501\) −2.90187e14 −0.410744
\(502\) −4.00724e14 −0.561015
\(503\) 4.51069e14 0.624624 0.312312 0.949980i \(-0.398897\pi\)
0.312312 + 0.949980i \(0.398897\pi\)
\(504\) 1.48691e14 0.203666
\(505\) −7.99695e12 −0.0108348
\(506\) −1.36565e13 −0.0183025
\(507\) 3.70680e14 0.491423
\(508\) −1.52698e14 −0.200255
\(509\) 7.55708e14 0.980407 0.490204 0.871608i \(-0.336922\pi\)
0.490204 + 0.871608i \(0.336922\pi\)
\(510\) −7.76788e11 −0.000996935 0
\(511\) 1.61487e14 0.205032
\(512\) −8.22495e14 −1.03312
\(513\) −2.77262e14 −0.344545
\(514\) −7.03058e14 −0.864358
\(515\) −1.11905e12 −0.00136116
\(516\) −1.61976e14 −0.194930
\(517\) −8.29162e13 −0.0987283
\(518\) −3.48149e14 −0.410158
\(519\) 1.16317e14 0.135588
\(520\) −6.15810e12 −0.00710278
\(521\) 1.42188e15 1.62276 0.811382 0.584517i \(-0.198716\pi\)
0.811382 + 0.584517i \(0.198716\pi\)
\(522\) 2.74229e14 0.309689
\(523\) −6.61743e14 −0.739486 −0.369743 0.929134i \(-0.620554\pi\)
−0.369743 + 0.929134i \(0.620554\pi\)
\(524\) 2.88835e13 0.0319394
\(525\) −2.96529e14 −0.324482
\(526\) −4.64084e14 −0.502545
\(527\) 3.84582e13 0.0412127
\(528\) 5.01782e13 0.0532143
\(529\) −9.31946e14 −0.978103
\(530\) 5.00239e12 0.00519590
\(531\) 4.22156e13 0.0433963
\(532\) 2.85134e14 0.290092
\(533\) −4.89625e13 −0.0493019
\(534\) −1.97618e14 −0.196947
\(535\) −2.92267e13 −0.0288293
\(536\) 1.90202e15 1.85699
\(537\) 5.27131e14 0.509401
\(538\) −1.30745e15 −1.25061
\(539\) −1.05903e14 −0.100270
\(540\) 1.00262e12 0.000939655 0
\(541\) −1.00319e15 −0.930675 −0.465338 0.885133i \(-0.654067\pi\)
−0.465338 + 0.885133i \(0.654067\pi\)
\(542\) 3.28562e14 0.301732
\(543\) −3.00324e14 −0.273018
\(544\) 7.47021e13 0.0672263
\(545\) −7.32216e12 −0.00652317
\(546\) 1.19782e14 0.105641
\(547\) −2.23450e15 −1.95097 −0.975486 0.220062i \(-0.929374\pi\)
−0.975486 + 0.220062i \(0.929374\pi\)
\(548\) 2.13548e14 0.184588
\(549\) 5.73093e14 0.490430
\(550\) −1.45944e14 −0.123649
\(551\) 2.35037e15 1.97153
\(552\) −1.11805e14 −0.0928532
\(553\) −9.85443e13 −0.0810295
\(554\) 1.00406e15 0.817442
\(555\) −1.04924e13 −0.00845789
\(556\) 3.31017e14 0.264203
\(557\) −1.28284e15 −1.01384 −0.506921 0.861992i \(-0.669217\pi\)
−0.506921 + 0.861992i \(0.669217\pi\)
\(558\) 1.22583e14 0.0959276
\(559\) −5.83202e14 −0.451912
\(560\) 7.80321e12 0.00598741
\(561\) −1.34591e13 −0.0102264
\(562\) −3.61245e14 −0.271802
\(563\) −1.02356e15 −0.762633 −0.381316 0.924445i \(-0.624529\pi\)
−0.381316 + 0.924445i \(0.624529\pi\)
\(564\) −1.51881e14 −0.112065
\(565\) −2.52976e13 −0.0184847
\(566\) −9.92235e14 −0.718000
\(567\) −8.71646e13 −0.0624645
\(568\) 1.73204e14 0.122925
\(569\) −8.08918e13 −0.0568574 −0.0284287 0.999596i \(-0.509050\pi\)
−0.0284287 + 0.999596i \(0.509050\pi\)
\(570\) −2.12211e13 −0.0147726
\(571\) −2.70125e15 −1.86237 −0.931185 0.364546i \(-0.881224\pi\)
−0.931185 + 0.364546i \(0.881224\pi\)
\(572\) −2.38726e13 −0.0163013
\(573\) −9.45294e14 −0.639317
\(574\) 9.04858e13 0.0606129
\(575\) 2.22968e14 0.147935
\(576\) 5.57003e14 0.366045
\(577\) −1.21212e15 −0.789000 −0.394500 0.918896i \(-0.629082\pi\)
−0.394500 + 0.918896i \(0.629082\pi\)
\(578\) −1.28940e15 −0.831352
\(579\) 7.42379e14 0.474126
\(580\) −8.49924e12 −0.00537683
\(581\) −2.99429e14 −0.187640
\(582\) 4.17295e14 0.259040
\(583\) 8.66746e13 0.0532985
\(584\) 6.50699e14 0.396378
\(585\) 3.60995e12 0.00217843
\(586\) 1.09330e15 0.653587
\(587\) 2.45076e15 1.45141 0.725706 0.688005i \(-0.241513\pi\)
0.725706 + 0.688005i \(0.241513\pi\)
\(588\) −1.93987e14 −0.113814
\(589\) 1.05064e15 0.610690
\(590\) 3.23110e12 0.00186065
\(591\) 1.31648e15 0.751074
\(592\) −9.61873e14 −0.543686
\(593\) 1.66723e15 0.933674 0.466837 0.884343i \(-0.345393\pi\)
0.466837 + 0.884343i \(0.345393\pi\)
\(594\) −4.29002e13 −0.0238031
\(595\) −2.09303e12 −0.00115062
\(596\) −8.75581e13 −0.0476916
\(597\) −6.15114e14 −0.331968
\(598\) −9.00676e13 −0.0481629
\(599\) −3.13480e15 −1.66097 −0.830485 0.557040i \(-0.811937\pi\)
−0.830485 + 0.557040i \(0.811937\pi\)
\(600\) −1.19484e15 −0.627304
\(601\) −2.87675e15 −1.49655 −0.748277 0.663387i \(-0.769118\pi\)
−0.748277 + 0.663387i \(0.769118\pi\)
\(602\) 1.07779e15 0.555591
\(603\) −1.11499e15 −0.569541
\(604\) −4.14810e14 −0.209965
\(605\) −3.30474e13 −0.0165761
\(606\) 6.26777e14 0.311540
\(607\) 1.38720e14 0.0683286 0.0341643 0.999416i \(-0.489123\pi\)
0.0341643 + 0.999416i \(0.489123\pi\)
\(608\) 2.04079e15 0.996160
\(609\) 7.38901e14 0.357430
\(610\) 4.38634e13 0.0210275
\(611\) −5.46852e14 −0.259803
\(612\) −2.46536e13 −0.0116077
\(613\) −2.58931e14 −0.120824 −0.0604118 0.998174i \(-0.519241\pi\)
−0.0604118 + 0.998174i \(0.519241\pi\)
\(614\) −1.40403e15 −0.649308
\(615\) 2.72702e12 0.00124990
\(616\) 1.97187e14 0.0895743
\(617\) 1.07625e15 0.484555 0.242277 0.970207i \(-0.422106\pi\)
0.242277 + 0.970207i \(0.422106\pi\)
\(618\) 8.77078e13 0.0391383
\(619\) 2.20936e15 0.977164 0.488582 0.872518i \(-0.337514\pi\)
0.488582 + 0.872518i \(0.337514\pi\)
\(620\) −3.79925e12 −0.00166550
\(621\) 6.55415e13 0.0284782
\(622\) 2.96480e15 1.27687
\(623\) −5.32474e14 −0.227308
\(624\) 3.30937e14 0.140033
\(625\) 2.38213e15 0.999139
\(626\) 3.29639e15 1.37050
\(627\) −3.67691e14 −0.151534
\(628\) 3.61830e14 0.147818
\(629\) 2.58000e14 0.104482
\(630\) −6.67142e12 −0.00267821
\(631\) −1.37298e14 −0.0546389 −0.0273195 0.999627i \(-0.508697\pi\)
−0.0273195 + 0.999627i \(0.508697\pi\)
\(632\) −3.97076e14 −0.156650
\(633\) 2.14514e15 0.838949
\(634\) 2.68524e15 1.04110
\(635\) 3.06215e13 0.0117698
\(636\) 1.58765e14 0.0604981
\(637\) −6.98456e14 −0.263859
\(638\) 3.63668e14 0.136204
\(639\) −1.01534e14 −0.0377014
\(640\) 1.70278e13 0.00626858
\(641\) 4.28931e15 1.56555 0.782777 0.622303i \(-0.213803\pi\)
0.782777 + 0.622303i \(0.213803\pi\)
\(642\) 2.29070e15 0.828945
\(643\) −4.12032e15 −1.47833 −0.739164 0.673525i \(-0.764779\pi\)
−0.739164 + 0.673525i \(0.764779\pi\)
\(644\) −6.74022e13 −0.0239774
\(645\) 3.24821e13 0.0114569
\(646\) 5.21813e14 0.182489
\(647\) −3.73194e15 −1.29408 −0.647040 0.762456i \(-0.723993\pi\)
−0.647040 + 0.762456i \(0.723993\pi\)
\(648\) −3.51223e14 −0.120759
\(649\) 5.59840e13 0.0190862
\(650\) −9.62535e14 −0.325382
\(651\) 3.30297e14 0.110716
\(652\) −3.57200e14 −0.118727
\(653\) 1.38669e14 0.0457042 0.0228521 0.999739i \(-0.492725\pi\)
0.0228521 + 0.999739i \(0.492725\pi\)
\(654\) 5.73890e14 0.187565
\(655\) −5.79217e12 −0.00187722
\(656\) 2.49996e14 0.0803456
\(657\) −3.81447e14 −0.121570
\(658\) 1.01062e15 0.319407
\(659\) −8.65288e14 −0.271201 −0.135600 0.990764i \(-0.543296\pi\)
−0.135600 + 0.990764i \(0.543296\pi\)
\(660\) 1.32961e12 0.000413270 0
\(661\) −5.98330e15 −1.84431 −0.922153 0.386825i \(-0.873572\pi\)
−0.922153 + 0.386825i \(0.873572\pi\)
\(662\) −4.85345e14 −0.148365
\(663\) −8.87662e13 −0.0269106
\(664\) −1.20652e15 −0.362753
\(665\) −5.71796e13 −0.0170499
\(666\) 8.22361e14 0.243195
\(667\) −5.55600e14 −0.162956
\(668\) −7.04910e14 −0.205052
\(669\) 3.59897e13 0.0103833
\(670\) −8.53390e13 −0.0244195
\(671\) 7.60006e14 0.215696
\(672\) 6.41576e14 0.180600
\(673\) −6.00946e15 −1.67785 −0.838924 0.544248i \(-0.816815\pi\)
−0.838924 + 0.544248i \(0.816815\pi\)
\(674\) −4.58981e14 −0.127106
\(675\) 7.00429e14 0.192395
\(676\) 9.00439e14 0.245328
\(677\) −2.25463e14 −0.0609309 −0.0304654 0.999536i \(-0.509699\pi\)
−0.0304654 + 0.999536i \(0.509699\pi\)
\(678\) 1.98275e15 0.531501
\(679\) 1.12439e15 0.298973
\(680\) −8.43370e12 −0.00222443
\(681\) 3.56215e15 0.931972
\(682\) 1.62564e14 0.0421900
\(683\) 2.92558e15 0.753178 0.376589 0.926381i \(-0.377097\pi\)
0.376589 + 0.926381i \(0.377097\pi\)
\(684\) −6.73513e14 −0.172004
\(685\) −4.28241e13 −0.0108490
\(686\) 3.17804e15 0.798688
\(687\) −2.83636e15 −0.707129
\(688\) 2.97775e15 0.736465
\(689\) 5.71640e14 0.140255
\(690\) 5.01642e12 0.00122102
\(691\) 5.93173e14 0.143236 0.0716180 0.997432i \(-0.477184\pi\)
0.0716180 + 0.997432i \(0.477184\pi\)
\(692\) 2.82552e14 0.0676885
\(693\) −1.15593e14 −0.0274726
\(694\) 6.16772e14 0.145428
\(695\) −6.63807e13 −0.0155284
\(696\) 2.97734e15 0.691000
\(697\) −6.70556e13 −0.0154403
\(698\) 4.04716e15 0.924582
\(699\) 1.82382e15 0.413388
\(700\) −7.20315e14 −0.161988
\(701\) 2.81888e15 0.628967 0.314483 0.949263i \(-0.398169\pi\)
0.314483 + 0.949263i \(0.398169\pi\)
\(702\) −2.82937e14 −0.0626377
\(703\) 7.04832e15 1.54822
\(704\) 7.38668e14 0.160990
\(705\) 3.04576e13 0.00658651
\(706\) 2.52815e15 0.542471
\(707\) 1.68883e15 0.359566
\(708\) 1.02548e14 0.0216643
\(709\) −7.68258e15 −1.61047 −0.805235 0.592955i \(-0.797961\pi\)
−0.805235 + 0.592955i \(0.797961\pi\)
\(710\) −7.77122e12 −0.00161647
\(711\) 2.32771e14 0.0480447
\(712\) −2.14556e15 −0.439442
\(713\) −2.48359e14 −0.0504764
\(714\) 1.64045e14 0.0330845
\(715\) 4.78732e12 0.000958097 0
\(716\) 1.28048e15 0.254303
\(717\) −1.08926e15 −0.214672
\(718\) −6.22367e14 −0.121720
\(719\) −6.63705e15 −1.28815 −0.644075 0.764962i \(-0.722758\pi\)
−0.644075 + 0.764962i \(0.722758\pi\)
\(720\) −1.84319e13 −0.00355011
\(721\) 2.36326e14 0.0451717
\(722\) 9.80784e15 1.86045
\(723\) −5.54521e15 −1.04390
\(724\) −7.29534e14 −0.136296
\(725\) −5.93759e15 −1.10091
\(726\) 2.59016e15 0.476623
\(727\) −4.84956e15 −0.885652 −0.442826 0.896608i \(-0.646024\pi\)
−0.442826 + 0.896608i \(0.646024\pi\)
\(728\) 1.30049e15 0.235714
\(729\) 2.05891e14 0.0370370
\(730\) −2.91953e13 −0.00521239
\(731\) −7.98712e14 −0.141529
\(732\) 1.39213e15 0.244833
\(733\) −3.68177e15 −0.642665 −0.321333 0.946966i \(-0.604131\pi\)
−0.321333 + 0.946966i \(0.604131\pi\)
\(734\) −2.41170e15 −0.417825
\(735\) 3.89013e13 0.00668936
\(736\) −4.82419e14 −0.0823372
\(737\) −1.47864e15 −0.250490
\(738\) −2.13736e14 −0.0359391
\(739\) −8.18857e14 −0.136667 −0.0683335 0.997663i \(-0.521768\pi\)
−0.0683335 + 0.997663i \(0.521768\pi\)
\(740\) −2.54876e13 −0.00422235
\(741\) −2.42501e15 −0.398761
\(742\) −1.05643e15 −0.172432
\(743\) −6.97943e15 −1.13079 −0.565394 0.824821i \(-0.691276\pi\)
−0.565394 + 0.824821i \(0.691276\pi\)
\(744\) 1.33091e15 0.214040
\(745\) 1.75585e13 0.00280304
\(746\) 5.92051e14 0.0938200
\(747\) 7.07279e14 0.111257
\(748\) −3.26943e13 −0.00510521
\(749\) 6.17221e15 0.956733
\(750\) 1.07234e14 0.0165005
\(751\) 8.78726e15 1.34225 0.671126 0.741343i \(-0.265811\pi\)
0.671126 + 0.741343i \(0.265811\pi\)
\(752\) 2.79216e15 0.423391
\(753\) 2.55044e15 0.383922
\(754\) 2.39848e15 0.358421
\(755\) 8.31844e13 0.0123405
\(756\) −2.11736e14 −0.0311836
\(757\) −8.53584e14 −0.124801 −0.0624006 0.998051i \(-0.519876\pi\)
−0.0624006 + 0.998051i \(0.519876\pi\)
\(758\) −3.38451e15 −0.491265
\(759\) 8.69177e13 0.0125250
\(760\) −2.30401e14 −0.0329617
\(761\) 4.67837e14 0.0664476 0.0332238 0.999448i \(-0.489423\pi\)
0.0332238 + 0.999448i \(0.489423\pi\)
\(762\) −2.40002e15 −0.338425
\(763\) 1.54633e15 0.216479
\(764\) −2.29626e15 −0.319160
\(765\) 4.94393e12 0.000682236 0
\(766\) −5.40309e14 −0.0740259
\(767\) 3.69228e14 0.0502250
\(768\) 3.35982e15 0.453764
\(769\) 1.20552e16 1.61652 0.808259 0.588828i \(-0.200410\pi\)
0.808259 + 0.588828i \(0.200410\pi\)
\(770\) −8.84728e12 −0.00117791
\(771\) 4.47467e15 0.591510
\(772\) 1.80335e15 0.236693
\(773\) −3.00579e15 −0.391717 −0.195858 0.980632i \(-0.562749\pi\)
−0.195858 + 0.980632i \(0.562749\pi\)
\(774\) −2.54585e15 −0.329426
\(775\) −2.65417e15 −0.341011
\(776\) 4.53063e15 0.577989
\(777\) 2.21582e15 0.280685
\(778\) −4.24959e15 −0.534514
\(779\) −1.83190e15 −0.228794
\(780\) 8.76913e12 0.00108752
\(781\) −1.34649e14 −0.0165815
\(782\) −1.23350e14 −0.0150835
\(783\) −1.74535e15 −0.211931
\(784\) 3.56623e15 0.430002
\(785\) −7.25599e13 −0.00868788
\(786\) 4.53973e14 0.0539767
\(787\) 4.13814e14 0.0488590 0.0244295 0.999702i \(-0.492223\pi\)
0.0244295 + 0.999702i \(0.492223\pi\)
\(788\) 3.19793e15 0.374952
\(789\) 2.95370e15 0.343909
\(790\) 1.78158e13 0.00205995
\(791\) 5.34245e15 0.613436
\(792\) −4.65773e14 −0.0531112
\(793\) 5.01242e15 0.567603
\(794\) 5.71430e15 0.642614
\(795\) −3.18381e13 −0.00355573
\(796\) −1.49421e15 −0.165725
\(797\) −1.22642e16 −1.35088 −0.675442 0.737413i \(-0.736047\pi\)
−0.675442 + 0.737413i \(0.736047\pi\)
\(798\) 4.48157e15 0.490246
\(799\) −7.48931e14 −0.0813644
\(800\) −5.15551e15 −0.556259
\(801\) 1.25775e15 0.134777
\(802\) −1.03127e16 −1.09753
\(803\) −5.05855e14 −0.0534676
\(804\) −2.70848e15 −0.284326
\(805\) 1.35166e13 0.00140925
\(806\) 1.07215e15 0.111023
\(807\) 8.32137e15 0.855835
\(808\) 6.80501e15 0.695130
\(809\) 4.94885e14 0.0502096 0.0251048 0.999685i \(-0.492008\pi\)
0.0251048 + 0.999685i \(0.492008\pi\)
\(810\) 1.57585e13 0.00158799
\(811\) −4.78177e15 −0.478602 −0.239301 0.970945i \(-0.576918\pi\)
−0.239301 + 0.970945i \(0.576918\pi\)
\(812\) 1.79490e15 0.178436
\(813\) −2.09116e15 −0.206485
\(814\) 1.09057e15 0.106960
\(815\) 7.16314e13 0.00697809
\(816\) 4.53228e14 0.0438552
\(817\) −2.18201e16 −2.09718
\(818\) −6.13482e15 −0.585678
\(819\) −7.62364e14 −0.0722938
\(820\) 6.62436e12 0.000623976 0
\(821\) 1.15726e16 1.08279 0.541393 0.840770i \(-0.317897\pi\)
0.541393 + 0.840770i \(0.317897\pi\)
\(822\) 3.35642e15 0.311948
\(823\) −9.85164e15 −0.909514 −0.454757 0.890615i \(-0.650274\pi\)
−0.454757 + 0.890615i \(0.650274\pi\)
\(824\) 9.52256e14 0.0873281
\(825\) 9.28872e14 0.0846173
\(826\) −6.82357e14 −0.0617478
\(827\) 7.54016e15 0.677798 0.338899 0.940823i \(-0.389945\pi\)
0.338899 + 0.940823i \(0.389945\pi\)
\(828\) 1.59210e14 0.0142169
\(829\) −1.77254e16 −1.57234 −0.786169 0.618012i \(-0.787938\pi\)
−0.786169 + 0.618012i \(0.787938\pi\)
\(830\) 5.41338e13 0.00477022
\(831\) −6.39044e15 −0.559403
\(832\) 4.87169e15 0.423645
\(833\) −9.56557e14 −0.0826349
\(834\) 5.20273e15 0.446496
\(835\) 1.41360e14 0.0120518
\(836\) −8.93178e14 −0.0756490
\(837\) −7.80192e14 −0.0656465
\(838\) 1.62075e16 1.35479
\(839\) −9.41894e15 −0.782188 −0.391094 0.920351i \(-0.627903\pi\)
−0.391094 + 0.920351i \(0.627903\pi\)
\(840\) −7.24325e13 −0.00597582
\(841\) 2.59497e15 0.212693
\(842\) −5.29072e15 −0.430822
\(843\) 2.29917e15 0.186003
\(844\) 5.21088e15 0.418820
\(845\) −1.80570e14 −0.0144190
\(846\) −2.38717e15 −0.189386
\(847\) 6.97909e15 0.550098
\(848\) −2.91872e15 −0.228568
\(849\) 6.31516e15 0.491352
\(850\) −1.31822e15 −0.101902
\(851\) −1.66614e15 −0.127967
\(852\) −2.46642e14 −0.0188213
\(853\) 1.95835e16 1.48481 0.742404 0.669952i \(-0.233685\pi\)
0.742404 + 0.669952i \(0.233685\pi\)
\(854\) −9.26327e15 −0.697824
\(855\) 1.35064e14 0.0101094
\(856\) 2.48704e16 1.84960
\(857\) −4.29029e15 −0.317024 −0.158512 0.987357i \(-0.550670\pi\)
−0.158512 + 0.987357i \(0.550670\pi\)
\(858\) −3.75216e14 −0.0275487
\(859\) 1.07177e16 0.781880 0.390940 0.920416i \(-0.372150\pi\)
0.390940 + 0.920416i \(0.372150\pi\)
\(860\) 7.89041e13 0.00571950
\(861\) −5.75904e14 −0.0414794
\(862\) 8.87554e15 0.635192
\(863\) −2.33202e16 −1.65833 −0.829167 0.559001i \(-0.811185\pi\)
−0.829167 + 0.559001i \(0.811185\pi\)
\(864\) −1.51546e15 −0.107083
\(865\) −5.66618e13 −0.00397834
\(866\) −1.37021e15 −0.0955961
\(867\) 8.20650e15 0.568922
\(868\) 8.02342e14 0.0552715
\(869\) 3.08688e14 0.0211306
\(870\) −1.33586e14 −0.00908668
\(871\) −9.75197e15 −0.659162
\(872\) 6.23080e15 0.418507
\(873\) −2.65591e15 −0.177270
\(874\) −3.36982e15 −0.223508
\(875\) 2.88939e14 0.0190442
\(876\) −9.26595e14 −0.0606900
\(877\) 2.18713e16 1.42356 0.711781 0.702402i \(-0.247889\pi\)
0.711781 + 0.702402i \(0.247889\pi\)
\(878\) 1.33282e16 0.862087
\(879\) −6.95841e15 −0.447271
\(880\) −2.44434e13 −0.00156138
\(881\) −1.76902e16 −1.12296 −0.561482 0.827489i \(-0.689769\pi\)
−0.561482 + 0.827489i \(0.689769\pi\)
\(882\) −3.04897e15 −0.192343
\(883\) 6.93163e15 0.434562 0.217281 0.976109i \(-0.430281\pi\)
0.217281 + 0.976109i \(0.430281\pi\)
\(884\) −2.15627e14 −0.0134343
\(885\) −2.05646e13 −0.00127330
\(886\) −5.31733e15 −0.327196
\(887\) 1.95770e16 1.19720 0.598600 0.801048i \(-0.295724\pi\)
0.598600 + 0.801048i \(0.295724\pi\)
\(888\) 8.92849e15 0.542633
\(889\) −6.46677e15 −0.390596
\(890\) 9.62661e13 0.00577868
\(891\) 2.73042e14 0.0162893
\(892\) 8.74245e13 0.00518354
\(893\) −2.04601e16 −1.20566
\(894\) −1.37619e15 −0.0805974
\(895\) −2.56783e14 −0.0149465
\(896\) −3.59600e15 −0.208030
\(897\) 5.73243e14 0.0329595
\(898\) 1.69396e16 0.968018
\(899\) 6.61374e15 0.375638
\(900\) 1.70145e15 0.0960474
\(901\) 7.82878e14 0.0439246
\(902\) −2.83445e14 −0.0158064
\(903\) −6.85971e15 −0.380209
\(904\) 2.15270e16 1.18592
\(905\) 1.46298e14 0.00801071
\(906\) −6.51975e15 −0.354834
\(907\) −7.72373e15 −0.417818 −0.208909 0.977935i \(-0.566991\pi\)
−0.208909 + 0.977935i \(0.566991\pi\)
\(908\) 8.65301e15 0.465259
\(909\) −3.98917e15 −0.213197
\(910\) −5.83499e13 −0.00309965
\(911\) 1.78083e14 0.00940311 0.00470156 0.999989i \(-0.498503\pi\)
0.00470156 + 0.999989i \(0.498503\pi\)
\(912\) 1.23818e16 0.649847
\(913\) 9.37955e14 0.0489320
\(914\) −1.05216e16 −0.545603
\(915\) −2.79173e14 −0.0143899
\(916\) −6.88995e15 −0.353013
\(917\) 1.22322e15 0.0622976
\(918\) −3.87491e14 −0.0196167
\(919\) 9.91685e14 0.0499043 0.0249522 0.999689i \(-0.492057\pi\)
0.0249522 + 0.999689i \(0.492057\pi\)
\(920\) 5.44640e13 0.00272443
\(921\) 8.93607e15 0.444343
\(922\) −2.60799e16 −1.28910
\(923\) −8.88043e14 −0.0436340
\(924\) −2.80794e14 −0.0137149
\(925\) −1.78057e16 −0.864529
\(926\) −2.24505e16 −1.08359
\(927\) −5.58223e14 −0.0267836
\(928\) 1.28467e16 0.612741
\(929\) −1.04273e16 −0.494409 −0.247205 0.968963i \(-0.579512\pi\)
−0.247205 + 0.968963i \(0.579512\pi\)
\(930\) −5.97144e13 −0.00281464
\(931\) −2.61322e16 −1.22449
\(932\) 4.43035e15 0.206372
\(933\) −1.88697e16 −0.873808
\(934\) −3.00975e15 −0.138555
\(935\) 6.55638e12 0.000300055 0
\(936\) −3.07189e15 −0.139762
\(937\) −2.29193e16 −1.03665 −0.518326 0.855183i \(-0.673445\pi\)
−0.518326 + 0.855183i \(0.673445\pi\)
\(938\) 1.80222e16 0.810389
\(939\) −2.09802e16 −0.937882
\(940\) 7.39862e13 0.00328812
\(941\) 1.18608e15 0.0524049 0.0262025 0.999657i \(-0.491659\pi\)
0.0262025 + 0.999657i \(0.491659\pi\)
\(942\) 5.68703e15 0.249808
\(943\) 4.33038e14 0.0189109
\(944\) −1.88523e15 −0.0818500
\(945\) 4.24608e13 0.00183279
\(946\) −3.37617e15 −0.144885
\(947\) −1.55576e16 −0.663769 −0.331885 0.943320i \(-0.607685\pi\)
−0.331885 + 0.943320i \(0.607685\pi\)
\(948\) 5.65437e14 0.0239849
\(949\) −3.33624e15 −0.140700
\(950\) −3.60126e16 −1.50999
\(951\) −1.70904e16 −0.712460
\(952\) 1.78106e15 0.0738204
\(953\) 9.11804e15 0.375743 0.187871 0.982194i \(-0.439841\pi\)
0.187871 + 0.982194i \(0.439841\pi\)
\(954\) 2.49538e15 0.102240
\(955\) 4.60484e14 0.0187584
\(956\) −2.64598e15 −0.107169
\(957\) −2.31459e15 −0.0932093
\(958\) −2.61722e15 −0.104793
\(959\) 9.04377e15 0.360037
\(960\) −2.71335e14 −0.0107402
\(961\) −2.24521e16 −0.883644
\(962\) 7.19258e15 0.281463
\(963\) −1.45794e16 −0.567275
\(964\) −1.34702e16 −0.521134
\(965\) −3.61637e14 −0.0139115
\(966\) −1.05939e15 −0.0405211
\(967\) −1.89068e16 −0.719071 −0.359535 0.933131i \(-0.617065\pi\)
−0.359535 + 0.933131i \(0.617065\pi\)
\(968\) 2.81217e16 1.06347
\(969\) −3.32112e15 −0.124883
\(970\) −2.03278e14 −0.00760058
\(971\) −4.50296e16 −1.67414 −0.837070 0.547095i \(-0.815734\pi\)
−0.837070 + 0.547095i \(0.815734\pi\)
\(972\) 5.00142e14 0.0184896
\(973\) 1.40186e16 0.515327
\(974\) −2.36686e16 −0.865163
\(975\) 6.12613e15 0.222670
\(976\) −2.55927e16 −0.925002
\(977\) 1.08082e16 0.388449 0.194224 0.980957i \(-0.437781\pi\)
0.194224 + 0.980957i \(0.437781\pi\)
\(978\) −5.61426e15 −0.200645
\(979\) 1.66797e15 0.0592765
\(980\) 9.44974e13 0.00333946
\(981\) −3.65257e15 −0.128357
\(982\) −5.96367e15 −0.208401
\(983\) −3.83117e16 −1.33133 −0.665666 0.746250i \(-0.731853\pi\)
−0.665666 + 0.746250i \(0.731853\pi\)
\(984\) −2.32056e15 −0.0801900
\(985\) −6.41301e14 −0.0220375
\(986\) 3.28479e15 0.112249
\(987\) −6.43216e15 −0.218581
\(988\) −5.89072e15 −0.199070
\(989\) 5.15800e15 0.173341
\(990\) 2.08981e13 0.000698415 0
\(991\) −1.85561e16 −0.616712 −0.308356 0.951271i \(-0.599779\pi\)
−0.308356 + 0.951271i \(0.599779\pi\)
\(992\) 5.74261e15 0.189800
\(993\) 3.08902e15 0.101531
\(994\) 1.64116e15 0.0536446
\(995\) 2.99642e14 0.00974039
\(996\) 1.71809e15 0.0555417
\(997\) 2.11779e16 0.680863 0.340431 0.940269i \(-0.389427\pi\)
0.340431 + 0.940269i \(0.389427\pi\)
\(998\) −2.19607e16 −0.702148
\(999\) −5.23398e15 −0.166426
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 177.12.a.a.1.19 26
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.12.a.a.1.19 26 1.1 even 1 trivial