Properties

Label 177.12.a.d.1.1
Level $177$
Weight $12$
Character 177.1
Self dual yes
Analytic conductor $135.997$
Analytic rank $0$
Dimension $28$
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: \(0\)
Dimension: \(28\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-88.1033 q^{2} +243.000 q^{3} +5714.20 q^{4} +1093.28 q^{5} -21409.1 q^{6} -20521.5 q^{7} -323004. q^{8} +59049.0 q^{9} +O(q^{10})\) \(q-88.1033 q^{2} +243.000 q^{3} +5714.20 q^{4} +1093.28 q^{5} -21409.1 q^{6} -20521.5 q^{7} -323004. q^{8} +59049.0 q^{9} -96322.0 q^{10} +619162. q^{11} +1.38855e6 q^{12} +1.84815e6 q^{13} +1.80802e6 q^{14} +265668. q^{15} +1.67551e7 q^{16} -6.92865e6 q^{17} -5.20241e6 q^{18} -1.82009e7 q^{19} +6.24724e6 q^{20} -4.98674e6 q^{21} -5.45502e7 q^{22} +3.56074e7 q^{23} -7.84900e7 q^{24} -4.76329e7 q^{25} -1.62828e8 q^{26} +1.43489e7 q^{27} -1.17264e8 q^{28} +1.95174e8 q^{29} -2.34063e7 q^{30} -2.04113e7 q^{31} -8.14665e8 q^{32} +1.50456e8 q^{33} +6.10437e8 q^{34} -2.24359e7 q^{35} +3.37418e8 q^{36} -1.33972e8 q^{37} +1.60356e9 q^{38} +4.49100e8 q^{39} -3.53136e8 q^{40} +5.40984e8 q^{41} +4.39348e8 q^{42} +6.33160e8 q^{43} +3.53801e9 q^{44} +6.45574e7 q^{45} -3.13713e9 q^{46} +1.05650e8 q^{47} +4.07148e9 q^{48} -1.55619e9 q^{49} +4.19661e9 q^{50} -1.68366e9 q^{51} +1.05607e10 q^{52} +2.39652e9 q^{53} -1.26419e9 q^{54} +6.76920e8 q^{55} +6.62855e9 q^{56} -4.42283e9 q^{57} -1.71955e10 q^{58} +7.14924e8 q^{59} +1.51808e9 q^{60} -5.42720e9 q^{61} +1.79830e9 q^{62} -1.21178e9 q^{63} +3.74603e10 q^{64} +2.02055e9 q^{65} -1.32557e10 q^{66} +3.68213e9 q^{67} -3.95917e10 q^{68} +8.65259e9 q^{69} +1.97668e9 q^{70} -1.73720e10 q^{71} -1.90731e10 q^{72} +3.22301e10 q^{73} +1.18034e10 q^{74} -1.15748e10 q^{75} -1.04004e11 q^{76} -1.27062e10 q^{77} -3.95672e10 q^{78} +3.35728e10 q^{79} +1.83181e10 q^{80} +3.48678e9 q^{81} -4.76625e10 q^{82} +4.98543e10 q^{83} -2.84952e10 q^{84} -7.57498e9 q^{85} -5.57835e10 q^{86} +4.74273e10 q^{87} -1.99992e11 q^{88} +4.19017e10 q^{89} -5.68772e9 q^{90} -3.79268e10 q^{91} +2.03468e11 q^{92} -4.95994e9 q^{93} -9.30808e9 q^{94} -1.98988e10 q^{95} -1.97964e11 q^{96} +5.06612e10 q^{97} +1.37106e11 q^{98} +3.65609e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 28q + 96q^{2} + 6804q^{3} + 29214q^{4} + 26562q^{5} + 23328q^{6} + 142333q^{7} + 332331q^{8} + 1653372q^{9} + O(q^{10}) \) \( 28q + 96q^{2} + 6804q^{3} + 29214q^{4} + 26562q^{5} + 23328q^{6} + 142333q^{7} + 332331q^{8} + 1653372q^{9} + 616281q^{10} + 1082362q^{11} + 7099002q^{12} + 503712q^{13} + 1321669q^{14} + 6454566q^{15} + 34870338q^{16} + 13513579q^{17} + 5668704q^{18} + 35971687q^{19} + 96105997q^{20} + 34586919q^{21} - 47598882q^{22} + 61380539q^{23} + 80756433q^{24} + 294744746q^{25} + 62820734q^{26} + 401769396q^{27} + 148068294q^{28} + 322339307q^{29} + 149756283q^{30} + 151247077q^{31} + 466383494q^{32} + 263013966q^{33} + 684479860q^{34} + 960297361q^{35} + 1725057486q^{36} + 863508437q^{37} + 992640509q^{38} + 122402016q^{39} + 3067680252q^{40} + 3081170377q^{41} + 321165567q^{42} + 2554238300q^{43} + 4350123570q^{44} + 1568459538q^{45} - 1987059155q^{46} + 6203398333q^{47} + 8473492134q^{48} + 10327857997q^{49} + 17577682253q^{50} + 3283799697q^{51} + 32137181618q^{52} + 14571770754q^{53} + 1377495072q^{54} + 18251419334q^{55} + 33498842836q^{56} + 8741119941q^{57} + 11860778276q^{58} + 20017880372q^{59} + 23353757271q^{60} + 2761613771q^{61} + 13785829526q^{62} + 8404621317q^{63} + 86547545293q^{64} + 32034985256q^{65} - 11566528326q^{66} + 39381333296q^{67} + 38995496621q^{68} + 14915470977q^{69} + 8551800364q^{70} + 26130020296q^{71} + 19623813219q^{72} + 41382402799q^{73} + 23815315058q^{74} + 71622973278q^{75} + 10611720128q^{76} - 8426124313q^{77} + 15265438362q^{78} + 59825111206q^{79} + 4009687655q^{80} + 97629963228q^{81} - 39592715115q^{82} + 35433122727q^{83} + 35980595442q^{84} - 8950496085q^{85} - 182032360688q^{86} + 78328451601q^{87} - 220003602335q^{88} + 102303043039q^{89} + 36390776769q^{90} - 111146323655q^{91} - 163000203526q^{92} + 36753039711q^{93} - 81314346008q^{94} + 208102168887q^{95} + 113331189042q^{96} - 171891031490q^{97} + 72304707792q^{98} + 63912393738q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −88.1033 −1.94683 −0.973414 0.229055i \(-0.926436\pi\)
−0.973414 + 0.229055i \(0.926436\pi\)
\(3\) 243.000 0.577350
\(4\) 5714.20 2.79014
\(5\) 1093.28 0.156458 0.0782291 0.996935i \(-0.475073\pi\)
0.0782291 + 0.996935i \(0.475073\pi\)
\(6\) −21409.1 −1.12400
\(7\) −20521.5 −0.461499 −0.230750 0.973013i \(-0.574118\pi\)
−0.230750 + 0.973013i \(0.574118\pi\)
\(8\) −323004. −3.48508
\(9\) 59049.0 0.333333
\(10\) −96322.0 −0.304597
\(11\) 619162. 1.15916 0.579581 0.814915i \(-0.303216\pi\)
0.579581 + 0.814915i \(0.303216\pi\)
\(12\) 1.38855e6 1.61089
\(13\) 1.84815e6 1.38054 0.690268 0.723553i \(-0.257492\pi\)
0.690268 + 0.723553i \(0.257492\pi\)
\(14\) 1.80802e6 0.898459
\(15\) 265668. 0.0903312
\(16\) 1.67551e7 3.99472
\(17\) −6.92865e6 −1.18353 −0.591765 0.806110i \(-0.701569\pi\)
−0.591765 + 0.806110i \(0.701569\pi\)
\(18\) −5.20241e6 −0.648942
\(19\) −1.82009e7 −1.68635 −0.843177 0.537636i \(-0.819318\pi\)
−0.843177 + 0.537636i \(0.819318\pi\)
\(20\) 6.24724e6 0.436539
\(21\) −4.98674e6 −0.266447
\(22\) −5.45502e7 −2.25669
\(23\) 3.56074e7 1.15355 0.576776 0.816903i \(-0.304311\pi\)
0.576776 + 0.816903i \(0.304311\pi\)
\(24\) −7.84900e7 −2.01211
\(25\) −4.76329e7 −0.975521
\(26\) −1.62828e8 −2.68767
\(27\) 1.43489e7 0.192450
\(28\) −1.17264e8 −1.28765
\(29\) 1.95174e8 1.76699 0.883494 0.468443i \(-0.155185\pi\)
0.883494 + 0.468443i \(0.155185\pi\)
\(30\) −2.34063e7 −0.175859
\(31\) −2.04113e7 −0.128050 −0.0640252 0.997948i \(-0.520394\pi\)
−0.0640252 + 0.997948i \(0.520394\pi\)
\(32\) −8.14665e8 −4.29195
\(33\) 1.50456e8 0.669243
\(34\) 6.10437e8 2.30413
\(35\) −2.24359e7 −0.0722053
\(36\) 3.37418e8 0.930045
\(37\) −1.33972e8 −0.317618 −0.158809 0.987309i \(-0.550765\pi\)
−0.158809 + 0.987309i \(0.550765\pi\)
\(38\) 1.60356e9 3.28304
\(39\) 4.49100e8 0.797053
\(40\) −3.53136e8 −0.545270
\(41\) 5.40984e8 0.729245 0.364623 0.931155i \(-0.381198\pi\)
0.364623 + 0.931155i \(0.381198\pi\)
\(42\) 4.39348e8 0.518726
\(43\) 6.33160e8 0.656805 0.328403 0.944538i \(-0.393490\pi\)
0.328403 + 0.944538i \(0.393490\pi\)
\(44\) 3.53801e9 3.23422
\(45\) 6.45574e7 0.0521527
\(46\) −3.13713e9 −2.24576
\(47\) 1.05650e8 0.0671938 0.0335969 0.999435i \(-0.489304\pi\)
0.0335969 + 0.999435i \(0.489304\pi\)
\(48\) 4.07148e9 2.30635
\(49\) −1.55619e9 −0.787019
\(50\) 4.19661e9 1.89917
\(51\) −1.68366e9 −0.683312
\(52\) 1.05607e10 3.85188
\(53\) 2.39652e9 0.787162 0.393581 0.919290i \(-0.371236\pi\)
0.393581 + 0.919290i \(0.371236\pi\)
\(54\) −1.26419e9 −0.374667
\(55\) 6.76920e8 0.181360
\(56\) 6.62855e9 1.60836
\(57\) −4.42283e9 −0.973617
\(58\) −1.71955e10 −3.44002
\(59\) 7.14924e8 0.130189
\(60\) 1.51808e9 0.252036
\(61\) −5.42720e9 −0.822739 −0.411369 0.911469i \(-0.634949\pi\)
−0.411369 + 0.911469i \(0.634949\pi\)
\(62\) 1.79830e9 0.249292
\(63\) −1.21178e9 −0.153833
\(64\) 3.74603e10 4.36095
\(65\) 2.02055e9 0.215996
\(66\) −1.32557e10 −1.30290
\(67\) 3.68213e9 0.333187 0.166593 0.986026i \(-0.446723\pi\)
0.166593 + 0.986026i \(0.446723\pi\)
\(68\) −3.95917e10 −3.30221
\(69\) 8.65259e9 0.666003
\(70\) 1.97668e9 0.140571
\(71\) −1.73720e10 −1.14269 −0.571345 0.820710i \(-0.693578\pi\)
−0.571345 + 0.820710i \(0.693578\pi\)
\(72\) −1.90731e10 −1.16169
\(73\) 3.22301e10 1.81964 0.909822 0.415000i \(-0.136218\pi\)
0.909822 + 0.415000i \(0.136218\pi\)
\(74\) 1.18034e10 0.618347
\(75\) −1.15748e10 −0.563217
\(76\) −1.04004e11 −4.70516
\(77\) −1.27062e10 −0.534952
\(78\) −3.95672e10 −1.55172
\(79\) 3.35728e10 1.22755 0.613775 0.789481i \(-0.289650\pi\)
0.613775 + 0.789481i \(0.289650\pi\)
\(80\) 1.83181e10 0.625007
\(81\) 3.48678e9 0.111111
\(82\) −4.76625e10 −1.41971
\(83\) 4.98543e10 1.38923 0.694613 0.719384i \(-0.255576\pi\)
0.694613 + 0.719384i \(0.255576\pi\)
\(84\) −2.84952e10 −0.743422
\(85\) −7.57498e9 −0.185173
\(86\) −5.57835e10 −1.27869
\(87\) 4.74273e10 1.02017
\(88\) −1.99992e11 −4.03978
\(89\) 4.19017e10 0.795403 0.397701 0.917515i \(-0.369808\pi\)
0.397701 + 0.917515i \(0.369808\pi\)
\(90\) −5.68772e9 −0.101532
\(91\) −3.79268e10 −0.637117
\(92\) 2.03468e11 3.21856
\(93\) −4.95994e9 −0.0739300
\(94\) −9.30808e9 −0.130815
\(95\) −1.98988e10 −0.263844
\(96\) −1.97964e11 −2.47796
\(97\) 5.06612e10 0.599005 0.299503 0.954095i \(-0.403179\pi\)
0.299503 + 0.954095i \(0.403179\pi\)
\(98\) 1.37106e11 1.53219
\(99\) 3.65609e10 0.386387
\(100\) −2.72184e11 −2.72184
\(101\) −1.58115e10 −0.149694 −0.0748472 0.997195i \(-0.523847\pi\)
−0.0748472 + 0.997195i \(0.523847\pi\)
\(102\) 1.48336e11 1.33029
\(103\) −4.41647e10 −0.375380 −0.187690 0.982228i \(-0.560100\pi\)
−0.187690 + 0.982228i \(0.560100\pi\)
\(104\) −5.96959e11 −4.81129
\(105\) −5.45192e9 −0.0416878
\(106\) −2.11142e11 −1.53247
\(107\) −2.25267e11 −1.55270 −0.776348 0.630305i \(-0.782930\pi\)
−0.776348 + 0.630305i \(0.782930\pi\)
\(108\) 8.19925e10 0.536962
\(109\) −2.67814e11 −1.66720 −0.833598 0.552371i \(-0.813723\pi\)
−0.833598 + 0.552371i \(0.813723\pi\)
\(110\) −5.96389e10 −0.353077
\(111\) −3.25552e10 −0.183377
\(112\) −3.43840e11 −1.84356
\(113\) −2.91433e11 −1.48802 −0.744008 0.668171i \(-0.767077\pi\)
−0.744008 + 0.668171i \(0.767077\pi\)
\(114\) 3.89666e11 1.89546
\(115\) 3.89290e10 0.180483
\(116\) 1.11526e12 4.93013
\(117\) 1.09131e11 0.460179
\(118\) −6.29872e10 −0.253455
\(119\) 1.42187e11 0.546198
\(120\) −8.58119e10 −0.314812
\(121\) 9.80494e10 0.343657
\(122\) 4.78155e11 1.60173
\(123\) 1.31459e11 0.421030
\(124\) −1.16634e11 −0.357278
\(125\) −1.05459e11 −0.309086
\(126\) 1.06762e11 0.299486
\(127\) 5.75720e10 0.154629 0.0773145 0.997007i \(-0.475365\pi\)
0.0773145 + 0.997007i \(0.475365\pi\)
\(128\) −1.63194e12 −4.19808
\(129\) 1.53858e11 0.379207
\(130\) −1.78017e11 −0.420507
\(131\) 4.17694e11 0.945945 0.472973 0.881077i \(-0.343181\pi\)
0.472973 + 0.881077i \(0.343181\pi\)
\(132\) 8.59737e11 1.86728
\(133\) 3.73511e11 0.778251
\(134\) −3.24408e11 −0.648657
\(135\) 1.56874e10 0.0301104
\(136\) 2.23798e12 4.12470
\(137\) −9.85876e10 −0.174526 −0.0872628 0.996185i \(-0.527812\pi\)
−0.0872628 + 0.996185i \(0.527812\pi\)
\(138\) −7.62322e11 −1.29659
\(139\) −1.00244e12 −1.63861 −0.819305 0.573358i \(-0.805640\pi\)
−0.819305 + 0.573358i \(0.805640\pi\)
\(140\) −1.28203e11 −0.201463
\(141\) 2.56728e10 0.0387944
\(142\) 1.53053e12 2.22462
\(143\) 1.14430e12 1.60027
\(144\) 9.89370e11 1.33157
\(145\) 2.13381e11 0.276460
\(146\) −2.83958e12 −3.54253
\(147\) −3.78155e11 −0.454385
\(148\) −7.65543e11 −0.886197
\(149\) 1.24570e12 1.38959 0.694797 0.719206i \(-0.255494\pi\)
0.694797 + 0.719206i \(0.255494\pi\)
\(150\) 1.01978e12 1.09649
\(151\) −1.32489e12 −1.37343 −0.686713 0.726929i \(-0.740947\pi\)
−0.686713 + 0.726929i \(0.740947\pi\)
\(152\) 5.87898e12 5.87709
\(153\) −4.09130e11 −0.394510
\(154\) 1.11945e12 1.04146
\(155\) −2.23154e10 −0.0200345
\(156\) 2.56624e12 2.22389
\(157\) −1.12257e12 −0.939215 −0.469607 0.882875i \(-0.655605\pi\)
−0.469607 + 0.882875i \(0.655605\pi\)
\(158\) −2.95788e12 −2.38983
\(159\) 5.82355e11 0.454468
\(160\) −8.90661e11 −0.671510
\(161\) −7.30719e11 −0.532363
\(162\) −3.07197e11 −0.216314
\(163\) 4.81822e11 0.327985 0.163993 0.986462i \(-0.447563\pi\)
0.163993 + 0.986462i \(0.447563\pi\)
\(164\) 3.09129e12 2.03469
\(165\) 1.64492e11 0.104708
\(166\) −4.39233e12 −2.70458
\(167\) 1.26248e12 0.752112 0.376056 0.926597i \(-0.377280\pi\)
0.376056 + 0.926597i \(0.377280\pi\)
\(168\) 1.61074e12 0.928589
\(169\) 1.62349e12 0.905882
\(170\) 6.67381e11 0.360500
\(171\) −1.07475e12 −0.562118
\(172\) 3.61800e12 1.83258
\(173\) 2.46270e12 1.20825 0.604126 0.796889i \(-0.293522\pi\)
0.604126 + 0.796889i \(0.293522\pi\)
\(174\) −4.17851e12 −1.98610
\(175\) 9.77500e11 0.450202
\(176\) 1.03741e13 4.63053
\(177\) 1.73727e11 0.0751646
\(178\) −3.69168e12 −1.54851
\(179\) 5.63806e11 0.229318 0.114659 0.993405i \(-0.463422\pi\)
0.114659 + 0.993405i \(0.463422\pi\)
\(180\) 3.68894e11 0.145513
\(181\) −3.73001e12 −1.42718 −0.713588 0.700566i \(-0.752931\pi\)
−0.713588 + 0.700566i \(0.752931\pi\)
\(182\) 3.34148e12 1.24036
\(183\) −1.31881e12 −0.475008
\(184\) −1.15013e13 −4.02022
\(185\) −1.46470e11 −0.0496939
\(186\) 4.36988e11 0.143929
\(187\) −4.28995e12 −1.37190
\(188\) 6.03702e11 0.187480
\(189\) −2.94462e11 −0.0888156
\(190\) 1.75315e12 0.513658
\(191\) 1.29232e12 0.367864 0.183932 0.982939i \(-0.441117\pi\)
0.183932 + 0.982939i \(0.441117\pi\)
\(192\) 9.10286e12 2.51780
\(193\) 6.90513e12 1.85612 0.928061 0.372428i \(-0.121475\pi\)
0.928061 + 0.372428i \(0.121475\pi\)
\(194\) −4.46342e12 −1.16616
\(195\) 4.90994e11 0.124706
\(196\) −8.89239e12 −2.19589
\(197\) 7.78475e12 1.86931 0.934653 0.355562i \(-0.115711\pi\)
0.934653 + 0.355562i \(0.115711\pi\)
\(198\) −3.22113e12 −0.752229
\(199\) −4.00111e12 −0.908843 −0.454421 0.890787i \(-0.650154\pi\)
−0.454421 + 0.890787i \(0.650154\pi\)
\(200\) 1.53856e13 3.39977
\(201\) 8.94757e11 0.192365
\(202\) 1.39305e12 0.291429
\(203\) −4.00528e12 −0.815463
\(204\) −9.62077e12 −1.90653
\(205\) 5.91450e11 0.114096
\(206\) 3.89106e12 0.730799
\(207\) 2.10258e12 0.384517
\(208\) 3.09658e13 5.51486
\(209\) −1.12693e13 −1.95476
\(210\) 4.80333e11 0.0811588
\(211\) 1.12159e13 1.84620 0.923101 0.384558i \(-0.125646\pi\)
0.923101 + 0.384558i \(0.125646\pi\)
\(212\) 1.36942e13 2.19629
\(213\) −4.22139e12 −0.659733
\(214\) 1.98468e13 3.02283
\(215\) 6.92224e11 0.102763
\(216\) −4.63476e12 −0.670705
\(217\) 4.18871e11 0.0590952
\(218\) 2.35953e13 3.24574
\(219\) 7.83192e12 1.05057
\(220\) 3.86805e12 0.506020
\(221\) −1.28052e13 −1.63391
\(222\) 2.86822e12 0.357003
\(223\) 1.87818e12 0.228065 0.114033 0.993477i \(-0.463623\pi\)
0.114033 + 0.993477i \(0.463623\pi\)
\(224\) 1.67182e13 1.98073
\(225\) −2.81267e12 −0.325174
\(226\) 2.56762e13 2.89691
\(227\) 6.00593e12 0.661360 0.330680 0.943743i \(-0.392722\pi\)
0.330680 + 0.943743i \(0.392722\pi\)
\(228\) −2.52729e13 −2.71652
\(229\) −7.32590e12 −0.768716 −0.384358 0.923184i \(-0.625577\pi\)
−0.384358 + 0.923184i \(0.625577\pi\)
\(230\) −3.42978e12 −0.351368
\(231\) −3.08760e12 −0.308855
\(232\) −6.30421e13 −6.15810
\(233\) −7.79070e11 −0.0743223 −0.0371611 0.999309i \(-0.511831\pi\)
−0.0371611 + 0.999309i \(0.511831\pi\)
\(234\) −9.61482e12 −0.895889
\(235\) 1.15505e11 0.0105130
\(236\) 4.08522e12 0.363245
\(237\) 8.15820e12 0.708726
\(238\) −1.25271e13 −1.06335
\(239\) 1.42626e13 1.18307 0.591536 0.806278i \(-0.298522\pi\)
0.591536 + 0.806278i \(0.298522\pi\)
\(240\) 4.45129e12 0.360848
\(241\) 1.12584e13 0.892038 0.446019 0.895024i \(-0.352841\pi\)
0.446019 + 0.895024i \(0.352841\pi\)
\(242\) −8.63848e12 −0.669041
\(243\) 8.47289e11 0.0641500
\(244\) −3.10121e13 −2.29555
\(245\) −1.70136e12 −0.123135
\(246\) −1.15820e13 −0.819672
\(247\) −3.36380e13 −2.32807
\(248\) 6.59293e12 0.446267
\(249\) 1.21146e13 0.802070
\(250\) 9.29132e12 0.601738
\(251\) 7.93703e12 0.502866 0.251433 0.967875i \(-0.419098\pi\)
0.251433 + 0.967875i \(0.419098\pi\)
\(252\) −6.92433e12 −0.429215
\(253\) 2.20467e13 1.33715
\(254\) −5.07228e12 −0.301036
\(255\) −1.84072e12 −0.106910
\(256\) 6.70610e13 3.81198
\(257\) −1.72639e13 −0.960523 −0.480262 0.877125i \(-0.659458\pi\)
−0.480262 + 0.877125i \(0.659458\pi\)
\(258\) −1.35554e13 −0.738250
\(259\) 2.74931e12 0.146580
\(260\) 1.15458e13 0.602659
\(261\) 1.15248e13 0.588996
\(262\) −3.68002e13 −1.84159
\(263\) −1.16164e13 −0.569265 −0.284632 0.958637i \(-0.591872\pi\)
−0.284632 + 0.958637i \(0.591872\pi\)
\(264\) −4.85980e13 −2.33237
\(265\) 2.62008e12 0.123158
\(266\) −3.29076e13 −1.51512
\(267\) 1.01821e13 0.459226
\(268\) 2.10404e13 0.929636
\(269\) −1.52243e13 −0.659020 −0.329510 0.944152i \(-0.606884\pi\)
−0.329510 + 0.944152i \(0.606884\pi\)
\(270\) −1.38212e12 −0.0586197
\(271\) 3.62900e13 1.50819 0.754094 0.656767i \(-0.228076\pi\)
0.754094 + 0.656767i \(0.228076\pi\)
\(272\) −1.16090e14 −4.72787
\(273\) −9.21622e12 −0.367839
\(274\) 8.68589e12 0.339771
\(275\) −2.94924e13 −1.13079
\(276\) 4.94426e13 1.85824
\(277\) −2.91834e13 −1.07522 −0.537610 0.843193i \(-0.680673\pi\)
−0.537610 + 0.843193i \(0.680673\pi\)
\(278\) 8.83180e13 3.19009
\(279\) −1.20527e12 −0.0426835
\(280\) 7.24689e12 0.251642
\(281\) −3.08384e12 −0.105004 −0.0525022 0.998621i \(-0.516720\pi\)
−0.0525022 + 0.998621i \(0.516720\pi\)
\(282\) −2.26186e12 −0.0755259
\(283\) 4.66033e13 1.52613 0.763064 0.646323i \(-0.223694\pi\)
0.763064 + 0.646323i \(0.223694\pi\)
\(284\) −9.92670e13 −3.18826
\(285\) −4.83541e12 −0.152330
\(286\) −1.00817e14 −3.11544
\(287\) −1.11018e13 −0.336546
\(288\) −4.81052e13 −1.43065
\(289\) 1.37343e13 0.400744
\(290\) −1.87996e13 −0.538219
\(291\) 1.23107e13 0.345836
\(292\) 1.84169e14 5.07705
\(293\) 2.51439e13 0.680238 0.340119 0.940382i \(-0.389533\pi\)
0.340119 + 0.940382i \(0.389533\pi\)
\(294\) 3.33167e13 0.884610
\(295\) 7.81616e11 0.0203691
\(296\) 4.32735e13 1.10692
\(297\) 8.88429e12 0.223081
\(298\) −1.09750e14 −2.70530
\(299\) 6.58077e13 1.59252
\(300\) −6.61406e13 −1.57145
\(301\) −1.29934e13 −0.303115
\(302\) 1.16727e14 2.67382
\(303\) −3.84219e12 −0.0864261
\(304\) −3.04958e14 −6.73651
\(305\) −5.93348e12 −0.128724
\(306\) 3.60457e13 0.768043
\(307\) −6.27605e13 −1.31349 −0.656743 0.754115i \(-0.728066\pi\)
−0.656743 + 0.754115i \(0.728066\pi\)
\(308\) −7.26055e13 −1.49259
\(309\) −1.07320e13 −0.216726
\(310\) 1.96606e12 0.0390038
\(311\) −5.92798e13 −1.15538 −0.577689 0.816257i \(-0.696045\pi\)
−0.577689 + 0.816257i \(0.696045\pi\)
\(312\) −1.45061e14 −2.77780
\(313\) 4.70084e13 0.884468 0.442234 0.896900i \(-0.354186\pi\)
0.442234 + 0.896900i \(0.354186\pi\)
\(314\) 9.89021e13 1.82849
\(315\) −1.32482e12 −0.0240684
\(316\) 1.91842e14 3.42503
\(317\) 7.22217e13 1.26719 0.633595 0.773665i \(-0.281579\pi\)
0.633595 + 0.773665i \(0.281579\pi\)
\(318\) −5.13074e13 −0.884771
\(319\) 1.20844e14 2.04823
\(320\) 4.09548e13 0.682307
\(321\) −5.47398e13 −0.896449
\(322\) 6.43787e13 1.03642
\(323\) 1.26108e14 1.99585
\(324\) 1.99242e13 0.310015
\(325\) −8.80325e13 −1.34674
\(326\) −4.24501e13 −0.638531
\(327\) −6.50787e13 −0.962556
\(328\) −1.74740e14 −2.54148
\(329\) −2.16809e12 −0.0310099
\(330\) −1.44923e13 −0.203849
\(331\) 9.66001e13 1.33636 0.668180 0.743999i \(-0.267073\pi\)
0.668180 + 0.743999i \(0.267073\pi\)
\(332\) 2.84877e14 3.87613
\(333\) −7.91092e12 −0.105873
\(334\) −1.11228e14 −1.46423
\(335\) 4.02562e12 0.0521298
\(336\) −8.35531e13 −1.06438
\(337\) 3.74070e13 0.468800 0.234400 0.972140i \(-0.424687\pi\)
0.234400 + 0.972140i \(0.424687\pi\)
\(338\) −1.43035e14 −1.76360
\(339\) −7.08183e13 −0.859106
\(340\) −4.32850e13 −0.516658
\(341\) −1.26379e13 −0.148431
\(342\) 9.46888e13 1.09435
\(343\) 7.25133e13 0.824708
\(344\) −2.04513e14 −2.28902
\(345\) 9.45975e12 0.104202
\(346\) −2.16972e14 −2.35226
\(347\) −5.26244e13 −0.561532 −0.280766 0.959776i \(-0.590589\pi\)
−0.280766 + 0.959776i \(0.590589\pi\)
\(348\) 2.71009e14 2.84641
\(349\) 6.09941e13 0.630592 0.315296 0.948993i \(-0.397896\pi\)
0.315296 + 0.948993i \(0.397896\pi\)
\(350\) −8.61210e13 −0.876465
\(351\) 2.65189e13 0.265684
\(352\) −5.04409e14 −4.97506
\(353\) 2.03250e14 1.97365 0.986824 0.161798i \(-0.0517292\pi\)
0.986824 + 0.161798i \(0.0517292\pi\)
\(354\) −1.53059e13 −0.146332
\(355\) −1.89925e13 −0.178783
\(356\) 2.39435e14 2.21928
\(357\) 3.45513e13 0.315348
\(358\) −4.96732e13 −0.446443
\(359\) 7.30453e13 0.646506 0.323253 0.946313i \(-0.395223\pi\)
0.323253 + 0.946313i \(0.395223\pi\)
\(360\) −2.08523e13 −0.181757
\(361\) 2.14784e14 1.84379
\(362\) 3.28626e14 2.77846
\(363\) 2.38260e13 0.198411
\(364\) −2.16721e14 −1.77764
\(365\) 3.52367e13 0.284698
\(366\) 1.16192e14 0.924759
\(367\) 1.07824e14 0.845379 0.422690 0.906275i \(-0.361086\pi\)
0.422690 + 0.906275i \(0.361086\pi\)
\(368\) 5.96604e14 4.60811
\(369\) 3.19446e13 0.243082
\(370\) 1.29045e13 0.0967454
\(371\) −4.91804e13 −0.363275
\(372\) −2.83421e13 −0.206275
\(373\) 1.82945e14 1.31197 0.655984 0.754775i \(-0.272254\pi\)
0.655984 + 0.754775i \(0.272254\pi\)
\(374\) 3.77959e14 2.67086
\(375\) −2.56266e13 −0.178451
\(376\) −3.41252e13 −0.234176
\(377\) 3.60711e14 2.43939
\(378\) 2.59431e13 0.172909
\(379\) 1.05180e14 0.690903 0.345451 0.938437i \(-0.387726\pi\)
0.345451 + 0.938437i \(0.387726\pi\)
\(380\) −1.13706e14 −0.736160
\(381\) 1.39900e13 0.0892751
\(382\) −1.13858e14 −0.716167
\(383\) −8.06324e13 −0.499938 −0.249969 0.968254i \(-0.580420\pi\)
−0.249969 + 0.968254i \(0.580420\pi\)
\(384\) −3.96563e14 −2.42376
\(385\) −1.38914e13 −0.0836977
\(386\) −6.08365e14 −3.61355
\(387\) 3.73874e13 0.218935
\(388\) 2.89488e14 1.67131
\(389\) 1.77980e14 1.01309 0.506546 0.862213i \(-0.330922\pi\)
0.506546 + 0.862213i \(0.330922\pi\)
\(390\) −4.32582e13 −0.242780
\(391\) −2.46711e14 −1.36526
\(392\) 5.02657e14 2.74283
\(393\) 1.01500e14 0.546142
\(394\) −6.85862e14 −3.63921
\(395\) 3.67047e13 0.192060
\(396\) 2.08916e14 1.07807
\(397\) −1.80915e13 −0.0920717 −0.0460359 0.998940i \(-0.514659\pi\)
−0.0460359 + 0.998940i \(0.514659\pi\)
\(398\) 3.52511e14 1.76936
\(399\) 9.07632e13 0.449323
\(400\) −7.98092e14 −3.89693
\(401\) −2.06313e14 −0.993649 −0.496825 0.867851i \(-0.665501\pi\)
−0.496825 + 0.867851i \(0.665501\pi\)
\(402\) −7.88311e13 −0.374502
\(403\) −3.77231e13 −0.176778
\(404\) −9.03500e13 −0.417668
\(405\) 3.81205e12 0.0173842
\(406\) 3.52878e14 1.58757
\(407\) −8.29504e13 −0.368171
\(408\) 5.43830e14 2.38140
\(409\) −2.21304e14 −0.956118 −0.478059 0.878328i \(-0.658660\pi\)
−0.478059 + 0.878328i \(0.658660\pi\)
\(410\) −5.21087e13 −0.222126
\(411\) −2.39568e13 −0.100762
\(412\) −2.52366e14 −1.04736
\(413\) −1.46714e13 −0.0600821
\(414\) −1.85244e14 −0.748588
\(415\) 5.45049e13 0.217356
\(416\) −1.50562e15 −5.92519
\(417\) −2.43592e14 −0.946052
\(418\) 9.92865e14 3.80558
\(419\) 6.95620e13 0.263145 0.131572 0.991307i \(-0.457997\pi\)
0.131572 + 0.991307i \(0.457997\pi\)
\(420\) −3.11534e13 −0.116314
\(421\) −2.67808e14 −0.986896 −0.493448 0.869775i \(-0.664264\pi\)
−0.493448 + 0.869775i \(0.664264\pi\)
\(422\) −9.88155e14 −3.59424
\(423\) 6.23850e12 0.0223979
\(424\) −7.74087e14 −2.74333
\(425\) 3.30031e14 1.15456
\(426\) 3.71919e14 1.28439
\(427\) 1.11375e14 0.379693
\(428\) −1.28722e15 −4.33223
\(429\) 2.78065e14 0.923914
\(430\) −6.09872e13 −0.200061
\(431\) 9.63333e13 0.311998 0.155999 0.987757i \(-0.450140\pi\)
0.155999 + 0.987757i \(0.450140\pi\)
\(432\) 2.40417e14 0.768784
\(433\) 2.29331e14 0.724070 0.362035 0.932165i \(-0.382082\pi\)
0.362035 + 0.932165i \(0.382082\pi\)
\(434\) −3.69040e13 −0.115048
\(435\) 5.18516e13 0.159614
\(436\) −1.53034e15 −4.65170
\(437\) −6.48088e14 −1.94530
\(438\) −6.90018e14 −2.04528
\(439\) 3.63338e14 1.06355 0.531773 0.846887i \(-0.321526\pi\)
0.531773 + 0.846887i \(0.321526\pi\)
\(440\) −2.18648e14 −0.632056
\(441\) −9.18916e13 −0.262340
\(442\) 1.12818e15 3.18093
\(443\) 1.99410e14 0.555298 0.277649 0.960683i \(-0.410445\pi\)
0.277649 + 0.960683i \(0.410445\pi\)
\(444\) −1.86027e14 −0.511646
\(445\) 4.58105e13 0.124447
\(446\) −1.65474e14 −0.444004
\(447\) 3.02704e14 0.802282
\(448\) −7.68744e14 −2.01258
\(449\) −3.12549e14 −0.808283 −0.404141 0.914697i \(-0.632430\pi\)
−0.404141 + 0.914697i \(0.632430\pi\)
\(450\) 2.47806e14 0.633057
\(451\) 3.34957e14 0.845314
\(452\) −1.66531e15 −4.15177
\(453\) −3.21947e14 −0.792947
\(454\) −5.29142e14 −1.28755
\(455\) −4.14648e13 −0.0996821
\(456\) 1.42859e15 3.39314
\(457\) −9.29028e13 −0.218017 −0.109008 0.994041i \(-0.534768\pi\)
−0.109008 + 0.994041i \(0.534768\pi\)
\(458\) 6.45436e14 1.49656
\(459\) −9.94185e13 −0.227771
\(460\) 2.22448e14 0.503571
\(461\) 9.54233e13 0.213452 0.106726 0.994288i \(-0.465963\pi\)
0.106726 + 0.994288i \(0.465963\pi\)
\(462\) 2.72027e14 0.601287
\(463\) 8.09112e13 0.176731 0.0883656 0.996088i \(-0.471836\pi\)
0.0883656 + 0.996088i \(0.471836\pi\)
\(464\) 3.27016e15 7.05862
\(465\) −5.42263e12 −0.0115670
\(466\) 6.86387e13 0.144693
\(467\) 4.40005e14 0.916674 0.458337 0.888778i \(-0.348445\pi\)
0.458337 + 0.888778i \(0.348445\pi\)
\(468\) 6.23597e14 1.28396
\(469\) −7.55630e13 −0.153765
\(470\) −1.01764e13 −0.0204670
\(471\) −2.72784e14 −0.542256
\(472\) −2.30924e14 −0.453719
\(473\) 3.92028e14 0.761344
\(474\) −7.18764e14 −1.37977
\(475\) 8.66962e14 1.64507
\(476\) 8.12482e14 1.52397
\(477\) 1.41512e14 0.262387
\(478\) −1.25659e15 −2.30324
\(479\) −2.58001e14 −0.467495 −0.233747 0.972297i \(-0.575099\pi\)
−0.233747 + 0.972297i \(0.575099\pi\)
\(480\) −2.16431e14 −0.387696
\(481\) −2.47600e14 −0.438483
\(482\) −9.91903e14 −1.73664
\(483\) −1.77565e14 −0.307360
\(484\) 5.60274e14 0.958850
\(485\) 5.53871e13 0.0937193
\(486\) −7.46490e13 −0.124889
\(487\) 1.06913e15 1.76856 0.884279 0.466958i \(-0.154650\pi\)
0.884279 + 0.466958i \(0.154650\pi\)
\(488\) 1.75301e15 2.86731
\(489\) 1.17083e14 0.189362
\(490\) 1.49896e14 0.239723
\(491\) −3.04828e14 −0.482066 −0.241033 0.970517i \(-0.577486\pi\)
−0.241033 + 0.970517i \(0.577486\pi\)
\(492\) 7.51184e14 1.17473
\(493\) −1.35229e15 −2.09128
\(494\) 2.96362e15 4.53236
\(495\) 3.99714e13 0.0604535
\(496\) −3.41993e14 −0.511526
\(497\) 3.56500e14 0.527351
\(498\) −1.06734e15 −1.56149
\(499\) 1.75672e12 0.00254185 0.00127092 0.999999i \(-0.499595\pi\)
0.00127092 + 0.999999i \(0.499595\pi\)
\(500\) −6.02615e14 −0.862393
\(501\) 3.06782e14 0.434232
\(502\) −6.99279e14 −0.978993
\(503\) −7.62058e14 −1.05527 −0.527636 0.849471i \(-0.676921\pi\)
−0.527636 + 0.849471i \(0.676921\pi\)
\(504\) 3.91409e14 0.536121
\(505\) −1.72865e13 −0.0234209
\(506\) −1.94239e15 −2.60321
\(507\) 3.94507e14 0.523011
\(508\) 3.28978e14 0.431436
\(509\) 6.14338e14 0.797003 0.398501 0.917168i \(-0.369530\pi\)
0.398501 + 0.917168i \(0.369530\pi\)
\(510\) 1.62174e14 0.208135
\(511\) −6.61412e14 −0.839764
\(512\) −2.56608e15 −3.22318
\(513\) −2.61163e14 −0.324539
\(514\) 1.52101e15 1.86997
\(515\) −4.82846e13 −0.0587312
\(516\) 8.79174e14 1.05804
\(517\) 6.54141e13 0.0778885
\(518\) −2.42224e14 −0.285367
\(519\) 5.98435e14 0.697584
\(520\) −6.52646e14 −0.752765
\(521\) 6.95169e14 0.793383 0.396692 0.917952i \(-0.370158\pi\)
0.396692 + 0.917952i \(0.370158\pi\)
\(522\) −1.01538e15 −1.14667
\(523\) 8.61346e14 0.962539 0.481270 0.876573i \(-0.340176\pi\)
0.481270 + 0.876573i \(0.340176\pi\)
\(524\) 2.38679e15 2.63932
\(525\) 2.37532e14 0.259924
\(526\) 1.02344e15 1.10826
\(527\) 1.41423e14 0.151552
\(528\) 2.52091e15 2.67344
\(529\) 3.15076e14 0.330681
\(530\) −2.30838e14 −0.239767
\(531\) 4.22156e13 0.0433963
\(532\) 2.13432e15 2.17143
\(533\) 9.99819e14 1.00675
\(534\) −8.97079e14 −0.894033
\(535\) −2.46281e14 −0.242932
\(536\) −1.18934e15 −1.16118
\(537\) 1.37005e14 0.132397
\(538\) 1.34131e15 1.28300
\(539\) −9.63535e14 −0.912282
\(540\) 8.96411e13 0.0840121
\(541\) −2.12299e15 −1.96953 −0.984766 0.173886i \(-0.944368\pi\)
−0.984766 + 0.173886i \(0.944368\pi\)
\(542\) −3.19727e15 −2.93618
\(543\) −9.06391e14 −0.823980
\(544\) 5.64453e15 5.07965
\(545\) −2.92796e14 −0.260847
\(546\) 8.11980e14 0.716120
\(547\) −7.34491e14 −0.641293 −0.320646 0.947199i \(-0.603900\pi\)
−0.320646 + 0.947199i \(0.603900\pi\)
\(548\) −5.63349e14 −0.486950
\(549\) −3.20471e14 −0.274246
\(550\) 2.59838e15 2.20145
\(551\) −3.55235e15 −2.97977
\(552\) −2.79482e15 −2.32108
\(553\) −6.88966e14 −0.566513
\(554\) 2.57116e15 2.09327
\(555\) −3.55921e13 −0.0286908
\(556\) −5.72812e15 −4.57194
\(557\) 8.26995e14 0.653581 0.326790 0.945097i \(-0.394033\pi\)
0.326790 + 0.945097i \(0.394033\pi\)
\(558\) 1.06188e14 0.0830974
\(559\) 1.17017e15 0.906744
\(560\) −3.75915e14 −0.288440
\(561\) −1.04246e15 −0.792069
\(562\) 2.71697e14 0.204425
\(563\) 1.57630e15 1.17447 0.587236 0.809416i \(-0.300216\pi\)
0.587236 + 0.809416i \(0.300216\pi\)
\(564\) 1.46700e14 0.108242
\(565\) −3.18619e14 −0.232812
\(566\) −4.10590e15 −2.97111
\(567\) −7.15542e13 −0.0512777
\(568\) 5.61123e15 3.98237
\(569\) −1.61475e14 −0.113498 −0.0567491 0.998388i \(-0.518074\pi\)
−0.0567491 + 0.998388i \(0.518074\pi\)
\(570\) 4.26016e14 0.296561
\(571\) 1.21281e15 0.836167 0.418083 0.908409i \(-0.362702\pi\)
0.418083 + 0.908409i \(0.362702\pi\)
\(572\) 6.53876e15 4.46496
\(573\) 3.14034e14 0.212386
\(574\) 9.78109e14 0.655197
\(575\) −1.69608e15 −1.12531
\(576\) 2.21199e15 1.45365
\(577\) 9.98453e14 0.649921 0.324961 0.945728i \(-0.394649\pi\)
0.324961 + 0.945728i \(0.394649\pi\)
\(578\) −1.21003e15 −0.780180
\(579\) 1.67795e15 1.07163
\(580\) 1.21930e15 0.771360
\(581\) −1.02309e15 −0.641127
\(582\) −1.08461e15 −0.673283
\(583\) 1.48384e15 0.912449
\(584\) −1.04105e16 −6.34161
\(585\) 1.19311e14 0.0719988
\(586\) −2.21526e15 −1.32430
\(587\) 1.48736e15 0.880857 0.440428 0.897788i \(-0.354827\pi\)
0.440428 + 0.897788i \(0.354827\pi\)
\(588\) −2.16085e15 −1.26780
\(589\) 3.71505e14 0.215939
\(590\) −6.88630e13 −0.0396552
\(591\) 1.89169e15 1.07924
\(592\) −2.24471e15 −1.26879
\(593\) −2.02826e15 −1.13585 −0.567927 0.823079i \(-0.692255\pi\)
−0.567927 + 0.823079i \(0.692255\pi\)
\(594\) −7.82736e14 −0.434300
\(595\) 1.55450e14 0.0854572
\(596\) 7.11816e15 3.87715
\(597\) −9.72270e14 −0.524721
\(598\) −5.79787e15 −3.10036
\(599\) −2.43769e15 −1.29161 −0.645804 0.763503i \(-0.723478\pi\)
−0.645804 + 0.763503i \(0.723478\pi\)
\(600\) 3.73870e15 1.96286
\(601\) 3.49824e15 1.81987 0.909935 0.414751i \(-0.136131\pi\)
0.909935 + 0.414751i \(0.136131\pi\)
\(602\) 1.14476e15 0.590113
\(603\) 2.17426e14 0.111062
\(604\) −7.57065e15 −3.83204
\(605\) 1.07196e14 0.0537680
\(606\) 3.38510e14 0.168257
\(607\) 3.57626e14 0.176153 0.0880766 0.996114i \(-0.471928\pi\)
0.0880766 + 0.996114i \(0.471928\pi\)
\(608\) 1.48277e16 7.23774
\(609\) −9.73282e14 −0.470808
\(610\) 5.22759e14 0.250604
\(611\) 1.95256e14 0.0927636
\(612\) −2.33785e15 −1.10074
\(613\) 8.75613e14 0.408582 0.204291 0.978910i \(-0.434511\pi\)
0.204291 + 0.978910i \(0.434511\pi\)
\(614\) 5.52941e15 2.55713
\(615\) 1.43722e14 0.0658736
\(616\) 4.10414e15 1.86435
\(617\) 3.35302e15 1.50962 0.754809 0.655944i \(-0.227729\pi\)
0.754809 + 0.655944i \(0.227729\pi\)
\(618\) 9.45527e14 0.421927
\(619\) 2.89676e15 1.28119 0.640595 0.767879i \(-0.278688\pi\)
0.640595 + 0.767879i \(0.278688\pi\)
\(620\) −1.27514e14 −0.0558991
\(621\) 5.10927e14 0.222001
\(622\) 5.22274e15 2.24932
\(623\) −8.59889e14 −0.367078
\(624\) 7.52470e15 3.18401
\(625\) 2.21053e15 0.927162
\(626\) −4.14160e15 −1.72191
\(627\) −2.73844e15 −1.12858
\(628\) −6.41458e15 −2.62054
\(629\) 9.28245e14 0.375910
\(630\) 1.16721e14 0.0468571
\(631\) −8.50201e14 −0.338345 −0.169173 0.985586i \(-0.554110\pi\)
−0.169173 + 0.985586i \(0.554110\pi\)
\(632\) −1.08442e16 −4.27811
\(633\) 2.72545e15 1.06591
\(634\) −6.36297e15 −2.46700
\(635\) 6.29426e13 0.0241930
\(636\) 3.32769e15 1.26803
\(637\) −2.87607e15 −1.08651
\(638\) −1.06468e16 −3.98754
\(639\) −1.02580e15 −0.380897
\(640\) −1.78418e15 −0.656824
\(641\) 2.26152e15 0.825432 0.412716 0.910860i \(-0.364580\pi\)
0.412716 + 0.910860i \(0.364580\pi\)
\(642\) 4.82276e15 1.74523
\(643\) 2.45342e15 0.880261 0.440130 0.897934i \(-0.354932\pi\)
0.440130 + 0.897934i \(0.354932\pi\)
\(644\) −4.17547e15 −1.48536
\(645\) 1.68210e14 0.0593300
\(646\) −1.11105e16 −3.88558
\(647\) −3.49515e15 −1.21197 −0.605986 0.795475i \(-0.707221\pi\)
−0.605986 + 0.795475i \(0.707221\pi\)
\(648\) −1.12625e15 −0.387232
\(649\) 4.42654e14 0.150910
\(650\) 7.75596e15 2.62187
\(651\) 1.01786e14 0.0341186
\(652\) 2.75322e15 0.915124
\(653\) −2.53333e15 −0.834967 −0.417483 0.908685i \(-0.637088\pi\)
−0.417483 + 0.908685i \(0.637088\pi\)
\(654\) 5.73365e15 1.87393
\(655\) 4.56658e14 0.148001
\(656\) 9.06423e15 2.91313
\(657\) 1.90316e15 0.606548
\(658\) 1.91016e14 0.0603709
\(659\) −4.97965e14 −0.156074 −0.0780368 0.996950i \(-0.524865\pi\)
−0.0780368 + 0.996950i \(0.524865\pi\)
\(660\) 9.39937e14 0.292151
\(661\) −2.54752e15 −0.785253 −0.392626 0.919698i \(-0.628433\pi\)
−0.392626 + 0.919698i \(0.628433\pi\)
\(662\) −8.51079e15 −2.60166
\(663\) −3.11165e15 −0.943337
\(664\) −1.61031e16 −4.84157
\(665\) 4.08354e14 0.121764
\(666\) 6.96978e14 0.206116
\(667\) 6.94964e15 2.03831
\(668\) 7.21404e15 2.09849
\(669\) 4.56397e14 0.131674
\(670\) −3.54670e14 −0.101488
\(671\) −3.36032e15 −0.953688
\(672\) 4.06252e15 1.14357
\(673\) −5.08841e15 −1.42069 −0.710345 0.703853i \(-0.751461\pi\)
−0.710345 + 0.703853i \(0.751461\pi\)
\(674\) −3.29568e15 −0.912673
\(675\) −6.83479e14 −0.187739
\(676\) 9.27692e15 2.52753
\(677\) 3.00230e15 0.811366 0.405683 0.914014i \(-0.367034\pi\)
0.405683 + 0.914014i \(0.367034\pi\)
\(678\) 6.23932e15 1.67253
\(679\) −1.03965e15 −0.276440
\(680\) 2.44675e15 0.645343
\(681\) 1.45944e15 0.381836
\(682\) 1.11344e15 0.288970
\(683\) −4.11320e14 −0.105893 −0.0529464 0.998597i \(-0.516861\pi\)
−0.0529464 + 0.998597i \(0.516861\pi\)
\(684\) −6.14132e15 −1.56839
\(685\) −1.07784e14 −0.0273060
\(686\) −6.38866e15 −1.60556
\(687\) −1.78019e15 −0.443818
\(688\) 1.06086e16 2.62375
\(689\) 4.42913e15 1.08671
\(690\) −8.33435e14 −0.202863
\(691\) −1.49559e14 −0.0361146 −0.0180573 0.999837i \(-0.505748\pi\)
−0.0180573 + 0.999837i \(0.505748\pi\)
\(692\) 1.40723e16 3.37118
\(693\) −7.50286e14 −0.178317
\(694\) 4.63638e15 1.09321
\(695\) −1.09595e15 −0.256374
\(696\) −1.53192e16 −3.55538
\(697\) −3.74829e15 −0.863084
\(698\) −5.37379e15 −1.22765
\(699\) −1.89314e14 −0.0429100
\(700\) 5.58563e15 1.25612
\(701\) 7.18063e14 0.160219 0.0801094 0.996786i \(-0.474473\pi\)
0.0801094 + 0.996786i \(0.474473\pi\)
\(702\) −2.33640e15 −0.517242
\(703\) 2.43842e15 0.535616
\(704\) 2.31940e16 5.05505
\(705\) 2.80677e13 0.00606970
\(706\) −1.79070e16 −3.84235
\(707\) 3.24476e14 0.0690838
\(708\) 9.92708e14 0.209719
\(709\) 3.52396e15 0.738716 0.369358 0.929287i \(-0.379578\pi\)
0.369358 + 0.929287i \(0.379578\pi\)
\(710\) 1.67331e15 0.348060
\(711\) 1.98244e15 0.409183
\(712\) −1.35344e16 −2.77205
\(713\) −7.26793e14 −0.147713
\(714\) −3.04409e15 −0.613927
\(715\) 1.25105e15 0.250375
\(716\) 3.22170e15 0.639828
\(717\) 3.46582e15 0.683047
\(718\) −6.43553e15 −1.25864
\(719\) 4.02584e15 0.781353 0.390677 0.920528i \(-0.372241\pi\)
0.390677 + 0.920528i \(0.372241\pi\)
\(720\) 1.08166e15 0.208336
\(721\) 9.06328e14 0.173237
\(722\) −1.89232e16 −3.58954
\(723\) 2.73579e15 0.515018
\(724\) −2.13140e16 −3.98201
\(725\) −9.29671e15 −1.72373
\(726\) −2.09915e15 −0.386271
\(727\) −2.36147e15 −0.431264 −0.215632 0.976475i \(-0.569181\pi\)
−0.215632 + 0.976475i \(0.569181\pi\)
\(728\) 1.22505e16 2.22040
\(729\) 2.05891e14 0.0370370
\(730\) −3.10447e15 −0.554258
\(731\) −4.38694e15 −0.777349
\(732\) −7.53594e15 −1.32534
\(733\) −4.82143e15 −0.841597 −0.420798 0.907154i \(-0.638250\pi\)
−0.420798 + 0.907154i \(0.638250\pi\)
\(734\) −9.49964e15 −1.64581
\(735\) −4.13431e14 −0.0710923
\(736\) −2.90081e16 −4.95098
\(737\) 2.27983e15 0.386217
\(738\) −2.81442e15 −0.473238
\(739\) −3.95517e15 −0.660117 −0.330058 0.943960i \(-0.607068\pi\)
−0.330058 + 0.943960i \(0.607068\pi\)
\(740\) −8.36956e14 −0.138653
\(741\) −8.17403e15 −1.34411
\(742\) 4.33295e15 0.707233
\(743\) 5.89583e15 0.955227 0.477614 0.878570i \(-0.341502\pi\)
0.477614 + 0.878570i \(0.341502\pi\)
\(744\) 1.60208e15 0.257652
\(745\) 1.36190e15 0.217413
\(746\) −1.61181e16 −2.55417
\(747\) 2.94385e15 0.463075
\(748\) −2.45136e16 −3.82780
\(749\) 4.62282e15 0.716568
\(750\) 2.25779e15 0.347413
\(751\) −3.65010e15 −0.557551 −0.278776 0.960356i \(-0.589929\pi\)
−0.278776 + 0.960356i \(0.589929\pi\)
\(752\) 1.77017e15 0.268421
\(753\) 1.92870e15 0.290330
\(754\) −3.17798e16 −4.74907
\(755\) −1.44848e15 −0.214884
\(756\) −1.68261e15 −0.247807
\(757\) 5.88657e15 0.860666 0.430333 0.902670i \(-0.358396\pi\)
0.430333 + 0.902670i \(0.358396\pi\)
\(758\) −9.26669e15 −1.34507
\(759\) 5.35735e15 0.772006
\(760\) 6.42740e15 0.919518
\(761\) 3.76203e14 0.0534327 0.0267164 0.999643i \(-0.491495\pi\)
0.0267164 + 0.999643i \(0.491495\pi\)
\(762\) −1.23257e15 −0.173803
\(763\) 5.49595e15 0.769410
\(764\) 7.38458e15 1.02639
\(765\) −4.47295e14 −0.0617243
\(766\) 7.10398e15 0.973293
\(767\) 1.32129e15 0.179731
\(768\) 1.62958e16 2.20085
\(769\) −4.85108e15 −0.650494 −0.325247 0.945629i \(-0.605447\pi\)
−0.325247 + 0.945629i \(0.605447\pi\)
\(770\) 1.22388e15 0.162945
\(771\) −4.19514e15 −0.554558
\(772\) 3.94573e16 5.17883
\(773\) −3.67949e15 −0.479513 −0.239757 0.970833i \(-0.577068\pi\)
−0.239757 + 0.970833i \(0.577068\pi\)
\(774\) −3.29396e15 −0.426229
\(775\) 9.72248e14 0.124916
\(776\) −1.63638e16 −2.08758
\(777\) 6.68083e14 0.0846282
\(778\) −1.56806e16 −1.97231
\(779\) −9.84642e15 −1.22977
\(780\) 2.80564e15 0.347945
\(781\) −1.07561e16 −1.32456
\(782\) 2.17361e16 2.65793
\(783\) 2.80054e15 0.340057
\(784\) −2.60741e16 −3.14392
\(785\) −1.22729e15 −0.146948
\(786\) −8.94245e15 −1.06324
\(787\) 6.11674e15 0.722203 0.361101 0.932527i \(-0.382401\pi\)
0.361101 + 0.932527i \(0.382401\pi\)
\(788\) 4.44836e16 5.21561
\(789\) −2.82278e15 −0.328665
\(790\) −3.23380e15 −0.373908
\(791\) 5.98066e15 0.686718
\(792\) −1.18093e16 −1.34659
\(793\) −1.00303e16 −1.13582
\(794\) 1.59392e15 0.179248
\(795\) 6.36680e14 0.0711053
\(796\) −2.28631e16 −2.53579
\(797\) −6.01045e15 −0.662043 −0.331021 0.943623i \(-0.607393\pi\)
−0.331021 + 0.943623i \(0.607393\pi\)
\(798\) −7.99654e15 −0.874755
\(799\) −7.32008e14 −0.0795259
\(800\) 3.88048e16 4.18688
\(801\) 2.47426e15 0.265134
\(802\) 1.81769e16 1.93446
\(803\) 1.99557e16 2.10926
\(804\) 5.11282e15 0.536726
\(805\) −7.98883e14 −0.0832925
\(806\) 3.32353e15 0.344157
\(807\) −3.69950e15 −0.380486
\(808\) 5.10718e15 0.521697
\(809\) 6.30966e15 0.640161 0.320080 0.947390i \(-0.396290\pi\)
0.320080 + 0.947390i \(0.396290\pi\)
\(810\) −3.35854e14 −0.0338441
\(811\) 7.52302e15 0.752970 0.376485 0.926423i \(-0.377133\pi\)
0.376485 + 0.926423i \(0.377133\pi\)
\(812\) −2.28869e16 −2.27525
\(813\) 8.81846e15 0.870752
\(814\) 7.30820e15 0.716764
\(815\) 5.26768e14 0.0513160
\(816\) −2.82099e16 −2.72964
\(817\) −1.15241e16 −1.10761
\(818\) 1.94976e16 1.86140
\(819\) −2.23954e15 −0.212372
\(820\) 3.37966e15 0.318344
\(821\) 1.66550e16 1.55832 0.779160 0.626825i \(-0.215646\pi\)
0.779160 + 0.626825i \(0.215646\pi\)
\(822\) 2.11067e15 0.196167
\(823\) −1.22782e16 −1.13354 −0.566768 0.823877i \(-0.691806\pi\)
−0.566768 + 0.823877i \(0.691806\pi\)
\(824\) 1.42654e16 1.30823
\(825\) −7.16666e15 −0.652860
\(826\) 1.29260e15 0.116969
\(827\) −2.18763e16 −1.96650 −0.983248 0.182270i \(-0.941655\pi\)
−0.983248 + 0.182270i \(0.941655\pi\)
\(828\) 1.20146e16 1.07285
\(829\) 1.22692e16 1.08835 0.544173 0.838973i \(-0.316843\pi\)
0.544173 + 0.838973i \(0.316843\pi\)
\(830\) −4.80206e15 −0.423154
\(831\) −7.09157e15 −0.620779
\(832\) 6.92322e16 6.02046
\(833\) 1.07823e16 0.931460
\(834\) 2.14613e16 1.84180
\(835\) 1.38025e15 0.117674
\(836\) −6.43951e16 −5.45404
\(837\) −2.92880e14 −0.0246433
\(838\) −6.12864e15 −0.512297
\(839\) −2.00159e15 −0.166220 −0.0831102 0.996540i \(-0.526485\pi\)
−0.0831102 + 0.996540i \(0.526485\pi\)
\(840\) 1.76099e15 0.145285
\(841\) 2.58925e16 2.12224
\(842\) 2.35948e16 1.92132
\(843\) −7.49373e14 −0.0606243
\(844\) 6.40896e16 5.15115
\(845\) 1.77493e15 0.141733
\(846\) −5.49633e14 −0.0436049
\(847\) −2.01213e15 −0.158598
\(848\) 4.01539e16 3.14449
\(849\) 1.13246e16 0.881111
\(850\) −2.90769e16 −2.24773
\(851\) −4.77040e15 −0.366388
\(852\) −2.41219e16 −1.84074
\(853\) 1.90656e16 1.44555 0.722773 0.691086i \(-0.242867\pi\)
0.722773 + 0.691086i \(0.242867\pi\)
\(854\) −9.81247e15 −0.739197
\(855\) −1.17500e15 −0.0879480
\(856\) 7.27621e16 5.41128
\(857\) −1.44564e16 −1.06823 −0.534117 0.845411i \(-0.679356\pi\)
−0.534117 + 0.845411i \(0.679356\pi\)
\(858\) −2.44985e16 −1.79870
\(859\) −1.39840e16 −1.02017 −0.510083 0.860125i \(-0.670385\pi\)
−0.510083 + 0.860125i \(0.670385\pi\)
\(860\) 3.95550e15 0.286722
\(861\) −2.69775e15 −0.194305
\(862\) −8.48728e15 −0.607405
\(863\) −2.45715e16 −1.74732 −0.873660 0.486538i \(-0.838260\pi\)
−0.873660 + 0.486538i \(0.838260\pi\)
\(864\) −1.16896e16 −0.825985
\(865\) 2.69243e15 0.189041
\(866\) −2.02049e16 −1.40964
\(867\) 3.33743e15 0.231370
\(868\) 2.39351e15 0.164884
\(869\) 2.07870e16 1.42293
\(870\) −4.56830e15 −0.310741
\(871\) 6.80512e15 0.459976
\(872\) 8.65049e16 5.81032
\(873\) 2.99149e15 0.199668
\(874\) 5.70987e16 3.78716
\(875\) 2.16419e15 0.142643
\(876\) 4.47531e16 2.93124
\(877\) 5.40079e15 0.351528 0.175764 0.984432i \(-0.443760\pi\)
0.175764 + 0.984432i \(0.443760\pi\)
\(878\) −3.20113e16 −2.07054
\(879\) 6.10997e15 0.392735
\(880\) 1.13418e16 0.724484
\(881\) −2.10009e16 −1.33312 −0.666562 0.745449i \(-0.732235\pi\)
−0.666562 + 0.745449i \(0.732235\pi\)
\(882\) 8.09596e15 0.510730
\(883\) −1.28931e16 −0.808303 −0.404152 0.914692i \(-0.632433\pi\)
−0.404152 + 0.914692i \(0.632433\pi\)
\(884\) −7.31712e16 −4.55882
\(885\) 1.89933e14 0.0117601
\(886\) −1.75687e16 −1.08107
\(887\) −5.76541e13 −0.00352574 −0.00176287 0.999998i \(-0.500561\pi\)
−0.00176287 + 0.999998i \(0.500561\pi\)
\(888\) 1.05155e16 0.639083
\(889\) −1.18147e15 −0.0713611
\(890\) −4.03606e15 −0.242277
\(891\) 2.15888e15 0.128796
\(892\) 1.07323e16 0.636333
\(893\) −1.92292e15 −0.113313
\(894\) −2.66693e16 −1.56190
\(895\) 6.16401e14 0.0358787
\(896\) 3.34900e16 1.93741
\(897\) 1.59913e16 0.919442
\(898\) 2.75366e16 1.57359
\(899\) −3.98376e15 −0.226264
\(900\) −1.60722e16 −0.907278
\(901\) −1.66047e16 −0.931631
\(902\) −2.95108e16 −1.64568
\(903\) −3.15740e15 −0.175004
\(904\) 9.41341e16 5.18586
\(905\) −4.07796e15 −0.223293
\(906\) 2.83646e16 1.54373
\(907\) −2.25546e16 −1.22010 −0.610049 0.792364i \(-0.708850\pi\)
−0.610049 + 0.792364i \(0.708850\pi\)
\(908\) 3.43191e16 1.84528
\(909\) −9.33653e14 −0.0498981
\(910\) 3.65319e15 0.194064
\(911\) 1.05516e16 0.557143 0.278571 0.960416i \(-0.410139\pi\)
0.278571 + 0.960416i \(0.410139\pi\)
\(912\) −7.41048e16 −3.88933
\(913\) 3.08679e16 1.61034
\(914\) 8.18505e15 0.424441
\(915\) −1.44184e15 −0.0743190
\(916\) −4.18616e16 −2.14482
\(917\) −8.57173e15 −0.436553
\(918\) 8.75910e15 0.443430
\(919\) 5.16429e15 0.259881 0.129941 0.991522i \(-0.458521\pi\)
0.129941 + 0.991522i \(0.458521\pi\)
\(920\) −1.25742e16 −0.628997
\(921\) −1.52508e16 −0.758341
\(922\) −8.40711e15 −0.415553
\(923\) −3.21060e16 −1.57753
\(924\) −1.76431e16 −0.861747
\(925\) 6.38147e15 0.309843
\(926\) −7.12855e15 −0.344065
\(927\) −2.60788e15 −0.125127
\(928\) −1.59002e17 −7.58381
\(929\) −1.35039e16 −0.640286 −0.320143 0.947369i \(-0.603731\pi\)
−0.320143 + 0.947369i \(0.603731\pi\)
\(930\) 4.77752e14 0.0225189
\(931\) 2.83242e16 1.32719
\(932\) −4.45176e15 −0.207369
\(933\) −1.44050e16 −0.667058
\(934\) −3.87659e16 −1.78461
\(935\) −4.69014e15 −0.214646
\(936\) −3.52498e16 −1.60376
\(937\) 4.28349e16 1.93745 0.968724 0.248142i \(-0.0798198\pi\)
0.968724 + 0.248142i \(0.0798198\pi\)
\(938\) 6.65735e15 0.299355
\(939\) 1.14230e16 0.510648
\(940\) 6.60018e14 0.0293328
\(941\) 2.49613e16 1.10287 0.551435 0.834218i \(-0.314081\pi\)
0.551435 + 0.834218i \(0.314081\pi\)
\(942\) 2.40332e16 1.05568
\(943\) 1.92630e16 0.841222
\(944\) 1.19786e16 0.520068
\(945\) −3.21931e14 −0.0138959
\(946\) −3.45390e16 −1.48221
\(947\) −1.25777e15 −0.0536630 −0.0268315 0.999640i \(-0.508542\pi\)
−0.0268315 + 0.999640i \(0.508542\pi\)
\(948\) 4.66175e16 1.97744
\(949\) 5.95660e16 2.51208
\(950\) −7.63823e16 −3.20267
\(951\) 1.75499e16 0.731613
\(952\) −4.59269e16 −1.90355
\(953\) 1.31585e16 0.542246 0.271123 0.962545i \(-0.412605\pi\)
0.271123 + 0.962545i \(0.412605\pi\)
\(954\) −1.24677e16 −0.510823
\(955\) 1.41287e15 0.0575553
\(956\) 8.14995e16 3.30093
\(957\) 2.93652e16 1.18254
\(958\) 2.27308e16 0.910132
\(959\) 2.02317e15 0.0805434
\(960\) 9.95201e15 0.393930
\(961\) −2.49919e16 −0.983603
\(962\) 2.18144e16 0.853651
\(963\) −1.33018e16 −0.517565
\(964\) 6.43328e16 2.48891
\(965\) 7.54927e15 0.290405
\(966\) 1.56440e16 0.598377
\(967\) −2.13925e16 −0.813609 −0.406804 0.913515i \(-0.633357\pi\)
−0.406804 + 0.913515i \(0.633357\pi\)
\(968\) −3.16704e16 −1.19767
\(969\) 3.06442e16 1.15231
\(970\) −4.87979e15 −0.182455
\(971\) −3.57391e16 −1.32873 −0.664366 0.747407i \(-0.731299\pi\)
−0.664366 + 0.747407i \(0.731299\pi\)
\(972\) 4.84157e15 0.178987
\(973\) 2.05716e16 0.756217
\(974\) −9.41935e16 −3.44308
\(975\) −2.13919e16 −0.777542
\(976\) −9.09332e16 −3.28661
\(977\) −2.16860e16 −0.779400 −0.389700 0.920942i \(-0.627421\pi\)
−0.389700 + 0.920942i \(0.627421\pi\)
\(978\) −1.03154e16 −0.368656
\(979\) 2.59439e16 0.922001
\(980\) −9.72192e15 −0.343565
\(981\) −1.58141e16 −0.555732
\(982\) 2.68563e16 0.938499
\(983\) 4.70805e16 1.63605 0.818025 0.575183i \(-0.195069\pi\)
0.818025 + 0.575183i \(0.195069\pi\)
\(984\) −4.24619e16 −1.46732
\(985\) 8.51094e15 0.292468
\(986\) 1.19142e17 4.07137
\(987\) −5.26846e14 −0.0179036
\(988\) −1.92214e17 −6.49564
\(989\) 2.25452e16 0.757659
\(990\) −3.52162e15 −0.117692
\(991\) 2.59802e14 0.00863452 0.00431726 0.999991i \(-0.498626\pi\)
0.00431726 + 0.999991i \(0.498626\pi\)
\(992\) 1.66284e16 0.549586
\(993\) 2.34738e16 0.771548
\(994\) −3.14088e16 −1.02666
\(995\) −4.37435e15 −0.142196
\(996\) 6.92252e16 2.23788
\(997\) −1.73914e16 −0.559128 −0.279564 0.960127i \(-0.590190\pi\)
−0.279564 + 0.960127i \(0.590190\pi\)
\(998\) −1.54773e14 −0.00494854
\(999\) −1.92235e15 −0.0611256
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.d.1.1 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.12.a.d.1.1 28 1.1 even 1 trivial