Properties

Label 177.12.a.d.1.9
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 1.9
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-39.1439 q^{2} +243.000 q^{3} -515.752 q^{4} +158.720 q^{5} -9511.98 q^{6} -70159.1 q^{7} +100355. q^{8} +59049.0 q^{9} +O(q^{10})\) \(q-39.1439 q^{2} +243.000 q^{3} -515.752 q^{4} +158.720 q^{5} -9511.98 q^{6} -70159.1 q^{7} +100355. q^{8} +59049.0 q^{9} -6212.92 q^{10} +492429. q^{11} -125328. q^{12} +2.38920e6 q^{13} +2.74630e6 q^{14} +38568.9 q^{15} -2.87204e6 q^{16} +4.34922e6 q^{17} -2.31141e6 q^{18} +1.74914e7 q^{19} -81860.1 q^{20} -1.70487e7 q^{21} -1.92756e7 q^{22} +3.13914e6 q^{23} +2.43864e7 q^{24} -4.88029e7 q^{25} -9.35227e7 q^{26} +1.43489e7 q^{27} +3.61847e7 q^{28} -4.79247e7 q^{29} -1.50974e6 q^{30} -6.76298e7 q^{31} -9.31047e7 q^{32} +1.19660e8 q^{33} -1.70246e8 q^{34} -1.11356e7 q^{35} -3.04546e7 q^{36} -2.35146e8 q^{37} -6.84681e8 q^{38} +5.80576e8 q^{39} +1.59284e7 q^{40} -2.65713e8 q^{41} +6.67352e8 q^{42} +5.47895e8 q^{43} -2.53971e8 q^{44} +9.37225e6 q^{45} -1.22878e8 q^{46} -7.16085e8 q^{47} -6.97907e8 q^{48} +2.94498e9 q^{49} +1.91034e9 q^{50} +1.05686e9 q^{51} -1.23224e9 q^{52} -1.02867e7 q^{53} -5.61673e8 q^{54} +7.81582e7 q^{55} -7.04084e9 q^{56} +4.25040e9 q^{57} +1.87596e9 q^{58} +7.14924e8 q^{59} -1.98920e7 q^{60} -9.31011e8 q^{61} +2.64730e9 q^{62} -4.14283e9 q^{63} +9.52643e9 q^{64} +3.79213e8 q^{65} -4.68397e9 q^{66} +7.59551e9 q^{67} -2.24312e9 q^{68} +7.62811e8 q^{69} +4.35893e8 q^{70} +1.44444e9 q^{71} +5.92588e9 q^{72} +1.34828e10 q^{73} +9.20456e9 q^{74} -1.18591e10 q^{75} -9.02121e9 q^{76} -3.45484e10 q^{77} -2.27260e10 q^{78} +2.11165e10 q^{79} -4.55850e8 q^{80} +3.48678e9 q^{81} +1.04010e10 q^{82} -4.40162e10 q^{83} +8.79289e9 q^{84} +6.90307e8 q^{85} -2.14468e10 q^{86} -1.16457e10 q^{87} +4.94179e10 q^{88} -6.64139e10 q^{89} -3.66867e8 q^{90} -1.67624e11 q^{91} -1.61902e9 q^{92} -1.64340e10 q^{93} +2.80304e10 q^{94} +2.77623e9 q^{95} -2.26244e10 q^{96} +6.49449e10 q^{97} -1.15278e11 q^{98} +2.90774e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 28q + 96q^{2} + 6804q^{3} + 29214q^{4} + 26562q^{5} + 23328q^{6} + 142333q^{7} + 332331q^{8} + 1653372q^{9} + O(q^{10}) \) \( 28q + 96q^{2} + 6804q^{3} + 29214q^{4} + 26562q^{5} + 23328q^{6} + 142333q^{7} + 332331q^{8} + 1653372q^{9} + 616281q^{10} + 1082362q^{11} + 7099002q^{12} + 503712q^{13} + 1321669q^{14} + 6454566q^{15} + 34870338q^{16} + 13513579q^{17} + 5668704q^{18} + 35971687q^{19} + 96105997q^{20} + 34586919q^{21} - 47598882q^{22} + 61380539q^{23} + 80756433q^{24} + 294744746q^{25} + 62820734q^{26} + 401769396q^{27} + 148068294q^{28} + 322339307q^{29} + 149756283q^{30} + 151247077q^{31} + 466383494q^{32} + 263013966q^{33} + 684479860q^{34} + 960297361q^{35} + 1725057486q^{36} + 863508437q^{37} + 992640509q^{38} + 122402016q^{39} + 3067680252q^{40} + 3081170377q^{41} + 321165567q^{42} + 2554238300q^{43} + 4350123570q^{44} + 1568459538q^{45} - 1987059155q^{46} + 6203398333q^{47} + 8473492134q^{48} + 10327857997q^{49} + 17577682253q^{50} + 3283799697q^{51} + 32137181618q^{52} + 14571770754q^{53} + 1377495072q^{54} + 18251419334q^{55} + 33498842836q^{56} + 8741119941q^{57} + 11860778276q^{58} + 20017880372q^{59} + 23353757271q^{60} + 2761613771q^{61} + 13785829526q^{62} + 8404621317q^{63} + 86547545293q^{64} + 32034985256q^{65} - 11566528326q^{66} + 39381333296q^{67} + 38995496621q^{68} + 14915470977q^{69} + 8551800364q^{70} + 26130020296q^{71} + 19623813219q^{72} + 41382402799q^{73} + 23815315058q^{74} + 71622973278q^{75} + 10611720128q^{76} - 8426124313q^{77} + 15265438362q^{78} + 59825111206q^{79} + 4009687655q^{80} + 97629963228q^{81} - 39592715115q^{82} + 35433122727q^{83} + 35980595442q^{84} - 8950496085q^{85} - 182032360688q^{86} + 78328451601q^{87} - 220003602335q^{88} + 102303043039q^{89} + 36390776769q^{90} - 111146323655q^{91} - 163000203526q^{92} + 36753039711q^{93} - 81314346008q^{94} + 208102168887q^{95} + 113331189042q^{96} - 171891031490q^{97} + 72304707792q^{98} + 63912393738q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −39.1439 −0.864967 −0.432483 0.901642i \(-0.642363\pi\)
−0.432483 + 0.901642i \(0.642363\pi\)
\(3\) 243.000 0.577350
\(4\) −515.752 −0.251832
\(5\) 158.720 0.0227141 0.0113571 0.999936i \(-0.496385\pi\)
0.0113571 + 0.999936i \(0.496385\pi\)
\(6\) −9511.98 −0.499389
\(7\) −70159.1 −1.57777 −0.788887 0.614538i \(-0.789342\pi\)
−0.788887 + 0.614538i \(0.789342\pi\)
\(8\) 100355. 1.08279
\(9\) 59049.0 0.333333
\(10\) −6212.92 −0.0196470
\(11\) 492429. 0.921900 0.460950 0.887426i \(-0.347509\pi\)
0.460950 + 0.887426i \(0.347509\pi\)
\(12\) −125328. −0.145395
\(13\) 2.38920e6 1.78470 0.892348 0.451348i \(-0.149057\pi\)
0.892348 + 0.451348i \(0.149057\pi\)
\(14\) 2.74630e6 1.36472
\(15\) 38568.9 0.0131140
\(16\) −2.87204e6 −0.684748
\(17\) 4.34922e6 0.742920 0.371460 0.928449i \(-0.378857\pi\)
0.371460 + 0.928449i \(0.378857\pi\)
\(18\) −2.31141e6 −0.288322
\(19\) 1.74914e7 1.62061 0.810305 0.586008i \(-0.199301\pi\)
0.810305 + 0.586008i \(0.199301\pi\)
\(20\) −81860.1 −0.00572015
\(21\) −1.70487e7 −0.910928
\(22\) −1.92756e7 −0.797413
\(23\) 3.13914e6 0.101697 0.0508484 0.998706i \(-0.483807\pi\)
0.0508484 + 0.998706i \(0.483807\pi\)
\(24\) 2.43864e7 0.625151
\(25\) −4.88029e7 −0.999484
\(26\) −9.35227e7 −1.54370
\(27\) 1.43489e7 0.192450
\(28\) 3.61847e7 0.397334
\(29\) −4.79247e7 −0.433881 −0.216940 0.976185i \(-0.569608\pi\)
−0.216940 + 0.976185i \(0.569608\pi\)
\(30\) −1.50974e6 −0.0113432
\(31\) −6.76298e7 −0.424276 −0.212138 0.977240i \(-0.568043\pi\)
−0.212138 + 0.977240i \(0.568043\pi\)
\(32\) −9.31047e7 −0.490509
\(33\) 1.19660e8 0.532259
\(34\) −1.70246e8 −0.642602
\(35\) −1.11356e7 −0.0358378
\(36\) −3.04546e7 −0.0839440
\(37\) −2.35146e8 −0.557480 −0.278740 0.960367i \(-0.589917\pi\)
−0.278740 + 0.960367i \(0.589917\pi\)
\(38\) −6.84681e8 −1.40177
\(39\) 5.80576e8 1.03039
\(40\) 1.59284e7 0.0245947
\(41\) −2.65713e8 −0.358180 −0.179090 0.983833i \(-0.557315\pi\)
−0.179090 + 0.983833i \(0.557315\pi\)
\(42\) 6.67352e8 0.787923
\(43\) 5.47895e8 0.568356 0.284178 0.958771i \(-0.408279\pi\)
0.284178 + 0.958771i \(0.408279\pi\)
\(44\) −2.53971e8 −0.232164
\(45\) 9.37225e6 0.00757138
\(46\) −1.22878e8 −0.0879644
\(47\) −7.16085e8 −0.455435 −0.227717 0.973727i \(-0.573126\pi\)
−0.227717 + 0.973727i \(0.573126\pi\)
\(48\) −6.97907e8 −0.395340
\(49\) 2.94498e9 1.48937
\(50\) 1.91034e9 0.864521
\(51\) 1.05686e9 0.428925
\(52\) −1.23224e9 −0.449444
\(53\) −1.02867e7 −0.00337878 −0.00168939 0.999999i \(-0.500538\pi\)
−0.00168939 + 0.999999i \(0.500538\pi\)
\(54\) −5.61673e8 −0.166463
\(55\) 7.81582e7 0.0209402
\(56\) −7.04084e9 −1.70840
\(57\) 4.25040e9 0.935660
\(58\) 1.87596e9 0.375293
\(59\) 7.14924e8 0.130189
\(60\) −1.98920e7 −0.00330253
\(61\) −9.31011e8 −0.141137 −0.0705685 0.997507i \(-0.522481\pi\)
−0.0705685 + 0.997507i \(0.522481\pi\)
\(62\) 2.64730e9 0.366985
\(63\) −4.14283e9 −0.525925
\(64\) 9.52643e9 1.10902
\(65\) 3.79213e8 0.0405378
\(66\) −4.68397e9 −0.460387
\(67\) 7.59551e9 0.687299 0.343649 0.939098i \(-0.388337\pi\)
0.343649 + 0.939098i \(0.388337\pi\)
\(68\) −2.24312e9 −0.187091
\(69\) 7.62811e8 0.0587147
\(70\) 4.35893e8 0.0309985
\(71\) 1.44444e9 0.0950121 0.0475060 0.998871i \(-0.484873\pi\)
0.0475060 + 0.998871i \(0.484873\pi\)
\(72\) 5.92588e9 0.360931
\(73\) 1.34828e10 0.761212 0.380606 0.924737i \(-0.375715\pi\)
0.380606 + 0.924737i \(0.375715\pi\)
\(74\) 9.20456e9 0.482201
\(75\) −1.18591e10 −0.577052
\(76\) −9.02121e9 −0.408122
\(77\) −3.45484e10 −1.45455
\(78\) −2.27260e10 −0.891257
\(79\) 2.11165e10 0.772099 0.386049 0.922478i \(-0.373839\pi\)
0.386049 + 0.922478i \(0.373839\pi\)
\(80\) −4.55850e8 −0.0155535
\(81\) 3.48678e9 0.111111
\(82\) 1.04010e10 0.309814
\(83\) −4.40162e10 −1.22654 −0.613272 0.789872i \(-0.710147\pi\)
−0.613272 + 0.789872i \(0.710147\pi\)
\(84\) 8.79289e9 0.229401
\(85\) 6.90307e8 0.0168748
\(86\) −2.14468e10 −0.491609
\(87\) −1.16457e10 −0.250501
\(88\) 4.94179e10 0.998227
\(89\) −6.64139e10 −1.26071 −0.630353 0.776308i \(-0.717090\pi\)
−0.630353 + 0.776308i \(0.717090\pi\)
\(90\) −3.66867e8 −0.00654899
\(91\) −1.67624e11 −2.81585
\(92\) −1.61902e9 −0.0256105
\(93\) −1.64340e10 −0.244956
\(94\) 2.80304e10 0.393936
\(95\) 2.77623e9 0.0368108
\(96\) −2.26244e10 −0.283195
\(97\) 6.49449e10 0.767893 0.383946 0.923355i \(-0.374565\pi\)
0.383946 + 0.923355i \(0.374565\pi\)
\(98\) −1.15278e11 −1.28826
\(99\) 2.90774e10 0.307300
\(100\) 2.51702e10 0.251702
\(101\) 1.33002e11 1.25918 0.629592 0.776926i \(-0.283222\pi\)
0.629592 + 0.776926i \(0.283222\pi\)
\(102\) −4.13697e10 −0.371006
\(103\) 1.52129e11 1.29303 0.646513 0.762903i \(-0.276227\pi\)
0.646513 + 0.762903i \(0.276227\pi\)
\(104\) 2.39769e11 1.93246
\(105\) −2.70596e9 −0.0206909
\(106\) 4.02663e8 0.00292253
\(107\) −1.25642e11 −0.866011 −0.433005 0.901391i \(-0.642547\pi\)
−0.433005 + 0.901391i \(0.642547\pi\)
\(108\) −7.40048e9 −0.0484651
\(109\) −2.26950e10 −0.141281 −0.0706405 0.997502i \(-0.522504\pi\)
−0.0706405 + 0.997502i \(0.522504\pi\)
\(110\) −3.05942e9 −0.0181125
\(111\) −5.71406e10 −0.321861
\(112\) 2.01500e11 1.08038
\(113\) 7.42906e10 0.379317 0.189659 0.981850i \(-0.439262\pi\)
0.189659 + 0.981850i \(0.439262\pi\)
\(114\) −1.66377e11 −0.809315
\(115\) 4.98244e8 0.00230996
\(116\) 2.47173e10 0.109265
\(117\) 1.41080e11 0.594899
\(118\) −2.79850e10 −0.112609
\(119\) −3.05137e11 −1.17216
\(120\) 3.87060e9 0.0141998
\(121\) −4.28255e10 −0.150101
\(122\) 3.64435e10 0.122079
\(123\) −6.45682e10 −0.206795
\(124\) 3.48802e10 0.106846
\(125\) −1.54960e10 −0.0454165
\(126\) 1.62167e11 0.454908
\(127\) −6.27051e11 −1.68415 −0.842077 0.539357i \(-0.818668\pi\)
−0.842077 + 0.539357i \(0.818668\pi\)
\(128\) −1.82224e11 −0.468759
\(129\) 1.33138e11 0.328141
\(130\) −1.48439e10 −0.0350639
\(131\) −1.87407e11 −0.424419 −0.212209 0.977224i \(-0.568066\pi\)
−0.212209 + 0.977224i \(0.568066\pi\)
\(132\) −6.17150e10 −0.134040
\(133\) −1.22718e12 −2.55696
\(134\) −2.97318e11 −0.594491
\(135\) 2.27746e9 0.00437134
\(136\) 4.36467e11 0.804429
\(137\) 7.43105e11 1.31549 0.657745 0.753241i \(-0.271511\pi\)
0.657745 + 0.753241i \(0.271511\pi\)
\(138\) −2.98594e10 −0.0507863
\(139\) 2.42816e11 0.396914 0.198457 0.980110i \(-0.436407\pi\)
0.198457 + 0.980110i \(0.436407\pi\)
\(140\) 5.74323e9 0.00902510
\(141\) −1.74009e11 −0.262946
\(142\) −5.65411e10 −0.0821823
\(143\) 1.17651e12 1.64531
\(144\) −1.69591e11 −0.228249
\(145\) −7.60660e9 −0.00985523
\(146\) −5.27771e11 −0.658423
\(147\) 7.15629e11 0.859889
\(148\) 1.21277e11 0.140391
\(149\) 1.53014e12 1.70690 0.853448 0.521178i \(-0.174507\pi\)
0.853448 + 0.521178i \(0.174507\pi\)
\(150\) 4.64212e11 0.499131
\(151\) −1.05358e12 −1.09219 −0.546093 0.837725i \(-0.683885\pi\)
−0.546093 + 0.837725i \(0.683885\pi\)
\(152\) 1.75535e12 1.75479
\(153\) 2.56817e11 0.247640
\(154\) 1.35236e12 1.25814
\(155\) −1.07342e10 −0.00963706
\(156\) −2.99433e11 −0.259486
\(157\) −1.53165e12 −1.28148 −0.640740 0.767758i \(-0.721372\pi\)
−0.640740 + 0.767758i \(0.721372\pi\)
\(158\) −8.26583e11 −0.667840
\(159\) −2.49967e9 −0.00195074
\(160\) −1.47776e10 −0.0111415
\(161\) −2.20239e11 −0.160455
\(162\) −1.36486e11 −0.0961074
\(163\) −5.32648e11 −0.362584 −0.181292 0.983429i \(-0.558028\pi\)
−0.181292 + 0.983429i \(0.558028\pi\)
\(164\) 1.37042e11 0.0902012
\(165\) 1.89924e10 0.0120898
\(166\) 1.72297e12 1.06092
\(167\) 2.20232e12 1.31202 0.656008 0.754754i \(-0.272244\pi\)
0.656008 + 0.754754i \(0.272244\pi\)
\(168\) −1.71092e12 −0.986347
\(169\) 3.91612e12 2.18514
\(170\) −2.70213e10 −0.0145961
\(171\) 1.03285e12 0.540204
\(172\) −2.82578e11 −0.143130
\(173\) 2.45988e12 1.20687 0.603434 0.797413i \(-0.293799\pi\)
0.603434 + 0.797413i \(0.293799\pi\)
\(174\) 4.55859e11 0.216675
\(175\) 3.42397e12 1.57696
\(176\) −1.41428e12 −0.631270
\(177\) 1.73727e11 0.0751646
\(178\) 2.59970e12 1.09047
\(179\) −1.26117e12 −0.512957 −0.256478 0.966550i \(-0.582562\pi\)
−0.256478 + 0.966550i \(0.582562\pi\)
\(180\) −4.83376e9 −0.00190672
\(181\) 3.98209e12 1.52363 0.761814 0.647796i \(-0.224309\pi\)
0.761814 + 0.647796i \(0.224309\pi\)
\(182\) 6.56147e12 2.43561
\(183\) −2.26236e11 −0.0814855
\(184\) 3.15029e11 0.110117
\(185\) −3.73224e10 −0.0126627
\(186\) 6.43293e11 0.211879
\(187\) 2.14168e12 0.684898
\(188\) 3.69322e11 0.114693
\(189\) −1.00671e12 −0.303643
\(190\) −1.08672e11 −0.0318401
\(191\) 6.26611e12 1.78367 0.891836 0.452360i \(-0.149418\pi\)
0.891836 + 0.452360i \(0.149418\pi\)
\(192\) 2.31492e12 0.640294
\(193\) 1.42924e12 0.384184 0.192092 0.981377i \(-0.438473\pi\)
0.192092 + 0.981377i \(0.438473\pi\)
\(194\) −2.54220e12 −0.664202
\(195\) 9.21489e10 0.0234045
\(196\) −1.51888e12 −0.375072
\(197\) −2.27297e12 −0.545795 −0.272897 0.962043i \(-0.587982\pi\)
−0.272897 + 0.962043i \(0.587982\pi\)
\(198\) −1.13821e12 −0.265804
\(199\) 8.10936e12 1.84202 0.921011 0.389537i \(-0.127365\pi\)
0.921011 + 0.389537i \(0.127365\pi\)
\(200\) −4.89764e12 −1.08223
\(201\) 1.84571e12 0.396812
\(202\) −5.20621e12 −1.08915
\(203\) 3.36236e12 0.684566
\(204\) −5.45078e11 −0.108017
\(205\) −4.21739e10 −0.00813575
\(206\) −5.95493e12 −1.11842
\(207\) 1.85363e11 0.0338989
\(208\) −6.86189e12 −1.22207
\(209\) 8.61325e12 1.49404
\(210\) 1.05922e11 0.0178970
\(211\) −8.72185e12 −1.43567 −0.717836 0.696212i \(-0.754867\pi\)
−0.717836 + 0.696212i \(0.754867\pi\)
\(212\) 5.30540e9 0.000850885 0
\(213\) 3.50999e11 0.0548552
\(214\) 4.91811e12 0.749071
\(215\) 8.69618e10 0.0129097
\(216\) 1.43999e12 0.208384
\(217\) 4.74484e12 0.669412
\(218\) 8.88371e11 0.122203
\(219\) 3.27633e12 0.439486
\(220\) −4.03103e10 −0.00527340
\(221\) 1.03912e13 1.32589
\(222\) 2.23671e12 0.278399
\(223\) −9.90424e12 −1.20266 −0.601332 0.798999i \(-0.705363\pi\)
−0.601332 + 0.798999i \(0.705363\pi\)
\(224\) 6.53214e12 0.773912
\(225\) −2.88176e12 −0.333161
\(226\) −2.90803e12 −0.328097
\(227\) 9.81507e12 1.08082 0.540408 0.841403i \(-0.318270\pi\)
0.540408 + 0.841403i \(0.318270\pi\)
\(228\) −2.19215e12 −0.235629
\(229\) 4.48537e12 0.470656 0.235328 0.971916i \(-0.424384\pi\)
0.235328 + 0.971916i \(0.424384\pi\)
\(230\) −1.95032e10 −0.00199804
\(231\) −8.39526e12 −0.839785
\(232\) −4.80950e12 −0.469804
\(233\) −1.92906e13 −1.84030 −0.920150 0.391567i \(-0.871933\pi\)
−0.920150 + 0.391567i \(0.871933\pi\)
\(234\) −5.52242e12 −0.514568
\(235\) −1.13657e11 −0.0103448
\(236\) −3.68724e11 −0.0327857
\(237\) 5.13131e12 0.445772
\(238\) 1.19443e13 1.01388
\(239\) 1.90066e13 1.57658 0.788289 0.615306i \(-0.210967\pi\)
0.788289 + 0.615306i \(0.210967\pi\)
\(240\) −1.10772e11 −0.00897980
\(241\) −1.83679e13 −1.45534 −0.727672 0.685925i \(-0.759398\pi\)
−0.727672 + 0.685925i \(0.759398\pi\)
\(242\) 1.67636e12 0.129832
\(243\) 8.47289e11 0.0641500
\(244\) 4.80171e11 0.0355428
\(245\) 4.67426e11 0.0338298
\(246\) 2.52746e12 0.178871
\(247\) 4.17904e13 2.89230
\(248\) −6.78701e12 −0.459403
\(249\) −1.06959e13 −0.708146
\(250\) 6.06574e11 0.0392838
\(251\) −1.47420e13 −0.934007 −0.467003 0.884256i \(-0.654666\pi\)
−0.467003 + 0.884256i \(0.654666\pi\)
\(252\) 2.13667e12 0.132445
\(253\) 1.54580e12 0.0937543
\(254\) 2.45452e13 1.45674
\(255\) 1.67745e11 0.00974266
\(256\) −1.23772e13 −0.703561
\(257\) −2.00386e13 −1.11490 −0.557449 0.830211i \(-0.688220\pi\)
−0.557449 + 0.830211i \(0.688220\pi\)
\(258\) −5.21156e12 −0.283831
\(259\) 1.64977e13 0.879577
\(260\) −1.95580e11 −0.0102087
\(261\) −2.82991e12 −0.144627
\(262\) 7.33586e12 0.367108
\(263\) 1.59440e13 0.781339 0.390670 0.920531i \(-0.372243\pi\)
0.390670 + 0.920531i \(0.372243\pi\)
\(264\) 1.20085e13 0.576327
\(265\) −1.63271e9 −7.67460e−5 0
\(266\) 4.80366e13 2.21168
\(267\) −1.61386e13 −0.727869
\(268\) −3.91740e12 −0.173084
\(269\) −6.18550e12 −0.267755 −0.133877 0.990998i \(-0.542743\pi\)
−0.133877 + 0.990998i \(0.542743\pi\)
\(270\) −8.91486e10 −0.00378106
\(271\) −9.42813e12 −0.391827 −0.195914 0.980621i \(-0.562767\pi\)
−0.195914 + 0.980621i \(0.562767\pi\)
\(272\) −1.24911e13 −0.508714
\(273\) −4.07327e13 −1.62573
\(274\) −2.90881e13 −1.13785
\(275\) −2.40320e13 −0.921424
\(276\) −3.93421e11 −0.0147862
\(277\) −1.50680e13 −0.555160 −0.277580 0.960703i \(-0.589532\pi\)
−0.277580 + 0.960703i \(0.589532\pi\)
\(278\) −9.50478e12 −0.343317
\(279\) −3.99347e12 −0.141425
\(280\) −1.11752e12 −0.0388049
\(281\) −3.61256e13 −1.23007 −0.615035 0.788500i \(-0.710858\pi\)
−0.615035 + 0.788500i \(0.710858\pi\)
\(282\) 6.81138e12 0.227439
\(283\) −7.38949e11 −0.0241985 −0.0120993 0.999927i \(-0.503851\pi\)
−0.0120993 + 0.999927i \(0.503851\pi\)
\(284\) −7.44973e11 −0.0239271
\(285\) 6.74623e11 0.0212527
\(286\) −4.60533e13 −1.42314
\(287\) 1.86422e13 0.565127
\(288\) −5.49774e12 −0.163503
\(289\) −1.53562e13 −0.448069
\(290\) 2.97752e11 0.00852445
\(291\) 1.57816e13 0.443343
\(292\) −6.95380e12 −0.191698
\(293\) 5.28326e13 1.42932 0.714661 0.699471i \(-0.246581\pi\)
0.714661 + 0.699471i \(0.246581\pi\)
\(294\) −2.80125e13 −0.743776
\(295\) 1.13473e11 0.00295713
\(296\) −2.35982e13 −0.603635
\(297\) 7.06582e12 0.177420
\(298\) −5.98958e13 −1.47641
\(299\) 7.50003e12 0.181498
\(300\) 6.11636e12 0.145320
\(301\) −3.84398e13 −0.896738
\(302\) 4.12415e13 0.944704
\(303\) 3.23194e13 0.726990
\(304\) −5.02359e13 −1.10971
\(305\) −1.47770e11 −0.00320580
\(306\) −1.00528e13 −0.214201
\(307\) 5.30662e13 1.11060 0.555299 0.831651i \(-0.312604\pi\)
0.555299 + 0.831651i \(0.312604\pi\)
\(308\) 1.78184e13 0.366302
\(309\) 3.69673e13 0.746529
\(310\) 4.20178e11 0.00833574
\(311\) 4.95431e13 0.965609 0.482804 0.875728i \(-0.339618\pi\)
0.482804 + 0.875728i \(0.339618\pi\)
\(312\) 5.82639e13 1.11570
\(313\) 4.02674e13 0.757634 0.378817 0.925472i \(-0.376331\pi\)
0.378817 + 0.925472i \(0.376331\pi\)
\(314\) 5.99549e13 1.10844
\(315\) −6.57549e11 −0.0119459
\(316\) −1.08909e13 −0.194439
\(317\) 2.96572e13 0.520360 0.260180 0.965560i \(-0.416218\pi\)
0.260180 + 0.965560i \(0.416218\pi\)
\(318\) 9.78470e10 0.00168732
\(319\) −2.35995e13 −0.399995
\(320\) 1.51203e12 0.0251905
\(321\) −3.05310e13 −0.499992
\(322\) 8.62103e12 0.138788
\(323\) 7.60738e13 1.20398
\(324\) −1.79832e12 −0.0279813
\(325\) −1.16600e14 −1.78377
\(326\) 2.08499e13 0.313623
\(327\) −5.51488e12 −0.0815687
\(328\) −2.66657e13 −0.387835
\(329\) 5.02399e13 0.718574
\(330\) −7.43439e11 −0.0104573
\(331\) −5.87214e13 −0.812349 −0.406174 0.913796i \(-0.633137\pi\)
−0.406174 + 0.913796i \(0.633137\pi\)
\(332\) 2.27015e13 0.308883
\(333\) −1.38852e13 −0.185827
\(334\) −8.62074e13 −1.13485
\(335\) 1.20556e12 0.0156114
\(336\) 4.89645e13 0.623757
\(337\) 8.21322e13 1.02932 0.514659 0.857395i \(-0.327919\pi\)
0.514659 + 0.857395i \(0.327919\pi\)
\(338\) −1.53292e14 −1.89007
\(339\) 1.80526e13 0.218999
\(340\) −3.56027e11 −0.00424961
\(341\) −3.33028e13 −0.391140
\(342\) −4.04297e13 −0.467258
\(343\) −6.78894e13 −0.772119
\(344\) 5.49842e13 0.615412
\(345\) 1.21073e11 0.00133365
\(346\) −9.62892e13 −1.04390
\(347\) 7.99022e13 0.852602 0.426301 0.904581i \(-0.359816\pi\)
0.426301 + 0.904581i \(0.359816\pi\)
\(348\) 6.00630e12 0.0630843
\(349\) 5.57237e13 0.576103 0.288051 0.957615i \(-0.406993\pi\)
0.288051 + 0.957615i \(0.406993\pi\)
\(350\) −1.34028e14 −1.36402
\(351\) 3.42824e13 0.343465
\(352\) −4.58474e13 −0.452200
\(353\) 2.40830e13 0.233857 0.116928 0.993140i \(-0.462695\pi\)
0.116928 + 0.993140i \(0.462695\pi\)
\(354\) −6.80034e12 −0.0650149
\(355\) 2.29261e11 0.00215812
\(356\) 3.42531e13 0.317486
\(357\) −7.41484e13 −0.676747
\(358\) 4.93670e13 0.443691
\(359\) −1.66898e14 −1.47717 −0.738587 0.674158i \(-0.764507\pi\)
−0.738587 + 0.674158i \(0.764507\pi\)
\(360\) 9.40555e11 0.00819824
\(361\) 1.89457e14 1.62638
\(362\) −1.55875e14 −1.31789
\(363\) −1.04066e13 −0.0866607
\(364\) 8.64525e13 0.709121
\(365\) 2.13999e12 0.0172903
\(366\) 8.85576e12 0.0704823
\(367\) 1.58476e13 0.124251 0.0621254 0.998068i \(-0.480212\pi\)
0.0621254 + 0.998068i \(0.480212\pi\)
\(368\) −9.01574e12 −0.0696368
\(369\) −1.56901e13 −0.119393
\(370\) 1.46095e12 0.0109528
\(371\) 7.21707e11 0.00533095
\(372\) 8.47589e12 0.0616878
\(373\) −3.37056e13 −0.241715 −0.120857 0.992670i \(-0.538564\pi\)
−0.120857 + 0.992670i \(0.538564\pi\)
\(374\) −8.38338e13 −0.592414
\(375\) −3.76552e12 −0.0262213
\(376\) −7.18630e13 −0.493142
\(377\) −1.14502e14 −0.774346
\(378\) 3.94065e13 0.262641
\(379\) −7.12872e13 −0.468270 −0.234135 0.972204i \(-0.575226\pi\)
−0.234135 + 0.972204i \(0.575226\pi\)
\(380\) −1.43184e12 −0.00927013
\(381\) −1.52373e14 −0.972347
\(382\) −2.45280e14 −1.54282
\(383\) 1.55879e14 0.966485 0.483243 0.875486i \(-0.339459\pi\)
0.483243 + 0.875486i \(0.339459\pi\)
\(384\) −4.42803e13 −0.270638
\(385\) −5.48351e12 −0.0330388
\(386\) −5.59460e13 −0.332307
\(387\) 3.23526e13 0.189452
\(388\) −3.34955e13 −0.193380
\(389\) 2.37267e13 0.135056 0.0675280 0.997717i \(-0.478489\pi\)
0.0675280 + 0.997717i \(0.478489\pi\)
\(390\) −3.60707e12 −0.0202441
\(391\) 1.36528e13 0.0755527
\(392\) 2.95544e14 1.61268
\(393\) −4.55400e13 −0.245038
\(394\) 8.89730e13 0.472094
\(395\) 3.35161e12 0.0175376
\(396\) −1.49967e13 −0.0773880
\(397\) 2.57979e14 1.31291 0.656457 0.754364i \(-0.272054\pi\)
0.656457 + 0.754364i \(0.272054\pi\)
\(398\) −3.17432e14 −1.59329
\(399\) −2.98204e14 −1.47626
\(400\) 1.40164e14 0.684395
\(401\) 3.04127e14 1.46474 0.732370 0.680907i \(-0.238414\pi\)
0.732370 + 0.680907i \(0.238414\pi\)
\(402\) −7.22483e13 −0.343229
\(403\) −1.61581e14 −0.757204
\(404\) −6.85958e13 −0.317103
\(405\) 5.53422e11 0.00252379
\(406\) −1.31616e14 −0.592127
\(407\) −1.15793e14 −0.513940
\(408\) 1.06062e14 0.464437
\(409\) 2.04430e14 0.883213 0.441606 0.897209i \(-0.354409\pi\)
0.441606 + 0.897209i \(0.354409\pi\)
\(410\) 1.65085e12 0.00703715
\(411\) 1.80575e14 0.759498
\(412\) −7.84609e13 −0.325625
\(413\) −5.01585e13 −0.205409
\(414\) −7.25584e12 −0.0293215
\(415\) −6.98625e12 −0.0278599
\(416\) −2.22446e14 −0.875409
\(417\) 5.90043e13 0.229158
\(418\) −3.37157e14 −1.29230
\(419\) −3.11752e14 −1.17932 −0.589660 0.807652i \(-0.700738\pi\)
−0.589660 + 0.807652i \(0.700738\pi\)
\(420\) 1.39561e12 0.00521064
\(421\) 2.19862e14 0.810211 0.405105 0.914270i \(-0.367235\pi\)
0.405105 + 0.914270i \(0.367235\pi\)
\(422\) 3.41408e14 1.24181
\(423\) −4.22841e13 −0.151812
\(424\) −1.03233e12 −0.00365852
\(425\) −2.12255e14 −0.742537
\(426\) −1.37395e13 −0.0474480
\(427\) 6.53189e13 0.222682
\(428\) 6.48000e13 0.218089
\(429\) 2.85892e14 0.949921
\(430\) −3.40403e12 −0.0111665
\(431\) 5.72745e14 1.85497 0.927484 0.373864i \(-0.121967\pi\)
0.927484 + 0.373864i \(0.121967\pi\)
\(432\) −4.12107e13 −0.131780
\(433\) −4.52318e14 −1.42811 −0.714054 0.700091i \(-0.753143\pi\)
−0.714054 + 0.700091i \(0.753143\pi\)
\(434\) −1.85732e14 −0.579019
\(435\) −1.84840e12 −0.00568992
\(436\) 1.17050e13 0.0355791
\(437\) 5.49078e13 0.164811
\(438\) −1.28248e14 −0.380141
\(439\) 2.80722e14 0.821716 0.410858 0.911699i \(-0.365229\pi\)
0.410858 + 0.911699i \(0.365229\pi\)
\(440\) 7.84360e12 0.0226739
\(441\) 1.73898e14 0.496457
\(442\) −4.06751e14 −1.14685
\(443\) 4.29346e14 1.19560 0.597802 0.801644i \(-0.296041\pi\)
0.597802 + 0.801644i \(0.296041\pi\)
\(444\) 2.94704e13 0.0810549
\(445\) −1.05412e13 −0.0286359
\(446\) 3.87691e14 1.04027
\(447\) 3.71824e14 0.985477
\(448\) −6.68366e14 −1.74979
\(449\) −3.52063e14 −0.910470 −0.455235 0.890371i \(-0.650445\pi\)
−0.455235 + 0.890371i \(0.650445\pi\)
\(450\) 1.12804e14 0.288174
\(451\) −1.30845e14 −0.330206
\(452\) −3.83155e13 −0.0955242
\(453\) −2.56021e14 −0.630573
\(454\) −3.84201e14 −0.934869
\(455\) −2.66053e13 −0.0639595
\(456\) 4.26550e14 1.01313
\(457\) −1.86469e14 −0.437590 −0.218795 0.975771i \(-0.570213\pi\)
−0.218795 + 0.975771i \(0.570213\pi\)
\(458\) −1.75575e14 −0.407102
\(459\) 6.24065e13 0.142975
\(460\) −2.56970e11 −0.000581721 0
\(461\) −2.25969e14 −0.505469 −0.252734 0.967536i \(-0.581330\pi\)
−0.252734 + 0.967536i \(0.581330\pi\)
\(462\) 3.28623e14 0.726386
\(463\) 6.99333e14 1.52753 0.763763 0.645497i \(-0.223350\pi\)
0.763763 + 0.645497i \(0.223350\pi\)
\(464\) 1.37642e14 0.297099
\(465\) −2.60841e12 −0.00556396
\(466\) 7.55111e14 1.59180
\(467\) 7.18639e13 0.149716 0.0748580 0.997194i \(-0.476150\pi\)
0.0748580 + 0.997194i \(0.476150\pi\)
\(468\) −7.27623e13 −0.149815
\(469\) −5.32894e14 −1.08440
\(470\) 4.44898e12 0.00894792
\(471\) −3.72191e14 −0.739863
\(472\) 7.17465e13 0.140968
\(473\) 2.69799e14 0.523968
\(474\) −2.00860e14 −0.385578
\(475\) −8.53630e14 −1.61977
\(476\) 1.57375e14 0.295188
\(477\) −6.07421e11 −0.00112626
\(478\) −7.43992e14 −1.36369
\(479\) 6.90841e13 0.125179 0.0625897 0.998039i \(-0.480064\pi\)
0.0625897 + 0.998039i \(0.480064\pi\)
\(480\) −3.59095e12 −0.00643253
\(481\) −5.61812e14 −0.994931
\(482\) 7.18992e14 1.25883
\(483\) −5.35181e13 −0.0926385
\(484\) 2.20873e13 0.0378002
\(485\) 1.03080e13 0.0174420
\(486\) −3.31662e13 −0.0554877
\(487\) 6.50107e14 1.07541 0.537707 0.843132i \(-0.319291\pi\)
0.537707 + 0.843132i \(0.319291\pi\)
\(488\) −9.34320e13 −0.152822
\(489\) −1.29433e14 −0.209338
\(490\) −1.82969e13 −0.0292617
\(491\) 1.02957e15 1.62820 0.814102 0.580722i \(-0.197230\pi\)
0.814102 + 0.580722i \(0.197230\pi\)
\(492\) 3.33012e13 0.0520777
\(493\) −2.08435e14 −0.322339
\(494\) −1.63584e15 −2.50174
\(495\) 4.61516e12 0.00698005
\(496\) 1.94236e14 0.290522
\(497\) −1.01341e14 −0.149908
\(498\) 4.18681e14 0.612523
\(499\) −7.05829e14 −1.02128 −0.510642 0.859793i \(-0.670592\pi\)
−0.510642 + 0.859793i \(0.670592\pi\)
\(500\) 7.99209e12 0.0114373
\(501\) 5.35163e14 0.757493
\(502\) 5.77059e14 0.807885
\(503\) −2.56115e14 −0.354659 −0.177330 0.984152i \(-0.556746\pi\)
−0.177330 + 0.984152i \(0.556746\pi\)
\(504\) −4.15755e14 −0.569468
\(505\) 2.11100e13 0.0286013
\(506\) −6.05088e13 −0.0810944
\(507\) 9.51617e14 1.26159
\(508\) 3.23403e14 0.424124
\(509\) 1.10840e15 1.43796 0.718982 0.695029i \(-0.244609\pi\)
0.718982 + 0.695029i \(0.244609\pi\)
\(510\) −6.56619e12 −0.00842708
\(511\) −9.45944e14 −1.20102
\(512\) 8.57685e14 1.07732
\(513\) 2.50982e14 0.311887
\(514\) 7.84389e14 0.964350
\(515\) 2.41459e13 0.0293700
\(516\) −6.86664e13 −0.0826364
\(517\) −3.52621e14 −0.419865
\(518\) −6.45784e14 −0.760805
\(519\) 5.97750e14 0.696785
\(520\) 3.80561e13 0.0438941
\(521\) −2.81370e14 −0.321122 −0.160561 0.987026i \(-0.551330\pi\)
−0.160561 + 0.987026i \(0.551330\pi\)
\(522\) 1.10774e14 0.125098
\(523\) 6.77347e14 0.756923 0.378462 0.925617i \(-0.376453\pi\)
0.378462 + 0.925617i \(0.376453\pi\)
\(524\) 9.66557e13 0.106882
\(525\) 8.32025e14 0.910459
\(526\) −6.24110e14 −0.675833
\(527\) −2.94137e14 −0.315203
\(528\) −3.43669e14 −0.364464
\(529\) −9.42956e14 −0.989658
\(530\) 6.39106e10 6.63828e−5 0
\(531\) 4.22156e13 0.0433963
\(532\) 6.32920e14 0.643924
\(533\) −6.34841e14 −0.639242
\(534\) 6.31728e14 0.629583
\(535\) −1.99418e13 −0.0196707
\(536\) 7.62250e14 0.744203
\(537\) −3.06464e14 −0.296156
\(538\) 2.42125e14 0.231599
\(539\) 1.45019e15 1.37305
\(540\) −1.17460e12 −0.00110084
\(541\) 1.38190e15 1.28201 0.641003 0.767538i \(-0.278519\pi\)
0.641003 + 0.767538i \(0.278519\pi\)
\(542\) 3.69054e14 0.338918
\(543\) 9.67648e14 0.879667
\(544\) −4.04933e14 −0.364409
\(545\) −3.60214e12 −0.00320908
\(546\) 1.59444e15 1.40620
\(547\) −6.01823e13 −0.0525458 −0.0262729 0.999655i \(-0.508364\pi\)
−0.0262729 + 0.999655i \(0.508364\pi\)
\(548\) −3.83258e14 −0.331282
\(549\) −5.49753e13 −0.0470457
\(550\) 9.40706e14 0.797002
\(551\) −8.38269e14 −0.703152
\(552\) 7.65521e13 0.0635759
\(553\) −1.48152e15 −1.21820
\(554\) 5.89822e14 0.480195
\(555\) −9.06934e12 −0.00731079
\(556\) −1.25233e14 −0.0999556
\(557\) −7.65629e14 −0.605083 −0.302541 0.953136i \(-0.597835\pi\)
−0.302541 + 0.953136i \(0.597835\pi\)
\(558\) 1.56320e14 0.122328
\(559\) 1.30903e15 1.01434
\(560\) 3.19820e13 0.0245399
\(561\) 5.20429e14 0.395426
\(562\) 1.41410e15 1.06397
\(563\) −7.33817e14 −0.546754 −0.273377 0.961907i \(-0.588141\pi\)
−0.273377 + 0.961907i \(0.588141\pi\)
\(564\) 8.97453e13 0.0662181
\(565\) 1.17914e13 0.00861586
\(566\) 2.89254e13 0.0209309
\(567\) −2.44630e14 −0.175308
\(568\) 1.44957e14 0.102878
\(569\) −9.71035e14 −0.682523 −0.341261 0.939968i \(-0.610854\pi\)
−0.341261 + 0.939968i \(0.610854\pi\)
\(570\) −2.64074e13 −0.0183829
\(571\) 1.96144e15 1.35231 0.676154 0.736761i \(-0.263646\pi\)
0.676154 + 0.736761i \(0.263646\pi\)
\(572\) −6.06788e14 −0.414342
\(573\) 1.52267e15 1.02980
\(574\) −7.29728e14 −0.488817
\(575\) −1.53199e14 −0.101644
\(576\) 5.62526e14 0.369674
\(577\) −1.81907e12 −0.00118408 −0.000592041 1.00000i \(-0.500188\pi\)
−0.000592041 1.00000i \(0.500188\pi\)
\(578\) 6.01102e14 0.387565
\(579\) 3.47305e14 0.221809
\(580\) 3.92312e12 0.00248186
\(581\) 3.08814e15 1.93521
\(582\) −6.17754e14 −0.383477
\(583\) −5.06548e12 −0.00311489
\(584\) 1.35308e15 0.824235
\(585\) 2.23922e13 0.0135126
\(586\) −2.06807e15 −1.23632
\(587\) 5.42865e14 0.321501 0.160751 0.986995i \(-0.448609\pi\)
0.160751 + 0.986995i \(0.448609\pi\)
\(588\) −3.69087e14 −0.216548
\(589\) −1.18294e15 −0.687586
\(590\) −4.44177e12 −0.00255782
\(591\) −5.52331e14 −0.315115
\(592\) 6.75351e14 0.381733
\(593\) −1.05930e15 −0.593226 −0.296613 0.954998i \(-0.595857\pi\)
−0.296613 + 0.954998i \(0.595857\pi\)
\(594\) −2.76584e14 −0.153462
\(595\) −4.84314e13 −0.0266246
\(596\) −7.89174e14 −0.429851
\(597\) 1.97057e15 1.06349
\(598\) −2.93581e14 −0.156990
\(599\) 7.90466e14 0.418828 0.209414 0.977827i \(-0.432844\pi\)
0.209414 + 0.977827i \(0.432844\pi\)
\(600\) −1.19013e15 −0.624829
\(601\) 1.21619e15 0.632692 0.316346 0.948644i \(-0.397544\pi\)
0.316346 + 0.948644i \(0.397544\pi\)
\(602\) 1.50469e15 0.775649
\(603\) 4.48507e14 0.229100
\(604\) 5.43389e14 0.275047
\(605\) −6.79725e12 −0.00340941
\(606\) −1.26511e15 −0.628823
\(607\) −1.68034e15 −0.827675 −0.413838 0.910351i \(-0.635812\pi\)
−0.413838 + 0.910351i \(0.635812\pi\)
\(608\) −1.62853e15 −0.794924
\(609\) 8.17053e14 0.395235
\(610\) 5.78430e12 0.00277291
\(611\) −1.71087e15 −0.812813
\(612\) −1.32454e14 −0.0623637
\(613\) −3.94994e15 −1.84314 −0.921569 0.388214i \(-0.873092\pi\)
−0.921569 + 0.388214i \(0.873092\pi\)
\(614\) −2.07722e15 −0.960631
\(615\) −1.02483e13 −0.00469718
\(616\) −3.46711e15 −1.57498
\(617\) −2.85141e15 −1.28378 −0.641892 0.766795i \(-0.721850\pi\)
−0.641892 + 0.766795i \(0.721850\pi\)
\(618\) −1.44705e15 −0.645723
\(619\) −4.06614e14 −0.179839 −0.0899194 0.995949i \(-0.528661\pi\)
−0.0899194 + 0.995949i \(0.528661\pi\)
\(620\) 5.53618e12 0.00242692
\(621\) 4.50432e13 0.0195716
\(622\) −1.93931e15 −0.835220
\(623\) 4.65954e15 1.98911
\(624\) −1.66744e15 −0.705561
\(625\) 2.38050e15 0.998452
\(626\) −1.57622e15 −0.655328
\(627\) 2.09302e15 0.862585
\(628\) 7.89953e14 0.322718
\(629\) −1.02270e15 −0.414163
\(630\) 2.57390e13 0.0103328
\(631\) 8.63879e14 0.343789 0.171894 0.985115i \(-0.445011\pi\)
0.171894 + 0.985115i \(0.445011\pi\)
\(632\) 2.11915e15 0.836024
\(633\) −2.11941e15 −0.828885
\(634\) −1.16090e15 −0.450094
\(635\) −9.95253e13 −0.0382541
\(636\) 1.28921e12 0.000491259 0
\(637\) 7.03614e15 2.65808
\(638\) 9.23778e14 0.345982
\(639\) 8.52928e13 0.0316707
\(640\) −2.89225e13 −0.0106475
\(641\) 1.73480e15 0.633184 0.316592 0.948562i \(-0.397461\pi\)
0.316592 + 0.948562i \(0.397461\pi\)
\(642\) 1.19510e15 0.432476
\(643\) 3.12183e15 1.12008 0.560039 0.828466i \(-0.310786\pi\)
0.560039 + 0.828466i \(0.310786\pi\)
\(644\) 1.13589e14 0.0404076
\(645\) 2.11317e13 0.00745343
\(646\) −2.97783e15 −1.04141
\(647\) 4.04414e15 1.40234 0.701168 0.712996i \(-0.252662\pi\)
0.701168 + 0.712996i \(0.252662\pi\)
\(648\) 3.49917e14 0.120310
\(649\) 3.52049e14 0.120021
\(650\) 4.56418e15 1.54291
\(651\) 1.15300e15 0.386485
\(652\) 2.74714e14 0.0913102
\(653\) 5.64810e15 1.86157 0.930787 0.365563i \(-0.119124\pi\)
0.930787 + 0.365563i \(0.119124\pi\)
\(654\) 2.15874e14 0.0705542
\(655\) −2.97453e13 −0.00964030
\(656\) 7.63139e14 0.245263
\(657\) 7.96148e14 0.253737
\(658\) −1.96659e15 −0.621542
\(659\) −4.96751e14 −0.155693 −0.0778465 0.996965i \(-0.524804\pi\)
−0.0778465 + 0.996965i \(0.524804\pi\)
\(660\) −9.79540e12 −0.00304460
\(661\) −5.54252e15 −1.70844 −0.854218 0.519914i \(-0.825964\pi\)
−0.854218 + 0.519914i \(0.825964\pi\)
\(662\) 2.29859e15 0.702655
\(663\) 2.52505e15 0.765501
\(664\) −4.41726e15 −1.32809
\(665\) −1.94778e14 −0.0580791
\(666\) 5.43520e14 0.160734
\(667\) −1.50442e14 −0.0441243
\(668\) −1.13585e15 −0.330408
\(669\) −2.40673e15 −0.694359
\(670\) −4.71903e13 −0.0135033
\(671\) −4.58457e14 −0.130114
\(672\) 1.58731e15 0.446818
\(673\) −1.45891e15 −0.407331 −0.203665 0.979041i \(-0.565285\pi\)
−0.203665 + 0.979041i \(0.565285\pi\)
\(674\) −3.21498e15 −0.890325
\(675\) −7.00269e14 −0.192351
\(676\) −2.01975e15 −0.550288
\(677\) −2.10073e14 −0.0567717 −0.0283858 0.999597i \(-0.509037\pi\)
−0.0283858 + 0.999597i \(0.509037\pi\)
\(678\) −7.06650e14 −0.189427
\(679\) −4.55648e15 −1.21156
\(680\) 6.92760e13 0.0182719
\(681\) 2.38506e15 0.624009
\(682\) 1.30360e15 0.338323
\(683\) −1.08298e15 −0.278810 −0.139405 0.990235i \(-0.544519\pi\)
−0.139405 + 0.990235i \(0.544519\pi\)
\(684\) −5.32693e14 −0.136041
\(685\) 1.17946e14 0.0298802
\(686\) 2.65746e15 0.667857
\(687\) 1.08995e15 0.271733
\(688\) −1.57358e15 −0.389181
\(689\) −2.45770e13 −0.00603009
\(690\) −4.73928e12 −0.00115357
\(691\) 2.04378e15 0.493520 0.246760 0.969077i \(-0.420634\pi\)
0.246760 + 0.969077i \(0.420634\pi\)
\(692\) −1.26869e15 −0.303928
\(693\) −2.04005e15 −0.484850
\(694\) −3.12769e15 −0.737473
\(695\) 3.85397e13 0.00901555
\(696\) −1.16871e15 −0.271241
\(697\) −1.15564e15 −0.266099
\(698\) −2.18124e15 −0.498310
\(699\) −4.68762e15 −1.06250
\(700\) −1.76592e15 −0.397129
\(701\) −6.80781e15 −1.51900 −0.759501 0.650507i \(-0.774557\pi\)
−0.759501 + 0.650507i \(0.774557\pi\)
\(702\) −1.34195e15 −0.297086
\(703\) −4.11303e15 −0.903457
\(704\) 4.69109e15 1.02241
\(705\) −2.76186e13 −0.00597258
\(706\) −9.42704e14 −0.202278
\(707\) −9.33127e15 −1.98671
\(708\) −8.95999e13 −0.0189289
\(709\) 2.71429e15 0.568987 0.284493 0.958678i \(-0.408175\pi\)
0.284493 + 0.958678i \(0.408175\pi\)
\(710\) −8.97419e12 −0.00186670
\(711\) 1.24691e15 0.257366
\(712\) −6.66499e15 −1.36509
\(713\) −2.12299e14 −0.0431475
\(714\) 2.90246e15 0.585364
\(715\) 1.86736e14 0.0373718
\(716\) 6.50450e14 0.129179
\(717\) 4.61860e15 0.910237
\(718\) 6.53305e15 1.27771
\(719\) −5.42045e15 −1.05203 −0.526013 0.850477i \(-0.676314\pi\)
−0.526013 + 0.850477i \(0.676314\pi\)
\(720\) −2.69175e13 −0.00518449
\(721\) −1.06732e16 −2.04010
\(722\) −7.41611e15 −1.40676
\(723\) −4.46340e15 −0.840244
\(724\) −2.05377e15 −0.383699
\(725\) 2.33887e15 0.433657
\(726\) 4.07355e14 0.0749586
\(727\) −4.59899e15 −0.839891 −0.419946 0.907549i \(-0.637951\pi\)
−0.419946 + 0.907549i \(0.637951\pi\)
\(728\) −1.68220e16 −3.04898
\(729\) 2.05891e14 0.0370370
\(730\) −8.37678e13 −0.0149555
\(731\) 2.38291e15 0.422243
\(732\) 1.16682e14 0.0205207
\(733\) −8.40465e15 −1.46706 −0.733530 0.679657i \(-0.762129\pi\)
−0.733530 + 0.679657i \(0.762129\pi\)
\(734\) −6.20336e14 −0.107473
\(735\) 1.13585e14 0.0195316
\(736\) −2.92269e14 −0.0498832
\(737\) 3.74025e15 0.633621
\(738\) 6.14172e14 0.103271
\(739\) 9.76215e15 1.62930 0.814650 0.579953i \(-0.196929\pi\)
0.814650 + 0.579953i \(0.196929\pi\)
\(740\) 1.92491e13 0.00318887
\(741\) 1.01551e16 1.66987
\(742\) −2.82505e13 −0.00461110
\(743\) −1.02184e16 −1.65556 −0.827781 0.561052i \(-0.810397\pi\)
−0.827781 + 0.561052i \(0.810397\pi\)
\(744\) −1.64924e15 −0.265237
\(745\) 2.42864e14 0.0387707
\(746\) 1.31937e15 0.209075
\(747\) −2.59911e15 −0.408848
\(748\) −1.10458e15 −0.172479
\(749\) 8.81492e15 1.36637
\(750\) 1.47397e14 0.0226805
\(751\) −8.36434e14 −0.127765 −0.0638825 0.997957i \(-0.520348\pi\)
−0.0638825 + 0.997957i \(0.520348\pi\)
\(752\) 2.05663e15 0.311858
\(753\) −3.58230e15 −0.539249
\(754\) 4.48205e15 0.669783
\(755\) −1.67225e14 −0.0248080
\(756\) 5.19211e14 0.0764670
\(757\) −3.94834e15 −0.577280 −0.288640 0.957438i \(-0.593203\pi\)
−0.288640 + 0.957438i \(0.593203\pi\)
\(758\) 2.79046e15 0.405038
\(759\) 3.75630e14 0.0541291
\(760\) 2.78609e14 0.0398585
\(761\) 3.51564e15 0.499332 0.249666 0.968332i \(-0.419679\pi\)
0.249666 + 0.968332i \(0.419679\pi\)
\(762\) 5.96449e15 0.841048
\(763\) 1.59226e15 0.222910
\(764\) −3.23176e15 −0.449186
\(765\) 4.07620e13 0.00562493
\(766\) −6.10173e15 −0.835978
\(767\) 1.70810e15 0.232348
\(768\) −3.00765e15 −0.406201
\(769\) −6.49784e15 −0.871313 −0.435657 0.900113i \(-0.643484\pi\)
−0.435657 + 0.900113i \(0.643484\pi\)
\(770\) 2.14646e14 0.0285775
\(771\) −4.86938e15 −0.643687
\(772\) −7.37133e14 −0.0967499
\(773\) 1.12344e16 1.46408 0.732039 0.681263i \(-0.238569\pi\)
0.732039 + 0.681263i \(0.238569\pi\)
\(774\) −1.26641e15 −0.163870
\(775\) 3.30053e15 0.424057
\(776\) 6.51757e15 0.831469
\(777\) 4.00893e15 0.507824
\(778\) −9.28756e14 −0.116819
\(779\) −4.64768e15 −0.580471
\(780\) −4.75260e13 −0.00589401
\(781\) 7.11284e14 0.0875916
\(782\) −5.34425e14 −0.0653506
\(783\) −6.87667e14 −0.0835004
\(784\) −8.45810e15 −1.01985
\(785\) −2.43104e14 −0.0291077
\(786\) 1.78261e15 0.211950
\(787\) −6.35498e15 −0.750331 −0.375166 0.926958i \(-0.622414\pi\)
−0.375166 + 0.926958i \(0.622414\pi\)
\(788\) 1.17229e15 0.137449
\(789\) 3.87438e15 0.451107
\(790\) −1.31195e14 −0.0151694
\(791\) −5.21216e15 −0.598477
\(792\) 2.91808e15 0.332742
\(793\) −2.22437e15 −0.251887
\(794\) −1.00983e16 −1.13563
\(795\) −3.96748e11 −4.43093e−5 0
\(796\) −4.18242e15 −0.463880
\(797\) 1.65088e16 1.81842 0.909209 0.416340i \(-0.136687\pi\)
0.909209 + 0.416340i \(0.136687\pi\)
\(798\) 1.16729e16 1.27692
\(799\) −3.11441e15 −0.338352
\(800\) 4.54378e15 0.490256
\(801\) −3.92168e15 −0.420236
\(802\) −1.19047e16 −1.26695
\(803\) 6.63934e15 0.701761
\(804\) −9.51928e14 −0.0999300
\(805\) −3.49563e13 −0.00364459
\(806\) 6.32492e15 0.654956
\(807\) −1.50308e15 −0.154588
\(808\) 1.33474e16 1.36344
\(809\) 1.16065e16 1.17757 0.588784 0.808290i \(-0.299607\pi\)
0.588784 + 0.808290i \(0.299607\pi\)
\(810\) −2.16631e13 −0.00218300
\(811\) −1.31655e16 −1.31772 −0.658859 0.752266i \(-0.728961\pi\)
−0.658859 + 0.752266i \(0.728961\pi\)
\(812\) −1.73414e15 −0.172396
\(813\) −2.29104e15 −0.226222
\(814\) 4.53259e15 0.444541
\(815\) −8.45417e13 −0.00823577
\(816\) −3.03535e15 −0.293706
\(817\) 9.58343e15 0.921084
\(818\) −8.00218e15 −0.763950
\(819\) −9.89804e15 −0.938616
\(820\) 2.17513e13 0.00204884
\(821\) 1.73884e16 1.62695 0.813473 0.581603i \(-0.197574\pi\)
0.813473 + 0.581603i \(0.197574\pi\)
\(822\) −7.06840e15 −0.656941
\(823\) −8.90289e15 −0.821925 −0.410962 0.911652i \(-0.634807\pi\)
−0.410962 + 0.911652i \(0.634807\pi\)
\(824\) 1.52670e16 1.40008
\(825\) −5.83977e15 −0.531985
\(826\) 1.96340e15 0.177672
\(827\) −4.62970e15 −0.416172 −0.208086 0.978111i \(-0.566723\pi\)
−0.208086 + 0.978111i \(0.566723\pi\)
\(828\) −9.56014e13 −0.00853684
\(829\) 1.14604e16 1.01660 0.508299 0.861181i \(-0.330275\pi\)
0.508299 + 0.861181i \(0.330275\pi\)
\(830\) 2.73469e14 0.0240979
\(831\) −3.66153e15 −0.320522
\(832\) 2.27605e16 1.97927
\(833\) 1.28083e16 1.10648
\(834\) −2.30966e15 −0.198214
\(835\) 3.49552e14 0.0298013
\(836\) −4.44230e15 −0.376247
\(837\) −9.70413e14 −0.0816520
\(838\) 1.22032e16 1.02007
\(839\) 7.27881e15 0.604463 0.302232 0.953235i \(-0.402268\pi\)
0.302232 + 0.953235i \(0.402268\pi\)
\(840\) −2.71558e14 −0.0224040
\(841\) −9.90373e15 −0.811747
\(842\) −8.60625e15 −0.700806
\(843\) −8.77852e15 −0.710182
\(844\) 4.49831e15 0.361548
\(845\) 6.21566e14 0.0496335
\(846\) 1.65517e15 0.131312
\(847\) 3.00460e15 0.236825
\(848\) 2.95439e13 0.00231361
\(849\) −1.79565e14 −0.0139710
\(850\) 8.30848e15 0.642270
\(851\) −7.38157e14 −0.0566939
\(852\) −1.81029e14 −0.0138143
\(853\) −2.24056e16 −1.69878 −0.849390 0.527766i \(-0.823030\pi\)
−0.849390 + 0.527766i \(0.823030\pi\)
\(854\) −2.55684e15 −0.192613
\(855\) 1.63933e14 0.0122703
\(856\) −1.26088e16 −0.937711
\(857\) 2.08862e16 1.54336 0.771678 0.636014i \(-0.219418\pi\)
0.771678 + 0.636014i \(0.219418\pi\)
\(858\) −1.11909e16 −0.821650
\(859\) 1.61887e16 1.18100 0.590500 0.807038i \(-0.298931\pi\)
0.590500 + 0.807038i \(0.298931\pi\)
\(860\) −4.48507e13 −0.00325108
\(861\) 4.53005e15 0.326276
\(862\) −2.24195e16 −1.60449
\(863\) −2.12574e16 −1.51165 −0.755823 0.654776i \(-0.772763\pi\)
−0.755823 + 0.654776i \(0.772763\pi\)
\(864\) −1.33595e15 −0.0943984
\(865\) 3.90431e14 0.0274129
\(866\) 1.77055e16 1.23527
\(867\) −3.73155e15 −0.258693
\(868\) −2.44716e15 −0.168579
\(869\) 1.03984e16 0.711798
\(870\) 7.23538e13 0.00492159
\(871\) 1.81472e16 1.22662
\(872\) −2.27756e15 −0.152978
\(873\) 3.83493e15 0.255964
\(874\) −2.14931e15 −0.142556
\(875\) 1.08718e15 0.0716571
\(876\) −1.68977e15 −0.110677
\(877\) 1.97651e16 1.28647 0.643237 0.765667i \(-0.277591\pi\)
0.643237 + 0.765667i \(0.277591\pi\)
\(878\) −1.09886e16 −0.710757
\(879\) 1.28383e16 0.825219
\(880\) −2.24474e14 −0.0143387
\(881\) −8.05942e14 −0.0511607 −0.0255803 0.999673i \(-0.508143\pi\)
−0.0255803 + 0.999673i \(0.508143\pi\)
\(882\) −6.80705e15 −0.429419
\(883\) 2.51360e16 1.57584 0.787922 0.615775i \(-0.211157\pi\)
0.787922 + 0.615775i \(0.211157\pi\)
\(884\) −5.35926e15 −0.333901
\(885\) 2.75739e13 0.00170730
\(886\) −1.68063e16 −1.03416
\(887\) −1.94110e16 −1.18705 −0.593524 0.804816i \(-0.702264\pi\)
−0.593524 + 0.804816i \(0.702264\pi\)
\(888\) −5.73436e15 −0.348509
\(889\) 4.39933e16 2.65722
\(890\) 4.12624e14 0.0247691
\(891\) 1.71699e15 0.102433
\(892\) 5.10814e15 0.302870
\(893\) −1.25253e16 −0.738083
\(894\) −1.45547e16 −0.852405
\(895\) −2.00172e14 −0.0116514
\(896\) 1.27846e16 0.739596
\(897\) 1.82251e15 0.104788
\(898\) 1.37811e16 0.787526
\(899\) 3.24114e15 0.184085
\(900\) 1.48628e15 0.0839007
\(901\) −4.47392e13 −0.00251016
\(902\) 5.12178e15 0.285617
\(903\) −9.34088e15 −0.517732
\(904\) 7.45546e15 0.410722
\(905\) 6.32037e14 0.0346079
\(906\) 1.00217e16 0.545425
\(907\) 2.00957e16 1.08708 0.543542 0.839382i \(-0.317083\pi\)
0.543542 + 0.839382i \(0.317083\pi\)
\(908\) −5.06215e15 −0.272184
\(909\) 7.85361e15 0.419728
\(910\) 1.04144e15 0.0553229
\(911\) 7.42610e15 0.392112 0.196056 0.980593i \(-0.437187\pi\)
0.196056 + 0.980593i \(0.437187\pi\)
\(912\) −1.22073e16 −0.640692
\(913\) −2.16749e16 −1.13075
\(914\) 7.29913e15 0.378501
\(915\) −3.59081e13 −0.00185087
\(916\) −2.31334e15 −0.118526
\(917\) 1.31483e16 0.669637
\(918\) −2.44284e15 −0.123669
\(919\) 6.17137e15 0.310561 0.155280 0.987870i \(-0.450372\pi\)
0.155280 + 0.987870i \(0.450372\pi\)
\(920\) 5.00014e13 0.00250120
\(921\) 1.28951e16 0.641204
\(922\) 8.84533e15 0.437214
\(923\) 3.45106e15 0.169568
\(924\) 4.32987e15 0.211485
\(925\) 1.14758e16 0.557192
\(926\) −2.73746e16 −1.32126
\(927\) 8.98307e15 0.431009
\(928\) 4.46202e15 0.212822
\(929\) −7.02023e15 −0.332863 −0.166431 0.986053i \(-0.553224\pi\)
−0.166431 + 0.986053i \(0.553224\pi\)
\(930\) 1.02103e14 0.00481264
\(931\) 5.15116e16 2.41369
\(932\) 9.94918e15 0.463446
\(933\) 1.20390e16 0.557494
\(934\) −2.81304e15 −0.129499
\(935\) 3.39927e14 0.0155569
\(936\) 1.41581e16 0.644152
\(937\) −2.63747e16 −1.19294 −0.596472 0.802634i \(-0.703431\pi\)
−0.596472 + 0.802634i \(0.703431\pi\)
\(938\) 2.08596e16 0.937972
\(939\) 9.78497e15 0.437420
\(940\) 5.86188e13 0.00260516
\(941\) −8.37641e15 −0.370096 −0.185048 0.982729i \(-0.559244\pi\)
−0.185048 + 0.982729i \(0.559244\pi\)
\(942\) 1.45690e16 0.639957
\(943\) −8.34110e14 −0.0364258
\(944\) −2.05329e15 −0.0891467
\(945\) −1.59784e14 −0.00689698
\(946\) −1.05610e16 −0.453215
\(947\) −1.64277e16 −0.700895 −0.350447 0.936582i \(-0.613970\pi\)
−0.350447 + 0.936582i \(0.613970\pi\)
\(948\) −2.64648e15 −0.112260
\(949\) 3.22132e16 1.35853
\(950\) 3.34144e16 1.40105
\(951\) 7.20669e15 0.300430
\(952\) −3.06222e16 −1.26921
\(953\) −1.19901e16 −0.494097 −0.247048 0.969003i \(-0.579461\pi\)
−0.247048 + 0.969003i \(0.579461\pi\)
\(954\) 2.37768e13 0.000974177 0
\(955\) 9.94557e14 0.0405145
\(956\) −9.80268e15 −0.397033
\(957\) −5.73468e15 −0.230937
\(958\) −2.70423e15 −0.108276
\(959\) −5.21356e16 −2.07555
\(960\) 3.67424e14 0.0145437
\(961\) −2.08347e16 −0.819990
\(962\) 2.19915e16 0.860583
\(963\) −7.41902e15 −0.288670
\(964\) 9.47329e15 0.366502
\(965\) 2.26848e14 0.00872641
\(966\) 2.09491e15 0.0801293
\(967\) 2.13763e16 0.812995 0.406497 0.913652i \(-0.366750\pi\)
0.406497 + 0.913652i \(0.366750\pi\)
\(968\) −4.29777e15 −0.162528
\(969\) 1.84859e16 0.695121
\(970\) −4.03497e14 −0.0150868
\(971\) 2.09662e16 0.779497 0.389749 0.920921i \(-0.372562\pi\)
0.389749 + 0.920921i \(0.372562\pi\)
\(972\) −4.36991e14 −0.0161550
\(973\) −1.70358e16 −0.626240
\(974\) −2.54478e16 −0.930198
\(975\) −2.83338e16 −1.02986
\(976\) 2.67391e15 0.0966434
\(977\) 4.44124e16 1.59619 0.798095 0.602532i \(-0.205841\pi\)
0.798095 + 0.602532i \(0.205841\pi\)
\(978\) 5.06653e15 0.181070
\(979\) −3.27041e16 −1.16225
\(980\) −2.41076e14 −0.00851943
\(981\) −1.34012e15 −0.0470937
\(982\) −4.03015e16 −1.40834
\(983\) 2.35985e16 0.820051 0.410025 0.912074i \(-0.365520\pi\)
0.410025 + 0.912074i \(0.365520\pi\)
\(984\) −6.47977e15 −0.223917
\(985\) −3.60765e14 −0.0123973
\(986\) 8.15897e15 0.278813
\(987\) 1.22083e16 0.414869
\(988\) −2.15535e16 −0.728373
\(989\) 1.71992e15 0.0578000
\(990\) −1.80656e14 −0.00603751
\(991\) 2.85391e16 0.948497 0.474248 0.880391i \(-0.342720\pi\)
0.474248 + 0.880391i \(0.342720\pi\)
\(992\) 6.29665e15 0.208111
\(993\) −1.42693e16 −0.469010
\(994\) 3.96687e15 0.129665
\(995\) 1.28712e15 0.0418399
\(996\) 5.51646e15 0.178334
\(997\) 7.94845e15 0.255540 0.127770 0.991804i \(-0.459218\pi\)
0.127770 + 0.991804i \(0.459218\pi\)
\(998\) 2.76289e16 0.883377
\(999\) −3.37409e15 −0.107287
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.1.9 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.12.a.d.1.9 28 1.1 even 1 trivial