Properties

Label 177.10.a.d.1.8
Level $177$
Weight $10$
Character 177.1
Self dual yes
Analytic conductor $91.161$
Analytic rank $0$
Dimension $22$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(91.1613430010\)
Analytic rank: \(0\)
Dimension: \(22\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-12.8477 q^{2} +81.0000 q^{3} -346.936 q^{4} -288.160 q^{5} -1040.66 q^{6} -11158.8 q^{7} +11035.4 q^{8} +6561.00 q^{9} +O(q^{10})\) \(q-12.8477 q^{2} +81.0000 q^{3} -346.936 q^{4} -288.160 q^{5} -1040.66 q^{6} -11158.8 q^{7} +11035.4 q^{8} +6561.00 q^{9} +3702.19 q^{10} -5354.27 q^{11} -28101.8 q^{12} -11968.9 q^{13} +143365. q^{14} -23340.9 q^{15} +35852.3 q^{16} -253224. q^{17} -84293.8 q^{18} -368392. q^{19} +99973.1 q^{20} -903863. q^{21} +68790.1 q^{22} -1.43580e6 q^{23} +893865. q^{24} -1.87009e6 q^{25} +153773. q^{26} +531441. q^{27} +3.87139e6 q^{28} -4.56562e6 q^{29} +299878. q^{30} -552087. q^{31} -6.11073e6 q^{32} -433696. q^{33} +3.25335e6 q^{34} +3.21552e6 q^{35} -2.27625e6 q^{36} +5.14812e6 q^{37} +4.73300e6 q^{38} -969482. q^{39} -3.17995e6 q^{40} -1.80732e7 q^{41} +1.16126e7 q^{42} -1.18322e7 q^{43} +1.85759e6 q^{44} -1.89062e6 q^{45} +1.84468e7 q^{46} +3.30018e7 q^{47} +2.90403e6 q^{48} +8.41652e7 q^{49} +2.40264e7 q^{50} -2.05111e7 q^{51} +4.15245e6 q^{52} -4.28341e7 q^{53} -6.82780e6 q^{54} +1.54289e6 q^{55} -1.23141e8 q^{56} -2.98398e7 q^{57} +5.86577e7 q^{58} -1.21174e7 q^{59} +8.09782e6 q^{60} +9.99497e7 q^{61} +7.09306e6 q^{62} -7.32129e7 q^{63} +6.01525e7 q^{64} +3.44896e6 q^{65} +5.57200e6 q^{66} -1.80984e8 q^{67} +8.78526e7 q^{68} -1.16300e8 q^{69} -4.13120e7 q^{70} -2.83834e8 q^{71} +7.24030e7 q^{72} -1.11541e8 q^{73} -6.61415e7 q^{74} -1.51477e8 q^{75} +1.27809e8 q^{76} +5.97473e7 q^{77} +1.24556e7 q^{78} +4.87439e8 q^{79} -1.03312e7 q^{80} +4.30467e7 q^{81} +2.32199e8 q^{82} +7.30631e7 q^{83} +3.13583e8 q^{84} +7.29690e7 q^{85} +1.52017e8 q^{86} -3.69815e8 q^{87} -5.90864e7 q^{88} +6.25788e8 q^{89} +2.42901e7 q^{90} +1.33559e8 q^{91} +4.98132e8 q^{92} -4.47191e7 q^{93} -4.23997e8 q^{94} +1.06156e8 q^{95} -4.94969e8 q^{96} +1.65134e8 q^{97} -1.08133e9 q^{98} -3.51294e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 22q + 46q^{2} + 1782q^{3} + 5974q^{4} + 5786q^{5} + 3726q^{6} + 7641q^{7} + 61395q^{8} + 144342q^{9} + O(q^{10}) \) \( 22q + 46q^{2} + 1782q^{3} + 5974q^{4} + 5786q^{5} + 3726q^{6} + 7641q^{7} + 61395q^{8} + 144342q^{9} + 45337q^{10} + 111769q^{11} + 483894q^{12} + 189121q^{13} + 251053q^{14} + 468666q^{15} + 2311074q^{16} + 1113841q^{17} + 301806q^{18} + 476068q^{19} - 42495q^{20} + 618921q^{21} - 2252022q^{22} + 7103062q^{23} + 4972995q^{24} + 10628442q^{25} + 6871048q^{26} + 11691702q^{27} + 8112650q^{28} + 15279316q^{29} + 3672297q^{30} + 17610338q^{31} + 32378276q^{32} + 9053289q^{33} + 29339436q^{34} + 7134904q^{35} + 39195414q^{36} + 21961411q^{37} + 65195131q^{38} + 15318801q^{39} + 75185084q^{40} + 52781575q^{41} + 20335293q^{42} + 76191313q^{43} + 61127768q^{44} + 37961946q^{45} + 290208769q^{46} + 160572396q^{47} + 187196994q^{48} + 156292703q^{49} + 169504821q^{50} + 90221121q^{51} + 65465920q^{52} - 8762038q^{53} + 24446286q^{54} + 147125140q^{55} + 9671794q^{56} + 38561508q^{57} - 37665424q^{58} - 266581942q^{59} - 3442095q^{60} + 120750754q^{61} - 152465186q^{62} + 50132601q^{63} - 40658803q^{64} + 331055798q^{65} - 182413782q^{66} + 41371828q^{67} + 145606631q^{68} + 575348022q^{69} - 920887614q^{70} + 261018751q^{71} + 402812595q^{72} + 178388q^{73} - 303908734q^{74} + 860903802q^{75} - 94541144q^{76} + 299640561q^{77} + 556554888q^{78} - 905381353q^{79} + 939128289q^{80} + 947027862q^{81} - 551739753q^{82} + 1173257869q^{83} + 657124650q^{84} - 1546633210q^{85} + 1384869460q^{86} + 1237624596q^{87} + 189740713q^{88} + 898004974q^{89} + 297456057q^{90} + 591272339q^{91} + 4328210270q^{92} + 1426437378q^{93} + 122568068q^{94} + 2487967134q^{95} + 2622640356q^{96} + 3175709684q^{97} + 5095778404q^{98} + 733316409q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −12.8477 −0.567794 −0.283897 0.958855i \(-0.591627\pi\)
−0.283897 + 0.958855i \(0.591627\pi\)
\(3\) 81.0000 0.577350
\(4\) −346.936 −0.677610
\(5\) −288.160 −0.206190 −0.103095 0.994671i \(-0.532875\pi\)
−0.103095 + 0.994671i \(0.532875\pi\)
\(6\) −1040.66 −0.327816
\(7\) −11158.8 −1.75661 −0.878307 0.478098i \(-0.841327\pi\)
−0.878307 + 0.478098i \(0.841327\pi\)
\(8\) 11035.4 0.952537
\(9\) 6561.00 0.333333
\(10\) 3702.19 0.117074
\(11\) −5354.27 −0.110264 −0.0551320 0.998479i \(-0.517558\pi\)
−0.0551320 + 0.998479i \(0.517558\pi\)
\(12\) −28101.8 −0.391218
\(13\) −11968.9 −0.116228 −0.0581139 0.998310i \(-0.518509\pi\)
−0.0581139 + 0.998310i \(0.518509\pi\)
\(14\) 143365. 0.997394
\(15\) −23340.9 −0.119044
\(16\) 35852.3 0.136766
\(17\) −253224. −0.735334 −0.367667 0.929957i \(-0.619843\pi\)
−0.367667 + 0.929957i \(0.619843\pi\)
\(18\) −84293.8 −0.189265
\(19\) −368392. −0.648514 −0.324257 0.945969i \(-0.605114\pi\)
−0.324257 + 0.945969i \(0.605114\pi\)
\(20\) 99973.1 0.139717
\(21\) −903863. −1.01418
\(22\) 68790.1 0.0626072
\(23\) −1.43580e6 −1.06984 −0.534920 0.844902i \(-0.679658\pi\)
−0.534920 + 0.844902i \(0.679658\pi\)
\(24\) 893865. 0.549947
\(25\) −1.87009e6 −0.957486
\(26\) 153773. 0.0659934
\(27\) 531441. 0.192450
\(28\) 3.87139e6 1.19030
\(29\) −4.56562e6 −1.19869 −0.599347 0.800489i \(-0.704573\pi\)
−0.599347 + 0.800489i \(0.704573\pi\)
\(30\) 299878. 0.0675925
\(31\) −552087. −0.107369 −0.0536847 0.998558i \(-0.517097\pi\)
−0.0536847 + 0.998558i \(0.517097\pi\)
\(32\) −6.11073e6 −1.03019
\(33\) −433696. −0.0636609
\(34\) 3.25335e6 0.417518
\(35\) 3.21552e6 0.362197
\(36\) −2.27625e6 −0.225870
\(37\) 5.14812e6 0.451586 0.225793 0.974175i \(-0.427503\pi\)
0.225793 + 0.974175i \(0.427503\pi\)
\(38\) 4.73300e6 0.368222
\(39\) −969482. −0.0671041
\(40\) −3.17995e6 −0.196404
\(41\) −1.80732e7 −0.998865 −0.499432 0.866353i \(-0.666458\pi\)
−0.499432 + 0.866353i \(0.666458\pi\)
\(42\) 1.16126e7 0.575846
\(43\) −1.18322e7 −0.527787 −0.263893 0.964552i \(-0.585007\pi\)
−0.263893 + 0.964552i \(0.585007\pi\)
\(44\) 1.85759e6 0.0747159
\(45\) −1.89062e6 −0.0687301
\(46\) 1.84468e7 0.607449
\(47\) 3.30018e7 0.986500 0.493250 0.869888i \(-0.335809\pi\)
0.493250 + 0.869888i \(0.335809\pi\)
\(48\) 2.90403e6 0.0789617
\(49\) 8.41652e7 2.08569
\(50\) 2.40264e7 0.543654
\(51\) −2.05111e7 −0.424545
\(52\) 4.15245e6 0.0787571
\(53\) −4.28341e7 −0.745673 −0.372836 0.927897i \(-0.621615\pi\)
−0.372836 + 0.927897i \(0.621615\pi\)
\(54\) −6.82780e6 −0.109272
\(55\) 1.54289e6 0.0227354
\(56\) −1.23141e8 −1.67324
\(57\) −2.98398e7 −0.374420
\(58\) 5.86577e7 0.680612
\(59\) −1.21174e7 −0.130189
\(60\) 8.09782e6 0.0806655
\(61\) 9.99497e7 0.924266 0.462133 0.886811i \(-0.347084\pi\)
0.462133 + 0.886811i \(0.347084\pi\)
\(62\) 7.09306e6 0.0609636
\(63\) −7.32129e7 −0.585538
\(64\) 6.01525e7 0.448171
\(65\) 3.44896e6 0.0239650
\(66\) 5.57200e6 0.0361463
\(67\) −1.80984e8 −1.09724 −0.548622 0.836070i \(-0.684848\pi\)
−0.548622 + 0.836070i \(0.684848\pi\)
\(68\) 8.78526e7 0.498270
\(69\) −1.16300e8 −0.617673
\(70\) −4.13120e7 −0.205653
\(71\) −2.83834e8 −1.32557 −0.662784 0.748811i \(-0.730625\pi\)
−0.662784 + 0.748811i \(0.730625\pi\)
\(72\) 7.24030e7 0.317512
\(73\) −1.11541e8 −0.459706 −0.229853 0.973225i \(-0.573825\pi\)
−0.229853 + 0.973225i \(0.573825\pi\)
\(74\) −6.61415e7 −0.256408
\(75\) −1.51477e8 −0.552805
\(76\) 1.27809e8 0.439440
\(77\) 5.97473e7 0.193691
\(78\) 1.24556e7 0.0381013
\(79\) 4.87439e8 1.40799 0.703993 0.710206i \(-0.251398\pi\)
0.703993 + 0.710206i \(0.251398\pi\)
\(80\) −1.03312e7 −0.0281997
\(81\) 4.30467e7 0.111111
\(82\) 2.32199e8 0.567149
\(83\) 7.30631e7 0.168984 0.0844922 0.996424i \(-0.473073\pi\)
0.0844922 + 0.996424i \(0.473073\pi\)
\(84\) 3.13583e8 0.687220
\(85\) 7.29690e7 0.151619
\(86\) 1.52017e8 0.299674
\(87\) −3.69815e8 −0.692067
\(88\) −5.90864e7 −0.105030
\(89\) 6.25788e8 1.05724 0.528619 0.848859i \(-0.322710\pi\)
0.528619 + 0.848859i \(0.322710\pi\)
\(90\) 2.42901e7 0.0390245
\(91\) 1.33559e8 0.204167
\(92\) 4.98132e8 0.724935
\(93\) −4.47191e7 −0.0619897
\(94\) −4.23997e8 −0.560129
\(95\) 1.06156e8 0.133717
\(96\) −4.94969e8 −0.594781
\(97\) 1.65134e8 0.189393 0.0946965 0.995506i \(-0.469812\pi\)
0.0946965 + 0.995506i \(0.469812\pi\)
\(98\) −1.08133e9 −1.18424
\(99\) −3.51294e7 −0.0367546
\(100\) 6.48802e8 0.648802
\(101\) 8.61651e7 0.0823920 0.0411960 0.999151i \(-0.486883\pi\)
0.0411960 + 0.999151i \(0.486883\pi\)
\(102\) 2.63521e8 0.241054
\(103\) −4.41594e8 −0.386595 −0.193297 0.981140i \(-0.561918\pi\)
−0.193297 + 0.981140i \(0.561918\pi\)
\(104\) −1.32081e8 −0.110711
\(105\) 2.60457e8 0.209114
\(106\) 5.50320e8 0.423388
\(107\) −1.38847e9 −1.02402 −0.512012 0.858978i \(-0.671100\pi\)
−0.512012 + 0.858978i \(0.671100\pi\)
\(108\) −1.84376e8 −0.130406
\(109\) −2.18603e9 −1.48333 −0.741664 0.670771i \(-0.765963\pi\)
−0.741664 + 0.670771i \(0.765963\pi\)
\(110\) −1.98226e7 −0.0129090
\(111\) 4.16998e8 0.260724
\(112\) −4.00068e8 −0.240244
\(113\) 2.86843e9 1.65498 0.827488 0.561483i \(-0.189769\pi\)
0.827488 + 0.561483i \(0.189769\pi\)
\(114\) 3.83373e8 0.212593
\(115\) 4.13740e8 0.220591
\(116\) 1.58398e9 0.812248
\(117\) −7.85281e7 −0.0387426
\(118\) 1.55680e8 0.0739205
\(119\) 2.82568e9 1.29170
\(120\) −2.57576e8 −0.113394
\(121\) −2.32928e9 −0.987842
\(122\) −1.28412e9 −0.524793
\(123\) −1.46393e9 −0.576695
\(124\) 1.91539e8 0.0727545
\(125\) 1.10170e9 0.403615
\(126\) 9.40618e8 0.332465
\(127\) 4.81144e9 1.64119 0.820595 0.571510i \(-0.193642\pi\)
0.820595 + 0.571510i \(0.193642\pi\)
\(128\) 2.35587e9 0.775723
\(129\) −9.58410e8 −0.304718
\(130\) −4.43112e7 −0.0136072
\(131\) 3.84589e9 1.14098 0.570488 0.821306i \(-0.306754\pi\)
0.570488 + 0.821306i \(0.306754\pi\)
\(132\) 1.50465e8 0.0431373
\(133\) 4.11082e9 1.13919
\(134\) 2.32523e9 0.623009
\(135\) −1.53140e8 −0.0396814
\(136\) −2.79442e9 −0.700433
\(137\) 4.88795e9 1.18545 0.592727 0.805404i \(-0.298051\pi\)
0.592727 + 0.805404i \(0.298051\pi\)
\(138\) 1.49419e9 0.350711
\(139\) 4.04703e9 0.919539 0.459769 0.888038i \(-0.347932\pi\)
0.459769 + 0.888038i \(0.347932\pi\)
\(140\) −1.11558e9 −0.245428
\(141\) 2.67314e9 0.569556
\(142\) 3.64662e9 0.752649
\(143\) 6.40848e7 0.0128157
\(144\) 2.35227e8 0.0455885
\(145\) 1.31563e9 0.247159
\(146\) 1.43304e9 0.261018
\(147\) 6.81738e9 1.20417
\(148\) −1.78607e9 −0.305999
\(149\) 4.92185e9 0.818070 0.409035 0.912519i \(-0.365865\pi\)
0.409035 + 0.912519i \(0.365865\pi\)
\(150\) 1.94613e9 0.313879
\(151\) 7.11467e9 1.11367 0.556837 0.830621i \(-0.312015\pi\)
0.556837 + 0.830621i \(0.312015\pi\)
\(152\) −4.06534e9 −0.617733
\(153\) −1.66140e9 −0.245111
\(154\) −7.67615e8 −0.109977
\(155\) 1.59089e8 0.0221385
\(156\) 3.36349e8 0.0454704
\(157\) 7.84897e9 1.03101 0.515507 0.856886i \(-0.327604\pi\)
0.515507 + 0.856886i \(0.327604\pi\)
\(158\) −6.26248e9 −0.799446
\(159\) −3.46956e9 −0.430514
\(160\) 1.76087e9 0.212416
\(161\) 1.60218e10 1.87930
\(162\) −5.53052e8 −0.0630882
\(163\) 1.00224e10 1.11206 0.556029 0.831163i \(-0.312324\pi\)
0.556029 + 0.831163i \(0.312324\pi\)
\(164\) 6.27023e9 0.676841
\(165\) 1.24974e8 0.0131263
\(166\) −9.38693e8 −0.0959483
\(167\) −5.55108e9 −0.552273 −0.276136 0.961118i \(-0.589054\pi\)
−0.276136 + 0.961118i \(0.589054\pi\)
\(168\) −9.97445e9 −0.966045
\(169\) −1.04612e10 −0.986491
\(170\) −9.37484e8 −0.0860882
\(171\) −2.41702e9 −0.216171
\(172\) 4.10503e9 0.357634
\(173\) −5.37563e9 −0.456270 −0.228135 0.973629i \(-0.573263\pi\)
−0.228135 + 0.973629i \(0.573263\pi\)
\(174\) 4.75128e9 0.392951
\(175\) 2.08679e10 1.68193
\(176\) −1.91963e8 −0.0150803
\(177\) −9.81506e8 −0.0751646
\(178\) −8.03995e9 −0.600293
\(179\) −1.07723e10 −0.784275 −0.392137 0.919907i \(-0.628264\pi\)
−0.392137 + 0.919907i \(0.628264\pi\)
\(180\) 6.55924e8 0.0465722
\(181\) −2.27451e10 −1.57520 −0.787598 0.616189i \(-0.788676\pi\)
−0.787598 + 0.616189i \(0.788676\pi\)
\(182\) −1.71592e9 −0.115925
\(183\) 8.09592e9 0.533625
\(184\) −1.58446e10 −1.01906
\(185\) −1.48348e9 −0.0931128
\(186\) 5.74538e8 0.0351974
\(187\) 1.35583e9 0.0810808
\(188\) −1.14495e10 −0.668462
\(189\) −5.93024e9 −0.338060
\(190\) −1.36386e9 −0.0759239
\(191\) −4.63754e9 −0.252138 −0.126069 0.992021i \(-0.540236\pi\)
−0.126069 + 0.992021i \(0.540236\pi\)
\(192\) 4.87235e9 0.258752
\(193\) 1.78697e10 0.927066 0.463533 0.886080i \(-0.346582\pi\)
0.463533 + 0.886080i \(0.346582\pi\)
\(194\) −2.12159e9 −0.107536
\(195\) 2.79366e8 0.0138362
\(196\) −2.92000e10 −1.41329
\(197\) 5.80897e9 0.274790 0.137395 0.990516i \(-0.456127\pi\)
0.137395 + 0.990516i \(0.456127\pi\)
\(198\) 4.51332e8 0.0208691
\(199\) 2.52274e9 0.114034 0.0570168 0.998373i \(-0.481841\pi\)
0.0570168 + 0.998373i \(0.481841\pi\)
\(200\) −2.06371e10 −0.912040
\(201\) −1.46597e10 −0.633495
\(202\) −1.10702e9 −0.0467817
\(203\) 5.09468e10 2.10564
\(204\) 7.11606e9 0.287676
\(205\) 5.20796e9 0.205956
\(206\) 5.67347e9 0.219506
\(207\) −9.42029e9 −0.356614
\(208\) −4.29113e8 −0.0158960
\(209\) 1.97247e9 0.0715077
\(210\) −3.34627e9 −0.118734
\(211\) −2.26437e10 −0.786459 −0.393230 0.919440i \(-0.628642\pi\)
−0.393230 + 0.919440i \(0.628642\pi\)
\(212\) 1.48607e10 0.505275
\(213\) −2.29906e10 −0.765317
\(214\) 1.78387e10 0.581435
\(215\) 3.40957e9 0.108825
\(216\) 5.86465e9 0.183316
\(217\) 6.16063e9 0.188606
\(218\) 2.80855e10 0.842225
\(219\) −9.03479e9 −0.265412
\(220\) −5.35283e8 −0.0154057
\(221\) 3.03082e9 0.0854662
\(222\) −5.35746e9 −0.148037
\(223\) −5.70112e10 −1.54379 −0.771895 0.635750i \(-0.780691\pi\)
−0.771895 + 0.635750i \(0.780691\pi\)
\(224\) 6.81884e10 1.80965
\(225\) −1.22697e10 −0.319162
\(226\) −3.68528e10 −0.939685
\(227\) 4.70965e9 0.117726 0.0588629 0.998266i \(-0.481253\pi\)
0.0588629 + 0.998266i \(0.481253\pi\)
\(228\) 1.03525e10 0.253711
\(229\) −9.30975e9 −0.223706 −0.111853 0.993725i \(-0.535679\pi\)
−0.111853 + 0.993725i \(0.535679\pi\)
\(230\) −5.31561e9 −0.125250
\(231\) 4.83953e9 0.111828
\(232\) −5.03833e10 −1.14180
\(233\) −2.22868e10 −0.495388 −0.247694 0.968838i \(-0.579673\pi\)
−0.247694 + 0.968838i \(0.579673\pi\)
\(234\) 1.00891e9 0.0219978
\(235\) −9.50979e9 −0.203407
\(236\) 4.20395e9 0.0882173
\(237\) 3.94826e10 0.812902
\(238\) −3.63035e10 −0.733418
\(239\) 6.22197e10 1.23349 0.616747 0.787161i \(-0.288450\pi\)
0.616747 + 0.787161i \(0.288450\pi\)
\(240\) −8.36826e8 −0.0162811
\(241\) −7.82772e10 −1.49471 −0.747357 0.664422i \(-0.768678\pi\)
−0.747357 + 0.664422i \(0.768678\pi\)
\(242\) 2.99259e10 0.560891
\(243\) 3.48678e9 0.0641500
\(244\) −3.46762e10 −0.626292
\(245\) −2.42530e10 −0.430049
\(246\) 1.88081e10 0.327444
\(247\) 4.40926e9 0.0753753
\(248\) −6.09248e9 −0.102273
\(249\) 5.91811e9 0.0975632
\(250\) −1.41543e10 −0.229170
\(251\) −4.20358e10 −0.668479 −0.334239 0.942488i \(-0.608479\pi\)
−0.334239 + 0.942488i \(0.608479\pi\)
\(252\) 2.54002e10 0.396766
\(253\) 7.68767e9 0.117965
\(254\) −6.18160e10 −0.931857
\(255\) 5.91049e9 0.0875372
\(256\) −6.10656e10 −0.888621
\(257\) 3.72649e10 0.532845 0.266423 0.963856i \(-0.414158\pi\)
0.266423 + 0.963856i \(0.414158\pi\)
\(258\) 1.23134e10 0.173017
\(259\) −5.74468e10 −0.793263
\(260\) −1.19657e9 −0.0162390
\(261\) −2.99550e10 −0.399565
\(262\) −4.94109e10 −0.647839
\(263\) −3.73389e9 −0.0481239 −0.0240619 0.999710i \(-0.507660\pi\)
−0.0240619 + 0.999710i \(0.507660\pi\)
\(264\) −4.78599e9 −0.0606394
\(265\) 1.23431e10 0.153751
\(266\) −5.28146e10 −0.646824
\(267\) 5.06889e10 0.610396
\(268\) 6.27899e10 0.743504
\(269\) −9.73102e10 −1.13311 −0.566556 0.824023i \(-0.691725\pi\)
−0.566556 + 0.824023i \(0.691725\pi\)
\(270\) 1.96750e9 0.0225308
\(271\) 1.33771e11 1.50661 0.753305 0.657671i \(-0.228458\pi\)
0.753305 + 0.657671i \(0.228458\pi\)
\(272\) −9.07866e9 −0.100568
\(273\) 1.08183e10 0.117876
\(274\) −6.27990e10 −0.673093
\(275\) 1.00130e10 0.105576
\(276\) 4.03487e10 0.418541
\(277\) 2.30470e10 0.235210 0.117605 0.993060i \(-0.462478\pi\)
0.117605 + 0.993060i \(0.462478\pi\)
\(278\) −5.19951e10 −0.522109
\(279\) −3.62225e9 −0.0357898
\(280\) 3.54844e10 0.345006
\(281\) 6.46954e10 0.619006 0.309503 0.950899i \(-0.399837\pi\)
0.309503 + 0.950899i \(0.399837\pi\)
\(282\) −3.43438e10 −0.323390
\(283\) −2.08643e10 −0.193359 −0.0966796 0.995316i \(-0.530822\pi\)
−0.0966796 + 0.995316i \(0.530822\pi\)
\(284\) 9.84723e10 0.898218
\(285\) 8.59862e9 0.0772017
\(286\) −8.23343e8 −0.00727669
\(287\) 2.01675e11 1.75462
\(288\) −4.00925e10 −0.343397
\(289\) −5.44655e10 −0.459284
\(290\) −1.69028e10 −0.140336
\(291\) 1.33759e10 0.109346
\(292\) 3.86975e10 0.311502
\(293\) −1.83552e11 −1.45497 −0.727486 0.686123i \(-0.759311\pi\)
−0.727486 + 0.686123i \(0.759311\pi\)
\(294\) −8.75877e10 −0.683723
\(295\) 3.49174e9 0.0268437
\(296\) 5.68114e10 0.430153
\(297\) −2.84548e9 −0.0212203
\(298\) −6.32345e10 −0.464495
\(299\) 1.71850e10 0.124345
\(300\) 5.25530e10 0.374586
\(301\) 1.32033e11 0.927117
\(302\) −9.14072e10 −0.632338
\(303\) 6.97937e9 0.0475690
\(304\) −1.32077e10 −0.0886944
\(305\) −2.88015e10 −0.190575
\(306\) 2.13452e10 0.139173
\(307\) 1.37976e10 0.0886503 0.0443251 0.999017i \(-0.485886\pi\)
0.0443251 + 0.999017i \(0.485886\pi\)
\(308\) −2.07285e10 −0.131247
\(309\) −3.57691e10 −0.223201
\(310\) −2.04393e9 −0.0125701
\(311\) −1.38946e10 −0.0842217 −0.0421109 0.999113i \(-0.513408\pi\)
−0.0421109 + 0.999113i \(0.513408\pi\)
\(312\) −1.06986e10 −0.0639191
\(313\) −2.49015e11 −1.46648 −0.733238 0.679972i \(-0.761992\pi\)
−0.733238 + 0.679972i \(0.761992\pi\)
\(314\) −1.00841e11 −0.585403
\(315\) 2.10970e10 0.120732
\(316\) −1.69110e11 −0.954066
\(317\) 2.53996e11 1.41273 0.706365 0.707847i \(-0.250334\pi\)
0.706365 + 0.707847i \(0.250334\pi\)
\(318\) 4.45759e10 0.244443
\(319\) 2.44456e10 0.132173
\(320\) −1.73335e10 −0.0924085
\(321\) −1.12466e11 −0.591221
\(322\) −2.05844e11 −1.06705
\(323\) 9.32858e10 0.476874
\(324\) −1.49345e10 −0.0752900
\(325\) 2.23829e10 0.111286
\(326\) −1.28765e11 −0.631419
\(327\) −1.77069e11 −0.856400
\(328\) −1.99444e11 −0.951455
\(329\) −3.68260e11 −1.73290
\(330\) −1.60563e9 −0.00745301
\(331\) 1.69491e11 0.776106 0.388053 0.921637i \(-0.373148\pi\)
0.388053 + 0.921637i \(0.373148\pi\)
\(332\) −2.53482e10 −0.114506
\(333\) 3.37768e10 0.150529
\(334\) 7.13187e10 0.313577
\(335\) 5.21523e10 0.226241
\(336\) −3.24055e10 −0.138705
\(337\) 2.08297e11 0.879728 0.439864 0.898064i \(-0.355027\pi\)
0.439864 + 0.898064i \(0.355027\pi\)
\(338\) 1.34403e11 0.560124
\(339\) 2.32343e11 0.955501
\(340\) −2.53156e10 −0.102738
\(341\) 2.95603e9 0.0118390
\(342\) 3.10532e10 0.122741
\(343\) −4.88884e11 −1.90714
\(344\) −1.30573e11 −0.502736
\(345\) 3.35130e10 0.127358
\(346\) 6.90645e10 0.259067
\(347\) 3.01426e11 1.11609 0.558043 0.829812i \(-0.311552\pi\)
0.558043 + 0.829812i \(0.311552\pi\)
\(348\) 1.28302e11 0.468951
\(349\) 2.59548e10 0.0936489 0.0468244 0.998903i \(-0.485090\pi\)
0.0468244 + 0.998903i \(0.485090\pi\)
\(350\) −2.68105e11 −0.954991
\(351\) −6.36077e9 −0.0223680
\(352\) 3.27185e10 0.113593
\(353\) −1.72509e11 −0.591325 −0.295662 0.955293i \(-0.595540\pi\)
−0.295662 + 0.955293i \(0.595540\pi\)
\(354\) 1.26101e10 0.0426780
\(355\) 8.17895e10 0.273319
\(356\) −2.17109e11 −0.716395
\(357\) 2.28880e11 0.745762
\(358\) 1.38399e11 0.445306
\(359\) −2.87460e11 −0.913383 −0.456692 0.889625i \(-0.650966\pi\)
−0.456692 + 0.889625i \(0.650966\pi\)
\(360\) −2.08636e10 −0.0654680
\(361\) −1.86975e11 −0.579430
\(362\) 2.92223e11 0.894387
\(363\) −1.88672e11 −0.570331
\(364\) −4.63364e10 −0.138346
\(365\) 3.21415e10 0.0947870
\(366\) −1.04014e11 −0.302989
\(367\) 6.06272e11 1.74450 0.872249 0.489062i \(-0.162661\pi\)
0.872249 + 0.489062i \(0.162661\pi\)
\(368\) −5.14768e10 −0.146317
\(369\) −1.18578e11 −0.332955
\(370\) 1.90593e10 0.0528688
\(371\) 4.77977e11 1.30986
\(372\) 1.55147e10 0.0420048
\(373\) −1.10006e11 −0.294256 −0.147128 0.989117i \(-0.547003\pi\)
−0.147128 + 0.989117i \(0.547003\pi\)
\(374\) −1.74193e10 −0.0460372
\(375\) 8.92374e10 0.233027
\(376\) 3.64187e11 0.939677
\(377\) 5.46455e10 0.139322
\(378\) 7.61900e10 0.191949
\(379\) −1.53320e11 −0.381701 −0.190850 0.981619i \(-0.561125\pi\)
−0.190850 + 0.981619i \(0.561125\pi\)
\(380\) −3.68293e10 −0.0906082
\(381\) 3.89727e11 0.947541
\(382\) 5.95818e10 0.143162
\(383\) −8.02971e8 −0.00190680 −0.000953400 1.00000i \(-0.500303\pi\)
−0.000953400 1.00000i \(0.500303\pi\)
\(384\) 1.90826e11 0.447864
\(385\) −1.72168e10 −0.0399372
\(386\) −2.29585e11 −0.526382
\(387\) −7.76312e10 −0.175929
\(388\) −5.72910e10 −0.128335
\(389\) 7.71042e11 1.70728 0.853640 0.520864i \(-0.174390\pi\)
0.853640 + 0.520864i \(0.174390\pi\)
\(390\) −3.58921e9 −0.00785612
\(391\) 3.63579e11 0.786691
\(392\) 9.28793e11 1.98670
\(393\) 3.11517e11 0.658743
\(394\) −7.46319e10 −0.156024
\(395\) −1.40460e11 −0.290313
\(396\) 1.21877e10 0.0249053
\(397\) −8.59617e11 −1.73679 −0.868396 0.495872i \(-0.834849\pi\)
−0.868396 + 0.495872i \(0.834849\pi\)
\(398\) −3.24114e10 −0.0647476
\(399\) 3.32976e11 0.657711
\(400\) −6.70470e10 −0.130951
\(401\) 5.09559e11 0.984113 0.492056 0.870563i \(-0.336245\pi\)
0.492056 + 0.870563i \(0.336245\pi\)
\(402\) 1.88344e11 0.359694
\(403\) 6.60789e9 0.0124793
\(404\) −2.98938e10 −0.0558297
\(405\) −1.24043e10 −0.0229100
\(406\) −6.54550e11 −1.19557
\(407\) −2.75644e10 −0.0497937
\(408\) −2.26348e11 −0.404395
\(409\) −2.01097e11 −0.355346 −0.177673 0.984090i \(-0.556857\pi\)
−0.177673 + 0.984090i \(0.556857\pi\)
\(410\) −6.69103e10 −0.116941
\(411\) 3.95924e11 0.684422
\(412\) 1.53205e11 0.261961
\(413\) 1.35215e11 0.228692
\(414\) 1.21029e11 0.202483
\(415\) −2.10538e10 −0.0348430
\(416\) 7.31388e10 0.119737
\(417\) 3.27810e11 0.530896
\(418\) −2.53418e10 −0.0406016
\(419\) 5.57874e11 0.884247 0.442123 0.896954i \(-0.354225\pi\)
0.442123 + 0.896954i \(0.354225\pi\)
\(420\) −9.03620e10 −0.141698
\(421\) −1.23812e12 −1.92085 −0.960424 0.278542i \(-0.910149\pi\)
−0.960424 + 0.278542i \(0.910149\pi\)
\(422\) 2.90920e11 0.446547
\(423\) 2.16525e11 0.328833
\(424\) −4.72690e11 −0.710281
\(425\) 4.73551e11 0.704072
\(426\) 2.95376e11 0.434542
\(427\) −1.11532e12 −1.62358
\(428\) 4.81712e11 0.693889
\(429\) 5.19087e9 0.00739916
\(430\) −4.38052e10 −0.0617899
\(431\) −6.13759e11 −0.856743 −0.428371 0.903603i \(-0.640913\pi\)
−0.428371 + 0.903603i \(0.640913\pi\)
\(432\) 1.90534e10 0.0263206
\(433\) −3.09324e11 −0.422881 −0.211440 0.977391i \(-0.567815\pi\)
−0.211440 + 0.977391i \(0.567815\pi\)
\(434\) −7.91500e10 −0.107090
\(435\) 1.06566e11 0.142698
\(436\) 7.58414e11 1.00512
\(437\) 5.28938e11 0.693807
\(438\) 1.16076e11 0.150699
\(439\) 2.58853e11 0.332632 0.166316 0.986073i \(-0.446813\pi\)
0.166316 + 0.986073i \(0.446813\pi\)
\(440\) 1.70263e10 0.0216563
\(441\) 5.52208e11 0.695230
\(442\) −3.89391e10 −0.0485272
\(443\) −6.08274e11 −0.750382 −0.375191 0.926948i \(-0.622423\pi\)
−0.375191 + 0.926948i \(0.622423\pi\)
\(444\) −1.44672e11 −0.176669
\(445\) −1.80327e11 −0.217992
\(446\) 7.32463e11 0.876554
\(447\) 3.98670e11 0.472313
\(448\) −6.71229e11 −0.787263
\(449\) −9.88861e11 −1.14822 −0.574112 0.818777i \(-0.694653\pi\)
−0.574112 + 0.818777i \(0.694653\pi\)
\(450\) 1.57637e11 0.181218
\(451\) 9.67686e10 0.110139
\(452\) −9.95164e11 −1.12143
\(453\) 5.76288e11 0.642981
\(454\) −6.05082e10 −0.0668440
\(455\) −3.84863e10 −0.0420973
\(456\) −3.29293e11 −0.356649
\(457\) 3.06396e11 0.328595 0.164297 0.986411i \(-0.447464\pi\)
0.164297 + 0.986411i \(0.447464\pi\)
\(458\) 1.19609e11 0.127019
\(459\) −1.34574e11 −0.141515
\(460\) −1.43542e11 −0.149475
\(461\) 4.36858e11 0.450491 0.225245 0.974302i \(-0.427682\pi\)
0.225245 + 0.974302i \(0.427682\pi\)
\(462\) −6.21768e10 −0.0634950
\(463\) −1.25176e12 −1.26592 −0.632960 0.774185i \(-0.718160\pi\)
−0.632960 + 0.774185i \(0.718160\pi\)
\(464\) −1.63688e11 −0.163940
\(465\) 1.28862e10 0.0127817
\(466\) 2.86334e11 0.281278
\(467\) −7.55685e11 −0.735216 −0.367608 0.929981i \(-0.619823\pi\)
−0.367608 + 0.929981i \(0.619823\pi\)
\(468\) 2.72442e10 0.0262524
\(469\) 2.01956e12 1.92744
\(470\) 1.22179e11 0.115493
\(471\) 6.35767e11 0.595256
\(472\) −1.33719e11 −0.124010
\(473\) 6.33530e10 0.0581958
\(474\) −5.07261e11 −0.461561
\(475\) 6.88926e11 0.620943
\(476\) −9.80330e11 −0.875268
\(477\) −2.81035e11 −0.248558
\(478\) −7.99380e11 −0.700371
\(479\) 1.35024e12 1.17193 0.585963 0.810338i \(-0.300716\pi\)
0.585963 + 0.810338i \(0.300716\pi\)
\(480\) 1.42630e11 0.122638
\(481\) −6.16174e10 −0.0524869
\(482\) 1.00568e12 0.848690
\(483\) 1.29777e12 1.08501
\(484\) 8.08112e11 0.669372
\(485\) −4.75850e10 −0.0390510
\(486\) −4.47972e10 −0.0364240
\(487\) 1.13263e12 0.912447 0.456224 0.889865i \(-0.349202\pi\)
0.456224 + 0.889865i \(0.349202\pi\)
\(488\) 1.10298e12 0.880398
\(489\) 8.11814e11 0.642047
\(490\) 3.11596e11 0.244179
\(491\) 9.25546e11 0.718673 0.359337 0.933208i \(-0.383003\pi\)
0.359337 + 0.933208i \(0.383003\pi\)
\(492\) 5.07889e11 0.390774
\(493\) 1.15612e12 0.881441
\(494\) −5.66488e10 −0.0427976
\(495\) 1.01229e10 0.00757845
\(496\) −1.97936e10 −0.0146844
\(497\) 3.16725e12 2.32851
\(498\) −7.60342e10 −0.0553958
\(499\) −2.72332e12 −1.96628 −0.983141 0.182847i \(-0.941469\pi\)
−0.983141 + 0.182847i \(0.941469\pi\)
\(500\) −3.82219e11 −0.273493
\(501\) −4.49638e11 −0.318855
\(502\) 5.40064e11 0.379558
\(503\) −2.02205e12 −1.40843 −0.704216 0.709985i \(-0.748701\pi\)
−0.704216 + 0.709985i \(0.748701\pi\)
\(504\) −8.07931e11 −0.557746
\(505\) −2.48293e10 −0.0169884
\(506\) −9.87690e10 −0.0669797
\(507\) −8.47361e11 −0.569551
\(508\) −1.66927e12 −1.11209
\(509\) 3.50255e10 0.0231289 0.0115644 0.999933i \(-0.496319\pi\)
0.0115644 + 0.999933i \(0.496319\pi\)
\(510\) −7.59362e10 −0.0497031
\(511\) 1.24466e12 0.807526
\(512\) −4.21653e11 −0.271169
\(513\) −1.95779e11 −0.124807
\(514\) −4.78769e11 −0.302546
\(515\) 1.27250e11 0.0797121
\(516\) 3.32507e11 0.206480
\(517\) −1.76701e11 −0.108775
\(518\) 7.38060e11 0.450410
\(519\) −4.35426e11 −0.263428
\(520\) 3.80605e10 0.0228276
\(521\) 6.72889e9 0.00400105 0.00200052 0.999998i \(-0.499363\pi\)
0.00200052 + 0.999998i \(0.499363\pi\)
\(522\) 3.84853e11 0.226871
\(523\) −1.94157e12 −1.13474 −0.567370 0.823463i \(-0.692039\pi\)
−0.567370 + 0.823463i \(0.692039\pi\)
\(524\) −1.33428e12 −0.773137
\(525\) 1.69030e12 0.971064
\(526\) 4.79719e10 0.0273244
\(527\) 1.39802e11 0.0789523
\(528\) −1.55490e10 −0.00870662
\(529\) 2.60374e11 0.144560
\(530\) −1.58580e11 −0.0872986
\(531\) −7.95020e10 −0.0433963
\(532\) −1.42619e12 −0.771926
\(533\) 2.16316e11 0.116096
\(534\) −6.51236e11 −0.346579
\(535\) 4.00102e11 0.211144
\(536\) −1.99722e12 −1.04517
\(537\) −8.72553e11 −0.452801
\(538\) 1.25021e12 0.643374
\(539\) −4.50643e11 −0.229976
\(540\) 5.31298e10 0.0268885
\(541\) −4.30477e11 −0.216054 −0.108027 0.994148i \(-0.534453\pi\)
−0.108027 + 0.994148i \(0.534453\pi\)
\(542\) −1.71865e12 −0.855444
\(543\) −1.84236e12 −0.909440
\(544\) 1.54738e12 0.757535
\(545\) 6.29927e11 0.305848
\(546\) −1.38990e11 −0.0669293
\(547\) 3.69233e12 1.76343 0.881714 0.471784i \(-0.156390\pi\)
0.881714 + 0.471784i \(0.156390\pi\)
\(548\) −1.69581e12 −0.803275
\(549\) 6.55770e11 0.308089
\(550\) −1.28644e11 −0.0599455
\(551\) 1.68194e12 0.777370
\(552\) −1.28341e12 −0.588356
\(553\) −5.43924e12 −2.47329
\(554\) −2.96101e11 −0.133551
\(555\) −1.20162e11 −0.0537587
\(556\) −1.40406e12 −0.623089
\(557\) 1.24067e12 0.546147 0.273073 0.961993i \(-0.411960\pi\)
0.273073 + 0.961993i \(0.411960\pi\)
\(558\) 4.65375e10 0.0203212
\(559\) 1.41619e11 0.0613434
\(560\) 1.15284e11 0.0495361
\(561\) 1.09822e11 0.0468120
\(562\) −8.31187e11 −0.351468
\(563\) 1.17319e12 0.492132 0.246066 0.969253i \(-0.420862\pi\)
0.246066 + 0.969253i \(0.420862\pi\)
\(564\) −9.27411e11 −0.385937
\(565\) −8.26567e11 −0.341240
\(566\) 2.68059e11 0.109788
\(567\) −4.80350e11 −0.195179
\(568\) −3.13221e12 −1.26265
\(569\) −2.91701e12 −1.16663 −0.583314 0.812247i \(-0.698244\pi\)
−0.583314 + 0.812247i \(0.698244\pi\)
\(570\) −1.10473e11 −0.0438347
\(571\) 2.19394e11 0.0863699 0.0431850 0.999067i \(-0.486250\pi\)
0.0431850 + 0.999067i \(0.486250\pi\)
\(572\) −2.22334e10 −0.00868406
\(573\) −3.75641e11 −0.145572
\(574\) −2.59106e12 −0.996262
\(575\) 2.68508e12 1.02436
\(576\) 3.94660e11 0.149390
\(577\) 3.01436e12 1.13215 0.566075 0.824353i \(-0.308461\pi\)
0.566075 + 0.824353i \(0.308461\pi\)
\(578\) 6.99757e11 0.260778
\(579\) 1.44745e12 0.535242
\(580\) −4.56439e11 −0.167478
\(581\) −8.15296e11 −0.296840
\(582\) −1.71849e11 −0.0620860
\(583\) 2.29346e11 0.0822208
\(584\) −1.23089e12 −0.437887
\(585\) 2.26286e10 0.00798835
\(586\) 2.35822e12 0.826124
\(587\) −3.73815e12 −1.29953 −0.649764 0.760136i \(-0.725132\pi\)
−0.649764 + 0.760136i \(0.725132\pi\)
\(588\) −2.36520e12 −0.815961
\(589\) 2.03385e11 0.0696305
\(590\) −4.48608e10 −0.0152417
\(591\) 4.70526e11 0.158650
\(592\) 1.84572e11 0.0617615
\(593\) 5.21655e12 1.73236 0.866179 0.499734i \(-0.166569\pi\)
0.866179 + 0.499734i \(0.166569\pi\)
\(594\) 3.65579e10 0.0120488
\(595\) −8.14246e11 −0.266336
\(596\) −1.70757e12 −0.554332
\(597\) 2.04342e11 0.0658373
\(598\) −2.20788e11 −0.0706024
\(599\) −3.22617e11 −0.102392 −0.0511960 0.998689i \(-0.516303\pi\)
−0.0511960 + 0.998689i \(0.516303\pi\)
\(600\) −1.67161e12 −0.526567
\(601\) 5.74234e11 0.179537 0.0897685 0.995963i \(-0.471387\pi\)
0.0897685 + 0.995963i \(0.471387\pi\)
\(602\) −1.69633e12 −0.526411
\(603\) −1.18744e12 −0.365748
\(604\) −2.46834e12 −0.754637
\(605\) 6.71205e11 0.203683
\(606\) −8.96689e10 −0.0270094
\(607\) −4.53684e11 −0.135645 −0.0678226 0.997697i \(-0.521605\pi\)
−0.0678226 + 0.997697i \(0.521605\pi\)
\(608\) 2.25114e12 0.668094
\(609\) 4.12669e12 1.21569
\(610\) 3.70033e11 0.108207
\(611\) −3.94996e11 −0.114659
\(612\) 5.76401e11 0.166090
\(613\) −3.50305e12 −1.00201 −0.501007 0.865443i \(-0.667037\pi\)
−0.501007 + 0.865443i \(0.667037\pi\)
\(614\) −1.77267e11 −0.0503351
\(615\) 4.21844e11 0.118909
\(616\) 6.59333e11 0.184498
\(617\) 3.03553e12 0.843239 0.421620 0.906773i \(-0.361462\pi\)
0.421620 + 0.906773i \(0.361462\pi\)
\(618\) 4.59551e11 0.126732
\(619\) −3.99709e12 −1.09430 −0.547149 0.837035i \(-0.684287\pi\)
−0.547149 + 0.837035i \(0.684287\pi\)
\(620\) −5.51939e10 −0.0150013
\(621\) −7.63044e11 −0.205891
\(622\) 1.78514e11 0.0478206
\(623\) −6.98305e12 −1.85716
\(624\) −3.47581e10 −0.00917753
\(625\) 3.33505e12 0.874264
\(626\) 3.19927e12 0.832657
\(627\) 1.59770e11 0.0412850
\(628\) −2.72309e12 −0.698625
\(629\) −1.30363e12 −0.332067
\(630\) −2.71048e11 −0.0685510
\(631\) −3.92587e12 −0.985834 −0.492917 0.870076i \(-0.664069\pi\)
−0.492917 + 0.870076i \(0.664069\pi\)
\(632\) 5.37907e12 1.34116
\(633\) −1.83414e12 −0.454063
\(634\) −3.26326e12 −0.802140
\(635\) −1.38646e12 −0.338397
\(636\) 1.20372e12 0.291721
\(637\) −1.00737e12 −0.242415
\(638\) −3.14070e11 −0.0750469
\(639\) −1.86223e12 −0.441856
\(640\) −6.78867e11 −0.159947
\(641\) 1.53691e12 0.359573 0.179787 0.983706i \(-0.442459\pi\)
0.179787 + 0.983706i \(0.442459\pi\)
\(642\) 1.44493e12 0.335691
\(643\) 9.66855e11 0.223055 0.111528 0.993761i \(-0.464426\pi\)
0.111528 + 0.993761i \(0.464426\pi\)
\(644\) −5.55855e12 −1.27343
\(645\) 2.76175e11 0.0628299
\(646\) −1.19851e12 −0.270766
\(647\) −5.92400e12 −1.32906 −0.664532 0.747260i \(-0.731369\pi\)
−0.664532 + 0.747260i \(0.731369\pi\)
\(648\) 4.75036e11 0.105837
\(649\) 6.48797e10 0.0143551
\(650\) −2.87569e11 −0.0631877
\(651\) 4.99011e11 0.108892
\(652\) −3.47713e12 −0.753541
\(653\) 1.50571e12 0.324065 0.162033 0.986785i \(-0.448195\pi\)
0.162033 + 0.986785i \(0.448195\pi\)
\(654\) 2.27493e12 0.486259
\(655\) −1.10823e12 −0.235258
\(656\) −6.47964e11 −0.136610
\(657\) −7.31818e11 −0.153235
\(658\) 4.73130e12 0.983929
\(659\) 1.30444e12 0.269426 0.134713 0.990885i \(-0.456989\pi\)
0.134713 + 0.990885i \(0.456989\pi\)
\(660\) −4.33580e10 −0.00889449
\(661\) 1.94983e12 0.397273 0.198637 0.980073i \(-0.436349\pi\)
0.198637 + 0.980073i \(0.436349\pi\)
\(662\) −2.17757e12 −0.440668
\(663\) 2.45496e11 0.0493439
\(664\) 8.06278e11 0.160964
\(665\) −1.18457e12 −0.234890
\(666\) −4.33955e11 −0.0854693
\(667\) 6.55532e12 1.28241
\(668\) 1.92587e12 0.374225
\(669\) −4.61790e12 −0.891307
\(670\) −6.70038e11 −0.128458
\(671\) −5.35158e11 −0.101913
\(672\) 5.52326e12 1.04480
\(673\) 1.57403e12 0.295765 0.147882 0.989005i \(-0.452754\pi\)
0.147882 + 0.989005i \(0.452754\pi\)
\(674\) −2.67614e12 −0.499504
\(675\) −9.93842e11 −0.184268
\(676\) 3.62939e12 0.668456
\(677\) −9.48856e11 −0.173601 −0.0868003 0.996226i \(-0.527664\pi\)
−0.0868003 + 0.996226i \(0.527664\pi\)
\(678\) −2.98508e12 −0.542527
\(679\) −1.84270e12 −0.332690
\(680\) 8.05239e11 0.144422
\(681\) 3.81481e11 0.0679691
\(682\) −3.79782e10 −0.00672209
\(683\) −6.32726e12 −1.11256 −0.556279 0.830996i \(-0.687771\pi\)
−0.556279 + 0.830996i \(0.687771\pi\)
\(684\) 8.38553e11 0.146480
\(685\) −1.40851e12 −0.244429
\(686\) 6.28104e12 1.08286
\(687\) −7.54090e11 −0.129157
\(688\) −4.24212e11 −0.0721830
\(689\) 5.12678e11 0.0866679
\(690\) −4.30565e11 −0.0723132
\(691\) 1.32805e12 0.221597 0.110798 0.993843i \(-0.464659\pi\)
0.110798 + 0.993843i \(0.464659\pi\)
\(692\) 1.86500e12 0.309173
\(693\) 3.92002e11 0.0645637
\(694\) −3.87263e12 −0.633707
\(695\) −1.16619e12 −0.189600
\(696\) −4.08104e12 −0.659219
\(697\) 4.57656e12 0.734499
\(698\) −3.33459e11 −0.0531732
\(699\) −1.80523e12 −0.286012
\(700\) −7.23985e12 −1.13969
\(701\) 2.84917e12 0.445643 0.222822 0.974859i \(-0.428473\pi\)
0.222822 + 0.974859i \(0.428473\pi\)
\(702\) 8.17214e10 0.0127004
\(703\) −1.89653e12 −0.292860
\(704\) −3.22073e11 −0.0494171
\(705\) −7.70293e11 −0.117437
\(706\) 2.21635e12 0.335750
\(707\) −9.61499e11 −0.144731
\(708\) 3.40520e11 0.0509323
\(709\) 1.69917e12 0.252540 0.126270 0.991996i \(-0.459700\pi\)
0.126270 + 0.991996i \(0.459700\pi\)
\(710\) −1.05081e12 −0.155189
\(711\) 3.19809e12 0.469329
\(712\) 6.90580e12 1.00706
\(713\) 7.92688e11 0.114868
\(714\) −2.94058e12 −0.423439
\(715\) −1.84667e10 −0.00264248
\(716\) 3.73729e12 0.531432
\(717\) 5.03979e12 0.712158
\(718\) 3.69321e12 0.518613
\(719\) 8.44614e12 1.17863 0.589316 0.807902i \(-0.299397\pi\)
0.589316 + 0.807902i \(0.299397\pi\)
\(720\) −6.77829e10 −0.00939992
\(721\) 4.92766e12 0.679098
\(722\) 2.40220e12 0.328997
\(723\) −6.34045e12 −0.862974
\(724\) 7.89111e12 1.06737
\(725\) 8.53811e12 1.14773
\(726\) 2.42400e12 0.323830
\(727\) −4.74564e12 −0.630071 −0.315036 0.949080i \(-0.602016\pi\)
−0.315036 + 0.949080i \(0.602016\pi\)
\(728\) 1.47387e12 0.194477
\(729\) 2.82430e11 0.0370370
\(730\) −4.12945e11 −0.0538195
\(731\) 2.99620e12 0.388100
\(732\) −2.80877e12 −0.361590
\(733\) 1.07965e13 1.38138 0.690692 0.723149i \(-0.257306\pi\)
0.690692 + 0.723149i \(0.257306\pi\)
\(734\) −7.78921e12 −0.990515
\(735\) −1.96449e12 −0.248289
\(736\) 8.77379e12 1.10214
\(737\) 9.69038e11 0.120987
\(738\) 1.52346e12 0.189050
\(739\) −2.90351e12 −0.358116 −0.179058 0.983839i \(-0.557305\pi\)
−0.179058 + 0.983839i \(0.557305\pi\)
\(740\) 5.14673e11 0.0630941
\(741\) 3.57150e11 0.0435180
\(742\) −6.14091e12 −0.743730
\(743\) 1.37305e13 1.65286 0.826429 0.563041i \(-0.190369\pi\)
0.826429 + 0.563041i \(0.190369\pi\)
\(744\) −4.93491e11 −0.0590475
\(745\) −1.41828e12 −0.168678
\(746\) 1.41332e12 0.167077
\(747\) 4.79367e11 0.0563281
\(748\) −4.70387e11 −0.0549412
\(749\) 1.54937e13 1.79881
\(750\) −1.14650e12 −0.132311
\(751\) 1.33843e13 1.53538 0.767691 0.640821i \(-0.221406\pi\)
0.767691 + 0.640821i \(0.221406\pi\)
\(752\) 1.18319e12 0.134919
\(753\) −3.40490e12 −0.385946
\(754\) −7.02070e11 −0.0791059
\(755\) −2.05016e12 −0.229629
\(756\) 2.05742e12 0.229073
\(757\) 6.94181e12 0.768319 0.384159 0.923267i \(-0.374491\pi\)
0.384159 + 0.923267i \(0.374491\pi\)
\(758\) 1.96981e12 0.216727
\(759\) 6.22702e11 0.0681070
\(760\) 1.17147e12 0.127371
\(761\) −1.81658e13 −1.96346 −0.981732 0.190269i \(-0.939064\pi\)
−0.981732 + 0.190269i \(0.939064\pi\)
\(762\) −5.00710e12 −0.538008
\(763\) 2.43935e13 2.60563
\(764\) 1.60893e12 0.170851
\(765\) 4.78749e11 0.0505396
\(766\) 1.03163e10 0.00108267
\(767\) 1.45032e11 0.0151316
\(768\) −4.94631e12 −0.513046
\(769\) 6.48177e12 0.668382 0.334191 0.942505i \(-0.391537\pi\)
0.334191 + 0.942505i \(0.391537\pi\)
\(770\) 2.21196e11 0.0226761
\(771\) 3.01846e12 0.307638
\(772\) −6.19966e12 −0.628189
\(773\) −3.31272e12 −0.333716 −0.166858 0.985981i \(-0.553362\pi\)
−0.166858 + 0.985981i \(0.553362\pi\)
\(774\) 9.97383e11 0.0998913
\(775\) 1.03245e12 0.102805
\(776\) 1.82231e12 0.180404
\(777\) −4.65319e12 −0.457990
\(778\) −9.90612e12 −0.969383
\(779\) 6.65801e12 0.647778
\(780\) −9.69222e10 −0.00937556
\(781\) 1.51972e12 0.146162
\(782\) −4.67116e12 −0.446678
\(783\) −2.42636e12 −0.230689
\(784\) 3.01751e12 0.285251
\(785\) −2.26176e12 −0.212585
\(786\) −4.00229e12 −0.374030
\(787\) 1.37042e13 1.27341 0.636704 0.771108i \(-0.280297\pi\)
0.636704 + 0.771108i \(0.280297\pi\)
\(788\) −2.01534e12 −0.186200
\(789\) −3.02445e11 −0.0277843
\(790\) 1.80459e12 0.164838
\(791\) −3.20083e13 −2.90715
\(792\) −3.87666e11 −0.0350101
\(793\) −1.19629e12 −0.107425
\(794\) 1.10441e13 0.986140
\(795\) 9.99789e11 0.0887679
\(796\) −8.75229e11 −0.0772703
\(797\) −1.12390e13 −0.986659 −0.493329 0.869843i \(-0.664220\pi\)
−0.493329 + 0.869843i \(0.664220\pi\)
\(798\) −4.27798e12 −0.373444
\(799\) −8.35684e12 −0.725407
\(800\) 1.14276e13 0.986393
\(801\) 4.10580e12 0.352412
\(802\) −6.54667e12 −0.558773
\(803\) 5.97219e11 0.0506890
\(804\) 5.08598e12 0.429262
\(805\) −4.61684e12 −0.387493
\(806\) −8.48962e10 −0.00708566
\(807\) −7.88212e12 −0.654203
\(808\) 9.50863e11 0.0784814
\(809\) 6.71071e12 0.550808 0.275404 0.961329i \(-0.411188\pi\)
0.275404 + 0.961329i \(0.411188\pi\)
\(810\) 1.59367e11 0.0130082
\(811\) 3.22992e12 0.262179 0.131090 0.991371i \(-0.458152\pi\)
0.131090 + 0.991371i \(0.458152\pi\)
\(812\) −1.76753e13 −1.42681
\(813\) 1.08355e13 0.869842
\(814\) 3.54140e11 0.0282725
\(815\) −2.88805e12 −0.229295
\(816\) −7.35371e11 −0.0580632
\(817\) 4.35890e12 0.342277
\(818\) 2.58364e12 0.201763
\(819\) 8.76279e11 0.0680557
\(820\) −1.80683e12 −0.139558
\(821\) −5.85236e12 −0.449559 −0.224780 0.974410i \(-0.572166\pi\)
−0.224780 + 0.974410i \(0.572166\pi\)
\(822\) −5.08672e12 −0.388611
\(823\) 1.73378e13 1.31733 0.658665 0.752437i \(-0.271122\pi\)
0.658665 + 0.752437i \(0.271122\pi\)
\(824\) −4.87315e12 −0.368246
\(825\) 8.11050e11 0.0609544
\(826\) −1.73721e12 −0.129850
\(827\) −1.21426e12 −0.0902683 −0.0451341 0.998981i \(-0.514372\pi\)
−0.0451341 + 0.998981i \(0.514372\pi\)
\(828\) 3.26824e12 0.241645
\(829\) 1.64938e13 1.21290 0.606449 0.795122i \(-0.292593\pi\)
0.606449 + 0.795122i \(0.292593\pi\)
\(830\) 2.70494e11 0.0197836
\(831\) 1.86681e12 0.135799
\(832\) −7.19960e11 −0.0520899
\(833\) −2.13126e13 −1.53368
\(834\) −4.21160e12 −0.301440
\(835\) 1.59960e12 0.113873
\(836\) −6.84323e11 −0.0484543
\(837\) −2.93402e11 −0.0206632
\(838\) −7.16741e12 −0.502070
\(839\) 2.54317e13 1.77193 0.885964 0.463754i \(-0.153498\pi\)
0.885964 + 0.463754i \(0.153498\pi\)
\(840\) 2.87424e12 0.199189
\(841\) 6.33773e12 0.436869
\(842\) 1.59070e13 1.09065
\(843\) 5.24032e12 0.357383
\(844\) 7.85592e12 0.532913
\(845\) 3.01451e12 0.203405
\(846\) −2.78185e12 −0.186710
\(847\) 2.59920e13 1.73526
\(848\) −1.53570e12 −0.101982
\(849\) −1.69001e12 −0.111636
\(850\) −6.08405e12 −0.399768
\(851\) −7.39168e12 −0.483126
\(852\) 7.97626e12 0.518586
\(853\) −2.00195e12 −0.129474 −0.0647370 0.997902i \(-0.520621\pi\)
−0.0647370 + 0.997902i \(0.520621\pi\)
\(854\) 1.43293e13 0.921858
\(855\) 6.96488e11 0.0445724
\(856\) −1.53223e13 −0.975421
\(857\) 1.17946e13 0.746913 0.373457 0.927648i \(-0.378172\pi\)
0.373457 + 0.927648i \(0.378172\pi\)
\(858\) −6.66908e10 −0.00420120
\(859\) −2.87741e12 −0.180315 −0.0901577 0.995927i \(-0.528737\pi\)
−0.0901577 + 0.995927i \(0.528737\pi\)
\(860\) −1.18290e12 −0.0737406
\(861\) 1.63356e13 1.01303
\(862\) 7.88540e12 0.486453
\(863\) 9.28264e12 0.569669 0.284835 0.958577i \(-0.408061\pi\)
0.284835 + 0.958577i \(0.408061\pi\)
\(864\) −3.24749e12 −0.198260
\(865\) 1.54904e12 0.0940785
\(866\) 3.97410e12 0.240109
\(867\) −4.41170e12 −0.265168
\(868\) −2.13735e12 −0.127802
\(869\) −2.60988e12 −0.155250
\(870\) −1.36913e12 −0.0810228
\(871\) 2.16618e12 0.127530
\(872\) −2.41237e13 −1.41292
\(873\) 1.08344e12 0.0631310
\(874\) −6.79564e12 −0.393939
\(875\) −1.22936e13 −0.708995
\(876\) 3.13450e12 0.179846
\(877\) 4.93768e12 0.281854 0.140927 0.990020i \(-0.454992\pi\)
0.140927 + 0.990020i \(0.454992\pi\)
\(878\) −3.32567e12 −0.188866
\(879\) −1.48677e13 −0.840028
\(880\) 5.53160e10 0.00310941
\(881\) −6.10283e11 −0.0341303 −0.0170651 0.999854i \(-0.505432\pi\)
−0.0170651 + 0.999854i \(0.505432\pi\)
\(882\) −7.09460e12 −0.394748
\(883\) −7.15626e11 −0.0396153 −0.0198076 0.999804i \(-0.506305\pi\)
−0.0198076 + 0.999804i \(0.506305\pi\)
\(884\) −1.05150e12 −0.0579128
\(885\) 2.82831e11 0.0154982
\(886\) 7.81493e12 0.426062
\(887\) −2.10952e13 −1.14427 −0.572134 0.820160i \(-0.693884\pi\)
−0.572134 + 0.820160i \(0.693884\pi\)
\(888\) 4.60172e12 0.248349
\(889\) −5.36899e13 −2.88294
\(890\) 2.31679e12 0.123775
\(891\) −2.30484e11 −0.0122515
\(892\) 1.97792e13 1.04609
\(893\) −1.21576e13 −0.639759
\(894\) −5.12199e12 −0.268176
\(895\) 3.10413e12 0.161710
\(896\) −2.62887e13 −1.36265
\(897\) 1.39198e12 0.0717907
\(898\) 1.27046e13 0.651955
\(899\) 2.52062e12 0.128703
\(900\) 4.25679e12 0.216267
\(901\) 1.08466e13 0.548319
\(902\) −1.24325e12 −0.0625361
\(903\) 1.06947e13 0.535271
\(904\) 3.16542e13 1.57643
\(905\) 6.55423e12 0.324790
\(906\) −7.40398e12 −0.365080
\(907\) −7.69464e12 −0.377533 −0.188767 0.982022i \(-0.560449\pi\)
−0.188767 + 0.982022i \(0.560449\pi\)
\(908\) −1.63395e12 −0.0797723
\(909\) 5.65329e11 0.0274640
\(910\) 4.94460e11 0.0239026
\(911\) −1.62851e13 −0.783353 −0.391676 0.920103i \(-0.628105\pi\)
−0.391676 + 0.920103i \(0.628105\pi\)
\(912\) −1.06982e12 −0.0512077
\(913\) −3.91200e11 −0.0186329
\(914\) −3.93649e12 −0.186574
\(915\) −2.33292e12 −0.110028
\(916\) 3.22989e12 0.151586
\(917\) −4.29156e13 −2.00425
\(918\) 1.72896e12 0.0803514
\(919\) 6.17320e12 0.285490 0.142745 0.989759i \(-0.454407\pi\)
0.142745 + 0.989759i \(0.454407\pi\)
\(920\) 4.56577e12 0.210121
\(921\) 1.11760e12 0.0511823
\(922\) −5.61262e12 −0.255786
\(923\) 3.39719e12 0.154068
\(924\) −1.67901e12 −0.0757755
\(925\) −9.62744e12 −0.432387
\(926\) 1.60822e13 0.718781
\(927\) −2.89730e12 −0.128865
\(928\) 2.78992e13 1.23489
\(929\) −2.36498e13 −1.04173 −0.520867 0.853638i \(-0.674391\pi\)
−0.520867 + 0.853638i \(0.674391\pi\)
\(930\) −1.65559e11 −0.00725736
\(931\) −3.10058e13 −1.35260
\(932\) 7.73209e12 0.335680
\(933\) −1.12546e12 −0.0486254
\(934\) 9.70882e12 0.417451
\(935\) −3.90696e11 −0.0167181
\(936\) −8.66586e11 −0.0369037
\(937\) −4.58998e13 −1.94528 −0.972642 0.232309i \(-0.925372\pi\)
−0.972642 + 0.232309i \(0.925372\pi\)
\(938\) −2.59468e13 −1.09439
\(939\) −2.01702e13 −0.846671
\(940\) 3.29929e12 0.137830
\(941\) 3.09088e13 1.28507 0.642537 0.766254i \(-0.277882\pi\)
0.642537 + 0.766254i \(0.277882\pi\)
\(942\) −8.16814e12 −0.337983
\(943\) 2.59495e13 1.06863
\(944\) −4.34435e11 −0.0178054
\(945\) 1.70886e12 0.0697048
\(946\) −8.13940e11 −0.0330432
\(947\) −3.03772e12 −0.122736 −0.0613682 0.998115i \(-0.519546\pi\)
−0.0613682 + 0.998115i \(0.519546\pi\)
\(948\) −1.36979e13 −0.550830
\(949\) 1.33502e12 0.0534306
\(950\) −8.85113e12 −0.352567
\(951\) 2.05736e13 0.815641
\(952\) 3.11824e13 1.23039
\(953\) 2.25232e13 0.884528 0.442264 0.896885i \(-0.354175\pi\)
0.442264 + 0.896885i \(0.354175\pi\)
\(954\) 3.61065e12 0.141129
\(955\) 1.33635e12 0.0519884
\(956\) −2.15863e13 −0.835828
\(957\) 1.98009e12 0.0763100
\(958\) −1.73474e13 −0.665412
\(959\) −5.45437e13 −2.08238
\(960\) −1.40402e12 −0.0533521
\(961\) −2.61348e13 −0.988472
\(962\) 7.91642e11 0.0298017
\(963\) −9.10977e12 −0.341341
\(964\) 2.71572e13 1.01283
\(965\) −5.14934e12 −0.191152
\(966\) −1.66733e13 −0.616064
\(967\) −2.80300e13 −1.03087 −0.515434 0.856929i \(-0.672369\pi\)
−0.515434 + 0.856929i \(0.672369\pi\)
\(968\) −2.57044e13 −0.940956
\(969\) 7.55615e12 0.275324
\(970\) 6.11358e11 0.0221729
\(971\) −1.48038e13 −0.534424 −0.267212 0.963638i \(-0.586102\pi\)
−0.267212 + 0.963638i \(0.586102\pi\)
\(972\) −1.20969e12 −0.0434687
\(973\) −4.51600e13 −1.61527
\(974\) −1.45517e13 −0.518082
\(975\) 1.81302e12 0.0642512
\(976\) 3.58342e12 0.126408
\(977\) −4.58267e13 −1.60914 −0.804569 0.593859i \(-0.797604\pi\)
−0.804569 + 0.593859i \(0.797604\pi\)
\(978\) −1.04299e13 −0.364550
\(979\) −3.35064e12 −0.116575
\(980\) 8.41425e12 0.291406
\(981\) −1.43426e13 −0.494443
\(982\) −1.18911e13 −0.408058
\(983\) 4.73679e13 1.61806 0.809028 0.587770i \(-0.199994\pi\)
0.809028 + 0.587770i \(0.199994\pi\)
\(984\) −1.61550e13 −0.549323
\(985\) −1.67391e12 −0.0566590
\(986\) −1.48535e13 −0.500477
\(987\) −2.98291e13 −1.00049
\(988\) −1.52973e12 −0.0510751
\(989\) 1.69887e13 0.564648
\(990\) −1.30056e11 −0.00430300
\(991\) 3.56521e13 1.17423 0.587116 0.809503i \(-0.300263\pi\)
0.587116 + 0.809503i \(0.300263\pi\)
\(992\) 3.37365e12 0.110611
\(993\) 1.37288e13 0.448085
\(994\) −4.06919e13 −1.32211
\(995\) −7.26951e11 −0.0235126
\(996\) −2.05321e12 −0.0661098
\(997\) 8.44601e11 0.0270722 0.0135361 0.999908i \(-0.495691\pi\)
0.0135361 + 0.999908i \(0.495691\pi\)
\(998\) 3.49884e13 1.11644
\(999\) 2.73592e12 0.0869078
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 177.10.a.d.1.8 22
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.10.a.d.1.8 22 1.1 even 1 trivial