Properties

Label 177.12.a.d.1.20
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 1.20
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+45.0643 q^{2} +243.000 q^{3} -17.2112 q^{4} -12444.4 q^{5} +10950.6 q^{6} -11322.2 q^{7} -93067.2 q^{8} +59049.0 q^{9} +O(q^{10})\) \(q+45.0643 q^{2} +243.000 q^{3} -17.2112 q^{4} -12444.4 q^{5} +10950.6 q^{6} -11322.2 q^{7} -93067.2 q^{8} +59049.0 q^{9} -560797. q^{10} -624551. q^{11} -4182.32 q^{12} -2.19482e6 q^{13} -510225. q^{14} -3.02398e6 q^{15} -4.15876e6 q^{16} -2.69636e6 q^{17} +2.66100e6 q^{18} -1.40825e7 q^{19} +214183. q^{20} -2.75128e6 q^{21} -2.81450e7 q^{22} +2.01395e7 q^{23} -2.26153e7 q^{24} +1.06034e8 q^{25} -9.89082e7 q^{26} +1.43489e7 q^{27} +194868. q^{28} -3.17284e6 q^{29} -1.36274e8 q^{30} +1.00377e8 q^{31} +3.19025e6 q^{32} -1.51766e8 q^{33} -1.21510e8 q^{34} +1.40897e8 q^{35} -1.01630e6 q^{36} +7.58547e8 q^{37} -6.34617e8 q^{38} -5.33342e8 q^{39} +1.15816e9 q^{40} -7.54684e6 q^{41} -1.23985e8 q^{42} -1.42501e9 q^{43} +1.07493e7 q^{44} -7.34828e8 q^{45} +9.07572e8 q^{46} -1.00068e9 q^{47} -1.01058e9 q^{48} -1.84914e9 q^{49} +4.77836e9 q^{50} -6.55216e8 q^{51} +3.77756e7 q^{52} -3.23368e9 q^{53} +6.46623e8 q^{54} +7.77215e9 q^{55} +1.05372e9 q^{56} -3.42204e9 q^{57} -1.42982e8 q^{58} +7.14924e8 q^{59} +5.20464e7 q^{60} -9.23337e9 q^{61} +4.52343e9 q^{62} -6.68562e8 q^{63} +8.66091e9 q^{64} +2.73132e10 q^{65} -6.83922e9 q^{66} -1.91218e9 q^{67} +4.64076e7 q^{68} +4.89390e9 q^{69} +6.34943e9 q^{70} -1.76446e10 q^{71} -5.49553e9 q^{72} +1.51898e10 q^{73} +3.41834e10 q^{74} +2.57664e10 q^{75} +2.42377e8 q^{76} +7.07127e9 q^{77} -2.40347e10 q^{78} +7.55668e9 q^{79} +5.17532e10 q^{80} +3.48678e9 q^{81} -3.40093e8 q^{82} +5.47937e10 q^{83} +4.73529e7 q^{84} +3.35545e10 q^{85} -6.42172e10 q^{86} -7.70999e8 q^{87} +5.81253e10 q^{88} -3.81766e10 q^{89} -3.31145e10 q^{90} +2.48501e10 q^{91} -3.46625e8 q^{92} +2.43917e10 q^{93} -4.50949e10 q^{94} +1.75248e11 q^{95} +7.75231e8 q^{96} -5.92325e10 q^{97} -8.33300e10 q^{98} -3.68791e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 28q + 96q^{2} + 6804q^{3} + 29214q^{4} + 26562q^{5} + 23328q^{6} + 142333q^{7} + 332331q^{8} + 1653372q^{9} + O(q^{10}) \) \( 28q + 96q^{2} + 6804q^{3} + 29214q^{4} + 26562q^{5} + 23328q^{6} + 142333q^{7} + 332331q^{8} + 1653372q^{9} + 616281q^{10} + 1082362q^{11} + 7099002q^{12} + 503712q^{13} + 1321669q^{14} + 6454566q^{15} + 34870338q^{16} + 13513579q^{17} + 5668704q^{18} + 35971687q^{19} + 96105997q^{20} + 34586919q^{21} - 47598882q^{22} + 61380539q^{23} + 80756433q^{24} + 294744746q^{25} + 62820734q^{26} + 401769396q^{27} + 148068294q^{28} + 322339307q^{29} + 149756283q^{30} + 151247077q^{31} + 466383494q^{32} + 263013966q^{33} + 684479860q^{34} + 960297361q^{35} + 1725057486q^{36} + 863508437q^{37} + 992640509q^{38} + 122402016q^{39} + 3067680252q^{40} + 3081170377q^{41} + 321165567q^{42} + 2554238300q^{43} + 4350123570q^{44} + 1568459538q^{45} - 1987059155q^{46} + 6203398333q^{47} + 8473492134q^{48} + 10327857997q^{49} + 17577682253q^{50} + 3283799697q^{51} + 32137181618q^{52} + 14571770754q^{53} + 1377495072q^{54} + 18251419334q^{55} + 33498842836q^{56} + 8741119941q^{57} + 11860778276q^{58} + 20017880372q^{59} + 23353757271q^{60} + 2761613771q^{61} + 13785829526q^{62} + 8404621317q^{63} + 86547545293q^{64} + 32034985256q^{65} - 11566528326q^{66} + 39381333296q^{67} + 38995496621q^{68} + 14915470977q^{69} + 8551800364q^{70} + 26130020296q^{71} + 19623813219q^{72} + 41382402799q^{73} + 23815315058q^{74} + 71622973278q^{75} + 10611720128q^{76} - 8426124313q^{77} + 15265438362q^{78} + 59825111206q^{79} + 4009687655q^{80} + 97629963228q^{81} - 39592715115q^{82} + 35433122727q^{83} + 35980595442q^{84} - 8950496085q^{85} - 182032360688q^{86} + 78328451601q^{87} - 220003602335q^{88} + 102303043039q^{89} + 36390776769q^{90} - 111146323655q^{91} - 163000203526q^{92} + 36753039711q^{93} - 81314346008q^{94} + 208102168887q^{95} + 113331189042q^{96} - 171891031490q^{97} + 72304707792q^{98} + 63912393738q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 45.0643 0.995789 0.497895 0.867238i \(-0.334107\pi\)
0.497895 + 0.867238i \(0.334107\pi\)
\(3\) 243.000 0.577350
\(4\) −17.2112 −0.00840391
\(5\) −12444.4 −1.78089 −0.890447 0.455086i \(-0.849608\pi\)
−0.890447 + 0.455086i \(0.849608\pi\)
\(6\) 10950.6 0.574919
\(7\) −11322.2 −0.254618 −0.127309 0.991863i \(-0.540634\pi\)
−0.127309 + 0.991863i \(0.540634\pi\)
\(8\) −93067.2 −1.00416
\(9\) 59049.0 0.333333
\(10\) −560797. −1.77340
\(11\) −624551. −1.16925 −0.584626 0.811303i \(-0.698759\pi\)
−0.584626 + 0.811303i \(0.698759\pi\)
\(12\) −4182.32 −0.00485200
\(13\) −2.19482e6 −1.63950 −0.819750 0.572722i \(-0.805888\pi\)
−0.819750 + 0.572722i \(0.805888\pi\)
\(14\) −510225. −0.253546
\(15\) −3.02398e6 −1.02820
\(16\) −4.15876e6 −0.991525
\(17\) −2.69636e6 −0.460584 −0.230292 0.973122i \(-0.573968\pi\)
−0.230292 + 0.973122i \(0.573968\pi\)
\(18\) 2.66100e6 0.331930
\(19\) −1.40825e7 −1.30477 −0.652386 0.757887i \(-0.726232\pi\)
−0.652386 + 0.757887i \(0.726232\pi\)
\(20\) 214183. 0.0149665
\(21\) −2.75128e6 −0.147004
\(22\) −2.81450e7 −1.16433
\(23\) 2.01395e7 0.652447 0.326224 0.945293i \(-0.394224\pi\)
0.326224 + 0.945293i \(0.394224\pi\)
\(24\) −2.26153e7 −0.579751
\(25\) 1.06034e8 2.17159
\(26\) −9.89082e7 −1.63260
\(27\) 1.43489e7 0.192450
\(28\) 194868. 0.00213979
\(29\) −3.17284e6 −0.0287249 −0.0143625 0.999897i \(-0.504572\pi\)
−0.0143625 + 0.999897i \(0.504572\pi\)
\(30\) −1.36274e8 −1.02387
\(31\) 1.00377e8 0.629718 0.314859 0.949138i \(-0.398043\pi\)
0.314859 + 0.949138i \(0.398043\pi\)
\(32\) 3.19025e6 0.0168074
\(33\) −1.51766e8 −0.675068
\(34\) −1.21510e8 −0.458645
\(35\) 1.40897e8 0.453449
\(36\) −1.01630e6 −0.00280130
\(37\) 7.58547e8 1.79834 0.899172 0.437595i \(-0.144170\pi\)
0.899172 + 0.437595i \(0.144170\pi\)
\(38\) −6.34617e8 −1.29928
\(39\) −5.33342e8 −0.946566
\(40\) 1.15816e9 1.78830
\(41\) −7.54684e6 −0.0101731 −0.00508656 0.999987i \(-0.501619\pi\)
−0.00508656 + 0.999987i \(0.501619\pi\)
\(42\) −1.23985e8 −0.146385
\(43\) −1.42501e9 −1.47823 −0.739116 0.673579i \(-0.764756\pi\)
−0.739116 + 0.673579i \(0.764756\pi\)
\(44\) 1.07493e7 0.00982629
\(45\) −7.34828e8 −0.593631
\(46\) 9.07572e8 0.649700
\(47\) −1.00068e9 −0.636440 −0.318220 0.948017i \(-0.603085\pi\)
−0.318220 + 0.948017i \(0.603085\pi\)
\(48\) −1.01058e9 −0.572457
\(49\) −1.84914e9 −0.935169
\(50\) 4.77836e9 2.16244
\(51\) −6.55216e8 −0.265918
\(52\) 3.77756e7 0.0137782
\(53\) −3.23368e9 −1.06213 −0.531067 0.847330i \(-0.678209\pi\)
−0.531067 + 0.847330i \(0.678209\pi\)
\(54\) 6.46623e8 0.191640
\(55\) 7.77215e9 2.08232
\(56\) 1.05372e9 0.255677
\(57\) −3.42204e9 −0.753310
\(58\) −1.42982e8 −0.0286040
\(59\) 7.14924e8 0.130189
\(60\) 5.20464e7 0.00864090
\(61\) −9.23337e9 −1.39974 −0.699868 0.714272i \(-0.746758\pi\)
−0.699868 + 0.714272i \(0.746758\pi\)
\(62\) 4.52343e9 0.627067
\(63\) −6.68562e8 −0.0848728
\(64\) 8.66091e9 1.00826
\(65\) 2.73132e10 2.91978
\(66\) −6.83922e9 −0.672226
\(67\) −1.91218e9 −0.173028 −0.0865141 0.996251i \(-0.527573\pi\)
−0.0865141 + 0.996251i \(0.527573\pi\)
\(68\) 4.64076e7 0.00387071
\(69\) 4.89390e9 0.376691
\(70\) 6.34943e9 0.451539
\(71\) −1.76446e10 −1.16062 −0.580311 0.814395i \(-0.697069\pi\)
−0.580311 + 0.814395i \(0.697069\pi\)
\(72\) −5.49553e9 −0.334719
\(73\) 1.51898e10 0.857584 0.428792 0.903403i \(-0.358939\pi\)
0.428792 + 0.903403i \(0.358939\pi\)
\(74\) 3.41834e10 1.79077
\(75\) 2.57664e10 1.25377
\(76\) 2.42377e8 0.0109652
\(77\) 7.07127e9 0.297713
\(78\) −2.40347e10 −0.942580
\(79\) 7.55668e9 0.276301 0.138150 0.990411i \(-0.455884\pi\)
0.138150 + 0.990411i \(0.455884\pi\)
\(80\) 5.17532e10 1.76580
\(81\) 3.48678e9 0.111111
\(82\) −3.40093e8 −0.0101303
\(83\) 5.47937e10 1.52687 0.763433 0.645886i \(-0.223512\pi\)
0.763433 + 0.645886i \(0.223512\pi\)
\(84\) 4.73529e7 0.00123541
\(85\) 3.35545e10 0.820252
\(86\) −6.42172e10 −1.47201
\(87\) −7.70999e8 −0.0165843
\(88\) 5.81253e10 1.17411
\(89\) −3.81766e10 −0.724690 −0.362345 0.932044i \(-0.618024\pi\)
−0.362345 + 0.932044i \(0.618024\pi\)
\(90\) −3.31145e10 −0.591132
\(91\) 2.48501e10 0.417447
\(92\) −3.46625e8 −0.00548311
\(93\) 2.43917e10 0.363568
\(94\) −4.50949e10 −0.633760
\(95\) 1.75248e11 2.32366
\(96\) 7.75231e8 0.00970374
\(97\) −5.92325e10 −0.700351 −0.350175 0.936684i \(-0.613878\pi\)
−0.350175 + 0.936684i \(0.613878\pi\)
\(98\) −8.33300e10 −0.931232
\(99\) −3.68791e10 −0.389751
\(100\) −1.82498e9 −0.0182498
\(101\) 1.10713e10 0.104817 0.0524085 0.998626i \(-0.483310\pi\)
0.0524085 + 0.998626i \(0.483310\pi\)
\(102\) −2.95268e10 −0.264799
\(103\) 1.09862e10 0.0933777 0.0466888 0.998909i \(-0.485133\pi\)
0.0466888 + 0.998909i \(0.485133\pi\)
\(104\) 2.04266e11 1.64632
\(105\) 3.42380e10 0.261799
\(106\) −1.45723e11 −1.05766
\(107\) −4.85869e10 −0.334895 −0.167447 0.985881i \(-0.553552\pi\)
−0.167447 + 0.985881i \(0.553552\pi\)
\(108\) −2.46962e8 −0.00161733
\(109\) −2.58166e11 −1.60714 −0.803571 0.595209i \(-0.797069\pi\)
−0.803571 + 0.595209i \(0.797069\pi\)
\(110\) 3.50247e11 2.07355
\(111\) 1.84327e11 1.03827
\(112\) 4.70861e10 0.252461
\(113\) 2.98929e11 1.52629 0.763144 0.646228i \(-0.223654\pi\)
0.763144 + 0.646228i \(0.223654\pi\)
\(114\) −1.54212e11 −0.750138
\(115\) −2.50623e11 −1.16194
\(116\) 5.46083e7 0.000241402 0
\(117\) −1.29602e11 −0.546500
\(118\) 3.22175e10 0.129641
\(119\) 3.05286e10 0.117273
\(120\) 2.81434e11 1.03247
\(121\) 1.04753e11 0.367152
\(122\) −4.16095e11 −1.39384
\(123\) −1.83388e9 −0.00587345
\(124\) −1.72761e9 −0.00529210
\(125\) −7.11897e11 −2.08647
\(126\) −3.01283e10 −0.0845154
\(127\) −1.68798e11 −0.453364 −0.226682 0.973969i \(-0.572788\pi\)
−0.226682 + 0.973969i \(0.572788\pi\)
\(128\) 3.83764e11 0.987209
\(129\) −3.46278e11 −0.853457
\(130\) 1.23085e12 2.90748
\(131\) 1.91861e11 0.434505 0.217252 0.976115i \(-0.430291\pi\)
0.217252 + 0.976115i \(0.430291\pi\)
\(132\) 2.61208e9 0.00567321
\(133\) 1.59444e11 0.332219
\(134\) −8.61709e10 −0.172300
\(135\) −1.78563e11 −0.342733
\(136\) 2.50943e11 0.462499
\(137\) −3.68526e11 −0.652386 −0.326193 0.945303i \(-0.605766\pi\)
−0.326193 + 0.945303i \(0.605766\pi\)
\(138\) 2.20540e11 0.375104
\(139\) 4.13605e11 0.676090 0.338045 0.941130i \(-0.390234\pi\)
0.338045 + 0.941130i \(0.390234\pi\)
\(140\) −2.42501e9 −0.00381074
\(141\) −2.43165e11 −0.367449
\(142\) −7.95141e11 −1.15573
\(143\) 1.37078e12 1.91699
\(144\) −2.45571e11 −0.330508
\(145\) 3.94840e10 0.0511560
\(146\) 6.84518e11 0.853973
\(147\) −4.49340e11 −0.539920
\(148\) −1.30555e10 −0.0151131
\(149\) 1.51131e12 1.68588 0.842942 0.538004i \(-0.180821\pi\)
0.842942 + 0.538004i \(0.180821\pi\)
\(150\) 1.16114e12 1.24849
\(151\) 5.95905e11 0.617737 0.308869 0.951105i \(-0.400050\pi\)
0.308869 + 0.951105i \(0.400050\pi\)
\(152\) 1.31062e12 1.31020
\(153\) −1.59217e11 −0.153528
\(154\) 3.18662e11 0.296460
\(155\) −1.24913e12 −1.12146
\(156\) 9.17946e9 0.00795485
\(157\) 7.47869e11 0.625716 0.312858 0.949800i \(-0.398714\pi\)
0.312858 + 0.949800i \(0.398714\pi\)
\(158\) 3.40536e11 0.275137
\(159\) −7.85784e11 −0.613223
\(160\) −3.97007e10 −0.0299322
\(161\) −2.28022e11 −0.166125
\(162\) 1.57129e11 0.110643
\(163\) −2.66878e12 −1.81669 −0.908345 0.418221i \(-0.862654\pi\)
−0.908345 + 0.418221i \(0.862654\pi\)
\(164\) 1.29890e8 8.54940e−5 0
\(165\) 1.88863e12 1.20223
\(166\) 2.46924e12 1.52044
\(167\) −1.02807e12 −0.612468 −0.306234 0.951956i \(-0.599069\pi\)
−0.306234 + 0.951956i \(0.599069\pi\)
\(168\) 2.56054e11 0.147615
\(169\) 3.02509e12 1.68796
\(170\) 1.51211e12 0.816798
\(171\) −8.31557e11 −0.434924
\(172\) 2.45262e10 0.0124229
\(173\) −1.72017e11 −0.0843951 −0.0421975 0.999109i \(-0.513436\pi\)
−0.0421975 + 0.999109i \(0.513436\pi\)
\(174\) −3.47445e10 −0.0165145
\(175\) −1.20054e12 −0.552926
\(176\) 2.59736e12 1.15934
\(177\) 1.73727e11 0.0751646
\(178\) −1.72040e12 −0.721638
\(179\) 5.62298e11 0.228705 0.114352 0.993440i \(-0.463521\pi\)
0.114352 + 0.993440i \(0.463521\pi\)
\(180\) 1.26473e10 0.00498882
\(181\) −3.52626e12 −1.34922 −0.674609 0.738175i \(-0.735688\pi\)
−0.674609 + 0.738175i \(0.735688\pi\)
\(182\) 1.11985e12 0.415689
\(183\) −2.24371e12 −0.808138
\(184\) −1.87433e12 −0.655160
\(185\) −9.43964e12 −3.20266
\(186\) 1.09919e12 0.362037
\(187\) 1.68402e12 0.538539
\(188\) 1.72229e10 0.00534858
\(189\) −1.62461e11 −0.0490013
\(190\) 7.89741e12 2.31388
\(191\) −2.61082e12 −0.743178 −0.371589 0.928397i \(-0.621187\pi\)
−0.371589 + 0.928397i \(0.621187\pi\)
\(192\) 2.10460e12 0.582120
\(193\) −6.52555e12 −1.75409 −0.877046 0.480407i \(-0.840489\pi\)
−0.877046 + 0.480407i \(0.840489\pi\)
\(194\) −2.66927e12 −0.697402
\(195\) 6.63711e12 1.68573
\(196\) 3.18258e10 0.00785908
\(197\) −4.64972e12 −1.11651 −0.558255 0.829670i \(-0.688529\pi\)
−0.558255 + 0.829670i \(0.688529\pi\)
\(198\) −1.66193e12 −0.388110
\(199\) 6.85711e12 1.55758 0.778788 0.627288i \(-0.215835\pi\)
0.778788 + 0.627288i \(0.215835\pi\)
\(200\) −9.86833e12 −2.18061
\(201\) −4.64659e11 −0.0998979
\(202\) 4.98921e11 0.104376
\(203\) 3.59234e10 0.00731389
\(204\) 1.12771e10 0.00223475
\(205\) 9.39158e10 0.0181173
\(206\) 4.95085e11 0.0929845
\(207\) 1.18922e12 0.217482
\(208\) 9.12775e12 1.62561
\(209\) 8.79524e12 1.52561
\(210\) 1.54291e12 0.260696
\(211\) −1.53005e12 −0.251856 −0.125928 0.992039i \(-0.540191\pi\)
−0.125928 + 0.992039i \(0.540191\pi\)
\(212\) 5.56555e10 0.00892608
\(213\) −4.28764e12 −0.670085
\(214\) −2.18953e12 −0.333485
\(215\) 1.77334e13 2.63257
\(216\) −1.33541e12 −0.193250
\(217\) −1.13649e12 −0.160338
\(218\) −1.16341e13 −1.60037
\(219\) 3.69112e12 0.495126
\(220\) −1.33768e11 −0.0174996
\(221\) 5.91804e12 0.755128
\(222\) 8.30655e12 1.03390
\(223\) −4.41839e11 −0.0536521 −0.0268261 0.999640i \(-0.508540\pi\)
−0.0268261 + 0.999640i \(0.508540\pi\)
\(224\) −3.61205e10 −0.00427947
\(225\) 6.26123e12 0.723862
\(226\) 1.34710e13 1.51986
\(227\) 6.18166e12 0.680711 0.340355 0.940297i \(-0.389453\pi\)
0.340355 + 0.940297i \(0.389453\pi\)
\(228\) 5.88975e10 0.00633075
\(229\) 8.41904e12 0.883421 0.441710 0.897158i \(-0.354372\pi\)
0.441710 + 0.897158i \(0.354372\pi\)
\(230\) −1.12942e13 −1.15705
\(231\) 1.71832e12 0.171885
\(232\) 2.95287e11 0.0288443
\(233\) 1.26195e13 1.20389 0.601944 0.798538i \(-0.294393\pi\)
0.601944 + 0.798538i \(0.294393\pi\)
\(234\) −5.84043e12 −0.544199
\(235\) 1.24528e13 1.13343
\(236\) −1.23047e10 −0.00109410
\(237\) 1.83627e12 0.159522
\(238\) 1.37575e12 0.116779
\(239\) −1.13206e13 −0.939037 −0.469518 0.882923i \(-0.655572\pi\)
−0.469518 + 0.882923i \(0.655572\pi\)
\(240\) 1.25760e13 1.01949
\(241\) 4.78681e12 0.379273 0.189637 0.981854i \(-0.439269\pi\)
0.189637 + 0.981854i \(0.439269\pi\)
\(242\) 4.72061e12 0.365606
\(243\) 8.47289e11 0.0641500
\(244\) 1.58917e11 0.0117633
\(245\) 2.30113e13 1.66544
\(246\) −8.26426e10 −0.00584872
\(247\) 3.09086e13 2.13917
\(248\) −9.34184e12 −0.632337
\(249\) 1.33149e13 0.881537
\(250\) −3.20811e13 −2.07768
\(251\) −4.01399e12 −0.254314 −0.127157 0.991883i \(-0.540585\pi\)
−0.127157 + 0.991883i \(0.540585\pi\)
\(252\) 1.15068e10 0.000713263 0
\(253\) −1.25781e13 −0.762876
\(254\) −7.60677e12 −0.451455
\(255\) 8.15375e12 0.473573
\(256\) −4.43498e11 −0.0252099
\(257\) −1.94695e13 −1.08323 −0.541617 0.840626i \(-0.682188\pi\)
−0.541617 + 0.840626i \(0.682188\pi\)
\(258\) −1.56048e13 −0.849864
\(259\) −8.58838e12 −0.457892
\(260\) −4.70093e11 −0.0245375
\(261\) −1.87353e11 −0.00957497
\(262\) 8.64607e12 0.432675
\(263\) −3.27832e13 −1.60655 −0.803276 0.595607i \(-0.796911\pi\)
−0.803276 + 0.595607i \(0.796911\pi\)
\(264\) 1.41244e13 0.677875
\(265\) 4.02411e13 1.89155
\(266\) 7.18523e12 0.330820
\(267\) −9.27691e12 −0.418400
\(268\) 3.29109e10 0.00145411
\(269\) −1.49518e11 −0.00647224 −0.00323612 0.999995i \(-0.501030\pi\)
−0.00323612 + 0.999995i \(0.501030\pi\)
\(270\) −8.04682e12 −0.341290
\(271\) −3.18363e12 −0.132310 −0.0661548 0.997809i \(-0.521073\pi\)
−0.0661548 + 0.997809i \(0.521073\pi\)
\(272\) 1.12135e13 0.456681
\(273\) 6.03858e12 0.241013
\(274\) −1.66073e13 −0.649639
\(275\) −6.62240e13 −2.53913
\(276\) −8.42298e10 −0.00316567
\(277\) −3.21889e13 −1.18595 −0.592977 0.805219i \(-0.702048\pi\)
−0.592977 + 0.805219i \(0.702048\pi\)
\(278\) 1.86388e13 0.673243
\(279\) 5.92718e12 0.209906
\(280\) −1.31129e13 −0.455334
\(281\) 4.16532e13 1.41828 0.709142 0.705065i \(-0.249082\pi\)
0.709142 + 0.705065i \(0.249082\pi\)
\(282\) −1.09581e13 −0.365901
\(283\) 2.20023e13 0.720515 0.360258 0.932853i \(-0.382689\pi\)
0.360258 + 0.932853i \(0.382689\pi\)
\(284\) 3.03685e11 0.00975376
\(285\) 4.25852e13 1.34157
\(286\) 6.17732e13 1.90892
\(287\) 8.54466e10 0.00259026
\(288\) 1.88381e11 0.00560246
\(289\) −2.70015e13 −0.787862
\(290\) 1.77932e12 0.0509406
\(291\) −1.43935e13 −0.404348
\(292\) −2.61435e11 −0.00720706
\(293\) −4.68622e11 −0.0126780 −0.00633900 0.999980i \(-0.502018\pi\)
−0.00633900 + 0.999980i \(0.502018\pi\)
\(294\) −2.02492e13 −0.537647
\(295\) −8.89679e12 −0.231853
\(296\) −7.05958e13 −1.80582
\(297\) −8.96163e12 −0.225023
\(298\) 6.81059e13 1.67879
\(299\) −4.42026e13 −1.06969
\(300\) −4.43470e11 −0.0105365
\(301\) 1.61342e13 0.376385
\(302\) 2.68540e13 0.615136
\(303\) 2.69033e12 0.0605161
\(304\) 5.85657e13 1.29371
\(305\) 1.14904e14 2.49278
\(306\) −7.17502e12 −0.152882
\(307\) 5.73318e13 1.19987 0.599935 0.800049i \(-0.295193\pi\)
0.599935 + 0.800049i \(0.295193\pi\)
\(308\) −1.21705e11 −0.00250196
\(309\) 2.66965e12 0.0539116
\(310\) −5.62913e13 −1.11674
\(311\) 2.08663e13 0.406690 0.203345 0.979107i \(-0.434819\pi\)
0.203345 + 0.979107i \(0.434819\pi\)
\(312\) 4.96367e13 0.950501
\(313\) −5.90856e13 −1.11170 −0.555850 0.831283i \(-0.687607\pi\)
−0.555850 + 0.831283i \(0.687607\pi\)
\(314\) 3.37022e13 0.623081
\(315\) 8.31984e12 0.151150
\(316\) −1.30060e11 −0.00232201
\(317\) −3.81528e13 −0.669422 −0.334711 0.942321i \(-0.608639\pi\)
−0.334711 + 0.942321i \(0.608639\pi\)
\(318\) −3.54108e13 −0.610641
\(319\) 1.98160e12 0.0335867
\(320\) −1.07780e14 −1.79561
\(321\) −1.18066e13 −0.193352
\(322\) −1.02757e13 −0.165426
\(323\) 3.79715e13 0.600957
\(324\) −6.00118e10 −0.000933768 0
\(325\) −2.32727e14 −3.56031
\(326\) −1.20267e14 −1.80904
\(327\) −6.27344e13 −0.927883
\(328\) 7.02364e11 0.0102154
\(329\) 1.13299e13 0.162049
\(330\) 8.51099e13 1.19716
\(331\) −9.76917e13 −1.35146 −0.675731 0.737148i \(-0.736172\pi\)
−0.675731 + 0.737148i \(0.736172\pi\)
\(332\) −9.43066e11 −0.0128316
\(333\) 4.47914e13 0.599448
\(334\) −4.63294e13 −0.609889
\(335\) 2.37959e13 0.308145
\(336\) 1.14419e13 0.145758
\(337\) −1.51543e14 −1.89920 −0.949600 0.313466i \(-0.898510\pi\)
−0.949600 + 0.313466i \(0.898510\pi\)
\(338\) 1.36324e14 1.68085
\(339\) 7.26398e13 0.881203
\(340\) −5.77514e11 −0.00689332
\(341\) −6.26908e13 −0.736300
\(342\) −3.74735e13 −0.433092
\(343\) 4.33238e13 0.492730
\(344\) 1.32622e14 1.48438
\(345\) −6.09015e13 −0.670846
\(346\) −7.75181e12 −0.0840397
\(347\) 9.52383e13 1.01625 0.508124 0.861284i \(-0.330339\pi\)
0.508124 + 0.861284i \(0.330339\pi\)
\(348\) 1.32698e10 0.000139373 0
\(349\) 8.82646e13 0.912529 0.456265 0.889844i \(-0.349187\pi\)
0.456265 + 0.889844i \(0.349187\pi\)
\(350\) −5.41014e13 −0.550597
\(351\) −3.14933e13 −0.315522
\(352\) −1.99247e12 −0.0196521
\(353\) 1.60371e13 0.155727 0.0778635 0.996964i \(-0.475190\pi\)
0.0778635 + 0.996964i \(0.475190\pi\)
\(354\) 7.82886e12 0.0748481
\(355\) 2.19576e14 2.06695
\(356\) 6.57065e11 0.00609023
\(357\) 7.41846e12 0.0677077
\(358\) 2.53395e13 0.227742
\(359\) −1.81807e14 −1.60913 −0.804566 0.593863i \(-0.797602\pi\)
−0.804566 + 0.593863i \(0.797602\pi\)
\(360\) 6.83884e13 0.596100
\(361\) 8.18261e13 0.702429
\(362\) −1.58908e14 −1.34354
\(363\) 2.54549e13 0.211976
\(364\) −4.27701e11 −0.00350819
\(365\) −1.89028e14 −1.52727
\(366\) −1.01111e14 −0.804735
\(367\) −3.54522e13 −0.277958 −0.138979 0.990295i \(-0.544382\pi\)
−0.138979 + 0.990295i \(0.544382\pi\)
\(368\) −8.37553e13 −0.646918
\(369\) −4.45634e11 −0.00339104
\(370\) −4.25391e14 −3.18918
\(371\) 3.66122e13 0.270439
\(372\) −4.19810e11 −0.00305539
\(373\) −1.39186e14 −0.998150 −0.499075 0.866559i \(-0.666327\pi\)
−0.499075 + 0.866559i \(0.666327\pi\)
\(374\) 7.58890e13 0.536272
\(375\) −1.72991e14 −1.20462
\(376\) 9.31306e13 0.639086
\(377\) 6.96382e12 0.0470945
\(378\) −7.32117e12 −0.0487950
\(379\) −3.23319e12 −0.0212381 −0.0106191 0.999944i \(-0.503380\pi\)
−0.0106191 + 0.999944i \(0.503380\pi\)
\(380\) −3.01623e12 −0.0195278
\(381\) −4.10180e13 −0.261750
\(382\) −1.17655e14 −0.740049
\(383\) 7.59209e13 0.470726 0.235363 0.971908i \(-0.424372\pi\)
0.235363 + 0.971908i \(0.424372\pi\)
\(384\) 9.32546e13 0.569965
\(385\) −8.79975e13 −0.530196
\(386\) −2.94069e14 −1.74670
\(387\) −8.41456e13 −0.492744
\(388\) 1.01946e12 0.00588568
\(389\) −8.62586e12 −0.0490998 −0.0245499 0.999699i \(-0.507815\pi\)
−0.0245499 + 0.999699i \(0.507815\pi\)
\(390\) 2.99097e14 1.67864
\(391\) −5.43033e13 −0.300507
\(392\) 1.72094e14 0.939058
\(393\) 4.66222e13 0.250861
\(394\) −2.09536e14 −1.11181
\(395\) −9.40382e13 −0.492063
\(396\) 6.34734e11 0.00327543
\(397\) −2.23278e13 −0.113631 −0.0568156 0.998385i \(-0.518095\pi\)
−0.0568156 + 0.998385i \(0.518095\pi\)
\(398\) 3.09011e14 1.55102
\(399\) 3.87449e13 0.191807
\(400\) −4.40972e14 −2.15318
\(401\) −2.96381e14 −1.42744 −0.713718 0.700433i \(-0.752990\pi\)
−0.713718 + 0.700433i \(0.752990\pi\)
\(402\) −2.09395e13 −0.0994772
\(403\) −2.20311e14 −1.03242
\(404\) −1.90551e11 −0.000880872 0
\(405\) −4.33909e13 −0.197877
\(406\) 1.61886e12 0.00728310
\(407\) −4.73751e14 −2.10272
\(408\) 6.09791e13 0.267024
\(409\) −2.90683e14 −1.25586 −0.627930 0.778270i \(-0.716098\pi\)
−0.627930 + 0.778270i \(0.716098\pi\)
\(410\) 4.23225e12 0.0180410
\(411\) −8.95517e13 −0.376655
\(412\) −1.89086e11 −0.000784737 0
\(413\) −8.09448e12 −0.0331485
\(414\) 5.35912e13 0.216567
\(415\) −6.81874e14 −2.71919
\(416\) −7.00204e12 −0.0275557
\(417\) 1.00506e14 0.390341
\(418\) 3.96351e14 1.51918
\(419\) 3.58199e14 1.35503 0.677513 0.735511i \(-0.263058\pi\)
0.677513 + 0.735511i \(0.263058\pi\)
\(420\) −5.89278e11 −0.00220013
\(421\) 1.75978e14 0.648497 0.324248 0.945972i \(-0.394889\pi\)
0.324248 + 0.945972i \(0.394889\pi\)
\(422\) −6.89506e13 −0.250795
\(423\) −5.90892e13 −0.212147
\(424\) 3.00949e14 1.06655
\(425\) −2.85907e14 −1.00020
\(426\) −1.93219e14 −0.667264
\(427\) 1.04542e14 0.356399
\(428\) 8.36239e11 0.00281443
\(429\) 3.33100e14 1.10677
\(430\) 7.99143e14 2.62149
\(431\) 1.45826e14 0.472291 0.236145 0.971718i \(-0.424116\pi\)
0.236145 + 0.971718i \(0.424116\pi\)
\(432\) −5.96736e13 −0.190819
\(433\) 3.15377e14 0.995741 0.497871 0.867251i \(-0.334115\pi\)
0.497871 + 0.867251i \(0.334115\pi\)
\(434\) −5.12150e13 −0.159663
\(435\) 9.59461e12 0.0295350
\(436\) 4.44336e12 0.0135063
\(437\) −2.83614e14 −0.851295
\(438\) 1.66338e14 0.493042
\(439\) 9.13881e13 0.267507 0.133753 0.991015i \(-0.457297\pi\)
0.133753 + 0.991015i \(0.457297\pi\)
\(440\) −7.23333e14 −2.09097
\(441\) −1.09190e14 −0.311723
\(442\) 2.66692e14 0.751948
\(443\) −3.80912e14 −1.06073 −0.530364 0.847770i \(-0.677945\pi\)
−0.530364 + 0.847770i \(0.677945\pi\)
\(444\) −3.17249e12 −0.00872556
\(445\) 4.75084e14 1.29060
\(446\) −1.99111e13 −0.0534262
\(447\) 3.67247e14 0.973346
\(448\) −9.80601e13 −0.256722
\(449\) 1.65064e14 0.426871 0.213436 0.976957i \(-0.431535\pi\)
0.213436 + 0.976957i \(0.431535\pi\)
\(450\) 2.82158e14 0.720814
\(451\) 4.71339e12 0.0118950
\(452\) −5.14493e12 −0.0128268
\(453\) 1.44805e14 0.356651
\(454\) 2.78572e14 0.677844
\(455\) −3.09245e14 −0.743429
\(456\) 3.18480e14 0.756442
\(457\) 7.84400e14 1.84077 0.920383 0.391018i \(-0.127877\pi\)
0.920383 + 0.391018i \(0.127877\pi\)
\(458\) 3.79398e14 0.879701
\(459\) −3.86898e13 −0.0886395
\(460\) 4.31353e12 0.00976483
\(461\) 4.60652e14 1.03043 0.515215 0.857061i \(-0.327712\pi\)
0.515215 + 0.857061i \(0.327712\pi\)
\(462\) 7.74348e13 0.171161
\(463\) 5.74513e14 1.25489 0.627443 0.778662i \(-0.284101\pi\)
0.627443 + 0.778662i \(0.284101\pi\)
\(464\) 1.31951e13 0.0284815
\(465\) −3.03539e14 −0.647476
\(466\) 5.68690e14 1.19882
\(467\) 6.03014e14 1.25627 0.628137 0.778102i \(-0.283818\pi\)
0.628137 + 0.778102i \(0.283818\pi\)
\(468\) 2.23061e12 0.00459273
\(469\) 2.16500e13 0.0440562
\(470\) 5.61178e14 1.12866
\(471\) 1.81732e14 0.361257
\(472\) −6.65360e13 −0.130730
\(473\) 8.89994e14 1.72843
\(474\) 8.27503e13 0.158851
\(475\) −1.49323e15 −2.83342
\(476\) −5.25434e11 −0.000985554 0
\(477\) −1.90945e14 −0.354045
\(478\) −5.10156e14 −0.935083
\(479\) 7.01199e14 1.27056 0.635282 0.772281i \(-0.280884\pi\)
0.635282 + 0.772281i \(0.280884\pi\)
\(480\) −9.64726e12 −0.0172813
\(481\) −1.66488e15 −2.94838
\(482\) 2.15714e14 0.377676
\(483\) −5.54095e13 −0.0959124
\(484\) −1.80292e12 −0.00308551
\(485\) 7.37112e14 1.24725
\(486\) 3.81824e13 0.0638799
\(487\) 8.80594e14 1.45669 0.728343 0.685212i \(-0.240291\pi\)
0.728343 + 0.685212i \(0.240291\pi\)
\(488\) 8.59324e14 1.40556
\(489\) −6.48513e14 −1.04887
\(490\) 1.03699e15 1.65843
\(491\) −9.75945e14 −1.54340 −0.771698 0.635989i \(-0.780592\pi\)
−0.771698 + 0.635989i \(0.780592\pi\)
\(492\) 3.15633e10 4.93600e−5 0
\(493\) 8.55512e12 0.0132302
\(494\) 1.39287e15 2.13016
\(495\) 4.58938e14 0.694105
\(496\) −4.17445e14 −0.624382
\(497\) 1.99775e14 0.295516
\(498\) 6.00025e14 0.877825
\(499\) −5.44945e14 −0.788496 −0.394248 0.919004i \(-0.628995\pi\)
−0.394248 + 0.919004i \(0.628995\pi\)
\(500\) 1.22526e13 0.0175345
\(501\) −2.49822e14 −0.353609
\(502\) −1.80887e14 −0.253243
\(503\) 4.94822e14 0.685212 0.342606 0.939479i \(-0.388690\pi\)
0.342606 + 0.939479i \(0.388690\pi\)
\(504\) 6.22212e13 0.0852257
\(505\) −1.37776e14 −0.186668
\(506\) −5.66825e14 −0.759663
\(507\) 7.35098e14 0.974544
\(508\) 2.90522e12 0.00381003
\(509\) 6.49641e14 0.842802 0.421401 0.906874i \(-0.361538\pi\)
0.421401 + 0.906874i \(0.361538\pi\)
\(510\) 3.67443e14 0.471579
\(511\) −1.71981e14 −0.218357
\(512\) −8.05934e14 −1.01231
\(513\) −2.02068e14 −0.251103
\(514\) −8.77378e14 −1.07867
\(515\) −1.36717e14 −0.166296
\(516\) 5.95987e12 0.00717238
\(517\) 6.24976e14 0.744159
\(518\) −3.87029e14 −0.455964
\(519\) −4.18001e13 −0.0487255
\(520\) −2.54197e15 −2.93192
\(521\) 1.36599e15 1.55898 0.779489 0.626416i \(-0.215479\pi\)
0.779489 + 0.626416i \(0.215479\pi\)
\(522\) −8.44292e12 −0.00953465
\(523\) −1.68812e15 −1.88644 −0.943221 0.332165i \(-0.892221\pi\)
−0.943221 + 0.332165i \(0.892221\pi\)
\(524\) −3.30216e12 −0.00365154
\(525\) −2.91731e14 −0.319232
\(526\) −1.47735e15 −1.59979
\(527\) −2.70654e14 −0.290038
\(528\) 6.31158e14 0.669348
\(529\) −5.47211e14 −0.574313
\(530\) 1.81344e15 1.88358
\(531\) 4.22156e13 0.0433963
\(532\) −2.74422e12 −0.00279194
\(533\) 1.65640e13 0.0166788
\(534\) −4.18057e14 −0.416638
\(535\) 6.04634e14 0.596412
\(536\) 1.77961e14 0.173748
\(537\) 1.36638e14 0.132043
\(538\) −6.73790e12 −0.00644499
\(539\) 1.15488e15 1.09345
\(540\) 3.07329e12 0.00288030
\(541\) 6.54086e14 0.606806 0.303403 0.952862i \(-0.401877\pi\)
0.303403 + 0.952862i \(0.401877\pi\)
\(542\) −1.43468e14 −0.131752
\(543\) −8.56882e14 −0.778972
\(544\) −8.60207e12 −0.00774121
\(545\) 3.21272e15 2.86215
\(546\) 2.72124e14 0.239998
\(547\) −2.23272e15 −1.94941 −0.974707 0.223488i \(-0.928256\pi\)
−0.974707 + 0.223488i \(0.928256\pi\)
\(548\) 6.34277e12 0.00548259
\(549\) −5.45221e14 −0.466579
\(550\) −2.98433e15 −2.52844
\(551\) 4.46814e13 0.0374794
\(552\) −4.55461e14 −0.378257
\(553\) −8.55579e13 −0.0703513
\(554\) −1.45057e15 −1.18096
\(555\) −2.29383e15 −1.84906
\(556\) −7.11864e12 −0.00568180
\(557\) 1.39841e15 1.10518 0.552589 0.833454i \(-0.313640\pi\)
0.552589 + 0.833454i \(0.313640\pi\)
\(558\) 2.67104e14 0.209022
\(559\) 3.12765e15 2.42356
\(560\) −5.85958e14 −0.449606
\(561\) 4.09216e14 0.310926
\(562\) 1.87707e15 1.41231
\(563\) −3.16839e14 −0.236071 −0.118035 0.993009i \(-0.537660\pi\)
−0.118035 + 0.993009i \(0.537660\pi\)
\(564\) 4.18517e12 0.00308800
\(565\) −3.71999e15 −2.71816
\(566\) 9.91519e14 0.717481
\(567\) −3.94779e13 −0.0282909
\(568\) 1.64213e15 1.16545
\(569\) 2.70314e15 1.89999 0.949996 0.312262i \(-0.101087\pi\)
0.949996 + 0.312262i \(0.101087\pi\)
\(570\) 1.91907e15 1.33592
\(571\) −4.19562e14 −0.289266 −0.144633 0.989485i \(-0.546200\pi\)
−0.144633 + 0.989485i \(0.546200\pi\)
\(572\) −2.35928e13 −0.0161102
\(573\) −6.34429e14 −0.429074
\(574\) 3.85059e12 0.00257936
\(575\) 2.13548e15 1.41684
\(576\) 5.11418e14 0.336087
\(577\) 1.04042e15 0.677239 0.338619 0.940923i \(-0.390040\pi\)
0.338619 + 0.940923i \(0.390040\pi\)
\(578\) −1.21680e15 −0.784545
\(579\) −1.58571e15 −1.01272
\(580\) −6.79567e11 −0.000429911 0
\(581\) −6.20383e14 −0.388769
\(582\) −6.48633e14 −0.402645
\(583\) 2.01960e15 1.24190
\(584\) −1.41367e15 −0.861150
\(585\) 1.61282e15 0.973259
\(586\) −2.11181e13 −0.0126246
\(587\) 1.53474e14 0.0908918 0.0454459 0.998967i \(-0.485529\pi\)
0.0454459 + 0.998967i \(0.485529\pi\)
\(588\) 7.73368e12 0.00453744
\(589\) −1.41356e15 −0.821639
\(590\) −4.00927e14 −0.230876
\(591\) −1.12988e15 −0.644617
\(592\) −3.15461e15 −1.78310
\(593\) 2.29184e15 1.28346 0.641731 0.766930i \(-0.278217\pi\)
0.641731 + 0.766930i \(0.278217\pi\)
\(594\) −4.03849e14 −0.224075
\(595\) −3.79910e14 −0.208851
\(596\) −2.60114e13 −0.0141680
\(597\) 1.66628e15 0.899266
\(598\) −1.99196e15 −1.06518
\(599\) 2.05085e15 1.08664 0.543322 0.839524i \(-0.317166\pi\)
0.543322 + 0.839524i \(0.317166\pi\)
\(600\) −2.39800e15 −1.25898
\(601\) 2.75368e15 1.43253 0.716264 0.697829i \(-0.245851\pi\)
0.716264 + 0.697829i \(0.245851\pi\)
\(602\) 7.27077e14 0.374800
\(603\) −1.12912e14 −0.0576761
\(604\) −1.02562e13 −0.00519141
\(605\) −1.30358e15 −0.653860
\(606\) 1.21238e14 0.0602613
\(607\) −2.93931e15 −1.44780 −0.723899 0.689906i \(-0.757652\pi\)
−0.723899 + 0.689906i \(0.757652\pi\)
\(608\) −4.49266e13 −0.0219298
\(609\) 8.72937e12 0.00422268
\(610\) 5.17804e15 2.48229
\(611\) 2.19632e15 1.04344
\(612\) 2.74032e12 0.00129024
\(613\) −1.72820e15 −0.806421 −0.403210 0.915107i \(-0.632106\pi\)
−0.403210 + 0.915107i \(0.632106\pi\)
\(614\) 2.58361e15 1.19482
\(615\) 2.28215e13 0.0104600
\(616\) −6.58104e14 −0.298951
\(617\) 2.86506e14 0.128993 0.0644964 0.997918i \(-0.479456\pi\)
0.0644964 + 0.997918i \(0.479456\pi\)
\(618\) 1.20306e14 0.0536846
\(619\) −2.04300e15 −0.903588 −0.451794 0.892122i \(-0.649216\pi\)
−0.451794 + 0.892122i \(0.649216\pi\)
\(620\) 2.14991e13 0.00942466
\(621\) 2.88980e14 0.125564
\(622\) 9.40326e14 0.404978
\(623\) 4.32241e14 0.184519
\(624\) 2.21804e15 0.938544
\(625\) 3.68165e15 1.54420
\(626\) −2.66265e15 −1.10702
\(627\) 2.13724e15 0.880810
\(628\) −1.28717e13 −0.00525846
\(629\) −2.04532e15 −0.828289
\(630\) 3.74927e14 0.150513
\(631\) −1.86553e15 −0.742405 −0.371203 0.928552i \(-0.621055\pi\)
−0.371203 + 0.928552i \(0.621055\pi\)
\(632\) −7.03280e14 −0.277450
\(633\) −3.71802e14 −0.145409
\(634\) −1.71933e15 −0.666603
\(635\) 2.10059e15 0.807394
\(636\) 1.35243e13 0.00515347
\(637\) 4.05853e15 1.53321
\(638\) 8.92994e13 0.0334453
\(639\) −1.04190e15 −0.386874
\(640\) −4.77570e15 −1.75812
\(641\) −2.20124e15 −0.803429 −0.401715 0.915765i \(-0.631586\pi\)
−0.401715 + 0.915765i \(0.631586\pi\)
\(642\) −5.32057e14 −0.192537
\(643\) −2.16711e15 −0.777534 −0.388767 0.921336i \(-0.627099\pi\)
−0.388767 + 0.921336i \(0.627099\pi\)
\(644\) 3.92454e12 0.00139610
\(645\) 4.30922e15 1.51992
\(646\) 1.71116e15 0.598427
\(647\) −3.64758e15 −1.26483 −0.632413 0.774631i \(-0.717936\pi\)
−0.632413 + 0.774631i \(0.717936\pi\)
\(648\) −3.24505e14 −0.111573
\(649\) −4.46507e14 −0.152224
\(650\) −1.04877e16 −3.54532
\(651\) −2.76167e14 −0.0925711
\(652\) 4.59329e13 0.0152673
\(653\) −8.51590e14 −0.280678 −0.140339 0.990104i \(-0.544819\pi\)
−0.140339 + 0.990104i \(0.544819\pi\)
\(654\) −2.82708e15 −0.923976
\(655\) −2.38759e15 −0.773807
\(656\) 3.13855e13 0.0100869
\(657\) 8.96943e14 0.285861
\(658\) 5.10572e14 0.161367
\(659\) −1.70154e15 −0.533302 −0.266651 0.963793i \(-0.585917\pi\)
−0.266651 + 0.963793i \(0.585917\pi\)
\(660\) −3.25057e13 −0.0101034
\(661\) −4.26270e15 −1.31394 −0.656972 0.753915i \(-0.728163\pi\)
−0.656972 + 0.753915i \(0.728163\pi\)
\(662\) −4.40241e15 −1.34577
\(663\) 1.43808e15 0.435973
\(664\) −5.09950e15 −1.53322
\(665\) −1.98418e15 −0.591647
\(666\) 2.01849e15 0.596924
\(667\) −6.38993e13 −0.0187415
\(668\) 1.76944e13 0.00514713
\(669\) −1.07367e14 −0.0309761
\(670\) 1.07234e15 0.306847
\(671\) 5.76671e15 1.63665
\(672\) −8.77728e12 −0.00247075
\(673\) 6.74193e15 1.88236 0.941178 0.337912i \(-0.109721\pi\)
0.941178 + 0.337912i \(0.109721\pi\)
\(674\) −6.82916e15 −1.89120
\(675\) 1.52148e15 0.417922
\(676\) −5.20655e13 −0.0141855
\(677\) 4.63491e15 1.25257 0.626287 0.779592i \(-0.284574\pi\)
0.626287 + 0.779592i \(0.284574\pi\)
\(678\) 3.27346e15 0.877493
\(679\) 6.70640e14 0.178322
\(680\) −3.12283e15 −0.823662
\(681\) 1.50214e15 0.393009
\(682\) −2.82512e15 −0.733200
\(683\) 7.01051e15 1.80483 0.902414 0.430870i \(-0.141793\pi\)
0.902414 + 0.430870i \(0.141793\pi\)
\(684\) 1.43121e13 0.00365506
\(685\) 4.58607e15 1.16183
\(686\) 1.95236e15 0.490655
\(687\) 2.04583e15 0.510043
\(688\) 5.92629e15 1.46570
\(689\) 7.09735e15 1.74137
\(690\) −2.74448e15 −0.668021
\(691\) −5.90367e15 −1.42558 −0.712792 0.701376i \(-0.752570\pi\)
−0.712792 + 0.701376i \(0.752570\pi\)
\(692\) 2.96062e12 0.000709249 0
\(693\) 4.17551e14 0.0992378
\(694\) 4.29184e15 1.01197
\(695\) −5.14706e15 −1.20404
\(696\) 7.17548e13 0.0166533
\(697\) 2.03490e13 0.00468558
\(698\) 3.97758e15 0.908687
\(699\) 3.06655e15 0.695065
\(700\) 2.06627e13 0.00464674
\(701\) 6.10899e15 1.36308 0.681539 0.731782i \(-0.261311\pi\)
0.681539 + 0.731782i \(0.261311\pi\)
\(702\) −1.41922e15 −0.314193
\(703\) −1.06822e16 −2.34643
\(704\) −5.40918e15 −1.17891
\(705\) 3.02604e15 0.654387
\(706\) 7.22698e14 0.155071
\(707\) −1.25351e14 −0.0266883
\(708\) −2.99004e12 −0.000631676 0
\(709\) 2.94639e15 0.617642 0.308821 0.951120i \(-0.400066\pi\)
0.308821 + 0.951120i \(0.400066\pi\)
\(710\) 9.89503e15 2.05824
\(711\) 4.46215e14 0.0921003
\(712\) 3.55299e15 0.727703
\(713\) 2.02155e15 0.410858
\(714\) 3.34307e14 0.0674226
\(715\) −1.70585e16 −3.41396
\(716\) −9.67782e12 −0.00192201
\(717\) −2.75092e15 −0.542153
\(718\) −8.19301e15 −1.60236
\(719\) −7.99512e15 −1.55173 −0.775865 0.630899i \(-0.782686\pi\)
−0.775865 + 0.630899i \(0.782686\pi\)
\(720\) 3.05597e15 0.588601
\(721\) −1.24388e14 −0.0237757
\(722\) 3.68743e15 0.699471
\(723\) 1.16319e15 0.218973
\(724\) 6.06912e13 0.0113387
\(725\) −3.36430e14 −0.0623786
\(726\) 1.14711e15 0.211083
\(727\) 2.89015e15 0.527815 0.263907 0.964548i \(-0.414989\pi\)
0.263907 + 0.964548i \(0.414989\pi\)
\(728\) −2.31273e15 −0.419183
\(729\) 2.05891e14 0.0370370
\(730\) −8.51840e15 −1.52084
\(731\) 3.84235e15 0.680850
\(732\) 3.86169e13 0.00679152
\(733\) 3.30445e15 0.576802 0.288401 0.957510i \(-0.406876\pi\)
0.288401 + 0.957510i \(0.406876\pi\)
\(734\) −1.59763e15 −0.276788
\(735\) 5.59176e15 0.961541
\(736\) 6.42500e13 0.0109659
\(737\) 1.19425e15 0.202314
\(738\) −2.00822e13 −0.00337676
\(739\) −9.69025e15 −1.61730 −0.808650 0.588290i \(-0.799801\pi\)
−0.808650 + 0.588290i \(0.799801\pi\)
\(740\) 1.62468e14 0.0269149
\(741\) 7.51078e15 1.23505
\(742\) 1.64990e15 0.269300
\(743\) −7.43546e15 −1.20467 −0.602336 0.798242i \(-0.705763\pi\)
−0.602336 + 0.798242i \(0.705763\pi\)
\(744\) −2.27007e15 −0.365080
\(745\) −1.88073e16 −3.00238
\(746\) −6.27230e15 −0.993947
\(747\) 3.23551e15 0.508956
\(748\) −2.89840e13 −0.00452584
\(749\) 5.50109e14 0.0852704
\(750\) −7.79571e15 −1.19955
\(751\) 1.07005e16 1.63450 0.817250 0.576283i \(-0.195497\pi\)
0.817250 + 0.576283i \(0.195497\pi\)
\(752\) 4.16159e15 0.631046
\(753\) −9.75399e14 −0.146828
\(754\) 3.13819e14 0.0468962
\(755\) −7.41567e15 −1.10012
\(756\) 2.79614e12 0.000411803 0
\(757\) 1.05616e16 1.54419 0.772094 0.635508i \(-0.219209\pi\)
0.772094 + 0.635508i \(0.219209\pi\)
\(758\) −1.45701e14 −0.0211487
\(759\) −3.05649e15 −0.440447
\(760\) −1.63098e16 −2.33332
\(761\) 2.67131e15 0.379410 0.189705 0.981841i \(-0.439247\pi\)
0.189705 + 0.981841i \(0.439247\pi\)
\(762\) −1.84845e15 −0.260648
\(763\) 2.92300e15 0.409208
\(764\) 4.49353e13 0.00624560
\(765\) 1.98136e15 0.273417
\(766\) 3.42132e15 0.468744
\(767\) −1.56913e15 −0.213445
\(768\) −1.07770e14 −0.0145550
\(769\) 5.22731e15 0.700944 0.350472 0.936573i \(-0.386021\pi\)
0.350472 + 0.936573i \(0.386021\pi\)
\(770\) −3.96555e15 −0.527964
\(771\) −4.73108e15 −0.625405
\(772\) 1.12313e14 0.0147412
\(773\) −2.43776e15 −0.317690 −0.158845 0.987304i \(-0.550777\pi\)
−0.158845 + 0.987304i \(0.550777\pi\)
\(774\) −3.79196e15 −0.490669
\(775\) 1.06435e16 1.36749
\(776\) 5.51261e15 0.703263
\(777\) −2.08698e15 −0.264364
\(778\) −3.88718e14 −0.0488931
\(779\) 1.06278e14 0.0132736
\(780\) −1.14233e14 −0.0141667
\(781\) 1.10200e16 1.35706
\(782\) −2.44714e15 −0.299242
\(783\) −4.55267e13 −0.00552811
\(784\) 7.69011e15 0.927244
\(785\) −9.30676e15 −1.11433
\(786\) 2.10100e15 0.249805
\(787\) 5.96374e15 0.704138 0.352069 0.935974i \(-0.385478\pi\)
0.352069 + 0.935974i \(0.385478\pi\)
\(788\) 8.00272e13 0.00938304
\(789\) −7.96631e15 −0.927543
\(790\) −4.23776e15 −0.489991
\(791\) −3.38452e15 −0.388621
\(792\) 3.43224e15 0.391371
\(793\) 2.02656e16 2.29487
\(794\) −1.00619e15 −0.113153
\(795\) 9.77859e15 1.09209
\(796\) −1.18019e14 −0.0130897
\(797\) −1.14517e16 −1.26139 −0.630693 0.776033i \(-0.717229\pi\)
−0.630693 + 0.776033i \(0.717229\pi\)
\(798\) 1.74601e15 0.190999
\(799\) 2.69820e15 0.293134
\(800\) 3.38276e14 0.0364986
\(801\) −2.25429e15 −0.241563
\(802\) −1.33562e16 −1.42143
\(803\) −9.48682e15 −1.00273
\(804\) 7.99734e12 0.000839532 0
\(805\) 2.83760e15 0.295851
\(806\) −9.92814e15 −1.02808
\(807\) −3.63328e13 −0.00373675
\(808\) −1.03038e15 −0.105253
\(809\) 1.32160e16 1.34086 0.670429 0.741974i \(-0.266110\pi\)
0.670429 + 0.741974i \(0.266110\pi\)
\(810\) −1.95538e15 −0.197044
\(811\) −2.80378e15 −0.280627 −0.140313 0.990107i \(-0.544811\pi\)
−0.140313 + 0.990107i \(0.544811\pi\)
\(812\) −6.18284e11 −6.14653e−5 0
\(813\) −7.73621e14 −0.0763889
\(814\) −2.13493e16 −2.09386
\(815\) 3.32113e16 3.23533
\(816\) 2.72489e15 0.263665
\(817\) 2.00677e16 1.92875
\(818\) −1.30994e16 −1.25057
\(819\) 1.46738e15 0.139149
\(820\) −1.61640e12 −0.000152256 0
\(821\) −1.80837e16 −1.69199 −0.845997 0.533188i \(-0.820994\pi\)
−0.845997 + 0.533188i \(0.820994\pi\)
\(822\) −4.03558e15 −0.375069
\(823\) −3.00103e15 −0.277058 −0.138529 0.990358i \(-0.544237\pi\)
−0.138529 + 0.990358i \(0.544237\pi\)
\(824\) −1.02246e15 −0.0937659
\(825\) −1.60924e16 −1.46597
\(826\) −3.64772e14 −0.0330089
\(827\) −1.21648e15 −0.109351 −0.0546756 0.998504i \(-0.517412\pi\)
−0.0546756 + 0.998504i \(0.517412\pi\)
\(828\) −2.04679e13 −0.00182770
\(829\) 1.13958e16 1.01087 0.505433 0.862866i \(-0.331333\pi\)
0.505433 + 0.862866i \(0.331333\pi\)
\(830\) −3.07281e16 −2.70774
\(831\) −7.82191e15 −0.684711
\(832\) −1.90092e16 −1.65305
\(833\) 4.98594e15 0.430724
\(834\) 4.52923e15 0.388697
\(835\) 1.27937e16 1.09074
\(836\) −1.51377e14 −0.0128211
\(837\) 1.44031e15 0.121189
\(838\) 1.61420e16 1.34932
\(839\) 8.39530e15 0.697181 0.348591 0.937275i \(-0.386660\pi\)
0.348591 + 0.937275i \(0.386660\pi\)
\(840\) −3.18644e15 −0.262887
\(841\) −1.21904e16 −0.999175
\(842\) 7.93034e15 0.645766
\(843\) 1.01217e16 0.818847
\(844\) 2.63340e13 0.00211657
\(845\) −3.76454e16 −3.00608
\(846\) −2.66281e15 −0.211253
\(847\) −1.18603e15 −0.0934838
\(848\) 1.34481e16 1.05313
\(849\) 5.34657e15 0.415990
\(850\) −1.28842e16 −0.995986
\(851\) 1.52767e16 1.17332
\(852\) 7.37954e13 0.00563134
\(853\) 3.40599e15 0.258240 0.129120 0.991629i \(-0.458785\pi\)
0.129120 + 0.991629i \(0.458785\pi\)
\(854\) 4.71109e15 0.354898
\(855\) 1.03482e16 0.774553
\(856\) 4.52185e15 0.336287
\(857\) −1.41524e16 −1.04577 −0.522886 0.852403i \(-0.675145\pi\)
−0.522886 + 0.852403i \(0.675145\pi\)
\(858\) 1.50109e16 1.10211
\(859\) −2.08355e16 −1.52000 −0.759998 0.649925i \(-0.774800\pi\)
−0.759998 + 0.649925i \(0.774800\pi\)
\(860\) −3.05213e14 −0.0221239
\(861\) 2.07635e13 0.00149549
\(862\) 6.57154e15 0.470302
\(863\) −1.88043e16 −1.33720 −0.668601 0.743621i \(-0.733107\pi\)
−0.668601 + 0.743621i \(0.733107\pi\)
\(864\) 4.57766e13 0.00323458
\(865\) 2.14064e15 0.150299
\(866\) 1.42122e16 0.991549
\(867\) −6.56137e15 −0.454872
\(868\) 1.95603e13 0.00134747
\(869\) −4.71954e15 −0.323066
\(870\) 4.32374e14 0.0294106
\(871\) 4.19689e15 0.283680
\(872\) 2.40268e16 1.61382
\(873\) −3.49762e15 −0.233450
\(874\) −1.27809e16 −0.847710
\(875\) 8.06021e15 0.531254
\(876\) −6.35287e13 −0.00416100
\(877\) 1.88287e16 1.22552 0.612761 0.790268i \(-0.290059\pi\)
0.612761 + 0.790268i \(0.290059\pi\)
\(878\) 4.11834e15 0.266380
\(879\) −1.13875e14 −0.00731965
\(880\) −3.23225e16 −2.06467
\(881\) −2.24685e16 −1.42629 −0.713144 0.701018i \(-0.752729\pi\)
−0.713144 + 0.701018i \(0.752729\pi\)
\(882\) −4.92055e15 −0.310411
\(883\) −3.23036e15 −0.202520 −0.101260 0.994860i \(-0.532287\pi\)
−0.101260 + 0.994860i \(0.532287\pi\)
\(884\) −1.01857e14 −0.00634602
\(885\) −2.16192e15 −0.133860
\(886\) −1.71655e16 −1.05626
\(887\) 1.42033e16 0.868581 0.434290 0.900773i \(-0.356999\pi\)
0.434290 + 0.900773i \(0.356999\pi\)
\(888\) −1.71548e16 −1.04259
\(889\) 1.91116e15 0.115435
\(890\) 2.14093e16 1.28516
\(891\) −2.17768e15 −0.129917
\(892\) 7.60457e12 0.000450887 0
\(893\) 1.40921e16 0.830408
\(894\) 1.65497e16 0.969247
\(895\) −6.99745e15 −0.407299
\(896\) −4.34503e15 −0.251362
\(897\) −1.07412e16 −0.617584
\(898\) 7.43848e15 0.425074
\(899\) −3.18481e14 −0.0180886
\(900\) −1.07763e14 −0.00608327
\(901\) 8.71916e15 0.489202
\(902\) 2.12406e14 0.0118449
\(903\) 3.92062e15 0.217306
\(904\) −2.78205e16 −1.53263
\(905\) 4.38821e16 2.40282
\(906\) 6.52553e15 0.355149
\(907\) −2.40970e16 −1.30354 −0.651769 0.758418i \(-0.725973\pi\)
−0.651769 + 0.758418i \(0.725973\pi\)
\(908\) −1.06394e14 −0.00572063
\(909\) 6.53750e14 0.0349390
\(910\) −1.39359e16 −0.740298
\(911\) −1.59187e16 −0.840536 −0.420268 0.907400i \(-0.638064\pi\)
−0.420268 + 0.907400i \(0.638064\pi\)
\(912\) 1.42315e16 0.746926
\(913\) −3.42215e16 −1.78529
\(914\) 3.53484e16 1.83302
\(915\) 2.79216e16 1.43921
\(916\) −1.44902e14 −0.00742419
\(917\) −2.17228e15 −0.110633
\(918\) −1.74353e15 −0.0882662
\(919\) 3.25248e16 1.63674 0.818370 0.574692i \(-0.194878\pi\)
0.818370 + 0.574692i \(0.194878\pi\)
\(920\) 2.33248e16 1.16677
\(921\) 1.39316e16 0.692745
\(922\) 2.07590e16 1.02609
\(923\) 3.87268e16 1.90284
\(924\) −2.95743e13 −0.00144450
\(925\) 8.04321e16 3.90526
\(926\) 2.58900e16 1.24960
\(927\) 6.48724e14 0.0311259
\(928\) −1.01221e13 −0.000482790 0
\(929\) −1.72874e16 −0.819677 −0.409838 0.912158i \(-0.634415\pi\)
−0.409838 + 0.912158i \(0.634415\pi\)
\(930\) −1.36788e16 −0.644750
\(931\) 2.60404e16 1.22018
\(932\) −2.17197e14 −0.0101174
\(933\) 5.07052e15 0.234803
\(934\) 2.71744e16 1.25099
\(935\) −2.09565e16 −0.959082
\(936\) 1.20617e16 0.548772
\(937\) 1.56294e16 0.706925 0.353462 0.935449i \(-0.385004\pi\)
0.353462 + 0.935449i \(0.385004\pi\)
\(938\) 9.75640e14 0.0438707
\(939\) −1.43578e16 −0.641840
\(940\) −2.14328e14 −0.00952526
\(941\) −1.99183e16 −0.880054 −0.440027 0.897985i \(-0.645031\pi\)
−0.440027 + 0.897985i \(0.645031\pi\)
\(942\) 8.18963e15 0.359736
\(943\) −1.51990e14 −0.00663742
\(944\) −2.97320e15 −0.129086
\(945\) 2.02172e15 0.0872662
\(946\) 4.01069e16 1.72115
\(947\) 1.45996e16 0.622899 0.311449 0.950263i \(-0.399186\pi\)
0.311449 + 0.950263i \(0.399186\pi\)
\(948\) −3.16045e13 −0.00134061
\(949\) −3.33390e16 −1.40601
\(950\) −6.72912e16 −2.82149
\(951\) −9.27112e15 −0.386491
\(952\) −2.84122e15 −0.117761
\(953\) 4.55783e15 0.187822 0.0939112 0.995581i \(-0.470063\pi\)
0.0939112 + 0.995581i \(0.470063\pi\)
\(954\) −8.60482e15 −0.352554
\(955\) 3.24900e16 1.32352
\(956\) 1.94842e14 0.00789158
\(957\) 4.81529e14 0.0193913
\(958\) 3.15990e16 1.26521
\(959\) 4.17250e15 0.166110
\(960\) −2.61904e16 −1.03669
\(961\) −1.53329e16 −0.603455
\(962\) −7.50265e16 −2.93597
\(963\) −2.86901e15 −0.111632
\(964\) −8.23867e13 −0.00318738
\(965\) 8.12065e16 3.12385
\(966\) −2.49699e15 −0.0955085
\(967\) −2.52536e16 −0.960457 −0.480228 0.877144i \(-0.659446\pi\)
−0.480228 + 0.877144i \(0.659446\pi\)
\(968\) −9.74906e15 −0.368679
\(969\) 9.22707e15 0.346963
\(970\) 3.32174e16 1.24200
\(971\) 7.21331e14 0.0268182 0.0134091 0.999910i \(-0.495732\pi\)
0.0134091 + 0.999910i \(0.495732\pi\)
\(972\) −1.45829e13 −0.000539111 0
\(973\) −4.68290e15 −0.172145
\(974\) 3.96833e16 1.45055
\(975\) −5.65526e16 −2.05555
\(976\) 3.83994e16 1.38787
\(977\) −1.43712e15 −0.0516503 −0.0258251 0.999666i \(-0.508221\pi\)
−0.0258251 + 0.999666i \(0.508221\pi\)
\(978\) −2.92248e16 −1.04445
\(979\) 2.38432e16 0.847346
\(980\) −3.96053e14 −0.0139962
\(981\) −1.52445e16 −0.535714
\(982\) −4.39803e16 −1.53690
\(983\) 3.62501e16 1.25969 0.629847 0.776719i \(-0.283118\pi\)
0.629847 + 0.776719i \(0.283118\pi\)
\(984\) 1.70674e14 0.00589787
\(985\) 5.78629e16 1.98839
\(986\) 3.85530e14 0.0131745
\(987\) 2.75316e15 0.0935592
\(988\) −5.31974e14 −0.0179774
\(989\) −2.86990e16 −0.964468
\(990\) 2.06817e16 0.691183
\(991\) 3.84107e16 1.27658 0.638289 0.769797i \(-0.279643\pi\)
0.638289 + 0.769797i \(0.279643\pi\)
\(992\) 3.20229e14 0.0105839
\(993\) −2.37391e16 −0.780267
\(994\) 9.00271e15 0.294271
\(995\) −8.53324e16 −2.77388
\(996\) −2.29165e14 −0.00740836
\(997\) −3.93768e16 −1.26595 −0.632975 0.774172i \(-0.718167\pi\)
−0.632975 + 0.774172i \(0.718167\pi\)
\(998\) −2.45576e16 −0.785176
\(999\) 1.08843e16 0.346091
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.1.20 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.12.a.d.1.20 28 1.1 even 1 trivial