Properties

Label 43.8.a.a.1.8
Level $43$
Weight $8$
Character 43.1
Self dual yes
Analytic conductor $13.433$
Analytic rank $1$
Dimension $11$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 43 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 43.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.4325560958\)
Analytic rank: \(1\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Defining polynomial: \(x^{11} - 2 x^{10} - 977 x^{9} + 2592 x^{8} + 344686 x^{7} - 1160956 x^{6} - 53409536 x^{5} + 209758592 x^{4} + 3410917248 x^{3} - 14180732672 x^{2} - 60918607872 x + 238240894976\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6}\cdot 7 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(-10.3097\) of defining polynomial
Character \(\chi\) \(=\) 43.1

$q$-expansion

\(f(q)\) \(=\) \(q+8.30971 q^{2} +47.1073 q^{3} -58.9487 q^{4} -187.284 q^{5} +391.449 q^{6} -1501.21 q^{7} -1553.49 q^{8} +32.1014 q^{9} +O(q^{10})\) \(q+8.30971 q^{2} +47.1073 q^{3} -58.9487 q^{4} -187.284 q^{5} +391.449 q^{6} -1501.21 q^{7} -1553.49 q^{8} +32.1014 q^{9} -1556.27 q^{10} +1942.03 q^{11} -2776.91 q^{12} -4434.78 q^{13} -12474.6 q^{14} -8822.44 q^{15} -5363.63 q^{16} +31834.8 q^{17} +266.753 q^{18} +8145.26 q^{19} +11040.1 q^{20} -70718.0 q^{21} +16137.7 q^{22} +11917.4 q^{23} -73180.8 q^{24} -43049.8 q^{25} -36851.8 q^{26} -101512. q^{27} +88494.3 q^{28} -115330. q^{29} -73311.9 q^{30} -100315. q^{31} +154276. q^{32} +91484.1 q^{33} +264538. q^{34} +281152. q^{35} -1892.33 q^{36} -455058. q^{37} +67684.8 q^{38} -208911. q^{39} +290943. q^{40} +396428. q^{41} -587646. q^{42} +79507.0 q^{43} -114480. q^{44} -6012.06 q^{45} +99029.9 q^{46} +249124. q^{47} -252666. q^{48} +1.43009e6 q^{49} -357732. q^{50} +1.49965e6 q^{51} +261424. q^{52} -363774. q^{53} -843532. q^{54} -363711. q^{55} +2.33211e6 q^{56} +383701. q^{57} -958356. q^{58} -705627. q^{59} +520071. q^{60} -831030. q^{61} -833586. q^{62} -48190.9 q^{63} +1.96854e6 q^{64} +830562. q^{65} +760206. q^{66} -2.27120e6 q^{67} -1.87662e6 q^{68} +561396. q^{69} +2.33629e6 q^{70} +4.30808e6 q^{71} -49869.1 q^{72} -3.27796e6 q^{73} -3.78140e6 q^{74} -2.02796e6 q^{75} -480152. q^{76} -2.91540e6 q^{77} -1.73599e6 q^{78} -8.41456e6 q^{79} +1.00452e6 q^{80} -4.85214e6 q^{81} +3.29420e6 q^{82} -3.46468e6 q^{83} +4.16873e6 q^{84} -5.96213e6 q^{85} +660680. q^{86} -5.43287e6 q^{87} -3.01693e6 q^{88} +5.16792e6 q^{89} -49958.5 q^{90} +6.65753e6 q^{91} -702513. q^{92} -4.72556e6 q^{93} +2.07015e6 q^{94} -1.52547e6 q^{95} +7.26755e6 q^{96} +1.54280e7 q^{97} +1.18836e7 q^{98} +62341.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11q - 24q^{2} - 68q^{3} + 602q^{4} - 752q^{5} - 681q^{6} - 12q^{7} - 3810q^{8} + 2721q^{9} + O(q^{10}) \) \( 11q - 24q^{2} - 68q^{3} + 602q^{4} - 752q^{5} - 681q^{6} - 12q^{7} - 3810q^{8} + 2721q^{9} - 1333q^{10} + 1333q^{11} + 5089q^{12} - 17967q^{13} - 22352q^{14} - 49504q^{15} - 34406q^{16} - 63095q^{17} - 165931q^{18} - 54524q^{19} - 280995q^{20} - 139788q^{21} - 289358q^{22} - 138139q^{23} - 429583q^{24} + 3455q^{25} - 132946q^{26} - 240356q^{27} - 12704q^{28} - 308658q^{29} + 421284q^{30} - 209523q^{31} - 644934q^{32} + 96814q^{33} + 762435q^{34} - 578892q^{35} + 426161q^{36} - 298472q^{37} - 369707q^{38} + 292298q^{39} + 2633173q^{40} - 1346735q^{41} + 1173266q^{42} + 874577q^{43} + 3134292q^{44} + 1893784q^{45} + 3588111q^{46} + 499284q^{47} + 5647533q^{48} + 2544563q^{49} + 3049745q^{50} + 1258424q^{51} + 983088q^{52} - 2210495q^{53} + 6789698q^{54} - 1855072q^{55} - 469976q^{56} - 1238444q^{57} + 4397067q^{58} - 5824216q^{59} - 2889372q^{60} - 4453034q^{61} + 1002789q^{62} - 6240564q^{63} + 4757538q^{64} - 2162872q^{65} - 258940q^{66} - 6859513q^{67} - 9397005q^{68} - 10040030q^{69} + 845078q^{70} - 10726554q^{71} + 1199517q^{72} - 4456898q^{73} + 1046637q^{74} - 3349114q^{75} + 5861267q^{76} - 17019816q^{77} + 1999122q^{78} - 15541320q^{79} - 15680911q^{80} - 12976697q^{81} + 20233655q^{82} - 11146767q^{83} + 12348278q^{84} - 12471976q^{85} - 1908168q^{86} - 18648900q^{87} - 24463544q^{88} - 13531356q^{89} + 20858990q^{90} - 19746448q^{91} - 26023161q^{92} - 21903110q^{93} + 20288857q^{94} - 12291624q^{95} - 13954503q^{96} - 10999901q^{97} + 29909168q^{98} + 29396057q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.30971 0.734482 0.367241 0.930126i \(-0.380302\pi\)
0.367241 + 0.930126i \(0.380302\pi\)
\(3\) 47.1073 1.00731 0.503656 0.863904i \(-0.331988\pi\)
0.503656 + 0.863904i \(0.331988\pi\)
\(4\) −58.9487 −0.460536
\(5\) −187.284 −0.670047 −0.335023 0.942210i \(-0.608744\pi\)
−0.335023 + 0.942210i \(0.608744\pi\)
\(6\) 391.449 0.739853
\(7\) −1501.21 −1.65424 −0.827119 0.562027i \(-0.810022\pi\)
−0.827119 + 0.562027i \(0.810022\pi\)
\(8\) −1553.49 −1.07274
\(9\) 32.1014 0.0146783
\(10\) −1556.27 −0.492137
\(11\) 1942.03 0.439929 0.219964 0.975508i \(-0.429406\pi\)
0.219964 + 0.975508i \(0.429406\pi\)
\(12\) −2776.91 −0.463904
\(13\) −4434.78 −0.559849 −0.279924 0.960022i \(-0.590309\pi\)
−0.279924 + 0.960022i \(0.590309\pi\)
\(14\) −12474.6 −1.21501
\(15\) −8822.44 −0.674946
\(16\) −5363.63 −0.327370
\(17\) 31834.8 1.57156 0.785779 0.618508i \(-0.212262\pi\)
0.785779 + 0.618508i \(0.212262\pi\)
\(18\) 266.753 0.0107809
\(19\) 8145.26 0.272438 0.136219 0.990679i \(-0.456505\pi\)
0.136219 + 0.990679i \(0.456505\pi\)
\(20\) 11040.1 0.308581
\(21\) −70718.0 −1.66633
\(22\) 16137.7 0.323120
\(23\) 11917.4 0.204236 0.102118 0.994772i \(-0.467438\pi\)
0.102118 + 0.994772i \(0.467438\pi\)
\(24\) −73180.8 −1.08058
\(25\) −43049.8 −0.551038
\(26\) −36851.8 −0.411199
\(27\) −101512. −0.992527
\(28\) 88494.3 0.761837
\(29\) −115330. −0.878108 −0.439054 0.898461i \(-0.644686\pi\)
−0.439054 + 0.898461i \(0.644686\pi\)
\(30\) −73311.9 −0.495736
\(31\) −100315. −0.604781 −0.302391 0.953184i \(-0.597785\pi\)
−0.302391 + 0.953184i \(0.597785\pi\)
\(32\) 154276. 0.832290
\(33\) 91484.1 0.443146
\(34\) 264538. 1.15428
\(35\) 281152. 1.10842
\(36\) −1892.33 −0.00675987
\(37\) −455058. −1.47693 −0.738466 0.674291i \(-0.764449\pi\)
−0.738466 + 0.674291i \(0.764449\pi\)
\(38\) 67684.8 0.200101
\(39\) −208911. −0.563942
\(40\) 290943. 0.718784
\(41\) 396428. 0.898298 0.449149 0.893457i \(-0.351727\pi\)
0.449149 + 0.893457i \(0.351727\pi\)
\(42\) −587646. −1.22389
\(43\) 79507.0 0.152499
\(44\) −114480. −0.202603
\(45\) −6012.06 −0.00983512
\(46\) 99029.9 0.150008
\(47\) 249124. 0.350004 0.175002 0.984568i \(-0.444007\pi\)
0.175002 + 0.984568i \(0.444007\pi\)
\(48\) −252666. −0.329764
\(49\) 1.43009e6 1.73650
\(50\) −357732. −0.404727
\(51\) 1.49965e6 1.58305
\(52\) 261424. 0.257831
\(53\) −363774. −0.335634 −0.167817 0.985818i \(-0.553672\pi\)
−0.167817 + 0.985818i \(0.553672\pi\)
\(54\) −843532. −0.728993
\(55\) −363711. −0.294773
\(56\) 2.33211e6 1.77456
\(57\) 383701. 0.274430
\(58\) −958356. −0.644954
\(59\) −705627. −0.447294 −0.223647 0.974670i \(-0.571796\pi\)
−0.223647 + 0.974670i \(0.571796\pi\)
\(60\) 520071. 0.310837
\(61\) −831030. −0.468773 −0.234386 0.972144i \(-0.575308\pi\)
−0.234386 + 0.972144i \(0.575308\pi\)
\(62\) −833586. −0.444201
\(63\) −48190.9 −0.0242813
\(64\) 1.96854e6 0.938672
\(65\) 830562. 0.375125
\(66\) 760206. 0.325483
\(67\) −2.27120e6 −0.922557 −0.461279 0.887255i \(-0.652609\pi\)
−0.461279 + 0.887255i \(0.652609\pi\)
\(68\) −1.87662e6 −0.723759
\(69\) 561396. 0.205730
\(70\) 2.33629e6 0.814112
\(71\) 4.30808e6 1.42850 0.714248 0.699893i \(-0.246769\pi\)
0.714248 + 0.699893i \(0.246769\pi\)
\(72\) −49869.1 −0.0157459
\(73\) −3.27796e6 −0.986220 −0.493110 0.869967i \(-0.664140\pi\)
−0.493110 + 0.869967i \(0.664140\pi\)
\(74\) −3.78140e6 −1.08478
\(75\) −2.02796e6 −0.555067
\(76\) −480152. −0.125467
\(77\) −2.91540e6 −0.727747
\(78\) −1.73599e6 −0.414205
\(79\) −8.41456e6 −1.92016 −0.960078 0.279732i \(-0.909754\pi\)
−0.960078 + 0.279732i \(0.909754\pi\)
\(80\) 1.00452e6 0.219353
\(81\) −4.85214e6 −1.01446
\(82\) 3.29420e6 0.659784
\(83\) −3.46468e6 −0.665104 −0.332552 0.943085i \(-0.607910\pi\)
−0.332552 + 0.943085i \(0.607910\pi\)
\(84\) 4.16873e6 0.767408
\(85\) −5.96213e6 −1.05302
\(86\) 660680. 0.112007
\(87\) −5.43287e6 −0.884529
\(88\) −3.01693e6 −0.471928
\(89\) 5.16792e6 0.777053 0.388526 0.921438i \(-0.372984\pi\)
0.388526 + 0.921438i \(0.372984\pi\)
\(90\) −49958.5 −0.00722372
\(91\) 6.65753e6 0.926123
\(92\) −702513. −0.0940583
\(93\) −4.72556e6 −0.609204
\(94\) 2.07015e6 0.257071
\(95\) −1.52547e6 −0.182546
\(96\) 7.26755e6 0.838376
\(97\) 1.54280e7 1.71637 0.858183 0.513344i \(-0.171593\pi\)
0.858183 + 0.513344i \(0.171593\pi\)
\(98\) 1.18836e7 1.27543
\(99\) 62341.9 0.00645739
\(100\) 2.53773e6 0.253773
\(101\) 4.87285e6 0.470606 0.235303 0.971922i \(-0.424392\pi\)
0.235303 + 0.971922i \(0.424392\pi\)
\(102\) 1.24617e7 1.16272
\(103\) −1.16758e7 −1.05283 −0.526413 0.850229i \(-0.676463\pi\)
−0.526413 + 0.850229i \(0.676463\pi\)
\(104\) 6.88939e6 0.600570
\(105\) 1.32443e7 1.11652
\(106\) −3.02286e6 −0.246517
\(107\) −1.74276e7 −1.37529 −0.687646 0.726046i \(-0.741356\pi\)
−0.687646 + 0.726046i \(0.741356\pi\)
\(108\) 5.98397e6 0.457095
\(109\) 1.50763e6 0.111507 0.0557535 0.998445i \(-0.482244\pi\)
0.0557535 + 0.998445i \(0.482244\pi\)
\(110\) −3.02234e6 −0.216505
\(111\) −2.14365e7 −1.48773
\(112\) 8.05193e6 0.541548
\(113\) 2.05709e7 1.34116 0.670579 0.741838i \(-0.266046\pi\)
0.670579 + 0.741838i \(0.266046\pi\)
\(114\) 3.18845e6 0.201564
\(115\) −2.23193e6 −0.136848
\(116\) 6.79852e6 0.404400
\(117\) −142363. −0.00821760
\(118\) −5.86356e6 −0.328529
\(119\) −4.77906e7 −2.59973
\(120\) 1.37056e7 0.724040
\(121\) −1.57157e7 −0.806463
\(122\) −6.90562e6 −0.344305
\(123\) 1.86747e7 0.904867
\(124\) 5.91341e6 0.278524
\(125\) 2.26941e7 1.03927
\(126\) −400452. −0.0178342
\(127\) 7.41435e6 0.321188 0.160594 0.987021i \(-0.448659\pi\)
0.160594 + 0.987021i \(0.448659\pi\)
\(128\) −3.38940e6 −0.142853
\(129\) 3.74536e6 0.153614
\(130\) 6.90174e6 0.275522
\(131\) 4.53106e7 1.76096 0.880482 0.474079i \(-0.157219\pi\)
0.880482 + 0.474079i \(0.157219\pi\)
\(132\) −5.39286e6 −0.204085
\(133\) −1.22277e7 −0.450677
\(134\) −1.88730e7 −0.677602
\(135\) 1.90115e7 0.665039
\(136\) −4.94550e7 −1.68587
\(137\) −4.06972e7 −1.35220 −0.676102 0.736808i \(-0.736332\pi\)
−0.676102 + 0.736808i \(0.736332\pi\)
\(138\) 4.66504e6 0.151105
\(139\) −4.72006e7 −1.49072 −0.745360 0.666662i \(-0.767722\pi\)
−0.745360 + 0.666662i \(0.767722\pi\)
\(140\) −1.65735e7 −0.510466
\(141\) 1.17356e7 0.352563
\(142\) 3.57989e7 1.04920
\(143\) −8.61250e6 −0.246294
\(144\) −172180. −0.00480522
\(145\) 2.15994e7 0.588373
\(146\) −2.72389e7 −0.724360
\(147\) 6.73675e7 1.74920
\(148\) 2.68250e7 0.680181
\(149\) −6.71032e7 −1.66185 −0.830924 0.556386i \(-0.812188\pi\)
−0.830924 + 0.556386i \(0.812188\pi\)
\(150\) −1.68518e7 −0.407687
\(151\) 5.69247e7 1.34549 0.672747 0.739873i \(-0.265114\pi\)
0.672747 + 0.739873i \(0.265114\pi\)
\(152\) −1.26536e7 −0.292254
\(153\) 1.02194e6 0.0230677
\(154\) −2.42261e7 −0.534517
\(155\) 1.87873e7 0.405232
\(156\) 1.23150e7 0.259716
\(157\) 7.10156e7 1.46455 0.732277 0.681007i \(-0.238458\pi\)
0.732277 + 0.681007i \(0.238458\pi\)
\(158\) −6.99226e7 −1.41032
\(159\) −1.71364e7 −0.338088
\(160\) −2.88935e7 −0.557673
\(161\) −1.78905e7 −0.337856
\(162\) −4.03199e7 −0.745105
\(163\) 4.61695e7 0.835024 0.417512 0.908671i \(-0.362902\pi\)
0.417512 + 0.908671i \(0.362902\pi\)
\(164\) −2.33689e7 −0.413699
\(165\) −1.71335e7 −0.296928
\(166\) −2.87905e7 −0.488507
\(167\) −4.64887e7 −0.772395 −0.386198 0.922416i \(-0.626212\pi\)
−0.386198 + 0.922416i \(0.626212\pi\)
\(168\) 1.09860e8 1.78754
\(169\) −4.30812e7 −0.686570
\(170\) −4.95436e7 −0.773422
\(171\) 261474. 0.00399891
\(172\) −4.68683e6 −0.0702311
\(173\) −3.94243e7 −0.578898 −0.289449 0.957193i \(-0.593472\pi\)
−0.289449 + 0.957193i \(0.593472\pi\)
\(174\) −4.51456e7 −0.649670
\(175\) 6.46268e7 0.911548
\(176\) −1.04163e7 −0.144019
\(177\) −3.32402e7 −0.450565
\(178\) 4.29439e7 0.570731
\(179\) 2.63135e7 0.342921 0.171460 0.985191i \(-0.445151\pi\)
0.171460 + 0.985191i \(0.445151\pi\)
\(180\) 354403. 0.00452943
\(181\) −3.37886e7 −0.423541 −0.211771 0.977319i \(-0.567923\pi\)
−0.211771 + 0.977319i \(0.567923\pi\)
\(182\) 5.53222e7 0.680220
\(183\) −3.91476e7 −0.472201
\(184\) −1.85135e7 −0.219092
\(185\) 8.52249e7 0.989613
\(186\) −3.92680e7 −0.447449
\(187\) 6.18242e7 0.691374
\(188\) −1.46855e7 −0.161189
\(189\) 1.52390e8 1.64188
\(190\) −1.26763e7 −0.134077
\(191\) −1.27960e8 −1.32880 −0.664398 0.747379i \(-0.731312\pi\)
−0.664398 + 0.747379i \(0.731312\pi\)
\(192\) 9.27326e7 0.945536
\(193\) −4.75377e7 −0.475979 −0.237990 0.971268i \(-0.576488\pi\)
−0.237990 + 0.971268i \(0.576488\pi\)
\(194\) 1.28203e8 1.26064
\(195\) 3.91256e7 0.377868
\(196\) −8.43016e7 −0.799723
\(197\) 1.75594e7 0.163636 0.0818178 0.996647i \(-0.473927\pi\)
0.0818178 + 0.996647i \(0.473927\pi\)
\(198\) 518044. 0.00474284
\(199\) 1.25178e8 1.12601 0.563004 0.826454i \(-0.309646\pi\)
0.563004 + 0.826454i \(0.309646\pi\)
\(200\) 6.68775e7 0.591119
\(201\) −1.06990e8 −0.929304
\(202\) 4.04919e7 0.345652
\(203\) 1.73134e8 1.45260
\(204\) −8.84024e7 −0.729052
\(205\) −7.42445e7 −0.601902
\(206\) −9.70225e7 −0.773281
\(207\) 382564. 0.00299783
\(208\) 2.37865e7 0.183278
\(209\) 1.58184e7 0.119853
\(210\) 1.10057e8 0.820065
\(211\) 8.23349e7 0.603386 0.301693 0.953405i \(-0.402448\pi\)
0.301693 + 0.953405i \(0.402448\pi\)
\(212\) 2.14440e7 0.154572
\(213\) 2.02942e8 1.43894
\(214\) −1.44819e8 −1.01013
\(215\) −1.48904e7 −0.102181
\(216\) 1.57697e8 1.06472
\(217\) 1.50593e8 1.00045
\(218\) 1.25280e7 0.0818999
\(219\) −1.54416e8 −0.993431
\(220\) 2.14403e7 0.135754
\(221\) −1.41180e8 −0.879834
\(222\) −1.78132e8 −1.09271
\(223\) 2.64599e8 1.59779 0.798897 0.601468i \(-0.205417\pi\)
0.798897 + 0.601468i \(0.205417\pi\)
\(224\) −2.31601e8 −1.37681
\(225\) −1.38196e6 −0.00808827
\(226\) 1.70939e8 0.985056
\(227\) −4.70117e7 −0.266757 −0.133378 0.991065i \(-0.542583\pi\)
−0.133378 + 0.991065i \(0.542583\pi\)
\(228\) −2.26187e7 −0.126385
\(229\) −1.76441e7 −0.0970900 −0.0485450 0.998821i \(-0.515458\pi\)
−0.0485450 + 0.998821i \(0.515458\pi\)
\(230\) −1.85467e7 −0.100512
\(231\) −1.37337e8 −0.733069
\(232\) 1.79163e8 0.941979
\(233\) −2.96873e8 −1.53754 −0.768768 0.639528i \(-0.779130\pi\)
−0.768768 + 0.639528i \(0.779130\pi\)
\(234\) −1.18299e6 −0.00603568
\(235\) −4.66568e7 −0.234519
\(236\) 4.15957e7 0.205995
\(237\) −3.96388e8 −1.93420
\(238\) −3.97126e8 −1.90946
\(239\) 7.90520e7 0.374559 0.187280 0.982307i \(-0.440033\pi\)
0.187280 + 0.982307i \(0.440033\pi\)
\(240\) 4.73203e7 0.220957
\(241\) 2.33364e8 1.07393 0.536963 0.843606i \(-0.319571\pi\)
0.536963 + 0.843606i \(0.319571\pi\)
\(242\) −1.30593e8 −0.592332
\(243\) −6.56586e6 −0.0293542
\(244\) 4.89881e7 0.215887
\(245\) −2.67832e8 −1.16354
\(246\) 1.55181e8 0.664608
\(247\) −3.61224e7 −0.152524
\(248\) 1.55838e8 0.648772
\(249\) −1.63212e8 −0.669967
\(250\) 1.88581e8 0.763323
\(251\) −8.80712e7 −0.351541 −0.175770 0.984431i \(-0.556242\pi\)
−0.175770 + 0.984431i \(0.556242\pi\)
\(252\) 2.84079e6 0.0111824
\(253\) 2.31439e7 0.0898495
\(254\) 6.16111e7 0.235907
\(255\) −2.80860e8 −1.06072
\(256\) −2.80138e8 −1.04359
\(257\) −1.29253e8 −0.474979 −0.237489 0.971390i \(-0.576325\pi\)
−0.237489 + 0.971390i \(0.576325\pi\)
\(258\) 3.11229e7 0.112826
\(259\) 6.83136e8 2.44320
\(260\) −4.89605e7 −0.172759
\(261\) −3.70224e6 −0.0128891
\(262\) 3.76518e8 1.29340
\(263\) −1.06033e8 −0.359415 −0.179708 0.983720i \(-0.557515\pi\)
−0.179708 + 0.983720i \(0.557515\pi\)
\(264\) −1.42120e8 −0.475379
\(265\) 6.81289e7 0.224890
\(266\) −1.01609e8 −0.331014
\(267\) 2.43447e8 0.782735
\(268\) 1.33884e8 0.424871
\(269\) −2.84361e8 −0.890711 −0.445356 0.895354i \(-0.646923\pi\)
−0.445356 + 0.895354i \(0.646923\pi\)
\(270\) 1.57980e8 0.488459
\(271\) −3.12787e8 −0.954677 −0.477339 0.878719i \(-0.658398\pi\)
−0.477339 + 0.878719i \(0.658398\pi\)
\(272\) −1.70750e8 −0.514481
\(273\) 3.13619e8 0.932895
\(274\) −3.38182e8 −0.993170
\(275\) −8.36042e7 −0.242417
\(276\) −3.30935e7 −0.0947461
\(277\) −1.35018e8 −0.381692 −0.190846 0.981620i \(-0.561123\pi\)
−0.190846 + 0.981620i \(0.561123\pi\)
\(278\) −3.92224e8 −1.09491
\(279\) −3.22024e6 −0.00887714
\(280\) −4.36767e8 −1.18904
\(281\) 2.08009e8 0.559255 0.279628 0.960109i \(-0.409789\pi\)
0.279628 + 0.960109i \(0.409789\pi\)
\(282\) 9.75191e7 0.258951
\(283\) 3.32152e8 0.871134 0.435567 0.900156i \(-0.356548\pi\)
0.435567 + 0.900156i \(0.356548\pi\)
\(284\) −2.53955e8 −0.657875
\(285\) −7.18610e7 −0.183881
\(286\) −7.15674e7 −0.180898
\(287\) −5.95121e8 −1.48600
\(288\) 4.95248e6 0.0122166
\(289\) 6.03113e8 1.46979
\(290\) 1.79484e8 0.432149
\(291\) 7.26774e8 1.72892
\(292\) 1.93231e8 0.454190
\(293\) 2.71797e8 0.631260 0.315630 0.948882i \(-0.397784\pi\)
0.315630 + 0.948882i \(0.397784\pi\)
\(294\) 5.59805e8 1.28476
\(295\) 1.32152e8 0.299708
\(296\) 7.06927e8 1.58436
\(297\) −1.97139e8 −0.436641
\(298\) −5.57608e8 −1.22060
\(299\) −5.28509e7 −0.114341
\(300\) 1.19546e8 0.255629
\(301\) −1.19357e8 −0.252269
\(302\) 4.73028e8 0.988241
\(303\) 2.29547e8 0.474048
\(304\) −4.36881e7 −0.0891879
\(305\) 1.55638e8 0.314100
\(306\) 8.49202e6 0.0169428
\(307\) −2.68192e7 −0.0529007 −0.0264503 0.999650i \(-0.508420\pi\)
−0.0264503 + 0.999650i \(0.508420\pi\)
\(308\) 1.71859e8 0.335154
\(309\) −5.50016e8 −1.06052
\(310\) 1.56117e8 0.297635
\(311\) 8.62107e8 1.62517 0.812587 0.582840i \(-0.198058\pi\)
0.812587 + 0.582840i \(0.198058\pi\)
\(312\) 3.24541e8 0.604962
\(313\) 1.68619e7 0.0310814 0.0155407 0.999879i \(-0.495053\pi\)
0.0155407 + 0.999879i \(0.495053\pi\)
\(314\) 5.90120e8 1.07569
\(315\) 9.02536e6 0.0162696
\(316\) 4.96027e8 0.884302
\(317\) −6.67505e8 −1.17692 −0.588461 0.808526i \(-0.700266\pi\)
−0.588461 + 0.808526i \(0.700266\pi\)
\(318\) −1.42399e8 −0.248320
\(319\) −2.23974e8 −0.386305
\(320\) −3.68675e8 −0.628954
\(321\) −8.20969e8 −1.38535
\(322\) −1.48665e8 −0.248149
\(323\) 2.59302e8 0.428152
\(324\) 2.86027e8 0.467197
\(325\) 1.90917e8 0.308498
\(326\) 3.83656e8 0.613310
\(327\) 7.10205e7 0.112322
\(328\) −6.15847e8 −0.963638
\(329\) −3.73987e8 −0.578989
\(330\) −1.42374e8 −0.218088
\(331\) −1.47144e8 −0.223021 −0.111510 0.993763i \(-0.535569\pi\)
−0.111510 + 0.993763i \(0.535569\pi\)
\(332\) 2.04238e8 0.306304
\(333\) −1.46080e7 −0.0216788
\(334\) −3.86308e8 −0.567310
\(335\) 4.25359e8 0.618156
\(336\) 3.79305e8 0.545508
\(337\) −1.09036e9 −1.55190 −0.775951 0.630793i \(-0.782730\pi\)
−0.775951 + 0.630793i \(0.782730\pi\)
\(338\) −3.57993e8 −0.504273
\(339\) 9.69043e8 1.35096
\(340\) 3.51460e8 0.484953
\(341\) −1.94814e8 −0.266061
\(342\) 2.17277e6 0.00293713
\(343\) −9.10548e8 −1.21835
\(344\) −1.23513e8 −0.163591
\(345\) −1.05140e8 −0.137849
\(346\) −3.27604e8 −0.425190
\(347\) 2.16834e8 0.278596 0.139298 0.990251i \(-0.455515\pi\)
0.139298 + 0.990251i \(0.455515\pi\)
\(348\) 3.20260e8 0.407358
\(349\) −1.24901e9 −1.57281 −0.786404 0.617712i \(-0.788060\pi\)
−0.786404 + 0.617712i \(0.788060\pi\)
\(350\) 5.37030e8 0.669515
\(351\) 4.50182e8 0.555665
\(352\) 2.99610e8 0.366149
\(353\) −4.43185e8 −0.536258 −0.268129 0.963383i \(-0.586405\pi\)
−0.268129 + 0.963383i \(0.586405\pi\)
\(354\) −2.76216e8 −0.330932
\(355\) −8.06832e8 −0.957159
\(356\) −3.04642e8 −0.357861
\(357\) −2.25129e9 −2.61874
\(358\) 2.18658e8 0.251869
\(359\) 1.42062e9 1.62049 0.810247 0.586089i \(-0.199333\pi\)
0.810247 + 0.586089i \(0.199333\pi\)
\(360\) 9.33968e6 0.0105505
\(361\) −8.27527e8 −0.925778
\(362\) −2.80774e8 −0.311083
\(363\) −7.40324e8 −0.812360
\(364\) −3.92453e8 −0.426513
\(365\) 6.13908e8 0.660813
\(366\) −3.25306e8 −0.346823
\(367\) −1.61585e9 −1.70636 −0.853180 0.521617i \(-0.825329\pi\)
−0.853180 + 0.521617i \(0.825329\pi\)
\(368\) −6.39203e7 −0.0668608
\(369\) 1.27259e7 0.0131855
\(370\) 7.08194e8 0.726852
\(371\) 5.46100e8 0.555218
\(372\) 2.78565e8 0.280561
\(373\) 1.89477e9 1.89049 0.945245 0.326361i \(-0.105822\pi\)
0.945245 + 0.326361i \(0.105822\pi\)
\(374\) 5.13741e8 0.507801
\(375\) 1.06906e9 1.04687
\(376\) −3.87011e8 −0.375462
\(377\) 5.11462e8 0.491607
\(378\) 1.26632e9 1.20593
\(379\) −6.09382e8 −0.574979 −0.287490 0.957784i \(-0.592821\pi\)
−0.287490 + 0.957784i \(0.592821\pi\)
\(380\) 8.99246e7 0.0840690
\(381\) 3.49270e8 0.323537
\(382\) −1.06331e9 −0.975977
\(383\) 1.23247e9 1.12094 0.560470 0.828175i \(-0.310620\pi\)
0.560470 + 0.828175i \(0.310620\pi\)
\(384\) −1.59666e8 −0.143897
\(385\) 5.46007e8 0.487625
\(386\) −3.95025e8 −0.349598
\(387\) 2.55228e6 0.00223841
\(388\) −9.09463e8 −0.790449
\(389\) −1.81226e9 −1.56098 −0.780491 0.625167i \(-0.785031\pi\)
−0.780491 + 0.625167i \(0.785031\pi\)
\(390\) 3.25122e8 0.277537
\(391\) 3.79387e8 0.320969
\(392\) −2.22162e9 −1.86281
\(393\) 2.13446e9 1.77384
\(394\) 1.45914e8 0.120187
\(395\) 1.57591e9 1.28659
\(396\) −3.67497e6 −0.00297386
\(397\) 1.53157e9 1.22848 0.614242 0.789117i \(-0.289462\pi\)
0.614242 + 0.789117i \(0.289462\pi\)
\(398\) 1.04019e9 0.827033
\(399\) −5.76016e8 −0.453972
\(400\) 2.30903e8 0.180393
\(401\) −1.62261e9 −1.25663 −0.628317 0.777958i \(-0.716256\pi\)
−0.628317 + 0.777958i \(0.716256\pi\)
\(402\) −8.89057e8 −0.682557
\(403\) 4.44874e8 0.338586
\(404\) −2.87248e8 −0.216731
\(405\) 9.08728e8 0.679737
\(406\) 1.43869e9 1.06691
\(407\) −8.83737e8 −0.649745
\(408\) −2.32969e9 −1.69820
\(409\) 1.42089e8 0.102690 0.0513452 0.998681i \(-0.483649\pi\)
0.0513452 + 0.998681i \(0.483649\pi\)
\(410\) −6.16950e8 −0.442086
\(411\) −1.91714e9 −1.36209
\(412\) 6.88273e8 0.484864
\(413\) 1.05929e9 0.739931
\(414\) 3.17900e6 0.00220185
\(415\) 6.48877e8 0.445650
\(416\) −6.84183e8 −0.465956
\(417\) −2.22350e9 −1.50162
\(418\) 1.31446e8 0.0880300
\(419\) −2.04493e8 −0.135809 −0.0679047 0.997692i \(-0.521631\pi\)
−0.0679047 + 0.997692i \(0.521631\pi\)
\(420\) −7.80735e8 −0.514199
\(421\) −1.14148e9 −0.745557 −0.372778 0.927920i \(-0.621595\pi\)
−0.372778 + 0.927920i \(0.621595\pi\)
\(422\) 6.84180e8 0.443176
\(423\) 7.99721e6 0.00513745
\(424\) 5.65119e8 0.360047
\(425\) −1.37048e9 −0.865987
\(426\) 1.68639e9 1.05688
\(427\) 1.24755e9 0.775462
\(428\) 1.02734e9 0.633372
\(429\) −4.05712e8 −0.248095
\(430\) −1.23735e8 −0.0750502
\(431\) 2.13524e9 1.28463 0.642314 0.766442i \(-0.277975\pi\)
0.642314 + 0.766442i \(0.277975\pi\)
\(432\) 5.44470e8 0.324923
\(433\) 1.78226e9 1.05503 0.527514 0.849547i \(-0.323124\pi\)
0.527514 + 0.849547i \(0.323124\pi\)
\(434\) 1.25139e9 0.734814
\(435\) 1.01749e9 0.592675
\(436\) −8.88728e7 −0.0513530
\(437\) 9.70700e7 0.0556417
\(438\) −1.28315e9 −0.729657
\(439\) −8.74976e8 −0.493595 −0.246797 0.969067i \(-0.579378\pi\)
−0.246797 + 0.969067i \(0.579378\pi\)
\(440\) 5.65022e8 0.316214
\(441\) 4.59077e7 0.0254889
\(442\) −1.17317e9 −0.646222
\(443\) 2.15446e9 1.17740 0.588702 0.808350i \(-0.299639\pi\)
0.588702 + 0.808350i \(0.299639\pi\)
\(444\) 1.26366e9 0.685154
\(445\) −9.67867e8 −0.520661
\(446\) 2.19874e9 1.17355
\(447\) −3.16105e9 −1.67400
\(448\) −2.95519e9 −1.55279
\(449\) 2.98017e9 1.55374 0.776870 0.629661i \(-0.216806\pi\)
0.776870 + 0.629661i \(0.216806\pi\)
\(450\) −1.14837e7 −0.00594069
\(451\) 7.69876e8 0.395187
\(452\) −1.21263e9 −0.617652
\(453\) 2.68157e9 1.35533
\(454\) −3.90654e8 −0.195928
\(455\) −1.24685e9 −0.620545
\(456\) −5.96076e8 −0.294391
\(457\) 2.85176e9 1.39768 0.698838 0.715280i \(-0.253701\pi\)
0.698838 + 0.715280i \(0.253701\pi\)
\(458\) −1.46617e8 −0.0713108
\(459\) −3.23160e9 −1.55981
\(460\) 1.31569e8 0.0630234
\(461\) 1.65358e8 0.0786091 0.0393045 0.999227i \(-0.487486\pi\)
0.0393045 + 0.999227i \(0.487486\pi\)
\(462\) −1.14123e9 −0.538426
\(463\) −1.65271e9 −0.773862 −0.386931 0.922109i \(-0.626465\pi\)
−0.386931 + 0.922109i \(0.626465\pi\)
\(464\) 6.18585e8 0.287466
\(465\) 8.85020e8 0.408195
\(466\) −2.46693e9 −1.12929
\(467\) 5.62317e8 0.255489 0.127744 0.991807i \(-0.459226\pi\)
0.127744 + 0.991807i \(0.459226\pi\)
\(468\) 8.39208e6 0.00378450
\(469\) 3.40954e9 1.52613
\(470\) −3.87705e8 −0.172250
\(471\) 3.34536e9 1.47526
\(472\) 1.09618e9 0.479829
\(473\) 1.54405e8 0.0670885
\(474\) −3.29387e9 −1.42063
\(475\) −3.50652e8 −0.150123
\(476\) 2.81719e9 1.19727
\(477\) −1.16776e7 −0.00492652
\(478\) 6.56900e8 0.275107
\(479\) 2.55353e9 1.06162 0.530808 0.847492i \(-0.321889\pi\)
0.530808 + 0.847492i \(0.321889\pi\)
\(480\) −1.36109e9 −0.561751
\(481\) 2.01808e9 0.826858
\(482\) 1.93919e9 0.788779
\(483\) −8.42772e8 −0.340326
\(484\) 9.26418e8 0.371405
\(485\) −2.88942e9 −1.15005
\(486\) −5.45604e7 −0.0215601
\(487\) −8.52710e8 −0.334542 −0.167271 0.985911i \(-0.553495\pi\)
−0.167271 + 0.985911i \(0.553495\pi\)
\(488\) 1.29100e9 0.502870
\(489\) 2.17492e9 0.841130
\(490\) −2.22561e9 −0.854598
\(491\) 4.91989e9 1.87573 0.937864 0.347002i \(-0.112800\pi\)
0.937864 + 0.347002i \(0.112800\pi\)
\(492\) −1.10085e9 −0.416724
\(493\) −3.67149e9 −1.38000
\(494\) −3.00167e8 −0.112026
\(495\) −1.16756e7 −0.00432675
\(496\) 5.38050e8 0.197987
\(497\) −6.46732e9 −2.36307
\(498\) −1.35624e9 −0.492079
\(499\) −2.15629e9 −0.776882 −0.388441 0.921474i \(-0.626986\pi\)
−0.388441 + 0.921474i \(0.626986\pi\)
\(500\) −1.33778e9 −0.478620
\(501\) −2.18996e9 −0.778043
\(502\) −7.31847e8 −0.258200
\(503\) −4.90685e9 −1.71915 −0.859577 0.511007i \(-0.829273\pi\)
−0.859577 + 0.511007i \(0.829273\pi\)
\(504\) 7.48640e7 0.0260475
\(505\) −9.12604e8 −0.315328
\(506\) 1.92320e8 0.0659928
\(507\) −2.02944e9 −0.691590
\(508\) −4.37066e8 −0.147919
\(509\) −1.30025e9 −0.437035 −0.218517 0.975833i \(-0.570122\pi\)
−0.218517 + 0.975833i \(0.570122\pi\)
\(510\) −2.33387e9 −0.779077
\(511\) 4.92090e9 1.63144
\(512\) −1.89402e9 −0.623649
\(513\) −8.26838e8 −0.270402
\(514\) −1.07405e9 −0.348863
\(515\) 2.18669e9 0.705442
\(516\) −2.20784e8 −0.0707447
\(517\) 4.83807e8 0.153977
\(518\) 5.67667e9 1.79448
\(519\) −1.85717e9 −0.583131
\(520\) −1.29027e9 −0.402410
\(521\) −3.07871e9 −0.953756 −0.476878 0.878970i \(-0.658232\pi\)
−0.476878 + 0.878970i \(0.658232\pi\)
\(522\) −3.07645e7 −0.00946680
\(523\) −3.18094e9 −0.972298 −0.486149 0.873876i \(-0.661599\pi\)
−0.486149 + 0.873876i \(0.661599\pi\)
\(524\) −2.67100e9 −0.810988
\(525\) 3.04440e9 0.918213
\(526\) −8.81105e8 −0.263984
\(527\) −3.19349e9 −0.950449
\(528\) −4.90686e8 −0.145073
\(529\) −3.26280e9 −0.958288
\(530\) 5.66132e8 0.165178
\(531\) −2.26516e7 −0.00656550
\(532\) 7.20808e8 0.207553
\(533\) −1.75807e9 −0.502911
\(534\) 2.02297e9 0.574905
\(535\) 3.26391e9 0.921510
\(536\) 3.52828e9 0.989662
\(537\) 1.23956e9 0.345428
\(538\) −2.36296e9 −0.654211
\(539\) 2.77728e9 0.763938
\(540\) −1.12070e9 −0.306275
\(541\) 1.47730e9 0.401123 0.200561 0.979681i \(-0.435723\pi\)
0.200561 + 0.979681i \(0.435723\pi\)
\(542\) −2.59917e9 −0.701193
\(543\) −1.59169e9 −0.426638
\(544\) 4.91135e9 1.30799
\(545\) −2.82355e8 −0.0747149
\(546\) 2.60608e9 0.685194
\(547\) 1.95085e9 0.509646 0.254823 0.966988i \(-0.417983\pi\)
0.254823 + 0.966988i \(0.417983\pi\)
\(548\) 2.39904e9 0.622739
\(549\) −2.66772e7 −0.00688077
\(550\) −6.94727e8 −0.178051
\(551\) −9.39389e8 −0.239230
\(552\) −8.72122e8 −0.220694
\(553\) 1.26320e10 3.17640
\(554\) −1.12196e9 −0.280346
\(555\) 4.01472e9 0.996849
\(556\) 2.78241e9 0.686531
\(557\) 3.18198e9 0.780197 0.390099 0.920773i \(-0.372441\pi\)
0.390099 + 0.920773i \(0.372441\pi\)
\(558\) −2.67592e7 −0.00652010
\(559\) −3.52596e8 −0.0853761
\(560\) −1.50799e9 −0.362862
\(561\) 2.91237e9 0.696429
\(562\) 1.72850e9 0.410763
\(563\) −5.93733e9 −1.40221 −0.701103 0.713060i \(-0.747309\pi\)
−0.701103 + 0.713060i \(0.747309\pi\)
\(564\) −6.91795e8 −0.162368
\(565\) −3.85260e9 −0.898638
\(566\) 2.76009e9 0.639832
\(567\) 7.28408e9 1.67816
\(568\) −6.69255e9 −1.53240
\(569\) −7.94607e9 −1.80825 −0.904127 0.427265i \(-0.859477\pi\)
−0.904127 + 0.427265i \(0.859477\pi\)
\(570\) −5.97144e8 −0.135057
\(571\) 7.59574e9 1.70743 0.853717 0.520737i \(-0.174343\pi\)
0.853717 + 0.520737i \(0.174343\pi\)
\(572\) 5.07695e8 0.113427
\(573\) −6.02787e9 −1.33851
\(574\) −4.94529e9 −1.09144
\(575\) −5.13041e8 −0.112542
\(576\) 6.31927e7 0.0137781
\(577\) −7.50510e9 −1.62645 −0.813226 0.581948i \(-0.802291\pi\)
−0.813226 + 0.581948i \(0.802291\pi\)
\(578\) 5.01170e9 1.07954
\(579\) −2.23938e9 −0.479460
\(580\) −1.27325e9 −0.270967
\(581\) 5.20120e9 1.10024
\(582\) 6.03929e9 1.26986
\(583\) −7.06461e8 −0.147655
\(584\) 5.09228e9 1.05795
\(585\) 2.66622e7 0.00550618
\(586\) 2.25856e9 0.463649
\(587\) 6.28252e8 0.128204 0.0641019 0.997943i \(-0.479582\pi\)
0.0641019 + 0.997943i \(0.479582\pi\)
\(588\) −3.97123e9 −0.805571
\(589\) −8.17089e8 −0.164765
\(590\) 1.09815e9 0.220130
\(591\) 8.27177e8 0.164832
\(592\) 2.44076e9 0.483503
\(593\) −4.09806e9 −0.807024 −0.403512 0.914974i \(-0.632211\pi\)
−0.403512 + 0.914974i \(0.632211\pi\)
\(594\) −1.63817e9 −0.320705
\(595\) 8.95041e9 1.74194
\(596\) 3.95564e9 0.765341
\(597\) 5.89679e9 1.13424
\(598\) −4.39176e8 −0.0839817
\(599\) 3.70675e8 0.0704692 0.0352346 0.999379i \(-0.488782\pi\)
0.0352346 + 0.999379i \(0.488782\pi\)
\(600\) 3.15042e9 0.595441
\(601\) −5.04786e9 −0.948520 −0.474260 0.880385i \(-0.657284\pi\)
−0.474260 + 0.880385i \(0.657284\pi\)
\(602\) −9.91819e8 −0.185287
\(603\) −7.29086e7 −0.0135415
\(604\) −3.35564e9 −0.619649
\(605\) 2.94329e9 0.540367
\(606\) 1.90747e9 0.348179
\(607\) 1.88528e9 0.342148 0.171074 0.985258i \(-0.445276\pi\)
0.171074 + 0.985258i \(0.445276\pi\)
\(608\) 1.25662e9 0.226747
\(609\) 8.15587e9 1.46322
\(610\) 1.29331e9 0.230700
\(611\) −1.10481e9 −0.195949
\(612\) −6.02419e7 −0.0106235
\(613\) −5.95226e9 −1.04369 −0.521843 0.853041i \(-0.674755\pi\)
−0.521843 + 0.853041i \(0.674755\pi\)
\(614\) −2.22860e8 −0.0388546
\(615\) −3.49746e9 −0.606303
\(616\) 4.52904e9 0.780682
\(617\) −5.79027e9 −0.992433 −0.496216 0.868199i \(-0.665278\pi\)
−0.496216 + 0.868199i \(0.665278\pi\)
\(618\) −4.57047e9 −0.778935
\(619\) 9.93652e7 0.0168390 0.00841951 0.999965i \(-0.497320\pi\)
0.00841951 + 0.999965i \(0.497320\pi\)
\(620\) −1.10749e9 −0.186624
\(621\) −1.20975e9 −0.202710
\(622\) 7.16386e9 1.19366
\(623\) −7.75812e9 −1.28543
\(624\) 1.12052e9 0.184618
\(625\) −8.86962e8 −0.145320
\(626\) 1.40117e8 0.0228287
\(627\) 7.45161e8 0.120730
\(628\) −4.18628e9 −0.674480
\(629\) −1.44866e10 −2.32108
\(630\) 7.49982e7 0.0119497
\(631\) 7.60492e9 1.20501 0.602507 0.798114i \(-0.294169\pi\)
0.602507 + 0.798114i \(0.294169\pi\)
\(632\) 1.30719e10 2.05982
\(633\) 3.87858e9 0.607799
\(634\) −5.54678e9 −0.864427
\(635\) −1.38859e9 −0.215211
\(636\) 1.01017e9 0.155702
\(637\) −6.34212e9 −0.972179
\(638\) −1.86116e9 −0.283734
\(639\) 1.38295e8 0.0209678
\(640\) 6.34780e8 0.0957180
\(641\) −7.05623e9 −1.05820 −0.529102 0.848558i \(-0.677471\pi\)
−0.529102 + 0.848558i \(0.677471\pi\)
\(642\) −6.82202e9 −1.01751
\(643\) −1.00382e10 −1.48908 −0.744539 0.667579i \(-0.767330\pi\)
−0.744539 + 0.667579i \(0.767330\pi\)
\(644\) 1.05462e9 0.155595
\(645\) −7.01445e8 −0.102928
\(646\) 2.15473e9 0.314470
\(647\) −1.05830e10 −1.53618 −0.768092 0.640339i \(-0.778794\pi\)
−0.768092 + 0.640339i \(0.778794\pi\)
\(648\) 7.53776e9 1.08825
\(649\) −1.37035e9 −0.196778
\(650\) 1.58646e9 0.226586
\(651\) 7.09405e9 1.00777
\(652\) −2.72163e9 −0.384559
\(653\) 5.20110e9 0.730969 0.365484 0.930818i \(-0.380903\pi\)
0.365484 + 0.930818i \(0.380903\pi\)
\(654\) 5.90160e8 0.0824988
\(655\) −8.48594e9 −1.17993
\(656\) −2.12629e9 −0.294076
\(657\) −1.05227e8 −0.0144760
\(658\) −3.10772e9 −0.425257
\(659\) −4.76603e9 −0.648721 −0.324361 0.945934i \(-0.605149\pi\)
−0.324361 + 0.945934i \(0.605149\pi\)
\(660\) 1.01000e9 0.136746
\(661\) 5.23533e9 0.705081 0.352541 0.935796i \(-0.385318\pi\)
0.352541 + 0.935796i \(0.385318\pi\)
\(662\) −1.22273e9 −0.163805
\(663\) −6.65062e9 −0.886268
\(664\) 5.38234e9 0.713482
\(665\) 2.29006e9 0.301974
\(666\) −1.21388e8 −0.0159227
\(667\) −1.37443e9 −0.179341
\(668\) 2.74044e9 0.355716
\(669\) 1.24645e10 1.60948
\(670\) 3.53461e9 0.454025
\(671\) −1.61389e9 −0.206227
\(672\) −1.09101e10 −1.38687
\(673\) −5.60424e8 −0.0708703 −0.0354351 0.999372i \(-0.511282\pi\)
−0.0354351 + 0.999372i \(0.511282\pi\)
\(674\) −9.06056e9 −1.13984
\(675\) 4.37005e9 0.546920
\(676\) 2.53958e9 0.316190
\(677\) −3.94896e9 −0.489128 −0.244564 0.969633i \(-0.578645\pi\)
−0.244564 + 0.969633i \(0.578645\pi\)
\(678\) 8.05247e9 0.992259
\(679\) −2.31607e10 −2.83928
\(680\) 9.26211e9 1.12961
\(681\) −2.21460e9 −0.268707
\(682\) −1.61885e9 −0.195417
\(683\) 1.30516e10 1.56744 0.783720 0.621115i \(-0.213320\pi\)
0.783720 + 0.621115i \(0.213320\pi\)
\(684\) −1.54135e7 −0.00184164
\(685\) 7.62192e9 0.906040
\(686\) −7.56639e9 −0.894858
\(687\) −8.31165e8 −0.0977999
\(688\) −4.26446e8 −0.0499234
\(689\) 1.61326e9 0.187904
\(690\) −8.73685e8 −0.101247
\(691\) 2.94742e9 0.339835 0.169918 0.985458i \(-0.445650\pi\)
0.169918 + 0.985458i \(0.445650\pi\)
\(692\) 2.32401e9 0.266604
\(693\) −9.35883e7 −0.0106821
\(694\) 1.80183e9 0.204624
\(695\) 8.83991e9 0.998852
\(696\) 8.43991e9 0.948867
\(697\) 1.26202e10 1.41173
\(698\) −1.03789e10 −1.15520
\(699\) −1.39849e10 −1.54878
\(700\) −3.80966e9 −0.419801
\(701\) −7.95719e9 −0.872462 −0.436231 0.899835i \(-0.643687\pi\)
−0.436231 + 0.899835i \(0.643687\pi\)
\(702\) 3.74088e9 0.408126
\(703\) −3.70656e9 −0.402372
\(704\) 3.82297e9 0.412949
\(705\) −2.19788e9 −0.236234
\(706\) −3.68274e9 −0.393872
\(707\) −7.31516e9 −0.778495
\(708\) 1.95946e9 0.207501
\(709\) 4.19832e9 0.442398 0.221199 0.975229i \(-0.429003\pi\)
0.221199 + 0.975229i \(0.429003\pi\)
\(710\) −6.70455e9 −0.703016
\(711\) −2.70119e8 −0.0281846
\(712\) −8.02831e9 −0.833574
\(713\) −1.19549e9 −0.123518
\(714\) −1.87076e10 −1.92342
\(715\) 1.61298e9 0.165028
\(716\) −1.55115e9 −0.157927
\(717\) 3.72393e9 0.377298
\(718\) 1.18049e10 1.19022
\(719\) 6.55620e9 0.657811 0.328906 0.944363i \(-0.393320\pi\)
0.328906 + 0.944363i \(0.393320\pi\)
\(720\) 3.22465e7 0.00321972
\(721\) 1.75278e10 1.74162
\(722\) −6.87651e9 −0.679967
\(723\) 1.09932e10 1.08178
\(724\) 1.99179e9 0.195056
\(725\) 4.96492e9 0.483870
\(726\) −6.15188e9 −0.596663
\(727\) −1.72739e9 −0.166732 −0.0833662 0.996519i \(-0.526567\pi\)
−0.0833662 + 0.996519i \(0.526567\pi\)
\(728\) −1.03424e10 −0.993487
\(729\) 1.03023e10 0.984894
\(730\) 5.10140e9 0.485355
\(731\) 2.53109e9 0.239660
\(732\) 2.30770e9 0.217466
\(733\) 1.67067e10 1.56685 0.783426 0.621486i \(-0.213471\pi\)
0.783426 + 0.621486i \(0.213471\pi\)
\(734\) −1.34273e10 −1.25329
\(735\) −1.26168e10 −1.17205
\(736\) 1.83857e9 0.169984
\(737\) −4.41075e9 −0.405860
\(738\) 1.05748e8 0.00968448
\(739\) 1.02748e10 0.936522 0.468261 0.883590i \(-0.344881\pi\)
0.468261 + 0.883590i \(0.344881\pi\)
\(740\) −5.02389e9 −0.455753
\(741\) −1.70163e9 −0.153639
\(742\) 4.53794e9 0.407798
\(743\) 1.44275e10 1.29042 0.645209 0.764006i \(-0.276770\pi\)
0.645209 + 0.764006i \(0.276770\pi\)
\(744\) 7.34110e9 0.653516
\(745\) 1.25673e10 1.11352
\(746\) 1.57450e10 1.38853
\(747\) −1.11221e8 −0.00976257
\(748\) −3.64445e9 −0.318403
\(749\) 2.61625e10 2.27506
\(750\) 8.88356e9 0.768905
\(751\) −7.52453e9 −0.648246 −0.324123 0.946015i \(-0.605069\pi\)
−0.324123 + 0.946015i \(0.605069\pi\)
\(752\) −1.33621e9 −0.114581
\(753\) −4.14880e9 −0.354112
\(754\) 4.25010e9 0.361077
\(755\) −1.06611e10 −0.901543
\(756\) −8.98319e9 −0.756144
\(757\) −2.21226e10 −1.85353 −0.926767 0.375636i \(-0.877424\pi\)
−0.926767 + 0.375636i \(0.877424\pi\)
\(758\) −5.06379e9 −0.422312
\(759\) 1.09025e9 0.0905065
\(760\) 2.36981e9 0.195824
\(761\) 4.50264e9 0.370357 0.185178 0.982705i \(-0.440714\pi\)
0.185178 + 0.982705i \(0.440714\pi\)
\(762\) 2.90233e9 0.237632
\(763\) −2.26327e9 −0.184459
\(764\) 7.54309e9 0.611959
\(765\) −1.91392e8 −0.0154565
\(766\) 1.02415e10 0.823310
\(767\) 3.12930e9 0.250417
\(768\) −1.31965e10 −1.05123
\(769\) −3.77262e9 −0.299158 −0.149579 0.988750i \(-0.547792\pi\)
−0.149579 + 0.988750i \(0.547792\pi\)
\(770\) 4.53716e9 0.358151
\(771\) −6.08876e9 −0.478452
\(772\) 2.80229e9 0.219206
\(773\) −6.76884e9 −0.527091 −0.263546 0.964647i \(-0.584892\pi\)
−0.263546 + 0.964647i \(0.584892\pi\)
\(774\) 2.12087e7 0.00164407
\(775\) 4.31853e9 0.333257
\(776\) −2.39673e10 −1.84121
\(777\) 3.21807e10 2.46106
\(778\) −1.50594e10 −1.14651
\(779\) 3.22901e9 0.244730
\(780\) −2.30640e9 −0.174022
\(781\) 8.36643e9 0.628437
\(782\) 3.15259e9 0.235746
\(783\) 1.17073e10 0.871545
\(784\) −7.67045e9 −0.568479
\(785\) −1.33001e10 −0.981319
\(786\) 1.77368e10 1.30285
\(787\) −6.17492e9 −0.451564 −0.225782 0.974178i \(-0.572494\pi\)
−0.225782 + 0.974178i \(0.572494\pi\)
\(788\) −1.03510e9 −0.0753602
\(789\) −4.99494e9 −0.362044
\(790\) 1.30954e10 0.944980
\(791\) −3.08813e10 −2.21859
\(792\) −9.68476e7 −0.00692709
\(793\) 3.68544e9 0.262442
\(794\) 1.27269e10 0.902300
\(795\) 3.20937e9 0.226535
\(796\) −7.37906e9 −0.518568
\(797\) 7.98475e8 0.0558672 0.0279336 0.999610i \(-0.491107\pi\)
0.0279336 + 0.999610i \(0.491107\pi\)
\(798\) −4.78653e9 −0.333434
\(799\) 7.93079e9 0.550051
\(800\) −6.64157e9 −0.458623
\(801\) 1.65897e8 0.0114058
\(802\) −1.34834e10 −0.922974
\(803\) −6.36591e9 −0.433867
\(804\) 6.30693e9 0.427978
\(805\) 3.35059e9 0.226379
\(806\) 3.69677e9 0.248685
\(807\) −1.33955e10 −0.897225
\(808\) −7.56992e9 −0.504837
\(809\) 1.09645e10 0.728064 0.364032 0.931386i \(-0.381400\pi\)
0.364032 + 0.931386i \(0.381400\pi\)
\(810\) 7.55127e9 0.499255
\(811\) −1.61849e10 −1.06546 −0.532729 0.846286i \(-0.678834\pi\)
−0.532729 + 0.846286i \(0.678834\pi\)
\(812\) −1.02060e10 −0.668975
\(813\) −1.47346e10 −0.961658
\(814\) −7.34360e9 −0.477226
\(815\) −8.64680e9 −0.559505
\(816\) −8.04357e9 −0.518243
\(817\) 6.47605e8 0.0415464
\(818\) 1.18072e9 0.0754243
\(819\) 2.13716e8 0.0135939
\(820\) 4.37661e9 0.277198
\(821\) 1.75120e10 1.10442 0.552212 0.833704i \(-0.313784\pi\)
0.552212 + 0.833704i \(0.313784\pi\)
\(822\) −1.59309e10 −1.00043
\(823\) 9.36396e9 0.585545 0.292773 0.956182i \(-0.405422\pi\)
0.292773 + 0.956182i \(0.405422\pi\)
\(824\) 1.81382e10 1.12940
\(825\) −3.93837e9 −0.244190
\(826\) 8.80242e9 0.543466
\(827\) 1.31384e10 0.807743 0.403872 0.914816i \(-0.367664\pi\)
0.403872 + 0.914816i \(0.367664\pi\)
\(828\) −2.25516e7 −0.00138061
\(829\) 5.48336e9 0.334277 0.167138 0.985933i \(-0.446547\pi\)
0.167138 + 0.985933i \(0.446547\pi\)
\(830\) 5.39199e9 0.327322
\(831\) −6.36035e9 −0.384483
\(832\) −8.73004e9 −0.525514
\(833\) 4.55264e10 2.72902
\(834\) −1.84766e10 −1.10291
\(835\) 8.70657e9 0.517541
\(836\) −9.32472e8 −0.0551968
\(837\) 1.01831e10 0.600262
\(838\) −1.69928e9 −0.0997495
\(839\) −2.31219e10 −1.35162 −0.675812 0.737074i \(-0.736207\pi\)
−0.675812 + 0.737074i \(0.736207\pi\)
\(840\) −2.05749e10 −1.19773
\(841\) −3.94896e9 −0.228927
\(842\) −9.48536e9 −0.547598
\(843\) 9.79875e9 0.563345
\(844\) −4.85353e9 −0.277881
\(845\) 8.06841e9 0.460034
\(846\) 6.64545e7 0.00377336
\(847\) 2.35925e10 1.33408
\(848\) 1.95115e9 0.109876
\(849\) 1.56468e10 0.877504
\(850\) −1.13883e10 −0.636052
\(851\) −5.42309e9 −0.301643
\(852\) −1.19632e10 −0.662685
\(853\) −2.88007e10 −1.58885 −0.794423 0.607365i \(-0.792227\pi\)
−0.794423 + 0.607365i \(0.792227\pi\)
\(854\) 1.03668e10 0.569563
\(855\) −4.89698e7 −0.00267946
\(856\) 2.70736e10 1.47533
\(857\) −9.62309e9 −0.522254 −0.261127 0.965304i \(-0.584094\pi\)
−0.261127 + 0.965304i \(0.584094\pi\)
\(858\) −3.37135e9 −0.182221
\(859\) −2.79141e10 −1.50261 −0.751306 0.659954i \(-0.770576\pi\)
−0.751306 + 0.659954i \(0.770576\pi\)
\(860\) 8.77767e8 0.0470581
\(861\) −2.80346e10 −1.49687
\(862\) 1.77433e10 0.943535
\(863\) −1.26248e10 −0.668633 −0.334317 0.942461i \(-0.608505\pi\)
−0.334317 + 0.942461i \(0.608505\pi\)
\(864\) −1.56608e10 −0.826070
\(865\) 7.38352e9 0.387889
\(866\) 1.48101e10 0.774898
\(867\) 2.84111e10 1.48054
\(868\) −8.87727e9 −0.460745
\(869\) −1.63414e10 −0.844732
\(870\) 8.45503e9 0.435309
\(871\) 1.00723e10 0.516492
\(872\) −2.34209e9 −0.119618
\(873\) 4.95261e8 0.0251933
\(874\) 8.06624e8 0.0408678
\(875\) −3.40685e10 −1.71920
\(876\) 9.10261e9 0.457511
\(877\) −1.54386e10 −0.772876 −0.386438 0.922315i \(-0.626295\pi\)
−0.386438 + 0.922315i \(0.626295\pi\)
\(878\) −7.27080e9 −0.362536
\(879\) 1.28036e10 0.635876
\(880\) 1.95081e9 0.0964997
\(881\) 7.19197e9 0.354350 0.177175 0.984179i \(-0.443304\pi\)
0.177175 + 0.984179i \(0.443304\pi\)
\(882\) 3.81480e8 0.0187211
\(883\) −2.42226e10 −1.18402 −0.592009 0.805931i \(-0.701665\pi\)
−0.592009 + 0.805931i \(0.701665\pi\)
\(884\) 8.32238e9 0.405196
\(885\) 6.22535e9 0.301899
\(886\) 1.79029e10 0.864781
\(887\) 2.30169e9 0.110742 0.0553712 0.998466i \(-0.482366\pi\)
0.0553712 + 0.998466i \(0.482366\pi\)
\(888\) 3.33015e10 1.59594
\(889\) −1.11305e10 −0.531322
\(890\) −8.04269e9 −0.382416
\(891\) −9.42303e9 −0.446292
\(892\) −1.55977e10 −0.735842
\(893\) 2.02918e9 0.0953542
\(894\) −2.62674e10 −1.22952
\(895\) −4.92810e9 −0.229773
\(896\) 5.08821e9 0.236312
\(897\) −2.48967e9 −0.115178
\(898\) 2.47643e10 1.14119
\(899\) 1.15692e10 0.531063
\(900\) 8.14645e7 0.00372494
\(901\) −1.15806e10 −0.527468
\(902\) 6.39745e9 0.290258
\(903\) −5.62257e9 −0.254114
\(904\) −3.19568e10 −1.43871
\(905\) 6.32806e9 0.283792
\(906\) 2.22831e10 0.995467
\(907\) −3.00170e10 −1.33580 −0.667899 0.744252i \(-0.732806\pi\)
−0.667899 + 0.744252i \(0.732806\pi\)
\(908\) 2.77128e9 0.122851
\(909\) 1.56425e8 0.00690768
\(910\) −1.03609e10 −0.455779
\(911\) 1.56381e10 0.685283 0.342642 0.939466i \(-0.388678\pi\)
0.342642 + 0.939466i \(0.388678\pi\)
\(912\) −2.05803e9 −0.0898401
\(913\) −6.72852e9 −0.292598
\(914\) 2.36973e10 1.02657
\(915\) 7.33171e9 0.316396
\(916\) 1.04009e9 0.0447135
\(917\) −6.80207e10 −2.91305
\(918\) −2.68536e10 −1.14565
\(919\) 2.98731e10 1.26963 0.634813 0.772666i \(-0.281077\pi\)
0.634813 + 0.772666i \(0.281077\pi\)
\(920\) 3.46728e9 0.146802
\(921\) −1.26338e9 −0.0532875
\(922\) 1.37408e9 0.0577370
\(923\) −1.91054e10 −0.799742
\(924\) 8.09582e9 0.337605
\(925\) 1.95901e10 0.813845
\(926\) −1.37336e10 −0.568387
\(927\) −3.74809e8 −0.0154536
\(928\) −1.77926e10 −0.730840
\(929\) −2.76307e10 −1.13067 −0.565337 0.824860i \(-0.691254\pi\)
−0.565337 + 0.824860i \(0.691254\pi\)
\(930\) 7.35426e9 0.299812
\(931\) 1.16484e10 0.473089
\(932\) 1.75003e10 0.708091
\(933\) 4.06116e10 1.63706
\(934\) 4.67269e9 0.187652
\(935\) −1.15787e10 −0.463253
\(936\) 2.21159e8 0.00881533
\(937\) 3.33320e10 1.32365 0.661825 0.749658i \(-0.269782\pi\)
0.661825 + 0.749658i \(0.269782\pi\)
\(938\) 2.83323e10 1.12091
\(939\) 7.94319e8 0.0313087
\(940\) 2.75036e9 0.108004
\(941\) 1.77120e10 0.692955 0.346477 0.938058i \(-0.387378\pi\)
0.346477 + 0.938058i \(0.387378\pi\)
\(942\) 2.77990e10 1.08355
\(943\) 4.72438e9 0.183465
\(944\) 3.78472e9 0.146431
\(945\) −2.85402e10 −1.10013
\(946\) 1.28306e9 0.0492753
\(947\) −3.48632e10 −1.33396 −0.666980 0.745076i \(-0.732413\pi\)
−0.666980 + 0.745076i \(0.732413\pi\)
\(948\) 2.33665e10 0.890768
\(949\) 1.45370e10 0.552134
\(950\) −2.91382e9 −0.110263
\(951\) −3.14444e10 −1.18553
\(952\) 7.42423e10 2.78883
\(953\) 2.76617e10 1.03527 0.517635 0.855602i \(-0.326813\pi\)
0.517635 + 0.855602i \(0.326813\pi\)
\(954\) −9.70378e7 −0.00361844
\(955\) 2.39649e10 0.890355
\(956\) −4.66001e9 −0.172498
\(957\) −1.05508e10 −0.389130
\(958\) 2.12191e10 0.779737
\(959\) 6.10950e10 2.23687
\(960\) −1.73673e10 −0.633553
\(961\) −1.74496e10 −0.634239
\(962\) 1.67697e10 0.607312
\(963\) −5.59451e8 −0.0201869
\(964\) −1.37565e10 −0.494582
\(965\) 8.90304e9 0.318928
\(966\) −7.00320e9 −0.249963
\(967\) 2.41546e9 0.0859028 0.0429514 0.999077i \(-0.486324\pi\)
0.0429514 + 0.999077i \(0.486324\pi\)
\(968\) 2.44141e10 0.865123
\(969\) 1.22150e10 0.431282
\(970\) −2.40103e10 −0.844687
\(971\) −3.04413e10 −1.06708 −0.533539 0.845776i \(-0.679138\pi\)
−0.533539 + 0.845776i \(0.679138\pi\)
\(972\) 3.87049e8 0.0135187
\(973\) 7.08580e10 2.46601
\(974\) −7.08578e9 −0.245715
\(975\) 8.99357e9 0.310753
\(976\) 4.45734e9 0.153462
\(977\) 2.20394e10 0.756082 0.378041 0.925789i \(-0.376598\pi\)
0.378041 + 0.925789i \(0.376598\pi\)
\(978\) 1.80730e10 0.617795
\(979\) 1.00363e10 0.341848
\(980\) 1.57883e10 0.535852
\(981\) 4.83970e7 0.00163673
\(982\) 4.08829e10 1.37769
\(983\) 9.14954e9 0.307229 0.153614 0.988131i \(-0.450909\pi\)
0.153614 + 0.988131i \(0.450909\pi\)
\(984\) −2.90109e10 −0.970685
\(985\) −3.28859e9 −0.109644
\(986\) −3.05090e10 −1.01358
\(987\) −1.76175e10 −0.583223
\(988\) 2.12937e9 0.0702428
\(989\) 9.47514e8 0.0311457
\(990\) −9.70211e7 −0.00317792
\(991\) 2.26100e10 0.737976 0.368988 0.929434i \(-0.379704\pi\)
0.368988 + 0.929434i \(0.379704\pi\)
\(992\) −1.54762e10 −0.503354
\(993\) −6.93157e9 −0.224651
\(994\) −5.37416e10 −1.73563
\(995\) −2.34438e10 −0.754478
\(996\) 9.62111e9 0.308544
\(997\) −5.32639e10 −1.70216 −0.851080 0.525036i \(-0.824052\pi\)
−0.851080 + 0.525036i \(0.824052\pi\)
\(998\) −1.79181e10 −0.570606
\(999\) 4.61936e10 1.46589
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 43.8.a.a.1.8 11
3.2 odd 2 387.8.a.b.1.4 11
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.8.a.a.1.8 11 1.1 even 1 trivial
387.8.a.b.1.4 11 3.2 odd 2