Properties

Label 2.40.a.a.1.1
Level 2
Weight 40
Character 2.1
Self dual yes
Analytic conductor 19.268
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 40 \)
Character orbit: \([\chi]\) \(=\) 2.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.2679102779\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2.1

$q$-expansion

\(f(q)\) \(=\) \(q+524288. q^{2} -7.35458e8 q^{3} +2.74878e11 q^{4} -1.62262e13 q^{5} -3.85592e14 q^{6} +1.60501e16 q^{7} +1.44115e17 q^{8} -3.51166e18 q^{9} +O(q^{10})\) \(q+524288. q^{2} -7.35458e8 q^{3} +2.74878e11 q^{4} -1.62262e13 q^{5} -3.85592e14 q^{6} +1.60501e16 q^{7} +1.44115e17 q^{8} -3.51166e18 q^{9} -8.50719e18 q^{10} -1.67430e20 q^{11} -2.02161e20 q^{12} -1.32369e21 q^{13} +8.41486e21 q^{14} +1.19337e22 q^{15} +7.55579e22 q^{16} -4.96481e23 q^{17} -1.84112e24 q^{18} -1.14998e25 q^{19} -4.46022e24 q^{20} -1.18042e25 q^{21} -8.77814e25 q^{22} -6.66242e26 q^{23} -1.05991e26 q^{24} -1.55570e27 q^{25} -6.93995e26 q^{26} +5.56316e27 q^{27} +4.41181e27 q^{28} +4.40183e28 q^{29} +6.25668e27 q^{30} +1.58310e28 q^{31} +3.96141e28 q^{32} +1.23138e29 q^{33} -2.60299e29 q^{34} -2.60431e29 q^{35} -9.65277e29 q^{36} -2.58657e30 q^{37} -6.02920e30 q^{38} +9.73520e29 q^{39} -2.33844e30 q^{40} +5.12373e31 q^{41} -6.18878e30 q^{42} +7.87984e31 q^{43} -4.60227e31 q^{44} +5.69808e31 q^{45} -3.49303e32 q^{46} +2.42751e32 q^{47} -5.55697e31 q^{48} -6.51939e32 q^{49} -8.15635e32 q^{50} +3.65141e32 q^{51} -3.63853e32 q^{52} +6.95517e32 q^{53} +2.91670e33 q^{54} +2.71675e33 q^{55} +2.31306e33 q^{56} +8.45761e33 q^{57} +2.30783e34 q^{58} -2.01815e34 q^{59} +3.28030e33 q^{60} -1.28433e35 q^{61} +8.30001e33 q^{62} -5.63623e34 q^{63} +2.07692e34 q^{64} +2.14784e34 q^{65} +6.45596e34 q^{66} +4.56918e35 q^{67} -1.36472e35 q^{68} +4.89993e35 q^{69} -1.36541e35 q^{70} -9.94586e34 q^{71} -5.06083e35 q^{72} +8.12888e35 q^{73} -1.35611e36 q^{74} +1.14415e36 q^{75} -3.16104e36 q^{76} -2.68726e36 q^{77} +5.10405e35 q^{78} -8.57275e36 q^{79} -1.22602e36 q^{80} +1.01397e37 q^{81} +2.68631e37 q^{82} +7.12589e36 q^{83} -3.24470e36 q^{84} +8.05599e36 q^{85} +4.13131e37 q^{86} -3.23736e37 q^{87} -2.41292e37 q^{88} -1.30970e38 q^{89} +2.98743e37 q^{90} -2.12453e37 q^{91} -1.83135e38 q^{92} -1.16430e37 q^{93} +1.27271e38 q^{94} +1.86598e38 q^{95} -2.91345e37 q^{96} +7.61043e38 q^{97} -3.41804e38 q^{98} +5.87956e38 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 524288. 0.707107
\(3\) −7.35458e8 −0.365337 −0.182668 0.983175i \(-0.558473\pi\)
−0.182668 + 0.983175i \(0.558473\pi\)
\(4\) 2.74878e11 0.500000
\(5\) −1.62262e13 −0.380453 −0.190227 0.981740i \(-0.560922\pi\)
−0.190227 + 0.981740i \(0.560922\pi\)
\(6\) −3.85592e14 −0.258332
\(7\) 1.60501e16 0.532188 0.266094 0.963947i \(-0.414267\pi\)
0.266094 + 0.963947i \(0.414267\pi\)
\(8\) 1.44115e17 0.353553
\(9\) −3.51166e18 −0.866529
\(10\) −8.50719e18 −0.269021
\(11\) −1.67430e20 −0.825421 −0.412710 0.910862i \(-0.635418\pi\)
−0.412710 + 0.910862i \(0.635418\pi\)
\(12\) −2.02161e20 −0.182668
\(13\) −1.32369e21 −0.251126 −0.125563 0.992086i \(-0.540074\pi\)
−0.125563 + 0.992086i \(0.540074\pi\)
\(14\) 8.41486e21 0.376314
\(15\) 1.19337e22 0.138994
\(16\) 7.55579e22 0.250000
\(17\) −4.96481e23 −0.503673 −0.251836 0.967770i \(-0.581034\pi\)
−0.251836 + 0.967770i \(0.581034\pi\)
\(18\) −1.84112e24 −0.612728
\(19\) −1.14998e25 −1.33350 −0.666752 0.745280i \(-0.732316\pi\)
−0.666752 + 0.745280i \(0.732316\pi\)
\(20\) −4.46022e24 −0.190227
\(21\) −1.18042e25 −0.194428
\(22\) −8.77814e25 −0.583660
\(23\) −6.66242e26 −1.86183 −0.930913 0.365241i \(-0.880987\pi\)
−0.930913 + 0.365241i \(0.880987\pi\)
\(24\) −1.05991e26 −0.129166
\(25\) −1.55570e27 −0.855255
\(26\) −6.93995e26 −0.177573
\(27\) 5.56316e27 0.681912
\(28\) 4.41181e27 0.266094
\(29\) 4.40183e28 1.33928 0.669641 0.742685i \(-0.266448\pi\)
0.669641 + 0.742685i \(0.266448\pi\)
\(30\) 6.25668e27 0.0982833
\(31\) 1.58310e28 0.131207 0.0656033 0.997846i \(-0.479103\pi\)
0.0656033 + 0.997846i \(0.479103\pi\)
\(32\) 3.96141e28 0.176777
\(33\) 1.23138e29 0.301557
\(34\) −2.60299e29 −0.356150
\(35\) −2.60431e29 −0.202473
\(36\) −9.65277e29 −0.433264
\(37\) −2.58657e30 −0.680440 −0.340220 0.940346i \(-0.610502\pi\)
−0.340220 + 0.940346i \(0.610502\pi\)
\(38\) −6.02920e30 −0.942930
\(39\) 9.73520e29 0.0917455
\(40\) −2.33844e30 −0.134510
\(41\) 5.12373e31 1.82096 0.910482 0.413550i \(-0.135711\pi\)
0.910482 + 0.413550i \(0.135711\pi\)
\(42\) −6.18878e30 −0.137481
\(43\) 7.87984e31 1.10632 0.553162 0.833074i \(-0.313421\pi\)
0.553162 + 0.833074i \(0.313421\pi\)
\(44\) −4.60227e31 −0.412710
\(45\) 5.69808e31 0.329674
\(46\) −3.49303e32 −1.31651
\(47\) 2.42751e32 0.601524 0.300762 0.953699i \(-0.402759\pi\)
0.300762 + 0.953699i \(0.402759\pi\)
\(48\) −5.55697e31 −0.0913342
\(49\) −6.51939e32 −0.716776
\(50\) −8.15635e32 −0.604757
\(51\) 3.65141e32 0.184010
\(52\) −3.63853e32 −0.125563
\(53\) 6.95517e32 0.165550 0.0827748 0.996568i \(-0.473622\pi\)
0.0827748 + 0.996568i \(0.473622\pi\)
\(54\) 2.91670e33 0.482185
\(55\) 2.71675e33 0.314034
\(56\) 2.31306e33 0.188157
\(57\) 8.45761e33 0.487178
\(58\) 2.30783e34 0.947016
\(59\) −2.01815e34 −0.593389 −0.296695 0.954972i \(-0.595884\pi\)
−0.296695 + 0.954972i \(0.595884\pi\)
\(60\) 3.28030e33 0.0694968
\(61\) −1.28433e35 −1.97127 −0.985635 0.168890i \(-0.945982\pi\)
−0.985635 + 0.168890i \(0.945982\pi\)
\(62\) 8.30001e33 0.0927770
\(63\) −5.63623e34 −0.461156
\(64\) 2.07692e34 0.125000
\(65\) 2.14784e34 0.0955416
\(66\) 6.45596e34 0.213233
\(67\) 4.56918e35 1.12559 0.562794 0.826597i \(-0.309726\pi\)
0.562794 + 0.826597i \(0.309726\pi\)
\(68\) −1.36472e35 −0.251836
\(69\) 4.89993e35 0.680194
\(70\) −1.36541e35 −0.143170
\(71\) −9.94586e34 −0.0790869 −0.0395434 0.999218i \(-0.512590\pi\)
−0.0395434 + 0.999218i \(0.512590\pi\)
\(72\) −5.06083e35 −0.306364
\(73\) 8.12888e35 0.376041 0.188020 0.982165i \(-0.439793\pi\)
0.188020 + 0.982165i \(0.439793\pi\)
\(74\) −1.35611e36 −0.481144
\(75\) 1.14415e36 0.312456
\(76\) −3.16104e36 −0.666752
\(77\) −2.68726e36 −0.439279
\(78\) 5.10405e35 0.0648739
\(79\) −8.57275e36 −0.849947 −0.424973 0.905206i \(-0.639717\pi\)
−0.424973 + 0.905206i \(0.639717\pi\)
\(80\) −1.22602e36 −0.0951133
\(81\) 1.01397e37 0.617401
\(82\) 2.68631e37 1.28762
\(83\) 7.12589e36 0.269660 0.134830 0.990869i \(-0.456951\pi\)
0.134830 + 0.990869i \(0.456951\pi\)
\(84\) −3.24470e36 −0.0972140
\(85\) 8.05599e36 0.191624
\(86\) 4.13131e37 0.782290
\(87\) −3.23736e37 −0.489290
\(88\) −2.41292e37 −0.291830
\(89\) −1.30970e38 −1.27076 −0.635382 0.772198i \(-0.719157\pi\)
−0.635382 + 0.772198i \(0.719157\pi\)
\(90\) 2.98743e37 0.233114
\(91\) −2.12453e37 −0.133646
\(92\) −1.83135e38 −0.930913
\(93\) −1.16430e37 −0.0479346
\(94\) 1.27271e38 0.425342
\(95\) 1.86598e38 0.507336
\(96\) −2.91345e37 −0.0645831
\(97\) 7.61043e38 1.37835 0.689176 0.724594i \(-0.257973\pi\)
0.689176 + 0.724594i \(0.257973\pi\)
\(98\) −3.41804e38 −0.506837
\(99\) 5.87956e38 0.715251
\(100\) −4.27628e38 −0.427628
\(101\) −7.14964e38 −0.588868 −0.294434 0.955672i \(-0.595131\pi\)
−0.294434 + 0.955672i \(0.595131\pi\)
\(102\) 1.91439e38 0.130115
\(103\) 6.55259e38 0.368203 0.184101 0.982907i \(-0.441063\pi\)
0.184101 + 0.982907i \(0.441063\pi\)
\(104\) −1.90764e38 −0.0887864
\(105\) 1.91536e38 0.0739707
\(106\) 3.64651e38 0.117061
\(107\) −5.91456e39 −1.58103 −0.790513 0.612445i \(-0.790186\pi\)
−0.790513 + 0.612445i \(0.790186\pi\)
\(108\) 1.52919e39 0.340956
\(109\) −4.18504e39 −0.779620 −0.389810 0.920895i \(-0.627459\pi\)
−0.389810 + 0.920895i \(0.627459\pi\)
\(110\) 1.42436e39 0.222055
\(111\) 1.90231e39 0.248590
\(112\) 1.21271e39 0.133047
\(113\) 3.82469e39 0.352831 0.176415 0.984316i \(-0.443550\pi\)
0.176415 + 0.984316i \(0.443550\pi\)
\(114\) 4.43422e39 0.344487
\(115\) 1.08106e40 0.708338
\(116\) 1.20997e40 0.669641
\(117\) 4.64835e39 0.217608
\(118\) −1.05809e40 −0.419590
\(119\) −7.96855e39 −0.268049
\(120\) 1.71982e39 0.0491417
\(121\) −1.31121e40 −0.318681
\(122\) −6.73361e40 −1.39390
\(123\) −3.76829e40 −0.665265
\(124\) 4.35159e39 0.0656033
\(125\) 5.47583e40 0.705838
\(126\) −2.95501e40 −0.326087
\(127\) 8.14150e40 0.770073 0.385037 0.922901i \(-0.374189\pi\)
0.385037 + 0.922901i \(0.374189\pi\)
\(128\) 1.08890e40 0.0883883
\(129\) −5.79529e40 −0.404181
\(130\) 1.12609e40 0.0675581
\(131\) 2.55072e41 1.31787 0.658936 0.752199i \(-0.271007\pi\)
0.658936 + 0.752199i \(0.271007\pi\)
\(132\) 3.38478e40 0.150778
\(133\) −1.84572e41 −0.709675
\(134\) 2.39556e41 0.795911
\(135\) −9.02689e40 −0.259436
\(136\) −7.15504e40 −0.178075
\(137\) −1.57560e41 −0.339933 −0.169967 0.985450i \(-0.554366\pi\)
−0.169967 + 0.985450i \(0.554366\pi\)
\(138\) 2.56898e41 0.480970
\(139\) −5.88829e41 −0.957637 −0.478819 0.877914i \(-0.658935\pi\)
−0.478819 + 0.877914i \(0.658935\pi\)
\(140\) −7.15868e40 −0.101236
\(141\) −1.78533e41 −0.219759
\(142\) −5.21449e40 −0.0559229
\(143\) 2.21625e41 0.207284
\(144\) −2.65333e41 −0.216632
\(145\) −7.14249e41 −0.509534
\(146\) 4.26187e41 0.265901
\(147\) 4.79474e41 0.261865
\(148\) −7.10990e41 −0.340220
\(149\) 1.20747e42 0.506691 0.253346 0.967376i \(-0.418469\pi\)
0.253346 + 0.967376i \(0.418469\pi\)
\(150\) 5.99866e41 0.220940
\(151\) 5.68508e42 1.83944 0.919721 0.392573i \(-0.128415\pi\)
0.919721 + 0.392573i \(0.128415\pi\)
\(152\) −1.65729e42 −0.471465
\(153\) 1.74347e42 0.436447
\(154\) −1.40890e42 −0.310617
\(155\) −2.56877e41 −0.0499179
\(156\) 2.67599e41 0.0458728
\(157\) −9.75064e42 −1.47567 −0.737836 0.674980i \(-0.764152\pi\)
−0.737836 + 0.674980i \(0.764152\pi\)
\(158\) −4.49459e42 −0.601003
\(159\) −5.11524e41 −0.0604814
\(160\) −6.42785e41 −0.0672552
\(161\) −1.06932e43 −0.990842
\(162\) 5.31613e42 0.436569
\(163\) 4.57034e42 0.332882 0.166441 0.986051i \(-0.446773\pi\)
0.166441 + 0.986051i \(0.446773\pi\)
\(164\) 1.40840e43 0.910482
\(165\) −1.99805e42 −0.114728
\(166\) 3.73602e42 0.190678
\(167\) 2.16180e43 0.981397 0.490699 0.871329i \(-0.336742\pi\)
0.490699 + 0.871329i \(0.336742\pi\)
\(168\) −1.70116e42 −0.0687406
\(169\) −2.60316e43 −0.936936
\(170\) 4.22366e42 0.135499
\(171\) 4.03833e43 1.15552
\(172\) 2.16599e43 0.553162
\(173\) −3.61908e43 −0.825467 −0.412733 0.910852i \(-0.635426\pi\)
−0.412733 + 0.910852i \(0.635426\pi\)
\(174\) −1.69731e43 −0.345980
\(175\) −2.49691e43 −0.455157
\(176\) −1.26506e43 −0.206355
\(177\) 1.48427e43 0.216787
\(178\) −6.86657e43 −0.898566
\(179\) 8.73010e43 1.02420 0.512100 0.858926i \(-0.328868\pi\)
0.512100 + 0.858926i \(0.328868\pi\)
\(180\) 1.56628e43 0.164837
\(181\) −1.51508e44 −1.43121 −0.715607 0.698504i \(-0.753850\pi\)
−0.715607 + 0.698504i \(0.753850\pi\)
\(182\) −1.11387e43 −0.0945021
\(183\) 9.44574e43 0.720178
\(184\) −9.60156e43 −0.658255
\(185\) 4.19701e43 0.258876
\(186\) −6.10431e42 −0.0338949
\(187\) 8.31257e43 0.415742
\(188\) 6.67269e43 0.300762
\(189\) 8.92891e43 0.362905
\(190\) 9.78308e43 0.358741
\(191\) 4.32591e44 1.43195 0.715974 0.698127i \(-0.245983\pi\)
0.715974 + 0.698127i \(0.245983\pi\)
\(192\) −1.52749e43 −0.0456671
\(193\) −7.04390e44 −1.90303 −0.951515 0.307601i \(-0.900474\pi\)
−0.951515 + 0.307601i \(0.900474\pi\)
\(194\) 3.99006e44 0.974643
\(195\) −1.57965e43 −0.0349049
\(196\) −1.79204e44 −0.358388
\(197\) −1.81044e44 −0.327864 −0.163932 0.986472i \(-0.552418\pi\)
−0.163932 + 0.986472i \(0.552418\pi\)
\(198\) 3.08258e44 0.505759
\(199\) −9.78460e44 −1.45515 −0.727575 0.686028i \(-0.759353\pi\)
−0.727575 + 0.686028i \(0.759353\pi\)
\(200\) −2.24200e44 −0.302378
\(201\) −3.36044e44 −0.411219
\(202\) −3.74847e44 −0.416392
\(203\) 7.06497e44 0.712750
\(204\) 1.00369e44 0.0920051
\(205\) −8.31386e44 −0.692791
\(206\) 3.43544e44 0.260359
\(207\) 2.33961e45 1.61333
\(208\) −1.00015e44 −0.0627814
\(209\) 1.92541e45 1.10070
\(210\) 1.00420e44 0.0523052
\(211\) 1.87844e44 0.0891843 0.0445922 0.999005i \(-0.485801\pi\)
0.0445922 + 0.999005i \(0.485801\pi\)
\(212\) 1.91182e44 0.0827748
\(213\) 7.31476e43 0.0288934
\(214\) −3.10093e45 −1.11795
\(215\) −1.27860e45 −0.420905
\(216\) 8.01736e44 0.241092
\(217\) 2.54089e44 0.0698265
\(218\) −2.19417e45 −0.551275
\(219\) −5.97845e44 −0.137382
\(220\) 7.46773e44 0.157017
\(221\) 6.57187e44 0.126485
\(222\) 9.97359e44 0.175780
\(223\) −7.17839e45 −1.15900 −0.579498 0.814974i \(-0.696751\pi\)
−0.579498 + 0.814974i \(0.696751\pi\)
\(224\) 6.35809e44 0.0940784
\(225\) 5.46309e45 0.741104
\(226\) 2.00524e45 0.249489
\(227\) −7.82184e45 −0.892904 −0.446452 0.894808i \(-0.647313\pi\)
−0.446452 + 0.894808i \(0.647313\pi\)
\(228\) 2.32481e45 0.243589
\(229\) −4.60714e45 −0.443240 −0.221620 0.975133i \(-0.571135\pi\)
−0.221620 + 0.975133i \(0.571135\pi\)
\(230\) 5.66785e45 0.500870
\(231\) 1.97637e45 0.160485
\(232\) 6.34371e45 0.473508
\(233\) 1.80888e46 1.24157 0.620784 0.783982i \(-0.286814\pi\)
0.620784 + 0.783982i \(0.286814\pi\)
\(234\) 2.43707e45 0.153872
\(235\) −3.93892e45 −0.228852
\(236\) −5.54746e45 −0.296695
\(237\) 6.30490e45 0.310517
\(238\) −4.17781e45 −0.189539
\(239\) −8.37506e45 −0.350129 −0.175065 0.984557i \(-0.556013\pi\)
−0.175065 + 0.984557i \(0.556013\pi\)
\(240\) 9.01683e44 0.0347484
\(241\) 1.67167e45 0.0594044 0.0297022 0.999559i \(-0.490544\pi\)
0.0297022 + 0.999559i \(0.490544\pi\)
\(242\) −6.87449e45 −0.225341
\(243\) −3.00024e46 −0.907471
\(244\) −3.53035e46 −0.985635
\(245\) 1.05785e46 0.272700
\(246\) −1.97567e46 −0.470413
\(247\) 1.52222e46 0.334877
\(248\) 2.28149e45 0.0463885
\(249\) −5.24080e45 −0.0985167
\(250\) 2.87091e46 0.499103
\(251\) −7.60545e45 −0.122317 −0.0611586 0.998128i \(-0.519480\pi\)
−0.0611586 + 0.998128i \(0.519480\pi\)
\(252\) −1.54928e46 −0.230578
\(253\) 1.11549e47 1.53679
\(254\) 4.26849e46 0.544524
\(255\) −5.92484e45 −0.0700073
\(256\) 5.70899e45 0.0625000
\(257\) 9.64642e46 0.978747 0.489373 0.872074i \(-0.337226\pi\)
0.489373 + 0.872074i \(0.337226\pi\)
\(258\) −3.03840e46 −0.285799
\(259\) −4.15146e46 −0.362122
\(260\) 5.90395e45 0.0477708
\(261\) −1.54577e47 −1.16053
\(262\) 1.33731e47 0.931876
\(263\) −2.38043e47 −1.53999 −0.769997 0.638048i \(-0.779742\pi\)
−0.769997 + 0.638048i \(0.779742\pi\)
\(264\) 1.77460e46 0.106616
\(265\) −1.12856e46 −0.0629839
\(266\) −9.67690e46 −0.501816
\(267\) 9.63226e46 0.464257
\(268\) 1.25597e47 0.562794
\(269\) −3.43730e47 −1.43235 −0.716173 0.697923i \(-0.754108\pi\)
−0.716173 + 0.697923i \(0.754108\pi\)
\(270\) −4.73269e46 −0.183449
\(271\) 2.34239e47 0.844809 0.422404 0.906408i \(-0.361186\pi\)
0.422404 + 0.906408i \(0.361186\pi\)
\(272\) −3.75130e46 −0.125918
\(273\) 1.56251e46 0.0488259
\(274\) −8.26066e46 −0.240369
\(275\) 2.60471e47 0.705945
\(276\) 1.34688e47 0.340097
\(277\) 4.97473e47 1.17062 0.585308 0.810811i \(-0.300974\pi\)
0.585308 + 0.810811i \(0.300974\pi\)
\(278\) −3.08716e47 −0.677152
\(279\) −5.55931e46 −0.113694
\(280\) −3.75321e46 −0.0715849
\(281\) −2.28343e47 −0.406270 −0.203135 0.979151i \(-0.565113\pi\)
−0.203135 + 0.979151i \(0.565113\pi\)
\(282\) −9.36029e46 −0.155393
\(283\) 6.09891e47 0.944968 0.472484 0.881339i \(-0.343358\pi\)
0.472484 + 0.881339i \(0.343358\pi\)
\(284\) −2.73390e46 −0.0395434
\(285\) −1.37235e47 −0.185349
\(286\) 1.16195e47 0.146572
\(287\) 8.22362e47 0.969095
\(288\) −1.39111e47 −0.153182
\(289\) −7.25153e47 −0.746314
\(290\) −3.74472e47 −0.360295
\(291\) −5.59715e47 −0.503563
\(292\) 2.23445e47 0.188020
\(293\) −2.18536e48 −1.72030 −0.860149 0.510043i \(-0.829630\pi\)
−0.860149 + 0.510043i \(0.829630\pi\)
\(294\) 2.51382e47 0.185166
\(295\) 3.27469e47 0.225757
\(296\) −3.72763e47 −0.240572
\(297\) −9.31439e47 −0.562864
\(298\) 6.33061e47 0.358285
\(299\) 8.81898e47 0.467553
\(300\) 3.14502e47 0.156228
\(301\) 1.26472e48 0.588773
\(302\) 2.98062e48 1.30068
\(303\) 5.25827e47 0.215135
\(304\) −8.68899e47 −0.333376
\(305\) 2.08398e48 0.749976
\(306\) 9.14080e47 0.308615
\(307\) −1.84711e48 −0.585187 −0.292594 0.956237i \(-0.594518\pi\)
−0.292594 + 0.956237i \(0.594518\pi\)
\(308\) −7.38668e47 −0.219639
\(309\) −4.81915e47 −0.134518
\(310\) −1.34677e47 −0.0352973
\(311\) 3.22674e48 0.794211 0.397105 0.917773i \(-0.370015\pi\)
0.397105 + 0.917773i \(0.370015\pi\)
\(312\) 1.40299e47 0.0324369
\(313\) −6.99390e48 −1.51916 −0.759582 0.650411i \(-0.774597\pi\)
−0.759582 + 0.650411i \(0.774597\pi\)
\(314\) −5.11215e48 −1.04346
\(315\) 9.14545e47 0.175448
\(316\) −2.35646e48 −0.424973
\(317\) 5.37916e48 0.912135 0.456067 0.889945i \(-0.349258\pi\)
0.456067 + 0.889945i \(0.349258\pi\)
\(318\) −2.68186e47 −0.0427668
\(319\) −7.36997e48 −1.10547
\(320\) −3.37005e47 −0.0475566
\(321\) 4.34991e48 0.577607
\(322\) −5.60633e48 −0.700631
\(323\) 5.70942e48 0.671650
\(324\) 2.78718e48 0.308701
\(325\) 2.05927e48 0.214777
\(326\) 2.39617e48 0.235383
\(327\) 3.07792e48 0.284824
\(328\) 7.38408e48 0.643808
\(329\) 3.89617e48 0.320124
\(330\) −1.04756e48 −0.0811251
\(331\) −1.40018e49 −1.02220 −0.511102 0.859520i \(-0.670762\pi\)
−0.511102 + 0.859520i \(0.670762\pi\)
\(332\) 1.95875e48 0.134830
\(333\) 9.08313e48 0.589621
\(334\) 1.13341e49 0.693953
\(335\) −7.41403e48 −0.428233
\(336\) −8.91897e47 −0.0486070
\(337\) −3.35577e48 −0.172587 −0.0862936 0.996270i \(-0.527502\pi\)
−0.0862936 + 0.996270i \(0.527502\pi\)
\(338\) −1.36480e49 −0.662514
\(339\) −2.81290e48 −0.128902
\(340\) 2.21441e48 0.0958119
\(341\) −2.65058e48 −0.108301
\(342\) 2.11725e49 0.817076
\(343\) −2.50619e49 −0.913647
\(344\) 1.13560e49 0.391145
\(345\) −7.95072e48 −0.258782
\(346\) −1.89744e49 −0.583693
\(347\) −2.60839e49 −0.758486 −0.379243 0.925297i \(-0.623816\pi\)
−0.379243 + 0.925297i \(0.623816\pi\)
\(348\) −8.89879e48 −0.244645
\(349\) 2.03635e49 0.529366 0.264683 0.964335i \(-0.414733\pi\)
0.264683 + 0.964335i \(0.414733\pi\)
\(350\) −1.30910e49 −0.321844
\(351\) −7.36391e48 −0.171246
\(352\) −6.63258e48 −0.145915
\(353\) 8.08319e49 1.68258 0.841291 0.540583i \(-0.181796\pi\)
0.841291 + 0.540583i \(0.181796\pi\)
\(354\) 7.78183e48 0.153292
\(355\) 1.61383e48 0.0300888
\(356\) −3.60006e49 −0.635382
\(357\) 5.86054e48 0.0979281
\(358\) 4.57709e49 0.724219
\(359\) 7.35363e49 1.10194 0.550972 0.834524i \(-0.314257\pi\)
0.550972 + 0.834524i \(0.314257\pi\)
\(360\) 8.21179e48 0.116557
\(361\) 5.78763e49 0.778234
\(362\) −7.94340e49 −1.01202
\(363\) 9.64337e48 0.116426
\(364\) −5.83987e48 −0.0668231
\(365\) −1.31901e49 −0.143066
\(366\) 4.95229e49 0.509243
\(367\) −9.99437e49 −0.974467 −0.487233 0.873272i \(-0.661994\pi\)
−0.487233 + 0.873272i \(0.661994\pi\)
\(368\) −5.03398e49 −0.465457
\(369\) −1.79928e50 −1.57792
\(370\) 2.20044e49 0.183053
\(371\) 1.11631e49 0.0881036
\(372\) −3.20042e48 −0.0239673
\(373\) −1.06368e50 −0.755944 −0.377972 0.925817i \(-0.623378\pi\)
−0.377972 + 0.925817i \(0.623378\pi\)
\(374\) 4.35818e49 0.293974
\(375\) −4.02725e49 −0.257869
\(376\) 3.49841e49 0.212671
\(377\) −5.82666e49 −0.336328
\(378\) 4.68132e49 0.256613
\(379\) 3.09568e50 1.61173 0.805864 0.592101i \(-0.201701\pi\)
0.805864 + 0.592101i \(0.201701\pi\)
\(380\) 5.12915e49 0.253668
\(381\) −5.98773e49 −0.281336
\(382\) 2.26802e50 1.01254
\(383\) 1.31362e50 0.557304 0.278652 0.960392i \(-0.410112\pi\)
0.278652 + 0.960392i \(0.410112\pi\)
\(384\) −8.00843e48 −0.0322915
\(385\) 4.36039e49 0.167125
\(386\) −3.69303e50 −1.34565
\(387\) −2.76713e50 −0.958662
\(388\) 2.09194e50 0.689176
\(389\) 1.27217e50 0.398592 0.199296 0.979939i \(-0.436134\pi\)
0.199296 + 0.979939i \(0.436134\pi\)
\(390\) −8.28192e48 −0.0246815
\(391\) 3.30776e50 0.937751
\(392\) −9.39543e49 −0.253419
\(393\) −1.87595e50 −0.481467
\(394\) −9.49193e49 −0.231835
\(395\) 1.39103e50 0.323365
\(396\) 1.61616e50 0.357625
\(397\) −4.10443e50 −0.864646 −0.432323 0.901719i \(-0.642306\pi\)
−0.432323 + 0.901719i \(0.642306\pi\)
\(398\) −5.12995e50 −1.02895
\(399\) 1.35745e50 0.259270
\(400\) −1.17545e50 −0.213814
\(401\) 7.58787e49 0.131463 0.0657317 0.997837i \(-0.479062\pi\)
0.0657317 + 0.997837i \(0.479062\pi\)
\(402\) −1.76184e50 −0.290776
\(403\) −2.09554e49 −0.0329493
\(404\) −1.96528e50 −0.294434
\(405\) −1.64529e50 −0.234892
\(406\) 3.70408e50 0.503991
\(407\) 4.33068e50 0.561649
\(408\) 5.26224e49 0.0650575
\(409\) 2.10087e50 0.247626 0.123813 0.992306i \(-0.460488\pi\)
0.123813 + 0.992306i \(0.460488\pi\)
\(410\) −4.35886e50 −0.489877
\(411\) 1.15879e50 0.124190
\(412\) 1.80116e50 0.184101
\(413\) −3.23915e50 −0.315795
\(414\) 1.22663e51 1.14079
\(415\) −1.15626e50 −0.102593
\(416\) −5.24368e49 −0.0443932
\(417\) 4.33060e50 0.349860
\(418\) 1.00947e51 0.778314
\(419\) 1.77773e50 0.130826 0.0654129 0.997858i \(-0.479164\pi\)
0.0654129 + 0.997858i \(0.479164\pi\)
\(420\) 5.26491e49 0.0369854
\(421\) −7.65653e50 −0.513489 −0.256744 0.966479i \(-0.582650\pi\)
−0.256744 + 0.966479i \(0.582650\pi\)
\(422\) 9.84841e49 0.0630628
\(423\) −8.52458e50 −0.521238
\(424\) 1.00235e50 0.0585307
\(425\) 7.72375e50 0.430769
\(426\) 3.83504e49 0.0204307
\(427\) −2.06136e51 −1.04909
\(428\) −1.62578e51 −0.790513
\(429\) −1.62996e50 −0.0757286
\(430\) −6.70353e50 −0.297625
\(431\) −9.06012e50 −0.384439 −0.192219 0.981352i \(-0.561569\pi\)
−0.192219 + 0.981352i \(0.561569\pi\)
\(432\) 4.20341e50 0.170478
\(433\) −2.95958e51 −1.14740 −0.573702 0.819064i \(-0.694493\pi\)
−0.573702 + 0.819064i \(0.694493\pi\)
\(434\) 1.33216e50 0.0493748
\(435\) 5.25300e50 0.186152
\(436\) −1.15038e51 −0.389810
\(437\) 7.66164e51 2.48275
\(438\) −3.13443e50 −0.0971434
\(439\) −1.93285e51 −0.572982 −0.286491 0.958083i \(-0.592489\pi\)
−0.286491 + 0.958083i \(0.592489\pi\)
\(440\) 3.91524e50 0.111028
\(441\) 2.28939e51 0.621107
\(442\) 3.44555e50 0.0894386
\(443\) 5.23473e51 1.30023 0.650117 0.759834i \(-0.274720\pi\)
0.650117 + 0.759834i \(0.274720\pi\)
\(444\) 5.22903e50 0.124295
\(445\) 2.12513e51 0.483466
\(446\) −3.76354e51 −0.819534
\(447\) −8.88042e50 −0.185113
\(448\) 3.33347e50 0.0665235
\(449\) 2.64498e51 0.505380 0.252690 0.967547i \(-0.418685\pi\)
0.252690 + 0.967547i \(0.418685\pi\)
\(450\) 2.86423e51 0.524039
\(451\) −8.57865e51 −1.50306
\(452\) 1.05132e51 0.176415
\(453\) −4.18114e51 −0.672016
\(454\) −4.10090e51 −0.631378
\(455\) 3.44731e50 0.0508461
\(456\) 1.21887e51 0.172244
\(457\) 3.37218e49 0.00456610 0.00228305 0.999997i \(-0.499273\pi\)
0.00228305 + 0.999997i \(0.499273\pi\)
\(458\) −2.41547e51 −0.313418
\(459\) −2.76200e51 −0.343460
\(460\) 2.97158e51 0.354169
\(461\) −7.93487e50 −0.0906508 −0.0453254 0.998972i \(-0.514432\pi\)
−0.0453254 + 0.998972i \(0.514432\pi\)
\(462\) 1.03619e51 0.113480
\(463\) −7.37575e51 −0.774420 −0.387210 0.921992i \(-0.626561\pi\)
−0.387210 + 0.921992i \(0.626561\pi\)
\(464\) 3.32593e51 0.334821
\(465\) 1.88922e50 0.0182369
\(466\) 9.48376e51 0.877921
\(467\) −8.02286e51 −0.712279 −0.356139 0.934433i \(-0.615907\pi\)
−0.356139 + 0.934433i \(0.615907\pi\)
\(468\) 1.27773e51 0.108804
\(469\) 7.33356e51 0.599024
\(470\) −2.06513e51 −0.161823
\(471\) 7.17119e51 0.539118
\(472\) −2.90846e51 −0.209795
\(473\) −1.31932e52 −0.913183
\(474\) 3.30558e51 0.219569
\(475\) 1.78902e52 1.14049
\(476\) −2.19038e51 −0.134024
\(477\) −2.44242e51 −0.143454
\(478\) −4.39094e51 −0.247579
\(479\) 2.10871e52 1.14150 0.570748 0.821125i \(-0.306653\pi\)
0.570748 + 0.821125i \(0.306653\pi\)
\(480\) 4.72742e50 0.0245708
\(481\) 3.42381e51 0.170876
\(482\) 8.76438e50 0.0420053
\(483\) 7.86442e51 0.361991
\(484\) −3.60421e51 −0.159340
\(485\) −1.23488e52 −0.524399
\(486\) −1.57299e52 −0.641679
\(487\) 2.17331e52 0.851738 0.425869 0.904785i \(-0.359968\pi\)
0.425869 + 0.904785i \(0.359968\pi\)
\(488\) −1.85092e52 −0.696949
\(489\) −3.36129e51 −0.121614
\(490\) 5.54617e51 0.192828
\(491\) −4.64580e52 −1.55228 −0.776142 0.630558i \(-0.782826\pi\)
−0.776142 + 0.630558i \(0.782826\pi\)
\(492\) −1.03582e52 −0.332633
\(493\) −2.18542e52 −0.674560
\(494\) 7.98080e51 0.236794
\(495\) −9.54027e51 −0.272119
\(496\) 1.19616e51 0.0328016
\(497\) −1.59632e51 −0.0420891
\(498\) −2.74769e51 −0.0696618
\(499\) 7.18834e52 1.75254 0.876270 0.481820i \(-0.160024\pi\)
0.876270 + 0.481820i \(0.160024\pi\)
\(500\) 1.50519e52 0.352919
\(501\) −1.58992e52 −0.358541
\(502\) −3.98745e51 −0.0864913
\(503\) −1.23248e52 −0.257161 −0.128581 0.991699i \(-0.541042\pi\)
−0.128581 + 0.991699i \(0.541042\pi\)
\(504\) −8.12267e51 −0.163043
\(505\) 1.16011e52 0.224037
\(506\) 5.84837e52 1.08667
\(507\) 1.91451e52 0.342297
\(508\) 2.23792e52 0.385037
\(509\) 2.18738e52 0.362182 0.181091 0.983466i \(-0.442037\pi\)
0.181091 + 0.983466i \(0.442037\pi\)
\(510\) −3.10632e51 −0.0495026
\(511\) 1.30469e52 0.200124
\(512\) 2.99316e51 0.0441942
\(513\) −6.39752e52 −0.909333
\(514\) 5.05750e52 0.692079
\(515\) −1.06323e52 −0.140084
\(516\) −1.59300e52 −0.202091
\(517\) −4.06438e52 −0.496510
\(518\) −2.17656e52 −0.256059
\(519\) 2.66169e52 0.301574
\(520\) 3.09537e51 0.0337791
\(521\) −7.50925e52 −0.789334 −0.394667 0.918824i \(-0.629140\pi\)
−0.394667 + 0.918824i \(0.629140\pi\)
\(522\) −8.10429e52 −0.820617
\(523\) −1.56095e53 −1.52267 −0.761337 0.648356i \(-0.775457\pi\)
−0.761337 + 0.648356i \(0.775457\pi\)
\(524\) 7.01138e52 0.658936
\(525\) 1.83637e52 0.166286
\(526\) −1.24803e53 −1.08894
\(527\) −7.85979e51 −0.0660852
\(528\) 9.30401e51 0.0753892
\(529\) 3.15827e53 2.46640
\(530\) −5.91690e51 −0.0445363
\(531\) 7.08706e52 0.514189
\(532\) −5.07348e52 −0.354837
\(533\) −6.78224e52 −0.457291
\(534\) 5.05008e52 0.328279
\(535\) 9.59708e52 0.601506
\(536\) 6.58488e52 0.397955
\(537\) −6.42063e52 −0.374178
\(538\) −1.80213e53 −1.01282
\(539\) 1.09154e53 0.591642
\(540\) −2.48129e52 −0.129718
\(541\) 2.61710e52 0.131969 0.0659847 0.997821i \(-0.478981\pi\)
0.0659847 + 0.997821i \(0.478981\pi\)
\(542\) 1.22809e53 0.597370
\(543\) 1.11428e53 0.522875
\(544\) −1.96676e52 −0.0890376
\(545\) 6.79072e52 0.296609
\(546\) 8.19203e51 0.0345251
\(547\) −4.07795e53 −1.65840 −0.829199 0.558953i \(-0.811203\pi\)
−0.829199 + 0.558953i \(0.811203\pi\)
\(548\) −4.33096e52 −0.169967
\(549\) 4.51014e53 1.70816
\(550\) 1.36562e53 0.499179
\(551\) −5.06201e53 −1.78594
\(552\) 7.06155e52 0.240485
\(553\) −1.37593e53 −0.452331
\(554\) 2.60819e53 0.827750
\(555\) −3.08672e52 −0.0945768
\(556\) −1.61856e53 −0.478819
\(557\) −6.72948e52 −0.192223 −0.0961115 0.995371i \(-0.530641\pi\)
−0.0961115 + 0.995371i \(0.530641\pi\)
\(558\) −2.91468e52 −0.0803940
\(559\) −1.04305e53 −0.277827
\(560\) −1.96776e52 −0.0506181
\(561\) −6.11355e52 −0.151886
\(562\) −1.19717e53 −0.287276
\(563\) −4.55904e53 −1.05672 −0.528360 0.849020i \(-0.677193\pi\)
−0.528360 + 0.849020i \(0.677193\pi\)
\(564\) −4.90749e52 −0.109879
\(565\) −6.20601e52 −0.134236
\(566\) 3.19759e53 0.668193
\(567\) 1.62743e53 0.328574
\(568\) −1.43335e52 −0.0279614
\(569\) 5.98039e53 1.12730 0.563651 0.826013i \(-0.309396\pi\)
0.563651 + 0.826013i \(0.309396\pi\)
\(570\) −7.19505e52 −0.131061
\(571\) 3.54056e52 0.0623258 0.0311629 0.999514i \(-0.490079\pi\)
0.0311629 + 0.999514i \(0.490079\pi\)
\(572\) 6.09199e52 0.103642
\(573\) −3.18153e53 −0.523143
\(574\) 4.31155e53 0.685253
\(575\) 1.03647e54 1.59234
\(576\) −7.29342e52 −0.108316
\(577\) 2.71624e53 0.389978 0.194989 0.980805i \(-0.437533\pi\)
0.194989 + 0.980805i \(0.437533\pi\)
\(578\) −3.80189e53 −0.527723
\(579\) 5.18050e53 0.695247
\(580\) −1.96331e53 −0.254767
\(581\) 1.14371e53 0.143510
\(582\) −2.93452e53 −0.356073
\(583\) −1.16450e53 −0.136648
\(584\) 1.17149e53 0.132950
\(585\) −7.54249e52 −0.0827895
\(586\) −1.14576e54 −1.21643
\(587\) −9.17025e53 −0.941755 −0.470878 0.882199i \(-0.656063\pi\)
−0.470878 + 0.882199i \(0.656063\pi\)
\(588\) 1.31797e53 0.130932
\(589\) −1.82053e53 −0.174964
\(590\) 1.71688e53 0.159634
\(591\) 1.33151e53 0.119781
\(592\) −1.95435e53 −0.170110
\(593\) −1.82661e54 −1.53844 −0.769219 0.638985i \(-0.779355\pi\)
−0.769219 + 0.638985i \(0.779355\pi\)
\(594\) −4.88342e53 −0.398005
\(595\) 1.29299e53 0.101980
\(596\) 3.31906e53 0.253346
\(597\) 7.19616e53 0.531620
\(598\) 4.62369e53 0.330610
\(599\) 2.02996e54 1.40496 0.702482 0.711702i \(-0.252075\pi\)
0.702482 + 0.711702i \(0.252075\pi\)
\(600\) 1.64890e53 0.110470
\(601\) −1.35173e54 −0.876673 −0.438337 0.898811i \(-0.644432\pi\)
−0.438337 + 0.898811i \(0.644432\pi\)
\(602\) 6.63077e53 0.416325
\(603\) −1.60454e54 −0.975354
\(604\) 1.56270e54 0.919721
\(605\) 2.12759e53 0.121243
\(606\) 2.75685e53 0.152123
\(607\) 2.31650e54 1.23781 0.618903 0.785468i \(-0.287578\pi\)
0.618903 + 0.785468i \(0.287578\pi\)
\(608\) −4.55553e53 −0.235732
\(609\) −5.19599e53 −0.260394
\(610\) 1.09261e54 0.530313
\(611\) −3.21327e53 −0.151058
\(612\) 4.79241e53 0.218224
\(613\) −2.04561e53 −0.0902284 −0.0451142 0.998982i \(-0.514365\pi\)
−0.0451142 + 0.998982i \(0.514365\pi\)
\(614\) −9.68418e53 −0.413790
\(615\) 6.11450e53 0.253102
\(616\) −3.87275e53 −0.155309
\(617\) −5.94190e53 −0.230869 −0.115434 0.993315i \(-0.536826\pi\)
−0.115434 + 0.993315i \(0.536826\pi\)
\(618\) −2.52662e53 −0.0951186
\(619\) 1.60053e54 0.583845 0.291922 0.956442i \(-0.405705\pi\)
0.291922 + 0.956442i \(0.405705\pi\)
\(620\) −7.06097e52 −0.0249590
\(621\) −3.70641e54 −1.26960
\(622\) 1.69174e54 0.561592
\(623\) −2.10207e54 −0.676286
\(624\) 7.35571e52 0.0229364
\(625\) 1.94128e54 0.586717
\(626\) −3.66682e54 −1.07421
\(627\) −1.41606e54 −0.402127
\(628\) −2.68024e54 −0.737836
\(629\) 1.28418e54 0.342719
\(630\) 4.79485e53 0.124061
\(631\) 3.92756e54 0.985260 0.492630 0.870239i \(-0.336036\pi\)
0.492630 + 0.870239i \(0.336036\pi\)
\(632\) −1.23546e54 −0.300502
\(633\) −1.38151e53 −0.0325823
\(634\) 2.82023e54 0.644977
\(635\) −1.32105e54 −0.292977
\(636\) −1.40607e53 −0.0302407
\(637\) 8.62966e53 0.180001
\(638\) −3.86399e54 −0.781686
\(639\) 3.49264e53 0.0685311
\(640\) −1.76687e53 −0.0336276
\(641\) 3.85006e54 0.710781 0.355390 0.934718i \(-0.384348\pi\)
0.355390 + 0.934718i \(0.384348\pi\)
\(642\) 2.28061e54 0.408430
\(643\) −3.53258e54 −0.613732 −0.306866 0.951753i \(-0.599280\pi\)
−0.306866 + 0.951753i \(0.599280\pi\)
\(644\) −2.93933e54 −0.495421
\(645\) 9.40355e53 0.153772
\(646\) 2.99338e54 0.474928
\(647\) 6.40779e54 0.986449 0.493225 0.869902i \(-0.335818\pi\)
0.493225 + 0.869902i \(0.335818\pi\)
\(648\) 1.46129e54 0.218284
\(649\) 3.37899e54 0.489796
\(650\) 1.07965e54 0.151870
\(651\) −1.86872e53 −0.0255102
\(652\) 1.25628e54 0.166441
\(653\) 1.99210e54 0.256156 0.128078 0.991764i \(-0.459119\pi\)
0.128078 + 0.991764i \(0.459119\pi\)
\(654\) 1.61372e54 0.201401
\(655\) −4.13885e54 −0.501389
\(656\) 3.87138e54 0.455241
\(657\) −2.85458e54 −0.325850
\(658\) 2.04272e54 0.226362
\(659\) −1.17248e55 −1.26137 −0.630684 0.776040i \(-0.717225\pi\)
−0.630684 + 0.776040i \(0.717225\pi\)
\(660\) −5.49221e53 −0.0573641
\(661\) −1.31406e55 −1.33256 −0.666282 0.745700i \(-0.732115\pi\)
−0.666282 + 0.745700i \(0.732115\pi\)
\(662\) −7.34096e54 −0.722807
\(663\) −4.83334e53 −0.0462097
\(664\) 1.02695e54 0.0953392
\(665\) 2.99490e54 0.269998
\(666\) 4.76218e54 0.416925
\(667\) −2.93268e55 −2.49351
\(668\) 5.94232e54 0.490699
\(669\) 5.27940e54 0.423424
\(670\) −3.88709e54 −0.302807
\(671\) 2.15036e55 1.62713
\(672\) −4.67611e53 −0.0343703
\(673\) 9.01981e54 0.644027 0.322013 0.946735i \(-0.395640\pi\)
0.322013 + 0.946735i \(0.395640\pi\)
\(674\) −1.75939e54 −0.122038
\(675\) −8.65461e54 −0.583209
\(676\) −7.15551e54 −0.468468
\(677\) 2.47399e55 1.57369 0.786844 0.617153i \(-0.211714\pi\)
0.786844 + 0.617153i \(0.211714\pi\)
\(678\) −1.47477e54 −0.0911476
\(679\) 1.22148e55 0.733543
\(680\) 1.16099e54 0.0677493
\(681\) 5.75264e54 0.326211
\(682\) −1.38967e54 −0.0765801
\(683\) 3.20867e54 0.171839 0.0859195 0.996302i \(-0.472617\pi\)
0.0859195 + 0.996302i \(0.472617\pi\)
\(684\) 1.11005e55 0.577760
\(685\) 2.55659e54 0.129329
\(686\) −1.31397e55 −0.646046
\(687\) 3.38836e54 0.161932
\(688\) 5.95384e54 0.276581
\(689\) −9.20650e53 −0.0415738
\(690\) −4.16847e54 −0.182986
\(691\) −2.16393e55 −0.923468 −0.461734 0.887019i \(-0.652772\pi\)
−0.461734 + 0.887019i \(0.652772\pi\)
\(692\) −9.94806e54 −0.412733
\(693\) 9.43673e54 0.380648
\(694\) −1.36755e55 −0.536331
\(695\) 9.55445e54 0.364336
\(696\) −4.66553e54 −0.172990
\(697\) −2.54383e55 −0.917170
\(698\) 1.06763e55 0.374319
\(699\) −1.33036e55 −0.453590
\(700\) −6.86345e54 −0.227578
\(701\) −1.92533e55 −0.620876 −0.310438 0.950594i \(-0.600476\pi\)
−0.310438 + 0.950594i \(0.600476\pi\)
\(702\) −3.86081e54 −0.121089
\(703\) 2.97449e55 0.907370
\(704\) −3.47738e54 −0.103178
\(705\) 2.89691e54 0.0836080
\(706\) 4.23792e55 1.18976
\(707\) −1.14752e55 −0.313388
\(708\) 4.07992e54 0.108394
\(709\) −5.00428e55 −1.29342 −0.646711 0.762735i \(-0.723856\pi\)
−0.646711 + 0.762735i \(0.723856\pi\)
\(710\) 8.46113e53 0.0212760
\(711\) 3.01045e55 0.736503
\(712\) −1.88747e55 −0.449283
\(713\) −1.05473e55 −0.244284
\(714\) 3.07261e54 0.0692456
\(715\) −3.59613e54 −0.0788620
\(716\) 2.39971e55 0.512100
\(717\) 6.15950e54 0.127915
\(718\) 3.85542e55 0.779192
\(719\) 2.94837e55 0.579921 0.289961 0.957039i \(-0.406358\pi\)
0.289961 + 0.957039i \(0.406358\pi\)
\(720\) 4.30534e54 0.0824184
\(721\) 1.05169e55 0.195953
\(722\) 3.03438e55 0.550294
\(723\) −1.22945e54 −0.0217026
\(724\) −4.16463e55 −0.715607
\(725\) −6.84793e55 −1.14543
\(726\) 5.05590e54 0.0823256
\(727\) 9.06551e55 1.43705 0.718524 0.695502i \(-0.244818\pi\)
0.718524 + 0.695502i \(0.244818\pi\)
\(728\) −3.06177e54 −0.0472510
\(729\) −1.90262e55 −0.285868
\(730\) −6.91539e54 −0.101163
\(731\) −3.91219e55 −0.557226
\(732\) 2.59642e55 0.360089
\(733\) 1.21737e56 1.64397 0.821987 0.569507i \(-0.192866\pi\)
0.821987 + 0.569507i \(0.192866\pi\)
\(734\) −5.23993e55 −0.689052
\(735\) −7.78003e54 −0.0996273
\(736\) −2.63926e55 −0.329128
\(737\) −7.65016e55 −0.929083
\(738\) −9.43340e55 −1.11576
\(739\) −7.42605e55 −0.855444 −0.427722 0.903910i \(-0.640684\pi\)
−0.427722 + 0.903910i \(0.640684\pi\)
\(740\) 1.15366e55 0.129438
\(741\) −1.11953e55 −0.122343
\(742\) 5.85268e54 0.0622986
\(743\) 9.86558e55 1.02292 0.511459 0.859308i \(-0.329105\pi\)
0.511459 + 0.859308i \(0.329105\pi\)
\(744\) −1.67794e54 −0.0169474
\(745\) −1.95926e55 −0.192772
\(746\) −5.57676e55 −0.534533
\(747\) −2.50237e55 −0.233668
\(748\) 2.28494e55 0.207871
\(749\) −9.49291e55 −0.841403
\(750\) −2.11144e55 −0.182341
\(751\) 1.97249e56 1.65972 0.829861 0.557971i \(-0.188420\pi\)
0.829861 + 0.557971i \(0.188420\pi\)
\(752\) 1.83418e55 0.150381
\(753\) 5.59349e54 0.0446870
\(754\) −3.05485e55 −0.237820
\(755\) −9.22471e55 −0.699821
\(756\) 2.45436e55 0.181453
\(757\) 1.98251e56 1.42839 0.714193 0.699949i \(-0.246794\pi\)
0.714193 + 0.699949i \(0.246794\pi\)
\(758\) 1.62303e56 1.13966
\(759\) −8.20394e55 −0.561446
\(760\) 2.68915e55 0.179370
\(761\) 1.83095e56 1.19035 0.595175 0.803596i \(-0.297083\pi\)
0.595175 + 0.803596i \(0.297083\pi\)
\(762\) −3.13930e55 −0.198935
\(763\) −6.71702e55 −0.414904
\(764\) 1.18910e56 0.715974
\(765\) −2.82899e55 −0.166048
\(766\) 6.88713e55 0.394074
\(767\) 2.67141e55 0.149015
\(768\) −4.19872e54 −0.0228336
\(769\) −2.06387e56 −1.09425 −0.547126 0.837050i \(-0.684278\pi\)
−0.547126 + 0.837050i \(0.684278\pi\)
\(770\) 2.28610e55 0.118175
\(771\) −7.09454e55 −0.357572
\(772\) −1.93621e56 −0.951515
\(773\) −1.30613e56 −0.625871 −0.312936 0.949774i \(-0.601312\pi\)
−0.312936 + 0.949774i \(0.601312\pi\)
\(774\) −1.45077e56 −0.677877
\(775\) −2.46283e55 −0.112215
\(776\) 1.09678e56 0.487321
\(777\) 3.05322e55 0.132297
\(778\) 6.66985e55 0.281847
\(779\) −5.89218e56 −2.42826
\(780\) −4.34211e54 −0.0174524
\(781\) 1.66523e55 0.0652799
\(782\) 1.73422e56 0.663090
\(783\) 2.44881e56 0.913273
\(784\) −4.92591e55 −0.179194
\(785\) 1.58216e56 0.561424
\(786\) −9.83539e55 −0.340449
\(787\) 1.04231e56 0.351956 0.175978 0.984394i \(-0.443691\pi\)
0.175978 + 0.984394i \(0.443691\pi\)
\(788\) −4.97651e55 −0.163932
\(789\) 1.75071e56 0.562617
\(790\) 7.29300e55 0.228653
\(791\) 6.13865e55 0.187772
\(792\) 8.47333e55 0.252879
\(793\) 1.70006e56 0.495037
\(794\) −2.15191e56 −0.611397
\(795\) 8.30008e54 0.0230103
\(796\) −2.68957e56 −0.727575
\(797\) 1.94067e56 0.512289 0.256145 0.966638i \(-0.417548\pi\)
0.256145 + 0.966638i \(0.417548\pi\)
\(798\) 7.11696e55 0.183332
\(799\) −1.20521e56 −0.302971
\(800\) −6.16276e55 −0.151189
\(801\) 4.59920e56 1.10115
\(802\) 3.97823e55 0.0929587
\(803\) −1.36102e56 −0.310392
\(804\) −9.23711e55 −0.205609
\(805\) 1.73510e56 0.376969
\(806\) −1.09866e55 −0.0232987
\(807\) 2.52799e56 0.523289
\(808\) −1.03037e56 −0.208196
\(809\) 2.23938e56 0.441705 0.220852 0.975307i \(-0.429116\pi\)
0.220852 + 0.975307i \(0.429116\pi\)
\(810\) −8.62604e55 −0.166094
\(811\) −5.47814e56 −1.02974 −0.514869 0.857269i \(-0.672159\pi\)
−0.514869 + 0.857269i \(0.672159\pi\)
\(812\) 1.94200e56 0.356375
\(813\) −1.72273e56 −0.308640
\(814\) 2.27052e56 0.397146
\(815\) −7.41591e55 −0.126646
\(816\) 2.75893e55 0.0460026
\(817\) −9.06165e56 −1.47529
\(818\) 1.10146e56 0.175098
\(819\) 7.46063e55 0.115808
\(820\) −2.28530e56 −0.346396
\(821\) −4.98980e56 −0.738569 −0.369285 0.929316i \(-0.620397\pi\)
−0.369285 + 0.929316i \(0.620397\pi\)
\(822\) 6.07537e55 0.0878157
\(823\) 4.03098e56 0.569003 0.284501 0.958676i \(-0.408172\pi\)
0.284501 + 0.958676i \(0.408172\pi\)
\(824\) 9.44327e55 0.130179
\(825\) −1.91565e56 −0.257908
\(826\) −1.69825e56 −0.223301
\(827\) 5.98393e54 0.00768474 0.00384237 0.999993i \(-0.498777\pi\)
0.00384237 + 0.999993i \(0.498777\pi\)
\(828\) 6.43108e56 0.806663
\(829\) −6.22265e56 −0.762363 −0.381182 0.924500i \(-0.624483\pi\)
−0.381182 + 0.924500i \(0.624483\pi\)
\(830\) −6.06213e55 −0.0725442
\(831\) −3.65871e56 −0.427669
\(832\) −2.74920e55 −0.0313907
\(833\) 3.23675e56 0.361021
\(834\) 2.27048e56 0.247389
\(835\) −3.50778e56 −0.373376
\(836\) 5.29251e56 0.550351
\(837\) 8.80705e55 0.0894713
\(838\) 9.32045e55 0.0925078
\(839\) 1.71925e57 1.66717 0.833586 0.552389i \(-0.186284\pi\)
0.833586 + 0.552389i \(0.186284\pi\)
\(840\) 2.76033e55 0.0261526
\(841\) 8.57367e56 0.793679
\(842\) −4.01423e56 −0.363092
\(843\) 1.67937e56 0.148425
\(844\) 5.16341e55 0.0445922
\(845\) 4.22393e56 0.356460
\(846\) −4.46934e56 −0.368571
\(847\) −2.10449e56 −0.169598
\(848\) 5.25518e55 0.0413874
\(849\) −4.48550e56 −0.345232
\(850\) 4.04947e56 0.304600
\(851\) 1.72328e57 1.26686
\(852\) 2.01067e55 0.0144467
\(853\) −1.47498e57 −1.03581 −0.517906 0.855438i \(-0.673288\pi\)
−0.517906 + 0.855438i \(0.673288\pi\)
\(854\) −1.08075e57 −0.741816
\(855\) −6.55266e56 −0.439621
\(856\) −8.52378e56 −0.558977
\(857\) −2.25922e57 −1.44822 −0.724108 0.689687i \(-0.757748\pi\)
−0.724108 + 0.689687i \(0.757748\pi\)
\(858\) −8.54569e55 −0.0535482
\(859\) 1.62602e57 0.996002 0.498001 0.867177i \(-0.334068\pi\)
0.498001 + 0.867177i \(0.334068\pi\)
\(860\) −3.51458e56 −0.210452
\(861\) −6.04813e56 −0.354046
\(862\) −4.75011e56 −0.271839
\(863\) −4.93931e56 −0.276347 −0.138174 0.990408i \(-0.544123\pi\)
−0.138174 + 0.990408i \(0.544123\pi\)
\(864\) 2.20380e56 0.120546
\(865\) 5.87239e56 0.314051
\(866\) −1.55167e57 −0.811336
\(867\) 5.33319e56 0.272656
\(868\) 6.98434e55 0.0349133
\(869\) 1.43533e57 0.701563
\(870\) 2.75409e56 0.131629
\(871\) −6.04818e56 −0.282664
\(872\) −6.03128e56 −0.275637
\(873\) −2.67252e57 −1.19438
\(874\) 4.01690e57 1.75557
\(875\) 8.78875e56 0.375638
\(876\) −1.64334e56 −0.0686908
\(877\) 3.26253e57 1.33371 0.666857 0.745186i \(-0.267639\pi\)
0.666857 + 0.745186i \(0.267639\pi\)
\(878\) −1.01337e57 −0.405160
\(879\) 1.60724e57 0.628488
\(880\) 2.05271e56 0.0785085
\(881\) 1.49144e57 0.557925 0.278963 0.960302i \(-0.410009\pi\)
0.278963 + 0.960302i \(0.410009\pi\)
\(882\) 1.20030e57 0.439189
\(883\) −1.82576e57 −0.653446 −0.326723 0.945120i \(-0.605944\pi\)
−0.326723 + 0.945120i \(0.605944\pi\)
\(884\) 1.80646e56 0.0632426
\(885\) −2.40840e56 −0.0824773
\(886\) 2.74451e57 0.919405
\(887\) −3.77183e57 −1.23607 −0.618034 0.786152i \(-0.712070\pi\)
−0.618034 + 0.786152i \(0.712070\pi\)
\(888\) 2.74152e56 0.0878898
\(889\) 1.30672e57 0.409824
\(890\) 1.11418e57 0.341862
\(891\) −1.69769e57 −0.509616
\(892\) −1.97318e57 −0.579498
\(893\) −2.79158e57 −0.802135
\(894\) −4.65590e56 −0.130895
\(895\) −1.41656e57 −0.389660
\(896\) 1.74770e56 0.0470392
\(897\) −6.48600e56 −0.170814
\(898\) 1.38673e57 0.357358
\(899\) 6.96854e56 0.175723
\(900\) 1.50168e57 0.370552
\(901\) −3.45311e56 −0.0833829
\(902\) −4.49768e57 −1.06282
\(903\) −9.30149e56 −0.215100
\(904\) 5.51196e56 0.124745
\(905\) 2.45840e57 0.544510
\(906\) −2.19212e57 −0.475187
\(907\) 3.05905e57 0.649000 0.324500 0.945886i \(-0.394804\pi\)
0.324500 + 0.945886i \(0.394804\pi\)
\(908\) −2.15005e57 −0.446452
\(909\) 2.51071e57 0.510271
\(910\) 1.80738e56 0.0359536
\(911\) −3.49012e57 −0.679566 −0.339783 0.940504i \(-0.610354\pi\)
−0.339783 + 0.940504i \(0.610354\pi\)
\(912\) 6.39039e56 0.121795
\(913\) −1.19309e57 −0.222583
\(914\) 1.76800e55 0.00322872
\(915\) −1.53268e57 −0.273994
\(916\) −1.26640e57 −0.221620
\(917\) 4.09393e57 0.701356
\(918\) −1.44809e57 −0.242863
\(919\) 1.50067e57 0.246395 0.123198 0.992382i \(-0.460685\pi\)
0.123198 + 0.992382i \(0.460685\pi\)
\(920\) 1.55797e57 0.250435
\(921\) 1.35847e57 0.213790
\(922\) −4.16015e56 −0.0640998
\(923\) 1.31652e56 0.0198608
\(924\) 5.43259e56 0.0802424
\(925\) 4.02392e57 0.581950
\(926\) −3.86702e57 −0.547598
\(927\) −2.30104e57 −0.319058
\(928\) 1.74374e57 0.236754
\(929\) 1.07229e58 1.42563 0.712817 0.701351i \(-0.247419\pi\)
0.712817 + 0.701351i \(0.247419\pi\)
\(930\) 9.90496e55 0.0128954
\(931\) 7.49716e57 0.955824
\(932\) 4.97222e57 0.620784
\(933\) −2.37313e57 −0.290155
\(934\) −4.20629e57 −0.503657
\(935\) −1.34881e57 −0.158170
\(936\) 6.69898e56 0.0769360
\(937\) −2.19037e57 −0.246374 −0.123187 0.992383i \(-0.539311\pi\)
−0.123187 + 0.992383i \(0.539311\pi\)
\(938\) 3.84490e57 0.423574
\(939\) 5.14372e57 0.555007
\(940\) −1.08272e57 −0.114426
\(941\) −1.08101e58 −1.11901 −0.559505 0.828827i \(-0.689009\pi\)
−0.559505 + 0.828827i \(0.689009\pi\)
\(942\) 3.75977e57 0.381214
\(943\) −3.41364e58 −3.39032
\(944\) −1.52487e57 −0.148347
\(945\) −1.44882e57 −0.138068
\(946\) −6.91703e57 −0.645718
\(947\) 1.01364e58 0.926956 0.463478 0.886108i \(-0.346601\pi\)
0.463478 + 0.886108i \(0.346601\pi\)
\(948\) 1.73308e57 0.155258
\(949\) −1.07601e57 −0.0944335
\(950\) 9.37963e57 0.806446
\(951\) −3.95615e57 −0.333237
\(952\) −1.14839e57 −0.0947695
\(953\) −3.31702e55 −0.00268186 −0.00134093 0.999999i \(-0.500427\pi\)
−0.00134093 + 0.999999i \(0.500427\pi\)
\(954\) −1.28053e57 −0.101437
\(955\) −7.01930e57 −0.544789
\(956\) −2.30212e57 −0.175065
\(957\) 5.42031e57 0.403870
\(958\) 1.10557e58 0.807160
\(959\) −2.52884e57 −0.180908
\(960\) 2.47853e56 0.0173742
\(961\) −1.43075e58 −0.982785
\(962\) 1.79506e57 0.120828
\(963\) 2.07699e58 1.37000
\(964\) 4.59506e56 0.0297022
\(965\) 1.14296e58 0.724014
\(966\) 4.12322e57 0.255966
\(967\) 2.19730e58 1.33682 0.668411 0.743793i \(-0.266975\pi\)
0.668411 + 0.743793i \(0.266975\pi\)
\(968\) −1.88965e57 −0.112671
\(969\) −4.19904e57 −0.245378
\(970\) −6.47434e57 −0.370806
\(971\) 1.76835e58 0.992646 0.496323 0.868138i \(-0.334683\pi\)
0.496323 + 0.868138i \(0.334683\pi\)
\(972\) −8.24698e57 −0.453736
\(973\) −9.45075e57 −0.509643
\(974\) 1.13944e58 0.602270
\(975\) −1.51450e57 −0.0784658
\(976\) −9.70415e57 −0.492817
\(977\) −5.66963e57 −0.282235 −0.141118 0.989993i \(-0.545070\pi\)
−0.141118 + 0.989993i \(0.545070\pi\)
\(978\) −1.76229e57 −0.0859941
\(979\) 2.19282e58 1.04892
\(980\) 2.90779e57 0.136350
\(981\) 1.46964e58 0.675563
\(982\) −2.43574e58 −1.09763
\(983\) −3.22545e58 −1.42494 −0.712469 0.701703i \(-0.752423\pi\)
−0.712469 + 0.701703i \(0.752423\pi\)
\(984\) −5.43068e57 −0.235207
\(985\) 2.93766e57 0.124737
\(986\) −1.14579e58 −0.476986
\(987\) −2.86547e57 −0.116953
\(988\) 4.18424e57 0.167439
\(989\) −5.24988e58 −2.05978
\(990\) −5.00185e57 −0.192417
\(991\) 2.73974e58 1.03341 0.516706 0.856163i \(-0.327158\pi\)
0.516706 + 0.856163i \(0.327158\pi\)
\(992\) 6.27131e56 0.0231943
\(993\) 1.02977e58 0.373449
\(994\) −8.36930e56 −0.0297615
\(995\) 1.58767e58 0.553617
\(996\) −1.44058e57 −0.0492584
\(997\) −2.71260e58 −0.909559 −0.454779 0.890604i \(-0.650282\pi\)
−0.454779 + 0.890604i \(0.650282\pi\)
\(998\) 3.76876e58 1.23923
\(999\) −1.43895e58 −0.464000
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 2.40.a.a.1.1 1
3.2 odd 2 18.40.a.a.1.1 1
4.3 odd 2 16.40.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.40.a.a.1.1 1 1.1 even 1 trivial
16.40.a.a.1.1 1 4.3 odd 2
18.40.a.a.1.1 1 3.2 odd 2