Properties

Label 5054.2.a.bh.1.1
Level $5054$
Weight $2$
Character 5054.1
Self dual yes
Analytic conductor $40.356$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5054 = 2 \cdot 7 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5054.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.3563931816\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: 8.8.19520000000.1
Defining polynomial: \( x^{8} - 2x^{7} - 12x^{6} + 16x^{5} + 50x^{4} - 24x^{3} - 72x^{2} - 32x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.04552\) of defining polynomial
Character \(\chi\) \(=\) 5054.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -3.13415 q^{3} +1.00000 q^{4} -0.529405 q^{5} -3.13415 q^{6} +1.00000 q^{7} +1.00000 q^{8} +6.82288 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -3.13415 q^{3} +1.00000 q^{4} -0.529405 q^{5} -3.13415 q^{6} +1.00000 q^{7} +1.00000 q^{8} +6.82288 q^{9} -0.529405 q^{10} -3.29795 q^{11} -3.13415 q^{12} -0.516798 q^{13} +1.00000 q^{14} +1.65923 q^{15} +1.00000 q^{16} +6.14185 q^{17} +6.82288 q^{18} -0.529405 q^{20} -3.13415 q^{21} -3.29795 q^{22} -5.69094 q^{23} -3.13415 q^{24} -4.71973 q^{25} -0.516798 q^{26} -11.9815 q^{27} +1.00000 q^{28} -0.202602 q^{29} +1.65923 q^{30} -0.736319 q^{31} +1.00000 q^{32} +10.3363 q^{33} +6.14185 q^{34} -0.529405 q^{35} +6.82288 q^{36} -10.8924 q^{37} +1.61972 q^{39} -0.529405 q^{40} +3.68913 q^{41} -3.13415 q^{42} +7.19952 q^{43} -3.29795 q^{44} -3.61206 q^{45} -5.69094 q^{46} +12.1388 q^{47} -3.13415 q^{48} +1.00000 q^{49} -4.71973 q^{50} -19.2495 q^{51} -0.516798 q^{52} +3.18512 q^{53} -11.9815 q^{54} +1.74595 q^{55} +1.00000 q^{56} -0.202602 q^{58} -3.91656 q^{59} +1.65923 q^{60} +3.73389 q^{61} -0.736319 q^{62} +6.82288 q^{63} +1.00000 q^{64} +0.273596 q^{65} +10.3363 q^{66} -9.77683 q^{67} +6.14185 q^{68} +17.8362 q^{69} -0.529405 q^{70} +6.75039 q^{71} +6.82288 q^{72} +16.3744 q^{73} -10.8924 q^{74} +14.7923 q^{75} -3.29795 q^{77} +1.61972 q^{78} -8.69327 q^{79} -0.529405 q^{80} +17.0830 q^{81} +3.68913 q^{82} -18.0349 q^{83} -3.13415 q^{84} -3.25153 q^{85} +7.19952 q^{86} +0.634986 q^{87} -3.29795 q^{88} -16.2090 q^{89} -3.61206 q^{90} -0.516798 q^{91} -5.69094 q^{92} +2.30773 q^{93} +12.1388 q^{94} -3.13415 q^{96} +2.63263 q^{97} +1.00000 q^{98} -22.5015 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} - 4 q^{3} + 8 q^{4} - 2 q^{5} - 4 q^{6} + 8 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{2} - 4 q^{3} + 8 q^{4} - 2 q^{5} - 4 q^{6} + 8 q^{7} + 8 q^{8} + 8 q^{9} - 2 q^{10} - 12 q^{11} - 4 q^{12} - 10 q^{13} + 8 q^{14} - 24 q^{15} + 8 q^{16} - 6 q^{17} + 8 q^{18} - 2 q^{20} - 4 q^{21} - 12 q^{22} - 20 q^{23} - 4 q^{24} + 8 q^{25} - 10 q^{26} - 22 q^{27} + 8 q^{28} - 8 q^{29} - 24 q^{30} - 18 q^{31} + 8 q^{32} + 16 q^{33} - 6 q^{34} - 2 q^{35} + 8 q^{36} - 36 q^{37} - 2 q^{40} - 4 q^{41} - 4 q^{42} - 16 q^{43} - 12 q^{44} + 8 q^{45} - 20 q^{46} - 10 q^{47} - 4 q^{48} + 8 q^{49} + 8 q^{50} - 12 q^{51} - 10 q^{52} - 32 q^{53} - 22 q^{54} - 22 q^{55} + 8 q^{56} - 8 q^{58} + 2 q^{59} - 24 q^{60} + 2 q^{61} - 18 q^{62} + 8 q^{63} + 8 q^{64} + 16 q^{66} - 44 q^{67} - 6 q^{68} - 2 q^{70} - 8 q^{71} + 8 q^{72} + 30 q^{73} - 36 q^{74} + 16 q^{75} - 12 q^{77} - 60 q^{79} - 2 q^{80} + 12 q^{81} - 4 q^{82} - 28 q^{83} - 4 q^{84} - 16 q^{85} - 16 q^{86} + 24 q^{87} - 12 q^{88} + 22 q^{89} + 8 q^{90} - 10 q^{91} - 20 q^{92} - 16 q^{93} - 10 q^{94} - 4 q^{96} + 6 q^{97} + 8 q^{98} - 52 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −3.13415 −1.80950 −0.904750 0.425942i \(-0.859943\pi\)
−0.904750 + 0.425942i \(0.859943\pi\)
\(4\) 1.00000 0.500000
\(5\) −0.529405 −0.236757 −0.118379 0.992969i \(-0.537770\pi\)
−0.118379 + 0.992969i \(0.537770\pi\)
\(6\) −3.13415 −1.27951
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 6.82288 2.27429
\(10\) −0.529405 −0.167413
\(11\) −3.29795 −0.994369 −0.497185 0.867645i \(-0.665633\pi\)
−0.497185 + 0.867645i \(0.665633\pi\)
\(12\) −3.13415 −0.904750
\(13\) −0.516798 −0.143334 −0.0716670 0.997429i \(-0.522832\pi\)
−0.0716670 + 0.997429i \(0.522832\pi\)
\(14\) 1.00000 0.267261
\(15\) 1.65923 0.428412
\(16\) 1.00000 0.250000
\(17\) 6.14185 1.48962 0.744809 0.667278i \(-0.232541\pi\)
0.744809 + 0.667278i \(0.232541\pi\)
\(18\) 6.82288 1.60817
\(19\) 0 0
\(20\) −0.529405 −0.118379
\(21\) −3.13415 −0.683927
\(22\) −3.29795 −0.703125
\(23\) −5.69094 −1.18664 −0.593321 0.804966i \(-0.702184\pi\)
−0.593321 + 0.804966i \(0.702184\pi\)
\(24\) −3.13415 −0.639755
\(25\) −4.71973 −0.943946
\(26\) −0.516798 −0.101352
\(27\) −11.9815 −2.30583
\(28\) 1.00000 0.188982
\(29\) −0.202602 −0.0376223 −0.0188112 0.999823i \(-0.505988\pi\)
−0.0188112 + 0.999823i \(0.505988\pi\)
\(30\) 1.65923 0.302933
\(31\) −0.736319 −0.132247 −0.0661234 0.997811i \(-0.521063\pi\)
−0.0661234 + 0.997811i \(0.521063\pi\)
\(32\) 1.00000 0.176777
\(33\) 10.3363 1.79931
\(34\) 6.14185 1.05332
\(35\) −0.529405 −0.0894858
\(36\) 6.82288 1.13715
\(37\) −10.8924 −1.79070 −0.895348 0.445368i \(-0.853073\pi\)
−0.895348 + 0.445368i \(0.853073\pi\)
\(38\) 0 0
\(39\) 1.61972 0.259363
\(40\) −0.529405 −0.0837063
\(41\) 3.68913 0.576145 0.288073 0.957609i \(-0.406986\pi\)
0.288073 + 0.957609i \(0.406986\pi\)
\(42\) −3.13415 −0.483609
\(43\) 7.19952 1.09792 0.548958 0.835850i \(-0.315025\pi\)
0.548958 + 0.835850i \(0.315025\pi\)
\(44\) −3.29795 −0.497185
\(45\) −3.61206 −0.538455
\(46\) −5.69094 −0.839083
\(47\) 12.1388 1.77062 0.885311 0.464999i \(-0.153945\pi\)
0.885311 + 0.464999i \(0.153945\pi\)
\(48\) −3.13415 −0.452375
\(49\) 1.00000 0.142857
\(50\) −4.71973 −0.667471
\(51\) −19.2495 −2.69546
\(52\) −0.516798 −0.0716670
\(53\) 3.18512 0.437509 0.218755 0.975780i \(-0.429801\pi\)
0.218755 + 0.975780i \(0.429801\pi\)
\(54\) −11.9815 −1.63047
\(55\) 1.74595 0.235424
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) −0.202602 −0.0266030
\(59\) −3.91656 −0.509893 −0.254947 0.966955i \(-0.582058\pi\)
−0.254947 + 0.966955i \(0.582058\pi\)
\(60\) 1.65923 0.214206
\(61\) 3.73389 0.478075 0.239038 0.971010i \(-0.423168\pi\)
0.239038 + 0.971010i \(0.423168\pi\)
\(62\) −0.736319 −0.0935126
\(63\) 6.82288 0.859602
\(64\) 1.00000 0.125000
\(65\) 0.273596 0.0339354
\(66\) 10.3363 1.27231
\(67\) −9.77683 −1.19443 −0.597215 0.802081i \(-0.703726\pi\)
−0.597215 + 0.802081i \(0.703726\pi\)
\(68\) 6.14185 0.744809
\(69\) 17.8362 2.14723
\(70\) −0.529405 −0.0632760
\(71\) 6.75039 0.801123 0.400562 0.916270i \(-0.368815\pi\)
0.400562 + 0.916270i \(0.368815\pi\)
\(72\) 6.82288 0.804084
\(73\) 16.3744 1.91648 0.958238 0.285971i \(-0.0923161\pi\)
0.958238 + 0.285971i \(0.0923161\pi\)
\(74\) −10.8924 −1.26621
\(75\) 14.7923 1.70807
\(76\) 0 0
\(77\) −3.29795 −0.375836
\(78\) 1.61972 0.183397
\(79\) −8.69327 −0.978069 −0.489035 0.872264i \(-0.662651\pi\)
−0.489035 + 0.872264i \(0.662651\pi\)
\(80\) −0.529405 −0.0591893
\(81\) 17.0830 1.89811
\(82\) 3.68913 0.407396
\(83\) −18.0349 −1.97959 −0.989796 0.142489i \(-0.954489\pi\)
−0.989796 + 0.142489i \(0.954489\pi\)
\(84\) −3.13415 −0.341963
\(85\) −3.25153 −0.352678
\(86\) 7.19952 0.776344
\(87\) 0.634986 0.0680776
\(88\) −3.29795 −0.351563
\(89\) −16.2090 −1.71815 −0.859074 0.511851i \(-0.828960\pi\)
−0.859074 + 0.511851i \(0.828960\pi\)
\(90\) −3.61206 −0.380745
\(91\) −0.516798 −0.0541752
\(92\) −5.69094 −0.593321
\(93\) 2.30773 0.239301
\(94\) 12.1388 1.25202
\(95\) 0 0
\(96\) −3.13415 −0.319878
\(97\) 2.63263 0.267303 0.133652 0.991028i \(-0.457330\pi\)
0.133652 + 0.991028i \(0.457330\pi\)
\(98\) 1.00000 0.101015
\(99\) −22.5015 −2.26149
\(100\) −4.71973 −0.471973
\(101\) 9.83739 0.978856 0.489428 0.872044i \(-0.337206\pi\)
0.489428 + 0.872044i \(0.337206\pi\)
\(102\) −19.2495 −1.90598
\(103\) 4.31612 0.425280 0.212640 0.977131i \(-0.431794\pi\)
0.212640 + 0.977131i \(0.431794\pi\)
\(104\) −0.516798 −0.0506762
\(105\) 1.65923 0.161925
\(106\) 3.18512 0.309366
\(107\) −18.9251 −1.82956 −0.914779 0.403954i \(-0.867636\pi\)
−0.914779 + 0.403954i \(0.867636\pi\)
\(108\) −11.9815 −1.15292
\(109\) 14.8991 1.42707 0.713536 0.700618i \(-0.247092\pi\)
0.713536 + 0.700618i \(0.247092\pi\)
\(110\) 1.74595 0.166470
\(111\) 34.1383 3.24026
\(112\) 1.00000 0.0944911
\(113\) −1.20410 −0.113272 −0.0566361 0.998395i \(-0.518037\pi\)
−0.0566361 + 0.998395i \(0.518037\pi\)
\(114\) 0 0
\(115\) 3.01281 0.280946
\(116\) −0.202602 −0.0188112
\(117\) −3.52605 −0.325983
\(118\) −3.91656 −0.360549
\(119\) 6.14185 0.563023
\(120\) 1.65923 0.151467
\(121\) −0.123528 −0.0112298
\(122\) 3.73389 0.338050
\(123\) −11.5623 −1.04254
\(124\) −0.736319 −0.0661234
\(125\) 5.14567 0.460243
\(126\) 6.82288 0.607830
\(127\) 0.707401 0.0627717 0.0313858 0.999507i \(-0.490008\pi\)
0.0313858 + 0.999507i \(0.490008\pi\)
\(128\) 1.00000 0.0883883
\(129\) −22.5644 −1.98668
\(130\) 0.273596 0.0239959
\(131\) −10.3358 −0.903044 −0.451522 0.892260i \(-0.649119\pi\)
−0.451522 + 0.892260i \(0.649119\pi\)
\(132\) 10.3363 0.899656
\(133\) 0 0
\(134\) −9.77683 −0.844590
\(135\) 6.34304 0.545922
\(136\) 6.14185 0.526659
\(137\) 4.56839 0.390304 0.195152 0.980773i \(-0.437480\pi\)
0.195152 + 0.980773i \(0.437480\pi\)
\(138\) 17.8362 1.51832
\(139\) −6.04179 −0.512458 −0.256229 0.966616i \(-0.582480\pi\)
−0.256229 + 0.966616i \(0.582480\pi\)
\(140\) −0.529405 −0.0447429
\(141\) −38.0447 −3.20394
\(142\) 6.75039 0.566480
\(143\) 1.70437 0.142527
\(144\) 6.82288 0.568573
\(145\) 0.107259 0.00890735
\(146\) 16.3744 1.35515
\(147\) −3.13415 −0.258500
\(148\) −10.8924 −0.895348
\(149\) −18.2013 −1.49111 −0.745553 0.666447i \(-0.767814\pi\)
−0.745553 + 0.666447i \(0.767814\pi\)
\(150\) 14.7923 1.20779
\(151\) 20.7582 1.68928 0.844641 0.535334i \(-0.179814\pi\)
0.844641 + 0.535334i \(0.179814\pi\)
\(152\) 0 0
\(153\) 41.9051 3.38783
\(154\) −3.29795 −0.265756
\(155\) 0.389811 0.0313104
\(156\) 1.61972 0.129682
\(157\) 10.4064 0.830519 0.415259 0.909703i \(-0.363691\pi\)
0.415259 + 0.909703i \(0.363691\pi\)
\(158\) −8.69327 −0.691599
\(159\) −9.98262 −0.791673
\(160\) −0.529405 −0.0418531
\(161\) −5.69094 −0.448509
\(162\) 17.0830 1.34217
\(163\) −17.8149 −1.39537 −0.697687 0.716402i \(-0.745787\pi\)
−0.697687 + 0.716402i \(0.745787\pi\)
\(164\) 3.68913 0.288073
\(165\) −5.47207 −0.426000
\(166\) −18.0349 −1.39978
\(167\) −7.93765 −0.614234 −0.307117 0.951672i \(-0.599364\pi\)
−0.307117 + 0.951672i \(0.599364\pi\)
\(168\) −3.13415 −0.241805
\(169\) −12.7329 −0.979455
\(170\) −3.25153 −0.249381
\(171\) 0 0
\(172\) 7.19952 0.548958
\(173\) −9.69338 −0.736974 −0.368487 0.929633i \(-0.620124\pi\)
−0.368487 + 0.929633i \(0.620124\pi\)
\(174\) 0.634986 0.0481382
\(175\) −4.71973 −0.356778
\(176\) −3.29795 −0.248592
\(177\) 12.2751 0.922652
\(178\) −16.2090 −1.21491
\(179\) 13.9458 1.04236 0.521178 0.853448i \(-0.325493\pi\)
0.521178 + 0.853448i \(0.325493\pi\)
\(180\) −3.61206 −0.269227
\(181\) −19.1572 −1.42395 −0.711973 0.702207i \(-0.752198\pi\)
−0.711973 + 0.702207i \(0.752198\pi\)
\(182\) −0.516798 −0.0383076
\(183\) −11.7025 −0.865077
\(184\) −5.69094 −0.419542
\(185\) 5.76648 0.423960
\(186\) 2.30773 0.169211
\(187\) −20.2555 −1.48123
\(188\) 12.1388 0.885311
\(189\) −11.9815 −0.871523
\(190\) 0 0
\(191\) −12.7895 −0.925418 −0.462709 0.886510i \(-0.653122\pi\)
−0.462709 + 0.886510i \(0.653122\pi\)
\(192\) −3.13415 −0.226188
\(193\) −10.4419 −0.751622 −0.375811 0.926696i \(-0.622636\pi\)
−0.375811 + 0.926696i \(0.622636\pi\)
\(194\) 2.63263 0.189012
\(195\) −0.857489 −0.0614060
\(196\) 1.00000 0.0714286
\(197\) 0.817360 0.0582345 0.0291173 0.999576i \(-0.490730\pi\)
0.0291173 + 0.999576i \(0.490730\pi\)
\(198\) −22.5015 −1.59911
\(199\) −7.81078 −0.553691 −0.276846 0.960914i \(-0.589289\pi\)
−0.276846 + 0.960914i \(0.589289\pi\)
\(200\) −4.71973 −0.333735
\(201\) 30.6420 2.16132
\(202\) 9.83739 0.692156
\(203\) −0.202602 −0.0142199
\(204\) −19.2495 −1.34773
\(205\) −1.95304 −0.136406
\(206\) 4.31612 0.300718
\(207\) −38.8286 −2.69877
\(208\) −0.516798 −0.0358335
\(209\) 0 0
\(210\) 1.65923 0.114498
\(211\) −24.5746 −1.69179 −0.845893 0.533353i \(-0.820932\pi\)
−0.845893 + 0.533353i \(0.820932\pi\)
\(212\) 3.18512 0.218755
\(213\) −21.1567 −1.44963
\(214\) −18.9251 −1.29369
\(215\) −3.81146 −0.259940
\(216\) −11.9815 −0.815235
\(217\) −0.736319 −0.0499846
\(218\) 14.8991 1.00909
\(219\) −51.3197 −3.46787
\(220\) 1.74595 0.117712
\(221\) −3.17410 −0.213513
\(222\) 34.1383 2.29121
\(223\) −18.4764 −1.23727 −0.618635 0.785679i \(-0.712314\pi\)
−0.618635 + 0.785679i \(0.712314\pi\)
\(224\) 1.00000 0.0668153
\(225\) −32.2021 −2.14681
\(226\) −1.20410 −0.0800956
\(227\) −3.85747 −0.256029 −0.128015 0.991772i \(-0.540860\pi\)
−0.128015 + 0.991772i \(0.540860\pi\)
\(228\) 0 0
\(229\) −11.4759 −0.758350 −0.379175 0.925325i \(-0.623792\pi\)
−0.379175 + 0.925325i \(0.623792\pi\)
\(230\) 3.01281 0.198659
\(231\) 10.3363 0.680076
\(232\) −0.202602 −0.0133015
\(233\) −10.2608 −0.672206 −0.336103 0.941825i \(-0.609109\pi\)
−0.336103 + 0.941825i \(0.609109\pi\)
\(234\) −3.52605 −0.230505
\(235\) −6.42633 −0.419208
\(236\) −3.91656 −0.254947
\(237\) 27.2460 1.76982
\(238\) 6.14185 0.398117
\(239\) −18.9095 −1.22315 −0.611577 0.791185i \(-0.709464\pi\)
−0.611577 + 0.791185i \(0.709464\pi\)
\(240\) 1.65923 0.107103
\(241\) −11.5763 −0.745698 −0.372849 0.927892i \(-0.621619\pi\)
−0.372849 + 0.927892i \(0.621619\pi\)
\(242\) −0.123528 −0.00794066
\(243\) −17.5963 −1.12880
\(244\) 3.73389 0.239038
\(245\) −0.529405 −0.0338224
\(246\) −11.5623 −0.737184
\(247\) 0 0
\(248\) −0.736319 −0.0467563
\(249\) 56.5242 3.58207
\(250\) 5.14567 0.325441
\(251\) 26.8853 1.69698 0.848492 0.529208i \(-0.177511\pi\)
0.848492 + 0.529208i \(0.177511\pi\)
\(252\) 6.82288 0.429801
\(253\) 18.7684 1.17996
\(254\) 0.707401 0.0443863
\(255\) 10.1908 0.638170
\(256\) 1.00000 0.0625000
\(257\) −13.2469 −0.826318 −0.413159 0.910659i \(-0.635575\pi\)
−0.413159 + 0.910659i \(0.635575\pi\)
\(258\) −22.5644 −1.40480
\(259\) −10.8924 −0.676819
\(260\) 0.273596 0.0169677
\(261\) −1.38233 −0.0855642
\(262\) −10.3358 −0.638548
\(263\) −1.67507 −0.103289 −0.0516445 0.998666i \(-0.516446\pi\)
−0.0516445 + 0.998666i \(0.516446\pi\)
\(264\) 10.3363 0.636153
\(265\) −1.68622 −0.103583
\(266\) 0 0
\(267\) 50.8013 3.10899
\(268\) −9.77683 −0.597215
\(269\) 13.7127 0.836077 0.418038 0.908429i \(-0.362718\pi\)
0.418038 + 0.908429i \(0.362718\pi\)
\(270\) 6.34304 0.386025
\(271\) −21.3184 −1.29500 −0.647501 0.762065i \(-0.724186\pi\)
−0.647501 + 0.762065i \(0.724186\pi\)
\(272\) 6.14185 0.372404
\(273\) 1.61972 0.0980300
\(274\) 4.56839 0.275986
\(275\) 15.5654 0.938631
\(276\) 17.8362 1.07362
\(277\) 12.9204 0.776312 0.388156 0.921594i \(-0.373112\pi\)
0.388156 + 0.921594i \(0.373112\pi\)
\(278\) −6.04179 −0.362362
\(279\) −5.02381 −0.300768
\(280\) −0.529405 −0.0316380
\(281\) −2.53217 −0.151057 −0.0755283 0.997144i \(-0.524064\pi\)
−0.0755283 + 0.997144i \(0.524064\pi\)
\(282\) −38.0447 −2.26553
\(283\) 7.18663 0.427201 0.213601 0.976921i \(-0.431481\pi\)
0.213601 + 0.976921i \(0.431481\pi\)
\(284\) 6.75039 0.400562
\(285\) 0 0
\(286\) 1.70437 0.100782
\(287\) 3.68913 0.217762
\(288\) 6.82288 0.402042
\(289\) 20.7223 1.21896
\(290\) 0.107259 0.00629845
\(291\) −8.25105 −0.483685
\(292\) 16.3744 0.958238
\(293\) −1.03791 −0.0606352 −0.0303176 0.999540i \(-0.509652\pi\)
−0.0303176 + 0.999540i \(0.509652\pi\)
\(294\) −3.13415 −0.182787
\(295\) 2.07345 0.120721
\(296\) −10.8924 −0.633106
\(297\) 39.5142 2.29285
\(298\) −18.2013 −1.05437
\(299\) 2.94107 0.170086
\(300\) 14.7923 0.854035
\(301\) 7.19952 0.414973
\(302\) 20.7582 1.19450
\(303\) −30.8318 −1.77124
\(304\) 0 0
\(305\) −1.97674 −0.113188
\(306\) 41.9051 2.39555
\(307\) 8.78461 0.501364 0.250682 0.968069i \(-0.419345\pi\)
0.250682 + 0.968069i \(0.419345\pi\)
\(308\) −3.29795 −0.187918
\(309\) −13.5273 −0.769544
\(310\) 0.389811 0.0221398
\(311\) −0.240874 −0.0136587 −0.00682936 0.999977i \(-0.502174\pi\)
−0.00682936 + 0.999977i \(0.502174\pi\)
\(312\) 1.61972 0.0916987
\(313\) 1.87814 0.106159 0.0530795 0.998590i \(-0.483096\pi\)
0.0530795 + 0.998590i \(0.483096\pi\)
\(314\) 10.4064 0.587266
\(315\) −3.61206 −0.203517
\(316\) −8.69327 −0.489035
\(317\) 33.6024 1.88730 0.943648 0.330950i \(-0.107369\pi\)
0.943648 + 0.330950i \(0.107369\pi\)
\(318\) −9.98262 −0.559798
\(319\) 0.668173 0.0374105
\(320\) −0.529405 −0.0295946
\(321\) 59.3140 3.31059
\(322\) −5.69094 −0.317144
\(323\) 0 0
\(324\) 17.0830 0.949056
\(325\) 2.43915 0.135300
\(326\) −17.8149 −0.986679
\(327\) −46.6959 −2.58229
\(328\) 3.68913 0.203698
\(329\) 12.1388 0.669233
\(330\) −5.47207 −0.301227
\(331\) 11.1959 0.615383 0.307691 0.951486i \(-0.400444\pi\)
0.307691 + 0.951486i \(0.400444\pi\)
\(332\) −18.0349 −0.989796
\(333\) −74.3173 −4.07256
\(334\) −7.93765 −0.434329
\(335\) 5.17590 0.282790
\(336\) −3.13415 −0.170982
\(337\) −15.5993 −0.849746 −0.424873 0.905253i \(-0.639681\pi\)
−0.424873 + 0.905253i \(0.639681\pi\)
\(338\) −12.7329 −0.692580
\(339\) 3.77383 0.204966
\(340\) −3.25153 −0.176339
\(341\) 2.42834 0.131502
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 7.19952 0.388172
\(345\) −9.44259 −0.508372
\(346\) −9.69338 −0.521119
\(347\) −17.2497 −0.926011 −0.463006 0.886355i \(-0.653229\pi\)
−0.463006 + 0.886355i \(0.653229\pi\)
\(348\) 0.634986 0.0340388
\(349\) 2.55529 0.136782 0.0683908 0.997659i \(-0.478214\pi\)
0.0683908 + 0.997659i \(0.478214\pi\)
\(350\) −4.71973 −0.252280
\(351\) 6.19199 0.330504
\(352\) −3.29795 −0.175781
\(353\) −27.4877 −1.46302 −0.731511 0.681829i \(-0.761185\pi\)
−0.731511 + 0.681829i \(0.761185\pi\)
\(354\) 12.2751 0.652414
\(355\) −3.57369 −0.189672
\(356\) −16.2090 −0.859074
\(357\) −19.2495 −1.01879
\(358\) 13.9458 0.737058
\(359\) −15.4937 −0.817727 −0.408864 0.912595i \(-0.634075\pi\)
−0.408864 + 0.912595i \(0.634075\pi\)
\(360\) −3.61206 −0.190373
\(361\) 0 0
\(362\) −19.1572 −1.00688
\(363\) 0.387154 0.0203203
\(364\) −0.516798 −0.0270876
\(365\) −8.66868 −0.453739
\(366\) −11.7025 −0.611702
\(367\) −26.5673 −1.38680 −0.693401 0.720552i \(-0.743889\pi\)
−0.693401 + 0.720552i \(0.743889\pi\)
\(368\) −5.69094 −0.296661
\(369\) 25.1705 1.31032
\(370\) 5.76648 0.299785
\(371\) 3.18512 0.165363
\(372\) 2.30773 0.119650
\(373\) −14.5329 −0.752487 −0.376244 0.926521i \(-0.622784\pi\)
−0.376244 + 0.926521i \(0.622784\pi\)
\(374\) −20.2555 −1.04739
\(375\) −16.1273 −0.832810
\(376\) 12.1388 0.626010
\(377\) 0.104705 0.00539256
\(378\) −11.9815 −0.616260
\(379\) −23.1135 −1.18726 −0.593631 0.804738i \(-0.702306\pi\)
−0.593631 + 0.804738i \(0.702306\pi\)
\(380\) 0 0
\(381\) −2.21710 −0.113585
\(382\) −12.7895 −0.654369
\(383\) 25.5356 1.30481 0.652405 0.757871i \(-0.273760\pi\)
0.652405 + 0.757871i \(0.273760\pi\)
\(384\) −3.13415 −0.159939
\(385\) 1.74595 0.0889819
\(386\) −10.4419 −0.531477
\(387\) 49.1214 2.49698
\(388\) 2.63263 0.133652
\(389\) −30.1536 −1.52885 −0.764424 0.644714i \(-0.776976\pi\)
−0.764424 + 0.644714i \(0.776976\pi\)
\(390\) −0.857489 −0.0434206
\(391\) −34.9529 −1.76764
\(392\) 1.00000 0.0505076
\(393\) 32.3939 1.63406
\(394\) 0.817360 0.0411780
\(395\) 4.60226 0.231565
\(396\) −22.5015 −1.13074
\(397\) −13.9802 −0.701647 −0.350824 0.936442i \(-0.614098\pi\)
−0.350824 + 0.936442i \(0.614098\pi\)
\(398\) −7.81078 −0.391519
\(399\) 0 0
\(400\) −4.71973 −0.235987
\(401\) 17.6953 0.883661 0.441830 0.897099i \(-0.354329\pi\)
0.441830 + 0.897099i \(0.354329\pi\)
\(402\) 30.6420 1.52829
\(403\) 0.380528 0.0189555
\(404\) 9.83739 0.489428
\(405\) −9.04383 −0.449392
\(406\) −0.202602 −0.0100550
\(407\) 35.9225 1.78061
\(408\) −19.2495 −0.952990
\(409\) −12.5454 −0.620329 −0.310165 0.950683i \(-0.600384\pi\)
−0.310165 + 0.950683i \(0.600384\pi\)
\(410\) −1.95304 −0.0964540
\(411\) −14.3180 −0.706255
\(412\) 4.31612 0.212640
\(413\) −3.91656 −0.192722
\(414\) −38.8286 −1.90832
\(415\) 9.54779 0.468683
\(416\) −0.516798 −0.0253381
\(417\) 18.9358 0.927292
\(418\) 0 0
\(419\) −7.81346 −0.381713 −0.190856 0.981618i \(-0.561126\pi\)
−0.190856 + 0.981618i \(0.561126\pi\)
\(420\) 1.65923 0.0809623
\(421\) 29.1874 1.42251 0.711253 0.702936i \(-0.248128\pi\)
0.711253 + 0.702936i \(0.248128\pi\)
\(422\) −24.5746 −1.19627
\(423\) 82.8214 4.02691
\(424\) 3.18512 0.154683
\(425\) −28.9879 −1.40612
\(426\) −21.1567 −1.02505
\(427\) 3.73389 0.180695
\(428\) −18.9251 −0.914779
\(429\) −5.34176 −0.257903
\(430\) −3.81146 −0.183805
\(431\) −1.27732 −0.0615266 −0.0307633 0.999527i \(-0.509794\pi\)
−0.0307633 + 0.999527i \(0.509794\pi\)
\(432\) −11.9815 −0.576458
\(433\) −15.5705 −0.748272 −0.374136 0.927374i \(-0.622061\pi\)
−0.374136 + 0.927374i \(0.622061\pi\)
\(434\) −0.736319 −0.0353444
\(435\) −0.336165 −0.0161179
\(436\) 14.8991 0.713536
\(437\) 0 0
\(438\) −51.3197 −2.45215
\(439\) −20.3046 −0.969084 −0.484542 0.874768i \(-0.661014\pi\)
−0.484542 + 0.874768i \(0.661014\pi\)
\(440\) 1.74595 0.0832349
\(441\) 6.82288 0.324899
\(442\) −3.17410 −0.150976
\(443\) −5.99965 −0.285052 −0.142526 0.989791i \(-0.545522\pi\)
−0.142526 + 0.989791i \(0.545522\pi\)
\(444\) 34.1383 1.62013
\(445\) 8.58112 0.406784
\(446\) −18.4764 −0.874882
\(447\) 57.0454 2.69816
\(448\) 1.00000 0.0472456
\(449\) 22.1087 1.04338 0.521688 0.853136i \(-0.325302\pi\)
0.521688 + 0.853136i \(0.325302\pi\)
\(450\) −32.2021 −1.51802
\(451\) −12.1666 −0.572901
\(452\) −1.20410 −0.0566361
\(453\) −65.0594 −3.05676
\(454\) −3.85747 −0.181040
\(455\) 0.273596 0.0128264
\(456\) 0 0
\(457\) −13.4648 −0.629859 −0.314929 0.949115i \(-0.601981\pi\)
−0.314929 + 0.949115i \(0.601981\pi\)
\(458\) −11.4759 −0.536234
\(459\) −73.5883 −3.43481
\(460\) 3.01281 0.140473
\(461\) 7.05129 0.328411 0.164206 0.986426i \(-0.447494\pi\)
0.164206 + 0.986426i \(0.447494\pi\)
\(462\) 10.3363 0.480886
\(463\) 8.47412 0.393826 0.196913 0.980421i \(-0.436908\pi\)
0.196913 + 0.980421i \(0.436908\pi\)
\(464\) −0.202602 −0.00940558
\(465\) −1.22172 −0.0566561
\(466\) −10.2608 −0.475321
\(467\) 30.8727 1.42862 0.714308 0.699832i \(-0.246742\pi\)
0.714308 + 0.699832i \(0.246742\pi\)
\(468\) −3.52605 −0.162992
\(469\) −9.77683 −0.451452
\(470\) −6.42633 −0.296424
\(471\) −32.6151 −1.50282
\(472\) −3.91656 −0.180275
\(473\) −23.7437 −1.09173
\(474\) 27.2460 1.25145
\(475\) 0 0
\(476\) 6.14185 0.281511
\(477\) 21.7317 0.995024
\(478\) −18.9095 −0.864900
\(479\) −28.5989 −1.30671 −0.653357 0.757050i \(-0.726640\pi\)
−0.653357 + 0.757050i \(0.726640\pi\)
\(480\) 1.65923 0.0757333
\(481\) 5.62916 0.256668
\(482\) −11.5763 −0.527288
\(483\) 17.8362 0.811577
\(484\) −0.123528 −0.00561489
\(485\) −1.39373 −0.0632859
\(486\) −17.5963 −0.798184
\(487\) −35.6285 −1.61448 −0.807240 0.590223i \(-0.799040\pi\)
−0.807240 + 0.590223i \(0.799040\pi\)
\(488\) 3.73389 0.169025
\(489\) 55.8347 2.52493
\(490\) −0.529405 −0.0239161
\(491\) 12.1881 0.550039 0.275020 0.961439i \(-0.411316\pi\)
0.275020 + 0.961439i \(0.411316\pi\)
\(492\) −11.5623 −0.521268
\(493\) −1.24435 −0.0560429
\(494\) 0 0
\(495\) 11.9124 0.535423
\(496\) −0.736319 −0.0330617
\(497\) 6.75039 0.302796
\(498\) 56.5242 2.53291
\(499\) 1.69496 0.0758766 0.0379383 0.999280i \(-0.487921\pi\)
0.0379383 + 0.999280i \(0.487921\pi\)
\(500\) 5.14567 0.230122
\(501\) 24.8778 1.11146
\(502\) 26.8853 1.19995
\(503\) 6.70132 0.298797 0.149398 0.988777i \(-0.452266\pi\)
0.149398 + 0.988777i \(0.452266\pi\)
\(504\) 6.82288 0.303915
\(505\) −5.20796 −0.231751
\(506\) 18.7684 0.834358
\(507\) 39.9068 1.77232
\(508\) 0.707401 0.0313858
\(509\) −16.9421 −0.750946 −0.375473 0.926833i \(-0.622520\pi\)
−0.375473 + 0.926833i \(0.622520\pi\)
\(510\) 10.1908 0.451255
\(511\) 16.3744 0.724360
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −13.2469 −0.584295
\(515\) −2.28497 −0.100688
\(516\) −22.5644 −0.993340
\(517\) −40.0331 −1.76065
\(518\) −10.8924 −0.478583
\(519\) 30.3805 1.33356
\(520\) 0.273596 0.0119980
\(521\) 11.8846 0.520675 0.260338 0.965518i \(-0.416166\pi\)
0.260338 + 0.965518i \(0.416166\pi\)
\(522\) −1.38233 −0.0605030
\(523\) 4.32787 0.189244 0.0946222 0.995513i \(-0.469836\pi\)
0.0946222 + 0.995513i \(0.469836\pi\)
\(524\) −10.3358 −0.451522
\(525\) 14.7923 0.645590
\(526\) −1.67507 −0.0730364
\(527\) −4.52236 −0.196997
\(528\) 10.3363 0.449828
\(529\) 9.38678 0.408121
\(530\) −1.68622 −0.0732446
\(531\) −26.7222 −1.15965
\(532\) 0 0
\(533\) −1.90654 −0.0825812
\(534\) 50.8013 2.19839
\(535\) 10.0190 0.433161
\(536\) −9.77683 −0.422295
\(537\) −43.7081 −1.88615
\(538\) 13.7127 0.591196
\(539\) −3.29795 −0.142053
\(540\) 6.34304 0.272961
\(541\) −24.5223 −1.05430 −0.527148 0.849773i \(-0.676739\pi\)
−0.527148 + 0.849773i \(0.676739\pi\)
\(542\) −21.3184 −0.915704
\(543\) 60.0416 2.57663
\(544\) 6.14185 0.263330
\(545\) −7.88765 −0.337870
\(546\) 1.61972 0.0693177
\(547\) 5.73732 0.245310 0.122655 0.992449i \(-0.460859\pi\)
0.122655 + 0.992449i \(0.460859\pi\)
\(548\) 4.56839 0.195152
\(549\) 25.4758 1.08728
\(550\) 15.5654 0.663712
\(551\) 0 0
\(552\) 17.8362 0.759161
\(553\) −8.69327 −0.369675
\(554\) 12.9204 0.548936
\(555\) −18.0730 −0.767155
\(556\) −6.04179 −0.256229
\(557\) 24.4382 1.03548 0.517741 0.855537i \(-0.326773\pi\)
0.517741 + 0.855537i \(0.326773\pi\)
\(558\) −5.02381 −0.212675
\(559\) −3.72070 −0.157369
\(560\) −0.529405 −0.0223714
\(561\) 63.4838 2.68029
\(562\) −2.53217 −0.106813
\(563\) 17.0563 0.718837 0.359419 0.933176i \(-0.382975\pi\)
0.359419 + 0.933176i \(0.382975\pi\)
\(564\) −38.0447 −1.60197
\(565\) 0.637457 0.0268180
\(566\) 7.18663 0.302077
\(567\) 17.0830 0.717419
\(568\) 6.75039 0.283240
\(569\) −12.1689 −0.510148 −0.255074 0.966922i \(-0.582100\pi\)
−0.255074 + 0.966922i \(0.582100\pi\)
\(570\) 0 0
\(571\) −28.5696 −1.19560 −0.597801 0.801644i \(-0.703959\pi\)
−0.597801 + 0.801644i \(0.703959\pi\)
\(572\) 1.70437 0.0712635
\(573\) 40.0842 1.67454
\(574\) 3.68913 0.153981
\(575\) 26.8597 1.12013
\(576\) 6.82288 0.284287
\(577\) 11.2721 0.469263 0.234632 0.972084i \(-0.424612\pi\)
0.234632 + 0.972084i \(0.424612\pi\)
\(578\) 20.7223 0.861936
\(579\) 32.7263 1.36006
\(580\) 0.107259 0.00445368
\(581\) −18.0349 −0.748216
\(582\) −8.25105 −0.342017
\(583\) −10.5044 −0.435046
\(584\) 16.3744 0.677577
\(585\) 1.86671 0.0771789
\(586\) −1.03791 −0.0428755
\(587\) 19.7831 0.816537 0.408269 0.912862i \(-0.366133\pi\)
0.408269 + 0.912862i \(0.366133\pi\)
\(588\) −3.13415 −0.129250
\(589\) 0 0
\(590\) 2.07345 0.0853625
\(591\) −2.56173 −0.105375
\(592\) −10.8924 −0.447674
\(593\) 23.3997 0.960911 0.480455 0.877019i \(-0.340471\pi\)
0.480455 + 0.877019i \(0.340471\pi\)
\(594\) 39.5142 1.62129
\(595\) −3.25153 −0.133300
\(596\) −18.2013 −0.745553
\(597\) 24.4801 1.00190
\(598\) 2.94107 0.120269
\(599\) −15.5005 −0.633334 −0.316667 0.948537i \(-0.602564\pi\)
−0.316667 + 0.948537i \(0.602564\pi\)
\(600\) 14.7923 0.603894
\(601\) −41.8017 −1.70513 −0.852564 0.522623i \(-0.824954\pi\)
−0.852564 + 0.522623i \(0.824954\pi\)
\(602\) 7.19952 0.293431
\(603\) −66.7061 −2.71648
\(604\) 20.7582 0.844641
\(605\) 0.0653962 0.00265873
\(606\) −30.8318 −1.25246
\(607\) 45.2892 1.83823 0.919117 0.393985i \(-0.128904\pi\)
0.919117 + 0.393985i \(0.128904\pi\)
\(608\) 0 0
\(609\) 0.634986 0.0257309
\(610\) −1.97674 −0.0800358
\(611\) −6.27330 −0.253791
\(612\) 41.9051 1.69391
\(613\) 26.3197 1.06304 0.531521 0.847045i \(-0.321621\pi\)
0.531521 + 0.847045i \(0.321621\pi\)
\(614\) 8.78461 0.354518
\(615\) 6.12113 0.246828
\(616\) −3.29795 −0.132878
\(617\) 40.1140 1.61493 0.807464 0.589917i \(-0.200840\pi\)
0.807464 + 0.589917i \(0.200840\pi\)
\(618\) −13.5273 −0.544150
\(619\) −28.6211 −1.15038 −0.575191 0.818019i \(-0.695072\pi\)
−0.575191 + 0.818019i \(0.695072\pi\)
\(620\) 0.389811 0.0156552
\(621\) 68.1857 2.73620
\(622\) −0.240874 −0.00965818
\(623\) −16.2090 −0.649399
\(624\) 1.61972 0.0648408
\(625\) 20.8745 0.834980
\(626\) 1.87814 0.0750658
\(627\) 0 0
\(628\) 10.4064 0.415259
\(629\) −66.8993 −2.66745
\(630\) −3.61206 −0.143908
\(631\) −13.9752 −0.556345 −0.278172 0.960531i \(-0.589729\pi\)
−0.278172 + 0.960531i \(0.589729\pi\)
\(632\) −8.69327 −0.345800
\(633\) 77.0204 3.06129
\(634\) 33.6024 1.33452
\(635\) −0.374502 −0.0148616
\(636\) −9.98262 −0.395837
\(637\) −0.516798 −0.0204763
\(638\) 0.668173 0.0264532
\(639\) 46.0571 1.82199
\(640\) −0.529405 −0.0209266
\(641\) 40.3587 1.59407 0.797037 0.603931i \(-0.206400\pi\)
0.797037 + 0.603931i \(0.206400\pi\)
\(642\) 59.3140 2.34094
\(643\) 0.0182292 0.000718890 0 0.000359445 1.00000i \(-0.499886\pi\)
0.000359445 1.00000i \(0.499886\pi\)
\(644\) −5.69094 −0.224254
\(645\) 11.9457 0.470361
\(646\) 0 0
\(647\) 19.8386 0.779937 0.389968 0.920828i \(-0.372486\pi\)
0.389968 + 0.920828i \(0.372486\pi\)
\(648\) 17.0830 0.671084
\(649\) 12.9166 0.507022
\(650\) 2.43915 0.0956713
\(651\) 2.30773 0.0904472
\(652\) −17.8149 −0.697687
\(653\) −42.0361 −1.64500 −0.822500 0.568765i \(-0.807421\pi\)
−0.822500 + 0.568765i \(0.807421\pi\)
\(654\) −46.6959 −1.82595
\(655\) 5.47183 0.213802
\(656\) 3.68913 0.144036
\(657\) 111.720 4.35863
\(658\) 12.1388 0.473219
\(659\) 16.5518 0.644765 0.322382 0.946610i \(-0.395516\pi\)
0.322382 + 0.946610i \(0.395516\pi\)
\(660\) −5.47207 −0.213000
\(661\) 29.7497 1.15713 0.578565 0.815636i \(-0.303613\pi\)
0.578565 + 0.815636i \(0.303613\pi\)
\(662\) 11.1959 0.435141
\(663\) 9.94809 0.386352
\(664\) −18.0349 −0.699892
\(665\) 0 0
\(666\) −74.3173 −2.87974
\(667\) 1.15300 0.0446443
\(668\) −7.93765 −0.307117
\(669\) 57.9077 2.23884
\(670\) 5.17590 0.199963
\(671\) −12.3142 −0.475383
\(672\) −3.13415 −0.120902
\(673\) 13.3441 0.514376 0.257188 0.966361i \(-0.417204\pi\)
0.257188 + 0.966361i \(0.417204\pi\)
\(674\) −15.5993 −0.600861
\(675\) 56.5492 2.17658
\(676\) −12.7329 −0.489728
\(677\) −18.0194 −0.692543 −0.346271 0.938134i \(-0.612552\pi\)
−0.346271 + 0.938134i \(0.612552\pi\)
\(678\) 3.77383 0.144933
\(679\) 2.63263 0.101031
\(680\) −3.25153 −0.124690
\(681\) 12.0899 0.463285
\(682\) 2.42834 0.0929861
\(683\) 11.8762 0.454431 0.227216 0.973845i \(-0.427038\pi\)
0.227216 + 0.973845i \(0.427038\pi\)
\(684\) 0 0
\(685\) −2.41853 −0.0924072
\(686\) 1.00000 0.0381802
\(687\) 35.9672 1.37223
\(688\) 7.19952 0.274479
\(689\) −1.64606 −0.0627100
\(690\) −9.44259 −0.359473
\(691\) 10.8249 0.411799 0.205899 0.978573i \(-0.433988\pi\)
0.205899 + 0.978573i \(0.433988\pi\)
\(692\) −9.69338 −0.368487
\(693\) −22.5015 −0.854761
\(694\) −17.2497 −0.654789
\(695\) 3.19855 0.121328
\(696\) 0.634986 0.0240691
\(697\) 22.6581 0.858236
\(698\) 2.55529 0.0967193
\(699\) 32.1588 1.21636
\(700\) −4.71973 −0.178389
\(701\) 13.6726 0.516408 0.258204 0.966090i \(-0.416869\pi\)
0.258204 + 0.966090i \(0.416869\pi\)
\(702\) 6.19199 0.233702
\(703\) 0 0
\(704\) −3.29795 −0.124296
\(705\) 20.1411 0.758556
\(706\) −27.4877 −1.03451
\(707\) 9.83739 0.369973
\(708\) 12.2751 0.461326
\(709\) −46.0283 −1.72863 −0.864314 0.502953i \(-0.832247\pi\)
−0.864314 + 0.502953i \(0.832247\pi\)
\(710\) −3.57369 −0.134118
\(711\) −59.3131 −2.22441
\(712\) −16.2090 −0.607457
\(713\) 4.19035 0.156930
\(714\) −19.2495 −0.720393
\(715\) −0.902304 −0.0337443
\(716\) 13.9458 0.521178
\(717\) 59.2651 2.21330
\(718\) −15.4937 −0.578221
\(719\) 19.4728 0.726214 0.363107 0.931748i \(-0.381716\pi\)
0.363107 + 0.931748i \(0.381716\pi\)
\(720\) −3.61206 −0.134614
\(721\) 4.31612 0.160741
\(722\) 0 0
\(723\) 36.2820 1.34934
\(724\) −19.1572 −0.711973
\(725\) 0.956229 0.0355135
\(726\) 0.387154 0.0143686
\(727\) 39.9706 1.48243 0.741214 0.671269i \(-0.234250\pi\)
0.741214 + 0.671269i \(0.234250\pi\)
\(728\) −0.516798 −0.0191538
\(729\) 3.90035 0.144458
\(730\) −8.66868 −0.320842
\(731\) 44.2184 1.63548
\(732\) −11.7025 −0.432539
\(733\) −21.1476 −0.781105 −0.390553 0.920581i \(-0.627716\pi\)
−0.390553 + 0.920581i \(0.627716\pi\)
\(734\) −26.5673 −0.980617
\(735\) 1.65923 0.0612017
\(736\) −5.69094 −0.209771
\(737\) 32.2435 1.18770
\(738\) 25.1705 0.926538
\(739\) −42.1012 −1.54872 −0.774359 0.632746i \(-0.781928\pi\)
−0.774359 + 0.632746i \(0.781928\pi\)
\(740\) 5.76648 0.211980
\(741\) 0 0
\(742\) 3.18512 0.116929
\(743\) 18.1350 0.665308 0.332654 0.943049i \(-0.392056\pi\)
0.332654 + 0.943049i \(0.392056\pi\)
\(744\) 2.30773 0.0846056
\(745\) 9.63584 0.353030
\(746\) −14.5329 −0.532089
\(747\) −123.050 −4.50217
\(748\) −20.2555 −0.740615
\(749\) −18.9251 −0.691508
\(750\) −16.1273 −0.588886
\(751\) −10.4898 −0.382780 −0.191390 0.981514i \(-0.561300\pi\)
−0.191390 + 0.981514i \(0.561300\pi\)
\(752\) 12.1388 0.442656
\(753\) −84.2625 −3.07069
\(754\) 0.104705 0.00381312
\(755\) −10.9895 −0.399949
\(756\) −11.9815 −0.435761
\(757\) 12.0758 0.438904 0.219452 0.975623i \(-0.429573\pi\)
0.219452 + 0.975623i \(0.429573\pi\)
\(758\) −23.1135 −0.839521
\(759\) −58.8230 −2.13514
\(760\) 0 0
\(761\) −28.0992 −1.01859 −0.509297 0.860591i \(-0.670094\pi\)
−0.509297 + 0.860591i \(0.670094\pi\)
\(762\) −2.21710 −0.0803170
\(763\) 14.8991 0.539383
\(764\) −12.7895 −0.462709
\(765\) −22.1848 −0.802092
\(766\) 25.5356 0.922639
\(767\) 2.02407 0.0730851
\(768\) −3.13415 −0.113094
\(769\) 34.2232 1.23412 0.617060 0.786916i \(-0.288324\pi\)
0.617060 + 0.786916i \(0.288324\pi\)
\(770\) 1.74595 0.0629197
\(771\) 41.5177 1.49522
\(772\) −10.4419 −0.375811
\(773\) 34.5711 1.24344 0.621719 0.783241i \(-0.286435\pi\)
0.621719 + 0.783241i \(0.286435\pi\)
\(774\) 49.1214 1.76563
\(775\) 3.47523 0.124834
\(776\) 2.63263 0.0945059
\(777\) 34.1383 1.22470
\(778\) −30.1536 −1.08106
\(779\) 0 0
\(780\) −0.857489 −0.0307030
\(781\) −22.2624 −0.796613
\(782\) −34.9529 −1.24991
\(783\) 2.42747 0.0867508
\(784\) 1.00000 0.0357143
\(785\) −5.50919 −0.196631
\(786\) 32.3939 1.15545
\(787\) −38.3673 −1.36765 −0.683823 0.729648i \(-0.739684\pi\)
−0.683823 + 0.729648i \(0.739684\pi\)
\(788\) 0.817360 0.0291173
\(789\) 5.24991 0.186902
\(790\) 4.60226 0.163741
\(791\) −1.20410 −0.0428129
\(792\) −22.5015 −0.799556
\(793\) −1.92967 −0.0685244
\(794\) −13.9802 −0.496139
\(795\) 5.28485 0.187434
\(796\) −7.81078 −0.276846
\(797\) −35.5206 −1.25820 −0.629102 0.777323i \(-0.716577\pi\)
−0.629102 + 0.777323i \(0.716577\pi\)
\(798\) 0 0
\(799\) 74.5546 2.63755
\(800\) −4.71973 −0.166868
\(801\) −110.592 −3.90757
\(802\) 17.6953 0.624843
\(803\) −54.0019 −1.90569
\(804\) 30.6420 1.08066
\(805\) 3.01281 0.106188
\(806\) 0.380528 0.0134035
\(807\) −42.9776 −1.51288
\(808\) 9.83739 0.346078
\(809\) 12.0795 0.424693 0.212347 0.977194i \(-0.431889\pi\)
0.212347 + 0.977194i \(0.431889\pi\)
\(810\) −9.04383 −0.317768
\(811\) −28.4964 −1.00065 −0.500323 0.865839i \(-0.666785\pi\)
−0.500323 + 0.865839i \(0.666785\pi\)
\(812\) −0.202602 −0.00710995
\(813\) 66.8150 2.34331
\(814\) 35.9225 1.25908
\(815\) 9.43132 0.330365
\(816\) −19.2495 −0.673866
\(817\) 0 0
\(818\) −12.5454 −0.438639
\(819\) −3.52605 −0.123210
\(820\) −1.95304 −0.0682032
\(821\) 37.6434 1.31376 0.656882 0.753993i \(-0.271875\pi\)
0.656882 + 0.753993i \(0.271875\pi\)
\(822\) −14.3180 −0.499398
\(823\) −9.09651 −0.317084 −0.158542 0.987352i \(-0.550679\pi\)
−0.158542 + 0.987352i \(0.550679\pi\)
\(824\) 4.31612 0.150359
\(825\) −48.7844 −1.69845
\(826\) −3.91656 −0.136275
\(827\) −11.1658 −0.388274 −0.194137 0.980974i \(-0.562191\pi\)
−0.194137 + 0.980974i \(0.562191\pi\)
\(828\) −38.8286 −1.34939
\(829\) −29.0197 −1.00790 −0.503948 0.863734i \(-0.668120\pi\)
−0.503948 + 0.863734i \(0.668120\pi\)
\(830\) 9.54779 0.331409
\(831\) −40.4945 −1.40474
\(832\) −0.516798 −0.0179168
\(833\) 6.14185 0.212803
\(834\) 18.9358 0.655695
\(835\) 4.20223 0.145424
\(836\) 0 0
\(837\) 8.82217 0.304939
\(838\) −7.81346 −0.269912
\(839\) −16.3525 −0.564552 −0.282276 0.959333i \(-0.591089\pi\)
−0.282276 + 0.959333i \(0.591089\pi\)
\(840\) 1.65923 0.0572490
\(841\) −28.9590 −0.998585
\(842\) 29.1874 1.00586
\(843\) 7.93619 0.273337
\(844\) −24.5746 −0.845893
\(845\) 6.74087 0.231893
\(846\) 82.8214 2.84746
\(847\) −0.123528 −0.00424446
\(848\) 3.18512 0.109377
\(849\) −22.5240 −0.773020
\(850\) −28.9879 −0.994276
\(851\) 61.9878 2.12492
\(852\) −21.1567 −0.724817
\(853\) 2.01194 0.0688873 0.0344437 0.999407i \(-0.489034\pi\)
0.0344437 + 0.999407i \(0.489034\pi\)
\(854\) 3.73389 0.127771
\(855\) 0 0
\(856\) −18.9251 −0.646847
\(857\) −24.2234 −0.827454 −0.413727 0.910401i \(-0.635773\pi\)
−0.413727 + 0.910401i \(0.635773\pi\)
\(858\) −5.34176 −0.182365
\(859\) 3.77776 0.128895 0.0644477 0.997921i \(-0.479471\pi\)
0.0644477 + 0.997921i \(0.479471\pi\)
\(860\) −3.81146 −0.129970
\(861\) −11.5623 −0.394041
\(862\) −1.27732 −0.0435059
\(863\) 14.3259 0.487658 0.243829 0.969818i \(-0.421596\pi\)
0.243829 + 0.969818i \(0.421596\pi\)
\(864\) −11.9815 −0.407617
\(865\) 5.13172 0.174484
\(866\) −15.5705 −0.529108
\(867\) −64.9469 −2.20571
\(868\) −0.736319 −0.0249923
\(869\) 28.6700 0.972562
\(870\) −0.336165 −0.0113971
\(871\) 5.05265 0.171203
\(872\) 14.8991 0.504546
\(873\) 17.9621 0.607925
\(874\) 0 0
\(875\) 5.14567 0.173956
\(876\) −51.3197 −1.73393
\(877\) 16.9707 0.573060 0.286530 0.958071i \(-0.407498\pi\)
0.286530 + 0.958071i \(0.407498\pi\)
\(878\) −20.3046 −0.685246
\(879\) 3.25295 0.109719
\(880\) 1.74595 0.0588560
\(881\) −13.8679 −0.467221 −0.233611 0.972330i \(-0.575054\pi\)
−0.233611 + 0.972330i \(0.575054\pi\)
\(882\) 6.82288 0.229738
\(883\) −12.6029 −0.424123 −0.212061 0.977256i \(-0.568018\pi\)
−0.212061 + 0.977256i \(0.568018\pi\)
\(884\) −3.17410 −0.106756
\(885\) −6.49849 −0.218444
\(886\) −5.99965 −0.201562
\(887\) 49.9001 1.67548 0.837740 0.546069i \(-0.183876\pi\)
0.837740 + 0.546069i \(0.183876\pi\)
\(888\) 34.1383 1.14561
\(889\) 0.707401 0.0237255
\(890\) 8.58112 0.287640
\(891\) −56.3389 −1.88742
\(892\) −18.4764 −0.618635
\(893\) 0 0
\(894\) 57.0454 1.90788
\(895\) −7.38297 −0.246785
\(896\) 1.00000 0.0334077
\(897\) −9.21773 −0.307771
\(898\) 22.1087 0.737778
\(899\) 0.149180 0.00497543
\(900\) −32.2021 −1.07340
\(901\) 19.5625 0.651722
\(902\) −12.1666 −0.405102
\(903\) −22.5644 −0.750895
\(904\) −1.20410 −0.0400478
\(905\) 10.1419 0.337129
\(906\) −65.0594 −2.16145
\(907\) 3.19849 0.106204 0.0531020 0.998589i \(-0.483089\pi\)
0.0531020 + 0.998589i \(0.483089\pi\)
\(908\) −3.85747 −0.128015
\(909\) 67.1193 2.22621
\(910\) 0.273596 0.00906960
\(911\) 29.5246 0.978195 0.489097 0.872229i \(-0.337326\pi\)
0.489097 + 0.872229i \(0.337326\pi\)
\(912\) 0 0
\(913\) 59.4784 1.96845
\(914\) −13.4648 −0.445378
\(915\) 6.19539 0.204813
\(916\) −11.4759 −0.379175
\(917\) −10.3358 −0.341318
\(918\) −73.5883 −2.42878
\(919\) 4.15733 0.137138 0.0685689 0.997646i \(-0.478157\pi\)
0.0685689 + 0.997646i \(0.478157\pi\)
\(920\) 3.01281 0.0993294
\(921\) −27.5323 −0.907219
\(922\) 7.05129 0.232222
\(923\) −3.48859 −0.114828
\(924\) 10.3363 0.340038
\(925\) 51.4091 1.69032
\(926\) 8.47412 0.278477
\(927\) 29.4483 0.967210
\(928\) −0.202602 −0.00665075
\(929\) 30.8217 1.01123 0.505614 0.862760i \(-0.331266\pi\)
0.505614 + 0.862760i \(0.331266\pi\)
\(930\) −1.22172 −0.0400619
\(931\) 0 0
\(932\) −10.2608 −0.336103
\(933\) 0.754935 0.0247155
\(934\) 30.8727 1.01018
\(935\) 10.7234 0.350692
\(936\) −3.52605 −0.115253
\(937\) 42.3822 1.38457 0.692284 0.721626i \(-0.256605\pi\)
0.692284 + 0.721626i \(0.256605\pi\)
\(938\) −9.77683 −0.319225
\(939\) −5.88638 −0.192095
\(940\) −6.42633 −0.209604
\(941\) 39.7048 1.29434 0.647170 0.762346i \(-0.275952\pi\)
0.647170 + 0.762346i \(0.275952\pi\)
\(942\) −32.6151 −1.06266
\(943\) −20.9946 −0.683679
\(944\) −3.91656 −0.127473
\(945\) 6.34304 0.206339
\(946\) −23.7437 −0.771973
\(947\) 52.6101 1.70960 0.854799 0.518959i \(-0.173680\pi\)
0.854799 + 0.518959i \(0.173680\pi\)
\(948\) 27.2460 0.884908
\(949\) −8.46225 −0.274696
\(950\) 0 0
\(951\) −105.315 −3.41506
\(952\) 6.14185 0.199059
\(953\) −12.3258 −0.399271 −0.199636 0.979870i \(-0.563976\pi\)
−0.199636 + 0.979870i \(0.563976\pi\)
\(954\) 21.7317 0.703588
\(955\) 6.77084 0.219099
\(956\) −18.9095 −0.611577
\(957\) −2.09415 −0.0676943
\(958\) −28.5989 −0.923987
\(959\) 4.56839 0.147521
\(960\) 1.65923 0.0535515
\(961\) −30.4578 −0.982511
\(962\) 5.62916 0.181491
\(963\) −129.124 −4.16095
\(964\) −11.5763 −0.372849
\(965\) 5.52798 0.177952
\(966\) 17.8362 0.573871
\(967\) −26.2329 −0.843594 −0.421797 0.906690i \(-0.638601\pi\)
−0.421797 + 0.906690i \(0.638601\pi\)
\(968\) −0.123528 −0.00397033
\(969\) 0 0
\(970\) −1.39373 −0.0447499
\(971\) 38.3207 1.22977 0.614884 0.788617i \(-0.289203\pi\)
0.614884 + 0.788617i \(0.289203\pi\)
\(972\) −17.5963 −0.564402
\(973\) −6.04179 −0.193691
\(974\) −35.6285 −1.14161
\(975\) −7.64465 −0.244825
\(976\) 3.73389 0.119519
\(977\) 30.1695 0.965209 0.482604 0.875838i \(-0.339691\pi\)
0.482604 + 0.875838i \(0.339691\pi\)
\(978\) 55.8347 1.78540
\(979\) 53.4564 1.70847
\(980\) −0.529405 −0.0169112
\(981\) 101.655 3.24558
\(982\) 12.1881 0.388937
\(983\) −10.5645 −0.336955 −0.168478 0.985705i \(-0.553885\pi\)
−0.168478 + 0.985705i \(0.553885\pi\)
\(984\) −11.5623 −0.368592
\(985\) −0.432715 −0.0137874
\(986\) −1.24435 −0.0396283
\(987\) −38.0447 −1.21098
\(988\) 0 0
\(989\) −40.9720 −1.30283
\(990\) 11.9124 0.378601
\(991\) −11.1078 −0.352850 −0.176425 0.984314i \(-0.556453\pi\)
−0.176425 + 0.984314i \(0.556453\pi\)
\(992\) −0.736319 −0.0233782
\(993\) −35.0896 −1.11354
\(994\) 6.75039 0.214109
\(995\) 4.13506 0.131090
\(996\) 56.5242 1.79104
\(997\) 34.4533 1.09115 0.545573 0.838063i \(-0.316312\pi\)
0.545573 + 0.838063i \(0.316312\pi\)
\(998\) 1.69496 0.0536529
\(999\) 130.506 4.12904
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 5054.2.a.bh.1.1 yes 8
19.18 odd 2 5054.2.a.bg.1.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5054.2.a.bg.1.8 8 19.18 odd 2
5054.2.a.bh.1.1 yes 8 1.1 even 1 trivial