Properties

Label 6039.2.a.i.1.1
Level $6039$
Weight $2$
Character 6039.1
Self dual yes
Analytic conductor $48.222$
Analytic rank $0$
Dimension $13$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6039.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.2216577807\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Defining polynomial: \(x^{13} - 2 x^{12} - 19 x^{11} + 35 x^{10} + 136 x^{9} - 220 x^{8} - 469 x^{7} + 610 x^{6} + 841 x^{5} - 760 x^{4} - 742 x^{3} + 366 x^{2} + 236 x - 47\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2013)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.73913\) of defining polynomial
Character \(\chi\) \(=\) 6039.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.73913 q^{2} +5.50285 q^{4} -0.604616 q^{5} -1.91298 q^{7} -9.59477 q^{8} +O(q^{10})\) \(q-2.73913 q^{2} +5.50285 q^{4} -0.604616 q^{5} -1.91298 q^{7} -9.59477 q^{8} +1.65612 q^{10} -1.00000 q^{11} +0.176875 q^{13} +5.23991 q^{14} +15.2757 q^{16} -2.74225 q^{17} +3.64728 q^{19} -3.32711 q^{20} +2.73913 q^{22} -1.61899 q^{23} -4.63444 q^{25} -0.484484 q^{26} -10.5269 q^{28} +7.55848 q^{29} +7.88196 q^{31} -22.6525 q^{32} +7.51138 q^{34} +1.15662 q^{35} +8.43648 q^{37} -9.99040 q^{38} +5.80115 q^{40} -8.93210 q^{41} -4.51816 q^{43} -5.50285 q^{44} +4.43462 q^{46} +2.48579 q^{47} -3.34050 q^{49} +12.6943 q^{50} +0.973317 q^{52} +7.57327 q^{53} +0.604616 q^{55} +18.3546 q^{56} -20.7037 q^{58} -1.33758 q^{59} -1.00000 q^{61} -21.5897 q^{62} +31.4969 q^{64} -0.106941 q^{65} -5.50196 q^{67} -15.0902 q^{68} -3.16813 q^{70} -0.330471 q^{71} -1.75560 q^{73} -23.1086 q^{74} +20.0705 q^{76} +1.91298 q^{77} -12.9780 q^{79} -9.23590 q^{80} +24.4662 q^{82} -12.9300 q^{83} +1.65801 q^{85} +12.3759 q^{86} +9.59477 q^{88} +4.11061 q^{89} -0.338359 q^{91} -8.90903 q^{92} -6.80891 q^{94} -2.20521 q^{95} -12.7524 q^{97} +9.15006 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q - 2q^{2} + 16q^{4} - 3q^{5} + 11q^{7} - 9q^{8} + O(q^{10}) \) \( 13q - 2q^{2} + 16q^{4} - 3q^{5} + 11q^{7} - 9q^{8} + 6q^{10} - 13q^{11} + 13q^{13} - q^{14} + 18q^{16} - 17q^{17} + 14q^{19} + 7q^{20} + 2q^{22} - 7q^{23} + 18q^{25} + 10q^{26} + 19q^{28} + 6q^{29} + 27q^{31} - 5q^{32} + 6q^{34} - 14q^{35} + 10q^{37} - 2q^{38} + 8q^{40} - 3q^{41} + 29q^{43} - 16q^{44} - 24q^{46} - 8q^{47} + 8q^{49} + 27q^{50} + 37q^{52} + 24q^{53} + 3q^{55} - 24q^{56} - 5q^{58} - 13q^{59} - 13q^{61} - 39q^{62} + 47q^{64} + 11q^{65} + 44q^{67} + 8q^{68} - 12q^{70} - 3q^{71} + 48q^{73} + 22q^{74} + 47q^{76} - 11q^{77} - 17q^{79} + 26q^{80} + 56q^{82} - 50q^{83} + 8q^{85} - 18q^{86} + 9q^{88} + 15q^{89} + 47q^{91} - 14q^{92} + 45q^{94} + q^{95} + 27q^{97} - 47q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.73913 −1.93686 −0.968430 0.249287i \(-0.919804\pi\)
−0.968430 + 0.249287i \(0.919804\pi\)
\(3\) 0 0
\(4\) 5.50285 2.75142
\(5\) −0.604616 −0.270392 −0.135196 0.990819i \(-0.543166\pi\)
−0.135196 + 0.990819i \(0.543166\pi\)
\(6\) 0 0
\(7\) −1.91298 −0.723040 −0.361520 0.932364i \(-0.617742\pi\)
−0.361520 + 0.932364i \(0.617742\pi\)
\(8\) −9.59477 −3.39226
\(9\) 0 0
\(10\) 1.65612 0.523712
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 0.176875 0.0490563 0.0245282 0.999699i \(-0.492192\pi\)
0.0245282 + 0.999699i \(0.492192\pi\)
\(14\) 5.23991 1.40043
\(15\) 0 0
\(16\) 15.2757 3.81891
\(17\) −2.74225 −0.665093 −0.332546 0.943087i \(-0.607908\pi\)
−0.332546 + 0.943087i \(0.607908\pi\)
\(18\) 0 0
\(19\) 3.64728 0.836745 0.418372 0.908276i \(-0.362601\pi\)
0.418372 + 0.908276i \(0.362601\pi\)
\(20\) −3.32711 −0.743964
\(21\) 0 0
\(22\) 2.73913 0.583985
\(23\) −1.61899 −0.337582 −0.168791 0.985652i \(-0.553986\pi\)
−0.168791 + 0.985652i \(0.553986\pi\)
\(24\) 0 0
\(25\) −4.63444 −0.926888
\(26\) −0.484484 −0.0950152
\(27\) 0 0
\(28\) −10.5269 −1.98939
\(29\) 7.55848 1.40357 0.701787 0.712387i \(-0.252386\pi\)
0.701787 + 0.712387i \(0.252386\pi\)
\(30\) 0 0
\(31\) 7.88196 1.41564 0.707821 0.706392i \(-0.249678\pi\)
0.707821 + 0.706392i \(0.249678\pi\)
\(32\) −22.6525 −4.00443
\(33\) 0 0
\(34\) 7.51138 1.28819
\(35\) 1.15662 0.195504
\(36\) 0 0
\(37\) 8.43648 1.38695 0.693475 0.720481i \(-0.256079\pi\)
0.693475 + 0.720481i \(0.256079\pi\)
\(38\) −9.99040 −1.62066
\(39\) 0 0
\(40\) 5.80115 0.917242
\(41\) −8.93210 −1.39496 −0.697480 0.716604i \(-0.745695\pi\)
−0.697480 + 0.716604i \(0.745695\pi\)
\(42\) 0 0
\(43\) −4.51816 −0.689013 −0.344507 0.938784i \(-0.611954\pi\)
−0.344507 + 0.938784i \(0.611954\pi\)
\(44\) −5.50285 −0.829586
\(45\) 0 0
\(46\) 4.43462 0.653848
\(47\) 2.48579 0.362590 0.181295 0.983429i \(-0.441971\pi\)
0.181295 + 0.983429i \(0.441971\pi\)
\(48\) 0 0
\(49\) −3.34050 −0.477214
\(50\) 12.6943 1.79525
\(51\) 0 0
\(52\) 0.973317 0.134975
\(53\) 7.57327 1.04027 0.520135 0.854084i \(-0.325882\pi\)
0.520135 + 0.854084i \(0.325882\pi\)
\(54\) 0 0
\(55\) 0.604616 0.0815264
\(56\) 18.3546 2.45274
\(57\) 0 0
\(58\) −20.7037 −2.71852
\(59\) −1.33758 −0.174138 −0.0870692 0.996202i \(-0.527750\pi\)
−0.0870692 + 0.996202i \(0.527750\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037
\(62\) −21.5897 −2.74190
\(63\) 0 0
\(64\) 31.4969 3.93711
\(65\) −0.106941 −0.0132645
\(66\) 0 0
\(67\) −5.50196 −0.672171 −0.336086 0.941831i \(-0.609103\pi\)
−0.336086 + 0.941831i \(0.609103\pi\)
\(68\) −15.0902 −1.82995
\(69\) 0 0
\(70\) −3.16813 −0.378665
\(71\) −0.330471 −0.0392197 −0.0196098 0.999808i \(-0.506242\pi\)
−0.0196098 + 0.999808i \(0.506242\pi\)
\(72\) 0 0
\(73\) −1.75560 −0.205477 −0.102739 0.994708i \(-0.532761\pi\)
−0.102739 + 0.994708i \(0.532761\pi\)
\(74\) −23.1086 −2.68633
\(75\) 0 0
\(76\) 20.0705 2.30224
\(77\) 1.91298 0.218005
\(78\) 0 0
\(79\) −12.9780 −1.46014 −0.730070 0.683372i \(-0.760513\pi\)
−0.730070 + 0.683372i \(0.760513\pi\)
\(80\) −9.23590 −1.03260
\(81\) 0 0
\(82\) 24.4662 2.70184
\(83\) −12.9300 −1.41925 −0.709625 0.704580i \(-0.751136\pi\)
−0.709625 + 0.704580i \(0.751136\pi\)
\(84\) 0 0
\(85\) 1.65801 0.179836
\(86\) 12.3759 1.33452
\(87\) 0 0
\(88\) 9.59477 1.02281
\(89\) 4.11061 0.435724 0.217862 0.975980i \(-0.430092\pi\)
0.217862 + 0.975980i \(0.430092\pi\)
\(90\) 0 0
\(91\) −0.338359 −0.0354697
\(92\) −8.90903 −0.928831
\(93\) 0 0
\(94\) −6.80891 −0.702285
\(95\) −2.20521 −0.226249
\(96\) 0 0
\(97\) −12.7524 −1.29481 −0.647404 0.762147i \(-0.724145\pi\)
−0.647404 + 0.762147i \(0.724145\pi\)
\(98\) 9.15006 0.924296
\(99\) 0 0
\(100\) −25.5026 −2.55026
\(101\) 10.0353 0.998553 0.499276 0.866443i \(-0.333599\pi\)
0.499276 + 0.866443i \(0.333599\pi\)
\(102\) 0 0
\(103\) −6.05237 −0.596357 −0.298179 0.954510i \(-0.596379\pi\)
−0.298179 + 0.954510i \(0.596379\pi\)
\(104\) −1.69708 −0.166412
\(105\) 0 0
\(106\) −20.7442 −2.01485
\(107\) −12.2595 −1.18517 −0.592587 0.805506i \(-0.701893\pi\)
−0.592587 + 0.805506i \(0.701893\pi\)
\(108\) 0 0
\(109\) −1.38440 −0.132602 −0.0663008 0.997800i \(-0.521120\pi\)
−0.0663008 + 0.997800i \(0.521120\pi\)
\(110\) −1.65612 −0.157905
\(111\) 0 0
\(112\) −29.2221 −2.76123
\(113\) −12.6874 −1.19353 −0.596765 0.802416i \(-0.703548\pi\)
−0.596765 + 0.802416i \(0.703548\pi\)
\(114\) 0 0
\(115\) 0.978864 0.0912795
\(116\) 41.5931 3.86183
\(117\) 0 0
\(118\) 3.66382 0.337282
\(119\) 5.24587 0.480888
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 2.73913 0.247989
\(123\) 0 0
\(124\) 43.3732 3.89503
\(125\) 5.82513 0.521016
\(126\) 0 0
\(127\) 7.42636 0.658983 0.329491 0.944159i \(-0.393123\pi\)
0.329491 + 0.944159i \(0.393123\pi\)
\(128\) −40.9692 −3.62120
\(129\) 0 0
\(130\) 0.292927 0.0256914
\(131\) 7.69981 0.672735 0.336368 0.941731i \(-0.390802\pi\)
0.336368 + 0.941731i \(0.390802\pi\)
\(132\) 0 0
\(133\) −6.97719 −0.604999
\(134\) 15.0706 1.30190
\(135\) 0 0
\(136\) 26.3112 2.25617
\(137\) 15.9210 1.36022 0.680112 0.733108i \(-0.261931\pi\)
0.680112 + 0.733108i \(0.261931\pi\)
\(138\) 0 0
\(139\) −2.40561 −0.204041 −0.102021 0.994782i \(-0.532531\pi\)
−0.102021 + 0.994782i \(0.532531\pi\)
\(140\) 6.36470 0.537916
\(141\) 0 0
\(142\) 0.905204 0.0759630
\(143\) −0.176875 −0.0147910
\(144\) 0 0
\(145\) −4.56997 −0.379516
\(146\) 4.80881 0.397980
\(147\) 0 0
\(148\) 46.4247 3.81609
\(149\) 14.5339 1.19066 0.595331 0.803480i \(-0.297021\pi\)
0.595331 + 0.803480i \(0.297021\pi\)
\(150\) 0 0
\(151\) −10.9430 −0.890525 −0.445262 0.895400i \(-0.646890\pi\)
−0.445262 + 0.895400i \(0.646890\pi\)
\(152\) −34.9949 −2.83846
\(153\) 0 0
\(154\) −5.23991 −0.422244
\(155\) −4.76556 −0.382779
\(156\) 0 0
\(157\) 16.6497 1.32879 0.664394 0.747383i \(-0.268690\pi\)
0.664394 + 0.747383i \(0.268690\pi\)
\(158\) 35.5485 2.82809
\(159\) 0 0
\(160\) 13.6961 1.08277
\(161\) 3.09709 0.244085
\(162\) 0 0
\(163\) −2.05976 −0.161333 −0.0806666 0.996741i \(-0.525705\pi\)
−0.0806666 + 0.996741i \(0.525705\pi\)
\(164\) −49.1520 −3.83813
\(165\) 0 0
\(166\) 35.4169 2.74889
\(167\) 15.3296 1.18624 0.593121 0.805113i \(-0.297895\pi\)
0.593121 + 0.805113i \(0.297895\pi\)
\(168\) 0 0
\(169\) −12.9687 −0.997593
\(170\) −4.54150 −0.348317
\(171\) 0 0
\(172\) −24.8628 −1.89577
\(173\) −13.0360 −0.991105 −0.495553 0.868578i \(-0.665034\pi\)
−0.495553 + 0.868578i \(0.665034\pi\)
\(174\) 0 0
\(175\) 8.86560 0.670177
\(176\) −15.2757 −1.15145
\(177\) 0 0
\(178\) −11.2595 −0.843936
\(179\) 11.6608 0.871571 0.435786 0.900050i \(-0.356471\pi\)
0.435786 + 0.900050i \(0.356471\pi\)
\(180\) 0 0
\(181\) −9.58234 −0.712250 −0.356125 0.934438i \(-0.615902\pi\)
−0.356125 + 0.934438i \(0.615902\pi\)
\(182\) 0.926810 0.0686997
\(183\) 0 0
\(184\) 15.5338 1.14517
\(185\) −5.10083 −0.375021
\(186\) 0 0
\(187\) 2.74225 0.200533
\(188\) 13.6789 0.997638
\(189\) 0 0
\(190\) 6.04035 0.438213
\(191\) 18.4200 1.33283 0.666414 0.745582i \(-0.267828\pi\)
0.666414 + 0.745582i \(0.267828\pi\)
\(192\) 0 0
\(193\) −5.40862 −0.389321 −0.194661 0.980871i \(-0.562361\pi\)
−0.194661 + 0.980871i \(0.562361\pi\)
\(194\) 34.9305 2.50786
\(195\) 0 0
\(196\) −18.3822 −1.31302
\(197\) 2.86614 0.204204 0.102102 0.994774i \(-0.467443\pi\)
0.102102 + 0.994774i \(0.467443\pi\)
\(198\) 0 0
\(199\) 11.9550 0.847469 0.423735 0.905786i \(-0.360719\pi\)
0.423735 + 0.905786i \(0.360719\pi\)
\(200\) 44.4664 3.14425
\(201\) 0 0
\(202\) −27.4881 −1.93406
\(203\) −14.4592 −1.01484
\(204\) 0 0
\(205\) 5.40049 0.377187
\(206\) 16.5782 1.15506
\(207\) 0 0
\(208\) 2.70188 0.187342
\(209\) −3.64728 −0.252288
\(210\) 0 0
\(211\) 24.3432 1.67585 0.837927 0.545783i \(-0.183768\pi\)
0.837927 + 0.545783i \(0.183768\pi\)
\(212\) 41.6746 2.86222
\(213\) 0 0
\(214\) 33.5805 2.29552
\(215\) 2.73175 0.186304
\(216\) 0 0
\(217\) −15.0781 −1.02357
\(218\) 3.79206 0.256831
\(219\) 0 0
\(220\) 3.32711 0.224314
\(221\) −0.485035 −0.0326270
\(222\) 0 0
\(223\) 15.7257 1.05307 0.526535 0.850154i \(-0.323491\pi\)
0.526535 + 0.850154i \(0.323491\pi\)
\(224\) 43.3338 2.89536
\(225\) 0 0
\(226\) 34.7525 2.31170
\(227\) 0.677571 0.0449720 0.0224860 0.999747i \(-0.492842\pi\)
0.0224860 + 0.999747i \(0.492842\pi\)
\(228\) 0 0
\(229\) 13.4531 0.889006 0.444503 0.895777i \(-0.353380\pi\)
0.444503 + 0.895777i \(0.353380\pi\)
\(230\) −2.68124 −0.176796
\(231\) 0 0
\(232\) −72.5218 −4.76129
\(233\) 12.1025 0.792862 0.396431 0.918064i \(-0.370249\pi\)
0.396431 + 0.918064i \(0.370249\pi\)
\(234\) 0 0
\(235\) −1.50295 −0.0980415
\(236\) −7.36052 −0.479129
\(237\) 0 0
\(238\) −14.3691 −0.931413
\(239\) 7.34142 0.474877 0.237438 0.971403i \(-0.423692\pi\)
0.237438 + 0.971403i \(0.423692\pi\)
\(240\) 0 0
\(241\) −26.9723 −1.73744 −0.868719 0.495306i \(-0.835056\pi\)
−0.868719 + 0.495306i \(0.835056\pi\)
\(242\) −2.73913 −0.176078
\(243\) 0 0
\(244\) −5.50285 −0.352284
\(245\) 2.01972 0.129035
\(246\) 0 0
\(247\) 0.645114 0.0410476
\(248\) −75.6256 −4.80223
\(249\) 0 0
\(250\) −15.9558 −1.00913
\(251\) 17.0405 1.07559 0.537794 0.843076i \(-0.319258\pi\)
0.537794 + 0.843076i \(0.319258\pi\)
\(252\) 0 0
\(253\) 1.61899 0.101785
\(254\) −20.3418 −1.27636
\(255\) 0 0
\(256\) 49.2263 3.07664
\(257\) −13.6010 −0.848410 −0.424205 0.905566i \(-0.639446\pi\)
−0.424205 + 0.905566i \(0.639446\pi\)
\(258\) 0 0
\(259\) −16.1388 −1.00282
\(260\) −0.588483 −0.0364961
\(261\) 0 0
\(262\) −21.0908 −1.30299
\(263\) −18.2508 −1.12539 −0.562696 0.826664i \(-0.690236\pi\)
−0.562696 + 0.826664i \(0.690236\pi\)
\(264\) 0 0
\(265\) −4.57892 −0.281281
\(266\) 19.1115 1.17180
\(267\) 0 0
\(268\) −30.2764 −1.84943
\(269\) 9.66539 0.589309 0.294655 0.955604i \(-0.404795\pi\)
0.294655 + 0.955604i \(0.404795\pi\)
\(270\) 0 0
\(271\) 30.1115 1.82914 0.914571 0.404425i \(-0.132528\pi\)
0.914571 + 0.404425i \(0.132528\pi\)
\(272\) −41.8896 −2.53993
\(273\) 0 0
\(274\) −43.6098 −2.63456
\(275\) 4.63444 0.279467
\(276\) 0 0
\(277\) −5.89926 −0.354452 −0.177226 0.984170i \(-0.556712\pi\)
−0.177226 + 0.984170i \(0.556712\pi\)
\(278\) 6.58929 0.395199
\(279\) 0 0
\(280\) −11.0975 −0.663202
\(281\) −6.40992 −0.382384 −0.191192 0.981553i \(-0.561235\pi\)
−0.191192 + 0.981553i \(0.561235\pi\)
\(282\) 0 0
\(283\) 2.51457 0.149476 0.0747380 0.997203i \(-0.476188\pi\)
0.0747380 + 0.997203i \(0.476188\pi\)
\(284\) −1.81853 −0.107910
\(285\) 0 0
\(286\) 0.484484 0.0286482
\(287\) 17.0870 1.00861
\(288\) 0 0
\(289\) −9.48008 −0.557652
\(290\) 12.5178 0.735068
\(291\) 0 0
\(292\) −9.66079 −0.565355
\(293\) 12.3742 0.722908 0.361454 0.932390i \(-0.382280\pi\)
0.361454 + 0.932390i \(0.382280\pi\)
\(294\) 0 0
\(295\) 0.808724 0.0470857
\(296\) −80.9461 −4.70490
\(297\) 0 0
\(298\) −39.8102 −2.30615
\(299\) −0.286358 −0.0165605
\(300\) 0 0
\(301\) 8.64317 0.498184
\(302\) 29.9742 1.72482
\(303\) 0 0
\(304\) 55.7146 3.19545
\(305\) 0.604616 0.0346202
\(306\) 0 0
\(307\) 21.2440 1.21246 0.606230 0.795290i \(-0.292681\pi\)
0.606230 + 0.795290i \(0.292681\pi\)
\(308\) 10.5269 0.599823
\(309\) 0 0
\(310\) 13.0535 0.741389
\(311\) −26.4545 −1.50010 −0.750049 0.661383i \(-0.769970\pi\)
−0.750049 + 0.661383i \(0.769970\pi\)
\(312\) 0 0
\(313\) 30.7427 1.73768 0.868841 0.495090i \(-0.164865\pi\)
0.868841 + 0.495090i \(0.164865\pi\)
\(314\) −45.6056 −2.57367
\(315\) 0 0
\(316\) −71.4160 −4.01747
\(317\) 21.9554 1.23314 0.616569 0.787301i \(-0.288522\pi\)
0.616569 + 0.787301i \(0.288522\pi\)
\(318\) 0 0
\(319\) −7.55848 −0.423193
\(320\) −19.0435 −1.06457
\(321\) 0 0
\(322\) −8.48334 −0.472758
\(323\) −10.0018 −0.556513
\(324\) 0 0
\(325\) −0.819717 −0.0454697
\(326\) 5.64197 0.312480
\(327\) 0 0
\(328\) 85.7015 4.73207
\(329\) −4.75527 −0.262167
\(330\) 0 0
\(331\) −15.8159 −0.869318 −0.434659 0.900595i \(-0.643131\pi\)
−0.434659 + 0.900595i \(0.643131\pi\)
\(332\) −71.1517 −3.90496
\(333\) 0 0
\(334\) −41.9899 −2.29759
\(335\) 3.32657 0.181750
\(336\) 0 0
\(337\) −33.1532 −1.80597 −0.902986 0.429671i \(-0.858630\pi\)
−0.902986 + 0.429671i \(0.858630\pi\)
\(338\) 35.5230 1.93220
\(339\) 0 0
\(340\) 9.12376 0.494805
\(341\) −7.88196 −0.426832
\(342\) 0 0
\(343\) 19.7812 1.06808
\(344\) 43.3507 2.33732
\(345\) 0 0
\(346\) 35.7072 1.91963
\(347\) −23.0402 −1.23686 −0.618430 0.785840i \(-0.712231\pi\)
−0.618430 + 0.785840i \(0.712231\pi\)
\(348\) 0 0
\(349\) −32.7544 −1.75330 −0.876651 0.481127i \(-0.840227\pi\)
−0.876651 + 0.481127i \(0.840227\pi\)
\(350\) −24.2841 −1.29804
\(351\) 0 0
\(352\) 22.6525 1.20738
\(353\) −0.349353 −0.0185942 −0.00929708 0.999957i \(-0.502959\pi\)
−0.00929708 + 0.999957i \(0.502959\pi\)
\(354\) 0 0
\(355\) 0.199808 0.0106047
\(356\) 22.6201 1.19886
\(357\) 0 0
\(358\) −31.9406 −1.68811
\(359\) −29.2479 −1.54365 −0.771823 0.635838i \(-0.780655\pi\)
−0.771823 + 0.635838i \(0.780655\pi\)
\(360\) 0 0
\(361\) −5.69731 −0.299859
\(362\) 26.2473 1.37953
\(363\) 0 0
\(364\) −1.86194 −0.0975921
\(365\) 1.06146 0.0555594
\(366\) 0 0
\(367\) −13.9617 −0.728795 −0.364397 0.931244i \(-0.618725\pi\)
−0.364397 + 0.931244i \(0.618725\pi\)
\(368\) −24.7311 −1.28920
\(369\) 0 0
\(370\) 13.9719 0.726362
\(371\) −14.4875 −0.752156
\(372\) 0 0
\(373\) 13.7140 0.710084 0.355042 0.934850i \(-0.384467\pi\)
0.355042 + 0.934850i \(0.384467\pi\)
\(374\) −7.51138 −0.388404
\(375\) 0 0
\(376\) −23.8506 −1.23000
\(377\) 1.33691 0.0688541
\(378\) 0 0
\(379\) 34.6378 1.77922 0.889612 0.456717i \(-0.150975\pi\)
0.889612 + 0.456717i \(0.150975\pi\)
\(380\) −12.1349 −0.622508
\(381\) 0 0
\(382\) −50.4550 −2.58150
\(383\) −7.28622 −0.372308 −0.186154 0.982521i \(-0.559602\pi\)
−0.186154 + 0.982521i \(0.559602\pi\)
\(384\) 0 0
\(385\) −1.15662 −0.0589468
\(386\) 14.8149 0.754060
\(387\) 0 0
\(388\) −70.1744 −3.56257
\(389\) 30.6669 1.55487 0.777437 0.628960i \(-0.216519\pi\)
0.777437 + 0.628960i \(0.216519\pi\)
\(390\) 0 0
\(391\) 4.43966 0.224523
\(392\) 32.0513 1.61883
\(393\) 0 0
\(394\) −7.85074 −0.395514
\(395\) 7.84671 0.394811
\(396\) 0 0
\(397\) −1.12627 −0.0565257 −0.0282628 0.999601i \(-0.508998\pi\)
−0.0282628 + 0.999601i \(0.508998\pi\)
\(398\) −32.7464 −1.64143
\(399\) 0 0
\(400\) −70.7941 −3.53970
\(401\) 9.38421 0.468625 0.234312 0.972161i \(-0.424716\pi\)
0.234312 + 0.972161i \(0.424716\pi\)
\(402\) 0 0
\(403\) 1.39412 0.0694462
\(404\) 55.2229 2.74744
\(405\) 0 0
\(406\) 39.6058 1.96560
\(407\) −8.43648 −0.418181
\(408\) 0 0
\(409\) 34.3783 1.69990 0.849948 0.526867i \(-0.176633\pi\)
0.849948 + 0.526867i \(0.176633\pi\)
\(410\) −14.7927 −0.730558
\(411\) 0 0
\(412\) −33.3053 −1.64083
\(413\) 2.55877 0.125909
\(414\) 0 0
\(415\) 7.81767 0.383754
\(416\) −4.00666 −0.196443
\(417\) 0 0
\(418\) 9.99040 0.488646
\(419\) 1.53252 0.0748685 0.0374343 0.999299i \(-0.488082\pi\)
0.0374343 + 0.999299i \(0.488082\pi\)
\(420\) 0 0
\(421\) 13.4022 0.653185 0.326593 0.945165i \(-0.394100\pi\)
0.326593 + 0.945165i \(0.394100\pi\)
\(422\) −66.6792 −3.24589
\(423\) 0 0
\(424\) −72.6638 −3.52887
\(425\) 12.7088 0.616466
\(426\) 0 0
\(427\) 1.91298 0.0925757
\(428\) −67.4624 −3.26092
\(429\) 0 0
\(430\) −7.48263 −0.360845
\(431\) 4.01874 0.193576 0.0967880 0.995305i \(-0.469143\pi\)
0.0967880 + 0.995305i \(0.469143\pi\)
\(432\) 0 0
\(433\) 19.2933 0.927175 0.463587 0.886051i \(-0.346562\pi\)
0.463587 + 0.886051i \(0.346562\pi\)
\(434\) 41.3008 1.98250
\(435\) 0 0
\(436\) −7.61815 −0.364843
\(437\) −5.90490 −0.282470
\(438\) 0 0
\(439\) −23.1285 −1.10386 −0.551930 0.833890i \(-0.686109\pi\)
−0.551930 + 0.833890i \(0.686109\pi\)
\(440\) −5.80115 −0.276559
\(441\) 0 0
\(442\) 1.32858 0.0631939
\(443\) 18.3618 0.872398 0.436199 0.899850i \(-0.356324\pi\)
0.436199 + 0.899850i \(0.356324\pi\)
\(444\) 0 0
\(445\) −2.48534 −0.117816
\(446\) −43.0747 −2.03965
\(447\) 0 0
\(448\) −60.2530 −2.84669
\(449\) 12.2717 0.579135 0.289568 0.957158i \(-0.406488\pi\)
0.289568 + 0.957158i \(0.406488\pi\)
\(450\) 0 0
\(451\) 8.93210 0.420596
\(452\) −69.8168 −3.28391
\(453\) 0 0
\(454\) −1.85596 −0.0871044
\(455\) 0.204577 0.00959073
\(456\) 0 0
\(457\) 17.9930 0.841679 0.420840 0.907135i \(-0.361736\pi\)
0.420840 + 0.907135i \(0.361736\pi\)
\(458\) −36.8498 −1.72188
\(459\) 0 0
\(460\) 5.38654 0.251149
\(461\) 1.26483 0.0589088 0.0294544 0.999566i \(-0.490623\pi\)
0.0294544 + 0.999566i \(0.490623\pi\)
\(462\) 0 0
\(463\) −10.3872 −0.482732 −0.241366 0.970434i \(-0.577595\pi\)
−0.241366 + 0.970434i \(0.577595\pi\)
\(464\) 115.461 5.36012
\(465\) 0 0
\(466\) −33.1504 −1.53566
\(467\) −10.9780 −0.507999 −0.253999 0.967204i \(-0.581746\pi\)
−0.253999 + 0.967204i \(0.581746\pi\)
\(468\) 0 0
\(469\) 10.5252 0.486006
\(470\) 4.11677 0.189893
\(471\) 0 0
\(472\) 12.8338 0.590724
\(473\) 4.51816 0.207745
\(474\) 0 0
\(475\) −16.9031 −0.775568
\(476\) 28.8672 1.32313
\(477\) 0 0
\(478\) −20.1091 −0.919770
\(479\) 17.8446 0.815339 0.407670 0.913130i \(-0.366342\pi\)
0.407670 + 0.913130i \(0.366342\pi\)
\(480\) 0 0
\(481\) 1.49220 0.0680386
\(482\) 73.8807 3.36517
\(483\) 0 0
\(484\) 5.50285 0.250130
\(485\) 7.71029 0.350106
\(486\) 0 0
\(487\) −19.6964 −0.892531 −0.446266 0.894901i \(-0.647246\pi\)
−0.446266 + 0.894901i \(0.647246\pi\)
\(488\) 9.59477 0.434335
\(489\) 0 0
\(490\) −5.53227 −0.249923
\(491\) 17.3750 0.784124 0.392062 0.919939i \(-0.371762\pi\)
0.392062 + 0.919939i \(0.371762\pi\)
\(492\) 0 0
\(493\) −20.7272 −0.933507
\(494\) −1.76705 −0.0795034
\(495\) 0 0
\(496\) 120.402 5.40621
\(497\) 0.632185 0.0283574
\(498\) 0 0
\(499\) 25.1532 1.12601 0.563005 0.826454i \(-0.309645\pi\)
0.563005 + 0.826454i \(0.309645\pi\)
\(500\) 32.0548 1.43354
\(501\) 0 0
\(502\) −46.6763 −2.08326
\(503\) −18.0309 −0.803958 −0.401979 0.915649i \(-0.631678\pi\)
−0.401979 + 0.915649i \(0.631678\pi\)
\(504\) 0 0
\(505\) −6.06752 −0.270001
\(506\) −4.43462 −0.197143
\(507\) 0 0
\(508\) 40.8661 1.81314
\(509\) 35.8893 1.59077 0.795383 0.606108i \(-0.207270\pi\)
0.795383 + 0.606108i \(0.207270\pi\)
\(510\) 0 0
\(511\) 3.35843 0.148568
\(512\) −52.8990 −2.33783
\(513\) 0 0
\(514\) 37.2551 1.64325
\(515\) 3.65936 0.161250
\(516\) 0 0
\(517\) −2.48579 −0.109325
\(518\) 44.2065 1.94232
\(519\) 0 0
\(520\) 1.02608 0.0449965
\(521\) −6.16062 −0.269902 −0.134951 0.990852i \(-0.543088\pi\)
−0.134951 + 0.990852i \(0.543088\pi\)
\(522\) 0 0
\(523\) −2.39141 −0.104569 −0.0522846 0.998632i \(-0.516650\pi\)
−0.0522846 + 0.998632i \(0.516650\pi\)
\(524\) 42.3709 1.85098
\(525\) 0 0
\(526\) 49.9914 2.17973
\(527\) −21.6143 −0.941533
\(528\) 0 0
\(529\) −20.3789 −0.886039
\(530\) 12.5423 0.544801
\(531\) 0 0
\(532\) −38.3944 −1.66461
\(533\) −1.57987 −0.0684316
\(534\) 0 0
\(535\) 7.41231 0.320462
\(536\) 52.7900 2.28018
\(537\) 0 0
\(538\) −26.4748 −1.14141
\(539\) 3.34050 0.143885
\(540\) 0 0
\(541\) 30.9269 1.32965 0.664827 0.746998i \(-0.268505\pi\)
0.664827 + 0.746998i \(0.268505\pi\)
\(542\) −82.4793 −3.54279
\(543\) 0 0
\(544\) 62.1188 2.66332
\(545\) 0.837031 0.0358545
\(546\) 0 0
\(547\) 31.4601 1.34514 0.672569 0.740035i \(-0.265191\pi\)
0.672569 + 0.740035i \(0.265191\pi\)
\(548\) 87.6109 3.74255
\(549\) 0 0
\(550\) −12.6943 −0.541289
\(551\) 27.5679 1.17443
\(552\) 0 0
\(553\) 24.8267 1.05574
\(554\) 16.1589 0.686524
\(555\) 0 0
\(556\) −13.2377 −0.561404
\(557\) 43.3173 1.83541 0.917706 0.397261i \(-0.130039\pi\)
0.917706 + 0.397261i \(0.130039\pi\)
\(558\) 0 0
\(559\) −0.799150 −0.0338005
\(560\) 17.6681 0.746614
\(561\) 0 0
\(562\) 17.5576 0.740623
\(563\) 12.7937 0.539192 0.269596 0.962974i \(-0.413110\pi\)
0.269596 + 0.962974i \(0.413110\pi\)
\(564\) 0 0
\(565\) 7.67100 0.322721
\(566\) −6.88775 −0.289514
\(567\) 0 0
\(568\) 3.17079 0.133044
\(569\) −16.2156 −0.679791 −0.339896 0.940463i \(-0.610392\pi\)
−0.339896 + 0.940463i \(0.610392\pi\)
\(570\) 0 0
\(571\) 15.4511 0.646607 0.323303 0.946295i \(-0.395207\pi\)
0.323303 + 0.946295i \(0.395207\pi\)
\(572\) −0.973317 −0.0406964
\(573\) 0 0
\(574\) −46.8035 −1.95354
\(575\) 7.50309 0.312900
\(576\) 0 0
\(577\) −13.7967 −0.574365 −0.287182 0.957876i \(-0.592719\pi\)
−0.287182 + 0.957876i \(0.592719\pi\)
\(578\) 25.9672 1.08009
\(579\) 0 0
\(580\) −25.1479 −1.04421
\(581\) 24.7348 1.02617
\(582\) 0 0
\(583\) −7.57327 −0.313653
\(584\) 16.8445 0.697032
\(585\) 0 0
\(586\) −33.8946 −1.40017
\(587\) 0.731957 0.0302111 0.0151056 0.999886i \(-0.495192\pi\)
0.0151056 + 0.999886i \(0.495192\pi\)
\(588\) 0 0
\(589\) 28.7478 1.18453
\(590\) −2.21520 −0.0911984
\(591\) 0 0
\(592\) 128.873 5.29664
\(593\) −16.9597 −0.696450 −0.348225 0.937411i \(-0.613215\pi\)
−0.348225 + 0.937411i \(0.613215\pi\)
\(594\) 0 0
\(595\) −3.17174 −0.130029
\(596\) 79.9778 3.27602
\(597\) 0 0
\(598\) 0.784373 0.0320754
\(599\) 11.3334 0.463070 0.231535 0.972827i \(-0.425625\pi\)
0.231535 + 0.972827i \(0.425625\pi\)
\(600\) 0 0
\(601\) 37.8868 1.54543 0.772717 0.634750i \(-0.218897\pi\)
0.772717 + 0.634750i \(0.218897\pi\)
\(602\) −23.6748 −0.964912
\(603\) 0 0
\(604\) −60.2174 −2.45021
\(605\) −0.604616 −0.0245811
\(606\) 0 0
\(607\) −9.55272 −0.387733 −0.193866 0.981028i \(-0.562103\pi\)
−0.193866 + 0.981028i \(0.562103\pi\)
\(608\) −82.6201 −3.35069
\(609\) 0 0
\(610\) −1.65612 −0.0670545
\(611\) 0.439674 0.0177873
\(612\) 0 0
\(613\) 18.7337 0.756646 0.378323 0.925674i \(-0.376501\pi\)
0.378323 + 0.925674i \(0.376501\pi\)
\(614\) −58.1902 −2.34836
\(615\) 0 0
\(616\) −18.3546 −0.739529
\(617\) 2.88614 0.116192 0.0580958 0.998311i \(-0.481497\pi\)
0.0580958 + 0.998311i \(0.481497\pi\)
\(618\) 0 0
\(619\) 28.4797 1.14470 0.572348 0.820011i \(-0.306033\pi\)
0.572348 + 0.820011i \(0.306033\pi\)
\(620\) −26.2241 −1.05319
\(621\) 0 0
\(622\) 72.4624 2.90548
\(623\) −7.86353 −0.315046
\(624\) 0 0
\(625\) 19.6502 0.786009
\(626\) −84.2085 −3.36565
\(627\) 0 0
\(628\) 91.6205 3.65606
\(629\) −23.1349 −0.922450
\(630\) 0 0
\(631\) 18.8981 0.752322 0.376161 0.926554i \(-0.377244\pi\)
0.376161 + 0.926554i \(0.377244\pi\)
\(632\) 124.521 4.95318
\(633\) 0 0
\(634\) −60.1388 −2.38842
\(635\) −4.49009 −0.178184
\(636\) 0 0
\(637\) −0.590850 −0.0234103
\(638\) 20.7037 0.819666
\(639\) 0 0
\(640\) 24.7706 0.979145
\(641\) −29.7753 −1.17605 −0.588026 0.808842i \(-0.700095\pi\)
−0.588026 + 0.808842i \(0.700095\pi\)
\(642\) 0 0
\(643\) −1.68829 −0.0665798 −0.0332899 0.999446i \(-0.510598\pi\)
−0.0332899 + 0.999446i \(0.510598\pi\)
\(644\) 17.0428 0.671581
\(645\) 0 0
\(646\) 27.3961 1.07789
\(647\) 31.7652 1.24882 0.624409 0.781098i \(-0.285340\pi\)
0.624409 + 0.781098i \(0.285340\pi\)
\(648\) 0 0
\(649\) 1.33758 0.0525047
\(650\) 2.24531 0.0880684
\(651\) 0 0
\(652\) −11.3346 −0.443896
\(653\) 12.0681 0.472261 0.236130 0.971721i \(-0.424121\pi\)
0.236130 + 0.971721i \(0.424121\pi\)
\(654\) 0 0
\(655\) −4.65542 −0.181902
\(656\) −136.444 −5.32723
\(657\) 0 0
\(658\) 13.0253 0.507780
\(659\) −17.5896 −0.685192 −0.342596 0.939483i \(-0.611306\pi\)
−0.342596 + 0.939483i \(0.611306\pi\)
\(660\) 0 0
\(661\) 18.9318 0.736361 0.368181 0.929754i \(-0.379981\pi\)
0.368181 + 0.929754i \(0.379981\pi\)
\(662\) 43.3217 1.68375
\(663\) 0 0
\(664\) 124.060 4.81447
\(665\) 4.21852 0.163587
\(666\) 0 0
\(667\) −12.2371 −0.473821
\(668\) 84.3567 3.26386
\(669\) 0 0
\(670\) −9.11192 −0.352024
\(671\) 1.00000 0.0386046
\(672\) 0 0
\(673\) 49.3060 1.90061 0.950303 0.311325i \(-0.100773\pi\)
0.950303 + 0.311325i \(0.100773\pi\)
\(674\) 90.8111 3.49791
\(675\) 0 0
\(676\) −71.3649 −2.74480
\(677\) −6.96645 −0.267742 −0.133871 0.990999i \(-0.542741\pi\)
−0.133871 + 0.990999i \(0.542741\pi\)
\(678\) 0 0
\(679\) 24.3951 0.936198
\(680\) −15.9082 −0.610051
\(681\) 0 0
\(682\) 21.5897 0.826714
\(683\) −17.4032 −0.665913 −0.332957 0.942942i \(-0.608046\pi\)
−0.332957 + 0.942942i \(0.608046\pi\)
\(684\) 0 0
\(685\) −9.62609 −0.367794
\(686\) −54.1833 −2.06873
\(687\) 0 0
\(688\) −69.0179 −2.63128
\(689\) 1.33952 0.0510318
\(690\) 0 0
\(691\) −16.3254 −0.621048 −0.310524 0.950566i \(-0.600505\pi\)
−0.310524 + 0.950566i \(0.600505\pi\)
\(692\) −71.7349 −2.72695
\(693\) 0 0
\(694\) 63.1100 2.39562
\(695\) 1.45447 0.0551712
\(696\) 0 0
\(697\) 24.4940 0.927778
\(698\) 89.7186 3.39590
\(699\) 0 0
\(700\) 48.7861 1.84394
\(701\) 1.25848 0.0475322 0.0237661 0.999718i \(-0.492434\pi\)
0.0237661 + 0.999718i \(0.492434\pi\)
\(702\) 0 0
\(703\) 30.7703 1.16052
\(704\) −31.4969 −1.18708
\(705\) 0 0
\(706\) 0.956923 0.0360143
\(707\) −19.1974 −0.721993
\(708\) 0 0
\(709\) 44.8435 1.68413 0.842067 0.539373i \(-0.181339\pi\)
0.842067 + 0.539373i \(0.181339\pi\)
\(710\) −0.547301 −0.0205398
\(711\) 0 0
\(712\) −39.4404 −1.47809
\(713\) −12.7608 −0.477895
\(714\) 0 0
\(715\) 0.106941 0.00399938
\(716\) 64.1678 2.39806
\(717\) 0 0
\(718\) 80.1139 2.98982
\(719\) −39.7590 −1.48276 −0.741381 0.671085i \(-0.765829\pi\)
−0.741381 + 0.671085i \(0.765829\pi\)
\(720\) 0 0
\(721\) 11.5781 0.431190
\(722\) 15.6057 0.580784
\(723\) 0 0
\(724\) −52.7302 −1.95970
\(725\) −35.0293 −1.30096
\(726\) 0 0
\(727\) 39.6673 1.47118 0.735589 0.677428i \(-0.236905\pi\)
0.735589 + 0.677428i \(0.236905\pi\)
\(728\) 3.24648 0.120322
\(729\) 0 0
\(730\) −2.90748 −0.107611
\(731\) 12.3899 0.458258
\(732\) 0 0
\(733\) −4.45224 −0.164447 −0.0822235 0.996614i \(-0.526202\pi\)
−0.0822235 + 0.996614i \(0.526202\pi\)
\(734\) 38.2430 1.41157
\(735\) 0 0
\(736\) 36.6741 1.35182
\(737\) 5.50196 0.202667
\(738\) 0 0
\(739\) −33.8922 −1.24674 −0.623372 0.781926i \(-0.714238\pi\)
−0.623372 + 0.781926i \(0.714238\pi\)
\(740\) −28.0691 −1.03184
\(741\) 0 0
\(742\) 39.6833 1.45682
\(743\) 9.90914 0.363531 0.181766 0.983342i \(-0.441819\pi\)
0.181766 + 0.983342i \(0.441819\pi\)
\(744\) 0 0
\(745\) −8.78742 −0.321946
\(746\) −37.5645 −1.37533
\(747\) 0 0
\(748\) 15.0902 0.551751
\(749\) 23.4523 0.856928
\(750\) 0 0
\(751\) 3.30472 0.120591 0.0602955 0.998181i \(-0.480796\pi\)
0.0602955 + 0.998181i \(0.480796\pi\)
\(752\) 37.9720 1.38470
\(753\) 0 0
\(754\) −3.66196 −0.133361
\(755\) 6.61628 0.240791
\(756\) 0 0
\(757\) 13.4977 0.490583 0.245292 0.969449i \(-0.421116\pi\)
0.245292 + 0.969449i \(0.421116\pi\)
\(758\) −94.8776 −3.44611
\(759\) 0 0
\(760\) 21.1584 0.767497
\(761\) 28.0515 1.01687 0.508433 0.861101i \(-0.330225\pi\)
0.508433 + 0.861101i \(0.330225\pi\)
\(762\) 0 0
\(763\) 2.64834 0.0958762
\(764\) 101.363 3.66718
\(765\) 0 0
\(766\) 19.9579 0.721109
\(767\) −0.236585 −0.00854259
\(768\) 0 0
\(769\) −29.0212 −1.04653 −0.523266 0.852170i \(-0.675286\pi\)
−0.523266 + 0.852170i \(0.675286\pi\)
\(770\) 3.16813 0.114172
\(771\) 0 0
\(772\) −29.7628 −1.07119
\(773\) 18.5820 0.668350 0.334175 0.942511i \(-0.391542\pi\)
0.334175 + 0.942511i \(0.391542\pi\)
\(774\) 0 0
\(775\) −36.5285 −1.31214
\(776\) 122.356 4.39233
\(777\) 0 0
\(778\) −84.0008 −3.01157
\(779\) −32.5779 −1.16723
\(780\) 0 0
\(781\) 0.330471 0.0118252
\(782\) −12.1608 −0.434870
\(783\) 0 0
\(784\) −51.0282 −1.82244
\(785\) −10.0666 −0.359294
\(786\) 0 0
\(787\) 34.2557 1.22108 0.610541 0.791985i \(-0.290952\pi\)
0.610541 + 0.791985i \(0.290952\pi\)
\(788\) 15.7719 0.561852
\(789\) 0 0
\(790\) −21.4932 −0.764693
\(791\) 24.2708 0.862969
\(792\) 0 0
\(793\) −0.176875 −0.00628102
\(794\) 3.08499 0.109482
\(795\) 0 0
\(796\) 65.7867 2.33175
\(797\) 11.0127 0.390088 0.195044 0.980794i \(-0.437515\pi\)
0.195044 + 0.980794i \(0.437515\pi\)
\(798\) 0 0
\(799\) −6.81665 −0.241156
\(800\) 104.982 3.71166
\(801\) 0 0
\(802\) −25.7046 −0.907660
\(803\) 1.75560 0.0619537
\(804\) 0 0
\(805\) −1.87255 −0.0659987
\(806\) −3.81869 −0.134507
\(807\) 0 0
\(808\) −96.2867 −3.38735
\(809\) −45.2337 −1.59033 −0.795165 0.606392i \(-0.792616\pi\)
−0.795165 + 0.606392i \(0.792616\pi\)
\(810\) 0 0
\(811\) 32.9688 1.15769 0.578846 0.815437i \(-0.303503\pi\)
0.578846 + 0.815437i \(0.303503\pi\)
\(812\) −79.5670 −2.79225
\(813\) 0 0
\(814\) 23.1086 0.809958
\(815\) 1.24537 0.0436232
\(816\) 0 0
\(817\) −16.4790 −0.576528
\(818\) −94.1666 −3.29246
\(819\) 0 0
\(820\) 29.7181 1.03780
\(821\) −17.6348 −0.615460 −0.307730 0.951474i \(-0.599569\pi\)
−0.307730 + 0.951474i \(0.599569\pi\)
\(822\) 0 0
\(823\) −6.60202 −0.230132 −0.115066 0.993358i \(-0.536708\pi\)
−0.115066 + 0.993358i \(0.536708\pi\)
\(824\) 58.0711 2.02300
\(825\) 0 0
\(826\) −7.00882 −0.243868
\(827\) 0.843089 0.0293171 0.0146585 0.999893i \(-0.495334\pi\)
0.0146585 + 0.999893i \(0.495334\pi\)
\(828\) 0 0
\(829\) −40.4340 −1.40433 −0.702166 0.712014i \(-0.747783\pi\)
−0.702166 + 0.712014i \(0.747783\pi\)
\(830\) −21.4136 −0.743278
\(831\) 0 0
\(832\) 5.57102 0.193140
\(833\) 9.16047 0.317391
\(834\) 0 0
\(835\) −9.26854 −0.320751
\(836\) −20.0705 −0.694151
\(837\) 0 0
\(838\) −4.19778 −0.145010
\(839\) −49.0996 −1.69511 −0.847553 0.530710i \(-0.821925\pi\)
−0.847553 + 0.530710i \(0.821925\pi\)
\(840\) 0 0
\(841\) 28.1305 0.970019
\(842\) −36.7105 −1.26513
\(843\) 0 0
\(844\) 133.957 4.61098
\(845\) 7.84109 0.269742
\(846\) 0 0
\(847\) −1.91298 −0.0657309
\(848\) 115.687 3.97270
\(849\) 0 0
\(850\) −34.8110 −1.19401
\(851\) −13.6585 −0.468209
\(852\) 0 0
\(853\) 35.0517 1.20015 0.600074 0.799944i \(-0.295138\pi\)
0.600074 + 0.799944i \(0.295138\pi\)
\(854\) −5.23991 −0.179306
\(855\) 0 0
\(856\) 117.627 4.02042
\(857\) −51.8553 −1.77134 −0.885672 0.464311i \(-0.846302\pi\)
−0.885672 + 0.464311i \(0.846302\pi\)
\(858\) 0 0
\(859\) −31.2016 −1.06458 −0.532292 0.846561i \(-0.678669\pi\)
−0.532292 + 0.846561i \(0.678669\pi\)
\(860\) 15.0324 0.512601
\(861\) 0 0
\(862\) −11.0079 −0.374929
\(863\) 21.2847 0.724539 0.362269 0.932073i \(-0.382002\pi\)
0.362269 + 0.932073i \(0.382002\pi\)
\(864\) 0 0
\(865\) 7.88174 0.267987
\(866\) −52.8468 −1.79581
\(867\) 0 0
\(868\) −82.9723 −2.81626
\(869\) 12.9780 0.440249
\(870\) 0 0
\(871\) −0.973159 −0.0329742
\(872\) 13.2830 0.449819
\(873\) 0 0
\(874\) 16.1743 0.547104
\(875\) −11.1434 −0.376715
\(876\) 0 0
\(877\) 56.4130 1.90493 0.952465 0.304647i \(-0.0985386\pi\)
0.952465 + 0.304647i \(0.0985386\pi\)
\(878\) 63.3519 2.13802
\(879\) 0 0
\(880\) 9.23590 0.311342
\(881\) 13.2697 0.447067 0.223534 0.974696i \(-0.428241\pi\)
0.223534 + 0.974696i \(0.428241\pi\)
\(882\) 0 0
\(883\) −30.1064 −1.01316 −0.506581 0.862192i \(-0.669091\pi\)
−0.506581 + 0.862192i \(0.669091\pi\)
\(884\) −2.66908 −0.0897707
\(885\) 0 0
\(886\) −50.2955 −1.68971
\(887\) −7.10163 −0.238449 −0.119225 0.992867i \(-0.538041\pi\)
−0.119225 + 0.992867i \(0.538041\pi\)
\(888\) 0 0
\(889\) −14.2065 −0.476471
\(890\) 6.80768 0.228194
\(891\) 0 0
\(892\) 86.5360 2.89744
\(893\) 9.06638 0.303395
\(894\) 0 0
\(895\) −7.05032 −0.235666
\(896\) 78.3734 2.61827
\(897\) 0 0
\(898\) −33.6137 −1.12170
\(899\) 59.5756 1.98696
\(900\) 0 0
\(901\) −20.7678 −0.691875
\(902\) −24.4662 −0.814636
\(903\) 0 0
\(904\) 121.733 4.04877
\(905\) 5.79363 0.192587
\(906\) 0 0
\(907\) 24.7879 0.823069 0.411535 0.911394i \(-0.364993\pi\)
0.411535 + 0.911394i \(0.364993\pi\)
\(908\) 3.72857 0.123737
\(909\) 0 0
\(910\) −0.560364 −0.0185759
\(911\) −41.3254 −1.36917 −0.684586 0.728932i \(-0.740017\pi\)
−0.684586 + 0.728932i \(0.740017\pi\)
\(912\) 0 0
\(913\) 12.9300 0.427920
\(914\) −49.2853 −1.63021
\(915\) 0 0
\(916\) 74.0304 2.44603
\(917\) −14.7296 −0.486414
\(918\) 0 0
\(919\) 36.7229 1.21138 0.605688 0.795702i \(-0.292898\pi\)
0.605688 + 0.795702i \(0.292898\pi\)
\(920\) −9.39197 −0.309644
\(921\) 0 0
\(922\) −3.46453 −0.114098
\(923\) −0.0584521 −0.00192397
\(924\) 0 0
\(925\) −39.0984 −1.28555
\(926\) 28.4518 0.934984
\(927\) 0 0
\(928\) −171.218 −5.62052
\(929\) −1.56310 −0.0512837 −0.0256418 0.999671i \(-0.508163\pi\)
−0.0256418 + 0.999671i \(0.508163\pi\)
\(930\) 0 0
\(931\) −12.1837 −0.399306
\(932\) 66.5983 2.18150
\(933\) 0 0
\(934\) 30.0701 0.983923
\(935\) −1.65801 −0.0542226
\(936\) 0 0
\(937\) 3.58638 0.117162 0.0585810 0.998283i \(-0.481342\pi\)
0.0585810 + 0.998283i \(0.481342\pi\)
\(938\) −28.8298 −0.941326
\(939\) 0 0
\(940\) −8.27049 −0.269754
\(941\) 41.9408 1.36723 0.683616 0.729842i \(-0.260407\pi\)
0.683616 + 0.729842i \(0.260407\pi\)
\(942\) 0 0
\(943\) 14.4609 0.470913
\(944\) −20.4325 −0.665020
\(945\) 0 0
\(946\) −12.3759 −0.402374
\(947\) −53.9167 −1.75206 −0.876028 0.482260i \(-0.839816\pi\)
−0.876028 + 0.482260i \(0.839816\pi\)
\(948\) 0 0
\(949\) −0.310521 −0.0100799
\(950\) 46.2999 1.50217
\(951\) 0 0
\(952\) −50.3329 −1.63130
\(953\) −6.24884 −0.202420 −0.101210 0.994865i \(-0.532271\pi\)
−0.101210 + 0.994865i \(0.532271\pi\)
\(954\) 0 0
\(955\) −11.1371 −0.360387
\(956\) 40.3987 1.30659
\(957\) 0 0
\(958\) −48.8786 −1.57920
\(959\) −30.4566 −0.983496
\(960\) 0 0
\(961\) 31.1253 1.00404
\(962\) −4.08734 −0.131781
\(963\) 0 0
\(964\) −148.424 −4.78043
\(965\) 3.27014 0.105269
\(966\) 0 0
\(967\) −11.7133 −0.376674 −0.188337 0.982104i \(-0.560310\pi\)
−0.188337 + 0.982104i \(0.560310\pi\)
\(968\) −9.59477 −0.308388
\(969\) 0 0
\(970\) −21.1195 −0.678107
\(971\) 20.8971 0.670620 0.335310 0.942108i \(-0.391159\pi\)
0.335310 + 0.942108i \(0.391159\pi\)
\(972\) 0 0
\(973\) 4.60189 0.147530
\(974\) 53.9512 1.72871
\(975\) 0 0
\(976\) −15.2757 −0.488962
\(977\) −15.7320 −0.503311 −0.251655 0.967817i \(-0.580975\pi\)
−0.251655 + 0.967817i \(0.580975\pi\)
\(978\) 0 0
\(979\) −4.11061 −0.131376
\(980\) 11.1142 0.355030
\(981\) 0 0
\(982\) −47.5925 −1.51874
\(983\) 40.6563 1.29674 0.648368 0.761327i \(-0.275452\pi\)
0.648368 + 0.761327i \(0.275452\pi\)
\(984\) 0 0
\(985\) −1.73291 −0.0552152
\(986\) 56.7746 1.80807
\(987\) 0 0
\(988\) 3.54996 0.112939
\(989\) 7.31484 0.232598
\(990\) 0 0
\(991\) 13.9691 0.443742 0.221871 0.975076i \(-0.428784\pi\)
0.221871 + 0.975076i \(0.428784\pi\)
\(992\) −178.546 −5.66885
\(993\) 0 0
\(994\) −1.73164 −0.0549243
\(995\) −7.22820 −0.229149
\(996\) 0 0
\(997\) −11.2639 −0.356731 −0.178365 0.983964i \(-0.557081\pi\)
−0.178365 + 0.983964i \(0.557081\pi\)
\(998\) −68.8978 −2.18092
\(999\) 0 0
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 6039.2.a.i.1.1 13
3.2 odd 2 2013.2.a.e.1.13 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2013.2.a.e.1.13 13 3.2 odd 2
6039.2.a.i.1.1 13 1.1 even 1 trivial