Properties

Label 6008.2.a.b.1.39
Level 6008
Weight 2
Character 6008.1
Self dual yes
Analytic conductor 47.974
Analytic rank 1
Dimension 44
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6008 = 2^{3} \cdot 751 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6008.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(47.9741215344\)
Analytic rank: \(1\)
Dimension: \(44\)
Coefficient ring index: multiple of None
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.39
Character \(\chi\) = 6008.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.94041 q^{3} +0.760740 q^{5} -0.889256 q^{7} +0.765188 q^{9} +O(q^{10})\) \(q+1.94041 q^{3} +0.760740 q^{5} -0.889256 q^{7} +0.765188 q^{9} +5.58913 q^{11} -2.54312 q^{13} +1.47615 q^{15} -2.48725 q^{17} -7.62571 q^{19} -1.72552 q^{21} -5.94283 q^{23} -4.42127 q^{25} -4.33645 q^{27} -0.846897 q^{29} +4.09083 q^{31} +10.8452 q^{33} -0.676492 q^{35} -1.60686 q^{37} -4.93470 q^{39} -3.27310 q^{41} -5.80394 q^{43} +0.582109 q^{45} -5.13868 q^{47} -6.20922 q^{49} -4.82627 q^{51} -3.26645 q^{53} +4.25188 q^{55} -14.7970 q^{57} +2.72175 q^{59} +0.913771 q^{61} -0.680448 q^{63} -1.93466 q^{65} +8.89055 q^{67} -11.5315 q^{69} -0.635627 q^{71} +1.47952 q^{73} -8.57908 q^{75} -4.97017 q^{77} +9.39602 q^{79} -10.7101 q^{81} -5.77044 q^{83} -1.89215 q^{85} -1.64333 q^{87} -11.2838 q^{89} +2.26149 q^{91} +7.93789 q^{93} -5.80118 q^{95} -1.62968 q^{97} +4.27674 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 44q - 14q^{3} + 7q^{5} - 20q^{7} + 38q^{9} + O(q^{10}) \) \( 44q - 14q^{3} + 7q^{5} - 20q^{7} + 38q^{9} - 19q^{11} - 10q^{13} - 17q^{15} - 16q^{17} - 25q^{19} + 16q^{21} - 29q^{23} + 29q^{25} - 50q^{27} + 35q^{29} - 49q^{31} - 28q^{33} - 37q^{35} - 30q^{37} - 28q^{39} - 14q^{41} - 35q^{43} + 6q^{45} - 45q^{47} + 20q^{49} - 17q^{51} + 18q^{53} - 53q^{55} - 31q^{57} - 57q^{59} + 27q^{61} - 77q^{63} - 21q^{65} - 56q^{67} + 36q^{69} - 52q^{71} - 68q^{73} - 77q^{75} + 37q^{77} - 55q^{79} + 28q^{81} - 51q^{83} - 16q^{85} - 67q^{87} - 21q^{89} - 51q^{91} - 14q^{93} - 56q^{95} - 67q^{97} - 58q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.94041 1.12030 0.560148 0.828393i \(-0.310744\pi\)
0.560148 + 0.828393i \(0.310744\pi\)
\(4\) 0 0
\(5\) 0.760740 0.340213 0.170107 0.985426i \(-0.445589\pi\)
0.170107 + 0.985426i \(0.445589\pi\)
\(6\) 0 0
\(7\) −0.889256 −0.336107 −0.168054 0.985778i \(-0.553748\pi\)
−0.168054 + 0.985778i \(0.553748\pi\)
\(8\) 0 0
\(9\) 0.765188 0.255063
\(10\) 0 0
\(11\) 5.58913 1.68519 0.842594 0.538550i \(-0.181028\pi\)
0.842594 + 0.538550i \(0.181028\pi\)
\(12\) 0 0
\(13\) −2.54312 −0.705336 −0.352668 0.935749i \(-0.614725\pi\)
−0.352668 + 0.935749i \(0.614725\pi\)
\(14\) 0 0
\(15\) 1.47615 0.381140
\(16\) 0 0
\(17\) −2.48725 −0.603246 −0.301623 0.953427i \(-0.597528\pi\)
−0.301623 + 0.953427i \(0.597528\pi\)
\(18\) 0 0
\(19\) −7.62571 −1.74946 −0.874728 0.484613i \(-0.838960\pi\)
−0.874728 + 0.484613i \(0.838960\pi\)
\(20\) 0 0
\(21\) −1.72552 −0.376539
\(22\) 0 0
\(23\) −5.94283 −1.23917 −0.619583 0.784931i \(-0.712698\pi\)
−0.619583 + 0.784931i \(0.712698\pi\)
\(24\) 0 0
\(25\) −4.42127 −0.884255
\(26\) 0 0
\(27\) −4.33645 −0.834550
\(28\) 0 0
\(29\) −0.846897 −0.157265 −0.0786325 0.996904i \(-0.525055\pi\)
−0.0786325 + 0.996904i \(0.525055\pi\)
\(30\) 0 0
\(31\) 4.09083 0.734735 0.367368 0.930076i \(-0.380259\pi\)
0.367368 + 0.930076i \(0.380259\pi\)
\(32\) 0 0
\(33\) 10.8452 1.88791
\(34\) 0 0
\(35\) −0.676492 −0.114348
\(36\) 0 0
\(37\) −1.60686 −0.264166 −0.132083 0.991239i \(-0.542167\pi\)
−0.132083 + 0.991239i \(0.542167\pi\)
\(38\) 0 0
\(39\) −4.93470 −0.790185
\(40\) 0 0
\(41\) −3.27310 −0.511173 −0.255586 0.966786i \(-0.582269\pi\)
−0.255586 + 0.966786i \(0.582269\pi\)
\(42\) 0 0
\(43\) −5.80394 −0.885093 −0.442547 0.896746i \(-0.645925\pi\)
−0.442547 + 0.896746i \(0.645925\pi\)
\(44\) 0 0
\(45\) 0.582109 0.0867757
\(46\) 0 0
\(47\) −5.13868 −0.749553 −0.374777 0.927115i \(-0.622281\pi\)
−0.374777 + 0.927115i \(0.622281\pi\)
\(48\) 0 0
\(49\) −6.20922 −0.887032
\(50\) 0 0
\(51\) −4.82627 −0.675814
\(52\) 0 0
\(53\) −3.26645 −0.448681 −0.224341 0.974511i \(-0.572023\pi\)
−0.224341 + 0.974511i \(0.572023\pi\)
\(54\) 0 0
\(55\) 4.25188 0.573323
\(56\) 0 0
\(57\) −14.7970 −1.95991
\(58\) 0 0
\(59\) 2.72175 0.354342 0.177171 0.984180i \(-0.443305\pi\)
0.177171 + 0.984180i \(0.443305\pi\)
\(60\) 0 0
\(61\) 0.913771 0.116996 0.0584982 0.998288i \(-0.481369\pi\)
0.0584982 + 0.998288i \(0.481369\pi\)
\(62\) 0 0
\(63\) −0.680448 −0.0857284
\(64\) 0 0
\(65\) −1.93466 −0.239965
\(66\) 0 0
\(67\) 8.89055 1.08615 0.543077 0.839683i \(-0.317259\pi\)
0.543077 + 0.839683i \(0.317259\pi\)
\(68\) 0 0
\(69\) −11.5315 −1.38823
\(70\) 0 0
\(71\) −0.635627 −0.0754350 −0.0377175 0.999288i \(-0.512009\pi\)
−0.0377175 + 0.999288i \(0.512009\pi\)
\(72\) 0 0
\(73\) 1.47952 0.173165 0.0865825 0.996245i \(-0.472405\pi\)
0.0865825 + 0.996245i \(0.472405\pi\)
\(74\) 0 0
\(75\) −8.57908 −0.990627
\(76\) 0 0
\(77\) −4.97017 −0.566403
\(78\) 0 0
\(79\) 9.39602 1.05714 0.528568 0.848891i \(-0.322729\pi\)
0.528568 + 0.848891i \(0.322729\pi\)
\(80\) 0 0
\(81\) −10.7101 −1.19001
\(82\) 0 0
\(83\) −5.77044 −0.633388 −0.316694 0.948528i \(-0.602573\pi\)
−0.316694 + 0.948528i \(0.602573\pi\)
\(84\) 0 0
\(85\) −1.89215 −0.205232
\(86\) 0 0
\(87\) −1.64333 −0.176183
\(88\) 0 0
\(89\) −11.2838 −1.19608 −0.598041 0.801466i \(-0.704054\pi\)
−0.598041 + 0.801466i \(0.704054\pi\)
\(90\) 0 0
\(91\) 2.26149 0.237068
\(92\) 0 0
\(93\) 7.93789 0.823121
\(94\) 0 0
\(95\) −5.80118 −0.595189
\(96\) 0 0
\(97\) −1.62968 −0.165469 −0.0827346 0.996572i \(-0.526365\pi\)
−0.0827346 + 0.996572i \(0.526365\pi\)
\(98\) 0 0
\(99\) 4.27674 0.429828
\(100\) 0 0
\(101\) −1.98318 −0.197333 −0.0986667 0.995121i \(-0.531458\pi\)
−0.0986667 + 0.995121i \(0.531458\pi\)
\(102\) 0 0
\(103\) −3.93741 −0.387964 −0.193982 0.981005i \(-0.562140\pi\)
−0.193982 + 0.981005i \(0.562140\pi\)
\(104\) 0 0
\(105\) −1.31267 −0.128104
\(106\) 0 0
\(107\) 8.39359 0.811439 0.405720 0.913998i \(-0.367021\pi\)
0.405720 + 0.913998i \(0.367021\pi\)
\(108\) 0 0
\(109\) −3.58941 −0.343803 −0.171901 0.985114i \(-0.554991\pi\)
−0.171901 + 0.985114i \(0.554991\pi\)
\(110\) 0 0
\(111\) −3.11796 −0.295944
\(112\) 0 0
\(113\) −6.02379 −0.566671 −0.283335 0.959021i \(-0.591441\pi\)
−0.283335 + 0.959021i \(0.591441\pi\)
\(114\) 0 0
\(115\) −4.52095 −0.421581
\(116\) 0 0
\(117\) −1.94597 −0.179905
\(118\) 0 0
\(119\) 2.21180 0.202755
\(120\) 0 0
\(121\) 20.2384 1.83986
\(122\) 0 0
\(123\) −6.35116 −0.572665
\(124\) 0 0
\(125\) −7.16714 −0.641049
\(126\) 0 0
\(127\) 0.546205 0.0484678 0.0242339 0.999706i \(-0.492285\pi\)
0.0242339 + 0.999706i \(0.492285\pi\)
\(128\) 0 0
\(129\) −11.2620 −0.991566
\(130\) 0 0
\(131\) 7.90479 0.690645 0.345322 0.938484i \(-0.387770\pi\)
0.345322 + 0.938484i \(0.387770\pi\)
\(132\) 0 0
\(133\) 6.78120 0.588005
\(134\) 0 0
\(135\) −3.29891 −0.283925
\(136\) 0 0
\(137\) 17.4011 1.48667 0.743336 0.668918i \(-0.233242\pi\)
0.743336 + 0.668918i \(0.233242\pi\)
\(138\) 0 0
\(139\) −3.16276 −0.268262 −0.134131 0.990964i \(-0.542824\pi\)
−0.134131 + 0.990964i \(0.542824\pi\)
\(140\) 0 0
\(141\) −9.97114 −0.839721
\(142\) 0 0
\(143\) −14.2139 −1.18862
\(144\) 0 0
\(145\) −0.644269 −0.0535036
\(146\) 0 0
\(147\) −12.0484 −0.993738
\(148\) 0 0
\(149\) 24.3175 1.99217 0.996085 0.0883961i \(-0.0281741\pi\)
0.996085 + 0.0883961i \(0.0281741\pi\)
\(150\) 0 0
\(151\) 21.7037 1.76622 0.883112 0.469163i \(-0.155444\pi\)
0.883112 + 0.469163i \(0.155444\pi\)
\(152\) 0 0
\(153\) −1.90321 −0.153866
\(154\) 0 0
\(155\) 3.11206 0.249967
\(156\) 0 0
\(157\) 5.62629 0.449027 0.224514 0.974471i \(-0.427921\pi\)
0.224514 + 0.974471i \(0.427921\pi\)
\(158\) 0 0
\(159\) −6.33825 −0.502656
\(160\) 0 0
\(161\) 5.28469 0.416492
\(162\) 0 0
\(163\) −0.216070 −0.0169239 −0.00846197 0.999964i \(-0.502694\pi\)
−0.00846197 + 0.999964i \(0.502694\pi\)
\(164\) 0 0
\(165\) 8.25038 0.642291
\(166\) 0 0
\(167\) 7.16553 0.554485 0.277243 0.960800i \(-0.410579\pi\)
0.277243 + 0.960800i \(0.410579\pi\)
\(168\) 0 0
\(169\) −6.53252 −0.502502
\(170\) 0 0
\(171\) −5.83510 −0.446221
\(172\) 0 0
\(173\) −11.1074 −0.844478 −0.422239 0.906485i \(-0.638756\pi\)
−0.422239 + 0.906485i \(0.638756\pi\)
\(174\) 0 0
\(175\) 3.93164 0.297204
\(176\) 0 0
\(177\) 5.28131 0.396968
\(178\) 0 0
\(179\) −15.7388 −1.17637 −0.588187 0.808725i \(-0.700158\pi\)
−0.588187 + 0.808725i \(0.700158\pi\)
\(180\) 0 0
\(181\) 16.3363 1.21427 0.607135 0.794598i \(-0.292319\pi\)
0.607135 + 0.794598i \(0.292319\pi\)
\(182\) 0 0
\(183\) 1.77309 0.131071
\(184\) 0 0
\(185\) −1.22240 −0.0898728
\(186\) 0 0
\(187\) −13.9015 −1.01658
\(188\) 0 0
\(189\) 3.85621 0.280498
\(190\) 0 0
\(191\) −14.6581 −1.06062 −0.530312 0.847803i \(-0.677925\pi\)
−0.530312 + 0.847803i \(0.677925\pi\)
\(192\) 0 0
\(193\) 11.7349 0.844694 0.422347 0.906434i \(-0.361206\pi\)
0.422347 + 0.906434i \(0.361206\pi\)
\(194\) 0 0
\(195\) −3.75402 −0.268831
\(196\) 0 0
\(197\) 17.2492 1.22895 0.614477 0.788934i \(-0.289367\pi\)
0.614477 + 0.788934i \(0.289367\pi\)
\(198\) 0 0
\(199\) 26.2438 1.86037 0.930186 0.367087i \(-0.119645\pi\)
0.930186 + 0.367087i \(0.119645\pi\)
\(200\) 0 0
\(201\) 17.2513 1.21681
\(202\) 0 0
\(203\) 0.753108 0.0528578
\(204\) 0 0
\(205\) −2.48998 −0.173908
\(206\) 0 0
\(207\) −4.54738 −0.316065
\(208\) 0 0
\(209\) −42.6211 −2.94816
\(210\) 0 0
\(211\) −11.9267 −0.821070 −0.410535 0.911845i \(-0.634658\pi\)
−0.410535 + 0.911845i \(0.634658\pi\)
\(212\) 0 0
\(213\) −1.23338 −0.0845095
\(214\) 0 0
\(215\) −4.41529 −0.301120
\(216\) 0 0
\(217\) −3.63780 −0.246950
\(218\) 0 0
\(219\) 2.87088 0.193996
\(220\) 0 0
\(221\) 6.32537 0.425491
\(222\) 0 0
\(223\) −2.42930 −0.162678 −0.0813390 0.996686i \(-0.525920\pi\)
−0.0813390 + 0.996686i \(0.525920\pi\)
\(224\) 0 0
\(225\) −3.38311 −0.225540
\(226\) 0 0
\(227\) −18.7788 −1.24639 −0.623197 0.782065i \(-0.714167\pi\)
−0.623197 + 0.782065i \(0.714167\pi\)
\(228\) 0 0
\(229\) −0.853762 −0.0564182 −0.0282091 0.999602i \(-0.508980\pi\)
−0.0282091 + 0.999602i \(0.508980\pi\)
\(230\) 0 0
\(231\) −9.64416 −0.634539
\(232\) 0 0
\(233\) −2.78016 −0.182134 −0.0910670 0.995845i \(-0.529028\pi\)
−0.0910670 + 0.995845i \(0.529028\pi\)
\(234\) 0 0
\(235\) −3.90920 −0.255008
\(236\) 0 0
\(237\) 18.2321 1.18430
\(238\) 0 0
\(239\) −18.3807 −1.18895 −0.594475 0.804114i \(-0.702640\pi\)
−0.594475 + 0.804114i \(0.702640\pi\)
\(240\) 0 0
\(241\) 1.39286 0.0897221 0.0448611 0.998993i \(-0.485715\pi\)
0.0448611 + 0.998993i \(0.485715\pi\)
\(242\) 0 0
\(243\) −7.77254 −0.498608
\(244\) 0 0
\(245\) −4.72361 −0.301780
\(246\) 0 0
\(247\) 19.3931 1.23395
\(248\) 0 0
\(249\) −11.1970 −0.709582
\(250\) 0 0
\(251\) −0.00595589 −0.000375933 0 −0.000187966 1.00000i \(-0.500060\pi\)
−0.000187966 1.00000i \(0.500060\pi\)
\(252\) 0 0
\(253\) −33.2153 −2.08823
\(254\) 0 0
\(255\) −3.67154 −0.229921
\(256\) 0 0
\(257\) 0.646536 0.0403298 0.0201649 0.999797i \(-0.493581\pi\)
0.0201649 + 0.999797i \(0.493581\pi\)
\(258\) 0 0
\(259\) 1.42891 0.0887881
\(260\) 0 0
\(261\) −0.648036 −0.0401124
\(262\) 0 0
\(263\) −8.75176 −0.539656 −0.269828 0.962908i \(-0.586967\pi\)
−0.269828 + 0.962908i \(0.586967\pi\)
\(264\) 0 0
\(265\) −2.48492 −0.152647
\(266\) 0 0
\(267\) −21.8952 −1.33997
\(268\) 0 0
\(269\) −24.7747 −1.51054 −0.755270 0.655414i \(-0.772494\pi\)
−0.755270 + 0.655414i \(0.772494\pi\)
\(270\) 0 0
\(271\) 0.947238 0.0575406 0.0287703 0.999586i \(-0.490841\pi\)
0.0287703 + 0.999586i \(0.490841\pi\)
\(272\) 0 0
\(273\) 4.38821 0.265587
\(274\) 0 0
\(275\) −24.7111 −1.49014
\(276\) 0 0
\(277\) 9.33287 0.560758 0.280379 0.959889i \(-0.409540\pi\)
0.280379 + 0.959889i \(0.409540\pi\)
\(278\) 0 0
\(279\) 3.13026 0.187404
\(280\) 0 0
\(281\) 6.54062 0.390181 0.195090 0.980785i \(-0.437500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(282\) 0 0
\(283\) −29.6081 −1.76002 −0.880010 0.474954i \(-0.842465\pi\)
−0.880010 + 0.474954i \(0.842465\pi\)
\(284\) 0 0
\(285\) −11.2567 −0.666787
\(286\) 0 0
\(287\) 2.91063 0.171809
\(288\) 0 0
\(289\) −10.8136 −0.636095
\(290\) 0 0
\(291\) −3.16225 −0.185374
\(292\) 0 0
\(293\) −14.4482 −0.844074 −0.422037 0.906579i \(-0.638685\pi\)
−0.422037 + 0.906579i \(0.638685\pi\)
\(294\) 0 0
\(295\) 2.07055 0.120552
\(296\) 0 0
\(297\) −24.2370 −1.40637
\(298\) 0 0
\(299\) 15.1133 0.874028
\(300\) 0 0
\(301\) 5.16119 0.297486
\(302\) 0 0
\(303\) −3.84817 −0.221072
\(304\) 0 0
\(305\) 0.695142 0.0398037
\(306\) 0 0
\(307\) −0.0600235 −0.00342572 −0.00171286 0.999999i \(-0.500545\pi\)
−0.00171286 + 0.999999i \(0.500545\pi\)
\(308\) 0 0
\(309\) −7.64018 −0.434635
\(310\) 0 0
\(311\) 6.48188 0.367554 0.183777 0.982968i \(-0.441168\pi\)
0.183777 + 0.982968i \(0.441168\pi\)
\(312\) 0 0
\(313\) 6.20051 0.350473 0.175237 0.984526i \(-0.443931\pi\)
0.175237 + 0.984526i \(0.443931\pi\)
\(314\) 0 0
\(315\) −0.517644 −0.0291659
\(316\) 0 0
\(317\) −8.47100 −0.475779 −0.237889 0.971292i \(-0.576456\pi\)
−0.237889 + 0.971292i \(0.576456\pi\)
\(318\) 0 0
\(319\) −4.73342 −0.265021
\(320\) 0 0
\(321\) 16.2870 0.909052
\(322\) 0 0
\(323\) 18.9670 1.05535
\(324\) 0 0
\(325\) 11.2438 0.623696
\(326\) 0 0
\(327\) −6.96492 −0.385161
\(328\) 0 0
\(329\) 4.56960 0.251930
\(330\) 0 0
\(331\) −22.4479 −1.23385 −0.616924 0.787023i \(-0.711621\pi\)
−0.616924 + 0.787023i \(0.711621\pi\)
\(332\) 0 0
\(333\) −1.22955 −0.0673789
\(334\) 0 0
\(335\) 6.76339 0.369524
\(336\) 0 0
\(337\) −33.7773 −1.83997 −0.919984 0.391956i \(-0.871799\pi\)
−0.919984 + 0.391956i \(0.871799\pi\)
\(338\) 0 0
\(339\) −11.6886 −0.634839
\(340\) 0 0
\(341\) 22.8642 1.23817
\(342\) 0 0
\(343\) 11.7464 0.634245
\(344\) 0 0
\(345\) −8.77249 −0.472295
\(346\) 0 0
\(347\) −17.8075 −0.955958 −0.477979 0.878371i \(-0.658630\pi\)
−0.477979 + 0.878371i \(0.658630\pi\)
\(348\) 0 0
\(349\) −4.37841 −0.234371 −0.117185 0.993110i \(-0.537387\pi\)
−0.117185 + 0.993110i \(0.537387\pi\)
\(350\) 0 0
\(351\) 11.0281 0.588638
\(352\) 0 0
\(353\) 28.6094 1.52272 0.761361 0.648328i \(-0.224531\pi\)
0.761361 + 0.648328i \(0.224531\pi\)
\(354\) 0 0
\(355\) −0.483547 −0.0256640
\(356\) 0 0
\(357\) 4.29179 0.227146
\(358\) 0 0
\(359\) 19.6096 1.03495 0.517477 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(360\) 0 0
\(361\) 39.1514 2.06060
\(362\) 0 0
\(363\) 39.2708 2.06118
\(364\) 0 0
\(365\) 1.12553 0.0589130
\(366\) 0 0
\(367\) −18.1778 −0.948871 −0.474436 0.880290i \(-0.657348\pi\)
−0.474436 + 0.880290i \(0.657348\pi\)
\(368\) 0 0
\(369\) −2.50454 −0.130381
\(370\) 0 0
\(371\) 2.90471 0.150805
\(372\) 0 0
\(373\) −18.6427 −0.965280 −0.482640 0.875819i \(-0.660322\pi\)
−0.482640 + 0.875819i \(0.660322\pi\)
\(374\) 0 0
\(375\) −13.9072 −0.718164
\(376\) 0 0
\(377\) 2.15377 0.110925
\(378\) 0 0
\(379\) 13.3055 0.683458 0.341729 0.939799i \(-0.388987\pi\)
0.341729 + 0.939799i \(0.388987\pi\)
\(380\) 0 0
\(381\) 1.05986 0.0542983
\(382\) 0 0
\(383\) 32.7040 1.67110 0.835548 0.549417i \(-0.185150\pi\)
0.835548 + 0.549417i \(0.185150\pi\)
\(384\) 0 0
\(385\) −3.78101 −0.192698
\(386\) 0 0
\(387\) −4.44111 −0.225754
\(388\) 0 0
\(389\) 3.35251 0.169979 0.0849894 0.996382i \(-0.472914\pi\)
0.0849894 + 0.996382i \(0.472914\pi\)
\(390\) 0 0
\(391\) 14.7813 0.747521
\(392\) 0 0
\(393\) 15.3385 0.773726
\(394\) 0 0
\(395\) 7.14793 0.359651
\(396\) 0 0
\(397\) 37.1617 1.86509 0.932547 0.361049i \(-0.117581\pi\)
0.932547 + 0.361049i \(0.117581\pi\)
\(398\) 0 0
\(399\) 13.1583 0.658739
\(400\) 0 0
\(401\) 16.4617 0.822058 0.411029 0.911622i \(-0.365170\pi\)
0.411029 + 0.911622i \(0.365170\pi\)
\(402\) 0 0
\(403\) −10.4035 −0.518235
\(404\) 0 0
\(405\) −8.14757 −0.404856
\(406\) 0 0
\(407\) −8.98095 −0.445169
\(408\) 0 0
\(409\) 22.7898 1.12688 0.563441 0.826157i \(-0.309477\pi\)
0.563441 + 0.826157i \(0.309477\pi\)
\(410\) 0 0
\(411\) 33.7652 1.66551
\(412\) 0 0
\(413\) −2.42033 −0.119097
\(414\) 0 0
\(415\) −4.38980 −0.215487
\(416\) 0 0
\(417\) −6.13705 −0.300533
\(418\) 0 0
\(419\) −26.4630 −1.29280 −0.646401 0.762997i \(-0.723727\pi\)
−0.646401 + 0.762997i \(0.723727\pi\)
\(420\) 0 0
\(421\) −18.9057 −0.921405 −0.460703 0.887555i \(-0.652403\pi\)
−0.460703 + 0.887555i \(0.652403\pi\)
\(422\) 0 0
\(423\) −3.93205 −0.191183
\(424\) 0 0
\(425\) 10.9968 0.533423
\(426\) 0 0
\(427\) −0.812576 −0.0393233
\(428\) 0 0
\(429\) −27.5807 −1.33161
\(430\) 0 0
\(431\) −40.0800 −1.93059 −0.965294 0.261166i \(-0.915893\pi\)
−0.965294 + 0.261166i \(0.915893\pi\)
\(432\) 0 0
\(433\) −16.2689 −0.781831 −0.390916 0.920427i \(-0.627842\pi\)
−0.390916 + 0.920427i \(0.627842\pi\)
\(434\) 0 0
\(435\) −1.25015 −0.0599399
\(436\) 0 0
\(437\) 45.3183 2.16787
\(438\) 0 0
\(439\) 20.3794 0.972658 0.486329 0.873776i \(-0.338336\pi\)
0.486329 + 0.873776i \(0.338336\pi\)
\(440\) 0 0
\(441\) −4.75123 −0.226249
\(442\) 0 0
\(443\) −28.1415 −1.33704 −0.668522 0.743692i \(-0.733073\pi\)
−0.668522 + 0.743692i \(0.733073\pi\)
\(444\) 0 0
\(445\) −8.58405 −0.406923
\(446\) 0 0
\(447\) 47.1860 2.23182
\(448\) 0 0
\(449\) 28.8287 1.36051 0.680256 0.732975i \(-0.261869\pi\)
0.680256 + 0.732975i \(0.261869\pi\)
\(450\) 0 0
\(451\) −18.2938 −0.861422
\(452\) 0 0
\(453\) 42.1141 1.97869
\(454\) 0 0
\(455\) 1.72040 0.0806538
\(456\) 0 0
\(457\) −28.3695 −1.32707 −0.663534 0.748146i \(-0.730944\pi\)
−0.663534 + 0.748146i \(0.730944\pi\)
\(458\) 0 0
\(459\) 10.7858 0.503439
\(460\) 0 0
\(461\) 8.27357 0.385339 0.192669 0.981264i \(-0.438286\pi\)
0.192669 + 0.981264i \(0.438286\pi\)
\(462\) 0 0
\(463\) 17.4972 0.813164 0.406582 0.913614i \(-0.366720\pi\)
0.406582 + 0.913614i \(0.366720\pi\)
\(464\) 0 0
\(465\) 6.03867 0.280037
\(466\) 0 0
\(467\) −2.72307 −0.126009 −0.0630043 0.998013i \(-0.520068\pi\)
−0.0630043 + 0.998013i \(0.520068\pi\)
\(468\) 0 0
\(469\) −7.90597 −0.365064
\(470\) 0 0
\(471\) 10.9173 0.503043
\(472\) 0 0
\(473\) −32.4390 −1.49155
\(474\) 0 0
\(475\) 33.7153 1.54697
\(476\) 0 0
\(477\) −2.49945 −0.114442
\(478\) 0 0
\(479\) −18.2038 −0.831755 −0.415878 0.909421i \(-0.636526\pi\)
−0.415878 + 0.909421i \(0.636526\pi\)
\(480\) 0 0
\(481\) 4.08644 0.186326
\(482\) 0 0
\(483\) 10.2545 0.466594
\(484\) 0 0
\(485\) −1.23976 −0.0562948
\(486\) 0 0
\(487\) 5.19719 0.235507 0.117754 0.993043i \(-0.462431\pi\)
0.117754 + 0.993043i \(0.462431\pi\)
\(488\) 0 0
\(489\) −0.419265 −0.0189598
\(490\) 0 0
\(491\) 20.8190 0.939550 0.469775 0.882786i \(-0.344335\pi\)
0.469775 + 0.882786i \(0.344335\pi\)
\(492\) 0 0
\(493\) 2.10644 0.0948694
\(494\) 0 0
\(495\) 3.25349 0.146233
\(496\) 0 0
\(497\) 0.565234 0.0253542
\(498\) 0 0
\(499\) −25.1995 −1.12808 −0.564042 0.825746i \(-0.690754\pi\)
−0.564042 + 0.825746i \(0.690754\pi\)
\(500\) 0 0
\(501\) 13.9041 0.621188
\(502\) 0 0
\(503\) −11.4413 −0.510142 −0.255071 0.966922i \(-0.582099\pi\)
−0.255071 + 0.966922i \(0.582099\pi\)
\(504\) 0 0
\(505\) −1.50868 −0.0671354
\(506\) 0 0
\(507\) −12.6758 −0.562951
\(508\) 0 0
\(509\) 7.73511 0.342853 0.171426 0.985197i \(-0.445162\pi\)
0.171426 + 0.985197i \(0.445162\pi\)
\(510\) 0 0
\(511\) −1.31567 −0.0582020
\(512\) 0 0
\(513\) 33.0685 1.46001
\(514\) 0 0
\(515\) −2.99534 −0.131991
\(516\) 0 0
\(517\) −28.7207 −1.26314
\(518\) 0 0
\(519\) −21.5528 −0.946065
\(520\) 0 0
\(521\) −16.0704 −0.704055 −0.352028 0.935990i \(-0.614508\pi\)
−0.352028 + 0.935990i \(0.614508\pi\)
\(522\) 0 0
\(523\) −13.4267 −0.587109 −0.293554 0.955942i \(-0.594838\pi\)
−0.293554 + 0.955942i \(0.594838\pi\)
\(524\) 0 0
\(525\) 7.62900 0.332957
\(526\) 0 0
\(527\) −10.1749 −0.443226
\(528\) 0 0
\(529\) 12.3172 0.535531
\(530\) 0 0
\(531\) 2.08265 0.0903794
\(532\) 0 0
\(533\) 8.32391 0.360548
\(534\) 0 0
\(535\) 6.38534 0.276062
\(536\) 0 0
\(537\) −30.5398 −1.31789
\(538\) 0 0
\(539\) −34.7042 −1.49482
\(540\) 0 0
\(541\) 9.55287 0.410710 0.205355 0.978688i \(-0.434165\pi\)
0.205355 + 0.978688i \(0.434165\pi\)
\(542\) 0 0
\(543\) 31.6992 1.36034
\(544\) 0 0
\(545\) −2.73061 −0.116966
\(546\) 0 0
\(547\) 15.3514 0.656380 0.328190 0.944612i \(-0.393561\pi\)
0.328190 + 0.944612i \(0.393561\pi\)
\(548\) 0 0
\(549\) 0.699207 0.0298414
\(550\) 0 0
\(551\) 6.45819 0.275128
\(552\) 0 0
\(553\) −8.35547 −0.355311
\(554\) 0 0
\(555\) −2.37196 −0.100684
\(556\) 0 0
\(557\) −9.75517 −0.413340 −0.206670 0.978411i \(-0.566263\pi\)
−0.206670 + 0.978411i \(0.566263\pi\)
\(558\) 0 0
\(559\) 14.7601 0.624288
\(560\) 0 0
\(561\) −26.9747 −1.13887
\(562\) 0 0
\(563\) −1.53778 −0.0648095 −0.0324048 0.999475i \(-0.510317\pi\)
−0.0324048 + 0.999475i \(0.510317\pi\)
\(564\) 0 0
\(565\) −4.58254 −0.192789
\(566\) 0 0
\(567\) 9.52397 0.399969
\(568\) 0 0
\(569\) 2.94761 0.123570 0.0617851 0.998089i \(-0.480321\pi\)
0.0617851 + 0.998089i \(0.480321\pi\)
\(570\) 0 0
\(571\) 7.30486 0.305699 0.152849 0.988249i \(-0.451155\pi\)
0.152849 + 0.988249i \(0.451155\pi\)
\(572\) 0 0
\(573\) −28.4427 −1.18821
\(574\) 0 0
\(575\) 26.2749 1.09574
\(576\) 0 0
\(577\) −25.3558 −1.05557 −0.527787 0.849377i \(-0.676978\pi\)
−0.527787 + 0.849377i \(0.676978\pi\)
\(578\) 0 0
\(579\) 22.7704 0.946307
\(580\) 0 0
\(581\) 5.13139 0.212886
\(582\) 0 0
\(583\) −18.2566 −0.756112
\(584\) 0 0
\(585\) −1.48038 −0.0612060
\(586\) 0 0
\(587\) −12.7695 −0.527054 −0.263527 0.964652i \(-0.584886\pi\)
−0.263527 + 0.964652i \(0.584886\pi\)
\(588\) 0 0
\(589\) −31.1955 −1.28539
\(590\) 0 0
\(591\) 33.4705 1.37679
\(592\) 0 0
\(593\) −20.3188 −0.834394 −0.417197 0.908816i \(-0.636987\pi\)
−0.417197 + 0.908816i \(0.636987\pi\)
\(594\) 0 0
\(595\) 1.68260 0.0689800
\(596\) 0 0
\(597\) 50.9237 2.08417
\(598\) 0 0
\(599\) −25.5588 −1.04430 −0.522152 0.852853i \(-0.674871\pi\)
−0.522152 + 0.852853i \(0.674871\pi\)
\(600\) 0 0
\(601\) 2.34401 0.0956143 0.0478072 0.998857i \(-0.484777\pi\)
0.0478072 + 0.998857i \(0.484777\pi\)
\(602\) 0 0
\(603\) 6.80294 0.277037
\(604\) 0 0
\(605\) 15.3962 0.625943
\(606\) 0 0
\(607\) −11.1210 −0.451386 −0.225693 0.974198i \(-0.572465\pi\)
−0.225693 + 0.974198i \(0.572465\pi\)
\(608\) 0 0
\(609\) 1.46134 0.0592164
\(610\) 0 0
\(611\) 13.0683 0.528686
\(612\) 0 0
\(613\) 26.2733 1.06117 0.530584 0.847632i \(-0.321973\pi\)
0.530584 + 0.847632i \(0.321973\pi\)
\(614\) 0 0
\(615\) −4.83158 −0.194828
\(616\) 0 0
\(617\) 13.3346 0.536831 0.268415 0.963303i \(-0.413500\pi\)
0.268415 + 0.963303i \(0.413500\pi\)
\(618\) 0 0
\(619\) 13.1131 0.527061 0.263531 0.964651i \(-0.415113\pi\)
0.263531 + 0.964651i \(0.415113\pi\)
\(620\) 0 0
\(621\) 25.7708 1.03415
\(622\) 0 0
\(623\) 10.0342 0.402011
\(624\) 0 0
\(625\) 16.6540 0.666162
\(626\) 0 0
\(627\) −82.7024 −3.30281
\(628\) 0 0
\(629\) 3.99665 0.159357
\(630\) 0 0
\(631\) 14.4912 0.576887 0.288444 0.957497i \(-0.406862\pi\)
0.288444 + 0.957497i \(0.406862\pi\)
\(632\) 0 0
\(633\) −23.1427 −0.919841
\(634\) 0 0
\(635\) 0.415520 0.0164894
\(636\) 0 0
\(637\) 15.7908 0.625655
\(638\) 0 0
\(639\) −0.486374 −0.0192407
\(640\) 0 0
\(641\) 40.0623 1.58237 0.791183 0.611579i \(-0.209466\pi\)
0.791183 + 0.611579i \(0.209466\pi\)
\(642\) 0 0
\(643\) −41.1960 −1.62461 −0.812305 0.583232i \(-0.801788\pi\)
−0.812305 + 0.583232i \(0.801788\pi\)
\(644\) 0 0
\(645\) −8.56747 −0.337344
\(646\) 0 0
\(647\) −22.6970 −0.892311 −0.446155 0.894955i \(-0.647207\pi\)
−0.446155 + 0.894955i \(0.647207\pi\)
\(648\) 0 0
\(649\) 15.2122 0.597133
\(650\) 0 0
\(651\) −7.05882 −0.276657
\(652\) 0 0
\(653\) −1.20015 −0.0469656 −0.0234828 0.999724i \(-0.507475\pi\)
−0.0234828 + 0.999724i \(0.507475\pi\)
\(654\) 0 0
\(655\) 6.01349 0.234966
\(656\) 0 0
\(657\) 1.13211 0.0441679
\(658\) 0 0
\(659\) 9.35596 0.364457 0.182228 0.983256i \(-0.441669\pi\)
0.182228 + 0.983256i \(0.441669\pi\)
\(660\) 0 0
\(661\) −46.8278 −1.82139 −0.910696 0.413077i \(-0.864454\pi\)
−0.910696 + 0.413077i \(0.864454\pi\)
\(662\) 0 0
\(663\) 12.2738 0.476675
\(664\) 0 0
\(665\) 5.15873 0.200047
\(666\) 0 0
\(667\) 5.03297 0.194877
\(668\) 0 0
\(669\) −4.71384 −0.182247
\(670\) 0 0
\(671\) 5.10719 0.197161
\(672\) 0 0
\(673\) −26.1467 −1.00788 −0.503940 0.863738i \(-0.668117\pi\)
−0.503940 + 0.863738i \(0.668117\pi\)
\(674\) 0 0
\(675\) 19.1726 0.737955
\(676\) 0 0
\(677\) −13.7646 −0.529016 −0.264508 0.964383i \(-0.585210\pi\)
−0.264508 + 0.964383i \(0.585210\pi\)
\(678\) 0 0
\(679\) 1.44920 0.0556154
\(680\) 0 0
\(681\) −36.4386 −1.39633
\(682\) 0 0
\(683\) −6.85441 −0.262277 −0.131138 0.991364i \(-0.541863\pi\)
−0.131138 + 0.991364i \(0.541863\pi\)
\(684\) 0 0
\(685\) 13.2377 0.505786
\(686\) 0 0
\(687\) −1.65665 −0.0632051
\(688\) 0 0
\(689\) 8.30698 0.316471
\(690\) 0 0
\(691\) −33.9403 −1.29115 −0.645574 0.763697i \(-0.723382\pi\)
−0.645574 + 0.763697i \(0.723382\pi\)
\(692\) 0 0
\(693\) −3.80311 −0.144468
\(694\) 0 0
\(695\) −2.40604 −0.0912663
\(696\) 0 0
\(697\) 8.14101 0.308363
\(698\) 0 0
\(699\) −5.39464 −0.204044
\(700\) 0 0
\(701\) 23.6825 0.894477 0.447239 0.894415i \(-0.352407\pi\)
0.447239 + 0.894415i \(0.352407\pi\)
\(702\) 0 0
\(703\) 12.2534 0.462147
\(704\) 0 0
\(705\) −7.58544 −0.285684
\(706\) 0 0
\(707\) 1.76355 0.0663251
\(708\) 0 0
\(709\) 47.5820 1.78698 0.893490 0.449083i \(-0.148249\pi\)
0.893490 + 0.449083i \(0.148249\pi\)
\(710\) 0 0
\(711\) 7.18973 0.269636
\(712\) 0 0
\(713\) −24.3111 −0.910459
\(714\) 0 0
\(715\) −10.8131 −0.404385
\(716\) 0 0
\(717\) −35.6661 −1.33198
\(718\) 0 0
\(719\) 3.27158 0.122009 0.0610046 0.998137i \(-0.480570\pi\)
0.0610046 + 0.998137i \(0.480570\pi\)
\(720\) 0 0
\(721\) 3.50136 0.130398
\(722\) 0 0
\(723\) 2.70272 0.100515
\(724\) 0 0
\(725\) 3.74437 0.139062
\(726\) 0 0
\(727\) −4.73998 −0.175796 −0.0878981 0.996129i \(-0.528015\pi\)
−0.0878981 + 0.996129i \(0.528015\pi\)
\(728\) 0 0
\(729\) 17.0483 0.631417
\(730\) 0 0
\(731\) 14.4358 0.533929
\(732\) 0 0
\(733\) 10.2577 0.378878 0.189439 0.981892i \(-0.439333\pi\)
0.189439 + 0.981892i \(0.439333\pi\)
\(734\) 0 0
\(735\) −9.16573 −0.338083
\(736\) 0 0
\(737\) 49.6904 1.83037
\(738\) 0 0
\(739\) −7.86962 −0.289489 −0.144744 0.989469i \(-0.546236\pi\)
−0.144744 + 0.989469i \(0.546236\pi\)
\(740\) 0 0
\(741\) 37.6306 1.38239
\(742\) 0 0
\(743\) −5.31215 −0.194884 −0.0974419 0.995241i \(-0.531066\pi\)
−0.0974419 + 0.995241i \(0.531066\pi\)
\(744\) 0 0
\(745\) 18.4993 0.677763
\(746\) 0 0
\(747\) −4.41547 −0.161554
\(748\) 0 0
\(749\) −7.46405 −0.272730
\(750\) 0 0
\(751\) −1.00000 −0.0364905
\(752\) 0 0
\(753\) −0.0115569 −0.000421156 0
\(754\) 0 0
\(755\) 16.5109 0.600893
\(756\) 0 0
\(757\) 32.3429 1.17552 0.587761 0.809035i \(-0.300010\pi\)
0.587761 + 0.809035i \(0.300010\pi\)
\(758\) 0 0
\(759\) −64.4512 −2.33943
\(760\) 0 0
\(761\) 35.6228 1.29133 0.645663 0.763622i \(-0.276581\pi\)
0.645663 + 0.763622i \(0.276581\pi\)
\(762\) 0 0
\(763\) 3.19190 0.115555
\(764\) 0 0
\(765\) −1.44785 −0.0523471
\(766\) 0 0
\(767\) −6.92175 −0.249930
\(768\) 0 0
\(769\) −6.76647 −0.244005 −0.122003 0.992530i \(-0.538932\pi\)
−0.122003 + 0.992530i \(0.538932\pi\)
\(770\) 0 0
\(771\) 1.25454 0.0451813
\(772\) 0 0
\(773\) −23.0179 −0.827897 −0.413949 0.910300i \(-0.635851\pi\)
−0.413949 + 0.910300i \(0.635851\pi\)
\(774\) 0 0
\(775\) −18.0867 −0.649693
\(776\) 0 0
\(777\) 2.77267 0.0994689
\(778\) 0 0
\(779\) 24.9597 0.894275
\(780\) 0 0
\(781\) −3.55260 −0.127122
\(782\) 0 0
\(783\) 3.67253 0.131245
\(784\) 0 0
\(785\) 4.28015 0.152765
\(786\) 0 0
\(787\) 1.19416 0.0425674 0.0212837 0.999773i \(-0.493225\pi\)
0.0212837 + 0.999773i \(0.493225\pi\)
\(788\) 0 0
\(789\) −16.9820 −0.604575
\(790\) 0 0
\(791\) 5.35669 0.190462
\(792\) 0 0
\(793\) −2.32383 −0.0825218
\(794\) 0 0
\(795\) −4.82176 −0.171010
\(796\) 0 0
\(797\) 10.3521 0.366691 0.183346 0.983049i \(-0.441307\pi\)
0.183346 + 0.983049i \(0.441307\pi\)
\(798\) 0 0
\(799\) 12.7812 0.452165
\(800\) 0 0
\(801\) −8.63424 −0.305076
\(802\) 0 0
\(803\) 8.26925 0.291815
\(804\) 0 0
\(805\) 4.02028 0.141696
\(806\) 0 0
\(807\) −48.0731 −1.69225
\(808\) 0 0
\(809\) 42.7756 1.50391 0.751956 0.659213i \(-0.229111\pi\)
0.751956 + 0.659213i \(0.229111\pi\)
\(810\) 0 0
\(811\) −40.0146 −1.40510 −0.702552 0.711632i \(-0.747956\pi\)
−0.702552 + 0.711632i \(0.747956\pi\)
\(812\) 0 0
\(813\) 1.83803 0.0644625
\(814\) 0 0
\(815\) −0.164373 −0.00575775
\(816\) 0 0
\(817\) 44.2592 1.54843
\(818\) 0 0
\(819\) 1.73046 0.0604673
\(820\) 0 0
\(821\) 37.6264 1.31317 0.656585 0.754252i \(-0.272000\pi\)
0.656585 + 0.754252i \(0.272000\pi\)
\(822\) 0 0
\(823\) −8.26833 −0.288216 −0.144108 0.989562i \(-0.546031\pi\)
−0.144108 + 0.989562i \(0.546031\pi\)
\(824\) 0 0
\(825\) −47.9496 −1.66939
\(826\) 0 0
\(827\) −26.4386 −0.919361 −0.459681 0.888084i \(-0.652036\pi\)
−0.459681 + 0.888084i \(0.652036\pi\)
\(828\) 0 0
\(829\) −32.3975 −1.12521 −0.562606 0.826725i \(-0.690201\pi\)
−0.562606 + 0.826725i \(0.690201\pi\)
\(830\) 0 0
\(831\) 18.1096 0.628215
\(832\) 0 0
\(833\) 15.4439 0.535098
\(834\) 0 0
\(835\) 5.45111 0.188643
\(836\) 0 0
\(837\) −17.7397 −0.613174
\(838\) 0 0
\(839\) −11.3566 −0.392072 −0.196036 0.980597i \(-0.562807\pi\)
−0.196036 + 0.980597i \(0.562807\pi\)
\(840\) 0 0
\(841\) −28.2828 −0.975268
\(842\) 0 0
\(843\) 12.6915 0.437118
\(844\) 0 0
\(845\) −4.96955 −0.170958
\(846\) 0 0
\(847\) −17.9971 −0.618388
\(848\) 0 0
\(849\) −57.4519 −1.97174
\(850\) 0 0
\(851\) 9.54929 0.327345
\(852\) 0 0
\(853\) 27.9316 0.956360 0.478180 0.878262i \(-0.341297\pi\)
0.478180 + 0.878262i \(0.341297\pi\)
\(854\) 0 0
\(855\) −4.43899 −0.151810
\(856\) 0 0
\(857\) −19.1162 −0.652996 −0.326498 0.945198i \(-0.605869\pi\)
−0.326498 + 0.945198i \(0.605869\pi\)
\(858\) 0 0
\(859\) −30.0727 −1.02607 −0.513033 0.858369i \(-0.671478\pi\)
−0.513033 + 0.858369i \(0.671478\pi\)
\(860\) 0 0
\(861\) 5.64781 0.192477
\(862\) 0 0
\(863\) −12.9678 −0.441429 −0.220715 0.975338i \(-0.570839\pi\)
−0.220715 + 0.975338i \(0.570839\pi\)
\(864\) 0 0
\(865\) −8.44982 −0.287303
\(866\) 0 0
\(867\) −20.9828 −0.712614
\(868\) 0 0
\(869\) 52.5156 1.78147
\(870\) 0 0
\(871\) −22.6098 −0.766102
\(872\) 0 0
\(873\) −1.24701 −0.0422050
\(874\) 0 0
\(875\) 6.37342 0.215461
\(876\) 0 0
\(877\) 24.3252 0.821402 0.410701 0.911770i \(-0.365284\pi\)
0.410701 + 0.911770i \(0.365284\pi\)
\(878\) 0 0
\(879\) −28.0355 −0.945613
\(880\) 0 0
\(881\) 11.4277 0.385009 0.192505 0.981296i \(-0.438339\pi\)
0.192505 + 0.981296i \(0.438339\pi\)
\(882\) 0 0
\(883\) 1.66153 0.0559150 0.0279575 0.999609i \(-0.491100\pi\)
0.0279575 + 0.999609i \(0.491100\pi\)
\(884\) 0 0
\(885\) 4.01771 0.135054
\(886\) 0 0
\(887\) −37.0453 −1.24386 −0.621929 0.783073i \(-0.713651\pi\)
−0.621929 + 0.783073i \(0.713651\pi\)
\(888\) 0 0
\(889\) −0.485715 −0.0162904
\(890\) 0 0
\(891\) −59.8599 −2.00538
\(892\) 0 0
\(893\) 39.1860 1.31131
\(894\) 0 0
\(895\) −11.9732 −0.400218
\(896\) 0 0
\(897\) 29.3261 0.979169
\(898\) 0 0
\(899\) −3.46452 −0.115548
\(900\) 0 0
\(901\) 8.12446 0.270665
\(902\) 0 0
\(903\) 10.0148 0.333272
\(904\) 0 0
\(905\) 12.4277 0.413111
\(906\) 0 0
\(907\) −40.7204 −1.35210 −0.676050 0.736856i \(-0.736310\pi\)
−0.676050 + 0.736856i \(0.736310\pi\)
\(908\) 0 0
\(909\) −1.51750 −0.0503324
\(910\) 0 0
\(911\) −29.1509 −0.965811 −0.482906 0.875672i \(-0.660419\pi\)
−0.482906 + 0.875672i \(0.660419\pi\)
\(912\) 0 0
\(913\) −32.2517 −1.06738
\(914\) 0 0
\(915\) 1.34886 0.0445920
\(916\) 0 0
\(917\) −7.02938 −0.232131
\(918\) 0 0
\(919\) −12.6637 −0.417736 −0.208868 0.977944i \(-0.566978\pi\)
−0.208868 + 0.977944i \(0.566978\pi\)
\(920\) 0 0
\(921\) −0.116470 −0.00383782
\(922\) 0 0
\(923\) 1.61648 0.0532070
\(924\) 0 0
\(925\) 7.10437 0.233590
\(926\) 0 0
\(927\) −3.01286 −0.0989552
\(928\) 0 0
\(929\) 6.35630 0.208543 0.104272 0.994549i \(-0.466749\pi\)
0.104272 + 0.994549i \(0.466749\pi\)
\(930\) 0 0
\(931\) 47.3497 1.55182
\(932\) 0 0
\(933\) 12.5775 0.411769
\(934\) 0 0
\(935\) −10.5755 −0.345855
\(936\) 0 0
\(937\) 42.0539 1.37384 0.686921 0.726732i \(-0.258962\pi\)
0.686921 + 0.726732i \(0.258962\pi\)
\(938\) 0 0
\(939\) 12.0315 0.392634
\(940\) 0 0
\(941\) 25.0966 0.818124 0.409062 0.912507i \(-0.365856\pi\)
0.409062 + 0.912507i \(0.365856\pi\)
\(942\) 0 0
\(943\) 19.4515 0.633428
\(944\) 0 0
\(945\) 2.93357 0.0954292
\(946\) 0 0
\(947\) 36.0835 1.17256 0.586278 0.810110i \(-0.300593\pi\)
0.586278 + 0.810110i \(0.300593\pi\)
\(948\) 0 0
\(949\) −3.76261 −0.122139
\(950\) 0 0
\(951\) −16.4372 −0.533013
\(952\) 0 0
\(953\) −39.7433 −1.28741 −0.643706 0.765273i \(-0.722604\pi\)
−0.643706 + 0.765273i \(0.722604\pi\)
\(954\) 0 0
\(955\) −11.1510 −0.360838
\(956\) 0 0
\(957\) −9.18478 −0.296902
\(958\) 0 0
\(959\) −15.4740 −0.499681
\(960\) 0 0
\(961\) −14.2651 −0.460164
\(962\) 0 0
\(963\) 6.42268 0.206968
\(964\) 0 0
\(965\) 8.92718 0.287376
\(966\) 0 0
\(967\) −8.76146 −0.281750 −0.140875 0.990027i \(-0.544991\pi\)
−0.140875 + 0.990027i \(0.544991\pi\)
\(968\) 0 0
\(969\) 36.8038 1.18231
\(970\) 0 0
\(971\) 11.5792 0.371593 0.185796 0.982588i \(-0.440514\pi\)
0.185796 + 0.982588i \(0.440514\pi\)
\(972\) 0 0
\(973\) 2.81250 0.0901647
\(974\) 0 0
\(975\) 21.8177 0.698725
\(976\) 0 0
\(977\) 17.7690 0.568479 0.284240 0.958753i \(-0.408259\pi\)
0.284240 + 0.958753i \(0.408259\pi\)
\(978\) 0 0
\(979\) −63.0667 −2.01562
\(980\) 0 0
\(981\) −2.74657 −0.0876913
\(982\) 0 0
\(983\) −5.88553 −0.187719 −0.0938597 0.995585i \(-0.529921\pi\)
−0.0938597 + 0.995585i \(0.529921\pi\)
\(984\) 0 0
\(985\) 13.1222 0.418107
\(986\) 0 0
\(987\) 8.86689 0.282236
\(988\) 0 0
\(989\) 34.4918 1.09678
\(990\) 0 0
\(991\) −48.5494 −1.54222 −0.771111 0.636700i \(-0.780299\pi\)
−0.771111 + 0.636700i \(0.780299\pi\)
\(992\) 0 0
\(993\) −43.5581 −1.38227
\(994\) 0 0
\(995\) 19.9647 0.632924
\(996\) 0 0
\(997\) −26.0283 −0.824324 −0.412162 0.911111i \(-0.635226\pi\)
−0.412162 + 0.911111i \(0.635226\pi\)
\(998\) 0 0
\(999\) 6.96806 0.220460
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 6008.2.a.b.1.39 44
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6008.2.a.b.1.39 44 1.1 even 1 trivial