Properties

Label 8016.2.a.be.1.3
Level 8016
Weight 2
Character 8016.1
Self dual yes
Analytic conductor 64.008
Analytic rank 0
Dimension 11
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8016.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.0080822603\)
Analytic rank: \(0\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Defining polynomial: \(x^{11} - x^{10} - 33 x^{9} + 22 x^{8} + 417 x^{7} - 151 x^{6} - 2470 x^{5} + 272 x^{4} + 6584 x^{3} + 292 x^{2} - 5687 x + 242\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 4008)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.80306\) of defining polynomial
Character \(\chi\) \(=\) 8016.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.80306 q^{5} +4.10861 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.80306 q^{5} +4.10861 q^{7} +1.00000 q^{9} +2.59144 q^{11} -3.46748 q^{13} -1.80306 q^{15} +1.68470 q^{17} +2.49774 q^{19} +4.10861 q^{21} +8.91932 q^{23} -1.74898 q^{25} +1.00000 q^{27} -4.42158 q^{29} +2.95084 q^{31} +2.59144 q^{33} -7.40807 q^{35} +0.918154 q^{37} -3.46748 q^{39} -1.83849 q^{41} -0.850997 q^{43} -1.80306 q^{45} +6.74458 q^{47} +9.88072 q^{49} +1.68470 q^{51} +4.65455 q^{53} -4.67252 q^{55} +2.49774 q^{57} -1.23850 q^{59} -7.97208 q^{61} +4.10861 q^{63} +6.25208 q^{65} +1.65708 q^{67} +8.91932 q^{69} +6.24057 q^{71} -4.76409 q^{73} -1.74898 q^{75} +10.6472 q^{77} +10.7193 q^{79} +1.00000 q^{81} +3.69008 q^{83} -3.03761 q^{85} -4.42158 q^{87} -12.0739 q^{89} -14.2466 q^{91} +2.95084 q^{93} -4.50357 q^{95} +14.8662 q^{97} +2.59144 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11q + 11q^{3} + 10q^{5} + q^{7} + 11q^{9} + O(q^{10}) \) \( 11q + 11q^{3} + 10q^{5} + q^{7} + 11q^{9} + q^{11} + 10q^{13} + 10q^{15} + 17q^{17} - 2q^{19} + q^{21} + 3q^{23} + 21q^{25} + 11q^{27} + 17q^{29} + 15q^{31} + q^{33} - 11q^{35} + 4q^{37} + 10q^{39} + 16q^{41} - 10q^{43} + 10q^{45} + 16q^{47} + 22q^{49} + 17q^{51} + 42q^{53} + 5q^{55} - 2q^{57} + 2q^{59} + 12q^{61} + q^{63} + 10q^{65} + q^{67} + 3q^{69} + 9q^{71} + 24q^{73} + 21q^{75} + 22q^{77} + 30q^{79} + 11q^{81} - 16q^{83} + 25q^{85} + 17q^{87} + 37q^{89} - q^{91} + 15q^{93} - 5q^{95} + 4q^{97} + q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −1.80306 −0.806352 −0.403176 0.915122i \(-0.632094\pi\)
−0.403176 + 0.915122i \(0.632094\pi\)
\(6\) 0 0
\(7\) 4.10861 1.55291 0.776455 0.630172i \(-0.217016\pi\)
0.776455 + 0.630172i \(0.217016\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.59144 0.781349 0.390675 0.920529i \(-0.372242\pi\)
0.390675 + 0.920529i \(0.372242\pi\)
\(12\) 0 0
\(13\) −3.46748 −0.961707 −0.480853 0.876801i \(-0.659673\pi\)
−0.480853 + 0.876801i \(0.659673\pi\)
\(14\) 0 0
\(15\) −1.80306 −0.465548
\(16\) 0 0
\(17\) 1.68470 0.408600 0.204300 0.978908i \(-0.434508\pi\)
0.204300 + 0.978908i \(0.434508\pi\)
\(18\) 0 0
\(19\) 2.49774 0.573021 0.286510 0.958077i \(-0.407505\pi\)
0.286510 + 0.958077i \(0.407505\pi\)
\(20\) 0 0
\(21\) 4.10861 0.896573
\(22\) 0 0
\(23\) 8.91932 1.85981 0.929904 0.367803i \(-0.119890\pi\)
0.929904 + 0.367803i \(0.119890\pi\)
\(24\) 0 0
\(25\) −1.74898 −0.349796
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −4.42158 −0.821067 −0.410534 0.911845i \(-0.634658\pi\)
−0.410534 + 0.911845i \(0.634658\pi\)
\(30\) 0 0
\(31\) 2.95084 0.529987 0.264993 0.964250i \(-0.414630\pi\)
0.264993 + 0.964250i \(0.414630\pi\)
\(32\) 0 0
\(33\) 2.59144 0.451112
\(34\) 0 0
\(35\) −7.40807 −1.25219
\(36\) 0 0
\(37\) 0.918154 0.150944 0.0754718 0.997148i \(-0.475954\pi\)
0.0754718 + 0.997148i \(0.475954\pi\)
\(38\) 0 0
\(39\) −3.46748 −0.555242
\(40\) 0 0
\(41\) −1.83849 −0.287125 −0.143562 0.989641i \(-0.545856\pi\)
−0.143562 + 0.989641i \(0.545856\pi\)
\(42\) 0 0
\(43\) −0.850997 −0.129776 −0.0648879 0.997893i \(-0.520669\pi\)
−0.0648879 + 0.997893i \(0.520669\pi\)
\(44\) 0 0
\(45\) −1.80306 −0.268784
\(46\) 0 0
\(47\) 6.74458 0.983798 0.491899 0.870652i \(-0.336303\pi\)
0.491899 + 0.870652i \(0.336303\pi\)
\(48\) 0 0
\(49\) 9.88072 1.41153
\(50\) 0 0
\(51\) 1.68470 0.235905
\(52\) 0 0
\(53\) 4.65455 0.639351 0.319676 0.947527i \(-0.396426\pi\)
0.319676 + 0.947527i \(0.396426\pi\)
\(54\) 0 0
\(55\) −4.67252 −0.630043
\(56\) 0 0
\(57\) 2.49774 0.330834
\(58\) 0 0
\(59\) −1.23850 −0.161239 −0.0806194 0.996745i \(-0.525690\pi\)
−0.0806194 + 0.996745i \(0.525690\pi\)
\(60\) 0 0
\(61\) −7.97208 −1.02072 −0.510360 0.859961i \(-0.670488\pi\)
−0.510360 + 0.859961i \(0.670488\pi\)
\(62\) 0 0
\(63\) 4.10861 0.517637
\(64\) 0 0
\(65\) 6.25208 0.775475
\(66\) 0 0
\(67\) 1.65708 0.202445 0.101223 0.994864i \(-0.467725\pi\)
0.101223 + 0.994864i \(0.467725\pi\)
\(68\) 0 0
\(69\) 8.91932 1.07376
\(70\) 0 0
\(71\) 6.24057 0.740620 0.370310 0.928908i \(-0.379251\pi\)
0.370310 + 0.928908i \(0.379251\pi\)
\(72\) 0 0
\(73\) −4.76409 −0.557594 −0.278797 0.960350i \(-0.589936\pi\)
−0.278797 + 0.960350i \(0.589936\pi\)
\(74\) 0 0
\(75\) −1.74898 −0.201955
\(76\) 0 0
\(77\) 10.6472 1.21337
\(78\) 0 0
\(79\) 10.7193 1.20601 0.603007 0.797736i \(-0.293969\pi\)
0.603007 + 0.797736i \(0.293969\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 3.69008 0.405039 0.202519 0.979278i \(-0.435087\pi\)
0.202519 + 0.979278i \(0.435087\pi\)
\(84\) 0 0
\(85\) −3.03761 −0.329475
\(86\) 0 0
\(87\) −4.42158 −0.474044
\(88\) 0 0
\(89\) −12.0739 −1.27983 −0.639916 0.768445i \(-0.721031\pi\)
−0.639916 + 0.768445i \(0.721031\pi\)
\(90\) 0 0
\(91\) −14.2466 −1.49344
\(92\) 0 0
\(93\) 2.95084 0.305988
\(94\) 0 0
\(95\) −4.50357 −0.462057
\(96\) 0 0
\(97\) 14.8662 1.50944 0.754719 0.656048i \(-0.227773\pi\)
0.754719 + 0.656048i \(0.227773\pi\)
\(98\) 0 0
\(99\) 2.59144 0.260450
\(100\) 0 0
\(101\) 1.27675 0.127041 0.0635206 0.997981i \(-0.479767\pi\)
0.0635206 + 0.997981i \(0.479767\pi\)
\(102\) 0 0
\(103\) −5.26805 −0.519077 −0.259538 0.965733i \(-0.583570\pi\)
−0.259538 + 0.965733i \(0.583570\pi\)
\(104\) 0 0
\(105\) −7.40807 −0.722954
\(106\) 0 0
\(107\) −7.81076 −0.755094 −0.377547 0.925990i \(-0.623232\pi\)
−0.377547 + 0.925990i \(0.623232\pi\)
\(108\) 0 0
\(109\) −2.01235 −0.192748 −0.0963739 0.995345i \(-0.530724\pi\)
−0.0963739 + 0.995345i \(0.530724\pi\)
\(110\) 0 0
\(111\) 0.918154 0.0871473
\(112\) 0 0
\(113\) 8.20964 0.772298 0.386149 0.922436i \(-0.373805\pi\)
0.386149 + 0.922436i \(0.373805\pi\)
\(114\) 0 0
\(115\) −16.0821 −1.49966
\(116\) 0 0
\(117\) −3.46748 −0.320569
\(118\) 0 0
\(119\) 6.92178 0.634519
\(120\) 0 0
\(121\) −4.28443 −0.389494
\(122\) 0 0
\(123\) −1.83849 −0.165771
\(124\) 0 0
\(125\) 12.1688 1.08841
\(126\) 0 0
\(127\) −9.38074 −0.832406 −0.416203 0.909272i \(-0.636639\pi\)
−0.416203 + 0.909272i \(0.636639\pi\)
\(128\) 0 0
\(129\) −0.850997 −0.0749261
\(130\) 0 0
\(131\) −19.8572 −1.73493 −0.867465 0.497498i \(-0.834252\pi\)
−0.867465 + 0.497498i \(0.834252\pi\)
\(132\) 0 0
\(133\) 10.2623 0.889850
\(134\) 0 0
\(135\) −1.80306 −0.155183
\(136\) 0 0
\(137\) 5.44295 0.465023 0.232511 0.972594i \(-0.425306\pi\)
0.232511 + 0.972594i \(0.425306\pi\)
\(138\) 0 0
\(139\) 13.0858 1.10992 0.554960 0.831877i \(-0.312734\pi\)
0.554960 + 0.831877i \(0.312734\pi\)
\(140\) 0 0
\(141\) 6.74458 0.567996
\(142\) 0 0
\(143\) −8.98578 −0.751429
\(144\) 0 0
\(145\) 7.97238 0.662070
\(146\) 0 0
\(147\) 9.88072 0.814948
\(148\) 0 0
\(149\) −0.961230 −0.0787470 −0.0393735 0.999225i \(-0.512536\pi\)
−0.0393735 + 0.999225i \(0.512536\pi\)
\(150\) 0 0
\(151\) 8.62272 0.701707 0.350853 0.936430i \(-0.385892\pi\)
0.350853 + 0.936430i \(0.385892\pi\)
\(152\) 0 0
\(153\) 1.68470 0.136200
\(154\) 0 0
\(155\) −5.32054 −0.427356
\(156\) 0 0
\(157\) 3.40121 0.271446 0.135723 0.990747i \(-0.456664\pi\)
0.135723 + 0.990747i \(0.456664\pi\)
\(158\) 0 0
\(159\) 4.65455 0.369130
\(160\) 0 0
\(161\) 36.6461 2.88811
\(162\) 0 0
\(163\) 2.70291 0.211708 0.105854 0.994382i \(-0.466242\pi\)
0.105854 + 0.994382i \(0.466242\pi\)
\(164\) 0 0
\(165\) −4.67252 −0.363755
\(166\) 0 0
\(167\) −1.00000 −0.0773823
\(168\) 0 0
\(169\) −0.976559 −0.0751200
\(170\) 0 0
\(171\) 2.49774 0.191007
\(172\) 0 0
\(173\) 13.7489 1.04531 0.522654 0.852545i \(-0.324942\pi\)
0.522654 + 0.852545i \(0.324942\pi\)
\(174\) 0 0
\(175\) −7.18588 −0.543201
\(176\) 0 0
\(177\) −1.23850 −0.0930912
\(178\) 0 0
\(179\) −0.148844 −0.0111251 −0.00556257 0.999985i \(-0.501771\pi\)
−0.00556257 + 0.999985i \(0.501771\pi\)
\(180\) 0 0
\(181\) 7.10022 0.527755 0.263877 0.964556i \(-0.414999\pi\)
0.263877 + 0.964556i \(0.414999\pi\)
\(182\) 0 0
\(183\) −7.97208 −0.589313
\(184\) 0 0
\(185\) −1.65549 −0.121714
\(186\) 0 0
\(187\) 4.36580 0.319259
\(188\) 0 0
\(189\) 4.10861 0.298858
\(190\) 0 0
\(191\) 5.56545 0.402702 0.201351 0.979519i \(-0.435467\pi\)
0.201351 + 0.979519i \(0.435467\pi\)
\(192\) 0 0
\(193\) −16.6812 −1.20074 −0.600371 0.799721i \(-0.704981\pi\)
−0.600371 + 0.799721i \(0.704981\pi\)
\(194\) 0 0
\(195\) 6.25208 0.447721
\(196\) 0 0
\(197\) 10.0656 0.717145 0.358573 0.933502i \(-0.383264\pi\)
0.358573 + 0.933502i \(0.383264\pi\)
\(198\) 0 0
\(199\) 14.2270 1.00853 0.504263 0.863550i \(-0.331764\pi\)
0.504263 + 0.863550i \(0.331764\pi\)
\(200\) 0 0
\(201\) 1.65708 0.116882
\(202\) 0 0
\(203\) −18.1666 −1.27504
\(204\) 0 0
\(205\) 3.31491 0.231524
\(206\) 0 0
\(207\) 8.91932 0.619936
\(208\) 0 0
\(209\) 6.47275 0.447729
\(210\) 0 0
\(211\) −19.4006 −1.33560 −0.667798 0.744343i \(-0.732763\pi\)
−0.667798 + 0.744343i \(0.732763\pi\)
\(212\) 0 0
\(213\) 6.24057 0.427597
\(214\) 0 0
\(215\) 1.53440 0.104645
\(216\) 0 0
\(217\) 12.1239 0.823022
\(218\) 0 0
\(219\) −4.76409 −0.321927
\(220\) 0 0
\(221\) −5.84167 −0.392953
\(222\) 0 0
\(223\) 5.50496 0.368640 0.184320 0.982866i \(-0.440992\pi\)
0.184320 + 0.982866i \(0.440992\pi\)
\(224\) 0 0
\(225\) −1.74898 −0.116599
\(226\) 0 0
\(227\) 24.7959 1.64576 0.822880 0.568215i \(-0.192366\pi\)
0.822880 + 0.568215i \(0.192366\pi\)
\(228\) 0 0
\(229\) −22.8293 −1.50860 −0.754302 0.656528i \(-0.772024\pi\)
−0.754302 + 0.656528i \(0.772024\pi\)
\(230\) 0 0
\(231\) 10.6472 0.700537
\(232\) 0 0
\(233\) 12.2217 0.800672 0.400336 0.916368i \(-0.368893\pi\)
0.400336 + 0.916368i \(0.368893\pi\)
\(234\) 0 0
\(235\) −12.1609 −0.793288
\(236\) 0 0
\(237\) 10.7193 0.696293
\(238\) 0 0
\(239\) −7.79185 −0.504013 −0.252007 0.967726i \(-0.581090\pi\)
−0.252007 + 0.967726i \(0.581090\pi\)
\(240\) 0 0
\(241\) −22.1396 −1.42613 −0.713067 0.701096i \(-0.752695\pi\)
−0.713067 + 0.701096i \(0.752695\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −17.8155 −1.13819
\(246\) 0 0
\(247\) −8.66087 −0.551078
\(248\) 0 0
\(249\) 3.69008 0.233849
\(250\) 0 0
\(251\) 16.1687 1.02056 0.510279 0.860009i \(-0.329542\pi\)
0.510279 + 0.860009i \(0.329542\pi\)
\(252\) 0 0
\(253\) 23.1139 1.45316
\(254\) 0 0
\(255\) −3.03761 −0.190223
\(256\) 0 0
\(257\) 22.3740 1.39565 0.697825 0.716268i \(-0.254151\pi\)
0.697825 + 0.716268i \(0.254151\pi\)
\(258\) 0 0
\(259\) 3.77234 0.234402
\(260\) 0 0
\(261\) −4.42158 −0.273689
\(262\) 0 0
\(263\) 20.4376 1.26023 0.630117 0.776500i \(-0.283007\pi\)
0.630117 + 0.776500i \(0.283007\pi\)
\(264\) 0 0
\(265\) −8.39242 −0.515542
\(266\) 0 0
\(267\) −12.0739 −0.738911
\(268\) 0 0
\(269\) 16.8017 1.02442 0.512209 0.858861i \(-0.328827\pi\)
0.512209 + 0.858861i \(0.328827\pi\)
\(270\) 0 0
\(271\) 8.49263 0.515890 0.257945 0.966160i \(-0.416955\pi\)
0.257945 + 0.966160i \(0.416955\pi\)
\(272\) 0 0
\(273\) −14.2466 −0.862241
\(274\) 0 0
\(275\) −4.53238 −0.273313
\(276\) 0 0
\(277\) −12.1608 −0.730670 −0.365335 0.930876i \(-0.619046\pi\)
−0.365335 + 0.930876i \(0.619046\pi\)
\(278\) 0 0
\(279\) 2.95084 0.176662
\(280\) 0 0
\(281\) −14.5701 −0.869178 −0.434589 0.900629i \(-0.643106\pi\)
−0.434589 + 0.900629i \(0.643106\pi\)
\(282\) 0 0
\(283\) −8.16542 −0.485384 −0.242692 0.970103i \(-0.578030\pi\)
−0.242692 + 0.970103i \(0.578030\pi\)
\(284\) 0 0
\(285\) −4.50357 −0.266769
\(286\) 0 0
\(287\) −7.55366 −0.445879
\(288\) 0 0
\(289\) −14.1618 −0.833046
\(290\) 0 0
\(291\) 14.8662 0.871474
\(292\) 0 0
\(293\) 29.2346 1.70791 0.853953 0.520350i \(-0.174199\pi\)
0.853953 + 0.520350i \(0.174199\pi\)
\(294\) 0 0
\(295\) 2.23309 0.130015
\(296\) 0 0
\(297\) 2.59144 0.150371
\(298\) 0 0
\(299\) −30.9276 −1.78859
\(300\) 0 0
\(301\) −3.49642 −0.201530
\(302\) 0 0
\(303\) 1.27675 0.0733473
\(304\) 0 0
\(305\) 14.3741 0.823061
\(306\) 0 0
\(307\) −23.8192 −1.35943 −0.679716 0.733475i \(-0.737897\pi\)
−0.679716 + 0.733475i \(0.737897\pi\)
\(308\) 0 0
\(309\) −5.26805 −0.299689
\(310\) 0 0
\(311\) 16.4270 0.931492 0.465746 0.884918i \(-0.345786\pi\)
0.465746 + 0.884918i \(0.345786\pi\)
\(312\) 0 0
\(313\) 12.4875 0.705832 0.352916 0.935655i \(-0.385190\pi\)
0.352916 + 0.935655i \(0.385190\pi\)
\(314\) 0 0
\(315\) −7.40807 −0.417398
\(316\) 0 0
\(317\) −3.38861 −0.190323 −0.0951617 0.995462i \(-0.530337\pi\)
−0.0951617 + 0.995462i \(0.530337\pi\)
\(318\) 0 0
\(319\) −11.4583 −0.641540
\(320\) 0 0
\(321\) −7.81076 −0.435954
\(322\) 0 0
\(323\) 4.20794 0.234136
\(324\) 0 0
\(325\) 6.06455 0.336401
\(326\) 0 0
\(327\) −2.01235 −0.111283
\(328\) 0 0
\(329\) 27.7109 1.52775
\(330\) 0 0
\(331\) −7.83764 −0.430795 −0.215398 0.976526i \(-0.569105\pi\)
−0.215398 + 0.976526i \(0.569105\pi\)
\(332\) 0 0
\(333\) 0.918154 0.0503145
\(334\) 0 0
\(335\) −2.98782 −0.163242
\(336\) 0 0
\(337\) −19.0047 −1.03525 −0.517627 0.855606i \(-0.673185\pi\)
−0.517627 + 0.855606i \(0.673185\pi\)
\(338\) 0 0
\(339\) 8.20964 0.445887
\(340\) 0 0
\(341\) 7.64694 0.414105
\(342\) 0 0
\(343\) 11.8358 0.639071
\(344\) 0 0
\(345\) −16.0821 −0.865829
\(346\) 0 0
\(347\) −25.2208 −1.35392 −0.676961 0.736019i \(-0.736703\pi\)
−0.676961 + 0.736019i \(0.736703\pi\)
\(348\) 0 0
\(349\) 1.06370 0.0569387 0.0284693 0.999595i \(-0.490937\pi\)
0.0284693 + 0.999595i \(0.490937\pi\)
\(350\) 0 0
\(351\) −3.46748 −0.185081
\(352\) 0 0
\(353\) 19.0049 1.01153 0.505764 0.862672i \(-0.331211\pi\)
0.505764 + 0.862672i \(0.331211\pi\)
\(354\) 0 0
\(355\) −11.2521 −0.597201
\(356\) 0 0
\(357\) 6.92178 0.366340
\(358\) 0 0
\(359\) −22.4615 −1.18547 −0.592736 0.805397i \(-0.701952\pi\)
−0.592736 + 0.805397i \(0.701952\pi\)
\(360\) 0 0
\(361\) −12.7613 −0.671647
\(362\) 0 0
\(363\) −4.28443 −0.224874
\(364\) 0 0
\(365\) 8.58993 0.449618
\(366\) 0 0
\(367\) −6.29449 −0.328569 −0.164285 0.986413i \(-0.552532\pi\)
−0.164285 + 0.986413i \(0.552532\pi\)
\(368\) 0 0
\(369\) −1.83849 −0.0957082
\(370\) 0 0
\(371\) 19.1237 0.992855
\(372\) 0 0
\(373\) −11.9702 −0.619794 −0.309897 0.950770i \(-0.600295\pi\)
−0.309897 + 0.950770i \(0.600295\pi\)
\(374\) 0 0
\(375\) 12.1688 0.628394
\(376\) 0 0
\(377\) 15.3318 0.789626
\(378\) 0 0
\(379\) 18.5730 0.954032 0.477016 0.878895i \(-0.341718\pi\)
0.477016 + 0.878895i \(0.341718\pi\)
\(380\) 0 0
\(381\) −9.38074 −0.480590
\(382\) 0 0
\(383\) 29.3120 1.49777 0.748886 0.662698i \(-0.230589\pi\)
0.748886 + 0.662698i \(0.230589\pi\)
\(384\) 0 0
\(385\) −19.1976 −0.978400
\(386\) 0 0
\(387\) −0.850997 −0.0432586
\(388\) 0 0
\(389\) 15.9476 0.808573 0.404287 0.914632i \(-0.367520\pi\)
0.404287 + 0.914632i \(0.367520\pi\)
\(390\) 0 0
\(391\) 15.0264 0.759917
\(392\) 0 0
\(393\) −19.8572 −1.00166
\(394\) 0 0
\(395\) −19.3275 −0.972473
\(396\) 0 0
\(397\) −22.9256 −1.15060 −0.575302 0.817941i \(-0.695116\pi\)
−0.575302 + 0.817941i \(0.695116\pi\)
\(398\) 0 0
\(399\) 10.2623 0.513755
\(400\) 0 0
\(401\) 21.0031 1.04884 0.524421 0.851459i \(-0.324282\pi\)
0.524421 + 0.851459i \(0.324282\pi\)
\(402\) 0 0
\(403\) −10.2320 −0.509692
\(404\) 0 0
\(405\) −1.80306 −0.0895947
\(406\) 0 0
\(407\) 2.37934 0.117940
\(408\) 0 0
\(409\) 34.4707 1.70447 0.852234 0.523160i \(-0.175247\pi\)
0.852234 + 0.523160i \(0.175247\pi\)
\(410\) 0 0
\(411\) 5.44295 0.268481
\(412\) 0 0
\(413\) −5.08851 −0.250389
\(414\) 0 0
\(415\) −6.65343 −0.326604
\(416\) 0 0
\(417\) 13.0858 0.640812
\(418\) 0 0
\(419\) 7.80419 0.381260 0.190630 0.981662i \(-0.438947\pi\)
0.190630 + 0.981662i \(0.438947\pi\)
\(420\) 0 0
\(421\) −28.5378 −1.39085 −0.695424 0.718599i \(-0.744783\pi\)
−0.695424 + 0.718599i \(0.744783\pi\)
\(422\) 0 0
\(423\) 6.74458 0.327933
\(424\) 0 0
\(425\) −2.94650 −0.142926
\(426\) 0 0
\(427\) −32.7542 −1.58509
\(428\) 0 0
\(429\) −8.98578 −0.433838
\(430\) 0 0
\(431\) −10.9732 −0.528559 −0.264279 0.964446i \(-0.585134\pi\)
−0.264279 + 0.964446i \(0.585134\pi\)
\(432\) 0 0
\(433\) 18.8851 0.907558 0.453779 0.891114i \(-0.350076\pi\)
0.453779 + 0.891114i \(0.350076\pi\)
\(434\) 0 0
\(435\) 7.97238 0.382246
\(436\) 0 0
\(437\) 22.2781 1.06571
\(438\) 0 0
\(439\) 25.2125 1.20333 0.601665 0.798749i \(-0.294504\pi\)
0.601665 + 0.798749i \(0.294504\pi\)
\(440\) 0 0
\(441\) 9.88072 0.470510
\(442\) 0 0
\(443\) 20.2876 0.963893 0.481947 0.876201i \(-0.339930\pi\)
0.481947 + 0.876201i \(0.339930\pi\)
\(444\) 0 0
\(445\) 21.7700 1.03200
\(446\) 0 0
\(447\) −0.961230 −0.0454646
\(448\) 0 0
\(449\) −26.3700 −1.24448 −0.622240 0.782827i \(-0.713777\pi\)
−0.622240 + 0.782827i \(0.713777\pi\)
\(450\) 0 0
\(451\) −4.76435 −0.224345
\(452\) 0 0
\(453\) 8.62272 0.405131
\(454\) 0 0
\(455\) 25.6874 1.20424
\(456\) 0 0
\(457\) −20.0483 −0.937818 −0.468909 0.883247i \(-0.655353\pi\)
−0.468909 + 0.883247i \(0.655353\pi\)
\(458\) 0 0
\(459\) 1.68470 0.0786351
\(460\) 0 0
\(461\) −28.0956 −1.30854 −0.654272 0.756260i \(-0.727025\pi\)
−0.654272 + 0.756260i \(0.727025\pi\)
\(462\) 0 0
\(463\) 23.8772 1.10967 0.554834 0.831961i \(-0.312782\pi\)
0.554834 + 0.831961i \(0.312782\pi\)
\(464\) 0 0
\(465\) −5.32054 −0.246734
\(466\) 0 0
\(467\) 33.4385 1.54735 0.773674 0.633584i \(-0.218417\pi\)
0.773674 + 0.633584i \(0.218417\pi\)
\(468\) 0 0
\(469\) 6.80832 0.314379
\(470\) 0 0
\(471\) 3.40121 0.156720
\(472\) 0 0
\(473\) −2.20531 −0.101400
\(474\) 0 0
\(475\) −4.36849 −0.200440
\(476\) 0 0
\(477\) 4.65455 0.213117
\(478\) 0 0
\(479\) 3.13119 0.143068 0.0715340 0.997438i \(-0.477211\pi\)
0.0715340 + 0.997438i \(0.477211\pi\)
\(480\) 0 0
\(481\) −3.18368 −0.145163
\(482\) 0 0
\(483\) 36.6461 1.66745
\(484\) 0 0
\(485\) −26.8047 −1.21714
\(486\) 0 0
\(487\) 26.2378 1.18895 0.594475 0.804114i \(-0.297360\pi\)
0.594475 + 0.804114i \(0.297360\pi\)
\(488\) 0 0
\(489\) 2.70291 0.122230
\(490\) 0 0
\(491\) 5.20065 0.234702 0.117351 0.993090i \(-0.462560\pi\)
0.117351 + 0.993090i \(0.462560\pi\)
\(492\) 0 0
\(493\) −7.44904 −0.335488
\(494\) 0 0
\(495\) −4.67252 −0.210014
\(496\) 0 0
\(497\) 25.6401 1.15012
\(498\) 0 0
\(499\) 33.0631 1.48011 0.740053 0.672548i \(-0.234800\pi\)
0.740053 + 0.672548i \(0.234800\pi\)
\(500\) 0 0
\(501\) −1.00000 −0.0446767
\(502\) 0 0
\(503\) −14.7807 −0.659040 −0.329520 0.944149i \(-0.606887\pi\)
−0.329520 + 0.944149i \(0.606887\pi\)
\(504\) 0 0
\(505\) −2.30205 −0.102440
\(506\) 0 0
\(507\) −0.976559 −0.0433705
\(508\) 0 0
\(509\) 31.0505 1.37629 0.688145 0.725573i \(-0.258425\pi\)
0.688145 + 0.725573i \(0.258425\pi\)
\(510\) 0 0
\(511\) −19.5738 −0.865894
\(512\) 0 0
\(513\) 2.49774 0.110278
\(514\) 0 0
\(515\) 9.49861 0.418559
\(516\) 0 0
\(517\) 17.4782 0.768689
\(518\) 0 0
\(519\) 13.7489 0.603509
\(520\) 0 0
\(521\) 3.63406 0.159211 0.0796056 0.996826i \(-0.474634\pi\)
0.0796056 + 0.996826i \(0.474634\pi\)
\(522\) 0 0
\(523\) −26.1659 −1.14416 −0.572078 0.820199i \(-0.693863\pi\)
−0.572078 + 0.820199i \(0.693863\pi\)
\(524\) 0 0
\(525\) −7.18588 −0.313617
\(526\) 0 0
\(527\) 4.97128 0.216553
\(528\) 0 0
\(529\) 56.5543 2.45888
\(530\) 0 0
\(531\) −1.23850 −0.0537462
\(532\) 0 0
\(533\) 6.37495 0.276130
\(534\) 0 0
\(535\) 14.0833 0.608872
\(536\) 0 0
\(537\) −0.148844 −0.00642310
\(538\) 0 0
\(539\) 25.6053 1.10290
\(540\) 0 0
\(541\) 0.260024 0.0111793 0.00558966 0.999984i \(-0.498221\pi\)
0.00558966 + 0.999984i \(0.498221\pi\)
\(542\) 0 0
\(543\) 7.10022 0.304699
\(544\) 0 0
\(545\) 3.62838 0.155423
\(546\) 0 0
\(547\) 27.8527 1.19090 0.595448 0.803394i \(-0.296975\pi\)
0.595448 + 0.803394i \(0.296975\pi\)
\(548\) 0 0
\(549\) −7.97208 −0.340240
\(550\) 0 0
\(551\) −11.0440 −0.470489
\(552\) 0 0
\(553\) 44.0414 1.87283
\(554\) 0 0
\(555\) −1.65549 −0.0702714
\(556\) 0 0
\(557\) 8.13555 0.344714 0.172357 0.985035i \(-0.444862\pi\)
0.172357 + 0.985035i \(0.444862\pi\)
\(558\) 0 0
\(559\) 2.95082 0.124806
\(560\) 0 0
\(561\) 4.36580 0.184324
\(562\) 0 0
\(563\) 44.9622 1.89493 0.947466 0.319856i \(-0.103634\pi\)
0.947466 + 0.319856i \(0.103634\pi\)
\(564\) 0 0
\(565\) −14.8025 −0.622745
\(566\) 0 0
\(567\) 4.10861 0.172546
\(568\) 0 0
\(569\) 8.14797 0.341581 0.170790 0.985307i \(-0.445368\pi\)
0.170790 + 0.985307i \(0.445368\pi\)
\(570\) 0 0
\(571\) −5.71162 −0.239024 −0.119512 0.992833i \(-0.538133\pi\)
−0.119512 + 0.992833i \(0.538133\pi\)
\(572\) 0 0
\(573\) 5.56545 0.232500
\(574\) 0 0
\(575\) −15.5997 −0.650553
\(576\) 0 0
\(577\) 8.28848 0.345054 0.172527 0.985005i \(-0.444807\pi\)
0.172527 + 0.985005i \(0.444807\pi\)
\(578\) 0 0
\(579\) −16.6812 −0.693249
\(580\) 0 0
\(581\) 15.1611 0.628989
\(582\) 0 0
\(583\) 12.0620 0.499556
\(584\) 0 0
\(585\) 6.25208 0.258492
\(586\) 0 0
\(587\) −32.2682 −1.33185 −0.665925 0.746019i \(-0.731963\pi\)
−0.665925 + 0.746019i \(0.731963\pi\)
\(588\) 0 0
\(589\) 7.37043 0.303693
\(590\) 0 0
\(591\) 10.0656 0.414044
\(592\) 0 0
\(593\) −26.2913 −1.07965 −0.539826 0.841776i \(-0.681510\pi\)
−0.539826 + 0.841776i \(0.681510\pi\)
\(594\) 0 0
\(595\) −12.4804 −0.511646
\(596\) 0 0
\(597\) 14.2270 0.582273
\(598\) 0 0
\(599\) −1.95363 −0.0798233 −0.0399117 0.999203i \(-0.512708\pi\)
−0.0399117 + 0.999203i \(0.512708\pi\)
\(600\) 0 0
\(601\) −0.855179 −0.0348835 −0.0174417 0.999848i \(-0.505552\pi\)
−0.0174417 + 0.999848i \(0.505552\pi\)
\(602\) 0 0
\(603\) 1.65708 0.0674817
\(604\) 0 0
\(605\) 7.72508 0.314069
\(606\) 0 0
\(607\) 17.8511 0.724553 0.362276 0.932071i \(-0.382000\pi\)
0.362276 + 0.932071i \(0.382000\pi\)
\(608\) 0 0
\(609\) −18.1666 −0.736147
\(610\) 0 0
\(611\) −23.3867 −0.946125
\(612\) 0 0
\(613\) 21.0354 0.849611 0.424806 0.905285i \(-0.360342\pi\)
0.424806 + 0.905285i \(0.360342\pi\)
\(614\) 0 0
\(615\) 3.31491 0.133670
\(616\) 0 0
\(617\) −29.7077 −1.19599 −0.597993 0.801501i \(-0.704035\pi\)
−0.597993 + 0.801501i \(0.704035\pi\)
\(618\) 0 0
\(619\) 7.12602 0.286419 0.143209 0.989692i \(-0.454258\pi\)
0.143209 + 0.989692i \(0.454258\pi\)
\(620\) 0 0
\(621\) 8.91932 0.357920
\(622\) 0 0
\(623\) −49.6070 −1.98746
\(624\) 0 0
\(625\) −13.1962 −0.527847
\(626\) 0 0
\(627\) 6.47275 0.258497
\(628\) 0 0
\(629\) 1.54681 0.0616755
\(630\) 0 0
\(631\) 10.8881 0.433447 0.216724 0.976233i \(-0.430463\pi\)
0.216724 + 0.976233i \(0.430463\pi\)
\(632\) 0 0
\(633\) −19.4006 −0.771106
\(634\) 0 0
\(635\) 16.9140 0.671213
\(636\) 0 0
\(637\) −34.2612 −1.35748
\(638\) 0 0
\(639\) 6.24057 0.246873
\(640\) 0 0
\(641\) 6.00715 0.237268 0.118634 0.992938i \(-0.462148\pi\)
0.118634 + 0.992938i \(0.462148\pi\)
\(642\) 0 0
\(643\) −22.8736 −0.902045 −0.451023 0.892513i \(-0.648941\pi\)
−0.451023 + 0.892513i \(0.648941\pi\)
\(644\) 0 0
\(645\) 1.53440 0.0604169
\(646\) 0 0
\(647\) −19.1587 −0.753205 −0.376602 0.926375i \(-0.622908\pi\)
−0.376602 + 0.926375i \(0.622908\pi\)
\(648\) 0 0
\(649\) −3.20950 −0.125984
\(650\) 0 0
\(651\) 12.1239 0.475172
\(652\) 0 0
\(653\) −29.0547 −1.13700 −0.568499 0.822684i \(-0.692475\pi\)
−0.568499 + 0.822684i \(0.692475\pi\)
\(654\) 0 0
\(655\) 35.8037 1.39897
\(656\) 0 0
\(657\) −4.76409 −0.185865
\(658\) 0 0
\(659\) −18.7083 −0.728771 −0.364386 0.931248i \(-0.618721\pi\)
−0.364386 + 0.931248i \(0.618721\pi\)
\(660\) 0 0
\(661\) −6.65833 −0.258979 −0.129490 0.991581i \(-0.541334\pi\)
−0.129490 + 0.991581i \(0.541334\pi\)
\(662\) 0 0
\(663\) −5.84167 −0.226872
\(664\) 0 0
\(665\) −18.5034 −0.717533
\(666\) 0 0
\(667\) −39.4375 −1.52703
\(668\) 0 0
\(669\) 5.50496 0.212834
\(670\) 0 0
\(671\) −20.6592 −0.797539
\(672\) 0 0
\(673\) −25.7547 −0.992772 −0.496386 0.868102i \(-0.665340\pi\)
−0.496386 + 0.868102i \(0.665340\pi\)
\(674\) 0 0
\(675\) −1.74898 −0.0673182
\(676\) 0 0
\(677\) 11.7352 0.451019 0.225510 0.974241i \(-0.427595\pi\)
0.225510 + 0.974241i \(0.427595\pi\)
\(678\) 0 0
\(679\) 61.0797 2.34402
\(680\) 0 0
\(681\) 24.7959 0.950180
\(682\) 0 0
\(683\) 30.4318 1.16444 0.582220 0.813031i \(-0.302184\pi\)
0.582220 + 0.813031i \(0.302184\pi\)
\(684\) 0 0
\(685\) −9.81397 −0.374972
\(686\) 0 0
\(687\) −22.8293 −0.870993
\(688\) 0 0
\(689\) −16.1396 −0.614868
\(690\) 0 0
\(691\) 35.5839 1.35368 0.676838 0.736132i \(-0.263350\pi\)
0.676838 + 0.736132i \(0.263350\pi\)
\(692\) 0 0
\(693\) 10.6472 0.404455
\(694\) 0 0
\(695\) −23.5944 −0.894986
\(696\) 0 0
\(697\) −3.09731 −0.117319
\(698\) 0 0
\(699\) 12.2217 0.462268
\(700\) 0 0
\(701\) −11.4767 −0.433468 −0.216734 0.976231i \(-0.569540\pi\)
−0.216734 + 0.976231i \(0.569540\pi\)
\(702\) 0 0
\(703\) 2.29331 0.0864938
\(704\) 0 0
\(705\) −12.1609 −0.458005
\(706\) 0 0
\(707\) 5.24567 0.197284
\(708\) 0 0
\(709\) 2.41968 0.0908729 0.0454365 0.998967i \(-0.485532\pi\)
0.0454365 + 0.998967i \(0.485532\pi\)
\(710\) 0 0
\(711\) 10.7193 0.402005
\(712\) 0 0
\(713\) 26.3195 0.985673
\(714\) 0 0
\(715\) 16.2019 0.605916
\(716\) 0 0
\(717\) −7.79185 −0.290992
\(718\) 0 0
\(719\) −26.9803 −1.00620 −0.503098 0.864229i \(-0.667806\pi\)
−0.503098 + 0.864229i \(0.667806\pi\)
\(720\) 0 0
\(721\) −21.6444 −0.806080
\(722\) 0 0
\(723\) −22.1396 −0.823379
\(724\) 0 0
\(725\) 7.73325 0.287206
\(726\) 0 0
\(727\) −8.04133 −0.298237 −0.149118 0.988819i \(-0.547644\pi\)
−0.149118 + 0.988819i \(0.547644\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −1.43368 −0.0530264
\(732\) 0 0
\(733\) −48.1812 −1.77961 −0.889807 0.456336i \(-0.849161\pi\)
−0.889807 + 0.456336i \(0.849161\pi\)
\(734\) 0 0
\(735\) −17.8155 −0.657135
\(736\) 0 0
\(737\) 4.29424 0.158180
\(738\) 0 0
\(739\) −21.9605 −0.807829 −0.403915 0.914797i \(-0.632351\pi\)
−0.403915 + 0.914797i \(0.632351\pi\)
\(740\) 0 0
\(741\) −8.66087 −0.318165
\(742\) 0 0
\(743\) −4.55455 −0.167090 −0.0835451 0.996504i \(-0.526624\pi\)
−0.0835451 + 0.996504i \(0.526624\pi\)
\(744\) 0 0
\(745\) 1.73315 0.0634979
\(746\) 0 0
\(747\) 3.69008 0.135013
\(748\) 0 0
\(749\) −32.0914 −1.17259
\(750\) 0 0
\(751\) −24.3222 −0.887530 −0.443765 0.896143i \(-0.646357\pi\)
−0.443765 + 0.896143i \(0.646357\pi\)
\(752\) 0 0
\(753\) 16.1687 0.589219
\(754\) 0 0
\(755\) −15.5473 −0.565823
\(756\) 0 0
\(757\) 9.86234 0.358453 0.179226 0.983808i \(-0.442641\pi\)
0.179226 + 0.983808i \(0.442641\pi\)
\(758\) 0 0
\(759\) 23.1139 0.838982
\(760\) 0 0
\(761\) −1.04767 −0.0379781 −0.0189891 0.999820i \(-0.506045\pi\)
−0.0189891 + 0.999820i \(0.506045\pi\)
\(762\) 0 0
\(763\) −8.26796 −0.299320
\(764\) 0 0
\(765\) −3.03761 −0.109825
\(766\) 0 0
\(767\) 4.29447 0.155064
\(768\) 0 0
\(769\) −11.2840 −0.406912 −0.203456 0.979084i \(-0.565217\pi\)
−0.203456 + 0.979084i \(0.565217\pi\)
\(770\) 0 0
\(771\) 22.3740 0.805779
\(772\) 0 0
\(773\) 2.32588 0.0836561 0.0418280 0.999125i \(-0.486682\pi\)
0.0418280 + 0.999125i \(0.486682\pi\)
\(774\) 0 0
\(775\) −5.16096 −0.185387
\(776\) 0 0
\(777\) 3.77234 0.135332
\(778\) 0 0
\(779\) −4.59208 −0.164528
\(780\) 0 0
\(781\) 16.1721 0.578683
\(782\) 0 0
\(783\) −4.42158 −0.158015
\(784\) 0 0
\(785\) −6.13259 −0.218882
\(786\) 0 0
\(787\) 9.04798 0.322526 0.161263 0.986912i \(-0.448443\pi\)
0.161263 + 0.986912i \(0.448443\pi\)
\(788\) 0 0
\(789\) 20.4376 0.727596
\(790\) 0 0
\(791\) 33.7303 1.19931
\(792\) 0 0
\(793\) 27.6431 0.981634
\(794\) 0 0
\(795\) −8.39242 −0.297649
\(796\) 0 0
\(797\) 5.67319 0.200955 0.100477 0.994939i \(-0.467963\pi\)
0.100477 + 0.994939i \(0.467963\pi\)
\(798\) 0 0
\(799\) 11.3626 0.401979
\(800\) 0 0
\(801\) −12.0739 −0.426611
\(802\) 0 0
\(803\) −12.3459 −0.435676
\(804\) 0 0
\(805\) −66.0750 −2.32884
\(806\) 0 0
\(807\) 16.8017 0.591448
\(808\) 0 0
\(809\) −37.7932 −1.32874 −0.664369 0.747405i \(-0.731300\pi\)
−0.664369 + 0.747405i \(0.731300\pi\)
\(810\) 0 0
\(811\) −17.2974 −0.607394 −0.303697 0.952769i \(-0.598221\pi\)
−0.303697 + 0.952769i \(0.598221\pi\)
\(812\) 0 0
\(813\) 8.49263 0.297849
\(814\) 0 0
\(815\) −4.87350 −0.170711
\(816\) 0 0
\(817\) −2.12557 −0.0743643
\(818\) 0 0
\(819\) −14.2466 −0.497815
\(820\) 0 0
\(821\) 13.1038 0.457325 0.228662 0.973506i \(-0.426565\pi\)
0.228662 + 0.973506i \(0.426565\pi\)
\(822\) 0 0
\(823\) 5.47594 0.190879 0.0954396 0.995435i \(-0.469574\pi\)
0.0954396 + 0.995435i \(0.469574\pi\)
\(824\) 0 0
\(825\) −4.53238 −0.157797
\(826\) 0 0
\(827\) 2.81367 0.0978411 0.0489205 0.998803i \(-0.484422\pi\)
0.0489205 + 0.998803i \(0.484422\pi\)
\(828\) 0 0
\(829\) 11.7545 0.408250 0.204125 0.978945i \(-0.434565\pi\)
0.204125 + 0.978945i \(0.434565\pi\)
\(830\) 0 0
\(831\) −12.1608 −0.421852
\(832\) 0 0
\(833\) 16.6460 0.576751
\(834\) 0 0
\(835\) 1.80306 0.0623974
\(836\) 0 0
\(837\) 2.95084 0.101996
\(838\) 0 0
\(839\) 35.1585 1.21381 0.606903 0.794776i \(-0.292412\pi\)
0.606903 + 0.794776i \(0.292412\pi\)
\(840\) 0 0
\(841\) −9.44960 −0.325848
\(842\) 0 0
\(843\) −14.5701 −0.501820
\(844\) 0 0
\(845\) 1.76079 0.0605732
\(846\) 0 0
\(847\) −17.6031 −0.604849
\(848\) 0 0
\(849\) −8.16542 −0.280236
\(850\) 0 0
\(851\) 8.18931 0.280726
\(852\) 0 0
\(853\) −5.19930 −0.178021 −0.0890104 0.996031i \(-0.528370\pi\)
−0.0890104 + 0.996031i \(0.528370\pi\)
\(854\) 0 0
\(855\) −4.50357 −0.154019
\(856\) 0 0
\(857\) 1.20592 0.0411933 0.0205966 0.999788i \(-0.493443\pi\)
0.0205966 + 0.999788i \(0.493443\pi\)
\(858\) 0 0
\(859\) 39.2254 1.33835 0.669176 0.743104i \(-0.266647\pi\)
0.669176 + 0.743104i \(0.266647\pi\)
\(860\) 0 0
\(861\) −7.55366 −0.257428
\(862\) 0 0
\(863\) −17.4008 −0.592329 −0.296164 0.955137i \(-0.595708\pi\)
−0.296164 + 0.955137i \(0.595708\pi\)
\(864\) 0 0
\(865\) −24.7900 −0.842886
\(866\) 0 0
\(867\) −14.1618 −0.480959
\(868\) 0 0
\(869\) 27.7784 0.942319
\(870\) 0 0
\(871\) −5.74591 −0.194693
\(872\) 0 0
\(873\) 14.8662 0.503146
\(874\) 0 0
\(875\) 49.9969 1.69020
\(876\) 0 0
\(877\) −51.8185 −1.74979 −0.874893 0.484316i \(-0.839068\pi\)
−0.874893 + 0.484316i \(0.839068\pi\)
\(878\) 0 0
\(879\) 29.2346 0.986060
\(880\) 0 0
\(881\) −55.7905 −1.87963 −0.939814 0.341686i \(-0.889002\pi\)
−0.939814 + 0.341686i \(0.889002\pi\)
\(882\) 0 0
\(883\) −34.6270 −1.16529 −0.582645 0.812727i \(-0.697982\pi\)
−0.582645 + 0.812727i \(0.697982\pi\)
\(884\) 0 0
\(885\) 2.23309 0.0750643
\(886\) 0 0
\(887\) 3.25650 0.109343 0.0546713 0.998504i \(-0.482589\pi\)
0.0546713 + 0.998504i \(0.482589\pi\)
\(888\) 0 0
\(889\) −38.5418 −1.29265
\(890\) 0 0
\(891\) 2.59144 0.0868166
\(892\) 0 0
\(893\) 16.8462 0.563736
\(894\) 0 0
\(895\) 0.268375 0.00897079
\(896\) 0 0
\(897\) −30.9276 −1.03264
\(898\) 0 0
\(899\) −13.0474 −0.435155
\(900\) 0 0
\(901\) 7.84151 0.261239
\(902\) 0 0
\(903\) −3.49642 −0.116354
\(904\) 0 0
\(905\) −12.8021 −0.425556
\(906\) 0 0
\(907\) −1.93813 −0.0643547 −0.0321773 0.999482i \(-0.510244\pi\)
−0.0321773 + 0.999482i \(0.510244\pi\)
\(908\) 0 0
\(909\) 1.27675 0.0423471
\(910\) 0 0
\(911\) 22.6161 0.749304 0.374652 0.927165i \(-0.377762\pi\)
0.374652 + 0.927165i \(0.377762\pi\)
\(912\) 0 0
\(913\) 9.56262 0.316477
\(914\) 0 0
\(915\) 14.3741 0.475194
\(916\) 0 0
\(917\) −81.5855 −2.69419
\(918\) 0 0
\(919\) 0.152045 0.00501549 0.00250774 0.999997i \(-0.499202\pi\)
0.00250774 + 0.999997i \(0.499202\pi\)
\(920\) 0 0
\(921\) −23.8192 −0.784869
\(922\) 0 0
\(923\) −21.6391 −0.712259
\(924\) 0 0
\(925\) −1.60583 −0.0527994
\(926\) 0 0
\(927\) −5.26805 −0.173026
\(928\) 0 0
\(929\) −40.4885 −1.32838 −0.664192 0.747562i \(-0.731224\pi\)
−0.664192 + 0.747562i \(0.731224\pi\)
\(930\) 0 0
\(931\) 24.6795 0.808836
\(932\) 0 0
\(933\) 16.4270 0.537797
\(934\) 0 0
\(935\) −7.87180 −0.257435
\(936\) 0 0
\(937\) −49.3072 −1.61079 −0.805397 0.592735i \(-0.798048\pi\)
−0.805397 + 0.592735i \(0.798048\pi\)
\(938\) 0 0
\(939\) 12.4875 0.407513
\(940\) 0 0
\(941\) −22.8417 −0.744618 −0.372309 0.928109i \(-0.621434\pi\)
−0.372309 + 0.928109i \(0.621434\pi\)
\(942\) 0 0
\(943\) −16.3981 −0.533996
\(944\) 0 0
\(945\) −7.40807 −0.240985
\(946\) 0 0
\(947\) −20.8527 −0.677622 −0.338811 0.940854i \(-0.610025\pi\)
−0.338811 + 0.940854i \(0.610025\pi\)
\(948\) 0 0
\(949\) 16.5194 0.536242
\(950\) 0 0
\(951\) −3.38861 −0.109883
\(952\) 0 0
\(953\) −20.6396 −0.668581 −0.334291 0.942470i \(-0.608497\pi\)
−0.334291 + 0.942470i \(0.608497\pi\)
\(954\) 0 0
\(955\) −10.0348 −0.324719
\(956\) 0 0
\(957\) −11.4583 −0.370393
\(958\) 0 0
\(959\) 22.3630 0.722139
\(960\) 0 0
\(961\) −22.2925 −0.719114
\(962\) 0 0
\(963\) −7.81076 −0.251698
\(964\) 0 0
\(965\) 30.0773 0.968222
\(966\) 0 0
\(967\) −43.8774 −1.41100 −0.705501 0.708709i \(-0.749278\pi\)
−0.705501 + 0.708709i \(0.749278\pi\)
\(968\) 0 0
\(969\) 4.20794 0.135179
\(970\) 0 0
\(971\) −9.54975 −0.306466 −0.153233 0.988190i \(-0.548969\pi\)
−0.153233 + 0.988190i \(0.548969\pi\)
\(972\) 0 0
\(973\) 53.7643 1.72361
\(974\) 0 0
\(975\) 6.06455 0.194221
\(976\) 0 0
\(977\) 11.9627 0.382721 0.191360 0.981520i \(-0.438710\pi\)
0.191360 + 0.981520i \(0.438710\pi\)
\(978\) 0 0
\(979\) −31.2888 −0.999995
\(980\) 0 0
\(981\) −2.01235 −0.0642493
\(982\) 0 0
\(983\) 37.2845 1.18919 0.594595 0.804025i \(-0.297312\pi\)
0.594595 + 0.804025i \(0.297312\pi\)
\(984\) 0 0
\(985\) −18.1489 −0.578272
\(986\) 0 0
\(987\) 27.7109 0.882047
\(988\) 0 0
\(989\) −7.59032 −0.241358
\(990\) 0 0
\(991\) 46.0389 1.46247 0.731237 0.682123i \(-0.238943\pi\)
0.731237 + 0.682123i \(0.238943\pi\)
\(992\) 0 0
\(993\) −7.83764 −0.248720
\(994\) 0 0
\(995\) −25.6521 −0.813228
\(996\) 0 0
\(997\) −6.30573 −0.199705 −0.0998523 0.995002i \(-0.531837\pi\)
−0.0998523 + 0.995002i \(0.531837\pi\)
\(998\) 0 0
\(999\) 0.918154 0.0290491
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 8016.2.a.be.1.3 11
4.3 odd 2 4008.2.a.k.1.3 11
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4008.2.a.k.1.3 11 4.3 odd 2
8016.2.a.be.1.3 11 1.1 even 1 trivial