Properties

Label 8016.2.a.q.1.1
Level 8016
Weight 2
Character 8016.1
Self dual yes
Analytic conductor 64.008
Analytic rank 1
Dimension 5
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: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.161121.1
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2004)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.544588\) of \(x^{5} - x^{4} - 6 x^{3} + 3 x^{2} + 5 x + 1\)
Character \(\chi\) \(=\) 8016.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -4.42860 q^{5} -1.88401 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -4.42860 q^{5} -1.88401 q^{7} +1.00000 q^{9} -1.84527 q^{11} -5.16786 q^{13} +4.42860 q^{15} +0.0891752 q^{17} -6.45461 q^{19} +1.88401 q^{21} +3.21151 q^{23} +14.6125 q^{25} -1.00000 q^{27} +3.02095 q^{29} +10.4636 q^{31} +1.84527 q^{33} +8.34353 q^{35} -1.51367 q^{37} +5.16786 q^{39} -7.82628 q^{41} +7.61344 q^{43} -4.42860 q^{45} +11.5711 q^{47} -3.45050 q^{49} -0.0891752 q^{51} -2.84132 q^{53} +8.17196 q^{55} +6.45461 q^{57} +5.92036 q^{59} +4.29853 q^{61} -1.88401 q^{63} +22.8864 q^{65} -13.3364 q^{67} -3.21151 q^{69} +11.9240 q^{71} -0.174243 q^{73} -14.6125 q^{75} +3.47651 q^{77} -2.45661 q^{79} +1.00000 q^{81} +11.2161 q^{83} -0.394921 q^{85} -3.02095 q^{87} -18.3673 q^{89} +9.73630 q^{91} -10.4636 q^{93} +28.5849 q^{95} -3.13202 q^{97} -1.84527 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - 5q^{3} - 7q^{5} + 2q^{7} + 5q^{9} + O(q^{10}) \) \( 5q - 5q^{3} - 7q^{5} + 2q^{7} + 5q^{9} + 5q^{11} - 8q^{13} + 7q^{15} - 7q^{17} - 2q^{19} - 2q^{21} + 13q^{23} + 2q^{25} - 5q^{27} - 11q^{29} + 12q^{31} - 5q^{33} + 12q^{35} - 7q^{37} + 8q^{39} - 12q^{41} - 7q^{45} + 19q^{47} - 9q^{49} + 7q^{51} - 21q^{53} + q^{55} + 2q^{57} + 7q^{59} - 6q^{61} + 2q^{63} + 14q^{65} - 10q^{67} - 13q^{69} + 35q^{71} - 8q^{73} - 2q^{75} - 6q^{77} + 5q^{81} + 11q^{83} + 5q^{85} + 11q^{87} - 32q^{89} - 5q^{91} - 12q^{93} + 19q^{95} + 11q^{97} + 5q^{99} + O(q^{100}) \)

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
\(5\) −4.42860 −1.98053 −0.990265 0.139196i \(-0.955548\pi\)
−0.990265 + 0.139196i \(0.955548\pi\)
\(6\) 0 0
\(7\) −1.88401 −0.712089 −0.356045 0.934469i \(-0.615875\pi\)
−0.356045 + 0.934469i \(0.615875\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.84527 −0.556370 −0.278185 0.960528i \(-0.589733\pi\)
−0.278185 + 0.960528i \(0.589733\pi\)
\(12\) 0 0
\(13\) −5.16786 −1.43331 −0.716653 0.697430i \(-0.754327\pi\)
−0.716653 + 0.697430i \(0.754327\pi\)
\(14\) 0 0
\(15\) 4.42860 1.14346
\(16\) 0 0
\(17\) 0.0891752 0.0216282 0.0108141 0.999942i \(-0.496558\pi\)
0.0108141 + 0.999942i \(0.496558\pi\)
\(18\) 0 0
\(19\) −6.45461 −1.48079 −0.740394 0.672173i \(-0.765361\pi\)
−0.740394 + 0.672173i \(0.765361\pi\)
\(20\) 0 0
\(21\) 1.88401 0.411125
\(22\) 0 0
\(23\) 3.21151 0.669646 0.334823 0.942281i \(-0.391324\pi\)
0.334823 + 0.942281i \(0.391324\pi\)
\(24\) 0 0
\(25\) 14.6125 2.92250
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 3.02095 0.560976 0.280488 0.959858i \(-0.409504\pi\)
0.280488 + 0.959858i \(0.409504\pi\)
\(30\) 0 0
\(31\) 10.4636 1.87932 0.939661 0.342106i \(-0.111140\pi\)
0.939661 + 0.342106i \(0.111140\pi\)
\(32\) 0 0
\(33\) 1.84527 0.321220
\(34\) 0 0
\(35\) 8.34353 1.41031
\(36\) 0 0
\(37\) −1.51367 −0.248845 −0.124423 0.992229i \(-0.539708\pi\)
−0.124423 + 0.992229i \(0.539708\pi\)
\(38\) 0 0
\(39\) 5.16786 0.827519
\(40\) 0 0
\(41\) −7.82628 −1.22226 −0.611130 0.791531i \(-0.709285\pi\)
−0.611130 + 0.791531i \(0.709285\pi\)
\(42\) 0 0
\(43\) 7.61344 1.16104 0.580520 0.814246i \(-0.302849\pi\)
0.580520 + 0.814246i \(0.302849\pi\)
\(44\) 0 0
\(45\) −4.42860 −0.660177
\(46\) 0 0
\(47\) 11.5711 1.68782 0.843910 0.536484i \(-0.180248\pi\)
0.843910 + 0.536484i \(0.180248\pi\)
\(48\) 0 0
\(49\) −3.45050 −0.492929
\(50\) 0 0
\(51\) −0.0891752 −0.0124870
\(52\) 0 0
\(53\) −2.84132 −0.390285 −0.195142 0.980775i \(-0.562517\pi\)
−0.195142 + 0.980775i \(0.562517\pi\)
\(54\) 0 0
\(55\) 8.17196 1.10191
\(56\) 0 0
\(57\) 6.45461 0.854934
\(58\) 0 0
\(59\) 5.92036 0.770766 0.385383 0.922757i \(-0.374069\pi\)
0.385383 + 0.922757i \(0.374069\pi\)
\(60\) 0 0
\(61\) 4.29853 0.550370 0.275185 0.961391i \(-0.411261\pi\)
0.275185 + 0.961391i \(0.411261\pi\)
\(62\) 0 0
\(63\) −1.88401 −0.237363
\(64\) 0 0
\(65\) 22.8864 2.83870
\(66\) 0 0
\(67\) −13.3364 −1.62930 −0.814649 0.579954i \(-0.803071\pi\)
−0.814649 + 0.579954i \(0.803071\pi\)
\(68\) 0 0
\(69\) −3.21151 −0.386620
\(70\) 0 0
\(71\) 11.9240 1.41511 0.707556 0.706657i \(-0.249798\pi\)
0.707556 + 0.706657i \(0.249798\pi\)
\(72\) 0 0
\(73\) −0.174243 −0.0203936 −0.0101968 0.999948i \(-0.503246\pi\)
−0.0101968 + 0.999948i \(0.503246\pi\)
\(74\) 0 0
\(75\) −14.6125 −1.68730
\(76\) 0 0
\(77\) 3.47651 0.396185
\(78\) 0 0
\(79\) −2.45661 −0.276390 −0.138195 0.990405i \(-0.544130\pi\)
−0.138195 + 0.990405i \(0.544130\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 11.2161 1.23113 0.615565 0.788086i \(-0.288928\pi\)
0.615565 + 0.788086i \(0.288928\pi\)
\(84\) 0 0
\(85\) −0.394921 −0.0428352
\(86\) 0 0
\(87\) −3.02095 −0.323879
\(88\) 0 0
\(89\) −18.3673 −1.94693 −0.973465 0.228835i \(-0.926508\pi\)
−0.973465 + 0.228835i \(0.926508\pi\)
\(90\) 0 0
\(91\) 9.73630 1.02064
\(92\) 0 0
\(93\) −10.4636 −1.08503
\(94\) 0 0
\(95\) 28.5849 2.93275
\(96\) 0 0
\(97\) −3.13202 −0.318009 −0.159004 0.987278i \(-0.550828\pi\)
−0.159004 + 0.987278i \(0.550828\pi\)
\(98\) 0 0
\(99\) −1.84527 −0.185457
\(100\) 0 0
\(101\) 1.76183 0.175309 0.0876544 0.996151i \(-0.472063\pi\)
0.0876544 + 0.996151i \(0.472063\pi\)
\(102\) 0 0
\(103\) −9.99153 −0.984494 −0.492247 0.870455i \(-0.663824\pi\)
−0.492247 + 0.870455i \(0.663824\pi\)
\(104\) 0 0
\(105\) −8.34353 −0.814245
\(106\) 0 0
\(107\) −3.65486 −0.353329 −0.176664 0.984271i \(-0.556531\pi\)
−0.176664 + 0.984271i \(0.556531\pi\)
\(108\) 0 0
\(109\) 8.58240 0.822045 0.411022 0.911625i \(-0.365172\pi\)
0.411022 + 0.911625i \(0.365172\pi\)
\(110\) 0 0
\(111\) 1.51367 0.143671
\(112\) 0 0
\(113\) 6.92953 0.651876 0.325938 0.945391i \(-0.394320\pi\)
0.325938 + 0.945391i \(0.394320\pi\)
\(114\) 0 0
\(115\) −14.2225 −1.32625
\(116\) 0 0
\(117\) −5.16786 −0.477768
\(118\) 0 0
\(119\) −0.168007 −0.0154012
\(120\) 0 0
\(121\) −7.59498 −0.690452
\(122\) 0 0
\(123\) 7.82628 0.705672
\(124\) 0 0
\(125\) −42.5699 −3.80756
\(126\) 0 0
\(127\) 12.7672 1.13291 0.566454 0.824094i \(-0.308315\pi\)
0.566454 + 0.824094i \(0.308315\pi\)
\(128\) 0 0
\(129\) −7.61344 −0.670326
\(130\) 0 0
\(131\) −1.75251 −0.153117 −0.0765587 0.997065i \(-0.524393\pi\)
−0.0765587 + 0.997065i \(0.524393\pi\)
\(132\) 0 0
\(133\) 12.1606 1.05445
\(134\) 0 0
\(135\) 4.42860 0.381153
\(136\) 0 0
\(137\) 11.7440 1.00335 0.501677 0.865055i \(-0.332716\pi\)
0.501677 + 0.865055i \(0.332716\pi\)
\(138\) 0 0
\(139\) 12.8695 1.09158 0.545789 0.837923i \(-0.316230\pi\)
0.545789 + 0.837923i \(0.316230\pi\)
\(140\) 0 0
\(141\) −11.5711 −0.974464
\(142\) 0 0
\(143\) 9.53609 0.797448
\(144\) 0 0
\(145\) −13.3786 −1.11103
\(146\) 0 0
\(147\) 3.45050 0.284593
\(148\) 0 0
\(149\) −8.74986 −0.716816 −0.358408 0.933565i \(-0.616680\pi\)
−0.358408 + 0.933565i \(0.616680\pi\)
\(150\) 0 0
\(151\) −0.761158 −0.0619422 −0.0309711 0.999520i \(-0.509860\pi\)
−0.0309711 + 0.999520i \(0.509860\pi\)
\(152\) 0 0
\(153\) 0.0891752 0.00720939
\(154\) 0 0
\(155\) −46.3392 −3.72205
\(156\) 0 0
\(157\) 22.5093 1.79643 0.898217 0.439552i \(-0.144863\pi\)
0.898217 + 0.439552i \(0.144863\pi\)
\(158\) 0 0
\(159\) 2.84132 0.225331
\(160\) 0 0
\(161\) −6.05052 −0.476848
\(162\) 0 0
\(163\) −18.9531 −1.48452 −0.742260 0.670112i \(-0.766246\pi\)
−0.742260 + 0.670112i \(0.766246\pi\)
\(164\) 0 0
\(165\) −8.17196 −0.636186
\(166\) 0 0
\(167\) 1.00000 0.0773823
\(168\) 0 0
\(169\) 13.7067 1.05436
\(170\) 0 0
\(171\) −6.45461 −0.493596
\(172\) 0 0
\(173\) −7.76587 −0.590428 −0.295214 0.955431i \(-0.595391\pi\)
−0.295214 + 0.955431i \(0.595391\pi\)
\(174\) 0 0
\(175\) −27.5301 −2.08108
\(176\) 0 0
\(177\) −5.92036 −0.445002
\(178\) 0 0
\(179\) 4.63802 0.346662 0.173331 0.984864i \(-0.444547\pi\)
0.173331 + 0.984864i \(0.444547\pi\)
\(180\) 0 0
\(181\) −11.9653 −0.889375 −0.444688 0.895686i \(-0.646685\pi\)
−0.444688 + 0.895686i \(0.646685\pi\)
\(182\) 0 0
\(183\) −4.29853 −0.317756
\(184\) 0 0
\(185\) 6.70342 0.492846
\(186\) 0 0
\(187\) −0.164552 −0.0120333
\(188\) 0 0
\(189\) 1.88401 0.137042
\(190\) 0 0
\(191\) −2.49863 −0.180794 −0.0903972 0.995906i \(-0.528814\pi\)
−0.0903972 + 0.995906i \(0.528814\pi\)
\(192\) 0 0
\(193\) −19.6727 −1.41607 −0.708036 0.706176i \(-0.750419\pi\)
−0.708036 + 0.706176i \(0.750419\pi\)
\(194\) 0 0
\(195\) −22.8864 −1.63893
\(196\) 0 0
\(197\) 21.0553 1.50012 0.750062 0.661367i \(-0.230023\pi\)
0.750062 + 0.661367i \(0.230023\pi\)
\(198\) 0 0
\(199\) 1.30427 0.0924572 0.0462286 0.998931i \(-0.485280\pi\)
0.0462286 + 0.998931i \(0.485280\pi\)
\(200\) 0 0
\(201\) 13.3364 0.940676
\(202\) 0 0
\(203\) −5.69150 −0.399465
\(204\) 0 0
\(205\) 34.6594 2.42072
\(206\) 0 0
\(207\) 3.21151 0.223215
\(208\) 0 0
\(209\) 11.9105 0.823866
\(210\) 0 0
\(211\) 18.2559 1.25679 0.628395 0.777894i \(-0.283712\pi\)
0.628395 + 0.777894i \(0.283712\pi\)
\(212\) 0 0
\(213\) −11.9240 −0.817016
\(214\) 0 0
\(215\) −33.7169 −2.29947
\(216\) 0 0
\(217\) −19.7136 −1.33825
\(218\) 0 0
\(219\) 0.174243 0.0117743
\(220\) 0 0
\(221\) −0.460844 −0.0309997
\(222\) 0 0
\(223\) −7.16390 −0.479730 −0.239865 0.970806i \(-0.577103\pi\)
−0.239865 + 0.970806i \(0.577103\pi\)
\(224\) 0 0
\(225\) 14.6125 0.974166
\(226\) 0 0
\(227\) −4.58453 −0.304286 −0.152143 0.988359i \(-0.548617\pi\)
−0.152143 + 0.988359i \(0.548617\pi\)
\(228\) 0 0
\(229\) −8.62848 −0.570186 −0.285093 0.958500i \(-0.592025\pi\)
−0.285093 + 0.958500i \(0.592025\pi\)
\(230\) 0 0
\(231\) −3.47651 −0.228738
\(232\) 0 0
\(233\) 11.3435 0.743139 0.371570 0.928405i \(-0.378820\pi\)
0.371570 + 0.928405i \(0.378820\pi\)
\(234\) 0 0
\(235\) −51.2438 −3.34278
\(236\) 0 0
\(237\) 2.45661 0.159574
\(238\) 0 0
\(239\) 7.95726 0.514712 0.257356 0.966317i \(-0.417149\pi\)
0.257356 + 0.966317i \(0.417149\pi\)
\(240\) 0 0
\(241\) −24.9868 −1.60954 −0.804770 0.593586i \(-0.797712\pi\)
−0.804770 + 0.593586i \(0.797712\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 15.2809 0.976260
\(246\) 0 0
\(247\) 33.3565 2.12242
\(248\) 0 0
\(249\) −11.2161 −0.710794
\(250\) 0 0
\(251\) 12.7487 0.804689 0.402344 0.915488i \(-0.368195\pi\)
0.402344 + 0.915488i \(0.368195\pi\)
\(252\) 0 0
\(253\) −5.92610 −0.372571
\(254\) 0 0
\(255\) 0.394921 0.0247309
\(256\) 0 0
\(257\) 14.1781 0.884404 0.442202 0.896916i \(-0.354197\pi\)
0.442202 + 0.896916i \(0.354197\pi\)
\(258\) 0 0
\(259\) 2.85177 0.177200
\(260\) 0 0
\(261\) 3.02095 0.186992
\(262\) 0 0
\(263\) 7.65792 0.472208 0.236104 0.971728i \(-0.424129\pi\)
0.236104 + 0.971728i \(0.424129\pi\)
\(264\) 0 0
\(265\) 12.5830 0.772970
\(266\) 0 0
\(267\) 18.3673 1.12406
\(268\) 0 0
\(269\) 19.2060 1.17101 0.585506 0.810668i \(-0.300896\pi\)
0.585506 + 0.810668i \(0.300896\pi\)
\(270\) 0 0
\(271\) 0.299771 0.0182098 0.00910489 0.999959i \(-0.497102\pi\)
0.00910489 + 0.999959i \(0.497102\pi\)
\(272\) 0 0
\(273\) −9.73630 −0.589268
\(274\) 0 0
\(275\) −26.9640 −1.62599
\(276\) 0 0
\(277\) −11.4831 −0.689955 −0.344978 0.938611i \(-0.612113\pi\)
−0.344978 + 0.938611i \(0.612113\pi\)
\(278\) 0 0
\(279\) 10.4636 0.626441
\(280\) 0 0
\(281\) 10.8670 0.648274 0.324137 0.946010i \(-0.394926\pi\)
0.324137 + 0.946010i \(0.394926\pi\)
\(282\) 0 0
\(283\) −9.59764 −0.570521 −0.285260 0.958450i \(-0.592080\pi\)
−0.285260 + 0.958450i \(0.592080\pi\)
\(284\) 0 0
\(285\) −28.5849 −1.69322
\(286\) 0 0
\(287\) 14.7448 0.870358
\(288\) 0 0
\(289\) −16.9920 −0.999532
\(290\) 0 0
\(291\) 3.13202 0.183602
\(292\) 0 0
\(293\) 31.9508 1.86659 0.933294 0.359114i \(-0.116921\pi\)
0.933294 + 0.359114i \(0.116921\pi\)
\(294\) 0 0
\(295\) −26.2189 −1.52652
\(296\) 0 0
\(297\) 1.84527 0.107073
\(298\) 0 0
\(299\) −16.5966 −0.959807
\(300\) 0 0
\(301\) −14.3438 −0.826764
\(302\) 0 0
\(303\) −1.76183 −0.101215
\(304\) 0 0
\(305\) −19.0365 −1.09002
\(306\) 0 0
\(307\) −11.5965 −0.661847 −0.330924 0.943658i \(-0.607360\pi\)
−0.330924 + 0.943658i \(0.607360\pi\)
\(308\) 0 0
\(309\) 9.99153 0.568398
\(310\) 0 0
\(311\) −5.68667 −0.322461 −0.161231 0.986917i \(-0.551546\pi\)
−0.161231 + 0.986917i \(0.551546\pi\)
\(312\) 0 0
\(313\) 6.50704 0.367800 0.183900 0.982945i \(-0.441128\pi\)
0.183900 + 0.982945i \(0.441128\pi\)
\(314\) 0 0
\(315\) 8.34353 0.470105
\(316\) 0 0
\(317\) 23.1953 1.30278 0.651390 0.758743i \(-0.274186\pi\)
0.651390 + 0.758743i \(0.274186\pi\)
\(318\) 0 0
\(319\) −5.57446 −0.312110
\(320\) 0 0
\(321\) 3.65486 0.203994
\(322\) 0 0
\(323\) −0.575591 −0.0320267
\(324\) 0 0
\(325\) −75.5152 −4.18883
\(326\) 0 0
\(327\) −8.58240 −0.474608
\(328\) 0 0
\(329\) −21.8001 −1.20188
\(330\) 0 0
\(331\) 19.7232 1.08409 0.542044 0.840350i \(-0.317651\pi\)
0.542044 + 0.840350i \(0.317651\pi\)
\(332\) 0 0
\(333\) −1.51367 −0.0829484
\(334\) 0 0
\(335\) 59.0615 3.22687
\(336\) 0 0
\(337\) −11.8567 −0.645874 −0.322937 0.946420i \(-0.604670\pi\)
−0.322937 + 0.946420i \(0.604670\pi\)
\(338\) 0 0
\(339\) −6.92953 −0.376361
\(340\) 0 0
\(341\) −19.3082 −1.04560
\(342\) 0 0
\(343\) 19.6889 1.06310
\(344\) 0 0
\(345\) 14.2225 0.765713
\(346\) 0 0
\(347\) 17.4703 0.937857 0.468928 0.883236i \(-0.344640\pi\)
0.468928 + 0.883236i \(0.344640\pi\)
\(348\) 0 0
\(349\) 30.2137 1.61730 0.808650 0.588290i \(-0.200199\pi\)
0.808650 + 0.588290i \(0.200199\pi\)
\(350\) 0 0
\(351\) 5.16786 0.275840
\(352\) 0 0
\(353\) −23.6819 −1.26046 −0.630229 0.776409i \(-0.717039\pi\)
−0.630229 + 0.776409i \(0.717039\pi\)
\(354\) 0 0
\(355\) −52.8064 −2.80267
\(356\) 0 0
\(357\) 0.168007 0.00889188
\(358\) 0 0
\(359\) −21.7084 −1.14573 −0.572863 0.819651i \(-0.694167\pi\)
−0.572863 + 0.819651i \(0.694167\pi\)
\(360\) 0 0
\(361\) 22.6620 1.19274
\(362\) 0 0
\(363\) 7.59498 0.398633
\(364\) 0 0
\(365\) 0.771653 0.0403902
\(366\) 0 0
\(367\) −30.4853 −1.59132 −0.795661 0.605742i \(-0.792876\pi\)
−0.795661 + 0.605742i \(0.792876\pi\)
\(368\) 0 0
\(369\) −7.82628 −0.407420
\(370\) 0 0
\(371\) 5.35307 0.277918
\(372\) 0 0
\(373\) 28.3119 1.46593 0.732967 0.680264i \(-0.238135\pi\)
0.732967 + 0.680264i \(0.238135\pi\)
\(374\) 0 0
\(375\) 42.5699 2.19830
\(376\) 0 0
\(377\) −15.6118 −0.804049
\(378\) 0 0
\(379\) −22.0469 −1.13247 −0.566237 0.824243i \(-0.691601\pi\)
−0.566237 + 0.824243i \(0.691601\pi\)
\(380\) 0 0
\(381\) −12.7672 −0.654084
\(382\) 0 0
\(383\) −20.3210 −1.03835 −0.519177 0.854667i \(-0.673761\pi\)
−0.519177 + 0.854667i \(0.673761\pi\)
\(384\) 0 0
\(385\) −15.3961 −0.784656
\(386\) 0 0
\(387\) 7.61344 0.387013
\(388\) 0 0
\(389\) −5.81405 −0.294784 −0.147392 0.989078i \(-0.547088\pi\)
−0.147392 + 0.989078i \(0.547088\pi\)
\(390\) 0 0
\(391\) 0.286387 0.0144832
\(392\) 0 0
\(393\) 1.75251 0.0884024
\(394\) 0 0
\(395\) 10.8793 0.547399
\(396\) 0 0
\(397\) −25.3084 −1.27019 −0.635096 0.772433i \(-0.719039\pi\)
−0.635096 + 0.772433i \(0.719039\pi\)
\(398\) 0 0
\(399\) −12.1606 −0.608789
\(400\) 0 0
\(401\) −21.2771 −1.06253 −0.531265 0.847206i \(-0.678283\pi\)
−0.531265 + 0.847206i \(0.678283\pi\)
\(402\) 0 0
\(403\) −54.0745 −2.69364
\(404\) 0 0
\(405\) −4.42860 −0.220059
\(406\) 0 0
\(407\) 2.79312 0.138450
\(408\) 0 0
\(409\) −39.4470 −1.95053 −0.975263 0.221047i \(-0.929053\pi\)
−0.975263 + 0.221047i \(0.929053\pi\)
\(410\) 0 0
\(411\) −11.7440 −0.579287
\(412\) 0 0
\(413\) −11.1540 −0.548854
\(414\) 0 0
\(415\) −49.6718 −2.43829
\(416\) 0 0
\(417\) −12.8695 −0.630223
\(418\) 0 0
\(419\) −15.0752 −0.736474 −0.368237 0.929732i \(-0.620038\pi\)
−0.368237 + 0.929732i \(0.620038\pi\)
\(420\) 0 0
\(421\) −20.2233 −0.985624 −0.492812 0.870136i \(-0.664031\pi\)
−0.492812 + 0.870136i \(0.664031\pi\)
\(422\) 0 0
\(423\) 11.5711 0.562607
\(424\) 0 0
\(425\) 1.30307 0.0632082
\(426\) 0 0
\(427\) −8.09848 −0.391913
\(428\) 0 0
\(429\) −9.53609 −0.460407
\(430\) 0 0
\(431\) −32.4054 −1.56091 −0.780456 0.625211i \(-0.785013\pi\)
−0.780456 + 0.625211i \(0.785013\pi\)
\(432\) 0 0
\(433\) −12.6597 −0.608387 −0.304194 0.952610i \(-0.598387\pi\)
−0.304194 + 0.952610i \(0.598387\pi\)
\(434\) 0 0
\(435\) 13.3786 0.641453
\(436\) 0 0
\(437\) −20.7290 −0.991604
\(438\) 0 0
\(439\) −33.4985 −1.59879 −0.799397 0.600803i \(-0.794848\pi\)
−0.799397 + 0.600803i \(0.794848\pi\)
\(440\) 0 0
\(441\) −3.45050 −0.164310
\(442\) 0 0
\(443\) 25.7875 1.22520 0.612601 0.790392i \(-0.290123\pi\)
0.612601 + 0.790392i \(0.290123\pi\)
\(444\) 0 0
\(445\) 81.3414 3.85595
\(446\) 0 0
\(447\) 8.74986 0.413854
\(448\) 0 0
\(449\) 0.134437 0.00634446 0.00317223 0.999995i \(-0.498990\pi\)
0.00317223 + 0.999995i \(0.498990\pi\)
\(450\) 0 0
\(451\) 14.4416 0.680028
\(452\) 0 0
\(453\) 0.761158 0.0357623
\(454\) 0 0
\(455\) −43.1182 −2.02141
\(456\) 0 0
\(457\) −10.7011 −0.500578 −0.250289 0.968171i \(-0.580526\pi\)
−0.250289 + 0.968171i \(0.580526\pi\)
\(458\) 0 0
\(459\) −0.0891752 −0.00416234
\(460\) 0 0
\(461\) −6.23604 −0.290441 −0.145221 0.989399i \(-0.546389\pi\)
−0.145221 + 0.989399i \(0.546389\pi\)
\(462\) 0 0
\(463\) 25.0329 1.16338 0.581689 0.813411i \(-0.302392\pi\)
0.581689 + 0.813411i \(0.302392\pi\)
\(464\) 0 0
\(465\) 46.3392 2.14893
\(466\) 0 0
\(467\) 6.25511 0.289452 0.144726 0.989472i \(-0.453770\pi\)
0.144726 + 0.989472i \(0.453770\pi\)
\(468\) 0 0
\(469\) 25.1259 1.16021
\(470\) 0 0
\(471\) −22.5093 −1.03717
\(472\) 0 0
\(473\) −14.0489 −0.645967
\(474\) 0 0
\(475\) −94.3179 −4.32760
\(476\) 0 0
\(477\) −2.84132 −0.130095
\(478\) 0 0
\(479\) 39.8726 1.82183 0.910913 0.412598i \(-0.135378\pi\)
0.910913 + 0.412598i \(0.135378\pi\)
\(480\) 0 0
\(481\) 7.82241 0.356671
\(482\) 0 0
\(483\) 6.05052 0.275308
\(484\) 0 0
\(485\) 13.8705 0.629826
\(486\) 0 0
\(487\) 3.81602 0.172921 0.0864603 0.996255i \(-0.472444\pi\)
0.0864603 + 0.996255i \(0.472444\pi\)
\(488\) 0 0
\(489\) 18.9531 0.857088
\(490\) 0 0
\(491\) 16.4535 0.742535 0.371267 0.928526i \(-0.378923\pi\)
0.371267 + 0.928526i \(0.378923\pi\)
\(492\) 0 0
\(493\) 0.269393 0.0121329
\(494\) 0 0
\(495\) 8.17196 0.367302
\(496\) 0 0
\(497\) −22.4649 −1.00769
\(498\) 0 0
\(499\) −14.3871 −0.644057 −0.322028 0.946730i \(-0.604365\pi\)
−0.322028 + 0.946730i \(0.604365\pi\)
\(500\) 0 0
\(501\) −1.00000 −0.0446767
\(502\) 0 0
\(503\) −38.7487 −1.72772 −0.863860 0.503732i \(-0.831960\pi\)
−0.863860 + 0.503732i \(0.831960\pi\)
\(504\) 0 0
\(505\) −7.80244 −0.347204
\(506\) 0 0
\(507\) −13.7067 −0.608737
\(508\) 0 0
\(509\) 13.9491 0.618282 0.309141 0.951016i \(-0.399958\pi\)
0.309141 + 0.951016i \(0.399958\pi\)
\(510\) 0 0
\(511\) 0.328276 0.0145221
\(512\) 0 0
\(513\) 6.45461 0.284978
\(514\) 0 0
\(515\) 44.2485 1.94982
\(516\) 0 0
\(517\) −21.3518 −0.939053
\(518\) 0 0
\(519\) 7.76587 0.340884
\(520\) 0 0
\(521\) 34.7198 1.52110 0.760550 0.649279i \(-0.224929\pi\)
0.760550 + 0.649279i \(0.224929\pi\)
\(522\) 0 0
\(523\) 7.18919 0.314362 0.157181 0.987570i \(-0.449759\pi\)
0.157181 + 0.987570i \(0.449759\pi\)
\(524\) 0 0
\(525\) 27.5301 1.20151
\(526\) 0 0
\(527\) 0.933096 0.0406463
\(528\) 0 0
\(529\) −12.6862 −0.551575
\(530\) 0 0
\(531\) 5.92036 0.256922
\(532\) 0 0
\(533\) 40.4451 1.75187
\(534\) 0 0
\(535\) 16.1859 0.699778
\(536\) 0 0
\(537\) −4.63802 −0.200145
\(538\) 0 0
\(539\) 6.36711 0.274251
\(540\) 0 0
\(541\) 12.5805 0.540878 0.270439 0.962737i \(-0.412831\pi\)
0.270439 + 0.962737i \(0.412831\pi\)
\(542\) 0 0
\(543\) 11.9653 0.513481
\(544\) 0 0
\(545\) −38.0080 −1.62808
\(546\) 0 0
\(547\) −1.54235 −0.0659462 −0.0329731 0.999456i \(-0.510498\pi\)
−0.0329731 + 0.999456i \(0.510498\pi\)
\(548\) 0 0
\(549\) 4.29853 0.183457
\(550\) 0 0
\(551\) −19.4990 −0.830686
\(552\) 0 0
\(553\) 4.62828 0.196815
\(554\) 0 0
\(555\) −6.70342 −0.284545
\(556\) 0 0
\(557\) −26.4661 −1.12140 −0.560702 0.828017i \(-0.689469\pi\)
−0.560702 + 0.828017i \(0.689469\pi\)
\(558\) 0 0
\(559\) −39.3452 −1.66412
\(560\) 0 0
\(561\) 0.164552 0.00694740
\(562\) 0 0
\(563\) −41.2657 −1.73914 −0.869572 0.493806i \(-0.835605\pi\)
−0.869572 + 0.493806i \(0.835605\pi\)
\(564\) 0 0
\(565\) −30.6881 −1.29106
\(566\) 0 0
\(567\) −1.88401 −0.0791210
\(568\) 0 0
\(569\) −24.6241 −1.03229 −0.516147 0.856500i \(-0.672634\pi\)
−0.516147 + 0.856500i \(0.672634\pi\)
\(570\) 0 0
\(571\) 35.5529 1.48784 0.743921 0.668268i \(-0.232964\pi\)
0.743921 + 0.668268i \(0.232964\pi\)
\(572\) 0 0
\(573\) 2.49863 0.104382
\(574\) 0 0
\(575\) 46.9281 1.95704
\(576\) 0 0
\(577\) −13.6068 −0.566459 −0.283229 0.959052i \(-0.591406\pi\)
−0.283229 + 0.959052i \(0.591406\pi\)
\(578\) 0 0
\(579\) 19.6727 0.817569
\(580\) 0 0
\(581\) −21.1313 −0.876675
\(582\) 0 0
\(583\) 5.24299 0.217143
\(584\) 0 0
\(585\) 22.8864 0.946234
\(586\) 0 0
\(587\) −4.46542 −0.184308 −0.0921538 0.995745i \(-0.529375\pi\)
−0.0921538 + 0.995745i \(0.529375\pi\)
\(588\) 0 0
\(589\) −67.5386 −2.78288
\(590\) 0 0
\(591\) −21.0553 −0.866097
\(592\) 0 0
\(593\) 35.7574 1.46838 0.734191 0.678943i \(-0.237562\pi\)
0.734191 + 0.678943i \(0.237562\pi\)
\(594\) 0 0
\(595\) 0.744036 0.0305025
\(596\) 0 0
\(597\) −1.30427 −0.0533802
\(598\) 0 0
\(599\) −17.4065 −0.711210 −0.355605 0.934636i \(-0.615725\pi\)
−0.355605 + 0.934636i \(0.615725\pi\)
\(600\) 0 0
\(601\) 34.6763 1.41447 0.707237 0.706976i \(-0.249941\pi\)
0.707237 + 0.706976i \(0.249941\pi\)
\(602\) 0 0
\(603\) −13.3364 −0.543099
\(604\) 0 0
\(605\) 33.6351 1.36746
\(606\) 0 0
\(607\) 22.9382 0.931034 0.465517 0.885039i \(-0.345868\pi\)
0.465517 + 0.885039i \(0.345868\pi\)
\(608\) 0 0
\(609\) 5.69150 0.230631
\(610\) 0 0
\(611\) −59.7979 −2.41916
\(612\) 0 0
\(613\) 19.1586 0.773810 0.386905 0.922120i \(-0.373544\pi\)
0.386905 + 0.922120i \(0.373544\pi\)
\(614\) 0 0
\(615\) −34.6594 −1.39760
\(616\) 0 0
\(617\) 17.9411 0.722283 0.361141 0.932511i \(-0.382387\pi\)
0.361141 + 0.932511i \(0.382387\pi\)
\(618\) 0 0
\(619\) 3.33510 0.134049 0.0670245 0.997751i \(-0.478649\pi\)
0.0670245 + 0.997751i \(0.478649\pi\)
\(620\) 0 0
\(621\) −3.21151 −0.128873
\(622\) 0 0
\(623\) 34.6042 1.38639
\(624\) 0 0
\(625\) 115.462 4.61850
\(626\) 0 0
\(627\) −11.9105 −0.475659
\(628\) 0 0
\(629\) −0.134982 −0.00538207
\(630\) 0 0
\(631\) −35.5155 −1.41385 −0.706925 0.707289i \(-0.749918\pi\)
−0.706925 + 0.707289i \(0.749918\pi\)
\(632\) 0 0
\(633\) −18.2559 −0.725609
\(634\) 0 0
\(635\) −56.5409 −2.24376
\(636\) 0 0
\(637\) 17.8317 0.706517
\(638\) 0 0
\(639\) 11.9240 0.471704
\(640\) 0 0
\(641\) 13.9112 0.549461 0.274730 0.961521i \(-0.411411\pi\)
0.274730 + 0.961521i \(0.411411\pi\)
\(642\) 0 0
\(643\) −8.57110 −0.338011 −0.169006 0.985615i \(-0.554056\pi\)
−0.169006 + 0.985615i \(0.554056\pi\)
\(644\) 0 0
\(645\) 33.7169 1.32760
\(646\) 0 0
\(647\) 44.5038 1.74963 0.874813 0.484461i \(-0.160984\pi\)
0.874813 + 0.484461i \(0.160984\pi\)
\(648\) 0 0
\(649\) −10.9247 −0.428831
\(650\) 0 0
\(651\) 19.7136 0.772637
\(652\) 0 0
\(653\) 4.37382 0.171161 0.0855805 0.996331i \(-0.472726\pi\)
0.0855805 + 0.996331i \(0.472726\pi\)
\(654\) 0 0
\(655\) 7.76116 0.303254
\(656\) 0 0
\(657\) −0.174243 −0.00679788
\(658\) 0 0
\(659\) −9.68235 −0.377171 −0.188585 0.982057i \(-0.560390\pi\)
−0.188585 + 0.982057i \(0.560390\pi\)
\(660\) 0 0
\(661\) 20.8248 0.809992 0.404996 0.914318i \(-0.367273\pi\)
0.404996 + 0.914318i \(0.367273\pi\)
\(662\) 0 0
\(663\) 0.460844 0.0178977
\(664\) 0 0
\(665\) −53.8542 −2.08838
\(666\) 0 0
\(667\) 9.70179 0.375655
\(668\) 0 0
\(669\) 7.16390 0.276972
\(670\) 0 0
\(671\) −7.93195 −0.306209
\(672\) 0 0
\(673\) 9.75633 0.376079 0.188039 0.982161i \(-0.439787\pi\)
0.188039 + 0.982161i \(0.439787\pi\)
\(674\) 0 0
\(675\) −14.6125 −0.562435
\(676\) 0 0
\(677\) −21.4827 −0.825647 −0.412823 0.910811i \(-0.635457\pi\)
−0.412823 + 0.910811i \(0.635457\pi\)
\(678\) 0 0
\(679\) 5.90077 0.226451
\(680\) 0 0
\(681\) 4.58453 0.175679
\(682\) 0 0
\(683\) 16.4925 0.631068 0.315534 0.948914i \(-0.397816\pi\)
0.315534 + 0.948914i \(0.397816\pi\)
\(684\) 0 0
\(685\) −52.0093 −1.98717
\(686\) 0 0
\(687\) 8.62848 0.329197
\(688\) 0 0
\(689\) 14.6835 0.559397
\(690\) 0 0
\(691\) −24.0700 −0.915666 −0.457833 0.889038i \(-0.651374\pi\)
−0.457833 + 0.889038i \(0.651374\pi\)
\(692\) 0 0
\(693\) 3.47651 0.132062
\(694\) 0 0
\(695\) −56.9939 −2.16190
\(696\) 0 0
\(697\) −0.697910 −0.0264352
\(698\) 0 0
\(699\) −11.3435 −0.429052
\(700\) 0 0
\(701\) −15.5056 −0.585640 −0.292820 0.956168i \(-0.594594\pi\)
−0.292820 + 0.956168i \(0.594594\pi\)
\(702\) 0 0
\(703\) 9.77013 0.368487
\(704\) 0 0
\(705\) 51.2438 1.92995
\(706\) 0 0
\(707\) −3.31931 −0.124835
\(708\) 0 0
\(709\) −32.9477 −1.23738 −0.618689 0.785636i \(-0.712336\pi\)
−0.618689 + 0.785636i \(0.712336\pi\)
\(710\) 0 0
\(711\) −2.45661 −0.0921301
\(712\) 0 0
\(713\) 33.6040 1.25848
\(714\) 0 0
\(715\) −42.2315 −1.57937
\(716\) 0 0
\(717\) −7.95726 −0.297169
\(718\) 0 0
\(719\) 33.1464 1.23615 0.618076 0.786119i \(-0.287913\pi\)
0.618076 + 0.786119i \(0.287913\pi\)
\(720\) 0 0
\(721\) 18.8242 0.701048
\(722\) 0 0
\(723\) 24.9868 0.929269
\(724\) 0 0
\(725\) 44.1435 1.63945
\(726\) 0 0
\(727\) 3.57131 0.132453 0.0662263 0.997805i \(-0.478904\pi\)
0.0662263 + 0.997805i \(0.478904\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0.678930 0.0251111
\(732\) 0 0
\(733\) −11.4545 −0.423081 −0.211541 0.977369i \(-0.567848\pi\)
−0.211541 + 0.977369i \(0.567848\pi\)
\(734\) 0 0
\(735\) −15.2809 −0.563644
\(736\) 0 0
\(737\) 24.6092 0.906493
\(738\) 0 0
\(739\) 1.93622 0.0712251 0.0356126 0.999366i \(-0.488662\pi\)
0.0356126 + 0.999366i \(0.488662\pi\)
\(740\) 0 0
\(741\) −33.3565 −1.22538
\(742\) 0 0
\(743\) 47.0852 1.72739 0.863695 0.504015i \(-0.168144\pi\)
0.863695 + 0.504015i \(0.168144\pi\)
\(744\) 0 0
\(745\) 38.7496 1.41968
\(746\) 0 0
\(747\) 11.2161 0.410377
\(748\) 0 0
\(749\) 6.88580 0.251602
\(750\) 0 0
\(751\) 38.9219 1.42028 0.710141 0.704060i \(-0.248631\pi\)
0.710141 + 0.704060i \(0.248631\pi\)
\(752\) 0 0
\(753\) −12.7487 −0.464587
\(754\) 0 0
\(755\) 3.37087 0.122678
\(756\) 0 0
\(757\) 25.1383 0.913669 0.456834 0.889552i \(-0.348983\pi\)
0.456834 + 0.889552i \(0.348983\pi\)
\(758\) 0 0
\(759\) 5.92610 0.215104
\(760\) 0 0
\(761\) −24.6330 −0.892945 −0.446473 0.894797i \(-0.647320\pi\)
−0.446473 + 0.894797i \(0.647320\pi\)
\(762\) 0 0
\(763\) −16.1693 −0.585369
\(764\) 0 0
\(765\) −0.394921 −0.0142784
\(766\) 0 0
\(767\) −30.5956 −1.10474
\(768\) 0 0
\(769\) 5.50988 0.198692 0.0993458 0.995053i \(-0.468325\pi\)
0.0993458 + 0.995053i \(0.468325\pi\)
\(770\) 0 0
\(771\) −14.1781 −0.510611
\(772\) 0 0
\(773\) −3.73543 −0.134354 −0.0671771 0.997741i \(-0.521399\pi\)
−0.0671771 + 0.997741i \(0.521399\pi\)
\(774\) 0 0
\(775\) 152.900 5.49232
\(776\) 0 0
\(777\) −2.85177 −0.102307
\(778\) 0 0
\(779\) 50.5156 1.80991
\(780\) 0 0
\(781\) −22.0029 −0.787326
\(782\) 0 0
\(783\) −3.02095 −0.107960
\(784\) 0 0
\(785\) −99.6845 −3.55789
\(786\) 0 0
\(787\) −20.4386 −0.728557 −0.364279 0.931290i \(-0.618684\pi\)
−0.364279 + 0.931290i \(0.618684\pi\)
\(788\) 0 0
\(789\) −7.65792 −0.272629
\(790\) 0 0
\(791\) −13.0553 −0.464194
\(792\) 0 0
\(793\) −22.2142 −0.788848
\(794\) 0 0
\(795\) −12.5830 −0.446275
\(796\) 0 0
\(797\) 12.0001 0.425065 0.212533 0.977154i \(-0.431829\pi\)
0.212533 + 0.977154i \(0.431829\pi\)
\(798\) 0 0
\(799\) 1.03186 0.0365045
\(800\) 0 0
\(801\) −18.3673 −0.648977
\(802\) 0 0
\(803\) 0.321526 0.0113464
\(804\) 0 0
\(805\) 26.7953 0.944411
\(806\) 0 0
\(807\) −19.2060 −0.676084
\(808\) 0 0
\(809\) 2.91890 0.102623 0.0513116 0.998683i \(-0.483660\pi\)
0.0513116 + 0.998683i \(0.483660\pi\)
\(810\) 0 0
\(811\) 10.1192 0.355333 0.177667 0.984091i \(-0.443145\pi\)
0.177667 + 0.984091i \(0.443145\pi\)
\(812\) 0 0
\(813\) −0.299771 −0.0105134
\(814\) 0 0
\(815\) 83.9356 2.94014
\(816\) 0 0
\(817\) −49.1418 −1.71925
\(818\) 0 0
\(819\) 9.73630 0.340214
\(820\) 0 0
\(821\) 49.3896 1.72371 0.861855 0.507155i \(-0.169303\pi\)
0.861855 + 0.507155i \(0.169303\pi\)
\(822\) 0 0
\(823\) −29.1619 −1.01652 −0.508261 0.861203i \(-0.669711\pi\)
−0.508261 + 0.861203i \(0.669711\pi\)
\(824\) 0 0
\(825\) 26.9640 0.938766
\(826\) 0 0
\(827\) 16.0347 0.557583 0.278791 0.960352i \(-0.410066\pi\)
0.278791 + 0.960352i \(0.410066\pi\)
\(828\) 0 0
\(829\) −25.0995 −0.871743 −0.435871 0.900009i \(-0.643560\pi\)
−0.435871 + 0.900009i \(0.643560\pi\)
\(830\) 0 0
\(831\) 11.4831 0.398346
\(832\) 0 0
\(833\) −0.307699 −0.0106611
\(834\) 0 0
\(835\) −4.42860 −0.153258
\(836\) 0 0
\(837\) −10.4636 −0.361676
\(838\) 0 0
\(839\) −30.9491 −1.06848 −0.534241 0.845332i \(-0.679402\pi\)
−0.534241 + 0.845332i \(0.679402\pi\)
\(840\) 0 0
\(841\) −19.8739 −0.685306
\(842\) 0 0
\(843\) −10.8670 −0.374281
\(844\) 0 0
\(845\) −60.7016 −2.08820
\(846\) 0 0
\(847\) 14.3090 0.491664
\(848\) 0 0
\(849\) 9.59764 0.329390
\(850\) 0 0
\(851\) −4.86115 −0.166638
\(852\) 0 0
\(853\) −38.7371 −1.32633 −0.663166 0.748472i \(-0.730788\pi\)
−0.663166 + 0.748472i \(0.730788\pi\)
\(854\) 0 0
\(855\) 28.5849 0.977582
\(856\) 0 0
\(857\) −45.5386 −1.55557 −0.777784 0.628532i \(-0.783656\pi\)
−0.777784 + 0.628532i \(0.783656\pi\)
\(858\) 0 0
\(859\) 20.4128 0.696477 0.348238 0.937406i \(-0.386780\pi\)
0.348238 + 0.937406i \(0.386780\pi\)
\(860\) 0 0
\(861\) −14.7448 −0.502501
\(862\) 0 0
\(863\) −27.2563 −0.927815 −0.463907 0.885884i \(-0.653553\pi\)
−0.463907 + 0.885884i \(0.653553\pi\)
\(864\) 0 0
\(865\) 34.3919 1.16936
\(866\) 0 0
\(867\) 16.9920 0.577080
\(868\) 0 0
\(869\) 4.53311 0.153775
\(870\) 0 0
\(871\) 68.9205 2.33528
\(872\) 0 0
\(873\) −3.13202 −0.106003
\(874\) 0 0
\(875\) 80.2021 2.71133
\(876\) 0 0
\(877\) −54.1311 −1.82788 −0.913939 0.405851i \(-0.866975\pi\)
−0.913939 + 0.405851i \(0.866975\pi\)
\(878\) 0 0
\(879\) −31.9508 −1.07767
\(880\) 0 0
\(881\) 49.1257 1.65509 0.827543 0.561403i \(-0.189738\pi\)
0.827543 + 0.561403i \(0.189738\pi\)
\(882\) 0 0
\(883\) 35.8325 1.20586 0.602930 0.797794i \(-0.294000\pi\)
0.602930 + 0.797794i \(0.294000\pi\)
\(884\) 0 0
\(885\) 26.2189 0.881339
\(886\) 0 0
\(887\) 32.2790 1.08382 0.541911 0.840436i \(-0.317701\pi\)
0.541911 + 0.840436i \(0.317701\pi\)
\(888\) 0 0
\(889\) −24.0536 −0.806731
\(890\) 0 0
\(891\) −1.84527 −0.0618189
\(892\) 0 0
\(893\) −74.6870 −2.49931
\(894\) 0 0
\(895\) −20.5399 −0.686574
\(896\) 0 0
\(897\) 16.5966 0.554145
\(898\) 0 0
\(899\) 31.6101 1.05425
\(900\) 0 0
\(901\) −0.253375 −0.00844114
\(902\) 0 0
\(903\) 14.3438 0.477332
\(904\) 0 0
\(905\) 52.9896 1.76143
\(906\) 0 0
\(907\) 12.9281 0.429269 0.214635 0.976694i \(-0.431144\pi\)
0.214635 + 0.976694i \(0.431144\pi\)
\(908\) 0 0
\(909\) 1.76183 0.0584362
\(910\) 0 0
\(911\) −41.4084 −1.37192 −0.685961 0.727638i \(-0.740618\pi\)
−0.685961 + 0.727638i \(0.740618\pi\)
\(912\) 0 0
\(913\) −20.6968 −0.684964
\(914\) 0 0
\(915\) 19.0365 0.629326
\(916\) 0 0
\(917\) 3.30175 0.109033
\(918\) 0 0
\(919\) 0.204782 0.00675514 0.00337757 0.999994i \(-0.498925\pi\)
0.00337757 + 0.999994i \(0.498925\pi\)
\(920\) 0 0
\(921\) 11.5965 0.382118
\(922\) 0 0
\(923\) −61.6212 −2.02829
\(924\) 0 0
\(925\) −22.1184 −0.727250
\(926\) 0 0
\(927\) −9.99153 −0.328165
\(928\) 0 0
\(929\) 29.8315 0.978741 0.489371 0.872076i \(-0.337227\pi\)
0.489371 + 0.872076i \(0.337227\pi\)
\(930\) 0 0
\(931\) 22.2716 0.729923
\(932\) 0 0
\(933\) 5.68667 0.186173
\(934\) 0 0
\(935\) 0.728736 0.0238322
\(936\) 0 0
\(937\) 27.7370 0.906128 0.453064 0.891478i \(-0.350331\pi\)
0.453064 + 0.891478i \(0.350331\pi\)
\(938\) 0 0
\(939\) −6.50704 −0.212349
\(940\) 0 0
\(941\) −47.2119 −1.53906 −0.769532 0.638609i \(-0.779510\pi\)
−0.769532 + 0.638609i \(0.779510\pi\)
\(942\) 0 0
\(943\) −25.1341 −0.818481
\(944\) 0 0
\(945\) −8.34353 −0.271415
\(946\) 0 0
\(947\) −16.6942 −0.542487 −0.271244 0.962511i \(-0.587435\pi\)
−0.271244 + 0.962511i \(0.587435\pi\)
\(948\) 0 0
\(949\) 0.900464 0.0292303
\(950\) 0 0
\(951\) −23.1953 −0.752160
\(952\) 0 0
\(953\) 12.0236 0.389482 0.194741 0.980855i \(-0.437613\pi\)
0.194741 + 0.980855i \(0.437613\pi\)
\(954\) 0 0
\(955\) 11.0654 0.358069
\(956\) 0 0
\(957\) 5.57446 0.180197
\(958\) 0 0
\(959\) −22.1258 −0.714478
\(960\) 0 0
\(961\) 78.4875 2.53185
\(962\) 0 0
\(963\) −3.65486 −0.117776
\(964\) 0 0
\(965\) 87.1225 2.80457
\(966\) 0 0
\(967\) 40.3542 1.29770 0.648852 0.760914i \(-0.275249\pi\)
0.648852 + 0.760914i \(0.275249\pi\)
\(968\) 0 0
\(969\) 0.575591 0.0184906
\(970\) 0 0
\(971\) −29.5121 −0.947088 −0.473544 0.880770i \(-0.657026\pi\)
−0.473544 + 0.880770i \(0.657026\pi\)
\(972\) 0 0
\(973\) −24.2463 −0.777301
\(974\) 0 0
\(975\) 75.5152 2.41842
\(976\) 0 0
\(977\) −0.577211 −0.0184666 −0.00923330 0.999957i \(-0.502939\pi\)
−0.00923330 + 0.999957i \(0.502939\pi\)
\(978\) 0 0
\(979\) 33.8926 1.08321
\(980\) 0 0
\(981\) 8.58240 0.274015
\(982\) 0 0
\(983\) 41.6804 1.32940 0.664698 0.747112i \(-0.268560\pi\)
0.664698 + 0.747112i \(0.268560\pi\)
\(984\) 0 0
\(985\) −93.2453 −2.97104
\(986\) 0 0
\(987\) 21.8001 0.693905
\(988\) 0 0
\(989\) 24.4506 0.777485
\(990\) 0 0
\(991\) 8.68185 0.275788 0.137894 0.990447i \(-0.455967\pi\)
0.137894 + 0.990447i \(0.455967\pi\)
\(992\) 0 0
\(993\) −19.7232 −0.625898
\(994\) 0 0
\(995\) −5.77609 −0.183114
\(996\) 0 0
\(997\) 33.2583 1.05330 0.526650 0.850082i \(-0.323448\pi\)
0.526650 + 0.850082i \(0.323448\pi\)
\(998\) 0 0
\(999\) 1.51367 0.0478903
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.q.1.1 5
4.3 odd 2 2004.2.a.b.1.1 5
12.11 even 2 6012.2.a.f.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2004.2.a.b.1.1 5 4.3 odd 2
6012.2.a.f.1.5 5 12.11 even 2
8016.2.a.q.1.1 5 1.1 even 1 trivial