Properties

Label 8001.2.a.g.1.1
Level 8001
Weight 2
Character 8001.1
Self dual yes
Analytic conductor 63.888
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8001.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.8883066572\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 8001.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000 q^{2} +2.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +O(q^{10})\) \(q+2.00000 q^{2} +2.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +2.00000 q^{10} -1.00000 q^{13} -2.00000 q^{14} -4.00000 q^{16} +6.00000 q^{17} -4.00000 q^{19} +2.00000 q^{20} +9.00000 q^{23} -4.00000 q^{25} -2.00000 q^{26} -2.00000 q^{28} +5.00000 q^{29} -3.00000 q^{31} -8.00000 q^{32} +12.0000 q^{34} -1.00000 q^{35} +1.00000 q^{37} -8.00000 q^{38} +10.0000 q^{41} -4.00000 q^{43} +18.0000 q^{46} +12.0000 q^{47} +1.00000 q^{49} -8.00000 q^{50} -2.00000 q^{52} +3.00000 q^{53} +10.0000 q^{58} +3.00000 q^{59} -3.00000 q^{61} -6.00000 q^{62} -8.00000 q^{64} -1.00000 q^{65} +12.0000 q^{67} +12.0000 q^{68} -2.00000 q^{70} +6.00000 q^{71} -7.00000 q^{73} +2.00000 q^{74} -8.00000 q^{76} -4.00000 q^{80} +20.0000 q^{82} -9.00000 q^{83} +6.00000 q^{85} -8.00000 q^{86} +1.00000 q^{89} +1.00000 q^{91} +18.0000 q^{92} +24.0000 q^{94} -4.00000 q^{95} +14.0000 q^{97} +2.00000 q^{98} +O(q^{100})\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(3\) 0 0
\(4\) 2.00000 1.00000
\(5\) 1.00000 0.447214 0.223607 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 2.00000 0.632456
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350 −0.138675 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 2.00000 0.447214
\(21\) 0 0
\(22\) 0 0
\(23\) 9.00000 1.87663 0.938315 0.345782i \(-0.112386\pi\)
0.938315 + 0.345782i \(0.112386\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) −2.00000 −0.377964
\(29\) 5.00000 0.928477 0.464238 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 0 0
\(31\) −3.00000 −0.538816 −0.269408 0.963026i \(-0.586828\pi\)
−0.269408 + 0.963026i \(0.586828\pi\)
\(32\) −8.00000 −1.41421
\(33\) 0 0
\(34\) 12.0000 2.05798
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) 1.00000 0.164399 0.0821995 0.996616i \(-0.473806\pi\)
0.0821995 + 0.996616i \(0.473806\pi\)
\(38\) −8.00000 −1.29777
\(39\) 0 0
\(40\) 0 0
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 18.0000 2.65396
\(47\) 12.0000 1.75038 0.875190 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −8.00000 −1.13137
\(51\) 0 0
\(52\) −2.00000 −0.277350
\(53\) 3.00000 0.412082 0.206041 0.978543i \(-0.433942\pi\)
0.206041 + 0.978543i \(0.433942\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 10.0000 1.31306
\(59\) 3.00000 0.390567 0.195283 0.980747i \(-0.437437\pi\)
0.195283 + 0.980747i \(0.437437\pi\)
\(60\) 0 0
\(61\) −3.00000 −0.384111 −0.192055 0.981384i \(-0.561515\pi\)
−0.192055 + 0.981384i \(0.561515\pi\)
\(62\) −6.00000 −0.762001
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) 12.0000 1.46603 0.733017 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) 12.0000 1.45521
\(69\) 0 0
\(70\) −2.00000 −0.239046
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 0 0
\(73\) −7.00000 −0.819288 −0.409644 0.912245i \(-0.634347\pi\)
−0.409644 + 0.912245i \(0.634347\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) −8.00000 −0.917663
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) −4.00000 −0.447214
\(81\) 0 0
\(82\) 20.0000 2.20863
\(83\) −9.00000 −0.987878 −0.493939 0.869496i \(-0.664443\pi\)
−0.493939 + 0.869496i \(0.664443\pi\)
\(84\) 0 0
\(85\) 6.00000 0.650791
\(86\) −8.00000 −0.862662
\(87\) 0 0
\(88\) 0 0
\(89\) 1.00000 0.106000 0.0529999 0.998595i \(-0.483122\pi\)
0.0529999 + 0.998595i \(0.483122\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 18.0000 1.87663
\(93\) 0 0
\(94\) 24.0000 2.47541
\(95\) −4.00000 −0.410391
\(96\) 0 0
\(97\) 14.0000 1.42148 0.710742 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 2.00000 0.202031
\(99\) 0 0
\(100\) −8.00000 −0.800000
\(101\) −15.0000 −1.49256 −0.746278 0.665635i \(-0.768161\pi\)
−0.746278 + 0.665635i \(0.768161\pi\)
\(102\) 0 0
\(103\) 16.0000 1.57653 0.788263 0.615338i \(-0.210980\pi\)
0.788263 + 0.615338i \(0.210980\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) 4.00000 0.383131 0.191565 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 4.00000 0.377964
\(113\) 10.0000 0.940721 0.470360 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) 0 0
\(115\) 9.00000 0.839254
\(116\) 10.0000 0.928477
\(117\) 0 0
\(118\) 6.00000 0.552345
\(119\) −6.00000 −0.550019
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) −6.00000 −0.543214
\(123\) 0 0
\(124\) −6.00000 −0.538816
\(125\) −9.00000 −0.804984
\(126\) 0 0
\(127\) 1.00000 0.0887357
\(128\) 0 0
\(129\) 0 0
\(130\) −2.00000 −0.175412
\(131\) 6.00000 0.524222 0.262111 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) 24.0000 2.07328
\(135\) 0 0
\(136\) 0 0
\(137\) 21.0000 1.79415 0.897076 0.441877i \(-0.145687\pi\)
0.897076 + 0.441877i \(0.145687\pi\)
\(138\) 0 0
\(139\) 2.00000 0.169638 0.0848189 0.996396i \(-0.472969\pi\)
0.0848189 + 0.996396i \(0.472969\pi\)
\(140\) −2.00000 −0.169031
\(141\) 0 0
\(142\) 12.0000 1.00702
\(143\) 0 0
\(144\) 0 0
\(145\) 5.00000 0.415227
\(146\) −14.0000 −1.15865
\(147\) 0 0
\(148\) 2.00000 0.164399
\(149\) −24.0000 −1.96616 −0.983078 0.183186i \(-0.941359\pi\)
−0.983078 + 0.183186i \(0.941359\pi\)
\(150\) 0 0
\(151\) −4.00000 −0.325515 −0.162758 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −3.00000 −0.240966
\(156\) 0 0
\(157\) 22.0000 1.75579 0.877896 0.478852i \(-0.158947\pi\)
0.877896 + 0.478852i \(0.158947\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) −8.00000 −0.632456
\(161\) −9.00000 −0.709299
\(162\) 0 0
\(163\) −13.0000 −1.01824 −0.509119 0.860696i \(-0.670029\pi\)
−0.509119 + 0.860696i \(0.670029\pi\)
\(164\) 20.0000 1.56174
\(165\) 0 0
\(166\) −18.0000 −1.39707
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 12.0000 0.920358
\(171\) 0 0
\(172\) −8.00000 −0.609994
\(173\) 14.0000 1.06440 0.532200 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) 0 0
\(175\) 4.00000 0.302372
\(176\) 0 0
\(177\) 0 0
\(178\) 2.00000 0.149906
\(179\) −16.0000 −1.19590 −0.597948 0.801535i \(-0.704017\pi\)
−0.597948 + 0.801535i \(0.704017\pi\)
\(180\) 0 0
\(181\) −8.00000 −0.594635 −0.297318 0.954779i \(-0.596092\pi\)
−0.297318 + 0.954779i \(0.596092\pi\)
\(182\) 2.00000 0.148250
\(183\) 0 0
\(184\) 0 0
\(185\) 1.00000 0.0735215
\(186\) 0 0
\(187\) 0 0
\(188\) 24.0000 1.75038
\(189\) 0 0
\(190\) −8.00000 −0.580381
\(191\) 6.00000 0.434145 0.217072 0.976156i \(-0.430349\pi\)
0.217072 + 0.976156i \(0.430349\pi\)
\(192\) 0 0
\(193\) 4.00000 0.287926 0.143963 0.989583i \(-0.454015\pi\)
0.143963 + 0.989583i \(0.454015\pi\)
\(194\) 28.0000 2.01028
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 0 0
\(199\) −11.0000 −0.779769 −0.389885 0.920864i \(-0.627485\pi\)
−0.389885 + 0.920864i \(0.627485\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −30.0000 −2.11079
\(203\) −5.00000 −0.350931
\(204\) 0 0
\(205\) 10.0000 0.698430
\(206\) 32.0000 2.22955
\(207\) 0 0
\(208\) 4.00000 0.277350
\(209\) 0 0
\(210\) 0 0
\(211\) 15.0000 1.03264 0.516321 0.856395i \(-0.327301\pi\)
0.516321 + 0.856395i \(0.327301\pi\)
\(212\) 6.00000 0.412082
\(213\) 0 0
\(214\) 24.0000 1.64061
\(215\) −4.00000 −0.272798
\(216\) 0 0
\(217\) 3.00000 0.203653
\(218\) 8.00000 0.541828
\(219\) 0 0
\(220\) 0 0
\(221\) −6.00000 −0.403604
\(222\) 0 0
\(223\) −10.0000 −0.669650 −0.334825 0.942280i \(-0.608677\pi\)
−0.334825 + 0.942280i \(0.608677\pi\)
\(224\) 8.00000 0.534522
\(225\) 0 0
\(226\) 20.0000 1.33038
\(227\) 18.0000 1.19470 0.597351 0.801980i \(-0.296220\pi\)
0.597351 + 0.801980i \(0.296220\pi\)
\(228\) 0 0
\(229\) −20.0000 −1.32164 −0.660819 0.750546i \(-0.729791\pi\)
−0.660819 + 0.750546i \(0.729791\pi\)
\(230\) 18.0000 1.18688
\(231\) 0 0
\(232\) 0 0
\(233\) −29.0000 −1.89985 −0.949927 0.312473i \(-0.898843\pi\)
−0.949927 + 0.312473i \(0.898843\pi\)
\(234\) 0 0
\(235\) 12.0000 0.782794
\(236\) 6.00000 0.390567
\(237\) 0 0
\(238\) −12.0000 −0.777844
\(239\) −7.00000 −0.452792 −0.226396 0.974035i \(-0.572694\pi\)
−0.226396 + 0.974035i \(0.572694\pi\)
\(240\) 0 0
\(241\) 18.0000 1.15948 0.579741 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) −22.0000 −1.41421
\(243\) 0 0
\(244\) −6.00000 −0.384111
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) 4.00000 0.254514
\(248\) 0 0
\(249\) 0 0
\(250\) −18.0000 −1.13842
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 2.00000 0.125491
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 3.00000 0.187135 0.0935674 0.995613i \(-0.470173\pi\)
0.0935674 + 0.995613i \(0.470173\pi\)
\(258\) 0 0
\(259\) −1.00000 −0.0621370
\(260\) −2.00000 −0.124035
\(261\) 0 0
\(262\) 12.0000 0.741362
\(263\) 26.0000 1.60323 0.801614 0.597841i \(-0.203975\pi\)
0.801614 + 0.597841i \(0.203975\pi\)
\(264\) 0 0
\(265\) 3.00000 0.184289
\(266\) 8.00000 0.490511
\(267\) 0 0
\(268\) 24.0000 1.46603
\(269\) −24.0000 −1.46331 −0.731653 0.681677i \(-0.761251\pi\)
−0.731653 + 0.681677i \(0.761251\pi\)
\(270\) 0 0
\(271\) 11.0000 0.668202 0.334101 0.942537i \(-0.391567\pi\)
0.334101 + 0.942537i \(0.391567\pi\)
\(272\) −24.0000 −1.45521
\(273\) 0 0
\(274\) 42.0000 2.53731
\(275\) 0 0
\(276\) 0 0
\(277\) −24.0000 −1.44202 −0.721010 0.692925i \(-0.756322\pi\)
−0.721010 + 0.692925i \(0.756322\pi\)
\(278\) 4.00000 0.239904
\(279\) 0 0
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) 16.0000 0.951101 0.475551 0.879688i \(-0.342249\pi\)
0.475551 + 0.879688i \(0.342249\pi\)
\(284\) 12.0000 0.712069
\(285\) 0 0
\(286\) 0 0
\(287\) −10.0000 −0.590281
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 10.0000 0.587220
\(291\) 0 0
\(292\) −14.0000 −0.819288
\(293\) 18.0000 1.05157 0.525786 0.850617i \(-0.323771\pi\)
0.525786 + 0.850617i \(0.323771\pi\)
\(294\) 0 0
\(295\) 3.00000 0.174667
\(296\) 0 0
\(297\) 0 0
\(298\) −48.0000 −2.78057
\(299\) −9.00000 −0.520483
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) −8.00000 −0.460348
\(303\) 0 0
\(304\) 16.0000 0.917663
\(305\) −3.00000 −0.171780
\(306\) 0 0
\(307\) 22.0000 1.25561 0.627803 0.778372i \(-0.283954\pi\)
0.627803 + 0.778372i \(0.283954\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −6.00000 −0.340777
\(311\) 15.0000 0.850572 0.425286 0.905059i \(-0.360174\pi\)
0.425286 + 0.905059i \(0.360174\pi\)
\(312\) 0 0
\(313\) −16.0000 −0.904373 −0.452187 0.891923i \(-0.649356\pi\)
−0.452187 + 0.891923i \(0.649356\pi\)
\(314\) 44.0000 2.48306
\(315\) 0 0
\(316\) 0 0
\(317\) −10.0000 −0.561656 −0.280828 0.959758i \(-0.590609\pi\)
−0.280828 + 0.959758i \(0.590609\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −8.00000 −0.447214
\(321\) 0 0
\(322\) −18.0000 −1.00310
\(323\) −24.0000 −1.33540
\(324\) 0 0
\(325\) 4.00000 0.221880
\(326\) −26.0000 −1.44001
\(327\) 0 0
\(328\) 0 0
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) −18.0000 −0.987878
\(333\) 0 0
\(334\) 16.0000 0.875481
\(335\) 12.0000 0.655630
\(336\) 0 0
\(337\) −18.0000 −0.980522 −0.490261 0.871576i \(-0.663099\pi\)
−0.490261 + 0.871576i \(0.663099\pi\)
\(338\) −24.0000 −1.30543
\(339\) 0 0
\(340\) 12.0000 0.650791
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 28.0000 1.50529
\(347\) −23.0000 −1.23470 −0.617352 0.786687i \(-0.711795\pi\)
−0.617352 + 0.786687i \(0.711795\pi\)
\(348\) 0 0
\(349\) −4.00000 −0.214115 −0.107058 0.994253i \(-0.534143\pi\)
−0.107058 + 0.994253i \(0.534143\pi\)
\(350\) 8.00000 0.427618
\(351\) 0 0
\(352\) 0 0
\(353\) 12.0000 0.638696 0.319348 0.947638i \(-0.396536\pi\)
0.319348 + 0.947638i \(0.396536\pi\)
\(354\) 0 0
\(355\) 6.00000 0.318447
\(356\) 2.00000 0.106000
\(357\) 0 0
\(358\) −32.0000 −1.69125
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) −16.0000 −0.840941
\(363\) 0 0
\(364\) 2.00000 0.104828
\(365\) −7.00000 −0.366397
\(366\) 0 0
\(367\) −7.00000 −0.365397 −0.182699 0.983169i \(-0.558483\pi\)
−0.182699 + 0.983169i \(0.558483\pi\)
\(368\) −36.0000 −1.87663
\(369\) 0 0
\(370\) 2.00000 0.103975
\(371\) −3.00000 −0.155752
\(372\) 0 0
\(373\) −22.0000 −1.13912 −0.569558 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −5.00000 −0.257513
\(378\) 0 0
\(379\) 12.0000 0.616399 0.308199 0.951322i \(-0.400274\pi\)
0.308199 + 0.951322i \(0.400274\pi\)
\(380\) −8.00000 −0.410391
\(381\) 0 0
\(382\) 12.0000 0.613973
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 8.00000 0.407189
\(387\) 0 0
\(388\) 28.0000 1.42148
\(389\) −12.0000 −0.608424 −0.304212 0.952604i \(-0.598393\pi\)
−0.304212 + 0.952604i \(0.598393\pi\)
\(390\) 0 0
\(391\) 54.0000 2.73090
\(392\) 0 0
\(393\) 0 0
\(394\) 24.0000 1.20910
\(395\) 0 0
\(396\) 0 0
\(397\) −13.0000 −0.652451 −0.326226 0.945292i \(-0.605777\pi\)
−0.326226 + 0.945292i \(0.605777\pi\)
\(398\) −22.0000 −1.10276
\(399\) 0 0
\(400\) 16.0000 0.800000
\(401\) −13.0000 −0.649189 −0.324595 0.945853i \(-0.605228\pi\)
−0.324595 + 0.945853i \(0.605228\pi\)
\(402\) 0 0
\(403\) 3.00000 0.149441
\(404\) −30.0000 −1.49256
\(405\) 0 0
\(406\) −10.0000 −0.496292
\(407\) 0 0
\(408\) 0 0
\(409\) −30.0000 −1.48340 −0.741702 0.670729i \(-0.765981\pi\)
−0.741702 + 0.670729i \(0.765981\pi\)
\(410\) 20.0000 0.987730
\(411\) 0 0
\(412\) 32.0000 1.57653
\(413\) −3.00000 −0.147620
\(414\) 0 0
\(415\) −9.00000 −0.441793
\(416\) 8.00000 0.392232
\(417\) 0 0
\(418\) 0 0
\(419\) −36.0000 −1.75872 −0.879358 0.476162i \(-0.842028\pi\)
−0.879358 + 0.476162i \(0.842028\pi\)
\(420\) 0 0
\(421\) 18.0000 0.877266 0.438633 0.898666i \(-0.355463\pi\)
0.438633 + 0.898666i \(0.355463\pi\)
\(422\) 30.0000 1.46038
\(423\) 0 0
\(424\) 0 0
\(425\) −24.0000 −1.16417
\(426\) 0 0
\(427\) 3.00000 0.145180
\(428\) 24.0000 1.16008
\(429\) 0 0
\(430\) −8.00000 −0.385794
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 38.0000 1.82616 0.913082 0.407777i \(-0.133696\pi\)
0.913082 + 0.407777i \(0.133696\pi\)
\(434\) 6.00000 0.288009
\(435\) 0 0
\(436\) 8.00000 0.383131
\(437\) −36.0000 −1.72211
\(438\) 0 0
\(439\) −10.0000 −0.477274 −0.238637 0.971109i \(-0.576701\pi\)
−0.238637 + 0.971109i \(0.576701\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −12.0000 −0.570782
\(443\) 14.0000 0.665160 0.332580 0.943075i \(-0.392081\pi\)
0.332580 + 0.943075i \(0.392081\pi\)
\(444\) 0 0
\(445\) 1.00000 0.0474045
\(446\) −20.0000 −0.947027
\(447\) 0 0
\(448\) 8.00000 0.377964
\(449\) 28.0000 1.32140 0.660701 0.750649i \(-0.270259\pi\)
0.660701 + 0.750649i \(0.270259\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 20.0000 0.940721
\(453\) 0 0
\(454\) 36.0000 1.68956
\(455\) 1.00000 0.0468807
\(456\) 0 0
\(457\) −29.0000 −1.35656 −0.678281 0.734802i \(-0.737275\pi\)
−0.678281 + 0.734802i \(0.737275\pi\)
\(458\) −40.0000 −1.86908
\(459\) 0 0
\(460\) 18.0000 0.839254
\(461\) 14.0000 0.652045 0.326023 0.945362i \(-0.394291\pi\)
0.326023 + 0.945362i \(0.394291\pi\)
\(462\) 0 0
\(463\) −19.0000 −0.883005 −0.441502 0.897260i \(-0.645554\pi\)
−0.441502 + 0.897260i \(0.645554\pi\)
\(464\) −20.0000 −0.928477
\(465\) 0 0
\(466\) −58.0000 −2.68680
\(467\) 9.00000 0.416470 0.208235 0.978079i \(-0.433228\pi\)
0.208235 + 0.978079i \(0.433228\pi\)
\(468\) 0 0
\(469\) −12.0000 −0.554109
\(470\) 24.0000 1.10704
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 16.0000 0.734130
\(476\) −12.0000 −0.550019
\(477\) 0 0
\(478\) −14.0000 −0.640345
\(479\) −34.0000 −1.55350 −0.776750 0.629809i \(-0.783133\pi\)
−0.776750 + 0.629809i \(0.783133\pi\)
\(480\) 0 0
\(481\) −1.00000 −0.0455961
\(482\) 36.0000 1.63976
\(483\) 0 0
\(484\) −22.0000 −1.00000
\(485\) 14.0000 0.635707
\(486\) 0 0
\(487\) −14.0000 −0.634401 −0.317200 0.948359i \(-0.602743\pi\)
−0.317200 + 0.948359i \(0.602743\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 2.00000 0.0903508
\(491\) 24.0000 1.08310 0.541552 0.840667i \(-0.317837\pi\)
0.541552 + 0.840667i \(0.317837\pi\)
\(492\) 0 0
\(493\) 30.0000 1.35113
\(494\) 8.00000 0.359937
\(495\) 0 0
\(496\) 12.0000 0.538816
\(497\) −6.00000 −0.269137
\(498\) 0 0
\(499\) −26.0000 −1.16392 −0.581960 0.813217i \(-0.697714\pi\)
−0.581960 + 0.813217i \(0.697714\pi\)
\(500\) −18.0000 −0.804984
\(501\) 0 0
\(502\) −24.0000 −1.07117
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) −15.0000 −0.667491
\(506\) 0 0
\(507\) 0 0
\(508\) 2.00000 0.0887357
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 7.00000 0.309662
\(512\) 32.0000 1.41421
\(513\) 0 0
\(514\) 6.00000 0.264649
\(515\) 16.0000 0.705044
\(516\) 0 0
\(517\) 0 0
\(518\) −2.00000 −0.0878750
\(519\) 0 0
\(520\) 0 0
\(521\) −4.00000 −0.175243 −0.0876216 0.996154i \(-0.527927\pi\)
−0.0876216 + 0.996154i \(0.527927\pi\)
\(522\) 0 0
\(523\) 33.0000 1.44299 0.721495 0.692420i \(-0.243455\pi\)
0.721495 + 0.692420i \(0.243455\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) 52.0000 2.26731
\(527\) −18.0000 −0.784092
\(528\) 0 0
\(529\) 58.0000 2.52174
\(530\) 6.00000 0.260623
\(531\) 0 0
\(532\) 8.00000 0.346844
\(533\) −10.0000 −0.433148
\(534\) 0 0
\(535\) 12.0000 0.518805
\(536\) 0 0
\(537\) 0 0
\(538\) −48.0000 −2.06943
\(539\) 0 0
\(540\) 0 0
\(541\) −32.0000 −1.37579 −0.687894 0.725811i \(-0.741464\pi\)
−0.687894 + 0.725811i \(0.741464\pi\)
\(542\) 22.0000 0.944981
\(543\) 0 0
\(544\) −48.0000 −2.05798
\(545\) 4.00000 0.171341
\(546\) 0 0
\(547\) −26.0000 −1.11168 −0.555840 0.831289i \(-0.687603\pi\)
−0.555840 + 0.831289i \(0.687603\pi\)
\(548\) 42.0000 1.79415
\(549\) 0 0
\(550\) 0 0
\(551\) −20.0000 −0.852029
\(552\) 0 0
\(553\) 0 0
\(554\) −48.0000 −2.03932
\(555\) 0 0
\(556\) 4.00000 0.169638
\(557\) −20.0000 −0.847427 −0.423714 0.905796i \(-0.639274\pi\)
−0.423714 + 0.905796i \(0.639274\pi\)
\(558\) 0 0
\(559\) 4.00000 0.169182
\(560\) 4.00000 0.169031
\(561\) 0 0
\(562\) 20.0000 0.843649
\(563\) 8.00000 0.337160 0.168580 0.985688i \(-0.446082\pi\)
0.168580 + 0.985688i \(0.446082\pi\)
\(564\) 0 0
\(565\) 10.0000 0.420703
\(566\) 32.0000 1.34506
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 16.0000 0.669579 0.334790 0.942293i \(-0.391335\pi\)
0.334790 + 0.942293i \(0.391335\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −20.0000 −0.834784
\(575\) −36.0000 −1.50130
\(576\) 0 0
\(577\) −37.0000 −1.54033 −0.770165 0.637845i \(-0.779826\pi\)
−0.770165 + 0.637845i \(0.779826\pi\)
\(578\) 38.0000 1.58059
\(579\) 0 0
\(580\) 10.0000 0.415227
\(581\) 9.00000 0.373383
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 36.0000 1.48715
\(587\) 14.0000 0.577842 0.288921 0.957353i \(-0.406704\pi\)
0.288921 + 0.957353i \(0.406704\pi\)
\(588\) 0 0
\(589\) 12.0000 0.494451
\(590\) 6.00000 0.247016
\(591\) 0 0
\(592\) −4.00000 −0.164399
\(593\) 23.0000 0.944497 0.472248 0.881466i \(-0.343443\pi\)
0.472248 + 0.881466i \(0.343443\pi\)
\(594\) 0 0
\(595\) −6.00000 −0.245976
\(596\) −48.0000 −1.96616
\(597\) 0 0
\(598\) −18.0000 −0.736075
\(599\) 19.0000 0.776319 0.388159 0.921592i \(-0.373111\pi\)
0.388159 + 0.921592i \(0.373111\pi\)
\(600\) 0 0
\(601\) −12.0000 −0.489490 −0.244745 0.969587i \(-0.578704\pi\)
−0.244745 + 0.969587i \(0.578704\pi\)
\(602\) 8.00000 0.326056
\(603\) 0 0
\(604\) −8.00000 −0.325515
\(605\) −11.0000 −0.447214
\(606\) 0 0
\(607\) 13.0000 0.527654 0.263827 0.964570i \(-0.415015\pi\)
0.263827 + 0.964570i \(0.415015\pi\)
\(608\) 32.0000 1.29777
\(609\) 0 0
\(610\) −6.00000 −0.242933
\(611\) −12.0000 −0.485468
\(612\) 0 0
\(613\) −2.00000 −0.0807792 −0.0403896 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) 44.0000 1.77570
\(615\) 0 0
\(616\) 0 0
\(617\) 13.0000 0.523360 0.261680 0.965155i \(-0.415723\pi\)
0.261680 + 0.965155i \(0.415723\pi\)
\(618\) 0 0
\(619\) 8.00000 0.321547 0.160774 0.986991i \(-0.448601\pi\)
0.160774 + 0.986991i \(0.448601\pi\)
\(620\) −6.00000 −0.240966
\(621\) 0 0
\(622\) 30.0000 1.20289
\(623\) −1.00000 −0.0400642
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) −32.0000 −1.27898
\(627\) 0 0
\(628\) 44.0000 1.75579
\(629\) 6.00000 0.239236
\(630\) 0 0
\(631\) −20.0000 −0.796187 −0.398094 0.917345i \(-0.630328\pi\)
−0.398094 + 0.917345i \(0.630328\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −20.0000 −0.794301
\(635\) 1.00000 0.0396838
\(636\) 0 0
\(637\) −1.00000 −0.0396214
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −34.0000 −1.34292 −0.671460 0.741041i \(-0.734332\pi\)
−0.671460 + 0.741041i \(0.734332\pi\)
\(642\) 0 0
\(643\) −28.0000 −1.10421 −0.552106 0.833774i \(-0.686176\pi\)
−0.552106 + 0.833774i \(0.686176\pi\)
\(644\) −18.0000 −0.709299
\(645\) 0 0
\(646\) −48.0000 −1.88853
\(647\) −9.00000 −0.353827 −0.176913 0.984226i \(-0.556611\pi\)
−0.176913 + 0.984226i \(0.556611\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 8.00000 0.313786
\(651\) 0 0
\(652\) −26.0000 −1.01824
\(653\) −2.00000 −0.0782660 −0.0391330 0.999234i \(-0.512460\pi\)
−0.0391330 + 0.999234i \(0.512460\pi\)
\(654\) 0 0
\(655\) 6.00000 0.234439
\(656\) −40.0000 −1.56174
\(657\) 0 0
\(658\) −24.0000 −0.935617
\(659\) 15.0000 0.584317 0.292159 0.956370i \(-0.405627\pi\)
0.292159 + 0.956370i \(0.405627\pi\)
\(660\) 0 0
\(661\) 49.0000 1.90588 0.952940 0.303160i \(-0.0980418\pi\)
0.952940 + 0.303160i \(0.0980418\pi\)
\(662\) −40.0000 −1.55464
\(663\) 0 0
\(664\) 0 0
\(665\) 4.00000 0.155113
\(666\) 0 0
\(667\) 45.0000 1.74241
\(668\) 16.0000 0.619059
\(669\) 0 0
\(670\) 24.0000 0.927201
\(671\) 0 0
\(672\) 0 0
\(673\) 3.00000 0.115642 0.0578208 0.998327i \(-0.481585\pi\)
0.0578208 + 0.998327i \(0.481585\pi\)
\(674\) −36.0000 −1.38667
\(675\) 0 0
\(676\) −24.0000 −0.923077
\(677\) −2.00000 −0.0768662 −0.0384331 0.999261i \(-0.512237\pi\)
−0.0384331 + 0.999261i \(0.512237\pi\)
\(678\) 0 0
\(679\) −14.0000 −0.537271
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) 0 0
\(685\) 21.0000 0.802369
\(686\) −2.00000 −0.0763604
\(687\) 0 0
\(688\) 16.0000 0.609994
\(689\) −3.00000 −0.114291
\(690\) 0 0
\(691\) −6.00000 −0.228251 −0.114125 0.993466i \(-0.536407\pi\)
−0.114125 + 0.993466i \(0.536407\pi\)
\(692\) 28.0000 1.06440
\(693\) 0 0
\(694\) −46.0000 −1.74614
\(695\) 2.00000 0.0758643
\(696\) 0 0
\(697\) 60.0000 2.27266
\(698\) −8.00000 −0.302804
\(699\) 0 0
\(700\) 8.00000 0.302372
\(701\) −39.0000 −1.47301 −0.736505 0.676432i \(-0.763525\pi\)
−0.736505 + 0.676432i \(0.763525\pi\)
\(702\) 0 0
\(703\) −4.00000 −0.150863
\(704\) 0 0
\(705\) 0 0
\(706\) 24.0000 0.903252
\(707\) 15.0000 0.564133
\(708\) 0 0
\(709\) −46.0000 −1.72757 −0.863783 0.503864i \(-0.831911\pi\)
−0.863783 + 0.503864i \(0.831911\pi\)
\(710\) 12.0000 0.450352
\(711\) 0 0
\(712\) 0 0
\(713\) −27.0000 −1.01116
\(714\) 0 0
\(715\) 0 0
\(716\) −32.0000 −1.19590
\(717\) 0 0
\(718\) 48.0000 1.79134
\(719\) 36.0000 1.34257 0.671287 0.741198i \(-0.265742\pi\)
0.671287 + 0.741198i \(0.265742\pi\)
\(720\) 0 0
\(721\) −16.0000 −0.595871
\(722\) −6.00000 −0.223297
\(723\) 0 0
\(724\) −16.0000 −0.594635
\(725\) −20.0000 −0.742781
\(726\) 0 0
\(727\) −16.0000 −0.593407 −0.296704 0.954970i \(-0.595887\pi\)
−0.296704 + 0.954970i \(0.595887\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −14.0000 −0.518163
\(731\) −24.0000 −0.887672
\(732\) 0 0
\(733\) −5.00000 −0.184679 −0.0923396 0.995728i \(-0.529435\pi\)
−0.0923396 + 0.995728i \(0.529435\pi\)
\(734\) −14.0000 −0.516749
\(735\) 0 0
\(736\) −72.0000 −2.65396
\(737\) 0 0
\(738\) 0 0
\(739\) −11.0000 −0.404642 −0.202321 0.979319i \(-0.564848\pi\)
−0.202321 + 0.979319i \(0.564848\pi\)
\(740\) 2.00000 0.0735215
\(741\) 0 0
\(742\) −6.00000 −0.220267
\(743\) −41.0000 −1.50414 −0.752072 0.659081i \(-0.770945\pi\)
−0.752072 + 0.659081i \(0.770945\pi\)
\(744\) 0 0
\(745\) −24.0000 −0.879292
\(746\) −44.0000 −1.61095
\(747\) 0 0
\(748\) 0 0
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) 28.0000 1.02173 0.510867 0.859660i \(-0.329324\pi\)
0.510867 + 0.859660i \(0.329324\pi\)
\(752\) −48.0000 −1.75038
\(753\) 0 0
\(754\) −10.0000 −0.364179
\(755\) −4.00000 −0.145575
\(756\) 0 0
\(757\) −7.00000 −0.254419 −0.127210 0.991876i \(-0.540602\pi\)
−0.127210 + 0.991876i \(0.540602\pi\)
\(758\) 24.0000 0.871719
\(759\) 0 0
\(760\) 0 0
\(761\) 3.00000 0.108750 0.0543750 0.998521i \(-0.482683\pi\)
0.0543750 + 0.998521i \(0.482683\pi\)
\(762\) 0 0
\(763\) −4.00000 −0.144810
\(764\) 12.0000 0.434145
\(765\) 0 0
\(766\) 48.0000 1.73431
\(767\) −3.00000 −0.108324
\(768\) 0 0
\(769\) 46.0000 1.65880 0.829401 0.558653i \(-0.188682\pi\)
0.829401 + 0.558653i \(0.188682\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 8.00000 0.287926
\(773\) −48.0000 −1.72644 −0.863220 0.504828i \(-0.831556\pi\)
−0.863220 + 0.504828i \(0.831556\pi\)
\(774\) 0 0
\(775\) 12.0000 0.431053
\(776\) 0 0
\(777\) 0 0
\(778\) −24.0000 −0.860442
\(779\) −40.0000 −1.43315
\(780\) 0 0
\(781\) 0 0
\(782\) 108.000 3.86207
\(783\) 0 0
\(784\) −4.00000 −0.142857
\(785\) 22.0000 0.785214
\(786\) 0 0
\(787\) 39.0000 1.39020 0.695100 0.718913i \(-0.255360\pi\)
0.695100 + 0.718913i \(0.255360\pi\)
\(788\) 24.0000 0.854965
\(789\) 0 0
\(790\) 0 0
\(791\) −10.0000 −0.355559
\(792\) 0 0
\(793\) 3.00000 0.106533
\(794\) −26.0000 −0.922705
\(795\) 0 0
\(796\) −22.0000 −0.779769
\(797\) −48.0000 −1.70025 −0.850124 0.526583i \(-0.823473\pi\)
−0.850124 + 0.526583i \(0.823473\pi\)
\(798\) 0 0
\(799\) 72.0000 2.54718
\(800\) 32.0000 1.13137
\(801\) 0 0
\(802\) −26.0000 −0.918092
\(803\) 0 0
\(804\) 0 0
\(805\) −9.00000 −0.317208
\(806\) 6.00000 0.211341
\(807\) 0 0
\(808\) 0 0
\(809\) 44.0000 1.54696 0.773479 0.633822i \(-0.218515\pi\)
0.773479 + 0.633822i \(0.218515\pi\)
\(810\) 0 0
\(811\) 19.0000 0.667180 0.333590 0.942718i \(-0.391740\pi\)
0.333590 + 0.942718i \(0.391740\pi\)
\(812\) −10.0000 −0.350931
\(813\) 0 0
\(814\) 0 0
\(815\) −13.0000 −0.455370
\(816\) 0 0
\(817\) 16.0000 0.559769
\(818\) −60.0000 −2.09785
\(819\) 0 0
\(820\) 20.0000 0.698430
\(821\) 31.0000 1.08191 0.540954 0.841052i \(-0.318063\pi\)
0.540954 + 0.841052i \(0.318063\pi\)
\(822\) 0 0
\(823\) −24.0000 −0.836587 −0.418294 0.908312i \(-0.637372\pi\)
−0.418294 + 0.908312i \(0.637372\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) −6.00000 −0.208767
\(827\) 15.0000 0.521601 0.260801 0.965393i \(-0.416014\pi\)
0.260801 + 0.965393i \(0.416014\pi\)
\(828\) 0 0
\(829\) −8.00000 −0.277851 −0.138926 0.990303i \(-0.544365\pi\)
−0.138926 + 0.990303i \(0.544365\pi\)
\(830\) −18.0000 −0.624789
\(831\) 0 0
\(832\) 8.00000 0.277350
\(833\) 6.00000 0.207888
\(834\) 0 0
\(835\) 8.00000 0.276851
\(836\) 0 0
\(837\) 0 0
\(838\) −72.0000 −2.48720
\(839\) −25.0000 −0.863096 −0.431548 0.902090i \(-0.642032\pi\)
−0.431548 + 0.902090i \(0.642032\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 36.0000 1.24064
\(843\) 0 0
\(844\) 30.0000 1.03264
\(845\) −12.0000 −0.412813
\(846\) 0 0
\(847\) 11.0000 0.377964
\(848\) −12.0000 −0.412082
\(849\) 0 0
\(850\) −48.0000 −1.64639
\(851\) 9.00000 0.308516
\(852\) 0 0
\(853\) 4.00000 0.136957 0.0684787 0.997653i \(-0.478185\pi\)
0.0684787 + 0.997653i \(0.478185\pi\)
\(854\) 6.00000 0.205316
\(855\) 0 0
\(856\) 0 0
\(857\) 51.0000 1.74213 0.871063 0.491171i \(-0.163431\pi\)
0.871063 + 0.491171i \(0.163431\pi\)
\(858\) 0 0
\(859\) −50.0000 −1.70598 −0.852989 0.521929i \(-0.825213\pi\)
−0.852989 + 0.521929i \(0.825213\pi\)
\(860\) −8.00000 −0.272798
\(861\) 0 0
\(862\) 0 0
\(863\) 31.0000 1.05525 0.527626 0.849477i \(-0.323082\pi\)
0.527626 + 0.849477i \(0.323082\pi\)
\(864\) 0 0
\(865\) 14.0000 0.476014
\(866\) 76.0000 2.58259
\(867\) 0 0
\(868\) 6.00000 0.203653
\(869\) 0 0
\(870\) 0 0
\(871\) −12.0000 −0.406604
\(872\) 0 0
\(873\) 0 0
\(874\) −72.0000 −2.43544
\(875\) 9.00000 0.304256
\(876\) 0 0
\(877\) 38.0000 1.28317 0.641584 0.767052i \(-0.278277\pi\)
0.641584 + 0.767052i \(0.278277\pi\)
\(878\) −20.0000 −0.674967
\(879\) 0 0
\(880\) 0 0
\(881\) 3.00000 0.101073 0.0505363 0.998722i \(-0.483907\pi\)
0.0505363 + 0.998722i \(0.483907\pi\)
\(882\) 0 0
\(883\) −43.0000 −1.44707 −0.723533 0.690290i \(-0.757483\pi\)
−0.723533 + 0.690290i \(0.757483\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) 28.0000 0.940678
\(887\) 47.0000 1.57811 0.789053 0.614325i \(-0.210572\pi\)
0.789053 + 0.614325i \(0.210572\pi\)
\(888\) 0 0
\(889\) −1.00000 −0.0335389
\(890\) 2.00000 0.0670402
\(891\) 0 0
\(892\) −20.0000 −0.669650
\(893\) −48.0000 −1.60626
\(894\) 0 0
\(895\) −16.0000 −0.534821
\(896\) 0 0
\(897\) 0 0
\(898\) 56.0000 1.86874
\(899\) −15.0000 −0.500278
\(900\) 0 0
\(901\) 18.0000 0.599667
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −8.00000 −0.265929
\(906\) 0 0
\(907\) −13.0000 −0.431658 −0.215829 0.976431i \(-0.569245\pi\)
−0.215829 + 0.976431i \(0.569245\pi\)
\(908\) 36.0000 1.19470
\(909\) 0 0
\(910\) 2.00000 0.0662994
\(911\) −48.0000 −1.59031 −0.795155 0.606406i \(-0.792611\pi\)
−0.795155 + 0.606406i \(0.792611\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −58.0000 −1.91847
\(915\) 0 0
\(916\) −40.0000 −1.32164
\(917\) −6.00000 −0.198137
\(918\) 0 0
\(919\) 45.0000 1.48441 0.742207 0.670171i \(-0.233779\pi\)
0.742207 + 0.670171i \(0.233779\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 28.0000 0.922131
\(923\) −6.00000 −0.197492
\(924\) 0 0
\(925\) −4.00000 −0.131519
\(926\) −38.0000 −1.24876
\(927\) 0 0
\(928\) −40.0000 −1.31306
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 0 0
\(931\) −4.00000 −0.131095
\(932\) −58.0000 −1.89985
\(933\) 0 0
\(934\) 18.0000 0.588978
\(935\) 0 0
\(936\) 0 0
\(937\) −8.00000 −0.261349 −0.130674 0.991425i \(-0.541714\pi\)
−0.130674 + 0.991425i \(0.541714\pi\)
\(938\) −24.0000 −0.783628
\(939\) 0 0
\(940\) 24.0000 0.782794
\(941\) −14.0000 −0.456387 −0.228193 0.973616i \(-0.573282\pi\)
−0.228193 + 0.973616i \(0.573282\pi\)
\(942\) 0 0
\(943\) 90.0000 2.93080
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) 0 0
\(947\) −12.0000 −0.389948 −0.194974 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) 0 0
\(949\) 7.00000 0.227230
\(950\) 32.0000 1.03822
\(951\) 0 0
\(952\) 0 0
\(953\) 56.0000 1.81402 0.907009 0.421111i \(-0.138360\pi\)
0.907009 + 0.421111i \(0.138360\pi\)
\(954\) 0 0
\(955\) 6.00000 0.194155
\(956\) −14.0000 −0.452792
\(957\) 0 0
\(958\) −68.0000 −2.19698
\(959\) −21.0000 −0.678125
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) −2.00000 −0.0644826
\(963\) 0 0
\(964\) 36.0000 1.15948
\(965\) 4.00000 0.128765
\(966\) 0 0
\(967\) 14.0000 0.450210 0.225105 0.974335i \(-0.427728\pi\)
0.225105 + 0.974335i \(0.427728\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 28.0000 0.899026
\(971\) −16.0000 −0.513464 −0.256732 0.966483i \(-0.582646\pi\)
−0.256732 + 0.966483i \(0.582646\pi\)
\(972\) 0 0
\(973\) −2.00000 −0.0641171
\(974\) −28.0000 −0.897178
\(975\) 0 0
\(976\) 12.0000 0.384111
\(977\) 16.0000 0.511885 0.255943 0.966692i \(-0.417614\pi\)
0.255943 + 0.966692i \(0.417614\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 2.00000 0.0638877
\(981\) 0 0
\(982\) 48.0000 1.53174
\(983\) 30.0000 0.956851 0.478426 0.878128i \(-0.341208\pi\)
0.478426 + 0.878128i \(0.341208\pi\)
\(984\) 0 0
\(985\) 12.0000 0.382352
\(986\) 60.0000 1.91079
\(987\) 0 0
\(988\) 8.00000 0.254514
\(989\) −36.0000 −1.14473
\(990\) 0 0
\(991\) −48.0000 −1.52477 −0.762385 0.647124i \(-0.775972\pi\)
−0.762385 + 0.647124i \(0.775972\pi\)
\(992\) 24.0000 0.762001
\(993\) 0 0
\(994\) −12.0000 −0.380617
\(995\) −11.0000 −0.348723
\(996\) 0 0
\(997\) 26.0000 0.823428 0.411714 0.911313i \(-0.364930\pi\)
0.411714 + 0.911313i \(0.364930\pi\)
\(998\) −52.0000 −1.64603
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 8001.2.a.g.1.1 yes 1
3.2 odd 2 8001.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8001.2.a.b.1.1 1 3.2 odd 2
8001.2.a.g.1.1 yes 1 1.1 even 1 trivial