Properties

Label 2016.3.g.a.1135.3
Level 2016
Weight 3
Character 2016.1135
Analytic conductor 54.932
Analytic rank 0
Dimension 4
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2016.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(54.9320212950\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-7})\)
Defining polynomial: \(x^{4} + 6 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1135.3
Root \(0.707107 + 1.87083i\) of defining polynomial
Character \(\chi\) \(=\) 2016.1135
Dual form 2016.3.g.a.1135.2

$q$-expansion

\(f(q)\) \(=\) \(q+1.54985i q^{5} +2.64575i q^{7} +O(q^{10})\) \(q+1.54985i q^{5} +2.64575i q^{7} -4.48528 q^{11} +1.54985i q^{13} -23.6569 q^{17} +24.8701 q^{19} -35.2248i q^{23} +22.5980 q^{25} -22.4499i q^{29} -46.7156i q^{31} -4.10051 q^{35} +58.5826i q^{37} +26.9706 q^{41} +17.1716 q^{43} +36.1326i q^{47} -7.00000 q^{49} +97.8149i q^{53} -6.95149i q^{55} +61.5563 q^{59} +37.6825i q^{61} -2.40202 q^{65} +33.3726 q^{67} -102.199i q^{71} +69.3137 q^{73} -11.8669i q^{77} +38.7005i q^{79} +3.61522 q^{83} -36.6645i q^{85} -44.0589 q^{89} -4.10051 q^{91} +38.5447i q^{95} +96.1076 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 16q^{11} - 72q^{17} - 8q^{19} - 68q^{25} - 56q^{35} + 40q^{41} + 80q^{43} - 28q^{49} + 184q^{59} - 168q^{65} + 224q^{67} + 232q^{73} + 88q^{83} - 312q^{89} - 56q^{91} - 136q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2016\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(1765\) \(1793\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 1.54985i 0.309969i 0.987917 + 0.154985i \(0.0495328\pi\)
−0.987917 + 0.154985i \(0.950467\pi\)
\(6\) 0 0
\(7\) 2.64575i 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −4.48528 −0.407753 −0.203876 0.978997i \(-0.565354\pi\)
−0.203876 + 0.978997i \(0.565354\pi\)
\(12\) 0 0
\(13\) 1.54985i 0.119219i 0.998222 + 0.0596094i \(0.0189855\pi\)
−0.998222 + 0.0596094i \(0.981014\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −23.6569 −1.39158 −0.695790 0.718245i \(-0.744945\pi\)
−0.695790 + 0.718245i \(0.744945\pi\)
\(18\) 0 0
\(19\) 24.8701 1.30895 0.654475 0.756083i \(-0.272890\pi\)
0.654475 + 0.756083i \(0.272890\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 35.2248i − 1.53151i −0.643132 0.765756i \(-0.722365\pi\)
0.643132 0.765756i \(-0.277635\pi\)
\(24\) 0 0
\(25\) 22.5980 0.903919
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 22.4499i − 0.774136i −0.922051 0.387068i \(-0.873488\pi\)
0.922051 0.387068i \(-0.126512\pi\)
\(30\) 0 0
\(31\) − 46.7156i − 1.50696i −0.657473 0.753478i \(-0.728375\pi\)
0.657473 0.753478i \(-0.271625\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −4.10051 −0.117157
\(36\) 0 0
\(37\) 58.5826i 1.58331i 0.610966 + 0.791657i \(0.290781\pi\)
−0.610966 + 0.791657i \(0.709219\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 26.9706 0.657819 0.328909 0.944362i \(-0.393319\pi\)
0.328909 + 0.944362i \(0.393319\pi\)
\(42\) 0 0
\(43\) 17.1716 0.399339 0.199669 0.979863i \(-0.436013\pi\)
0.199669 + 0.979863i \(0.436013\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 36.1326i 0.768780i 0.923171 + 0.384390i \(0.125588\pi\)
−0.923171 + 0.384390i \(0.874412\pi\)
\(48\) 0 0
\(49\) −7.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 97.8149i 1.84556i 0.385322 + 0.922782i \(0.374090\pi\)
−0.385322 + 0.922782i \(0.625910\pi\)
\(54\) 0 0
\(55\) − 6.95149i − 0.126391i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 61.5563 1.04333 0.521664 0.853151i \(-0.325311\pi\)
0.521664 + 0.853151i \(0.325311\pi\)
\(60\) 0 0
\(61\) 37.6825i 0.617746i 0.951103 + 0.308873i \(0.0999518\pi\)
−0.951103 + 0.308873i \(0.900048\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.40202 −0.0369542
\(66\) 0 0
\(67\) 33.3726 0.498098 0.249049 0.968491i \(-0.419882\pi\)
0.249049 + 0.968491i \(0.419882\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 102.199i − 1.43942i −0.694277 0.719708i \(-0.744276\pi\)
0.694277 0.719708i \(-0.255724\pi\)
\(72\) 0 0
\(73\) 69.3137 0.949503 0.474751 0.880120i \(-0.342538\pi\)
0.474751 + 0.880120i \(0.342538\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 11.8669i − 0.154116i
\(78\) 0 0
\(79\) 38.7005i 0.489880i 0.969538 + 0.244940i \(0.0787682\pi\)
−0.969538 + 0.244940i \(0.921232\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 3.61522 0.0435569 0.0217785 0.999763i \(-0.493067\pi\)
0.0217785 + 0.999763i \(0.493067\pi\)
\(84\) 0 0
\(85\) − 36.6645i − 0.431347i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −44.0589 −0.495044 −0.247522 0.968882i \(-0.579616\pi\)
−0.247522 + 0.968882i \(0.579616\pi\)
\(90\) 0 0
\(91\) −4.10051 −0.0450605
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 38.5447i 0.405734i
\(96\) 0 0
\(97\) 96.1076 0.990800 0.495400 0.868665i \(-0.335021\pi\)
0.495400 + 0.868665i \(0.335021\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 19.6162i 0.194219i 0.995274 + 0.0971097i \(0.0309598\pi\)
−0.995274 + 0.0971097i \(0.969040\pi\)
\(102\) 0 0
\(103\) 43.0841i 0.418293i 0.977884 + 0.209146i \(0.0670685\pi\)
−0.977884 + 0.209146i \(0.932932\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 15.5980 0.145776 0.0728878 0.997340i \(-0.476779\pi\)
0.0728878 + 0.997340i \(0.476779\pi\)
\(108\) 0 0
\(109\) − 3.85180i − 0.0353376i −0.999844 0.0176688i \(-0.994376\pi\)
0.999844 0.0176688i \(-0.00562445\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 13.7746 0.121899 0.0609496 0.998141i \(-0.480587\pi\)
0.0609496 + 0.998141i \(0.480587\pi\)
\(114\) 0 0
\(115\) 54.5929 0.474721
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 62.5902i − 0.525968i
\(120\) 0 0
\(121\) −100.882 −0.833738
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 73.7695i 0.590156i
\(126\) 0 0
\(127\) 125.025i 0.984445i 0.870469 + 0.492223i \(0.163815\pi\)
−0.870469 + 0.492223i \(0.836185\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 100.350 0.766033 0.383016 0.923742i \(-0.374885\pi\)
0.383016 + 0.923742i \(0.374885\pi\)
\(132\) 0 0
\(133\) 65.8000i 0.494737i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −57.3137 −0.418348 −0.209174 0.977878i \(-0.567078\pi\)
−0.209174 + 0.977878i \(0.567078\pi\)
\(138\) 0 0
\(139\) 183.664 1.32132 0.660662 0.750684i \(-0.270276\pi\)
0.660662 + 0.750684i \(0.270276\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) − 6.95149i − 0.0486118i
\(144\) 0 0
\(145\) 34.7939 0.239958
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 192.310i 1.29067i 0.763900 + 0.645335i \(0.223282\pi\)
−0.763900 + 0.645335i \(0.776718\pi\)
\(150\) 0 0
\(151\) − 114.753i − 0.759954i −0.924996 0.379977i \(-0.875932\pi\)
0.924996 0.379977i \(-0.124068\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 72.4020 0.467110
\(156\) 0 0
\(157\) 212.146i 1.35125i 0.737245 + 0.675625i \(0.236126\pi\)
−0.737245 + 0.675625i \(0.763874\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 93.1960 0.578857
\(162\) 0 0
\(163\) 240.534 1.47567 0.737835 0.674982i \(-0.235848\pi\)
0.737835 + 0.674982i \(0.235848\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 212.101i 1.27006i 0.772486 + 0.635032i \(0.219013\pi\)
−0.772486 + 0.635032i \(0.780987\pi\)
\(168\) 0 0
\(169\) 166.598 0.985787
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 182.213i − 1.05325i −0.850096 0.526627i \(-0.823456\pi\)
0.850096 0.526627i \(-0.176544\pi\)
\(174\) 0 0
\(175\) 59.7886i 0.341649i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 57.2061 0.319587 0.159793 0.987150i \(-0.448917\pi\)
0.159793 + 0.987150i \(0.448917\pi\)
\(180\) 0 0
\(181\) − 326.212i − 1.80228i −0.433533 0.901138i \(-0.642733\pi\)
0.433533 0.901138i \(-0.357267\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −90.7939 −0.490778
\(186\) 0 0
\(187\) 106.108 0.567421
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 97.0628i − 0.508182i −0.967180 0.254091i \(-0.918224\pi\)
0.967180 0.254091i \(-0.0817763\pi\)
\(192\) 0 0
\(193\) 157.304 0.815045 0.407522 0.913195i \(-0.366393\pi\)
0.407522 + 0.913195i \(0.366393\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 124.117i − 0.630034i −0.949086 0.315017i \(-0.897990\pi\)
0.949086 0.315017i \(-0.102010\pi\)
\(198\) 0 0
\(199\) 180.975i 0.909421i 0.890639 + 0.454710i \(0.150257\pi\)
−0.890639 + 0.454710i \(0.849743\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 59.3970 0.292596
\(204\) 0 0
\(205\) 41.8002i 0.203903i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −111.549 −0.533728
\(210\) 0 0
\(211\) −164.049 −0.777482 −0.388741 0.921347i \(-0.627090\pi\)
−0.388741 + 0.921347i \(0.627090\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 26.6133i 0.123783i
\(216\) 0 0
\(217\) 123.598 0.569576
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 36.6645i − 0.165903i
\(222\) 0 0
\(223\) 10.5830i 0.0474574i 0.999718 + 0.0237287i \(0.00755379\pi\)
−0.999718 + 0.0237287i \(0.992446\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 105.806 0.466106 0.233053 0.972464i \(-0.425128\pi\)
0.233053 + 0.972464i \(0.425128\pi\)
\(228\) 0 0
\(229\) − 74.8788i − 0.326982i −0.986545 0.163491i \(-0.947725\pi\)
0.986545 0.163491i \(-0.0522754\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 419.137 1.79887 0.899436 0.437053i \(-0.143978\pi\)
0.899436 + 0.437053i \(0.143978\pi\)
\(234\) 0 0
\(235\) −56.0000 −0.238298
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 148.318i 0.620577i 0.950642 + 0.310288i \(0.100426\pi\)
−0.950642 + 0.310288i \(0.899574\pi\)
\(240\) 0 0
\(241\) −459.872 −1.90818 −0.954092 0.299515i \(-0.903175\pi\)
−0.954092 + 0.299515i \(0.903175\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 10.8489i − 0.0442813i
\(246\) 0 0
\(247\) 38.5447i 0.156052i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 124.919 0.497685 0.248842 0.968544i \(-0.419950\pi\)
0.248842 + 0.968544i \(0.419950\pi\)
\(252\) 0 0
\(253\) 157.993i 0.624478i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 427.352 1.66285 0.831425 0.555637i \(-0.187526\pi\)
0.831425 + 0.555637i \(0.187526\pi\)
\(258\) 0 0
\(259\) −154.995 −0.598436
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 257.624i 0.979558i 0.871847 + 0.489779i \(0.162922\pi\)
−0.871847 + 0.489779i \(0.837078\pi\)
\(264\) 0 0
\(265\) −151.598 −0.572068
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 215.246i 0.800171i 0.916478 + 0.400085i \(0.131020\pi\)
−0.916478 + 0.400085i \(0.868980\pi\)
\(270\) 0 0
\(271\) − 378.549i − 1.39686i −0.715678 0.698431i \(-0.753882\pi\)
0.715678 0.698431i \(-0.246118\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −101.358 −0.368576
\(276\) 0 0
\(277\) − 166.449i − 0.600898i −0.953798 0.300449i \(-0.902864\pi\)
0.953798 0.300449i \(-0.0971365\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 421.765 1.50094 0.750471 0.660904i \(-0.229827\pi\)
0.750471 + 0.660904i \(0.229827\pi\)
\(282\) 0 0
\(283\) 345.439 1.22063 0.610316 0.792158i \(-0.291043\pi\)
0.610316 + 0.792158i \(0.291043\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 71.3574i 0.248632i
\(288\) 0 0
\(289\) 270.647 0.936494
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 511.038i − 1.74416i −0.489365 0.872079i \(-0.662771\pi\)
0.489365 0.872079i \(-0.337229\pi\)
\(294\) 0 0
\(295\) 95.4028i 0.323399i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 54.5929 0.182585
\(300\) 0 0
\(301\) 45.4317i 0.150936i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −58.4020 −0.191482
\(306\) 0 0
\(307\) −223.331 −0.727462 −0.363731 0.931504i \(-0.618497\pi\)
−0.363731 + 0.931504i \(0.618497\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 12.3988i − 0.0398674i −0.999801 0.0199337i \(-0.993654\pi\)
0.999801 0.0199337i \(-0.00634551\pi\)
\(312\) 0 0
\(313\) 410.049 1.31006 0.655030 0.755603i \(-0.272656\pi\)
0.655030 + 0.755603i \(0.272656\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 130.316i 0.411092i 0.978647 + 0.205546i \(0.0658970\pi\)
−0.978647 + 0.205546i \(0.934103\pi\)
\(318\) 0 0
\(319\) 100.694i 0.315656i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −588.347 −1.82151
\(324\) 0 0
\(325\) 35.0234i 0.107764i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −95.5980 −0.290571
\(330\) 0 0
\(331\) −214.260 −0.647311 −0.323655 0.946175i \(-0.604912\pi\)
−0.323655 + 0.946175i \(0.604912\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 51.7223i 0.154395i
\(336\) 0 0
\(337\) 164.049 0.486792 0.243396 0.969927i \(-0.421739\pi\)
0.243396 + 0.969927i \(0.421739\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 209.533i 0.614466i
\(342\) 0 0
\(343\) − 18.5203i − 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −109.691 −0.316113 −0.158057 0.987430i \(-0.550523\pi\)
−0.158057 + 0.987430i \(0.550523\pi\)
\(348\) 0 0
\(349\) 463.479i 1.32802i 0.747723 + 0.664010i \(0.231147\pi\)
−0.747723 + 0.664010i \(0.768853\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −78.0975 −0.221240 −0.110620 0.993863i \(-0.535284\pi\)
−0.110620 + 0.993863i \(0.535284\pi\)
\(354\) 0 0
\(355\) 158.392 0.446174
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 365.114i − 1.01703i −0.861053 0.508515i \(-0.830195\pi\)
0.861053 0.508515i \(-0.169805\pi\)
\(360\) 0 0
\(361\) 257.520 0.713351
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 107.426i 0.294316i
\(366\) 0 0
\(367\) − 220.739i − 0.601468i −0.953708 0.300734i \(-0.902768\pi\)
0.953708 0.300734i \(-0.0972317\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −258.794 −0.697558
\(372\) 0 0
\(373\) 251.553i 0.674406i 0.941432 + 0.337203i \(0.109481\pi\)
−0.941432 + 0.337203i \(0.890519\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 34.7939 0.0922916
\(378\) 0 0
\(379\) −286.024 −0.754682 −0.377341 0.926074i \(-0.623161\pi\)
−0.377341 + 0.926074i \(0.623161\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 106.894i − 0.279096i −0.990215 0.139548i \(-0.955435\pi\)
0.990215 0.139548i \(-0.0445649\pi\)
\(384\) 0 0
\(385\) 18.3919 0.0477712
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 77.1807i − 0.198408i −0.995067 0.0992040i \(-0.968370\pi\)
0.995067 0.0992040i \(-0.0316296\pi\)
\(390\) 0 0
\(391\) 833.307i 2.13122i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −59.9798 −0.151848
\(396\) 0 0
\(397\) − 657.514i − 1.65621i −0.560576 0.828103i \(-0.689420\pi\)
0.560576 0.828103i \(-0.310580\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −318.794 −0.794997 −0.397499 0.917603i \(-0.630122\pi\)
−0.397499 + 0.917603i \(0.630122\pi\)
\(402\) 0 0
\(403\) 72.4020 0.179658
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 262.759i − 0.645600i
\(408\) 0 0
\(409\) 145.265 0.355171 0.177585 0.984105i \(-0.443171\pi\)
0.177585 + 0.984105i \(0.443171\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 162.863i 0.394341i
\(414\) 0 0
\(415\) 5.60304i 0.0135013i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 707.012 1.68738 0.843690 0.536831i \(-0.180379\pi\)
0.843690 + 0.536831i \(0.180379\pi\)
\(420\) 0 0
\(421\) − 121.989i − 0.289761i −0.989449 0.144880i \(-0.953720\pi\)
0.989449 0.144880i \(-0.0462798\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −534.597 −1.25788
\(426\) 0 0
\(427\) −99.6985 −0.233486
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 588.861i 1.36627i 0.730294 + 0.683133i \(0.239383\pi\)
−0.730294 + 0.683133i \(0.760617\pi\)
\(432\) 0 0
\(433\) 137.696 0.318004 0.159002 0.987278i \(-0.449172\pi\)
0.159002 + 0.987278i \(0.449172\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 876.042i − 2.00467i
\(438\) 0 0
\(439\) − 440.543i − 1.00352i −0.865008 0.501758i \(-0.832687\pi\)
0.865008 0.501758i \(-0.167313\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −487.058 −1.09945 −0.549727 0.835344i \(-0.685268\pi\)
−0.549727 + 0.835344i \(0.685268\pi\)
\(444\) 0 0
\(445\) − 68.2844i − 0.153448i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −264.039 −0.588059 −0.294030 0.955796i \(-0.594996\pi\)
−0.294030 + 0.955796i \(0.594996\pi\)
\(450\) 0 0
\(451\) −120.971 −0.268227
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) − 6.35515i − 0.0139674i
\(456\) 0 0
\(457\) −514.323 −1.12543 −0.562717 0.826650i \(-0.690244\pi\)
−0.562717 + 0.826650i \(0.690244\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 202.224i − 0.438664i −0.975650 0.219332i \(-0.929612\pi\)
0.975650 0.219332i \(-0.0703878\pi\)
\(462\) 0 0
\(463\) − 722.653i − 1.56081i −0.625277 0.780403i \(-0.715014\pi\)
0.625277 0.780403i \(-0.284986\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −347.282 −0.743645 −0.371822 0.928304i \(-0.621267\pi\)
−0.371822 + 0.928304i \(0.621267\pi\)
\(468\) 0 0
\(469\) 88.2956i 0.188263i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −77.0193 −0.162832
\(474\) 0 0
\(475\) 562.013 1.18319
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) − 29.1811i − 0.0609210i −0.999536 0.0304605i \(-0.990303\pi\)
0.999536 0.0304605i \(-0.00969737\pi\)
\(480\) 0 0
\(481\) −90.7939 −0.188761
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 148.952i 0.307117i
\(486\) 0 0
\(487\) 701.643i 1.44074i 0.693588 + 0.720372i \(0.256029\pi\)
−0.693588 + 0.720372i \(0.743971\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −59.9512 −0.122100 −0.0610501 0.998135i \(-0.519445\pi\)
−0.0610501 + 0.998135i \(0.519445\pi\)
\(492\) 0 0
\(493\) 531.095i 1.07727i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 270.392 0.544048
\(498\) 0 0
\(499\) 84.2843 0.168906 0.0844532 0.996427i \(-0.473086\pi\)
0.0844532 + 0.996427i \(0.473086\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 409.987i − 0.815083i −0.913187 0.407542i \(-0.866386\pi\)
0.913187 0.407542i \(-0.133614\pi\)
\(504\) 0 0
\(505\) −30.4020 −0.0602020
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 477.033i 0.937196i 0.883411 + 0.468598i \(0.155241\pi\)
−0.883411 + 0.468598i \(0.844759\pi\)
\(510\) 0 0
\(511\) 183.387i 0.358878i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −66.7737 −0.129658
\(516\) 0 0
\(517\) − 162.065i − 0.313472i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −210.873 −0.404747 −0.202373 0.979308i \(-0.564865\pi\)
−0.202373 + 0.979308i \(0.564865\pi\)
\(522\) 0 0
\(523\) −511.566 −0.978139 −0.489069 0.872245i \(-0.662663\pi\)
−0.489069 + 0.872245i \(0.662663\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1105.15i 2.09705i
\(528\) 0 0
\(529\) −711.784 −1.34553
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 41.8002i 0.0784244i
\(534\) 0 0
\(535\) 24.1745i 0.0451859i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 31.3970 0.0582504
\(540\) 0 0
\(541\) − 342.417i − 0.632933i −0.948604 0.316466i \(-0.897504\pi\)
0.948604 0.316466i \(-0.102496\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 5.96970 0.0109536
\(546\) 0 0
\(547\) 441.976 0.807999 0.404000 0.914759i \(-0.367620\pi\)
0.404000 + 0.914759i \(0.367620\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) − 558.331i − 1.01331i
\(552\) 0 0
\(553\) −102.392 −0.185157
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 365.710i 0.656571i 0.944579 + 0.328285i \(0.106471\pi\)
−0.944579 + 0.328285i \(0.893529\pi\)
\(558\) 0 0
\(559\) 26.6133i 0.0476087i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 806.389 1.43231 0.716154 0.697943i \(-0.245901\pi\)
0.716154 + 0.697943i \(0.245901\pi\)
\(564\) 0 0
\(565\) 21.3485i 0.0377850i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −222.891 −0.391725 −0.195862 0.980631i \(-0.562751\pi\)
−0.195862 + 0.980631i \(0.562751\pi\)
\(570\) 0 0
\(571\) −573.082 −1.00365 −0.501823 0.864970i \(-0.667337\pi\)
−0.501823 + 0.864970i \(0.667337\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 796.008i − 1.38436i
\(576\) 0 0
\(577\) −723.901 −1.25459 −0.627297 0.778780i \(-0.715839\pi\)
−0.627297 + 0.778780i \(0.715839\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 9.56498i 0.0164630i
\(582\) 0 0
\(583\) − 438.727i − 0.752534i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 21.1198 0.0359793 0.0179896 0.999838i \(-0.494273\pi\)
0.0179896 + 0.999838i \(0.494273\pi\)
\(588\) 0 0
\(589\) − 1161.82i − 1.97253i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 128.745 0.217108 0.108554 0.994091i \(-0.465378\pi\)
0.108554 + 0.994091i \(0.465378\pi\)
\(594\) 0 0
\(595\) 97.0051 0.163034
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 324.130i 0.541119i 0.962703 + 0.270559i \(0.0872086\pi\)
−0.962703 + 0.270559i \(0.912791\pi\)
\(600\) 0 0
\(601\) −721.862 −1.20110 −0.600551 0.799587i \(-0.705052\pi\)
−0.600551 + 0.799587i \(0.705052\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 156.352i − 0.258433i
\(606\) 0 0
\(607\) − 705.999i − 1.16310i −0.813512 0.581548i \(-0.802447\pi\)
0.813512 0.581548i \(-0.197553\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −56.0000 −0.0916530
\(612\) 0 0
\(613\) 21.8269i 0.0356066i 0.999842 + 0.0178033i \(0.00566727\pi\)
−0.999842 + 0.0178033i \(0.994333\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 699.578 1.13384 0.566919 0.823774i \(-0.308135\pi\)
0.566919 + 0.823774i \(0.308135\pi\)
\(618\) 0 0
\(619\) −96.1981 −0.155409 −0.0777044 0.996976i \(-0.524759\pi\)
−0.0777044 + 0.996976i \(0.524759\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 116.569i − 0.187109i
\(624\) 0 0
\(625\) 450.618 0.720989
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 1385.88i − 2.20331i
\(630\) 0 0
\(631\) − 269.399i − 0.426940i −0.976950 0.213470i \(-0.931523\pi\)
0.976950 0.213470i \(-0.0684766\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −193.769 −0.305148
\(636\) 0 0
\(637\) − 10.8489i − 0.0170313i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −635.813 −0.991908 −0.495954 0.868349i \(-0.665182\pi\)
−0.495954 + 0.868349i \(0.665182\pi\)
\(642\) 0 0
\(643\) −1281.70 −1.99332 −0.996658 0.0816828i \(-0.973971\pi\)
−0.996658 + 0.0816828i \(0.973971\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 260.761i − 0.403031i −0.979485 0.201516i \(-0.935413\pi\)
0.979485 0.201516i \(-0.0645867\pi\)
\(648\) 0 0
\(649\) −276.098 −0.425420
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1090.58i 1.67011i 0.550167 + 0.835055i \(0.314564\pi\)
−0.550167 + 0.835055i \(0.685436\pi\)
\(654\) 0 0
\(655\) 155.527i 0.237446i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −362.780 −0.550500 −0.275250 0.961373i \(-0.588761\pi\)
−0.275250 + 0.961373i \(0.588761\pi\)
\(660\) 0 0
\(661\) 117.834i 0.178266i 0.996020 + 0.0891330i \(0.0284096\pi\)
−0.996020 + 0.0891330i \(0.971590\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −101.980 −0.153353
\(666\) 0 0
\(667\) −790.794 −1.18560
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) − 169.017i − 0.251888i
\(672\) 0 0
\(673\) −6.56854 −0.00976009 −0.00488005 0.999988i \(-0.501553\pi\)
−0.00488005 + 0.999988i \(0.501553\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 125.796i − 0.185813i −0.995675 0.0929066i \(-0.970384\pi\)
0.995675 0.0929066i \(-0.0296158\pi\)
\(678\) 0 0
\(679\) 254.277i 0.374487i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −553.775 −0.810797 −0.405399 0.914140i \(-0.632867\pi\)
−0.405399 + 0.914140i \(0.632867\pi\)
\(684\) 0 0
\(685\) − 88.8274i − 0.129675i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −151.598 −0.220026
\(690\) 0 0
\(691\) 1046.83 1.51494 0.757471 0.652868i \(-0.226435\pi\)
0.757471 + 0.652868i \(0.226435\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 284.651i 0.409569i
\(696\) 0 0
\(697\) −638.039 −0.915407
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 625.993i 0.893000i 0.894784 + 0.446500i \(0.147330\pi\)
−0.894784 + 0.446500i \(0.852670\pi\)
\(702\) 0 0
\(703\) 1456.95i 2.07248i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −51.8995 −0.0734081
\(708\) 0 0
\(709\) 593.492i 0.837083i 0.908198 + 0.418541i \(0.137459\pi\)
−0.908198 + 0.418541i \(0.862541\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −1645.55 −2.30792
\(714\) 0 0
\(715\) 10.7737 0.0150682
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) − 611.505i − 0.850493i −0.905078 0.425247i \(-0.860187\pi\)
0.905078 0.425247i \(-0.139813\pi\)
\(720\) 0 0
\(721\) −113.990 −0.158100
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 507.323i − 0.699756i
\(726\) 0 0
\(727\) − 944.144i − 1.29868i −0.760496 0.649342i \(-0.775044\pi\)
0.760496 0.649342i \(-0.224956\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −406.225 −0.555712
\(732\) 0 0
\(733\) − 218.254i − 0.297755i −0.988856 0.148878i \(-0.952434\pi\)
0.988856 0.148878i \(-0.0475660\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −149.685 −0.203101
\(738\) 0 0
\(739\) 7.29942 0.00987743 0.00493872 0.999988i \(-0.498428\pi\)
0.00493872 + 0.999988i \(0.498428\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 106.867i − 0.143832i −0.997411 0.0719159i \(-0.977089\pi\)
0.997411 0.0719159i \(-0.0229113\pi\)
\(744\) 0 0
\(745\) −298.051 −0.400068
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 41.2684i 0.0550980i
\(750\) 0 0
\(751\) − 127.463i − 0.169725i −0.996393 0.0848624i \(-0.972955\pi\)
0.996393 0.0848624i \(-0.0270451\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 177.849 0.235562
\(756\) 0 0
\(757\) 704.275i 0.930350i 0.885219 + 0.465175i \(0.154009\pi\)
−0.885219 + 0.465175i \(0.845991\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1002.93 1.31791 0.658955 0.752182i \(-0.270999\pi\)
0.658955 + 0.752182i \(0.270999\pi\)
\(762\) 0 0
\(763\) 10.1909 0.0133564
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 95.4028i 0.124384i
\(768\) 0 0
\(769\) −646.950 −0.841288 −0.420644 0.907226i \(-0.638196\pi\)
−0.420644 + 0.907226i \(0.638196\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 564.265i 0.729968i 0.931014 + 0.364984i \(0.118925\pi\)
−0.931014 + 0.364984i \(0.881075\pi\)
\(774\) 0 0
\(775\) − 1055.68i − 1.36217i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 670.759 0.861052
\(780\) 0 0
\(781\) 458.389i 0.586926i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −328.794 −0.418846
\(786\) 0 0
\(787\) −923.345 −1.17325 −0.586623 0.809860i \(-0.699543\pi\)
−0.586623 + 0.809860i \(0.699543\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 36.4442i 0.0460735i
\(792\) 0 0
\(793\) −58.4020 −0.0736469
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 207.983i 0.260957i 0.991451 + 0.130479i \(0.0416514\pi\)
−0.991451 + 0.130479i \(0.958349\pi\)
\(798\) 0 0
\(799\) − 854.785i − 1.06982i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −310.891 −0.387162
\(804\) 0 0
\(805\) 144.439i 0.179428i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −340.540 −0.420939 −0.210470 0.977600i \(-0.567499\pi\)
−0.210470 + 0.977600i \(0.567499\pi\)
\(810\) 0 0
\(811\) 907.380 1.11884 0.559420 0.828884i \(-0.311024\pi\)
0.559420 + 0.828884i \(0.311024\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 372.791i 0.457412i
\(816\) 0 0
\(817\) 427.058 0.522715
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 633.423i − 0.771526i −0.922598 0.385763i \(-0.873938\pi\)
0.922598 0.385763i \(-0.126062\pi\)
\(822\) 0 0
\(823\) 143.649i 0.174544i 0.996185 + 0.0872718i \(0.0278149\pi\)
−0.996185 + 0.0872718i \(0.972185\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1545.57 1.86888 0.934442 0.356114i \(-0.115899\pi\)
0.934442 + 0.356114i \(0.115899\pi\)
\(828\) 0 0
\(829\) − 743.956i − 0.897413i −0.893679 0.448707i \(-0.851885\pi\)
0.893679 0.448707i \(-0.148115\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 165.598 0.198797
\(834\) 0 0
\(835\) −328.723 −0.393681
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 96.3107i 0.114792i 0.998351 + 0.0573961i \(0.0182798\pi\)
−0.998351 + 0.0573961i \(0.981720\pi\)
\(840\) 0 0
\(841\) 337.000 0.400713
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 258.201i 0.305563i
\(846\) 0 0
\(847\) − 266.909i − 0.315123i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 2063.56 2.42486
\(852\) 0 0
\(853\) − 904.866i − 1.06080i −0.847746 0.530402i \(-0.822041\pi\)
0.847746 0.530402i \(-0.177959\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −160.932 −0.187785 −0.0938926 0.995582i \(-0.529931\pi\)
−0.0938926 + 0.995582i \(0.529931\pi\)
\(858\) 0 0
\(859\) −231.693 −0.269724 −0.134862 0.990864i \(-0.543059\pi\)
−0.134862 + 0.990864i \(0.543059\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 1337.35i − 1.54965i −0.632176 0.774825i \(-0.717838\pi\)
0.632176 0.774825i \(-0.282162\pi\)
\(864\) 0 0
\(865\) 282.402 0.326476
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 173.583i − 0.199750i
\(870\) 0 0
\(871\) 51.7223i 0.0593827i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −195.176 −0.223058
\(876\) 0 0
\(877\) 1436.14i 1.63755i 0.574111 + 0.818777i \(0.305348\pi\)
−0.574111 + 0.818777i \(0.694652\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 186.706 0.211926 0.105963 0.994370i \(-0.466208\pi\)
0.105963 + 0.994370i \(0.466208\pi\)
\(882\) 0 0
\(883\) −1277.99 −1.44733 −0.723664 0.690153i \(-0.757543\pi\)
−0.723664 + 0.690153i \(0.757543\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 980.717i − 1.10566i −0.833295 0.552828i \(-0.813549\pi\)
0.833295 0.552828i \(-0.186451\pi\)
\(888\) 0 0
\(889\) −330.784 −0.372085
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 898.621i 1.00629i
\(894\) 0 0
\(895\) 88.6605i 0.0990621i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −1048.76 −1.16659
\(900\) 0 0
\(901\) − 2313.99i − 2.56825i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 505.578 0.558649
\(906\) 0 0
\(907\) 658.372 0.725878 0.362939 0.931813i \(-0.381773\pi\)
0.362939 + 0.931813i \(0.381773\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 276.507i − 0.303520i −0.988417 0.151760i \(-0.951506\pi\)
0.988417 0.151760i \(-0.0484941\pi\)
\(912\) 0 0
\(913\) −16.2153 −0.0177605
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 265.502i 0.289533i
\(918\) 0 0
\(919\) 1339.73i 1.45782i 0.684611 + 0.728908i \(0.259972\pi\)
−0.684611 + 0.728908i \(0.740028\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 158.392 0.171606
\(924\) 0 0
\(925\) 1323.85i 1.43119i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −35.4012 −0.0381067 −0.0190534 0.999818i \(-0.506065\pi\)
−0.0190534 + 0.999818i \(0.506065\pi\)
\(930\) 0 0
\(931\) −174.090 −0.186993
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 164.450i 0.175883i
\(936\) 0 0
\(937\) 610.235 0.651265 0.325633 0.945496i \(-0.394423\pi\)
0.325633 + 0.945496i \(0.394423\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 1852.90i 1.96907i 0.175175 + 0.984537i \(0.443951\pi\)
−0.175175 + 0.984537i \(0.556049\pi\)
\(942\) 0 0
\(943\) − 950.032i − 1.00746i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −1832.90 −1.93548 −0.967738 0.251959i \(-0.918925\pi\)
−0.967738 + 0.251959i \(0.918925\pi\)
\(948\) 0 0
\(949\) 107.426i 0.113199i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −349.687 −0.366933 −0.183467 0.983026i \(-0.558732\pi\)
−0.183467 + 0.983026i \(0.558732\pi\)
\(954\) 0 0
\(955\) 150.432 0.157521
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) − 151.638i − 0.158121i
\(960\) 0 0
\(961\) −1221.35 −1.27092
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 243.796i 0.252639i
\(966\) 0 0
\(967\) 632.128i 0.653700i 0.945076 + 0.326850i \(0.105987\pi\)
−0.945076 + 0.326850i \(0.894013\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 656.497 0.676104 0.338052 0.941128i \(-0.390232\pi\)
0.338052 + 0.941128i \(0.390232\pi\)
\(972\) 0 0
\(973\) 485.929i 0.499413i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −169.314 −0.173300 −0.0866498 0.996239i \(-0.527616\pi\)
−0.0866498 + 0.996239i \(0.527616\pi\)
\(978\) 0 0
\(979\) 197.616 0.201855
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 698.607i 0.710689i 0.934735 + 0.355345i \(0.115636\pi\)
−0.934735 + 0.355345i \(0.884364\pi\)
\(984\) 0 0
\(985\) 192.362 0.195291
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 604.865i − 0.611592i
\(990\) 0 0
\(991\) − 429.702i − 0.433605i −0.976216 0.216802i \(-0.930437\pi\)
0.976216 0.216802i \(-0.0695627\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −280.483 −0.281892
\(996\) 0 0
\(997\) − 52.3910i − 0.0525487i −0.999655 0.0262743i \(-0.991636\pi\)
0.999655 0.0262743i \(-0.00836434\pi\)
\(998\) 0 0
\(999\) 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 2016.3.g.a.1135.3 4
3.2 odd 2 224.3.g.a.15.1 4
4.3 odd 2 504.3.g.a.379.1 4
8.3 odd 2 inner 2016.3.g.a.1135.2 4
8.5 even 2 504.3.g.a.379.2 4
12.11 even 2 56.3.g.a.43.4 yes 4
21.20 even 2 1568.3.g.h.687.4 4
24.5 odd 2 56.3.g.a.43.3 4
24.11 even 2 224.3.g.a.15.2 4
48.5 odd 4 1792.3.d.g.1023.8 8
48.11 even 4 1792.3.d.g.1023.2 8
48.29 odd 4 1792.3.d.g.1023.1 8
48.35 even 4 1792.3.d.g.1023.7 8
84.11 even 6 392.3.k.i.275.1 8
84.23 even 6 392.3.k.i.67.3 8
84.47 odd 6 392.3.k.j.67.3 8
84.59 odd 6 392.3.k.j.275.1 8
84.83 odd 2 392.3.g.h.99.4 4
168.5 even 6 392.3.k.j.67.1 8
168.53 odd 6 392.3.k.i.275.3 8
168.83 odd 2 1568.3.g.h.687.3 4
168.101 even 6 392.3.k.j.275.3 8
168.125 even 2 392.3.g.h.99.3 4
168.149 odd 6 392.3.k.i.67.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.3.g.a.43.3 4 24.5 odd 2
56.3.g.a.43.4 yes 4 12.11 even 2
224.3.g.a.15.1 4 3.2 odd 2
224.3.g.a.15.2 4 24.11 even 2
392.3.g.h.99.3 4 168.125 even 2
392.3.g.h.99.4 4 84.83 odd 2
392.3.k.i.67.1 8 168.149 odd 6
392.3.k.i.67.3 8 84.23 even 6
392.3.k.i.275.1 8 84.11 even 6
392.3.k.i.275.3 8 168.53 odd 6
392.3.k.j.67.1 8 168.5 even 6
392.3.k.j.67.3 8 84.47 odd 6
392.3.k.j.275.1 8 84.59 odd 6
392.3.k.j.275.3 8 168.101 even 6
504.3.g.a.379.1 4 4.3 odd 2
504.3.g.a.379.2 4 8.5 even 2
1568.3.g.h.687.3 4 168.83 odd 2
1568.3.g.h.687.4 4 21.20 even 2
1792.3.d.g.1023.1 8 48.29 odd 4
1792.3.d.g.1023.2 8 48.11 even 4
1792.3.d.g.1023.7 8 48.35 even 4
1792.3.d.g.1023.8 8 48.5 odd 4
2016.3.g.a.1135.2 4 8.3 odd 2 inner
2016.3.g.a.1135.3 4 1.1 even 1 trivial