Properties

Label 224.3.g.a.15.1
Level 224
Weight 3
Character 224.15
Analytic conductor 6.104
Analytic rank 0
Dimension 4
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 224.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.10355792167\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-7})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 15.1
Root \(0.707107 - 1.87083i\) of \(x^{4} + 6 x^{2} + 16\)
Character \(\chi\) \(=\) 224.15
Dual form 224.3.g.a.15.2

$q$-expansion

\(f(q)\) \(=\) \(q-3.41421 q^{3} -1.54985i q^{5} +2.64575i q^{7} +2.65685 q^{9} +O(q^{10})\) \(q-3.41421 q^{3} -1.54985i q^{5} +2.64575i q^{7} +2.65685 q^{9} +4.48528 q^{11} +1.54985i q^{13} +5.29150i q^{15} +23.6569 q^{17} +24.8701 q^{19} -9.03316i q^{21} +35.2248i q^{23} +22.5980 q^{25} +21.6569 q^{27} +22.4499i q^{29} -46.7156i q^{31} -15.3137 q^{33} +4.10051 q^{35} +58.5826i q^{37} -5.29150i q^{39} -26.9706 q^{41} +17.1716 q^{43} -4.11771i q^{45} -36.1326i q^{47} -7.00000 q^{49} -80.7696 q^{51} -97.8149i q^{53} -6.95149i q^{55} -84.9117 q^{57} -61.5563 q^{59} +37.6825i q^{61} +7.02938i q^{63} +2.40202 q^{65} +33.3726 q^{67} -120.265i q^{69} +102.199i q^{71} +69.3137 q^{73} -77.1543 q^{75} +11.8669i q^{77} +38.7005i q^{79} -97.8528 q^{81} -3.61522 q^{83} -36.6645i q^{85} -76.6489i q^{87} +44.0589 q^{89} -4.10051 q^{91} +159.497i q^{93} -38.5447i q^{95} +96.1076 q^{97} +11.9167 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{3} - 12q^{9} + O(q^{10}) \) \( 4q - 8q^{3} - 12q^{9} - 16q^{11} + 72q^{17} - 8q^{19} - 68q^{25} + 64q^{27} - 16q^{33} + 56q^{35} - 40q^{41} + 80q^{43} - 28q^{49} - 176q^{51} - 136q^{57} - 184q^{59} + 168q^{65} + 224q^{67} + 232q^{73} - 88q^{75} - 52q^{81} - 88q^{83} + 312q^{89} - 56q^{91} - 136q^{97} + 240q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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
\(3\) −3.41421 −1.13807 −0.569036 0.822313i \(-0.692683\pi\)
−0.569036 + 0.822313i \(0.692683\pi\)
\(4\) 0 0
\(5\) − 1.54985i − 0.309969i −0.987917 0.154985i \(-0.950467\pi\)
0.987917 0.154985i \(-0.0495328\pi\)
\(6\) 0 0
\(7\) 2.64575i 0.377964i
\(8\) 0 0
\(9\) 2.65685 0.295206
\(10\) 0 0
\(11\) 4.48528 0.407753 0.203876 0.978997i \(-0.434646\pi\)
0.203876 + 0.978997i \(0.434646\pi\)
\(12\) 0 0
\(13\) 1.54985i 0.119219i 0.998222 + 0.0596094i \(0.0189855\pi\)
−0.998222 + 0.0596094i \(0.981014\pi\)
\(14\) 0 0
\(15\) 5.29150i 0.352767i
\(16\) 0 0
\(17\) 23.6569 1.39158 0.695790 0.718245i \(-0.255055\pi\)
0.695790 + 0.718245i \(0.255055\pi\)
\(18\) 0 0
\(19\) 24.8701 1.30895 0.654475 0.756083i \(-0.272890\pi\)
0.654475 + 0.756083i \(0.272890\pi\)
\(20\) 0 0
\(21\) − 9.03316i − 0.430150i
\(22\) 0 0
\(23\) 35.2248i 1.53151i 0.643132 + 0.765756i \(0.277635\pi\)
−0.643132 + 0.765756i \(0.722365\pi\)
\(24\) 0 0
\(25\) 22.5980 0.903919
\(26\) 0 0
\(27\) 21.6569 0.802106
\(28\) 0 0
\(29\) 22.4499i 0.774136i 0.922051 + 0.387068i \(0.126512\pi\)
−0.922051 + 0.387068i \(0.873488\pi\)
\(30\) 0 0
\(31\) − 46.7156i − 1.50696i −0.657473 0.753478i \(-0.728375\pi\)
0.657473 0.753478i \(-0.271625\pi\)
\(32\) 0 0
\(33\) −15.3137 −0.464052
\(34\) 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
\(39\) − 5.29150i − 0.135680i
\(40\) 0 0
\(41\) −26.9706 −0.657819 −0.328909 0.944362i \(-0.606681\pi\)
−0.328909 + 0.944362i \(0.606681\pi\)
\(42\) 0 0
\(43\) 17.1716 0.399339 0.199669 0.979863i \(-0.436013\pi\)
0.199669 + 0.979863i \(0.436013\pi\)
\(44\) 0 0
\(45\) − 4.11771i − 0.0915047i
\(46\) 0 0
\(47\) − 36.1326i − 0.768780i −0.923171 0.384390i \(-0.874412\pi\)
0.923171 0.384390i \(-0.125588\pi\)
\(48\) 0 0
\(49\) −7.00000 −0.142857
\(50\) 0 0
\(51\) −80.7696 −1.58372
\(52\) 0 0
\(53\) − 97.8149i − 1.84556i −0.385322 0.922782i \(-0.625910\pi\)
0.385322 0.922782i \(-0.374090\pi\)
\(54\) 0 0
\(55\) − 6.95149i − 0.126391i
\(56\) 0 0
\(57\) −84.9117 −1.48968
\(58\) 0 0
\(59\) −61.5563 −1.04333 −0.521664 0.853151i \(-0.674689\pi\)
−0.521664 + 0.853151i \(0.674689\pi\)
\(60\) 0 0
\(61\) 37.6825i 0.617746i 0.951103 + 0.308873i \(0.0999518\pi\)
−0.951103 + 0.308873i \(0.900048\pi\)
\(62\) 0 0
\(63\) 7.02938i 0.111577i
\(64\) 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
\(69\) − 120.265i − 1.74297i
\(70\) 0 0
\(71\) 102.199i 1.43942i 0.694277 + 0.719708i \(0.255724\pi\)
−0.694277 + 0.719708i \(0.744276\pi\)
\(72\) 0 0
\(73\) 69.3137 0.949503 0.474751 0.880120i \(-0.342538\pi\)
0.474751 + 0.880120i \(0.342538\pi\)
\(74\) 0 0
\(75\) −77.1543 −1.02872
\(76\) 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
\(81\) −97.8528 −1.20806
\(82\) 0 0
\(83\) −3.61522 −0.0435569 −0.0217785 0.999763i \(-0.506933\pi\)
−0.0217785 + 0.999763i \(0.506933\pi\)
\(84\) 0 0
\(85\) − 36.6645i − 0.431347i
\(86\) 0 0
\(87\) − 76.6489i − 0.881022i
\(88\) 0 0
\(89\) 44.0589 0.495044 0.247522 0.968882i \(-0.420384\pi\)
0.247522 + 0.968882i \(0.420384\pi\)
\(90\) 0 0
\(91\) −4.10051 −0.0450605
\(92\) 0 0
\(93\) 159.497i 1.71502i
\(94\) 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
\(99\) 11.9167 0.120371
\(100\) 0 0
\(101\) − 19.6162i − 0.194219i −0.995274 0.0971097i \(-0.969040\pi\)
0.995274 0.0971097i \(-0.0309598\pi\)
\(102\) 0 0
\(103\) 43.0841i 0.418293i 0.977884 + 0.209146i \(0.0670685\pi\)
−0.977884 + 0.209146i \(0.932932\pi\)
\(104\) 0 0
\(105\) −14.0000 −0.133333
\(106\) 0 0
\(107\) −15.5980 −0.145776 −0.0728878 0.997340i \(-0.523221\pi\)
−0.0728878 + 0.997340i \(0.523221\pi\)
\(108\) 0 0
\(109\) − 3.85180i − 0.0353376i −0.999844 0.0176688i \(-0.994376\pi\)
0.999844 0.0176688i \(-0.00562445\pi\)
\(110\) 0 0
\(111\) − 200.013i − 1.80192i
\(112\) 0 0
\(113\) −13.7746 −0.121899 −0.0609496 0.998141i \(-0.519413\pi\)
−0.0609496 + 0.998141i \(0.519413\pi\)
\(114\) 0 0
\(115\) 54.5929 0.474721
\(116\) 0 0
\(117\) 4.11771i 0.0351941i
\(118\) 0 0
\(119\) 62.5902i 0.525968i
\(120\) 0 0
\(121\) −100.882 −0.833738
\(122\) 0 0
\(123\) 92.0833 0.748644
\(124\) 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
\(129\) −58.6274 −0.454476
\(130\) 0 0
\(131\) −100.350 −0.766033 −0.383016 0.923742i \(-0.625115\pi\)
−0.383016 + 0.923742i \(0.625115\pi\)
\(132\) 0 0
\(133\) 65.8000i 0.494737i
\(134\) 0 0
\(135\) − 33.5648i − 0.248628i
\(136\) 0 0
\(137\) 57.3137 0.418348 0.209174 0.977878i \(-0.432922\pi\)
0.209174 + 0.977878i \(0.432922\pi\)
\(138\) 0 0
\(139\) 183.664 1.32132 0.660662 0.750684i \(-0.270276\pi\)
0.660662 + 0.750684i \(0.270276\pi\)
\(140\) 0 0
\(141\) 123.365i 0.874926i
\(142\) 0 0
\(143\) 6.95149i 0.0486118i
\(144\) 0 0
\(145\) 34.7939 0.239958
\(146\) 0 0
\(147\) 23.8995 0.162582
\(148\) 0 0
\(149\) − 192.310i − 1.29067i −0.763900 0.645335i \(-0.776718\pi\)
0.763900 0.645335i \(-0.223282\pi\)
\(150\) 0 0
\(151\) − 114.753i − 0.759954i −0.924996 0.379977i \(-0.875932\pi\)
0.924996 0.379977i \(-0.124068\pi\)
\(152\) 0 0
\(153\) 62.8528 0.410803
\(154\) 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
\(159\) 333.961i 2.10038i
\(160\) 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
\(165\) 23.7339i 0.143842i
\(166\) 0 0
\(167\) − 212.101i − 1.27006i −0.772486 0.635032i \(-0.780987\pi\)
0.772486 0.635032i \(-0.219013\pi\)
\(168\) 0 0
\(169\) 166.598 0.985787
\(170\) 0 0
\(171\) 66.0761 0.386410
\(172\) 0 0
\(173\) 182.213i 1.05325i 0.850096 + 0.526627i \(0.176544\pi\)
−0.850096 + 0.526627i \(0.823456\pi\)
\(174\) 0 0
\(175\) 59.7886i 0.341649i
\(176\) 0 0
\(177\) 210.167 1.18738
\(178\) 0 0
\(179\) −57.2061 −0.319587 −0.159793 0.987150i \(-0.551083\pi\)
−0.159793 + 0.987150i \(0.551083\pi\)
\(180\) 0 0
\(181\) − 326.212i − 1.80228i −0.433533 0.901138i \(-0.642733\pi\)
0.433533 0.901138i \(-0.357267\pi\)
\(182\) 0 0
\(183\) − 128.656i − 0.703039i
\(184\) 0 0
\(185\) 90.7939 0.490778
\(186\) 0 0
\(187\) 106.108 0.567421
\(188\) 0 0
\(189\) 57.2987i 0.303167i
\(190\) 0 0
\(191\) 97.0628i 0.508182i 0.967180 + 0.254091i \(0.0817763\pi\)
−0.967180 + 0.254091i \(0.918224\pi\)
\(192\) 0 0
\(193\) 157.304 0.815045 0.407522 0.913195i \(-0.366393\pi\)
0.407522 + 0.913195i \(0.366393\pi\)
\(194\) 0 0
\(195\) −8.20101 −0.0420565
\(196\) 0 0
\(197\) 124.117i 0.630034i 0.949086 + 0.315017i \(0.102010\pi\)
−0.949086 + 0.315017i \(0.897990\pi\)
\(198\) 0 0
\(199\) 180.975i 0.909421i 0.890639 + 0.454710i \(0.150257\pi\)
−0.890639 + 0.454710i \(0.849743\pi\)
\(200\) 0 0
\(201\) −113.941 −0.566871
\(202\) 0 0
\(203\) −59.3970 −0.292596
\(204\) 0 0
\(205\) 41.8002i 0.203903i
\(206\) 0 0
\(207\) 93.5871i 0.452111i
\(208\) 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
\(213\) − 348.928i − 1.63816i
\(214\) 0 0
\(215\) − 26.6133i − 0.123783i
\(216\) 0 0
\(217\) 123.598 0.569576
\(218\) 0 0
\(219\) −236.652 −1.08060
\(220\) 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
\(225\) 60.0395 0.266842
\(226\) 0 0
\(227\) −105.806 −0.466106 −0.233053 0.972464i \(-0.574872\pi\)
−0.233053 + 0.972464i \(0.574872\pi\)
\(228\) 0 0
\(229\) − 74.8788i − 0.326982i −0.986545 0.163491i \(-0.947725\pi\)
0.986545 0.163491i \(-0.0522754\pi\)
\(230\) 0 0
\(231\) − 40.5163i − 0.175395i
\(232\) 0 0
\(233\) −419.137 −1.79887 −0.899436 0.437053i \(-0.856022\pi\)
−0.899436 + 0.437053i \(0.856022\pi\)
\(234\) 0 0
\(235\) −56.0000 −0.238298
\(236\) 0 0
\(237\) − 132.132i − 0.557518i
\(238\) 0 0
\(239\) − 148.318i − 0.620577i −0.950642 0.310288i \(-0.899574\pi\)
0.950642 0.310288i \(-0.100426\pi\)
\(240\) 0 0
\(241\) −459.872 −1.90818 −0.954092 0.299515i \(-0.903175\pi\)
−0.954092 + 0.299515i \(0.903175\pi\)
\(242\) 0 0
\(243\) 139.179 0.572752
\(244\) 0 0
\(245\) 10.8489i 0.0442813i
\(246\) 0 0
\(247\) 38.5447i 0.156052i
\(248\) 0 0
\(249\) 12.3431 0.0495709
\(250\) 0 0
\(251\) −124.919 −0.497685 −0.248842 0.968544i \(-0.580050\pi\)
−0.248842 + 0.968544i \(0.580050\pi\)
\(252\) 0 0
\(253\) 157.993i 0.624478i
\(254\) 0 0
\(255\) 125.180i 0.490903i
\(256\) 0 0
\(257\) −427.352 −1.66285 −0.831425 0.555637i \(-0.812474\pi\)
−0.831425 + 0.555637i \(0.812474\pi\)
\(258\) 0 0
\(259\) −154.995 −0.598436
\(260\) 0 0
\(261\) 59.6462i 0.228530i
\(262\) 0 0
\(263\) − 257.624i − 0.979558i −0.871847 0.489779i \(-0.837078\pi\)
0.871847 0.489779i \(-0.162922\pi\)
\(264\) 0 0
\(265\) −151.598 −0.572068
\(266\) 0 0
\(267\) −150.426 −0.563395
\(268\) 0 0
\(269\) − 215.246i − 0.800171i −0.916478 0.400085i \(-0.868980\pi\)
0.916478 0.400085i \(-0.131020\pi\)
\(270\) 0 0
\(271\) − 378.549i − 1.39686i −0.715678 0.698431i \(-0.753882\pi\)
0.715678 0.698431i \(-0.246118\pi\)
\(272\) 0 0
\(273\) 14.0000 0.0512821
\(274\) 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
\(279\) − 124.117i − 0.444863i
\(280\) 0 0
\(281\) −421.765 −1.50094 −0.750471 0.660904i \(-0.770173\pi\)
−0.750471 + 0.660904i \(0.770173\pi\)
\(282\) 0 0
\(283\) 345.439 1.22063 0.610316 0.792158i \(-0.291043\pi\)
0.610316 + 0.792158i \(0.291043\pi\)
\(284\) 0 0
\(285\) 131.600i 0.461754i
\(286\) 0 0
\(287\) − 71.3574i − 0.248632i
\(288\) 0 0
\(289\) 270.647 0.936494
\(290\) 0 0
\(291\) −328.132 −1.12760
\(292\) 0 0
\(293\) 511.038i 1.74416i 0.489365 + 0.872079i \(0.337229\pi\)
−0.489365 + 0.872079i \(0.662771\pi\)
\(294\) 0 0
\(295\) 95.4028i 0.323399i
\(296\) 0 0
\(297\) 97.1371 0.327061
\(298\) 0 0
\(299\) −54.5929 −0.182585
\(300\) 0 0
\(301\) 45.4317i 0.150936i
\(302\) 0 0
\(303\) 66.9738i 0.221036i
\(304\) 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
\(309\) − 147.098i − 0.476047i
\(310\) 0 0
\(311\) 12.3988i 0.0398674i 0.999801 + 0.0199337i \(0.00634551\pi\)
−0.999801 + 0.0199337i \(0.993654\pi\)
\(312\) 0 0
\(313\) 410.049 1.31006 0.655030 0.755603i \(-0.272656\pi\)
0.655030 + 0.755603i \(0.272656\pi\)
\(314\) 0 0
\(315\) 10.8944 0.0345855
\(316\) 0 0
\(317\) − 130.316i − 0.411092i −0.978647 0.205546i \(-0.934103\pi\)
0.978647 0.205546i \(-0.0658970\pi\)
\(318\) 0 0
\(319\) 100.694i 0.315656i
\(320\) 0 0
\(321\) 53.2548 0.165903
\(322\) 0 0
\(323\) 588.347 1.82151
\(324\) 0 0
\(325\) 35.0234i 0.107764i
\(326\) 0 0
\(327\) 13.1509i 0.0402167i
\(328\) 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
\(333\) 155.645i 0.467404i
\(334\) 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
\(339\) 47.0294 0.138730
\(340\) 0 0
\(341\) − 209.533i − 0.614466i
\(342\) 0 0
\(343\) − 18.5203i − 0.0539949i
\(344\) 0 0
\(345\) −186.392 −0.540266
\(346\) 0 0
\(347\) 109.691 0.316113 0.158057 0.987430i \(-0.449477\pi\)
0.158057 + 0.987430i \(0.449477\pi\)
\(348\) 0 0
\(349\) 463.479i 1.32802i 0.747723 + 0.664010i \(0.231147\pi\)
−0.747723 + 0.664010i \(0.768853\pi\)
\(350\) 0 0
\(351\) 33.5648i 0.0956261i
\(352\) 0 0
\(353\) 78.0975 0.221240 0.110620 0.993863i \(-0.464716\pi\)
0.110620 + 0.993863i \(0.464716\pi\)
\(354\) 0 0
\(355\) 158.392 0.446174
\(356\) 0 0
\(357\) − 213.696i − 0.598589i
\(358\) 0 0
\(359\) 365.114i 1.01703i 0.861053 + 0.508515i \(0.169805\pi\)
−0.861053 + 0.508515i \(0.830195\pi\)
\(360\) 0 0
\(361\) 257.520 0.713351
\(362\) 0 0
\(363\) 344.434 0.948853
\(364\) 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
\(369\) −71.6569 −0.194192
\(370\) 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
\(375\) 251.865i 0.671640i
\(376\) 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
\(381\) − 426.860i − 1.12037i
\(382\) 0 0
\(383\) 106.894i 0.279096i 0.990215 + 0.139548i \(0.0445649\pi\)
−0.990215 + 0.139548i \(0.955435\pi\)
\(384\) 0 0
\(385\) 18.3919 0.0477712
\(386\) 0 0
\(387\) 45.6224 0.117887
\(388\) 0 0
\(389\) 77.1807i 0.198408i 0.995067 + 0.0992040i \(0.0316296\pi\)
−0.995067 + 0.0992040i \(0.968370\pi\)
\(390\) 0 0
\(391\) 833.307i 2.13122i
\(392\) 0 0
\(393\) 342.617 0.871800
\(394\) 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
\(399\) − 224.655i − 0.563046i
\(400\) 0 0
\(401\) 318.794 0.794997 0.397499 0.917603i \(-0.369878\pi\)
0.397499 + 0.917603i \(0.369878\pi\)
\(402\) 0 0
\(403\) 72.4020 0.179658
\(404\) 0 0
\(405\) 151.657i 0.374461i
\(406\) 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
\(411\) −195.681 −0.476110
\(412\) 0 0
\(413\) − 162.863i − 0.394341i
\(414\) 0 0
\(415\) 5.60304i 0.0135013i
\(416\) 0 0
\(417\) −627.068 −1.50376
\(418\) 0 0
\(419\) −707.012 −1.68738 −0.843690 0.536831i \(-0.819621\pi\)
−0.843690 + 0.536831i \(0.819621\pi\)
\(420\) 0 0
\(421\) − 121.989i − 0.289761i −0.989449 0.144880i \(-0.953720\pi\)
0.989449 0.144880i \(-0.0462798\pi\)
\(422\) 0 0
\(423\) − 95.9992i − 0.226948i
\(424\) 0 0
\(425\) 534.597 1.25788
\(426\) 0 0
\(427\) −99.6985 −0.233486
\(428\) 0 0
\(429\) − 23.7339i − 0.0553237i
\(430\) 0 0
\(431\) − 588.861i − 1.36627i −0.730294 0.683133i \(-0.760617\pi\)
0.730294 0.683133i \(-0.239383\pi\)
\(432\) 0 0
\(433\) 137.696 0.318004 0.159002 0.987278i \(-0.449172\pi\)
0.159002 + 0.987278i \(0.449172\pi\)
\(434\) 0 0
\(435\) −118.794 −0.273090
\(436\) 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
\(441\) −18.5980 −0.0421723
\(442\) 0 0
\(443\) 487.058 1.09945 0.549727 0.835344i \(-0.314732\pi\)
0.549727 + 0.835344i \(0.314732\pi\)
\(444\) 0 0
\(445\) − 68.2844i − 0.153448i
\(446\) 0 0
\(447\) 656.587i 1.46887i
\(448\) 0 0
\(449\) 264.039 0.588059 0.294030 0.955796i \(-0.405004\pi\)
0.294030 + 0.955796i \(0.405004\pi\)
\(450\) 0 0
\(451\) −120.971 −0.268227
\(452\) 0 0
\(453\) 391.791i 0.864882i
\(454\) 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
\(459\) 512.333 1.11619
\(460\) 0 0
\(461\) 202.224i 0.438664i 0.975650 + 0.219332i \(0.0703878\pi\)
−0.975650 + 0.219332i \(0.929612\pi\)
\(462\) 0 0
\(463\) − 722.653i − 1.56081i −0.625277 0.780403i \(-0.715014\pi\)
0.625277 0.780403i \(-0.284986\pi\)
\(464\) 0 0
\(465\) 247.196 0.531604
\(466\) 0 0
\(467\) 347.282 0.743645 0.371822 0.928304i \(-0.378733\pi\)
0.371822 + 0.928304i \(0.378733\pi\)
\(468\) 0 0
\(469\) 88.2956i 0.188263i
\(470\) 0 0
\(471\) − 724.313i − 1.53782i
\(472\) 0 0
\(473\) 77.0193 0.162832
\(474\) 0 0
\(475\) 562.013 1.18319
\(476\) 0 0
\(477\) − 259.880i − 0.544822i
\(478\) 0 0
\(479\) 29.1811i 0.0609210i 0.999536 + 0.0304605i \(0.00969737\pi\)
−0.999536 + 0.0304605i \(0.990303\pi\)
\(480\) 0 0
\(481\) −90.7939 −0.188761
\(482\) 0 0
\(483\) 318.191 0.658780
\(484\) 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
\(489\) −821.235 −1.67942
\(490\) 0 0
\(491\) 59.9512 0.122100 0.0610501 0.998135i \(-0.480555\pi\)
0.0610501 + 0.998135i \(0.480555\pi\)
\(492\) 0 0
\(493\) 531.095i 1.07727i
\(494\) 0 0
\(495\) − 18.4691i − 0.0373113i
\(496\) 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
\(501\) 724.157i 1.44542i
\(502\) 0 0
\(503\) 409.987i 0.815083i 0.913187 + 0.407542i \(0.133614\pi\)
−0.913187 + 0.407542i \(0.866386\pi\)
\(504\) 0 0
\(505\) −30.4020 −0.0602020
\(506\) 0 0
\(507\) −568.801 −1.12190
\(508\) 0 0
\(509\) − 477.033i − 0.937196i −0.883411 0.468598i \(-0.844759\pi\)
0.883411 0.468598i \(-0.155241\pi\)
\(510\) 0 0
\(511\) 183.387i 0.358878i
\(512\) 0 0
\(513\) 538.607 1.04992
\(514\) 0 0
\(515\) 66.7737 0.129658
\(516\) 0 0
\(517\) − 162.065i − 0.313472i
\(518\) 0 0
\(519\) − 622.114i − 1.19868i
\(520\) 0 0
\(521\) 210.873 0.404747 0.202373 0.979308i \(-0.435135\pi\)
0.202373 + 0.979308i \(0.435135\pi\)
\(522\) 0 0
\(523\) −511.566 −0.978139 −0.489069 0.872245i \(-0.662663\pi\)
−0.489069 + 0.872245i \(0.662663\pi\)
\(524\) 0 0
\(525\) − 204.131i − 0.388821i
\(526\) 0 0
\(527\) − 1105.15i − 2.09705i
\(528\) 0 0
\(529\) −711.784 −1.34553
\(530\) 0 0
\(531\) −163.546 −0.307997
\(532\) 0 0
\(533\) − 41.8002i − 0.0784244i
\(534\) 0 0
\(535\) 24.1745i 0.0451859i
\(536\) 0 0
\(537\) 195.314 0.363713
\(538\) 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
\(543\) 1113.76i 2.05112i
\(544\) 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
\(549\) 100.117i 0.182362i
\(550\) 0 0
\(551\) 558.331i 1.01331i
\(552\) 0 0
\(553\) −102.392 −0.185157
\(554\) 0 0
\(555\) −309.990 −0.558540
\(556\) 0 0
\(557\) − 365.710i − 0.656571i −0.944579 0.328285i \(-0.893529\pi\)
0.944579 0.328285i \(-0.106471\pi\)
\(558\) 0 0
\(559\) 26.6133i 0.0476087i
\(560\) 0 0
\(561\) −362.274 −0.645765
\(562\) 0 0
\(563\) −806.389 −1.43231 −0.716154 0.697943i \(-0.754099\pi\)
−0.716154 + 0.697943i \(0.754099\pi\)
\(564\) 0 0
\(565\) 21.3485i 0.0377850i
\(566\) 0 0
\(567\) − 258.894i − 0.456604i
\(568\) 0 0
\(569\) 222.891 0.391725 0.195862 0.980631i \(-0.437249\pi\)
0.195862 + 0.980631i \(0.437249\pi\)
\(570\) 0 0
\(571\) −573.082 −1.00365 −0.501823 0.864970i \(-0.667337\pi\)
−0.501823 + 0.864970i \(0.667337\pi\)
\(572\) 0 0
\(573\) − 331.393i − 0.578348i
\(574\) 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
\(579\) −537.068 −0.927579
\(580\) 0 0
\(581\) − 9.56498i − 0.0164630i
\(582\) 0 0
\(583\) − 438.727i − 0.752534i
\(584\) 0 0
\(585\) 6.38182 0.0109091
\(586\) 0 0
\(587\) −21.1198 −0.0359793 −0.0179896 0.999838i \(-0.505727\pi\)
−0.0179896 + 0.999838i \(0.505727\pi\)
\(588\) 0 0
\(589\) − 1161.82i − 1.97253i
\(590\) 0 0
\(591\) − 423.761i − 0.717023i
\(592\) 0 0
\(593\) −128.745 −0.217108 −0.108554 0.994091i \(-0.534622\pi\)
−0.108554 + 0.994091i \(0.534622\pi\)
\(594\) 0 0
\(595\) 97.0051 0.163034
\(596\) 0 0
\(597\) − 617.886i − 1.03499i
\(598\) 0 0
\(599\) − 324.130i − 0.541119i −0.962703 0.270559i \(-0.912791\pi\)
0.962703 0.270559i \(-0.0872086\pi\)
\(600\) 0 0
\(601\) −721.862 −1.20110 −0.600551 0.799587i \(-0.705052\pi\)
−0.600551 + 0.799587i \(0.705052\pi\)
\(602\) 0 0
\(603\) 88.6661 0.147042
\(604\) 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
\(609\) 202.794 0.332995
\(610\) 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
\(615\) − 142.715i − 0.232057i
\(616\) 0 0
\(617\) −699.578 −1.13384 −0.566919 0.823774i \(-0.691865\pi\)
−0.566919 + 0.823774i \(0.691865\pi\)
\(618\) 0 0
\(619\) −96.1981 −0.155409 −0.0777044 0.996976i \(-0.524759\pi\)
−0.0777044 + 0.996976i \(0.524759\pi\)
\(620\) 0 0
\(621\) 762.858i 1.22843i
\(622\) 0 0
\(623\) 116.569i 0.187109i
\(624\) 0 0
\(625\) 450.618 0.720989
\(626\) 0 0
\(627\) −380.853 −0.607421
\(628\) 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
\(633\) 560.098 0.884830
\(634\) 0 0
\(635\) 193.769 0.305148
\(636\) 0 0
\(637\) − 10.8489i − 0.0170313i
\(638\) 0 0
\(639\) 271.527i 0.424924i
\(640\) 0 0
\(641\) 635.813 0.991908 0.495954 0.868349i \(-0.334818\pi\)
0.495954 + 0.868349i \(0.334818\pi\)
\(642\) 0 0
\(643\) −1281.70 −1.99332 −0.996658 0.0816828i \(-0.973971\pi\)
−0.996658 + 0.0816828i \(0.973971\pi\)
\(644\) 0 0
\(645\) 90.8634i 0.140874i
\(646\) 0 0
\(647\) 260.761i 0.403031i 0.979485 + 0.201516i \(0.0645867\pi\)
−0.979485 + 0.201516i \(0.935413\pi\)
\(648\) 0 0
\(649\) −276.098 −0.425420
\(650\) 0 0
\(651\) −421.990 −0.648218
\(652\) 0 0
\(653\) − 1090.58i − 1.67011i −0.550167 0.835055i \(-0.685436\pi\)
0.550167 0.835055i \(-0.314564\pi\)
\(654\) 0 0
\(655\) 155.527i 0.237446i
\(656\) 0 0
\(657\) 184.156 0.280299
\(658\) 0 0
\(659\) 362.780 0.550500 0.275250 0.961373i \(-0.411239\pi\)
0.275250 + 0.961373i \(0.411239\pi\)
\(660\) 0 0
\(661\) 117.834i 0.178266i 0.996020 + 0.0891330i \(0.0284096\pi\)
−0.996020 + 0.0891330i \(0.971590\pi\)
\(662\) 0 0
\(663\) − 125.180i − 0.188809i
\(664\) 0 0
\(665\) 101.980 0.153353
\(666\) 0 0
\(667\) −790.794 −1.18560
\(668\) 0 0
\(669\) − 36.1326i − 0.0540099i
\(670\) 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
\(675\) 489.401 0.725039
\(676\) 0 0
\(677\) 125.796i 0.185813i 0.995675 + 0.0929066i \(0.0296158\pi\)
−0.995675 + 0.0929066i \(0.970384\pi\)
\(678\) 0 0
\(679\) 254.277i 0.374487i
\(680\) 0 0
\(681\) 361.245 0.530462
\(682\) 0 0
\(683\) 553.775 0.810797 0.405399 0.914140i \(-0.367133\pi\)
0.405399 + 0.914140i \(0.367133\pi\)
\(684\) 0 0
\(685\) − 88.8274i − 0.129675i
\(686\) 0 0
\(687\) 255.652i 0.372128i
\(688\) 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
\(693\) 31.5287i 0.0454960i
\(694\) 0 0
\(695\) − 284.651i − 0.409569i
\(696\) 0 0
\(697\) −638.039 −0.915407
\(698\) 0 0
\(699\) 1431.02 2.04724
\(700\) 0 0
\(701\) − 625.993i − 0.893000i −0.894784 0.446500i \(-0.852670\pi\)
0.894784 0.446500i \(-0.147330\pi\)
\(702\) 0 0
\(703\) 1456.95i 2.07248i
\(704\) 0 0
\(705\) 191.196 0.271200
\(706\) 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
\(711\) 102.822i 0.144615i
\(712\) 0 0
\(713\) 1645.55 2.30792
\(714\) 0 0
\(715\) 10.7737 0.0150682
\(716\) 0 0
\(717\) 506.389i 0.706261i
\(718\) 0 0
\(719\) 611.505i 0.850493i 0.905078 + 0.425247i \(0.139813\pi\)
−0.905078 + 0.425247i \(0.860187\pi\)
\(720\) 0 0
\(721\) −113.990 −0.158100
\(722\) 0 0
\(723\) 1570.10 2.17165
\(724\) 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
\(729\) 405.489 0.556227
\(730\) 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
\(735\) − 37.0405i − 0.0503953i
\(736\) 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
\(741\) − 131.600i − 0.177598i
\(742\) 0 0
\(743\) 106.867i 0.143832i 0.997411 + 0.0719159i \(0.0229113\pi\)
−0.997411 + 0.0719159i \(0.977089\pi\)
\(744\) 0 0
\(745\) −298.051 −0.400068
\(746\) 0 0
\(747\) −9.60512 −0.0128583
\(748\) 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
\(753\) 426.500 0.566400
\(754\) 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
\(759\) − 539.422i − 0.710701i
\(760\) 0 0
\(761\) −1002.93 −1.31791 −0.658955 0.752182i \(-0.729001\pi\)
−0.658955 + 0.752182i \(0.729001\pi\)
\(762\) 0 0
\(763\) 10.1909 0.0133564
\(764\) 0 0
\(765\) − 97.4121i − 0.127336i
\(766\) 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
\(771\) 1459.07 1.89244
\(772\) 0 0
\(773\) − 564.265i − 0.729968i −0.931014 0.364984i \(-0.881075\pi\)
0.931014 0.364984i \(-0.118925\pi\)
\(774\) 0 0
\(775\) − 1055.68i − 1.36217i
\(776\) 0 0
\(777\) 529.186 0.681063
\(778\) 0 0
\(779\) −670.759 −0.861052
\(780\) 0 0
\(781\) 458.389i 0.586926i
\(782\) 0 0
\(783\) 486.195i 0.620939i
\(784\) 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
\(789\) 879.582i 1.11481i
\(790\) 0 0
\(791\) − 36.4442i − 0.0460735i
\(792\) 0 0
\(793\) −58.4020 −0.0736469
\(794\) 0 0
\(795\) 517.588 0.651054
\(796\) 0 0
\(797\) − 207.983i − 0.260957i −0.991451 0.130479i \(-0.958349\pi\)
0.991451 0.130479i \(-0.0416514\pi\)
\(798\) 0 0
\(799\) − 854.785i − 1.06982i
\(800\) 0 0
\(801\) 117.058 0.146140
\(802\) 0 0
\(803\) 310.891 0.387162
\(804\) 0 0
\(805\) 144.439i 0.179428i
\(806\) 0 0
\(807\) 734.896i 0.910652i
\(808\) 0 0
\(809\) 340.540 0.420939 0.210470 0.977600i \(-0.432501\pi\)
0.210470 + 0.977600i \(0.432501\pi\)
\(810\) 0 0
\(811\) 907.380 1.11884 0.559420 0.828884i \(-0.311024\pi\)
0.559420 + 0.828884i \(0.311024\pi\)
\(812\) 0 0
\(813\) 1292.45i 1.58973i
\(814\) 0 0
\(815\) − 372.791i − 0.457412i
\(816\) 0 0
\(817\) 427.058 0.522715
\(818\) 0 0
\(819\) −10.8944 −0.0133021
\(820\) 0 0
\(821\) 633.423i 0.771526i 0.922598 + 0.385763i \(0.126062\pi\)
−0.922598 + 0.385763i \(0.873938\pi\)
\(822\) 0 0
\(823\) 143.649i 0.174544i 0.996185 + 0.0872718i \(0.0278149\pi\)
−0.996185 + 0.0872718i \(0.972185\pi\)
\(824\) 0 0
\(825\) −346.059 −0.419465
\(826\) 0 0
\(827\) −1545.57 −1.86888 −0.934442 0.356114i \(-0.884101\pi\)
−0.934442 + 0.356114i \(0.884101\pi\)
\(828\) 0 0
\(829\) − 743.956i − 0.897413i −0.893679 0.448707i \(-0.851885\pi\)
0.893679 0.448707i \(-0.148115\pi\)
\(830\) 0 0
\(831\) 568.291i 0.683864i
\(832\) 0 0
\(833\) −165.598 −0.198797
\(834\) 0 0
\(835\) −328.723 −0.393681
\(836\) 0 0
\(837\) − 1011.71i − 1.20874i
\(838\) 0 0
\(839\) − 96.3107i − 0.114792i −0.998351 0.0573961i \(-0.981720\pi\)
0.998351 0.0573961i \(-0.0182798\pi\)
\(840\) 0 0
\(841\) 337.000 0.400713
\(842\) 0 0
\(843\) 1439.99 1.70818
\(844\) 0 0
\(845\) − 258.201i − 0.305563i
\(846\) 0 0
\(847\) − 266.909i − 0.315123i
\(848\) 0 0
\(849\) −1179.40 −1.38917
\(850\) 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
\(855\) − 102.408i − 0.119775i
\(856\) 0 0
\(857\) 160.932 0.187785 0.0938926 0.995582i \(-0.470069\pi\)
0.0938926 + 0.995582i \(0.470069\pi\)
\(858\) 0 0
\(859\) −231.693 −0.269724 −0.134862 0.990864i \(-0.543059\pi\)
−0.134862 + 0.990864i \(0.543059\pi\)
\(860\) 0 0
\(861\) 243.629i 0.282961i
\(862\) 0 0
\(863\) 1337.35i 1.54965i 0.632176 + 0.774825i \(0.282162\pi\)
−0.632176 + 0.774825i \(0.717838\pi\)
\(864\) 0 0
\(865\) 282.402 0.326476
\(866\) 0 0
\(867\) −924.046 −1.06580
\(868\) 0 0
\(869\) 173.583i 0.199750i
\(870\) 0 0
\(871\) 51.7223i 0.0593827i
\(872\) 0 0
\(873\) 255.344 0.292490
\(874\) 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
\(879\) − 1744.79i − 1.98498i
\(880\) 0 0
\(881\) −186.706 −0.211926 −0.105963 0.994370i \(-0.533792\pi\)
−0.105963 + 0.994370i \(0.533792\pi\)
\(882\) 0 0
\(883\) −1277.99 −1.44733 −0.723664 0.690153i \(-0.757543\pi\)
−0.723664 + 0.690153i \(0.757543\pi\)
\(884\) 0 0
\(885\) − 325.726i − 0.368052i
\(886\) 0 0
\(887\) 980.717i 1.10566i 0.833295 + 0.552828i \(0.186451\pi\)
−0.833295 + 0.552828i \(0.813549\pi\)
\(888\) 0 0
\(889\) −330.784 −0.372085
\(890\) 0 0
\(891\) −438.897 −0.492590
\(892\) 0 0
\(893\) − 898.621i − 1.00629i
\(894\) 0 0
\(895\) 88.6605i 0.0990621i
\(896\) 0 0
\(897\) 186.392 0.207795
\(898\) 0 0
\(899\) 1048.76 1.16659
\(900\) 0 0
\(901\) − 2313.99i − 2.56825i
\(902\) 0 0
\(903\) − 155.114i − 0.171776i
\(904\) 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
\(909\) − 52.1173i − 0.0573348i
\(910\) 0 0
\(911\) 276.507i 0.303520i 0.988417 + 0.151760i \(0.0484941\pi\)
−0.988417 + 0.151760i \(0.951506\pi\)
\(912\) 0 0
\(913\) −16.2153 −0.0177605
\(914\) 0 0
\(915\) −199.397 −0.217920
\(916\) 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
\(921\) 762.500 0.827904
\(922\) 0 0
\(923\) −158.392 −0.171606
\(924\) 0 0
\(925\) 1323.85i 1.43119i
\(926\) 0 0
\(927\) 114.468i 0.123482i
\(928\) 0 0
\(929\) 35.4012 0.0381067 0.0190534 0.999818i \(-0.493935\pi\)
0.0190534 + 0.999818i \(0.493935\pi\)
\(930\) 0 0
\(931\) −174.090 −0.186993
\(932\) 0 0
\(933\) − 42.3320i − 0.0453719i
\(934\) 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
\(939\) −1399.99 −1.49094
\(940\) 0 0
\(941\) − 1852.90i − 1.96907i −0.175175 0.984537i \(-0.556049\pi\)
0.175175 0.984537i \(-0.443951\pi\)
\(942\) 0 0
\(943\) − 950.032i − 1.00746i
\(944\) 0 0
\(945\) 88.8040 0.0939725
\(946\) 0 0
\(947\) 1832.90 1.93548 0.967738 0.251959i \(-0.0810748\pi\)
0.967738 + 0.251959i \(0.0810748\pi\)
\(948\) 0 0
\(949\) 107.426i 0.113199i
\(950\) 0 0
\(951\) 444.927i 0.467852i
\(952\) 0 0
\(953\) 349.687 0.366933 0.183467 0.983026i \(-0.441268\pi\)
0.183467 + 0.983026i \(0.441268\pi\)
\(954\) 0 0
\(955\) 150.432 0.157521
\(956\) 0 0
\(957\) − 343.792i − 0.359239i
\(958\) 0 0
\(959\) 151.638i 0.158121i
\(960\) 0 0
\(961\) −1221.35 −1.27092
\(962\) 0 0
\(963\) −41.4416 −0.0430338
\(964\) 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
\(969\) −2008.74 −2.07301
\(970\) 0 0
\(971\) −656.497 −0.676104 −0.338052 0.941128i \(-0.609768\pi\)
−0.338052 + 0.941128i \(0.609768\pi\)
\(972\) 0 0
\(973\) 485.929i 0.499413i
\(974\) 0 0
\(975\) − 119.577i − 0.122643i
\(976\) 0 0
\(977\) 169.314 0.173300 0.0866498 0.996239i \(-0.472384\pi\)
0.0866498 + 0.996239i \(0.472384\pi\)
\(978\) 0 0
\(979\) 197.616 0.201855
\(980\) 0 0
\(981\) − 10.2337i − 0.0104319i
\(982\) 0 0
\(983\) − 698.607i − 0.710689i −0.934735 0.355345i \(-0.884364\pi\)
0.934735 0.355345i \(-0.115636\pi\)
\(984\) 0 0
\(985\) 192.362 0.195291
\(986\) 0 0
\(987\) −326.392 −0.330691
\(988\) 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
\(993\) 731.529 0.736686
\(994\) 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
\(999\) 1268.71i 1.26998i
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 224.3.g.a.15.1 4
3.2 odd 2 2016.3.g.a.1135.3 4
4.3 odd 2 56.3.g.a.43.4 yes 4
7.6 odd 2 1568.3.g.h.687.4 4
8.3 odd 2 inner 224.3.g.a.15.2 4
8.5 even 2 56.3.g.a.43.3 4
12.11 even 2 504.3.g.a.379.1 4
16.3 odd 4 1792.3.d.g.1023.7 8
16.5 even 4 1792.3.d.g.1023.8 8
16.11 odd 4 1792.3.d.g.1023.2 8
16.13 even 4 1792.3.d.g.1023.1 8
24.5 odd 2 504.3.g.a.379.2 4
24.11 even 2 2016.3.g.a.1135.2 4
28.3 even 6 392.3.k.j.275.1 8
28.11 odd 6 392.3.k.i.275.1 8
28.19 even 6 392.3.k.j.67.3 8
28.23 odd 6 392.3.k.i.67.3 8
28.27 even 2 392.3.g.h.99.4 4
56.5 odd 6 392.3.k.j.67.1 8
56.13 odd 2 392.3.g.h.99.3 4
56.27 even 2 1568.3.g.h.687.3 4
56.37 even 6 392.3.k.i.67.1 8
56.45 odd 6 392.3.k.j.275.3 8
56.53 even 6 392.3.k.i.275.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.3.g.a.43.3 4 8.5 even 2
56.3.g.a.43.4 yes 4 4.3 odd 2
224.3.g.a.15.1 4 1.1 even 1 trivial
224.3.g.a.15.2 4 8.3 odd 2 inner
392.3.g.h.99.3 4 56.13 odd 2
392.3.g.h.99.4 4 28.27 even 2
392.3.k.i.67.1 8 56.37 even 6
392.3.k.i.67.3 8 28.23 odd 6
392.3.k.i.275.1 8 28.11 odd 6
392.3.k.i.275.3 8 56.53 even 6
392.3.k.j.67.1 8 56.5 odd 6
392.3.k.j.67.3 8 28.19 even 6
392.3.k.j.275.1 8 28.3 even 6
392.3.k.j.275.3 8 56.45 odd 6
504.3.g.a.379.1 4 12.11 even 2
504.3.g.a.379.2 4 24.5 odd 2
1568.3.g.h.687.3 4 56.27 even 2
1568.3.g.h.687.4 4 7.6 odd 2
1792.3.d.g.1023.1 8 16.13 even 4
1792.3.d.g.1023.2 8 16.11 odd 4
1792.3.d.g.1023.7 8 16.3 odd 4
1792.3.d.g.1023.8 8 16.5 even 4
2016.3.g.a.1135.2 4 24.11 even 2
2016.3.g.a.1135.3 4 3.2 odd 2