Properties

Label 1840.2.m.g.1839.4
Level $1840$
Weight $2$
Character 1840.1839
Analytic conductor $14.692$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1840.m (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.6924739719\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1839.4
Character \(\chi\) \(=\) 1840.1839
Dual form 1840.2.m.g.1839.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.28652 q^{3} +(1.54120 - 1.62009i) q^{5} -2.73174i q^{7} +7.80121 q^{9} +O(q^{10})\) \(q-3.28652 q^{3} +(1.54120 - 1.62009i) q^{5} -2.73174i q^{7} +7.80121 q^{9} -2.28639 q^{11} +4.92586i q^{13} +(-5.06519 + 5.32446i) q^{15} -5.97341 q^{17} +0.377877 q^{19} +8.97791i q^{21} +(-0.582385 + 4.76034i) q^{23} +(-0.249397 - 4.99378i) q^{25} -15.7793 q^{27} +3.70400 q^{29} -2.89324i q^{31} +7.51426 q^{33} +(-4.42567 - 4.21016i) q^{35} -7.23320 q^{37} -16.1889i q^{39} +0.451624 q^{41} +0.359476i q^{43} +(12.0232 - 12.6387i) q^{45} +12.1733 q^{47} -0.462395 q^{49} +19.6317 q^{51} -9.69298 q^{53} +(-3.52378 + 3.70416i) q^{55} -1.24190 q^{57} +10.7427i q^{59} +6.73249i q^{61} -21.3109i q^{63} +(7.98034 + 7.59174i) q^{65} -7.06480i q^{67} +(1.91402 - 15.6449i) q^{69} +7.34667i q^{71} -1.24543i q^{73} +(0.819647 + 16.4121i) q^{75} +6.24581i q^{77} -13.6417 q^{79} +28.4553 q^{81} +15.1841i q^{83} +(-9.20623 + 9.67748i) q^{85} -12.1733 q^{87} -12.1526i q^{89} +13.4562 q^{91} +9.50868i q^{93} +(0.582385 - 0.612196i) q^{95} +4.34220 q^{97} -17.8366 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40q + 80q^{9} + O(q^{10}) \) \( 40q + 80q^{9} - 24q^{25} + 24q^{41} - 16q^{49} + 80q^{69} + 40q^{81} - 8q^{85} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times\).

\(n\) \(737\) \(1151\) \(1201\) \(1381\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

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
\(3\) −3.28652 −1.89747 −0.948737 0.316068i \(-0.897637\pi\)
−0.948737 + 0.316068i \(0.897637\pi\)
\(4\) 0 0
\(5\) 1.54120 1.62009i 0.689246 0.724527i
\(6\) 0 0
\(7\) 2.73174i 1.03250i −0.856438 0.516250i \(-0.827328\pi\)
0.856438 0.516250i \(-0.172672\pi\)
\(8\) 0 0
\(9\) 7.80121 2.60040
\(10\) 0 0
\(11\) −2.28639 −0.689372 −0.344686 0.938718i \(-0.612015\pi\)
−0.344686 + 0.938718i \(0.612015\pi\)
\(12\) 0 0
\(13\) 4.92586i 1.36619i 0.730331 + 0.683094i \(0.239366\pi\)
−0.730331 + 0.683094i \(0.760634\pi\)
\(14\) 0 0
\(15\) −5.06519 + 5.32446i −1.30783 + 1.37477i
\(16\) 0 0
\(17\) −5.97341 −1.44876 −0.724382 0.689398i \(-0.757875\pi\)
−0.724382 + 0.689398i \(0.757875\pi\)
\(18\) 0 0
\(19\) 0.377877 0.0866910 0.0433455 0.999060i \(-0.486198\pi\)
0.0433455 + 0.999060i \(0.486198\pi\)
\(20\) 0 0
\(21\) 8.97791i 1.95914i
\(22\) 0 0
\(23\) −0.582385 + 4.76034i −0.121436 + 0.992599i
\(24\) 0 0
\(25\) −0.249397 4.99378i −0.0498794 0.998755i
\(26\) 0 0
\(27\) −15.7793 −3.03672
\(28\) 0 0
\(29\) 3.70400 0.687815 0.343907 0.939004i \(-0.388249\pi\)
0.343907 + 0.939004i \(0.388249\pi\)
\(30\) 0 0
\(31\) 2.89324i 0.519641i −0.965657 0.259820i \(-0.916337\pi\)
0.965657 0.259820i \(-0.0836634\pi\)
\(32\) 0 0
\(33\) 7.51426 1.30806
\(34\) 0 0
\(35\) −4.42567 4.21016i −0.748074 0.711647i
\(36\) 0 0
\(37\) −7.23320 −1.18913 −0.594566 0.804047i \(-0.702676\pi\)
−0.594566 + 0.804047i \(0.702676\pi\)
\(38\) 0 0
\(39\) 16.1889i 2.59230i
\(40\) 0 0
\(41\) 0.451624 0.0705318 0.0352659 0.999378i \(-0.488772\pi\)
0.0352659 + 0.999378i \(0.488772\pi\)
\(42\) 0 0
\(43\) 0.359476i 0.0548196i 0.999624 + 0.0274098i \(0.00872591\pi\)
−0.999624 + 0.0274098i \(0.991274\pi\)
\(44\) 0 0
\(45\) 12.0232 12.6387i 1.79232 1.88406i
\(46\) 0 0
\(47\) 12.1733 1.77565 0.887826 0.460179i \(-0.152215\pi\)
0.887826 + 0.460179i \(0.152215\pi\)
\(48\) 0 0
\(49\) −0.462395 −0.0660565
\(50\) 0 0
\(51\) 19.6317 2.74899
\(52\) 0 0
\(53\) −9.69298 −1.33143 −0.665716 0.746205i \(-0.731874\pi\)
−0.665716 + 0.746205i \(0.731874\pi\)
\(54\) 0 0
\(55\) −3.52378 + 3.70416i −0.475147 + 0.499468i
\(56\) 0 0
\(57\) −1.24190 −0.164494
\(58\) 0 0
\(59\) 10.7427i 1.39858i 0.714838 + 0.699290i \(0.246500\pi\)
−0.714838 + 0.699290i \(0.753500\pi\)
\(60\) 0 0
\(61\) 6.73249i 0.862007i 0.902350 + 0.431003i \(0.141840\pi\)
−0.902350 + 0.431003i \(0.858160\pi\)
\(62\) 0 0
\(63\) 21.3109i 2.68492i
\(64\) 0 0
\(65\) 7.98034 + 7.59174i 0.989840 + 0.941639i
\(66\) 0 0
\(67\) 7.06480i 0.863103i −0.902088 0.431551i \(-0.857966\pi\)
0.902088 0.431551i \(-0.142034\pi\)
\(68\) 0 0
\(69\) 1.91402 15.6449i 0.230421 1.88343i
\(70\) 0 0
\(71\) 7.34667i 0.871889i 0.899974 + 0.435945i \(0.143586\pi\)
−0.899974 + 0.435945i \(0.856414\pi\)
\(72\) 0 0
\(73\) 1.24543i 0.145767i −0.997340 0.0728834i \(-0.976780\pi\)
0.997340 0.0728834i \(-0.0232201\pi\)
\(74\) 0 0
\(75\) 0.819647 + 16.4121i 0.0946447 + 1.89511i
\(76\) 0 0
\(77\) 6.24581i 0.711776i
\(78\) 0 0
\(79\) −13.6417 −1.53481 −0.767405 0.641163i \(-0.778452\pi\)
−0.767405 + 0.641163i \(0.778452\pi\)
\(80\) 0 0
\(81\) 28.4553 3.16170
\(82\) 0 0
\(83\) 15.1841i 1.66667i 0.552768 + 0.833335i \(0.313571\pi\)
−0.552768 + 0.833335i \(0.686429\pi\)
\(84\) 0 0
\(85\) −9.20623 + 9.67748i −0.998556 + 1.04967i
\(86\) 0 0
\(87\) −12.1733 −1.30511
\(88\) 0 0
\(89\) 12.1526i 1.28818i −0.764951 0.644089i \(-0.777237\pi\)
0.764951 0.644089i \(-0.222763\pi\)
\(90\) 0 0
\(91\) 13.4562 1.41059
\(92\) 0 0
\(93\) 9.50868i 0.986004i
\(94\) 0 0
\(95\) 0.582385 0.612196i 0.0597514 0.0628100i
\(96\) 0 0
\(97\) 4.34220 0.440883 0.220442 0.975400i \(-0.429250\pi\)
0.220442 + 0.975400i \(0.429250\pi\)
\(98\) 0 0
\(99\) −17.8366 −1.79264
\(100\) 0 0
\(101\) 3.11922 0.310374 0.155187 0.987885i \(-0.450402\pi\)
0.155187 + 0.987885i \(0.450402\pi\)
\(102\) 0 0
\(103\) 3.38893i 0.333921i 0.985964 + 0.166960i \(0.0533952\pi\)
−0.985964 + 0.166960i \(0.946605\pi\)
\(104\) 0 0
\(105\) 14.5450 + 13.8368i 1.41945 + 1.35033i
\(106\) 0 0
\(107\) 7.94818i 0.768380i 0.923254 + 0.384190i \(0.125519\pi\)
−0.923254 + 0.384190i \(0.874481\pi\)
\(108\) 0 0
\(109\) 14.1195i 1.35240i 0.736718 + 0.676200i \(0.236375\pi\)
−0.736718 + 0.676200i \(0.763625\pi\)
\(110\) 0 0
\(111\) 23.7721 2.25634
\(112\) 0 0
\(113\) 0.322974 0.0303828 0.0151914 0.999885i \(-0.495164\pi\)
0.0151914 + 0.999885i \(0.495164\pi\)
\(114\) 0 0
\(115\) 6.81462 + 8.28016i 0.635466 + 0.772129i
\(116\) 0 0
\(117\) 38.4277i 3.55264i
\(118\) 0 0
\(119\) 16.3178i 1.49585i
\(120\) 0 0
\(121\) −5.77244 −0.524767
\(122\) 0 0
\(123\) −1.48427 −0.133832
\(124\) 0 0
\(125\) −8.47475 7.29237i −0.758005 0.652249i
\(126\) 0 0
\(127\) 10.5340 0.934738 0.467369 0.884062i \(-0.345202\pi\)
0.467369 + 0.884062i \(0.345202\pi\)
\(128\) 0 0
\(129\) 1.18143i 0.104019i
\(130\) 0 0
\(131\) 11.3581i 0.992362i 0.868219 + 0.496181i \(0.165265\pi\)
−0.868219 + 0.496181i \(0.834735\pi\)
\(132\) 0 0
\(133\) 1.03226i 0.0895084i
\(134\) 0 0
\(135\) −24.3190 + 25.5639i −2.09305 + 2.20019i
\(136\) 0 0
\(137\) 19.0252 1.62543 0.812716 0.582660i \(-0.197988\pi\)
0.812716 + 0.582660i \(0.197988\pi\)
\(138\) 0 0
\(139\) 9.12171i 0.773693i 0.922144 + 0.386847i \(0.126436\pi\)
−0.922144 + 0.386847i \(0.873564\pi\)
\(140\) 0 0
\(141\) −40.0076 −3.36925
\(142\) 0 0
\(143\) 11.2624i 0.941810i
\(144\) 0 0
\(145\) 5.70860 6.00081i 0.474074 0.498341i
\(146\) 0 0
\(147\) 1.51967 0.125340
\(148\) 0 0
\(149\) 5.95344i 0.487725i −0.969810 0.243863i \(-0.921585\pi\)
0.969810 0.243863i \(-0.0784146\pi\)
\(150\) 0 0
\(151\) 9.94683i 0.809462i 0.914436 + 0.404731i \(0.132635\pi\)
−0.914436 + 0.404731i \(0.867365\pi\)
\(152\) 0 0
\(153\) −46.5998 −3.76737
\(154\) 0 0
\(155\) −4.68731 4.45906i −0.376494 0.358160i
\(156\) 0 0
\(157\) 16.2147 1.29407 0.647036 0.762460i \(-0.276008\pi\)
0.647036 + 0.762460i \(0.276008\pi\)
\(158\) 0 0
\(159\) 31.8562 2.52636
\(160\) 0 0
\(161\) 13.0040 + 1.59092i 1.02486 + 0.125382i
\(162\) 0 0
\(163\) 14.1754 1.11030 0.555151 0.831750i \(-0.312661\pi\)
0.555151 + 0.831750i \(0.312661\pi\)
\(164\) 0 0
\(165\) 11.5810 12.1738i 0.901578 0.947728i
\(166\) 0 0
\(167\) 8.85134 0.684937 0.342468 0.939529i \(-0.388737\pi\)
0.342468 + 0.939529i \(0.388737\pi\)
\(168\) 0 0
\(169\) −11.2641 −0.866467
\(170\) 0 0
\(171\) 2.94790 0.225432
\(172\) 0 0
\(173\) 24.2863i 1.84645i 0.384260 + 0.923225i \(0.374457\pi\)
−0.384260 + 0.923225i \(0.625543\pi\)
\(174\) 0 0
\(175\) −13.6417 + 0.681287i −1.03121 + 0.0515004i
\(176\) 0 0
\(177\) 35.3061i 2.65377i
\(178\) 0 0
\(179\) 14.8358i 1.10888i −0.832223 0.554441i \(-0.812932\pi\)
0.832223 0.554441i \(-0.187068\pi\)
\(180\) 0 0
\(181\) 3.42685i 0.254716i 0.991857 + 0.127358i \(0.0406497\pi\)
−0.991857 + 0.127358i \(0.959350\pi\)
\(182\) 0 0
\(183\) 22.1265i 1.63563i
\(184\) 0 0
\(185\) −11.1478 + 11.7185i −0.819604 + 0.861558i
\(186\) 0 0
\(187\) 13.6575 0.998737
\(188\) 0 0
\(189\) 43.1049i 3.13542i
\(190\) 0 0
\(191\) 9.37462 0.678324 0.339162 0.940728i \(-0.389857\pi\)
0.339162 + 0.940728i \(0.389857\pi\)
\(192\) 0 0
\(193\) 1.72975i 0.124510i 0.998060 + 0.0622550i \(0.0198292\pi\)
−0.998060 + 0.0622550i \(0.980171\pi\)
\(194\) 0 0
\(195\) −26.2276 24.9504i −1.87819 1.78673i
\(196\) 0 0
\(197\) 15.8158i 1.12683i −0.826174 0.563415i \(-0.809487\pi\)
0.826174 0.563415i \(-0.190513\pi\)
\(198\) 0 0
\(199\) −19.9550 −1.41458 −0.707288 0.706926i \(-0.750082\pi\)
−0.707288 + 0.706926i \(0.750082\pi\)
\(200\) 0 0
\(201\) 23.2186i 1.63771i
\(202\) 0 0
\(203\) 10.1183i 0.710169i
\(204\) 0 0
\(205\) 0.696043 0.731672i 0.0486138 0.0511022i
\(206\) 0 0
\(207\) −4.54331 + 37.1364i −0.315782 + 2.58116i
\(208\) 0 0
\(209\) −0.863973 −0.0597623
\(210\) 0 0
\(211\) 19.0013i 1.30810i −0.756449 0.654052i \(-0.773068\pi\)
0.756449 0.654052i \(-0.226932\pi\)
\(212\) 0 0
\(213\) 24.1450i 1.65439i
\(214\) 0 0
\(215\) 0.582385 + 0.554025i 0.0397183 + 0.0377842i
\(216\) 0 0
\(217\) −7.90357 −0.536529
\(218\) 0 0
\(219\) 4.09314i 0.276588i
\(220\) 0 0
\(221\) 29.4242i 1.97928i
\(222\) 0 0
\(223\) −1.87457 −0.125531 −0.0627653 0.998028i \(-0.519992\pi\)
−0.0627653 + 0.998028i \(0.519992\pi\)
\(224\) 0 0
\(225\) −1.94560 38.9575i −0.129706 2.59717i
\(226\) 0 0
\(227\) 5.18877i 0.344391i −0.985063 0.172195i \(-0.944914\pi\)
0.985063 0.172195i \(-0.0550860\pi\)
\(228\) 0 0
\(229\) 15.8023i 1.04425i 0.852870 + 0.522124i \(0.174860\pi\)
−0.852870 + 0.522124i \(0.825140\pi\)
\(230\) 0 0
\(231\) 20.5270i 1.35058i
\(232\) 0 0
\(233\) 14.6477i 0.959603i 0.877377 + 0.479802i \(0.159291\pi\)
−0.877377 + 0.479802i \(0.840709\pi\)
\(234\) 0 0
\(235\) 18.7614 19.7218i 1.22386 1.28651i
\(236\) 0 0
\(237\) 44.8337 2.91226
\(238\) 0 0
\(239\) 9.60182i 0.621090i 0.950559 + 0.310545i \(0.100512\pi\)
−0.950559 + 0.310545i \(0.899488\pi\)
\(240\) 0 0
\(241\) 18.8553i 1.21458i 0.794481 + 0.607289i \(0.207743\pi\)
−0.794481 + 0.607289i \(0.792257\pi\)
\(242\) 0 0
\(243\) −46.1810 −2.96251
\(244\) 0 0
\(245\) −0.712644 + 0.749123i −0.0455292 + 0.0478597i
\(246\) 0 0
\(247\) 1.86137i 0.118436i
\(248\) 0 0
\(249\) 49.9028i 3.16246i
\(250\) 0 0
\(251\) 24.9057 1.57203 0.786017 0.618205i \(-0.212140\pi\)
0.786017 + 0.618205i \(0.212140\pi\)
\(252\) 0 0
\(253\) 1.33156 10.8840i 0.0837143 0.684270i
\(254\) 0 0
\(255\) 30.2565 31.8052i 1.89473 1.99172i
\(256\) 0 0
\(257\) 16.5764i 1.03401i 0.855983 + 0.517004i \(0.172953\pi\)
−0.855983 + 0.517004i \(0.827047\pi\)
\(258\) 0 0
\(259\) 19.7592i 1.22778i
\(260\) 0 0
\(261\) 28.8957 1.78860
\(262\) 0 0
\(263\) 17.7754i 1.09608i −0.836453 0.548039i \(-0.815374\pi\)
0.836453 0.548039i \(-0.184626\pi\)
\(264\) 0 0
\(265\) −14.9388 + 15.7035i −0.917685 + 0.964659i
\(266\) 0 0
\(267\) 39.9399i 2.44428i
\(268\) 0 0
\(269\) −18.2884 −1.11506 −0.557532 0.830155i \(-0.688252\pi\)
−0.557532 + 0.830155i \(0.688252\pi\)
\(270\) 0 0
\(271\) 8.44481i 0.512986i 0.966546 + 0.256493i \(0.0825670\pi\)
−0.966546 + 0.256493i \(0.917433\pi\)
\(272\) 0 0
\(273\) −44.2239 −2.67655
\(274\) 0 0
\(275\) 0.570218 + 11.4177i 0.0343854 + 0.688513i
\(276\) 0 0
\(277\) 27.9612i 1.68003i −0.542565 0.840014i \(-0.682547\pi\)
0.542565 0.840014i \(-0.317453\pi\)
\(278\) 0 0
\(279\) 22.5708i 1.35128i
\(280\) 0 0
\(281\) 28.3900i 1.69360i 0.531909 + 0.846802i \(0.321475\pi\)
−0.531909 + 0.846802i \(0.678525\pi\)
\(282\) 0 0
\(283\) 9.11525i 0.541845i 0.962601 + 0.270923i \(0.0873288\pi\)
−0.962601 + 0.270923i \(0.912671\pi\)
\(284\) 0 0
\(285\) −1.91402 + 2.01199i −0.113377 + 0.119180i
\(286\) 0 0
\(287\) 1.23372i 0.0728241i
\(288\) 0 0
\(289\) 18.6816 1.09892
\(290\) 0 0
\(291\) −14.2707 −0.836564
\(292\) 0 0
\(293\) −5.14021 −0.300294 −0.150147 0.988664i \(-0.547975\pi\)
−0.150147 + 0.988664i \(0.547975\pi\)
\(294\) 0 0
\(295\) 17.4042 + 16.5567i 1.01331 + 0.963966i
\(296\) 0 0
\(297\) 36.0775 2.09343
\(298\) 0 0
\(299\) −23.4488 2.86874i −1.35608 0.165904i
\(300\) 0 0
\(301\) 0.981995 0.0566013
\(302\) 0 0
\(303\) −10.2514 −0.588927
\(304\) 0 0
\(305\) 10.9072 + 10.3761i 0.624547 + 0.594135i
\(306\) 0 0
\(307\) −16.0988 −0.918806 −0.459403 0.888228i \(-0.651937\pi\)
−0.459403 + 0.888228i \(0.651937\pi\)
\(308\) 0 0
\(309\) 11.1378i 0.633606i
\(310\) 0 0
\(311\) 13.6114i 0.771834i 0.922534 + 0.385917i \(0.126115\pi\)
−0.922534 + 0.385917i \(0.873885\pi\)
\(312\) 0 0
\(313\) 16.7024 0.944073 0.472037 0.881579i \(-0.343519\pi\)
0.472037 + 0.881579i \(0.343519\pi\)
\(314\) 0 0
\(315\) −34.5256 32.8443i −1.94530 1.85057i
\(316\) 0 0
\(317\) 24.1290i 1.35522i 0.735423 + 0.677609i \(0.236984\pi\)
−0.735423 + 0.677609i \(0.763016\pi\)
\(318\) 0 0
\(319\) −8.46877 −0.474160
\(320\) 0 0
\(321\) 26.1219i 1.45798i
\(322\) 0 0
\(323\) −2.25722 −0.125595
\(324\) 0 0
\(325\) 24.5986 1.22849i 1.36449 0.0681445i
\(326\) 0 0
\(327\) 46.4039i 2.56614i
\(328\) 0 0
\(329\) 33.2542i 1.83336i
\(330\) 0 0
\(331\) 9.49579i 0.521936i −0.965348 0.260968i \(-0.915958\pi\)
0.965348 0.260968i \(-0.0840416\pi\)
\(332\) 0 0
\(333\) −56.4278 −3.09222
\(334\) 0 0
\(335\) −11.4456 10.8883i −0.625341 0.594890i
\(336\) 0 0
\(337\) −9.95485 −0.542275 −0.271138 0.962541i \(-0.587400\pi\)
−0.271138 + 0.962541i \(0.587400\pi\)
\(338\) 0 0
\(339\) −1.06146 −0.0576506
\(340\) 0 0
\(341\) 6.61506i 0.358226i
\(342\) 0 0
\(343\) 17.8590i 0.964297i
\(344\) 0 0
\(345\) −22.3964 27.2129i −1.20578 1.46509i
\(346\) 0 0
\(347\) −28.5546 −1.53289 −0.766446 0.642309i \(-0.777977\pi\)
−0.766446 + 0.642309i \(0.777977\pi\)
\(348\) 0 0
\(349\) 11.7060 0.626610 0.313305 0.949652i \(-0.398564\pi\)
0.313305 + 0.949652i \(0.398564\pi\)
\(350\) 0 0
\(351\) 77.7265i 4.14873i
\(352\) 0 0
\(353\) 10.6128i 0.564864i −0.959287 0.282432i \(-0.908859\pi\)
0.959287 0.282432i \(-0.0911411\pi\)
\(354\) 0 0
\(355\) 11.9023 + 11.3227i 0.631707 + 0.600946i
\(356\) 0 0
\(357\) 53.6288i 2.83833i
\(358\) 0 0
\(359\) 16.7030 0.881548 0.440774 0.897618i \(-0.354704\pi\)
0.440774 + 0.897618i \(0.354704\pi\)
\(360\) 0 0
\(361\) −18.8572 −0.992485
\(362\) 0 0
\(363\) 18.9712 0.995731
\(364\) 0 0
\(365\) −2.01771 1.91946i −0.105612 0.100469i
\(366\) 0 0
\(367\) 1.96080i 0.102353i −0.998690 0.0511764i \(-0.983703\pi\)
0.998690 0.0511764i \(-0.0162971\pi\)
\(368\) 0 0
\(369\) 3.52321 0.183411
\(370\) 0 0
\(371\) 26.4787i 1.37470i
\(372\) 0 0
\(373\) −31.3228 −1.62183 −0.810916 0.585163i \(-0.801031\pi\)
−0.810916 + 0.585163i \(0.801031\pi\)
\(374\) 0 0
\(375\) 27.8524 + 23.9665i 1.43829 + 1.23763i
\(376\) 0 0
\(377\) 18.2454i 0.939684i
\(378\) 0 0
\(379\) −0.853853 −0.0438595 −0.0219297 0.999760i \(-0.506981\pi\)
−0.0219297 + 0.999760i \(0.506981\pi\)
\(380\) 0 0
\(381\) −34.6201 −1.77364
\(382\) 0 0
\(383\) 2.33128i 0.119123i −0.998225 0.0595614i \(-0.981030\pi\)
0.998225 0.0595614i \(-0.0189702\pi\)
\(384\) 0 0
\(385\) 10.1188 + 9.62605i 0.515701 + 0.490589i
\(386\) 0 0
\(387\) 2.80435i 0.142553i
\(388\) 0 0
\(389\) 10.4152i 0.528074i 0.964513 + 0.264037i \(0.0850541\pi\)
−0.964513 + 0.264037i \(0.914946\pi\)
\(390\) 0 0
\(391\) 3.47882 28.4355i 0.175932 1.43804i
\(392\) 0 0
\(393\) 37.3286i 1.88298i
\(394\) 0 0
\(395\) −21.0246 + 22.1008i −1.05786 + 1.11201i
\(396\) 0 0
\(397\) 9.63854i 0.483745i −0.970308 0.241872i \(-0.922238\pi\)
0.970308 0.241872i \(-0.0777615\pi\)
\(398\) 0 0
\(399\) 3.39255i 0.169840i
\(400\) 0 0
\(401\) 18.4820i 0.922947i 0.887154 + 0.461473i \(0.152679\pi\)
−0.887154 + 0.461473i \(0.847321\pi\)
\(402\) 0 0
\(403\) 14.2517 0.709927
\(404\) 0 0
\(405\) 43.8553 46.1002i 2.17919 2.29074i
\(406\) 0 0
\(407\) 16.5379 0.819753
\(408\) 0 0
\(409\) −26.0001 −1.28562 −0.642811 0.766025i \(-0.722232\pi\)
−0.642811 + 0.766025i \(0.722232\pi\)
\(410\) 0 0
\(411\) −62.5267 −3.08421
\(412\) 0 0
\(413\) 29.3462 1.44403
\(414\) 0 0
\(415\) 24.5996 + 23.4017i 1.20755 + 1.14875i
\(416\) 0 0
\(417\) 29.9787i 1.46806i
\(418\) 0 0
\(419\) −16.0811 −0.785614 −0.392807 0.919621i \(-0.628496\pi\)
−0.392807 + 0.919621i \(0.628496\pi\)
\(420\) 0 0
\(421\) 23.5031i 1.14547i −0.819740 0.572735i \(-0.805882\pi\)
0.819740 0.572735i \(-0.194118\pi\)
\(422\) 0 0
\(423\) 94.9662 4.61741
\(424\) 0 0
\(425\) 1.48975 + 29.8299i 0.0722635 + 1.44696i
\(426\) 0 0
\(427\) 18.3914 0.890022
\(428\) 0 0
\(429\) 37.0141i 1.78706i
\(430\) 0 0
\(431\) −6.61906 −0.318829 −0.159415 0.987212i \(-0.550961\pi\)
−0.159415 + 0.987212i \(0.550961\pi\)
\(432\) 0 0
\(433\) 20.3334 0.977163 0.488581 0.872518i \(-0.337515\pi\)
0.488581 + 0.872518i \(0.337515\pi\)
\(434\) 0 0
\(435\) −18.7614 + 19.7218i −0.899542 + 0.945588i
\(436\) 0 0
\(437\) −0.220070 + 1.79882i −0.0105274 + 0.0860494i
\(438\) 0 0
\(439\) 15.1971i 0.725317i −0.931922 0.362659i \(-0.881869\pi\)
0.931922 0.362659i \(-0.118131\pi\)
\(440\) 0 0
\(441\) −3.60724 −0.171774
\(442\) 0 0
\(443\) −8.20593 −0.389875 −0.194938 0.980816i \(-0.562450\pi\)
−0.194938 + 0.980816i \(0.562450\pi\)
\(444\) 0 0
\(445\) −19.6884 18.7297i −0.933320 0.887872i
\(446\) 0 0
\(447\) 19.5661i 0.925445i
\(448\) 0 0
\(449\) −23.0265 −1.08669 −0.543343 0.839511i \(-0.682842\pi\)
−0.543343 + 0.839511i \(0.682842\pi\)
\(450\) 0 0
\(451\) −1.03259 −0.0486226
\(452\) 0 0
\(453\) 32.6905i 1.53593i
\(454\) 0 0
\(455\) 20.7386 21.8002i 0.972243 1.02201i
\(456\) 0 0
\(457\) −38.0523 −1.78001 −0.890005 0.455951i \(-0.849299\pi\)
−0.890005 + 0.455951i \(0.849299\pi\)
\(458\) 0 0
\(459\) 94.2561 4.39950
\(460\) 0 0
\(461\) 28.7920 1.34098 0.670489 0.741919i \(-0.266084\pi\)
0.670489 + 0.741919i \(0.266084\pi\)
\(462\) 0 0
\(463\) 1.34881 0.0626843 0.0313421 0.999509i \(-0.490022\pi\)
0.0313421 + 0.999509i \(0.490022\pi\)
\(464\) 0 0
\(465\) 15.4049 + 14.6548i 0.714387 + 0.679600i
\(466\) 0 0
\(467\) 8.23187i 0.380926i −0.981694 0.190463i \(-0.939001\pi\)
0.981694 0.190463i \(-0.0609988\pi\)
\(468\) 0 0
\(469\) −19.2992 −0.891154
\(470\) 0 0
\(471\) −53.2898 −2.45547
\(472\) 0 0
\(473\) 0.821902i 0.0377911i
\(474\) 0 0
\(475\) −0.0942413 1.88703i −0.00432409 0.0865831i
\(476\) 0 0
\(477\) −75.6170 −3.46226
\(478\) 0 0
\(479\) −2.78805 −0.127389 −0.0636946 0.997969i \(-0.520288\pi\)
−0.0636946 + 0.997969i \(0.520288\pi\)
\(480\) 0 0
\(481\) 35.6297i 1.62458i
\(482\) 0 0
\(483\) −42.7379 5.22860i −1.94464 0.237910i
\(484\) 0 0
\(485\) 6.69220 7.03476i 0.303877 0.319432i
\(486\) 0 0
\(487\) −1.35672 −0.0614787 −0.0307394 0.999527i \(-0.509786\pi\)
−0.0307394 + 0.999527i \(0.509786\pi\)
\(488\) 0 0
\(489\) −46.5877 −2.10677
\(490\) 0 0
\(491\) 0.539939i 0.0243671i 0.999926 + 0.0121835i \(0.00387824\pi\)
−0.999926 + 0.0121835i \(0.996122\pi\)
\(492\) 0 0
\(493\) −22.1255 −0.996482
\(494\) 0 0
\(495\) −27.4898 + 28.8969i −1.23557 + 1.29882i
\(496\) 0 0
\(497\) 20.0692 0.900226
\(498\) 0 0
\(499\) 23.4902i 1.05157i 0.850619 + 0.525783i \(0.176228\pi\)
−0.850619 + 0.525783i \(0.823772\pi\)
\(500\) 0 0
\(501\) −29.0901 −1.29965
\(502\) 0 0
\(503\) 17.7835i 0.792929i 0.918050 + 0.396464i \(0.129763\pi\)
−0.918050 + 0.396464i \(0.870237\pi\)
\(504\) 0 0
\(505\) 4.80735 5.05343i 0.213924 0.224875i
\(506\) 0 0
\(507\) 37.0196 1.64410
\(508\) 0 0
\(509\) 29.3504 1.30094 0.650468 0.759534i \(-0.274573\pi\)
0.650468 + 0.759534i \(0.274573\pi\)
\(510\) 0 0
\(511\) −3.40219 −0.150504
\(512\) 0 0
\(513\) −5.96263 −0.263256
\(514\) 0 0
\(515\) 5.49037 + 5.22302i 0.241935 + 0.230154i
\(516\) 0 0
\(517\) −27.8328 −1.22408
\(518\) 0 0
\(519\) 79.8173i 3.50359i
\(520\) 0 0
\(521\) 15.4545i 0.677074i 0.940953 + 0.338537i \(0.109932\pi\)
−0.940953 + 0.338537i \(0.890068\pi\)
\(522\) 0 0
\(523\) 31.5873i 1.38122i −0.723229 0.690608i \(-0.757343\pi\)
0.723229 0.690608i \(-0.242657\pi\)
\(524\) 0 0
\(525\) 44.8337 2.23906i 1.95670 0.0977207i
\(526\) 0 0
\(527\) 17.2825i 0.752837i
\(528\) 0 0
\(529\) −22.3217 5.54470i −0.970507 0.241074i
\(530\) 0 0
\(531\) 83.8060i 3.63687i
\(532\) 0 0
\(533\) 2.22463i 0.0963596i
\(534\) 0 0
\(535\) 12.8768 + 12.2497i 0.556712 + 0.529603i
\(536\) 0 0
\(537\) 48.7582i 2.10407i
\(538\) 0 0
\(539\) 1.05721 0.0455375
\(540\) 0 0
\(541\) −20.7309 −0.891292 −0.445646 0.895209i \(-0.647026\pi\)
−0.445646 + 0.895209i \(0.647026\pi\)
\(542\) 0 0
\(543\) 11.2624i 0.483316i
\(544\) 0 0
\(545\) 22.8749 + 21.7610i 0.979851 + 0.932137i
\(546\) 0 0
\(547\) −9.88064 −0.422466 −0.211233 0.977436i \(-0.567748\pi\)
−0.211233 + 0.977436i \(0.567748\pi\)
\(548\) 0 0
\(549\) 52.5216i 2.24157i
\(550\) 0 0
\(551\) 1.39966 0.0596273
\(552\) 0 0
\(553\) 37.2655i 1.58469i
\(554\) 0 0
\(555\) 36.6375 38.5129i 1.55518 1.63478i
\(556\) 0 0
\(557\) −13.3481 −0.565576 −0.282788 0.959182i \(-0.591259\pi\)
−0.282788 + 0.959182i \(0.591259\pi\)
\(558\) 0 0
\(559\) −1.77073 −0.0748939
\(560\) 0 0
\(561\) −44.8857 −1.89508
\(562\) 0 0
\(563\) 24.9591i 1.05190i −0.850516 0.525950i \(-0.823710\pi\)
0.850516 0.525950i \(-0.176290\pi\)
\(564\) 0 0
\(565\) 0.497768 0.523247i 0.0209412 0.0220132i
\(566\) 0 0
\(567\) 77.7324i 3.26445i
\(568\) 0 0
\(569\) 33.7602i 1.41530i 0.706562 + 0.707651i \(0.250245\pi\)
−0.706562 + 0.707651i \(0.749755\pi\)
\(570\) 0 0
\(571\) −38.3767 −1.60602 −0.803008 0.595968i \(-0.796768\pi\)
−0.803008 + 0.595968i \(0.796768\pi\)
\(572\) 0 0
\(573\) −30.8099 −1.28710
\(574\) 0 0
\(575\) 23.9173 + 1.72109i 0.997421 + 0.0717743i
\(576\) 0 0
\(577\) 25.7420i 1.07166i 0.844327 + 0.535828i \(0.180000\pi\)
−0.844327 + 0.535828i \(0.820000\pi\)
\(578\) 0 0
\(579\) 5.68485i 0.236254i
\(580\) 0 0
\(581\) 41.4790 1.72084
\(582\) 0 0
\(583\) 22.1619 0.917852
\(584\) 0 0
\(585\) 62.2563 + 59.2248i 2.57398 + 2.44864i
\(586\) 0 0
\(587\) 16.8165 0.694091 0.347045 0.937848i \(-0.387185\pi\)
0.347045 + 0.937848i \(0.387185\pi\)
\(588\) 0 0
\(589\) 1.09329i 0.0450482i
\(590\) 0 0
\(591\) 51.9790i 2.13813i
\(592\) 0 0
\(593\) 26.0486i 1.06969i 0.844951 + 0.534843i \(0.179629\pi\)
−0.844951 + 0.534843i \(0.820371\pi\)
\(594\) 0 0
\(595\) 26.4363 + 25.1490i 1.08378 + 1.03101i
\(596\) 0 0
\(597\) 65.5827 2.68412
\(598\) 0 0
\(599\) 20.6703i 0.844567i 0.906464 + 0.422283i \(0.138771\pi\)
−0.906464 + 0.422283i \(0.861229\pi\)
\(600\) 0 0
\(601\) 8.32601 0.339625 0.169813 0.985476i \(-0.445684\pi\)
0.169813 + 0.985476i \(0.445684\pi\)
\(602\) 0 0
\(603\) 55.1140i 2.24442i
\(604\) 0 0
\(605\) −8.89649 + 9.35188i −0.361694 + 0.380208i
\(606\) 0 0
\(607\) −15.9501 −0.647396 −0.323698 0.946160i \(-0.604926\pi\)
−0.323698 + 0.946160i \(0.604926\pi\)
\(608\) 0 0
\(609\) 33.2542i 1.34753i
\(610\) 0 0
\(611\) 59.9637i 2.42587i
\(612\) 0 0
\(613\) 23.1501 0.935023 0.467512 0.883987i \(-0.345151\pi\)
0.467512 + 0.883987i \(0.345151\pi\)
\(614\) 0 0
\(615\) −2.28756 + 2.40465i −0.0922433 + 0.0969650i
\(616\) 0 0
\(617\) −2.88371 −0.116094 −0.0580468 0.998314i \(-0.518487\pi\)
−0.0580468 + 0.998314i \(0.518487\pi\)
\(618\) 0 0
\(619\) −6.49465 −0.261042 −0.130521 0.991446i \(-0.541665\pi\)
−0.130521 + 0.991446i \(0.541665\pi\)
\(620\) 0 0
\(621\) 9.18961 75.1147i 0.368766 3.01425i
\(622\) 0 0
\(623\) −33.1978 −1.33004
\(624\) 0 0
\(625\) −24.8756 + 2.49086i −0.995024 + 0.0996345i
\(626\) 0 0
\(627\) 2.83947 0.113397
\(628\) 0 0
\(629\) 43.2069 1.72277
\(630\) 0 0
\(631\) 28.6830 1.14185 0.570927 0.821001i \(-0.306584\pi\)
0.570927 + 0.821001i \(0.306584\pi\)
\(632\) 0 0
\(633\) 62.4482i 2.48209i
\(634\) 0 0
\(635\) 16.2350 17.0660i 0.644265 0.677243i
\(636\) 0 0
\(637\) 2.27769i 0.0902455i
\(638\) 0 0
\(639\) 57.3129i 2.26726i
\(640\) 0 0
\(641\) 31.5938i 1.24788i −0.781473 0.623939i \(-0.785531\pi\)
0.781473 0.623939i \(-0.214469\pi\)
\(642\) 0 0
\(643\) 47.3512i 1.86735i 0.358120 + 0.933675i \(0.383418\pi\)
−0.358120 + 0.933675i \(0.616582\pi\)
\(644\) 0 0
\(645\) −1.91402 1.82082i −0.0753644 0.0716945i
\(646\) 0 0
\(647\) 21.8763 0.860045 0.430022 0.902818i \(-0.358506\pi\)
0.430022 + 0.902818i \(0.358506\pi\)
\(648\) 0 0
\(649\) 24.5619i 0.964141i
\(650\) 0 0
\(651\) 25.9752 1.01805
\(652\) 0 0
\(653\) 0.359578i 0.0140714i −0.999975 0.00703568i \(-0.997760\pi\)
0.999975 0.00703568i \(-0.00223954\pi\)
\(654\) 0 0
\(655\) 18.4012 + 17.5051i 0.718993 + 0.683982i
\(656\) 0 0
\(657\) 9.71588i 0.379052i
\(658\) 0 0
\(659\) −1.91275 −0.0745104 −0.0372552 0.999306i \(-0.511861\pi\)
−0.0372552 + 0.999306i \(0.511861\pi\)
\(660\) 0 0
\(661\) 4.73207i 0.184056i 0.995756 + 0.0920280i \(0.0293349\pi\)
−0.995756 + 0.0920280i \(0.970665\pi\)
\(662\) 0 0
\(663\) 96.7031i 3.75564i
\(664\) 0 0
\(665\) −1.67236 1.59092i −0.0648513 0.0616933i
\(666\) 0 0
\(667\) −2.15715 + 17.6323i −0.0835252 + 0.682725i
\(668\) 0 0
\(669\) 6.16082 0.238191
\(670\) 0 0
\(671\) 15.3931i 0.594243i
\(672\) 0 0
\(673\) 42.7608i 1.64831i −0.566366 0.824154i \(-0.691651\pi\)
0.566366 0.824154i \(-0.308349\pi\)
\(674\) 0 0
\(675\) 3.93530 + 78.7982i 0.151470 + 3.03294i
\(676\) 0 0
\(677\) −1.30895 −0.0503072 −0.0251536 0.999684i \(-0.508007\pi\)
−0.0251536 + 0.999684i \(0.508007\pi\)
\(678\) 0 0
\(679\) 11.8617i 0.455212i
\(680\) 0 0
\(681\) 17.0530i 0.653472i
\(682\) 0 0
\(683\) 16.4311 0.628718 0.314359 0.949304i \(-0.398210\pi\)
0.314359 + 0.949304i \(0.398210\pi\)
\(684\) 0 0
\(685\) 29.3217 30.8226i 1.12032 1.17767i
\(686\) 0 0
\(687\) 51.9347i 1.98143i
\(688\) 0 0
\(689\) 47.7462i 1.81899i
\(690\) 0 0
\(691\) 50.1615i 1.90823i −0.299433 0.954117i \(-0.596797\pi\)
0.299433 0.954117i \(-0.403203\pi\)
\(692\) 0 0
\(693\) 48.7249i 1.85091i
\(694\) 0 0
\(695\) 14.7780 + 14.0584i 0.560562 + 0.533265i
\(696\) 0 0
\(697\) −2.69773 −0.102184
\(698\) 0 0
\(699\) 48.1400i 1.82082i
\(700\) 0 0
\(701\) 0.199633i 0.00754004i 0.999993 + 0.00377002i \(0.00120004\pi\)
−0.999993 + 0.00377002i \(0.998800\pi\)
\(702\) 0 0
\(703\) −2.73326 −0.103087
\(704\) 0 0
\(705\) −61.6598 + 64.8161i −2.32224 + 2.44111i
\(706\) 0 0
\(707\) 8.52090i 0.320461i
\(708\) 0 0
\(709\) 40.1194i 1.50671i −0.657611 0.753357i \(-0.728433\pi\)
0.657611 0.753357i \(-0.271567\pi\)
\(710\) 0 0
\(711\) −106.422 −3.99113
\(712\) 0 0
\(713\) 13.7728 + 1.68498i 0.515795 + 0.0631029i
\(714\) 0 0
\(715\) −18.2461 17.3576i −0.682367 0.649139i
\(716\) 0 0
\(717\) 31.5566i 1.17850i
\(718\) 0 0
\(719\) 1.40670i 0.0524610i 0.999656 + 0.0262305i \(0.00835039\pi\)
−0.999656 + 0.0262305i \(0.991650\pi\)
\(720\) 0 0
\(721\) 9.25766 0.344773
\(722\) 0 0
\(723\) 61.9684i 2.30463i
\(724\) 0 0
\(725\) −0.923765 18.4969i −0.0343078 0.686959i
\(726\) 0 0
\(727\) 36.3686i 1.34884i −0.738349 0.674419i \(-0.764394\pi\)
0.738349 0.674419i \(-0.235606\pi\)
\(728\) 0 0
\(729\) 66.4089 2.45959
\(730\) 0 0
\(731\) 2.14730i 0.0794208i
\(732\) 0 0
\(733\) 35.2005 1.30016 0.650081 0.759865i \(-0.274735\pi\)
0.650081 + 0.759865i \(0.274735\pi\)
\(734\) 0 0
\(735\) 2.34212 2.46201i 0.0863904 0.0908125i
\(736\) 0 0
\(737\) 16.1529i 0.594998i
\(738\) 0 0
\(739\) 6.59702i 0.242675i −0.992611 0.121338i \(-0.961282\pi\)
0.992611 0.121338i \(-0.0387184\pi\)
\(740\) 0 0
\(741\) 6.11743i 0.224729i
\(742\) 0 0
\(743\) 21.9968i 0.806985i 0.914983 + 0.403493i \(0.132204\pi\)
−0.914983 + 0.403493i \(0.867796\pi\)
\(744\) 0 0
\(745\) −9.64513 9.17546i −0.353370 0.336163i
\(746\) 0 0
\(747\) 118.454i 4.33402i
\(748\) 0 0
\(749\) 21.7124 0.793352
\(750\) 0 0
\(751\) −10.8072 −0.394359 −0.197179 0.980367i \(-0.563178\pi\)
−0.197179 + 0.980367i \(0.563178\pi\)
\(752\) 0 0
\(753\) −81.8531 −2.98289
\(754\) 0 0
\(755\) 16.1148 + 15.3301i 0.586477 + 0.557919i
\(756\) 0 0
\(757\) 9.47219 0.344273 0.172136 0.985073i \(-0.444933\pi\)
0.172136 + 0.985073i \(0.444933\pi\)
\(758\) 0 0
\(759\) −4.37619 + 35.7704i −0.158846 + 1.29838i
\(760\) 0 0
\(761\) −48.3960 −1.75436 −0.877178 0.480166i \(-0.840576\pi\)
−0.877178 + 0.480166i \(0.840576\pi\)
\(762\) 0 0
\(763\) 38.5707 1.39635
\(764\) 0 0
\(765\) −71.8197 + 75.4960i −2.59665 + 2.72956i
\(766\) 0 0
\(767\) −52.9170 −1.91072
\(768\) 0 0
\(769\) 5.14793i 0.185639i −0.995683 0.0928195i \(-0.970412\pi\)
0.995683 0.0928195i \(-0.0295880\pi\)
\(770\) 0 0
\(771\) 54.4786i 1.96200i
\(772\) 0 0
\(773\) −35.7974 −1.28754 −0.643771 0.765218i \(-0.722631\pi\)
−0.643771 + 0.765218i \(0.722631\pi\)
\(774\) 0 0
\(775\) −14.4482 + 0.721564i −0.518994 + 0.0259193i
\(776\) 0 0
\(777\) 64.9391i 2.32968i
\(778\) 0 0
\(779\) 0.170658 0.00611447
\(780\) 0 0
\(781\) 16.7973i 0.601056i
\(782\) 0 0
\(783\) −58.4464 −2.08870
\(784\) 0 0
\(785\) 24.9901 26.2693i 0.891934 0.937590i
\(786\) 0 0
\(787\) 29.7273i 1.05966i 0.848103 + 0.529832i \(0.177745\pi\)
−0.848103 + 0.529832i \(0.822255\pi\)
\(788\) 0 0
\(789\) 58.4192i 2.07978i
\(790\) 0 0
\(791\) 0.882280i 0.0313703i
\(792\) 0 0
\(793\) −33.1633 −1.17766
\(794\) 0 0
\(795\) 49.0967 51.6099i 1.74128 1.83041i
\(796\) 0 0
\(797\) 22.5547 0.798927 0.399464 0.916749i \(-0.369196\pi\)
0.399464 + 0.916749i \(0.369196\pi\)
\(798\) 0 0
\(799\) −72.7159 −2.57250
\(800\) 0 0
\(801\) 94.8054i 3.34978i
\(802\) 0 0
\(803\) 2.84754i 0.100487i