Properties

Label 1260.2.s.b.361.1
Level $1260$
Weight $2$
Character 1260.361
Analytic conductor $10.061$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1260.s (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.0611506547\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 140)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 361.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1260.361
Dual form 1260.2.s.b.541.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{5} +(0.500000 + 2.59808i) q^{7} +O(q^{10})\) \(q+(-0.500000 - 0.866025i) q^{5} +(0.500000 + 2.59808i) q^{7} +(-1.00000 + 1.73205i) q^{11} -6.00000 q^{13} +(1.00000 - 1.73205i) q^{17} +(-4.50000 - 7.79423i) q^{23} +(-0.500000 + 0.866025i) q^{25} -3.00000 q^{29} +(-1.00000 + 1.73205i) q^{31} +(2.00000 - 1.73205i) q^{35} +(-4.00000 - 6.92820i) q^{37} -5.00000 q^{41} +1.00000 q^{43} +(4.00000 + 6.92820i) q^{47} +(-6.50000 + 2.59808i) q^{49} +(2.00000 - 3.46410i) q^{53} +2.00000 q^{55} +(-4.00000 + 6.92820i) q^{59} +(-3.50000 - 6.06218i) q^{61} +(3.00000 + 5.19615i) q^{65} +(1.50000 - 2.59808i) q^{67} -8.00000 q^{71} +(-7.00000 + 12.1244i) q^{73} +(-5.00000 - 1.73205i) q^{77} +(-2.00000 - 3.46410i) q^{79} +1.00000 q^{83} -2.00000 q^{85} +(6.50000 + 11.2583i) q^{89} +(-3.00000 - 15.5885i) q^{91} -10.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{5} + q^{7} + O(q^{10}) \) \( 2q - q^{5} + q^{7} - 2q^{11} - 12q^{13} + 2q^{17} - 9q^{23} - q^{25} - 6q^{29} - 2q^{31} + 4q^{35} - 8q^{37} - 10q^{41} + 2q^{43} + 8q^{47} - 13q^{49} + 4q^{53} + 4q^{55} - 8q^{59} - 7q^{61} + 6q^{65} + 3q^{67} - 16q^{71} - 14q^{73} - 10q^{77} - 4q^{79} + 2q^{83} - 4q^{85} + 13q^{89} - 6q^{91} - 20q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1260\mathbb{Z}\right)^\times\).

\(n\) \(281\) \(631\) \(757\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(5\) −0.500000 0.866025i −0.223607 0.387298i
\(6\) 0 0
\(7\) 0.500000 + 2.59808i 0.188982 + 0.981981i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.00000 + 1.73205i −0.301511 + 0.522233i −0.976478 0.215615i \(-0.930824\pi\)
0.674967 + 0.737848i \(0.264158\pi\)
\(12\) 0 0
\(13\) −6.00000 −1.66410 −0.832050 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.00000 1.73205i 0.242536 0.420084i −0.718900 0.695113i \(-0.755354\pi\)
0.961436 + 0.275029i \(0.0886875\pi\)
\(18\) 0 0
\(19\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.50000 7.79423i −0.938315 1.62521i −0.768613 0.639713i \(-0.779053\pi\)
−0.169701 0.985496i \(-0.554280\pi\)
\(24\) 0 0
\(25\) −0.500000 + 0.866025i −0.100000 + 0.173205i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −3.00000 −0.557086 −0.278543 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 0 0
\(31\) −1.00000 + 1.73205i −0.179605 + 0.311086i −0.941745 0.336327i \(-0.890815\pi\)
0.762140 + 0.647412i \(0.224149\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.00000 1.73205i 0.338062 0.292770i
\(36\) 0 0
\(37\) −4.00000 6.92820i −0.657596 1.13899i −0.981236 0.192809i \(-0.938240\pi\)
0.323640 0.946180i \(-0.395093\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −5.00000 −0.780869 −0.390434 0.920631i \(-0.627675\pi\)
−0.390434 + 0.920631i \(0.627675\pi\)
\(42\) 0 0
\(43\) 1.00000 0.152499 0.0762493 0.997089i \(-0.475706\pi\)
0.0762493 + 0.997089i \(0.475706\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.00000 + 6.92820i 0.583460 + 1.01058i 0.995066 + 0.0992202i \(0.0316348\pi\)
−0.411606 + 0.911362i \(0.635032\pi\)
\(48\) 0 0
\(49\) −6.50000 + 2.59808i −0.928571 + 0.371154i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.00000 3.46410i 0.274721 0.475831i −0.695344 0.718677i \(-0.744748\pi\)
0.970065 + 0.242846i \(0.0780811\pi\)
\(54\) 0 0
\(55\) 2.00000 0.269680
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.00000 + 6.92820i −0.520756 + 0.901975i 0.478953 + 0.877841i \(0.341016\pi\)
−0.999709 + 0.0241347i \(0.992317\pi\)
\(60\) 0 0
\(61\) −3.50000 6.06218i −0.448129 0.776182i 0.550135 0.835076i \(-0.314576\pi\)
−0.998264 + 0.0588933i \(0.981243\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.00000 + 5.19615i 0.372104 + 0.644503i
\(66\) 0 0
\(67\) 1.50000 2.59808i 0.183254 0.317406i −0.759733 0.650236i \(-0.774670\pi\)
0.942987 + 0.332830i \(0.108004\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) −7.00000 + 12.1244i −0.819288 + 1.41905i 0.0869195 + 0.996215i \(0.472298\pi\)
−0.906208 + 0.422833i \(0.861036\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −5.00000 1.73205i −0.569803 0.197386i
\(78\) 0 0
\(79\) −2.00000 3.46410i −0.225018 0.389742i 0.731307 0.682048i \(-0.238911\pi\)
−0.956325 + 0.292306i \(0.905577\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1.00000 0.109764 0.0548821 0.998493i \(-0.482522\pi\)
0.0548821 + 0.998493i \(0.482522\pi\)
\(84\) 0 0
\(85\) −2.00000 −0.216930
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.50000 + 11.2583i 0.688999 + 1.19338i 0.972162 + 0.234309i \(0.0752827\pi\)
−0.283164 + 0.959072i \(0.591384\pi\)
\(90\) 0 0
\(91\) −3.00000 15.5885i −0.314485 1.63411i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1.50000 + 2.59808i −0.149256 + 0.258518i −0.930953 0.365140i \(-0.881021\pi\)
0.781697 + 0.623658i \(0.214354\pi\)
\(102\) 0 0
\(103\) −6.50000 11.2583i −0.640464 1.10932i −0.985329 0.170664i \(-0.945409\pi\)
0.344865 0.938652i \(-0.387925\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −7.50000 12.9904i −0.725052 1.25583i −0.958952 0.283567i \(-0.908482\pi\)
0.233900 0.972261i \(-0.424851\pi\)
\(108\) 0 0
\(109\) −4.50000 + 7.79423i −0.431022 + 0.746552i −0.996962 0.0778949i \(-0.975180\pi\)
0.565940 + 0.824447i \(0.308513\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 4.00000 0.376288 0.188144 0.982141i \(-0.439753\pi\)
0.188144 + 0.982141i \(0.439753\pi\)
\(114\) 0 0
\(115\) −4.50000 + 7.79423i −0.419627 + 0.726816i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 5.00000 + 1.73205i 0.458349 + 0.158777i
\(120\) 0 0
\(121\) 3.50000 + 6.06218i 0.318182 + 0.551107i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 16.0000 1.41977 0.709885 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −2.00000 3.46410i −0.174741 0.302660i 0.765331 0.643637i \(-0.222575\pi\)
−0.940072 + 0.340977i \(0.889242\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −6.00000 + 10.3923i −0.512615 + 0.887875i 0.487278 + 0.873247i \(0.337990\pi\)
−0.999893 + 0.0146279i \(0.995344\pi\)
\(138\) 0 0
\(139\) 10.0000 0.848189 0.424094 0.905618i \(-0.360592\pi\)
0.424094 + 0.905618i \(0.360592\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 6.00000 10.3923i 0.501745 0.869048i
\(144\) 0 0
\(145\) 1.50000 + 2.59808i 0.124568 + 0.215758i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 4.50000 + 7.79423i 0.368654 + 0.638528i 0.989355 0.145519i \(-0.0464853\pi\)
−0.620701 + 0.784047i \(0.713152\pi\)
\(150\) 0 0
\(151\) 5.00000 8.66025i 0.406894 0.704761i −0.587646 0.809118i \(-0.699945\pi\)
0.994540 + 0.104357i \(0.0332784\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 2.00000 0.160644
\(156\) 0 0
\(157\) 1.00000 1.73205i 0.0798087 0.138233i −0.823359 0.567521i \(-0.807902\pi\)
0.903167 + 0.429289i \(0.141236\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 18.0000 15.5885i 1.41860 1.22854i
\(162\) 0 0
\(163\) −4.00000 6.92820i −0.313304 0.542659i 0.665771 0.746156i \(-0.268103\pi\)
−0.979076 + 0.203497i \(0.934769\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 9.00000 0.696441 0.348220 0.937413i \(-0.386786\pi\)
0.348220 + 0.937413i \(0.386786\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −8.00000 13.8564i −0.608229 1.05348i −0.991532 0.129861i \(-0.958547\pi\)
0.383304 0.923622i \(-0.374786\pi\)
\(174\) 0 0
\(175\) −2.50000 0.866025i −0.188982 0.0654654i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −3.00000 + 5.19615i −0.224231 + 0.388379i −0.956088 0.293079i \(-0.905320\pi\)
0.731858 + 0.681457i \(0.238654\pi\)
\(180\) 0 0
\(181\) 1.00000 0.0743294 0.0371647 0.999309i \(-0.488167\pi\)
0.0371647 + 0.999309i \(0.488167\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −4.00000 + 6.92820i −0.294086 + 0.509372i
\(186\) 0 0
\(187\) 2.00000 + 3.46410i 0.146254 + 0.253320i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 3.00000 + 5.19615i 0.217072 + 0.375980i 0.953912 0.300088i \(-0.0970159\pi\)
−0.736839 + 0.676068i \(0.763683\pi\)
\(192\) 0 0
\(193\) −1.00000 + 1.73205i −0.0719816 + 0.124676i −0.899770 0.436365i \(-0.856266\pi\)
0.827788 + 0.561041i \(0.189599\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −14.0000 −0.997459 −0.498729 0.866758i \(-0.666200\pi\)
−0.498729 + 0.866758i \(0.666200\pi\)
\(198\) 0 0
\(199\) −10.0000 + 17.3205i −0.708881 + 1.22782i 0.256391 + 0.966573i \(0.417466\pi\)
−0.965272 + 0.261245i \(0.915867\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −1.50000 7.79423i −0.105279 0.547048i
\(204\) 0 0
\(205\) 2.50000 + 4.33013i 0.174608 + 0.302429i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −0.500000 0.866025i −0.0340997 0.0590624i
\(216\) 0 0
\(217\) −5.00000 1.73205i −0.339422 0.117579i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −6.00000 + 10.3923i −0.403604 + 0.699062i
\(222\) 0 0
\(223\) 16.0000 1.07144 0.535720 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −2.00000 + 3.46410i −0.132745 + 0.229920i −0.924734 0.380615i \(-0.875712\pi\)
0.791989 + 0.610535i \(0.209046\pi\)
\(228\) 0 0
\(229\) 7.00000 + 12.1244i 0.462573 + 0.801200i 0.999088 0.0426906i \(-0.0135930\pi\)
−0.536515 + 0.843891i \(0.680260\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −9.00000 15.5885i −0.589610 1.02123i −0.994283 0.106773i \(-0.965948\pi\)
0.404674 0.914461i \(-0.367385\pi\)
\(234\) 0 0
\(235\) 4.00000 6.92820i 0.260931 0.451946i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 26.0000 1.68180 0.840900 0.541190i \(-0.182026\pi\)
0.840900 + 0.541190i \(0.182026\pi\)
\(240\) 0 0
\(241\) −5.00000 + 8.66025i −0.322078 + 0.557856i −0.980917 0.194429i \(-0.937715\pi\)
0.658838 + 0.752285i \(0.271048\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 5.50000 + 4.33013i 0.351382 + 0.276642i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −30.0000 −1.89358 −0.946792 0.321847i \(-0.895696\pi\)
−0.946792 + 0.321847i \(0.895696\pi\)
\(252\) 0 0
\(253\) 18.0000 1.13165
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 4.00000 + 6.92820i 0.249513 + 0.432169i 0.963391 0.268101i \(-0.0863961\pi\)
−0.713878 + 0.700270i \(0.753063\pi\)
\(258\) 0 0
\(259\) 16.0000 13.8564i 0.994192 0.860995i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 8.50000 14.7224i 0.524132 0.907824i −0.475473 0.879730i \(-0.657723\pi\)
0.999605 0.0280936i \(-0.00894366\pi\)
\(264\) 0 0
\(265\) −4.00000 −0.245718
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 4.50000 7.79423i 0.274370 0.475223i −0.695606 0.718423i \(-0.744864\pi\)
0.969976 + 0.243201i \(0.0781974\pi\)
\(270\) 0 0
\(271\) 12.0000 + 20.7846i 0.728948 + 1.26258i 0.957328 + 0.289003i \(0.0933238\pi\)
−0.228380 + 0.973572i \(0.573343\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.00000 1.73205i −0.0603023 0.104447i
\(276\) 0 0
\(277\) −9.00000 + 15.5885i −0.540758 + 0.936620i 0.458103 + 0.888899i \(0.348529\pi\)
−0.998861 + 0.0477206i \(0.984804\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 22.0000 1.31241 0.656205 0.754583i \(-0.272161\pi\)
0.656205 + 0.754583i \(0.272161\pi\)
\(282\) 0 0
\(283\) 2.00000 3.46410i 0.118888 0.205919i −0.800439 0.599414i \(-0.795400\pi\)
0.919327 + 0.393494i \(0.128734\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −2.50000 12.9904i −0.147570 0.766798i
\(288\) 0 0
\(289\) 6.50000 + 11.2583i 0.382353 + 0.662255i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −4.00000 −0.233682 −0.116841 0.993151i \(-0.537277\pi\)
−0.116841 + 0.993151i \(0.537277\pi\)
\(294\) 0 0
\(295\) 8.00000 0.465778
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 27.0000 + 46.7654i 1.56145 + 2.70451i
\(300\) 0 0
\(301\) 0.500000 + 2.59808i 0.0288195 + 0.149751i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −3.50000 + 6.06218i −0.200409 + 0.347119i
\(306\) 0 0
\(307\) 1.00000 0.0570730 0.0285365 0.999593i \(-0.490915\pi\)
0.0285365 + 0.999593i \(0.490915\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 12.0000 20.7846i 0.680458 1.17859i −0.294384 0.955687i \(-0.595114\pi\)
0.974841 0.222900i \(-0.0715523\pi\)
\(312\) 0 0
\(313\) 2.00000 + 3.46410i 0.113047 + 0.195803i 0.916997 0.398894i \(-0.130606\pi\)
−0.803951 + 0.594696i \(0.797272\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 5.00000 + 8.66025i 0.280828 + 0.486408i 0.971589 0.236675i \(-0.0760576\pi\)
−0.690761 + 0.723083i \(0.742724\pi\)
\(318\) 0 0
\(319\) 3.00000 5.19615i 0.167968 0.290929i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 3.00000 5.19615i 0.166410 0.288231i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −16.0000 + 13.8564i −0.882109 + 0.763928i
\(330\) 0 0
\(331\) −5.00000 8.66025i −0.274825 0.476011i 0.695266 0.718752i \(-0.255287\pi\)
−0.970091 + 0.242742i \(0.921953\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −3.00000 −0.163908
\(336\) 0 0
\(337\) 28.0000 1.52526 0.762629 0.646837i \(-0.223908\pi\)
0.762629 + 0.646837i \(0.223908\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −2.00000 3.46410i −0.108306 0.187592i
\(342\) 0 0
\(343\) −10.0000 15.5885i −0.539949 0.841698i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 12.5000 21.6506i 0.671035 1.16227i −0.306576 0.951846i \(-0.599183\pi\)
0.977611 0.210421i \(-0.0674834\pi\)
\(348\) 0 0
\(349\) −17.0000 −0.909989 −0.454995 0.890494i \(-0.650359\pi\)
−0.454995 + 0.890494i \(0.650359\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −18.0000 + 31.1769i −0.958043 + 1.65938i −0.230799 + 0.973002i \(0.574134\pi\)
−0.727245 + 0.686378i \(0.759200\pi\)
\(354\) 0 0
\(355\) 4.00000 + 6.92820i 0.212298 + 0.367711i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −5.00000 8.66025i −0.263890 0.457071i 0.703382 0.710812i \(-0.251672\pi\)
−0.967272 + 0.253741i \(0.918339\pi\)
\(360\) 0 0
\(361\) 9.50000 16.4545i 0.500000 0.866025i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 14.0000 0.732793
\(366\) 0 0
\(367\) −0.500000 + 0.866025i −0.0260998 + 0.0452062i −0.878780 0.477227i \(-0.841642\pi\)
0.852680 + 0.522433i \(0.174975\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 10.0000 + 3.46410i 0.519174 + 0.179847i
\(372\) 0 0
\(373\) −16.0000 27.7128i −0.828449 1.43492i −0.899255 0.437425i \(-0.855891\pi\)
0.0708063 0.997490i \(-0.477443\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 18.0000 0.927047
\(378\) 0 0
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −4.50000 7.79423i −0.229939 0.398266i 0.727851 0.685736i \(-0.240519\pi\)
−0.957790 + 0.287469i \(0.907186\pi\)
\(384\) 0 0
\(385\) 1.00000 + 5.19615i 0.0509647 + 0.264820i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −5.00000 + 8.66025i −0.253510 + 0.439092i −0.964490 0.264120i \(-0.914918\pi\)
0.710980 + 0.703213i \(0.248252\pi\)
\(390\) 0 0
\(391\) −18.0000 −0.910299
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −2.00000 + 3.46410i −0.100631 + 0.174298i
\(396\) 0 0
\(397\) −5.00000 8.66025i −0.250943 0.434646i 0.712843 0.701324i \(-0.247407\pi\)
−0.963786 + 0.266678i \(0.914074\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −14.5000 25.1147i −0.724095 1.25417i −0.959345 0.282235i \(-0.908924\pi\)
0.235250 0.971935i \(-0.424409\pi\)
\(402\) 0 0
\(403\) 6.00000 10.3923i 0.298881 0.517678i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 16.0000 0.793091
\(408\) 0 0
\(409\) −1.50000 + 2.59808i −0.0741702 + 0.128467i −0.900725 0.434389i \(-0.856964\pi\)
0.826555 + 0.562856i \(0.190297\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −20.0000 6.92820i −0.984136 0.340915i
\(414\) 0 0
\(415\) −0.500000 0.866025i −0.0245440 0.0425115i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 24.0000 1.17248 0.586238 0.810139i \(-0.300608\pi\)
0.586238 + 0.810139i \(0.300608\pi\)
\(420\) 0 0
\(421\) −1.00000 −0.0487370 −0.0243685 0.999703i \(-0.507758\pi\)
−0.0243685 + 0.999703i \(0.507758\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 1.00000 + 1.73205i 0.0485071 + 0.0840168i
\(426\) 0 0
\(427\) 14.0000 12.1244i 0.677507 0.586739i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 3.00000 5.19615i 0.144505 0.250290i −0.784683 0.619897i \(-0.787174\pi\)
0.929188 + 0.369607i \(0.120508\pi\)
\(432\) 0 0
\(433\) −4.00000 −0.192228 −0.0961139 0.995370i \(-0.530641\pi\)
−0.0961139 + 0.995370i \(0.530641\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −2.00000 3.46410i −0.0954548 0.165333i 0.814344 0.580383i \(-0.197097\pi\)
−0.909798 + 0.415051i \(0.863764\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 18.5000 + 32.0429i 0.878962 + 1.52241i 0.852482 + 0.522757i \(0.175096\pi\)
0.0264796 + 0.999649i \(0.491570\pi\)
\(444\) 0 0
\(445\) 6.50000 11.2583i 0.308130 0.533696i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −15.0000 −0.707894 −0.353947 0.935266i \(-0.615161\pi\)
−0.353947 + 0.935266i \(0.615161\pi\)
\(450\) 0 0
\(451\) 5.00000 8.66025i 0.235441 0.407795i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −12.0000 + 10.3923i −0.562569 + 0.487199i
\(456\) 0 0
\(457\) −14.0000 24.2487i −0.654892 1.13431i −0.981921 0.189292i \(-0.939381\pi\)
0.327028 0.945015i \(-0.393953\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 2.00000 0.0931493 0.0465746 0.998915i \(-0.485169\pi\)
0.0465746 + 0.998915i \(0.485169\pi\)
\(462\) 0 0
\(463\) 17.0000 0.790057 0.395029 0.918669i \(-0.370735\pi\)
0.395029 + 0.918669i \(0.370735\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 2.50000 + 4.33013i 0.115686 + 0.200374i 0.918054 0.396456i \(-0.129760\pi\)
−0.802368 + 0.596830i \(0.796427\pi\)
\(468\) 0 0
\(469\) 7.50000 + 2.59808i 0.346318 + 0.119968i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −1.00000 + 1.73205i −0.0459800 + 0.0796398i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −15.0000 + 25.9808i −0.685367 + 1.18709i 0.287954 + 0.957644i \(0.407025\pi\)
−0.973321 + 0.229447i \(0.926308\pi\)
\(480\) 0 0
\(481\) 24.0000 + 41.5692i 1.09431 + 1.89539i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 5.00000 + 8.66025i 0.227038 + 0.393242i
\(486\) 0 0
\(487\) −16.0000 + 27.7128i −0.725029 + 1.25579i 0.233933 + 0.972253i \(0.424840\pi\)
−0.958962 + 0.283535i \(0.908493\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −6.00000 −0.270776 −0.135388 0.990793i \(-0.543228\pi\)
−0.135388 + 0.990793i \(0.543228\pi\)
\(492\) 0 0
\(493\) −3.00000 + 5.19615i −0.135113 + 0.234023i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −4.00000 20.7846i −0.179425 0.932317i
\(498\) 0 0
\(499\) 3.00000 + 5.19615i 0.134298 + 0.232612i 0.925329 0.379165i \(-0.123789\pi\)
−0.791031 + 0.611776i \(0.790455\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −27.0000 −1.20387 −0.601935 0.798545i \(-0.705603\pi\)
−0.601935 + 0.798545i \(0.705603\pi\)
\(504\) 0 0
\(505\) 3.00000 0.133498
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −20.5000 35.5070i −0.908647 1.57382i −0.815946 0.578128i \(-0.803783\pi\)
−0.0927004 0.995694i \(-0.529550\pi\)
\(510\) 0 0
\(511\) −35.0000 12.1244i −1.54831 0.536350i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −6.50000 + 11.2583i −0.286424 + 0.496101i
\(516\) 0 0
\(517\) −16.0000 −0.703679
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 7.00000 12.1244i 0.306676 0.531178i −0.670957 0.741496i \(-0.734117\pi\)
0.977633 + 0.210318i \(0.0674500\pi\)
\(522\) 0 0
\(523\) −2.00000 3.46410i −0.0874539 0.151475i 0.818980 0.573822i \(-0.194540\pi\)
−0.906434 + 0.422347i \(0.861206\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 2.00000 + 3.46410i 0.0871214 + 0.150899i
\(528\) 0 0
\(529\) −29.0000 + 50.2295i −1.26087 + 2.18389i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 30.0000 1.29944
\(534\) 0 0
\(535\) −7.50000 + 12.9904i −0.324253 + 0.561623i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 2.00000 13.8564i 0.0861461 0.596838i
\(540\) 0 0
\(541\) −16.5000 28.5788i −0.709390 1.22870i −0.965084 0.261942i \(-0.915637\pi\)
0.255693 0.966758i \(-0.417696\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 9.00000 0.385518
\(546\) 0 0
\(547\) −15.0000 −0.641354 −0.320677 0.947189i \(-0.603910\pi\)
−0.320677 + 0.947189i \(0.603910\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 8.00000 6.92820i 0.340195 0.294617i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −1.00000 + 1.73205i −0.0423714 + 0.0733893i −0.886433 0.462856i \(-0.846825\pi\)
0.844062 + 0.536246i \(0.180158\pi\)
\(558\) 0 0
\(559\) −6.00000 −0.253773
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −6.50000 + 11.2583i −0.273942 + 0.474482i −0.969868 0.243632i \(-0.921661\pi\)
0.695925 + 0.718114i \(0.254994\pi\)
\(564\) 0 0
\(565\) −2.00000 3.46410i −0.0841406 0.145736i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 15.0000 + 25.9808i 0.628833 + 1.08917i 0.987786 + 0.155815i \(0.0498003\pi\)
−0.358954 + 0.933355i \(0.616866\pi\)
\(570\) 0 0
\(571\) −12.0000 + 20.7846i −0.502184 + 0.869809i 0.497812 + 0.867285i \(0.334137\pi\)
−0.999997 + 0.00252413i \(0.999197\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 9.00000 0.375326
\(576\) 0 0
\(577\) −11.0000 + 19.0526i −0.457936 + 0.793168i −0.998852 0.0479084i \(-0.984744\pi\)
0.540916 + 0.841077i \(0.318078\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0.500000 + 2.59808i 0.0207435 + 0.107786i
\(582\) 0 0
\(583\) 4.00000 + 6.92820i 0.165663 + 0.286937i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −20.0000 −0.825488 −0.412744 0.910847i \(-0.635430\pi\)
−0.412744 + 0.910847i \(0.635430\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −18.0000 31.1769i −0.739171 1.28028i −0.952869 0.303383i \(-0.901884\pi\)
0.213697 0.976900i \(-0.431449\pi\)
\(594\) 0 0
\(595\) −1.00000 5.19615i −0.0409960 0.213021i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(600\) 0 0
\(601\) −6.00000 −0.244745 −0.122373 0.992484i \(-0.539050\pi\)
−0.122373 + 0.992484i \(0.539050\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 3.50000 6.06218i 0.142295 0.246463i
\(606\) 0 0
\(607\) −0.500000 0.866025i −0.0202944 0.0351509i 0.855700 0.517472i \(-0.173127\pi\)
−0.875994 + 0.482322i \(0.839794\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −24.0000 41.5692i −0.970936 1.68171i
\(612\) 0 0
\(613\) 3.00000 5.19615i 0.121169 0.209871i −0.799060 0.601251i \(-0.794669\pi\)
0.920229 + 0.391381i \(0.128002\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −20.0000 −0.805170 −0.402585 0.915383i \(-0.631888\pi\)
−0.402585 + 0.915383i \(0.631888\pi\)
\(618\) 0 0
\(619\) 5.00000 8.66025i 0.200967 0.348085i −0.747873 0.663842i \(-0.768925\pi\)
0.948840 + 0.315757i \(0.102258\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −26.0000 + 22.5167i −1.04167 + 0.902111i
\(624\) 0 0
\(625\) −0.500000 0.866025i −0.0200000 0.0346410i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −16.0000 −0.637962
\(630\) 0 0
\(631\) −34.0000 −1.35352 −0.676759 0.736204i \(-0.736616\pi\)
−0.676759 + 0.736204i \(0.736616\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −8.00000 13.8564i −0.317470 0.549875i
\(636\) 0 0
\(637\) 39.0000 15.5885i 1.54524 0.617637i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −15.5000 + 26.8468i −0.612213 + 1.06038i 0.378653 + 0.925539i \(0.376387\pi\)
−0.990867 + 0.134846i \(0.956946\pi\)
\(642\) 0 0
\(643\) −20.0000 −0.788723 −0.394362 0.918955i \(-0.629034\pi\)
−0.394362 + 0.918955i \(0.629034\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0.500000 0.866025i 0.0196570 0.0340470i −0.856030 0.516927i \(-0.827076\pi\)
0.875687 + 0.482880i \(0.160409\pi\)
\(648\) 0 0
\(649\) −8.00000 13.8564i −0.314027 0.543912i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −17.0000 29.4449i −0.665261 1.15227i −0.979214 0.202828i \(-0.934987\pi\)
0.313953 0.949439i \(-0.398347\pi\)
\(654\) 0 0
\(655\) −2.00000 + 3.46410i −0.0781465 + 0.135354i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −4.00000 −0.155818 −0.0779089 0.996960i \(-0.524824\pi\)
−0.0779089 + 0.996960i \(0.524824\pi\)
\(660\) 0 0
\(661\) −15.5000 + 26.8468i −0.602880 + 1.04422i 0.389503 + 0.921025i \(0.372647\pi\)
−0.992383 + 0.123194i \(0.960686\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 13.5000 + 23.3827i 0.522722 + 0.905381i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 14.0000 0.540464
\(672\) 0 0
\(673\) 16.0000 0.616755 0.308377 0.951264i \(-0.400214\pi\)
0.308377 + 0.951264i \(0.400214\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 3.00000 + 5.19615i 0.115299 + 0.199704i 0.917899 0.396813i \(-0.129884\pi\)
−0.802600 + 0.596518i \(0.796551\pi\)
\(678\) 0 0
\(679\) −5.00000 25.9808i −0.191882 0.997050i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 14.5000 25.1147i 0.554827 0.960989i −0.443090 0.896477i \(-0.646118\pi\)
0.997917 0.0645115i \(-0.0205489\pi\)
\(684\) 0 0
\(685\) 12.0000 0.458496
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −12.0000 + 20.7846i −0.457164 + 0.791831i
\(690\) 0 0
\(691\) 13.0000 + 22.5167i 0.494543 + 0.856574i 0.999980 0.00628943i \(-0.00200200\pi\)
−0.505437 + 0.862864i \(0.668669\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −5.00000 8.66025i −0.189661 0.328502i
\(696\) 0 0
\(697\) −5.00000 + 8.66025i −0.189389 + 0.328031i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 29.0000 1.09531 0.547657 0.836703i \(-0.315520\pi\)
0.547657 + 0.836703i \(0.315520\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −7.50000 2.59808i −0.282067 0.0977107i
\(708\) 0 0
\(709\) −15.5000 26.8468i −0.582115 1.00825i −0.995228 0.0975728i \(-0.968892\pi\)
0.413114 0.910679i \(-0.364441\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 18.0000 0.674105
\(714\) 0 0
\(715\) −12.0000 −0.448775
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −3.00000 5.19615i −0.111881 0.193784i 0.804648 0.593753i \(-0.202354\pi\)
−0.916529 + 0.399969i \(0.869021\pi\)
\(720\) 0 0
\(721\) 26.0000 22.5167i 0.968291 0.838564i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 1.50000 2.59808i 0.0557086 0.0964901i
\(726\) 0 0
\(727\) 3.00000 0.111264 0.0556319 0.998451i \(-0.482283\pi\)
0.0556319 + 0.998451i \(0.482283\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1.00000 1.73205i 0.0369863 0.0640622i
\(732\) 0 0
\(733\) −17.0000 29.4449i −0.627909 1.08757i −0.987971 0.154642i \(-0.950578\pi\)
0.360061 0.932929i \(-0.382756\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 3.00000 + 5.19615i 0.110506 + 0.191403i
\(738\) 0 0
\(739\) −5.00000 + 8.66025i −0.183928 + 0.318573i −0.943215 0.332184i \(-0.892215\pi\)
0.759287 + 0.650756i \(0.225548\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 3.00000 0.110059 0.0550297 0.998485i \(-0.482475\pi\)
0.0550297 + 0.998485i \(0.482475\pi\)
\(744\) 0 0
\(745\) 4.50000 7.79423i 0.164867 0.285558i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 30.0000 25.9808i 1.09618 0.949316i
\(750\) 0 0
\(751\) −10.0000 17.3205i −0.364905 0.632034i 0.623856 0.781540i \(-0.285565\pi\)
−0.988761 + 0.149505i \(0.952232\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −10.0000 −0.363937
\(756\) 0 0
\(757\) −38.0000 −1.38113 −0.690567 0.723269i \(-0.742639\pi\)
−0.690567 + 0.723269i \(0.742639\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 15.0000 + 25.9808i 0.543750 + 0.941802i 0.998684 + 0.0512772i \(0.0163292\pi\)
−0.454935 + 0.890525i \(0.650337\pi\)
\(762\) 0 0
\(763\) −22.5000 7.79423i −0.814555 0.282170i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 24.0000 41.5692i 0.866590 1.50098i
\(768\) 0 0
\(769\) −2.00000 −0.0721218 −0.0360609 0.999350i \(-0.511481\pi\)
−0.0360609 + 0.999350i \(0.511481\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −9.00000 + 15.5885i −0.323708 + 0.560678i −0.981250 0.192740i \(-0.938263\pi\)
0.657542 + 0.753418i \(0.271596\pi\)
\(774\) 0 0
\(775\) −1.00000 1.73205i −0.0359211 0.0622171i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 8.00000 13.8564i 0.286263 0.495821i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −2.00000 −0.0713831
\(786\) 0 0
\(787\) 5.50000 9.52628i 0.196054 0.339575i −0.751192 0.660084i \(-0.770521\pi\)
0.947245 + 0.320509i \(0.103854\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 2.00000 + 10.3923i 0.0711118 + 0.369508i
\(792\) 0 0
\(793\) 21.0000 + 36.3731i 0.745732 + 1.29165i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −36.0000 −1.27519 −0.637593 0.770374i \(-0.720070\pi\)
−0.637593 + 0.770374i \(0.720070\pi\)
\(798\) 0 0
\(799\) 16.0000 0.566039
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −14.0000 24.2487i −0.494049