Properties

Label 1232.2.e.c.769.1
Level $1232$
Weight $2$
Character 1232.769
Analytic conductor $9.838$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1232.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.83756952902\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-5})\)
Defining polynomial: \( x^{4} - 4x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 77)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 769.1
Root \(-1.58114 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1232.769
Dual form 1232.2.e.c.769.3

$q$-expansion

\(f(q)\) \(=\) \(q-2.23607i q^{3} -2.23607i q^{5} +(-1.58114 - 2.12132i) q^{7} -2.00000 q^{9} +O(q^{10})\) \(q-2.23607i q^{3} -2.23607i q^{5} +(-1.58114 - 2.12132i) q^{7} -2.00000 q^{9} +(3.00000 - 1.41421i) q^{11} -6.32456 q^{13} -5.00000 q^{15} -3.16228 q^{19} +(-4.74342 + 3.53553i) q^{21} +3.00000 q^{23} -2.23607i q^{27} -1.41421i q^{29} +6.70820i q^{31} +(-3.16228 - 6.70820i) q^{33} +(-4.74342 + 3.53553i) q^{35} -1.00000 q^{37} +14.1421i q^{39} +9.48683 q^{41} -4.24264i q^{43} +4.47214i q^{45} -4.47214i q^{47} +(-2.00000 + 6.70820i) q^{49} +(-3.16228 - 6.70820i) q^{55} +7.07107i q^{57} +2.23607i q^{59} +3.16228 q^{61} +(3.16228 + 4.24264i) q^{63} +14.1421i q^{65} -11.0000 q^{67} -6.70820i q^{69} -9.00000 q^{71} +3.16228 q^{73} +(-7.74342 - 4.12789i) q^{77} +8.48528i q^{79} -11.0000 q^{81} +9.48683 q^{83} -3.16228 q^{87} -2.23607i q^{89} +(10.0000 + 13.4164i) q^{91} +15.0000 q^{93} +7.07107i q^{95} +6.70820i q^{97} +(-6.00000 + 2.82843i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{9} + 12 q^{11} - 20 q^{15} + 12 q^{23} - 4 q^{37} - 8 q^{49} - 44 q^{67} - 36 q^{71} - 12 q^{77} - 44 q^{81} + 40 q^{91} + 60 q^{93} - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1232\mathbb{Z}\right)^\times\).

\(n\) \(309\) \(353\) \(463\) \(673\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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
\(3\) 2.23607i 1.29099i −0.763763 0.645497i \(-0.776650\pi\)
0.763763 0.645497i \(-0.223350\pi\)
\(4\) 0 0
\(5\) 2.23607i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(6\) 0 0
\(7\) −1.58114 2.12132i −0.597614 0.801784i
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) 3.00000 1.41421i 0.904534 0.426401i
\(12\) 0 0
\(13\) −6.32456 −1.75412 −0.877058 0.480384i \(-0.840497\pi\)
−0.877058 + 0.480384i \(0.840497\pi\)
\(14\) 0 0
\(15\) −5.00000 −1.29099
\(16\) 0 0
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −3.16228 −0.725476 −0.362738 0.931891i \(-0.618158\pi\)
−0.362738 + 0.931891i \(0.618158\pi\)
\(20\) 0 0
\(21\) −4.74342 + 3.53553i −1.03510 + 0.771517i
\(22\) 0 0
\(23\) 3.00000 0.625543 0.312772 0.949828i \(-0.398743\pi\)
0.312772 + 0.949828i \(0.398743\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 2.23607i 0.430331i
\(28\) 0 0
\(29\) 1.41421i 0.262613i −0.991342 0.131306i \(-0.958083\pi\)
0.991342 0.131306i \(-0.0419172\pi\)
\(30\) 0 0
\(31\) 6.70820i 1.20483i 0.798183 + 0.602414i \(0.205795\pi\)
−0.798183 + 0.602414i \(0.794205\pi\)
\(32\) 0 0
\(33\) −3.16228 6.70820i −0.550482 1.16775i
\(34\) 0 0
\(35\) −4.74342 + 3.53553i −0.801784 + 0.597614i
\(36\) 0 0
\(37\) −1.00000 −0.164399 −0.0821995 0.996616i \(-0.526194\pi\)
−0.0821995 + 0.996616i \(0.526194\pi\)
\(38\) 0 0
\(39\) 14.1421i 2.26455i
\(40\) 0 0
\(41\) 9.48683 1.48159 0.740797 0.671729i \(-0.234448\pi\)
0.740797 + 0.671729i \(0.234448\pi\)
\(42\) 0 0
\(43\) 4.24264i 0.646997i −0.946229 0.323498i \(-0.895141\pi\)
0.946229 0.323498i \(-0.104859\pi\)
\(44\) 0 0
\(45\) 4.47214i 0.666667i
\(46\) 0 0
\(47\) 4.47214i 0.652328i −0.945313 0.326164i \(-0.894244\pi\)
0.945313 0.326164i \(-0.105756\pi\)
\(48\) 0 0
\(49\) −2.00000 + 6.70820i −0.285714 + 0.958315i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) −3.16228 6.70820i −0.426401 0.904534i
\(56\) 0 0
\(57\) 7.07107i 0.936586i
\(58\) 0 0
\(59\) 2.23607i 0.291111i 0.989350 + 0.145556i \(0.0464970\pi\)
−0.989350 + 0.145556i \(0.953503\pi\)
\(60\) 0 0
\(61\) 3.16228 0.404888 0.202444 0.979294i \(-0.435112\pi\)
0.202444 + 0.979294i \(0.435112\pi\)
\(62\) 0 0
\(63\) 3.16228 + 4.24264i 0.398410 + 0.534522i
\(64\) 0 0
\(65\) 14.1421i 1.75412i
\(66\) 0 0
\(67\) −11.0000 −1.34386 −0.671932 0.740613i \(-0.734535\pi\)
−0.671932 + 0.740613i \(0.734535\pi\)
\(68\) 0 0
\(69\) 6.70820i 0.807573i
\(70\) 0 0
\(71\) −9.00000 −1.06810 −0.534052 0.845452i \(-0.679331\pi\)
−0.534052 + 0.845452i \(0.679331\pi\)
\(72\) 0 0
\(73\) 3.16228 0.370117 0.185058 0.982728i \(-0.440753\pi\)
0.185058 + 0.982728i \(0.440753\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −7.74342 4.12789i −0.882444 0.470417i
\(78\) 0 0
\(79\) 8.48528i 0.954669i 0.878722 + 0.477334i \(0.158397\pi\)
−0.878722 + 0.477334i \(0.841603\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) 9.48683 1.04132 0.520658 0.853766i \(-0.325687\pi\)
0.520658 + 0.853766i \(0.325687\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −3.16228 −0.339032
\(88\) 0 0
\(89\) 2.23607i 0.237023i −0.992953 0.118511i \(-0.962188\pi\)
0.992953 0.118511i \(-0.0378122\pi\)
\(90\) 0 0
\(91\) 10.0000 + 13.4164i 1.04828 + 1.40642i
\(92\) 0 0
\(93\) 15.0000 1.55543
\(94\) 0 0
\(95\) 7.07107i 0.725476i
\(96\) 0 0
\(97\) 6.70820i 0.681115i 0.940224 + 0.340557i \(0.110616\pi\)
−0.940224 + 0.340557i \(0.889384\pi\)
\(98\) 0 0
\(99\) −6.00000 + 2.82843i −0.603023 + 0.284268i
\(100\) 0 0
\(101\) −9.48683 −0.943975 −0.471988 0.881605i \(-0.656463\pi\)
−0.471988 + 0.881605i \(0.656463\pi\)
\(102\) 0 0
\(103\) 13.4164i 1.32196i −0.750404 0.660979i \(-0.770141\pi\)
0.750404 0.660979i \(-0.229859\pi\)
\(104\) 0 0
\(105\) 7.90569 + 10.6066i 0.771517 + 1.03510i
\(106\) 0 0
\(107\) 2.82843i 0.273434i −0.990610 0.136717i \(-0.956345\pi\)
0.990610 0.136717i \(-0.0436552\pi\)
\(108\) 0 0
\(109\) 12.7279i 1.21911i −0.792742 0.609557i \(-0.791347\pi\)
0.792742 0.609557i \(-0.208653\pi\)
\(110\) 0 0
\(111\) 2.23607i 0.212238i
\(112\) 0 0
\(113\) −3.00000 −0.282216 −0.141108 0.989994i \(-0.545067\pi\)
−0.141108 + 0.989994i \(0.545067\pi\)
\(114\) 0 0
\(115\) 6.70820i 0.625543i
\(116\) 0 0
\(117\) 12.6491 1.16941
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 7.00000 8.48528i 0.636364 0.771389i
\(122\) 0 0
\(123\) 21.2132i 1.91273i
\(124\) 0 0
\(125\) 11.1803i 1.00000i
\(126\) 0 0
\(127\) 12.7279i 1.12942i 0.825289 + 0.564710i \(0.191012\pi\)
−0.825289 + 0.564710i \(0.808988\pi\)
\(128\) 0 0
\(129\) −9.48683 −0.835269
\(130\) 0 0
\(131\) 18.9737 1.65774 0.828868 0.559444i \(-0.188985\pi\)
0.828868 + 0.559444i \(0.188985\pi\)
\(132\) 0 0
\(133\) 5.00000 + 6.70820i 0.433555 + 0.581675i
\(134\) 0 0
\(135\) −5.00000 −0.430331
\(136\) 0 0
\(137\) −15.0000 −1.28154 −0.640768 0.767734i \(-0.721384\pi\)
−0.640768 + 0.767734i \(0.721384\pi\)
\(138\) 0 0
\(139\) −12.6491 −1.07288 −0.536442 0.843937i \(-0.680232\pi\)
−0.536442 + 0.843937i \(0.680232\pi\)
\(140\) 0 0
\(141\) −10.0000 −0.842152
\(142\) 0 0
\(143\) −18.9737 + 8.94427i −1.58666 + 0.747958i
\(144\) 0 0
\(145\) −3.16228 −0.262613
\(146\) 0 0
\(147\) 15.0000 + 4.47214i 1.23718 + 0.368856i
\(148\) 0 0
\(149\) 19.7990i 1.62200i 0.585049 + 0.810998i \(0.301075\pi\)
−0.585049 + 0.810998i \(0.698925\pi\)
\(150\) 0 0
\(151\) 16.9706i 1.38104i −0.723311 0.690522i \(-0.757381\pi\)
0.723311 0.690522i \(-0.242619\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 15.0000 1.20483
\(156\) 0 0
\(157\) 6.70820i 0.535373i −0.963506 0.267686i \(-0.913741\pi\)
0.963506 0.267686i \(-0.0862591\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −4.74342 6.36396i −0.373834 0.501550i
\(162\) 0 0
\(163\) 10.0000 0.783260 0.391630 0.920123i \(-0.371911\pi\)
0.391630 + 0.920123i \(0.371911\pi\)
\(164\) 0 0
\(165\) −15.0000 + 7.07107i −1.16775 + 0.550482i
\(166\) 0 0
\(167\) −9.48683 −0.734113 −0.367057 0.930199i \(-0.619634\pi\)
−0.367057 + 0.930199i \(0.619634\pi\)
\(168\) 0 0
\(169\) 27.0000 2.07692
\(170\) 0 0
\(171\) 6.32456 0.483651
\(172\) 0 0
\(173\) −9.48683 −0.721271 −0.360635 0.932707i \(-0.617440\pi\)
−0.360635 + 0.932707i \(0.617440\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 5.00000 0.375823
\(178\) 0 0
\(179\) −9.00000 −0.672692 −0.336346 0.941739i \(-0.609191\pi\)
−0.336346 + 0.941739i \(0.609191\pi\)
\(180\) 0 0
\(181\) 20.1246i 1.49585i −0.663783 0.747925i \(-0.731050\pi\)
0.663783 0.747925i \(-0.268950\pi\)
\(182\) 0 0
\(183\) 7.07107i 0.522708i
\(184\) 0 0
\(185\) 2.23607i 0.164399i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −4.74342 + 3.53553i −0.345033 + 0.257172i
\(190\) 0 0
\(191\) −3.00000 −0.217072 −0.108536 0.994092i \(-0.534616\pi\)
−0.108536 + 0.994092i \(0.534616\pi\)
\(192\) 0 0
\(193\) 4.24264i 0.305392i −0.988273 0.152696i \(-0.951204\pi\)
0.988273 0.152696i \(-0.0487955\pi\)
\(194\) 0 0
\(195\) 31.6228 2.26455
\(196\) 0 0
\(197\) 5.65685i 0.403034i −0.979485 0.201517i \(-0.935413\pi\)
0.979485 0.201517i \(-0.0645872\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 24.5967i 1.73492i
\(202\) 0 0
\(203\) −3.00000 + 2.23607i −0.210559 + 0.156941i
\(204\) 0 0
\(205\) 21.2132i 1.48159i
\(206\) 0 0
\(207\) −6.00000 −0.417029
\(208\) 0 0
\(209\) −9.48683 + 4.47214i −0.656218 + 0.309344i
\(210\) 0 0
\(211\) 21.2132i 1.46038i −0.683246 0.730189i \(-0.739432\pi\)
0.683246 0.730189i \(-0.260568\pi\)
\(212\) 0 0
\(213\) 20.1246i 1.37892i
\(214\) 0 0
\(215\) −9.48683 −0.646997
\(216\) 0 0
\(217\) 14.2302 10.6066i 0.966012 0.720023i
\(218\) 0 0
\(219\) 7.07107i 0.477818i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 6.70820i 0.449215i −0.974449 0.224607i \(-0.927890\pi\)
0.974449 0.224607i \(-0.0721099\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 9.48683 0.629663 0.314832 0.949148i \(-0.398052\pi\)
0.314832 + 0.949148i \(0.398052\pi\)
\(228\) 0 0
\(229\) 6.70820i 0.443291i −0.975127 0.221645i \(-0.928857\pi\)
0.975127 0.221645i \(-0.0711427\pi\)
\(230\) 0 0
\(231\) −9.23025 + 17.3148i −0.607306 + 1.13923i
\(232\) 0 0
\(233\) 18.3848i 1.20443i −0.798335 0.602213i \(-0.794286\pi\)
0.798335 0.602213i \(-0.205714\pi\)
\(234\) 0 0
\(235\) −10.0000 −0.652328
\(236\) 0 0
\(237\) 18.9737 1.23247
\(238\) 0 0
\(239\) 7.07107i 0.457389i −0.973498 0.228695i \(-0.926554\pi\)
0.973498 0.228695i \(-0.0734457\pi\)
\(240\) 0 0
\(241\) 12.6491 0.814801 0.407400 0.913250i \(-0.366435\pi\)
0.407400 + 0.913250i \(0.366435\pi\)
\(242\) 0 0
\(243\) 17.8885i 1.14755i
\(244\) 0 0
\(245\) 15.0000 + 4.47214i 0.958315 + 0.285714i
\(246\) 0 0
\(247\) 20.0000 1.27257
\(248\) 0 0
\(249\) 21.2132i 1.34433i
\(250\) 0 0
\(251\) 11.1803i 0.705697i −0.935681 0.352848i \(-0.885213\pi\)
0.935681 0.352848i \(-0.114787\pi\)
\(252\) 0 0
\(253\) 9.00000 4.24264i 0.565825 0.266733i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 22.3607i 1.39482i −0.716672 0.697410i \(-0.754335\pi\)
0.716672 0.697410i \(-0.245665\pi\)
\(258\) 0 0
\(259\) 1.58114 + 2.12132i 0.0982472 + 0.131812i
\(260\) 0 0
\(261\) 2.82843i 0.175075i
\(262\) 0 0
\(263\) 11.3137i 0.697633i −0.937191 0.348817i \(-0.886584\pi\)
0.937191 0.348817i \(-0.113416\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −5.00000 −0.305995
\(268\) 0 0
\(269\) 17.8885i 1.09068i 0.838214 + 0.545342i \(0.183600\pi\)
−0.838214 + 0.545342i \(0.816400\pi\)
\(270\) 0 0
\(271\) 25.2982 1.53676 0.768379 0.639995i \(-0.221064\pi\)
0.768379 + 0.639995i \(0.221064\pi\)
\(272\) 0 0
\(273\) 30.0000 22.3607i 1.81568 1.35333i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 4.24264i 0.254916i 0.991844 + 0.127458i \(0.0406817\pi\)
−0.991844 + 0.127458i \(0.959318\pi\)
\(278\) 0 0
\(279\) 13.4164i 0.803219i
\(280\) 0 0
\(281\) 24.0416i 1.43420i 0.696969 + 0.717102i \(0.254532\pi\)
−0.696969 + 0.717102i \(0.745468\pi\)
\(282\) 0 0
\(283\) 6.32456 0.375956 0.187978 0.982173i \(-0.439807\pi\)
0.187978 + 0.982173i \(0.439807\pi\)
\(284\) 0 0
\(285\) 15.8114 0.936586
\(286\) 0 0
\(287\) −15.0000 20.1246i −0.885422 1.18792i
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 15.0000 0.879316
\(292\) 0 0
\(293\) 9.48683 0.554227 0.277113 0.960837i \(-0.410622\pi\)
0.277113 + 0.960837i \(0.410622\pi\)
\(294\) 0 0
\(295\) 5.00000 0.291111
\(296\) 0 0
\(297\) −3.16228 6.70820i −0.183494 0.389249i
\(298\) 0 0
\(299\) −18.9737 −1.09728
\(300\) 0 0
\(301\) −9.00000 + 6.70820i −0.518751 + 0.386654i
\(302\) 0 0
\(303\) 21.2132i 1.21867i
\(304\) 0 0
\(305\) 7.07107i 0.404888i
\(306\) 0 0
\(307\) 6.32456 0.360961 0.180481 0.983579i \(-0.442235\pi\)
0.180481 + 0.983579i \(0.442235\pi\)
\(308\) 0 0
\(309\) −30.0000 −1.70664
\(310\) 0 0
\(311\) 4.47214i 0.253592i −0.991929 0.126796i \(-0.959531\pi\)
0.991929 0.126796i \(-0.0404693\pi\)
\(312\) 0 0
\(313\) 6.70820i 0.379170i 0.981864 + 0.189585i \(0.0607143\pi\)
−0.981864 + 0.189585i \(0.939286\pi\)
\(314\) 0 0
\(315\) 9.48683 7.07107i 0.534522 0.398410i
\(316\) 0 0
\(317\) 21.0000 1.17948 0.589739 0.807594i \(-0.299231\pi\)
0.589739 + 0.807594i \(0.299231\pi\)
\(318\) 0 0
\(319\) −2.00000 4.24264i −0.111979 0.237542i
\(320\) 0 0
\(321\) −6.32456 −0.353002
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −28.4605 −1.57387
\(328\) 0 0
\(329\) −9.48683 + 7.07107i −0.523026 + 0.389841i
\(330\) 0 0
\(331\) 19.0000 1.04433 0.522167 0.852843i \(-0.325124\pi\)
0.522167 + 0.852843i \(0.325124\pi\)
\(332\) 0 0
\(333\) 2.00000 0.109599
\(334\) 0 0
\(335\) 24.5967i 1.34386i
\(336\) 0 0
\(337\) 16.9706i 0.924445i −0.886764 0.462223i \(-0.847052\pi\)
0.886764 0.462223i \(-0.152948\pi\)
\(338\) 0 0
\(339\) 6.70820i 0.364340i
\(340\) 0 0
\(341\) 9.48683 + 20.1246i 0.513741 + 1.08981i
\(342\) 0 0
\(343\) 17.3925 6.36396i 0.939108 0.343622i
\(344\) 0 0
\(345\) −15.0000 −0.807573
\(346\) 0 0
\(347\) 1.41421i 0.0759190i 0.999279 + 0.0379595i \(0.0120858\pi\)
−0.999279 + 0.0379595i \(0.987914\pi\)
\(348\) 0 0
\(349\) 12.6491 0.677091 0.338546 0.940950i \(-0.390065\pi\)
0.338546 + 0.940950i \(0.390065\pi\)
\(350\) 0 0
\(351\) 14.1421i 0.754851i
\(352\) 0 0
\(353\) 15.6525i 0.833097i −0.909113 0.416549i \(-0.863240\pi\)
0.909113 0.416549i \(-0.136760\pi\)
\(354\) 0 0
\(355\) 20.1246i 1.06810i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 5.65685i 0.298557i 0.988795 + 0.149279i \(0.0476951\pi\)
−0.988795 + 0.149279i \(0.952305\pi\)
\(360\) 0 0
\(361\) −9.00000 −0.473684
\(362\) 0 0
\(363\) −18.9737 15.6525i −0.995859 0.821542i
\(364\) 0 0
\(365\) 7.07107i 0.370117i
\(366\) 0 0
\(367\) 33.5410i 1.75083i 0.483375 + 0.875413i \(0.339411\pi\)
−0.483375 + 0.875413i \(0.660589\pi\)
\(368\) 0 0
\(369\) −18.9737 −0.987730
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 4.24264i 0.219676i −0.993950 0.109838i \(-0.964967\pi\)
0.993950 0.109838i \(-0.0350331\pi\)
\(374\) 0 0
\(375\) −25.0000 −1.29099
\(376\) 0 0
\(377\) 8.94427i 0.460653i
\(378\) 0 0
\(379\) −17.0000 −0.873231 −0.436616 0.899648i \(-0.643823\pi\)
−0.436616 + 0.899648i \(0.643823\pi\)
\(380\) 0 0
\(381\) 28.4605 1.45808
\(382\) 0 0
\(383\) 24.5967i 1.25684i −0.777876 0.628418i \(-0.783703\pi\)
0.777876 0.628418i \(-0.216297\pi\)
\(384\) 0 0
\(385\) −9.23025 + 17.3148i −0.470417 + 0.882444i
\(386\) 0 0
\(387\) 8.48528i 0.431331i
\(388\) 0 0
\(389\) 3.00000 0.152106 0.0760530 0.997104i \(-0.475768\pi\)
0.0760530 + 0.997104i \(0.475768\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 42.4264i 2.14013i
\(394\) 0 0
\(395\) 18.9737 0.954669
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 15.0000 11.1803i 0.750939 0.559717i
\(400\) 0 0
\(401\) 24.0000 1.19850 0.599251 0.800561i \(-0.295465\pi\)
0.599251 + 0.800561i \(0.295465\pi\)
\(402\) 0 0
\(403\) 42.4264i 2.11341i
\(404\) 0 0
\(405\) 24.5967i 1.22222i
\(406\) 0 0
\(407\) −3.00000 + 1.41421i −0.148704 + 0.0701000i
\(408\) 0 0
\(409\) 12.6491 0.625458 0.312729 0.949842i \(-0.398757\pi\)
0.312729 + 0.949842i \(0.398757\pi\)
\(410\) 0 0
\(411\) 33.5410i 1.65446i
\(412\) 0 0
\(413\) 4.74342 3.53553i 0.233408 0.173972i
\(414\) 0 0
\(415\) 21.2132i 1.04132i
\(416\) 0 0
\(417\) 28.2843i 1.38509i
\(418\) 0 0
\(419\) 22.3607i 1.09239i 0.837658 + 0.546195i \(0.183924\pi\)
−0.837658 + 0.546195i \(0.816076\pi\)
\(420\) 0 0
\(421\) −4.00000 −0.194948 −0.0974740 0.995238i \(-0.531076\pi\)
−0.0974740 + 0.995238i \(0.531076\pi\)
\(422\) 0 0
\(423\) 8.94427i 0.434885i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −5.00000 6.70820i −0.241967 0.324633i
\(428\) 0 0
\(429\) 20.0000 + 42.4264i 0.965609 + 2.04837i
\(430\) 0 0
\(431\) 35.3553i 1.70301i 0.524349 + 0.851503i \(0.324309\pi\)
−0.524349 + 0.851503i \(0.675691\pi\)
\(432\) 0 0
\(433\) 20.1246i 0.967127i 0.875309 + 0.483564i \(0.160658\pi\)
−0.875309 + 0.483564i \(0.839342\pi\)
\(434\) 0 0
\(435\) 7.07107i 0.339032i
\(436\) 0 0
\(437\) −9.48683 −0.453817
\(438\) 0 0
\(439\) 15.8114 0.754636 0.377318 0.926084i \(-0.376846\pi\)
0.377318 + 0.926084i \(0.376846\pi\)
\(440\) 0 0
\(441\) 4.00000 13.4164i 0.190476 0.638877i
\(442\) 0 0
\(443\) 15.0000 0.712672 0.356336 0.934358i \(-0.384026\pi\)
0.356336 + 0.934358i \(0.384026\pi\)
\(444\) 0 0
\(445\) −5.00000 −0.237023
\(446\) 0 0
\(447\) 44.2719 2.09399
\(448\) 0 0
\(449\) −27.0000 −1.27421 −0.637104 0.770778i \(-0.719868\pi\)
−0.637104 + 0.770778i \(0.719868\pi\)
\(450\) 0 0
\(451\) 28.4605 13.4164i 1.34015 0.631754i
\(452\) 0 0
\(453\) −37.9473 −1.78292
\(454\) 0 0
\(455\) 30.0000 22.3607i 1.40642 1.04828i
\(456\) 0 0
\(457\) 8.48528i 0.396925i −0.980109 0.198462i \(-0.936405\pi\)
0.980109 0.198462i \(-0.0635948\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −9.48683 −0.441846 −0.220923 0.975291i \(-0.570907\pi\)
−0.220923 + 0.975291i \(0.570907\pi\)
\(462\) 0 0
\(463\) 25.0000 1.16185 0.580924 0.813958i \(-0.302691\pi\)
0.580924 + 0.813958i \(0.302691\pi\)
\(464\) 0 0
\(465\) 33.5410i 1.55543i
\(466\) 0 0
\(467\) 11.1803i 0.517364i −0.965962 0.258682i \(-0.916712\pi\)
0.965962 0.258682i \(-0.0832882\pi\)
\(468\) 0 0
\(469\) 17.3925 + 23.3345i 0.803112 + 1.07749i
\(470\) 0 0
\(471\) −15.0000 −0.691164
\(472\) 0 0
\(473\) −6.00000 12.7279i −0.275880 0.585230i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −18.9737 −0.866929 −0.433464 0.901171i \(-0.642709\pi\)
−0.433464 + 0.901171i \(0.642709\pi\)
\(480\) 0 0
\(481\) 6.32456 0.288375
\(482\) 0 0
\(483\) −14.2302 + 10.6066i −0.647499 + 0.482617i
\(484\) 0 0
\(485\) 15.0000 0.681115
\(486\) 0 0
\(487\) −29.0000 −1.31412 −0.657058 0.753840i \(-0.728199\pi\)
−0.657058 + 0.753840i \(0.728199\pi\)
\(488\) 0 0
\(489\) 22.3607i 1.01118i
\(490\) 0 0
\(491\) 32.5269i 1.46792i −0.679193 0.733959i \(-0.737670\pi\)
0.679193 0.733959i \(-0.262330\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 6.32456 + 13.4164i 0.284268 + 0.603023i
\(496\) 0 0
\(497\) 14.2302 + 19.0919i 0.638314 + 0.856388i
\(498\) 0 0
\(499\) 16.0000 0.716258 0.358129 0.933672i \(-0.383415\pi\)
0.358129 + 0.933672i \(0.383415\pi\)
\(500\) 0 0
\(501\) 21.2132i 0.947736i
\(502\) 0 0
\(503\) −28.4605 −1.26899 −0.634495 0.772927i \(-0.718792\pi\)
−0.634495 + 0.772927i \(0.718792\pi\)
\(504\) 0 0
\(505\) 21.2132i 0.943975i
\(506\) 0 0
\(507\) 60.3738i 2.68130i
\(508\) 0 0
\(509\) 15.6525i 0.693784i −0.937905 0.346892i \(-0.887237\pi\)
0.937905 0.346892i \(-0.112763\pi\)
\(510\) 0 0
\(511\) −5.00000 6.70820i −0.221187 0.296753i
\(512\) 0 0
\(513\) 7.07107i 0.312195i
\(514\) 0 0
\(515\) −30.0000 −1.32196
\(516\) 0 0
\(517\) −6.32456 13.4164i −0.278154 0.590053i
\(518\) 0 0
\(519\) 21.2132i 0.931156i
\(520\) 0 0
\(521\) 11.1803i 0.489820i 0.969546 + 0.244910i \(0.0787583\pi\)
−0.969546 + 0.244910i \(0.921242\pi\)
\(522\) 0 0
\(523\) −31.6228 −1.38277 −0.691384 0.722488i \(-0.742999\pi\)
−0.691384 + 0.722488i \(0.742999\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) 4.47214i 0.194074i
\(532\) 0 0
\(533\) −60.0000 −2.59889
\(534\) 0 0
\(535\) −6.32456 −0.273434
\(536\) 0 0
\(537\) 20.1246i 0.868441i
\(538\) 0 0
\(539\) 3.48683 + 22.9530i 0.150189 + 0.988657i
\(540\) 0 0
\(541\) 25.4558i 1.09443i 0.836991 + 0.547216i \(0.184312\pi\)
−0.836991 + 0.547216i \(0.815688\pi\)
\(542\) 0 0
\(543\) −45.0000 −1.93113
\(544\) 0 0
\(545\) −28.4605 −1.21911
\(546\) 0 0
\(547\) 8.48528i 0.362804i 0.983409 + 0.181402i \(0.0580636\pi\)
−0.983409 + 0.181402i \(0.941936\pi\)
\(548\) 0 0
\(549\) −6.32456 −0.269925
\(550\) 0 0
\(551\) 4.47214i 0.190519i
\(552\) 0 0
\(553\) 18.0000 13.4164i 0.765438 0.570524i
\(554\) 0 0
\(555\) 5.00000 0.212238
\(556\) 0 0
\(557\) 22.6274i 0.958754i −0.877609 0.479377i \(-0.840863\pi\)
0.877609 0.479377i \(-0.159137\pi\)
\(558\) 0 0
\(559\) 26.8328i 1.13491i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −37.9473 −1.59929 −0.799645 0.600473i \(-0.794979\pi\)
−0.799645 + 0.600473i \(0.794979\pi\)
\(564\) 0 0
\(565\) 6.70820i 0.282216i
\(566\) 0 0
\(567\) 17.3925 + 23.3345i 0.730417 + 0.979958i
\(568\) 0 0
\(569\) 15.5563i 0.652156i 0.945343 + 0.326078i \(0.105727\pi\)
−0.945343 + 0.326078i \(0.894273\pi\)
\(570\) 0 0
\(571\) 16.9706i 0.710196i 0.934829 + 0.355098i \(0.115552\pi\)
−0.934829 + 0.355098i \(0.884448\pi\)
\(572\) 0 0
\(573\) 6.70820i 0.280239i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 20.1246i 0.837799i 0.908033 + 0.418899i \(0.137584\pi\)
−0.908033 + 0.418899i \(0.862416\pi\)
\(578\) 0 0
\(579\) −9.48683 −0.394259
\(580\) 0 0
\(581\) −15.0000 20.1246i −0.622305 0.834910i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 28.2843i 1.16941i
\(586\) 0 0
\(587\) 4.47214i 0.184585i −0.995732 0.0922924i \(-0.970581\pi\)
0.995732 0.0922924i \(-0.0294195\pi\)
\(588\) 0 0
\(589\) 21.2132i 0.874075i
\(590\) 0 0
\(591\) −12.6491 −0.520315
\(592\) 0 0
\(593\) 28.4605 1.16873 0.584366 0.811490i \(-0.301343\pi\)
0.584366 + 0.811490i \(0.301343\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 12.0000 0.490307 0.245153 0.969484i \(-0.421162\pi\)
0.245153 + 0.969484i \(0.421162\pi\)
\(600\) 0 0
\(601\) −15.8114 −0.644960 −0.322480 0.946576i \(-0.604517\pi\)
−0.322480 + 0.946576i \(0.604517\pi\)
\(602\) 0 0
\(603\) 22.0000 0.895909
\(604\) 0 0
\(605\) −18.9737 15.6525i −0.771389 0.636364i
\(606\) 0 0
\(607\) 6.32456 0.256706 0.128353 0.991729i \(-0.459031\pi\)
0.128353 + 0.991729i \(0.459031\pi\)
\(608\) 0 0
\(609\) 5.00000 + 6.70820i 0.202610 + 0.271830i
\(610\) 0 0
\(611\) 28.2843i 1.14426i
\(612\) 0 0
\(613\) 25.4558i 1.02815i −0.857745 0.514076i \(-0.828135\pi\)
0.857745 0.514076i \(-0.171865\pi\)
\(614\) 0 0
\(615\) −47.4342 −1.91273
\(616\) 0 0
\(617\) 24.0000 0.966204 0.483102 0.875564i \(-0.339510\pi\)
0.483102 + 0.875564i \(0.339510\pi\)
\(618\) 0 0
\(619\) 46.9574i 1.88738i −0.330833 0.943689i \(-0.607330\pi\)
0.330833 0.943689i \(-0.392670\pi\)
\(620\) 0 0
\(621\) 6.70820i 0.269191i
\(622\) 0 0
\(623\) −4.74342 + 3.53553i −0.190041 + 0.141648i
\(624\) 0 0
\(625\) −25.0000 −1.00000
\(626\) 0 0
\(627\) 10.0000 + 21.2132i 0.399362 + 0.847174i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −11.0000 −0.437903 −0.218952 0.975736i \(-0.570264\pi\)
−0.218952 + 0.975736i \(0.570264\pi\)
\(632\) 0 0
\(633\) −47.4342 −1.88534
\(634\) 0 0
\(635\) 28.4605 1.12942
\(636\) 0 0
\(637\) 12.6491 42.4264i 0.501176 1.68100i
\(638\) 0 0
\(639\) 18.0000 0.712069
\(640\) 0 0
\(641\) 21.0000 0.829450 0.414725 0.909947i \(-0.363878\pi\)
0.414725 + 0.909947i \(0.363878\pi\)
\(642\) 0 0
\(643\) 6.70820i 0.264546i 0.991213 + 0.132273i \(0.0422275\pi\)
−0.991213 + 0.132273i \(0.957772\pi\)
\(644\) 0 0
\(645\) 21.2132i 0.835269i
\(646\) 0 0
\(647\) 29.0689i 1.14282i 0.820666 + 0.571408i \(0.193603\pi\)
−0.820666 + 0.571408i \(0.806397\pi\)
\(648\) 0 0
\(649\) 3.16228 + 6.70820i 0.124130 + 0.263320i
\(650\) 0 0
\(651\) −23.7171 31.8198i −0.929546 1.24712i
\(652\) 0 0
\(653\) 15.0000 0.586995 0.293498 0.955960i \(-0.405181\pi\)
0.293498 + 0.955960i \(0.405181\pi\)
\(654\) 0 0
\(655\) 42.4264i 1.65774i
\(656\) 0 0
\(657\) −6.32456 −0.246744
\(658\) 0 0
\(659\) 14.1421i 0.550899i 0.961315 + 0.275450i \(0.0888267\pi\)
−0.961315 + 0.275450i \(0.911173\pi\)
\(660\) 0 0
\(661\) 46.9574i 1.82643i −0.407476 0.913216i \(-0.633591\pi\)
0.407476 0.913216i \(-0.366409\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 15.0000 11.1803i 0.581675 0.433555i
\(666\) 0 0
\(667\) 4.24264i 0.164276i
\(668\) 0 0
\(669\) −15.0000 −0.579934
\(670\) 0 0
\(671\) 9.48683 4.47214i 0.366235 0.172645i
\(672\) 0 0
\(673\) 29.6985i 1.14479i 0.819977 + 0.572396i \(0.193986\pi\)
−0.819977 + 0.572396i \(0.806014\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −37.9473 −1.45843 −0.729217 0.684282i \(-0.760116\pi\)
−0.729217 + 0.684282i \(0.760116\pi\)
\(678\) 0 0
\(679\) 14.2302 10.6066i 0.546107 0.407044i
\(680\) 0 0
\(681\) 21.2132i 0.812892i
\(682\) 0 0
\(683\) −18.0000 −0.688751 −0.344375 0.938832i \(-0.611909\pi\)
−0.344375 + 0.938832i \(0.611909\pi\)
\(684\) 0 0
\(685\) 33.5410i 1.28154i
\(686\) 0 0
\(687\) −15.0000 −0.572286
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 6.70820i 0.255192i −0.991826 0.127596i \(-0.959274\pi\)
0.991826 0.127596i \(-0.0407261\pi\)
\(692\) 0 0
\(693\) 15.4868 + 8.25579i 0.588296 + 0.313611i
\(694\) 0 0
\(695\) 28.2843i 1.07288i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −41.1096 −1.55491
\(700\) 0 0
\(701\) 7.07107i 0.267071i 0.991044 + 0.133535i \(0.0426329\pi\)
−0.991044 + 0.133535i \(0.957367\pi\)
\(702\) 0 0
\(703\) 3.16228 0.119268
\(704\) 0 0
\(705\) 22.3607i 0.842152i
\(706\) 0 0
\(707\) 15.0000 + 20.1246i 0.564133 + 0.756864i
\(708\) 0 0
\(709\) 41.0000 1.53979 0.769894 0.638172i \(-0.220309\pi\)
0.769894 + 0.638172i \(0.220309\pi\)
\(710\) 0 0
\(711\) 16.9706i 0.636446i
\(712\) 0 0
\(713\) 20.1246i 0.753673i
\(714\) 0 0
\(715\) 20.0000 + 42.4264i 0.747958 + 1.58666i
\(716\) 0 0
\(717\) −15.8114 −0.590487
\(718\) 0 0
\(719\) 51.4296i 1.91800i −0.283408 0.959000i \(-0.591465\pi\)
0.283408 0.959000i \(-0.408535\pi\)
\(720\) 0 0
\(721\) −28.4605 + 21.2132i −1.05992 + 0.790021i
\(722\) 0 0
\(723\) 28.2843i 1.05190i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 33.5410i 1.24397i 0.783030 + 0.621984i \(0.213673\pi\)
−0.783030 + 0.621984i \(0.786327\pi\)
\(728\) 0 0
\(729\) 7.00000 0.259259
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 12.6491 0.467206 0.233603 0.972332i \(-0.424948\pi\)
0.233603 + 0.972332i \(0.424948\pi\)
\(734\) 0 0
\(735\) 10.0000 33.5410i 0.368856 1.23718i
\(736\) 0 0
\(737\) −33.0000 + 15.5563i −1.21557 + 0.573025i
\(738\) 0 0
\(739\) 50.9117i 1.87282i 0.350912 + 0.936408i \(0.385872\pi\)
−0.350912 + 0.936408i \(0.614128\pi\)
\(740\) 0 0
\(741\) 44.7214i 1.64288i
\(742\) 0 0
\(743\) 19.7990i 0.726354i −0.931720 0.363177i \(-0.881692\pi\)
0.931720 0.363177i \(-0.118308\pi\)
\(744\) 0 0
\(745\) 44.2719 1.62200
\(746\) 0 0
\(747\) −18.9737 −0.694210
\(748\) 0 0
\(749\) −6.00000 + 4.47214i −0.219235 + 0.163408i
\(750\) 0 0
\(751\) 7.00000 0.255434 0.127717 0.991811i \(-0.459235\pi\)
0.127717 + 0.991811i \(0.459235\pi\)
\(752\) 0 0
\(753\) −25.0000 −0.911051
\(754\) 0 0
\(755\) −37.9473 −1.38104
\(756\) 0 0
\(757\) 44.0000 1.59921 0.799604 0.600528i \(-0.205043\pi\)
0.799604 + 0.600528i \(0.205043\pi\)
\(758\) 0 0
\(759\) −9.48683 20.1246i −0.344350 0.730477i
\(760\) 0 0
\(761\) 37.9473 1.37559 0.687795 0.725905i \(-0.258579\pi\)
0.687795 + 0.725905i \(0.258579\pi\)
\(762\) 0 0
\(763\) −27.0000 + 20.1246i −0.977466 + 0.728560i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 14.1421i 0.510643i
\(768\) 0 0
\(769\) −6.32456 −0.228069 −0.114035 0.993477i \(-0.536377\pi\)
−0.114035 + 0.993477i \(0.536377\pi\)
\(770\) 0 0
\(771\) −50.0000 −1.80071
\(772\) 0 0
\(773\) 8.94427i 0.321703i −0.986979 0.160852i \(-0.948576\pi\)
0.986979 0.160852i \(-0.0514240\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 4.74342 3.53553i 0.170169 0.126837i
\(778\) 0 0
\(779\) −30.0000 −1.07486
\(780\) 0 0
\(781\) −27.0000 + 12.7279i −0.966136 + 0.455441i
\(782\) 0 0
\(783\) −3.16228 −0.113011
\(784\) 0 0
\(785\) −15.0000 −0.535373
\(786\) 0 0
\(787\) −22.1359 −0.789061 −0.394531 0.918883i \(-0.629093\pi\)
−0.394531 + 0.918883i \(0.629093\pi\)
\(788\) 0 0
\(789\) −25.2982 −0.900641
\(790\) 0 0
\(791\) 4.74342 + 6.36396i 0.168656 + 0.226276i
\(792\) 0 0
\(793\) −20.0000 −0.710221
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 38.0132i 1.34650i 0.739417 + 0.673248i \(0.235101\pi\)
−0.739417 + 0.673248i \(0.764899\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 4.47214i 0.158015i