Properties

Label 63.4.c.b.62.4
Level $63$
Weight $4$
Character 63.62
Analytic conductor $3.717$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 63.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.71712033036\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{111})\)
Defining polynomial: \(x^{4} + 112 x^{2} + 3025\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 62.4
Root \(6.74273i\) of defining polynomial
Character \(\chi\) \(=\) 63.62
Dual form 63.4.c.b.62.2

$q$-expansion

\(f(q)\) \(=\) \(q+2.82843i q^{2} +21.0713 q^{5} +(-11.0000 - 14.8997i) q^{7} +22.6274i q^{8} +O(q^{10})\) \(q+2.82843i q^{2} +21.0713 q^{5} +(-11.0000 - 14.8997i) q^{7} +22.6274i q^{8} +59.5987i q^{10} +15.5563i q^{11} +29.7993i q^{13} +(42.1426 - 31.1127i) q^{14} -64.0000 q^{16} -63.2139 q^{17} -89.3980i q^{19} -44.0000 q^{22} -77.7817i q^{23} +319.000 q^{25} -84.2852 q^{26} +125.865i q^{29} -238.395i q^{31} -178.796i q^{34} +(-231.784 - 313.955i) q^{35} -184.000 q^{37} +252.856 q^{38} +476.789i q^{40} -105.357 q^{41} -190.000 q^{43} +220.000 q^{46} +42.1426 q^{47} +(-101.000 + 327.793i) q^{49} +902.268i q^{50} -357.796i q^{53} +327.793i q^{55} +(337.141 - 248.902i) q^{56} -356.000 q^{58} +84.2852 q^{59} -655.585i q^{61} +674.282 q^{62} -512.000 q^{64} +627.911i q^{65} +296.000 q^{67} +(888.000 - 655.585i) q^{70} +329.512i q^{71} +804.582i q^{73} -520.431i q^{74} +(231.784 - 171.120i) q^{77} +836.000 q^{79} -1348.56 q^{80} -297.993i q^{82} -1222.14 q^{83} -1332.00 q^{85} -537.401i q^{86} -352.000 q^{88} +695.353 q^{89} +(444.000 - 327.793i) q^{91} +119.197i q^{94} -1883.73i q^{95} -566.187i q^{97} +(-927.138 - 285.671i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 44q^{7} + O(q^{10}) \) \( 4q - 44q^{7} - 256q^{16} - 176q^{22} + 1276q^{25} - 736q^{37} - 760q^{43} + 880q^{46} - 404q^{49} - 1424q^{58} - 2048q^{64} + 1184q^{67} + 3552q^{70} + 3344q^{79} - 5328q^{85} - 1408q^{88} + 1776q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/63\mathbb{Z}\right)^\times\).

\(n\) \(10\) \(29\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.82843i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(3\) 0 0
\(4\) 0 0
\(5\) 21.0713 1.88468 0.942338 0.334664i \(-0.108623\pi\)
0.942338 + 0.334664i \(0.108623\pi\)
\(6\) 0 0
\(7\) −11.0000 14.8997i −0.593944 0.804506i
\(8\) 22.6274i 1.00000i
\(9\) 0 0
\(10\) 59.5987i 1.88468i
\(11\) 15.5563i 0.426401i 0.977008 + 0.213201i \(0.0683888\pi\)
−0.977008 + 0.213201i \(0.931611\pi\)
\(12\) 0 0
\(13\) 29.7993i 0.635757i 0.948131 + 0.317879i \(0.102970\pi\)
−0.948131 + 0.317879i \(0.897030\pi\)
\(14\) 42.1426 31.1127i 0.804506 0.593944i
\(15\) 0 0
\(16\) −64.0000 −1.00000
\(17\) −63.2139 −0.901860 −0.450930 0.892559i \(-0.648908\pi\)
−0.450930 + 0.892559i \(0.648908\pi\)
\(18\) 0 0
\(19\) 89.3980i 1.07944i −0.841846 0.539719i \(-0.818531\pi\)
0.841846 0.539719i \(-0.181469\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −44.0000 −0.426401
\(23\) 77.7817i 0.705157i −0.935782 0.352579i \(-0.885305\pi\)
0.935782 0.352579i \(-0.114695\pi\)
\(24\) 0 0
\(25\) 319.000 2.55200
\(26\) −84.2852 −0.635757
\(27\) 0 0
\(28\) 0 0
\(29\) 125.865i 0.805950i 0.915211 + 0.402975i \(0.132024\pi\)
−0.915211 + 0.402975i \(0.867976\pi\)
\(30\) 0 0
\(31\) 238.395i 1.38119i −0.723241 0.690596i \(-0.757348\pi\)
0.723241 0.690596i \(-0.242652\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 178.796i 0.901860i
\(35\) −231.784 313.955i −1.11939 1.51623i
\(36\) 0 0
\(37\) −184.000 −0.817552 −0.408776 0.912635i \(-0.634044\pi\)
−0.408776 + 0.912635i \(0.634044\pi\)
\(38\) 252.856 1.07944
\(39\) 0 0
\(40\) 476.789i 1.88468i
\(41\) −105.357 −0.401315 −0.200658 0.979661i \(-0.564308\pi\)
−0.200658 + 0.979661i \(0.564308\pi\)
\(42\) 0 0
\(43\) −190.000 −0.673831 −0.336915 0.941535i \(-0.609384\pi\)
−0.336915 + 0.941535i \(0.609384\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 220.000 0.705157
\(47\) 42.1426 0.130790 0.0653950 0.997859i \(-0.479169\pi\)
0.0653950 + 0.997859i \(0.479169\pi\)
\(48\) 0 0
\(49\) −101.000 + 327.793i −0.294461 + 0.955664i
\(50\) 902.268i 2.55200i
\(51\) 0 0
\(52\) 0 0
\(53\) 357.796i 0.927303i −0.886018 0.463652i \(-0.846539\pi\)
0.886018 0.463652i \(-0.153461\pi\)
\(54\) 0 0
\(55\) 327.793i 0.803628i
\(56\) 337.141 248.902i 0.804506 0.593944i
\(57\) 0 0
\(58\) −356.000 −0.805950
\(59\) 84.2852 0.185983 0.0929915 0.995667i \(-0.470357\pi\)
0.0929915 + 0.995667i \(0.470357\pi\)
\(60\) 0 0
\(61\) 655.585i 1.37605i −0.725687 0.688025i \(-0.758478\pi\)
0.725687 0.688025i \(-0.241522\pi\)
\(62\) 674.282 1.38119
\(63\) 0 0
\(64\) −512.000 −1.00000
\(65\) 627.911i 1.19820i
\(66\) 0 0
\(67\) 296.000 0.539734 0.269867 0.962898i \(-0.413020\pi\)
0.269867 + 0.962898i \(0.413020\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 888.000 655.585i 1.51623 1.11939i
\(71\) 329.512i 0.550787i 0.961332 + 0.275393i \(0.0888081\pi\)
−0.961332 + 0.275393i \(0.911192\pi\)
\(72\) 0 0
\(73\) 804.582i 1.28999i 0.764187 + 0.644994i \(0.223140\pi\)
−0.764187 + 0.644994i \(0.776860\pi\)
\(74\) 520.431i 0.817552i
\(75\) 0 0
\(76\) 0 0
\(77\) 231.784 171.120i 0.343043 0.253259i
\(78\) 0 0
\(79\) 836.000 1.19060 0.595300 0.803504i \(-0.297033\pi\)
0.595300 + 0.803504i \(0.297033\pi\)
\(80\) −1348.56 −1.88468
\(81\) 0 0
\(82\) 297.993i 0.401315i
\(83\) −1222.14 −1.61623 −0.808113 0.589027i \(-0.799511\pi\)
−0.808113 + 0.589027i \(0.799511\pi\)
\(84\) 0 0
\(85\) −1332.00 −1.69971
\(86\) 537.401i 0.673831i
\(87\) 0 0
\(88\) −352.000 −0.426401
\(89\) 695.353 0.828172 0.414086 0.910238i \(-0.364101\pi\)
0.414086 + 0.910238i \(0.364101\pi\)
\(90\) 0 0
\(91\) 444.000 327.793i 0.511471 0.377604i
\(92\) 0 0
\(93\) 0 0
\(94\) 119.197i 0.130790i
\(95\) 1883.73i 2.03439i
\(96\) 0 0
\(97\) 566.187i 0.592656i −0.955086 0.296328i \(-0.904238\pi\)
0.955086 0.296328i \(-0.0957621\pi\)
\(98\) −927.138 285.671i −0.955664 0.294461i
\(99\) 0 0
\(100\) 0 0
\(101\) 737.496 0.726570 0.363285 0.931678i \(-0.381655\pi\)
0.363285 + 0.931678i \(0.381655\pi\)
\(102\) 0 0
\(103\) 655.585i 0.627153i 0.949563 + 0.313576i \(0.101527\pi\)
−0.949563 + 0.313576i \(0.898473\pi\)
\(104\) −674.282 −0.635757
\(105\) 0 0
\(106\) 1012.00 0.927303
\(107\) 1984.14i 1.79266i 0.443391 + 0.896328i \(0.353775\pi\)
−0.443391 + 0.896328i \(0.646225\pi\)
\(108\) 0 0
\(109\) −844.000 −0.741656 −0.370828 0.928702i \(-0.620926\pi\)
−0.370828 + 0.928702i \(0.620926\pi\)
\(110\) −927.138 −0.803628
\(111\) 0 0
\(112\) 704.000 + 953.579i 0.593944 + 0.804506i
\(113\) 575.585i 0.479172i 0.970875 + 0.239586i \(0.0770118\pi\)
−0.970875 + 0.239586i \(0.922988\pi\)
\(114\) 0 0
\(115\) 1638.96i 1.32899i
\(116\) 0 0
\(117\) 0 0
\(118\) 238.395i 0.185983i
\(119\) 695.353 + 941.866i 0.535655 + 0.725552i
\(120\) 0 0
\(121\) 1089.00 0.818182
\(122\) 1854.28 1.37605
\(123\) 0 0
\(124\) 0 0
\(125\) 4087.83 2.92502
\(126\) 0 0
\(127\) −220.000 −0.153715 −0.0768577 0.997042i \(-0.524489\pi\)
−0.0768577 + 0.997042i \(0.524489\pi\)
\(128\) 1448.15i 1.00000i
\(129\) 0 0
\(130\) −1776.00 −1.19820
\(131\) −1306.42 −0.871317 −0.435659 0.900112i \(-0.643484\pi\)
−0.435659 + 0.900112i \(0.643484\pi\)
\(132\) 0 0
\(133\) −1332.00 + 983.378i −0.868414 + 0.641125i
\(134\) 837.214i 0.539734i
\(135\) 0 0
\(136\) 1430.37i 0.901860i
\(137\) 1568.36i 0.978060i 0.872267 + 0.489030i \(0.162649\pi\)
−0.872267 + 0.489030i \(0.837351\pi\)
\(138\) 0 0
\(139\) 655.585i 0.400043i 0.979791 + 0.200022i \(0.0641012\pi\)
−0.979791 + 0.200022i \(0.935899\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −932.000 −0.550787
\(143\) −463.569 −0.271088
\(144\) 0 0
\(145\) 2652.14i 1.51895i
\(146\) −2275.70 −1.28999
\(147\) 0 0
\(148\) 0 0
\(149\) 1466.54i 0.806333i 0.915127 + 0.403166i \(0.132090\pi\)
−0.915127 + 0.403166i \(0.867910\pi\)
\(150\) 0 0
\(151\) 1970.00 1.06170 0.530849 0.847467i \(-0.321873\pi\)
0.530849 + 0.847467i \(0.321873\pi\)
\(152\) 2022.85 1.07944
\(153\) 0 0
\(154\) 484.000 + 655.585i 0.253259 + 0.343043i
\(155\) 5023.29i 2.60310i
\(156\) 0 0
\(157\) 2622.34i 1.33303i 0.745492 + 0.666515i \(0.232215\pi\)
−0.745492 + 0.666515i \(0.767785\pi\)
\(158\) 2364.57i 1.19060i
\(159\) 0 0
\(160\) 0 0
\(161\) −1158.92 + 855.599i −0.567303 + 0.418824i
\(162\) 0 0
\(163\) −1336.00 −0.641985 −0.320993 0.947082i \(-0.604016\pi\)
−0.320993 + 0.947082i \(0.604016\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 3456.72i 1.61623i
\(167\) −2317.84 −1.07401 −0.537006 0.843578i \(-0.680445\pi\)
−0.537006 + 0.843578i \(0.680445\pi\)
\(168\) 0 0
\(169\) 1309.00 0.595812
\(170\) 3767.46i 1.69971i
\(171\) 0 0
\(172\) 0 0
\(173\) 231.784 0.101863 0.0509313 0.998702i \(-0.483781\pi\)
0.0509313 + 0.998702i \(0.483781\pi\)
\(174\) 0 0
\(175\) −3509.00 4752.99i −1.51575 2.05310i
\(176\) 995.606i 0.426401i
\(177\) 0 0
\(178\) 1966.76i 0.828172i
\(179\) 108.894i 0.0454701i 0.999742 + 0.0227351i \(0.00723742\pi\)
−0.999742 + 0.0227351i \(0.992763\pi\)
\(180\) 0 0
\(181\) 2592.54i 1.06465i −0.846539 0.532326i \(-0.821318\pi\)
0.846539 0.532326i \(-0.178682\pi\)
\(182\) 927.138 + 1255.82i 0.377604 + 0.511471i
\(183\) 0 0
\(184\) 1760.00 0.705157
\(185\) −3877.12 −1.54082
\(186\) 0 0
\(187\) 983.378i 0.384555i
\(188\) 0 0
\(189\) 0 0
\(190\) 5328.00 2.03439
\(191\) 2224.56i 0.842740i −0.906889 0.421370i \(-0.861549\pi\)
0.906889 0.421370i \(-0.138451\pi\)
\(192\) 0 0
\(193\) 3740.00 1.39488 0.697438 0.716645i \(-0.254323\pi\)
0.697438 + 0.716645i \(0.254323\pi\)
\(194\) 1601.42 0.592656
\(195\) 0 0
\(196\) 0 0
\(197\) 1197.84i 0.433211i −0.976259 0.216605i \(-0.930502\pi\)
0.976259 0.216605i \(-0.0694985\pi\)
\(198\) 0 0
\(199\) 804.582i 0.286610i −0.989679 0.143305i \(-0.954227\pi\)
0.989679 0.143305i \(-0.0457729\pi\)
\(200\) 7218.15i 2.55200i
\(201\) 0 0
\(202\) 2085.95i 0.726570i
\(203\) 1875.35 1384.52i 0.648392 0.478689i
\(204\) 0 0
\(205\) −2220.00 −0.756349
\(206\) −1854.28 −0.627153
\(207\) 0 0
\(208\) 1907.16i 0.635757i
\(209\) 1390.71 0.460274
\(210\) 0 0
\(211\) 3590.00 1.17131 0.585654 0.810561i \(-0.300838\pi\)
0.585654 + 0.810561i \(0.300838\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) −5612.00 −1.79266
\(215\) −4003.55 −1.26995
\(216\) 0 0
\(217\) −3552.00 + 2622.34i −1.11118 + 0.820351i
\(218\) 2387.19i 0.741656i
\(219\) 0 0
\(220\) 0 0
\(221\) 1883.73i 0.573365i
\(222\) 0 0
\(223\) 3009.73i 0.903796i 0.892070 + 0.451898i \(0.149253\pi\)
−0.892070 + 0.451898i \(0.850747\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −1628.00 −0.479172
\(227\) 3708.55 1.08434 0.542170 0.840269i \(-0.317603\pi\)
0.542170 + 0.840269i \(0.317603\pi\)
\(228\) 0 0
\(229\) 327.793i 0.0945902i −0.998881 0.0472951i \(-0.984940\pi\)
0.998881 0.0472951i \(-0.0150601\pi\)
\(230\) 4635.69 1.32899
\(231\) 0 0
\(232\) −2848.00 −0.805950
\(233\) 643.467i 0.180922i 0.995900 + 0.0904612i \(0.0288341\pi\)
−0.995900 + 0.0904612i \(0.971166\pi\)
\(234\) 0 0
\(235\) 888.000 0.246497
\(236\) 0 0
\(237\) 0 0
\(238\) −2664.00 + 1966.76i −0.725552 + 0.535655i
\(239\) 646.296i 0.174918i −0.996168 0.0874590i \(-0.972125\pi\)
0.996168 0.0874590i \(-0.0278747\pi\)
\(240\) 0 0
\(241\) 387.391i 0.103544i −0.998659 0.0517719i \(-0.983513\pi\)
0.998659 0.0517719i \(-0.0164869\pi\)
\(242\) 3080.16i 0.818182i
\(243\) 0 0
\(244\) 0 0
\(245\) −2128.20 + 6907.02i −0.554963 + 1.80112i
\(246\) 0 0
\(247\) 2664.00 0.686260
\(248\) 5394.25 1.38119
\(249\) 0 0
\(250\) 11562.1i 2.92502i
\(251\) −6194.96 −1.55786 −0.778930 0.627111i \(-0.784237\pi\)
−0.778930 + 0.627111i \(0.784237\pi\)
\(252\) 0 0
\(253\) 1210.00 0.300680
\(254\) 622.254i 0.153715i
\(255\) 0 0
\(256\) 0 0
\(257\) 3561.05 0.864328 0.432164 0.901795i \(-0.357750\pi\)
0.432164 + 0.901795i \(0.357750\pi\)
\(258\) 0 0
\(259\) 2024.00 + 2741.54i 0.485580 + 0.657725i
\(260\) 0 0
\(261\) 0 0
\(262\) 3695.12i 0.871317i
\(263\) 601.041i 0.140919i 0.997515 + 0.0704596i \(0.0224466\pi\)
−0.997515 + 0.0704596i \(0.977553\pi\)
\(264\) 0 0
\(265\) 7539.23i 1.74767i
\(266\) −2781.41 3767.46i −0.641125 0.868414i
\(267\) 0 0
\(268\) 0 0
\(269\) 4867.47 1.10325 0.551626 0.834091i \(-0.314007\pi\)
0.551626 + 0.834091i \(0.314007\pi\)
\(270\) 0 0
\(271\) 1787.96i 0.400778i −0.979716 0.200389i \(-0.935779\pi\)
0.979716 0.200389i \(-0.0642206\pi\)
\(272\) 4045.69 0.901860
\(273\) 0 0
\(274\) −4436.00 −0.978060
\(275\) 4962.48i 1.08818i
\(276\) 0 0
\(277\) −1126.00 −0.244241 −0.122121 0.992515i \(-0.538969\pi\)
−0.122121 + 0.992515i \(0.538969\pi\)
\(278\) −1854.28 −0.400043
\(279\) 0 0
\(280\) 7104.00 5244.68i 1.51623 1.11939i
\(281\) 5075.61i 1.07753i −0.842456 0.538765i \(-0.818891\pi\)
0.842456 0.538765i \(-0.181109\pi\)
\(282\) 0 0
\(283\) 7122.04i 1.49598i −0.663712 0.747988i \(-0.731020\pi\)
0.663712 0.747988i \(-0.268980\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 1311.17i 0.271088i
\(287\) 1158.92 + 1569.78i 0.238359 + 0.322861i
\(288\) 0 0
\(289\) −917.000 −0.186648
\(290\) −7501.39 −1.51895
\(291\) 0 0
\(292\) 0 0
\(293\) −231.784 −0.0462150 −0.0231075 0.999733i \(-0.507356\pi\)
−0.0231075 + 0.999733i \(0.507356\pi\)
\(294\) 0 0
\(295\) 1776.00 0.350518
\(296\) 4163.44i 0.817552i
\(297\) 0 0
\(298\) −4148.00 −0.806333
\(299\) 2317.84 0.448309
\(300\) 0 0
\(301\) 2090.00 + 2830.94i 0.400218 + 0.542101i
\(302\) 5572.00i 1.06170i
\(303\) 0 0
\(304\) 5721.47i 1.07944i
\(305\) 13814.0i 2.59341i
\(306\) 0 0
\(307\) 8850.40i 1.64534i 0.568520 + 0.822669i \(0.307516\pi\)
−0.568520 + 0.822669i \(0.692484\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 14208.0 2.60310
\(311\) −6489.96 −1.18332 −0.591659 0.806188i \(-0.701527\pi\)
−0.591659 + 0.806188i \(0.701527\pi\)
\(312\) 0 0
\(313\) 10847.0i 1.95881i −0.201916 0.979403i \(-0.564717\pi\)
0.201916 0.979403i \(-0.435283\pi\)
\(314\) −7417.10 −1.33303
\(315\) 0 0
\(316\) 0 0
\(317\) 1780.49i 0.315465i 0.987482 + 0.157733i \(0.0504184\pi\)
−0.987482 + 0.157733i \(0.949582\pi\)
\(318\) 0 0
\(319\) −1958.00 −0.343658
\(320\) −10788.5 −1.88468
\(321\) 0 0
\(322\) −2420.00 3277.93i −0.418824 0.567303i
\(323\) 5651.20i 0.973502i
\(324\) 0 0
\(325\) 9505.99i 1.62245i
\(326\) 3778.78i 0.641985i
\(327\) 0 0
\(328\) 2383.95i 0.401315i
\(329\) −463.569 627.911i −0.0776820 0.105221i
\(330\) 0 0
\(331\) −9526.00 −1.58186 −0.790931 0.611905i \(-0.790403\pi\)
−0.790931 + 0.611905i \(0.790403\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 6555.85i 1.07401i
\(335\) 6237.11 1.01722
\(336\) 0 0
\(337\) −8272.00 −1.33711 −0.668553 0.743665i \(-0.733086\pi\)
−0.668553 + 0.743665i \(0.733086\pi\)
\(338\) 3702.41i 0.595812i
\(339\) 0 0
\(340\) 0 0
\(341\) 3708.55 0.588942
\(342\) 0 0
\(343\) 5995.00 2100.85i 0.943731 0.330715i
\(344\) 4299.21i 0.673831i
\(345\) 0 0
\(346\) 655.585i 0.101863i
\(347\) 9784.94i 1.51378i −0.653540 0.756892i \(-0.726717\pi\)
0.653540 0.756892i \(-0.273283\pi\)
\(348\) 0 0
\(349\) 6317.46i 0.968956i 0.874803 + 0.484478i \(0.160990\pi\)
−0.874803 + 0.484478i \(0.839010\pi\)
\(350\) 13443.5 9924.95i 2.05310 1.51575i
\(351\) 0 0
\(352\) 0 0
\(353\) −5457.47 −0.822866 −0.411433 0.911440i \(-0.634972\pi\)
−0.411433 + 0.911440i \(0.634972\pi\)
\(354\) 0 0
\(355\) 6943.24i 1.03805i
\(356\) 0 0
\(357\) 0 0
\(358\) −308.000 −0.0454701
\(359\) 10842.8i 1.59404i 0.603954 + 0.797019i \(0.293591\pi\)
−0.603954 + 0.797019i \(0.706409\pi\)
\(360\) 0 0
\(361\) −1133.00 −0.165184
\(362\) 7332.82 1.06465
\(363\) 0 0
\(364\) 0 0
\(365\) 16953.6i 2.43121i
\(366\) 0 0
\(367\) 3724.92i 0.529807i 0.964275 + 0.264903i \(0.0853400\pi\)
−0.964275 + 0.264903i \(0.914660\pi\)
\(368\) 4978.03i 0.705157i
\(369\) 0 0
\(370\) 10966.2i 1.54082i
\(371\) −5331.04 + 3935.76i −0.746021 + 0.550766i
\(372\) 0 0
\(373\) 4202.00 0.583301 0.291651 0.956525i \(-0.405796\pi\)
0.291651 + 0.956525i \(0.405796\pi\)
\(374\) 2781.41 0.384555
\(375\) 0 0
\(376\) 953.579i 0.130790i
\(377\) −3750.69 −0.512389
\(378\) 0 0
\(379\) −2506.00 −0.339643 −0.169821 0.985475i \(-0.554319\pi\)
−0.169821 + 0.985475i \(0.554319\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 6292.00 0.842740
\(383\) 13106.4 1.74857 0.874286 0.485410i \(-0.161330\pi\)
0.874286 + 0.485410i \(0.161330\pi\)
\(384\) 0 0
\(385\) 4884.00 3605.72i 0.646524 0.477310i
\(386\) 10578.3i 1.39488i
\(387\) 0 0
\(388\) 0 0
\(389\) 4631.55i 0.603673i 0.953360 + 0.301837i \(0.0975997\pi\)
−0.953360 + 0.301837i \(0.902400\pi\)
\(390\) 0 0
\(391\) 4916.89i 0.635953i
\(392\) −7417.10 2285.37i −0.955664 0.294461i
\(393\) 0 0
\(394\) 3388.00 0.433211
\(395\) 17615.6 2.24389
\(396\) 0 0
\(397\) 8045.82i 1.01715i 0.861018 + 0.508574i \(0.169827\pi\)
−0.861018 + 0.508574i \(0.830173\pi\)
\(398\) 2275.70 0.286610
\(399\) 0 0
\(400\) −20416.0 −2.55200
\(401\) 13552.4i 1.68772i −0.536565 0.843859i \(-0.680278\pi\)
0.536565 0.843859i \(-0.319722\pi\)
\(402\) 0 0
\(403\) 7104.00 0.878103
\(404\) 0 0
\(405\) 0 0
\(406\) 3916.00 + 5304.28i 0.478689 + 0.648392i
\(407\) 2862.37i 0.348605i
\(408\) 0 0
\(409\) 10161.6i 1.22850i −0.789111 0.614251i \(-0.789458\pi\)
0.789111 0.614251i \(-0.210542\pi\)
\(410\) 6279.11i 0.756349i
\(411\) 0 0
\(412\) 0 0
\(413\) −927.138 1255.82i −0.110464 0.149625i
\(414\) 0 0
\(415\) −25752.0 −3.04606
\(416\) 0 0
\(417\) 0 0
\(418\) 3933.51i 0.460274i
\(419\) 8934.23 1.04168 0.520842 0.853653i \(-0.325618\pi\)
0.520842 + 0.853653i \(0.325618\pi\)
\(420\) 0 0
\(421\) 5606.00 0.648978 0.324489 0.945889i \(-0.394808\pi\)
0.324489 + 0.945889i \(0.394808\pi\)
\(422\) 10154.1i 1.17131i
\(423\) 0 0
\(424\) 8096.00 0.927303
\(425\) −20165.2 −2.30155
\(426\) 0 0
\(427\) −9768.00 + 7211.44i −1.10704 + 0.817297i
\(428\) 0 0
\(429\) 0 0
\(430\) 11323.7i 1.26995i
\(431\) 883.883i 0.0987823i −0.998780 0.0493911i \(-0.984272\pi\)
0.998780 0.0493911i \(-0.0157281\pi\)
\(432\) 0 0
\(433\) 1966.76i 0.218282i −0.994026 0.109141i \(-0.965190\pi\)
0.994026 0.109141i \(-0.0348101\pi\)
\(434\) −7417.10 10046.6i −0.820351 1.11118i
\(435\) 0 0
\(436\) 0 0
\(437\) −6953.53 −0.761173
\(438\) 0 0
\(439\) 387.391i 0.0421166i −0.999778 0.0210583i \(-0.993296\pi\)
0.999778 0.0210583i \(-0.00670356\pi\)
\(440\) −7417.10 −0.803628
\(441\) 0 0
\(442\) 5328.00 0.573365
\(443\) 2623.37i 0.281354i −0.990056 0.140677i \(-0.955072\pi\)
0.990056 0.140677i \(-0.0449279\pi\)
\(444\) 0 0
\(445\) 14652.0 1.56083
\(446\) −8512.81 −0.903796
\(447\) 0 0
\(448\) 5632.00 + 7628.63i 0.593944 + 0.804506i
\(449\) 5140.67i 0.540319i 0.962816 + 0.270159i \(0.0870764\pi\)
−0.962816 + 0.270159i \(0.912924\pi\)
\(450\) 0 0
\(451\) 1638.96i 0.171121i
\(452\) 0 0
\(453\) 0 0
\(454\) 10489.4i 1.08434i
\(455\) 9355.66 6907.02i 0.963956 0.711662i
\(456\) 0 0
\(457\) 3608.00 0.369311 0.184655 0.982803i \(-0.440883\pi\)
0.184655 + 0.982803i \(0.440883\pi\)
\(458\) 927.138 0.0945902
\(459\) 0 0
\(460\) 0 0
\(461\) −1538.21 −0.155404 −0.0777021 0.996977i \(-0.524758\pi\)
−0.0777021 + 0.996977i \(0.524758\pi\)
\(462\) 0 0
\(463\) 1772.00 0.177866 0.0889329 0.996038i \(-0.471654\pi\)
0.0889329 + 0.996038i \(0.471654\pi\)
\(464\) 8055.36i 0.805950i
\(465\) 0 0
\(466\) −1820.00 −0.180922
\(467\) −12769.2 −1.26529 −0.632643 0.774443i \(-0.718030\pi\)
−0.632643 + 0.774443i \(0.718030\pi\)
\(468\) 0 0
\(469\) −3256.00 4410.30i −0.320572 0.434219i
\(470\) 2511.64i 0.246497i
\(471\) 0 0
\(472\) 1907.16i 0.185983i
\(473\) 2955.71i 0.287322i
\(474\) 0 0
\(475\) 28518.0i 2.75472i
\(476\) 0 0
\(477\) 0 0
\(478\) 1828.00 0.174918
\(479\) −4635.69 −0.442192 −0.221096 0.975252i \(-0.570963\pi\)
−0.221096 + 0.975252i \(0.570963\pi\)
\(480\) 0 0
\(481\) 5483.08i 0.519765i
\(482\) 1095.71 0.103544
\(483\) 0 0
\(484\) 0 0
\(485\) 11930.3i 1.11696i
\(486\) 0 0
\(487\) 1958.00 0.182188 0.0910939 0.995842i \(-0.470964\pi\)
0.0910939 + 0.995842i \(0.470964\pi\)
\(488\) 14834.2 1.37605
\(489\) 0 0
\(490\) −19536.0 6019.46i −1.80112 0.554963i
\(491\) 7126.22i 0.654994i 0.944852 + 0.327497i \(0.106205\pi\)
−0.944852 + 0.327497i \(0.893795\pi\)
\(492\) 0 0
\(493\) 7956.42i 0.726854i
\(494\) 7534.93i 0.686260i
\(495\) 0 0
\(496\) 15257.3i 1.38119i
\(497\) 4909.61 3624.63i 0.443111 0.327137i
\(498\) 0 0
\(499\) 10310.0 0.924928 0.462464 0.886638i \(-0.346965\pi\)
0.462464 + 0.886638i \(0.346965\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 17522.0i 1.55786i
\(503\) −126.428 −0.0112070 −0.00560352 0.999984i \(-0.501784\pi\)
−0.00560352 + 0.999984i \(0.501784\pi\)
\(504\) 0 0
\(505\) 15540.0 1.36935
\(506\) 3422.40i 0.300680i
\(507\) 0 0
\(508\) 0 0
\(509\) 7101.03 0.618365 0.309182 0.951003i \(-0.399945\pi\)
0.309182 + 0.951003i \(0.399945\pi\)
\(510\) 0 0
\(511\) 11988.0 8850.40i 1.03780 0.766181i
\(512\) 11585.2i 1.00000i
\(513\) 0 0
\(514\) 10072.2i 0.864328i
\(515\) 13814.0i 1.18198i
\(516\) 0 0
\(517\) 655.585i 0.0557691i
\(518\) −7754.24 + 5724.74i −0.657725 + 0.485580i
\(519\) 0 0
\(520\) −14208.0 −1.19820
\(521\) −4488.19 −0.377411 −0.188705 0.982034i \(-0.560429\pi\)
−0.188705 + 0.982034i \(0.560429\pi\)
\(522\) 0 0
\(523\) 9297.39i 0.777336i 0.921378 + 0.388668i \(0.127065\pi\)
−0.921378 + 0.388668i \(0.872935\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −1700.00 −0.140919
\(527\) 15069.9i 1.24564i
\(528\) 0 0
\(529\) 6117.00 0.502753
\(530\) 21324.2 1.74767
\(531\) 0 0
\(532\) 0 0
\(533\) 3139.55i 0.255139i
\(534\) 0 0
\(535\) 41808.5i 3.37857i
\(536\) 6697.72i 0.539734i
\(537\) 0 0
\(538\) 13767.3i 1.10325i
\(539\) −5099.26 1571.19i −0.407496 0.125558i
\(540\) 0 0
\(541\) −15646.0 −1.24339 −0.621695 0.783259i \(-0.713556\pi\)
−0.621695 + 0.783259i \(0.713556\pi\)
\(542\) 5057.11 0.400778
\(543\) 0 0
\(544\) 0 0
\(545\) −17784.2 −1.39778
\(546\) 0 0
\(547\) 1880.00 0.146952 0.0734762 0.997297i \(-0.476591\pi\)
0.0734762 + 0.997297i \(0.476591\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) −14036.0 −1.08818
\(551\) 11252.1 0.869972
\(552\) 0 0
\(553\) −9196.00 12456.1i −0.707150 0.957845i
\(554\) 3184.81i 0.244241i
\(555\) 0 0
\(556\) 0 0
\(557\) 7001.77i 0.532629i −0.963886 0.266315i \(-0.914194\pi\)
0.963886 0.266315i \(-0.0858060\pi\)
\(558\) 0 0
\(559\) 5661.87i 0.428393i
\(560\) 14834.2 + 20093.1i 1.11939 + 1.51623i
\(561\) 0 0
\(562\) 14356.0 1.07753
\(563\) −8386.38 −0.627786 −0.313893 0.949458i \(-0.601633\pi\)
−0.313893 + 0.949458i \(0.601633\pi\)
\(564\) 0 0
\(565\) 12128.3i 0.903084i
\(566\) 20144.2 1.49598
\(567\) 0 0
\(568\) −7456.00 −0.550787
\(569\) 17905.4i 1.31921i −0.751612 0.659606i \(-0.770723\pi\)
0.751612 0.659606i \(-0.229277\pi\)
\(570\) 0 0
\(571\) −24736.0 −1.81291 −0.906453 0.422307i \(-0.861221\pi\)
−0.906453 + 0.422307i \(0.861221\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −4440.00 + 3277.93i −0.322861 + 0.238359i
\(575\) 24812.4i 1.79956i
\(576\) 0 0
\(577\) 17343.2i 1.25131i −0.780099 0.625656i \(-0.784831\pi\)
0.780099 0.625656i \(-0.215169\pi\)
\(578\) 2593.67i 0.186648i
\(579\) 0 0
\(580\) 0 0
\(581\) 13443.5 + 18209.4i 0.959949 + 1.30026i
\(582\) 0 0
\(583\) 5566.00 0.395403
\(584\) −18205.6 −1.28999
\(585\) 0 0
\(586\) 655.585i 0.0462150i
\(587\) 3329.27 0.234095 0.117047 0.993126i \(-0.462657\pi\)
0.117047 + 0.993126i \(0.462657\pi\)
\(588\) 0 0
\(589\) −21312.0 −1.49091
\(590\) 5023.29i 0.350518i
\(591\) 0 0
\(592\) 11776.0 0.817552
\(593\) −22567.4 −1.56278 −0.781392 0.624041i \(-0.785490\pi\)
−0.781392 + 0.624041i \(0.785490\pi\)
\(594\) 0 0
\(595\) 14652.0 + 19846.4i 1.00954 + 1.36743i
\(596\) 0 0
\(597\) 0 0
\(598\) 6555.85i 0.448309i
\(599\) 25216.8i 1.72009i 0.510221 + 0.860044i \(0.329564\pi\)
−0.510221 + 0.860044i \(0.670436\pi\)
\(600\) 0 0
\(601\) 13290.5i 0.902048i 0.892512 + 0.451024i \(0.148941\pi\)
−0.892512 + 0.451024i \(0.851059\pi\)
\(602\) −8007.10 + 5911.41i −0.542101 + 0.400218i
\(603\) 0 0
\(604\) 0 0
\(605\) 22946.7 1.54201
\(606\) 0 0
\(607\) 18386.2i 1.22944i −0.788744 0.614722i \(-0.789268\pi\)
0.788744 0.614722i \(-0.210732\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 39072.0 2.59341
\(611\) 1255.82i 0.0831507i
\(612\) 0 0
\(613\) −26554.0 −1.74960 −0.874801 0.484483i \(-0.839008\pi\)
−0.874801 + 0.484483i \(0.839008\pi\)
\(614\) −25032.7 −1.64534
\(615\) 0 0
\(616\) 3872.00 + 5244.68i 0.253259 + 0.343043i
\(617\) 26475.5i 1.72749i −0.503927 0.863746i \(-0.668112\pi\)
0.503927 0.863746i \(-0.331888\pi\)
\(618\) 0 0
\(619\) 16389.6i 1.06422i −0.846674 0.532112i \(-0.821398\pi\)
0.846674 0.532112i \(-0.178602\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 18356.4i 1.18332i
\(623\) −7648.88 10360.5i −0.491888 0.666269i
\(624\) 0 0
\(625\) 46261.0 2.96070
\(626\) 30679.8 1.95881
\(627\) 0 0
\(628\) 0 0
\(629\) 11631.4 0.737318
\(630\) 0 0
\(631\) 24860.0 1.56840 0.784200 0.620508i \(-0.213073\pi\)
0.784200 + 0.620508i \(0.213073\pi\)
\(632\) 18916.5i 1.19060i
\(633\) 0 0
\(634\) −5036.00 −0.315465
\(635\) −4635.69 −0.289703
\(636\) 0 0
\(637\) −9768.00 3009.73i −0.607570 0.187206i
\(638\) 5538.06i 0.343658i
\(639\) 0 0
\(640\) 30514.5i 1.88468i
\(641\) 5279.26i 0.325301i −0.986684 0.162651i \(-0.947996\pi\)
0.986684 0.162651i \(-0.0520044\pi\)
\(642\) 0 0
\(643\) 2652.14i 0.162660i −0.996687 0.0813299i \(-0.974083\pi\)
0.996687 0.0813299i \(-0.0259167\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −15984.0 −0.973502
\(647\) −23009.9 −1.39816 −0.699081 0.715042i \(-0.746407\pi\)
−0.699081 + 0.715042i \(0.746407\pi\)
\(648\) 0 0
\(649\) 1311.17i 0.0793035i
\(650\) −26887.0 −1.62245
\(651\) 0 0
\(652\) 0 0
\(653\) 5864.74i 0.351463i −0.984438 0.175731i \(-0.943771\pi\)
0.984438 0.175731i \(-0.0562290\pi\)
\(654\) 0 0
\(655\) −27528.0 −1.64215
\(656\) 6742.82 0.401315
\(657\) 0 0
\(658\) 1776.00 1311.17i 0.105221 0.0776820i
\(659\) 2759.13i 0.163096i −0.996669 0.0815482i \(-0.974014\pi\)
0.996669 0.0815482i \(-0.0259864\pi\)
\(660\) 0 0
\(661\) 20978.7i 1.23446i −0.786783 0.617230i \(-0.788255\pi\)
0.786783 0.617230i \(-0.211745\pi\)
\(662\) 26943.6i 1.58186i
\(663\) 0 0
\(664\) 27653.8i 1.61623i
\(665\) −28067.0 + 20721.1i −1.63668 + 1.20831i
\(666\) 0 0
\(667\) 9790.00 0.568321
\(668\) 0 0
\(669\) 0 0
\(670\) 17641.2i 1.01722i
\(671\) 10198.5 0.586750
\(672\) 0 0
\(673\) −13636.0 −0.781024 −0.390512 0.920598i \(-0.627702\pi\)
−0.390512 + 0.920598i \(0.627702\pi\)
\(674\) 23396.7i 1.33711i
\(675\) 0 0
\(676\) 0 0
\(677\) 9966.73 0.565809 0.282904 0.959148i \(-0.408702\pi\)
0.282904 + 0.959148i \(0.408702\pi\)
\(678\) 0 0
\(679\) −8436.00 + 6228.06i −0.476795 + 0.352004i
\(680\) 30139.7i 1.69971i
\(681\) 0 0
\(682\) 10489.4i 0.588942i
\(683\) 202.233i 0.0113297i 0.999984 + 0.00566487i \(0.00180319\pi\)
−0.999984 + 0.00566487i \(0.998197\pi\)
\(684\) 0 0
\(685\) 33047.5i 1.84333i
\(686\) 5942.11 + 16956.4i 0.330715 + 0.943731i
\(687\) 0 0
\(688\) 12160.0 0.673831
\(689\) 10662.1 0.589540
\(690\) 0 0
\(691\) 20084.7i 1.10573i 0.833271 + 0.552865i \(0.186466\pi\)
−0.833271 + 0.552865i \(0.813534\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 27676.0 1.51378
\(695\) 13814.0i 0.753952i
\(696\) 0 0
\(697\) 6660.00 0.361930
\(698\) −17868.5 −0.968956
\(699\) 0 0
\(700\) 0 0
\(701\) 22883.4i 1.23294i 0.787377 + 0.616472i \(0.211439\pi\)
−0.787377 + 0.616472i \(0.788561\pi\)
\(702\) 0 0
\(703\) 16449.2i 0.882496i
\(704\) 7964.85i 0.426401i
\(705\) 0 0
\(706\) 15436.1i 0.822866i
\(707\) −8112.45 10988.4i −0.431542 0.584530i
\(708\) 0 0
\(709\) 12212.0 0.646871 0.323435 0.946250i \(-0.395162\pi\)
0.323435 + 0.946250i \(0.395162\pi\)
\(710\) −19638.5 −1.03805
\(711\) 0 0
\(712\) 15734.0i 0.828172i
\(713\) −18542.8 −0.973957
\(714\) 0 0
\(715\) −9768.00 −0.510913
\(716\) 0 0
\(717\) 0 0
\(718\) −30668.0 −1.59404
\(719\) 28825.5 1.49515 0.747574 0.664178i \(-0.231218\pi\)
0.747574 + 0.664178i \(0.231218\pi\)
\(720\) 0 0
\(721\) 9768.00 7211.44i 0.504548 0.372494i
\(722\) 3204.61i 0.165184i
\(723\) 0 0
\(724\) 0 0
\(725\) 40150.9i 2.05678i
\(726\) 0 0
\(727\) 3277.93i 0.167224i 0.996498 + 0.0836118i \(0.0266456\pi\)
−0.996498 + 0.0836118i \(0.973354\pi\)
\(728\) 7417.10 + 10046.6i 0.377604 + 0.511471i
\(729\) 0 0
\(730\) −47952.0 −2.43121
\(731\) 12010.6 0.607701
\(732\) 0 0
\(733\) 12128.3i 0.611146i −0.952169 0.305573i \(-0.901152\pi\)
0.952169 0.305573i \(-0.0988480\pi\)
\(734\) −10535.7 −0.529807
\(735\) 0 0
\(736\) 0 0
\(737\) 4604.68i 0.230143i
\(738\) 0 0
\(739\) 1760.00 0.0876085 0.0438042 0.999040i \(-0.486052\pi\)
0.0438042 + 0.999040i \(0.486052\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −11132.0 15078.5i −0.550766 0.746021i
\(743\) 22436.5i 1.10783i −0.832574 0.553913i \(-0.813134\pi\)
0.832574 0.553913i \(-0.186866\pi\)
\(744\) 0 0
\(745\) 30901.9i 1.51968i
\(746\) 11885.1i 0.583301i
\(747\) 0 0
\(748\) 0 0
\(749\) 29563.0 21825.6i 1.44220 1.06474i
\(750\) 0 0
\(751\) −5122.00 −0.248874 −0.124437 0.992228i \(-0.539712\pi\)
−0.124437 + 0.992228i \(0.539712\pi\)
\(752\) −2697.13 −0.130790
\(753\) 0 0
\(754\) 10608.6i 0.512389i
\(755\) 41510.5 2.00095
\(756\) 0 0
\(757\) −18772.0 −0.901295 −0.450647 0.892702i \(-0.648807\pi\)
−0.450647 + 0.892702i \(0.648807\pi\)
\(758\) 7088.04i 0.339643i
\(759\) 0 0
\(760\) 42624.0 2.03439
\(761\) 28973.0 1.38012 0.690061 0.723752i \(-0.257584\pi\)
0.690061 + 0.723752i \(0.257584\pi\)
\(762\) 0 0
\(763\) 9284.00 + 12575.3i 0.440502 + 0.596667i
\(764\) 0 0
\(765\) 0 0
\(766\) 37070.4i 1.74857i
\(767\) 2511.64i 0.118240i
\(768\) 0 0
\(769\) 11740.9i 0.550571i 0.961363 + 0.275285i \(0.0887724\pi\)
−0.961363 + 0.275285i \(0.911228\pi\)
\(770\) 10198.5 + 13814.0i 0.477310 + 0.646524i
\(771\) 0 0
\(772\) 0 0
\(773\) 12706.0 0.591207 0.295603 0.955311i \(-0.404479\pi\)
0.295603 + 0.955311i \(0.404479\pi\)
\(774\) 0 0
\(775\) 76047.9i 3.52480i
\(776\) 12811.4 0.592656
\(777\) 0 0
\(778\) −13100.0 −0.603673
\(779\) 9418.66i 0.433195i
\(780\) 0 0
\(781\) −5126.00 −0.234856
\(782\) −13907.1 −0.635953
\(783\) 0 0
\(784\) 6464.00 20978.7i 0.294461 0.955664i
\(785\) 55256.2i 2.51233i
\(786\) 0 0
\(787\) 26342.6i 1.19315i −0.802556 0.596577i \(-0.796527\pi\)
0.802556 0.596577i \(-0.203473\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 49824.5i 2.24389i
\(791\) 8576.02 6331.43i 0.385497 0.284602i
\(792\) 0 0
\(793\) 19536.0 0.874834
\(794\) −22757.0 −1.01715
\(795\) 0 0
\(796\) 0 0
\(797\) −19448.8 −0.864382 −0.432191 0.901782i \(-0.642259\pi\)