Properties

Label 189.2.c.c.188.2
Level $189$
Weight $2$
Character 189.188
Analytic conductor $1.509$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 189 = 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 189.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.50917259820\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{3})\)
Defining polynomial: \(x^{4} + 4 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 188.2
Root \(-1.93185i\) of defining polynomial
Character \(\chi\) \(=\) 189.188
Dual form 189.2.c.c.188.4

$q$-expansion

\(f(q)\) \(=\) \(q-1.41421i q^{2} +1.73205 q^{5} +(1.00000 + 2.44949i) q^{7} -2.82843i q^{8} +O(q^{10})\) \(q-1.41421i q^{2} +1.73205 q^{5} +(1.00000 + 2.44949i) q^{7} -2.82843i q^{8} -2.44949i q^{10} -1.41421i q^{11} +2.44949i q^{13} +(3.46410 - 1.41421i) q^{14} -4.00000 q^{16} -5.19615 q^{17} -7.34847i q^{19} -2.00000 q^{22} +2.82843i q^{23} -2.00000 q^{25} +3.46410 q^{26} +7.07107i q^{29} +2.44949i q^{31} +7.34847i q^{34} +(1.73205 + 4.24264i) q^{35} +5.00000 q^{37} -10.3923 q^{38} -4.89898i q^{40} -8.66025 q^{41} +5.00000 q^{43} +4.00000 q^{46} +8.66025 q^{47} +(-5.00000 + 4.89898i) q^{49} +2.82843i q^{50} +11.3137i q^{53} -2.44949i q^{55} +(6.92820 - 2.82843i) q^{56} +10.0000 q^{58} -8.66025 q^{59} -2.44949i q^{61} +3.46410 q^{62} -8.00000 q^{64} +4.24264i q^{65} +2.00000 q^{67} +(6.00000 - 2.44949i) q^{70} -14.1421i q^{71} -7.07107i q^{74} +(3.46410 - 1.41421i) q^{77} -13.0000 q^{79} -6.92820 q^{80} +12.2474i q^{82} -1.73205 q^{83} -9.00000 q^{85} -7.07107i q^{86} -4.00000 q^{88} +10.3923 q^{89} +(-6.00000 + 2.44949i) q^{91} -12.2474i q^{94} -12.7279i q^{95} -17.1464i q^{97} +(6.92820 + 7.07107i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{7} + O(q^{10}) \) \( 4q + 4q^{7} - 16q^{16} - 8q^{22} - 8q^{25} + 20q^{37} + 20q^{43} + 16q^{46} - 20q^{49} + 40q^{58} - 32q^{64} + 8q^{67} + 24q^{70} - 52q^{79} - 36q^{85} - 16q^{88} - 24q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/189\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(136\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(3\) 0 0
\(4\) 0 0
\(5\) 1.73205 0.774597 0.387298 0.921954i \(-0.373408\pi\)
0.387298 + 0.921954i \(0.373408\pi\)
\(6\) 0 0
\(7\) 1.00000 + 2.44949i 0.377964 + 0.925820i
\(8\) 2.82843i 1.00000i
\(9\) 0 0
\(10\) 2.44949i 0.774597i
\(11\) 1.41421i 0.426401i −0.977008 0.213201i \(-0.931611\pi\)
0.977008 0.213201i \(-0.0683888\pi\)
\(12\) 0 0
\(13\) 2.44949i 0.679366i 0.940540 + 0.339683i \(0.110320\pi\)
−0.940540 + 0.339683i \(0.889680\pi\)
\(14\) 3.46410 1.41421i 0.925820 0.377964i
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) −5.19615 −1.26025 −0.630126 0.776493i \(-0.716997\pi\)
−0.630126 + 0.776493i \(0.716997\pi\)
\(18\) 0 0
\(19\) 7.34847i 1.68585i −0.538028 0.842927i \(-0.680830\pi\)
0.538028 0.842927i \(-0.319170\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −2.00000 −0.426401
\(23\) 2.82843i 0.589768i 0.955533 + 0.294884i \(0.0952810\pi\)
−0.955533 + 0.294884i \(0.904719\pi\)
\(24\) 0 0
\(25\) −2.00000 −0.400000
\(26\) 3.46410 0.679366
\(27\) 0 0
\(28\) 0 0
\(29\) 7.07107i 1.31306i 0.754298 + 0.656532i \(0.227977\pi\)
−0.754298 + 0.656532i \(0.772023\pi\)
\(30\) 0 0
\(31\) 2.44949i 0.439941i 0.975506 + 0.219971i \(0.0705962\pi\)
−0.975506 + 0.219971i \(0.929404\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 7.34847i 1.26025i
\(35\) 1.73205 + 4.24264i 0.292770 + 0.717137i
\(36\) 0 0
\(37\) 5.00000 0.821995 0.410997 0.911636i \(-0.365181\pi\)
0.410997 + 0.911636i \(0.365181\pi\)
\(38\) −10.3923 −1.68585
\(39\) 0 0
\(40\) 4.89898i 0.774597i
\(41\) −8.66025 −1.35250 −0.676252 0.736670i \(-0.736397\pi\)
−0.676252 + 0.736670i \(0.736397\pi\)
\(42\) 0 0
\(43\) 5.00000 0.762493 0.381246 0.924473i \(-0.375495\pi\)
0.381246 + 0.924473i \(0.375495\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) 8.66025 1.26323 0.631614 0.775283i \(-0.282393\pi\)
0.631614 + 0.775283i \(0.282393\pi\)
\(48\) 0 0
\(49\) −5.00000 + 4.89898i −0.714286 + 0.699854i
\(50\) 2.82843i 0.400000i
\(51\) 0 0
\(52\) 0 0
\(53\) 11.3137i 1.55406i 0.629465 + 0.777029i \(0.283274\pi\)
−0.629465 + 0.777029i \(0.716726\pi\)
\(54\) 0 0
\(55\) 2.44949i 0.330289i
\(56\) 6.92820 2.82843i 0.925820 0.377964i
\(57\) 0 0
\(58\) 10.0000 1.31306
\(59\) −8.66025 −1.12747 −0.563735 0.825956i \(-0.690636\pi\)
−0.563735 + 0.825956i \(0.690636\pi\)
\(60\) 0 0
\(61\) 2.44949i 0.313625i −0.987628 0.156813i \(-0.949878\pi\)
0.987628 0.156813i \(-0.0501218\pi\)
\(62\) 3.46410 0.439941
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 4.24264i 0.526235i
\(66\) 0 0
\(67\) 2.00000 0.244339 0.122169 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 6.00000 2.44949i 0.717137 0.292770i
\(71\) 14.1421i 1.67836i −0.543852 0.839181i \(-0.683035\pi\)
0.543852 0.839181i \(-0.316965\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 7.07107i 0.821995i
\(75\) 0 0
\(76\) 0 0
\(77\) 3.46410 1.41421i 0.394771 0.161165i
\(78\) 0 0
\(79\) −13.0000 −1.46261 −0.731307 0.682048i \(-0.761089\pi\)
−0.731307 + 0.682048i \(0.761089\pi\)
\(80\) −6.92820 −0.774597
\(81\) 0 0
\(82\) 12.2474i 1.35250i
\(83\) −1.73205 −0.190117 −0.0950586 0.995472i \(-0.530304\pi\)
−0.0950586 + 0.995472i \(0.530304\pi\)
\(84\) 0 0
\(85\) −9.00000 −0.976187
\(86\) 7.07107i 0.762493i
\(87\) 0 0
\(88\) −4.00000 −0.426401
\(89\) 10.3923 1.10158 0.550791 0.834643i \(-0.314326\pi\)
0.550791 + 0.834643i \(0.314326\pi\)
\(90\) 0 0
\(91\) −6.00000 + 2.44949i −0.628971 + 0.256776i
\(92\) 0 0
\(93\) 0 0
\(94\) 12.2474i 1.26323i
\(95\) 12.7279i 1.30586i
\(96\) 0 0
\(97\) 17.1464i 1.74096i −0.492207 0.870478i \(-0.663810\pi\)
0.492207 0.870478i \(-0.336190\pi\)
\(98\) 6.92820 + 7.07107i 0.699854 + 0.714286i
\(99\) 0 0
\(100\) 0 0
\(101\) 3.46410 0.344691 0.172345 0.985037i \(-0.444865\pi\)
0.172345 + 0.985037i \(0.444865\pi\)
\(102\) 0 0
\(103\) 9.79796i 0.965422i 0.875780 + 0.482711i \(0.160348\pi\)
−0.875780 + 0.482711i \(0.839652\pi\)
\(104\) 6.92820 0.679366
\(105\) 0 0
\(106\) 16.0000 1.55406
\(107\) 14.1421i 1.36717i −0.729870 0.683586i \(-0.760419\pi\)
0.729870 0.683586i \(-0.239581\pi\)
\(108\) 0 0
\(109\) −7.00000 −0.670478 −0.335239 0.942133i \(-0.608817\pi\)
−0.335239 + 0.942133i \(0.608817\pi\)
\(110\) −3.46410 −0.330289
\(111\) 0 0
\(112\) −4.00000 9.79796i −0.377964 0.925820i
\(113\) 7.07107i 0.665190i 0.943070 + 0.332595i \(0.107924\pi\)
−0.943070 + 0.332595i \(0.892076\pi\)
\(114\) 0 0
\(115\) 4.89898i 0.456832i
\(116\) 0 0
\(117\) 0 0
\(118\) 12.2474i 1.12747i
\(119\) −5.19615 12.7279i −0.476331 1.16677i
\(120\) 0 0
\(121\) 9.00000 0.818182
\(122\) −3.46410 −0.313625
\(123\) 0 0
\(124\) 0 0
\(125\) −12.1244 −1.08444
\(126\) 0 0
\(127\) 5.00000 0.443678 0.221839 0.975083i \(-0.428794\pi\)
0.221839 + 0.975083i \(0.428794\pi\)
\(128\) 11.3137i 1.00000i
\(129\) 0 0
\(130\) 6.00000 0.526235
\(131\) 17.3205 1.51330 0.756650 0.653820i \(-0.226835\pi\)
0.756650 + 0.653820i \(0.226835\pi\)
\(132\) 0 0
\(133\) 18.0000 7.34847i 1.56080 0.637193i
\(134\) 2.82843i 0.244339i
\(135\) 0 0
\(136\) 14.6969i 1.26025i
\(137\) 1.41421i 0.120824i −0.998174 0.0604122i \(-0.980758\pi\)
0.998174 0.0604122i \(-0.0192415\pi\)
\(138\) 0 0
\(139\) 12.2474i 1.03882i −0.854527 0.519408i \(-0.826153\pi\)
0.854527 0.519408i \(-0.173847\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −20.0000 −1.67836
\(143\) 3.46410 0.289683
\(144\) 0 0
\(145\) 12.2474i 1.01710i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 7.07107i 0.579284i 0.957135 + 0.289642i \(0.0935363\pi\)
−0.957135 + 0.289642i \(0.906464\pi\)
\(150\) 0 0
\(151\) 5.00000 0.406894 0.203447 0.979086i \(-0.434786\pi\)
0.203447 + 0.979086i \(0.434786\pi\)
\(152\) −20.7846 −1.68585
\(153\) 0 0
\(154\) −2.00000 4.89898i −0.161165 0.394771i
\(155\) 4.24264i 0.340777i
\(156\) 0 0
\(157\) 17.1464i 1.36843i 0.729279 + 0.684217i \(0.239856\pi\)
−0.729279 + 0.684217i \(0.760144\pi\)
\(158\) 18.3848i 1.46261i
\(159\) 0 0
\(160\) 0 0
\(161\) −6.92820 + 2.82843i −0.546019 + 0.222911i
\(162\) 0 0
\(163\) 5.00000 0.391630 0.195815 0.980641i \(-0.437265\pi\)
0.195815 + 0.980641i \(0.437265\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 2.44949i 0.190117i
\(167\) 12.1244 0.938211 0.469105 0.883142i \(-0.344576\pi\)
0.469105 + 0.883142i \(0.344576\pi\)
\(168\) 0 0
\(169\) 7.00000 0.538462
\(170\) 12.7279i 0.976187i
\(171\) 0 0
\(172\) 0 0
\(173\) −6.92820 −0.526742 −0.263371 0.964695i \(-0.584834\pi\)
−0.263371 + 0.964695i \(0.584834\pi\)
\(174\) 0 0
\(175\) −2.00000 4.89898i −0.151186 0.370328i
\(176\) 5.65685i 0.426401i
\(177\) 0 0
\(178\) 14.6969i 1.10158i
\(179\) 7.07107i 0.528516i 0.964452 + 0.264258i \(0.0851271\pi\)
−0.964452 + 0.264258i \(0.914873\pi\)
\(180\) 0 0
\(181\) 14.6969i 1.09241i −0.837650 0.546207i \(-0.816071\pi\)
0.837650 0.546207i \(-0.183929\pi\)
\(182\) 3.46410 + 8.48528i 0.256776 + 0.628971i
\(183\) 0 0
\(184\) 8.00000 0.589768
\(185\) 8.66025 0.636715
\(186\) 0 0
\(187\) 7.34847i 0.537373i
\(188\) 0 0
\(189\) 0 0
\(190\) −18.0000 −1.30586
\(191\) 14.1421i 1.02329i −0.859197 0.511645i \(-0.829036\pi\)
0.859197 0.511645i \(-0.170964\pi\)
\(192\) 0 0
\(193\) −1.00000 −0.0719816 −0.0359908 0.999352i \(-0.511459\pi\)
−0.0359908 + 0.999352i \(0.511459\pi\)
\(194\) −24.2487 −1.74096
\(195\) 0 0
\(196\) 0 0
\(197\) 1.41421i 0.100759i −0.998730 0.0503793i \(-0.983957\pi\)
0.998730 0.0503793i \(-0.0160430\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 5.65685i 0.400000i
\(201\) 0 0
\(202\) 4.89898i 0.344691i
\(203\) −17.3205 + 7.07107i −1.21566 + 0.496292i
\(204\) 0 0
\(205\) −15.0000 −1.04765
\(206\) 13.8564 0.965422
\(207\) 0 0
\(208\) 9.79796i 0.679366i
\(209\) −10.3923 −0.718851
\(210\) 0 0
\(211\) −16.0000 −1.10149 −0.550743 0.834675i \(-0.685655\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) −20.0000 −1.36717
\(215\) 8.66025 0.590624
\(216\) 0 0
\(217\) −6.00000 + 2.44949i −0.407307 + 0.166282i
\(218\) 9.89949i 0.670478i
\(219\) 0 0
\(220\) 0 0
\(221\) 12.7279i 0.856173i
\(222\) 0 0
\(223\) 12.2474i 0.820150i 0.912052 + 0.410075i \(0.134497\pi\)
−0.912052 + 0.410075i \(0.865503\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 10.0000 0.665190
\(227\) 13.8564 0.919682 0.459841 0.888001i \(-0.347906\pi\)
0.459841 + 0.888001i \(0.347906\pi\)
\(228\) 0 0
\(229\) 4.89898i 0.323734i −0.986813 0.161867i \(-0.948248\pi\)
0.986813 0.161867i \(-0.0517515\pi\)
\(230\) 6.92820 0.456832
\(231\) 0 0
\(232\) 20.0000 1.31306
\(233\) 7.07107i 0.463241i 0.972806 + 0.231621i \(0.0744028\pi\)
−0.972806 + 0.231621i \(0.925597\pi\)
\(234\) 0 0
\(235\) 15.0000 0.978492
\(236\) 0 0
\(237\) 0 0
\(238\) −18.0000 + 7.34847i −1.16677 + 0.476331i
\(239\) 18.3848i 1.18921i −0.804017 0.594606i \(-0.797308\pi\)
0.804017 0.594606i \(-0.202692\pi\)
\(240\) 0 0
\(241\) 2.44949i 0.157786i −0.996883 0.0788928i \(-0.974862\pi\)
0.996883 0.0788928i \(-0.0251385\pi\)
\(242\) 12.7279i 0.818182i
\(243\) 0 0
\(244\) 0 0
\(245\) −8.66025 + 8.48528i −0.553283 + 0.542105i
\(246\) 0 0
\(247\) 18.0000 1.14531
\(248\) 6.92820 0.439941
\(249\) 0 0
\(250\) 17.1464i 1.08444i
\(251\) 5.19615 0.327978 0.163989 0.986462i \(-0.447564\pi\)
0.163989 + 0.986462i \(0.447564\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) 7.07107i 0.443678i
\(255\) 0 0
\(256\) 0 0
\(257\) −3.46410 −0.216085 −0.108042 0.994146i \(-0.534458\pi\)
−0.108042 + 0.994146i \(0.534458\pi\)
\(258\) 0 0
\(259\) 5.00000 + 12.2474i 0.310685 + 0.761019i
\(260\) 0 0
\(261\) 0 0
\(262\) 24.4949i 1.51330i
\(263\) 1.41421i 0.0872041i −0.999049 0.0436021i \(-0.986117\pi\)
0.999049 0.0436021i \(-0.0138834\pi\)
\(264\) 0 0
\(265\) 19.5959i 1.20377i
\(266\) −10.3923 25.4558i −0.637193 1.56080i
\(267\) 0 0
\(268\) 0 0
\(269\) 25.9808 1.58408 0.792038 0.610472i \(-0.209020\pi\)
0.792038 + 0.610472i \(0.209020\pi\)
\(270\) 0 0
\(271\) 14.6969i 0.892775i −0.894840 0.446388i \(-0.852710\pi\)
0.894840 0.446388i \(-0.147290\pi\)
\(272\) 20.7846 1.26025
\(273\) 0 0
\(274\) −2.00000 −0.120824
\(275\) 2.82843i 0.170561i
\(276\) 0 0
\(277\) 5.00000 0.300421 0.150210 0.988654i \(-0.452005\pi\)
0.150210 + 0.988654i \(0.452005\pi\)
\(278\) −17.3205 −1.03882
\(279\) 0 0
\(280\) 12.0000 4.89898i 0.717137 0.292770i
\(281\) 14.1421i 0.843649i −0.906677 0.421825i \(-0.861390\pi\)
0.906677 0.421825i \(-0.138610\pi\)
\(282\) 0 0
\(283\) 12.2474i 0.728035i −0.931392 0.364018i \(-0.881405\pi\)
0.931392 0.364018i \(-0.118595\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 4.89898i 0.289683i
\(287\) −8.66025 21.2132i −0.511199 1.25218i
\(288\) 0 0
\(289\) 10.0000 0.588235
\(290\) 17.3205 1.01710
\(291\) 0 0
\(292\) 0 0
\(293\) −19.0526 −1.11306 −0.556531 0.830827i \(-0.687868\pi\)
−0.556531 + 0.830827i \(0.687868\pi\)
\(294\) 0 0
\(295\) −15.0000 −0.873334
\(296\) 14.1421i 0.821995i
\(297\) 0 0
\(298\) 10.0000 0.579284
\(299\) −6.92820 −0.400668
\(300\) 0 0
\(301\) 5.00000 + 12.2474i 0.288195 + 0.705931i
\(302\) 7.07107i 0.406894i
\(303\) 0 0
\(304\) 29.3939i 1.68585i
\(305\) 4.24264i 0.242933i
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 6.00000 0.340777
\(311\) 12.1244 0.687509 0.343755 0.939060i \(-0.388301\pi\)
0.343755 + 0.939060i \(0.388301\pi\)
\(312\) 0 0
\(313\) 24.4949i 1.38453i −0.721642 0.692267i \(-0.756612\pi\)
0.721642 0.692267i \(-0.243388\pi\)
\(314\) 24.2487 1.36843
\(315\) 0 0
\(316\) 0 0
\(317\) 5.65685i 0.317721i −0.987301 0.158860i \(-0.949218\pi\)
0.987301 0.158860i \(-0.0507819\pi\)
\(318\) 0 0
\(319\) 10.0000 0.559893
\(320\) −13.8564 −0.774597
\(321\) 0 0
\(322\) 4.00000 + 9.79796i 0.222911 + 0.546019i
\(323\) 38.1838i 2.12460i
\(324\) 0 0
\(325\) 4.89898i 0.271746i
\(326\) 7.07107i 0.391630i
\(327\) 0 0
\(328\) 24.4949i 1.35250i
\(329\) 8.66025 + 21.2132i 0.477455 + 1.16952i
\(330\) 0 0
\(331\) −25.0000 −1.37412 −0.687062 0.726599i \(-0.741100\pi\)
−0.687062 + 0.726599i \(0.741100\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 17.1464i 0.938211i
\(335\) 3.46410 0.189264
\(336\) 0 0
\(337\) −7.00000 −0.381314 −0.190657 0.981657i \(-0.561062\pi\)
−0.190657 + 0.981657i \(0.561062\pi\)
\(338\) 9.89949i 0.538462i
\(339\) 0 0
\(340\) 0 0
\(341\) 3.46410 0.187592
\(342\) 0 0
\(343\) −17.0000 7.34847i −0.917914 0.396780i
\(344\) 14.1421i 0.762493i
\(345\) 0 0
\(346\) 9.79796i 0.526742i
\(347\) 7.07107i 0.379595i 0.981823 + 0.189797i \(0.0607831\pi\)
−0.981823 + 0.189797i \(0.939217\pi\)
\(348\) 0 0
\(349\) 19.5959i 1.04895i 0.851427 + 0.524473i \(0.175738\pi\)
−0.851427 + 0.524473i \(0.824262\pi\)
\(350\) −6.92820 + 2.82843i −0.370328 + 0.151186i
\(351\) 0 0
\(352\) 0 0
\(353\) 8.66025 0.460939 0.230469 0.973080i \(-0.425974\pi\)
0.230469 + 0.973080i \(0.425974\pi\)
\(354\) 0 0
\(355\) 24.4949i 1.30005i
\(356\) 0 0
\(357\) 0 0
\(358\) 10.0000 0.528516
\(359\) 11.3137i 0.597115i 0.954392 + 0.298557i \(0.0965054\pi\)
−0.954392 + 0.298557i \(0.903495\pi\)
\(360\) 0 0
\(361\) −35.0000 −1.84211
\(362\) −20.7846 −1.09241
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 4.89898i 0.255725i 0.991792 + 0.127862i \(0.0408116\pi\)
−0.991792 + 0.127862i \(0.959188\pi\)
\(368\) 11.3137i 0.589768i
\(369\) 0 0
\(370\) 12.2474i 0.636715i
\(371\) −27.7128 + 11.3137i −1.43878 + 0.587378i
\(372\) 0 0
\(373\) −25.0000 −1.29445 −0.647225 0.762299i \(-0.724071\pi\)
−0.647225 + 0.762299i \(0.724071\pi\)
\(374\) 10.3923 0.537373
\(375\) 0 0
\(376\) 24.4949i 1.26323i
\(377\) −17.3205 −0.892052
\(378\) 0 0
\(379\) 23.0000 1.18143 0.590715 0.806880i \(-0.298846\pi\)
0.590715 + 0.806880i \(0.298846\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −20.0000 −1.02329
\(383\) 1.73205 0.0885037 0.0442518 0.999020i \(-0.485910\pi\)
0.0442518 + 0.999020i \(0.485910\pi\)
\(384\) 0 0
\(385\) 6.00000 2.44949i 0.305788 0.124838i
\(386\) 1.41421i 0.0719816i
\(387\) 0 0
\(388\) 0 0
\(389\) 32.5269i 1.64918i 0.565731 + 0.824590i \(0.308594\pi\)
−0.565731 + 0.824590i \(0.691406\pi\)
\(390\) 0 0
\(391\) 14.6969i 0.743256i
\(392\) 13.8564 + 14.1421i 0.699854 + 0.714286i
\(393\) 0 0
\(394\) −2.00000 −0.100759
\(395\) −22.5167 −1.13294
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 8.00000 0.400000
\(401\) 19.7990i 0.988714i 0.869259 + 0.494357i \(0.164597\pi\)
−0.869259 + 0.494357i \(0.835403\pi\)
\(402\) 0 0
\(403\) −6.00000 −0.298881
\(404\) 0 0
\(405\) 0 0
\(406\) 10.0000 + 24.4949i 0.496292 + 1.21566i
\(407\) 7.07107i 0.350500i
\(408\) 0 0
\(409\) 19.5959i 0.968956i −0.874804 0.484478i \(-0.839010\pi\)
0.874804 0.484478i \(-0.160990\pi\)
\(410\) 21.2132i 1.04765i
\(411\) 0 0
\(412\) 0 0
\(413\) −8.66025 21.2132i −0.426143 1.04383i
\(414\) 0 0
\(415\) −3.00000 −0.147264
\(416\) 0 0
\(417\) 0 0
\(418\) 14.6969i 0.718851i
\(419\) 1.73205 0.0846162 0.0423081 0.999105i \(-0.486529\pi\)
0.0423081 + 0.999105i \(0.486529\pi\)
\(420\) 0 0
\(421\) −4.00000 −0.194948 −0.0974740 0.995238i \(-0.531076\pi\)
−0.0974740 + 0.995238i \(0.531076\pi\)
\(422\) 22.6274i 1.10149i
\(423\) 0 0
\(424\) 32.0000 1.55406
\(425\) 10.3923 0.504101
\(426\) 0 0
\(427\) 6.00000 2.44949i 0.290360 0.118539i
\(428\) 0 0
\(429\) 0 0
\(430\) 12.2474i 0.590624i
\(431\) 5.65685i 0.272481i −0.990676 0.136241i \(-0.956498\pi\)
0.990676 0.136241i \(-0.0435020\pi\)
\(432\) 0 0
\(433\) 36.7423i 1.76572i 0.469632 + 0.882862i \(0.344387\pi\)
−0.469632 + 0.882862i \(0.655613\pi\)
\(434\) 3.46410 + 8.48528i 0.166282 + 0.407307i
\(435\) 0 0
\(436\) 0 0
\(437\) 20.7846 0.994263
\(438\) 0 0
\(439\) 19.5959i 0.935262i 0.883924 + 0.467631i \(0.154892\pi\)
−0.883924 + 0.467631i \(0.845108\pi\)
\(440\) −6.92820 −0.330289
\(441\) 0 0
\(442\) −18.0000 −0.856173
\(443\) 32.5269i 1.54540i 0.634771 + 0.772700i \(0.281094\pi\)
−0.634771 + 0.772700i \(0.718906\pi\)
\(444\) 0 0
\(445\) 18.0000 0.853282
\(446\) 17.3205 0.820150
\(447\) 0 0
\(448\) −8.00000 19.5959i −0.377964 0.925820i
\(449\) 24.0416i 1.13459i 0.823513 + 0.567297i \(0.192011\pi\)
−0.823513 + 0.567297i \(0.807989\pi\)
\(450\) 0 0
\(451\) 12.2474i 0.576710i
\(452\) 0 0
\(453\) 0 0
\(454\) 19.5959i 0.919682i
\(455\) −10.3923 + 4.24264i −0.487199 + 0.198898i
\(456\) 0 0
\(457\) 32.0000 1.49690 0.748448 0.663193i \(-0.230799\pi\)
0.748448 + 0.663193i \(0.230799\pi\)
\(458\) −6.92820 −0.323734
\(459\) 0 0
\(460\) 0 0
\(461\) −12.1244 −0.564688 −0.282344 0.959313i \(-0.591112\pi\)
−0.282344 + 0.959313i \(0.591112\pi\)
\(462\) 0 0
\(463\) 11.0000 0.511213 0.255607 0.966781i \(-0.417725\pi\)
0.255607 + 0.966781i \(0.417725\pi\)
\(464\) 28.2843i 1.31306i
\(465\) 0 0
\(466\) 10.0000 0.463241
\(467\) −20.7846 −0.961797 −0.480899 0.876776i \(-0.659689\pi\)
−0.480899 + 0.876776i \(0.659689\pi\)
\(468\) 0 0
\(469\) 2.00000 + 4.89898i 0.0923514 + 0.226214i
\(470\) 21.2132i 0.978492i
\(471\) 0 0
\(472\) 24.4949i 1.12747i
\(473\) 7.07107i 0.325128i
\(474\) 0 0
\(475\) 14.6969i 0.674342i
\(476\) 0 0
\(477\) 0 0
\(478\) −26.0000 −1.18921
\(479\) −32.9090 −1.50365 −0.751825 0.659363i \(-0.770826\pi\)
−0.751825 + 0.659363i \(0.770826\pi\)
\(480\) 0 0
\(481\) 12.2474i 0.558436i
\(482\) −3.46410 −0.157786
\(483\) 0 0
\(484\) 0 0
\(485\) 29.6985i 1.34854i
\(486\) 0 0
\(487\) −22.0000 −0.996915 −0.498458 0.866914i \(-0.666100\pi\)
−0.498458 + 0.866914i \(0.666100\pi\)
\(488\) −6.92820 −0.313625
\(489\) 0 0
\(490\) 12.0000 + 12.2474i 0.542105 + 0.553283i
\(491\) 15.5563i 0.702048i 0.936366 + 0.351024i \(0.114166\pi\)
−0.936366 + 0.351024i \(0.885834\pi\)
\(492\) 0 0
\(493\) 36.7423i 1.65479i
\(494\) 25.4558i 1.14531i
\(495\) 0 0
\(496\) 9.79796i 0.439941i
\(497\) 34.6410 14.1421i 1.55386 0.634361i
\(498\) 0 0
\(499\) 17.0000 0.761025 0.380512 0.924776i \(-0.375748\pi\)
0.380512 + 0.924776i \(0.375748\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 7.34847i 0.327978i
\(503\) 36.3731 1.62179 0.810897 0.585188i \(-0.198979\pi\)
0.810897 + 0.585188i \(0.198979\pi\)
\(504\) 0 0
\(505\) 6.00000 0.266996
\(506\) 5.65685i 0.251478i
\(507\) 0 0
\(508\) 0 0
\(509\) −8.66025 −0.383859 −0.191930 0.981409i \(-0.561474\pi\)
−0.191930 + 0.981409i \(0.561474\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 22.6274i 1.00000i
\(513\) 0 0
\(514\) 4.89898i 0.216085i
\(515\) 16.9706i 0.747812i
\(516\) 0 0
\(517\) 12.2474i 0.538642i
\(518\) 17.3205 7.07107i 0.761019 0.310685i
\(519\) 0 0
\(520\) 12.0000 0.526235
\(521\) 5.19615 0.227648 0.113824 0.993501i \(-0.463690\pi\)
0.113824 + 0.993501i \(0.463690\pi\)
\(522\) 0 0
\(523\) 36.7423i 1.60663i 0.595554 + 0.803315i \(0.296933\pi\)
−0.595554 + 0.803315i \(0.703067\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −2.00000 −0.0872041
\(527\) 12.7279i 0.554437i
\(528\) 0 0
\(529\) 15.0000 0.652174
\(530\) 27.7128 1.20377
\(531\) 0 0
\(532\) 0 0
\(533\) 21.2132i 0.918846i
\(534\) 0 0
\(535\) 24.4949i 1.05901i
\(536\) 5.65685i 0.244339i
\(537\) 0 0
\(538\) 36.7423i 1.58408i
\(539\) 6.92820 + 7.07107i 0.298419 + 0.304572i
\(540\) 0 0
\(541\) 5.00000 0.214967 0.107483 0.994207i \(-0.465721\pi\)
0.107483 + 0.994207i \(0.465721\pi\)
\(542\) −20.7846 −0.892775
\(543\) 0 0
\(544\) 0 0
\(545\) −12.1244 −0.519350
\(546\) 0 0
\(547\) −13.0000 −0.555840 −0.277920 0.960604i \(-0.589645\pi\)
−0.277920 + 0.960604i \(0.589645\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 4.00000 0.170561
\(551\) 51.9615 2.21364
\(552\) 0 0
\(553\) −13.0000 31.8434i −0.552816 1.35412i
\(554\) 7.07107i 0.300421i
\(555\) 0 0
\(556\) 0 0
\(557\) 14.1421i 0.599222i −0.954062 0.299611i \(-0.903143\pi\)
0.954062 0.299611i \(-0.0968568\pi\)
\(558\) 0 0
\(559\) 12.2474i 0.518012i
\(560\) −6.92820 16.9706i −0.292770 0.717137i
\(561\) 0 0
\(562\) −20.0000 −0.843649
\(563\) −34.6410 −1.45994 −0.729972 0.683477i \(-0.760467\pi\)
−0.729972 + 0.683477i \(0.760467\pi\)
\(564\) 0 0
\(565\) 12.2474i 0.515254i
\(566\) −17.3205 −0.728035
\(567\) 0 0
\(568\) −40.0000 −1.67836
\(569\) 39.5980i 1.66003i −0.557738 0.830017i \(-0.688331\pi\)
0.557738 0.830017i \(-0.311669\pi\)
\(570\) 0 0
\(571\) −19.0000 −0.795125 −0.397563 0.917575i \(-0.630144\pi\)
−0.397563 + 0.917575i \(0.630144\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −30.0000 + 12.2474i −1.25218 + 0.511199i
\(575\) 5.65685i 0.235907i
\(576\) 0 0
\(577\) 29.3939i 1.22368i 0.790980 + 0.611842i \(0.209571\pi\)
−0.790980 + 0.611842i \(0.790429\pi\)
\(578\) 14.1421i 0.588235i
\(579\) 0 0
\(580\) 0 0
\(581\) −1.73205 4.24264i −0.0718576 0.176014i
\(582\) 0 0
\(583\) 16.0000 0.662652
\(584\) 0 0
\(585\) 0 0
\(586\) 26.9444i 1.11306i
\(587\) 13.8564 0.571915 0.285958 0.958242i \(-0.407688\pi\)
0.285958 + 0.958242i \(0.407688\pi\)
\(588\) 0 0
\(589\) 18.0000 0.741677
\(590\) 21.2132i 0.873334i
\(591\) 0 0
\(592\) −20.0000 −0.821995
\(593\) 15.5885 0.640141 0.320071 0.947394i \(-0.396293\pi\)
0.320071 + 0.947394i \(0.396293\pi\)
\(594\) 0 0
\(595\) −9.00000 22.0454i −0.368964 0.903774i
\(596\) 0 0
\(597\) 0 0
\(598\) 9.79796i 0.400668i
\(599\) 18.3848i 0.751182i −0.926786 0.375591i \(-0.877440\pi\)
0.926786 0.375591i \(-0.122560\pi\)
\(600\) 0 0
\(601\) 46.5403i 1.89842i −0.314645 0.949209i \(-0.601886\pi\)
0.314645 0.949209i \(-0.398114\pi\)
\(602\) 17.3205 7.07107i 0.705931 0.288195i
\(603\) 0 0
\(604\) 0 0
\(605\) 15.5885 0.633761
\(606\) 0 0
\(607\) 12.2474i 0.497109i −0.968618 0.248554i \(-0.920045\pi\)
0.968618 0.248554i \(-0.0799554\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) −6.00000 −0.242933
\(611\) 21.2132i 0.858194i
\(612\) 0 0
\(613\) −4.00000 −0.161558 −0.0807792 0.996732i \(-0.525741\pi\)
−0.0807792 + 0.996732i \(0.525741\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) −4.00000 9.79796i −0.161165 0.394771i
\(617\) 48.0833i 1.93576i −0.251414 0.967880i \(-0.580896\pi\)
0.251414 0.967880i \(-0.419104\pi\)
\(618\) 0 0
\(619\) 26.9444i 1.08299i 0.840705 + 0.541493i \(0.182141\pi\)
−0.840705 + 0.541493i \(0.817859\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 17.1464i 0.687509i
\(623\) 10.3923 + 25.4558i 0.416359 + 1.01987i
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) −34.6410 −1.38453
\(627\) 0 0
\(628\) 0 0
\(629\) −25.9808 −1.03592
\(630\) 0 0
\(631\) 29.0000 1.15447 0.577236 0.816577i \(-0.304131\pi\)
0.577236 + 0.816577i \(0.304131\pi\)
\(632\) 36.7696i 1.46261i
\(633\) 0 0
\(634\) −8.00000 −0.317721
\(635\) 8.66025 0.343672
\(636\) 0 0
\(637\) −12.0000 12.2474i −0.475457 0.485262i
\(638\) 14.1421i 0.559893i
\(639\) 0 0
\(640\) 19.5959i 0.774597i
\(641\) 1.41421i 0.0558581i −0.999610 0.0279290i \(-0.991109\pi\)
0.999610 0.0279290i \(-0.00889125\pi\)
\(642\) 0 0
\(643\) 24.4949i 0.965984i 0.875625 + 0.482992i \(0.160450\pi\)
−0.875625 + 0.482992i \(0.839550\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 54.0000 2.12460
\(647\) −20.7846 −0.817127 −0.408564 0.912730i \(-0.633970\pi\)
−0.408564 + 0.912730i \(0.633970\pi\)
\(648\) 0 0
\(649\) 12.2474i 0.480754i
\(650\) −6.92820 −0.271746
\(651\) 0 0
\(652\) 0 0
\(653\) 18.3848i 0.719452i −0.933058 0.359726i \(-0.882870\pi\)
0.933058 0.359726i \(-0.117130\pi\)
\(654\) 0 0
\(655\) 30.0000 1.17220
\(656\) 34.6410 1.35250
\(657\) 0 0
\(658\) 30.0000 12.2474i 1.16952 0.477455i
\(659\) 11.3137i 0.440720i 0.975419 + 0.220360i \(0.0707231\pi\)
−0.975419 + 0.220360i \(0.929277\pi\)
\(660\) 0 0
\(661\) 17.1464i 0.666919i 0.942764 + 0.333459i \(0.108216\pi\)
−0.942764 + 0.333459i \(0.891784\pi\)
\(662\) 35.3553i 1.37412i
\(663\) 0 0
\(664\) 4.89898i 0.190117i
\(665\) 31.1769 12.7279i 1.20899 0.493568i
\(666\) 0 0
\(667\) −20.0000 −0.774403
\(668\) 0 0
\(669\) 0 0
\(670\) 4.89898i 0.189264i
\(671\) −3.46410 −0.133730
\(672\) 0 0
\(673\) −40.0000 −1.54189 −0.770943 0.636904i \(-0.780215\pi\)
−0.770943 + 0.636904i \(0.780215\pi\)
\(674\) 9.89949i 0.381314i
\(675\) 0 0
\(676\) 0 0
\(677\) −17.3205 −0.665681 −0.332841 0.942983i \(-0.608007\pi\)
−0.332841 + 0.942983i \(0.608007\pi\)
\(678\) 0 0
\(679\) 42.0000 17.1464i 1.61181 0.658020i
\(680\) 25.4558i 0.976187i
\(681\) 0 0
\(682\) 4.89898i 0.187592i
\(683\) 24.0416i 0.919927i 0.887938 + 0.459964i \(0.152138\pi\)
−0.887938 + 0.459964i \(0.847862\pi\)
\(684\) 0 0
\(685\) 2.44949i 0.0935902i
\(686\) −10.3923 + 24.0416i −0.396780 + 0.917914i
\(687\) 0 0
\(688\) −20.0000 −0.762493
\(689\) −27.7128 −1.05577
\(690\) 0 0
\(691\) 26.9444i 1.02501i 0.858683 + 0.512506i \(0.171283\pi\)
−0.858683 + 0.512506i \(0.828717\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 10.0000 0.379595
\(695\) 21.2132i 0.804663i
\(696\) 0 0
\(697\) 45.0000 1.70450
\(698\) 27.7128 1.04895
\(699\) 0 0
\(700\) 0 0
\(701\) 1.41421i 0.0534141i −0.999643 0.0267071i \(-0.991498\pi\)
0.999643 0.0267071i \(-0.00850213\pi\)
\(702\) 0 0
\(703\) 36.7423i 1.38576i
\(704\) 11.3137i 0.426401i
\(705\) 0 0
\(706\) 12.2474i 0.460939i
\(707\) 3.46410 + 8.48528i 0.130281 + 0.319122i
\(708\) 0 0
\(709\) 5.00000 0.187779 0.0938895 0.995583i \(-0.470070\pi\)
0.0938895 + 0.995583i \(0.470070\pi\)
\(710\) −34.6410 −1.30005
\(711\) 0 0
\(712\) 29.3939i 1.10158i
\(713\) −6.92820 −0.259463
\(714\) 0 0
\(715\) 6.00000 0.224387
\(716\) 0 0
\(717\) 0 0
\(718\) 16.0000 0.597115
\(719\) −15.5885 −0.581351 −0.290676 0.956822i \(-0.593880\pi\)
−0.290676 + 0.956822i \(0.593880\pi\)
\(720\) 0 0
\(721\) −24.0000 + 9.79796i −0.893807 + 0.364895i
\(722\) 49.4975i 1.84211i
\(723\) 0 0
\(724\) 0 0
\(725\) 14.1421i 0.525226i
\(726\) 0 0
\(727\) 12.2474i 0.454233i 0.973868 + 0.227116i \(0.0729298\pi\)
−0.973868 + 0.227116i \(0.927070\pi\)
\(728\) 6.92820 + 16.9706i 0.256776 + 0.628971i
\(729\) 0 0
\(730\) 0 0
\(731\) −25.9808 −0.960933
\(732\) 0 0
\(733\) 39.1918i 1.44758i 0.690018 + 0.723792i \(0.257602\pi\)
−0.690018 + 0.723792i \(0.742398\pi\)
\(734\) 6.92820 0.255725
\(735\) 0 0
\(736\) 0 0
\(737\) 2.82843i 0.104186i
\(738\) 0 0
\(739\) −22.0000 −0.809283 −0.404642 0.914475i \(-0.632604\pi\)
−0.404642 + 0.914475i \(0.632604\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 16.0000 + 39.1918i 0.587378 + 1.43878i
\(743\) 7.07107i 0.259412i 0.991552 + 0.129706i \(0.0414034\pi\)
−0.991552 + 0.129706i \(0.958597\pi\)
\(744\) 0 0
\(745\) 12.2474i 0.448712i
\(746\) 35.3553i 1.29445i
\(747\) 0 0
\(748\) 0 0
\(749\) 34.6410 14.1421i 1.26576 0.516742i
\(750\) 0 0
\(751\) −34.0000 −1.24068 −0.620339 0.784334i \(-0.713005\pi\)
−0.620339 + 0.784334i \(0.713005\pi\)
\(752\) −34.6410 −1.26323
\(753\) 0 0
\(754\) 24.4949i 0.892052i
\(755\) 8.66025 0.315179
\(756\) 0 0
\(757\) 47.0000 1.70824 0.854122 0.520073i \(-0.174095\pi\)
0.854122 + 0.520073i \(0.174095\pi\)
\(758\) 32.5269i 1.18143i
\(759\) 0 0
\(760\) −36.0000 −1.30586
\(761\) 12.1244 0.439508 0.219754 0.975555i \(-0.429475\pi\)
0.219754 + 0.975555i \(0.429475\pi\)
\(762\) 0 0
\(763\) −7.00000 17.1464i −0.253417 0.620742i
\(764\) 0 0
\(765\) 0 0
\(766\) 2.44949i 0.0885037i
\(767\) 21.2132i 0.765964i
\(768\) 0 0
\(769\) 4.89898i 0.176662i −0.996091 0.0883309i \(-0.971847\pi\)
0.996091 0.0883309i \(-0.0281533\pi\)
\(770\) −3.46410 8.48528i −0.124838 0.305788i
\(771\) 0 0
\(772\) 0 0
\(773\) 15.5885 0.560678 0.280339 0.959901i \(-0.409553\pi\)
0.280339 + 0.959901i \(0.409553\pi\)
\(774\) 0 0
\(775\) 4.89898i 0.175977i
\(776\) −48.4974 −1.74096
\(777\) 0 0
\(778\) 46.0000 1.64918
\(779\) 63.6396i 2.28013i
\(780\) 0 0
\(781\) −20.0000 −0.715656
\(782\) −20.7846 −0.743256
\(783\) 0 0
\(784\) 20.0000 19.5959i 0.714286 0.699854i
\(785\) 29.6985i 1.05998i
\(786\) 0 0
\(787\) 4.89898i 0.174630i −0.996181 0.0873149i \(-0.972171\pi\)
0.996181 0.0873149i \(-0.0278286\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 31.8434i 1.13294i
\(791\) −17.3205 + 7.07107i −0.615846 + 0.251418i
\(792\) 0 0
\(793\) 6.00000 0.213066
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 38.1051 1.34975 0.674876 0.737931i \(-0.264197\pi\)
0.674876 + 0.737931i \(0.264197\pi\)
\(798\) 0 0
\(799\) −45.0000 −1.59199
\(800\) 0 0
\(801\) 0 0
\(802\) 28.0000 0.988714