Properties

Label 560.2.g.e.449.4
Level 560
Weight 2
Character 560.449
Analytic conductor 4.472
Analytic rank 0
Dimension 4
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 560.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.47162251319\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.4
Root \(1.22474 + 1.22474i\)
Character \(\chi\) = 560.449
Dual form 560.2.g.e.449.2

$q$-expansion

\(f(q)\) \(=\) \(q+2.44949i q^{3} +(2.22474 + 0.224745i) q^{5} -1.00000i q^{7} -3.00000 q^{9} +O(q^{10})\) \(q+2.44949i q^{3} +(2.22474 + 0.224745i) q^{5} -1.00000i q^{7} -3.00000 q^{9} +4.89898 q^{11} +4.44949i q^{13} +(-0.550510 + 5.44949i) q^{15} -2.00000i q^{17} +1.55051 q^{19} +2.44949 q^{21} -2.89898i q^{23} +(4.89898 + 1.00000i) q^{25} -6.89898 q^{29} -8.89898 q^{31} +12.0000i q^{33} +(0.224745 - 2.22474i) q^{35} -2.00000i q^{37} -10.8990 q^{39} -1.10102 q^{41} +0.898979i q^{43} +(-6.67423 - 0.674235i) q^{45} +8.89898i q^{47} -1.00000 q^{49} +4.89898 q^{51} -10.8990i q^{53} +(10.8990 + 1.10102i) q^{55} +3.79796i q^{57} -1.55051 q^{59} +3.55051 q^{61} +3.00000i q^{63} +(-1.00000 + 9.89898i) q^{65} -8.00000i q^{67} +7.10102 q^{69} +1.10102 q^{71} +2.89898i q^{73} +(-2.44949 + 12.0000i) q^{75} -4.89898i q^{77} +6.89898 q^{79} -9.00000 q^{81} +2.44949i q^{83} +(0.449490 - 4.44949i) q^{85} -16.8990i q^{87} +10.0000 q^{89} +4.44949 q^{91} -21.7980i q^{93} +(3.44949 + 0.348469i) q^{95} -15.7980i q^{97} -14.6969 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{5} - 12q^{9} + O(q^{10}) \) \( 4q + 4q^{5} - 12q^{9} - 12q^{15} + 16q^{19} - 8q^{29} - 16q^{31} - 4q^{35} - 24q^{39} - 24q^{41} - 12q^{45} - 4q^{49} + 24q^{55} - 16q^{59} + 24q^{61} - 4q^{65} + 48q^{69} + 24q^{71} + 8q^{79} - 36q^{81} - 8q^{85} + 40q^{89} + 8q^{91} + 4q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/560\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(337\) \(351\) \(421\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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
\(3\) 2.44949i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(4\) 0 0
\(5\) 2.22474 + 0.224745i 0.994936 + 0.100509i
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) 4.89898 1.47710 0.738549 0.674200i \(-0.235511\pi\)
0.738549 + 0.674200i \(0.235511\pi\)
\(12\) 0 0
\(13\) 4.44949i 1.23407i 0.786937 + 0.617033i \(0.211666\pi\)
−0.786937 + 0.617033i \(0.788334\pi\)
\(14\) 0 0
\(15\) −0.550510 + 5.44949i −0.142141 + 1.40705i
\(16\) 0 0
\(17\) 2.00000i 0.485071i −0.970143 0.242536i \(-0.922021\pi\)
0.970143 0.242536i \(-0.0779791\pi\)
\(18\) 0 0
\(19\) 1.55051 0.355711 0.177856 0.984057i \(-0.443084\pi\)
0.177856 + 0.984057i \(0.443084\pi\)
\(20\) 0 0
\(21\) 2.44949 0.534522
\(22\) 0 0
\(23\) 2.89898i 0.604479i −0.953232 0.302240i \(-0.902266\pi\)
0.953232 0.302240i \(-0.0977342\pi\)
\(24\) 0 0
\(25\) 4.89898 + 1.00000i 0.979796 + 0.200000i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −6.89898 −1.28111 −0.640554 0.767913i \(-0.721295\pi\)
−0.640554 + 0.767913i \(0.721295\pi\)
\(30\) 0 0
\(31\) −8.89898 −1.59830 −0.799152 0.601129i \(-0.794718\pi\)
−0.799152 + 0.601129i \(0.794718\pi\)
\(32\) 0 0
\(33\) 12.0000i 2.08893i
\(34\) 0 0
\(35\) 0.224745 2.22474i 0.0379888 0.376051i
\(36\) 0 0
\(37\) 2.00000i 0.328798i −0.986394 0.164399i \(-0.947432\pi\)
0.986394 0.164399i \(-0.0525685\pi\)
\(38\) 0 0
\(39\) −10.8990 −1.74523
\(40\) 0 0
\(41\) −1.10102 −0.171951 −0.0859753 0.996297i \(-0.527401\pi\)
−0.0859753 + 0.996297i \(0.527401\pi\)
\(42\) 0 0
\(43\) 0.898979i 0.137093i 0.997648 + 0.0685465i \(0.0218362\pi\)
−0.997648 + 0.0685465i \(0.978164\pi\)
\(44\) 0 0
\(45\) −6.67423 0.674235i −0.994936 0.100509i
\(46\) 0 0
\(47\) 8.89898i 1.29805i 0.760767 + 0.649025i \(0.224823\pi\)
−0.760767 + 0.649025i \(0.775177\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 4.89898 0.685994
\(52\) 0 0
\(53\) 10.8990i 1.49709i −0.663084 0.748545i \(-0.730753\pi\)
0.663084 0.748545i \(-0.269247\pi\)
\(54\) 0 0
\(55\) 10.8990 + 1.10102i 1.46962 + 0.148462i
\(56\) 0 0
\(57\) 3.79796i 0.503052i
\(58\) 0 0
\(59\) −1.55051 −0.201859 −0.100930 0.994894i \(-0.532182\pi\)
−0.100930 + 0.994894i \(0.532182\pi\)
\(60\) 0 0
\(61\) 3.55051 0.454596 0.227298 0.973825i \(-0.427011\pi\)
0.227298 + 0.973825i \(0.427011\pi\)
\(62\) 0 0
\(63\) 3.00000i 0.377964i
\(64\) 0 0
\(65\) −1.00000 + 9.89898i −0.124035 + 1.22782i
\(66\) 0 0
\(67\) 8.00000i 0.977356i −0.872464 0.488678i \(-0.837479\pi\)
0.872464 0.488678i \(-0.162521\pi\)
\(68\) 0 0
\(69\) 7.10102 0.854862
\(70\) 0 0
\(71\) 1.10102 0.130667 0.0653335 0.997863i \(-0.479189\pi\)
0.0653335 + 0.997863i \(0.479189\pi\)
\(72\) 0 0
\(73\) 2.89898i 0.339300i 0.985504 + 0.169650i \(0.0542637\pi\)
−0.985504 + 0.169650i \(0.945736\pi\)
\(74\) 0 0
\(75\) −2.44949 + 12.0000i −0.282843 + 1.38564i
\(76\) 0 0
\(77\) 4.89898i 0.558291i
\(78\) 0 0
\(79\) 6.89898 0.776196 0.388098 0.921618i \(-0.373132\pi\)
0.388098 + 0.921618i \(0.373132\pi\)
\(80\) 0 0
\(81\) −9.00000 −1.00000
\(82\) 0 0
\(83\) 2.44949i 0.268866i 0.990923 + 0.134433i \(0.0429214\pi\)
−0.990923 + 0.134433i \(0.957079\pi\)
\(84\) 0 0
\(85\) 0.449490 4.44949i 0.0487540 0.482615i
\(86\) 0 0
\(87\) 16.8990i 1.81176i
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 4.44949 0.466433
\(92\) 0 0
\(93\) 21.7980i 2.26034i
\(94\) 0 0
\(95\) 3.44949 + 0.348469i 0.353910 + 0.0357522i
\(96\) 0 0
\(97\) 15.7980i 1.60404i −0.597297 0.802020i \(-0.703759\pi\)
0.597297 0.802020i \(-0.296241\pi\)
\(98\) 0 0
\(99\) −14.6969 −1.47710
\(100\) 0 0
\(101\) 3.55051 0.353289 0.176644 0.984275i \(-0.443476\pi\)
0.176644 + 0.984275i \(0.443476\pi\)
\(102\) 0 0
\(103\) 12.8990i 1.27097i −0.772111 0.635487i \(-0.780799\pi\)
0.772111 0.635487i \(-0.219201\pi\)
\(104\) 0 0
\(105\) 5.44949 + 0.550510i 0.531816 + 0.0537243i
\(106\) 0 0
\(107\) 8.00000i 0.773389i −0.922208 0.386695i \(-0.873617\pi\)
0.922208 0.386695i \(-0.126383\pi\)
\(108\) 0 0
\(109\) −6.89898 −0.660802 −0.330401 0.943841i \(-0.607184\pi\)
−0.330401 + 0.943841i \(0.607184\pi\)
\(110\) 0 0
\(111\) 4.89898 0.464991
\(112\) 0 0
\(113\) 19.7980i 1.86244i 0.364464 + 0.931218i \(0.381252\pi\)
−0.364464 + 0.931218i \(0.618748\pi\)
\(114\) 0 0
\(115\) 0.651531 6.44949i 0.0607556 0.601418i
\(116\) 0 0
\(117\) 13.3485i 1.23407i
\(118\) 0 0
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) 13.0000 1.18182
\(122\) 0 0
\(123\) 2.69694i 0.243175i
\(124\) 0 0
\(125\) 10.6742 + 3.32577i 0.954733 + 0.297465i
\(126\) 0 0
\(127\) 14.8990i 1.32207i −0.750355 0.661035i \(-0.770117\pi\)
0.750355 0.661035i \(-0.229883\pi\)
\(128\) 0 0
\(129\) −2.20204 −0.193879
\(130\) 0 0
\(131\) 6.44949 0.563495 0.281747 0.959489i \(-0.409086\pi\)
0.281747 + 0.959489i \(0.409086\pi\)
\(132\) 0 0
\(133\) 1.55051i 0.134446i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1.79796i 0.153610i 0.997046 + 0.0768050i \(0.0244719\pi\)
−0.997046 + 0.0768050i \(0.975528\pi\)
\(138\) 0 0
\(139\) 1.55051 0.131513 0.0657563 0.997836i \(-0.479054\pi\)
0.0657563 + 0.997836i \(0.479054\pi\)
\(140\) 0 0
\(141\) −21.7980 −1.83572
\(142\) 0 0
\(143\) 21.7980i 1.82284i
\(144\) 0 0
\(145\) −15.3485 1.55051i −1.27462 0.128763i
\(146\) 0 0
\(147\) 2.44949i 0.202031i
\(148\) 0 0
\(149\) 3.79796 0.311141 0.155570 0.987825i \(-0.450278\pi\)
0.155570 + 0.987825i \(0.450278\pi\)
\(150\) 0 0
\(151\) −19.5959 −1.59469 −0.797347 0.603522i \(-0.793764\pi\)
−0.797347 + 0.603522i \(0.793764\pi\)
\(152\) 0 0
\(153\) 6.00000i 0.485071i
\(154\) 0 0
\(155\) −19.7980 2.00000i −1.59021 0.160644i
\(156\) 0 0
\(157\) 3.55051i 0.283362i −0.989912 0.141681i \(-0.954749\pi\)
0.989912 0.141681i \(-0.0452507\pi\)
\(158\) 0 0
\(159\) 26.6969 2.11720
\(160\) 0 0
\(161\) −2.89898 −0.228472
\(162\) 0 0
\(163\) 7.10102i 0.556195i 0.960553 + 0.278097i \(0.0897038\pi\)
−0.960553 + 0.278097i \(0.910296\pi\)
\(164\) 0 0
\(165\) −2.69694 + 26.6969i −0.209956 + 2.07835i
\(166\) 0 0
\(167\) 4.89898i 0.379094i −0.981872 0.189547i \(-0.939298\pi\)
0.981872 0.189547i \(-0.0607020\pi\)
\(168\) 0 0
\(169\) −6.79796 −0.522920
\(170\) 0 0
\(171\) −4.65153 −0.355711
\(172\) 0 0
\(173\) 6.24745i 0.474985i −0.971389 0.237492i \(-0.923675\pi\)
0.971389 0.237492i \(-0.0763255\pi\)
\(174\) 0 0
\(175\) 1.00000 4.89898i 0.0755929 0.370328i
\(176\) 0 0
\(177\) 3.79796i 0.285472i
\(178\) 0 0
\(179\) 13.7980 1.03131 0.515654 0.856797i \(-0.327549\pi\)
0.515654 + 0.856797i \(0.327549\pi\)
\(180\) 0 0
\(181\) −10.2474 −0.761687 −0.380843 0.924640i \(-0.624366\pi\)
−0.380843 + 0.924640i \(0.624366\pi\)
\(182\) 0 0
\(183\) 8.69694i 0.642896i
\(184\) 0 0
\(185\) 0.449490 4.44949i 0.0330471 0.327133i
\(186\) 0 0
\(187\) 9.79796i 0.716498i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −12.6969 −0.918718 −0.459359 0.888251i \(-0.651921\pi\)
−0.459359 + 0.888251i \(0.651921\pi\)
\(192\) 0 0
\(193\) 21.5959i 1.55451i −0.629187 0.777254i \(-0.716612\pi\)
0.629187 0.777254i \(-0.283388\pi\)
\(194\) 0 0
\(195\) −24.2474 2.44949i −1.73640 0.175412i
\(196\) 0 0
\(197\) 18.8990i 1.34650i −0.739417 0.673248i \(-0.764899\pi\)
0.739417 0.673248i \(-0.235101\pi\)
\(198\) 0 0
\(199\) 16.8990 1.19794 0.598968 0.800773i \(-0.295577\pi\)
0.598968 + 0.800773i \(0.295577\pi\)
\(200\) 0 0
\(201\) 19.5959 1.38219
\(202\) 0 0
\(203\) 6.89898i 0.484213i
\(204\) 0 0
\(205\) −2.44949 0.247449i −0.171080 0.0172826i
\(206\) 0 0
\(207\) 8.69694i 0.604479i
\(208\) 0 0
\(209\) 7.59592 0.525421
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 0 0
\(213\) 2.69694i 0.184791i
\(214\) 0 0
\(215\) −0.202041 + 2.00000i −0.0137791 + 0.136399i
\(216\) 0 0
\(217\) 8.89898i 0.604102i
\(218\) 0 0
\(219\) −7.10102 −0.479842
\(220\) 0 0
\(221\) 8.89898 0.598610
\(222\) 0 0
\(223\) 4.00000i 0.267860i 0.990991 + 0.133930i \(0.0427597\pi\)
−0.990991 + 0.133930i \(0.957240\pi\)
\(224\) 0 0
\(225\) −14.6969 3.00000i −0.979796 0.200000i
\(226\) 0 0
\(227\) 7.34847i 0.487735i 0.969809 + 0.243868i \(0.0784162\pi\)
−0.969809 + 0.243868i \(0.921584\pi\)
\(228\) 0 0
\(229\) 19.1464 1.26523 0.632616 0.774466i \(-0.281981\pi\)
0.632616 + 0.774466i \(0.281981\pi\)
\(230\) 0 0
\(231\) 12.0000 0.789542
\(232\) 0 0
\(233\) 29.7980i 1.95213i 0.217481 + 0.976065i \(0.430216\pi\)
−0.217481 + 0.976065i \(0.569784\pi\)
\(234\) 0 0
\(235\) −2.00000 + 19.7980i −0.130466 + 1.29148i
\(236\) 0 0
\(237\) 16.8990i 1.09771i
\(238\) 0 0
\(239\) −6.20204 −0.401177 −0.200588 0.979676i \(-0.564285\pi\)
−0.200588 + 0.979676i \(0.564285\pi\)
\(240\) 0 0
\(241\) −8.69694 −0.560219 −0.280110 0.959968i \(-0.590371\pi\)
−0.280110 + 0.959968i \(0.590371\pi\)
\(242\) 0 0
\(243\) 22.0454i 1.41421i
\(244\) 0 0
\(245\) −2.22474 0.224745i −0.142134 0.0143584i
\(246\) 0 0
\(247\) 6.89898i 0.438972i
\(248\) 0 0
\(249\) −6.00000 −0.380235
\(250\) 0 0
\(251\) 6.44949 0.407088 0.203544 0.979066i \(-0.434754\pi\)
0.203544 + 0.979066i \(0.434754\pi\)
\(252\) 0 0
\(253\) 14.2020i 0.892875i
\(254\) 0 0
\(255\) 10.8990 + 1.10102i 0.682521 + 0.0689486i
\(256\) 0 0
\(257\) 8.69694i 0.542500i 0.962509 + 0.271250i \(0.0874370\pi\)
−0.962509 + 0.271250i \(0.912563\pi\)
\(258\) 0 0
\(259\) −2.00000 −0.124274
\(260\) 0 0
\(261\) 20.6969 1.28111
\(262\) 0 0
\(263\) 9.79796i 0.604168i −0.953281 0.302084i \(-0.902318\pi\)
0.953281 0.302084i \(-0.0976823\pi\)
\(264\) 0 0
\(265\) 2.44949 24.2474i 0.150471 1.48951i
\(266\) 0 0
\(267\) 24.4949i 1.49906i
\(268\) 0 0
\(269\) −19.1464 −1.16738 −0.583689 0.811977i \(-0.698391\pi\)
−0.583689 + 0.811977i \(0.698391\pi\)
\(270\) 0 0
\(271\) −12.0000 −0.728948 −0.364474 0.931214i \(-0.618751\pi\)
−0.364474 + 0.931214i \(0.618751\pi\)
\(272\) 0 0
\(273\) 10.8990i 0.659636i
\(274\) 0 0
\(275\) 24.0000 + 4.89898i 1.44725 + 0.295420i
\(276\) 0 0
\(277\) 14.8990i 0.895193i 0.894236 + 0.447596i \(0.147720\pi\)
−0.894236 + 0.447596i \(0.852280\pi\)
\(278\) 0 0
\(279\) 26.6969 1.59830
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 0 0
\(283\) 3.75255i 0.223066i −0.993761 0.111533i \(-0.964424\pi\)
0.993761 0.111533i \(-0.0355761\pi\)
\(284\) 0 0
\(285\) −0.853572 + 8.44949i −0.0505612 + 0.500505i
\(286\) 0 0
\(287\) 1.10102i 0.0649912i
\(288\) 0 0
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) 38.6969 2.26845
\(292\) 0 0
\(293\) 18.2474i 1.06603i 0.846107 + 0.533014i \(0.178941\pi\)
−0.846107 + 0.533014i \(0.821059\pi\)
\(294\) 0 0
\(295\) −3.44949 0.348469i −0.200837 0.0202887i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 12.8990 0.745967
\(300\) 0 0
\(301\) 0.898979 0.0518163
\(302\) 0 0
\(303\) 8.69694i 0.499626i
\(304\) 0 0
\(305\) 7.89898 + 0.797959i 0.452294 + 0.0456910i
\(306\) 0 0
\(307\) 20.2474i 1.15558i −0.816184 0.577791i \(-0.803915\pi\)
0.816184 0.577791i \(-0.196085\pi\)
\(308\) 0 0
\(309\) 31.5959 1.79743
\(310\) 0 0
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) 21.5959i 1.22067i −0.792142 0.610337i \(-0.791034\pi\)
0.792142 0.610337i \(-0.208966\pi\)
\(314\) 0 0
\(315\) −0.674235 + 6.67423i −0.0379888 + 0.376051i
\(316\) 0 0
\(317\) 22.4949i 1.26344i 0.775197 + 0.631720i \(0.217651\pi\)
−0.775197 + 0.631720i \(0.782349\pi\)
\(318\) 0 0
\(319\) −33.7980 −1.89232
\(320\) 0 0
\(321\) 19.5959 1.09374
\(322\) 0 0
\(323\) 3.10102i 0.172545i
\(324\) 0 0
\(325\) −4.44949 + 21.7980i −0.246813 + 1.20913i
\(326\) 0 0
\(327\) 16.8990i 0.934516i
\(328\) 0 0
\(329\) 8.89898 0.490617
\(330\) 0 0
\(331\) 18.6969 1.02768 0.513838 0.857887i \(-0.328223\pi\)
0.513838 + 0.857887i \(0.328223\pi\)
\(332\) 0 0
\(333\) 6.00000i 0.328798i
\(334\) 0 0
\(335\) 1.79796 17.7980i 0.0982330 0.972406i
\(336\) 0 0
\(337\) 9.59592i 0.522723i −0.965241 0.261361i \(-0.915829\pi\)
0.965241 0.261361i \(-0.0841715\pi\)
\(338\) 0 0
\(339\) −48.4949 −2.63388
\(340\) 0 0
\(341\) −43.5959 −2.36085
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 15.7980 + 1.59592i 0.850534 + 0.0859213i
\(346\) 0 0
\(347\) 28.8990i 1.55138i 0.631115 + 0.775689i \(0.282598\pi\)
−0.631115 + 0.775689i \(0.717402\pi\)
\(348\) 0 0
\(349\) 8.44949 0.452291 0.226145 0.974094i \(-0.427388\pi\)
0.226145 + 0.974094i \(0.427388\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 22.8990i 1.21879i 0.792867 + 0.609395i \(0.208588\pi\)
−0.792867 + 0.609395i \(0.791412\pi\)
\(354\) 0 0
\(355\) 2.44949 + 0.247449i 0.130005 + 0.0131332i
\(356\) 0 0
\(357\) 4.89898i 0.259281i
\(358\) 0 0
\(359\) −27.5959 −1.45646 −0.728228 0.685334i \(-0.759656\pi\)
−0.728228 + 0.685334i \(0.759656\pi\)
\(360\) 0 0
\(361\) −16.5959 −0.873469
\(362\) 0 0
\(363\) 31.8434i 1.67134i
\(364\) 0 0
\(365\) −0.651531 + 6.44949i −0.0341027 + 0.337582i
\(366\) 0 0
\(367\) 32.0000i 1.67039i 0.549957 + 0.835193i \(0.314644\pi\)
−0.549957 + 0.835193i \(0.685356\pi\)
\(368\) 0 0
\(369\) 3.30306 0.171951
\(370\) 0 0
\(371\) −10.8990 −0.565847
\(372\) 0 0
\(373\) 4.69694i 0.243198i −0.992579 0.121599i \(-0.961198\pi\)
0.992579 0.121599i \(-0.0388022\pi\)
\(374\) 0 0
\(375\) −8.14643 + 26.1464i −0.420680 + 1.35020i
\(376\) 0 0
\(377\) 30.6969i 1.58097i
\(378\) 0 0
\(379\) 30.6969 1.57680 0.788398 0.615166i \(-0.210911\pi\)
0.788398 + 0.615166i \(0.210911\pi\)
\(380\) 0 0
\(381\) 36.4949 1.86969
\(382\) 0 0
\(383\) 7.10102i 0.362845i 0.983405 + 0.181423i \(0.0580702\pi\)
−0.983405 + 0.181423i \(0.941930\pi\)
\(384\) 0 0
\(385\) 1.10102 10.8990i 0.0561132 0.555463i
\(386\) 0 0
\(387\) 2.69694i 0.137093i
\(388\) 0 0
\(389\) −13.1010 −0.664248 −0.332124 0.943236i \(-0.607765\pi\)
−0.332124 + 0.943236i \(0.607765\pi\)
\(390\) 0 0
\(391\) −5.79796 −0.293215
\(392\) 0 0
\(393\) 15.7980i 0.796902i
\(394\) 0 0
\(395\) 15.3485 + 1.55051i 0.772265 + 0.0780146i
\(396\) 0 0
\(397\) 2.65153i 0.133077i 0.997784 + 0.0665383i \(0.0211954\pi\)
−0.997784 + 0.0665383i \(0.978805\pi\)
\(398\) 0 0
\(399\) 3.79796 0.190136
\(400\) 0 0
\(401\) −29.3939 −1.46786 −0.733930 0.679225i \(-0.762316\pi\)
−0.733930 + 0.679225i \(0.762316\pi\)
\(402\) 0 0
\(403\) 39.5959i 1.97241i
\(404\) 0 0
\(405\) −20.0227 2.02270i −0.994936 0.100509i
\(406\) 0 0
\(407\) 9.79796i 0.485667i
\(408\) 0 0
\(409\) −34.4949 −1.70566 −0.852831 0.522186i \(-0.825117\pi\)
−0.852831 + 0.522186i \(0.825117\pi\)
\(410\) 0 0
\(411\) −4.40408 −0.217237
\(412\) 0 0
\(413\) 1.55051i 0.0762956i
\(414\) 0 0
\(415\) −0.550510 + 5.44949i −0.0270235 + 0.267505i
\(416\) 0 0
\(417\) 3.79796i 0.185987i
\(418\) 0 0
\(419\) −1.55051 −0.0757474 −0.0378737 0.999283i \(-0.512058\pi\)
−0.0378737 + 0.999283i \(0.512058\pi\)
\(420\) 0 0
\(421\) −4.20204 −0.204795 −0.102397 0.994744i \(-0.532651\pi\)
−0.102397 + 0.994744i \(0.532651\pi\)
\(422\) 0 0
\(423\) 26.6969i 1.29805i
\(424\) 0 0
\(425\) 2.00000 9.79796i 0.0970143 0.475271i
\(426\) 0 0
\(427\) 3.55051i 0.171821i
\(428\) 0 0
\(429\) −53.3939 −2.57788
\(430\) 0 0
\(431\) 1.79796 0.0866046 0.0433023 0.999062i \(-0.486212\pi\)
0.0433023 + 0.999062i \(0.486212\pi\)
\(432\) 0 0
\(433\) 0.202041i 0.00970947i −0.999988 0.00485474i \(-0.998455\pi\)
0.999988 0.00485474i \(-0.00154532\pi\)
\(434\) 0 0
\(435\) 3.79796 37.5959i 0.182098 1.80259i
\(436\) 0 0
\(437\) 4.49490i 0.215020i
\(438\) 0 0
\(439\) −21.3939 −1.02107 −0.510537 0.859856i \(-0.670553\pi\)
−0.510537 + 0.859856i \(0.670553\pi\)
\(440\) 0 0
\(441\) 3.00000 0.142857
\(442\) 0 0
\(443\) 9.79796i 0.465515i −0.972535 0.232758i \(-0.925225\pi\)
0.972535 0.232758i \(-0.0747749\pi\)
\(444\) 0 0
\(445\) 22.2474 + 2.24745i 1.05463 + 0.106539i
\(446\) 0 0
\(447\) 9.30306i 0.440020i
\(448\) 0 0
\(449\) 10.0000 0.471929 0.235965 0.971762i \(-0.424175\pi\)
0.235965 + 0.971762i \(0.424175\pi\)
\(450\) 0 0
\(451\) −5.39388 −0.253988
\(452\) 0 0
\(453\) 48.0000i 2.25524i
\(454\) 0 0
\(455\) 9.89898 + 1.00000i 0.464071 + 0.0468807i
\(456\) 0 0
\(457\) 29.5959i 1.38444i −0.721687 0.692219i \(-0.756633\pi\)
0.721687 0.692219i \(-0.243367\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 17.3485 0.807999 0.403999 0.914759i \(-0.367620\pi\)
0.403999 + 0.914759i \(0.367620\pi\)
\(462\) 0 0
\(463\) 3.59592i 0.167116i −0.996503 0.0835582i \(-0.973372\pi\)
0.996503 0.0835582i \(-0.0266285\pi\)
\(464\) 0 0
\(465\) 4.89898 48.4949i 0.227185 2.24890i
\(466\) 0 0
\(467\) 10.4495i 0.483545i 0.970333 + 0.241772i \(0.0777287\pi\)
−0.970333 + 0.241772i \(0.922271\pi\)
\(468\) 0 0
\(469\) −8.00000 −0.369406
\(470\) 0 0
\(471\) 8.69694 0.400734
\(472\) 0 0
\(473\) 4.40408i 0.202500i
\(474\) 0 0
\(475\) 7.59592 + 1.55051i 0.348525 + 0.0711423i
\(476\) 0 0
\(477\) 32.6969i 1.49709i
\(478\) 0 0
\(479\) 9.30306 0.425068 0.212534 0.977154i \(-0.431828\pi\)
0.212534 + 0.977154i \(0.431828\pi\)
\(480\) 0 0
\(481\) 8.89898 0.405759
\(482\) 0 0
\(483\) 7.10102i 0.323108i
\(484\) 0 0
\(485\) 3.55051 35.1464i 0.161220 1.59592i
\(486\) 0 0
\(487\) 7.30306i 0.330933i −0.986215 0.165467i \(-0.947087\pi\)
0.986215 0.165467i \(-0.0529130\pi\)
\(488\) 0 0
\(489\) −17.3939 −0.786578
\(490\) 0 0
\(491\) −19.5959 −0.884351 −0.442176 0.896928i \(-0.645793\pi\)
−0.442176 + 0.896928i \(0.645793\pi\)
\(492\) 0 0
\(493\) 13.7980i 0.621429i
\(494\) 0 0
\(495\) −32.6969 3.30306i −1.46962 0.148462i
\(496\) 0 0
\(497\) 1.10102i 0.0493875i
\(498\) 0 0
\(499\) 6.20204 0.277641 0.138821 0.990318i \(-0.455669\pi\)
0.138821 + 0.990318i \(0.455669\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) 0 0
\(503\) 4.00000i 0.178351i 0.996016 + 0.0891756i \(0.0284232\pi\)
−0.996016 + 0.0891756i \(0.971577\pi\)
\(504\) 0 0
\(505\) 7.89898 + 0.797959i 0.351500 + 0.0355087i
\(506\) 0 0
\(507\) 16.6515i 0.739520i
\(508\) 0 0
\(509\) 31.5505 1.39845 0.699226 0.714901i \(-0.253528\pi\)
0.699226 + 0.714901i \(0.253528\pi\)
\(510\) 0 0
\(511\) 2.89898 0.128243
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 2.89898 28.6969i 0.127744 1.26454i
\(516\) 0 0
\(517\) 43.5959i 1.91735i
\(518\) 0 0
\(519\) 15.3031 0.671730
\(520\) 0 0
\(521\) 32.6969 1.43248 0.716239 0.697855i \(-0.245862\pi\)
0.716239 + 0.697855i \(0.245862\pi\)
\(522\) 0 0
\(523\) 33.1464i 1.44939i 0.689069 + 0.724696i \(0.258020\pi\)
−0.689069 + 0.724696i \(0.741980\pi\)
\(524\) 0 0
\(525\) 12.0000 + 2.44949i 0.523723 + 0.106904i
\(526\) 0 0
\(527\) 17.7980i 0.775291i
\(528\) 0 0
\(529\) 14.5959 0.634605
\(530\) 0 0
\(531\) 4.65153 0.201859
\(532\) 0 0
\(533\) 4.89898i 0.212198i
\(534\) 0 0
\(535\) 1.79796 17.7980i 0.0777325 0.769473i
\(536\) 0 0
\(537\) 33.7980i 1.45849i
\(538\) 0 0
\(539\) −4.89898 −0.211014
\(540\) 0 0
\(541\) 9.59592 0.412561 0.206280 0.978493i \(-0.433864\pi\)
0.206280 + 0.978493i \(0.433864\pi\)
\(542\) 0 0
\(543\) 25.1010i 1.07719i
\(544\) 0 0
\(545\) −15.3485 1.55051i −0.657456 0.0664166i
\(546\) 0 0
\(547\) 18.6969i 0.799423i −0.916641 0.399712i \(-0.869110\pi\)
0.916641 0.399712i \(-0.130890\pi\)
\(548\) 0 0
\(549\) −10.6515 −0.454596
\(550\) 0 0
\(551\) −10.6969 −0.455705
\(552\) 0 0
\(553\) 6.89898i 0.293374i
\(554\) 0 0
\(555\) 10.8990 + 1.10102i 0.462636 + 0.0467357i
\(556\) 0 0
\(557\) 12.6969i 0.537987i −0.963142 0.268993i \(-0.913309\pi\)
0.963142 0.268993i \(-0.0866909\pi\)
\(558\) 0 0
\(559\) −4.00000 −0.169182
\(560\) 0 0
\(561\) 24.0000 1.01328
\(562\) 0 0
\(563\) 30.0454i 1.26626i 0.774044 + 0.633131i \(0.218231\pi\)
−0.774044 + 0.633131i \(0.781769\pi\)
\(564\) 0 0
\(565\) −4.44949 + 44.0454i −0.187191 + 1.85300i
\(566\) 0 0
\(567\) 9.00000i 0.377964i
\(568\) 0 0
\(569\) −33.7980 −1.41688 −0.708442 0.705769i \(-0.750602\pi\)
−0.708442 + 0.705769i \(0.750602\pi\)
\(570\) 0 0
\(571\) 11.1010 0.464563 0.232282 0.972649i \(-0.425381\pi\)
0.232282 + 0.972649i \(0.425381\pi\)
\(572\) 0 0
\(573\) 31.1010i 1.29926i
\(574\) 0 0
\(575\) 2.89898 14.2020i 0.120896 0.592266i
\(576\) 0 0
\(577\) 2.49490i 0.103864i 0.998651 + 0.0519320i \(0.0165379\pi\)
−0.998651 + 0.0519320i \(0.983462\pi\)
\(578\) 0 0
\(579\) 52.8990 2.19841
\(580\) 0 0
\(581\) 2.44949 0.101622
\(582\) 0 0
\(583\) 53.3939i 2.21135i
\(584\) 0 0
\(585\) 3.00000 29.6969i 0.124035 1.22782i
\(586\) 0 0
\(587\) 1.14643i 0.0473182i 0.999720 + 0.0236591i \(0.00753162\pi\)
−0.999720 + 0.0236591i \(0.992468\pi\)
\(588\) 0 0
\(589\) −13.7980 −0.568535
\(590\) 0 0
\(591\) 46.2929 1.90423
\(592\) 0 0
\(593\) 10.8990i 0.447567i −0.974639 0.223784i \(-0.928159\pi\)
0.974639 0.223784i \(-0.0718409\pi\)
\(594\) 0 0
\(595\) −4.44949 0.449490i −0.182411 0.0184273i
\(596\) 0 0
\(597\) 41.3939i 1.69414i
\(598\) 0 0
\(599\) −13.1010 −0.535293 −0.267647 0.963517i \(-0.586246\pi\)
−0.267647 + 0.963517i \(0.586246\pi\)
\(600\) 0 0
\(601\) −39.3939 −1.60691 −0.803455 0.595366i \(-0.797007\pi\)
−0.803455 + 0.595366i \(0.797007\pi\)
\(602\) 0 0
\(603\) 24.0000i 0.977356i
\(604\) 0 0
\(605\) 28.9217 + 2.92168i 1.17583 + 0.118783i
\(606\) 0 0
\(607\) 33.3939i 1.35542i 0.735331 + 0.677708i \(0.237027\pi\)
−0.735331 + 0.677708i \(0.762973\pi\)
\(608\) 0 0
\(609\) −16.8990 −0.684781
\(610\) 0 0
\(611\) −39.5959 −1.60188
\(612\) 0 0
\(613\) 27.7980i 1.12275i −0.827562 0.561374i \(-0.810273\pi\)
0.827562 0.561374i \(-0.189727\pi\)
\(614\) 0 0
\(615\) 0.606123 6.00000i 0.0244412 0.241943i
\(616\) 0 0
\(617\) 29.5959i 1.19149i −0.803175 0.595743i \(-0.796858\pi\)
0.803175 0.595743i \(-0.203142\pi\)
\(618\) 0 0
\(619\) −41.5505 −1.67006 −0.835028 0.550207i \(-0.814549\pi\)
−0.835028 + 0.550207i \(0.814549\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 10.0000i 0.400642i
\(624\) 0 0
\(625\) 23.0000 + 9.79796i 0.920000 + 0.391918i
\(626\) 0 0
\(627\) 18.6061i 0.743057i
\(628\) 0 0
\(629\) −4.00000 −0.159490
\(630\) 0 0
\(631\) 42.4949 1.69170 0.845848 0.533425i \(-0.179095\pi\)
0.845848 + 0.533425i \(0.179095\pi\)
\(632\) 0 0
\(633\) 29.3939i 1.16830i
\(634\) 0 0
\(635\) 3.34847 33.1464i 0.132880 1.31538i
\(636\) 0 0
\(637\) 4.44949i 0.176295i
\(638\) 0 0
\(639\) −3.30306 −0.130667
\(640\) 0 0
\(641\) 25.7980 1.01896 0.509479 0.860483i \(-0.329838\pi\)
0.509479 + 0.860483i \(0.329838\pi\)
\(642\) 0 0
\(643\) 25.1464i 0.991678i −0.868414 0.495839i \(-0.834861\pi\)
0.868414 0.495839i \(-0.165139\pi\)
\(644\) 0 0
\(645\) −4.89898 0.494897i −0.192897 0.0194866i
\(646\) 0 0
\(647\) 46.2929i 1.81996i −0.414652 0.909980i \(-0.636097\pi\)
0.414652 0.909980i \(-0.363903\pi\)
\(648\) 0 0
\(649\) −7.59592 −0.298166
\(650\) 0 0
\(651\) −21.7980 −0.854329
\(652\) 0 0
\(653\) 20.2020i 0.790567i −0.918559 0.395283i \(-0.870646\pi\)
0.918559 0.395283i \(-0.129354\pi\)
\(654\) 0 0
\(655\) 14.3485 + 1.44949i 0.560641 + 0.0566363i
\(656\) 0 0
\(657\) 8.69694i 0.339300i
\(658\) 0 0
\(659\) 16.8990 0.658291 0.329145 0.944279i \(-0.393239\pi\)
0.329145 + 0.944279i \(0.393239\pi\)
\(660\) 0 0
\(661\) −40.9444 −1.59255 −0.796276 0.604933i \(-0.793200\pi\)
−0.796276 + 0.604933i \(0.793200\pi\)
\(662\) 0 0
\(663\) 21.7980i 0.846563i
\(664\) 0 0
\(665\) 0.348469 3.44949i 0.0135131 0.133765i
\(666\) 0 0
\(667\) 20.0000i 0.774403i
\(668\) 0 0
\(669\) −9.79796 −0.378811
\(670\) 0 0
\(671\) 17.3939 0.671483
\(672\) 0 0
\(673\) 17.7980i 0.686061i −0.939324 0.343030i \(-0.888547\pi\)
0.939324 0.343030i \(-0.111453\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 36.4495i 1.40087i 0.713717 + 0.700434i \(0.247010\pi\)
−0.713717 + 0.700434i \(0.752990\pi\)
\(678\) 0 0
\(679\) −15.7980 −0.606270
\(680\) 0 0
\(681\) −18.0000 −0.689761
\(682\) 0 0
\(683\) 3.59592i 0.137594i −0.997631 0.0687970i \(-0.978084\pi\)
0.997631 0.0687970i \(-0.0219161\pi\)
\(684\) 0 0
\(685\) −0.404082 + 4.00000i −0.0154392 + 0.152832i
\(686\) 0 0
\(687\) 46.8990i 1.78931i
\(688\) 0 0
\(689\) 48.4949 1.84751
\(690\) 0 0
\(691\) −21.1464 −0.804448 −0.402224 0.915541i \(-0.631763\pi\)
−0.402224 + 0.915541i \(0.631763\pi\)
\(692\) 0 0
\(693\) 14.6969i 0.558291i
\(694\) 0 0
\(695\) 3.44949 + 0.348469i 0.130847 + 0.0132182i
\(696\) 0 0
\(697\) 2.20204i 0.0834083i
\(698\) 0 0
\(699\) −72.9898 −2.76073
\(700\) 0 0
\(701\) 11.3031 0.426911 0.213455 0.976953i \(-0.431528\pi\)
0.213455 + 0.976953i \(0.431528\pi\)
\(702\) 0 0
\(703\) 3.10102i 0.116957i
\(704\) 0 0
\(705\) −48.4949 4.89898i −1.82642 0.184506i
\(706\) 0 0
\(707\) 3.55051i 0.133531i
\(708\) 0 0
\(709\) 28.2929 1.06256 0.531280 0.847196i \(-0.321711\pi\)
0.531280 + 0.847196i \(0.321711\pi\)
\(710\) 0 0
\(711\) −20.6969 −0.776196
\(712\) 0 0
\(713\) 25.7980i 0.966141i
\(714\) 0 0
\(715\) −4.89898 + 48.4949i −0.183211 + 1.81361i
\(716\) 0 0
\(717\) 15.1918i 0.567350i
\(718\) 0 0
\(719\) −4.49490 −0.167631 −0.0838157 0.996481i \(-0.526711\pi\)
−0.0838157 + 0.996481i \(0.526711\pi\)
\(720\) 0 0
\(721\) −12.8990 −0.480383
\(722\) 0 0
\(723\) 21.3031i 0.792269i
\(724\) 0 0
\(725\) −33.7980 6.89898i −1.25522 0.256222i
\(726\) 0 0
\(727\) 22.6969i 0.841783i 0.907111 + 0.420891i \(0.138283\pi\)
−0.907111 + 0.420891i \(0.861717\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) 1.79796 0.0664999
\(732\) 0 0
\(733\) 39.6413i 1.46419i 0.681205 + 0.732093i \(0.261456\pi\)
−0.681205 + 0.732093i \(0.738544\pi\)
\(734\) 0 0
\(735\) 0.550510 5.44949i 0.0203059 0.201007i
\(736\) 0 0
\(737\) 39.1918i 1.44365i
\(738\) 0 0
\(739\) 4.49490 0.165347 0.0826737 0.996577i \(-0.473654\pi\)
0.0826737 + 0.996577i \(0.473654\pi\)
\(740\) 0 0
\(741\) −16.8990 −0.620800
\(742\) 0 0
\(743\) 44.6969i 1.63977i 0.572527 + 0.819886i \(0.305963\pi\)
−0.572527 + 0.819886i \(0.694037\pi\)
\(744\) 0 0
\(745\) 8.44949 + 0.853572i 0.309565 + 0.0312725i
\(746\) 0 0
\(747\) 7.34847i 0.268866i
\(748\) 0 0
\(749\) −8.00000 −0.292314
\(750\) 0 0
\(751\) 41.7980 1.52523 0.762615 0.646853i \(-0.223915\pi\)
0.762615 + 0.646853i \(0.223915\pi\)
\(752\) 0 0
\(753\) 15.7980i 0.575710i
\(754\) 0 0
\(755\) −43.5959 4.40408i −1.58662 0.160281i
\(756\) 0 0
\(757\) 51.7980i 1.88263i 0.337531 + 0.941314i \(0.390408\pi\)
−0.337531 + 0.941314i \(0.609592\pi\)
\(758\) 0 0
\(759\) 34.7878 1.26272
\(760\) 0 0
\(761\) −21.1010 −0.764911 −0.382456 0.923974i \(-0.624922\pi\)
−0.382456 + 0.923974i \(0.624922\pi\)
\(762\) 0 0
\(763\) 6.89898i 0.249760i
\(764\) 0 0
\(765\) −1.34847 + 13.3485i −0.0487540 + 0.482615i
\(766\) 0 0
\(767\) 6.89898i 0.249108i
\(768\) 0 0
\(769\) 40.6969 1.46757 0.733785 0.679382i \(-0.237752\pi\)
0.733785 + 0.679382i \(0.237752\pi\)
\(770\) 0 0
\(771\) −21.3031 −0.767211
\(772\) 0 0
\(773\) 1.34847i 0.0485011i 0.999706 + 0.0242505i \(0.00771994\pi\)
−0.999706 + 0.0242505i \(0.992280\pi\)
\(774\) 0 0
\(775\) −43.5959 8.89898i −1.56601 0.319661i
\(776\) 0 0
\(777\) 4.89898i 0.175750i
\(778\) 0 0
\(779\) −1.70714 −0.0611648
\(780\) 0 0
\(781\) 5.39388 0.193008
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0.797959 7.89898i 0.0284804 0.281927i
\(786\) 0 0
\(787\) 50.4495i 1.79833i 0.437610 + 0.899165i \(0.355825\pi\)
−0.437610 + 0.899165i \(0.644175\pi\)
\(788\) 0 0
\(789\) 24.0000 0.854423
\(790\) 0 0
\(791\) 19.7980 0.703934
\(792\) 0 0
\(793\) 15.7980i 0.561002i
\(794\) 0 0
\(795\) 59.3939 + 6.00000i 2.10648 + 0.212798i
\(796\) 0 0
\(797\) 0.944387i 0.0334519i 0.999860 + 0.0167260i \(0.00532429\pi\)
−0.999860 + 0.0167260i \(0.994676\pi\)
\(798\) 0 0
\(799\) 17.7980 0.629647
\(800\) 0 0
\(801\) −30.0000 −1.06000
\(802\) 0 0
\(803\) 14.2020i 0.501179i
\(804\) 0 0
\(805\) −6.44949 0.651531i −0.227315 0.0229634i
\(806\) 0 0
\(807\) 46.8990i 1.65092i
\(808\) 0 0
\(809\) −47.5959 −1.67338 −0.836692 0.547674i \(-0.815513\pi\)
−0.836692 + 0.547674