Properties

Label 320.2.f.b.289.1
Level $320$
Weight $2$
Character 320.289
Analytic conductor $2.555$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 320.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.55521286468\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 289.1
Root \(0.965926 - 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 320.289
Dual form 320.2.f.b.289.2

$q$-expansion

\(f(q)\) \(=\) \(q-2.44949 q^{3} +(-1.41421 - 1.73205i) q^{5} +1.41421i q^{7} +3.00000 q^{9} +O(q^{10})\) \(q-2.44949 q^{3} +(-1.41421 - 1.73205i) q^{5} +1.41421i q^{7} +3.00000 q^{9} +2.00000i q^{11} +5.65685 q^{13} +(3.46410 + 4.24264i) q^{15} +4.89898i q^{17} +6.00000i q^{19} -3.46410i q^{21} -7.07107i q^{23} +(-1.00000 + 4.89898i) q^{25} +6.92820i q^{29} +6.92820 q^{31} -4.89898i q^{33} +(2.44949 - 2.00000i) q^{35} -2.82843 q^{37} -13.8564 q^{39} -4.00000 q^{41} -2.44949 q^{43} +(-4.24264 - 5.19615i) q^{45} +4.24264i q^{47} +5.00000 q^{49} -12.0000i q^{51} +(3.46410 - 2.82843i) q^{55} -14.6969i q^{57} -2.00000i q^{59} -3.46410i q^{61} +4.24264i q^{63} +(-8.00000 - 9.79796i) q^{65} +2.44949 q^{67} +17.3205i q^{69} -6.92820 q^{71} +4.89898i q^{73} +(2.44949 - 12.0000i) q^{75} -2.82843 q^{77} -6.92820 q^{79} -9.00000 q^{81} +12.2474 q^{83} +(8.48528 - 6.92820i) q^{85} -16.9706i q^{87} -2.00000 q^{89} +8.00000i q^{91} -16.9706 q^{93} +(10.3923 - 8.48528i) q^{95} +14.6969i q^{97} +6.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 24q^{9} + O(q^{10}) \) \( 8q + 24q^{9} - 8q^{25} - 32q^{41} + 40q^{49} - 64q^{65} - 72q^{81} - 16q^{89} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(257\) \(261\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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
\(3\) −2.44949 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(4\) 0 0
\(5\) −1.41421 1.73205i −0.632456 0.774597i
\(6\) 0 0
\(7\) 1.41421i 0.534522i 0.963624 + 0.267261i \(0.0861187\pi\)
−0.963624 + 0.267261i \(0.913881\pi\)
\(8\) 0 0
\(9\) 3.00000 1.00000
\(10\) 0 0
\(11\) 2.00000i 0.603023i 0.953463 + 0.301511i \(0.0974911\pi\)
−0.953463 + 0.301511i \(0.902509\pi\)
\(12\) 0 0
\(13\) 5.65685 1.56893 0.784465 0.620174i \(-0.212938\pi\)
0.784465 + 0.620174i \(0.212938\pi\)
\(14\) 0 0
\(15\) 3.46410 + 4.24264i 0.894427 + 1.09545i
\(16\) 0 0
\(17\) 4.89898i 1.18818i 0.804400 + 0.594089i \(0.202487\pi\)
−0.804400 + 0.594089i \(0.797513\pi\)
\(18\) 0 0
\(19\) 6.00000i 1.37649i 0.725476 + 0.688247i \(0.241620\pi\)
−0.725476 + 0.688247i \(0.758380\pi\)
\(20\) 0 0
\(21\) 3.46410i 0.755929i
\(22\) 0 0
\(23\) 7.07107i 1.47442i −0.675664 0.737210i \(-0.736143\pi\)
0.675664 0.737210i \(-0.263857\pi\)
\(24\) 0 0
\(25\) −1.00000 + 4.89898i −0.200000 + 0.979796i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.92820i 1.28654i 0.765641 + 0.643268i \(0.222422\pi\)
−0.765641 + 0.643268i \(0.777578\pi\)
\(30\) 0 0
\(31\) 6.92820 1.24434 0.622171 0.782881i \(-0.286251\pi\)
0.622171 + 0.782881i \(0.286251\pi\)
\(32\) 0 0
\(33\) 4.89898i 0.852803i
\(34\) 0 0
\(35\) 2.44949 2.00000i 0.414039 0.338062i
\(36\) 0 0
\(37\) −2.82843 −0.464991 −0.232495 0.972598i \(-0.574689\pi\)
−0.232495 + 0.972598i \(0.574689\pi\)
\(38\) 0 0
\(39\) −13.8564 −2.21880
\(40\) 0 0
\(41\) −4.00000 −0.624695 −0.312348 0.949968i \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(42\) 0 0
\(43\) −2.44949 −0.373544 −0.186772 0.982403i \(-0.559803\pi\)
−0.186772 + 0.982403i \(0.559803\pi\)
\(44\) 0 0
\(45\) −4.24264 5.19615i −0.632456 0.774597i
\(46\) 0 0
\(47\) 4.24264i 0.618853i 0.950923 + 0.309426i \(0.100137\pi\)
−0.950923 + 0.309426i \(0.899863\pi\)
\(48\) 0 0
\(49\) 5.00000 0.714286
\(50\) 0 0
\(51\) 12.0000i 1.68034i
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 3.46410 2.82843i 0.467099 0.381385i
\(56\) 0 0
\(57\) 14.6969i 1.94666i
\(58\) 0 0
\(59\) 2.00000i 0.260378i −0.991489 0.130189i \(-0.958442\pi\)
0.991489 0.130189i \(-0.0415584\pi\)
\(60\) 0 0
\(61\) 3.46410i 0.443533i −0.975100 0.221766i \(-0.928818\pi\)
0.975100 0.221766i \(-0.0711822\pi\)
\(62\) 0 0
\(63\) 4.24264i 0.534522i
\(64\) 0 0
\(65\) −8.00000 9.79796i −0.992278 1.21529i
\(66\) 0 0
\(67\) 2.44949 0.299253 0.149626 0.988743i \(-0.452193\pi\)
0.149626 + 0.988743i \(0.452193\pi\)
\(68\) 0 0
\(69\) 17.3205i 2.08514i
\(70\) 0 0
\(71\) −6.92820 −0.822226 −0.411113 0.911584i \(-0.634860\pi\)
−0.411113 + 0.911584i \(0.634860\pi\)
\(72\) 0 0
\(73\) 4.89898i 0.573382i 0.958023 + 0.286691i \(0.0925553\pi\)
−0.958023 + 0.286691i \(0.907445\pi\)
\(74\) 0 0
\(75\) 2.44949 12.0000i 0.282843 1.38564i
\(76\) 0 0
\(77\) −2.82843 −0.322329
\(78\) 0 0
\(79\) −6.92820 −0.779484 −0.389742 0.920924i \(-0.627436\pi\)
−0.389742 + 0.920924i \(0.627436\pi\)
\(80\) 0 0
\(81\) −9.00000 −1.00000
\(82\) 0 0
\(83\) 12.2474 1.34433 0.672166 0.740400i \(-0.265364\pi\)
0.672166 + 0.740400i \(0.265364\pi\)
\(84\) 0 0
\(85\) 8.48528 6.92820i 0.920358 0.751469i
\(86\) 0 0
\(87\) 16.9706i 1.81944i
\(88\) 0 0
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) 8.00000i 0.838628i
\(92\) 0 0
\(93\) −16.9706 −1.75977
\(94\) 0 0
\(95\) 10.3923 8.48528i 1.06623 0.870572i
\(96\) 0 0
\(97\) 14.6969i 1.49225i 0.665807 + 0.746124i \(0.268087\pi\)
−0.665807 + 0.746124i \(0.731913\pi\)
\(98\) 0 0
\(99\) 6.00000i 0.603023i
\(100\) 0 0
\(101\) 6.92820i 0.689382i 0.938716 + 0.344691i \(0.112016\pi\)
−0.938716 + 0.344691i \(0.887984\pi\)
\(102\) 0 0
\(103\) 9.89949i 0.975426i 0.873004 + 0.487713i \(0.162169\pi\)
−0.873004 + 0.487713i \(0.837831\pi\)
\(104\) 0 0
\(105\) −6.00000 + 4.89898i −0.585540 + 0.478091i
\(106\) 0 0
\(107\) 2.44949 0.236801 0.118401 0.992966i \(-0.462223\pi\)
0.118401 + 0.992966i \(0.462223\pi\)
\(108\) 0 0
\(109\) 3.46410i 0.331801i 0.986143 + 0.165900i \(0.0530530\pi\)
−0.986143 + 0.165900i \(0.946947\pi\)
\(110\) 0 0
\(111\) 6.92820 0.657596
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) −12.2474 + 10.0000i −1.14208 + 0.932505i
\(116\) 0 0
\(117\) 16.9706 1.56893
\(118\) 0 0
\(119\) −6.92820 −0.635107
\(120\) 0 0
\(121\) 7.00000 0.636364
\(122\) 0 0
\(123\) 9.79796 0.883452
\(124\) 0 0
\(125\) 9.89949 5.19615i 0.885438 0.464758i
\(126\) 0 0
\(127\) 7.07107i 0.627456i 0.949513 + 0.313728i \(0.101578\pi\)
−0.949513 + 0.313728i \(0.898422\pi\)
\(128\) 0 0
\(129\) 6.00000 0.528271
\(130\) 0 0
\(131\) 10.0000i 0.873704i −0.899533 0.436852i \(-0.856093\pi\)
0.899533 0.436852i \(-0.143907\pi\)
\(132\) 0 0
\(133\) −8.48528 −0.735767
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 9.79796i 0.837096i −0.908195 0.418548i \(-0.862539\pi\)
0.908195 0.418548i \(-0.137461\pi\)
\(138\) 0 0
\(139\) 10.0000i 0.848189i 0.905618 + 0.424094i \(0.139408\pi\)
−0.905618 + 0.424094i \(0.860592\pi\)
\(140\) 0 0
\(141\) 10.3923i 0.875190i
\(142\) 0 0
\(143\) 11.3137i 0.946100i
\(144\) 0 0
\(145\) 12.0000 9.79796i 0.996546 0.813676i
\(146\) 0 0
\(147\) −12.2474 −1.01015
\(148\) 0 0
\(149\) 10.3923i 0.851371i −0.904871 0.425685i \(-0.860033\pi\)
0.904871 0.425685i \(-0.139967\pi\)
\(150\) 0 0
\(151\) 13.8564 1.12762 0.563809 0.825905i \(-0.309335\pi\)
0.563809 + 0.825905i \(0.309335\pi\)
\(152\) 0 0
\(153\) 14.6969i 1.18818i
\(154\) 0 0
\(155\) −9.79796 12.0000i −0.786991 0.963863i
\(156\) 0 0
\(157\) 2.82843 0.225733 0.112867 0.993610i \(-0.463997\pi\)
0.112867 + 0.993610i \(0.463997\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 10.0000 0.788110
\(162\) 0 0
\(163\) −22.0454 −1.72673 −0.863365 0.504580i \(-0.831647\pi\)
−0.863365 + 0.504580i \(0.831647\pi\)
\(164\) 0 0
\(165\) −8.48528 + 6.92820i −0.660578 + 0.539360i
\(166\) 0 0
\(167\) 9.89949i 0.766046i 0.923739 + 0.383023i \(0.125117\pi\)
−0.923739 + 0.383023i \(0.874883\pi\)
\(168\) 0 0
\(169\) 19.0000 1.46154
\(170\) 0 0
\(171\) 18.0000i 1.37649i
\(172\) 0 0
\(173\) 8.48528 0.645124 0.322562 0.946548i \(-0.395456\pi\)
0.322562 + 0.946548i \(0.395456\pi\)
\(174\) 0 0
\(175\) −6.92820 1.41421i −0.523723 0.106904i
\(176\) 0 0
\(177\) 4.89898i 0.368230i
\(178\) 0 0
\(179\) 2.00000i 0.149487i 0.997203 + 0.0747435i \(0.0238138\pi\)
−0.997203 + 0.0747435i \(0.976186\pi\)
\(180\) 0 0
\(181\) 20.7846i 1.54491i −0.635071 0.772454i \(-0.719029\pi\)
0.635071 0.772454i \(-0.280971\pi\)
\(182\) 0 0
\(183\) 8.48528i 0.627250i
\(184\) 0 0
\(185\) 4.00000 + 4.89898i 0.294086 + 0.360180i
\(186\) 0 0
\(187\) −9.79796 −0.716498
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −13.8564 −1.00261 −0.501307 0.865269i \(-0.667147\pi\)
−0.501307 + 0.865269i \(0.667147\pi\)
\(192\) 0 0
\(193\) 4.89898i 0.352636i 0.984333 + 0.176318i \(0.0564187\pi\)
−0.984333 + 0.176318i \(0.943581\pi\)
\(194\) 0 0
\(195\) 19.5959 + 24.0000i 1.40329 + 1.71868i
\(196\) 0 0
\(197\) −22.6274 −1.61214 −0.806068 0.591822i \(-0.798409\pi\)
−0.806068 + 0.591822i \(0.798409\pi\)
\(198\) 0 0
\(199\) 20.7846 1.47338 0.736691 0.676230i \(-0.236387\pi\)
0.736691 + 0.676230i \(0.236387\pi\)
\(200\) 0 0
\(201\) −6.00000 −0.423207
\(202\) 0 0
\(203\) −9.79796 −0.687682
\(204\) 0 0
\(205\) 5.65685 + 6.92820i 0.395092 + 0.483887i
\(206\) 0 0
\(207\) 21.2132i 1.47442i
\(208\) 0 0
\(209\) −12.0000 −0.830057
\(210\) 0 0
\(211\) 10.0000i 0.688428i −0.938891 0.344214i \(-0.888145\pi\)
0.938891 0.344214i \(-0.111855\pi\)
\(212\) 0 0
\(213\) 16.9706 1.16280
\(214\) 0 0
\(215\) 3.46410 + 4.24264i 0.236250 + 0.289346i
\(216\) 0 0
\(217\) 9.79796i 0.665129i
\(218\) 0 0
\(219\) 12.0000i 0.810885i
\(220\) 0 0
\(221\) 27.7128i 1.86417i
\(222\) 0 0
\(223\) 26.8701i 1.79935i −0.436558 0.899676i \(-0.643803\pi\)
0.436558 0.899676i \(-0.356197\pi\)
\(224\) 0 0
\(225\) −3.00000 + 14.6969i −0.200000 + 0.979796i
\(226\) 0 0
\(227\) 17.1464 1.13805 0.569024 0.822321i \(-0.307321\pi\)
0.569024 + 0.822321i \(0.307321\pi\)
\(228\) 0 0
\(229\) 6.92820i 0.457829i −0.973447 0.228914i \(-0.926482\pi\)
0.973447 0.228914i \(-0.0735176\pi\)
\(230\) 0 0
\(231\) 6.92820 0.455842
\(232\) 0 0
\(233\) 14.6969i 0.962828i −0.876493 0.481414i \(-0.840123\pi\)
0.876493 0.481414i \(-0.159877\pi\)
\(234\) 0 0
\(235\) 7.34847 6.00000i 0.479361 0.391397i
\(236\) 0 0
\(237\) 16.9706 1.10236
\(238\) 0 0
\(239\) 6.92820 0.448148 0.224074 0.974572i \(-0.428064\pi\)
0.224074 + 0.974572i \(0.428064\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) 0 0
\(243\) 22.0454 1.41421
\(244\) 0 0
\(245\) −7.07107 8.66025i −0.451754 0.553283i
\(246\) 0 0
\(247\) 33.9411i 2.15962i
\(248\) 0 0
\(249\) −30.0000 −1.90117
\(250\) 0 0
\(251\) 26.0000i 1.64111i 0.571571 + 0.820553i \(0.306334\pi\)
−0.571571 + 0.820553i \(0.693666\pi\)
\(252\) 0 0
\(253\) 14.1421 0.889108
\(254\) 0 0
\(255\) −20.7846 + 16.9706i −1.30158 + 1.06274i
\(256\) 0 0
\(257\) 9.79796i 0.611180i −0.952163 0.305590i \(-0.901146\pi\)
0.952163 0.305590i \(-0.0988537\pi\)
\(258\) 0 0
\(259\) 4.00000i 0.248548i
\(260\) 0 0
\(261\) 20.7846i 1.28654i
\(262\) 0 0
\(263\) 26.8701i 1.65688i −0.560079 0.828439i \(-0.689229\pi\)
0.560079 0.828439i \(-0.310771\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 4.89898 0.299813
\(268\) 0 0
\(269\) 17.3205i 1.05605i −0.849229 0.528025i \(-0.822933\pi\)
0.849229 0.528025i \(-0.177067\pi\)
\(270\) 0 0
\(271\) −20.7846 −1.26258 −0.631288 0.775549i \(-0.717473\pi\)
−0.631288 + 0.775549i \(0.717473\pi\)
\(272\) 0 0
\(273\) 19.5959i 1.18600i
\(274\) 0 0
\(275\) −9.79796 2.00000i −0.590839 0.120605i
\(276\) 0 0
\(277\) −14.1421 −0.849719 −0.424859 0.905259i \(-0.639676\pi\)
−0.424859 + 0.905259i \(0.639676\pi\)
\(278\) 0 0
\(279\) 20.7846 1.24434
\(280\) 0 0
\(281\) 8.00000 0.477240 0.238620 0.971113i \(-0.423305\pi\)
0.238620 + 0.971113i \(0.423305\pi\)
\(282\) 0 0
\(283\) 2.44949 0.145607 0.0728035 0.997346i \(-0.476805\pi\)
0.0728035 + 0.997346i \(0.476805\pi\)
\(284\) 0 0
\(285\) −25.4558 + 20.7846i −1.50787 + 1.23117i
\(286\) 0 0
\(287\) 5.65685i 0.333914i
\(288\) 0 0
\(289\) −7.00000 −0.411765
\(290\) 0 0
\(291\) 36.0000i 2.11036i
\(292\) 0 0
\(293\) −2.82843 −0.165238 −0.0826192 0.996581i \(-0.526329\pi\)
−0.0826192 + 0.996581i \(0.526329\pi\)
\(294\) 0 0
\(295\) −3.46410 + 2.82843i −0.201688 + 0.164677i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 40.0000i 2.31326i
\(300\) 0 0
\(301\) 3.46410i 0.199667i
\(302\) 0 0
\(303\) 16.9706i 0.974933i
\(304\) 0 0
\(305\) −6.00000 + 4.89898i −0.343559 + 0.280515i
\(306\) 0 0
\(307\) 2.44949 0.139800 0.0698999 0.997554i \(-0.477732\pi\)
0.0698999 + 0.997554i \(0.477732\pi\)
\(308\) 0 0
\(309\) 24.2487i 1.37946i
\(310\) 0 0
\(311\) −27.7128 −1.57145 −0.785725 0.618576i \(-0.787710\pi\)
−0.785725 + 0.618576i \(0.787710\pi\)
\(312\) 0 0
\(313\) 29.3939i 1.66144i −0.556690 0.830720i \(-0.687929\pi\)
0.556690 0.830720i \(-0.312071\pi\)
\(314\) 0 0
\(315\) 7.34847 6.00000i 0.414039 0.338062i
\(316\) 0 0
\(317\) −22.6274 −1.27088 −0.635441 0.772149i \(-0.719182\pi\)
−0.635441 + 0.772149i \(0.719182\pi\)
\(318\) 0 0
\(319\) −13.8564 −0.775810
\(320\) 0 0
\(321\) −6.00000 −0.334887
\(322\) 0 0
\(323\) −29.3939 −1.63552
\(324\) 0 0
\(325\) −5.65685 + 27.7128i −0.313786 + 1.53723i
\(326\) 0 0
\(327\) 8.48528i 0.469237i
\(328\) 0 0
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) 6.00000i 0.329790i 0.986311 + 0.164895i \(0.0527285\pi\)
−0.986311 + 0.164895i \(0.947272\pi\)
\(332\) 0 0
\(333\) −8.48528 −0.464991
\(334\) 0 0
\(335\) −3.46410 4.24264i −0.189264 0.231800i
\(336\) 0 0
\(337\) 19.5959i 1.06746i −0.845656 0.533729i \(-0.820790\pi\)
0.845656 0.533729i \(-0.179210\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 13.8564i 0.750366i
\(342\) 0 0
\(343\) 16.9706i 0.916324i
\(344\) 0 0
\(345\) 30.0000 24.4949i 1.61515 1.31876i
\(346\) 0 0
\(347\) 17.1464 0.920468 0.460234 0.887798i \(-0.347765\pi\)
0.460234 + 0.887798i \(0.347765\pi\)
\(348\) 0 0
\(349\) 6.92820i 0.370858i −0.982658 0.185429i \(-0.940632\pi\)
0.982658 0.185429i \(-0.0593675\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 9.79796i 0.521493i 0.965407 + 0.260746i \(0.0839686\pi\)
−0.965407 + 0.260746i \(0.916031\pi\)
\(354\) 0 0
\(355\) 9.79796 + 12.0000i 0.520022 + 0.636894i
\(356\) 0 0
\(357\) 16.9706 0.898177
\(358\) 0 0
\(359\) 27.7128 1.46263 0.731313 0.682042i \(-0.238908\pi\)
0.731313 + 0.682042i \(0.238908\pi\)
\(360\) 0 0
\(361\) −17.0000 −0.894737
\(362\) 0 0
\(363\) −17.1464 −0.899954
\(364\) 0 0
\(365\) 8.48528 6.92820i 0.444140 0.362639i
\(366\) 0 0
\(367\) 9.89949i 0.516749i −0.966045 0.258375i \(-0.916813\pi\)
0.966045 0.258375i \(-0.0831869\pi\)
\(368\) 0 0
\(369\) −12.0000 −0.624695
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 19.7990 1.02515 0.512576 0.858642i \(-0.328691\pi\)
0.512576 + 0.858642i \(0.328691\pi\)
\(374\) 0 0
\(375\) −24.2487 + 12.7279i −1.25220 + 0.657267i
\(376\) 0 0
\(377\) 39.1918i 2.01848i
\(378\) 0 0
\(379\) 14.0000i 0.719132i −0.933120 0.359566i \(-0.882925\pi\)
0.933120 0.359566i \(-0.117075\pi\)
\(380\) 0 0
\(381\) 17.3205i 0.887357i
\(382\) 0 0
\(383\) 12.7279i 0.650366i 0.945651 + 0.325183i \(0.105426\pi\)
−0.945651 + 0.325183i \(0.894574\pi\)
\(384\) 0 0
\(385\) 4.00000 + 4.89898i 0.203859 + 0.249675i
\(386\) 0 0
\(387\) −7.34847 −0.373544
\(388\) 0 0
\(389\) 3.46410i 0.175637i 0.996136 + 0.0878185i \(0.0279895\pi\)
−0.996136 + 0.0878185i \(0.972010\pi\)
\(390\) 0 0
\(391\) 34.6410 1.75187
\(392\) 0 0
\(393\) 24.4949i 1.23560i
\(394\) 0 0
\(395\) 9.79796 + 12.0000i 0.492989 + 0.603786i
\(396\) 0 0
\(397\) −11.3137 −0.567819 −0.283909 0.958851i \(-0.591631\pi\)
−0.283909 + 0.958851i \(0.591631\pi\)
\(398\) 0 0
\(399\) 20.7846 1.04053
\(400\) 0 0
\(401\) 22.0000 1.09863 0.549314 0.835616i \(-0.314889\pi\)
0.549314 + 0.835616i \(0.314889\pi\)
\(402\) 0 0
\(403\) 39.1918 1.95228
\(404\) 0 0
\(405\) 12.7279 + 15.5885i 0.632456 + 0.774597i
\(406\) 0 0
\(407\) 5.65685i 0.280400i
\(408\) 0 0
\(409\) −20.0000 −0.988936 −0.494468 0.869196i \(-0.664637\pi\)
−0.494468 + 0.869196i \(0.664637\pi\)
\(410\) 0 0
\(411\) 24.0000i 1.18383i
\(412\) 0 0
\(413\) 2.82843 0.139178
\(414\) 0 0
\(415\) −17.3205 21.2132i −0.850230 1.04132i
\(416\) 0 0
\(417\) 24.4949i 1.19952i
\(418\) 0 0
\(419\) 26.0000i 1.27018i 0.772437 + 0.635092i \(0.219038\pi\)
−0.772437 + 0.635092i \(0.780962\pi\)
\(420\) 0 0
\(421\) 3.46410i 0.168830i 0.996431 + 0.0844150i \(0.0269021\pi\)
−0.996431 + 0.0844150i \(0.973098\pi\)
\(422\) 0 0
\(423\) 12.7279i 0.618853i
\(424\) 0 0
\(425\) −24.0000 4.89898i −1.16417 0.237635i
\(426\) 0 0
\(427\) 4.89898 0.237078
\(428\) 0 0
\(429\) 27.7128i 1.33799i
\(430\) 0 0
\(431\) −6.92820 −0.333720 −0.166860 0.985981i \(-0.553363\pi\)
−0.166860 + 0.985981i \(0.553363\pi\)
\(432\) 0 0
\(433\) 34.2929i 1.64801i 0.566583 + 0.824005i \(0.308265\pi\)
−0.566583 + 0.824005i \(0.691735\pi\)
\(434\) 0 0
\(435\) −29.3939 + 24.0000i −1.40933 + 1.15071i
\(436\) 0 0
\(437\) 42.4264 2.02953
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 15.0000 0.714286
\(442\) 0 0
\(443\) 22.0454 1.04741 0.523704 0.851900i \(-0.324550\pi\)
0.523704 + 0.851900i \(0.324550\pi\)
\(444\) 0 0
\(445\) 2.82843 + 3.46410i 0.134080 + 0.164214i
\(446\) 0 0
\(447\) 25.4558i 1.20402i
\(448\) 0 0
\(449\) 16.0000 0.755087 0.377543 0.925992i \(-0.376769\pi\)
0.377543 + 0.925992i \(0.376769\pi\)
\(450\) 0 0
\(451\) 8.00000i 0.376705i
\(452\) 0 0
\(453\) −33.9411 −1.59469
\(454\) 0 0
\(455\) 13.8564 11.3137i 0.649598 0.530395i
\(456\) 0 0
\(457\) 19.5959i 0.916658i 0.888783 + 0.458329i \(0.151552\pi\)
−0.888783 + 0.458329i \(0.848448\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 20.7846i 0.968036i 0.875058 + 0.484018i \(0.160823\pi\)
−0.875058 + 0.484018i \(0.839177\pi\)
\(462\) 0 0
\(463\) 1.41421i 0.0657241i −0.999460 0.0328620i \(-0.989538\pi\)
0.999460 0.0328620i \(-0.0104622\pi\)
\(464\) 0 0
\(465\) 24.0000 + 29.3939i 1.11297 + 1.36311i
\(466\) 0 0
\(467\) −31.8434 −1.47354 −0.736768 0.676146i \(-0.763649\pi\)
−0.736768 + 0.676146i \(0.763649\pi\)
\(468\) 0 0
\(469\) 3.46410i 0.159957i
\(470\) 0 0
\(471\) −6.92820 −0.319235
\(472\) 0 0
\(473\) 4.89898i 0.225255i
\(474\) 0 0
\(475\) −29.3939 6.00000i −1.34868 0.275299i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 41.5692 1.89935 0.949673 0.313243i \(-0.101415\pi\)
0.949673 + 0.313243i \(0.101415\pi\)
\(480\) 0 0
\(481\) −16.0000 −0.729537
\(482\) 0 0
\(483\) −24.4949 −1.11456
\(484\) 0 0
\(485\) 25.4558 20.7846i 1.15589 0.943781i
\(486\) 0 0
\(487\) 26.8701i 1.21760i −0.793324 0.608799i \(-0.791651\pi\)
0.793324 0.608799i \(-0.208349\pi\)
\(488\) 0 0
\(489\) 54.0000 2.44196
\(490\) 0 0
\(491\) 38.0000i 1.71492i 0.514554 + 0.857458i \(0.327958\pi\)
−0.514554 + 0.857458i \(0.672042\pi\)
\(492\) 0 0
\(493\) −33.9411 −1.52863
\(494\) 0 0
\(495\) 10.3923 8.48528i 0.467099 0.381385i
\(496\) 0 0
\(497\) 9.79796i 0.439499i
\(498\) 0 0
\(499\) 6.00000i 0.268597i −0.990941 0.134298i \(-0.957122\pi\)
0.990941 0.134298i \(-0.0428781\pi\)
\(500\) 0 0
\(501\) 24.2487i 1.08335i
\(502\) 0 0
\(503\) 21.2132i 0.945850i 0.881103 + 0.472925i \(0.156802\pi\)
−0.881103 + 0.472925i \(0.843198\pi\)
\(504\) 0 0
\(505\) 12.0000 9.79796i 0.533993 0.436003i
\(506\) 0 0
\(507\) −46.5403 −2.06693
\(508\) 0 0
\(509\) 20.7846i 0.921262i −0.887592 0.460631i \(-0.847623\pi\)
0.887592 0.460631i \(-0.152377\pi\)
\(510\) 0 0
\(511\) −6.92820 −0.306486
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 17.1464 14.0000i 0.755562 0.616914i
\(516\) 0 0
\(517\) −8.48528 −0.373182
\(518\) 0 0
\(519\) −20.7846 −0.912343
\(520\) 0 0
\(521\) 34.0000 1.48957 0.744784 0.667306i \(-0.232553\pi\)
0.744784 + 0.667306i \(0.232553\pi\)
\(522\) 0 0
\(523\) 2.44949 0.107109 0.0535544 0.998565i \(-0.482945\pi\)
0.0535544 + 0.998565i \(0.482945\pi\)
\(524\) 0 0
\(525\) 16.9706 + 3.46410i 0.740656 + 0.151186i
\(526\) 0 0
\(527\) 33.9411i 1.47850i
\(528\) 0 0
\(529\) −27.0000 −1.17391
\(530\) 0 0
\(531\) 6.00000i 0.260378i
\(532\) 0 0
\(533\) −22.6274 −0.980102
\(534\) 0 0
\(535\) −3.46410 4.24264i −0.149766 0.183425i
\(536\) 0 0
\(537\) 4.89898i 0.211407i
\(538\) 0 0
\(539\) 10.0000i 0.430730i
\(540\) 0 0
\(541\) 6.92820i 0.297867i −0.988847 0.148933i \(-0.952416\pi\)
0.988847 0.148933i \(-0.0475840\pi\)
\(542\) 0 0
\(543\) 50.9117i 2.18483i
\(544\) 0 0
\(545\) 6.00000 4.89898i 0.257012 0.209849i
\(546\) 0 0
\(547\) 26.9444 1.15206 0.576029 0.817429i \(-0.304601\pi\)
0.576029 + 0.817429i \(0.304601\pi\)
\(548\) 0 0
\(549\) 10.3923i 0.443533i
\(550\) 0 0
\(551\) −41.5692 −1.77091
\(552\) 0 0
\(553\) 9.79796i 0.416652i
\(554\) 0 0
\(555\) −9.79796 12.0000i −0.415900 0.509372i
\(556\) 0 0
\(557\) 25.4558 1.07860 0.539299 0.842114i \(-0.318689\pi\)
0.539299 + 0.842114i \(0.318689\pi\)
\(558\) 0 0
\(559\) −13.8564 −0.586064
\(560\) 0 0
\(561\) 24.0000 1.01328
\(562\) 0 0
\(563\) 7.34847 0.309701 0.154851 0.987938i \(-0.450510\pi\)
0.154851 + 0.987938i \(0.450510\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 12.7279i 0.534522i
\(568\) 0 0
\(569\) −28.0000 −1.17382 −0.586911 0.809652i \(-0.699656\pi\)
−0.586911 + 0.809652i \(0.699656\pi\)
\(570\) 0 0
\(571\) 30.0000i 1.25546i −0.778431 0.627730i \(-0.783984\pi\)
0.778431 0.627730i \(-0.216016\pi\)
\(572\) 0 0
\(573\) 33.9411 1.41791
\(574\) 0 0
\(575\) 34.6410 + 7.07107i 1.44463 + 0.294884i
\(576\) 0 0
\(577\) 29.3939i 1.22368i −0.790980 0.611842i \(-0.790429\pi\)
0.790980 0.611842i \(-0.209571\pi\)
\(578\) 0 0
\(579\) 12.0000i 0.498703i
\(580\) 0 0
\(581\) 17.3205i 0.718576i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −24.0000 29.3939i −0.992278 1.21529i
\(586\) 0 0
\(587\) −12.2474 −0.505506 −0.252753 0.967531i \(-0.581336\pi\)
−0.252753 + 0.967531i \(0.581336\pi\)
\(588\) 0 0
\(589\) 41.5692i 1.71283i
\(590\) 0 0
\(591\) 55.4256 2.27991
\(592\) 0 0
\(593\) 29.3939i 1.20706i 0.797340 + 0.603531i \(0.206240\pi\)
−0.797340 + 0.603531i \(0.793760\pi\)
\(594\) 0 0
\(595\) 9.79796 + 12.0000i 0.401677 + 0.491952i
\(596\) 0 0
\(597\) −50.9117 −2.08368
\(598\) 0 0
\(599\) −20.7846 −0.849236 −0.424618 0.905373i \(-0.639592\pi\)
−0.424618 + 0.905373i \(0.639592\pi\)
\(600\) 0 0
\(601\) −24.0000 −0.978980 −0.489490 0.872009i \(-0.662817\pi\)
−0.489490 + 0.872009i \(0.662817\pi\)
\(602\) 0 0
\(603\) 7.34847 0.299253
\(604\) 0 0
\(605\) −9.89949 12.1244i −0.402472 0.492925i
\(606\) 0 0
\(607\) 41.0122i 1.66463i −0.554300 0.832317i \(-0.687014\pi\)
0.554300 0.832317i \(-0.312986\pi\)
\(608\) 0 0
\(609\) 24.0000 0.972529
\(610\) 0 0
\(611\) 24.0000i 0.970936i
\(612\) 0 0
\(613\) 5.65685 0.228478 0.114239 0.993453i \(-0.463557\pi\)
0.114239 + 0.993453i \(0.463557\pi\)
\(614\) 0 0
\(615\) −13.8564 16.9706i −0.558744 0.684319i
\(616\) 0 0
\(617\) 24.4949i 0.986127i 0.869993 + 0.493064i \(0.164123\pi\)
−0.869993 + 0.493064i \(0.835877\pi\)
\(618\) 0 0
\(619\) 14.0000i 0.562708i 0.959604 + 0.281354i \(0.0907834\pi\)
−0.959604 + 0.281354i \(0.909217\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 2.82843i 0.113319i
\(624\) 0 0
\(625\) −23.0000 9.79796i −0.920000 0.391918i
\(626\) 0 0
\(627\) 29.3939 1.17388
\(628\) 0 0
\(629\) 13.8564i 0.552491i
\(630\) 0 0
\(631\) 13.8564 0.551615 0.275807 0.961213i \(-0.411055\pi\)
0.275807 + 0.961213i \(0.411055\pi\)
\(632\) 0 0
\(633\) 24.4949i 0.973585i
\(634\) 0 0
\(635\) 12.2474 10.0000i 0.486025 0.396838i
\(636\) 0 0
\(637\) 28.2843 1.12066
\(638\) 0 0
\(639\) −20.7846 −0.822226
\(640\) 0 0
\(641\) −40.0000 −1.57991 −0.789953 0.613168i \(-0.789895\pi\)
−0.789953 + 0.613168i \(0.789895\pi\)
\(642\) 0 0
\(643\) 2.44949 0.0965984 0.0482992 0.998833i \(-0.484620\pi\)
0.0482992 + 0.998833i \(0.484620\pi\)
\(644\) 0 0
\(645\) −8.48528 10.3923i −0.334108 0.409197i
\(646\) 0 0
\(647\) 4.24264i 0.166795i −0.996516 0.0833977i \(-0.973423\pi\)
0.996516 0.0833977i \(-0.0265772\pi\)
\(648\) 0 0
\(649\) 4.00000 0.157014
\(650\) 0 0
\(651\) 24.0000i 0.940634i
\(652\) 0 0
\(653\) 5.65685 0.221370 0.110685 0.993856i \(-0.464696\pi\)
0.110685 + 0.993856i \(0.464696\pi\)
\(654\) 0 0
\(655\) −17.3205 + 14.1421i −0.676768 + 0.552579i
\(656\) 0 0
\(657\) 14.6969i 0.573382i
\(658\) 0 0
\(659\) 2.00000i 0.0779089i 0.999241 + 0.0389545i \(0.0124027\pi\)
−0.999241 + 0.0389545i \(0.987597\pi\)
\(660\) 0 0
\(661\) 10.3923i 0.404214i 0.979363 + 0.202107i \(0.0647788\pi\)
−0.979363 + 0.202107i \(0.935221\pi\)
\(662\) 0 0
\(663\) 67.8823i 2.63633i
\(664\) 0 0
\(665\) 12.0000 + 14.6969i 0.465340 + 0.569923i
\(666\) 0 0
\(667\) 48.9898 1.89689
\(668\) 0 0
\(669\) 65.8179i 2.54467i
\(670\) 0 0
\(671\) 6.92820 0.267460
\(672\) 0 0
\(673\) 24.4949i 0.944209i −0.881543 0.472104i \(-0.843495\pi\)
0.881543 0.472104i \(-0.156505\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 5.65685 0.217411 0.108705 0.994074i \(-0.465330\pi\)
0.108705 + 0.994074i \(0.465330\pi\)
\(678\) 0 0
\(679\) −20.7846 −0.797640
\(680\) 0 0
\(681\) −42.0000 −1.60944
\(682\) 0 0
\(683\) −2.44949 −0.0937271 −0.0468636 0.998901i \(-0.514923\pi\)
−0.0468636 + 0.998901i \(0.514923\pi\)
\(684\) 0 0
\(685\) −16.9706 + 13.8564i −0.648412 + 0.529426i
\(686\) 0 0
\(687\) 16.9706i 0.647467i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 42.0000i 1.59776i −0.601494 0.798878i \(-0.705427\pi\)
0.601494 0.798878i \(-0.294573\pi\)
\(692\) 0 0
\(693\) −8.48528 −0.322329
\(694\) 0 0
\(695\) 17.3205 14.1421i 0.657004 0.536442i
\(696\) 0 0
\(697\) 19.5959i 0.742248i
\(698\) 0 0
\(699\) 36.0000i 1.36165i
\(700\) 0 0
\(701\) 3.46410i 0.130837i 0.997858 + 0.0654187i \(0.0208383\pi\)
−0.997858 + 0.0654187i \(0.979162\pi\)
\(702\) 0 0
\(703\) 16.9706i 0.640057i
\(704\) 0 0
\(705\) −18.0000 + 14.6969i −0.677919 + 0.553519i
\(706\) 0 0
\(707\) −9.79796 −0.368490
\(708\) 0 0
\(709\) 34.6410i 1.30097i 0.759519 + 0.650485i \(0.225434\pi\)
−0.759519 + 0.650485i \(0.774566\pi\)
\(710\) 0 0
\(711\) −20.7846 −0.779484
\(712\) 0 0
\(713\) 48.9898i 1.83468i
\(714\) 0 0
\(715\) 19.5959 16.0000i 0.732846 0.598366i
\(716\) 0 0
\(717\) −16.9706 −0.633777
\(718\) 0 0
\(719\) −6.92820 −0.258378 −0.129189 0.991620i \(-0.541237\pi\)
−0.129189 + 0.991620i \(0.541237\pi\)
\(720\) 0 0
\(721\) −14.0000 −0.521387
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −33.9411 6.92820i −1.26054 0.257307i
\(726\) 0 0
\(727\) 18.3848i 0.681854i 0.940090 + 0.340927i \(0.110741\pi\)
−0.940090 + 0.340927i \(0.889259\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 12.0000i 0.443836i
\(732\) 0 0
\(733\) −19.7990 −0.731292 −0.365646 0.930754i \(-0.619152\pi\)
−0.365646 + 0.930754i \(0.619152\pi\)
\(734\) 0 0
\(735\) 17.3205 + 21.2132i 0.638877 + 0.782461i
\(736\) 0 0
\(737\) 4.89898i 0.180456i
\(738\) 0 0
\(739\) 30.0000i 1.10357i −0.833987 0.551784i \(-0.813947\pi\)
0.833987 0.551784i \(-0.186053\pi\)
\(740\) 0 0
\(741\) 83.1384i 3.05417i
\(742\) 0 0
\(743\) 4.24264i 0.155647i −0.996967 0.0778237i \(-0.975203\pi\)
0.996967 0.0778237i \(-0.0247971\pi\)
\(744\) 0 0
\(745\) −18.0000 + 14.6969i −0.659469 + 0.538454i
\(746\) 0 0
\(747\) 36.7423 1.34433
\(748\) 0 0
\(749\) 3.46410i 0.126576i
\(750\) 0 0
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) 63.6867i 2.32087i
\(754\) 0 0
\(755\) −19.5959 24.0000i −0.713168 0.873449i
\(756\) 0 0
\(757\) 2.82843 0.102801 0.0514005 0.998678i \(-0.483632\pi\)
0.0514005 + 0.998678i \(0.483632\pi\)
\(758\) 0 0
\(759\) −34.6410 −1.25739
\(760\) 0 0
\(761\) 38.0000 1.37750 0.688749 0.724999i \(-0.258160\pi\)
0.688749 + 0.724999i \(0.258160\pi\)
\(762\) 0 0
\(763\) −4.89898 −0.177355
\(764\) 0 0
\(765\) 25.4558 20.7846i 0.920358 0.751469i
\(766\) 0 0
\(767\) 11.3137i 0.408514i
\(768\) 0 0
\(769\) 42.0000 1.51456 0.757279 0.653091i \(-0.226528\pi\)
0.757279 + 0.653091i \(0.226528\pi\)
\(770\) 0 0
\(771\) 24.0000i 0.864339i
\(772\) 0 0
\(773\) 45.2548 1.62770 0.813852 0.581073i \(-0.197367\pi\)
0.813852 + 0.581073i \(0.197367\pi\)
\(774\) 0 0
\(775\) −6.92820 + 33.9411i −0.248868 + 1.21920i
\(776\) 0 0
\(777\) 9.79796i 0.351500i
\(778\) 0 0
\(779\) 24.0000i 0.859889i
\(780\) 0 0
\(781\) 13.8564i 0.495821i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −4.00000 4.89898i −0.142766 0.174852i
\(786\) 0 0
\(787\) −46.5403 −1.65898 −0.829491 0.558520i \(-0.811370\pi\)
−0.829491 + 0.558520i \(0.811370\pi\)
\(788\) 0 0
\(789\) 65.8179i 2.34318i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 19.5959i 0.695871i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −33.9411 −1.20226 −0.601128 0.799153i \(-0.705282\pi\)
−0.601128 + 0.799153i \(0.705282\pi\)
\(798\) 0 0
\(799\) −20.7846 −0.735307
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 0 0
\(803\) −9.79796