Properties

Label 1764.2.e.f.1079.4
Level $1764$
Weight $2$
Character 1764.1079
Analytic conductor $14.086$
Analytic rank $0$
Dimension $8$
CM discriminant -7
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1764.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.0856109166\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.157351936.1
Defining polynomial: \(x^{8} + x^{4} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 1079.4
Root \(-1.28897 - 0.581861i\) of defining polynomial
Character \(\chi\) \(=\) 1764.1079
Dual form 1764.2.e.f.1079.3

$q$-expansion

\(f(q)\) \(=\) \(q+(-0.581861 + 1.28897i) q^{2} +(-1.32288 - 1.50000i) q^{4} +(2.70318 - 0.832353i) q^{8} +O(q^{10})\) \(q+(-0.581861 + 1.28897i) q^{2} +(-1.32288 - 1.50000i) q^{4} +(2.70318 - 0.832353i) q^{8} -6.57008 q^{11} +(-0.500000 + 3.96863i) q^{16} +(3.82288 - 8.46863i) q^{22} +1.91520 q^{23} +5.00000 q^{25} +6.06910i q^{29} +(-4.82450 - 2.95367i) q^{32} +10.5830 q^{37} -12.0000i q^{43} +(8.69140 + 9.85513i) q^{44} +(-1.11438 + 2.46863i) q^{46} +(-2.90930 + 6.44484i) q^{50} -14.5544i q^{53} +(-7.82288 - 3.53137i) q^{58} +(6.61438 - 4.50000i) q^{64} -15.8745i q^{67} +15.0554 q^{71} +(-6.15784 + 13.6412i) q^{74} -15.8745i q^{79} +(15.4676 + 6.98233i) q^{86} +(-17.7601 + 5.46863i) q^{88} +(-2.53357 - 2.87280i) q^{92} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 4q^{16} + 20q^{22} + 40q^{25} + 44q^{46} - 52q^{58} - 68q^{88} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.581861 + 1.28897i −0.411438 + 0.911438i
\(3\) 0 0
\(4\) −1.32288 1.50000i −0.661438 0.750000i
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 2.70318 0.832353i 0.955719 0.294281i
\(9\) 0 0
\(10\) 0 0
\(11\) −6.57008 −1.98096 −0.990478 0.137675i \(-0.956037\pi\)
−0.990478 + 0.137675i \(0.956037\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −0.500000 + 3.96863i −0.125000 + 0.992157i
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 3.82288 8.46863i 0.815040 1.80552i
\(23\) 1.91520 0.399346 0.199673 0.979863i \(-0.436012\pi\)
0.199673 + 0.979863i \(0.436012\pi\)
\(24\) 0 0
\(25\) 5.00000 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.06910i 1.12700i 0.826115 + 0.563502i \(0.190546\pi\)
−0.826115 + 0.563502i \(0.809454\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −4.82450 2.95367i −0.852859 0.522141i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 10.5830 1.73984 0.869918 0.493197i \(-0.164172\pi\)
0.869918 + 0.493197i \(0.164172\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 12.0000i 1.82998i −0.403473 0.914991i \(-0.632197\pi\)
0.403473 0.914991i \(-0.367803\pi\)
\(44\) 8.69140 + 9.85513i 1.31028 + 1.48572i
\(45\) 0 0
\(46\) −1.11438 + 2.46863i −0.164306 + 0.363979i
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −2.90930 + 6.44484i −0.411438 + 0.911438i
\(51\) 0 0
\(52\) 0 0
\(53\) 14.5544i 1.99920i −0.0283132 0.999599i \(-0.509014\pi\)
0.0283132 0.999599i \(-0.490986\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −7.82288 3.53137i −1.02719 0.463692i
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 6.61438 4.50000i 0.826797 0.562500i
\(65\) 0 0
\(66\) 0 0
\(67\) 15.8745i 1.93938i −0.244339 0.969690i \(-0.578571\pi\)
0.244339 0.969690i \(-0.421429\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 15.0554 1.78674 0.893372 0.449319i \(-0.148333\pi\)
0.893372 + 0.449319i \(0.148333\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) −6.15784 + 13.6412i −0.715834 + 1.58575i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 15.8745i 1.78602i −0.450035 0.893011i \(-0.648589\pi\)
0.450035 0.893011i \(-0.351411\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 15.4676 + 6.98233i 1.66792 + 0.752924i
\(87\) 0 0
\(88\) −17.7601 + 5.46863i −1.89324 + 0.582958i
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −2.53357 2.87280i −0.264143 0.299510i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −6.61438 7.50000i −0.661438 0.750000i
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 18.7601 + 8.46863i 1.82214 + 0.822546i
\(107\) 10.4005 1.00545 0.502726 0.864446i \(-0.332330\pi\)
0.502726 + 0.864446i \(0.332330\pi\)
\(108\) 0 0
\(109\) 10.5830 1.01367 0.506834 0.862044i \(-0.330816\pi\)
0.506834 + 0.862044i \(0.330816\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 16.3808i 1.54098i 0.637452 + 0.770490i \(0.279988\pi\)
−0.637452 + 0.770490i \(0.720012\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 9.10365 8.02867i 0.845253 0.745443i
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 32.1660 2.92418
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 15.8745i 1.40863i 0.709885 + 0.704317i \(0.248747\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 1.95171 + 11.1441i 0.172508 + 0.985008i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 20.4617 + 9.23676i 1.76762 + 0.797934i
\(135\) 0 0
\(136\) 0 0
\(137\) 7.89556i 0.674563i −0.941404 0.337282i \(-0.890493\pi\)
0.941404 0.337282i \(-0.109507\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −8.76013 + 19.4059i −0.735134 + 1.62851i
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) −14.0000 15.8745i −1.15079 1.30488i
\(149\) 23.0397i 1.88748i −0.330684 0.943741i \(-0.607280\pi\)
0.330684 0.943741i \(-0.392720\pi\)
\(150\) 0 0
\(151\) 24.0000i 1.95309i 0.215308 + 0.976546i \(0.430924\pi\)
−0.215308 + 0.976546i \(0.569076\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) 20.4617 + 9.23676i 1.62785 + 0.734837i
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 15.8745i 1.24339i −0.783260 0.621694i \(-0.786445\pi\)
0.783260 0.621694i \(-0.213555\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) −18.0000 + 15.8745i −1.37249 + 1.21042i
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 3.28504 26.0742i 0.247619 1.96542i
\(177\) 0 0
\(178\) 0 0
\(179\) −15.8799 −1.18692 −0.593458 0.804865i \(-0.702238\pi\)
−0.593458 + 0.804865i \(0.702238\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 5.17712 1.59412i 0.381663 0.117520i
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −24.3651 −1.76300 −0.881500 0.472184i \(-0.843466\pi\)
−0.881500 + 0.472184i \(0.843466\pi\)
\(192\) 0 0
\(193\) 21.1660 1.52356 0.761781 0.647834i \(-0.224325\pi\)
0.761781 + 0.647834i \(0.224325\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 10.9015i 0.776697i 0.921513 + 0.388348i \(0.126954\pi\)
−0.921513 + 0.388348i \(0.873046\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 13.5159 4.16176i 0.955719 0.294281i
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 12.0000i 0.826114i 0.910705 + 0.413057i \(0.135539\pi\)
−0.910705 + 0.413057i \(0.864461\pi\)
\(212\) −21.8316 + 19.2536i −1.49940 + 1.32235i
\(213\) 0 0
\(214\) −6.05163 + 13.4059i −0.413681 + 0.916407i
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −6.15784 + 13.6412i −0.417061 + 0.923895i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −21.1144 9.53137i −1.40451 0.634018i
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 5.05163 + 16.4059i 0.331656 + 1.07710i
\(233\) 0.589720i 0.0386338i 0.999813 + 0.0193169i \(0.00614915\pi\)
−0.999813 + 0.0193169i \(0.993851\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 7.39458 0.478316 0.239158 0.970981i \(-0.423129\pi\)
0.239158 + 0.970981i \(0.423129\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) −18.7161 + 41.4609i −1.20312 + 2.66521i
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) −12.5830 −0.791087
\(254\) −20.4617 9.23676i −1.28388 0.579566i
\(255\) 0 0
\(256\) −15.5000 3.96863i −0.968750 0.248039i
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 18.8858 1.16455 0.582273 0.812993i \(-0.302164\pi\)
0.582273 + 0.812993i \(0.302164\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −23.8118 + 21.0000i −1.45453 + 1.28278i
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 10.1771 + 4.59412i 0.614823 + 0.277541i
\(275\) −32.8504 −1.98096
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 33.3514i 1.98958i −0.101955 0.994789i \(-0.532510\pi\)
0.101955 0.994789i \(-0.467490\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) −19.9164 22.5830i −1.18182 1.34006i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 17.0000 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 28.6078 8.80879i 1.66279 0.512001i
\(297\) 0 0
\(298\) 29.6974 + 13.4059i 1.72032 + 0.776582i
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) −30.9352 13.9647i −1.78012 0.803576i
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −23.8118 + 21.0000i −1.33952 + 1.18134i
\(317\) 31.5249i 1.77062i −0.465004 0.885309i \(-0.653947\pi\)
0.465004 0.885309i \(-0.346053\pi\)
\(318\) 0 0
\(319\) 39.8745i 2.23254i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 20.4617 + 9.23676i 1.13327 + 0.511577i
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 36.0000i 1.97874i 0.145424 + 0.989369i \(0.453545\pi\)
−0.145424 + 0.989369i \(0.546455\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −21.1660 −1.15299 −0.576493 0.817102i \(-0.695579\pi\)
−0.576493 + 0.817102i \(0.695579\pi\)
\(338\) 7.56419 16.7566i 0.411438 0.911438i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) −9.98823 32.4382i −0.538529 1.74895i
\(345\) 0 0
\(346\) 0 0
\(347\) 29.0200 1.55788 0.778938 0.627100i \(-0.215758\pi\)
0.778938 + 0.627100i \(0.215758\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 31.6974 + 19.4059i 1.68948 + 1.03434i
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 9.23987 20.4686i 0.488342 1.08180i
\(359\) 20.5347 1.08378 0.541891 0.840449i \(-0.317708\pi\)
0.541891 + 0.840449i \(0.317708\pi\)
\(360\) 0 0
\(361\) 19.0000 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) −0.957598 + 7.60070i −0.0499183 + 0.396214i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −22.0000 −1.13912 −0.569558 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 12.0000i 0.616399i 0.951322 + 0.308199i \(0.0997264\pi\)
−0.951322 + 0.308199i \(0.900274\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 14.1771 31.4059i 0.725365 1.60686i
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −12.3157 + 27.2823i −0.626851 + 1.38863i
\(387\) 0 0
\(388\) 0 0
\(389\) 19.3867i 0.982947i −0.870893 0.491473i \(-0.836458\pi\)
0.870893 0.491473i \(-0.163542\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) −14.0516 6.34313i −0.707911 0.319563i
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −2.50000 + 19.8431i −0.125000 + 0.992157i
\(401\) 9.07500i 0.453184i 0.973990 + 0.226592i \(0.0727584\pi\)
−0.973990 + 0.226592i \(0.927242\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −69.5312 −3.44654
\(408\) 0 0
\(409\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 26.0000 1.26716 0.633581 0.773676i \(-0.281584\pi\)
0.633581 + 0.773676i \(0.281584\pi\)
\(422\) −15.4676 6.98233i −0.752952 0.339895i
\(423\) 0 0
\(424\) −12.1144 39.3431i −0.588326 1.91067i
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −13.7585 15.6007i −0.665044 0.754089i
\(429\) 0 0
\(430\) 0 0
\(431\) −41.3357 −1.99107 −0.995535 0.0943889i \(-0.969910\pi\)
−0.995535 + 0.0943889i \(0.969910\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −14.0000 15.8745i −0.670478 0.760251i
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 12.0495 0.572487 0.286244 0.958157i \(-0.407593\pi\)
0.286244 + 0.958157i \(0.407593\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 28.5190i 1.34590i −0.739689 0.672948i \(-0.765028\pi\)
0.739689 0.672948i \(-0.234972\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 24.5713 21.6698i 1.15574 1.01926i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −42.3320 −1.98021 −0.990104 0.140334i \(-0.955182\pi\)
−0.990104 + 0.140334i \(0.955182\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 15.8745i 0.737751i −0.929479 0.368875i \(-0.879743\pi\)
0.929479 0.368875i \(-0.120257\pi\)
\(464\) −24.0860 3.03455i −1.11816 0.140875i
\(465\) 0 0
\(466\) −0.760130 0.343135i −0.0352123 0.0158954i
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 78.8410i 3.62511i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) −4.30262 + 9.53137i −0.196797 + 0.435955i
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −42.5516 48.2490i −1.93417 2.19314i
\(485\) 0 0
\(486\) 0 0
\(487\) 24.0000i 1.08754i −0.839233 0.543772i \(-0.816996\pi\)
0.839233 0.543772i \(-0.183004\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −27.3710 −1.23524 −0.617619 0.786478i \(-0.711903\pi\)
−0.617619 + 0.786478i \(0.711903\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 36.0000i 1.61158i −0.592200 0.805791i \(-0.701741\pi\)
0.592200 0.805791i \(-0.298259\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 7.32156 16.2191i 0.325483 0.721026i
\(507\) 0 0
\(508\) 23.8118 21.0000i 1.05648 0.931724i
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 14.1343 17.6698i 0.624653 0.780903i
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −10.9889 + 24.3431i −0.479138 + 1.06141i
\(527\) 0 0
\(528\) 0 0
\(529\) −19.3320 −0.840523
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) −13.2132 42.9117i −0.570723 1.85350i
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −34.0000 −1.46177 −0.730887 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 15.8745i 0.678745i 0.940652 + 0.339372i \(0.110215\pi\)
−0.940652 + 0.339372i \(0.889785\pi\)
\(548\) −11.8433 + 10.4448i −0.505923 + 0.446182i
\(549\) 0 0
\(550\) 19.1144 42.3431i 0.815040 1.80552i
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 5.81861 12.8897i 0.247209 0.547630i
\(555\) 0 0
\(556\) 0 0
\(557\) 40.0102i 1.69529i −0.530566 0.847644i \(-0.678020\pi\)
0.530566 0.847644i \(-0.321980\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 42.9889 + 19.4059i 1.81338 + 0.818588i
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 40.6974 12.5314i 1.70762 0.525805i
\(569\) 45.4896i 1.90702i 0.301356 + 0.953512i \(0.402561\pi\)
−0.301356 + 0.953512i \(0.597439\pi\)
\(570\) 0 0
\(571\) 47.6235i 1.99298i 0.0836974 + 0.996491i \(0.473327\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 9.57598 0.399346
\(576\) 0 0
\(577\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(578\) −9.89164 + 21.9125i −0.411438 + 0.911438i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 95.6235i 3.96032i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −5.29150 + 42.0000i −0.217479 + 1.72619i
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −34.5595 + 30.4786i −1.41561 + 1.24845i
\(597\) 0 0
\(598\) 0 0
\(599\) 3.56418 0.145629 0.0728143 0.997346i \(-0.476802\pi\)
0.0728143 + 0.997346i \(0.476802\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 36.0000 31.7490i 1.46482 1.29185i
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −38.0000 −1.53481 −0.767403 0.641165i \(-0.778451\pi\)
−0.767403 + 0.641165i \(0.778451\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 11.5485i 0.464924i 0.972605 + 0.232462i \(0.0746782\pi\)
−0.972605 + 0.232462i \(0.925322\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 47.6235i 1.89586i −0.318475 0.947931i \(-0.603171\pi\)
0.318475 0.947931i \(-0.396829\pi\)
\(632\) −13.2132 42.9117i −0.525592 1.70693i
\(633\) 0 0
\(634\) 40.6346 + 18.3431i 1.61381 + 0.728499i
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 51.3970 + 23.2014i 2.03482 + 0.918553i
\(639\) 0 0
\(640\) 0 0
\(641\) 17.5603i 0.693589i 0.937941 + 0.346795i \(0.112730\pi\)
−0.937941 + 0.346795i \(0.887270\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −23.8118 + 21.0000i −0.932541 + 0.822423i
\(653\) 27.8720i 1.09072i −0.838203 0.545358i \(-0.816394\pi\)
0.838203 0.545358i \(-0.183606\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 49.8210 1.94075 0.970375 0.241604i \(-0.0776734\pi\)
0.970375 + 0.241604i \(0.0776734\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(662\) −46.4028 20.9470i −1.80350 0.814128i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 11.6235i 0.450065i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 42.3320 1.63178 0.815890 0.578208i \(-0.196248\pi\)
0.815890 + 0.578208i \(0.196248\pi\)
\(674\) 12.3157 27.2823i 0.474382 1.05088i
\(675\) 0 0
\(676\) 17.1974 + 19.5000i 0.661438 + 0.750000i
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 40.5112 1.55012 0.775059 0.631889i \(-0.217720\pi\)
0.775059 + 0.631889i \(0.217720\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 47.6235 + 6.00000i 1.81563 + 0.228748i
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −16.8856 + 37.4059i −0.640969 + 1.41991i
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 38.8308i 1.46662i −0.679895 0.733309i \(-0.737975\pi\)
0.679895 0.733309i \(-0.262025\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −43.4570 + 29.5654i −1.63785 + 1.11429i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −52.9150 −1.98727 −0.993633 0.112667i \(-0.964061\pi\)
−0.993633 + 0.112667i \(0.964061\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 21.0071 + 23.8198i 0.785071 + 0.890187i
\(717\) 0 0
\(718\) −11.9484 + 26.4686i −0.445909 + 0.987800i
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −11.0554 + 24.4904i −0.411438 + 0.911438i
\(723\) 0 0
\(724\) 0 0
\(725\) 30.3455i 1.12700i
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) −9.23987 5.65687i −0.340586 0.208515i
\(737\) 104.297i 3.84182i
\(738\) 0 0
\(739\) 15.8745i 0.583953i 0.956425 + 0.291977i \(0.0943129\pi\)
−0.956425 + 0.291977i \(0.905687\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 54.4759 1.99853 0.999263 0.0383863i \(-0.0122217\pi\)
0.999263 + 0.0383863i \(0.0122217\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 12.8009 28.3573i 0.468676 1.03823i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 48.0000i 1.75154i −0.482724 0.875772i \(-0.660353\pi\)
0.482724 0.875772i \(-0.339647\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 10.5830 0.384646 0.192323 0.981332i \(-0.438398\pi\)
0.192323 + 0.981332i \(0.438398\pi\)
\(758\) −15.4676 6.98233i −0.561809 0.253610i
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 32.2321 + 36.5477i 1.16611 + 1.32225i
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −28.0000 31.7490i −1.00774 1.14267i
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 24.9889 + 11.2804i 0.895895 + 0.404422i
\(779\) 0 0
\(780\) 0 0
\(781\) −98.9150 −3.53946
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 16.3522 14.4213i 0.582523 0.513737i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −24.1225 14.7684i −0.852859 0.522141i
\(801\) 0 0
\(802\) −11.6974 5.28039i −0.413049 0.186457i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 3.06320i 0.107696i −0.998549 0.0538482i \(-0.982851\pi\)
0.998549 0.0538482i \(-0.0171487\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\)