Properties

Label 4680.2.l.g.2809.7
Level $4680$
Weight $2$
Character 4680.2809
Analytic conductor $37.370$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4680.l (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(37.3699881460\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.57815240704.2
Defining polynomial: \(x^{8} - 2 x^{7} + 2 x^{6} + 89 x^{4} - 170 x^{3} + 162 x^{2} - 72 x + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1560)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2809.7
Root \(2.20793 + 2.20793i\) of defining polynomial
Character \(\chi\) \(=\) 4680.2809
Dual form 4680.2.l.g.2809.8

$q$-expansion

\(f(q)\) \(=\) \(q+(2.20793 - 0.353624i) q^{5} +1.65573i q^{7} +O(q^{10})\) \(q+(2.20793 - 0.353624i) q^{5} +1.65573i q^{7} +2.94848 q^{11} -1.00000i q^{13} +1.46738i q^{17} -0.532621i q^{23} +(4.74990 - 1.56155i) q^{25} +5.70861 q^{29} +(0.585504 + 3.65573i) q^{35} -8.77883i q^{37} -1.23987 q^{41} -1.70861i q^{43} +2.70725i q^{47} +4.25857 q^{49} +8.77883i q^{53} +(6.51003 - 1.04265i) q^{55} -3.83035 q^{59} -0.241231 q^{61} +(-0.353624 - 2.20793i) q^{65} -2.58550i q^{67} +2.55132 q^{71} +0.188347i q^{73} +4.88187i q^{77} +11.0729 q^{79} +7.91566i q^{83} +(0.518900 + 3.23987i) q^{85} +15.9685 q^{89} +1.65573 q^{91} -16.8641i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{5} + O(q^{10}) \) \( 8 q + 2 q^{5} - 2 q^{11} + 16 q^{29} + 8 q^{35} - 14 q^{41} - 18 q^{49} + 10 q^{55} + 4 q^{59} + 22 q^{61} - 2 q^{65} - 30 q^{71} + 2 q^{79} + 24 q^{85} + 18 q^{89} - 14 q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4680\mathbb{Z}\right)^\times\).

\(n\) \(937\) \(1081\) \(2081\) \(2341\) \(3511\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 2.20793 0.353624i 0.987416 0.158145i
\(6\) 0 0
\(7\) 1.65573i 0.625806i 0.949785 + 0.312903i \(0.101301\pi\)
−0.949785 + 0.312903i \(0.898699\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.94848 0.889000 0.444500 0.895779i \(-0.353381\pi\)
0.444500 + 0.895779i \(0.353381\pi\)
\(12\) 0 0
\(13\) 1.00000i 0.277350i
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.46738i 0.355892i 0.984040 + 0.177946i \(0.0569452\pi\)
−0.984040 + 0.177946i \(0.943055\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.532621i 0.111059i −0.998457 0.0555296i \(-0.982315\pi\)
0.998457 0.0555296i \(-0.0176847\pi\)
\(24\) 0 0
\(25\) 4.74990 1.56155i 0.949980 0.312311i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 5.70861 1.06006 0.530031 0.847978i \(-0.322180\pi\)
0.530031 + 0.847978i \(0.322180\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.585504 + 3.65573i 0.0989683 + 0.617931i
\(36\) 0 0
\(37\) 8.77883i 1.44323i −0.692294 0.721616i \(-0.743400\pi\)
0.692294 0.721616i \(-0.256600\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −1.23987 −0.193635 −0.0968175 0.995302i \(-0.530866\pi\)
−0.0968175 + 0.995302i \(0.530866\pi\)
\(42\) 0 0
\(43\) 1.70861i 0.260561i −0.991477 0.130280i \(-0.958412\pi\)
0.991477 0.130280i \(-0.0415877\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.70725i 0.394893i 0.980314 + 0.197446i \(0.0632648\pi\)
−0.980314 + 0.197446i \(0.936735\pi\)
\(48\) 0 0
\(49\) 4.25857 0.608367
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 8.77883i 1.20587i 0.797792 + 0.602933i \(0.206001\pi\)
−0.797792 + 0.602933i \(0.793999\pi\)
\(54\) 0 0
\(55\) 6.51003 1.04265i 0.877812 0.140591i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −3.83035 −0.498670 −0.249335 0.968417i \(-0.580212\pi\)
−0.249335 + 0.968417i \(0.580212\pi\)
\(60\) 0 0
\(61\) −0.241231 −0.0308865 −0.0154432 0.999881i \(-0.504916\pi\)
−0.0154432 + 0.999881i \(0.504916\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −0.353624 2.20793i −0.0438616 0.273860i
\(66\) 0 0
\(67\) 2.58550i 0.315870i −0.987450 0.157935i \(-0.949516\pi\)
0.987450 0.157935i \(-0.0504836\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.55132 0.302786 0.151393 0.988474i \(-0.451624\pi\)
0.151393 + 0.988474i \(0.451624\pi\)
\(72\) 0 0
\(73\) 0.188347i 0.0220444i 0.999939 + 0.0110222i \(0.00350855\pi\)
−0.999939 + 0.0110222i \(0.996491\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.88187i 0.556341i
\(78\) 0 0
\(79\) 11.0729 1.24580 0.622902 0.782300i \(-0.285954\pi\)
0.622902 + 0.782300i \(0.285954\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 7.91566i 0.868856i 0.900706 + 0.434428i \(0.143050\pi\)
−0.900706 + 0.434428i \(0.856950\pi\)
\(84\) 0 0
\(85\) 0.518900 + 3.23987i 0.0562826 + 0.351413i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 15.9685 1.69266 0.846331 0.532657i \(-0.178807\pi\)
0.846331 + 0.532657i \(0.178807\pi\)
\(90\) 0 0
\(91\) 1.65573 0.173567
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 16.8641i 1.71229i −0.516733 0.856147i \(-0.672852\pi\)
0.516733 0.856147i \(-0.327148\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 4.64337 0.462032 0.231016 0.972950i \(-0.425795\pi\)
0.231016 + 0.972950i \(0.425795\pi\)
\(102\) 0 0
\(103\) 9.66071i 0.951898i 0.879473 + 0.475949i \(0.157895\pi\)
−0.879473 + 0.475949i \(0.842105\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 11.9498i 1.15523i 0.816308 + 0.577617i \(0.196017\pi\)
−0.816308 + 0.577617i \(0.803983\pi\)
\(108\) 0 0
\(109\) −11.8970 −1.13952 −0.569761 0.821810i \(-0.692964\pi\)
−0.569761 + 0.821810i \(0.692964\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 2.37669i 0.223581i −0.993732 0.111790i \(-0.964341\pi\)
0.993732 0.111790i \(-0.0356585\pi\)
\(114\) 0 0
\(115\) −0.188347 1.17599i −0.0175635 0.109662i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −2.42958 −0.222719
\(120\) 0 0
\(121\) −2.30647 −0.209679
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 9.93524 5.12748i 0.888635 0.458615i
\(126\) 0 0
\(127\) 2.87689i 0.255283i −0.991820 0.127642i \(-0.959259\pi\)
0.991820 0.127642i \(-0.0407407\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −19.3968 −1.69470 −0.847351 0.531033i \(-0.821804\pi\)
−0.847351 + 0.531033i \(0.821804\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0.357994i 0.0305855i 0.999883 + 0.0152927i \(0.00486802\pi\)
−0.999883 + 0.0152927i \(0.995132\pi\)
\(138\) 0 0
\(139\) 4.88187 0.414075 0.207038 0.978333i \(-0.433618\pi\)
0.207038 + 0.978333i \(0.433618\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 2.94848i 0.246564i
\(144\) 0 0
\(145\) 12.6042 2.01870i 1.04672 0.167644i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 20.6092 1.68837 0.844185 0.536052i \(-0.180085\pi\)
0.844185 + 0.536052i \(0.180085\pi\)
\(150\) 0 0
\(151\) −3.89423 −0.316908 −0.158454 0.987366i \(-0.550651\pi\)
−0.158454 + 0.987366i \(0.550651\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 9.22887i 0.736544i −0.929718 0.368272i \(-0.879949\pi\)
0.929718 0.368272i \(-0.120051\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0.881875 0.0695015
\(162\) 0 0
\(163\) 16.5700i 1.29786i 0.760846 + 0.648932i \(0.224784\pi\)
−0.760846 + 0.648932i \(0.775216\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 15.5390i 1.20244i 0.799083 + 0.601221i \(0.205319\pi\)
−0.799083 + 0.601221i \(0.794681\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 3.17101i 0.241087i −0.992708 0.120544i \(-0.961536\pi\)
0.992708 0.120544i \(-0.0384638\pi\)
\(174\) 0 0
\(175\) 2.58550 + 7.86454i 0.195446 + 0.594503i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 2.29411 0.171470 0.0857351 0.996318i \(-0.472676\pi\)
0.0857351 + 0.996318i \(0.472676\pi\)
\(180\) 0 0
\(181\) −17.6105 −1.30898 −0.654491 0.756070i \(-0.727117\pi\)
−0.654491 + 0.756070i \(0.727117\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −3.10440 19.3830i −0.228240 1.42507i
\(186\) 0 0
\(187\) 4.32654i 0.316388i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0.105767 0.00765304 0.00382652 0.999993i \(-0.498782\pi\)
0.00382652 + 0.999993i \(0.498782\pi\)
\(192\) 0 0
\(193\) 3.27903i 0.236030i −0.993012 0.118015i \(-0.962347\pi\)
0.993012 0.118015i \(-0.0376531\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 6.01598i 0.428621i 0.976766 + 0.214310i \(0.0687504\pi\)
−0.976766 + 0.214310i \(0.931250\pi\)
\(198\) 0 0
\(199\) 5.33192 0.377969 0.188985 0.981980i \(-0.439480\pi\)
0.188985 + 0.981980i \(0.439480\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 9.45190i 0.663393i
\(204\) 0 0
\(205\) −2.73754 + 0.438447i −0.191198 + 0.0306225i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 14.5777 1.00357 0.501786 0.864992i \(-0.332676\pi\)
0.501786 + 0.864992i \(0.332676\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −0.604205 3.77249i −0.0412065 0.257282i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1.46738 0.0987066
\(222\) 0 0
\(223\) 19.8518i 1.32937i 0.747122 + 0.664687i \(0.231435\pi\)
−0.747122 + 0.664687i \(0.768565\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 6.85042i 0.454678i −0.973816 0.227339i \(-0.926997\pi\)
0.973816 0.227339i \(-0.0730026\pi\)
\(228\) 0 0
\(229\) −18.0374 −1.19195 −0.595973 0.803005i \(-0.703233\pi\)
−0.595973 + 0.803005i \(0.703233\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 14.4049i 0.943694i −0.881680 0.471847i \(-0.843587\pi\)
0.881680 0.471847i \(-0.156413\pi\)
\(234\) 0 0
\(235\) 0.957347 + 5.97741i 0.0624505 + 0.389923i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 21.7350 1.40592 0.702961 0.711229i \(-0.251861\pi\)
0.702961 + 0.711229i \(0.251861\pi\)
\(240\) 0 0
\(241\) 17.5604 1.13116 0.565582 0.824692i \(-0.308652\pi\)
0.565582 + 0.824692i \(0.308652\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 9.40262 1.50593i 0.600711 0.0962105i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 22.7082 1.43333 0.716665 0.697418i \(-0.245668\pi\)
0.716665 + 0.697418i \(0.245668\pi\)
\(252\) 0 0
\(253\) 1.57042i 0.0987316i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 6.58278i 0.410623i 0.978697 + 0.205311i \(0.0658207\pi\)
−0.978697 + 0.205311i \(0.934179\pi\)
\(258\) 0 0
\(259\) 14.5353 0.903182
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 4.24349i 0.261665i −0.991405 0.130832i \(-0.958235\pi\)
0.991405 0.130832i \(-0.0417649\pi\)
\(264\) 0 0
\(265\) 3.10440 + 19.3830i 0.190702 + 1.19069i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 12.6434 0.770880 0.385440 0.922733i \(-0.374050\pi\)
0.385440 + 0.922733i \(0.374050\pi\)
\(270\) 0 0
\(271\) −23.7665 −1.44371 −0.721855 0.692044i \(-0.756710\pi\)
−0.721855 + 0.692044i \(0.756710\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 14.0050 4.60421i 0.844532 0.277644i
\(276\) 0 0
\(277\) 17.8765i 1.07409i −0.843552 0.537047i \(-0.819540\pi\)
0.843552 0.537047i \(-0.180460\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −8.39540 −0.500827 −0.250414 0.968139i \(-0.580567\pi\)
−0.250414 + 0.968139i \(0.580567\pi\)
\(282\) 0 0
\(283\) 13.9344i 0.828312i −0.910206 0.414156i \(-0.864077\pi\)
0.910206 0.414156i \(-0.135923\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 2.05288i 0.121178i
\(288\) 0 0
\(289\) 14.8468 0.873341
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 11.4085i 0.666491i −0.942840 0.333245i \(-0.891856\pi\)
0.942840 0.333245i \(-0.108144\pi\)
\(294\) 0 0
\(295\) −8.45715 + 1.35450i −0.492394 + 0.0788623i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −0.532621 −0.0308023
\(300\) 0 0
\(301\) 2.82899 0.163060
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −0.532621 + 0.0853050i −0.0304978 + 0.00488455i
\(306\) 0 0
\(307\) 2.50790i 0.143134i 0.997436 + 0.0715668i \(0.0227999\pi\)
−0.997436 + 0.0715668i \(0.977200\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 6.96220 0.394790 0.197395 0.980324i \(-0.436752\pi\)
0.197395 + 0.980324i \(0.436752\pi\)
\(312\) 0 0
\(313\) 16.0880i 0.909349i −0.890658 0.454675i \(-0.849756\pi\)
0.890658 0.454675i \(-0.150244\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.25263i 0.0703545i 0.999381 + 0.0351772i \(0.0111996\pi\)
−0.999381 + 0.0351772i \(0.988800\pi\)
\(318\) 0 0
\(319\) 16.8317 0.942395
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −1.56155 4.74990i −0.0866194 0.263477i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −4.48246 −0.247126
\(330\) 0 0
\(331\) −8.24349 −0.453103 −0.226552 0.973999i \(-0.572745\pi\)
−0.226552 + 0.973999i \(0.572745\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −0.914296 5.70861i −0.0499533 0.311895i
\(336\) 0 0
\(337\) 22.1227i 1.20510i 0.798081 + 0.602550i \(0.205849\pi\)
−0.798081 + 0.602550i \(0.794151\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 18.6411i 1.00653i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 18.4395i 0.989886i 0.868925 + 0.494943i \(0.164811\pi\)
−0.868925 + 0.494943i \(0.835189\pi\)
\(348\) 0 0
\(349\) −19.6634 −1.05256 −0.526280 0.850312i \(-0.676414\pi\)
−0.526280 + 0.850312i \(0.676414\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 1.18971i 0.0633219i 0.999499 + 0.0316609i \(0.0100797\pi\)
−0.999499 + 0.0316609i \(0.989920\pi\)
\(354\) 0 0
\(355\) 5.63314 0.902208i 0.298976 0.0478842i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 4.98090 0.262882 0.131441 0.991324i \(-0.458040\pi\)
0.131441 + 0.991324i \(0.458040\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0.0666042 + 0.415858i 0.00348622 + 0.0217670i
\(366\) 0 0
\(367\) 10.7160i 0.559370i −0.960092 0.279685i \(-0.909770\pi\)
0.960092 0.279685i \(-0.0902300\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −14.5353 −0.754638
\(372\) 0 0
\(373\) 22.8691i 1.18412i 0.805895 + 0.592059i \(0.201685\pi\)
−0.805895 + 0.592059i \(0.798315\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 5.70861i 0.294008i
\(378\) 0 0
\(379\) −26.5325 −1.36289 −0.681443 0.731871i \(-0.738647\pi\)
−0.681443 + 0.731871i \(0.738647\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 38.3325i 1.95870i −0.202176 0.979349i \(-0.564801\pi\)
0.202176 0.979349i \(-0.435199\pi\)
\(384\) 0 0
\(385\) 1.72635 + 10.7788i 0.0879828 + 0.549340i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −7.87650 −0.399354 −0.199677 0.979862i \(-0.563989\pi\)
−0.199677 + 0.979862i \(0.563989\pi\)
\(390\) 0 0
\(391\) 0.781557 0.0395250
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 24.4483 3.91566i 1.23013 0.197018i
\(396\) 0 0
\(397\) 0.673065i 0.0337802i 0.999857 + 0.0168901i \(0.00537654\pi\)
−0.999857 + 0.0168901i \(0.994623\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 26.3981 1.31826 0.659130 0.752029i \(-0.270925\pi\)
0.659130 + 0.752029i \(0.270925\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 25.8842i 1.28303i
\(408\) 0 0
\(409\) 33.4646 1.65472 0.827359 0.561674i \(-0.189843\pi\)
0.827359 + 0.561674i \(0.189843\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 6.34202i 0.312070i
\(414\) 0 0
\(415\) 2.79917 + 17.4772i 0.137406 + 0.857923i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −32.3068 −1.57829 −0.789145 0.614207i \(-0.789476\pi\)
−0.789145 + 0.614207i \(0.789476\pi\)
\(420\) 0 0
\(421\) 2.33929 0.114010 0.0570051 0.998374i \(-0.481845\pi\)
0.0570051 + 0.998374i \(0.481845\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 2.29139 + 6.96990i 0.111149 + 0.338090i
\(426\) 0 0
\(427\) 0.399413i 0.0193289i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 20.5138 0.988117 0.494059 0.869429i \(-0.335513\pi\)
0.494059 + 0.869429i \(0.335513\pi\)
\(432\) 0 0
\(433\) 24.8514i 1.19428i −0.802137 0.597141i \(-0.796303\pi\)
0.802137 0.597141i \(-0.203697\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 8.18337 0.390571 0.195285 0.980746i \(-0.437437\pi\)
0.195285 + 0.980746i \(0.437437\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 26.1961i 1.24461i −0.782774 0.622306i \(-0.786196\pi\)
0.782774 0.622306i \(-0.213804\pi\)
\(444\) 0 0
\(445\) 35.2574 5.64686i 1.67136 0.267687i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −11.8034 −0.557036 −0.278518 0.960431i \(-0.589843\pi\)
−0.278518 + 0.960431i \(0.589843\pi\)
\(450\) 0 0
\(451\) −3.65573 −0.172141
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 3.65573 0.585504i 0.171383 0.0274489i
\(456\) 0 0
\(457\) 1.86454i 0.0872193i −0.999049 0.0436097i \(-0.986114\pi\)
0.999049 0.0436097i \(-0.0138858\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −7.42822 −0.345967 −0.172983 0.984925i \(-0.555341\pi\)
−0.172983 + 0.984925i \(0.555341\pi\)
\(462\) 0 0
\(463\) 29.0829i 1.35160i 0.737086 + 0.675799i \(0.236201\pi\)
−0.737086 + 0.675799i \(0.763799\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 10.5727i 0.489248i 0.969618 + 0.244624i \(0.0786646\pi\)
−0.969618 + 0.244624i \(0.921335\pi\)
\(468\) 0 0
\(469\) 4.28089 0.197673
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 5.03780i 0.231638i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −19.6166 −0.896304 −0.448152 0.893957i \(-0.647918\pi\)
−0.448152 + 0.893957i \(0.647918\pi\)
\(480\) 0 0
\(481\) −8.77883 −0.400280
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −5.96356 37.2348i −0.270791 1.69075i
\(486\) 0 0
\(487\) 0.235782i 0.0106843i −0.999986 0.00534215i \(-0.998300\pi\)
0.999986 0.00534215i \(-0.00170047\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 23.6183 1.06588 0.532938 0.846154i \(-0.321088\pi\)
0.532938 + 0.846154i \(0.321088\pi\)
\(492\) 0 0
\(493\) 8.37669i 0.377267i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 4.22429i 0.189485i
\(498\) 0 0
\(499\) 18.2737 0.818041 0.409021 0.912525i \(-0.365870\pi\)
0.409021 + 0.912525i \(0.365870\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 27.8722i 1.24276i 0.783509 + 0.621381i \(0.213428\pi\)
−0.783509 + 0.621381i \(0.786572\pi\)
\(504\) 0 0
\(505\) 10.2522 1.64201i 0.456218 0.0730683i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 17.2878 0.766267 0.383134 0.923693i \(-0.374845\pi\)
0.383134 + 0.923693i \(0.374845\pi\)
\(510\) 0 0
\(511\) −0.311852 −0.0137955
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 3.41626 + 21.3302i 0.150538 + 0.939919i
\(516\) 0 0
\(517\) 7.98226i 0.351060i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −26.9474 −1.18059 −0.590294 0.807188i \(-0.700988\pi\)
−0.590294 + 0.807188i \(0.700988\pi\)
\(522\) 0 0
\(523\) 0.150946i 0.00660040i 0.999995 + 0.00330020i \(0.00105049\pi\)
−0.999995 + 0.00330020i \(0.998950\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 22.7163 0.987666
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 1.23987i 0.0537047i
\(534\) 0 0
\(535\) 4.22575 + 26.3844i 0.182695 + 1.14070i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 12.5563 0.540838
\(540\) 0 0
\(541\) −1.97256 −0.0848069 −0.0424035 0.999101i \(-0.513501\pi\)
−0.0424035 + 0.999101i \(0.513501\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −26.2676 + 4.20705i −1.12518 + 0.180210i
\(546\) 0 0
\(547\) 21.6128i 0.924097i 0.886855 + 0.462048i \(0.152885\pi\)
−0.886855 + 0.462048i \(0.847115\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 18.3338i 0.779631i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 36.1619i 1.53223i 0.642705 + 0.766114i \(0.277812\pi\)
−0.642705 + 0.766114i \(0.722188\pi\)
\(558\) 0 0
\(559\) −1.70861 −0.0722665
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 9.05248i 0.381517i 0.981637 + 0.190758i \(0.0610947\pi\)
−0.981637 + 0.190758i \(0.938905\pi\)
\(564\) 0 0
\(565\) −0.840456 5.24757i −0.0353583 0.220767i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 17.7255 0.743094 0.371547 0.928414i \(-0.378828\pi\)
0.371547 + 0.928414i \(0.378828\pi\)
\(570\) 0 0
\(571\) −17.4095 −0.728566 −0.364283 0.931288i \(-0.618686\pi\)
−0.364283 + 0.931288i \(0.618686\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −0.831716 2.52990i −0.0346849 0.105504i
\(576\) 0 0
\(577\) 27.0729i 1.12706i −0.826095 0.563531i \(-0.809443\pi\)
0.826095 0.563531i \(-0.190557\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −13.1062 −0.543735
\(582\) 0 0
\(583\) 25.8842i 1.07201i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 21.5417i 0.889121i −0.895749 0.444560i \(-0.853360\pi\)
0.895749 0.444560i \(-0.146640\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0.542087i 0.0222608i 0.999938 + 0.0111304i \(0.00354300\pi\)
−0.999938 + 0.0111304i \(0.996457\pi\)
\(594\) 0 0
\(595\) −5.36434 + 0.859157i −0.219916 + 0.0352220i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −4.72595 −0.193097 −0.0965485 0.995328i \(-0.530780\pi\)
−0.0965485 + 0.995328i \(0.530780\pi\)
\(600\) 0 0
\(601\) −2.47203 −0.100836 −0.0504182 0.998728i \(-0.516055\pi\)
−0.0504182 + 0.998728i \(0.516055\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −5.09253 + 0.815624i −0.207041 + 0.0331598i
\(606\) 0 0
\(607\) 42.8966i 1.74112i −0.492064 0.870559i \(-0.663758\pi\)
0.492064 0.870559i \(-0.336242\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 2.70725 0.109524
\(612\) 0 0
\(613\) 39.5175i 1.59610i −0.602594 0.798048i \(-0.705866\pi\)
0.602594 0.798048i \(-0.294134\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 32.2951i 1.30015i 0.759870 + 0.650075i \(0.225263\pi\)
−0.759870 + 0.650075i \(0.774737\pi\)
\(618\) 0 0
\(619\) 17.4793 0.702554 0.351277 0.936272i \(-0.385748\pi\)
0.351277 + 0.936272i \(0.385748\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 26.4395i 1.05928i
\(624\) 0 0
\(625\) 20.1231 14.8344i 0.804924 0.593378i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 12.8819 0.513634
\(630\) 0 0
\(631\) −0.376695 −0.0149960 −0.00749799 0.999972i \(-0.502387\pi\)
−0.00749799 + 0.999972i \(0.502387\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −1.01734 6.35198i −0.0403718 0.252071i
\(636\) 0 0
\(637\) 4.25857i 0.168731i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −30.3921 −1.20042 −0.600208 0.799844i \(-0.704916\pi\)
−0.600208 + 0.799844i \(0.704916\pi\)
\(642\) 0 0
\(643\) 17.3918i 0.685865i −0.939360 0.342932i \(-0.888580\pi\)
0.939360 0.342932i \(-0.111420\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 34.1331i 1.34191i 0.741497 + 0.670956i \(0.234116\pi\)
−0.741497 + 0.670956i \(0.765884\pi\)
\(648\) 0 0
\(649\) −11.2937 −0.443317
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 20.6284i 0.807250i 0.914925 + 0.403625i \(0.132250\pi\)
−0.914925 + 0.403625i \(0.867750\pi\)
\(654\) 0 0
\(655\) −42.8267 + 6.85916i −1.67338 + 0.268009i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −17.3940 −0.677575 −0.338788 0.940863i \(-0.610017\pi\)
−0.338788 + 0.940863i \(0.610017\pi\)
\(660\) 0 0
\(661\) −9.04053 −0.351636 −0.175818 0.984423i \(-0.556257\pi\)
−0.175818 + 0.984423i \(0.556257\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 3.04053i 0.117730i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −0.711265 −0.0274581
\(672\) 0 0
\(673\) 12.7634i 0.491991i −0.969271 0.245996i \(-0.920885\pi\)
0.969271 0.245996i \(-0.0791148\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 15.9251i 0.612052i 0.952023 + 0.306026i \(0.0989995\pi\)
−0.952023 + 0.306026i \(0.901001\pi\)
\(678\) 0 0
\(679\) 27.9224 1.07156
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 9.04927i 0.346261i 0.984899 + 0.173130i \(0.0553882\pi\)
−0.984899 + 0.173130i \(0.944612\pi\)
\(684\) 0 0
\(685\) 0.126595 + 0.790425i 0.00483696 + 0.0302006i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 8.77883 0.334447
\(690\) 0 0
\(691\) −41.4848 −1.57816 −0.789078 0.614293i \(-0.789441\pi\)
−0.789078 + 0.614293i \(0.789441\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 10.7788 1.72635i 0.408864 0.0654841i
\(696\) 0 0
\(697\) 1.81936i 0.0689131i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −1.08298 −0.0409036 −0.0204518 0.999791i \(-0.506510\pi\)
−0.0204518 + 0.999791i \(0.506510\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 7.68815i 0.289143i
\(708\) 0 0
\(709\) 29.9371 1.12431 0.562155 0.827032i \(-0.309972\pi\)
0.562155 + 0.827032i \(0.309972\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) −1.04265 6.51003i −0.0389930 0.243461i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −29.4793 −1.09939 −0.549697 0.835364i \(-0.685257\pi\)
−0.549697 + 0.835364i \(0.685257\pi\)
\(720\) 0 0
\(721\) −15.9955 −0.595703
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 27.1153 8.91430i 1.00704 0.331069i
\(726\) 0 0
\(727\) 8.43456i 0.312820i 0.987692 + 0.156410i \(0.0499922\pi\)
−0.987692 + 0.156410i \(0.950008\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 2.50718 0.0927314
\(732\) 0 0
\(733\) 11.5449i 0.426421i −0.977006 0.213210i \(-0.931608\pi\)
0.977006 0.213210i \(-0.0683920\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 7.62331i 0.280808i
\(738\) 0 0
\(739\) −14.2891 −0.525632 −0.262816 0.964846i \(-0.584651\pi\)
−0.262816 + 0.964846i \(0.584651\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 2.60693i 0.0956390i −0.998856 0.0478195i \(-0.984773\pi\)
0.998856 0.0478195i \(-0.0152272\pi\)
\(744\) 0 0
\(745\) 45.5036 7.28790i 1.66712 0.267008i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −19.7857 −0.722953
\(750\) 0 0
\(751\) −46.1277 −1.68322 −0.841612 0.540083i \(-0.818393\pi\)
−0.841612 + 0.540083i \(0.818393\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −8.59819 + 1.37709i −0.312920 + 0.0501176i
\(756\) 0 0
\(757\) 20.6229i 0.749552i −0.927115 0.374776i \(-0.877720\pi\)
0.927115 0.374776i \(-0.122280\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −52.6872 −1.90991 −0.954954 0.296752i \(-0.904096\pi\)
−0.954954 + 0.296752i \(0.904096\pi\)
\(762\) 0 0
\(763\) 19.6981i 0.713119i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 3.83035i 0.138306i
\(768\) 0 0
\(769\) −31.2458 −1.12675 −0.563376 0.826200i \(-0.690498\pi\)
−0.563376 + 0.826200i \(0.690498\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 41.0584i 1.47677i −0.674380 0.738385i \(-0.735589\pi\)
0.674380 0.738385i \(-0.264411\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 7.52252 0.269177
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −3.26355 20.3767i −0.116481 0.727275i
\(786\) 0 0
\(787\) 26.7067i 0.951990i −0.879448 0.475995i \(-0.842088\pi\)
0.879448 0.475995i \(-0.157912\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 3.93516 0.139918
\(792\) 0 0
\(793\) 0.241231i 0.00856636i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 15.8293i 0.560703i −0.959897 0.280352i \(-0.909549\pi\)
0.959897 0.280352i \(-0.0904511\pi\)
\(798\) 0 0
\(799\) −3.97256 −0.140539
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0.555339i 0.0195975i
\(804\) 0 0
\(805\) 1.94712 0.311852i 0.0686268 0.0109913i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −20.3894 −0.716852 −0.358426 0.933558i \(-0.616687\pi\)