Properties

Label 1560.2.l.d.1249.1
Level $1560$
Weight $2$
Character 1560.1249
Analytic conductor $12.457$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1560.l (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.4566627153\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1249.1
Root \(2.20793 - 2.20793i\) of defining polynomial
Character \(\chi\) \(=\) 1560.1249
Dual form 1560.2.l.d.1249.5

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{3} +(-2.20793 - 0.353624i) q^{5} -1.65573i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} +(-2.20793 - 0.353624i) q^{5} -1.65573i q^{7} -1.00000 q^{9} -2.94848 q^{11} +1.00000i q^{13} +(-0.353624 + 2.20793i) q^{15} +1.46738i q^{17} -1.65573 q^{21} -0.532621i q^{23} +(4.74990 + 1.56155i) q^{25} +1.00000i q^{27} -5.70861 q^{29} +2.94848i q^{33} +(-0.585504 + 3.65573i) q^{35} +8.77883i q^{37} +1.00000 q^{39} +1.23987 q^{41} +1.70861i q^{43} +(2.20793 + 0.353624i) q^{45} +2.70725i q^{47} +4.25857 q^{49} +1.46738 q^{51} +8.77883i q^{53} +(6.51003 + 1.04265i) q^{55} +3.83035 q^{59} -0.241231 q^{61} +1.65573i q^{63} +(0.353624 - 2.20793i) q^{65} +2.58550i q^{67} -0.532621 q^{69} -2.55132 q^{71} -0.188347i q^{73} +(1.56155 - 4.74990i) q^{75} +4.88187i q^{77} +11.0729 q^{79} +1.00000 q^{81} +7.91566i q^{83} +(0.518900 - 3.23987i) q^{85} +5.70861i q^{87} -15.9685 q^{89} +1.65573 q^{91} +16.8641i q^{97} +2.94848 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 2q^{5} - 8q^{9} + O(q^{10}) \) \( 8q - 2q^{5} - 8q^{9} + 2q^{11} - 2q^{15} + 14q^{21} - 16q^{29} - 8q^{35} + 8q^{39} + 14q^{41} + 2q^{45} - 18q^{49} + 6q^{51} + 10q^{55} - 4q^{59} + 22q^{61} + 2q^{65} - 10q^{69} + 30q^{71} - 4q^{75} + 2q^{79} + 8q^{81} + 24q^{85} - 18q^{89} - 14q^{91} - 2q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1560\mathbb{Z}\right)^\times\).

\(n\) \(391\) \(521\) \(781\) \(937\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\) \(1\)

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
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) −2.20793 0.353624i −0.987416 0.158145i
\(6\) 0 0
\(7\) 1.65573i 0.625806i −0.949785 0.312903i \(-0.898699\pi\)
0.949785 0.312903i \(-0.101301\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −2.94848 −0.889000 −0.444500 0.895779i \(-0.646619\pi\)
−0.444500 + 0.895779i \(0.646619\pi\)
\(12\) 0 0
\(13\) 1.00000i 0.277350i
\(14\) 0 0
\(15\) −0.353624 + 2.20793i −0.0913053 + 0.570085i
\(16\) 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
\(21\) −1.65573 −0.361309
\(22\) 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
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) −5.70861 −1.06006 −0.530031 0.847978i \(-0.677820\pi\)
−0.530031 + 0.847978i \(0.677820\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 2.94848i 0.513264i
\(34\) 0 0
\(35\) −0.585504 + 3.65573i −0.0989683 + 0.617931i
\(36\) 0 0
\(37\) 8.77883i 1.44323i 0.692294 + 0.721616i \(0.256600\pi\)
−0.692294 + 0.721616i \(0.743400\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 1.23987 0.193635 0.0968175 0.995302i \(-0.469134\pi\)
0.0968175 + 0.995302i \(0.469134\pi\)
\(42\) 0 0
\(43\) 1.70861i 0.260561i 0.991477 + 0.130280i \(0.0415877\pi\)
−0.991477 + 0.130280i \(0.958412\pi\)
\(44\) 0 0
\(45\) 2.20793 + 0.353624i 0.329139 + 0.0527151i
\(46\) 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
\(51\) 1.46738 0.205474
\(52\) 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 \(-0.419788\pi\)
0.249335 + 0.968417i \(0.419788\pi\)
\(60\) 0 0
\(61\) −0.241231 −0.0308865 −0.0154432 0.999881i \(-0.504916\pi\)
−0.0154432 + 0.999881i \(0.504916\pi\)
\(62\) 0 0
\(63\) 1.65573i 0.208602i
\(64\) 0 0
\(65\) 0.353624 2.20793i 0.0438616 0.273860i
\(66\) 0 0
\(67\) 2.58550i 0.315870i 0.987450 + 0.157935i \(0.0504836\pi\)
−0.987450 + 0.157935i \(0.949516\pi\)
\(68\) 0 0
\(69\) −0.532621 −0.0641200
\(70\) 0 0
\(71\) −2.55132 −0.302786 −0.151393 0.988474i \(-0.548376\pi\)
−0.151393 + 0.988474i \(0.548376\pi\)
\(72\) 0 0
\(73\) 0.188347i 0.0220444i −0.999939 0.0110222i \(-0.996491\pi\)
0.999939 0.0110222i \(-0.00350855\pi\)
\(74\) 0 0
\(75\) 1.56155 4.74990i 0.180313 0.548471i
\(76\) 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
\(81\) 1.00000 0.111111
\(82\) 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
\(87\) 5.70861i 0.612027i
\(88\) 0 0
\(89\) −15.9685 −1.69266 −0.846331 0.532657i \(-0.821193\pi\)
−0.846331 + 0.532657i \(0.821193\pi\)
\(90\) 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 \(0.327148\pi\)
−0.516733 + 0.856147i \(0.672852\pi\)
\(98\) 0 0
\(99\) 2.94848 0.296333
\(100\) 0 0
\(101\) −4.64337 −0.462032 −0.231016 0.972950i \(-0.574205\pi\)
−0.231016 + 0.972950i \(0.574205\pi\)
\(102\) 0 0
\(103\) 9.66071i 0.951898i −0.879473 0.475949i \(-0.842105\pi\)
0.879473 0.475949i \(-0.157895\pi\)
\(104\) 0 0
\(105\) 3.65573 + 0.585504i 0.356762 + 0.0571394i
\(106\) 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
\(111\) 8.77883 0.833250
\(112\) 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
\(117\) 1.00000i 0.0924500i
\(118\) 0 0
\(119\) 2.42958 0.222719
\(120\) 0 0
\(121\) −2.30647 −0.209679
\(122\) 0 0
\(123\) 1.23987i 0.111795i
\(124\) 0 0
\(125\) −9.93524 5.12748i −0.888635 0.458615i
\(126\) 0 0
\(127\) 2.87689i 0.255283i 0.991820 + 0.127642i \(0.0407407\pi\)
−0.991820 + 0.127642i \(0.959259\pi\)
\(128\) 0 0
\(129\) 1.70861 0.150435
\(130\) 0 0
\(131\) 19.3968 1.69470 0.847351 0.531033i \(-0.178196\pi\)
0.847351 + 0.531033i \(0.178196\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0.353624 2.20793i 0.0304351 0.190028i
\(136\) 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
\(141\) 2.70725 0.227991
\(142\) 0 0
\(143\) 2.94848i 0.246564i
\(144\) 0 0
\(145\) 12.6042 + 2.01870i 1.04672 + 0.167644i
\(146\) 0 0
\(147\) 4.25857i 0.351241i
\(148\) 0 0
\(149\) −20.6092 −1.68837 −0.844185 0.536052i \(-0.819915\pi\)
−0.844185 + 0.536052i \(0.819915\pi\)
\(150\) 0 0
\(151\) −3.89423 −0.316908 −0.158454 0.987366i \(-0.550651\pi\)
−0.158454 + 0.987366i \(0.550651\pi\)
\(152\) 0 0
\(153\) 1.46738i 0.118631i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 9.22887i 0.736544i 0.929718 + 0.368272i \(0.120051\pi\)
−0.929718 + 0.368272i \(0.879949\pi\)
\(158\) 0 0
\(159\) 8.77883 0.696207
\(160\) 0 0
\(161\) −0.881875 −0.0695015
\(162\) 0 0
\(163\) 16.5700i 1.29786i −0.760846 0.648932i \(-0.775216\pi\)
0.760846 0.648932i \(-0.224784\pi\)
\(164\) 0 0
\(165\) 1.04265 6.51003i 0.0811704 0.506805i
\(166\) 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
\(177\) 3.83035i 0.287907i
\(178\) 0 0
\(179\) −2.29411 −0.171470 −0.0857351 0.996318i \(-0.527324\pi\)
−0.0857351 + 0.996318i \(0.527324\pi\)
\(180\) 0 0
\(181\) −17.6105 −1.30898 −0.654491 0.756070i \(-0.727117\pi\)
−0.654491 + 0.756070i \(0.727117\pi\)
\(182\) 0 0
\(183\) 0.241231i 0.0178323i
\(184\) 0 0
\(185\) 3.10440 19.3830i 0.228240 1.42507i
\(186\) 0 0
\(187\) 4.32654i 0.316388i
\(188\) 0 0
\(189\) 1.65573 0.120436
\(190\) 0 0
\(191\) −0.105767 −0.00765304 −0.00382652 0.999993i \(-0.501218\pi\)
−0.00382652 + 0.999993i \(0.501218\pi\)
\(192\) 0 0
\(193\) 3.27903i 0.236030i 0.993012 + 0.118015i \(0.0376531\pi\)
−0.993012 + 0.118015i \(0.962347\pi\)
\(194\) 0 0
\(195\) −2.20793 0.353624i −0.158113 0.0253235i
\(196\) 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
\(201\) 2.58550 0.182367
\(202\) 0 0
\(203\) 9.45190i 0.663393i
\(204\) 0 0
\(205\) −2.73754 0.438447i −0.191198 0.0306225i
\(206\) 0 0
\(207\) 0.532621i 0.0370197i
\(208\) 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
\(213\) 2.55132i 0.174814i
\(214\) 0 0
\(215\) 0.604205 3.77249i 0.0412065 0.257282i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −0.188347 −0.0127273
\(220\) 0 0
\(221\) −1.46738 −0.0987066
\(222\) 0 0
\(223\) 19.8518i 1.32937i −0.747122 0.664687i \(-0.768565\pi\)
0.747122 0.664687i \(-0.231435\pi\)
\(224\) 0 0
\(225\) −4.74990 1.56155i −0.316660 0.104104i
\(226\) 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
\(231\) 4.88187 0.321204
\(232\) 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
\(237\) 11.0729i 0.719265i
\(238\) 0 0
\(239\) −21.7350 −1.40592 −0.702961 0.711229i \(-0.748139\pi\)
−0.702961 + 0.711229i \(0.748139\pi\)
\(240\) 0 0
\(241\) 17.5604 1.13116 0.565582 0.824692i \(-0.308652\pi\)
0.565582 + 0.824692i \(0.308652\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) −9.40262 1.50593i −0.600711 0.0962105i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 7.91566 0.501634
\(250\) 0 0
\(251\) −22.7082 −1.43333 −0.716665 0.697418i \(-0.754332\pi\)
−0.716665 + 0.697418i \(0.754332\pi\)
\(252\) 0 0
\(253\) 1.57042i 0.0987316i
\(254\) 0 0
\(255\) −3.23987 0.518900i −0.202888 0.0324948i
\(256\) 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
\(261\) 5.70861 0.353354
\(262\) 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
\(267\) 15.9685i 0.977259i
\(268\) 0 0
\(269\) −12.6434 −0.770880 −0.385440 0.922733i \(-0.625950\pi\)
−0.385440 + 0.922733i \(0.625950\pi\)
\(270\) 0 0
\(271\) −23.7665 −1.44371 −0.721855 0.692044i \(-0.756710\pi\)
−0.721855 + 0.692044i \(0.756710\pi\)
\(272\) 0 0
\(273\) 1.65573i 0.100209i
\(274\) 0 0
\(275\) −14.0050 4.60421i −0.844532 0.277644i
\(276\) 0 0
\(277\) 17.8765i 1.07409i 0.843552 + 0.537047i \(0.180460\pi\)
−0.843552 + 0.537047i \(0.819540\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 8.39540 0.500827 0.250414 0.968139i \(-0.419433\pi\)
0.250414 + 0.968139i \(0.419433\pi\)
\(282\) 0 0
\(283\) 13.9344i 0.828312i 0.910206 + 0.414156i \(0.135923\pi\)
−0.910206 + 0.414156i \(0.864077\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 2.05288i 0.121178i
\(288\) 0 0
\(289\) 14.8468 0.873341
\(290\) 0 0
\(291\) 16.8641 0.988593
\(292\) 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
\(297\) 2.94848i 0.171088i
\(298\) 0 0
\(299\) 0.532621 0.0308023
\(300\) 0 0
\(301\) 2.82899 0.163060
\(302\) 0 0
\(303\) 4.64337i 0.266755i
\(304\) 0 0
\(305\) 0.532621 + 0.0853050i 0.0304978 + 0.00488455i
\(306\) 0 0
\(307\) 2.50790i 0.143134i −0.997436 0.0715668i \(-0.977200\pi\)
0.997436 0.0715668i \(-0.0227999\pi\)
\(308\) 0 0
\(309\) −9.66071 −0.549578
\(310\) 0 0
\(311\) −6.96220 −0.394790 −0.197395 0.980324i \(-0.563248\pi\)
−0.197395 + 0.980324i \(0.563248\pi\)
\(312\) 0 0
\(313\) 16.0880i 0.909349i 0.890658 + 0.454675i \(0.150244\pi\)
−0.890658 + 0.454675i \(0.849756\pi\)
\(314\) 0 0
\(315\) 0.585504 3.65573i 0.0329894 0.205977i
\(316\) 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
\(321\) 11.9498 0.666975
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −1.56155 + 4.74990i −0.0866194 + 0.263477i
\(326\) 0 0
\(327\) 11.8970i 0.657903i
\(328\) 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
\(333\) 8.77883i 0.481077i
\(334\) 0 0
\(335\) 0.914296 5.70861i 0.0499533 0.311895i
\(336\) 0 0
\(337\) 22.1227i 1.20510i −0.798081 0.602550i \(-0.794151\pi\)
0.798081 0.602550i \(-0.205849\pi\)
\(338\) 0 0
\(339\) −2.37669 −0.129084
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 18.6411i 1.00653i
\(344\) 0 0
\(345\) 1.17599 + 0.188347i 0.0633131 + 0.0101403i
\(346\) 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
\(351\) −1.00000 −0.0533761
\(352\) 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
\(357\) 2.42958i 0.128587i
\(358\) 0 0
\(359\) −4.98090 −0.262882 −0.131441 0.991324i \(-0.541960\pi\)
−0.131441 + 0.991324i \(0.541960\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) 2.30647i 0.121058i
\(364\) 0 0
\(365\) −0.0666042 + 0.415858i −0.00348622 + 0.0217670i
\(366\) 0 0
\(367\) 10.7160i 0.559370i 0.960092 + 0.279685i \(0.0902300\pi\)
−0.960092 + 0.279685i \(0.909770\pi\)
\(368\) 0 0
\(369\) −1.23987 −0.0645450
\(370\) 0 0
\(371\) 14.5353 0.754638
\(372\) 0 0
\(373\) 22.8691i 1.18412i −0.805895 0.592059i \(-0.798315\pi\)
0.805895 0.592059i \(-0.201685\pi\)
\(374\) 0 0
\(375\) −5.12748 + 9.93524i −0.264782 + 0.513054i
\(376\) 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
\(381\) 2.87689 0.147388
\(382\) 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
\(387\) 1.70861i 0.0868535i
\(388\) 0 0
\(389\) 7.87650 0.399354 0.199677 0.979862i \(-0.436011\pi\)
0.199677 + 0.979862i \(0.436011\pi\)
\(390\) 0 0
\(391\) 0.781557 0.0395250
\(392\) 0 0
\(393\) 19.3968i 0.978437i
\(394\) 0 0
\(395\) −24.4483 3.91566i −1.23013 0.197018i
\(396\) 0 0
\(397\) 0.673065i 0.0337802i −0.999857 0.0168901i \(-0.994623\pi\)
0.999857 0.0168901i \(-0.00537654\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −26.3981 −1.31826 −0.659130 0.752029i \(-0.729075\pi\)
−0.659130 + 0.752029i \(0.729075\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −2.20793 0.353624i −0.109713 0.0175717i
\(406\) 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
\(411\) 0.357994 0.0176585
\(412\) 0 0
\(413\) 6.34202i 0.312070i
\(414\) 0 0
\(415\) 2.79917 17.4772i 0.137406 0.857923i
\(416\) 0 0
\(417\) 4.88187i 0.239066i
\(418\) 0 0
\(419\) 32.3068 1.57829 0.789145 0.614207i \(-0.210524\pi\)
0.789145 + 0.614207i \(0.210524\pi\)
\(420\) 0 0
\(421\) 2.33929 0.114010 0.0570051 0.998374i \(-0.481845\pi\)
0.0570051 + 0.998374i \(0.481845\pi\)
\(422\) 0 0
\(423\) 2.70725i 0.131631i
\(424\) 0 0
\(425\) −2.29139 + 6.96990i −0.111149 + 0.338090i
\(426\) 0 0
\(427\) 0.399413i 0.0193289i
\(428\) 0 0
\(429\) −2.94848 −0.142354
\(430\) 0 0
\(431\) −20.5138 −0.988117 −0.494059 0.869429i \(-0.664487\pi\)
−0.494059 + 0.869429i \(0.664487\pi\)
\(432\) 0 0
\(433\) 24.8514i 1.19428i 0.802137 + 0.597141i \(0.203697\pi\)
−0.802137 + 0.597141i \(0.796303\pi\)
\(434\) 0 0
\(435\) 2.01870 12.6042i 0.0967893 0.604325i
\(436\) 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
\(441\) −4.25857 −0.202789
\(442\) 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
\(447\) 20.6092i 0.974781i
\(448\) 0 0
\(449\) 11.8034 0.557036 0.278518 0.960431i \(-0.410157\pi\)
0.278518 + 0.960431i \(0.410157\pi\)
\(450\) 0 0
\(451\) −3.65573 −0.172141
\(452\) 0 0
\(453\) 3.89423i 0.182967i
\(454\) 0 0
\(455\) −3.65573 0.585504i −0.171383 0.0274489i
\(456\) 0 0
\(457\) 1.86454i 0.0872193i 0.999049 + 0.0436097i \(0.0138858\pi\)
−0.999049 + 0.0436097i \(0.986114\pi\)
\(458\) 0 0
\(459\) −1.46738 −0.0684914
\(460\) 0 0
\(461\) 7.42822 0.345967 0.172983 0.984925i \(-0.444659\pi\)
0.172983 + 0.984925i \(0.444659\pi\)
\(462\) 0 0
\(463\) 29.0829i 1.35160i −0.737086 0.675799i \(-0.763799\pi\)
0.737086 0.675799i \(-0.236201\pi\)
\(464\) 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
\(471\) 9.22887 0.425244
\(472\) 0 0
\(473\) 5.03780i 0.231638i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 8.77883i 0.401955i
\(478\) 0 0
\(479\) 19.6166 0.896304 0.448152 0.893957i \(-0.352082\pi\)
0.448152 + 0.893957i \(0.352082\pi\)
\(480\) 0 0
\(481\) −8.77883 −0.400280
\(482\) 0 0
\(483\) 0.881875i 0.0401267i
\(484\) 0 0
\(485\) 5.96356 37.2348i 0.270791 1.69075i
\(486\) 0 0
\(487\) 0.235782i 0.0106843i 0.999986 + 0.00534215i \(0.00170047\pi\)
−0.999986 + 0.00534215i \(0.998300\pi\)
\(488\) 0 0
\(489\) −16.5700 −0.749322
\(490\) 0 0
\(491\) −23.6183 −1.06588 −0.532938 0.846154i \(-0.678912\pi\)
−0.532938 + 0.846154i \(0.678912\pi\)
\(492\) 0 0
\(493\) 8.37669i 0.377267i
\(494\) 0 0
\(495\) −6.51003 1.04265i −0.292604 0.0468637i
\(496\) 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
\(501\) 15.5390 0.694230
\(502\) 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
\(507\) 1.00000i 0.0444116i
\(508\) 0 0
\(509\) −17.2878 −0.766267 −0.383134 0.923693i \(-0.625155\pi\)
−0.383134 + 0.923693i \(0.625155\pi\)
\(510\) 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
\(519\) −3.17101 −0.139192
\(520\) 0 0
\(521\) 26.9474 1.18059 0.590294 0.807188i \(-0.299012\pi\)
0.590294 + 0.807188i \(0.299012\pi\)
\(522\) 0 0
\(523\) 0.150946i 0.00660040i −0.999995 0.00330020i \(-0.998950\pi\)
0.999995 0.00330020i \(-0.00105049\pi\)
\(524\) 0 0
\(525\) −7.86454 2.58550i −0.343236 0.112841i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 22.7163 0.987666
\(530\) 0 0
\(531\) −3.83035 −0.166223
\(532\) 0 0
\(533\) 1.23987i 0.0537047i
\(534\) 0 0
\(535\) 4.22575 26.3844i 0.182695 1.14070i
\(536\) 0 0
\(537\) 2.29411i 0.0989983i
\(538\) 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
\(543\) 17.6105i 0.755741i
\(544\) 0 0
\(545\) 26.2676 + 4.20705i 1.12518 + 0.180210i
\(546\) 0 0
\(547\) 21.6128i 0.924097i −0.886855 0.462048i \(-0.847115\pi\)
0.886855 0.462048i \(-0.152885\pi\)
\(548\) 0 0
\(549\) 0.241231 0.0102955
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 18.3338i 0.779631i
\(554\) 0 0
\(555\) −19.3830 3.10440i −0.822764 0.131775i
\(556\) 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
\(561\) −4.32654 −0.182666
\(562\) 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
\(567\) 1.65573i 0.0695340i
\(568\) 0 0
\(569\) −17.7255 −0.743094 −0.371547 0.928414i \(-0.621172\pi\)
−0.371547 + 0.928414i \(0.621172\pi\)
\(570\) 0 0
\(571\) −17.4095 −0.728566 −0.364283 0.931288i \(-0.618686\pi\)
−0.364283 + 0.931288i \(0.618686\pi\)
\(572\) 0 0
\(573\) 0.105767i 0.00441848i
\(574\) 0 0
\(575\) 0.831716 2.52990i 0.0346849 0.105504i
\(576\) 0 0
\(577\) 27.0729i 1.12706i 0.826095 + 0.563531i \(0.190557\pi\)
−0.826095 + 0.563531i \(0.809443\pi\)
\(578\) 0 0
\(579\) 3.27903 0.136272
\(580\) 0 0
\(581\) 13.1062 0.543735
\(582\) 0 0
\(583\) 25.8842i 1.07201i
\(584\) 0 0
\(585\) −0.353624 + 2.20793i −0.0146205 + 0.0912866i
\(586\) 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
\(591\) 6.01598 0.247464
\(592\) 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
\(597\) 5.33192i 0.218221i
\(598\) 0 0
\(599\) 4.72595 0.193097 0.0965485 0.995328i \(-0.469220\pi\)
0.0965485 + 0.995328i \(0.469220\pi\)
\(600\) 0 0
\(601\) −2.47203 −0.100836 −0.0504182 0.998728i \(-0.516055\pi\)
−0.0504182 + 0.998728i \(0.516055\pi\)
\(602\) 0 0
\(603\) 2.58550i 0.105290i
\(604\) 0 0
\(605\) 5.09253 + 0.815624i 0.207041 + 0.0331598i
\(606\) 0 0
\(607\) 42.8966i 1.74112i 0.492064 + 0.870559i \(0.336242\pi\)
−0.492064 + 0.870559i \(0.663758\pi\)
\(608\) 0 0
\(609\) 9.45190 0.383010
\(610\) 0 0
\(611\) −2.70725 −0.109524
\(612\) 0 0
\(613\) 39.5175i 1.59610i 0.602594 + 0.798048i \(0.294134\pi\)
−0.602594 + 0.798048i \(0.705866\pi\)
\(614\) 0 0
\(615\) −0.438447 + 2.73754i −0.0176799 + 0.110388i
\(616\) 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
\(621\) 0.532621 0.0213733
\(622\) 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
\(633\) 14.5777i 0.579413i
\(634\) 0 0
\(635\) 1.01734 6.35198i 0.0403718 0.252071i
\(636\) 0 0
\(637\) 4.25857i 0.168731i
\(638\) 0 0
\(639\) 2.55132 0.100929
\(640\) 0 0
\(641\) 30.3921 1.20042 0.600208 0.799844i \(-0.295084\pi\)
0.600208 + 0.799844i \(0.295084\pi\)
\(642\) 0 0
\(643\) 17.3918i 0.685865i 0.939360 + 0.342932i \(0.111420\pi\)
−0.939360 + 0.342932i \(0.888580\pi\)
\(644\) 0 0
\(645\) −3.77249 0.604205i −0.148542 0.0237906i
\(646\) 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
\(657\) 0.188347i 0.00734814i
\(658\) 0 0
\(659\) 17.3940 0.677575 0.338788 0.940863i \(-0.389983\pi\)
0.338788 + 0.940863i \(0.389983\pi\)
\(660\) 0 0
\(661\) −9.04053 −0.351636 −0.175818 0.984423i \(-0.556257\pi\)
−0.175818 + 0.984423i \(0.556257\pi\)
\(662\) 0 0
\(663\) 1.46738i 0.0569883i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 3.04053i 0.117730i
\(668\) 0 0
\(669\) −19.8518 −0.767514
\(670\) 0 0
\(671\) 0.711265 0.0274581
\(672\) 0 0
\(673\) 12.7634i 0.491991i 0.969271 + 0.245996i \(0.0791148\pi\)
−0.969271 + 0.245996i \(0.920885\pi\)
\(674\) 0 0
\(675\) −1.56155 + 4.74990i −0.0601042 + 0.182824i
\(676\) 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
\(681\) −6.85042 −0.262509
\(682\) 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
\(687\) 18.0374i 0.688170i
\(688\) 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
\(693\) 4.88187i 0.185447i
\(694\) 0 0
\(695\) −10.7788 1.72635i −0.408864 0.0654841i
\(696\) 0 0
\(697\) 1.81936i 0.0689131i
\(698\) 0 0
\(699\) −14.4049 −0.544842
\(700\) 0 0
\(701\) 1.08298 0.0409036 0.0204518 0.999791i \(-0.493490\pi\)
0.0204518 + 0.999791i \(0.493490\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) −5.97741 0.957347i −0.225122 0.0360558i
\(706\) 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
\(711\) −11.0729 −0.415268
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) −1.04265 + 6.51003i −0.0389930 + 0.243461i
\(716\) 0 0
\(717\) 21.7350i 0.811709i
\(718\) 0 0
\(719\) 29.4793 1.09939 0.549697 0.835364i \(-0.314743\pi\)
0.549697 + 0.835364i \(0.314743\pi\)
\(720\) 0 0
\(721\) −15.9955 −0.595703
\(722\) 0 0
\(723\) 17.5604i 0.653078i
\(724\) 0 0
\(725\) −27.1153 8.91430i −1.00704 0.331069i
\(726\) 0 0
\(727\) 8.43456i 0.312820i −0.987692 0.156410i \(-0.950008\pi\)
0.987692 0.156410i \(-0.0499922\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −2.50718 −0.0927314
\(732\) 0 0
\(733\) 11.5449i 0.426421i 0.977006 + 0.213210i \(0.0683920\pi\)
−0.977006 + 0.213210i \(0.931608\pi\)
\(734\) 0 0
\(735\) −1.50593 + 9.40262i −0.0555471 + 0.346821i
\(736\) 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
\(747\) 7.91566i 0.289619i
\(748\) 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
\(753\) 22.7082i 0.827533i
\(754\) 0 0
\(755\) 8.59819 + 1.37709i 0.312920 + 0.0501176i
\(756\) 0 0
\(757\) 20.6229i 0.749552i 0.927115 + 0.374776i \(0.122280\pi\)
−0.927115 + 0.374776i \(0.877720\pi\)
\(758\) 0 0
\(759\) 1.57042 0.0570027
\(760\) 0 0
\(761\) 52.6872 1.90991 0.954954 0.296752i \(-0.0959036\pi\)
0.954954 + 0.296752i \(0.0959036\pi\)
\(762\) 0 0
\(763\) 19.6981i 0.713119i
\(764\) 0 0
\(765\) −0.518900 + 3.23987i −0.0187609 + 0.117138i
\(766\) 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
\(771\) 6.58278 0.237073
\(772\) 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
\(777\) 14.5353i 0.521453i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 7.52252 0.269177
\(782\) 0 0
\(783\) 5.70861i 0.204009i
\(784\) 0 0
\(785\) 3.26355 20.3767i 0.116481 0.727275i
\(786\) 0 0
\(787\) 26.7067i 0.951990i 0.879448 + 0.475995i \(0.157912\pi\)
−0.879448 + 0.475995i \(0.842088\pi\)
\(788\) 0 0
\(789\) −4.24349 −0.151072
\(790\) 0 0
\(791\) −3.93516 −0.139918
\(792\) 0 0
\(793\) 0.241231i 0.00856636i
\(794\) 0 0
\(795\) −19.3830 3.10440i −0.687445 0.110102i
\(796\) 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
\(801\) 15.9685 0.564221
\(802\) 0 0
\(803\) 0.555339i 0.0195975i
\(804\) 0 0