Properties

Label 8280.2.p.a.1241.37
Level $8280$
Weight $2$
Character 8280.1241
Analytic conductor $66.116$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8280.p (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(66.1161328736\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1241.37
Character \(\chi\) \(=\) 8280.1241
Dual form 8280.2.p.a.1241.12

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{5} +2.72943i q^{7} +O(q^{10})\) \(q-1.00000 q^{5} +2.72943i q^{7} -6.18590 q^{11} -1.56797 q^{13} -7.34000 q^{17} -2.41299i q^{19} +(-4.64138 - 1.20731i) q^{23} +1.00000 q^{25} +7.31830i q^{29} +5.07406 q^{31} -2.72943i q^{35} +8.91623i q^{37} +1.74173i q^{41} +11.9685i q^{43} -0.0866043i q^{47} -0.449787 q^{49} -2.22559 q^{53} +6.18590 q^{55} +0.102805i q^{59} -10.4353i q^{61} +1.56797 q^{65} -8.46561i q^{67} +12.4723i q^{71} -11.2668 q^{73} -16.8840i q^{77} -1.00635i q^{79} +4.30642 q^{83} +7.34000 q^{85} -3.93965 q^{89} -4.27966i q^{91} +2.41299i q^{95} -14.9674i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q - 48 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 48 q - 48 q^{5} - 8 q^{11} + 4 q^{23} + 48 q^{25} + 8 q^{31} - 32 q^{49} + 8 q^{55} - 16 q^{73} - 32 q^{83} - 8 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8280\mathbb{Z}\right)^\times\).

\(n\) \(1657\) \(2071\) \(3961\) \(4141\) \(4601\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\) \(-1\)

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
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 2.72943i 1.03163i 0.856701 + 0.515814i \(0.172510\pi\)
−0.856701 + 0.515814i \(0.827490\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −6.18590 −1.86512 −0.932559 0.361016i \(-0.882430\pi\)
−0.932559 + 0.361016i \(0.882430\pi\)
\(12\) 0 0
\(13\) −1.56797 −0.434876 −0.217438 0.976074i \(-0.569770\pi\)
−0.217438 + 0.976074i \(0.569770\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −7.34000 −1.78021 −0.890105 0.455755i \(-0.849369\pi\)
−0.890105 + 0.455755i \(0.849369\pi\)
\(18\) 0 0
\(19\) 2.41299i 0.553579i −0.960931 0.276789i \(-0.910730\pi\)
0.960931 0.276789i \(-0.0892704\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.64138 1.20731i −0.967794 0.251742i
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 7.31830i 1.35897i 0.733687 + 0.679487i \(0.237798\pi\)
−0.733687 + 0.679487i \(0.762202\pi\)
\(30\) 0 0
\(31\) 5.07406 0.911329 0.455664 0.890152i \(-0.349402\pi\)
0.455664 + 0.890152i \(0.349402\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.72943i 0.461358i
\(36\) 0 0
\(37\) 8.91623i 1.46582i 0.680326 + 0.732910i \(0.261838\pi\)
−0.680326 + 0.732910i \(0.738162\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.74173i 0.272012i 0.990708 + 0.136006i \(0.0434266\pi\)
−0.990708 + 0.136006i \(0.956573\pi\)
\(42\) 0 0
\(43\) 11.9685i 1.82517i 0.408884 + 0.912587i \(0.365918\pi\)
−0.408884 + 0.912587i \(0.634082\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0.0866043i 0.0126325i −0.999980 0.00631627i \(-0.997989\pi\)
0.999980 0.00631627i \(-0.00201054\pi\)
\(48\) 0 0
\(49\) −0.449787 −0.0642553
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −2.22559 −0.305709 −0.152854 0.988249i \(-0.548847\pi\)
−0.152854 + 0.988249i \(0.548847\pi\)
\(54\) 0 0
\(55\) 6.18590 0.834107
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0.102805i 0.0133841i 0.999978 + 0.00669203i \(0.00213016\pi\)
−0.999978 + 0.00669203i \(0.997870\pi\)
\(60\) 0 0
\(61\) 10.4353i 1.33610i −0.744115 0.668051i \(-0.767129\pi\)
0.744115 0.668051i \(-0.232871\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.56797 0.194483
\(66\) 0 0
\(67\) 8.46561i 1.03424i −0.855913 0.517120i \(-0.827004\pi\)
0.855913 0.517120i \(-0.172996\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 12.4723i 1.48019i 0.672505 + 0.740093i \(0.265219\pi\)
−0.672505 + 0.740093i \(0.734781\pi\)
\(72\) 0 0
\(73\) −11.2668 −1.31868 −0.659339 0.751846i \(-0.729164\pi\)
−0.659339 + 0.751846i \(0.729164\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 16.8840i 1.92411i
\(78\) 0 0
\(79\) 1.00635i 0.113223i −0.998396 0.0566113i \(-0.981970\pi\)
0.998396 0.0566113i \(-0.0180296\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.30642 0.472692 0.236346 0.971669i \(-0.424050\pi\)
0.236346 + 0.971669i \(0.424050\pi\)
\(84\) 0 0
\(85\) 7.34000 0.796134
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −3.93965 −0.417602 −0.208801 0.977958i \(-0.566956\pi\)
−0.208801 + 0.977958i \(0.566956\pi\)
\(90\) 0 0
\(91\) 4.27966i 0.448630i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.41299i 0.247568i
\(96\) 0 0
\(97\) 14.9674i 1.51971i −0.650095 0.759853i \(-0.725271\pi\)
0.650095 0.759853i \(-0.274729\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 12.1022i 1.20421i −0.798415 0.602107i \(-0.794328\pi\)
0.798415 0.602107i \(-0.205672\pi\)
\(102\) 0 0
\(103\) 9.74573i 0.960276i −0.877193 0.480138i \(-0.840587\pi\)
0.877193 0.480138i \(-0.159413\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −10.0549 −0.972048 −0.486024 0.873945i \(-0.661553\pi\)
−0.486024 + 0.873945i \(0.661553\pi\)
\(108\) 0 0
\(109\) 15.9444i 1.52719i −0.645694 0.763596i \(-0.723432\pi\)
0.645694 0.763596i \(-0.276568\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 10.4113 0.979410 0.489705 0.871888i \(-0.337104\pi\)
0.489705 + 0.871888i \(0.337104\pi\)
\(114\) 0 0
\(115\) 4.64138 + 1.20731i 0.432811 + 0.112582i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 20.0340i 1.83651i
\(120\) 0 0
\(121\) 27.2654 2.47867
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −3.05363 −0.270966 −0.135483 0.990780i \(-0.543259\pi\)
−0.135483 + 0.990780i \(0.543259\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 7.35861i 0.642925i 0.946922 + 0.321462i \(0.104174\pi\)
−0.946922 + 0.321462i \(0.895826\pi\)
\(132\) 0 0
\(133\) 6.58609 0.571087
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −7.68321 −0.656421 −0.328211 0.944605i \(-0.606446\pi\)
−0.328211 + 0.944605i \(0.606446\pi\)
\(138\) 0 0
\(139\) −17.6844 −1.49997 −0.749986 0.661453i \(-0.769940\pi\)
−0.749986 + 0.661453i \(0.769940\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 9.69930 0.811096
\(144\) 0 0
\(145\) 7.31830i 0.607752i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 24.2027 1.98276 0.991380 0.131014i \(-0.0418234\pi\)
0.991380 + 0.131014i \(0.0418234\pi\)
\(150\) 0 0
\(151\) 10.3560 0.842759 0.421379 0.906885i \(-0.361546\pi\)
0.421379 + 0.906885i \(0.361546\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −5.07406 −0.407559
\(156\) 0 0
\(157\) 5.29703i 0.422749i 0.977405 + 0.211375i \(0.0677940\pi\)
−0.977405 + 0.211375i \(0.932206\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 3.29527 12.6683i 0.259704 0.998403i
\(162\) 0 0
\(163\) −4.95215 −0.387882 −0.193941 0.981013i \(-0.562127\pi\)
−0.193941 + 0.981013i \(0.562127\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 8.20101i 0.634613i 0.948323 + 0.317307i \(0.102778\pi\)
−0.948323 + 0.317307i \(0.897222\pi\)
\(168\) 0 0
\(169\) −10.5415 −0.810883
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 18.3258i 1.39328i −0.717420 0.696641i \(-0.754677\pi\)
0.717420 0.696641i \(-0.245323\pi\)
\(174\) 0 0
\(175\) 2.72943i 0.206325i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 6.21672i 0.464659i −0.972637 0.232330i \(-0.925365\pi\)
0.972637 0.232330i \(-0.0746348\pi\)
\(180\) 0 0
\(181\) 10.4391i 0.775932i 0.921674 + 0.387966i \(0.126822\pi\)
−0.921674 + 0.387966i \(0.873178\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 8.91623i 0.655534i
\(186\) 0 0
\(187\) 45.4045 3.32030
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 22.7744 1.64790 0.823949 0.566664i \(-0.191766\pi\)
0.823949 + 0.566664i \(0.191766\pi\)
\(192\) 0 0
\(193\) −16.6166 −1.19609 −0.598046 0.801462i \(-0.704056\pi\)
−0.598046 + 0.801462i \(0.704056\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 18.7600i 1.33660i −0.743894 0.668298i \(-0.767023\pi\)
0.743894 0.668298i \(-0.232977\pi\)
\(198\) 0 0
\(199\) 22.2158i 1.57483i 0.616420 + 0.787417i \(0.288582\pi\)
−0.616420 + 0.787417i \(0.711418\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −19.9748 −1.40196
\(204\) 0 0
\(205\) 1.74173i 0.121647i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 14.9265i 1.03249i
\(210\) 0 0
\(211\) 24.8053 1.70766 0.853832 0.520549i \(-0.174273\pi\)
0.853832 + 0.520549i \(0.174273\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 11.9685i 0.816242i
\(216\) 0 0
\(217\) 13.8493i 0.940152i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 11.5089 0.774172
\(222\) 0 0
\(223\) 25.9505 1.73777 0.868887 0.495010i \(-0.164836\pi\)
0.868887 + 0.495010i \(0.164836\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 11.0258 0.731808 0.365904 0.930653i \(-0.380760\pi\)
0.365904 + 0.930653i \(0.380760\pi\)
\(228\) 0 0
\(229\) 17.6864i 1.16875i 0.811483 + 0.584376i \(0.198661\pi\)
−0.811483 + 0.584376i \(0.801339\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 21.3444i 1.39832i 0.714965 + 0.699160i \(0.246443\pi\)
−0.714965 + 0.699160i \(0.753557\pi\)
\(234\) 0 0
\(235\) 0.0866043i 0.00564944i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 4.47318i 0.289346i 0.989480 + 0.144673i \(0.0462130\pi\)
−0.989480 + 0.144673i \(0.953787\pi\)
\(240\) 0 0
\(241\) 5.95063i 0.383314i −0.981462 0.191657i \(-0.938614\pi\)
0.981462 0.191657i \(-0.0613861\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0.449787 0.0287358
\(246\) 0 0
\(247\) 3.78350i 0.240738i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −25.0983 −1.58419 −0.792094 0.610399i \(-0.791009\pi\)
−0.792094 + 0.610399i \(0.791009\pi\)
\(252\) 0 0
\(253\) 28.7111 + 7.46830i 1.80505 + 0.469528i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 6.12015i 0.381764i −0.981613 0.190882i \(-0.938865\pi\)
0.981613 0.190882i \(-0.0611348\pi\)
\(258\) 0 0
\(259\) −24.3362 −1.51218
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 16.5265 1.01907 0.509534 0.860451i \(-0.329818\pi\)
0.509534 + 0.860451i \(0.329818\pi\)
\(264\) 0 0
\(265\) 2.22559 0.136717
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 16.7090i 1.01876i −0.860541 0.509382i \(-0.829874\pi\)
0.860541 0.509382i \(-0.170126\pi\)
\(270\) 0 0
\(271\) −13.5676 −0.824173 −0.412086 0.911145i \(-0.635200\pi\)
−0.412086 + 0.911145i \(0.635200\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −6.18590 −0.373024
\(276\) 0 0
\(277\) −14.8253 −0.890768 −0.445384 0.895340i \(-0.646933\pi\)
−0.445384 + 0.895340i \(0.646933\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −3.77942 −0.225461 −0.112731 0.993626i \(-0.535960\pi\)
−0.112731 + 0.993626i \(0.535960\pi\)
\(282\) 0 0
\(283\) 20.5953i 1.22427i −0.790755 0.612133i \(-0.790312\pi\)
0.790755 0.612133i \(-0.209688\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −4.75392 −0.280615
\(288\) 0 0
\(289\) 36.8756 2.16915
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −30.0552 −1.75585 −0.877923 0.478803i \(-0.841071\pi\)
−0.877923 + 0.478803i \(0.841071\pi\)
\(294\) 0 0
\(295\) 0.102805i 0.00598554i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 7.27754 + 1.89303i 0.420871 + 0.109477i
\(300\) 0 0
\(301\) −32.6671 −1.88290
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 10.4353i 0.597523i
\(306\) 0 0
\(307\) 34.0696 1.94445 0.972227 0.234041i \(-0.0751949\pi\)
0.972227 + 0.234041i \(0.0751949\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 9.51737i 0.539681i 0.962905 + 0.269840i \(0.0869709\pi\)
−0.962905 + 0.269840i \(0.913029\pi\)
\(312\) 0 0
\(313\) 7.53633i 0.425979i −0.977055 0.212989i \(-0.931680\pi\)
0.977055 0.212989i \(-0.0683199\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 14.3174i 0.804147i −0.915607 0.402074i \(-0.868290\pi\)
0.915607 0.402074i \(-0.131710\pi\)
\(318\) 0 0
\(319\) 45.2703i 2.53465i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 17.7114i 0.985486i
\(324\) 0 0
\(325\) −1.56797 −0.0869753
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0.236380 0.0130321
\(330\) 0 0
\(331\) −0.171170 −0.00940834 −0.00470417 0.999989i \(-0.501497\pi\)
−0.00470417 + 0.999989i \(0.501497\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 8.46561i 0.462526i
\(336\) 0 0
\(337\) 16.9754i 0.924710i 0.886695 + 0.462355i \(0.152995\pi\)
−0.886695 + 0.462355i \(0.847005\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −31.3876 −1.69974
\(342\) 0 0
\(343\) 17.8783i 0.965340i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 4.13762i 0.222119i −0.993814 0.111059i \(-0.964576\pi\)
0.993814 0.111059i \(-0.0354244\pi\)
\(348\) 0 0
\(349\) 5.48433 0.293569 0.146785 0.989168i \(-0.453108\pi\)
0.146785 + 0.989168i \(0.453108\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 30.8604i 1.64253i −0.570545 0.821267i \(-0.693268\pi\)
0.570545 0.821267i \(-0.306732\pi\)
\(354\) 0 0
\(355\) 12.4723i 0.661959i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 16.0049 0.844707 0.422353 0.906431i \(-0.361204\pi\)
0.422353 + 0.906431i \(0.361204\pi\)
\(360\) 0 0
\(361\) 13.1775 0.693551
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 11.2668 0.589731
\(366\) 0 0
\(367\) 12.5743i 0.656374i −0.944613 0.328187i \(-0.893562\pi\)
0.944613 0.328187i \(-0.106438\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 6.07460i 0.315377i
\(372\) 0 0
\(373\) 10.7448i 0.556345i 0.960531 + 0.278173i \(0.0897287\pi\)
−0.960531 + 0.278173i \(0.910271\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 11.4749i 0.590986i
\(378\) 0 0
\(379\) 4.11204i 0.211221i 0.994408 + 0.105611i \(0.0336797\pi\)
−0.994408 + 0.105611i \(0.966320\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 4.62303 0.236226 0.118113 0.993000i \(-0.462316\pi\)
0.118113 + 0.993000i \(0.462316\pi\)
\(384\) 0 0
\(385\) 16.8840i 0.860487i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 5.17964 0.262618 0.131309 0.991341i \(-0.458082\pi\)
0.131309 + 0.991341i \(0.458082\pi\)
\(390\) 0 0
\(391\) 34.0677 + 8.86166i 1.72288 + 0.448153i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 1.00635i 0.0506347i
\(396\) 0 0
\(397\) −20.4023 −1.02396 −0.511980 0.858997i \(-0.671088\pi\)
−0.511980 + 0.858997i \(0.671088\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 24.7937 1.23814 0.619069 0.785336i \(-0.287510\pi\)
0.619069 + 0.785336i \(0.287510\pi\)
\(402\) 0 0
\(403\) −7.95597 −0.396315
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 55.1549i 2.73393i
\(408\) 0 0
\(409\) 10.9243 0.540174 0.270087 0.962836i \(-0.412948\pi\)
0.270087 + 0.962836i \(0.412948\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −0.280599 −0.0138074
\(414\) 0 0
\(415\) −4.30642 −0.211394
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −31.0588 −1.51732 −0.758661 0.651486i \(-0.774146\pi\)
−0.758661 + 0.651486i \(0.774146\pi\)
\(420\) 0 0
\(421\) 35.6421i 1.73709i 0.495610 + 0.868545i \(0.334944\pi\)
−0.495610 + 0.868545i \(0.665056\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −7.34000 −0.356042
\(426\) 0 0
\(427\) 28.4824 1.37836
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 30.5109 1.46966 0.734830 0.678251i \(-0.237262\pi\)
0.734830 + 0.678251i \(0.237262\pi\)
\(432\) 0 0
\(433\) 31.4222i 1.51006i −0.655693 0.755028i \(-0.727623\pi\)
0.655693 0.755028i \(-0.272377\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −2.91323 + 11.1996i −0.139359 + 0.535750i
\(438\) 0 0
\(439\) −18.5846 −0.886993 −0.443497 0.896276i \(-0.646262\pi\)
−0.443497 + 0.896276i \(0.646262\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 22.6267i 1.07503i 0.843255 + 0.537513i \(0.180636\pi\)
−0.843255 + 0.537513i \(0.819364\pi\)
\(444\) 0 0
\(445\) 3.93965 0.186757
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 18.5270i 0.874344i −0.899378 0.437172i \(-0.855980\pi\)
0.899378 0.437172i \(-0.144020\pi\)
\(450\) 0 0
\(451\) 10.7741i 0.507335i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 4.27966i 0.200634i
\(456\) 0 0
\(457\) 18.8954i 0.883890i 0.897042 + 0.441945i \(0.145711\pi\)
−0.897042 + 0.441945i \(0.854289\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 15.0747i 0.702098i −0.936357 0.351049i \(-0.885825\pi\)
0.936357 0.351049i \(-0.114175\pi\)
\(462\) 0 0
\(463\) 12.1878 0.566413 0.283207 0.959059i \(-0.408602\pi\)
0.283207 + 0.959059i \(0.408602\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −31.2902 −1.44794 −0.723968 0.689833i \(-0.757684\pi\)
−0.723968 + 0.689833i \(0.757684\pi\)
\(468\) 0 0
\(469\) 23.1063 1.06695
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 74.0357i 3.40416i
\(474\) 0 0
\(475\) 2.41299i 0.110716i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 7.04788 0.322026 0.161013 0.986952i \(-0.448524\pi\)
0.161013 + 0.986952i \(0.448524\pi\)
\(480\) 0 0
\(481\) 13.9804i 0.637450i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 14.9674i 0.679633i
\(486\) 0 0
\(487\) 19.3153 0.875260 0.437630 0.899155i \(-0.355818\pi\)
0.437630 + 0.899155i \(0.355818\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 22.3584i 1.00902i 0.863405 + 0.504511i \(0.168327\pi\)
−0.863405 + 0.504511i \(0.831673\pi\)
\(492\) 0 0
\(493\) 53.7163i 2.41926i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −34.0422 −1.52700
\(498\) 0 0
\(499\) 3.51343 0.157283 0.0786413 0.996903i \(-0.474942\pi\)
0.0786413 + 0.996903i \(0.474942\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 16.8774 0.752527 0.376264 0.926513i \(-0.377209\pi\)
0.376264 + 0.926513i \(0.377209\pi\)
\(504\) 0 0
\(505\) 12.1022i 0.538541i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 8.62970i 0.382505i 0.981541 + 0.191252i \(0.0612549\pi\)
−0.981541 + 0.191252i \(0.938745\pi\)
\(510\) 0 0
\(511\) 30.7519i 1.36038i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 9.74573i 0.429448i
\(516\) 0 0
\(517\) 0.535725i 0.0235612i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 6.09279 0.266930 0.133465 0.991054i \(-0.457390\pi\)
0.133465 + 0.991054i \(0.457390\pi\)
\(522\) 0 0
\(523\) 18.5658i 0.811826i −0.913912 0.405913i \(-0.866954\pi\)
0.913912 0.405913i \(-0.133046\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −37.2436 −1.62236
\(528\) 0 0
\(529\) 20.0848 + 11.2072i 0.873252 + 0.487268i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 2.73097i 0.118292i
\(534\) 0 0
\(535\) 10.0549 0.434713
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 2.78234 0.119844
\(540\) 0 0
\(541\) −19.1292 −0.822427 −0.411213 0.911539i \(-0.634895\pi\)
−0.411213 + 0.911539i \(0.634895\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 15.9444i 0.682981i
\(546\) 0 0
\(547\) −16.3021 −0.697028 −0.348514 0.937304i \(-0.613314\pi\)
−0.348514 + 0.937304i \(0.613314\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 17.6590 0.752299
\(552\) 0 0
\(553\) 2.74675 0.116804
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −33.7685 −1.43082 −0.715409 0.698706i \(-0.753760\pi\)
−0.715409 + 0.698706i \(0.753760\pi\)
\(558\) 0 0
\(559\) 18.7662i 0.793725i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 22.4345 0.945501 0.472750 0.881196i \(-0.343261\pi\)
0.472750 + 0.881196i \(0.343261\pi\)
\(564\) 0 0
\(565\) −10.4113 −0.438005
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 27.0265 1.13301 0.566505 0.824059i \(-0.308295\pi\)
0.566505 + 0.824059i \(0.308295\pi\)
\(570\) 0 0
\(571\) 10.7398i 0.449449i −0.974422 0.224724i \(-0.927852\pi\)
0.974422 0.224724i \(-0.0721482\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −4.64138 1.20731i −0.193559 0.0503483i
\(576\) 0 0
\(577\) −8.54243 −0.355626 −0.177813 0.984064i \(-0.556902\pi\)
−0.177813 + 0.984064i \(0.556902\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 11.7541i 0.487642i
\(582\) 0 0
\(583\) 13.7673 0.570183
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 29.1545i 1.20334i 0.798746 + 0.601668i \(0.205497\pi\)
−0.798746 + 0.601668i \(0.794503\pi\)
\(588\) 0 0
\(589\) 12.2437i 0.504492i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 3.40263i 0.139729i 0.997556 + 0.0698645i \(0.0222567\pi\)
−0.997556 + 0.0698645i \(0.977743\pi\)
\(594\) 0 0
\(595\) 20.0340i 0.821314i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 4.55038i 0.185924i −0.995670 0.0929618i \(-0.970367\pi\)
0.995670 0.0929618i \(-0.0296334\pi\)
\(600\) 0 0
\(601\) −4.01391 −0.163731 −0.0818655 0.996643i \(-0.526088\pi\)
−0.0818655 + 0.996643i \(0.526088\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −27.2654 −1.10849
\(606\) 0 0
\(607\) −24.0608 −0.976599 −0.488299 0.872676i \(-0.662383\pi\)
−0.488299 + 0.872676i \(0.662383\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0.135793i 0.00549359i
\(612\) 0 0
\(613\) 24.2044i 0.977606i 0.872394 + 0.488803i \(0.162566\pi\)
−0.872394 + 0.488803i \(0.837434\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 44.3709 1.78631 0.893153 0.449753i \(-0.148488\pi\)
0.893153 + 0.449753i \(0.148488\pi\)
\(618\) 0 0
\(619\) 33.0294i 1.32756i −0.747926 0.663782i \(-0.768950\pi\)
0.747926 0.663782i \(-0.231050\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 10.7530i 0.430810i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 65.4451i 2.60947i
\(630\) 0 0
\(631\) 2.46903i 0.0982905i 0.998792 + 0.0491452i \(0.0156497\pi\)
−0.998792 + 0.0491452i \(0.984350\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 3.05363 0.121180
\(636\) 0 0
\(637\) 0.705252 0.0279431
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −34.7810 −1.37377 −0.686883 0.726768i \(-0.741021\pi\)
−0.686883 + 0.726768i \(0.741021\pi\)
\(642\) 0 0
\(643\) 34.0505i 1.34282i 0.741087 + 0.671409i \(0.234311\pi\)
−0.741087 + 0.671409i \(0.765689\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 8.32463i 0.327275i 0.986521 + 0.163637i \(0.0523227\pi\)
−0.986521 + 0.163637i \(0.947677\pi\)
\(648\) 0 0
\(649\) 0.635941i 0.0249629i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 8.54152i 0.334256i −0.985935 0.167128i \(-0.946551\pi\)
0.985935 0.167128i \(-0.0534493\pi\)
\(654\) 0 0
\(655\) 7.35861i 0.287525i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 3.56783 0.138983 0.0694914 0.997583i \(-0.477862\pi\)
0.0694914 + 0.997583i \(0.477862\pi\)
\(660\) 0 0
\(661\) 0.357224i 0.0138944i −0.999976 0.00694721i \(-0.997789\pi\)
0.999976 0.00694721i \(-0.00221138\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −6.58609 −0.255398
\(666\) 0 0
\(667\) 8.83547 33.9670i 0.342111 1.31521i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 64.5517i 2.49199i
\(672\) 0 0
\(673\) 11.4213 0.440260 0.220130 0.975471i \(-0.429352\pi\)
0.220130 + 0.975471i \(0.429352\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −29.2218 −1.12308 −0.561542 0.827449i \(-0.689792\pi\)
−0.561542 + 0.827449i \(0.689792\pi\)
\(678\) 0 0
\(679\) 40.8524 1.56777
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0.467688i 0.0178956i −0.999960 0.00894780i \(-0.997152\pi\)
0.999960 0.00894780i \(-0.00284821\pi\)
\(684\) 0 0
\(685\) 7.68321 0.293561
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 3.48966 0.132945
\(690\) 0 0
\(691\) −19.9850 −0.760265 −0.380133 0.924932i \(-0.624122\pi\)
−0.380133 + 0.924932i \(0.624122\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 17.6844 0.670808
\(696\) 0 0
\(697\) 12.7843i 0.484239i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −13.4329 −0.507354 −0.253677 0.967289i \(-0.581640\pi\)
−0.253677 + 0.967289i \(0.581640\pi\)
\(702\) 0 0
\(703\) 21.5148 0.811446
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 33.0321 1.24230
\(708\) 0 0
\(709\) 29.9396i 1.12440i −0.827000 0.562202i \(-0.809954\pi\)
0.827000 0.562202i \(-0.190046\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −23.5506 6.12597i −0.881979 0.229419i
\(714\) 0 0
\(715\) −9.69930 −0.362733
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 29.0706i 1.08415i −0.840330 0.542075i \(-0.817639\pi\)
0.840330 0.542075i \(-0.182361\pi\)
\(720\) 0 0
\(721\) 26.6003 0.990647
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 7.31830i 0.271795i
\(726\) 0 0
\(727\) 28.5122i 1.05746i 0.848791 + 0.528729i \(0.177331\pi\)
−0.848791 + 0.528729i \(0.822669\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 87.8485i 3.24919i
\(732\) 0 0
\(733\) 21.0285i 0.776705i −0.921511 0.388353i \(-0.873044\pi\)
0.921511 0.388353i \(-0.126956\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 52.3674i 1.92898i
\(738\) 0 0
\(739\) 24.6546 0.906932 0.453466 0.891273i \(-0.350187\pi\)
0.453466 + 0.891273i \(0.350187\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 11.5216 0.422685 0.211342 0.977412i \(-0.432216\pi\)
0.211342 + 0.977412i \(0.432216\pi\)
\(744\) 0 0
\(745\) −24.2027 −0.886718
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 27.4443i 1.00279i
\(750\) 0 0
\(751\) 17.5858i 0.641715i 0.947127 + 0.320858i \(0.103971\pi\)
−0.947127 + 0.320858i \(0.896029\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −10.3560 −0.376893
\(756\) 0 0
\(757\) 4.47669i 0.162708i −0.996685 0.0813540i \(-0.974076\pi\)
0.996685 0.0813540i \(-0.0259244\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 28.4348i 1.03076i 0.856962 + 0.515380i \(0.172349\pi\)
−0.856962 + 0.515380i \(0.827651\pi\)
\(762\) 0 0
\(763\) 43.5190 1.57549
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0.161195i 0.00582041i
\(768\) 0 0
\(769\) 1.96762i 0.0709542i 0.999370 + 0.0354771i \(0.0112951\pi\)
−0.999370 + 0.0354771i \(0.988705\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 9.57418 0.344359 0.172180 0.985066i \(-0.444919\pi\)
0.172180 + 0.985066i \(0.444919\pi\)
\(774\) 0 0
\(775\) 5.07406 0.182266
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 4.20277 0.150580
\(780\) 0 0
\(781\) 77.1522i 2.76072i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 5.29703i 0.189059i
\(786\) 0 0
\(787\) 24.5145i 0.873846i 0.899499 + 0.436923i \(0.143932\pi\)
−0.899499 + 0.436923i \(0.856068\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 28.4168i 1.01039i
\(792\) 0 0
\(793\) 16.3622i 0.581040i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −23.7348 −0.840730 −0.420365 0.907355i \(-0.638098\pi\)
−0.420365 + 0.907355i \(0.638098\pi\)
\(798\) 0 0
\(799\) 0.635675i 0.0224886i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 69.6952 2.45949
\(804\) 0 0
\(805\) −3.29527 + 12.6683i −0.116143 + 0.446500i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 20.3700i 0.716173i 0.933688 + 0.358086i \(0.116571\pi\)