Properties

Label 8280.2.p.a.1241.34
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 1241.34
Character \(\chi\) \(=\) 8280.1241
Dual form 8280.2.p.a.1241.15

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{5} +2.09276i q^{7} +O(q^{10})\) \(q-1.00000 q^{5} +2.09276i q^{7} -3.07645 q^{11} +3.07361 q^{13} +2.79768 q^{17} +7.09530i q^{19} +(-2.07759 - 4.32246i) q^{23} +1.00000 q^{25} +7.02820i q^{29} +2.54854 q^{31} -2.09276i q^{35} +11.8247i q^{37} +1.34987i q^{41} -7.71826i q^{43} +2.64576i q^{47} +2.62034 q^{49} -5.21522 q^{53} +3.07645 q^{55} -9.91273i q^{59} -7.34076i q^{61} -3.07361 q^{65} +1.63371i q^{67} -13.2280i q^{71} -10.0650 q^{73} -6.43829i q^{77} -0.0188146i q^{79} -1.02131 q^{83} -2.79768 q^{85} +8.49568 q^{89} +6.43233i q^{91} -7.09530i q^{95} +12.5278i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q - 48 q^{5}+O(q^{10}) \) 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
\(7\) 2.09276i 0.790991i 0.918468 + 0.395495i \(0.129427\pi\)
−0.918468 + 0.395495i \(0.870573\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −3.07645 −0.927585 −0.463793 0.885944i \(-0.653512\pi\)
−0.463793 + 0.885944i \(0.653512\pi\)
\(12\) 0 0
\(13\) 3.07361 0.852465 0.426232 0.904614i \(-0.359841\pi\)
0.426232 + 0.904614i \(0.359841\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.79768 0.678537 0.339268 0.940690i \(-0.389821\pi\)
0.339268 + 0.940690i \(0.389821\pi\)
\(18\) 0 0
\(19\) 7.09530i 1.62777i 0.581024 + 0.813886i \(0.302652\pi\)
−0.581024 + 0.813886i \(0.697348\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.07759 4.32246i −0.433207 0.901294i
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 7.02820i 1.30510i 0.757744 + 0.652552i \(0.226302\pi\)
−0.757744 + 0.652552i \(0.773698\pi\)
\(30\) 0 0
\(31\) 2.54854 0.457731 0.228865 0.973458i \(-0.426498\pi\)
0.228865 + 0.973458i \(0.426498\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.09276i 0.353742i
\(36\) 0 0
\(37\) 11.8247i 1.94397i 0.235049 + 0.971984i \(0.424475\pi\)
−0.235049 + 0.971984i \(0.575525\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.34987i 0.210814i 0.994429 + 0.105407i \(0.0336145\pi\)
−0.994429 + 0.105407i \(0.966386\pi\)
\(42\) 0 0
\(43\) 7.71826i 1.17702i −0.808489 0.588512i \(-0.799714\pi\)
0.808489 0.588512i \(-0.200286\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.64576i 0.385923i 0.981206 + 0.192962i \(0.0618093\pi\)
−0.981206 + 0.192962i \(0.938191\pi\)
\(48\) 0 0
\(49\) 2.62034 0.374334
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −5.21522 −0.716365 −0.358182 0.933652i \(-0.616603\pi\)
−0.358182 + 0.933652i \(0.616603\pi\)
\(54\) 0 0
\(55\) 3.07645 0.414829
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 9.91273i 1.29053i −0.763960 0.645264i \(-0.776748\pi\)
0.763960 0.645264i \(-0.223252\pi\)
\(60\) 0 0
\(61\) 7.34076i 0.939888i −0.882696 0.469944i \(-0.844274\pi\)
0.882696 0.469944i \(-0.155726\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −3.07361 −0.381234
\(66\) 0 0
\(67\) 1.63371i 0.199589i 0.995008 + 0.0997947i \(0.0318186\pi\)
−0.995008 + 0.0997947i \(0.968181\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 13.2280i 1.56987i −0.619579 0.784934i \(-0.712697\pi\)
0.619579 0.784934i \(-0.287303\pi\)
\(72\) 0 0
\(73\) −10.0650 −1.17802 −0.589008 0.808127i \(-0.700481\pi\)
−0.589008 + 0.808127i \(0.700481\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.43829i 0.733711i
\(78\) 0 0
\(79\) 0.0188146i 0.00211681i −0.999999 0.00105840i \(-0.999663\pi\)
0.999999 0.00105840i \(-0.000336901\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −1.02131 −0.112103 −0.0560515 0.998428i \(-0.517851\pi\)
−0.0560515 + 0.998428i \(0.517851\pi\)
\(84\) 0 0
\(85\) −2.79768 −0.303451
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 8.49568 0.900540 0.450270 0.892892i \(-0.351328\pi\)
0.450270 + 0.892892i \(0.351328\pi\)
\(90\) 0 0
\(91\) 6.43233i 0.674292i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 7.09530i 0.727962i
\(96\) 0 0
\(97\) 12.5278i 1.27201i 0.771685 + 0.636004i \(0.219414\pi\)
−0.771685 + 0.636004i \(0.780586\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 16.5341i 1.64520i 0.568619 + 0.822601i \(0.307478\pi\)
−0.568619 + 0.822601i \(0.692522\pi\)
\(102\) 0 0
\(103\) 3.79697i 0.374126i −0.982348 0.187063i \(-0.940103\pi\)
0.982348 0.187063i \(-0.0598969\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 6.62880 0.640831 0.320415 0.947277i \(-0.396178\pi\)
0.320415 + 0.947277i \(0.396178\pi\)
\(108\) 0 0
\(109\) 10.7152i 1.02633i 0.858291 + 0.513163i \(0.171526\pi\)
−0.858291 + 0.513163i \(0.828474\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 3.32932 0.313196 0.156598 0.987662i \(-0.449947\pi\)
0.156598 + 0.987662i \(0.449947\pi\)
\(114\) 0 0
\(115\) 2.07759 + 4.32246i 0.193736 + 0.403071i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 5.85488i 0.536716i
\(120\) 0 0
\(121\) −1.53544 −0.139586
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −6.87892 −0.610405 −0.305203 0.952287i \(-0.598724\pi\)
−0.305203 + 0.952287i \(0.598724\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 8.90544i 0.778072i 0.921223 + 0.389036i \(0.127192\pi\)
−0.921223 + 0.389036i \(0.872808\pi\)
\(132\) 0 0
\(133\) −14.8488 −1.28755
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −10.0328 −0.857160 −0.428580 0.903504i \(-0.640986\pi\)
−0.428580 + 0.903504i \(0.640986\pi\)
\(138\) 0 0
\(139\) 13.7064 1.16256 0.581280 0.813703i \(-0.302552\pi\)
0.581280 + 0.813703i \(0.302552\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −9.45580 −0.790734
\(144\) 0 0
\(145\) 7.02820i 0.583660i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −9.27123 −0.759528 −0.379764 0.925083i \(-0.623995\pi\)
−0.379764 + 0.925083i \(0.623995\pi\)
\(150\) 0 0
\(151\) 1.24400 0.101235 0.0506177 0.998718i \(-0.483881\pi\)
0.0506177 + 0.998718i \(0.483881\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −2.54854 −0.204703
\(156\) 0 0
\(157\) 4.74224i 0.378472i 0.981932 + 0.189236i \(0.0606011\pi\)
−0.981932 + 0.189236i \(0.939399\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 9.04588 4.34790i 0.712916 0.342663i
\(162\) 0 0
\(163\) 9.12442 0.714680 0.357340 0.933974i \(-0.383684\pi\)
0.357340 + 0.933974i \(0.383684\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 3.76985i 0.291719i −0.989305 0.145860i \(-0.953405\pi\)
0.989305 0.145860i \(-0.0465948\pi\)
\(168\) 0 0
\(169\) −3.55295 −0.273304
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 4.32929i 0.329150i −0.986365 0.164575i \(-0.947375\pi\)
0.986365 0.164575i \(-0.0526253\pi\)
\(174\) 0 0
\(175\) 2.09276i 0.158198i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 10.7196i 0.801221i 0.916248 + 0.400610i \(0.131202\pi\)
−0.916248 + 0.400610i \(0.868798\pi\)
\(180\) 0 0
\(181\) 3.63105i 0.269894i 0.990853 + 0.134947i \(0.0430864\pi\)
−0.990853 + 0.134947i \(0.956914\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 11.8247i 0.869369i
\(186\) 0 0
\(187\) −8.60692 −0.629400
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −10.9287 −0.790776 −0.395388 0.918514i \(-0.629390\pi\)
−0.395388 + 0.918514i \(0.629390\pi\)
\(192\) 0 0
\(193\) −11.6237 −0.836691 −0.418346 0.908288i \(-0.637390\pi\)
−0.418346 + 0.908288i \(0.637390\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 10.7721i 0.767478i −0.923442 0.383739i \(-0.874636\pi\)
0.923442 0.383739i \(-0.125364\pi\)
\(198\) 0 0
\(199\) 12.9289i 0.916503i −0.888823 0.458252i \(-0.848476\pi\)
0.888823 0.458252i \(-0.151524\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −14.7084 −1.03233
\(204\) 0 0
\(205\) 1.34987i 0.0942788i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 21.8283i 1.50990i
\(210\) 0 0
\(211\) −12.6022 −0.867571 −0.433785 0.901016i \(-0.642822\pi\)
−0.433785 + 0.901016i \(0.642822\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 7.71826i 0.526381i
\(216\) 0 0
\(217\) 5.33349i 0.362061i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 8.59896 0.578429
\(222\) 0 0
\(223\) 0.837321 0.0560711 0.0280356 0.999607i \(-0.491075\pi\)
0.0280356 + 0.999607i \(0.491075\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −15.6146 −1.03638 −0.518189 0.855266i \(-0.673393\pi\)
−0.518189 + 0.855266i \(0.673393\pi\)
\(228\) 0 0
\(229\) 4.41086i 0.291478i 0.989323 + 0.145739i \(0.0465560\pi\)
−0.989323 + 0.145739i \(0.953444\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 23.8541i 1.56274i 0.624070 + 0.781368i \(0.285478\pi\)
−0.624070 + 0.781368i \(0.714522\pi\)
\(234\) 0 0
\(235\) 2.64576i 0.172590i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 20.1853i 1.30568i 0.757497 + 0.652839i \(0.226422\pi\)
−0.757497 + 0.652839i \(0.773578\pi\)
\(240\) 0 0
\(241\) 13.0408i 0.840034i 0.907516 + 0.420017i \(0.137976\pi\)
−0.907516 + 0.420017i \(0.862024\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −2.62034 −0.167407
\(246\) 0 0
\(247\) 21.8081i 1.38762i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −27.2149 −1.71779 −0.858896 0.512150i \(-0.828849\pi\)
−0.858896 + 0.512150i \(0.828849\pi\)
\(252\) 0 0
\(253\) 6.39160 + 13.2978i 0.401836 + 0.836027i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 17.3463i 1.08203i −0.841012 0.541017i \(-0.818039\pi\)
0.841012 0.541017i \(-0.181961\pi\)
\(258\) 0 0
\(259\) −24.7463 −1.53766
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 8.47133 0.522364 0.261182 0.965290i \(-0.415888\pi\)
0.261182 + 0.965290i \(0.415888\pi\)
\(264\) 0 0
\(265\) 5.21522 0.320368
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 8.35318i 0.509302i −0.967033 0.254651i \(-0.918039\pi\)
0.967033 0.254651i \(-0.0819606\pi\)
\(270\) 0 0
\(271\) −11.7954 −0.716519 −0.358259 0.933622i \(-0.616630\pi\)
−0.358259 + 0.933622i \(0.616630\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −3.07645 −0.185517
\(276\) 0 0
\(277\) −24.9189 −1.49723 −0.748617 0.663003i \(-0.769282\pi\)
−0.748617 + 0.663003i \(0.769282\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 5.94998 0.354946 0.177473 0.984126i \(-0.443208\pi\)
0.177473 + 0.984126i \(0.443208\pi\)
\(282\) 0 0
\(283\) 16.1981i 0.962879i 0.876479 + 0.481440i \(0.159886\pi\)
−0.876479 + 0.481440i \(0.840114\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −2.82495 −0.166752
\(288\) 0 0
\(289\) −9.17300 −0.539588
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −2.80011 −0.163584 −0.0817922 0.996649i \(-0.526064\pi\)
−0.0817922 + 0.996649i \(0.526064\pi\)
\(294\) 0 0
\(295\) 9.91273i 0.577141i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −6.38569 13.2855i −0.369294 0.768322i
\(300\) 0 0
\(301\) 16.1525 0.931015
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 7.34076i 0.420331i
\(306\) 0 0
\(307\) 15.4933 0.884248 0.442124 0.896954i \(-0.354225\pi\)
0.442124 + 0.896954i \(0.354225\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 25.3465i 1.43727i −0.695389 0.718633i \(-0.744768\pi\)
0.695389 0.718633i \(-0.255232\pi\)
\(312\) 0 0
\(313\) 19.3753i 1.09516i −0.836754 0.547579i \(-0.815549\pi\)
0.836754 0.547579i \(-0.184451\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 17.9788i 1.00979i 0.863181 + 0.504895i \(0.168469\pi\)
−0.863181 + 0.504895i \(0.831531\pi\)
\(318\) 0 0
\(319\) 21.6219i 1.21059i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 19.8504i 1.10450i
\(324\) 0 0
\(325\) 3.07361 0.170493
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −5.53695 −0.305262
\(330\) 0 0
\(331\) −3.38098 −0.185835 −0.0929176 0.995674i \(-0.529619\pi\)
−0.0929176 + 0.995674i \(0.529619\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 1.63371i 0.0892591i
\(336\) 0 0
\(337\) 2.27395i 0.123870i −0.998080 0.0619351i \(-0.980273\pi\)
0.998080 0.0619351i \(-0.0197272\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −7.84045 −0.424584
\(342\) 0 0
\(343\) 20.1331i 1.08709i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 14.0479i 0.754133i −0.926186 0.377066i \(-0.876933\pi\)
0.926186 0.377066i \(-0.123067\pi\)
\(348\) 0 0
\(349\) −14.1740 −0.758714 −0.379357 0.925250i \(-0.623855\pi\)
−0.379357 + 0.925250i \(0.623855\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 7.13650i 0.379838i −0.981800 0.189919i \(-0.939178\pi\)
0.981800 0.189919i \(-0.0608225\pi\)
\(354\) 0 0
\(355\) 13.2280i 0.702067i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −2.04176 −0.107760 −0.0538799 0.998547i \(-0.517159\pi\)
−0.0538799 + 0.998547i \(0.517159\pi\)
\(360\) 0 0
\(361\) −31.3433 −1.64964
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 10.0650 0.526825
\(366\) 0 0
\(367\) 13.4289i 0.700985i 0.936566 + 0.350493i \(0.113986\pi\)
−0.936566 + 0.350493i \(0.886014\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 10.9142i 0.566638i
\(372\) 0 0
\(373\) 27.2918i 1.41312i 0.707655 + 0.706558i \(0.249753\pi\)
−0.707655 + 0.706558i \(0.750247\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 21.6019i 1.11256i
\(378\) 0 0
\(379\) 24.0862i 1.23723i 0.785695 + 0.618614i \(0.212305\pi\)
−0.785695 + 0.618614i \(0.787695\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −11.5208 −0.588685 −0.294343 0.955700i \(-0.595101\pi\)
−0.294343 + 0.955700i \(0.595101\pi\)
\(384\) 0 0
\(385\) 6.43829i 0.328126i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −10.8756 −0.551417 −0.275708 0.961241i \(-0.588912\pi\)
−0.275708 + 0.961241i \(0.588912\pi\)
\(390\) 0 0
\(391\) −5.81242 12.0928i −0.293947 0.611561i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0.0188146i 0.000946666i
\(396\) 0 0
\(397\) 20.2022 1.01392 0.506960 0.861970i \(-0.330769\pi\)
0.506960 + 0.861970i \(0.330769\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −27.3337 −1.36498 −0.682489 0.730896i \(-0.739103\pi\)
−0.682489 + 0.730896i \(0.739103\pi\)
\(402\) 0 0
\(403\) 7.83320 0.390199
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 36.3781i 1.80319i
\(408\) 0 0
\(409\) −38.2315 −1.89043 −0.945213 0.326456i \(-0.894146\pi\)
−0.945213 + 0.326456i \(0.894146\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 20.7450 1.02080
\(414\) 0 0
\(415\) 1.02131 0.0501340
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −2.11365 −0.103259 −0.0516294 0.998666i \(-0.516441\pi\)
−0.0516294 + 0.998666i \(0.516441\pi\)
\(420\) 0 0
\(421\) 7.36189i 0.358797i −0.983777 0.179398i \(-0.942585\pi\)
0.983777 0.179398i \(-0.0574151\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 2.79768 0.135707
\(426\) 0 0
\(427\) 15.3625 0.743442
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −11.8122 −0.568972 −0.284486 0.958680i \(-0.591823\pi\)
−0.284486 + 0.958680i \(0.591823\pi\)
\(432\) 0 0
\(433\) 21.0382i 1.01103i 0.862817 + 0.505516i \(0.168698\pi\)
−0.862817 + 0.505516i \(0.831302\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 30.6691 14.7411i 1.46710 0.705163i
\(438\) 0 0
\(439\) −2.22130 −0.106017 −0.0530085 0.998594i \(-0.516881\pi\)
−0.0530085 + 0.998594i \(0.516881\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 26.9357i 1.27975i −0.768478 0.639876i \(-0.778986\pi\)
0.768478 0.639876i \(-0.221014\pi\)
\(444\) 0 0
\(445\) −8.49568 −0.402734
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 5.74044i 0.270908i 0.990784 + 0.135454i \(0.0432493\pi\)
−0.990784 + 0.135454i \(0.956751\pi\)
\(450\) 0 0
\(451\) 4.15280i 0.195548i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 6.43233i 0.301552i
\(456\) 0 0
\(457\) 12.1356i 0.567682i 0.958871 + 0.283841i \(0.0916087\pi\)
−0.958871 + 0.283841i \(0.908391\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 25.2556i 1.17627i −0.808763 0.588135i \(-0.799863\pi\)
0.808763 0.588135i \(-0.200137\pi\)
\(462\) 0 0
\(463\) −14.6393 −0.680344 −0.340172 0.940363i \(-0.610485\pi\)
−0.340172 + 0.940363i \(0.610485\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −10.1415 −0.469293 −0.234647 0.972081i \(-0.575393\pi\)
−0.234647 + 0.972081i \(0.575393\pi\)
\(468\) 0 0
\(469\) −3.41897 −0.157873
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 23.7449i 1.09179i
\(474\) 0 0
\(475\) 7.09530i 0.325555i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 15.4928 0.707884 0.353942 0.935267i \(-0.384841\pi\)
0.353942 + 0.935267i \(0.384841\pi\)
\(480\) 0 0
\(481\) 36.3444i 1.65716i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 12.5278i 0.568860i
\(486\) 0 0
\(487\) −21.5590 −0.976933 −0.488467 0.872583i \(-0.662444\pi\)
−0.488467 + 0.872583i \(0.662444\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0.416288i 0.0187868i −0.999956 0.00939340i \(-0.997010\pi\)
0.999956 0.00939340i \(-0.00299006\pi\)
\(492\) 0 0
\(493\) 19.6626i 0.885561i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 27.6830 1.24175
\(498\) 0 0
\(499\) 34.6425 1.55081 0.775405 0.631464i \(-0.217546\pi\)
0.775405 + 0.631464i \(0.217546\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −30.5155 −1.36062 −0.680310 0.732925i \(-0.738155\pi\)
−0.680310 + 0.732925i \(0.738155\pi\)
\(504\) 0 0
\(505\) 16.5341i 0.735757i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 30.4217i 1.34842i 0.738541 + 0.674208i \(0.235515\pi\)
−0.738541 + 0.674208i \(0.764485\pi\)
\(510\) 0 0
\(511\) 21.0636i 0.931799i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 3.79697i 0.167314i
\(516\) 0 0
\(517\) 8.13955i 0.357977i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −10.4148 −0.456282 −0.228141 0.973628i \(-0.573265\pi\)
−0.228141 + 0.973628i \(0.573265\pi\)
\(522\) 0 0
\(523\) 25.5943i 1.11916i −0.828776 0.559580i \(-0.810962\pi\)
0.828776 0.559580i \(-0.189038\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 7.12998 0.310587
\(528\) 0 0
\(529\) −14.3673 + 17.9606i −0.624663 + 0.780894i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 4.14896i 0.179711i
\(534\) 0 0
\(535\) −6.62880 −0.286588
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −8.06134 −0.347226
\(540\) 0 0
\(541\) 36.2724 1.55947 0.779737 0.626108i \(-0.215353\pi\)
0.779737 + 0.626108i \(0.215353\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 10.7152i 0.458987i
\(546\) 0 0
\(547\) −28.7474 −1.22915 −0.614575 0.788859i \(-0.710672\pi\)
−0.614575 + 0.788859i \(0.710672\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −49.8672 −2.12441
\(552\) 0 0
\(553\) 0.0393746 0.00167438
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −4.42657 −0.187560 −0.0937799 0.995593i \(-0.529895\pi\)
−0.0937799 + 0.995593i \(0.529895\pi\)
\(558\) 0 0
\(559\) 23.7229i 1.00337i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 22.0078 0.927520 0.463760 0.885961i \(-0.346500\pi\)
0.463760 + 0.885961i \(0.346500\pi\)
\(564\) 0 0
\(565\) −3.32932 −0.140066
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −0.339615 −0.0142374 −0.00711869 0.999975i \(-0.502266\pi\)
−0.00711869 + 0.999975i \(0.502266\pi\)
\(570\) 0 0
\(571\) 7.06642i 0.295720i −0.989008 0.147860i \(-0.952761\pi\)
0.989008 0.147860i \(-0.0472386\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −2.07759 4.32246i −0.0866414 0.180259i
\(576\) 0 0
\(577\) 37.3067 1.55310 0.776550 0.630056i \(-0.216968\pi\)
0.776550 + 0.630056i \(0.216968\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 2.13735i 0.0886724i
\(582\) 0 0
\(583\) 16.0444 0.664489
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 26.1344i 1.07868i −0.842088 0.539340i \(-0.818674\pi\)
0.842088 0.539340i \(-0.181326\pi\)
\(588\) 0 0
\(589\) 18.0826i 0.745082i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 22.2360i 0.913123i −0.889692 0.456561i \(-0.849081\pi\)
0.889692 0.456561i \(-0.150919\pi\)
\(594\) 0 0
\(595\) 5.85488i 0.240027i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 40.5867i 1.65833i 0.559004 + 0.829165i \(0.311183\pi\)
−0.559004 + 0.829165i \(0.688817\pi\)
\(600\) 0 0
\(601\) 3.25425 0.132744 0.0663718 0.997795i \(-0.478858\pi\)
0.0663718 + 0.997795i \(0.478858\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1.53544 0.0624247
\(606\) 0 0
\(607\) 30.7935 1.24987 0.624935 0.780677i \(-0.285126\pi\)
0.624935 + 0.780677i \(0.285126\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 8.13202i 0.328986i
\(612\) 0 0
\(613\) 38.9495i 1.57315i 0.617492 + 0.786577i \(0.288149\pi\)
−0.617492 + 0.786577i \(0.711851\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −11.4150 −0.459551 −0.229775 0.973244i \(-0.573799\pi\)
−0.229775 + 0.973244i \(0.573799\pi\)
\(618\) 0 0
\(619\) 17.9167i 0.720135i 0.932926 + 0.360067i \(0.117246\pi\)
−0.932926 + 0.360067i \(0.882754\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 17.7795i 0.712319i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 33.0817i 1.31905i
\(630\) 0 0
\(631\) 15.9609i 0.635392i −0.948193 0.317696i \(-0.897091\pi\)
0.948193 0.317696i \(-0.102909\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 6.87892 0.272982
\(636\) 0 0
\(637\) 8.05388 0.319106
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −36.5256 −1.44267 −0.721337 0.692585i \(-0.756472\pi\)
−0.721337 + 0.692585i \(0.756472\pi\)
\(642\) 0 0
\(643\) 46.5350i 1.83516i −0.397548 0.917581i \(-0.630139\pi\)
0.397548 0.917581i \(-0.369861\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 9.05104i 0.355833i 0.984046 + 0.177917i \(0.0569357\pi\)
−0.984046 + 0.177917i \(0.943064\pi\)
\(648\) 0 0
\(649\) 30.4960i 1.19707i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 28.8899i 1.13055i 0.824903 + 0.565274i \(0.191230\pi\)
−0.824903 + 0.565274i \(0.808770\pi\)
\(654\) 0 0
\(655\) 8.90544i 0.347964i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −27.9777 −1.08986 −0.544929 0.838482i \(-0.683443\pi\)
−0.544929 + 0.838482i \(0.683443\pi\)
\(660\) 0 0
\(661\) 19.5857i 0.761795i 0.924617 + 0.380898i \(0.124385\pi\)
−0.924617 + 0.380898i \(0.875615\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 14.8488 0.575811
\(666\) 0 0
\(667\) 30.3791 14.6017i 1.17628 0.565380i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 22.5835i 0.871826i
\(672\) 0 0
\(673\) 18.2582 0.703802 0.351901 0.936037i \(-0.385535\pi\)
0.351901 + 0.936037i \(0.385535\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 31.3125 1.20344 0.601718 0.798709i \(-0.294483\pi\)
0.601718 + 0.798709i \(0.294483\pi\)
\(678\) 0 0
\(679\) −26.2178 −1.00615
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 2.45873i 0.0940807i 0.998893 + 0.0470404i \(0.0149789\pi\)
−0.998893 + 0.0470404i \(0.985021\pi\)
\(684\) 0 0
\(685\) 10.0328 0.383333
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −16.0295 −0.610676
\(690\) 0 0
\(691\) 47.9601 1.82449 0.912244 0.409647i \(-0.134348\pi\)
0.912244 + 0.409647i \(0.134348\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −13.7064 −0.519913
\(696\) 0 0
\(697\) 3.77649i 0.143045i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 4.31302 0.162901 0.0814503 0.996677i \(-0.474045\pi\)
0.0814503 + 0.996677i \(0.474045\pi\)
\(702\) 0 0
\(703\) −83.8997 −3.16434
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −34.6019 −1.30134
\(708\) 0 0
\(709\) 30.4757i 1.14454i −0.820065 0.572270i \(-0.806063\pi\)
0.820065 0.572270i \(-0.193937\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −5.29481 11.0159i −0.198292 0.412550i
\(714\) 0 0
\(715\) 9.45580 0.353627
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 42.7482i 1.59424i 0.603821 + 0.797120i \(0.293644\pi\)
−0.603821 + 0.797120i \(0.706356\pi\)
\(720\) 0 0
\(721\) 7.94616 0.295930
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 7.02820i 0.261021i
\(726\) 0 0
\(727\) 31.1931i 1.15689i 0.815722 + 0.578444i \(0.196340\pi\)
−0.815722 + 0.578444i \(0.803660\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 21.5932i 0.798654i
\(732\) 0 0
\(733\) 24.5956i 0.908459i 0.890885 + 0.454229i \(0.150085\pi\)
−0.890885 + 0.454229i \(0.849915\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 5.02603i 0.185136i
\(738\) 0 0
\(739\) 14.8144 0.544955 0.272478 0.962162i \(-0.412157\pi\)
0.272478 + 0.962162i \(0.412157\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −48.7808 −1.78959 −0.894797 0.446472i \(-0.852680\pi\)
−0.894797 + 0.446472i \(0.852680\pi\)
\(744\) 0 0
\(745\) 9.27123 0.339671
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 13.8725i 0.506891i
\(750\) 0 0
\(751\) 24.6795i 0.900568i 0.892885 + 0.450284i \(0.148677\pi\)
−0.892885 + 0.450284i \(0.851323\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −1.24400 −0.0452738
\(756\) 0 0
\(757\) 16.4648i 0.598423i −0.954187 0.299212i \(-0.903276\pi\)
0.954187 0.299212i \(-0.0967237\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 24.2547i 0.879231i 0.898186 + 0.439615i \(0.144885\pi\)
−0.898186 + 0.439615i \(0.855115\pi\)
\(762\) 0 0
\(763\) −22.4243 −0.811814
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 30.4678i 1.10013i
\(768\) 0 0
\(769\) 10.0716i 0.363192i −0.983373 0.181596i \(-0.941874\pi\)
0.983373 0.181596i \(-0.0581263\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 43.6499 1.56998 0.784990 0.619509i \(-0.212668\pi\)
0.784990 + 0.619509i \(0.212668\pi\)
\(774\) 0 0
\(775\) 2.54854 0.0915461
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −9.57771 −0.343157
\(780\) 0 0
\(781\) 40.6952i 1.45619i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 4.74224i 0.169258i
\(786\) 0 0
\(787\) 17.2423i 0.614623i 0.951609 + 0.307312i \(0.0994294\pi\)
−0.951609 + 0.307312i \(0.900571\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 6.96748i 0.247735i
\(792\) 0 0
\(793\) 22.5626i 0.801221i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 47.7562 1.69161 0.845805 0.533492i \(-0.179121\pi\)
0.845805 + 0.533492i \(0.179121\pi\)
\(798\) 0 0
\(799\) 7.40198i 0.261863i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 30.9644 1.09271
\(804\) 0 0
\(805\) −9.04588 + 4.34790i −0.318826 + 0.153243i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 23.9844i 0.843248i −0.906771 0.421624i