Properties

Label 819.2.a.f.1.1
Level $819$
Weight $2$
Character 819.1
Self dual yes
Analytic conductor $6.540$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.53974792554\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 819.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000 q^{2} +2.00000 q^{4} +3.00000 q^{5} -1.00000 q^{7} +O(q^{10})\) \(q+2.00000 q^{2} +2.00000 q^{4} +3.00000 q^{5} -1.00000 q^{7} +6.00000 q^{10} +6.00000 q^{11} -1.00000 q^{13} -2.00000 q^{14} -4.00000 q^{16} -4.00000 q^{17} +5.00000 q^{19} +6.00000 q^{20} +12.0000 q^{22} -3.00000 q^{23} +4.00000 q^{25} -2.00000 q^{26} -2.00000 q^{28} +5.00000 q^{29} -3.00000 q^{31} -8.00000 q^{32} -8.00000 q^{34} -3.00000 q^{35} -4.00000 q^{37} +10.0000 q^{38} +6.00000 q^{41} -1.00000 q^{43} +12.0000 q^{44} -6.00000 q^{46} -7.00000 q^{47} +1.00000 q^{49} +8.00000 q^{50} -2.00000 q^{52} +9.00000 q^{53} +18.0000 q^{55} +10.0000 q^{58} -8.00000 q^{59} -10.0000 q^{61} -6.00000 q^{62} -8.00000 q^{64} -3.00000 q^{65} -6.00000 q^{67} -8.00000 q^{68} -6.00000 q^{70} +8.00000 q^{71} -13.0000 q^{73} -8.00000 q^{74} +10.0000 q^{76} -6.00000 q^{77} +3.00000 q^{79} -12.0000 q^{80} +12.0000 q^{82} -15.0000 q^{83} -12.0000 q^{85} -2.00000 q^{86} -3.00000 q^{89} +1.00000 q^{91} -6.00000 q^{92} -14.0000 q^{94} +15.0000 q^{95} +7.00000 q^{97} +2.00000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(3\) 0 0
\(4\) 2.00000 1.00000
\(5\) 3.00000 1.34164 0.670820 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 6.00000 1.89737
\(11\) 6.00000 1.80907 0.904534 0.426401i \(-0.140219\pi\)
0.904534 + 0.426401i \(0.140219\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 0 0
\(19\) 5.00000 1.14708 0.573539 0.819178i \(-0.305570\pi\)
0.573539 + 0.819178i \(0.305570\pi\)
\(20\) 6.00000 1.34164
\(21\) 0 0
\(22\) 12.0000 2.55841
\(23\) −3.00000 −0.625543 −0.312772 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) −2.00000 −0.377964
\(29\) 5.00000 0.928477 0.464238 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 0 0
\(31\) −3.00000 −0.538816 −0.269408 0.963026i \(-0.586828\pi\)
−0.269408 + 0.963026i \(0.586828\pi\)
\(32\) −8.00000 −1.41421
\(33\) 0 0
\(34\) −8.00000 −1.37199
\(35\) −3.00000 −0.507093
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 10.0000 1.62221
\(39\) 0 0
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499 −0.0762493 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) 12.0000 1.80907
\(45\) 0 0
\(46\) −6.00000 −0.884652
\(47\) −7.00000 −1.02105 −0.510527 0.859861i \(-0.670550\pi\)
−0.510527 + 0.859861i \(0.670550\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 8.00000 1.13137
\(51\) 0 0
\(52\) −2.00000 −0.277350
\(53\) 9.00000 1.23625 0.618123 0.786082i \(-0.287894\pi\)
0.618123 + 0.786082i \(0.287894\pi\)
\(54\) 0 0
\(55\) 18.0000 2.42712
\(56\) 0 0
\(57\) 0 0
\(58\) 10.0000 1.31306
\(59\) −8.00000 −1.04151 −0.520756 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) −6.00000 −0.762001
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) −3.00000 −0.372104
\(66\) 0 0
\(67\) −6.00000 −0.733017 −0.366508 0.930415i \(-0.619447\pi\)
−0.366508 + 0.930415i \(0.619447\pi\)
\(68\) −8.00000 −0.970143
\(69\) 0 0
\(70\) −6.00000 −0.717137
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) −13.0000 −1.52153 −0.760767 0.649025i \(-0.775177\pi\)
−0.760767 + 0.649025i \(0.775177\pi\)
\(74\) −8.00000 −0.929981
\(75\) 0 0
\(76\) 10.0000 1.14708
\(77\) −6.00000 −0.683763
\(78\) 0 0
\(79\) 3.00000 0.337526 0.168763 0.985657i \(-0.446023\pi\)
0.168763 + 0.985657i \(0.446023\pi\)
\(80\) −12.0000 −1.34164
\(81\) 0 0
\(82\) 12.0000 1.32518
\(83\) −15.0000 −1.64646 −0.823232 0.567705i \(-0.807831\pi\)
−0.823232 + 0.567705i \(0.807831\pi\)
\(84\) 0 0
\(85\) −12.0000 −1.30158
\(86\) −2.00000 −0.215666
\(87\) 0 0
\(88\) 0 0
\(89\) −3.00000 −0.317999 −0.159000 0.987279i \(-0.550827\pi\)
−0.159000 + 0.987279i \(0.550827\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) −6.00000 −0.625543
\(93\) 0 0
\(94\) −14.0000 −1.44399
\(95\) 15.0000 1.53897
\(96\) 0 0
\(97\) 7.00000 0.710742 0.355371 0.934725i \(-0.384354\pi\)
0.355371 + 0.934725i \(0.384354\pi\)
\(98\) 2.00000 0.202031
\(99\) 0 0
\(100\) 8.00000 0.800000
\(101\) 14.0000 1.39305 0.696526 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 18.0000 1.74831
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 36.0000 3.43247
\(111\) 0 0
\(112\) 4.00000 0.377964
\(113\) 3.00000 0.282216 0.141108 0.989994i \(-0.454933\pi\)
0.141108 + 0.989994i \(0.454933\pi\)
\(114\) 0 0
\(115\) −9.00000 −0.839254
\(116\) 10.0000 0.928477
\(117\) 0 0
\(118\) −16.0000 −1.47292
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) −20.0000 −1.81071
\(123\) 0 0
\(124\) −6.00000 −0.538816
\(125\) −3.00000 −0.268328
\(126\) 0 0
\(127\) −4.00000 −0.354943 −0.177471 0.984126i \(-0.556792\pi\)
−0.177471 + 0.984126i \(0.556792\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) −6.00000 −0.526235
\(131\) −8.00000 −0.698963 −0.349482 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) 0 0
\(133\) −5.00000 −0.433555
\(134\) −12.0000 −1.03664
\(135\) 0 0
\(136\) 0 0
\(137\) −4.00000 −0.341743 −0.170872 0.985293i \(-0.554658\pi\)
−0.170872 + 0.985293i \(0.554658\pi\)
\(138\) 0 0
\(139\) −18.0000 −1.52674 −0.763370 0.645961i \(-0.776457\pi\)
−0.763370 + 0.645961i \(0.776457\pi\)
\(140\) −6.00000 −0.507093
\(141\) 0 0
\(142\) 16.0000 1.34269
\(143\) −6.00000 −0.501745
\(144\) 0 0
\(145\) 15.0000 1.24568
\(146\) −26.0000 −2.15178
\(147\) 0 0
\(148\) −8.00000 −0.657596
\(149\) 18.0000 1.47462 0.737309 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −12.0000 −0.966988
\(155\) −9.00000 −0.722897
\(156\) 0 0
\(157\) −8.00000 −0.638470 −0.319235 0.947676i \(-0.603426\pi\)
−0.319235 + 0.947676i \(0.603426\pi\)
\(158\) 6.00000 0.477334
\(159\) 0 0
\(160\) −24.0000 −1.89737
\(161\) 3.00000 0.236433
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 12.0000 0.937043
\(165\) 0 0
\(166\) −30.0000 −2.32845
\(167\) −5.00000 −0.386912 −0.193456 0.981109i \(-0.561970\pi\)
−0.193456 + 0.981109i \(0.561970\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) −24.0000 −1.84072
\(171\) 0 0
\(172\) −2.00000 −0.152499
\(173\) 8.00000 0.608229 0.304114 0.952636i \(-0.401639\pi\)
0.304114 + 0.952636i \(0.401639\pi\)
\(174\) 0 0
\(175\) −4.00000 −0.302372
\(176\) −24.0000 −1.80907
\(177\) 0 0
\(178\) −6.00000 −0.449719
\(179\) −23.0000 −1.71910 −0.859550 0.511051i \(-0.829256\pi\)
−0.859550 + 0.511051i \(0.829256\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 2.00000 0.148250
\(183\) 0 0
\(184\) 0 0
\(185\) −12.0000 −0.882258
\(186\) 0 0
\(187\) −24.0000 −1.75505
\(188\) −14.0000 −1.02105
\(189\) 0 0
\(190\) 30.0000 2.17643
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 0 0
\(193\) 22.0000 1.58359 0.791797 0.610784i \(-0.209146\pi\)
0.791797 + 0.610784i \(0.209146\pi\)
\(194\) 14.0000 1.00514
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 0 0
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 28.0000 1.97007
\(203\) −5.00000 −0.350931
\(204\) 0 0
\(205\) 18.0000 1.25717
\(206\) −8.00000 −0.557386
\(207\) 0 0
\(208\) 4.00000 0.277350
\(209\) 30.0000 2.07514
\(210\) 0 0
\(211\) −5.00000 −0.344214 −0.172107 0.985078i \(-0.555058\pi\)
−0.172107 + 0.985078i \(0.555058\pi\)
\(212\) 18.0000 1.23625
\(213\) 0 0
\(214\) 8.00000 0.546869
\(215\) −3.00000 −0.204598
\(216\) 0 0
\(217\) 3.00000 0.203653
\(218\) −4.00000 −0.270914
\(219\) 0 0
\(220\) 36.0000 2.42712
\(221\) 4.00000 0.269069
\(222\) 0 0
\(223\) 15.0000 1.00447 0.502237 0.864730i \(-0.332510\pi\)
0.502237 + 0.864730i \(0.332510\pi\)
\(224\) 8.00000 0.534522
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) −20.0000 −1.32745 −0.663723 0.747978i \(-0.731025\pi\)
−0.663723 + 0.747978i \(0.731025\pi\)
\(228\) 0 0
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) −18.0000 −1.18688
\(231\) 0 0
\(232\) 0 0
\(233\) −15.0000 −0.982683 −0.491341 0.870967i \(-0.663493\pi\)
−0.491341 + 0.870967i \(0.663493\pi\)
\(234\) 0 0
\(235\) −21.0000 −1.36989
\(236\) −16.0000 −1.04151
\(237\) 0 0
\(238\) 8.00000 0.518563
\(239\) 4.00000 0.258738 0.129369 0.991596i \(-0.458705\pi\)
0.129369 + 0.991596i \(0.458705\pi\)
\(240\) 0 0
\(241\) −17.0000 −1.09507 −0.547533 0.836784i \(-0.684433\pi\)
−0.547533 + 0.836784i \(0.684433\pi\)
\(242\) 50.0000 3.21412
\(243\) 0 0
\(244\) −20.0000 −1.28037
\(245\) 3.00000 0.191663
\(246\) 0 0
\(247\) −5.00000 −0.318142
\(248\) 0 0
\(249\) 0 0
\(250\) −6.00000 −0.379473
\(251\) 26.0000 1.64111 0.820553 0.571571i \(-0.193666\pi\)
0.820553 + 0.571571i \(0.193666\pi\)
\(252\) 0 0
\(253\) −18.0000 −1.13165
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 2.00000 0.124757 0.0623783 0.998053i \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) 0 0
\(259\) 4.00000 0.248548
\(260\) −6.00000 −0.372104
\(261\) 0 0
\(262\) −16.0000 −0.988483
\(263\) 15.0000 0.924940 0.462470 0.886635i \(-0.346963\pi\)
0.462470 + 0.886635i \(0.346963\pi\)
\(264\) 0 0
\(265\) 27.0000 1.65860
\(266\) −10.0000 −0.613139
\(267\) 0 0
\(268\) −12.0000 −0.733017
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 16.0000 0.970143
\(273\) 0 0
\(274\) −8.00000 −0.483298
\(275\) 24.0000 1.44725
\(276\) 0 0
\(277\) 1.00000 0.0600842 0.0300421 0.999549i \(-0.490436\pi\)
0.0300421 + 0.999549i \(0.490436\pi\)
\(278\) −36.0000 −2.15914
\(279\) 0 0
\(280\) 0 0
\(281\) 30.0000 1.78965 0.894825 0.446417i \(-0.147300\pi\)
0.894825 + 0.446417i \(0.147300\pi\)
\(282\) 0 0
\(283\) 16.0000 0.951101 0.475551 0.879688i \(-0.342249\pi\)
0.475551 + 0.879688i \(0.342249\pi\)
\(284\) 16.0000 0.949425
\(285\) 0 0
\(286\) −12.0000 −0.709575
\(287\) −6.00000 −0.354169
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 30.0000 1.76166
\(291\) 0 0
\(292\) −26.0000 −1.52153
\(293\) 19.0000 1.10999 0.554996 0.831853i \(-0.312720\pi\)
0.554996 + 0.831853i \(0.312720\pi\)
\(294\) 0 0
\(295\) −24.0000 −1.39733
\(296\) 0 0
\(297\) 0 0
\(298\) 36.0000 2.08542
\(299\) 3.00000 0.173494
\(300\) 0 0
\(301\) 1.00000 0.0576390
\(302\) 0 0
\(303\) 0 0
\(304\) −20.0000 −1.14708
\(305\) −30.0000 −1.71780
\(306\) 0 0
\(307\) −33.0000 −1.88341 −0.941705 0.336440i \(-0.890777\pi\)
−0.941705 + 0.336440i \(0.890777\pi\)
\(308\) −12.0000 −0.683763
\(309\) 0 0
\(310\) −18.0000 −1.02233
\(311\) 6.00000 0.340229 0.170114 0.985424i \(-0.445586\pi\)
0.170114 + 0.985424i \(0.445586\pi\)
\(312\) 0 0
\(313\) 22.0000 1.24351 0.621757 0.783210i \(-0.286419\pi\)
0.621757 + 0.783210i \(0.286419\pi\)
\(314\) −16.0000 −0.902932
\(315\) 0 0
\(316\) 6.00000 0.337526
\(317\) 24.0000 1.34797 0.673987 0.738743i \(-0.264580\pi\)
0.673987 + 0.738743i \(0.264580\pi\)
\(318\) 0 0
\(319\) 30.0000 1.67968
\(320\) −24.0000 −1.34164
\(321\) 0 0
\(322\) 6.00000 0.334367
\(323\) −20.0000 −1.11283
\(324\) 0 0
\(325\) −4.00000 −0.221880
\(326\) −8.00000 −0.443079
\(327\) 0 0
\(328\) 0 0
\(329\) 7.00000 0.385922
\(330\) 0 0
\(331\) 22.0000 1.20923 0.604615 0.796518i \(-0.293327\pi\)
0.604615 + 0.796518i \(0.293327\pi\)
\(332\) −30.0000 −1.64646
\(333\) 0 0
\(334\) −10.0000 −0.547176
\(335\) −18.0000 −0.983445
\(336\) 0 0
\(337\) 17.0000 0.926049 0.463025 0.886345i \(-0.346764\pi\)
0.463025 + 0.886345i \(0.346764\pi\)
\(338\) 2.00000 0.108786
\(339\) 0 0
\(340\) −24.0000 −1.30158
\(341\) −18.0000 −0.974755
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 16.0000 0.860165
\(347\) 32.0000 1.71785 0.858925 0.512101i \(-0.171133\pi\)
0.858925 + 0.512101i \(0.171133\pi\)
\(348\) 0 0
\(349\) 11.0000 0.588817 0.294408 0.955680i \(-0.404877\pi\)
0.294408 + 0.955680i \(0.404877\pi\)
\(350\) −8.00000 −0.427618
\(351\) 0 0
\(352\) −48.0000 −2.55841
\(353\) 10.0000 0.532246 0.266123 0.963939i \(-0.414257\pi\)
0.266123 + 0.963939i \(0.414257\pi\)
\(354\) 0 0
\(355\) 24.0000 1.27379
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) −46.0000 −2.43118
\(359\) −20.0000 −1.05556 −0.527780 0.849381i \(-0.676975\pi\)
−0.527780 + 0.849381i \(0.676975\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) 28.0000 1.47165
\(363\) 0 0
\(364\) 2.00000 0.104828
\(365\) −39.0000 −2.04135
\(366\) 0 0
\(367\) 14.0000 0.730794 0.365397 0.930852i \(-0.380933\pi\)
0.365397 + 0.930852i \(0.380933\pi\)
\(368\) 12.0000 0.625543
\(369\) 0 0
\(370\) −24.0000 −1.24770
\(371\) −9.00000 −0.467257
\(372\) 0 0
\(373\) 30.0000 1.55334 0.776671 0.629907i \(-0.216907\pi\)
0.776671 + 0.629907i \(0.216907\pi\)
\(374\) −48.0000 −2.48202
\(375\) 0 0
\(376\) 0 0
\(377\) −5.00000 −0.257513
\(378\) 0 0
\(379\) −6.00000 −0.308199 −0.154100 0.988055i \(-0.549248\pi\)
−0.154100 + 0.988055i \(0.549248\pi\)
\(380\) 30.0000 1.53897
\(381\) 0 0
\(382\) 16.0000 0.818631
\(383\) 36.0000 1.83951 0.919757 0.392488i \(-0.128386\pi\)
0.919757 + 0.392488i \(0.128386\pi\)
\(384\) 0 0
\(385\) −18.0000 −0.917365
\(386\) 44.0000 2.23954
\(387\) 0 0
\(388\) 14.0000 0.710742
\(389\) −30.0000 −1.52106 −0.760530 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 0 0
\(391\) 12.0000 0.606866
\(392\) 0 0
\(393\) 0 0
\(394\) −4.00000 −0.201517
\(395\) 9.00000 0.452839
\(396\) 0 0
\(397\) −13.0000 −0.652451 −0.326226 0.945292i \(-0.605777\pi\)
−0.326226 + 0.945292i \(0.605777\pi\)
\(398\) 8.00000 0.401004
\(399\) 0 0
\(400\) −16.0000 −0.800000
\(401\) 32.0000 1.59800 0.799002 0.601329i \(-0.205362\pi\)
0.799002 + 0.601329i \(0.205362\pi\)
\(402\) 0 0
\(403\) 3.00000 0.149441
\(404\) 28.0000 1.39305
\(405\) 0 0
\(406\) −10.0000 −0.496292
\(407\) −24.0000 −1.18964
\(408\) 0 0
\(409\) −13.0000 −0.642809 −0.321404 0.946942i \(-0.604155\pi\)
−0.321404 + 0.946942i \(0.604155\pi\)
\(410\) 36.0000 1.77791
\(411\) 0 0
\(412\) −8.00000 −0.394132
\(413\) 8.00000 0.393654
\(414\) 0 0
\(415\) −45.0000 −2.20896
\(416\) 8.00000 0.392232
\(417\) 0 0
\(418\) 60.0000 2.93470
\(419\) 10.0000 0.488532 0.244266 0.969708i \(-0.421453\pi\)
0.244266 + 0.969708i \(0.421453\pi\)
\(420\) 0 0
\(421\) −12.0000 −0.584844 −0.292422 0.956289i \(-0.594461\pi\)
−0.292422 + 0.956289i \(0.594461\pi\)
\(422\) −10.0000 −0.486792
\(423\) 0 0
\(424\) 0 0
\(425\) −16.0000 −0.776114
\(426\) 0 0
\(427\) 10.0000 0.483934
\(428\) 8.00000 0.386695
\(429\) 0 0
\(430\) −6.00000 −0.289346
\(431\) −6.00000 −0.289010 −0.144505 0.989504i \(-0.546159\pi\)
−0.144505 + 0.989504i \(0.546159\pi\)
\(432\) 0 0
\(433\) 12.0000 0.576683 0.288342 0.957528i \(-0.406896\pi\)
0.288342 + 0.957528i \(0.406896\pi\)
\(434\) 6.00000 0.288009
\(435\) 0 0
\(436\) −4.00000 −0.191565
\(437\) −15.0000 −0.717547
\(438\) 0 0
\(439\) −22.0000 −1.05000 −0.525001 0.851101i \(-0.675935\pi\)
−0.525001 + 0.851101i \(0.675935\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 8.00000 0.380521
\(443\) −19.0000 −0.902717 −0.451359 0.892343i \(-0.649060\pi\)
−0.451359 + 0.892343i \(0.649060\pi\)
\(444\) 0 0
\(445\) −9.00000 −0.426641
\(446\) 30.0000 1.42054
\(447\) 0 0
\(448\) 8.00000 0.377964
\(449\) −36.0000 −1.69895 −0.849473 0.527633i \(-0.823080\pi\)
−0.849473 + 0.527633i \(0.823080\pi\)
\(450\) 0 0
\(451\) 36.0000 1.69517
\(452\) 6.00000 0.282216
\(453\) 0 0
\(454\) −40.0000 −1.87729
\(455\) 3.00000 0.140642
\(456\) 0 0
\(457\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(458\) 28.0000 1.30835
\(459\) 0 0
\(460\) −18.0000 −0.839254
\(461\) 22.0000 1.02464 0.512321 0.858794i \(-0.328786\pi\)
0.512321 + 0.858794i \(0.328786\pi\)
\(462\) 0 0
\(463\) −14.0000 −0.650635 −0.325318 0.945605i \(-0.605471\pi\)
−0.325318 + 0.945605i \(0.605471\pi\)
\(464\) −20.0000 −0.928477
\(465\) 0 0
\(466\) −30.0000 −1.38972
\(467\) 22.0000 1.01804 0.509019 0.860755i \(-0.330008\pi\)
0.509019 + 0.860755i \(0.330008\pi\)
\(468\) 0 0
\(469\) 6.00000 0.277054
\(470\) −42.0000 −1.93732
\(471\) 0 0
\(472\) 0 0
\(473\) −6.00000 −0.275880
\(474\) 0 0
\(475\) 20.0000 0.917663
\(476\) 8.00000 0.366679
\(477\) 0 0
\(478\) 8.00000 0.365911
\(479\) 11.0000 0.502603 0.251301 0.967909i \(-0.419141\pi\)
0.251301 + 0.967909i \(0.419141\pi\)
\(480\) 0 0
\(481\) 4.00000 0.182384
\(482\) −34.0000 −1.54866
\(483\) 0 0
\(484\) 50.0000 2.27273
\(485\) 21.0000 0.953561
\(486\) 0 0
\(487\) −26.0000 −1.17817 −0.589086 0.808070i \(-0.700512\pi\)
−0.589086 + 0.808070i \(0.700512\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 6.00000 0.271052
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) −20.0000 −0.900755
\(494\) −10.0000 −0.449921
\(495\) 0 0
\(496\) 12.0000 0.538816
\(497\) −8.00000 −0.358849
\(498\) 0 0
\(499\) −16.0000 −0.716258 −0.358129 0.933672i \(-0.616585\pi\)
−0.358129 + 0.933672i \(0.616585\pi\)
\(500\) −6.00000 −0.268328
\(501\) 0 0
\(502\) 52.0000 2.32087
\(503\) −2.00000 −0.0891756 −0.0445878 0.999005i \(-0.514197\pi\)
−0.0445878 + 0.999005i \(0.514197\pi\)
\(504\) 0 0
\(505\) 42.0000 1.86898
\(506\) −36.0000 −1.60040
\(507\) 0 0
\(508\) −8.00000 −0.354943
\(509\) 19.0000 0.842160 0.421080 0.907023i \(-0.361651\pi\)
0.421080 + 0.907023i \(0.361651\pi\)
\(510\) 0 0
\(511\) 13.0000 0.575086
\(512\) 32.0000 1.41421
\(513\) 0 0
\(514\) 4.00000 0.176432
\(515\) −12.0000 −0.528783
\(516\) 0 0
\(517\) −42.0000 −1.84716
\(518\) 8.00000 0.351500
\(519\) 0 0
\(520\) 0 0
\(521\) −40.0000 −1.75243 −0.876216 0.481919i \(-0.839940\pi\)
−0.876216 + 0.481919i \(0.839940\pi\)
\(522\) 0 0
\(523\) 10.0000 0.437269 0.218635 0.975807i \(-0.429840\pi\)
0.218635 + 0.975807i \(0.429840\pi\)
\(524\) −16.0000 −0.698963
\(525\) 0 0
\(526\) 30.0000 1.30806
\(527\) 12.0000 0.522728
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) 54.0000 2.34561
\(531\) 0 0
\(532\) −10.0000 −0.433555
\(533\) −6.00000 −0.259889
\(534\) 0 0
\(535\) 12.0000 0.518805
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 6.00000 0.258438
\(540\) 0 0
\(541\) −40.0000 −1.71973 −0.859867 0.510518i \(-0.829454\pi\)
−0.859867 + 0.510518i \(0.829454\pi\)
\(542\) 16.0000 0.687259
\(543\) 0 0
\(544\) 32.0000 1.37199
\(545\) −6.00000 −0.257012
\(546\) 0 0
\(547\) −7.00000 −0.299298 −0.149649 0.988739i \(-0.547814\pi\)
−0.149649 + 0.988739i \(0.547814\pi\)
\(548\) −8.00000 −0.341743
\(549\) 0 0
\(550\) 48.0000 2.04673
\(551\) 25.0000 1.06504
\(552\) 0 0
\(553\) −3.00000 −0.127573
\(554\) 2.00000 0.0849719
\(555\) 0 0
\(556\) −36.0000 −1.52674
\(557\) 12.0000 0.508456 0.254228 0.967144i \(-0.418179\pi\)
0.254228 + 0.967144i \(0.418179\pi\)
\(558\) 0 0
\(559\) 1.00000 0.0422955
\(560\) 12.0000 0.507093
\(561\) 0 0
\(562\) 60.0000 2.53095
\(563\) −4.00000 −0.168580 −0.0842900 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) 0 0
\(565\) 9.00000 0.378633
\(566\) 32.0000 1.34506
\(567\) 0 0
\(568\) 0 0
\(569\) −7.00000 −0.293455 −0.146728 0.989177i \(-0.546874\pi\)
−0.146728 + 0.989177i \(0.546874\pi\)
\(570\) 0 0
\(571\) −17.0000 −0.711428 −0.355714 0.934595i \(-0.615762\pi\)
−0.355714 + 0.934595i \(0.615762\pi\)
\(572\) −12.0000 −0.501745
\(573\) 0 0
\(574\) −12.0000 −0.500870
\(575\) −12.0000 −0.500435
\(576\) 0 0
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) −2.00000 −0.0831890
\(579\) 0 0
\(580\) 30.0000 1.24568
\(581\) 15.0000 0.622305
\(582\) 0 0
\(583\) 54.0000 2.23645
\(584\) 0 0
\(585\) 0 0
\(586\) 38.0000 1.56977
\(587\) −39.0000 −1.60970 −0.804851 0.593477i \(-0.797755\pi\)
−0.804851 + 0.593477i \(0.797755\pi\)
\(588\) 0 0
\(589\) −15.0000 −0.618064
\(590\) −48.0000 −1.97613
\(591\) 0 0
\(592\) 16.0000 0.657596
\(593\) 27.0000 1.10876 0.554379 0.832265i \(-0.312956\pi\)
0.554379 + 0.832265i \(0.312956\pi\)
\(594\) 0 0
\(595\) 12.0000 0.491952
\(596\) 36.0000 1.47462
\(597\) 0 0
\(598\) 6.00000 0.245358
\(599\) −11.0000 −0.449448 −0.224724 0.974422i \(-0.572148\pi\)
−0.224724 + 0.974422i \(0.572148\pi\)
\(600\) 0 0
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 2.00000 0.0815139
\(603\) 0 0
\(604\) 0 0
\(605\) 75.0000 3.04918
\(606\) 0 0
\(607\) 2.00000 0.0811775 0.0405887 0.999176i \(-0.487077\pi\)
0.0405887 + 0.999176i \(0.487077\pi\)
\(608\) −40.0000 −1.62221
\(609\) 0 0
\(610\) −60.0000 −2.42933
\(611\) 7.00000 0.283190
\(612\) 0 0
\(613\) 8.00000 0.323117 0.161558 0.986863i \(-0.448348\pi\)
0.161558 + 0.986863i \(0.448348\pi\)
\(614\) −66.0000 −2.66354
\(615\) 0 0
\(616\) 0 0
\(617\) 30.0000 1.20775 0.603877 0.797077i \(-0.293622\pi\)
0.603877 + 0.797077i \(0.293622\pi\)
\(618\) 0 0
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) −18.0000 −0.722897
\(621\) 0 0
\(622\) 12.0000 0.481156
\(623\) 3.00000 0.120192
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 44.0000 1.75859
\(627\) 0 0
\(628\) −16.0000 −0.638470
\(629\) 16.0000 0.637962
\(630\) 0 0
\(631\) 22.0000 0.875806 0.437903 0.899022i \(-0.355721\pi\)
0.437903 + 0.899022i \(0.355721\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 48.0000 1.90632
\(635\) −12.0000 −0.476205
\(636\) 0 0
\(637\) −1.00000 −0.0396214
\(638\) 60.0000 2.37542
\(639\) 0 0
\(640\) 0 0
\(641\) −9.00000 −0.355479 −0.177739 0.984078i \(-0.556878\pi\)
−0.177739 + 0.984078i \(0.556878\pi\)
\(642\) 0 0
\(643\) 8.00000 0.315489 0.157745 0.987480i \(-0.449578\pi\)
0.157745 + 0.987480i \(0.449578\pi\)
\(644\) 6.00000 0.236433
\(645\) 0 0
\(646\) −40.0000 −1.57378
\(647\) −18.0000 −0.707653 −0.353827 0.935311i \(-0.615120\pi\)
−0.353827 + 0.935311i \(0.615120\pi\)
\(648\) 0 0
\(649\) −48.0000 −1.88416
\(650\) −8.00000 −0.313786
\(651\) 0 0
\(652\) −8.00000 −0.313304
\(653\) −18.0000 −0.704394 −0.352197 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) 0 0
\(655\) −24.0000 −0.937758
\(656\) −24.0000 −0.937043
\(657\) 0 0
\(658\) 14.0000 0.545777
\(659\) −17.0000 −0.662226 −0.331113 0.943591i \(-0.607424\pi\)
−0.331113 + 0.943591i \(0.607424\pi\)
\(660\) 0 0
\(661\) −33.0000 −1.28355 −0.641776 0.766892i \(-0.721802\pi\)
−0.641776 + 0.766892i \(0.721802\pi\)
\(662\) 44.0000 1.71011
\(663\) 0 0
\(664\) 0 0
\(665\) −15.0000 −0.581675
\(666\) 0 0
\(667\) −15.0000 −0.580802
\(668\) −10.0000 −0.386912
\(669\) 0 0
\(670\) −36.0000 −1.39080
\(671\) −60.0000 −2.31627
\(672\) 0 0
\(673\) 1.00000 0.0385472 0.0192736 0.999814i \(-0.493865\pi\)
0.0192736 + 0.999814i \(0.493865\pi\)
\(674\) 34.0000 1.30963
\(675\) 0 0
\(676\) 2.00000 0.0769231
\(677\) −22.0000 −0.845529 −0.422764 0.906240i \(-0.638940\pi\)
−0.422764 + 0.906240i \(0.638940\pi\)
\(678\) 0 0
\(679\) −7.00000 −0.268635
\(680\) 0 0
\(681\) 0 0
\(682\) −36.0000 −1.37851
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 0 0
\(685\) −12.0000 −0.458496
\(686\) −2.00000 −0.0763604
\(687\) 0 0
\(688\) 4.00000 0.152499
\(689\) −9.00000 −0.342873
\(690\) 0 0
\(691\) 11.0000 0.418460 0.209230 0.977866i \(-0.432904\pi\)
0.209230 + 0.977866i \(0.432904\pi\)
\(692\) 16.0000 0.608229
\(693\) 0 0
\(694\) 64.0000 2.42941
\(695\) −54.0000 −2.04834
\(696\) 0 0
\(697\) −24.0000 −0.909065
\(698\) 22.0000 0.832712
\(699\) 0 0
\(700\) −8.00000 −0.302372
\(701\) 27.0000 1.01978 0.509888 0.860241i \(-0.329687\pi\)
0.509888 + 0.860241i \(0.329687\pi\)
\(702\) 0 0
\(703\) −20.0000 −0.754314
\(704\) −48.0000 −1.80907
\(705\) 0 0
\(706\) 20.0000 0.752710
\(707\) −14.0000 −0.526524
\(708\) 0 0
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 48.0000 1.80141
\(711\) 0 0
\(712\) 0 0
\(713\) 9.00000 0.337053
\(714\) 0 0
\(715\) −18.0000 −0.673162
\(716\) −46.0000 −1.71910
\(717\) 0 0
\(718\) −40.0000 −1.49279
\(719\) −18.0000 −0.671287 −0.335643 0.941989i \(-0.608954\pi\)
−0.335643 + 0.941989i \(0.608954\pi\)
\(720\) 0 0
\(721\) 4.00000 0.148968
\(722\) 12.0000 0.446594
\(723\) 0 0
\(724\) 28.0000 1.04061
\(725\) 20.0000 0.742781
\(726\) 0 0
\(727\) 46.0000 1.70605 0.853023 0.521874i \(-0.174767\pi\)
0.853023 + 0.521874i \(0.174767\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −78.0000 −2.88691
\(731\) 4.00000 0.147945
\(732\) 0 0
\(733\) 51.0000 1.88373 0.941864 0.335994i \(-0.109072\pi\)
0.941864 + 0.335994i \(0.109072\pi\)
\(734\) 28.0000 1.03350
\(735\) 0 0
\(736\) 24.0000 0.884652
\(737\) −36.0000 −1.32608
\(738\) 0 0
\(739\) −26.0000 −0.956425 −0.478213 0.878244i \(-0.658715\pi\)
−0.478213 + 0.878244i \(0.658715\pi\)
\(740\) −24.0000 −0.882258
\(741\) 0 0
\(742\) −18.0000 −0.660801
\(743\) −36.0000 −1.32071 −0.660356 0.750953i \(-0.729595\pi\)
−0.660356 + 0.750953i \(0.729595\pi\)
\(744\) 0 0
\(745\) 54.0000 1.97841
\(746\) 60.0000 2.19676
\(747\) 0 0
\(748\) −48.0000 −1.75505
\(749\) −4.00000 −0.146157
\(750\) 0 0
\(751\) −17.0000 −0.620339 −0.310169 0.950681i \(-0.600386\pi\)
−0.310169 + 0.950681i \(0.600386\pi\)
\(752\) 28.0000 1.02105
\(753\) 0 0
\(754\) −10.0000 −0.364179
\(755\) 0 0
\(756\) 0 0
\(757\) −15.0000 −0.545184 −0.272592 0.962130i \(-0.587881\pi\)
−0.272592 + 0.962130i \(0.587881\pi\)
\(758\) −12.0000 −0.435860
\(759\) 0 0
\(760\) 0 0
\(761\) −9.00000 −0.326250 −0.163125 0.986605i \(-0.552157\pi\)
−0.163125 + 0.986605i \(0.552157\pi\)
\(762\) 0 0
\(763\) 2.00000 0.0724049
\(764\) 16.0000 0.578860
\(765\) 0 0
\(766\) 72.0000 2.60147
\(767\) 8.00000 0.288863
\(768\) 0 0
\(769\) −35.0000 −1.26213 −0.631066 0.775729i \(-0.717382\pi\)
−0.631066 + 0.775729i \(0.717382\pi\)
\(770\) −36.0000 −1.29735
\(771\) 0 0
\(772\) 44.0000 1.58359
\(773\) −54.0000 −1.94225 −0.971123 0.238581i \(-0.923318\pi\)
−0.971123 + 0.238581i \(0.923318\pi\)
\(774\) 0 0
\(775\) −12.0000 −0.431053
\(776\) 0 0
\(777\) 0 0
\(778\) −60.0000 −2.15110
\(779\) 30.0000 1.07486
\(780\) 0 0
\(781\) 48.0000 1.71758
\(782\) 24.0000 0.858238
\(783\) 0 0
\(784\) −4.00000 −0.142857
\(785\) −24.0000 −0.856597
\(786\) 0 0
\(787\) 37.0000 1.31891 0.659454 0.751745i \(-0.270788\pi\)
0.659454 + 0.751745i \(0.270788\pi\)
\(788\) −4.00000 −0.142494
\(789\) 0 0
\(790\) 18.0000 0.640411
\(791\) −3.00000 −0.106668
\(792\) 0 0
\(793\) 10.0000 0.355110
\(794\) −26.0000 −0.922705
\(795\) 0 0
\(796\) 8.00000 0.283552
\(797\) −18.0000 −0.637593 −0.318796 0.947823i \(-0.603279\pi\)
−0.318796 + 0.947823i \(0.603279\pi\)
\(798\) 0 0
\(799\) 28.0000 0.990569
\(800\) −32.0000 −1.13137
\(801\) 0 0
\(802\) 64.0000 2.25992
\(803\) −78.0000 −2.75256
\(804\) 0 0
\(805\) 9.00000 0.317208
\(806\) 6.00000 0.211341
\(807\) 0 0
\(808\) 0 0
\(809\) 31.0000 1.08990 0.544951 0.838468i \(-0.316548\pi\)
0.544951 + 0.838468i \(0.316548\pi\)
\(810\) 0 0
\(811\) −52.0000 −1.82597 −0.912983 0.407997i \(-0.866228\pi\)
−0.912983 + 0.407997i \(0.866228\pi\)
\(812\) −10.0000 −0.350931
\(813\) 0 0
\(814\) −48.0000 −1.68240
\(815\) −12.0000 −0.420342
\(816\) 0 0
\(817\) −5.00000 −0.174928
\(818\) −26.0000 −0.909069
\(819\) 0 0
\(820\) 36.0000 1.25717
\(821\) 6.00000 0.209401 0.104701 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) 0 0
\(823\) −32.0000 −1.11545 −0.557725 0.830026i \(-0.688326\pi\)
−0.557725 + 0.830026i \(0.688326\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 16.0000 0.556711
\(827\) −4.00000 −0.139094 −0.0695468 0.997579i \(-0.522155\pi\)
−0.0695468 + 0.997579i \(0.522155\pi\)
\(828\) 0 0
\(829\) −10.0000 −0.347314 −0.173657 0.984806i \(-0.555558\pi\)
−0.173657 + 0.984806i \(0.555558\pi\)
\(830\) −90.0000 −3.12395
\(831\) 0 0
\(832\) 8.00000 0.277350
\(833\) −4.00000 −0.138592
\(834\) 0 0
\(835\) −15.0000 −0.519096
\(836\) 60.0000 2.07514
\(837\) 0 0
\(838\) 20.0000 0.690889
\(839\) 8.00000 0.276191 0.138095 0.990419i \(-0.455902\pi\)
0.138095 + 0.990419i \(0.455902\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) −24.0000 −0.827095
\(843\) 0 0
\(844\) −10.0000 −0.344214
\(845\) 3.00000 0.103203
\(846\) 0 0
\(847\) −25.0000 −0.859010
\(848\) −36.0000 −1.23625
\(849\) 0 0
\(850\) −32.0000 −1.09759
\(851\) 12.0000 0.411355
\(852\) 0 0
\(853\) 45.0000 1.54077 0.770385 0.637579i \(-0.220064\pi\)
0.770385 + 0.637579i \(0.220064\pi\)
\(854\) 20.0000 0.684386
\(855\) 0 0
\(856\) 0 0
\(857\) 6.00000 0.204956 0.102478 0.994735i \(-0.467323\pi\)
0.102478 + 0.994735i \(0.467323\pi\)
\(858\) 0 0
\(859\) −2.00000 −0.0682391 −0.0341196 0.999418i \(-0.510863\pi\)
−0.0341196 + 0.999418i \(0.510863\pi\)
\(860\) −6.00000 −0.204598
\(861\) 0 0
\(862\) −12.0000 −0.408722
\(863\) 32.0000 1.08929 0.544646 0.838666i \(-0.316664\pi\)
0.544646 + 0.838666i \(0.316664\pi\)
\(864\) 0 0
\(865\) 24.0000 0.816024
\(866\) 24.0000 0.815553
\(867\) 0 0
\(868\) 6.00000 0.203653
\(869\) 18.0000 0.610608
\(870\) 0 0
\(871\) 6.00000 0.203302
\(872\) 0 0
\(873\) 0 0
\(874\) −30.0000 −1.01477
\(875\) 3.00000 0.101419
\(876\) 0 0
\(877\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(878\) −44.0000 −1.48493
\(879\) 0 0
\(880\) −72.0000 −2.42712
\(881\) 30.0000 1.01073 0.505363 0.862907i \(-0.331359\pi\)
0.505363 + 0.862907i \(0.331359\pi\)
\(882\) 0 0
\(883\) −4.00000 −0.134611 −0.0673054 0.997732i \(-0.521440\pi\)
−0.0673054 + 0.997732i \(0.521440\pi\)
\(884\) 8.00000 0.269069
\(885\) 0 0
\(886\) −38.0000 −1.27663
\(887\) 12.0000 0.402921 0.201460 0.979497i \(-0.435431\pi\)
0.201460 + 0.979497i \(0.435431\pi\)
\(888\) 0 0
\(889\) 4.00000 0.134156
\(890\) −18.0000 −0.603361
\(891\) 0 0
\(892\) 30.0000 1.00447
\(893\) −35.0000 −1.17123
\(894\) 0 0
\(895\) −69.0000 −2.30642
\(896\) 0 0
\(897\) 0 0
\(898\) −72.0000 −2.40267
\(899\) −15.0000 −0.500278
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) 72.0000 2.39734
\(903\) 0 0
\(904\) 0 0
\(905\) 42.0000 1.39613
\(906\) 0 0
\(907\) 7.00000 0.232431 0.116216 0.993224i \(-0.462924\pi\)
0.116216 + 0.993224i \(0.462924\pi\)
\(908\) −40.0000 −1.32745
\(909\) 0 0
\(910\) 6.00000 0.198898
\(911\) 15.0000 0.496972 0.248486 0.968635i \(-0.420067\pi\)
0.248486 + 0.968635i \(0.420067\pi\)
\(912\) 0 0
\(913\) −90.0000 −2.97857
\(914\) 0 0
\(915\) 0 0
\(916\) 28.0000 0.925146
\(917\) 8.00000 0.264183
\(918\) 0 0
\(919\) 8.00000 0.263896 0.131948 0.991257i \(-0.457877\pi\)
0.131948 + 0.991257i \(0.457877\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 44.0000 1.44906
\(923\) −8.00000 −0.263323
\(924\) 0 0
\(925\) −16.0000 −0.526077
\(926\) −28.0000 −0.920137
\(927\) 0 0
\(928\) −40.0000 −1.31306
\(929\) −5.00000 −0.164045 −0.0820223 0.996630i \(-0.526138\pi\)
−0.0820223 + 0.996630i \(0.526138\pi\)
\(930\) 0 0
\(931\) 5.00000 0.163868
\(932\) −30.0000 −0.982683
\(933\) 0 0
\(934\) 44.0000 1.43972
\(935\) −72.0000 −2.35465
\(936\) 0 0
\(937\) −8.00000 −0.261349 −0.130674 0.991425i \(-0.541714\pi\)
−0.130674 + 0.991425i \(0.541714\pi\)
\(938\) 12.0000 0.391814
\(939\) 0 0
\(940\) −42.0000 −1.36989
\(941\) −55.0000 −1.79295 −0.896474 0.443096i \(-0.853880\pi\)
−0.896474 + 0.443096i \(0.853880\pi\)
\(942\) 0 0
\(943\) −18.0000 −0.586161
\(944\) 32.0000 1.04151
\(945\) 0 0
\(946\) −12.0000 −0.390154
\(947\) 18.0000 0.584921 0.292461 0.956278i \(-0.405526\pi\)
0.292461 + 0.956278i \(0.405526\pi\)
\(948\) 0 0
\(949\) 13.0000 0.421998
\(950\) 40.0000 1.29777
\(951\) 0 0
\(952\) 0 0
\(953\) 39.0000 1.26333 0.631667 0.775240i \(-0.282371\pi\)
0.631667 + 0.775240i \(0.282371\pi\)
\(954\) 0 0
\(955\) 24.0000 0.776622
\(956\) 8.00000 0.258738
\(957\) 0 0
\(958\) 22.0000 0.710788
\(959\) 4.00000 0.129167
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) 8.00000 0.257930
\(963\) 0 0
\(964\) −34.0000 −1.09507
\(965\) 66.0000 2.12462
\(966\) 0 0
\(967\) 22.0000 0.707472 0.353736 0.935345i \(-0.384911\pi\)
0.353736 + 0.935345i \(0.384911\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 42.0000 1.34854
\(971\) −38.0000 −1.21948 −0.609739 0.792602i \(-0.708726\pi\)
−0.609739 + 0.792602i \(0.708726\pi\)
\(972\) 0 0
\(973\) 18.0000 0.577054
\(974\) −52.0000 −1.66619
\(975\) 0 0
\(976\) 40.0000 1.28037
\(977\) −10.0000 −0.319928 −0.159964 0.987123i \(-0.551138\pi\)
−0.159964 + 0.987123i \(0.551138\pi\)
\(978\) 0 0
\(979\) −18.0000 −0.575282
\(980\) 6.00000 0.191663
\(981\) 0 0
\(982\) 24.0000 0.765871
\(983\) −17.0000 −0.542216 −0.271108 0.962549i \(-0.587390\pi\)
−0.271108 + 0.962549i \(0.587390\pi\)
\(984\) 0 0
\(985\) −6.00000 −0.191176
\(986\) −40.0000 −1.27386
\(987\) 0 0
\(988\) −10.0000 −0.318142
\(989\) 3.00000 0.0953945
\(990\) 0 0
\(991\) 4.00000 0.127064 0.0635321 0.997980i \(-0.479763\pi\)
0.0635321 + 0.997980i \(0.479763\pi\)
\(992\) 24.0000 0.762001
\(993\) 0 0
\(994\) −16.0000 −0.507489
\(995\) 12.0000 0.380426
\(996\) 0 0
\(997\) −28.0000 −0.886769 −0.443384 0.896332i \(-0.646222\pi\)
−0.443384 + 0.896332i \(0.646222\pi\)
\(998\) −32.0000 −1.01294
\(999\) 0 0
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 \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 819.2.a.f.1.1 1
3.2 odd 2 91.2.a.a.1.1 1
7.6 odd 2 5733.2.a.l.1.1 1
12.11 even 2 1456.2.a.g.1.1 1
15.14 odd 2 2275.2.a.h.1.1 1
21.2 odd 6 637.2.e.e.508.1 2
21.5 even 6 637.2.e.d.508.1 2
21.11 odd 6 637.2.e.e.79.1 2
21.17 even 6 637.2.e.d.79.1 2
21.20 even 2 637.2.a.a.1.1 1
24.5 odd 2 5824.2.a.s.1.1 1
24.11 even 2 5824.2.a.t.1.1 1
39.5 even 4 1183.2.c.b.337.2 2
39.8 even 4 1183.2.c.b.337.1 2
39.38 odd 2 1183.2.a.b.1.1 1
273.272 even 2 8281.2.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.a.a.1.1 1 3.2 odd 2
637.2.a.a.1.1 1 21.20 even 2
637.2.e.d.79.1 2 21.17 even 6
637.2.e.d.508.1 2 21.5 even 6
637.2.e.e.79.1 2 21.11 odd 6
637.2.e.e.508.1 2 21.2 odd 6
819.2.a.f.1.1 1 1.1 even 1 trivial
1183.2.a.b.1.1 1 39.38 odd 2
1183.2.c.b.337.1 2 39.8 even 4
1183.2.c.b.337.2 2 39.5 even 4
1456.2.a.g.1.1 1 12.11 even 2
2275.2.a.h.1.1 1 15.14 odd 2
5733.2.a.l.1.1 1 7.6 odd 2
5824.2.a.s.1.1 1 24.5 odd 2
5824.2.a.t.1.1 1 24.11 even 2
8281.2.a.l.1.1 1 273.272 even 2