Properties

Label 8550.2.a.z.1.1
Level $8550$
Weight $2$
Character 8550.1
Self dual yes
Analytic conductor $68.272$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8550.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(68.2720937282\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2850)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 8550.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{8} -1.00000 q^{11} +4.00000 q^{13} +1.00000 q^{16} -4.00000 q^{17} -1.00000 q^{19} -1.00000 q^{22} -5.00000 q^{23} +4.00000 q^{26} -3.00000 q^{29} -5.00000 q^{31} +1.00000 q^{32} -4.00000 q^{34} -6.00000 q^{37} -1.00000 q^{38} +2.00000 q^{41} +4.00000 q^{43} -1.00000 q^{44} -5.00000 q^{46} -7.00000 q^{49} +4.00000 q^{52} -9.00000 q^{53} -3.00000 q^{58} -11.0000 q^{61} -5.00000 q^{62} +1.00000 q^{64} -1.00000 q^{67} -4.00000 q^{68} -2.00000 q^{71} -3.00000 q^{73} -6.00000 q^{74} -1.00000 q^{76} +17.0000 q^{79} +2.00000 q^{82} -3.00000 q^{83} +4.00000 q^{86} -1.00000 q^{88} -7.00000 q^{89} -5.00000 q^{92} +10.0000 q^{97} -7.00000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) −1.00000 −0.301511 −0.150756 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) −5.00000 −1.04257 −0.521286 0.853382i \(-0.674548\pi\)
−0.521286 + 0.853382i \(0.674548\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 4.00000 0.784465
\(27\) 0 0
\(28\) 0 0
\(29\) −3.00000 −0.557086 −0.278543 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 0 0
\(31\) −5.00000 −0.898027 −0.449013 0.893525i \(-0.648224\pi\)
−0.449013 + 0.893525i \(0.648224\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −4.00000 −0.685994
\(35\) 0 0
\(36\) 0 0
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) −1.00000 −0.162221
\(39\) 0 0
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) −5.00000 −0.737210
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 4.00000 0.554700
\(53\) −9.00000 −1.23625 −0.618123 0.786082i \(-0.712106\pi\)
−0.618123 + 0.786082i \(0.712106\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −3.00000 −0.393919
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −11.0000 −1.40841 −0.704203 0.709999i \(-0.748695\pi\)
−0.704203 + 0.709999i \(0.748695\pi\)
\(62\) −5.00000 −0.635001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −1.00000 −0.122169 −0.0610847 0.998133i \(-0.519456\pi\)
−0.0610847 + 0.998133i \(0.519456\pi\)
\(68\) −4.00000 −0.485071
\(69\) 0 0
\(70\) 0 0
\(71\) −2.00000 −0.237356 −0.118678 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) 0 0
\(73\) −3.00000 −0.351123 −0.175562 0.984468i \(-0.556174\pi\)
−0.175562 + 0.984468i \(0.556174\pi\)
\(74\) −6.00000 −0.697486
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) 0 0
\(78\) 0 0
\(79\) 17.0000 1.91265 0.956325 0.292306i \(-0.0944227\pi\)
0.956325 + 0.292306i \(0.0944227\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 2.00000 0.220863
\(83\) −3.00000 −0.329293 −0.164646 0.986353i \(-0.552648\pi\)
−0.164646 + 0.986353i \(0.552648\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) −1.00000 −0.106600
\(89\) −7.00000 −0.741999 −0.370999 0.928633i \(-0.620985\pi\)
−0.370999 + 0.928633i \(0.620985\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −5.00000 −0.521286
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 10.0000 1.01535 0.507673 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) −7.00000 −0.707107
\(99\) 0 0
\(100\) 0 0
\(101\) 8.00000 0.796030 0.398015 0.917379i \(-0.369699\pi\)
0.398015 + 0.917379i \(0.369699\pi\)
\(102\) 0 0
\(103\) 13.0000 1.28093 0.640464 0.767988i \(-0.278742\pi\)
0.640464 + 0.767988i \(0.278742\pi\)
\(104\) 4.00000 0.392232
\(105\) 0 0
\(106\) −9.00000 −0.874157
\(107\) −10.0000 −0.966736 −0.483368 0.875417i \(-0.660587\pi\)
−0.483368 + 0.875417i \(0.660587\pi\)
\(108\) 0 0
\(109\) 6.00000 0.574696 0.287348 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −5.00000 −0.470360 −0.235180 0.971952i \(-0.575568\pi\)
−0.235180 + 0.971952i \(0.575568\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −3.00000 −0.278543
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) −11.0000 −0.995893
\(123\) 0 0
\(124\) −5.00000 −0.449013
\(125\) 0 0
\(126\) 0 0
\(127\) −5.00000 −0.443678 −0.221839 0.975083i \(-0.571206\pi\)
−0.221839 + 0.975083i \(0.571206\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −15.0000 −1.31056 −0.655278 0.755388i \(-0.727449\pi\)
−0.655278 + 0.755388i \(0.727449\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −1.00000 −0.0863868
\(135\) 0 0
\(136\) −4.00000 −0.342997
\(137\) 4.00000 0.341743 0.170872 0.985293i \(-0.445342\pi\)
0.170872 + 0.985293i \(0.445342\pi\)
\(138\) 0 0
\(139\) −2.00000 −0.169638 −0.0848189 0.996396i \(-0.527031\pi\)
−0.0848189 + 0.996396i \(0.527031\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −2.00000 −0.167836
\(143\) −4.00000 −0.334497
\(144\) 0 0
\(145\) 0 0
\(146\) −3.00000 −0.248282
\(147\) 0 0
\(148\) −6.00000 −0.493197
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) −24.0000 −1.95309 −0.976546 0.215308i \(-0.930924\pi\)
−0.976546 + 0.215308i \(0.930924\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 14.0000 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 17.0000 1.35245
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 10.0000 0.783260 0.391630 0.920123i \(-0.371911\pi\)
0.391630 + 0.920123i \(0.371911\pi\)
\(164\) 2.00000 0.156174
\(165\) 0 0
\(166\) −3.00000 −0.232845
\(167\) −18.0000 −1.39288 −0.696441 0.717614i \(-0.745234\pi\)
−0.696441 + 0.717614i \(0.745234\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) 0 0
\(172\) 4.00000 0.304997
\(173\) −7.00000 −0.532200 −0.266100 0.963945i \(-0.585735\pi\)
−0.266100 + 0.963945i \(0.585735\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) −7.00000 −0.524672
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −5.00000 −0.368605
\(185\) 0 0
\(186\) 0 0
\(187\) 4.00000 0.292509
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −11.0000 −0.795932 −0.397966 0.917400i \(-0.630284\pi\)
−0.397966 + 0.917400i \(0.630284\pi\)
\(192\) 0 0
\(193\) 10.0000 0.719816 0.359908 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) 10.0000 0.717958
\(195\) 0 0
\(196\) −7.00000 −0.500000
\(197\) 20.0000 1.42494 0.712470 0.701702i \(-0.247576\pi\)
0.712470 + 0.701702i \(0.247576\pi\)
\(198\) 0 0
\(199\) 6.00000 0.425329 0.212664 0.977125i \(-0.431786\pi\)
0.212664 + 0.977125i \(0.431786\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 8.00000 0.562878
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 13.0000 0.905753
\(207\) 0 0
\(208\) 4.00000 0.277350
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) −13.0000 −0.894957 −0.447478 0.894295i \(-0.647678\pi\)
−0.447478 + 0.894295i \(0.647678\pi\)
\(212\) −9.00000 −0.618123
\(213\) 0 0
\(214\) −10.0000 −0.683586
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 6.00000 0.406371
\(219\) 0 0
\(220\) 0 0
\(221\) −16.0000 −1.07628
\(222\) 0 0
\(223\) −15.0000 −1.00447 −0.502237 0.864730i \(-0.667490\pi\)
−0.502237 + 0.864730i \(0.667490\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −5.00000 −0.332595
\(227\) −2.00000 −0.132745 −0.0663723 0.997795i \(-0.521143\pi\)
−0.0663723 + 0.997795i \(0.521143\pi\)
\(228\) 0 0
\(229\) 5.00000 0.330409 0.165205 0.986259i \(-0.447172\pi\)
0.165205 + 0.986259i \(0.447172\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −3.00000 −0.196960
\(233\) 24.0000 1.57229 0.786146 0.618041i \(-0.212073\pi\)
0.786146 + 0.618041i \(0.212073\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −4.00000 −0.257663 −0.128831 0.991667i \(-0.541123\pi\)
−0.128831 + 0.991667i \(0.541123\pi\)
\(242\) −10.0000 −0.642824
\(243\) 0 0
\(244\) −11.0000 −0.704203
\(245\) 0 0
\(246\) 0 0
\(247\) −4.00000 −0.254514
\(248\) −5.00000 −0.317500
\(249\) 0 0
\(250\) 0 0
\(251\) −16.0000 −1.00991 −0.504956 0.863145i \(-0.668491\pi\)
−0.504956 + 0.863145i \(0.668491\pi\)
\(252\) 0 0
\(253\) 5.00000 0.314347
\(254\) −5.00000 −0.313728
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −7.00000 −0.436648 −0.218324 0.975876i \(-0.570059\pi\)
−0.218324 + 0.975876i \(0.570059\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) −15.0000 −0.926703
\(263\) 11.0000 0.678289 0.339145 0.940734i \(-0.389862\pi\)
0.339145 + 0.940734i \(0.389862\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −1.00000 −0.0610847
\(269\) 2.00000 0.121942 0.0609711 0.998140i \(-0.480580\pi\)
0.0609711 + 0.998140i \(0.480580\pi\)
\(270\) 0 0
\(271\) 2.00000 0.121491 0.0607457 0.998153i \(-0.480652\pi\)
0.0607457 + 0.998153i \(0.480652\pi\)
\(272\) −4.00000 −0.242536
\(273\) 0 0
\(274\) 4.00000 0.241649
\(275\) 0 0
\(276\) 0 0
\(277\) −1.00000 −0.0600842 −0.0300421 0.999549i \(-0.509564\pi\)
−0.0300421 + 0.999549i \(0.509564\pi\)
\(278\) −2.00000 −0.119952
\(279\) 0 0
\(280\) 0 0
\(281\) 3.00000 0.178965 0.0894825 0.995988i \(-0.471479\pi\)
0.0894825 + 0.995988i \(0.471479\pi\)
\(282\) 0 0
\(283\) −8.00000 −0.475551 −0.237775 0.971320i \(-0.576418\pi\)
−0.237775 + 0.971320i \(0.576418\pi\)
\(284\) −2.00000 −0.118678
\(285\) 0 0
\(286\) −4.00000 −0.236525
\(287\) 0 0
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 0 0
\(292\) −3.00000 −0.175562
\(293\) −11.0000 −0.642627 −0.321313 0.946973i \(-0.604124\pi\)
−0.321313 + 0.946973i \(0.604124\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −6.00000 −0.348743
\(297\) 0 0
\(298\) 0 0
\(299\) −20.0000 −1.15663
\(300\) 0 0
\(301\) 0 0
\(302\) −24.0000 −1.38104
\(303\) 0 0
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) 0 0
\(307\) −25.0000 −1.42683 −0.713413 0.700744i \(-0.752851\pi\)
−0.713413 + 0.700744i \(0.752851\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −32.0000 −1.81455 −0.907277 0.420534i \(-0.861843\pi\)
−0.907277 + 0.420534i \(0.861843\pi\)
\(312\) 0 0
\(313\) 21.0000 1.18699 0.593495 0.804838i \(-0.297748\pi\)
0.593495 + 0.804838i \(0.297748\pi\)
\(314\) 14.0000 0.790066
\(315\) 0 0
\(316\) 17.0000 0.956325
\(317\) 13.0000 0.730153 0.365076 0.930978i \(-0.381043\pi\)
0.365076 + 0.930978i \(0.381043\pi\)
\(318\) 0 0
\(319\) 3.00000 0.167968
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 4.00000 0.222566
\(324\) 0 0
\(325\) 0 0
\(326\) 10.0000 0.553849
\(327\) 0 0
\(328\) 2.00000 0.110432
\(329\) 0 0
\(330\) 0 0
\(331\) 13.0000 0.714545 0.357272 0.934000i \(-0.383707\pi\)
0.357272 + 0.934000i \(0.383707\pi\)
\(332\) −3.00000 −0.164646
\(333\) 0 0
\(334\) −18.0000 −0.984916
\(335\) 0 0
\(336\) 0 0
\(337\) −8.00000 −0.435788 −0.217894 0.975972i \(-0.569919\pi\)
−0.217894 + 0.975972i \(0.569919\pi\)
\(338\) 3.00000 0.163178
\(339\) 0 0
\(340\) 0 0
\(341\) 5.00000 0.270765
\(342\) 0 0
\(343\) 0 0
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) −7.00000 −0.376322
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 0 0
\(349\) −1.00000 −0.0535288 −0.0267644 0.999642i \(-0.508520\pi\)
−0.0267644 + 0.999642i \(0.508520\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1.00000 −0.0533002
\(353\) −28.0000 −1.49029 −0.745145 0.666903i \(-0.767620\pi\)
−0.745145 + 0.666903i \(0.767620\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −7.00000 −0.370999
\(357\) 0 0
\(358\) 4.00000 0.211407
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −18.0000 −0.939592 −0.469796 0.882775i \(-0.655673\pi\)
−0.469796 + 0.882775i \(0.655673\pi\)
\(368\) −5.00000 −0.260643
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 28.0000 1.44979 0.724893 0.688862i \(-0.241889\pi\)
0.724893 + 0.688862i \(0.241889\pi\)
\(374\) 4.00000 0.206835
\(375\) 0 0
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) −8.00000 −0.410932 −0.205466 0.978664i \(-0.565871\pi\)
−0.205466 + 0.978664i \(0.565871\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −11.0000 −0.562809
\(383\) 6.00000 0.306586 0.153293 0.988181i \(-0.451012\pi\)
0.153293 + 0.988181i \(0.451012\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 10.0000 0.508987
\(387\) 0 0
\(388\) 10.0000 0.507673
\(389\) −26.0000 −1.31825 −0.659126 0.752032i \(-0.729074\pi\)
−0.659126 + 0.752032i \(0.729074\pi\)
\(390\) 0 0
\(391\) 20.0000 1.01144
\(392\) −7.00000 −0.353553
\(393\) 0 0
\(394\) 20.0000 1.00759
\(395\) 0 0
\(396\) 0 0
\(397\) −3.00000 −0.150566 −0.0752828 0.997162i \(-0.523986\pi\)
−0.0752828 + 0.997162i \(0.523986\pi\)
\(398\) 6.00000 0.300753
\(399\) 0 0
\(400\) 0 0
\(401\) 17.0000 0.848939 0.424470 0.905442i \(-0.360461\pi\)
0.424470 + 0.905442i \(0.360461\pi\)
\(402\) 0 0
\(403\) −20.0000 −0.996271
\(404\) 8.00000 0.398015
\(405\) 0 0
\(406\) 0 0
\(407\) 6.00000 0.297409
\(408\) 0 0
\(409\) 6.00000 0.296681 0.148340 0.988936i \(-0.452607\pi\)
0.148340 + 0.988936i \(0.452607\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 13.0000 0.640464
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 4.00000 0.196116
\(417\) 0 0
\(418\) 1.00000 0.0489116
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) −34.0000 −1.65706 −0.828529 0.559946i \(-0.810822\pi\)
−0.828529 + 0.559946i \(0.810822\pi\)
\(422\) −13.0000 −0.632830
\(423\) 0 0
\(424\) −9.00000 −0.437079
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −10.0000 −0.483368
\(429\) 0 0
\(430\) 0 0
\(431\) −38.0000 −1.83040 −0.915198 0.403005i \(-0.867966\pi\)
−0.915198 + 0.403005i \(0.867966\pi\)
\(432\) 0 0
\(433\) −22.0000 −1.05725 −0.528626 0.848855i \(-0.677293\pi\)
−0.528626 + 0.848855i \(0.677293\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 6.00000 0.287348
\(437\) 5.00000 0.239182
\(438\) 0 0
\(439\) 21.0000 1.00228 0.501138 0.865368i \(-0.332915\pi\)
0.501138 + 0.865368i \(0.332915\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −16.0000 −0.761042
\(443\) 13.0000 0.617649 0.308824 0.951119i \(-0.400064\pi\)
0.308824 + 0.951119i \(0.400064\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −15.0000 −0.710271
\(447\) 0 0
\(448\) 0 0
\(449\) 27.0000 1.27421 0.637104 0.770778i \(-0.280132\pi\)
0.637104 + 0.770778i \(0.280132\pi\)
\(450\) 0 0
\(451\) −2.00000 −0.0941763
\(452\) −5.00000 −0.235180
\(453\) 0 0
\(454\) −2.00000 −0.0938647
\(455\) 0 0
\(456\) 0 0
\(457\) −18.0000 −0.842004 −0.421002 0.907060i \(-0.638322\pi\)
−0.421002 + 0.907060i \(0.638322\pi\)
\(458\) 5.00000 0.233635
\(459\) 0 0
\(460\) 0 0
\(461\) −4.00000 −0.186299 −0.0931493 0.995652i \(-0.529693\pi\)
−0.0931493 + 0.995652i \(0.529693\pi\)
\(462\) 0 0
\(463\) 4.00000 0.185896 0.0929479 0.995671i \(-0.470371\pi\)
0.0929479 + 0.995671i \(0.470371\pi\)
\(464\) −3.00000 −0.139272
\(465\) 0 0
\(466\) 24.0000 1.11178
\(467\) 11.0000 0.509019 0.254510 0.967070i \(-0.418086\pi\)
0.254510 + 0.967070i \(0.418086\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −4.00000 −0.183920
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 37.0000 1.69057 0.845287 0.534313i \(-0.179430\pi\)
0.845287 + 0.534313i \(0.179430\pi\)
\(480\) 0 0
\(481\) −24.0000 −1.09431
\(482\) −4.00000 −0.182195
\(483\) 0 0
\(484\) −10.0000 −0.454545
\(485\) 0 0
\(486\) 0 0
\(487\) 8.00000 0.362515 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) −11.0000 −0.497947
\(489\) 0 0
\(490\) 0 0
\(491\) −36.0000 −1.62466 −0.812329 0.583200i \(-0.801800\pi\)
−0.812329 + 0.583200i \(0.801800\pi\)
\(492\) 0 0
\(493\) 12.0000 0.540453
\(494\) −4.00000 −0.179969
\(495\) 0 0
\(496\) −5.00000 −0.224507
\(497\) 0 0
\(498\) 0 0
\(499\) −24.0000 −1.07439 −0.537194 0.843459i \(-0.680516\pi\)
−0.537194 + 0.843459i \(0.680516\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −16.0000 −0.714115
\(503\) 36.0000 1.60516 0.802580 0.596544i \(-0.203460\pi\)
0.802580 + 0.596544i \(0.203460\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 5.00000 0.222277
\(507\) 0 0
\(508\) −5.00000 −0.221839
\(509\) 27.0000 1.19675 0.598377 0.801215i \(-0.295813\pi\)
0.598377 + 0.801215i \(0.295813\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −7.00000 −0.308757
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −3.00000 −0.131432 −0.0657162 0.997838i \(-0.520933\pi\)
−0.0657162 + 0.997838i \(0.520933\pi\)
\(522\) 0 0
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) −15.0000 −0.655278
\(525\) 0 0
\(526\) 11.0000 0.479623
\(527\) 20.0000 0.871214
\(528\) 0 0
\(529\) 2.00000 0.0869565
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 8.00000 0.346518
\(534\) 0 0
\(535\) 0 0
\(536\) −1.00000 −0.0431934
\(537\) 0 0
\(538\) 2.00000 0.0862261
\(539\) 7.00000 0.301511
\(540\) 0 0
\(541\) −13.0000 −0.558914 −0.279457 0.960158i \(-0.590154\pi\)
−0.279457 + 0.960158i \(0.590154\pi\)
\(542\) 2.00000 0.0859074
\(543\) 0 0
\(544\) −4.00000 −0.171499
\(545\) 0 0
\(546\) 0 0
\(547\) 1.00000 0.0427569 0.0213785 0.999771i \(-0.493195\pi\)
0.0213785 + 0.999771i \(0.493195\pi\)
\(548\) 4.00000 0.170872
\(549\) 0 0
\(550\) 0 0
\(551\) 3.00000 0.127804
\(552\) 0 0
\(553\) 0 0
\(554\) −1.00000 −0.0424859
\(555\) 0 0
\(556\) −2.00000 −0.0848189
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 16.0000 0.676728
\(560\) 0 0
\(561\) 0 0
\(562\) 3.00000 0.126547
\(563\) −26.0000 −1.09577 −0.547885 0.836554i \(-0.684567\pi\)
−0.547885 + 0.836554i \(0.684567\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −8.00000 −0.336265
\(567\) 0 0
\(568\) −2.00000 −0.0839181
\(569\) −22.0000 −0.922288 −0.461144 0.887325i \(-0.652561\pi\)
−0.461144 + 0.887325i \(0.652561\pi\)
\(570\) 0 0
\(571\) −26.0000 −1.08807 −0.544033 0.839064i \(-0.683103\pi\)
−0.544033 + 0.839064i \(0.683103\pi\)
\(572\) −4.00000 −0.167248
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −27.0000 −1.12402 −0.562012 0.827129i \(-0.689973\pi\)
−0.562012 + 0.827129i \(0.689973\pi\)
\(578\) −1.00000 −0.0415945
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 9.00000 0.372742
\(584\) −3.00000 −0.124141
\(585\) 0 0
\(586\) −11.0000 −0.454406
\(587\) 17.0000 0.701665 0.350833 0.936438i \(-0.385899\pi\)
0.350833 + 0.936438i \(0.385899\pi\)
\(588\) 0 0
\(589\) 5.00000 0.206021
\(590\) 0 0
\(591\) 0 0
\(592\) −6.00000 −0.246598
\(593\) 28.0000 1.14982 0.574911 0.818216i \(-0.305037\pi\)
0.574911 + 0.818216i \(0.305037\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) −20.0000 −0.817861
\(599\) −28.0000 −1.14405 −0.572024 0.820237i \(-0.693842\pi\)
−0.572024 + 0.820237i \(0.693842\pi\)
\(600\) 0 0
\(601\) −2.00000 −0.0815817 −0.0407909 0.999168i \(-0.512988\pi\)
−0.0407909 + 0.999168i \(0.512988\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −24.0000 −0.976546
\(605\) 0 0
\(606\) 0 0
\(607\) −25.0000 −1.01472 −0.507359 0.861735i \(-0.669378\pi\)
−0.507359 + 0.861735i \(0.669378\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 26.0000 1.05013 0.525065 0.851062i \(-0.324041\pi\)
0.525065 + 0.851062i \(0.324041\pi\)
\(614\) −25.0000 −1.00892
\(615\) 0 0
\(616\) 0 0
\(617\) 46.0000 1.85189 0.925945 0.377658i \(-0.123271\pi\)
0.925945 + 0.377658i \(0.123271\pi\)
\(618\) 0 0
\(619\) −8.00000 −0.321547 −0.160774 0.986991i \(-0.551399\pi\)
−0.160774 + 0.986991i \(0.551399\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −32.0000 −1.28308
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 21.0000 0.839329
\(627\) 0 0
\(628\) 14.0000 0.558661
\(629\) 24.0000 0.956943
\(630\) 0 0
\(631\) 28.0000 1.11466 0.557331 0.830290i \(-0.311825\pi\)
0.557331 + 0.830290i \(0.311825\pi\)
\(632\) 17.0000 0.676224
\(633\) 0 0
\(634\) 13.0000 0.516296
\(635\) 0 0
\(636\) 0 0
\(637\) −28.0000 −1.10940
\(638\) 3.00000 0.118771
\(639\) 0 0
\(640\) 0 0
\(641\) −6.00000 −0.236986 −0.118493 0.992955i \(-0.537806\pi\)
−0.118493 + 0.992955i \(0.537806\pi\)
\(642\) 0 0
\(643\) −14.0000 −0.552106 −0.276053 0.961142i \(-0.589027\pi\)
−0.276053 + 0.961142i \(0.589027\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 4.00000 0.157378
\(647\) 21.0000 0.825595 0.412798 0.910823i \(-0.364552\pi\)
0.412798 + 0.910823i \(0.364552\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 10.0000 0.391630
\(653\) 48.0000 1.87839 0.939193 0.343391i \(-0.111576\pi\)
0.939193 + 0.343391i \(0.111576\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 2.00000 0.0780869
\(657\) 0 0
\(658\) 0 0
\(659\) 36.0000 1.40236 0.701180 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) 0 0
\(661\) −32.0000 −1.24466 −0.622328 0.782757i \(-0.713813\pi\)
−0.622328 + 0.782757i \(0.713813\pi\)
\(662\) 13.0000 0.505259
\(663\) 0 0
\(664\) −3.00000 −0.116423
\(665\) 0 0
\(666\) 0 0
\(667\) 15.0000 0.580802
\(668\) −18.0000 −0.696441
\(669\) 0 0
\(670\) 0 0
\(671\) 11.0000 0.424650
\(672\) 0 0
\(673\) 44.0000 1.69608 0.848038 0.529936i \(-0.177784\pi\)
0.848038 + 0.529936i \(0.177784\pi\)
\(674\) −8.00000 −0.308148
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) 15.0000 0.576497 0.288248 0.957556i \(-0.406927\pi\)
0.288248 + 0.957556i \(0.406927\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 5.00000 0.191460
\(683\) 18.0000 0.688751 0.344375 0.938832i \(-0.388091\pi\)
0.344375 + 0.938832i \(0.388091\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 4.00000 0.152499
\(689\) −36.0000 −1.37149
\(690\) 0 0
\(691\) 26.0000 0.989087 0.494543 0.869153i \(-0.335335\pi\)
0.494543 + 0.869153i \(0.335335\pi\)
\(692\) −7.00000 −0.266100
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) 0 0
\(696\) 0 0
\(697\) −8.00000 −0.303022
\(698\) −1.00000 −0.0378506
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 6.00000 0.226294
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) −28.0000 −1.05379
\(707\) 0 0
\(708\) 0 0
\(709\) −43.0000 −1.61490 −0.807449 0.589937i \(-0.799153\pi\)
−0.807449 + 0.589937i \(0.799153\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −7.00000 −0.262336
\(713\) 25.0000 0.936257
\(714\) 0 0
\(715\) 0 0
\(716\) 4.00000 0.149487
\(717\) 0 0
\(718\) −24.0000 −0.895672
\(719\) 29.0000 1.08152 0.540759 0.841178i \(-0.318137\pi\)
0.540759 + 0.841178i \(0.318137\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 1.00000 0.0372161
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −10.0000 −0.370879 −0.185440 0.982656i \(-0.559371\pi\)
−0.185440 + 0.982656i \(0.559371\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −16.0000 −0.591781
\(732\) 0 0
\(733\) 41.0000 1.51437 0.757185 0.653201i \(-0.226574\pi\)
0.757185 + 0.653201i \(0.226574\pi\)
\(734\) −18.0000 −0.664392
\(735\) 0 0
\(736\) −5.00000 −0.184302
\(737\) 1.00000 0.0368355
\(738\) 0 0
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 16.0000 0.586983 0.293492 0.955962i \(-0.405183\pi\)
0.293492 + 0.955962i \(0.405183\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 28.0000 1.02515
\(747\) 0 0
\(748\) 4.00000 0.146254
\(749\) 0 0
\(750\) 0 0
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) −12.0000 −0.437014
\(755\) 0 0
\(756\) 0 0
\(757\) −23.0000 −0.835949 −0.417975 0.908459i \(-0.637260\pi\)
−0.417975 + 0.908459i \(0.637260\pi\)
\(758\) −8.00000 −0.290573
\(759\) 0 0
\(760\) 0 0
\(761\) 30.0000 1.08750 0.543750 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −11.0000 −0.397966
\(765\) 0 0
\(766\) 6.00000 0.216789
\(767\) 0 0
\(768\) 0 0
\(769\) 35.0000 1.26213 0.631066 0.775729i \(-0.282618\pi\)
0.631066 + 0.775729i \(0.282618\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 10.0000 0.359908
\(773\) −18.0000 −0.647415 −0.323708 0.946157i \(-0.604929\pi\)
−0.323708 + 0.946157i \(0.604929\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 10.0000 0.358979
\(777\) 0 0
\(778\) −26.0000 −0.932145
\(779\) −2.00000 −0.0716574
\(780\) 0 0
\(781\) 2.00000 0.0715656
\(782\) 20.0000 0.715199
\(783\) 0 0
\(784\) −7.00000 −0.250000
\(785\) 0 0
\(786\) 0 0
\(787\) 37.0000 1.31891 0.659454 0.751745i \(-0.270788\pi\)
0.659454 + 0.751745i \(0.270788\pi\)
\(788\) 20.0000 0.712470
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −44.0000 −1.56249
\(794\) −3.00000 −0.106466
\(795\) 0 0
\(796\) 6.00000 0.212664
\(797\) 18.0000 0.637593 0.318796 0.947823i \(-0.396721\pi\)
0.318796 + 0.947823i \(0.396721\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 17.0000 0.600291
\(803\) 3.00000 0.105868
\(804\) 0 0
\(805\) 0 0
\(806\) −20.0000 −0.704470
\(807\) 0 0
\(808\) 8.00000 0.281439
\(809\) 44.0000 1.54696 0.773479 0.633822i \(-0.218515\pi\)
0.773479 + 0.633822i \(0.218515\pi\)
\(810\) 0 0
\(811\) −35.0000 −1.22902 −0.614508 0.788911i \(-0.710645\pi\)
−0.614508 + 0.788911i \(0.710645\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 6.00000 0.210300
\(815\) 0 0
\(816\) 0 0
\(817\) −4.00000 −0.139942
\(818\) 6.00000 0.209785
\(819\) 0 0
\(820\) 0 0
\(821\) −26.0000 −0.907406 −0.453703 0.891153i \(-0.649897\pi\)
−0.453703 + 0.891153i \(0.649897\pi\)
\(822\) 0 0
\(823\) −26.0000 −0.906303 −0.453152 0.891434i \(-0.649700\pi\)
−0.453152 + 0.891434i \(0.649700\pi\)
\(824\) 13.0000 0.452876
\(825\) 0 0
\(826\) 0 0
\(827\) −48.0000 −1.66912 −0.834562 0.550914i \(-0.814279\pi\)
−0.834562 + 0.550914i \(0.814279\pi\)
\(828\) 0 0
\(829\) −50.0000 −1.73657 −0.868286 0.496064i \(-0.834778\pi\)
−0.868286 + 0.496064i \(0.834778\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 4.00000 0.138675
\(833\) 28.0000 0.970143
\(834\) 0 0
\(835\) 0 0
\(836\) 1.00000 0.0345857
\(837\) 0 0
\(838\) 12.0000 0.414533
\(839\) 12.0000 0.414286 0.207143 0.978311i \(-0.433583\pi\)
0.207143 + 0.978311i \(0.433583\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) −34.0000 −1.17172
\(843\) 0 0
\(844\) −13.0000 −0.447478
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) −9.00000 −0.309061
\(849\) 0 0
\(850\) 0 0
\(851\) 30.0000 1.02839
\(852\) 0 0
\(853\) 10.0000 0.342393 0.171197 0.985237i \(-0.445237\pi\)
0.171197 + 0.985237i \(0.445237\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −10.0000 −0.341793
\(857\) 38.0000 1.29806 0.649028 0.760765i \(-0.275176\pi\)
0.649028 + 0.760765i \(0.275176\pi\)
\(858\) 0 0
\(859\) −18.0000 −0.614152 −0.307076 0.951685i \(-0.599351\pi\)
−0.307076 + 0.951685i \(0.599351\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −38.0000 −1.29429
\(863\) −4.00000 −0.136162 −0.0680808 0.997680i \(-0.521688\pi\)
−0.0680808 + 0.997680i \(0.521688\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −22.0000 −0.747590
\(867\) 0 0
\(868\) 0 0
\(869\) −17.0000 −0.576686
\(870\) 0 0
\(871\) −4.00000 −0.135535
\(872\) 6.00000 0.203186
\(873\) 0 0
\(874\) 5.00000 0.169128
\(875\) 0 0
\(876\) 0 0
\(877\) −2.00000 −0.0675352 −0.0337676 0.999430i \(-0.510751\pi\)
−0.0337676 + 0.999430i \(0.510751\pi\)
\(878\) 21.0000 0.708716
\(879\) 0 0
\(880\) 0 0
\(881\) 48.0000 1.61716 0.808581 0.588386i \(-0.200236\pi\)
0.808581 + 0.588386i \(0.200236\pi\)
\(882\) 0 0
\(883\) −38.0000 −1.27880 −0.639401 0.768874i \(-0.720818\pi\)
−0.639401 + 0.768874i \(0.720818\pi\)
\(884\) −16.0000 −0.538138
\(885\) 0 0
\(886\) 13.0000 0.436744
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) −15.0000 −0.502237
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 27.0000 0.901002
\(899\) 15.0000 0.500278
\(900\) 0 0
\(901\) 36.0000 1.19933
\(902\) −2.00000 −0.0665927
\(903\) 0 0
\(904\) −5.00000 −0.166298
\(905\) 0 0
\(906\) 0 0
\(907\) 8.00000 0.265636 0.132818 0.991140i \(-0.457597\pi\)
0.132818 + 0.991140i \(0.457597\pi\)
\(908\) −2.00000 −0.0663723
\(909\) 0 0
\(910\) 0 0
\(911\) 18.0000 0.596367 0.298183 0.954509i \(-0.403619\pi\)
0.298183 + 0.954509i \(0.403619\pi\)
\(912\) 0 0
\(913\) 3.00000 0.0992855
\(914\) −18.0000 −0.595387
\(915\) 0 0
\(916\) 5.00000 0.165205
\(917\) 0 0
\(918\) 0 0
\(919\) 14.0000 0.461817 0.230909 0.972975i \(-0.425830\pi\)
0.230909 + 0.972975i \(0.425830\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −4.00000 −0.131733
\(923\) −8.00000 −0.263323
\(924\) 0 0
\(925\) 0 0
\(926\) 4.00000 0.131448
\(927\) 0 0
\(928\) −3.00000 −0.0984798
\(929\) 12.0000 0.393707 0.196854 0.980433i \(-0.436928\pi\)
0.196854 + 0.980433i \(0.436928\pi\)
\(930\) 0 0
\(931\) 7.00000 0.229416
\(932\) 24.0000 0.786146
\(933\) 0 0
\(934\) 11.0000 0.359931
\(935\) 0 0
\(936\) 0 0
\(937\) −10.0000 −0.326686 −0.163343 0.986569i \(-0.552228\pi\)
−0.163343 + 0.986569i \(0.552228\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −33.0000 −1.07577 −0.537885 0.843018i \(-0.680776\pi\)
−0.537885 + 0.843018i \(0.680776\pi\)
\(942\) 0 0
\(943\) −10.0000 −0.325645
\(944\) 0 0
\(945\) 0 0
\(946\) −4.00000 −0.130051
\(947\) −20.0000 −0.649913 −0.324956 0.945729i \(-0.605350\pi\)
−0.324956 + 0.945729i \(0.605350\pi\)
\(948\) 0 0
\(949\) −12.0000 −0.389536
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −3.00000 −0.0971795 −0.0485898 0.998819i \(-0.515473\pi\)
−0.0485898 + 0.998819i \(0.515473\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 37.0000 1.19542
\(959\) 0 0
\(960\) 0 0
\(961\) −6.00000 −0.193548
\(962\) −24.0000 −0.773791
\(963\) 0 0
\(964\) −4.00000 −0.128831
\(965\) 0 0
\(966\) 0 0
\(967\) 14.0000 0.450210 0.225105 0.974335i \(-0.427728\pi\)
0.225105 + 0.974335i \(0.427728\pi\)
\(968\) −10.0000 −0.321412
\(969\) 0 0
\(970\) 0 0
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 8.00000 0.256337
\(975\) 0 0
\(976\) −11.0000 −0.352101
\(977\) −46.0000 −1.47167 −0.735835 0.677161i \(-0.763210\pi\)
−0.735835 + 0.677161i \(0.763210\pi\)
\(978\) 0 0
\(979\) 7.00000 0.223721
\(980\) 0 0
\(981\) 0 0
\(982\) −36.0000 −1.14881
\(983\) −34.0000 −1.08443 −0.542216 0.840239i \(-0.682414\pi\)
−0.542216 + 0.840239i \(0.682414\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 12.0000 0.382158
\(987\) 0 0
\(988\) −4.00000 −0.127257
\(989\) −20.0000 −0.635963
\(990\) 0 0
\(991\) 13.0000 0.412959 0.206479 0.978451i \(-0.433799\pi\)
0.206479 + 0.978451i \(0.433799\pi\)
\(992\) −5.00000 −0.158750
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 37.0000 1.17180 0.585901 0.810383i \(-0.300741\pi\)
0.585901 + 0.810383i \(0.300741\pi\)
\(998\) −24.0000 −0.759707
\(999\) 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 8550.2.a.z.1.1 1
3.2 odd 2 2850.2.a.l.1.1 1
5.4 even 2 8550.2.a.h.1.1 1
15.2 even 4 2850.2.d.g.799.1 2
15.8 even 4 2850.2.d.g.799.2 2
15.14 odd 2 2850.2.a.u.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2850.2.a.l.1.1 1 3.2 odd 2
2850.2.a.u.1.1 yes 1 15.14 odd 2
2850.2.d.g.799.1 2 15.2 even 4
2850.2.d.g.799.2 2 15.8 even 4
8550.2.a.h.1.1 1 5.4 even 2
8550.2.a.z.1.1 1 1.1 even 1 trivial