Properties

Label 8820.2.a.bk.1.2
Level $8820$
Weight $2$
Character 8820.1
Self dual yes
Analytic conductor $70.428$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8820 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8820.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(70.4280545828\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 420)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 8820.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{5} +O(q^{10})\) \(q+1.00000 q^{5} +4.24264 q^{11} +3.24264 q^{13} -4.24264 q^{17} -7.00000 q^{19} +4.24264 q^{23} +1.00000 q^{25} +1.75736 q^{29} -9.48528 q^{31} +3.24264 q^{37} +4.24264 q^{41} +3.24264 q^{43} +6.00000 q^{47} -8.48528 q^{53} +4.24264 q^{55} +10.2426 q^{59} +4.48528 q^{61} +3.24264 q^{65} -5.24264 q^{67} +12.7279 q^{71} +9.24264 q^{73} +11.0000 q^{79} +10.2426 q^{83} -4.24264 q^{85} +10.2426 q^{89} -7.00000 q^{95} -0.485281 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} + O(q^{10}) \) \( 2 q + 2 q^{5} - 2 q^{13} - 14 q^{19} + 2 q^{25} + 12 q^{29} - 2 q^{31} - 2 q^{37} - 2 q^{43} + 12 q^{47} + 12 q^{59} - 8 q^{61} - 2 q^{65} - 2 q^{67} + 10 q^{73} + 22 q^{79} + 12 q^{83} + 12 q^{89} - 14 q^{95} + 16 q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.24264 1.27920 0.639602 0.768706i \(-0.279099\pi\)
0.639602 + 0.768706i \(0.279099\pi\)
\(12\) 0 0
\(13\) 3.24264 0.899347 0.449673 0.893193i \(-0.351540\pi\)
0.449673 + 0.893193i \(0.351540\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.24264 −1.02899 −0.514496 0.857493i \(-0.672021\pi\)
−0.514496 + 0.857493i \(0.672021\pi\)
\(18\) 0 0
\(19\) −7.00000 −1.60591 −0.802955 0.596040i \(-0.796740\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.24264 0.884652 0.442326 0.896854i \(-0.354153\pi\)
0.442326 + 0.896854i \(0.354153\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.75736 0.326333 0.163167 0.986599i \(-0.447829\pi\)
0.163167 + 0.986599i \(0.447829\pi\)
\(30\) 0 0
\(31\) −9.48528 −1.70361 −0.851803 0.523862i \(-0.824491\pi\)
−0.851803 + 0.523862i \(0.824491\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 3.24264 0.533087 0.266543 0.963823i \(-0.414118\pi\)
0.266543 + 0.963823i \(0.414118\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.24264 0.662589 0.331295 0.943527i \(-0.392515\pi\)
0.331295 + 0.943527i \(0.392515\pi\)
\(42\) 0 0
\(43\) 3.24264 0.494498 0.247249 0.968952i \(-0.420473\pi\)
0.247249 + 0.968952i \(0.420473\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −8.48528 −1.16554 −0.582772 0.812636i \(-0.698032\pi\)
−0.582772 + 0.812636i \(0.698032\pi\)
\(54\) 0 0
\(55\) 4.24264 0.572078
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 10.2426 1.33348 0.666739 0.745291i \(-0.267690\pi\)
0.666739 + 0.745291i \(0.267690\pi\)
\(60\) 0 0
\(61\) 4.48528 0.574281 0.287141 0.957888i \(-0.407295\pi\)
0.287141 + 0.957888i \(0.407295\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.24264 0.402200
\(66\) 0 0
\(67\) −5.24264 −0.640490 −0.320245 0.947335i \(-0.603765\pi\)
−0.320245 + 0.947335i \(0.603765\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 12.7279 1.51053 0.755263 0.655422i \(-0.227509\pi\)
0.755263 + 0.655422i \(0.227509\pi\)
\(72\) 0 0
\(73\) 9.24264 1.08177 0.540885 0.841097i \(-0.318090\pi\)
0.540885 + 0.841097i \(0.318090\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 11.0000 1.23760 0.618798 0.785550i \(-0.287620\pi\)
0.618798 + 0.785550i \(0.287620\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 10.2426 1.12428 0.562138 0.827043i \(-0.309979\pi\)
0.562138 + 0.827043i \(0.309979\pi\)
\(84\) 0 0
\(85\) −4.24264 −0.460179
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 10.2426 1.08572 0.542859 0.839824i \(-0.317342\pi\)
0.542859 + 0.839824i \(0.317342\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −7.00000 −0.718185
\(96\) 0 0
\(97\) −0.485281 −0.0492729 −0.0246364 0.999696i \(-0.507843\pi\)
−0.0246364 + 0.999696i \(0.507843\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 7.75736 0.771886 0.385943 0.922523i \(-0.373876\pi\)
0.385943 + 0.922523i \(0.373876\pi\)
\(102\) 0 0
\(103\) −17.2426 −1.69897 −0.849484 0.527614i \(-0.823087\pi\)
−0.849484 + 0.527614i \(0.823087\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −12.7279 −1.23045 −0.615227 0.788350i \(-0.710936\pi\)
−0.615227 + 0.788350i \(0.710936\pi\)
\(108\) 0 0
\(109\) −9.48528 −0.908525 −0.454263 0.890868i \(-0.650097\pi\)
−0.454263 + 0.890868i \(0.650097\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 18.0000 1.69330 0.846649 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) 0 0
\(115\) 4.24264 0.395628
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 7.00000 0.636364
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −2.75736 −0.244676 −0.122338 0.992488i \(-0.539039\pi\)
−0.122338 + 0.992488i \(0.539039\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −14.4853 −1.26558 −0.632792 0.774321i \(-0.718091\pi\)
−0.632792 + 0.774321i \(0.718091\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −4.24264 −0.362473 −0.181237 0.983440i \(-0.558010\pi\)
−0.181237 + 0.983440i \(0.558010\pi\)
\(138\) 0 0
\(139\) −15.4853 −1.31344 −0.656722 0.754133i \(-0.728058\pi\)
−0.656722 + 0.754133i \(0.728058\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 13.7574 1.15045
\(144\) 0 0
\(145\) 1.75736 0.145941
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 12.0000 0.983078 0.491539 0.870855i \(-0.336434\pi\)
0.491539 + 0.870855i \(0.336434\pi\)
\(150\) 0 0
\(151\) 22.4853 1.82983 0.914913 0.403651i \(-0.132259\pi\)
0.914913 + 0.403651i \(0.132259\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −9.48528 −0.761876
\(156\) 0 0
\(157\) −6.48528 −0.517582 −0.258791 0.965933i \(-0.583324\pi\)
−0.258791 + 0.965933i \(0.583324\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 8.00000 0.626608 0.313304 0.949653i \(-0.398564\pi\)
0.313304 + 0.949653i \(0.398564\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −18.7279 −1.44921 −0.724605 0.689164i \(-0.757978\pi\)
−0.724605 + 0.689164i \(0.757978\pi\)
\(168\) 0 0
\(169\) −2.48528 −0.191175
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 20.4853 1.55747 0.778734 0.627355i \(-0.215862\pi\)
0.778734 + 0.627355i \(0.215862\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 6.00000 0.448461 0.224231 0.974536i \(-0.428013\pi\)
0.224231 + 0.974536i \(0.428013\pi\)
\(180\) 0 0
\(181\) −13.0000 −0.966282 −0.483141 0.875542i \(-0.660504\pi\)
−0.483141 + 0.875542i \(0.660504\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 3.24264 0.238404
\(186\) 0 0
\(187\) −18.0000 −1.31629
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −6.00000 −0.434145 −0.217072 0.976156i \(-0.569651\pi\)
−0.217072 + 0.976156i \(0.569651\pi\)
\(192\) 0 0
\(193\) 6.75736 0.486405 0.243203 0.969975i \(-0.421802\pi\)
0.243203 + 0.969975i \(0.421802\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 16.2426 1.15724 0.578620 0.815597i \(-0.303591\pi\)
0.578620 + 0.815597i \(0.303591\pi\)
\(198\) 0 0
\(199\) 10.4853 0.743282 0.371641 0.928377i \(-0.378795\pi\)
0.371641 + 0.928377i \(0.378795\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 4.24264 0.296319
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −29.6985 −2.05429
\(210\) 0 0
\(211\) 22.4853 1.54795 0.773975 0.633216i \(-0.218265\pi\)
0.773975 + 0.633216i \(0.218265\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 3.24264 0.221146
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −13.7574 −0.925420
\(222\) 0 0
\(223\) −7.51472 −0.503223 −0.251611 0.967828i \(-0.580960\pi\)
−0.251611 + 0.967828i \(0.580960\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 15.2132 1.00974 0.504868 0.863197i \(-0.331541\pi\)
0.504868 + 0.863197i \(0.331541\pi\)
\(228\) 0 0
\(229\) −7.00000 −0.462573 −0.231287 0.972886i \(-0.574293\pi\)
−0.231287 + 0.972886i \(0.574293\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −14.4853 −0.948962 −0.474481 0.880266i \(-0.657364\pi\)
−0.474481 + 0.880266i \(0.657364\pi\)
\(234\) 0 0
\(235\) 6.00000 0.391397
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −10.9706 −0.709627 −0.354813 0.934937i \(-0.615456\pi\)
−0.354813 + 0.934937i \(0.615456\pi\)
\(240\) 0 0
\(241\) −4.00000 −0.257663 −0.128831 0.991667i \(-0.541123\pi\)
−0.128831 + 0.991667i \(0.541123\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −22.6985 −1.44427
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 18.7279 1.18210 0.591048 0.806636i \(-0.298714\pi\)
0.591048 + 0.806636i \(0.298714\pi\)
\(252\) 0 0
\(253\) 18.0000 1.13165
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −10.2426 −0.638918 −0.319459 0.947600i \(-0.603501\pi\)
−0.319459 + 0.947600i \(0.603501\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 8.48528 0.523225 0.261612 0.965173i \(-0.415746\pi\)
0.261612 + 0.965173i \(0.415746\pi\)
\(264\) 0 0
\(265\) −8.48528 −0.521247
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 2.48528 0.151530 0.0757651 0.997126i \(-0.475860\pi\)
0.0757651 + 0.997126i \(0.475860\pi\)
\(270\) 0 0
\(271\) −23.4558 −1.42484 −0.712421 0.701753i \(-0.752401\pi\)
−0.712421 + 0.701753i \(0.752401\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 4.24264 0.255841
\(276\) 0 0
\(277\) 17.7279 1.06517 0.532584 0.846377i \(-0.321221\pi\)
0.532584 + 0.846377i \(0.321221\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −28.9706 −1.72824 −0.864119 0.503287i \(-0.832124\pi\)
−0.864119 + 0.503287i \(0.832124\pi\)
\(282\) 0 0
\(283\) 23.7279 1.41048 0.705239 0.708969i \(-0.250840\pi\)
0.705239 + 0.708969i \(0.250840\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 4.97056 0.290383 0.145192 0.989404i \(-0.453620\pi\)
0.145192 + 0.989404i \(0.453620\pi\)
\(294\) 0 0
\(295\) 10.2426 0.596350
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 13.7574 0.795609
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 4.48528 0.256826
\(306\) 0 0
\(307\) 3.24264 0.185067 0.0925336 0.995710i \(-0.470503\pi\)
0.0925336 + 0.995710i \(0.470503\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −21.2132 −1.20289 −0.601445 0.798914i \(-0.705408\pi\)
−0.601445 + 0.798914i \(0.705408\pi\)
\(312\) 0 0
\(313\) 23.7279 1.34118 0.670591 0.741828i \(-0.266041\pi\)
0.670591 + 0.741828i \(0.266041\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 24.7279 1.38886 0.694429 0.719561i \(-0.255657\pi\)
0.694429 + 0.719561i \(0.255657\pi\)
\(318\) 0 0
\(319\) 7.45584 0.417447
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 29.6985 1.65247
\(324\) 0 0
\(325\) 3.24264 0.179869
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 17.0000 0.934405 0.467202 0.884150i \(-0.345262\pi\)
0.467202 + 0.884150i \(0.345262\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −5.24264 −0.286436
\(336\) 0 0
\(337\) −13.7279 −0.747808 −0.373904 0.927467i \(-0.621981\pi\)
−0.373904 + 0.927467i \(0.621981\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −40.2426 −2.17926
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 24.0000 1.28839 0.644194 0.764862i \(-0.277193\pi\)
0.644194 + 0.764862i \(0.277193\pi\)
\(348\) 0 0
\(349\) −10.0000 −0.535288 −0.267644 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 10.2426 0.545161 0.272580 0.962133i \(-0.412123\pi\)
0.272580 + 0.962133i \(0.412123\pi\)
\(354\) 0 0
\(355\) 12.7279 0.675528
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −1.75736 −0.0927499 −0.0463749 0.998924i \(-0.514767\pi\)
−0.0463749 + 0.998924i \(0.514767\pi\)
\(360\) 0 0
\(361\) 30.0000 1.57895
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 9.24264 0.483782
\(366\) 0 0
\(367\) 35.7279 1.86498 0.932491 0.361193i \(-0.117630\pi\)
0.932491 + 0.361193i \(0.117630\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 17.7279 0.917917 0.458959 0.888458i \(-0.348223\pi\)
0.458959 + 0.888458i \(0.348223\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 5.69848 0.293487
\(378\) 0 0
\(379\) −20.4558 −1.05075 −0.525373 0.850872i \(-0.676074\pi\)
−0.525373 + 0.850872i \(0.676074\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −25.4558 −1.30073 −0.650366 0.759621i \(-0.725385\pi\)
−0.650366 + 0.759621i \(0.725385\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 21.2132 1.07555 0.537776 0.843088i \(-0.319265\pi\)
0.537776 + 0.843088i \(0.319265\pi\)
\(390\) 0 0
\(391\) −18.0000 −0.910299
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 11.0000 0.553470
\(396\) 0 0
\(397\) −8.75736 −0.439519 −0.219760 0.975554i \(-0.570527\pi\)
−0.219760 + 0.975554i \(0.570527\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 3.51472 0.175517 0.0877583 0.996142i \(-0.472030\pi\)
0.0877583 + 0.996142i \(0.472030\pi\)
\(402\) 0 0
\(403\) −30.7574 −1.53213
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 13.7574 0.681927
\(408\) 0 0
\(409\) 17.0000 0.840596 0.420298 0.907386i \(-0.361926\pi\)
0.420298 + 0.907386i \(0.361926\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 10.2426 0.502791
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 14.4853 0.707652 0.353826 0.935311i \(-0.384880\pi\)
0.353826 + 0.935311i \(0.384880\pi\)
\(420\) 0 0
\(421\) 31.4853 1.53450 0.767249 0.641349i \(-0.221625\pi\)
0.767249 + 0.641349i \(0.221625\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −4.24264 −0.205798
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −19.4558 −0.937155 −0.468578 0.883422i \(-0.655233\pi\)
−0.468578 + 0.883422i \(0.655233\pi\)
\(432\) 0 0
\(433\) 33.2426 1.59754 0.798770 0.601637i \(-0.205485\pi\)
0.798770 + 0.601637i \(0.205485\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −29.6985 −1.42067
\(438\) 0 0
\(439\) −10.0000 −0.477274 −0.238637 0.971109i \(-0.576701\pi\)
−0.238637 + 0.971109i \(0.576701\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −20.4853 −0.973285 −0.486643 0.873601i \(-0.661779\pi\)
−0.486643 + 0.873601i \(0.661779\pi\)
\(444\) 0 0
\(445\) 10.2426 0.485548
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 0 0
\(451\) 18.0000 0.847587
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 32.2132 1.50687 0.753435 0.657522i \(-0.228395\pi\)
0.753435 + 0.657522i \(0.228395\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −18.7279 −0.872246 −0.436123 0.899887i \(-0.643649\pi\)
−0.436123 + 0.899887i \(0.643649\pi\)
\(462\) 0 0
\(463\) 10.2721 0.477384 0.238692 0.971095i \(-0.423281\pi\)
0.238692 + 0.971095i \(0.423281\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 10.9706 0.507657 0.253829 0.967249i \(-0.418310\pi\)
0.253829 + 0.967249i \(0.418310\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 13.7574 0.632564
\(474\) 0 0
\(475\) −7.00000 −0.321182
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −12.0000 −0.548294 −0.274147 0.961688i \(-0.588395\pi\)
−0.274147 + 0.961688i \(0.588395\pi\)
\(480\) 0 0
\(481\) 10.5147 0.479430
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −0.485281 −0.0220355
\(486\) 0 0
\(487\) −37.7279 −1.70962 −0.854808 0.518945i \(-0.826325\pi\)
−0.854808 + 0.518945i \(0.826325\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −15.5147 −0.700169 −0.350085 0.936718i \(-0.613847\pi\)
−0.350085 + 0.936718i \(0.613847\pi\)
\(492\) 0 0
\(493\) −7.45584 −0.335794
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 19.4853 0.872281 0.436140 0.899879i \(-0.356345\pi\)
0.436140 + 0.899879i \(0.356345\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 26.4853 1.18092 0.590460 0.807067i \(-0.298946\pi\)
0.590460 + 0.807067i \(0.298946\pi\)
\(504\) 0 0
\(505\) 7.75736 0.345198
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 19.4558 0.862365 0.431183 0.902265i \(-0.358096\pi\)
0.431183 + 0.902265i \(0.358096\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −17.2426 −0.759802
\(516\) 0 0
\(517\) 25.4558 1.11955
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 13.4558 0.589511 0.294756 0.955573i \(-0.404762\pi\)
0.294756 + 0.955573i \(0.404762\pi\)
\(522\) 0 0
\(523\) −5.24264 −0.229245 −0.114622 0.993409i \(-0.536566\pi\)
−0.114622 + 0.993409i \(0.536566\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 40.2426 1.75300
\(528\) 0 0
\(529\) −5.00000 −0.217391
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 13.7574 0.595897
\(534\) 0 0
\(535\) −12.7279 −0.550276
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −40.9411 −1.76020 −0.880098 0.474792i \(-0.842523\pi\)
−0.880098 + 0.474792i \(0.842523\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −9.48528 −0.406305
\(546\) 0 0
\(547\) 33.4558 1.43047 0.715234 0.698885i \(-0.246320\pi\)
0.715234 + 0.698885i \(0.246320\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −12.3015 −0.524062
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 30.0000 1.27114 0.635570 0.772043i \(-0.280765\pi\)
0.635570 + 0.772043i \(0.280765\pi\)
\(558\) 0 0
\(559\) 10.5147 0.444725
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 6.00000 0.252870 0.126435 0.991975i \(-0.459647\pi\)
0.126435 + 0.991975i \(0.459647\pi\)
\(564\) 0 0
\(565\) 18.0000 0.757266
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 41.6985 1.74809 0.874046 0.485844i \(-0.161488\pi\)
0.874046 + 0.485844i \(0.161488\pi\)
\(570\) 0 0
\(571\) −28.9411 −1.21115 −0.605574 0.795789i \(-0.707057\pi\)
−0.605574 + 0.795789i \(0.707057\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 4.24264 0.176930
\(576\) 0 0
\(577\) 12.7574 0.531096 0.265548 0.964098i \(-0.414447\pi\)
0.265548 + 0.964098i \(0.414447\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −36.0000 −1.49097
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −45.2132 −1.86615 −0.933074 0.359684i \(-0.882885\pi\)
−0.933074 + 0.359684i \(0.882885\pi\)
\(588\) 0 0
\(589\) 66.3970 2.73584
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −3.21320 −0.131950 −0.0659752 0.997821i \(-0.521016\pi\)
−0.0659752 + 0.997821i \(0.521016\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 32.4853 1.32731 0.663656 0.748038i \(-0.269004\pi\)
0.663656 + 0.748038i \(0.269004\pi\)
\(600\) 0 0
\(601\) −3.48528 −0.142168 −0.0710838 0.997470i \(-0.522646\pi\)
−0.0710838 + 0.997470i \(0.522646\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 7.00000 0.284590
\(606\) 0 0
\(607\) −29.2426 −1.18692 −0.593461 0.804863i \(-0.702239\pi\)
−0.593461 + 0.804863i \(0.702239\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 19.4558 0.787099
\(612\) 0 0
\(613\) −5.45584 −0.220359 −0.110180 0.993912i \(-0.535143\pi\)
−0.110180 + 0.993912i \(0.535143\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 26.4853 1.06626 0.533129 0.846034i \(-0.321016\pi\)
0.533129 + 0.846034i \(0.321016\pi\)
\(618\) 0 0
\(619\) −11.9706 −0.481138 −0.240569 0.970632i \(-0.577334\pi\)
−0.240569 + 0.970632i \(0.577334\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −13.7574 −0.548542
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −2.75736 −0.109422
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −34.2426 −1.35250 −0.676251 0.736671i \(-0.736397\pi\)
−0.676251 + 0.736671i \(0.736397\pi\)
\(642\) 0 0
\(643\) −19.7279 −0.777993 −0.388997 0.921239i \(-0.627178\pi\)
−0.388997 + 0.921239i \(0.627178\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −10.2426 −0.402680 −0.201340 0.979521i \(-0.564530\pi\)
−0.201340 + 0.979521i \(0.564530\pi\)
\(648\) 0 0
\(649\) 43.4558 1.70579
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −10.2426 −0.400826 −0.200413 0.979712i \(-0.564228\pi\)
−0.200413 + 0.979712i \(0.564228\pi\)
\(654\) 0 0
\(655\) −14.4853 −0.565987
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −40.9706 −1.59599 −0.797993 0.602666i \(-0.794105\pi\)
−0.797993 + 0.602666i \(0.794105\pi\)
\(660\) 0 0
\(661\) −2.02944 −0.0789360 −0.0394680 0.999221i \(-0.512566\pi\)
−0.0394680 + 0.999221i \(0.512566\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 7.45584 0.288691
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 19.0294 0.734623
\(672\) 0 0
\(673\) 29.7279 1.14593 0.572964 0.819581i \(-0.305794\pi\)
0.572964 + 0.819581i \(0.305794\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −12.7279 −0.489174 −0.244587 0.969627i \(-0.578652\pi\)
−0.244587 + 0.969627i \(0.578652\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −33.2132 −1.27087 −0.635434 0.772155i \(-0.719179\pi\)
−0.635434 + 0.772155i \(0.719179\pi\)
\(684\) 0 0
\(685\) −4.24264 −0.162103
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −27.5147 −1.04823
\(690\) 0 0
\(691\) 26.9411 1.02489 0.512444 0.858720i \(-0.328740\pi\)
0.512444 + 0.858720i \(0.328740\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −15.4853 −0.587390
\(696\) 0 0
\(697\) −18.0000 −0.681799
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −8.78680 −0.331873 −0.165936 0.986136i \(-0.553065\pi\)
−0.165936 + 0.986136i \(0.553065\pi\)
\(702\) 0 0
\(703\) −22.6985 −0.856090
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −36.4853 −1.37023 −0.685117 0.728433i \(-0.740249\pi\)
−0.685117 + 0.728433i \(0.740249\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −40.2426 −1.50710
\(714\) 0 0
\(715\) 13.7574 0.514496
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −26.4853 −0.987734 −0.493867 0.869537i \(-0.664417\pi\)
−0.493867 + 0.869537i \(0.664417\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 1.75736 0.0652667
\(726\) 0 0
\(727\) 0.757359 0.0280889 0.0140445 0.999901i \(-0.495529\pi\)
0.0140445 + 0.999901i \(0.495529\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −13.7574 −0.508834
\(732\) 0 0
\(733\) 1.78680 0.0659968 0.0329984 0.999455i \(-0.489494\pi\)
0.0329984 + 0.999455i \(0.489494\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −22.2426 −0.819318
\(738\) 0 0
\(739\) −15.4853 −0.569635 −0.284818 0.958582i \(-0.591933\pi\)
−0.284818 + 0.958582i \(0.591933\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −15.2132 −0.558118 −0.279059 0.960274i \(-0.590023\pi\)
−0.279059 + 0.960274i \(0.590023\pi\)
\(744\) 0 0
\(745\) 12.0000 0.439646
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 44.9411 1.63992 0.819962 0.572417i \(-0.193994\pi\)
0.819962 + 0.572417i \(0.193994\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 22.4853 0.818323
\(756\) 0 0
\(757\) 9.02944 0.328180 0.164090 0.986445i \(-0.447531\pi\)
0.164090 + 0.986445i \(0.447531\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −46.2426 −1.67629 −0.838147 0.545444i \(-0.816361\pi\)
−0.838147 + 0.545444i \(0.816361\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 33.2132 1.19926
\(768\) 0 0
\(769\) 5.00000 0.180305 0.0901523 0.995928i \(-0.471265\pi\)
0.0901523 + 0.995928i \(0.471265\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −11.6985 −0.420765 −0.210383 0.977619i \(-0.567471\pi\)
−0.210383 + 0.977619i \(0.567471\pi\)
\(774\) 0 0
\(775\) −9.48528 −0.340721
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −29.6985 −1.06406
\(780\) 0 0
\(781\) 54.0000 1.93227
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −6.48528 −0.231470
\(786\) 0 0
\(787\) 28.4853 1.01539 0.507695 0.861537i \(-0.330498\pi\)
0.507695 + 0.861537i \(0.330498\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 14.5442 0.516478
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 31.7574 1.12490 0.562452 0.826830i \(-0.309858\pi\)
0.562452 + 0.826830i \(0.309858\pi\)
\(798\) 0 0
\(799\) −25.4558 −0.900563
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 39.2132 1.38380
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −27.9411 −0.982358 −0.491179 0.871059i \(-0.663434\pi\)
−0.491179 + 0.871059i \(0.663434\pi\)
\(810\) 0 0
\(811\) 11.9411 0.419310 0.209655 0.977775i \(-0.432766\pi\)
0.209655 + 0.977775i \(0.432766\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 8.00000 0.280228
\(816\) 0 0
\(817\) −22.6985 −0.794119
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −5.69848 −0.198878 −0.0994392 0.995044i \(-0.531705\pi\)
−0.0994392 + 0.995044i \(0.531705\pi\)
\(822\) 0 0
\(823\) −38.9706 −1.35843 −0.679214 0.733940i \(-0.737679\pi\)
−0.679214 + 0.733940i \(0.737679\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −43.4558 −1.51111 −0.755554 0.655087i \(-0.772632\pi\)
−0.755554 + 0.655087i \(0.772632\pi\)
\(828\) 0 0
\(829\) 11.0000 0.382046 0.191023 0.981586i \(-0.438820\pi\)
0.191023 + 0.981586i \(0.438820\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −18.7279 −0.648106
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −25.7574 −0.889243 −0.444621 0.895719i \(-0.646662\pi\)
−0.444621 + 0.895719i \(0.646662\pi\)
\(840\) 0 0
\(841\) −25.9117 −0.893506
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −2.48528 −0.0854963
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 13.7574 0.471596
\(852\) 0 0
\(853\) −25.7279 −0.880907 −0.440454 0.897775i \(-0.645182\pi\)
−0.440454 + 0.897775i \(0.645182\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −4.24264 −0.144926 −0.0724629 0.997371i \(-0.523086\pi\)
−0.0724629 + 0.997371i \(0.523086\pi\)
\(858\) 0 0
\(859\) −22.0000 −0.750630 −0.375315 0.926897i \(-0.622466\pi\)
−0.375315 + 0.926897i \(0.622466\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 21.5147 0.732370 0.366185 0.930542i \(-0.380664\pi\)
0.366185 + 0.930542i \(0.380664\pi\)
\(864\) 0 0
\(865\) 20.4853 0.696520
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 46.6690 1.58314
\(870\) 0 0
\(871\) −17.0000 −0.576023
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −10.0000 −0.337676 −0.168838 0.985644i \(-0.554001\pi\)
−0.168838 + 0.985644i \(0.554001\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −40.9706 −1.38033 −0.690167 0.723650i \(-0.742463\pi\)
−0.690167 + 0.723650i \(0.742463\pi\)
\(882\) 0 0
\(883\) 5.72792 0.192760 0.0963800 0.995345i \(-0.469274\pi\)
0.0963800 + 0.995345i \(0.469274\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −15.2132 −0.510809 −0.255405 0.966834i \(-0.582209\pi\)
−0.255405 + 0.966834i \(0.582209\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −42.0000 −1.40548
\(894\) 0 0
\(895\) 6.00000 0.200558
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −16.6690 −0.555944
\(900\) 0 0
\(901\) 36.0000 1.19933
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −13.0000 −0.432135
\(906\) 0 0
\(907\) −30.2721 −1.00517 −0.502584 0.864528i \(-0.667617\pi\)
−0.502584 + 0.864528i \(0.667617\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −51.2132 −1.69677 −0.848385 0.529380i \(-0.822424\pi\)
−0.848385 + 0.529380i \(0.822424\pi\)
\(912\) 0 0
\(913\) 43.4558 1.43818
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 33.9706 1.12059 0.560293 0.828295i \(-0.310689\pi\)
0.560293 + 0.828295i \(0.310689\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 41.2721 1.35849
\(924\) 0 0
\(925\) 3.24264 0.106617
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −36.7279 −1.20500 −0.602502 0.798117i \(-0.705829\pi\)
−0.602502 + 0.798117i \(0.705829\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −18.0000 −0.588663
\(936\) 0 0
\(937\) −18.6985 −0.610853 −0.305426 0.952216i \(-0.598799\pi\)
−0.305426 + 0.952216i \(0.598799\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 47.6985 1.55493 0.777463 0.628929i \(-0.216506\pi\)
0.777463 + 0.628929i \(0.216506\pi\)
\(942\) 0 0
\(943\) 18.0000 0.586161
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 3.21320 0.104415 0.0522075 0.998636i \(-0.483374\pi\)
0.0522075 + 0.998636i \(0.483374\pi\)
\(948\) 0 0
\(949\) 29.9706 0.972886
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 27.5147 0.891289 0.445645 0.895210i \(-0.352975\pi\)
0.445645 + 0.895210i \(0.352975\pi\)
\(954\) 0 0
\(955\) −6.00000 −0.194155
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 58.9706 1.90228
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 6.75736 0.217527
\(966\) 0 0
\(967\) −35.2426 −1.13333 −0.566663 0.823949i \(-0.691766\pi\)
−0.566663 + 0.823949i \(0.691766\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −52.9706 −1.69991 −0.849953 0.526858i \(-0.823370\pi\)
−0.849953 + 0.526858i \(0.823370\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 8.78680 0.281115 0.140557 0.990073i \(-0.455111\pi\)
0.140557 + 0.990073i \(0.455111\pi\)
\(978\) 0 0
\(979\) 43.4558 1.38885
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 41.6985 1.32998 0.664988 0.746854i \(-0.268437\pi\)
0.664988 + 0.746854i \(0.268437\pi\)
\(984\) 0 0
\(985\) 16.2426 0.517534
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 13.7574 0.437459
\(990\) 0 0
\(991\) 14.9411 0.474620 0.237310 0.971434i \(-0.423734\pi\)
0.237310 + 0.971434i \(0.423734\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 10.4853 0.332406
\(996\) 0 0
\(997\) −25.7279 −0.814811 −0.407406 0.913247i \(-0.633567\pi\)
−0.407406 + 0.913247i \(0.633567\pi\)
\(998\) 0 0
\(999\) 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 8820.2.a.bk.1.2 2
3.2 odd 2 2940.2.a.r.1.1 2
7.2 even 3 1260.2.s.e.361.2 4
7.4 even 3 1260.2.s.e.541.2 4
7.6 odd 2 8820.2.a.bf.1.2 2
21.2 odd 6 420.2.q.d.361.2 yes 4
21.5 even 6 2940.2.q.q.361.2 4
21.11 odd 6 420.2.q.d.121.2 4
21.17 even 6 2940.2.q.q.961.2 4
21.20 even 2 2940.2.a.p.1.1 2
84.11 even 6 1680.2.bg.t.961.1 4
84.23 even 6 1680.2.bg.t.1201.1 4
105.2 even 12 2100.2.bc.f.949.4 8
105.23 even 12 2100.2.bc.f.949.1 8
105.32 even 12 2100.2.bc.f.1549.1 8
105.44 odd 6 2100.2.q.k.1201.1 4
105.53 even 12 2100.2.bc.f.1549.4 8
105.74 odd 6 2100.2.q.k.1801.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
420.2.q.d.121.2 4 21.11 odd 6
420.2.q.d.361.2 yes 4 21.2 odd 6
1260.2.s.e.361.2 4 7.2 even 3
1260.2.s.e.541.2 4 7.4 even 3
1680.2.bg.t.961.1 4 84.11 even 6
1680.2.bg.t.1201.1 4 84.23 even 6
2100.2.q.k.1201.1 4 105.44 odd 6
2100.2.q.k.1801.1 4 105.74 odd 6
2100.2.bc.f.949.1 8 105.23 even 12
2100.2.bc.f.949.4 8 105.2 even 12
2100.2.bc.f.1549.1 8 105.32 even 12
2100.2.bc.f.1549.4 8 105.53 even 12
2940.2.a.p.1.1 2 21.20 even 2
2940.2.a.r.1.1 2 3.2 odd 2
2940.2.q.q.361.2 4 21.5 even 6
2940.2.q.q.961.2 4 21.17 even 6
8820.2.a.bf.1.2 2 7.6 odd 2
8820.2.a.bk.1.2 2 1.1 even 1 trivial