Properties

Label 1456.2.a.g.1.1
Level $1456$
Weight $2$
Character 1456.1
Self dual yes
Analytic conductor $11.626$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1456.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.6262185343\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 1456.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000 q^{5} +1.00000 q^{7} -3.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{5} +1.00000 q^{7} -3.00000 q^{9} +6.00000 q^{11} -1.00000 q^{13} +4.00000 q^{17} -5.00000 q^{19} -3.00000 q^{23} +4.00000 q^{25} -5.00000 q^{29} +3.00000 q^{31} -3.00000 q^{35} -4.00000 q^{37} -6.00000 q^{41} +1.00000 q^{43} +9.00000 q^{45} -7.00000 q^{47} +1.00000 q^{49} -9.00000 q^{53} -18.0000 q^{55} -8.00000 q^{59} -10.0000 q^{61} -3.00000 q^{63} +3.00000 q^{65} +6.00000 q^{67} +8.00000 q^{71} -13.0000 q^{73} +6.00000 q^{77} -3.00000 q^{79} +9.00000 q^{81} -15.0000 q^{83} -12.0000 q^{85} +3.00000 q^{89} -1.00000 q^{91} +15.0000 q^{95} +7.00000 q^{97} -18.0000 q^{99} +O(q^{100})\)

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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 0 0
\(5\) −3.00000 −1.34164 −0.670820 0.741620i \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) 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
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) −5.00000 −1.14708 −0.573539 0.819178i \(-0.694430\pi\)
−0.573539 + 0.819178i \(0.694430\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −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
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −5.00000 −0.928477 −0.464238 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) 0 0
\(31\) 3.00000 0.538816 0.269408 0.963026i \(-0.413172\pi\)
0.269408 + 0.963026i \(0.413172\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.00000 −0.507093
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 1.00000 0.152499 0.0762493 0.997089i \(-0.475706\pi\)
0.0762493 + 0.997089i \(0.475706\pi\)
\(44\) 0 0
\(45\) 9.00000 1.34164
\(46\) 0 0
\(47\) −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
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −9.00000 −1.23625 −0.618123 0.786082i \(-0.712106\pi\)
−0.618123 + 0.786082i \(0.712106\pi\)
\(54\) 0 0
\(55\) −18.0000 −2.42712
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −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 \(0.721142\pi\)
\(62\) 0 0
\(63\) −3.00000 −0.377964
\(64\) 0 0
\(65\) 3.00000 0.372104
\(66\) 0 0
\(67\) 6.00000 0.733017 0.366508 0.930415i \(-0.380553\pi\)
0.366508 + 0.930415i \(0.380553\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 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 \(0.775177\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.00000 0.683763
\(78\) 0 0
\(79\) −3.00000 −0.337526 −0.168763 0.985657i \(-0.553977\pi\)
−0.168763 + 0.985657i \(0.553977\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) −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
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.00000 0.317999 0.159000 0.987279i \(-0.449173\pi\)
0.159000 + 0.987279i \(0.449173\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 15.0000 1.53897
\(96\) 0 0
\(97\) 7.00000 0.710742 0.355371 0.934725i \(-0.384354\pi\)
0.355371 + 0.934725i \(0.384354\pi\)
\(98\) 0 0
\(99\) −18.0000 −1.80907
\(100\) 0 0
\(101\) −14.0000 −1.39305 −0.696526 0.717532i \(-0.745272\pi\)
−0.696526 + 0.717532i \(0.745272\pi\)
\(102\) 0 0
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 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 \(0.530535\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −3.00000 −0.282216 −0.141108 0.989994i \(-0.545067\pi\)
−0.141108 + 0.989994i \(0.545067\pi\)
\(114\) 0 0
\(115\) 9.00000 0.839254
\(116\) 0 0
\(117\) 3.00000 0.277350
\(118\) 0 0
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 3.00000 0.268328
\(126\) 0 0
\(127\) 4.00000 0.354943 0.177471 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −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
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 4.00000 0.341743 0.170872 0.985293i \(-0.445342\pi\)
0.170872 + 0.985293i \(0.445342\pi\)
\(138\) 0 0
\(139\) 18.0000 1.52674 0.763370 0.645961i \(-0.223543\pi\)
0.763370 + 0.645961i \(0.223543\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −6.00000 −0.501745
\(144\) 0 0
\(145\) 15.0000 1.24568
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) −12.0000 −0.970143
\(154\) 0 0
\(155\) −9.00000 −0.722897
\(156\) 0 0
\(157\) −8.00000 −0.638470 −0.319235 0.947676i \(-0.603426\pi\)
−0.319235 + 0.947676i \(0.603426\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −3.00000 −0.236433
\(162\) 0 0
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −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
\(170\) 0 0
\(171\) 15.0000 1.14708
\(172\) 0 0
\(173\) −8.00000 −0.608229 −0.304114 0.952636i \(-0.598361\pi\)
−0.304114 + 0.952636i \(0.598361\pi\)
\(174\) 0 0
\(175\) 4.00000 0.302372
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −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 \(0.325818\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 12.0000 0.882258
\(186\) 0 0
\(187\) 24.0000 1.75505
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 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 \(0.209146\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) 0 0
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −5.00000 −0.350931
\(204\) 0 0
\(205\) 18.0000 1.25717
\(206\) 0 0
\(207\) 9.00000 0.625543
\(208\) 0 0
\(209\) −30.0000 −2.07514
\(210\) 0 0
\(211\) 5.00000 0.344214 0.172107 0.985078i \(-0.444942\pi\)
0.172107 + 0.985078i \(0.444942\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −3.00000 −0.204598
\(216\) 0 0
\(217\) 3.00000 0.203653
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −4.00000 −0.269069
\(222\) 0 0
\(223\) −15.0000 −1.00447 −0.502237 0.864730i \(-0.667490\pi\)
−0.502237 + 0.864730i \(0.667490\pi\)
\(224\) 0 0
\(225\) −12.0000 −0.800000
\(226\) 0 0
\(227\) −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 \(0.346926\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 15.0000 0.982683 0.491341 0.870967i \(-0.336507\pi\)
0.491341 + 0.870967i \(0.336507\pi\)
\(234\) 0 0
\(235\) 21.0000 1.36989
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 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 \(0.684433\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −3.00000 −0.191663
\(246\) 0 0
\(247\) 5.00000 0.318142
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 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
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −2.00000 −0.124757 −0.0623783 0.998053i \(-0.519869\pi\)
−0.0623783 + 0.998053i \(0.519869\pi\)
\(258\) 0 0
\(259\) −4.00000 −0.248548
\(260\) 0 0
\(261\) 15.0000 0.928477
\(262\) 0 0
\(263\) 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
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 24.0000 1.44725
\(276\) 0 0
\(277\) 1.00000 0.0600842 0.0300421 0.999549i \(-0.490436\pi\)
0.0300421 + 0.999549i \(0.490436\pi\)
\(278\) 0 0
\(279\) −9.00000 −0.538816
\(280\) 0 0
\(281\) −30.0000 −1.78965 −0.894825 0.446417i \(-0.852700\pi\)
−0.894825 + 0.446417i \(0.852700\pi\)
\(282\) 0 0
\(283\) −16.0000 −0.951101 −0.475551 0.879688i \(-0.657751\pi\)
−0.475551 + 0.879688i \(0.657751\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −6.00000 −0.354169
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −19.0000 −1.10999 −0.554996 0.831853i \(-0.687280\pi\)
−0.554996 + 0.831853i \(0.687280\pi\)
\(294\) 0 0
\(295\) 24.0000 1.39733
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 3.00000 0.173494
\(300\) 0 0
\(301\) 1.00000 0.0576390
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 30.0000 1.71780
\(306\) 0 0
\(307\) 33.0000 1.88341 0.941705 0.336440i \(-0.109223\pi\)
0.941705 + 0.336440i \(0.109223\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 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 \(0.286419\pi\)
\(314\) 0 0
\(315\) 9.00000 0.507093
\(316\) 0 0
\(317\) −24.0000 −1.34797 −0.673987 0.738743i \(-0.735420\pi\)
−0.673987 + 0.738743i \(0.735420\pi\)
\(318\) 0 0
\(319\) −30.0000 −1.67968
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −20.0000 −1.11283
\(324\) 0 0
\(325\) −4.00000 −0.221880
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −7.00000 −0.385922
\(330\) 0 0
\(331\) −22.0000 −1.20923 −0.604615 0.796518i \(-0.706673\pi\)
−0.604615 + 0.796518i \(0.706673\pi\)
\(332\) 0 0
\(333\) 12.0000 0.657596
\(334\) 0 0
\(335\) −18.0000 −0.983445
\(336\) 0 0
\(337\) 17.0000 0.926049 0.463025 0.886345i \(-0.346764\pi\)
0.463025 + 0.886345i \(0.346764\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 18.0000 0.974755
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 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 \(0.404877\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −10.0000 −0.532246 −0.266123 0.963939i \(-0.585743\pi\)
−0.266123 + 0.963939i \(0.585743\pi\)
\(354\) 0 0
\(355\) −24.0000 −1.27379
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −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
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 39.0000 2.04135
\(366\) 0 0
\(367\) −14.0000 −0.730794 −0.365397 0.930852i \(-0.619067\pi\)
−0.365397 + 0.930852i \(0.619067\pi\)
\(368\) 0 0
\(369\) 18.0000 0.937043
\(370\) 0 0
\(371\) −9.00000 −0.467257
\(372\) 0 0
\(373\) 30.0000 1.55334 0.776671 0.629907i \(-0.216907\pi\)
0.776671 + 0.629907i \(0.216907\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 5.00000 0.257513
\(378\) 0 0
\(379\) 6.00000 0.308199 0.154100 0.988055i \(-0.450752\pi\)
0.154100 + 0.988055i \(0.450752\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 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
\(386\) 0 0
\(387\) −3.00000 −0.152499
\(388\) 0 0
\(389\) 30.0000 1.52106 0.760530 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) 0 0
\(391\) −12.0000 −0.606866
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 9.00000 0.452839
\(396\) 0 0
\(397\) −13.0000 −0.652451 −0.326226 0.945292i \(-0.605777\pi\)
−0.326226 + 0.945292i \(0.605777\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −32.0000 −1.59800 −0.799002 0.601329i \(-0.794638\pi\)
−0.799002 + 0.601329i \(0.794638\pi\)
\(402\) 0 0
\(403\) −3.00000 −0.149441
\(404\) 0 0
\(405\) −27.0000 −1.34164
\(406\) 0 0
\(407\) −24.0000 −1.18964
\(408\) 0 0
\(409\) −13.0000 −0.642809 −0.321404 0.946942i \(-0.604155\pi\)
−0.321404 + 0.946942i \(0.604155\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −8.00000 −0.393654
\(414\) 0 0
\(415\) 45.0000 2.20896
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 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 \(0.594461\pi\)
\(422\) 0 0
\(423\) 21.0000 1.02105
\(424\) 0 0
\(425\) 16.0000 0.776114
\(426\) 0 0
\(427\) −10.0000 −0.483934
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −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 \(0.406896\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 15.0000 0.717547
\(438\) 0 0
\(439\) 22.0000 1.05000 0.525001 0.851101i \(-0.324065\pi\)
0.525001 + 0.851101i \(0.324065\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) −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
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 36.0000 1.69895 0.849473 0.527633i \(-0.176920\pi\)
0.849473 + 0.527633i \(0.176920\pi\)
\(450\) 0 0
\(451\) −36.0000 −1.69517
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 3.00000 0.140642
\(456\) 0 0
\(457\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −22.0000 −1.02464 −0.512321 0.858794i \(-0.671214\pi\)
−0.512321 + 0.858794i \(0.671214\pi\)
\(462\) 0 0
\(463\) 14.0000 0.650635 0.325318 0.945605i \(-0.394529\pi\)
0.325318 + 0.945605i \(0.394529\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 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
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 6.00000 0.275880
\(474\) 0 0
\(475\) −20.0000 −0.917663
\(476\) 0 0
\(477\) 27.0000 1.23625
\(478\) 0 0
\(479\) 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
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −21.0000 −0.953561
\(486\) 0 0
\(487\) 26.0000 1.17817 0.589086 0.808070i \(-0.299488\pi\)
0.589086 + 0.808070i \(0.299488\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 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
\(494\) 0 0
\(495\) 54.0000 2.42712
\(496\) 0 0
\(497\) 8.00000 0.358849
\(498\) 0 0
\(499\) 16.0000 0.716258 0.358129 0.933672i \(-0.383415\pi\)
0.358129 + 0.933672i \(0.383415\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −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
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −19.0000 −0.842160 −0.421080 0.907023i \(-0.638349\pi\)
−0.421080 + 0.907023i \(0.638349\pi\)
\(510\) 0 0
\(511\) −13.0000 −0.575086
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −12.0000 −0.528783
\(516\) 0 0
\(517\) −42.0000 −1.84716
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 40.0000 1.75243 0.876216 0.481919i \(-0.160060\pi\)
0.876216 + 0.481919i \(0.160060\pi\)
\(522\) 0 0
\(523\) −10.0000 −0.437269 −0.218635 0.975807i \(-0.570160\pi\)
−0.218635 + 0.975807i \(0.570160\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 12.0000 0.522728
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) 24.0000 1.04151
\(532\) 0 0
\(533\) 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 \(0.829454\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 6.00000 0.257012
\(546\) 0 0
\(547\) 7.00000 0.299298 0.149649 0.988739i \(-0.452186\pi\)
0.149649 + 0.988739i \(0.452186\pi\)
\(548\) 0 0
\(549\) 30.0000 1.28037
\(550\) 0 0
\(551\) 25.0000 1.06504
\(552\) 0 0
\(553\) −3.00000 −0.127573
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −12.0000 −0.508456 −0.254228 0.967144i \(-0.581821\pi\)
−0.254228 + 0.967144i \(0.581821\pi\)
\(558\) 0 0
\(559\) −1.00000 −0.0422955
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −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
\(566\) 0 0
\(567\) 9.00000 0.377964
\(568\) 0 0
\(569\) 7.00000 0.293455 0.146728 0.989177i \(-0.453126\pi\)
0.146728 + 0.989177i \(0.453126\pi\)
\(570\) 0 0
\(571\) 17.0000 0.711428 0.355714 0.934595i \(-0.384238\pi\)
0.355714 + 0.934595i \(0.384238\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −12.0000 −0.500435
\(576\) 0 0
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −15.0000 −0.622305
\(582\) 0 0
\(583\) −54.0000 −2.23645
\(584\) 0 0
\(585\) −9.00000 −0.372104
\(586\) 0 0
\(587\) −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
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −27.0000 −1.10876 −0.554379 0.832265i \(-0.687044\pi\)
−0.554379 + 0.832265i \(0.687044\pi\)
\(594\) 0 0
\(595\) −12.0000 −0.491952
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −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 \(0.565379\pi\)
\(602\) 0 0
\(603\) −18.0000 −0.733017
\(604\) 0 0
\(605\) −75.0000 −3.04918
\(606\) 0 0
\(607\) −2.00000 −0.0811775 −0.0405887 0.999176i \(-0.512923\pi\)
−0.0405887 + 0.999176i \(0.512923\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 7.00000 0.283190
\(612\) 0 0
\(613\) 8.00000 0.323117 0.161558 0.986863i \(-0.448348\pi\)
0.161558 + 0.986863i \(0.448348\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −30.0000 −1.20775 −0.603877 0.797077i \(-0.706378\pi\)
−0.603877 + 0.797077i \(0.706378\pi\)
\(618\) 0 0
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 3.00000 0.120192
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −16.0000 −0.637962
\(630\) 0 0
\(631\) −22.0000 −0.875806 −0.437903 0.899022i \(-0.644279\pi\)
−0.437903 + 0.899022i \(0.644279\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −12.0000 −0.476205
\(636\) 0 0
\(637\) −1.00000 −0.0396214
\(638\) 0 0
\(639\) −24.0000 −0.949425
\(640\) 0 0
\(641\) 9.00000 0.355479 0.177739 0.984078i \(-0.443122\pi\)
0.177739 + 0.984078i \(0.443122\pi\)
\(642\) 0 0
\(643\) −8.00000 −0.315489 −0.157745 0.987480i \(-0.550422\pi\)
−0.157745 + 0.987480i \(0.550422\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −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
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 18.0000 0.704394 0.352197 0.935926i \(-0.385435\pi\)
0.352197 + 0.935926i \(0.385435\pi\)
\(654\) 0 0
\(655\) 24.0000 0.937758
\(656\) 0 0
\(657\) 39.0000 1.52153
\(658\) 0 0
\(659\) −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 \(0.721802\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 15.0000 0.581675
\(666\) 0 0
\(667\) 15.0000 0.580802
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −60.0000 −2.31627
\(672\) 0 0
\(673\) 1.00000 0.0385472 0.0192736 0.999814i \(-0.493865\pi\)
0.0192736 + 0.999814i \(0.493865\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 22.0000 0.845529 0.422764 0.906240i \(-0.361060\pi\)
0.422764 + 0.906240i \(0.361060\pi\)
\(678\) 0 0
\(679\) 7.00000 0.268635
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 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
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 9.00000 0.342873
\(690\) 0 0
\(691\) −11.0000 −0.418460 −0.209230 0.977866i \(-0.567096\pi\)
−0.209230 + 0.977866i \(0.567096\pi\)
\(692\) 0 0
\(693\) −18.0000 −0.683763
\(694\) 0 0
\(695\) −54.0000 −2.04834
\(696\) 0 0
\(697\) −24.0000 −0.909065
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −27.0000 −1.01978 −0.509888 0.860241i \(-0.670313\pi\)
−0.509888 + 0.860241i \(0.670313\pi\)
\(702\) 0 0
\(703\) 20.0000 0.754314
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −14.0000 −0.526524
\(708\) 0 0
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 0 0
\(711\) 9.00000 0.337526
\(712\) 0 0
\(713\) −9.00000 −0.337053
\(714\) 0 0
\(715\) 18.0000 0.673162
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −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
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −20.0000 −0.742781
\(726\) 0 0
\(727\) −46.0000 −1.70605 −0.853023 0.521874i \(-0.825233\pi\)
−0.853023 + 0.521874i \(0.825233\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 4.00000 0.147945
\(732\) 0 0
\(733\) 51.0000 1.88373 0.941864 0.335994i \(-0.109072\pi\)
0.941864 + 0.335994i \(0.109072\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 36.0000 1.32608
\(738\) 0 0
\(739\) 26.0000 0.956425 0.478213 0.878244i \(-0.341285\pi\)
0.478213 + 0.878244i \(0.341285\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −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
\(746\) 0 0
\(747\) 45.0000 1.64646
\(748\) 0 0
\(749\) 4.00000 0.146157
\(750\) 0 0
\(751\) 17.0000 0.620339 0.310169 0.950681i \(-0.399614\pi\)
0.310169 + 0.950681i \(0.399614\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −15.0000 −0.545184 −0.272592 0.962130i \(-0.587881\pi\)
−0.272592 + 0.962130i \(0.587881\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 9.00000 0.326250 0.163125 0.986605i \(-0.447843\pi\)
0.163125 + 0.986605i \(0.447843\pi\)
\(762\) 0 0
\(763\) −2.00000 −0.0724049
\(764\) 0 0
\(765\) 36.0000 1.30158
\(766\) 0 0
\(767\) 8.00000 0.288863
\(768\) 0 0
\(769\) −35.0000 −1.26213 −0.631066 0.775729i \(-0.717382\pi\)
−0.631066 + 0.775729i \(0.717382\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 54.0000 1.94225 0.971123 0.238581i \(-0.0766824\pi\)
0.971123 + 0.238581i \(0.0766824\pi\)
\(774\) 0 0
\(775\) 12.0000 0.431053
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 30.0000 1.07486
\(780\) 0 0
\(781\) 48.0000 1.71758
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 24.0000 0.856597
\(786\) 0 0
\(787\) −37.0000 −1.31891 −0.659454 0.751745i \(-0.729212\pi\)
−0.659454 + 0.751745i \(0.729212\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −3.00000 −0.106668
\(792\) 0 0
\(793\) 10.0000 0.355110
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 18.0000 0.637593 0.318796 0.947823i \(-0.396721\pi\)
0.318796 + 0.947823i \(0.396721\pi\)
\(798\) 0 0
\(799\) −28.0000 −0.990569
\(800\) 0 0
\(801\) −9.00000 −0.317999
\(802\) 0 0
\(803\) −78.0000 −2.75256
\(804\) 0 0
\(805\) 9.00000 0.317208
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −31.0000 −1.08990 −0.544951 0.838468i \(-0.683452\pi\)
−0.544951 + 0.838468i \(0.683452\pi\)
\(810\) 0 0
\(811\) 52.0000 1.82597 0.912983 0.407997i \(-0.133772\pi\)
0.912983 + 0.407997i \(0.133772\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −12.0000 −0.420342
\(816\) 0 0
\(817\) −5.00000 −0.174928
\(818\) 0 0
\(819\) 3.00000 0.104828
\(820\) 0 0
\(821\) −6.00000 −0.209401 −0.104701 0.994504i \(-0.533388\pi\)
−0.104701 + 0.994504i \(0.533388\pi\)
\(822\) 0 0
\(823\) 32.0000 1.11545 0.557725 0.830026i \(-0.311674\pi\)
0.557725 + 0.830026i \(0.311674\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −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 \(0.555558\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 4.00000 0.138592
\(834\) 0 0
\(835\) 15.0000 0.519096
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 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
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −3.00000 −0.103203
\(846\) 0 0
\(847\) 25.0000 0.859010
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 12.0000 0.411355
\(852\) 0 0
\(853\) 45.0000 1.54077 0.770385 0.637579i \(-0.220064\pi\)
0.770385 + 0.637579i \(0.220064\pi\)
\(854\) 0 0
\(855\) −45.0000 −1.53897
\(856\) 0 0
\(857\) −6.00000 −0.204956 −0.102478 0.994735i \(-0.532677\pi\)
−0.102478 + 0.994735i \(0.532677\pi\)
\(858\) 0 0
\(859\) 2.00000 0.0682391 0.0341196 0.999418i \(-0.489137\pi\)
0.0341196 + 0.999418i \(0.489137\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 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
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −18.0000 −0.610608
\(870\) 0 0
\(871\) −6.00000 −0.203302
\(872\) 0 0
\(873\) −21.0000 −0.710742
\(874\) 0 0
\(875\) 3.00000 0.101419
\(876\) 0 0
\(877\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) 0 0
\(883\) 4.00000 0.134611 0.0673054 0.997732i \(-0.478560\pi\)
0.0673054 + 0.997732i \(0.478560\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 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
\(890\) 0 0
\(891\) 54.0000 1.80907
\(892\) 0 0
\(893\) 35.0000 1.17123
\(894\) 0 0
\(895\) 69.0000 2.30642
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −15.0000 −0.500278
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −42.0000 −1.39613
\(906\) 0 0
\(907\) −7.00000 −0.232431 −0.116216 0.993224i \(-0.537076\pi\)
−0.116216 + 0.993224i \(0.537076\pi\)
\(908\) 0 0
\(909\) 42.0000 1.39305
\(910\) 0 0
\(911\) 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
\(916\) 0 0
\(917\) −8.00000 −0.264183
\(918\) 0 0
\(919\) −8.00000 −0.263896 −0.131948 0.991257i \(-0.542123\pi\)
−0.131948 + 0.991257i \(0.542123\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −8.00000 −0.263323
\(924\) 0 0
\(925\) −16.0000 −0.526077
\(926\) 0 0
\(927\) −12.0000 −0.394132
\(928\) 0 0
\(929\) 5.00000 0.164045 0.0820223 0.996630i \(-0.473862\pi\)
0.0820223 + 0.996630i \(0.473862\pi\)
\(930\) 0 0
\(931\) −5.00000 −0.163868
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −72.0000 −2.35465
\(936\) 0 0
\(937\) −8.00000 −0.261349 −0.130674 0.991425i \(-0.541714\pi\)
−0.130674 + 0.991425i \(0.541714\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 55.0000 1.79295 0.896474 0.443096i \(-0.146120\pi\)
0.896474 + 0.443096i \(0.146120\pi\)
\(942\) 0 0
\(943\) 18.0000 0.586161
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 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
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −39.0000 −1.26333 −0.631667 0.775240i \(-0.717629\pi\)
−0.631667 + 0.775240i \(0.717629\pi\)
\(954\) 0 0
\(955\) −24.0000 −0.776622
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 4.00000 0.129167
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) 0 0
\(963\) −12.0000 −0.386695
\(964\) 0 0
\(965\) −66.0000 −2.12462
\(966\) 0 0
\(967\) −22.0000 −0.707472 −0.353736 0.935345i \(-0.615089\pi\)
−0.353736 + 0.935345i \(0.615089\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −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
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 10.0000 0.319928 0.159964 0.987123i \(-0.448862\pi\)
0.159964 + 0.987123i \(0.448862\pi\)
\(978\) 0 0
\(979\) 18.0000 0.575282
\(980\) 0 0
\(981\) 6.00000 0.191565
\(982\) 0 0
\(983\) −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
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −3.00000 −0.0953945
\(990\) 0 0
\(991\) −4.00000 −0.127064 −0.0635321 0.997980i \(-0.520237\pi\)
−0.0635321 + 0.997980i \(0.520237\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 12.0000 0.380426
\(996\) 0 0
\(997\) −28.0000 −0.886769 −0.443384 0.896332i \(-0.646222\pi\)
−0.443384 + 0.896332i \(0.646222\pi\)
\(998\) 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 1456.2.a.g.1.1 1
4.3 odd 2 91.2.a.a.1.1 1
8.3 odd 2 5824.2.a.s.1.1 1
8.5 even 2 5824.2.a.t.1.1 1
12.11 even 2 819.2.a.f.1.1 1
20.19 odd 2 2275.2.a.h.1.1 1
28.3 even 6 637.2.e.d.79.1 2
28.11 odd 6 637.2.e.e.79.1 2
28.19 even 6 637.2.e.d.508.1 2
28.23 odd 6 637.2.e.e.508.1 2
28.27 even 2 637.2.a.a.1.1 1
52.31 even 4 1183.2.c.b.337.2 2
52.47 even 4 1183.2.c.b.337.1 2
52.51 odd 2 1183.2.a.b.1.1 1
84.83 odd 2 5733.2.a.l.1.1 1
364.363 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 4.3 odd 2
637.2.a.a.1.1 1 28.27 even 2
637.2.e.d.79.1 2 28.3 even 6
637.2.e.d.508.1 2 28.19 even 6
637.2.e.e.79.1 2 28.11 odd 6
637.2.e.e.508.1 2 28.23 odd 6
819.2.a.f.1.1 1 12.11 even 2
1183.2.a.b.1.1 1 52.51 odd 2
1183.2.c.b.337.1 2 52.47 even 4
1183.2.c.b.337.2 2 52.31 even 4
1456.2.a.g.1.1 1 1.1 even 1 trivial
2275.2.a.h.1.1 1 20.19 odd 2
5733.2.a.l.1.1 1 84.83 odd 2
5824.2.a.s.1.1 1 8.3 odd 2
5824.2.a.t.1.1 1 8.5 even 2
8281.2.a.l.1.1 1 364.363 even 2