Properties

Label 637.2.a.c.1.1
Level $637$
Weight $2$
Character 637.1
Self dual yes
Analytic conductor $5.086$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.08647060876\)
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 \(\chi\) \(=\) 637.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{4} -3.00000 q^{8} -3.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{4} -3.00000 q^{8} -3.00000 q^{9} -3.00000 q^{11} -1.00000 q^{13} -1.00000 q^{16} +7.00000 q^{17} -3.00000 q^{18} -7.00000 q^{19} -3.00000 q^{22} -6.00000 q^{23} -5.00000 q^{25} -1.00000 q^{26} -5.00000 q^{29} +5.00000 q^{32} +7.00000 q^{34} +3.00000 q^{36} +8.00000 q^{37} -7.00000 q^{38} +2.00000 q^{43} +3.00000 q^{44} -6.00000 q^{46} +7.00000 q^{47} -5.00000 q^{50} +1.00000 q^{52} -3.00000 q^{53} -5.00000 q^{58} -7.00000 q^{59} -7.00000 q^{61} +7.00000 q^{64} -3.00000 q^{67} -7.00000 q^{68} -5.00000 q^{71} +9.00000 q^{72} +14.0000 q^{73} +8.00000 q^{74} +7.00000 q^{76} -6.00000 q^{79} +9.00000 q^{81} +2.00000 q^{86} +9.00000 q^{88} +6.00000 q^{92} +7.00000 q^{94} -14.0000 q^{97} +9.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107 0.353553 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −3.00000 −1.06066
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 7.00000 1.69775 0.848875 0.528594i \(-0.177281\pi\)
0.848875 + 0.528594i \(0.177281\pi\)
\(18\) −3.00000 −0.707107
\(19\) −7.00000 −1.60591 −0.802955 0.596040i \(-0.796740\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −3.00000 −0.639602
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) −1.00000 −0.196116
\(27\) 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
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 5.00000 0.883883
\(33\) 0 0
\(34\) 7.00000 1.20049
\(35\) 0 0
\(36\) 3.00000 0.500000
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) −7.00000 −1.13555
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 2.00000 0.304997 0.152499 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) 3.00000 0.452267
\(45\) 0 0
\(46\) −6.00000 −0.884652
\(47\) 7.00000 1.02105 0.510527 0.859861i \(-0.329450\pi\)
0.510527 + 0.859861i \(0.329450\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −5.00000 −0.707107
\(51\) 0 0
\(52\) 1.00000 0.138675
\(53\) −3.00000 −0.412082 −0.206041 0.978543i \(-0.566058\pi\)
−0.206041 + 0.978543i \(0.566058\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −5.00000 −0.656532
\(59\) −7.00000 −0.911322 −0.455661 0.890153i \(-0.650597\pi\)
−0.455661 + 0.890153i \(0.650597\pi\)
\(60\) 0 0
\(61\) −7.00000 −0.896258 −0.448129 0.893969i \(-0.647910\pi\)
−0.448129 + 0.893969i \(0.647910\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) 0 0
\(66\) 0 0
\(67\) −3.00000 −0.366508 −0.183254 0.983066i \(-0.558663\pi\)
−0.183254 + 0.983066i \(0.558663\pi\)
\(68\) −7.00000 −0.848875
\(69\) 0 0
\(70\) 0 0
\(71\) −5.00000 −0.593391 −0.296695 0.954972i \(-0.595885\pi\)
−0.296695 + 0.954972i \(0.595885\pi\)
\(72\) 9.00000 1.06066
\(73\) 14.0000 1.63858 0.819288 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) 8.00000 0.929981
\(75\) 0 0
\(76\) 7.00000 0.802955
\(77\) 0 0
\(78\) 0 0
\(79\) −6.00000 −0.675053 −0.337526 0.941316i \(-0.609590\pi\)
−0.337526 + 0.941316i \(0.609590\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 2.00000 0.215666
\(87\) 0 0
\(88\) 9.00000 0.959403
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 6.00000 0.625543
\(93\) 0 0
\(94\) 7.00000 0.721995
\(95\) 0 0
\(96\) 0 0
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 0 0
\(99\) 9.00000 0.904534
\(100\) 5.00000 0.500000
\(101\) −14.0000 −1.39305 −0.696526 0.717532i \(-0.745272\pi\)
−0.696526 + 0.717532i \(0.745272\pi\)
\(102\) 0 0
\(103\) 14.0000 1.37946 0.689730 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) 3.00000 0.294174
\(105\) 0 0
\(106\) −3.00000 −0.291386
\(107\) 8.00000 0.773389 0.386695 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) 0 0
\(109\) 4.00000 0.383131 0.191565 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 9.00000 0.846649 0.423324 0.905978i \(-0.360863\pi\)
0.423324 + 0.905978i \(0.360863\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 5.00000 0.464238
\(117\) 3.00000 0.277350
\(118\) −7.00000 −0.644402
\(119\) 0 0
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) −7.00000 −0.633750
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) −3.00000 −0.265165
\(129\) 0 0
\(130\) 0 0
\(131\) 14.0000 1.22319 0.611593 0.791173i \(-0.290529\pi\)
0.611593 + 0.791173i \(0.290529\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −3.00000 −0.259161
\(135\) 0 0
\(136\) −21.0000 −1.80074
\(137\) 4.00000 0.341743 0.170872 0.985293i \(-0.445342\pi\)
0.170872 + 0.985293i \(0.445342\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −5.00000 −0.419591
\(143\) 3.00000 0.250873
\(144\) 3.00000 0.250000
\(145\) 0 0
\(146\) 14.0000 1.15865
\(147\) 0 0
\(148\) −8.00000 −0.657596
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) −3.00000 −0.244137 −0.122068 0.992522i \(-0.538953\pi\)
−0.122068 + 0.992522i \(0.538953\pi\)
\(152\) 21.0000 1.70332
\(153\) −21.0000 −1.69775
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 7.00000 0.558661 0.279330 0.960195i \(-0.409888\pi\)
0.279330 + 0.960195i \(0.409888\pi\)
\(158\) −6.00000 −0.477334
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 9.00000 0.707107
\(163\) −13.0000 −1.01824 −0.509119 0.860696i \(-0.670029\pi\)
−0.509119 + 0.860696i \(0.670029\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −7.00000 −0.541676 −0.270838 0.962625i \(-0.587301\pi\)
−0.270838 + 0.962625i \(0.587301\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 21.0000 1.60591
\(172\) −2.00000 −0.152499
\(173\) 7.00000 0.532200 0.266100 0.963945i \(-0.414265\pi\)
0.266100 + 0.963945i \(0.414265\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 3.00000 0.226134
\(177\) 0 0
\(178\) 0 0
\(179\) −10.0000 −0.747435 −0.373718 0.927543i \(-0.621917\pi\)
−0.373718 + 0.927543i \(0.621917\pi\)
\(180\) 0 0
\(181\) −7.00000 −0.520306 −0.260153 0.965567i \(-0.583773\pi\)
−0.260153 + 0.965567i \(0.583773\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 18.0000 1.32698
\(185\) 0 0
\(186\) 0 0
\(187\) −21.0000 −1.53567
\(188\) −7.00000 −0.510527
\(189\) 0 0
\(190\) 0 0
\(191\) −20.0000 −1.44715 −0.723575 0.690246i \(-0.757502\pi\)
−0.723575 + 0.690246i \(0.757502\pi\)
\(192\) 0 0
\(193\) 4.00000 0.287926 0.143963 0.989583i \(-0.454015\pi\)
0.143963 + 0.989583i \(0.454015\pi\)
\(194\) −14.0000 −1.00514
\(195\) 0 0
\(196\) 0 0
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) 9.00000 0.639602
\(199\) −14.0000 −0.992434 −0.496217 0.868199i \(-0.665278\pi\)
−0.496217 + 0.868199i \(0.665278\pi\)
\(200\) 15.0000 1.06066
\(201\) 0 0
\(202\) −14.0000 −0.985037
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 14.0000 0.975426
\(207\) 18.0000 1.25109
\(208\) 1.00000 0.0693375
\(209\) 21.0000 1.45260
\(210\) 0 0
\(211\) −26.0000 −1.78991 −0.894957 0.446153i \(-0.852794\pi\)
−0.894957 + 0.446153i \(0.852794\pi\)
\(212\) 3.00000 0.206041
\(213\) 0 0
\(214\) 8.00000 0.546869
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 4.00000 0.270914
\(219\) 0 0
\(220\) 0 0
\(221\) −7.00000 −0.470871
\(222\) 0 0
\(223\) 21.0000 1.40626 0.703132 0.711059i \(-0.251784\pi\)
0.703132 + 0.711059i \(0.251784\pi\)
\(224\) 0 0
\(225\) 15.0000 1.00000
\(226\) 9.00000 0.598671
\(227\) −28.0000 −1.85843 −0.929213 0.369546i \(-0.879513\pi\)
−0.929213 + 0.369546i \(0.879513\pi\)
\(228\) 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
\(232\) 15.0000 0.984798
\(233\) −27.0000 −1.76883 −0.884414 0.466702i \(-0.845442\pi\)
−0.884414 + 0.466702i \(0.845442\pi\)
\(234\) 3.00000 0.196116
\(235\) 0 0
\(236\) 7.00000 0.455661
\(237\) 0 0
\(238\) 0 0
\(239\) −19.0000 −1.22901 −0.614504 0.788914i \(-0.710644\pi\)
−0.614504 + 0.788914i \(0.710644\pi\)
\(240\) 0 0
\(241\) 28.0000 1.80364 0.901819 0.432113i \(-0.142232\pi\)
0.901819 + 0.432113i \(0.142232\pi\)
\(242\) −2.00000 −0.128565
\(243\) 0 0
\(244\) 7.00000 0.448129
\(245\) 0 0
\(246\) 0 0
\(247\) 7.00000 0.445399
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −14.0000 −0.883672 −0.441836 0.897096i \(-0.645673\pi\)
−0.441836 + 0.897096i \(0.645673\pi\)
\(252\) 0 0
\(253\) 18.0000 1.13165
\(254\) 2.00000 0.125491
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) −14.0000 −0.873296 −0.436648 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 15.0000 0.928477
\(262\) 14.0000 0.864923
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 3.00000 0.183254
\(269\) −21.0000 −1.28039 −0.640196 0.768211i \(-0.721147\pi\)
−0.640196 + 0.768211i \(0.721147\pi\)
\(270\) 0 0
\(271\) −7.00000 −0.425220 −0.212610 0.977137i \(-0.568196\pi\)
−0.212610 + 0.977137i \(0.568196\pi\)
\(272\) −7.00000 −0.424437
\(273\) 0 0
\(274\) 4.00000 0.241649
\(275\) 15.0000 0.904534
\(276\) 0 0
\(277\) −17.0000 −1.02143 −0.510716 0.859750i \(-0.670619\pi\)
−0.510716 + 0.859750i \(0.670619\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −12.0000 −0.715860 −0.357930 0.933748i \(-0.616517\pi\)
−0.357930 + 0.933748i \(0.616517\pi\)
\(282\) 0 0
\(283\) −14.0000 −0.832214 −0.416107 0.909316i \(-0.636606\pi\)
−0.416107 + 0.909316i \(0.636606\pi\)
\(284\) 5.00000 0.296695
\(285\) 0 0
\(286\) 3.00000 0.177394
\(287\) 0 0
\(288\) −15.0000 −0.883883
\(289\) 32.0000 1.88235
\(290\) 0 0
\(291\) 0 0
\(292\) −14.0000 −0.819288
\(293\) 14.0000 0.817889 0.408944 0.912559i \(-0.365897\pi\)
0.408944 + 0.912559i \(0.365897\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −24.0000 −1.39497
\(297\) 0 0
\(298\) −6.00000 −0.347571
\(299\) 6.00000 0.346989
\(300\) 0 0
\(301\) 0 0
\(302\) −3.00000 −0.172631
\(303\) 0 0
\(304\) 7.00000 0.401478
\(305\) 0 0
\(306\) −21.0000 −1.20049
\(307\) 21.0000 1.19853 0.599267 0.800549i \(-0.295459\pi\)
0.599267 + 0.800549i \(0.295459\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −14.0000 −0.791327 −0.395663 0.918396i \(-0.629485\pi\)
−0.395663 + 0.918396i \(0.629485\pi\)
\(314\) 7.00000 0.395033
\(315\) 0 0
\(316\) 6.00000 0.337526
\(317\) −6.00000 −0.336994 −0.168497 0.985702i \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) 0 0
\(319\) 15.0000 0.839839
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −49.0000 −2.72643
\(324\) −9.00000 −0.500000
\(325\) 5.00000 0.277350
\(326\) −13.0000 −0.720003
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) 0 0
\(333\) −24.0000 −1.31519
\(334\) −7.00000 −0.383023
\(335\) 0 0
\(336\) 0 0
\(337\) 23.0000 1.25289 0.626445 0.779466i \(-0.284509\pi\)
0.626445 + 0.779466i \(0.284509\pi\)
\(338\) 1.00000 0.0543928
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 21.0000 1.13555
\(343\) 0 0
\(344\) −6.00000 −0.323498
\(345\) 0 0
\(346\) 7.00000 0.376322
\(347\) 4.00000 0.214731 0.107366 0.994220i \(-0.465758\pi\)
0.107366 + 0.994220i \(0.465758\pi\)
\(348\) 0 0
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −15.0000 −0.799503
\(353\) 14.0000 0.745145 0.372572 0.928003i \(-0.378476\pi\)
0.372572 + 0.928003i \(0.378476\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) −10.0000 −0.528516
\(359\) 8.00000 0.422224 0.211112 0.977462i \(-0.432292\pi\)
0.211112 + 0.977462i \(0.432292\pi\)
\(360\) 0 0
\(361\) 30.0000 1.57895
\(362\) −7.00000 −0.367912
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 14.0000 0.730794 0.365397 0.930852i \(-0.380933\pi\)
0.365397 + 0.930852i \(0.380933\pi\)
\(368\) 6.00000 0.312772
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 15.0000 0.776671 0.388335 0.921518i \(-0.373050\pi\)
0.388335 + 0.921518i \(0.373050\pi\)
\(374\) −21.0000 −1.08588
\(375\) 0 0
\(376\) −21.0000 −1.08299
\(377\) 5.00000 0.257513
\(378\) 0 0
\(379\) −12.0000 −0.616399 −0.308199 0.951322i \(-0.599726\pi\)
−0.308199 + 0.951322i \(0.599726\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −20.0000 −1.02329
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 4.00000 0.203595
\(387\) −6.00000 −0.304997
\(388\) 14.0000 0.710742
\(389\) −3.00000 −0.152106 −0.0760530 0.997104i \(-0.524232\pi\)
−0.0760530 + 0.997104i \(0.524232\pi\)
\(390\) 0 0
\(391\) −42.0000 −2.12403
\(392\) 0 0
\(393\) 0 0
\(394\) 2.00000 0.100759
\(395\) 0 0
\(396\) −9.00000 −0.452267
\(397\) 14.0000 0.702640 0.351320 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) −14.0000 −0.701757
\(399\) 0 0
\(400\) 5.00000 0.250000
\(401\) 22.0000 1.09863 0.549314 0.835616i \(-0.314889\pi\)
0.549314 + 0.835616i \(0.314889\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 14.0000 0.696526
\(405\) 0 0
\(406\) 0 0
\(407\) −24.0000 −1.18964
\(408\) 0 0
\(409\) −28.0000 −1.38451 −0.692255 0.721653i \(-0.743383\pi\)
−0.692255 + 0.721653i \(0.743383\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −14.0000 −0.689730
\(413\) 0 0
\(414\) 18.0000 0.884652
\(415\) 0 0
\(416\) −5.00000 −0.245145
\(417\) 0 0
\(418\) 21.0000 1.02714
\(419\) 14.0000 0.683945 0.341972 0.939710i \(-0.388905\pi\)
0.341972 + 0.939710i \(0.388905\pi\)
\(420\) 0 0
\(421\) 30.0000 1.46211 0.731055 0.682318i \(-0.239028\pi\)
0.731055 + 0.682318i \(0.239028\pi\)
\(422\) −26.0000 −1.26566
\(423\) −21.0000 −1.02105
\(424\) 9.00000 0.437079
\(425\) −35.0000 −1.69775
\(426\) 0 0
\(427\) 0 0
\(428\) −8.00000 −0.386695
\(429\) 0 0
\(430\) 0 0
\(431\) −24.0000 −1.15604 −0.578020 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 0 0
\(433\) 21.0000 1.00920 0.504598 0.863355i \(-0.331641\pi\)
0.504598 + 0.863355i \(0.331641\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −4.00000 −0.191565
\(437\) 42.0000 2.00913
\(438\) 0 0
\(439\) 14.0000 0.668184 0.334092 0.942541i \(-0.391570\pi\)
0.334092 + 0.942541i \(0.391570\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −7.00000 −0.332956
\(443\) −20.0000 −0.950229 −0.475114 0.879924i \(-0.657593\pi\)
−0.475114 + 0.879924i \(0.657593\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 21.0000 0.994379
\(447\) 0 0
\(448\) 0 0
\(449\) −12.0000 −0.566315 −0.283158 0.959073i \(-0.591382\pi\)
−0.283158 + 0.959073i \(0.591382\pi\)
\(450\) 15.0000 0.707107
\(451\) 0 0
\(452\) −9.00000 −0.423324
\(453\) 0 0
\(454\) −28.0000 −1.31411
\(455\) 0 0
\(456\) 0 0
\(457\) −6.00000 −0.280668 −0.140334 0.990104i \(-0.544818\pi\)
−0.140334 + 0.990104i \(0.544818\pi\)
\(458\) 14.0000 0.654177
\(459\) 0 0
\(460\) 0 0
\(461\) −28.0000 −1.30409 −0.652045 0.758180i \(-0.726089\pi\)
−0.652045 + 0.758180i \(0.726089\pi\)
\(462\) 0 0
\(463\) 16.0000 0.743583 0.371792 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) 5.00000 0.232119
\(465\) 0 0
\(466\) −27.0000 −1.25075
\(467\) 14.0000 0.647843 0.323921 0.946084i \(-0.394999\pi\)
0.323921 + 0.946084i \(0.394999\pi\)
\(468\) −3.00000 −0.138675
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 21.0000 0.966603
\(473\) −6.00000 −0.275880
\(474\) 0 0
\(475\) 35.0000 1.60591
\(476\) 0 0
\(477\) 9.00000 0.412082
\(478\) −19.0000 −0.869040
\(479\) 7.00000 0.319838 0.159919 0.987130i \(-0.448877\pi\)
0.159919 + 0.987130i \(0.448877\pi\)
\(480\) 0 0
\(481\) −8.00000 −0.364769
\(482\) 28.0000 1.27537
\(483\) 0 0
\(484\) 2.00000 0.0909091
\(485\) 0 0
\(486\) 0 0
\(487\) 25.0000 1.13286 0.566429 0.824110i \(-0.308325\pi\)
0.566429 + 0.824110i \(0.308325\pi\)
\(488\) 21.0000 0.950625
\(489\) 0 0
\(490\) 0 0
\(491\) 30.0000 1.35388 0.676941 0.736038i \(-0.263305\pi\)
0.676941 + 0.736038i \(0.263305\pi\)
\(492\) 0 0
\(493\) −35.0000 −1.57632
\(494\) 7.00000 0.314945
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 8.00000 0.358129 0.179065 0.983837i \(-0.442693\pi\)
0.179065 + 0.983837i \(0.442693\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −14.0000 −0.624851
\(503\) −28.0000 −1.24846 −0.624229 0.781241i \(-0.714587\pi\)
−0.624229 + 0.781241i \(0.714587\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 18.0000 0.800198
\(507\) 0 0
\(508\) −2.00000 −0.0887357
\(509\) −28.0000 −1.24108 −0.620539 0.784176i \(-0.713086\pi\)
−0.620539 + 0.784176i \(0.713086\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −11.0000 −0.486136
\(513\) 0 0
\(514\) −14.0000 −0.617514
\(515\) 0 0
\(516\) 0 0
\(517\) −21.0000 −0.923579
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −14.0000 −0.613351 −0.306676 0.951814i \(-0.599217\pi\)
−0.306676 + 0.951814i \(0.599217\pi\)
\(522\) 15.0000 0.656532
\(523\) −14.0000 −0.612177 −0.306089 0.952003i \(-0.599020\pi\)
−0.306089 + 0.952003i \(0.599020\pi\)
\(524\) −14.0000 −0.611593
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) 0 0
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 21.0000 0.911322
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 9.00000 0.388741
\(537\) 0 0
\(538\) −21.0000 −0.905374
\(539\) 0 0
\(540\) 0 0
\(541\) 8.00000 0.343947 0.171973 0.985102i \(-0.444986\pi\)
0.171973 + 0.985102i \(0.444986\pi\)
\(542\) −7.00000 −0.300676
\(543\) 0 0
\(544\) 35.0000 1.50061
\(545\) 0 0
\(546\) 0 0
\(547\) 2.00000 0.0855138 0.0427569 0.999086i \(-0.486386\pi\)
0.0427569 + 0.999086i \(0.486386\pi\)
\(548\) −4.00000 −0.170872
\(549\) 21.0000 0.896258
\(550\) 15.0000 0.639602
\(551\) 35.0000 1.49105
\(552\) 0 0
\(553\) 0 0
\(554\) −17.0000 −0.722261
\(555\) 0 0
\(556\) 0 0
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) 0 0
\(559\) −2.00000 −0.0845910
\(560\) 0 0
\(561\) 0 0
\(562\) −12.0000 −0.506189
\(563\) 28.0000 1.18006 0.590030 0.807382i \(-0.299116\pi\)
0.590030 + 0.807382i \(0.299116\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −14.0000 −0.588464
\(567\) 0 0
\(568\) 15.0000 0.629386
\(569\) 1.00000 0.0419222 0.0209611 0.999780i \(-0.493327\pi\)
0.0209611 + 0.999780i \(0.493327\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) −3.00000 −0.125436
\(573\) 0 0
\(574\) 0 0
\(575\) 30.0000 1.25109
\(576\) −21.0000 −0.875000
\(577\) 14.0000 0.582828 0.291414 0.956597i \(-0.405874\pi\)
0.291414 + 0.956597i \(0.405874\pi\)
\(578\) 32.0000 1.33102
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 9.00000 0.372742
\(584\) −42.0000 −1.73797
\(585\) 0 0
\(586\) 14.0000 0.578335
\(587\) 21.0000 0.866763 0.433381 0.901211i \(-0.357320\pi\)
0.433381 + 0.901211i \(0.357320\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −8.00000 −0.328798
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 6.00000 0.245770
\(597\) 0 0
\(598\) 6.00000 0.245358
\(599\) 32.0000 1.30748 0.653742 0.756717i \(-0.273198\pi\)
0.653742 + 0.756717i \(0.273198\pi\)
\(600\) 0 0
\(601\) −7.00000 −0.285536 −0.142768 0.989756i \(-0.545600\pi\)
−0.142768 + 0.989756i \(0.545600\pi\)
\(602\) 0 0
\(603\) 9.00000 0.366508
\(604\) 3.00000 0.122068
\(605\) 0 0
\(606\) 0 0
\(607\) 14.0000 0.568242 0.284121 0.958788i \(-0.408298\pi\)
0.284121 + 0.958788i \(0.408298\pi\)
\(608\) −35.0000 −1.41944
\(609\) 0 0
\(610\) 0 0
\(611\) −7.00000 −0.283190
\(612\) 21.0000 0.848875
\(613\) 32.0000 1.29247 0.646234 0.763139i \(-0.276343\pi\)
0.646234 + 0.763139i \(0.276343\pi\)
\(614\) 21.0000 0.847491
\(615\) 0 0
\(616\) 0 0
\(617\) 30.0000 1.20775 0.603877 0.797077i \(-0.293622\pi\)
0.603877 + 0.797077i \(0.293622\pi\)
\(618\) 0 0
\(619\) −28.0000 −1.12542 −0.562708 0.826656i \(-0.690240\pi\)
−0.562708 + 0.826656i \(0.690240\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) −14.0000 −0.559553
\(627\) 0 0
\(628\) −7.00000 −0.279330
\(629\) 56.0000 2.23287
\(630\) 0 0
\(631\) 16.0000 0.636950 0.318475 0.947931i \(-0.396829\pi\)
0.318475 + 0.947931i \(0.396829\pi\)
\(632\) 18.0000 0.716002
\(633\) 0 0
\(634\) −6.00000 −0.238290
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 15.0000 0.593856
\(639\) 15.0000 0.593391
\(640\) 0 0
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 0 0
\(643\) −7.00000 −0.276053 −0.138027 0.990429i \(-0.544076\pi\)
−0.138027 + 0.990429i \(0.544076\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −49.0000 −1.92788
\(647\) −42.0000 −1.65119 −0.825595 0.564263i \(-0.809160\pi\)
−0.825595 + 0.564263i \(0.809160\pi\)
\(648\) −27.0000 −1.06066
\(649\) 21.0000 0.824322
\(650\) 5.00000 0.196116
\(651\) 0 0
\(652\) 13.0000 0.509119
\(653\) −6.00000 −0.234798 −0.117399 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −42.0000 −1.63858
\(658\) 0 0
\(659\) −40.0000 −1.55818 −0.779089 0.626913i \(-0.784318\pi\)
−0.779089 + 0.626913i \(0.784318\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(662\) −20.0000 −0.777322
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) −24.0000 −0.929981
\(667\) 30.0000 1.16160
\(668\) 7.00000 0.270838
\(669\) 0 0
\(670\) 0 0
\(671\) 21.0000 0.810696
\(672\) 0 0
\(673\) −26.0000 −1.00223 −0.501113 0.865382i \(-0.667076\pi\)
−0.501113 + 0.865382i \(0.667076\pi\)
\(674\) 23.0000 0.885927
\(675\) 0 0
\(676\) −1.00000 −0.0384615
\(677\) −35.0000 −1.34516 −0.672580 0.740025i \(-0.734814\pi\)
−0.672580 + 0.740025i \(0.734814\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −24.0000 −0.918334 −0.459167 0.888350i \(-0.651852\pi\)
−0.459167 + 0.888350i \(0.651852\pi\)
\(684\) −21.0000 −0.802955
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −2.00000 −0.0762493
\(689\) 3.00000 0.114291
\(690\) 0 0
\(691\) 35.0000 1.33146 0.665731 0.746191i \(-0.268120\pi\)
0.665731 + 0.746191i \(0.268120\pi\)
\(692\) −7.00000 −0.266100
\(693\) 0 0
\(694\) 4.00000 0.151838
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 14.0000 0.529908
\(699\) 0 0
\(700\) 0 0
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) 0 0
\(703\) −56.0000 −2.11208
\(704\) −21.0000 −0.791467
\(705\) 0 0
\(706\) 14.0000 0.526897
\(707\) 0 0
\(708\) 0 0
\(709\) 50.0000 1.87779 0.938895 0.344204i \(-0.111851\pi\)
0.938895 + 0.344204i \(0.111851\pi\)
\(710\) 0 0
\(711\) 18.0000 0.675053
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 10.0000 0.373718
\(717\) 0 0
\(718\) 8.00000 0.298557
\(719\) −42.0000 −1.56634 −0.783168 0.621810i \(-0.786397\pi\)
−0.783168 + 0.621810i \(0.786397\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 30.0000 1.11648
\(723\) 0 0
\(724\) 7.00000 0.260153
\(725\) 25.0000 0.928477
\(726\) 0 0
\(727\) 28.0000 1.03846 0.519231 0.854634i \(-0.326218\pi\)
0.519231 + 0.854634i \(0.326218\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 14.0000 0.517809
\(732\) 0 0
\(733\) 42.0000 1.55131 0.775653 0.631160i \(-0.217421\pi\)
0.775653 + 0.631160i \(0.217421\pi\)
\(734\) 14.0000 0.516749
\(735\) 0 0
\(736\) −30.0000 −1.10581
\(737\) 9.00000 0.331519
\(738\) 0 0
\(739\) 4.00000 0.147142 0.0735712 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 9.00000 0.330178 0.165089 0.986279i \(-0.447209\pi\)
0.165089 + 0.986279i \(0.447209\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 15.0000 0.549189
\(747\) 0 0
\(748\) 21.0000 0.767836
\(749\) 0 0
\(750\) 0 0
\(751\) −20.0000 −0.729810 −0.364905 0.931045i \(-0.618899\pi\)
−0.364905 + 0.931045i \(0.618899\pi\)
\(752\) −7.00000 −0.255264
\(753\) 0 0
\(754\) 5.00000 0.182089
\(755\) 0 0
\(756\) 0 0
\(757\) 9.00000 0.327111 0.163555 0.986534i \(-0.447704\pi\)
0.163555 + 0.986534i \(0.447704\pi\)
\(758\) −12.0000 −0.435860
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 20.0000 0.723575
\(765\) 0 0
\(766\) 0 0
\(767\) 7.00000 0.252755
\(768\) 0 0
\(769\) −14.0000 −0.504853 −0.252426 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −4.00000 −0.143963
\(773\) 42.0000 1.51064 0.755318 0.655359i \(-0.227483\pi\)
0.755318 + 0.655359i \(0.227483\pi\)
\(774\) −6.00000 −0.215666
\(775\) 0 0
\(776\) 42.0000 1.50771
\(777\) 0 0
\(778\) −3.00000 −0.107555
\(779\) 0 0
\(780\) 0 0
\(781\) 15.0000 0.536742
\(782\) −42.0000 −1.50192
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 7.00000 0.249523 0.124762 0.992187i \(-0.460183\pi\)
0.124762 + 0.992187i \(0.460183\pi\)
\(788\) −2.00000 −0.0712470
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) −27.0000 −0.959403
\(793\) 7.00000 0.248577
\(794\) 14.0000 0.496841
\(795\) 0 0
\(796\) 14.0000 0.496217
\(797\) −42.0000 −1.48772 −0.743858 0.668338i \(-0.767006\pi\)
−0.743858 + 0.668338i \(0.767006\pi\)
\(798\) 0 0
\(799\) 49.0000 1.73350
\(800\) −25.0000 −0.883883
\(801\) 0 0
\(802\) 22.0000 0.776847
\(803\) −42.0000 −1.48215
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 42.0000 1.47755
\(809\) −31.0000 −1.08990 −0.544951 0.838468i \(-0.683452\pi\)
−0.544951 + 0.838468i \(0.683452\pi\)
\(810\) 0 0
\(811\) −28.0000 −0.983213 −0.491606 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −24.0000 −0.841200
\(815\) 0 0
\(816\) 0 0
\(817\) −14.0000 −0.489798
\(818\) −28.0000 −0.978997
\(819\) 0 0
\(820\) 0 0
\(821\) 36.0000 1.25641 0.628204 0.778048i \(-0.283790\pi\)
0.628204 + 0.778048i \(0.283790\pi\)
\(822\) 0 0
\(823\) 4.00000 0.139431 0.0697156 0.997567i \(-0.477791\pi\)
0.0697156 + 0.997567i \(0.477791\pi\)
\(824\) −42.0000 −1.46314
\(825\) 0 0
\(826\) 0 0
\(827\) −5.00000 −0.173867 −0.0869335 0.996214i \(-0.527707\pi\)
−0.0869335 + 0.996214i \(0.527707\pi\)
\(828\) −18.0000 −0.625543
\(829\) 35.0000 1.21560 0.607800 0.794090i \(-0.292052\pi\)
0.607800 + 0.794090i \(0.292052\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −7.00000 −0.242681
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) −21.0000 −0.726300
\(837\) 0 0
\(838\) 14.0000 0.483622
\(839\) 7.00000 0.241667 0.120833 0.992673i \(-0.461443\pi\)
0.120833 + 0.992673i \(0.461443\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 30.0000 1.03387
\(843\) 0 0
\(844\) 26.0000 0.894957
\(845\) 0 0
\(846\) −21.0000 −0.721995
\(847\) 0 0
\(848\) 3.00000 0.103020
\(849\) 0 0
\(850\) −35.0000 −1.20049
\(851\) −48.0000 −1.64542
\(852\) 0 0
\(853\) −42.0000 −1.43805 −0.719026 0.694983i \(-0.755412\pi\)
−0.719026 + 0.694983i \(0.755412\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −24.0000 −0.820303
\(857\) 21.0000 0.717346 0.358673 0.933463i \(-0.383229\pi\)
0.358673 + 0.933463i \(0.383229\pi\)
\(858\) 0 0
\(859\) −56.0000 −1.91070 −0.955348 0.295484i \(-0.904519\pi\)
−0.955348 + 0.295484i \(0.904519\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −24.0000 −0.817443
\(863\) −20.0000 −0.680808 −0.340404 0.940279i \(-0.610564\pi\)
−0.340404 + 0.940279i \(0.610564\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 21.0000 0.713609
\(867\) 0 0
\(868\) 0 0
\(869\) 18.0000 0.610608
\(870\) 0 0
\(871\) 3.00000 0.101651
\(872\) −12.0000 −0.406371
\(873\) 42.0000 1.42148
\(874\) 42.0000 1.42067
\(875\) 0 0
\(876\) 0 0
\(877\) −48.0000 −1.62084 −0.810422 0.585846i \(-0.800762\pi\)
−0.810422 + 0.585846i \(0.800762\pi\)
\(878\) 14.0000 0.472477
\(879\) 0 0
\(880\) 0 0
\(881\) 42.0000 1.41502 0.707508 0.706705i \(-0.249819\pi\)
0.707508 + 0.706705i \(0.249819\pi\)
\(882\) 0 0
\(883\) −40.0000 −1.34611 −0.673054 0.739594i \(-0.735018\pi\)
−0.673054 + 0.739594i \(0.735018\pi\)
\(884\) 7.00000 0.235435
\(885\) 0 0
\(886\) −20.0000 −0.671913
\(887\) 42.0000 1.41022 0.705111 0.709097i \(-0.250897\pi\)
0.705111 + 0.709097i \(0.250897\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −27.0000 −0.904534
\(892\) −21.0000 −0.703132
\(893\) −49.0000 −1.63972
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −12.0000 −0.400445
\(899\) 0 0
\(900\) −15.0000 −0.500000
\(901\) −21.0000 −0.699611
\(902\) 0 0
\(903\) 0 0
\(904\) −27.0000 −0.898007
\(905\) 0 0
\(906\) 0 0
\(907\) −38.0000 −1.26177 −0.630885 0.775877i \(-0.717308\pi\)
−0.630885 + 0.775877i \(0.717308\pi\)
\(908\) 28.0000 0.929213
\(909\) 42.0000 1.39305
\(910\) 0 0
\(911\) −54.0000 −1.78910 −0.894550 0.446968i \(-0.852504\pi\)
−0.894550 + 0.446968i \(0.852504\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −6.00000 −0.198462
\(915\) 0 0
\(916\) −14.0000 −0.462573
\(917\) 0 0
\(918\) 0 0
\(919\) 50.0000 1.64935 0.824674 0.565608i \(-0.191359\pi\)
0.824674 + 0.565608i \(0.191359\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −28.0000 −0.922131
\(923\) 5.00000 0.164577
\(924\) 0 0
\(925\) −40.0000 −1.31519
\(926\) 16.0000 0.525793
\(927\) −42.0000 −1.37946
\(928\) −25.0000 −0.820665
\(929\) 14.0000 0.459325 0.229663 0.973270i \(-0.426238\pi\)
0.229663 + 0.973270i \(0.426238\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 27.0000 0.884414
\(933\) 0 0
\(934\) 14.0000 0.458094
\(935\) 0 0
\(936\) −9.00000 −0.294174
\(937\) 7.00000 0.228680 0.114340 0.993442i \(-0.463525\pi\)
0.114340 + 0.993442i \(0.463525\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 28.0000 0.912774 0.456387 0.889781i \(-0.349143\pi\)
0.456387 + 0.889781i \(0.349143\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 7.00000 0.227831
\(945\) 0 0
\(946\) −6.00000 −0.195077
\(947\) −27.0000 −0.877382 −0.438691 0.898638i \(-0.644558\pi\)
−0.438691 + 0.898638i \(0.644558\pi\)
\(948\) 0 0
\(949\) −14.0000 −0.454459
\(950\) 35.0000 1.13555
\(951\) 0 0
\(952\) 0 0
\(953\) 9.00000 0.291539 0.145769 0.989319i \(-0.453434\pi\)
0.145769 + 0.989319i \(0.453434\pi\)
\(954\) 9.00000 0.291386
\(955\) 0 0
\(956\) 19.0000 0.614504
\(957\) 0 0
\(958\) 7.00000 0.226160
\(959\) 0 0
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) −8.00000 −0.257930
\(963\) −24.0000 −0.773389
\(964\) −28.0000 −0.901819
\(965\) 0 0
\(966\) 0 0
\(967\) −5.00000 −0.160789 −0.0803946 0.996763i \(-0.525618\pi\)
−0.0803946 + 0.996763i \(0.525618\pi\)
\(968\) 6.00000 0.192847
\(969\) 0 0
\(970\) 0 0
\(971\) −28.0000 −0.898563 −0.449281 0.893390i \(-0.648320\pi\)
−0.449281 + 0.893390i \(0.648320\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 25.0000 0.801052
\(975\) 0 0
\(976\) 7.00000 0.224065
\(977\) −38.0000 −1.21573 −0.607864 0.794041i \(-0.707973\pi\)
−0.607864 + 0.794041i \(0.707973\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −12.0000 −0.383131
\(982\) 30.0000 0.957338
\(983\) −7.00000 −0.223265 −0.111633 0.993750i \(-0.535608\pi\)
−0.111633 + 0.993750i \(0.535608\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −35.0000 −1.11463
\(987\) 0 0
\(988\) −7.00000 −0.222700
\(989\) −12.0000 −0.381578
\(990\) 0 0
\(991\) 4.00000 0.127064 0.0635321 0.997980i \(-0.479763\pi\)
0.0635321 + 0.997980i \(0.479763\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −7.00000 −0.221692 −0.110846 0.993838i \(-0.535356\pi\)
−0.110846 + 0.993838i \(0.535356\pi\)
\(998\) 8.00000 0.253236
\(999\) 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 637.2.a.c.1.1 1
3.2 odd 2 5733.2.a.c.1.1 1
7.2 even 3 91.2.e.a.53.1 2
7.3 odd 6 637.2.e.a.79.1 2
7.4 even 3 91.2.e.a.79.1 yes 2
7.5 odd 6 637.2.e.a.508.1 2
7.6 odd 2 637.2.a.d.1.1 1
13.12 even 2 8281.2.a.f.1.1 1
21.2 odd 6 819.2.j.b.235.1 2
21.11 odd 6 819.2.j.b.352.1 2
21.20 even 2 5733.2.a.d.1.1 1
28.11 odd 6 1456.2.r.g.625.1 2
28.23 odd 6 1456.2.r.g.417.1 2
91.25 even 6 1183.2.e.b.170.1 2
91.51 even 6 1183.2.e.b.508.1 2
91.90 odd 2 8281.2.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.e.a.53.1 2 7.2 even 3
91.2.e.a.79.1 yes 2 7.4 even 3
637.2.a.c.1.1 1 1.1 even 1 trivial
637.2.a.d.1.1 1 7.6 odd 2
637.2.e.a.79.1 2 7.3 odd 6
637.2.e.a.508.1 2 7.5 odd 6
819.2.j.b.235.1 2 21.2 odd 6
819.2.j.b.352.1 2 21.11 odd 6
1183.2.e.b.170.1 2 91.25 even 6
1183.2.e.b.508.1 2 91.51 even 6
1456.2.r.g.417.1 2 28.23 odd 6
1456.2.r.g.625.1 2 28.11 odd 6
5733.2.a.c.1.1 1 3.2 odd 2
5733.2.a.d.1.1 1 21.20 even 2
8281.2.a.e.1.1 1 91.90 odd 2
8281.2.a.f.1.1 1 13.12 even 2