Properties

Label 637.2.a.d.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 \(-0.822719\pi\)
−0.848875 + 0.528594i \(0.822719\pi\)
\(18\) −3.00000 −0.707107
\(19\) 7.00000 1.60591 0.802955 0.596040i \(-0.203260\pi\)
0.802955 + 0.596040i \(0.203260\pi\)
\(20\) 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 \(-0.670550\pi\)
−0.510527 + 0.859861i \(0.670550\pi\)
\(48\) 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 \(-0.349403\pi\)
0.455661 + 0.890153i \(0.349403\pi\)
\(60\) 0 0
\(61\) 7.00000 0.896258 0.448129 0.893969i \(-0.352090\pi\)
0.448129 + 0.893969i \(0.352090\pi\)
\(62\) 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 \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) 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 \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 0 0
\(99\) 9.00000 0.904534
\(100\) 5.00000 0.500000
\(101\) 14.0000 1.39305 0.696526 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) 0 0
\(103\) −14.0000 −1.37946 −0.689730 0.724066i \(-0.742271\pi\)
−0.689730 + 0.724066i \(0.742271\pi\)
\(104\) −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 \(-0.709471\pi\)
−0.611593 + 0.791173i \(0.709471\pi\)
\(132\) 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 \(-0.590112\pi\)
−0.279330 + 0.960195i \(0.590112\pi\)
\(158\) −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 \(-0.412699\pi\)
0.270838 + 0.962625i \(0.412699\pi\)
\(168\) 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 \(-0.585735\pi\)
−0.266100 + 0.963945i \(0.585735\pi\)
\(174\) 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 \(-0.416227\pi\)
0.260153 + 0.965567i \(0.416227\pi\)
\(182\) 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 \(-0.334722\pi\)
0.496217 + 0.868199i \(0.334722\pi\)
\(200\) 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 \(-0.748216\pi\)
−0.703132 + 0.711059i \(0.748216\pi\)
\(224\) 0 0
\(225\) 15.0000 1.00000
\(226\) 9.00000 0.598671
\(227\) 28.0000 1.85843 0.929213 0.369546i \(-0.120487\pi\)
0.929213 + 0.369546i \(0.120487\pi\)
\(228\) 0 0
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 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 \(-0.857768\pi\)
−0.901819 + 0.432113i \(0.857768\pi\)
\(242\) −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 \(-0.354327\pi\)
0.441836 + 0.897096i \(0.354327\pi\)
\(252\) 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 \(-0.356166\pi\)
0.436648 + 0.899632i \(0.356166\pi\)
\(258\) 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 \(-0.278853\pi\)
0.640196 + 0.768211i \(0.278853\pi\)
\(270\) 0 0
\(271\) 7.00000 0.425220 0.212610 0.977137i \(-0.431804\pi\)
0.212610 + 0.977137i \(0.431804\pi\)
\(272\) 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 \(-0.363394\pi\)
0.416107 + 0.909316i \(0.363394\pi\)
\(284\) 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 \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(294\) 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 \(-0.704541\pi\)
−0.599267 + 0.800549i \(0.704541\pi\)
\(308\) 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 \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) −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 \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −15.0000 −0.799503
\(353\) −14.0000 −0.745145 −0.372572 0.928003i \(-0.621524\pi\)
−0.372572 + 0.928003i \(0.621524\pi\)
\(354\) 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 \(-0.619067\pi\)
−0.365397 + 0.930852i \(0.619067\pi\)
\(368\) 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 \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) 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 \(-0.256617\pi\)
0.692255 + 0.721653i \(0.256617\pi\)
\(410\) 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 \(-0.611095\pi\)
−0.341972 + 0.939710i \(0.611095\pi\)
\(420\) 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 \(-0.668359\pi\)
−0.504598 + 0.863355i \(0.668359\pi\)
\(434\) 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 \(-0.608430\pi\)
−0.334092 + 0.942541i \(0.608430\pi\)
\(440\) 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 \(-0.273911\pi\)
0.652045 + 0.758180i \(0.273911\pi\)
\(462\) 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 \(-0.605001\pi\)
−0.323921 + 0.946084i \(0.605001\pi\)
\(468\) 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 \(-0.551123\pi\)
−0.159919 + 0.987130i \(0.551123\pi\)
\(480\) 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 \(-0.285413\pi\)
0.624229 + 0.781241i \(0.285413\pi\)
\(504\) 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 \(-0.286914\pi\)
0.620539 + 0.784176i \(0.286914\pi\)
\(510\) 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 \(-0.400783\pi\)
0.306676 + 0.951814i \(0.400783\pi\)
\(522\) 15.0000 0.656532
\(523\) 14.0000 0.612177 0.306089 0.952003i \(-0.400980\pi\)
0.306089 + 0.952003i \(0.400980\pi\)
\(524\) 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 \(-0.700884\pi\)
−0.590030 + 0.807382i \(0.700884\pi\)
\(564\) 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 \(-0.594126\pi\)
−0.291414 + 0.956597i \(0.594126\pi\)
\(578\) 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 \(-0.642680\pi\)
−0.433381 + 0.901211i \(0.642680\pi\)
\(588\) 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 \(-0.454400\pi\)
0.142768 + 0.989756i \(0.454400\pi\)
\(602\) 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 \(-0.591702\pi\)
−0.284121 + 0.958788i \(0.591702\pi\)
\(608\) 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 \(-0.309760\pi\)
0.562708 + 0.826656i \(0.309760\pi\)
\(620\) 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 \(-0.455924\pi\)
0.138027 + 0.990429i \(0.455924\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −49.0000 −1.92788
\(647\) 42.0000 1.65119 0.825595 0.564263i \(-0.190840\pi\)
0.825595 + 0.564263i \(0.190840\pi\)
\(648\) −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 \(-0.265186\pi\)
0.672580 + 0.740025i \(0.265186\pi\)
\(678\) 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 \(-0.731880\pi\)
−0.665731 + 0.746191i \(0.731880\pi\)
\(692\) 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 \(-0.213603\pi\)
0.783168 + 0.621810i \(0.213603\pi\)
\(720\) 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 \(-0.673782\pi\)
−0.519231 + 0.854634i \(0.673782\pi\)
\(728\) 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 \(-0.782579\pi\)
−0.775653 + 0.631160i \(0.782579\pi\)
\(734\) −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 \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −4.00000 −0.143963
\(773\) −42.0000 −1.51064 −0.755318 0.655359i \(-0.772517\pi\)
−0.755318 + 0.655359i \(0.772517\pi\)
\(774\) −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 \(-0.539817\pi\)
−0.124762 + 0.992187i \(0.539817\pi\)
\(788\) −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 \(-0.232994\pi\)
0.743858 + 0.668338i \(0.232994\pi\)
\(798\) 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 \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) 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 \(-0.707948\pi\)
−0.607800 + 0.794090i \(0.707948\pi\)
\(830\) 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 \(-0.538557\pi\)
−0.120833 + 0.992673i \(0.538557\pi\)
\(840\) 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 \(-0.244588\pi\)
0.719026 + 0.694983i \(0.244588\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −24.0000 −0.820303
\(857\) −21.0000 −0.717346 −0.358673 0.933463i \(-0.616771\pi\)
−0.358673 + 0.933463i \(0.616771\pi\)
\(858\) 0 0
\(859\) 56.0000 1.91070 0.955348 0.295484i \(-0.0954809\pi\)
0.955348 + 0.295484i \(0.0954809\pi\)
\(860\) 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 \(-0.750181\pi\)
−0.707508 + 0.706705i \(0.750181\pi\)
\(882\) 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 \(-0.749103\pi\)
−0.705111 + 0.709097i \(0.749103\pi\)
\(888\) 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 \(-0.573762\pi\)
−0.229663 + 0.973270i \(0.573762\pi\)
\(930\) 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 \(-0.536475\pi\)
−0.114340 + 0.993442i \(0.536475\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −28.0000 −0.912774 −0.456387 0.889781i \(-0.650857\pi\)
−0.456387 + 0.889781i \(0.650857\pi\)
\(942\) 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 \(-0.351680\pi\)
0.449281 + 0.893390i \(0.351680\pi\)
\(972\) 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 \(-0.464392\pi\)
0.111633 + 0.993750i \(0.464392\pi\)
\(984\) 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 \(-0.464644\pi\)
0.110846 + 0.993838i \(0.464644\pi\)
\(998\) 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.d.1.1 1
3.2 odd 2 5733.2.a.d.1.1 1
7.2 even 3 637.2.e.a.508.1 2
7.3 odd 6 91.2.e.a.79.1 yes 2
7.4 even 3 637.2.e.a.79.1 2
7.5 odd 6 91.2.e.a.53.1 2
7.6 odd 2 637.2.a.c.1.1 1
13.12 even 2 8281.2.a.e.1.1 1
21.5 even 6 819.2.j.b.235.1 2
21.17 even 6 819.2.j.b.352.1 2
21.20 even 2 5733.2.a.c.1.1 1
28.3 even 6 1456.2.r.g.625.1 2
28.19 even 6 1456.2.r.g.417.1 2
91.12 odd 6 1183.2.e.b.508.1 2
91.38 odd 6 1183.2.e.b.170.1 2
91.90 odd 2 8281.2.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.e.a.53.1 2 7.5 odd 6
91.2.e.a.79.1 yes 2 7.3 odd 6
637.2.a.c.1.1 1 7.6 odd 2
637.2.a.d.1.1 1 1.1 even 1 trivial
637.2.e.a.79.1 2 7.4 even 3
637.2.e.a.508.1 2 7.2 even 3
819.2.j.b.235.1 2 21.5 even 6
819.2.j.b.352.1 2 21.17 even 6
1183.2.e.b.170.1 2 91.38 odd 6
1183.2.e.b.508.1 2 91.12 odd 6
1456.2.r.g.417.1 2 28.19 even 6
1456.2.r.g.625.1 2 28.3 even 6
5733.2.a.c.1.1 1 21.20 even 2
5733.2.a.d.1.1 1 3.2 odd 2
8281.2.a.e.1.1 1 13.12 even 2
8281.2.a.f.1.1 1 91.90 odd 2