Properties

Label 4598.2.a.h.1.1
Level $4598$
Weight $2$
Character 4598.1
Self dual yes
Analytic conductor $36.715$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4598 = 2 \cdot 11^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4598.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.7152148494\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4598.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +1.00000 q^{12} +2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +6.00000 q^{17} +2.00000 q^{18} +1.00000 q^{19} -1.00000 q^{21} -9.00000 q^{23} -1.00000 q^{24} -5.00000 q^{25} -2.00000 q^{26} -5.00000 q^{27} -1.00000 q^{28} +3.00000 q^{29} +8.00000 q^{31} -1.00000 q^{32} -6.00000 q^{34} -2.00000 q^{36} -7.00000 q^{37} -1.00000 q^{38} +2.00000 q^{39} +1.00000 q^{42} -10.0000 q^{43} +9.00000 q^{46} +9.00000 q^{47} +1.00000 q^{48} -6.00000 q^{49} +5.00000 q^{50} +6.00000 q^{51} +2.00000 q^{52} -9.00000 q^{53} +5.00000 q^{54} +1.00000 q^{56} +1.00000 q^{57} -3.00000 q^{58} -9.00000 q^{59} +8.00000 q^{61} -8.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} -4.00000 q^{67} +6.00000 q^{68} -9.00000 q^{69} +12.0000 q^{71} +2.00000 q^{72} +2.00000 q^{73} +7.00000 q^{74} -5.00000 q^{75} +1.00000 q^{76} -2.00000 q^{78} +14.0000 q^{79} +1.00000 q^{81} -12.0000 q^{83} -1.00000 q^{84} +10.0000 q^{86} +3.00000 q^{87} +6.00000 q^{89} -2.00000 q^{91} -9.00000 q^{92} +8.00000 q^{93} -9.00000 q^{94} -1.00000 q^{96} -16.0000 q^{97} +6.00000 q^{98} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350 0.288675 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) −1.00000 −0.408248
\(7\) −1.00000 −0.377964 −0.188982 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) 0 0
\(12\) 1.00000 0.288675
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 2.00000 0.471405
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) −9.00000 −1.87663 −0.938315 0.345782i \(-0.887614\pi\)
−0.938315 + 0.345782i \(0.887614\pi\)
\(24\) −1.00000 −0.204124
\(25\) −5.00000 −1.00000
\(26\) −2.00000 −0.392232
\(27\) −5.00000 −0.962250
\(28\) −1.00000 −0.188982
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) −2.00000 −0.333333
\(37\) −7.00000 −1.15079 −0.575396 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) −1.00000 −0.162221
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 1.00000 0.154303
\(43\) −10.0000 −1.52499 −0.762493 0.646997i \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 9.00000 1.32698
\(47\) 9.00000 1.31278 0.656392 0.754420i \(-0.272082\pi\)
0.656392 + 0.754420i \(0.272082\pi\)
\(48\) 1.00000 0.144338
\(49\) −6.00000 −0.857143
\(50\) 5.00000 0.707107
\(51\) 6.00000 0.840168
\(52\) 2.00000 0.277350
\(53\) −9.00000 −1.23625 −0.618123 0.786082i \(-0.712106\pi\)
−0.618123 + 0.786082i \(0.712106\pi\)
\(54\) 5.00000 0.680414
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 1.00000 0.132453
\(58\) −3.00000 −0.393919
\(59\) −9.00000 −1.17170 −0.585850 0.810419i \(-0.699239\pi\)
−0.585850 + 0.810419i \(0.699239\pi\)
\(60\) 0 0
\(61\) 8.00000 1.02430 0.512148 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) −8.00000 −1.01600
\(63\) 2.00000 0.251976
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 6.00000 0.727607
\(69\) −9.00000 −1.08347
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 2.00000 0.235702
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 7.00000 0.813733
\(75\) −5.00000 −0.577350
\(76\) 1.00000 0.114708
\(77\) 0 0
\(78\) −2.00000 −0.226455
\(79\) 14.0000 1.57512 0.787562 0.616236i \(-0.211343\pi\)
0.787562 + 0.616236i \(0.211343\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) −1.00000 −0.109109
\(85\) 0 0
\(86\) 10.0000 1.07833
\(87\) 3.00000 0.321634
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) −9.00000 −0.938315
\(93\) 8.00000 0.829561
\(94\) −9.00000 −0.928279
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −16.0000 −1.62455 −0.812277 0.583272i \(-0.801772\pi\)
−0.812277 + 0.583272i \(0.801772\pi\)
\(98\) 6.00000 0.606092
\(99\) 0 0
\(100\) −5.00000 −0.500000
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) −6.00000 −0.594089
\(103\) 14.0000 1.37946 0.689730 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) 9.00000 0.874157
\(107\) −15.0000 −1.45010 −0.725052 0.688694i \(-0.758184\pi\)
−0.725052 + 0.688694i \(0.758184\pi\)
\(108\) −5.00000 −0.481125
\(109\) −19.0000 −1.81987 −0.909935 0.414751i \(-0.863869\pi\)
−0.909935 + 0.414751i \(0.863869\pi\)
\(110\) 0 0
\(111\) −7.00000 −0.664411
\(112\) −1.00000 −0.0944911
\(113\) 12.0000 1.12887 0.564433 0.825479i \(-0.309095\pi\)
0.564433 + 0.825479i \(0.309095\pi\)
\(114\) −1.00000 −0.0936586
\(115\) 0 0
\(116\) 3.00000 0.278543
\(117\) −4.00000 −0.369800
\(118\) 9.00000 0.828517
\(119\) −6.00000 −0.550019
\(120\) 0 0
\(121\) 0 0
\(122\) −8.00000 −0.724286
\(123\) 0 0
\(124\) 8.00000 0.718421
\(125\) 0 0
\(126\) −2.00000 −0.178174
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −10.0000 −0.880451
\(130\) 0 0
\(131\) −18.0000 −1.57267 −0.786334 0.617802i \(-0.788023\pi\)
−0.786334 + 0.617802i \(0.788023\pi\)
\(132\) 0 0
\(133\) −1.00000 −0.0867110
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 9.00000 0.766131
\(139\) 8.00000 0.678551 0.339276 0.940687i \(-0.389818\pi\)
0.339276 + 0.940687i \(0.389818\pi\)
\(140\) 0 0
\(141\) 9.00000 0.757937
\(142\) −12.0000 −1.00702
\(143\) 0 0
\(144\) −2.00000 −0.166667
\(145\) 0 0
\(146\) −2.00000 −0.165521
\(147\) −6.00000 −0.494872
\(148\) −7.00000 −0.575396
\(149\) −12.0000 −0.983078 −0.491539 0.870855i \(-0.663566\pi\)
−0.491539 + 0.870855i \(0.663566\pi\)
\(150\) 5.00000 0.408248
\(151\) −10.0000 −0.813788 −0.406894 0.913475i \(-0.633388\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(152\) −1.00000 −0.0811107
\(153\) −12.0000 −0.970143
\(154\) 0 0
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) −22.0000 −1.75579 −0.877896 0.478852i \(-0.841053\pi\)
−0.877896 + 0.478852i \(0.841053\pi\)
\(158\) −14.0000 −1.11378
\(159\) −9.00000 −0.713746
\(160\) 0 0
\(161\) 9.00000 0.709299
\(162\) −1.00000 −0.0785674
\(163\) −10.0000 −0.783260 −0.391630 0.920123i \(-0.628089\pi\)
−0.391630 + 0.920123i \(0.628089\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) 6.00000 0.464294 0.232147 0.972681i \(-0.425425\pi\)
0.232147 + 0.972681i \(0.425425\pi\)
\(168\) 1.00000 0.0771517
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −2.00000 −0.152944
\(172\) −10.0000 −0.762493
\(173\) 3.00000 0.228086 0.114043 0.993476i \(-0.463620\pi\)
0.114043 + 0.993476i \(0.463620\pi\)
\(174\) −3.00000 −0.227429
\(175\) 5.00000 0.377964
\(176\) 0 0
\(177\) −9.00000 −0.676481
\(178\) −6.00000 −0.449719
\(179\) 15.0000 1.12115 0.560576 0.828103i \(-0.310580\pi\)
0.560576 + 0.828103i \(0.310580\pi\)
\(180\) 0 0
\(181\) −22.0000 −1.63525 −0.817624 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) 2.00000 0.148250
\(183\) 8.00000 0.591377
\(184\) 9.00000 0.663489
\(185\) 0 0
\(186\) −8.00000 −0.586588
\(187\) 0 0
\(188\) 9.00000 0.656392
\(189\) 5.00000 0.363696
\(190\) 0 0
\(191\) −9.00000 −0.651217 −0.325609 0.945505i \(-0.605569\pi\)
−0.325609 + 0.945505i \(0.605569\pi\)
\(192\) 1.00000 0.0721688
\(193\) −22.0000 −1.58359 −0.791797 0.610784i \(-0.790854\pi\)
−0.791797 + 0.610784i \(0.790854\pi\)
\(194\) 16.0000 1.14873
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) −12.0000 −0.854965 −0.427482 0.904024i \(-0.640599\pi\)
−0.427482 + 0.904024i \(0.640599\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 5.00000 0.353553
\(201\) −4.00000 −0.282138
\(202\) 0 0
\(203\) −3.00000 −0.210559
\(204\) 6.00000 0.420084
\(205\) 0 0
\(206\) −14.0000 −0.975426
\(207\) 18.0000 1.25109
\(208\) 2.00000 0.138675
\(209\) 0 0
\(210\) 0 0
\(211\) 5.00000 0.344214 0.172107 0.985078i \(-0.444942\pi\)
0.172107 + 0.985078i \(0.444942\pi\)
\(212\) −9.00000 −0.618123
\(213\) 12.0000 0.822226
\(214\) 15.0000 1.02538
\(215\) 0 0
\(216\) 5.00000 0.340207
\(217\) −8.00000 −0.543075
\(218\) 19.0000 1.28684
\(219\) 2.00000 0.135147
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) 7.00000 0.469809
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 1.00000 0.0668153
\(225\) 10.0000 0.666667
\(226\) −12.0000 −0.798228
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 1.00000 0.0662266
\(229\) −4.00000 −0.264327 −0.132164 0.991228i \(-0.542192\pi\)
−0.132164 + 0.991228i \(0.542192\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −3.00000 −0.196960
\(233\) 27.0000 1.76883 0.884414 0.466702i \(-0.154558\pi\)
0.884414 + 0.466702i \(0.154558\pi\)
\(234\) 4.00000 0.261488
\(235\) 0 0
\(236\) −9.00000 −0.585850
\(237\) 14.0000 0.909398
\(238\) 6.00000 0.388922
\(239\) 9.00000 0.582162 0.291081 0.956698i \(-0.405985\pi\)
0.291081 + 0.956698i \(0.405985\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 0 0
\(243\) 16.0000 1.02640
\(244\) 8.00000 0.512148
\(245\) 0 0
\(246\) 0 0
\(247\) 2.00000 0.127257
\(248\) −8.00000 −0.508001
\(249\) −12.0000 −0.760469
\(250\) 0 0
\(251\) 18.0000 1.13615 0.568075 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) 2.00000 0.125988
\(253\) 0 0
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −12.0000 −0.748539 −0.374270 0.927320i \(-0.622107\pi\)
−0.374270 + 0.927320i \(0.622107\pi\)
\(258\) 10.0000 0.622573
\(259\) 7.00000 0.434959
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 18.0000 1.11204
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 1.00000 0.0613139
\(267\) 6.00000 0.367194
\(268\) −4.00000 −0.244339
\(269\) −3.00000 −0.182913 −0.0914566 0.995809i \(-0.529152\pi\)
−0.0914566 + 0.995809i \(0.529152\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 6.00000 0.363803
\(273\) −2.00000 −0.121046
\(274\) 6.00000 0.362473
\(275\) 0 0
\(276\) −9.00000 −0.541736
\(277\) −4.00000 −0.240337 −0.120168 0.992754i \(-0.538343\pi\)
−0.120168 + 0.992754i \(0.538343\pi\)
\(278\) −8.00000 −0.479808
\(279\) −16.0000 −0.957895
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) −9.00000 −0.535942
\(283\) −16.0000 −0.951101 −0.475551 0.879688i \(-0.657751\pi\)
−0.475551 + 0.879688i \(0.657751\pi\)
\(284\) 12.0000 0.712069
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 2.00000 0.117851
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) −16.0000 −0.937937
\(292\) 2.00000 0.117041
\(293\) 33.0000 1.92788 0.963940 0.266119i \(-0.0857413\pi\)
0.963940 + 0.266119i \(0.0857413\pi\)
\(294\) 6.00000 0.349927
\(295\) 0 0
\(296\) 7.00000 0.406867
\(297\) 0 0
\(298\) 12.0000 0.695141
\(299\) −18.0000 −1.04097
\(300\) −5.00000 −0.288675
\(301\) 10.0000 0.576390
\(302\) 10.0000 0.575435
\(303\) 0 0
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) 12.0000 0.685994
\(307\) 17.0000 0.970241 0.485121 0.874447i \(-0.338776\pi\)
0.485121 + 0.874447i \(0.338776\pi\)
\(308\) 0 0
\(309\) 14.0000 0.796432
\(310\) 0 0
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) −2.00000 −0.113228
\(313\) 17.0000 0.960897 0.480448 0.877023i \(-0.340474\pi\)
0.480448 + 0.877023i \(0.340474\pi\)
\(314\) 22.0000 1.24153
\(315\) 0 0
\(316\) 14.0000 0.787562
\(317\) −18.0000 −1.01098 −0.505490 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 9.00000 0.504695
\(319\) 0 0
\(320\) 0 0
\(321\) −15.0000 −0.837218
\(322\) −9.00000 −0.501550
\(323\) 6.00000 0.333849
\(324\) 1.00000 0.0555556
\(325\) −10.0000 −0.554700
\(326\) 10.0000 0.553849
\(327\) −19.0000 −1.05070
\(328\) 0 0
\(329\) −9.00000 −0.496186
\(330\) 0 0
\(331\) 5.00000 0.274825 0.137412 0.990514i \(-0.456121\pi\)
0.137412 + 0.990514i \(0.456121\pi\)
\(332\) −12.0000 −0.658586
\(333\) 14.0000 0.767195
\(334\) −6.00000 −0.328305
\(335\) 0 0
\(336\) −1.00000 −0.0545545
\(337\) 32.0000 1.74315 0.871576 0.490261i \(-0.163099\pi\)
0.871576 + 0.490261i \(0.163099\pi\)
\(338\) 9.00000 0.489535
\(339\) 12.0000 0.651751
\(340\) 0 0
\(341\) 0 0
\(342\) 2.00000 0.108148
\(343\) 13.0000 0.701934
\(344\) 10.0000 0.539164
\(345\) 0 0
\(346\) −3.00000 −0.161281
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 3.00000 0.160817
\(349\) −10.0000 −0.535288 −0.267644 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) −5.00000 −0.267261
\(351\) −10.0000 −0.533761
\(352\) 0 0
\(353\) −15.0000 −0.798369 −0.399185 0.916871i \(-0.630707\pi\)
−0.399185 + 0.916871i \(0.630707\pi\)
\(354\) 9.00000 0.478345
\(355\) 0 0
\(356\) 6.00000 0.317999
\(357\) −6.00000 −0.317554
\(358\) −15.0000 −0.792775
\(359\) −12.0000 −0.633336 −0.316668 0.948536i \(-0.602564\pi\)
−0.316668 + 0.948536i \(0.602564\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 22.0000 1.15629
\(363\) 0 0
\(364\) −2.00000 −0.104828
\(365\) 0 0
\(366\) −8.00000 −0.418167
\(367\) 23.0000 1.20059 0.600295 0.799779i \(-0.295050\pi\)
0.600295 + 0.799779i \(0.295050\pi\)
\(368\) −9.00000 −0.469157
\(369\) 0 0
\(370\) 0 0
\(371\) 9.00000 0.467257
\(372\) 8.00000 0.414781
\(373\) −19.0000 −0.983783 −0.491891 0.870657i \(-0.663694\pi\)
−0.491891 + 0.870657i \(0.663694\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −9.00000 −0.464140
\(377\) 6.00000 0.309016
\(378\) −5.00000 −0.257172
\(379\) −13.0000 −0.667765 −0.333883 0.942615i \(-0.608359\pi\)
−0.333883 + 0.942615i \(0.608359\pi\)
\(380\) 0 0
\(381\) 8.00000 0.409852
\(382\) 9.00000 0.460480
\(383\) −12.0000 −0.613171 −0.306586 0.951843i \(-0.599187\pi\)
−0.306586 + 0.951843i \(0.599187\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 22.0000 1.11977
\(387\) 20.0000 1.01666
\(388\) −16.0000 −0.812277
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) −54.0000 −2.73090
\(392\) 6.00000 0.303046
\(393\) −18.0000 −0.907980
\(394\) 12.0000 0.604551
\(395\) 0 0
\(396\) 0 0
\(397\) −34.0000 −1.70641 −0.853206 0.521575i \(-0.825345\pi\)
−0.853206 + 0.521575i \(0.825345\pi\)
\(398\) 16.0000 0.802008
\(399\) −1.00000 −0.0500626
\(400\) −5.00000 −0.250000
\(401\) −12.0000 −0.599251 −0.299626 0.954057i \(-0.596862\pi\)
−0.299626 + 0.954057i \(0.596862\pi\)
\(402\) 4.00000 0.199502
\(403\) 16.0000 0.797017
\(404\) 0 0
\(405\) 0 0
\(406\) 3.00000 0.148888
\(407\) 0 0
\(408\) −6.00000 −0.297044
\(409\) −4.00000 −0.197787 −0.0988936 0.995098i \(-0.531530\pi\)
−0.0988936 + 0.995098i \(0.531530\pi\)
\(410\) 0 0
\(411\) −6.00000 −0.295958
\(412\) 14.0000 0.689730
\(413\) 9.00000 0.442861
\(414\) −18.0000 −0.884652
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) 8.00000 0.391762
\(418\) 0 0
\(419\) 24.0000 1.17248 0.586238 0.810139i \(-0.300608\pi\)
0.586238 + 0.810139i \(0.300608\pi\)
\(420\) 0 0
\(421\) −1.00000 −0.0487370 −0.0243685 0.999703i \(-0.507758\pi\)
−0.0243685 + 0.999703i \(0.507758\pi\)
\(422\) −5.00000 −0.243396
\(423\) −18.0000 −0.875190
\(424\) 9.00000 0.437079
\(425\) −30.0000 −1.45521
\(426\) −12.0000 −0.581402
\(427\) −8.00000 −0.387147
\(428\) −15.0000 −0.725052
\(429\) 0 0
\(430\) 0 0
\(431\) −18.0000 −0.867029 −0.433515 0.901146i \(-0.642727\pi\)
−0.433515 + 0.901146i \(0.642727\pi\)
\(432\) −5.00000 −0.240563
\(433\) −34.0000 −1.63394 −0.816968 0.576683i \(-0.804347\pi\)
−0.816968 + 0.576683i \(0.804347\pi\)
\(434\) 8.00000 0.384012
\(435\) 0 0
\(436\) −19.0000 −0.909935
\(437\) −9.00000 −0.430528
\(438\) −2.00000 −0.0955637
\(439\) −16.0000 −0.763638 −0.381819 0.924237i \(-0.624702\pi\)
−0.381819 + 0.924237i \(0.624702\pi\)
\(440\) 0 0
\(441\) 12.0000 0.571429
\(442\) −12.0000 −0.570782
\(443\) −6.00000 −0.285069 −0.142534 0.989790i \(-0.545525\pi\)
−0.142534 + 0.989790i \(0.545525\pi\)
\(444\) −7.00000 −0.332205
\(445\) 0 0
\(446\) 16.0000 0.757622
\(447\) −12.0000 −0.567581
\(448\) −1.00000 −0.0472456
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) −10.0000 −0.471405
\(451\) 0 0
\(452\) 12.0000 0.564433
\(453\) −10.0000 −0.469841
\(454\) −12.0000 −0.563188
\(455\) 0 0
\(456\) −1.00000 −0.0468293
\(457\) 17.0000 0.795226 0.397613 0.917553i \(-0.369839\pi\)
0.397613 + 0.917553i \(0.369839\pi\)
\(458\) 4.00000 0.186908
\(459\) −30.0000 −1.40028
\(460\) 0 0
\(461\) 24.0000 1.11779 0.558896 0.829238i \(-0.311225\pi\)
0.558896 + 0.829238i \(0.311225\pi\)
\(462\) 0 0
\(463\) −16.0000 −0.743583 −0.371792 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) 3.00000 0.139272
\(465\) 0 0
\(466\) −27.0000 −1.25075
\(467\) −6.00000 −0.277647 −0.138823 0.990317i \(-0.544332\pi\)
−0.138823 + 0.990317i \(0.544332\pi\)
\(468\) −4.00000 −0.184900
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) −22.0000 −1.01371
\(472\) 9.00000 0.414259
\(473\) 0 0
\(474\) −14.0000 −0.643041
\(475\) −5.00000 −0.229416
\(476\) −6.00000 −0.275010
\(477\) 18.0000 0.824163
\(478\) −9.00000 −0.411650
\(479\) −21.0000 −0.959514 −0.479757 0.877401i \(-0.659275\pi\)
−0.479757 + 0.877401i \(0.659275\pi\)
\(480\) 0 0
\(481\) −14.0000 −0.638345
\(482\) −2.00000 −0.0910975
\(483\) 9.00000 0.409514
\(484\) 0 0
\(485\) 0 0
\(486\) −16.0000 −0.725775
\(487\) 2.00000 0.0906287 0.0453143 0.998973i \(-0.485571\pi\)
0.0453143 + 0.998973i \(0.485571\pi\)
\(488\) −8.00000 −0.362143
\(489\) −10.0000 −0.452216
\(490\) 0 0
\(491\) 24.0000 1.08310 0.541552 0.840667i \(-0.317837\pi\)
0.541552 + 0.840667i \(0.317837\pi\)
\(492\) 0 0
\(493\) 18.0000 0.810679
\(494\) −2.00000 −0.0899843
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) −12.0000 −0.538274
\(498\) 12.0000 0.537733
\(499\) −10.0000 −0.447661 −0.223831 0.974628i \(-0.571856\pi\)
−0.223831 + 0.974628i \(0.571856\pi\)
\(500\) 0 0
\(501\) 6.00000 0.268060
\(502\) −18.0000 −0.803379
\(503\) −9.00000 −0.401290 −0.200645 0.979664i \(-0.564304\pi\)
−0.200645 + 0.979664i \(0.564304\pi\)
\(504\) −2.00000 −0.0890871
\(505\) 0 0
\(506\) 0 0
\(507\) −9.00000 −0.399704
\(508\) 8.00000 0.354943
\(509\) −27.0000 −1.19675 −0.598377 0.801215i \(-0.704187\pi\)
−0.598377 + 0.801215i \(0.704187\pi\)
\(510\) 0 0
\(511\) −2.00000 −0.0884748
\(512\) −1.00000 −0.0441942
\(513\) −5.00000 −0.220755
\(514\) 12.0000 0.529297
\(515\) 0 0
\(516\) −10.0000 −0.440225
\(517\) 0 0
\(518\) −7.00000 −0.307562
\(519\) 3.00000 0.131685
\(520\) 0 0
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 6.00000 0.262613
\(523\) −7.00000 −0.306089 −0.153044 0.988219i \(-0.548908\pi\)
−0.153044 + 0.988219i \(0.548908\pi\)
\(524\) −18.0000 −0.786334
\(525\) 5.00000 0.218218
\(526\) 24.0000 1.04645
\(527\) 48.0000 2.09091
\(528\) 0 0
\(529\) 58.0000 2.52174
\(530\) 0 0
\(531\) 18.0000 0.781133
\(532\) −1.00000 −0.0433555
\(533\) 0 0
\(534\) −6.00000 −0.259645
\(535\) 0 0
\(536\) 4.00000 0.172774
\(537\) 15.0000 0.647298
\(538\) 3.00000 0.129339
\(539\) 0 0
\(540\) 0 0
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) 16.0000 0.687259
\(543\) −22.0000 −0.944110
\(544\) −6.00000 −0.257248
\(545\) 0 0
\(546\) 2.00000 0.0855921
\(547\) 23.0000 0.983409 0.491704 0.870762i \(-0.336374\pi\)
0.491704 + 0.870762i \(0.336374\pi\)
\(548\) −6.00000 −0.256307
\(549\) −16.0000 −0.682863
\(550\) 0 0
\(551\) 3.00000 0.127804
\(552\) 9.00000 0.383065
\(553\) −14.0000 −0.595341
\(554\) 4.00000 0.169944
\(555\) 0 0
\(556\) 8.00000 0.339276
\(557\) 30.0000 1.27114 0.635570 0.772043i \(-0.280765\pi\)
0.635570 + 0.772043i \(0.280765\pi\)
\(558\) 16.0000 0.677334
\(559\) −20.0000 −0.845910
\(560\) 0 0
\(561\) 0 0
\(562\) −6.00000 −0.253095
\(563\) 9.00000 0.379305 0.189652 0.981851i \(-0.439264\pi\)
0.189652 + 0.981851i \(0.439264\pi\)
\(564\) 9.00000 0.378968
\(565\) 0 0
\(566\) 16.0000 0.672530
\(567\) −1.00000 −0.0419961
\(568\) −12.0000 −0.503509
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 38.0000 1.59025 0.795125 0.606445i \(-0.207405\pi\)
0.795125 + 0.606445i \(0.207405\pi\)
\(572\) 0 0
\(573\) −9.00000 −0.375980
\(574\) 0 0
\(575\) 45.0000 1.87663
\(576\) −2.00000 −0.0833333
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) −19.0000 −0.790296
\(579\) −22.0000 −0.914289
\(580\) 0 0
\(581\) 12.0000 0.497844
\(582\) 16.0000 0.663221
\(583\) 0 0
\(584\) −2.00000 −0.0827606
\(585\) 0 0
\(586\) −33.0000 −1.36322
\(587\) 24.0000 0.990586 0.495293 0.868726i \(-0.335061\pi\)
0.495293 + 0.868726i \(0.335061\pi\)
\(588\) −6.00000 −0.247436
\(589\) 8.00000 0.329634
\(590\) 0 0
\(591\) −12.0000 −0.493614
\(592\) −7.00000 −0.287698
\(593\) 15.0000 0.615976 0.307988 0.951390i \(-0.400344\pi\)
0.307988 + 0.951390i \(0.400344\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −12.0000 −0.491539
\(597\) −16.0000 −0.654836
\(598\) 18.0000 0.736075
\(599\) −12.0000 −0.490307 −0.245153 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) 5.00000 0.204124
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) −10.0000 −0.407570
\(603\) 8.00000 0.325785
\(604\) −10.0000 −0.406894
\(605\) 0 0
\(606\) 0 0
\(607\) −10.0000 −0.405887 −0.202944 0.979190i \(-0.565051\pi\)
−0.202944 + 0.979190i \(0.565051\pi\)
\(608\) −1.00000 −0.0405554
\(609\) −3.00000 −0.121566
\(610\) 0 0
\(611\) 18.0000 0.728202
\(612\) −12.0000 −0.485071
\(613\) 14.0000 0.565455 0.282727 0.959200i \(-0.408761\pi\)
0.282727 + 0.959200i \(0.408761\pi\)
\(614\) −17.0000 −0.686064
\(615\) 0 0
\(616\) 0 0
\(617\) 45.0000 1.81163 0.905816 0.423672i \(-0.139259\pi\)
0.905816 + 0.423672i \(0.139259\pi\)
\(618\) −14.0000 −0.563163
\(619\) 14.0000 0.562708 0.281354 0.959604i \(-0.409217\pi\)
0.281354 + 0.959604i \(0.409217\pi\)
\(620\) 0 0
\(621\) 45.0000 1.80579
\(622\) 12.0000 0.481156
\(623\) −6.00000 −0.240385
\(624\) 2.00000 0.0800641
\(625\) 25.0000 1.00000
\(626\) −17.0000 −0.679457
\(627\) 0 0
\(628\) −22.0000 −0.877896
\(629\) −42.0000 −1.67465
\(630\) 0 0
\(631\) 35.0000 1.39333 0.696664 0.717398i \(-0.254667\pi\)
0.696664 + 0.717398i \(0.254667\pi\)
\(632\) −14.0000 −0.556890
\(633\) 5.00000 0.198732
\(634\) 18.0000 0.714871
\(635\) 0 0
\(636\) −9.00000 −0.356873
\(637\) −12.0000 −0.475457
\(638\) 0 0
\(639\) −24.0000 −0.949425
\(640\) 0 0
\(641\) 24.0000 0.947943 0.473972 0.880540i \(-0.342820\pi\)
0.473972 + 0.880540i \(0.342820\pi\)
\(642\) 15.0000 0.592003
\(643\) 26.0000 1.02534 0.512670 0.858586i \(-0.328656\pi\)
0.512670 + 0.858586i \(0.328656\pi\)
\(644\) 9.00000 0.354650
\(645\) 0 0
\(646\) −6.00000 −0.236067
\(647\) 45.0000 1.76913 0.884566 0.466415i \(-0.154454\pi\)
0.884566 + 0.466415i \(0.154454\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) 10.0000 0.392232
\(651\) −8.00000 −0.313545
\(652\) −10.0000 −0.391630
\(653\) −6.00000 −0.234798 −0.117399 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) 19.0000 0.742959
\(655\) 0 0
\(656\) 0 0
\(657\) −4.00000 −0.156055
\(658\) 9.00000 0.350857
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) −19.0000 −0.739014 −0.369507 0.929228i \(-0.620473\pi\)
−0.369507 + 0.929228i \(0.620473\pi\)
\(662\) −5.00000 −0.194331
\(663\) 12.0000 0.466041
\(664\) 12.0000 0.465690
\(665\) 0 0
\(666\) −14.0000 −0.542489
\(667\) −27.0000 −1.04544
\(668\) 6.00000 0.232147
\(669\) −16.0000 −0.618596
\(670\) 0 0
\(671\) 0 0
\(672\) 1.00000 0.0385758
\(673\) −46.0000 −1.77317 −0.886585 0.462566i \(-0.846929\pi\)
−0.886585 + 0.462566i \(0.846929\pi\)
\(674\) −32.0000 −1.23259
\(675\) 25.0000 0.962250
\(676\) −9.00000 −0.346154
\(677\) −3.00000 −0.115299 −0.0576497 0.998337i \(-0.518361\pi\)
−0.0576497 + 0.998337i \(0.518361\pi\)
\(678\) −12.0000 −0.460857
\(679\) 16.0000 0.614024
\(680\) 0 0
\(681\) 12.0000 0.459841
\(682\) 0 0
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) −2.00000 −0.0764719
\(685\) 0 0
\(686\) −13.0000 −0.496342
\(687\) −4.00000 −0.152610
\(688\) −10.0000 −0.381246
\(689\) −18.0000 −0.685745
\(690\) 0 0
\(691\) −40.0000 −1.52167 −0.760836 0.648944i \(-0.775211\pi\)
−0.760836 + 0.648944i \(0.775211\pi\)
\(692\) 3.00000 0.114043
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) 0 0
\(696\) −3.00000 −0.113715
\(697\) 0 0
\(698\) 10.0000 0.378506
\(699\) 27.0000 1.02123
\(700\) 5.00000 0.188982
\(701\) 24.0000 0.906467 0.453234 0.891392i \(-0.350270\pi\)
0.453234 + 0.891392i \(0.350270\pi\)
\(702\) 10.0000 0.377426
\(703\) −7.00000 −0.264010
\(704\) 0 0
\(705\) 0 0
\(706\) 15.0000 0.564532
\(707\) 0 0
\(708\) −9.00000 −0.338241
\(709\) −16.0000 −0.600893 −0.300446 0.953799i \(-0.597136\pi\)
−0.300446 + 0.953799i \(0.597136\pi\)
\(710\) 0 0
\(711\) −28.0000 −1.05008
\(712\) −6.00000 −0.224860
\(713\) −72.0000 −2.69642
\(714\) 6.00000 0.224544
\(715\) 0 0
\(716\) 15.0000 0.560576
\(717\) 9.00000 0.336111
\(718\) 12.0000 0.447836
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 0 0
\(721\) −14.0000 −0.521387
\(722\) −1.00000 −0.0372161
\(723\) 2.00000 0.0743808
\(724\) −22.0000 −0.817624
\(725\) −15.0000 −0.557086
\(726\) 0 0
\(727\) −19.0000 −0.704671 −0.352335 0.935874i \(-0.614612\pi\)
−0.352335 + 0.935874i \(0.614612\pi\)
\(728\) 2.00000 0.0741249
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −60.0000 −2.21918
\(732\) 8.00000 0.295689
\(733\) −4.00000 −0.147743 −0.0738717 0.997268i \(-0.523536\pi\)
−0.0738717 + 0.997268i \(0.523536\pi\)
\(734\) −23.0000 −0.848945
\(735\) 0 0
\(736\) 9.00000 0.331744
\(737\) 0 0
\(738\) 0 0
\(739\) 14.0000 0.514998 0.257499 0.966279i \(-0.417102\pi\)
0.257499 + 0.966279i \(0.417102\pi\)
\(740\) 0 0
\(741\) 2.00000 0.0734718
\(742\) −9.00000 −0.330400
\(743\) 30.0000 1.10059 0.550297 0.834969i \(-0.314515\pi\)
0.550297 + 0.834969i \(0.314515\pi\)
\(744\) −8.00000 −0.293294
\(745\) 0 0
\(746\) 19.0000 0.695639
\(747\) 24.0000 0.878114
\(748\) 0 0
\(749\) 15.0000 0.548088
\(750\) 0 0
\(751\) 20.0000 0.729810 0.364905 0.931045i \(-0.381101\pi\)
0.364905 + 0.931045i \(0.381101\pi\)
\(752\) 9.00000 0.328196
\(753\) 18.0000 0.655956
\(754\) −6.00000 −0.218507
\(755\) 0 0
\(756\) 5.00000 0.181848
\(757\) −46.0000 −1.67190 −0.835949 0.548807i \(-0.815082\pi\)
−0.835949 + 0.548807i \(0.815082\pi\)
\(758\) 13.0000 0.472181
\(759\) 0 0
\(760\) 0 0
\(761\) −21.0000 −0.761249 −0.380625 0.924730i \(-0.624291\pi\)
−0.380625 + 0.924730i \(0.624291\pi\)
\(762\) −8.00000 −0.289809
\(763\) 19.0000 0.687846
\(764\) −9.00000 −0.325609
\(765\) 0 0
\(766\) 12.0000 0.433578
\(767\) −18.0000 −0.649942
\(768\) 1.00000 0.0360844
\(769\) −7.00000 −0.252426 −0.126213 0.992003i \(-0.540282\pi\)
−0.126213 + 0.992003i \(0.540282\pi\)
\(770\) 0 0
\(771\) −12.0000 −0.432169
\(772\) −22.0000 −0.791797
\(773\) 21.0000 0.755318 0.377659 0.925945i \(-0.376729\pi\)
0.377659 + 0.925945i \(0.376729\pi\)
\(774\) −20.0000 −0.718885
\(775\) −40.0000 −1.43684
\(776\) 16.0000 0.574367
\(777\) 7.00000 0.251124
\(778\) −6.00000 −0.215110
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 54.0000 1.93104
\(783\) −15.0000 −0.536056
\(784\) −6.00000 −0.214286
\(785\) 0 0
\(786\) 18.0000 0.642039
\(787\) −7.00000 −0.249523 −0.124762 0.992187i \(-0.539817\pi\)
−0.124762 + 0.992187i \(0.539817\pi\)
\(788\) −12.0000 −0.427482
\(789\) −24.0000 −0.854423
\(790\) 0 0
\(791\) −12.0000 −0.426671
\(792\) 0 0
\(793\) 16.0000 0.568177
\(794\) 34.0000 1.20661
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) 30.0000 1.06265 0.531327 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) 1.00000 0.0353996
\(799\) 54.0000 1.91038
\(800\) 5.00000 0.176777
\(801\) −12.0000 −0.423999
\(802\) 12.0000 0.423735
\(803\) 0 0
\(804\) −4.00000 −0.141069
\(805\) 0 0
\(806\) −16.0000 −0.563576
\(807\) −3.00000 −0.105605
\(808\) 0 0
\(809\) 45.0000 1.58212 0.791058 0.611741i \(-0.209531\pi\)
0.791058 + 0.611741i \(0.209531\pi\)
\(810\) 0 0
\(811\) 35.0000 1.22902 0.614508 0.788911i \(-0.289355\pi\)
0.614508 + 0.788911i \(0.289355\pi\)
\(812\) −3.00000 −0.105279
\(813\) −16.0000 −0.561144
\(814\) 0 0
\(815\) 0 0
\(816\) 6.00000 0.210042
\(817\) −10.0000 −0.349856
\(818\) 4.00000 0.139857
\(819\) 4.00000 0.139771
\(820\) 0 0
\(821\) −48.0000 −1.67521 −0.837606 0.546275i \(-0.816045\pi\)
−0.837606 + 0.546275i \(0.816045\pi\)
\(822\) 6.00000 0.209274
\(823\) 23.0000 0.801730 0.400865 0.916137i \(-0.368710\pi\)
0.400865 + 0.916137i \(0.368710\pi\)
\(824\) −14.0000 −0.487713
\(825\) 0 0
\(826\) −9.00000 −0.313150
\(827\) 36.0000 1.25184 0.625921 0.779886i \(-0.284723\pi\)
0.625921 + 0.779886i \(0.284723\pi\)
\(828\) 18.0000 0.625543
\(829\) 38.0000 1.31979 0.659897 0.751356i \(-0.270600\pi\)
0.659897 + 0.751356i \(0.270600\pi\)
\(830\) 0 0
\(831\) −4.00000 −0.138758
\(832\) 2.00000 0.0693375
\(833\) −36.0000 −1.24733
\(834\) −8.00000 −0.277017
\(835\) 0 0
\(836\) 0 0
\(837\) −40.0000 −1.38260
\(838\) −24.0000 −0.829066
\(839\) 24.0000 0.828572 0.414286 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 1.00000 0.0344623
\(843\) 6.00000 0.206651
\(844\) 5.00000 0.172107
\(845\) 0 0
\(846\) 18.0000 0.618853
\(847\) 0 0
\(848\) −9.00000 −0.309061
\(849\) −16.0000 −0.549119
\(850\) 30.0000 1.02899
\(851\) 63.0000 2.15961
\(852\) 12.0000 0.411113
\(853\) −46.0000 −1.57501 −0.787505 0.616308i \(-0.788628\pi\)
−0.787505 + 0.616308i \(0.788628\pi\)
\(854\) 8.00000 0.273754
\(855\) 0 0
\(856\) 15.0000 0.512689
\(857\) −42.0000 −1.43469 −0.717346 0.696717i \(-0.754643\pi\)
−0.717346 + 0.696717i \(0.754643\pi\)
\(858\) 0 0
\(859\) −40.0000 −1.36478 −0.682391 0.730987i \(-0.739060\pi\)
−0.682391 + 0.730987i \(0.739060\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 18.0000 0.613082
\(863\) 36.0000 1.22545 0.612727 0.790295i \(-0.290072\pi\)
0.612727 + 0.790295i \(0.290072\pi\)
\(864\) 5.00000 0.170103
\(865\) 0 0
\(866\) 34.0000 1.15537
\(867\) 19.0000 0.645274
\(868\) −8.00000 −0.271538
\(869\) 0 0
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) 19.0000 0.643421
\(873\) 32.0000 1.08304
\(874\) 9.00000 0.304430
\(875\) 0 0
\(876\) 2.00000 0.0675737
\(877\) 14.0000 0.472746 0.236373 0.971662i \(-0.424041\pi\)
0.236373 + 0.971662i \(0.424041\pi\)
\(878\) 16.0000 0.539974
\(879\) 33.0000 1.11306
\(880\) 0 0
\(881\) 45.0000 1.51609 0.758044 0.652203i \(-0.226155\pi\)
0.758044 + 0.652203i \(0.226155\pi\)
\(882\) −12.0000 −0.404061
\(883\) −10.0000 −0.336527 −0.168263 0.985742i \(-0.553816\pi\)
−0.168263 + 0.985742i \(0.553816\pi\)
\(884\) 12.0000 0.403604
\(885\) 0 0
\(886\) 6.00000 0.201574
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 7.00000 0.234905
\(889\) −8.00000 −0.268311
\(890\) 0 0
\(891\) 0 0
\(892\) −16.0000 −0.535720
\(893\) 9.00000 0.301174
\(894\) 12.0000 0.401340
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) −18.0000 −0.601003
\(898\) −6.00000 −0.200223
\(899\) 24.0000 0.800445
\(900\) 10.0000 0.333333
\(901\) −54.0000 −1.79900
\(902\) 0 0
\(903\) 10.0000 0.332779
\(904\) −12.0000 −0.399114
\(905\) 0 0
\(906\) 10.0000 0.332228
\(907\) −19.0000 −0.630885 −0.315442 0.948945i \(-0.602153\pi\)
−0.315442 + 0.948945i \(0.602153\pi\)
\(908\) 12.0000 0.398234
\(909\) 0 0
\(910\) 0 0
\(911\) 18.0000 0.596367 0.298183 0.954509i \(-0.403619\pi\)
0.298183 + 0.954509i \(0.403619\pi\)
\(912\) 1.00000 0.0331133
\(913\) 0 0
\(914\) −17.0000 −0.562310
\(915\) 0 0
\(916\) −4.00000 −0.132164
\(917\) 18.0000 0.594412
\(918\) 30.0000 0.990148
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 0 0
\(921\) 17.0000 0.560169
\(922\) −24.0000 −0.790398
\(923\) 24.0000 0.789970
\(924\) 0 0
\(925\) 35.0000 1.15079
\(926\) 16.0000 0.525793
\(927\) −28.0000 −0.919641
\(928\) −3.00000 −0.0984798
\(929\) 45.0000 1.47640 0.738201 0.674581i \(-0.235676\pi\)
0.738201 + 0.674581i \(0.235676\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) 27.0000 0.884414
\(933\) −12.0000 −0.392862
\(934\) 6.00000 0.196326
\(935\) 0 0
\(936\) 4.00000 0.130744
\(937\) −19.0000 −0.620703 −0.310351 0.950622i \(-0.600447\pi\)
−0.310351 + 0.950622i \(0.600447\pi\)
\(938\) −4.00000 −0.130605
\(939\) 17.0000 0.554774
\(940\) 0 0
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) 22.0000 0.716799
\(943\) 0 0
\(944\) −9.00000 −0.292925
\(945\) 0 0
\(946\) 0 0
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) 14.0000 0.454699
\(949\) 4.00000 0.129845
\(950\) 5.00000 0.162221
\(951\) −18.0000 −0.583690
\(952\) 6.00000 0.194461
\(953\) −30.0000 −0.971795 −0.485898 0.874016i \(-0.661507\pi\)
−0.485898 + 0.874016i \(0.661507\pi\)
\(954\) −18.0000 −0.582772
\(955\) 0 0
\(956\) 9.00000 0.291081
\(957\) 0 0
\(958\) 21.0000 0.678479
\(959\) 6.00000 0.193750
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 14.0000 0.451378
\(963\) 30.0000 0.966736
\(964\) 2.00000 0.0644157
\(965\) 0 0
\(966\) −9.00000 −0.289570
\(967\) 59.0000 1.89731 0.948656 0.316310i \(-0.102444\pi\)
0.948656 + 0.316310i \(0.102444\pi\)
\(968\) 0 0
\(969\) 6.00000 0.192748
\(970\) 0 0
\(971\) 39.0000 1.25157 0.625785 0.779996i \(-0.284779\pi\)
0.625785 + 0.779996i \(0.284779\pi\)
\(972\) 16.0000 0.513200
\(973\) −8.00000 −0.256468
\(974\) −2.00000 −0.0640841
\(975\) −10.0000 −0.320256
\(976\) 8.00000 0.256074
\(977\) −30.0000 −0.959785 −0.479893 0.877327i \(-0.659324\pi\)
−0.479893 + 0.877327i \(0.659324\pi\)
\(978\) 10.0000 0.319765
\(979\) 0 0
\(980\) 0 0
\(981\) 38.0000 1.21325
\(982\) −24.0000 −0.765871
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −18.0000 −0.573237
\(987\) −9.00000 −0.286473
\(988\) 2.00000 0.0636285
\(989\) 90.0000 2.86183
\(990\) 0 0
\(991\) −28.0000 −0.889449 −0.444725 0.895667i \(-0.646698\pi\)
−0.444725 + 0.895667i \(0.646698\pi\)
\(992\) −8.00000 −0.254000
\(993\) 5.00000 0.158670
\(994\) 12.0000 0.380617
\(995\) 0 0
\(996\) −12.0000 −0.380235
\(997\) −52.0000 −1.64686 −0.823428 0.567420i \(-0.807941\pi\)
−0.823428 + 0.567420i \(0.807941\pi\)
\(998\) 10.0000 0.316544
\(999\) 35.0000 1.10735
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 4598.2.a.h.1.1 1
11.10 odd 2 4598.2.a.q.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4598.2.a.h.1.1 1 1.1 even 1 trivial
4598.2.a.q.1.1 yes 1 11.10 odd 2