Properties

Label 3822.2.a.e.1.1
Level $3822$
Weight $2$
Character 3822.1
Self dual yes
Analytic conductor $30.519$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3822.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.5188236525\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3822.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +3.00000 q^{11} -1.00000 q^{12} +1.00000 q^{13} +1.00000 q^{15} +1.00000 q^{16} -5.00000 q^{17} -1.00000 q^{18} -1.00000 q^{19} -1.00000 q^{20} -3.00000 q^{22} +3.00000 q^{23} +1.00000 q^{24} -4.00000 q^{25} -1.00000 q^{26} -1.00000 q^{27} +5.00000 q^{29} -1.00000 q^{30} -4.00000 q^{31} -1.00000 q^{32} -3.00000 q^{33} +5.00000 q^{34} +1.00000 q^{36} -5.00000 q^{37} +1.00000 q^{38} -1.00000 q^{39} +1.00000 q^{40} +8.00000 q^{41} -1.00000 q^{43} +3.00000 q^{44} -1.00000 q^{45} -3.00000 q^{46} -8.00000 q^{47} -1.00000 q^{48} +4.00000 q^{50} +5.00000 q^{51} +1.00000 q^{52} +6.00000 q^{53} +1.00000 q^{54} -3.00000 q^{55} +1.00000 q^{57} -5.00000 q^{58} +1.00000 q^{60} -13.0000 q^{61} +4.00000 q^{62} +1.00000 q^{64} -1.00000 q^{65} +3.00000 q^{66} -10.0000 q^{67} -5.00000 q^{68} -3.00000 q^{69} +8.00000 q^{71} -1.00000 q^{72} +15.0000 q^{73} +5.00000 q^{74} +4.00000 q^{75} -1.00000 q^{76} +1.00000 q^{78} +6.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -8.00000 q^{82} +2.00000 q^{83} +5.00000 q^{85} +1.00000 q^{86} -5.00000 q^{87} -3.00000 q^{88} +2.00000 q^{89} +1.00000 q^{90} +3.00000 q^{92} +4.00000 q^{93} +8.00000 q^{94} +1.00000 q^{95} +1.00000 q^{96} +2.00000 q^{97} +3.00000 q^{99} +O(q^{100})\)

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
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214 −0.223607 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) −1.00000 −0.288675
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) −5.00000 −1.21268 −0.606339 0.795206i \(-0.707363\pi\)
−0.606339 + 0.795206i \(0.707363\pi\)
\(18\) −1.00000 −0.235702
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) −3.00000 −0.639602
\(23\) 3.00000 0.625543 0.312772 0.949828i \(-0.398743\pi\)
0.312772 + 0.949828i \(0.398743\pi\)
\(24\) 1.00000 0.204124
\(25\) −4.00000 −0.800000
\(26\) −1.00000 −0.196116
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 5.00000 0.928477 0.464238 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) −1.00000 −0.182574
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −1.00000 −0.176777
\(33\) −3.00000 −0.522233
\(34\) 5.00000 0.857493
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −5.00000 −0.821995 −0.410997 0.911636i \(-0.634819\pi\)
−0.410997 + 0.911636i \(0.634819\pi\)
\(38\) 1.00000 0.162221
\(39\) −1.00000 −0.160128
\(40\) 1.00000 0.158114
\(41\) 8.00000 1.24939 0.624695 0.780869i \(-0.285223\pi\)
0.624695 + 0.780869i \(0.285223\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499 −0.0762493 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) 3.00000 0.452267
\(45\) −1.00000 −0.149071
\(46\) −3.00000 −0.442326
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) 4.00000 0.565685
\(51\) 5.00000 0.700140
\(52\) 1.00000 0.138675
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 1.00000 0.136083
\(55\) −3.00000 −0.404520
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) −5.00000 −0.656532
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 1.00000 0.129099
\(61\) −13.0000 −1.66448 −0.832240 0.554416i \(-0.812942\pi\)
−0.832240 + 0.554416i \(0.812942\pi\)
\(62\) 4.00000 0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −1.00000 −0.124035
\(66\) 3.00000 0.369274
\(67\) −10.0000 −1.22169 −0.610847 0.791748i \(-0.709171\pi\)
−0.610847 + 0.791748i \(0.709171\pi\)
\(68\) −5.00000 −0.606339
\(69\) −3.00000 −0.361158
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) −1.00000 −0.117851
\(73\) 15.0000 1.75562 0.877809 0.479012i \(-0.159005\pi\)
0.877809 + 0.479012i \(0.159005\pi\)
\(74\) 5.00000 0.581238
\(75\) 4.00000 0.461880
\(76\) −1.00000 −0.114708
\(77\) 0 0
\(78\) 1.00000 0.113228
\(79\) 6.00000 0.675053 0.337526 0.941316i \(-0.390410\pi\)
0.337526 + 0.941316i \(0.390410\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −8.00000 −0.883452
\(83\) 2.00000 0.219529 0.109764 0.993958i \(-0.464990\pi\)
0.109764 + 0.993958i \(0.464990\pi\)
\(84\) 0 0
\(85\) 5.00000 0.542326
\(86\) 1.00000 0.107833
\(87\) −5.00000 −0.536056
\(88\) −3.00000 −0.319801
\(89\) 2.00000 0.212000 0.106000 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) 3.00000 0.312772
\(93\) 4.00000 0.414781
\(94\) 8.00000 0.825137
\(95\) 1.00000 0.102598
\(96\) 1.00000 0.102062
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) 3.00000 0.301511
\(100\) −4.00000 −0.400000
\(101\) 16.0000 1.59206 0.796030 0.605257i \(-0.206930\pi\)
0.796030 + 0.605257i \(0.206930\pi\)
\(102\) −5.00000 −0.495074
\(103\) −1.00000 −0.0985329 −0.0492665 0.998786i \(-0.515688\pi\)
−0.0492665 + 0.998786i \(0.515688\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −7.00000 −0.670478 −0.335239 0.942133i \(-0.608817\pi\)
−0.335239 + 0.942133i \(0.608817\pi\)
\(110\) 3.00000 0.286039
\(111\) 5.00000 0.474579
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) −1.00000 −0.0936586
\(115\) −3.00000 −0.279751
\(116\) 5.00000 0.464238
\(117\) 1.00000 0.0924500
\(118\) 0 0
\(119\) 0 0
\(120\) −1.00000 −0.0912871
\(121\) −2.00000 −0.181818
\(122\) 13.0000 1.17696
\(123\) −8.00000 −0.721336
\(124\) −4.00000 −0.359211
\(125\) 9.00000 0.804984
\(126\) 0 0
\(127\) −18.0000 −1.59724 −0.798621 0.601834i \(-0.794437\pi\)
−0.798621 + 0.601834i \(0.794437\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 1.00000 0.0880451
\(130\) 1.00000 0.0877058
\(131\) −17.0000 −1.48530 −0.742648 0.669681i \(-0.766431\pi\)
−0.742648 + 0.669681i \(0.766431\pi\)
\(132\) −3.00000 −0.261116
\(133\) 0 0
\(134\) 10.0000 0.863868
\(135\) 1.00000 0.0860663
\(136\) 5.00000 0.428746
\(137\) −15.0000 −1.28154 −0.640768 0.767734i \(-0.721384\pi\)
−0.640768 + 0.767734i \(0.721384\pi\)
\(138\) 3.00000 0.255377
\(139\) 16.0000 1.35710 0.678551 0.734553i \(-0.262608\pi\)
0.678551 + 0.734553i \(0.262608\pi\)
\(140\) 0 0
\(141\) 8.00000 0.673722
\(142\) −8.00000 −0.671345
\(143\) 3.00000 0.250873
\(144\) 1.00000 0.0833333
\(145\) −5.00000 −0.415227
\(146\) −15.0000 −1.24141
\(147\) 0 0
\(148\) −5.00000 −0.410997
\(149\) −16.0000 −1.31077 −0.655386 0.755295i \(-0.727494\pi\)
−0.655386 + 0.755295i \(0.727494\pi\)
\(150\) −4.00000 −0.326599
\(151\) −5.00000 −0.406894 −0.203447 0.979086i \(-0.565214\pi\)
−0.203447 + 0.979086i \(0.565214\pi\)
\(152\) 1.00000 0.0811107
\(153\) −5.00000 −0.404226
\(154\) 0 0
\(155\) 4.00000 0.321288
\(156\) −1.00000 −0.0800641
\(157\) −9.00000 −0.718278 −0.359139 0.933284i \(-0.616930\pi\)
−0.359139 + 0.933284i \(0.616930\pi\)
\(158\) −6.00000 −0.477334
\(159\) −6.00000 −0.475831
\(160\) 1.00000 0.0790569
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 10.0000 0.783260 0.391630 0.920123i \(-0.371911\pi\)
0.391630 + 0.920123i \(0.371911\pi\)
\(164\) 8.00000 0.624695
\(165\) 3.00000 0.233550
\(166\) −2.00000 −0.155230
\(167\) 15.0000 1.16073 0.580367 0.814355i \(-0.302909\pi\)
0.580367 + 0.814355i \(0.302909\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) −5.00000 −0.383482
\(171\) −1.00000 −0.0764719
\(172\) −1.00000 −0.0762493
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 5.00000 0.379049
\(175\) 0 0
\(176\) 3.00000 0.226134
\(177\) 0 0
\(178\) −2.00000 −0.149906
\(179\) −16.0000 −1.19590 −0.597948 0.801535i \(-0.704017\pi\)
−0.597948 + 0.801535i \(0.704017\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −6.00000 −0.445976 −0.222988 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) 0 0
\(183\) 13.0000 0.960988
\(184\) −3.00000 −0.221163
\(185\) 5.00000 0.367607
\(186\) −4.00000 −0.293294
\(187\) −15.0000 −1.09691
\(188\) −8.00000 −0.583460
\(189\) 0 0
\(190\) −1.00000 −0.0725476
\(191\) 11.0000 0.795932 0.397966 0.917400i \(-0.369716\pi\)
0.397966 + 0.917400i \(0.369716\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −20.0000 −1.43963 −0.719816 0.694165i \(-0.755774\pi\)
−0.719816 + 0.694165i \(0.755774\pi\)
\(194\) −2.00000 −0.143592
\(195\) 1.00000 0.0716115
\(196\) 0 0
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) −3.00000 −0.213201
\(199\) 5.00000 0.354441 0.177220 0.984171i \(-0.443289\pi\)
0.177220 + 0.984171i \(0.443289\pi\)
\(200\) 4.00000 0.282843
\(201\) 10.0000 0.705346
\(202\) −16.0000 −1.12576
\(203\) 0 0
\(204\) 5.00000 0.350070
\(205\) −8.00000 −0.558744
\(206\) 1.00000 0.0696733
\(207\) 3.00000 0.208514
\(208\) 1.00000 0.0693375
\(209\) −3.00000 −0.207514
\(210\) 0 0
\(211\) 5.00000 0.344214 0.172107 0.985078i \(-0.444942\pi\)
0.172107 + 0.985078i \(0.444942\pi\)
\(212\) 6.00000 0.412082
\(213\) −8.00000 −0.548151
\(214\) 6.00000 0.410152
\(215\) 1.00000 0.0681994
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) 7.00000 0.474100
\(219\) −15.0000 −1.01361
\(220\) −3.00000 −0.202260
\(221\) −5.00000 −0.336336
\(222\) −5.00000 −0.335578
\(223\) −26.0000 −1.74109 −0.870544 0.492090i \(-0.836233\pi\)
−0.870544 + 0.492090i \(0.836233\pi\)
\(224\) 0 0
\(225\) −4.00000 −0.266667
\(226\) 6.00000 0.399114
\(227\) −2.00000 −0.132745 −0.0663723 0.997795i \(-0.521143\pi\)
−0.0663723 + 0.997795i \(0.521143\pi\)
\(228\) 1.00000 0.0662266
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 3.00000 0.197814
\(231\) 0 0
\(232\) −5.00000 −0.328266
\(233\) −4.00000 −0.262049 −0.131024 0.991379i \(-0.541827\pi\)
−0.131024 + 0.991379i \(0.541827\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 8.00000 0.521862
\(236\) 0 0
\(237\) −6.00000 −0.389742
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 1.00000 0.0645497
\(241\) −22.0000 −1.41714 −0.708572 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) 2.00000 0.128565
\(243\) −1.00000 −0.0641500
\(244\) −13.0000 −0.832240
\(245\) 0 0
\(246\) 8.00000 0.510061
\(247\) −1.00000 −0.0636285
\(248\) 4.00000 0.254000
\(249\) −2.00000 −0.126745
\(250\) −9.00000 −0.569210
\(251\) −21.0000 −1.32551 −0.662754 0.748837i \(-0.730613\pi\)
−0.662754 + 0.748837i \(0.730613\pi\)
\(252\) 0 0
\(253\) 9.00000 0.565825
\(254\) 18.0000 1.12942
\(255\) −5.00000 −0.313112
\(256\) 1.00000 0.0625000
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) −1.00000 −0.0622573
\(259\) 0 0
\(260\) −1.00000 −0.0620174
\(261\) 5.00000 0.309492
\(262\) 17.0000 1.05026
\(263\) 12.0000 0.739952 0.369976 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) 3.00000 0.184637
\(265\) −6.00000 −0.368577
\(266\) 0 0
\(267\) −2.00000 −0.122398
\(268\) −10.0000 −0.610847
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) −1.00000 −0.0608581
\(271\) −2.00000 −0.121491 −0.0607457 0.998153i \(-0.519348\pi\)
−0.0607457 + 0.998153i \(0.519348\pi\)
\(272\) −5.00000 −0.303170
\(273\) 0 0
\(274\) 15.0000 0.906183
\(275\) −12.0000 −0.723627
\(276\) −3.00000 −0.180579
\(277\) −18.0000 −1.08152 −0.540758 0.841178i \(-0.681862\pi\)
−0.540758 + 0.841178i \(0.681862\pi\)
\(278\) −16.0000 −0.959616
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) −8.00000 −0.476393
\(283\) 8.00000 0.475551 0.237775 0.971320i \(-0.423582\pi\)
0.237775 + 0.971320i \(0.423582\pi\)
\(284\) 8.00000 0.474713
\(285\) −1.00000 −0.0592349
\(286\) −3.00000 −0.177394
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) 8.00000 0.470588
\(290\) 5.00000 0.293610
\(291\) −2.00000 −0.117242
\(292\) 15.0000 0.877809
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 5.00000 0.290619
\(297\) −3.00000 −0.174078
\(298\) 16.0000 0.926855
\(299\) 3.00000 0.173494
\(300\) 4.00000 0.230940
\(301\) 0 0
\(302\) 5.00000 0.287718
\(303\) −16.0000 −0.919176
\(304\) −1.00000 −0.0573539
\(305\) 13.0000 0.744378
\(306\) 5.00000 0.285831
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 1.00000 0.0568880
\(310\) −4.00000 −0.227185
\(311\) −26.0000 −1.47432 −0.737162 0.675716i \(-0.763835\pi\)
−0.737162 + 0.675716i \(0.763835\pi\)
\(312\) 1.00000 0.0566139
\(313\) −28.0000 −1.58265 −0.791327 0.611393i \(-0.790609\pi\)
−0.791327 + 0.611393i \(0.790609\pi\)
\(314\) 9.00000 0.507899
\(315\) 0 0
\(316\) 6.00000 0.337526
\(317\) −24.0000 −1.34797 −0.673987 0.738743i \(-0.735420\pi\)
−0.673987 + 0.738743i \(0.735420\pi\)
\(318\) 6.00000 0.336463
\(319\) 15.0000 0.839839
\(320\) −1.00000 −0.0559017
\(321\) 6.00000 0.334887
\(322\) 0 0
\(323\) 5.00000 0.278207
\(324\) 1.00000 0.0555556
\(325\) −4.00000 −0.221880
\(326\) −10.0000 −0.553849
\(327\) 7.00000 0.387101
\(328\) −8.00000 −0.441726
\(329\) 0 0
\(330\) −3.00000 −0.165145
\(331\) 34.0000 1.86881 0.934405 0.356214i \(-0.115932\pi\)
0.934405 + 0.356214i \(0.115932\pi\)
\(332\) 2.00000 0.109764
\(333\) −5.00000 −0.273998
\(334\) −15.0000 −0.820763
\(335\) 10.0000 0.546358
\(336\) 0 0
\(337\) −29.0000 −1.57973 −0.789865 0.613280i \(-0.789850\pi\)
−0.789865 + 0.613280i \(0.789850\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 6.00000 0.325875
\(340\) 5.00000 0.271163
\(341\) −12.0000 −0.649836
\(342\) 1.00000 0.0540738
\(343\) 0 0
\(344\) 1.00000 0.0539164
\(345\) 3.00000 0.161515
\(346\) 6.00000 0.322562
\(347\) 32.0000 1.71785 0.858925 0.512101i \(-0.171133\pi\)
0.858925 + 0.512101i \(0.171133\pi\)
\(348\) −5.00000 −0.268028
\(349\) −6.00000 −0.321173 −0.160586 0.987022i \(-0.551338\pi\)
−0.160586 + 0.987022i \(0.551338\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) −3.00000 −0.159901
\(353\) −2.00000 −0.106449 −0.0532246 0.998583i \(-0.516950\pi\)
−0.0532246 + 0.998583i \(0.516950\pi\)
\(354\) 0 0
\(355\) −8.00000 −0.424596
\(356\) 2.00000 0.106000
\(357\) 0 0
\(358\) 16.0000 0.845626
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 1.00000 0.0527046
\(361\) −18.0000 −0.947368
\(362\) 6.00000 0.315353
\(363\) 2.00000 0.104973
\(364\) 0 0
\(365\) −15.0000 −0.785136
\(366\) −13.0000 −0.679521
\(367\) −8.00000 −0.417597 −0.208798 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) 3.00000 0.156386
\(369\) 8.00000 0.416463
\(370\) −5.00000 −0.259938
\(371\) 0 0
\(372\) 4.00000 0.207390
\(373\) −32.0000 −1.65690 −0.828449 0.560065i \(-0.810776\pi\)
−0.828449 + 0.560065i \(0.810776\pi\)
\(374\) 15.0000 0.775632
\(375\) −9.00000 −0.464758
\(376\) 8.00000 0.412568
\(377\) 5.00000 0.257513
\(378\) 0 0
\(379\) −8.00000 −0.410932 −0.205466 0.978664i \(-0.565871\pi\)
−0.205466 + 0.978664i \(0.565871\pi\)
\(380\) 1.00000 0.0512989
\(381\) 18.0000 0.922168
\(382\) −11.0000 −0.562809
\(383\) −1.00000 −0.0510976 −0.0255488 0.999674i \(-0.508133\pi\)
−0.0255488 + 0.999674i \(0.508133\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 20.0000 1.01797
\(387\) −1.00000 −0.0508329
\(388\) 2.00000 0.101535
\(389\) −18.0000 −0.912636 −0.456318 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) −1.00000 −0.0506370
\(391\) −15.0000 −0.758583
\(392\) 0 0
\(393\) 17.0000 0.857537
\(394\) 0 0
\(395\) −6.00000 −0.301893
\(396\) 3.00000 0.150756
\(397\) −30.0000 −1.50566 −0.752828 0.658217i \(-0.771311\pi\)
−0.752828 + 0.658217i \(0.771311\pi\)
\(398\) −5.00000 −0.250627
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) −10.0000 −0.498755
\(403\) −4.00000 −0.199254
\(404\) 16.0000 0.796030
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −15.0000 −0.743522
\(408\) −5.00000 −0.247537
\(409\) −3.00000 −0.148340 −0.0741702 0.997246i \(-0.523631\pi\)
−0.0741702 + 0.997246i \(0.523631\pi\)
\(410\) 8.00000 0.395092
\(411\) 15.0000 0.739895
\(412\) −1.00000 −0.0492665
\(413\) 0 0
\(414\) −3.00000 −0.147442
\(415\) −2.00000 −0.0981761
\(416\) −1.00000 −0.0490290
\(417\) −16.0000 −0.783523
\(418\) 3.00000 0.146735
\(419\) 5.00000 0.244266 0.122133 0.992514i \(-0.461027\pi\)
0.122133 + 0.992514i \(0.461027\pi\)
\(420\) 0 0
\(421\) −22.0000 −1.07221 −0.536107 0.844150i \(-0.680106\pi\)
−0.536107 + 0.844150i \(0.680106\pi\)
\(422\) −5.00000 −0.243396
\(423\) −8.00000 −0.388973
\(424\) −6.00000 −0.291386
\(425\) 20.0000 0.970143
\(426\) 8.00000 0.387601
\(427\) 0 0
\(428\) −6.00000 −0.290021
\(429\) −3.00000 −0.144841
\(430\) −1.00000 −0.0482243
\(431\) 10.0000 0.481683 0.240842 0.970564i \(-0.422577\pi\)
0.240842 + 0.970564i \(0.422577\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −20.0000 −0.961139 −0.480569 0.876957i \(-0.659570\pi\)
−0.480569 + 0.876957i \(0.659570\pi\)
\(434\) 0 0
\(435\) 5.00000 0.239732
\(436\) −7.00000 −0.335239
\(437\) −3.00000 −0.143509
\(438\) 15.0000 0.716728
\(439\) 11.0000 0.525001 0.262501 0.964932i \(-0.415453\pi\)
0.262501 + 0.964932i \(0.415453\pi\)
\(440\) 3.00000 0.143019
\(441\) 0 0
\(442\) 5.00000 0.237826
\(443\) −18.0000 −0.855206 −0.427603 0.903967i \(-0.640642\pi\)
−0.427603 + 0.903967i \(0.640642\pi\)
\(444\) 5.00000 0.237289
\(445\) −2.00000 −0.0948091
\(446\) 26.0000 1.23114
\(447\) 16.0000 0.756774
\(448\) 0 0
\(449\) 13.0000 0.613508 0.306754 0.951789i \(-0.400757\pi\)
0.306754 + 0.951789i \(0.400757\pi\)
\(450\) 4.00000 0.188562
\(451\) 24.0000 1.13012
\(452\) −6.00000 −0.282216
\(453\) 5.00000 0.234920
\(454\) 2.00000 0.0938647
\(455\) 0 0
\(456\) −1.00000 −0.0468293
\(457\) 8.00000 0.374224 0.187112 0.982339i \(-0.440087\pi\)
0.187112 + 0.982339i \(0.440087\pi\)
\(458\) −6.00000 −0.280362
\(459\) 5.00000 0.233380
\(460\) −3.00000 −0.139876
\(461\) 13.0000 0.605470 0.302735 0.953075i \(-0.402100\pi\)
0.302735 + 0.953075i \(0.402100\pi\)
\(462\) 0 0
\(463\) −13.0000 −0.604161 −0.302081 0.953282i \(-0.597681\pi\)
−0.302081 + 0.953282i \(0.597681\pi\)
\(464\) 5.00000 0.232119
\(465\) −4.00000 −0.185496
\(466\) 4.00000 0.185296
\(467\) −21.0000 −0.971764 −0.485882 0.874024i \(-0.661502\pi\)
−0.485882 + 0.874024i \(0.661502\pi\)
\(468\) 1.00000 0.0462250
\(469\) 0 0
\(470\) −8.00000 −0.369012
\(471\) 9.00000 0.414698
\(472\) 0 0
\(473\) −3.00000 −0.137940
\(474\) 6.00000 0.275589
\(475\) 4.00000 0.183533
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) 12.0000 0.548867
\(479\) −27.0000 −1.23366 −0.616831 0.787096i \(-0.711584\pi\)
−0.616831 + 0.787096i \(0.711584\pi\)
\(480\) −1.00000 −0.0456435
\(481\) −5.00000 −0.227980
\(482\) 22.0000 1.00207
\(483\) 0 0
\(484\) −2.00000 −0.0909091
\(485\) −2.00000 −0.0908153
\(486\) 1.00000 0.0453609
\(487\) 8.00000 0.362515 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) 13.0000 0.588482
\(489\) −10.0000 −0.452216
\(490\) 0 0
\(491\) −38.0000 −1.71492 −0.857458 0.514554i \(-0.827958\pi\)
−0.857458 + 0.514554i \(0.827958\pi\)
\(492\) −8.00000 −0.360668
\(493\) −25.0000 −1.12594
\(494\) 1.00000 0.0449921
\(495\) −3.00000 −0.134840
\(496\) −4.00000 −0.179605
\(497\) 0 0
\(498\) 2.00000 0.0896221
\(499\) 8.00000 0.358129 0.179065 0.983837i \(-0.442693\pi\)
0.179065 + 0.983837i \(0.442693\pi\)
\(500\) 9.00000 0.402492
\(501\) −15.0000 −0.670151
\(502\) 21.0000 0.937276
\(503\) 6.00000 0.267527 0.133763 0.991013i \(-0.457294\pi\)
0.133763 + 0.991013i \(0.457294\pi\)
\(504\) 0 0
\(505\) −16.0000 −0.711991
\(506\) −9.00000 −0.400099
\(507\) −1.00000 −0.0444116
\(508\) −18.0000 −0.798621
\(509\) −11.0000 −0.487566 −0.243783 0.969830i \(-0.578389\pi\)
−0.243783 + 0.969830i \(0.578389\pi\)
\(510\) 5.00000 0.221404
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 1.00000 0.0441511
\(514\) −18.0000 −0.793946
\(515\) 1.00000 0.0440653
\(516\) 1.00000 0.0440225
\(517\) −24.0000 −1.05552
\(518\) 0 0
\(519\) 6.00000 0.263371
\(520\) 1.00000 0.0438529
\(521\) −39.0000 −1.70862 −0.854311 0.519763i \(-0.826020\pi\)
−0.854311 + 0.519763i \(0.826020\pi\)
\(522\) −5.00000 −0.218844
\(523\) 24.0000 1.04945 0.524723 0.851273i \(-0.324169\pi\)
0.524723 + 0.851273i \(0.324169\pi\)
\(524\) −17.0000 −0.742648
\(525\) 0 0
\(526\) −12.0000 −0.523225
\(527\) 20.0000 0.871214
\(528\) −3.00000 −0.130558
\(529\) −14.0000 −0.608696
\(530\) 6.00000 0.260623
\(531\) 0 0
\(532\) 0 0
\(533\) 8.00000 0.346518
\(534\) 2.00000 0.0865485
\(535\) 6.00000 0.259403
\(536\) 10.0000 0.431934
\(537\) 16.0000 0.690451
\(538\) 6.00000 0.258678
\(539\) 0 0
\(540\) 1.00000 0.0430331
\(541\) 41.0000 1.76273 0.881364 0.472438i \(-0.156626\pi\)
0.881364 + 0.472438i \(0.156626\pi\)
\(542\) 2.00000 0.0859074
\(543\) 6.00000 0.257485
\(544\) 5.00000 0.214373
\(545\) 7.00000 0.299847
\(546\) 0 0
\(547\) −44.0000 −1.88130 −0.940652 0.339372i \(-0.889785\pi\)
−0.940652 + 0.339372i \(0.889785\pi\)
\(548\) −15.0000 −0.640768
\(549\) −13.0000 −0.554826
\(550\) 12.0000 0.511682
\(551\) −5.00000 −0.213007
\(552\) 3.00000 0.127688
\(553\) 0 0
\(554\) 18.0000 0.764747
\(555\) −5.00000 −0.212238
\(556\) 16.0000 0.678551
\(557\) −24.0000 −1.01691 −0.508456 0.861088i \(-0.669784\pi\)
−0.508456 + 0.861088i \(0.669784\pi\)
\(558\) 4.00000 0.169334
\(559\) −1.00000 −0.0422955
\(560\) 0 0
\(561\) 15.0000 0.633300
\(562\) −10.0000 −0.421825
\(563\) −9.00000 −0.379305 −0.189652 0.981851i \(-0.560736\pi\)
−0.189652 + 0.981851i \(0.560736\pi\)
\(564\) 8.00000 0.336861
\(565\) 6.00000 0.252422
\(566\) −8.00000 −0.336265
\(567\) 0 0
\(568\) −8.00000 −0.335673
\(569\) 34.0000 1.42535 0.712677 0.701492i \(-0.247483\pi\)
0.712677 + 0.701492i \(0.247483\pi\)
\(570\) 1.00000 0.0418854
\(571\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(572\) 3.00000 0.125436
\(573\) −11.0000 −0.459532
\(574\) 0 0
\(575\) −12.0000 −0.500435
\(576\) 1.00000 0.0416667
\(577\) 34.0000 1.41544 0.707719 0.706494i \(-0.249724\pi\)
0.707719 + 0.706494i \(0.249724\pi\)
\(578\) −8.00000 −0.332756
\(579\) 20.0000 0.831172
\(580\) −5.00000 −0.207614
\(581\) 0 0
\(582\) 2.00000 0.0829027
\(583\) 18.0000 0.745484
\(584\) −15.0000 −0.620704
\(585\) −1.00000 −0.0413449
\(586\) −6.00000 −0.247858
\(587\) 26.0000 1.07313 0.536567 0.843857i \(-0.319721\pi\)
0.536567 + 0.843857i \(0.319721\pi\)
\(588\) 0 0
\(589\) 4.00000 0.164817
\(590\) 0 0
\(591\) 0 0
\(592\) −5.00000 −0.205499
\(593\) −38.0000 −1.56047 −0.780236 0.625485i \(-0.784901\pi\)
−0.780236 + 0.625485i \(0.784901\pi\)
\(594\) 3.00000 0.123091
\(595\) 0 0
\(596\) −16.0000 −0.655386
\(597\) −5.00000 −0.204636
\(598\) −3.00000 −0.122679
\(599\) 37.0000 1.51178 0.755890 0.654699i \(-0.227205\pi\)
0.755890 + 0.654699i \(0.227205\pi\)
\(600\) −4.00000 −0.163299
\(601\) −20.0000 −0.815817 −0.407909 0.913023i \(-0.633742\pi\)
−0.407909 + 0.913023i \(0.633742\pi\)
\(602\) 0 0
\(603\) −10.0000 −0.407231
\(604\) −5.00000 −0.203447
\(605\) 2.00000 0.0813116
\(606\) 16.0000 0.649956
\(607\) 27.0000 1.09590 0.547948 0.836512i \(-0.315409\pi\)
0.547948 + 0.836512i \(0.315409\pi\)
\(608\) 1.00000 0.0405554
\(609\) 0 0
\(610\) −13.0000 −0.526355
\(611\) −8.00000 −0.323645
\(612\) −5.00000 −0.202113
\(613\) 41.0000 1.65597 0.827987 0.560747i \(-0.189486\pi\)
0.827987 + 0.560747i \(0.189486\pi\)
\(614\) 0 0
\(615\) 8.00000 0.322591
\(616\) 0 0
\(617\) 43.0000 1.73111 0.865557 0.500810i \(-0.166964\pi\)
0.865557 + 0.500810i \(0.166964\pi\)
\(618\) −1.00000 −0.0402259
\(619\) 31.0000 1.24600 0.622998 0.782224i \(-0.285915\pi\)
0.622998 + 0.782224i \(0.285915\pi\)
\(620\) 4.00000 0.160644
\(621\) −3.00000 −0.120386
\(622\) 26.0000 1.04251
\(623\) 0 0
\(624\) −1.00000 −0.0400320
\(625\) 11.0000 0.440000
\(626\) 28.0000 1.11911
\(627\) 3.00000 0.119808
\(628\) −9.00000 −0.359139
\(629\) 25.0000 0.996815
\(630\) 0 0
\(631\) −9.00000 −0.358284 −0.179142 0.983823i \(-0.557332\pi\)
−0.179142 + 0.983823i \(0.557332\pi\)
\(632\) −6.00000 −0.238667
\(633\) −5.00000 −0.198732
\(634\) 24.0000 0.953162
\(635\) 18.0000 0.714308
\(636\) −6.00000 −0.237915
\(637\) 0 0
\(638\) −15.0000 −0.593856
\(639\) 8.00000 0.316475
\(640\) 1.00000 0.0395285
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) −6.00000 −0.236801
\(643\) 1.00000 0.0394362 0.0197181 0.999806i \(-0.493723\pi\)
0.0197181 + 0.999806i \(0.493723\pi\)
\(644\) 0 0
\(645\) −1.00000 −0.0393750
\(646\) −5.00000 −0.196722
\(647\) −28.0000 −1.10079 −0.550397 0.834903i \(-0.685524\pi\)
−0.550397 + 0.834903i \(0.685524\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) 4.00000 0.156893
\(651\) 0 0
\(652\) 10.0000 0.391630
\(653\) 3.00000 0.117399 0.0586995 0.998276i \(-0.481305\pi\)
0.0586995 + 0.998276i \(0.481305\pi\)
\(654\) −7.00000 −0.273722
\(655\) 17.0000 0.664245
\(656\) 8.00000 0.312348
\(657\) 15.0000 0.585206
\(658\) 0 0
\(659\) 6.00000 0.233727 0.116863 0.993148i \(-0.462716\pi\)
0.116863 + 0.993148i \(0.462716\pi\)
\(660\) 3.00000 0.116775
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) −34.0000 −1.32145
\(663\) 5.00000 0.194184
\(664\) −2.00000 −0.0776151
\(665\) 0 0
\(666\) 5.00000 0.193746
\(667\) 15.0000 0.580802
\(668\) 15.0000 0.580367
\(669\) 26.0000 1.00522
\(670\) −10.0000 −0.386334
\(671\) −39.0000 −1.50558
\(672\) 0 0
\(673\) −1.00000 −0.0385472 −0.0192736 0.999814i \(-0.506135\pi\)
−0.0192736 + 0.999814i \(0.506135\pi\)
\(674\) 29.0000 1.11704
\(675\) 4.00000 0.153960
\(676\) 1.00000 0.0384615
\(677\) 18.0000 0.691796 0.345898 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) −6.00000 −0.230429
\(679\) 0 0
\(680\) −5.00000 −0.191741
\(681\) 2.00000 0.0766402
\(682\) 12.0000 0.459504
\(683\) 33.0000 1.26271 0.631355 0.775494i \(-0.282499\pi\)
0.631355 + 0.775494i \(0.282499\pi\)
\(684\) −1.00000 −0.0382360
\(685\) 15.0000 0.573121
\(686\) 0 0
\(687\) −6.00000 −0.228914
\(688\) −1.00000 −0.0381246
\(689\) 6.00000 0.228582
\(690\) −3.00000 −0.114208
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) −6.00000 −0.228086
\(693\) 0 0
\(694\) −32.0000 −1.21470
\(695\) −16.0000 −0.606915
\(696\) 5.00000 0.189525
\(697\) −40.0000 −1.51511
\(698\) 6.00000 0.227103
\(699\) 4.00000 0.151294
\(700\) 0 0
\(701\) −14.0000 −0.528773 −0.264386 0.964417i \(-0.585169\pi\)
−0.264386 + 0.964417i \(0.585169\pi\)
\(702\) 1.00000 0.0377426
\(703\) 5.00000 0.188579
\(704\) 3.00000 0.113067
\(705\) −8.00000 −0.301297
\(706\) 2.00000 0.0752710
\(707\) 0 0
\(708\) 0 0
\(709\) 50.0000 1.87779 0.938895 0.344204i \(-0.111851\pi\)
0.938895 + 0.344204i \(0.111851\pi\)
\(710\) 8.00000 0.300235
\(711\) 6.00000 0.225018
\(712\) −2.00000 −0.0749532
\(713\) −12.0000 −0.449404
\(714\) 0 0
\(715\) −3.00000 −0.112194
\(716\) −16.0000 −0.597948
\(717\) 12.0000 0.448148
\(718\) −24.0000 −0.895672
\(719\) −30.0000 −1.11881 −0.559406 0.828894i \(-0.688971\pi\)
−0.559406 + 0.828894i \(0.688971\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 0 0
\(722\) 18.0000 0.669891
\(723\) 22.0000 0.818189
\(724\) −6.00000 −0.222988
\(725\) −20.0000 −0.742781
\(726\) −2.00000 −0.0742270
\(727\) 27.0000 1.00137 0.500687 0.865628i \(-0.333081\pi\)
0.500687 + 0.865628i \(0.333081\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 15.0000 0.555175
\(731\) 5.00000 0.184932
\(732\) 13.0000 0.480494
\(733\) −20.0000 −0.738717 −0.369358 0.929287i \(-0.620423\pi\)
−0.369358 + 0.929287i \(0.620423\pi\)
\(734\) 8.00000 0.295285
\(735\) 0 0
\(736\) −3.00000 −0.110581
\(737\) −30.0000 −1.10506
\(738\) −8.00000 −0.294484
\(739\) −26.0000 −0.956425 −0.478213 0.878244i \(-0.658715\pi\)
−0.478213 + 0.878244i \(0.658715\pi\)
\(740\) 5.00000 0.183804
\(741\) 1.00000 0.0367359
\(742\) 0 0
\(743\) −42.0000 −1.54083 −0.770415 0.637542i \(-0.779951\pi\)
−0.770415 + 0.637542i \(0.779951\pi\)
\(744\) −4.00000 −0.146647
\(745\) 16.0000 0.586195
\(746\) 32.0000 1.17160
\(747\) 2.00000 0.0731762
\(748\) −15.0000 −0.548454
\(749\) 0 0
\(750\) 9.00000 0.328634
\(751\) 24.0000 0.875772 0.437886 0.899030i \(-0.355727\pi\)
0.437886 + 0.899030i \(0.355727\pi\)
\(752\) −8.00000 −0.291730
\(753\) 21.0000 0.765283
\(754\) −5.00000 −0.182089
\(755\) 5.00000 0.181969
\(756\) 0 0
\(757\) −34.0000 −1.23575 −0.617876 0.786276i \(-0.712006\pi\)
−0.617876 + 0.786276i \(0.712006\pi\)
\(758\) 8.00000 0.290573
\(759\) −9.00000 −0.326679
\(760\) −1.00000 −0.0362738
\(761\) 24.0000 0.869999 0.435000 0.900431i \(-0.356748\pi\)
0.435000 + 0.900431i \(0.356748\pi\)
\(762\) −18.0000 −0.652071
\(763\) 0 0
\(764\) 11.0000 0.397966
\(765\) 5.00000 0.180775
\(766\) 1.00000 0.0361315
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) −43.0000 −1.55062 −0.775310 0.631581i \(-0.782406\pi\)
−0.775310 + 0.631581i \(0.782406\pi\)
\(770\) 0 0
\(771\) −18.0000 −0.648254
\(772\) −20.0000 −0.719816
\(773\) −11.0000 −0.395643 −0.197821 0.980238i \(-0.563387\pi\)
−0.197821 + 0.980238i \(0.563387\pi\)
\(774\) 1.00000 0.0359443
\(775\) 16.0000 0.574737
\(776\) −2.00000 −0.0717958
\(777\) 0 0
\(778\) 18.0000 0.645331
\(779\) −8.00000 −0.286630
\(780\) 1.00000 0.0358057
\(781\) 24.0000 0.858788
\(782\) 15.0000 0.536399
\(783\) −5.00000 −0.178685
\(784\) 0 0
\(785\) 9.00000 0.321224
\(786\) −17.0000 −0.606370
\(787\) 47.0000 1.67537 0.837685 0.546154i \(-0.183909\pi\)
0.837685 + 0.546154i \(0.183909\pi\)
\(788\) 0 0
\(789\) −12.0000 −0.427211
\(790\) 6.00000 0.213470
\(791\) 0 0
\(792\) −3.00000 −0.106600
\(793\) −13.0000 −0.461644
\(794\) 30.0000 1.06466
\(795\) 6.00000 0.212798
\(796\) 5.00000 0.177220
\(797\) 28.0000 0.991811 0.495905 0.868377i \(-0.334836\pi\)
0.495905 + 0.868377i \(0.334836\pi\)
\(798\) 0 0
\(799\) 40.0000 1.41510
\(800\) 4.00000 0.141421
\(801\) 2.00000 0.0706665
\(802\) 30.0000 1.05934
\(803\) 45.0000 1.58802
\(804\) 10.0000 0.352673
\(805\) 0 0
\(806\) 4.00000 0.140894
\(807\) 6.00000 0.211210
\(808\) −16.0000 −0.562878
\(809\) 10.0000 0.351581 0.175791 0.984428i \(-0.443752\pi\)
0.175791 + 0.984428i \(0.443752\pi\)
\(810\) 1.00000 0.0351364
\(811\) 19.0000 0.667180 0.333590 0.942718i \(-0.391740\pi\)
0.333590 + 0.942718i \(0.391740\pi\)
\(812\) 0 0
\(813\) 2.00000 0.0701431
\(814\) 15.0000 0.525750
\(815\) −10.0000 −0.350285
\(816\) 5.00000 0.175035
\(817\) 1.00000 0.0349856
\(818\) 3.00000 0.104893
\(819\) 0 0
\(820\) −8.00000 −0.279372
\(821\) 24.0000 0.837606 0.418803 0.908077i \(-0.362450\pi\)
0.418803 + 0.908077i \(0.362450\pi\)
\(822\) −15.0000 −0.523185
\(823\) 24.0000 0.836587 0.418294 0.908312i \(-0.362628\pi\)
0.418294 + 0.908312i \(0.362628\pi\)
\(824\) 1.00000 0.0348367
\(825\) 12.0000 0.417786
\(826\) 0 0
\(827\) 43.0000 1.49526 0.747628 0.664117i \(-0.231193\pi\)
0.747628 + 0.664117i \(0.231193\pi\)
\(828\) 3.00000 0.104257
\(829\) 19.0000 0.659897 0.329949 0.943999i \(-0.392969\pi\)
0.329949 + 0.943999i \(0.392969\pi\)
\(830\) 2.00000 0.0694210
\(831\) 18.0000 0.624413
\(832\) 1.00000 0.0346688
\(833\) 0 0
\(834\) 16.0000 0.554035
\(835\) −15.0000 −0.519096
\(836\) −3.00000 −0.103757
\(837\) 4.00000 0.138260
\(838\) −5.00000 −0.172722
\(839\) 12.0000 0.414286 0.207143 0.978311i \(-0.433583\pi\)
0.207143 + 0.978311i \(0.433583\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 22.0000 0.758170
\(843\) −10.0000 −0.344418
\(844\) 5.00000 0.172107
\(845\) −1.00000 −0.0344010
\(846\) 8.00000 0.275046
\(847\) 0 0
\(848\) 6.00000 0.206041
\(849\) −8.00000 −0.274559
\(850\) −20.0000 −0.685994
\(851\) −15.0000 −0.514193
\(852\) −8.00000 −0.274075
\(853\) 28.0000 0.958702 0.479351 0.877623i \(-0.340872\pi\)
0.479351 + 0.877623i \(0.340872\pi\)
\(854\) 0 0
\(855\) 1.00000 0.0341993
\(856\) 6.00000 0.205076
\(857\) −42.0000 −1.43469 −0.717346 0.696717i \(-0.754643\pi\)
−0.717346 + 0.696717i \(0.754643\pi\)
\(858\) 3.00000 0.102418
\(859\) 28.0000 0.955348 0.477674 0.878537i \(-0.341480\pi\)
0.477674 + 0.878537i \(0.341480\pi\)
\(860\) 1.00000 0.0340997
\(861\) 0 0
\(862\) −10.0000 −0.340601
\(863\) −2.00000 −0.0680808 −0.0340404 0.999420i \(-0.510837\pi\)
−0.0340404 + 0.999420i \(0.510837\pi\)
\(864\) 1.00000 0.0340207
\(865\) 6.00000 0.204006
\(866\) 20.0000 0.679628
\(867\) −8.00000 −0.271694
\(868\) 0 0
\(869\) 18.0000 0.610608
\(870\) −5.00000 −0.169516
\(871\) −10.0000 −0.338837
\(872\) 7.00000 0.237050
\(873\) 2.00000 0.0676897
\(874\) 3.00000 0.101477
\(875\) 0 0
\(876\) −15.0000 −0.506803
\(877\) 34.0000 1.14810 0.574049 0.818821i \(-0.305372\pi\)
0.574049 + 0.818821i \(0.305372\pi\)
\(878\) −11.0000 −0.371232
\(879\) −6.00000 −0.202375
\(880\) −3.00000 −0.101130
\(881\) −37.0000 −1.24656 −0.623281 0.781998i \(-0.714201\pi\)
−0.623281 + 0.781998i \(0.714201\pi\)
\(882\) 0 0
\(883\) −41.0000 −1.37976 −0.689880 0.723924i \(-0.742337\pi\)
−0.689880 + 0.723924i \(0.742337\pi\)
\(884\) −5.00000 −0.168168
\(885\) 0 0
\(886\) 18.0000 0.604722
\(887\) 20.0000 0.671534 0.335767 0.941945i \(-0.391004\pi\)
0.335767 + 0.941945i \(0.391004\pi\)
\(888\) −5.00000 −0.167789
\(889\) 0 0
\(890\) 2.00000 0.0670402
\(891\) 3.00000 0.100504
\(892\) −26.0000 −0.870544
\(893\) 8.00000 0.267710
\(894\) −16.0000 −0.535120
\(895\) 16.0000 0.534821
\(896\) 0 0
\(897\) −3.00000 −0.100167
\(898\) −13.0000 −0.433816
\(899\) −20.0000 −0.667037
\(900\) −4.00000 −0.133333
\(901\) −30.0000 −0.999445
\(902\) −24.0000 −0.799113
\(903\) 0 0
\(904\) 6.00000 0.199557
\(905\) 6.00000 0.199447
\(906\) −5.00000 −0.166114
\(907\) −20.0000 −0.664089 −0.332045 0.943264i \(-0.607738\pi\)
−0.332045 + 0.943264i \(0.607738\pi\)
\(908\) −2.00000 −0.0663723
\(909\) 16.0000 0.530687
\(910\) 0 0
\(911\) 29.0000 0.960813 0.480406 0.877046i \(-0.340489\pi\)
0.480406 + 0.877046i \(0.340489\pi\)
\(912\) 1.00000 0.0331133
\(913\) 6.00000 0.198571
\(914\) −8.00000 −0.264616
\(915\) −13.0000 −0.429767
\(916\) 6.00000 0.198246
\(917\) 0 0
\(918\) −5.00000 −0.165025
\(919\) −50.0000 −1.64935 −0.824674 0.565608i \(-0.808641\pi\)
−0.824674 + 0.565608i \(0.808641\pi\)
\(920\) 3.00000 0.0989071
\(921\) 0 0
\(922\) −13.0000 −0.428132
\(923\) 8.00000 0.263323
\(924\) 0 0
\(925\) 20.0000 0.657596
\(926\) 13.0000 0.427207
\(927\) −1.00000 −0.0328443
\(928\) −5.00000 −0.164133
\(929\) 4.00000 0.131236 0.0656179 0.997845i \(-0.479098\pi\)
0.0656179 + 0.997845i \(0.479098\pi\)
\(930\) 4.00000 0.131165
\(931\) 0 0
\(932\) −4.00000 −0.131024
\(933\) 26.0000 0.851202
\(934\) 21.0000 0.687141
\(935\) 15.0000 0.490552
\(936\) −1.00000 −0.0326860
\(937\) 14.0000 0.457360 0.228680 0.973502i \(-0.426559\pi\)
0.228680 + 0.973502i \(0.426559\pi\)
\(938\) 0 0
\(939\) 28.0000 0.913745
\(940\) 8.00000 0.260931
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) −9.00000 −0.293236
\(943\) 24.0000 0.781548
\(944\) 0 0
\(945\) 0 0
\(946\) 3.00000 0.0975384
\(947\) 47.0000 1.52729 0.763647 0.645634i \(-0.223407\pi\)
0.763647 + 0.645634i \(0.223407\pi\)
\(948\) −6.00000 −0.194871
\(949\) 15.0000 0.486921
\(950\) −4.00000 −0.129777
\(951\) 24.0000 0.778253
\(952\) 0 0
\(953\) −44.0000 −1.42530 −0.712650 0.701520i \(-0.752505\pi\)
−0.712650 + 0.701520i \(0.752505\pi\)
\(954\) −6.00000 −0.194257
\(955\) −11.0000 −0.355952
\(956\) −12.0000 −0.388108
\(957\) −15.0000 −0.484881
\(958\) 27.0000 0.872330
\(959\) 0 0
\(960\) 1.00000 0.0322749
\(961\) −15.0000 −0.483871
\(962\) 5.00000 0.161206
\(963\) −6.00000 −0.193347
\(964\) −22.0000 −0.708572
\(965\) 20.0000 0.643823
\(966\) 0 0
\(967\) −31.0000 −0.996893 −0.498446 0.866921i \(-0.666096\pi\)
−0.498446 + 0.866921i \(0.666096\pi\)
\(968\) 2.00000 0.0642824
\(969\) −5.00000 −0.160623
\(970\) 2.00000 0.0642161
\(971\) 32.0000 1.02693 0.513464 0.858111i \(-0.328362\pi\)
0.513464 + 0.858111i \(0.328362\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) −8.00000 −0.256337
\(975\) 4.00000 0.128103
\(976\) −13.0000 −0.416120
\(977\) −25.0000 −0.799821 −0.399910 0.916554i \(-0.630959\pi\)
−0.399910 + 0.916554i \(0.630959\pi\)
\(978\) 10.0000 0.319765
\(979\) 6.00000 0.191761
\(980\) 0 0
\(981\) −7.00000 −0.223493
\(982\) 38.0000 1.21263
\(983\) 5.00000 0.159475 0.0797376 0.996816i \(-0.474592\pi\)
0.0797376 + 0.996816i \(0.474592\pi\)
\(984\) 8.00000 0.255031
\(985\) 0 0
\(986\) 25.0000 0.796162
\(987\) 0 0
\(988\) −1.00000 −0.0318142
\(989\) −3.00000 −0.0953945
\(990\) 3.00000 0.0953463
\(991\) −30.0000 −0.952981 −0.476491 0.879180i \(-0.658091\pi\)
−0.476491 + 0.879180i \(0.658091\pi\)
\(992\) 4.00000 0.127000
\(993\) −34.0000 −1.07896
\(994\) 0 0
\(995\) −5.00000 −0.158511
\(996\) −2.00000 −0.0633724
\(997\) 50.0000 1.58352 0.791758 0.610835i \(-0.209166\pi\)
0.791758 + 0.610835i \(0.209166\pi\)
\(998\) −8.00000 −0.253236
\(999\) 5.00000 0.158193
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 3822.2.a.e.1.1 1
7.6 odd 2 546.2.a.c.1.1 1
21.20 even 2 1638.2.a.o.1.1 1
28.27 even 2 4368.2.a.h.1.1 1
91.90 odd 2 7098.2.a.z.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.a.c.1.1 1 7.6 odd 2
1638.2.a.o.1.1 1 21.20 even 2
3822.2.a.e.1.1 1 1.1 even 1 trivial
4368.2.a.h.1.1 1 28.27 even 2
7098.2.a.z.1.1 1 91.90 odd 2