Properties

Label 2166.2.a.c.1.1
Level $2166$
Weight $2$
Character 2166.1
Self dual yes
Analytic conductor $17.296$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2166 = 2 \cdot 3 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2166.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.2955970778\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 114)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2166.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -4.00000 q^{5} -1.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -4.00000 q^{5} -1.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +4.00000 q^{10} +2.00000 q^{11} +1.00000 q^{12} +7.00000 q^{13} +3.00000 q^{14} -4.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} -4.00000 q^{20} -3.00000 q^{21} -2.00000 q^{22} -4.00000 q^{23} -1.00000 q^{24} +11.0000 q^{25} -7.00000 q^{26} +1.00000 q^{27} -3.00000 q^{28} -4.00000 q^{29} +4.00000 q^{30} -1.00000 q^{31} -1.00000 q^{32} +2.00000 q^{33} +12.0000 q^{35} +1.00000 q^{36} -7.00000 q^{37} +7.00000 q^{39} +4.00000 q^{40} -4.00000 q^{41} +3.00000 q^{42} +7.00000 q^{43} +2.00000 q^{44} -4.00000 q^{45} +4.00000 q^{46} +2.00000 q^{47} +1.00000 q^{48} +2.00000 q^{49} -11.0000 q^{50} +7.00000 q^{52} +4.00000 q^{53} -1.00000 q^{54} -8.00000 q^{55} +3.00000 q^{56} +4.00000 q^{58} +6.00000 q^{59} -4.00000 q^{60} -1.00000 q^{61} +1.00000 q^{62} -3.00000 q^{63} +1.00000 q^{64} -28.0000 q^{65} -2.00000 q^{66} -3.00000 q^{67} -4.00000 q^{69} -12.0000 q^{70} -2.00000 q^{71} -1.00000 q^{72} -3.00000 q^{73} +7.00000 q^{74} +11.0000 q^{75} -6.00000 q^{77} -7.00000 q^{78} -5.00000 q^{79} -4.00000 q^{80} +1.00000 q^{81} +4.00000 q^{82} -12.0000 q^{83} -3.00000 q^{84} -7.00000 q^{86} -4.00000 q^{87} -2.00000 q^{88} -18.0000 q^{89} +4.00000 q^{90} -21.0000 q^{91} -4.00000 q^{92} -1.00000 q^{93} -2.00000 q^{94} -1.00000 q^{96} -10.0000 q^{97} -2.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

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
\(5\) −4.00000 −1.78885 −0.894427 0.447214i \(-0.852416\pi\)
−0.894427 + 0.447214i \(0.852416\pi\)
\(6\) −1.00000 −0.408248
\(7\) −3.00000 −1.13389 −0.566947 0.823754i \(-0.691875\pi\)
−0.566947 + 0.823754i \(0.691875\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 4.00000 1.26491
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 1.00000 0.288675
\(13\) 7.00000 1.94145 0.970725 0.240192i \(-0.0772105\pi\)
0.970725 + 0.240192i \(0.0772105\pi\)
\(14\) 3.00000 0.801784
\(15\) −4.00000 −1.03280
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −1.00000 −0.235702
\(19\) 0 0
\(20\) −4.00000 −0.894427
\(21\) −3.00000 −0.654654
\(22\) −2.00000 −0.426401
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) −1.00000 −0.204124
\(25\) 11.0000 2.20000
\(26\) −7.00000 −1.37281
\(27\) 1.00000 0.192450
\(28\) −3.00000 −0.566947
\(29\) −4.00000 −0.742781 −0.371391 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 4.00000 0.730297
\(31\) −1.00000 −0.179605 −0.0898027 0.995960i \(-0.528624\pi\)
−0.0898027 + 0.995960i \(0.528624\pi\)
\(32\) −1.00000 −0.176777
\(33\) 2.00000 0.348155
\(34\) 0 0
\(35\) 12.0000 2.02837
\(36\) 1.00000 0.166667
\(37\) −7.00000 −1.15079 −0.575396 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) 0 0
\(39\) 7.00000 1.12090
\(40\) 4.00000 0.632456
\(41\) −4.00000 −0.624695 −0.312348 0.949968i \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(42\) 3.00000 0.462910
\(43\) 7.00000 1.06749 0.533745 0.845645i \(-0.320784\pi\)
0.533745 + 0.845645i \(0.320784\pi\)
\(44\) 2.00000 0.301511
\(45\) −4.00000 −0.596285
\(46\) 4.00000 0.589768
\(47\) 2.00000 0.291730 0.145865 0.989305i \(-0.453403\pi\)
0.145865 + 0.989305i \(0.453403\pi\)
\(48\) 1.00000 0.144338
\(49\) 2.00000 0.285714
\(50\) −11.0000 −1.55563
\(51\) 0 0
\(52\) 7.00000 0.970725
\(53\) 4.00000 0.549442 0.274721 0.961524i \(-0.411414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) −1.00000 −0.136083
\(55\) −8.00000 −1.07872
\(56\) 3.00000 0.400892
\(57\) 0 0
\(58\) 4.00000 0.525226
\(59\) 6.00000 0.781133 0.390567 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) −4.00000 −0.516398
\(61\) −1.00000 −0.128037 −0.0640184 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) 1.00000 0.127000
\(63\) −3.00000 −0.377964
\(64\) 1.00000 0.125000
\(65\) −28.0000 −3.47297
\(66\) −2.00000 −0.246183
\(67\) −3.00000 −0.366508 −0.183254 0.983066i \(-0.558663\pi\)
−0.183254 + 0.983066i \(0.558663\pi\)
\(68\) 0 0
\(69\) −4.00000 −0.481543
\(70\) −12.0000 −1.43427
\(71\) −2.00000 −0.237356 −0.118678 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) −1.00000 −0.117851
\(73\) −3.00000 −0.351123 −0.175562 0.984468i \(-0.556174\pi\)
−0.175562 + 0.984468i \(0.556174\pi\)
\(74\) 7.00000 0.813733
\(75\) 11.0000 1.27017
\(76\) 0 0
\(77\) −6.00000 −0.683763
\(78\) −7.00000 −0.792594
\(79\) −5.00000 −0.562544 −0.281272 0.959628i \(-0.590756\pi\)
−0.281272 + 0.959628i \(0.590756\pi\)
\(80\) −4.00000 −0.447214
\(81\) 1.00000 0.111111
\(82\) 4.00000 0.441726
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) −3.00000 −0.327327
\(85\) 0 0
\(86\) −7.00000 −0.754829
\(87\) −4.00000 −0.428845
\(88\) −2.00000 −0.213201
\(89\) −18.0000 −1.90800 −0.953998 0.299813i \(-0.903076\pi\)
−0.953998 + 0.299813i \(0.903076\pi\)
\(90\) 4.00000 0.421637
\(91\) −21.0000 −2.20140
\(92\) −4.00000 −0.417029
\(93\) −1.00000 −0.103695
\(94\) −2.00000 −0.206284
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) −2.00000 −0.202031
\(99\) 2.00000 0.201008
\(100\) 11.0000 1.10000
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) 0 0
\(103\) −9.00000 −0.886796 −0.443398 0.896325i \(-0.646227\pi\)
−0.443398 + 0.896325i \(0.646227\pi\)
\(104\) −7.00000 −0.686406
\(105\) 12.0000 1.17108
\(106\) −4.00000 −0.388514
\(107\) −2.00000 −0.193347 −0.0966736 0.995316i \(-0.530820\pi\)
−0.0966736 + 0.995316i \(0.530820\pi\)
\(108\) 1.00000 0.0962250
\(109\) −6.00000 −0.574696 −0.287348 0.957826i \(-0.592774\pi\)
−0.287348 + 0.957826i \(0.592774\pi\)
\(110\) 8.00000 0.762770
\(111\) −7.00000 −0.664411
\(112\) −3.00000 −0.283473
\(113\) −2.00000 −0.188144 −0.0940721 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 0 0
\(115\) 16.0000 1.49201
\(116\) −4.00000 −0.371391
\(117\) 7.00000 0.647150
\(118\) −6.00000 −0.552345
\(119\) 0 0
\(120\) 4.00000 0.365148
\(121\) −7.00000 −0.636364
\(122\) 1.00000 0.0905357
\(123\) −4.00000 −0.360668
\(124\) −1.00000 −0.0898027
\(125\) −24.0000 −2.14663
\(126\) 3.00000 0.267261
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 7.00000 0.616316
\(130\) 28.0000 2.45576
\(131\) −18.0000 −1.57267 −0.786334 0.617802i \(-0.788023\pi\)
−0.786334 + 0.617802i \(0.788023\pi\)
\(132\) 2.00000 0.174078
\(133\) 0 0
\(134\) 3.00000 0.259161
\(135\) −4.00000 −0.344265
\(136\) 0 0
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) 4.00000 0.340503
\(139\) 21.0000 1.78120 0.890598 0.454791i \(-0.150286\pi\)
0.890598 + 0.454791i \(0.150286\pi\)
\(140\) 12.0000 1.01419
\(141\) 2.00000 0.168430
\(142\) 2.00000 0.167836
\(143\) 14.0000 1.17074
\(144\) 1.00000 0.0833333
\(145\) 16.0000 1.32873
\(146\) 3.00000 0.248282
\(147\) 2.00000 0.164957
\(148\) −7.00000 −0.575396
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) −11.0000 −0.898146
\(151\) −20.0000 −1.62758 −0.813788 0.581161i \(-0.802599\pi\)
−0.813788 + 0.581161i \(0.802599\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 6.00000 0.483494
\(155\) 4.00000 0.321288
\(156\) 7.00000 0.560449
\(157\) 7.00000 0.558661 0.279330 0.960195i \(-0.409888\pi\)
0.279330 + 0.960195i \(0.409888\pi\)
\(158\) 5.00000 0.397779
\(159\) 4.00000 0.317221
\(160\) 4.00000 0.316228
\(161\) 12.0000 0.945732
\(162\) −1.00000 −0.0785674
\(163\) 11.0000 0.861586 0.430793 0.902451i \(-0.358234\pi\)
0.430793 + 0.902451i \(0.358234\pi\)
\(164\) −4.00000 −0.312348
\(165\) −8.00000 −0.622799
\(166\) 12.0000 0.931381
\(167\) −6.00000 −0.464294 −0.232147 0.972681i \(-0.574575\pi\)
−0.232147 + 0.972681i \(0.574575\pi\)
\(168\) 3.00000 0.231455
\(169\) 36.0000 2.76923
\(170\) 0 0
\(171\) 0 0
\(172\) 7.00000 0.533745
\(173\) −12.0000 −0.912343 −0.456172 0.889892i \(-0.650780\pi\)
−0.456172 + 0.889892i \(0.650780\pi\)
\(174\) 4.00000 0.303239
\(175\) −33.0000 −2.49457
\(176\) 2.00000 0.150756
\(177\) 6.00000 0.450988
\(178\) 18.0000 1.34916
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) −4.00000 −0.298142
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 21.0000 1.55662
\(183\) −1.00000 −0.0739221
\(184\) 4.00000 0.294884
\(185\) 28.0000 2.05860
\(186\) 1.00000 0.0733236
\(187\) 0 0
\(188\) 2.00000 0.145865
\(189\) −3.00000 −0.218218
\(190\) 0 0
\(191\) −20.0000 −1.44715 −0.723575 0.690246i \(-0.757502\pi\)
−0.723575 + 0.690246i \(0.757502\pi\)
\(192\) 1.00000 0.0721688
\(193\) 21.0000 1.51161 0.755807 0.654795i \(-0.227245\pi\)
0.755807 + 0.654795i \(0.227245\pi\)
\(194\) 10.0000 0.717958
\(195\) −28.0000 −2.00512
\(196\) 2.00000 0.142857
\(197\) −10.0000 −0.712470 −0.356235 0.934396i \(-0.615940\pi\)
−0.356235 + 0.934396i \(0.615940\pi\)
\(198\) −2.00000 −0.142134
\(199\) −7.00000 −0.496217 −0.248108 0.968732i \(-0.579809\pi\)
−0.248108 + 0.968732i \(0.579809\pi\)
\(200\) −11.0000 −0.777817
\(201\) −3.00000 −0.211604
\(202\) −2.00000 −0.140720
\(203\) 12.0000 0.842235
\(204\) 0 0
\(205\) 16.0000 1.11749
\(206\) 9.00000 0.627060
\(207\) −4.00000 −0.278019
\(208\) 7.00000 0.485363
\(209\) 0 0
\(210\) −12.0000 −0.828079
\(211\) −9.00000 −0.619586 −0.309793 0.950804i \(-0.600260\pi\)
−0.309793 + 0.950804i \(0.600260\pi\)
\(212\) 4.00000 0.274721
\(213\) −2.00000 −0.137038
\(214\) 2.00000 0.136717
\(215\) −28.0000 −1.90958
\(216\) −1.00000 −0.0680414
\(217\) 3.00000 0.203653
\(218\) 6.00000 0.406371
\(219\) −3.00000 −0.202721
\(220\) −8.00000 −0.539360
\(221\) 0 0
\(222\) 7.00000 0.469809
\(223\) −5.00000 −0.334825 −0.167412 0.985887i \(-0.553541\pi\)
−0.167412 + 0.985887i \(0.553541\pi\)
\(224\) 3.00000 0.200446
\(225\) 11.0000 0.733333
\(226\) 2.00000 0.133038
\(227\) −4.00000 −0.265489 −0.132745 0.991150i \(-0.542379\pi\)
−0.132745 + 0.991150i \(0.542379\pi\)
\(228\) 0 0
\(229\) −7.00000 −0.462573 −0.231287 0.972886i \(-0.574293\pi\)
−0.231287 + 0.972886i \(0.574293\pi\)
\(230\) −16.0000 −1.05501
\(231\) −6.00000 −0.394771
\(232\) 4.00000 0.262613
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) −7.00000 −0.457604
\(235\) −8.00000 −0.521862
\(236\) 6.00000 0.390567
\(237\) −5.00000 −0.324785
\(238\) 0 0
\(239\) 24.0000 1.55243 0.776215 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) −4.00000 −0.258199
\(241\) −1.00000 −0.0644157 −0.0322078 0.999481i \(-0.510254\pi\)
−0.0322078 + 0.999481i \(0.510254\pi\)
\(242\) 7.00000 0.449977
\(243\) 1.00000 0.0641500
\(244\) −1.00000 −0.0640184
\(245\) −8.00000 −0.511101
\(246\) 4.00000 0.255031
\(247\) 0 0
\(248\) 1.00000 0.0635001
\(249\) −12.0000 −0.760469
\(250\) 24.0000 1.51789
\(251\) 14.0000 0.883672 0.441836 0.897096i \(-0.354327\pi\)
0.441836 + 0.897096i \(0.354327\pi\)
\(252\) −3.00000 −0.188982
\(253\) −8.00000 −0.502956
\(254\) 0 0
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 28.0000 1.74659 0.873296 0.487190i \(-0.161978\pi\)
0.873296 + 0.487190i \(0.161978\pi\)
\(258\) −7.00000 −0.435801
\(259\) 21.0000 1.30488
\(260\) −28.0000 −1.73649
\(261\) −4.00000 −0.247594
\(262\) 18.0000 1.11204
\(263\) −18.0000 −1.10993 −0.554964 0.831875i \(-0.687268\pi\)
−0.554964 + 0.831875i \(0.687268\pi\)
\(264\) −2.00000 −0.123091
\(265\) −16.0000 −0.982872
\(266\) 0 0
\(267\) −18.0000 −1.10158
\(268\) −3.00000 −0.183254
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 4.00000 0.243432
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) 0 0
\(273\) −21.0000 −1.27098
\(274\) 2.00000 0.120824
\(275\) 22.0000 1.32665
\(276\) −4.00000 −0.240772
\(277\) −26.0000 −1.56219 −0.781094 0.624413i \(-0.785338\pi\)
−0.781094 + 0.624413i \(0.785338\pi\)
\(278\) −21.0000 −1.25950
\(279\) −1.00000 −0.0598684
\(280\) −12.0000 −0.717137
\(281\) −4.00000 −0.238620 −0.119310 0.992857i \(-0.538068\pi\)
−0.119310 + 0.992857i \(0.538068\pi\)
\(282\) −2.00000 −0.119098
\(283\) −12.0000 −0.713326 −0.356663 0.934233i \(-0.616086\pi\)
−0.356663 + 0.934233i \(0.616086\pi\)
\(284\) −2.00000 −0.118678
\(285\) 0 0
\(286\) −14.0000 −0.827837
\(287\) 12.0000 0.708338
\(288\) −1.00000 −0.0589256
\(289\) −17.0000 −1.00000
\(290\) −16.0000 −0.939552
\(291\) −10.0000 −0.586210
\(292\) −3.00000 −0.175562
\(293\) 18.0000 1.05157 0.525786 0.850617i \(-0.323771\pi\)
0.525786 + 0.850617i \(0.323771\pi\)
\(294\) −2.00000 −0.116642
\(295\) −24.0000 −1.39733
\(296\) 7.00000 0.406867
\(297\) 2.00000 0.116052
\(298\) 18.0000 1.04271
\(299\) −28.0000 −1.61928
\(300\) 11.0000 0.635085
\(301\) −21.0000 −1.21042
\(302\) 20.0000 1.15087
\(303\) 2.00000 0.114897
\(304\) 0 0
\(305\) 4.00000 0.229039
\(306\) 0 0
\(307\) −12.0000 −0.684876 −0.342438 0.939540i \(-0.611253\pi\)
−0.342438 + 0.939540i \(0.611253\pi\)
\(308\) −6.00000 −0.341882
\(309\) −9.00000 −0.511992
\(310\) −4.00000 −0.227185
\(311\) 34.0000 1.92796 0.963982 0.265969i \(-0.0856919\pi\)
0.963982 + 0.265969i \(0.0856919\pi\)
\(312\) −7.00000 −0.396297
\(313\) 14.0000 0.791327 0.395663 0.918396i \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) −7.00000 −0.395033
\(315\) 12.0000 0.676123
\(316\) −5.00000 −0.281272
\(317\) 24.0000 1.34797 0.673987 0.738743i \(-0.264580\pi\)
0.673987 + 0.738743i \(0.264580\pi\)
\(318\) −4.00000 −0.224309
\(319\) −8.00000 −0.447914
\(320\) −4.00000 −0.223607
\(321\) −2.00000 −0.111629
\(322\) −12.0000 −0.668734
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 77.0000 4.27119
\(326\) −11.0000 −0.609234
\(327\) −6.00000 −0.331801
\(328\) 4.00000 0.220863
\(329\) −6.00000 −0.330791
\(330\) 8.00000 0.440386
\(331\) −23.0000 −1.26419 −0.632097 0.774889i \(-0.717806\pi\)
−0.632097 + 0.774889i \(0.717806\pi\)
\(332\) −12.0000 −0.658586
\(333\) −7.00000 −0.383598
\(334\) 6.00000 0.328305
\(335\) 12.0000 0.655630
\(336\) −3.00000 −0.163663
\(337\) −13.0000 −0.708155 −0.354078 0.935216i \(-0.615205\pi\)
−0.354078 + 0.935216i \(0.615205\pi\)
\(338\) −36.0000 −1.95814
\(339\) −2.00000 −0.108625
\(340\) 0 0
\(341\) −2.00000 −0.108306
\(342\) 0 0
\(343\) 15.0000 0.809924
\(344\) −7.00000 −0.377415
\(345\) 16.0000 0.861411
\(346\) 12.0000 0.645124
\(347\) 28.0000 1.50312 0.751559 0.659665i \(-0.229302\pi\)
0.751559 + 0.659665i \(0.229302\pi\)
\(348\) −4.00000 −0.214423
\(349\) 31.0000 1.65939 0.829696 0.558216i \(-0.188514\pi\)
0.829696 + 0.558216i \(0.188514\pi\)
\(350\) 33.0000 1.76392
\(351\) 7.00000 0.373632
\(352\) −2.00000 −0.106600
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) −6.00000 −0.318896
\(355\) 8.00000 0.424596
\(356\) −18.0000 −0.953998
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) −12.0000 −0.633336 −0.316668 0.948536i \(-0.602564\pi\)
−0.316668 + 0.948536i \(0.602564\pi\)
\(360\) 4.00000 0.210819
\(361\) 0 0
\(362\) −2.00000 −0.105118
\(363\) −7.00000 −0.367405
\(364\) −21.0000 −1.10070
\(365\) 12.0000 0.628109
\(366\) 1.00000 0.0522708
\(367\) 19.0000 0.991792 0.495896 0.868382i \(-0.334840\pi\)
0.495896 + 0.868382i \(0.334840\pi\)
\(368\) −4.00000 −0.208514
\(369\) −4.00000 −0.208232
\(370\) −28.0000 −1.45565
\(371\) −12.0000 −0.623009
\(372\) −1.00000 −0.0518476
\(373\) 22.0000 1.13912 0.569558 0.821951i \(-0.307114\pi\)
0.569558 + 0.821951i \(0.307114\pi\)
\(374\) 0 0
\(375\) −24.0000 −1.23935
\(376\) −2.00000 −0.103142
\(377\) −28.0000 −1.44207
\(378\) 3.00000 0.154303
\(379\) 21.0000 1.07870 0.539349 0.842082i \(-0.318670\pi\)
0.539349 + 0.842082i \(0.318670\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 20.0000 1.02329
\(383\) 14.0000 0.715367 0.357683 0.933843i \(-0.383567\pi\)
0.357683 + 0.933843i \(0.383567\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 24.0000 1.22315
\(386\) −21.0000 −1.06887
\(387\) 7.00000 0.355830
\(388\) −10.0000 −0.507673
\(389\) −24.0000 −1.21685 −0.608424 0.793612i \(-0.708198\pi\)
−0.608424 + 0.793612i \(0.708198\pi\)
\(390\) 28.0000 1.41784
\(391\) 0 0
\(392\) −2.00000 −0.101015
\(393\) −18.0000 −0.907980
\(394\) 10.0000 0.503793
\(395\) 20.0000 1.00631
\(396\) 2.00000 0.100504
\(397\) −13.0000 −0.652451 −0.326226 0.945292i \(-0.605777\pi\)
−0.326226 + 0.945292i \(0.605777\pi\)
\(398\) 7.00000 0.350878
\(399\) 0 0
\(400\) 11.0000 0.550000
\(401\) −10.0000 −0.499376 −0.249688 0.968326i \(-0.580328\pi\)
−0.249688 + 0.968326i \(0.580328\pi\)
\(402\) 3.00000 0.149626
\(403\) −7.00000 −0.348695
\(404\) 2.00000 0.0995037
\(405\) −4.00000 −0.198762
\(406\) −12.0000 −0.595550
\(407\) −14.0000 −0.693954
\(408\) 0 0
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) −16.0000 −0.790184
\(411\) −2.00000 −0.0986527
\(412\) −9.00000 −0.443398
\(413\) −18.0000 −0.885722
\(414\) 4.00000 0.196589
\(415\) 48.0000 2.35623
\(416\) −7.00000 −0.343203
\(417\) 21.0000 1.02837
\(418\) 0 0
\(419\) 30.0000 1.46560 0.732798 0.680446i \(-0.238214\pi\)
0.732798 + 0.680446i \(0.238214\pi\)
\(420\) 12.0000 0.585540
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) 9.00000 0.438113
\(423\) 2.00000 0.0972433
\(424\) −4.00000 −0.194257
\(425\) 0 0
\(426\) 2.00000 0.0969003
\(427\) 3.00000 0.145180
\(428\) −2.00000 −0.0966736
\(429\) 14.0000 0.675926
\(430\) 28.0000 1.35028
\(431\) −6.00000 −0.289010 −0.144505 0.989504i \(-0.546159\pi\)
−0.144505 + 0.989504i \(0.546159\pi\)
\(432\) 1.00000 0.0481125
\(433\) −23.0000 −1.10531 −0.552655 0.833410i \(-0.686385\pi\)
−0.552655 + 0.833410i \(0.686385\pi\)
\(434\) −3.00000 −0.144005
\(435\) 16.0000 0.767141
\(436\) −6.00000 −0.287348
\(437\) 0 0
\(438\) 3.00000 0.143346
\(439\) −23.0000 −1.09773 −0.548865 0.835911i \(-0.684940\pi\)
−0.548865 + 0.835911i \(0.684940\pi\)
\(440\) 8.00000 0.381385
\(441\) 2.00000 0.0952381
\(442\) 0 0
\(443\) 4.00000 0.190046 0.0950229 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) −7.00000 −0.332205
\(445\) 72.0000 3.41313
\(446\) 5.00000 0.236757
\(447\) −18.0000 −0.851371
\(448\) −3.00000 −0.141737
\(449\) −16.0000 −0.755087 −0.377543 0.925992i \(-0.623231\pi\)
−0.377543 + 0.925992i \(0.623231\pi\)
\(450\) −11.0000 −0.518545
\(451\) −8.00000 −0.376705
\(452\) −2.00000 −0.0940721
\(453\) −20.0000 −0.939682
\(454\) 4.00000 0.187729
\(455\) 84.0000 3.93798
\(456\) 0 0
\(457\) 29.0000 1.35656 0.678281 0.734802i \(-0.262725\pi\)
0.678281 + 0.734802i \(0.262725\pi\)
\(458\) 7.00000 0.327089
\(459\) 0 0
\(460\) 16.0000 0.746004
\(461\) −34.0000 −1.58354 −0.791769 0.610821i \(-0.790840\pi\)
−0.791769 + 0.610821i \(0.790840\pi\)
\(462\) 6.00000 0.279145
\(463\) 19.0000 0.883005 0.441502 0.897260i \(-0.354446\pi\)
0.441502 + 0.897260i \(0.354446\pi\)
\(464\) −4.00000 −0.185695
\(465\) 4.00000 0.185496
\(466\) −6.00000 −0.277945
\(467\) −8.00000 −0.370196 −0.185098 0.982720i \(-0.559260\pi\)
−0.185098 + 0.982720i \(0.559260\pi\)
\(468\) 7.00000 0.323575
\(469\) 9.00000 0.415581
\(470\) 8.00000 0.369012
\(471\) 7.00000 0.322543
\(472\) −6.00000 −0.276172
\(473\) 14.0000 0.643721
\(474\) 5.00000 0.229658
\(475\) 0 0
\(476\) 0 0
\(477\) 4.00000 0.183147
\(478\) −24.0000 −1.09773
\(479\) 34.0000 1.55350 0.776750 0.629809i \(-0.216867\pi\)
0.776750 + 0.629809i \(0.216867\pi\)
\(480\) 4.00000 0.182574
\(481\) −49.0000 −2.23421
\(482\) 1.00000 0.0455488
\(483\) 12.0000 0.546019
\(484\) −7.00000 −0.318182
\(485\) 40.0000 1.81631
\(486\) −1.00000 −0.0453609
\(487\) −16.0000 −0.725029 −0.362515 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) 1.00000 0.0452679
\(489\) 11.0000 0.497437
\(490\) 8.00000 0.361403
\(491\) 26.0000 1.17336 0.586682 0.809818i \(-0.300434\pi\)
0.586682 + 0.809818i \(0.300434\pi\)
\(492\) −4.00000 −0.180334
\(493\) 0 0
\(494\) 0 0
\(495\) −8.00000 −0.359573
\(496\) −1.00000 −0.0449013
\(497\) 6.00000 0.269137
\(498\) 12.0000 0.537733
\(499\) −11.0000 −0.492428 −0.246214 0.969216i \(-0.579187\pi\)
−0.246214 + 0.969216i \(0.579187\pi\)
\(500\) −24.0000 −1.07331
\(501\) −6.00000 −0.268060
\(502\) −14.0000 −0.624851
\(503\) −12.0000 −0.535054 −0.267527 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) 3.00000 0.133631
\(505\) −8.00000 −0.355995
\(506\) 8.00000 0.355643
\(507\) 36.0000 1.59882
\(508\) 0 0
\(509\) 6.00000 0.265945 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 0 0
\(511\) 9.00000 0.398137
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −28.0000 −1.23503
\(515\) 36.0000 1.58635
\(516\) 7.00000 0.308158
\(517\) 4.00000 0.175920
\(518\) −21.0000 −0.922687
\(519\) −12.0000 −0.526742
\(520\) 28.0000 1.22788
\(521\) −38.0000 −1.66481 −0.832405 0.554168i \(-0.813037\pi\)
−0.832405 + 0.554168i \(0.813037\pi\)
\(522\) 4.00000 0.175075
\(523\) −29.0000 −1.26808 −0.634041 0.773300i \(-0.718605\pi\)
−0.634041 + 0.773300i \(0.718605\pi\)
\(524\) −18.0000 −0.786334
\(525\) −33.0000 −1.44024
\(526\) 18.0000 0.784837
\(527\) 0 0
\(528\) 2.00000 0.0870388
\(529\) −7.00000 −0.304348
\(530\) 16.0000 0.694996
\(531\) 6.00000 0.260378
\(532\) 0 0
\(533\) −28.0000 −1.21281
\(534\) 18.0000 0.778936
\(535\) 8.00000 0.345870
\(536\) 3.00000 0.129580
\(537\) 12.0000 0.517838
\(538\) 18.0000 0.776035
\(539\) 4.00000 0.172292
\(540\) −4.00000 −0.172133
\(541\) −19.0000 −0.816874 −0.408437 0.912787i \(-0.633926\pi\)
−0.408437 + 0.912787i \(0.633926\pi\)
\(542\) 8.00000 0.343629
\(543\) 2.00000 0.0858282
\(544\) 0 0
\(545\) 24.0000 1.02805
\(546\) 21.0000 0.898717
\(547\) 29.0000 1.23995 0.619975 0.784621i \(-0.287143\pi\)
0.619975 + 0.784621i \(0.287143\pi\)
\(548\) −2.00000 −0.0854358
\(549\) −1.00000 −0.0426790
\(550\) −22.0000 −0.938083
\(551\) 0 0
\(552\) 4.00000 0.170251
\(553\) 15.0000 0.637865
\(554\) 26.0000 1.10463
\(555\) 28.0000 1.18853
\(556\) 21.0000 0.890598
\(557\) 22.0000 0.932170 0.466085 0.884740i \(-0.345664\pi\)
0.466085 + 0.884740i \(0.345664\pi\)
\(558\) 1.00000 0.0423334
\(559\) 49.0000 2.07248
\(560\) 12.0000 0.507093
\(561\) 0 0
\(562\) 4.00000 0.168730
\(563\) 42.0000 1.77009 0.885044 0.465506i \(-0.154128\pi\)
0.885044 + 0.465506i \(0.154128\pi\)
\(564\) 2.00000 0.0842152
\(565\) 8.00000 0.336563
\(566\) 12.0000 0.504398
\(567\) −3.00000 −0.125988
\(568\) 2.00000 0.0839181
\(569\) −16.0000 −0.670755 −0.335377 0.942084i \(-0.608864\pi\)
−0.335377 + 0.942084i \(0.608864\pi\)
\(570\) 0 0
\(571\) −7.00000 −0.292941 −0.146470 0.989215i \(-0.546791\pi\)
−0.146470 + 0.989215i \(0.546791\pi\)
\(572\) 14.0000 0.585369
\(573\) −20.0000 −0.835512
\(574\) −12.0000 −0.500870
\(575\) −44.0000 −1.83493
\(576\) 1.00000 0.0416667
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 17.0000 0.707107
\(579\) 21.0000 0.872730
\(580\) 16.0000 0.664364
\(581\) 36.0000 1.49353
\(582\) 10.0000 0.414513
\(583\) 8.00000 0.331326
\(584\) 3.00000 0.124141
\(585\) −28.0000 −1.15766
\(586\) −18.0000 −0.743573
\(587\) −18.0000 −0.742940 −0.371470 0.928445i \(-0.621146\pi\)
−0.371470 + 0.928445i \(0.621146\pi\)
\(588\) 2.00000 0.0824786
\(589\) 0 0
\(590\) 24.0000 0.988064
\(591\) −10.0000 −0.411345
\(592\) −7.00000 −0.287698
\(593\) 34.0000 1.39621 0.698106 0.715994i \(-0.254026\pi\)
0.698106 + 0.715994i \(0.254026\pi\)
\(594\) −2.00000 −0.0820610
\(595\) 0 0
\(596\) −18.0000 −0.737309
\(597\) −7.00000 −0.286491
\(598\) 28.0000 1.14501
\(599\) −30.0000 −1.22577 −0.612883 0.790173i \(-0.709990\pi\)
−0.612883 + 0.790173i \(0.709990\pi\)
\(600\) −11.0000 −0.449073
\(601\) −19.0000 −0.775026 −0.387513 0.921864i \(-0.626666\pi\)
−0.387513 + 0.921864i \(0.626666\pi\)
\(602\) 21.0000 0.855896
\(603\) −3.00000 −0.122169
\(604\) −20.0000 −0.813788
\(605\) 28.0000 1.13836
\(606\) −2.00000 −0.0812444
\(607\) −41.0000 −1.66414 −0.832069 0.554672i \(-0.812844\pi\)
−0.832069 + 0.554672i \(0.812844\pi\)
\(608\) 0 0
\(609\) 12.0000 0.486265
\(610\) −4.00000 −0.161955
\(611\) 14.0000 0.566379
\(612\) 0 0
\(613\) −38.0000 −1.53481 −0.767403 0.641165i \(-0.778451\pi\)
−0.767403 + 0.641165i \(0.778451\pi\)
\(614\) 12.0000 0.484281
\(615\) 16.0000 0.645182
\(616\) 6.00000 0.241747
\(617\) 18.0000 0.724653 0.362326 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) 9.00000 0.362033
\(619\) −23.0000 −0.924448 −0.462224 0.886763i \(-0.652948\pi\)
−0.462224 + 0.886763i \(0.652948\pi\)
\(620\) 4.00000 0.160644
\(621\) −4.00000 −0.160514
\(622\) −34.0000 −1.36328
\(623\) 54.0000 2.16346
\(624\) 7.00000 0.280224
\(625\) 41.0000 1.64000
\(626\) −14.0000 −0.559553
\(627\) 0 0
\(628\) 7.00000 0.279330
\(629\) 0 0
\(630\) −12.0000 −0.478091
\(631\) −19.0000 −0.756378 −0.378189 0.925728i \(-0.623453\pi\)
−0.378189 + 0.925728i \(0.623453\pi\)
\(632\) 5.00000 0.198889
\(633\) −9.00000 −0.357718
\(634\) −24.0000 −0.953162
\(635\) 0 0
\(636\) 4.00000 0.158610
\(637\) 14.0000 0.554700
\(638\) 8.00000 0.316723
\(639\) −2.00000 −0.0791188
\(640\) 4.00000 0.158114
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 2.00000 0.0789337
\(643\) 13.0000 0.512670 0.256335 0.966588i \(-0.417485\pi\)
0.256335 + 0.966588i \(0.417485\pi\)
\(644\) 12.0000 0.472866
\(645\) −28.0000 −1.10250
\(646\) 0 0
\(647\) −28.0000 −1.10079 −0.550397 0.834903i \(-0.685524\pi\)
−0.550397 + 0.834903i \(0.685524\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 12.0000 0.471041
\(650\) −77.0000 −3.02019
\(651\) 3.00000 0.117579
\(652\) 11.0000 0.430793
\(653\) −30.0000 −1.17399 −0.586995 0.809590i \(-0.699689\pi\)
−0.586995 + 0.809590i \(0.699689\pi\)
\(654\) 6.00000 0.234619
\(655\) 72.0000 2.81327
\(656\) −4.00000 −0.156174
\(657\) −3.00000 −0.117041
\(658\) 6.00000 0.233904
\(659\) 10.0000 0.389545 0.194772 0.980848i \(-0.437603\pi\)
0.194772 + 0.980848i \(0.437603\pi\)
\(660\) −8.00000 −0.311400
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) 23.0000 0.893920
\(663\) 0 0
\(664\) 12.0000 0.465690
\(665\) 0 0
\(666\) 7.00000 0.271244
\(667\) 16.0000 0.619522
\(668\) −6.00000 −0.232147
\(669\) −5.00000 −0.193311
\(670\) −12.0000 −0.463600
\(671\) −2.00000 −0.0772091
\(672\) 3.00000 0.115728
\(673\) 1.00000 0.0385472 0.0192736 0.999814i \(-0.493865\pi\)
0.0192736 + 0.999814i \(0.493865\pi\)
\(674\) 13.0000 0.500741
\(675\) 11.0000 0.423390
\(676\) 36.0000 1.38462
\(677\) 38.0000 1.46046 0.730229 0.683202i \(-0.239413\pi\)
0.730229 + 0.683202i \(0.239413\pi\)
\(678\) 2.00000 0.0768095
\(679\) 30.0000 1.15129
\(680\) 0 0
\(681\) −4.00000 −0.153280
\(682\) 2.00000 0.0765840
\(683\) −40.0000 −1.53056 −0.765279 0.643699i \(-0.777399\pi\)
−0.765279 + 0.643699i \(0.777399\pi\)
\(684\) 0 0
\(685\) 8.00000 0.305664
\(686\) −15.0000 −0.572703
\(687\) −7.00000 −0.267067
\(688\) 7.00000 0.266872
\(689\) 28.0000 1.06672
\(690\) −16.0000 −0.609110
\(691\) 12.0000 0.456502 0.228251 0.973602i \(-0.426699\pi\)
0.228251 + 0.973602i \(0.426699\pi\)
\(692\) −12.0000 −0.456172
\(693\) −6.00000 −0.227921
\(694\) −28.0000 −1.06287
\(695\) −84.0000 −3.18630
\(696\) 4.00000 0.151620
\(697\) 0 0
\(698\) −31.0000 −1.17337
\(699\) 6.00000 0.226941
\(700\) −33.0000 −1.24728
\(701\) 8.00000 0.302156 0.151078 0.988522i \(-0.451726\pi\)
0.151078 + 0.988522i \(0.451726\pi\)
\(702\) −7.00000 −0.264198
\(703\) 0 0
\(704\) 2.00000 0.0753778
\(705\) −8.00000 −0.301297
\(706\) 0 0
\(707\) −6.00000 −0.225653
\(708\) 6.00000 0.225494
\(709\) −3.00000 −0.112667 −0.0563337 0.998412i \(-0.517941\pi\)
−0.0563337 + 0.998412i \(0.517941\pi\)
\(710\) −8.00000 −0.300235
\(711\) −5.00000 −0.187515
\(712\) 18.0000 0.674579
\(713\) 4.00000 0.149801
\(714\) 0 0
\(715\) −56.0000 −2.09428
\(716\) 12.0000 0.448461
\(717\) 24.0000 0.896296
\(718\) 12.0000 0.447836
\(719\) 16.0000 0.596699 0.298350 0.954457i \(-0.403564\pi\)
0.298350 + 0.954457i \(0.403564\pi\)
\(720\) −4.00000 −0.149071
\(721\) 27.0000 1.00553
\(722\) 0 0
\(723\) −1.00000 −0.0371904
\(724\) 2.00000 0.0743294
\(725\) −44.0000 −1.63412
\(726\) 7.00000 0.259794
\(727\) 37.0000 1.37225 0.686127 0.727482i \(-0.259309\pi\)
0.686127 + 0.727482i \(0.259309\pi\)
\(728\) 21.0000 0.778312
\(729\) 1.00000 0.0370370
\(730\) −12.0000 −0.444140
\(731\) 0 0
\(732\) −1.00000 −0.0369611
\(733\) 30.0000 1.10808 0.554038 0.832492i \(-0.313086\pi\)
0.554038 + 0.832492i \(0.313086\pi\)
\(734\) −19.0000 −0.701303
\(735\) −8.00000 −0.295084
\(736\) 4.00000 0.147442
\(737\) −6.00000 −0.221013
\(738\) 4.00000 0.147242
\(739\) 35.0000 1.28750 0.643748 0.765238i \(-0.277379\pi\)
0.643748 + 0.765238i \(0.277379\pi\)
\(740\) 28.0000 1.02930
\(741\) 0 0
\(742\) 12.0000 0.440534
\(743\) −26.0000 −0.953847 −0.476924 0.878945i \(-0.658248\pi\)
−0.476924 + 0.878945i \(0.658248\pi\)
\(744\) 1.00000 0.0366618
\(745\) 72.0000 2.63788
\(746\) −22.0000 −0.805477
\(747\) −12.0000 −0.439057
\(748\) 0 0
\(749\) 6.00000 0.219235
\(750\) 24.0000 0.876356
\(751\) 31.0000 1.13121 0.565603 0.824678i \(-0.308643\pi\)
0.565603 + 0.824678i \(0.308643\pi\)
\(752\) 2.00000 0.0729325
\(753\) 14.0000 0.510188
\(754\) 28.0000 1.01970
\(755\) 80.0000 2.91150
\(756\) −3.00000 −0.109109
\(757\) −23.0000 −0.835949 −0.417975 0.908459i \(-0.637260\pi\)
−0.417975 + 0.908459i \(0.637260\pi\)
\(758\) −21.0000 −0.762754
\(759\) −8.00000 −0.290382
\(760\) 0 0
\(761\) −42.0000 −1.52250 −0.761249 0.648459i \(-0.775414\pi\)
−0.761249 + 0.648459i \(0.775414\pi\)
\(762\) 0 0
\(763\) 18.0000 0.651644
\(764\) −20.0000 −0.723575
\(765\) 0 0
\(766\) −14.0000 −0.505841
\(767\) 42.0000 1.51653
\(768\) 1.00000 0.0360844
\(769\) −5.00000 −0.180305 −0.0901523 0.995928i \(-0.528735\pi\)
−0.0901523 + 0.995928i \(0.528735\pi\)
\(770\) −24.0000 −0.864900
\(771\) 28.0000 1.00840
\(772\) 21.0000 0.755807
\(773\) −6.00000 −0.215805 −0.107903 0.994161i \(-0.534413\pi\)
−0.107903 + 0.994161i \(0.534413\pi\)
\(774\) −7.00000 −0.251610
\(775\) −11.0000 −0.395132
\(776\) 10.0000 0.358979
\(777\) 21.0000 0.753371
\(778\) 24.0000 0.860442
\(779\) 0 0
\(780\) −28.0000 −1.00256
\(781\) −4.00000 −0.143131
\(782\) 0 0
\(783\) −4.00000 −0.142948
\(784\) 2.00000 0.0714286
\(785\) −28.0000 −0.999363
\(786\) 18.0000 0.642039
\(787\) −49.0000 −1.74666 −0.873331 0.487128i \(-0.838045\pi\)
−0.873331 + 0.487128i \(0.838045\pi\)
\(788\) −10.0000 −0.356235
\(789\) −18.0000 −0.640817
\(790\) −20.0000 −0.711568
\(791\) 6.00000 0.213335
\(792\) −2.00000 −0.0710669
\(793\) −7.00000 −0.248577
\(794\) 13.0000 0.461353
\(795\) −16.0000 −0.567462
\(796\) −7.00000 −0.248108
\(797\) −6.00000 −0.212531 −0.106265 0.994338i \(-0.533889\pi\)
−0.106265 + 0.994338i \(0.533889\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −11.0000 −0.388909
\(801\) −18.0000 −0.635999
\(802\) 10.0000 0.353112
\(803\) −6.00000 −0.211735
\(804\) −3.00000 −0.105802
\(805\) −48.0000 −1.69178
\(806\) 7.00000 0.246564
\(807\) −18.0000 −0.633630
\(808\) −2.00000 −0.0703598
\(809\) 36.0000 1.26569 0.632846 0.774277i \(-0.281886\pi\)
0.632846 + 0.774277i \(0.281886\pi\)
\(810\) 4.00000 0.140546
\(811\) −36.0000 −1.26413 −0.632065 0.774915i \(-0.717793\pi\)
−0.632065 + 0.774915i \(0.717793\pi\)
\(812\) 12.0000 0.421117
\(813\) −8.00000 −0.280572
\(814\) 14.0000 0.490700
\(815\) −44.0000 −1.54125
\(816\) 0 0
\(817\) 0 0
\(818\) 10.0000 0.349642
\(819\) −21.0000 −0.733799
\(820\) 16.0000 0.558744
\(821\) 12.0000 0.418803 0.209401 0.977830i \(-0.432848\pi\)
0.209401 + 0.977830i \(0.432848\pi\)
\(822\) 2.00000 0.0697580
\(823\) 44.0000 1.53374 0.766872 0.641800i \(-0.221812\pi\)
0.766872 + 0.641800i \(0.221812\pi\)
\(824\) 9.00000 0.313530
\(825\) 22.0000 0.765942
\(826\) 18.0000 0.626300
\(827\) −8.00000 −0.278187 −0.139094 0.990279i \(-0.544419\pi\)
−0.139094 + 0.990279i \(0.544419\pi\)
\(828\) −4.00000 −0.139010
\(829\) −45.0000 −1.56291 −0.781457 0.623959i \(-0.785523\pi\)
−0.781457 + 0.623959i \(0.785523\pi\)
\(830\) −48.0000 −1.66610
\(831\) −26.0000 −0.901930
\(832\) 7.00000 0.242681
\(833\) 0 0
\(834\) −21.0000 −0.727171
\(835\) 24.0000 0.830554
\(836\) 0 0
\(837\) −1.00000 −0.0345651
\(838\) −30.0000 −1.03633
\(839\) 40.0000 1.38095 0.690477 0.723355i \(-0.257401\pi\)
0.690477 + 0.723355i \(0.257401\pi\)
\(840\) −12.0000 −0.414039
\(841\) −13.0000 −0.448276
\(842\) 26.0000 0.896019
\(843\) −4.00000 −0.137767
\(844\) −9.00000 −0.309793
\(845\) −144.000 −4.95375
\(846\) −2.00000 −0.0687614
\(847\) 21.0000 0.721569
\(848\) 4.00000 0.137361
\(849\) −12.0000 −0.411839
\(850\) 0 0
\(851\) 28.0000 0.959828
\(852\) −2.00000 −0.0685189
\(853\) −9.00000 −0.308154 −0.154077 0.988059i \(-0.549240\pi\)
−0.154077 + 0.988059i \(0.549240\pi\)
\(854\) −3.00000 −0.102658
\(855\) 0 0
\(856\) 2.00000 0.0683586
\(857\) 6.00000 0.204956 0.102478 0.994735i \(-0.467323\pi\)
0.102478 + 0.994735i \(0.467323\pi\)
\(858\) −14.0000 −0.477952
\(859\) 7.00000 0.238837 0.119418 0.992844i \(-0.461897\pi\)
0.119418 + 0.992844i \(0.461897\pi\)
\(860\) −28.0000 −0.954792
\(861\) 12.0000 0.408959
\(862\) 6.00000 0.204361
\(863\) 6.00000 0.204242 0.102121 0.994772i \(-0.467437\pi\)
0.102121 + 0.994772i \(0.467437\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 48.0000 1.63205
\(866\) 23.0000 0.781572
\(867\) −17.0000 −0.577350
\(868\) 3.00000 0.101827
\(869\) −10.0000 −0.339227
\(870\) −16.0000 −0.542451
\(871\) −21.0000 −0.711558
\(872\) 6.00000 0.203186
\(873\) −10.0000 −0.338449
\(874\) 0 0
\(875\) 72.0000 2.43404
\(876\) −3.00000 −0.101361
\(877\) −19.0000 −0.641584 −0.320792 0.947150i \(-0.603949\pi\)
−0.320792 + 0.947150i \(0.603949\pi\)
\(878\) 23.0000 0.776212
\(879\) 18.0000 0.607125
\(880\) −8.00000 −0.269680
\(881\) −24.0000 −0.808581 −0.404290 0.914631i \(-0.632481\pi\)
−0.404290 + 0.914631i \(0.632481\pi\)
\(882\) −2.00000 −0.0673435
\(883\) −41.0000 −1.37976 −0.689880 0.723924i \(-0.742337\pi\)
−0.689880 + 0.723924i \(0.742337\pi\)
\(884\) 0 0
\(885\) −24.0000 −0.806751
\(886\) −4.00000 −0.134383
\(887\) −10.0000 −0.335767 −0.167884 0.985807i \(-0.553693\pi\)
−0.167884 + 0.985807i \(0.553693\pi\)
\(888\) 7.00000 0.234905
\(889\) 0 0
\(890\) −72.0000 −2.41345
\(891\) 2.00000 0.0670025
\(892\) −5.00000 −0.167412
\(893\) 0 0
\(894\) 18.0000 0.602010
\(895\) −48.0000 −1.60446
\(896\) 3.00000 0.100223
\(897\) −28.0000 −0.934893
\(898\) 16.0000 0.533927
\(899\) 4.00000 0.133407
\(900\) 11.0000 0.366667
\(901\) 0 0
\(902\) 8.00000 0.266371
\(903\) −21.0000 −0.698836
\(904\) 2.00000 0.0665190
\(905\) −8.00000 −0.265929
\(906\) 20.0000 0.664455
\(907\) −20.0000 −0.664089 −0.332045 0.943264i \(-0.607738\pi\)
−0.332045 + 0.943264i \(0.607738\pi\)
\(908\) −4.00000 −0.132745
\(909\) 2.00000 0.0663358
\(910\) −84.0000 −2.78457
\(911\) 12.0000 0.397578 0.198789 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) 0 0
\(913\) −24.0000 −0.794284
\(914\) −29.0000 −0.959235
\(915\) 4.00000 0.132236
\(916\) −7.00000 −0.231287
\(917\) 54.0000 1.78324
\(918\) 0 0
\(919\) 11.0000 0.362857 0.181428 0.983404i \(-0.441928\pi\)
0.181428 + 0.983404i \(0.441928\pi\)
\(920\) −16.0000 −0.527504
\(921\) −12.0000 −0.395413
\(922\) 34.0000 1.11973
\(923\) −14.0000 −0.460816
\(924\) −6.00000 −0.197386
\(925\) −77.0000 −2.53174
\(926\) −19.0000 −0.624379
\(927\) −9.00000 −0.295599
\(928\) 4.00000 0.131306
\(929\) 8.00000 0.262471 0.131236 0.991351i \(-0.458106\pi\)
0.131236 + 0.991351i \(0.458106\pi\)
\(930\) −4.00000 −0.131165
\(931\) 0 0
\(932\) 6.00000 0.196537
\(933\) 34.0000 1.11311
\(934\) 8.00000 0.261768
\(935\) 0 0
\(936\) −7.00000 −0.228802
\(937\) −25.0000 −0.816714 −0.408357 0.912822i \(-0.633898\pi\)
−0.408357 + 0.912822i \(0.633898\pi\)
\(938\) −9.00000 −0.293860
\(939\) 14.0000 0.456873
\(940\) −8.00000 −0.260931
\(941\) −16.0000 −0.521585 −0.260793 0.965395i \(-0.583984\pi\)
−0.260793 + 0.965395i \(0.583984\pi\)
\(942\) −7.00000 −0.228072
\(943\) 16.0000 0.521032
\(944\) 6.00000 0.195283
\(945\) 12.0000 0.390360
\(946\) −14.0000 −0.455179
\(947\) 54.0000 1.75476 0.877382 0.479792i \(-0.159288\pi\)
0.877382 + 0.479792i \(0.159288\pi\)
\(948\) −5.00000 −0.162392
\(949\) −21.0000 −0.681689
\(950\) 0 0
\(951\) 24.0000 0.778253
\(952\) 0 0
\(953\) 10.0000 0.323932 0.161966 0.986796i \(-0.448217\pi\)
0.161966 + 0.986796i \(0.448217\pi\)
\(954\) −4.00000 −0.129505
\(955\) 80.0000 2.58874
\(956\) 24.0000 0.776215
\(957\) −8.00000 −0.258603
\(958\) −34.0000 −1.09849
\(959\) 6.00000 0.193750
\(960\) −4.00000 −0.129099
\(961\) −30.0000 −0.967742
\(962\) 49.0000 1.57982
\(963\) −2.00000 −0.0644491
\(964\) −1.00000 −0.0322078
\(965\) −84.0000 −2.70406
\(966\) −12.0000 −0.386094
\(967\) −9.00000 −0.289420 −0.144710 0.989474i \(-0.546225\pi\)
−0.144710 + 0.989474i \(0.546225\pi\)
\(968\) 7.00000 0.224989
\(969\) 0 0
\(970\) −40.0000 −1.28432
\(971\) −14.0000 −0.449281 −0.224641 0.974442i \(-0.572121\pi\)
−0.224641 + 0.974442i \(0.572121\pi\)
\(972\) 1.00000 0.0320750
\(973\) −63.0000 −2.01969
\(974\) 16.0000 0.512673
\(975\) 77.0000 2.46597
\(976\) −1.00000 −0.0320092
\(977\) −14.0000 −0.447900 −0.223950 0.974601i \(-0.571895\pi\)
−0.223950 + 0.974601i \(0.571895\pi\)
\(978\) −11.0000 −0.351741
\(979\) −36.0000 −1.15056
\(980\) −8.00000 −0.255551
\(981\) −6.00000 −0.191565
\(982\) −26.0000 −0.829693
\(983\) −6.00000 −0.191370 −0.0956851 0.995412i \(-0.530504\pi\)
−0.0956851 + 0.995412i \(0.530504\pi\)
\(984\) 4.00000 0.127515
\(985\) 40.0000 1.27451
\(986\) 0 0
\(987\) −6.00000 −0.190982
\(988\) 0 0
\(989\) −28.0000 −0.890348
\(990\) 8.00000 0.254257
\(991\) 29.0000 0.921215 0.460608 0.887604i \(-0.347632\pi\)
0.460608 + 0.887604i \(0.347632\pi\)
\(992\) 1.00000 0.0317500
\(993\) −23.0000 −0.729883
\(994\) −6.00000 −0.190308
\(995\) 28.0000 0.887660
\(996\) −12.0000 −0.380235
\(997\) −53.0000 −1.67853 −0.839263 0.543725i \(-0.817013\pi\)
−0.839263 + 0.543725i \(0.817013\pi\)
\(998\) 11.0000 0.348199
\(999\) −7.00000 −0.221470
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 2166.2.a.c.1.1 1
3.2 odd 2 6498.2.a.x.1.1 1
19.8 odd 6 114.2.e.a.7.1 2
19.12 odd 6 114.2.e.a.49.1 yes 2
19.18 odd 2 2166.2.a.f.1.1 1
57.8 even 6 342.2.g.d.235.1 2
57.50 even 6 342.2.g.d.163.1 2
57.56 even 2 6498.2.a.l.1.1 1
76.27 even 6 912.2.q.d.577.1 2
76.31 even 6 912.2.q.d.49.1 2
228.107 odd 6 2736.2.s.c.1873.1 2
228.179 odd 6 2736.2.s.c.577.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.e.a.7.1 2 19.8 odd 6
114.2.e.a.49.1 yes 2 19.12 odd 6
342.2.g.d.163.1 2 57.50 even 6
342.2.g.d.235.1 2 57.8 even 6
912.2.q.d.49.1 2 76.31 even 6
912.2.q.d.577.1 2 76.27 even 6
2166.2.a.c.1.1 1 1.1 even 1 trivial
2166.2.a.f.1.1 1 19.18 odd 2
2736.2.s.c.577.1 2 228.179 odd 6
2736.2.s.c.1873.1 2 228.107 odd 6
6498.2.a.l.1.1 1 57.56 even 2
6498.2.a.x.1.1 1 3.2 odd 2