Properties

Label 5586.2.a.y.1.1
Level $5586$
Weight $2$
Character 5586.1
Self dual yes
Analytic conductor $44.604$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5586 = 2 \cdot 3 \cdot 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5586.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(44.6044345691\)
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 \(\chi\) \(=\) 5586.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} -4.00000 q^{11} +1.00000 q^{12} -2.00000 q^{13} -2.00000 q^{15} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} +1.00000 q^{19} -2.00000 q^{20} -4.00000 q^{22} -4.00000 q^{23} +1.00000 q^{24} -1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} -2.00000 q^{29} -2.00000 q^{30} -4.00000 q^{31} +1.00000 q^{32} -4.00000 q^{33} +6.00000 q^{34} +1.00000 q^{36} +10.0000 q^{37} +1.00000 q^{38} -2.00000 q^{39} -2.00000 q^{40} -10.0000 q^{41} +4.00000 q^{43} -4.00000 q^{44} -2.00000 q^{45} -4.00000 q^{46} +4.00000 q^{47} +1.00000 q^{48} -1.00000 q^{50} +6.00000 q^{51} -2.00000 q^{52} -10.0000 q^{53} +1.00000 q^{54} +8.00000 q^{55} +1.00000 q^{57} -2.00000 q^{58} -12.0000 q^{59} -2.00000 q^{60} -14.0000 q^{61} -4.00000 q^{62} +1.00000 q^{64} +4.00000 q^{65} -4.00000 q^{66} -12.0000 q^{67} +6.00000 q^{68} -4.00000 q^{69} +8.00000 q^{71} +1.00000 q^{72} +6.00000 q^{73} +10.0000 q^{74} -1.00000 q^{75} +1.00000 q^{76} -2.00000 q^{78} -4.00000 q^{79} -2.00000 q^{80} +1.00000 q^{81} -10.0000 q^{82} -12.0000 q^{83} -12.0000 q^{85} +4.00000 q^{86} -2.00000 q^{87} -4.00000 q^{88} +6.00000 q^{89} -2.00000 q^{90} -4.00000 q^{92} -4.00000 q^{93} +4.00000 q^{94} -2.00000 q^{95} +1.00000 q^{96} -10.0000 q^{97} -4.00000 q^{99} +O(q^{100})\)

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
\(5\) −2.00000 −0.894427 −0.447214 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −2.00000 −0.632456
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 1.00000 0.288675
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) −2.00000 −0.516398
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 1.00000 0.235702
\(19\) 1.00000 0.229416
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) −4.00000 −0.852803
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 1.00000 0.204124
\(25\) −1.00000 −0.200000
\(26\) −2.00000 −0.392232
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) −2.00000 −0.365148
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 1.00000 0.176777
\(33\) −4.00000 −0.696311
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 10.0000 1.64399 0.821995 0.569495i \(-0.192861\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(38\) 1.00000 0.162221
\(39\) −2.00000 −0.320256
\(40\) −2.00000 −0.316228
\(41\) −10.0000 −1.56174 −0.780869 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) −4.00000 −0.603023
\(45\) −2.00000 −0.298142
\(46\) −4.00000 −0.589768
\(47\) 4.00000 0.583460 0.291730 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) −1.00000 −0.141421
\(51\) 6.00000 0.840168
\(52\) −2.00000 −0.277350
\(53\) −10.0000 −1.37361 −0.686803 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) 1.00000 0.136083
\(55\) 8.00000 1.07872
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) −2.00000 −0.262613
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) −2.00000 −0.258199
\(61\) −14.0000 −1.79252 −0.896258 0.443533i \(-0.853725\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) −4.00000 −0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 4.00000 0.496139
\(66\) −4.00000 −0.492366
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 6.00000 0.727607
\(69\) −4.00000 −0.481543
\(70\) 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
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 10.0000 1.16248
\(75\) −1.00000 −0.115470
\(76\) 1.00000 0.114708
\(77\) 0 0
\(78\) −2.00000 −0.226455
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) −2.00000 −0.223607
\(81\) 1.00000 0.111111
\(82\) −10.0000 −1.10432
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) −12.0000 −1.30158
\(86\) 4.00000 0.431331
\(87\) −2.00000 −0.214423
\(88\) −4.00000 −0.426401
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) −2.00000 −0.210819
\(91\) 0 0
\(92\) −4.00000 −0.417029
\(93\) −4.00000 −0.414781
\(94\) 4.00000 0.412568
\(95\) −2.00000 −0.205196
\(96\) 1.00000 0.102062
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 0 0
\(99\) −4.00000 −0.402015
\(100\) −1.00000 −0.100000
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) 6.00000 0.594089
\(103\) 12.0000 1.18240 0.591198 0.806527i \(-0.298655\pi\)
0.591198 + 0.806527i \(0.298655\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) −10.0000 −0.971286
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 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
\(111\) 10.0000 0.949158
\(112\) 0 0
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) 1.00000 0.0936586
\(115\) 8.00000 0.746004
\(116\) −2.00000 −0.185695
\(117\) −2.00000 −0.184900
\(118\) −12.0000 −1.10469
\(119\) 0 0
\(120\) −2.00000 −0.182574
\(121\) 5.00000 0.454545
\(122\) −14.0000 −1.26750
\(123\) −10.0000 −0.901670
\(124\) −4.00000 −0.359211
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) −12.0000 −1.06483 −0.532414 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) 1.00000 0.0883883
\(129\) 4.00000 0.352180
\(130\) 4.00000 0.350823
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) −4.00000 −0.348155
\(133\) 0 0
\(134\) −12.0000 −1.03664
\(135\) −2.00000 −0.172133
\(136\) 6.00000 0.514496
\(137\) −14.0000 −1.19610 −0.598050 0.801459i \(-0.704058\pi\)
−0.598050 + 0.801459i \(0.704058\pi\)
\(138\) −4.00000 −0.340503
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 0 0
\(141\) 4.00000 0.336861
\(142\) 8.00000 0.671345
\(143\) 8.00000 0.668994
\(144\) 1.00000 0.0833333
\(145\) 4.00000 0.332182
\(146\) 6.00000 0.496564
\(147\) 0 0
\(148\) 10.0000 0.821995
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 20.0000 1.62758 0.813788 0.581161i \(-0.197401\pi\)
0.813788 + 0.581161i \(0.197401\pi\)
\(152\) 1.00000 0.0811107
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) 8.00000 0.642575
\(156\) −2.00000 −0.160128
\(157\) −22.0000 −1.75579 −0.877896 0.478852i \(-0.841053\pi\)
−0.877896 + 0.478852i \(0.841053\pi\)
\(158\) −4.00000 −0.318223
\(159\) −10.0000 −0.793052
\(160\) −2.00000 −0.158114
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 20.0000 1.56652 0.783260 0.621694i \(-0.213555\pi\)
0.783260 + 0.621694i \(0.213555\pi\)
\(164\) −10.0000 −0.780869
\(165\) 8.00000 0.622799
\(166\) −12.0000 −0.931381
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −12.0000 −0.920358
\(171\) 1.00000 0.0764719
\(172\) 4.00000 0.304997
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) −2.00000 −0.151620
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) −12.0000 −0.901975
\(178\) 6.00000 0.449719
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) −2.00000 −0.149071
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 0 0
\(183\) −14.0000 −1.03491
\(184\) −4.00000 −0.294884
\(185\) −20.0000 −1.47043
\(186\) −4.00000 −0.293294
\(187\) −24.0000 −1.75505
\(188\) 4.00000 0.291730
\(189\) 0 0
\(190\) −2.00000 −0.145095
\(191\) 4.00000 0.289430 0.144715 0.989473i \(-0.453773\pi\)
0.144715 + 0.989473i \(0.453773\pi\)
\(192\) 1.00000 0.0721688
\(193\) −6.00000 −0.431889 −0.215945 0.976406i \(-0.569283\pi\)
−0.215945 + 0.976406i \(0.569283\pi\)
\(194\) −10.0000 −0.717958
\(195\) 4.00000 0.286446
\(196\) 0 0
\(197\) −22.0000 −1.56744 −0.783718 0.621117i \(-0.786679\pi\)
−0.783718 + 0.621117i \(0.786679\pi\)
\(198\) −4.00000 −0.284268
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −12.0000 −0.846415
\(202\) −2.00000 −0.140720
\(203\) 0 0
\(204\) 6.00000 0.420084
\(205\) 20.0000 1.39686
\(206\) 12.0000 0.836080
\(207\) −4.00000 −0.278019
\(208\) −2.00000 −0.138675
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) −10.0000 −0.686803
\(213\) 8.00000 0.548151
\(214\) −4.00000 −0.273434
\(215\) −8.00000 −0.545595
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) −6.00000 −0.406371
\(219\) 6.00000 0.405442
\(220\) 8.00000 0.539360
\(221\) −12.0000 −0.807207
\(222\) 10.0000 0.671156
\(223\) 28.0000 1.87502 0.937509 0.347960i \(-0.113126\pi\)
0.937509 + 0.347960i \(0.113126\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) 2.00000 0.133038
\(227\) −28.0000 −1.85843 −0.929213 0.369546i \(-0.879513\pi\)
−0.929213 + 0.369546i \(0.879513\pi\)
\(228\) 1.00000 0.0662266
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 8.00000 0.527504
\(231\) 0 0
\(232\) −2.00000 −0.131306
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) −2.00000 −0.130744
\(235\) −8.00000 −0.521862
\(236\) −12.0000 −0.781133
\(237\) −4.00000 −0.259828
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) −2.00000 −0.129099
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 5.00000 0.321412
\(243\) 1.00000 0.0641500
\(244\) −14.0000 −0.896258
\(245\) 0 0
\(246\) −10.0000 −0.637577
\(247\) −2.00000 −0.127257
\(248\) −4.00000 −0.254000
\(249\) −12.0000 −0.760469
\(250\) 12.0000 0.758947
\(251\) 28.0000 1.76734 0.883672 0.468106i \(-0.155064\pi\)
0.883672 + 0.468106i \(0.155064\pi\)
\(252\) 0 0
\(253\) 16.0000 1.00591
\(254\) −12.0000 −0.752947
\(255\) −12.0000 −0.751469
\(256\) 1.00000 0.0625000
\(257\) −2.00000 −0.124757 −0.0623783 0.998053i \(-0.519869\pi\)
−0.0623783 + 0.998053i \(0.519869\pi\)
\(258\) 4.00000 0.249029
\(259\) 0 0
\(260\) 4.00000 0.248069
\(261\) −2.00000 −0.123797
\(262\) −12.0000 −0.741362
\(263\) −12.0000 −0.739952 −0.369976 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) −4.00000 −0.246183
\(265\) 20.0000 1.22859
\(266\) 0 0
\(267\) 6.00000 0.367194
\(268\) −12.0000 −0.733017
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) −2.00000 −0.121716
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 6.00000 0.363803
\(273\) 0 0
\(274\) −14.0000 −0.845771
\(275\) 4.00000 0.241209
\(276\) −4.00000 −0.240772
\(277\) −26.0000 −1.56219 −0.781094 0.624413i \(-0.785338\pi\)
−0.781094 + 0.624413i \(0.785338\pi\)
\(278\) −12.0000 −0.719712
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 4.00000 0.238197
\(283\) −12.0000 −0.713326 −0.356663 0.934233i \(-0.616086\pi\)
−0.356663 + 0.934233i \(0.616086\pi\)
\(284\) 8.00000 0.474713
\(285\) −2.00000 −0.118470
\(286\) 8.00000 0.473050
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) 19.0000 1.11765
\(290\) 4.00000 0.234888
\(291\) −10.0000 −0.586210
\(292\) 6.00000 0.351123
\(293\) 18.0000 1.05157 0.525786 0.850617i \(-0.323771\pi\)
0.525786 + 0.850617i \(0.323771\pi\)
\(294\) 0 0
\(295\) 24.0000 1.39733
\(296\) 10.0000 0.581238
\(297\) −4.00000 −0.232104
\(298\) −6.00000 −0.347571
\(299\) 8.00000 0.462652
\(300\) −1.00000 −0.0577350
\(301\) 0 0
\(302\) 20.0000 1.15087
\(303\) −2.00000 −0.114897
\(304\) 1.00000 0.0573539
\(305\) 28.0000 1.60328
\(306\) 6.00000 0.342997
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) 0 0
\(309\) 12.0000 0.682656
\(310\) 8.00000 0.454369
\(311\) −4.00000 −0.226819 −0.113410 0.993548i \(-0.536177\pi\)
−0.113410 + 0.993548i \(0.536177\pi\)
\(312\) −2.00000 −0.113228
\(313\) 22.0000 1.24351 0.621757 0.783210i \(-0.286419\pi\)
0.621757 + 0.783210i \(0.286419\pi\)
\(314\) −22.0000 −1.24153
\(315\) 0 0
\(316\) −4.00000 −0.225018
\(317\) 6.00000 0.336994 0.168497 0.985702i \(-0.446109\pi\)
0.168497 + 0.985702i \(0.446109\pi\)
\(318\) −10.0000 −0.560772
\(319\) 8.00000 0.447914
\(320\) −2.00000 −0.111803
\(321\) −4.00000 −0.223258
\(322\) 0 0
\(323\) 6.00000 0.333849
\(324\) 1.00000 0.0555556
\(325\) 2.00000 0.110940
\(326\) 20.0000 1.10770
\(327\) −6.00000 −0.331801
\(328\) −10.0000 −0.552158
\(329\) 0 0
\(330\) 8.00000 0.440386
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) −12.0000 −0.658586
\(333\) 10.0000 0.547997
\(334\) 0 0
\(335\) 24.0000 1.31126
\(336\) 0 0
\(337\) −14.0000 −0.762629 −0.381314 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) −9.00000 −0.489535
\(339\) 2.00000 0.108625
\(340\) −12.0000 −0.650791
\(341\) 16.0000 0.866449
\(342\) 1.00000 0.0540738
\(343\) 0 0
\(344\) 4.00000 0.215666
\(345\) 8.00000 0.430706
\(346\) −6.00000 −0.322562
\(347\) −20.0000 −1.07366 −0.536828 0.843692i \(-0.680378\pi\)
−0.536828 + 0.843692i \(0.680378\pi\)
\(348\) −2.00000 −0.107211
\(349\) 26.0000 1.39175 0.695874 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) −4.00000 −0.213201
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) −12.0000 −0.637793
\(355\) −16.0000 −0.849192
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) 12.0000 0.633336 0.316668 0.948536i \(-0.397436\pi\)
0.316668 + 0.948536i \(0.397436\pi\)
\(360\) −2.00000 −0.105409
\(361\) 1.00000 0.0526316
\(362\) 14.0000 0.735824
\(363\) 5.00000 0.262432
\(364\) 0 0
\(365\) −12.0000 −0.628109
\(366\) −14.0000 −0.731792
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) −4.00000 −0.208514
\(369\) −10.0000 −0.520579
\(370\) −20.0000 −1.03975
\(371\) 0 0
\(372\) −4.00000 −0.207390
\(373\) 26.0000 1.34623 0.673114 0.739538i \(-0.264956\pi\)
0.673114 + 0.739538i \(0.264956\pi\)
\(374\) −24.0000 −1.24101
\(375\) 12.0000 0.619677
\(376\) 4.00000 0.206284
\(377\) 4.00000 0.206010
\(378\) 0 0
\(379\) −36.0000 −1.84920 −0.924598 0.380945i \(-0.875599\pi\)
−0.924598 + 0.380945i \(0.875599\pi\)
\(380\) −2.00000 −0.102598
\(381\) −12.0000 −0.614779
\(382\) 4.00000 0.204658
\(383\) −16.0000 −0.817562 −0.408781 0.912633i \(-0.634046\pi\)
−0.408781 + 0.912633i \(0.634046\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −6.00000 −0.305392
\(387\) 4.00000 0.203331
\(388\) −10.0000 −0.507673
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 4.00000 0.202548
\(391\) −24.0000 −1.21373
\(392\) 0 0
\(393\) −12.0000 −0.605320
\(394\) −22.0000 −1.10834
\(395\) 8.00000 0.402524
\(396\) −4.00000 −0.201008
\(397\) 10.0000 0.501886 0.250943 0.968002i \(-0.419259\pi\)
0.250943 + 0.968002i \(0.419259\pi\)
\(398\) 16.0000 0.802008
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) −14.0000 −0.699127 −0.349563 0.936913i \(-0.613670\pi\)
−0.349563 + 0.936913i \(0.613670\pi\)
\(402\) −12.0000 −0.598506
\(403\) 8.00000 0.398508
\(404\) −2.00000 −0.0995037
\(405\) −2.00000 −0.0993808
\(406\) 0 0
\(407\) −40.0000 −1.98273
\(408\) 6.00000 0.297044
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 20.0000 0.987730
\(411\) −14.0000 −0.690569
\(412\) 12.0000 0.591198
\(413\) 0 0
\(414\) −4.00000 −0.196589
\(415\) 24.0000 1.17811
\(416\) −2.00000 −0.0980581
\(417\) −12.0000 −0.587643
\(418\) −4.00000 −0.195646
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) 26.0000 1.26716 0.633581 0.773676i \(-0.281584\pi\)
0.633581 + 0.773676i \(0.281584\pi\)
\(422\) 12.0000 0.584151
\(423\) 4.00000 0.194487
\(424\) −10.0000 −0.485643
\(425\) −6.00000 −0.291043
\(426\) 8.00000 0.387601
\(427\) 0 0
\(428\) −4.00000 −0.193347
\(429\) 8.00000 0.386244
\(430\) −8.00000 −0.385794
\(431\) −24.0000 −1.15604 −0.578020 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 1.00000 0.0481125
\(433\) −26.0000 −1.24948 −0.624740 0.780833i \(-0.714795\pi\)
−0.624740 + 0.780833i \(0.714795\pi\)
\(434\) 0 0
\(435\) 4.00000 0.191785
\(436\) −6.00000 −0.287348
\(437\) −4.00000 −0.191346
\(438\) 6.00000 0.286691
\(439\) 4.00000 0.190910 0.0954548 0.995434i \(-0.469569\pi\)
0.0954548 + 0.995434i \(0.469569\pi\)
\(440\) 8.00000 0.381385
\(441\) 0 0
\(442\) −12.0000 −0.570782
\(443\) −20.0000 −0.950229 −0.475114 0.879924i \(-0.657593\pi\)
−0.475114 + 0.879924i \(0.657593\pi\)
\(444\) 10.0000 0.474579
\(445\) −12.0000 −0.568855
\(446\) 28.0000 1.32584
\(447\) −6.00000 −0.283790
\(448\) 0 0
\(449\) 34.0000 1.60456 0.802280 0.596948i \(-0.203620\pi\)
0.802280 + 0.596948i \(0.203620\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 40.0000 1.88353
\(452\) 2.00000 0.0940721
\(453\) 20.0000 0.939682
\(454\) −28.0000 −1.31411
\(455\) 0 0
\(456\) 1.00000 0.0468293
\(457\) 26.0000 1.21623 0.608114 0.793849i \(-0.291926\pi\)
0.608114 + 0.793849i \(0.291926\pi\)
\(458\) 10.0000 0.467269
\(459\) 6.00000 0.280056
\(460\) 8.00000 0.373002
\(461\) −2.00000 −0.0931493 −0.0465746 0.998915i \(-0.514831\pi\)
−0.0465746 + 0.998915i \(0.514831\pi\)
\(462\) 0 0
\(463\) −8.00000 −0.371792 −0.185896 0.982569i \(-0.559519\pi\)
−0.185896 + 0.982569i \(0.559519\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 8.00000 0.370991
\(466\) −6.00000 −0.277945
\(467\) 20.0000 0.925490 0.462745 0.886492i \(-0.346865\pi\)
0.462745 + 0.886492i \(0.346865\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 0 0
\(470\) −8.00000 −0.369012
\(471\) −22.0000 −1.01371
\(472\) −12.0000 −0.552345
\(473\) −16.0000 −0.735681
\(474\) −4.00000 −0.183726
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) −10.0000 −0.457869
\(478\) 12.0000 0.548867
\(479\) −4.00000 −0.182765 −0.0913823 0.995816i \(-0.529129\pi\)
−0.0913823 + 0.995816i \(0.529129\pi\)
\(480\) −2.00000 −0.0912871
\(481\) −20.0000 −0.911922
\(482\) −10.0000 −0.455488
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) 20.0000 0.908153
\(486\) 1.00000 0.0453609
\(487\) 4.00000 0.181257 0.0906287 0.995885i \(-0.471112\pi\)
0.0906287 + 0.995885i \(0.471112\pi\)
\(488\) −14.0000 −0.633750
\(489\) 20.0000 0.904431
\(490\) 0 0
\(491\) 20.0000 0.902587 0.451294 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) −10.0000 −0.450835
\(493\) −12.0000 −0.540453
\(494\) −2.00000 −0.0899843
\(495\) 8.00000 0.359573
\(496\) −4.00000 −0.179605
\(497\) 0 0
\(498\) −12.0000 −0.537733
\(499\) −20.0000 −0.895323 −0.447661 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) 12.0000 0.536656
\(501\) 0 0
\(502\) 28.0000 1.24970
\(503\) −36.0000 −1.60516 −0.802580 0.596544i \(-0.796540\pi\)
−0.802580 + 0.596544i \(0.796540\pi\)
\(504\) 0 0
\(505\) 4.00000 0.177998
\(506\) 16.0000 0.711287
\(507\) −9.00000 −0.399704
\(508\) −12.0000 −0.532414
\(509\) −6.00000 −0.265945 −0.132973 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) −12.0000 −0.531369
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 1.00000 0.0441511
\(514\) −2.00000 −0.0882162
\(515\) −24.0000 −1.05757
\(516\) 4.00000 0.176090
\(517\) −16.0000 −0.703679
\(518\) 0 0
\(519\) −6.00000 −0.263371
\(520\) 4.00000 0.175412
\(521\) −26.0000 −1.13908 −0.569540 0.821963i \(-0.692879\pi\)
−0.569540 + 0.821963i \(0.692879\pi\)
\(522\) −2.00000 −0.0875376
\(523\) 4.00000 0.174908 0.0874539 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) −12.0000 −0.523225
\(527\) −24.0000 −1.04546
\(528\) −4.00000 −0.174078
\(529\) −7.00000 −0.304348
\(530\) 20.0000 0.868744
\(531\) −12.0000 −0.520756
\(532\) 0 0
\(533\) 20.0000 0.866296
\(534\) 6.00000 0.259645
\(535\) 8.00000 0.345870
\(536\) −12.0000 −0.518321
\(537\) 12.0000 0.517838
\(538\) −6.00000 −0.258678
\(539\) 0 0
\(540\) −2.00000 −0.0860663
\(541\) 38.0000 1.63375 0.816874 0.576816i \(-0.195705\pi\)
0.816874 + 0.576816i \(0.195705\pi\)
\(542\) 8.00000 0.343629
\(543\) 14.0000 0.600798
\(544\) 6.00000 0.257248
\(545\) 12.0000 0.514024
\(546\) 0 0
\(547\) 4.00000 0.171028 0.0855138 0.996337i \(-0.472747\pi\)
0.0855138 + 0.996337i \(0.472747\pi\)
\(548\) −14.0000 −0.598050
\(549\) −14.0000 −0.597505
\(550\) 4.00000 0.170561
\(551\) −2.00000 −0.0852029
\(552\) −4.00000 −0.170251
\(553\) 0 0
\(554\) −26.0000 −1.10463
\(555\) −20.0000 −0.848953
\(556\) −12.0000 −0.508913
\(557\) 34.0000 1.44063 0.720313 0.693649i \(-0.243998\pi\)
0.720313 + 0.693649i \(0.243998\pi\)
\(558\) −4.00000 −0.169334
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) −24.0000 −1.01328
\(562\) 10.0000 0.421825
\(563\) 12.0000 0.505740 0.252870 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 4.00000 0.168430
\(565\) −4.00000 −0.168281
\(566\) −12.0000 −0.504398
\(567\) 0 0
\(568\) 8.00000 0.335673
\(569\) −38.0000 −1.59304 −0.796521 0.604610i \(-0.793329\pi\)
−0.796521 + 0.604610i \(0.793329\pi\)
\(570\) −2.00000 −0.0837708
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 8.00000 0.334497
\(573\) 4.00000 0.167102
\(574\) 0 0
\(575\) 4.00000 0.166812
\(576\) 1.00000 0.0416667
\(577\) −2.00000 −0.0832611 −0.0416305 0.999133i \(-0.513255\pi\)
−0.0416305 + 0.999133i \(0.513255\pi\)
\(578\) 19.0000 0.790296
\(579\) −6.00000 −0.249351
\(580\) 4.00000 0.166091
\(581\) 0 0
\(582\) −10.0000 −0.414513
\(583\) 40.0000 1.65663
\(584\) 6.00000 0.248282
\(585\) 4.00000 0.165380
\(586\) 18.0000 0.743573
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 0 0
\(589\) −4.00000 −0.164817
\(590\) 24.0000 0.988064
\(591\) −22.0000 −0.904959
\(592\) 10.0000 0.410997
\(593\) −34.0000 −1.39621 −0.698106 0.715994i \(-0.745974\pi\)
−0.698106 + 0.715994i \(0.745974\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) 16.0000 0.654836
\(598\) 8.00000 0.327144
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 0 0
\(603\) −12.0000 −0.488678
\(604\) 20.0000 0.813788
\(605\) −10.0000 −0.406558
\(606\) −2.00000 −0.0812444
\(607\) 4.00000 0.162355 0.0811775 0.996700i \(-0.474132\pi\)
0.0811775 + 0.996700i \(0.474132\pi\)
\(608\) 1.00000 0.0405554
\(609\) 0 0
\(610\) 28.0000 1.13369
\(611\) −8.00000 −0.323645
\(612\) 6.00000 0.242536
\(613\) −2.00000 −0.0807792 −0.0403896 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) 12.0000 0.484281
\(615\) 20.0000 0.806478
\(616\) 0 0
\(617\) 42.0000 1.69086 0.845428 0.534089i \(-0.179345\pi\)
0.845428 + 0.534089i \(0.179345\pi\)
\(618\) 12.0000 0.482711
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) 8.00000 0.321288
\(621\) −4.00000 −0.160514
\(622\) −4.00000 −0.160385
\(623\) 0 0
\(624\) −2.00000 −0.0800641
\(625\) −19.0000 −0.760000
\(626\) 22.0000 0.879297
\(627\) −4.00000 −0.159745
\(628\) −22.0000 −0.877896
\(629\) 60.0000 2.39236
\(630\) 0 0
\(631\) −40.0000 −1.59237 −0.796187 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) −4.00000 −0.159111
\(633\) 12.0000 0.476957
\(634\) 6.00000 0.238290
\(635\) 24.0000 0.952411
\(636\) −10.0000 −0.396526
\(637\) 0 0
\(638\) 8.00000 0.316723
\(639\) 8.00000 0.316475
\(640\) −2.00000 −0.0790569
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) −4.00000 −0.157867
\(643\) −4.00000 −0.157745 −0.0788723 0.996885i \(-0.525132\pi\)
−0.0788723 + 0.996885i \(0.525132\pi\)
\(644\) 0 0
\(645\) −8.00000 −0.315000
\(646\) 6.00000 0.236067
\(647\) 4.00000 0.157256 0.0786281 0.996904i \(-0.474946\pi\)
0.0786281 + 0.996904i \(0.474946\pi\)
\(648\) 1.00000 0.0392837
\(649\) 48.0000 1.88416
\(650\) 2.00000 0.0784465
\(651\) 0 0
\(652\) 20.0000 0.783260
\(653\) 18.0000 0.704394 0.352197 0.935926i \(-0.385435\pi\)
0.352197 + 0.935926i \(0.385435\pi\)
\(654\) −6.00000 −0.234619
\(655\) 24.0000 0.937758
\(656\) −10.0000 −0.390434
\(657\) 6.00000 0.234082
\(658\) 0 0
\(659\) −4.00000 −0.155818 −0.0779089 0.996960i \(-0.524824\pi\)
−0.0779089 + 0.996960i \(0.524824\pi\)
\(660\) 8.00000 0.311400
\(661\) 38.0000 1.47803 0.739014 0.673690i \(-0.235292\pi\)
0.739014 + 0.673690i \(0.235292\pi\)
\(662\) −4.00000 −0.155464
\(663\) −12.0000 −0.466041
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) 10.0000 0.387492
\(667\) 8.00000 0.309761
\(668\) 0 0
\(669\) 28.0000 1.08254
\(670\) 24.0000 0.927201
\(671\) 56.0000 2.16186
\(672\) 0 0
\(673\) 26.0000 1.00223 0.501113 0.865382i \(-0.332924\pi\)
0.501113 + 0.865382i \(0.332924\pi\)
\(674\) −14.0000 −0.539260
\(675\) −1.00000 −0.0384900
\(676\) −9.00000 −0.346154
\(677\) 26.0000 0.999261 0.499631 0.866239i \(-0.333469\pi\)
0.499631 + 0.866239i \(0.333469\pi\)
\(678\) 2.00000 0.0768095
\(679\) 0 0
\(680\) −12.0000 −0.460179
\(681\) −28.0000 −1.07296
\(682\) 16.0000 0.612672
\(683\) 4.00000 0.153056 0.0765279 0.997067i \(-0.475617\pi\)
0.0765279 + 0.997067i \(0.475617\pi\)
\(684\) 1.00000 0.0382360
\(685\) 28.0000 1.06983
\(686\) 0 0
\(687\) 10.0000 0.381524
\(688\) 4.00000 0.152499
\(689\) 20.0000 0.761939
\(690\) 8.00000 0.304555
\(691\) 36.0000 1.36950 0.684752 0.728776i \(-0.259910\pi\)
0.684752 + 0.728776i \(0.259910\pi\)
\(692\) −6.00000 −0.228086
\(693\) 0 0
\(694\) −20.0000 −0.759190
\(695\) 24.0000 0.910372
\(696\) −2.00000 −0.0758098
\(697\) −60.0000 −2.27266
\(698\) 26.0000 0.984115
\(699\) −6.00000 −0.226941
\(700\) 0 0
\(701\) −22.0000 −0.830929 −0.415464 0.909610i \(-0.636381\pi\)
−0.415464 + 0.909610i \(0.636381\pi\)
\(702\) −2.00000 −0.0754851
\(703\) 10.0000 0.377157
\(704\) −4.00000 −0.150756
\(705\) −8.00000 −0.301297
\(706\) −18.0000 −0.677439
\(707\) 0 0
\(708\) −12.0000 −0.450988
\(709\) −42.0000 −1.57734 −0.788672 0.614815i \(-0.789231\pi\)
−0.788672 + 0.614815i \(0.789231\pi\)
\(710\) −16.0000 −0.600469
\(711\) −4.00000 −0.150012
\(712\) 6.00000 0.224860
\(713\) 16.0000 0.599205
\(714\) 0 0
\(715\) −16.0000 −0.598366
\(716\) 12.0000 0.448461
\(717\) 12.0000 0.448148
\(718\) 12.0000 0.447836
\(719\) −28.0000 −1.04422 −0.522112 0.852877i \(-0.674856\pi\)
−0.522112 + 0.852877i \(0.674856\pi\)
\(720\) −2.00000 −0.0745356
\(721\) 0 0
\(722\) 1.00000 0.0372161
\(723\) −10.0000 −0.371904
\(724\) 14.0000 0.520306
\(725\) 2.00000 0.0742781
\(726\) 5.00000 0.185567
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −12.0000 −0.444140
\(731\) 24.0000 0.887672
\(732\) −14.0000 −0.517455
\(733\) −6.00000 −0.221615 −0.110808 0.993842i \(-0.535344\pi\)
−0.110808 + 0.993842i \(0.535344\pi\)
\(734\) 8.00000 0.295285
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) 48.0000 1.76810
\(738\) −10.0000 −0.368105
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) −20.0000 −0.735215
\(741\) −2.00000 −0.0734718
\(742\) 0 0
\(743\) 32.0000 1.17397 0.586983 0.809599i \(-0.300316\pi\)
0.586983 + 0.809599i \(0.300316\pi\)
\(744\) −4.00000 −0.146647
\(745\) 12.0000 0.439646
\(746\) 26.0000 0.951928
\(747\) −12.0000 −0.439057
\(748\) −24.0000 −0.877527
\(749\) 0 0
\(750\) 12.0000 0.438178
\(751\) 44.0000 1.60558 0.802791 0.596260i \(-0.203347\pi\)
0.802791 + 0.596260i \(0.203347\pi\)
\(752\) 4.00000 0.145865
\(753\) 28.0000 1.02038
\(754\) 4.00000 0.145671
\(755\) −40.0000 −1.45575
\(756\) 0 0
\(757\) −2.00000 −0.0726912 −0.0363456 0.999339i \(-0.511572\pi\)
−0.0363456 + 0.999339i \(0.511572\pi\)
\(758\) −36.0000 −1.30758
\(759\) 16.0000 0.580763
\(760\) −2.00000 −0.0725476
\(761\) 30.0000 1.08750 0.543750 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) −12.0000 −0.434714
\(763\) 0 0
\(764\) 4.00000 0.144715
\(765\) −12.0000 −0.433861
\(766\) −16.0000 −0.578103
\(767\) 24.0000 0.866590
\(768\) 1.00000 0.0360844
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) −2.00000 −0.0720282
\(772\) −6.00000 −0.215945
\(773\) −30.0000 −1.07903 −0.539513 0.841978i \(-0.681391\pi\)
−0.539513 + 0.841978i \(0.681391\pi\)
\(774\) 4.00000 0.143777
\(775\) 4.00000 0.143684
\(776\) −10.0000 −0.358979
\(777\) 0 0
\(778\) 18.0000 0.645331
\(779\) −10.0000 −0.358287
\(780\) 4.00000 0.143223
\(781\) −32.0000 −1.14505
\(782\) −24.0000 −0.858238
\(783\) −2.00000 −0.0714742
\(784\) 0 0
\(785\) 44.0000 1.57043
\(786\) −12.0000 −0.428026
\(787\) 20.0000 0.712923 0.356462 0.934310i \(-0.383983\pi\)
0.356462 + 0.934310i \(0.383983\pi\)
\(788\) −22.0000 −0.783718
\(789\) −12.0000 −0.427211
\(790\) 8.00000 0.284627
\(791\) 0 0
\(792\) −4.00000 −0.142134
\(793\) 28.0000 0.994309
\(794\) 10.0000 0.354887
\(795\) 20.0000 0.709327
\(796\) 16.0000 0.567105
\(797\) −6.00000 −0.212531 −0.106265 0.994338i \(-0.533889\pi\)
−0.106265 + 0.994338i \(0.533889\pi\)
\(798\) 0 0
\(799\) 24.0000 0.849059
\(800\) −1.00000 −0.0353553
\(801\) 6.00000 0.212000
\(802\) −14.0000 −0.494357
\(803\) −24.0000 −0.846942
\(804\) −12.0000 −0.423207
\(805\) 0 0
\(806\) 8.00000 0.281788
\(807\) −6.00000 −0.211210
\(808\) −2.00000 −0.0703598
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) −2.00000 −0.0702728
\(811\) −12.0000 −0.421377 −0.210688 0.977553i \(-0.567571\pi\)
−0.210688 + 0.977553i \(0.567571\pi\)
\(812\) 0 0
\(813\) 8.00000 0.280572
\(814\) −40.0000 −1.40200
\(815\) −40.0000 −1.40114
\(816\) 6.00000 0.210042
\(817\) 4.00000 0.139942
\(818\) 14.0000 0.489499
\(819\) 0 0
\(820\) 20.0000 0.698430
\(821\) −30.0000 −1.04701 −0.523504 0.852023i \(-0.675375\pi\)
−0.523504 + 0.852023i \(0.675375\pi\)
\(822\) −14.0000 −0.488306
\(823\) 8.00000 0.278862 0.139431 0.990232i \(-0.455473\pi\)
0.139431 + 0.990232i \(0.455473\pi\)
\(824\) 12.0000 0.418040
\(825\) 4.00000 0.139262
\(826\) 0 0
\(827\) −28.0000 −0.973655 −0.486828 0.873498i \(-0.661846\pi\)
−0.486828 + 0.873498i \(0.661846\pi\)
\(828\) −4.00000 −0.139010
\(829\) −42.0000 −1.45872 −0.729360 0.684130i \(-0.760182\pi\)
−0.729360 + 0.684130i \(0.760182\pi\)
\(830\) 24.0000 0.833052
\(831\) −26.0000 −0.901930
\(832\) −2.00000 −0.0693375
\(833\) 0 0
\(834\) −12.0000 −0.415526
\(835\) 0 0
\(836\) −4.00000 −0.138343
\(837\) −4.00000 −0.138260
\(838\) 12.0000 0.414533
\(839\) 16.0000 0.552381 0.276191 0.961103i \(-0.410928\pi\)
0.276191 + 0.961103i \(0.410928\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 26.0000 0.896019
\(843\) 10.0000 0.344418
\(844\) 12.0000 0.413057
\(845\) 18.0000 0.619219
\(846\) 4.00000 0.137523
\(847\) 0 0
\(848\) −10.0000 −0.343401
\(849\) −12.0000 −0.411839
\(850\) −6.00000 −0.205798
\(851\) −40.0000 −1.37118
\(852\) 8.00000 0.274075
\(853\) −6.00000 −0.205436 −0.102718 0.994711i \(-0.532754\pi\)
−0.102718 + 0.994711i \(0.532754\pi\)
\(854\) 0 0
\(855\) −2.00000 −0.0683986
\(856\) −4.00000 −0.136717
\(857\) −42.0000 −1.43469 −0.717346 0.696717i \(-0.754643\pi\)
−0.717346 + 0.696717i \(0.754643\pi\)
\(858\) 8.00000 0.273115
\(859\) −52.0000 −1.77422 −0.887109 0.461561i \(-0.847290\pi\)
−0.887109 + 0.461561i \(0.847290\pi\)
\(860\) −8.00000 −0.272798
\(861\) 0 0
\(862\) −24.0000 −0.817443
\(863\) −24.0000 −0.816970 −0.408485 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) 1.00000 0.0340207
\(865\) 12.0000 0.408012
\(866\) −26.0000 −0.883516
\(867\) 19.0000 0.645274
\(868\) 0 0
\(869\) 16.0000 0.542763
\(870\) 4.00000 0.135613
\(871\) 24.0000 0.813209
\(872\) −6.00000 −0.203186
\(873\) −10.0000 −0.338449
\(874\) −4.00000 −0.135302
\(875\) 0 0
\(876\) 6.00000 0.202721
\(877\) 34.0000 1.14810 0.574049 0.818821i \(-0.305372\pi\)
0.574049 + 0.818821i \(0.305372\pi\)
\(878\) 4.00000 0.134993
\(879\) 18.0000 0.607125
\(880\) 8.00000 0.269680
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 0 0
\(883\) −20.0000 −0.673054 −0.336527 0.941674i \(-0.609252\pi\)
−0.336527 + 0.941674i \(0.609252\pi\)
\(884\) −12.0000 −0.403604
\(885\) 24.0000 0.806751
\(886\) −20.0000 −0.671913
\(887\) 32.0000 1.07445 0.537227 0.843437i \(-0.319472\pi\)
0.537227 + 0.843437i \(0.319472\pi\)
\(888\) 10.0000 0.335578
\(889\) 0 0
\(890\) −12.0000 −0.402241
\(891\) −4.00000 −0.134005
\(892\) 28.0000 0.937509
\(893\) 4.00000 0.133855
\(894\) −6.00000 −0.200670
\(895\) −24.0000 −0.802232
\(896\) 0 0
\(897\) 8.00000 0.267112
\(898\) 34.0000 1.13459
\(899\) 8.00000 0.266815
\(900\) −1.00000 −0.0333333
\(901\) −60.0000 −1.99889
\(902\) 40.0000 1.33185
\(903\) 0 0
\(904\) 2.00000 0.0665190
\(905\) −28.0000 −0.930751
\(906\) 20.0000 0.664455
\(907\) 44.0000 1.46100 0.730498 0.682915i \(-0.239288\pi\)
0.730498 + 0.682915i \(0.239288\pi\)
\(908\) −28.0000 −0.929213
\(909\) −2.00000 −0.0663358
\(910\) 0 0
\(911\) 24.0000 0.795155 0.397578 0.917568i \(-0.369851\pi\)
0.397578 + 0.917568i \(0.369851\pi\)
\(912\) 1.00000 0.0331133
\(913\) 48.0000 1.58857
\(914\) 26.0000 0.860004
\(915\) 28.0000 0.925651
\(916\) 10.0000 0.330409
\(917\) 0 0
\(918\) 6.00000 0.198030
\(919\) 8.00000 0.263896 0.131948 0.991257i \(-0.457877\pi\)
0.131948 + 0.991257i \(0.457877\pi\)
\(920\) 8.00000 0.263752
\(921\) 12.0000 0.395413
\(922\) −2.00000 −0.0658665
\(923\) −16.0000 −0.526646
\(924\) 0 0
\(925\) −10.0000 −0.328798
\(926\) −8.00000 −0.262896
\(927\) 12.0000 0.394132
\(928\) −2.00000 −0.0656532
\(929\) 22.0000 0.721797 0.360898 0.932605i \(-0.382470\pi\)
0.360898 + 0.932605i \(0.382470\pi\)
\(930\) 8.00000 0.262330
\(931\) 0 0
\(932\) −6.00000 −0.196537
\(933\) −4.00000 −0.130954
\(934\) 20.0000 0.654420
\(935\) 48.0000 1.56977
\(936\) −2.00000 −0.0653720
\(937\) −26.0000 −0.849383 −0.424691 0.905338i \(-0.639617\pi\)
−0.424691 + 0.905338i \(0.639617\pi\)
\(938\) 0 0
\(939\) 22.0000 0.717943
\(940\) −8.00000 −0.260931
\(941\) −22.0000 −0.717180 −0.358590 0.933495i \(-0.616742\pi\)
−0.358590 + 0.933495i \(0.616742\pi\)
\(942\) −22.0000 −0.716799
\(943\) 40.0000 1.30258
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) −16.0000 −0.520205
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) −4.00000 −0.129914
\(949\) −12.0000 −0.389536
\(950\) −1.00000 −0.0324443
\(951\) 6.00000 0.194563
\(952\) 0 0
\(953\) −22.0000 −0.712650 −0.356325 0.934362i \(-0.615970\pi\)
−0.356325 + 0.934362i \(0.615970\pi\)
\(954\) −10.0000 −0.323762
\(955\) −8.00000 −0.258874
\(956\) 12.0000 0.388108
\(957\) 8.00000 0.258603
\(958\) −4.00000 −0.129234
\(959\) 0 0
\(960\) −2.00000 −0.0645497
\(961\) −15.0000 −0.483871
\(962\) −20.0000 −0.644826
\(963\) −4.00000 −0.128898
\(964\) −10.0000 −0.322078
\(965\) 12.0000 0.386294
\(966\) 0 0
\(967\) 48.0000 1.54358 0.771788 0.635880i \(-0.219363\pi\)
0.771788 + 0.635880i \(0.219363\pi\)
\(968\) 5.00000 0.160706
\(969\) 6.00000 0.192748
\(970\) 20.0000 0.642161
\(971\) 28.0000 0.898563 0.449281 0.893390i \(-0.351680\pi\)
0.449281 + 0.893390i \(0.351680\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) 4.00000 0.128168
\(975\) 2.00000 0.0640513
\(976\) −14.0000 −0.448129
\(977\) −46.0000 −1.47167 −0.735835 0.677161i \(-0.763210\pi\)
−0.735835 + 0.677161i \(0.763210\pi\)
\(978\) 20.0000 0.639529
\(979\) −24.0000 −0.767043
\(980\) 0 0
\(981\) −6.00000 −0.191565
\(982\) 20.0000 0.638226
\(983\) 24.0000 0.765481 0.382741 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) −10.0000 −0.318788
\(985\) 44.0000 1.40196
\(986\) −12.0000 −0.382158
\(987\) 0 0
\(988\) −2.00000 −0.0636285
\(989\) −16.0000 −0.508770
\(990\) 8.00000 0.254257
\(991\) 4.00000 0.127064 0.0635321 0.997980i \(-0.479763\pi\)
0.0635321 + 0.997980i \(0.479763\pi\)
\(992\) −4.00000 −0.127000
\(993\) −4.00000 −0.126936
\(994\) 0 0
\(995\) −32.0000 −1.01447
\(996\) −12.0000 −0.380235
\(997\) 2.00000 0.0633406 0.0316703 0.999498i \(-0.489917\pi\)
0.0316703 + 0.999498i \(0.489917\pi\)
\(998\) −20.0000 −0.633089
\(999\) 10.0000 0.316386
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 5586.2.a.y.1.1 1
7.6 odd 2 114.2.a.b.1.1 1
21.20 even 2 342.2.a.b.1.1 1
28.27 even 2 912.2.a.k.1.1 1
35.13 even 4 2850.2.d.b.799.1 2
35.27 even 4 2850.2.d.b.799.2 2
35.34 odd 2 2850.2.a.j.1.1 1
56.13 odd 2 3648.2.a.x.1.1 1
56.27 even 2 3648.2.a.c.1.1 1
84.83 odd 2 2736.2.a.d.1.1 1
105.104 even 2 8550.2.a.ba.1.1 1
133.132 even 2 2166.2.a.d.1.1 1
399.398 odd 2 6498.2.a.p.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.a.b.1.1 1 7.6 odd 2
342.2.a.b.1.1 1 21.20 even 2
912.2.a.k.1.1 1 28.27 even 2
2166.2.a.d.1.1 1 133.132 even 2
2736.2.a.d.1.1 1 84.83 odd 2
2850.2.a.j.1.1 1 35.34 odd 2
2850.2.d.b.799.1 2 35.13 even 4
2850.2.d.b.799.2 2 35.27 even 4
3648.2.a.c.1.1 1 56.27 even 2
3648.2.a.x.1.1 1 56.13 odd 2
5586.2.a.y.1.1 1 1.1 even 1 trivial
6498.2.a.p.1.1 1 399.398 odd 2
8550.2.a.ba.1.1 1 105.104 even 2