Properties

Label 1386.2.a.r.1.2
Level $1386$
Weight $2$
Character 1386.1
Self dual yes
Analytic conductor $11.067$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1386.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.0672657201\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1304.1
Defining polynomial: \(x^{3} - 11 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.40405\) of defining polynomial
Character \(\chi\) \(=\) 1386.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.09174 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.09174 q^{5} +1.00000 q^{7} +1.00000 q^{8} +1.09174 q^{10} +1.00000 q^{11} +6.80809 q^{13} +1.00000 q^{14} +1.00000 q^{16} +1.09174 q^{17} -3.89984 q^{19} +1.09174 q^{20} +1.00000 q^{22} -6.99158 q^{23} -3.80809 q^{25} +6.80809 q^{26} +1.00000 q^{28} +4.80809 q^{29} -1.27523 q^{31} +1.00000 q^{32} +1.09174 q^{34} +1.09174 q^{35} +4.18349 q^{37} -3.89984 q^{38} +1.09174 q^{40} -1.09174 q^{41} +10.6246 q^{43} +1.00000 q^{44} -6.99158 q^{46} -9.71635 q^{47} +1.00000 q^{49} -3.80809 q^{50} +6.80809 q^{52} +3.81651 q^{53} +1.09174 q^{55} +1.00000 q^{56} +4.80809 q^{58} +11.7997 q^{59} +11.1751 q^{61} -1.27523 q^{62} +1.00000 q^{64} +7.43270 q^{65} +8.99158 q^{67} +1.09174 q^{68} +1.09174 q^{70} -9.17507 q^{71} -5.09174 q^{73} +4.18349 q^{74} -3.89984 q^{76} +1.00000 q^{77} -4.99158 q^{79} +1.09174 q^{80} -1.09174 q^{82} -7.09174 q^{83} +1.19191 q^{85} +10.6246 q^{86} +1.00000 q^{88} -12.9916 q^{89} +6.80809 q^{91} -6.99158 q^{92} -9.71635 q^{94} -4.25763 q^{95} -7.61619 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} + 3q^{4} + 2q^{5} + 3q^{7} + 3q^{8} + O(q^{10}) \) \( 3q + 3q^{2} + 3q^{4} + 2q^{5} + 3q^{7} + 3q^{8} + 2q^{10} + 3q^{11} + 3q^{14} + 3q^{16} + 2q^{17} + 10q^{19} + 2q^{20} + 3q^{22} + 2q^{23} + 9q^{25} + 3q^{28} - 6q^{29} + 3q^{32} + 2q^{34} + 2q^{35} + 10q^{37} + 10q^{38} + 2q^{40} - 2q^{41} + 14q^{43} + 3q^{44} + 2q^{46} - 10q^{47} + 3q^{49} + 9q^{50} + 14q^{53} + 2q^{55} + 3q^{56} - 6q^{58} - 8q^{59} + 8q^{61} + 3q^{64} - 16q^{65} + 4q^{67} + 2q^{68} + 2q^{70} - 2q^{71} - 14q^{73} + 10q^{74} + 10q^{76} + 3q^{77} + 8q^{79} + 2q^{80} - 2q^{82} - 20q^{83} + 24q^{85} + 14q^{86} + 3q^{88} - 16q^{89} + 2q^{92} - 10q^{94} + 18q^{97} + 3q^{98} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 1.09174 0.488243 0.244121 0.969745i \(-0.421500\pi\)
0.244121 + 0.969745i \(0.421500\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 1.09174 0.345240
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 6.80809 1.88823 0.944113 0.329623i \(-0.106922\pi\)
0.944113 + 0.329623i \(0.106922\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 1.09174 0.264787 0.132393 0.991197i \(-0.457734\pi\)
0.132393 + 0.991197i \(0.457734\pi\)
\(18\) 0 0
\(19\) −3.89984 −0.894684 −0.447342 0.894363i \(-0.647629\pi\)
−0.447342 + 0.894363i \(0.647629\pi\)
\(20\) 1.09174 0.244121
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) −6.99158 −1.45785 −0.728923 0.684596i \(-0.759979\pi\)
−0.728923 + 0.684596i \(0.759979\pi\)
\(24\) 0 0
\(25\) −3.80809 −0.761619
\(26\) 6.80809 1.33518
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) 4.80809 0.892841 0.446420 0.894823i \(-0.352699\pi\)
0.446420 + 0.894823i \(0.352699\pi\)
\(30\) 0 0
\(31\) −1.27523 −0.229039 −0.114519 0.993421i \(-0.536533\pi\)
−0.114519 + 0.993421i \(0.536533\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 1.09174 0.187233
\(35\) 1.09174 0.184538
\(36\) 0 0
\(37\) 4.18349 0.687761 0.343881 0.939013i \(-0.388258\pi\)
0.343881 + 0.939013i \(0.388258\pi\)
\(38\) −3.89984 −0.632637
\(39\) 0 0
\(40\) 1.09174 0.172620
\(41\) −1.09174 −0.170502 −0.0852509 0.996360i \(-0.527169\pi\)
−0.0852509 + 0.996360i \(0.527169\pi\)
\(42\) 0 0
\(43\) 10.6246 1.62024 0.810119 0.586266i \(-0.199403\pi\)
0.810119 + 0.586266i \(0.199403\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) −6.99158 −1.03085
\(47\) −9.71635 −1.41728 −0.708638 0.705573i \(-0.750690\pi\)
−0.708638 + 0.705573i \(0.750690\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −3.80809 −0.538546
\(51\) 0 0
\(52\) 6.80809 0.944113
\(53\) 3.81651 0.524238 0.262119 0.965036i \(-0.415579\pi\)
0.262119 + 0.965036i \(0.415579\pi\)
\(54\) 0 0
\(55\) 1.09174 0.147211
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) 4.80809 0.631334
\(59\) 11.7997 1.53619 0.768094 0.640338i \(-0.221206\pi\)
0.768094 + 0.640338i \(0.221206\pi\)
\(60\) 0 0
\(61\) 11.1751 1.43082 0.715411 0.698704i \(-0.246240\pi\)
0.715411 + 0.698704i \(0.246240\pi\)
\(62\) −1.27523 −0.161955
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 7.43270 0.921913
\(66\) 0 0
\(67\) 8.99158 1.09850 0.549248 0.835659i \(-0.314914\pi\)
0.549248 + 0.835659i \(0.314914\pi\)
\(68\) 1.09174 0.132393
\(69\) 0 0
\(70\) 1.09174 0.130488
\(71\) −9.17507 −1.08888 −0.544440 0.838800i \(-0.683258\pi\)
−0.544440 + 0.838800i \(0.683258\pi\)
\(72\) 0 0
\(73\) −5.09174 −0.595944 −0.297972 0.954575i \(-0.596310\pi\)
−0.297972 + 0.954575i \(0.596310\pi\)
\(74\) 4.18349 0.486321
\(75\) 0 0
\(76\) −3.89984 −0.447342
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) −4.99158 −0.561597 −0.280798 0.959767i \(-0.590599\pi\)
−0.280798 + 0.959767i \(0.590599\pi\)
\(80\) 1.09174 0.122061
\(81\) 0 0
\(82\) −1.09174 −0.120563
\(83\) −7.09174 −0.778420 −0.389210 0.921149i \(-0.627252\pi\)
−0.389210 + 0.921149i \(0.627252\pi\)
\(84\) 0 0
\(85\) 1.19191 0.129280
\(86\) 10.6246 1.14568
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) −12.9916 −1.37711 −0.688553 0.725186i \(-0.741754\pi\)
−0.688553 + 0.725186i \(0.741754\pi\)
\(90\) 0 0
\(91\) 6.80809 0.713682
\(92\) −6.99158 −0.728923
\(93\) 0 0
\(94\) −9.71635 −1.00216
\(95\) −4.25763 −0.436823
\(96\) 0 0
\(97\) −7.61619 −0.773307 −0.386653 0.922225i \(-0.626369\pi\)
−0.386653 + 0.922225i \(0.626369\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) −3.80809 −0.380809
\(101\) −15.6162 −1.55387 −0.776934 0.629582i \(-0.783226\pi\)
−0.776934 + 0.629582i \(0.783226\pi\)
\(102\) 0 0
\(103\) 3.09174 0.304639 0.152319 0.988331i \(-0.451326\pi\)
0.152319 + 0.988331i \(0.451326\pi\)
\(104\) 6.80809 0.667589
\(105\) 0 0
\(106\) 3.81651 0.370692
\(107\) −2.18349 −0.211086 −0.105543 0.994415i \(-0.533658\pi\)
−0.105543 + 0.994415i \(0.533658\pi\)
\(108\) 0 0
\(109\) 8.00000 0.766261 0.383131 0.923694i \(-0.374846\pi\)
0.383131 + 0.923694i \(0.374846\pi\)
\(110\) 1.09174 0.104094
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) 3.17507 0.298686 0.149343 0.988785i \(-0.452284\pi\)
0.149343 + 0.988785i \(0.452284\pi\)
\(114\) 0 0
\(115\) −7.63302 −0.711783
\(116\) 4.80809 0.446420
\(117\) 0 0
\(118\) 11.7997 1.08625
\(119\) 1.09174 0.100080
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 11.1751 1.01174
\(123\) 0 0
\(124\) −1.27523 −0.114519
\(125\) −9.61619 −0.860098
\(126\) 0 0
\(127\) −4.99158 −0.442931 −0.221466 0.975168i \(-0.571084\pi\)
−0.221466 + 0.975168i \(0.571084\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 7.43270 0.651891
\(131\) −16.9083 −1.47728 −0.738641 0.674099i \(-0.764532\pi\)
−0.738641 + 0.674099i \(0.764532\pi\)
\(132\) 0 0
\(133\) −3.89984 −0.338159
\(134\) 8.99158 0.776754
\(135\) 0 0
\(136\) 1.09174 0.0936163
\(137\) 15.1751 1.29649 0.648247 0.761430i \(-0.275502\pi\)
0.648247 + 0.761430i \(0.275502\pi\)
\(138\) 0 0
\(139\) −8.26682 −0.701182 −0.350591 0.936529i \(-0.614019\pi\)
−0.350591 + 0.936529i \(0.614019\pi\)
\(140\) 1.09174 0.0922692
\(141\) 0 0
\(142\) −9.17507 −0.769955
\(143\) 6.80809 0.569321
\(144\) 0 0
\(145\) 5.24921 0.435923
\(146\) −5.09174 −0.421396
\(147\) 0 0
\(148\) 4.18349 0.343881
\(149\) −15.6162 −1.27933 −0.639664 0.768655i \(-0.720926\pi\)
−0.639664 + 0.768655i \(0.720926\pi\)
\(150\) 0 0
\(151\) 21.9832 1.78896 0.894482 0.447103i \(-0.147544\pi\)
0.894482 + 0.447103i \(0.147544\pi\)
\(152\) −3.89984 −0.316319
\(153\) 0 0
\(154\) 1.00000 0.0805823
\(155\) −1.39223 −0.111827
\(156\) 0 0
\(157\) −14.9083 −1.18981 −0.594904 0.803797i \(-0.702810\pi\)
−0.594904 + 0.803797i \(0.702810\pi\)
\(158\) −4.99158 −0.397109
\(159\) 0 0
\(160\) 1.09174 0.0863100
\(161\) −6.99158 −0.551014
\(162\) 0 0
\(163\) 9.98317 0.781942 0.390971 0.920403i \(-0.372139\pi\)
0.390971 + 0.920403i \(0.372139\pi\)
\(164\) −1.09174 −0.0852509
\(165\) 0 0
\(166\) −7.09174 −0.550426
\(167\) 7.43270 0.575160 0.287580 0.957757i \(-0.407149\pi\)
0.287580 + 0.957757i \(0.407149\pi\)
\(168\) 0 0
\(169\) 33.3501 2.56540
\(170\) 1.19191 0.0914150
\(171\) 0 0
\(172\) 10.6246 0.810119
\(173\) 5.79968 0.440941 0.220471 0.975394i \(-0.429241\pi\)
0.220471 + 0.975394i \(0.429241\pi\)
\(174\) 0 0
\(175\) −3.80809 −0.287865
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) −12.9916 −0.973760
\(179\) 4.25763 0.318230 0.159115 0.987260i \(-0.449136\pi\)
0.159115 + 0.987260i \(0.449136\pi\)
\(180\) 0 0
\(181\) −5.09174 −0.378466 −0.189233 0.981932i \(-0.560600\pi\)
−0.189233 + 0.981932i \(0.560600\pi\)
\(182\) 6.80809 0.504650
\(183\) 0 0
\(184\) −6.99158 −0.515426
\(185\) 4.56730 0.335795
\(186\) 0 0
\(187\) 1.09174 0.0798363
\(188\) −9.71635 −0.708638
\(189\) 0 0
\(190\) −4.25763 −0.308881
\(191\) −24.2408 −1.75400 −0.877001 0.480488i \(-0.840459\pi\)
−0.877001 + 0.480488i \(0.840459\pi\)
\(192\) 0 0
\(193\) −5.63302 −0.405474 −0.202737 0.979233i \(-0.564984\pi\)
−0.202737 + 0.979233i \(0.564984\pi\)
\(194\) −7.61619 −0.546810
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −26.4243 −1.88265 −0.941326 0.337498i \(-0.890419\pi\)
−0.941326 + 0.337498i \(0.890419\pi\)
\(198\) 0 0
\(199\) −11.0917 −0.786273 −0.393136 0.919480i \(-0.628610\pi\)
−0.393136 + 0.919480i \(0.628610\pi\)
\(200\) −3.80809 −0.269273
\(201\) 0 0
\(202\) −15.6162 −1.09875
\(203\) 4.80809 0.337462
\(204\) 0 0
\(205\) −1.19191 −0.0832463
\(206\) 3.09174 0.215412
\(207\) 0 0
\(208\) 6.80809 0.472056
\(209\) −3.89984 −0.269757
\(210\) 0 0
\(211\) −20.6078 −1.41870 −0.709349 0.704858i \(-0.751011\pi\)
−0.709349 + 0.704858i \(0.751011\pi\)
\(212\) 3.81651 0.262119
\(213\) 0 0
\(214\) −2.18349 −0.149260
\(215\) 11.5994 0.791069
\(216\) 0 0
\(217\) −1.27523 −0.0865685
\(218\) 8.00000 0.541828
\(219\) 0 0
\(220\) 1.09174 0.0736054
\(221\) 7.43270 0.499977
\(222\) 0 0
\(223\) −5.64221 −0.377830 −0.188915 0.981993i \(-0.560497\pi\)
−0.188915 + 0.981993i \(0.560497\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) 3.17507 0.211203
\(227\) −11.4587 −0.760542 −0.380271 0.924875i \(-0.624169\pi\)
−0.380271 + 0.924875i \(0.624169\pi\)
\(228\) 0 0
\(229\) −12.5244 −0.827639 −0.413819 0.910359i \(-0.635805\pi\)
−0.413819 + 0.910359i \(0.635805\pi\)
\(230\) −7.63302 −0.503307
\(231\) 0 0
\(232\) 4.80809 0.315667
\(233\) 25.2324 1.65303 0.826514 0.562916i \(-0.190321\pi\)
0.826514 + 0.562916i \(0.190321\pi\)
\(234\) 0 0
\(235\) −10.6078 −0.691975
\(236\) 11.7997 0.768094
\(237\) 0 0
\(238\) 1.09174 0.0707673
\(239\) −13.9832 −0.904496 −0.452248 0.891892i \(-0.649378\pi\)
−0.452248 + 0.891892i \(0.649378\pi\)
\(240\) 0 0
\(241\) −7.27523 −0.468639 −0.234319 0.972160i \(-0.575286\pi\)
−0.234319 + 0.972160i \(0.575286\pi\)
\(242\) 1.00000 0.0642824
\(243\) 0 0
\(244\) 11.1751 0.715411
\(245\) 1.09174 0.0697490
\(246\) 0 0
\(247\) −26.5505 −1.68937
\(248\) −1.27523 −0.0809774
\(249\) 0 0
\(250\) −9.61619 −0.608181
\(251\) 28.1667 1.77786 0.888932 0.458040i \(-0.151448\pi\)
0.888932 + 0.458040i \(0.151448\pi\)
\(252\) 0 0
\(253\) −6.99158 −0.439557
\(254\) −4.99158 −0.313200
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 8.62461 0.537988 0.268994 0.963142i \(-0.413309\pi\)
0.268994 + 0.963142i \(0.413309\pi\)
\(258\) 0 0
\(259\) 4.18349 0.259949
\(260\) 7.43270 0.460956
\(261\) 0 0
\(262\) −16.9083 −1.04460
\(263\) −22.6078 −1.39405 −0.697027 0.717044i \(-0.745494\pi\)
−0.697027 + 0.717044i \(0.745494\pi\)
\(264\) 0 0
\(265\) 4.16665 0.255956
\(266\) −3.89984 −0.239114
\(267\) 0 0
\(268\) 8.99158 0.549248
\(269\) 22.9083 1.39674 0.698370 0.715736i \(-0.253909\pi\)
0.698370 + 0.715736i \(0.253909\pi\)
\(270\) 0 0
\(271\) −13.8165 −0.839293 −0.419647 0.907688i \(-0.637846\pi\)
−0.419647 + 0.907688i \(0.637846\pi\)
\(272\) 1.09174 0.0661967
\(273\) 0 0
\(274\) 15.1751 0.916760
\(275\) −3.80809 −0.229637
\(276\) 0 0
\(277\) 17.6162 1.05845 0.529227 0.848480i \(-0.322482\pi\)
0.529227 + 0.848480i \(0.322482\pi\)
\(278\) −8.26682 −0.495811
\(279\) 0 0
\(280\) 1.09174 0.0652442
\(281\) −19.9832 −1.19210 −0.596048 0.802949i \(-0.703263\pi\)
−0.596048 + 0.802949i \(0.703263\pi\)
\(282\) 0 0
\(283\) −23.5329 −1.39888 −0.699442 0.714690i \(-0.746568\pi\)
−0.699442 + 0.714690i \(0.746568\pi\)
\(284\) −9.17507 −0.544440
\(285\) 0 0
\(286\) 6.80809 0.402571
\(287\) −1.09174 −0.0644436
\(288\) 0 0
\(289\) −15.8081 −0.929888
\(290\) 5.24921 0.308244
\(291\) 0 0
\(292\) −5.09174 −0.297972
\(293\) 3.81651 0.222963 0.111481 0.993767i \(-0.464440\pi\)
0.111481 + 0.993767i \(0.464440\pi\)
\(294\) 0 0
\(295\) 12.8822 0.750033
\(296\) 4.18349 0.243160
\(297\) 0 0
\(298\) −15.6162 −0.904621
\(299\) −47.5994 −2.75274
\(300\) 0 0
\(301\) 10.6246 0.612392
\(302\) 21.9832 1.26499
\(303\) 0 0
\(304\) −3.89984 −0.223671
\(305\) 12.2003 0.698588
\(306\) 0 0
\(307\) 29.5160 1.68457 0.842284 0.539034i \(-0.181210\pi\)
0.842284 + 0.539034i \(0.181210\pi\)
\(308\) 1.00000 0.0569803
\(309\) 0 0
\(310\) −1.39223 −0.0790733
\(311\) −9.71635 −0.550964 −0.275482 0.961306i \(-0.588837\pi\)
−0.275482 + 0.961306i \(0.588837\pi\)
\(312\) 0 0
\(313\) −21.7997 −1.23219 −0.616095 0.787672i \(-0.711286\pi\)
−0.616095 + 0.787672i \(0.711286\pi\)
\(314\) −14.9083 −0.841322
\(315\) 0 0
\(316\) −4.99158 −0.280798
\(317\) 29.7997 1.67372 0.836858 0.547420i \(-0.184390\pi\)
0.836858 + 0.547420i \(0.184390\pi\)
\(318\) 0 0
\(319\) 4.80809 0.269202
\(320\) 1.09174 0.0610304
\(321\) 0 0
\(322\) −6.99158 −0.389626
\(323\) −4.25763 −0.236901
\(324\) 0 0
\(325\) −25.9259 −1.43811
\(326\) 9.98317 0.552916
\(327\) 0 0
\(328\) −1.09174 −0.0602815
\(329\) −9.71635 −0.535680
\(330\) 0 0
\(331\) −12.6246 −0.693911 −0.346956 0.937882i \(-0.612785\pi\)
−0.346956 + 0.937882i \(0.612785\pi\)
\(332\) −7.09174 −0.389210
\(333\) 0 0
\(334\) 7.43270 0.406699
\(335\) 9.81651 0.536333
\(336\) 0 0
\(337\) −2.56730 −0.139850 −0.0699249 0.997552i \(-0.522276\pi\)
−0.0699249 + 0.997552i \(0.522276\pi\)
\(338\) 33.3501 1.81401
\(339\) 0 0
\(340\) 1.19191 0.0646402
\(341\) −1.27523 −0.0690578
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 10.6246 0.572840
\(345\) 0 0
\(346\) 5.79968 0.311793
\(347\) −2.18349 −0.117216 −0.0586079 0.998281i \(-0.518666\pi\)
−0.0586079 + 0.998281i \(0.518666\pi\)
\(348\) 0 0
\(349\) 7.00842 0.375152 0.187576 0.982250i \(-0.439937\pi\)
0.187576 + 0.982250i \(0.439937\pi\)
\(350\) −3.80809 −0.203551
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) 8.62461 0.459041 0.229521 0.973304i \(-0.426284\pi\)
0.229521 + 0.973304i \(0.426284\pi\)
\(354\) 0 0
\(355\) −10.0168 −0.531638
\(356\) −12.9916 −0.688553
\(357\) 0 0
\(358\) 4.25763 0.225023
\(359\) 26.9747 1.42367 0.711836 0.702345i \(-0.247864\pi\)
0.711836 + 0.702345i \(0.247864\pi\)
\(360\) 0 0
\(361\) −3.79126 −0.199540
\(362\) −5.09174 −0.267616
\(363\) 0 0
\(364\) 6.80809 0.356841
\(365\) −5.55888 −0.290965
\(366\) 0 0
\(367\) −6.72477 −0.351030 −0.175515 0.984477i \(-0.556159\pi\)
−0.175515 + 0.984477i \(0.556159\pi\)
\(368\) −6.99158 −0.364461
\(369\) 0 0
\(370\) 4.56730 0.237443
\(371\) 3.81651 0.198143
\(372\) 0 0
\(373\) −10.4411 −0.540620 −0.270310 0.962773i \(-0.587126\pi\)
−0.270310 + 0.962773i \(0.587126\pi\)
\(374\) 1.09174 0.0564528
\(375\) 0 0
\(376\) −9.71635 −0.501082
\(377\) 32.7340 1.68588
\(378\) 0 0
\(379\) 14.3501 0.737117 0.368559 0.929604i \(-0.379851\pi\)
0.368559 + 0.929604i \(0.379851\pi\)
\(380\) −4.25763 −0.218412
\(381\) 0 0
\(382\) −24.2408 −1.24027
\(383\) 9.91667 0.506718 0.253359 0.967372i \(-0.418465\pi\)
0.253359 + 0.967372i \(0.418465\pi\)
\(384\) 0 0
\(385\) 1.09174 0.0556405
\(386\) −5.63302 −0.286713
\(387\) 0 0
\(388\) −7.61619 −0.386653
\(389\) 33.9663 1.72216 0.861081 0.508468i \(-0.169788\pi\)
0.861081 + 0.508468i \(0.169788\pi\)
\(390\) 0 0
\(391\) −7.63302 −0.386019
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) −26.4243 −1.33124
\(395\) −5.44953 −0.274196
\(396\) 0 0
\(397\) −26.7079 −1.34043 −0.670216 0.742166i \(-0.733799\pi\)
−0.670216 + 0.742166i \(0.733799\pi\)
\(398\) −11.0917 −0.555979
\(399\) 0 0
\(400\) −3.80809 −0.190405
\(401\) 15.1751 0.757807 0.378903 0.925436i \(-0.376301\pi\)
0.378903 + 0.925436i \(0.376301\pi\)
\(402\) 0 0
\(403\) −8.68191 −0.432477
\(404\) −15.6162 −0.776934
\(405\) 0 0
\(406\) 4.80809 0.238622
\(407\) 4.18349 0.207368
\(408\) 0 0
\(409\) −12.7248 −0.629199 −0.314600 0.949224i \(-0.601870\pi\)
−0.314600 + 0.949224i \(0.601870\pi\)
\(410\) −1.19191 −0.0588640
\(411\) 0 0
\(412\) 3.09174 0.152319
\(413\) 11.7997 0.580624
\(414\) 0 0
\(415\) −7.74237 −0.380058
\(416\) 6.80809 0.333794
\(417\) 0 0
\(418\) −3.89984 −0.190747
\(419\) −31.2324 −1.52580 −0.762901 0.646516i \(-0.776225\pi\)
−0.762901 + 0.646516i \(0.776225\pi\)
\(420\) 0 0
\(421\) 18.3670 0.895152 0.447576 0.894246i \(-0.352287\pi\)
0.447576 + 0.894246i \(0.352287\pi\)
\(422\) −20.6078 −1.00317
\(423\) 0 0
\(424\) 3.81651 0.185346
\(425\) −4.15747 −0.201667
\(426\) 0 0
\(427\) 11.1751 0.540800
\(428\) −2.18349 −0.105543
\(429\) 0 0
\(430\) 11.5994 0.559371
\(431\) 3.37539 0.162587 0.0812935 0.996690i \(-0.474095\pi\)
0.0812935 + 0.996690i \(0.474095\pi\)
\(432\) 0 0
\(433\) 5.26604 0.253070 0.126535 0.991962i \(-0.459614\pi\)
0.126535 + 0.991962i \(0.459614\pi\)
\(434\) −1.27523 −0.0612132
\(435\) 0 0
\(436\) 8.00000 0.383131
\(437\) 27.2660 1.30431
\(438\) 0 0
\(439\) −3.79968 −0.181349 −0.0906743 0.995881i \(-0.528902\pi\)
−0.0906743 + 0.995881i \(0.528902\pi\)
\(440\) 1.09174 0.0520469
\(441\) 0 0
\(442\) 7.43270 0.353537
\(443\) −20.6246 −0.979905 −0.489952 0.871749i \(-0.662986\pi\)
−0.489952 + 0.871749i \(0.662986\pi\)
\(444\) 0 0
\(445\) −14.1835 −0.672362
\(446\) −5.64221 −0.267166
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) 27.1751 1.28247 0.641235 0.767344i \(-0.278422\pi\)
0.641235 + 0.767344i \(0.278422\pi\)
\(450\) 0 0
\(451\) −1.09174 −0.0514082
\(452\) 3.17507 0.149343
\(453\) 0 0
\(454\) −11.4587 −0.537784
\(455\) 7.43270 0.348450
\(456\) 0 0
\(457\) −15.2492 −0.713328 −0.356664 0.934233i \(-0.616086\pi\)
−0.356664 + 0.934233i \(0.616086\pi\)
\(458\) −12.5244 −0.585229
\(459\) 0 0
\(460\) −7.63302 −0.355891
\(461\) 13.2324 0.616293 0.308147 0.951339i \(-0.400291\pi\)
0.308147 + 0.951339i \(0.400291\pi\)
\(462\) 0 0
\(463\) 3.63302 0.168841 0.0844204 0.996430i \(-0.473096\pi\)
0.0844204 + 0.996430i \(0.473096\pi\)
\(464\) 4.80809 0.223210
\(465\) 0 0
\(466\) 25.2324 1.16887
\(467\) −13.9832 −0.647064 −0.323532 0.946217i \(-0.604870\pi\)
−0.323532 + 0.946217i \(0.604870\pi\)
\(468\) 0 0
\(469\) 8.99158 0.415193
\(470\) −10.6078 −0.489300
\(471\) 0 0
\(472\) 11.7997 0.543124
\(473\) 10.6246 0.488520
\(474\) 0 0
\(475\) 14.8510 0.681408
\(476\) 1.09174 0.0500400
\(477\) 0 0
\(478\) −13.9832 −0.639575
\(479\) −16.3670 −0.747826 −0.373913 0.927464i \(-0.621984\pi\)
−0.373913 + 0.927464i \(0.621984\pi\)
\(480\) 0 0
\(481\) 28.4816 1.29865
\(482\) −7.27523 −0.331378
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −8.31493 −0.377562
\(486\) 0 0
\(487\) 24.1667 1.09510 0.547548 0.836774i \(-0.315561\pi\)
0.547548 + 0.836774i \(0.315561\pi\)
\(488\) 11.1751 0.505872
\(489\) 0 0
\(490\) 1.09174 0.0493200
\(491\) −7.63302 −0.344473 −0.172237 0.985056i \(-0.555099\pi\)
−0.172237 + 0.985056i \(0.555099\pi\)
\(492\) 0 0
\(493\) 5.24921 0.236413
\(494\) −26.5505 −1.19456
\(495\) 0 0
\(496\) −1.27523 −0.0572597
\(497\) −9.17507 −0.411558
\(498\) 0 0
\(499\) 12.2576 0.548727 0.274363 0.961626i \(-0.411533\pi\)
0.274363 + 0.961626i \(0.411533\pi\)
\(500\) −9.61619 −0.430049
\(501\) 0 0
\(502\) 28.1667 1.25714
\(503\) 31.4327 1.40151 0.700757 0.713400i \(-0.252846\pi\)
0.700757 + 0.713400i \(0.252846\pi\)
\(504\) 0 0
\(505\) −17.0489 −0.758666
\(506\) −6.99158 −0.310814
\(507\) 0 0
\(508\) −4.99158 −0.221466
\(509\) 37.7737 1.67429 0.837144 0.546983i \(-0.184224\pi\)
0.837144 + 0.546983i \(0.184224\pi\)
\(510\) 0 0
\(511\) −5.09174 −0.225246
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 8.62461 0.380415
\(515\) 3.37539 0.148738
\(516\) 0 0
\(517\) −9.71635 −0.427325
\(518\) 4.18349 0.183812
\(519\) 0 0
\(520\) 7.43270 0.325945
\(521\) 4.25763 0.186530 0.0932650 0.995641i \(-0.470270\pi\)
0.0932650 + 0.995641i \(0.470270\pi\)
\(522\) 0 0
\(523\) 4.83412 0.211381 0.105691 0.994399i \(-0.466295\pi\)
0.105691 + 0.994399i \(0.466295\pi\)
\(524\) −16.9083 −0.738641
\(525\) 0 0
\(526\) −22.6078 −0.985746
\(527\) −1.39223 −0.0606465
\(528\) 0 0
\(529\) 25.8822 1.12531
\(530\) 4.16665 0.180988
\(531\) 0 0
\(532\) −3.89984 −0.169079
\(533\) −7.43270 −0.321946
\(534\) 0 0
\(535\) −2.38381 −0.103061
\(536\) 8.99158 0.388377
\(537\) 0 0
\(538\) 22.9083 0.987645
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) −20.0573 −0.862331 −0.431165 0.902273i \(-0.641897\pi\)
−0.431165 + 0.902273i \(0.641897\pi\)
\(542\) −13.8165 −0.593470
\(543\) 0 0
\(544\) 1.09174 0.0468082
\(545\) 8.73396 0.374122
\(546\) 0 0
\(547\) 25.9090 1.10779 0.553895 0.832587i \(-0.313141\pi\)
0.553895 + 0.832587i \(0.313141\pi\)
\(548\) 15.1751 0.648247
\(549\) 0 0
\(550\) −3.80809 −0.162378
\(551\) −18.7508 −0.798811
\(552\) 0 0
\(553\) −4.99158 −0.212264
\(554\) 17.6162 0.748440
\(555\) 0 0
\(556\) −8.26682 −0.350591
\(557\) −1.54205 −0.0653387 −0.0326694 0.999466i \(-0.510401\pi\)
−0.0326694 + 0.999466i \(0.510401\pi\)
\(558\) 0 0
\(559\) 72.3333 3.05937
\(560\) 1.09174 0.0461346
\(561\) 0 0
\(562\) −19.9832 −0.842939
\(563\) −31.0917 −1.31036 −0.655180 0.755472i \(-0.727407\pi\)
−0.655180 + 0.755472i \(0.727407\pi\)
\(564\) 0 0
\(565\) 3.46637 0.145831
\(566\) −23.5329 −0.989160
\(567\) 0 0
\(568\) −9.17507 −0.384977
\(569\) 16.0168 0.671461 0.335730 0.941958i \(-0.391017\pi\)
0.335730 + 0.941958i \(0.391017\pi\)
\(570\) 0 0
\(571\) 12.8081 0.536002 0.268001 0.963419i \(-0.413637\pi\)
0.268001 + 0.963419i \(0.413637\pi\)
\(572\) 6.80809 0.284661
\(573\) 0 0
\(574\) −1.09174 −0.0455685
\(575\) 26.6246 1.11032
\(576\) 0 0
\(577\) −21.7997 −0.907532 −0.453766 0.891121i \(-0.649920\pi\)
−0.453766 + 0.891121i \(0.649920\pi\)
\(578\) −15.8081 −0.657530
\(579\) 0 0
\(580\) 5.24921 0.217962
\(581\) −7.09174 −0.294215
\(582\) 0 0
\(583\) 3.81651 0.158064
\(584\) −5.09174 −0.210698
\(585\) 0 0
\(586\) 3.81651 0.157659
\(587\) 9.81651 0.405171 0.202585 0.979265i \(-0.435066\pi\)
0.202585 + 0.979265i \(0.435066\pi\)
\(588\) 0 0
\(589\) 4.97320 0.204917
\(590\) 12.8822 0.530353
\(591\) 0 0
\(592\) 4.18349 0.171940
\(593\) −9.82570 −0.403493 −0.201747 0.979438i \(-0.564662\pi\)
−0.201747 + 0.979438i \(0.564662\pi\)
\(594\) 0 0
\(595\) 1.19191 0.0488634
\(596\) −15.6162 −0.639664
\(597\) 0 0
\(598\) −47.5994 −1.94648
\(599\) 2.82493 0.115423 0.0577117 0.998333i \(-0.481620\pi\)
0.0577117 + 0.998333i \(0.481620\pi\)
\(600\) 0 0
\(601\) 39.2416 1.60070 0.800348 0.599535i \(-0.204648\pi\)
0.800348 + 0.599535i \(0.204648\pi\)
\(602\) 10.6246 0.433027
\(603\) 0 0
\(604\) 21.9832 0.894482
\(605\) 1.09174 0.0443857
\(606\) 0 0
\(607\) −16.8822 −0.685229 −0.342614 0.939476i \(-0.611312\pi\)
−0.342614 + 0.939476i \(0.611312\pi\)
\(608\) −3.89984 −0.158159
\(609\) 0 0
\(610\) 12.2003 0.493977
\(611\) −66.1498 −2.67614
\(612\) 0 0
\(613\) 44.7913 1.80910 0.904551 0.426365i \(-0.140206\pi\)
0.904551 + 0.426365i \(0.140206\pi\)
\(614\) 29.5160 1.19117
\(615\) 0 0
\(616\) 1.00000 0.0402911
\(617\) −37.9832 −1.52914 −0.764572 0.644538i \(-0.777050\pi\)
−0.764572 + 0.644538i \(0.777050\pi\)
\(618\) 0 0
\(619\) 9.98317 0.401257 0.200629 0.979667i \(-0.435702\pi\)
0.200629 + 0.979667i \(0.435702\pi\)
\(620\) −1.39223 −0.0559133
\(621\) 0 0
\(622\) −9.71635 −0.389590
\(623\) −12.9916 −0.520497
\(624\) 0 0
\(625\) 8.54205 0.341682
\(626\) −21.7997 −0.871290
\(627\) 0 0
\(628\) −14.9083 −0.594904
\(629\) 4.56730 0.182110
\(630\) 0 0
\(631\) −1.41586 −0.0563647 −0.0281823 0.999603i \(-0.508972\pi\)
−0.0281823 + 0.999603i \(0.508972\pi\)
\(632\) −4.99158 −0.198555
\(633\) 0 0
\(634\) 29.7997 1.18350
\(635\) −5.44953 −0.216258
\(636\) 0 0
\(637\) 6.80809 0.269747
\(638\) 4.80809 0.190354
\(639\) 0 0
\(640\) 1.09174 0.0431550
\(641\) −33.2155 −1.31194 −0.655968 0.754789i \(-0.727739\pi\)
−0.655968 + 0.754789i \(0.727739\pi\)
\(642\) 0 0
\(643\) −27.7997 −1.09631 −0.548156 0.836376i \(-0.684670\pi\)
−0.548156 + 0.836376i \(0.684670\pi\)
\(644\) −6.99158 −0.275507
\(645\) 0 0
\(646\) −4.25763 −0.167514
\(647\) 14.2837 0.561548 0.280774 0.959774i \(-0.409409\pi\)
0.280774 + 0.959774i \(0.409409\pi\)
\(648\) 0 0
\(649\) 11.7997 0.463178
\(650\) −25.9259 −1.01690
\(651\) 0 0
\(652\) 9.98317 0.390971
\(653\) −37.2324 −1.45702 −0.728508 0.685038i \(-0.759786\pi\)
−0.728508 + 0.685038i \(0.759786\pi\)
\(654\) 0 0
\(655\) −18.4595 −0.721272
\(656\) −1.09174 −0.0426255
\(657\) 0 0
\(658\) −9.71635 −0.378783
\(659\) 23.3990 0.911497 0.455748 0.890109i \(-0.349372\pi\)
0.455748 + 0.890109i \(0.349372\pi\)
\(660\) 0 0
\(661\) −36.1238 −1.40505 −0.702526 0.711658i \(-0.747945\pi\)
−0.702526 + 0.711658i \(0.747945\pi\)
\(662\) −12.6246 −0.490669
\(663\) 0 0
\(664\) −7.09174 −0.275213
\(665\) −4.25763 −0.165104
\(666\) 0 0
\(667\) −33.6162 −1.30162
\(668\) 7.43270 0.287580
\(669\) 0 0
\(670\) 9.81651 0.379245
\(671\) 11.1751 0.431409
\(672\) 0 0
\(673\) 30.1667 1.16284 0.581420 0.813604i \(-0.302498\pi\)
0.581420 + 0.813604i \(0.302498\pi\)
\(674\) −2.56730 −0.0988887
\(675\) 0 0
\(676\) 33.3501 1.28270
\(677\) 2.93428 0.112773 0.0563867 0.998409i \(-0.482042\pi\)
0.0563867 + 0.998409i \(0.482042\pi\)
\(678\) 0 0
\(679\) −7.61619 −0.292282
\(680\) 1.19191 0.0457075
\(681\) 0 0
\(682\) −1.27523 −0.0488312
\(683\) −17.3586 −0.664207 −0.332103 0.943243i \(-0.607758\pi\)
−0.332103 + 0.943243i \(0.607758\pi\)
\(684\) 0 0
\(685\) 16.5673 0.633004
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) 10.6246 0.405059
\(689\) 25.9832 0.989880
\(690\) 0 0
\(691\) 50.1498 1.90779 0.953895 0.300142i \(-0.0970341\pi\)
0.953895 + 0.300142i \(0.0970341\pi\)
\(692\) 5.79968 0.220471
\(693\) 0 0
\(694\) −2.18349 −0.0828841
\(695\) −9.02525 −0.342347
\(696\) 0 0
\(697\) −1.19191 −0.0451467
\(698\) 7.00842 0.265272
\(699\) 0 0
\(700\) −3.80809 −0.143932
\(701\) 43.5825 1.64609 0.823045 0.567977i \(-0.192274\pi\)
0.823045 + 0.567977i \(0.192274\pi\)
\(702\) 0 0
\(703\) −16.3149 −0.615329
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) 8.62461 0.324591
\(707\) −15.6162 −0.587307
\(708\) 0 0
\(709\) 41.7660 1.56856 0.784278 0.620410i \(-0.213034\pi\)
0.784278 + 0.620410i \(0.213034\pi\)
\(710\) −10.0168 −0.375925
\(711\) 0 0
\(712\) −12.9916 −0.486880
\(713\) 8.91590 0.333903
\(714\) 0 0
\(715\) 7.43270 0.277967
\(716\) 4.25763 0.159115
\(717\) 0 0
\(718\) 26.9747 1.00669
\(719\) 28.2668 1.05417 0.527087 0.849811i \(-0.323284\pi\)
0.527087 + 0.849811i \(0.323284\pi\)
\(720\) 0 0
\(721\) 3.09174 0.115143
\(722\) −3.79126 −0.141096
\(723\) 0 0
\(724\) −5.09174 −0.189233
\(725\) −18.3097 −0.680004
\(726\) 0 0
\(727\) −29.2416 −1.08451 −0.542255 0.840214i \(-0.682429\pi\)
−0.542255 + 0.840214i \(0.682429\pi\)
\(728\) 6.80809 0.252325
\(729\) 0 0
\(730\) −5.55888 −0.205744
\(731\) 11.5994 0.429018
\(732\) 0 0
\(733\) −36.4243 −1.34536 −0.672681 0.739933i \(-0.734857\pi\)
−0.672681 + 0.739933i \(0.734857\pi\)
\(734\) −6.72477 −0.248216
\(735\) 0 0
\(736\) −6.99158 −0.257713
\(737\) 8.99158 0.331209
\(738\) 0 0
\(739\) 46.4243 1.70774 0.853872 0.520482i \(-0.174248\pi\)
0.853872 + 0.520482i \(0.174248\pi\)
\(740\) 4.56730 0.167897
\(741\) 0 0
\(742\) 3.81651 0.140109
\(743\) 4.36698 0.160209 0.0801044 0.996786i \(-0.474475\pi\)
0.0801044 + 0.996786i \(0.474475\pi\)
\(744\) 0 0
\(745\) −17.0489 −0.624623
\(746\) −10.4411 −0.382276
\(747\) 0 0
\(748\) 1.09174 0.0399181
\(749\) −2.18349 −0.0797830
\(750\) 0 0
\(751\) −34.1498 −1.24614 −0.623072 0.782164i \(-0.714116\pi\)
−0.623072 + 0.782164i \(0.714116\pi\)
\(752\) −9.71635 −0.354319
\(753\) 0 0
\(754\) 32.7340 1.19210
\(755\) 24.0000 0.873449
\(756\) 0 0
\(757\) −6.93428 −0.252031 −0.126015 0.992028i \(-0.540219\pi\)
−0.126015 + 0.992028i \(0.540219\pi\)
\(758\) 14.3501 0.521221
\(759\) 0 0
\(760\) −4.25763 −0.154440
\(761\) −40.1575 −1.45571 −0.727853 0.685733i \(-0.759482\pi\)
−0.727853 + 0.685733i \(0.759482\pi\)
\(762\) 0 0
\(763\) 8.00000 0.289619
\(764\) −24.2408 −0.877001
\(765\) 0 0
\(766\) 9.91667 0.358304
\(767\) 80.3333 2.90067
\(768\) 0 0
\(769\) −10.1406 −0.365681 −0.182840 0.983143i \(-0.558529\pi\)
−0.182840 + 0.983143i \(0.558529\pi\)
\(770\) 1.09174 0.0393437
\(771\) 0 0
\(772\) −5.63302 −0.202737
\(773\) −2.17430 −0.0782041 −0.0391021 0.999235i \(-0.512450\pi\)
−0.0391021 + 0.999235i \(0.512450\pi\)
\(774\) 0 0
\(775\) 4.85621 0.174440
\(776\) −7.61619 −0.273405
\(777\) 0 0
\(778\) 33.9663 1.21775
\(779\) 4.25763 0.152545
\(780\) 0 0
\(781\) −9.17507 −0.328310
\(782\) −7.63302 −0.272956
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) −16.2760 −0.580916
\(786\) 0 0
\(787\) −8.66746 −0.308962 −0.154481 0.987996i \(-0.549370\pi\)
−0.154481 + 0.987996i \(0.549370\pi\)
\(788\) −26.4243 −0.941326
\(789\) 0 0
\(790\) −5.44953 −0.193886
\(791\) 3.17507 0.112893
\(792\) 0 0
\(793\) 76.0809 2.70171
\(794\) −26.7079 −0.947829
\(795\) 0 0
\(796\) −11.0917 −0.393136
\(797\) 3.47556 0.123111 0.0615553 0.998104i \(-0.480394\pi\)
0.0615553 + 0.998104i \(0.480394\pi\)
\(798\) 0 0
\(799\) −10.6078 −0.375276
\(800\) −3.80809 −0.134636
\(801\) 0 0
\(802\) 15.1751 0.535850
\(803\) −5.09174 −0.179684
\(804\) 0 0
\(805\) −7.63302 −0.269029
\(806\) −8.68191 −0.305807
\(807\) 0 0
\(808\) −15.6162 −0.549376
\(809\) 10.1667 0.357441 0.178720 0.983900i \(-0.442804\pi\)
0.178720 + 0.983900i \(0.442804\pi\)
\(810\) 0 0
\(811\) −45.8493 −1.60999 −0.804994 0.593283i \(-0.797832\pi\)
−0.804994 + 0.593283i \(0.797832\pi\)
\(812\) 4.80809 0.168731
\(813\) 0 0
\(814\) 4.18349 0.146631
\(815\) 10.8991 0.381778
\(816\) 0 0
\(817\) −41.4342 −1.44960
\(818\) −12.7248 −0.444911
\(819\) 0 0
\(820\) −1.19191 −0.0416232
\(821\) −18.0000 −0.628204 −0.314102 0.949389i \(-0.601703\pi\)
−0.314102 + 0.949389i \(0.601703\pi\)
\(822\) 0 0
\(823\) −1.41586 −0.0493539 −0.0246770 0.999695i \(-0.507856\pi\)
−0.0246770 + 0.999695i \(0.507856\pi\)
\(824\) 3.09174 0.107706
\(825\) 0 0
\(826\) 11.7997 0.410563
\(827\) −16.3670 −0.569136 −0.284568 0.958656i \(-0.591850\pi\)
−0.284568 + 0.958656i \(0.591850\pi\)
\(828\) 0 0
\(829\) 52.1238 1.81033 0.905167 0.425056i \(-0.139746\pi\)
0.905167 + 0.425056i \(0.139746\pi\)
\(830\) −7.74237 −0.268742
\(831\) 0 0
\(832\) 6.80809 0.236028
\(833\) 1.09174 0.0378267
\(834\) 0 0
\(835\) 8.11461 0.280818
\(836\) −3.89984 −0.134879
\(837\) 0 0
\(838\) −31.2324 −1.07890
\(839\) 28.2668 0.975879 0.487939 0.872877i \(-0.337749\pi\)
0.487939 + 0.872877i \(0.337749\pi\)
\(840\) 0 0
\(841\) −5.88223 −0.202836
\(842\) 18.3670 0.632968
\(843\) 0 0
\(844\) −20.6078 −0.709349
\(845\) 36.4098 1.25254
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) 3.81651 0.131060
\(849\) 0 0
\(850\) −4.15747 −0.142600
\(851\) −29.2492 −1.00265
\(852\) 0 0
\(853\) 17.5252 0.600052 0.300026 0.953931i \(-0.403005\pi\)
0.300026 + 0.953931i \(0.403005\pi\)
\(854\) 11.1751 0.382403
\(855\) 0 0
\(856\) −2.18349 −0.0746301
\(857\) −3.95714 −0.135173 −0.0675867 0.997713i \(-0.521530\pi\)
−0.0675867 + 0.997713i \(0.521530\pi\)
\(858\) 0 0
\(859\) 53.4159 1.82253 0.911263 0.411825i \(-0.135109\pi\)
0.911263 + 0.411825i \(0.135109\pi\)
\(860\) 11.5994 0.395535
\(861\) 0 0
\(862\) 3.37539 0.114966
\(863\) −6.10935 −0.207965 −0.103982 0.994579i \(-0.533159\pi\)
−0.103982 + 0.994579i \(0.533159\pi\)
\(864\) 0 0
\(865\) 6.33177 0.215286
\(866\) 5.26604 0.178947
\(867\) 0 0
\(868\) −1.27523 −0.0432842
\(869\) −4.99158 −0.169328
\(870\) 0 0
\(871\) 61.2155 2.07421
\(872\) 8.00000 0.270914
\(873\) 0 0
\(874\) 27.2660 0.922288
\(875\) −9.61619 −0.325086
\(876\) 0 0
\(877\) 0.366978 0.0123920 0.00619598 0.999981i \(-0.498028\pi\)
0.00619598 + 0.999981i \(0.498028\pi\)
\(878\) −3.79968 −0.128233
\(879\) 0 0
\(880\) 1.09174 0.0368027
\(881\) 36.3906 1.22603 0.613015 0.790071i \(-0.289956\pi\)
0.613015 + 0.790071i \(0.289956\pi\)
\(882\) 0 0
\(883\) 23.0841 0.776842 0.388421 0.921482i \(-0.373021\pi\)
0.388421 + 0.921482i \(0.373021\pi\)
\(884\) 7.43270 0.249989
\(885\) 0 0
\(886\) −20.6246 −0.692897
\(887\) 29.2492 0.982092 0.491046 0.871134i \(-0.336615\pi\)
0.491046 + 0.871134i \(0.336615\pi\)
\(888\) 0 0
\(889\) −4.99158 −0.167412
\(890\) −14.1835 −0.475432
\(891\) 0 0
\(892\) −5.64221 −0.188915
\(893\) 37.8922 1.26801
\(894\) 0 0
\(895\) 4.64824 0.155374
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) 27.1751 0.906844
\(899\) −6.13144 −0.204495
\(900\) 0 0
\(901\) 4.16665 0.138811
\(902\) −1.09174 −0.0363511
\(903\) 0 0
\(904\) 3.17507 0.105601
\(905\) −5.55888 −0.184784
\(906\) 0 0
\(907\) −24.6246 −0.817647 −0.408823 0.912614i \(-0.634061\pi\)
−0.408823 + 0.912614i \(0.634061\pi\)
\(908\) −11.4587 −0.380271
\(909\) 0 0
\(910\) 7.43270 0.246392
\(911\) 17.8906 0.592744 0.296372 0.955073i \(-0.404223\pi\)
0.296372 + 0.955073i \(0.404223\pi\)
\(912\) 0 0
\(913\) −7.09174 −0.234702
\(914\) −15.2492 −0.504399
\(915\) 0 0
\(916\) −12.5244 −0.413819
\(917\) −16.9083 −0.558360
\(918\) 0 0
\(919\) −1.61619 −0.0533131 −0.0266566 0.999645i \(-0.508486\pi\)
−0.0266566 + 0.999645i \(0.508486\pi\)
\(920\) −7.63302 −0.251653
\(921\) 0 0
\(922\) 13.2324 0.435785
\(923\) −62.4648 −2.05605
\(924\) 0 0
\(925\) −15.9311 −0.523812
\(926\) 3.63302 0.119389
\(927\) 0 0
\(928\) 4.80809 0.157833
\(929\) 38.9747 1.27872 0.639360 0.768908i \(-0.279199\pi\)
0.639360 + 0.768908i \(0.279199\pi\)
\(930\) 0 0
\(931\) −3.89984 −0.127812
\(932\) 25.2324 0.826514
\(933\) 0 0
\(934\) −13.9832 −0.457543
\(935\) 1.19191 0.0389795
\(936\) 0 0
\(937\) 4.92509 0.160896 0.0804478 0.996759i \(-0.474365\pi\)
0.0804478 + 0.996759i \(0.474365\pi\)
\(938\) 8.99158 0.293586
\(939\) 0 0
\(940\) −10.6078 −0.345987
\(941\) 45.7660 1.49193 0.745965 0.665986i \(-0.231989\pi\)
0.745965 + 0.665986i \(0.231989\pi\)
\(942\) 0 0
\(943\) 7.63302 0.248565
\(944\) 11.7997 0.384047
\(945\) 0 0
\(946\) 10.6246 0.345436
\(947\) 25.5825 0.831320 0.415660 0.909520i \(-0.363551\pi\)
0.415660 + 0.909520i \(0.363551\pi\)
\(948\) 0 0
\(949\) −34.6651 −1.12528
\(950\) 14.8510 0.481828
\(951\) 0 0
\(952\) 1.09174 0.0353836
\(953\) 32.1835 1.04253 0.521263 0.853396i \(-0.325461\pi\)
0.521263 + 0.853396i \(0.325461\pi\)
\(954\) 0 0
\(955\) −26.4648 −0.856379
\(956\) −13.9832 −0.452248
\(957\) 0 0
\(958\) −16.3670 −0.528793
\(959\) 15.1751 0.490029
\(960\) 0 0
\(961\) −29.3738 −0.947541
\(962\) 28.4816 0.918283
\(963\) 0 0
\(964\) −7.27523 −0.234319
\(965\) −6.14982 −0.197970
\(966\) 0 0
\(967\) 2.35014 0.0755755 0.0377878 0.999286i \(-0.487969\pi\)
0.0377878 + 0.999286i \(0.487969\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) −8.31493 −0.266976
\(971\) −1.98317 −0.0636428 −0.0318214 0.999494i \(-0.510131\pi\)
−0.0318214 + 0.999494i \(0.510131\pi\)
\(972\) 0 0
\(973\) −8.26682 −0.265022
\(974\) 24.1667 0.774350
\(975\) 0 0
\(976\) 11.1751 0.357705
\(977\) 1.98317 0.0634471 0.0317235 0.999497i \(-0.489900\pi\)
0.0317235 + 0.999497i \(0.489900\pi\)
\(978\) 0 0
\(979\) −12.9916 −0.415213
\(980\) 1.09174 0.0348745
\(981\) 0 0
\(982\) −7.63302 −0.243580
\(983\) −48.5817 −1.54952 −0.774759 0.632257i \(-0.782129\pi\)
−0.774759 + 0.632257i \(0.782129\pi\)
\(984\) 0 0
\(985\) −28.8486 −0.919192
\(986\) 5.24921 0.167169
\(987\) 0 0
\(988\) −26.5505 −0.844683
\(989\) −74.2828 −2.36206
\(990\) 0 0
\(991\) 14.3501 0.455847 0.227924 0.973679i \(-0.426806\pi\)
0.227924 + 0.973679i \(0.426806\pi\)
\(992\) −1.27523 −0.0404887
\(993\) 0 0
\(994\) −9.17507 −0.291016
\(995\) −12.1094 −0.383892
\(996\) 0 0
\(997\) −36.6246 −1.15991 −0.579956 0.814647i \(-0.696930\pi\)
−0.579956 + 0.814647i \(0.696930\pi\)
\(998\) 12.2576 0.388008
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1386.2.a.r.1.2 yes 3
3.2 odd 2 1386.2.a.q.1.2 3
7.6 odd 2 9702.2.a.dy.1.2 3
21.20 even 2 9702.2.a.dx.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1386.2.a.q.1.2 3 3.2 odd 2
1386.2.a.r.1.2 yes 3 1.1 even 1 trivial
9702.2.a.dx.1.2 3 21.20 even 2
9702.2.a.dy.1.2 3 7.6 odd 2