Properties

Label 4864.2.a.bh.1.2
Level $4864$
Weight $2$
Character 4864.1
Self dual yes
Analytic conductor $38.839$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4864 = 2^{8} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4864.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(38.8392355432\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
Defining polynomial: \(x^{3} - x^{2} - 4 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2432)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.34292\) of defining polynomial
Character \(\chi\) \(=\) 4864.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.14637 q^{3} +3.34292 q^{5} -3.34292 q^{7} -1.68585 q^{9} +O(q^{10})\) \(q+1.14637 q^{3} +3.34292 q^{5} -3.34292 q^{7} -1.68585 q^{9} +0.489289 q^{11} -2.29273 q^{13} +3.83221 q^{15} +0.196558 q^{17} -1.00000 q^{19} -3.83221 q^{21} -4.00000 q^{23} +6.17513 q^{25} -5.37169 q^{27} -6.12494 q^{29} +8.22533 q^{31} +0.560904 q^{33} -11.1751 q^{35} -3.83221 q^{37} -2.62831 q^{39} +4.12494 q^{41} -7.17513 q^{43} -5.63565 q^{45} -11.0073 q^{47} +4.17513 q^{49} +0.225327 q^{51} +4.81079 q^{53} +1.63565 q^{55} -1.14637 q^{57} +5.31415 q^{59} +7.73604 q^{61} +5.63565 q^{63} -7.66442 q^{65} -15.4966 q^{67} -4.58546 q^{69} -11.2713 q^{71} +15.4679 q^{73} +7.07896 q^{75} -1.63565 q^{77} -10.5181 q^{79} -1.10038 q^{81} -0.728692 q^{83} +0.657077 q^{85} -7.02142 q^{87} -8.51806 q^{89} +7.66442 q^{91} +9.42923 q^{93} -3.34292 q^{95} -3.31415 q^{97} -0.824865 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 2q^{3} + 4q^{5} - 4q^{7} + 7q^{9} + O(q^{10}) \) \( 3q + 2q^{3} + 4q^{5} - 4q^{7} + 7q^{9} - 6q^{11} - 4q^{13} - 2q^{15} - 4q^{17} - 3q^{19} + 2q^{21} - 12q^{23} - q^{25} + 8q^{27} - 2q^{29} + 2q^{31} + 6q^{33} - 14q^{35} + 2q^{37} - 32q^{39} - 4q^{41} - 2q^{43} - 8q^{45} - 7q^{49} - 22q^{51} - 14q^{53} - 4q^{55} - 2q^{57} + 28q^{59} + 8q^{61} + 8q^{63} + 4q^{65} - 6q^{67} - 8q^{69} - 16q^{71} + 24q^{73} + 4q^{77} - 6q^{79} + 3q^{81} - 20q^{83} + 8q^{85} - 36q^{87} - 4q^{91} - 32q^{93} - 4q^{95} - 22q^{97} - 22q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.14637 0.661854 0.330927 0.943656i \(-0.392639\pi\)
0.330927 + 0.943656i \(0.392639\pi\)
\(4\) 0 0
\(5\) 3.34292 1.49500 0.747500 0.664261i \(-0.231254\pi\)
0.747500 + 0.664261i \(0.231254\pi\)
\(6\) 0 0
\(7\) −3.34292 −1.26351 −0.631753 0.775170i \(-0.717664\pi\)
−0.631753 + 0.775170i \(0.717664\pi\)
\(8\) 0 0
\(9\) −1.68585 −0.561949
\(10\) 0 0
\(11\) 0.489289 0.147526 0.0737630 0.997276i \(-0.476499\pi\)
0.0737630 + 0.997276i \(0.476499\pi\)
\(12\) 0 0
\(13\) −2.29273 −0.635889 −0.317945 0.948109i \(-0.602993\pi\)
−0.317945 + 0.948109i \(0.602993\pi\)
\(14\) 0 0
\(15\) 3.83221 0.989473
\(16\) 0 0
\(17\) 0.196558 0.0476722 0.0238361 0.999716i \(-0.492412\pi\)
0.0238361 + 0.999716i \(0.492412\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −3.83221 −0.836257
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) 6.17513 1.23503
\(26\) 0 0
\(27\) −5.37169 −1.03378
\(28\) 0 0
\(29\) −6.12494 −1.13737 −0.568687 0.822554i \(-0.692548\pi\)
−0.568687 + 0.822554i \(0.692548\pi\)
\(30\) 0 0
\(31\) 8.22533 1.47731 0.738656 0.674082i \(-0.235461\pi\)
0.738656 + 0.674082i \(0.235461\pi\)
\(32\) 0 0
\(33\) 0.560904 0.0976408
\(34\) 0 0
\(35\) −11.1751 −1.88894
\(36\) 0 0
\(37\) −3.83221 −0.630012 −0.315006 0.949090i \(-0.602006\pi\)
−0.315006 + 0.949090i \(0.602006\pi\)
\(38\) 0 0
\(39\) −2.62831 −0.420866
\(40\) 0 0
\(41\) 4.12494 0.644208 0.322104 0.946704i \(-0.395610\pi\)
0.322104 + 0.946704i \(0.395610\pi\)
\(42\) 0 0
\(43\) −7.17513 −1.09420 −0.547099 0.837068i \(-0.684268\pi\)
−0.547099 + 0.837068i \(0.684268\pi\)
\(44\) 0 0
\(45\) −5.63565 −0.840114
\(46\) 0 0
\(47\) −11.0073 −1.60559 −0.802793 0.596258i \(-0.796654\pi\)
−0.802793 + 0.596258i \(0.796654\pi\)
\(48\) 0 0
\(49\) 4.17513 0.596448
\(50\) 0 0
\(51\) 0.225327 0.0315521
\(52\) 0 0
\(53\) 4.81079 0.660813 0.330406 0.943839i \(-0.392814\pi\)
0.330406 + 0.943839i \(0.392814\pi\)
\(54\) 0 0
\(55\) 1.63565 0.220552
\(56\) 0 0
\(57\) −1.14637 −0.151840
\(58\) 0 0
\(59\) 5.31415 0.691844 0.345922 0.938263i \(-0.387566\pi\)
0.345922 + 0.938263i \(0.387566\pi\)
\(60\) 0 0
\(61\) 7.73604 0.990498 0.495249 0.868751i \(-0.335077\pi\)
0.495249 + 0.868751i \(0.335077\pi\)
\(62\) 0 0
\(63\) 5.63565 0.710026
\(64\) 0 0
\(65\) −7.66442 −0.950655
\(66\) 0 0
\(67\) −15.4966 −1.89322 −0.946608 0.322388i \(-0.895514\pi\)
−0.946608 + 0.322388i \(0.895514\pi\)
\(68\) 0 0
\(69\) −4.58546 −0.552025
\(70\) 0 0
\(71\) −11.2713 −1.33766 −0.668829 0.743416i \(-0.733204\pi\)
−0.668829 + 0.743416i \(0.733204\pi\)
\(72\) 0 0
\(73\) 15.4679 1.81038 0.905188 0.425011i \(-0.139730\pi\)
0.905188 + 0.425011i \(0.139730\pi\)
\(74\) 0 0
\(75\) 7.07896 0.817408
\(76\) 0 0
\(77\) −1.63565 −0.186400
\(78\) 0 0
\(79\) −10.5181 −1.18337 −0.591687 0.806168i \(-0.701538\pi\)
−0.591687 + 0.806168i \(0.701538\pi\)
\(80\) 0 0
\(81\) −1.10038 −0.122265
\(82\) 0 0
\(83\) −0.728692 −0.0799843 −0.0399922 0.999200i \(-0.512733\pi\)
−0.0399922 + 0.999200i \(0.512733\pi\)
\(84\) 0 0
\(85\) 0.657077 0.0712700
\(86\) 0 0
\(87\) −7.02142 −0.752776
\(88\) 0 0
\(89\) −8.51806 −0.902912 −0.451456 0.892293i \(-0.649095\pi\)
−0.451456 + 0.892293i \(0.649095\pi\)
\(90\) 0 0
\(91\) 7.66442 0.803450
\(92\) 0 0
\(93\) 9.42923 0.977766
\(94\) 0 0
\(95\) −3.34292 −0.342977
\(96\) 0 0
\(97\) −3.31415 −0.336501 −0.168251 0.985744i \(-0.553812\pi\)
−0.168251 + 0.985744i \(0.553812\pi\)
\(98\) 0 0
\(99\) −0.824865 −0.0829021
\(100\) 0 0
\(101\) −10.6858 −1.06328 −0.531641 0.846970i \(-0.678424\pi\)
−0.531641 + 0.846970i \(0.678424\pi\)
\(102\) 0 0
\(103\) −12.0575 −1.18806 −0.594032 0.804441i \(-0.702465\pi\)
−0.594032 + 0.804441i \(0.702465\pi\)
\(104\) 0 0
\(105\) −12.8108 −1.25020
\(106\) 0 0
\(107\) 8.39312 0.811393 0.405697 0.914008i \(-0.367029\pi\)
0.405697 + 0.914008i \(0.367029\pi\)
\(108\) 0 0
\(109\) −8.22533 −0.787843 −0.393922 0.919144i \(-0.628882\pi\)
−0.393922 + 0.919144i \(0.628882\pi\)
\(110\) 0 0
\(111\) −4.39312 −0.416976
\(112\) 0 0
\(113\) 13.4966 1.26966 0.634828 0.772653i \(-0.281071\pi\)
0.634828 + 0.772653i \(0.281071\pi\)
\(114\) 0 0
\(115\) −13.3717 −1.24692
\(116\) 0 0
\(117\) 3.86519 0.357337
\(118\) 0 0
\(119\) −0.657077 −0.0602341
\(120\) 0 0
\(121\) −10.7606 −0.978236
\(122\) 0 0
\(123\) 4.72869 0.426372
\(124\) 0 0
\(125\) 3.92839 0.351365
\(126\) 0 0
\(127\) −1.53948 −0.136607 −0.0683034 0.997665i \(-0.521759\pi\)
−0.0683034 + 0.997665i \(0.521759\pi\)
\(128\) 0 0
\(129\) −8.22533 −0.724200
\(130\) 0 0
\(131\) −3.56825 −0.311759 −0.155880 0.987776i \(-0.549821\pi\)
−0.155880 + 0.987776i \(0.549821\pi\)
\(132\) 0 0
\(133\) 3.34292 0.289868
\(134\) 0 0
\(135\) −17.9572 −1.54551
\(136\) 0 0
\(137\) 10.1537 0.867490 0.433745 0.901036i \(-0.357192\pi\)
0.433745 + 0.901036i \(0.357192\pi\)
\(138\) 0 0
\(139\) −3.90383 −0.331118 −0.165559 0.986200i \(-0.552943\pi\)
−0.165559 + 0.986200i \(0.552943\pi\)
\(140\) 0 0
\(141\) −12.6184 −1.06266
\(142\) 0 0
\(143\) −1.12181 −0.0938102
\(144\) 0 0
\(145\) −20.4752 −1.70037
\(146\) 0 0
\(147\) 4.78623 0.394762
\(148\) 0 0
\(149\) −17.3576 −1.42199 −0.710996 0.703196i \(-0.751755\pi\)
−0.710996 + 0.703196i \(0.751755\pi\)
\(150\) 0 0
\(151\) −4.05754 −0.330198 −0.165099 0.986277i \(-0.552794\pi\)
−0.165099 + 0.986277i \(0.552794\pi\)
\(152\) 0 0
\(153\) −0.331366 −0.0267893
\(154\) 0 0
\(155\) 27.4966 2.20858
\(156\) 0 0
\(157\) −2.35027 −0.187572 −0.0937860 0.995592i \(-0.529897\pi\)
−0.0937860 + 0.995592i \(0.529897\pi\)
\(158\) 0 0
\(159\) 5.51492 0.437362
\(160\) 0 0
\(161\) 13.3717 1.05384
\(162\) 0 0
\(163\) 13.7648 1.07814 0.539071 0.842260i \(-0.318775\pi\)
0.539071 + 0.842260i \(0.318775\pi\)
\(164\) 0 0
\(165\) 1.87506 0.145973
\(166\) 0 0
\(167\) 0.417674 0.0323206 0.0161603 0.999869i \(-0.494856\pi\)
0.0161603 + 0.999869i \(0.494856\pi\)
\(168\) 0 0
\(169\) −7.74338 −0.595645
\(170\) 0 0
\(171\) 1.68585 0.128920
\(172\) 0 0
\(173\) −3.60688 −0.274226 −0.137113 0.990555i \(-0.543782\pi\)
−0.137113 + 0.990555i \(0.543782\pi\)
\(174\) 0 0
\(175\) −20.6430 −1.56046
\(176\) 0 0
\(177\) 6.09196 0.457900
\(178\) 0 0
\(179\) −6.51806 −0.487183 −0.243591 0.969878i \(-0.578326\pi\)
−0.243591 + 0.969878i \(0.578326\pi\)
\(180\) 0 0
\(181\) 1.31415 0.0976803 0.0488401 0.998807i \(-0.484448\pi\)
0.0488401 + 0.998807i \(0.484448\pi\)
\(182\) 0 0
\(183\) 8.86833 0.655566
\(184\) 0 0
\(185\) −12.8108 −0.941868
\(186\) 0 0
\(187\) 0.0961734 0.00703289
\(188\) 0 0
\(189\) 17.9572 1.30619
\(190\) 0 0
\(191\) −9.24254 −0.668767 −0.334383 0.942437i \(-0.608528\pi\)
−0.334383 + 0.942437i \(0.608528\pi\)
\(192\) 0 0
\(193\) 8.85363 0.637299 0.318649 0.947873i \(-0.396771\pi\)
0.318649 + 0.947873i \(0.396771\pi\)
\(194\) 0 0
\(195\) −8.78623 −0.629195
\(196\) 0 0
\(197\) 27.3288 1.94710 0.973550 0.228475i \(-0.0733738\pi\)
0.973550 + 0.228475i \(0.0733738\pi\)
\(198\) 0 0
\(199\) −18.3643 −1.30181 −0.650907 0.759157i \(-0.725611\pi\)
−0.650907 + 0.759157i \(0.725611\pi\)
\(200\) 0 0
\(201\) −17.7648 −1.25303
\(202\) 0 0
\(203\) 20.4752 1.43708
\(204\) 0 0
\(205\) 13.7894 0.963091
\(206\) 0 0
\(207\) 6.74338 0.468698
\(208\) 0 0
\(209\) −0.489289 −0.0338448
\(210\) 0 0
\(211\) 18.3503 1.26328 0.631642 0.775260i \(-0.282381\pi\)
0.631642 + 0.775260i \(0.282381\pi\)
\(212\) 0 0
\(213\) −12.9210 −0.885335
\(214\) 0 0
\(215\) −23.9859 −1.63583
\(216\) 0 0
\(217\) −27.4966 −1.86659
\(218\) 0 0
\(219\) 17.7318 1.19821
\(220\) 0 0
\(221\) −0.450654 −0.0303142
\(222\) 0 0
\(223\) 18.3748 1.23047 0.615235 0.788344i \(-0.289061\pi\)
0.615235 + 0.788344i \(0.289061\pi\)
\(224\) 0 0
\(225\) −10.4103 −0.694022
\(226\) 0 0
\(227\) 18.9933 1.26063 0.630314 0.776340i \(-0.282926\pi\)
0.630314 + 0.776340i \(0.282926\pi\)
\(228\) 0 0
\(229\) −9.05019 −0.598054 −0.299027 0.954245i \(-0.596662\pi\)
−0.299027 + 0.954245i \(0.596662\pi\)
\(230\) 0 0
\(231\) −1.87506 −0.123370
\(232\) 0 0
\(233\) 9.76060 0.639438 0.319719 0.947512i \(-0.396411\pi\)
0.319719 + 0.947512i \(0.396411\pi\)
\(234\) 0 0
\(235\) −36.7967 −2.40035
\(236\) 0 0
\(237\) −12.0575 −0.783221
\(238\) 0 0
\(239\) 12.9645 0.838604 0.419302 0.907847i \(-0.362275\pi\)
0.419302 + 0.907847i \(0.362275\pi\)
\(240\) 0 0
\(241\) −20.3257 −1.30929 −0.654647 0.755935i \(-0.727183\pi\)
−0.654647 + 0.755935i \(0.727183\pi\)
\(242\) 0 0
\(243\) 14.8536 0.952861
\(244\) 0 0
\(245\) 13.9572 0.891690
\(246\) 0 0
\(247\) 2.29273 0.145883
\(248\) 0 0
\(249\) −0.835347 −0.0529380
\(250\) 0 0
\(251\) −6.05333 −0.382083 −0.191041 0.981582i \(-0.561186\pi\)
−0.191041 + 0.981582i \(0.561186\pi\)
\(252\) 0 0
\(253\) −1.95715 −0.123045
\(254\) 0 0
\(255\) 0.753250 0.0471704
\(256\) 0 0
\(257\) 14.0575 0.876885 0.438443 0.898759i \(-0.355530\pi\)
0.438443 + 0.898759i \(0.355530\pi\)
\(258\) 0 0
\(259\) 12.8108 0.796024
\(260\) 0 0
\(261\) 10.3257 0.639145
\(262\) 0 0
\(263\) −8.26396 −0.509578 −0.254789 0.966997i \(-0.582006\pi\)
−0.254789 + 0.966997i \(0.582006\pi\)
\(264\) 0 0
\(265\) 16.0821 0.987915
\(266\) 0 0
\(267\) −9.76481 −0.597597
\(268\) 0 0
\(269\) −3.18921 −0.194450 −0.0972248 0.995262i \(-0.530997\pi\)
−0.0972248 + 0.995262i \(0.530997\pi\)
\(270\) 0 0
\(271\) −9.89962 −0.601359 −0.300679 0.953725i \(-0.597213\pi\)
−0.300679 + 0.953725i \(0.597213\pi\)
\(272\) 0 0
\(273\) 8.78623 0.531767
\(274\) 0 0
\(275\) 3.02142 0.182199
\(276\) 0 0
\(277\) −12.7722 −0.767404 −0.383702 0.923457i \(-0.625351\pi\)
−0.383702 + 0.923457i \(0.625351\pi\)
\(278\) 0 0
\(279\) −13.8666 −0.830174
\(280\) 0 0
\(281\) 29.5296 1.76159 0.880795 0.473499i \(-0.157009\pi\)
0.880795 + 0.473499i \(0.157009\pi\)
\(282\) 0 0
\(283\) −25.3246 −1.50539 −0.752697 0.658367i \(-0.771247\pi\)
−0.752697 + 0.658367i \(0.771247\pi\)
\(284\) 0 0
\(285\) −3.83221 −0.227001
\(286\) 0 0
\(287\) −13.7894 −0.813961
\(288\) 0 0
\(289\) −16.9614 −0.997727
\(290\) 0 0
\(291\) −3.79923 −0.222715
\(292\) 0 0
\(293\) 16.8353 0.983531 0.491766 0.870728i \(-0.336352\pi\)
0.491766 + 0.870728i \(0.336352\pi\)
\(294\) 0 0
\(295\) 17.7648 1.03431
\(296\) 0 0
\(297\) −2.62831 −0.152510
\(298\) 0 0
\(299\) 9.17092 0.530368
\(300\) 0 0
\(301\) 23.9859 1.38253
\(302\) 0 0
\(303\) −12.2499 −0.703738
\(304\) 0 0
\(305\) 25.8610 1.48080
\(306\) 0 0
\(307\) 17.7894 1.01529 0.507646 0.861566i \(-0.330516\pi\)
0.507646 + 0.861566i \(0.330516\pi\)
\(308\) 0 0
\(309\) −13.8223 −0.786326
\(310\) 0 0
\(311\) −20.3215 −1.15233 −0.576163 0.817335i \(-0.695451\pi\)
−0.576163 + 0.817335i \(0.695451\pi\)
\(312\) 0 0
\(313\) 34.8929 1.97226 0.986131 0.165967i \(-0.0530746\pi\)
0.986131 + 0.165967i \(0.0530746\pi\)
\(314\) 0 0
\(315\) 18.8396 1.06149
\(316\) 0 0
\(317\) −12.7287 −0.714915 −0.357457 0.933929i \(-0.616356\pi\)
−0.357457 + 0.933929i \(0.616356\pi\)
\(318\) 0 0
\(319\) −2.99686 −0.167792
\(320\) 0 0
\(321\) 9.62158 0.537024
\(322\) 0 0
\(323\) −0.196558 −0.0109368
\(324\) 0 0
\(325\) −14.1579 −0.785340
\(326\) 0 0
\(327\) −9.42923 −0.521438
\(328\) 0 0
\(329\) 36.7967 2.02867
\(330\) 0 0
\(331\) −18.4605 −1.01468 −0.507341 0.861745i \(-0.669372\pi\)
−0.507341 + 0.861745i \(0.669372\pi\)
\(332\) 0 0
\(333\) 6.46052 0.354034
\(334\) 0 0
\(335\) −51.8041 −2.83036
\(336\) 0 0
\(337\) 28.5510 1.55527 0.777637 0.628713i \(-0.216418\pi\)
0.777637 + 0.628713i \(0.216418\pi\)
\(338\) 0 0
\(339\) 15.4721 0.840328
\(340\) 0 0
\(341\) 4.02456 0.217942
\(342\) 0 0
\(343\) 9.44331 0.509891
\(344\) 0 0
\(345\) −15.3288 −0.825277
\(346\) 0 0
\(347\) 30.3116 1.62721 0.813607 0.581415i \(-0.197501\pi\)
0.813607 + 0.581415i \(0.197501\pi\)
\(348\) 0 0
\(349\) 19.6503 1.05186 0.525929 0.850528i \(-0.323718\pi\)
0.525929 + 0.850528i \(0.323718\pi\)
\(350\) 0 0
\(351\) 12.3158 0.657371
\(352\) 0 0
\(353\) 1.80765 0.0962117 0.0481058 0.998842i \(-0.484682\pi\)
0.0481058 + 0.998842i \(0.484682\pi\)
\(354\) 0 0
\(355\) −37.6791 −1.99980
\(356\) 0 0
\(357\) −0.753250 −0.0398662
\(358\) 0 0
\(359\) 23.4496 1.23762 0.618811 0.785540i \(-0.287615\pi\)
0.618811 + 0.785540i \(0.287615\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −12.3356 −0.647450
\(364\) 0 0
\(365\) 51.7079 2.70651
\(366\) 0 0
\(367\) 16.2499 0.848237 0.424119 0.905607i \(-0.360584\pi\)
0.424119 + 0.905607i \(0.360584\pi\)
\(368\) 0 0
\(369\) −6.95402 −0.362012
\(370\) 0 0
\(371\) −16.0821 −0.834941
\(372\) 0 0
\(373\) 16.8353 0.871701 0.435851 0.900019i \(-0.356448\pi\)
0.435851 + 0.900019i \(0.356448\pi\)
\(374\) 0 0
\(375\) 4.50337 0.232553
\(376\) 0 0
\(377\) 14.0428 0.723243
\(378\) 0 0
\(379\) 2.68585 0.137963 0.0689813 0.997618i \(-0.478025\pi\)
0.0689813 + 0.997618i \(0.478025\pi\)
\(380\) 0 0
\(381\) −1.76481 −0.0904138
\(382\) 0 0
\(383\) −0.110250 −0.00563350 −0.00281675 0.999996i \(-0.500897\pi\)
−0.00281675 + 0.999996i \(0.500897\pi\)
\(384\) 0 0
\(385\) −5.46787 −0.278668
\(386\) 0 0
\(387\) 12.0962 0.614883
\(388\) 0 0
\(389\) 9.24254 0.468615 0.234308 0.972162i \(-0.424718\pi\)
0.234308 + 0.972162i \(0.424718\pi\)
\(390\) 0 0
\(391\) −0.786230 −0.0397614
\(392\) 0 0
\(393\) −4.09052 −0.206339
\(394\) 0 0
\(395\) −35.1611 −1.76914
\(396\) 0 0
\(397\) −32.9070 −1.65155 −0.825777 0.563997i \(-0.809263\pi\)
−0.825777 + 0.563997i \(0.809263\pi\)
\(398\) 0 0
\(399\) 3.83221 0.191851
\(400\) 0 0
\(401\) −23.9817 −1.19759 −0.598795 0.800902i \(-0.704353\pi\)
−0.598795 + 0.800902i \(0.704353\pi\)
\(402\) 0 0
\(403\) −18.8585 −0.939407
\(404\) 0 0
\(405\) −3.67850 −0.182786
\(406\) 0 0
\(407\) −1.87506 −0.0929431
\(408\) 0 0
\(409\) 0.182481 0.00902311 0.00451156 0.999990i \(-0.498564\pi\)
0.00451156 + 0.999990i \(0.498564\pi\)
\(410\) 0 0
\(411\) 11.6399 0.574152
\(412\) 0 0
\(413\) −17.7648 −0.874149
\(414\) 0 0
\(415\) −2.43596 −0.119577
\(416\) 0 0
\(417\) −4.47521 −0.219152
\(418\) 0 0
\(419\) −13.6497 −0.666833 −0.333416 0.942780i \(-0.608202\pi\)
−0.333416 + 0.942780i \(0.608202\pi\)
\(420\) 0 0
\(421\) 14.4324 0.703390 0.351695 0.936115i \(-0.385605\pi\)
0.351695 + 0.936115i \(0.385605\pi\)
\(422\) 0 0
\(423\) 18.5567 0.902257
\(424\) 0 0
\(425\) 1.21377 0.0588765
\(426\) 0 0
\(427\) −25.8610 −1.25150
\(428\) 0 0
\(429\) −1.28600 −0.0620887
\(430\) 0 0
\(431\) −17.1218 −0.824728 −0.412364 0.911019i \(-0.635297\pi\)
−0.412364 + 0.911019i \(0.635297\pi\)
\(432\) 0 0
\(433\) −28.3748 −1.36361 −0.681804 0.731535i \(-0.738804\pi\)
−0.681804 + 0.731535i \(0.738804\pi\)
\(434\) 0 0
\(435\) −23.4721 −1.12540
\(436\) 0 0
\(437\) 4.00000 0.191346
\(438\) 0 0
\(439\) 15.5823 0.743704 0.371852 0.928292i \(-0.378723\pi\)
0.371852 + 0.928292i \(0.378723\pi\)
\(440\) 0 0
\(441\) −7.03863 −0.335173
\(442\) 0 0
\(443\) −24.7476 −1.17579 −0.587897 0.808936i \(-0.700044\pi\)
−0.587897 + 0.808936i \(0.700044\pi\)
\(444\) 0 0
\(445\) −28.4752 −1.34985
\(446\) 0 0
\(447\) −19.8982 −0.941151
\(448\) 0 0
\(449\) −37.7220 −1.78021 −0.890105 0.455756i \(-0.849369\pi\)
−0.890105 + 0.455756i \(0.849369\pi\)
\(450\) 0 0
\(451\) 2.01829 0.0950374
\(452\) 0 0
\(453\) −4.65142 −0.218543
\(454\) 0 0
\(455\) 25.6216 1.20116
\(456\) 0 0
\(457\) −21.6258 −1.01161 −0.505806 0.862647i \(-0.668805\pi\)
−0.505806 + 0.862647i \(0.668805\pi\)
\(458\) 0 0
\(459\) −1.05585 −0.0492827
\(460\) 0 0
\(461\) 11.5353 0.537251 0.268626 0.963245i \(-0.413431\pi\)
0.268626 + 0.963245i \(0.413431\pi\)
\(462\) 0 0
\(463\) −40.4282 −1.87886 −0.939428 0.342747i \(-0.888643\pi\)
−0.939428 + 0.342747i \(0.888643\pi\)
\(464\) 0 0
\(465\) 31.5212 1.46176
\(466\) 0 0
\(467\) 27.7606 1.28461 0.642304 0.766450i \(-0.277979\pi\)
0.642304 + 0.766450i \(0.277979\pi\)
\(468\) 0 0
\(469\) 51.8041 2.39209
\(470\) 0 0
\(471\) −2.69427 −0.124145
\(472\) 0 0
\(473\) −3.51071 −0.161423
\(474\) 0 0
\(475\) −6.17513 −0.283335
\(476\) 0 0
\(477\) −8.11025 −0.371343
\(478\) 0 0
\(479\) 10.4935 0.479460 0.239730 0.970840i \(-0.422941\pi\)
0.239730 + 0.970840i \(0.422941\pi\)
\(480\) 0 0
\(481\) 8.78623 0.400618
\(482\) 0 0
\(483\) 15.3288 0.697487
\(484\) 0 0
\(485\) −11.0790 −0.503070
\(486\) 0 0
\(487\) −18.2646 −0.827647 −0.413824 0.910357i \(-0.635807\pi\)
−0.413824 + 0.910357i \(0.635807\pi\)
\(488\) 0 0
\(489\) 15.7795 0.713574
\(490\) 0 0
\(491\) 41.6791 1.88095 0.940476 0.339860i \(-0.110380\pi\)
0.940476 + 0.339860i \(0.110380\pi\)
\(492\) 0 0
\(493\) −1.20390 −0.0542211
\(494\) 0 0
\(495\) −2.75746 −0.123939
\(496\) 0 0
\(497\) 37.6791 1.69014
\(498\) 0 0
\(499\) 13.0748 0.585306 0.292653 0.956219i \(-0.405462\pi\)
0.292653 + 0.956219i \(0.405462\pi\)
\(500\) 0 0
\(501\) 0.478807 0.0213915
\(502\) 0 0
\(503\) 18.4935 0.824584 0.412292 0.911052i \(-0.364728\pi\)
0.412292 + 0.911052i \(0.364728\pi\)
\(504\) 0 0
\(505\) −35.7220 −1.58961
\(506\) 0 0
\(507\) −8.87675 −0.394230
\(508\) 0 0
\(509\) 11.8652 0.525915 0.262958 0.964807i \(-0.415302\pi\)
0.262958 + 0.964807i \(0.415302\pi\)
\(510\) 0 0
\(511\) −51.7079 −2.28742
\(512\) 0 0
\(513\) 5.37169 0.237166
\(514\) 0 0
\(515\) −40.3074 −1.77616
\(516\) 0 0
\(517\) −5.38577 −0.236866
\(518\) 0 0
\(519\) −4.13481 −0.181498
\(520\) 0 0
\(521\) −24.3832 −1.06825 −0.534125 0.845406i \(-0.679359\pi\)
−0.534125 + 0.845406i \(0.679359\pi\)
\(522\) 0 0
\(523\) −38.5510 −1.68572 −0.842860 0.538134i \(-0.819130\pi\)
−0.842860 + 0.538134i \(0.819130\pi\)
\(524\) 0 0
\(525\) −23.6644 −1.03280
\(526\) 0 0
\(527\) 1.61675 0.0704268
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −8.95885 −0.388781
\(532\) 0 0
\(533\) −9.45738 −0.409645
\(534\) 0 0
\(535\) 28.0575 1.21303
\(536\) 0 0
\(537\) −7.47208 −0.322444
\(538\) 0 0
\(539\) 2.04285 0.0879916
\(540\) 0 0
\(541\) −0.857845 −0.0368817 −0.0184408 0.999830i \(-0.505870\pi\)
−0.0184408 + 0.999830i \(0.505870\pi\)
\(542\) 0 0
\(543\) 1.50650 0.0646501
\(544\) 0 0
\(545\) −27.4966 −1.17783
\(546\) 0 0
\(547\) −10.8291 −0.463018 −0.231509 0.972833i \(-0.574366\pi\)
−0.231509 + 0.972833i \(0.574366\pi\)
\(548\) 0 0
\(549\) −13.0418 −0.556609
\(550\) 0 0
\(551\) 6.12494 0.260931
\(552\) 0 0
\(553\) 35.1611 1.49520
\(554\) 0 0
\(555\) −14.6858 −0.623379
\(556\) 0 0
\(557\) 36.7722 1.55809 0.779043 0.626970i \(-0.215705\pi\)
0.779043 + 0.626970i \(0.215705\pi\)
\(558\) 0 0
\(559\) 16.4507 0.695789
\(560\) 0 0
\(561\) 0.110250 0.00465475
\(562\) 0 0
\(563\) 29.8469 1.25790 0.628949 0.777447i \(-0.283486\pi\)
0.628949 + 0.777447i \(0.283486\pi\)
\(564\) 0 0
\(565\) 45.1182 1.89814
\(566\) 0 0
\(567\) 3.67850 0.154482
\(568\) 0 0
\(569\) −36.2646 −1.52029 −0.760145 0.649753i \(-0.774872\pi\)
−0.760145 + 0.649753i \(0.774872\pi\)
\(570\) 0 0
\(571\) −11.3570 −0.475276 −0.237638 0.971354i \(-0.576373\pi\)
−0.237638 + 0.971354i \(0.576373\pi\)
\(572\) 0 0
\(573\) −10.5953 −0.442626
\(574\) 0 0
\(575\) −24.7005 −1.03008
\(576\) 0 0
\(577\) 18.2113 0.758144 0.379072 0.925367i \(-0.376243\pi\)
0.379072 + 0.925367i \(0.376243\pi\)
\(578\) 0 0
\(579\) 10.1495 0.421799
\(580\) 0 0
\(581\) 2.43596 0.101061
\(582\) 0 0
\(583\) 2.35386 0.0974871
\(584\) 0 0
\(585\) 12.9210 0.534219
\(586\) 0 0
\(587\) −26.4464 −1.09156 −0.545781 0.837928i \(-0.683767\pi\)
−0.545781 + 0.837928i \(0.683767\pi\)
\(588\) 0 0
\(589\) −8.22533 −0.338919
\(590\) 0 0
\(591\) 31.3288 1.28870
\(592\) 0 0
\(593\) −26.8929 −1.10436 −0.552179 0.833725i \(-0.686204\pi\)
−0.552179 + 0.833725i \(0.686204\pi\)
\(594\) 0 0
\(595\) −2.19656 −0.0900501
\(596\) 0 0
\(597\) −21.0523 −0.861611
\(598\) 0 0
\(599\) −34.5181 −1.41037 −0.705185 0.709024i \(-0.749136\pi\)
−0.705185 + 0.709024i \(0.749136\pi\)
\(600\) 0 0
\(601\) −27.8715 −1.13690 −0.568450 0.822718i \(-0.692457\pi\)
−0.568450 + 0.822718i \(0.692457\pi\)
\(602\) 0 0
\(603\) 26.1249 1.06389
\(604\) 0 0
\(605\) −35.9718 −1.46246
\(606\) 0 0
\(607\) −27.5787 −1.11939 −0.559693 0.828700i \(-0.689081\pi\)
−0.559693 + 0.828700i \(0.689081\pi\)
\(608\) 0 0
\(609\) 23.4721 0.951137
\(610\) 0 0
\(611\) 25.2369 1.02098
\(612\) 0 0
\(613\) −2.67177 −0.107912 −0.0539559 0.998543i \(-0.517183\pi\)
−0.0539559 + 0.998543i \(0.517183\pi\)
\(614\) 0 0
\(615\) 15.8077 0.637426
\(616\) 0 0
\(617\) 15.0832 0.607226 0.303613 0.952795i \(-0.401807\pi\)
0.303613 + 0.952795i \(0.401807\pi\)
\(618\) 0 0
\(619\) −31.7220 −1.27501 −0.637507 0.770445i \(-0.720034\pi\)
−0.637507 + 0.770445i \(0.720034\pi\)
\(620\) 0 0
\(621\) 21.4868 0.862234
\(622\) 0 0
\(623\) 28.4752 1.14084
\(624\) 0 0
\(625\) −17.7434 −0.709735
\(626\) 0 0
\(627\) −0.560904 −0.0224003
\(628\) 0 0
\(629\) −0.753250 −0.0300341
\(630\) 0 0
\(631\) 47.1800 1.87820 0.939102 0.343638i \(-0.111659\pi\)
0.939102 + 0.343638i \(0.111659\pi\)
\(632\) 0 0
\(633\) 21.0361 0.836111
\(634\) 0 0
\(635\) −5.14637 −0.204227
\(636\) 0 0
\(637\) −9.57246 −0.379275
\(638\) 0 0
\(639\) 19.0017 0.751695
\(640\) 0 0
\(641\) 34.1067 1.34713 0.673566 0.739127i \(-0.264762\pi\)
0.673566 + 0.739127i \(0.264762\pi\)
\(642\) 0 0
\(643\) 21.6686 0.854528 0.427264 0.904127i \(-0.359478\pi\)
0.427264 + 0.904127i \(0.359478\pi\)
\(644\) 0 0
\(645\) −27.4966 −1.08268
\(646\) 0 0
\(647\) −35.8427 −1.40912 −0.704561 0.709644i \(-0.748856\pi\)
−0.704561 + 0.709644i \(0.748856\pi\)
\(648\) 0 0
\(649\) 2.60015 0.102065
\(650\) 0 0
\(651\) −31.5212 −1.23541
\(652\) 0 0
\(653\) −21.7507 −0.851172 −0.425586 0.904918i \(-0.639932\pi\)
−0.425586 + 0.904918i \(0.639932\pi\)
\(654\) 0 0
\(655\) −11.9284 −0.466081
\(656\) 0 0
\(657\) −26.0764 −1.01734
\(658\) 0 0
\(659\) 10.3012 0.401276 0.200638 0.979665i \(-0.435699\pi\)
0.200638 + 0.979665i \(0.435699\pi\)
\(660\) 0 0
\(661\) −1.95715 −0.0761245 −0.0380622 0.999275i \(-0.512119\pi\)
−0.0380622 + 0.999275i \(0.512119\pi\)
\(662\) 0 0
\(663\) −0.516614 −0.0200636
\(664\) 0 0
\(665\) 11.1751 0.433353
\(666\) 0 0
\(667\) 24.4998 0.948635
\(668\) 0 0
\(669\) 21.0643 0.814392
\(670\) 0 0
\(671\) 3.78516 0.146124
\(672\) 0 0
\(673\) −0.0428457 −0.00165158 −0.000825790 1.00000i \(-0.500263\pi\)
−0.000825790 1.00000i \(0.500263\pi\)
\(674\) 0 0
\(675\) −33.1709 −1.27675
\(676\) 0 0
\(677\) −42.1543 −1.62012 −0.810061 0.586345i \(-0.800566\pi\)
−0.810061 + 0.586345i \(0.800566\pi\)
\(678\) 0 0
\(679\) 11.0790 0.425172
\(680\) 0 0
\(681\) 21.7732 0.834352
\(682\) 0 0
\(683\) −28.8353 −1.10335 −0.551677 0.834058i \(-0.686012\pi\)
−0.551677 + 0.834058i \(0.686012\pi\)
\(684\) 0 0
\(685\) 33.9431 1.29690
\(686\) 0 0
\(687\) −10.3748 −0.395824
\(688\) 0 0
\(689\) −11.0298 −0.420204
\(690\) 0 0
\(691\) 21.4103 0.814487 0.407244 0.913320i \(-0.366490\pi\)
0.407244 + 0.913320i \(0.366490\pi\)
\(692\) 0 0
\(693\) 2.75746 0.104747
\(694\) 0 0
\(695\) −13.0502 −0.495022
\(696\) 0 0
\(697\) 0.810789 0.0307108
\(698\) 0 0
\(699\) 11.1892 0.423215
\(700\) 0 0
\(701\) 3.07054 0.115973 0.0579863 0.998317i \(-0.481532\pi\)
0.0579863 + 0.998317i \(0.481532\pi\)
\(702\) 0 0
\(703\) 3.83221 0.144535
\(704\) 0 0
\(705\) −42.1825 −1.58868
\(706\) 0 0
\(707\) 35.7220 1.34346
\(708\) 0 0
\(709\) −42.8500 −1.60927 −0.804634 0.593772i \(-0.797638\pi\)
−0.804634 + 0.593772i \(0.797638\pi\)
\(710\) 0 0
\(711\) 17.7318 0.664995
\(712\) 0 0
\(713\) −32.9013 −1.23216
\(714\) 0 0
\(715\) −3.75011 −0.140246
\(716\) 0 0
\(717\) 14.8621 0.555034
\(718\) 0 0
\(719\) 34.5285 1.28770 0.643849 0.765153i \(-0.277337\pi\)
0.643849 + 0.765153i \(0.277337\pi\)
\(720\) 0 0
\(721\) 40.3074 1.50113
\(722\) 0 0
\(723\) −23.3007 −0.866562
\(724\) 0 0
\(725\) −37.8223 −1.40469
\(726\) 0 0
\(727\) 44.0350 1.63317 0.816585 0.577226i \(-0.195865\pi\)
0.816585 + 0.577226i \(0.195865\pi\)
\(728\) 0 0
\(729\) 20.3288 0.752920
\(730\) 0 0
\(731\) −1.41033 −0.0521628
\(732\) 0 0
\(733\) 35.4721 1.31019 0.655096 0.755546i \(-0.272628\pi\)
0.655096 + 0.755546i \(0.272628\pi\)
\(734\) 0 0
\(735\) 16.0000 0.590169
\(736\) 0 0
\(737\) −7.58233 −0.279299
\(738\) 0 0
\(739\) 34.4956 1.26894 0.634470 0.772948i \(-0.281218\pi\)
0.634470 + 0.772948i \(0.281218\pi\)
\(740\) 0 0
\(741\) 2.62831 0.0965533
\(742\) 0 0
\(743\) −2.40298 −0.0881568 −0.0440784 0.999028i \(-0.514035\pi\)
−0.0440784 + 0.999028i \(0.514035\pi\)
\(744\) 0 0
\(745\) −58.0252 −2.12588
\(746\) 0 0
\(747\) 1.22846 0.0449471
\(748\) 0 0
\(749\) −28.0575 −1.02520
\(750\) 0 0
\(751\) 46.3221 1.69032 0.845159 0.534515i \(-0.179506\pi\)
0.845159 + 0.534515i \(0.179506\pi\)
\(752\) 0 0
\(753\) −6.93933 −0.252883
\(754\) 0 0
\(755\) −13.5640 −0.493646
\(756\) 0 0
\(757\) 28.5798 1.03875 0.519375 0.854546i \(-0.326165\pi\)
0.519375 + 0.854546i \(0.326165\pi\)
\(758\) 0 0
\(759\) −2.24361 −0.0814380
\(760\) 0 0
\(761\) 21.5401 0.780828 0.390414 0.920639i \(-0.372332\pi\)
0.390414 + 0.920639i \(0.372332\pi\)
\(762\) 0 0
\(763\) 27.4966 0.995445
\(764\) 0 0
\(765\) −1.10773 −0.0400501
\(766\) 0 0
\(767\) −12.1839 −0.439936
\(768\) 0 0
\(769\) 16.6472 0.600314 0.300157 0.953890i \(-0.402961\pi\)
0.300157 + 0.953890i \(0.402961\pi\)
\(770\) 0 0
\(771\) 16.1151 0.580370
\(772\) 0 0
\(773\) 38.8009 1.39557 0.697786 0.716306i \(-0.254169\pi\)
0.697786 + 0.716306i \(0.254169\pi\)
\(774\) 0 0
\(775\) 50.7925 1.82452
\(776\) 0 0
\(777\) 14.6858 0.526852
\(778\) 0 0
\(779\) −4.12494 −0.147791
\(780\) 0 0
\(781\) −5.51492 −0.197339
\(782\) 0 0
\(783\) 32.9013 1.17580
\(784\) 0 0
\(785\) −7.85677 −0.280420
\(786\) 0 0
\(787\) 23.8652 0.850702 0.425351 0.905028i \(-0.360151\pi\)
0.425351 + 0.905028i \(0.360151\pi\)
\(788\) 0 0
\(789\) −9.47352 −0.337266
\(790\) 0 0
\(791\) −45.1182 −1.60422
\(792\) 0 0
\(793\) −17.7367 −0.629847
\(794\) 0 0
\(795\) 18.4360 0.653856
\(796\) 0 0
\(797\) 2.01829 0.0714914 0.0357457 0.999361i \(-0.488619\pi\)
0.0357457 + 0.999361i \(0.488619\pi\)
\(798\) 0 0
\(799\) −2.16358 −0.0765419
\(800\) 0 0
\(801\) 14.3601 0.507390
\(802\) 0 0
\(803\) 7.56825 0.267078
\(804\) 0 0
\(805\) 44.7005 1.57549
\(806\) 0 0
\(807\) −3.65600 −0.128697
\(808\) 0 0
\(809\) −52.0533 −1.83010 −0.915049 0.403343i \(-0.867848\pi\)
−0.915049 + 0.403343i \(0.867848\pi\)
\(810\) 0 0
\(811\) −34.1579 −1.19945 −0.599723 0.800207i \(-0.704723\pi\)
−0.599723 + 0.800207i \(0.704723\pi\)
\(812\) 0 0
\(813\) −11.3486 −0.398012
\(814\) 0 0
\(815\) 46.0147 1.61182
\(816\) 0 0
\(817\) 7.17513 0.251026
\(818\) 0 0
\(819\) −12.9210 −0.451498
\(820\) 0 0
\(821\) −30.2787 −1.05673 −0.528366 0.849017i \(-0.677195\pi\)
−0.528366 + 0.849017i \(0.677195\pi\)
\(822\) 0 0
\(823\) 39.9284 1.39182 0.695908 0.718131i \(-0.255002\pi\)
0.695908 + 0.718131i \(0.255002\pi\)
\(824\) 0 0
\(825\) 3.46365 0.120589
\(826\) 0 0
\(827\) 12.0330 0.418428 0.209214 0.977870i \(-0.432910\pi\)
0.209214 + 0.977870i \(0.432910\pi\)
\(828\) 0 0
\(829\) −9.84208 −0.341829 −0.170915 0.985286i \(-0.554672\pi\)
−0.170915 + 0.985286i \(0.554672\pi\)
\(830\) 0 0
\(831\) −14.6416 −0.507910
\(832\) 0 0
\(833\) 0.820654 0.0284340
\(834\) 0 0
\(835\) 1.39625 0.0483192
\(836\) 0 0
\(837\) −44.1839 −1.52722
\(838\) 0 0
\(839\) −19.2713 −0.665319 −0.332660 0.943047i \(-0.607946\pi\)
−0.332660 + 0.943047i \(0.607946\pi\)
\(840\) 0 0
\(841\) 8.51492 0.293618
\(842\) 0 0
\(843\) 33.8517 1.16592
\(844\) 0 0
\(845\) −25.8855 −0.890490
\(846\) 0 0
\(847\) 35.9718 1.23601
\(848\) 0 0
\(849\) −29.0313 −0.996351
\(850\) 0 0
\(851\) 15.3288 0.525466
\(852\) 0 0
\(853\) −27.4721 −0.940626 −0.470313 0.882500i \(-0.655859\pi\)
−0.470313 + 0.882500i \(0.655859\pi\)
\(854\) 0 0
\(855\) 5.63565 0.192735
\(856\) 0 0
\(857\) 0.292731 0.00999950 0.00499975 0.999988i \(-0.498409\pi\)
0.00499975 + 0.999988i \(0.498409\pi\)
\(858\) 0 0
\(859\) 23.9614 0.817551 0.408776 0.912635i \(-0.365956\pi\)
0.408776 + 0.912635i \(0.365956\pi\)
\(860\) 0 0
\(861\) −15.8077 −0.538723
\(862\) 0 0
\(863\) 6.12494 0.208495 0.104248 0.994551i \(-0.466757\pi\)
0.104248 + 0.994551i \(0.466757\pi\)
\(864\) 0 0
\(865\) −12.0575 −0.409969
\(866\) 0 0
\(867\) −19.4439 −0.660350
\(868\) 0 0
\(869\) −5.14637 −0.174578
\(870\) 0 0
\(871\) 35.5296 1.20388
\(872\) 0 0
\(873\) 5.58715 0.189096
\(874\) 0 0
\(875\) −13.1323 −0.443952
\(876\) 0 0
\(877\) −22.3503 −0.754715 −0.377357 0.926068i \(-0.623167\pi\)
−0.377357 + 0.926068i \(0.623167\pi\)
\(878\) 0 0
\(879\) 19.2995 0.650955
\(880\) 0 0
\(881\) 5.70306 0.192141 0.0960705 0.995375i \(-0.469373\pi\)
0.0960705 + 0.995375i \(0.469373\pi\)
\(882\) 0 0
\(883\) 8.29694 0.279214 0.139607 0.990207i \(-0.455416\pi\)
0.139607 + 0.990207i \(0.455416\pi\)
\(884\) 0 0
\(885\) 20.3650 0.684561
\(886\) 0 0
\(887\) −25.4783 −0.855479 −0.427740 0.903902i \(-0.640690\pi\)
−0.427740 + 0.903902i \(0.640690\pi\)
\(888\) 0 0
\(889\) 5.14637 0.172604
\(890\) 0 0
\(891\) −0.538405 −0.0180373
\(892\) 0 0
\(893\) 11.0073 0.368347
\(894\) 0 0
\(895\) −21.7894 −0.728338
\(896\) 0 0
\(897\) 10.5132 0.351027
\(898\) 0 0
\(899\) −50.3797 −1.68026
\(900\) 0 0
\(901\) 0.945597 0.0315024
\(902\) 0 0
\(903\) 27.4966 0.915031
\(904\) 0 0
\(905\) 4.39312 0.146032
\(906\) 0 0
\(907\) 39.9964 1.32806 0.664029 0.747706i \(-0.268845\pi\)
0.664029 + 0.747706i \(0.268845\pi\)
\(908\) 0 0
\(909\) 18.0147 0.597510
\(910\) 0 0
\(911\) 19.3864 0.642300 0.321150 0.947028i \(-0.395931\pi\)
0.321150 + 0.947028i \(0.395931\pi\)
\(912\) 0 0
\(913\) −0.356541 −0.0117998
\(914\) 0 0
\(915\) 29.6461 0.980071
\(916\) 0 0
\(917\) 11.9284 0.393910
\(918\) 0 0
\(919\) −38.1213 −1.25751 −0.628754 0.777605i \(-0.716435\pi\)
−0.628754 + 0.777605i \(0.716435\pi\)
\(920\) 0 0
\(921\) 20.3931 0.671976
\(922\) 0 0
\(923\) 25.8421 0.850602
\(924\) 0 0
\(925\) −23.6644 −0.778081
\(926\) 0 0
\(927\) 20.3272 0.667631
\(928\) 0 0
\(929\) −16.8500 −0.552832 −0.276416 0.961038i \(-0.589147\pi\)
−0.276416 + 0.961038i \(0.589147\pi\)
\(930\) 0 0
\(931\) −4.17513 −0.136835
\(932\) 0 0
\(933\) −23.2959 −0.762672
\(934\) 0 0
\(935\) 0.321500 0.0105142
\(936\) 0 0
\(937\) −39.4973 −1.29032 −0.645159 0.764048i \(-0.723209\pi\)
−0.645159 + 0.764048i \(0.723209\pi\)
\(938\) 0 0
\(939\) 40.0000 1.30535
\(940\) 0 0
\(941\) 55.9718 1.82463 0.912315 0.409489i \(-0.134293\pi\)
0.912315 + 0.409489i \(0.134293\pi\)
\(942\) 0 0
\(943\) −16.4998 −0.537306
\(944\) 0 0
\(945\) 60.0294 1.95276
\(946\) 0 0
\(947\) 13.5725 0.441046 0.220523 0.975382i \(-0.429224\pi\)
0.220523 + 0.975382i \(0.429224\pi\)
\(948\) 0 0
\(949\) −35.4637 −1.15120
\(950\) 0 0
\(951\) −14.5917 −0.473169
\(952\) 0 0
\(953\) 26.0246 0.843018 0.421509 0.906824i \(-0.361501\pi\)
0.421509 + 0.906824i \(0.361501\pi\)
\(954\) 0 0
\(955\) −30.8971 −0.999807
\(956\) 0 0
\(957\) −3.43550 −0.111054
\(958\) 0 0
\(959\) −33.9431 −1.09608
\(960\) 0 0
\(961\) 36.6560 1.18245
\(962\) 0 0
\(963\) −14.1495 −0.455961
\(964\) 0 0
\(965\) 29.5970 0.952762
\(966\) 0 0
\(967\) 13.0277 0.418942 0.209471 0.977815i \(-0.432826\pi\)
0.209471 + 0.977815i \(0.432826\pi\)
\(968\) 0 0
\(969\) −0.225327 −0.00723854
\(970\) 0 0
\(971\) −60.6148 −1.94522 −0.972612 0.232437i \(-0.925330\pi\)
−0.972612 + 0.232437i \(0.925330\pi\)
\(972\) 0 0
\(973\) 13.0502 0.418370
\(974\) 0 0
\(975\) −16.2302 −0.519781
\(976\) 0 0
\(977\) −62.1396 −1.98802 −0.994012 0.109275i \(-0.965147\pi\)
−0.994012 + 0.109275i \(0.965147\pi\)
\(978\) 0 0
\(979\) −4.16779 −0.133203
\(980\) 0 0
\(981\) 13.8666 0.442728
\(982\) 0 0
\(983\) −6.37842 −0.203440 −0.101720 0.994813i \(-0.532435\pi\)
−0.101720 + 0.994813i \(0.532435\pi\)
\(984\) 0 0
\(985\) 91.3582 2.91092
\(986\) 0 0
\(987\) 42.1825 1.34268
\(988\) 0 0
\(989\) 28.7005 0.912624
\(990\) 0 0
\(991\) 14.1249 0.448694 0.224347 0.974509i \(-0.427975\pi\)
0.224347 + 0.974509i \(0.427975\pi\)
\(992\) 0 0
\(993\) −21.1625 −0.671572
\(994\) 0 0
\(995\) −61.3906 −1.94621
\(996\) 0 0
\(997\) −13.1568 −0.416682 −0.208341 0.978056i \(-0.566806\pi\)
−0.208341 + 0.978056i \(0.566806\pi\)
\(998\) 0 0
\(999\) 20.5855 0.651295
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 4864.2.a.bh.1.2 3
4.3 odd 2 4864.2.a.bb.1.2 3
8.3 odd 2 4864.2.a.bg.1.2 3
8.5 even 2 4864.2.a.ba.1.2 3
16.3 odd 4 2432.2.c.e.1217.3 6
16.5 even 4 2432.2.c.h.1217.3 yes 6
16.11 odd 4 2432.2.c.e.1217.4 yes 6
16.13 even 4 2432.2.c.h.1217.4 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2432.2.c.e.1217.3 6 16.3 odd 4
2432.2.c.e.1217.4 yes 6 16.11 odd 4
2432.2.c.h.1217.3 yes 6 16.5 even 4
2432.2.c.h.1217.4 yes 6 16.13 even 4
4864.2.a.ba.1.2 3 8.5 even 2
4864.2.a.bb.1.2 3 4.3 odd 2
4864.2.a.bg.1.2 3 8.3 odd 2
4864.2.a.bh.1.2 3 1.1 even 1 trivial