Properties

Label 4864.2.a.bl.1.2
Level $4864$
Weight $2$
Character 4864.1
Self dual yes
Analytic conductor $38.839$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more about

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: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{11})\)
Defining polynomial: \(x^{4} - 7 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1216)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.792287\) of defining polynomial
Character \(\chi\) \(=\) 4864.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.37228 q^{3} +0.792287 q^{5} -3.31662 q^{7} +2.62772 q^{9} +O(q^{10})\) \(q-2.37228 q^{3} +0.792287 q^{5} -3.31662 q^{7} +2.62772 q^{9} -3.37228 q^{11} -4.10891 q^{13} -1.87953 q^{15} +5.00000 q^{17} -1.00000 q^{19} +7.86797 q^{21} -2.52434 q^{23} -4.37228 q^{25} +0.883156 q^{27} -5.98844 q^{29} -3.46410 q^{31} +8.00000 q^{33} -2.62772 q^{35} +3.16915 q^{37} +9.74749 q^{39} -9.37228 q^{43} +2.08191 q^{45} -9.30506 q^{47} +4.00000 q^{49} -11.8614 q^{51} -9.45254 q^{53} -2.67181 q^{55} +2.37228 q^{57} -4.37228 q^{59} -4.55134 q^{61} -8.71516 q^{63} -3.25544 q^{65} -8.37228 q^{67} +5.98844 q^{69} +6.63325 q^{71} +14.4891 q^{73} +10.3723 q^{75} +11.1846 q^{77} +13.5615 q^{79} -9.97825 q^{81} +8.00000 q^{83} +3.96143 q^{85} +14.2063 q^{87} -7.48913 q^{89} +13.6277 q^{91} +8.21782 q^{93} -0.792287 q^{95} +8.74456 q^{97} -8.86141 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{3} + 22q^{9} + O(q^{10}) \) \( 4q + 2q^{3} + 22q^{9} - 2q^{11} + 20q^{17} - 4q^{19} - 6q^{25} + 38q^{27} + 32q^{33} - 22q^{35} - 26q^{43} + 16q^{49} + 10q^{51} - 2q^{57} - 6q^{59} - 36q^{65} - 22q^{67} + 12q^{73} + 30q^{75} + 52q^{81} + 32q^{83} + 16q^{89} + 66q^{91} + 12q^{97} + 22q^{99} + O(q^{100}) \)

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
\(3\) −2.37228 −1.36964 −0.684819 0.728714i \(-0.740119\pi\)
−0.684819 + 0.728714i \(0.740119\pi\)
\(4\) 0 0
\(5\) 0.792287 0.354322 0.177161 0.984182i \(-0.443309\pi\)
0.177161 + 0.984182i \(0.443309\pi\)
\(6\) 0 0
\(7\) −3.31662 −1.25357 −0.626783 0.779194i \(-0.715629\pi\)
−0.626783 + 0.779194i \(0.715629\pi\)
\(8\) 0 0
\(9\) 2.62772 0.875906
\(10\) 0 0
\(11\) −3.37228 −1.01678 −0.508391 0.861127i \(-0.669759\pi\)
−0.508391 + 0.861127i \(0.669759\pi\)
\(12\) 0 0
\(13\) −4.10891 −1.13961 −0.569804 0.821781i \(-0.692981\pi\)
−0.569804 + 0.821781i \(0.692981\pi\)
\(14\) 0 0
\(15\) −1.87953 −0.485292
\(16\) 0 0
\(17\) 5.00000 1.21268 0.606339 0.795206i \(-0.292637\pi\)
0.606339 + 0.795206i \(0.292637\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 7.86797 1.71693
\(22\) 0 0
\(23\) −2.52434 −0.526361 −0.263180 0.964747i \(-0.584771\pi\)
−0.263180 + 0.964747i \(0.584771\pi\)
\(24\) 0 0
\(25\) −4.37228 −0.874456
\(26\) 0 0
\(27\) 0.883156 0.169963
\(28\) 0 0
\(29\) −5.98844 −1.11203 −0.556013 0.831174i \(-0.687669\pi\)
−0.556013 + 0.831174i \(0.687669\pi\)
\(30\) 0 0
\(31\) −3.46410 −0.622171 −0.311086 0.950382i \(-0.600693\pi\)
−0.311086 + 0.950382i \(0.600693\pi\)
\(32\) 0 0
\(33\) 8.00000 1.39262
\(34\) 0 0
\(35\) −2.62772 −0.444166
\(36\) 0 0
\(37\) 3.16915 0.521005 0.260502 0.965473i \(-0.416112\pi\)
0.260502 + 0.965473i \(0.416112\pi\)
\(38\) 0 0
\(39\) 9.74749 1.56085
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −9.37228 −1.42926 −0.714630 0.699503i \(-0.753405\pi\)
−0.714630 + 0.699503i \(0.753405\pi\)
\(44\) 0 0
\(45\) 2.08191 0.310352
\(46\) 0 0
\(47\) −9.30506 −1.35728 −0.678642 0.734470i \(-0.737431\pi\)
−0.678642 + 0.734470i \(0.737431\pi\)
\(48\) 0 0
\(49\) 4.00000 0.571429
\(50\) 0 0
\(51\) −11.8614 −1.66093
\(52\) 0 0
\(53\) −9.45254 −1.29841 −0.649203 0.760615i \(-0.724898\pi\)
−0.649203 + 0.760615i \(0.724898\pi\)
\(54\) 0 0
\(55\) −2.67181 −0.360267
\(56\) 0 0
\(57\) 2.37228 0.314216
\(58\) 0 0
\(59\) −4.37228 −0.569223 −0.284611 0.958643i \(-0.591865\pi\)
−0.284611 + 0.958643i \(0.591865\pi\)
\(60\) 0 0
\(61\) −4.55134 −0.582740 −0.291370 0.956610i \(-0.594111\pi\)
−0.291370 + 0.956610i \(0.594111\pi\)
\(62\) 0 0
\(63\) −8.71516 −1.09801
\(64\) 0 0
\(65\) −3.25544 −0.403787
\(66\) 0 0
\(67\) −8.37228 −1.02284 −0.511418 0.859332i \(-0.670880\pi\)
−0.511418 + 0.859332i \(0.670880\pi\)
\(68\) 0 0
\(69\) 5.98844 0.720923
\(70\) 0 0
\(71\) 6.63325 0.787222 0.393611 0.919277i \(-0.371226\pi\)
0.393611 + 0.919277i \(0.371226\pi\)
\(72\) 0 0
\(73\) 14.4891 1.69582 0.847912 0.530137i \(-0.177860\pi\)
0.847912 + 0.530137i \(0.177860\pi\)
\(74\) 0 0
\(75\) 10.3723 1.19769
\(76\) 0 0
\(77\) 11.1846 1.27460
\(78\) 0 0
\(79\) 13.5615 1.52578 0.762891 0.646527i \(-0.223779\pi\)
0.762891 + 0.646527i \(0.223779\pi\)
\(80\) 0 0
\(81\) −9.97825 −1.10869
\(82\) 0 0
\(83\) 8.00000 0.878114 0.439057 0.898459i \(-0.355313\pi\)
0.439057 + 0.898459i \(0.355313\pi\)
\(84\) 0 0
\(85\) 3.96143 0.429678
\(86\) 0 0
\(87\) 14.2063 1.52307
\(88\) 0 0
\(89\) −7.48913 −0.793846 −0.396923 0.917852i \(-0.629922\pi\)
−0.396923 + 0.917852i \(0.629922\pi\)
\(90\) 0 0
\(91\) 13.6277 1.42857
\(92\) 0 0
\(93\) 8.21782 0.852149
\(94\) 0 0
\(95\) −0.792287 −0.0812869
\(96\) 0 0
\(97\) 8.74456 0.887876 0.443938 0.896058i \(-0.353581\pi\)
0.443938 + 0.896058i \(0.353581\pi\)
\(98\) 0 0
\(99\) −8.86141 −0.890605
\(100\) 0 0
\(101\) 13.8564 1.37876 0.689382 0.724398i \(-0.257882\pi\)
0.689382 + 0.724398i \(0.257882\pi\)
\(102\) 0 0
\(103\) 5.34363 0.526523 0.263262 0.964724i \(-0.415202\pi\)
0.263262 + 0.964724i \(0.415202\pi\)
\(104\) 0 0
\(105\) 6.23369 0.608346
\(106\) 0 0
\(107\) −7.62772 −0.737399 −0.368700 0.929549i \(-0.620197\pi\)
−0.368700 + 0.929549i \(0.620197\pi\)
\(108\) 0 0
\(109\) −12.6217 −1.20894 −0.604469 0.796628i \(-0.706615\pi\)
−0.604469 + 0.796628i \(0.706615\pi\)
\(110\) 0 0
\(111\) −7.51811 −0.713587
\(112\) 0 0
\(113\) −3.25544 −0.306246 −0.153123 0.988207i \(-0.548933\pi\)
−0.153123 + 0.988207i \(0.548933\pi\)
\(114\) 0 0
\(115\) −2.00000 −0.186501
\(116\) 0 0
\(117\) −10.7971 −0.998189
\(118\) 0 0
\(119\) −16.5831 −1.52017
\(120\) 0 0
\(121\) 0.372281 0.0338438
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −7.42554 −0.664160
\(126\) 0 0
\(127\) −18.9051 −1.67755 −0.838777 0.544475i \(-0.816729\pi\)
−0.838777 + 0.544475i \(0.816729\pi\)
\(128\) 0 0
\(129\) 22.2337 1.95757
\(130\) 0 0
\(131\) −1.37228 −0.119897 −0.0599484 0.998201i \(-0.519094\pi\)
−0.0599484 + 0.998201i \(0.519094\pi\)
\(132\) 0 0
\(133\) 3.31662 0.287588
\(134\) 0 0
\(135\) 0.699713 0.0602217
\(136\) 0 0
\(137\) −5.00000 −0.427179 −0.213589 0.976924i \(-0.568515\pi\)
−0.213589 + 0.976924i \(0.568515\pi\)
\(138\) 0 0
\(139\) −9.37228 −0.794947 −0.397473 0.917614i \(-0.630113\pi\)
−0.397473 + 0.917614i \(0.630113\pi\)
\(140\) 0 0
\(141\) 22.0742 1.85899
\(142\) 0 0
\(143\) 13.8564 1.15873
\(144\) 0 0
\(145\) −4.74456 −0.394014
\(146\) 0 0
\(147\) −9.48913 −0.782650
\(148\) 0 0
\(149\) −21.2819 −1.74348 −0.871742 0.489965i \(-0.837010\pi\)
−0.871742 + 0.489965i \(0.837010\pi\)
\(150\) 0 0
\(151\) 22.3692 1.82038 0.910189 0.414193i \(-0.135936\pi\)
0.910189 + 0.414193i \(0.135936\pi\)
\(152\) 0 0
\(153\) 13.1386 1.06219
\(154\) 0 0
\(155\) −2.74456 −0.220449
\(156\) 0 0
\(157\) 1.28962 0.102923 0.0514615 0.998675i \(-0.483612\pi\)
0.0514615 + 0.998675i \(0.483612\pi\)
\(158\) 0 0
\(159\) 22.4241 1.77835
\(160\) 0 0
\(161\) 8.37228 0.659828
\(162\) 0 0
\(163\) 0.744563 0.0583186 0.0291593 0.999575i \(-0.490717\pi\)
0.0291593 + 0.999575i \(0.490717\pi\)
\(164\) 0 0
\(165\) 6.33830 0.493436
\(166\) 0 0
\(167\) −20.1947 −1.56271 −0.781356 0.624085i \(-0.785472\pi\)
−0.781356 + 0.624085i \(0.785472\pi\)
\(168\) 0 0
\(169\) 3.88316 0.298704
\(170\) 0 0
\(171\) −2.62772 −0.200947
\(172\) 0 0
\(173\) −4.75372 −0.361419 −0.180709 0.983537i \(-0.557839\pi\)
−0.180709 + 0.983537i \(0.557839\pi\)
\(174\) 0 0
\(175\) 14.5012 1.09619
\(176\) 0 0
\(177\) 10.3723 0.779628
\(178\) 0 0
\(179\) 22.9783 1.71748 0.858738 0.512416i \(-0.171249\pi\)
0.858738 + 0.512416i \(0.171249\pi\)
\(180\) 0 0
\(181\) 11.6819 0.868311 0.434155 0.900838i \(-0.357047\pi\)
0.434155 + 0.900838i \(0.357047\pi\)
\(182\) 0 0
\(183\) 10.7971 0.798142
\(184\) 0 0
\(185\) 2.51087 0.184603
\(186\) 0 0
\(187\) −16.8614 −1.23303
\(188\) 0 0
\(189\) −2.92910 −0.213060
\(190\) 0 0
\(191\) 13.7089 0.991943 0.495972 0.868339i \(-0.334812\pi\)
0.495972 + 0.868339i \(0.334812\pi\)
\(192\) 0 0
\(193\) 14.7446 1.06134 0.530668 0.847580i \(-0.321941\pi\)
0.530668 + 0.847580i \(0.321941\pi\)
\(194\) 0 0
\(195\) 7.72281 0.553042
\(196\) 0 0
\(197\) −20.1947 −1.43881 −0.719406 0.694589i \(-0.755586\pi\)
−0.719406 + 0.694589i \(0.755586\pi\)
\(198\) 0 0
\(199\) −6.78073 −0.480673 −0.240336 0.970690i \(-0.577258\pi\)
−0.240336 + 0.970690i \(0.577258\pi\)
\(200\) 0 0
\(201\) 19.8614 1.40092
\(202\) 0 0
\(203\) 19.8614 1.39400
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −6.63325 −0.461043
\(208\) 0 0
\(209\) 3.37228 0.233266
\(210\) 0 0
\(211\) 8.37228 0.576372 0.288186 0.957575i \(-0.406948\pi\)
0.288186 + 0.957575i \(0.406948\pi\)
\(212\) 0 0
\(213\) −15.7359 −1.07821
\(214\) 0 0
\(215\) −7.42554 −0.506417
\(216\) 0 0
\(217\) 11.4891 0.779933
\(218\) 0 0
\(219\) −34.3723 −2.32266
\(220\) 0 0
\(221\) −20.5446 −1.38198
\(222\) 0 0
\(223\) 6.92820 0.463947 0.231973 0.972722i \(-0.425482\pi\)
0.231973 + 0.972722i \(0.425482\pi\)
\(224\) 0 0
\(225\) −11.4891 −0.765942
\(226\) 0 0
\(227\) −3.11684 −0.206872 −0.103436 0.994636i \(-0.532984\pi\)
−0.103436 + 0.994636i \(0.532984\pi\)
\(228\) 0 0
\(229\) 3.96143 0.261779 0.130889 0.991397i \(-0.458217\pi\)
0.130889 + 0.991397i \(0.458217\pi\)
\(230\) 0 0
\(231\) −26.5330 −1.74574
\(232\) 0 0
\(233\) 28.1168 1.84200 0.920998 0.389568i \(-0.127376\pi\)
0.920998 + 0.389568i \(0.127376\pi\)
\(234\) 0 0
\(235\) −7.37228 −0.480915
\(236\) 0 0
\(237\) −32.1716 −2.08977
\(238\) 0 0
\(239\) 24.1012 1.55898 0.779490 0.626415i \(-0.215478\pi\)
0.779490 + 0.626415i \(0.215478\pi\)
\(240\) 0 0
\(241\) 8.00000 0.515325 0.257663 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) 0 0
\(243\) 21.0217 1.34855
\(244\) 0 0
\(245\) 3.16915 0.202469
\(246\) 0 0
\(247\) 4.10891 0.261444
\(248\) 0 0
\(249\) −18.9783 −1.20270
\(250\) 0 0
\(251\) 10.8614 0.685566 0.342783 0.939415i \(-0.388630\pi\)
0.342783 + 0.939415i \(0.388630\pi\)
\(252\) 0 0
\(253\) 8.51278 0.535194
\(254\) 0 0
\(255\) −9.39764 −0.588503
\(256\) 0 0
\(257\) 18.2337 1.13739 0.568693 0.822550i \(-0.307449\pi\)
0.568693 + 0.822550i \(0.307449\pi\)
\(258\) 0 0
\(259\) −10.5109 −0.653114
\(260\) 0 0
\(261\) −15.7359 −0.974030
\(262\) 0 0
\(263\) −24.1561 −1.48953 −0.744766 0.667326i \(-0.767439\pi\)
−0.744766 + 0.667326i \(0.767439\pi\)
\(264\) 0 0
\(265\) −7.48913 −0.460053
\(266\) 0 0
\(267\) 17.7663 1.08728
\(268\) 0 0
\(269\) 8.51278 0.519033 0.259517 0.965739i \(-0.416437\pi\)
0.259517 + 0.965739i \(0.416437\pi\)
\(270\) 0 0
\(271\) 7.57301 0.460028 0.230014 0.973187i \(-0.426123\pi\)
0.230014 + 0.973187i \(0.426123\pi\)
\(272\) 0 0
\(273\) −32.3288 −1.95663
\(274\) 0 0
\(275\) 14.7446 0.889131
\(276\) 0 0
\(277\) 13.0641 0.784947 0.392473 0.919763i \(-0.371619\pi\)
0.392473 + 0.919763i \(0.371619\pi\)
\(278\) 0 0
\(279\) −9.10268 −0.544963
\(280\) 0 0
\(281\) 28.2337 1.68428 0.842140 0.539258i \(-0.181295\pi\)
0.842140 + 0.539258i \(0.181295\pi\)
\(282\) 0 0
\(283\) 23.3723 1.38934 0.694669 0.719330i \(-0.255551\pi\)
0.694669 + 0.719330i \(0.255551\pi\)
\(284\) 0 0
\(285\) 1.87953 0.111334
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) −20.7446 −1.21607
\(292\) 0 0
\(293\) −28.0627 −1.63944 −0.819719 0.572765i \(-0.805871\pi\)
−0.819719 + 0.572765i \(0.805871\pi\)
\(294\) 0 0
\(295\) −3.46410 −0.201688
\(296\) 0 0
\(297\) −2.97825 −0.172816
\(298\) 0 0
\(299\) 10.3723 0.599845
\(300\) 0 0
\(301\) 31.0843 1.79167
\(302\) 0 0
\(303\) −32.8713 −1.88841
\(304\) 0 0
\(305\) −3.60597 −0.206477
\(306\) 0 0
\(307\) 26.7446 1.52639 0.763196 0.646167i \(-0.223629\pi\)
0.763196 + 0.646167i \(0.223629\pi\)
\(308\) 0 0
\(309\) −12.6766 −0.721146
\(310\) 0 0
\(311\) −23.2164 −1.31648 −0.658240 0.752808i \(-0.728699\pi\)
−0.658240 + 0.752808i \(0.728699\pi\)
\(312\) 0 0
\(313\) −21.1168 −1.19359 −0.596797 0.802392i \(-0.703560\pi\)
−0.596797 + 0.802392i \(0.703560\pi\)
\(314\) 0 0
\(315\) −6.90491 −0.389047
\(316\) 0 0
\(317\) 18.2603 1.02560 0.512800 0.858508i \(-0.328608\pi\)
0.512800 + 0.858508i \(0.328608\pi\)
\(318\) 0 0
\(319\) 20.1947 1.13069
\(320\) 0 0
\(321\) 18.0951 1.00997
\(322\) 0 0
\(323\) −5.00000 −0.278207
\(324\) 0 0
\(325\) 17.9653 0.996537
\(326\) 0 0
\(327\) 29.9422 1.65581
\(328\) 0 0
\(329\) 30.8614 1.70144
\(330\) 0 0
\(331\) 17.8614 0.981752 0.490876 0.871230i \(-0.336677\pi\)
0.490876 + 0.871230i \(0.336677\pi\)
\(332\) 0 0
\(333\) 8.32763 0.456351
\(334\) 0 0
\(335\) −6.63325 −0.362413
\(336\) 0 0
\(337\) −24.7446 −1.34792 −0.673961 0.738767i \(-0.735408\pi\)
−0.673961 + 0.738767i \(0.735408\pi\)
\(338\) 0 0
\(339\) 7.72281 0.419446
\(340\) 0 0
\(341\) 11.6819 0.632612
\(342\) 0 0
\(343\) 9.94987 0.537243
\(344\) 0 0
\(345\) 4.74456 0.255439
\(346\) 0 0
\(347\) −19.8832 −1.06738 −0.533692 0.845679i \(-0.679196\pi\)
−0.533692 + 0.845679i \(0.679196\pi\)
\(348\) 0 0
\(349\) 23.1615 1.23981 0.619903 0.784679i \(-0.287172\pi\)
0.619903 + 0.784679i \(0.287172\pi\)
\(350\) 0 0
\(351\) −3.62881 −0.193692
\(352\) 0 0
\(353\) 21.1168 1.12394 0.561968 0.827159i \(-0.310044\pi\)
0.561968 + 0.827159i \(0.310044\pi\)
\(354\) 0 0
\(355\) 5.25544 0.278930
\(356\) 0 0
\(357\) 39.3398 2.08208
\(358\) 0 0
\(359\) 16.5831 0.875224 0.437612 0.899164i \(-0.355824\pi\)
0.437612 + 0.899164i \(0.355824\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −0.883156 −0.0463537
\(364\) 0 0
\(365\) 11.4795 0.600867
\(366\) 0 0
\(367\) −30.5870 −1.59663 −0.798314 0.602241i \(-0.794275\pi\)
−0.798314 + 0.602241i \(0.794275\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 31.3505 1.62764
\(372\) 0 0
\(373\) −24.8935 −1.28894 −0.644469 0.764631i \(-0.722921\pi\)
−0.644469 + 0.764631i \(0.722921\pi\)
\(374\) 0 0
\(375\) 17.6155 0.909659
\(376\) 0 0
\(377\) 24.6060 1.26727
\(378\) 0 0
\(379\) 15.3505 0.788504 0.394252 0.919002i \(-0.371004\pi\)
0.394252 + 0.919002i \(0.371004\pi\)
\(380\) 0 0
\(381\) 44.8482 2.29764
\(382\) 0 0
\(383\) −7.92287 −0.404840 −0.202420 0.979299i \(-0.564881\pi\)
−0.202420 + 0.979299i \(0.564881\pi\)
\(384\) 0 0
\(385\) 8.86141 0.451619
\(386\) 0 0
\(387\) −24.6277 −1.25190
\(388\) 0 0
\(389\) 12.1793 0.617513 0.308756 0.951141i \(-0.400087\pi\)
0.308756 + 0.951141i \(0.400087\pi\)
\(390\) 0 0
\(391\) −12.6217 −0.638306
\(392\) 0 0
\(393\) 3.25544 0.164215
\(394\) 0 0
\(395\) 10.7446 0.540618
\(396\) 0 0
\(397\) −9.30506 −0.467008 −0.233504 0.972356i \(-0.575019\pi\)
−0.233504 + 0.972356i \(0.575019\pi\)
\(398\) 0 0
\(399\) −7.86797 −0.393891
\(400\) 0 0
\(401\) −21.7228 −1.08479 −0.542393 0.840125i \(-0.682482\pi\)
−0.542393 + 0.840125i \(0.682482\pi\)
\(402\) 0 0
\(403\) 14.2337 0.709030
\(404\) 0 0
\(405\) −7.90564 −0.392834
\(406\) 0 0
\(407\) −10.6873 −0.529748
\(408\) 0 0
\(409\) 6.00000 0.296681 0.148340 0.988936i \(-0.452607\pi\)
0.148340 + 0.988936i \(0.452607\pi\)
\(410\) 0 0
\(411\) 11.8614 0.585080
\(412\) 0 0
\(413\) 14.5012 0.713558
\(414\) 0 0
\(415\) 6.33830 0.311135
\(416\) 0 0
\(417\) 22.2337 1.08879
\(418\) 0 0
\(419\) 34.2337 1.67243 0.836213 0.548405i \(-0.184765\pi\)
0.836213 + 0.548405i \(0.184765\pi\)
\(420\) 0 0
\(421\) −35.9855 −1.75383 −0.876914 0.480647i \(-0.840402\pi\)
−0.876914 + 0.480647i \(0.840402\pi\)
\(422\) 0 0
\(423\) −24.4511 −1.18885
\(424\) 0 0
\(425\) −21.8614 −1.06043
\(426\) 0 0
\(427\) 15.0951 0.730503
\(428\) 0 0
\(429\) −32.8713 −1.58704
\(430\) 0 0
\(431\) −11.3870 −0.548491 −0.274246 0.961660i \(-0.588428\pi\)
−0.274246 + 0.961660i \(0.588428\pi\)
\(432\) 0 0
\(433\) 21.4891 1.03270 0.516351 0.856377i \(-0.327290\pi\)
0.516351 + 0.856377i \(0.327290\pi\)
\(434\) 0 0
\(435\) 11.2554 0.539657
\(436\) 0 0
\(437\) 2.52434 0.120755
\(438\) 0 0
\(439\) −6.92820 −0.330665 −0.165333 0.986238i \(-0.552870\pi\)
−0.165333 + 0.986238i \(0.552870\pi\)
\(440\) 0 0
\(441\) 10.5109 0.500518
\(442\) 0 0
\(443\) 10.3505 0.491769 0.245884 0.969299i \(-0.420922\pi\)
0.245884 + 0.969299i \(0.420922\pi\)
\(444\) 0 0
\(445\) −5.93354 −0.281277
\(446\) 0 0
\(447\) 50.4868 2.38794
\(448\) 0 0
\(449\) 14.5109 0.684811 0.342405 0.939552i \(-0.388758\pi\)
0.342405 + 0.939552i \(0.388758\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −53.0660 −2.49326
\(454\) 0 0
\(455\) 10.7971 0.506174
\(456\) 0 0
\(457\) −31.0000 −1.45012 −0.725059 0.688686i \(-0.758188\pi\)
−0.725059 + 0.688686i \(0.758188\pi\)
\(458\) 0 0
\(459\) 4.41578 0.206111
\(460\) 0 0
\(461\) 11.1846 0.520918 0.260459 0.965485i \(-0.416126\pi\)
0.260459 + 0.965485i \(0.416126\pi\)
\(462\) 0 0
\(463\) −8.31040 −0.386217 −0.193108 0.981177i \(-0.561857\pi\)
−0.193108 + 0.981177i \(0.561857\pi\)
\(464\) 0 0
\(465\) 6.51087 0.301935
\(466\) 0 0
\(467\) −28.1168 −1.30109 −0.650546 0.759467i \(-0.725460\pi\)
−0.650546 + 0.759467i \(0.725460\pi\)
\(468\) 0 0
\(469\) 27.7677 1.28219
\(470\) 0 0
\(471\) −3.05934 −0.140967
\(472\) 0 0
\(473\) 31.6060 1.45324
\(474\) 0 0
\(475\) 4.37228 0.200614
\(476\) 0 0
\(477\) −24.8386 −1.13728
\(478\) 0 0
\(479\) −14.1514 −0.646592 −0.323296 0.946298i \(-0.604791\pi\)
−0.323296 + 0.946298i \(0.604791\pi\)
\(480\) 0 0
\(481\) −13.0217 −0.593741
\(482\) 0 0
\(483\) −19.8614 −0.903725
\(484\) 0 0
\(485\) 6.92820 0.314594
\(486\) 0 0
\(487\) 29.2974 1.32759 0.663796 0.747914i \(-0.268944\pi\)
0.663796 + 0.747914i \(0.268944\pi\)
\(488\) 0 0
\(489\) −1.76631 −0.0798754
\(490\) 0 0
\(491\) 14.9783 0.675959 0.337979 0.941153i \(-0.390257\pi\)
0.337979 + 0.941153i \(0.390257\pi\)
\(492\) 0 0
\(493\) −29.9422 −1.34853
\(494\) 0 0
\(495\) −7.02078 −0.315560
\(496\) 0 0
\(497\) −22.0000 −0.986835
\(498\) 0 0
\(499\) −1.37228 −0.0614317 −0.0307159 0.999528i \(-0.509779\pi\)
−0.0307159 + 0.999528i \(0.509779\pi\)
\(500\) 0 0
\(501\) 47.9075 2.14035
\(502\) 0 0
\(503\) −21.4294 −0.955491 −0.477745 0.878498i \(-0.658546\pi\)
−0.477745 + 0.878498i \(0.658546\pi\)
\(504\) 0 0
\(505\) 10.9783 0.488526
\(506\) 0 0
\(507\) −9.21194 −0.409117
\(508\) 0 0
\(509\) −24.5437 −1.08788 −0.543939 0.839124i \(-0.683068\pi\)
−0.543939 + 0.839124i \(0.683068\pi\)
\(510\) 0 0
\(511\) −48.0550 −2.12583
\(512\) 0 0
\(513\) −0.883156 −0.0389923
\(514\) 0 0
\(515\) 4.23369 0.186559
\(516\) 0 0
\(517\) 31.3793 1.38006
\(518\) 0 0
\(519\) 11.2772 0.495013
\(520\) 0 0
\(521\) 40.2337 1.76267 0.881335 0.472492i \(-0.156645\pi\)
0.881335 + 0.472492i \(0.156645\pi\)
\(522\) 0 0
\(523\) 19.6277 0.858260 0.429130 0.903243i \(-0.358820\pi\)
0.429130 + 0.903243i \(0.358820\pi\)
\(524\) 0 0
\(525\) −34.4010 −1.50138
\(526\) 0 0
\(527\) −17.3205 −0.754493
\(528\) 0 0
\(529\) −16.6277 −0.722944
\(530\) 0 0
\(531\) −11.4891 −0.498586
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −6.04334 −0.261276
\(536\) 0 0
\(537\) −54.5109 −2.35232
\(538\) 0 0
\(539\) −13.4891 −0.581018
\(540\) 0 0
\(541\) −33.5538 −1.44259 −0.721295 0.692628i \(-0.756453\pi\)
−0.721295 + 0.692628i \(0.756453\pi\)
\(542\) 0 0
\(543\) −27.7128 −1.18927
\(544\) 0 0
\(545\) −10.0000 −0.428353
\(546\) 0 0
\(547\) 25.4891 1.08984 0.544918 0.838489i \(-0.316561\pi\)
0.544918 + 0.838489i \(0.316561\pi\)
\(548\) 0 0
\(549\) −11.9596 −0.510425
\(550\) 0 0
\(551\) 5.98844 0.255116
\(552\) 0 0
\(553\) −44.9783 −1.91267
\(554\) 0 0
\(555\) −5.95650 −0.252839
\(556\) 0 0
\(557\) 28.5051 1.20780 0.603900 0.797060i \(-0.293613\pi\)
0.603900 + 0.797060i \(0.293613\pi\)
\(558\) 0 0
\(559\) 38.5099 1.62879
\(560\) 0 0
\(561\) 40.0000 1.68880
\(562\) 0 0
\(563\) −9.48913 −0.399919 −0.199959 0.979804i \(-0.564081\pi\)
−0.199959 + 0.979804i \(0.564081\pi\)
\(564\) 0 0
\(565\) −2.57924 −0.108509
\(566\) 0 0
\(567\) 33.0941 1.38982
\(568\) 0 0
\(569\) −18.9783 −0.795610 −0.397805 0.917470i \(-0.630228\pi\)
−0.397805 + 0.917470i \(0.630228\pi\)
\(570\) 0 0
\(571\) 2.97825 0.124636 0.0623180 0.998056i \(-0.480151\pi\)
0.0623180 + 0.998056i \(0.480151\pi\)
\(572\) 0 0
\(573\) −32.5214 −1.35860
\(574\) 0 0
\(575\) 11.0371 0.460280
\(576\) 0 0
\(577\) −35.4674 −1.47653 −0.738263 0.674513i \(-0.764354\pi\)
−0.738263 + 0.674513i \(0.764354\pi\)
\(578\) 0 0
\(579\) −34.9783 −1.45365
\(580\) 0 0
\(581\) −26.5330 −1.10077
\(582\) 0 0
\(583\) 31.8766 1.32020
\(584\) 0 0
\(585\) −8.55437 −0.353680
\(586\) 0 0
\(587\) 3.13859 0.129544 0.0647718 0.997900i \(-0.479368\pi\)
0.0647718 + 0.997900i \(0.479368\pi\)
\(588\) 0 0
\(589\) 3.46410 0.142736
\(590\) 0 0
\(591\) 47.9075 1.97065
\(592\) 0 0
\(593\) −40.9783 −1.68278 −0.841388 0.540432i \(-0.818261\pi\)
−0.841388 + 0.540432i \(0.818261\pi\)
\(594\) 0 0
\(595\) −13.1386 −0.538630
\(596\) 0 0
\(597\) 16.0858 0.658348
\(598\) 0 0
\(599\) −33.0564 −1.35065 −0.675325 0.737520i \(-0.735997\pi\)
−0.675325 + 0.737520i \(0.735997\pi\)
\(600\) 0 0
\(601\) −46.4674 −1.89544 −0.947722 0.319097i \(-0.896620\pi\)
−0.947722 + 0.319097i \(0.896620\pi\)
\(602\) 0 0
\(603\) −22.0000 −0.895909
\(604\) 0 0
\(605\) 0.294954 0.0119916
\(606\) 0 0
\(607\) −7.62792 −0.309608 −0.154804 0.987945i \(-0.549475\pi\)
−0.154804 + 0.987945i \(0.549475\pi\)
\(608\) 0 0
\(609\) −47.1168 −1.90927
\(610\) 0 0
\(611\) 38.2337 1.54677
\(612\) 0 0
\(613\) 35.1383 1.41922 0.709612 0.704592i \(-0.248870\pi\)
0.709612 + 0.704592i \(0.248870\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −33.8397 −1.36233 −0.681167 0.732128i \(-0.738527\pi\)
−0.681167 + 0.732128i \(0.738527\pi\)
\(618\) 0 0
\(619\) −18.2337 −0.732874 −0.366437 0.930443i \(-0.619422\pi\)
−0.366437 + 0.930443i \(0.619422\pi\)
\(620\) 0 0
\(621\) −2.22938 −0.0894621
\(622\) 0 0
\(623\) 24.8386 0.995138
\(624\) 0 0
\(625\) 15.9783 0.639130
\(626\) 0 0
\(627\) −8.00000 −0.319489
\(628\) 0 0
\(629\) 15.8457 0.631811
\(630\) 0 0
\(631\) 11.8843 0.473107 0.236553 0.971619i \(-0.423982\pi\)
0.236553 + 0.971619i \(0.423982\pi\)
\(632\) 0 0
\(633\) −19.8614 −0.789420
\(634\) 0 0
\(635\) −14.9783 −0.594394
\(636\) 0 0
\(637\) −16.4356 −0.651204
\(638\) 0 0
\(639\) 17.4303 0.689533
\(640\) 0 0
\(641\) 34.9783 1.38156 0.690779 0.723066i \(-0.257268\pi\)
0.690779 + 0.723066i \(0.257268\pi\)
\(642\) 0 0
\(643\) −30.6277 −1.20784 −0.603920 0.797045i \(-0.706395\pi\)
−0.603920 + 0.797045i \(0.706395\pi\)
\(644\) 0 0
\(645\) 17.6155 0.693608
\(646\) 0 0
\(647\) 16.5831 0.651950 0.325975 0.945378i \(-0.394307\pi\)
0.325975 + 0.945378i \(0.394307\pi\)
\(648\) 0 0
\(649\) 14.7446 0.578775
\(650\) 0 0
\(651\) −27.2554 −1.06822
\(652\) 0 0
\(653\) 47.5200 1.85960 0.929800 0.368064i \(-0.119979\pi\)
0.929800 + 0.368064i \(0.119979\pi\)
\(654\) 0 0
\(655\) −1.08724 −0.0424820
\(656\) 0 0
\(657\) 38.0733 1.48538
\(658\) 0 0
\(659\) 18.1386 0.706579 0.353290 0.935514i \(-0.385063\pi\)
0.353290 + 0.935514i \(0.385063\pi\)
\(660\) 0 0
\(661\) 19.2549 0.748930 0.374465 0.927241i \(-0.377826\pi\)
0.374465 + 0.927241i \(0.377826\pi\)
\(662\) 0 0
\(663\) 48.7375 1.89281
\(664\) 0 0
\(665\) 2.62772 0.101899
\(666\) 0 0
\(667\) 15.1168 0.585327
\(668\) 0 0
\(669\) −16.4356 −0.635439
\(670\) 0 0
\(671\) 15.3484 0.592519
\(672\) 0 0
\(673\) 5.25544 0.202582 0.101291 0.994857i \(-0.467703\pi\)
0.101291 + 0.994857i \(0.467703\pi\)
\(674\) 0 0
\(675\) −3.86141 −0.148626
\(676\) 0 0
\(677\) −4.10891 −0.157918 −0.0789592 0.996878i \(-0.525160\pi\)
−0.0789592 + 0.996878i \(0.525160\pi\)
\(678\) 0 0
\(679\) −29.0024 −1.11301
\(680\) 0 0
\(681\) 7.39403 0.283340
\(682\) 0 0
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 0 0
\(685\) −3.96143 −0.151359
\(686\) 0 0
\(687\) −9.39764 −0.358542
\(688\) 0 0
\(689\) 38.8397 1.47967
\(690\) 0 0
\(691\) 10.8614 0.413187 0.206594 0.978427i \(-0.433762\pi\)
0.206594 + 0.978427i \(0.433762\pi\)
\(692\) 0 0
\(693\) 29.3900 1.11643
\(694\) 0 0
\(695\) −7.42554 −0.281667
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −66.7011 −2.52287
\(700\) 0 0
\(701\) 27.7128 1.04670 0.523349 0.852118i \(-0.324682\pi\)
0.523349 + 0.852118i \(0.324682\pi\)
\(702\) 0 0
\(703\) −3.16915 −0.119527
\(704\) 0 0
\(705\) 17.4891 0.658679
\(706\) 0 0
\(707\) −45.9565 −1.72837
\(708\) 0 0
\(709\) −0.699713 −0.0262783 −0.0131391 0.999914i \(-0.504182\pi\)
−0.0131391 + 0.999914i \(0.504182\pi\)
\(710\) 0 0
\(711\) 35.6357 1.33644
\(712\) 0 0
\(713\) 8.74456 0.327486
\(714\) 0 0
\(715\) 10.9783 0.410563
\(716\) 0 0
\(717\) −57.1749 −2.13524
\(718\) 0 0
\(719\) −5.89587 −0.219879 −0.109939 0.993938i \(-0.535066\pi\)
−0.109939 + 0.993938i \(0.535066\pi\)
\(720\) 0 0
\(721\) −17.7228 −0.660032
\(722\) 0 0
\(723\) −18.9783 −0.705809
\(724\) 0 0
\(725\) 26.1831 0.972417
\(726\) 0 0
\(727\) −20.6371 −0.765389 −0.382694 0.923875i \(-0.625004\pi\)
−0.382694 + 0.923875i \(0.625004\pi\)
\(728\) 0 0
\(729\) −19.9348 −0.738324
\(730\) 0 0
\(731\) −46.8614 −1.73323
\(732\) 0 0
\(733\) 38.5099 1.42239 0.711197 0.702992i \(-0.248153\pi\)
0.711197 + 0.702992i \(0.248153\pi\)
\(734\) 0 0
\(735\) −7.51811 −0.277310
\(736\) 0 0
\(737\) 28.2337 1.04000
\(738\) 0 0
\(739\) −27.6060 −1.01550 −0.507751 0.861504i \(-0.669523\pi\)
−0.507751 + 0.861504i \(0.669523\pi\)
\(740\) 0 0
\(741\) −9.74749 −0.358083
\(742\) 0 0
\(743\) 6.63325 0.243350 0.121675 0.992570i \(-0.461173\pi\)
0.121675 + 0.992570i \(0.461173\pi\)
\(744\) 0 0
\(745\) −16.8614 −0.617754
\(746\) 0 0
\(747\) 21.0217 0.769146
\(748\) 0 0
\(749\) 25.2983 0.924379
\(750\) 0 0
\(751\) −30.2921 −1.10537 −0.552686 0.833389i \(-0.686397\pi\)
−0.552686 + 0.833389i \(0.686397\pi\)
\(752\) 0 0
\(753\) −25.7663 −0.938977
\(754\) 0 0
\(755\) 17.7228 0.644999
\(756\) 0 0
\(757\) −48.6998 −1.77002 −0.885012 0.465568i \(-0.845850\pi\)
−0.885012 + 0.465568i \(0.845850\pi\)
\(758\) 0 0
\(759\) −20.1947 −0.733021
\(760\) 0 0
\(761\) 11.0000 0.398750 0.199375 0.979923i \(-0.436109\pi\)
0.199375 + 0.979923i \(0.436109\pi\)
\(762\) 0 0
\(763\) 41.8614 1.51548
\(764\) 0 0
\(765\) 10.4095 0.376358
\(766\) 0 0
\(767\) 17.9653 0.648690
\(768\) 0 0
\(769\) −31.4674 −1.13474 −0.567371 0.823462i \(-0.692040\pi\)
−0.567371 + 0.823462i \(0.692040\pi\)
\(770\) 0 0
\(771\) −43.2554 −1.55781
\(772\) 0 0
\(773\) −18.9600 −0.681943 −0.340972 0.940074i \(-0.610756\pi\)
−0.340972 + 0.940074i \(0.610756\pi\)
\(774\) 0 0
\(775\) 15.1460 0.544061
\(776\) 0 0
\(777\) 24.9348 0.894529
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −22.3692 −0.800432
\(782\) 0 0
\(783\) −5.28873 −0.189004
\(784\) 0 0
\(785\) 1.02175 0.0364678
\(786\) 0 0
\(787\) −20.6060 −0.734523 −0.367262 0.930118i \(-0.619705\pi\)
−0.367262 + 0.930118i \(0.619705\pi\)
\(788\) 0 0
\(789\) 57.3052 2.04012
\(790\) 0 0
\(791\) 10.7971 0.383899
\(792\) 0 0
\(793\) 18.7011 0.664094
\(794\) 0 0
\(795\) 17.7663 0.630106
\(796\) 0 0
\(797\) −11.3321 −0.401402 −0.200701 0.979652i \(-0.564322\pi\)
−0.200701 + 0.979652i \(0.564322\pi\)
\(798\) 0 0
\(799\) −46.5253 −1.64595
\(800\) 0 0
\(801\) −19.6793 −0.695334
\(802\) 0 0
\(803\) −48.8614 −1.72428
\(804\) 0 0
\(805\) 6.63325 0.233791
\(806\) 0 0
\(807\) −20.1947 −0.710887
\(808\) 0 0
\(809\) −7.97825 −0.280500 −0.140250 0.990116i \(-0.544791\pi\)
−0.140250 + 0.990116i \(0.544791\pi\)
\(810\) 0 0
\(811\) −0.138593 −0.00486667 −0.00243334 0.999997i \(-0.500775\pi\)
−0.00243334 + 0.999997i \(0.500775\pi\)
\(812\) 0 0
\(813\) −17.9653 −0.630071
\(814\) 0 0
\(815\) 0.589907 0.0206636
\(816\) 0 0
\(817\) 9.37228 0.327895
\(818\) 0 0
\(819\) 35.8098 1.25130
\(820\) 0 0
\(821\) −37.0179 −1.29193 −0.645966 0.763366i \(-0.723545\pi\)
−0.645966 + 0.763366i \(0.723545\pi\)
\(822\) 0 0
\(823\) 27.2704 0.950586 0.475293 0.879828i \(-0.342342\pi\)
0.475293 + 0.879828i \(0.342342\pi\)
\(824\) 0 0
\(825\) −34.9783 −1.21779
\(826\) 0 0
\(827\) 11.3505 0.394697 0.197348 0.980333i \(-0.436767\pi\)
0.197348 + 0.980333i \(0.436767\pi\)
\(828\) 0 0
\(829\) 33.1113 1.15000 0.575002 0.818152i \(-0.305001\pi\)
0.575002 + 0.818152i \(0.305001\pi\)
\(830\) 0 0
\(831\) −30.9918 −1.07509
\(832\) 0 0
\(833\) 20.0000 0.692959
\(834\) 0 0
\(835\) −16.0000 −0.553703
\(836\) 0 0
\(837\) −3.05934 −0.105746
\(838\) 0 0
\(839\) 16.7306 0.577604 0.288802 0.957389i \(-0.406743\pi\)
0.288802 + 0.957389i \(0.406743\pi\)
\(840\) 0 0
\(841\) 6.86141 0.236600
\(842\) 0 0
\(843\) −66.9783 −2.30685
\(844\) 0 0
\(845\) 3.07657 0.105837
\(846\) 0 0
\(847\) −1.23472 −0.0424254
\(848\) 0 0
\(849\) −55.4456 −1.90289
\(850\) 0 0
\(851\) −8.00000 −0.274236
\(852\) 0 0
\(853\) −12.6766 −0.434038 −0.217019 0.976167i \(-0.569633\pi\)
−0.217019 + 0.976167i \(0.569633\pi\)
\(854\) 0 0
\(855\) −2.08191 −0.0711997
\(856\) 0 0
\(857\) 12.2337 0.417895 0.208947 0.977927i \(-0.432996\pi\)
0.208947 + 0.977927i \(0.432996\pi\)
\(858\) 0 0
\(859\) −31.6060 −1.07838 −0.539191 0.842184i \(-0.681270\pi\)
−0.539191 + 0.842184i \(0.681270\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −49.1971 −1.67469 −0.837345 0.546675i \(-0.815893\pi\)
−0.837345 + 0.546675i \(0.815893\pi\)
\(864\) 0 0
\(865\) −3.76631 −0.128058
\(866\) 0 0
\(867\) −18.9783 −0.644535
\(868\) 0 0
\(869\) −45.7330 −1.55139
\(870\) 0 0
\(871\) 34.4010 1.16563
\(872\) 0 0
\(873\) 22.9783 0.777696
\(874\) 0 0
\(875\) 24.6277 0.832569
\(876\) 0 0
\(877\) −24.5986 −0.830635 −0.415317 0.909677i \(-0.636329\pi\)
−0.415317 + 0.909677i \(0.636329\pi\)
\(878\) 0 0
\(879\) 66.5725 2.24544
\(880\) 0 0
\(881\) 1.37228 0.0462333 0.0231167 0.999733i \(-0.492641\pi\)
0.0231167 + 0.999733i \(0.492641\pi\)
\(882\) 0 0
\(883\) 36.1168 1.21543 0.607714 0.794156i \(-0.292087\pi\)
0.607714 + 0.794156i \(0.292087\pi\)
\(884\) 0 0
\(885\) 8.21782 0.276239
\(886\) 0 0
\(887\) 45.0333 1.51207 0.756035 0.654531i \(-0.227134\pi\)
0.756035 + 0.654531i \(0.227134\pi\)
\(888\) 0 0
\(889\) 62.7011 2.10293
\(890\) 0 0
\(891\) 33.6495 1.12730
\(892\) 0 0
\(893\) 9.30506 0.311382
\(894\) 0 0
\(895\) 18.2054 0.608538
\(896\) 0 0
\(897\) −24.6060 −0.821569
\(898\) 0 0
\(899\) 20.7446 0.691870
\(900\) 0 0
\(901\) −47.2627 −1.57455
\(902\) 0 0
\(903\) −73.7408 −2.45394
\(904\) 0 0
\(905\) 9.25544 0.307661
\(906\) 0 0
\(907\) 18.1386 0.602282 0.301141 0.953580i \(-0.402632\pi\)
0.301141 + 0.953580i \(0.402632\pi\)
\(908\) 0 0
\(909\) 36.4107 1.20767
\(910\) 0 0
\(911\) −7.81306 −0.258858 −0.129429 0.991589i \(-0.541314\pi\)
−0.129429 + 0.991589i \(0.541314\pi\)
\(912\) 0 0
\(913\) −26.9783 −0.892850
\(914\) 0 0
\(915\) 8.55437 0.282799
\(916\) 0 0
\(917\) 4.55134 0.150299
\(918\) 0 0
\(919\) 27.7677 0.915972 0.457986 0.888959i \(-0.348571\pi\)
0.457986 + 0.888959i \(0.348571\pi\)
\(920\) 0 0
\(921\) −63.4456 −2.09060
\(922\) 0 0
\(923\) −27.2554 −0.897124
\(924\) 0 0
\(925\) −13.8564 −0.455596
\(926\) 0 0
\(927\) 14.0416 0.461185
\(928\) 0 0
\(929\) −45.1168 −1.48024 −0.740118 0.672477i \(-0.765230\pi\)
−0.740118 + 0.672477i \(0.765230\pi\)
\(930\) 0 0
\(931\) −4.00000 −0.131095
\(932\) 0 0
\(933\) 55.0758 1.80310
\(934\) 0 0
\(935\) −13.3591 −0.436888
\(936\) 0 0
\(937\) −23.9783 −0.783335 −0.391668 0.920107i \(-0.628102\pi\)
−0.391668 + 0.920107i \(0.628102\pi\)
\(938\) 0 0
\(939\) 50.0951 1.63479
\(940\) 0 0
\(941\) −5.69349 −0.185602 −0.0928012 0.995685i \(-0.529582\pi\)
−0.0928012 + 0.995685i \(0.529582\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −2.32069 −0.0754919
\(946\) 0 0
\(947\) −37.2119 −1.20923 −0.604613 0.796520i \(-0.706672\pi\)
−0.604613 + 0.796520i \(0.706672\pi\)
\(948\) 0 0
\(949\) −59.5345 −1.93257
\(950\) 0 0
\(951\) −43.3185 −1.40470
\(952\) 0 0
\(953\) 8.23369 0.266715 0.133358 0.991068i \(-0.457424\pi\)
0.133358 + 0.991068i \(0.457424\pi\)
\(954\) 0 0
\(955\) 10.8614 0.351467
\(956\) 0 0
\(957\) −47.9075 −1.54863
\(958\) 0 0
\(959\) 16.5831 0.535497
\(960\) 0 0
\(961\) −19.0000 −0.612903
\(962\) 0 0
\(963\) −20.0435 −0.645893
\(964\) 0 0
\(965\) 11.6819 0.376054
\(966\) 0 0
\(967\) −41.8642 −1.34626 −0.673131 0.739524i \(-0.735051\pi\)
−0.673131 + 0.739524i \(0.735051\pi\)
\(968\) 0 0
\(969\) 11.8614 0.381043
\(970\) 0 0
\(971\) −4.00000 −0.128366 −0.0641831 0.997938i \(-0.520444\pi\)
−0.0641831 + 0.997938i \(0.520444\pi\)
\(972\) 0 0
\(973\) 31.0843 0.996518
\(974\) 0 0
\(975\) −42.6188 −1.36489
\(976\) 0 0
\(977\) 4.51087 0.144316 0.0721578 0.997393i \(-0.477011\pi\)
0.0721578 + 0.997393i \(0.477011\pi\)
\(978\) 0 0
\(979\) 25.2554 0.807167
\(980\) 0 0
\(981\) −33.1662 −1.05892
\(982\) 0 0
\(983\) −6.22849 −0.198658 −0.0993290 0.995055i \(-0.531670\pi\)
−0.0993290 + 0.995055i \(0.531670\pi\)
\(984\) 0 0
\(985\) −16.0000 −0.509802
\(986\) 0 0
\(987\) −73.2119 −2.33036
\(988\) 0 0
\(989\) 23.6588 0.752306
\(990\) 0 0
\(991\) 25.2434 0.801882 0.400941 0.916104i \(-0.368683\pi\)
0.400941 + 0.916104i \(0.368683\pi\)
\(992\) 0 0
\(993\) −42.3723 −1.34464
\(994\) 0 0
\(995\) −5.37228 −0.170313
\(996\) 0 0
\(997\) −10.2997 −0.326196 −0.163098 0.986610i \(-0.552149\pi\)
−0.163098 + 0.986610i \(0.552149\pi\)
\(998\) 0 0
\(999\) 2.79885 0.0885518
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.bl.1.2 4
4.3 odd 2 4864.2.a.bi.1.4 4
8.3 odd 2 inner 4864.2.a.bl.1.1 4
8.5 even 2 4864.2.a.bi.1.3 4
16.3 odd 4 1216.2.c.h.609.5 yes 8
16.5 even 4 1216.2.c.h.609.6 yes 8
16.11 odd 4 1216.2.c.h.609.4 yes 8
16.13 even 4 1216.2.c.h.609.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1216.2.c.h.609.3 8 16.13 even 4
1216.2.c.h.609.4 yes 8 16.11 odd 4
1216.2.c.h.609.5 yes 8 16.3 odd 4
1216.2.c.h.609.6 yes 8 16.5 even 4
4864.2.a.bi.1.3 4 8.5 even 2
4864.2.a.bi.1.4 4 4.3 odd 2
4864.2.a.bl.1.1 4 8.3 odd 2 inner
4864.2.a.bl.1.2 4 1.1 even 1 trivial