Properties

Label 6864.2.a.cf.1.4
Level $6864$
Weight $2$
Character 6864.1
Self dual yes
Analytic conductor $54.809$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6864.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(54.8093159474\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.2172244.1
Defining polynomial: \(x^{5} - 2 x^{4} - 8 x^{3} + 16 x^{2} + 5 x - 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 3432)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-0.758834\) of defining polynomial
Character \(\chi\) \(=\) 6864.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.82899 q^{5} +0.862883 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.82899 q^{5} +0.862883 q^{7} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{13} +1.82899 q^{15} +0.551559 q^{17} +0.300773 q^{19} +0.862883 q^{21} +7.71122 q^{23} -1.65479 q^{25} +1.00000 q^{27} -4.56356 q^{29} -0.391103 q^{31} +1.00000 q^{33} +1.57821 q^{35} +9.60944 q^{37} +1.00000 q^{39} +5.29667 q^{41} +11.7603 q^{43} +1.82899 q^{45} -6.50633 q^{47} -6.25543 q^{49} +0.551559 q^{51} -9.76446 q^{53} +1.82899 q^{55} +0.300773 q^{57} +6.49433 q^{59} -7.39575 q^{61} +0.862883 q^{63} +1.82899 q^{65} -8.53702 q^{67} +7.71122 q^{69} +3.58545 q^{71} +0.795102 q^{73} -1.65479 q^{75} +0.862883 q^{77} +9.81445 q^{79} +1.00000 q^{81} -5.49023 q^{83} +1.00880 q^{85} -4.56356 q^{87} +13.6585 q^{89} +0.862883 q^{91} -0.391103 q^{93} +0.550112 q^{95} +2.57411 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 5q^{3} + q^{5} + 5q^{7} + 5q^{9} + O(q^{10}) \) \( 5q + 5q^{3} + q^{5} + 5q^{7} + 5q^{9} + 5q^{11} + 5q^{13} + q^{15} + 4q^{19} + 5q^{21} + 5q^{23} + 4q^{25} + 5q^{27} + 11q^{29} + 8q^{31} + 5q^{33} + 5q^{35} - 8q^{37} + 5q^{39} - q^{41} - q^{43} + q^{45} + 18q^{47} + 10q^{49} + 2q^{53} + q^{55} + 4q^{57} + 13q^{59} + 9q^{61} + 5q^{63} + q^{65} - 5q^{67} + 5q^{69} + 24q^{71} - 13q^{73} + 4q^{75} + 5q^{77} + 6q^{79} + 5q^{81} + 22q^{83} - 22q^{85} + 11q^{87} - 14q^{89} + 5q^{91} + 8q^{93} + 32q^{95} - 20q^{97} + 5q^{99} + O(q^{100}) \)

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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 1.82899 0.817950 0.408975 0.912545i \(-0.365886\pi\)
0.408975 + 0.912545i \(0.365886\pi\)
\(6\) 0 0
\(7\) 0.862883 0.326139 0.163070 0.986615i \(-0.447860\pi\)
0.163070 + 0.986615i \(0.447860\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 1.82899 0.472244
\(16\) 0 0
\(17\) 0.551559 0.133773 0.0668863 0.997761i \(-0.478694\pi\)
0.0668863 + 0.997761i \(0.478694\pi\)
\(18\) 0 0
\(19\) 0.300773 0.0690021 0.0345011 0.999405i \(-0.489016\pi\)
0.0345011 + 0.999405i \(0.489016\pi\)
\(20\) 0 0
\(21\) 0.862883 0.188297
\(22\) 0 0
\(23\) 7.71122 1.60790 0.803951 0.594696i \(-0.202728\pi\)
0.803951 + 0.594696i \(0.202728\pi\)
\(24\) 0 0
\(25\) −1.65479 −0.330957
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −4.56356 −0.847431 −0.423716 0.905795i \(-0.639274\pi\)
−0.423716 + 0.905795i \(0.639274\pi\)
\(30\) 0 0
\(31\) −0.391103 −0.0702441 −0.0351221 0.999383i \(-0.511182\pi\)
−0.0351221 + 0.999383i \(0.511182\pi\)
\(32\) 0 0
\(33\) 1.00000 0.174078
\(34\) 0 0
\(35\) 1.57821 0.266766
\(36\) 0 0
\(37\) 9.60944 1.57978 0.789891 0.613247i \(-0.210137\pi\)
0.789891 + 0.613247i \(0.210137\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 5.29667 0.827201 0.413601 0.910458i \(-0.364271\pi\)
0.413601 + 0.910458i \(0.364271\pi\)
\(42\) 0 0
\(43\) 11.7603 1.79343 0.896715 0.442608i \(-0.145947\pi\)
0.896715 + 0.442608i \(0.145947\pi\)
\(44\) 0 0
\(45\) 1.82899 0.272650
\(46\) 0 0
\(47\) −6.50633 −0.949045 −0.474523 0.880243i \(-0.657379\pi\)
−0.474523 + 0.880243i \(0.657379\pi\)
\(48\) 0 0
\(49\) −6.25543 −0.893633
\(50\) 0 0
\(51\) 0.551559 0.0772337
\(52\) 0 0
\(53\) −9.76446 −1.34125 −0.670626 0.741796i \(-0.733974\pi\)
−0.670626 + 0.741796i \(0.733974\pi\)
\(54\) 0 0
\(55\) 1.82899 0.246621
\(56\) 0 0
\(57\) 0.300773 0.0398384
\(58\) 0 0
\(59\) 6.49433 0.845490 0.422745 0.906249i \(-0.361067\pi\)
0.422745 + 0.906249i \(0.361067\pi\)
\(60\) 0 0
\(61\) −7.39575 −0.946929 −0.473464 0.880813i \(-0.656997\pi\)
−0.473464 + 0.880813i \(0.656997\pi\)
\(62\) 0 0
\(63\) 0.862883 0.108713
\(64\) 0 0
\(65\) 1.82899 0.226859
\(66\) 0 0
\(67\) −8.53702 −1.04296 −0.521481 0.853263i \(-0.674620\pi\)
−0.521481 + 0.853263i \(0.674620\pi\)
\(68\) 0 0
\(69\) 7.71122 0.928322
\(70\) 0 0
\(71\) 3.58545 0.425515 0.212757 0.977105i \(-0.431756\pi\)
0.212757 + 0.977105i \(0.431756\pi\)
\(72\) 0 0
\(73\) 0.795102 0.0930597 0.0465298 0.998917i \(-0.485184\pi\)
0.0465298 + 0.998917i \(0.485184\pi\)
\(74\) 0 0
\(75\) −1.65479 −0.191078
\(76\) 0 0
\(77\) 0.862883 0.0983347
\(78\) 0 0
\(79\) 9.81445 1.10421 0.552106 0.833774i \(-0.313824\pi\)
0.552106 + 0.833774i \(0.313824\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −5.49023 −0.602631 −0.301315 0.953525i \(-0.597426\pi\)
−0.301315 + 0.953525i \(0.597426\pi\)
\(84\) 0 0
\(85\) 1.00880 0.109419
\(86\) 0 0
\(87\) −4.56356 −0.489265
\(88\) 0 0
\(89\) 13.6585 1.44780 0.723901 0.689904i \(-0.242347\pi\)
0.723901 + 0.689904i \(0.242347\pi\)
\(90\) 0 0
\(91\) 0.862883 0.0904547
\(92\) 0 0
\(93\) −0.391103 −0.0405555
\(94\) 0 0
\(95\) 0.550112 0.0564403
\(96\) 0 0
\(97\) 2.57411 0.261361 0.130681 0.991425i \(-0.458284\pi\)
0.130681 + 0.991425i \(0.458284\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) 14.8709 1.47971 0.739854 0.672767i \(-0.234894\pi\)
0.739854 + 0.672767i \(0.234894\pi\)
\(102\) 0 0
\(103\) 2.83634 0.279473 0.139737 0.990189i \(-0.455374\pi\)
0.139737 + 0.990189i \(0.455374\pi\)
\(104\) 0 0
\(105\) 1.57821 0.154017
\(106\) 0 0
\(107\) 15.2024 1.46967 0.734833 0.678248i \(-0.237260\pi\)
0.734833 + 0.678248i \(0.237260\pi\)
\(108\) 0 0
\(109\) −2.11832 −0.202898 −0.101449 0.994841i \(-0.532348\pi\)
−0.101449 + 0.994841i \(0.532348\pi\)
\(110\) 0 0
\(111\) 9.60944 0.912088
\(112\) 0 0
\(113\) −14.5650 −1.37016 −0.685080 0.728468i \(-0.740233\pi\)
−0.685080 + 0.728468i \(0.740233\pi\)
\(114\) 0 0
\(115\) 14.1038 1.31518
\(116\) 0 0
\(117\) 1.00000 0.0924500
\(118\) 0 0
\(119\) 0.475931 0.0436285
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 5.29667 0.477585
\(124\) 0 0
\(125\) −12.1716 −1.08866
\(126\) 0 0
\(127\) −0.638797 −0.0566840 −0.0283420 0.999598i \(-0.509023\pi\)
−0.0283420 + 0.999598i \(0.509023\pi\)
\(128\) 0 0
\(129\) 11.7603 1.03544
\(130\) 0 0
\(131\) −8.58611 −0.750171 −0.375086 0.926990i \(-0.622387\pi\)
−0.375086 + 0.926990i \(0.622387\pi\)
\(132\) 0 0
\(133\) 0.259532 0.0225043
\(134\) 0 0
\(135\) 1.82899 0.157415
\(136\) 0 0
\(137\) −14.9652 −1.27856 −0.639282 0.768972i \(-0.720768\pi\)
−0.639282 + 0.768972i \(0.720768\pi\)
\(138\) 0 0
\(139\) 9.26144 0.785546 0.392773 0.919636i \(-0.371516\pi\)
0.392773 + 0.919636i \(0.371516\pi\)
\(140\) 0 0
\(141\) −6.50633 −0.547932
\(142\) 0 0
\(143\) 1.00000 0.0836242
\(144\) 0 0
\(145\) −8.34671 −0.693157
\(146\) 0 0
\(147\) −6.25543 −0.515939
\(148\) 0 0
\(149\) −8.73456 −0.715563 −0.357782 0.933805i \(-0.616467\pi\)
−0.357782 + 0.933805i \(0.616467\pi\)
\(150\) 0 0
\(151\) 8.46033 0.688492 0.344246 0.938880i \(-0.388135\pi\)
0.344246 + 0.938880i \(0.388135\pi\)
\(152\) 0 0
\(153\) 0.551559 0.0445909
\(154\) 0 0
\(155\) −0.715324 −0.0574562
\(156\) 0 0
\(157\) 16.0020 1.27710 0.638549 0.769581i \(-0.279535\pi\)
0.638549 + 0.769581i \(0.279535\pi\)
\(158\) 0 0
\(159\) −9.76446 −0.774372
\(160\) 0 0
\(161\) 6.65389 0.524400
\(162\) 0 0
\(163\) 5.89513 0.461742 0.230871 0.972984i \(-0.425842\pi\)
0.230871 + 0.972984i \(0.425842\pi\)
\(164\) 0 0
\(165\) 1.82899 0.142387
\(166\) 0 0
\(167\) 0.402553 0.0311505 0.0155752 0.999879i \(-0.495042\pi\)
0.0155752 + 0.999879i \(0.495042\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 0.300773 0.0230007
\(172\) 0 0
\(173\) −22.1794 −1.68627 −0.843134 0.537703i \(-0.819292\pi\)
−0.843134 + 0.537703i \(0.819292\pi\)
\(174\) 0 0
\(175\) −1.42789 −0.107938
\(176\) 0 0
\(177\) 6.49433 0.488144
\(178\) 0 0
\(179\) 6.18156 0.462031 0.231016 0.972950i \(-0.425795\pi\)
0.231016 + 0.972950i \(0.425795\pi\)
\(180\) 0 0
\(181\) −17.7958 −1.32275 −0.661374 0.750056i \(-0.730026\pi\)
−0.661374 + 0.750056i \(0.730026\pi\)
\(182\) 0 0
\(183\) −7.39575 −0.546710
\(184\) 0 0
\(185\) 17.5756 1.29218
\(186\) 0 0
\(187\) 0.551559 0.0403340
\(188\) 0 0
\(189\) 0.862883 0.0627655
\(190\) 0 0
\(191\) 22.2377 1.60906 0.804531 0.593910i \(-0.202417\pi\)
0.804531 + 0.593910i \(0.202417\pi\)
\(192\) 0 0
\(193\) −1.95876 −0.140995 −0.0704973 0.997512i \(-0.522459\pi\)
−0.0704973 + 0.997512i \(0.522459\pi\)
\(194\) 0 0
\(195\) 1.82899 0.130977
\(196\) 0 0
\(197\) 18.0652 1.28709 0.643547 0.765406i \(-0.277462\pi\)
0.643547 + 0.765406i \(0.277462\pi\)
\(198\) 0 0
\(199\) −22.7521 −1.61285 −0.806425 0.591336i \(-0.798601\pi\)
−0.806425 + 0.591336i \(0.798601\pi\)
\(200\) 0 0
\(201\) −8.53702 −0.602155
\(202\) 0 0
\(203\) −3.93782 −0.276381
\(204\) 0 0
\(205\) 9.68758 0.676610
\(206\) 0 0
\(207\) 7.71122 0.535967
\(208\) 0 0
\(209\) 0.300773 0.0208049
\(210\) 0 0
\(211\) −26.7562 −1.84197 −0.920987 0.389594i \(-0.872615\pi\)
−0.920987 + 0.389594i \(0.872615\pi\)
\(212\) 0 0
\(213\) 3.58545 0.245671
\(214\) 0 0
\(215\) 21.5095 1.46694
\(216\) 0 0
\(217\) −0.337476 −0.0229094
\(218\) 0 0
\(219\) 0.795102 0.0537280
\(220\) 0 0
\(221\) 0.551559 0.0371019
\(222\) 0 0
\(223\) 3.84264 0.257322 0.128661 0.991689i \(-0.458932\pi\)
0.128661 + 0.991689i \(0.458932\pi\)
\(224\) 0 0
\(225\) −1.65479 −0.110319
\(226\) 0 0
\(227\) 22.8941 1.51954 0.759768 0.650194i \(-0.225312\pi\)
0.759768 + 0.650194i \(0.225312\pi\)
\(228\) 0 0
\(229\) −5.22979 −0.345594 −0.172797 0.984957i \(-0.555281\pi\)
−0.172797 + 0.984957i \(0.555281\pi\)
\(230\) 0 0
\(231\) 0.862883 0.0567735
\(232\) 0 0
\(233\) 6.48832 0.425064 0.212532 0.977154i \(-0.431829\pi\)
0.212532 + 0.977154i \(0.431829\pi\)
\(234\) 0 0
\(235\) −11.9000 −0.776272
\(236\) 0 0
\(237\) 9.81445 0.637517
\(238\) 0 0
\(239\) 28.1498 1.82086 0.910429 0.413666i \(-0.135752\pi\)
0.910429 + 0.413666i \(0.135752\pi\)
\(240\) 0 0
\(241\) 6.31293 0.406652 0.203326 0.979111i \(-0.434825\pi\)
0.203326 + 0.979111i \(0.434825\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −11.4411 −0.730948
\(246\) 0 0
\(247\) 0.300773 0.0191377
\(248\) 0 0
\(249\) −5.49023 −0.347929
\(250\) 0 0
\(251\) −4.48709 −0.283222 −0.141611 0.989922i \(-0.545228\pi\)
−0.141611 + 0.989922i \(0.545228\pi\)
\(252\) 0 0
\(253\) 7.71122 0.484801
\(254\) 0 0
\(255\) 1.00880 0.0631733
\(256\) 0 0
\(257\) −22.4505 −1.40043 −0.700213 0.713934i \(-0.746912\pi\)
−0.700213 + 0.713934i \(0.746912\pi\)
\(258\) 0 0
\(259\) 8.29183 0.515229
\(260\) 0 0
\(261\) −4.56356 −0.282477
\(262\) 0 0
\(263\) 12.0854 0.745216 0.372608 0.927989i \(-0.378464\pi\)
0.372608 + 0.927989i \(0.378464\pi\)
\(264\) 0 0
\(265\) −17.8591 −1.09708
\(266\) 0 0
\(267\) 13.6585 0.835889
\(268\) 0 0
\(269\) 21.7885 1.32847 0.664233 0.747526i \(-0.268758\pi\)
0.664233 + 0.747526i \(0.268758\pi\)
\(270\) 0 0
\(271\) −9.38761 −0.570257 −0.285128 0.958489i \(-0.592036\pi\)
−0.285128 + 0.958489i \(0.592036\pi\)
\(272\) 0 0
\(273\) 0.862883 0.0522241
\(274\) 0 0
\(275\) −1.65479 −0.0997873
\(276\) 0 0
\(277\) 4.07524 0.244857 0.122429 0.992477i \(-0.460932\pi\)
0.122429 + 0.992477i \(0.460932\pi\)
\(278\) 0 0
\(279\) −0.391103 −0.0234147
\(280\) 0 0
\(281\) −24.7933 −1.47904 −0.739522 0.673132i \(-0.764949\pi\)
−0.739522 + 0.673132i \(0.764949\pi\)
\(282\) 0 0
\(283\) 11.3247 0.673181 0.336591 0.941651i \(-0.390726\pi\)
0.336591 + 0.941651i \(0.390726\pi\)
\(284\) 0 0
\(285\) 0.550112 0.0325858
\(286\) 0 0
\(287\) 4.57041 0.269783
\(288\) 0 0
\(289\) −16.6958 −0.982105
\(290\) 0 0
\(291\) 2.57411 0.150897
\(292\) 0 0
\(293\) 7.05389 0.412093 0.206046 0.978542i \(-0.433940\pi\)
0.206046 + 0.978542i \(0.433940\pi\)
\(294\) 0 0
\(295\) 11.8781 0.691569
\(296\) 0 0
\(297\) 1.00000 0.0580259
\(298\) 0 0
\(299\) 7.71122 0.445952
\(300\) 0 0
\(301\) 10.1478 0.584908
\(302\) 0 0
\(303\) 14.8709 0.854310
\(304\) 0 0
\(305\) −13.5268 −0.774541
\(306\) 0 0
\(307\) 20.4440 1.16680 0.583402 0.812184i \(-0.301721\pi\)
0.583402 + 0.812184i \(0.301721\pi\)
\(308\) 0 0
\(309\) 2.83634 0.161354
\(310\) 0 0
\(311\) −15.9342 −0.903546 −0.451773 0.892133i \(-0.649208\pi\)
−0.451773 + 0.892133i \(0.649208\pi\)
\(312\) 0 0
\(313\) 26.1754 1.47952 0.739761 0.672870i \(-0.234939\pi\)
0.739761 + 0.672870i \(0.234939\pi\)
\(314\) 0 0
\(315\) 1.57821 0.0889219
\(316\) 0 0
\(317\) 16.2142 0.910683 0.455341 0.890317i \(-0.349517\pi\)
0.455341 + 0.890317i \(0.349517\pi\)
\(318\) 0 0
\(319\) −4.56356 −0.255510
\(320\) 0 0
\(321\) 15.2024 0.848513
\(322\) 0 0
\(323\) 0.165894 0.00923060
\(324\) 0 0
\(325\) −1.65479 −0.0917910
\(326\) 0 0
\(327\) −2.11832 −0.117143
\(328\) 0 0
\(329\) −5.61420 −0.309521
\(330\) 0 0
\(331\) −22.9140 −1.25947 −0.629733 0.776812i \(-0.716836\pi\)
−0.629733 + 0.776812i \(0.716836\pi\)
\(332\) 0 0
\(333\) 9.60944 0.526594
\(334\) 0 0
\(335\) −15.6141 −0.853092
\(336\) 0 0
\(337\) −17.1335 −0.933322 −0.466661 0.884436i \(-0.654543\pi\)
−0.466661 + 0.884436i \(0.654543\pi\)
\(338\) 0 0
\(339\) −14.5650 −0.791062
\(340\) 0 0
\(341\) −0.391103 −0.0211794
\(342\) 0 0
\(343\) −11.4379 −0.617588
\(344\) 0 0
\(345\) 14.1038 0.759322
\(346\) 0 0
\(347\) 17.4563 0.937103 0.468551 0.883436i \(-0.344776\pi\)
0.468551 + 0.883436i \(0.344776\pi\)
\(348\) 0 0
\(349\) −21.9240 −1.17357 −0.586783 0.809744i \(-0.699606\pi\)
−0.586783 + 0.809744i \(0.699606\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) −32.8729 −1.74965 −0.874824 0.484442i \(-0.839023\pi\)
−0.874824 + 0.484442i \(0.839023\pi\)
\(354\) 0 0
\(355\) 6.55776 0.348050
\(356\) 0 0
\(357\) 0.475931 0.0251889
\(358\) 0 0
\(359\) −8.28533 −0.437283 −0.218642 0.975805i \(-0.570163\pi\)
−0.218642 + 0.975805i \(0.570163\pi\)
\(360\) 0 0
\(361\) −18.9095 −0.995239
\(362\) 0 0
\(363\) 1.00000 0.0524864
\(364\) 0 0
\(365\) 1.45424 0.0761182
\(366\) 0 0
\(367\) 17.5967 0.918541 0.459270 0.888297i \(-0.348111\pi\)
0.459270 + 0.888297i \(0.348111\pi\)
\(368\) 0 0
\(369\) 5.29667 0.275734
\(370\) 0 0
\(371\) −8.42559 −0.437435
\(372\) 0 0
\(373\) −18.5890 −0.962502 −0.481251 0.876583i \(-0.659817\pi\)
−0.481251 + 0.876583i \(0.659817\pi\)
\(374\) 0 0
\(375\) −12.1716 −0.628536
\(376\) 0 0
\(377\) −4.56356 −0.235035
\(378\) 0 0
\(379\) −33.3018 −1.71060 −0.855300 0.518134i \(-0.826627\pi\)
−0.855300 + 0.518134i \(0.826627\pi\)
\(380\) 0 0
\(381\) −0.638797 −0.0327265
\(382\) 0 0
\(383\) 13.9082 0.710677 0.355339 0.934738i \(-0.384366\pi\)
0.355339 + 0.934738i \(0.384366\pi\)
\(384\) 0 0
\(385\) 1.57821 0.0804329
\(386\) 0 0
\(387\) 11.7603 0.597810
\(388\) 0 0
\(389\) 28.0627 1.42283 0.711417 0.702770i \(-0.248054\pi\)
0.711417 + 0.702770i \(0.248054\pi\)
\(390\) 0 0
\(391\) 4.25319 0.215093
\(392\) 0 0
\(393\) −8.58611 −0.433112
\(394\) 0 0
\(395\) 17.9506 0.903191
\(396\) 0 0
\(397\) −23.7618 −1.19257 −0.596284 0.802773i \(-0.703357\pi\)
−0.596284 + 0.802773i \(0.703357\pi\)
\(398\) 0 0
\(399\) 0.259532 0.0129929
\(400\) 0 0
\(401\) −29.0470 −1.45054 −0.725270 0.688465i \(-0.758285\pi\)
−0.725270 + 0.688465i \(0.758285\pi\)
\(402\) 0 0
\(403\) −0.391103 −0.0194822
\(404\) 0 0
\(405\) 1.82899 0.0908834
\(406\) 0 0
\(407\) 9.60944 0.476322
\(408\) 0 0
\(409\) 21.0997 1.04331 0.521657 0.853155i \(-0.325314\pi\)
0.521657 + 0.853155i \(0.325314\pi\)
\(410\) 0 0
\(411\) −14.9652 −0.738179
\(412\) 0 0
\(413\) 5.60385 0.275747
\(414\) 0 0
\(415\) −10.0416 −0.492922
\(416\) 0 0
\(417\) 9.26144 0.453535
\(418\) 0 0
\(419\) −23.5151 −1.14879 −0.574394 0.818579i \(-0.694762\pi\)
−0.574394 + 0.818579i \(0.694762\pi\)
\(420\) 0 0
\(421\) 34.0322 1.65863 0.829315 0.558782i \(-0.188731\pi\)
0.829315 + 0.558782i \(0.188731\pi\)
\(422\) 0 0
\(423\) −6.50633 −0.316348
\(424\) 0 0
\(425\) −0.912712 −0.0442730
\(426\) 0 0
\(427\) −6.38167 −0.308831
\(428\) 0 0
\(429\) 1.00000 0.0482805
\(430\) 0 0
\(431\) 0.920660 0.0443466 0.0221733 0.999754i \(-0.492941\pi\)
0.0221733 + 0.999754i \(0.492941\pi\)
\(432\) 0 0
\(433\) 28.7393 1.38112 0.690560 0.723275i \(-0.257364\pi\)
0.690560 + 0.723275i \(0.257364\pi\)
\(434\) 0 0
\(435\) −8.34671 −0.400194
\(436\) 0 0
\(437\) 2.31933 0.110949
\(438\) 0 0
\(439\) −8.91779 −0.425623 −0.212811 0.977093i \(-0.568262\pi\)
−0.212811 + 0.977093i \(0.568262\pi\)
\(440\) 0 0
\(441\) −6.25543 −0.297878
\(442\) 0 0
\(443\) −1.42885 −0.0678867 −0.0339433 0.999424i \(-0.510807\pi\)
−0.0339433 + 0.999424i \(0.510807\pi\)
\(444\) 0 0
\(445\) 24.9814 1.18423
\(446\) 0 0
\(447\) −8.73456 −0.413131
\(448\) 0 0
\(449\) −31.4742 −1.48536 −0.742680 0.669647i \(-0.766446\pi\)
−0.742680 + 0.669647i \(0.766446\pi\)
\(450\) 0 0
\(451\) 5.29667 0.249411
\(452\) 0 0
\(453\) 8.46033 0.397501
\(454\) 0 0
\(455\) 1.57821 0.0739875
\(456\) 0 0
\(457\) 23.6132 1.10458 0.552290 0.833652i \(-0.313754\pi\)
0.552290 + 0.833652i \(0.313754\pi\)
\(458\) 0 0
\(459\) 0.551559 0.0257446
\(460\) 0 0
\(461\) −4.90904 −0.228637 −0.114318 0.993444i \(-0.536468\pi\)
−0.114318 + 0.993444i \(0.536468\pi\)
\(462\) 0 0
\(463\) −13.4072 −0.623085 −0.311543 0.950232i \(-0.600846\pi\)
−0.311543 + 0.950232i \(0.600846\pi\)
\(464\) 0 0
\(465\) −0.715324 −0.0331724
\(466\) 0 0
\(467\) 29.0824 1.34577 0.672887 0.739745i \(-0.265054\pi\)
0.672887 + 0.739745i \(0.265054\pi\)
\(468\) 0 0
\(469\) −7.36645 −0.340151
\(470\) 0 0
\(471\) 16.0020 0.737333
\(472\) 0 0
\(473\) 11.7603 0.540740
\(474\) 0 0
\(475\) −0.497715 −0.0228367
\(476\) 0 0
\(477\) −9.76446 −0.447084
\(478\) 0 0
\(479\) 34.8915 1.59423 0.797116 0.603826i \(-0.206358\pi\)
0.797116 + 0.603826i \(0.206358\pi\)
\(480\) 0 0
\(481\) 9.60944 0.438153
\(482\) 0 0
\(483\) 6.65389 0.302762
\(484\) 0 0
\(485\) 4.70802 0.213780
\(486\) 0 0
\(487\) 31.3699 1.42151 0.710753 0.703441i \(-0.248354\pi\)
0.710753 + 0.703441i \(0.248354\pi\)
\(488\) 0 0
\(489\) 5.89513 0.266587
\(490\) 0 0
\(491\) 28.4504 1.28395 0.641974 0.766726i \(-0.278116\pi\)
0.641974 + 0.766726i \(0.278116\pi\)
\(492\) 0 0
\(493\) −2.51707 −0.113363
\(494\) 0 0
\(495\) 1.82899 0.0822071
\(496\) 0 0
\(497\) 3.09382 0.138777
\(498\) 0 0
\(499\) −26.7818 −1.19892 −0.599460 0.800405i \(-0.704618\pi\)
−0.599460 + 0.800405i \(0.704618\pi\)
\(500\) 0 0
\(501\) 0.402553 0.0179847
\(502\) 0 0
\(503\) −0.685123 −0.0305481 −0.0152741 0.999883i \(-0.504862\pi\)
−0.0152741 + 0.999883i \(0.504862\pi\)
\(504\) 0 0
\(505\) 27.1987 1.21033
\(506\) 0 0
\(507\) 1.00000 0.0444116
\(508\) 0 0
\(509\) 7.33920 0.325304 0.162652 0.986683i \(-0.447995\pi\)
0.162652 + 0.986683i \(0.447995\pi\)
\(510\) 0 0
\(511\) 0.686080 0.0303504
\(512\) 0 0
\(513\) 0.300773 0.0132795
\(514\) 0 0
\(515\) 5.18765 0.228595
\(516\) 0 0
\(517\) −6.50633 −0.286148
\(518\) 0 0
\(519\) −22.1794 −0.973568
\(520\) 0 0
\(521\) 4.64738 0.203606 0.101803 0.994805i \(-0.467539\pi\)
0.101803 + 0.994805i \(0.467539\pi\)
\(522\) 0 0
\(523\) −29.4724 −1.28874 −0.644370 0.764714i \(-0.722880\pi\)
−0.644370 + 0.764714i \(0.722880\pi\)
\(524\) 0 0
\(525\) −1.42789 −0.0623181
\(526\) 0 0
\(527\) −0.215716 −0.00939674
\(528\) 0 0
\(529\) 36.4630 1.58535
\(530\) 0 0
\(531\) 6.49433 0.281830
\(532\) 0 0
\(533\) 5.29667 0.229424
\(534\) 0 0
\(535\) 27.8050 1.20211
\(536\) 0 0
\(537\) 6.18156 0.266754
\(538\) 0 0
\(539\) −6.25543 −0.269441
\(540\) 0 0
\(541\) 10.9737 0.471798 0.235899 0.971778i \(-0.424197\pi\)
0.235899 + 0.971778i \(0.424197\pi\)
\(542\) 0 0
\(543\) −17.7958 −0.763689
\(544\) 0 0
\(545\) −3.87438 −0.165960
\(546\) 0 0
\(547\) 10.7231 0.458487 0.229244 0.973369i \(-0.426375\pi\)
0.229244 + 0.973369i \(0.426375\pi\)
\(548\) 0 0
\(549\) −7.39575 −0.315643
\(550\) 0 0
\(551\) −1.37260 −0.0584746
\(552\) 0 0
\(553\) 8.46872 0.360127
\(554\) 0 0
\(555\) 17.5756 0.746043
\(556\) 0 0
\(557\) −12.8392 −0.544016 −0.272008 0.962295i \(-0.587688\pi\)
−0.272008 + 0.962295i \(0.587688\pi\)
\(558\) 0 0
\(559\) 11.7603 0.497408
\(560\) 0 0
\(561\) 0.551559 0.0232868
\(562\) 0 0
\(563\) −8.32091 −0.350685 −0.175342 0.984508i \(-0.556103\pi\)
−0.175342 + 0.984508i \(0.556103\pi\)
\(564\) 0 0
\(565\) −26.6393 −1.12072
\(566\) 0 0
\(567\) 0.862883 0.0362377
\(568\) 0 0
\(569\) −12.9917 −0.544642 −0.272321 0.962206i \(-0.587791\pi\)
−0.272321 + 0.962206i \(0.587791\pi\)
\(570\) 0 0
\(571\) −33.9797 −1.42200 −0.711002 0.703190i \(-0.751759\pi\)
−0.711002 + 0.703190i \(0.751759\pi\)
\(572\) 0 0
\(573\) 22.2377 0.928993
\(574\) 0 0
\(575\) −12.7604 −0.532146
\(576\) 0 0
\(577\) 33.9415 1.41300 0.706502 0.707711i \(-0.250272\pi\)
0.706502 + 0.707711i \(0.250272\pi\)
\(578\) 0 0
\(579\) −1.95876 −0.0814032
\(580\) 0 0
\(581\) −4.73743 −0.196542
\(582\) 0 0
\(583\) −9.76446 −0.404403
\(584\) 0 0
\(585\) 1.82899 0.0756195
\(586\) 0 0
\(587\) −36.7614 −1.51730 −0.758652 0.651496i \(-0.774142\pi\)
−0.758652 + 0.651496i \(0.774142\pi\)
\(588\) 0 0
\(589\) −0.117633 −0.00484699
\(590\) 0 0
\(591\) 18.0652 0.743105
\(592\) 0 0
\(593\) 35.4759 1.45682 0.728410 0.685142i \(-0.240260\pi\)
0.728410 + 0.685142i \(0.240260\pi\)
\(594\) 0 0
\(595\) 0.870474 0.0356860
\(596\) 0 0
\(597\) −22.7521 −0.931180
\(598\) 0 0
\(599\) −5.54787 −0.226680 −0.113340 0.993556i \(-0.536155\pi\)
−0.113340 + 0.993556i \(0.536155\pi\)
\(600\) 0 0
\(601\) 21.0593 0.859028 0.429514 0.903060i \(-0.358685\pi\)
0.429514 + 0.903060i \(0.358685\pi\)
\(602\) 0 0
\(603\) −8.53702 −0.347654
\(604\) 0 0
\(605\) 1.82899 0.0743591
\(606\) 0 0
\(607\) −0.638797 −0.0259280 −0.0129640 0.999916i \(-0.504127\pi\)
−0.0129640 + 0.999916i \(0.504127\pi\)
\(608\) 0 0
\(609\) −3.93782 −0.159568
\(610\) 0 0
\(611\) −6.50633 −0.263218
\(612\) 0 0
\(613\) −15.2683 −0.616682 −0.308341 0.951276i \(-0.599774\pi\)
−0.308341 + 0.951276i \(0.599774\pi\)
\(614\) 0 0
\(615\) 9.68758 0.390641
\(616\) 0 0
\(617\) 17.2360 0.693895 0.346948 0.937885i \(-0.387218\pi\)
0.346948 + 0.937885i \(0.387218\pi\)
\(618\) 0 0
\(619\) −15.8099 −0.635452 −0.317726 0.948183i \(-0.602919\pi\)
−0.317726 + 0.948183i \(0.602919\pi\)
\(620\) 0 0
\(621\) 7.71122 0.309441
\(622\) 0 0
\(623\) 11.7857 0.472185
\(624\) 0 0
\(625\) −13.9878 −0.559510
\(626\) 0 0
\(627\) 0.300773 0.0120117
\(628\) 0 0
\(629\) 5.30018 0.211332
\(630\) 0 0
\(631\) −38.8232 −1.54553 −0.772765 0.634693i \(-0.781127\pi\)
−0.772765 + 0.634693i \(0.781127\pi\)
\(632\) 0 0
\(633\) −26.7562 −1.06346
\(634\) 0 0
\(635\) −1.16835 −0.0463647
\(636\) 0 0
\(637\) −6.25543 −0.247849
\(638\) 0 0
\(639\) 3.58545 0.141838
\(640\) 0 0
\(641\) 41.4165 1.63585 0.817927 0.575322i \(-0.195123\pi\)
0.817927 + 0.575322i \(0.195123\pi\)
\(642\) 0 0
\(643\) −29.0908 −1.14723 −0.573615 0.819125i \(-0.694459\pi\)
−0.573615 + 0.819125i \(0.694459\pi\)
\(644\) 0 0
\(645\) 21.5095 0.846937
\(646\) 0 0
\(647\) −15.2691 −0.600289 −0.300145 0.953894i \(-0.597035\pi\)
−0.300145 + 0.953894i \(0.597035\pi\)
\(648\) 0 0
\(649\) 6.49433 0.254925
\(650\) 0 0
\(651\) −0.337476 −0.0132267
\(652\) 0 0
\(653\) −0.714425 −0.0279576 −0.0139788 0.999902i \(-0.504450\pi\)
−0.0139788 + 0.999902i \(0.504450\pi\)
\(654\) 0 0
\(655\) −15.7039 −0.613603
\(656\) 0 0
\(657\) 0.795102 0.0310199
\(658\) 0 0
\(659\) 15.6563 0.609884 0.304942 0.952371i \(-0.401363\pi\)
0.304942 + 0.952371i \(0.401363\pi\)
\(660\) 0 0
\(661\) 2.83528 0.110280 0.0551399 0.998479i \(-0.482440\pi\)
0.0551399 + 0.998479i \(0.482440\pi\)
\(662\) 0 0
\(663\) 0.551559 0.0214208
\(664\) 0 0
\(665\) 0.474683 0.0184074
\(666\) 0 0
\(667\) −35.1906 −1.36259
\(668\) 0 0
\(669\) 3.84264 0.148565
\(670\) 0 0
\(671\) −7.39575 −0.285510
\(672\) 0 0
\(673\) −37.6316 −1.45059 −0.725297 0.688437i \(-0.758297\pi\)
−0.725297 + 0.688437i \(0.758297\pi\)
\(674\) 0 0
\(675\) −1.65479 −0.0636927
\(676\) 0 0
\(677\) 36.0092 1.38394 0.691972 0.721924i \(-0.256742\pi\)
0.691972 + 0.721924i \(0.256742\pi\)
\(678\) 0 0
\(679\) 2.22115 0.0852401
\(680\) 0 0
\(681\) 22.8941 0.877305
\(682\) 0 0
\(683\) −19.9680 −0.764054 −0.382027 0.924151i \(-0.624774\pi\)
−0.382027 + 0.924151i \(0.624774\pi\)
\(684\) 0 0
\(685\) −27.3713 −1.04580
\(686\) 0 0
\(687\) −5.22979 −0.199529
\(688\) 0 0
\(689\) −9.76446 −0.371996
\(690\) 0 0
\(691\) −35.6854 −1.35754 −0.678769 0.734352i \(-0.737486\pi\)
−0.678769 + 0.734352i \(0.737486\pi\)
\(692\) 0 0
\(693\) 0.862883 0.0327782
\(694\) 0 0
\(695\) 16.9391 0.642537
\(696\) 0 0
\(697\) 2.92143 0.110657
\(698\) 0 0
\(699\) 6.48832 0.245411
\(700\) 0 0
\(701\) −23.4890 −0.887166 −0.443583 0.896233i \(-0.646293\pi\)
−0.443583 + 0.896233i \(0.646293\pi\)
\(702\) 0 0
\(703\) 2.89026 0.109008
\(704\) 0 0
\(705\) −11.9000 −0.448181
\(706\) 0 0
\(707\) 12.8318 0.482591
\(708\) 0 0
\(709\) −30.6568 −1.15134 −0.575671 0.817681i \(-0.695259\pi\)
−0.575671 + 0.817681i \(0.695259\pi\)
\(710\) 0 0
\(711\) 9.81445 0.368071
\(712\) 0 0
\(713\) −3.01588 −0.112946
\(714\) 0 0
\(715\) 1.82899 0.0684005
\(716\) 0 0
\(717\) 28.1498 1.05127
\(718\) 0 0
\(719\) −40.5661 −1.51286 −0.756430 0.654075i \(-0.773058\pi\)
−0.756430 + 0.654075i \(0.773058\pi\)
\(720\) 0 0
\(721\) 2.44743 0.0911472
\(722\) 0 0
\(723\) 6.31293 0.234780
\(724\) 0 0
\(725\) 7.55171 0.280463
\(726\) 0 0
\(727\) 8.18066 0.303404 0.151702 0.988426i \(-0.451525\pi\)
0.151702 + 0.988426i \(0.451525\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 6.48651 0.239912
\(732\) 0 0
\(733\) 22.2907 0.823325 0.411662 0.911336i \(-0.364948\pi\)
0.411662 + 0.911336i \(0.364948\pi\)
\(734\) 0 0
\(735\) −11.4411 −0.422013
\(736\) 0 0
\(737\) −8.53702 −0.314465
\(738\) 0 0
\(739\) −20.2328 −0.744275 −0.372137 0.928178i \(-0.621375\pi\)
−0.372137 + 0.928178i \(0.621375\pi\)
\(740\) 0 0
\(741\) 0.300773 0.0110492
\(742\) 0 0
\(743\) 10.5586 0.387358 0.193679 0.981065i \(-0.437958\pi\)
0.193679 + 0.981065i \(0.437958\pi\)
\(744\) 0 0
\(745\) −15.9755 −0.585295
\(746\) 0 0
\(747\) −5.49023 −0.200877
\(748\) 0 0
\(749\) 13.1179 0.479316
\(750\) 0 0
\(751\) 14.5954 0.532594 0.266297 0.963891i \(-0.414200\pi\)
0.266297 + 0.963891i \(0.414200\pi\)
\(752\) 0 0
\(753\) −4.48709 −0.163519
\(754\) 0 0
\(755\) 15.4739 0.563152
\(756\) 0 0
\(757\) 3.76356 0.136789 0.0683945 0.997658i \(-0.478212\pi\)
0.0683945 + 0.997658i \(0.478212\pi\)
\(758\) 0 0
\(759\) 7.71122 0.279900
\(760\) 0 0
\(761\) −2.80319 −0.101615 −0.0508077 0.998708i \(-0.516180\pi\)
−0.0508077 + 0.998708i \(0.516180\pi\)
\(762\) 0 0
\(763\) −1.82786 −0.0661729
\(764\) 0 0
\(765\) 1.00880 0.0364731
\(766\) 0 0
\(767\) 6.49433 0.234497
\(768\) 0 0
\(769\) −12.3119 −0.443977 −0.221989 0.975049i \(-0.571255\pi\)
−0.221989 + 0.975049i \(0.571255\pi\)
\(770\) 0 0
\(771\) −22.4505 −0.808537
\(772\) 0 0
\(773\) 41.4627 1.49131 0.745655 0.666333i \(-0.232137\pi\)
0.745655 + 0.666333i \(0.232137\pi\)
\(774\) 0 0
\(775\) 0.647191 0.0232478
\(776\) 0 0
\(777\) 8.29183 0.297468
\(778\) 0 0
\(779\) 1.59310 0.0570787
\(780\) 0 0
\(781\) 3.58545 0.128297
\(782\) 0 0
\(783\) −4.56356 −0.163088
\(784\) 0 0
\(785\) 29.2675 1.04460
\(786\) 0 0
\(787\) −34.1807 −1.21841 −0.609205 0.793013i \(-0.708511\pi\)
−0.609205 + 0.793013i \(0.708511\pi\)
\(788\) 0 0
\(789\) 12.0854 0.430251
\(790\) 0 0
\(791\) −12.5679 −0.446863
\(792\) 0 0
\(793\) −7.39575 −0.262631
\(794\) 0 0
\(795\) −17.8591 −0.633398
\(796\) 0 0
\(797\) 16.5951 0.587830 0.293915 0.955832i \(-0.405042\pi\)
0.293915 + 0.955832i \(0.405042\pi\)
\(798\) 0 0
\(799\) −3.58862 −0.126956
\(800\) 0 0
\(801\) 13.6585 0.482601
\(802\) 0 0
\(803\) 0.795102 0.0280586
\(804\) 0 0
\(805\) 12.1699 0.428933
\(806\) 0 0
\(807\) 21.7885 0.766990
\(808\) 0 0
\(809\) 16.8533 0.592530 0.296265 0.955106i \(-0.404259\pi\)
0.296265 + 0.955106i \(0.404259\pi\)
\(810\) 0 0
\(811\) −5.97346 −0.209757 −0.104878 0.994485i \(-0.533445\pi\)
−0.104878 + 0.994485i \(0.533445\pi\)
\(812\) 0 0
\(813\) −9.38761 −0.329238
\(814\) 0 0
\(815\) 10.7821 0.377682
\(816\) 0 0
\(817\) 3.53719 0.123751
\(818\) 0 0
\(819\) 0.862883 0.0301516
\(820\) 0 0
\(821\) 23.5104 0.820518 0.410259 0.911969i \(-0.365438\pi\)
0.410259 + 0.911969i \(0.365438\pi\)
\(822\) 0 0
\(823\) −18.2908 −0.637576 −0.318788 0.947826i \(-0.603276\pi\)
−0.318788 + 0.947826i \(0.603276\pi\)
\(824\) 0 0
\(825\) −1.65479 −0.0576122
\(826\) 0 0
\(827\) −41.8892 −1.45663 −0.728315 0.685243i \(-0.759696\pi\)
−0.728315 + 0.685243i \(0.759696\pi\)
\(828\) 0 0
\(829\) 39.5062 1.37211 0.686054 0.727550i \(-0.259341\pi\)
0.686054 + 0.727550i \(0.259341\pi\)
\(830\) 0 0
\(831\) 4.07524 0.141369
\(832\) 0 0
\(833\) −3.45024 −0.119544
\(834\) 0 0
\(835\) 0.736267 0.0254796
\(836\) 0 0
\(837\) −0.391103 −0.0135185
\(838\) 0 0
\(839\) 54.9994 1.89879 0.949394 0.314086i \(-0.101698\pi\)
0.949394 + 0.314086i \(0.101698\pi\)
\(840\) 0 0
\(841\) −8.17395 −0.281860
\(842\) 0 0
\(843\) −24.7933 −0.853927
\(844\) 0 0
\(845\) 1.82899 0.0629193
\(846\) 0 0
\(847\) 0.862883 0.0296490
\(848\) 0 0
\(849\) 11.3247 0.388661
\(850\) 0 0
\(851\) 74.1006 2.54014
\(852\) 0 0
\(853\) −6.02579 −0.206319 −0.103160 0.994665i \(-0.532895\pi\)
−0.103160 + 0.994665i \(0.532895\pi\)
\(854\) 0 0
\(855\) 0.550112 0.0188134
\(856\) 0 0
\(857\) −9.12058 −0.311553 −0.155776 0.987792i \(-0.549788\pi\)
−0.155776 + 0.987792i \(0.549788\pi\)
\(858\) 0 0
\(859\) 20.9607 0.715169 0.357584 0.933881i \(-0.383600\pi\)
0.357584 + 0.933881i \(0.383600\pi\)
\(860\) 0 0
\(861\) 4.57041 0.155759
\(862\) 0 0
\(863\) 19.6380 0.668485 0.334242 0.942487i \(-0.391520\pi\)
0.334242 + 0.942487i \(0.391520\pi\)
\(864\) 0 0
\(865\) −40.5660 −1.37928
\(866\) 0 0
\(867\) −16.6958 −0.567019
\(868\) 0 0
\(869\) 9.81445 0.332932
\(870\) 0 0
\(871\) −8.53702 −0.289266
\(872\) 0 0
\(873\) 2.57411 0.0871203
\(874\) 0 0
\(875\) −10.5026 −0.355054
\(876\) 0 0
\(877\) −21.2011 −0.715910 −0.357955 0.933739i \(-0.616526\pi\)
−0.357955 + 0.933739i \(0.616526\pi\)
\(878\) 0 0
\(879\) 7.05389 0.237922
\(880\) 0 0
\(881\) 11.7783 0.396822 0.198411 0.980119i \(-0.436422\pi\)
0.198411 + 0.980119i \(0.436422\pi\)
\(882\) 0 0
\(883\) 27.0887 0.911609 0.455804 0.890080i \(-0.349352\pi\)
0.455804 + 0.890080i \(0.349352\pi\)
\(884\) 0 0
\(885\) 11.8781 0.399277
\(886\) 0 0
\(887\) 4.58252 0.153866 0.0769330 0.997036i \(-0.475487\pi\)
0.0769330 + 0.997036i \(0.475487\pi\)
\(888\) 0 0
\(889\) −0.551207 −0.0184869
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 0 0
\(893\) −1.95693 −0.0654862
\(894\) 0 0
\(895\) 11.3060 0.377919
\(896\) 0 0
\(897\) 7.71122 0.257470
\(898\) 0 0
\(899\) 1.78482 0.0595271
\(900\) 0 0
\(901\) −5.38568 −0.179423
\(902\) 0 0
\(903\) 10.1478 0.337697
\(904\) 0 0
\(905\) −32.5483 −1.08194
\(906\) 0 0
\(907\) −45.8736 −1.52321 −0.761604 0.648042i \(-0.775588\pi\)
−0.761604 + 0.648042i \(0.775588\pi\)
\(908\) 0 0
\(909\) 14.8709 0.493236
\(910\) 0 0
\(911\) −11.7205 −0.388316 −0.194158 0.980970i \(-0.562197\pi\)
−0.194158 + 0.980970i \(0.562197\pi\)
\(912\) 0 0
\(913\) −5.49023 −0.181700
\(914\) 0 0
\(915\) −13.5268 −0.447181
\(916\) 0 0
\(917\) −7.40881 −0.244660
\(918\) 0 0
\(919\) −6.18872 −0.204147 −0.102074 0.994777i \(-0.532548\pi\)
−0.102074 + 0.994777i \(0.532548\pi\)
\(920\) 0 0
\(921\) 20.4440 0.673654
\(922\) 0 0
\(923\) 3.58545 0.118017
\(924\) 0 0
\(925\) −15.9016 −0.522840
\(926\) 0 0
\(927\) 2.83634 0.0931577
\(928\) 0 0
\(929\) −0.788059 −0.0258554 −0.0129277 0.999916i \(-0.504115\pi\)
−0.0129277 + 0.999916i \(0.504115\pi\)
\(930\) 0 0
\(931\) −1.88147 −0.0616626
\(932\) 0 0
\(933\) −15.9342 −0.521663
\(934\) 0 0
\(935\) 1.00880 0.0329912
\(936\) 0 0
\(937\) 22.7726 0.743948 0.371974 0.928243i \(-0.378681\pi\)
0.371974 + 0.928243i \(0.378681\pi\)
\(938\) 0 0
\(939\) 26.1754 0.854202
\(940\) 0 0
\(941\) −28.1370 −0.917240 −0.458620 0.888632i \(-0.651656\pi\)
−0.458620 + 0.888632i \(0.651656\pi\)
\(942\) 0 0
\(943\) 40.8438 1.33006
\(944\) 0 0
\(945\) 1.57821 0.0513391
\(946\) 0 0
\(947\) 6.73345 0.218808 0.109404 0.993997i \(-0.465106\pi\)
0.109404 + 0.993997i \(0.465106\pi\)
\(948\) 0 0
\(949\) 0.795102 0.0258101
\(950\) 0 0
\(951\) 16.2142 0.525783
\(952\) 0 0
\(953\) −0.990105 −0.0320726 −0.0160363 0.999871i \(-0.505105\pi\)
−0.0160363 + 0.999871i \(0.505105\pi\)
\(954\) 0 0
\(955\) 40.6726 1.31613
\(956\) 0 0
\(957\) −4.56356 −0.147519
\(958\) 0 0
\(959\) −12.9132 −0.416990
\(960\) 0 0
\(961\) −30.8470 −0.995066
\(962\) 0 0
\(963\) 15.2024 0.489889
\(964\) 0 0
\(965\) −3.58256 −0.115327
\(966\) 0 0
\(967\) −28.6223 −0.920430 −0.460215 0.887808i \(-0.652228\pi\)
−0.460215 + 0.887808i \(0.652228\pi\)
\(968\) 0 0
\(969\) 0.165894 0.00532929
\(970\) 0 0
\(971\) 4.46377 0.143249 0.0716247 0.997432i \(-0.477182\pi\)
0.0716247 + 0.997432i \(0.477182\pi\)
\(972\) 0 0
\(973\) 7.99155 0.256197
\(974\) 0 0
\(975\) −1.65479 −0.0529955
\(976\) 0 0
\(977\) 0.969686 0.0310230 0.0155115 0.999880i \(-0.495062\pi\)
0.0155115 + 0.999880i \(0.495062\pi\)
\(978\) 0 0
\(979\) 13.6585 0.436529
\(980\) 0 0
\(981\) −2.11832 −0.0676326
\(982\) 0 0
\(983\) 27.6060 0.880495 0.440247 0.897877i \(-0.354891\pi\)
0.440247 + 0.897877i \(0.354891\pi\)
\(984\) 0 0
\(985\) 33.0412 1.05278
\(986\) 0 0
\(987\) −5.61420 −0.178702
\(988\) 0 0
\(989\) 90.6864 2.88366
\(990\) 0 0
\(991\) −14.7750 −0.469342 −0.234671 0.972075i \(-0.575401\pi\)
−0.234671 + 0.972075i \(0.575401\pi\)
\(992\) 0 0
\(993\) −22.9140 −0.727153
\(994\) 0 0
\(995\) −41.6134 −1.31923
\(996\) 0 0
\(997\) −26.8328 −0.849804 −0.424902 0.905239i \(-0.639692\pi\)
−0.424902 + 0.905239i \(0.639692\pi\)
\(998\) 0 0
\(999\) 9.60944 0.304029
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 6864.2.a.cf.1.4 5
4.3 odd 2 3432.2.a.w.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3432.2.a.w.1.4 5 4.3 odd 2
6864.2.a.cf.1.4 5 1.1 even 1 trivial