Properties

Label 1368.2.a.k.1.2
Level $1368$
Weight $2$
Character 1368.1
Self dual yes
Analytic conductor $10.924$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1368.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.9235349965\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 456)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 1368.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.56155 q^{5} -1.56155 q^{7} +O(q^{10})\) \(q+1.56155 q^{5} -1.56155 q^{7} -5.56155 q^{11} +3.12311 q^{13} -6.68466 q^{17} +1.00000 q^{19} -9.12311 q^{23} -2.56155 q^{25} +8.24621 q^{29} -2.00000 q^{31} -2.43845 q^{35} -8.00000 q^{37} +5.12311 q^{41} -1.56155 q^{43} +6.68466 q^{47} -4.56155 q^{49} -4.24621 q^{53} -8.68466 q^{55} -12.0000 q^{59} +6.68466 q^{61} +4.87689 q^{65} +6.24621 q^{67} -16.9309 q^{73} +8.68466 q^{77} +11.3693 q^{79} -4.00000 q^{83} -10.4384 q^{85} +6.00000 q^{89} -4.87689 q^{91} +1.56155 q^{95} +4.24621 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{5} + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{5} + q^{7} - 7 q^{11} - 2 q^{13} - q^{17} + 2 q^{19} - 10 q^{23} - q^{25} - 4 q^{31} - 9 q^{35} - 16 q^{37} + 2 q^{41} + q^{43} + q^{47} - 5 q^{49} + 8 q^{53} - 5 q^{55} - 24 q^{59} + q^{61} + 18 q^{65} - 4 q^{67} - 5 q^{73} + 5 q^{77} - 2 q^{79} - 8 q^{83} - 25 q^{85} + 12 q^{89} - 18 q^{91} - q^{95} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 0 0
\(4\) 0 0
\(5\) 1.56155 0.698348 0.349174 0.937058i \(-0.386462\pi\)
0.349174 + 0.937058i \(0.386462\pi\)
\(6\) 0 0
\(7\) −1.56155 −0.590211 −0.295106 0.955465i \(-0.595355\pi\)
−0.295106 + 0.955465i \(0.595355\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −5.56155 −1.67687 −0.838436 0.545001i \(-0.816529\pi\)
−0.838436 + 0.545001i \(0.816529\pi\)
\(12\) 0 0
\(13\) 3.12311 0.866194 0.433097 0.901347i \(-0.357421\pi\)
0.433097 + 0.901347i \(0.357421\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6.68466 −1.62127 −0.810634 0.585553i \(-0.800877\pi\)
−0.810634 + 0.585553i \(0.800877\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −9.12311 −1.90230 −0.951150 0.308731i \(-0.900096\pi\)
−0.951150 + 0.308731i \(0.900096\pi\)
\(24\) 0 0
\(25\) −2.56155 −0.512311
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 8.24621 1.53128 0.765641 0.643268i \(-0.222422\pi\)
0.765641 + 0.643268i \(0.222422\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.43845 −0.412173
\(36\) 0 0
\(37\) −8.00000 −1.31519 −0.657596 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 5.12311 0.800095 0.400047 0.916494i \(-0.368994\pi\)
0.400047 + 0.916494i \(0.368994\pi\)
\(42\) 0 0
\(43\) −1.56155 −0.238135 −0.119067 0.992886i \(-0.537990\pi\)
−0.119067 + 0.992886i \(0.537990\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.68466 0.975058 0.487529 0.873107i \(-0.337898\pi\)
0.487529 + 0.873107i \(0.337898\pi\)
\(48\) 0 0
\(49\) −4.56155 −0.651650
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −4.24621 −0.583262 −0.291631 0.956531i \(-0.594198\pi\)
−0.291631 + 0.956531i \(0.594198\pi\)
\(54\) 0 0
\(55\) −8.68466 −1.17104
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) 6.68466 0.855883 0.427941 0.903806i \(-0.359239\pi\)
0.427941 + 0.903806i \(0.359239\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.87689 0.604904
\(66\) 0 0
\(67\) 6.24621 0.763096 0.381548 0.924349i \(-0.375391\pi\)
0.381548 + 0.924349i \(0.375391\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −16.9309 −1.98161 −0.990804 0.135303i \(-0.956799\pi\)
−0.990804 + 0.135303i \(0.956799\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 8.68466 0.989709
\(78\) 0 0
\(79\) 11.3693 1.27915 0.639574 0.768729i \(-0.279111\pi\)
0.639574 + 0.768729i \(0.279111\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) −10.4384 −1.13221
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −4.87689 −0.511237
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.56155 0.160212
\(96\) 0 0
\(97\) 4.24621 0.431137 0.215569 0.976489i \(-0.430839\pi\)
0.215569 + 0.976489i \(0.430839\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −7.12311 −0.708776 −0.354388 0.935099i \(-0.615311\pi\)
−0.354388 + 0.935099i \(0.615311\pi\)
\(102\) 0 0
\(103\) −6.00000 −0.591198 −0.295599 0.955312i \(-0.595519\pi\)
−0.295599 + 0.955312i \(0.595519\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 13.3693 1.29246 0.646230 0.763142i \(-0.276345\pi\)
0.646230 + 0.763142i \(0.276345\pi\)
\(108\) 0 0
\(109\) −2.24621 −0.215148 −0.107574 0.994197i \(-0.534308\pi\)
−0.107574 + 0.994197i \(0.534308\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) 0 0
\(115\) −14.2462 −1.32847
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 10.4384 0.956891
\(120\) 0 0
\(121\) 19.9309 1.81190
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −11.8078 −1.05612
\(126\) 0 0
\(127\) −18.0000 −1.59724 −0.798621 0.601834i \(-0.794437\pi\)
−0.798621 + 0.601834i \(0.794437\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −20.6847 −1.80723 −0.903613 0.428349i \(-0.859095\pi\)
−0.903613 + 0.428349i \(0.859095\pi\)
\(132\) 0 0
\(133\) −1.56155 −0.135404
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 13.8078 1.17968 0.589838 0.807521i \(-0.299191\pi\)
0.589838 + 0.807521i \(0.299191\pi\)
\(138\) 0 0
\(139\) −7.80776 −0.662246 −0.331123 0.943588i \(-0.607427\pi\)
−0.331123 + 0.943588i \(0.607427\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −17.3693 −1.45250
\(144\) 0 0
\(145\) 12.8769 1.06937
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −3.80776 −0.311944 −0.155972 0.987761i \(-0.549851\pi\)
−0.155972 + 0.987761i \(0.549851\pi\)
\(150\) 0 0
\(151\) −17.1231 −1.39346 −0.696729 0.717334i \(-0.745362\pi\)
−0.696729 + 0.717334i \(0.745362\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −3.12311 −0.250854
\(156\) 0 0
\(157\) −12.2462 −0.977354 −0.488677 0.872465i \(-0.662520\pi\)
−0.488677 + 0.872465i \(0.662520\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 14.2462 1.12276
\(162\) 0 0
\(163\) −2.24621 −0.175937 −0.0879684 0.996123i \(-0.528037\pi\)
−0.0879684 + 0.996123i \(0.528037\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 8.87689 0.686915 0.343457 0.939168i \(-0.388402\pi\)
0.343457 + 0.939168i \(0.388402\pi\)
\(168\) 0 0
\(169\) −3.24621 −0.249709
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 4.24621 0.322833 0.161417 0.986886i \(-0.448394\pi\)
0.161417 + 0.986886i \(0.448394\pi\)
\(174\) 0 0
\(175\) 4.00000 0.302372
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 15.1231 1.13035 0.565177 0.824970i \(-0.308808\pi\)
0.565177 + 0.824970i \(0.308808\pi\)
\(180\) 0 0
\(181\) 15.1231 1.12409 0.562046 0.827106i \(-0.310014\pi\)
0.562046 + 0.827106i \(0.310014\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −12.4924 −0.918461
\(186\) 0 0
\(187\) 37.1771 2.71866
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1.31534 0.0951748 0.0475874 0.998867i \(-0.484847\pi\)
0.0475874 + 0.998867i \(0.484847\pi\)
\(192\) 0 0
\(193\) 18.8769 1.35879 0.679394 0.733773i \(-0.262243\pi\)
0.679394 + 0.733773i \(0.262243\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1.36932 0.0975598 0.0487799 0.998810i \(-0.484467\pi\)
0.0487799 + 0.998810i \(0.484467\pi\)
\(198\) 0 0
\(199\) 2.93087 0.207764 0.103882 0.994590i \(-0.466874\pi\)
0.103882 + 0.994590i \(0.466874\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −12.8769 −0.903781
\(204\) 0 0
\(205\) 8.00000 0.558744
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −5.56155 −0.384701
\(210\) 0 0
\(211\) 5.75379 0.396107 0.198054 0.980191i \(-0.436538\pi\)
0.198054 + 0.980191i \(0.436538\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −2.43845 −0.166301
\(216\) 0 0
\(217\) 3.12311 0.212010
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −20.8769 −1.40433
\(222\) 0 0
\(223\) −11.3693 −0.761346 −0.380673 0.924710i \(-0.624308\pi\)
−0.380673 + 0.924710i \(0.624308\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 8.87689 0.589180 0.294590 0.955624i \(-0.404817\pi\)
0.294590 + 0.955624i \(0.404817\pi\)
\(228\) 0 0
\(229\) 3.56155 0.235354 0.117677 0.993052i \(-0.462455\pi\)
0.117677 + 0.993052i \(0.462455\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 14.6847 0.962024 0.481012 0.876714i \(-0.340269\pi\)
0.481012 + 0.876714i \(0.340269\pi\)
\(234\) 0 0
\(235\) 10.4384 0.680929
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −6.19224 −0.400542 −0.200271 0.979740i \(-0.564182\pi\)
−0.200271 + 0.979740i \(0.564182\pi\)
\(240\) 0 0
\(241\) −11.3693 −0.732362 −0.366181 0.930544i \(-0.619335\pi\)
−0.366181 + 0.930544i \(0.619335\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −7.12311 −0.455079
\(246\) 0 0
\(247\) 3.12311 0.198718
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 7.80776 0.492822 0.246411 0.969165i \(-0.420749\pi\)
0.246411 + 0.969165i \(0.420749\pi\)
\(252\) 0 0
\(253\) 50.7386 3.18991
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −13.1231 −0.818597 −0.409298 0.912401i \(-0.634227\pi\)
−0.409298 + 0.912401i \(0.634227\pi\)
\(258\) 0 0
\(259\) 12.4924 0.776241
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 14.6847 0.905495 0.452747 0.891639i \(-0.350444\pi\)
0.452747 + 0.891639i \(0.350444\pi\)
\(264\) 0 0
\(265\) −6.63068 −0.407320
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −10.0000 −0.609711 −0.304855 0.952399i \(-0.598608\pi\)
−0.304855 + 0.952399i \(0.598608\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 14.2462 0.859079
\(276\) 0 0
\(277\) −18.6847 −1.12265 −0.561326 0.827595i \(-0.689709\pi\)
−0.561326 + 0.827595i \(0.689709\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 24.2462 1.44641 0.723204 0.690635i \(-0.242669\pi\)
0.723204 + 0.690635i \(0.242669\pi\)
\(282\) 0 0
\(283\) 28.6847 1.70513 0.852563 0.522625i \(-0.175047\pi\)
0.852563 + 0.522625i \(0.175047\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −8.00000 −0.472225
\(288\) 0 0
\(289\) 27.6847 1.62851
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1.12311 0.0656125 0.0328063 0.999462i \(-0.489556\pi\)
0.0328063 + 0.999462i \(0.489556\pi\)
\(294\) 0 0
\(295\) −18.7386 −1.09101
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −28.4924 −1.64776
\(300\) 0 0
\(301\) 2.43845 0.140550
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 10.4384 0.597704
\(306\) 0 0
\(307\) −4.49242 −0.256396 −0.128198 0.991749i \(-0.540919\pi\)
−0.128198 + 0.991749i \(0.540919\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 20.9309 1.18688 0.593440 0.804878i \(-0.297769\pi\)
0.593440 + 0.804878i \(0.297769\pi\)
\(312\) 0 0
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 18.8769 1.06023 0.530116 0.847925i \(-0.322148\pi\)
0.530116 + 0.847925i \(0.322148\pi\)
\(318\) 0 0
\(319\) −45.8617 −2.56776
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −6.68466 −0.371944
\(324\) 0 0
\(325\) −8.00000 −0.443760
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −10.4384 −0.575490
\(330\) 0 0
\(331\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 9.75379 0.532906
\(336\) 0 0
\(337\) 20.7386 1.12971 0.564853 0.825192i \(-0.308933\pi\)
0.564853 + 0.825192i \(0.308933\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 11.1231 0.602350
\(342\) 0 0
\(343\) 18.0540 0.974823
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −27.4233 −1.47216 −0.736080 0.676895i \(-0.763325\pi\)
−0.736080 + 0.676895i \(0.763325\pi\)
\(348\) 0 0
\(349\) 17.3153 0.926869 0.463434 0.886131i \(-0.346617\pi\)
0.463434 + 0.886131i \(0.346617\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 28.2462 1.50339 0.751697 0.659509i \(-0.229236\pi\)
0.751697 + 0.659509i \(0.229236\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −0.438447 −0.0231404 −0.0115702 0.999933i \(-0.503683\pi\)
−0.0115702 + 0.999933i \(0.503683\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −26.4384 −1.38385
\(366\) 0 0
\(367\) 2.24621 0.117251 0.0586256 0.998280i \(-0.481328\pi\)
0.0586256 + 0.998280i \(0.481328\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 6.63068 0.344248
\(372\) 0 0
\(373\) −36.4924 −1.88951 −0.944753 0.327783i \(-0.893698\pi\)
−0.944753 + 0.327783i \(0.893698\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 25.7538 1.32639
\(378\) 0 0
\(379\) −20.8769 −1.07237 −0.536187 0.844099i \(-0.680136\pi\)
−0.536187 + 0.844099i \(0.680136\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −12.8769 −0.657979 −0.328989 0.944334i \(-0.606708\pi\)
−0.328989 + 0.944334i \(0.606708\pi\)
\(384\) 0 0
\(385\) 13.5616 0.691161
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −18.9309 −0.959833 −0.479917 0.877314i \(-0.659333\pi\)
−0.479917 + 0.877314i \(0.659333\pi\)
\(390\) 0 0
\(391\) 60.9848 3.08414
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 17.7538 0.893290
\(396\) 0 0
\(397\) −22.6847 −1.13851 −0.569255 0.822161i \(-0.692768\pi\)
−0.569255 + 0.822161i \(0.692768\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0.630683 0.0314948 0.0157474 0.999876i \(-0.494987\pi\)
0.0157474 + 0.999876i \(0.494987\pi\)
\(402\) 0 0
\(403\) −6.24621 −0.311146
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 44.4924 2.20541
\(408\) 0 0
\(409\) −0.246211 −0.0121744 −0.00608718 0.999981i \(-0.501938\pi\)
−0.00608718 + 0.999981i \(0.501938\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 18.7386 0.922068
\(414\) 0 0
\(415\) −6.24621 −0.306614
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 26.7386 1.30627 0.653134 0.757242i \(-0.273454\pi\)
0.653134 + 0.757242i \(0.273454\pi\)
\(420\) 0 0
\(421\) 4.87689 0.237685 0.118843 0.992913i \(-0.462082\pi\)
0.118843 + 0.992913i \(0.462082\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 17.1231 0.830593
\(426\) 0 0
\(427\) −10.4384 −0.505152
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 11.6155 0.559500 0.279750 0.960073i \(-0.409748\pi\)
0.279750 + 0.960073i \(0.409748\pi\)
\(432\) 0 0
\(433\) −8.24621 −0.396288 −0.198144 0.980173i \(-0.563491\pi\)
−0.198144 + 0.980173i \(0.563491\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −9.12311 −0.436417
\(438\) 0 0
\(439\) 28.2462 1.34812 0.674059 0.738677i \(-0.264549\pi\)
0.674059 + 0.738677i \(0.264549\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −12.1922 −0.579271 −0.289635 0.957137i \(-0.593534\pi\)
−0.289635 + 0.957137i \(0.593534\pi\)
\(444\) 0 0
\(445\) 9.36932 0.444148
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −35.3693 −1.66918 −0.834591 0.550871i \(-0.814296\pi\)
−0.834591 + 0.550871i \(0.814296\pi\)
\(450\) 0 0
\(451\) −28.4924 −1.34166
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −7.61553 −0.357021
\(456\) 0 0
\(457\) −40.0540 −1.87365 −0.936823 0.349804i \(-0.886248\pi\)
−0.936823 + 0.349804i \(0.886248\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −32.3002 −1.50437 −0.752185 0.658952i \(-0.771000\pi\)
−0.752185 + 0.658952i \(0.771000\pi\)
\(462\) 0 0
\(463\) 31.8078 1.47823 0.739116 0.673578i \(-0.235243\pi\)
0.739116 + 0.673578i \(0.235243\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −10.9309 −0.505820 −0.252910 0.967490i \(-0.581388\pi\)
−0.252910 + 0.967490i \(0.581388\pi\)
\(468\) 0 0
\(469\) −9.75379 −0.450388
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 8.68466 0.399321
\(474\) 0 0
\(475\) −2.56155 −0.117532
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −15.3693 −0.702242 −0.351121 0.936330i \(-0.614199\pi\)
−0.351121 + 0.936330i \(0.614199\pi\)
\(480\) 0 0
\(481\) −24.9848 −1.13921
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 6.63068 0.301084
\(486\) 0 0
\(487\) −33.1231 −1.50095 −0.750476 0.660898i \(-0.770176\pi\)
−0.750476 + 0.660898i \(0.770176\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 14.7386 0.665145 0.332573 0.943078i \(-0.392083\pi\)
0.332573 + 0.943078i \(0.392083\pi\)
\(492\) 0 0
\(493\) −55.1231 −2.48262
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −9.17708 −0.410823 −0.205411 0.978676i \(-0.565853\pi\)
−0.205411 + 0.978676i \(0.565853\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −25.6155 −1.14214 −0.571070 0.820901i \(-0.693472\pi\)
−0.571070 + 0.820901i \(0.693472\pi\)
\(504\) 0 0
\(505\) −11.1231 −0.494972
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −26.0000 −1.15243 −0.576215 0.817298i \(-0.695471\pi\)
−0.576215 + 0.817298i \(0.695471\pi\)
\(510\) 0 0
\(511\) 26.4384 1.16957
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −9.36932 −0.412861
\(516\) 0 0
\(517\) −37.1771 −1.63505
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −9.61553 −0.421264 −0.210632 0.977565i \(-0.567552\pi\)
−0.210632 + 0.977565i \(0.567552\pi\)
\(522\) 0 0
\(523\) 17.3693 0.759507 0.379754 0.925088i \(-0.376009\pi\)
0.379754 + 0.925088i \(0.376009\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 13.3693 0.582377
\(528\) 0 0
\(529\) 60.2311 2.61874
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 16.0000 0.693037
\(534\) 0 0
\(535\) 20.8769 0.902587
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 25.3693 1.09273
\(540\) 0 0
\(541\) −17.8078 −0.765616 −0.382808 0.923828i \(-0.625043\pi\)
−0.382808 + 0.923828i \(0.625043\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −3.50758 −0.150248
\(546\) 0 0
\(547\) −0.384472 −0.0164388 −0.00821942 0.999966i \(-0.502616\pi\)
−0.00821942 + 0.999966i \(0.502616\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 8.24621 0.351300
\(552\) 0 0
\(553\) −17.7538 −0.754968
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −45.5616 −1.93050 −0.965252 0.261319i \(-0.915843\pi\)
−0.965252 + 0.261319i \(0.915843\pi\)
\(558\) 0 0
\(559\) −4.87689 −0.206271
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −4.00000 −0.168580 −0.0842900 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) 0 0
\(565\) 3.12311 0.131390
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 18.4924 0.775243 0.387621 0.921819i \(-0.373297\pi\)
0.387621 + 0.921819i \(0.373297\pi\)
\(570\) 0 0
\(571\) 6.73863 0.282003 0.141002 0.990009i \(-0.454968\pi\)
0.141002 + 0.990009i \(0.454968\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 23.3693 0.974568
\(576\) 0 0
\(577\) −11.5616 −0.481314 −0.240657 0.970610i \(-0.577363\pi\)
−0.240657 + 0.970610i \(0.577363\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 6.24621 0.259137
\(582\) 0 0
\(583\) 23.6155 0.978055
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 18.0540 0.745167 0.372584 0.927999i \(-0.378472\pi\)
0.372584 + 0.927999i \(0.378472\pi\)
\(588\) 0 0
\(589\) −2.00000 −0.0824086
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −38.4924 −1.58069 −0.790347 0.612659i \(-0.790100\pi\)
−0.790347 + 0.612659i \(0.790100\pi\)
\(594\) 0 0
\(595\) 16.3002 0.668242
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −15.5076 −0.633622 −0.316811 0.948489i \(-0.602612\pi\)
−0.316811 + 0.948489i \(0.602612\pi\)
\(600\) 0 0
\(601\) 39.8617 1.62599 0.812997 0.582268i \(-0.197834\pi\)
0.812997 + 0.582268i \(0.197834\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 31.1231 1.26533
\(606\) 0 0
\(607\) −22.9848 −0.932926 −0.466463 0.884541i \(-0.654472\pi\)
−0.466463 + 0.884541i \(0.654472\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 20.8769 0.844589
\(612\) 0 0
\(613\) 38.7926 1.56682 0.783409 0.621506i \(-0.213479\pi\)
0.783409 + 0.621506i \(0.213479\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 14.3002 0.575704 0.287852 0.957675i \(-0.407059\pi\)
0.287852 + 0.957675i \(0.407059\pi\)
\(618\) 0 0
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −9.36932 −0.375374
\(624\) 0 0
\(625\) −5.63068 −0.225227
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 53.4773 2.13228
\(630\) 0 0
\(631\) 22.9309 0.912864 0.456432 0.889758i \(-0.349127\pi\)
0.456432 + 0.889758i \(0.349127\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −28.1080 −1.11543
\(636\) 0 0
\(637\) −14.2462 −0.564455
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 40.7386 1.60908 0.804540 0.593899i \(-0.202412\pi\)
0.804540 + 0.593899i \(0.202412\pi\)
\(642\) 0 0
\(643\) 9.94602 0.392233 0.196116 0.980581i \(-0.437167\pi\)
0.196116 + 0.980581i \(0.437167\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 21.8078 0.857352 0.428676 0.903458i \(-0.358980\pi\)
0.428676 + 0.903458i \(0.358980\pi\)
\(648\) 0 0
\(649\) 66.7386 2.61972
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −40.3002 −1.57707 −0.788534 0.614991i \(-0.789160\pi\)
−0.788534 + 0.614991i \(0.789160\pi\)
\(654\) 0 0
\(655\) −32.3002 −1.26207
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −21.3693 −0.832430 −0.416215 0.909266i \(-0.636644\pi\)
−0.416215 + 0.909266i \(0.636644\pi\)
\(660\) 0 0
\(661\) −42.7386 −1.66234 −0.831170 0.556018i \(-0.812328\pi\)
−0.831170 + 0.556018i \(0.812328\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −2.43845 −0.0945589
\(666\) 0 0
\(667\) −75.2311 −2.91296
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −37.1771 −1.43521
\(672\) 0 0
\(673\) −31.8617 −1.22818 −0.614090 0.789236i \(-0.710477\pi\)
−0.614090 + 0.789236i \(0.710477\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −38.0000 −1.46046 −0.730229 0.683202i \(-0.760587\pi\)
−0.730229 + 0.683202i \(0.760587\pi\)
\(678\) 0 0
\(679\) −6.63068 −0.254462
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 9.86174 0.377349 0.188674 0.982040i \(-0.439581\pi\)
0.188674 + 0.982040i \(0.439581\pi\)
\(684\) 0 0
\(685\) 21.5616 0.823825
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −13.2614 −0.505218
\(690\) 0 0
\(691\) 21.0691 0.801507 0.400754 0.916186i \(-0.368748\pi\)
0.400754 + 0.916186i \(0.368748\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −12.1922 −0.462478
\(696\) 0 0
\(697\) −34.2462 −1.29717
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −23.1231 −0.873348 −0.436674 0.899620i \(-0.643844\pi\)
−0.436674 + 0.899620i \(0.643844\pi\)
\(702\) 0 0
\(703\) −8.00000 −0.301726
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 11.1231 0.418327
\(708\) 0 0
\(709\) 9.50758 0.357065 0.178532 0.983934i \(-0.442865\pi\)
0.178532 + 0.983934i \(0.442865\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 18.2462 0.683326
\(714\) 0 0
\(715\) −27.1231 −1.01435
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −37.4233 −1.39565 −0.697827 0.716267i \(-0.745849\pi\)
−0.697827 + 0.716267i \(0.745849\pi\)
\(720\) 0 0
\(721\) 9.36932 0.348932
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −21.1231 −0.784492
\(726\) 0 0
\(727\) 13.5616 0.502970 0.251485 0.967861i \(-0.419081\pi\)
0.251485 + 0.967861i \(0.419081\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 10.4384 0.386080
\(732\) 0 0
\(733\) −18.4924 −0.683033 −0.341517 0.939876i \(-0.610941\pi\)
−0.341517 + 0.939876i \(0.610941\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −34.7386 −1.27961
\(738\) 0 0
\(739\) 25.1771 0.926154 0.463077 0.886318i \(-0.346745\pi\)
0.463077 + 0.886318i \(0.346745\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −40.4924 −1.48552 −0.742761 0.669556i \(-0.766484\pi\)
−0.742761 + 0.669556i \(0.766484\pi\)
\(744\) 0 0
\(745\) −5.94602 −0.217845
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −20.8769 −0.762825
\(750\) 0 0
\(751\) 33.6155 1.22665 0.613324 0.789831i \(-0.289832\pi\)
0.613324 + 0.789831i \(0.289832\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −26.7386 −0.973119
\(756\) 0 0
\(757\) −32.0540 −1.16502 −0.582511 0.812823i \(-0.697930\pi\)
−0.582511 + 0.812823i \(0.697930\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −47.6695 −1.72802 −0.864009 0.503476i \(-0.832054\pi\)
−0.864009 + 0.503476i \(0.832054\pi\)
\(762\) 0 0
\(763\) 3.50758 0.126983
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −37.4773 −1.35323
\(768\) 0 0
\(769\) −34.3002 −1.23690 −0.618448 0.785826i \(-0.712238\pi\)
−0.618448 + 0.785826i \(0.712238\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −17.6155 −0.633587 −0.316793 0.948495i \(-0.602606\pi\)
−0.316793 + 0.948495i \(0.602606\pi\)
\(774\) 0 0
\(775\) 5.12311 0.184027
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 5.12311 0.183554
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −19.1231 −0.682533
\(786\) 0 0
\(787\) 10.2462 0.365238 0.182619 0.983184i \(-0.441543\pi\)
0.182619 + 0.983184i \(0.441543\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −3.12311 −0.111045
\(792\) 0 0
\(793\) 20.8769 0.741360
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −41.6155 −1.47410 −0.737049 0.675840i \(-0.763781\pi\)
−0.737049 + 0.675840i \(0.763781\pi\)
\(798\) 0 0
\(799\) −44.6847 −1.58083
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 94.1619 3.32290
\(804\) 0 0
\(805\) 22.2462 0.784076
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 7.94602 0.279367 0.139684 0.990196i \(-0.455391\pi\)
0.139684 + 0.990196i \(0.455391\pi\)
\(810\) 0 0
\(811\) −35.1231 −1.23334 −0.616670 0.787222i \(-0.711519\pi\)
−0.616670 + 0.787222i \(0.711519\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −3.50758 −0.122865
\(816\) 0 0
\(817\) −1.56155 −0.0546318
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1.94602 0.0679167 0.0339584 0.999423i \(-0.489189\pi\)
0.0339584 + 0.999423i \(0.489189\pi\)
\(822\) 0 0
\(823\) −40.3002 −1.40478 −0.702388 0.711794i \(-0.747883\pi\)
−0.702388 + 0.711794i \(0.747883\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 14.7386 0.512513 0.256256 0.966609i \(-0.417511\pi\)
0.256256 + 0.966609i \(0.417511\pi\)
\(828\) 0 0
\(829\) 4.00000 0.138926 0.0694629 0.997585i \(-0.477871\pi\)
0.0694629 + 0.997585i \(0.477871\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 30.4924 1.05650
\(834\) 0 0
\(835\) 13.8617 0.479705
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 49.8617 1.72142 0.860709 0.509097i \(-0.170021\pi\)
0.860709 + 0.509097i \(0.170021\pi\)
\(840\) 0 0
\(841\) 39.0000 1.34483
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −5.06913 −0.174383
\(846\) 0 0
\(847\) −31.1231 −1.06940
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 72.9848 2.50189
\(852\) 0 0
\(853\) −7.26137 −0.248624 −0.124312 0.992243i \(-0.539672\pi\)
−0.124312 + 0.992243i \(0.539672\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −15.8617 −0.541827 −0.270913 0.962604i \(-0.587326\pi\)
−0.270913 + 0.962604i \(0.587326\pi\)
\(858\) 0 0
\(859\) −28.3002 −0.965590 −0.482795 0.875733i \(-0.660378\pi\)
−0.482795 + 0.875733i \(0.660378\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 12.0000 0.408485 0.204242 0.978920i \(-0.434527\pi\)
0.204242 + 0.978920i \(0.434527\pi\)
\(864\) 0 0
\(865\) 6.63068 0.225450
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −63.2311 −2.14497
\(870\) 0 0
\(871\) 19.5076 0.660989
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 18.4384 0.623333
\(876\) 0 0
\(877\) −22.6307 −0.764184 −0.382092 0.924124i \(-0.624796\pi\)
−0.382092 + 0.924124i \(0.624796\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 30.3002 1.02084 0.510420 0.859925i \(-0.329490\pi\)
0.510420 + 0.859925i \(0.329490\pi\)
\(882\) 0 0
\(883\) −35.3153 −1.18846 −0.594228 0.804297i \(-0.702542\pi\)
−0.594228 + 0.804297i \(0.702542\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −51.2311 −1.72017 −0.860085 0.510150i \(-0.829590\pi\)
−0.860085 + 0.510150i \(0.829590\pi\)
\(888\) 0 0
\(889\) 28.1080 0.942710
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 6.68466 0.223694
\(894\) 0 0
\(895\) 23.6155 0.789380
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −16.4924 −0.550053
\(900\) 0 0
\(901\) 28.3845 0.945624
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 23.6155 0.785007
\(906\) 0 0
\(907\) −43.1231 −1.43188 −0.715940 0.698162i \(-0.754001\pi\)
−0.715940 + 0.698162i \(0.754001\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 7.12311 0.235999 0.118000 0.993014i \(-0.462352\pi\)
0.118000 + 0.993014i \(0.462352\pi\)
\(912\) 0 0
\(913\) 22.2462 0.736242
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 32.3002 1.06665
\(918\) 0 0
\(919\) 11.5076 0.379600 0.189800 0.981823i \(-0.439216\pi\)
0.189800 + 0.981823i \(0.439216\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 20.4924 0.673787
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −14.0000 −0.459325 −0.229663 0.973270i \(-0.573762\pi\)
−0.229663 + 0.973270i \(0.573762\pi\)
\(930\) 0 0
\(931\) −4.56155 −0.149499
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 58.0540 1.89857
\(936\) 0 0
\(937\) −12.4384 −0.406346 −0.203173 0.979143i \(-0.565125\pi\)
−0.203173 + 0.979143i \(0.565125\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 30.0000 0.977972 0.488986 0.872292i \(-0.337367\pi\)
0.488986 + 0.872292i \(0.337367\pi\)
\(942\) 0 0
\(943\) −46.7386 −1.52202
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −4.00000 −0.129983 −0.0649913 0.997886i \(-0.520702\pi\)
−0.0649913 + 0.997886i \(0.520702\pi\)
\(948\) 0 0
\(949\) −52.8769 −1.71646
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −29.6155 −0.959341 −0.479671 0.877449i \(-0.659244\pi\)
−0.479671 + 0.877449i \(0.659244\pi\)
\(954\) 0 0
\(955\) 2.05398 0.0664651
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −21.5616 −0.696259
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 29.4773 0.948907
\(966\) 0 0
\(967\) 22.7386 0.731225 0.365613 0.930767i \(-0.380860\pi\)
0.365613 + 0.930767i \(0.380860\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −20.0000 −0.641831 −0.320915 0.947108i \(-0.603990\pi\)
−0.320915 + 0.947108i \(0.603990\pi\)
\(972\) 0 0
\(973\) 12.1922 0.390865
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 58.9848 1.88709 0.943546 0.331241i \(-0.107467\pi\)
0.943546 + 0.331241i \(0.107467\pi\)
\(978\) 0 0
\(979\) −33.3693 −1.06649
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 18.7386 0.597670 0.298835 0.954305i \(-0.403402\pi\)
0.298835 + 0.954305i \(0.403402\pi\)
\(984\) 0 0
\(985\) 2.13826 0.0681306
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 14.2462 0.453003
\(990\) 0 0
\(991\) 35.8617 1.13919 0.569593 0.821927i \(-0.307101\pi\)
0.569593 + 0.821927i \(0.307101\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 4.57671 0.145091
\(996\) 0 0
\(997\) 30.7926 0.975212 0.487606 0.873064i \(-0.337870\pi\)
0.487606 + 0.873064i \(0.337870\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1368.2.a.k.1.2 2
3.2 odd 2 456.2.a.f.1.1 2
4.3 odd 2 2736.2.a.z.1.2 2
12.11 even 2 912.2.a.m.1.1 2
24.5 odd 2 3648.2.a.bl.1.2 2
24.11 even 2 3648.2.a.br.1.2 2
57.56 even 2 8664.2.a.r.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
456.2.a.f.1.1 2 3.2 odd 2
912.2.a.m.1.1 2 12.11 even 2
1368.2.a.k.1.2 2 1.1 even 1 trivial
2736.2.a.z.1.2 2 4.3 odd 2
3648.2.a.bl.1.2 2 24.5 odd 2
3648.2.a.br.1.2 2 24.11 even 2
8664.2.a.r.1.1 2 57.56 even 2