Properties

Label 1216.2.a.v.1.3
Level $1216$
Weight $2$
Character 1216.1
Self dual yes
Analytic conductor $9.710$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.70980888579\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.961.1
Defining polynomial: \(x^{3} - x^{2} - 10 x + 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 152)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(3.29707\) of defining polynomial
Character \(\chi\) \(=\) 1216.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.29707 q^{3} +3.08387 q^{5} -1.78680 q^{7} +7.87067 q^{9} +O(q^{10})\) \(q+3.29707 q^{3} +3.08387 q^{5} -1.78680 q^{7} +7.87067 q^{9} -5.08387 q^{11} +1.29707 q^{13} +10.1677 q^{15} +0.213198 q^{17} -1.00000 q^{19} -5.89121 q^{21} +3.72347 q^{23} +4.51027 q^{25} +16.0590 q^{27} -0.870674 q^{29} -16.7619 q^{33} -5.51027 q^{35} +2.00000 q^{37} +4.27653 q^{39} +8.59414 q^{41} -3.67801 q^{43} +24.2722 q^{45} +4.65748 q^{47} -3.80734 q^{49} +0.702929 q^{51} -11.0384 q^{53} -15.6780 q^{55} -3.29707 q^{57} -4.70293 q^{59} -3.51027 q^{61} -14.0633 q^{63} +4.00000 q^{65} +1.12933 q^{67} +12.2765 q^{69} -8.76189 q^{71} +6.80734 q^{73} +14.8707 q^{75} +9.08387 q^{77} -14.5941 q^{79} +29.3355 q^{81} -9.74135 q^{83} +0.657476 q^{85} -2.87067 q^{87} -6.76189 q^{89} -2.31761 q^{91} -3.08387 q^{95} +4.16774 q^{97} -40.0135 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{3} - q^{5} - 4 q^{7} + 12 q^{9} + O(q^{10}) \) \( 3 q + q^{3} - q^{5} - 4 q^{7} + 12 q^{9} - 5 q^{11} - 5 q^{13} + 10 q^{15} + 2 q^{17} - 3 q^{19} + 9 q^{21} + 5 q^{23} + 6 q^{25} + q^{27} + 9 q^{29} - 12 q^{33} - 9 q^{35} + 6 q^{37} + 19 q^{39} + 8 q^{41} + 17 q^{43} + 27 q^{45} + q^{47} + 5 q^{49} + 11 q^{51} - q^{53} - 19 q^{55} - q^{57} - 23 q^{59} - 3 q^{61} - 47 q^{63} + 12 q^{65} + 15 q^{67} + 43 q^{69} + 12 q^{71} + 4 q^{73} + 33 q^{75} + 17 q^{77} - 26 q^{79} + 47 q^{81} - 6 q^{83} - 11 q^{85} + 3 q^{87} + 18 q^{89} + 17 q^{91} + q^{95} - 8 q^{97} - 51 q^{99} + O(q^{100}) \)

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
\(3\) 3.29707 1.90356 0.951782 0.306774i \(-0.0992497\pi\)
0.951782 + 0.306774i \(0.0992497\pi\)
\(4\) 0 0
\(5\) 3.08387 1.37915 0.689575 0.724214i \(-0.257797\pi\)
0.689575 + 0.724214i \(0.257797\pi\)
\(6\) 0 0
\(7\) −1.78680 −0.675348 −0.337674 0.941263i \(-0.609640\pi\)
−0.337674 + 0.941263i \(0.609640\pi\)
\(8\) 0 0
\(9\) 7.87067 2.62356
\(10\) 0 0
\(11\) −5.08387 −1.53285 −0.766423 0.642337i \(-0.777965\pi\)
−0.766423 + 0.642337i \(0.777965\pi\)
\(12\) 0 0
\(13\) 1.29707 0.359743 0.179871 0.983690i \(-0.442432\pi\)
0.179871 + 0.983690i \(0.442432\pi\)
\(14\) 0 0
\(15\) 10.1677 2.62530
\(16\) 0 0
\(17\) 0.213198 0.0517082 0.0258541 0.999666i \(-0.491769\pi\)
0.0258541 + 0.999666i \(0.491769\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −5.89121 −1.28557
\(22\) 0 0
\(23\) 3.72347 0.776397 0.388198 0.921576i \(-0.373098\pi\)
0.388198 + 0.921576i \(0.373098\pi\)
\(24\) 0 0
\(25\) 4.51027 0.902054
\(26\) 0 0
\(27\) 16.0590 3.09055
\(28\) 0 0
\(29\) −0.870674 −0.161680 −0.0808401 0.996727i \(-0.525760\pi\)
−0.0808401 + 0.996727i \(0.525760\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) −16.7619 −2.91787
\(34\) 0 0
\(35\) −5.51027 −0.931405
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) 4.27653 0.684793
\(40\) 0 0
\(41\) 8.59414 1.34218 0.671090 0.741376i \(-0.265827\pi\)
0.671090 + 0.741376i \(0.265827\pi\)
\(42\) 0 0
\(43\) −3.67801 −0.560892 −0.280446 0.959870i \(-0.590482\pi\)
−0.280446 + 0.959870i \(0.590482\pi\)
\(44\) 0 0
\(45\) 24.2722 3.61828
\(46\) 0 0
\(47\) 4.65748 0.679363 0.339681 0.940541i \(-0.389681\pi\)
0.339681 + 0.940541i \(0.389681\pi\)
\(48\) 0 0
\(49\) −3.80734 −0.543906
\(50\) 0 0
\(51\) 0.702929 0.0984298
\(52\) 0 0
\(53\) −11.0384 −1.51624 −0.758122 0.652113i \(-0.773883\pi\)
−0.758122 + 0.652113i \(0.773883\pi\)
\(54\) 0 0
\(55\) −15.6780 −2.11402
\(56\) 0 0
\(57\) −3.29707 −0.436708
\(58\) 0 0
\(59\) −4.70293 −0.612269 −0.306135 0.951988i \(-0.599036\pi\)
−0.306135 + 0.951988i \(0.599036\pi\)
\(60\) 0 0
\(61\) −3.51027 −0.449444 −0.224722 0.974423i \(-0.572147\pi\)
−0.224722 + 0.974423i \(0.572147\pi\)
\(62\) 0 0
\(63\) −14.0633 −1.77181
\(64\) 0 0
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) 1.12933 0.137969 0.0689846 0.997618i \(-0.478024\pi\)
0.0689846 + 0.997618i \(0.478024\pi\)
\(68\) 0 0
\(69\) 12.2765 1.47792
\(70\) 0 0
\(71\) −8.76189 −1.03984 −0.519922 0.854214i \(-0.674039\pi\)
−0.519922 + 0.854214i \(0.674039\pi\)
\(72\) 0 0
\(73\) 6.80734 0.796739 0.398369 0.917225i \(-0.369576\pi\)
0.398369 + 0.917225i \(0.369576\pi\)
\(74\) 0 0
\(75\) 14.8707 1.71712
\(76\) 0 0
\(77\) 9.08387 1.03520
\(78\) 0 0
\(79\) −14.5941 −1.64197 −0.820985 0.570950i \(-0.806575\pi\)
−0.820985 + 0.570950i \(0.806575\pi\)
\(80\) 0 0
\(81\) 29.3355 3.25950
\(82\) 0 0
\(83\) −9.74135 −1.06925 −0.534626 0.845089i \(-0.679547\pi\)
−0.534626 + 0.845089i \(0.679547\pi\)
\(84\) 0 0
\(85\) 0.657476 0.0713133
\(86\) 0 0
\(87\) −2.87067 −0.307769
\(88\) 0 0
\(89\) −6.76189 −0.716758 −0.358379 0.933576i \(-0.616671\pi\)
−0.358379 + 0.933576i \(0.616671\pi\)
\(90\) 0 0
\(91\) −2.31761 −0.242951
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −3.08387 −0.316399
\(96\) 0 0
\(97\) 4.16774 0.423170 0.211585 0.977360i \(-0.432137\pi\)
0.211585 + 0.977360i \(0.432137\pi\)
\(98\) 0 0
\(99\) −40.0135 −4.02151
\(100\) 0 0
\(101\) −19.1883 −1.90931 −0.954653 0.297722i \(-0.903773\pi\)
−0.954653 + 0.297722i \(0.903773\pi\)
\(102\) 0 0
\(103\) 15.9091 1.56757 0.783785 0.621033i \(-0.213287\pi\)
0.783785 + 0.621033i \(0.213287\pi\)
\(104\) 0 0
\(105\) −18.1677 −1.77299
\(106\) 0 0
\(107\) −3.29707 −0.318740 −0.159370 0.987219i \(-0.550946\pi\)
−0.159370 + 0.987219i \(0.550946\pi\)
\(108\) 0 0
\(109\) −4.44428 −0.425685 −0.212842 0.977087i \(-0.568272\pi\)
−0.212842 + 0.977087i \(0.568272\pi\)
\(110\) 0 0
\(111\) 6.59414 0.625888
\(112\) 0 0
\(113\) −11.1883 −1.05251 −0.526253 0.850328i \(-0.676403\pi\)
−0.526253 + 0.850328i \(0.676403\pi\)
\(114\) 0 0
\(115\) 11.4827 1.07077
\(116\) 0 0
\(117\) 10.2088 0.943806
\(118\) 0 0
\(119\) −0.380943 −0.0349210
\(120\) 0 0
\(121\) 14.8458 1.34961
\(122\) 0 0
\(123\) 28.3355 2.55493
\(124\) 0 0
\(125\) −1.51027 −0.135083
\(126\) 0 0
\(127\) −16.7619 −1.48738 −0.743688 0.668526i \(-0.766925\pi\)
−0.743688 + 0.668526i \(0.766925\pi\)
\(128\) 0 0
\(129\) −12.1267 −1.06769
\(130\) 0 0
\(131\) 3.67801 0.321350 0.160675 0.987007i \(-0.448633\pi\)
0.160675 + 0.987007i \(0.448633\pi\)
\(132\) 0 0
\(133\) 1.78680 0.154935
\(134\) 0 0
\(135\) 49.5238 4.26233
\(136\) 0 0
\(137\) 21.4015 1.82845 0.914226 0.405205i \(-0.132800\pi\)
0.914226 + 0.405205i \(0.132800\pi\)
\(138\) 0 0
\(139\) −10.8252 −0.918183 −0.459092 0.888389i \(-0.651825\pi\)
−0.459092 + 0.888389i \(0.651825\pi\)
\(140\) 0 0
\(141\) 15.3560 1.29321
\(142\) 0 0
\(143\) −6.59414 −0.551430
\(144\) 0 0
\(145\) −2.68505 −0.222981
\(146\) 0 0
\(147\) −12.5531 −1.03536
\(148\) 0 0
\(149\) 21.2516 1.74100 0.870500 0.492168i \(-0.163795\pi\)
0.870500 + 0.492168i \(0.163795\pi\)
\(150\) 0 0
\(151\) 15.9091 1.29466 0.647332 0.762208i \(-0.275885\pi\)
0.647332 + 0.762208i \(0.275885\pi\)
\(152\) 0 0
\(153\) 1.67801 0.135659
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −14.0000 −1.11732 −0.558661 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) 0 0
\(159\) −36.3944 −2.88627
\(160\) 0 0
\(161\) −6.65310 −0.524338
\(162\) 0 0
\(163\) 9.18828 0.719682 0.359841 0.933014i \(-0.382831\pi\)
0.359841 + 0.933014i \(0.382831\pi\)
\(164\) 0 0
\(165\) −51.6915 −4.02418
\(166\) 0 0
\(167\) 5.83226 0.451313 0.225657 0.974207i \(-0.427547\pi\)
0.225657 + 0.974207i \(0.427547\pi\)
\(168\) 0 0
\(169\) −11.3176 −0.870585
\(170\) 0 0
\(171\) −7.87067 −0.601885
\(172\) 0 0
\(173\) 7.18828 0.546515 0.273257 0.961941i \(-0.411899\pi\)
0.273257 + 0.961941i \(0.411899\pi\)
\(174\) 0 0
\(175\) −8.05896 −0.609200
\(176\) 0 0
\(177\) −15.5059 −1.16549
\(178\) 0 0
\(179\) −7.87333 −0.588480 −0.294240 0.955732i \(-0.595067\pi\)
−0.294240 + 0.955732i \(0.595067\pi\)
\(180\) 0 0
\(181\) 22.3355 1.66018 0.830092 0.557627i \(-0.188288\pi\)
0.830092 + 0.557627i \(0.188288\pi\)
\(182\) 0 0
\(183\) −11.5736 −0.855545
\(184\) 0 0
\(185\) 6.16774 0.453462
\(186\) 0 0
\(187\) −1.08387 −0.0792606
\(188\) 0 0
\(189\) −28.6942 −2.08719
\(190\) 0 0
\(191\) 23.7370 1.71755 0.858773 0.512356i \(-0.171227\pi\)
0.858773 + 0.512356i \(0.171227\pi\)
\(192\) 0 0
\(193\) −7.31495 −0.526542 −0.263271 0.964722i \(-0.584801\pi\)
−0.263271 + 0.964722i \(0.584801\pi\)
\(194\) 0 0
\(195\) 13.1883 0.944433
\(196\) 0 0
\(197\) 8.59414 0.612307 0.306154 0.951982i \(-0.400958\pi\)
0.306154 + 0.951982i \(0.400958\pi\)
\(198\) 0 0
\(199\) 7.06599 0.500895 0.250447 0.968130i \(-0.419422\pi\)
0.250447 + 0.968130i \(0.419422\pi\)
\(200\) 0 0
\(201\) 3.72347 0.262633
\(202\) 0 0
\(203\) 1.55572 0.109190
\(204\) 0 0
\(205\) 26.5032 1.85107
\(206\) 0 0
\(207\) 29.3062 2.03692
\(208\) 0 0
\(209\) 5.08387 0.351659
\(210\) 0 0
\(211\) −5.55572 −0.382472 −0.191236 0.981544i \(-0.561250\pi\)
−0.191236 + 0.981544i \(0.561250\pi\)
\(212\) 0 0
\(213\) −28.8886 −1.97941
\(214\) 0 0
\(215\) −11.3425 −0.773554
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 22.4443 1.51664
\(220\) 0 0
\(221\) 0.276533 0.0186016
\(222\) 0 0
\(223\) 16.7619 1.12246 0.561229 0.827660i \(-0.310329\pi\)
0.561229 + 0.827660i \(0.310329\pi\)
\(224\) 0 0
\(225\) 35.4989 2.36659
\(226\) 0 0
\(227\) 8.27653 0.549333 0.274666 0.961540i \(-0.411433\pi\)
0.274666 + 0.961540i \(0.411433\pi\)
\(228\) 0 0
\(229\) 12.6986 0.839144 0.419572 0.907722i \(-0.362180\pi\)
0.419572 + 0.907722i \(0.362180\pi\)
\(230\) 0 0
\(231\) 29.9502 1.97058
\(232\) 0 0
\(233\) 8.36306 0.547882 0.273941 0.961746i \(-0.411673\pi\)
0.273941 + 0.961746i \(0.411673\pi\)
\(234\) 0 0
\(235\) 14.3631 0.936943
\(236\) 0 0
\(237\) −48.1179 −3.12559
\(238\) 0 0
\(239\) −6.97508 −0.451181 −0.225590 0.974222i \(-0.572431\pi\)
−0.225590 + 0.974222i \(0.572431\pi\)
\(240\) 0 0
\(241\) −19.7413 −1.27165 −0.635826 0.771832i \(-0.719340\pi\)
−0.635826 + 0.771832i \(0.719340\pi\)
\(242\) 0 0
\(243\) 48.5443 3.11412
\(244\) 0 0
\(245\) −11.7413 −0.750127
\(246\) 0 0
\(247\) −1.29707 −0.0825306
\(248\) 0 0
\(249\) −32.1179 −2.03539
\(250\) 0 0
\(251\) 10.2722 0.648373 0.324186 0.945993i \(-0.394910\pi\)
0.324186 + 0.945993i \(0.394910\pi\)
\(252\) 0 0
\(253\) −18.9296 −1.19010
\(254\) 0 0
\(255\) 2.16774 0.135749
\(256\) 0 0
\(257\) 9.90909 0.618112 0.309056 0.951044i \(-0.399987\pi\)
0.309056 + 0.951044i \(0.399987\pi\)
\(258\) 0 0
\(259\) −3.57360 −0.222053
\(260\) 0 0
\(261\) −6.85279 −0.424177
\(262\) 0 0
\(263\) −23.6780 −1.46005 −0.730024 0.683421i \(-0.760491\pi\)
−0.730024 + 0.683421i \(0.760491\pi\)
\(264\) 0 0
\(265\) −34.0411 −2.09113
\(266\) 0 0
\(267\) −22.2944 −1.36440
\(268\) 0 0
\(269\) 5.02054 0.306108 0.153054 0.988218i \(-0.451089\pi\)
0.153054 + 0.988218i \(0.451089\pi\)
\(270\) 0 0
\(271\) −4.27653 −0.259781 −0.129890 0.991528i \(-0.541463\pi\)
−0.129890 + 0.991528i \(0.541463\pi\)
\(272\) 0 0
\(273\) −7.64132 −0.462474
\(274\) 0 0
\(275\) −22.9296 −1.38271
\(276\) 0 0
\(277\) −7.93667 −0.476868 −0.238434 0.971159i \(-0.576634\pi\)
−0.238434 + 0.971159i \(0.576634\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 22.8886 1.36542 0.682708 0.730691i \(-0.260802\pi\)
0.682708 + 0.730691i \(0.260802\pi\)
\(282\) 0 0
\(283\) −5.08387 −0.302205 −0.151102 0.988518i \(-0.548282\pi\)
−0.151102 + 0.988518i \(0.548282\pi\)
\(284\) 0 0
\(285\) −10.1677 −0.602285
\(286\) 0 0
\(287\) −15.3560 −0.906438
\(288\) 0 0
\(289\) −16.9545 −0.997326
\(290\) 0 0
\(291\) 13.7413 0.805532
\(292\) 0 0
\(293\) 3.55572 0.207728 0.103864 0.994592i \(-0.466879\pi\)
0.103864 + 0.994592i \(0.466879\pi\)
\(294\) 0 0
\(295\) −14.5032 −0.844411
\(296\) 0 0
\(297\) −81.6417 −4.73733
\(298\) 0 0
\(299\) 4.82960 0.279303
\(300\) 0 0
\(301\) 6.57188 0.378797
\(302\) 0 0
\(303\) −63.2651 −3.63449
\(304\) 0 0
\(305\) −10.8252 −0.619850
\(306\) 0 0
\(307\) 16.3355 0.932316 0.466158 0.884702i \(-0.345638\pi\)
0.466158 + 0.884702i \(0.345638\pi\)
\(308\) 0 0
\(309\) 52.4534 2.98397
\(310\) 0 0
\(311\) 16.3809 0.928878 0.464439 0.885605i \(-0.346256\pi\)
0.464439 + 0.885605i \(0.346256\pi\)
\(312\) 0 0
\(313\) −30.0590 −1.69903 −0.849516 0.527562i \(-0.823106\pi\)
−0.849516 + 0.527562i \(0.823106\pi\)
\(314\) 0 0
\(315\) −43.3695 −2.44360
\(316\) 0 0
\(317\) 12.0179 0.674991 0.337496 0.941327i \(-0.390420\pi\)
0.337496 + 0.941327i \(0.390420\pi\)
\(318\) 0 0
\(319\) 4.42640 0.247831
\(320\) 0 0
\(321\) −10.8707 −0.606742
\(322\) 0 0
\(323\) −0.213198 −0.0118627
\(324\) 0 0
\(325\) 5.85014 0.324507
\(326\) 0 0
\(327\) −14.6531 −0.810318
\(328\) 0 0
\(329\) −8.32199 −0.458806
\(330\) 0 0
\(331\) −14.3176 −0.786967 −0.393483 0.919332i \(-0.628730\pi\)
−0.393483 + 0.919332i \(0.628730\pi\)
\(332\) 0 0
\(333\) 15.7413 0.862621
\(334\) 0 0
\(335\) 3.48270 0.190280
\(336\) 0 0
\(337\) 11.6147 0.632692 0.316346 0.948644i \(-0.397544\pi\)
0.316346 + 0.948644i \(0.397544\pi\)
\(338\) 0 0
\(339\) −36.8886 −2.00351
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 19.3106 1.04267
\(344\) 0 0
\(345\) 37.8593 2.03827
\(346\) 0 0
\(347\) −2.91613 −0.156546 −0.0782730 0.996932i \(-0.524941\pi\)
−0.0782730 + 0.996932i \(0.524941\pi\)
\(348\) 0 0
\(349\) −12.8252 −0.686518 −0.343259 0.939241i \(-0.611531\pi\)
−0.343259 + 0.939241i \(0.611531\pi\)
\(350\) 0 0
\(351\) 20.8296 1.11180
\(352\) 0 0
\(353\) 25.7592 1.37103 0.685513 0.728061i \(-0.259578\pi\)
0.685513 + 0.728061i \(0.259578\pi\)
\(354\) 0 0
\(355\) −27.0205 −1.43410
\(356\) 0 0
\(357\) −1.25600 −0.0664743
\(358\) 0 0
\(359\) −22.4220 −1.18339 −0.591694 0.806162i \(-0.701541\pi\)
−0.591694 + 0.806162i \(0.701541\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 48.9475 2.56908
\(364\) 0 0
\(365\) 20.9930 1.09882
\(366\) 0 0
\(367\) −5.18828 −0.270826 −0.135413 0.990789i \(-0.543236\pi\)
−0.135413 + 0.990789i \(0.543236\pi\)
\(368\) 0 0
\(369\) 67.6417 3.52129
\(370\) 0 0
\(371\) 19.7235 1.02399
\(372\) 0 0
\(373\) −36.0091 −1.86448 −0.932241 0.361838i \(-0.882149\pi\)
−0.932241 + 0.361838i \(0.882149\pi\)
\(374\) 0 0
\(375\) −4.97946 −0.257138
\(376\) 0 0
\(377\) −1.12933 −0.0581632
\(378\) 0 0
\(379\) −25.0384 −1.28614 −0.643069 0.765809i \(-0.722339\pi\)
−0.643069 + 0.765809i \(0.722339\pi\)
\(380\) 0 0
\(381\) −55.2651 −2.83132
\(382\) 0 0
\(383\) 21.7413 1.11093 0.555466 0.831540i \(-0.312540\pi\)
0.555466 + 0.831540i \(0.312540\pi\)
\(384\) 0 0
\(385\) 28.0135 1.42770
\(386\) 0 0
\(387\) −28.9484 −1.47153
\(388\) 0 0
\(389\) −11.5103 −0.583594 −0.291797 0.956480i \(-0.594253\pi\)
−0.291797 + 0.956480i \(0.594253\pi\)
\(390\) 0 0
\(391\) 0.793836 0.0401460
\(392\) 0 0
\(393\) 12.1267 0.611710
\(394\) 0 0
\(395\) −45.0065 −2.26452
\(396\) 0 0
\(397\) 7.21054 0.361887 0.180943 0.983494i \(-0.442085\pi\)
0.180943 + 0.983494i \(0.442085\pi\)
\(398\) 0 0
\(399\) 5.89121 0.294929
\(400\) 0 0
\(401\) 1.23811 0.0618285 0.0309142 0.999522i \(-0.490158\pi\)
0.0309142 + 0.999522i \(0.490158\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 90.4669 4.49534
\(406\) 0 0
\(407\) −10.1677 −0.503996
\(408\) 0 0
\(409\) 32.8030 1.62200 0.811001 0.585045i \(-0.198923\pi\)
0.811001 + 0.585045i \(0.198923\pi\)
\(410\) 0 0
\(411\) 70.5622 3.48058
\(412\) 0 0
\(413\) 8.40320 0.413495
\(414\) 0 0
\(415\) −30.0411 −1.47466
\(416\) 0 0
\(417\) −35.6915 −1.74782
\(418\) 0 0
\(419\) −17.1883 −0.839703 −0.419851 0.907593i \(-0.637918\pi\)
−0.419851 + 0.907593i \(0.637918\pi\)
\(420\) 0 0
\(421\) −19.8912 −0.969438 −0.484719 0.874670i \(-0.661078\pi\)
−0.484719 + 0.874670i \(0.661078\pi\)
\(422\) 0 0
\(423\) 36.6575 1.78235
\(424\) 0 0
\(425\) 0.961581 0.0466435
\(426\) 0 0
\(427\) 6.27215 0.303531
\(428\) 0 0
\(429\) −21.7413 −1.04968
\(430\) 0 0
\(431\) −8.55307 −0.411987 −0.205993 0.978553i \(-0.566043\pi\)
−0.205993 + 0.978553i \(0.566043\pi\)
\(432\) 0 0
\(433\) −31.5238 −1.51494 −0.757468 0.652872i \(-0.773564\pi\)
−0.757468 + 0.652872i \(0.773564\pi\)
\(434\) 0 0
\(435\) −8.85279 −0.424459
\(436\) 0 0
\(437\) −3.72347 −0.178118
\(438\) 0 0
\(439\) 4.12667 0.196955 0.0984776 0.995139i \(-0.468603\pi\)
0.0984776 + 0.995139i \(0.468603\pi\)
\(440\) 0 0
\(441\) −29.9663 −1.42697
\(442\) 0 0
\(443\) 37.7549 1.79379 0.896894 0.442246i \(-0.145818\pi\)
0.896894 + 0.442246i \(0.145818\pi\)
\(444\) 0 0
\(445\) −20.8528 −0.988517
\(446\) 0 0
\(447\) 70.0681 3.31411
\(448\) 0 0
\(449\) 33.3560 1.57417 0.787084 0.616846i \(-0.211590\pi\)
0.787084 + 0.616846i \(0.211590\pi\)
\(450\) 0 0
\(451\) −43.6915 −2.05735
\(452\) 0 0
\(453\) 52.4534 2.46448
\(454\) 0 0
\(455\) −7.14721 −0.335066
\(456\) 0 0
\(457\) 32.1223 1.50262 0.751309 0.659951i \(-0.229423\pi\)
0.751309 + 0.659951i \(0.229423\pi\)
\(458\) 0 0
\(459\) 3.42374 0.159807
\(460\) 0 0
\(461\) 7.81000 0.363748 0.181874 0.983322i \(-0.441784\pi\)
0.181874 + 0.983322i \(0.441784\pi\)
\(462\) 0 0
\(463\) −18.4897 −0.859291 −0.429645 0.902998i \(-0.641361\pi\)
−0.429645 + 0.902998i \(0.641361\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 21.0839 0.975645 0.487823 0.872943i \(-0.337791\pi\)
0.487823 + 0.872943i \(0.337791\pi\)
\(468\) 0 0
\(469\) −2.01788 −0.0931771
\(470\) 0 0
\(471\) −46.1590 −2.12689
\(472\) 0 0
\(473\) 18.6986 0.859760
\(474\) 0 0
\(475\) −4.51027 −0.206945
\(476\) 0 0
\(477\) −86.8798 −3.97795
\(478\) 0 0
\(479\) 25.5238 1.16621 0.583105 0.812396i \(-0.301837\pi\)
0.583105 + 0.812396i \(0.301837\pi\)
\(480\) 0 0
\(481\) 2.59414 0.118283
\(482\) 0 0
\(483\) −21.9357 −0.998110
\(484\) 0 0
\(485\) 12.8528 0.583615
\(486\) 0 0
\(487\) 10.4675 0.474326 0.237163 0.971470i \(-0.423782\pi\)
0.237163 + 0.971470i \(0.423782\pi\)
\(488\) 0 0
\(489\) 30.2944 1.36996
\(490\) 0 0
\(491\) 12.6710 0.571833 0.285917 0.958254i \(-0.407702\pi\)
0.285917 + 0.958254i \(0.407702\pi\)
\(492\) 0 0
\(493\) −0.185626 −0.00836018
\(494\) 0 0
\(495\) −123.397 −5.54626
\(496\) 0 0
\(497\) 15.6558 0.702257
\(498\) 0 0
\(499\) −13.8458 −0.619821 −0.309911 0.950766i \(-0.600299\pi\)
−0.309911 + 0.950766i \(0.600299\pi\)
\(500\) 0 0
\(501\) 19.2294 0.859104
\(502\) 0 0
\(503\) 9.76454 0.435379 0.217690 0.976018i \(-0.430148\pi\)
0.217690 + 0.976018i \(0.430148\pi\)
\(504\) 0 0
\(505\) −59.1742 −2.63322
\(506\) 0 0
\(507\) −37.3150 −1.65722
\(508\) 0 0
\(509\) 28.1677 1.24851 0.624257 0.781219i \(-0.285402\pi\)
0.624257 + 0.781219i \(0.285402\pi\)
\(510\) 0 0
\(511\) −12.1634 −0.538076
\(512\) 0 0
\(513\) −16.0590 −0.709020
\(514\) 0 0
\(515\) 49.0616 2.16191
\(516\) 0 0
\(517\) −23.6780 −1.04136
\(518\) 0 0
\(519\) 23.7003 1.04033
\(520\) 0 0
\(521\) −28.2035 −1.23562 −0.617809 0.786328i \(-0.711980\pi\)
−0.617809 + 0.786328i \(0.711980\pi\)
\(522\) 0 0
\(523\) 1.22023 0.0533571 0.0266785 0.999644i \(-0.491507\pi\)
0.0266785 + 0.999644i \(0.491507\pi\)
\(524\) 0 0
\(525\) −26.5709 −1.15965
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −9.13579 −0.397208
\(530\) 0 0
\(531\) −37.0152 −1.60632
\(532\) 0 0
\(533\) 11.1472 0.482839
\(534\) 0 0
\(535\) −10.1677 −0.439590
\(536\) 0 0
\(537\) −25.9589 −1.12021
\(538\) 0 0
\(539\) 19.3560 0.833723
\(540\) 0 0
\(541\) −12.1812 −0.523713 −0.261856 0.965107i \(-0.584335\pi\)
−0.261856 + 0.965107i \(0.584335\pi\)
\(542\) 0 0
\(543\) 73.6417 3.16027
\(544\) 0 0
\(545\) −13.7056 −0.587083
\(546\) 0 0
\(547\) 12.0000 0.513083 0.256541 0.966533i \(-0.417417\pi\)
0.256541 + 0.966533i \(0.417417\pi\)
\(548\) 0 0
\(549\) −27.6282 −1.17914
\(550\) 0 0
\(551\) 0.870674 0.0370920
\(552\) 0 0
\(553\) 26.0768 1.10890
\(554\) 0 0
\(555\) 20.3355 0.863193
\(556\) 0 0
\(557\) −8.60764 −0.364718 −0.182359 0.983232i \(-0.558373\pi\)
−0.182359 + 0.983232i \(0.558373\pi\)
\(558\) 0 0
\(559\) −4.77064 −0.201777
\(560\) 0 0
\(561\) −3.57360 −0.150878
\(562\) 0 0
\(563\) −19.3560 −0.815759 −0.407880 0.913036i \(-0.633732\pi\)
−0.407880 + 0.913036i \(0.633732\pi\)
\(564\) 0 0
\(565\) −34.5032 −1.45156
\(566\) 0 0
\(567\) −52.4167 −2.20129
\(568\) 0 0
\(569\) 9.23811 0.387282 0.193641 0.981072i \(-0.437970\pi\)
0.193641 + 0.981072i \(0.437970\pi\)
\(570\) 0 0
\(571\) 14.3766 0.601640 0.300820 0.953681i \(-0.402740\pi\)
0.300820 + 0.953681i \(0.402740\pi\)
\(572\) 0 0
\(573\) 78.2625 3.26946
\(574\) 0 0
\(575\) 16.7938 0.700351
\(576\) 0 0
\(577\) 5.19266 0.216173 0.108087 0.994141i \(-0.465528\pi\)
0.108087 + 0.994141i \(0.465528\pi\)
\(578\) 0 0
\(579\) −24.1179 −1.00231
\(580\) 0 0
\(581\) 17.4059 0.722117
\(582\) 0 0
\(583\) 56.1179 2.32417
\(584\) 0 0
\(585\) 31.4827 1.30165
\(586\) 0 0
\(587\) 30.3079 1.25094 0.625471 0.780248i \(-0.284907\pi\)
0.625471 + 0.780248i \(0.284907\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 28.3355 1.16557
\(592\) 0 0
\(593\) −4.04107 −0.165947 −0.0829735 0.996552i \(-0.526442\pi\)
−0.0829735 + 0.996552i \(0.526442\pi\)
\(594\) 0 0
\(595\) −1.17478 −0.0481613
\(596\) 0 0
\(597\) 23.2971 0.953486
\(598\) 0 0
\(599\) 6.29441 0.257183 0.128591 0.991698i \(-0.458954\pi\)
0.128591 + 0.991698i \(0.458954\pi\)
\(600\) 0 0
\(601\) −10.0358 −0.409367 −0.204684 0.978828i \(-0.565617\pi\)
−0.204684 + 0.978828i \(0.565617\pi\)
\(602\) 0 0
\(603\) 8.88856 0.361970
\(604\) 0 0
\(605\) 45.7824 1.86132
\(606\) 0 0
\(607\) 45.3062 1.83892 0.919461 0.393182i \(-0.128626\pi\)
0.919461 + 0.393182i \(0.128626\pi\)
\(608\) 0 0
\(609\) 5.12933 0.207851
\(610\) 0 0
\(611\) 6.04107 0.244396
\(612\) 0 0
\(613\) −25.1607 −1.01623 −0.508116 0.861289i \(-0.669658\pi\)
−0.508116 + 0.861289i \(0.669658\pi\)
\(614\) 0 0
\(615\) 87.3830 3.52362
\(616\) 0 0
\(617\) 6.01350 0.242095 0.121047 0.992647i \(-0.461375\pi\)
0.121047 + 0.992647i \(0.461375\pi\)
\(618\) 0 0
\(619\) 37.2240 1.49616 0.748080 0.663608i \(-0.230976\pi\)
0.748080 + 0.663608i \(0.230976\pi\)
\(620\) 0 0
\(621\) 59.7950 2.39949
\(622\) 0 0
\(623\) 12.0821 0.484061
\(624\) 0 0
\(625\) −27.2088 −1.08835
\(626\) 0 0
\(627\) 16.7619 0.669405
\(628\) 0 0
\(629\) 0.426396 0.0170015
\(630\) 0 0
\(631\) 12.0135 0.478250 0.239125 0.970989i \(-0.423139\pi\)
0.239125 + 0.970989i \(0.423139\pi\)
\(632\) 0 0
\(633\) −18.3176 −0.728060
\(634\) 0 0
\(635\) −51.6915 −2.05132
\(636\) 0 0
\(637\) −4.93839 −0.195666
\(638\) 0 0
\(639\) −68.9619 −2.72809
\(640\) 0 0
\(641\) 6.97946 0.275672 0.137836 0.990455i \(-0.455985\pi\)
0.137836 + 0.990455i \(0.455985\pi\)
\(642\) 0 0
\(643\) 40.7754 1.60802 0.804012 0.594613i \(-0.202695\pi\)
0.804012 + 0.594613i \(0.202695\pi\)
\(644\) 0 0
\(645\) −37.3971 −1.47251
\(646\) 0 0
\(647\) −24.9340 −0.980257 −0.490129 0.871650i \(-0.663050\pi\)
−0.490129 + 0.871650i \(0.663050\pi\)
\(648\) 0 0
\(649\) 23.9091 0.938514
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 36.3988 1.42440 0.712198 0.701979i \(-0.247700\pi\)
0.712198 + 0.701979i \(0.247700\pi\)
\(654\) 0 0
\(655\) 11.3425 0.443189
\(656\) 0 0
\(657\) 53.5783 2.09029
\(658\) 0 0
\(659\) 11.8501 0.461616 0.230808 0.972999i \(-0.425863\pi\)
0.230808 + 0.972999i \(0.425863\pi\)
\(660\) 0 0
\(661\) −34.4854 −1.34132 −0.670662 0.741763i \(-0.733990\pi\)
−0.670662 + 0.741763i \(0.733990\pi\)
\(662\) 0 0
\(663\) 0.911749 0.0354094
\(664\) 0 0
\(665\) 5.51027 0.213679
\(666\) 0 0
\(667\) −3.24193 −0.125528
\(668\) 0 0
\(669\) 55.2651 2.13667
\(670\) 0 0
\(671\) 17.8458 0.688928
\(672\) 0 0
\(673\) 17.7003 0.682295 0.341148 0.940010i \(-0.389184\pi\)
0.341148 + 0.940010i \(0.389184\pi\)
\(674\) 0 0
\(675\) 72.4302 2.78784
\(676\) 0 0
\(677\) 26.0590 1.00153 0.500764 0.865584i \(-0.333053\pi\)
0.500764 + 0.865584i \(0.333053\pi\)
\(678\) 0 0
\(679\) −7.44693 −0.285787
\(680\) 0 0
\(681\) 27.2883 1.04569
\(682\) 0 0
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 0 0
\(685\) 65.9994 2.52171
\(686\) 0 0
\(687\) 41.8680 1.59736
\(688\) 0 0
\(689\) −14.3176 −0.545457
\(690\) 0 0
\(691\) 13.9367 0.530176 0.265088 0.964224i \(-0.414599\pi\)
0.265088 + 0.964224i \(0.414599\pi\)
\(692\) 0 0
\(693\) 71.4962 2.71592
\(694\) 0 0
\(695\) −33.3836 −1.26631
\(696\) 0 0
\(697\) 1.83226 0.0694016
\(698\) 0 0
\(699\) 27.5736 1.04293
\(700\) 0 0
\(701\) −6.55307 −0.247506 −0.123753 0.992313i \(-0.539493\pi\)
−0.123753 + 0.992313i \(0.539493\pi\)
\(702\) 0 0
\(703\) −2.00000 −0.0754314
\(704\) 0 0
\(705\) 47.3560 1.78353
\(706\) 0 0
\(707\) 34.2857 1.28944
\(708\) 0 0
\(709\) −23.5238 −0.883454 −0.441727 0.897150i \(-0.645634\pi\)
−0.441727 + 0.897150i \(0.645634\pi\)
\(710\) 0 0
\(711\) −114.866 −4.30780
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) −20.3355 −0.760504
\(716\) 0 0
\(717\) −22.9973 −0.858852
\(718\) 0 0
\(719\) 14.8842 0.555086 0.277543 0.960713i \(-0.410480\pi\)
0.277543 + 0.960713i \(0.410480\pi\)
\(720\) 0 0
\(721\) −28.4264 −1.05865
\(722\) 0 0
\(723\) −65.0886 −2.42067
\(724\) 0 0
\(725\) −3.92697 −0.145844
\(726\) 0 0
\(727\) −1.32464 −0.0491283 −0.0245641 0.999698i \(-0.507820\pi\)
−0.0245641 + 0.999698i \(0.507820\pi\)
\(728\) 0 0
\(729\) 72.0475 2.66843
\(730\) 0 0
\(731\) −0.784146 −0.0290027
\(732\) 0 0
\(733\) 13.7824 0.509065 0.254533 0.967064i \(-0.418078\pi\)
0.254533 + 0.967064i \(0.418078\pi\)
\(734\) 0 0
\(735\) −38.7121 −1.42792
\(736\) 0 0
\(737\) −5.74135 −0.211485
\(738\) 0 0
\(739\) 23.5513 0.866350 0.433175 0.901310i \(-0.357393\pi\)
0.433175 + 0.901310i \(0.357393\pi\)
\(740\) 0 0
\(741\) −4.27653 −0.157102
\(742\) 0 0
\(743\) 14.0411 0.515117 0.257559 0.966263i \(-0.417082\pi\)
0.257559 + 0.966263i \(0.417082\pi\)
\(744\) 0 0
\(745\) 65.5373 2.40110
\(746\) 0 0
\(747\) −76.6710 −2.80524
\(748\) 0 0
\(749\) 5.89121 0.215260
\(750\) 0 0
\(751\) 17.7056 0.646086 0.323043 0.946384i \(-0.395294\pi\)
0.323043 + 0.946384i \(0.395294\pi\)
\(752\) 0 0
\(753\) 33.8680 1.23422
\(754\) 0 0
\(755\) 49.0616 1.78553
\(756\) 0 0
\(757\) −30.3220 −1.10207 −0.551036 0.834482i \(-0.685767\pi\)
−0.551036 + 0.834482i \(0.685767\pi\)
\(758\) 0 0
\(759\) −62.4123 −2.26542
\(760\) 0 0
\(761\) 12.5487 0.454890 0.227445 0.973791i \(-0.426963\pi\)
0.227445 + 0.973791i \(0.426963\pi\)
\(762\) 0 0
\(763\) 7.94104 0.287485
\(764\) 0 0
\(765\) 5.17478 0.187095
\(766\) 0 0
\(767\) −6.10003 −0.220259
\(768\) 0 0
\(769\) −50.0725 −1.80566 −0.902830 0.429999i \(-0.858514\pi\)
−0.902830 + 0.429999i \(0.858514\pi\)
\(770\) 0 0
\(771\) 32.6710 1.17662
\(772\) 0 0
\(773\) −11.8003 −0.424427 −0.212214 0.977223i \(-0.568067\pi\)
−0.212214 + 0.977223i \(0.568067\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −11.7824 −0.422692
\(778\) 0 0
\(779\) −8.59414 −0.307917
\(780\) 0 0
\(781\) 44.5443 1.59392
\(782\) 0 0
\(783\) −13.9821 −0.499680
\(784\) 0 0
\(785\) −43.1742 −1.54095
\(786\) 0 0
\(787\) 32.9475 1.17445 0.587226 0.809423i \(-0.300220\pi\)
0.587226 + 0.809423i \(0.300220\pi\)
\(788\) 0 0
\(789\) −78.0681 −2.77930
\(790\) 0 0
\(791\) 19.9912 0.710807
\(792\) 0 0
\(793\) −4.55307 −0.161684
\(794\) 0 0
\(795\) −112.236 −3.98059
\(796\) 0 0
\(797\) 18.8208 0.666668 0.333334 0.942809i \(-0.391826\pi\)
0.333334 + 0.942809i \(0.391826\pi\)
\(798\) 0 0
\(799\) 0.992965 0.0351286
\(800\) 0 0
\(801\) −53.2206 −1.88046
\(802\) 0 0
\(803\) −34.6076 −1.22128
\(804\) 0 0
\(805\) −20.5173 −0.723140
\(806\) 0 0
\(807\) 16.5531 0.582696
\(808\) 0 0
\(809\) −39.2607 −1.38033 −0.690167 0.723650i \(-0.742463\pi\)
−0.690167 + 0.723650i \(0.742463\pi\)
\(810\) 0 0
\(811\) 33.9182 1.19103 0.595515 0.803344i \(-0.296948\pi\)
0.595515 + 0.803344i \(0.296948\pi\)
\(812\) 0 0
\(813\) −14.1000 −0.494510
\(814\) 0 0
\(815\) 28.3355 0.992549
\(816\) 0 0
\(817\) 3.67801 0.128677
\(818\) 0 0
\(819\) −18.2411 −0.637397
\(820\) 0 0
\(821\) −38.6487 −1.34885 −0.674425 0.738344i \(-0.735608\pi\)
−0.674425 + 0.738344i \(0.735608\pi\)
\(822\) 0 0
\(823\) 8.47185 0.295310 0.147655 0.989039i \(-0.452827\pi\)
0.147655 + 0.989039i \(0.452827\pi\)
\(824\) 0 0
\(825\) −75.6006 −2.63208
\(826\) 0 0
\(827\) −44.7299 −1.55541 −0.777706 0.628628i \(-0.783617\pi\)
−0.777706 + 0.628628i \(0.783617\pi\)
\(828\) 0 0
\(829\) 38.8566 1.34955 0.674773 0.738025i \(-0.264242\pi\)
0.674773 + 0.738025i \(0.264242\pi\)
\(830\) 0 0
\(831\) −26.1677 −0.907749
\(832\) 0 0
\(833\) −0.811718 −0.0281244
\(834\) 0 0
\(835\) 17.9859 0.622429
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 7.53784 0.260235 0.130118 0.991499i \(-0.458465\pi\)
0.130118 + 0.991499i \(0.458465\pi\)
\(840\) 0 0
\(841\) −28.2419 −0.973860
\(842\) 0 0
\(843\) 75.4652 2.59916
\(844\) 0 0
\(845\) −34.9021 −1.20067
\(846\) 0 0
\(847\) −26.5264 −0.911459
\(848\) 0 0
\(849\) −16.7619 −0.575266
\(850\) 0 0
\(851\) 7.44693 0.255278
\(852\) 0 0
\(853\) 43.5238 1.49023 0.745113 0.666938i \(-0.232396\pi\)
0.745113 + 0.666938i \(0.232396\pi\)
\(854\) 0 0
\(855\) −24.2722 −0.830090
\(856\) 0 0
\(857\) −14.5531 −0.497123 −0.248562 0.968616i \(-0.579958\pi\)
−0.248562 + 0.968616i \(0.579958\pi\)
\(858\) 0 0
\(859\) −24.3132 −0.829557 −0.414778 0.909922i \(-0.636141\pi\)
−0.414778 + 0.909922i \(0.636141\pi\)
\(860\) 0 0
\(861\) −50.6299 −1.72546
\(862\) 0 0
\(863\) 12.4264 0.422999 0.211500 0.977378i \(-0.432165\pi\)
0.211500 + 0.977378i \(0.432165\pi\)
\(864\) 0 0
\(865\) 22.1677 0.753726
\(866\) 0 0
\(867\) −55.9003 −1.89847
\(868\) 0 0
\(869\) 74.1948 2.51688
\(870\) 0 0
\(871\) 1.46482 0.0496334
\(872\) 0 0
\(873\) 32.8030 1.11021
\(874\) 0 0
\(875\) 2.69855 0.0912277
\(876\) 0 0
\(877\) −54.6120 −1.84412 −0.922058 0.387051i \(-0.873494\pi\)
−0.922058 + 0.387051i \(0.873494\pi\)
\(878\) 0 0
\(879\) 11.7235 0.395423
\(880\) 0 0
\(881\) 35.4551 1.19451 0.597257 0.802050i \(-0.296257\pi\)
0.597257 + 0.802050i \(0.296257\pi\)
\(882\) 0 0
\(883\) 21.5460 0.725082 0.362541 0.931968i \(-0.381909\pi\)
0.362541 + 0.931968i \(0.381909\pi\)
\(884\) 0 0
\(885\) −47.8182 −1.60739
\(886\) 0 0
\(887\) −49.6147 −1.66590 −0.832949 0.553350i \(-0.813349\pi\)
−0.832949 + 0.553350i \(0.813349\pi\)
\(888\) 0 0
\(889\) 29.9502 1.00450
\(890\) 0 0
\(891\) −149.138 −4.99631
\(892\) 0 0
\(893\) −4.65748 −0.155856
\(894\) 0 0
\(895\) −24.2803 −0.811602
\(896\) 0 0
\(897\) 15.9235 0.531671
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −2.35337 −0.0784022
\(902\) 0 0
\(903\) 21.6680 0.721064
\(904\) 0 0
\(905\) 68.8798 2.28964
\(906\) 0 0
\(907\) −11.0883 −0.368179 −0.184090 0.982909i \(-0.558934\pi\)
−0.184090 + 0.982909i \(0.558934\pi\)
\(908\) 0 0
\(909\) −151.025 −5.00917
\(910\) 0 0
\(911\) −12.5443 −0.415612 −0.207806 0.978170i \(-0.566632\pi\)
−0.207806 + 0.978170i \(0.566632\pi\)
\(912\) 0 0
\(913\) 49.5238 1.63900
\(914\) 0 0
\(915\) −35.6915 −1.17993
\(916\) 0 0
\(917\) −6.57188 −0.217023
\(918\) 0 0
\(919\) −42.8707 −1.41417 −0.707087 0.707127i \(-0.749991\pi\)
−0.707087 + 0.707127i \(0.749991\pi\)
\(920\) 0 0
\(921\) 53.8593 1.77472
\(922\) 0 0
\(923\) −11.3648 −0.374076
\(924\) 0 0
\(925\) 9.02054 0.296593
\(926\) 0 0
\(927\) 125.215 4.11261
\(928\) 0 0
\(929\) −27.9182 −0.915967 −0.457984 0.888961i \(-0.651428\pi\)
−0.457984 + 0.888961i \(0.651428\pi\)
\(930\) 0 0
\(931\) 3.80734 0.124781
\(932\) 0 0
\(933\) 54.0091 1.76818
\(934\) 0 0
\(935\) −3.34252 −0.109312
\(936\) 0 0
\(937\) −10.8982 −0.356030 −0.178015 0.984028i \(-0.556968\pi\)
−0.178015 + 0.984028i \(0.556968\pi\)
\(938\) 0 0
\(939\) −99.1065 −3.23422
\(940\) 0 0
\(941\) 30.9475 1.00886 0.504430 0.863453i \(-0.331703\pi\)
0.504430 + 0.863453i \(0.331703\pi\)
\(942\) 0 0
\(943\) 32.0000 1.04206
\(944\) 0 0
\(945\) −88.4892 −2.87855
\(946\) 0 0
\(947\) −18.2944 −0.594489 −0.297244 0.954801i \(-0.596068\pi\)
−0.297244 + 0.954801i \(0.596068\pi\)
\(948\) 0 0
\(949\) 8.82960 0.286621
\(950\) 0 0
\(951\) 39.6238 1.28489
\(952\) 0 0
\(953\) −23.0616 −0.747039 −0.373519 0.927622i \(-0.621849\pi\)
−0.373519 + 0.927622i \(0.621849\pi\)
\(954\) 0 0
\(955\) 73.2018 2.36875
\(956\) 0 0
\(957\) 14.5941 0.471762
\(958\) 0 0
\(959\) −38.2402 −1.23484
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) −25.9502 −0.836232
\(964\) 0 0
\(965\) −22.5584 −0.726180
\(966\) 0 0
\(967\) −26.3766 −0.848213 −0.424107 0.905612i \(-0.639412\pi\)
−0.424107 + 0.905612i \(0.639412\pi\)
\(968\) 0 0
\(969\) −0.702929 −0.0225813
\(970\) 0 0
\(971\) 20.0000 0.641831 0.320915 0.947108i \(-0.396010\pi\)
0.320915 + 0.947108i \(0.396010\pi\)
\(972\) 0 0
\(973\) 19.3425 0.620093
\(974\) 0 0
\(975\) 19.2883 0.617720
\(976\) 0 0
\(977\) −26.4264 −0.845455 −0.422728 0.906257i \(-0.638927\pi\)
−0.422728 + 0.906257i \(0.638927\pi\)
\(978\) 0 0
\(979\) 34.3766 1.09868
\(980\) 0 0
\(981\) −34.9795 −1.11681
\(982\) 0 0
\(983\) −25.4059 −0.810321 −0.405161 0.914246i \(-0.632784\pi\)
−0.405161 + 0.914246i \(0.632784\pi\)
\(984\) 0 0
\(985\) 26.5032 0.844463
\(986\) 0 0
\(987\) −27.4382 −0.873367
\(988\) 0 0
\(989\) −13.6950 −0.435474
\(990\) 0 0
\(991\) −29.6504 −0.941877 −0.470939 0.882166i \(-0.656085\pi\)
−0.470939 + 0.882166i \(0.656085\pi\)
\(992\) 0 0
\(993\) −47.2062 −1.49804
\(994\) 0 0
\(995\) 21.7906 0.690809
\(996\) 0 0
\(997\) 38.7754 1.22803 0.614014 0.789295i \(-0.289554\pi\)
0.614014 + 0.789295i \(0.289554\pi\)
\(998\) 0 0
\(999\) 32.1179 1.01617
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 1216.2.a.v.1.3 3
4.3 odd 2 1216.2.a.u.1.1 3
8.3 odd 2 152.2.a.c.1.3 3
8.5 even 2 304.2.a.g.1.1 3
24.5 odd 2 2736.2.a.bd.1.3 3
24.11 even 2 1368.2.a.n.1.3 3
40.3 even 4 3800.2.d.j.3649.6 6
40.19 odd 2 3800.2.a.r.1.1 3
40.27 even 4 3800.2.d.j.3649.1 6
40.29 even 2 7600.2.a.bv.1.3 3
56.27 even 2 7448.2.a.bf.1.1 3
152.37 odd 2 5776.2.a.bp.1.3 3
152.75 even 2 2888.2.a.o.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
152.2.a.c.1.3 3 8.3 odd 2
304.2.a.g.1.1 3 8.5 even 2
1216.2.a.u.1.1 3 4.3 odd 2
1216.2.a.v.1.3 3 1.1 even 1 trivial
1368.2.a.n.1.3 3 24.11 even 2
2736.2.a.bd.1.3 3 24.5 odd 2
2888.2.a.o.1.1 3 152.75 even 2
3800.2.a.r.1.1 3 40.19 odd 2
3800.2.d.j.3649.1 6 40.27 even 4
3800.2.d.j.3649.6 6 40.3 even 4
5776.2.a.bp.1.3 3 152.37 odd 2
7448.2.a.bf.1.1 3 56.27 even 2
7600.2.a.bv.1.3 3 40.29 even 2