Properties

Label 6048.2.h.h.2591.2
Level 6048
Weight 2
Character 6048.2591
Analytic conductor 48.294
Analytic rank 0
Dimension 16
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6048.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.2935231425\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2591.2
Root \(1.53379 + 1.53379i\)
Character \(\chi\) = 6048.2591
Dual form 6048.2.h.h.2591.16

$q$-expansion

\(f(q)\) \(=\) \(q-1.93185i q^{5} -1.00000i q^{7} +O(q^{10})\) \(q-1.93185i q^{5} -1.00000i q^{7} +3.58521 q^{11} +5.92609 q^{13} -6.96655i q^{17} +5.33820i q^{19} +2.96713 q^{23} +1.26795 q^{25} -3.06757i q^{29} +1.27006i q^{31} -1.93185 q^{35} +9.65815 q^{37} +8.10227i q^{41} +2.46199i q^{43} +6.45299 q^{47} -1.00000 q^{49} +11.2092i q^{53} -6.92609i q^{55} +6.34847 q^{59} -10.7263 q^{61} -11.4483i q^{65} -6.26430i q^{67} -1.92887 q^{71} -2.12379 q^{73} -3.58521i q^{77} +8.71379i q^{79} +7.96655 q^{83} -13.4583 q^{85} -9.04118i q^{89} -5.92609i q^{91} +10.3126 q^{95} -4.33820 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q + 8q^{13} + 48q^{25} + 40q^{37} - 16q^{49} - 56q^{73} - 16q^{85} - 16q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6048\mathbb{Z}\right)^\times\).

\(n\) \(2593\) \(3781\) \(3809\) \(4159\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\)

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
\(5\) − 1.93185i − 0.863950i −0.901886 0.431975i \(-0.857817\pi\)
0.901886 0.431975i \(-0.142183\pi\)
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.58521 1.08098 0.540491 0.841350i \(-0.318239\pi\)
0.540491 + 0.841350i \(0.318239\pi\)
\(12\) 0 0
\(13\) 5.92609 1.64360 0.821801 0.569774i \(-0.192969\pi\)
0.821801 + 0.569774i \(0.192969\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 6.96655i − 1.68964i −0.535053 0.844818i \(-0.679709\pi\)
0.535053 0.844818i \(-0.320291\pi\)
\(18\) 0 0
\(19\) 5.33820i 1.22467i 0.790599 + 0.612334i \(0.209769\pi\)
−0.790599 + 0.612334i \(0.790231\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.96713 0.618689 0.309344 0.950950i \(-0.399890\pi\)
0.309344 + 0.950950i \(0.399890\pi\)
\(24\) 0 0
\(25\) 1.26795 0.253590
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 3.06757i − 0.569634i −0.958582 0.284817i \(-0.908067\pi\)
0.958582 0.284817i \(-0.0919328\pi\)
\(30\) 0 0
\(31\) 1.27006i 0.228109i 0.993474 + 0.114055i \(0.0363839\pi\)
−0.993474 + 0.114055i \(0.963616\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.93185 −0.326543
\(36\) 0 0
\(37\) 9.65815 1.58779 0.793895 0.608055i \(-0.208050\pi\)
0.793895 + 0.608055i \(0.208050\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 8.10227i 1.26536i 0.774413 + 0.632681i \(0.218046\pi\)
−0.774413 + 0.632681i \(0.781954\pi\)
\(42\) 0 0
\(43\) 2.46199i 0.375450i 0.982222 + 0.187725i \(0.0601114\pi\)
−0.982222 + 0.187725i \(0.939889\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.45299 0.941265 0.470632 0.882329i \(-0.344026\pi\)
0.470632 + 0.882329i \(0.344026\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 11.2092i 1.53970i 0.638224 + 0.769850i \(0.279669\pi\)
−0.638224 + 0.769850i \(0.720331\pi\)
\(54\) 0 0
\(55\) − 6.92609i − 0.933914i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 6.34847 0.826500 0.413250 0.910618i \(-0.364394\pi\)
0.413250 + 0.910618i \(0.364394\pi\)
\(60\) 0 0
\(61\) −10.7263 −1.37336 −0.686680 0.726960i \(-0.740933\pi\)
−0.686680 + 0.726960i \(0.740933\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 11.4483i − 1.41999i
\(66\) 0 0
\(67\) − 6.26430i − 0.765306i −0.923892 0.382653i \(-0.875011\pi\)
0.923892 0.382653i \(-0.124989\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −1.92887 −0.228915 −0.114457 0.993428i \(-0.536513\pi\)
−0.114457 + 0.993428i \(0.536513\pi\)
\(72\) 0 0
\(73\) −2.12379 −0.248571 −0.124285 0.992247i \(-0.539664\pi\)
−0.124285 + 0.992247i \(0.539664\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 3.58521i − 0.408573i
\(78\) 0 0
\(79\) 8.71379i 0.980378i 0.871616 + 0.490189i \(0.163072\pi\)
−0.871616 + 0.490189i \(0.836928\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 7.96655 0.874443 0.437221 0.899354i \(-0.355963\pi\)
0.437221 + 0.899354i \(0.355963\pi\)
\(84\) 0 0
\(85\) −13.4583 −1.45976
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 9.04118i − 0.958363i −0.877716 0.479181i \(-0.840934\pi\)
0.877716 0.479181i \(-0.159066\pi\)
\(90\) 0 0
\(91\) − 5.92609i − 0.621223i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 10.3126 1.05805
\(96\) 0 0
\(97\) −4.33820 −0.440478 −0.220239 0.975446i \(-0.570684\pi\)
−0.220239 + 0.975446i \(0.570684\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 6.13216i 0.610173i 0.952325 + 0.305086i \(0.0986854\pi\)
−0.952325 + 0.305086i \(0.901315\pi\)
\(102\) 0 0
\(103\) 13.9963i 1.37910i 0.724238 + 0.689551i \(0.242192\pi\)
−0.724238 + 0.689551i \(0.757808\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −9.34251 −0.903174 −0.451587 0.892227i \(-0.649142\pi\)
−0.451587 + 0.892227i \(0.649142\pi\)
\(108\) 0 0
\(109\) 19.0666 1.82625 0.913125 0.407681i \(-0.133662\pi\)
0.913125 + 0.407681i \(0.133662\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1.68863i 0.158853i 0.996841 + 0.0794266i \(0.0253089\pi\)
−0.996841 + 0.0794266i \(0.974691\pi\)
\(114\) 0 0
\(115\) − 5.73205i − 0.534516i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −6.96655 −0.638623
\(120\) 0 0
\(121\) 1.85373 0.168521
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 12.1087i − 1.08304i
\(126\) 0 0
\(127\) 15.2424i 1.35254i 0.736653 + 0.676271i \(0.236405\pi\)
−0.736653 + 0.676271i \(0.763595\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −16.0699 −1.40403 −0.702017 0.712160i \(-0.747717\pi\)
−0.702017 + 0.712160i \(0.747717\pi\)
\(132\) 0 0
\(133\) 5.33820 0.462881
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 3.82545i 0.326830i 0.986557 + 0.163415i \(0.0522509\pi\)
−0.986557 + 0.163415i \(0.947749\pi\)
\(138\) 0 0
\(139\) 15.9021i 1.34880i 0.738368 + 0.674398i \(0.235597\pi\)
−0.738368 + 0.674398i \(0.764403\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 21.2463 1.77670
\(144\) 0 0
\(145\) −5.92609 −0.492135
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 8.17987i − 0.670121i −0.942197 0.335061i \(-0.891243\pi\)
0.942197 0.335061i \(-0.108757\pi\)
\(150\) 0 0
\(151\) − 8.04988i − 0.655090i −0.944836 0.327545i \(-0.893779\pi\)
0.944836 0.327545i \(-0.106221\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 2.45356 0.197075
\(156\) 0 0
\(157\) −17.6420 −1.40798 −0.703992 0.710208i \(-0.748601\pi\)
−0.703992 + 0.710208i \(0.748601\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 2.96713i − 0.233842i
\(162\) 0 0
\(163\) − 2.46621i − 0.193169i −0.995325 0.0965843i \(-0.969208\pi\)
0.995325 0.0965843i \(-0.0307917\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 16.1340 1.24849 0.624243 0.781230i \(-0.285408\pi\)
0.624243 + 0.781230i \(0.285408\pi\)
\(168\) 0 0
\(169\) 22.1186 1.70143
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 12.0042i − 0.912665i −0.889809 0.456332i \(-0.849163\pi\)
0.889809 0.456332i \(-0.150837\pi\)
\(174\) 0 0
\(175\) − 1.26795i − 0.0958479i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −14.4871 −1.08282 −0.541408 0.840760i \(-0.682109\pi\)
−0.541408 + 0.840760i \(0.682109\pi\)
\(180\) 0 0
\(181\) −8.72629 −0.648620 −0.324310 0.945951i \(-0.605132\pi\)
−0.324310 + 0.945951i \(0.605132\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 18.6581i − 1.37177i
\(186\) 0 0
\(187\) − 24.9765i − 1.82647i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 12.6628 0.916245 0.458122 0.888889i \(-0.348522\pi\)
0.458122 + 0.888889i \(0.348522\pi\)
\(192\) 0 0
\(193\) −3.51399 −0.252942 −0.126471 0.991970i \(-0.540365\pi\)
−0.126471 + 0.991970i \(0.540365\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 11.3056i − 0.805488i −0.915313 0.402744i \(-0.868057\pi\)
0.915313 0.402744i \(-0.131943\pi\)
\(198\) 0 0
\(199\) 2.53801i 0.179915i 0.995946 + 0.0899573i \(0.0286731\pi\)
−0.995946 + 0.0899573i \(0.971327\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −3.06757 −0.215301
\(204\) 0 0
\(205\) 15.6524 1.09321
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 19.1386i 1.32384i
\(210\) 0 0
\(211\) − 22.8303i − 1.57170i −0.618416 0.785851i \(-0.712225\pi\)
0.618416 0.785851i \(-0.287775\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 4.75620 0.324370
\(216\) 0 0
\(217\) 1.27006 0.0862172
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 41.2844i − 2.77709i
\(222\) 0 0
\(223\) 19.4048i 1.29944i 0.760173 + 0.649721i \(0.225114\pi\)
−0.760173 + 0.649721i \(0.774886\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 25.6287 1.70104 0.850519 0.525944i \(-0.176288\pi\)
0.850519 + 0.525944i \(0.176288\pi\)
\(228\) 0 0
\(229\) −21.8023 −1.44074 −0.720368 0.693592i \(-0.756027\pi\)
−0.720368 + 0.693592i \(0.756027\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 28.1070i − 1.84135i −0.390327 0.920676i \(-0.627638\pi\)
0.390327 0.920676i \(-0.372362\pi\)
\(234\) 0 0
\(235\) − 12.4662i − 0.813206i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −1.79017 −0.115796 −0.0578982 0.998322i \(-0.518440\pi\)
−0.0578982 + 0.998322i \(0.518440\pi\)
\(240\) 0 0
\(241\) −19.6785 −1.26760 −0.633802 0.773495i \(-0.718507\pi\)
−0.633802 + 0.773495i \(0.718507\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1.93185i 0.123421i
\(246\) 0 0
\(247\) 31.6347i 2.01287i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 22.9672 1.44968 0.724839 0.688918i \(-0.241914\pi\)
0.724839 + 0.688918i \(0.241914\pi\)
\(252\) 0 0
\(253\) 10.6378 0.668791
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 28.7563i − 1.79377i −0.442265 0.896884i \(-0.645825\pi\)
0.442265 0.896884i \(-0.354175\pi\)
\(258\) 0 0
\(259\) − 9.65815i − 0.600128i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 30.8415 1.90177 0.950883 0.309549i \(-0.100178\pi\)
0.950883 + 0.309549i \(0.100178\pi\)
\(264\) 0 0
\(265\) 21.6545 1.33022
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 8.16635i 0.497911i 0.968515 + 0.248956i \(0.0800873\pi\)
−0.968515 + 0.248956i \(0.919913\pi\)
\(270\) 0 0
\(271\) 1.46832i 0.0891941i 0.999005 + 0.0445970i \(0.0142004\pi\)
−0.999005 + 0.0445970i \(0.985800\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 4.54586 0.274126
\(276\) 0 0
\(277\) −6.53225 −0.392485 −0.196242 0.980555i \(-0.562874\pi\)
−0.196242 + 0.980555i \(0.562874\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 13.2415i 0.789921i 0.918698 + 0.394960i \(0.129242\pi\)
−0.918698 + 0.394960i \(0.870758\pi\)
\(282\) 0 0
\(283\) − 6.44219i − 0.382948i −0.981498 0.191474i \(-0.938673\pi\)
0.981498 0.191474i \(-0.0613268\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 8.10227 0.478262
\(288\) 0 0
\(289\) −31.5328 −1.85487
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 13.5623i − 0.792319i −0.918182 0.396159i \(-0.870343\pi\)
0.918182 0.396159i \(-0.129657\pi\)
\(294\) 0 0
\(295\) − 12.2643i − 0.714055i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 17.5835 1.01688
\(300\) 0 0
\(301\) 2.46199 0.141907
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 20.7216i 1.18652i
\(306\) 0 0
\(307\) − 29.8710i − 1.70483i −0.522867 0.852414i \(-0.675138\pi\)
0.522867 0.852414i \(-0.324862\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −27.6551 −1.56817 −0.784087 0.620651i \(-0.786869\pi\)
−0.784087 + 0.620651i \(0.786869\pi\)
\(312\) 0 0
\(313\) 16.7430 0.946371 0.473185 0.880963i \(-0.343104\pi\)
0.473185 + 0.880963i \(0.343104\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0.136810i 0.00768401i 0.999993 + 0.00384201i \(0.00122295\pi\)
−0.999993 + 0.00384201i \(0.998777\pi\)
\(318\) 0 0
\(319\) − 10.9979i − 0.615764i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 37.1888 2.06924
\(324\) 0 0
\(325\) 7.51399 0.416801
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 6.45299i − 0.355765i
\(330\) 0 0
\(331\) − 21.9927i − 1.20883i −0.796670 0.604414i \(-0.793407\pi\)
0.796670 0.604414i \(-0.206593\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −12.1017 −0.661186
\(336\) 0 0
\(337\) −8.11859 −0.442248 −0.221124 0.975246i \(-0.570973\pi\)
−0.221124 + 0.975246i \(0.570973\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 4.55343i 0.246582i
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 6.86909 0.368752 0.184376 0.982856i \(-0.440974\pi\)
0.184376 + 0.982856i \(0.440974\pi\)
\(348\) 0 0
\(349\) −0.140506 −0.00752111 −0.00376055 0.999993i \(-0.501197\pi\)
−0.00376055 + 0.999993i \(0.501197\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 24.6711i 1.31311i 0.754279 + 0.656554i \(0.227987\pi\)
−0.754279 + 0.656554i \(0.772013\pi\)
\(354\) 0 0
\(355\) 3.72629i 0.197771i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −30.8985 −1.63076 −0.815380 0.578926i \(-0.803472\pi\)
−0.815380 + 0.578926i \(0.803472\pi\)
\(360\) 0 0
\(361\) −9.49640 −0.499811
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 4.10285i 0.214753i
\(366\) 0 0
\(367\) 22.7326i 1.18663i 0.804969 + 0.593316i \(0.202182\pi\)
−0.804969 + 0.593316i \(0.797818\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 11.2092 0.581952
\(372\) 0 0
\(373\) 11.9084 0.616594 0.308297 0.951290i \(-0.400241\pi\)
0.308297 + 0.951290i \(0.400241\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 18.1787i − 0.936252i
\(378\) 0 0
\(379\) − 27.2466i − 1.39956i −0.714356 0.699782i \(-0.753280\pi\)
0.714356 0.699782i \(-0.246720\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −16.2385 −0.829749 −0.414875 0.909879i \(-0.636175\pi\)
−0.414875 + 0.909879i \(0.636175\pi\)
\(384\) 0 0
\(385\) −6.92609 −0.352986
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 11.0002i − 0.557730i −0.960330 0.278865i \(-0.910042\pi\)
0.960330 0.278865i \(-0.0899582\pi\)
\(390\) 0 0
\(391\) − 20.6706i − 1.04536i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 16.8338 0.846998
\(396\) 0 0
\(397\) −27.5608 −1.38324 −0.691618 0.722263i \(-0.743102\pi\)
−0.691618 + 0.722263i \(0.743102\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 5.66929i − 0.283111i −0.989930 0.141555i \(-0.954790\pi\)
0.989930 0.141555i \(-0.0452103\pi\)
\(402\) 0 0
\(403\) 7.52648i 0.374921i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 34.6265 1.71637
\(408\) 0 0
\(409\) −15.2382 −0.753479 −0.376739 0.926319i \(-0.622955\pi\)
−0.376739 + 0.926319i \(0.622955\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 6.34847i − 0.312388i
\(414\) 0 0
\(415\) − 15.3902i − 0.755475i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −23.7569 −1.16060 −0.580300 0.814403i \(-0.697065\pi\)
−0.580300 + 0.814403i \(0.697065\pi\)
\(420\) 0 0
\(421\) 33.6010 1.63761 0.818805 0.574072i \(-0.194637\pi\)
0.818805 + 0.574072i \(0.194637\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 8.83323i − 0.428475i
\(426\) 0 0
\(427\) 10.7263i 0.519082i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 3.41960 0.164716 0.0823581 0.996603i \(-0.473755\pi\)
0.0823581 + 0.996603i \(0.473755\pi\)
\(432\) 0 0
\(433\) 9.42250 0.452816 0.226408 0.974033i \(-0.427302\pi\)
0.226408 + 0.974033i \(0.427302\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 15.8391i 0.757688i
\(438\) 0 0
\(439\) 14.1040i 0.673147i 0.941657 + 0.336573i \(0.109268\pi\)
−0.941657 + 0.336573i \(0.890732\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −5.06481 −0.240636 −0.120318 0.992735i \(-0.538391\pi\)
−0.120318 + 0.992735i \(0.538391\pi\)
\(444\) 0 0
\(445\) −17.4662 −0.827978
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 7.52055i − 0.354917i −0.984128 0.177458i \(-0.943212\pi\)
0.984128 0.177458i \(-0.0567875\pi\)
\(450\) 0 0
\(451\) 29.0483i 1.36783i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −11.4483 −0.536706
\(456\) 0 0
\(457\) −12.3620 −0.578268 −0.289134 0.957289i \(-0.593367\pi\)
−0.289134 + 0.957289i \(0.593367\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 12.2111i − 0.568727i −0.958717 0.284363i \(-0.908218\pi\)
0.958717 0.284363i \(-0.0917822\pi\)
\(462\) 0 0
\(463\) 8.00422i 0.371988i 0.982551 + 0.185994i \(0.0595504\pi\)
−0.982551 + 0.185994i \(0.940450\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −34.9725 −1.61834 −0.809168 0.587577i \(-0.800082\pi\)
−0.809168 + 0.587577i \(0.800082\pi\)
\(468\) 0 0
\(469\) −6.26430 −0.289258
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 8.82676i 0.405855i
\(474\) 0 0
\(475\) 6.76857i 0.310563i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −0.141096 −0.00644683 −0.00322342 0.999995i \(-0.501026\pi\)
−0.00322342 + 0.999995i \(0.501026\pi\)
\(480\) 0 0
\(481\) 57.2351 2.60969
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 8.38076i 0.380551i
\(486\) 0 0
\(487\) 18.5161i 0.839044i 0.907745 + 0.419522i \(0.137802\pi\)
−0.907745 + 0.419522i \(0.862198\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 33.4544 1.50978 0.754888 0.655854i \(-0.227691\pi\)
0.754888 + 0.655854i \(0.227691\pi\)
\(492\) 0 0
\(493\) −21.3704 −0.962474
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 1.92887i 0.0865216i
\(498\) 0 0
\(499\) 37.8647i 1.69506i 0.530751 + 0.847528i \(0.321910\pi\)
−0.530751 + 0.847528i \(0.678090\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −19.9336 −0.888796 −0.444398 0.895829i \(-0.646582\pi\)
−0.444398 + 0.895829i \(0.646582\pi\)
\(504\) 0 0
\(505\) 11.8464 0.527159
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 6.33087i 0.280611i 0.990108 + 0.140305i \(0.0448084\pi\)
−0.990108 + 0.140305i \(0.955192\pi\)
\(510\) 0 0
\(511\) 2.12379i 0.0939510i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 27.0389 1.19147
\(516\) 0 0
\(517\) 23.1353 1.01749
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 29.1404i − 1.27666i −0.769761 0.638332i \(-0.779625\pi\)
0.769761 0.638332i \(-0.220375\pi\)
\(522\) 0 0
\(523\) − 7.13474i − 0.311981i −0.987759 0.155990i \(-0.950143\pi\)
0.987759 0.155990i \(-0.0498569\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 8.84792 0.385422
\(528\) 0 0
\(529\) −14.1962 −0.617224
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 48.0148i 2.07975i
\(534\) 0 0
\(535\) 18.0483i 0.780298i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −3.58521 −0.154426
\(540\) 0 0
\(541\) −29.0243 −1.24785 −0.623926 0.781483i \(-0.714464\pi\)
−0.623926 + 0.781483i \(0.714464\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 36.8338i − 1.57779i
\(546\) 0 0
\(547\) 16.8759i 0.721563i 0.932650 + 0.360782i \(0.117490\pi\)
−0.932650 + 0.360782i \(0.882510\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 16.3753 0.697612
\(552\) 0 0
\(553\) 8.71379 0.370548
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 6.41255i 0.271708i 0.990729 + 0.135854i \(0.0433779\pi\)
−0.990729 + 0.135854i \(0.956622\pi\)
\(558\) 0 0
\(559\) 14.5900i 0.617091i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −34.1803 −1.44053 −0.720263 0.693701i \(-0.755979\pi\)
−0.720263 + 0.693701i \(0.755979\pi\)
\(564\) 0 0
\(565\) 3.26219 0.137241
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 16.4841i 0.691050i 0.938409 + 0.345525i \(0.112299\pi\)
−0.938409 + 0.345525i \(0.887701\pi\)
\(570\) 0 0
\(571\) − 23.9105i − 1.00062i −0.865845 0.500312i \(-0.833219\pi\)
0.865845 0.500312i \(-0.166781\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 3.76217 0.156893
\(576\) 0 0
\(577\) 30.0967 1.25294 0.626471 0.779445i \(-0.284499\pi\)
0.626471 + 0.779445i \(0.284499\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 7.96655i − 0.330508i
\(582\) 0 0
\(583\) 40.1873i 1.66439i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −11.6853 −0.482304 −0.241152 0.970487i \(-0.577525\pi\)
−0.241152 + 0.970487i \(0.577525\pi\)
\(588\) 0 0
\(589\) −6.77983 −0.279358
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 16.6788i 0.684914i 0.939533 + 0.342457i \(0.111259\pi\)
−0.939533 + 0.342457i \(0.888741\pi\)
\(594\) 0 0
\(595\) 13.4583i 0.551738i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −8.64980 −0.353421 −0.176711 0.984263i \(-0.556546\pi\)
−0.176711 + 0.984263i \(0.556546\pi\)
\(600\) 0 0
\(601\) −15.6171 −0.637035 −0.318518 0.947917i \(-0.603185\pi\)
−0.318518 + 0.947917i \(0.603185\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 3.58114i − 0.145594i
\(606\) 0 0
\(607\) − 20.8668i − 0.846957i −0.905906 0.423479i \(-0.860809\pi\)
0.905906 0.423479i \(-0.139191\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 38.2410 1.54707
\(612\) 0 0
\(613\) 35.5526 1.43596 0.717978 0.696066i \(-0.245068\pi\)
0.717978 + 0.696066i \(0.245068\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 27.3777i − 1.10218i −0.834445 0.551091i \(-0.814212\pi\)
0.834445 0.551091i \(-0.185788\pi\)
\(618\) 0 0
\(619\) 27.5800i 1.10853i 0.832339 + 0.554267i \(0.187001\pi\)
−0.832339 + 0.554267i \(0.812999\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −9.04118 −0.362227
\(624\) 0 0
\(625\) −17.0526 −0.682102
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 67.2839i − 2.68279i
\(630\) 0 0
\(631\) 18.1207i 0.721374i 0.932687 + 0.360687i \(0.117458\pi\)
−0.932687 + 0.360687i \(0.882542\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 29.4460 1.16853
\(636\) 0 0
\(637\) −5.92609 −0.234800
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 10.9215i − 0.431372i −0.976463 0.215686i \(-0.930801\pi\)
0.976463 0.215686i \(-0.0691987\pi\)
\(642\) 0 0
\(643\) − 43.2372i − 1.70511i −0.522639 0.852554i \(-0.675052\pi\)
0.522639 0.852554i \(-0.324948\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 19.4472 0.764547 0.382274 0.924049i \(-0.375141\pi\)
0.382274 + 0.924049i \(0.375141\pi\)
\(648\) 0 0
\(649\) 22.7606 0.893431
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 27.7057i 1.08421i 0.840311 + 0.542105i \(0.182372\pi\)
−0.840311 + 0.542105i \(0.817628\pi\)
\(654\) 0 0
\(655\) 31.0447i 1.21302i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 20.5359 0.799966 0.399983 0.916523i \(-0.369016\pi\)
0.399983 + 0.916523i \(0.369016\pi\)
\(660\) 0 0
\(661\) 0.763677 0.0297036 0.0148518 0.999890i \(-0.495272\pi\)
0.0148518 + 0.999890i \(0.495272\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 10.3126i − 0.399906i
\(666\) 0 0
\(667\) − 9.10188i − 0.352426i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −38.4560 −1.48458
\(672\) 0 0
\(673\) −26.2944 −1.01357 −0.506787 0.862071i \(-0.669167\pi\)
−0.506787 + 0.862071i \(0.669167\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 17.6603i 0.678740i 0.940653 + 0.339370i \(0.110214\pi\)
−0.940653 + 0.339370i \(0.889786\pi\)
\(678\) 0 0
\(679\) 4.33820i 0.166485i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −41.9049 −1.60345 −0.801723 0.597695i \(-0.796083\pi\)
−0.801723 + 0.597695i \(0.796083\pi\)
\(684\) 0 0
\(685\) 7.39020 0.282365
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 66.4267i 2.53066i
\(690\) 0 0
\(691\) − 32.1831i − 1.22430i −0.790741 0.612151i \(-0.790304\pi\)
0.790741 0.612151i \(-0.209696\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 30.7204 1.16529
\(696\) 0 0
\(697\) 56.4449 2.13800
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 40.8362i 1.54236i 0.636615 + 0.771182i \(0.280334\pi\)
−0.636615 + 0.771182i \(0.719666\pi\)
\(702\) 0 0
\(703\) 51.5571i 1.94451i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 6.13216 0.230624
\(708\) 0 0
\(709\) 24.3986 0.916310 0.458155 0.888872i \(-0.348510\pi\)
0.458155 + 0.888872i \(0.348510\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 3.76842i 0.141129i
\(714\) 0 0
\(715\) − 41.0447i − 1.53498i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −13.9008 −0.518413 −0.259206 0.965822i \(-0.583461\pi\)
−0.259206 + 0.965822i \(0.583461\pi\)
\(720\) 0 0
\(721\) 13.9963 0.521251
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 3.88953i − 0.144453i
\(726\) 0 0
\(727\) 2.98652i 0.110764i 0.998465 + 0.0553820i \(0.0176377\pi\)
−0.998465 + 0.0553820i \(0.982362\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 17.1516 0.634375
\(732\) 0 0
\(733\) −2.16874 −0.0801044 −0.0400522 0.999198i \(-0.512752\pi\)
−0.0400522 + 0.999198i \(0.512752\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 22.4588i − 0.827281i
\(738\) 0 0
\(739\) − 19.1144i − 0.703134i −0.936163 0.351567i \(-0.885649\pi\)
0.936163 0.351567i \(-0.114351\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 42.1582 1.54663 0.773317 0.634020i \(-0.218596\pi\)
0.773317 + 0.634020i \(0.218596\pi\)
\(744\) 0 0
\(745\) −15.8023 −0.578952
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 9.34251i 0.341368i
\(750\) 0 0
\(751\) − 19.2016i − 0.700678i −0.936623 0.350339i \(-0.886066\pi\)
0.936623 0.350339i \(-0.113934\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −15.5512 −0.565966
\(756\) 0 0
\(757\) −3.84797 −0.139857 −0.0699284 0.997552i \(-0.522277\pi\)
−0.0699284 + 0.997552i \(0.522277\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 45.3580i − 1.64423i −0.569325 0.822113i \(-0.692795\pi\)
0.569325 0.822113i \(-0.307205\pi\)
\(762\) 0 0
\(763\) − 19.0666i − 0.690257i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 37.6216 1.35844
\(768\) 0 0
\(769\) −5.83239 −0.210321 −0.105161 0.994455i \(-0.533536\pi\)
−0.105161 + 0.994455i \(0.533536\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 34.9259i − 1.25620i −0.778133 0.628099i \(-0.783833\pi\)
0.778133 0.628099i \(-0.216167\pi\)
\(774\) 0 0
\(775\) 1.61037i 0.0578462i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −43.2516 −1.54965
\(780\) 0 0
\(781\) −6.91540 −0.247453
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 34.0817i 1.21643i
\(786\) 0 0
\(787\) − 51.7877i − 1.84603i −0.384762 0.923016i \(-0.625716\pi\)
0.384762 0.923016i \(-0.374284\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 1.68863 0.0600409
\(792\) 0 0
\(793\) −63.5650 −2.25726
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 43.7027i − 1.54803i −0.633167 0.774015i \(-0.718246\pi\)
0.633167 0.774015i \(-0.281754\pi\)
\(798\) 0 0
\(799\) − 44.9550i − 1.59040i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −7.61424 −0.268701
\(804\) 0 0
\(805\) −5.73205 −0.202028
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 19.6717i − 0.691620i −0.938304 0.345810i \(-0.887604\pi\)
0.938304 0.345810i \(-0.112396\pi\)
\(810\) 0 0
\(811\) − 3.16694i − 0.111206i −0.998453 0.0556031i \(-0.982292\pi\)
0.998453 0.0556031i \(-0.0177082\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −4.76435 −0.166888
\(816\) 0 0
\(817\) −13.1426 −0.459802
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 7.54805i 0.263429i 0.991288 + 0.131714i \(0.0420482\pi\)
−0.991288 + 0.131714i \(0.957952\pi\)
\(822\) 0 0
\(823\) 25.0280i 0.872420i 0.899845 + 0.436210i \(0.143680\pi\)
−0.899845 + 0.436210i \(0.856320\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −26.9477 −0.937062 −0.468531 0.883447i \(-0.655217\pi\)
−0.468531 + 0.883447i \(0.655217\pi\)
\(828\) 0 0
\(829\) −19.2466 −0.668462 −0.334231 0.942491i \(-0.608477\pi\)
−0.334231 + 0.942491i \(0.608477\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 6.96655i 0.241377i
\(834\) 0 0
\(835\) − 31.1685i − 1.07863i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −25.4979 −0.880285 −0.440142 0.897928i \(-0.645072\pi\)
−0.440142 + 0.897928i \(0.645072\pi\)
\(840\) 0 0
\(841\) 19.5900 0.675517
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 42.7298i − 1.46995i
\(846\) 0 0
\(847\) − 1.85373i − 0.0636950i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 28.6570 0.982348
\(852\) 0 0
\(853\) 16.4464 0.563114 0.281557 0.959544i \(-0.409149\pi\)
0.281557 + 0.959544i \(0.409149\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 37.0695i 1.26627i 0.774041 + 0.633135i \(0.218232\pi\)
−0.774041 + 0.633135i \(0.781768\pi\)
\(858\) 0 0
\(859\) 17.6284i 0.601472i 0.953707 + 0.300736i \(0.0972323\pi\)
−0.953707 + 0.300736i \(0.902768\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −21.5272 −0.732794 −0.366397 0.930459i \(-0.619409\pi\)
−0.366397 + 0.930459i \(0.619409\pi\)
\(864\) 0 0
\(865\) −23.1904 −0.788497
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 31.2408i 1.05977i
\(870\) 0 0
\(871\) − 37.1228i − 1.25786i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −12.1087 −0.409350
\(876\) 0 0
\(877\) −28.1862 −0.951779 −0.475890 0.879505i \(-0.657874\pi\)
−0.475890 + 0.879505i \(0.657874\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 15.3854i 0.518346i 0.965831 + 0.259173i \(0.0834500\pi\)
−0.965831 + 0.259173i \(0.916550\pi\)
\(882\) 0 0
\(883\) 29.4069i 0.989621i 0.869001 + 0.494811i \(0.164763\pi\)
−0.869001 + 0.494811i \(0.835237\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −28.1982 −0.946801 −0.473401 0.880847i \(-0.656974\pi\)
−0.473401 + 0.880847i \(0.656974\pi\)
\(888\) 0 0
\(889\) 15.2424 0.511213
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 34.4473i 1.15274i
\(894\) 0 0
\(895\) 27.9869i 0.935500i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 3.89599 0.129939
\(900\) 0 0
\(901\) 78.0894 2.60153
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 16.8579i 0.560375i
\(906\) 0 0
\(907\) 15.0718i 0.500451i 0.968188 + 0.250225i \(0.0805047\pi\)
−0.968188 + 0.250225i \(0.919495\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 5.49292 0.181989 0.0909943 0.995851i \(-0.470996\pi\)
0.0909943 + 0.995851i \(0.470996\pi\)
\(912\) 0 0
\(913\) 28.5618 0.945256
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 16.0699i 0.530675i
\(918\) 0 0
\(919\) − 18.6898i − 0.616519i −0.951302 0.308259i \(-0.900254\pi\)
0.951302 0.308259i \(-0.0997464\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −11.4307 −0.376245
\(924\) 0 0
\(925\) 12.2460 0.402647
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 35.6186i 1.16861i 0.811534 + 0.584305i \(0.198633\pi\)
−0.811534 + 0.584305i \(0.801367\pi\)
\(930\) 0 0
\(931\) − 5.33820i − 0.174953i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −48.2510 −1.57798
\(936\) 0 0
\(937\) −36.3017 −1.18592 −0.592962 0.805230i \(-0.702042\pi\)
−0.592962 + 0.805230i \(0.702042\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 31.7581i − 1.03529i −0.855597 0.517643i \(-0.826810\pi\)
0.855597 0.517643i \(-0.173190\pi\)
\(942\) 0 0
\(943\) 24.0405i 0.782865i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −30.0513 −0.976536 −0.488268 0.872694i \(-0.662371\pi\)
−0.488268 + 0.872694i \(0.662371\pi\)
\(948\) 0 0
\(949\) −12.5858 −0.408552
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 52.4957i 1.70050i 0.526376 + 0.850252i \(0.323550\pi\)
−0.526376 + 0.850252i \(0.676450\pi\)
\(954\) 0 0
\(955\) − 24.4626i − 0.791590i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 3.82545 0.123530
\(960\) 0 0
\(961\) 29.3870 0.947966
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 6.78850i 0.218530i
\(966\) 0 0
\(967\) 47.4411i 1.52560i 0.646634 + 0.762801i \(0.276176\pi\)
−0.646634 + 0.762801i \(0.723824\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −33.9492 −1.08948 −0.544741 0.838605i \(-0.683372\pi\)
−0.544741 + 0.838605i \(0.683372\pi\)
\(972\) 0 0
\(973\) 15.9021 0.509797
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 12.2747i − 0.392701i −0.980534 0.196351i \(-0.937091\pi\)
0.980534 0.196351i \(-0.0629090\pi\)
\(978\) 0 0
\(979\) − 32.4145i − 1.03597i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −5.43320 −0.173292 −0.0866461 0.996239i \(-0.527615\pi\)
−0.0866461 + 0.996239i \(0.527615\pi\)
\(984\) 0 0
\(985\) −21.8407 −0.695901
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 7.30505i 0.232287i
\(990\) 0 0
\(991\) 49.4370i 1.57042i 0.619231 + 0.785209i \(0.287444\pi\)
−0.619231 + 0.785209i \(0.712556\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 4.90305 0.155437
\(996\) 0 0
\(997\) 28.0925 0.889697 0.444849 0.895606i \(-0.353257\pi\)
0.444849 + 0.895606i \(0.353257\pi\)
\(998\) 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 6048.2.h.h.2591.2 16
3.2 odd 2 inner 6048.2.h.h.2591.13 yes 16
4.3 odd 2 inner 6048.2.h.h.2591.3 yes 16
12.11 even 2 inner 6048.2.h.h.2591.16 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6048.2.h.h.2591.2 16 1.1 even 1 trivial
6048.2.h.h.2591.3 yes 16 4.3 odd 2 inner
6048.2.h.h.2591.13 yes 16 3.2 odd 2 inner
6048.2.h.h.2591.16 yes 16 12.11 even 2 inner