Properties

Label 6048.2.c.g.3025.21
Level 6048
Weight 2
Character 6048.3025
Analytic conductor 48.294
Analytic rank 0
Dimension 24
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.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.2935231425\)
Analytic rank: \(0\)
Dimension: \(24\)
Coefficient ring index: multiple of None
Twist minimal: no (minimal twist has level 1512)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3025.21
Character \(\chi\) = 6048.3025
Dual form 6048.2.c.g.3025.4

$q$-expansion

\(f(q)\) \(=\) \(q+3.11390i q^{5} -1.00000 q^{7} +O(q^{10})\) \(q+3.11390i q^{5} -1.00000 q^{7} -2.37402i q^{11} -1.09044i q^{13} +3.69300 q^{17} +1.08263i q^{19} -4.87366 q^{23} -4.69640 q^{25} +1.59607i q^{29} -7.45516 q^{31} -3.11390i q^{35} +4.61410i q^{37} -0.0380616 q^{41} -11.0608i q^{43} -0.337175 q^{47} +1.00000 q^{49} -8.14672i q^{53} +7.39246 q^{55} -15.0761i q^{59} -5.53745i q^{61} +3.39553 q^{65} +7.70421i q^{67} +13.5472 q^{71} -14.6727 q^{73} +2.37402i q^{77} -5.18122 q^{79} +0.138321i q^{83} +11.4996i q^{85} -11.9531 q^{89} +1.09044i q^{91} -3.37121 q^{95} -5.30202 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24q - 24q^{7} + O(q^{10}) \) \( 24q - 24q^{7} - 24q^{25} + 16q^{31} + 24q^{49} + 8q^{55} - 32q^{79} + O(q^{100}) \)

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
\(5\) 3.11390i 1.39258i 0.717760 + 0.696290i \(0.245167\pi\)
−0.717760 + 0.696290i \(0.754833\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 2.37402i − 0.715793i −0.933761 0.357897i \(-0.883494\pi\)
0.933761 0.357897i \(-0.116506\pi\)
\(12\) 0 0
\(13\) − 1.09044i − 0.302434i −0.988501 0.151217i \(-0.951681\pi\)
0.988501 0.151217i \(-0.0483192\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.69300 0.895684 0.447842 0.894113i \(-0.352193\pi\)
0.447842 + 0.894113i \(0.352193\pi\)
\(18\) 0 0
\(19\) 1.08263i 0.248372i 0.992259 + 0.124186i \(0.0396320\pi\)
−0.992259 + 0.124186i \(0.960368\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.87366 −1.01623 −0.508114 0.861290i \(-0.669657\pi\)
−0.508114 + 0.861290i \(0.669657\pi\)
\(24\) 0 0
\(25\) −4.69640 −0.939280
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.59607i 0.296382i 0.988959 + 0.148191i \(0.0473451\pi\)
−0.988959 + 0.148191i \(0.952655\pi\)
\(30\) 0 0
\(31\) −7.45516 −1.33899 −0.669493 0.742818i \(-0.733489\pi\)
−0.669493 + 0.742818i \(0.733489\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 3.11390i − 0.526346i
\(36\) 0 0
\(37\) 4.61410i 0.758554i 0.925283 + 0.379277i \(0.123827\pi\)
−0.925283 + 0.379277i \(0.876173\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −0.0380616 −0.00594422 −0.00297211 0.999996i \(-0.500946\pi\)
−0.00297211 + 0.999996i \(0.500946\pi\)
\(42\) 0 0
\(43\) − 11.0608i − 1.68675i −0.537323 0.843377i \(-0.680564\pi\)
0.537323 0.843377i \(-0.319436\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −0.337175 −0.0491821 −0.0245910 0.999698i \(-0.507828\pi\)
−0.0245910 + 0.999698i \(0.507828\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 8.14672i − 1.11904i −0.828818 0.559519i \(-0.810986\pi\)
0.828818 0.559519i \(-0.189014\pi\)
\(54\) 0 0
\(55\) 7.39246 0.996799
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 15.0761i − 1.96274i −0.192129 0.981370i \(-0.561539\pi\)
0.192129 0.981370i \(-0.438461\pi\)
\(60\) 0 0
\(61\) − 5.53745i − 0.708998i −0.935056 0.354499i \(-0.884651\pi\)
0.935056 0.354499i \(-0.115349\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.39553 0.421163
\(66\) 0 0
\(67\) 7.70421i 0.941219i 0.882342 + 0.470609i \(0.155966\pi\)
−0.882342 + 0.470609i \(0.844034\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 13.5472 1.60776 0.803878 0.594794i \(-0.202766\pi\)
0.803878 + 0.594794i \(0.202766\pi\)
\(72\) 0 0
\(73\) −14.6727 −1.71731 −0.858656 0.512553i \(-0.828700\pi\)
−0.858656 + 0.512553i \(0.828700\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.37402i 0.270544i
\(78\) 0 0
\(79\) −5.18122 −0.582932 −0.291466 0.956581i \(-0.594143\pi\)
−0.291466 + 0.956581i \(0.594143\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0.138321i 0.0151827i 0.999971 + 0.00759136i \(0.00241643\pi\)
−0.999971 + 0.00759136i \(0.997584\pi\)
\(84\) 0 0
\(85\) 11.4996i 1.24731i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −11.9531 −1.26703 −0.633514 0.773731i \(-0.718388\pi\)
−0.633514 + 0.773731i \(0.718388\pi\)
\(90\) 0 0
\(91\) 1.09044i 0.114309i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −3.37121 −0.345879
\(96\) 0 0
\(97\) −5.30202 −0.538339 −0.269169 0.963093i \(-0.586749\pi\)
−0.269169 + 0.963093i \(0.586749\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0.0753369i 0.00749630i 0.999993 + 0.00374815i \(0.00119308\pi\)
−0.999993 + 0.00374815i \(0.998807\pi\)
\(102\) 0 0
\(103\) −15.3347 −1.51097 −0.755487 0.655163i \(-0.772600\pi\)
−0.755487 + 0.655163i \(0.772600\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.03726i 0.390297i 0.980774 + 0.195148i \(0.0625189\pi\)
−0.980774 + 0.195148i \(0.937481\pi\)
\(108\) 0 0
\(109\) 12.2335i 1.17176i 0.810399 + 0.585879i \(0.199251\pi\)
−0.810399 + 0.585879i \(0.800749\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −11.1934 −1.05299 −0.526495 0.850178i \(-0.676494\pi\)
−0.526495 + 0.850178i \(0.676494\pi\)
\(114\) 0 0
\(115\) − 15.1761i − 1.41518i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −3.69300 −0.338537
\(120\) 0 0
\(121\) 5.36404 0.487640
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0.945386i 0.0845579i
\(126\) 0 0
\(127\) 2.43139 0.215751 0.107876 0.994164i \(-0.465595\pi\)
0.107876 + 0.994164i \(0.465595\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 8.83723i − 0.772112i −0.922475 0.386056i \(-0.873837\pi\)
0.922475 0.386056i \(-0.126163\pi\)
\(132\) 0 0
\(133\) − 1.08263i − 0.0938760i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 11.1478 0.952422 0.476211 0.879331i \(-0.342010\pi\)
0.476211 + 0.879331i \(0.342010\pi\)
\(138\) 0 0
\(139\) − 17.2724i − 1.46502i −0.680755 0.732511i \(-0.738348\pi\)
0.680755 0.732511i \(-0.261652\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −2.58872 −0.216480
\(144\) 0 0
\(145\) −4.97000 −0.412736
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 13.7236i − 1.12428i −0.827042 0.562140i \(-0.809978\pi\)
0.827042 0.562140i \(-0.190022\pi\)
\(150\) 0 0
\(151\) 12.8195 1.04324 0.521620 0.853178i \(-0.325328\pi\)
0.521620 + 0.853178i \(0.325328\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 23.2146i − 1.86465i
\(156\) 0 0
\(157\) 21.1418i 1.68730i 0.536892 + 0.843651i \(0.319598\pi\)
−0.536892 + 0.843651i \(0.680402\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 4.87366 0.384098
\(162\) 0 0
\(163\) 0.386568i 0.0302784i 0.999885 + 0.0151392i \(0.00481914\pi\)
−0.999885 + 0.0151392i \(0.995181\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 14.4336 1.11690 0.558451 0.829537i \(-0.311396\pi\)
0.558451 + 0.829537i \(0.311396\pi\)
\(168\) 0 0
\(169\) 11.8109 0.908534
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 13.9493i − 1.06055i −0.847826 0.530274i \(-0.822089\pi\)
0.847826 0.530274i \(-0.177911\pi\)
\(174\) 0 0
\(175\) 4.69640 0.355014
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 10.6569i − 0.796537i −0.917269 0.398269i \(-0.869611\pi\)
0.917269 0.398269i \(-0.130389\pi\)
\(180\) 0 0
\(181\) 6.63866i 0.493448i 0.969086 + 0.246724i \(0.0793540\pi\)
−0.969086 + 0.246724i \(0.920646\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −14.3679 −1.05635
\(186\) 0 0
\(187\) − 8.76724i − 0.641124i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −9.81148 −0.709934 −0.354967 0.934879i \(-0.615508\pi\)
−0.354967 + 0.934879i \(0.615508\pi\)
\(192\) 0 0
\(193\) 17.4539 1.25636 0.628180 0.778068i \(-0.283800\pi\)
0.628180 + 0.778068i \(0.283800\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 27.6871i − 1.97262i −0.164888 0.986312i \(-0.552726\pi\)
0.164888 0.986312i \(-0.447274\pi\)
\(198\) 0 0
\(199\) −9.29804 −0.659121 −0.329560 0.944134i \(-0.606901\pi\)
−0.329560 + 0.944134i \(0.606901\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 1.59607i − 0.112022i
\(204\) 0 0
\(205\) − 0.118520i − 0.00827780i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 2.57018 0.177783
\(210\) 0 0
\(211\) − 16.3068i − 1.12260i −0.827611 0.561302i \(-0.810301\pi\)
0.827611 0.561302i \(-0.189699\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 34.4422 2.34894
\(216\) 0 0
\(217\) 7.45516 0.506089
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 4.02699i − 0.270885i
\(222\) 0 0
\(223\) 18.8846 1.26461 0.632304 0.774720i \(-0.282109\pi\)
0.632304 + 0.774720i \(0.282109\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 12.9709i − 0.860907i −0.902613 0.430454i \(-0.858354\pi\)
0.902613 0.430454i \(-0.141646\pi\)
\(228\) 0 0
\(229\) − 5.92552i − 0.391569i −0.980647 0.195785i \(-0.937275\pi\)
0.980647 0.195785i \(-0.0627253\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 12.2093 0.799861 0.399930 0.916546i \(-0.369034\pi\)
0.399930 + 0.916546i \(0.369034\pi\)
\(234\) 0 0
\(235\) − 1.04993i − 0.0684900i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −22.5273 −1.45717 −0.728585 0.684956i \(-0.759822\pi\)
−0.728585 + 0.684956i \(0.759822\pi\)
\(240\) 0 0
\(241\) −16.1826 −1.04241 −0.521206 0.853431i \(-0.674518\pi\)
−0.521206 + 0.853431i \(0.674518\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 3.11390i 0.198940i
\(246\) 0 0
\(247\) 1.18054 0.0751162
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 6.88454i − 0.434548i −0.976111 0.217274i \(-0.930283\pi\)
0.976111 0.217274i \(-0.0697166\pi\)
\(252\) 0 0
\(253\) 11.5702i 0.727409i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −2.70139 −0.168508 −0.0842541 0.996444i \(-0.526851\pi\)
−0.0842541 + 0.996444i \(0.526851\pi\)
\(258\) 0 0
\(259\) − 4.61410i − 0.286707i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −13.5500 −0.835527 −0.417763 0.908556i \(-0.637186\pi\)
−0.417763 + 0.908556i \(0.637186\pi\)
\(264\) 0 0
\(265\) 25.3681 1.55835
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 6.41522i − 0.391143i −0.980689 0.195571i \(-0.937344\pi\)
0.980689 0.195571i \(-0.0626562\pi\)
\(270\) 0 0
\(271\) 14.5096 0.881397 0.440698 0.897655i \(-0.354731\pi\)
0.440698 + 0.897655i \(0.354731\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 11.1493i 0.672330i
\(276\) 0 0
\(277\) − 25.3175i − 1.52118i −0.649231 0.760591i \(-0.724909\pi\)
0.649231 0.760591i \(-0.275091\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −0.719979 −0.0429504 −0.0214752 0.999769i \(-0.506836\pi\)
−0.0214752 + 0.999769i \(0.506836\pi\)
\(282\) 0 0
\(283\) − 11.3262i − 0.673275i −0.941634 0.336637i \(-0.890710\pi\)
0.941634 0.336637i \(-0.109290\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0.0380616 0.00224670
\(288\) 0 0
\(289\) −3.36176 −0.197751
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 15.2567i − 0.891304i −0.895206 0.445652i \(-0.852972\pi\)
0.895206 0.445652i \(-0.147028\pi\)
\(294\) 0 0
\(295\) 46.9455 2.73327
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 5.31443i 0.307342i
\(300\) 0 0
\(301\) 11.0608i 0.637533i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 17.2431 0.987337
\(306\) 0 0
\(307\) 25.5497i 1.45820i 0.684409 + 0.729098i \(0.260060\pi\)
−0.684409 + 0.729098i \(0.739940\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −3.00328 −0.170300 −0.0851501 0.996368i \(-0.527137\pi\)
−0.0851501 + 0.996368i \(0.527137\pi\)
\(312\) 0 0
\(313\) −5.53962 −0.313118 −0.156559 0.987669i \(-0.550040\pi\)
−0.156559 + 0.987669i \(0.550040\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 32.1705i − 1.80688i −0.428718 0.903439i \(-0.641034\pi\)
0.428718 0.903439i \(-0.358966\pi\)
\(318\) 0 0
\(319\) 3.78909 0.212148
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 3.99815i 0.222463i
\(324\) 0 0
\(325\) 5.12114i 0.284070i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0.337175 0.0185891
\(330\) 0 0
\(331\) 24.9617i 1.37202i 0.727592 + 0.686010i \(0.240639\pi\)
−0.727592 + 0.686010i \(0.759361\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −23.9902 −1.31072
\(336\) 0 0
\(337\) 17.7899 0.969076 0.484538 0.874770i \(-0.338988\pi\)
0.484538 + 0.874770i \(0.338988\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 17.6987i 0.958437i
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 2.08108i − 0.111718i −0.998439 0.0558591i \(-0.982210\pi\)
0.998439 0.0558591i \(-0.0177898\pi\)
\(348\) 0 0
\(349\) 7.11683i 0.380955i 0.981692 + 0.190477i \(0.0610036\pi\)
−0.981692 + 0.190477i \(0.938996\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 28.9449 1.54058 0.770290 0.637694i \(-0.220111\pi\)
0.770290 + 0.637694i \(0.220111\pi\)
\(354\) 0 0
\(355\) 42.1847i 2.23893i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 10.0190 0.528780 0.264390 0.964416i \(-0.414829\pi\)
0.264390 + 0.964416i \(0.414829\pi\)
\(360\) 0 0
\(361\) 17.8279 0.938311
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 45.6894i − 2.39149i
\(366\) 0 0
\(367\) −31.4662 −1.64252 −0.821261 0.570552i \(-0.806729\pi\)
−0.821261 + 0.570552i \(0.806729\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 8.14672i 0.422956i
\(372\) 0 0
\(373\) − 31.8121i − 1.64717i −0.567193 0.823585i \(-0.691970\pi\)
0.567193 0.823585i \(-0.308030\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1.74042 0.0896360
\(378\) 0 0
\(379\) − 15.2806i − 0.784911i −0.919771 0.392456i \(-0.871626\pi\)
0.919771 0.392456i \(-0.128374\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −19.0555 −0.973691 −0.486845 0.873488i \(-0.661853\pi\)
−0.486845 + 0.873488i \(0.661853\pi\)
\(384\) 0 0
\(385\) −7.39246 −0.376755
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 3.69055i 0.187118i 0.995614 + 0.0935591i \(0.0298244\pi\)
−0.995614 + 0.0935591i \(0.970176\pi\)
\(390\) 0 0
\(391\) −17.9984 −0.910219
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 16.1338i − 0.811780i
\(396\) 0 0
\(397\) − 14.8184i − 0.743715i −0.928290 0.371857i \(-0.878721\pi\)
0.928290 0.371857i \(-0.121279\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −1.80006 −0.0898908 −0.0449454 0.998989i \(-0.514311\pi\)
−0.0449454 + 0.998989i \(0.514311\pi\)
\(402\) 0 0
\(403\) 8.12940i 0.404954i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 10.9540 0.542968
\(408\) 0 0
\(409\) −25.6731 −1.26945 −0.634727 0.772736i \(-0.718888\pi\)
−0.634727 + 0.772736i \(0.718888\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 15.0761i 0.741846i
\(414\) 0 0
\(415\) −0.430719 −0.0211431
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 17.3661i 0.848392i 0.905571 + 0.424196i \(0.139443\pi\)
−0.905571 + 0.424196i \(0.860557\pi\)
\(420\) 0 0
\(421\) 30.4837i 1.48568i 0.669468 + 0.742841i \(0.266522\pi\)
−0.669468 + 0.742841i \(0.733478\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −17.3438 −0.841298
\(426\) 0 0
\(427\) 5.53745i 0.267976i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 9.49698 0.457453 0.228727 0.973491i \(-0.426544\pi\)
0.228727 + 0.973491i \(0.426544\pi\)
\(432\) 0 0
\(433\) 25.6935 1.23475 0.617376 0.786668i \(-0.288196\pi\)
0.617376 + 0.786668i \(0.288196\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 5.27637i − 0.252403i
\(438\) 0 0
\(439\) −11.0047 −0.525227 −0.262614 0.964901i \(-0.584584\pi\)
−0.262614 + 0.964901i \(0.584584\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 32.0650i 1.52346i 0.647897 + 0.761728i \(0.275649\pi\)
−0.647897 + 0.761728i \(0.724351\pi\)
\(444\) 0 0
\(445\) − 37.2208i − 1.76444i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 16.5474 0.780922 0.390461 0.920619i \(-0.372316\pi\)
0.390461 + 0.920619i \(0.372316\pi\)
\(450\) 0 0
\(451\) 0.0903588i 0.00425483i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −3.39553 −0.159185
\(456\) 0 0
\(457\) −26.9067 −1.25864 −0.629321 0.777145i \(-0.716667\pi\)
−0.629321 + 0.777145i \(0.716667\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 28.9351i − 1.34764i −0.738895 0.673821i \(-0.764652\pi\)
0.738895 0.673821i \(-0.235348\pi\)
\(462\) 0 0
\(463\) −10.3287 −0.480017 −0.240009 0.970771i \(-0.577150\pi\)
−0.240009 + 0.970771i \(0.577150\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 22.1113i − 1.02319i −0.859227 0.511595i \(-0.829055\pi\)
0.859227 0.511595i \(-0.170945\pi\)
\(468\) 0 0
\(469\) − 7.70421i − 0.355747i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −26.2585 −1.20737
\(474\) 0 0
\(475\) − 5.08446i − 0.233291i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 5.77499 0.263866 0.131933 0.991259i \(-0.457882\pi\)
0.131933 + 0.991259i \(0.457882\pi\)
\(480\) 0 0
\(481\) 5.03140 0.229412
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 16.5100i − 0.749680i
\(486\) 0 0
\(487\) −12.5566 −0.568993 −0.284497 0.958677i \(-0.591826\pi\)
−0.284497 + 0.958677i \(0.591826\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 20.9548i − 0.945677i −0.881149 0.472839i \(-0.843229\pi\)
0.881149 0.472839i \(-0.156771\pi\)
\(492\) 0 0
\(493\) 5.89428i 0.265465i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −13.5472 −0.607675
\(498\) 0 0
\(499\) − 13.0152i − 0.582641i −0.956626 0.291320i \(-0.905905\pi\)
0.956626 0.291320i \(-0.0940946\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 22.9100 1.02150 0.510752 0.859728i \(-0.329367\pi\)
0.510752 + 0.859728i \(0.329367\pi\)
\(504\) 0 0
\(505\) −0.234592 −0.0104392
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 24.5703i 1.08906i 0.838741 + 0.544530i \(0.183292\pi\)
−0.838741 + 0.544530i \(0.816708\pi\)
\(510\) 0 0
\(511\) 14.6727 0.649083
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 47.7508i − 2.10415i
\(516\) 0 0
\(517\) 0.800460i 0.0352042i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 1.51316 0.0662929 0.0331465 0.999451i \(-0.489447\pi\)
0.0331465 + 0.999451i \(0.489447\pi\)
\(522\) 0 0
\(523\) 0.202190i 0.00884117i 0.999990 + 0.00442059i \(0.00140712\pi\)
−0.999990 + 0.00442059i \(0.998593\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −27.5319 −1.19931
\(528\) 0 0
\(529\) 0.752559 0.0327199
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0.0415038i 0.00179773i
\(534\) 0 0
\(535\) −12.5716 −0.543520
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 2.37402i − 0.102256i
\(540\) 0 0
\(541\) − 28.5045i − 1.22550i −0.790275 0.612752i \(-0.790062\pi\)
0.790275 0.612752i \(-0.209938\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −38.0940 −1.63177
\(546\) 0 0
\(547\) 37.9150i 1.62113i 0.585648 + 0.810565i \(0.300840\pi\)
−0.585648 + 0.810565i \(0.699160\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −1.72795 −0.0736132
\(552\) 0 0
\(553\) 5.18122 0.220328
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 16.6109i 0.703828i 0.936032 + 0.351914i \(0.114469\pi\)
−0.936032 + 0.351914i \(0.885531\pi\)
\(558\) 0 0
\(559\) −12.0611 −0.510131
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 6.84219i − 0.288364i −0.989551 0.144182i \(-0.953945\pi\)
0.989551 0.144182i \(-0.0460551\pi\)
\(564\) 0 0
\(565\) − 34.8553i − 1.46637i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −15.3184 −0.642180 −0.321090 0.947049i \(-0.604049\pi\)
−0.321090 + 0.947049i \(0.604049\pi\)
\(570\) 0 0
\(571\) 36.6853i 1.53523i 0.640911 + 0.767616i \(0.278557\pi\)
−0.640911 + 0.767616i \(0.721443\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 22.8886 0.954523
\(576\) 0 0
\(577\) −23.9150 −0.995596 −0.497798 0.867293i \(-0.665858\pi\)
−0.497798 + 0.867293i \(0.665858\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 0.138321i − 0.00573852i
\(582\) 0 0
\(583\) −19.3404 −0.800999
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 45.0433i 1.85914i 0.368651 + 0.929568i \(0.379820\pi\)
−0.368651 + 0.929568i \(0.620180\pi\)
\(588\) 0 0
\(589\) − 8.07118i − 0.332567i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 13.4524 0.552425 0.276212 0.961097i \(-0.410921\pi\)
0.276212 + 0.961097i \(0.410921\pi\)
\(594\) 0 0
\(595\) − 11.4996i − 0.471439i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 21.8377 0.892263 0.446132 0.894967i \(-0.352801\pi\)
0.446132 + 0.894967i \(0.352801\pi\)
\(600\) 0 0
\(601\) −16.3704 −0.667764 −0.333882 0.942615i \(-0.608359\pi\)
−0.333882 + 0.942615i \(0.608359\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 16.7031i 0.679078i
\(606\) 0 0
\(607\) −28.8256 −1.17000 −0.584998 0.811034i \(-0.698905\pi\)
−0.584998 + 0.811034i \(0.698905\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0.367669i 0.0148743i
\(612\) 0 0
\(613\) − 34.6397i − 1.39908i −0.714592 0.699541i \(-0.753388\pi\)
0.714592 0.699541i \(-0.246612\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 38.6359 1.55542 0.777711 0.628622i \(-0.216381\pi\)
0.777711 + 0.628622i \(0.216381\pi\)
\(618\) 0 0
\(619\) − 25.9642i − 1.04359i −0.853071 0.521794i \(-0.825263\pi\)
0.853071 0.521794i \(-0.174737\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 11.9531 0.478891
\(624\) 0 0
\(625\) −26.4258 −1.05703
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 17.0399i 0.679425i
\(630\) 0 0
\(631\) −5.94731 −0.236759 −0.118379 0.992968i \(-0.537770\pi\)
−0.118379 + 0.992968i \(0.537770\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 7.57112i 0.300451i
\(636\) 0 0
\(637\) − 1.09044i − 0.0432048i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −41.7290 −1.64820 −0.824098 0.566447i \(-0.808318\pi\)
−0.824098 + 0.566447i \(0.808318\pi\)
\(642\) 0 0
\(643\) − 36.9825i − 1.45845i −0.684275 0.729224i \(-0.739881\pi\)
0.684275 0.729224i \(-0.260119\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 2.39044 0.0939778 0.0469889 0.998895i \(-0.485037\pi\)
0.0469889 + 0.998895i \(0.485037\pi\)
\(648\) 0 0
\(649\) −35.7909 −1.40492
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 27.1588i 1.06280i 0.847120 + 0.531402i \(0.178335\pi\)
−0.847120 + 0.531402i \(0.821665\pi\)
\(654\) 0 0
\(655\) 27.5183 1.07523
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 41.3343i − 1.61016i −0.593169 0.805078i \(-0.702123\pi\)
0.593169 0.805078i \(-0.297877\pi\)
\(660\) 0 0
\(661\) 6.44768i 0.250786i 0.992107 + 0.125393i \(0.0400192\pi\)
−0.992107 + 0.125393i \(0.959981\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 3.37121 0.130730
\(666\) 0 0
\(667\) − 7.77869i − 0.301192i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −13.1460 −0.507496
\(672\) 0 0
\(673\) −36.1564 −1.39373 −0.696863 0.717204i \(-0.745421\pi\)
−0.696863 + 0.717204i \(0.745421\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 47.4088i 1.82207i 0.412330 + 0.911035i \(0.364715\pi\)
−0.412330 + 0.911035i \(0.635285\pi\)
\(678\) 0 0
\(679\) 5.30202 0.203473
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 3.55646i − 0.136084i −0.997682 0.0680420i \(-0.978325\pi\)
0.997682 0.0680420i \(-0.0216752\pi\)
\(684\) 0 0
\(685\) 34.7132i 1.32632i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −8.88350 −0.338435
\(690\) 0 0
\(691\) 41.6181i 1.58323i 0.611023 + 0.791613i \(0.290758\pi\)
−0.611023 + 0.791613i \(0.709242\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 53.7845 2.04016
\(696\) 0 0
\(697\) −0.140561 −0.00532414
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 34.8945i − 1.31795i −0.752166 0.658973i \(-0.770991\pi\)
0.752166 0.658973i \(-0.229009\pi\)
\(702\) 0 0
\(703\) −4.99537 −0.188404
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 0.0753369i − 0.00283334i
\(708\) 0 0
\(709\) 49.2296i 1.84886i 0.381357 + 0.924428i \(0.375457\pi\)
−0.381357 + 0.924428i \(0.624543\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 36.3339 1.36072
\(714\) 0 0
\(715\) − 8.06103i − 0.301466i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −10.9969 −0.410117 −0.205058 0.978750i \(-0.565738\pi\)
−0.205058 + 0.978750i \(0.565738\pi\)
\(720\) 0 0
\(721\) 15.3347 0.571095
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 7.49577i − 0.278386i
\(726\) 0 0
\(727\) −9.03008 −0.334907 −0.167454 0.985880i \(-0.553554\pi\)
−0.167454 + 0.985880i \(0.553554\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 40.8474i − 1.51080i
\(732\) 0 0
\(733\) 41.6840i 1.53963i 0.638265 + 0.769817i \(0.279653\pi\)
−0.638265 + 0.769817i \(0.720347\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 18.2899 0.673718
\(738\) 0 0
\(739\) 34.7320i 1.27764i 0.769358 + 0.638818i \(0.220576\pi\)
−0.769358 + 0.638818i \(0.779424\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −15.9173 −0.583951 −0.291975 0.956426i \(-0.594313\pi\)
−0.291975 + 0.956426i \(0.594313\pi\)
\(744\) 0 0
\(745\) 42.7339 1.56565
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 4.03726i − 0.147518i
\(750\) 0 0
\(751\) −13.7001 −0.499925 −0.249963 0.968255i \(-0.580418\pi\)
−0.249963 + 0.968255i \(0.580418\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 39.9188i 1.45279i
\(756\) 0 0
\(757\) − 27.3072i − 0.992496i −0.868181 0.496248i \(-0.834711\pi\)
0.868181 0.496248i \(-0.165289\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 18.3078 0.663657 0.331829 0.943340i \(-0.392334\pi\)
0.331829 + 0.943340i \(0.392334\pi\)
\(762\) 0 0
\(763\) − 12.2335i − 0.442883i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −16.4396 −0.593598
\(768\) 0 0
\(769\) −13.3631 −0.481884 −0.240942 0.970540i \(-0.577456\pi\)
−0.240942 + 0.970540i \(0.577456\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 39.3918i − 1.41682i −0.705799 0.708412i \(-0.749412\pi\)
0.705799 0.708412i \(-0.250588\pi\)
\(774\) 0 0
\(775\) 35.0124 1.25768
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 0.0412066i − 0.00147638i
\(780\) 0 0
\(781\) − 32.1613i − 1.15082i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −65.8336 −2.34970
\(786\) 0 0
\(787\) 36.3862i 1.29703i 0.761203 + 0.648514i \(0.224609\pi\)
−0.761203 + 0.648514i \(0.775391\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 11.1934 0.397993
\(792\) 0 0
\(793\) −6.03826 −0.214425
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 33.1598i 1.17458i 0.809377 + 0.587290i \(0.199805\pi\)
−0.809377 + 0.587290i \(0.800195\pi\)
\(798\) 0 0
\(799\) −1.24519 −0.0440516
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 34.8333i 1.22924i
\(804\) 0 0
\(805\) 15.1761i 0.534888i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 44.9173 1.57921 0.789604 0.613616i \(-0.210286\pi\)
0.789604 + 0.613616i \(0.210286\pi\)
\(810\) 0 0
\(811\) − 37.5222i − 1.31758i −0.752325 0.658792i \(-0.771068\pi\)
0.752325 0.658792i \(-0.228932\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −1.20374 −0.0421650
\(816\) 0 0
\(817\) 11.9747 0.418943
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0.214362i 0.00748129i 0.999993 + 0.00374065i \(0.00119069\pi\)
−0.999993 + 0.00374065i \(0.998809\pi\)
\(822\) 0 0
\(823\) 4.84153 0.168765 0.0843825 0.996433i \(-0.473108\pi\)
0.0843825 + 0.996433i \(0.473108\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 9.13088i 0.317512i 0.987318 + 0.158756i \(0.0507483\pi\)
−0.987318 + 0.158756i \(0.949252\pi\)
\(828\) 0 0
\(829\) − 22.8115i − 0.792275i −0.918191 0.396138i \(-0.870350\pi\)
0.918191 0.396138i \(-0.129650\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 3.69300 0.127955
\(834\) 0 0
\(835\) 44.9447i 1.55538i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −37.9537 −1.31031 −0.655154 0.755496i \(-0.727396\pi\)
−0.655154 + 0.755496i \(0.727396\pi\)
\(840\) 0 0
\(841\) 26.4526 0.912157
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 36.7781i 1.26521i
\(846\) 0 0
\(847\) −5.36404 −0.184311
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 22.4876i − 0.770864i
\(852\) 0 0
\(853\) 35.9465i 1.23078i 0.788221 + 0.615392i \(0.211002\pi\)
−0.788221 + 0.615392i \(0.788998\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −36.9904 −1.26357 −0.631783 0.775145i \(-0.717677\pi\)
−0.631783 + 0.775145i \(0.717677\pi\)
\(858\) 0 0
\(859\) − 0.924367i − 0.0315390i −0.999876 0.0157695i \(-0.994980\pi\)
0.999876 0.0157695i \(-0.00501979\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −11.2057 −0.381445 −0.190723 0.981644i \(-0.561083\pi\)
−0.190723 + 0.981644i \(0.561083\pi\)
\(864\) 0 0
\(865\) 43.4369 1.47690
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 12.3003i 0.417259i
\(870\) 0 0
\(871\) 8.40098 0.284656
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 0.945386i − 0.0319599i
\(876\) 0 0
\(877\) − 38.6354i − 1.30463i −0.757950 0.652313i \(-0.773799\pi\)
0.757950 0.652313i \(-0.226201\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 47.0860 1.58637 0.793184 0.608982i \(-0.208422\pi\)
0.793184 + 0.608982i \(0.208422\pi\)
\(882\) 0 0
\(883\) − 2.23864i − 0.0753362i −0.999290 0.0376681i \(-0.988007\pi\)
0.999290 0.0376681i \(-0.0119930\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −56.5911 −1.90014 −0.950071 0.312032i \(-0.898990\pi\)
−0.950071 + 0.312032i \(0.898990\pi\)
\(888\) 0 0
\(889\) −2.43139 −0.0815463
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 0.365036i − 0.0122155i
\(894\) 0 0
\(895\) 33.1847 1.10924
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 11.8989i − 0.396852i
\(900\) 0 0
\(901\) − 30.0858i − 1.00230i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −20.6721 −0.687165
\(906\) 0 0
\(907\) 37.0996i 1.23187i 0.787797 + 0.615935i \(0.211222\pi\)
−0.787797 + 0.615935i \(0.788778\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −45.3701 −1.50318 −0.751589 0.659632i \(-0.770712\pi\)
−0.751589 + 0.659632i \(0.770712\pi\)
\(912\) 0 0
\(913\) 0.328377 0.0108677
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 8.83723i 0.291831i
\(918\) 0 0
\(919\) −28.8038 −0.950148 −0.475074 0.879946i \(-0.657579\pi\)
−0.475074 + 0.879946i \(0.657579\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 14.7724i − 0.486240i
\(924\) 0 0
\(925\) − 21.6697i − 0.712495i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 45.5518 1.49451 0.747254 0.664539i \(-0.231372\pi\)
0.747254 + 0.664539i \(0.231372\pi\)
\(930\) 0 0
\(931\) 1.08263i 0.0354818i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 27.3003 0.892817
\(936\) 0 0
\(937\) −43.3973 −1.41773 −0.708864 0.705346i \(-0.750792\pi\)
−0.708864 + 0.705346i \(0.750792\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 17.5629i − 0.572534i −0.958150 0.286267i \(-0.907586\pi\)
0.958150 0.286267i \(-0.0924145\pi\)
\(942\) 0 0
\(943\) 0.185499 0.00604068
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 33.8968i 1.10150i 0.834671 + 0.550749i \(0.185658\pi\)
−0.834671 + 0.550749i \(0.814342\pi\)
\(948\) 0 0
\(949\) 15.9997i 0.519373i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 42.7021 1.38326 0.691628 0.722254i \(-0.256894\pi\)
0.691628 + 0.722254i \(0.256894\pi\)
\(954\) 0 0
\(955\) − 30.5520i − 0.988639i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −11.1478 −0.359982
\(960\) 0 0
\(961\) 24.5794 0.792883
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 54.3498i 1.74958i
\(966\) 0 0
\(967\) −23.9796 −0.771132 −0.385566 0.922680i \(-0.625994\pi\)
−0.385566 + 0.922680i \(0.625994\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 13.2383i 0.424837i 0.977179 + 0.212418i \(0.0681340\pi\)
−0.977179 + 0.212418i \(0.931866\pi\)
\(972\) 0 0
\(973\) 17.2724i 0.553726i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −49.7224 −1.59076 −0.795380 0.606110i \(-0.792729\pi\)
−0.795380 + 0.606110i \(0.792729\pi\)
\(978\) 0 0
\(979\) 28.3769i 0.906929i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 33.8135 1.07848 0.539241 0.842151i \(-0.318711\pi\)
0.539241 + 0.842151i \(0.318711\pi\)
\(984\) 0 0
\(985\) 86.2150 2.74704
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 53.9065i 1.71413i
\(990\) 0 0
\(991\) −36.2781 −1.15241 −0.576206 0.817304i \(-0.695468\pi\)
−0.576206 + 0.817304i \(0.695468\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 28.9532i − 0.917879i
\(996\) 0 0
\(997\) − 22.7054i − 0.719088i −0.933128 0.359544i \(-0.882932\pi\)
0.933128 0.359544i \(-0.117068\pi\)
\(998\) 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.c.g.3025.21 24
3.2 odd 2 inner 6048.2.c.g.3025.3 24
4.3 odd 2 1512.2.c.f.757.2 yes 24
8.3 odd 2 1512.2.c.f.757.1 24
8.5 even 2 inner 6048.2.c.g.3025.4 24
12.11 even 2 1512.2.c.f.757.23 yes 24
24.5 odd 2 inner 6048.2.c.g.3025.22 24
24.11 even 2 1512.2.c.f.757.24 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.c.f.757.1 24 8.3 odd 2
1512.2.c.f.757.2 yes 24 4.3 odd 2
1512.2.c.f.757.23 yes 24 12.11 even 2
1512.2.c.f.757.24 yes 24 24.11 even 2
6048.2.c.g.3025.3 24 3.2 odd 2 inner
6048.2.c.g.3025.4 24 8.5 even 2 inner
6048.2.c.g.3025.21 24 1.1 even 1 trivial
6048.2.c.g.3025.22 24 24.5 odd 2 inner