Properties

Label 6048.2.j.c.5615.21
Level 6048
Weight 2
Character 6048.5615
Analytic conductor 48.294
Analytic rank 0
Dimension 32
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.j (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.2935231425\)
Analytic rank: \(0\)
Dimension: \(32\)
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 5615.21
Character \(\chi\) = 6048.5615
Dual form 6048.2.j.c.5615.22

$q$-expansion

\(f(q)\) \(=\) \(q+0.436221 q^{5} -1.00000i q^{7} +O(q^{10})\) \(q+0.436221 q^{5} -1.00000i q^{7} +2.73863i q^{11} -1.64657i q^{13} +5.22231i q^{17} +5.99963 q^{19} +7.68296 q^{23} -4.80971 q^{25} -4.94791 q^{29} -9.77950i q^{31} -0.436221i q^{35} +3.81676i q^{37} -9.74668i q^{41} +8.84195 q^{43} -4.54668 q^{47} -1.00000 q^{49} +9.94845 q^{53} +1.19465i q^{55} +13.2892i q^{59} +6.26145i q^{61} -0.718269i q^{65} -9.42171 q^{67} +9.76751 q^{71} -9.14079 q^{73} +2.73863 q^{77} +5.19984i q^{79} -0.227071i q^{83} +2.27808i q^{85} -2.46567i q^{89} -1.64657 q^{91} +2.61717 q^{95} -3.37287 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32q + O(q^{10}) \) \( 32q + 64q^{19} - 16q^{25} + 48q^{43} - 32q^{49} - 16q^{67} - 16q^{73} - 16q^{91} + 64q^{97} + O(q^{100}) \)

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
\(5\) 0.436221 0.195084 0.0975421 0.995231i \(-0.468902\pi\)
0.0975421 + 0.995231i \(0.468902\pi\)
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.73863i 0.825727i 0.910793 + 0.412863i \(0.135471\pi\)
−0.910793 + 0.412863i \(0.864529\pi\)
\(12\) 0 0
\(13\) − 1.64657i − 0.456676i −0.973582 0.228338i \(-0.926671\pi\)
0.973582 0.228338i \(-0.0733291\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.22231i 1.26660i 0.773908 + 0.633298i \(0.218299\pi\)
−0.773908 + 0.633298i \(0.781701\pi\)
\(18\) 0 0
\(19\) 5.99963 1.37641 0.688204 0.725517i \(-0.258399\pi\)
0.688204 + 0.725517i \(0.258399\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 7.68296 1.60201 0.801004 0.598659i \(-0.204300\pi\)
0.801004 + 0.598659i \(0.204300\pi\)
\(24\) 0 0
\(25\) −4.80971 −0.961942
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −4.94791 −0.918804 −0.459402 0.888228i \(-0.651936\pi\)
−0.459402 + 0.888228i \(0.651936\pi\)
\(30\) 0 0
\(31\) − 9.77950i − 1.75645i −0.478248 0.878225i \(-0.658728\pi\)
0.478248 0.878225i \(-0.341272\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 0.436221i − 0.0737349i
\(36\) 0 0
\(37\) 3.81676i 0.627472i 0.949510 + 0.313736i \(0.101581\pi\)
−0.949510 + 0.313736i \(0.898419\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 9.74668i − 1.52218i −0.648649 0.761088i \(-0.724666\pi\)
0.648649 0.761088i \(-0.275334\pi\)
\(42\) 0 0
\(43\) 8.84195 1.34838 0.674192 0.738556i \(-0.264492\pi\)
0.674192 + 0.738556i \(0.264492\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.54668 −0.663202 −0.331601 0.943420i \(-0.607589\pi\)
−0.331601 + 0.943420i \(0.607589\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 9.94845 1.36652 0.683262 0.730173i \(-0.260561\pi\)
0.683262 + 0.730173i \(0.260561\pi\)
\(54\) 0 0
\(55\) 1.19465i 0.161086i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 13.2892i 1.73011i 0.501681 + 0.865053i \(0.332715\pi\)
−0.501681 + 0.865053i \(0.667285\pi\)
\(60\) 0 0
\(61\) 6.26145i 0.801696i 0.916145 + 0.400848i \(0.131284\pi\)
−0.916145 + 0.400848i \(0.868716\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 0.718269i − 0.0890903i
\(66\) 0 0
\(67\) −9.42171 −1.15104 −0.575522 0.817786i \(-0.695201\pi\)
−0.575522 + 0.817786i \(0.695201\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 9.76751 1.15919 0.579595 0.814905i \(-0.303211\pi\)
0.579595 + 0.814905i \(0.303211\pi\)
\(72\) 0 0
\(73\) −9.14079 −1.06985 −0.534924 0.844900i \(-0.679660\pi\)
−0.534924 + 0.844900i \(0.679660\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.73863 0.312095
\(78\) 0 0
\(79\) 5.19984i 0.585028i 0.956261 + 0.292514i \(0.0944917\pi\)
−0.956261 + 0.292514i \(0.905508\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 0.227071i − 0.0249243i −0.999922 0.0124622i \(-0.996033\pi\)
0.999922 0.0124622i \(-0.00396693\pi\)
\(84\) 0 0
\(85\) 2.27808i 0.247093i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 2.46567i − 0.261360i −0.991425 0.130680i \(-0.958284\pi\)
0.991425 0.130680i \(-0.0417161\pi\)
\(90\) 0 0
\(91\) −1.64657 −0.172607
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.61717 0.268516
\(96\) 0 0
\(97\) −3.37287 −0.342463 −0.171232 0.985231i \(-0.554775\pi\)
−0.171232 + 0.985231i \(0.554775\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1.20021 −0.119425 −0.0597125 0.998216i \(-0.519018\pi\)
−0.0597125 + 0.998216i \(0.519018\pi\)
\(102\) 0 0
\(103\) 10.7004i 1.05434i 0.849761 + 0.527169i \(0.176746\pi\)
−0.849761 + 0.527169i \(0.823254\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.49655i 0.144676i 0.997380 + 0.0723382i \(0.0230461\pi\)
−0.997380 + 0.0723382i \(0.976954\pi\)
\(108\) 0 0
\(109\) 18.4352i 1.76578i 0.469583 + 0.882888i \(0.344404\pi\)
−0.469583 + 0.882888i \(0.655596\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 5.25692i 0.494529i 0.968948 + 0.247265i \(0.0795317\pi\)
−0.968948 + 0.247265i \(0.920468\pi\)
\(114\) 0 0
\(115\) 3.35147 0.312526
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 5.22231 0.478729
\(120\) 0 0
\(121\) 3.49993 0.318175
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −4.27921 −0.382744
\(126\) 0 0
\(127\) − 19.7677i − 1.75410i −0.480396 0.877051i \(-0.659507\pi\)
0.480396 0.877051i \(-0.340493\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 9.25252i 0.808397i 0.914671 + 0.404198i \(0.132449\pi\)
−0.914671 + 0.404198i \(0.867551\pi\)
\(132\) 0 0
\(133\) − 5.99963i − 0.520234i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 2.53255i − 0.216370i −0.994131 0.108185i \(-0.965496\pi\)
0.994131 0.108185i \(-0.0345039\pi\)
\(138\) 0 0
\(139\) 19.7831 1.67798 0.838991 0.544146i \(-0.183146\pi\)
0.838991 + 0.544146i \(0.183146\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 4.50934 0.377090
\(144\) 0 0
\(145\) −2.15839 −0.179244
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 8.16461 0.668871 0.334435 0.942419i \(-0.391454\pi\)
0.334435 + 0.942419i \(0.391454\pi\)
\(150\) 0 0
\(151\) − 14.8741i − 1.21044i −0.796058 0.605220i \(-0.793085\pi\)
0.796058 0.605220i \(-0.206915\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 4.26603i − 0.342656i
\(156\) 0 0
\(157\) 4.92207i 0.392824i 0.980521 + 0.196412i \(0.0629290\pi\)
−0.980521 + 0.196412i \(0.937071\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 7.68296i − 0.605502i
\(162\) 0 0
\(163\) 5.87250 0.459970 0.229985 0.973194i \(-0.426132\pi\)
0.229985 + 0.973194i \(0.426132\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 22.1181 1.71155 0.855777 0.517345i \(-0.173080\pi\)
0.855777 + 0.517345i \(0.173080\pi\)
\(168\) 0 0
\(169\) 10.2888 0.791447
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 24.1920 1.83928 0.919642 0.392758i \(-0.128479\pi\)
0.919642 + 0.392758i \(0.128479\pi\)
\(174\) 0 0
\(175\) 4.80971i 0.363580i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 5.41287i − 0.404577i −0.979326 0.202288i \(-0.935162\pi\)
0.979326 0.202288i \(-0.0648378\pi\)
\(180\) 0 0
\(181\) − 6.19672i − 0.460598i −0.973120 0.230299i \(-0.926030\pi\)
0.973120 0.230299i \(-0.0739705\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 1.66495i 0.122410i
\(186\) 0 0
\(187\) −14.3020 −1.04586
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 13.6191 0.985444 0.492722 0.870187i \(-0.336002\pi\)
0.492722 + 0.870187i \(0.336002\pi\)
\(192\) 0 0
\(193\) 5.89278 0.424172 0.212086 0.977251i \(-0.431974\pi\)
0.212086 + 0.977251i \(0.431974\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 19.0323 1.35600 0.677998 0.735064i \(-0.262848\pi\)
0.677998 + 0.735064i \(0.262848\pi\)
\(198\) 0 0
\(199\) 5.91398i 0.419231i 0.977784 + 0.209616i \(0.0672212\pi\)
−0.977784 + 0.209616i \(0.932779\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 4.94791i 0.347275i
\(204\) 0 0
\(205\) − 4.25171i − 0.296952i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 16.4307i 1.13654i
\(210\) 0 0
\(211\) −12.9334 −0.890369 −0.445185 0.895439i \(-0.646862\pi\)
−0.445185 + 0.895439i \(0.646862\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 3.85705 0.263049
\(216\) 0 0
\(217\) −9.77950 −0.663876
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 8.59890 0.578424
\(222\) 0 0
\(223\) 0.758557i 0.0507967i 0.999677 + 0.0253984i \(0.00808542\pi\)
−0.999677 + 0.0253984i \(0.991915\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 3.60986i 0.239595i 0.992798 + 0.119797i \(0.0382245\pi\)
−0.992798 + 0.119797i \(0.961776\pi\)
\(228\) 0 0
\(229\) − 3.47625i − 0.229717i −0.993382 0.114859i \(-0.963358\pi\)
0.993382 0.114859i \(-0.0366415\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 22.9786i 1.50538i 0.658376 + 0.752689i \(0.271244\pi\)
−0.658376 + 0.752689i \(0.728756\pi\)
\(234\) 0 0
\(235\) −1.98336 −0.129380
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 10.4478 0.675813 0.337906 0.941180i \(-0.390281\pi\)
0.337906 + 0.941180i \(0.390281\pi\)
\(240\) 0 0
\(241\) −10.0939 −0.650208 −0.325104 0.945678i \(-0.605399\pi\)
−0.325104 + 0.945678i \(0.605399\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −0.436221 −0.0278692
\(246\) 0 0
\(247\) − 9.87880i − 0.628573i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 18.0568i − 1.13974i −0.821736 0.569868i \(-0.806994\pi\)
0.821736 0.569868i \(-0.193006\pi\)
\(252\) 0 0
\(253\) 21.0408i 1.32282i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 3.77643i 0.235567i 0.993039 + 0.117784i \(0.0375789\pi\)
−0.993039 + 0.117784i \(0.962421\pi\)
\(258\) 0 0
\(259\) 3.81676 0.237162
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −19.1460 −1.18060 −0.590298 0.807185i \(-0.700990\pi\)
−0.590298 + 0.807185i \(0.700990\pi\)
\(264\) 0 0
\(265\) 4.33973 0.266587
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 11.9031 0.725747 0.362873 0.931838i \(-0.381796\pi\)
0.362873 + 0.931838i \(0.381796\pi\)
\(270\) 0 0
\(271\) − 26.4765i − 1.60833i −0.594403 0.804167i \(-0.702612\pi\)
0.594403 0.804167i \(-0.297388\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 13.1720i − 0.794302i
\(276\) 0 0
\(277\) − 9.49587i − 0.570552i −0.958445 0.285276i \(-0.907915\pi\)
0.958445 0.285276i \(-0.0920852\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 16.7754i 1.00073i 0.865813 + 0.500367i \(0.166802\pi\)
−0.865813 + 0.500367i \(0.833198\pi\)
\(282\) 0 0
\(283\) −13.6841 −0.813437 −0.406719 0.913554i \(-0.633327\pi\)
−0.406719 + 0.913554i \(0.633327\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −9.74668 −0.575328
\(288\) 0 0
\(289\) −10.2725 −0.604268
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 9.46088 0.552710 0.276355 0.961056i \(-0.410873\pi\)
0.276355 + 0.961056i \(0.410873\pi\)
\(294\) 0 0
\(295\) 5.79703i 0.337516i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 12.6505i − 0.731599i
\(300\) 0 0
\(301\) − 8.84195i − 0.509641i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 2.73138i 0.156398i
\(306\) 0 0
\(307\) 12.4055 0.708019 0.354009 0.935242i \(-0.384818\pi\)
0.354009 + 0.935242i \(0.384818\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −1.70862 −0.0968870 −0.0484435 0.998826i \(-0.515426\pi\)
−0.0484435 + 0.998826i \(0.515426\pi\)
\(312\) 0 0
\(313\) 11.9889 0.677652 0.338826 0.940849i \(-0.389970\pi\)
0.338826 + 0.940849i \(0.389970\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −24.6470 −1.38431 −0.692155 0.721748i \(-0.743339\pi\)
−0.692155 + 0.721748i \(0.743339\pi\)
\(318\) 0 0
\(319\) − 13.5505i − 0.758681i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 31.3319i 1.74336i
\(324\) 0 0
\(325\) 7.91952i 0.439296i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 4.54668i 0.250667i
\(330\) 0 0
\(331\) 5.66221 0.311223 0.155612 0.987818i \(-0.450265\pi\)
0.155612 + 0.987818i \(0.450265\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −4.10995 −0.224551
\(336\) 0 0
\(337\) −32.2480 −1.75666 −0.878331 0.478053i \(-0.841343\pi\)
−0.878331 + 0.478053i \(0.841343\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 26.7824 1.45035
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 20.3357i 1.09168i 0.837889 + 0.545840i \(0.183789\pi\)
−0.837889 + 0.545840i \(0.816211\pi\)
\(348\) 0 0
\(349\) − 8.27539i − 0.442971i −0.975164 0.221486i \(-0.928909\pi\)
0.975164 0.221486i \(-0.0710906\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 8.23298i − 0.438197i −0.975703 0.219099i \(-0.929688\pi\)
0.975703 0.219099i \(-0.0703117\pi\)
\(354\) 0 0
\(355\) 4.26080 0.226140
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 10.5802 0.558403 0.279202 0.960232i \(-0.409930\pi\)
0.279202 + 0.960232i \(0.409930\pi\)
\(360\) 0 0
\(361\) 16.9955 0.894502
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −3.98741 −0.208710
\(366\) 0 0
\(367\) 23.7595i 1.24023i 0.784509 + 0.620117i \(0.212915\pi\)
−0.784509 + 0.620117i \(0.787085\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 9.94845i − 0.516498i
\(372\) 0 0
\(373\) 12.1442i 0.628802i 0.949290 + 0.314401i \(0.101804\pi\)
−0.949290 + 0.314401i \(0.898196\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 8.14708i 0.419596i
\(378\) 0 0
\(379\) 32.1386 1.65085 0.825426 0.564511i \(-0.190935\pi\)
0.825426 + 0.564511i \(0.190935\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 30.0530 1.53564 0.767819 0.640667i \(-0.221342\pi\)
0.767819 + 0.640667i \(0.221342\pi\)
\(384\) 0 0
\(385\) 1.19465 0.0608849
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −12.8723 −0.652652 −0.326326 0.945257i \(-0.605811\pi\)
−0.326326 + 0.945257i \(0.605811\pi\)
\(390\) 0 0
\(391\) 40.1228i 2.02910i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 2.26828i 0.114130i
\(396\) 0 0
\(397\) − 20.8323i − 1.04554i −0.852473 0.522771i \(-0.824898\pi\)
0.852473 0.522771i \(-0.175102\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 15.8270i − 0.790361i −0.918604 0.395180i \(-0.870682\pi\)
0.918604 0.395180i \(-0.129318\pi\)
\(402\) 0 0
\(403\) −16.1026 −0.802129
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −10.4527 −0.518120
\(408\) 0 0
\(409\) −3.95136 −0.195382 −0.0976910 0.995217i \(-0.531146\pi\)
−0.0976910 + 0.995217i \(0.531146\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 13.2892 0.653919
\(414\) 0 0
\(415\) − 0.0990534i − 0.00486234i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 3.56977i − 0.174395i −0.996191 0.0871974i \(-0.972209\pi\)
0.996191 0.0871974i \(-0.0277911\pi\)
\(420\) 0 0
\(421\) 16.3767i 0.798152i 0.916918 + 0.399076i \(0.130669\pi\)
−0.916918 + 0.399076i \(0.869331\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 25.1178i − 1.21839i
\(426\) 0 0
\(427\) 6.26145 0.303013
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 28.7503 1.38485 0.692426 0.721489i \(-0.256542\pi\)
0.692426 + 0.721489i \(0.256542\pi\)
\(432\) 0 0
\(433\) −23.6276 −1.13547 −0.567735 0.823212i \(-0.692180\pi\)
−0.567735 + 0.823212i \(0.692180\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 46.0949 2.20502
\(438\) 0 0
\(439\) 15.4891i 0.739255i 0.929180 + 0.369628i \(0.120515\pi\)
−0.929180 + 0.369628i \(0.879485\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 21.7640i − 1.03404i −0.855974 0.517019i \(-0.827042\pi\)
0.855974 0.517019i \(-0.172958\pi\)
\(444\) 0 0
\(445\) − 1.07558i − 0.0509872i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 28.9358i 1.36557i 0.730621 + 0.682783i \(0.239231\pi\)
−0.730621 + 0.682783i \(0.760769\pi\)
\(450\) 0 0
\(451\) 26.6925 1.25690
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −0.718269 −0.0336730
\(456\) 0 0
\(457\) 26.2466 1.22776 0.613882 0.789397i \(-0.289607\pi\)
0.613882 + 0.789397i \(0.289607\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 7.47471 0.348132 0.174066 0.984734i \(-0.444309\pi\)
0.174066 + 0.984734i \(0.444309\pi\)
\(462\) 0 0
\(463\) 8.09410i 0.376165i 0.982153 + 0.188082i \(0.0602272\pi\)
−0.982153 + 0.188082i \(0.939773\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 16.0617i − 0.743246i −0.928384 0.371623i \(-0.878801\pi\)
0.928384 0.371623i \(-0.121199\pi\)
\(468\) 0 0
\(469\) 9.42171i 0.435054i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 24.2148i 1.11340i
\(474\) 0 0
\(475\) −28.8565 −1.32403
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −31.6989 −1.44836 −0.724181 0.689610i \(-0.757782\pi\)
−0.724181 + 0.689610i \(0.757782\pi\)
\(480\) 0 0
\(481\) 6.28456 0.286551
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −1.47132 −0.0668092
\(486\) 0 0
\(487\) − 30.5363i − 1.38373i −0.722025 0.691867i \(-0.756789\pi\)
0.722025 0.691867i \(-0.243211\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 41.2201i 1.86024i 0.367259 + 0.930119i \(0.380296\pi\)
−0.367259 + 0.930119i \(0.619704\pi\)
\(492\) 0 0
\(493\) − 25.8395i − 1.16375i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 9.76751i − 0.438133i
\(498\) 0 0
\(499\) −4.03676 −0.180710 −0.0903552 0.995910i \(-0.528800\pi\)
−0.0903552 + 0.995910i \(0.528800\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 40.2393 1.79418 0.897092 0.441845i \(-0.145676\pi\)
0.897092 + 0.441845i \(0.145676\pi\)
\(504\) 0 0
\(505\) −0.523556 −0.0232979
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 9.68050 0.429081 0.214540 0.976715i \(-0.431175\pi\)
0.214540 + 0.976715i \(0.431175\pi\)
\(510\) 0 0
\(511\) 9.14079i 0.404365i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 4.66773i 0.205685i
\(516\) 0 0
\(517\) − 12.4517i − 0.547624i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 3.92629i 0.172014i 0.996295 + 0.0860069i \(0.0274107\pi\)
−0.996295 + 0.0860069i \(0.972589\pi\)
\(522\) 0 0
\(523\) −20.8709 −0.912623 −0.456311 0.889820i \(-0.650830\pi\)
−0.456311 + 0.889820i \(0.650830\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 51.0716 2.22471
\(528\) 0 0
\(529\) 36.0279 1.56643
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −16.0486 −0.695141
\(534\) 0 0
\(535\) 0.652825i 0.0282241i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 2.73863i − 0.117961i
\(540\) 0 0
\(541\) − 3.18975i − 0.137138i −0.997646 0.0685691i \(-0.978157\pi\)
0.997646 0.0685691i \(-0.0218434\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 8.04185i 0.344475i
\(546\) 0 0
\(547\) −27.9137 −1.19350 −0.596751 0.802426i \(-0.703542\pi\)
−0.596751 + 0.802426i \(0.703542\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −29.6856 −1.26465
\(552\) 0 0
\(553\) 5.19984 0.221120
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 14.0025 0.593304 0.296652 0.954986i \(-0.404130\pi\)
0.296652 + 0.954986i \(0.404130\pi\)
\(558\) 0 0
\(559\) − 14.5589i − 0.615775i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 13.8839i − 0.585137i −0.956245 0.292569i \(-0.905490\pi\)
0.956245 0.292569i \(-0.0945100\pi\)
\(564\) 0 0
\(565\) 2.29318i 0.0964749i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 14.7512i 0.618401i 0.950997 + 0.309200i \(0.100061\pi\)
−0.950997 + 0.309200i \(0.899939\pi\)
\(570\) 0 0
\(571\) −4.28135 −0.179169 −0.0895845 0.995979i \(-0.528554\pi\)
−0.0895845 + 0.995979i \(0.528554\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −36.9528 −1.54104
\(576\) 0 0
\(577\) 11.1951 0.466059 0.233030 0.972470i \(-0.425136\pi\)
0.233030 + 0.972470i \(0.425136\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −0.227071 −0.00942051
\(582\) 0 0
\(583\) 27.2451i 1.12838i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 1.45377i − 0.0600035i −0.999550 0.0300018i \(-0.990449\pi\)
0.999550 0.0300018i \(-0.00955129\pi\)
\(588\) 0 0
\(589\) − 58.6733i − 2.41759i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 13.1827i − 0.541347i −0.962671 0.270674i \(-0.912754\pi\)
0.962671 0.270674i \(-0.0872464\pi\)
\(594\) 0 0
\(595\) 2.27808 0.0933924
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −14.4852 −0.591848 −0.295924 0.955211i \(-0.595628\pi\)
−0.295924 + 0.955211i \(0.595628\pi\)
\(600\) 0 0
\(601\) −27.7364 −1.13139 −0.565695 0.824614i \(-0.691392\pi\)
−0.565695 + 0.824614i \(0.691392\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1.52674 0.0620709
\(606\) 0 0
\(607\) − 18.5137i − 0.751447i −0.926732 0.375723i \(-0.877394\pi\)
0.926732 0.375723i \(-0.122606\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 7.48643i 0.302868i
\(612\) 0 0
\(613\) 25.3069i 1.02213i 0.859541 + 0.511067i \(0.170750\pi\)
−0.859541 + 0.511067i \(0.829250\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 47.8249i 1.92536i 0.270647 + 0.962679i \(0.412762\pi\)
−0.270647 + 0.962679i \(0.587238\pi\)
\(618\) 0 0
\(619\) −3.22411 −0.129588 −0.0647939 0.997899i \(-0.520639\pi\)
−0.0647939 + 0.997899i \(0.520639\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −2.46567 −0.0987848
\(624\) 0 0
\(625\) 22.1819 0.887275
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −19.9323 −0.794754
\(630\) 0 0
\(631\) 22.6025i 0.899791i 0.893081 + 0.449896i \(0.148539\pi\)
−0.893081 + 0.449896i \(0.851461\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 8.62311i − 0.342198i
\(636\) 0 0
\(637\) 1.64657i 0.0652394i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 15.4383i − 0.609775i −0.952388 0.304888i \(-0.901381\pi\)
0.952388 0.304888i \(-0.0986189\pi\)
\(642\) 0 0
\(643\) −1.41036 −0.0556191 −0.0278095 0.999613i \(-0.508853\pi\)
−0.0278095 + 0.999613i \(0.508853\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −0.811574 −0.0319063 −0.0159531 0.999873i \(-0.505078\pi\)
−0.0159531 + 0.999873i \(0.505078\pi\)
\(648\) 0 0
\(649\) −36.3941 −1.42860
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −9.82168 −0.384352 −0.192176 0.981360i \(-0.561554\pi\)
−0.192176 + 0.981360i \(0.561554\pi\)
\(654\) 0 0
\(655\) 4.03615i 0.157705i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 11.1456i 0.434170i 0.976153 + 0.217085i \(0.0696548\pi\)
−0.976153 + 0.217085i \(0.930345\pi\)
\(660\) 0 0
\(661\) 15.3989i 0.598947i 0.954105 + 0.299474i \(0.0968111\pi\)
−0.954105 + 0.299474i \(0.903189\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 2.61717i − 0.101489i
\(666\) 0 0
\(667\) −38.0146 −1.47193
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −17.1478 −0.661982
\(672\) 0 0
\(673\) 20.4112 0.786795 0.393398 0.919368i \(-0.371300\pi\)
0.393398 + 0.919368i \(0.371300\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −50.5124 −1.94135 −0.970674 0.240401i \(-0.922721\pi\)
−0.970674 + 0.240401i \(0.922721\pi\)
\(678\) 0 0
\(679\) 3.37287i 0.129439i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 33.3625i 1.27658i 0.769796 + 0.638291i \(0.220358\pi\)
−0.769796 + 0.638291i \(0.779642\pi\)
\(684\) 0 0
\(685\) − 1.10475i − 0.0422104i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 16.3808i − 0.624059i
\(690\) 0 0
\(691\) −24.3714 −0.927133 −0.463567 0.886062i \(-0.653431\pi\)
−0.463567 + 0.886062i \(0.653431\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 8.62982 0.327348
\(696\) 0 0
\(697\) 50.9002 1.92798
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −34.1271 −1.28896 −0.644481 0.764620i \(-0.722927\pi\)
−0.644481 + 0.764620i \(0.722927\pi\)
\(702\) 0 0
\(703\) 22.8991i 0.863658i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1.20021i 0.0451384i
\(708\) 0 0
\(709\) − 22.8145i − 0.856818i −0.903585 0.428409i \(-0.859074\pi\)
0.903585 0.428409i \(-0.140926\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 75.1355i − 2.81385i
\(714\) 0 0
\(715\) 1.96707 0.0735642
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 2.18175 0.0813654 0.0406827 0.999172i \(-0.487047\pi\)
0.0406827 + 0.999172i \(0.487047\pi\)
\(720\) 0 0
\(721\) 10.7004 0.398502
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 23.7980 0.883837
\(726\) 0 0
\(727\) 15.3017i 0.567507i 0.958897 + 0.283754i \(0.0915798\pi\)
−0.958897 + 0.283754i \(0.908420\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 46.1754i 1.70786i
\(732\) 0 0
\(733\) 15.8711i 0.586213i 0.956080 + 0.293106i \(0.0946890\pi\)
−0.956080 + 0.293106i \(0.905311\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 25.8025i − 0.950449i
\(738\) 0 0
\(739\) −31.1319 −1.14521 −0.572603 0.819833i \(-0.694066\pi\)
−0.572603 + 0.819833i \(0.694066\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 14.6629 0.537930 0.268965 0.963150i \(-0.413318\pi\)
0.268965 + 0.963150i \(0.413318\pi\)
\(744\) 0 0
\(745\) 3.56158 0.130486
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 1.49655 0.0546826
\(750\) 0 0
\(751\) − 8.46100i − 0.308746i −0.988013 0.154373i \(-0.950664\pi\)
0.988013 0.154373i \(-0.0493358\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 6.48842i − 0.236138i
\(756\) 0 0
\(757\) − 39.6473i − 1.44101i −0.693451 0.720504i \(-0.743911\pi\)
0.693451 0.720504i \(-0.256089\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 49.7301i − 1.80271i −0.433076 0.901357i \(-0.642572\pi\)
0.433076 0.901357i \(-0.357428\pi\)
\(762\) 0 0
\(763\) 18.4352 0.667401
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 21.8816 0.790098
\(768\) 0 0
\(769\) 5.62877 0.202979 0.101489 0.994837i \(-0.467639\pi\)
0.101489 + 0.994837i \(0.467639\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −25.3999 −0.913570 −0.456785 0.889577i \(-0.650999\pi\)
−0.456785 + 0.889577i \(0.650999\pi\)
\(774\) 0 0
\(775\) 47.0366i 1.68960i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 58.4764i − 2.09514i
\(780\) 0 0
\(781\) 26.7495i 0.957174i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 2.14711i 0.0766338i
\(786\) 0 0
\(787\) 31.5311 1.12396 0.561982 0.827150i \(-0.310039\pi\)
0.561982 + 0.827150i \(0.310039\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 5.25692 0.186915
\(792\) 0 0
\(793\) 10.3099 0.366115
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 37.3724 1.32380 0.661899 0.749593i \(-0.269751\pi\)
0.661899 + 0.749593i \(0.269751\pi\)
\(798\) 0 0
\(799\) − 23.7442i − 0.840010i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 25.0332i − 0.883402i
\(804\) 0 0
\(805\) − 3.35147i − 0.118124i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 4.37230i 0.153722i 0.997042 + 0.0768610i \(0.0244898\pi\)
−0.997042 + 0.0768610i \(0.975510\pi\)
\(810\) 0 0
\(811\) 29.1067 1.02208 0.511038 0.859558i \(-0.329261\pi\)
0.511038 + 0.859558i \(0.329261\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 2.56171 0.0897328
\(816\) 0 0
\(817\) 53.0484 1.85593
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −45.1785 −1.57674 −0.788370 0.615201i \(-0.789075\pi\)
−0.788370 + 0.615201i \(0.789075\pi\)
\(822\) 0 0
\(823\) 30.6242i 1.06749i 0.845644 + 0.533747i \(0.179217\pi\)
−0.845644 + 0.533747i \(0.820783\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 12.0993i − 0.420734i −0.977623 0.210367i \(-0.932534\pi\)
0.977623 0.210367i \(-0.0674658\pi\)
\(828\) 0 0
\(829\) − 15.2105i − 0.528283i −0.964484 0.264142i \(-0.914911\pi\)
0.964484 0.264142i \(-0.0850886\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 5.22231i − 0.180942i
\(834\) 0 0
\(835\) 9.64841 0.333897
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −42.2155 −1.45744 −0.728721 0.684811i \(-0.759885\pi\)
−0.728721 + 0.684811i \(0.759885\pi\)
\(840\) 0 0
\(841\) −4.51816 −0.155799
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 4.48820 0.154399
\(846\) 0 0
\(847\) − 3.49993i − 0.120259i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 29.3240i 1.00521i
\(852\) 0 0
\(853\) − 49.2536i − 1.68641i −0.537592 0.843205i \(-0.680666\pi\)
0.537592 0.843205i \(-0.319334\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 25.9243i 0.885557i 0.896631 + 0.442779i \(0.146007\pi\)
−0.896631 + 0.442779i \(0.853993\pi\)
\(858\) 0 0
\(859\) −9.25940 −0.315927 −0.157963 0.987445i \(-0.550493\pi\)
−0.157963 + 0.987445i \(0.550493\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −6.21443 −0.211542 −0.105771 0.994391i \(-0.533731\pi\)
−0.105771 + 0.994391i \(0.533731\pi\)
\(864\) 0 0
\(865\) 10.5531 0.358815
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −14.2404 −0.483073
\(870\) 0 0
\(871\) 15.5135i 0.525655i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 4.27921i 0.144664i
\(876\) 0 0
\(877\) 15.1038i 0.510020i 0.966938 + 0.255010i \(0.0820787\pi\)
−0.966938 + 0.255010i \(0.917921\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 2.59677i − 0.0874874i −0.999043 0.0437437i \(-0.986072\pi\)
0.999043 0.0437437i \(-0.0139285\pi\)
\(882\) 0 0
\(883\) −11.0661 −0.372404 −0.186202 0.982511i \(-0.559618\pi\)
−0.186202 + 0.982511i \(0.559618\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −7.38595 −0.247996 −0.123998 0.992282i \(-0.539572\pi\)
−0.123998 + 0.992282i \(0.539572\pi\)
\(888\) 0 0
\(889\) −19.7677 −0.662989
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −27.2784 −0.912837
\(894\) 0 0
\(895\) − 2.36121i − 0.0789265i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 48.3881i 1.61383i
\(900\) 0 0
\(901\) 51.9539i 1.73084i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 2.70314i − 0.0898555i
\(906\) 0 0
\(907\) −5.58114 −0.185319 −0.0926594 0.995698i \(-0.529537\pi\)
−0.0926594 + 0.995698i \(0.529537\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −7.02450 −0.232732 −0.116366 0.993206i \(-0.537125\pi\)
−0.116366 + 0.993206i \(0.537125\pi\)
\(912\) 0 0
\(913\) 0.621864 0.0205807
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 9.25252 0.305545
\(918\) 0 0
\(919\) − 25.7961i − 0.850935i −0.904974 0.425468i \(-0.860110\pi\)
0.904974 0.425468i \(-0.139890\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 16.0829i − 0.529374i
\(924\) 0 0
\(925\) − 18.3575i − 0.603591i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 10.1558i 0.333201i 0.986024 + 0.166600i \(0.0532790\pi\)
−0.986024 + 0.166600i \(0.946721\pi\)
\(930\) 0 0
\(931\) −5.99963 −0.196630
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −6.23882 −0.204031
\(936\) 0 0
\(937\) 19.1504 0.625616 0.312808 0.949816i \(-0.398730\pi\)
0.312808 + 0.949816i \(0.398730\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −14.3260 −0.467016 −0.233508 0.972355i \(-0.575020\pi\)
−0.233508 + 0.972355i \(0.575020\pi\)
\(942\) 0 0
\(943\) − 74.8833i − 2.43854i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 7.67662i − 0.249456i −0.992191 0.124728i \(-0.960194\pi\)
0.992191 0.124728i \(-0.0398059\pi\)
\(948\) 0 0
\(949\) 15.0509i 0.488574i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 52.6546i − 1.70565i −0.522197 0.852825i \(-0.674887\pi\)
0.522197 0.852825i \(-0.325113\pi\)
\(954\) 0 0
\(955\) 5.94095 0.192245
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −2.53255 −0.0817803
\(960\) 0 0
\(961\) −64.6386 −2.08512
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 2.57056 0.0827492
\(966\) 0 0
\(967\) 50.1578i 1.61297i 0.591257 + 0.806483i \(0.298632\pi\)
−0.591257 + 0.806483i \(0.701368\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 54.8144i 1.75908i 0.475828 + 0.879538i \(0.342148\pi\)
−0.475828 + 0.879538i \(0.657852\pi\)
\(972\) 0 0
\(973\) − 19.7831i − 0.634217i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0.0603583i 0.00193103i 1.00000 0.000965516i \(0.000307333\pi\)
−1.00000 0.000965516i \(0.999693\pi\)
\(978\) 0 0
\(979\) 6.75254 0.215812
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −7.40354 −0.236136 −0.118068 0.993005i \(-0.537670\pi\)
−0.118068 + 0.993005i \(0.537670\pi\)
\(984\) 0 0
\(985\) 8.30230 0.264533
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 67.9323 2.16012
\(990\) 0 0
\(991\) − 35.6861i − 1.13361i −0.823853 0.566803i \(-0.808180\pi\)
0.823853 0.566803i \(-0.191820\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 2.57981i 0.0817853i
\(996\) 0 0
\(997\) − 52.4345i − 1.66062i −0.557304 0.830308i \(-0.688164\pi\)
0.557304 0.830308i \(-0.311836\pi\)
\(998\) 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.j.c.5615.21 32
3.2 odd 2 inner 6048.2.j.c.5615.12 32
4.3 odd 2 1512.2.j.c.323.20 yes 32
8.3 odd 2 inner 6048.2.j.c.5615.11 32
8.5 even 2 1512.2.j.c.323.14 yes 32
12.11 even 2 1512.2.j.c.323.13 32
24.5 odd 2 1512.2.j.c.323.19 yes 32
24.11 even 2 inner 6048.2.j.c.5615.22 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.j.c.323.13 32 12.11 even 2
1512.2.j.c.323.14 yes 32 8.5 even 2
1512.2.j.c.323.19 yes 32 24.5 odd 2
1512.2.j.c.323.20 yes 32 4.3 odd 2
6048.2.j.c.5615.11 32 8.3 odd 2 inner
6048.2.j.c.5615.12 32 3.2 odd 2 inner
6048.2.j.c.5615.21 32 1.1 even 1 trivial
6048.2.j.c.5615.22 32 24.11 even 2 inner