Properties

Label 6048.2.j.c.5615.12
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 5615.12
Character \(\chi\) = 6048.5615
Dual form 6048.2.j.c.5615.11

$q$-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 \(-0.531098\pi\)
−0.0975421 + 0.995231i \(0.531098\pi\)
\(6\) 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 \(-0.864529\pi\)
0.910793 0.412863i \(-0.135471\pi\)
\(12\) 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 \(-0.781701\pi\)
0.773908 0.633298i \(-0.218299\pi\)
\(18\) 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 \(-0.795700\pi\)
−0.801004 + 0.598659i \(0.795700\pi\)
\(24\) 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 \(-0.348064\pi\)
0.459402 + 0.888228i \(0.348064\pi\)
\(30\) 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 \(0.275334\pi\)
−0.648649 + 0.761088i \(0.724666\pi\)
\(42\) 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 \(-0.392411\pi\)
0.331601 + 0.943420i \(0.392411\pi\)
\(48\) 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 \(-0.739439\pi\)
−0.683262 + 0.730173i \(0.739439\pi\)
\(54\) 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 \(-0.667285\pi\)
0.501681 0.865053i \(-0.332715\pi\)
\(60\) 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 \(-0.696789\pi\)
−0.579595 + 0.814905i \(0.696789\pi\)
\(72\) 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 \(0.00396693\pi\)
−0.999922 + 0.0124622i \(0.996033\pi\)
\(84\) 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 \(0.0417161\pi\)
−0.991425 + 0.130680i \(0.958284\pi\)
\(90\) 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 \(-0.480982\pi\)
0.0597125 + 0.998216i \(0.480982\pi\)
\(102\) 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 \(-0.976954\pi\)
0.997380 0.0723382i \(-0.0230461\pi\)
\(108\) 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 \(-0.920468\pi\)
0.968948 0.247265i \(-0.0795317\pi\)
\(114\) 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 \(-0.867551\pi\)
0.914671 0.404198i \(-0.132449\pi\)
\(132\) 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 \(0.0345039\pi\)
−0.994131 + 0.108185i \(0.965496\pi\)
\(138\) 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 \(-0.608546\pi\)
−0.334435 + 0.942419i \(0.608546\pi\)
\(150\) 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 \(-0.826920\pi\)
−0.855777 + 0.517345i \(0.826920\pi\)
\(168\) 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 \(-0.871521\pi\)
−0.919642 + 0.392758i \(0.871521\pi\)
\(174\) 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 \(0.0648378\pi\)
−0.979326 + 0.202288i \(0.935162\pi\)
\(180\) 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 \(-0.663998\pi\)
−0.492722 + 0.870187i \(0.663998\pi\)
\(192\) 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 \(-0.737152\pi\)
−0.677998 + 0.735064i \(0.737152\pi\)
\(198\) 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 \(-0.961776\pi\)
0.992798 0.119797i \(-0.0382245\pi\)
\(228\) 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 \(-0.728756\pi\)
0.658376 0.752689i \(-0.271244\pi\)
\(234\) 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 \(-0.609719\pi\)
−0.337906 + 0.941180i \(0.609719\pi\)
\(240\) 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 \(0.193006\pi\)
−0.821736 + 0.569868i \(0.806994\pi\)
\(252\) 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 \(-0.962421\pi\)
0.993039 0.117784i \(-0.0375789\pi\)
\(258\) 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 \(-0.299010\pi\)
0.590298 + 0.807185i \(0.299010\pi\)
\(264\) 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 \(-0.618204\pi\)
−0.362873 + 0.931838i \(0.618204\pi\)
\(270\) 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 \(-0.833198\pi\)
0.865813 0.500367i \(-0.166802\pi\)
\(282\) 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 \(-0.589127\pi\)
−0.276355 + 0.961056i \(0.589127\pi\)
\(294\) 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 \(-0.484574\pi\)
0.0484435 + 0.998826i \(0.484574\pi\)
\(312\) 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 \(-0.256661\pi\)
0.692155 + 0.721748i \(0.256661\pi\)
\(318\) 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 \(-0.816211\pi\)
0.837889 0.545840i \(-0.183789\pi\)
\(348\) 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 \(0.0703117\pi\)
−0.975703 + 0.219099i \(0.929688\pi\)
\(354\) 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 \(-0.590070\pi\)
−0.279202 + 0.960232i \(0.590070\pi\)
\(360\) 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 \(-0.778658\pi\)
−0.767819 + 0.640667i \(0.778658\pi\)
\(384\) 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 \(-0.394189\pi\)
0.326326 + 0.945257i \(0.394189\pi\)
\(390\) 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 \(0.129318\pi\)
−0.918604 + 0.395180i \(0.870682\pi\)
\(402\) 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 \(0.0277911\pi\)
−0.996191 + 0.0871974i \(0.972209\pi\)
\(420\) 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 \(-0.743458\pi\)
−0.692426 + 0.721489i \(0.743458\pi\)
\(432\) 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 \(0.172958\pi\)
−0.855974 + 0.517019i \(0.827042\pi\)
\(444\) 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 \(-0.760769\pi\)
0.730621 0.682783i \(-0.239231\pi\)
\(450\) 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 \(-0.555691\pi\)
−0.174066 + 0.984734i \(0.555691\pi\)
\(462\) 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 \(0.121199\pi\)
−0.928384 + 0.371623i \(0.878801\pi\)
\(468\) 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 \(-0.242218\pi\)
0.724181 + 0.689610i \(0.242218\pi\)
\(480\) 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 \(-0.619704\pi\)
0.367259 0.930119i \(-0.380296\pi\)
\(492\) 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 \(-0.854324\pi\)
−0.897092 + 0.441845i \(0.854324\pi\)
\(504\) 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 \(-0.568825\pi\)
−0.214540 + 0.976715i \(0.568825\pi\)
\(510\) 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 \(-0.972589\pi\)
0.996295 0.0860069i \(-0.0274107\pi\)
\(522\) 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 \(-0.595870\pi\)
−0.296652 + 0.954986i \(0.595870\pi\)
\(558\) 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 \(0.0945100\pi\)
−0.956245 + 0.292569i \(0.905490\pi\)
\(564\) 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 \(-0.899939\pi\)
0.950997 0.309200i \(-0.100061\pi\)
\(570\) 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 \(0.00955129\pi\)
−0.999550 + 0.0300018i \(0.990449\pi\)
\(588\) 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 \(0.0872464\pi\)
−0.962671 + 0.270674i \(0.912754\pi\)
\(594\) 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 \(-0.404372\pi\)
0.295924 + 0.955211i \(0.404372\pi\)
\(600\) 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 \(-0.587238\pi\)
0.270647 0.962679i \(-0.412762\pi\)
\(618\) 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 \(0.0986189\pi\)
−0.952388 + 0.304888i \(0.901381\pi\)
\(642\) 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 \(-0.494922\pi\)
0.0159531 + 0.999873i \(0.494922\pi\)
\(648\) 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 \(-0.438446\pi\)
0.192176 + 0.981360i \(0.438446\pi\)
\(654\) 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 \(-0.930345\pi\)
0.976153 0.217085i \(-0.0696548\pi\)
\(660\) 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 \(-0.0772789\pi\)
0.970674 + 0.240401i \(0.0772789\pi\)
\(678\) 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 \(-0.779642\pi\)
0.769796 0.638291i \(-0.220358\pi\)
\(684\) 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 \(-0.277073\pi\)
0.644481 + 0.764620i \(0.277073\pi\)
\(702\) 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 \(-0.512953\pi\)
−0.0406827 + 0.999172i \(0.512953\pi\)
\(720\) 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 \(-0.586682\pi\)
−0.268965 + 0.963150i \(0.586682\pi\)
\(744\) 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 \(0.357428\pi\)
−0.433076 + 0.901357i \(0.642572\pi\)
\(762\) 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 \(-0.349001\pi\)
0.456785 + 0.889577i \(0.349001\pi\)
\(774\) 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 \(-0.730249\pi\)
−0.661899 + 0.749593i \(0.730249\pi\)
\(798\) 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 \(-0.975510\pi\)
0.997042 0.0768610i \(-0.0244898\pi\)
\(810\) 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 \(-0.210925\pi\)
0.788370 + 0.615201i \(0.210925\pi\)
\(822\) 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 \(0.0674658\pi\)
−0.977623 + 0.210367i \(0.932534\pi\)
\(828\) 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 \(-0.240115\pi\)
0.728721 + 0.684811i \(0.240115\pi\)
\(840\) 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 \(-0.853993\pi\)
0.896631 0.442779i \(-0.146007\pi\)
\(858\) 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 \(-0.466269\pi\)
0.105771 + 0.994391i \(0.466269\pi\)
\(864\) 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 \(0.0139285\pi\)
−0.999043 + 0.0437437i \(0.986072\pi\)
\(882\) 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 \(-0.460428\pi\)
0.123998 + 0.992282i \(0.460428\pi\)
\(888\) 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 \(-0.462875\pi\)
0.116366 + 0.993206i \(0.462875\pi\)
\(912\) 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 \(-0.946721\pi\)
0.986024 0.166600i \(-0.0532790\pi\)
\(930\) 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 \(-0.424980\pi\)
0.233508 + 0.972355i \(0.424980\pi\)
\(942\) 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 \(0.0398059\pi\)
−0.992191 + 0.124728i \(0.960194\pi\)
\(948\) 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 \(0.325113\pi\)
−0.522197 + 0.852825i \(0.674887\pi\)
\(954\) 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 \(-0.657852\pi\)
0.475828 0.879538i \(-0.342148\pi\)
\(972\) 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 \(-0.999693\pi\)
1.00000 0.000965516i \(-0.000307333\pi\)
\(978\) 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 \(-0.462330\pi\)
0.118068 + 0.993005i \(0.462330\pi\)
\(984\) 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.5615.12 32
3.2 odd 2 inner 6048.2.j.c.5615.21 32
4.3 odd 2 1512.2.j.c.323.13 32
8.3 odd 2 inner 6048.2.j.c.5615.22 32
8.5 even 2 1512.2.j.c.323.19 yes 32
12.11 even 2 1512.2.j.c.323.20 yes 32
24.5 odd 2 1512.2.j.c.323.14 yes 32
24.11 even 2 inner 6048.2.j.c.5615.11 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.j.c.323.13 32 4.3 odd 2
1512.2.j.c.323.14 yes 32 24.5 odd 2
1512.2.j.c.323.19 yes 32 8.5 even 2
1512.2.j.c.323.20 yes 32 12.11 even 2
6048.2.j.c.5615.11 32 24.11 even 2 inner
6048.2.j.c.5615.12 32 1.1 even 1 trivial
6048.2.j.c.5615.21 32 3.2 odd 2 inner
6048.2.j.c.5615.22 32 8.3 odd 2 inner