Properties

Label 6048.2.h.h.2591.5
Level 6048
Weight 2
Character 6048.2591
Analytic conductor 48.294
Analytic rank 0
Dimension 16
CM no
Inner twists 4

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Newspace parameters

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

Newform invariants

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

Embedding invariants

Embedding label 2591.5
Root \(-2.55743 - 2.55743i\)
Character \(\chi\) = 6048.2591
Dual form 6048.2.h.h.2591.11

$q$-expansion

\(f(q)\) \(=\) \(q-0.517638i q^{5} -1.00000i q^{7} +O(q^{10})\) \(q-0.517638i q^{5} -1.00000i q^{7} -3.18300 q^{11} -2.64764 q^{13} -5.15854i q^{17} +8.23349i q^{19} +4.38134 q^{23} +4.73205 q^{25} +5.11485i q^{29} -0.548511i q^{31} -0.517638 q^{35} -2.37969 q^{37} -0.473945i q^{41} +0.816460i q^{43} +0.493272 q^{47} -1.00000 q^{49} +0.915903i q^{53} +1.64764i q^{55} +12.7229 q^{59} -3.40231 q^{61} +1.37052i q^{65} -0.585849i q^{67} +6.95020 q^{71} +2.41703 q^{73} +3.18300i q^{77} -12.5787i q^{79} -0.215871 q^{83} -2.67026 q^{85} -10.1525i q^{89} +2.64764i q^{91} +4.26197 q^{95} -7.23349 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q + 8q^{13} + 48q^{25} + 40q^{37} - 16q^{49} - 56q^{73} - 16q^{85} - 16q^{97} + O(q^{100}) \)

Character values

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

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

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) − 0.517638i − 0.231495i −0.993279 0.115747i \(-0.963074\pi\)
0.993279 0.115747i \(-0.0369263\pi\)
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −3.18300 −0.959710 −0.479855 0.877348i \(-0.659311\pi\)
−0.479855 + 0.877348i \(0.659311\pi\)
\(12\) 0 0
\(13\) −2.64764 −0.734324 −0.367162 0.930157i \(-0.619670\pi\)
−0.367162 + 0.930157i \(0.619670\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 5.15854i − 1.25113i −0.780172 0.625565i \(-0.784868\pi\)
0.780172 0.625565i \(-0.215132\pi\)
\(18\) 0 0
\(19\) 8.23349i 1.88889i 0.328666 + 0.944446i \(0.393401\pi\)
−0.328666 + 0.944446i \(0.606599\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.38134 0.913573 0.456786 0.889576i \(-0.349000\pi\)
0.456786 + 0.889576i \(0.349000\pi\)
\(24\) 0 0
\(25\) 4.73205 0.946410
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 5.11485i 0.949804i 0.880039 + 0.474902i \(0.157516\pi\)
−0.880039 + 0.474902i \(0.842484\pi\)
\(30\) 0 0
\(31\) − 0.548511i − 0.0985155i −0.998786 0.0492577i \(-0.984314\pi\)
0.998786 0.0492577i \(-0.0156856\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.517638 −0.0874968
\(36\) 0 0
\(37\) −2.37969 −0.391219 −0.195609 0.980682i \(-0.562669\pi\)
−0.195609 + 0.980682i \(0.562669\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 0.473945i − 0.0740177i −0.999315 0.0370089i \(-0.988217\pi\)
0.999315 0.0370089i \(-0.0117830\pi\)
\(42\) 0 0
\(43\) 0.816460i 0.124509i 0.998060 + 0.0622545i \(0.0198290\pi\)
−0.998060 + 0.0622545i \(0.980171\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0.493272 0.0719512 0.0359756 0.999353i \(-0.488546\pi\)
0.0359756 + 0.999353i \(0.488546\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0.915903i 0.125809i 0.998020 + 0.0629045i \(0.0200363\pi\)
−0.998020 + 0.0629045i \(0.979964\pi\)
\(54\) 0 0
\(55\) 1.64764i 0.222168i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 12.7229 1.65638 0.828189 0.560449i \(-0.189371\pi\)
0.828189 + 0.560449i \(0.189371\pi\)
\(60\) 0 0
\(61\) −3.40231 −0.435621 −0.217811 0.975991i \(-0.569891\pi\)
−0.217811 + 0.975991i \(0.569891\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.37052i 0.169992i
\(66\) 0 0
\(67\) − 0.585849i − 0.0715729i −0.999359 0.0357864i \(-0.988606\pi\)
0.999359 0.0357864i \(-0.0113936\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 6.95020 0.824837 0.412419 0.910994i \(-0.364684\pi\)
0.412419 + 0.910994i \(0.364684\pi\)
\(72\) 0 0
\(73\) 2.41703 0.282892 0.141446 0.989946i \(-0.454825\pi\)
0.141446 + 0.989946i \(0.454825\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.18300i 0.362736i
\(78\) 0 0
\(79\) − 12.5787i − 1.41522i −0.706605 0.707609i \(-0.749774\pi\)
0.706605 0.707609i \(-0.250226\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −0.215871 −0.0236949 −0.0118475 0.999930i \(-0.503771\pi\)
−0.0118475 + 0.999930i \(0.503771\pi\)
\(84\) 0 0
\(85\) −2.67026 −0.289630
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 10.1525i − 1.07617i −0.842892 0.538083i \(-0.819149\pi\)
0.842892 0.538083i \(-0.180851\pi\)
\(90\) 0 0
\(91\) 2.64764i 0.277548i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 4.26197 0.437269
\(96\) 0 0
\(97\) −7.23349 −0.734450 −0.367225 0.930132i \(-0.619692\pi\)
−0.367225 + 0.930132i \(0.619692\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 17.6975i − 1.76097i −0.474073 0.880486i \(-0.657217\pi\)
0.474073 0.880486i \(-0.342783\pi\)
\(102\) 0 0
\(103\) 4.85380i 0.478259i 0.970988 + 0.239129i \(0.0768620\pi\)
−0.970988 + 0.239129i \(0.923138\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 18.3360 1.77261 0.886306 0.463100i \(-0.153263\pi\)
0.886306 + 0.463100i \(0.153263\pi\)
\(108\) 0 0
\(109\) 9.35524 0.896069 0.448035 0.894016i \(-0.352124\pi\)
0.448035 + 0.894016i \(0.352124\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 5.53748i 0.520922i 0.965484 + 0.260461i \(0.0838746\pi\)
−0.965484 + 0.260461i \(0.916125\pi\)
\(114\) 0 0
\(115\) − 2.26795i − 0.211487i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −5.15854 −0.472883
\(120\) 0 0
\(121\) −0.868519 −0.0789563
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 5.03768i − 0.450584i
\(126\) 0 0
\(127\) − 17.4070i − 1.54462i −0.635244 0.772312i \(-0.719100\pi\)
0.635244 0.772312i \(-0.280900\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 10.8911 0.951558 0.475779 0.879565i \(-0.342166\pi\)
0.475779 + 0.879565i \(0.342166\pi\)
\(132\) 0 0
\(133\) 8.23349 0.713934
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 15.6707i − 1.33884i −0.742886 0.669418i \(-0.766543\pi\)
0.742886 0.669418i \(-0.233457\pi\)
\(138\) 0 0
\(139\) − 14.3600i − 1.21800i −0.793172 0.608998i \(-0.791572\pi\)
0.793172 0.608998i \(-0.208428\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 8.42744 0.704738
\(144\) 0 0
\(145\) 2.64764 0.219875
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 22.7367i − 1.86266i −0.364171 0.931332i \(-0.618648\pi\)
0.364171 0.931332i \(-0.381352\pi\)
\(150\) 0 0
\(151\) 5.06467i 0.412157i 0.978535 + 0.206079i \(0.0660703\pi\)
−0.978535 + 0.206079i \(0.933930\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −0.283930 −0.0228058
\(156\) 0 0
\(157\) 17.5069 1.39720 0.698602 0.715510i \(-0.253806\pi\)
0.698602 + 0.715510i \(0.253806\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 4.38134i − 0.345298i
\(162\) 0 0
\(163\) 9.74466i 0.763261i 0.924315 + 0.381630i \(0.124637\pi\)
−0.924315 + 0.381630i \(0.875363\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −19.4241 −1.50308 −0.751542 0.659685i \(-0.770690\pi\)
−0.751542 + 0.659685i \(0.770690\pi\)
\(168\) 0 0
\(169\) −5.99000 −0.460769
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 17.2673i − 1.31281i −0.754410 0.656404i \(-0.772077\pi\)
0.754410 0.656404i \(-0.227923\pi\)
\(174\) 0 0
\(175\) − 4.73205i − 0.357709i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −11.2858 −0.843540 −0.421770 0.906703i \(-0.638591\pi\)
−0.421770 + 0.906703i \(0.638591\pi\)
\(180\) 0 0
\(181\) −1.40231 −0.104233 −0.0521164 0.998641i \(-0.516597\pi\)
−0.0521164 + 0.998641i \(0.516597\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 1.23182i 0.0905652i
\(186\) 0 0
\(187\) 16.4196i 1.20072i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 6.00639 0.434607 0.217303 0.976104i \(-0.430274\pi\)
0.217303 + 0.976104i \(0.430274\pi\)
\(192\) 0 0
\(193\) 16.5288 1.18977 0.594884 0.803812i \(-0.297198\pi\)
0.594884 + 0.803812i \(0.297198\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 5.84687i 0.416572i 0.978068 + 0.208286i \(0.0667885\pi\)
−0.978068 + 0.208286i \(0.933211\pi\)
\(198\) 0 0
\(199\) 4.18354i 0.296563i 0.988945 + 0.148282i \(0.0473742\pi\)
−0.988945 + 0.148282i \(0.952626\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 5.11485 0.358992
\(204\) 0 0
\(205\) −0.245332 −0.0171347
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 26.2072i − 1.81279i
\(210\) 0 0
\(211\) 21.2882i 1.46554i 0.680478 + 0.732769i \(0.261772\pi\)
−0.680478 + 0.732769i \(0.738228\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0.422631 0.0288232
\(216\) 0 0
\(217\) −0.548511 −0.0372354
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 13.6580i 0.918735i
\(222\) 0 0
\(223\) 12.5887i 0.843004i 0.906828 + 0.421502i \(0.138497\pi\)
−0.906828 + 0.421502i \(0.861503\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 1.19330 0.0792024 0.0396012 0.999216i \(-0.487391\pi\)
0.0396012 + 0.999216i \(0.487391\pi\)
\(228\) 0 0
\(229\) −17.7694 −1.17423 −0.587117 0.809502i \(-0.699737\pi\)
−0.587117 + 0.809502i \(0.699737\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 14.1284i 0.925580i 0.886468 + 0.462790i \(0.153152\pi\)
−0.886468 + 0.462790i \(0.846848\pi\)
\(234\) 0 0
\(235\) − 0.255337i − 0.0166563i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 14.1600 0.915932 0.457966 0.888970i \(-0.348578\pi\)
0.457966 + 0.888970i \(0.348578\pi\)
\(240\) 0 0
\(241\) −20.1864 −1.30032 −0.650161 0.759797i \(-0.725298\pi\)
−0.650161 + 0.759797i \(0.725298\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0.517638i 0.0330707i
\(246\) 0 0
\(247\) − 21.7993i − 1.38706i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 15.7352 0.993197 0.496598 0.867980i \(-0.334582\pi\)
0.496598 + 0.867980i \(0.334582\pi\)
\(252\) 0 0
\(253\) −13.9458 −0.876765
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 9.72117i 0.606390i 0.952929 + 0.303195i \(0.0980534\pi\)
−0.952929 + 0.303195i \(0.901947\pi\)
\(258\) 0 0
\(259\) 2.37969i 0.147867i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 19.5487 1.20542 0.602712 0.797959i \(-0.294087\pi\)
0.602712 + 0.797959i \(0.294087\pi\)
\(264\) 0 0
\(265\) 0.474106 0.0291241
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 9.00699i − 0.549166i −0.961563 0.274583i \(-0.911460\pi\)
0.961563 0.274583i \(-0.0885398\pi\)
\(270\) 0 0
\(271\) − 16.0252i − 0.973463i −0.873552 0.486732i \(-0.838189\pi\)
0.873552 0.486732i \(-0.161811\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −15.0621 −0.908279
\(276\) 0 0
\(277\) −4.31790 −0.259437 −0.129719 0.991551i \(-0.541407\pi\)
−0.129719 + 0.991551i \(0.541407\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 8.06265i − 0.480977i −0.970652 0.240489i \(-0.922692\pi\)
0.970652 0.240489i \(-0.0773077\pi\)
\(282\) 0 0
\(283\) 27.4570i 1.63215i 0.577948 + 0.816074i \(0.303854\pi\)
−0.577948 + 0.816074i \(0.696146\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −0.473945 −0.0279761
\(288\) 0 0
\(289\) −9.61057 −0.565328
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 10.1282i − 0.591694i −0.955235 0.295847i \(-0.904398\pi\)
0.955235 0.295847i \(-0.0956018\pi\)
\(294\) 0 0
\(295\) − 6.58585i − 0.383443i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −11.6002 −0.670858
\(300\) 0 0
\(301\) 0.816460 0.0470600
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 1.76117i 0.100844i
\(306\) 0 0
\(307\) − 10.8441i − 0.618904i −0.950915 0.309452i \(-0.899854\pi\)
0.950915 0.309452i \(-0.100146\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 22.7209 1.28839 0.644193 0.764863i \(-0.277193\pi\)
0.644193 + 0.764863i \(0.277193\pi\)
\(312\) 0 0
\(313\) 12.8222 0.724755 0.362377 0.932031i \(-0.381965\pi\)
0.362377 + 0.932031i \(0.381965\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 10.4593i − 0.587455i −0.955889 0.293727i \(-0.905104\pi\)
0.955889 0.293727i \(-0.0948958\pi\)
\(318\) 0 0
\(319\) − 16.2806i − 0.911536i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 42.4728 2.36325
\(324\) 0 0
\(325\) −12.5288 −0.694971
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 0.493272i − 0.0271950i
\(330\) 0 0
\(331\) − 3.70760i − 0.203788i −0.994795 0.101894i \(-0.967510\pi\)
0.994795 0.101894i \(-0.0324902\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −0.303258 −0.0165688
\(336\) 0 0
\(337\) 19.9900 1.08892 0.544462 0.838785i \(-0.316734\pi\)
0.544462 + 0.838785i \(0.316734\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 1.74591i 0.0945463i
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 22.1226 1.18760 0.593801 0.804612i \(-0.297627\pi\)
0.593801 + 0.804612i \(0.297627\pi\)
\(348\) 0 0
\(349\) 0.997120 0.0533746 0.0266873 0.999644i \(-0.491504\pi\)
0.0266873 + 0.999644i \(0.491504\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 20.5403i 1.09325i 0.837378 + 0.546624i \(0.184087\pi\)
−0.837378 + 0.546624i \(0.815913\pi\)
\(354\) 0 0
\(355\) − 3.59769i − 0.190946i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −6.28121 −0.331509 −0.165755 0.986167i \(-0.553006\pi\)
−0.165755 + 0.986167i \(0.553006\pi\)
\(360\) 0 0
\(361\) −48.7904 −2.56791
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 1.25115i − 0.0654880i
\(366\) 0 0
\(367\) − 0.439376i − 0.0229352i −0.999934 0.0114676i \(-0.996350\pi\)
0.999934 0.0114676i \(-0.00365034\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0.915903 0.0475513
\(372\) 0 0
\(373\) −34.2016 −1.77089 −0.885447 0.464741i \(-0.846148\pi\)
−0.885447 + 0.464741i \(0.846148\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 13.5423i − 0.697463i
\(378\) 0 0
\(379\) 15.9681i 0.820229i 0.912034 + 0.410114i \(0.134511\pi\)
−0.912034 + 0.410114i \(0.865489\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 31.6537 1.61743 0.808715 0.588200i \(-0.200163\pi\)
0.808715 + 0.588200i \(0.200163\pi\)
\(384\) 0 0
\(385\) 1.64764 0.0839716
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 25.3751i − 1.28657i −0.765627 0.643285i \(-0.777571\pi\)
0.765627 0.643285i \(-0.222429\pi\)
\(390\) 0 0
\(391\) − 22.6013i − 1.14300i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −6.51123 −0.327615
\(396\) 0 0
\(397\) 34.4470 1.72884 0.864422 0.502767i \(-0.167685\pi\)
0.864422 + 0.502767i \(0.167685\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 36.2881i − 1.81214i −0.423126 0.906071i \(-0.639067\pi\)
0.423126 0.906071i \(-0.360933\pi\)
\(402\) 0 0
\(403\) 1.45226i 0.0723423i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 7.57456 0.375457
\(408\) 0 0
\(409\) 6.84590 0.338508 0.169254 0.985572i \(-0.445864\pi\)
0.169254 + 0.985572i \(0.445864\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 12.7229i − 0.626052i
\(414\) 0 0
\(415\) 0.111743i 0.00548525i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 5.12396 0.250322 0.125161 0.992136i \(-0.460055\pi\)
0.125161 + 0.992136i \(0.460055\pi\)
\(420\) 0 0
\(421\) 16.3926 0.798925 0.399463 0.916749i \(-0.369197\pi\)
0.399463 + 0.916749i \(0.369197\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 24.4105i − 1.18408i
\(426\) 0 0
\(427\) 3.40231i 0.164649i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 25.0475 1.20650 0.603248 0.797554i \(-0.293873\pi\)
0.603248 + 0.797554i \(0.293873\pi\)
\(432\) 0 0
\(433\) 40.1427 1.92914 0.964568 0.263834i \(-0.0849870\pi\)
0.964568 + 0.263834i \(0.0849870\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 36.0737i 1.72564i
\(438\) 0 0
\(439\) − 22.6905i − 1.08296i −0.840715 0.541478i \(-0.817865\pi\)
0.840715 0.541478i \(-0.182135\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −26.8137 −1.27396 −0.636979 0.770881i \(-0.719816\pi\)
−0.636979 + 0.770881i \(0.719816\pi\)
\(444\) 0 0
\(445\) −5.25534 −0.249127
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 6.12725i − 0.289163i −0.989493 0.144581i \(-0.953816\pi\)
0.989493 0.144581i \(-0.0461836\pi\)
\(450\) 0 0
\(451\) 1.50856i 0.0710355i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 1.37052 0.0642510
\(456\) 0 0
\(457\) 14.2629 0.667192 0.333596 0.942716i \(-0.391738\pi\)
0.333596 + 0.942716i \(0.391738\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 13.2106i − 0.615279i −0.951503 0.307639i \(-0.900461\pi\)
0.951503 0.307639i \(-0.0995390\pi\)
\(462\) 0 0
\(463\) − 2.56112i − 0.119026i −0.998228 0.0595128i \(-0.981045\pi\)
0.998228 0.0595128i \(-0.0189547\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −22.8012 −1.05511 −0.527557 0.849519i \(-0.676892\pi\)
−0.527557 + 0.849519i \(0.676892\pi\)
\(468\) 0 0
\(469\) −0.585849 −0.0270520
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 2.59879i − 0.119493i
\(474\) 0 0
\(475\) 38.9613i 1.78767i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −36.9525 −1.68840 −0.844201 0.536027i \(-0.819924\pi\)
−0.844201 + 0.536027i \(0.819924\pi\)
\(480\) 0 0
\(481\) 6.30057 0.287281
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 3.74433i 0.170021i
\(486\) 0 0
\(487\) − 6.80934i − 0.308560i −0.988027 0.154280i \(-0.950694\pi\)
0.988027 0.154280i \(-0.0493059\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 14.1702 0.639492 0.319746 0.947503i \(-0.396402\pi\)
0.319746 + 0.947503i \(0.396402\pi\)
\(492\) 0 0
\(493\) 26.3852 1.18833
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 6.95020i − 0.311759i
\(498\) 0 0
\(499\) 34.6858i 1.55275i 0.630273 + 0.776374i \(0.282943\pi\)
−0.630273 + 0.776374i \(0.717057\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 9.85580 0.439448 0.219724 0.975562i \(-0.429484\pi\)
0.219724 + 0.975562i \(0.429484\pi\)
\(504\) 0 0
\(505\) −9.16092 −0.407656
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 16.2874i − 0.721926i −0.932580 0.360963i \(-0.882448\pi\)
0.932580 0.360963i \(-0.117552\pi\)
\(510\) 0 0
\(511\) − 2.41703i − 0.106923i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 2.51251 0.110714
\(516\) 0 0
\(517\) −1.57008 −0.0690523
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 27.9338i 1.22380i 0.790935 + 0.611900i \(0.209595\pi\)
−0.790935 + 0.611900i \(0.790405\pi\)
\(522\) 0 0
\(523\) − 2.13724i − 0.0934550i −0.998908 0.0467275i \(-0.985121\pi\)
0.998908 0.0467275i \(-0.0148792\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −2.82952 −0.123256
\(528\) 0 0
\(529\) −3.80385 −0.165385
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 1.25484i 0.0543529i
\(534\) 0 0
\(535\) − 9.49144i − 0.410351i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 3.18300 0.137101
\(540\) 0 0
\(541\) 20.2037 0.868627 0.434313 0.900762i \(-0.356991\pi\)
0.434313 + 0.900762i \(0.356991\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 4.84263i − 0.207435i
\(546\) 0 0
\(547\) − 29.7917i − 1.27380i −0.770946 0.636901i \(-0.780216\pi\)
0.770946 0.636901i \(-0.219784\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −42.1131 −1.79408
\(552\) 0 0
\(553\) −12.5787 −0.534902
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 4.18984i 0.177529i 0.996053 + 0.0887645i \(0.0282919\pi\)
−0.996053 + 0.0887645i \(0.971708\pi\)
\(558\) 0 0
\(559\) − 2.16169i − 0.0914299i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 36.2275 1.52681 0.763405 0.645921i \(-0.223526\pi\)
0.763405 + 0.645921i \(0.223526\pi\)
\(564\) 0 0
\(565\) 2.86641 0.120591
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 6.93088i − 0.290557i −0.989391 0.145279i \(-0.953592\pi\)
0.989391 0.145279i \(-0.0464079\pi\)
\(570\) 0 0
\(571\) 27.4822i 1.15009i 0.818120 + 0.575047i \(0.195016\pi\)
−0.818120 + 0.575047i \(0.804984\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 20.7327 0.864615
\(576\) 0 0
\(577\) −24.9829 −1.04005 −0.520025 0.854151i \(-0.674077\pi\)
−0.520025 + 0.854151i \(0.674077\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0.215871i 0.00895583i
\(582\) 0 0
\(583\) − 2.91532i − 0.120740i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −16.7456 −0.691164 −0.345582 0.938389i \(-0.612319\pi\)
−0.345582 + 0.938389i \(0.612319\pi\)
\(588\) 0 0
\(589\) 4.51616 0.186085
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 2.78732i − 0.114461i −0.998361 0.0572307i \(-0.981773\pi\)
0.998361 0.0572307i \(-0.0182270\pi\)
\(594\) 0 0
\(595\) 2.67026i 0.109470i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 26.5145 1.08335 0.541677 0.840587i \(-0.317790\pi\)
0.541677 + 0.840587i \(0.317790\pi\)
\(600\) 0 0
\(601\) −21.5198 −0.877811 −0.438906 0.898533i \(-0.644634\pi\)
−0.438906 + 0.898533i \(0.644634\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0.449579i 0.0182780i
\(606\) 0 0
\(607\) − 12.4052i − 0.503511i −0.967791 0.251756i \(-0.918992\pi\)
0.967791 0.251756i \(-0.0810079\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −1.30601 −0.0528354
\(612\) 0 0
\(613\) 45.8840 1.85324 0.926619 0.376002i \(-0.122701\pi\)
0.926619 + 0.376002i \(0.122701\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 37.1405i 1.49522i 0.664139 + 0.747610i \(0.268798\pi\)
−0.664139 + 0.747610i \(0.731202\pi\)
\(618\) 0 0
\(619\) 17.5338i 0.704743i 0.935860 + 0.352371i \(0.114625\pi\)
−0.935860 + 0.352371i \(0.885375\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −10.1525 −0.406753
\(624\) 0 0
\(625\) 21.0526 0.842102
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 12.2757i 0.489466i
\(630\) 0 0
\(631\) − 15.2706i − 0.607911i −0.952686 0.303956i \(-0.901693\pi\)
0.952686 0.303956i \(-0.0983075\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −9.01054 −0.357572
\(636\) 0 0
\(637\) 2.64764 0.104903
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 12.3657i − 0.488417i −0.969723 0.244208i \(-0.921472\pi\)
0.969723 0.244208i \(-0.0785281\pi\)
\(642\) 0 0
\(643\) 12.9800i 0.511881i 0.966693 + 0.255940i \(0.0823851\pi\)
−0.966693 + 0.255940i \(0.917615\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0.183885 0.00722926 0.00361463 0.999993i \(-0.498849\pi\)
0.00361463 + 0.999993i \(0.498849\pi\)
\(648\) 0 0
\(649\) −40.4969 −1.58964
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 35.9300i 1.40605i 0.711165 + 0.703025i \(0.248168\pi\)
−0.711165 + 0.703025i \(0.751832\pi\)
\(654\) 0 0
\(655\) − 5.63764i − 0.220281i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 20.0211 0.779913 0.389956 0.920833i \(-0.372490\pi\)
0.389956 + 0.920833i \(0.372490\pi\)
\(660\) 0 0
\(661\) −33.6434 −1.30858 −0.654288 0.756245i \(-0.727032\pi\)
−0.654288 + 0.756245i \(0.727032\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 4.26197i − 0.165272i
\(666\) 0 0
\(667\) 22.4099i 0.867715i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 10.8295 0.418070
\(672\) 0 0
\(673\) 24.7523 0.954129 0.477065 0.878868i \(-0.341701\pi\)
0.477065 + 0.878868i \(0.341701\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 16.2599i − 0.624920i −0.949931 0.312460i \(-0.898847\pi\)
0.949931 0.312460i \(-0.101153\pi\)
\(678\) 0 0
\(679\) 7.23349i 0.277596i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −8.52039 −0.326024 −0.163012 0.986624i \(-0.552121\pi\)
−0.163012 + 0.986624i \(0.552121\pi\)
\(684\) 0 0
\(685\) −8.11174 −0.309934
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 2.42498i − 0.0923845i
\(690\) 0 0
\(691\) 0.354195i 0.0134742i 0.999977 + 0.00673710i \(0.00214450\pi\)
−0.999977 + 0.00673710i \(0.997855\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −7.43326 −0.281960
\(696\) 0 0
\(697\) −2.44486 −0.0926058
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 13.0877i 0.494315i 0.968975 + 0.247157i \(0.0794964\pi\)
−0.968975 + 0.247157i \(0.920504\pi\)
\(702\) 0 0
\(703\) − 19.5932i − 0.738971i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −17.6975 −0.665585
\(708\) 0 0
\(709\) −12.2340 −0.459457 −0.229729 0.973255i \(-0.573784\pi\)
−0.229729 + 0.973255i \(0.573784\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 2.40321i − 0.0900011i
\(714\) 0 0
\(715\) − 4.36236i − 0.163143i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −8.54681 −0.318742 −0.159371 0.987219i \(-0.550947\pi\)
−0.159371 + 0.987219i \(0.550947\pi\)
\(720\) 0 0
\(721\) 4.85380 0.180765
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 24.2037i 0.898904i
\(726\) 0 0
\(727\) − 45.1151i − 1.67323i −0.547793 0.836614i \(-0.684532\pi\)
0.547793 0.836614i \(-0.315468\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 4.21175 0.155777
\(732\) 0 0
\(733\) −12.1541 −0.448921 −0.224460 0.974483i \(-0.572062\pi\)
−0.224460 + 0.974483i \(0.572062\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.86476i 0.0686892i
\(738\) 0 0
\(739\) − 1.57113i − 0.0577949i −0.999582 0.0288974i \(-0.990800\pi\)
0.999582 0.0288974i \(-0.00919962\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 15.7028 0.576080 0.288040 0.957618i \(-0.406996\pi\)
0.288040 + 0.957618i \(0.406996\pi\)
\(744\) 0 0
\(745\) −11.7694 −0.431197
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 18.3360i − 0.669984i
\(750\) 0 0
\(751\) 38.5392i 1.40632i 0.711033 + 0.703159i \(0.248228\pi\)
−0.711033 + 0.703159i \(0.751772\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 2.62167 0.0954122
\(756\) 0 0
\(757\) 2.73416 0.0993747 0.0496873 0.998765i \(-0.484178\pi\)
0.0496873 + 0.998765i \(0.484178\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 9.21155i 0.333919i 0.985964 + 0.166959i \(0.0533949\pi\)
−0.985964 + 0.166959i \(0.946605\pi\)
\(762\) 0 0
\(763\) − 9.35524i − 0.338682i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −33.6856 −1.21632
\(768\) 0 0
\(769\) 43.5687 1.57113 0.785564 0.618780i \(-0.212373\pi\)
0.785564 + 0.618780i \(0.212373\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 38.5831i 1.38774i 0.720101 + 0.693869i \(0.244095\pi\)
−0.720101 + 0.693869i \(0.755905\pi\)
\(774\) 0 0
\(775\) − 2.59558i − 0.0932361i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 3.90222 0.139811
\(780\) 0 0
\(781\) −22.1225 −0.791605
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 9.06225i − 0.323446i
\(786\) 0 0
\(787\) − 11.1846i − 0.398687i −0.979930 0.199344i \(-0.936119\pi\)
0.979930 0.199344i \(-0.0638810\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 5.53748 0.196890
\(792\) 0 0
\(793\) 9.00810 0.319887
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 23.1800i 0.821079i 0.911843 + 0.410539i \(0.134660\pi\)
−0.911843 + 0.410539i \(0.865340\pi\)
\(798\) 0 0
\(799\) − 2.54457i − 0.0900203i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −7.69340 −0.271494
\(804\) 0 0
\(805\) −2.26795 −0.0799347
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 23.6450i 0.831313i 0.909522 + 0.415657i \(0.136448\pi\)
−0.909522 + 0.415657i \(0.863552\pi\)
\(810\) 0 0
\(811\) 24.5971i 0.863721i 0.901940 + 0.431860i \(0.142143\pi\)
−0.901940 + 0.431860i \(0.857857\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 5.04421 0.176691
\(816\) 0 0
\(817\) −6.72232 −0.235184
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 51.5879i − 1.80043i −0.435446 0.900215i \(-0.643409\pi\)
0.435446 0.900215i \(-0.356591\pi\)
\(822\) 0 0
\(823\) − 15.0575i − 0.524873i −0.964949 0.262437i \(-0.915474\pi\)
0.964949 0.262437i \(-0.0845260\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 3.65940 0.127250 0.0636249 0.997974i \(-0.479734\pi\)
0.0636249 + 0.997974i \(0.479734\pi\)
\(828\) 0 0
\(829\) 23.9681 0.832448 0.416224 0.909262i \(-0.363353\pi\)
0.416224 + 0.909262i \(0.363353\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 5.15854i 0.178733i
\(834\) 0 0
\(835\) 10.0547i 0.347956i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −26.5883 −0.917931 −0.458965 0.888454i \(-0.651780\pi\)
−0.458965 + 0.888454i \(0.651780\pi\)
\(840\) 0 0
\(841\) 2.83831 0.0978726
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 3.10065i 0.106666i
\(846\) 0 0
\(847\) 0.868519i 0.0298427i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −10.4262 −0.357407
\(852\) 0 0
\(853\) −28.0181 −0.959321 −0.479661 0.877454i \(-0.659240\pi\)
−0.479661 + 0.877454i \(0.659240\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 16.9852i − 0.580205i −0.956996 0.290102i \(-0.906311\pi\)
0.956996 0.290102i \(-0.0936893\pi\)
\(858\) 0 0
\(859\) − 19.9576i − 0.680946i −0.940254 0.340473i \(-0.889413\pi\)
0.940254 0.340473i \(-0.110587\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −42.3884 −1.44292 −0.721459 0.692457i \(-0.756528\pi\)
−0.721459 + 0.692457i \(0.756528\pi\)
\(864\) 0 0
\(865\) −8.93821 −0.303908
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 40.0381i 1.35820i
\(870\) 0 0
\(871\) 1.55112i 0.0525577i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −5.03768 −0.170305
\(876\) 0 0
\(877\) −24.4993 −0.827284 −0.413642 0.910440i \(-0.635743\pi\)
−0.413642 + 0.910440i \(0.635743\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 24.5364i − 0.826652i −0.910583 0.413326i \(-0.864367\pi\)
0.910583 0.413326i \(-0.135633\pi\)
\(882\) 0 0
\(883\) 17.3082i 0.582466i 0.956652 + 0.291233i \(0.0940655\pi\)
−0.956652 + 0.291233i \(0.905934\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −40.8260 −1.37080 −0.685401 0.728166i \(-0.740373\pi\)
−0.685401 + 0.728166i \(0.740373\pi\)
\(888\) 0 0
\(889\) −17.4070 −0.583813
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 4.06135i 0.135908i
\(894\) 0 0
\(895\) 5.84196i 0.195275i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 2.80555 0.0935704
\(900\) 0 0
\(901\) 4.72473 0.157403
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0.725889i 0.0241294i
\(906\) 0 0
\(907\) 28.9282i 0.960545i 0.877119 + 0.480273i \(0.159462\pi\)
−0.877119 + 0.480273i \(0.840538\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −9.90248 −0.328084 −0.164042 0.986453i \(-0.552453\pi\)
−0.164042 + 0.986453i \(0.552453\pi\)
\(912\) 0 0
\(913\) 0.687117 0.0227402
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 10.8911i − 0.359655i
\(918\) 0 0
\(919\) 24.2910i 0.801287i 0.916234 + 0.400644i \(0.131213\pi\)
−0.916234 + 0.400644i \(0.868787\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −18.4016 −0.605698
\(924\) 0 0
\(925\) −11.2608 −0.370254
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 30.4046i − 0.997544i −0.866733 0.498772i \(-0.833784\pi\)
0.866733 0.498772i \(-0.166216\pi\)
\(930\) 0 0
\(931\) − 8.23349i − 0.269842i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 8.49943 0.277961
\(936\) 0 0
\(937\) −3.54014 −0.115651 −0.0578257 0.998327i \(-0.518417\pi\)
−0.0578257 + 0.998327i \(0.518417\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 26.1731i − 0.853220i −0.904436 0.426610i \(-0.859708\pi\)
0.904436 0.426610i \(-0.140292\pi\)
\(942\) 0 0
\(943\) − 2.07651i − 0.0676206i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −28.3342 −0.920739 −0.460369 0.887727i \(-0.652283\pi\)
−0.460369 + 0.887727i \(0.652283\pi\)
\(948\) 0 0
\(949\) −6.39943 −0.207734
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 55.0723i 1.78397i 0.452069 + 0.891983i \(0.350686\pi\)
−0.452069 + 0.891983i \(0.649314\pi\)
\(954\) 0 0
\(955\) − 3.10914i − 0.100609i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −15.6707 −0.506033
\(960\) 0 0
\(961\) 30.6991 0.990295
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 8.55592i − 0.275425i
\(966\) 0 0
\(967\) 25.0733i 0.806304i 0.915133 + 0.403152i \(0.132085\pi\)
−0.915133 + 0.403152i \(0.867915\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −48.8089 −1.56635 −0.783176 0.621800i \(-0.786402\pi\)
−0.783176 + 0.621800i \(0.786402\pi\)
\(972\) 0 0
\(973\) −14.3600 −0.460359
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 61.2645i 1.96002i 0.198940 + 0.980012i \(0.436250\pi\)
−0.198940 + 0.980012i \(0.563750\pi\)
\(978\) 0 0
\(979\) 32.3155i 1.03281i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 2.74007 0.0873948 0.0436974 0.999045i \(-0.486086\pi\)
0.0436974 + 0.999045i \(0.486086\pi\)
\(984\) 0 0
\(985\) 3.02656 0.0964343
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 3.57719i 0.113748i
\(990\) 0 0
\(991\) − 8.02994i − 0.255079i −0.991833 0.127540i \(-0.959292\pi\)
0.991833 0.127540i \(-0.0407080\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 2.16556 0.0686529
\(996\) 0 0
\(997\) −16.4217 −0.520082 −0.260041 0.965598i \(-0.583736\pi\)
−0.260041 + 0.965598i \(0.583736\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 6048.2.h.h.2591.5 16
3.2 odd 2 inner 6048.2.h.h.2591.10 yes 16
4.3 odd 2 inner 6048.2.h.h.2591.8 yes 16
12.11 even 2 inner 6048.2.h.h.2591.11 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6048.2.h.h.2591.5 16 1.1 even 1 trivial
6048.2.h.h.2591.8 yes 16 4.3 odd 2 inner
6048.2.h.h.2591.10 yes 16 3.2 odd 2 inner
6048.2.h.h.2591.11 yes 16 12.11 even 2 inner