Properties

Label 6048.2.a.bf.1.2
Level 6048
Weight 2
Character 6048.1
Self dual yes
Analytic conductor 48.294
Analytic rank 1
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6048.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.2935231425\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.56155\)
Character \(\chi\) = 6048.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.56155 q^{5} +1.00000 q^{7} +O(q^{10})\) \(q+2.56155 q^{5} +1.00000 q^{7} -5.12311 q^{11} -2.43845 q^{13} +4.12311 q^{17} -3.12311 q^{19} -8.68466 q^{23} +1.56155 q^{25} +9.56155 q^{29} -1.56155 q^{31} +2.56155 q^{35} -0.561553 q^{37} -7.43845 q^{41} +7.24621 q^{43} -0.561553 q^{47} +1.00000 q^{49} -6.68466 q^{53} -13.1231 q^{55} +8.12311 q^{59} -8.24621 q^{61} -6.24621 q^{65} -3.31534 q^{67} -1.56155 q^{71} -7.12311 q^{73} -5.12311 q^{77} -8.56155 q^{79} -2.56155 q^{83} +10.5616 q^{85} -1.31534 q^{89} -2.43845 q^{91} -8.00000 q^{95} -1.75379 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{5} + 2q^{7} + O(q^{10}) \) \( 2q + q^{5} + 2q^{7} - 2q^{11} - 9q^{13} + 2q^{19} - 5q^{23} - q^{25} + 15q^{29} + q^{31} + q^{35} + 3q^{37} - 19q^{41} - 2q^{43} + 3q^{47} + 2q^{49} - q^{53} - 18q^{55} + 8q^{59} + 4q^{65} - 19q^{67} + q^{71} - 6q^{73} - 2q^{77} - 13q^{79} - q^{83} + 17q^{85} - 15q^{89} - 9q^{91} - 16q^{95} - 20q^{97} + O(q^{100}) \)

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
\(5\) 2.56155 1.14556 0.572781 0.819709i \(-0.305865\pi\)
0.572781 + 0.819709i \(0.305865\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −5.12311 −1.54467 −0.772337 0.635213i \(-0.780912\pi\)
−0.772337 + 0.635213i \(0.780912\pi\)
\(12\) 0 0
\(13\) −2.43845 −0.676304 −0.338152 0.941092i \(-0.609802\pi\)
−0.338152 + 0.941092i \(0.609802\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.12311 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(18\) 0 0
\(19\) −3.12311 −0.716490 −0.358245 0.933628i \(-0.616625\pi\)
−0.358245 + 0.933628i \(0.616625\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −8.68466 −1.81088 −0.905438 0.424478i \(-0.860458\pi\)
−0.905438 + 0.424478i \(0.860458\pi\)
\(24\) 0 0
\(25\) 1.56155 0.312311
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 9.56155 1.77554 0.887768 0.460291i \(-0.152255\pi\)
0.887768 + 0.460291i \(0.152255\pi\)
\(30\) 0 0
\(31\) −1.56155 −0.280463 −0.140232 0.990119i \(-0.544785\pi\)
−0.140232 + 0.990119i \(0.544785\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.56155 0.432981
\(36\) 0 0
\(37\) −0.561553 −0.0923187 −0.0461594 0.998934i \(-0.514698\pi\)
−0.0461594 + 0.998934i \(0.514698\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −7.43845 −1.16169 −0.580845 0.814014i \(-0.697278\pi\)
−0.580845 + 0.814014i \(0.697278\pi\)
\(42\) 0 0
\(43\) 7.24621 1.10504 0.552518 0.833501i \(-0.313667\pi\)
0.552518 + 0.833501i \(0.313667\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −0.561553 −0.0819109 −0.0409554 0.999161i \(-0.513040\pi\)
−0.0409554 + 0.999161i \(0.513040\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −6.68466 −0.918208 −0.459104 0.888382i \(-0.651830\pi\)
−0.459104 + 0.888382i \(0.651830\pi\)
\(54\) 0 0
\(55\) −13.1231 −1.76952
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 8.12311 1.05754 0.528769 0.848766i \(-0.322654\pi\)
0.528769 + 0.848766i \(0.322654\pi\)
\(60\) 0 0
\(61\) −8.24621 −1.05582 −0.527910 0.849301i \(-0.677024\pi\)
−0.527910 + 0.849301i \(0.677024\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −6.24621 −0.774747
\(66\) 0 0
\(67\) −3.31534 −0.405033 −0.202517 0.979279i \(-0.564912\pi\)
−0.202517 + 0.979279i \(0.564912\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −1.56155 −0.185322 −0.0926611 0.995698i \(-0.529537\pi\)
−0.0926611 + 0.995698i \(0.529537\pi\)
\(72\) 0 0
\(73\) −7.12311 −0.833696 −0.416848 0.908976i \(-0.636865\pi\)
−0.416848 + 0.908976i \(0.636865\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −5.12311 −0.583832
\(78\) 0 0
\(79\) −8.56155 −0.963250 −0.481625 0.876377i \(-0.659953\pi\)
−0.481625 + 0.876377i \(0.659953\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −2.56155 −0.281167 −0.140583 0.990069i \(-0.544898\pi\)
−0.140583 + 0.990069i \(0.544898\pi\)
\(84\) 0 0
\(85\) 10.5616 1.14556
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1.31534 −0.139426 −0.0697130 0.997567i \(-0.522208\pi\)
−0.0697130 + 0.997567i \(0.522208\pi\)
\(90\) 0 0
\(91\) −2.43845 −0.255619
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −8.00000 −0.820783
\(96\) 0 0
\(97\) −1.75379 −0.178070 −0.0890351 0.996028i \(-0.528378\pi\)
−0.0890351 + 0.996028i \(0.528378\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 0 0
\(103\) −6.43845 −0.634399 −0.317200 0.948359i \(-0.602742\pi\)
−0.317200 + 0.948359i \(0.602742\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −16.4924 −1.59438 −0.797191 0.603727i \(-0.793682\pi\)
−0.797191 + 0.603727i \(0.793682\pi\)
\(108\) 0 0
\(109\) −16.8078 −1.60989 −0.804946 0.593348i \(-0.797806\pi\)
−0.804946 + 0.593348i \(0.797806\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −5.12311 −0.481941 −0.240971 0.970532i \(-0.577466\pi\)
−0.240971 + 0.970532i \(0.577466\pi\)
\(114\) 0 0
\(115\) −22.2462 −2.07447
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 4.12311 0.377964
\(120\) 0 0
\(121\) 15.2462 1.38602
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −8.80776 −0.787790
\(126\) 0 0
\(127\) −18.5616 −1.64707 −0.823536 0.567264i \(-0.808002\pi\)
−0.823536 + 0.567264i \(0.808002\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −3.31534 −0.289663 −0.144831 0.989456i \(-0.546264\pi\)
−0.144831 + 0.989456i \(0.546264\pi\)
\(132\) 0 0
\(133\) −3.12311 −0.270808
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −7.12311 −0.608568 −0.304284 0.952581i \(-0.598417\pi\)
−0.304284 + 0.952581i \(0.598417\pi\)
\(138\) 0 0
\(139\) 18.4924 1.56851 0.784253 0.620441i \(-0.213046\pi\)
0.784253 + 0.620441i \(0.213046\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 12.4924 1.04467
\(144\) 0 0
\(145\) 24.4924 2.03398
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 12.9309 1.05934 0.529669 0.848204i \(-0.322316\pi\)
0.529669 + 0.848204i \(0.322316\pi\)
\(150\) 0 0
\(151\) 0.561553 0.0456985 0.0228493 0.999739i \(-0.492726\pi\)
0.0228493 + 0.999739i \(0.492726\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) 18.9309 1.51085 0.755424 0.655236i \(-0.227431\pi\)
0.755424 + 0.655236i \(0.227431\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −8.68466 −0.684447
\(162\) 0 0
\(163\) −14.3693 −1.12549 −0.562746 0.826630i \(-0.690255\pi\)
−0.562746 + 0.826630i \(0.690255\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0.315342 0.0244019 0.0122009 0.999926i \(-0.496116\pi\)
0.0122009 + 0.999926i \(0.496116\pi\)
\(168\) 0 0
\(169\) −7.05398 −0.542613
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 19.3693 1.47262 0.736311 0.676643i \(-0.236566\pi\)
0.736311 + 0.676643i \(0.236566\pi\)
\(174\) 0 0
\(175\) 1.56155 0.118042
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 14.8769 1.11195 0.555976 0.831199i \(-0.312345\pi\)
0.555976 + 0.831199i \(0.312345\pi\)
\(180\) 0 0
\(181\) −19.5616 −1.45400 −0.726999 0.686638i \(-0.759086\pi\)
−0.726999 + 0.686638i \(0.759086\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −1.43845 −0.105757
\(186\) 0 0
\(187\) −21.1231 −1.54467
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −9.12311 −0.660125 −0.330062 0.943959i \(-0.607070\pi\)
−0.330062 + 0.943959i \(0.607070\pi\)
\(192\) 0 0
\(193\) 22.6155 1.62790 0.813951 0.580934i \(-0.197313\pi\)
0.813951 + 0.580934i \(0.197313\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −15.1231 −1.07748 −0.538738 0.842473i \(-0.681099\pi\)
−0.538738 + 0.842473i \(0.681099\pi\)
\(198\) 0 0
\(199\) 6.68466 0.473863 0.236931 0.971526i \(-0.423858\pi\)
0.236931 + 0.971526i \(0.423858\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 9.56155 0.671089
\(204\) 0 0
\(205\) −19.0540 −1.33079
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 16.0000 1.10674
\(210\) 0 0
\(211\) −26.0540 −1.79363 −0.896815 0.442406i \(-0.854125\pi\)
−0.896815 + 0.442406i \(0.854125\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 18.5616 1.26589
\(216\) 0 0
\(217\) −1.56155 −0.106005
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −10.0540 −0.676304
\(222\) 0 0
\(223\) −10.2462 −0.686137 −0.343069 0.939310i \(-0.611466\pi\)
−0.343069 + 0.939310i \(0.611466\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 23.8078 1.58018 0.790088 0.612993i \(-0.210035\pi\)
0.790088 + 0.612993i \(0.210035\pi\)
\(228\) 0 0
\(229\) −26.4924 −1.75067 −0.875334 0.483518i \(-0.839359\pi\)
−0.875334 + 0.483518i \(0.839359\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −5.75379 −0.376943 −0.188472 0.982079i \(-0.560353\pi\)
−0.188472 + 0.982079i \(0.560353\pi\)
\(234\) 0 0
\(235\) −1.43845 −0.0938339
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 6.87689 0.444829 0.222415 0.974952i \(-0.428606\pi\)
0.222415 + 0.974952i \(0.428606\pi\)
\(240\) 0 0
\(241\) 0.876894 0.0564857 0.0282429 0.999601i \(-0.491009\pi\)
0.0282429 + 0.999601i \(0.491009\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 2.56155 0.163652
\(246\) 0 0
\(247\) 7.61553 0.484564
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −12.3153 −0.777337 −0.388669 0.921378i \(-0.627065\pi\)
−0.388669 + 0.921378i \(0.627065\pi\)
\(252\) 0 0
\(253\) 44.4924 2.79721
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2.00000 0.124757 0.0623783 0.998053i \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) 0 0
\(259\) −0.561553 −0.0348932
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 13.1771 0.812534 0.406267 0.913754i \(-0.366830\pi\)
0.406267 + 0.913754i \(0.366830\pi\)
\(264\) 0 0
\(265\) −17.1231 −1.05186
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 21.3002 1.29870 0.649348 0.760492i \(-0.275042\pi\)
0.649348 + 0.760492i \(0.275042\pi\)
\(270\) 0 0
\(271\) 20.0540 1.21819 0.609096 0.793096i \(-0.291532\pi\)
0.609096 + 0.793096i \(0.291532\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −8.00000 −0.482418
\(276\) 0 0
\(277\) 2.56155 0.153909 0.0769544 0.997035i \(-0.475480\pi\)
0.0769544 + 0.997035i \(0.475480\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −29.3693 −1.75203 −0.876013 0.482287i \(-0.839806\pi\)
−0.876013 + 0.482287i \(0.839806\pi\)
\(282\) 0 0
\(283\) −21.6155 −1.28491 −0.642455 0.766324i \(-0.722084\pi\)
−0.642455 + 0.766324i \(0.722084\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −7.43845 −0.439078
\(288\) 0 0
\(289\) 0 0
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 24.5616 1.43490 0.717451 0.696609i \(-0.245309\pi\)
0.717451 + 0.696609i \(0.245309\pi\)
\(294\) 0 0
\(295\) 20.8078 1.21147
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 21.1771 1.22470
\(300\) 0 0
\(301\) 7.24621 0.417665
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −21.1231 −1.20951
\(306\) 0 0
\(307\) −12.8769 −0.734923 −0.367462 0.930039i \(-0.619773\pi\)
−0.367462 + 0.930039i \(0.619773\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −15.6847 −0.889395 −0.444698 0.895681i \(-0.646689\pi\)
−0.444698 + 0.895681i \(0.646689\pi\)
\(312\) 0 0
\(313\) −24.2462 −1.37048 −0.685238 0.728319i \(-0.740302\pi\)
−0.685238 + 0.728319i \(0.740302\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 31.1231 1.74805 0.874024 0.485883i \(-0.161502\pi\)
0.874024 + 0.485883i \(0.161502\pi\)
\(318\) 0 0
\(319\) −48.9848 −2.74262
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −12.8769 −0.716490
\(324\) 0 0
\(325\) −3.80776 −0.211217
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −0.561553 −0.0309594
\(330\) 0 0
\(331\) −22.1231 −1.21600 −0.607998 0.793939i \(-0.708027\pi\)
−0.607998 + 0.793939i \(0.708027\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −8.49242 −0.463991
\(336\) 0 0
\(337\) 0.753789 0.0410615 0.0205307 0.999789i \(-0.493464\pi\)
0.0205307 + 0.999789i \(0.493464\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 8.00000 0.433224
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 29.8617 1.60306 0.801531 0.597953i \(-0.204019\pi\)
0.801531 + 0.597953i \(0.204019\pi\)
\(348\) 0 0
\(349\) 29.1771 1.56181 0.780907 0.624648i \(-0.214757\pi\)
0.780907 + 0.624648i \(0.214757\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 33.9848 1.80883 0.904415 0.426653i \(-0.140307\pi\)
0.904415 + 0.426653i \(0.140307\pi\)
\(354\) 0 0
\(355\) −4.00000 −0.212298
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 25.8078 1.36208 0.681041 0.732245i \(-0.261528\pi\)
0.681041 + 0.732245i \(0.261528\pi\)
\(360\) 0 0
\(361\) −9.24621 −0.486643
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −18.2462 −0.955050
\(366\) 0 0
\(367\) 0.0539753 0.00281749 0.00140874 0.999999i \(-0.499552\pi\)
0.00140874 + 0.999999i \(0.499552\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −6.68466 −0.347050
\(372\) 0 0
\(373\) 2.80776 0.145381 0.0726903 0.997355i \(-0.476842\pi\)
0.0726903 + 0.997355i \(0.476842\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −23.3153 −1.20080
\(378\) 0 0
\(379\) −4.31534 −0.221664 −0.110832 0.993839i \(-0.535352\pi\)
−0.110832 + 0.993839i \(0.535352\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 19.0540 0.973613 0.486806 0.873510i \(-0.338162\pi\)
0.486806 + 0.873510i \(0.338162\pi\)
\(384\) 0 0
\(385\) −13.1231 −0.668815
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 32.2462 1.63495 0.817474 0.575966i \(-0.195374\pi\)
0.817474 + 0.575966i \(0.195374\pi\)
\(390\) 0 0
\(391\) −35.8078 −1.81088
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −21.9309 −1.10346
\(396\) 0 0
\(397\) 7.61553 0.382212 0.191106 0.981569i \(-0.438793\pi\)
0.191106 + 0.981569i \(0.438793\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 14.4924 0.723717 0.361859 0.932233i \(-0.382142\pi\)
0.361859 + 0.932233i \(0.382142\pi\)
\(402\) 0 0
\(403\) 3.80776 0.189678
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 2.87689 0.142602
\(408\) 0 0
\(409\) −5.36932 −0.265496 −0.132748 0.991150i \(-0.542380\pi\)
−0.132748 + 0.991150i \(0.542380\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 8.12311 0.399712
\(414\) 0 0
\(415\) −6.56155 −0.322094
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 4.75379 0.232238 0.116119 0.993235i \(-0.462955\pi\)
0.116119 + 0.993235i \(0.462955\pi\)
\(420\) 0 0
\(421\) −2.00000 −0.0974740 −0.0487370 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 6.43845 0.312311
\(426\) 0 0
\(427\) −8.24621 −0.399062
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 16.0000 0.770693 0.385346 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) 0 0
\(433\) −38.1080 −1.83135 −0.915676 0.401918i \(-0.868344\pi\)
−0.915676 + 0.401918i \(0.868344\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 27.1231 1.29747
\(438\) 0 0
\(439\) −16.0540 −0.766214 −0.383107 0.923704i \(-0.625146\pi\)
−0.383107 + 0.923704i \(0.625146\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0.876894 0.0416625 0.0208313 0.999783i \(-0.493369\pi\)
0.0208313 + 0.999783i \(0.493369\pi\)
\(444\) 0 0
\(445\) −3.36932 −0.159721
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −32.4924 −1.53341 −0.766706 0.641998i \(-0.778106\pi\)
−0.766706 + 0.641998i \(0.778106\pi\)
\(450\) 0 0
\(451\) 38.1080 1.79443
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −6.24621 −0.292827
\(456\) 0 0
\(457\) −29.8078 −1.39435 −0.697174 0.716902i \(-0.745560\pi\)
−0.697174 + 0.716902i \(0.745560\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −32.4233 −1.51010 −0.755052 0.655665i \(-0.772388\pi\)
−0.755052 + 0.655665i \(0.772388\pi\)
\(462\) 0 0
\(463\) −11.4384 −0.531590 −0.265795 0.964030i \(-0.585634\pi\)
−0.265795 + 0.964030i \(0.585634\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −2.24621 −0.103942 −0.0519711 0.998649i \(-0.516550\pi\)
−0.0519711 + 0.998649i \(0.516550\pi\)
\(468\) 0 0
\(469\) −3.31534 −0.153088
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −37.1231 −1.70692
\(474\) 0 0
\(475\) −4.87689 −0.223767
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −19.0540 −0.870598 −0.435299 0.900286i \(-0.643357\pi\)
−0.435299 + 0.900286i \(0.643357\pi\)
\(480\) 0 0
\(481\) 1.36932 0.0624355
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −4.49242 −0.203990
\(486\) 0 0
\(487\) −6.63068 −0.300465 −0.150232 0.988651i \(-0.548002\pi\)
−0.150232 + 0.988651i \(0.548002\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 1.75379 0.0791474 0.0395737 0.999217i \(-0.487400\pi\)
0.0395737 + 0.999217i \(0.487400\pi\)
\(492\) 0 0
\(493\) 39.4233 1.77554
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −1.56155 −0.0700452
\(498\) 0 0
\(499\) 25.9309 1.16083 0.580413 0.814323i \(-0.302891\pi\)
0.580413 + 0.814323i \(0.302891\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −7.19224 −0.320686 −0.160343 0.987061i \(-0.551260\pi\)
−0.160343 + 0.987061i \(0.551260\pi\)
\(504\) 0 0
\(505\) 25.6155 1.13988
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −17.0540 −0.755904 −0.377952 0.925825i \(-0.623372\pi\)
−0.377952 + 0.925825i \(0.623372\pi\)
\(510\) 0 0
\(511\) −7.12311 −0.315108
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −16.4924 −0.726743
\(516\) 0 0
\(517\) 2.87689 0.126526
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −40.3693 −1.76861 −0.884306 0.466908i \(-0.845368\pi\)
−0.884306 + 0.466908i \(0.845368\pi\)
\(522\) 0 0
\(523\) −4.63068 −0.202486 −0.101243 0.994862i \(-0.532282\pi\)
−0.101243 + 0.994862i \(0.532282\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −6.43845 −0.280463
\(528\) 0 0
\(529\) 52.4233 2.27927
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 18.1383 0.785655
\(534\) 0 0
\(535\) −42.2462 −1.82646
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −5.12311 −0.220668
\(540\) 0 0
\(541\) −11.1922 −0.481192 −0.240596 0.970625i \(-0.577343\pi\)
−0.240596 + 0.970625i \(0.577343\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −43.0540 −1.84423
\(546\) 0 0
\(547\) −31.6847 −1.35474 −0.677369 0.735643i \(-0.736880\pi\)
−0.677369 + 0.735643i \(0.736880\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −29.8617 −1.27215
\(552\) 0 0
\(553\) −8.56155 −0.364074
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −12.1922 −0.516602 −0.258301 0.966065i \(-0.583163\pi\)
−0.258301 + 0.966065i \(0.583163\pi\)
\(558\) 0 0
\(559\) −17.6695 −0.747340
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −10.9309 −0.460681 −0.230341 0.973110i \(-0.573984\pi\)
−0.230341 + 0.973110i \(0.573984\pi\)
\(564\) 0 0
\(565\) −13.1231 −0.552093
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 10.2462 0.429544 0.214772 0.976664i \(-0.431099\pi\)
0.214772 + 0.976664i \(0.431099\pi\)
\(570\) 0 0
\(571\) 36.1231 1.51170 0.755852 0.654742i \(-0.227223\pi\)
0.755852 + 0.654742i \(0.227223\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −13.5616 −0.565556
\(576\) 0 0
\(577\) 6.00000 0.249783 0.124892 0.992170i \(-0.460142\pi\)
0.124892 + 0.992170i \(0.460142\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −2.56155 −0.106271
\(582\) 0 0
\(583\) 34.2462 1.41833
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −28.6847 −1.18394 −0.591971 0.805959i \(-0.701650\pi\)
−0.591971 + 0.805959i \(0.701650\pi\)
\(588\) 0 0
\(589\) 4.87689 0.200949
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −41.0540 −1.68588 −0.842942 0.538004i \(-0.819179\pi\)
−0.842942 + 0.538004i \(0.819179\pi\)
\(594\) 0 0
\(595\) 10.5616 0.432981
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −36.9309 −1.50895 −0.754477 0.656326i \(-0.772110\pi\)
−0.754477 + 0.656326i \(0.772110\pi\)
\(600\) 0 0
\(601\) 16.4924 0.672740 0.336370 0.941730i \(-0.390801\pi\)
0.336370 + 0.941730i \(0.390801\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 39.0540 1.58777
\(606\) 0 0
\(607\) −4.43845 −0.180151 −0.0900755 0.995935i \(-0.528711\pi\)
−0.0900755 + 0.995935i \(0.528711\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1.36932 0.0553966
\(612\) 0 0
\(613\) −8.73863 −0.352950 −0.176475 0.984305i \(-0.556469\pi\)
−0.176475 + 0.984305i \(0.556469\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −7.12311 −0.286765 −0.143383 0.989667i \(-0.545798\pi\)
−0.143383 + 0.989667i \(0.545798\pi\)
\(618\) 0 0
\(619\) 31.6155 1.27074 0.635368 0.772210i \(-0.280849\pi\)
0.635368 + 0.772210i \(0.280849\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −1.31534 −0.0526980
\(624\) 0 0
\(625\) −30.3693 −1.21477
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −2.31534 −0.0923187
\(630\) 0 0
\(631\) 27.1922 1.08251 0.541253 0.840860i \(-0.317950\pi\)
0.541253 + 0.840860i \(0.317950\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −47.5464 −1.88682
\(636\) 0 0
\(637\) −2.43845 −0.0966148
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 16.6307 0.656872 0.328436 0.944526i \(-0.393478\pi\)
0.328436 + 0.944526i \(0.393478\pi\)
\(642\) 0 0
\(643\) 35.3693 1.39483 0.697415 0.716668i \(-0.254334\pi\)
0.697415 + 0.716668i \(0.254334\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −12.8769 −0.506243 −0.253121 0.967435i \(-0.581457\pi\)
−0.253121 + 0.967435i \(0.581457\pi\)
\(648\) 0 0
\(649\) −41.6155 −1.63355
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 4.05398 0.158644 0.0793222 0.996849i \(-0.474724\pi\)
0.0793222 + 0.996849i \(0.474724\pi\)
\(654\) 0 0
\(655\) −8.49242 −0.331826
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −22.8769 −0.891157 −0.445579 0.895243i \(-0.647002\pi\)
−0.445579 + 0.895243i \(0.647002\pi\)
\(660\) 0 0
\(661\) −2.49242 −0.0969440 −0.0484720 0.998825i \(-0.515435\pi\)
−0.0484720 + 0.998825i \(0.515435\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −8.00000 −0.310227
\(666\) 0 0
\(667\) −83.0388 −3.21528
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 42.2462 1.63090
\(672\) 0 0
\(673\) 30.1922 1.16383 0.581913 0.813251i \(-0.302305\pi\)
0.581913 + 0.813251i \(0.302305\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 30.8769 1.18670 0.593348 0.804946i \(-0.297806\pi\)
0.593348 + 0.804946i \(0.297806\pi\)
\(678\) 0 0
\(679\) −1.75379 −0.0673042
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −4.49242 −0.171898 −0.0859489 0.996300i \(-0.527392\pi\)
−0.0859489 + 0.996300i \(0.527392\pi\)
\(684\) 0 0
\(685\) −18.2462 −0.697152
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 16.3002 0.620988
\(690\) 0 0
\(691\) −5.61553 −0.213625 −0.106812 0.994279i \(-0.534064\pi\)
−0.106812 + 0.994279i \(0.534064\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 47.3693 1.79682
\(696\) 0 0
\(697\) −30.6695 −1.16169
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 10.0000 0.377695 0.188847 0.982006i \(-0.439525\pi\)
0.188847 + 0.982006i \(0.439525\pi\)
\(702\) 0 0
\(703\) 1.75379 0.0661454
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 10.0000 0.376089
\(708\) 0 0
\(709\) 9.93087 0.372962 0.186481 0.982459i \(-0.440292\pi\)
0.186481 + 0.982459i \(0.440292\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 13.5616 0.507884
\(714\) 0 0
\(715\) 32.0000 1.19673
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −44.1771 −1.64753 −0.823764 0.566934i \(-0.808130\pi\)
−0.823764 + 0.566934i \(0.808130\pi\)
\(720\) 0 0
\(721\) −6.43845 −0.239780
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 14.9309 0.554519
\(726\) 0 0
\(727\) 10.4384 0.387141 0.193570 0.981086i \(-0.437993\pi\)
0.193570 + 0.981086i \(0.437993\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 29.8769 1.10504
\(732\) 0 0
\(733\) −10.1922 −0.376459 −0.188229 0.982125i \(-0.560275\pi\)
−0.188229 + 0.982125i \(0.560275\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 16.9848 0.625645
\(738\) 0 0
\(739\) 4.00000 0.147142 0.0735712 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 29.5616 1.08451 0.542254 0.840215i \(-0.317571\pi\)
0.542254 + 0.840215i \(0.317571\pi\)
\(744\) 0 0
\(745\) 33.1231 1.21354
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −16.4924 −0.602620
\(750\) 0 0
\(751\) 52.1080 1.90145 0.950723 0.310041i \(-0.100343\pi\)
0.950723 + 0.310041i \(0.100343\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 1.43845 0.0523505
\(756\) 0 0
\(757\) 23.1922 0.842936 0.421468 0.906843i \(-0.361515\pi\)
0.421468 + 0.906843i \(0.361515\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −29.8769 −1.08304 −0.541518 0.840689i \(-0.682150\pi\)
−0.541518 + 0.840689i \(0.682150\pi\)
\(762\) 0 0
\(763\) −16.8078 −0.608482
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −19.8078 −0.715217
\(768\) 0 0
\(769\) −39.8617 −1.43745 −0.718726 0.695294i \(-0.755274\pi\)
−0.718726 + 0.695294i \(0.755274\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 9.43845 0.339477 0.169739 0.985489i \(-0.445708\pi\)
0.169739 + 0.985489i \(0.445708\pi\)
\(774\) 0 0
\(775\) −2.43845 −0.0875916
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 23.2311 0.832339
\(780\) 0 0
\(781\) 8.00000 0.286263
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 48.4924 1.73077
\(786\) 0 0
\(787\) 29.3693 1.04690 0.523452 0.852055i \(-0.324644\pi\)
0.523452 + 0.852055i \(0.324644\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −5.12311 −0.182157
\(792\) 0 0
\(793\) 20.1080 0.714054
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 22.9848 0.814165 0.407082 0.913391i \(-0.366546\pi\)
0.407082 + 0.913391i \(0.366546\pi\)
\(798\) 0 0
\(799\) −2.31534 −0.0819109
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 36.4924 1.28779
\(804\) 0 0
\(805\) −22.2462 −0.784076
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 19.3693 0.680989 0.340494 0.940247i \(-0.389406\pi\)
0.340494 + 0.940247i \(0.389406\pi\)
\(810\) 0 0
\(811\) 34.4924 1.21119 0.605596 0.795772i \(-0.292935\pi\)
0.605596 + 0.795772i \(0.292935\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −36.8078 −1.28932
\(816\) 0 0
\(817\) −22.6307 −0.791747
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 38.9309 1.35870 0.679348 0.733816i \(-0.262263\pi\)
0.679348 + 0.733816i \(0.262263\pi\)
\(822\) 0 0
\(823\) 4.56155 0.159006 0.0795029 0.996835i \(-0.474667\pi\)
0.0795029 + 0.996835i \(0.474667\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 34.4924 1.19942 0.599710 0.800218i \(-0.295283\pi\)
0.599710 + 0.800218i \(0.295283\pi\)
\(828\) 0 0
\(829\) 43.1231 1.49773 0.748864 0.662724i \(-0.230600\pi\)
0.748864 + 0.662724i \(0.230600\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 4.12311 0.142857
\(834\) 0 0
\(835\) 0.807764 0.0279538
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 44.9157 1.55066 0.775331 0.631555i \(-0.217583\pi\)
0.775331 + 0.631555i \(0.217583\pi\)
\(840\) 0 0
\(841\) 62.4233 2.15253
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −18.0691 −0.621597
\(846\) 0 0
\(847\) 15.2462 0.523866
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 4.87689 0.167178
\(852\) 0 0
\(853\) 36.0540 1.23446 0.617232 0.786781i \(-0.288254\pi\)
0.617232 + 0.786781i \(0.288254\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 42.1231 1.43890 0.719449 0.694545i \(-0.244394\pi\)
0.719449 + 0.694545i \(0.244394\pi\)
\(858\) 0 0
\(859\) 13.5076 0.460873 0.230436 0.973087i \(-0.425985\pi\)
0.230436 + 0.973087i \(0.425985\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 29.8078 1.01467 0.507334 0.861749i \(-0.330631\pi\)
0.507334 + 0.861749i \(0.330631\pi\)
\(864\) 0 0
\(865\) 49.6155 1.68698
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 43.8617 1.48791
\(870\) 0 0
\(871\) 8.08429 0.273926
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −8.80776 −0.297757
\(876\) 0 0
\(877\) −31.6847 −1.06991 −0.534957 0.844879i \(-0.679672\pi\)
−0.534957 + 0.844879i \(0.679672\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −33.4233 −1.12606 −0.563030 0.826437i \(-0.690364\pi\)
−0.563030 + 0.826437i \(0.690364\pi\)
\(882\) 0 0
\(883\) −48.2311 −1.62310 −0.811552 0.584280i \(-0.801377\pi\)
−0.811552 + 0.584280i \(0.801377\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −21.4384 −0.719833 −0.359916 0.932985i \(-0.617195\pi\)
−0.359916 + 0.932985i \(0.617195\pi\)
\(888\) 0 0
\(889\) −18.5616 −0.622535
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 1.75379 0.0586883
\(894\) 0 0
\(895\) 38.1080 1.27381
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −14.9309 −0.497972
\(900\) 0 0
\(901\) −27.5616 −0.918208
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −50.1080 −1.66564
\(906\) 0 0
\(907\) −26.5616 −0.881962 −0.440981 0.897516i \(-0.645369\pi\)
−0.440981 + 0.897516i \(0.645369\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −0.630683 −0.0208955 −0.0104477 0.999945i \(-0.503326\pi\)
−0.0104477 + 0.999945i \(0.503326\pi\)
\(912\) 0 0
\(913\) 13.1231 0.434311
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −3.31534 −0.109482
\(918\) 0 0
\(919\) −30.1771 −0.995450 −0.497725 0.867335i \(-0.665831\pi\)
−0.497725 + 0.867335i \(0.665831\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 3.80776 0.125334
\(924\) 0 0
\(925\) −0.876894 −0.0288321
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 4.56155 0.149660 0.0748298 0.997196i \(-0.476159\pi\)
0.0748298 + 0.997196i \(0.476159\pi\)
\(930\) 0 0
\(931\) −3.12311 −0.102356
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −54.1080 −1.76952
\(936\) 0 0
\(937\) 9.75379 0.318642 0.159321 0.987227i \(-0.449069\pi\)
0.159321 + 0.987227i \(0.449069\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −30.8078 −1.00430 −0.502152 0.864779i \(-0.667458\pi\)
−0.502152 + 0.864779i \(0.667458\pi\)
\(942\) 0 0
\(943\) 64.6004 2.10368
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −14.6307 −0.475433 −0.237717 0.971335i \(-0.576399\pi\)
−0.237717 + 0.971335i \(0.576399\pi\)
\(948\) 0 0
\(949\) 17.3693 0.563832
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −23.5076 −0.761485 −0.380743 0.924681i \(-0.624332\pi\)
−0.380743 + 0.924681i \(0.624332\pi\)
\(954\) 0 0
\(955\) −23.3693 −0.756213
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −7.12311 −0.230017
\(960\) 0 0
\(961\) −28.5616 −0.921340
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 57.9309 1.86486
\(966\) 0 0
\(967\) 8.98485 0.288933 0.144467 0.989510i \(-0.453853\pi\)
0.144467 + 0.989510i \(0.453853\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 30.1231 0.966696 0.483348 0.875428i \(-0.339421\pi\)
0.483348 + 0.875428i \(0.339421\pi\)
\(972\) 0 0
\(973\) 18.4924 0.592840
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 22.0000 0.703842 0.351921 0.936030i \(-0.385529\pi\)
0.351921 + 0.936030i \(0.385529\pi\)
\(978\) 0 0
\(979\) 6.73863 0.215368
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 12.8078 0.408504 0.204252 0.978918i \(-0.434524\pi\)
0.204252 + 0.978918i \(0.434524\pi\)
\(984\) 0 0
\(985\) −38.7386 −1.23432
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −62.9309 −2.00109
\(990\) 0 0
\(991\) 56.4233 1.79234 0.896172 0.443706i \(-0.146337\pi\)
0.896172 + 0.443706i \(0.146337\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 17.1231 0.542839
\(996\) 0 0
\(997\) −45.6695 −1.44637 −0.723184 0.690656i \(-0.757322\pi\)
−0.723184 + 0.690656i \(0.757322\pi\)
\(998\) 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.a.bf.1.2 yes 2
3.2 odd 2 6048.2.a.bd.1.1 yes 2
4.3 odd 2 6048.2.a.be.1.2 yes 2
12.11 even 2 6048.2.a.bc.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6048.2.a.bc.1.1 2 12.11 even 2
6048.2.a.bd.1.1 yes 2 3.2 odd 2
6048.2.a.be.1.2 yes 2 4.3 odd 2
6048.2.a.bf.1.2 yes 2 1.1 even 1 trivial