Properties

Label 6048.2.a.bc.1.2
Level 6048
Weight 2
Character 6048.1
Self dual yes
Analytic conductor 48.294
Analytic rank 0
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: \(0\)
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 \(-1.56155\)
Character \(\chi\) = 6048.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.56155 q^{5} -1.00000 q^{7} +O(q^{10})\) \(q+1.56155 q^{5} -1.00000 q^{7} +3.12311 q^{11} -6.56155 q^{13} +4.12311 q^{17} -5.12311 q^{19} +3.68466 q^{23} -2.56155 q^{25} -5.43845 q^{29} -2.56155 q^{31} -1.56155 q^{35} +3.56155 q^{37} +11.5616 q^{41} +9.24621 q^{43} +3.56155 q^{47} +1.00000 q^{49} -5.68466 q^{53} +4.87689 q^{55} -0.123106 q^{59} +8.24621 q^{61} -10.2462 q^{65} +15.6847 q^{67} +2.56155 q^{71} +1.12311 q^{73} -3.12311 q^{77} +4.43845 q^{79} +1.56155 q^{83} +6.43845 q^{85} +13.6847 q^{89} +6.56155 q^{91} -8.00000 q^{95} -18.2462 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{5} - 2q^{7} + O(q^{10}) \) \( 2q - q^{5} - 2q^{7} - 2q^{11} - 9q^{13} - 2q^{19} - 5q^{23} - q^{25} - 15q^{29} - q^{31} + q^{35} + 3q^{37} + 19q^{41} + 2q^{43} + 3q^{47} + 2q^{49} + q^{53} + 18q^{55} + 8q^{59} - 4q^{65} + 19q^{67} + q^{71} - 6q^{73} + 2q^{77} + 13q^{79} - q^{83} + 17q^{85} + 15q^{89} + 9q^{91} - 16q^{95} - 20q^{97} + O(q^{100}) \)

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
\(5\) 1.56155 0.698348 0.349174 0.937058i \(-0.386462\pi\)
0.349174 + 0.937058i \(0.386462\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.12311 0.941652 0.470826 0.882226i \(-0.343956\pi\)
0.470826 + 0.882226i \(0.343956\pi\)
\(12\) 0 0
\(13\) −6.56155 −1.81985 −0.909924 0.414776i \(-0.863860\pi\)
−0.909924 + 0.414776i \(0.863860\pi\)
\(14\) 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
\(19\) −5.12311 −1.17532 −0.587661 0.809108i \(-0.699951\pi\)
−0.587661 + 0.809108i \(0.699951\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.68466 0.768304 0.384152 0.923270i \(-0.374494\pi\)
0.384152 + 0.923270i \(0.374494\pi\)
\(24\) 0 0
\(25\) −2.56155 −0.512311
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −5.43845 −1.00989 −0.504947 0.863150i \(-0.668488\pi\)
−0.504947 + 0.863150i \(0.668488\pi\)
\(30\) 0 0
\(31\) −2.56155 −0.460068 −0.230034 0.973183i \(-0.573884\pi\)
−0.230034 + 0.973183i \(0.573884\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.56155 −0.263951
\(36\) 0 0
\(37\) 3.56155 0.585516 0.292758 0.956187i \(-0.405427\pi\)
0.292758 + 0.956187i \(0.405427\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 11.5616 1.80561 0.902806 0.430049i \(-0.141504\pi\)
0.902806 + 0.430049i \(0.141504\pi\)
\(42\) 0 0
\(43\) 9.24621 1.41003 0.705017 0.709190i \(-0.250939\pi\)
0.705017 + 0.709190i \(0.250939\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.56155 0.519506 0.259753 0.965675i \(-0.416359\pi\)
0.259753 + 0.965675i \(0.416359\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −5.68466 −0.780848 −0.390424 0.920635i \(-0.627672\pi\)
−0.390424 + 0.920635i \(0.627672\pi\)
\(54\) 0 0
\(55\) 4.87689 0.657600
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −0.123106 −0.0160270 −0.00801349 0.999968i \(-0.502551\pi\)
−0.00801349 + 0.999968i \(0.502551\pi\)
\(60\) 0 0
\(61\) 8.24621 1.05582 0.527910 0.849301i \(-0.322976\pi\)
0.527910 + 0.849301i \(0.322976\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −10.2462 −1.27089
\(66\) 0 0
\(67\) 15.6847 1.91619 0.958093 0.286457i \(-0.0924777\pi\)
0.958093 + 0.286457i \(0.0924777\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.56155 0.304000 0.152000 0.988380i \(-0.451429\pi\)
0.152000 + 0.988380i \(0.451429\pi\)
\(72\) 0 0
\(73\) 1.12311 0.131450 0.0657248 0.997838i \(-0.479064\pi\)
0.0657248 + 0.997838i \(0.479064\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −3.12311 −0.355911
\(78\) 0 0
\(79\) 4.43845 0.499364 0.249682 0.968328i \(-0.419674\pi\)
0.249682 + 0.968328i \(0.419674\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1.56155 0.171403 0.0857013 0.996321i \(-0.472687\pi\)
0.0857013 + 0.996321i \(0.472687\pi\)
\(84\) 0 0
\(85\) 6.43845 0.698348
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 13.6847 1.45057 0.725285 0.688448i \(-0.241708\pi\)
0.725285 + 0.688448i \(0.241708\pi\)
\(90\) 0 0
\(91\) 6.56155 0.687838
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −8.00000 −0.820783
\(96\) 0 0
\(97\) −18.2462 −1.85262 −0.926311 0.376760i \(-0.877038\pi\)
−0.926311 + 0.376760i \(0.877038\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 0 0
\(103\) 10.5616 1.04066 0.520330 0.853965i \(-0.325809\pi\)
0.520330 + 0.853965i \(0.325809\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 16.4924 1.59438 0.797191 0.603727i \(-0.206318\pi\)
0.797191 + 0.603727i \(0.206318\pi\)
\(108\) 0 0
\(109\) 3.80776 0.364718 0.182359 0.983232i \(-0.441627\pi\)
0.182359 + 0.983232i \(0.441627\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −3.12311 −0.293797 −0.146899 0.989152i \(-0.546929\pi\)
−0.146899 + 0.989152i \(0.546929\pi\)
\(114\) 0 0
\(115\) 5.75379 0.536544
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −4.12311 −0.377964
\(120\) 0 0
\(121\) −1.24621 −0.113292
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −11.8078 −1.05612
\(126\) 0 0
\(127\) 14.4384 1.28121 0.640603 0.767873i \(-0.278685\pi\)
0.640603 + 0.767873i \(0.278685\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −15.6847 −1.37037 −0.685187 0.728367i \(-0.740280\pi\)
−0.685187 + 0.728367i \(0.740280\pi\)
\(132\) 0 0
\(133\) 5.12311 0.444230
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1.12311 −0.0959534 −0.0479767 0.998848i \(-0.515277\pi\)
−0.0479767 + 0.998848i \(0.515277\pi\)
\(138\) 0 0
\(139\) 14.4924 1.22923 0.614616 0.788827i \(-0.289311\pi\)
0.614616 + 0.788827i \(0.289311\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −20.4924 −1.71366
\(144\) 0 0
\(145\) −8.49242 −0.705257
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 15.9309 1.30511 0.652554 0.757742i \(-0.273698\pi\)
0.652554 + 0.757742i \(0.273698\pi\)
\(150\) 0 0
\(151\) 3.56155 0.289835 0.144918 0.989444i \(-0.453708\pi\)
0.144918 + 0.989444i \(0.453708\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) −9.93087 −0.792570 −0.396285 0.918128i \(-0.629701\pi\)
−0.396285 + 0.918128i \(0.629701\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −3.68466 −0.290392
\(162\) 0 0
\(163\) −10.3693 −0.812188 −0.406094 0.913831i \(-0.633109\pi\)
−0.406094 + 0.913831i \(0.633109\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 12.6847 0.981568 0.490784 0.871281i \(-0.336710\pi\)
0.490784 + 0.871281i \(0.336710\pi\)
\(168\) 0 0
\(169\) 30.0540 2.31184
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 5.36932 0.408222 0.204111 0.978948i \(-0.434570\pi\)
0.204111 + 0.978948i \(0.434570\pi\)
\(174\) 0 0
\(175\) 2.56155 0.193635
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 23.1231 1.72830 0.864151 0.503233i \(-0.167856\pi\)
0.864151 + 0.503233i \(0.167856\pi\)
\(180\) 0 0
\(181\) −15.4384 −1.14753 −0.573765 0.819020i \(-0.694518\pi\)
−0.573765 + 0.819020i \(0.694518\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 5.56155 0.408893
\(186\) 0 0
\(187\) 12.8769 0.941652
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −0.876894 −0.0634499 −0.0317249 0.999497i \(-0.510100\pi\)
−0.0317249 + 0.999497i \(0.510100\pi\)
\(192\) 0 0
\(193\) −18.6155 −1.33998 −0.669988 0.742372i \(-0.733701\pi\)
−0.669988 + 0.742372i \(0.733701\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 6.87689 0.489958 0.244979 0.969528i \(-0.421219\pi\)
0.244979 + 0.969528i \(0.421219\pi\)
\(198\) 0 0
\(199\) 5.68466 0.402975 0.201487 0.979491i \(-0.435422\pi\)
0.201487 + 0.979491i \(0.435422\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 5.43845 0.381704
\(204\) 0 0
\(205\) 18.0540 1.26094
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −16.0000 −1.10674
\(210\) 0 0
\(211\) −11.0540 −0.760987 −0.380494 0.924784i \(-0.624246\pi\)
−0.380494 + 0.924784i \(0.624246\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 14.4384 0.984694
\(216\) 0 0
\(217\) 2.56155 0.173890
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −27.0540 −1.81985
\(222\) 0 0
\(223\) −6.24621 −0.418277 −0.209139 0.977886i \(-0.567066\pi\)
−0.209139 + 0.977886i \(0.567066\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 3.19224 0.211876 0.105938 0.994373i \(-0.466215\pi\)
0.105938 + 0.994373i \(0.466215\pi\)
\(228\) 0 0
\(229\) 6.49242 0.429031 0.214516 0.976721i \(-0.431183\pi\)
0.214516 + 0.976721i \(0.431183\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 22.2462 1.45740 0.728699 0.684834i \(-0.240125\pi\)
0.728699 + 0.684834i \(0.240125\pi\)
\(234\) 0 0
\(235\) 5.56155 0.362796
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 15.1231 0.978232 0.489116 0.872219i \(-0.337319\pi\)
0.489116 + 0.872219i \(0.337319\pi\)
\(240\) 0 0
\(241\) 9.12311 0.587671 0.293835 0.955856i \(-0.405068\pi\)
0.293835 + 0.955856i \(0.405068\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1.56155 0.0997639
\(246\) 0 0
\(247\) 33.6155 2.13890
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −24.6847 −1.55808 −0.779041 0.626973i \(-0.784294\pi\)
−0.779041 + 0.626973i \(0.784294\pi\)
\(252\) 0 0
\(253\) 11.5076 0.723475
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −2.00000 −0.124757 −0.0623783 0.998053i \(-0.519869\pi\)
−0.0623783 + 0.998053i \(0.519869\pi\)
\(258\) 0 0
\(259\) −3.56155 −0.221304
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −32.1771 −1.98412 −0.992062 0.125750i \(-0.959866\pi\)
−0.992062 + 0.125750i \(0.959866\pi\)
\(264\) 0 0
\(265\) −8.87689 −0.545303
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 32.3002 1.96938 0.984689 0.174323i \(-0.0557736\pi\)
0.984689 + 0.174323i \(0.0557736\pi\)
\(270\) 0 0
\(271\) 17.0540 1.03596 0.517978 0.855394i \(-0.326685\pi\)
0.517978 + 0.855394i \(0.326685\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −8.00000 −0.482418
\(276\) 0 0
\(277\) −1.56155 −0.0938246 −0.0469123 0.998899i \(-0.514938\pi\)
−0.0469123 + 0.998899i \(0.514938\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 4.63068 0.276243 0.138122 0.990415i \(-0.455893\pi\)
0.138122 + 0.990415i \(0.455893\pi\)
\(282\) 0 0
\(283\) −19.6155 −1.16602 −0.583011 0.812464i \(-0.698126\pi\)
−0.583011 + 0.812464i \(0.698126\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −11.5616 −0.682457
\(288\) 0 0
\(289\) 0 0
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −20.4384 −1.19403 −0.597013 0.802231i \(-0.703646\pi\)
−0.597013 + 0.802231i \(0.703646\pi\)
\(294\) 0 0
\(295\) −0.192236 −0.0111924
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −24.1771 −1.39820
\(300\) 0 0
\(301\) −9.24621 −0.532943
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 12.8769 0.737329
\(306\) 0 0
\(307\) 21.1231 1.20556 0.602780 0.797908i \(-0.294060\pi\)
0.602780 + 0.797908i \(0.294060\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −3.31534 −0.187996 −0.0939979 0.995572i \(-0.529965\pi\)
−0.0939979 + 0.995572i \(0.529965\pi\)
\(312\) 0 0
\(313\) −7.75379 −0.438270 −0.219135 0.975695i \(-0.570324\pi\)
−0.219135 + 0.975695i \(0.570324\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −22.8769 −1.28489 −0.642447 0.766330i \(-0.722081\pi\)
−0.642447 + 0.766330i \(0.722081\pi\)
\(318\) 0 0
\(319\) −16.9848 −0.950969
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −21.1231 −1.17532
\(324\) 0 0
\(325\) 16.8078 0.932327
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −3.56155 −0.196355
\(330\) 0 0
\(331\) 13.8769 0.762743 0.381372 0.924422i \(-0.375452\pi\)
0.381372 + 0.924422i \(0.375452\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 24.4924 1.33816
\(336\) 0 0
\(337\) 17.2462 0.939461 0.469730 0.882810i \(-0.344351\pi\)
0.469730 + 0.882810i \(0.344351\pi\)
\(338\) 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
\(347\) −27.8617 −1.49570 −0.747848 0.663870i \(-0.768913\pi\)
−0.747848 + 0.663870i \(0.768913\pi\)
\(348\) 0 0
\(349\) −16.1771 −0.865939 −0.432970 0.901409i \(-0.642534\pi\)
−0.432970 + 0.901409i \(0.642534\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 31.9848 1.70238 0.851191 0.524856i \(-0.175881\pi\)
0.851191 + 0.524856i \(0.175881\pi\)
\(354\) 0 0
\(355\) 4.00000 0.212298
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 5.19224 0.274036 0.137018 0.990569i \(-0.456248\pi\)
0.137018 + 0.990569i \(0.456248\pi\)
\(360\) 0 0
\(361\) 7.24621 0.381380
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 1.75379 0.0917975
\(366\) 0 0
\(367\) 37.0540 1.93420 0.967101 0.254393i \(-0.0818757\pi\)
0.967101 + 0.254393i \(0.0818757\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 5.68466 0.295133
\(372\) 0 0
\(373\) −17.8078 −0.922051 −0.461026 0.887387i \(-0.652518\pi\)
−0.461026 + 0.887387i \(0.652518\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 35.6847 1.83785
\(378\) 0 0
\(379\) 16.6847 0.857033 0.428517 0.903534i \(-0.359036\pi\)
0.428517 + 0.903534i \(0.359036\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −18.0540 −0.922515 −0.461258 0.887266i \(-0.652602\pi\)
−0.461258 + 0.887266i \(0.652602\pi\)
\(384\) 0 0
\(385\) −4.87689 −0.248550
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −15.7538 −0.798749 −0.399374 0.916788i \(-0.630773\pi\)
−0.399374 + 0.916788i \(0.630773\pi\)
\(390\) 0 0
\(391\) 15.1922 0.768304
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 6.93087 0.348730
\(396\) 0 0
\(397\) −33.6155 −1.68711 −0.843557 0.537039i \(-0.819543\pi\)
−0.843557 + 0.537039i \(0.819543\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 18.4924 0.923468 0.461734 0.887019i \(-0.347228\pi\)
0.461734 + 0.887019i \(0.347228\pi\)
\(402\) 0 0
\(403\) 16.8078 0.837254
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 11.1231 0.551352
\(408\) 0 0
\(409\) 19.3693 0.957751 0.478876 0.877883i \(-0.341045\pi\)
0.478876 + 0.877883i \(0.341045\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0.123106 0.00605763
\(414\) 0 0
\(415\) 2.43845 0.119699
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 21.2462 1.03795 0.518973 0.854791i \(-0.326315\pi\)
0.518973 + 0.854791i \(0.326315\pi\)
\(420\) 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
\(425\) −10.5616 −0.512311
\(426\) 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
\(433\) 36.1080 1.73524 0.867619 0.497230i \(-0.165650\pi\)
0.867619 + 0.497230i \(0.165650\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −18.8769 −0.903004
\(438\) 0 0
\(439\) −21.0540 −1.00485 −0.502426 0.864620i \(-0.667559\pi\)
−0.502426 + 0.864620i \(0.667559\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 9.12311 0.433452 0.216726 0.976232i \(-0.430462\pi\)
0.216726 + 0.976232i \(0.430462\pi\)
\(444\) 0 0
\(445\) 21.3693 1.01300
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −0.492423 −0.0232389 −0.0116194 0.999932i \(-0.503699\pi\)
−0.0116194 + 0.999932i \(0.503699\pi\)
\(450\) 0 0
\(451\) 36.1080 1.70026
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 10.2462 0.480350
\(456\) 0 0
\(457\) −9.19224 −0.429995 −0.214997 0.976615i \(-0.568974\pi\)
−0.214997 + 0.976615i \(0.568974\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −29.4233 −1.37038 −0.685190 0.728365i \(-0.740281\pi\)
−0.685190 + 0.728365i \(0.740281\pi\)
\(462\) 0 0
\(463\) 15.5616 0.723207 0.361603 0.932332i \(-0.382229\pi\)
0.361603 + 0.932332i \(0.382229\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 14.2462 0.659236 0.329618 0.944114i \(-0.393080\pi\)
0.329618 + 0.944114i \(0.393080\pi\)
\(468\) 0 0
\(469\) −15.6847 −0.724250
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 28.8769 1.32776
\(474\) 0 0
\(475\) 13.1231 0.602129
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 18.0540 0.824907 0.412454 0.910979i \(-0.364672\pi\)
0.412454 + 0.910979i \(0.364672\pi\)
\(480\) 0 0
\(481\) −23.3693 −1.06555
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −28.4924 −1.29377
\(486\) 0 0
\(487\) 31.3693 1.42148 0.710740 0.703455i \(-0.248360\pi\)
0.710740 + 0.703455i \(0.248360\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 18.2462 0.823440 0.411720 0.911310i \(-0.364928\pi\)
0.411720 + 0.911310i \(0.364928\pi\)
\(492\) 0 0
\(493\) −22.4233 −1.00989
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −2.56155 −0.114901
\(498\) 0 0
\(499\) 2.93087 0.131204 0.0656019 0.997846i \(-0.479103\pi\)
0.0656019 + 0.997846i \(0.479103\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −27.8078 −1.23989 −0.619943 0.784646i \(-0.712844\pi\)
−0.619943 + 0.784646i \(0.712844\pi\)
\(504\) 0 0
\(505\) −15.6155 −0.694882
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −20.0540 −0.888877 −0.444438 0.895809i \(-0.646597\pi\)
−0.444438 + 0.895809i \(0.646597\pi\)
\(510\) 0 0
\(511\) −1.12311 −0.0496833
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 16.4924 0.726743
\(516\) 0 0
\(517\) 11.1231 0.489194
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 15.6307 0.684793 0.342396 0.939556i \(-0.388761\pi\)
0.342396 + 0.939556i \(0.388761\pi\)
\(522\) 0 0
\(523\) 29.3693 1.28423 0.642115 0.766608i \(-0.278057\pi\)
0.642115 + 0.766608i \(0.278057\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −10.5616 −0.460068
\(528\) 0 0
\(529\) −9.42329 −0.409708
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −75.8617 −3.28594
\(534\) 0 0
\(535\) 25.7538 1.11343
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 3.12311 0.134522
\(540\) 0 0
\(541\) −31.8078 −1.36752 −0.683761 0.729706i \(-0.739657\pi\)
−0.683761 + 0.729706i \(0.739657\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 5.94602 0.254700
\(546\) 0 0
\(547\) 19.3153 0.825864 0.412932 0.910762i \(-0.364505\pi\)
0.412932 + 0.910762i \(0.364505\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 27.8617 1.18695
\(552\) 0 0
\(553\) −4.43845 −0.188742
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 32.8078 1.39011 0.695055 0.718957i \(-0.255380\pi\)
0.695055 + 0.718957i \(0.255380\pi\)
\(558\) 0 0
\(559\) −60.6695 −2.56605
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 17.9309 0.755696 0.377848 0.925868i \(-0.376664\pi\)
0.377848 + 0.925868i \(0.376664\pi\)
\(564\) 0 0
\(565\) −4.87689 −0.205172
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 6.24621 0.261855 0.130927 0.991392i \(-0.458205\pi\)
0.130927 + 0.991392i \(0.458205\pi\)
\(570\) 0 0
\(571\) −27.8769 −1.16661 −0.583306 0.812253i \(-0.698241\pi\)
−0.583306 + 0.812253i \(0.698241\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −9.43845 −0.393610
\(576\) 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
\(581\) −1.56155 −0.0647841
\(582\) 0 0
\(583\) −17.7538 −0.735287
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −16.3153 −0.673406 −0.336703 0.941611i \(-0.609312\pi\)
−0.336703 + 0.941611i \(0.609312\pi\)
\(588\) 0 0
\(589\) 13.1231 0.540728
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 3.94602 0.162044 0.0810219 0.996712i \(-0.474182\pi\)
0.0810219 + 0.996712i \(0.474182\pi\)
\(594\) 0 0
\(595\) −6.43845 −0.263951
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −8.06913 −0.329696 −0.164848 0.986319i \(-0.552713\pi\)
−0.164848 + 0.986319i \(0.552713\pi\)
\(600\) 0 0
\(601\) −16.4924 −0.672740 −0.336370 0.941730i \(-0.609199\pi\)
−0.336370 + 0.941730i \(0.609199\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −1.94602 −0.0791172
\(606\) 0 0
\(607\) 8.56155 0.347503 0.173751 0.984790i \(-0.444411\pi\)
0.173751 + 0.984790i \(0.444411\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −23.3693 −0.945421
\(612\) 0 0
\(613\) 40.7386 1.64542 0.822709 0.568463i \(-0.192462\pi\)
0.822709 + 0.568463i \(0.192462\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −1.12311 −0.0452145 −0.0226073 0.999744i \(-0.507197\pi\)
−0.0226073 + 0.999744i \(0.507197\pi\)
\(618\) 0 0
\(619\) 9.61553 0.386481 0.193240 0.981151i \(-0.438100\pi\)
0.193240 + 0.981151i \(0.438100\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −13.6847 −0.548264
\(624\) 0 0
\(625\) −5.63068 −0.225227
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 14.6847 0.585516
\(630\) 0 0
\(631\) −47.8078 −1.90320 −0.951599 0.307344i \(-0.900560\pi\)
−0.951599 + 0.307344i \(0.900560\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 22.5464 0.894727
\(636\) 0 0
\(637\) −6.56155 −0.259978
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −41.3693 −1.63399 −0.816995 0.576645i \(-0.804362\pi\)
−0.816995 + 0.576645i \(0.804362\pi\)
\(642\) 0 0
\(643\) −10.6307 −0.419233 −0.209617 0.977784i \(-0.567222\pi\)
−0.209617 + 0.977784i \(0.567222\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −21.1231 −0.830435 −0.415217 0.909722i \(-0.636295\pi\)
−0.415217 + 0.909722i \(0.636295\pi\)
\(648\) 0 0
\(649\) −0.384472 −0.0150918
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 33.0540 1.29350 0.646751 0.762701i \(-0.276127\pi\)
0.646751 + 0.762701i \(0.276127\pi\)
\(654\) 0 0
\(655\) −24.4924 −0.956998
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −31.1231 −1.21238 −0.606192 0.795318i \(-0.707304\pi\)
−0.606192 + 0.795318i \(0.707304\pi\)
\(660\) 0 0
\(661\) 30.4924 1.18602 0.593009 0.805196i \(-0.297940\pi\)
0.593009 + 0.805196i \(0.297940\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 8.00000 0.310227
\(666\) 0 0
\(667\) −20.0388 −0.775906
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 25.7538 0.994214
\(672\) 0 0
\(673\) 50.8078 1.95850 0.979248 0.202667i \(-0.0649609\pi\)
0.979248 + 0.202667i \(0.0649609\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −39.1231 −1.50362 −0.751812 0.659378i \(-0.770820\pi\)
−0.751812 + 0.659378i \(0.770820\pi\)
\(678\) 0 0
\(679\) 18.2462 0.700225
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 28.4924 1.09023 0.545116 0.838361i \(-0.316486\pi\)
0.545116 + 0.838361i \(0.316486\pi\)
\(684\) 0 0
\(685\) −1.75379 −0.0670088
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 37.3002 1.42102
\(690\) 0 0
\(691\) −35.6155 −1.35488 −0.677439 0.735579i \(-0.736910\pi\)
−0.677439 + 0.735579i \(0.736910\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 22.6307 0.858431
\(696\) 0 0
\(697\) 47.6695 1.80561
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −10.0000 −0.377695 −0.188847 0.982006i \(-0.560475\pi\)
−0.188847 + 0.982006i \(0.560475\pi\)
\(702\) 0 0
\(703\) −18.2462 −0.688169
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 10.0000 0.376089
\(708\) 0 0
\(709\) −18.9309 −0.710964 −0.355482 0.934683i \(-0.615683\pi\)
−0.355482 + 0.934683i \(0.615683\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −9.43845 −0.353473
\(714\) 0 0
\(715\) −32.0000 −1.19673
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 1.17708 0.0438977 0.0219489 0.999759i \(-0.493013\pi\)
0.0219489 + 0.999759i \(0.493013\pi\)
\(720\) 0 0
\(721\) −10.5616 −0.393333
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 13.9309 0.517380
\(726\) 0 0
\(727\) −14.5616 −0.540058 −0.270029 0.962852i \(-0.587033\pi\)
−0.270029 + 0.962852i \(0.587033\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 38.1231 1.41003
\(732\) 0 0
\(733\) −30.8078 −1.13791 −0.568955 0.822368i \(-0.692652\pi\)
−0.568955 + 0.822368i \(0.692652\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 48.9848 1.80438
\(738\) 0 0
\(739\) −4.00000 −0.147142 −0.0735712 0.997290i \(-0.523440\pi\)
−0.0735712 + 0.997290i \(0.523440\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 25.4384 0.933246 0.466623 0.884456i \(-0.345470\pi\)
0.466623 + 0.884456i \(0.345470\pi\)
\(744\) 0 0
\(745\) 24.8769 0.911419
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −16.4924 −0.602620
\(750\) 0 0
\(751\) 22.1080 0.806731 0.403365 0.915039i \(-0.367840\pi\)
0.403365 + 0.915039i \(0.367840\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 5.56155 0.202406
\(756\) 0 0
\(757\) 43.8078 1.59222 0.796110 0.605152i \(-0.206888\pi\)
0.796110 + 0.605152i \(0.206888\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 38.1231 1.38196 0.690981 0.722873i \(-0.257179\pi\)
0.690981 + 0.722873i \(0.257179\pi\)
\(762\) 0 0
\(763\) −3.80776 −0.137850
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0.807764 0.0291667
\(768\) 0 0
\(769\) 17.8617 0.644111 0.322055 0.946721i \(-0.395626\pi\)
0.322055 + 0.946721i \(0.395626\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −13.5616 −0.487775 −0.243888 0.969804i \(-0.578423\pi\)
−0.243888 + 0.969804i \(0.578423\pi\)
\(774\) 0 0
\(775\) 6.56155 0.235698
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −59.2311 −2.12217
\(780\) 0 0
\(781\) 8.00000 0.286263
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −15.5076 −0.553489
\(786\) 0 0
\(787\) −4.63068 −0.165066 −0.0825330 0.996588i \(-0.526301\pi\)
−0.0825330 + 0.996588i \(0.526301\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 3.12311 0.111045
\(792\) 0 0
\(793\) −54.1080 −1.92143
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 42.9848 1.52260 0.761301 0.648399i \(-0.224561\pi\)
0.761301 + 0.648399i \(0.224561\pi\)
\(798\) 0 0
\(799\) 14.6847 0.519506
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 3.50758 0.123780
\(804\) 0 0
\(805\) −5.75379 −0.202794
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 5.36932 0.188775 0.0943876 0.995536i \(-0.469911\pi\)
0.0943876 + 0.995536i \(0.469911\pi\)
\(810\) 0 0
\(811\) −1.50758 −0.0529382 −0.0264691 0.999650i \(-0.508426\pi\)
−0.0264691 + 0.999650i \(0.508426\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −16.1922 −0.567189
\(816\) 0 0
\(817\) −47.3693 −1.65724
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −10.0691 −0.351415 −0.175708 0.984442i \(-0.556221\pi\)
−0.175708 + 0.984442i \(0.556221\pi\)
\(822\) 0 0
\(823\) −0.438447 −0.0152833 −0.00764165 0.999971i \(-0.502432\pi\)
−0.00764165 + 0.999971i \(0.502432\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1.50758 0.0524236 0.0262118 0.999656i \(-0.491656\pi\)
0.0262118 + 0.999656i \(0.491656\pi\)
\(828\) 0 0
\(829\) 34.8769 1.21132 0.605662 0.795722i \(-0.292908\pi\)
0.605662 + 0.795722i \(0.292908\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 4.12311 0.142857
\(834\) 0 0
\(835\) 19.8078 0.685476
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −49.9157 −1.72328 −0.861641 0.507518i \(-0.830563\pi\)
−0.861641 + 0.507518i \(0.830563\pi\)
\(840\) 0 0
\(841\) 0.576708 0.0198865
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 46.9309 1.61447
\(846\) 0 0
\(847\) 1.24621 0.0428203
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 13.1231 0.449854
\(852\) 0 0
\(853\) −1.05398 −0.0360874 −0.0180437 0.999837i \(-0.505744\pi\)
−0.0180437 + 0.999837i \(0.505744\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −33.8769 −1.15721 −0.578606 0.815607i \(-0.696403\pi\)
−0.578606 + 0.815607i \(0.696403\pi\)
\(858\) 0 0
\(859\) −46.4924 −1.58630 −0.793150 0.609026i \(-0.791561\pi\)
−0.793150 + 0.609026i \(0.791561\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 9.19224 0.312907 0.156454 0.987685i \(-0.449994\pi\)
0.156454 + 0.987685i \(0.449994\pi\)
\(864\) 0 0
\(865\) 8.38447 0.285081
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 13.8617 0.470227
\(870\) 0 0
\(871\) −102.916 −3.48717
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 11.8078 0.399175
\(876\) 0 0
\(877\) −19.3153 −0.652233 −0.326116 0.945330i \(-0.605740\pi\)
−0.326116 + 0.945330i \(0.605740\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −28.4233 −0.957605 −0.478803 0.877923i \(-0.658929\pi\)
−0.478803 + 0.877923i \(0.658929\pi\)
\(882\) 0 0
\(883\) −34.2311 −1.15197 −0.575983 0.817461i \(-0.695381\pi\)
−0.575983 + 0.817461i \(0.695381\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −25.5616 −0.858273 −0.429136 0.903240i \(-0.641182\pi\)
−0.429136 + 0.903240i \(0.641182\pi\)
\(888\) 0 0
\(889\) −14.4384 −0.484250
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −18.2462 −0.610586
\(894\) 0 0
\(895\) 36.1080 1.20696
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 13.9309 0.464621
\(900\) 0 0
\(901\) −23.4384 −0.780848
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −24.1080 −0.801375
\(906\) 0 0
\(907\) 22.4384 0.745056 0.372528 0.928021i \(-0.378491\pi\)
0.372528 + 0.928021i \(0.378491\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −25.3693 −0.840523 −0.420261 0.907403i \(-0.638062\pi\)
−0.420261 + 0.907403i \(0.638062\pi\)
\(912\) 0 0
\(913\) 4.87689 0.161402
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 15.6847 0.517953
\(918\) 0 0
\(919\) −15.1771 −0.500646 −0.250323 0.968162i \(-0.580537\pi\)
−0.250323 + 0.968162i \(0.580537\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −16.8078 −0.553234
\(924\) 0 0
\(925\) −9.12311 −0.299966
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −0.438447 −0.0143850 −0.00719249 0.999974i \(-0.502289\pi\)
−0.00719249 + 0.999974i \(0.502289\pi\)
\(930\) 0 0
\(931\) −5.12311 −0.167903
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 20.1080 0.657600
\(936\) 0 0
\(937\) 26.2462 0.857426 0.428713 0.903441i \(-0.358967\pi\)
0.428713 + 0.903441i \(0.358967\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 10.1922 0.332257 0.166129 0.986104i \(-0.446873\pi\)
0.166129 + 0.986104i \(0.446873\pi\)
\(942\) 0 0
\(943\) 42.6004 1.38726
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −39.3693 −1.27933 −0.639665 0.768653i \(-0.720927\pi\)
−0.639665 + 0.768653i \(0.720927\pi\)
\(948\) 0 0
\(949\) −7.36932 −0.239218
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 56.4924 1.82997 0.914985 0.403489i \(-0.132203\pi\)
0.914985 + 0.403489i \(0.132203\pi\)
\(954\) 0 0
\(955\) −1.36932 −0.0443101
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 1.12311 0.0362670
\(960\) 0 0
\(961\) −24.4384 −0.788337
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −29.0691 −0.935768
\(966\) 0 0
\(967\) 56.9848 1.83251 0.916255 0.400597i \(-0.131197\pi\)
0.916255 + 0.400597i \(0.131197\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 21.8769 0.702063 0.351031 0.936364i \(-0.385831\pi\)
0.351031 + 0.936364i \(0.385831\pi\)
\(972\) 0 0
\(973\) −14.4924 −0.464606
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −22.0000 −0.703842 −0.351921 0.936030i \(-0.614471\pi\)
−0.351921 + 0.936030i \(0.614471\pi\)
\(978\) 0 0
\(979\) 42.7386 1.36593
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −7.80776 −0.249029 −0.124514 0.992218i \(-0.539737\pi\)
−0.124514 + 0.992218i \(0.539737\pi\)
\(984\) 0 0
\(985\) 10.7386 0.342161
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 34.0691 1.08334
\(990\) 0 0
\(991\) 5.42329 0.172277 0.0861383 0.996283i \(-0.472547\pi\)
0.0861383 + 0.996283i \(0.472547\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 8.87689 0.281416
\(996\) 0 0
\(997\) 32.6695 1.03465 0.517327 0.855788i \(-0.326927\pi\)
0.517327 + 0.855788i \(0.326927\pi\)
\(998\) 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.bc.1.2 2
3.2 odd 2 6048.2.a.be.1.1 yes 2
4.3 odd 2 6048.2.a.bd.1.2 yes 2
12.11 even 2 6048.2.a.bf.1.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6048.2.a.bc.1.2 2 1.1 even 1 trivial
6048.2.a.bd.1.2 yes 2 4.3 odd 2
6048.2.a.be.1.1 yes 2 3.2 odd 2
6048.2.a.bf.1.1 yes 2 12.11 even 2