Properties

Label 6048.2.c.e.3025.3
Level 6048
Weight 2
Character 6048.3025
Analytic conductor 48.294
Analytic rank 0
Dimension 20
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.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.2935231425\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{18}\cdot 3^{8} \)
Twist minimal: no (minimal twist has level 1512)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3025.3
Root \(1.19566 - 0.755240i\)
Character \(\chi\) = 6048.3025
Dual form 6048.2.c.e.3025.18

$q$-expansion

\(f(q)\) \(=\) \(q-3.16969i q^{5} +1.00000 q^{7} +O(q^{10})\) \(q-3.16969i q^{5} +1.00000 q^{7} +5.44696i q^{11} -3.61205i q^{13} +3.27628 q^{17} +3.20627i q^{19} -0.673274 q^{23} -5.04692 q^{25} +2.85127i q^{29} -3.71845 q^{31} -3.16969i q^{35} +11.9988i q^{37} -7.44602 q^{41} +12.5741i q^{43} +4.06341 q^{47} +1.00000 q^{49} +0.291601i q^{53} +17.2652 q^{55} -0.0209587i q^{59} +5.34034i q^{61} -11.4491 q^{65} -6.20714i q^{67} -15.5050 q^{71} +1.35371 q^{73} +5.44696i q^{77} +11.0846 q^{79} +13.6442i q^{83} -10.3848i q^{85} -5.93965 q^{89} -3.61205i q^{91} +10.1629 q^{95} +9.60073 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 20q^{7} + O(q^{10}) \) \( 20q + 20q^{7} - 28q^{25} - 36q^{31} + 20q^{49} - 48q^{55} - 64q^{79} + 56q^{97} + O(q^{100}) \)

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
\(5\) − 3.16969i − 1.41753i −0.705446 0.708764i \(-0.749253\pi\)
0.705446 0.708764i \(-0.250747\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.44696i 1.64232i 0.570699 + 0.821160i \(0.306673\pi\)
−0.570699 + 0.821160i \(0.693327\pi\)
\(12\) 0 0
\(13\) − 3.61205i − 1.00180i −0.865504 0.500902i \(-0.833002\pi\)
0.865504 0.500902i \(-0.166998\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.27628 0.794616 0.397308 0.917685i \(-0.369945\pi\)
0.397308 + 0.917685i \(0.369945\pi\)
\(18\) 0 0
\(19\) 3.20627i 0.735569i 0.929911 + 0.367785i \(0.119884\pi\)
−0.929911 + 0.367785i \(0.880116\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −0.673274 −0.140387 −0.0701936 0.997533i \(-0.522362\pi\)
−0.0701936 + 0.997533i \(0.522362\pi\)
\(24\) 0 0
\(25\) −5.04692 −1.00938
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.85127i 0.529468i 0.964322 + 0.264734i \(0.0852841\pi\)
−0.964322 + 0.264734i \(0.914716\pi\)
\(30\) 0 0
\(31\) −3.71845 −0.667854 −0.333927 0.942599i \(-0.608374\pi\)
−0.333927 + 0.942599i \(0.608374\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 3.16969i − 0.535775i
\(36\) 0 0
\(37\) 11.9988i 1.97260i 0.164971 + 0.986298i \(0.447247\pi\)
−0.164971 + 0.986298i \(0.552753\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −7.44602 −1.16287 −0.581436 0.813592i \(-0.697509\pi\)
−0.581436 + 0.813592i \(0.697509\pi\)
\(42\) 0 0
\(43\) 12.5741i 1.91753i 0.284195 + 0.958766i \(0.408274\pi\)
−0.284195 + 0.958766i \(0.591726\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.06341 0.592710 0.296355 0.955078i \(-0.404229\pi\)
0.296355 + 0.955078i \(0.404229\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0.291601i 0.0400545i 0.999799 + 0.0200272i \(0.00637529\pi\)
−0.999799 + 0.0200272i \(0.993625\pi\)
\(54\) 0 0
\(55\) 17.2652 2.32803
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 0.0209587i − 0.00272859i −0.999999 0.00136430i \(-0.999566\pi\)
0.999999 0.00136430i \(-0.000434269\pi\)
\(60\) 0 0
\(61\) 5.34034i 0.683760i 0.939744 + 0.341880i \(0.111064\pi\)
−0.939744 + 0.341880i \(0.888936\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −11.4491 −1.42008
\(66\) 0 0
\(67\) − 6.20714i − 0.758323i −0.925331 0.379161i \(-0.876213\pi\)
0.925331 0.379161i \(-0.123787\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −15.5050 −1.84010 −0.920050 0.391801i \(-0.871852\pi\)
−0.920050 + 0.391801i \(0.871852\pi\)
\(72\) 0 0
\(73\) 1.35371 0.158440 0.0792198 0.996857i \(-0.474757\pi\)
0.0792198 + 0.996857i \(0.474757\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.44696i 0.620738i
\(78\) 0 0
\(79\) 11.0846 1.24711 0.623555 0.781779i \(-0.285688\pi\)
0.623555 + 0.781779i \(0.285688\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 13.6442i 1.49765i 0.662768 + 0.748824i \(0.269381\pi\)
−0.662768 + 0.748824i \(0.730619\pi\)
\(84\) 0 0
\(85\) − 10.3848i − 1.12639i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −5.93965 −0.629601 −0.314801 0.949158i \(-0.601938\pi\)
−0.314801 + 0.949158i \(0.601938\pi\)
\(90\) 0 0
\(91\) − 3.61205i − 0.378646i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 10.1629 1.04269
\(96\) 0 0
\(97\) 9.60073 0.974807 0.487403 0.873177i \(-0.337944\pi\)
0.487403 + 0.873177i \(0.337944\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 14.2123i − 1.41418i −0.707124 0.707090i \(-0.750008\pi\)
0.707124 0.707090i \(-0.249992\pi\)
\(102\) 0 0
\(103\) −1.69321 −0.166837 −0.0834187 0.996515i \(-0.526584\pi\)
−0.0834187 + 0.996515i \(0.526584\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 11.6644i 1.12764i 0.825897 + 0.563821i \(0.190669\pi\)
−0.825897 + 0.563821i \(0.809331\pi\)
\(108\) 0 0
\(109\) 14.0521i 1.34595i 0.739667 + 0.672973i \(0.234983\pi\)
−0.739667 + 0.672973i \(0.765017\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −14.9455 −1.40596 −0.702979 0.711211i \(-0.748147\pi\)
−0.702979 + 0.711211i \(0.748147\pi\)
\(114\) 0 0
\(115\) 2.13407i 0.199003i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 3.27628 0.300336
\(120\) 0 0
\(121\) −18.6693 −1.69721
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0.148729i 0.0133028i
\(126\) 0 0
\(127\) −4.07216 −0.361346 −0.180673 0.983543i \(-0.557828\pi\)
−0.180673 + 0.983543i \(0.557828\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 17.8643i − 1.56081i −0.625275 0.780404i \(-0.715013\pi\)
0.625275 0.780404i \(-0.284987\pi\)
\(132\) 0 0
\(133\) 3.20627i 0.278019i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −21.1351 −1.80569 −0.902847 0.429963i \(-0.858527\pi\)
−0.902847 + 0.429963i \(0.858527\pi\)
\(138\) 0 0
\(139\) 7.39359i 0.627116i 0.949569 + 0.313558i \(0.101521\pi\)
−0.949569 + 0.313558i \(0.898479\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 19.6747 1.64528
\(144\) 0 0
\(145\) 9.03764 0.750535
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 13.0541i 1.06944i 0.845030 + 0.534719i \(0.179582\pi\)
−0.845030 + 0.534719i \(0.820418\pi\)
\(150\) 0 0
\(151\) 17.6477 1.43615 0.718073 0.695968i \(-0.245024\pi\)
0.718073 + 0.695968i \(0.245024\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 11.7863i 0.946701i
\(156\) 0 0
\(157\) − 3.00087i − 0.239495i −0.992804 0.119748i \(-0.961791\pi\)
0.992804 0.119748i \(-0.0382085\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −0.673274 −0.0530614
\(162\) 0 0
\(163\) 0.871148i 0.0682335i 0.999418 + 0.0341168i \(0.0108618\pi\)
−0.999418 + 0.0341168i \(0.989138\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −7.49952 −0.580330 −0.290165 0.956977i \(-0.593710\pi\)
−0.290165 + 0.956977i \(0.593710\pi\)
\(168\) 0 0
\(169\) −0.0469224 −0.00360941
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 0.652919i − 0.0496405i −0.999692 0.0248203i \(-0.992099\pi\)
0.999692 0.0248203i \(-0.00790135\pi\)
\(174\) 0 0
\(175\) −5.04692 −0.381511
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 4.43263i 0.331310i 0.986184 + 0.165655i \(0.0529738\pi\)
−0.986184 + 0.165655i \(0.947026\pi\)
\(180\) 0 0
\(181\) 21.9779i 1.63360i 0.576920 + 0.816801i \(0.304254\pi\)
−0.576920 + 0.816801i \(0.695746\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 38.0326 2.79621
\(186\) 0 0
\(187\) 17.8458i 1.30501i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 6.67327 0.482861 0.241430 0.970418i \(-0.422383\pi\)
0.241430 + 0.970418i \(0.422383\pi\)
\(192\) 0 0
\(193\) 25.0305 1.80174 0.900868 0.434092i \(-0.142931\pi\)
0.900868 + 0.434092i \(0.142931\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 3.77971i − 0.269293i −0.990894 0.134646i \(-0.957010\pi\)
0.990894 0.134646i \(-0.0429899\pi\)
\(198\) 0 0
\(199\) 7.18059 0.509019 0.254509 0.967070i \(-0.418086\pi\)
0.254509 + 0.967070i \(0.418086\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 2.85127i 0.200120i
\(204\) 0 0
\(205\) 23.6016i 1.64840i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −17.4644 −1.20804
\(210\) 0 0
\(211\) − 5.30442i − 0.365171i −0.983190 0.182586i \(-0.941553\pi\)
0.983190 0.182586i \(-0.0584467\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 39.8560 2.71816
\(216\) 0 0
\(217\) −3.71845 −0.252425
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 11.8341i − 0.796048i
\(222\) 0 0
\(223\) −4.62678 −0.309832 −0.154916 0.987928i \(-0.549511\pi\)
−0.154916 + 0.987928i \(0.549511\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 9.92356i − 0.658650i −0.944217 0.329325i \(-0.893179\pi\)
0.944217 0.329325i \(-0.106821\pi\)
\(228\) 0 0
\(229\) − 2.32546i − 0.153671i −0.997044 0.0768355i \(-0.975518\pi\)
0.997044 0.0768355i \(-0.0244816\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −3.23031 −0.211625 −0.105812 0.994386i \(-0.533744\pi\)
−0.105812 + 0.994386i \(0.533744\pi\)
\(234\) 0 0
\(235\) − 12.8797i − 0.840182i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 3.48961 0.225724 0.112862 0.993611i \(-0.463998\pi\)
0.112862 + 0.993611i \(0.463998\pi\)
\(240\) 0 0
\(241\) 9.77778 0.629842 0.314921 0.949118i \(-0.398022\pi\)
0.314921 + 0.949118i \(0.398022\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 3.16969i − 0.202504i
\(246\) 0 0
\(247\) 11.5812 0.736896
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 25.2169i − 1.59168i −0.605509 0.795838i \(-0.707031\pi\)
0.605509 0.795838i \(-0.292969\pi\)
\(252\) 0 0
\(253\) − 3.66729i − 0.230561i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 6.12920 0.382329 0.191165 0.981558i \(-0.438774\pi\)
0.191165 + 0.981558i \(0.438774\pi\)
\(258\) 0 0
\(259\) 11.9988i 0.745572i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 1.27084 0.0783635 0.0391818 0.999232i \(-0.487525\pi\)
0.0391818 + 0.999232i \(0.487525\pi\)
\(264\) 0 0
\(265\) 0.924284 0.0567783
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 19.8081i − 1.20772i −0.797091 0.603859i \(-0.793629\pi\)
0.797091 0.603859i \(-0.206371\pi\)
\(270\) 0 0
\(271\) −24.1532 −1.46720 −0.733600 0.679581i \(-0.762162\pi\)
−0.733600 + 0.679581i \(0.762162\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 27.4904i − 1.65773i
\(276\) 0 0
\(277\) − 1.31816i − 0.0792005i −0.999216 0.0396003i \(-0.987392\pi\)
0.999216 0.0396003i \(-0.0126084\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 24.7284 1.47517 0.737587 0.675252i \(-0.235965\pi\)
0.737587 + 0.675252i \(0.235965\pi\)
\(282\) 0 0
\(283\) 18.6573i 1.10906i 0.832163 + 0.554532i \(0.187103\pi\)
−0.832163 + 0.554532i \(0.812897\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −7.44602 −0.439525
\(288\) 0 0
\(289\) −6.26596 −0.368586
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 14.4030i 0.841431i 0.907193 + 0.420715i \(0.138221\pi\)
−0.907193 + 0.420715i \(0.861779\pi\)
\(294\) 0 0
\(295\) −0.0664326 −0.00386786
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 2.43190i 0.140640i
\(300\) 0 0
\(301\) 12.5741i 0.724759i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 16.9272 0.969249
\(306\) 0 0
\(307\) 27.3734i 1.56228i 0.624353 + 0.781142i \(0.285363\pi\)
−0.624353 + 0.781142i \(0.714637\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 27.9767 1.58641 0.793206 0.608953i \(-0.208410\pi\)
0.793206 + 0.608953i \(0.208410\pi\)
\(312\) 0 0
\(313\) 27.5228 1.55568 0.777841 0.628461i \(-0.216315\pi\)
0.777841 + 0.628461i \(0.216315\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 1.73928i − 0.0976879i −0.998806 0.0488440i \(-0.984446\pi\)
0.998806 0.0488440i \(-0.0155537\pi\)
\(318\) 0 0
\(319\) −15.5307 −0.869555
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 10.5047i 0.584495i
\(324\) 0 0
\(325\) 18.2297i 1.01120i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 4.06341 0.224023
\(330\) 0 0
\(331\) 13.7965i 0.758323i 0.925331 + 0.379161i \(0.123788\pi\)
−0.925331 + 0.379161i \(0.876212\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −19.6747 −1.07494
\(336\) 0 0
\(337\) −1.96903 −0.107260 −0.0536300 0.998561i \(-0.517079\pi\)
−0.0536300 + 0.998561i \(0.517079\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 20.2542i − 1.09683i
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 21.4183i 1.14979i 0.818226 + 0.574897i \(0.194958\pi\)
−0.818226 + 0.574897i \(0.805042\pi\)
\(348\) 0 0
\(349\) − 24.2315i − 1.29708i −0.761180 0.648541i \(-0.775380\pi\)
0.761180 0.648541i \(-0.224620\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 20.1659 1.07332 0.536662 0.843797i \(-0.319685\pi\)
0.536662 + 0.843797i \(0.319685\pi\)
\(354\) 0 0
\(355\) 49.1459i 2.60839i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 19.8643 1.04840 0.524198 0.851597i \(-0.324365\pi\)
0.524198 + 0.851597i \(0.324365\pi\)
\(360\) 0 0
\(361\) 8.71982 0.458938
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 4.29083i − 0.224592i
\(366\) 0 0
\(367\) −37.1713 −1.94033 −0.970163 0.242454i \(-0.922048\pi\)
−0.970163 + 0.242454i \(0.922048\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0.291601i 0.0151392i
\(372\) 0 0
\(373\) − 23.6106i − 1.22251i −0.791433 0.611256i \(-0.790665\pi\)
0.791433 0.611256i \(-0.209335\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 10.2989 0.530422
\(378\) 0 0
\(379\) − 15.2051i − 0.781034i −0.920596 0.390517i \(-0.872296\pi\)
0.920596 0.390517i \(-0.127704\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 1.26331 0.0645522 0.0322761 0.999479i \(-0.489724\pi\)
0.0322761 + 0.999479i \(0.489724\pi\)
\(384\) 0 0
\(385\) 17.2652 0.879914
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 12.3813i − 0.627754i −0.949464 0.313877i \(-0.898372\pi\)
0.949464 0.313877i \(-0.101628\pi\)
\(390\) 0 0
\(391\) −2.20584 −0.111554
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 35.1346i − 1.76781i
\(396\) 0 0
\(397\) − 19.0272i − 0.954948i −0.878646 0.477474i \(-0.841553\pi\)
0.878646 0.477474i \(-0.158447\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 6.06885 0.303064 0.151532 0.988452i \(-0.451579\pi\)
0.151532 + 0.988452i \(0.451579\pi\)
\(402\) 0 0
\(403\) 13.4312i 0.669058i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −65.3572 −3.23963
\(408\) 0 0
\(409\) 28.9491 1.43144 0.715720 0.698387i \(-0.246099\pi\)
0.715720 + 0.698387i \(0.246099\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 0.0209587i − 0.00103131i
\(414\) 0 0
\(415\) 43.2480 2.12296
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 4.30387i 0.210258i 0.994459 + 0.105129i \(0.0335255\pi\)
−0.994459 + 0.105129i \(0.966474\pi\)
\(420\) 0 0
\(421\) 18.0243i 0.878453i 0.898376 + 0.439226i \(0.144747\pi\)
−0.898376 + 0.439226i \(0.855253\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −16.5351 −0.802073
\(426\) 0 0
\(427\) 5.34034i 0.258437i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 11.0189 0.530760 0.265380 0.964144i \(-0.414503\pi\)
0.265380 + 0.964144i \(0.414503\pi\)
\(432\) 0 0
\(433\) 11.0881 0.532861 0.266430 0.963854i \(-0.414156\pi\)
0.266430 + 0.963854i \(0.414156\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 2.15870i − 0.103265i
\(438\) 0 0
\(439\) 32.6928 1.56034 0.780171 0.625567i \(-0.215132\pi\)
0.780171 + 0.625567i \(0.215132\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 23.6225i 1.12234i 0.827700 + 0.561171i \(0.189649\pi\)
−0.827700 + 0.561171i \(0.810351\pi\)
\(444\) 0 0
\(445\) 18.8268i 0.892477i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 8.75599 0.413220 0.206610 0.978423i \(-0.433757\pi\)
0.206610 + 0.978423i \(0.433757\pi\)
\(450\) 0 0
\(451\) − 40.5581i − 1.90981i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −11.4491 −0.536741
\(456\) 0 0
\(457\) 15.8326 0.740618 0.370309 0.928909i \(-0.379252\pi\)
0.370309 + 0.928909i \(0.379252\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 36.1188i 1.68222i 0.540865 + 0.841109i \(0.318097\pi\)
−0.540865 + 0.841109i \(0.681903\pi\)
\(462\) 0 0
\(463\) −8.28903 −0.385224 −0.192612 0.981275i \(-0.561696\pi\)
−0.192612 + 0.981275i \(0.561696\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 22.7640i 1.05339i 0.850053 + 0.526697i \(0.176570\pi\)
−0.850053 + 0.526697i \(0.823430\pi\)
\(468\) 0 0
\(469\) − 6.20714i − 0.286619i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −68.4906 −3.14920
\(474\) 0 0
\(475\) − 16.1818i − 0.742472i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −38.1931 −1.74509 −0.872543 0.488537i \(-0.837531\pi\)
−0.872543 + 0.488537i \(0.837531\pi\)
\(480\) 0 0
\(481\) 43.3404 1.97615
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 30.4313i − 1.38182i
\(486\) 0 0
\(487\) −8.84028 −0.400591 −0.200296 0.979735i \(-0.564190\pi\)
−0.200296 + 0.979735i \(0.564190\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 28.9660i 1.30722i 0.756833 + 0.653608i \(0.226746\pi\)
−0.756833 + 0.653608i \(0.773254\pi\)
\(492\) 0 0
\(493\) 9.34157i 0.420723i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −15.5050 −0.695492
\(498\) 0 0
\(499\) − 0.417929i − 0.0187091i −0.999956 0.00935453i \(-0.997022\pi\)
0.999956 0.00935453i \(-0.00297768\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −10.3133 −0.459848 −0.229924 0.973209i \(-0.573848\pi\)
−0.229924 + 0.973209i \(0.573848\pi\)
\(504\) 0 0
\(505\) −45.0487 −2.00464
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 22.2340i 0.985505i 0.870169 + 0.492753i \(0.164009\pi\)
−0.870169 + 0.492753i \(0.835991\pi\)
\(510\) 0 0
\(511\) 1.35371 0.0598845
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 5.36696i 0.236497i
\(516\) 0 0
\(517\) 22.1332i 0.973418i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 29.3079 1.28400 0.642001 0.766704i \(-0.278104\pi\)
0.642001 + 0.766704i \(0.278104\pi\)
\(522\) 0 0
\(523\) 1.15302i 0.0504181i 0.999682 + 0.0252090i \(0.00802514\pi\)
−0.999682 + 0.0252090i \(0.991975\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −12.1827 −0.530687
\(528\) 0 0
\(529\) −22.5467 −0.980291
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 26.8954i 1.16497i
\(534\) 0 0
\(535\) 36.9726 1.59846
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 5.44696i 0.234617i
\(540\) 0 0
\(541\) 34.3825i 1.47822i 0.673586 + 0.739109i \(0.264753\pi\)
−0.673586 + 0.739109i \(0.735247\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 44.5407 1.90792
\(546\) 0 0
\(547\) − 17.1388i − 0.732802i −0.930457 0.366401i \(-0.880590\pi\)
0.930457 0.366401i \(-0.119410\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −9.14195 −0.389460
\(552\) 0 0
\(553\) 11.0846 0.471363
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 29.0976i 1.23290i 0.787393 + 0.616452i \(0.211430\pi\)
−0.787393 + 0.616452i \(0.788570\pi\)
\(558\) 0 0
\(559\) 45.4183 1.92099
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 31.6652i − 1.33453i −0.744821 0.667264i \(-0.767465\pi\)
0.744821 0.667264i \(-0.232535\pi\)
\(564\) 0 0
\(565\) 47.3727i 1.99298i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −6.16080 −0.258274 −0.129137 0.991627i \(-0.541221\pi\)
−0.129137 + 0.991627i \(0.541221\pi\)
\(570\) 0 0
\(571\) 41.0313i 1.71711i 0.512724 + 0.858553i \(0.328636\pi\)
−0.512724 + 0.858553i \(0.671364\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 3.39796 0.141705
\(576\) 0 0
\(577\) −25.4290 −1.05862 −0.529311 0.848428i \(-0.677550\pi\)
−0.529311 + 0.848428i \(0.677550\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 13.6442i 0.566058i
\(582\) 0 0
\(583\) −1.58834 −0.0657822
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 3.55639i 0.146788i 0.997303 + 0.0733939i \(0.0233830\pi\)
−0.997303 + 0.0733939i \(0.976617\pi\)
\(588\) 0 0
\(589\) − 11.9224i − 0.491253i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −2.57142 −0.105596 −0.0527978 0.998605i \(-0.516814\pi\)
−0.0527978 + 0.998605i \(0.516814\pi\)
\(594\) 0 0
\(595\) − 10.3848i − 0.425735i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −21.8400 −0.892357 −0.446178 0.894944i \(-0.647215\pi\)
−0.446178 + 0.894944i \(0.647215\pi\)
\(600\) 0 0
\(601\) −43.8140 −1.78721 −0.893606 0.448853i \(-0.851833\pi\)
−0.893606 + 0.448853i \(0.851833\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 59.1760i 2.40585i
\(606\) 0 0
\(607\) −17.9012 −0.726588 −0.363294 0.931675i \(-0.618348\pi\)
−0.363294 + 0.931675i \(0.618348\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 14.6773i − 0.593778i
\(612\) 0 0
\(613\) − 15.8902i − 0.641798i −0.947113 0.320899i \(-0.896015\pi\)
0.947113 0.320899i \(-0.103985\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −20.8402 −0.838994 −0.419497 0.907757i \(-0.637794\pi\)
−0.419497 + 0.907757i \(0.637794\pi\)
\(618\) 0 0
\(619\) − 31.4914i − 1.26575i −0.774255 0.632873i \(-0.781875\pi\)
0.774255 0.632873i \(-0.218125\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −5.93965 −0.237967
\(624\) 0 0
\(625\) −24.7632 −0.990527
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 39.3116i 1.56746i
\(630\) 0 0
\(631\) 10.6786 0.425109 0.212555 0.977149i \(-0.431822\pi\)
0.212555 + 0.977149i \(0.431822\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 12.9075i 0.512218i
\(636\) 0 0
\(637\) − 3.61205i − 0.143115i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −24.7669 −0.978232 −0.489116 0.872219i \(-0.662681\pi\)
−0.489116 + 0.872219i \(0.662681\pi\)
\(642\) 0 0
\(643\) 11.5834i 0.456805i 0.973567 + 0.228402i \(0.0733502\pi\)
−0.973567 + 0.228402i \(0.926650\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −30.1155 −1.18396 −0.591981 0.805952i \(-0.701654\pi\)
−0.591981 + 0.805952i \(0.701654\pi\)
\(648\) 0 0
\(649\) 0.114161 0.00448122
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 45.7814i − 1.79157i −0.444491 0.895783i \(-0.646616\pi\)
0.444491 0.895783i \(-0.353384\pi\)
\(654\) 0 0
\(655\) −56.6242 −2.21249
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 34.3446i − 1.33788i −0.743318 0.668938i \(-0.766749\pi\)
0.743318 0.668938i \(-0.233251\pi\)
\(660\) 0 0
\(661\) − 26.5119i − 1.03120i −0.856831 0.515598i \(-0.827570\pi\)
0.856831 0.515598i \(-0.172430\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 10.1629 0.394100
\(666\) 0 0
\(667\) − 1.91969i − 0.0743305i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −29.0886 −1.12295
\(672\) 0 0
\(673\) −26.2427 −1.01158 −0.505790 0.862657i \(-0.668799\pi\)
−0.505790 + 0.862657i \(0.668799\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 3.23458i − 0.124315i −0.998066 0.0621576i \(-0.980202\pi\)
0.998066 0.0621576i \(-0.0197981\pi\)
\(678\) 0 0
\(679\) 9.60073 0.368442
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 0.708911i − 0.0271257i −0.999908 0.0135629i \(-0.995683\pi\)
0.999908 0.0135629i \(-0.00431733\pi\)
\(684\) 0 0
\(685\) 66.9917i 2.55962i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 1.05328 0.0401267
\(690\) 0 0
\(691\) − 39.0071i − 1.48390i −0.670454 0.741951i \(-0.733901\pi\)
0.670454 0.741951i \(-0.266099\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 23.4354 0.888955
\(696\) 0 0
\(697\) −24.3953 −0.924037
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 16.2448i − 0.613559i −0.951781 0.306779i \(-0.900749\pi\)
0.951781 0.306779i \(-0.0992514\pi\)
\(702\) 0 0
\(703\) −38.4715 −1.45098
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 14.2123i − 0.534510i
\(708\) 0 0
\(709\) − 25.1720i − 0.945354i −0.881236 0.472677i \(-0.843288\pi\)
0.881236 0.472677i \(-0.156712\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 2.50354 0.0937581
\(714\) 0 0
\(715\) − 62.3626i − 2.33223i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −7.92289 −0.295474 −0.147737 0.989027i \(-0.547199\pi\)
−0.147737 + 0.989027i \(0.547199\pi\)
\(720\) 0 0
\(721\) −1.69321 −0.0630586
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 14.3901i − 0.534436i
\(726\) 0 0
\(727\) 6.10313 0.226353 0.113176 0.993575i \(-0.463898\pi\)
0.113176 + 0.993575i \(0.463898\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 41.1963i 1.52370i
\(732\) 0 0
\(733\) 26.6568i 0.984591i 0.870428 + 0.492296i \(0.163842\pi\)
−0.870428 + 0.492296i \(0.836158\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 33.8100 1.24541
\(738\) 0 0
\(739\) 9.27736i 0.341273i 0.985334 + 0.170637i \(0.0545824\pi\)
−0.985334 + 0.170637i \(0.945418\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 12.6494 0.464062 0.232031 0.972708i \(-0.425463\pi\)
0.232031 + 0.972708i \(0.425463\pi\)
\(744\) 0 0
\(745\) 41.3776 1.51596
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 11.6644i 0.426209i
\(750\) 0 0
\(751\) 7.10625 0.259311 0.129655 0.991559i \(-0.458613\pi\)
0.129655 + 0.991559i \(0.458613\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 55.9376i − 2.03578i
\(756\) 0 0
\(757\) − 34.6019i − 1.25763i −0.777556 0.628814i \(-0.783541\pi\)
0.777556 0.628814i \(-0.216459\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 21.7114 0.787039 0.393520 0.919316i \(-0.371257\pi\)
0.393520 + 0.919316i \(0.371257\pi\)
\(762\) 0 0
\(763\) 14.0521i 0.508720i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −0.0757040 −0.00273351
\(768\) 0 0
\(769\) −5.94161 −0.214260 −0.107130 0.994245i \(-0.534166\pi\)
−0.107130 + 0.994245i \(0.534166\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 1.14102i 0.0410398i 0.999789 + 0.0205199i \(0.00653215\pi\)
−0.999789 + 0.0205199i \(0.993468\pi\)
\(774\) 0 0
\(775\) 18.7667 0.674121
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 23.8740i − 0.855374i
\(780\) 0 0
\(781\) − 84.4548i − 3.02203i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −9.51181 −0.339491
\(786\) 0 0
\(787\) 42.2224i 1.50507i 0.658555 + 0.752533i \(0.271168\pi\)
−0.658555 + 0.752533i \(0.728832\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −14.9455 −0.531402
\(792\) 0 0
\(793\) 19.2896 0.684993
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 37.0441i − 1.31217i −0.754687 0.656085i \(-0.772211\pi\)
0.754687 0.656085i \(-0.227789\pi\)
\(798\) 0 0
\(799\) 13.3129 0.470976
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 7.37359i 0.260208i
\(804\) 0 0
\(805\) 2.13407i 0.0752160i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −18.7803 −0.660279 −0.330140 0.943932i \(-0.607096\pi\)
−0.330140 + 0.943932i \(0.607096\pi\)
\(810\) 0 0
\(811\) 54.4054i 1.91043i 0.295909 + 0.955216i \(0.404377\pi\)
−0.295909 + 0.955216i \(0.595623\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 2.76127 0.0967229
\(816\) 0 0
\(817\) −40.3160 −1.41048
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 26.7178i 0.932458i 0.884664 + 0.466229i \(0.154388\pi\)
−0.884664 + 0.466229i \(0.845612\pi\)
\(822\) 0 0
\(823\) 15.5033 0.540412 0.270206 0.962802i \(-0.412908\pi\)
0.270206 + 0.962802i \(0.412908\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 16.0053i − 0.556559i −0.960500 0.278279i \(-0.910236\pi\)
0.960500 0.278279i \(-0.0897641\pi\)
\(828\) 0 0
\(829\) − 46.1590i − 1.60317i −0.597881 0.801585i \(-0.703991\pi\)
0.597881 0.801585i \(-0.296009\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 3.27628 0.113517
\(834\) 0 0
\(835\) 23.7711i 0.822634i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −4.38306 −0.151320 −0.0756601 0.997134i \(-0.524106\pi\)
−0.0756601 + 0.997134i \(0.524106\pi\)
\(840\) 0 0
\(841\) 20.8703 0.719664
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0.148729i 0.00511644i
\(846\) 0 0
\(847\) −18.6693 −0.641486
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 8.07850i − 0.276927i
\(852\) 0 0
\(853\) 15.5653i 0.532946i 0.963842 + 0.266473i \(0.0858583\pi\)
−0.963842 + 0.266473i \(0.914142\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −30.2023 −1.03169 −0.515846 0.856681i \(-0.672522\pi\)
−0.515846 + 0.856681i \(0.672522\pi\)
\(858\) 0 0
\(859\) 6.32139i 0.215683i 0.994168 + 0.107841i \(0.0343939\pi\)
−0.994168 + 0.107841i \(0.965606\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 35.8902 1.22172 0.610858 0.791740i \(-0.290825\pi\)
0.610858 + 0.791740i \(0.290825\pi\)
\(864\) 0 0
\(865\) −2.06955 −0.0703668
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 60.3771i 2.04815i
\(870\) 0 0
\(871\) −22.4205 −0.759690
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0.148729i 0.00502797i
\(876\) 0 0
\(877\) − 32.3677i − 1.09298i −0.837466 0.546490i \(-0.815964\pi\)
0.837466 0.546490i \(-0.184036\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 33.1052 1.11534 0.557670 0.830062i \(-0.311695\pi\)
0.557670 + 0.830062i \(0.311695\pi\)
\(882\) 0 0
\(883\) − 19.7891i − 0.665957i −0.942934 0.332979i \(-0.891946\pi\)
0.942934 0.332979i \(-0.108054\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0.843244 0.0283133 0.0141567 0.999900i \(-0.495494\pi\)
0.0141567 + 0.999900i \(0.495494\pi\)
\(888\) 0 0
\(889\) −4.07216 −0.136576
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 13.0284i 0.435979i
\(894\) 0 0
\(895\) 14.0500 0.469641
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 10.6023i − 0.353607i
\(900\) 0 0
\(901\) 0.955367i 0.0318279i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 69.6629 2.31567
\(906\) 0 0
\(907\) − 9.01198i − 0.299238i −0.988744 0.149619i \(-0.952195\pi\)
0.988744 0.149619i \(-0.0478047\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 1.53535 0.0508685 0.0254343 0.999676i \(-0.491903\pi\)
0.0254343 + 0.999676i \(0.491903\pi\)
\(912\) 0 0
\(913\) −74.3195 −2.45962
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 17.8643i − 0.589930i
\(918\) 0 0
\(919\) 22.4581 0.740826 0.370413 0.928867i \(-0.379216\pi\)
0.370413 + 0.928867i \(0.379216\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 56.0047i 1.84342i
\(924\) 0 0
\(925\) − 60.5572i − 1.99111i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −50.2178 −1.64759 −0.823797 0.566885i \(-0.808148\pi\)
−0.823797 + 0.566885i \(0.808148\pi\)
\(930\) 0 0
\(931\) 3.20627i 0.105081i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 56.5655 1.84989
\(936\) 0 0
\(937\) 11.9114 0.389130 0.194565 0.980890i \(-0.437670\pi\)
0.194565 + 0.980890i \(0.437670\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 59.5916i − 1.94263i −0.237795 0.971315i \(-0.576425\pi\)
0.237795 0.971315i \(-0.423575\pi\)
\(942\) 0 0
\(943\) 5.01321 0.163252
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 4.69613i − 0.152604i −0.997085 0.0763018i \(-0.975689\pi\)
0.997085 0.0763018i \(-0.0243113\pi\)
\(948\) 0 0
\(949\) − 4.88966i − 0.158725i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 38.9727 1.26245 0.631224 0.775600i \(-0.282553\pi\)
0.631224 + 0.775600i \(0.282553\pi\)
\(954\) 0 0
\(955\) − 21.1522i − 0.684469i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −21.1351 −0.682488
\(960\) 0 0
\(961\) −17.1731 −0.553971
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 79.3390i − 2.55401i
\(966\) 0 0
\(967\) −40.5025 −1.30247 −0.651236 0.758875i \(-0.725749\pi\)
−0.651236 + 0.758875i \(0.725749\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 5.76542i − 0.185021i −0.995712 0.0925105i \(-0.970511\pi\)
0.995712 0.0925105i \(-0.0294892\pi\)
\(972\) 0 0
\(973\) 7.39359i 0.237028i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −10.3362 −0.330684 −0.165342 0.986236i \(-0.552873\pi\)
−0.165342 + 0.986236i \(0.552873\pi\)
\(978\) 0 0
\(979\) − 32.3530i − 1.03401i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −22.9663 −0.732512 −0.366256 0.930514i \(-0.619361\pi\)
−0.366256 + 0.930514i \(0.619361\pi\)
\(984\) 0 0
\(985\) −11.9805 −0.381730
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 8.46581i − 0.269197i
\(990\) 0 0
\(991\) 5.74450 0.182480 0.0912400 0.995829i \(-0.470917\pi\)
0.0912400 + 0.995829i \(0.470917\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 22.7602i − 0.721548i
\(996\) 0 0
\(997\) − 59.7098i − 1.89103i −0.325580 0.945515i \(-0.605559\pi\)
0.325580 0.945515i \(-0.394441\pi\)
\(998\) 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.c.e.3025.3 20
3.2 odd 2 inner 6048.2.c.e.3025.17 20
4.3 odd 2 1512.2.c.e.757.3 20
8.3 odd 2 1512.2.c.e.757.4 yes 20
8.5 even 2 inner 6048.2.c.e.3025.18 20
12.11 even 2 1512.2.c.e.757.18 yes 20
24.5 odd 2 inner 6048.2.c.e.3025.4 20
24.11 even 2 1512.2.c.e.757.17 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.c.e.757.3 20 4.3 odd 2
1512.2.c.e.757.4 yes 20 8.3 odd 2
1512.2.c.e.757.17 yes 20 24.11 even 2
1512.2.c.e.757.18 yes 20 12.11 even 2
6048.2.c.e.3025.3 20 1.1 even 1 trivial
6048.2.c.e.3025.4 20 24.5 odd 2 inner
6048.2.c.e.3025.17 20 3.2 odd 2 inner
6048.2.c.e.3025.18 20 8.5 even 2 inner