Properties

Label 6048.2.h.i.2591.19
Level 6048
Weight 2
Character 6048.2591
Analytic conductor 48.294
Analytic rank 0
Dimension 24
CM no
Inner twists 4

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(48.2935231425\)
Analytic rank: \(0\)
Dimension: \(24\)
Coefficient ring index: multiple of None
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2591.19
Character \(\chi\) = 6048.2591
Dual form 6048.2.h.i.2591.20

$q$-expansion

\(f(q)\) \(=\) \(q-2.64388i q^{5} +1.00000i q^{7} +O(q^{10})\) \(q-2.64388i q^{5} +1.00000i q^{7} -4.12083 q^{11} -4.86886 q^{13} +2.03749i q^{17} -4.16827i q^{19} -9.48754 q^{23} -1.99011 q^{25} -1.54751i q^{29} -0.248395i q^{31} +2.64388 q^{35} +7.88909 q^{37} -2.00821i q^{41} +0.547808i q^{43} -2.59422 q^{47} -1.00000 q^{49} +11.5632i q^{53} +10.8950i q^{55} +12.4330 q^{59} +14.2962 q^{61} +12.8727i q^{65} -6.52832i q^{67} -5.95273 q^{71} +11.7079 q^{73} -4.12083i q^{77} +7.47832i q^{79} -3.94011 q^{83} +5.38690 q^{85} +12.9230i q^{89} -4.86886i q^{91} -11.0204 q^{95} -5.64630 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24q + O(q^{10}) \) \( 24q - 8q^{13} - 24q^{25} + 48q^{37} - 24q^{49} - 48q^{61} - 32q^{73} - 80q^{85} + 64q^{97} + O(q^{100}) \)

Character values

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

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

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) − 2.64388i − 1.18238i −0.806532 0.591190i \(-0.798658\pi\)
0.806532 0.591190i \(-0.201342\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −4.12083 −1.24248 −0.621238 0.783622i \(-0.713370\pi\)
−0.621238 + 0.783622i \(0.713370\pi\)
\(12\) 0 0
\(13\) −4.86886 −1.35038 −0.675190 0.737644i \(-0.735938\pi\)
−0.675190 + 0.737644i \(0.735938\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.03749i 0.494165i 0.968994 + 0.247083i \(0.0794719\pi\)
−0.968994 + 0.247083i \(0.920528\pi\)
\(18\) 0 0
\(19\) − 4.16827i − 0.956268i −0.878287 0.478134i \(-0.841313\pi\)
0.878287 0.478134i \(-0.158687\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −9.48754 −1.97829 −0.989145 0.146945i \(-0.953056\pi\)
−0.989145 + 0.146945i \(0.953056\pi\)
\(24\) 0 0
\(25\) −1.99011 −0.398022
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 1.54751i − 0.287365i −0.989624 0.143682i \(-0.954106\pi\)
0.989624 0.143682i \(-0.0458943\pi\)
\(30\) 0 0
\(31\) − 0.248395i − 0.0446131i −0.999751 0.0223066i \(-0.992899\pi\)
0.999751 0.0223066i \(-0.00710099\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.64388 0.446898
\(36\) 0 0
\(37\) 7.88909 1.29696 0.648479 0.761232i \(-0.275405\pi\)
0.648479 + 0.761232i \(0.275405\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 2.00821i − 0.313629i −0.987628 0.156815i \(-0.949877\pi\)
0.987628 0.156815i \(-0.0501225\pi\)
\(42\) 0 0
\(43\) 0.547808i 0.0835399i 0.999127 + 0.0417700i \(0.0132997\pi\)
−0.999127 + 0.0417700i \(0.986700\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.59422 −0.378406 −0.189203 0.981938i \(-0.560590\pi\)
−0.189203 + 0.981938i \(0.560590\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 11.5632i 1.58832i 0.607707 + 0.794161i \(0.292090\pi\)
−0.607707 + 0.794161i \(0.707910\pi\)
\(54\) 0 0
\(55\) 10.8950i 1.46908i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 12.4330 1.61864 0.809322 0.587365i \(-0.199835\pi\)
0.809322 + 0.587365i \(0.199835\pi\)
\(60\) 0 0
\(61\) 14.2962 1.83044 0.915218 0.402959i \(-0.132018\pi\)
0.915218 + 0.402959i \(0.132018\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 12.8727i 1.59666i
\(66\) 0 0
\(67\) − 6.52832i − 0.797561i −0.917046 0.398781i \(-0.869433\pi\)
0.917046 0.398781i \(-0.130567\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −5.95273 −0.706459 −0.353229 0.935537i \(-0.614916\pi\)
−0.353229 + 0.935537i \(0.614916\pi\)
\(72\) 0 0
\(73\) 11.7079 1.37031 0.685156 0.728396i \(-0.259734\pi\)
0.685156 + 0.728396i \(0.259734\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 4.12083i − 0.469612i
\(78\) 0 0
\(79\) 7.47832i 0.841376i 0.907205 + 0.420688i \(0.138211\pi\)
−0.907205 + 0.420688i \(0.861789\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −3.94011 −0.432483 −0.216241 0.976340i \(-0.569380\pi\)
−0.216241 + 0.976340i \(0.569380\pi\)
\(84\) 0 0
\(85\) 5.38690 0.584291
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 12.9230i 1.36984i 0.728619 + 0.684920i \(0.240163\pi\)
−0.728619 + 0.684920i \(0.759837\pi\)
\(90\) 0 0
\(91\) − 4.86886i − 0.510395i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −11.0204 −1.13067
\(96\) 0 0
\(97\) −5.64630 −0.573295 −0.286648 0.958036i \(-0.592541\pi\)
−0.286648 + 0.958036i \(0.592541\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 4.71066i 0.468729i 0.972149 + 0.234364i \(0.0753008\pi\)
−0.972149 + 0.234364i \(0.924699\pi\)
\(102\) 0 0
\(103\) 19.6158i 1.93280i 0.257038 + 0.966401i \(0.417253\pi\)
−0.257038 + 0.966401i \(0.582747\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.10360 0.396710 0.198355 0.980130i \(-0.436440\pi\)
0.198355 + 0.980130i \(0.436440\pi\)
\(108\) 0 0
\(109\) 8.48792 0.812995 0.406497 0.913652i \(-0.366750\pi\)
0.406497 + 0.913652i \(0.366750\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 5.60719i − 0.527480i −0.964594 0.263740i \(-0.915044\pi\)
0.964594 0.263740i \(-0.0849561\pi\)
\(114\) 0 0
\(115\) 25.0839i 2.33909i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −2.03749 −0.186777
\(120\) 0 0
\(121\) 5.98124 0.543749
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 7.95779i − 0.711766i
\(126\) 0 0
\(127\) − 16.0322i − 1.42263i −0.702875 0.711313i \(-0.748101\pi\)
0.702875 0.711313i \(-0.251899\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −2.05016 −0.179123 −0.0895615 0.995981i \(-0.528547\pi\)
−0.0895615 + 0.995981i \(0.528547\pi\)
\(132\) 0 0
\(133\) 4.16827 0.361435
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 10.4115i − 0.889517i −0.895650 0.444759i \(-0.853289\pi\)
0.895650 0.444759i \(-0.146711\pi\)
\(138\) 0 0
\(139\) 13.0177i 1.10415i 0.833795 + 0.552074i \(0.186163\pi\)
−0.833795 + 0.552074i \(0.813837\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 20.0637 1.67781
\(144\) 0 0
\(145\) −4.09142 −0.339774
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 22.5842i 1.85017i 0.379758 + 0.925086i \(0.376007\pi\)
−0.379758 + 0.925086i \(0.623993\pi\)
\(150\) 0 0
\(151\) 23.7533i 1.93302i 0.256638 + 0.966508i \(0.417385\pi\)
−0.256638 + 0.966508i \(0.582615\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −0.656728 −0.0527497
\(156\) 0 0
\(157\) 1.78100 0.142139 0.0710696 0.997471i \(-0.477359\pi\)
0.0710696 + 0.997471i \(0.477359\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 9.48754i − 0.747723i
\(162\) 0 0
\(163\) − 8.78773i − 0.688308i −0.938913 0.344154i \(-0.888166\pi\)
0.938913 0.344154i \(-0.111834\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −3.35719 −0.259787 −0.129893 0.991528i \(-0.541464\pi\)
−0.129893 + 0.991528i \(0.541464\pi\)
\(168\) 0 0
\(169\) 10.7058 0.823524
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 16.7408i − 1.27278i −0.771369 0.636388i \(-0.780428\pi\)
0.771369 0.636388i \(-0.219572\pi\)
\(174\) 0 0
\(175\) − 1.99011i − 0.150438i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 13.1125 0.980074 0.490037 0.871702i \(-0.336983\pi\)
0.490037 + 0.871702i \(0.336983\pi\)
\(180\) 0 0
\(181\) −10.1190 −0.752136 −0.376068 0.926592i \(-0.622724\pi\)
−0.376068 + 0.926592i \(0.622724\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 20.8578i − 1.53350i
\(186\) 0 0
\(187\) − 8.39617i − 0.613989i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 7.11278 0.514663 0.257331 0.966323i \(-0.417157\pi\)
0.257331 + 0.966323i \(0.417157\pi\)
\(192\) 0 0
\(193\) 9.21790 0.663519 0.331759 0.943364i \(-0.392358\pi\)
0.331759 + 0.943364i \(0.392358\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 8.43379i − 0.600883i −0.953800 0.300441i \(-0.902866\pi\)
0.953800 0.300441i \(-0.0971340\pi\)
\(198\) 0 0
\(199\) 0.742006i 0.0525994i 0.999654 + 0.0262997i \(0.00837242\pi\)
−0.999654 + 0.0262997i \(0.991628\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1.54751 0.108614
\(204\) 0 0
\(205\) −5.30946 −0.370829
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 17.1767i 1.18814i
\(210\) 0 0
\(211\) 7.66439i 0.527639i 0.964572 + 0.263819i \(0.0849823\pi\)
−0.964572 + 0.263819i \(0.915018\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 1.44834 0.0987759
\(216\) 0 0
\(217\) 0.248395 0.0168622
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 9.92028i − 0.667310i
\(222\) 0 0
\(223\) 16.3982i 1.09810i 0.835788 + 0.549052i \(0.185011\pi\)
−0.835788 + 0.549052i \(0.814989\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 14.6099 0.969692 0.484846 0.874599i \(-0.338876\pi\)
0.484846 + 0.874599i \(0.338876\pi\)
\(228\) 0 0
\(229\) −20.1027 −1.32842 −0.664212 0.747544i \(-0.731233\pi\)
−0.664212 + 0.747544i \(0.731233\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 4.91623i 0.322073i 0.986948 + 0.161036i \(0.0514837\pi\)
−0.986948 + 0.161036i \(0.948516\pi\)
\(234\) 0 0
\(235\) 6.85881i 0.447419i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −21.3618 −1.38178 −0.690890 0.722960i \(-0.742781\pi\)
−0.690890 + 0.722960i \(0.742781\pi\)
\(240\) 0 0
\(241\) 5.87880 0.378687 0.189344 0.981911i \(-0.439364\pi\)
0.189344 + 0.981911i \(0.439364\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 2.64388i 0.168911i
\(246\) 0 0
\(247\) 20.2947i 1.29132i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −11.0745 −0.699016 −0.349508 0.936933i \(-0.613651\pi\)
−0.349508 + 0.936933i \(0.613651\pi\)
\(252\) 0 0
\(253\) 39.0965 2.45798
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 11.7076i − 0.730299i −0.930949 0.365150i \(-0.881018\pi\)
0.930949 0.365150i \(-0.118982\pi\)
\(258\) 0 0
\(259\) 7.88909i 0.490204i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 11.5702 0.713451 0.356726 0.934209i \(-0.383893\pi\)
0.356726 + 0.934209i \(0.383893\pi\)
\(264\) 0 0
\(265\) 30.5716 1.87800
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 18.6358i 1.13625i 0.822944 + 0.568123i \(0.192330\pi\)
−0.822944 + 0.568123i \(0.807670\pi\)
\(270\) 0 0
\(271\) 11.2036i 0.680571i 0.940322 + 0.340286i \(0.110524\pi\)
−0.940322 + 0.340286i \(0.889476\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 8.20091 0.494533
\(276\) 0 0
\(277\) −29.2418 −1.75697 −0.878485 0.477769i \(-0.841446\pi\)
−0.878485 + 0.477769i \(0.841446\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 11.6928i 0.697534i 0.937209 + 0.348767i \(0.113400\pi\)
−0.937209 + 0.348767i \(0.886600\pi\)
\(282\) 0 0
\(283\) − 18.3784i − 1.09248i −0.837628 0.546242i \(-0.816058\pi\)
0.837628 0.546242i \(-0.183942\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 2.00821 0.118541
\(288\) 0 0
\(289\) 12.8486 0.755801
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 22.2299i − 1.29868i −0.760497 0.649341i \(-0.775045\pi\)
0.760497 0.649341i \(-0.224955\pi\)
\(294\) 0 0
\(295\) − 32.8715i − 1.91385i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 46.1935 2.67144
\(300\) 0 0
\(301\) −0.547808 −0.0315751
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 37.7974i − 2.16427i
\(306\) 0 0
\(307\) 17.2640i 0.985306i 0.870226 + 0.492653i \(0.163973\pi\)
−0.870226 + 0.492653i \(0.836027\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −3.80882 −0.215978 −0.107989 0.994152i \(-0.534441\pi\)
−0.107989 + 0.994152i \(0.534441\pi\)
\(312\) 0 0
\(313\) 25.6882 1.45198 0.725991 0.687705i \(-0.241382\pi\)
0.725991 + 0.687705i \(0.241382\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 6.32114i − 0.355030i −0.984118 0.177515i \(-0.943194\pi\)
0.984118 0.177515i \(-0.0568059\pi\)
\(318\) 0 0
\(319\) 6.37701i 0.357044i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 8.49284 0.472554
\(324\) 0 0
\(325\) 9.68958 0.537481
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 2.59422i − 0.143024i
\(330\) 0 0
\(331\) − 3.46340i − 0.190366i −0.995460 0.0951829i \(-0.969656\pi\)
0.995460 0.0951829i \(-0.0303436\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −17.2601 −0.943020
\(336\) 0 0
\(337\) −5.11720 −0.278752 −0.139376 0.990240i \(-0.544510\pi\)
−0.139376 + 0.990240i \(0.544510\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 1.02359i 0.0554308i
\(342\) 0 0
\(343\) − 1.00000i − 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −26.8628 −1.44207 −0.721034 0.692900i \(-0.756333\pi\)
−0.721034 + 0.692900i \(0.756333\pi\)
\(348\) 0 0
\(349\) −19.9241 −1.06651 −0.533256 0.845954i \(-0.679032\pi\)
−0.533256 + 0.845954i \(0.679032\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 28.8604i 1.53608i 0.640401 + 0.768041i \(0.278768\pi\)
−0.640401 + 0.768041i \(0.721232\pi\)
\(354\) 0 0
\(355\) 15.7383i 0.835302i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −26.1252 −1.37883 −0.689417 0.724365i \(-0.742133\pi\)
−0.689417 + 0.724365i \(0.742133\pi\)
\(360\) 0 0
\(361\) 1.62549 0.0855522
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 30.9544i − 1.62023i
\(366\) 0 0
\(367\) − 17.4489i − 0.910827i −0.890280 0.455414i \(-0.849491\pi\)
0.890280 0.455414i \(-0.150509\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −11.5632 −0.600330
\(372\) 0 0
\(373\) 21.0226 1.08851 0.544254 0.838920i \(-0.316813\pi\)
0.544254 + 0.838920i \(0.316813\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 7.53459i 0.388051i
\(378\) 0 0
\(379\) 28.6392i 1.47110i 0.677471 + 0.735549i \(0.263076\pi\)
−0.677471 + 0.735549i \(0.736924\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −8.16718 −0.417323 −0.208662 0.977988i \(-0.566911\pi\)
−0.208662 + 0.977988i \(0.566911\pi\)
\(384\) 0 0
\(385\) −10.8950 −0.555260
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 26.2176i − 1.32929i −0.747161 0.664643i \(-0.768584\pi\)
0.747161 0.664643i \(-0.231416\pi\)
\(390\) 0 0
\(391\) − 19.3308i − 0.977601i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 19.7718 0.994827
\(396\) 0 0
\(397\) 16.3245 0.819304 0.409652 0.912242i \(-0.365650\pi\)
0.409652 + 0.912242i \(0.365650\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 20.4029i − 1.01887i −0.860509 0.509435i \(-0.829854\pi\)
0.860509 0.509435i \(-0.170146\pi\)
\(402\) 0 0
\(403\) 1.20940i 0.0602446i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −32.5096 −1.61144
\(408\) 0 0
\(409\) 24.0523 1.18931 0.594654 0.803982i \(-0.297289\pi\)
0.594654 + 0.803982i \(0.297289\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 12.4330i 0.611790i
\(414\) 0 0
\(415\) 10.4172i 0.511359i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 7.79478 0.380800 0.190400 0.981707i \(-0.439022\pi\)
0.190400 + 0.981707i \(0.439022\pi\)
\(420\) 0 0
\(421\) 30.7294 1.49766 0.748830 0.662762i \(-0.230616\pi\)
0.748830 + 0.662762i \(0.230616\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 4.05484i − 0.196689i
\(426\) 0 0
\(427\) 14.2962i 0.691840i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −16.3139 −0.785814 −0.392907 0.919578i \(-0.628531\pi\)
−0.392907 + 0.919578i \(0.628531\pi\)
\(432\) 0 0
\(433\) −6.02836 −0.289705 −0.144852 0.989453i \(-0.546271\pi\)
−0.144852 + 0.989453i \(0.546271\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 39.5467i 1.89177i
\(438\) 0 0
\(439\) 25.2529i 1.20526i 0.798022 + 0.602629i \(0.205880\pi\)
−0.798022 + 0.602629i \(0.794120\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 26.6419 1.26580 0.632898 0.774235i \(-0.281865\pi\)
0.632898 + 0.774235i \(0.281865\pi\)
\(444\) 0 0
\(445\) 34.1670 1.61967
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 7.22348i − 0.340897i −0.985367 0.170449i \(-0.945478\pi\)
0.985367 0.170449i \(-0.0545217\pi\)
\(450\) 0 0
\(451\) 8.27548i 0.389677i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −12.8727 −0.603481
\(456\) 0 0
\(457\) −39.2788 −1.83738 −0.918692 0.394974i \(-0.870754\pi\)
−0.918692 + 0.394974i \(0.870754\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 18.0976i 0.842892i 0.906854 + 0.421446i \(0.138477\pi\)
−0.906854 + 0.421446i \(0.861523\pi\)
\(462\) 0 0
\(463\) 28.4913i 1.32410i 0.749458 + 0.662052i \(0.230314\pi\)
−0.749458 + 0.662052i \(0.769686\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 31.9125 1.47673 0.738366 0.674400i \(-0.235598\pi\)
0.738366 + 0.674400i \(0.235598\pi\)
\(468\) 0 0
\(469\) 6.52832 0.301450
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 2.25742i − 0.103796i
\(474\) 0 0
\(475\) 8.29533i 0.380616i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −3.73828 −0.170806 −0.0854031 0.996346i \(-0.527218\pi\)
−0.0854031 + 0.996346i \(0.527218\pi\)
\(480\) 0 0
\(481\) −38.4109 −1.75139
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 14.9282i 0.677853i
\(486\) 0 0
\(487\) − 3.85953i − 0.174892i −0.996169 0.0874461i \(-0.972129\pi\)
0.996169 0.0874461i \(-0.0278706\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0.428753 0.0193494 0.00967468 0.999953i \(-0.496920\pi\)
0.00967468 + 0.999953i \(0.496920\pi\)
\(492\) 0 0
\(493\) 3.15303 0.142006
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 5.95273i − 0.267016i
\(498\) 0 0
\(499\) 39.9296i 1.78749i 0.448571 + 0.893747i \(0.351933\pi\)
−0.448571 + 0.893747i \(0.648067\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 2.19899 0.0980479 0.0490239 0.998798i \(-0.484389\pi\)
0.0490239 + 0.998798i \(0.484389\pi\)
\(504\) 0 0
\(505\) 12.4544 0.554215
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 16.5978i 0.735684i 0.929888 + 0.367842i \(0.119903\pi\)
−0.929888 + 0.367842i \(0.880097\pi\)
\(510\) 0 0
\(511\) 11.7079i 0.517929i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 51.8619 2.28531
\(516\) 0 0
\(517\) 10.6903 0.470161
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 41.1482i 1.80274i 0.433053 + 0.901368i \(0.357436\pi\)
−0.433053 + 0.901368i \(0.642564\pi\)
\(522\) 0 0
\(523\) 14.5857i 0.637790i 0.947790 + 0.318895i \(0.103312\pi\)
−0.947790 + 0.318895i \(0.896688\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0.506104 0.0220462
\(528\) 0 0
\(529\) 67.0135 2.91363
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 9.77768i 0.423518i
\(534\) 0 0
\(535\) − 10.8494i − 0.469062i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 4.12083 0.177497
\(540\) 0 0
\(541\) 27.2868 1.17315 0.586575 0.809895i \(-0.300476\pi\)
0.586575 + 0.809895i \(0.300476\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 22.4410i − 0.961269i
\(546\) 0 0
\(547\) − 31.2511i − 1.33620i −0.744071 0.668100i \(-0.767108\pi\)
0.744071 0.668100i \(-0.232892\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −6.45043 −0.274797
\(552\) 0 0
\(553\) −7.47832 −0.318010
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 32.1278i 1.36130i 0.732610 + 0.680648i \(0.238302\pi\)
−0.732610 + 0.680648i \(0.761698\pi\)
\(558\) 0 0
\(559\) − 2.66720i − 0.112811i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 4.62528 0.194933 0.0974663 0.995239i \(-0.468926\pi\)
0.0974663 + 0.995239i \(0.468926\pi\)
\(564\) 0 0
\(565\) −14.8248 −0.623682
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 26.5383i 1.11254i 0.831000 + 0.556272i \(0.187769\pi\)
−0.831000 + 0.556272i \(0.812231\pi\)
\(570\) 0 0
\(571\) − 30.6374i − 1.28213i −0.767485 0.641067i \(-0.778492\pi\)
0.767485 0.641067i \(-0.221508\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 18.8813 0.787403
\(576\) 0 0
\(577\) −39.0588 −1.62604 −0.813019 0.582238i \(-0.802177\pi\)
−0.813019 + 0.582238i \(0.802177\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 3.94011i − 0.163463i
\(582\) 0 0
\(583\) − 47.6498i − 1.97345i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −7.80587 −0.322183 −0.161091 0.986940i \(-0.551501\pi\)
−0.161091 + 0.986940i \(0.551501\pi\)
\(588\) 0 0
\(589\) −1.03538 −0.0426621
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 41.6924i 1.71210i 0.516892 + 0.856051i \(0.327089\pi\)
−0.516892 + 0.856051i \(0.672911\pi\)
\(594\) 0 0
\(595\) 5.38690i 0.220841i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −8.38483 −0.342595 −0.171297 0.985219i \(-0.554796\pi\)
−0.171297 + 0.985219i \(0.554796\pi\)
\(600\) 0 0
\(601\) 20.3188 0.828820 0.414410 0.910090i \(-0.363988\pi\)
0.414410 + 0.910090i \(0.363988\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 15.8137i − 0.642918i
\(606\) 0 0
\(607\) − 27.8127i − 1.12888i −0.825473 0.564442i \(-0.809091\pi\)
0.825473 0.564442i \(-0.190909\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 12.6309 0.510991
\(612\) 0 0
\(613\) −3.65547 −0.147643 −0.0738214 0.997271i \(-0.523519\pi\)
−0.0738214 + 0.997271i \(0.523519\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 21.3637i 0.860070i 0.902812 + 0.430035i \(0.141499\pi\)
−0.902812 + 0.430035i \(0.858501\pi\)
\(618\) 0 0
\(619\) 1.48661i 0.0597520i 0.999554 + 0.0298760i \(0.00951124\pi\)
−0.999554 + 0.0298760i \(0.990489\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −12.9230 −0.517750
\(624\) 0 0
\(625\) −30.9900 −1.23960
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 16.0740i 0.640912i
\(630\) 0 0
\(631\) − 10.2236i − 0.406995i −0.979075 0.203497i \(-0.934769\pi\)
0.979075 0.203497i \(-0.0652308\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −42.3872 −1.68208
\(636\) 0 0
\(637\) 4.86886 0.192911
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 7.36460i − 0.290884i −0.989367 0.145442i \(-0.953540\pi\)
0.989367 0.145442i \(-0.0464605\pi\)
\(642\) 0 0
\(643\) 20.0823i 0.791967i 0.918258 + 0.395983i \(0.129596\pi\)
−0.918258 + 0.395983i \(0.870404\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −21.1276 −0.830610 −0.415305 0.909682i \(-0.636325\pi\)
−0.415305 + 0.909682i \(0.636325\pi\)
\(648\) 0 0
\(649\) −51.2345 −2.01113
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 48.3880i − 1.89357i −0.321867 0.946785i \(-0.604310\pi\)
0.321867 0.946785i \(-0.395690\pi\)
\(654\) 0 0
\(655\) 5.42037i 0.211791i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −16.8453 −0.656198 −0.328099 0.944643i \(-0.606408\pi\)
−0.328099 + 0.944643i \(0.606408\pi\)
\(660\) 0 0
\(661\) −25.7271 −1.00067 −0.500334 0.865832i \(-0.666790\pi\)
−0.500334 + 0.865832i \(0.666790\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 11.0204i − 0.427354i
\(666\) 0 0
\(667\) 14.6820i 0.568490i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −58.9120 −2.27427
\(672\) 0 0
\(673\) −0.545897 −0.0210428 −0.0105214 0.999945i \(-0.503349\pi\)
−0.0105214 + 0.999945i \(0.503349\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 40.6066i 1.56064i 0.625382 + 0.780318i \(0.284943\pi\)
−0.625382 + 0.780318i \(0.715057\pi\)
\(678\) 0 0
\(679\) − 5.64630i − 0.216685i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 3.25283 0.124466 0.0622330 0.998062i \(-0.480178\pi\)
0.0622330 + 0.998062i \(0.480178\pi\)
\(684\) 0 0
\(685\) −27.5269 −1.05175
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 56.2994i − 2.14484i
\(690\) 0 0
\(691\) 34.9406i 1.32920i 0.747198 + 0.664602i \(0.231399\pi\)
−0.747198 + 0.664602i \(0.768601\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 34.4173 1.30552
\(696\) 0 0
\(697\) 4.09171 0.154985
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 7.09751i − 0.268069i −0.990977 0.134035i \(-0.957207\pi\)
0.990977 0.134035i \(-0.0427934\pi\)
\(702\) 0 0
\(703\) − 32.8839i − 1.24024i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −4.71066 −0.177163
\(708\) 0 0
\(709\) −43.0565 −1.61702 −0.808510 0.588482i \(-0.799726\pi\)
−0.808510 + 0.588482i \(0.799726\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 2.35666i 0.0882577i
\(714\) 0 0
\(715\) − 53.0462i − 1.98381i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −10.9861 −0.409714 −0.204857 0.978792i \(-0.565673\pi\)
−0.204857 + 0.978792i \(0.565673\pi\)
\(720\) 0 0
\(721\) −19.6158 −0.730531
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 3.07971i 0.114377i
\(726\) 0 0
\(727\) − 10.6945i − 0.396637i −0.980138 0.198319i \(-0.936452\pi\)
0.980138 0.198319i \(-0.0635481\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −1.11616 −0.0412825
\(732\) 0 0
\(733\) 16.3535 0.604029 0.302015 0.953303i \(-0.402341\pi\)
0.302015 + 0.953303i \(0.402341\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 26.9021i 0.990951i
\(738\) 0 0
\(739\) − 33.0891i − 1.21720i −0.793476 0.608601i \(-0.791731\pi\)
0.793476 0.608601i \(-0.208269\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −28.9244 −1.06113 −0.530566 0.847644i \(-0.678021\pi\)
−0.530566 + 0.847644i \(0.678021\pi\)
\(744\) 0 0
\(745\) 59.7100 2.18761
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 4.10360i 0.149942i
\(750\) 0 0
\(751\) 18.3341i 0.669020i 0.942392 + 0.334510i \(0.108571\pi\)
−0.942392 + 0.334510i \(0.891429\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 62.8009 2.28556
\(756\) 0 0
\(757\) 14.8508 0.539761 0.269880 0.962894i \(-0.413016\pi\)
0.269880 + 0.962894i \(0.413016\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 7.52709i − 0.272857i −0.990650 0.136428i \(-0.956438\pi\)
0.990650 0.136428i \(-0.0435624\pi\)
\(762\) 0 0
\(763\) 8.48792i 0.307283i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −60.5348 −2.18578
\(768\) 0 0
\(769\) 49.6002 1.78863 0.894314 0.447440i \(-0.147664\pi\)
0.894314 + 0.447440i \(0.147664\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 45.5457i 1.63817i 0.573676 + 0.819083i \(0.305517\pi\)
−0.573676 + 0.819083i \(0.694483\pi\)
\(774\) 0 0
\(775\) 0.494334i 0.0177570i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −8.37076 −0.299914
\(780\) 0 0
\(781\) 24.5302 0.877758
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 4.70875i − 0.168063i
\(786\) 0 0
\(787\) − 24.5359i − 0.874610i −0.899313 0.437305i \(-0.855933\pi\)
0.899313 0.437305i \(-0.144067\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 5.60719 0.199369
\(792\) 0 0
\(793\) −69.6060 −2.47178
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 25.2974i − 0.896080i −0.894013 0.448040i \(-0.852122\pi\)
0.894013 0.448040i \(-0.147878\pi\)
\(798\) 0 0
\(799\) − 5.28571i − 0.186995i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −48.2465 −1.70258
\(804\) 0 0
\(805\) −25.0839 −0.884093
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 23.3260i 0.820099i 0.912063 + 0.410050i \(0.134489\pi\)
−0.912063 + 0.410050i \(0.865511\pi\)
\(810\) 0 0
\(811\) 36.8400i 1.29363i 0.762648 + 0.646814i \(0.223899\pi\)
−0.762648 + 0.646814i \(0.776101\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −23.2337 −0.813841
\(816\) 0 0
\(817\) 2.28341 0.0798865
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 54.7705i − 1.91150i −0.294175 0.955752i \(-0.595045\pi\)
0.294175 0.955752i \(-0.404955\pi\)
\(822\) 0 0
\(823\) − 43.9701i − 1.53270i −0.642423 0.766350i \(-0.722071\pi\)
0.642423 0.766350i \(-0.277929\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 24.0167 0.835142 0.417571 0.908644i \(-0.362882\pi\)
0.417571 + 0.908644i \(0.362882\pi\)
\(828\) 0 0
\(829\) 6.23391 0.216513 0.108256 0.994123i \(-0.465473\pi\)
0.108256 + 0.994123i \(0.465473\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 2.03749i − 0.0705950i
\(834\) 0 0
\(835\) 8.87601i 0.307167i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 13.7221 0.473738 0.236869 0.971542i \(-0.423879\pi\)
0.236869 + 0.971542i \(0.423879\pi\)
\(840\) 0 0
\(841\) 26.6052 0.917422
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 28.3049i − 0.973718i
\(846\) 0 0
\(847\) 5.98124i 0.205518i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −74.8481 −2.56576
\(852\) 0 0
\(853\) −36.8815 −1.26280 −0.631400 0.775457i \(-0.717519\pi\)
−0.631400 + 0.775457i \(0.717519\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 6.35611i − 0.217120i −0.994090 0.108560i \(-0.965376\pi\)
0.994090 0.108560i \(-0.0346240\pi\)
\(858\) 0 0
\(859\) 25.3449i 0.864755i 0.901693 + 0.432378i \(0.142325\pi\)
−0.901693 + 0.432378i \(0.857675\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 17.9486 0.610976 0.305488 0.952196i \(-0.401180\pi\)
0.305488 + 0.952196i \(0.401180\pi\)
\(864\) 0 0
\(865\) −44.2606 −1.50491
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 30.8169i − 1.04539i
\(870\) 0 0
\(871\) 31.7855i 1.07701i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 7.95779 0.269022
\(876\) 0 0
\(877\) −24.0001 −0.810427 −0.405213 0.914222i \(-0.632803\pi\)
−0.405213 + 0.914222i \(0.632803\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 37.1921i − 1.25303i −0.779408 0.626516i \(-0.784480\pi\)
0.779408 0.626516i \(-0.215520\pi\)
\(882\) 0 0
\(883\) − 48.2212i − 1.62277i −0.584511 0.811386i \(-0.698714\pi\)
0.584511 0.811386i \(-0.301286\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0.900487 0.0302354 0.0151177 0.999886i \(-0.495188\pi\)
0.0151177 + 0.999886i \(0.495188\pi\)
\(888\) 0 0
\(889\) 16.0322 0.537702
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 10.8134i 0.361857i
\(894\) 0 0
\(895\) − 34.6679i − 1.15882i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −0.384393 −0.0128202
\(900\) 0 0
\(901\) −23.5599 −0.784894
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 26.7533i 0.889311i
\(906\) 0 0
\(907\) − 7.11861i − 0.236370i −0.992992 0.118185i \(-0.962292\pi\)
0.992992 0.118185i \(-0.0377075\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 53.7843 1.78195 0.890977 0.454049i \(-0.150021\pi\)
0.890977 + 0.454049i \(0.150021\pi\)
\(912\) 0 0
\(913\) 16.2365 0.537350
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 2.05016i − 0.0677021i
\(918\) 0 0
\(919\) 59.4104i 1.95977i 0.199563 + 0.979885i \(0.436048\pi\)
−0.199563 + 0.979885i \(0.563952\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 28.9830 0.953987
\(924\) 0 0
\(925\) −15.7002 −0.516218
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 32.3649i − 1.06186i −0.847416 0.530930i \(-0.821843\pi\)
0.847416 0.530930i \(-0.178157\pi\)
\(930\) 0 0
\(931\) 4.16827i 0.136610i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −22.1985 −0.725968
\(936\) 0 0
\(937\) −14.8314 −0.484521 −0.242260 0.970211i \(-0.577889\pi\)
−0.242260 + 0.970211i \(0.577889\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 22.1324i − 0.721494i −0.932664 0.360747i \(-0.882522\pi\)
0.932664 0.360747i \(-0.117478\pi\)
\(942\) 0 0
\(943\) 19.0530i 0.620450i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 9.12874 0.296644 0.148322 0.988939i \(-0.452613\pi\)
0.148322 + 0.988939i \(0.452613\pi\)
\(948\) 0 0
\(949\) −57.0044 −1.85044
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 4.40579i 0.142718i 0.997451 + 0.0713588i \(0.0227335\pi\)
−0.997451 + 0.0713588i \(0.977266\pi\)
\(954\) 0 0
\(955\) − 18.8054i − 0.608527i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 10.4115 0.336206
\(960\) 0 0
\(961\) 30.9383 0.998010
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 24.3710i − 0.784531i
\(966\) 0 0
\(967\) 19.7941i 0.636536i 0.948001 + 0.318268i \(0.103101\pi\)
−0.948001 + 0.318268i \(0.896899\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 4.61141 0.147987 0.0739936 0.997259i \(-0.476426\pi\)
0.0739936 + 0.997259i \(0.476426\pi\)
\(972\) 0 0
\(973\) −13.0177 −0.417328
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 15.0782i 0.482395i 0.970476 + 0.241197i \(0.0775401\pi\)
−0.970476 + 0.241197i \(0.922460\pi\)
\(978\) 0 0
\(979\) − 53.2536i − 1.70199i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 35.7616 1.14062 0.570308 0.821431i \(-0.306824\pi\)
0.570308 + 0.821431i \(0.306824\pi\)
\(984\) 0 0
\(985\) −22.2980 −0.710472
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 5.19735i − 0.165266i
\(990\) 0 0
\(991\) 51.8448i 1.64690i 0.567387 + 0.823451i \(0.307954\pi\)
−0.567387 + 0.823451i \(0.692046\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 1.96178 0.0621925
\(996\) 0 0
\(997\) 42.7850 1.35501 0.677507 0.735516i \(-0.263060\pi\)
0.677507 + 0.735516i \(0.263060\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 6048.2.h.i.2591.19 yes 24
3.2 odd 2 inner 6048.2.h.i.2591.6 yes 24
4.3 odd 2 inner 6048.2.h.i.2591.5 24
12.11 even 2 inner 6048.2.h.i.2591.20 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6048.2.h.i.2591.5 24 4.3 odd 2 inner
6048.2.h.i.2591.6 yes 24 3.2 odd 2 inner
6048.2.h.i.2591.19 yes 24 1.1 even 1 trivial
6048.2.h.i.2591.20 yes 24 12.11 even 2 inner