Properties

Label 6048.2.h.i.2591.6
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 2591.6
Character \(\chi\) = 6048.2591
Dual form 6048.2.h.i.2591.5

$q$-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 \(0.201342\pi\)
−0.806532 + 0.591190i \(0.798658\pi\)
\(6\) 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 \(-0.286630\pi\)
0.621238 + 0.783622i \(0.286630\pi\)
\(12\) 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 \(-0.920528\pi\)
0.968994 0.247083i \(-0.0794719\pi\)
\(18\) 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 \(-0.0469440\pi\)
0.989145 + 0.146945i \(0.0469440\pi\)
\(24\) 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 \(0.0458943\pi\)
−0.989624 + 0.143682i \(0.954106\pi\)
\(30\) 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 \(0.0501225\pi\)
−0.987628 + 0.156815i \(0.949877\pi\)
\(42\) 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 \(-0.439410\pi\)
0.189203 + 0.981938i \(0.439410\pi\)
\(48\) 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 \(-0.707910\pi\)
0.607707 0.794161i \(-0.292090\pi\)
\(54\) 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 \(-0.800165\pi\)
−0.809322 + 0.587365i \(0.800165\pi\)
\(60\) 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 \(-0.385084\pi\)
0.353229 + 0.935537i \(0.385084\pi\)
\(72\) 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 \(-0.430620\pi\)
0.216241 + 0.976340i \(0.430620\pi\)
\(84\) 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 \(-0.759837\pi\)
0.728619 0.684920i \(-0.240163\pi\)
\(90\) 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 \(-0.924699\pi\)
0.972149 0.234364i \(-0.0753008\pi\)
\(102\) 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 \(-0.563560\pi\)
−0.198355 + 0.980130i \(0.563560\pi\)
\(108\) 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 \(0.0849561\pi\)
−0.964594 + 0.263740i \(0.915044\pi\)
\(114\) 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 \(-0.471453\pi\)
0.0895615 + 0.995981i \(0.471453\pi\)
\(132\) 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 \(0.146711\pi\)
−0.895650 + 0.444759i \(0.853289\pi\)
\(138\) 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 \(-0.623993\pi\)
0.379758 0.925086i \(-0.376007\pi\)
\(150\) 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 \(-0.458536\pi\)
0.129893 + 0.991528i \(0.458536\pi\)
\(168\) 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 \(0.219572\pi\)
−0.771369 + 0.636388i \(0.780428\pi\)
\(174\) 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 \(-0.663017\pi\)
−0.490037 + 0.871702i \(0.663017\pi\)
\(180\) 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 \(-0.582843\pi\)
−0.257331 + 0.966323i \(0.582843\pi\)
\(192\) 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 \(0.0971340\pi\)
−0.953800 + 0.300441i \(0.902866\pi\)
\(198\) 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 \(-0.661124\pi\)
−0.484846 + 0.874599i \(0.661124\pi\)
\(228\) 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 \(-0.948516\pi\)
0.986948 0.161036i \(-0.0514837\pi\)
\(234\) 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 \(-0.257219\pi\)
0.690890 + 0.722960i \(0.257219\pi\)
\(240\) 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 \(-0.386349\pi\)
0.349508 + 0.936933i \(0.386349\pi\)
\(252\) 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 \(0.118982\pi\)
−0.930949 + 0.365150i \(0.881018\pi\)
\(258\) 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 \(-0.616107\pi\)
−0.356726 + 0.934209i \(0.616107\pi\)
\(264\) 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 \(-0.807670\pi\)
0.822944 0.568123i \(-0.192330\pi\)
\(270\) 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 \(-0.886600\pi\)
0.937209 0.348767i \(-0.113400\pi\)
\(282\) 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 \(0.224955\pi\)
−0.760497 + 0.649341i \(0.775045\pi\)
\(294\) 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 \(-0.465559\pi\)
0.107989 + 0.994152i \(0.465559\pi\)
\(312\) 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 \(0.0568059\pi\)
−0.984118 + 0.177515i \(0.943194\pi\)
\(318\) 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 \(-0.243667\pi\)
0.721034 + 0.692900i \(0.243667\pi\)
\(348\) 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 \(-0.721232\pi\)
0.640401 0.768041i \(-0.278768\pi\)
\(354\) 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 \(-0.257867\pi\)
0.689417 + 0.724365i \(0.257867\pi\)
\(360\) 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 \(-0.433089\pi\)
0.208662 + 0.977988i \(0.433089\pi\)
\(384\) 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 \(0.231416\pi\)
−0.747161 + 0.664643i \(0.768584\pi\)
\(390\) 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 \(0.170146\pi\)
−0.860509 + 0.509435i \(0.829854\pi\)
\(402\) 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 \(-0.560978\pi\)
−0.190400 + 0.981707i \(0.560978\pi\)
\(420\) 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 \(-0.371469\pi\)
0.392907 + 0.919578i \(0.371469\pi\)
\(432\) 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 \(-0.718135\pi\)
−0.632898 + 0.774235i \(0.718135\pi\)
\(444\) 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 \(0.0545217\pi\)
−0.985367 + 0.170449i \(0.945478\pi\)
\(450\) 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 \(-0.861523\pi\)
0.906854 0.421446i \(-0.138477\pi\)
\(462\) 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 \(-0.764402\pi\)
−0.738366 + 0.674400i \(0.764402\pi\)
\(468\) 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 \(-0.472782\pi\)
0.0854031 + 0.996346i \(0.472782\pi\)
\(480\) 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 \(-0.503080\pi\)
−0.00967468 + 0.999953i \(0.503080\pi\)
\(492\) 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 \(-0.515611\pi\)
−0.0490239 + 0.998798i \(0.515611\pi\)
\(504\) 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 \(-0.880097\pi\)
0.929888 0.367842i \(-0.119903\pi\)
\(510\) 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 \(-0.642564\pi\)
0.433053 0.901368i \(-0.357436\pi\)
\(522\) 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 \(-0.761698\pi\)
0.732610 0.680648i \(-0.238302\pi\)
\(558\) 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 \(-0.531074\pi\)
−0.0974663 + 0.995239i \(0.531074\pi\)
\(564\) 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 \(-0.812231\pi\)
0.831000 0.556272i \(-0.187769\pi\)
\(570\) 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 \(-0.448499\pi\)
0.161091 + 0.986940i \(0.448499\pi\)
\(588\) 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 \(-0.672911\pi\)
0.516892 0.856051i \(-0.327089\pi\)
\(594\) 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 \(-0.445204\pi\)
0.171297 + 0.985219i \(0.445204\pi\)
\(600\) 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 \(-0.858501\pi\)
0.902812 0.430035i \(-0.141499\pi\)
\(618\) 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 \(0.0464605\pi\)
−0.989367 + 0.145442i \(0.953540\pi\)
\(642\) 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 \(-0.363675\pi\)
0.415305 + 0.909682i \(0.363675\pi\)
\(648\) 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 \(0.395690\pi\)
−0.321867 + 0.946785i \(0.604310\pi\)
\(654\) 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 \(-0.393592\pi\)
0.328099 + 0.944643i \(0.393592\pi\)
\(660\) 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 \(-0.715057\pi\)
0.625382 0.780318i \(-0.284943\pi\)
\(678\) 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 \(-0.519822\pi\)
−0.0622330 + 0.998062i \(0.519822\pi\)
\(684\) 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 \(0.0427934\pi\)
−0.990977 + 0.134035i \(0.957207\pi\)
\(702\) 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 \(-0.434327\pi\)
0.204857 + 0.978792i \(0.434327\pi\)
\(720\) 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 \(-0.321979\pi\)
0.530566 + 0.847644i \(0.321979\pi\)
\(744\) 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 \(0.0435624\pi\)
−0.990650 + 0.136428i \(0.956438\pi\)
\(762\) 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 \(-0.694483\pi\)
0.573676 0.819083i \(-0.305517\pi\)
\(774\) 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 \(0.147878\pi\)
−0.894013 + 0.448040i \(0.852122\pi\)
\(798\) 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 \(-0.865511\pi\)
0.912063 0.410050i \(-0.134489\pi\)
\(810\) 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 \(0.404955\pi\)
−0.294175 + 0.955752i \(0.595045\pi\)
\(822\) 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 \(-0.637118\pi\)
−0.417571 + 0.908644i \(0.637118\pi\)
\(828\) 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 \(-0.576121\pi\)
−0.236869 + 0.971542i \(0.576121\pi\)
\(840\) 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 \(0.0346240\pi\)
−0.994090 + 0.108560i \(0.965376\pi\)
\(858\) 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 \(-0.598820\pi\)
−0.305488 + 0.952196i \(0.598820\pi\)
\(864\) 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 \(0.215520\pi\)
−0.779408 + 0.626516i \(0.784480\pi\)
\(882\) 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 \(-0.504812\pi\)
−0.0151177 + 0.999886i \(0.504812\pi\)
\(888\) 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 \(-0.849979\pi\)
−0.890977 + 0.454049i \(0.849979\pi\)
\(912\) 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 \(0.178157\pi\)
−0.847416 + 0.530930i \(0.821843\pi\)
\(930\) 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 \(0.117478\pi\)
−0.932664 + 0.360747i \(0.882522\pi\)
\(942\) 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 \(-0.547387\pi\)
−0.148322 + 0.988939i \(0.547387\pi\)
\(948\) 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 \(-0.977266\pi\)
0.997451 0.0713588i \(-0.0227335\pi\)
\(954\) 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 \(-0.523574\pi\)
−0.0739936 + 0.997259i \(0.523574\pi\)
\(972\) 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 \(-0.922460\pi\)
0.970476 0.241197i \(-0.0775401\pi\)
\(978\) 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 \(-0.693176\pi\)
−0.570308 + 0.821431i \(0.693176\pi\)
\(984\) 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.2591.6 yes 24
3.2 odd 2 inner 6048.2.h.i.2591.19 yes 24
4.3 odd 2 inner 6048.2.h.i.2591.20 yes 24
12.11 even 2 inner 6048.2.h.i.2591.5 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6048.2.h.i.2591.5 24 12.11 even 2 inner
6048.2.h.i.2591.6 yes 24 1.1 even 1 trivial
6048.2.h.i.2591.19 yes 24 3.2 odd 2 inner
6048.2.h.i.2591.20 yes 24 4.3 odd 2 inner