Properties

Label 6048.2.h.d.2591.6
Level 6048
Weight 2
Character 6048.2591
Analytic conductor 48.294
Analytic rank 0
Dimension 8
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: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2591.6
Root \(0.965926 - 0.258819i\)
Character \(\chi\) = 6048.2591
Dual form 6048.2.h.d.2591.3

$q$-expansion

\(f(q)\) \(=\) \(q+0.317837i q^{5} +1.00000i q^{7} +O(q^{10})\) \(q+0.317837i q^{5} +1.00000i q^{7} +4.87832 q^{11} -2.44949 q^{13} +0.317837i q^{17} -0.449490i q^{19} -2.82843 q^{23} +4.89898 q^{25} -7.70674i q^{29} -4.44949i q^{31} -0.317837 q^{35} -7.00000 q^{37} -5.97469i q^{41} -10.3485i q^{43} -3.92480 q^{47} -1.00000 q^{49} +0.635674i q^{53} +1.55051i q^{55} +11.4887 q^{59} +9.34847 q^{61} -0.778539i q^{65} +8.00000i q^{67} +3.10102 q^{73} +4.87832i q^{77} -8.55051i q^{79} -10.2173 q^{83} -0.101021 q^{85} +14.1421i q^{89} -2.44949i q^{91} +0.142865 q^{95} +12.2474 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 56q^{37} - 8q^{49} + 16q^{61} + 64q^{73} - 40q^{85} + O(q^{100}) \)

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
\(5\) 0.317837i 0.142141i 0.997471 + 0.0710706i \(0.0226416\pi\)
−0.997471 + 0.0710706i \(0.977358\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.87832 1.47087 0.735434 0.677597i \(-0.236978\pi\)
0.735434 + 0.677597i \(0.236978\pi\)
\(12\) 0 0
\(13\) −2.44949 −0.679366 −0.339683 0.940540i \(-0.610320\pi\)
−0.339683 + 0.940540i \(0.610320\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.317837i 0.0770869i 0.999257 + 0.0385434i \(0.0122718\pi\)
−0.999257 + 0.0385434i \(0.987728\pi\)
\(18\) 0 0
\(19\) − 0.449490i − 0.103120i −0.998670 0.0515600i \(-0.983581\pi\)
0.998670 0.0515600i \(-0.0164193\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.82843 −0.589768 −0.294884 0.955533i \(-0.595281\pi\)
−0.294884 + 0.955533i \(0.595281\pi\)
\(24\) 0 0
\(25\) 4.89898 0.979796
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 7.70674i − 1.43111i −0.698558 0.715553i \(-0.746175\pi\)
0.698558 0.715553i \(-0.253825\pi\)
\(30\) 0 0
\(31\) − 4.44949i − 0.799152i −0.916700 0.399576i \(-0.869157\pi\)
0.916700 0.399576i \(-0.130843\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.317837 −0.0537243
\(36\) 0 0
\(37\) −7.00000 −1.15079 −0.575396 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 5.97469i − 0.933090i −0.884497 0.466545i \(-0.845499\pi\)
0.884497 0.466545i \(-0.154501\pi\)
\(42\) 0 0
\(43\) − 10.3485i − 1.57813i −0.614312 0.789063i \(-0.710566\pi\)
0.614312 0.789063i \(-0.289434\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.92480 −0.572491 −0.286246 0.958156i \(-0.592407\pi\)
−0.286246 + 0.958156i \(0.592407\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0.635674i 0.0873166i 0.999047 + 0.0436583i \(0.0139013\pi\)
−0.999047 + 0.0436583i \(0.986099\pi\)
\(54\) 0 0
\(55\) 1.55051i 0.209071i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 11.4887 1.49570 0.747849 0.663868i \(-0.231086\pi\)
0.747849 + 0.663868i \(0.231086\pi\)
\(60\) 0 0
\(61\) 9.34847 1.19695 0.598474 0.801142i \(-0.295774\pi\)
0.598474 + 0.801142i \(0.295774\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 0.778539i − 0.0965659i
\(66\) 0 0
\(67\) 8.00000i 0.977356i 0.872464 + 0.488678i \(0.162521\pi\)
−0.872464 + 0.488678i \(0.837479\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 3.10102 0.362947 0.181473 0.983396i \(-0.441913\pi\)
0.181473 + 0.983396i \(0.441913\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.87832i 0.555936i
\(78\) 0 0
\(79\) − 8.55051i − 0.962008i −0.876719 0.481004i \(-0.840272\pi\)
0.876719 0.481004i \(-0.159728\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −10.2173 −1.12150 −0.560749 0.827986i \(-0.689487\pi\)
−0.560749 + 0.827986i \(0.689487\pi\)
\(84\) 0 0
\(85\) −0.101021 −0.0109572
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 14.1421i 1.49906i 0.661968 + 0.749532i \(0.269721\pi\)
−0.661968 + 0.749532i \(0.730279\pi\)
\(90\) 0 0
\(91\) − 2.44949i − 0.256776i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0.142865 0.0146576
\(96\) 0 0
\(97\) 12.2474 1.24354 0.621770 0.783200i \(-0.286414\pi\)
0.621770 + 0.783200i \(0.286414\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 4.38551i 0.436374i 0.975907 + 0.218187i \(0.0700143\pi\)
−0.975907 + 0.218187i \(0.929986\pi\)
\(102\) 0 0
\(103\) − 3.10102i − 0.305553i −0.988261 0.152776i \(-0.951179\pi\)
0.988261 0.152776i \(-0.0488214\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −14.1421 −1.36717 −0.683586 0.729870i \(-0.739581\pi\)
−0.683586 + 0.729870i \(0.739581\pi\)
\(108\) 0 0
\(109\) 7.00000 0.670478 0.335239 0.942133i \(-0.391183\pi\)
0.335239 + 0.942133i \(0.391183\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 2.68556i 0.252636i 0.991990 + 0.126318i \(0.0403160\pi\)
−0.991990 + 0.126318i \(0.959684\pi\)
\(114\) 0 0
\(115\) − 0.898979i − 0.0838303i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −0.317837 −0.0291361
\(120\) 0 0
\(121\) 12.7980 1.16345
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 3.14626i 0.281410i
\(126\) 0 0
\(127\) − 15.2474i − 1.35299i −0.736446 0.676496i \(-0.763498\pi\)
0.736446 0.676496i \(-0.236502\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 16.0492 1.40222 0.701111 0.713052i \(-0.252688\pi\)
0.701111 + 0.713052i \(0.252688\pi\)
\(132\) 0 0
\(133\) 0.449490 0.0389757
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 8.97809i 0.767050i 0.923530 + 0.383525i \(0.125290\pi\)
−0.923530 + 0.383525i \(0.874710\pi\)
\(138\) 0 0
\(139\) − 12.2474i − 1.03882i −0.854527 0.519408i \(-0.826153\pi\)
0.854527 0.519408i \(-0.173847\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −11.9494 −0.999258
\(144\) 0 0
\(145\) 2.44949 0.203419
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 16.8277i − 1.37858i −0.724486 0.689289i \(-0.757923\pi\)
0.724486 0.689289i \(-0.242077\pi\)
\(150\) 0 0
\(151\) − 17.2474i − 1.40358i −0.712385 0.701789i \(-0.752385\pi\)
0.712385 0.701789i \(-0.247615\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1.41421 0.113592
\(156\) 0 0
\(157\) 15.3485 1.22494 0.612471 0.790493i \(-0.290176\pi\)
0.612471 + 0.790493i \(0.290176\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 2.82843i − 0.222911i
\(162\) 0 0
\(163\) − 11.2474i − 0.880968i −0.897760 0.440484i \(-0.854807\pi\)
0.897760 0.440484i \(-0.145193\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −9.93160 −0.768531 −0.384265 0.923223i \(-0.625545\pi\)
−0.384265 + 0.923223i \(0.625545\pi\)
\(168\) 0 0
\(169\) −7.00000 −0.538462
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 9.12096i 0.693453i 0.937966 + 0.346727i \(0.112707\pi\)
−0.937966 + 0.346727i \(0.887293\pi\)
\(174\) 0 0
\(175\) 4.89898i 0.370328i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 4.87832 0.364622 0.182311 0.983241i \(-0.441642\pi\)
0.182311 + 0.983241i \(0.441642\pi\)
\(180\) 0 0
\(181\) 24.6969 1.83571 0.917854 0.396917i \(-0.129920\pi\)
0.917854 + 0.396917i \(0.129920\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 2.22486i − 0.163575i
\(186\) 0 0
\(187\) 1.55051i 0.113385i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −16.0492 −1.16128 −0.580638 0.814162i \(-0.697197\pi\)
−0.580638 + 0.814162i \(0.697197\pi\)
\(192\) 0 0
\(193\) −11.0000 −0.791797 −0.395899 0.918294i \(-0.629567\pi\)
−0.395899 + 0.918294i \(0.629567\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0.778539i 0.0554686i 0.999615 + 0.0277343i \(0.00882924\pi\)
−0.999615 + 0.0277343i \(0.991171\pi\)
\(198\) 0 0
\(199\) 10.0000i 0.708881i 0.935079 + 0.354441i \(0.115329\pi\)
−0.935079 + 0.354441i \(0.884671\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 7.70674 0.540907
\(204\) 0 0
\(205\) 1.89898 0.132630
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 2.19275i − 0.151676i
\(210\) 0 0
\(211\) 6.00000i 0.413057i 0.978441 + 0.206529i \(0.0662166\pi\)
−0.978441 + 0.206529i \(0.933783\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 3.28913 0.224317
\(216\) 0 0
\(217\) 4.44949 0.302051
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 0.778539i − 0.0523702i
\(222\) 0 0
\(223\) 18.2474i 1.22194i 0.791654 + 0.610970i \(0.209220\pi\)
−0.791654 + 0.610970i \(0.790780\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 4.09978 0.272112 0.136056 0.990701i \(-0.456557\pi\)
0.136056 + 0.990701i \(0.456557\pi\)
\(228\) 0 0
\(229\) 12.6969 0.839037 0.419519 0.907747i \(-0.362199\pi\)
0.419519 + 0.907747i \(0.362199\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 18.7347i − 1.22735i −0.789558 0.613676i \(-0.789690\pi\)
0.789558 0.613676i \(-0.210310\pi\)
\(234\) 0 0
\(235\) − 1.24745i − 0.0813746i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −3.32124 −0.214833 −0.107416 0.994214i \(-0.534258\pi\)
−0.107416 + 0.994214i \(0.534258\pi\)
\(240\) 0 0
\(241\) −3.75255 −0.241723 −0.120862 0.992669i \(-0.538566\pi\)
−0.120862 + 0.992669i \(0.538566\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 0.317837i − 0.0203059i
\(246\) 0 0
\(247\) 1.10102i 0.0700563i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 18.4169 1.16246 0.581232 0.813738i \(-0.302571\pi\)
0.581232 + 0.813738i \(0.302571\pi\)
\(252\) 0 0
\(253\) −13.7980 −0.867470
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 11.0280i − 0.687906i −0.938987 0.343953i \(-0.888234\pi\)
0.938987 0.343953i \(-0.111766\pi\)
\(258\) 0 0
\(259\) − 7.00000i − 0.434959i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 7.07107 0.436021 0.218010 0.975946i \(-0.430043\pi\)
0.218010 + 0.975946i \(0.430043\pi\)
\(264\) 0 0
\(265\) −0.202041 −0.0124113
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 27.0450i 1.64896i 0.565888 + 0.824482i \(0.308534\pi\)
−0.565888 + 0.824482i \(0.691466\pi\)
\(270\) 0 0
\(271\) − 16.6969i − 1.01427i −0.861868 0.507133i \(-0.830705\pi\)
0.861868 0.507133i \(-0.169295\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 23.8988 1.44115
\(276\) 0 0
\(277\) 8.79796 0.528618 0.264309 0.964438i \(-0.414856\pi\)
0.264309 + 0.964438i \(0.414856\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 8.48528i − 0.506189i −0.967442 0.253095i \(-0.918552\pi\)
0.967442 0.253095i \(-0.0814484\pi\)
\(282\) 0 0
\(283\) − 32.2474i − 1.91691i −0.285241 0.958456i \(-0.592074\pi\)
0.285241 0.958456i \(-0.407926\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 5.97469 0.352675
\(288\) 0 0
\(289\) 16.8990 0.994058
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 20.7525i − 1.21237i −0.795322 0.606187i \(-0.792698\pi\)
0.795322 0.606187i \(-0.207302\pi\)
\(294\) 0 0
\(295\) 3.65153i 0.212600i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 6.92820 0.400668
\(300\) 0 0
\(301\) 10.3485 0.596476
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 2.97129i 0.170136i
\(306\) 0 0
\(307\) 4.89898i 0.279600i 0.990180 + 0.139800i \(0.0446459\pi\)
−0.990180 + 0.139800i \(0.955354\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 9.58166 0.543326 0.271663 0.962392i \(-0.412426\pi\)
0.271663 + 0.962392i \(0.412426\pi\)
\(312\) 0 0
\(313\) 6.89898 0.389953 0.194977 0.980808i \(-0.437537\pi\)
0.194977 + 0.980808i \(0.437537\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0.285729i 0.0160481i 0.999968 + 0.00802407i \(0.00255417\pi\)
−0.999968 + 0.00802407i \(0.997446\pi\)
\(318\) 0 0
\(319\) − 37.5959i − 2.10497i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0.142865 0.00794920
\(324\) 0 0
\(325\) −12.0000 −0.665640
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 3.92480i − 0.216381i
\(330\) 0 0
\(331\) 7.24745i 0.398356i 0.979963 + 0.199178i \(0.0638272\pi\)
−0.979963 + 0.199178i \(0.936173\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −2.54270 −0.138922
\(336\) 0 0
\(337\) 3.00000 0.163420 0.0817102 0.996656i \(-0.473962\pi\)
0.0817102 + 0.996656i \(0.473962\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 21.7060i − 1.17545i
\(342\) 0 0
\(343\) − 1.00000i − 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 1.69994 0.0912577 0.0456289 0.998958i \(-0.485471\pi\)
0.0456289 + 0.998958i \(0.485471\pi\)
\(348\) 0 0
\(349\) 14.2020 0.760218 0.380109 0.924942i \(-0.375887\pi\)
0.380109 + 0.924942i \(0.375887\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 10.3602i 0.551418i 0.961241 + 0.275709i \(0.0889126\pi\)
−0.961241 + 0.275709i \(0.911087\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −19.1633 −1.01140 −0.505701 0.862709i \(-0.668766\pi\)
−0.505701 + 0.862709i \(0.668766\pi\)
\(360\) 0 0
\(361\) 18.7980 0.989366
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0.985620i 0.0515897i
\(366\) 0 0
\(367\) − 12.6969i − 0.662775i −0.943495 0.331387i \(-0.892483\pi\)
0.943495 0.331387i \(-0.107517\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −0.635674 −0.0330026
\(372\) 0 0
\(373\) 33.2929 1.72384 0.861919 0.507045i \(-0.169262\pi\)
0.861919 + 0.507045i \(0.169262\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 18.8776i 0.972245i
\(378\) 0 0
\(379\) − 33.0454i − 1.69743i −0.528852 0.848714i \(-0.677377\pi\)
0.528852 0.848714i \(-0.322623\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −5.83183 −0.297992 −0.148996 0.988838i \(-0.547604\pi\)
−0.148996 + 0.988838i \(0.547604\pi\)
\(384\) 0 0
\(385\) −1.55051 −0.0790213
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 7.07107i 0.358517i 0.983802 + 0.179259i \(0.0573699\pi\)
−0.983802 + 0.179259i \(0.942630\pi\)
\(390\) 0 0
\(391\) − 0.898979i − 0.0454633i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 2.71767 0.136741
\(396\) 0 0
\(397\) −15.1010 −0.757898 −0.378949 0.925417i \(-0.623715\pi\)
−0.378949 + 0.925417i \(0.623715\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 11.6637i 0.582455i 0.956654 + 0.291228i \(0.0940637\pi\)
−0.956654 + 0.291228i \(0.905936\pi\)
\(402\) 0 0
\(403\) 10.8990i 0.542917i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −34.1482 −1.69266
\(408\) 0 0
\(409\) −5.79796 −0.286691 −0.143345 0.989673i \(-0.545786\pi\)
−0.143345 + 0.989673i \(0.545786\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 11.4887i 0.565321i
\(414\) 0 0
\(415\) − 3.24745i − 0.159411i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 20.6096 1.00685 0.503423 0.864040i \(-0.332074\pi\)
0.503423 + 0.864040i \(0.332074\pi\)
\(420\) 0 0
\(421\) 7.10102 0.346083 0.173041 0.984915i \(-0.444641\pi\)
0.173041 + 0.984915i \(0.444641\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 1.55708i 0.0755294i
\(426\) 0 0
\(427\) 9.34847i 0.452404i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −28.9842 −1.39612 −0.698059 0.716040i \(-0.745953\pi\)
−0.698059 + 0.716040i \(0.745953\pi\)
\(432\) 0 0
\(433\) 25.8434 1.24195 0.620976 0.783829i \(-0.286736\pi\)
0.620976 + 0.783829i \(0.286736\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.27135i 0.0608169i
\(438\) 0 0
\(439\) 2.00000i 0.0954548i 0.998860 + 0.0477274i \(0.0151979\pi\)
−0.998860 + 0.0477274i \(0.984802\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 10.8851 0.517167 0.258584 0.965989i \(-0.416744\pi\)
0.258584 + 0.965989i \(0.416744\pi\)
\(444\) 0 0
\(445\) −4.49490 −0.213079
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 31.6055i 1.49156i 0.666194 + 0.745778i \(0.267922\pi\)
−0.666194 + 0.745778i \(0.732078\pi\)
\(450\) 0 0
\(451\) − 29.1464i − 1.37245i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0.778539 0.0364985
\(456\) 0 0
\(457\) −21.3939 −1.00076 −0.500382 0.865805i \(-0.666807\pi\)
−0.500382 + 0.865805i \(0.666807\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 9.43879i − 0.439608i −0.975544 0.219804i \(-0.929458\pi\)
0.975544 0.219804i \(-0.0705419\pi\)
\(462\) 0 0
\(463\) − 28.8434i − 1.34046i −0.742151 0.670232i \(-0.766194\pi\)
0.742151 0.670232i \(-0.233806\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −12.8708 −0.595589 −0.297794 0.954630i \(-0.596251\pi\)
−0.297794 + 0.954630i \(0.596251\pi\)
\(468\) 0 0
\(469\) −8.00000 −0.369406
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 50.4831i − 2.32122i
\(474\) 0 0
\(475\) − 2.20204i − 0.101037i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −5.48188 −0.250474 −0.125237 0.992127i \(-0.539969\pi\)
−0.125237 + 0.992127i \(0.539969\pi\)
\(480\) 0 0
\(481\) 17.1464 0.781810
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 3.89270i 0.176758i
\(486\) 0 0
\(487\) − 6.20204i − 0.281041i −0.990078 0.140521i \(-0.955122\pi\)
0.990078 0.140521i \(-0.0448776\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 30.6841 1.38475 0.692377 0.721536i \(-0.256563\pi\)
0.692377 + 0.721536i \(0.256563\pi\)
\(492\) 0 0
\(493\) 2.44949 0.110319
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 21.4495i 0.960211i 0.877211 + 0.480106i \(0.159402\pi\)
−0.877211 + 0.480106i \(0.840598\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −34.1161 −1.52116 −0.760581 0.649243i \(-0.775086\pi\)
−0.760581 + 0.649243i \(0.775086\pi\)
\(504\) 0 0
\(505\) −1.39388 −0.0620267
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 2.79632i − 0.123945i −0.998078 0.0619723i \(-0.980261\pi\)
0.998078 0.0619723i \(-0.0197391\pi\)
\(510\) 0 0
\(511\) 3.10102i 0.137181i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0.985620 0.0434316
\(516\) 0 0
\(517\) −19.1464 −0.842059
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 33.0518i − 1.44803i −0.689786 0.724013i \(-0.742295\pi\)
0.689786 0.724013i \(-0.257705\pi\)
\(522\) 0 0
\(523\) 32.7423i 1.43172i 0.698242 + 0.715861i \(0.253966\pi\)
−0.698242 + 0.715861i \(0.746034\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1.41421 0.0616041
\(528\) 0 0
\(529\) −15.0000 −0.652174
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 14.6349i 0.633910i
\(534\) 0 0
\(535\) − 4.49490i − 0.194331i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −4.87832 −0.210124
\(540\) 0 0
\(541\) 2.59592 0.111607 0.0558036 0.998442i \(-0.482228\pi\)
0.0558036 + 0.998442i \(0.482228\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 2.22486i 0.0953026i
\(546\) 0 0
\(547\) − 41.0454i − 1.75497i −0.479600 0.877487i \(-0.659218\pi\)
0.479600 0.877487i \(-0.340782\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −3.46410 −0.147576
\(552\) 0 0
\(553\) 8.55051 0.363605
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 39.0265i 1.65361i 0.562491 + 0.826803i \(0.309843\pi\)
−0.562491 + 0.826803i \(0.690157\pi\)
\(558\) 0 0
\(559\) 25.3485i 1.07213i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −8.77101 −0.369654 −0.184827 0.982771i \(-0.559172\pi\)
−0.184827 + 0.982771i \(0.559172\pi\)
\(564\) 0 0
\(565\) −0.853572 −0.0359100
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 41.8549i 1.75465i 0.479896 + 0.877325i \(0.340674\pi\)
−0.479896 + 0.877325i \(0.659326\pi\)
\(570\) 0 0
\(571\) − 2.55051i − 0.106736i −0.998575 0.0533678i \(-0.983004\pi\)
0.998575 0.0533678i \(-0.0169956\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −13.8564 −0.577852
\(576\) 0 0
\(577\) 6.89898 0.287208 0.143604 0.989635i \(-0.454131\pi\)
0.143604 + 0.989635i \(0.454131\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 10.2173i − 0.423886i
\(582\) 0 0
\(583\) 3.10102i 0.128431i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 8.48528 0.350225 0.175113 0.984548i \(-0.443971\pi\)
0.175113 + 0.984548i \(0.443971\pi\)
\(588\) 0 0
\(589\) −2.00000 −0.0824086
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 16.0813i 0.660378i 0.943915 + 0.330189i \(0.107113\pi\)
−0.943915 + 0.330189i \(0.892887\pi\)
\(594\) 0 0
\(595\) − 0.101021i − 0.00414144i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −6.43539 −0.262943 −0.131472 0.991320i \(-0.541970\pi\)
−0.131472 + 0.991320i \(0.541970\pi\)
\(600\) 0 0
\(601\) −11.7526 −0.479397 −0.239698 0.970847i \(-0.577049\pi\)
−0.239698 + 0.970847i \(0.577049\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 4.06767i 0.165374i
\(606\) 0 0
\(607\) 39.8434i 1.61719i 0.588364 + 0.808596i \(0.299772\pi\)
−0.588364 + 0.808596i \(0.700228\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 9.61377 0.388931
\(612\) 0 0
\(613\) −20.0000 −0.807792 −0.403896 0.914805i \(-0.632344\pi\)
−0.403896 + 0.914805i \(0.632344\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 39.3123i − 1.58265i −0.611395 0.791326i \(-0.709391\pi\)
0.611395 0.791326i \(-0.290609\pi\)
\(618\) 0 0
\(619\) − 3.14643i − 0.126466i −0.997999 0.0632328i \(-0.979859\pi\)
0.997999 0.0632328i \(-0.0201411\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −14.1421 −0.566593
\(624\) 0 0
\(625\) 23.4949 0.939796
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 2.22486i − 0.0887110i
\(630\) 0 0
\(631\) − 0.550510i − 0.0219155i −0.999940 0.0109577i \(-0.996512\pi\)
0.999940 0.0109577i \(-0.00348802\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 4.84621 0.192316
\(636\) 0 0
\(637\) 2.44949 0.0970523
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 1.69994i − 0.0671437i −0.999436 0.0335719i \(-0.989312\pi\)
0.999436 0.0335719i \(-0.0106883\pi\)
\(642\) 0 0
\(643\) − 44.6969i − 1.76268i −0.472487 0.881338i \(-0.656644\pi\)
0.472487 0.881338i \(-0.343356\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −24.1845 −0.950791 −0.475395 0.879772i \(-0.657695\pi\)
−0.475395 + 0.879772i \(0.657695\pi\)
\(648\) 0 0
\(649\) 56.0454 2.19997
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 5.51399i − 0.215779i −0.994163 0.107890i \(-0.965591\pi\)
0.994163 0.107890i \(-0.0344093\pi\)
\(654\) 0 0
\(655\) 5.10102i 0.199313i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 18.8776 0.735366 0.367683 0.929951i \(-0.380151\pi\)
0.367683 + 0.929951i \(0.380151\pi\)
\(660\) 0 0
\(661\) −46.7423 −1.81807 −0.909033 0.416724i \(-0.863178\pi\)
−0.909033 + 0.416724i \(0.863178\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0.142865i 0.00554005i
\(666\) 0 0
\(667\) 21.7980i 0.844020i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 45.6048 1.76055
\(672\) 0 0
\(673\) −22.2929 −0.859326 −0.429663 0.902989i \(-0.641368\pi\)
−0.429663 + 0.902989i \(0.641368\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 33.6554i 1.29348i 0.762710 + 0.646741i \(0.223869\pi\)
−0.762710 + 0.646741i \(0.776131\pi\)
\(678\) 0 0
\(679\) 12.2474i 0.470014i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 29.1270 1.11451 0.557257 0.830340i \(-0.311854\pi\)
0.557257 + 0.830340i \(0.311854\pi\)
\(684\) 0 0
\(685\) −2.85357 −0.109029
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 1.55708i − 0.0593200i
\(690\) 0 0
\(691\) 3.75255i 0.142754i 0.997449 + 0.0713769i \(0.0227393\pi\)
−0.997449 + 0.0713769i \(0.977261\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 3.89270 0.147658
\(696\) 0 0
\(697\) 1.89898 0.0719290
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 15.5563i − 0.587555i −0.955874 0.293778i \(-0.905087\pi\)
0.955874 0.293778i \(-0.0949125\pi\)
\(702\) 0 0
\(703\) 3.14643i 0.118670i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −4.38551 −0.164934
\(708\) 0 0
\(709\) 35.0000 1.31445 0.657226 0.753693i \(-0.271730\pi\)
0.657226 + 0.753693i \(0.271730\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 12.5851i 0.471314i
\(714\) 0 0
\(715\) − 3.79796i − 0.142036i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −47.3368 −1.76537 −0.882683 0.469969i \(-0.844265\pi\)
−0.882683 + 0.469969i \(0.844265\pi\)
\(720\) 0 0
\(721\) 3.10102 0.115488
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 37.7552i − 1.40219i
\(726\) 0 0
\(727\) − 17.7526i − 0.658406i −0.944259 0.329203i \(-0.893220\pi\)
0.944259 0.329203i \(-0.106780\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 3.28913 0.121653
\(732\) 0 0
\(733\) 33.1010 1.22261 0.611307 0.791394i \(-0.290644\pi\)
0.611307 + 0.791394i \(0.290644\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 39.0265i 1.43756i
\(738\) 0 0
\(739\) − 0.404082i − 0.0148644i −0.999972 0.00743220i \(-0.997634\pi\)
0.999972 0.00743220i \(-0.00236576\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 45.1120 1.65500 0.827499 0.561467i \(-0.189763\pi\)
0.827499 + 0.561467i \(0.189763\pi\)
\(744\) 0 0
\(745\) 5.34847 0.195953
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 14.1421i − 0.516742i
\(750\) 0 0
\(751\) 25.3939i 0.926636i 0.886192 + 0.463318i \(0.153341\pi\)
−0.886192 + 0.463318i \(0.846659\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 5.48188 0.199506
\(756\) 0 0
\(757\) −53.6969 −1.95165 −0.975824 0.218557i \(-0.929865\pi\)
−0.975824 + 0.218557i \(0.929865\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 33.9732i − 1.23153i −0.787930 0.615764i \(-0.788847\pi\)
0.787930 0.615764i \(-0.211153\pi\)
\(762\) 0 0
\(763\) 7.00000i 0.253417i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −28.1414 −1.01613
\(768\) 0 0
\(769\) −34.8990 −1.25849 −0.629245 0.777207i \(-0.716636\pi\)
−0.629245 + 0.777207i \(0.716636\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 21.6739i − 0.779556i −0.920909 0.389778i \(-0.872552\pi\)
0.920909 0.389778i \(-0.127448\pi\)
\(774\) 0 0
\(775\) − 21.7980i − 0.783006i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −2.68556 −0.0962203
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 4.87832i 0.174115i
\(786\) 0 0
\(787\) 44.4949i 1.58607i 0.609175 + 0.793036i \(0.291501\pi\)
−0.609175 + 0.793036i \(0.708499\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −2.68556 −0.0954876
\(792\) 0 0
\(793\) −22.8990 −0.813167
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 19.2275i 0.681074i 0.940231 + 0.340537i \(0.110609\pi\)
−0.940231 + 0.340537i \(0.889391\pi\)
\(798\) 0 0
\(799\) − 1.24745i − 0.0441316i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 15.1278 0.533847
\(804\) 0 0
\(805\) 0.898979 0.0316849
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 27.0771i − 0.951981i −0.879450 0.475991i \(-0.842090\pi\)
0.879450 0.475991i \(-0.157910\pi\)
\(810\) 0 0
\(811\) 36.7423i 1.29020i 0.764099 + 0.645099i \(0.223184\pi\)
−0.764099 + 0.645099i \(0.776816\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 3.57486 0.125222
\(816\) 0 0
\(817\) −4.65153 −0.162736
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 33.3697i − 1.16461i −0.812971 0.582305i \(-0.802151\pi\)
0.812971 0.582305i \(-0.197849\pi\)
\(822\) 0 0
\(823\) 9.65153i 0.336431i 0.985750 + 0.168216i \(0.0538005\pi\)
−0.985750 + 0.168216i \(0.946200\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 11.5208 0.400617 0.200309 0.979733i \(-0.435806\pi\)
0.200309 + 0.979733i \(0.435806\pi\)
\(828\) 0 0
\(829\) −6.04541 −0.209966 −0.104983 0.994474i \(-0.533479\pi\)
−0.104983 + 0.994474i \(0.533479\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 0.317837i − 0.0110124i
\(834\) 0 0
\(835\) − 3.15663i − 0.109240i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −27.1879 −0.938630 −0.469315 0.883031i \(-0.655499\pi\)
−0.469315 + 0.883031i \(0.655499\pi\)
\(840\) 0 0
\(841\) −30.3939 −1.04806
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 2.22486i − 0.0765375i
\(846\) 0 0
\(847\) 12.7980i 0.439743i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 19.7990 0.678701
\(852\) 0 0
\(853\) 46.8990 1.60579 0.802895 0.596120i \(-0.203292\pi\)
0.802895 + 0.596120i \(0.203292\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 23.0095i − 0.785989i −0.919541 0.392994i \(-0.871439\pi\)
0.919541 0.392994i \(-0.128561\pi\)
\(858\) 0 0
\(859\) 26.4949i 0.903994i 0.892019 + 0.451997i \(0.149288\pi\)
−0.892019 + 0.451997i \(0.850712\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 19.1633 0.652327 0.326163 0.945313i \(-0.394244\pi\)
0.326163 + 0.945313i \(0.394244\pi\)
\(864\) 0 0
\(865\) −2.89898 −0.0985683
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 41.7121i − 1.41499i
\(870\) 0 0
\(871\) − 19.5959i − 0.663982i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −3.14626 −0.106363
\(876\) 0 0
\(877\) −38.7980 −1.31011 −0.655057 0.755579i \(-0.727355\pi\)
−0.655057 + 0.755579i \(0.727355\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 49.0689i 1.65317i 0.562810 + 0.826586i \(0.309720\pi\)
−0.562810 + 0.826586i \(0.690280\pi\)
\(882\) 0 0
\(883\) − 7.04541i − 0.237097i −0.992948 0.118548i \(-0.962176\pi\)
0.992948 0.118548i \(-0.0378241\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 9.58166 0.321721 0.160860 0.986977i \(-0.448573\pi\)
0.160860 + 0.986977i \(0.448573\pi\)
\(888\) 0 0
\(889\) 15.2474 0.511383
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 1.76416i 0.0590353i
\(894\) 0 0
\(895\) 1.55051i 0.0518278i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −34.2911 −1.14367
\(900\) 0 0
\(901\) −0.202041 −0.00673096
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 7.84961i 0.260930i
\(906\) 0 0
\(907\) 2.05561i 0.0682555i 0.999417 + 0.0341278i \(0.0108653\pi\)
−0.999417 + 0.0341278i \(0.989135\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −0.778539 −0.0257942 −0.0128971 0.999917i \(-0.504105\pi\)
−0.0128971 + 0.999917i \(0.504105\pi\)
\(912\) 0 0
\(913\) −49.8434 −1.64957
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 16.0492i 0.529990i
\(918\) 0 0
\(919\) − 12.7526i − 0.420668i −0.977630 0.210334i \(-0.932545\pi\)
0.977630 0.210334i \(-0.0674551\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −34.2929 −1.12754
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 45.0012i − 1.47644i −0.674559 0.738221i \(-0.735666\pi\)
0.674559 0.738221i \(-0.264334\pi\)
\(930\) 0 0
\(931\) 0.449490i 0.0147314i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −0.492810 −0.0161166
\(936\) 0 0
\(937\) 15.3485 0.501413 0.250706 0.968063i \(-0.419337\pi\)
0.250706 + 0.968063i \(0.419337\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 25.7737i − 0.840198i −0.907478 0.420099i \(-0.861995\pi\)
0.907478 0.420099i \(-0.138005\pi\)
\(942\) 0 0
\(943\) 16.8990i 0.550306i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 40.2979 1.30950 0.654752 0.755843i \(-0.272773\pi\)
0.654752 + 0.755843i \(0.272773\pi\)
\(948\) 0 0
\(949\) −7.59592 −0.246574
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 53.1687i 1.72230i 0.508349 + 0.861151i \(0.330256\pi\)
−0.508349 + 0.861151i \(0.669744\pi\)
\(954\) 0 0
\(955\) − 5.10102i − 0.165065i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −8.97809 −0.289918
\(960\) 0 0
\(961\) 11.2020 0.361356
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 3.49621i − 0.112547i
\(966\) 0 0
\(967\) 4.89898i 0.157541i 0.996893 + 0.0787703i \(0.0250994\pi\)
−0.996893 + 0.0787703i \(0.974901\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −17.7170 −0.568565 −0.284283 0.958741i \(-0.591755\pi\)
−0.284283 + 0.958741i \(0.591755\pi\)
\(972\) 0 0
\(973\) 12.2474 0.392635
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 32.4483i 1.03811i 0.854740 + 0.519056i \(0.173716\pi\)
−0.854740 + 0.519056i \(0.826284\pi\)
\(978\) 0 0
\(979\) 68.9898i 2.20492i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −16.8598 −0.537744 −0.268872 0.963176i \(-0.586651\pi\)
−0.268872 + 0.963176i \(0.586651\pi\)
\(984\) 0 0
\(985\) −0.247449 −0.00788437
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 29.2699i 0.930728i
\(990\) 0 0
\(991\) 19.4495i 0.617833i 0.951089 + 0.308917i \(0.0999664\pi\)
−0.951089 + 0.308917i \(0.900034\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −3.17837 −0.100761
\(996\) 0 0
\(997\) 7.75255 0.245526 0.122763 0.992436i \(-0.460825\pi\)
0.122763 + 0.992436i \(0.460825\pi\)
\(998\) 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.d.2591.6 yes 8
3.2 odd 2 inner 6048.2.h.d.2591.4 yes 8
4.3 odd 2 inner 6048.2.h.d.2591.5 yes 8
12.11 even 2 inner 6048.2.h.d.2591.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6048.2.h.d.2591.3 8 12.11 even 2 inner
6048.2.h.d.2591.4 yes 8 3.2 odd 2 inner
6048.2.h.d.2591.5 yes 8 4.3 odd 2 inner
6048.2.h.d.2591.6 yes 8 1.1 even 1 trivial