Properties

Label 6048.2.h.g.2591.3
Level 6048
Weight 2
Character 6048.2591
Analytic conductor 48.294
Analytic rank 0
Dimension 8
CM no
Inner twists 2

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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: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

\(f(q)\) \(=\) \(q-2.21441i q^{5} -1.00000i q^{7} +O(q^{10})\) \(q-2.21441i q^{5} -1.00000i q^{7} -0.614014 q^{11} -5.56048 q^{13} +0.585786i q^{17} +3.89898i q^{19} -0.850087 q^{23} +0.0963763 q^{25} +4.74202i q^{29} -3.07055i q^{31} -2.21441 q^{35} +2.17157 q^{37} -6.07812i q^{41} +9.21733i q^{43} -3.41421 q^{47} -1.00000 q^{49} +4.47015i q^{53} +1.35968i q^{55} +13.9136 q^{59} -8.92820 q^{61} +12.3132i q^{65} +4.83183i q^{67} -5.33610 q^{71} -9.28788 q^{73} +0.614014i q^{77} -1.29717i q^{79} +14.5400 q^{83} +1.29717 q^{85} +11.4059i q^{89} +5.56048i q^{91} +8.63395 q^{95} +7.27135 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 8q^{11} - 8q^{13} + 8q^{23} - 8q^{25} - 8q^{35} + 40q^{37} - 16q^{47} - 8q^{49} + 64q^{59} - 16q^{61} - 72q^{71} + 24q^{73} - 32q^{83} + 8q^{85} - 56q^{95} + 48q^{97} + O(q^{100}) \)

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
\(5\) − 2.21441i − 0.990315i −0.868803 0.495158i \(-0.835110\pi\)
0.868803 0.495158i \(-0.164890\pi\)
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −0.614014 −0.185132 −0.0925662 0.995707i \(-0.529507\pi\)
−0.0925662 + 0.995707i \(0.529507\pi\)
\(12\) 0 0
\(13\) −5.56048 −1.54220 −0.771100 0.636715i \(-0.780293\pi\)
−0.771100 + 0.636715i \(0.780293\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.585786i 0.142074i 0.997474 + 0.0710370i \(0.0226309\pi\)
−0.997474 + 0.0710370i \(0.977369\pi\)
\(18\) 0 0
\(19\) 3.89898i 0.894487i 0.894412 + 0.447244i \(0.147594\pi\)
−0.894412 + 0.447244i \(0.852406\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −0.850087 −0.177255 −0.0886277 0.996065i \(-0.528248\pi\)
−0.0886277 + 0.996065i \(0.528248\pi\)
\(24\) 0 0
\(25\) 0.0963763 0.0192753
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.74202i 0.880571i 0.897858 + 0.440285i \(0.145123\pi\)
−0.897858 + 0.440285i \(0.854877\pi\)
\(30\) 0 0
\(31\) − 3.07055i − 0.551487i −0.961231 0.275744i \(-0.911076\pi\)
0.961231 0.275744i \(-0.0889241\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.21441 −0.374304
\(36\) 0 0
\(37\) 2.17157 0.357004 0.178502 0.983940i \(-0.442875\pi\)
0.178502 + 0.983940i \(0.442875\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 6.07812i − 0.949242i −0.880190 0.474621i \(-0.842585\pi\)
0.880190 0.474621i \(-0.157415\pi\)
\(42\) 0 0
\(43\) 9.21733i 1.40563i 0.711373 + 0.702815i \(0.248074\pi\)
−0.711373 + 0.702815i \(0.751926\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.41421 −0.498014 −0.249007 0.968502i \(-0.580104\pi\)
−0.249007 + 0.968502i \(0.580104\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4.47015i 0.614023i 0.951706 + 0.307011i \(0.0993290\pi\)
−0.951706 + 0.307011i \(0.900671\pi\)
\(54\) 0 0
\(55\) 1.35968i 0.183339i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 13.9136 1.81140 0.905698 0.423924i \(-0.139348\pi\)
0.905698 + 0.423924i \(0.139348\pi\)
\(60\) 0 0
\(61\) −8.92820 −1.14314 −0.571570 0.820554i \(-0.693665\pi\)
−0.571570 + 0.820554i \(0.693665\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 12.3132i 1.52726i
\(66\) 0 0
\(67\) 4.83183i 0.590302i 0.955451 + 0.295151i \(0.0953699\pi\)
−0.955451 + 0.295151i \(0.904630\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −5.33610 −0.633278 −0.316639 0.948546i \(-0.602554\pi\)
−0.316639 + 0.948546i \(0.602554\pi\)
\(72\) 0 0
\(73\) −9.28788 −1.08706 −0.543532 0.839388i \(-0.682913\pi\)
−0.543532 + 0.839388i \(0.682913\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.614014i 0.0699734i
\(78\) 0 0
\(79\) − 1.29717i − 0.145943i −0.997334 0.0729717i \(-0.976752\pi\)
0.997334 0.0729717i \(-0.0232483\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 14.5400 1.59597 0.797985 0.602677i \(-0.205899\pi\)
0.797985 + 0.602677i \(0.205899\pi\)
\(84\) 0 0
\(85\) 1.29717 0.140698
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 11.4059i 1.20903i 0.796596 + 0.604513i \(0.206632\pi\)
−0.796596 + 0.604513i \(0.793368\pi\)
\(90\) 0 0
\(91\) 5.56048i 0.582896i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 8.63395 0.885825
\(96\) 0 0
\(97\) 7.27135 0.738294 0.369147 0.929371i \(-0.379650\pi\)
0.369147 + 0.929371i \(0.379650\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 19.1416i 1.90466i 0.305066 + 0.952331i \(0.401321\pi\)
−0.305066 + 0.952331i \(0.598679\pi\)
\(102\) 0 0
\(103\) − 10.3843i − 1.02319i −0.859226 0.511596i \(-0.829055\pi\)
0.859226 0.511596i \(-0.170945\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 6.65634 0.643493 0.321746 0.946826i \(-0.395730\pi\)
0.321746 + 0.946826i \(0.395730\pi\)
\(108\) 0 0
\(109\) 5.31835 0.509406 0.254703 0.967019i \(-0.418022\pi\)
0.254703 + 0.967019i \(0.418022\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 13.8977i 1.30739i 0.756759 + 0.653694i \(0.226782\pi\)
−0.756759 + 0.653694i \(0.773218\pi\)
\(114\) 0 0
\(115\) 1.88244i 0.175539i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0.585786 0.0536990
\(120\) 0 0
\(121\) −10.6230 −0.965726
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 11.2855i − 1.00940i
\(126\) 0 0
\(127\) 2.16693i 0.192284i 0.995368 + 0.0961419i \(0.0306503\pi\)
−0.995368 + 0.0961419i \(0.969350\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 7.98414 0.697578 0.348789 0.937201i \(-0.386593\pi\)
0.348789 + 0.937201i \(0.386593\pi\)
\(132\) 0 0
\(133\) 3.89898 0.338084
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 10.1862i − 0.870264i −0.900367 0.435132i \(-0.856702\pi\)
0.900367 0.435132i \(-0.143298\pi\)
\(138\) 0 0
\(139\) − 2.53590i − 0.215092i −0.994200 0.107546i \(-0.965701\pi\)
0.994200 0.107546i \(-0.0342993\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 3.41421 0.285511
\(144\) 0 0
\(145\) 10.5008 0.872043
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 7.02798i 0.575755i 0.957667 + 0.287877i \(0.0929495\pi\)
−0.957667 + 0.287877i \(0.907050\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −6.79947 −0.546147
\(156\) 0 0
\(157\) −3.21733 −0.256771 −0.128386 0.991724i \(-0.540979\pi\)
−0.128386 + 0.991724i \(0.540979\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0.850087i 0.0669963i
\(162\) 0 0
\(163\) − 8.23748i − 0.645209i −0.946534 0.322605i \(-0.895442\pi\)
0.946534 0.322605i \(-0.104558\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 19.8836 1.53864 0.769321 0.638862i \(-0.220595\pi\)
0.769321 + 0.638862i \(0.220595\pi\)
\(168\) 0 0
\(169\) 17.9189 1.37838
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 11.2855i 0.858019i 0.903300 + 0.429010i \(0.141137\pi\)
−0.903300 + 0.429010i \(0.858863\pi\)
\(174\) 0 0
\(175\) − 0.0963763i − 0.00728536i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 18.8126 1.40612 0.703059 0.711132i \(-0.251817\pi\)
0.703059 + 0.711132i \(0.251817\pi\)
\(180\) 0 0
\(181\) 11.4548 0.851430 0.425715 0.904857i \(-0.360023\pi\)
0.425715 + 0.904857i \(0.360023\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 4.80876i − 0.353547i
\(186\) 0 0
\(187\) − 0.359681i − 0.0263025i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −0.336613 −0.0243565 −0.0121782 0.999926i \(-0.503877\pi\)
−0.0121782 + 0.999926i \(0.503877\pi\)
\(192\) 0 0
\(193\) −9.80725 −0.705941 −0.352971 0.935634i \(-0.614828\pi\)
−0.352971 + 0.935634i \(0.614828\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 4.38750i − 0.312596i −0.987710 0.156298i \(-0.950044\pi\)
0.987710 0.156298i \(-0.0499561\pi\)
\(198\) 0 0
\(199\) − 5.21609i − 0.369759i −0.982761 0.184879i \(-0.940811\pi\)
0.982761 0.184879i \(-0.0591894\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 4.74202 0.332824
\(204\) 0 0
\(205\) −13.4595 −0.940049
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 2.39403i − 0.165598i
\(210\) 0 0
\(211\) 3.12096i 0.214855i 0.994213 + 0.107428i \(0.0342614\pi\)
−0.994213 + 0.107428i \(0.965739\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 20.4110 1.39202
\(216\) 0 0
\(217\) −3.07055 −0.208443
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 3.25725i − 0.219107i
\(222\) 0 0
\(223\) 7.21858i 0.483392i 0.970352 + 0.241696i \(0.0777036\pi\)
−0.970352 + 0.241696i \(0.922296\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −8.85765 −0.587903 −0.293951 0.955820i \(-0.594970\pi\)
−0.293951 + 0.955820i \(0.594970\pi\)
\(228\) 0 0
\(229\) −0.677003 −0.0447376 −0.0223688 0.999750i \(-0.507121\pi\)
−0.0223688 + 0.999750i \(0.507121\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 8.89898i 0.582992i 0.956572 + 0.291496i \(0.0941529\pi\)
−0.956572 + 0.291496i \(0.905847\pi\)
\(234\) 0 0
\(235\) 7.56048i 0.493191i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −10.0134 −0.647711 −0.323855 0.946107i \(-0.604979\pi\)
−0.323855 + 0.946107i \(0.604979\pi\)
\(240\) 0 0
\(241\) −13.8306 −0.890906 −0.445453 0.895305i \(-0.646957\pi\)
−0.445453 + 0.895305i \(0.646957\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 2.21441i 0.141474i
\(246\) 0 0
\(247\) − 21.6802i − 1.37948i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 11.5546 0.729319 0.364660 0.931141i \(-0.381185\pi\)
0.364660 + 0.931141i \(0.381185\pi\)
\(252\) 0 0
\(253\) 0.521966 0.0328157
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 8.80604i − 0.549306i −0.961543 0.274653i \(-0.911437\pi\)
0.961543 0.274653i \(-0.0885629\pi\)
\(258\) 0 0
\(259\) − 2.17157i − 0.134935i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −15.9662 −0.984520 −0.492260 0.870448i \(-0.663829\pi\)
−0.492260 + 0.870448i \(0.663829\pi\)
\(264\) 0 0
\(265\) 9.89877 0.608076
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 5.05870i − 0.308434i −0.988037 0.154217i \(-0.950714\pi\)
0.988037 0.154217i \(-0.0492855\pi\)
\(270\) 0 0
\(271\) − 2.05845i − 0.125042i −0.998044 0.0625209i \(-0.980086\pi\)
0.998044 0.0625209i \(-0.0199140\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −0.0591764 −0.00356847
\(276\) 0 0
\(277\) 28.1598 1.69196 0.845979 0.533216i \(-0.179017\pi\)
0.845979 + 0.533216i \(0.179017\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 30.1676i 1.79965i 0.436252 + 0.899824i \(0.356305\pi\)
−0.436252 + 0.899824i \(0.643695\pi\)
\(282\) 0 0
\(283\) − 23.3044i − 1.38530i −0.721272 0.692652i \(-0.756442\pi\)
0.721272 0.692652i \(-0.243558\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −6.07812 −0.358780
\(288\) 0 0
\(289\) 16.6569 0.979815
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 22.3273i 1.30437i 0.758058 + 0.652187i \(0.226148\pi\)
−0.758058 + 0.652187i \(0.773852\pi\)
\(294\) 0 0
\(295\) − 30.8104i − 1.79385i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 4.72689 0.273363
\(300\) 0 0
\(301\) 9.21733 0.531278
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 19.7707i 1.13207i
\(306\) 0 0
\(307\) 16.0985i 0.918792i 0.888232 + 0.459396i \(0.151934\pi\)
−0.888232 + 0.459396i \(0.848066\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −0.486010 −0.0275591 −0.0137796 0.999905i \(-0.504386\pi\)
−0.0137796 + 0.999905i \(0.504386\pi\)
\(312\) 0 0
\(313\) 11.4899 0.649449 0.324724 0.945809i \(-0.394728\pi\)
0.324724 + 0.945809i \(0.394728\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 8.77021i − 0.492584i −0.969196 0.246292i \(-0.920788\pi\)
0.969196 0.246292i \(-0.0792122\pi\)
\(318\) 0 0
\(319\) − 2.91167i − 0.163022i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −2.28397 −0.127083
\(324\) 0 0
\(325\) −0.535898 −0.0297263
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 3.41421i 0.188232i
\(330\) 0 0
\(331\) 22.3246i 1.22707i 0.789668 + 0.613535i \(0.210253\pi\)
−0.789668 + 0.613535i \(0.789747\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 10.6997 0.584585
\(336\) 0 0
\(337\) −7.19275 −0.391814 −0.195907 0.980622i \(-0.562765\pi\)
−0.195907 + 0.980622i \(0.562765\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 1.88536i 0.102098i
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −6.99776 −0.375659 −0.187830 0.982202i \(-0.560145\pi\)
−0.187830 + 0.982202i \(0.560145\pi\)
\(348\) 0 0
\(349\) −6.47496 −0.346597 −0.173298 0.984869i \(-0.555442\pi\)
−0.173298 + 0.984869i \(0.555442\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 22.9605i 1.22206i 0.791606 + 0.611032i \(0.209245\pi\)
−0.791606 + 0.611032i \(0.790755\pi\)
\(354\) 0 0
\(355\) 11.8163i 0.627145i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 28.7819 1.51905 0.759525 0.650478i \(-0.225431\pi\)
0.759525 + 0.650478i \(0.225431\pi\)
\(360\) 0 0
\(361\) 3.79796 0.199893
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 20.5672i 1.07654i
\(366\) 0 0
\(367\) − 34.1867i − 1.78453i −0.451513 0.892264i \(-0.649116\pi\)
0.451513 0.892264i \(-0.350884\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 4.47015 0.232079
\(372\) 0 0
\(373\) 16.1457 0.835995 0.417998 0.908448i \(-0.362732\pi\)
0.417998 + 0.908448i \(0.362732\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 26.3679i − 1.35802i
\(378\) 0 0
\(379\) 14.4370i 0.741581i 0.928717 + 0.370790i \(0.120913\pi\)
−0.928717 + 0.370790i \(0.879087\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 19.5546 0.999193 0.499596 0.866258i \(-0.333482\pi\)
0.499596 + 0.866258i \(0.333482\pi\)
\(384\) 0 0
\(385\) 1.35968 0.0692958
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 26.9116i 1.36447i 0.731131 + 0.682237i \(0.238993\pi\)
−0.731131 + 0.682237i \(0.761007\pi\)
\(390\) 0 0
\(391\) − 0.497970i − 0.0251834i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −2.87248 −0.144530
\(396\) 0 0
\(397\) −39.6383 −1.98939 −0.994694 0.102877i \(-0.967195\pi\)
−0.994694 + 0.102877i \(0.967195\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 2.79744i 0.139698i 0.997558 + 0.0698488i \(0.0222517\pi\)
−0.997558 + 0.0698488i \(0.977748\pi\)
\(402\) 0 0
\(403\) 17.0737i 0.850504i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −1.33338 −0.0660930
\(408\) 0 0
\(409\) −23.7318 −1.17346 −0.586732 0.809781i \(-0.699586\pi\)
−0.586732 + 0.809781i \(0.699586\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 13.9136i − 0.684643i
\(414\) 0 0
\(415\) − 32.1975i − 1.58051i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 16.4268 0.802503 0.401252 0.915968i \(-0.368575\pi\)
0.401252 + 0.915968i \(0.368575\pi\)
\(420\) 0 0
\(421\) −19.9695 −0.973255 −0.486628 0.873609i \(-0.661773\pi\)
−0.486628 + 0.873609i \(0.661773\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0.0564559i 0.00273852i
\(426\) 0 0
\(427\) 8.92820i 0.432066i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 29.4332 1.41775 0.708873 0.705337i \(-0.249204\pi\)
0.708873 + 0.705337i \(0.249204\pi\)
\(432\) 0 0
\(433\) 3.23873 0.155643 0.0778216 0.996967i \(-0.475204\pi\)
0.0778216 + 0.996967i \(0.475204\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 3.31447i − 0.158553i
\(438\) 0 0
\(439\) − 7.94835i − 0.379354i −0.981847 0.189677i \(-0.939256\pi\)
0.981847 0.189677i \(-0.0607441\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −3.67171 −0.174448 −0.0872242 0.996189i \(-0.527800\pi\)
−0.0872242 + 0.996189i \(0.527800\pi\)
\(444\) 0 0
\(445\) 25.2574 1.19732
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 23.6392i 1.11560i 0.829974 + 0.557802i \(0.188355\pi\)
−0.829974 + 0.557802i \(0.811645\pi\)
\(450\) 0 0
\(451\) 3.73205i 0.175735i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 12.3132 0.577251
\(456\) 0 0
\(457\) −9.57826 −0.448052 −0.224026 0.974583i \(-0.571920\pi\)
−0.224026 + 0.974583i \(0.571920\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 38.7836i 1.80633i 0.429290 + 0.903166i \(0.358764\pi\)
−0.429290 + 0.903166i \(0.641236\pi\)
\(462\) 0 0
\(463\) 25.7060i 1.19466i 0.801996 + 0.597330i \(0.203772\pi\)
−0.801996 + 0.597330i \(0.796228\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 11.0983 0.513568 0.256784 0.966469i \(-0.417337\pi\)
0.256784 + 0.966469i \(0.417337\pi\)
\(468\) 0 0
\(469\) 4.83183 0.223113
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 5.65957i − 0.260228i
\(474\) 0 0
\(475\) 0.375769i 0.0172415i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 13.2713 0.606383 0.303192 0.952930i \(-0.401948\pi\)
0.303192 + 0.952930i \(0.401948\pi\)
\(480\) 0 0
\(481\) −12.0750 −0.550572
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 16.1018i − 0.731144i
\(486\) 0 0
\(487\) 8.20929i 0.371998i 0.982550 + 0.185999i \(0.0595522\pi\)
−0.982550 + 0.185999i \(0.940448\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 28.7413 1.29708 0.648539 0.761182i \(-0.275380\pi\)
0.648539 + 0.761182i \(0.275380\pi\)
\(492\) 0 0
\(493\) −2.77781 −0.125106
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 5.33610i 0.239357i
\(498\) 0 0
\(499\) 29.7600i 1.33224i 0.745844 + 0.666121i \(0.232047\pi\)
−0.745844 + 0.666121i \(0.767953\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −8.75715 −0.390462 −0.195231 0.980757i \(-0.562546\pi\)
−0.195231 + 0.980757i \(0.562546\pi\)
\(504\) 0 0
\(505\) 42.3874 1.88622
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 29.6843i − 1.31573i −0.753134 0.657867i \(-0.771459\pi\)
0.753134 0.657867i \(-0.228541\pi\)
\(510\) 0 0
\(511\) 9.28788i 0.410872i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −22.9950 −1.01328
\(516\) 0 0
\(517\) 2.09638 0.0921985
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 10.8766i 0.476512i 0.971202 + 0.238256i \(0.0765757\pi\)
−0.971202 + 0.238256i \(0.923424\pi\)
\(522\) 0 0
\(523\) 17.7812i 0.777518i 0.921339 + 0.388759i \(0.127096\pi\)
−0.921339 + 0.388759i \(0.872904\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1.79869 0.0783521
\(528\) 0 0
\(529\) −22.2774 −0.968581
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 33.7972i 1.46392i
\(534\) 0 0
\(535\) − 14.7399i − 0.637261i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0.614014 0.0264475
\(540\) 0 0
\(541\) −15.7433 −0.676857 −0.338428 0.940992i \(-0.609895\pi\)
−0.338428 + 0.940992i \(0.609895\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 11.7770i − 0.504472i
\(546\) 0 0
\(547\) 41.8963i 1.79136i 0.444704 + 0.895678i \(0.353309\pi\)
−0.444704 + 0.895678i \(0.646691\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −18.4890 −0.787659
\(552\) 0 0
\(553\) −1.29717 −0.0551614
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 30.2681i 1.28250i 0.767332 + 0.641251i \(0.221584\pi\)
−0.767332 + 0.641251i \(0.778416\pi\)
\(558\) 0 0
\(559\) − 51.2528i − 2.16776i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −30.8126 −1.29860 −0.649298 0.760535i \(-0.724937\pi\)
−0.649298 + 0.760535i \(0.724937\pi\)
\(564\) 0 0
\(565\) 30.7753 1.29473
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 15.8252i − 0.663427i −0.943380 0.331713i \(-0.892373\pi\)
0.943380 0.331713i \(-0.107627\pi\)
\(570\) 0 0
\(571\) − 6.00929i − 0.251481i −0.992063 0.125740i \(-0.959869\pi\)
0.992063 0.125740i \(-0.0401307\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −0.0819283 −0.00341665
\(576\) 0 0
\(577\) −26.5148 −1.10383 −0.551913 0.833901i \(-0.686102\pi\)
−0.551913 + 0.833901i \(0.686102\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 14.5400i − 0.603220i
\(582\) 0 0
\(583\) − 2.74474i − 0.113675i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −39.9572 −1.64921 −0.824605 0.565708i \(-0.808603\pi\)
−0.824605 + 0.565708i \(0.808603\pi\)
\(588\) 0 0
\(589\) 11.9720 0.493299
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 3.10931i 0.127684i 0.997960 + 0.0638421i \(0.0203354\pi\)
−0.997960 + 0.0638421i \(0.979665\pi\)
\(594\) 0 0
\(595\) − 1.29717i − 0.0531789i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 30.1517 1.23196 0.615982 0.787760i \(-0.288759\pi\)
0.615982 + 0.787760i \(0.288759\pi\)
\(600\) 0 0
\(601\) 14.9024 0.607881 0.303940 0.952691i \(-0.401698\pi\)
0.303940 + 0.952691i \(0.401698\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 23.5237i 0.956373i
\(606\) 0 0
\(607\) − 21.7060i − 0.881020i −0.897748 0.440510i \(-0.854798\pi\)
0.897748 0.440510i \(-0.145202\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 18.9847 0.768037
\(612\) 0 0
\(613\) −42.5127 −1.71707 −0.858535 0.512754i \(-0.828625\pi\)
−0.858535 + 0.512754i \(0.828625\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 2.68460i − 0.108078i −0.998539 0.0540390i \(-0.982790\pi\)
0.998539 0.0540390i \(-0.0172095\pi\)
\(618\) 0 0
\(619\) 20.9631i 0.842578i 0.906926 + 0.421289i \(0.138422\pi\)
−0.906926 + 0.421289i \(0.861578\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 11.4059 0.456969
\(624\) 0 0
\(625\) −24.5088 −0.980353
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 1.27208i 0.0507211i
\(630\) 0 0
\(631\) − 0.458427i − 0.0182497i −0.999958 0.00912484i \(-0.997095\pi\)
0.999958 0.00912484i \(-0.00290457\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 4.79847 0.190422
\(636\) 0 0
\(637\) 5.56048 0.220314
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 13.4573i 0.531531i 0.964038 + 0.265766i \(0.0856248\pi\)
−0.964038 + 0.265766i \(0.914375\pi\)
\(642\) 0 0
\(643\) 44.4655i 1.75355i 0.480904 + 0.876773i \(0.340309\pi\)
−0.480904 + 0.876773i \(0.659691\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −40.4377 −1.58977 −0.794885 0.606760i \(-0.792469\pi\)
−0.794885 + 0.606760i \(0.792469\pi\)
\(648\) 0 0
\(649\) −8.54315 −0.335348
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 50.2226i − 1.96536i −0.185304 0.982681i \(-0.559327\pi\)
0.185304 0.982681i \(-0.440673\pi\)
\(654\) 0 0
\(655\) − 17.6802i − 0.690822i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −49.3533 −1.92253 −0.961266 0.275622i \(-0.911116\pi\)
−0.961266 + 0.275622i \(0.911116\pi\)
\(660\) 0 0
\(661\) 30.3476 1.18038 0.590192 0.807263i \(-0.299052\pi\)
0.590192 + 0.807263i \(0.299052\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 8.63395i − 0.334810i
\(666\) 0 0
\(667\) − 4.03113i − 0.156086i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 5.48205 0.211632
\(672\) 0 0
\(673\) −10.0000 −0.385472 −0.192736 0.981251i \(-0.561736\pi\)
−0.192736 + 0.981251i \(0.561736\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 34.2805i − 1.31751i −0.752359 0.658753i \(-0.771084\pi\)
0.752359 0.658753i \(-0.228916\pi\)
\(678\) 0 0
\(679\) − 7.27135i − 0.279049i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −20.5843 −0.787636 −0.393818 0.919188i \(-0.628846\pi\)
−0.393818 + 0.919188i \(0.628846\pi\)
\(684\) 0 0
\(685\) −22.5564 −0.861836
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 24.8562i − 0.946946i
\(690\) 0 0
\(691\) − 36.7778i − 1.39909i −0.714587 0.699547i \(-0.753385\pi\)
0.714587 0.699547i \(-0.246615\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −5.61553 −0.213009
\(696\) 0 0
\(697\) 3.56048 0.134863
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 29.0711i − 1.09800i −0.835823 0.549000i \(-0.815009\pi\)
0.835823 0.549000i \(-0.184991\pi\)
\(702\) 0 0
\(703\) 8.46692i 0.319336i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 19.1416 0.719895
\(708\) 0 0
\(709\) −37.4663 −1.40707 −0.703537 0.710658i \(-0.748397\pi\)
−0.703537 + 0.710658i \(0.748397\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 2.61024i 0.0977542i
\(714\) 0 0
\(715\) − 7.56048i − 0.282746i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 26.0552 0.971696 0.485848 0.874043i \(-0.338511\pi\)
0.485848 + 0.874043i \(0.338511\pi\)
\(720\) 0 0
\(721\) −10.3843 −0.386730
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0.457018i 0.0169732i
\(726\) 0 0
\(727\) 41.9975i 1.55760i 0.627271 + 0.778801i \(0.284172\pi\)
−0.627271 + 0.778801i \(0.715828\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −5.39939 −0.199704
\(732\) 0 0
\(733\) 32.2375 1.19072 0.595359 0.803460i \(-0.297010\pi\)
0.595359 + 0.803460i \(0.297010\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 2.96681i − 0.109284i
\(738\) 0 0
\(739\) 36.1810i 1.33094i 0.746425 + 0.665469i \(0.231769\pi\)
−0.746425 + 0.665469i \(0.768231\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 18.0113 0.660771 0.330386 0.943846i \(-0.392821\pi\)
0.330386 + 0.943846i \(0.392821\pi\)
\(744\) 0 0
\(745\) 15.5628 0.570179
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 6.65634i − 0.243217i
\(750\) 0 0
\(751\) 32.1484i 1.17311i 0.809909 + 0.586555i \(0.199516\pi\)
−0.809909 + 0.586555i \(0.800484\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 4.40159 0.159979 0.0799893 0.996796i \(-0.474511\pi\)
0.0799893 + 0.996796i \(0.474511\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 35.7247i − 1.29502i −0.762057 0.647509i \(-0.775811\pi\)
0.762057 0.647509i \(-0.224189\pi\)
\(762\) 0 0
\(763\) − 5.31835i − 0.192537i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −77.3662 −2.79353
\(768\) 0 0
\(769\) −21.1561 −0.762907 −0.381454 0.924388i \(-0.624576\pi\)
−0.381454 + 0.924388i \(0.624576\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 30.9557i 1.11340i 0.830714 + 0.556700i \(0.187933\pi\)
−0.830714 + 0.556700i \(0.812067\pi\)
\(774\) 0 0
\(775\) − 0.295929i − 0.0106301i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 23.6984 0.849085
\(780\) 0 0
\(781\) 3.27644 0.117240
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 7.12450i 0.254284i
\(786\) 0 0
\(787\) 32.2326i 1.14897i 0.818515 + 0.574484i \(0.194797\pi\)
−0.818515 + 0.574484i \(0.805203\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 13.8977 0.494147
\(792\) 0 0
\(793\) 49.6451 1.76295
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 11.4686i − 0.406240i −0.979154 0.203120i \(-0.934892\pi\)
0.979154 0.203120i \(-0.0651082\pi\)
\(798\) 0 0
\(799\) − 2.00000i − 0.0707549i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 5.70289 0.201251
\(804\) 0 0
\(805\) 1.88244 0.0663474
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 6.12628i 0.215389i 0.994184 + 0.107694i \(0.0343468\pi\)
−0.994184 + 0.107694i \(0.965653\pi\)
\(810\) 0 0
\(811\) − 3.23645i − 0.113647i −0.998384 0.0568236i \(-0.981903\pi\)
0.998384 0.0568236i \(-0.0180973\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −18.2412 −0.638961
\(816\) 0 0
\(817\) −35.9382 −1.25732
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 22.7411i 0.793668i 0.917890 + 0.396834i \(0.129891\pi\)
−0.917890 + 0.396834i \(0.870109\pi\)
\(822\) 0 0
\(823\) 16.9282i 0.590080i 0.955485 + 0.295040i \(0.0953330\pi\)
−0.955485 + 0.295040i \(0.904667\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −54.0092 −1.87808 −0.939041 0.343805i \(-0.888284\pi\)
−0.939041 + 0.343805i \(0.888284\pi\)
\(828\) 0 0
\(829\) −36.8581 −1.28013 −0.640067 0.768319i \(-0.721094\pi\)
−0.640067 + 0.768319i \(0.721094\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 0.585786i − 0.0202963i
\(834\) 0 0
\(835\) − 44.0306i − 1.52374i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −0.700690 −0.0241905 −0.0120953 0.999927i \(-0.503850\pi\)
−0.0120953 + 0.999927i \(0.503850\pi\)
\(840\) 0 0
\(841\) 6.51326 0.224595
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 39.6799i − 1.36503i
\(846\) 0 0
\(847\) 10.6230i 0.365010i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −1.84603 −0.0632810
\(852\) 0 0
\(853\) −41.2951 −1.41392 −0.706959 0.707254i \(-0.749934\pi\)
−0.706959 + 0.707254i \(0.749934\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 17.9935i 0.614645i 0.951605 + 0.307323i \(0.0994331\pi\)
−0.951605 + 0.307323i \(0.900567\pi\)
\(858\) 0 0
\(859\) 21.6877i 0.739973i 0.929037 + 0.369987i \(0.120638\pi\)
−0.929037 + 0.369987i \(0.879362\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 22.6116 0.769706 0.384853 0.922978i \(-0.374252\pi\)
0.384853 + 0.922978i \(0.374252\pi\)
\(864\) 0 0
\(865\) 24.9907 0.849710
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0.796483i 0.0270188i
\(870\) 0 0
\(871\) − 26.8673i − 0.910363i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −11.2855 −0.381519
\(876\) 0 0
\(877\) −34.7821 −1.17451 −0.587254 0.809402i \(-0.699791\pi\)
−0.587254 + 0.809402i \(0.699791\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 33.6493i − 1.13367i −0.823830 0.566837i \(-0.808167\pi\)
0.823830 0.566837i \(-0.191833\pi\)
\(882\) 0 0
\(883\) − 56.1132i − 1.88836i −0.329429 0.944180i \(-0.606856\pi\)
0.329429 0.944180i \(-0.393144\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 25.7126 0.863344 0.431672 0.902031i \(-0.357924\pi\)
0.431672 + 0.902031i \(0.357924\pi\)
\(888\) 0 0
\(889\) 2.16693 0.0726765
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 13.3119i − 0.445467i
\(894\) 0 0
\(895\) − 41.6588i − 1.39250i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 14.5606 0.485624
\(900\) 0 0
\(901\) −2.61856 −0.0872367
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 25.3657i − 0.843184i
\(906\) 0 0
\(907\) − 7.05322i − 0.234198i −0.993120 0.117099i \(-0.962640\pi\)
0.993120 0.117099i \(-0.0373595\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −33.1805 −1.09932 −0.549659 0.835389i \(-0.685242\pi\)
−0.549659 + 0.835389i \(0.685242\pi\)
\(912\) 0 0
\(913\) −8.92776 −0.295466
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 7.98414i − 0.263660i
\(918\) 0 0
\(919\) − 14.7710i − 0.487251i −0.969869 0.243625i \(-0.921663\pi\)
0.969869 0.243625i \(-0.0783367\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 29.6713 0.976641
\(924\) 0 0
\(925\) 0.209288 0.00688135
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 21.9025i 0.718598i 0.933222 + 0.359299i \(0.116984\pi\)
−0.933222 + 0.359299i \(0.883016\pi\)
\(930\) 0 0
\(931\) − 3.89898i − 0.127784i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −0.796483 −0.0260478
\(936\) 0 0
\(937\) 52.8854 1.72769 0.863846 0.503757i \(-0.168049\pi\)
0.863846 + 0.503757i \(0.168049\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 0.929998i − 0.0303171i −0.999885 0.0151585i \(-0.995175\pi\)
0.999885 0.0151585i \(-0.00482530\pi\)
\(942\) 0 0
\(943\) 5.16693i 0.168258i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 12.7277 0.413593 0.206797 0.978384i \(-0.433696\pi\)
0.206797 + 0.978384i \(0.433696\pi\)
\(948\) 0 0
\(949\) 51.6451 1.67647
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 3.98965i − 0.129238i −0.997910 0.0646188i \(-0.979417\pi\)
0.997910 0.0646188i \(-0.0205831\pi\)
\(954\) 0 0
\(955\) 0.745400i 0.0241206i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −10.1862 −0.328929
\(960\) 0 0
\(961\) 21.5717 0.695862
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 21.7173i 0.699104i
\(966\) 0 0
\(967\) 2.37871i 0.0764940i 0.999268 + 0.0382470i \(0.0121774\pi\)
−0.999268 + 0.0382470i \(0.987823\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 40.5224 1.30042 0.650212 0.759753i \(-0.274680\pi\)
0.650212 + 0.759753i \(0.274680\pi\)
\(972\) 0 0
\(973\) −2.53590 −0.0812972
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 14.2117i − 0.454671i −0.973816 0.227336i \(-0.926999\pi\)
0.973816 0.227336i \(-0.0730014\pi\)
\(978\) 0 0
\(979\) − 7.00340i − 0.223830i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −33.6919 −1.07461 −0.537303 0.843389i \(-0.680557\pi\)
−0.537303 + 0.843389i \(0.680557\pi\)
\(984\) 0 0
\(985\) −9.71573 −0.309569
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 7.83554i − 0.249156i
\(990\) 0 0
\(991\) − 25.9339i − 0.823817i −0.911225 0.411908i \(-0.864862\pi\)
0.911225 0.411908i \(-0.135138\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −11.5506 −0.366178
\(996\) 0 0
\(997\) 31.3697 0.993487 0.496744 0.867897i \(-0.334529\pi\)
0.496744 + 0.867897i \(0.334529\pi\)
\(998\) 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.g.2591.3 yes 8
3.2 odd 2 6048.2.h.a.2591.6 yes 8
4.3 odd 2 6048.2.h.a.2591.3 8
12.11 even 2 inner 6048.2.h.g.2591.6 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6048.2.h.a.2591.3 8 4.3 odd 2
6048.2.h.a.2591.6 yes 8 3.2 odd 2
6048.2.h.g.2591.3 yes 8 1.1 even 1 trivial
6048.2.h.g.2591.6 yes 8 12.11 even 2 inner