Properties

Label 6048.2.c.g.3025.24
Level 6048
Weight 2
Character 6048.3025
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.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.2935231425\)
Analytic rank: \(0\)
Dimension: \(24\)
Coefficient ring index: multiple of None
Twist minimal: no (minimal twist has level 1512)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3025.24
Character \(\chi\) = 6048.3025
Dual form 6048.2.c.g.3025.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.66698i q^{5} -1.00000 q^{7} +O(q^{10})\) \(q+3.66698i q^{5} -1.00000 q^{7} -4.45794i q^{11} +1.51546i q^{13} +3.45516 q^{17} -2.29933i q^{19} -8.76326 q^{23} -8.44677 q^{25} -1.62037i q^{29} +9.26498 q^{31} -3.66698i q^{35} -5.69381i q^{37} +6.86926 q^{41} -0.880042i q^{43} +10.5955 q^{47} +1.00000 q^{49} +11.5036i q^{53} +16.3472 q^{55} +2.99426i q^{59} -8.51202i q^{61} -5.55718 q^{65} -10.6629i q^{67} +10.6172 q^{71} +7.85523 q^{73} +4.45794i q^{77} -4.57730 q^{79} -2.60500i q^{83} +12.6700i q^{85} +2.14381 q^{89} -1.51546i q^{91} +8.43162 q^{95} -13.8317 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24q - 24q^{7} + O(q^{10}) \) \( 24q - 24q^{7} - 24q^{25} + 16q^{31} + 24q^{49} + 8q^{55} - 32q^{79} + O(q^{100}) \)

Character values

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

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

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 3.66698i 1.63993i 0.572417 + 0.819963i \(0.306006\pi\)
−0.572417 + 0.819963i \(0.693994\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 4.45794i − 1.34412i −0.740498 0.672059i \(-0.765410\pi\)
0.740498 0.672059i \(-0.234590\pi\)
\(12\) 0 0
\(13\) 1.51546i 0.420314i 0.977668 + 0.210157i \(0.0673975\pi\)
−0.977668 + 0.210157i \(0.932602\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.45516 0.838000 0.419000 0.907986i \(-0.362381\pi\)
0.419000 + 0.907986i \(0.362381\pi\)
\(18\) 0 0
\(19\) − 2.29933i − 0.527503i −0.964591 0.263752i \(-0.915040\pi\)
0.964591 0.263752i \(-0.0849600\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −8.76326 −1.82727 −0.913633 0.406539i \(-0.866735\pi\)
−0.913633 + 0.406539i \(0.866735\pi\)
\(24\) 0 0
\(25\) −8.44677 −1.68935
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 1.62037i − 0.300895i −0.988618 0.150448i \(-0.951928\pi\)
0.988618 0.150448i \(-0.0480715\pi\)
\(30\) 0 0
\(31\) 9.26498 1.66404 0.832020 0.554746i \(-0.187184\pi\)
0.832020 + 0.554746i \(0.187184\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 3.66698i − 0.619833i
\(36\) 0 0
\(37\) − 5.69381i − 0.936056i −0.883714 0.468028i \(-0.844965\pi\)
0.883714 0.468028i \(-0.155035\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.86926 1.07280 0.536399 0.843965i \(-0.319784\pi\)
0.536399 + 0.843965i \(0.319784\pi\)
\(42\) 0 0
\(43\) − 0.880042i − 0.134205i −0.997746 0.0671026i \(-0.978625\pi\)
0.997746 0.0671026i \(-0.0213755\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 10.5955 1.54551 0.772753 0.634707i \(-0.218879\pi\)
0.772753 + 0.634707i \(0.218879\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 11.5036i 1.58014i 0.613014 + 0.790072i \(0.289957\pi\)
−0.613014 + 0.790072i \(0.710043\pi\)
\(54\) 0 0
\(55\) 16.3472 2.20425
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.99426i 0.389819i 0.980821 + 0.194909i \(0.0624413\pi\)
−0.980821 + 0.194909i \(0.937559\pi\)
\(60\) 0 0
\(61\) − 8.51202i − 1.08985i −0.838484 0.544926i \(-0.816558\pi\)
0.838484 0.544926i \(-0.183442\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −5.55718 −0.689284
\(66\) 0 0
\(67\) − 10.6629i − 1.30268i −0.758786 0.651341i \(-0.774207\pi\)
0.758786 0.651341i \(-0.225793\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 10.6172 1.26003 0.630015 0.776583i \(-0.283049\pi\)
0.630015 + 0.776583i \(0.283049\pi\)
\(72\) 0 0
\(73\) 7.85523 0.919385 0.459692 0.888078i \(-0.347960\pi\)
0.459692 + 0.888078i \(0.347960\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.45794i 0.508029i
\(78\) 0 0
\(79\) −4.57730 −0.514986 −0.257493 0.966280i \(-0.582896\pi\)
−0.257493 + 0.966280i \(0.582896\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 2.60500i − 0.285936i −0.989727 0.142968i \(-0.954335\pi\)
0.989727 0.142968i \(-0.0456645\pi\)
\(84\) 0 0
\(85\) 12.6700i 1.37426i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2.14381 0.227244 0.113622 0.993524i \(-0.463755\pi\)
0.113622 + 0.993524i \(0.463755\pi\)
\(90\) 0 0
\(91\) − 1.51546i − 0.158864i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 8.43162 0.865066
\(96\) 0 0
\(97\) −13.8317 −1.40440 −0.702199 0.711981i \(-0.747798\pi\)
−0.702199 + 0.711981i \(0.747798\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 10.2333i 1.01826i 0.860691 + 0.509128i \(0.170032\pi\)
−0.860691 + 0.509128i \(0.829968\pi\)
\(102\) 0 0
\(103\) 12.7223 1.25357 0.626784 0.779193i \(-0.284371\pi\)
0.626784 + 0.779193i \(0.284371\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 6.40575i − 0.619268i −0.950856 0.309634i \(-0.899794\pi\)
0.950856 0.309634i \(-0.100206\pi\)
\(108\) 0 0
\(109\) − 3.38839i − 0.324549i −0.986746 0.162274i \(-0.948117\pi\)
0.986746 0.162274i \(-0.0518829\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1.40668 0.132329 0.0661644 0.997809i \(-0.478924\pi\)
0.0661644 + 0.997809i \(0.478924\pi\)
\(114\) 0 0
\(115\) − 32.1347i − 2.99658i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −3.45516 −0.316734
\(120\) 0 0
\(121\) −8.87319 −0.806653
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 12.6393i − 1.13049i
\(126\) 0 0
\(127\) −10.4597 −0.928152 −0.464076 0.885795i \(-0.653614\pi\)
−0.464076 + 0.885795i \(0.653614\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 12.6638i 1.10644i 0.833034 + 0.553221i \(0.186602\pi\)
−0.833034 + 0.553221i \(0.813398\pi\)
\(132\) 0 0
\(133\) 2.29933i 0.199378i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 6.87026 0.586966 0.293483 0.955964i \(-0.405186\pi\)
0.293483 + 0.955964i \(0.405186\pi\)
\(138\) 0 0
\(139\) 13.4362i 1.13964i 0.821768 + 0.569822i \(0.192988\pi\)
−0.821768 + 0.569822i \(0.807012\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 6.75584 0.564952
\(144\) 0 0
\(145\) 5.94188 0.493446
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 6.35294i 0.520453i 0.965548 + 0.260227i \(0.0837973\pi\)
−0.965548 + 0.260227i \(0.916203\pi\)
\(150\) 0 0
\(151\) −19.5918 −1.59436 −0.797179 0.603743i \(-0.793675\pi\)
−0.797179 + 0.603743i \(0.793675\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 33.9745i 2.72890i
\(156\) 0 0
\(157\) − 22.6978i − 1.81148i −0.423834 0.905740i \(-0.639316\pi\)
0.423834 0.905740i \(-0.360684\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 8.76326 0.690642
\(162\) 0 0
\(163\) 3.60107i 0.282058i 0.990005 + 0.141029i \(0.0450410\pi\)
−0.990005 + 0.141029i \(0.954959\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −5.46351 −0.422779 −0.211390 0.977402i \(-0.567799\pi\)
−0.211390 + 0.977402i \(0.567799\pi\)
\(168\) 0 0
\(169\) 10.7034 0.823336
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1.71944i 0.130727i 0.997862 + 0.0653634i \(0.0208207\pi\)
−0.997862 + 0.0653634i \(0.979179\pi\)
\(174\) 0 0
\(175\) 8.44677 0.638516
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 10.9956i − 0.821846i −0.911670 0.410923i \(-0.865206\pi\)
0.911670 0.410923i \(-0.134794\pi\)
\(180\) 0 0
\(181\) 26.6227i 1.97885i 0.145040 + 0.989426i \(0.453669\pi\)
−0.145040 + 0.989426i \(0.546331\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 20.8791 1.53506
\(186\) 0 0
\(187\) − 15.4029i − 1.12637i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 22.1581 1.60330 0.801652 0.597791i \(-0.203955\pi\)
0.801652 + 0.597791i \(0.203955\pi\)
\(192\) 0 0
\(193\) 11.5599 0.832098 0.416049 0.909342i \(-0.363414\pi\)
0.416049 + 0.909342i \(0.363414\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 13.1740i − 0.938611i −0.883036 0.469305i \(-0.844504\pi\)
0.883036 0.469305i \(-0.155496\pi\)
\(198\) 0 0
\(199\) 12.1012 0.857828 0.428914 0.903345i \(-0.358896\pi\)
0.428914 + 0.903345i \(0.358896\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1.62037i 0.113728i
\(204\) 0 0
\(205\) 25.1895i 1.75931i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −10.2503 −0.709027
\(210\) 0 0
\(211\) 21.7501i 1.49734i 0.662944 + 0.748669i \(0.269307\pi\)
−0.662944 + 0.748669i \(0.730693\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 3.22710 0.220086
\(216\) 0 0
\(217\) −9.26498 −0.628948
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 5.23617i 0.352223i
\(222\) 0 0
\(223\) 3.00739 0.201390 0.100695 0.994917i \(-0.467893\pi\)
0.100695 + 0.994917i \(0.467893\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 11.1508i − 0.740102i −0.929011 0.370051i \(-0.879340\pi\)
0.929011 0.370051i \(-0.120660\pi\)
\(228\) 0 0
\(229\) 2.46317i 0.162771i 0.996683 + 0.0813853i \(0.0259344\pi\)
−0.996683 + 0.0813853i \(0.974066\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 26.1143 1.71080 0.855401 0.517966i \(-0.173311\pi\)
0.855401 + 0.517966i \(0.173311\pi\)
\(234\) 0 0
\(235\) 38.8534i 2.53451i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 24.4496 1.58151 0.790757 0.612130i \(-0.209687\pi\)
0.790757 + 0.612130i \(0.209687\pi\)
\(240\) 0 0
\(241\) −9.01348 −0.580609 −0.290305 0.956934i \(-0.593757\pi\)
−0.290305 + 0.956934i \(0.593757\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 3.66698i 0.234275i
\(246\) 0 0
\(247\) 3.48456 0.221717
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 24.7649i 1.56315i 0.623814 + 0.781573i \(0.285582\pi\)
−0.623814 + 0.781573i \(0.714418\pi\)
\(252\) 0 0
\(253\) 39.0661i 2.45606i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −17.1618 −1.07052 −0.535262 0.844686i \(-0.679787\pi\)
−0.535262 + 0.844686i \(0.679787\pi\)
\(258\) 0 0
\(259\) 5.69381i 0.353796i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 4.02861 0.248415 0.124207 0.992256i \(-0.460361\pi\)
0.124207 + 0.992256i \(0.460361\pi\)
\(264\) 0 0
\(265\) −42.1836 −2.59132
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 24.4193i 1.48887i 0.667693 + 0.744436i \(0.267282\pi\)
−0.667693 + 0.744436i \(0.732718\pi\)
\(270\) 0 0
\(271\) 1.21510 0.0738120 0.0369060 0.999319i \(-0.488250\pi\)
0.0369060 + 0.999319i \(0.488250\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 37.6552i 2.27069i
\(276\) 0 0
\(277\) 11.1239i 0.668371i 0.942507 + 0.334185i \(0.108461\pi\)
−0.942507 + 0.334185i \(0.891539\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 29.4678 1.75790 0.878952 0.476910i \(-0.158243\pi\)
0.878952 + 0.476910i \(0.158243\pi\)
\(282\) 0 0
\(283\) − 13.0312i − 0.774624i −0.921949 0.387312i \(-0.873404\pi\)
0.921949 0.387312i \(-0.126596\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −6.86926 −0.405479
\(288\) 0 0
\(289\) −5.06186 −0.297756
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 4.86503i − 0.284218i −0.989851 0.142109i \(-0.954612\pi\)
0.989851 0.142109i \(-0.0453883\pi\)
\(294\) 0 0
\(295\) −10.9799 −0.639274
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 13.2804i − 0.768026i
\(300\) 0 0
\(301\) 0.880042i 0.0507248i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 31.2134 1.78728
\(306\) 0 0
\(307\) − 9.07611i − 0.518001i −0.965877 0.259000i \(-0.916607\pi\)
0.965877 0.259000i \(-0.0833931\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 1.21020 0.0686240 0.0343120 0.999411i \(-0.489076\pi\)
0.0343120 + 0.999411i \(0.489076\pi\)
\(312\) 0 0
\(313\) −3.15697 −0.178443 −0.0892214 0.996012i \(-0.528438\pi\)
−0.0892214 + 0.996012i \(0.528438\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 7.08197i − 0.397763i −0.980024 0.198882i \(-0.936269\pi\)
0.980024 0.198882i \(-0.0637309\pi\)
\(318\) 0 0
\(319\) −7.22351 −0.404439
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 7.94457i − 0.442048i
\(324\) 0 0
\(325\) − 12.8008i − 0.710060i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −10.5955 −0.584146
\(330\) 0 0
\(331\) 7.34765i 0.403863i 0.979400 + 0.201932i \(0.0647219\pi\)
−0.979400 + 0.201932i \(0.935278\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 39.1007 2.13630
\(336\) 0 0
\(337\) −26.9873 −1.47009 −0.735046 0.678017i \(-0.762840\pi\)
−0.735046 + 0.678017i \(0.762840\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 41.3027i − 2.23667i
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 9.06776i − 0.486783i −0.969928 0.243391i \(-0.921740\pi\)
0.969928 0.243391i \(-0.0782599\pi\)
\(348\) 0 0
\(349\) 13.6784i 0.732191i 0.930577 + 0.366095i \(0.119306\pi\)
−0.930577 + 0.366095i \(0.880694\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0.949067 0.0505138 0.0252569 0.999681i \(-0.491960\pi\)
0.0252569 + 0.999681i \(0.491960\pi\)
\(354\) 0 0
\(355\) 38.9331i 2.06636i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −15.0508 −0.794351 −0.397175 0.917743i \(-0.630009\pi\)
−0.397175 + 0.917743i \(0.630009\pi\)
\(360\) 0 0
\(361\) 13.7131 0.721740
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 28.8050i 1.50772i
\(366\) 0 0
\(367\) 3.70674 0.193490 0.0967451 0.995309i \(-0.469157\pi\)
0.0967451 + 0.995309i \(0.469157\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 11.5036i − 0.597239i
\(372\) 0 0
\(373\) − 31.3688i − 1.62421i −0.583508 0.812107i \(-0.698320\pi\)
0.583508 0.812107i \(-0.301680\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 2.45562 0.126471
\(378\) 0 0
\(379\) − 17.6560i − 0.906930i −0.891274 0.453465i \(-0.850188\pi\)
0.891274 0.453465i \(-0.149812\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −17.7974 −0.909402 −0.454701 0.890644i \(-0.650254\pi\)
−0.454701 + 0.890644i \(0.650254\pi\)
\(384\) 0 0
\(385\) −16.3472 −0.833129
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 25.1886i 1.27711i 0.769574 + 0.638557i \(0.220468\pi\)
−0.769574 + 0.638557i \(0.779532\pi\)
\(390\) 0 0
\(391\) −30.2785 −1.53125
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 16.7849i − 0.844539i
\(396\) 0 0
\(397\) − 34.3710i − 1.72503i −0.506032 0.862515i \(-0.668888\pi\)
0.506032 0.862515i \(-0.331112\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −9.65560 −0.482178 −0.241089 0.970503i \(-0.577505\pi\)
−0.241089 + 0.970503i \(0.577505\pi\)
\(402\) 0 0
\(403\) 14.0408i 0.699420i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −25.3826 −1.25817
\(408\) 0 0
\(409\) 28.4161 1.40509 0.702543 0.711642i \(-0.252048\pi\)
0.702543 + 0.711642i \(0.252048\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 2.99426i − 0.147338i
\(414\) 0 0
\(415\) 9.55248 0.468913
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 11.5381i 0.563673i 0.959462 + 0.281837i \(0.0909436\pi\)
−0.959462 + 0.281837i \(0.909056\pi\)
\(420\) 0 0
\(421\) − 6.84814i − 0.333758i −0.985977 0.166879i \(-0.946631\pi\)
0.985977 0.166879i \(-0.0533689\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −29.1850 −1.41568
\(426\) 0 0
\(427\) 8.51202i 0.411925i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 10.4500 0.503357 0.251679 0.967811i \(-0.419017\pi\)
0.251679 + 0.967811i \(0.419017\pi\)
\(432\) 0 0
\(433\) 18.1818 0.873763 0.436882 0.899519i \(-0.356083\pi\)
0.436882 + 0.899519i \(0.356083\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 20.1497i 0.963889i
\(438\) 0 0
\(439\) −7.91836 −0.377923 −0.188961 0.981985i \(-0.560512\pi\)
−0.188961 + 0.981985i \(0.560512\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 18.2916i 0.869062i 0.900657 + 0.434531i \(0.143086\pi\)
−0.900657 + 0.434531i \(0.856914\pi\)
\(444\) 0 0
\(445\) 7.86132i 0.372662i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 15.2172 0.718144 0.359072 0.933310i \(-0.383093\pi\)
0.359072 + 0.933310i \(0.383093\pi\)
\(450\) 0 0
\(451\) − 30.6227i − 1.44197i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 5.55718 0.260525
\(456\) 0 0
\(457\) 38.6658 1.80871 0.904354 0.426783i \(-0.140353\pi\)
0.904354 + 0.426783i \(0.140353\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 5.18888i − 0.241670i −0.992673 0.120835i \(-0.961443\pi\)
0.992673 0.120835i \(-0.0385572\pi\)
\(462\) 0 0
\(463\) 3.81583 0.177337 0.0886683 0.996061i \(-0.471739\pi\)
0.0886683 + 0.996061i \(0.471739\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 7.29246i 0.337455i 0.985663 + 0.168727i \(0.0539657\pi\)
−0.985663 + 0.168727i \(0.946034\pi\)
\(468\) 0 0
\(469\) 10.6629i 0.492367i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −3.92317 −0.180388
\(474\) 0 0
\(475\) 19.4220i 0.891140i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 16.2695 0.743371 0.371686 0.928359i \(-0.378780\pi\)
0.371686 + 0.928359i \(0.378780\pi\)
\(480\) 0 0
\(481\) 8.62876 0.393438
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 50.7207i − 2.30311i
\(486\) 0 0
\(487\) −7.16667 −0.324753 −0.162376 0.986729i \(-0.551916\pi\)
−0.162376 + 0.986729i \(0.551916\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 17.8247i 0.804418i 0.915548 + 0.402209i \(0.131757\pi\)
−0.915548 + 0.402209i \(0.868243\pi\)
\(492\) 0 0
\(493\) − 5.59864i − 0.252150i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −10.6172 −0.476247
\(498\) 0 0
\(499\) − 37.6121i − 1.68375i −0.539673 0.841875i \(-0.681452\pi\)
0.539673 0.841875i \(-0.318548\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −20.3707 −0.908286 −0.454143 0.890929i \(-0.650055\pi\)
−0.454143 + 0.890929i \(0.650055\pi\)
\(504\) 0 0
\(505\) −37.5255 −1.66986
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 26.6481i 1.18116i 0.806980 + 0.590578i \(0.201100\pi\)
−0.806980 + 0.590578i \(0.798900\pi\)
\(510\) 0 0
\(511\) −7.85523 −0.347495
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 46.6526i 2.05576i
\(516\) 0 0
\(517\) − 47.2339i − 2.07734i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 13.2614 0.580992 0.290496 0.956876i \(-0.406180\pi\)
0.290496 + 0.956876i \(0.406180\pi\)
\(522\) 0 0
\(523\) − 21.3786i − 0.934821i −0.884040 0.467411i \(-0.845187\pi\)
0.884040 0.467411i \(-0.154813\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 32.0120 1.39447
\(528\) 0 0
\(529\) 53.7948 2.33890
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 10.4101i 0.450912i
\(534\) 0 0
\(535\) 23.4898 1.01555
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 4.45794i − 0.192017i
\(540\) 0 0
\(541\) − 31.4969i − 1.35416i −0.735910 0.677080i \(-0.763245\pi\)
0.735910 0.677080i \(-0.236755\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 12.4252 0.532235
\(546\) 0 0
\(547\) − 1.38840i − 0.0593637i −0.999559 0.0296819i \(-0.990551\pi\)
0.999559 0.0296819i \(-0.00944942\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −3.72578 −0.158723
\(552\) 0 0
\(553\) 4.57730 0.194647
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 10.8246i − 0.458654i −0.973349 0.229327i \(-0.926347\pi\)
0.973349 0.229327i \(-0.0736525\pi\)
\(558\) 0 0
\(559\) 1.33367 0.0564083
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 7.56473i 0.318816i 0.987213 + 0.159408i \(0.0509585\pi\)
−0.987213 + 0.159408i \(0.949042\pi\)
\(564\) 0 0
\(565\) 5.15826i 0.217009i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 21.9063 0.918360 0.459180 0.888343i \(-0.348143\pi\)
0.459180 + 0.888343i \(0.348143\pi\)
\(570\) 0 0
\(571\) 43.6061i 1.82486i 0.409235 + 0.912429i \(0.365796\pi\)
−0.409235 + 0.912429i \(0.634204\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 74.0213 3.08690
\(576\) 0 0
\(577\) 12.6116 0.525028 0.262514 0.964928i \(-0.415448\pi\)
0.262514 + 0.964928i \(0.415448\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 2.60500i 0.108073i
\(582\) 0 0
\(583\) 51.2824 2.12390
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 10.5118i 0.433866i 0.976186 + 0.216933i \(0.0696054\pi\)
−0.976186 + 0.216933i \(0.930395\pi\)
\(588\) 0 0
\(589\) − 21.3033i − 0.877787i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 23.9765 0.984598 0.492299 0.870426i \(-0.336157\pi\)
0.492299 + 0.870426i \(0.336157\pi\)
\(594\) 0 0
\(595\) − 12.6700i − 0.519420i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −32.2302 −1.31689 −0.658446 0.752628i \(-0.728786\pi\)
−0.658446 + 0.752628i \(0.728786\pi\)
\(600\) 0 0
\(601\) 43.3405 1.76790 0.883949 0.467584i \(-0.154875\pi\)
0.883949 + 0.467584i \(0.154875\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 32.5378i − 1.32285i
\(606\) 0 0
\(607\) 13.2372 0.537283 0.268642 0.963240i \(-0.413425\pi\)
0.268642 + 0.963240i \(0.413425\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 16.0570i 0.649598i
\(612\) 0 0
\(613\) − 6.23612i − 0.251874i −0.992038 0.125937i \(-0.959806\pi\)
0.992038 0.125937i \(-0.0401938\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −7.66713 −0.308667 −0.154334 0.988019i \(-0.549323\pi\)
−0.154334 + 0.988019i \(0.549323\pi\)
\(618\) 0 0
\(619\) − 27.9974i − 1.12531i −0.826692 0.562655i \(-0.809780\pi\)
0.826692 0.562655i \(-0.190220\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −2.14381 −0.0858900
\(624\) 0 0
\(625\) 4.11412 0.164565
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 19.6730i − 0.784415i
\(630\) 0 0
\(631\) −18.2806 −0.727739 −0.363869 0.931450i \(-0.618545\pi\)
−0.363869 + 0.931450i \(0.618545\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 38.3557i − 1.52210i
\(636\) 0 0
\(637\) 1.51546i 0.0600449i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −5.99361 −0.236733 −0.118367 0.992970i \(-0.537766\pi\)
−0.118367 + 0.992970i \(0.537766\pi\)
\(642\) 0 0
\(643\) − 29.0547i − 1.14581i −0.819623 0.572903i \(-0.805817\pi\)
0.819623 0.572903i \(-0.194183\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 26.7288 1.05082 0.525408 0.850850i \(-0.323913\pi\)
0.525408 + 0.850850i \(0.323913\pi\)
\(648\) 0 0
\(649\) 13.3482 0.523963
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 41.0011i − 1.60450i −0.596991 0.802248i \(-0.703637\pi\)
0.596991 0.802248i \(-0.296363\pi\)
\(654\) 0 0
\(655\) −46.4380 −1.81448
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 1.91818i − 0.0747219i −0.999302 0.0373609i \(-0.988105\pi\)
0.999302 0.0373609i \(-0.0118951\pi\)
\(660\) 0 0
\(661\) 10.9347i 0.425312i 0.977127 + 0.212656i \(0.0682114\pi\)
−0.977127 + 0.212656i \(0.931789\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −8.43162 −0.326964
\(666\) 0 0
\(667\) 14.1997i 0.549816i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −37.9460 −1.46489
\(672\) 0 0
\(673\) 10.0071 0.385745 0.192872 0.981224i \(-0.438220\pi\)
0.192872 + 0.981224i \(0.438220\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 35.4441i 1.36223i 0.732178 + 0.681113i \(0.238504\pi\)
−0.732178 + 0.681113i \(0.761496\pi\)
\(678\) 0 0
\(679\) 13.8317 0.530812
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 34.8397i − 1.33310i −0.745459 0.666551i \(-0.767770\pi\)
0.745459 0.666551i \(-0.232230\pi\)
\(684\) 0 0
\(685\) 25.1931i 0.962580i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −17.4333 −0.664157
\(690\) 0 0
\(691\) − 16.2381i − 0.617727i −0.951106 0.308864i \(-0.900051\pi\)
0.951106 0.308864i \(-0.0999486\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −49.2704 −1.86893
\(696\) 0 0
\(697\) 23.7344 0.899004
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 28.8081i 1.08807i 0.839064 + 0.544033i \(0.183103\pi\)
−0.839064 + 0.544033i \(0.816897\pi\)
\(702\) 0 0
\(703\) −13.0920 −0.493773
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 10.2333i − 0.384864i
\(708\) 0 0
\(709\) − 18.1039i − 0.679906i −0.940442 0.339953i \(-0.889589\pi\)
0.940442 0.339953i \(-0.110411\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −81.1915 −3.04064
\(714\) 0 0
\(715\) 24.7736i 0.926479i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −12.7731 −0.476357 −0.238179 0.971221i \(-0.576550\pi\)
−0.238179 + 0.971221i \(0.576550\pi\)
\(720\) 0 0
\(721\) −12.7223 −0.473804
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 13.6869i 0.508319i
\(726\) 0 0
\(727\) 10.1654 0.377012 0.188506 0.982072i \(-0.439635\pi\)
0.188506 + 0.982072i \(0.439635\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 3.04069i − 0.112464i
\(732\) 0 0
\(733\) − 26.1041i − 0.964178i −0.876122 0.482089i \(-0.839878\pi\)
0.876122 0.482089i \(-0.160122\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −47.5345 −1.75096
\(738\) 0 0
\(739\) − 9.90517i − 0.364368i −0.983264 0.182184i \(-0.941683\pi\)
0.983264 0.182184i \(-0.0583166\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 33.5353 1.23029 0.615146 0.788413i \(-0.289097\pi\)
0.615146 + 0.788413i \(0.289097\pi\)
\(744\) 0 0
\(745\) −23.2961 −0.853504
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 6.40575i 0.234061i
\(750\) 0 0
\(751\) −53.8436 −1.96478 −0.982391 0.186839i \(-0.940176\pi\)
−0.982391 + 0.186839i \(0.940176\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 71.8428i − 2.61463i
\(756\) 0 0
\(757\) 1.18921i 0.0432225i 0.999766 + 0.0216113i \(0.00687962\pi\)
−0.999766 + 0.0216113i \(0.993120\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 38.0810 1.38043 0.690217 0.723602i \(-0.257515\pi\)
0.690217 + 0.723602i \(0.257515\pi\)
\(762\) 0 0
\(763\) 3.38839i 0.122668i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −4.53769 −0.163846
\(768\) 0 0
\(769\) −38.6053 −1.39214 −0.696071 0.717973i \(-0.745070\pi\)
−0.696071 + 0.717973i \(0.745070\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 18.5902i 0.668642i 0.942459 + 0.334321i \(0.108507\pi\)
−0.942459 + 0.334321i \(0.891493\pi\)
\(774\) 0 0
\(775\) −78.2592 −2.81115
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 15.7947i − 0.565905i
\(780\) 0 0
\(781\) − 47.3308i − 1.69363i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 83.2324 2.97069
\(786\) 0 0
\(787\) 17.3947i 0.620056i 0.950727 + 0.310028i \(0.100338\pi\)
−0.950727 + 0.310028i \(0.899662\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −1.40668 −0.0500156
\(792\) 0 0
\(793\) 12.8997 0.458080
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 15.3683i 0.544374i 0.962244 + 0.272187i \(0.0877470\pi\)
−0.962244 + 0.272187i \(0.912253\pi\)
\(798\) 0 0
\(799\) 36.6090 1.29513
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 35.0181i − 1.23576i
\(804\) 0 0
\(805\) 32.1347i 1.13260i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −29.7888 −1.04732 −0.523659 0.851928i \(-0.675433\pi\)
−0.523659 + 0.851928i \(0.675433\pi\)
\(810\) 0 0
\(811\) − 6.50515i − 0.228427i −0.993456 0.114213i \(-0.963565\pi\)
0.993456 0.114213i \(-0.0364347\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −13.2051 −0.462553
\(816\) 0 0
\(817\) −2.02351 −0.0707937
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 3.89581i 0.135965i 0.997687 + 0.0679823i \(0.0216562\pi\)
−0.997687 + 0.0679823i \(0.978344\pi\)
\(822\) 0 0
\(823\) −47.1577 −1.64381 −0.821907 0.569621i \(-0.807090\pi\)
−0.821907 + 0.569621i \(0.807090\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 30.9915i − 1.07768i −0.842408 0.538841i \(-0.818862\pi\)
0.842408 0.538841i \(-0.181138\pi\)
\(828\) 0 0
\(829\) − 34.2090i − 1.18813i −0.804417 0.594065i \(-0.797522\pi\)
0.804417 0.594065i \(-0.202478\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 3.45516 0.119714
\(834\) 0 0
\(835\) − 20.0346i − 0.693326i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −40.9980 −1.41541 −0.707704 0.706509i \(-0.750269\pi\)
−0.707704 + 0.706509i \(0.750269\pi\)
\(840\) 0 0
\(841\) 26.3744 0.909462
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 39.2491i 1.35021i
\(846\) 0 0
\(847\) 8.87319 0.304886
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 49.8963i 1.71042i
\(852\) 0 0
\(853\) 26.3408i 0.901891i 0.892551 + 0.450946i \(0.148913\pi\)
−0.892551 + 0.450946i \(0.851087\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 56.4187 1.92723 0.963613 0.267300i \(-0.0861316\pi\)
0.963613 + 0.267300i \(0.0861316\pi\)
\(858\) 0 0
\(859\) 34.5693i 1.17949i 0.807589 + 0.589745i \(0.200772\pi\)
−0.807589 + 0.589745i \(0.799228\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1.52709 0.0519826 0.0259913 0.999662i \(-0.491726\pi\)
0.0259913 + 0.999662i \(0.491726\pi\)
\(864\) 0 0
\(865\) −6.30517 −0.214382
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 20.4053i 0.692202i
\(870\) 0 0
\(871\) 16.1593 0.547535
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 12.6393i 0.427285i
\(876\) 0 0
\(877\) − 20.5638i − 0.694389i −0.937793 0.347195i \(-0.887134\pi\)
0.937793 0.347195i \(-0.112866\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 16.8073 0.566252 0.283126 0.959083i \(-0.408629\pi\)
0.283126 + 0.959083i \(0.408629\pi\)
\(882\) 0 0
\(883\) − 13.7849i − 0.463900i −0.972728 0.231950i \(-0.925489\pi\)
0.972728 0.231950i \(-0.0745105\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −25.0037 −0.839541 −0.419771 0.907630i \(-0.637889\pi\)
−0.419771 + 0.907630i \(0.637889\pi\)
\(888\) 0 0
\(889\) 10.4597 0.350808
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 24.3625i − 0.815260i
\(894\) 0 0
\(895\) 40.3205 1.34777
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 15.0127i − 0.500702i
\(900\) 0 0
\(901\) 39.7469i 1.32416i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −97.6251 −3.24517
\(906\) 0 0
\(907\) 23.5827i 0.783052i 0.920167 + 0.391526i \(0.128053\pi\)
−0.920167 + 0.391526i \(0.871947\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −29.1770 −0.966677 −0.483338 0.875434i \(-0.660576\pi\)
−0.483338 + 0.875434i \(0.660576\pi\)
\(912\) 0 0
\(913\) −11.6129 −0.384331
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 12.6638i − 0.418196i
\(918\) 0 0
\(919\) −51.6854 −1.70494 −0.852472 0.522772i \(-0.824898\pi\)
−0.852472 + 0.522772i \(0.824898\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 16.0900i 0.529609i
\(924\) 0 0
\(925\) 48.0943i 1.58133i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 20.0819 0.658865 0.329432 0.944179i \(-0.393143\pi\)
0.329432 + 0.944179i \(0.393143\pi\)
\(930\) 0 0
\(931\) − 2.29933i − 0.0753576i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 56.4821 1.84716
\(936\) 0 0
\(937\) −19.1583 −0.625875 −0.312937 0.949774i \(-0.601313\pi\)
−0.312937 + 0.949774i \(0.601313\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 54.2409i 1.76820i 0.467296 + 0.884101i \(0.345228\pi\)
−0.467296 + 0.884101i \(0.654772\pi\)
\(942\) 0 0
\(943\) −60.1971 −1.96029
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 47.7355i − 1.55120i −0.631228 0.775598i \(-0.717449\pi\)
0.631228 0.775598i \(-0.282551\pi\)
\(948\) 0 0
\(949\) 11.9043i 0.386430i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −11.2075 −0.363046 −0.181523 0.983387i \(-0.558103\pi\)
−0.181523 + 0.983387i \(0.558103\pi\)
\(954\) 0 0
\(955\) 81.2534i 2.62930i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −6.87026 −0.221852
\(960\) 0 0
\(961\) 54.8399 1.76903
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 42.3899i 1.36458i
\(966\) 0 0
\(967\) 22.5979 0.726701 0.363350 0.931653i \(-0.381633\pi\)
0.363350 + 0.931653i \(0.381633\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 46.5586i − 1.49414i −0.664747 0.747068i \(-0.731461\pi\)
0.664747 0.747068i \(-0.268539\pi\)
\(972\) 0 0
\(973\) − 13.4362i − 0.430745i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 43.5968 1.39479 0.697393 0.716689i \(-0.254343\pi\)
0.697393 + 0.716689i \(0.254343\pi\)
\(978\) 0 0
\(979\) − 9.55697i − 0.305442i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 22.5407 0.718937 0.359469 0.933157i \(-0.382958\pi\)
0.359469 + 0.933157i \(0.382958\pi\)
\(984\) 0 0
\(985\) 48.3090 1.53925
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 7.71204i 0.245229i
\(990\) 0 0
\(991\) −24.2444 −0.770148 −0.385074 0.922886i \(-0.625824\pi\)
−0.385074 + 0.922886i \(0.625824\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 44.3747i 1.40677i
\(996\) 0 0
\(997\) 49.3412i 1.56265i 0.624124 + 0.781325i \(0.285456\pi\)
−0.624124 + 0.781325i \(0.714544\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 6048.2.c.g.3025.24 24
3.2 odd 2 inner 6048.2.c.g.3025.2 24
4.3 odd 2 1512.2.c.f.757.9 24
8.3 odd 2 1512.2.c.f.757.10 yes 24
8.5 even 2 inner 6048.2.c.g.3025.1 24
12.11 even 2 1512.2.c.f.757.16 yes 24
24.5 odd 2 inner 6048.2.c.g.3025.23 24
24.11 even 2 1512.2.c.f.757.15 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.c.f.757.9 24 4.3 odd 2
1512.2.c.f.757.10 yes 24 8.3 odd 2
1512.2.c.f.757.15 yes 24 24.11 even 2
1512.2.c.f.757.16 yes 24 12.11 even 2
6048.2.c.g.3025.1 24 8.5 even 2 inner
6048.2.c.g.3025.2 24 3.2 odd 2 inner
6048.2.c.g.3025.23 24 24.5 odd 2 inner
6048.2.c.g.3025.24 24 1.1 even 1 trivial