Properties

Label 6048.2.c.g.3025.18
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 3025.18
Character \(\chi\) = 6048.3025
Dual form 6048.2.c.g.3025.7

$q$-expansion

\(f(q)\) \(=\) \(q+1.58470i q^{5} -1.00000 q^{7} +O(q^{10})\) \(q+1.58470i q^{5} -1.00000 q^{7} +0.790533i q^{11} +0.494366i q^{13} -4.46962 q^{17} -7.55705i q^{19} +0.839644 q^{23} +2.48872 q^{25} +2.76437i q^{29} +0.568664 q^{31} -1.58470i q^{35} +0.343650i q^{37} +5.93014 q^{41} -3.16166i q^{43} +4.80964 q^{47} +1.00000 q^{49} -4.36792i q^{53} -1.25276 q^{55} +4.50798i q^{59} +5.40103i q^{61} -0.783423 q^{65} +7.57396i q^{67} +9.52886 q^{71} +14.1344 q^{73} -0.790533i q^{77} +3.71341 q^{79} -7.30905i q^{83} -7.08301i q^{85} -11.5305 q^{89} -0.494366i q^{91} +11.9757 q^{95} +1.75839 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24q - 24q^{7} + O(q^{10}) \) \( 24q - 24q^{7} - 24q^{25} + 16q^{31} + 24q^{49} + 8q^{55} - 32q^{79} + O(q^{100}) \)

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
\(5\) 1.58470i 0.708700i 0.935113 + 0.354350i \(0.115298\pi\)
−0.935113 + 0.354350i \(0.884702\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.790533i 0.238355i 0.992873 + 0.119177i \(0.0380257\pi\)
−0.992873 + 0.119177i \(0.961974\pi\)
\(12\) 0 0
\(13\) 0.494366i 0.137112i 0.997647 + 0.0685562i \(0.0218392\pi\)
−0.997647 + 0.0685562i \(0.978161\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.46962 −1.08404 −0.542021 0.840365i \(-0.682340\pi\)
−0.542021 + 0.840365i \(0.682340\pi\)
\(18\) 0 0
\(19\) − 7.55705i − 1.73371i −0.498564 0.866853i \(-0.666139\pi\)
0.498564 0.866853i \(-0.333861\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.839644 0.175078 0.0875390 0.996161i \(-0.472100\pi\)
0.0875390 + 0.996161i \(0.472100\pi\)
\(24\) 0 0
\(25\) 2.48872 0.497744
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.76437i 0.513331i 0.966500 + 0.256665i \(0.0826238\pi\)
−0.966500 + 0.256665i \(0.917376\pi\)
\(30\) 0 0
\(31\) 0.568664 0.102135 0.0510675 0.998695i \(-0.483738\pi\)
0.0510675 + 0.998695i \(0.483738\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 1.58470i − 0.267864i
\(36\) 0 0
\(37\) 0.343650i 0.0564957i 0.999601 + 0.0282479i \(0.00899277\pi\)
−0.999601 + 0.0282479i \(0.991007\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 5.93014 0.926132 0.463066 0.886324i \(-0.346749\pi\)
0.463066 + 0.886324i \(0.346749\pi\)
\(42\) 0 0
\(43\) − 3.16166i − 0.482149i −0.970507 0.241074i \(-0.922500\pi\)
0.970507 0.241074i \(-0.0774998\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.80964 0.701558 0.350779 0.936458i \(-0.385917\pi\)
0.350779 + 0.936458i \(0.385917\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 4.36792i − 0.599980i −0.953942 0.299990i \(-0.903017\pi\)
0.953942 0.299990i \(-0.0969833\pi\)
\(54\) 0 0
\(55\) −1.25276 −0.168922
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.50798i 0.586889i 0.955976 + 0.293445i \(0.0948016\pi\)
−0.955976 + 0.293445i \(0.905198\pi\)
\(60\) 0 0
\(61\) 5.40103i 0.691531i 0.938321 + 0.345766i \(0.112381\pi\)
−0.938321 + 0.345766i \(0.887619\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −0.783423 −0.0971716
\(66\) 0 0
\(67\) 7.57396i 0.925307i 0.886539 + 0.462653i \(0.153103\pi\)
−0.886539 + 0.462653i \(0.846897\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 9.52886 1.13087 0.565434 0.824794i \(-0.308709\pi\)
0.565434 + 0.824794i \(0.308709\pi\)
\(72\) 0 0
\(73\) 14.1344 1.65431 0.827155 0.561975i \(-0.189958\pi\)
0.827155 + 0.561975i \(0.189958\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 0.790533i − 0.0900896i
\(78\) 0 0
\(79\) 3.71341 0.417791 0.208896 0.977938i \(-0.433013\pi\)
0.208896 + 0.977938i \(0.433013\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 7.30905i − 0.802273i −0.916018 0.401136i \(-0.868615\pi\)
0.916018 0.401136i \(-0.131385\pi\)
\(84\) 0 0
\(85\) − 7.08301i − 0.768260i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −11.5305 −1.22223 −0.611114 0.791543i \(-0.709278\pi\)
−0.611114 + 0.791543i \(0.709278\pi\)
\(90\) 0 0
\(91\) − 0.494366i − 0.0518236i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 11.9757 1.22868
\(96\) 0 0
\(97\) 1.75839 0.178538 0.0892689 0.996008i \(-0.471547\pi\)
0.0892689 + 0.996008i \(0.471547\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 17.5623i 1.74752i 0.486362 + 0.873758i \(0.338324\pi\)
−0.486362 + 0.873758i \(0.661676\pi\)
\(102\) 0 0
\(103\) −8.30641 −0.818454 −0.409227 0.912432i \(-0.634202\pi\)
−0.409227 + 0.912432i \(0.634202\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 18.7750i 1.81505i 0.419996 + 0.907526i \(0.362032\pi\)
−0.419996 + 0.907526i \(0.637968\pi\)
\(108\) 0 0
\(109\) − 0.165074i − 0.0158113i −0.999969 0.00790563i \(-0.997484\pi\)
0.999969 0.00790563i \(-0.00251646\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 10.8486 1.02055 0.510275 0.860012i \(-0.329544\pi\)
0.510275 + 0.860012i \(0.329544\pi\)
\(114\) 0 0
\(115\) 1.33059i 0.124078i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 4.46962 0.409729
\(120\) 0 0
\(121\) 10.3751 0.943187
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 11.8674i 1.06145i
\(126\) 0 0
\(127\) −21.0320 −1.86629 −0.933145 0.359499i \(-0.882948\pi\)
−0.933145 + 0.359499i \(0.882948\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 5.17978i − 0.452560i −0.974062 0.226280i \(-0.927344\pi\)
0.974062 0.226280i \(-0.0726564\pi\)
\(132\) 0 0
\(133\) 7.55705i 0.655279i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 8.19905 0.700492 0.350246 0.936658i \(-0.386098\pi\)
0.350246 + 0.936658i \(0.386098\pi\)
\(138\) 0 0
\(139\) − 3.89763i − 0.330593i −0.986244 0.165296i \(-0.947142\pi\)
0.986244 0.165296i \(-0.0528581\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −0.390813 −0.0326814
\(144\) 0 0
\(145\) −4.38070 −0.363798
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 3.71034i 0.303963i 0.988383 + 0.151981i \(0.0485654\pi\)
−0.988383 + 0.151981i \(0.951435\pi\)
\(150\) 0 0
\(151\) 4.08172 0.332165 0.166083 0.986112i \(-0.446888\pi\)
0.166083 + 0.986112i \(0.446888\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0.901163i 0.0723832i
\(156\) 0 0
\(157\) − 9.64250i − 0.769555i −0.923009 0.384778i \(-0.874278\pi\)
0.923009 0.384778i \(-0.125722\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −0.839644 −0.0661732
\(162\) 0 0
\(163\) − 6.79301i − 0.532069i −0.963963 0.266035i \(-0.914286\pi\)
0.963963 0.266035i \(-0.0857136\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 5.35614 0.414471 0.207235 0.978291i \(-0.433553\pi\)
0.207235 + 0.978291i \(0.433553\pi\)
\(168\) 0 0
\(169\) 12.7556 0.981200
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 15.3862i 1.16979i 0.811110 + 0.584894i \(0.198864\pi\)
−0.811110 + 0.584894i \(0.801136\pi\)
\(174\) 0 0
\(175\) −2.48872 −0.188129
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 19.5360i 1.46019i 0.683347 + 0.730094i \(0.260524\pi\)
−0.683347 + 0.730094i \(0.739476\pi\)
\(180\) 0 0
\(181\) 1.07045i 0.0795657i 0.999208 + 0.0397828i \(0.0126666\pi\)
−0.999208 + 0.0397828i \(0.987333\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −0.544583 −0.0400386
\(186\) 0 0
\(187\) − 3.53338i − 0.258386i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −17.7019 −1.28086 −0.640431 0.768016i \(-0.721244\pi\)
−0.640431 + 0.768016i \(0.721244\pi\)
\(192\) 0 0
\(193\) −10.5405 −0.758719 −0.379359 0.925249i \(-0.623856\pi\)
−0.379359 + 0.925249i \(0.623856\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 15.1573i 1.07991i 0.841692 + 0.539957i \(0.181560\pi\)
−0.841692 + 0.539957i \(0.818440\pi\)
\(198\) 0 0
\(199\) −19.8834 −1.40950 −0.704750 0.709456i \(-0.748941\pi\)
−0.704750 + 0.709456i \(0.748941\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 2.76437i − 0.194021i
\(204\) 0 0
\(205\) 9.39751i 0.656350i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 5.97410 0.413237
\(210\) 0 0
\(211\) 26.0319i 1.79211i 0.443942 + 0.896055i \(0.353580\pi\)
−0.443942 + 0.896055i \(0.646420\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 5.01029 0.341699
\(216\) 0 0
\(217\) −0.568664 −0.0386034
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 2.20963i − 0.148635i
\(222\) 0 0
\(223\) −21.1231 −1.41451 −0.707255 0.706959i \(-0.750067\pi\)
−0.707255 + 0.706959i \(0.750067\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 9.07343i 0.602224i 0.953589 + 0.301112i \(0.0973579\pi\)
−0.953589 + 0.301112i \(0.902642\pi\)
\(228\) 0 0
\(229\) − 15.8950i − 1.05037i −0.850987 0.525187i \(-0.823995\pi\)
0.850987 0.525187i \(-0.176005\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 25.2633 1.65505 0.827527 0.561426i \(-0.189747\pi\)
0.827527 + 0.561426i \(0.189747\pi\)
\(234\) 0 0
\(235\) 7.62185i 0.497194i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 3.13797 0.202979 0.101489 0.994837i \(-0.467639\pi\)
0.101489 + 0.994837i \(0.467639\pi\)
\(240\) 0 0
\(241\) 7.05736 0.454604 0.227302 0.973824i \(-0.427009\pi\)
0.227302 + 0.973824i \(0.427009\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1.58470i 0.101243i
\(246\) 0 0
\(247\) 3.73595 0.237713
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 12.8195i 0.809158i 0.914503 + 0.404579i \(0.132582\pi\)
−0.914503 + 0.404579i \(0.867418\pi\)
\(252\) 0 0
\(253\) 0.663767i 0.0417307i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 11.1680 0.696639 0.348319 0.937376i \(-0.386752\pi\)
0.348319 + 0.937376i \(0.386752\pi\)
\(258\) 0 0
\(259\) − 0.343650i − 0.0213534i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 15.1277 0.932817 0.466408 0.884570i \(-0.345548\pi\)
0.466408 + 0.884570i \(0.345548\pi\)
\(264\) 0 0
\(265\) 6.92185 0.425206
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 4.70567i 0.286910i 0.989657 + 0.143455i \(0.0458212\pi\)
−0.989657 + 0.143455i \(0.954179\pi\)
\(270\) 0 0
\(271\) 19.6194 1.19179 0.595897 0.803061i \(-0.296797\pi\)
0.595897 + 0.803061i \(0.296797\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.96742i 0.118640i
\(276\) 0 0
\(277\) 10.0661i 0.604811i 0.953179 + 0.302406i \(0.0977897\pi\)
−0.953179 + 0.302406i \(0.902210\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 28.6669 1.71013 0.855063 0.518525i \(-0.173519\pi\)
0.855063 + 0.518525i \(0.173519\pi\)
\(282\) 0 0
\(283\) − 8.69308i − 0.516750i −0.966045 0.258375i \(-0.916813\pi\)
0.966045 0.258375i \(-0.0831870\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −5.93014 −0.350045
\(288\) 0 0
\(289\) 2.97746 0.175145
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 23.9254i − 1.39774i −0.715250 0.698868i \(-0.753687\pi\)
0.715250 0.698868i \(-0.246313\pi\)
\(294\) 0 0
\(295\) −7.14381 −0.415928
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0.415091i 0.0240054i
\(300\) 0 0
\(301\) 3.16166i 0.182235i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −8.55903 −0.490089
\(306\) 0 0
\(307\) − 7.88720i − 0.450146i −0.974342 0.225073i \(-0.927738\pi\)
0.974342 0.225073i \(-0.0722621\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 34.7041 1.96789 0.983944 0.178480i \(-0.0571179\pi\)
0.983944 + 0.178480i \(0.0571179\pi\)
\(312\) 0 0
\(313\) −11.2387 −0.635249 −0.317624 0.948217i \(-0.602885\pi\)
−0.317624 + 0.948217i \(0.602885\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 30.2902i − 1.70127i −0.525761 0.850633i \(-0.676219\pi\)
0.525761 0.850633i \(-0.323781\pi\)
\(318\) 0 0
\(319\) −2.18533 −0.122355
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 33.7771i 1.87941i
\(324\) 0 0
\(325\) 1.23034i 0.0682469i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −4.80964 −0.265164
\(330\) 0 0
\(331\) − 20.2855i − 1.11499i −0.830180 0.557495i \(-0.811763\pi\)
0.830180 0.557495i \(-0.188237\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −12.0025 −0.655765
\(336\) 0 0
\(337\) 2.73774 0.149134 0.0745671 0.997216i \(-0.476242\pi\)
0.0745671 + 0.997216i \(0.476242\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0.449548i 0.0243444i
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 5.91592i 0.317583i 0.987312 + 0.158792i \(0.0507598\pi\)
−0.987312 + 0.158792i \(0.949240\pi\)
\(348\) 0 0
\(349\) 26.5968i 1.42370i 0.702333 + 0.711848i \(0.252142\pi\)
−0.702333 + 0.711848i \(0.747858\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 21.8800 1.16455 0.582277 0.812990i \(-0.302162\pi\)
0.582277 + 0.812990i \(0.302162\pi\)
\(354\) 0 0
\(355\) 15.1004i 0.801446i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −18.4834 −0.975519 −0.487759 0.872978i \(-0.662186\pi\)
−0.487759 + 0.872978i \(0.662186\pi\)
\(360\) 0 0
\(361\) −38.1090 −2.00573
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 22.3989i 1.17241i
\(366\) 0 0
\(367\) 32.8686 1.71573 0.857863 0.513878i \(-0.171792\pi\)
0.857863 + 0.513878i \(0.171792\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 4.36792i 0.226771i
\(372\) 0 0
\(373\) − 4.11202i − 0.212912i −0.994317 0.106456i \(-0.966050\pi\)
0.994317 0.106456i \(-0.0339504\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −1.36661 −0.0703840
\(378\) 0 0
\(379\) 10.6063i 0.544811i 0.962183 + 0.272406i \(0.0878192\pi\)
−0.962183 + 0.272406i \(0.912181\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −0.193380 −0.00988125 −0.00494063 0.999988i \(-0.501573\pi\)
−0.00494063 + 0.999988i \(0.501573\pi\)
\(384\) 0 0
\(385\) 1.25276 0.0638466
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 30.2102i 1.53172i 0.643008 + 0.765859i \(0.277686\pi\)
−0.643008 + 0.765859i \(0.722314\pi\)
\(390\) 0 0
\(391\) −3.75289 −0.189792
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 5.88465i 0.296089i
\(396\) 0 0
\(397\) 19.1773i 0.962480i 0.876589 + 0.481240i \(0.159814\pi\)
−0.876589 + 0.481240i \(0.840186\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 33.6813 1.68196 0.840981 0.541065i \(-0.181979\pi\)
0.840981 + 0.541065i \(0.181979\pi\)
\(402\) 0 0
\(403\) 0.281128i 0.0140040i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −0.271667 −0.0134660
\(408\) 0 0
\(409\) 15.0414 0.743751 0.371876 0.928283i \(-0.378715\pi\)
0.371876 + 0.928283i \(0.378715\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 4.50798i − 0.221823i
\(414\) 0 0
\(415\) 11.5827 0.568571
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 15.8239i 0.773048i 0.922279 + 0.386524i \(0.126324\pi\)
−0.922279 + 0.386524i \(0.873676\pi\)
\(420\) 0 0
\(421\) 2.62118i 0.127748i 0.997958 + 0.0638742i \(0.0203457\pi\)
−0.997958 + 0.0638742i \(0.979654\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −11.1236 −0.539575
\(426\) 0 0
\(427\) − 5.40103i − 0.261374i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −21.5464 −1.03785 −0.518927 0.854819i \(-0.673668\pi\)
−0.518927 + 0.854819i \(0.673668\pi\)
\(432\) 0 0
\(433\) 12.4258 0.597145 0.298572 0.954387i \(-0.403490\pi\)
0.298572 + 0.954387i \(0.403490\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 6.34523i − 0.303534i
\(438\) 0 0
\(439\) 24.2735 1.15851 0.579256 0.815146i \(-0.303343\pi\)
0.579256 + 0.815146i \(0.303343\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 0.931584i − 0.0442609i −0.999755 0.0221304i \(-0.992955\pi\)
0.999755 0.0221304i \(-0.00704492\pi\)
\(444\) 0 0
\(445\) − 18.2724i − 0.866193i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −20.8693 −0.984883 −0.492441 0.870346i \(-0.663895\pi\)
−0.492441 + 0.870346i \(0.663895\pi\)
\(450\) 0 0
\(451\) 4.68797i 0.220748i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0.783423 0.0367274
\(456\) 0 0
\(457\) −31.3346 −1.46577 −0.732885 0.680353i \(-0.761827\pi\)
−0.732885 + 0.680353i \(0.761827\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 1.90910i 0.0889156i 0.999011 + 0.0444578i \(0.0141560\pi\)
−0.999011 + 0.0444578i \(0.985844\pi\)
\(462\) 0 0
\(463\) −18.6081 −0.864793 −0.432397 0.901683i \(-0.642332\pi\)
−0.432397 + 0.901683i \(0.642332\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 17.7972i − 0.823554i −0.911285 0.411777i \(-0.864908\pi\)
0.911285 0.411777i \(-0.135092\pi\)
\(468\) 0 0
\(469\) − 7.57396i − 0.349733i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 2.49940 0.114922
\(474\) 0 0
\(475\) − 18.8074i − 0.862941i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 21.6737 0.990294 0.495147 0.868809i \(-0.335114\pi\)
0.495147 + 0.868809i \(0.335114\pi\)
\(480\) 0 0
\(481\) −0.169889 −0.00774627
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 2.78653i 0.126530i
\(486\) 0 0
\(487\) 38.6259 1.75030 0.875152 0.483848i \(-0.160761\pi\)
0.875152 + 0.483848i \(0.160761\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 17.9010i 0.807863i 0.914789 + 0.403931i \(0.132357\pi\)
−0.914789 + 0.403931i \(0.867643\pi\)
\(492\) 0 0
\(493\) − 12.3557i − 0.556472i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −9.52886 −0.427428
\(498\) 0 0
\(499\) 24.3201i 1.08872i 0.838853 + 0.544359i \(0.183227\pi\)
−0.838853 + 0.544359i \(0.816773\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −36.3076 −1.61888 −0.809438 0.587205i \(-0.800228\pi\)
−0.809438 + 0.587205i \(0.800228\pi\)
\(504\) 0 0
\(505\) −27.8310 −1.23846
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 3.12375i − 0.138458i −0.997601 0.0692290i \(-0.977946\pi\)
0.997601 0.0692290i \(-0.0220539\pi\)
\(510\) 0 0
\(511\) −14.1344 −0.625270
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 13.1632i − 0.580039i
\(516\) 0 0
\(517\) 3.80218i 0.167220i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −1.68044 −0.0736213 −0.0368107 0.999322i \(-0.511720\pi\)
−0.0368107 + 0.999322i \(0.511720\pi\)
\(522\) 0 0
\(523\) − 26.5738i − 1.16199i −0.813906 0.580996i \(-0.802663\pi\)
0.813906 0.580996i \(-0.197337\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −2.54171 −0.110719
\(528\) 0 0
\(529\) −22.2950 −0.969348
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 2.93166i 0.126984i
\(534\) 0 0
\(535\) −29.7529 −1.28633
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0.790533i 0.0340507i
\(540\) 0 0
\(541\) − 26.0022i − 1.11792i −0.829194 0.558962i \(-0.811200\pi\)
0.829194 0.558962i \(-0.188800\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0.261594 0.0112054
\(546\) 0 0
\(547\) − 13.4108i − 0.573406i −0.958019 0.286703i \(-0.907441\pi\)
0.958019 0.286703i \(-0.0925594\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 20.8905 0.889964
\(552\) 0 0
\(553\) −3.71341 −0.157910
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 38.9728i 1.65133i 0.564161 + 0.825665i \(0.309200\pi\)
−0.564161 + 0.825665i \(0.690800\pi\)
\(558\) 0 0
\(559\) 1.56302 0.0661086
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 15.8415i − 0.667642i −0.942637 0.333821i \(-0.891662\pi\)
0.942637 0.333821i \(-0.108338\pi\)
\(564\) 0 0
\(565\) 17.1918i 0.723264i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −17.4790 −0.732758 −0.366379 0.930466i \(-0.619403\pi\)
−0.366379 + 0.930466i \(0.619403\pi\)
\(570\) 0 0
\(571\) − 11.4195i − 0.477893i −0.971033 0.238946i \(-0.923198\pi\)
0.971033 0.238946i \(-0.0768020\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 2.08964 0.0871439
\(576\) 0 0
\(577\) 27.4108 1.14113 0.570564 0.821253i \(-0.306725\pi\)
0.570564 + 0.821253i \(0.306725\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 7.30905i 0.303231i
\(582\) 0 0
\(583\) 3.45299 0.143008
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 23.8334i − 0.983709i −0.870677 0.491855i \(-0.836319\pi\)
0.870677 0.491855i \(-0.163681\pi\)
\(588\) 0 0
\(589\) − 4.29742i − 0.177072i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −28.0780 −1.15302 −0.576512 0.817088i \(-0.695587\pi\)
−0.576512 + 0.817088i \(0.695587\pi\)
\(594\) 0 0
\(595\) 7.08301i 0.290375i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 21.6780 0.885738 0.442869 0.896586i \(-0.353961\pi\)
0.442869 + 0.896586i \(0.353961\pi\)
\(600\) 0 0
\(601\) 5.83369 0.237961 0.118981 0.992897i \(-0.462037\pi\)
0.118981 + 0.992897i \(0.462037\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 16.4414i 0.668437i
\(606\) 0 0
\(607\) 29.5358 1.19882 0.599410 0.800442i \(-0.295402\pi\)
0.599410 + 0.800442i \(0.295402\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 2.37772i 0.0961923i
\(612\) 0 0
\(613\) − 19.5561i − 0.789861i −0.918711 0.394931i \(-0.870769\pi\)
0.918711 0.394931i \(-0.129231\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 33.8435 1.36249 0.681244 0.732056i \(-0.261439\pi\)
0.681244 + 0.732056i \(0.261439\pi\)
\(618\) 0 0
\(619\) − 20.6581i − 0.830320i −0.909748 0.415160i \(-0.863726\pi\)
0.909748 0.415160i \(-0.136274\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 11.5305 0.461959
\(624\) 0 0
\(625\) −6.36268 −0.254507
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 1.53598i − 0.0612437i
\(630\) 0 0
\(631\) 42.3648 1.68651 0.843257 0.537510i \(-0.180635\pi\)
0.843257 + 0.537510i \(0.180635\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 33.3295i − 1.32264i
\(636\) 0 0
\(637\) 0.494366i 0.0195875i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 11.6219 0.459039 0.229519 0.973304i \(-0.426285\pi\)
0.229519 + 0.973304i \(0.426285\pi\)
\(642\) 0 0
\(643\) 40.4455i 1.59502i 0.603309 + 0.797508i \(0.293849\pi\)
−0.603309 + 0.797508i \(0.706151\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 29.9913 1.17908 0.589540 0.807739i \(-0.299309\pi\)
0.589540 + 0.807739i \(0.299309\pi\)
\(648\) 0 0
\(649\) −3.56371 −0.139888
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 14.9263i − 0.584110i −0.956402 0.292055i \(-0.905661\pi\)
0.956402 0.292055i \(-0.0943390\pi\)
\(654\) 0 0
\(655\) 8.20842 0.320729
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 27.3581i 1.06572i 0.846204 + 0.532859i \(0.178883\pi\)
−0.846204 + 0.532859i \(0.821117\pi\)
\(660\) 0 0
\(661\) − 14.3560i − 0.558385i −0.960235 0.279192i \(-0.909933\pi\)
0.960235 0.279192i \(-0.0900667\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −11.9757 −0.464397
\(666\) 0 0
\(667\) 2.32109i 0.0898729i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −4.26970 −0.164830
\(672\) 0 0
\(673\) 20.5132 0.790727 0.395364 0.918525i \(-0.370619\pi\)
0.395364 + 0.918525i \(0.370619\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 43.5656i − 1.67436i −0.546926 0.837181i \(-0.684202\pi\)
0.546926 0.837181i \(-0.315798\pi\)
\(678\) 0 0
\(679\) −1.75839 −0.0674810
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 16.9585i − 0.648898i −0.945903 0.324449i \(-0.894821\pi\)
0.945903 0.324449i \(-0.105179\pi\)
\(684\) 0 0
\(685\) 12.9931i 0.496439i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 2.15935 0.0822647
\(690\) 0 0
\(691\) 27.7324i 1.05499i 0.849558 + 0.527495i \(0.176869\pi\)
−0.849558 + 0.527495i \(0.823131\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 6.17659 0.234291
\(696\) 0 0
\(697\) −26.5054 −1.00397
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 29.5427i 1.11581i 0.829905 + 0.557905i \(0.188395\pi\)
−0.829905 + 0.557905i \(0.811605\pi\)
\(702\) 0 0
\(703\) 2.59698 0.0979470
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 17.5623i − 0.660499i
\(708\) 0 0
\(709\) − 45.9348i − 1.72512i −0.505955 0.862560i \(-0.668860\pi\)
0.505955 0.862560i \(-0.331140\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0.477476 0.0178816
\(714\) 0 0
\(715\) − 0.619322i − 0.0231613i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −16.3756 −0.610708 −0.305354 0.952239i \(-0.598775\pi\)
−0.305354 + 0.952239i \(0.598775\pi\)
\(720\) 0 0
\(721\) 8.30641 0.309347
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 6.87974i 0.255507i
\(726\) 0 0
\(727\) −3.43696 −0.127470 −0.0637348 0.997967i \(-0.520301\pi\)
−0.0637348 + 0.997967i \(0.520301\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 14.1314i 0.522669i
\(732\) 0 0
\(733\) − 35.3171i − 1.30447i −0.758019 0.652233i \(-0.773833\pi\)
0.758019 0.652233i \(-0.226167\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −5.98747 −0.220551
\(738\) 0 0
\(739\) 41.3212i 1.52003i 0.649908 + 0.760013i \(0.274807\pi\)
−0.649908 + 0.760013i \(0.725193\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 22.9110 0.840524 0.420262 0.907403i \(-0.361938\pi\)
0.420262 + 0.907403i \(0.361938\pi\)
\(744\) 0 0
\(745\) −5.87978 −0.215418
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 18.7750i − 0.686025i
\(750\) 0 0
\(751\) 48.2764 1.76163 0.880815 0.473461i \(-0.156995\pi\)
0.880815 + 0.473461i \(0.156995\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 6.46830i 0.235406i
\(756\) 0 0
\(757\) 26.9389i 0.979112i 0.871972 + 0.489556i \(0.162841\pi\)
−0.871972 + 0.489556i \(0.837159\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −2.98366 −0.108158 −0.0540789 0.998537i \(-0.517222\pi\)
−0.0540789 + 0.998537i \(0.517222\pi\)
\(762\) 0 0
\(763\) 0.165074i 0.00597609i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −2.22859 −0.0804698
\(768\) 0 0
\(769\) 1.13907 0.0410760 0.0205380 0.999789i \(-0.493462\pi\)
0.0205380 + 0.999789i \(0.493462\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 8.40051i 0.302145i 0.988523 + 0.151073i \(0.0482728\pi\)
−0.988523 + 0.151073i \(0.951727\pi\)
\(774\) 0 0
\(775\) 1.41525 0.0508371
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 44.8143i − 1.60564i
\(780\) 0 0
\(781\) 7.53288i 0.269548i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 15.2805 0.545384
\(786\) 0 0
\(787\) − 13.6837i − 0.487771i −0.969804 0.243886i \(-0.921578\pi\)
0.969804 0.243886i \(-0.0784222\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −10.8486 −0.385731
\(792\) 0 0
\(793\) −2.67009 −0.0948175
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 43.1786i − 1.52946i −0.644349 0.764731i \(-0.722872\pi\)
0.644349 0.764731i \(-0.277128\pi\)
\(798\) 0 0
\(799\) −21.4972 −0.760518
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 11.1737i 0.394312i
\(804\) 0 0
\(805\) − 1.33059i − 0.0468970i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −24.5341 −0.862573 −0.431287 0.902215i \(-0.641940\pi\)
−0.431287 + 0.902215i \(0.641940\pi\)
\(810\) 0 0
\(811\) − 0.566589i − 0.0198956i −0.999951 0.00994781i \(-0.996833\pi\)
0.999951 0.00994781i \(-0.00316654\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 10.7649 0.377078
\(816\) 0 0
\(817\) −23.8928 −0.835904
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1.22205i 0.0426499i 0.999773 + 0.0213250i \(0.00678846\pi\)
−0.999773 + 0.0213250i \(0.993212\pi\)
\(822\) 0 0
\(823\) 46.7218 1.62862 0.814309 0.580431i \(-0.197116\pi\)
0.814309 + 0.580431i \(0.197116\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 35.2221i 1.22479i 0.790551 + 0.612396i \(0.209794\pi\)
−0.790551 + 0.612396i \(0.790206\pi\)
\(828\) 0 0
\(829\) 1.09826i 0.0381443i 0.999818 + 0.0190721i \(0.00607122\pi\)
−0.999818 + 0.0190721i \(0.993929\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −4.46962 −0.154863
\(834\) 0 0
\(835\) 8.48789i 0.293735i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 7.78246 0.268680 0.134340 0.990935i \(-0.457109\pi\)
0.134340 + 0.990935i \(0.457109\pi\)
\(840\) 0 0
\(841\) 21.3583 0.736492
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 20.2138i 0.695377i
\(846\) 0 0
\(847\) −10.3751 −0.356491
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0.288544i 0.00989116i
\(852\) 0 0
\(853\) − 35.3178i − 1.20926i −0.796507 0.604630i \(-0.793321\pi\)
0.796507 0.604630i \(-0.206679\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −48.1812 −1.64584 −0.822919 0.568159i \(-0.807656\pi\)
−0.822919 + 0.568159i \(0.807656\pi\)
\(858\) 0 0
\(859\) 23.2897i 0.794633i 0.917682 + 0.397317i \(0.130059\pi\)
−0.917682 + 0.397317i \(0.869941\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −22.2727 −0.758173 −0.379086 0.925361i \(-0.623762\pi\)
−0.379086 + 0.925361i \(0.623762\pi\)
\(864\) 0 0
\(865\) −24.3825 −0.829029
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 2.93557i 0.0995825i
\(870\) 0 0
\(871\) −3.74431 −0.126871
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 11.8674i − 0.401191i
\(876\) 0 0
\(877\) 52.5939i 1.77597i 0.459872 + 0.887985i \(0.347895\pi\)
−0.459872 + 0.887985i \(0.652105\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −25.9486 −0.874229 −0.437115 0.899406i \(-0.644000\pi\)
−0.437115 + 0.899406i \(0.644000\pi\)
\(882\) 0 0
\(883\) − 6.37782i − 0.214631i −0.994225 0.107315i \(-0.965775\pi\)
0.994225 0.107315i \(-0.0342255\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −7.60567 −0.255373 −0.127687 0.991815i \(-0.540755\pi\)
−0.127687 + 0.991815i \(0.540755\pi\)
\(888\) 0 0
\(889\) 21.0320 0.705392
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 36.3467i − 1.21629i
\(894\) 0 0
\(895\) −30.9587 −1.03484
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 1.57200i 0.0524291i
\(900\) 0 0
\(901\) 19.5229i 0.650403i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −1.69634 −0.0563882
\(906\) 0 0
\(907\) − 21.2861i − 0.706794i −0.935473 0.353397i \(-0.885026\pi\)
0.935473 0.353397i \(-0.114974\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 48.0894 1.59327 0.796637 0.604458i \(-0.206610\pi\)
0.796637 + 0.604458i \(0.206610\pi\)
\(912\) 0 0
\(913\) 5.77805 0.191226
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 5.17978i 0.171052i
\(918\) 0 0
\(919\) −33.6403 −1.10969 −0.554846 0.831953i \(-0.687223\pi\)
−0.554846 + 0.831953i \(0.687223\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 4.71074i 0.155056i
\(924\) 0 0
\(925\) 0.855249i 0.0281204i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −35.9109 −1.17820 −0.589100 0.808060i \(-0.700517\pi\)
−0.589100 + 0.808060i \(0.700517\pi\)
\(930\) 0 0
\(931\) − 7.55705i − 0.247672i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 5.59936 0.183119
\(936\) 0 0
\(937\) 6.70862 0.219161 0.109581 0.993978i \(-0.465049\pi\)
0.109581 + 0.993978i \(0.465049\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 5.81896i 0.189693i 0.995492 + 0.0948464i \(0.0302360\pi\)
−0.995492 + 0.0948464i \(0.969764\pi\)
\(942\) 0 0
\(943\) 4.97921 0.162145
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 22.8143i 0.741365i 0.928760 + 0.370682i \(0.120876\pi\)
−0.928760 + 0.370682i \(0.879124\pi\)
\(948\) 0 0
\(949\) 6.98758i 0.226826i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0.659700 0.0213698 0.0106849 0.999943i \(-0.496599\pi\)
0.0106849 + 0.999943i \(0.496599\pi\)
\(954\) 0 0
\(955\) − 28.0522i − 0.907747i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −8.19905 −0.264761
\(960\) 0 0
\(961\) −30.6766 −0.989568
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 16.7035i − 0.537704i
\(966\) 0 0
\(967\) 3.46722 0.111498 0.0557491 0.998445i \(-0.482245\pi\)
0.0557491 + 0.998445i \(0.482245\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 2.84056i 0.0911580i 0.998961 + 0.0455790i \(0.0145133\pi\)
−0.998961 + 0.0455790i \(0.985487\pi\)
\(972\) 0 0
\(973\) 3.89763i 0.124952i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −17.3811 −0.556071 −0.278035 0.960571i \(-0.589683\pi\)
−0.278035 + 0.960571i \(0.589683\pi\)
\(978\) 0 0
\(979\) − 9.11522i − 0.291324i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −54.8615 −1.74981 −0.874905 0.484294i \(-0.839076\pi\)
−0.874905 + 0.484294i \(0.839076\pi\)
\(984\) 0 0
\(985\) −24.0198 −0.765336
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 2.65467i − 0.0844136i
\(990\) 0 0
\(991\) 20.9952 0.666933 0.333467 0.942762i \(-0.391782\pi\)
0.333467 + 0.942762i \(0.391782\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 31.5093i − 0.998913i
\(996\) 0 0
\(997\) 44.0239i 1.39425i 0.716950 + 0.697125i \(0.245538\pi\)
−0.716950 + 0.697125i \(0.754462\pi\)
\(998\) 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.3025.18 24
3.2 odd 2 inner 6048.2.c.g.3025.8 24
4.3 odd 2 1512.2.c.f.757.20 yes 24
8.3 odd 2 1512.2.c.f.757.19 yes 24
8.5 even 2 inner 6048.2.c.g.3025.7 24
12.11 even 2 1512.2.c.f.757.5 24
24.5 odd 2 inner 6048.2.c.g.3025.17 24
24.11 even 2 1512.2.c.f.757.6 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.c.f.757.5 24 12.11 even 2
1512.2.c.f.757.6 yes 24 24.11 even 2
1512.2.c.f.757.19 yes 24 8.3 odd 2
1512.2.c.f.757.20 yes 24 4.3 odd 2
6048.2.c.g.3025.7 24 8.5 even 2 inner
6048.2.c.g.3025.8 24 3.2 odd 2 inner
6048.2.c.g.3025.17 24 24.5 odd 2 inner
6048.2.c.g.3025.18 24 1.1 even 1 trivial