Properties

Label 6048.2.h.g.2591.2
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 2591.2
Root \(0.965926 - 0.258819i\)
Character \(\chi\) = 6048.2591
Dual form 6048.2.h.g.2591.7

$q$-expansion

\(f(q)\) \(=\) \(q-2.66390i q^{5} +1.00000i q^{7} +O(q^{10})\) \(q-2.66390i q^{5} +1.00000i q^{7} +0.164525 q^{11} +3.56048 q^{13} -3.41421i q^{17} -3.89898i q^{19} +5.29958 q^{23} -2.09638 q^{25} +7.84304i q^{29} +8.72741i q^{31} +2.66390 q^{35} +7.82843 q^{37} -1.62863i q^{41} +11.2173i q^{43} -0.585786 q^{47} -1.00000 q^{49} -14.2268i q^{53} -0.438278i q^{55} +6.98539 q^{59} +4.92820 q^{61} -9.48477i q^{65} +6.83183i q^{67} -10.2144 q^{71} +5.48993 q^{73} +0.164525i q^{77} -9.09513i q^{79} +1.95492 q^{83} -9.09513 q^{85} +6.05745i q^{89} +3.56048i q^{91} -10.3865 q^{95} +4.72865 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 8q^{11} - 8q^{13} + 8q^{23} - 8q^{25} - 8q^{35} + 40q^{37} - 16q^{47} - 8q^{49} + 64q^{59} - 16q^{61} - 72q^{71} + 24q^{73} - 32q^{83} + 8q^{85} - 56q^{95} + 48q^{97} + O(q^{100}) \)

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
\(5\) − 2.66390i − 1.19133i −0.803232 0.595667i \(-0.796888\pi\)
0.803232 0.595667i \(-0.203112\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.164525 0.0496061 0.0248030 0.999692i \(-0.492104\pi\)
0.0248030 + 0.999692i \(0.492104\pi\)
\(12\) 0 0
\(13\) 3.56048 0.987499 0.493749 0.869604i \(-0.335626\pi\)
0.493749 + 0.869604i \(0.335626\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 3.41421i − 0.828068i −0.910261 0.414034i \(-0.864119\pi\)
0.910261 0.414034i \(-0.135881\pi\)
\(18\) 0 0
\(19\) − 3.89898i − 0.894487i −0.894412 0.447244i \(-0.852406\pi\)
0.894412 0.447244i \(-0.147594\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.29958 1.10504 0.552519 0.833500i \(-0.313667\pi\)
0.552519 + 0.833500i \(0.313667\pi\)
\(24\) 0 0
\(25\) −2.09638 −0.419275
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 7.84304i 1.45642i 0.685356 + 0.728208i \(0.259646\pi\)
−0.685356 + 0.728208i \(0.740354\pi\)
\(30\) 0 0
\(31\) 8.72741i 1.56749i 0.621084 + 0.783744i \(0.286693\pi\)
−0.621084 + 0.783744i \(0.713307\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.66390 0.450282
\(36\) 0 0
\(37\) 7.82843 1.28699 0.643493 0.765452i \(-0.277485\pi\)
0.643493 + 0.765452i \(0.277485\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 1.62863i − 0.254349i −0.991880 0.127174i \(-0.959409\pi\)
0.991880 0.127174i \(-0.0405908\pi\)
\(42\) 0 0
\(43\) 11.2173i 1.71063i 0.518111 + 0.855314i \(0.326636\pi\)
−0.518111 + 0.855314i \(0.673364\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −0.585786 −0.0854457 −0.0427229 0.999087i \(-0.513603\pi\)
−0.0427229 + 0.999087i \(0.513603\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 14.2268i − 1.95420i −0.212783 0.977100i \(-0.568253\pi\)
0.212783 0.977100i \(-0.431747\pi\)
\(54\) 0 0
\(55\) − 0.438278i − 0.0590973i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 6.98539 0.909420 0.454710 0.890640i \(-0.349743\pi\)
0.454710 + 0.890640i \(0.349743\pi\)
\(60\) 0 0
\(61\) 4.92820 0.630992 0.315496 0.948927i \(-0.397829\pi\)
0.315496 + 0.948927i \(0.397829\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 9.48477i − 1.17644i
\(66\) 0 0
\(67\) 6.83183i 0.834640i 0.908759 + 0.417320i \(0.137031\pi\)
−0.908759 + 0.417320i \(0.862969\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −10.2144 −1.21223 −0.606114 0.795378i \(-0.707272\pi\)
−0.606114 + 0.795378i \(0.707272\pi\)
\(72\) 0 0
\(73\) 5.48993 0.642547 0.321274 0.946986i \(-0.395889\pi\)
0.321274 + 0.946986i \(0.395889\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.164525i 0.0187493i
\(78\) 0 0
\(79\) − 9.09513i − 1.02328i −0.859199 0.511641i \(-0.829038\pi\)
0.859199 0.511641i \(-0.170962\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1.95492 0.214580 0.107290 0.994228i \(-0.465783\pi\)
0.107290 + 0.994228i \(0.465783\pi\)
\(84\) 0 0
\(85\) −9.09513 −0.986506
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.05745i 0.642089i 0.947064 + 0.321044i \(0.104034\pi\)
−0.947064 + 0.321044i \(0.895966\pi\)
\(90\) 0 0
\(91\) 3.56048i 0.373240i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −10.3865 −1.06563
\(96\) 0 0
\(97\) 4.72865 0.480122 0.240061 0.970758i \(-0.422833\pi\)
0.240061 + 0.970758i \(0.422833\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 10.6563i − 1.06035i −0.847890 0.530173i \(-0.822127\pi\)
0.847890 0.530173i \(-0.177873\pi\)
\(102\) 0 0
\(103\) − 6.58630i − 0.648968i −0.945891 0.324484i \(-0.894809\pi\)
0.945891 0.324484i \(-0.105191\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 15.1416 1.46380 0.731898 0.681414i \(-0.238635\pi\)
0.731898 + 0.681414i \(0.238635\pi\)
\(108\) 0 0
\(109\) −15.1163 −1.44788 −0.723940 0.689863i \(-0.757671\pi\)
−0.723940 + 0.689863i \(0.757671\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 5.69818i − 0.536040i −0.963413 0.268020i \(-0.913631\pi\)
0.963413 0.268020i \(-0.0863693\pi\)
\(114\) 0 0
\(115\) − 14.1176i − 1.31647i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 3.41421 0.312980
\(120\) 0 0
\(121\) −10.9729 −0.997539
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 7.73497i − 0.691837i
\(126\) 0 0
\(127\) − 5.63103i − 0.499673i −0.968288 0.249837i \(-0.919623\pi\)
0.968288 0.249837i \(-0.0803769\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 6.71279 0.586500 0.293250 0.956036i \(-0.405263\pi\)
0.293250 + 0.956036i \(0.405263\pi\)
\(132\) 0 0
\(133\) 3.89898 0.338084
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 8.91484i 0.761646i 0.924648 + 0.380823i \(0.124359\pi\)
−0.924648 + 0.380823i \(0.875641\pi\)
\(138\) 0 0
\(139\) 9.46410i 0.802735i 0.915917 + 0.401367i \(0.131465\pi\)
−0.915917 + 0.401367i \(0.868535\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0.585786 0.0489859
\(144\) 0 0
\(145\) 20.8931 1.73508
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 15.0280i 1.23114i 0.788082 + 0.615570i \(0.211074\pi\)
−0.788082 + 0.615570i \(0.788926\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 23.2490 1.86740
\(156\) 0 0
\(157\) 17.2173 1.37409 0.687046 0.726614i \(-0.258907\pi\)
0.687046 + 0.726614i \(0.258907\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 5.29958i 0.417665i
\(162\) 0 0
\(163\) 17.3584i 1.35962i 0.733389 + 0.679809i \(0.237937\pi\)
−0.733389 + 0.679809i \(0.762063\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −1.18670 −0.0918296 −0.0459148 0.998945i \(-0.514620\pi\)
−0.0459148 + 0.998945i \(0.514620\pi\)
\(168\) 0 0
\(169\) −0.322997 −0.0248459
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 7.73497i 0.588079i 0.955793 + 0.294039i \(0.0949997\pi\)
−0.955793 + 0.294039i \(0.905000\pi\)
\(174\) 0 0
\(175\) − 2.09638i − 0.158471i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 11.8844 0.888279 0.444140 0.895958i \(-0.353509\pi\)
0.444140 + 0.895958i \(0.353509\pi\)
\(180\) 0 0
\(181\) 0.141105 0.0104882 0.00524412 0.999986i \(-0.498331\pi\)
0.00524412 + 0.999986i \(0.498331\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 20.8542i − 1.53323i
\(186\) 0 0
\(187\) − 0.561722i − 0.0410772i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 14.5841 1.05527 0.527633 0.849473i \(-0.323080\pi\)
0.527633 + 0.849473i \(0.323080\pi\)
\(192\) 0 0
\(193\) −14.1928 −1.02162 −0.510808 0.859695i \(-0.670654\pi\)
−0.510808 + 0.859695i \(0.670654\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 24.8824i − 1.77280i −0.462923 0.886399i \(-0.653199\pi\)
0.462923 0.886399i \(-0.346801\pi\)
\(198\) 0 0
\(199\) − 23.4181i − 1.66007i −0.557713 0.830034i \(-0.688321\pi\)
0.557713 0.830034i \(-0.311679\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −7.84304 −0.550473
\(204\) 0 0
\(205\) −4.33850 −0.303014
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 0.641478i − 0.0443720i
\(210\) 0 0
\(211\) 15.1210i 1.04097i 0.853871 + 0.520485i \(0.174249\pi\)
−0.853871 + 0.520485i \(0.825751\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 29.8819 2.03793
\(216\) 0 0
\(217\) −8.72741 −0.592455
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 12.1562i − 0.817717i
\(222\) 0 0
\(223\) 5.01654i 0.335932i 0.985793 + 0.167966i \(0.0537199\pi\)
−0.985793 + 0.167966i \(0.946280\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 10.6556 0.707238 0.353619 0.935390i \(-0.384951\pi\)
0.353619 + 0.935390i \(0.384951\pi\)
\(228\) 0 0
\(229\) −18.9189 −1.25020 −0.625099 0.780546i \(-0.714941\pi\)
−0.625099 + 0.780546i \(0.714941\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 8.89898i − 0.582992i −0.956572 0.291496i \(-0.905847\pi\)
0.956572 0.291496i \(-0.0941529\pi\)
\(234\) 0 0
\(235\) 1.56048i 0.101794i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 5.11439 0.330822 0.165411 0.986225i \(-0.447105\pi\)
0.165411 + 0.986225i \(0.447105\pi\)
\(240\) 0 0
\(241\) 6.03262 0.388595 0.194298 0.980943i \(-0.437757\pi\)
0.194298 + 0.980943i \(0.437757\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 2.66390i 0.170190i
\(246\) 0 0
\(247\) − 13.8822i − 0.883305i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −7.95867 −0.502347 −0.251173 0.967942i \(-0.580816\pi\)
−0.251173 + 0.967942i \(0.580816\pi\)
\(252\) 0 0
\(253\) 0.871911 0.0548166
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 24.3565i − 1.51932i −0.650321 0.759660i \(-0.725365\pi\)
0.650321 0.759660i \(-0.274635\pi\)
\(258\) 0 0
\(259\) 7.82843i 0.486435i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 16.9106 1.04275 0.521377 0.853327i \(-0.325419\pi\)
0.521377 + 0.853327i \(0.325419\pi\)
\(264\) 0 0
\(265\) −37.8988 −2.32810
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 4.20512i − 0.256391i −0.991749 0.128195i \(-0.959082\pi\)
0.991749 0.128195i \(-0.0409185\pi\)
\(270\) 0 0
\(271\) − 25.6544i − 1.55839i −0.626781 0.779196i \(-0.715628\pi\)
0.626781 0.779196i \(-0.284372\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −0.344906 −0.0207986
\(276\) 0 0
\(277\) 13.0320 0.783019 0.391510 0.920174i \(-0.371953\pi\)
0.391510 + 0.920174i \(0.371953\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 20.1252i − 1.20057i −0.799786 0.600286i \(-0.795053\pi\)
0.799786 0.600286i \(-0.204947\pi\)
\(282\) 0 0
\(283\) − 3.70850i − 0.220448i −0.993907 0.110224i \(-0.964843\pi\)
0.993907 0.110224i \(-0.0351568\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.62863 0.0961348
\(288\) 0 0
\(289\) 5.34315 0.314303
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 32.3696i − 1.89106i −0.325542 0.945528i \(-0.605547\pi\)
0.325542 0.945528i \(-0.394453\pi\)
\(294\) 0 0
\(295\) − 18.6084i − 1.08342i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 18.8690 1.09122
\(300\) 0 0
\(301\) −11.2173 −0.646556
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 13.1283i − 0.751722i
\(306\) 0 0
\(307\) 0.300573i 0.0171546i 0.999963 + 0.00857730i \(0.00273027\pi\)
−0.999963 + 0.00857730i \(0.997270\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −11.5140 −0.652898 −0.326449 0.945215i \(-0.605852\pi\)
−0.326449 + 0.945215i \(0.605852\pi\)
\(312\) 0 0
\(313\) −3.28788 −0.185842 −0.0929211 0.995673i \(-0.529620\pi\)
−0.0929211 + 0.995673i \(0.529620\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 21.9267i 1.23153i 0.787931 + 0.615764i \(0.211152\pi\)
−0.787931 + 0.615764i \(0.788848\pi\)
\(318\) 0 0
\(319\) 1.29037i 0.0722470i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −13.3119 −0.740697
\(324\) 0 0
\(325\) −7.46410 −0.414034
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 0.585786i − 0.0322955i
\(330\) 0 0
\(331\) − 24.8673i − 1.36683i −0.730031 0.683414i \(-0.760494\pi\)
0.730031 0.683414i \(-0.239506\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 18.1993 0.994335
\(336\) 0 0
\(337\) −2.80725 −0.152921 −0.0764603 0.997073i \(-0.524362\pi\)
−0.0764603 + 0.997073i \(0.524362\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 1.43587i 0.0777569i
\(342\) 0 0
\(343\) − 1.00000i − 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −9.04765 −0.485703 −0.242852 0.970063i \(-0.578083\pi\)
−0.242852 + 0.970063i \(0.578083\pi\)
\(348\) 0 0
\(349\) −24.7169 −1.32306 −0.661532 0.749917i \(-0.730093\pi\)
−0.661532 + 0.749917i \(0.730093\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 14.0161i 0.746003i 0.927831 + 0.373001i \(0.121671\pi\)
−0.927831 + 0.373001i \(0.878329\pi\)
\(354\) 0 0
\(355\) 27.2102i 1.44417i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −20.2870 −1.07071 −0.535353 0.844628i \(-0.679822\pi\)
−0.535353 + 0.844628i \(0.679822\pi\)
\(360\) 0 0
\(361\) 3.79796 0.199893
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 14.6246i − 0.765488i
\(366\) 0 0
\(367\) − 28.3887i − 1.48188i −0.671573 0.740939i \(-0.734381\pi\)
0.671573 0.740939i \(-0.265619\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 14.2268 0.738618
\(372\) 0 0
\(373\) 29.6522 1.53533 0.767667 0.640849i \(-0.221418\pi\)
0.767667 + 0.640849i \(0.221418\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 27.9250i 1.43821i
\(378\) 0 0
\(379\) − 7.15888i − 0.367727i −0.982952 0.183864i \(-0.941140\pi\)
0.982952 0.183864i \(-0.0588605\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0.0413286 0.00211179 0.00105590 0.999999i \(-0.499664\pi\)
0.00105590 + 0.999999i \(0.499664\pi\)
\(384\) 0 0
\(385\) 0.438278 0.0223367
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 16.2147i 0.822117i 0.911609 + 0.411058i \(0.134841\pi\)
−0.911609 + 0.411058i \(0.865159\pi\)
\(390\) 0 0
\(391\) − 18.0939i − 0.915047i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −24.2285 −1.21907
\(396\) 0 0
\(397\) −19.5536 −0.981365 −0.490682 0.871338i \(-0.663252\pi\)
−0.490682 + 0.871338i \(0.663252\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 22.5964i − 1.12841i −0.825634 0.564206i \(-0.809182\pi\)
0.825634 0.564206i \(-0.190818\pi\)
\(402\) 0 0
\(403\) 31.0737i 1.54789i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.28797 0.0638423
\(408\) 0 0
\(409\) 27.5298 1.36126 0.680630 0.732627i \(-0.261706\pi\)
0.680630 + 0.732627i \(0.261706\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 6.98539i 0.343728i
\(414\) 0 0
\(415\) − 5.20772i − 0.255637i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 27.1691 1.32730 0.663648 0.748045i \(-0.269007\pi\)
0.663648 + 0.748045i \(0.269007\pi\)
\(420\) 0 0
\(421\) −25.6264 −1.24895 −0.624477 0.781043i \(-0.714688\pi\)
−0.624477 + 0.781043i \(0.714688\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 7.15748i 0.347189i
\(426\) 0 0
\(427\) 4.92820i 0.238492i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 18.6123 0.896521 0.448260 0.893903i \(-0.352044\pi\)
0.448260 + 0.893903i \(0.352044\pi\)
\(432\) 0 0
\(433\) 20.5592 0.988014 0.494007 0.869458i \(-0.335532\pi\)
0.494007 + 0.869458i \(0.335532\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 20.6629i − 0.988443i
\(438\) 0 0
\(439\) 23.6476i 1.12864i 0.825557 + 0.564318i \(0.190861\pi\)
−0.825557 + 0.564318i \(0.809139\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −3.87880 −0.184287 −0.0921436 0.995746i \(-0.529372\pi\)
−0.0921436 + 0.995746i \(0.529372\pi\)
\(444\) 0 0
\(445\) 16.1365 0.764942
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 22.6536i − 1.06909i −0.845140 0.534545i \(-0.820483\pi\)
0.845140 0.534545i \(-0.179517\pi\)
\(450\) 0 0
\(451\) − 0.267949i − 0.0126172i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 9.48477 0.444653
\(456\) 0 0
\(457\) 3.57826 0.167384 0.0836919 0.996492i \(-0.473329\pi\)
0.0836919 + 0.996492i \(0.473329\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 7.46383i − 0.347625i −0.984779 0.173813i \(-0.944391\pi\)
0.984779 0.173813i \(-0.0556087\pi\)
\(462\) 0 0
\(463\) 17.7060i 0.822868i 0.911440 + 0.411434i \(0.134972\pi\)
−0.911440 + 0.411434i \(0.865028\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 3.59864 0.166525 0.0832626 0.996528i \(-0.473466\pi\)
0.0832626 + 0.996528i \(0.473466\pi\)
\(468\) 0 0
\(469\) −6.83183 −0.315464
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1.84553i 0.0848575i
\(474\) 0 0
\(475\) 8.17373i 0.375036i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 10.7287 0.490205 0.245102 0.969497i \(-0.421178\pi\)
0.245102 + 0.969497i \(0.421178\pi\)
\(480\) 0 0
\(481\) 27.8729 1.27090
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 12.5967i − 0.571985i
\(486\) 0 0
\(487\) 8.41133i 0.381154i 0.981672 + 0.190577i \(0.0610358\pi\)
−0.981672 + 0.190577i \(0.938964\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −41.1908 −1.85892 −0.929458 0.368929i \(-0.879725\pi\)
−0.929458 + 0.368929i \(0.879725\pi\)
\(492\) 0 0
\(493\) 26.7778 1.20601
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 10.2144i − 0.458179i
\(498\) 0 0
\(499\) − 4.23997i − 0.189807i −0.995486 0.0949036i \(-0.969746\pi\)
0.995486 0.0949036i \(-0.0302543\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 30.5551 1.36238 0.681192 0.732104i \(-0.261462\pi\)
0.681192 + 0.732104i \(0.261462\pi\)
\(504\) 0 0
\(505\) −28.3874 −1.26322
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 16.1136i 0.714225i 0.934061 + 0.357112i \(0.116239\pi\)
−0.934061 + 0.357112i \(0.883761\pi\)
\(510\) 0 0
\(511\) 5.48993i 0.242860i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −17.5453 −0.773137
\(516\) 0 0
\(517\) −0.0963763 −0.00423863
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 16.6291i 0.728536i 0.931294 + 0.364268i \(0.118681\pi\)
−0.931294 + 0.364268i \(0.881319\pi\)
\(522\) 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
\(527\) 29.7972 1.29799
\(528\) 0 0
\(529\) 5.08552 0.221109
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 5.79869i − 0.251169i
\(534\) 0 0
\(535\) − 40.3358i − 1.74387i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −0.164525 −0.00708658
\(540\) 0 0
\(541\) 45.3392 1.94928 0.974642 0.223769i \(-0.0718361\pi\)
0.974642 + 0.223769i \(0.0718361\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 40.2684i 1.72491i
\(546\) 0 0
\(547\) 22.3004i 0.953495i 0.879040 + 0.476747i \(0.158184\pi\)
−0.879040 + 0.476747i \(0.841816\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 30.5798 1.30275
\(552\) 0 0
\(553\) 9.09513 0.386764
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 40.0247i − 1.69590i −0.530074 0.847951i \(-0.677836\pi\)
0.530074 0.847951i \(-0.322164\pi\)
\(558\) 0 0
\(559\) 39.9391i 1.68924i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −23.8844 −1.00661 −0.503303 0.864110i \(-0.667882\pi\)
−0.503303 + 0.864110i \(0.667882\pi\)
\(564\) 0 0
\(565\) −15.1794 −0.638602
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 22.4677i 0.941894i 0.882162 + 0.470947i \(0.156088\pi\)
−0.882162 + 0.470947i \(0.843912\pi\)
\(570\) 0 0
\(571\) 10.3948i 0.435009i 0.976059 + 0.217504i \(0.0697916\pi\)
−0.976059 + 0.217504i \(0.930208\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −11.1099 −0.463315
\(576\) 0 0
\(577\) −8.27292 −0.344406 −0.172203 0.985061i \(-0.555089\pi\)
−0.172203 + 0.985061i \(0.555089\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 1.95492i 0.0811037i
\(582\) 0 0
\(583\) − 2.34066i − 0.0969401i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 35.5531 1.46743 0.733717 0.679455i \(-0.237784\pi\)
0.733717 + 0.679455i \(0.237784\pi\)
\(588\) 0 0
\(589\) 34.0280 1.40210
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 17.7443i − 0.728669i −0.931268 0.364335i \(-0.881296\pi\)
0.931268 0.364335i \(-0.118704\pi\)
\(594\) 0 0
\(595\) − 9.09513i − 0.372864i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −31.9951 −1.30728 −0.653641 0.756804i \(-0.726759\pi\)
−0.653641 + 0.756804i \(0.726759\pi\)
\(600\) 0 0
\(601\) 8.89558 0.362858 0.181429 0.983404i \(-0.441928\pi\)
0.181429 + 0.983404i \(0.441928\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 29.2308i 1.18840i
\(606\) 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
\(611\) −2.08568 −0.0843776
\(612\) 0 0
\(613\) −38.4771 −1.55408 −0.777038 0.629454i \(-0.783279\pi\)
−0.777038 + 0.629454i \(0.783279\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 21.7856i − 0.877056i −0.898717 0.438528i \(-0.855500\pi\)
0.898717 0.438528i \(-0.144500\pi\)
\(618\) 0 0
\(619\) 44.1549i 1.77474i 0.461061 + 0.887368i \(0.347469\pi\)
−0.461061 + 0.887368i \(0.652531\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −6.05745 −0.242687
\(624\) 0 0
\(625\) −31.0871 −1.24348
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 26.7279i − 1.06571i
\(630\) 0 0
\(631\) 30.9355i 1.23152i 0.787933 + 0.615760i \(0.211151\pi\)
−0.787933 + 0.615760i \(0.788849\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −15.0005 −0.595277
\(636\) 0 0
\(637\) −3.56048 −0.141071
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 18.5427i − 0.732393i −0.930538 0.366196i \(-0.880660\pi\)
0.930538 0.366196i \(-0.119340\pi\)
\(642\) 0 0
\(643\) − 10.5243i − 0.415039i −0.978231 0.207520i \(-0.933461\pi\)
0.978231 0.207520i \(-0.0665391\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −19.6531 −0.772643 −0.386322 0.922364i \(-0.626255\pi\)
−0.386322 + 0.922364i \(0.626255\pi\)
\(648\) 0 0
\(649\) 1.14927 0.0451127
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 23.0308i − 0.901264i −0.892710 0.450632i \(-0.851199\pi\)
0.892710 0.450632i \(-0.148801\pi\)
\(654\) 0 0
\(655\) − 17.8822i − 0.698717i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −28.4901 −1.10981 −0.554907 0.831912i \(-0.687246\pi\)
−0.554907 + 0.831912i \(0.687246\pi\)
\(660\) 0 0
\(661\) −3.94349 −0.153384 −0.0766921 0.997055i \(-0.524436\pi\)
−0.0766921 + 0.997055i \(0.524436\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 10.3865i − 0.402771i
\(666\) 0 0
\(667\) 41.5648i 1.60940i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0.810811 0.0313010
\(672\) 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
\(677\) 39.1588i 1.50499i 0.658595 + 0.752497i \(0.271151\pi\)
−0.658595 + 0.752497i \(0.728849\pi\)
\(678\) 0 0
\(679\) 4.72865i 0.181469i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 3.23582 0.123815 0.0619077 0.998082i \(-0.480282\pi\)
0.0619077 + 0.998082i \(0.480282\pi\)
\(684\) 0 0
\(685\) 23.7483 0.907374
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 50.6542i − 1.92977i
\(690\) 0 0
\(691\) 7.22219i 0.274745i 0.990519 + 0.137373i \(0.0438657\pi\)
−0.990519 + 0.137373i \(0.956134\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 25.2114 0.956325
\(696\) 0 0
\(697\) −5.56048 −0.210618
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 14.9289i 0.563858i 0.959435 + 0.281929i \(0.0909743\pi\)
−0.959435 + 0.281929i \(0.909026\pi\)
\(702\) 0 0
\(703\) − 30.5229i − 1.15119i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 10.6563 0.400773
\(708\) 0 0
\(709\) −16.3317 −0.613350 −0.306675 0.951814i \(-0.599216\pi\)
−0.306675 + 0.951814i \(0.599216\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 46.2516i 1.73213i
\(714\) 0 0
\(715\) − 1.56048i − 0.0583586i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −46.0552 −1.71757 −0.858785 0.512336i \(-0.828780\pi\)
−0.858785 + 0.512336i \(0.828780\pi\)
\(720\) 0 0
\(721\) 6.58630 0.245287
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 16.4420i − 0.610639i
\(726\) 0 0
\(727\) − 25.5984i − 0.949392i −0.880150 0.474696i \(-0.842558\pi\)
0.880150 0.474696i \(-0.157442\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 38.2984 1.41652
\(732\) 0 0
\(733\) 41.3584 1.52761 0.763804 0.645448i \(-0.223329\pi\)
0.763804 + 0.645448i \(0.223329\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.12400i 0.0414032i
\(738\) 0 0
\(739\) − 11.0109i − 0.405041i −0.979278 0.202521i \(-0.935087\pi\)
0.979278 0.202521i \(-0.0649133\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −27.4506 −1.00706 −0.503532 0.863976i \(-0.667967\pi\)
−0.503532 + 0.863976i \(0.667967\pi\)
\(744\) 0 0
\(745\) 40.0331 1.46670
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 15.1416i 0.553263i
\(750\) 0 0
\(751\) − 26.8414i − 0.979458i −0.871875 0.489729i \(-0.837096\pi\)
0.871875 0.489729i \(-0.162904\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −11.9975 −0.436057 −0.218028 0.975942i \(-0.569963\pi\)
−0.218028 + 0.975942i \(0.569963\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 22.5682i 0.818096i 0.912513 + 0.409048i \(0.134139\pi\)
−0.912513 + 0.409048i \(0.865861\pi\)
\(762\) 0 0
\(763\) − 15.1163i − 0.547247i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 24.8713 0.898051
\(768\) 0 0
\(769\) 0.549945 0.0198315 0.00991576 0.999951i \(-0.496844\pi\)
0.00991576 + 0.999951i \(0.496844\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 14.5062i 0.521752i 0.965372 + 0.260876i \(0.0840114\pi\)
−0.965372 + 0.260876i \(0.915989\pi\)
\(774\) 0 0
\(775\) − 18.2959i − 0.657209i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −6.34998 −0.227512
\(780\) 0 0
\(781\) −1.68052 −0.0601338
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 45.8653i − 1.63700i
\(786\) 0 0
\(787\) 8.63671i 0.307865i 0.988081 + 0.153933i \(0.0491939\pi\)
−0.988081 + 0.153933i \(0.950806\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 5.69818 0.202604
\(792\) 0 0
\(793\) 17.5468 0.623104
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 51.2737i 1.81621i 0.418745 + 0.908104i \(0.362470\pi\)
−0.418745 + 0.908104i \(0.637530\pi\)
\(798\) 0 0
\(799\) 2.00000i 0.0707549i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0.903228 0.0318742
\(804\) 0 0
\(805\) 14.1176 0.497578
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 23.4293i 0.823732i 0.911244 + 0.411866i \(0.135123\pi\)
−0.911244 + 0.411866i \(0.864877\pi\)
\(810\) 0 0
\(811\) 51.9554i 1.82440i 0.409745 + 0.912200i \(0.365618\pi\)
−0.409745 + 0.912200i \(0.634382\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 46.2412 1.61976
\(816\) 0 0
\(817\) 43.7361 1.53013
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 8.94310i 0.312116i 0.987748 + 0.156058i \(0.0498787\pi\)
−0.987748 + 0.156058i \(0.950121\pi\)
\(822\) 0 0
\(823\) − 3.07180i − 0.107076i −0.998566 0.0535381i \(-0.982950\pi\)
0.998566 0.0535381i \(-0.0170498\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −4.73319 −0.164589 −0.0822946 0.996608i \(-0.526225\pi\)
−0.0822946 + 0.996608i \(0.526225\pi\)
\(828\) 0 0
\(829\) −12.7378 −0.442403 −0.221201 0.975228i \(-0.570998\pi\)
−0.221201 + 0.975228i \(0.570998\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 3.41421i 0.118295i
\(834\) 0 0
\(835\) 3.16125i 0.109400i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 31.3976 1.08397 0.541983 0.840389i \(-0.317674\pi\)
0.541983 + 0.840389i \(0.317674\pi\)
\(840\) 0 0
\(841\) −32.5133 −1.12115
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0.860432i 0.0295998i
\(846\) 0 0
\(847\) − 10.9729i − 0.377034i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 41.4874 1.42217
\(852\) 0 0
\(853\) −9.89670 −0.338857 −0.169428 0.985543i \(-0.554192\pi\)
−0.169428 + 0.985543i \(0.554192\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 8.24092i 0.281504i 0.990045 + 0.140752i \(0.0449521\pi\)
−0.990045 + 0.140752i \(0.955048\pi\)
\(858\) 0 0
\(859\) − 17.3021i − 0.590342i −0.955445 0.295171i \(-0.904623\pi\)
0.955445 0.295171i \(-0.0953765\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −23.9146 −0.814063 −0.407032 0.913414i \(-0.633436\pi\)
−0.407032 + 0.913414i \(0.633436\pi\)
\(864\) 0 0
\(865\) 20.6052 0.700598
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 1.49637i − 0.0507610i
\(870\) 0 0
\(871\) 24.3246i 0.824207i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 7.73497 0.261490
\(876\) 0 0
\(877\) 23.1862 0.782942 0.391471 0.920190i \(-0.371966\pi\)
0.391471 + 0.920190i \(0.371966\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 35.6992i 1.20274i 0.798972 + 0.601368i \(0.205377\pi\)
−0.798972 + 0.601368i \(0.794623\pi\)
\(882\) 0 0
\(883\) 54.2704i 1.82635i 0.407573 + 0.913173i \(0.366375\pi\)
−0.407573 + 0.913173i \(0.633625\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −20.8136 −0.698852 −0.349426 0.936964i \(-0.613623\pi\)
−0.349426 + 0.936964i \(0.613623\pi\)
\(888\) 0 0
\(889\) 5.63103 0.188859
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 2.28397i 0.0764301i
\(894\) 0 0
\(895\) − 31.6588i − 1.05824i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −68.4494 −2.28291
\(900\) 0 0
\(901\) −48.5733 −1.61821
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 0.375889i − 0.0124950i
\(906\) 0 0
\(907\) 12.1386i 0.403056i 0.979483 + 0.201528i \(0.0645907\pi\)
−0.979483 + 0.201528i \(0.935409\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −27.8093 −0.921364 −0.460682 0.887565i \(-0.652395\pi\)
−0.460682 + 0.887565i \(0.652395\pi\)
\(912\) 0 0
\(913\) 0.321633 0.0106445
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 6.71279i 0.221676i
\(918\) 0 0
\(919\) − 2.77101i − 0.0914072i −0.998955 0.0457036i \(-0.985447\pi\)
0.998955 0.0457036i \(-0.0145530\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −36.3682 −1.19707
\(924\) 0 0
\(925\) −16.4113 −0.539601
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 57.9933i 1.90270i 0.308113 + 0.951350i \(0.400303\pi\)
−0.308113 + 0.951350i \(0.599697\pi\)
\(930\) 0 0
\(931\) 3.89898i 0.127784i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −1.49637 −0.0489366
\(936\) 0 0
\(937\) −36.4813 −1.19179 −0.595897 0.803061i \(-0.703203\pi\)
−0.595897 + 0.803061i \(0.703203\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 53.8744i − 1.75625i −0.478427 0.878127i \(-0.658793\pi\)
0.478427 0.878127i \(-0.341207\pi\)
\(942\) 0 0
\(943\) − 8.63103i − 0.281065i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 25.5198 0.829282 0.414641 0.909985i \(-0.363907\pi\)
0.414641 + 0.909985i \(0.363907\pi\)
\(948\) 0 0
\(949\) 19.5468 0.634515
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 40.9794i − 1.32745i −0.747975 0.663727i \(-0.768974\pi\)
0.747975 0.663727i \(-0.231026\pi\)
\(954\) 0 0
\(955\) − 38.8505i − 1.25717i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −8.91484 −0.287875
\(960\) 0 0
\(961\) −45.1676 −1.45702
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 37.8081i 1.21709i
\(966\) 0 0
\(967\) − 5.62129i − 0.180769i −0.995907 0.0903843i \(-0.971190\pi\)
0.995907 0.0903843i \(-0.0288095\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 38.2654 1.22799 0.613997 0.789308i \(-0.289561\pi\)
0.613997 + 0.789308i \(0.289561\pi\)
\(972\) 0 0
\(973\) −9.46410 −0.303405
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 31.1822i 0.997608i 0.866715 + 0.498804i \(0.166227\pi\)
−0.866715 + 0.498804i \(0.833773\pi\)
\(978\) 0 0
\(979\) 0.996600i 0.0318515i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −57.5907 −1.83686 −0.918429 0.395585i \(-0.870542\pi\)
−0.918429 + 0.395585i \(0.870542\pi\)
\(984\) 0 0
\(985\) −66.2843 −2.11199
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 59.4471i 1.89031i
\(990\) 0 0
\(991\) − 25.3278i − 0.804563i −0.915516 0.402281i \(-0.868217\pi\)
0.915516 0.402281i \(-0.131783\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −62.3836 −1.97769
\(996\) 0 0
\(997\) −35.3697 −1.12017 −0.560084 0.828436i \(-0.689231\pi\)
−0.560084 + 0.828436i \(0.689231\pi\)
\(998\) 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.2591.2 yes 8
3.2 odd 2 6048.2.h.a.2591.7 yes 8
4.3 odd 2 6048.2.h.a.2591.2 8
12.11 even 2 inner 6048.2.h.g.2591.7 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6048.2.h.a.2591.2 8 4.3 odd 2
6048.2.h.a.2591.7 yes 8 3.2 odd 2
6048.2.h.g.2591.2 yes 8 1.1 even 1 trivial
6048.2.h.g.2591.7 yes 8 12.11 even 2 inner