Properties

Label 8016.2.a.o.1.3
Level $8016$
Weight $2$
Character 8016.1
Self dual yes
Analytic conductor $64.008$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8016.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.0080822603\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.2777.1
Defining polynomial: \(x^{4} - x^{3} - 4 x^{2} + x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1002)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.825785\) of defining polynomial
Character \(\chi\) \(=\) 8016.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +2.77037 q^{5} -1.86579 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +2.77037 q^{5} -1.86579 q^{7} +1.00000 q^{9} -2.28773 q^{13} +2.77037 q^{15} +3.65157 q^{17} -4.84388 q^{19} -1.86579 q^{21} +4.84388 q^{23} +2.67495 q^{25} +1.00000 q^{27} +0.459260 q^{29} +6.70967 q^{31} -5.16893 q^{35} +9.26582 q^{37} -2.28773 q^{39} -6.03619 q^{41} +2.63616 q^{43} +2.77037 q^{45} -0.325049 q^{47} -3.51883 q^{49} +3.65157 q^{51} +7.61425 q^{53} -4.84388 q^{57} -7.69426 q^{59} +3.54074 q^{61} -1.86579 q^{63} -6.33786 q^{65} +11.1082 q^{67} +4.84388 q^{69} -8.14702 q^{71} +12.1162 q^{73} +2.67495 q^{75} -4.73305 q^{79} +1.00000 q^{81} +5.50342 q^{83} +10.1162 q^{85} +0.459260 q^{87} +7.16893 q^{89} +4.26842 q^{91} +6.70967 q^{93} -13.4193 q^{95} -3.21569 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} + 5q^{5} - q^{7} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} + 5q^{5} - q^{7} + 4q^{9} + 8q^{13} + 5q^{15} + 10q^{17} + 2q^{19} - q^{21} - 2q^{23} + 5q^{25} + 4q^{27} + 14q^{29} - q^{31} - 5q^{35} + 5q^{37} + 8q^{39} + 14q^{41} - 2q^{43} + 5q^{45} - 7q^{47} + 13q^{49} + 10q^{51} + 3q^{53} + 2q^{57} + 5q^{59} + 2q^{61} - q^{63} + 6q^{65} + 7q^{67} - 2q^{69} - 2q^{71} + 2q^{73} + 5q^{75} + 10q^{79} + 4q^{81} - 13q^{83} - 6q^{85} + 14q^{87} + 13q^{89} + 30q^{91} - q^{93} + 2q^{95} + 5q^{97} + O(q^{100}) \)

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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 2.77037 1.23895 0.619474 0.785017i \(-0.287346\pi\)
0.619474 + 0.785017i \(0.287346\pi\)
\(6\) 0 0
\(7\) −1.86579 −0.705202 −0.352601 0.935774i \(-0.614703\pi\)
−0.352601 + 0.935774i \(0.614703\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −2.28773 −0.634502 −0.317251 0.948342i \(-0.602760\pi\)
−0.317251 + 0.948342i \(0.602760\pi\)
\(14\) 0 0
\(15\) 2.77037 0.715306
\(16\) 0 0
\(17\) 3.65157 0.885636 0.442818 0.896612i \(-0.353979\pi\)
0.442818 + 0.896612i \(0.353979\pi\)
\(18\) 0 0
\(19\) −4.84388 −1.11126 −0.555631 0.831429i \(-0.687523\pi\)
−0.555631 + 0.831429i \(0.687523\pi\)
\(20\) 0 0
\(21\) −1.86579 −0.407149
\(22\) 0 0
\(23\) 4.84388 1.01002 0.505009 0.863114i \(-0.331489\pi\)
0.505009 + 0.863114i \(0.331489\pi\)
\(24\) 0 0
\(25\) 2.67495 0.534990
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 0.459260 0.0852824 0.0426412 0.999090i \(-0.486423\pi\)
0.0426412 + 0.999090i \(0.486423\pi\)
\(30\) 0 0
\(31\) 6.70967 1.20509 0.602546 0.798084i \(-0.294153\pi\)
0.602546 + 0.798084i \(0.294153\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −5.16893 −0.873708
\(36\) 0 0
\(37\) 9.26582 1.52329 0.761646 0.647994i \(-0.224392\pi\)
0.761646 + 0.647994i \(0.224392\pi\)
\(38\) 0 0
\(39\) −2.28773 −0.366330
\(40\) 0 0
\(41\) −6.03619 −0.942694 −0.471347 0.881948i \(-0.656232\pi\)
−0.471347 + 0.881948i \(0.656232\pi\)
\(42\) 0 0
\(43\) 2.63616 0.402011 0.201005 0.979590i \(-0.435579\pi\)
0.201005 + 0.979590i \(0.435579\pi\)
\(44\) 0 0
\(45\) 2.77037 0.412982
\(46\) 0 0
\(47\) −0.325049 −0.0474133 −0.0237067 0.999719i \(-0.507547\pi\)
−0.0237067 + 0.999719i \(0.507547\pi\)
\(48\) 0 0
\(49\) −3.51883 −0.502690
\(50\) 0 0
\(51\) 3.65157 0.511322
\(52\) 0 0
\(53\) 7.61425 1.04590 0.522949 0.852364i \(-0.324832\pi\)
0.522949 + 0.852364i \(0.324832\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −4.84388 −0.641587
\(58\) 0 0
\(59\) −7.69426 −1.00171 −0.500854 0.865532i \(-0.666980\pi\)
−0.500854 + 0.865532i \(0.666980\pi\)
\(60\) 0 0
\(61\) 3.54074 0.453345 0.226673 0.973971i \(-0.427215\pi\)
0.226673 + 0.973971i \(0.427215\pi\)
\(62\) 0 0
\(63\) −1.86579 −0.235067
\(64\) 0 0
\(65\) −6.33786 −0.786114
\(66\) 0 0
\(67\) 11.1082 1.35709 0.678543 0.734561i \(-0.262612\pi\)
0.678543 + 0.734561i \(0.262612\pi\)
\(68\) 0 0
\(69\) 4.84388 0.583134
\(70\) 0 0
\(71\) −8.14702 −0.966873 −0.483436 0.875379i \(-0.660612\pi\)
−0.483436 + 0.875379i \(0.660612\pi\)
\(72\) 0 0
\(73\) 12.1162 1.41809 0.709047 0.705161i \(-0.249125\pi\)
0.709047 + 0.705161i \(0.249125\pi\)
\(74\) 0 0
\(75\) 2.67495 0.308877
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −4.73305 −0.532510 −0.266255 0.963903i \(-0.585786\pi\)
−0.266255 + 0.963903i \(0.585786\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 5.50342 0.604079 0.302039 0.953295i \(-0.402333\pi\)
0.302039 + 0.953295i \(0.402333\pi\)
\(84\) 0 0
\(85\) 10.1162 1.09726
\(86\) 0 0
\(87\) 0.459260 0.0492378
\(88\) 0 0
\(89\) 7.16893 0.759905 0.379952 0.925006i \(-0.375940\pi\)
0.379952 + 0.925006i \(0.375940\pi\)
\(90\) 0 0
\(91\) 4.26842 0.447452
\(92\) 0 0
\(93\) 6.70967 0.695760
\(94\) 0 0
\(95\) −13.4193 −1.37679
\(96\) 0 0
\(97\) −3.21569 −0.326504 −0.163252 0.986584i \(-0.552198\pi\)
−0.163252 + 0.986584i \(0.552198\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −9.53667 −0.948934 −0.474467 0.880273i \(-0.657359\pi\)
−0.474467 + 0.880273i \(0.657359\pi\)
\(102\) 0 0
\(103\) 13.1615 1.29684 0.648420 0.761283i \(-0.275430\pi\)
0.648420 + 0.761283i \(0.275430\pi\)
\(104\) 0 0
\(105\) −5.16893 −0.504436
\(106\) 0 0
\(107\) 9.22556 0.891868 0.445934 0.895066i \(-0.352872\pi\)
0.445934 + 0.895066i \(0.352872\pi\)
\(108\) 0 0
\(109\) −5.90605 −0.565697 −0.282849 0.959165i \(-0.591279\pi\)
−0.282849 + 0.959165i \(0.591279\pi\)
\(110\) 0 0
\(111\) 9.26582 0.879472
\(112\) 0 0
\(113\) −1.41397 −0.133015 −0.0665074 0.997786i \(-0.521186\pi\)
−0.0665074 + 0.997786i \(0.521186\pi\)
\(114\) 0 0
\(115\) 13.4193 1.25136
\(116\) 0 0
\(117\) −2.28773 −0.211501
\(118\) 0 0
\(119\) −6.81306 −0.624552
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) −6.03619 −0.544265
\(124\) 0 0
\(125\) −6.44125 −0.576123
\(126\) 0 0
\(127\) 1.71877 0.152516 0.0762582 0.997088i \(-0.475703\pi\)
0.0762582 + 0.997088i \(0.475703\pi\)
\(128\) 0 0
\(129\) 2.63616 0.232101
\(130\) 0 0
\(131\) 6.47857 0.566035 0.283018 0.959115i \(-0.408665\pi\)
0.283018 + 0.959115i \(0.408665\pi\)
\(132\) 0 0
\(133\) 9.03766 0.783664
\(134\) 0 0
\(135\) 2.77037 0.238435
\(136\) 0 0
\(137\) 20.1598 1.72237 0.861185 0.508292i \(-0.169723\pi\)
0.861185 + 0.508292i \(0.169723\pi\)
\(138\) 0 0
\(139\) 11.4234 0.968921 0.484460 0.874813i \(-0.339016\pi\)
0.484460 + 0.874813i \(0.339016\pi\)
\(140\) 0 0
\(141\) −0.325049 −0.0273741
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 1.27232 0.105660
\(146\) 0 0
\(147\) −3.51883 −0.290228
\(148\) 0 0
\(149\) 1.22963 0.100735 0.0503676 0.998731i \(-0.483961\pi\)
0.0503676 + 0.998731i \(0.483961\pi\)
\(150\) 0 0
\(151\) 21.3085 1.73406 0.867031 0.498254i \(-0.166025\pi\)
0.867031 + 0.498254i \(0.166025\pi\)
\(152\) 0 0
\(153\) 3.65157 0.295212
\(154\) 0 0
\(155\) 18.5883 1.49305
\(156\) 0 0
\(157\) 7.68776 0.613550 0.306775 0.951782i \(-0.400750\pi\)
0.306775 + 0.951782i \(0.400750\pi\)
\(158\) 0 0
\(159\) 7.61425 0.603849
\(160\) 0 0
\(161\) −9.03766 −0.712267
\(162\) 0 0
\(163\) 23.3020 1.82515 0.912577 0.408905i \(-0.134089\pi\)
0.912577 + 0.408905i \(0.134089\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.00000 0.0773823
\(168\) 0 0
\(169\) −7.76630 −0.597407
\(170\) 0 0
\(171\) −4.84388 −0.370421
\(172\) 0 0
\(173\) −3.54074 −0.269197 −0.134599 0.990900i \(-0.542975\pi\)
−0.134599 + 0.990900i \(0.542975\pi\)
\(174\) 0 0
\(175\) −4.99090 −0.377276
\(176\) 0 0
\(177\) −7.69426 −0.578336
\(178\) 0 0
\(179\) 13.6878 1.02307 0.511535 0.859262i \(-0.329077\pi\)
0.511535 + 0.859262i \(0.329077\pi\)
\(180\) 0 0
\(181\) −21.4969 −1.59785 −0.798927 0.601428i \(-0.794598\pi\)
−0.798927 + 0.601428i \(0.794598\pi\)
\(182\) 0 0
\(183\) 3.54074 0.261739
\(184\) 0 0
\(185\) 25.6697 1.88728
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −1.86579 −0.135716
\(190\) 0 0
\(191\) −20.9729 −1.51754 −0.758772 0.651356i \(-0.774200\pi\)
−0.758772 + 0.651356i \(0.774200\pi\)
\(192\) 0 0
\(193\) 2.57546 0.185386 0.0926928 0.995695i \(-0.470453\pi\)
0.0926928 + 0.995695i \(0.470453\pi\)
\(194\) 0 0
\(195\) −6.33786 −0.453863
\(196\) 0 0
\(197\) −3.36384 −0.239664 −0.119832 0.992794i \(-0.538236\pi\)
−0.119832 + 0.992794i \(0.538236\pi\)
\(198\) 0 0
\(199\) 8.57546 0.607898 0.303949 0.952688i \(-0.401695\pi\)
0.303949 + 0.952688i \(0.401695\pi\)
\(200\) 0 0
\(201\) 11.1082 0.783514
\(202\) 0 0
\(203\) −0.856882 −0.0601413
\(204\) 0 0
\(205\) −16.7225 −1.16795
\(206\) 0 0
\(207\) 4.84388 0.336673
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 11.1561 0.768019 0.384010 0.923329i \(-0.374543\pi\)
0.384010 + 0.923329i \(0.374543\pi\)
\(212\) 0 0
\(213\) −8.14702 −0.558224
\(214\) 0 0
\(215\) 7.30314 0.498070
\(216\) 0 0
\(217\) −12.5188 −0.849833
\(218\) 0 0
\(219\) 12.1162 0.818737
\(220\) 0 0
\(221\) −8.35380 −0.561937
\(222\) 0 0
\(223\) 1.86579 0.124943 0.0624713 0.998047i \(-0.480102\pi\)
0.0624713 + 0.998047i \(0.480102\pi\)
\(224\) 0 0
\(225\) 2.67495 0.178330
\(226\) 0 0
\(227\) −14.2259 −0.944206 −0.472103 0.881543i \(-0.656505\pi\)
−0.472103 + 0.881543i \(0.656505\pi\)
\(228\) 0 0
\(229\) −8.65304 −0.571809 −0.285904 0.958258i \(-0.592294\pi\)
−0.285904 + 0.958258i \(0.592294\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 7.81903 0.512241 0.256121 0.966645i \(-0.417556\pi\)
0.256121 + 0.966645i \(0.417556\pi\)
\(234\) 0 0
\(235\) −0.900507 −0.0587426
\(236\) 0 0
\(237\) −4.73305 −0.307445
\(238\) 0 0
\(239\) −8.57546 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(240\) 0 0
\(241\) 4.19084 0.269956 0.134978 0.990849i \(-0.456904\pi\)
0.134978 + 0.990849i \(0.456904\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −9.74846 −0.622806
\(246\) 0 0
\(247\) 11.0815 0.705098
\(248\) 0 0
\(249\) 5.50342 0.348765
\(250\) 0 0
\(251\) −4.46220 −0.281652 −0.140826 0.990034i \(-0.544976\pi\)
−0.140826 + 0.990034i \(0.544976\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 10.1162 0.633501
\(256\) 0 0
\(257\) 19.5769 1.22118 0.610588 0.791949i \(-0.290933\pi\)
0.610588 + 0.791949i \(0.290933\pi\)
\(258\) 0 0
\(259\) −17.2881 −1.07423
\(260\) 0 0
\(261\) 0.459260 0.0284275
\(262\) 0 0
\(263\) 17.4013 1.07301 0.536506 0.843897i \(-0.319744\pi\)
0.536506 + 0.843897i \(0.319744\pi\)
\(264\) 0 0
\(265\) 21.0943 1.29581
\(266\) 0 0
\(267\) 7.16893 0.438731
\(268\) 0 0
\(269\) 7.37371 0.449583 0.224791 0.974407i \(-0.427830\pi\)
0.224791 + 0.974407i \(0.427830\pi\)
\(270\) 0 0
\(271\) −14.9963 −0.910958 −0.455479 0.890246i \(-0.650532\pi\)
−0.455479 + 0.890246i \(0.650532\pi\)
\(272\) 0 0
\(273\) 4.26842 0.258337
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 10.4379 0.627152 0.313576 0.949563i \(-0.398473\pi\)
0.313576 + 0.949563i \(0.398473\pi\)
\(278\) 0 0
\(279\) 6.70967 0.401697
\(280\) 0 0
\(281\) 7.91255 0.472023 0.236012 0.971750i \(-0.424160\pi\)
0.236012 + 0.971750i \(0.424160\pi\)
\(282\) 0 0
\(283\) −21.0507 −1.25133 −0.625666 0.780091i \(-0.715173\pi\)
−0.625666 + 0.780091i \(0.715173\pi\)
\(284\) 0 0
\(285\) −13.4193 −0.794893
\(286\) 0 0
\(287\) 11.2623 0.664790
\(288\) 0 0
\(289\) −3.66604 −0.215650
\(290\) 0 0
\(291\) −3.21569 −0.188507
\(292\) 0 0
\(293\) 19.5407 1.14158 0.570791 0.821095i \(-0.306637\pi\)
0.570791 + 0.821095i \(0.306637\pi\)
\(294\) 0 0
\(295\) −21.3159 −1.24106
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −11.0815 −0.640859
\(300\) 0 0
\(301\) −4.91852 −0.283499
\(302\) 0 0
\(303\) −9.53667 −0.547867
\(304\) 0 0
\(305\) 9.80916 0.561671
\(306\) 0 0
\(307\) −2.99497 −0.170932 −0.0854659 0.996341i \(-0.527238\pi\)
−0.0854659 + 0.996341i \(0.527238\pi\)
\(308\) 0 0
\(309\) 13.1615 0.748731
\(310\) 0 0
\(311\) 13.2543 0.751583 0.375791 0.926704i \(-0.377371\pi\)
0.375791 + 0.926704i \(0.377371\pi\)
\(312\) 0 0
\(313\) 25.8318 1.46010 0.730051 0.683393i \(-0.239496\pi\)
0.730051 + 0.683393i \(0.239496\pi\)
\(314\) 0 0
\(315\) −5.16893 −0.291236
\(316\) 0 0
\(317\) −28.6917 −1.61148 −0.805742 0.592267i \(-0.798233\pi\)
−0.805742 + 0.592267i \(0.798233\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 9.22556 0.514920
\(322\) 0 0
\(323\) −17.6878 −0.984173
\(324\) 0 0
\(325\) −6.11956 −0.339452
\(326\) 0 0
\(327\) −5.90605 −0.326605
\(328\) 0 0
\(329\) 0.606474 0.0334360
\(330\) 0 0
\(331\) −0.445321 −0.0244770 −0.0122385 0.999925i \(-0.503896\pi\)
−0.0122385 + 0.999925i \(0.503896\pi\)
\(332\) 0 0
\(333\) 9.26582 0.507764
\(334\) 0 0
\(335\) 30.7739 1.68136
\(336\) 0 0
\(337\) −33.6259 −1.83172 −0.915860 0.401497i \(-0.868490\pi\)
−0.915860 + 0.401497i \(0.868490\pi\)
\(338\) 0 0
\(339\) −1.41397 −0.0767961
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 19.6259 1.05970
\(344\) 0 0
\(345\) 13.4193 0.722473
\(346\) 0 0
\(347\) −9.08798 −0.487868 −0.243934 0.969792i \(-0.578438\pi\)
−0.243934 + 0.969792i \(0.578438\pi\)
\(348\) 0 0
\(349\) −2.18434 −0.116925 −0.0584624 0.998290i \(-0.518620\pi\)
−0.0584624 + 0.998290i \(0.518620\pi\)
\(350\) 0 0
\(351\) −2.28773 −0.122110
\(352\) 0 0
\(353\) 15.9126 0.846940 0.423470 0.905910i \(-0.360812\pi\)
0.423470 + 0.905910i \(0.360812\pi\)
\(354\) 0 0
\(355\) −22.5703 −1.19790
\(356\) 0 0
\(357\) −6.81306 −0.360585
\(358\) 0 0
\(359\) −10.9601 −0.578451 −0.289225 0.957261i \(-0.593398\pi\)
−0.289225 + 0.957261i \(0.593398\pi\)
\(360\) 0 0
\(361\) 4.46316 0.234903
\(362\) 0 0
\(363\) −11.0000 −0.577350
\(364\) 0 0
\(365\) 33.5664 1.75694
\(366\) 0 0
\(367\) −13.2723 −0.692809 −0.346405 0.938085i \(-0.612598\pi\)
−0.346405 + 0.938085i \(0.612598\pi\)
\(368\) 0 0
\(369\) −6.03619 −0.314231
\(370\) 0 0
\(371\) −14.2066 −0.737569
\(372\) 0 0
\(373\) −31.2736 −1.61929 −0.809643 0.586922i \(-0.800339\pi\)
−0.809643 + 0.586922i \(0.800339\pi\)
\(374\) 0 0
\(375\) −6.44125 −0.332625
\(376\) 0 0
\(377\) −1.05066 −0.0541118
\(378\) 0 0
\(379\) 11.9989 0.616340 0.308170 0.951331i \(-0.400283\pi\)
0.308170 + 0.951331i \(0.400283\pi\)
\(380\) 0 0
\(381\) 1.71877 0.0880554
\(382\) 0 0
\(383\) −10.9215 −0.558061 −0.279030 0.960282i \(-0.590013\pi\)
−0.279030 + 0.960282i \(0.590013\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 2.63616 0.134004
\(388\) 0 0
\(389\) 27.3148 1.38492 0.692458 0.721458i \(-0.256528\pi\)
0.692458 + 0.721458i \(0.256528\pi\)
\(390\) 0 0
\(391\) 17.6878 0.894508
\(392\) 0 0
\(393\) 6.47857 0.326801
\(394\) 0 0
\(395\) −13.1123 −0.659751
\(396\) 0 0
\(397\) 9.57156 0.480383 0.240191 0.970726i \(-0.422790\pi\)
0.240191 + 0.970726i \(0.422790\pi\)
\(398\) 0 0
\(399\) 9.03766 0.452449
\(400\) 0 0
\(401\) 3.22313 0.160955 0.0804777 0.996756i \(-0.474355\pi\)
0.0804777 + 0.996756i \(0.474355\pi\)
\(402\) 0 0
\(403\) −15.3499 −0.764633
\(404\) 0 0
\(405\) 2.77037 0.137661
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −4.41250 −0.218184 −0.109092 0.994032i \(-0.534794\pi\)
−0.109092 + 0.994032i \(0.534794\pi\)
\(410\) 0 0
\(411\) 20.1598 0.994411
\(412\) 0 0
\(413\) 14.3559 0.706406
\(414\) 0 0
\(415\) 15.2465 0.748422
\(416\) 0 0
\(417\) 11.4234 0.559407
\(418\) 0 0
\(419\) 1.11230 0.0543394 0.0271697 0.999631i \(-0.491351\pi\)
0.0271697 + 0.999631i \(0.491351\pi\)
\(420\) 0 0
\(421\) −9.88674 −0.481850 −0.240925 0.970544i \(-0.577451\pi\)
−0.240925 + 0.970544i \(0.577451\pi\)
\(422\) 0 0
\(423\) −0.325049 −0.0158044
\(424\) 0 0
\(425\) 9.76777 0.473806
\(426\) 0 0
\(427\) −6.60628 −0.319700
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0.165029 0.00794917 0.00397459 0.999992i \(-0.498735\pi\)
0.00397459 + 0.999992i \(0.498735\pi\)
\(432\) 0 0
\(433\) 1.78335 0.0857024 0.0428512 0.999081i \(-0.486356\pi\)
0.0428512 + 0.999081i \(0.486356\pi\)
\(434\) 0 0
\(435\) 1.27232 0.0610031
\(436\) 0 0
\(437\) −23.4632 −1.12240
\(438\) 0 0
\(439\) −12.1914 −0.581861 −0.290931 0.956744i \(-0.593965\pi\)
−0.290931 + 0.956744i \(0.593965\pi\)
\(440\) 0 0
\(441\) −3.51883 −0.167563
\(442\) 0 0
\(443\) −0.919815 −0.0437017 −0.0218509 0.999761i \(-0.506956\pi\)
−0.0218509 + 0.999761i \(0.506956\pi\)
\(444\) 0 0
\(445\) 19.8606 0.941482
\(446\) 0 0
\(447\) 1.22963 0.0581595
\(448\) 0 0
\(449\) 5.25951 0.248212 0.124106 0.992269i \(-0.460394\pi\)
0.124106 + 0.992269i \(0.460394\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 21.3085 1.00116
\(454\) 0 0
\(455\) 11.8251 0.554369
\(456\) 0 0
\(457\) −23.1847 −1.08453 −0.542267 0.840206i \(-0.682434\pi\)
−0.542267 + 0.840206i \(0.682434\pi\)
\(458\) 0 0
\(459\) 3.65157 0.170441
\(460\) 0 0
\(461\) −26.5008 −1.23427 −0.617133 0.786859i \(-0.711706\pi\)
−0.617133 + 0.786859i \(0.711706\pi\)
\(462\) 0 0
\(463\) −33.3553 −1.55015 −0.775075 0.631869i \(-0.782288\pi\)
−0.775075 + 0.631869i \(0.782288\pi\)
\(464\) 0 0
\(465\) 18.5883 0.862010
\(466\) 0 0
\(467\) 30.1886 1.39696 0.698480 0.715629i \(-0.253860\pi\)
0.698480 + 0.715629i \(0.253860\pi\)
\(468\) 0 0
\(469\) −20.7256 −0.957020
\(470\) 0 0
\(471\) 7.68776 0.354233
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −12.9571 −0.594514
\(476\) 0 0
\(477\) 7.61425 0.348632
\(478\) 0 0
\(479\) 10.5398 0.481575 0.240787 0.970578i \(-0.422594\pi\)
0.240787 + 0.970578i \(0.422594\pi\)
\(480\) 0 0
\(481\) −21.1977 −0.966531
\(482\) 0 0
\(483\) −9.03766 −0.411228
\(484\) 0 0
\(485\) −8.90865 −0.404521
\(486\) 0 0
\(487\) 9.00147 0.407896 0.203948 0.978982i \(-0.434623\pi\)
0.203948 + 0.978982i \(0.434623\pi\)
\(488\) 0 0
\(489\) 23.3020 1.05375
\(490\) 0 0
\(491\) −36.6200 −1.65264 −0.826318 0.563204i \(-0.809568\pi\)
−0.826318 + 0.563204i \(0.809568\pi\)
\(492\) 0 0
\(493\) 1.67702 0.0755291
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 15.2006 0.681841
\(498\) 0 0
\(499\) 3.13924 0.140532 0.0702659 0.997528i \(-0.477615\pi\)
0.0702659 + 0.997528i \(0.477615\pi\)
\(500\) 0 0
\(501\) 1.00000 0.0446767
\(502\) 0 0
\(503\) −22.0029 −0.981063 −0.490531 0.871424i \(-0.663197\pi\)
−0.490531 + 0.871424i \(0.663197\pi\)
\(504\) 0 0
\(505\) −26.4201 −1.17568
\(506\) 0 0
\(507\) −7.76630 −0.344913
\(508\) 0 0
\(509\) −3.18694 −0.141259 −0.0706293 0.997503i \(-0.522501\pi\)
−0.0706293 + 0.997503i \(0.522501\pi\)
\(510\) 0 0
\(511\) −22.6063 −1.00004
\(512\) 0 0
\(513\) −4.84388 −0.213862
\(514\) 0 0
\(515\) 36.4622 1.60672
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −3.54074 −0.155421
\(520\) 0 0
\(521\) 15.6983 0.687756 0.343878 0.939014i \(-0.388259\pi\)
0.343878 + 0.939014i \(0.388259\pi\)
\(522\) 0 0
\(523\) 23.0325 1.00714 0.503569 0.863955i \(-0.332020\pi\)
0.503569 + 0.863955i \(0.332020\pi\)
\(524\) 0 0
\(525\) −4.99090 −0.217821
\(526\) 0 0
\(527\) 24.5008 1.06727
\(528\) 0 0
\(529\) 0.463158 0.0201373
\(530\) 0 0
\(531\) −7.69426 −0.333902
\(532\) 0 0
\(533\) 13.8092 0.598141
\(534\) 0 0
\(535\) 25.5582 1.10498
\(536\) 0 0
\(537\) 13.6878 0.590670
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 34.0269 1.46293 0.731466 0.681878i \(-0.238837\pi\)
0.731466 + 0.681878i \(0.238837\pi\)
\(542\) 0 0
\(543\) −21.4969 −0.922521
\(544\) 0 0
\(545\) −16.3619 −0.700869
\(546\) 0 0
\(547\) −16.6620 −0.712414 −0.356207 0.934407i \(-0.615930\pi\)
−0.356207 + 0.934407i \(0.615930\pi\)
\(548\) 0 0
\(549\) 3.54074 0.151115
\(550\) 0 0
\(551\) −2.22460 −0.0947711
\(552\) 0 0
\(553\) 8.83087 0.375527
\(554\) 0 0
\(555\) 25.6697 1.08962
\(556\) 0 0
\(557\) −8.87470 −0.376033 −0.188016 0.982166i \(-0.560206\pi\)
−0.188016 + 0.982166i \(0.560206\pi\)
\(558\) 0 0
\(559\) −6.03082 −0.255076
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 11.3885 0.479969 0.239984 0.970777i \(-0.422858\pi\)
0.239984 + 0.970777i \(0.422858\pi\)
\(564\) 0 0
\(565\) −3.91721 −0.164798
\(566\) 0 0
\(567\) −1.86579 −0.0783558
\(568\) 0 0
\(569\) −23.6304 −0.990639 −0.495319 0.868711i \(-0.664949\pi\)
−0.495319 + 0.868711i \(0.664949\pi\)
\(570\) 0 0
\(571\) 4.46333 0.186785 0.0933923 0.995629i \(-0.470229\pi\)
0.0933923 + 0.995629i \(0.470229\pi\)
\(572\) 0 0
\(573\) −20.9729 −0.876155
\(574\) 0 0
\(575\) 12.9571 0.540350
\(576\) 0 0
\(577\) 28.1319 1.17115 0.585574 0.810619i \(-0.300869\pi\)
0.585574 + 0.810619i \(0.300869\pi\)
\(578\) 0 0
\(579\) 2.57546 0.107032
\(580\) 0 0
\(581\) −10.2682 −0.425998
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −6.33786 −0.262038
\(586\) 0 0
\(587\) 7.58100 0.312901 0.156451 0.987686i \(-0.449995\pi\)
0.156451 + 0.987686i \(0.449995\pi\)
\(588\) 0 0
\(589\) −32.5008 −1.33917
\(590\) 0 0
\(591\) −3.36384 −0.138370
\(592\) 0 0
\(593\) −14.1832 −0.582434 −0.291217 0.956657i \(-0.594060\pi\)
−0.291217 + 0.956657i \(0.594060\pi\)
\(594\) 0 0
\(595\) −18.8747 −0.773787
\(596\) 0 0
\(597\) 8.57546 0.350970
\(598\) 0 0
\(599\) −5.91255 −0.241580 −0.120790 0.992678i \(-0.538543\pi\)
−0.120790 + 0.992678i \(0.538543\pi\)
\(600\) 0 0
\(601\) 20.3150 0.828666 0.414333 0.910125i \(-0.364015\pi\)
0.414333 + 0.910125i \(0.364015\pi\)
\(602\) 0 0
\(603\) 11.1082 0.452362
\(604\) 0 0
\(605\) −30.4741 −1.23895
\(606\) 0 0
\(607\) −18.3403 −0.744409 −0.372205 0.928151i \(-0.621398\pi\)
−0.372205 + 0.928151i \(0.621398\pi\)
\(608\) 0 0
\(609\) −0.856882 −0.0347226
\(610\) 0 0
\(611\) 0.743625 0.0300838
\(612\) 0 0
\(613\) 9.45830 0.382017 0.191009 0.981588i \(-0.438824\pi\)
0.191009 + 0.981588i \(0.438824\pi\)
\(614\) 0 0
\(615\) −16.7225 −0.674315
\(616\) 0 0
\(617\) 15.5537 0.626170 0.313085 0.949725i \(-0.398637\pi\)
0.313085 + 0.949725i \(0.398637\pi\)
\(618\) 0 0
\(619\) −17.7841 −0.714805 −0.357402 0.933951i \(-0.616338\pi\)
−0.357402 + 0.933951i \(0.616338\pi\)
\(620\) 0 0
\(621\) 4.84388 0.194378
\(622\) 0 0
\(623\) −13.3757 −0.535887
\(624\) 0 0
\(625\) −31.2194 −1.24878
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 33.8348 1.34908
\(630\) 0 0
\(631\) −8.32505 −0.331415 −0.165707 0.986175i \(-0.552991\pi\)
−0.165707 + 0.986175i \(0.552991\pi\)
\(632\) 0 0
\(633\) 11.1561 0.443416
\(634\) 0 0
\(635\) 4.76164 0.188960
\(636\) 0 0
\(637\) 8.05013 0.318958
\(638\) 0 0
\(639\) −8.14702 −0.322291
\(640\) 0 0
\(641\) −46.4286 −1.83382 −0.916910 0.399094i \(-0.869325\pi\)
−0.916910 + 0.399094i \(0.869325\pi\)
\(642\) 0 0
\(643\) −24.4839 −0.965552 −0.482776 0.875744i \(-0.660372\pi\)
−0.482776 + 0.875744i \(0.660372\pi\)
\(644\) 0 0
\(645\) 7.30314 0.287561
\(646\) 0 0
\(647\) −32.1110 −1.26241 −0.631207 0.775615i \(-0.717440\pi\)
−0.631207 + 0.775615i \(0.717440\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −12.5188 −0.490652
\(652\) 0 0
\(653\) −32.3100 −1.26439 −0.632194 0.774810i \(-0.717845\pi\)
−0.632194 + 0.774810i \(0.717845\pi\)
\(654\) 0 0
\(655\) 17.9480 0.701288
\(656\) 0 0
\(657\) 12.1162 0.472698
\(658\) 0 0
\(659\) 17.6634 0.688070 0.344035 0.938957i \(-0.388206\pi\)
0.344035 + 0.938957i \(0.388206\pi\)
\(660\) 0 0
\(661\) 48.1800 1.87398 0.936992 0.349350i \(-0.113597\pi\)
0.936992 + 0.349350i \(0.113597\pi\)
\(662\) 0 0
\(663\) −8.35380 −0.324435
\(664\) 0 0
\(665\) 25.0377 0.970919
\(666\) 0 0
\(667\) 2.22460 0.0861368
\(668\) 0 0
\(669\) 1.86579 0.0721356
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 37.5634 1.44796 0.723982 0.689819i \(-0.242310\pi\)
0.723982 + 0.689819i \(0.242310\pi\)
\(674\) 0 0
\(675\) 2.67495 0.102959
\(676\) 0 0
\(677\) −16.0776 −0.617912 −0.308956 0.951076i \(-0.599980\pi\)
−0.308956 + 0.951076i \(0.599980\pi\)
\(678\) 0 0
\(679\) 5.99980 0.230251
\(680\) 0 0
\(681\) −14.2259 −0.545137
\(682\) 0 0
\(683\) −30.5847 −1.17029 −0.585146 0.810928i \(-0.698963\pi\)
−0.585146 + 0.810928i \(0.698963\pi\)
\(684\) 0 0
\(685\) 55.8502 2.13393
\(686\) 0 0
\(687\) −8.65304 −0.330134
\(688\) 0 0
\(689\) −17.4193 −0.663624
\(690\) 0 0
\(691\) 8.16390 0.310569 0.155285 0.987870i \(-0.450371\pi\)
0.155285 + 0.987870i \(0.450371\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 31.6471 1.20044
\(696\) 0 0
\(697\) −22.0416 −0.834884
\(698\) 0 0
\(699\) 7.81903 0.295743
\(700\) 0 0
\(701\) −10.7948 −0.407716 −0.203858 0.979000i \(-0.565348\pi\)
−0.203858 + 0.979000i \(0.565348\pi\)
\(702\) 0 0
\(703\) −44.8825 −1.69278
\(704\) 0 0
\(705\) −0.900507 −0.0339151
\(706\) 0 0
\(707\) 17.7934 0.669190
\(708\) 0 0
\(709\) 36.6462 1.37628 0.688138 0.725580i \(-0.258428\pi\)
0.688138 + 0.725580i \(0.258428\pi\)
\(710\) 0 0
\(711\) −4.73305 −0.177503
\(712\) 0 0
\(713\) 32.5008 1.21717
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −8.57546 −0.320256
\(718\) 0 0
\(719\) 0.797115 0.0297274 0.0148637 0.999890i \(-0.495269\pi\)
0.0148637 + 0.999890i \(0.495269\pi\)
\(720\) 0 0
\(721\) −24.5566 −0.914535
\(722\) 0 0
\(723\) 4.19084 0.155859
\(724\) 0 0
\(725\) 1.22850 0.0456252
\(726\) 0 0
\(727\) −43.8401 −1.62594 −0.812970 0.582305i \(-0.802151\pi\)
−0.812970 + 0.582305i \(0.802151\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 9.62612 0.356035
\(732\) 0 0
\(733\) 47.4917 1.75415 0.877073 0.480357i \(-0.159493\pi\)
0.877073 + 0.480357i \(0.159493\pi\)
\(734\) 0 0
\(735\) −9.74846 −0.359577
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 36.7600 1.35224 0.676118 0.736793i \(-0.263661\pi\)
0.676118 + 0.736793i \(0.263661\pi\)
\(740\) 0 0
\(741\) 11.0815 0.407088
\(742\) 0 0
\(743\) −37.7830 −1.38612 −0.693062 0.720878i \(-0.743739\pi\)
−0.693062 + 0.720878i \(0.743739\pi\)
\(744\) 0 0
\(745\) 3.40653 0.124806
\(746\) 0 0
\(747\) 5.50342 0.201360
\(748\) 0 0
\(749\) −17.2129 −0.628947
\(750\) 0 0
\(751\) 0.746053 0.0272239 0.0136119 0.999907i \(-0.495667\pi\)
0.0136119 + 0.999907i \(0.495667\pi\)
\(752\) 0 0
\(753\) −4.46220 −0.162612
\(754\) 0 0
\(755\) 59.0325 2.14841
\(756\) 0 0
\(757\) −43.9458 −1.59724 −0.798618 0.601838i \(-0.794435\pi\)
−0.798618 + 0.601838i \(0.794435\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 42.4360 1.53831 0.769153 0.639065i \(-0.220679\pi\)
0.769153 + 0.639065i \(0.220679\pi\)
\(762\) 0 0
\(763\) 11.0194 0.398931
\(764\) 0 0
\(765\) 10.1162 0.365752
\(766\) 0 0
\(767\) 17.6024 0.635585
\(768\) 0 0
\(769\) −10.4006 −0.375054 −0.187527 0.982259i \(-0.560047\pi\)
−0.187527 + 0.982259i \(0.560047\pi\)
\(770\) 0 0
\(771\) 19.5769 0.705046
\(772\) 0 0
\(773\) −29.5960 −1.06450 −0.532248 0.846589i \(-0.678653\pi\)
−0.532248 + 0.846589i \(0.678653\pi\)
\(774\) 0 0
\(775\) 17.9480 0.644712
\(776\) 0 0
\(777\) −17.2881 −0.620206
\(778\) 0 0
\(779\) 29.2386 1.04758
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0.459260 0.0164126
\(784\) 0 0
\(785\) 21.2979 0.760156
\(786\) 0 0
\(787\) −34.4531 −1.22812 −0.614061 0.789259i \(-0.710465\pi\)
−0.614061 + 0.789259i \(0.710465\pi\)
\(788\) 0 0
\(789\) 17.4013 0.619504
\(790\) 0 0
\(791\) 2.63817 0.0938024
\(792\) 0 0
\(793\) −8.10025 −0.287648
\(794\) 0 0
\(795\) 21.0943 0.748137
\(796\) 0 0
\(797\) 19.5239 0.691571 0.345785 0.938314i \(-0.387613\pi\)
0.345785 + 0.938314i \(0.387613\pi\)
\(798\) 0 0
\(799\) −1.18694 −0.0419909
\(800\) 0 0
\(801\) 7.16893 0.253302
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −25.0377 −0.882461
\(806\) 0 0
\(807\) 7.37371 0.259567
\(808\) 0 0
\(809\) −13.8220 −0.485954 −0.242977 0.970032i \(-0.578124\pi\)
−0.242977 + 0.970032i \(0.578124\pi\)
\(810\) 0 0
\(811\) 14.3579 0.504173 0.252087 0.967705i \(-0.418883\pi\)
0.252087 + 0.967705i \(0.418883\pi\)
\(812\) 0 0
\(813\) −14.9963 −0.525942
\(814\) 0 0
\(815\) 64.5552 2.26127
\(816\) 0 0
\(817\) −12.7692 −0.446739
\(818\) 0 0
\(819\) 4.26842 0.149151
\(820\) 0 0
\(821\) −0.819201 −0.0285903 −0.0142952 0.999898i \(-0.504550\pi\)
−0.0142952 + 0.999898i \(0.504550\pi\)
\(822\) 0 0
\(823\) −2.98553 −0.104069 −0.0520344 0.998645i \(-0.516571\pi\)
−0.0520344 + 0.998645i \(0.516571\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 5.14461 0.178896 0.0894478 0.995992i \(-0.471490\pi\)
0.0894478 + 0.995992i \(0.471490\pi\)
\(828\) 0 0
\(829\) 36.5878 1.27075 0.635374 0.772204i \(-0.280846\pi\)
0.635374 + 0.772204i \(0.280846\pi\)
\(830\) 0 0
\(831\) 10.4379 0.362086
\(832\) 0 0
\(833\) −12.8492 −0.445200
\(834\) 0 0
\(835\) 2.77037 0.0958726
\(836\) 0 0
\(837\) 6.70967 0.231920
\(838\) 0 0
\(839\) −40.3484 −1.39298 −0.696491 0.717566i \(-0.745256\pi\)
−0.696491 + 0.717566i \(0.745256\pi\)
\(840\) 0 0
\(841\) −28.7891 −0.992727
\(842\) 0 0
\(843\) 7.91255 0.272523
\(844\) 0 0
\(845\) −21.5155 −0.740156
\(846\) 0 0
\(847\) 20.5237 0.705202
\(848\) 0 0
\(849\) −21.0507 −0.722457
\(850\) 0 0
\(851\) 44.8825 1.53855
\(852\) 0 0
\(853\) 23.1977 0.794273 0.397137 0.917759i \(-0.370004\pi\)
0.397137 + 0.917759i \(0.370004\pi\)
\(854\) 0 0
\(855\) −13.4193 −0.458932
\(856\) 0 0
\(857\) 18.2095 0.622026 0.311013 0.950406i \(-0.399332\pi\)
0.311013 + 0.950406i \(0.399332\pi\)
\(858\) 0 0
\(859\) −26.6170 −0.908161 −0.454080 0.890961i \(-0.650032\pi\)
−0.454080 + 0.890961i \(0.650032\pi\)
\(860\) 0 0
\(861\) 11.2623 0.383817
\(862\) 0 0
\(863\) −43.9617 −1.49647 −0.748237 0.663432i \(-0.769099\pi\)
−0.748237 + 0.663432i \(0.769099\pi\)
\(864\) 0 0
\(865\) −9.80916 −0.333521
\(866\) 0 0
\(867\) −3.66604 −0.124505
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −25.4126 −0.861073
\(872\) 0 0
\(873\) −3.21569 −0.108835
\(874\) 0 0
\(875\) 12.0180 0.406283
\(876\) 0 0
\(877\) −9.53780 −0.322069 −0.161034 0.986949i \(-0.551483\pi\)
−0.161034 + 0.986949i \(0.551483\pi\)
\(878\) 0 0
\(879\) 19.5407 0.659093
\(880\) 0 0
\(881\) 37.0430 1.24801 0.624006 0.781420i \(-0.285504\pi\)
0.624006 + 0.781420i \(0.285504\pi\)
\(882\) 0 0
\(883\) 19.3629 0.651614 0.325807 0.945436i \(-0.394364\pi\)
0.325807 + 0.945436i \(0.394364\pi\)
\(884\) 0 0
\(885\) −21.3159 −0.716528
\(886\) 0 0
\(887\) 44.5673 1.49642 0.748212 0.663460i \(-0.230913\pi\)
0.748212 + 0.663460i \(0.230913\pi\)
\(888\) 0 0
\(889\) −3.20687 −0.107555
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 1.57450 0.0526886
\(894\) 0 0
\(895\) 37.9202 1.26753
\(896\) 0 0
\(897\) −11.0815 −0.370000
\(898\) 0 0
\(899\) 3.08148 0.102773
\(900\) 0 0
\(901\) 27.8040 0.926284
\(902\) 0 0
\(903\) −4.91852 −0.163678
\(904\) 0 0
\(905\) −59.5544 −1.97966
\(906\) 0 0
\(907\) −14.3846 −0.477633 −0.238817 0.971065i \(-0.576760\pi\)
−0.238817 + 0.971065i \(0.576760\pi\)
\(908\) 0 0
\(909\) −9.53667 −0.316311
\(910\) 0 0
\(911\) 8.95988 0.296854 0.148427 0.988923i \(-0.452579\pi\)
0.148427 + 0.988923i \(0.452579\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 9.80916 0.324281
\(916\) 0 0
\(917\) −12.0876 −0.399169
\(918\) 0 0
\(919\) 1.95932 0.0646319 0.0323159 0.999478i \(-0.489712\pi\)
0.0323159 + 0.999478i \(0.489712\pi\)
\(920\) 0 0
\(921\) −2.99497 −0.0986876
\(922\) 0 0
\(923\) 18.6382 0.613483
\(924\) 0 0
\(925\) 24.7856 0.814946
\(926\) 0 0
\(927\) 13.1615 0.432280
\(928\) 0 0
\(929\) −5.17187 −0.169684 −0.0848418 0.996394i \(-0.527038\pi\)
−0.0848418 + 0.996394i \(0.527038\pi\)
\(930\) 0 0
\(931\) 17.0448 0.558620
\(932\) 0 0
\(933\) 13.2543 0.433927
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −18.5625 −0.606409 −0.303204 0.952926i \(-0.598057\pi\)
−0.303204 + 0.952926i \(0.598057\pi\)
\(938\) 0 0
\(939\) 25.8318 0.842990
\(940\) 0 0
\(941\) −31.7244 −1.03418 −0.517092 0.855930i \(-0.672986\pi\)
−0.517092 + 0.855930i \(0.672986\pi\)
\(942\) 0 0
\(943\) −29.2386 −0.952139
\(944\) 0 0
\(945\) −5.16893 −0.168145
\(946\) 0 0
\(947\) 29.5686 0.960851 0.480425 0.877036i \(-0.340482\pi\)
0.480425 + 0.877036i \(0.340482\pi\)
\(948\) 0 0
\(949\) −27.7186 −0.899783
\(950\) 0 0
\(951\) −28.6917 −0.930391
\(952\) 0 0
\(953\) 45.9337 1.48794 0.743969 0.668214i \(-0.232941\pi\)
0.743969 + 0.668214i \(0.232941\pi\)
\(954\) 0 0
\(955\) −58.1026 −1.88016
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −37.6140 −1.21462
\(960\) 0 0
\(961\) 14.0196 0.452247
\(962\) 0 0
\(963\) 9.22556 0.297289
\(964\) 0 0
\(965\) 7.13497 0.229683
\(966\) 0 0
\(967\) −58.4340 −1.87911 −0.939555 0.342398i \(-0.888761\pi\)
−0.939555 + 0.342398i \(0.888761\pi\)
\(968\) 0 0
\(969\) −17.6878 −0.568213
\(970\) 0 0
\(971\) 41.4158 1.32910 0.664548 0.747246i \(-0.268624\pi\)
0.664548 + 0.747246i \(0.268624\pi\)
\(972\) 0 0
\(973\) −21.3137 −0.683285
\(974\) 0 0
\(975\) −6.11956 −0.195983
\(976\) 0 0
\(977\) 1.11179 0.0355692 0.0177846 0.999842i \(-0.494339\pi\)
0.0177846 + 0.999842i \(0.494339\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −5.90605 −0.188566
\(982\) 0 0
\(983\) 54.5294 1.73922 0.869608 0.493742i \(-0.164371\pi\)
0.869608 + 0.493742i \(0.164371\pi\)
\(984\) 0 0
\(985\) −9.31908 −0.296931
\(986\) 0 0
\(987\) 0.606474 0.0193043
\(988\) 0 0
\(989\) 12.7692 0.406038
\(990\) 0 0
\(991\) 33.2060 1.05482 0.527411 0.849610i \(-0.323163\pi\)
0.527411 + 0.849610i \(0.323163\pi\)
\(992\) 0 0
\(993\) −0.445321 −0.0141318
\(994\) 0 0
\(995\) 23.7572 0.753154
\(996\) 0 0
\(997\) −6.03376 −0.191091 −0.0955455 0.995425i \(-0.530460\pi\)
−0.0955455 + 0.995425i \(0.530460\pi\)
\(998\) 0 0
\(999\) 9.26582 0.293157
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 8016.2.a.o.1.3 4
4.3 odd 2 1002.2.a.i.1.3 4
12.11 even 2 3006.2.a.s.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1002.2.a.i.1.3 4 4.3 odd 2
3006.2.a.s.1.2 4 12.11 even 2
8016.2.a.o.1.3 4 1.1 even 1 trivial