Properties

Label 8550.2.a.cm.1.2
Level $8550$
Weight $2$
Character 8550.1
Self dual yes
Analytic conductor $68.272$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8550.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(68.2720937282\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.788.1
Defining polynomial: \(x^{3} - x^{2} - 7 x - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.476452\) of defining polynomial
Character \(\chi\) \(=\) 8550.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.47645 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.47645 q^{7} +1.00000 q^{8} +2.29654 q^{11} -5.29654 q^{13} -1.47645 q^{14} +1.00000 q^{16} +1.47645 q^{17} -1.00000 q^{19} +2.29654 q^{22} -1.86719 q^{23} -5.29654 q^{26} -1.47645 q^{28} -7.16373 q^{29} +0.0470959 q^{31} +1.00000 q^{32} +1.47645 q^{34} +6.59308 q^{37} -1.00000 q^{38} -0.179911 q^{41} +7.98382 q^{43} +2.29654 q^{44} -1.86719 q^{46} +12.4132 q^{47} -4.82009 q^{49} -5.29654 q^{52} +11.9367 q^{53} -1.47645 q^{56} -7.16373 q^{58} -6.34364 q^{59} -9.93672 q^{61} +0.0470959 q^{62} +1.00000 q^{64} -14.8039 q^{67} +1.47645 q^{68} -14.0695 q^{71} -14.4603 q^{73} +6.59308 q^{74} -1.00000 q^{76} -3.39073 q^{77} -12.5460 q^{79} -0.179911 q^{82} +6.11663 q^{83} +7.98382 q^{86} +2.29654 q^{88} +6.46027 q^{89} +7.82009 q^{91} -1.86719 q^{92} +12.4132 q^{94} +4.42936 q^{97} -4.82009 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} + 3q^{4} - 2q^{7} + 3q^{8} + O(q^{10}) \) \( 3q + 3q^{2} + 3q^{4} - 2q^{7} + 3q^{8} - 5q^{11} - 4q^{13} - 2q^{14} + 3q^{16} + 2q^{17} - 3q^{19} - 5q^{22} - q^{23} - 4q^{26} - 2q^{28} - 5q^{29} + 5q^{31} + 3q^{32} + 2q^{34} - 4q^{37} - 3q^{38} - 10q^{41} - 2q^{43} - 5q^{44} - q^{46} + 4q^{47} - 5q^{49} - 4q^{52} + 5q^{53} - 2q^{56} - 5q^{58} - 12q^{59} + q^{61} + 5q^{62} + 3q^{64} - 9q^{67} + 2q^{68} - 16q^{71} - 15q^{73} - 4q^{74} - 3q^{76} - 8q^{77} - 9q^{79} - 10q^{82} - 3q^{83} - 2q^{86} - 5q^{88} - 9q^{89} + 14q^{91} - q^{92} + 4q^{94} + 6q^{97} - 5q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) −1.47645 −0.558046 −0.279023 0.960284i \(-0.590011\pi\)
−0.279023 + 0.960284i \(0.590011\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 2.29654 0.692433 0.346217 0.938155i \(-0.387466\pi\)
0.346217 + 0.938155i \(0.387466\pi\)
\(12\) 0 0
\(13\) −5.29654 −1.46900 −0.734498 0.678611i \(-0.762582\pi\)
−0.734498 + 0.678611i \(0.762582\pi\)
\(14\) −1.47645 −0.394598
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 1.47645 0.358092 0.179046 0.983841i \(-0.442699\pi\)
0.179046 + 0.983841i \(0.442699\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 2.29654 0.489624
\(23\) −1.86719 −0.389335 −0.194668 0.980869i \(-0.562363\pi\)
−0.194668 + 0.980869i \(0.562363\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −5.29654 −1.03874
\(27\) 0 0
\(28\) −1.47645 −0.279023
\(29\) −7.16373 −1.33027 −0.665135 0.746723i \(-0.731626\pi\)
−0.665135 + 0.746723i \(0.731626\pi\)
\(30\) 0 0
\(31\) 0.0470959 0.00845868 0.00422934 0.999991i \(-0.498654\pi\)
0.00422934 + 0.999991i \(0.498654\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 1.47645 0.253209
\(35\) 0 0
\(36\) 0 0
\(37\) 6.59308 1.08390 0.541948 0.840412i \(-0.317687\pi\)
0.541948 + 0.840412i \(0.317687\pi\)
\(38\) −1.00000 −0.162221
\(39\) 0 0
\(40\) 0 0
\(41\) −0.179911 −0.0280973 −0.0140487 0.999901i \(-0.504472\pi\)
−0.0140487 + 0.999901i \(0.504472\pi\)
\(42\) 0 0
\(43\) 7.98382 1.21752 0.608760 0.793354i \(-0.291667\pi\)
0.608760 + 0.793354i \(0.291667\pi\)
\(44\) 2.29654 0.346217
\(45\) 0 0
\(46\) −1.86719 −0.275301
\(47\) 12.4132 1.81065 0.905324 0.424722i \(-0.139628\pi\)
0.905324 + 0.424722i \(0.139628\pi\)
\(48\) 0 0
\(49\) −4.82009 −0.688584
\(50\) 0 0
\(51\) 0 0
\(52\) −5.29654 −0.734498
\(53\) 11.9367 1.63963 0.819817 0.572625i \(-0.194075\pi\)
0.819817 + 0.572625i \(0.194075\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.47645 −0.197299
\(57\) 0 0
\(58\) −7.16373 −0.940643
\(59\) −6.34364 −0.825871 −0.412936 0.910760i \(-0.635497\pi\)
−0.412936 + 0.910760i \(0.635497\pi\)
\(60\) 0 0
\(61\) −9.93672 −1.27227 −0.636133 0.771579i \(-0.719467\pi\)
−0.636133 + 0.771579i \(0.719467\pi\)
\(62\) 0.0470959 0.00598119
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −14.8039 −1.80858 −0.904292 0.426914i \(-0.859601\pi\)
−0.904292 + 0.426914i \(0.859601\pi\)
\(68\) 1.47645 0.179046
\(69\) 0 0
\(70\) 0 0
\(71\) −14.0695 −1.66975 −0.834873 0.550442i \(-0.814459\pi\)
−0.834873 + 0.550442i \(0.814459\pi\)
\(72\) 0 0
\(73\) −14.4603 −1.69245 −0.846223 0.532829i \(-0.821129\pi\)
−0.846223 + 0.532829i \(0.821129\pi\)
\(74\) 6.59308 0.766430
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) −3.39073 −0.386410
\(78\) 0 0
\(79\) −12.5460 −1.41153 −0.705767 0.708444i \(-0.749397\pi\)
−0.705767 + 0.708444i \(0.749397\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −0.179911 −0.0198678
\(83\) 6.11663 0.671387 0.335694 0.941971i \(-0.391029\pi\)
0.335694 + 0.941971i \(0.391029\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 7.98382 0.860917
\(87\) 0 0
\(88\) 2.29654 0.244812
\(89\) 6.46027 0.684787 0.342394 0.939557i \(-0.388762\pi\)
0.342394 + 0.939557i \(0.388762\pi\)
\(90\) 0 0
\(91\) 7.82009 0.819768
\(92\) −1.86719 −0.194668
\(93\) 0 0
\(94\) 12.4132 1.28032
\(95\) 0 0
\(96\) 0 0
\(97\) 4.42936 0.449733 0.224866 0.974390i \(-0.427805\pi\)
0.224866 + 0.974390i \(0.427805\pi\)
\(98\) −4.82009 −0.486903
\(99\) 0 0
\(100\) 0 0
\(101\) −13.2965 −1.32306 −0.661528 0.749921i \(-0.730092\pi\)
−0.661528 + 0.749921i \(0.730092\pi\)
\(102\) 0 0
\(103\) −0.227007 −0.0223676 −0.0111838 0.999937i \(-0.503560\pi\)
−0.0111838 + 0.999937i \(0.503560\pi\)
\(104\) −5.29654 −0.519369
\(105\) 0 0
\(106\) 11.9367 1.15940
\(107\) −7.20235 −0.696277 −0.348139 0.937443i \(-0.613186\pi\)
−0.348139 + 0.937443i \(0.613186\pi\)
\(108\) 0 0
\(109\) −7.04710 −0.674989 −0.337495 0.941327i \(-0.609580\pi\)
−0.337495 + 0.941327i \(0.609580\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −1.47645 −0.139512
\(113\) −20.1004 −1.89089 −0.945445 0.325780i \(-0.894373\pi\)
−0.945445 + 0.325780i \(0.894373\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −7.16373 −0.665135
\(117\) 0 0
\(118\) −6.34364 −0.583979
\(119\) −2.17991 −0.199832
\(120\) 0 0
\(121\) −5.72590 −0.520536
\(122\) −9.93672 −0.899628
\(123\) 0 0
\(124\) 0.0470959 0.00422934
\(125\) 0 0
\(126\) 0 0
\(127\) 5.68727 0.504664 0.252332 0.967641i \(-0.418802\pi\)
0.252332 + 0.967641i \(0.418802\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −12.2108 −1.06686 −0.533432 0.845843i \(-0.679098\pi\)
−0.533432 + 0.845843i \(0.679098\pi\)
\(132\) 0 0
\(133\) 1.47645 0.128025
\(134\) −14.8039 −1.27886
\(135\) 0 0
\(136\) 1.47645 0.126605
\(137\) −8.23326 −0.703415 −0.351708 0.936110i \(-0.614399\pi\)
−0.351708 + 0.936110i \(0.614399\pi\)
\(138\) 0 0
\(139\) −13.2023 −1.11981 −0.559904 0.828557i \(-0.689162\pi\)
−0.559904 + 0.828557i \(0.689162\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −14.0695 −1.18069
\(143\) −12.1637 −1.01718
\(144\) 0 0
\(145\) 0 0
\(146\) −14.4603 −1.19674
\(147\) 0 0
\(148\) 6.59308 0.541948
\(149\) −6.85871 −0.561888 −0.280944 0.959724i \(-0.590647\pi\)
−0.280944 + 0.959724i \(0.590647\pi\)
\(150\) 0 0
\(151\) 10.6788 0.869029 0.434514 0.900665i \(-0.356920\pi\)
0.434514 + 0.900665i \(0.356920\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 0 0
\(154\) −3.39073 −0.273233
\(155\) 0 0
\(156\) 0 0
\(157\) 3.22701 0.257543 0.128772 0.991674i \(-0.458897\pi\)
0.128772 + 0.991674i \(0.458897\pi\)
\(158\) −12.5460 −0.998105
\(159\) 0 0
\(160\) 0 0
\(161\) 2.75681 0.217267
\(162\) 0 0
\(163\) −6.00000 −0.469956 −0.234978 0.972001i \(-0.575502\pi\)
−0.234978 + 0.972001i \(0.575502\pi\)
\(164\) −0.179911 −0.0140487
\(165\) 0 0
\(166\) 6.11663 0.474743
\(167\) 20.9205 1.61888 0.809440 0.587203i \(-0.199771\pi\)
0.809440 + 0.587203i \(0.199771\pi\)
\(168\) 0 0
\(169\) 15.0534 1.15795
\(170\) 0 0
\(171\) 0 0
\(172\) 7.98382 0.608760
\(173\) 1.25792 0.0956378 0.0478189 0.998856i \(-0.484773\pi\)
0.0478189 + 0.998856i \(0.484773\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 2.29654 0.173108
\(177\) 0 0
\(178\) 6.46027 0.484218
\(179\) 6.49889 0.485750 0.242875 0.970058i \(-0.421910\pi\)
0.242875 + 0.970058i \(0.421910\pi\)
\(180\) 0 0
\(181\) 19.6240 1.45864 0.729320 0.684173i \(-0.239837\pi\)
0.729320 + 0.684173i \(0.239837\pi\)
\(182\) 7.82009 0.579664
\(183\) 0 0
\(184\) −1.86719 −0.137651
\(185\) 0 0
\(186\) 0 0
\(187\) 3.39073 0.247955
\(188\) 12.4132 0.905324
\(189\) 0 0
\(190\) 0 0
\(191\) −0.406917 −0.0294435 −0.0147217 0.999892i \(-0.504686\pi\)
−0.0147217 + 0.999892i \(0.504686\pi\)
\(192\) 0 0
\(193\) 12.3970 0.892355 0.446177 0.894945i \(-0.352785\pi\)
0.446177 + 0.894945i \(0.352785\pi\)
\(194\) 4.42936 0.318009
\(195\) 0 0
\(196\) −4.82009 −0.344292
\(197\) −2.24945 −0.160266 −0.0801332 0.996784i \(-0.525535\pi\)
−0.0801332 + 0.996784i \(0.525535\pi\)
\(198\) 0 0
\(199\) 18.4665 1.30906 0.654529 0.756037i \(-0.272867\pi\)
0.654529 + 0.756037i \(0.272867\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −13.2965 −0.935541
\(203\) 10.5769 0.742353
\(204\) 0 0
\(205\) 0 0
\(206\) −0.227007 −0.0158163
\(207\) 0 0
\(208\) −5.29654 −0.367249
\(209\) −2.29654 −0.158855
\(210\) 0 0
\(211\) 24.3584 1.67690 0.838450 0.544979i \(-0.183462\pi\)
0.838450 + 0.544979i \(0.183462\pi\)
\(212\) 11.9367 0.819817
\(213\) 0 0
\(214\) −7.20235 −0.492342
\(215\) 0 0
\(216\) 0 0
\(217\) −0.0695349 −0.00472033
\(218\) −7.04710 −0.477290
\(219\) 0 0
\(220\) 0 0
\(221\) −7.82009 −0.526036
\(222\) 0 0
\(223\) −20.1946 −1.35233 −0.676167 0.736749i \(-0.736360\pi\)
−0.676167 + 0.736749i \(0.736360\pi\)
\(224\) −1.47645 −0.0986496
\(225\) 0 0
\(226\) −20.1004 −1.33706
\(227\) 6.35982 0.422116 0.211058 0.977474i \(-0.432309\pi\)
0.211058 + 0.977474i \(0.432309\pi\)
\(228\) 0 0
\(229\) −11.8510 −0.783136 −0.391568 0.920149i \(-0.628067\pi\)
−0.391568 + 0.920149i \(0.628067\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −7.16373 −0.470322
\(233\) −27.6155 −1.80915 −0.904576 0.426311i \(-0.859813\pi\)
−0.904576 + 0.426311i \(0.859813\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −6.34364 −0.412936
\(237\) 0 0
\(238\) −2.17991 −0.141303
\(239\) −28.2248 −1.82571 −0.912855 0.408284i \(-0.866127\pi\)
−0.912855 + 0.408284i \(0.866127\pi\)
\(240\) 0 0
\(241\) 7.38226 0.475533 0.237767 0.971322i \(-0.423585\pi\)
0.237767 + 0.971322i \(0.423585\pi\)
\(242\) −5.72590 −0.368075
\(243\) 0 0
\(244\) −9.93672 −0.636133
\(245\) 0 0
\(246\) 0 0
\(247\) 5.29654 0.337011
\(248\) 0.0470959 0.00299059
\(249\) 0 0
\(250\) 0 0
\(251\) −15.6317 −0.986665 −0.493332 0.869841i \(-0.664221\pi\)
−0.493332 + 0.869841i \(0.664221\pi\)
\(252\) 0 0
\(253\) −4.28807 −0.269589
\(254\) 5.68727 0.356851
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 8.13281 0.507311 0.253656 0.967295i \(-0.418367\pi\)
0.253656 + 0.967295i \(0.418367\pi\)
\(258\) 0 0
\(259\) −9.73437 −0.604864
\(260\) 0 0
\(261\) 0 0
\(262\) −12.2108 −0.754387
\(263\) −10.2270 −0.630624 −0.315312 0.948988i \(-0.602109\pi\)
−0.315312 + 0.948988i \(0.602109\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 1.47645 0.0905271
\(267\) 0 0
\(268\) −14.8039 −0.904292
\(269\) 8.59308 0.523930 0.261965 0.965077i \(-0.415630\pi\)
0.261965 + 0.965077i \(0.415630\pi\)
\(270\) 0 0
\(271\) 8.78918 0.533905 0.266952 0.963710i \(-0.413983\pi\)
0.266952 + 0.963710i \(0.413983\pi\)
\(272\) 1.47645 0.0895231
\(273\) 0 0
\(274\) −8.23326 −0.497390
\(275\) 0 0
\(276\) 0 0
\(277\) 24.7954 1.48981 0.744907 0.667169i \(-0.232494\pi\)
0.744907 + 0.667169i \(0.232494\pi\)
\(278\) −13.2023 −0.791824
\(279\) 0 0
\(280\) 0 0
\(281\) −23.5931 −1.40745 −0.703723 0.710475i \(-0.748480\pi\)
−0.703723 + 0.710475i \(0.748480\pi\)
\(282\) 0 0
\(283\) −13.5460 −0.805225 −0.402613 0.915370i \(-0.631898\pi\)
−0.402613 + 0.915370i \(0.631898\pi\)
\(284\) −14.0695 −0.834873
\(285\) 0 0
\(286\) −12.1637 −0.719256
\(287\) 0.265629 0.0156796
\(288\) 0 0
\(289\) −14.8201 −0.871770
\(290\) 0 0
\(291\) 0 0
\(292\) −14.4603 −0.846223
\(293\) −24.7954 −1.44856 −0.724282 0.689504i \(-0.757829\pi\)
−0.724282 + 0.689504i \(0.757829\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 6.59308 0.383215
\(297\) 0 0
\(298\) −6.85871 −0.397315
\(299\) 9.88962 0.571932
\(300\) 0 0
\(301\) −11.7877 −0.679433
\(302\) 10.6788 0.614496
\(303\) 0 0
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) 0 0
\(307\) −11.6626 −0.665621 −0.332810 0.942994i \(-0.607997\pi\)
−0.332810 + 0.942994i \(0.607997\pi\)
\(308\) −3.39073 −0.193205
\(309\) 0 0
\(310\) 0 0
\(311\) 3.72590 0.211276 0.105638 0.994405i \(-0.466311\pi\)
0.105638 + 0.994405i \(0.466311\pi\)
\(312\) 0 0
\(313\) 5.00847 0.283096 0.141548 0.989931i \(-0.454792\pi\)
0.141548 + 0.989931i \(0.454792\pi\)
\(314\) 3.22701 0.182111
\(315\) 0 0
\(316\) −12.5460 −0.705767
\(317\) 27.1637 1.52567 0.762833 0.646595i \(-0.223808\pi\)
0.762833 + 0.646595i \(0.223808\pi\)
\(318\) 0 0
\(319\) −16.4518 −0.921124
\(320\) 0 0
\(321\) 0 0
\(322\) 2.75681 0.153631
\(323\) −1.47645 −0.0821520
\(324\) 0 0
\(325\) 0 0
\(326\) −6.00000 −0.332309
\(327\) 0 0
\(328\) −0.179911 −0.00993390
\(329\) −18.3275 −1.01043
\(330\) 0 0
\(331\) −2.80391 −0.154117 −0.0770583 0.997027i \(-0.524553\pi\)
−0.0770583 + 0.997027i \(0.524553\pi\)
\(332\) 6.11663 0.335694
\(333\) 0 0
\(334\) 20.9205 1.14472
\(335\) 0 0
\(336\) 0 0
\(337\) 6.85871 0.373618 0.186809 0.982396i \(-0.440185\pi\)
0.186809 + 0.982396i \(0.440185\pi\)
\(338\) 15.0534 0.818794
\(339\) 0 0
\(340\) 0 0
\(341\) 0.108158 0.00585707
\(342\) 0 0
\(343\) 17.4518 0.942308
\(344\) 7.98382 0.430459
\(345\) 0 0
\(346\) 1.25792 0.0676261
\(347\) −19.7792 −1.06181 −0.530903 0.847433i \(-0.678147\pi\)
−0.530903 + 0.847433i \(0.678147\pi\)
\(348\) 0 0
\(349\) −7.39699 −0.395952 −0.197976 0.980207i \(-0.563437\pi\)
−0.197976 + 0.980207i \(0.563437\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 2.29654 0.122406
\(353\) −36.0372 −1.91806 −0.959032 0.283296i \(-0.908572\pi\)
−0.959032 + 0.283296i \(0.908572\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 6.46027 0.342394
\(357\) 0 0
\(358\) 6.49889 0.343477
\(359\) −24.4132 −1.28848 −0.644239 0.764824i \(-0.722826\pi\)
−0.644239 + 0.764824i \(0.722826\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 19.6240 1.03141
\(363\) 0 0
\(364\) 7.82009 0.409884
\(365\) 0 0
\(366\) 0 0
\(367\) 10.0695 0.525625 0.262813 0.964847i \(-0.415350\pi\)
0.262813 + 0.964847i \(0.415350\pi\)
\(368\) −1.86719 −0.0973338
\(369\) 0 0
\(370\) 0 0
\(371\) −17.6240 −0.914992
\(372\) 0 0
\(373\) −24.5769 −1.27254 −0.636272 0.771465i \(-0.719524\pi\)
−0.636272 + 0.771465i \(0.719524\pi\)
\(374\) 3.39073 0.175331
\(375\) 0 0
\(376\) 12.4132 0.640160
\(377\) 37.9430 1.95416
\(378\) 0 0
\(379\) −3.50111 −0.179840 −0.0899199 0.995949i \(-0.528661\pi\)
−0.0899199 + 0.995949i \(0.528661\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −0.406917 −0.0208197
\(383\) 18.9901 0.970347 0.485174 0.874418i \(-0.338756\pi\)
0.485174 + 0.874418i \(0.338756\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 12.3970 0.630990
\(387\) 0 0
\(388\) 4.42936 0.224866
\(389\) −14.2333 −0.721655 −0.360828 0.932633i \(-0.617506\pi\)
−0.360828 + 0.932633i \(0.617506\pi\)
\(390\) 0 0
\(391\) −2.75681 −0.139418
\(392\) −4.82009 −0.243451
\(393\) 0 0
\(394\) −2.24945 −0.113325
\(395\) 0 0
\(396\) 0 0
\(397\) 29.3885 1.47497 0.737484 0.675365i \(-0.236014\pi\)
0.737484 + 0.675365i \(0.236014\pi\)
\(398\) 18.4665 0.925643
\(399\) 0 0
\(400\) 0 0
\(401\) 14.5460 0.726392 0.363196 0.931713i \(-0.381686\pi\)
0.363196 + 0.931713i \(0.381686\pi\)
\(402\) 0 0
\(403\) −0.249445 −0.0124258
\(404\) −13.2965 −0.661528
\(405\) 0 0
\(406\) 10.5769 0.524923
\(407\) 15.1413 0.750526
\(408\) 0 0
\(409\) 18.5931 0.919369 0.459684 0.888082i \(-0.347963\pi\)
0.459684 + 0.888082i \(0.347963\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −0.227007 −0.0111838
\(413\) 9.36608 0.460874
\(414\) 0 0
\(415\) 0 0
\(416\) −5.29654 −0.259684
\(417\) 0 0
\(418\) −2.29654 −0.112328
\(419\) 10.2333 0.499928 0.249964 0.968255i \(-0.419581\pi\)
0.249964 + 0.968255i \(0.419581\pi\)
\(420\) 0 0
\(421\) 9.82856 0.479015 0.239507 0.970895i \(-0.423014\pi\)
0.239507 + 0.970895i \(0.423014\pi\)
\(422\) 24.3584 1.18575
\(423\) 0 0
\(424\) 11.9367 0.579698
\(425\) 0 0
\(426\) 0 0
\(427\) 14.6711 0.709984
\(428\) −7.20235 −0.348139
\(429\) 0 0
\(430\) 0 0
\(431\) −36.0449 −1.73622 −0.868110 0.496371i \(-0.834665\pi\)
−0.868110 + 0.496371i \(0.834665\pi\)
\(432\) 0 0
\(433\) 19.2881 0.926925 0.463463 0.886116i \(-0.346607\pi\)
0.463463 + 0.886116i \(0.346607\pi\)
\(434\) −0.0695349 −0.00333778
\(435\) 0 0
\(436\) −7.04710 −0.337495
\(437\) 1.86719 0.0893196
\(438\) 0 0
\(439\) 24.4194 1.16548 0.582738 0.812660i \(-0.301981\pi\)
0.582738 + 0.812660i \(0.301981\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −7.82009 −0.371964
\(443\) −13.8834 −0.659619 −0.329809 0.944048i \(-0.606984\pi\)
−0.329809 + 0.944048i \(0.606984\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −20.1946 −0.956244
\(447\) 0 0
\(448\) −1.47645 −0.0697558
\(449\) −13.6016 −0.641897 −0.320949 0.947097i \(-0.604002\pi\)
−0.320949 + 0.947097i \(0.604002\pi\)
\(450\) 0 0
\(451\) −0.413172 −0.0194555
\(452\) −20.1004 −0.945445
\(453\) 0 0
\(454\) 6.35982 0.298481
\(455\) 0 0
\(456\) 0 0
\(457\) −25.5051 −1.19308 −0.596540 0.802583i \(-0.703458\pi\)
−0.596540 + 0.802583i \(0.703458\pi\)
\(458\) −11.8510 −0.553761
\(459\) 0 0
\(460\) 0 0
\(461\) −16.3598 −0.761953 −0.380976 0.924585i \(-0.624412\pi\)
−0.380976 + 0.924585i \(0.624412\pi\)
\(462\) 0 0
\(463\) −14.7815 −0.686953 −0.343477 0.939161i \(-0.611605\pi\)
−0.343477 + 0.939161i \(0.611605\pi\)
\(464\) −7.16373 −0.332568
\(465\) 0 0
\(466\) −27.6155 −1.27926
\(467\) −13.4055 −0.620331 −0.310165 0.950683i \(-0.600384\pi\)
−0.310165 + 0.950683i \(0.600384\pi\)
\(468\) 0 0
\(469\) 21.8573 1.00927
\(470\) 0 0
\(471\) 0 0
\(472\) −6.34364 −0.291990
\(473\) 18.3352 0.843052
\(474\) 0 0
\(475\) 0 0
\(476\) −2.17991 −0.0999160
\(477\) 0 0
\(478\) −28.2248 −1.29097
\(479\) 37.0534 1.69301 0.846505 0.532380i \(-0.178702\pi\)
0.846505 + 0.532380i \(0.178702\pi\)
\(480\) 0 0
\(481\) −34.9205 −1.59224
\(482\) 7.38226 0.336253
\(483\) 0 0
\(484\) −5.72590 −0.260268
\(485\) 0 0
\(486\) 0 0
\(487\) −13.0471 −0.591220 −0.295610 0.955309i \(-0.595523\pi\)
−0.295610 + 0.955309i \(0.595523\pi\)
\(488\) −9.93672 −0.449814
\(489\) 0 0
\(490\) 0 0
\(491\) 30.7322 1.38692 0.693461 0.720494i \(-0.256085\pi\)
0.693461 + 0.720494i \(0.256085\pi\)
\(492\) 0 0
\(493\) −10.5769 −0.476360
\(494\) 5.29654 0.238303
\(495\) 0 0
\(496\) 0.0470959 0.00211467
\(497\) 20.7730 0.931796
\(498\) 0 0
\(499\) −17.3414 −0.776309 −0.388154 0.921594i \(-0.626887\pi\)
−0.388154 + 0.921594i \(0.626887\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −15.6317 −0.697677
\(503\) −16.0000 −0.713405 −0.356702 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −4.28807 −0.190628
\(507\) 0 0
\(508\) 5.68727 0.252332
\(509\) 25.3885 1.12533 0.562663 0.826686i \(-0.309777\pi\)
0.562663 + 0.826686i \(0.309777\pi\)
\(510\) 0 0
\(511\) 21.3499 0.944464
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 8.13281 0.358723
\(515\) 0 0
\(516\) 0 0
\(517\) 28.5074 1.25375
\(518\) −9.73437 −0.427704
\(519\) 0 0
\(520\) 0 0
\(521\) −13.7645 −0.603035 −0.301517 0.953461i \(-0.597493\pi\)
−0.301517 + 0.953461i \(0.597493\pi\)
\(522\) 0 0
\(523\) 18.5931 0.813019 0.406509 0.913647i \(-0.366746\pi\)
0.406509 + 0.913647i \(0.366746\pi\)
\(524\) −12.2108 −0.533432
\(525\) 0 0
\(526\) −10.2270 −0.445919
\(527\) 0.0695349 0.00302899
\(528\) 0 0
\(529\) −19.5136 −0.848418
\(530\) 0 0
\(531\) 0 0
\(532\) 1.47645 0.0640123
\(533\) 0.952904 0.0412749
\(534\) 0 0
\(535\) 0 0
\(536\) −14.8039 −0.639431
\(537\) 0 0
\(538\) 8.59308 0.370474
\(539\) −11.0695 −0.476799
\(540\) 0 0
\(541\) 34.2557 1.47277 0.736384 0.676564i \(-0.236532\pi\)
0.736384 + 0.676564i \(0.236532\pi\)
\(542\) 8.78918 0.377527
\(543\) 0 0
\(544\) 1.47645 0.0633024
\(545\) 0 0
\(546\) 0 0
\(547\) −7.93672 −0.339350 −0.169675 0.985500i \(-0.554272\pi\)
−0.169675 + 0.985500i \(0.554272\pi\)
\(548\) −8.23326 −0.351708
\(549\) 0 0
\(550\) 0 0
\(551\) 7.16373 0.305185
\(552\) 0 0
\(553\) 18.5235 0.787701
\(554\) 24.7954 1.05346
\(555\) 0 0
\(556\) −13.2023 −0.559904
\(557\) −1.46798 −0.0622003 −0.0311001 0.999516i \(-0.509901\pi\)
−0.0311001 + 0.999516i \(0.509901\pi\)
\(558\) 0 0
\(559\) −42.2866 −1.78853
\(560\) 0 0
\(561\) 0 0
\(562\) −23.5931 −0.995214
\(563\) −32.7160 −1.37881 −0.689407 0.724374i \(-0.742129\pi\)
−0.689407 + 0.724374i \(0.742129\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −13.5460 −0.569380
\(567\) 0 0
\(568\) −14.0695 −0.590345
\(569\) 22.6873 0.951100 0.475550 0.879689i \(-0.342249\pi\)
0.475550 + 0.879689i \(0.342249\pi\)
\(570\) 0 0
\(571\) 8.34364 0.349170 0.174585 0.984642i \(-0.444142\pi\)
0.174585 + 0.984642i \(0.444142\pi\)
\(572\) −12.1637 −0.508591
\(573\) 0 0
\(574\) 0.265629 0.0110872
\(575\) 0 0
\(576\) 0 0
\(577\) −0.451795 −0.0188085 −0.00940424 0.999956i \(-0.502994\pi\)
−0.00940424 + 0.999956i \(0.502994\pi\)
\(578\) −14.8201 −0.616434
\(579\) 0 0
\(580\) 0 0
\(581\) −9.03091 −0.374665
\(582\) 0 0
\(583\) 27.4132 1.13534
\(584\) −14.4603 −0.598370
\(585\) 0 0
\(586\) −24.7954 −1.02429
\(587\) 32.8812 1.35715 0.678575 0.734531i \(-0.262598\pi\)
0.678575 + 0.734531i \(0.262598\pi\)
\(588\) 0 0
\(589\) −0.0470959 −0.00194055
\(590\) 0 0
\(591\) 0 0
\(592\) 6.59308 0.270974
\(593\) 36.8882 1.51482 0.757408 0.652942i \(-0.226466\pi\)
0.757408 + 0.652942i \(0.226466\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −6.85871 −0.280944
\(597\) 0 0
\(598\) 9.88962 0.404417
\(599\) 43.7715 1.78846 0.894228 0.447611i \(-0.147725\pi\)
0.894228 + 0.447611i \(0.147725\pi\)
\(600\) 0 0
\(601\) 15.9430 0.650328 0.325164 0.945658i \(-0.394581\pi\)
0.325164 + 0.945658i \(0.394581\pi\)
\(602\) −11.7877 −0.480432
\(603\) 0 0
\(604\) 10.6788 0.434514
\(605\) 0 0
\(606\) 0 0
\(607\) −19.9444 −0.809519 −0.404760 0.914423i \(-0.632645\pi\)
−0.404760 + 0.914423i \(0.632645\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 0 0
\(610\) 0 0
\(611\) −65.7469 −2.65983
\(612\) 0 0
\(613\) 36.9205 1.49121 0.745603 0.666390i \(-0.232161\pi\)
0.745603 + 0.666390i \(0.232161\pi\)
\(614\) −11.6626 −0.470665
\(615\) 0 0
\(616\) −3.39073 −0.136617
\(617\) −23.1538 −0.932137 −0.466068 0.884749i \(-0.654330\pi\)
−0.466068 + 0.884749i \(0.654330\pi\)
\(618\) 0 0
\(619\) −5.84475 −0.234920 −0.117460 0.993078i \(-0.537475\pi\)
−0.117460 + 0.993078i \(0.537475\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 3.72590 0.149395
\(623\) −9.53828 −0.382143
\(624\) 0 0
\(625\) 0 0
\(626\) 5.00847 0.200179
\(627\) 0 0
\(628\) 3.22701 0.128772
\(629\) 9.73437 0.388135
\(630\) 0 0
\(631\) 44.5607 1.77393 0.886967 0.461833i \(-0.152808\pi\)
0.886967 + 0.461833i \(0.152808\pi\)
\(632\) −12.5460 −0.499053
\(633\) 0 0
\(634\) 27.1637 1.07881
\(635\) 0 0
\(636\) 0 0
\(637\) 25.5298 1.01153
\(638\) −16.4518 −0.651333
\(639\) 0 0
\(640\) 0 0
\(641\) 4.50736 0.178030 0.0890151 0.996030i \(-0.471628\pi\)
0.0890151 + 0.996030i \(0.471628\pi\)
\(642\) 0 0
\(643\) 19.0596 0.751637 0.375819 0.926693i \(-0.377362\pi\)
0.375819 + 0.926693i \(0.377362\pi\)
\(644\) 2.75681 0.108634
\(645\) 0 0
\(646\) −1.47645 −0.0580902
\(647\) −21.0857 −0.828965 −0.414483 0.910057i \(-0.636037\pi\)
−0.414483 + 0.910057i \(0.636037\pi\)
\(648\) 0 0
\(649\) −14.5684 −0.571861
\(650\) 0 0
\(651\) 0 0
\(652\) −6.00000 −0.234978
\(653\) −11.3745 −0.445121 −0.222560 0.974919i \(-0.571441\pi\)
−0.222560 + 0.974919i \(0.571441\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −0.179911 −0.00702433
\(657\) 0 0
\(658\) −18.3275 −0.714479
\(659\) −1.48493 −0.0578445 −0.0289222 0.999582i \(-0.509208\pi\)
−0.0289222 + 0.999582i \(0.509208\pi\)
\(660\) 0 0
\(661\) −20.4216 −0.794310 −0.397155 0.917752i \(-0.630002\pi\)
−0.397155 + 0.917752i \(0.630002\pi\)
\(662\) −2.80391 −0.108977
\(663\) 0 0
\(664\) 6.11663 0.237371
\(665\) 0 0
\(666\) 0 0
\(667\) 13.3760 0.517921
\(668\) 20.9205 0.809440
\(669\) 0 0
\(670\) 0 0
\(671\) −22.8201 −0.880960
\(672\) 0 0
\(673\) 23.0224 0.887450 0.443725 0.896163i \(-0.353657\pi\)
0.443725 + 0.896163i \(0.353657\pi\)
\(674\) 6.85871 0.264188
\(675\) 0 0
\(676\) 15.0534 0.578975
\(677\) −25.9198 −0.996178 −0.498089 0.867126i \(-0.665965\pi\)
−0.498089 + 0.867126i \(0.665965\pi\)
\(678\) 0 0
\(679\) −6.53973 −0.250972
\(680\) 0 0
\(681\) 0 0
\(682\) 0.108158 0.00414157
\(683\) −3.56217 −0.136303 −0.0681513 0.997675i \(-0.521710\pi\)
−0.0681513 + 0.997675i \(0.521710\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 17.4518 0.666313
\(687\) 0 0
\(688\) 7.98382 0.304380
\(689\) −63.2233 −2.40862
\(690\) 0 0
\(691\) −50.6218 −1.92574 −0.962872 0.269960i \(-0.912990\pi\)
−0.962872 + 0.269960i \(0.912990\pi\)
\(692\) 1.25792 0.0478189
\(693\) 0 0
\(694\) −19.7792 −0.750810
\(695\) 0 0
\(696\) 0 0
\(697\) −0.265629 −0.0100614
\(698\) −7.39699 −0.279980
\(699\) 0 0
\(700\) 0 0
\(701\) 6.82634 0.257827 0.128914 0.991656i \(-0.458851\pi\)
0.128914 + 0.991656i \(0.458851\pi\)
\(702\) 0 0
\(703\) −6.59308 −0.248663
\(704\) 2.29654 0.0865542
\(705\) 0 0
\(706\) −36.0372 −1.35628
\(707\) 19.6317 0.738326
\(708\) 0 0
\(709\) 38.3499 1.44026 0.720130 0.693839i \(-0.244082\pi\)
0.720130 + 0.693839i \(0.244082\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 6.46027 0.242109
\(713\) −0.0879368 −0.00329326
\(714\) 0 0
\(715\) 0 0
\(716\) 6.49889 0.242875
\(717\) 0 0
\(718\) −24.4132 −0.911091
\(719\) −28.7020 −1.07040 −0.535202 0.844724i \(-0.679765\pi\)
−0.535202 + 0.844724i \(0.679765\pi\)
\(720\) 0 0
\(721\) 0.335164 0.0124822
\(722\) 1.00000 0.0372161
\(723\) 0 0
\(724\) 19.6240 0.729320
\(725\) 0 0
\(726\) 0 0
\(727\) −1.51362 −0.0561370 −0.0280685 0.999606i \(-0.508936\pi\)
−0.0280685 + 0.999606i \(0.508936\pi\)
\(728\) 7.82009 0.289832
\(729\) 0 0
\(730\) 0 0
\(731\) 11.7877 0.435985
\(732\) 0 0
\(733\) −18.8488 −0.696196 −0.348098 0.937458i \(-0.613172\pi\)
−0.348098 + 0.937458i \(0.613172\pi\)
\(734\) 10.0695 0.371673
\(735\) 0 0
\(736\) −1.86719 −0.0688254
\(737\) −33.9978 −1.25232
\(738\) 0 0
\(739\) −28.4989 −1.04835 −0.524174 0.851611i \(-0.675626\pi\)
−0.524174 + 0.851611i \(0.675626\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −17.6240 −0.646997
\(743\) −11.1089 −0.407547 −0.203773 0.979018i \(-0.565321\pi\)
−0.203773 + 0.979018i \(0.565321\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −24.5769 −0.899824
\(747\) 0 0
\(748\) 3.39073 0.123977
\(749\) 10.6339 0.388555
\(750\) 0 0
\(751\) −21.6487 −0.789971 −0.394985 0.918687i \(-0.629250\pi\)
−0.394985 + 0.918687i \(0.629250\pi\)
\(752\) 12.4132 0.452662
\(753\) 0 0
\(754\) 37.9430 1.38180
\(755\) 0 0
\(756\) 0 0
\(757\) −5.99007 −0.217713 −0.108856 0.994057i \(-0.534719\pi\)
−0.108856 + 0.994057i \(0.534719\pi\)
\(758\) −3.50111 −0.127166
\(759\) 0 0
\(760\) 0 0
\(761\) −19.1615 −0.694604 −0.347302 0.937753i \(-0.612902\pi\)
−0.347302 + 0.937753i \(0.612902\pi\)
\(762\) 0 0
\(763\) 10.4047 0.376675
\(764\) −0.406917 −0.0147217
\(765\) 0 0
\(766\) 18.9901 0.686139
\(767\) 33.5993 1.21320
\(768\) 0 0
\(769\) −18.9761 −0.684296 −0.342148 0.939646i \(-0.611154\pi\)
−0.342148 + 0.939646i \(0.611154\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 12.3970 0.446177
\(773\) 36.5438 1.31439 0.657194 0.753721i \(-0.271743\pi\)
0.657194 + 0.753721i \(0.271743\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 4.42936 0.159005
\(777\) 0 0
\(778\) −14.2333 −0.510287
\(779\) 0.179911 0.00644597
\(780\) 0 0
\(781\) −32.3113 −1.15619
\(782\) −2.75681 −0.0985833
\(783\) 0 0
\(784\) −4.82009 −0.172146
\(785\) 0 0
\(786\) 0 0
\(787\) 32.5213 1.15926 0.579630 0.814880i \(-0.303197\pi\)
0.579630 + 0.814880i \(0.303197\pi\)
\(788\) −2.24945 −0.0801332
\(789\) 0 0
\(790\) 0 0
\(791\) 29.6773 1.05520
\(792\) 0 0
\(793\) 52.6302 1.86895
\(794\) 29.3885 1.04296
\(795\) 0 0
\(796\) 18.4665 0.654529
\(797\) 41.4110 1.46685 0.733426 0.679770i \(-0.237920\pi\)
0.733426 + 0.679770i \(0.237920\pi\)
\(798\) 0 0
\(799\) 18.3275 0.648379
\(800\) 0 0
\(801\) 0 0
\(802\) 14.5460 0.513637
\(803\) −33.2086 −1.17191
\(804\) 0 0
\(805\) 0 0
\(806\) −0.249445 −0.00878634
\(807\) 0 0
\(808\) −13.2965 −0.467771
\(809\) 8.72444 0.306735 0.153368 0.988169i \(-0.450988\pi\)
0.153368 + 0.988169i \(0.450988\pi\)
\(810\) 0 0
\(811\) −11.8425 −0.415847 −0.207924 0.978145i \(-0.566671\pi\)
−0.207924 + 0.978145i \(0.566671\pi\)
\(812\) 10.5769 0.371176
\(813\) 0 0
\(814\) 15.1413 0.530702
\(815\) 0 0
\(816\) 0 0
\(817\) −7.98382 −0.279318
\(818\) 18.5931 0.650092
\(819\) 0 0
\(820\) 0 0
\(821\) −5.03091 −0.175580 −0.0877900 0.996139i \(-0.527980\pi\)
−0.0877900 + 0.996139i \(0.527980\pi\)
\(822\) 0 0
\(823\) −7.81162 −0.272296 −0.136148 0.990689i \(-0.543472\pi\)
−0.136148 + 0.990689i \(0.543472\pi\)
\(824\) −0.227007 −0.00790815
\(825\) 0 0
\(826\) 9.36608 0.325887
\(827\) −2.78070 −0.0966946 −0.0483473 0.998831i \(-0.515395\pi\)
−0.0483473 + 0.998831i \(0.515395\pi\)
\(828\) 0 0
\(829\) 47.9963 1.66698 0.833491 0.552534i \(-0.186339\pi\)
0.833491 + 0.552534i \(0.186339\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −5.29654 −0.183625
\(833\) −7.11663 −0.246577
\(834\) 0 0
\(835\) 0 0
\(836\) −2.29654 −0.0794275
\(837\) 0 0
\(838\) 10.2333 0.353502
\(839\) 0.290286 0.0100218 0.00501090 0.999987i \(-0.498405\pi\)
0.00501090 + 0.999987i \(0.498405\pi\)
\(840\) 0 0
\(841\) 22.3190 0.769620
\(842\) 9.82856 0.338715
\(843\) 0 0
\(844\) 24.3584 0.838450
\(845\) 0 0
\(846\) 0 0
\(847\) 8.45401 0.290483
\(848\) 11.9367 0.409909
\(849\) 0 0
\(850\) 0 0
\(851\) −12.3105 −0.421999
\(852\) 0 0
\(853\) 13.6725 0.468139 0.234070 0.972220i \(-0.424796\pi\)
0.234070 + 0.972220i \(0.424796\pi\)
\(854\) 14.6711 0.502034
\(855\) 0 0
\(856\) −7.20235 −0.246171
\(857\) 48.9569 1.67234 0.836169 0.548472i \(-0.184790\pi\)
0.836169 + 0.548472i \(0.184790\pi\)
\(858\) 0 0
\(859\) −0.985272 −0.0336170 −0.0168085 0.999859i \(-0.505351\pi\)
−0.0168085 + 0.999859i \(0.505351\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −36.0449 −1.22769
\(863\) −9.31273 −0.317009 −0.158504 0.987358i \(-0.550667\pi\)
−0.158504 + 0.987358i \(0.550667\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 19.2881 0.655435
\(867\) 0 0
\(868\) −0.0695349 −0.00236017
\(869\) −28.8124 −0.977393
\(870\) 0 0
\(871\) 78.4095 2.65680
\(872\) −7.04710 −0.238645
\(873\) 0 0
\(874\) 1.86719 0.0631585
\(875\) 0 0
\(876\) 0 0
\(877\) −31.7469 −1.07202 −0.536008 0.844213i \(-0.680068\pi\)
−0.536008 + 0.844213i \(0.680068\pi\)
\(878\) 24.4194 0.824116
\(879\) 0 0
\(880\) 0 0
\(881\) −8.78147 −0.295855 −0.147928 0.988998i \(-0.547260\pi\)
−0.147928 + 0.988998i \(0.547260\pi\)
\(882\) 0 0
\(883\) −16.9044 −0.568877 −0.284438 0.958694i \(-0.591807\pi\)
−0.284438 + 0.958694i \(0.591807\pi\)
\(884\) −7.82009 −0.263018
\(885\) 0 0
\(886\) −13.8834 −0.466421
\(887\) 17.6851 0.593806 0.296903 0.954908i \(-0.404046\pi\)
0.296903 + 0.954908i \(0.404046\pi\)
\(888\) 0 0
\(889\) −8.39699 −0.281626
\(890\) 0 0
\(891\) 0 0
\(892\) −20.1946 −0.676167
\(893\) −12.4132 −0.415391
\(894\) 0 0
\(895\) 0 0
\(896\) −1.47645 −0.0493248
\(897\) 0 0
\(898\) −13.6016 −0.453890
\(899\) −0.337382 −0.0112523
\(900\) 0 0
\(901\) 17.6240 0.587140
\(902\) −0.413172 −0.0137571
\(903\) 0 0
\(904\) −20.1004 −0.668531
\(905\) 0 0
\(906\) 0 0
\(907\) 46.0210 1.52810 0.764051 0.645156i \(-0.223208\pi\)
0.764051 + 0.645156i \(0.223208\pi\)
\(908\) 6.35982 0.211058
\(909\) 0 0
\(910\) 0 0
\(911\) −10.1560 −0.336484 −0.168242 0.985746i \(-0.553809\pi\)
−0.168242 + 0.985746i \(0.553809\pi\)
\(912\) 0 0
\(913\) 14.0471 0.464891
\(914\) −25.5051 −0.843635
\(915\) 0 0
\(916\) −11.8510 −0.391568
\(917\) 18.0287 0.595360
\(918\) 0 0
\(919\) −40.0820 −1.32218 −0.661092 0.750305i \(-0.729907\pi\)
−0.661092 + 0.750305i \(0.729907\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −16.3598 −0.538782
\(923\) 74.5199 2.45285
\(924\) 0 0
\(925\) 0 0
\(926\) −14.7815 −0.485749
\(927\) 0 0
\(928\) −7.16373 −0.235161
\(929\) −29.2634 −0.960101 −0.480051 0.877241i \(-0.659382\pi\)
−0.480051 + 0.877241i \(0.659382\pi\)
\(930\) 0 0
\(931\) 4.82009 0.157972
\(932\) −27.6155 −0.904576
\(933\) 0 0
\(934\) −13.4055 −0.438640
\(935\) 0 0
\(936\) 0 0
\(937\) 26.4989 0.865681 0.432841 0.901471i \(-0.357511\pi\)
0.432841 + 0.901471i \(0.357511\pi\)
\(938\) 21.8573 0.713665
\(939\) 0 0
\(940\) 0 0
\(941\) 19.8749 0.647903 0.323952 0.946074i \(-0.394989\pi\)
0.323952 + 0.946074i \(0.394989\pi\)
\(942\) 0 0
\(943\) 0.335926 0.0109393
\(944\) −6.34364 −0.206468
\(945\) 0 0
\(946\) 18.3352 0.596128
\(947\) −31.0766 −1.00985 −0.504926 0.863163i \(-0.668480\pi\)
−0.504926 + 0.863163i \(0.668480\pi\)
\(948\) 0 0
\(949\) 76.5894 2.48620
\(950\) 0 0
\(951\) 0 0
\(952\) −2.17991 −0.0706513
\(953\) −17.3275 −0.561291 −0.280646 0.959811i \(-0.590549\pi\)
−0.280646 + 0.959811i \(0.590549\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −28.2248 −0.912855
\(957\) 0 0
\(958\) 37.0534 1.19714
\(959\) 12.1560 0.392538
\(960\) 0 0
\(961\) −30.9978 −0.999928
\(962\) −34.9205 −1.12588
\(963\) 0 0
\(964\) 7.38226 0.237767
\(965\) 0 0
\(966\) 0 0
\(967\) −12.4989 −0.401937 −0.200969 0.979598i \(-0.564409\pi\)
−0.200969 + 0.979598i \(0.564409\pi\)
\(968\) −5.72590 −0.184037
\(969\) 0 0
\(970\) 0 0
\(971\) 50.3113 1.61457 0.807283 0.590165i \(-0.200937\pi\)
0.807283 + 0.590165i \(0.200937\pi\)
\(972\) 0 0
\(973\) 19.4926 0.624905
\(974\) −13.0471 −0.418056
\(975\) 0 0
\(976\) −9.93672 −0.318067
\(977\) −30.9121 −0.988965 −0.494482 0.869188i \(-0.664642\pi\)
−0.494482 + 0.869188i \(0.664642\pi\)
\(978\) 0 0
\(979\) 14.8363 0.474169
\(980\) 0 0
\(981\) 0 0
\(982\) 30.7322 0.980702
\(983\) 40.9128 1.30492 0.652458 0.757825i \(-0.273738\pi\)
0.652458 + 0.757825i \(0.273738\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −10.5769 −0.336837
\(987\) 0 0
\(988\) 5.29654 0.168505
\(989\) −14.9073 −0.474023
\(990\) 0 0
\(991\) 4.25937 0.135303 0.0676517 0.997709i \(-0.478449\pi\)
0.0676517 + 0.997709i \(0.478449\pi\)
\(992\) 0.0470959 0.00149530
\(993\) 0 0
\(994\) 20.7730 0.658879
\(995\) 0 0
\(996\) 0 0
\(997\) 50.6070 1.60274 0.801371 0.598168i \(-0.204104\pi\)
0.801371 + 0.598168i \(0.204104\pi\)
\(998\) −17.3414 −0.548933
\(999\) 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 8550.2.a.cm.1.2 yes 3
3.2 odd 2 8550.2.a.cc.1.2 3
5.4 even 2 8550.2.a.ch.1.2 yes 3
15.14 odd 2 8550.2.a.ct.1.2 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8550.2.a.cc.1.2 3 3.2 odd 2
8550.2.a.ch.1.2 yes 3 5.4 even 2
8550.2.a.cm.1.2 yes 3 1.1 even 1 trivial
8550.2.a.ct.1.2 yes 3 15.14 odd 2