Properties

Label 8550.2.a.cm.1.1
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 1.1
Root \(-1.87740\) of defining polynomial
Character \(\chi\) \(=\) 8550.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -2.87740 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -2.87740 q^{7} +1.00000 q^{8} -2.40205 q^{11} -0.597951 q^{13} -2.87740 q^{14} +1.00000 q^{16} +2.87740 q^{17} -1.00000 q^{19} -2.40205 q^{22} +7.03426 q^{23} -0.597951 q^{26} -2.87740 q^{28} +6.43631 q^{29} -2.75481 q^{31} +1.00000 q^{32} +2.87740 q^{34} -2.80410 q^{37} -1.00000 q^{38} -6.27945 q^{41} -11.7158 q^{43} -2.40205 q^{44} +7.03426 q^{46} -3.08355 q^{47} +1.27945 q^{49} -0.597951 q^{52} -4.96095 q^{53} -2.87740 q^{56} +6.43631 q^{58} +1.15686 q^{59} +6.96095 q^{61} -2.75481 q^{62} +1.00000 q^{64} +10.9952 q^{67} +2.87740 q^{68} -6.07331 q^{71} +3.83836 q^{73} -2.80410 q^{74} -1.00000 q^{76} +6.91166 q^{77} -5.95071 q^{79} -6.27945 q^{82} -4.68150 q^{83} -11.7158 q^{86} -2.40205 q^{88} -11.8384 q^{89} +1.72055 q^{91} +7.03426 q^{92} -3.08355 q^{94} +8.63221 q^{97} +1.27945 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} + 3q^{4} - 2q^{7} + 3q^{8} + O(q^{10}) \) \( 3q + 3q^{2} + 3q^{4} - 2q^{7} + 3q^{8} - 5q^{11} - 4q^{13} - 2q^{14} + 3q^{16} + 2q^{17} - 3q^{19} - 5q^{22} - q^{23} - 4q^{26} - 2q^{28} - 5q^{29} + 5q^{31} + 3q^{32} + 2q^{34} - 4q^{37} - 3q^{38} - 10q^{41} - 2q^{43} - 5q^{44} - q^{46} + 4q^{47} - 5q^{49} - 4q^{52} + 5q^{53} - 2q^{56} - 5q^{58} - 12q^{59} + q^{61} + 5q^{62} + 3q^{64} - 9q^{67} + 2q^{68} - 16q^{71} - 15q^{73} - 4q^{74} - 3q^{76} - 8q^{77} - 9q^{79} - 10q^{82} - 3q^{83} - 2q^{86} - 5q^{88} - 9q^{89} + 14q^{91} - q^{92} + 4q^{94} + 6q^{97} - 5q^{98} + O(q^{100}) \)

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
\(7\) −2.87740 −1.08756 −0.543778 0.839229i \(-0.683007\pi\)
−0.543778 + 0.839229i \(0.683007\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) −2.40205 −0.724245 −0.362122 0.932131i \(-0.617948\pi\)
−0.362122 + 0.932131i \(0.617948\pi\)
\(12\) 0 0
\(13\) −0.597951 −0.165842 −0.0829209 0.996556i \(-0.526425\pi\)
−0.0829209 + 0.996556i \(0.526425\pi\)
\(14\) −2.87740 −0.769018
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.87740 0.697873 0.348936 0.937146i \(-0.386543\pi\)
0.348936 + 0.937146i \(0.386543\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) −2.40205 −0.512118
\(23\) 7.03426 1.46674 0.733372 0.679827i \(-0.237945\pi\)
0.733372 + 0.679827i \(0.237945\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −0.597951 −0.117268
\(27\) 0 0
\(28\) −2.87740 −0.543778
\(29\) 6.43631 1.19519 0.597596 0.801797i \(-0.296123\pi\)
0.597596 + 0.801797i \(0.296123\pi\)
\(30\) 0 0
\(31\) −2.75481 −0.494778 −0.247389 0.968916i \(-0.579573\pi\)
−0.247389 + 0.968916i \(0.579573\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 2.87740 0.493471
\(35\) 0 0
\(36\) 0 0
\(37\) −2.80410 −0.460991 −0.230495 0.973073i \(-0.574035\pi\)
−0.230495 + 0.973073i \(0.574035\pi\)
\(38\) −1.00000 −0.162221
\(39\) 0 0
\(40\) 0 0
\(41\) −6.27945 −0.980686 −0.490343 0.871530i \(-0.663128\pi\)
−0.490343 + 0.871530i \(0.663128\pi\)
\(42\) 0 0
\(43\) −11.7158 −1.78664 −0.893318 0.449424i \(-0.851629\pi\)
−0.893318 + 0.449424i \(0.851629\pi\)
\(44\) −2.40205 −0.362122
\(45\) 0 0
\(46\) 7.03426 1.03715
\(47\) −3.08355 −0.449782 −0.224891 0.974384i \(-0.572203\pi\)
−0.224891 + 0.974384i \(0.572203\pi\)
\(48\) 0 0
\(49\) 1.27945 0.182779
\(50\) 0 0
\(51\) 0 0
\(52\) −0.597951 −0.0829209
\(53\) −4.96095 −0.681439 −0.340720 0.940165i \(-0.610671\pi\)
−0.340720 + 0.940165i \(0.610671\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −2.87740 −0.384509
\(57\) 0 0
\(58\) 6.43631 0.845129
\(59\) 1.15686 0.150610 0.0753049 0.997161i \(-0.476007\pi\)
0.0753049 + 0.997161i \(0.476007\pi\)
\(60\) 0 0
\(61\) 6.96095 0.891259 0.445629 0.895218i \(-0.352980\pi\)
0.445629 + 0.895218i \(0.352980\pi\)
\(62\) −2.75481 −0.349861
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 10.9952 1.34328 0.671640 0.740878i \(-0.265590\pi\)
0.671640 + 0.740878i \(0.265590\pi\)
\(68\) 2.87740 0.348936
\(69\) 0 0
\(70\) 0 0
\(71\) −6.07331 −0.720769 −0.360384 0.932804i \(-0.617354\pi\)
−0.360384 + 0.932804i \(0.617354\pi\)
\(72\) 0 0
\(73\) 3.83836 0.449246 0.224623 0.974446i \(-0.427885\pi\)
0.224623 + 0.974446i \(0.427885\pi\)
\(74\) −2.80410 −0.325970
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) 6.91166 0.787657
\(78\) 0 0
\(79\) −5.95071 −0.669507 −0.334754 0.942306i \(-0.608653\pi\)
−0.334754 + 0.942306i \(0.608653\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −6.27945 −0.693450
\(83\) −4.68150 −0.513861 −0.256931 0.966430i \(-0.582711\pi\)
−0.256931 + 0.966430i \(0.582711\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −11.7158 −1.26334
\(87\) 0 0
\(88\) −2.40205 −0.256059
\(89\) −11.8384 −1.25486 −0.627432 0.778672i \(-0.715894\pi\)
−0.627432 + 0.778672i \(0.715894\pi\)
\(90\) 0 0
\(91\) 1.72055 0.180362
\(92\) 7.03426 0.733372
\(93\) 0 0
\(94\) −3.08355 −0.318044
\(95\) 0 0
\(96\) 0 0
\(97\) 8.63221 0.876468 0.438234 0.898861i \(-0.355604\pi\)
0.438234 + 0.898861i \(0.355604\pi\)
\(98\) 1.27945 0.129244
\(99\) 0 0
\(100\) 0 0
\(101\) −8.59795 −0.855528 −0.427764 0.903890i \(-0.640699\pi\)
−0.427764 + 0.903890i \(0.640699\pi\)
\(102\) 0 0
\(103\) −3.52464 −0.347294 −0.173647 0.984808i \(-0.555555\pi\)
−0.173647 + 0.984808i \(0.555555\pi\)
\(104\) −0.597951 −0.0586340
\(105\) 0 0
\(106\) −4.96095 −0.481850
\(107\) −8.10757 −0.783788 −0.391894 0.920010i \(-0.628180\pi\)
−0.391894 + 0.920010i \(0.628180\pi\)
\(108\) 0 0
\(109\) −4.24519 −0.406616 −0.203308 0.979115i \(-0.565169\pi\)
−0.203308 + 0.979115i \(0.565169\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −2.87740 −0.271889
\(113\) 10.3973 0.978092 0.489046 0.872258i \(-0.337345\pi\)
0.489046 + 0.872258i \(0.337345\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 6.43631 0.597596
\(117\) 0 0
\(118\) 1.15686 0.106497
\(119\) −8.27945 −0.758976
\(120\) 0 0
\(121\) −5.23016 −0.475469
\(122\) 6.96095 0.630215
\(123\) 0 0
\(124\) −2.75481 −0.247389
\(125\) 0 0
\(126\) 0 0
\(127\) −9.31371 −0.826458 −0.413229 0.910627i \(-0.635599\pi\)
−0.413229 + 0.910627i \(0.635599\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) 4.19112 0.366179 0.183090 0.983096i \(-0.441390\pi\)
0.183090 + 0.983096i \(0.441390\pi\)
\(132\) 0 0
\(133\) 2.87740 0.249503
\(134\) 10.9952 0.949842
\(135\) 0 0
\(136\) 2.87740 0.246735
\(137\) 13.3630 1.14168 0.570839 0.821062i \(-0.306618\pi\)
0.570839 + 0.821062i \(0.306618\pi\)
\(138\) 0 0
\(139\) −14.1076 −1.19659 −0.598294 0.801277i \(-0.704155\pi\)
−0.598294 + 0.801277i \(0.704155\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −6.07331 −0.509661
\(143\) 1.43631 0.120110
\(144\) 0 0
\(145\) 0 0
\(146\) 3.83836 0.317665
\(147\) 0 0
\(148\) −2.80410 −0.230495
\(149\) −15.2644 −1.25051 −0.625255 0.780420i \(-0.715005\pi\)
−0.625255 + 0.780420i \(0.715005\pi\)
\(150\) 0 0
\(151\) 12.9850 1.05670 0.528351 0.849026i \(-0.322811\pi\)
0.528351 + 0.849026i \(0.322811\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 0 0
\(154\) 6.91166 0.556958
\(155\) 0 0
\(156\) 0 0
\(157\) 6.52464 0.520723 0.260362 0.965511i \(-0.416158\pi\)
0.260362 + 0.965511i \(0.416158\pi\)
\(158\) −5.95071 −0.473413
\(159\) 0 0
\(160\) 0 0
\(161\) −20.2404 −1.59517
\(162\) 0 0
\(163\) −6.00000 −0.469956 −0.234978 0.972001i \(-0.575502\pi\)
−0.234978 + 0.972001i \(0.575502\pi\)
\(164\) −6.27945 −0.490343
\(165\) 0 0
\(166\) −4.68150 −0.363355
\(167\) −15.6767 −1.21310 −0.606550 0.795045i \(-0.707447\pi\)
−0.606550 + 0.795045i \(0.707447\pi\)
\(168\) 0 0
\(169\) −12.6425 −0.972496
\(170\) 0 0
\(171\) 0 0
\(172\) −11.7158 −0.893318
\(173\) −17.9459 −1.36440 −0.682202 0.731164i \(-0.738977\pi\)
−0.682202 + 0.731164i \(0.738977\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −2.40205 −0.181061
\(177\) 0 0
\(178\) −11.8384 −0.887322
\(179\) 2.70552 0.202220 0.101110 0.994875i \(-0.467761\pi\)
0.101110 + 0.994875i \(0.467761\pi\)
\(180\) 0 0
\(181\) −12.2747 −0.912369 −0.456184 0.889885i \(-0.650784\pi\)
−0.456184 + 0.889885i \(0.650784\pi\)
\(182\) 1.72055 0.127535
\(183\) 0 0
\(184\) 7.03426 0.518573
\(185\) 0 0
\(186\) 0 0
\(187\) −6.91166 −0.505431
\(188\) −3.08355 −0.224891
\(189\) 0 0
\(190\) 0 0
\(191\) −9.80410 −0.709400 −0.354700 0.934980i \(-0.615417\pi\)
−0.354700 + 0.934980i \(0.615417\pi\)
\(192\) 0 0
\(193\) −22.7993 −1.64113 −0.820565 0.571553i \(-0.806341\pi\)
−0.820565 + 0.571553i \(0.806341\pi\)
\(194\) 8.63221 0.619757
\(195\) 0 0
\(196\) 1.27945 0.0913895
\(197\) −0.352759 −0.0251330 −0.0125665 0.999921i \(-0.504000\pi\)
−0.0125665 + 0.999921i \(0.504000\pi\)
\(198\) 0 0
\(199\) −24.7260 −1.75278 −0.876390 0.481602i \(-0.840055\pi\)
−0.876390 + 0.481602i \(0.840055\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −8.59795 −0.604950
\(203\) −18.5199 −1.29984
\(204\) 0 0
\(205\) 0 0
\(206\) −3.52464 −0.245574
\(207\) 0 0
\(208\) −0.597951 −0.0414605
\(209\) 2.40205 0.166153
\(210\) 0 0
\(211\) −25.3432 −1.74470 −0.872348 0.488885i \(-0.837404\pi\)
−0.872348 + 0.488885i \(0.837404\pi\)
\(212\) −4.96095 −0.340720
\(213\) 0 0
\(214\) −8.10757 −0.554222
\(215\) 0 0
\(216\) 0 0
\(217\) 7.92669 0.538099
\(218\) −4.24519 −0.287521
\(219\) 0 0
\(220\) 0 0
\(221\) −1.72055 −0.115737
\(222\) 0 0
\(223\) 15.9069 1.06520 0.532602 0.846366i \(-0.321214\pi\)
0.532602 + 0.846366i \(0.321214\pi\)
\(224\) −2.87740 −0.192255
\(225\) 0 0
\(226\) 10.3973 0.691616
\(227\) 18.5589 1.23180 0.615899 0.787825i \(-0.288793\pi\)
0.615899 + 0.787825i \(0.288793\pi\)
\(228\) 0 0
\(229\) 16.7500 1.10687 0.553436 0.832892i \(-0.313316\pi\)
0.553436 + 0.832892i \(0.313316\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 6.43631 0.422564
\(233\) −13.0240 −0.853232 −0.426616 0.904433i \(-0.640294\pi\)
−0.426616 + 0.904433i \(0.640294\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 1.15686 0.0753049
\(237\) 0 0
\(238\) −8.27945 −0.536677
\(239\) −23.9357 −1.54827 −0.774135 0.633020i \(-0.781815\pi\)
−0.774135 + 0.633020i \(0.781815\pi\)
\(240\) 0 0
\(241\) 14.3870 0.926749 0.463375 0.886162i \(-0.346638\pi\)
0.463375 + 0.886162i \(0.346638\pi\)
\(242\) −5.23016 −0.336208
\(243\) 0 0
\(244\) 6.96095 0.445629
\(245\) 0 0
\(246\) 0 0
\(247\) 0.597951 0.0380467
\(248\) −2.75481 −0.174930
\(249\) 0 0
\(250\) 0 0
\(251\) −20.7398 −1.30908 −0.654542 0.756026i \(-0.727138\pi\)
−0.654542 + 0.756026i \(0.727138\pi\)
\(252\) 0 0
\(253\) −16.8966 −1.06228
\(254\) −9.31371 −0.584394
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 17.0343 1.06257 0.531284 0.847194i \(-0.321710\pi\)
0.531284 + 0.847194i \(0.321710\pi\)
\(258\) 0 0
\(259\) 8.06852 0.501353
\(260\) 0 0
\(261\) 0 0
\(262\) 4.19112 0.258928
\(263\) −13.5246 −0.833965 −0.416983 0.908914i \(-0.636912\pi\)
−0.416983 + 0.908914i \(0.636912\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 2.87740 0.176425
\(267\) 0 0
\(268\) 10.9952 0.671640
\(269\) −0.804097 −0.0490267 −0.0245133 0.999700i \(-0.507804\pi\)
−0.0245133 + 0.999700i \(0.507804\pi\)
\(270\) 0 0
\(271\) 25.1911 1.53025 0.765126 0.643881i \(-0.222677\pi\)
0.765126 + 0.643881i \(0.222677\pi\)
\(272\) 2.87740 0.174468
\(273\) 0 0
\(274\) 13.3630 0.807288
\(275\) 0 0
\(276\) 0 0
\(277\) 16.3035 0.979581 0.489790 0.871840i \(-0.337073\pi\)
0.489790 + 0.871840i \(0.337073\pi\)
\(278\) −14.1076 −0.846116
\(279\) 0 0
\(280\) 0 0
\(281\) −14.1959 −0.846857 −0.423428 0.905930i \(-0.639173\pi\)
−0.423428 + 0.905930i \(0.639173\pi\)
\(282\) 0 0
\(283\) −6.95071 −0.413177 −0.206588 0.978428i \(-0.566236\pi\)
−0.206588 + 0.978428i \(0.566236\pi\)
\(284\) −6.07331 −0.360384
\(285\) 0 0
\(286\) 1.43631 0.0849307
\(287\) 18.0685 1.06655
\(288\) 0 0
\(289\) −8.72055 −0.512973
\(290\) 0 0
\(291\) 0 0
\(292\) 3.83836 0.224623
\(293\) −16.3035 −0.952459 −0.476229 0.879321i \(-0.657997\pi\)
−0.476229 + 0.879321i \(0.657997\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −2.80410 −0.162985
\(297\) 0 0
\(298\) −15.2644 −0.884244
\(299\) −4.20615 −0.243248
\(300\) 0 0
\(301\) 33.7110 1.94307
\(302\) 12.9850 0.747201
\(303\) 0 0
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) 0 0
\(307\) 5.73079 0.327073 0.163537 0.986537i \(-0.447710\pi\)
0.163537 + 0.986537i \(0.447710\pi\)
\(308\) 6.91166 0.393829
\(309\) 0 0
\(310\) 0 0
\(311\) 3.23016 0.183166 0.0915829 0.995797i \(-0.470807\pi\)
0.0915829 + 0.995797i \(0.470807\pi\)
\(312\) 0 0
\(313\) −12.2987 −0.695163 −0.347581 0.937650i \(-0.612997\pi\)
−0.347581 + 0.937650i \(0.612997\pi\)
\(314\) 6.52464 0.368207
\(315\) 0 0
\(316\) −5.95071 −0.334754
\(317\) 13.5637 0.761813 0.380906 0.924614i \(-0.375612\pi\)
0.380906 + 0.924614i \(0.375612\pi\)
\(318\) 0 0
\(319\) −15.4603 −0.865612
\(320\) 0 0
\(321\) 0 0
\(322\) −20.2404 −1.12795
\(323\) −2.87740 −0.160103
\(324\) 0 0
\(325\) 0 0
\(326\) −6.00000 −0.332309
\(327\) 0 0
\(328\) −6.27945 −0.346725
\(329\) 8.87262 0.489163
\(330\) 0 0
\(331\) 22.9952 1.26393 0.631966 0.774996i \(-0.282248\pi\)
0.631966 + 0.774996i \(0.282248\pi\)
\(332\) −4.68150 −0.256931
\(333\) 0 0
\(334\) −15.6767 −0.857792
\(335\) 0 0
\(336\) 0 0
\(337\) 15.2644 0.831506 0.415753 0.909478i \(-0.363518\pi\)
0.415753 + 0.909478i \(0.363518\pi\)
\(338\) −12.6425 −0.687659
\(339\) 0 0
\(340\) 0 0
\(341\) 6.61718 0.358340
\(342\) 0 0
\(343\) 16.4603 0.888774
\(344\) −11.7158 −0.631671
\(345\) 0 0
\(346\) −17.9459 −0.964779
\(347\) 8.41229 0.451595 0.225798 0.974174i \(-0.427501\pi\)
0.225798 + 0.974174i \(0.427501\pi\)
\(348\) 0 0
\(349\) 27.7993 1.48806 0.744031 0.668145i \(-0.232911\pi\)
0.744031 + 0.668145i \(0.232911\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −2.40205 −0.128030
\(353\) 11.3582 0.604537 0.302268 0.953223i \(-0.402256\pi\)
0.302268 + 0.953223i \(0.402256\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −11.8384 −0.627432
\(357\) 0 0
\(358\) 2.70552 0.142991
\(359\) −8.91645 −0.470592 −0.235296 0.971924i \(-0.575606\pi\)
−0.235296 + 0.971924i \(0.575606\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −12.2747 −0.645142
\(363\) 0 0
\(364\) 1.72055 0.0901812
\(365\) 0 0
\(366\) 0 0
\(367\) 2.07331 0.108226 0.0541129 0.998535i \(-0.482767\pi\)
0.0541129 + 0.998535i \(0.482767\pi\)
\(368\) 7.03426 0.366686
\(369\) 0 0
\(370\) 0 0
\(371\) 14.2747 0.741104
\(372\) 0 0
\(373\) 4.51986 0.234029 0.117015 0.993130i \(-0.462668\pi\)
0.117015 + 0.993130i \(0.462668\pi\)
\(374\) −6.91166 −0.357394
\(375\) 0 0
\(376\) −3.08355 −0.159022
\(377\) −3.84860 −0.198213
\(378\) 0 0
\(379\) −7.29448 −0.374692 −0.187346 0.982294i \(-0.559989\pi\)
−0.187346 + 0.982294i \(0.559989\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −9.80410 −0.501621
\(383\) −25.6034 −1.30827 −0.654136 0.756377i \(-0.726968\pi\)
−0.654136 + 0.756377i \(0.726968\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −22.7993 −1.16045
\(387\) 0 0
\(388\) 8.63221 0.438234
\(389\) 7.36300 0.373319 0.186660 0.982425i \(-0.440234\pi\)
0.186660 + 0.982425i \(0.440234\pi\)
\(390\) 0 0
\(391\) 20.2404 1.02360
\(392\) 1.27945 0.0646221
\(393\) 0 0
\(394\) −0.352759 −0.0177717
\(395\) 0 0
\(396\) 0 0
\(397\) 11.4994 0.577137 0.288568 0.957459i \(-0.406821\pi\)
0.288568 + 0.957459i \(0.406821\pi\)
\(398\) −24.7260 −1.23940
\(399\) 0 0
\(400\) 0 0
\(401\) 7.95071 0.397040 0.198520 0.980097i \(-0.436387\pi\)
0.198520 + 0.980097i \(0.436387\pi\)
\(402\) 0 0
\(403\) 1.64724 0.0820549
\(404\) −8.59795 −0.427764
\(405\) 0 0
\(406\) −18.5199 −0.919125
\(407\) 6.73558 0.333870
\(408\) 0 0
\(409\) 9.19590 0.454708 0.227354 0.973812i \(-0.426993\pi\)
0.227354 + 0.973812i \(0.426993\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −3.52464 −0.173647
\(413\) −3.32874 −0.163797
\(414\) 0 0
\(415\) 0 0
\(416\) −0.597951 −0.0293170
\(417\) 0 0
\(418\) 2.40205 0.117488
\(419\) −11.3630 −0.555119 −0.277559 0.960708i \(-0.589526\pi\)
−0.277559 + 0.960708i \(0.589526\pi\)
\(420\) 0 0
\(421\) −13.5781 −0.661758 −0.330879 0.943673i \(-0.607345\pi\)
−0.330879 + 0.943673i \(0.607345\pi\)
\(422\) −25.3432 −1.23369
\(423\) 0 0
\(424\) −4.96095 −0.240925
\(425\) 0 0
\(426\) 0 0
\(427\) −20.0295 −0.969294
\(428\) −8.10757 −0.391894
\(429\) 0 0
\(430\) 0 0
\(431\) −25.6562 −1.23582 −0.617909 0.786250i \(-0.712020\pi\)
−0.617909 + 0.786250i \(0.712020\pi\)
\(432\) 0 0
\(433\) 31.8966 1.53285 0.766427 0.642331i \(-0.222033\pi\)
0.766427 + 0.642331i \(0.222033\pi\)
\(434\) 7.92669 0.380493
\(435\) 0 0
\(436\) −4.24519 −0.203308
\(437\) −7.03426 −0.336494
\(438\) 0 0
\(439\) −15.9712 −0.762264 −0.381132 0.924521i \(-0.624466\pi\)
−0.381132 + 0.924521i \(0.624466\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −1.72055 −0.0818381
\(443\) −24.6815 −1.17265 −0.586327 0.810075i \(-0.699427\pi\)
−0.586327 + 0.810075i \(0.699427\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 15.9069 0.753212
\(447\) 0 0
\(448\) −2.87740 −0.135945
\(449\) 13.1028 0.618358 0.309179 0.951004i \(-0.399946\pi\)
0.309179 + 0.951004i \(0.399946\pi\)
\(450\) 0 0
\(451\) 15.0835 0.710257
\(452\) 10.3973 0.489046
\(453\) 0 0
\(454\) 18.5589 0.871013
\(455\) 0 0
\(456\) 0 0
\(457\) 3.18213 0.148854 0.0744269 0.997226i \(-0.476287\pi\)
0.0744269 + 0.997226i \(0.476287\pi\)
\(458\) 16.7500 0.782677
\(459\) 0 0
\(460\) 0 0
\(461\) −28.5589 −1.33012 −0.665060 0.746790i \(-0.731594\pi\)
−0.665060 + 0.746790i \(0.731594\pi\)
\(462\) 0 0
\(463\) 5.82333 0.270633 0.135316 0.990802i \(-0.456795\pi\)
0.135316 + 0.990802i \(0.456795\pi\)
\(464\) 6.43631 0.298798
\(465\) 0 0
\(466\) −13.0240 −0.603326
\(467\) 39.0980 1.80924 0.904620 0.426220i \(-0.140155\pi\)
0.904620 + 0.426220i \(0.140155\pi\)
\(468\) 0 0
\(469\) −31.6377 −1.46089
\(470\) 0 0
\(471\) 0 0
\(472\) 1.15686 0.0532486
\(473\) 28.1418 1.29396
\(474\) 0 0
\(475\) 0 0
\(476\) −8.27945 −0.379488
\(477\) 0 0
\(478\) −23.9357 −1.09479
\(479\) 9.35755 0.427557 0.213779 0.976882i \(-0.431423\pi\)
0.213779 + 0.976882i \(0.431423\pi\)
\(480\) 0 0
\(481\) 1.67671 0.0764516
\(482\) 14.3870 0.655311
\(483\) 0 0
\(484\) −5.23016 −0.237735
\(485\) 0 0
\(486\) 0 0
\(487\) −10.2452 −0.464254 −0.232127 0.972685i \(-0.574568\pi\)
−0.232127 + 0.972685i \(0.574568\pi\)
\(488\) 6.96095 0.315108
\(489\) 0 0
\(490\) 0 0
\(491\) 5.34252 0.241104 0.120552 0.992707i \(-0.461533\pi\)
0.120552 + 0.992707i \(0.461533\pi\)
\(492\) 0 0
\(493\) 18.5199 0.834092
\(494\) 0.597951 0.0269031
\(495\) 0 0
\(496\) −2.75481 −0.123695
\(497\) 17.4754 0.783877
\(498\) 0 0
\(499\) −2.25418 −0.100911 −0.0504555 0.998726i \(-0.516067\pi\)
−0.0504555 + 0.998726i \(0.516067\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −20.7398 −0.925662
\(503\) −16.0000 −0.713405 −0.356702 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −16.8966 −0.751147
\(507\) 0 0
\(508\) −9.31371 −0.413229
\(509\) 7.49937 0.332404 0.166202 0.986092i \(-0.446850\pi\)
0.166202 + 0.986092i \(0.446850\pi\)
\(510\) 0 0
\(511\) −11.0445 −0.488580
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 17.0343 0.751349
\(515\) 0 0
\(516\) 0 0
\(517\) 7.40684 0.325752
\(518\) 8.06852 0.354510
\(519\) 0 0
\(520\) 0 0
\(521\) −27.7740 −1.21680 −0.608401 0.793630i \(-0.708189\pi\)
−0.608401 + 0.793630i \(0.708189\pi\)
\(522\) 0 0
\(523\) 9.19590 0.402109 0.201054 0.979580i \(-0.435563\pi\)
0.201054 + 0.979580i \(0.435563\pi\)
\(524\) 4.19112 0.183090
\(525\) 0 0
\(526\) −13.5246 −0.589703
\(527\) −7.92669 −0.345292
\(528\) 0 0
\(529\) 26.4808 1.15134
\(530\) 0 0
\(531\) 0 0
\(532\) 2.87740 0.124751
\(533\) 3.75481 0.162639
\(534\) 0 0
\(535\) 0 0
\(536\) 10.9952 0.474921
\(537\) 0 0
\(538\) −0.804097 −0.0346671
\(539\) −3.07331 −0.132377
\(540\) 0 0
\(541\) 7.46511 0.320950 0.160475 0.987040i \(-0.448697\pi\)
0.160475 + 0.987040i \(0.448697\pi\)
\(542\) 25.1911 1.08205
\(543\) 0 0
\(544\) 2.87740 0.123368
\(545\) 0 0
\(546\) 0 0
\(547\) 8.96095 0.383143 0.191571 0.981479i \(-0.438642\pi\)
0.191571 + 0.981479i \(0.438642\pi\)
\(548\) 13.3630 0.570839
\(549\) 0 0
\(550\) 0 0
\(551\) −6.43631 −0.274196
\(552\) 0 0
\(553\) 17.1226 0.728127
\(554\) 16.3035 0.692668
\(555\) 0 0
\(556\) −14.1076 −0.598294
\(557\) −20.1761 −0.854888 −0.427444 0.904042i \(-0.640586\pi\)
−0.427444 + 0.904042i \(0.640586\pi\)
\(558\) 0 0
\(559\) 7.00546 0.296299
\(560\) 0 0
\(561\) 0 0
\(562\) −14.1959 −0.598818
\(563\) 12.3732 0.521470 0.260735 0.965410i \(-0.416035\pi\)
0.260735 + 0.965410i \(0.416035\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −6.95071 −0.292160
\(567\) 0 0
\(568\) −6.07331 −0.254830
\(569\) 7.68629 0.322226 0.161113 0.986936i \(-0.448492\pi\)
0.161113 + 0.986936i \(0.448492\pi\)
\(570\) 0 0
\(571\) 0.843144 0.0352845 0.0176422 0.999844i \(-0.494384\pi\)
0.0176422 + 0.999844i \(0.494384\pi\)
\(572\) 1.43631 0.0600551
\(573\) 0 0
\(574\) 18.0685 0.754165
\(575\) 0 0
\(576\) 0 0
\(577\) 0.539675 0.0224670 0.0112335 0.999937i \(-0.496424\pi\)
0.0112335 + 0.999937i \(0.496424\pi\)
\(578\) −8.72055 −0.362727
\(579\) 0 0
\(580\) 0 0
\(581\) 13.4706 0.558853
\(582\) 0 0
\(583\) 11.9165 0.493529
\(584\) 3.83836 0.158832
\(585\) 0 0
\(586\) −16.3035 −0.673490
\(587\) 36.0925 1.48970 0.744849 0.667233i \(-0.232521\pi\)
0.744849 + 0.667233i \(0.232521\pi\)
\(588\) 0 0
\(589\) 2.75481 0.113510
\(590\) 0 0
\(591\) 0 0
\(592\) −2.80410 −0.115248
\(593\) −39.1082 −1.60598 −0.802991 0.595991i \(-0.796760\pi\)
−0.802991 + 0.595991i \(0.796760\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −15.2644 −0.625255
\(597\) 0 0
\(598\) −4.20615 −0.172002
\(599\) −21.4267 −0.875473 −0.437736 0.899103i \(-0.644220\pi\)
−0.437736 + 0.899103i \(0.644220\pi\)
\(600\) 0 0
\(601\) −25.8486 −1.05439 −0.527193 0.849745i \(-0.676756\pi\)
−0.527193 + 0.849745i \(0.676756\pi\)
\(602\) 33.7110 1.37396
\(603\) 0 0
\(604\) 12.9850 0.528351
\(605\) 0 0
\(606\) 0 0
\(607\) −40.0535 −1.62572 −0.812860 0.582458i \(-0.802091\pi\)
−0.812860 + 0.582458i \(0.802091\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 0 0
\(610\) 0 0
\(611\) 1.84381 0.0745927
\(612\) 0 0
\(613\) 0.323286 0.0130574 0.00652870 0.999979i \(-0.497922\pi\)
0.00652870 + 0.999979i \(0.497922\pi\)
\(614\) 5.73079 0.231276
\(615\) 0 0
\(616\) 6.91166 0.278479
\(617\) 35.0397 1.41065 0.705323 0.708886i \(-0.250802\pi\)
0.705323 + 0.708886i \(0.250802\pi\)
\(618\) 0 0
\(619\) −2.13763 −0.0859184 −0.0429592 0.999077i \(-0.513679\pi\)
−0.0429592 + 0.999077i \(0.513679\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 3.23016 0.129518
\(623\) 34.0637 1.36473
\(624\) 0 0
\(625\) 0 0
\(626\) −12.2987 −0.491554
\(627\) 0 0
\(628\) 6.52464 0.260362
\(629\) −8.06852 −0.321713
\(630\) 0 0
\(631\) −4.23562 −0.168617 −0.0843087 0.996440i \(-0.526868\pi\)
−0.0843087 + 0.996440i \(0.526868\pi\)
\(632\) −5.95071 −0.236707
\(633\) 0 0
\(634\) 13.5637 0.538683
\(635\) 0 0
\(636\) 0 0
\(637\) −0.765050 −0.0303124
\(638\) −15.4603 −0.612080
\(639\) 0 0
\(640\) 0 0
\(641\) −16.5932 −0.655391 −0.327695 0.944783i \(-0.606272\pi\)
−0.327695 + 0.944783i \(0.606272\pi\)
\(642\) 0 0
\(643\) −33.5301 −1.32230 −0.661149 0.750255i \(-0.729931\pi\)
−0.661149 + 0.750255i \(0.729931\pi\)
\(644\) −20.2404 −0.797584
\(645\) 0 0
\(646\) −2.87740 −0.113210
\(647\) −32.7891 −1.28907 −0.644536 0.764574i \(-0.722949\pi\)
−0.644536 + 0.764574i \(0.722949\pi\)
\(648\) 0 0
\(649\) −2.77882 −0.109078
\(650\) 0 0
\(651\) 0 0
\(652\) −6.00000 −0.234978
\(653\) 18.6274 0.728947 0.364474 0.931214i \(-0.381249\pi\)
0.364474 + 0.931214i \(0.381249\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −6.27945 −0.245171
\(657\) 0 0
\(658\) 8.87262 0.345891
\(659\) 14.4213 0.561773 0.280887 0.959741i \(-0.409372\pi\)
0.280887 + 0.959741i \(0.409372\pi\)
\(660\) 0 0
\(661\) 12.3822 0.481613 0.240806 0.970573i \(-0.422588\pi\)
0.240806 + 0.970573i \(0.422588\pi\)
\(662\) 22.9952 0.893734
\(663\) 0 0
\(664\) −4.68150 −0.181677
\(665\) 0 0
\(666\) 0 0
\(667\) 45.2747 1.75304
\(668\) −15.6767 −0.606550
\(669\) 0 0
\(670\) 0 0
\(671\) −16.7205 −0.645490
\(672\) 0 0
\(673\) 17.8281 0.687223 0.343612 0.939112i \(-0.388350\pi\)
0.343612 + 0.939112i \(0.388350\pi\)
\(674\) 15.2644 0.587964
\(675\) 0 0
\(676\) −12.6425 −0.486248
\(677\) −43.6364 −1.67708 −0.838542 0.544837i \(-0.816591\pi\)
−0.838542 + 0.544837i \(0.816591\pi\)
\(678\) 0 0
\(679\) −24.8384 −0.953209
\(680\) 0 0
\(681\) 0 0
\(682\) 6.61718 0.253385
\(683\) −16.6665 −0.637725 −0.318862 0.947801i \(-0.603301\pi\)
−0.318862 + 0.947801i \(0.603301\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 16.4603 0.628458
\(687\) 0 0
\(688\) −11.7158 −0.446659
\(689\) 2.96641 0.113011
\(690\) 0 0
\(691\) −11.1364 −0.423647 −0.211824 0.977308i \(-0.567940\pi\)
−0.211824 + 0.977308i \(0.567940\pi\)
\(692\) −17.9459 −0.682202
\(693\) 0 0
\(694\) 8.41229 0.319326
\(695\) 0 0
\(696\) 0 0
\(697\) −18.0685 −0.684394
\(698\) 27.7993 1.05222
\(699\) 0 0
\(700\) 0 0
\(701\) −24.1671 −0.912779 −0.456389 0.889780i \(-0.650858\pi\)
−0.456389 + 0.889780i \(0.650858\pi\)
\(702\) 0 0
\(703\) 2.80410 0.105759
\(704\) −2.40205 −0.0905306
\(705\) 0 0
\(706\) 11.3582 0.427472
\(707\) 24.7398 0.930435
\(708\) 0 0
\(709\) 5.95550 0.223663 0.111832 0.993727i \(-0.464328\pi\)
0.111832 + 0.993727i \(0.464328\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −11.8384 −0.443661
\(713\) −19.3780 −0.725713
\(714\) 0 0
\(715\) 0 0
\(716\) 2.70552 0.101110
\(717\) 0 0
\(718\) −8.91645 −0.332759
\(719\) 28.5000 1.06287 0.531436 0.847098i \(-0.321653\pi\)
0.531436 + 0.847098i \(0.321653\pi\)
\(720\) 0 0
\(721\) 10.1418 0.377701
\(722\) 1.00000 0.0372161
\(723\) 0 0
\(724\) −12.2747 −0.456184
\(725\) 0 0
\(726\) 0 0
\(727\) 44.4808 1.64970 0.824851 0.565350i \(-0.191259\pi\)
0.824851 + 0.565350i \(0.191259\pi\)
\(728\) 1.72055 0.0637677
\(729\) 0 0
\(730\) 0 0
\(731\) −33.7110 −1.24685
\(732\) 0 0
\(733\) 17.3390 0.640430 0.320215 0.947345i \(-0.396245\pi\)
0.320215 + 0.947345i \(0.396245\pi\)
\(734\) 2.07331 0.0765271
\(735\) 0 0
\(736\) 7.03426 0.259286
\(737\) −26.4110 −0.972863
\(738\) 0 0
\(739\) −24.7055 −0.908807 −0.454404 0.890796i \(-0.650148\pi\)
−0.454404 + 0.890796i \(0.650148\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 14.2747 0.524039
\(743\) 36.6959 1.34624 0.673122 0.739532i \(-0.264953\pi\)
0.673122 + 0.739532i \(0.264953\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 4.51986 0.165484
\(747\) 0 0
\(748\) −6.91166 −0.252715
\(749\) 23.3287 0.852414
\(750\) 0 0
\(751\) 7.85759 0.286727 0.143364 0.989670i \(-0.454208\pi\)
0.143364 + 0.989670i \(0.454208\pi\)
\(752\) −3.08355 −0.112445
\(753\) 0 0
\(754\) −3.84860 −0.140158
\(755\) 0 0
\(756\) 0 0
\(757\) 38.6034 1.40306 0.701532 0.712638i \(-0.252500\pi\)
0.701532 + 0.712638i \(0.252500\pi\)
\(758\) −7.29448 −0.264948
\(759\) 0 0
\(760\) 0 0
\(761\) 2.02527 0.0734161 0.0367080 0.999326i \(-0.488313\pi\)
0.0367080 + 0.999326i \(0.488313\pi\)
\(762\) 0 0
\(763\) 12.2151 0.442217
\(764\) −9.80410 −0.354700
\(765\) 0 0
\(766\) −25.6034 −0.925089
\(767\) −0.691744 −0.0249774
\(768\) 0 0
\(769\) 37.7302 1.36059 0.680293 0.732940i \(-0.261853\pi\)
0.680293 + 0.732940i \(0.261853\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −22.7993 −0.820565
\(773\) 22.3617 0.804296 0.402148 0.915575i \(-0.368264\pi\)
0.402148 + 0.915575i \(0.368264\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 8.63221 0.309878
\(777\) 0 0
\(778\) 7.36300 0.263976
\(779\) 6.27945 0.224985
\(780\) 0 0
\(781\) 14.5884 0.522013
\(782\) 20.2404 0.723795
\(783\) 0 0
\(784\) 1.27945 0.0456947
\(785\) 0 0
\(786\) 0 0
\(787\) 23.5336 0.838883 0.419442 0.907782i \(-0.362226\pi\)
0.419442 + 0.907782i \(0.362226\pi\)
\(788\) −0.352759 −0.0125665
\(789\) 0 0
\(790\) 0 0
\(791\) −29.9171 −1.06373
\(792\) 0 0
\(793\) −4.16231 −0.147808
\(794\) 11.4994 0.408097
\(795\) 0 0
\(796\) −24.7260 −0.876390
\(797\) 18.3275 0.649193 0.324596 0.945853i \(-0.394772\pi\)
0.324596 + 0.945853i \(0.394772\pi\)
\(798\) 0 0
\(799\) −8.87262 −0.313891
\(800\) 0 0
\(801\) 0 0
\(802\) 7.95071 0.280749
\(803\) −9.21992 −0.325364
\(804\) 0 0
\(805\) 0 0
\(806\) 1.64724 0.0580216
\(807\) 0 0
\(808\) −8.59795 −0.302475
\(809\) −53.6719 −1.88700 −0.943502 0.331366i \(-0.892490\pi\)
−0.943502 + 0.331366i \(0.892490\pi\)
\(810\) 0 0
\(811\) −0.548662 −0.0192661 −0.00963306 0.999954i \(-0.503066\pi\)
−0.00963306 + 0.999954i \(0.503066\pi\)
\(812\) −18.5199 −0.649920
\(813\) 0 0
\(814\) 6.73558 0.236082
\(815\) 0 0
\(816\) 0 0
\(817\) 11.7158 0.409883
\(818\) 9.19590 0.321527
\(819\) 0 0
\(820\) 0 0
\(821\) 17.4706 0.609727 0.304864 0.952396i \(-0.401389\pi\)
0.304864 + 0.952396i \(0.401389\pi\)
\(822\) 0 0
\(823\) −19.0192 −0.662969 −0.331484 0.943461i \(-0.607549\pi\)
−0.331484 + 0.943461i \(0.607549\pi\)
\(824\) −3.52464 −0.122787
\(825\) 0 0
\(826\) −3.32874 −0.115822
\(827\) −36.4898 −1.26887 −0.634437 0.772974i \(-0.718768\pi\)
−0.634437 + 0.772974i \(0.718768\pi\)
\(828\) 0 0
\(829\) −21.4911 −0.746415 −0.373208 0.927748i \(-0.621742\pi\)
−0.373208 + 0.927748i \(0.621742\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −0.597951 −0.0207302
\(833\) 3.68150 0.127556
\(834\) 0 0
\(835\) 0 0
\(836\) 2.40205 0.0830766
\(837\) 0 0
\(838\) −11.3630 −0.392528
\(839\) 20.4856 0.707241 0.353621 0.935389i \(-0.384950\pi\)
0.353621 + 0.935389i \(0.384950\pi\)
\(840\) 0 0
\(841\) 12.4261 0.428485
\(842\) −13.5781 −0.467933
\(843\) 0 0
\(844\) −25.3432 −0.872348
\(845\) 0 0
\(846\) 0 0
\(847\) 15.0493 0.517100
\(848\) −4.96095 −0.170360
\(849\) 0 0
\(850\) 0 0
\(851\) −19.7247 −0.676156
\(852\) 0 0
\(853\) 40.8726 1.39945 0.699726 0.714411i \(-0.253305\pi\)
0.699726 + 0.714411i \(0.253305\pi\)
\(854\) −20.0295 −0.685394
\(855\) 0 0
\(856\) −8.10757 −0.277111
\(857\) 19.2782 0.658531 0.329265 0.944237i \(-0.393199\pi\)
0.329265 + 0.944237i \(0.393199\pi\)
\(858\) 0 0
\(859\) −43.1863 −1.47350 −0.736749 0.676166i \(-0.763640\pi\)
−0.736749 + 0.676166i \(0.763640\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −25.6562 −0.873855
\(863\) −24.3137 −0.827648 −0.413824 0.910357i \(-0.635807\pi\)
−0.413824 + 0.910357i \(0.635807\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 31.8966 1.08389
\(867\) 0 0
\(868\) 7.92669 0.269050
\(869\) 14.2939 0.484887
\(870\) 0 0
\(871\) −6.57460 −0.222772
\(872\) −4.24519 −0.143760
\(873\) 0 0
\(874\) −7.03426 −0.237937
\(875\) 0 0
\(876\) 0 0
\(877\) 35.8438 1.21036 0.605180 0.796089i \(-0.293101\pi\)
0.605180 + 0.796089i \(0.293101\pi\)
\(878\) −15.9712 −0.539002
\(879\) 0 0
\(880\) 0 0
\(881\) 11.8233 0.398338 0.199169 0.979965i \(-0.436176\pi\)
0.199169 + 0.979965i \(0.436176\pi\)
\(882\) 0 0
\(883\) 39.3925 1.32566 0.662831 0.748769i \(-0.269355\pi\)
0.662831 + 0.748769i \(0.269355\pi\)
\(884\) −1.72055 −0.0578683
\(885\) 0 0
\(886\) −24.6815 −0.829191
\(887\) −4.90268 −0.164616 −0.0823079 0.996607i \(-0.526229\pi\)
−0.0823079 + 0.996607i \(0.526229\pi\)
\(888\) 0 0
\(889\) 26.7993 0.898820
\(890\) 0 0
\(891\) 0 0
\(892\) 15.9069 0.532602
\(893\) 3.08355 0.103187
\(894\) 0 0
\(895\) 0 0
\(896\) −2.87740 −0.0961273
\(897\) 0 0
\(898\) 13.1028 0.437245
\(899\) −17.7308 −0.591355
\(900\) 0 0
\(901\) −14.2747 −0.475558
\(902\) 15.0835 0.502227
\(903\) 0 0
\(904\) 10.3973 0.345808
\(905\) 0 0
\(906\) 0 0
\(907\) −21.0740 −0.699750 −0.349875 0.936796i \(-0.613776\pi\)
−0.349875 + 0.936796i \(0.613776\pi\)
\(908\) 18.5589 0.615899
\(909\) 0 0
\(910\) 0 0
\(911\) 40.4508 1.34019 0.670097 0.742274i \(-0.266253\pi\)
0.670097 + 0.742274i \(0.266253\pi\)
\(912\) 0 0
\(913\) 11.2452 0.372162
\(914\) 3.18213 0.105255
\(915\) 0 0
\(916\) 16.7500 0.553436
\(917\) −12.0595 −0.398241
\(918\) 0 0
\(919\) 17.7020 0.583935 0.291967 0.956428i \(-0.405690\pi\)
0.291967 + 0.956428i \(0.405690\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −28.5589 −0.940537
\(923\) 3.63154 0.119534
\(924\) 0 0
\(925\) 0 0
\(926\) 5.82333 0.191366
\(927\) 0 0
\(928\) 6.43631 0.211282
\(929\) −39.4796 −1.29528 −0.647641 0.761946i \(-0.724244\pi\)
−0.647641 + 0.761946i \(0.724244\pi\)
\(930\) 0 0
\(931\) −1.27945 −0.0419324
\(932\) −13.0240 −0.426616
\(933\) 0 0
\(934\) 39.0980 1.27933
\(935\) 0 0
\(936\) 0 0
\(937\) 22.7055 0.741757 0.370878 0.928681i \(-0.379057\pi\)
0.370878 + 0.928681i \(0.379057\pi\)
\(938\) −31.6377 −1.03301
\(939\) 0 0
\(940\) 0 0
\(941\) 47.9802 1.56411 0.782055 0.623210i \(-0.214172\pi\)
0.782055 + 0.623210i \(0.214172\pi\)
\(942\) 0 0
\(943\) −44.1713 −1.43842
\(944\) 1.15686 0.0376525
\(945\) 0 0
\(946\) 28.1418 0.914970
\(947\) 56.1275 1.82390 0.911949 0.410304i \(-0.134577\pi\)
0.911949 + 0.410304i \(0.134577\pi\)
\(948\) 0 0
\(949\) −2.29515 −0.0745038
\(950\) 0 0
\(951\) 0 0
\(952\) −8.27945 −0.268339
\(953\) 9.87262 0.319805 0.159903 0.987133i \(-0.448882\pi\)
0.159903 + 0.987133i \(0.448882\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −23.9357 −0.774135
\(957\) 0 0
\(958\) 9.35755 0.302329
\(959\) −38.4508 −1.24164
\(960\) 0 0
\(961\) −23.4110 −0.755195
\(962\) 1.67671 0.0540594
\(963\) 0 0
\(964\) 14.3870 0.463375
\(965\) 0 0
\(966\) 0 0
\(967\) −8.70552 −0.279951 −0.139975 0.990155i \(-0.544702\pi\)
−0.139975 + 0.990155i \(0.544702\pi\)
\(968\) −5.23016 −0.168104
\(969\) 0 0
\(970\) 0 0
\(971\) 3.41162 0.109484 0.0547421 0.998501i \(-0.482566\pi\)
0.0547421 + 0.998501i \(0.482566\pi\)
\(972\) 0 0
\(973\) 40.5932 1.30136
\(974\) −10.2452 −0.328277
\(975\) 0 0
\(976\) 6.96095 0.222815
\(977\) −11.6220 −0.371820 −0.185910 0.982567i \(-0.559523\pi\)
−0.185910 + 0.982567i \(0.559523\pi\)
\(978\) 0 0
\(979\) 28.4363 0.908828
\(980\) 0 0
\(981\) 0 0
\(982\) 5.34252 0.170487
\(983\) −32.6912 −1.04269 −0.521343 0.853347i \(-0.674569\pi\)
−0.521343 + 0.853347i \(0.674569\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 18.5199 0.589792
\(987\) 0 0
\(988\) 0.597951 0.0190234
\(989\) −82.4117 −2.62054
\(990\) 0 0
\(991\) 46.9562 1.49161 0.745806 0.666163i \(-0.232065\pi\)
0.745806 + 0.666163i \(0.232065\pi\)
\(992\) −2.75481 −0.0874652
\(993\) 0 0
\(994\) 17.4754 0.554285
\(995\) 0 0
\(996\) 0 0
\(997\) 53.3227 1.68875 0.844373 0.535755i \(-0.179973\pi\)
0.844373 + 0.535755i \(0.179973\pi\)
\(998\) −2.25418 −0.0713548
\(999\) 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.1.1 yes 3
3.2 odd 2 8550.2.a.cc.1.1 3
5.4 even 2 8550.2.a.ch.1.3 yes 3
15.14 odd 2 8550.2.a.ct.1.3 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8550.2.a.cc.1.1 3 3.2 odd 2
8550.2.a.ch.1.3 yes 3 5.4 even 2
8550.2.a.cm.1.1 yes 3 1.1 even 1 trivial
8550.2.a.ct.1.3 yes 3 15.14 odd 2