Properties

Label 8550.2.a.cg.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: 3.3.564.1
Defining polynomial: \(x^{3} - x^{2} - 5 x + 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.08613\) of defining polynomial
Character \(\chi\) \(=\) 8550.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -2.43807 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -2.43807 q^{7} -1.00000 q^{8} -4.08613 q^{11} +3.79001 q^{13} +2.43807 q^{14} +1.00000 q^{16} +3.73419 q^{17} +1.00000 q^{19} +4.08613 q^{22} +0.351939 q^{23} -3.79001 q^{26} -2.43807 q^{28} -6.73419 q^{29} +9.34452 q^{31} -1.00000 q^{32} -3.73419 q^{34} -6.17226 q^{37} -1.00000 q^{38} -4.05582 q^{41} +0.913870 q^{43} -4.08613 q^{44} -0.351939 q^{46} -5.52420 q^{47} -1.05582 q^{49} +3.79001 q^{52} +6.96227 q^{53} +2.43807 q^{56} +6.73419 q^{58} +14.3068 q^{59} -13.1345 q^{61} -9.34452 q^{62} +1.00000 q^{64} -7.61033 q^{67} +3.73419 q^{68} -9.73419 q^{71} +11.9926 q^{73} +6.17226 q^{74} +1.00000 q^{76} +9.96227 q^{77} -3.17226 q^{79} +4.05582 q^{82} +2.90645 q^{83} -0.913870 q^{86} +4.08613 q^{88} -15.2887 q^{89} -9.24030 q^{91} +0.351939 q^{92} +5.52420 q^{94} +16.0787 q^{97} +1.05582 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 3 q^{4} + 2 q^{7} - 3 q^{8} + O(q^{10}) \) \( 3 q - 3 q^{2} + 3 q^{4} + 2 q^{7} - 3 q^{8} - 5 q^{11} - 2 q^{14} + 3 q^{16} + 6 q^{17} + 3 q^{19} + 5 q^{22} - q^{23} + 2 q^{28} - 15 q^{29} - q^{31} - 3 q^{32} - 6 q^{34} - 4 q^{37} - 3 q^{38} - 6 q^{41} + 10 q^{43} - 5 q^{44} + q^{46} + 3 q^{49} - 5 q^{53} - 2 q^{56} + 15 q^{58} - 12 q^{59} + q^{61} + q^{62} + 3 q^{64} + q^{67} + 6 q^{68} - 24 q^{71} + 9 q^{73} + 4 q^{74} + 3 q^{76} + 4 q^{77} + 5 q^{79} + 6 q^{82} - 11 q^{83} - 10 q^{86} + 5 q^{88} - 23 q^{89} - 38 q^{91} - q^{92} + 14 q^{97} - 3 q^{98} + O(q^{100}) \)

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
\(7\) −2.43807 −0.921504 −0.460752 0.887529i \(-0.652420\pi\)
−0.460752 + 0.887529i \(0.652420\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) −4.08613 −1.23201 −0.616007 0.787740i \(-0.711251\pi\)
−0.616007 + 0.787740i \(0.711251\pi\)
\(12\) 0 0
\(13\) 3.79001 1.05116 0.525580 0.850744i \(-0.323848\pi\)
0.525580 + 0.850744i \(0.323848\pi\)
\(14\) 2.43807 0.651601
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.73419 0.905674 0.452837 0.891593i \(-0.350412\pi\)
0.452837 + 0.891593i \(0.350412\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 4.08613 0.871166
\(23\) 0.351939 0.0733844 0.0366922 0.999327i \(-0.488318\pi\)
0.0366922 + 0.999327i \(0.488318\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −3.79001 −0.743282
\(27\) 0 0
\(28\) −2.43807 −0.460752
\(29\) −6.73419 −1.25051 −0.625254 0.780421i \(-0.715005\pi\)
−0.625254 + 0.780421i \(0.715005\pi\)
\(30\) 0 0
\(31\) 9.34452 1.67833 0.839163 0.543880i \(-0.183045\pi\)
0.839163 + 0.543880i \(0.183045\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −3.73419 −0.640408
\(35\) 0 0
\(36\) 0 0
\(37\) −6.17226 −1.01471 −0.507357 0.861736i \(-0.669377\pi\)
−0.507357 + 0.861736i \(0.669377\pi\)
\(38\) −1.00000 −0.162221
\(39\) 0 0
\(40\) 0 0
\(41\) −4.05582 −0.633412 −0.316706 0.948524i \(-0.602577\pi\)
−0.316706 + 0.948524i \(0.602577\pi\)
\(42\) 0 0
\(43\) 0.913870 0.139364 0.0696819 0.997569i \(-0.477802\pi\)
0.0696819 + 0.997569i \(0.477802\pi\)
\(44\) −4.08613 −0.616007
\(45\) 0 0
\(46\) −0.351939 −0.0518906
\(47\) −5.52420 −0.805787 −0.402894 0.915247i \(-0.631996\pi\)
−0.402894 + 0.915247i \(0.631996\pi\)
\(48\) 0 0
\(49\) −1.05582 −0.150831
\(50\) 0 0
\(51\) 0 0
\(52\) 3.79001 0.525580
\(53\) 6.96227 0.956341 0.478171 0.878267i \(-0.341300\pi\)
0.478171 + 0.878267i \(0.341300\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 2.43807 0.325801
\(57\) 0 0
\(58\) 6.73419 0.884243
\(59\) 14.3068 1.86259 0.931293 0.364272i \(-0.118682\pi\)
0.931293 + 0.364272i \(0.118682\pi\)
\(60\) 0 0
\(61\) −13.1345 −1.68170 −0.840852 0.541265i \(-0.817946\pi\)
−0.840852 + 0.541265i \(0.817946\pi\)
\(62\) −9.34452 −1.18676
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −7.61033 −0.929750 −0.464875 0.885376i \(-0.653901\pi\)
−0.464875 + 0.885376i \(0.653901\pi\)
\(68\) 3.73419 0.452837
\(69\) 0 0
\(70\) 0 0
\(71\) −9.73419 −1.15524 −0.577618 0.816307i \(-0.696018\pi\)
−0.577618 + 0.816307i \(0.696018\pi\)
\(72\) 0 0
\(73\) 11.9926 1.40363 0.701813 0.712361i \(-0.252374\pi\)
0.701813 + 0.712361i \(0.252374\pi\)
\(74\) 6.17226 0.717511
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) 9.96227 1.13531
\(78\) 0 0
\(79\) −3.17226 −0.356907 −0.178454 0.983948i \(-0.557109\pi\)
−0.178454 + 0.983948i \(0.557109\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 4.05582 0.447890
\(83\) 2.90645 0.319024 0.159512 0.987196i \(-0.449008\pi\)
0.159512 + 0.987196i \(0.449008\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −0.913870 −0.0985451
\(87\) 0 0
\(88\) 4.08613 0.435583
\(89\) −15.2887 −1.62060 −0.810300 0.586016i \(-0.800696\pi\)
−0.810300 + 0.586016i \(0.800696\pi\)
\(90\) 0 0
\(91\) −9.24030 −0.968647
\(92\) 0.351939 0.0366922
\(93\) 0 0
\(94\) 5.52420 0.569778
\(95\) 0 0
\(96\) 0 0
\(97\) 16.0787 1.63255 0.816273 0.577666i \(-0.196037\pi\)
0.816273 + 0.577666i \(0.196037\pi\)
\(98\) 1.05582 0.106654
\(99\) 0 0
\(100\) 0 0
\(101\) −6.38225 −0.635058 −0.317529 0.948249i \(-0.602853\pi\)
−0.317529 + 0.948249i \(0.602853\pi\)
\(102\) 0 0
\(103\) 8.35194 0.822941 0.411471 0.911423i \(-0.365015\pi\)
0.411471 + 0.911423i \(0.365015\pi\)
\(104\) −3.79001 −0.371641
\(105\) 0 0
\(106\) −6.96227 −0.676235
\(107\) 17.4307 1.68508 0.842542 0.538630i \(-0.181058\pi\)
0.842542 + 0.538630i \(0.181058\pi\)
\(108\) 0 0
\(109\) 10.8761 1.04175 0.520873 0.853634i \(-0.325607\pi\)
0.520873 + 0.853634i \(0.325607\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −2.43807 −0.230376
\(113\) 1.05582 0.0993230 0.0496615 0.998766i \(-0.484186\pi\)
0.0496615 + 0.998766i \(0.484186\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −6.73419 −0.625254
\(117\) 0 0
\(118\) −14.3068 −1.31705
\(119\) −9.10422 −0.834582
\(120\) 0 0
\(121\) 5.69646 0.517860
\(122\) 13.1345 1.18914
\(123\) 0 0
\(124\) 9.34452 0.839163
\(125\) 0 0
\(126\) 0 0
\(127\) 8.64064 0.766733 0.383367 0.923596i \(-0.374765\pi\)
0.383367 + 0.923596i \(0.374765\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) 0.0303126 0.00264842 0.00132421 0.999999i \(-0.499578\pi\)
0.00132421 + 0.999999i \(0.499578\pi\)
\(132\) 0 0
\(133\) −2.43807 −0.211407
\(134\) 7.61033 0.657432
\(135\) 0 0
\(136\) −3.73419 −0.320204
\(137\) −1.46838 −0.125452 −0.0627262 0.998031i \(-0.519979\pi\)
−0.0627262 + 0.998031i \(0.519979\pi\)
\(138\) 0 0
\(139\) 18.6661 1.58324 0.791621 0.611012i \(-0.209237\pi\)
0.791621 + 0.611012i \(0.209237\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 9.73419 0.816875
\(143\) −15.4865 −1.29504
\(144\) 0 0
\(145\) 0 0
\(146\) −11.9926 −0.992513
\(147\) 0 0
\(148\) −6.17226 −0.507357
\(149\) −14.8761 −1.21870 −0.609350 0.792901i \(-0.708570\pi\)
−0.609350 + 0.792901i \(0.708570\pi\)
\(150\) 0 0
\(151\) 12.5726 1.02314 0.511572 0.859241i \(-0.329063\pi\)
0.511572 + 0.859241i \(0.329063\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 0 0
\(154\) −9.96227 −0.802783
\(155\) 0 0
\(156\) 0 0
\(157\) 8.22808 0.656672 0.328336 0.944561i \(-0.393512\pi\)
0.328336 + 0.944561i \(0.393512\pi\)
\(158\) 3.17226 0.252371
\(159\) 0 0
\(160\) 0 0
\(161\) −0.858052 −0.0676240
\(162\) 0 0
\(163\) 3.40776 0.266916 0.133458 0.991054i \(-0.457392\pi\)
0.133458 + 0.991054i \(0.457392\pi\)
\(164\) −4.05582 −0.316706
\(165\) 0 0
\(166\) −2.90645 −0.225584
\(167\) 18.4562 1.42818 0.714090 0.700054i \(-0.246841\pi\)
0.714090 + 0.700054i \(0.246841\pi\)
\(168\) 0 0
\(169\) 1.36417 0.104936
\(170\) 0 0
\(171\) 0 0
\(172\) 0.913870 0.0696819
\(173\) −25.9549 −1.97331 −0.986655 0.162823i \(-0.947940\pi\)
−0.986655 + 0.162823i \(0.947940\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −4.08613 −0.308004
\(177\) 0 0
\(178\) 15.2887 1.14594
\(179\) −5.23550 −0.391319 −0.195660 0.980672i \(-0.562685\pi\)
−0.195660 + 0.980672i \(0.562685\pi\)
\(180\) 0 0
\(181\) −25.2584 −1.87744 −0.938721 0.344679i \(-0.887988\pi\)
−0.938721 + 0.344679i \(0.887988\pi\)
\(182\) 9.24030 0.684937
\(183\) 0 0
\(184\) −0.351939 −0.0259453
\(185\) 0 0
\(186\) 0 0
\(187\) −15.2584 −1.11580
\(188\) −5.52420 −0.402894
\(189\) 0 0
\(190\) 0 0
\(191\) −3.95160 −0.285928 −0.142964 0.989728i \(-0.545663\pi\)
−0.142964 + 0.989728i \(0.545663\pi\)
\(192\) 0 0
\(193\) −2.43807 −0.175496 −0.0877480 0.996143i \(-0.527967\pi\)
−0.0877480 + 0.996143i \(0.527967\pi\)
\(194\) −16.0787 −1.15438
\(195\) 0 0
\(196\) −1.05582 −0.0754155
\(197\) −13.6029 −0.969167 −0.484584 0.874745i \(-0.661029\pi\)
−0.484584 + 0.874745i \(0.661029\pi\)
\(198\) 0 0
\(199\) 4.81551 0.341363 0.170681 0.985326i \(-0.445403\pi\)
0.170681 + 0.985326i \(0.445403\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 6.38225 0.449054
\(203\) 16.4184 1.15235
\(204\) 0 0
\(205\) 0 0
\(206\) −8.35194 −0.581907
\(207\) 0 0
\(208\) 3.79001 0.262790
\(209\) −4.08613 −0.282644
\(210\) 0 0
\(211\) −18.4307 −1.26882 −0.634409 0.772997i \(-0.718757\pi\)
−0.634409 + 0.772997i \(0.718757\pi\)
\(212\) 6.96227 0.478171
\(213\) 0 0
\(214\) −17.4307 −1.19153
\(215\) 0 0
\(216\) 0 0
\(217\) −22.7826 −1.54658
\(218\) −10.8761 −0.736625
\(219\) 0 0
\(220\) 0 0
\(221\) 14.1526 0.952008
\(222\) 0 0
\(223\) −7.22808 −0.484028 −0.242014 0.970273i \(-0.577808\pi\)
−0.242014 + 0.970273i \(0.577808\pi\)
\(224\) 2.43807 0.162900
\(225\) 0 0
\(226\) −1.05582 −0.0702319
\(227\) −21.3929 −1.41990 −0.709949 0.704253i \(-0.751282\pi\)
−0.709949 + 0.704253i \(0.751282\pi\)
\(228\) 0 0
\(229\) 1.96969 0.130161 0.0650803 0.997880i \(-0.479270\pi\)
0.0650803 + 0.997880i \(0.479270\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 6.73419 0.442121
\(233\) −5.08132 −0.332889 −0.166444 0.986051i \(-0.553229\pi\)
−0.166444 + 0.986051i \(0.553229\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 14.3068 0.931293
\(237\) 0 0
\(238\) 9.10422 0.590139
\(239\) −15.1648 −0.980932 −0.490466 0.871460i \(-0.663173\pi\)
−0.490466 + 0.871460i \(0.663173\pi\)
\(240\) 0 0
\(241\) 5.79482 0.373277 0.186638 0.982429i \(-0.440241\pi\)
0.186638 + 0.982429i \(0.440241\pi\)
\(242\) −5.69646 −0.366182
\(243\) 0 0
\(244\) −13.1345 −0.840852
\(245\) 0 0
\(246\) 0 0
\(247\) 3.79001 0.241152
\(248\) −9.34452 −0.593378
\(249\) 0 0
\(250\) 0 0
\(251\) −16.0558 −1.01343 −0.506717 0.862112i \(-0.669141\pi\)
−0.506717 + 0.862112i \(0.669141\pi\)
\(252\) 0 0
\(253\) −1.43807 −0.0904106
\(254\) −8.64064 −0.542162
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −3.22808 −0.201362 −0.100681 0.994919i \(-0.532102\pi\)
−0.100681 + 0.994919i \(0.532102\pi\)
\(258\) 0 0
\(259\) 15.0484 0.935062
\(260\) 0 0
\(261\) 0 0
\(262\) −0.0303126 −0.00187272
\(263\) −25.5120 −1.57314 −0.786568 0.617504i \(-0.788144\pi\)
−0.786568 + 0.617504i \(0.788144\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 2.43807 0.149488
\(267\) 0 0
\(268\) −7.61033 −0.464875
\(269\) 18.2691 1.11388 0.556942 0.830551i \(-0.311975\pi\)
0.556942 + 0.830551i \(0.311975\pi\)
\(270\) 0 0
\(271\) −24.0787 −1.46268 −0.731339 0.682014i \(-0.761104\pi\)
−0.731339 + 0.682014i \(0.761104\pi\)
\(272\) 3.73419 0.226419
\(273\) 0 0
\(274\) 1.46838 0.0887082
\(275\) 0 0
\(276\) 0 0
\(277\) −3.49389 −0.209927 −0.104964 0.994476i \(-0.533473\pi\)
−0.104964 + 0.994476i \(0.533473\pi\)
\(278\) −18.6661 −1.11952
\(279\) 0 0
\(280\) 0 0
\(281\) −18.5800 −1.10839 −0.554195 0.832387i \(-0.686974\pi\)
−0.554195 + 0.832387i \(0.686974\pi\)
\(282\) 0 0
\(283\) −1.64064 −0.0975261 −0.0487630 0.998810i \(-0.515528\pi\)
−0.0487630 + 0.998810i \(0.515528\pi\)
\(284\) −9.73419 −0.577618
\(285\) 0 0
\(286\) 15.4865 0.915734
\(287\) 9.88836 0.583692
\(288\) 0 0
\(289\) −3.05582 −0.179754
\(290\) 0 0
\(291\) 0 0
\(292\) 11.9926 0.701813
\(293\) −15.8384 −0.925290 −0.462645 0.886544i \(-0.653099\pi\)
−0.462645 + 0.886544i \(0.653099\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 6.17226 0.358755
\(297\) 0 0
\(298\) 14.8761 0.861752
\(299\) 1.33385 0.0771387
\(300\) 0 0
\(301\) −2.22808 −0.128424
\(302\) −12.5726 −0.723472
\(303\) 0 0
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) 0 0
\(307\) −9.32643 −0.532288 −0.266144 0.963933i \(-0.585750\pi\)
−0.266144 + 0.963933i \(0.585750\pi\)
\(308\) 9.96227 0.567653
\(309\) 0 0
\(310\) 0 0
\(311\) −19.6965 −1.11688 −0.558442 0.829544i \(-0.688601\pi\)
−0.558442 + 0.829544i \(0.688601\pi\)
\(312\) 0 0
\(313\) −2.94418 −0.166415 −0.0832075 0.996532i \(-0.526516\pi\)
−0.0832075 + 0.996532i \(0.526516\pi\)
\(314\) −8.22808 −0.464337
\(315\) 0 0
\(316\) −3.17226 −0.178454
\(317\) 18.7342 1.05222 0.526108 0.850417i \(-0.323651\pi\)
0.526108 + 0.850417i \(0.323651\pi\)
\(318\) 0 0
\(319\) 27.5168 1.54064
\(320\) 0 0
\(321\) 0 0
\(322\) 0.858052 0.0478174
\(323\) 3.73419 0.207776
\(324\) 0 0
\(325\) 0 0
\(326\) −3.40776 −0.188738
\(327\) 0 0
\(328\) 4.05582 0.223945
\(329\) 13.4684 0.742536
\(330\) 0 0
\(331\) 15.8942 0.873626 0.436813 0.899552i \(-0.356107\pi\)
0.436813 + 0.899552i \(0.356107\pi\)
\(332\) 2.90645 0.159512
\(333\) 0 0
\(334\) −18.4562 −1.00988
\(335\) 0 0
\(336\) 0 0
\(337\) −21.8129 −1.18822 −0.594112 0.804382i \(-0.702497\pi\)
−0.594112 + 0.804382i \(0.702497\pi\)
\(338\) −1.36417 −0.0742008
\(339\) 0 0
\(340\) 0 0
\(341\) −38.1829 −2.06772
\(342\) 0 0
\(343\) 19.6406 1.06050
\(344\) −0.913870 −0.0492726
\(345\) 0 0
\(346\) 25.9549 1.39534
\(347\) 18.5168 0.994033 0.497016 0.867741i \(-0.334429\pi\)
0.497016 + 0.867741i \(0.334429\pi\)
\(348\) 0 0
\(349\) −35.7678 −1.91460 −0.957302 0.289090i \(-0.906647\pi\)
−0.957302 + 0.289090i \(0.906647\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 4.08613 0.217791
\(353\) −18.5497 −0.987301 −0.493651 0.869660i \(-0.664338\pi\)
−0.493651 + 0.869660i \(0.664338\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −15.2887 −0.810300
\(357\) 0 0
\(358\) 5.23550 0.276705
\(359\) 29.0288 1.53208 0.766040 0.642793i \(-0.222225\pi\)
0.766040 + 0.642793i \(0.222225\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 25.2584 1.32755
\(363\) 0 0
\(364\) −9.24030 −0.484324
\(365\) 0 0
\(366\) 0 0
\(367\) 8.36261 0.436525 0.218262 0.975890i \(-0.429961\pi\)
0.218262 + 0.975890i \(0.429961\pi\)
\(368\) 0.351939 0.0183461
\(369\) 0 0
\(370\) 0 0
\(371\) −16.9745 −0.881272
\(372\) 0 0
\(373\) −33.1223 −1.71501 −0.857504 0.514477i \(-0.827986\pi\)
−0.857504 + 0.514477i \(0.827986\pi\)
\(374\) 15.2584 0.788993
\(375\) 0 0
\(376\) 5.52420 0.284889
\(377\) −25.5226 −1.31448
\(378\) 0 0
\(379\) 37.1090 1.90616 0.953081 0.302715i \(-0.0978929\pi\)
0.953081 + 0.302715i \(0.0978929\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 3.95160 0.202181
\(383\) 17.5981 0.899221 0.449611 0.893225i \(-0.351563\pi\)
0.449611 + 0.893225i \(0.351563\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 2.43807 0.124094
\(387\) 0 0
\(388\) 16.0787 0.816273
\(389\) −8.87614 −0.450038 −0.225019 0.974354i \(-0.572244\pi\)
−0.225019 + 0.974354i \(0.572244\pi\)
\(390\) 0 0
\(391\) 1.31421 0.0664624
\(392\) 1.05582 0.0533268
\(393\) 0 0
\(394\) 13.6029 0.685305
\(395\) 0 0
\(396\) 0 0
\(397\) −37.1952 −1.86677 −0.933386 0.358875i \(-0.883160\pi\)
−0.933386 + 0.358875i \(0.883160\pi\)
\(398\) −4.81551 −0.241380
\(399\) 0 0
\(400\) 0 0
\(401\) −17.1116 −0.854514 −0.427257 0.904130i \(-0.640520\pi\)
−0.427257 + 0.904130i \(0.640520\pi\)
\(402\) 0 0
\(403\) 35.4158 1.76419
\(404\) −6.38225 −0.317529
\(405\) 0 0
\(406\) −16.4184 −0.814833
\(407\) 25.2207 1.25014
\(408\) 0 0
\(409\) −16.2691 −0.804453 −0.402227 0.915540i \(-0.631764\pi\)
−0.402227 + 0.915540i \(0.631764\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 8.35194 0.411471
\(413\) −34.8809 −1.71638
\(414\) 0 0
\(415\) 0 0
\(416\) −3.79001 −0.185820
\(417\) 0 0
\(418\) 4.08613 0.199859
\(419\) 3.12386 0.152611 0.0763053 0.997085i \(-0.475688\pi\)
0.0763053 + 0.997085i \(0.475688\pi\)
\(420\) 0 0
\(421\) −11.8639 −0.578212 −0.289106 0.957297i \(-0.593358\pi\)
−0.289106 + 0.957297i \(0.593358\pi\)
\(422\) 18.4307 0.897190
\(423\) 0 0
\(424\) −6.96227 −0.338118
\(425\) 0 0
\(426\) 0 0
\(427\) 32.0229 1.54970
\(428\) 17.4307 0.842542
\(429\) 0 0
\(430\) 0 0
\(431\) −5.69165 −0.274157 −0.137079 0.990560i \(-0.543771\pi\)
−0.137079 + 0.990560i \(0.543771\pi\)
\(432\) 0 0
\(433\) 8.36261 0.401881 0.200941 0.979603i \(-0.435600\pi\)
0.200941 + 0.979603i \(0.435600\pi\)
\(434\) 22.7826 1.09360
\(435\) 0 0
\(436\) 10.8761 0.520873
\(437\) 0.351939 0.0168355
\(438\) 0 0
\(439\) 13.9368 0.665165 0.332583 0.943074i \(-0.392080\pi\)
0.332583 + 0.943074i \(0.392080\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −14.1526 −0.673171
\(443\) −32.7194 −1.55454 −0.777272 0.629165i \(-0.783397\pi\)
−0.777272 + 0.629165i \(0.783397\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 7.22808 0.342259
\(447\) 0 0
\(448\) −2.43807 −0.115188
\(449\) −32.0288 −1.51153 −0.755765 0.654843i \(-0.772735\pi\)
−0.755765 + 0.654843i \(0.772735\pi\)
\(450\) 0 0
\(451\) 16.5726 0.780373
\(452\) 1.05582 0.0496615
\(453\) 0 0
\(454\) 21.3929 1.00402
\(455\) 0 0
\(456\) 0 0
\(457\) 4.70869 0.220263 0.110132 0.993917i \(-0.464873\pi\)
0.110132 + 0.993917i \(0.464873\pi\)
\(458\) −1.96969 −0.0920374
\(459\) 0 0
\(460\) 0 0
\(461\) 30.2084 1.40695 0.703474 0.710721i \(-0.251631\pi\)
0.703474 + 0.710721i \(0.251631\pi\)
\(462\) 0 0
\(463\) 5.82774 0.270838 0.135419 0.990788i \(-0.456762\pi\)
0.135419 + 0.990788i \(0.456762\pi\)
\(464\) −6.73419 −0.312627
\(465\) 0 0
\(466\) 5.08132 0.235388
\(467\) −17.0591 −0.789399 −0.394700 0.918810i \(-0.629151\pi\)
−0.394700 + 0.918810i \(0.629151\pi\)
\(468\) 0 0
\(469\) 18.5545 0.856768
\(470\) 0 0
\(471\) 0 0
\(472\) −14.3068 −0.658523
\(473\) −3.73419 −0.171698
\(474\) 0 0
\(475\) 0 0
\(476\) −9.10422 −0.417291
\(477\) 0 0
\(478\) 15.1648 0.693624
\(479\) −4.94418 −0.225905 −0.112953 0.993600i \(-0.536031\pi\)
−0.112953 + 0.993600i \(0.536031\pi\)
\(480\) 0 0
\(481\) −23.3929 −1.06663
\(482\) −5.79482 −0.263947
\(483\) 0 0
\(484\) 5.69646 0.258930
\(485\) 0 0
\(486\) 0 0
\(487\) 17.9097 0.811566 0.405783 0.913969i \(-0.366999\pi\)
0.405783 + 0.913969i \(0.366999\pi\)
\(488\) 13.1345 0.594572
\(489\) 0 0
\(490\) 0 0
\(491\) −25.9852 −1.17269 −0.586347 0.810060i \(-0.699434\pi\)
−0.586347 + 0.810060i \(0.699434\pi\)
\(492\) 0 0
\(493\) −25.1468 −1.13255
\(494\) −3.79001 −0.170521
\(495\) 0 0
\(496\) 9.34452 0.419581
\(497\) 23.7326 1.06455
\(498\) 0 0
\(499\) 3.54229 0.158575 0.0792873 0.996852i \(-0.474736\pi\)
0.0792873 + 0.996852i \(0.474736\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 16.0558 0.716606
\(503\) −7.16003 −0.319250 −0.159625 0.987178i \(-0.551029\pi\)
−0.159625 + 0.987178i \(0.551029\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 1.43807 0.0639300
\(507\) 0 0
\(508\) 8.64064 0.383367
\(509\) −15.9597 −0.707399 −0.353700 0.935359i \(-0.615077\pi\)
−0.353700 + 0.935359i \(0.615077\pi\)
\(510\) 0 0
\(511\) −29.2387 −1.29345
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 3.22808 0.142384
\(515\) 0 0
\(516\) 0 0
\(517\) 22.5726 0.992742
\(518\) −15.0484 −0.661189
\(519\) 0 0
\(520\) 0 0
\(521\) 5.62842 0.246585 0.123293 0.992370i \(-0.460655\pi\)
0.123293 + 0.992370i \(0.460655\pi\)
\(522\) 0 0
\(523\) −34.0213 −1.48765 −0.743825 0.668375i \(-0.766990\pi\)
−0.743825 + 0.668375i \(0.766990\pi\)
\(524\) 0.0303126 0.00132421
\(525\) 0 0
\(526\) 25.5120 1.11237
\(527\) 34.8942 1.52002
\(528\) 0 0
\(529\) −22.8761 −0.994615
\(530\) 0 0
\(531\) 0 0
\(532\) −2.43807 −0.105704
\(533\) −15.3716 −0.665817
\(534\) 0 0
\(535\) 0 0
\(536\) 7.61033 0.328716
\(537\) 0 0
\(538\) −18.2691 −0.787635
\(539\) 4.31421 0.185826
\(540\) 0 0
\(541\) 27.7071 1.19122 0.595611 0.803273i \(-0.296910\pi\)
0.595611 + 0.803273i \(0.296910\pi\)
\(542\) 24.0787 1.03427
\(543\) 0 0
\(544\) −3.73419 −0.160102
\(545\) 0 0
\(546\) 0 0
\(547\) −45.8236 −1.95927 −0.979637 0.200776i \(-0.935654\pi\)
−0.979637 + 0.200776i \(0.935654\pi\)
\(548\) −1.46838 −0.0627262
\(549\) 0 0
\(550\) 0 0
\(551\) −6.73419 −0.286886
\(552\) 0 0
\(553\) 7.73419 0.328891
\(554\) 3.49389 0.148441
\(555\) 0 0
\(556\) 18.6661 0.791621
\(557\) 10.4184 0.441443 0.220721 0.975337i \(-0.429159\pi\)
0.220721 + 0.975337i \(0.429159\pi\)
\(558\) 0 0
\(559\) 3.46357 0.146494
\(560\) 0 0
\(561\) 0 0
\(562\) 18.5800 0.783751
\(563\) 0.899033 0.0378897 0.0189449 0.999821i \(-0.493969\pi\)
0.0189449 + 0.999821i \(0.493969\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 1.64064 0.0689613
\(567\) 0 0
\(568\) 9.73419 0.408438
\(569\) 34.6136 1.45108 0.725538 0.688182i \(-0.241591\pi\)
0.725538 + 0.688182i \(0.241591\pi\)
\(570\) 0 0
\(571\) 14.8990 0.623505 0.311753 0.950163i \(-0.399084\pi\)
0.311753 + 0.950163i \(0.399084\pi\)
\(572\) −15.4865 −0.647522
\(573\) 0 0
\(574\) −9.88836 −0.412732
\(575\) 0 0
\(576\) 0 0
\(577\) −21.1116 −0.878889 −0.439444 0.898270i \(-0.644825\pi\)
−0.439444 + 0.898270i \(0.644825\pi\)
\(578\) 3.05582 0.127105
\(579\) 0 0
\(580\) 0 0
\(581\) −7.08613 −0.293982
\(582\) 0 0
\(583\) −28.4487 −1.17823
\(584\) −11.9926 −0.496257
\(585\) 0 0
\(586\) 15.8384 0.654279
\(587\) 13.6858 0.564873 0.282437 0.959286i \(-0.408857\pi\)
0.282437 + 0.959286i \(0.408857\pi\)
\(588\) 0 0
\(589\) 9.34452 0.385034
\(590\) 0 0
\(591\) 0 0
\(592\) −6.17226 −0.253678
\(593\) 42.7252 1.75451 0.877257 0.480021i \(-0.159371\pi\)
0.877257 + 0.480021i \(0.159371\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −14.8761 −0.609350
\(597\) 0 0
\(598\) −1.33385 −0.0545453
\(599\) −24.5497 −1.00307 −0.501537 0.865136i \(-0.667232\pi\)
−0.501537 + 0.865136i \(0.667232\pi\)
\(600\) 0 0
\(601\) −28.1755 −1.14930 −0.574652 0.818398i \(-0.694862\pi\)
−0.574652 + 0.818398i \(0.694862\pi\)
\(602\) 2.22808 0.0908097
\(603\) 0 0
\(604\) 12.5726 0.511572
\(605\) 0 0
\(606\) 0 0
\(607\) 15.3855 0.624478 0.312239 0.950004i \(-0.398921\pi\)
0.312239 + 0.950004i \(0.398921\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 0 0
\(610\) 0 0
\(611\) −20.9368 −0.847011
\(612\) 0 0
\(613\) 17.3929 0.702493 0.351247 0.936283i \(-0.385758\pi\)
0.351247 + 0.936283i \(0.385758\pi\)
\(614\) 9.32643 0.376384
\(615\) 0 0
\(616\) −9.96227 −0.401391
\(617\) −6.76450 −0.272329 −0.136164 0.990686i \(-0.543478\pi\)
−0.136164 + 0.990686i \(0.543478\pi\)
\(618\) 0 0
\(619\) 25.1223 1.00975 0.504875 0.863192i \(-0.331538\pi\)
0.504875 + 0.863192i \(0.331538\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 19.6965 0.789756
\(623\) 37.2749 1.49339
\(624\) 0 0
\(625\) 0 0
\(626\) 2.94418 0.117673
\(627\) 0 0
\(628\) 8.22808 0.328336
\(629\) −23.0484 −0.919000
\(630\) 0 0
\(631\) 16.9368 0.674242 0.337121 0.941461i \(-0.390547\pi\)
0.337121 + 0.941461i \(0.390547\pi\)
\(632\) 3.17226 0.126186
\(633\) 0 0
\(634\) −18.7342 −0.744030
\(635\) 0 0
\(636\) 0 0
\(637\) −4.00156 −0.158547
\(638\) −27.5168 −1.08940
\(639\) 0 0
\(640\) 0 0
\(641\) 13.0532 0.515571 0.257785 0.966202i \(-0.417007\pi\)
0.257785 + 0.966202i \(0.417007\pi\)
\(642\) 0 0
\(643\) 2.51678 0.0992522 0.0496261 0.998768i \(-0.484197\pi\)
0.0496261 + 0.998768i \(0.484197\pi\)
\(644\) −0.858052 −0.0338120
\(645\) 0 0
\(646\) −3.73419 −0.146920
\(647\) −28.5094 −1.12082 −0.560409 0.828216i \(-0.689356\pi\)
−0.560409 + 0.828216i \(0.689356\pi\)
\(648\) 0 0
\(649\) −58.4594 −2.29473
\(650\) 0 0
\(651\) 0 0
\(652\) 3.40776 0.133458
\(653\) −3.18449 −0.124619 −0.0623093 0.998057i \(-0.519847\pi\)
−0.0623093 + 0.998057i \(0.519847\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −4.05582 −0.158353
\(657\) 0 0
\(658\) −13.4684 −0.525052
\(659\) 25.9984 1.01276 0.506378 0.862312i \(-0.330984\pi\)
0.506378 + 0.862312i \(0.330984\pi\)
\(660\) 0 0
\(661\) −2.65287 −0.103185 −0.0515923 0.998668i \(-0.516430\pi\)
−0.0515923 + 0.998668i \(0.516430\pi\)
\(662\) −15.8942 −0.617747
\(663\) 0 0
\(664\) −2.90645 −0.112792
\(665\) 0 0
\(666\) 0 0
\(667\) −2.37003 −0.0917678
\(668\) 18.4562 0.714090
\(669\) 0 0
\(670\) 0 0
\(671\) 53.6694 2.07188
\(672\) 0 0
\(673\) −7.27803 −0.280548 −0.140274 0.990113i \(-0.544798\pi\)
−0.140274 + 0.990113i \(0.544798\pi\)
\(674\) 21.8129 0.840202
\(675\) 0 0
\(676\) 1.36417 0.0524679
\(677\) 46.9984 1.80630 0.903148 0.429328i \(-0.141250\pi\)
0.903148 + 0.429328i \(0.141250\pi\)
\(678\) 0 0
\(679\) −39.2010 −1.50440
\(680\) 0 0
\(681\) 0 0
\(682\) 38.1829 1.46210
\(683\) 2.03773 0.0779716 0.0389858 0.999240i \(-0.487587\pi\)
0.0389858 + 0.999240i \(0.487587\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −19.6406 −0.749883
\(687\) 0 0
\(688\) 0.913870 0.0348410
\(689\) 26.3871 1.00527
\(690\) 0 0
\(691\) −46.6513 −1.77470 −0.887350 0.461097i \(-0.847456\pi\)
−0.887350 + 0.461097i \(0.847456\pi\)
\(692\) −25.9549 −0.986655
\(693\) 0 0
\(694\) −18.5168 −0.702887
\(695\) 0 0
\(696\) 0 0
\(697\) −15.1452 −0.573665
\(698\) 35.7678 1.35383
\(699\) 0 0
\(700\) 0 0
\(701\) −46.3052 −1.74892 −0.874462 0.485094i \(-0.838786\pi\)
−0.874462 + 0.485094i \(0.838786\pi\)
\(702\) 0 0
\(703\) −6.17226 −0.232791
\(704\) −4.08613 −0.154002
\(705\) 0 0
\(706\) 18.5497 0.698127
\(707\) 15.5604 0.585208
\(708\) 0 0
\(709\) 10.2026 0.383166 0.191583 0.981476i \(-0.438638\pi\)
0.191583 + 0.981476i \(0.438638\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 15.2887 0.572968
\(713\) 3.28870 0.123163
\(714\) 0 0
\(715\) 0 0
\(716\) −5.23550 −0.195660
\(717\) 0 0
\(718\) −29.0288 −1.08334
\(719\) −7.83997 −0.292381 −0.146191 0.989256i \(-0.546701\pi\)
−0.146191 + 0.989256i \(0.546701\pi\)
\(720\) 0 0
\(721\) −20.3626 −0.758343
\(722\) −1.00000 −0.0372161
\(723\) 0 0
\(724\) −25.2584 −0.938721
\(725\) 0 0
\(726\) 0 0
\(727\) −29.3419 −1.08823 −0.544116 0.839010i \(-0.683135\pi\)
−0.544116 + 0.839010i \(0.683135\pi\)
\(728\) 9.24030 0.342468
\(729\) 0 0
\(730\) 0 0
\(731\) 3.41256 0.126218
\(732\) 0 0
\(733\) −35.1903 −1.29979 −0.649893 0.760026i \(-0.725186\pi\)
−0.649893 + 0.760026i \(0.725186\pi\)
\(734\) −8.36261 −0.308669
\(735\) 0 0
\(736\) −0.351939 −0.0129727
\(737\) 31.0968 1.14547
\(738\) 0 0
\(739\) −10.5168 −0.386866 −0.193433 0.981113i \(-0.561962\pi\)
−0.193433 + 0.981113i \(0.561962\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 16.9745 0.623153
\(743\) 13.9852 0.513066 0.256533 0.966535i \(-0.417420\pi\)
0.256533 + 0.966535i \(0.417420\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 33.1223 1.21269
\(747\) 0 0
\(748\) −15.2584 −0.557902
\(749\) −42.4971 −1.55281
\(750\) 0 0
\(751\) −38.9681 −1.42197 −0.710984 0.703209i \(-0.751750\pi\)
−0.710984 + 0.703209i \(0.751750\pi\)
\(752\) −5.52420 −0.201447
\(753\) 0 0
\(754\) 25.5226 0.929480
\(755\) 0 0
\(756\) 0 0
\(757\) −32.5619 −1.18348 −0.591742 0.806128i \(-0.701559\pi\)
−0.591742 + 0.806128i \(0.701559\pi\)
\(758\) −37.1090 −1.34786
\(759\) 0 0
\(760\) 0 0
\(761\) −26.5955 −0.964086 −0.482043 0.876148i \(-0.660105\pi\)
−0.482043 + 0.876148i \(0.660105\pi\)
\(762\) 0 0
\(763\) −26.5168 −0.959972
\(764\) −3.95160 −0.142964
\(765\) 0 0
\(766\) −17.5981 −0.635845
\(767\) 54.2229 1.95787
\(768\) 0 0
\(769\) 4.64545 0.167519 0.0837596 0.996486i \(-0.473307\pi\)
0.0837596 + 0.996486i \(0.473307\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −2.43807 −0.0877480
\(773\) −18.0820 −0.650363 −0.325181 0.945652i \(-0.605425\pi\)
−0.325181 + 0.945652i \(0.605425\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −16.0787 −0.577192
\(777\) 0 0
\(778\) 8.87614 0.318225
\(779\) −4.05582 −0.145315
\(780\) 0 0
\(781\) 39.7752 1.42327
\(782\) −1.31421 −0.0469960
\(783\) 0 0
\(784\) −1.05582 −0.0377078
\(785\) 0 0
\(786\) 0 0
\(787\) −9.07871 −0.323621 −0.161811 0.986822i \(-0.551733\pi\)
−0.161811 + 0.986822i \(0.551733\pi\)
\(788\) −13.6029 −0.484584
\(789\) 0 0
\(790\) 0 0
\(791\) −2.57416 −0.0915265
\(792\) 0 0
\(793\) −49.7800 −1.76774
\(794\) 37.1952 1.32001
\(795\) 0 0
\(796\) 4.81551 0.170681
\(797\) 5.64545 0.199972 0.0999860 0.994989i \(-0.468120\pi\)
0.0999860 + 0.994989i \(0.468120\pi\)
\(798\) 0 0
\(799\) −20.6284 −0.729781
\(800\) 0 0
\(801\) 0 0
\(802\) 17.1116 0.604233
\(803\) −49.0032 −1.72929
\(804\) 0 0
\(805\) 0 0
\(806\) −35.4158 −1.24747
\(807\) 0 0
\(808\) 6.38225 0.224527
\(809\) 33.7555 1.18678 0.593391 0.804915i \(-0.297789\pi\)
0.593391 + 0.804915i \(0.297789\pi\)
\(810\) 0 0
\(811\) −13.3823 −0.469914 −0.234957 0.972006i \(-0.575495\pi\)
−0.234957 + 0.972006i \(0.575495\pi\)
\(812\) 16.4184 0.576174
\(813\) 0 0
\(814\) −25.2207 −0.883984
\(815\) 0 0
\(816\) 0 0
\(817\) 0.913870 0.0319723
\(818\) 16.2691 0.568834
\(819\) 0 0
\(820\) 0 0
\(821\) −41.1829 −1.43729 −0.718647 0.695375i \(-0.755238\pi\)
−0.718647 + 0.695375i \(0.755238\pi\)
\(822\) 0 0
\(823\) −4.00000 −0.139431 −0.0697156 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) −8.35194 −0.290954
\(825\) 0 0
\(826\) 34.8809 1.21366
\(827\) −40.7268 −1.41621 −0.708104 0.706108i \(-0.750449\pi\)
−0.708104 + 0.706108i \(0.750449\pi\)
\(828\) 0 0
\(829\) 23.9016 0.830138 0.415069 0.909790i \(-0.363757\pi\)
0.415069 + 0.909790i \(0.363757\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 3.79001 0.131395
\(833\) −3.94262 −0.136604
\(834\) 0 0
\(835\) 0 0
\(836\) −4.08613 −0.141322
\(837\) 0 0
\(838\) −3.12386 −0.107912
\(839\) −22.0639 −0.761730 −0.380865 0.924631i \(-0.624374\pi\)
−0.380865 + 0.924631i \(0.624374\pi\)
\(840\) 0 0
\(841\) 16.3493 0.563770
\(842\) 11.8639 0.408857
\(843\) 0 0
\(844\) −18.4307 −0.634409
\(845\) 0 0
\(846\) 0 0
\(847\) −13.8884 −0.477210
\(848\) 6.96227 0.239085
\(849\) 0 0
\(850\) 0 0
\(851\) −2.17226 −0.0744641
\(852\) 0 0
\(853\) −20.7793 −0.711471 −0.355736 0.934587i \(-0.615770\pi\)
−0.355736 + 0.934587i \(0.615770\pi\)
\(854\) −32.0229 −1.09580
\(855\) 0 0
\(856\) −17.4307 −0.595767
\(857\) −13.9655 −0.477053 −0.238527 0.971136i \(-0.576664\pi\)
−0.238527 + 0.971136i \(0.576664\pi\)
\(858\) 0 0
\(859\) −21.0484 −0.718162 −0.359081 0.933306i \(-0.616910\pi\)
−0.359081 + 0.933306i \(0.616910\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 5.69165 0.193858
\(863\) 57.2058 1.94731 0.973654 0.228029i \(-0.0732280\pi\)
0.973654 + 0.228029i \(0.0732280\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −8.36261 −0.284173
\(867\) 0 0
\(868\) −22.7826 −0.773291
\(869\) 12.9623 0.439715
\(870\) 0 0
\(871\) −28.8432 −0.977315
\(872\) −10.8761 −0.368313
\(873\) 0 0
\(874\) −0.351939 −0.0119045
\(875\) 0 0
\(876\) 0 0
\(877\) −2.47099 −0.0834395 −0.0417197 0.999129i \(-0.513284\pi\)
−0.0417197 + 0.999129i \(0.513284\pi\)
\(878\) −13.9368 −0.470343
\(879\) 0 0
\(880\) 0 0
\(881\) 13.9245 0.469130 0.234565 0.972100i \(-0.424633\pi\)
0.234565 + 0.972100i \(0.424633\pi\)
\(882\) 0 0
\(883\) 14.3823 0.484001 0.242001 0.970276i \(-0.422196\pi\)
0.242001 + 0.970276i \(0.422196\pi\)
\(884\) 14.1526 0.476004
\(885\) 0 0
\(886\) 32.7194 1.09923
\(887\) −24.7645 −0.831511 −0.415755 0.909477i \(-0.636483\pi\)
−0.415755 + 0.909477i \(0.636483\pi\)
\(888\) 0 0
\(889\) −21.0665 −0.706547
\(890\) 0 0
\(891\) 0 0
\(892\) −7.22808 −0.242014
\(893\) −5.52420 −0.184860
\(894\) 0 0
\(895\) 0 0
\(896\) 2.43807 0.0814502
\(897\) 0 0
\(898\) 32.0288 1.06881
\(899\) −62.9278 −2.09876
\(900\) 0 0
\(901\) 25.9984 0.866134
\(902\) −16.5726 −0.551807
\(903\) 0 0
\(904\) −1.05582 −0.0351160
\(905\) 0 0
\(906\) 0 0
\(907\) −4.16745 −0.138378 −0.0691890 0.997604i \(-0.522041\pi\)
−0.0691890 + 0.997604i \(0.522041\pi\)
\(908\) −21.3929 −0.709949
\(909\) 0 0
\(910\) 0 0
\(911\) 27.4535 0.909577 0.454788 0.890600i \(-0.349715\pi\)
0.454788 + 0.890600i \(0.349715\pi\)
\(912\) 0 0
\(913\) −11.8761 −0.393043
\(914\) −4.70869 −0.155749
\(915\) 0 0
\(916\) 1.96969 0.0650803
\(917\) −0.0739042 −0.00244053
\(918\) 0 0
\(919\) 23.0303 0.759700 0.379850 0.925048i \(-0.375976\pi\)
0.379850 + 0.925048i \(0.375976\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −30.2084 −0.994862
\(923\) −36.8927 −1.21434
\(924\) 0 0
\(925\) 0 0
\(926\) −5.82774 −0.191511
\(927\) 0 0
\(928\) 6.73419 0.221061
\(929\) 38.1116 1.25040 0.625201 0.780464i \(-0.285017\pi\)
0.625201 + 0.780464i \(0.285017\pi\)
\(930\) 0 0
\(931\) −1.05582 −0.0346030
\(932\) −5.08132 −0.166444
\(933\) 0 0
\(934\) 17.0591 0.558190
\(935\) 0 0
\(936\) 0 0
\(937\) −44.7401 −1.46159 −0.730797 0.682595i \(-0.760851\pi\)
−0.730797 + 0.682595i \(0.760851\pi\)
\(938\) −18.5545 −0.605826
\(939\) 0 0
\(940\) 0 0
\(941\) 13.7268 0.447480 0.223740 0.974649i \(-0.428173\pi\)
0.223740 + 0.974649i \(0.428173\pi\)
\(942\) 0 0
\(943\) −1.42740 −0.0464826
\(944\) 14.3068 0.465646
\(945\) 0 0
\(946\) 3.73419 0.121409
\(947\) −29.5652 −0.960739 −0.480370 0.877066i \(-0.659497\pi\)
−0.480370 + 0.877066i \(0.659497\pi\)
\(948\) 0 0
\(949\) 45.4520 1.47543
\(950\) 0 0
\(951\) 0 0
\(952\) 9.10422 0.295069
\(953\) 35.6284 1.15412 0.577059 0.816703i \(-0.304200\pi\)
0.577059 + 0.816703i \(0.304200\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −15.1648 −0.490466
\(957\) 0 0
\(958\) 4.94418 0.159739
\(959\) 3.58002 0.115605
\(960\) 0 0
\(961\) 56.3201 1.81678
\(962\) 23.3929 0.754218
\(963\) 0 0
\(964\) 5.79482 0.186638
\(965\) 0 0
\(966\) 0 0
\(967\) 52.9581 1.70302 0.851509 0.524340i \(-0.175688\pi\)
0.851509 + 0.524340i \(0.175688\pi\)
\(968\) −5.69646 −0.183091
\(969\) 0 0
\(970\) 0 0
\(971\) −25.5423 −0.819691 −0.409845 0.912155i \(-0.634417\pi\)
−0.409845 + 0.912155i \(0.634417\pi\)
\(972\) 0 0
\(973\) −45.5094 −1.45896
\(974\) −17.9097 −0.573864
\(975\) 0 0
\(976\) −13.1345 −0.420426
\(977\) −12.6629 −0.405122 −0.202561 0.979270i \(-0.564926\pi\)
−0.202561 + 0.979270i \(0.564926\pi\)
\(978\) 0 0
\(979\) 62.4716 1.99660
\(980\) 0 0
\(981\) 0 0
\(982\) 25.9852 0.829220
\(983\) −8.26581 −0.263638 −0.131819 0.991274i \(-0.542082\pi\)
−0.131819 + 0.991274i \(0.542082\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 25.1468 0.800836
\(987\) 0 0
\(988\) 3.79001 0.120576
\(989\) 0.321627 0.0102271
\(990\) 0 0
\(991\) −44.9655 −1.42838 −0.714188 0.699954i \(-0.753204\pi\)
−0.714188 + 0.699954i \(0.753204\pi\)
\(992\) −9.34452 −0.296689
\(993\) 0 0
\(994\) −23.7326 −0.752753
\(995\) 0 0
\(996\) 0 0
\(997\) −21.4939 −0.680718 −0.340359 0.940296i \(-0.610549\pi\)
−0.340359 + 0.940296i \(0.610549\pi\)
\(998\) −3.54229 −0.112129
\(999\) 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.cg.1.1 yes 3
3.2 odd 2 8550.2.a.cs.1.1 yes 3
5.4 even 2 8550.2.a.cn.1.3 yes 3
15.14 odd 2 8550.2.a.cd.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8550.2.a.cd.1.3 3 15.14 odd 2
8550.2.a.cg.1.1 yes 3 1.1 even 1 trivial
8550.2.a.cn.1.3 yes 3 5.4 even 2
8550.2.a.cs.1.1 yes 3 3.2 odd 2