Properties

Label 8550.2.a.cn.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: 3.3.564.1
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 1.2
Root \(2.51414\) of defining polynomial
Character \(\chi\) \(=\) 8550.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -0.193252 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -0.193252 q^{7} +1.00000 q^{8} +0.514137 q^{11} -3.12763 q^{13} -0.193252 q^{14} +1.00000 q^{16} +2.83502 q^{17} +1.00000 q^{19} +0.514137 q^{22} -2.32088 q^{23} -3.12763 q^{26} -0.193252 q^{28} -0.164979 q^{29} -9.05655 q^{31} +1.00000 q^{32} +2.83502 q^{34} -3.02827 q^{37} +1.00000 q^{38} -9.96265 q^{41} -5.51414 q^{43} +0.514137 q^{44} -2.32088 q^{46} -1.70739 q^{47} -6.96265 q^{49} -3.12763 q^{52} +2.90064 q^{53} -0.193252 q^{56} -0.164979 q^{58} -13.9572 q^{59} +5.92892 q^{61} -9.05655 q^{62} +1.00000 q^{64} -4.22153 q^{67} +2.83502 q^{68} -3.16498 q^{71} +8.37743 q^{73} -3.02827 q^{74} +1.00000 q^{76} -0.0993582 q^{77} +6.02827 q^{79} -9.96265 q^{82} +12.8633 q^{83} -5.51414 q^{86} +0.514137 q^{88} +9.01920 q^{89} +0.604422 q^{91} -2.32088 q^{92} -1.70739 q^{94} +8.89157 q^{97} -6.96265 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} + 3q^{4} - 2q^{7} + 3q^{8} + O(q^{10}) \) \( 3q + 3q^{2} + 3q^{4} - 2q^{7} + 3q^{8} - 5q^{11} - 2q^{14} + 3q^{16} - 6q^{17} + 3q^{19} - 5q^{22} + q^{23} - 2q^{28} - 15q^{29} - q^{31} + 3q^{32} - 6q^{34} + 4q^{37} + 3q^{38} - 6q^{41} - 10q^{43} - 5q^{44} + q^{46} + 3q^{49} + 5q^{53} - 2q^{56} - 15q^{58} - 12q^{59} + q^{61} - q^{62} + 3q^{64} - q^{67} - 6q^{68} - 24q^{71} - 9q^{73} + 4q^{74} + 3q^{76} - 4q^{77} + 5q^{79} - 6q^{82} + 11q^{83} - 10q^{86} - 5q^{88} - 23q^{89} - 38q^{91} + q^{92} - 14q^{97} + 3q^{98} + O(q^{100}) \)

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
\(7\) −0.193252 −0.0730426 −0.0365213 0.999333i \(-0.511628\pi\)
−0.0365213 + 0.999333i \(0.511628\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 0.514137 0.155018 0.0775091 0.996992i \(-0.475303\pi\)
0.0775091 + 0.996992i \(0.475303\pi\)
\(12\) 0 0
\(13\) −3.12763 −0.867449 −0.433725 0.901046i \(-0.642801\pi\)
−0.433725 + 0.901046i \(0.642801\pi\)
\(14\) −0.193252 −0.0516489
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.83502 0.687594 0.343797 0.939044i \(-0.388287\pi\)
0.343797 + 0.939044i \(0.388287\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0.514137 0.109614
\(23\) −2.32088 −0.483938 −0.241969 0.970284i \(-0.577793\pi\)
−0.241969 + 0.970284i \(0.577793\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −3.12763 −0.613379
\(27\) 0 0
\(28\) −0.193252 −0.0365213
\(29\) −0.164979 −0.0306358 −0.0153179 0.999883i \(-0.504876\pi\)
−0.0153179 + 0.999883i \(0.504876\pi\)
\(30\) 0 0
\(31\) −9.05655 −1.62660 −0.813302 0.581842i \(-0.802332\pi\)
−0.813302 + 0.581842i \(0.802332\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 2.83502 0.486202
\(35\) 0 0
\(36\) 0 0
\(37\) −3.02827 −0.497845 −0.248923 0.968523i \(-0.580076\pi\)
−0.248923 + 0.968523i \(0.580076\pi\)
\(38\) 1.00000 0.162221
\(39\) 0 0
\(40\) 0 0
\(41\) −9.96265 −1.55591 −0.777953 0.628323i \(-0.783742\pi\)
−0.777953 + 0.628323i \(0.783742\pi\)
\(42\) 0 0
\(43\) −5.51414 −0.840898 −0.420449 0.907316i \(-0.638127\pi\)
−0.420449 + 0.907316i \(0.638127\pi\)
\(44\) 0.514137 0.0775091
\(45\) 0 0
\(46\) −2.32088 −0.342196
\(47\) −1.70739 −0.249048 −0.124524 0.992217i \(-0.539740\pi\)
−0.124524 + 0.992217i \(0.539740\pi\)
\(48\) 0 0
\(49\) −6.96265 −0.994665
\(50\) 0 0
\(51\) 0 0
\(52\) −3.12763 −0.433725
\(53\) 2.90064 0.398434 0.199217 0.979955i \(-0.436160\pi\)
0.199217 + 0.979955i \(0.436160\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −0.193252 −0.0258244
\(57\) 0 0
\(58\) −0.164979 −0.0216627
\(59\) −13.9572 −1.81707 −0.908536 0.417807i \(-0.862799\pi\)
−0.908536 + 0.417807i \(0.862799\pi\)
\(60\) 0 0
\(61\) 5.92892 0.759120 0.379560 0.925167i \(-0.376075\pi\)
0.379560 + 0.925167i \(0.376075\pi\)
\(62\) −9.05655 −1.15018
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −4.22153 −0.515742 −0.257871 0.966179i \(-0.583021\pi\)
−0.257871 + 0.966179i \(0.583021\pi\)
\(68\) 2.83502 0.343797
\(69\) 0 0
\(70\) 0 0
\(71\) −3.16498 −0.375614 −0.187807 0.982206i \(-0.560138\pi\)
−0.187807 + 0.982206i \(0.560138\pi\)
\(72\) 0 0
\(73\) 8.37743 0.980504 0.490252 0.871581i \(-0.336905\pi\)
0.490252 + 0.871581i \(0.336905\pi\)
\(74\) −3.02827 −0.352030
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) −0.0993582 −0.0113229
\(78\) 0 0
\(79\) 6.02827 0.678234 0.339117 0.940744i \(-0.389872\pi\)
0.339117 + 0.940744i \(0.389872\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −9.96265 −1.10019
\(83\) 12.8633 1.41193 0.705965 0.708247i \(-0.250514\pi\)
0.705965 + 0.708247i \(0.250514\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −5.51414 −0.594605
\(87\) 0 0
\(88\) 0.514137 0.0548072
\(89\) 9.01920 0.956033 0.478017 0.878351i \(-0.341356\pi\)
0.478017 + 0.878351i \(0.341356\pi\)
\(90\) 0 0
\(91\) 0.604422 0.0633607
\(92\) −2.32088 −0.241969
\(93\) 0 0
\(94\) −1.70739 −0.176104
\(95\) 0 0
\(96\) 0 0
\(97\) 8.89157 0.902802 0.451401 0.892321i \(-0.350924\pi\)
0.451401 + 0.892321i \(0.350924\pi\)
\(98\) −6.96265 −0.703334
\(99\) 0 0
\(100\) 0 0
\(101\) 2.15591 0.214521 0.107260 0.994231i \(-0.465792\pi\)
0.107260 + 0.994231i \(0.465792\pi\)
\(102\) 0 0
\(103\) −10.3209 −1.01695 −0.508473 0.861078i \(-0.669790\pi\)
−0.508473 + 0.861078i \(0.669790\pi\)
\(104\) −3.12763 −0.306690
\(105\) 0 0
\(106\) 2.90064 0.281735
\(107\) 5.57068 0.538538 0.269269 0.963065i \(-0.413218\pi\)
0.269269 + 0.963065i \(0.413218\pi\)
\(108\) 0 0
\(109\) 5.61350 0.537675 0.268838 0.963186i \(-0.413360\pi\)
0.268838 + 0.963186i \(0.413360\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −0.193252 −0.0182606
\(113\) −6.96265 −0.654991 −0.327496 0.944853i \(-0.606205\pi\)
−0.327496 + 0.944853i \(0.606205\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −0.164979 −0.0153179
\(117\) 0 0
\(118\) −13.9572 −1.28486
\(119\) −0.547875 −0.0502236
\(120\) 0 0
\(121\) −10.7357 −0.975969
\(122\) 5.92892 0.536779
\(123\) 0 0
\(124\) −9.05655 −0.813302
\(125\) 0 0
\(126\) 0 0
\(127\) 13.6983 1.21553 0.607765 0.794117i \(-0.292066\pi\)
0.607765 + 0.794117i \(0.292066\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −10.4768 −0.915361 −0.457681 0.889117i \(-0.651320\pi\)
−0.457681 + 0.889117i \(0.651320\pi\)
\(132\) 0 0
\(133\) −0.193252 −0.0167571
\(134\) −4.22153 −0.364684
\(135\) 0 0
\(136\) 2.83502 0.243101
\(137\) −11.6700 −0.997039 −0.498520 0.866878i \(-0.666123\pi\)
−0.498520 + 0.866878i \(0.666123\pi\)
\(138\) 0 0
\(139\) 12.7411 1.08069 0.540344 0.841444i \(-0.318294\pi\)
0.540344 + 0.841444i \(0.318294\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −3.16498 −0.265599
\(143\) −1.60803 −0.134470
\(144\) 0 0
\(145\) 0 0
\(146\) 8.37743 0.693321
\(147\) 0 0
\(148\) −3.02827 −0.248923
\(149\) −9.61350 −0.787568 −0.393784 0.919203i \(-0.628834\pi\)
−0.393784 + 0.919203i \(0.628834\pi\)
\(150\) 0 0
\(151\) −9.12217 −0.742352 −0.371176 0.928563i \(-0.621045\pi\)
−0.371176 + 0.928563i \(0.621045\pi\)
\(152\) 1.00000 0.0811107
\(153\) 0 0
\(154\) −0.0993582 −0.00800651
\(155\) 0 0
\(156\) 0 0
\(157\) −4.93438 −0.393806 −0.196903 0.980423i \(-0.563088\pi\)
−0.196903 + 0.980423i \(0.563088\pi\)
\(158\) 6.02827 0.479584
\(159\) 0 0
\(160\) 0 0
\(161\) 0.448517 0.0353481
\(162\) 0 0
\(163\) −11.2835 −0.883795 −0.441897 0.897066i \(-0.645694\pi\)
−0.441897 + 0.897066i \(0.645694\pi\)
\(164\) −9.96265 −0.777953
\(165\) 0 0
\(166\) 12.8633 0.998385
\(167\) −11.8688 −0.918432 −0.459216 0.888325i \(-0.651870\pi\)
−0.459216 + 0.888325i \(0.651870\pi\)
\(168\) 0 0
\(169\) −3.21792 −0.247532
\(170\) 0 0
\(171\) 0 0
\(172\) −5.51414 −0.420449
\(173\) −4.27807 −0.325256 −0.162628 0.986687i \(-0.551997\pi\)
−0.162628 + 0.986687i \(0.551997\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0.514137 0.0387545
\(177\) 0 0
\(178\) 9.01920 0.676018
\(179\) −22.3118 −1.66766 −0.833832 0.552019i \(-0.813858\pi\)
−0.833832 + 0.552019i \(0.813858\pi\)
\(180\) 0 0
\(181\) −11.4576 −0.851636 −0.425818 0.904809i \(-0.640014\pi\)
−0.425818 + 0.904809i \(0.640014\pi\)
\(182\) 0.604422 0.0448028
\(183\) 0 0
\(184\) −2.32088 −0.171098
\(185\) 0 0
\(186\) 0 0
\(187\) 1.45759 0.106589
\(188\) −1.70739 −0.124524
\(189\) 0 0
\(190\) 0 0
\(191\) −18.4148 −1.33245 −0.666223 0.745752i \(-0.732090\pi\)
−0.666223 + 0.745752i \(0.732090\pi\)
\(192\) 0 0
\(193\) −0.193252 −0.0139106 −0.00695531 0.999976i \(-0.502214\pi\)
−0.00695531 + 0.999976i \(0.502214\pi\)
\(194\) 8.89157 0.638377
\(195\) 0 0
\(196\) −6.96265 −0.497332
\(197\) −18.5990 −1.32512 −0.662560 0.749008i \(-0.730530\pi\)
−0.662560 + 0.749008i \(0.730530\pi\)
\(198\) 0 0
\(199\) 20.5671 1.45796 0.728981 0.684534i \(-0.239994\pi\)
0.728981 + 0.684534i \(0.239994\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 2.15591 0.151689
\(203\) 0.0318825 0.00223771
\(204\) 0 0
\(205\) 0 0
\(206\) −10.3209 −0.719090
\(207\) 0 0
\(208\) −3.12763 −0.216862
\(209\) 0.514137 0.0355636
\(210\) 0 0
\(211\) 4.57068 0.314659 0.157329 0.987546i \(-0.449712\pi\)
0.157329 + 0.987546i \(0.449712\pi\)
\(212\) 2.90064 0.199217
\(213\) 0 0
\(214\) 5.57068 0.380804
\(215\) 0 0
\(216\) 0 0
\(217\) 1.75020 0.118811
\(218\) 5.61350 0.380194
\(219\) 0 0
\(220\) 0 0
\(221\) −8.86690 −0.596453
\(222\) 0 0
\(223\) 3.93438 0.263466 0.131733 0.991285i \(-0.457946\pi\)
0.131733 + 0.991285i \(0.457946\pi\)
\(224\) −0.193252 −0.0129122
\(225\) 0 0
\(226\) −6.96265 −0.463149
\(227\) −11.4713 −0.761379 −0.380689 0.924703i \(-0.624313\pi\)
−0.380689 + 0.924703i \(0.624313\pi\)
\(228\) 0 0
\(229\) 12.4768 0.824490 0.412245 0.911073i \(-0.364745\pi\)
0.412245 + 0.911073i \(0.364745\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −0.164979 −0.0108314
\(233\) 27.4021 1.79517 0.897586 0.440840i \(-0.145319\pi\)
0.897586 + 0.440840i \(0.145319\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −13.9572 −0.908536
\(237\) 0 0
\(238\) −0.547875 −0.0355134
\(239\) 14.4057 0.931828 0.465914 0.884830i \(-0.345726\pi\)
0.465914 + 0.884830i \(0.345726\pi\)
\(240\) 0 0
\(241\) −21.7886 −1.40353 −0.701764 0.712410i \(-0.747604\pi\)
−0.701764 + 0.712410i \(0.747604\pi\)
\(242\) −10.7357 −0.690115
\(243\) 0 0
\(244\) 5.92892 0.379560
\(245\) 0 0
\(246\) 0 0
\(247\) −3.12763 −0.199006
\(248\) −9.05655 −0.575091
\(249\) 0 0
\(250\) 0 0
\(251\) −21.9627 −1.38627 −0.693135 0.720808i \(-0.743771\pi\)
−0.693135 + 0.720808i \(0.743771\pi\)
\(252\) 0 0
\(253\) −1.19325 −0.0750191
\(254\) 13.6983 0.859509
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −0.0656204 −0.00409329 −0.00204664 0.999998i \(-0.500651\pi\)
−0.00204664 + 0.999998i \(0.500651\pi\)
\(258\) 0 0
\(259\) 0.585221 0.0363639
\(260\) 0 0
\(261\) 0 0
\(262\) −10.4768 −0.647258
\(263\) 24.8314 1.53117 0.765585 0.643334i \(-0.222450\pi\)
0.765585 + 0.643334i \(0.222450\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −0.193252 −0.0118491
\(267\) 0 0
\(268\) −4.22153 −0.257871
\(269\) −19.8578 −1.21075 −0.605377 0.795939i \(-0.706978\pi\)
−0.605377 + 0.795939i \(0.706978\pi\)
\(270\) 0 0
\(271\) 0.891569 0.0541590 0.0270795 0.999633i \(-0.491379\pi\)
0.0270795 + 0.999633i \(0.491379\pi\)
\(272\) 2.83502 0.171898
\(273\) 0 0
\(274\) −11.6700 −0.705013
\(275\) 0 0
\(276\) 0 0
\(277\) 6.76940 0.406734 0.203367 0.979103i \(-0.434812\pi\)
0.203367 + 0.979103i \(0.434812\pi\)
\(278\) 12.7411 0.764162
\(279\) 0 0
\(280\) 0 0
\(281\) −17.2553 −1.02936 −0.514681 0.857382i \(-0.672090\pi\)
−0.514681 + 0.857382i \(0.672090\pi\)
\(282\) 0 0
\(283\) −20.6983 −1.23039 −0.615194 0.788376i \(-0.710922\pi\)
−0.615194 + 0.788376i \(0.710922\pi\)
\(284\) −3.16498 −0.187807
\(285\) 0 0
\(286\) −1.60803 −0.0950849
\(287\) 1.92531 0.113647
\(288\) 0 0
\(289\) −8.96265 −0.527215
\(290\) 0 0
\(291\) 0 0
\(292\) 8.37743 0.490252
\(293\) 0.712853 0.0416453 0.0208227 0.999783i \(-0.493371\pi\)
0.0208227 + 0.999783i \(0.493371\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −3.02827 −0.176015
\(297\) 0 0
\(298\) −9.61350 −0.556895
\(299\) 7.25887 0.419791
\(300\) 0 0
\(301\) 1.06562 0.0614213
\(302\) −9.12217 −0.524922
\(303\) 0 0
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) 0 0
\(307\) −5.11856 −0.292132 −0.146066 0.989275i \(-0.546661\pi\)
−0.146066 + 0.989275i \(0.546661\pi\)
\(308\) −0.0993582 −0.00566146
\(309\) 0 0
\(310\) 0 0
\(311\) −3.26434 −0.185104 −0.0925518 0.995708i \(-0.529502\pi\)
−0.0925518 + 0.995708i \(0.529502\pi\)
\(312\) 0 0
\(313\) −2.96265 −0.167459 −0.0837295 0.996489i \(-0.526683\pi\)
−0.0837295 + 0.996489i \(0.526683\pi\)
\(314\) −4.93438 −0.278463
\(315\) 0 0
\(316\) 6.02827 0.339117
\(317\) −12.1650 −0.683253 −0.341627 0.939836i \(-0.610978\pi\)
−0.341627 + 0.939836i \(0.610978\pi\)
\(318\) 0 0
\(319\) −0.0848216 −0.00474910
\(320\) 0 0
\(321\) 0 0
\(322\) 0.448517 0.0249949
\(323\) 2.83502 0.157745
\(324\) 0 0
\(325\) 0 0
\(326\) −11.2835 −0.624937
\(327\) 0 0
\(328\) −9.96265 −0.550096
\(329\) 0.329957 0.0181911
\(330\) 0 0
\(331\) 6.67551 0.366919 0.183460 0.983027i \(-0.441270\pi\)
0.183460 + 0.983027i \(0.441270\pi\)
\(332\) 12.8633 0.705965
\(333\) 0 0
\(334\) −11.8688 −0.649430
\(335\) 0 0
\(336\) 0 0
\(337\) −9.72659 −0.529841 −0.264921 0.964270i \(-0.585346\pi\)
−0.264921 + 0.964270i \(0.585346\pi\)
\(338\) −3.21792 −0.175032
\(339\) 0 0
\(340\) 0 0
\(341\) −4.65631 −0.252153
\(342\) 0 0
\(343\) 2.69832 0.145695
\(344\) −5.51414 −0.297302
\(345\) 0 0
\(346\) −4.27807 −0.229991
\(347\) 9.08482 0.487699 0.243849 0.969813i \(-0.421590\pi\)
0.243849 + 0.969813i \(0.421590\pi\)
\(348\) 0 0
\(349\) 26.0047 1.39200 0.695999 0.718043i \(-0.254962\pi\)
0.695999 + 0.718043i \(0.254962\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0.514137 0.0274036
\(353\) 27.7321 1.47603 0.738014 0.674785i \(-0.235764\pi\)
0.738014 + 0.674785i \(0.235764\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 9.01920 0.478017
\(357\) 0 0
\(358\) −22.3118 −1.17922
\(359\) 0.746591 0.0394036 0.0197018 0.999806i \(-0.493728\pi\)
0.0197018 + 0.999806i \(0.493728\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −11.4576 −0.602198
\(363\) 0 0
\(364\) 0.604422 0.0316803
\(365\) 0 0
\(366\) 0 0
\(367\) 13.9945 0.730509 0.365254 0.930908i \(-0.380982\pi\)
0.365254 + 0.930908i \(0.380982\pi\)
\(368\) −2.32088 −0.120984
\(369\) 0 0
\(370\) 0 0
\(371\) −0.560556 −0.0291026
\(372\) 0 0
\(373\) 20.6099 1.06714 0.533570 0.845756i \(-0.320850\pi\)
0.533570 + 0.845756i \(0.320850\pi\)
\(374\) 1.45759 0.0753701
\(375\) 0 0
\(376\) −1.70739 −0.0880519
\(377\) 0.515992 0.0265750
\(378\) 0 0
\(379\) 1.63164 0.0838117 0.0419059 0.999122i \(-0.486657\pi\)
0.0419059 + 0.999122i \(0.486657\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −18.4148 −0.942182
\(383\) −12.3173 −0.629383 −0.314692 0.949194i \(-0.601901\pi\)
−0.314692 + 0.949194i \(0.601901\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −0.193252 −0.00983629
\(387\) 0 0
\(388\) 8.89157 0.451401
\(389\) −3.61350 −0.183211 −0.0916057 0.995795i \(-0.529200\pi\)
−0.0916057 + 0.995795i \(0.529200\pi\)
\(390\) 0 0
\(391\) −6.57976 −0.332753
\(392\) −6.96265 −0.351667
\(393\) 0 0
\(394\) −18.5990 −0.937002
\(395\) 0 0
\(396\) 0 0
\(397\) −2.88250 −0.144668 −0.0723342 0.997380i \(-0.523045\pi\)
−0.0723342 + 0.997380i \(0.523045\pi\)
\(398\) 20.5671 1.03093
\(399\) 0 0
\(400\) 0 0
\(401\) −28.9253 −1.44446 −0.722230 0.691652i \(-0.756883\pi\)
−0.722230 + 0.691652i \(0.756883\pi\)
\(402\) 0 0
\(403\) 28.3255 1.41100
\(404\) 2.15591 0.107260
\(405\) 0 0
\(406\) 0.0318825 0.00158230
\(407\) −1.55695 −0.0771750
\(408\) 0 0
\(409\) 21.8578 1.08080 0.540400 0.841408i \(-0.318273\pi\)
0.540400 + 0.841408i \(0.318273\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −10.3209 −0.508473
\(413\) 2.69726 0.132724
\(414\) 0 0
\(415\) 0 0
\(416\) −3.12763 −0.153345
\(417\) 0 0
\(418\) 0.514137 0.0251473
\(419\) 8.38650 0.409708 0.204854 0.978793i \(-0.434328\pi\)
0.204854 + 0.978793i \(0.434328\pi\)
\(420\) 0 0
\(421\) −13.1523 −0.641004 −0.320502 0.947248i \(-0.603852\pi\)
−0.320502 + 0.947248i \(0.603852\pi\)
\(422\) 4.57068 0.222497
\(423\) 0 0
\(424\) 2.90064 0.140868
\(425\) 0 0
\(426\) 0 0
\(427\) −1.14578 −0.0554481
\(428\) 5.57068 0.269269
\(429\) 0 0
\(430\) 0 0
\(431\) −16.1806 −0.779391 −0.389695 0.920944i \(-0.627420\pi\)
−0.389695 + 0.920944i \(0.627420\pi\)
\(432\) 0 0
\(433\) 13.9945 0.672534 0.336267 0.941767i \(-0.390835\pi\)
0.336267 + 0.941767i \(0.390835\pi\)
\(434\) 1.75020 0.0840123
\(435\) 0 0
\(436\) 5.61350 0.268838
\(437\) −2.32088 −0.111023
\(438\) 0 0
\(439\) −12.3401 −0.588960 −0.294480 0.955658i \(-0.595146\pi\)
−0.294480 + 0.955658i \(0.595146\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −8.86690 −0.421756
\(443\) −14.5899 −0.693186 −0.346593 0.938016i \(-0.612662\pi\)
−0.346593 + 0.938016i \(0.612662\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 3.93438 0.186298
\(447\) 0 0
\(448\) −0.193252 −0.00913032
\(449\) −3.74659 −0.176813 −0.0884063 0.996084i \(-0.528177\pi\)
−0.0884063 + 0.996084i \(0.528177\pi\)
\(450\) 0 0
\(451\) −5.12217 −0.241193
\(452\) −6.96265 −0.327496
\(453\) 0 0
\(454\) −11.4713 −0.538376
\(455\) 0 0
\(456\) 0 0
\(457\) 18.2745 0.854843 0.427422 0.904052i \(-0.359422\pi\)
0.427422 + 0.904052i \(0.359422\pi\)
\(458\) 12.4768 0.583002
\(459\) 0 0
\(460\) 0 0
\(461\) 13.0957 0.609930 0.304965 0.952364i \(-0.401355\pi\)
0.304965 + 0.952364i \(0.401355\pi\)
\(462\) 0 0
\(463\) −15.0283 −0.698423 −0.349212 0.937044i \(-0.613551\pi\)
−0.349212 + 0.937044i \(0.613551\pi\)
\(464\) −0.164979 −0.00765894
\(465\) 0 0
\(466\) 27.4021 1.26938
\(467\) −21.7302 −1.00555 −0.502777 0.864416i \(-0.667688\pi\)
−0.502777 + 0.864416i \(0.667688\pi\)
\(468\) 0 0
\(469\) 0.815820 0.0376711
\(470\) 0 0
\(471\) 0 0
\(472\) −13.9572 −0.642432
\(473\) −2.83502 −0.130354
\(474\) 0 0
\(475\) 0 0
\(476\) −0.547875 −0.0251118
\(477\) 0 0
\(478\) 14.4057 0.658902
\(479\) 0.962653 0.0439848 0.0219924 0.999758i \(-0.492999\pi\)
0.0219924 + 0.999758i \(0.492999\pi\)
\(480\) 0 0
\(481\) 9.47133 0.431855
\(482\) −21.7886 −0.992444
\(483\) 0 0
\(484\) −10.7357 −0.487985
\(485\) 0 0
\(486\) 0 0
\(487\) 42.5561 1.92840 0.964202 0.265170i \(-0.0854282\pi\)
0.964202 + 0.265170i \(0.0854282\pi\)
\(488\) 5.92892 0.268389
\(489\) 0 0
\(490\) 0 0
\(491\) 14.7549 0.665878 0.332939 0.942948i \(-0.391960\pi\)
0.332939 + 0.942948i \(0.391960\pi\)
\(492\) 0 0
\(493\) −0.467718 −0.0210649
\(494\) −3.12763 −0.140719
\(495\) 0 0
\(496\) −9.05655 −0.406651
\(497\) 0.611640 0.0274358
\(498\) 0 0
\(499\) −7.64538 −0.342254 −0.171127 0.985249i \(-0.554741\pi\)
−0.171127 + 0.985249i \(0.554741\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −21.9627 −0.980241
\(503\) 4.51053 0.201115 0.100557 0.994931i \(-0.467937\pi\)
0.100557 + 0.994931i \(0.467937\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −1.19325 −0.0530465
\(507\) 0 0
\(508\) 13.6983 0.607765
\(509\) 41.1943 1.82591 0.912953 0.408065i \(-0.133796\pi\)
0.912953 + 0.408065i \(0.133796\pi\)
\(510\) 0 0
\(511\) −1.61896 −0.0716185
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −0.0656204 −0.00289439
\(515\) 0 0
\(516\) 0 0
\(517\) −0.877832 −0.0386070
\(518\) 0.585221 0.0257131
\(519\) 0 0
\(520\) 0 0
\(521\) −10.1595 −0.445096 −0.222548 0.974922i \(-0.571437\pi\)
−0.222548 + 0.974922i \(0.571437\pi\)
\(522\) 0 0
\(523\) −14.6308 −0.639762 −0.319881 0.947458i \(-0.603643\pi\)
−0.319881 + 0.947458i \(0.603643\pi\)
\(524\) −10.4768 −0.457681
\(525\) 0 0
\(526\) 24.8314 1.08270
\(527\) −25.6755 −1.11844
\(528\) 0 0
\(529\) −17.6135 −0.765804
\(530\) 0 0
\(531\) 0 0
\(532\) −0.193252 −0.00837856
\(533\) 31.1595 1.34967
\(534\) 0 0
\(535\) 0 0
\(536\) −4.22153 −0.182342
\(537\) 0 0
\(538\) −19.8578 −0.856132
\(539\) −3.57976 −0.154191
\(540\) 0 0
\(541\) −13.0511 −0.561110 −0.280555 0.959838i \(-0.590518\pi\)
−0.280555 + 0.959838i \(0.590518\pi\)
\(542\) 0.891569 0.0382962
\(543\) 0 0
\(544\) 2.83502 0.121551
\(545\) 0 0
\(546\) 0 0
\(547\) −10.0420 −0.429365 −0.214683 0.976684i \(-0.568872\pi\)
−0.214683 + 0.976684i \(0.568872\pi\)
\(548\) −11.6700 −0.498520
\(549\) 0 0
\(550\) 0 0
\(551\) −0.164979 −0.00702832
\(552\) 0 0
\(553\) −1.16498 −0.0495399
\(554\) 6.76940 0.287604
\(555\) 0 0
\(556\) 12.7411 0.540344
\(557\) 6.03188 0.255579 0.127790 0.991801i \(-0.459212\pi\)
0.127790 + 0.991801i \(0.459212\pi\)
\(558\) 0 0
\(559\) 17.2462 0.729436
\(560\) 0 0
\(561\) 0 0
\(562\) −17.2553 −0.727869
\(563\) 35.2407 1.48522 0.742610 0.669724i \(-0.233588\pi\)
0.742610 + 0.669724i \(0.233588\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −20.6983 −0.870015
\(567\) 0 0
\(568\) −3.16498 −0.132800
\(569\) −21.9144 −0.918699 −0.459349 0.888256i \(-0.651917\pi\)
−0.459349 + 0.888256i \(0.651917\pi\)
\(570\) 0 0
\(571\) −21.2407 −0.888897 −0.444448 0.895804i \(-0.646600\pi\)
−0.444448 + 0.895804i \(0.646600\pi\)
\(572\) −1.60803 −0.0672352
\(573\) 0 0
\(574\) 1.92531 0.0803608
\(575\) 0 0
\(576\) 0 0
\(577\) 32.9253 1.37070 0.685349 0.728215i \(-0.259650\pi\)
0.685349 + 0.728215i \(0.259650\pi\)
\(578\) −8.96265 −0.372797
\(579\) 0 0
\(580\) 0 0
\(581\) −2.48586 −0.103131
\(582\) 0 0
\(583\) 1.49133 0.0617645
\(584\) 8.37743 0.346661
\(585\) 0 0
\(586\) 0.712853 0.0294477
\(587\) −21.5798 −0.890692 −0.445346 0.895359i \(-0.646919\pi\)
−0.445346 + 0.895359i \(0.646919\pi\)
\(588\) 0 0
\(589\) −9.05655 −0.373169
\(590\) 0 0
\(591\) 0 0
\(592\) −3.02827 −0.124461
\(593\) 1.98907 0.0816814 0.0408407 0.999166i \(-0.486996\pi\)
0.0408407 + 0.999166i \(0.486996\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −9.61350 −0.393784
\(597\) 0 0
\(598\) 7.25887 0.296837
\(599\) −33.7321 −1.37825 −0.689127 0.724640i \(-0.742006\pi\)
−0.689127 + 0.724640i \(0.742006\pi\)
\(600\) 0 0
\(601\) 25.7211 1.04919 0.524593 0.851353i \(-0.324217\pi\)
0.524593 + 0.851353i \(0.324217\pi\)
\(602\) 1.06562 0.0434314
\(603\) 0 0
\(604\) −9.12217 −0.371176
\(605\) 0 0
\(606\) 0 0
\(607\) 37.8488 1.53623 0.768117 0.640310i \(-0.221194\pi\)
0.768117 + 0.640310i \(0.221194\pi\)
\(608\) 1.00000 0.0405554
\(609\) 0 0
\(610\) 0 0
\(611\) 5.34009 0.216037
\(612\) 0 0
\(613\) 15.4713 0.624881 0.312440 0.949937i \(-0.398854\pi\)
0.312440 + 0.949937i \(0.398854\pi\)
\(614\) −5.11856 −0.206568
\(615\) 0 0
\(616\) −0.0993582 −0.00400326
\(617\) −10.3118 −0.415138 −0.207569 0.978220i \(-0.566555\pi\)
−0.207569 + 0.978220i \(0.566555\pi\)
\(618\) 0 0
\(619\) 12.6099 0.506834 0.253417 0.967357i \(-0.418446\pi\)
0.253417 + 0.967357i \(0.418446\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −3.26434 −0.130888
\(623\) −1.74298 −0.0698311
\(624\) 0 0
\(625\) 0 0
\(626\) −2.96265 −0.118411
\(627\) 0 0
\(628\) −4.93438 −0.196903
\(629\) −8.58522 −0.342315
\(630\) 0 0
\(631\) −9.34009 −0.371823 −0.185911 0.982566i \(-0.559524\pi\)
−0.185911 + 0.982566i \(0.559524\pi\)
\(632\) 6.02827 0.239792
\(633\) 0 0
\(634\) −12.1650 −0.483133
\(635\) 0 0
\(636\) 0 0
\(637\) 21.7766 0.862821
\(638\) −0.0848216 −0.00335812
\(639\) 0 0
\(640\) 0 0
\(641\) −28.3310 −1.11901 −0.559504 0.828828i \(-0.689008\pi\)
−0.559504 + 0.828828i \(0.689008\pi\)
\(642\) 0 0
\(643\) 25.0848 0.989249 0.494624 0.869107i \(-0.335306\pi\)
0.494624 + 0.869107i \(0.335306\pi\)
\(644\) 0.448517 0.0176740
\(645\) 0 0
\(646\) 2.83502 0.111542
\(647\) −19.4623 −0.765140 −0.382570 0.923926i \(-0.624961\pi\)
−0.382570 + 0.923926i \(0.624961\pi\)
\(648\) 0 0
\(649\) −7.17591 −0.281679
\(650\) 0 0
\(651\) 0 0
\(652\) −11.2835 −0.441897
\(653\) −12.5671 −0.491788 −0.245894 0.969297i \(-0.579081\pi\)
−0.245894 + 0.969297i \(0.579081\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −9.96265 −0.388976
\(657\) 0 0
\(658\) 0.329957 0.0128631
\(659\) 8.22338 0.320337 0.160169 0.987090i \(-0.448796\pi\)
0.160169 + 0.987090i \(0.448796\pi\)
\(660\) 0 0
\(661\) 26.2371 1.02051 0.510253 0.860024i \(-0.329552\pi\)
0.510253 + 0.860024i \(0.329552\pi\)
\(662\) 6.67551 0.259451
\(663\) 0 0
\(664\) 12.8633 0.499193
\(665\) 0 0
\(666\) 0 0
\(667\) 0.382896 0.0148258
\(668\) −11.8688 −0.459216
\(669\) 0 0
\(670\) 0 0
\(671\) 3.04827 0.117677
\(672\) 0 0
\(673\) 7.29622 0.281249 0.140624 0.990063i \(-0.455089\pi\)
0.140624 + 0.990063i \(0.455089\pi\)
\(674\) −9.72659 −0.374654
\(675\) 0 0
\(676\) −3.21792 −0.123766
\(677\) −29.2234 −1.12315 −0.561573 0.827427i \(-0.689804\pi\)
−0.561573 + 0.827427i \(0.689804\pi\)
\(678\) 0 0
\(679\) −1.71832 −0.0659430
\(680\) 0 0
\(681\) 0 0
\(682\) −4.65631 −0.178299
\(683\) −11.9006 −0.455365 −0.227683 0.973735i \(-0.573115\pi\)
−0.227683 + 0.973735i \(0.573115\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 2.69832 0.103022
\(687\) 0 0
\(688\) −5.51414 −0.210224
\(689\) −9.07214 −0.345621
\(690\) 0 0
\(691\) 0.0137372 0.000522589 0 0.000261294 1.00000i \(-0.499917\pi\)
0.000261294 1.00000i \(0.499917\pi\)
\(692\) −4.27807 −0.162628
\(693\) 0 0
\(694\) 9.08482 0.344855
\(695\) 0 0
\(696\) 0 0
\(697\) −28.2443 −1.06983
\(698\) 26.0047 0.984291
\(699\) 0 0
\(700\) 0 0
\(701\) −0.266192 −0.0100539 −0.00502697 0.999987i \(-0.501600\pi\)
−0.00502697 + 0.999987i \(0.501600\pi\)
\(702\) 0 0
\(703\) −3.02827 −0.114214
\(704\) 0.514137 0.0193773
\(705\) 0 0
\(706\) 27.7321 1.04371
\(707\) −0.416634 −0.0156691
\(708\) 0 0
\(709\) −9.50506 −0.356970 −0.178485 0.983943i \(-0.557120\pi\)
−0.178485 + 0.983943i \(0.557120\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 9.01920 0.338009
\(713\) 21.0192 0.787175
\(714\) 0 0
\(715\) 0 0
\(716\) −22.3118 −0.833832
\(717\) 0 0
\(718\) 0.746591 0.0278625
\(719\) −10.4895 −0.391191 −0.195596 0.980685i \(-0.562664\pi\)
−0.195596 + 0.980685i \(0.562664\pi\)
\(720\) 0 0
\(721\) 1.99454 0.0742804
\(722\) 1.00000 0.0372161
\(723\) 0 0
\(724\) −11.4576 −0.425818
\(725\) 0 0
\(726\) 0 0
\(727\) −36.3502 −1.34815 −0.674077 0.738661i \(-0.735459\pi\)
−0.674077 + 0.738661i \(0.735459\pi\)
\(728\) 0.604422 0.0224014
\(729\) 0 0
\(730\) 0 0
\(731\) −15.6327 −0.578196
\(732\) 0 0
\(733\) 22.0337 0.813835 0.406917 0.913465i \(-0.366604\pi\)
0.406917 + 0.913465i \(0.366604\pi\)
\(734\) 13.9945 0.516548
\(735\) 0 0
\(736\) −2.32088 −0.0855489
\(737\) −2.17044 −0.0799493
\(738\) 0 0
\(739\) 17.0848 0.628475 0.314238 0.949344i \(-0.398251\pi\)
0.314238 + 0.949344i \(0.398251\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −0.560556 −0.0205787
\(743\) 26.7549 0.981541 0.490770 0.871289i \(-0.336715\pi\)
0.490770 + 0.871289i \(0.336715\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 20.6099 0.754582
\(747\) 0 0
\(748\) 1.45759 0.0532947
\(749\) −1.07655 −0.0393362
\(750\) 0 0
\(751\) −31.7002 −1.15676 −0.578378 0.815769i \(-0.696314\pi\)
−0.578378 + 0.815769i \(0.696314\pi\)
\(752\) −1.70739 −0.0622621
\(753\) 0 0
\(754\) 0.515992 0.0187913
\(755\) 0 0
\(756\) 0 0
\(757\) 35.1933 1.27912 0.639560 0.768741i \(-0.279116\pi\)
0.639560 + 0.768741i \(0.279116\pi\)
\(758\) 1.63164 0.0592638
\(759\) 0 0
\(760\) 0 0
\(761\) 25.9764 0.941643 0.470822 0.882228i \(-0.343958\pi\)
0.470822 + 0.882228i \(0.343958\pi\)
\(762\) 0 0
\(763\) −1.08482 −0.0392732
\(764\) −18.4148 −0.666223
\(765\) 0 0
\(766\) −12.3173 −0.445041
\(767\) 43.6530 1.57622
\(768\) 0 0
\(769\) −44.6146 −1.60884 −0.804421 0.594060i \(-0.797524\pi\)
−0.804421 + 0.594060i \(0.797524\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −0.193252 −0.00695531
\(773\) −51.5844 −1.85536 −0.927681 0.373373i \(-0.878201\pi\)
−0.927681 + 0.373373i \(0.878201\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 8.89157 0.319189
\(777\) 0 0
\(778\) −3.61350 −0.129550
\(779\) −9.96265 −0.356949
\(780\) 0 0
\(781\) −1.62723 −0.0582270
\(782\) −6.57976 −0.235292
\(783\) 0 0
\(784\) −6.96265 −0.248666
\(785\) 0 0
\(786\) 0 0
\(787\) −15.8916 −0.566473 −0.283237 0.959050i \(-0.591408\pi\)
−0.283237 + 0.959050i \(0.591408\pi\)
\(788\) −18.5990 −0.662560
\(789\) 0 0
\(790\) 0 0
\(791\) 1.34555 0.0478422
\(792\) 0 0
\(793\) −18.5435 −0.658498
\(794\) −2.88250 −0.102296
\(795\) 0 0
\(796\) 20.5671 0.728981
\(797\) 43.6146 1.54491 0.772453 0.635072i \(-0.219029\pi\)
0.772453 + 0.635072i \(0.219029\pi\)
\(798\) 0 0
\(799\) −4.84049 −0.171244
\(800\) 0 0
\(801\) 0 0
\(802\) −28.9253 −1.02139
\(803\) 4.30715 0.151996
\(804\) 0 0
\(805\) 0 0
\(806\) 28.3255 0.997725
\(807\) 0 0
\(808\) 2.15591 0.0758445
\(809\) −21.4659 −0.754700 −0.377350 0.926071i \(-0.623165\pi\)
−0.377350 + 0.926071i \(0.623165\pi\)
\(810\) 0 0
\(811\) −4.84409 −0.170099 −0.0850496 0.996377i \(-0.527105\pi\)
−0.0850496 + 0.996377i \(0.527105\pi\)
\(812\) 0.0318825 0.00111886
\(813\) 0 0
\(814\) −1.55695 −0.0545710
\(815\) 0 0
\(816\) 0 0
\(817\) −5.51414 −0.192915
\(818\) 21.8578 0.764241
\(819\) 0 0
\(820\) 0 0
\(821\) −7.65631 −0.267207 −0.133603 0.991035i \(-0.542655\pi\)
−0.133603 + 0.991035i \(0.542655\pi\)
\(822\) 0 0
\(823\) 4.00000 0.139431 0.0697156 0.997567i \(-0.477791\pi\)
0.0697156 + 0.997567i \(0.477791\pi\)
\(824\) −10.3209 −0.359545
\(825\) 0 0
\(826\) 2.69726 0.0938497
\(827\) 13.7875 0.479440 0.239720 0.970842i \(-0.422944\pi\)
0.239720 + 0.970842i \(0.422944\pi\)
\(828\) 0 0
\(829\) 35.0529 1.21744 0.608719 0.793386i \(-0.291684\pi\)
0.608719 + 0.793386i \(0.291684\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −3.12763 −0.108431
\(833\) −19.7393 −0.683925
\(834\) 0 0
\(835\) 0 0
\(836\) 0.514137 0.0177818
\(837\) 0 0
\(838\) 8.38650 0.289707
\(839\) 43.6464 1.50684 0.753421 0.657538i \(-0.228402\pi\)
0.753421 + 0.657538i \(0.228402\pi\)
\(840\) 0 0
\(841\) −28.9728 −0.999061
\(842\) −13.1523 −0.453258
\(843\) 0 0
\(844\) 4.57068 0.157329
\(845\) 0 0
\(846\) 0 0
\(847\) 2.07469 0.0712873
\(848\) 2.90064 0.0996084
\(849\) 0 0
\(850\) 0 0
\(851\) 7.02827 0.240926
\(852\) 0 0
\(853\) 44.4431 1.52170 0.760851 0.648927i \(-0.224782\pi\)
0.760851 + 0.648927i \(0.224782\pi\)
\(854\) −1.14578 −0.0392077
\(855\) 0 0
\(856\) 5.57068 0.190402
\(857\) −40.5935 −1.38665 −0.693324 0.720626i \(-0.743854\pi\)
−0.693324 + 0.720626i \(0.743854\pi\)
\(858\) 0 0
\(859\) −6.58522 −0.224685 −0.112342 0.993670i \(-0.535835\pi\)
−0.112342 + 0.993670i \(0.535835\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −16.1806 −0.551112
\(863\) 7.19792 0.245020 0.122510 0.992467i \(-0.460906\pi\)
0.122510 + 0.992467i \(0.460906\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 13.9945 0.475554
\(867\) 0 0
\(868\) 1.75020 0.0594057
\(869\) 3.09936 0.105139
\(870\) 0 0
\(871\) 13.2034 0.447379
\(872\) 5.61350 0.190097
\(873\) 0 0
\(874\) −2.32088 −0.0785051
\(875\) 0 0
\(876\) 0 0
\(877\) 36.6236 1.23669 0.618346 0.785906i \(-0.287803\pi\)
0.618346 + 0.785906i \(0.287803\pi\)
\(878\) −12.3401 −0.416458
\(879\) 0 0
\(880\) 0 0
\(881\) −5.80128 −0.195450 −0.0977251 0.995213i \(-0.531157\pi\)
−0.0977251 + 0.995213i \(0.531157\pi\)
\(882\) 0 0
\(883\) −5.84409 −0.196669 −0.0983347 0.995153i \(-0.531352\pi\)
−0.0983347 + 0.995153i \(0.531352\pi\)
\(884\) −8.86690 −0.298226
\(885\) 0 0
\(886\) −14.5899 −0.490157
\(887\) 7.68819 0.258144 0.129072 0.991635i \(-0.458800\pi\)
0.129072 + 0.991635i \(0.458800\pi\)
\(888\) 0 0
\(889\) −2.64723 −0.0887853
\(890\) 0 0
\(891\) 0 0
\(892\) 3.93438 0.131733
\(893\) −1.70739 −0.0571356
\(894\) 0 0
\(895\) 0 0
\(896\) −0.193252 −0.00645611
\(897\) 0 0
\(898\) −3.74659 −0.125025
\(899\) 1.49414 0.0498322
\(900\) 0 0
\(901\) 8.22338 0.273961
\(902\) −5.12217 −0.170550
\(903\) 0 0
\(904\) −6.96265 −0.231574
\(905\) 0 0
\(906\) 0 0
\(907\) 21.8880 0.726778 0.363389 0.931638i \(-0.381620\pi\)
0.363389 + 0.931638i \(0.381620\pi\)
\(908\) −11.4713 −0.380689
\(909\) 0 0
\(910\) 0 0
\(911\) −26.4249 −0.875496 −0.437748 0.899098i \(-0.644224\pi\)
−0.437748 + 0.899098i \(0.644224\pi\)
\(912\) 0 0
\(913\) 6.61350 0.218875
\(914\) 18.2745 0.604466
\(915\) 0 0
\(916\) 12.4768 0.412245
\(917\) 2.02467 0.0668603
\(918\) 0 0
\(919\) 12.5232 0.413103 0.206551 0.978436i \(-0.433776\pi\)
0.206551 + 0.978436i \(0.433776\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 13.0957 0.431286
\(923\) 9.89889 0.325826
\(924\) 0 0
\(925\) 0 0
\(926\) −15.0283 −0.493860
\(927\) 0 0
\(928\) −0.164979 −0.00541569
\(929\) 49.9253 1.63800 0.818998 0.573796i \(-0.194530\pi\)
0.818998 + 0.573796i \(0.194530\pi\)
\(930\) 0 0
\(931\) −6.96265 −0.228192
\(932\) 27.4021 0.897586
\(933\) 0 0
\(934\) −21.7302 −0.711034
\(935\) 0 0
\(936\) 0 0
\(937\) 40.7658 1.33176 0.665880 0.746059i \(-0.268056\pi\)
0.665880 + 0.746059i \(0.268056\pi\)
\(938\) 0.815820 0.0266375
\(939\) 0 0
\(940\) 0 0
\(941\) −13.2125 −0.430714 −0.215357 0.976535i \(-0.569091\pi\)
−0.215357 + 0.976535i \(0.569091\pi\)
\(942\) 0 0
\(943\) 23.1222 0.752961
\(944\) −13.9572 −0.454268
\(945\) 0 0
\(946\) −2.83502 −0.0921745
\(947\) −12.4996 −0.406182 −0.203091 0.979160i \(-0.565099\pi\)
−0.203091 + 0.979160i \(0.565099\pi\)
\(948\) 0 0
\(949\) −26.2015 −0.850538
\(950\) 0 0
\(951\) 0 0
\(952\) −0.547875 −0.0177567
\(953\) −19.8405 −0.642696 −0.321348 0.946961i \(-0.604136\pi\)
−0.321348 + 0.946961i \(0.604136\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 14.4057 0.465914
\(957\) 0 0
\(958\) 0.962653 0.0311019
\(959\) 2.25526 0.0728263
\(960\) 0 0
\(961\) 51.0211 1.64584
\(962\) 9.47133 0.305368
\(963\) 0 0
\(964\) −21.7886 −0.701764
\(965\) 0 0
\(966\) 0 0
\(967\) 21.9709 0.706537 0.353269 0.935522i \(-0.385070\pi\)
0.353269 + 0.935522i \(0.385070\pi\)
\(968\) −10.7357 −0.345057
\(969\) 0 0
\(970\) 0 0
\(971\) −14.3546 −0.460662 −0.230331 0.973112i \(-0.573981\pi\)
−0.230331 + 0.973112i \(0.573981\pi\)
\(972\) 0 0
\(973\) −2.46225 −0.0789362
\(974\) 42.5561 1.36359
\(975\) 0 0
\(976\) 5.92892 0.189780
\(977\) 51.4340 1.64552 0.822759 0.568390i \(-0.192433\pi\)
0.822759 + 0.568390i \(0.192433\pi\)
\(978\) 0 0
\(979\) 4.63710 0.148202
\(980\) 0 0
\(981\) 0 0
\(982\) 14.7549 0.470847
\(983\) 14.8350 0.473164 0.236582 0.971612i \(-0.423973\pi\)
0.236582 + 0.971612i \(0.423973\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −0.467718 −0.0148952
\(987\) 0 0
\(988\) −3.12763 −0.0995032
\(989\) 12.7977 0.406942
\(990\) 0 0
\(991\) 9.59349 0.304747 0.152374 0.988323i \(-0.451308\pi\)
0.152374 + 0.988323i \(0.451308\pi\)
\(992\) −9.05655 −0.287546
\(993\) 0 0
\(994\) 0.611640 0.0194000
\(995\) 0 0
\(996\) 0 0
\(997\) 24.7694 0.784455 0.392227 0.919868i \(-0.371705\pi\)
0.392227 + 0.919868i \(0.371705\pi\)
\(998\) −7.64538 −0.242010
\(999\) 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.cn.1.2 yes 3
3.2 odd 2 8550.2.a.cd.1.2 3
5.4 even 2 8550.2.a.cg.1.2 yes 3
15.14 odd 2 8550.2.a.cs.1.2 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8550.2.a.cd.1.2 3 3.2 odd 2
8550.2.a.cg.1.2 yes 3 5.4 even 2
8550.2.a.cn.1.2 yes 3 1.1 even 1 trivial
8550.2.a.cs.1.2 yes 3 15.14 odd 2