Properties

Label 6160.2.a.r.1.2
Level $6160$
Weight $2$
Character 6160.1
Self dual yes
Analytic conductor $49.188$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6160.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(49.1878476451\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \(x^{2} - x - 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.37228\) of defining polynomial
Character \(\chi\) \(=\) 6160.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} +6.74456 q^{13} -2.00000 q^{15} -6.74456 q^{17} +6.74456 q^{19} +2.00000 q^{21} +6.74456 q^{23} +1.00000 q^{25} +4.00000 q^{27} +8.74456 q^{29} -4.74456 q^{31} -2.00000 q^{33} -1.00000 q^{35} +0.744563 q^{37} -13.4891 q^{39} -4.00000 q^{41} -4.00000 q^{43} +1.00000 q^{45} -4.74456 q^{47} +1.00000 q^{49} +13.4891 q^{51} -12.7446 q^{53} +1.00000 q^{55} -13.4891 q^{57} +8.74456 q^{59} +1.25544 q^{61} -1.00000 q^{63} +6.74456 q^{65} +4.00000 q^{67} -13.4891 q^{69} +4.00000 q^{71} +10.7446 q^{73} -2.00000 q^{75} -1.00000 q^{77} -6.74456 q^{79} -11.0000 q^{81} -8.00000 q^{83} -6.74456 q^{85} -17.4891 q^{87} +15.4891 q^{89} -6.74456 q^{91} +9.48913 q^{93} +6.74456 q^{95} -16.7446 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9} + O(q^{10}) \) \( 2 q - 4 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{13} - 4 q^{15} - 2 q^{17} + 2 q^{19} + 4 q^{21} + 2 q^{23} + 2 q^{25} + 8 q^{27} + 6 q^{29} + 2 q^{31} - 4 q^{33} - 2 q^{35} - 10 q^{37} - 4 q^{39} - 8 q^{41} - 8 q^{43} + 2 q^{45} + 2 q^{47} + 2 q^{49} + 4 q^{51} - 14 q^{53} + 2 q^{55} - 4 q^{57} + 6 q^{59} + 14 q^{61} - 2 q^{63} + 2 q^{65} + 8 q^{67} - 4 q^{69} + 8 q^{71} + 10 q^{73} - 4 q^{75} - 2 q^{77} - 2 q^{79} - 22 q^{81} - 16 q^{83} - 2 q^{85} - 12 q^{87} + 8 q^{89} - 2 q^{91} - 4 q^{93} + 2 q^{95} - 22 q^{97} + 2 q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −2.00000 −1.15470 −0.577350 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 6.74456 1.87061 0.935303 0.353849i \(-0.115127\pi\)
0.935303 + 0.353849i \(0.115127\pi\)
\(14\) 0 0
\(15\) −2.00000 −0.516398
\(16\) 0 0
\(17\) −6.74456 −1.63580 −0.817898 0.575363i \(-0.804861\pi\)
−0.817898 + 0.575363i \(0.804861\pi\)
\(18\) 0 0
\(19\) 6.74456 1.54731 0.773654 0.633608i \(-0.218427\pi\)
0.773654 + 0.633608i \(0.218427\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 0 0
\(23\) 6.74456 1.40634 0.703169 0.711022i \(-0.251768\pi\)
0.703169 + 0.711022i \(0.251768\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 4.00000 0.769800
\(28\) 0 0
\(29\) 8.74456 1.62382 0.811912 0.583779i \(-0.198427\pi\)
0.811912 + 0.583779i \(0.198427\pi\)
\(30\) 0 0
\(31\) −4.74456 −0.852149 −0.426074 0.904688i \(-0.640104\pi\)
−0.426074 + 0.904688i \(0.640104\pi\)
\(32\) 0 0
\(33\) −2.00000 −0.348155
\(34\) 0 0
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) 0.744563 0.122405 0.0612027 0.998125i \(-0.480506\pi\)
0.0612027 + 0.998125i \(0.480506\pi\)
\(38\) 0 0
\(39\) −13.4891 −2.15999
\(40\) 0 0
\(41\) −4.00000 −0.624695 −0.312348 0.949968i \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) −4.74456 −0.692066 −0.346033 0.938222i \(-0.612471\pi\)
−0.346033 + 0.938222i \(0.612471\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 13.4891 1.88886
\(52\) 0 0
\(53\) −12.7446 −1.75060 −0.875300 0.483580i \(-0.839336\pi\)
−0.875300 + 0.483580i \(0.839336\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) 0 0
\(57\) −13.4891 −1.78668
\(58\) 0 0
\(59\) 8.74456 1.13845 0.569223 0.822183i \(-0.307244\pi\)
0.569223 + 0.822183i \(0.307244\pi\)
\(60\) 0 0
\(61\) 1.25544 0.160742 0.0803711 0.996765i \(-0.474389\pi\)
0.0803711 + 0.996765i \(0.474389\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) 6.74456 0.836560
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) −13.4891 −1.62390
\(70\) 0 0
\(71\) 4.00000 0.474713 0.237356 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) 0 0
\(73\) 10.7446 1.25756 0.628778 0.777585i \(-0.283555\pi\)
0.628778 + 0.777585i \(0.283555\pi\)
\(74\) 0 0
\(75\) −2.00000 −0.230940
\(76\) 0 0
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) −6.74456 −0.758823 −0.379411 0.925228i \(-0.623873\pi\)
−0.379411 + 0.925228i \(0.623873\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) −8.00000 −0.878114 −0.439057 0.898459i \(-0.644687\pi\)
−0.439057 + 0.898459i \(0.644687\pi\)
\(84\) 0 0
\(85\) −6.74456 −0.731551
\(86\) 0 0
\(87\) −17.4891 −1.87503
\(88\) 0 0
\(89\) 15.4891 1.64184 0.820922 0.571040i \(-0.193460\pi\)
0.820922 + 0.571040i \(0.193460\pi\)
\(90\) 0 0
\(91\) −6.74456 −0.707022
\(92\) 0 0
\(93\) 9.48913 0.983976
\(94\) 0 0
\(95\) 6.74456 0.691978
\(96\) 0 0
\(97\) −16.7446 −1.70015 −0.850076 0.526659i \(-0.823444\pi\)
−0.850076 + 0.526659i \(0.823444\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) 2.74456 0.273094 0.136547 0.990634i \(-0.456399\pi\)
0.136547 + 0.990634i \(0.456399\pi\)
\(102\) 0 0
\(103\) 0.744563 0.0733639 0.0366820 0.999327i \(-0.488321\pi\)
0.0366820 + 0.999327i \(0.488321\pi\)
\(104\) 0 0
\(105\) 2.00000 0.195180
\(106\) 0 0
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) −18.2337 −1.74647 −0.873235 0.487299i \(-0.837982\pi\)
−0.873235 + 0.487299i \(0.837982\pi\)
\(110\) 0 0
\(111\) −1.48913 −0.141342
\(112\) 0 0
\(113\) 10.0000 0.940721 0.470360 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) 0 0
\(115\) 6.74456 0.628934
\(116\) 0 0
\(117\) 6.74456 0.623535
\(118\) 0 0
\(119\) 6.74456 0.618273
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 8.00000 0.721336
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) 2.74456 0.239794 0.119897 0.992786i \(-0.461744\pi\)
0.119897 + 0.992786i \(0.461744\pi\)
\(132\) 0 0
\(133\) −6.74456 −0.584828
\(134\) 0 0
\(135\) 4.00000 0.344265
\(136\) 0 0
\(137\) 3.48913 0.298096 0.149048 0.988830i \(-0.452379\pi\)
0.149048 + 0.988830i \(0.452379\pi\)
\(138\) 0 0
\(139\) −14.7446 −1.25062 −0.625309 0.780377i \(-0.715027\pi\)
−0.625309 + 0.780377i \(0.715027\pi\)
\(140\) 0 0
\(141\) 9.48913 0.799129
\(142\) 0 0
\(143\) 6.74456 0.564009
\(144\) 0 0
\(145\) 8.74456 0.726196
\(146\) 0 0
\(147\) −2.00000 −0.164957
\(148\) 0 0
\(149\) −0.744563 −0.0609969 −0.0304985 0.999535i \(-0.509709\pi\)
−0.0304985 + 0.999535i \(0.509709\pi\)
\(150\) 0 0
\(151\) 9.25544 0.753197 0.376598 0.926377i \(-0.377094\pi\)
0.376598 + 0.926377i \(0.377094\pi\)
\(152\) 0 0
\(153\) −6.74456 −0.545266
\(154\) 0 0
\(155\) −4.74456 −0.381092
\(156\) 0 0
\(157\) 0.510875 0.0407722 0.0203861 0.999792i \(-0.493510\pi\)
0.0203861 + 0.999792i \(0.493510\pi\)
\(158\) 0 0
\(159\) 25.4891 2.02142
\(160\) 0 0
\(161\) −6.74456 −0.531546
\(162\) 0 0
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 0 0
\(165\) −2.00000 −0.155700
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 32.4891 2.49916
\(170\) 0 0
\(171\) 6.74456 0.515770
\(172\) 0 0
\(173\) 20.2337 1.53834 0.769169 0.639045i \(-0.220670\pi\)
0.769169 + 0.639045i \(0.220670\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) −17.4891 −1.31456
\(178\) 0 0
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) −19.4891 −1.44862 −0.724308 0.689477i \(-0.757840\pi\)
−0.724308 + 0.689477i \(0.757840\pi\)
\(182\) 0 0
\(183\) −2.51087 −0.185609
\(184\) 0 0
\(185\) 0.744563 0.0547413
\(186\) 0 0
\(187\) −6.74456 −0.493211
\(188\) 0 0
\(189\) −4.00000 −0.290957
\(190\) 0 0
\(191\) 16.0000 1.15772 0.578860 0.815427i \(-0.303498\pi\)
0.578860 + 0.815427i \(0.303498\pi\)
\(192\) 0 0
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) 0 0
\(195\) −13.4891 −0.965976
\(196\) 0 0
\(197\) 10.0000 0.712470 0.356235 0.934396i \(-0.384060\pi\)
0.356235 + 0.934396i \(0.384060\pi\)
\(198\) 0 0
\(199\) −3.25544 −0.230772 −0.115386 0.993321i \(-0.536810\pi\)
−0.115386 + 0.993321i \(0.536810\pi\)
\(200\) 0 0
\(201\) −8.00000 −0.564276
\(202\) 0 0
\(203\) −8.74456 −0.613748
\(204\) 0 0
\(205\) −4.00000 −0.279372
\(206\) 0 0
\(207\) 6.74456 0.468780
\(208\) 0 0
\(209\) 6.74456 0.466531
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 0 0
\(213\) −8.00000 −0.548151
\(214\) 0 0
\(215\) −4.00000 −0.272798
\(216\) 0 0
\(217\) 4.74456 0.322082
\(218\) 0 0
\(219\) −21.4891 −1.45210
\(220\) 0 0
\(221\) −45.4891 −3.05993
\(222\) 0 0
\(223\) −15.2554 −1.02158 −0.510790 0.859706i \(-0.670647\pi\)
−0.510790 + 0.859706i \(0.670647\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −20.0000 −1.32745 −0.663723 0.747978i \(-0.731025\pi\)
−0.663723 + 0.747978i \(0.731025\pi\)
\(228\) 0 0
\(229\) −6.00000 −0.396491 −0.198246 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) 0 0
\(231\) 2.00000 0.131590
\(232\) 0 0
\(233\) −24.9783 −1.63638 −0.818190 0.574948i \(-0.805022\pi\)
−0.818190 + 0.574948i \(0.805022\pi\)
\(234\) 0 0
\(235\) −4.74456 −0.309501
\(236\) 0 0
\(237\) 13.4891 0.876213
\(238\) 0 0
\(239\) 14.7446 0.953746 0.476873 0.878972i \(-0.341770\pi\)
0.476873 + 0.878972i \(0.341770\pi\)
\(240\) 0 0
\(241\) 20.0000 1.28831 0.644157 0.764894i \(-0.277208\pi\)
0.644157 + 0.764894i \(0.277208\pi\)
\(242\) 0 0
\(243\) 10.0000 0.641500
\(244\) 0 0
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) 45.4891 2.89440
\(248\) 0 0
\(249\) 16.0000 1.01396
\(250\) 0 0
\(251\) 26.2337 1.65586 0.827928 0.560835i \(-0.189520\pi\)
0.827928 + 0.560835i \(0.189520\pi\)
\(252\) 0 0
\(253\) 6.74456 0.424027
\(254\) 0 0
\(255\) 13.4891 0.844722
\(256\) 0 0
\(257\) −10.2337 −0.638360 −0.319180 0.947694i \(-0.603407\pi\)
−0.319180 + 0.947694i \(0.603407\pi\)
\(258\) 0 0
\(259\) −0.744563 −0.0462649
\(260\) 0 0
\(261\) 8.74456 0.541275
\(262\) 0 0
\(263\) −26.9783 −1.66355 −0.831775 0.555113i \(-0.812675\pi\)
−0.831775 + 0.555113i \(0.812675\pi\)
\(264\) 0 0
\(265\) −12.7446 −0.782892
\(266\) 0 0
\(267\) −30.9783 −1.89584
\(268\) 0 0
\(269\) 20.9783 1.27907 0.639533 0.768763i \(-0.279128\pi\)
0.639533 + 0.768763i \(0.279128\pi\)
\(270\) 0 0
\(271\) −14.9783 −0.909864 −0.454932 0.890526i \(-0.650336\pi\)
−0.454932 + 0.890526i \(0.650336\pi\)
\(272\) 0 0
\(273\) 13.4891 0.816399
\(274\) 0 0
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) 15.4891 0.930651 0.465326 0.885140i \(-0.345937\pi\)
0.465326 + 0.885140i \(0.345937\pi\)
\(278\) 0 0
\(279\) −4.74456 −0.284050
\(280\) 0 0
\(281\) 14.0000 0.835170 0.417585 0.908638i \(-0.362877\pi\)
0.417585 + 0.908638i \(0.362877\pi\)
\(282\) 0 0
\(283\) −28.0000 −1.66443 −0.832214 0.554455i \(-0.812927\pi\)
−0.832214 + 0.554455i \(0.812927\pi\)
\(284\) 0 0
\(285\) −13.4891 −0.799027
\(286\) 0 0
\(287\) 4.00000 0.236113
\(288\) 0 0
\(289\) 28.4891 1.67583
\(290\) 0 0
\(291\) 33.4891 1.96317
\(292\) 0 0
\(293\) 13.2554 0.774391 0.387195 0.921998i \(-0.373444\pi\)
0.387195 + 0.921998i \(0.373444\pi\)
\(294\) 0 0
\(295\) 8.74456 0.509128
\(296\) 0 0
\(297\) 4.00000 0.232104
\(298\) 0 0
\(299\) 45.4891 2.63070
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) 0 0
\(303\) −5.48913 −0.315342
\(304\) 0 0
\(305\) 1.25544 0.0718861
\(306\) 0 0
\(307\) 1.48913 0.0849889 0.0424944 0.999097i \(-0.486470\pi\)
0.0424944 + 0.999097i \(0.486470\pi\)
\(308\) 0 0
\(309\) −1.48913 −0.0847134
\(310\) 0 0
\(311\) 20.7446 1.17632 0.588158 0.808746i \(-0.299853\pi\)
0.588158 + 0.808746i \(0.299853\pi\)
\(312\) 0 0
\(313\) −2.23369 −0.126256 −0.0631278 0.998005i \(-0.520108\pi\)
−0.0631278 + 0.998005i \(0.520108\pi\)
\(314\) 0 0
\(315\) −1.00000 −0.0563436
\(316\) 0 0
\(317\) 2.23369 0.125456 0.0627282 0.998031i \(-0.480020\pi\)
0.0627282 + 0.998031i \(0.480020\pi\)
\(318\) 0 0
\(319\) 8.74456 0.489602
\(320\) 0 0
\(321\) −24.0000 −1.33955
\(322\) 0 0
\(323\) −45.4891 −2.53108
\(324\) 0 0
\(325\) 6.74456 0.374121
\(326\) 0 0
\(327\) 36.4674 2.01665
\(328\) 0 0
\(329\) 4.74456 0.261576
\(330\) 0 0
\(331\) −14.9783 −0.823279 −0.411640 0.911347i \(-0.635044\pi\)
−0.411640 + 0.911347i \(0.635044\pi\)
\(332\) 0 0
\(333\) 0.744563 0.0408018
\(334\) 0 0
\(335\) 4.00000 0.218543
\(336\) 0 0
\(337\) 26.0000 1.41631 0.708155 0.706057i \(-0.249528\pi\)
0.708155 + 0.706057i \(0.249528\pi\)
\(338\) 0 0
\(339\) −20.0000 −1.08625
\(340\) 0 0
\(341\) −4.74456 −0.256932
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −13.4891 −0.726230
\(346\) 0 0
\(347\) 22.9783 1.23354 0.616769 0.787145i \(-0.288441\pi\)
0.616769 + 0.787145i \(0.288441\pi\)
\(348\) 0 0
\(349\) 30.7446 1.64572 0.822859 0.568245i \(-0.192377\pi\)
0.822859 + 0.568245i \(0.192377\pi\)
\(350\) 0 0
\(351\) 26.9783 1.43999
\(352\) 0 0
\(353\) 8.74456 0.465426 0.232713 0.972545i \(-0.425240\pi\)
0.232713 + 0.972545i \(0.425240\pi\)
\(354\) 0 0
\(355\) 4.00000 0.212298
\(356\) 0 0
\(357\) −13.4891 −0.713920
\(358\) 0 0
\(359\) 1.25544 0.0662594 0.0331297 0.999451i \(-0.489453\pi\)
0.0331297 + 0.999451i \(0.489453\pi\)
\(360\) 0 0
\(361\) 26.4891 1.39416
\(362\) 0 0
\(363\) −2.00000 −0.104973
\(364\) 0 0
\(365\) 10.7446 0.562396
\(366\) 0 0
\(367\) 8.74456 0.456462 0.228231 0.973607i \(-0.426706\pi\)
0.228231 + 0.973607i \(0.426706\pi\)
\(368\) 0 0
\(369\) −4.00000 −0.208232
\(370\) 0 0
\(371\) 12.7446 0.661665
\(372\) 0 0
\(373\) −16.9783 −0.879100 −0.439550 0.898218i \(-0.644862\pi\)
−0.439550 + 0.898218i \(0.644862\pi\)
\(374\) 0 0
\(375\) −2.00000 −0.103280
\(376\) 0 0
\(377\) 58.9783 3.03753
\(378\) 0 0
\(379\) 37.4891 1.92569 0.962844 0.270060i \(-0.0870435\pi\)
0.962844 + 0.270060i \(0.0870435\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −19.7228 −1.00779 −0.503894 0.863765i \(-0.668100\pi\)
−0.503894 + 0.863765i \(0.668100\pi\)
\(384\) 0 0
\(385\) −1.00000 −0.0509647
\(386\) 0 0
\(387\) −4.00000 −0.203331
\(388\) 0 0
\(389\) −7.48913 −0.379714 −0.189857 0.981812i \(-0.560802\pi\)
−0.189857 + 0.981812i \(0.560802\pi\)
\(390\) 0 0
\(391\) −45.4891 −2.30048
\(392\) 0 0
\(393\) −5.48913 −0.276890
\(394\) 0 0
\(395\) −6.74456 −0.339356
\(396\) 0 0
\(397\) 35.4891 1.78115 0.890574 0.454838i \(-0.150303\pi\)
0.890574 + 0.454838i \(0.150303\pi\)
\(398\) 0 0
\(399\) 13.4891 0.675301
\(400\) 0 0
\(401\) 23.4891 1.17299 0.586495 0.809953i \(-0.300507\pi\)
0.586495 + 0.809953i \(0.300507\pi\)
\(402\) 0 0
\(403\) −32.0000 −1.59403
\(404\) 0 0
\(405\) −11.0000 −0.546594
\(406\) 0 0
\(407\) 0.744563 0.0369066
\(408\) 0 0
\(409\) −21.4891 −1.06257 −0.531284 0.847194i \(-0.678290\pi\)
−0.531284 + 0.847194i \(0.678290\pi\)
\(410\) 0 0
\(411\) −6.97825 −0.344212
\(412\) 0 0
\(413\) −8.74456 −0.430292
\(414\) 0 0
\(415\) −8.00000 −0.392705
\(416\) 0 0
\(417\) 29.4891 1.44409
\(418\) 0 0
\(419\) 0.744563 0.0363743 0.0181871 0.999835i \(-0.494211\pi\)
0.0181871 + 0.999835i \(0.494211\pi\)
\(420\) 0 0
\(421\) 4.51087 0.219847 0.109923 0.993940i \(-0.464939\pi\)
0.109923 + 0.993940i \(0.464939\pi\)
\(422\) 0 0
\(423\) −4.74456 −0.230689
\(424\) 0 0
\(425\) −6.74456 −0.327159
\(426\) 0 0
\(427\) −1.25544 −0.0607549
\(428\) 0 0
\(429\) −13.4891 −0.651261
\(430\) 0 0
\(431\) 17.2554 0.831165 0.415583 0.909555i \(-0.363578\pi\)
0.415583 + 0.909555i \(0.363578\pi\)
\(432\) 0 0
\(433\) 36.7446 1.76583 0.882915 0.469532i \(-0.155577\pi\)
0.882915 + 0.469532i \(0.155577\pi\)
\(434\) 0 0
\(435\) −17.4891 −0.838539
\(436\) 0 0
\(437\) 45.4891 2.17604
\(438\) 0 0
\(439\) −14.9783 −0.714873 −0.357436 0.933937i \(-0.616349\pi\)
−0.357436 + 0.933937i \(0.616349\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 29.4891 1.40107 0.700535 0.713618i \(-0.252945\pi\)
0.700535 + 0.713618i \(0.252945\pi\)
\(444\) 0 0
\(445\) 15.4891 0.734255
\(446\) 0 0
\(447\) 1.48913 0.0704332
\(448\) 0 0
\(449\) −4.97825 −0.234938 −0.117469 0.993077i \(-0.537478\pi\)
−0.117469 + 0.993077i \(0.537478\pi\)
\(450\) 0 0
\(451\) −4.00000 −0.188353
\(452\) 0 0
\(453\) −18.5109 −0.869717
\(454\) 0 0
\(455\) −6.74456 −0.316190
\(456\) 0 0
\(457\) 3.48913 0.163214 0.0816072 0.996665i \(-0.473995\pi\)
0.0816072 + 0.996665i \(0.473995\pi\)
\(458\) 0 0
\(459\) −26.9783 −1.25924
\(460\) 0 0
\(461\) 33.7228 1.57063 0.785314 0.619098i \(-0.212502\pi\)
0.785314 + 0.619098i \(0.212502\pi\)
\(462\) 0 0
\(463\) −1.25544 −0.0583451 −0.0291726 0.999574i \(-0.509287\pi\)
−0.0291726 + 0.999574i \(0.509287\pi\)
\(464\) 0 0
\(465\) 9.48913 0.440048
\(466\) 0 0
\(467\) −16.9783 −0.785660 −0.392830 0.919611i \(-0.628504\pi\)
−0.392830 + 0.919611i \(0.628504\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) 0 0
\(471\) −1.02175 −0.0470797
\(472\) 0 0
\(473\) −4.00000 −0.183920
\(474\) 0 0
\(475\) 6.74456 0.309462
\(476\) 0 0
\(477\) −12.7446 −0.583533
\(478\) 0 0
\(479\) 41.4891 1.89569 0.947843 0.318737i \(-0.103259\pi\)
0.947843 + 0.318737i \(0.103259\pi\)
\(480\) 0 0
\(481\) 5.02175 0.228972
\(482\) 0 0
\(483\) 13.4891 0.613776
\(484\) 0 0
\(485\) −16.7446 −0.760331
\(486\) 0 0
\(487\) −9.25544 −0.419404 −0.209702 0.977765i \(-0.567249\pi\)
−0.209702 + 0.977765i \(0.567249\pi\)
\(488\) 0 0
\(489\) −8.00000 −0.361773
\(490\) 0 0
\(491\) −30.9783 −1.39803 −0.699014 0.715108i \(-0.746378\pi\)
−0.699014 + 0.715108i \(0.746378\pi\)
\(492\) 0 0
\(493\) −58.9783 −2.65625
\(494\) 0 0
\(495\) 1.00000 0.0449467
\(496\) 0 0
\(497\) −4.00000 −0.179425
\(498\) 0 0
\(499\) 13.4891 0.603856 0.301928 0.953331i \(-0.402370\pi\)
0.301928 + 0.953331i \(0.402370\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −6.51087 −0.290306 −0.145153 0.989409i \(-0.546367\pi\)
−0.145153 + 0.989409i \(0.546367\pi\)
\(504\) 0 0
\(505\) 2.74456 0.122131
\(506\) 0 0
\(507\) −64.9783 −2.88579
\(508\) 0 0
\(509\) −35.4891 −1.57303 −0.786514 0.617573i \(-0.788116\pi\)
−0.786514 + 0.617573i \(0.788116\pi\)
\(510\) 0 0
\(511\) −10.7446 −0.475311
\(512\) 0 0
\(513\) 26.9783 1.19112
\(514\) 0 0
\(515\) 0.744563 0.0328094
\(516\) 0 0
\(517\) −4.74456 −0.208666
\(518\) 0 0
\(519\) −40.4674 −1.77632
\(520\) 0 0
\(521\) −2.00000 −0.0876216 −0.0438108 0.999040i \(-0.513950\pi\)
−0.0438108 + 0.999040i \(0.513950\pi\)
\(522\) 0 0
\(523\) 17.4891 0.764746 0.382373 0.924008i \(-0.375107\pi\)
0.382373 + 0.924008i \(0.375107\pi\)
\(524\) 0 0
\(525\) 2.00000 0.0872872
\(526\) 0 0
\(527\) 32.0000 1.39394
\(528\) 0 0
\(529\) 22.4891 0.977788
\(530\) 0 0
\(531\) 8.74456 0.379482
\(532\) 0 0
\(533\) −26.9783 −1.16856
\(534\) 0 0
\(535\) 12.0000 0.518805
\(536\) 0 0
\(537\) −8.00000 −0.345225
\(538\) 0 0
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) 1.76631 0.0759397 0.0379698 0.999279i \(-0.487911\pi\)
0.0379698 + 0.999279i \(0.487911\pi\)
\(542\) 0 0
\(543\) 38.9783 1.67272
\(544\) 0 0
\(545\) −18.2337 −0.781045
\(546\) 0 0
\(547\) −14.9783 −0.640424 −0.320212 0.947346i \(-0.603754\pi\)
−0.320212 + 0.947346i \(0.603754\pi\)
\(548\) 0 0
\(549\) 1.25544 0.0535808
\(550\) 0 0
\(551\) 58.9783 2.51256
\(552\) 0 0
\(553\) 6.74456 0.286808
\(554\) 0 0
\(555\) −1.48913 −0.0632098
\(556\) 0 0
\(557\) −0.978251 −0.0414498 −0.0207249 0.999785i \(-0.506597\pi\)
−0.0207249 + 0.999785i \(0.506597\pi\)
\(558\) 0 0
\(559\) −26.9783 −1.14106
\(560\) 0 0
\(561\) 13.4891 0.569511
\(562\) 0 0
\(563\) −5.48913 −0.231339 −0.115670 0.993288i \(-0.536901\pi\)
−0.115670 + 0.993288i \(0.536901\pi\)
\(564\) 0 0
\(565\) 10.0000 0.420703
\(566\) 0 0
\(567\) 11.0000 0.461957
\(568\) 0 0
\(569\) 16.5109 0.692172 0.346086 0.938203i \(-0.387511\pi\)
0.346086 + 0.938203i \(0.387511\pi\)
\(570\) 0 0
\(571\) 17.4891 0.731897 0.365949 0.930635i \(-0.380745\pi\)
0.365949 + 0.930635i \(0.380745\pi\)
\(572\) 0 0
\(573\) −32.0000 −1.33682
\(574\) 0 0
\(575\) 6.74456 0.281268
\(576\) 0 0
\(577\) −8.74456 −0.364041 −0.182020 0.983295i \(-0.558264\pi\)
−0.182020 + 0.983295i \(0.558264\pi\)
\(578\) 0 0
\(579\) 4.00000 0.166234
\(580\) 0 0
\(581\) 8.00000 0.331896
\(582\) 0 0
\(583\) −12.7446 −0.527826
\(584\) 0 0
\(585\) 6.74456 0.278853
\(586\) 0 0
\(587\) 4.97825 0.205474 0.102737 0.994709i \(-0.467240\pi\)
0.102737 + 0.994709i \(0.467240\pi\)
\(588\) 0 0
\(589\) −32.0000 −1.31854
\(590\) 0 0
\(591\) −20.0000 −0.822690
\(592\) 0 0
\(593\) −40.2337 −1.65220 −0.826100 0.563524i \(-0.809445\pi\)
−0.826100 + 0.563524i \(0.809445\pi\)
\(594\) 0 0
\(595\) 6.74456 0.276500
\(596\) 0 0
\(597\) 6.51087 0.266472
\(598\) 0 0
\(599\) 12.0000 0.490307 0.245153 0.969484i \(-0.421162\pi\)
0.245153 + 0.969484i \(0.421162\pi\)
\(600\) 0 0
\(601\) −33.4891 −1.36605 −0.683025 0.730395i \(-0.739336\pi\)
−0.683025 + 0.730395i \(0.739336\pi\)
\(602\) 0 0
\(603\) 4.00000 0.162893
\(604\) 0 0
\(605\) 1.00000 0.0406558
\(606\) 0 0
\(607\) −8.00000 −0.324710 −0.162355 0.986732i \(-0.551909\pi\)
−0.162355 + 0.986732i \(0.551909\pi\)
\(608\) 0 0
\(609\) 17.4891 0.708695
\(610\) 0 0
\(611\) −32.0000 −1.29458
\(612\) 0 0
\(613\) 11.4891 0.464041 0.232021 0.972711i \(-0.425466\pi\)
0.232021 + 0.972711i \(0.425466\pi\)
\(614\) 0 0
\(615\) 8.00000 0.322591
\(616\) 0 0
\(617\) −10.0000 −0.402585 −0.201292 0.979531i \(-0.564514\pi\)
−0.201292 + 0.979531i \(0.564514\pi\)
\(618\) 0 0
\(619\) −0.744563 −0.0299265 −0.0149632 0.999888i \(-0.504763\pi\)
−0.0149632 + 0.999888i \(0.504763\pi\)
\(620\) 0 0
\(621\) 26.9783 1.08260
\(622\) 0 0
\(623\) −15.4891 −0.620559
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −13.4891 −0.538704
\(628\) 0 0
\(629\) −5.02175 −0.200230
\(630\) 0 0
\(631\) 2.97825 0.118562 0.0592811 0.998241i \(-0.481119\pi\)
0.0592811 + 0.998241i \(0.481119\pi\)
\(632\) 0 0
\(633\) −24.0000 −0.953914
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 6.74456 0.267229
\(638\) 0 0
\(639\) 4.00000 0.158238
\(640\) 0 0
\(641\) 11.4891 0.453793 0.226897 0.973919i \(-0.427142\pi\)
0.226897 + 0.973919i \(0.427142\pi\)
\(642\) 0 0
\(643\) −46.4674 −1.83249 −0.916247 0.400613i \(-0.868797\pi\)
−0.916247 + 0.400613i \(0.868797\pi\)
\(644\) 0 0
\(645\) 8.00000 0.315000
\(646\) 0 0
\(647\) −8.74456 −0.343784 −0.171892 0.985116i \(-0.554988\pi\)
−0.171892 + 0.985116i \(0.554988\pi\)
\(648\) 0 0
\(649\) 8.74456 0.343254
\(650\) 0 0
\(651\) −9.48913 −0.371908
\(652\) 0 0
\(653\) 15.7228 0.615281 0.307641 0.951503i \(-0.400461\pi\)
0.307641 + 0.951503i \(0.400461\pi\)
\(654\) 0 0
\(655\) 2.74456 0.107239
\(656\) 0 0
\(657\) 10.7446 0.419185
\(658\) 0 0
\(659\) 41.4891 1.61619 0.808093 0.589054i \(-0.200500\pi\)
0.808093 + 0.589054i \(0.200500\pi\)
\(660\) 0 0
\(661\) 22.0000 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) 0 0
\(663\) 90.9783 3.53330
\(664\) 0 0
\(665\) −6.74456 −0.261543
\(666\) 0 0
\(667\) 58.9783 2.28365
\(668\) 0 0
\(669\) 30.5109 1.17962
\(670\) 0 0
\(671\) 1.25544 0.0484656
\(672\) 0 0
\(673\) 20.9783 0.808652 0.404326 0.914615i \(-0.367506\pi\)
0.404326 + 0.914615i \(0.367506\pi\)
\(674\) 0 0
\(675\) 4.00000 0.153960
\(676\) 0 0
\(677\) 36.2337 1.39257 0.696287 0.717764i \(-0.254834\pi\)
0.696287 + 0.717764i \(0.254834\pi\)
\(678\) 0 0
\(679\) 16.7446 0.642597
\(680\) 0 0
\(681\) 40.0000 1.53280
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) 3.48913 0.133313
\(686\) 0 0
\(687\) 12.0000 0.457829
\(688\) 0 0
\(689\) −85.9565 −3.27468
\(690\) 0 0
\(691\) −1.76631 −0.0671937 −0.0335968 0.999435i \(-0.510696\pi\)
−0.0335968 + 0.999435i \(0.510696\pi\)
\(692\) 0 0
\(693\) −1.00000 −0.0379869
\(694\) 0 0
\(695\) −14.7446 −0.559293
\(696\) 0 0
\(697\) 26.9783 1.02187
\(698\) 0 0
\(699\) 49.9565 1.88953
\(700\) 0 0
\(701\) 22.2337 0.839755 0.419877 0.907581i \(-0.362073\pi\)
0.419877 + 0.907581i \(0.362073\pi\)
\(702\) 0 0
\(703\) 5.02175 0.189399
\(704\) 0 0
\(705\) 9.48913 0.357381
\(706\) 0 0
\(707\) −2.74456 −0.103220
\(708\) 0 0
\(709\) 0.510875 0.0191863 0.00959315 0.999954i \(-0.496946\pi\)
0.00959315 + 0.999954i \(0.496946\pi\)
\(710\) 0 0
\(711\) −6.74456 −0.252941
\(712\) 0 0
\(713\) −32.0000 −1.19841
\(714\) 0 0
\(715\) 6.74456 0.252232
\(716\) 0 0
\(717\) −29.4891 −1.10129
\(718\) 0 0
\(719\) −7.72281 −0.288012 −0.144006 0.989577i \(-0.545999\pi\)
−0.144006 + 0.989577i \(0.545999\pi\)
\(720\) 0 0
\(721\) −0.744563 −0.0277290
\(722\) 0 0
\(723\) −40.0000 −1.48762
\(724\) 0 0
\(725\) 8.74456 0.324765
\(726\) 0 0
\(727\) −14.2337 −0.527898 −0.263949 0.964537i \(-0.585025\pi\)
−0.263949 + 0.964537i \(0.585025\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 26.9783 0.997827
\(732\) 0 0
\(733\) −16.2337 −0.599605 −0.299802 0.954001i \(-0.596921\pi\)
−0.299802 + 0.954001i \(0.596921\pi\)
\(734\) 0 0
\(735\) −2.00000 −0.0737711
\(736\) 0 0
\(737\) 4.00000 0.147342
\(738\) 0 0
\(739\) 30.9783 1.13955 0.569777 0.821800i \(-0.307030\pi\)
0.569777 + 0.821800i \(0.307030\pi\)
\(740\) 0 0
\(741\) −90.9783 −3.34217
\(742\) 0 0
\(743\) −26.9783 −0.989736 −0.494868 0.868968i \(-0.664784\pi\)
−0.494868 + 0.868968i \(0.664784\pi\)
\(744\) 0 0
\(745\) −0.744563 −0.0272787
\(746\) 0 0
\(747\) −8.00000 −0.292705
\(748\) 0 0
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) −20.0000 −0.729810 −0.364905 0.931045i \(-0.618899\pi\)
−0.364905 + 0.931045i \(0.618899\pi\)
\(752\) 0 0
\(753\) −52.4674 −1.91202
\(754\) 0 0
\(755\) 9.25544 0.336840
\(756\) 0 0
\(757\) −19.2554 −0.699851 −0.349925 0.936778i \(-0.613793\pi\)
−0.349925 + 0.936778i \(0.613793\pi\)
\(758\) 0 0
\(759\) −13.4891 −0.489624
\(760\) 0 0
\(761\) −29.4891 −1.06898 −0.534490 0.845175i \(-0.679496\pi\)
−0.534490 + 0.845175i \(0.679496\pi\)
\(762\) 0 0
\(763\) 18.2337 0.660104
\(764\) 0 0
\(765\) −6.74456 −0.243850
\(766\) 0 0
\(767\) 58.9783 2.12958
\(768\) 0 0
\(769\) −8.00000 −0.288487 −0.144244 0.989542i \(-0.546075\pi\)
−0.144244 + 0.989542i \(0.546075\pi\)
\(770\) 0 0
\(771\) 20.4674 0.737115
\(772\) 0 0
\(773\) −51.4891 −1.85194 −0.925968 0.377603i \(-0.876748\pi\)
−0.925968 + 0.377603i \(0.876748\pi\)
\(774\) 0 0
\(775\) −4.74456 −0.170430
\(776\) 0 0
\(777\) 1.48913 0.0534221
\(778\) 0 0
\(779\) −26.9783 −0.966596
\(780\) 0 0
\(781\) 4.00000 0.143131
\(782\) 0 0
\(783\) 34.9783 1.25002
\(784\) 0 0
\(785\) 0.510875 0.0182339
\(786\) 0 0
\(787\) −5.48913 −0.195666 −0.0978331 0.995203i \(-0.531191\pi\)
−0.0978331 + 0.995203i \(0.531191\pi\)
\(788\) 0 0
\(789\) 53.9565 1.92090
\(790\) 0 0
\(791\) −10.0000 −0.355559
\(792\) 0 0
\(793\) 8.46738 0.300685
\(794\) 0 0
\(795\) 25.4891 0.904006
\(796\) 0 0
\(797\) −42.4674 −1.50427 −0.752136 0.659008i \(-0.770976\pi\)
−0.752136 + 0.659008i \(0.770976\pi\)
\(798\) 0 0
\(799\) 32.0000 1.13208
\(800\) 0 0
\(801\) 15.4891 0.547281
\(802\) 0 0
\(803\) 10.7446 0.379167
\(804\) 0 0
\(805\) −6.74456 −0.237715
\(806\) 0 0
\(807\) −41.9565 −1.47694
\(808\) 0 0
\(809\) 34.4674 1.21181 0.605904 0.795538i \(-0.292811\pi\)
0.605904 + 0.795538i \(0.292811\pi\)
\(810\) 0 0
\(811\) 18.7446 0.658211 0.329105 0.944293i \(-0.393253\pi\)
0.329105 + 0.944293i \(0.393253\pi\)
\(812\) 0 0
\(813\) 29.9565 1.05062
\(814\) 0 0
\(815\) 4.00000 0.140114
\(816\) 0 0
\(817\) −26.9783 −0.943850
\(818\) 0 0
\(819\) −6.74456 −0.235674
\(820\) 0 0
\(821\) 7.72281 0.269528 0.134764 0.990878i \(-0.456972\pi\)
0.134764 + 0.990878i \(0.456972\pi\)
\(822\) 0 0
\(823\) −7.76631 −0.270717 −0.135358 0.990797i \(-0.543219\pi\)
−0.135358 + 0.990797i \(0.543219\pi\)
\(824\) 0 0
\(825\) −2.00000 −0.0696311
\(826\) 0 0
\(827\) −4.00000 −0.139094 −0.0695468 0.997579i \(-0.522155\pi\)
−0.0695468 + 0.997579i \(0.522155\pi\)
\(828\) 0 0
\(829\) 14.4674 0.502473 0.251236 0.967926i \(-0.419163\pi\)
0.251236 + 0.967926i \(0.419163\pi\)
\(830\) 0 0
\(831\) −30.9783 −1.07462
\(832\) 0 0
\(833\) −6.74456 −0.233685
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −18.9783 −0.655984
\(838\) 0 0
\(839\) 3.25544 0.112390 0.0561951 0.998420i \(-0.482103\pi\)
0.0561951 + 0.998420i \(0.482103\pi\)
\(840\) 0 0
\(841\) 47.4674 1.63681
\(842\) 0 0
\(843\) −28.0000 −0.964371
\(844\) 0 0
\(845\) 32.4891 1.11766
\(846\) 0 0
\(847\) −1.00000 −0.0343604
\(848\) 0 0
\(849\) 56.0000 1.92192
\(850\) 0 0
\(851\) 5.02175 0.172143
\(852\) 0 0
\(853\) −22.7446 −0.778759 −0.389379 0.921077i \(-0.627311\pi\)
−0.389379 + 0.921077i \(0.627311\pi\)
\(854\) 0 0
\(855\) 6.74456 0.230659
\(856\) 0 0
\(857\) −43.2119 −1.47609 −0.738046 0.674751i \(-0.764251\pi\)
−0.738046 + 0.674751i \(0.764251\pi\)
\(858\) 0 0
\(859\) 29.2119 0.996698 0.498349 0.866976i \(-0.333940\pi\)
0.498349 + 0.866976i \(0.333940\pi\)
\(860\) 0 0
\(861\) −8.00000 −0.272639
\(862\) 0 0
\(863\) 40.2337 1.36957 0.684785 0.728745i \(-0.259896\pi\)
0.684785 + 0.728745i \(0.259896\pi\)
\(864\) 0 0
\(865\) 20.2337 0.687966
\(866\) 0 0
\(867\) −56.9783 −1.93508
\(868\) 0 0
\(869\) −6.74456 −0.228794
\(870\) 0 0
\(871\) 26.9783 0.914123
\(872\) 0 0
\(873\) −16.7446 −0.566718
\(874\) 0 0
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) −26.4674 −0.893740 −0.446870 0.894599i \(-0.647461\pi\)
−0.446870 + 0.894599i \(0.647461\pi\)
\(878\) 0 0
\(879\) −26.5109 −0.894190
\(880\) 0 0
\(881\) −55.4891 −1.86948 −0.934738 0.355338i \(-0.884366\pi\)
−0.934738 + 0.355338i \(0.884366\pi\)
\(882\) 0 0
\(883\) 8.00000 0.269221 0.134611 0.990899i \(-0.457022\pi\)
0.134611 + 0.990899i \(0.457022\pi\)
\(884\) 0 0
\(885\) −17.4891 −0.587891
\(886\) 0 0
\(887\) 41.4891 1.39307 0.696534 0.717524i \(-0.254724\pi\)
0.696534 + 0.717524i \(0.254724\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −11.0000 −0.368514
\(892\) 0 0
\(893\) −32.0000 −1.07084
\(894\) 0 0
\(895\) 4.00000 0.133705
\(896\) 0 0
\(897\) −90.9783 −3.03768
\(898\) 0 0
\(899\) −41.4891 −1.38374
\(900\) 0 0
\(901\) 85.9565 2.86363
\(902\) 0 0
\(903\) −8.00000 −0.266223
\(904\) 0 0
\(905\) −19.4891 −0.647840
\(906\) 0 0
\(907\) 14.5109 0.481826 0.240913 0.970547i \(-0.422553\pi\)
0.240913 + 0.970547i \(0.422553\pi\)
\(908\) 0 0
\(909\) 2.74456 0.0910314
\(910\) 0 0
\(911\) −20.0000 −0.662630 −0.331315 0.943520i \(-0.607492\pi\)
−0.331315 + 0.943520i \(0.607492\pi\)
\(912\) 0 0
\(913\) −8.00000 −0.264761
\(914\) 0 0
\(915\) −2.51087 −0.0830070
\(916\) 0 0
\(917\) −2.74456 −0.0906334
\(918\) 0 0
\(919\) 55.2119 1.82127 0.910637 0.413207i \(-0.135592\pi\)
0.910637 + 0.413207i \(0.135592\pi\)
\(920\) 0 0
\(921\) −2.97825 −0.0981367
\(922\) 0 0
\(923\) 26.9783 0.888000
\(924\) 0 0
\(925\) 0.744563 0.0244811
\(926\) 0 0
\(927\) 0.744563 0.0244546
\(928\) 0 0
\(929\) −28.9783 −0.950746 −0.475373 0.879784i \(-0.657687\pi\)
−0.475373 + 0.879784i \(0.657687\pi\)
\(930\) 0 0
\(931\) 6.74456 0.221044
\(932\) 0 0
\(933\) −41.4891 −1.35829
\(934\) 0 0
\(935\) −6.74456 −0.220571
\(936\) 0 0
\(937\) −18.7446 −0.612358 −0.306179 0.951974i \(-0.599051\pi\)
−0.306179 + 0.951974i \(0.599051\pi\)
\(938\) 0 0
\(939\) 4.46738 0.145787
\(940\) 0 0
\(941\) 12.2337 0.398807 0.199403 0.979917i \(-0.436100\pi\)
0.199403 + 0.979917i \(0.436100\pi\)
\(942\) 0 0
\(943\) −26.9783 −0.878533
\(944\) 0 0
\(945\) −4.00000 −0.130120
\(946\) 0 0
\(947\) 8.00000 0.259965 0.129983 0.991516i \(-0.458508\pi\)
0.129983 + 0.991516i \(0.458508\pi\)
\(948\) 0 0
\(949\) 72.4674 2.35239
\(950\) 0 0
\(951\) −4.46738 −0.144865
\(952\) 0 0
\(953\) 6.00000 0.194359 0.0971795 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) 0 0
\(955\) 16.0000 0.517748
\(956\) 0 0
\(957\) −17.4891 −0.565343
\(958\) 0 0
\(959\) −3.48913 −0.112670
\(960\) 0 0
\(961\) −8.48913 −0.273843
\(962\) 0 0
\(963\) 12.0000 0.386695
\(964\) 0 0
\(965\) −2.00000 −0.0643823
\(966\) 0 0
\(967\) 50.9783 1.63935 0.819675 0.572829i \(-0.194154\pi\)
0.819675 + 0.572829i \(0.194154\pi\)
\(968\) 0 0
\(969\) 90.9783 2.92264
\(970\) 0 0
\(971\) −19.7228 −0.632935 −0.316468 0.948603i \(-0.602497\pi\)
−0.316468 + 0.948603i \(0.602497\pi\)
\(972\) 0 0
\(973\) 14.7446 0.472689
\(974\) 0 0
\(975\) −13.4891 −0.431998
\(976\) 0 0
\(977\) 12.9783 0.415211 0.207606 0.978213i \(-0.433433\pi\)
0.207606 + 0.978213i \(0.433433\pi\)
\(978\) 0 0
\(979\) 15.4891 0.495035
\(980\) 0 0
\(981\) −18.2337 −0.582157
\(982\) 0 0
\(983\) −2.23369 −0.0712436 −0.0356218 0.999365i \(-0.511341\pi\)
−0.0356218 + 0.999365i \(0.511341\pi\)
\(984\) 0 0
\(985\) 10.0000 0.318626
\(986\) 0 0
\(987\) −9.48913 −0.302042
\(988\) 0 0
\(989\) −26.9783 −0.857858
\(990\) 0 0
\(991\) −1.48913 −0.0473036 −0.0236518 0.999720i \(-0.507529\pi\)
−0.0236518 + 0.999720i \(0.507529\pi\)
\(992\) 0 0
\(993\) 29.9565 0.950641
\(994\) 0 0
\(995\) −3.25544 −0.103204
\(996\) 0 0
\(997\) 39.2119 1.24185 0.620927 0.783868i \(-0.286756\pi\)
0.620927 + 0.783868i \(0.286756\pi\)
\(998\) 0 0
\(999\) 2.97825 0.0942277
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 6160.2.a.r.1.2 2
4.3 odd 2 770.2.a.k.1.2 2
12.11 even 2 6930.2.a.bo.1.2 2
20.3 even 4 3850.2.c.y.1849.2 4
20.7 even 4 3850.2.c.y.1849.3 4
20.19 odd 2 3850.2.a.bc.1.1 2
28.27 even 2 5390.2.a.bq.1.1 2
44.43 even 2 8470.2.a.bu.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.k.1.2 2 4.3 odd 2
3850.2.a.bc.1.1 2 20.19 odd 2
3850.2.c.y.1849.2 4 20.3 even 4
3850.2.c.y.1849.3 4 20.7 even 4
5390.2.a.bq.1.1 2 28.27 even 2
6160.2.a.r.1.2 2 1.1 even 1 trivial
6930.2.a.bo.1.2 2 12.11 even 2
8470.2.a.bu.1.1 2 44.43 even 2