Properties

Label 6080.2.a.bj.1.1
Level $6080$
Weight $2$
Character 6080.1
Self dual yes
Analytic conductor $48.549$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6080 = 2^{6} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6080.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.5490444289\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 380)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 6080.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.732051 q^{3} -1.00000 q^{5} -2.00000 q^{7} -2.46410 q^{9} +O(q^{10})\) \(q-0.732051 q^{3} -1.00000 q^{5} -2.00000 q^{7} -2.46410 q^{9} +3.46410 q^{11} -0.732051 q^{13} +0.732051 q^{15} +3.46410 q^{17} +1.00000 q^{19} +1.46410 q^{21} -3.46410 q^{23} +1.00000 q^{25} +4.00000 q^{27} +3.46410 q^{29} -5.46410 q^{31} -2.53590 q^{33} +2.00000 q^{35} -3.26795 q^{37} +0.535898 q^{39} -6.00000 q^{41} +8.92820 q^{43} +2.46410 q^{45} +0.928203 q^{47} -3.00000 q^{49} -2.53590 q^{51} +7.26795 q^{53} -3.46410 q^{55} -0.732051 q^{57} -6.92820 q^{59} +8.39230 q^{61} +4.92820 q^{63} +0.732051 q^{65} +3.26795 q^{67} +2.53590 q^{69} +9.46410 q^{71} -7.46410 q^{73} -0.732051 q^{75} -6.92820 q^{77} +10.9282 q^{79} +4.46410 q^{81} -3.46410 q^{83} -3.46410 q^{85} -2.53590 q^{87} -8.53590 q^{89} +1.46410 q^{91} +4.00000 q^{93} -1.00000 q^{95} +14.5885 q^{97} -8.53590 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{5} - 4 q^{7} + 2 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{3} - 2 q^{5} - 4 q^{7} + 2 q^{9} + 2 q^{13} - 2 q^{15} + 2 q^{19} - 4 q^{21} + 2 q^{25} + 8 q^{27} - 4 q^{31} - 12 q^{33} + 4 q^{35} - 10 q^{37} + 8 q^{39} - 12 q^{41} + 4 q^{43} - 2 q^{45} - 12 q^{47} - 6 q^{49} - 12 q^{51} + 18 q^{53} + 2 q^{57} - 4 q^{61} - 4 q^{63} - 2 q^{65} + 10 q^{67} + 12 q^{69} + 12 q^{71} - 8 q^{73} + 2 q^{75} + 8 q^{79} + 2 q^{81} - 12 q^{87} - 24 q^{89} - 4 q^{91} + 8 q^{93} - 2 q^{95} - 2 q^{97} - 24 q^{99} + O(q^{100}) \)

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
\(3\) −0.732051 −0.422650 −0.211325 0.977416i \(-0.567778\pi\)
−0.211325 + 0.977416i \(0.567778\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 0 0
\(9\) −2.46410 −0.821367
\(10\) 0 0
\(11\) 3.46410 1.04447 0.522233 0.852803i \(-0.325099\pi\)
0.522233 + 0.852803i \(0.325099\pi\)
\(12\) 0 0
\(13\) −0.732051 −0.203034 −0.101517 0.994834i \(-0.532370\pi\)
−0.101517 + 0.994834i \(0.532370\pi\)
\(14\) 0 0
\(15\) 0.732051 0.189015
\(16\) 0 0
\(17\) 3.46410 0.840168 0.420084 0.907485i \(-0.362001\pi\)
0.420084 + 0.907485i \(0.362001\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 1.46410 0.319493
\(22\) 0 0
\(23\) −3.46410 −0.722315 −0.361158 0.932505i \(-0.617618\pi\)
−0.361158 + 0.932505i \(0.617618\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 4.00000 0.769800
\(28\) 0 0
\(29\) 3.46410 0.643268 0.321634 0.946864i \(-0.395768\pi\)
0.321634 + 0.946864i \(0.395768\pi\)
\(30\) 0 0
\(31\) −5.46410 −0.981382 −0.490691 0.871334i \(-0.663256\pi\)
−0.490691 + 0.871334i \(0.663256\pi\)
\(32\) 0 0
\(33\) −2.53590 −0.441443
\(34\) 0 0
\(35\) 2.00000 0.338062
\(36\) 0 0
\(37\) −3.26795 −0.537248 −0.268624 0.963245i \(-0.586569\pi\)
−0.268624 + 0.963245i \(0.586569\pi\)
\(38\) 0 0
\(39\) 0.535898 0.0858124
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 8.92820 1.36154 0.680769 0.732498i \(-0.261646\pi\)
0.680769 + 0.732498i \(0.261646\pi\)
\(44\) 0 0
\(45\) 2.46410 0.367327
\(46\) 0 0
\(47\) 0.928203 0.135392 0.0676962 0.997706i \(-0.478435\pi\)
0.0676962 + 0.997706i \(0.478435\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) −2.53590 −0.355097
\(52\) 0 0
\(53\) 7.26795 0.998330 0.499165 0.866507i \(-0.333640\pi\)
0.499165 + 0.866507i \(0.333640\pi\)
\(54\) 0 0
\(55\) −3.46410 −0.467099
\(56\) 0 0
\(57\) −0.732051 −0.0969625
\(58\) 0 0
\(59\) −6.92820 −0.901975 −0.450988 0.892530i \(-0.648928\pi\)
−0.450988 + 0.892530i \(0.648928\pi\)
\(60\) 0 0
\(61\) 8.39230 1.07452 0.537262 0.843415i \(-0.319459\pi\)
0.537262 + 0.843415i \(0.319459\pi\)
\(62\) 0 0
\(63\) 4.92820 0.620895
\(64\) 0 0
\(65\) 0.732051 0.0907997
\(66\) 0 0
\(67\) 3.26795 0.399244 0.199622 0.979873i \(-0.436029\pi\)
0.199622 + 0.979873i \(0.436029\pi\)
\(68\) 0 0
\(69\) 2.53590 0.305286
\(70\) 0 0
\(71\) 9.46410 1.12318 0.561591 0.827415i \(-0.310189\pi\)
0.561591 + 0.827415i \(0.310189\pi\)
\(72\) 0 0
\(73\) −7.46410 −0.873607 −0.436804 0.899557i \(-0.643889\pi\)
−0.436804 + 0.899557i \(0.643889\pi\)
\(74\) 0 0
\(75\) −0.732051 −0.0845299
\(76\) 0 0
\(77\) −6.92820 −0.789542
\(78\) 0 0
\(79\) 10.9282 1.22952 0.614759 0.788715i \(-0.289253\pi\)
0.614759 + 0.788715i \(0.289253\pi\)
\(80\) 0 0
\(81\) 4.46410 0.496011
\(82\) 0 0
\(83\) −3.46410 −0.380235 −0.190117 0.981761i \(-0.560887\pi\)
−0.190117 + 0.981761i \(0.560887\pi\)
\(84\) 0 0
\(85\) −3.46410 −0.375735
\(86\) 0 0
\(87\) −2.53590 −0.271877
\(88\) 0 0
\(89\) −8.53590 −0.904803 −0.452402 0.891814i \(-0.649433\pi\)
−0.452402 + 0.891814i \(0.649433\pi\)
\(90\) 0 0
\(91\) 1.46410 0.153480
\(92\) 0 0
\(93\) 4.00000 0.414781
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) 14.5885 1.48123 0.740617 0.671928i \(-0.234533\pi\)
0.740617 + 0.671928i \(0.234533\pi\)
\(98\) 0 0
\(99\) −8.53590 −0.857890
\(100\) 0 0
\(101\) −4.39230 −0.437051 −0.218525 0.975831i \(-0.570125\pi\)
−0.218525 + 0.975831i \(0.570125\pi\)
\(102\) 0 0
\(103\) 3.66025 0.360656 0.180328 0.983607i \(-0.442284\pi\)
0.180328 + 0.983607i \(0.442284\pi\)
\(104\) 0 0
\(105\) −1.46410 −0.142882
\(106\) 0 0
\(107\) 11.6603 1.12724 0.563620 0.826034i \(-0.309408\pi\)
0.563620 + 0.826034i \(0.309408\pi\)
\(108\) 0 0
\(109\) −6.39230 −0.612272 −0.306136 0.951988i \(-0.599036\pi\)
−0.306136 + 0.951988i \(0.599036\pi\)
\(110\) 0 0
\(111\) 2.39230 0.227068
\(112\) 0 0
\(113\) −18.5885 −1.74865 −0.874327 0.485336i \(-0.838697\pi\)
−0.874327 + 0.485336i \(0.838697\pi\)
\(114\) 0 0
\(115\) 3.46410 0.323029
\(116\) 0 0
\(117\) 1.80385 0.166766
\(118\) 0 0
\(119\) −6.92820 −0.635107
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 4.39230 0.396041
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 10.5885 0.939574 0.469787 0.882780i \(-0.344331\pi\)
0.469787 + 0.882780i \(0.344331\pi\)
\(128\) 0 0
\(129\) −6.53590 −0.575454
\(130\) 0 0
\(131\) −5.07180 −0.443125 −0.221562 0.975146i \(-0.571116\pi\)
−0.221562 + 0.975146i \(0.571116\pi\)
\(132\) 0 0
\(133\) −2.00000 −0.173422
\(134\) 0 0
\(135\) −4.00000 −0.344265
\(136\) 0 0
\(137\) −17.3205 −1.47979 −0.739895 0.672722i \(-0.765125\pi\)
−0.739895 + 0.672722i \(0.765125\pi\)
\(138\) 0 0
\(139\) 4.53590 0.384730 0.192365 0.981323i \(-0.438384\pi\)
0.192365 + 0.981323i \(0.438384\pi\)
\(140\) 0 0
\(141\) −0.679492 −0.0572235
\(142\) 0 0
\(143\) −2.53590 −0.212062
\(144\) 0 0
\(145\) −3.46410 −0.287678
\(146\) 0 0
\(147\) 2.19615 0.181136
\(148\) 0 0
\(149\) −16.3923 −1.34291 −0.671455 0.741045i \(-0.734330\pi\)
−0.671455 + 0.741045i \(0.734330\pi\)
\(150\) 0 0
\(151\) −12.3923 −1.00847 −0.504236 0.863566i \(-0.668226\pi\)
−0.504236 + 0.863566i \(0.668226\pi\)
\(152\) 0 0
\(153\) −8.53590 −0.690086
\(154\) 0 0
\(155\) 5.46410 0.438887
\(156\) 0 0
\(157\) −16.5359 −1.31971 −0.659854 0.751394i \(-0.729382\pi\)
−0.659854 + 0.751394i \(0.729382\pi\)
\(158\) 0 0
\(159\) −5.32051 −0.421944
\(160\) 0 0
\(161\) 6.92820 0.546019
\(162\) 0 0
\(163\) 20.9282 1.63922 0.819612 0.572919i \(-0.194189\pi\)
0.819612 + 0.572919i \(0.194189\pi\)
\(164\) 0 0
\(165\) 2.53590 0.197419
\(166\) 0 0
\(167\) 7.26795 0.562411 0.281205 0.959648i \(-0.409266\pi\)
0.281205 + 0.959648i \(0.409266\pi\)
\(168\) 0 0
\(169\) −12.4641 −0.958777
\(170\) 0 0
\(171\) −2.46410 −0.188435
\(172\) 0 0
\(173\) −14.1962 −1.07931 −0.539657 0.841885i \(-0.681446\pi\)
−0.539657 + 0.841885i \(0.681446\pi\)
\(174\) 0 0
\(175\) −2.00000 −0.151186
\(176\) 0 0
\(177\) 5.07180 0.381220
\(178\) 0 0
\(179\) −20.7846 −1.55351 −0.776757 0.629800i \(-0.783137\pi\)
−0.776757 + 0.629800i \(0.783137\pi\)
\(180\) 0 0
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) 0 0
\(183\) −6.14359 −0.454148
\(184\) 0 0
\(185\) 3.26795 0.240264
\(186\) 0 0
\(187\) 12.0000 0.877527
\(188\) 0 0
\(189\) −8.00000 −0.581914
\(190\) 0 0
\(191\) 6.92820 0.501307 0.250654 0.968077i \(-0.419354\pi\)
0.250654 + 0.968077i \(0.419354\pi\)
\(192\) 0 0
\(193\) −18.1962 −1.30979 −0.654894 0.755721i \(-0.727287\pi\)
−0.654894 + 0.755721i \(0.727287\pi\)
\(194\) 0 0
\(195\) −0.535898 −0.0383765
\(196\) 0 0
\(197\) 12.9282 0.921096 0.460548 0.887635i \(-0.347653\pi\)
0.460548 + 0.887635i \(0.347653\pi\)
\(198\) 0 0
\(199\) −13.0718 −0.926635 −0.463318 0.886192i \(-0.653341\pi\)
−0.463318 + 0.886192i \(0.653341\pi\)
\(200\) 0 0
\(201\) −2.39230 −0.168740
\(202\) 0 0
\(203\) −6.92820 −0.486265
\(204\) 0 0
\(205\) 6.00000 0.419058
\(206\) 0 0
\(207\) 8.53590 0.593286
\(208\) 0 0
\(209\) 3.46410 0.239617
\(210\) 0 0
\(211\) −15.3205 −1.05471 −0.527354 0.849646i \(-0.676816\pi\)
−0.527354 + 0.849646i \(0.676816\pi\)
\(212\) 0 0
\(213\) −6.92820 −0.474713
\(214\) 0 0
\(215\) −8.92820 −0.608898
\(216\) 0 0
\(217\) 10.9282 0.741855
\(218\) 0 0
\(219\) 5.46410 0.369230
\(220\) 0 0
\(221\) −2.53590 −0.170583
\(222\) 0 0
\(223\) −7.66025 −0.512969 −0.256484 0.966548i \(-0.582564\pi\)
−0.256484 + 0.966548i \(0.582564\pi\)
\(224\) 0 0
\(225\) −2.46410 −0.164273
\(226\) 0 0
\(227\) −2.19615 −0.145764 −0.0728819 0.997341i \(-0.523220\pi\)
−0.0728819 + 0.997341i \(0.523220\pi\)
\(228\) 0 0
\(229\) −10.5359 −0.696232 −0.348116 0.937452i \(-0.613178\pi\)
−0.348116 + 0.937452i \(0.613178\pi\)
\(230\) 0 0
\(231\) 5.07180 0.333700
\(232\) 0 0
\(233\) 12.9282 0.846955 0.423477 0.905907i \(-0.360809\pi\)
0.423477 + 0.905907i \(0.360809\pi\)
\(234\) 0 0
\(235\) −0.928203 −0.0605493
\(236\) 0 0
\(237\) −8.00000 −0.519656
\(238\) 0 0
\(239\) −13.8564 −0.896296 −0.448148 0.893959i \(-0.647916\pi\)
−0.448148 + 0.893959i \(0.647916\pi\)
\(240\) 0 0
\(241\) 15.8564 1.02140 0.510700 0.859759i \(-0.329386\pi\)
0.510700 + 0.859759i \(0.329386\pi\)
\(242\) 0 0
\(243\) −15.2679 −0.979439
\(244\) 0 0
\(245\) 3.00000 0.191663
\(246\) 0 0
\(247\) −0.732051 −0.0465793
\(248\) 0 0
\(249\) 2.53590 0.160706
\(250\) 0 0
\(251\) −24.0000 −1.51487 −0.757433 0.652913i \(-0.773547\pi\)
−0.757433 + 0.652913i \(0.773547\pi\)
\(252\) 0 0
\(253\) −12.0000 −0.754434
\(254\) 0 0
\(255\) 2.53590 0.158804
\(256\) 0 0
\(257\) 9.12436 0.569162 0.284581 0.958652i \(-0.408146\pi\)
0.284581 + 0.958652i \(0.408146\pi\)
\(258\) 0 0
\(259\) 6.53590 0.406121
\(260\) 0 0
\(261\) −8.53590 −0.528359
\(262\) 0 0
\(263\) −15.4641 −0.953557 −0.476779 0.879023i \(-0.658196\pi\)
−0.476779 + 0.879023i \(0.658196\pi\)
\(264\) 0 0
\(265\) −7.26795 −0.446467
\(266\) 0 0
\(267\) 6.24871 0.382415
\(268\) 0 0
\(269\) 10.3923 0.633630 0.316815 0.948487i \(-0.397387\pi\)
0.316815 + 0.948487i \(0.397387\pi\)
\(270\) 0 0
\(271\) −18.3923 −1.11725 −0.558626 0.829419i \(-0.688671\pi\)
−0.558626 + 0.829419i \(0.688671\pi\)
\(272\) 0 0
\(273\) −1.07180 −0.0648681
\(274\) 0 0
\(275\) 3.46410 0.208893
\(276\) 0 0
\(277\) 2.39230 0.143740 0.0718698 0.997414i \(-0.477103\pi\)
0.0718698 + 0.997414i \(0.477103\pi\)
\(278\) 0 0
\(279\) 13.4641 0.806075
\(280\) 0 0
\(281\) −12.9282 −0.771232 −0.385616 0.922659i \(-0.626011\pi\)
−0.385616 + 0.922659i \(0.626011\pi\)
\(282\) 0 0
\(283\) −2.39230 −0.142208 −0.0711039 0.997469i \(-0.522652\pi\)
−0.0711039 + 0.997469i \(0.522652\pi\)
\(284\) 0 0
\(285\) 0.732051 0.0433629
\(286\) 0 0
\(287\) 12.0000 0.708338
\(288\) 0 0
\(289\) −5.00000 −0.294118
\(290\) 0 0
\(291\) −10.6795 −0.626043
\(292\) 0 0
\(293\) −11.6603 −0.681199 −0.340600 0.940208i \(-0.610630\pi\)
−0.340600 + 0.940208i \(0.610630\pi\)
\(294\) 0 0
\(295\) 6.92820 0.403376
\(296\) 0 0
\(297\) 13.8564 0.804030
\(298\) 0 0
\(299\) 2.53590 0.146655
\(300\) 0 0
\(301\) −17.8564 −1.02923
\(302\) 0 0
\(303\) 3.21539 0.184719
\(304\) 0 0
\(305\) −8.39230 −0.480542
\(306\) 0 0
\(307\) −18.1962 −1.03851 −0.519255 0.854620i \(-0.673790\pi\)
−0.519255 + 0.854620i \(0.673790\pi\)
\(308\) 0 0
\(309\) −2.67949 −0.152431
\(310\) 0 0
\(311\) 22.3923 1.26975 0.634876 0.772614i \(-0.281051\pi\)
0.634876 + 0.772614i \(0.281051\pi\)
\(312\) 0 0
\(313\) −30.7846 −1.74005 −0.870025 0.493008i \(-0.835897\pi\)
−0.870025 + 0.493008i \(0.835897\pi\)
\(314\) 0 0
\(315\) −4.92820 −0.277673
\(316\) 0 0
\(317\) 21.1244 1.18646 0.593231 0.805032i \(-0.297852\pi\)
0.593231 + 0.805032i \(0.297852\pi\)
\(318\) 0 0
\(319\) 12.0000 0.671871
\(320\) 0 0
\(321\) −8.53590 −0.476427
\(322\) 0 0
\(323\) 3.46410 0.192748
\(324\) 0 0
\(325\) −0.732051 −0.0406069
\(326\) 0 0
\(327\) 4.67949 0.258776
\(328\) 0 0
\(329\) −1.85641 −0.102347
\(330\) 0 0
\(331\) 26.2487 1.44276 0.721380 0.692540i \(-0.243508\pi\)
0.721380 + 0.692540i \(0.243508\pi\)
\(332\) 0 0
\(333\) 8.05256 0.441278
\(334\) 0 0
\(335\) −3.26795 −0.178547
\(336\) 0 0
\(337\) −20.7321 −1.12935 −0.564673 0.825314i \(-0.690998\pi\)
−0.564673 + 0.825314i \(0.690998\pi\)
\(338\) 0 0
\(339\) 13.6077 0.739069
\(340\) 0 0
\(341\) −18.9282 −1.02502
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 0 0
\(345\) −2.53590 −0.136528
\(346\) 0 0
\(347\) −31.1769 −1.67366 −0.836832 0.547459i \(-0.815595\pi\)
−0.836832 + 0.547459i \(0.815595\pi\)
\(348\) 0 0
\(349\) −7.07180 −0.378545 −0.189272 0.981925i \(-0.560613\pi\)
−0.189272 + 0.981925i \(0.560613\pi\)
\(350\) 0 0
\(351\) −2.92820 −0.156296
\(352\) 0 0
\(353\) 22.3923 1.19182 0.595911 0.803050i \(-0.296791\pi\)
0.595911 + 0.803050i \(0.296791\pi\)
\(354\) 0 0
\(355\) −9.46410 −0.502302
\(356\) 0 0
\(357\) 5.07180 0.268428
\(358\) 0 0
\(359\) 15.4641 0.816164 0.408082 0.912945i \(-0.366198\pi\)
0.408082 + 0.912945i \(0.366198\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −0.732051 −0.0384227
\(364\) 0 0
\(365\) 7.46410 0.390689
\(366\) 0 0
\(367\) 23.8564 1.24529 0.622647 0.782503i \(-0.286057\pi\)
0.622647 + 0.782503i \(0.286057\pi\)
\(368\) 0 0
\(369\) 14.7846 0.769656
\(370\) 0 0
\(371\) −14.5359 −0.754666
\(372\) 0 0
\(373\) −14.5885 −0.755362 −0.377681 0.925936i \(-0.623278\pi\)
−0.377681 + 0.925936i \(0.623278\pi\)
\(374\) 0 0
\(375\) 0.732051 0.0378029
\(376\) 0 0
\(377\) −2.53590 −0.130605
\(378\) 0 0
\(379\) 1.07180 0.0550545 0.0275273 0.999621i \(-0.491237\pi\)
0.0275273 + 0.999621i \(0.491237\pi\)
\(380\) 0 0
\(381\) −7.75129 −0.397111
\(382\) 0 0
\(383\) −23.6603 −1.20898 −0.604491 0.796612i \(-0.706624\pi\)
−0.604491 + 0.796612i \(0.706624\pi\)
\(384\) 0 0
\(385\) 6.92820 0.353094
\(386\) 0 0
\(387\) −22.0000 −1.11832
\(388\) 0 0
\(389\) −18.0000 −0.912636 −0.456318 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) 0 0
\(391\) −12.0000 −0.606866
\(392\) 0 0
\(393\) 3.71281 0.187287
\(394\) 0 0
\(395\) −10.9282 −0.549858
\(396\) 0 0
\(397\) 12.5359 0.629159 0.314579 0.949231i \(-0.398137\pi\)
0.314579 + 0.949231i \(0.398137\pi\)
\(398\) 0 0
\(399\) 1.46410 0.0732968
\(400\) 0 0
\(401\) −2.78461 −0.139057 −0.0695284 0.997580i \(-0.522149\pi\)
−0.0695284 + 0.997580i \(0.522149\pi\)
\(402\) 0 0
\(403\) 4.00000 0.199254
\(404\) 0 0
\(405\) −4.46410 −0.221823
\(406\) 0 0
\(407\) −11.3205 −0.561137
\(408\) 0 0
\(409\) 6.39230 0.316079 0.158040 0.987433i \(-0.449483\pi\)
0.158040 + 0.987433i \(0.449483\pi\)
\(410\) 0 0
\(411\) 12.6795 0.625433
\(412\) 0 0
\(413\) 13.8564 0.681829
\(414\) 0 0
\(415\) 3.46410 0.170046
\(416\) 0 0
\(417\) −3.32051 −0.162606
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) 34.0000 1.65706 0.828529 0.559946i \(-0.189178\pi\)
0.828529 + 0.559946i \(0.189178\pi\)
\(422\) 0 0
\(423\) −2.28719 −0.111207
\(424\) 0 0
\(425\) 3.46410 0.168034
\(426\) 0 0
\(427\) −16.7846 −0.812264
\(428\) 0 0
\(429\) 1.85641 0.0896281
\(430\) 0 0
\(431\) 26.5359 1.27819 0.639095 0.769128i \(-0.279309\pi\)
0.639095 + 0.769128i \(0.279309\pi\)
\(432\) 0 0
\(433\) −39.6603 −1.90595 −0.952975 0.303049i \(-0.901996\pi\)
−0.952975 + 0.303049i \(0.901996\pi\)
\(434\) 0 0
\(435\) 2.53590 0.121587
\(436\) 0 0
\(437\) −3.46410 −0.165710
\(438\) 0 0
\(439\) −23.7128 −1.13175 −0.565875 0.824491i \(-0.691462\pi\)
−0.565875 + 0.824491i \(0.691462\pi\)
\(440\) 0 0
\(441\) 7.39230 0.352015
\(442\) 0 0
\(443\) 34.3923 1.63403 0.817014 0.576618i \(-0.195628\pi\)
0.817014 + 0.576618i \(0.195628\pi\)
\(444\) 0 0
\(445\) 8.53590 0.404640
\(446\) 0 0
\(447\) 12.0000 0.567581
\(448\) 0 0
\(449\) 3.46410 0.163481 0.0817405 0.996654i \(-0.473952\pi\)
0.0817405 + 0.996654i \(0.473952\pi\)
\(450\) 0 0
\(451\) −20.7846 −0.978709
\(452\) 0 0
\(453\) 9.07180 0.426230
\(454\) 0 0
\(455\) −1.46410 −0.0686381
\(456\) 0 0
\(457\) −6.78461 −0.317371 −0.158685 0.987329i \(-0.550726\pi\)
−0.158685 + 0.987329i \(0.550726\pi\)
\(458\) 0 0
\(459\) 13.8564 0.646762
\(460\) 0 0
\(461\) −33.7128 −1.57016 −0.785081 0.619393i \(-0.787379\pi\)
−0.785081 + 0.619393i \(0.787379\pi\)
\(462\) 0 0
\(463\) −11.4641 −0.532782 −0.266391 0.963865i \(-0.585831\pi\)
−0.266391 + 0.963865i \(0.585831\pi\)
\(464\) 0 0
\(465\) −4.00000 −0.185496
\(466\) 0 0
\(467\) 5.32051 0.246204 0.123102 0.992394i \(-0.460716\pi\)
0.123102 + 0.992394i \(0.460716\pi\)
\(468\) 0 0
\(469\) −6.53590 −0.301800
\(470\) 0 0
\(471\) 12.1051 0.557774
\(472\) 0 0
\(473\) 30.9282 1.42208
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) −17.9090 −0.819995
\(478\) 0 0
\(479\) 1.60770 0.0734575 0.0367287 0.999325i \(-0.488306\pi\)
0.0367287 + 0.999325i \(0.488306\pi\)
\(480\) 0 0
\(481\) 2.39230 0.109080
\(482\) 0 0
\(483\) −5.07180 −0.230775
\(484\) 0 0
\(485\) −14.5885 −0.662428
\(486\) 0 0
\(487\) 1.80385 0.0817401 0.0408701 0.999164i \(-0.486987\pi\)
0.0408701 + 0.999164i \(0.486987\pi\)
\(488\) 0 0
\(489\) −15.3205 −0.692817
\(490\) 0 0
\(491\) 5.07180 0.228887 0.114443 0.993430i \(-0.463492\pi\)
0.114443 + 0.993430i \(0.463492\pi\)
\(492\) 0 0
\(493\) 12.0000 0.540453
\(494\) 0 0
\(495\) 8.53590 0.383660
\(496\) 0 0
\(497\) −18.9282 −0.849046
\(498\) 0 0
\(499\) 16.5359 0.740248 0.370124 0.928982i \(-0.379315\pi\)
0.370124 + 0.928982i \(0.379315\pi\)
\(500\) 0 0
\(501\) −5.32051 −0.237703
\(502\) 0 0
\(503\) −8.53590 −0.380597 −0.190298 0.981726i \(-0.560946\pi\)
−0.190298 + 0.981726i \(0.560946\pi\)
\(504\) 0 0
\(505\) 4.39230 0.195455
\(506\) 0 0
\(507\) 9.12436 0.405227
\(508\) 0 0
\(509\) −29.3205 −1.29961 −0.649804 0.760102i \(-0.725149\pi\)
−0.649804 + 0.760102i \(0.725149\pi\)
\(510\) 0 0
\(511\) 14.9282 0.660385
\(512\) 0 0
\(513\) 4.00000 0.176604
\(514\) 0 0
\(515\) −3.66025 −0.161290
\(516\) 0 0
\(517\) 3.21539 0.141413
\(518\) 0 0
\(519\) 10.3923 0.456172
\(520\) 0 0
\(521\) 26.7846 1.17346 0.586728 0.809784i \(-0.300416\pi\)
0.586728 + 0.809784i \(0.300416\pi\)
\(522\) 0 0
\(523\) 35.3731 1.54676 0.773378 0.633945i \(-0.218565\pi\)
0.773378 + 0.633945i \(0.218565\pi\)
\(524\) 0 0
\(525\) 1.46410 0.0638986
\(526\) 0 0
\(527\) −18.9282 −0.824525
\(528\) 0 0
\(529\) −11.0000 −0.478261
\(530\) 0 0
\(531\) 17.0718 0.740853
\(532\) 0 0
\(533\) 4.39230 0.190252
\(534\) 0 0
\(535\) −11.6603 −0.504117
\(536\) 0 0
\(537\) 15.2154 0.656593
\(538\) 0 0
\(539\) −10.3923 −0.447628
\(540\) 0 0
\(541\) −33.1769 −1.42639 −0.713193 0.700967i \(-0.752752\pi\)
−0.713193 + 0.700967i \(0.752752\pi\)
\(542\) 0 0
\(543\) 10.2487 0.439814
\(544\) 0 0
\(545\) 6.39230 0.273816
\(546\) 0 0
\(547\) 3.26795 0.139727 0.0698637 0.997557i \(-0.477744\pi\)
0.0698637 + 0.997557i \(0.477744\pi\)
\(548\) 0 0
\(549\) −20.6795 −0.882579
\(550\) 0 0
\(551\) 3.46410 0.147576
\(552\) 0 0
\(553\) −21.8564 −0.929429
\(554\) 0 0
\(555\) −2.39230 −0.101548
\(556\) 0 0
\(557\) 29.3205 1.24235 0.621175 0.783672i \(-0.286656\pi\)
0.621175 + 0.783672i \(0.286656\pi\)
\(558\) 0 0
\(559\) −6.53590 −0.276439
\(560\) 0 0
\(561\) −8.78461 −0.370887
\(562\) 0 0
\(563\) 33.8038 1.42466 0.712331 0.701844i \(-0.247639\pi\)
0.712331 + 0.701844i \(0.247639\pi\)
\(564\) 0 0
\(565\) 18.5885 0.782022
\(566\) 0 0
\(567\) −8.92820 −0.374949
\(568\) 0 0
\(569\) 5.32051 0.223047 0.111524 0.993762i \(-0.464427\pi\)
0.111524 + 0.993762i \(0.464427\pi\)
\(570\) 0 0
\(571\) −2.39230 −0.100115 −0.0500574 0.998746i \(-0.515940\pi\)
−0.0500574 + 0.998746i \(0.515940\pi\)
\(572\) 0 0
\(573\) −5.07180 −0.211877
\(574\) 0 0
\(575\) −3.46410 −0.144463
\(576\) 0 0
\(577\) 34.7846 1.44810 0.724051 0.689746i \(-0.242278\pi\)
0.724051 + 0.689746i \(0.242278\pi\)
\(578\) 0 0
\(579\) 13.3205 0.553581
\(580\) 0 0
\(581\) 6.92820 0.287430
\(582\) 0 0
\(583\) 25.1769 1.04272
\(584\) 0 0
\(585\) −1.80385 −0.0745799
\(586\) 0 0
\(587\) −19.1769 −0.791516 −0.395758 0.918355i \(-0.629518\pi\)
−0.395758 + 0.918355i \(0.629518\pi\)
\(588\) 0 0
\(589\) −5.46410 −0.225144
\(590\) 0 0
\(591\) −9.46410 −0.389301
\(592\) 0 0
\(593\) 4.14359 0.170157 0.0850785 0.996374i \(-0.472886\pi\)
0.0850785 + 0.996374i \(0.472886\pi\)
\(594\) 0 0
\(595\) 6.92820 0.284029
\(596\) 0 0
\(597\) 9.56922 0.391642
\(598\) 0 0
\(599\) −20.7846 −0.849236 −0.424618 0.905373i \(-0.639592\pi\)
−0.424618 + 0.905373i \(0.639592\pi\)
\(600\) 0 0
\(601\) −44.6410 −1.82095 −0.910473 0.413570i \(-0.864282\pi\)
−0.910473 + 0.413570i \(0.864282\pi\)
\(602\) 0 0
\(603\) −8.05256 −0.327926
\(604\) 0 0
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) −25.9090 −1.05161 −0.525806 0.850604i \(-0.676236\pi\)
−0.525806 + 0.850604i \(0.676236\pi\)
\(608\) 0 0
\(609\) 5.07180 0.205520
\(610\) 0 0
\(611\) −0.679492 −0.0274893
\(612\) 0 0
\(613\) 14.3923 0.581300 0.290650 0.956829i \(-0.406129\pi\)
0.290650 + 0.956829i \(0.406129\pi\)
\(614\) 0 0
\(615\) −4.39230 −0.177115
\(616\) 0 0
\(617\) 22.3923 0.901480 0.450740 0.892655i \(-0.351160\pi\)
0.450740 + 0.892655i \(0.351160\pi\)
\(618\) 0 0
\(619\) 18.3923 0.739249 0.369625 0.929181i \(-0.379486\pi\)
0.369625 + 0.929181i \(0.379486\pi\)
\(620\) 0 0
\(621\) −13.8564 −0.556038
\(622\) 0 0
\(623\) 17.0718 0.683967
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −2.53590 −0.101274
\(628\) 0 0
\(629\) −11.3205 −0.451378
\(630\) 0 0
\(631\) −4.53590 −0.180571 −0.0902856 0.995916i \(-0.528778\pi\)
−0.0902856 + 0.995916i \(0.528778\pi\)
\(632\) 0 0
\(633\) 11.2154 0.445772
\(634\) 0 0
\(635\) −10.5885 −0.420190
\(636\) 0 0
\(637\) 2.19615 0.0870147
\(638\) 0 0
\(639\) −23.3205 −0.922545
\(640\) 0 0
\(641\) −11.0718 −0.437310 −0.218655 0.975802i \(-0.570167\pi\)
−0.218655 + 0.975802i \(0.570167\pi\)
\(642\) 0 0
\(643\) 6.39230 0.252088 0.126044 0.992025i \(-0.459772\pi\)
0.126044 + 0.992025i \(0.459772\pi\)
\(644\) 0 0
\(645\) 6.53590 0.257351
\(646\) 0 0
\(647\) −8.53590 −0.335581 −0.167790 0.985823i \(-0.553663\pi\)
−0.167790 + 0.985823i \(0.553663\pi\)
\(648\) 0 0
\(649\) −24.0000 −0.942082
\(650\) 0 0
\(651\) −8.00000 −0.313545
\(652\) 0 0
\(653\) −20.5359 −0.803632 −0.401816 0.915720i \(-0.631621\pi\)
−0.401816 + 0.915720i \(0.631621\pi\)
\(654\) 0 0
\(655\) 5.07180 0.198171
\(656\) 0 0
\(657\) 18.3923 0.717552
\(658\) 0 0
\(659\) −32.7846 −1.27711 −0.638554 0.769577i \(-0.720467\pi\)
−0.638554 + 0.769577i \(0.720467\pi\)
\(660\) 0 0
\(661\) −46.7846 −1.81971 −0.909855 0.414926i \(-0.863807\pi\)
−0.909855 + 0.414926i \(0.863807\pi\)
\(662\) 0 0
\(663\) 1.85641 0.0720969
\(664\) 0 0
\(665\) 2.00000 0.0775567
\(666\) 0 0
\(667\) −12.0000 −0.464642
\(668\) 0 0
\(669\) 5.60770 0.216806
\(670\) 0 0
\(671\) 29.0718 1.12230
\(672\) 0 0
\(673\) −47.2679 −1.82205 −0.911023 0.412356i \(-0.864706\pi\)
−0.911023 + 0.412356i \(0.864706\pi\)
\(674\) 0 0
\(675\) 4.00000 0.153960
\(676\) 0 0
\(677\) 34.9808 1.34442 0.672210 0.740361i \(-0.265345\pi\)
0.672210 + 0.740361i \(0.265345\pi\)
\(678\) 0 0
\(679\) −29.1769 −1.11971
\(680\) 0 0
\(681\) 1.60770 0.0616070
\(682\) 0 0
\(683\) −0.339746 −0.0130000 −0.00650001 0.999979i \(-0.502069\pi\)
−0.00650001 + 0.999979i \(0.502069\pi\)
\(684\) 0 0
\(685\) 17.3205 0.661783
\(686\) 0 0
\(687\) 7.71281 0.294262
\(688\) 0 0
\(689\) −5.32051 −0.202695
\(690\) 0 0
\(691\) −11.1769 −0.425190 −0.212595 0.977140i \(-0.568191\pi\)
−0.212595 + 0.977140i \(0.568191\pi\)
\(692\) 0 0
\(693\) 17.0718 0.648504
\(694\) 0 0
\(695\) −4.53590 −0.172056
\(696\) 0 0
\(697\) −20.7846 −0.787273
\(698\) 0 0
\(699\) −9.46410 −0.357965
\(700\) 0 0
\(701\) 33.4641 1.26392 0.631961 0.775000i \(-0.282250\pi\)
0.631961 + 0.775000i \(0.282250\pi\)
\(702\) 0 0
\(703\) −3.26795 −0.123253
\(704\) 0 0
\(705\) 0.679492 0.0255911
\(706\) 0 0
\(707\) 8.78461 0.330379
\(708\) 0 0
\(709\) −10.7846 −0.405025 −0.202512 0.979280i \(-0.564911\pi\)
−0.202512 + 0.979280i \(0.564911\pi\)
\(710\) 0 0
\(711\) −26.9282 −1.00989
\(712\) 0 0
\(713\) 18.9282 0.708867
\(714\) 0 0
\(715\) 2.53590 0.0948372
\(716\) 0 0
\(717\) 10.1436 0.378819
\(718\) 0 0
\(719\) 17.3205 0.645946 0.322973 0.946408i \(-0.395318\pi\)
0.322973 + 0.946408i \(0.395318\pi\)
\(720\) 0 0
\(721\) −7.32051 −0.272630
\(722\) 0 0
\(723\) −11.6077 −0.431695
\(724\) 0 0
\(725\) 3.46410 0.128654
\(726\) 0 0
\(727\) −27.8564 −1.03314 −0.516568 0.856246i \(-0.672791\pi\)
−0.516568 + 0.856246i \(0.672791\pi\)
\(728\) 0 0
\(729\) −2.21539 −0.0820515
\(730\) 0 0
\(731\) 30.9282 1.14392
\(732\) 0 0
\(733\) 30.7846 1.13706 0.568528 0.822664i \(-0.307513\pi\)
0.568528 + 0.822664i \(0.307513\pi\)
\(734\) 0 0
\(735\) −2.19615 −0.0810063
\(736\) 0 0
\(737\) 11.3205 0.416996
\(738\) 0 0
\(739\) −28.0000 −1.03000 −0.514998 0.857191i \(-0.672207\pi\)
−0.514998 + 0.857191i \(0.672207\pi\)
\(740\) 0 0
\(741\) 0.535898 0.0196867
\(742\) 0 0
\(743\) −45.1244 −1.65545 −0.827726 0.561132i \(-0.810366\pi\)
−0.827726 + 0.561132i \(0.810366\pi\)
\(744\) 0 0
\(745\) 16.3923 0.600568
\(746\) 0 0
\(747\) 8.53590 0.312312
\(748\) 0 0
\(749\) −23.3205 −0.852113
\(750\) 0 0
\(751\) 18.5359 0.676385 0.338192 0.941077i \(-0.390185\pi\)
0.338192 + 0.941077i \(0.390185\pi\)
\(752\) 0 0
\(753\) 17.5692 0.640258
\(754\) 0 0
\(755\) 12.3923 0.451002
\(756\) 0 0
\(757\) 10.0000 0.363456 0.181728 0.983349i \(-0.441831\pi\)
0.181728 + 0.983349i \(0.441831\pi\)
\(758\) 0 0
\(759\) 8.78461 0.318861
\(760\) 0 0
\(761\) 40.3923 1.46422 0.732110 0.681186i \(-0.238536\pi\)
0.732110 + 0.681186i \(0.238536\pi\)
\(762\) 0 0
\(763\) 12.7846 0.462834
\(764\) 0 0
\(765\) 8.53590 0.308616
\(766\) 0 0
\(767\) 5.07180 0.183132
\(768\) 0 0
\(769\) −37.4641 −1.35099 −0.675495 0.737365i \(-0.736070\pi\)
−0.675495 + 0.737365i \(0.736070\pi\)
\(770\) 0 0
\(771\) −6.67949 −0.240556
\(772\) 0 0
\(773\) 16.0526 0.577370 0.288685 0.957424i \(-0.406782\pi\)
0.288685 + 0.957424i \(0.406782\pi\)
\(774\) 0 0
\(775\) −5.46410 −0.196276
\(776\) 0 0
\(777\) −4.78461 −0.171647
\(778\) 0 0
\(779\) −6.00000 −0.214972
\(780\) 0 0
\(781\) 32.7846 1.17313
\(782\) 0 0
\(783\) 13.8564 0.495188
\(784\) 0 0
\(785\) 16.5359 0.590192
\(786\) 0 0
\(787\) −1.80385 −0.0643002 −0.0321501 0.999483i \(-0.510235\pi\)
−0.0321501 + 0.999483i \(0.510235\pi\)
\(788\) 0 0
\(789\) 11.3205 0.403021
\(790\) 0 0
\(791\) 37.1769 1.32186
\(792\) 0 0
\(793\) −6.14359 −0.218165
\(794\) 0 0
\(795\) 5.32051 0.188699
\(796\) 0 0
\(797\) −11.6603 −0.413027 −0.206514 0.978444i \(-0.566212\pi\)
−0.206514 + 0.978444i \(0.566212\pi\)
\(798\) 0 0
\(799\) 3.21539 0.113752
\(800\) 0 0
\(801\) 21.0333 0.743176
\(802\) 0 0
\(803\) −25.8564 −0.912453
\(804\) 0 0
\(805\) −6.92820 −0.244187
\(806\) 0 0
\(807\) −7.60770 −0.267804
\(808\) 0 0
\(809\) −45.7128 −1.60718 −0.803588 0.595185i \(-0.797079\pi\)
−0.803588 + 0.595185i \(0.797079\pi\)
\(810\) 0 0
\(811\) 36.3923 1.27791 0.638953 0.769246i \(-0.279368\pi\)
0.638953 + 0.769246i \(0.279368\pi\)
\(812\) 0 0
\(813\) 13.4641 0.472207
\(814\) 0 0
\(815\) −20.9282 −0.733083
\(816\) 0 0
\(817\) 8.92820 0.312358
\(818\) 0 0
\(819\) −3.60770 −0.126063
\(820\) 0 0
\(821\) 43.8564 1.53060 0.765300 0.643674i \(-0.222591\pi\)
0.765300 + 0.643674i \(0.222591\pi\)
\(822\) 0 0
\(823\) −43.5692 −1.51873 −0.759364 0.650666i \(-0.774490\pi\)
−0.759364 + 0.650666i \(0.774490\pi\)
\(824\) 0 0
\(825\) −2.53590 −0.0882886
\(826\) 0 0
\(827\) −12.3397 −0.429095 −0.214548 0.976714i \(-0.568828\pi\)
−0.214548 + 0.976714i \(0.568828\pi\)
\(828\) 0 0
\(829\) 40.2487 1.39790 0.698948 0.715173i \(-0.253652\pi\)
0.698948 + 0.715173i \(0.253652\pi\)
\(830\) 0 0
\(831\) −1.75129 −0.0607515
\(832\) 0 0
\(833\) −10.3923 −0.360072
\(834\) 0 0
\(835\) −7.26795 −0.251518
\(836\) 0 0
\(837\) −21.8564 −0.755468
\(838\) 0 0
\(839\) −5.07180 −0.175098 −0.0875489 0.996160i \(-0.527903\pi\)
−0.0875489 + 0.996160i \(0.527903\pi\)
\(840\) 0 0
\(841\) −17.0000 −0.586207
\(842\) 0 0
\(843\) 9.46410 0.325961
\(844\) 0 0
\(845\) 12.4641 0.428778
\(846\) 0 0
\(847\) −2.00000 −0.0687208
\(848\) 0 0
\(849\) 1.75129 0.0601041
\(850\) 0 0
\(851\) 11.3205 0.388062
\(852\) 0 0
\(853\) −24.6410 −0.843692 −0.421846 0.906667i \(-0.638618\pi\)
−0.421846 + 0.906667i \(0.638618\pi\)
\(854\) 0 0
\(855\) 2.46410 0.0842705
\(856\) 0 0
\(857\) 14.1962 0.484931 0.242466 0.970160i \(-0.422044\pi\)
0.242466 + 0.970160i \(0.422044\pi\)
\(858\) 0 0
\(859\) −7.71281 −0.263158 −0.131579 0.991306i \(-0.542005\pi\)
−0.131579 + 0.991306i \(0.542005\pi\)
\(860\) 0 0
\(861\) −8.78461 −0.299379
\(862\) 0 0
\(863\) 40.7321 1.38654 0.693268 0.720680i \(-0.256170\pi\)
0.693268 + 0.720680i \(0.256170\pi\)
\(864\) 0 0
\(865\) 14.1962 0.482684
\(866\) 0 0
\(867\) 3.66025 0.124309
\(868\) 0 0
\(869\) 37.8564 1.28419
\(870\) 0 0
\(871\) −2.39230 −0.0810602
\(872\) 0 0
\(873\) −35.9474 −1.21664
\(874\) 0 0
\(875\) 2.00000 0.0676123
\(876\) 0 0
\(877\) 50.9808 1.72150 0.860749 0.509030i \(-0.169996\pi\)
0.860749 + 0.509030i \(0.169996\pi\)
\(878\) 0 0
\(879\) 8.53590 0.287909
\(880\) 0 0
\(881\) −35.3205 −1.18998 −0.594989 0.803734i \(-0.702844\pi\)
−0.594989 + 0.803734i \(0.702844\pi\)
\(882\) 0 0
\(883\) −22.0000 −0.740359 −0.370179 0.928960i \(-0.620704\pi\)
−0.370179 + 0.928960i \(0.620704\pi\)
\(884\) 0 0
\(885\) −5.07180 −0.170487
\(886\) 0 0
\(887\) −54.5885 −1.83290 −0.916451 0.400148i \(-0.868959\pi\)
−0.916451 + 0.400148i \(0.868959\pi\)
\(888\) 0 0
\(889\) −21.1769 −0.710251
\(890\) 0 0
\(891\) 15.4641 0.518067
\(892\) 0 0
\(893\) 0.928203 0.0310611
\(894\) 0 0
\(895\) 20.7846 0.694753
\(896\) 0 0
\(897\) −1.85641 −0.0619836
\(898\) 0 0
\(899\) −18.9282 −0.631291
\(900\) 0 0
\(901\) 25.1769 0.838765
\(902\) 0 0
\(903\) 13.0718 0.435002
\(904\) 0 0
\(905\) 14.0000 0.465376
\(906\) 0 0
\(907\) 38.5885 1.28131 0.640654 0.767829i \(-0.278663\pi\)
0.640654 + 0.767829i \(0.278663\pi\)
\(908\) 0 0
\(909\) 10.8231 0.358979
\(910\) 0 0
\(911\) −21.4641 −0.711137 −0.355569 0.934650i \(-0.615713\pi\)
−0.355569 + 0.934650i \(0.615713\pi\)
\(912\) 0 0
\(913\) −12.0000 −0.397142
\(914\) 0 0
\(915\) 6.14359 0.203101
\(916\) 0 0
\(917\) 10.1436 0.334971
\(918\) 0 0
\(919\) 52.4974 1.73173 0.865865 0.500278i \(-0.166769\pi\)
0.865865 + 0.500278i \(0.166769\pi\)
\(920\) 0 0
\(921\) 13.3205 0.438926
\(922\) 0 0
\(923\) −6.92820 −0.228045
\(924\) 0 0
\(925\) −3.26795 −0.107450
\(926\) 0 0
\(927\) −9.01924 −0.296231
\(928\) 0 0
\(929\) 35.5692 1.16699 0.583494 0.812117i \(-0.301685\pi\)
0.583494 + 0.812117i \(0.301685\pi\)
\(930\) 0 0
\(931\) −3.00000 −0.0983210
\(932\) 0 0
\(933\) −16.3923 −0.536660
\(934\) 0 0
\(935\) −12.0000 −0.392442
\(936\) 0 0
\(937\) −11.1769 −0.365134 −0.182567 0.983193i \(-0.558441\pi\)
−0.182567 + 0.983193i \(0.558441\pi\)
\(938\) 0 0
\(939\) 22.5359 0.735431
\(940\) 0 0
\(941\) 16.6410 0.542482 0.271241 0.962512i \(-0.412566\pi\)
0.271241 + 0.962512i \(0.412566\pi\)
\(942\) 0 0
\(943\) 20.7846 0.676840
\(944\) 0 0
\(945\) 8.00000 0.260240
\(946\) 0 0
\(947\) 41.3205 1.34274 0.671368 0.741124i \(-0.265707\pi\)
0.671368 + 0.741124i \(0.265707\pi\)
\(948\) 0 0
\(949\) 5.46410 0.177372
\(950\) 0 0
\(951\) −15.4641 −0.501458
\(952\) 0 0
\(953\) 29.4115 0.952733 0.476367 0.879247i \(-0.341953\pi\)
0.476367 + 0.879247i \(0.341953\pi\)
\(954\) 0 0
\(955\) −6.92820 −0.224191
\(956\) 0 0
\(957\) −8.78461 −0.283966
\(958\) 0 0
\(959\) 34.6410 1.11862
\(960\) 0 0
\(961\) −1.14359 −0.0368901
\(962\) 0 0
\(963\) −28.7321 −0.925877
\(964\) 0 0
\(965\) 18.1962 0.585755
\(966\) 0 0
\(967\) 10.6795 0.343429 0.171715 0.985147i \(-0.445069\pi\)
0.171715 + 0.985147i \(0.445069\pi\)
\(968\) 0 0
\(969\) −2.53590 −0.0814648
\(970\) 0 0
\(971\) −21.4641 −0.688816 −0.344408 0.938820i \(-0.611920\pi\)
−0.344408 + 0.938820i \(0.611920\pi\)
\(972\) 0 0
\(973\) −9.07180 −0.290828
\(974\) 0 0
\(975\) 0.535898 0.0171625
\(976\) 0 0
\(977\) 21.8038 0.697567 0.348783 0.937203i \(-0.386595\pi\)
0.348783 + 0.937203i \(0.386595\pi\)
\(978\) 0 0
\(979\) −29.5692 −0.945036
\(980\) 0 0
\(981\) 15.7513 0.502900
\(982\) 0 0
\(983\) −56.4449 −1.80031 −0.900156 0.435568i \(-0.856548\pi\)
−0.900156 + 0.435568i \(0.856548\pi\)
\(984\) 0 0
\(985\) −12.9282 −0.411927
\(986\) 0 0
\(987\) 1.35898 0.0432569
\(988\) 0 0
\(989\) −30.9282 −0.983460
\(990\) 0 0
\(991\) −12.3923 −0.393655 −0.196827 0.980438i \(-0.563064\pi\)
−0.196827 + 0.980438i \(0.563064\pi\)
\(992\) 0 0
\(993\) −19.2154 −0.609782
\(994\) 0 0
\(995\) 13.0718 0.414404
\(996\) 0 0
\(997\) −9.60770 −0.304279 −0.152139 0.988359i \(-0.548616\pi\)
−0.152139 + 0.988359i \(0.548616\pi\)
\(998\) 0 0
\(999\) −13.0718 −0.413573
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 6080.2.a.bj.1.1 2
4.3 odd 2 6080.2.a.z.1.2 2
8.3 odd 2 380.2.a.d.1.1 2
8.5 even 2 1520.2.a.l.1.2 2
24.11 even 2 3420.2.a.h.1.1 2
40.3 even 4 1900.2.c.e.1749.2 4
40.19 odd 2 1900.2.a.d.1.2 2
40.27 even 4 1900.2.c.e.1749.3 4
40.29 even 2 7600.2.a.bf.1.1 2
152.75 even 2 7220.2.a.h.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.a.d.1.1 2 8.3 odd 2
1520.2.a.l.1.2 2 8.5 even 2
1900.2.a.d.1.2 2 40.19 odd 2
1900.2.c.e.1749.2 4 40.3 even 4
1900.2.c.e.1749.3 4 40.27 even 4
3420.2.a.h.1.1 2 24.11 even 2
6080.2.a.z.1.2 2 4.3 odd 2
6080.2.a.bj.1.1 2 1.1 even 1 trivial
7220.2.a.h.1.2 2 152.75 even 2
7600.2.a.bf.1.1 2 40.29 even 2