Properties

Label 8112.2.a.by.1.2
Level $8112$
Weight $2$
Character 8112.1
Self dual yes
Analytic conductor $64.775$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8112 = 2^{4} \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8112.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.7746461197\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \(x^{3} - x^{2} - 2 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 507)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.445042\) of defining polynomial
Character \(\chi\) \(=\) 8112.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.44504 q^{5} +3.44504 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.44504 q^{5} +3.44504 q^{7} +1.00000 q^{9} -5.18598 q^{11} +1.44504 q^{15} -0.753020 q^{17} +7.96077 q^{19} -3.44504 q^{21} +2.82908 q^{23} -2.91185 q^{25} -1.00000 q^{27} -3.91185 q^{29} -4.89977 q^{31} +5.18598 q^{33} -4.97823 q^{35} -6.24698 q^{37} -1.80194 q^{41} +7.09783 q^{43} -1.44504 q^{45} +10.5526 q^{47} +4.86831 q^{49} +0.753020 q^{51} -3.08815 q^{53} +7.49396 q^{55} -7.96077 q^{57} +1.87800 q^{59} +3.34481 q^{61} +3.44504 q^{63} -4.54288 q^{67} -2.82908 q^{69} -9.11960 q^{71} -2.95108 q^{73} +2.91185 q^{75} -17.8659 q^{77} +9.43296 q^{79} +1.00000 q^{81} -6.46681 q^{83} +1.08815 q^{85} +3.91185 q^{87} -1.15883 q^{89} +4.89977 q^{93} -11.5036 q^{95} -8.65817 q^{97} -5.18598 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{3} - 4q^{5} + 10q^{7} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{3} - 4q^{5} + 10q^{7} + 3q^{9} - q^{11} + 4q^{15} - 7q^{17} + 11q^{19} - 10q^{21} - 2q^{23} - 5q^{25} - 3q^{27} - 8q^{29} + 8q^{31} + q^{33} - 18q^{35} - 14q^{37} - q^{41} + 3q^{43} - 4q^{45} - 9q^{47} + 17q^{49} + 7q^{51} - 13q^{53} + 13q^{55} - 11q^{57} - 14q^{59} - 13q^{61} + 10q^{63} + 5q^{67} + 2q^{69} - 6q^{71} - 18q^{73} + 5q^{75} - 15q^{77} + 9q^{79} + 3q^{81} - 16q^{83} + 7q^{85} + 8q^{87} + 5q^{89} - 8q^{93} - 3q^{95} - 5q^{97} - q^{99} + O(q^{100}) \)

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −1.44504 −0.646242 −0.323121 0.946358i \(-0.604732\pi\)
−0.323121 + 0.946358i \(0.604732\pi\)
\(6\) 0 0
\(7\) 3.44504 1.30210 0.651052 0.759033i \(-0.274328\pi\)
0.651052 + 0.759033i \(0.274328\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −5.18598 −1.56363 −0.781816 0.623509i \(-0.785706\pi\)
−0.781816 + 0.623509i \(0.785706\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) 1.44504 0.373108
\(16\) 0 0
\(17\) −0.753020 −0.182634 −0.0913171 0.995822i \(-0.529108\pi\)
−0.0913171 + 0.995822i \(0.529108\pi\)
\(18\) 0 0
\(19\) 7.96077 1.82633 0.913163 0.407594i \(-0.133632\pi\)
0.913163 + 0.407594i \(0.133632\pi\)
\(20\) 0 0
\(21\) −3.44504 −0.751770
\(22\) 0 0
\(23\) 2.82908 0.589905 0.294952 0.955512i \(-0.404696\pi\)
0.294952 + 0.955512i \(0.404696\pi\)
\(24\) 0 0
\(25\) −2.91185 −0.582371
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −3.91185 −0.726413 −0.363207 0.931709i \(-0.618318\pi\)
−0.363207 + 0.931709i \(0.618318\pi\)
\(30\) 0 0
\(31\) −4.89977 −0.880025 −0.440013 0.897992i \(-0.645026\pi\)
−0.440013 + 0.897992i \(0.645026\pi\)
\(32\) 0 0
\(33\) 5.18598 0.902763
\(34\) 0 0
\(35\) −4.97823 −0.841474
\(36\) 0 0
\(37\) −6.24698 −1.02700 −0.513499 0.858090i \(-0.671651\pi\)
−0.513499 + 0.858090i \(0.671651\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −1.80194 −0.281415 −0.140708 0.990051i \(-0.544938\pi\)
−0.140708 + 0.990051i \(0.544938\pi\)
\(42\) 0 0
\(43\) 7.09783 1.08241 0.541205 0.840891i \(-0.317968\pi\)
0.541205 + 0.840891i \(0.317968\pi\)
\(44\) 0 0
\(45\) −1.44504 −0.215414
\(46\) 0 0
\(47\) 10.5526 1.53925 0.769625 0.638496i \(-0.220443\pi\)
0.769625 + 0.638496i \(0.220443\pi\)
\(48\) 0 0
\(49\) 4.86831 0.695473
\(50\) 0 0
\(51\) 0.753020 0.105444
\(52\) 0 0
\(53\) −3.08815 −0.424189 −0.212095 0.977249i \(-0.568029\pi\)
−0.212095 + 0.977249i \(0.568029\pi\)
\(54\) 0 0
\(55\) 7.49396 1.01049
\(56\) 0 0
\(57\) −7.96077 −1.05443
\(58\) 0 0
\(59\) 1.87800 0.244495 0.122248 0.992500i \(-0.460990\pi\)
0.122248 + 0.992500i \(0.460990\pi\)
\(60\) 0 0
\(61\) 3.34481 0.428260 0.214130 0.976805i \(-0.431308\pi\)
0.214130 + 0.976805i \(0.431308\pi\)
\(62\) 0 0
\(63\) 3.44504 0.434034
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −4.54288 −0.555001 −0.277500 0.960726i \(-0.589506\pi\)
−0.277500 + 0.960726i \(0.589506\pi\)
\(68\) 0 0
\(69\) −2.82908 −0.340582
\(70\) 0 0
\(71\) −9.11960 −1.08230 −0.541149 0.840927i \(-0.682011\pi\)
−0.541149 + 0.840927i \(0.682011\pi\)
\(72\) 0 0
\(73\) −2.95108 −0.345398 −0.172699 0.984975i \(-0.555249\pi\)
−0.172699 + 0.984975i \(0.555249\pi\)
\(74\) 0 0
\(75\) 2.91185 0.336232
\(76\) 0 0
\(77\) −17.8659 −2.03601
\(78\) 0 0
\(79\) 9.43296 1.06129 0.530645 0.847594i \(-0.321950\pi\)
0.530645 + 0.847594i \(0.321950\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −6.46681 −0.709825 −0.354912 0.934900i \(-0.615489\pi\)
−0.354912 + 0.934900i \(0.615489\pi\)
\(84\) 0 0
\(85\) 1.08815 0.118026
\(86\) 0 0
\(87\) 3.91185 0.419395
\(88\) 0 0
\(89\) −1.15883 −0.122836 −0.0614181 0.998112i \(-0.519562\pi\)
−0.0614181 + 0.998112i \(0.519562\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 4.89977 0.508083
\(94\) 0 0
\(95\) −11.5036 −1.18025
\(96\) 0 0
\(97\) −8.65817 −0.879104 −0.439552 0.898217i \(-0.644863\pi\)
−0.439552 + 0.898217i \(0.644863\pi\)
\(98\) 0 0
\(99\) −5.18598 −0.521211
\(100\) 0 0
\(101\) −8.47650 −0.843443 −0.421722 0.906725i \(-0.638574\pi\)
−0.421722 + 0.906725i \(0.638574\pi\)
\(102\) 0 0
\(103\) −5.64742 −0.556456 −0.278228 0.960515i \(-0.589747\pi\)
−0.278228 + 0.960515i \(0.589747\pi\)
\(104\) 0 0
\(105\) 4.97823 0.485825
\(106\) 0 0
\(107\) 6.73556 0.651151 0.325576 0.945516i \(-0.394442\pi\)
0.325576 + 0.945516i \(0.394442\pi\)
\(108\) 0 0
\(109\) 2.07606 0.198851 0.0994255 0.995045i \(-0.468300\pi\)
0.0994255 + 0.995045i \(0.468300\pi\)
\(110\) 0 0
\(111\) 6.24698 0.592937
\(112\) 0 0
\(113\) 6.16852 0.580286 0.290143 0.956983i \(-0.406297\pi\)
0.290143 + 0.956983i \(0.406297\pi\)
\(114\) 0 0
\(115\) −4.08815 −0.381222
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −2.59419 −0.237809
\(120\) 0 0
\(121\) 15.8944 1.44495
\(122\) 0 0
\(123\) 1.80194 0.162475
\(124\) 0 0
\(125\) 11.4330 1.02260
\(126\) 0 0
\(127\) 14.2620 1.26555 0.632776 0.774335i \(-0.281915\pi\)
0.632776 + 0.774335i \(0.281915\pi\)
\(128\) 0 0
\(129\) −7.09783 −0.624929
\(130\) 0 0
\(131\) −22.6015 −1.97470 −0.987350 0.158554i \(-0.949317\pi\)
−0.987350 + 0.158554i \(0.949317\pi\)
\(132\) 0 0
\(133\) 27.4252 2.37807
\(134\) 0 0
\(135\) 1.44504 0.124369
\(136\) 0 0
\(137\) 13.6353 1.16495 0.582473 0.812850i \(-0.302085\pi\)
0.582473 + 0.812850i \(0.302085\pi\)
\(138\) 0 0
\(139\) −17.6015 −1.49294 −0.746469 0.665420i \(-0.768252\pi\)
−0.746469 + 0.665420i \(0.768252\pi\)
\(140\) 0 0
\(141\) −10.5526 −0.888686
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 5.65279 0.469439
\(146\) 0 0
\(147\) −4.86831 −0.401532
\(148\) 0 0
\(149\) −12.7385 −1.04358 −0.521791 0.853073i \(-0.674736\pi\)
−0.521791 + 0.853073i \(0.674736\pi\)
\(150\) 0 0
\(151\) 15.6407 1.27282 0.636412 0.771350i \(-0.280418\pi\)
0.636412 + 0.771350i \(0.280418\pi\)
\(152\) 0 0
\(153\) −0.753020 −0.0608781
\(154\) 0 0
\(155\) 7.08038 0.568710
\(156\) 0 0
\(157\) −0.823708 −0.0657391 −0.0328695 0.999460i \(-0.510465\pi\)
−0.0328695 + 0.999460i \(0.510465\pi\)
\(158\) 0 0
\(159\) 3.08815 0.244906
\(160\) 0 0
\(161\) 9.74632 0.768117
\(162\) 0 0
\(163\) 6.26875 0.491006 0.245503 0.969396i \(-0.421047\pi\)
0.245503 + 0.969396i \(0.421047\pi\)
\(164\) 0 0
\(165\) −7.49396 −0.583404
\(166\) 0 0
\(167\) −7.45042 −0.576531 −0.288265 0.957551i \(-0.593079\pi\)
−0.288265 + 0.957551i \(0.593079\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 7.96077 0.608775
\(172\) 0 0
\(173\) −2.00969 −0.152794 −0.0763969 0.997077i \(-0.524342\pi\)
−0.0763969 + 0.997077i \(0.524342\pi\)
\(174\) 0 0
\(175\) −10.0315 −0.758307
\(176\) 0 0
\(177\) −1.87800 −0.141159
\(178\) 0 0
\(179\) −20.0368 −1.49762 −0.748812 0.662783i \(-0.769375\pi\)
−0.748812 + 0.662783i \(0.769375\pi\)
\(180\) 0 0
\(181\) 24.1226 1.79302 0.896509 0.443026i \(-0.146095\pi\)
0.896509 + 0.443026i \(0.146095\pi\)
\(182\) 0 0
\(183\) −3.34481 −0.247256
\(184\) 0 0
\(185\) 9.02715 0.663689
\(186\) 0 0
\(187\) 3.90515 0.285573
\(188\) 0 0
\(189\) −3.44504 −0.250590
\(190\) 0 0
\(191\) 7.08038 0.512318 0.256159 0.966635i \(-0.417543\pi\)
0.256159 + 0.966635i \(0.417543\pi\)
\(192\) 0 0
\(193\) −9.76809 −0.703122 −0.351561 0.936165i \(-0.614349\pi\)
−0.351561 + 0.936165i \(0.614349\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −23.4112 −1.66798 −0.833989 0.551781i \(-0.813948\pi\)
−0.833989 + 0.551781i \(0.813948\pi\)
\(198\) 0 0
\(199\) −4.02475 −0.285307 −0.142654 0.989773i \(-0.545563\pi\)
−0.142654 + 0.989773i \(0.545563\pi\)
\(200\) 0 0
\(201\) 4.54288 0.320430
\(202\) 0 0
\(203\) −13.4765 −0.945865
\(204\) 0 0
\(205\) 2.60388 0.181863
\(206\) 0 0
\(207\) 2.82908 0.196635
\(208\) 0 0
\(209\) −41.2844 −2.85570
\(210\) 0 0
\(211\) 3.91185 0.269303 0.134652 0.990893i \(-0.457008\pi\)
0.134652 + 0.990893i \(0.457008\pi\)
\(212\) 0 0
\(213\) 9.11960 0.624865
\(214\) 0 0
\(215\) −10.2567 −0.699499
\(216\) 0 0
\(217\) −16.8799 −1.14588
\(218\) 0 0
\(219\) 2.95108 0.199416
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 7.44935 0.498846 0.249423 0.968395i \(-0.419759\pi\)
0.249423 + 0.968395i \(0.419759\pi\)
\(224\) 0 0
\(225\) −2.91185 −0.194124
\(226\) 0 0
\(227\) −21.2500 −1.41041 −0.705205 0.709004i \(-0.749145\pi\)
−0.705205 + 0.709004i \(0.749145\pi\)
\(228\) 0 0
\(229\) −9.29590 −0.614290 −0.307145 0.951663i \(-0.599374\pi\)
−0.307145 + 0.951663i \(0.599374\pi\)
\(230\) 0 0
\(231\) 17.8659 1.17549
\(232\) 0 0
\(233\) 16.2107 1.06200 0.531000 0.847372i \(-0.321816\pi\)
0.531000 + 0.847372i \(0.321816\pi\)
\(234\) 0 0
\(235\) −15.2489 −0.994728
\(236\) 0 0
\(237\) −9.43296 −0.612737
\(238\) 0 0
\(239\) −13.5090 −0.873826 −0.436913 0.899504i \(-0.643928\pi\)
−0.436913 + 0.899504i \(0.643928\pi\)
\(240\) 0 0
\(241\) −6.26875 −0.403806 −0.201903 0.979406i \(-0.564713\pi\)
−0.201903 + 0.979406i \(0.564713\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −7.03492 −0.449444
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 6.46681 0.409818
\(250\) 0 0
\(251\) 0.753020 0.0475302 0.0237651 0.999718i \(-0.492435\pi\)
0.0237651 + 0.999718i \(0.492435\pi\)
\(252\) 0 0
\(253\) −14.6716 −0.922394
\(254\) 0 0
\(255\) −1.08815 −0.0681423
\(256\) 0 0
\(257\) 19.7265 1.23050 0.615252 0.788331i \(-0.289054\pi\)
0.615252 + 0.788331i \(0.289054\pi\)
\(258\) 0 0
\(259\) −21.5211 −1.33726
\(260\) 0 0
\(261\) −3.91185 −0.242138
\(262\) 0 0
\(263\) 17.6093 1.08583 0.542917 0.839787i \(-0.317320\pi\)
0.542917 + 0.839787i \(0.317320\pi\)
\(264\) 0 0
\(265\) 4.46250 0.274129
\(266\) 0 0
\(267\) 1.15883 0.0709195
\(268\) 0 0
\(269\) −16.3870 −0.999135 −0.499567 0.866275i \(-0.666508\pi\)
−0.499567 + 0.866275i \(0.666508\pi\)
\(270\) 0 0
\(271\) −0.795233 −0.0483070 −0.0241535 0.999708i \(-0.507689\pi\)
−0.0241535 + 0.999708i \(0.507689\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 15.1008 0.910614
\(276\) 0 0
\(277\) −4.83340 −0.290411 −0.145205 0.989402i \(-0.546384\pi\)
−0.145205 + 0.989402i \(0.546384\pi\)
\(278\) 0 0
\(279\) −4.89977 −0.293342
\(280\) 0 0
\(281\) −18.7748 −1.12001 −0.560005 0.828489i \(-0.689201\pi\)
−0.560005 + 0.828489i \(0.689201\pi\)
\(282\) 0 0
\(283\) 7.91723 0.470631 0.235315 0.971919i \(-0.424388\pi\)
0.235315 + 0.971919i \(0.424388\pi\)
\(284\) 0 0
\(285\) 11.5036 0.681417
\(286\) 0 0
\(287\) −6.20775 −0.366432
\(288\) 0 0
\(289\) −16.4330 −0.966645
\(290\) 0 0
\(291\) 8.65817 0.507551
\(292\) 0 0
\(293\) −6.57912 −0.384356 −0.192178 0.981360i \(-0.561555\pi\)
−0.192178 + 0.981360i \(0.561555\pi\)
\(294\) 0 0
\(295\) −2.71379 −0.158003
\(296\) 0 0
\(297\) 5.18598 0.300921
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 24.4523 1.40941
\(302\) 0 0
\(303\) 8.47650 0.486962
\(304\) 0 0
\(305\) −4.83340 −0.276759
\(306\) 0 0
\(307\) −24.8649 −1.41911 −0.709556 0.704649i \(-0.751105\pi\)
−0.709556 + 0.704649i \(0.751105\pi\)
\(308\) 0 0
\(309\) 5.64742 0.321270
\(310\) 0 0
\(311\) 17.0804 0.968539 0.484270 0.874919i \(-0.339085\pi\)
0.484270 + 0.874919i \(0.339085\pi\)
\(312\) 0 0
\(313\) 15.6974 0.887269 0.443635 0.896208i \(-0.353689\pi\)
0.443635 + 0.896208i \(0.353689\pi\)
\(314\) 0 0
\(315\) −4.97823 −0.280491
\(316\) 0 0
\(317\) −32.7821 −1.84123 −0.920613 0.390477i \(-0.872310\pi\)
−0.920613 + 0.390477i \(0.872310\pi\)
\(318\) 0 0
\(319\) 20.2868 1.13584
\(320\) 0 0
\(321\) −6.73556 −0.375942
\(322\) 0 0
\(323\) −5.99462 −0.333550
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −2.07606 −0.114807
\(328\) 0 0
\(329\) 36.3540 2.00426
\(330\) 0 0
\(331\) −29.1618 −1.60288 −0.801439 0.598076i \(-0.795932\pi\)
−0.801439 + 0.598076i \(0.795932\pi\)
\(332\) 0 0
\(333\) −6.24698 −0.342332
\(334\) 0 0
\(335\) 6.56465 0.358665
\(336\) 0 0
\(337\) −33.2911 −1.81348 −0.906741 0.421688i \(-0.861438\pi\)
−0.906741 + 0.421688i \(0.861438\pi\)
\(338\) 0 0
\(339\) −6.16852 −0.335028
\(340\) 0 0
\(341\) 25.4101 1.37604
\(342\) 0 0
\(343\) −7.34375 −0.396525
\(344\) 0 0
\(345\) 4.08815 0.220098
\(346\) 0 0
\(347\) 0.873690 0.0469022 0.0234511 0.999725i \(-0.492535\pi\)
0.0234511 + 0.999725i \(0.492535\pi\)
\(348\) 0 0
\(349\) 3.23191 0.173000 0.0865002 0.996252i \(-0.472432\pi\)
0.0865002 + 0.996252i \(0.472432\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −8.14675 −0.433608 −0.216804 0.976215i \(-0.569563\pi\)
−0.216804 + 0.976215i \(0.569563\pi\)
\(354\) 0 0
\(355\) 13.1782 0.699427
\(356\) 0 0
\(357\) 2.59419 0.137299
\(358\) 0 0
\(359\) 2.64071 0.139371 0.0696857 0.997569i \(-0.477800\pi\)
0.0696857 + 0.997569i \(0.477800\pi\)
\(360\) 0 0
\(361\) 44.3739 2.33547
\(362\) 0 0
\(363\) −15.8944 −0.834239
\(364\) 0 0
\(365\) 4.26444 0.223211
\(366\) 0 0
\(367\) 2.90408 0.151592 0.0757960 0.997123i \(-0.475850\pi\)
0.0757960 + 0.997123i \(0.475850\pi\)
\(368\) 0 0
\(369\) −1.80194 −0.0938051
\(370\) 0 0
\(371\) −10.6388 −0.552339
\(372\) 0 0
\(373\) −8.39852 −0.434859 −0.217429 0.976076i \(-0.569767\pi\)
−0.217429 + 0.976076i \(0.569767\pi\)
\(374\) 0 0
\(375\) −11.4330 −0.590396
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 15.7482 0.808932 0.404466 0.914553i \(-0.367457\pi\)
0.404466 + 0.914553i \(0.367457\pi\)
\(380\) 0 0
\(381\) −14.2620 −0.730667
\(382\) 0 0
\(383\) 12.7385 0.650909 0.325455 0.945558i \(-0.394483\pi\)
0.325455 + 0.945558i \(0.394483\pi\)
\(384\) 0 0
\(385\) 25.8170 1.31576
\(386\) 0 0
\(387\) 7.09783 0.360803
\(388\) 0 0
\(389\) −0.310371 −0.0157365 −0.00786823 0.999969i \(-0.502505\pi\)
−0.00786823 + 0.999969i \(0.502505\pi\)
\(390\) 0 0
\(391\) −2.13036 −0.107737
\(392\) 0 0
\(393\) 22.6015 1.14009
\(394\) 0 0
\(395\) −13.6310 −0.685851
\(396\) 0 0
\(397\) −1.49098 −0.0748299 −0.0374150 0.999300i \(-0.511912\pi\)
−0.0374150 + 0.999300i \(0.511912\pi\)
\(398\) 0 0
\(399\) −27.4252 −1.37298
\(400\) 0 0
\(401\) 23.8334 1.19018 0.595092 0.803658i \(-0.297116\pi\)
0.595092 + 0.803658i \(0.297116\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −1.44504 −0.0718047
\(406\) 0 0
\(407\) 32.3967 1.60585
\(408\) 0 0
\(409\) −4.26742 −0.211010 −0.105505 0.994419i \(-0.533646\pi\)
−0.105505 + 0.994419i \(0.533646\pi\)
\(410\) 0 0
\(411\) −13.6353 −0.672581
\(412\) 0 0
\(413\) 6.46980 0.318358
\(414\) 0 0
\(415\) 9.34481 0.458719
\(416\) 0 0
\(417\) 17.6015 0.861948
\(418\) 0 0
\(419\) −29.6896 −1.45043 −0.725217 0.688521i \(-0.758260\pi\)
−0.725217 + 0.688521i \(0.758260\pi\)
\(420\) 0 0
\(421\) −29.3991 −1.43282 −0.716412 0.697677i \(-0.754217\pi\)
−0.716412 + 0.697677i \(0.754217\pi\)
\(422\) 0 0
\(423\) 10.5526 0.513083
\(424\) 0 0
\(425\) 2.19269 0.106361
\(426\) 0 0
\(427\) 11.5230 0.557638
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −33.0562 −1.59226 −0.796131 0.605124i \(-0.793123\pi\)
−0.796131 + 0.605124i \(0.793123\pi\)
\(432\) 0 0
\(433\) −29.2664 −1.40645 −0.703226 0.710967i \(-0.748258\pi\)
−0.703226 + 0.710967i \(0.748258\pi\)
\(434\) 0 0
\(435\) −5.65279 −0.271031
\(436\) 0 0
\(437\) 22.5217 1.07736
\(438\) 0 0
\(439\) 2.13169 0.101740 0.0508699 0.998705i \(-0.483801\pi\)
0.0508699 + 0.998705i \(0.483801\pi\)
\(440\) 0 0
\(441\) 4.86831 0.231824
\(442\) 0 0
\(443\) 22.9922 1.09239 0.546197 0.837657i \(-0.316075\pi\)
0.546197 + 0.837657i \(0.316075\pi\)
\(444\) 0 0
\(445\) 1.67456 0.0793819
\(446\) 0 0
\(447\) 12.7385 0.602513
\(448\) 0 0
\(449\) 12.9379 0.610579 0.305289 0.952260i \(-0.401247\pi\)
0.305289 + 0.952260i \(0.401247\pi\)
\(450\) 0 0
\(451\) 9.34481 0.440030
\(452\) 0 0
\(453\) −15.6407 −0.734865
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 4.85325 0.227025 0.113513 0.993537i \(-0.463790\pi\)
0.113513 + 0.993537i \(0.463790\pi\)
\(458\) 0 0
\(459\) 0.753020 0.0351480
\(460\) 0 0
\(461\) −18.8345 −0.877208 −0.438604 0.898680i \(-0.644527\pi\)
−0.438604 + 0.898680i \(0.644527\pi\)
\(462\) 0 0
\(463\) −22.8767 −1.06317 −0.531585 0.847005i \(-0.678403\pi\)
−0.531585 + 0.847005i \(0.678403\pi\)
\(464\) 0 0
\(465\) −7.08038 −0.328345
\(466\) 0 0
\(467\) −13.0000 −0.601568 −0.300784 0.953692i \(-0.597248\pi\)
−0.300784 + 0.953692i \(0.597248\pi\)
\(468\) 0 0
\(469\) −15.6504 −0.722668
\(470\) 0 0
\(471\) 0.823708 0.0379545
\(472\) 0 0
\(473\) −36.8092 −1.69249
\(474\) 0 0
\(475\) −23.1806 −1.06360
\(476\) 0 0
\(477\) −3.08815 −0.141396
\(478\) 0 0
\(479\) −38.0901 −1.74038 −0.870190 0.492717i \(-0.836004\pi\)
−0.870190 + 0.492717i \(0.836004\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) −9.74632 −0.443473
\(484\) 0 0
\(485\) 12.5114 0.568114
\(486\) 0 0
\(487\) −21.2500 −0.962928 −0.481464 0.876466i \(-0.659895\pi\)
−0.481464 + 0.876466i \(0.659895\pi\)
\(488\) 0 0
\(489\) −6.26875 −0.283483
\(490\) 0 0
\(491\) 6.35019 0.286580 0.143290 0.989681i \(-0.454232\pi\)
0.143290 + 0.989681i \(0.454232\pi\)
\(492\) 0 0
\(493\) 2.94571 0.132668
\(494\) 0 0
\(495\) 7.49396 0.336828
\(496\) 0 0
\(497\) −31.4174 −1.40926
\(498\) 0 0
\(499\) −4.65087 −0.208202 −0.104101 0.994567i \(-0.533196\pi\)
−0.104101 + 0.994567i \(0.533196\pi\)
\(500\) 0 0
\(501\) 7.45042 0.332860
\(502\) 0 0
\(503\) −15.4752 −0.690004 −0.345002 0.938602i \(-0.612122\pi\)
−0.345002 + 0.938602i \(0.612122\pi\)
\(504\) 0 0
\(505\) 12.2489 0.545069
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 20.5047 0.908855 0.454428 0.890784i \(-0.349844\pi\)
0.454428 + 0.890784i \(0.349844\pi\)
\(510\) 0 0
\(511\) −10.1666 −0.449744
\(512\) 0 0
\(513\) −7.96077 −0.351477
\(514\) 0 0
\(515\) 8.16075 0.359606
\(516\) 0 0
\(517\) −54.7254 −2.40682
\(518\) 0 0
\(519\) 2.00969 0.0882155
\(520\) 0 0
\(521\) −42.0267 −1.84122 −0.920611 0.390481i \(-0.872309\pi\)
−0.920611 + 0.390481i \(0.872309\pi\)
\(522\) 0 0
\(523\) 29.9885 1.31131 0.655653 0.755062i \(-0.272393\pi\)
0.655653 + 0.755062i \(0.272393\pi\)
\(524\) 0 0
\(525\) 10.0315 0.437809
\(526\) 0 0
\(527\) 3.68963 0.160723
\(528\) 0 0
\(529\) −14.9963 −0.652012
\(530\) 0 0
\(531\) 1.87800 0.0814984
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −9.73317 −0.420802
\(536\) 0 0
\(537\) 20.0368 0.864653
\(538\) 0 0
\(539\) −25.2470 −1.08746
\(540\) 0 0
\(541\) −36.3803 −1.56411 −0.782056 0.623208i \(-0.785829\pi\)
−0.782056 + 0.623208i \(0.785829\pi\)
\(542\) 0 0
\(543\) −24.1226 −1.03520
\(544\) 0 0
\(545\) −3.00000 −0.128506
\(546\) 0 0
\(547\) 25.8159 1.10381 0.551905 0.833907i \(-0.313901\pi\)
0.551905 + 0.833907i \(0.313901\pi\)
\(548\) 0 0
\(549\) 3.34481 0.142753
\(550\) 0 0
\(551\) −31.1414 −1.32667
\(552\) 0 0
\(553\) 32.4969 1.38191
\(554\) 0 0
\(555\) −9.02715 −0.383181
\(556\) 0 0
\(557\) −17.9903 −0.762274 −0.381137 0.924519i \(-0.624467\pi\)
−0.381137 + 0.924519i \(0.624467\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −3.90515 −0.164876
\(562\) 0 0
\(563\) 39.1323 1.64923 0.824614 0.565695i \(-0.191392\pi\)
0.824614 + 0.565695i \(0.191392\pi\)
\(564\) 0 0
\(565\) −8.91377 −0.375005
\(566\) 0 0
\(567\) 3.44504 0.144678
\(568\) 0 0
\(569\) 30.6002 1.28283 0.641413 0.767196i \(-0.278349\pi\)
0.641413 + 0.767196i \(0.278349\pi\)
\(570\) 0 0
\(571\) 2.96184 0.123949 0.0619745 0.998078i \(-0.480260\pi\)
0.0619745 + 0.998078i \(0.480260\pi\)
\(572\) 0 0
\(573\) −7.08038 −0.295787
\(574\) 0 0
\(575\) −8.23788 −0.343543
\(576\) 0 0
\(577\) 0.819396 0.0341119 0.0170560 0.999855i \(-0.494571\pi\)
0.0170560 + 0.999855i \(0.494571\pi\)
\(578\) 0 0
\(579\) 9.76809 0.405948
\(580\) 0 0
\(581\) −22.2784 −0.924265
\(582\) 0 0
\(583\) 16.0151 0.663276
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −31.7995 −1.31251 −0.656254 0.754540i \(-0.727860\pi\)
−0.656254 + 0.754540i \(0.727860\pi\)
\(588\) 0 0
\(589\) −39.0060 −1.60721
\(590\) 0 0
\(591\) 23.4112 0.963008
\(592\) 0 0
\(593\) 4.26337 0.175076 0.0875379 0.996161i \(-0.472100\pi\)
0.0875379 + 0.996161i \(0.472100\pi\)
\(594\) 0 0
\(595\) 3.74871 0.153682
\(596\) 0 0
\(597\) 4.02475 0.164722
\(598\) 0 0
\(599\) −24.7278 −1.01035 −0.505175 0.863017i \(-0.668572\pi\)
−0.505175 + 0.863017i \(0.668572\pi\)
\(600\) 0 0
\(601\) 6.82371 0.278345 0.139172 0.990268i \(-0.455556\pi\)
0.139172 + 0.990268i \(0.455556\pi\)
\(602\) 0 0
\(603\) −4.54288 −0.185000
\(604\) 0 0
\(605\) −22.9681 −0.933785
\(606\) 0 0
\(607\) −31.9963 −1.29869 −0.649344 0.760494i \(-0.724957\pi\)
−0.649344 + 0.760494i \(0.724957\pi\)
\(608\) 0 0
\(609\) 13.4765 0.546095
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −33.5875 −1.35659 −0.678293 0.734792i \(-0.737280\pi\)
−0.678293 + 0.734792i \(0.737280\pi\)
\(614\) 0 0
\(615\) −2.60388 −0.104998
\(616\) 0 0
\(617\) 26.5870 1.07035 0.535176 0.844740i \(-0.320245\pi\)
0.535176 + 0.844740i \(0.320245\pi\)
\(618\) 0 0
\(619\) −9.17928 −0.368946 −0.184473 0.982838i \(-0.559058\pi\)
−0.184473 + 0.982838i \(0.559058\pi\)
\(620\) 0 0
\(621\) −2.82908 −0.113527
\(622\) 0 0
\(623\) −3.99223 −0.159945
\(624\) 0 0
\(625\) −1.96184 −0.0784735
\(626\) 0 0
\(627\) 41.2844 1.64874
\(628\) 0 0
\(629\) 4.70410 0.187565
\(630\) 0 0
\(631\) 17.0043 0.676931 0.338465 0.940979i \(-0.390092\pi\)
0.338465 + 0.940979i \(0.390092\pi\)
\(632\) 0 0
\(633\) −3.91185 −0.155482
\(634\) 0 0
\(635\) −20.6093 −0.817853
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −9.11960 −0.360766
\(640\) 0 0
\(641\) −21.6649 −0.855711 −0.427856 0.903847i \(-0.640731\pi\)
−0.427856 + 0.903847i \(0.640731\pi\)
\(642\) 0 0
\(643\) 9.35557 0.368948 0.184474 0.982837i \(-0.440942\pi\)
0.184474 + 0.982837i \(0.440942\pi\)
\(644\) 0 0
\(645\) 10.2567 0.403856
\(646\) 0 0
\(647\) 0.702775 0.0276289 0.0138145 0.999905i \(-0.495603\pi\)
0.0138145 + 0.999905i \(0.495603\pi\)
\(648\) 0 0
\(649\) −9.73928 −0.382300
\(650\) 0 0
\(651\) 16.8799 0.661576
\(652\) 0 0
\(653\) −37.3411 −1.46127 −0.730635 0.682768i \(-0.760776\pi\)
−0.730635 + 0.682768i \(0.760776\pi\)
\(654\) 0 0
\(655\) 32.6601 1.27614
\(656\) 0 0
\(657\) −2.95108 −0.115133
\(658\) 0 0
\(659\) 0.735562 0.0286534 0.0143267 0.999897i \(-0.495440\pi\)
0.0143267 + 0.999897i \(0.495440\pi\)
\(660\) 0 0
\(661\) −13.8485 −0.538643 −0.269321 0.963050i \(-0.586799\pi\)
−0.269321 + 0.963050i \(0.586799\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −39.6305 −1.53681
\(666\) 0 0
\(667\) −11.0670 −0.428515
\(668\) 0 0
\(669\) −7.44935 −0.288009
\(670\) 0 0
\(671\) −17.3461 −0.669640
\(672\) 0 0
\(673\) −6.35019 −0.244782 −0.122391 0.992482i \(-0.539056\pi\)
−0.122391 + 0.992482i \(0.539056\pi\)
\(674\) 0 0
\(675\) 2.91185 0.112077
\(676\) 0 0
\(677\) 33.7241 1.29612 0.648061 0.761589i \(-0.275580\pi\)
0.648061 + 0.761589i \(0.275580\pi\)
\(678\) 0 0
\(679\) −29.8278 −1.14468
\(680\) 0 0
\(681\) 21.2500 0.814300
\(682\) 0 0
\(683\) 19.2687 0.737298 0.368649 0.929569i \(-0.379820\pi\)
0.368649 + 0.929569i \(0.379820\pi\)
\(684\) 0 0
\(685\) −19.7036 −0.752837
\(686\) 0 0
\(687\) 9.29590 0.354661
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −39.4010 −1.49889 −0.749443 0.662069i \(-0.769679\pi\)
−0.749443 + 0.662069i \(0.769679\pi\)
\(692\) 0 0
\(693\) −17.8659 −0.678670
\(694\) 0 0
\(695\) 25.4349 0.964800
\(696\) 0 0
\(697\) 1.35690 0.0513961
\(698\) 0 0
\(699\) −16.2107 −0.613146
\(700\) 0 0
\(701\) 18.3985 0.694902 0.347451 0.937698i \(-0.387047\pi\)
0.347451 + 0.937698i \(0.387047\pi\)
\(702\) 0 0
\(703\) −49.7308 −1.87563
\(704\) 0 0
\(705\) 15.2489 0.574307
\(706\) 0 0
\(707\) −29.2019 −1.09825
\(708\) 0 0
\(709\) 38.4553 1.44422 0.722110 0.691778i \(-0.243172\pi\)
0.722110 + 0.691778i \(0.243172\pi\)
\(710\) 0 0
\(711\) 9.43296 0.353764
\(712\) 0 0
\(713\) −13.8619 −0.519131
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 13.5090 0.504504
\(718\) 0 0
\(719\) −48.4999 −1.80874 −0.904371 0.426747i \(-0.859659\pi\)
−0.904371 + 0.426747i \(0.859659\pi\)
\(720\) 0 0
\(721\) −19.4556 −0.724564
\(722\) 0 0
\(723\) 6.26875 0.233137
\(724\) 0 0
\(725\) 11.3907 0.423042
\(726\) 0 0
\(727\) −19.0344 −0.705948 −0.352974 0.935633i \(-0.614830\pi\)
−0.352974 + 0.935633i \(0.614830\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −5.34481 −0.197685
\(732\) 0 0
\(733\) −24.6213 −0.909410 −0.454705 0.890642i \(-0.650255\pi\)
−0.454705 + 0.890642i \(0.650255\pi\)
\(734\) 0 0
\(735\) 7.03492 0.259487
\(736\) 0 0
\(737\) 23.5593 0.867817
\(738\) 0 0
\(739\) −44.5115 −1.63738 −0.818692 0.574233i \(-0.805300\pi\)
−0.818692 + 0.574233i \(0.805300\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 10.4112 0.381950 0.190975 0.981595i \(-0.438835\pi\)
0.190975 + 0.981595i \(0.438835\pi\)
\(744\) 0 0
\(745\) 18.4077 0.674407
\(746\) 0 0
\(747\) −6.46681 −0.236608
\(748\) 0 0
\(749\) 23.2043 0.847866
\(750\) 0 0
\(751\) −1.69979 −0.0620263 −0.0310131 0.999519i \(-0.509873\pi\)
−0.0310131 + 0.999519i \(0.509873\pi\)
\(752\) 0 0
\(753\) −0.753020 −0.0274416
\(754\) 0 0
\(755\) −22.6015 −0.822552
\(756\) 0 0
\(757\) 27.4252 0.996786 0.498393 0.866951i \(-0.333924\pi\)
0.498393 + 0.866951i \(0.333924\pi\)
\(758\) 0 0
\(759\) 14.6716 0.532545
\(760\) 0 0
\(761\) 5.02608 0.182195 0.0910977 0.995842i \(-0.470962\pi\)
0.0910977 + 0.995842i \(0.470962\pi\)
\(762\) 0 0
\(763\) 7.15213 0.258924
\(764\) 0 0
\(765\) 1.08815 0.0393420
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −42.4456 −1.53063 −0.765314 0.643657i \(-0.777416\pi\)
−0.765314 + 0.643657i \(0.777416\pi\)
\(770\) 0 0
\(771\) −19.7265 −0.710431
\(772\) 0 0
\(773\) −26.3593 −0.948078 −0.474039 0.880504i \(-0.657204\pi\)
−0.474039 + 0.880504i \(0.657204\pi\)
\(774\) 0 0
\(775\) 14.2674 0.512501
\(776\) 0 0
\(777\) 21.5211 0.772065
\(778\) 0 0
\(779\) −14.3448 −0.513956
\(780\) 0 0
\(781\) 47.2941 1.69232
\(782\) 0 0
\(783\) 3.91185 0.139798
\(784\) 0 0
\(785\) 1.19029 0.0424834
\(786\) 0 0
\(787\) 17.1424 0.611062 0.305531 0.952182i \(-0.401166\pi\)
0.305531 + 0.952182i \(0.401166\pi\)
\(788\) 0 0
\(789\) −17.6093 −0.626906
\(790\) 0 0
\(791\) 21.2508 0.755592
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −4.46250 −0.158269
\(796\) 0 0
\(797\) 30.1629 1.06842 0.534212 0.845351i \(-0.320608\pi\)
0.534212 + 0.845351i \(0.320608\pi\)
\(798\) 0 0
\(799\) −7.94630 −0.281120
\(800\) 0 0
\(801\) −1.15883 −0.0409454
\(802\) 0 0
\(803\) 15.3043 0.540076
\(804\) 0 0
\(805\) −14.0838 −0.496390
\(806\) 0 0
\(807\) 16.3870 0.576851
\(808\) 0 0
\(809\) −29.8504 −1.04948 −0.524742 0.851261i \(-0.675838\pi\)
−0.524742 + 0.851261i \(0.675838\pi\)
\(810\) 0 0
\(811\) 47.7362 1.67624 0.838122 0.545484i \(-0.183654\pi\)
0.838122 + 0.545484i \(0.183654\pi\)
\(812\) 0 0
\(813\) 0.795233 0.0278900
\(814\) 0 0
\(815\) −9.05861 −0.317309
\(816\) 0 0
\(817\) 56.5042 1.97683
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 17.9299 0.625758 0.312879 0.949793i \(-0.398707\pi\)
0.312879 + 0.949793i \(0.398707\pi\)
\(822\) 0 0
\(823\) −54.3196 −1.89346 −0.946731 0.322026i \(-0.895636\pi\)
−0.946731 + 0.322026i \(0.895636\pi\)
\(824\) 0 0
\(825\) −15.1008 −0.525743
\(826\) 0 0
\(827\) 49.1041 1.70752 0.853758 0.520670i \(-0.174318\pi\)
0.853758 + 0.520670i \(0.174318\pi\)
\(828\) 0 0
\(829\) −7.35796 −0.255553 −0.127776 0.991803i \(-0.540784\pi\)
−0.127776 + 0.991803i \(0.540784\pi\)
\(830\) 0 0
\(831\) 4.83340 0.167669
\(832\) 0 0
\(833\) −3.66594 −0.127017
\(834\) 0 0
\(835\) 10.7662 0.372579
\(836\) 0 0
\(837\) 4.89977 0.169361
\(838\) 0 0
\(839\) −36.5013 −1.26016 −0.630082 0.776529i \(-0.716979\pi\)
−0.630082 + 0.776529i \(0.716979\pi\)
\(840\) 0 0
\(841\) −13.6974 −0.472324
\(842\) 0 0
\(843\) 18.7748 0.646638
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 54.7569 1.88147
\(848\) 0 0
\(849\) −7.91723 −0.271719
\(850\) 0 0
\(851\) −17.6732 −0.605831
\(852\) 0 0
\(853\) 9.73855 0.333441 0.166721 0.986004i \(-0.446682\pi\)
0.166721 + 0.986004i \(0.446682\pi\)
\(854\) 0 0
\(855\) −11.5036 −0.393416
\(856\) 0 0
\(857\) −15.2030 −0.519323 −0.259662 0.965700i \(-0.583611\pi\)
−0.259662 + 0.965700i \(0.583611\pi\)
\(858\) 0 0
\(859\) 31.9885 1.09143 0.545717 0.837970i \(-0.316257\pi\)
0.545717 + 0.837970i \(0.316257\pi\)
\(860\) 0 0
\(861\) 6.20775 0.211560
\(862\) 0 0
\(863\) −35.2905 −1.20130 −0.600652 0.799511i \(-0.705092\pi\)
−0.600652 + 0.799511i \(0.705092\pi\)
\(864\) 0 0
\(865\) 2.90408 0.0987418
\(866\) 0 0
\(867\) 16.4330 0.558093
\(868\) 0 0
\(869\) −48.9191 −1.65947
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −8.65817 −0.293035
\(874\) 0 0
\(875\) 39.3870 1.33152
\(876\) 0 0
\(877\) 15.3263 0.517532 0.258766 0.965940i \(-0.416684\pi\)
0.258766 + 0.965940i \(0.416684\pi\)
\(878\) 0 0
\(879\) 6.57912 0.221908
\(880\) 0 0
\(881\) −36.4306 −1.22738 −0.613689 0.789548i \(-0.710315\pi\)
−0.613689 + 0.789548i \(0.710315\pi\)
\(882\) 0 0
\(883\) 37.6819 1.26810 0.634048 0.773294i \(-0.281392\pi\)
0.634048 + 0.773294i \(0.281392\pi\)
\(884\) 0 0
\(885\) 2.71379 0.0912231
\(886\) 0 0
\(887\) −4.24890 −0.142664 −0.0713320 0.997453i \(-0.522725\pi\)
−0.0713320 + 0.997453i \(0.522725\pi\)
\(888\) 0 0
\(889\) 49.1333 1.64788
\(890\) 0 0
\(891\) −5.18598 −0.173737
\(892\) 0 0
\(893\) 84.0066 2.81117
\(894\) 0 0
\(895\) 28.9541 0.967828
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 19.1672 0.639262
\(900\) 0 0
\(901\) 2.32544 0.0774715
\(902\) 0 0
\(903\) −24.4523 −0.813723
\(904\) 0 0
\(905\) −34.8582 −1.15872
\(906\) 0 0
\(907\) 12.6183 0.418985 0.209493 0.977810i \(-0.432819\pi\)
0.209493 + 0.977810i \(0.432819\pi\)
\(908\) 0 0
\(909\) −8.47650 −0.281148
\(910\) 0 0
\(911\) −6.77777 −0.224558 −0.112279 0.993677i \(-0.535815\pi\)
−0.112279 + 0.993677i \(0.535815\pi\)
\(912\) 0 0
\(913\) 33.5368 1.10990
\(914\) 0 0
\(915\) 4.83340 0.159787
\(916\) 0 0
\(917\) −77.8631 −2.57126
\(918\) 0 0
\(919\) 20.4674 0.675157 0.337579 0.941297i \(-0.390392\pi\)
0.337579 + 0.941297i \(0.390392\pi\)
\(920\) 0 0
\(921\) 24.8649 0.819325
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 18.1903 0.598093
\(926\) 0 0
\(927\) −5.64742 −0.185485
\(928\) 0 0
\(929\) 4.65220 0.152634 0.0763169 0.997084i \(-0.475684\pi\)
0.0763169 + 0.997084i \(0.475684\pi\)
\(930\) 0 0
\(931\) 38.7555 1.27016
\(932\) 0 0
\(933\) −17.0804 −0.559186
\(934\) 0 0
\(935\) −5.64310 −0.184549
\(936\) 0 0
\(937\) −41.8544 −1.36732 −0.683662 0.729798i \(-0.739614\pi\)
−0.683662 + 0.729798i \(0.739614\pi\)
\(938\) 0 0
\(939\) −15.6974 −0.512265
\(940\) 0 0
\(941\) −30.3454 −0.989232 −0.494616 0.869112i \(-0.664691\pi\)
−0.494616 + 0.869112i \(0.664691\pi\)
\(942\) 0 0
\(943\) −5.09783 −0.166008
\(944\) 0 0
\(945\) 4.97823 0.161942
\(946\) 0 0
\(947\) 12.0325 0.391004 0.195502 0.980703i \(-0.437366\pi\)
0.195502 + 0.980703i \(0.437366\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 32.7821 1.06303
\(952\) 0 0
\(953\) 22.9825 0.744478 0.372239 0.928137i \(-0.378590\pi\)
0.372239 + 0.928137i \(0.378590\pi\)
\(954\) 0 0
\(955\) −10.2314 −0.331082
\(956\) 0 0
\(957\) −20.2868 −0.655779
\(958\) 0 0
\(959\) 46.9743 1.51688
\(960\) 0 0
\(961\) −6.99223 −0.225556
\(962\) 0 0
\(963\) 6.73556 0.217050
\(964\) 0 0
\(965\) 14.1153 0.454387
\(966\) 0 0
\(967\) −38.8883 −1.25056 −0.625281 0.780399i \(-0.715016\pi\)
−0.625281 + 0.780399i \(0.715016\pi\)
\(968\) 0 0
\(969\) 5.99462 0.192575
\(970\) 0 0
\(971\) 57.5133 1.84569 0.922845 0.385171i \(-0.125857\pi\)
0.922845 + 0.385171i \(0.125857\pi\)
\(972\) 0 0
\(973\) −60.6378 −1.94396
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −16.3690 −0.523690 −0.261845 0.965110i \(-0.584331\pi\)
−0.261845 + 0.965110i \(0.584331\pi\)
\(978\) 0 0
\(979\) 6.00969 0.192070
\(980\) 0 0
\(981\) 2.07606 0.0662836
\(982\) 0 0
\(983\) −15.6963 −0.500635 −0.250318 0.968164i \(-0.580535\pi\)
−0.250318 + 0.968164i \(0.580535\pi\)
\(984\) 0 0
\(985\) 33.8301 1.07792
\(986\) 0 0
\(987\) −36.3540 −1.15716
\(988\) 0 0
\(989\) 20.0804 0.638519
\(990\) 0 0
\(991\) 11.2644 0.357827 0.178913 0.983865i \(-0.442742\pi\)
0.178913 + 0.983865i \(0.442742\pi\)
\(992\) 0 0
\(993\) 29.1618 0.925422
\(994\) 0 0
\(995\) 5.81594 0.184378
\(996\) 0 0
\(997\) −7.70112 −0.243897 −0.121948 0.992536i \(-0.538914\pi\)
−0.121948 + 0.992536i \(0.538914\pi\)
\(998\) 0 0
\(999\) 6.24698 0.197646
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 8112.2.a.by.1.2 3
4.3 odd 2 507.2.a.j.1.1 3
12.11 even 2 1521.2.a.q.1.3 3
13.12 even 2 8112.2.a.cf.1.2 3
52.3 odd 6 507.2.e.k.22.3 6
52.7 even 12 507.2.j.h.361.1 12
52.11 even 12 507.2.j.h.316.6 12
52.15 even 12 507.2.j.h.316.1 12
52.19 even 12 507.2.j.h.361.6 12
52.23 odd 6 507.2.e.j.22.1 6
52.31 even 4 507.2.b.g.337.6 6
52.35 odd 6 507.2.e.k.484.3 6
52.43 odd 6 507.2.e.j.484.1 6
52.47 even 4 507.2.b.g.337.1 6
52.51 odd 2 507.2.a.k.1.3 yes 3
156.47 odd 4 1521.2.b.m.1351.6 6
156.83 odd 4 1521.2.b.m.1351.1 6
156.155 even 2 1521.2.a.p.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.2.a.j.1.1 3 4.3 odd 2
507.2.a.k.1.3 yes 3 52.51 odd 2
507.2.b.g.337.1 6 52.47 even 4
507.2.b.g.337.6 6 52.31 even 4
507.2.e.j.22.1 6 52.23 odd 6
507.2.e.j.484.1 6 52.43 odd 6
507.2.e.k.22.3 6 52.3 odd 6
507.2.e.k.484.3 6 52.35 odd 6
507.2.j.h.316.1 12 52.15 even 12
507.2.j.h.316.6 12 52.11 even 12
507.2.j.h.361.1 12 52.7 even 12
507.2.j.h.361.6 12 52.19 even 12
1521.2.a.p.1.1 3 156.155 even 2
1521.2.a.q.1.3 3 12.11 even 2
1521.2.b.m.1351.1 6 156.83 odd 4
1521.2.b.m.1351.6 6 156.47 odd 4
8112.2.a.by.1.2 3 1.1 even 1 trivial
8112.2.a.cf.1.2 3 13.12 even 2