Properties

Label 4864.2.a.bm.1.2
Level $4864$
Weight $2$
Character 4864.1
Self dual yes
Analytic conductor $38.839$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4864 = 2^{8} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4864.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(38.8392355432\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: 8.8.34309996544.1
Defining polynomial: \(x^{8} - 4 x^{7} - 8 x^{6} + 28 x^{5} + 31 x^{4} - 36 x^{3} - 22 x^{2} + 12 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 2432)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.139869\) of defining polynomial
Character \(\chi\) \(=\) 4864.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.63640 q^{3} +2.63403 q^{5} +3.88603 q^{7} +3.95063 q^{9} +O(q^{10})\) \(q-2.63640 q^{3} +2.63403 q^{5} +3.88603 q^{7} +3.95063 q^{9} -5.95063 q^{11} +4.14940 q^{13} -6.94438 q^{15} +1.82843 q^{17} +1.00000 q^{19} -10.2452 q^{21} -4.14940 q^{23} +1.93813 q^{25} -2.50625 q^{27} -8.19638 q^{29} +1.67631 q^{31} +15.6883 q^{33} +10.2359 q^{35} -8.14452 q^{37} -10.9395 q^{39} -0.514201 q^{41} -10.9631 q^{43} +10.4061 q^{45} -13.3035 q^{47} +8.10124 q^{49} -4.82047 q^{51} +1.62446 q^{53} -15.6742 q^{55} -2.63640 q^{57} -12.2234 q^{59} -5.98666 q^{61} +15.3523 q^{63} +10.9297 q^{65} +4.09328 q^{67} +10.9395 q^{69} +11.8696 q^{71} -9.28531 q^{73} -5.10971 q^{75} -23.1243 q^{77} -2.42124 q^{79} -5.24441 q^{81} -0.987504 q^{83} +4.81614 q^{85} +21.6090 q^{87} -2.25389 q^{89} +16.1247 q^{91} -4.41944 q^{93} +2.63403 q^{95} +9.27281 q^{97} -23.5087 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 4q^{3} + 4q^{9} + O(q^{10}) \) \( 8q - 4q^{3} + 4q^{9} - 20q^{11} - 8q^{17} + 8q^{19} + 20q^{25} - 4q^{27} + 24q^{33} - 12q^{35} + 8q^{41} - 28q^{43} + 8q^{49} - 12q^{51} - 4q^{57} - 36q^{59} + 8q^{65} - 28q^{67} - 8q^{73} - 68q^{75} - 32q^{81} - 40q^{83} - 8q^{89} + 12q^{91} + 40q^{97} - 60q^{99} + O(q^{100}) \)

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
\(3\) −2.63640 −1.52213 −0.761065 0.648676i \(-0.775323\pi\)
−0.761065 + 0.648676i \(0.775323\pi\)
\(4\) 0 0
\(5\) 2.63403 1.17798 0.588988 0.808142i \(-0.299527\pi\)
0.588988 + 0.808142i \(0.299527\pi\)
\(6\) 0 0
\(7\) 3.88603 1.46878 0.734391 0.678727i \(-0.237468\pi\)
0.734391 + 0.678727i \(0.237468\pi\)
\(8\) 0 0
\(9\) 3.95063 1.31688
\(10\) 0 0
\(11\) −5.95063 −1.79418 −0.897091 0.441845i \(-0.854324\pi\)
−0.897091 + 0.441845i \(0.854324\pi\)
\(12\) 0 0
\(13\) 4.14940 1.15084 0.575418 0.817859i \(-0.304839\pi\)
0.575418 + 0.817859i \(0.304839\pi\)
\(14\) 0 0
\(15\) −6.94438 −1.79303
\(16\) 0 0
\(17\) 1.82843 0.443459 0.221729 0.975108i \(-0.428830\pi\)
0.221729 + 0.975108i \(0.428830\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −10.2452 −2.23568
\(22\) 0 0
\(23\) −4.14940 −0.865210 −0.432605 0.901584i \(-0.642406\pi\)
−0.432605 + 0.901584i \(0.642406\pi\)
\(24\) 0 0
\(25\) 1.93813 0.387627
\(26\) 0 0
\(27\) −2.50625 −0.482328
\(28\) 0 0
\(29\) −8.19638 −1.52203 −0.761015 0.648735i \(-0.775298\pi\)
−0.761015 + 0.648735i \(0.775298\pi\)
\(30\) 0 0
\(31\) 1.67631 0.301075 0.150537 0.988604i \(-0.451900\pi\)
0.150537 + 0.988604i \(0.451900\pi\)
\(32\) 0 0
\(33\) 15.6883 2.73098
\(34\) 0 0
\(35\) 10.2359 1.73019
\(36\) 0 0
\(37\) −8.14452 −1.33895 −0.669476 0.742834i \(-0.733481\pi\)
−0.669476 + 0.742834i \(0.733481\pi\)
\(38\) 0 0
\(39\) −10.9395 −1.75172
\(40\) 0 0
\(41\) −0.514201 −0.0803047 −0.0401524 0.999194i \(-0.512784\pi\)
−0.0401524 + 0.999194i \(0.512784\pi\)
\(42\) 0 0
\(43\) −10.9631 −1.67186 −0.835931 0.548835i \(-0.815071\pi\)
−0.835931 + 0.548835i \(0.815071\pi\)
\(44\) 0 0
\(45\) 10.4061 1.55125
\(46\) 0 0
\(47\) −13.3035 −1.94051 −0.970257 0.242075i \(-0.922172\pi\)
−0.970257 + 0.242075i \(0.922172\pi\)
\(48\) 0 0
\(49\) 8.10124 1.15732
\(50\) 0 0
\(51\) −4.82047 −0.675001
\(52\) 0 0
\(53\) 1.62446 0.223137 0.111568 0.993757i \(-0.464413\pi\)
0.111568 + 0.993757i \(0.464413\pi\)
\(54\) 0 0
\(55\) −15.6742 −2.11350
\(56\) 0 0
\(57\) −2.63640 −0.349200
\(58\) 0 0
\(59\) −12.2234 −1.59136 −0.795678 0.605720i \(-0.792885\pi\)
−0.795678 + 0.605720i \(0.792885\pi\)
\(60\) 0 0
\(61\) −5.98666 −0.766513 −0.383257 0.923642i \(-0.625197\pi\)
−0.383257 + 0.923642i \(0.625197\pi\)
\(62\) 0 0
\(63\) 15.3523 1.93420
\(64\) 0 0
\(65\) 10.9297 1.35566
\(66\) 0 0
\(67\) 4.09328 0.500074 0.250037 0.968236i \(-0.419557\pi\)
0.250037 + 0.968236i \(0.419557\pi\)
\(68\) 0 0
\(69\) 10.9395 1.31696
\(70\) 0 0
\(71\) 11.8696 1.40866 0.704332 0.709871i \(-0.251247\pi\)
0.704332 + 0.709871i \(0.251247\pi\)
\(72\) 0 0
\(73\) −9.28531 −1.08676 −0.543381 0.839486i \(-0.682856\pi\)
−0.543381 + 0.839486i \(0.682856\pi\)
\(74\) 0 0
\(75\) −5.10971 −0.590018
\(76\) 0 0
\(77\) −23.1243 −2.63526
\(78\) 0 0
\(79\) −2.42124 −0.272410 −0.136205 0.990681i \(-0.543491\pi\)
−0.136205 + 0.990681i \(0.543491\pi\)
\(80\) 0 0
\(81\) −5.24441 −0.582712
\(82\) 0 0
\(83\) −0.987504 −0.108393 −0.0541963 0.998530i \(-0.517260\pi\)
−0.0541963 + 0.998530i \(0.517260\pi\)
\(84\) 0 0
\(85\) 4.81614 0.522384
\(86\) 0 0
\(87\) 21.6090 2.31672
\(88\) 0 0
\(89\) −2.25389 −0.238912 −0.119456 0.992840i \(-0.538115\pi\)
−0.119456 + 0.992840i \(0.538115\pi\)
\(90\) 0 0
\(91\) 16.1247 1.69033
\(92\) 0 0
\(93\) −4.41944 −0.458274
\(94\) 0 0
\(95\) 2.63403 0.270246
\(96\) 0 0
\(97\) 9.27281 0.941511 0.470756 0.882264i \(-0.343981\pi\)
0.470756 + 0.882264i \(0.343981\pi\)
\(98\) 0 0
\(99\) −23.5087 −2.36272
\(100\) 0 0
\(101\) −13.3620 −1.32957 −0.664785 0.747034i \(-0.731477\pi\)
−0.664785 + 0.747034i \(0.731477\pi\)
\(102\) 0 0
\(103\) −10.0094 −0.986255 −0.493127 0.869957i \(-0.664146\pi\)
−0.493127 + 0.869957i \(0.664146\pi\)
\(104\) 0 0
\(105\) −26.9861 −2.63357
\(106\) 0 0
\(107\) 4.85189 0.469050 0.234525 0.972110i \(-0.424646\pi\)
0.234525 + 0.972110i \(0.424646\pi\)
\(108\) 0 0
\(109\) 15.1198 1.44822 0.724108 0.689687i \(-0.242252\pi\)
0.724108 + 0.689687i \(0.242252\pi\)
\(110\) 0 0
\(111\) 21.4723 2.03806
\(112\) 0 0
\(113\) −3.10175 −0.291789 −0.145894 0.989300i \(-0.546606\pi\)
−0.145894 + 0.989300i \(0.546606\pi\)
\(114\) 0 0
\(115\) −10.9297 −1.01920
\(116\) 0 0
\(117\) 16.3928 1.51551
\(118\) 0 0
\(119\) 7.10532 0.651344
\(120\) 0 0
\(121\) 24.4100 2.21909
\(122\) 0 0
\(123\) 1.35564 0.122234
\(124\) 0 0
\(125\) −8.06506 −0.721361
\(126\) 0 0
\(127\) 3.59176 0.318717 0.159358 0.987221i \(-0.449057\pi\)
0.159358 + 0.987221i \(0.449057\pi\)
\(128\) 0 0
\(129\) 28.9032 2.54479
\(130\) 0 0
\(131\) 9.76906 0.853527 0.426763 0.904363i \(-0.359654\pi\)
0.426763 + 0.904363i \(0.359654\pi\)
\(132\) 0 0
\(133\) 3.88603 0.336962
\(134\) 0 0
\(135\) −6.60154 −0.568170
\(136\) 0 0
\(137\) 4.25690 0.363692 0.181846 0.983327i \(-0.441793\pi\)
0.181846 + 0.983327i \(0.441793\pi\)
\(138\) 0 0
\(139\) 9.50875 0.806521 0.403261 0.915085i \(-0.367877\pi\)
0.403261 + 0.915085i \(0.367877\pi\)
\(140\) 0 0
\(141\) 35.0734 2.95371
\(142\) 0 0
\(143\) −24.6916 −2.06481
\(144\) 0 0
\(145\) −21.5895 −1.79291
\(146\) 0 0
\(147\) −21.3581 −1.76159
\(148\) 0 0
\(149\) 5.13803 0.420924 0.210462 0.977602i \(-0.432503\pi\)
0.210462 + 0.977602i \(0.432503\pi\)
\(150\) 0 0
\(151\) 5.26807 0.428709 0.214355 0.976756i \(-0.431235\pi\)
0.214355 + 0.976756i \(0.431235\pi\)
\(152\) 0 0
\(153\) 7.22344 0.583981
\(154\) 0 0
\(155\) 4.41546 0.354659
\(156\) 0 0
\(157\) 7.12828 0.568899 0.284449 0.958691i \(-0.408189\pi\)
0.284449 + 0.958691i \(0.408189\pi\)
\(158\) 0 0
\(159\) −4.28273 −0.339643
\(160\) 0 0
\(161\) −16.1247 −1.27080
\(162\) 0 0
\(163\) 12.4060 0.971712 0.485856 0.874039i \(-0.338508\pi\)
0.485856 + 0.874039i \(0.338508\pi\)
\(164\) 0 0
\(165\) 41.3234 3.21703
\(166\) 0 0
\(167\) −8.80460 −0.681320 −0.340660 0.940187i \(-0.610651\pi\)
−0.340660 + 0.940187i \(0.610651\pi\)
\(168\) 0 0
\(169\) 4.21753 0.324426
\(170\) 0 0
\(171\) 3.95063 0.302112
\(172\) 0 0
\(173\) 1.52203 0.115718 0.0578590 0.998325i \(-0.481573\pi\)
0.0578590 + 0.998325i \(0.481573\pi\)
\(174\) 0 0
\(175\) 7.53165 0.569339
\(176\) 0 0
\(177\) 32.2259 2.42225
\(178\) 0 0
\(179\) −18.6599 −1.39470 −0.697352 0.716729i \(-0.745639\pi\)
−0.697352 + 0.716729i \(0.745639\pi\)
\(180\) 0 0
\(181\) 11.3132 0.840907 0.420453 0.907314i \(-0.361871\pi\)
0.420453 + 0.907314i \(0.361871\pi\)
\(182\) 0 0
\(183\) 15.7833 1.16673
\(184\) 0 0
\(185\) −21.4530 −1.57725
\(186\) 0 0
\(187\) −10.8803 −0.795646
\(188\) 0 0
\(189\) −9.73936 −0.708434
\(190\) 0 0
\(191\) 11.5963 0.839078 0.419539 0.907737i \(-0.362192\pi\)
0.419539 + 0.907737i \(0.362192\pi\)
\(192\) 0 0
\(193\) −3.78701 −0.272595 −0.136298 0.990668i \(-0.543520\pi\)
−0.136298 + 0.990668i \(0.543520\pi\)
\(194\) 0 0
\(195\) −28.8150 −2.06349
\(196\) 0 0
\(197\) −26.0803 −1.85814 −0.929071 0.369902i \(-0.879391\pi\)
−0.929071 + 0.369902i \(0.879391\pi\)
\(198\) 0 0
\(199\) −3.53452 −0.250555 −0.125278 0.992122i \(-0.539982\pi\)
−0.125278 + 0.992122i \(0.539982\pi\)
\(200\) 0 0
\(201\) −10.7916 −0.761178
\(202\) 0 0
\(203\) −31.8514 −2.23553
\(204\) 0 0
\(205\) −1.35442 −0.0945970
\(206\) 0 0
\(207\) −16.3928 −1.13938
\(208\) 0 0
\(209\) −5.95063 −0.411614
\(210\) 0 0
\(211\) −2.84939 −0.196160 −0.0980802 0.995179i \(-0.531270\pi\)
−0.0980802 + 0.995179i \(0.531270\pi\)
\(212\) 0 0
\(213\) −31.2931 −2.14417
\(214\) 0 0
\(215\) −28.8772 −1.96941
\(216\) 0 0
\(217\) 6.51420 0.442213
\(218\) 0 0
\(219\) 24.4798 1.65419
\(220\) 0 0
\(221\) 7.58688 0.510349
\(222\) 0 0
\(223\) −23.4383 −1.56954 −0.784772 0.619785i \(-0.787220\pi\)
−0.784772 + 0.619785i \(0.787220\pi\)
\(224\) 0 0
\(225\) 7.65685 0.510457
\(226\) 0 0
\(227\) −7.43341 −0.493373 −0.246686 0.969095i \(-0.579342\pi\)
−0.246686 + 0.969095i \(0.579342\pi\)
\(228\) 0 0
\(229\) −19.2316 −1.27086 −0.635431 0.772157i \(-0.719178\pi\)
−0.635431 + 0.772157i \(0.719178\pi\)
\(230\) 0 0
\(231\) 60.9651 4.01121
\(232\) 0 0
\(233\) −29.9306 −1.96082 −0.980411 0.196965i \(-0.936892\pi\)
−0.980411 + 0.196965i \(0.936892\pi\)
\(234\) 0 0
\(235\) −35.0419 −2.28588
\(236\) 0 0
\(237\) 6.38336 0.414644
\(238\) 0 0
\(239\) 16.0479 1.03805 0.519026 0.854758i \(-0.326295\pi\)
0.519026 + 0.854758i \(0.326295\pi\)
\(240\) 0 0
\(241\) 23.9168 1.54061 0.770307 0.637673i \(-0.220103\pi\)
0.770307 + 0.637673i \(0.220103\pi\)
\(242\) 0 0
\(243\) 21.3451 1.36929
\(244\) 0 0
\(245\) 21.3389 1.36329
\(246\) 0 0
\(247\) 4.14940 0.264020
\(248\) 0 0
\(249\) 2.60346 0.164988
\(250\) 0 0
\(251\) −10.1950 −0.643505 −0.321753 0.946824i \(-0.604272\pi\)
−0.321753 + 0.946824i \(0.604272\pi\)
\(252\) 0 0
\(253\) 24.6916 1.55234
\(254\) 0 0
\(255\) −12.6973 −0.795135
\(256\) 0 0
\(257\) 20.1866 1.25920 0.629602 0.776918i \(-0.283218\pi\)
0.629602 + 0.776918i \(0.283218\pi\)
\(258\) 0 0
\(259\) −31.6499 −1.96663
\(260\) 0 0
\(261\) −32.3809 −2.00432
\(262\) 0 0
\(263\) 5.32659 0.328451 0.164226 0.986423i \(-0.447487\pi\)
0.164226 + 0.986423i \(0.447487\pi\)
\(264\) 0 0
\(265\) 4.27888 0.262850
\(266\) 0 0
\(267\) 5.94216 0.363654
\(268\) 0 0
\(269\) −14.0564 −0.857032 −0.428516 0.903534i \(-0.640963\pi\)
−0.428516 + 0.903534i \(0.640963\pi\)
\(270\) 0 0
\(271\) 8.56884 0.520520 0.260260 0.965539i \(-0.416192\pi\)
0.260260 + 0.965539i \(0.416192\pi\)
\(272\) 0 0
\(273\) −42.5112 −2.57290
\(274\) 0 0
\(275\) −11.5331 −0.695474
\(276\) 0 0
\(277\) −22.7013 −1.36399 −0.681995 0.731357i \(-0.738887\pi\)
−0.681995 + 0.731357i \(0.738887\pi\)
\(278\) 0 0
\(279\) 6.62249 0.396478
\(280\) 0 0
\(281\) 8.26940 0.493311 0.246656 0.969103i \(-0.420668\pi\)
0.246656 + 0.969103i \(0.420668\pi\)
\(282\) 0 0
\(283\) −21.7941 −1.29552 −0.647761 0.761844i \(-0.724294\pi\)
−0.647761 + 0.761844i \(0.724294\pi\)
\(284\) 0 0
\(285\) −6.94438 −0.411350
\(286\) 0 0
\(287\) −1.99820 −0.117950
\(288\) 0 0
\(289\) −13.6569 −0.803344
\(290\) 0 0
\(291\) −24.4469 −1.43310
\(292\) 0 0
\(293\) 8.64036 0.504775 0.252388 0.967626i \(-0.418784\pi\)
0.252388 + 0.967626i \(0.418784\pi\)
\(294\) 0 0
\(295\) −32.1970 −1.87458
\(296\) 0 0
\(297\) 14.9138 0.865384
\(298\) 0 0
\(299\) −17.2175 −0.995716
\(300\) 0 0
\(301\) −42.6031 −2.45560
\(302\) 0 0
\(303\) 35.2277 2.02378
\(304\) 0 0
\(305\) −15.7691 −0.902934
\(306\) 0 0
\(307\) 11.3871 0.649894 0.324947 0.945732i \(-0.394654\pi\)
0.324947 + 0.945732i \(0.394654\pi\)
\(308\) 0 0
\(309\) 26.3888 1.50121
\(310\) 0 0
\(311\) −11.3199 −0.641893 −0.320947 0.947097i \(-0.604001\pi\)
−0.320947 + 0.947097i \(0.604001\pi\)
\(312\) 0 0
\(313\) 18.3052 1.03467 0.517337 0.855782i \(-0.326924\pi\)
0.517337 + 0.855782i \(0.326924\pi\)
\(314\) 0 0
\(315\) 40.4384 2.27845
\(316\) 0 0
\(317\) 26.0999 1.46592 0.732959 0.680273i \(-0.238139\pi\)
0.732959 + 0.680273i \(0.238139\pi\)
\(318\) 0 0
\(319\) 48.7736 2.73080
\(320\) 0 0
\(321\) −12.7916 −0.713955
\(322\) 0 0
\(323\) 1.82843 0.101736
\(324\) 0 0
\(325\) 8.04210 0.446095
\(326\) 0 0
\(327\) −39.8619 −2.20437
\(328\) 0 0
\(329\) −51.6978 −2.85019
\(330\) 0 0
\(331\) 12.2799 0.674962 0.337481 0.941332i \(-0.390425\pi\)
0.337481 + 0.941332i \(0.390425\pi\)
\(332\) 0 0
\(333\) −32.1760 −1.76323
\(334\) 0 0
\(335\) 10.7818 0.589075
\(336\) 0 0
\(337\) 15.5263 0.845772 0.422886 0.906183i \(-0.361017\pi\)
0.422886 + 0.906183i \(0.361017\pi\)
\(338\) 0 0
\(339\) 8.17748 0.444140
\(340\) 0 0
\(341\) −9.97512 −0.540183
\(342\) 0 0
\(343\) 4.27944 0.231068
\(344\) 0 0
\(345\) 28.8150 1.55135
\(346\) 0 0
\(347\) 12.5212 0.672176 0.336088 0.941831i \(-0.390896\pi\)
0.336088 + 0.941831i \(0.390896\pi\)
\(348\) 0 0
\(349\) 1.36237 0.0729259 0.0364630 0.999335i \(-0.488391\pi\)
0.0364630 + 0.999335i \(0.488391\pi\)
\(350\) 0 0
\(351\) −10.3994 −0.555081
\(352\) 0 0
\(353\) −22.3247 −1.18822 −0.594111 0.804383i \(-0.702496\pi\)
−0.594111 + 0.804383i \(0.702496\pi\)
\(354\) 0 0
\(355\) 31.2650 1.65937
\(356\) 0 0
\(357\) −18.7325 −0.991430
\(358\) 0 0
\(359\) 16.2824 0.859351 0.429676 0.902983i \(-0.358628\pi\)
0.429676 + 0.902983i \(0.358628\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −64.3547 −3.37774
\(364\) 0 0
\(365\) −24.4578 −1.28018
\(366\) 0 0
\(367\) −29.6214 −1.54623 −0.773113 0.634268i \(-0.781302\pi\)
−0.773113 + 0.634268i \(0.781302\pi\)
\(368\) 0 0
\(369\) −2.03142 −0.105751
\(370\) 0 0
\(371\) 6.31270 0.327739
\(372\) 0 0
\(373\) 5.18659 0.268551 0.134276 0.990944i \(-0.457129\pi\)
0.134276 + 0.990944i \(0.457129\pi\)
\(374\) 0 0
\(375\) 21.2628 1.09800
\(376\) 0 0
\(377\) −34.0101 −1.75161
\(378\) 0 0
\(379\) −11.6485 −0.598344 −0.299172 0.954199i \(-0.596710\pi\)
−0.299172 + 0.954199i \(0.596710\pi\)
\(380\) 0 0
\(381\) −9.46932 −0.485128
\(382\) 0 0
\(383\) 33.3465 1.70393 0.851964 0.523600i \(-0.175411\pi\)
0.851964 + 0.523600i \(0.175411\pi\)
\(384\) 0 0
\(385\) −60.9103 −3.10428
\(386\) 0 0
\(387\) −43.3113 −2.20164
\(388\) 0 0
\(389\) −1.51874 −0.0770033 −0.0385016 0.999259i \(-0.512258\pi\)
−0.0385016 + 0.999259i \(0.512258\pi\)
\(390\) 0 0
\(391\) −7.58688 −0.383685
\(392\) 0 0
\(393\) −25.7552 −1.29918
\(394\) 0 0
\(395\) −6.37762 −0.320893
\(396\) 0 0
\(397\) 29.4592 1.47852 0.739258 0.673422i \(-0.235176\pi\)
0.739258 + 0.673422i \(0.235176\pi\)
\(398\) 0 0
\(399\) −10.2452 −0.512899
\(400\) 0 0
\(401\) −32.0523 −1.60062 −0.800308 0.599589i \(-0.795331\pi\)
−0.800308 + 0.599589i \(0.795331\pi\)
\(402\) 0 0
\(403\) 6.95569 0.346488
\(404\) 0 0
\(405\) −13.8139 −0.686421
\(406\) 0 0
\(407\) 48.4651 2.40232
\(408\) 0 0
\(409\) −10.7427 −0.531192 −0.265596 0.964084i \(-0.585569\pi\)
−0.265596 + 0.964084i \(0.585569\pi\)
\(410\) 0 0
\(411\) −11.2229 −0.553586
\(412\) 0 0
\(413\) −47.5007 −2.33736
\(414\) 0 0
\(415\) −2.60112 −0.127684
\(416\) 0 0
\(417\) −25.0689 −1.22763
\(418\) 0 0
\(419\) −19.9990 −0.977013 −0.488507 0.872560i \(-0.662458\pi\)
−0.488507 + 0.872560i \(0.662458\pi\)
\(420\) 0 0
\(421\) −10.5670 −0.515006 −0.257503 0.966277i \(-0.582900\pi\)
−0.257503 + 0.966277i \(0.582900\pi\)
\(422\) 0 0
\(423\) −52.5572 −2.55542
\(424\) 0 0
\(425\) 3.54374 0.171897
\(426\) 0 0
\(427\) −23.2643 −1.12584
\(428\) 0 0
\(429\) 65.0969 3.14291
\(430\) 0 0
\(431\) −3.24892 −0.156495 −0.0782474 0.996934i \(-0.524932\pi\)
−0.0782474 + 0.996934i \(0.524932\pi\)
\(432\) 0 0
\(433\) −25.4848 −1.22472 −0.612360 0.790579i \(-0.709780\pi\)
−0.612360 + 0.790579i \(0.709780\pi\)
\(434\) 0 0
\(435\) 56.9188 2.72905
\(436\) 0 0
\(437\) −4.14940 −0.198493
\(438\) 0 0
\(439\) −17.9520 −0.856804 −0.428402 0.903588i \(-0.640923\pi\)
−0.428402 + 0.903588i \(0.640923\pi\)
\(440\) 0 0
\(441\) 32.0050 1.52405
\(442\) 0 0
\(443\) 1.68123 0.0798777 0.0399388 0.999202i \(-0.487284\pi\)
0.0399388 + 0.999202i \(0.487284\pi\)
\(444\) 0 0
\(445\) −5.93682 −0.281432
\(446\) 0 0
\(447\) −13.5459 −0.640700
\(448\) 0 0
\(449\) −21.8275 −1.03010 −0.515052 0.857159i \(-0.672227\pi\)
−0.515052 + 0.857159i \(0.672227\pi\)
\(450\) 0 0
\(451\) 3.05982 0.144081
\(452\) 0 0
\(453\) −13.8888 −0.652551
\(454\) 0 0
\(455\) 42.4730 1.99117
\(456\) 0 0
\(457\) 8.04431 0.376297 0.188148 0.982141i \(-0.439751\pi\)
0.188148 + 0.982141i \(0.439751\pi\)
\(458\) 0 0
\(459\) −4.58249 −0.213892
\(460\) 0 0
\(461\) −6.79339 −0.316400 −0.158200 0.987407i \(-0.550569\pi\)
−0.158200 + 0.987407i \(0.550569\pi\)
\(462\) 0 0
\(463\) −31.2479 −1.45221 −0.726107 0.687582i \(-0.758672\pi\)
−0.726107 + 0.687582i \(0.758672\pi\)
\(464\) 0 0
\(465\) −11.6409 −0.539836
\(466\) 0 0
\(467\) 36.9113 1.70805 0.854026 0.520230i \(-0.174154\pi\)
0.854026 + 0.520230i \(0.174154\pi\)
\(468\) 0 0
\(469\) 15.9066 0.734500
\(470\) 0 0
\(471\) −18.7930 −0.865938
\(472\) 0 0
\(473\) 65.2375 2.99962
\(474\) 0 0
\(475\) 1.93813 0.0889277
\(476\) 0 0
\(477\) 6.41764 0.293844
\(478\) 0 0
\(479\) 7.26161 0.331792 0.165896 0.986143i \(-0.446948\pi\)
0.165896 + 0.986143i \(0.446948\pi\)
\(480\) 0 0
\(481\) −33.7949 −1.54091
\(482\) 0 0
\(483\) 42.5112 1.93433
\(484\) 0 0
\(485\) 24.4249 1.10908
\(486\) 0 0
\(487\) −1.00500 −0.0455411 −0.0227705 0.999741i \(-0.507249\pi\)
−0.0227705 + 0.999741i \(0.507249\pi\)
\(488\) 0 0
\(489\) −32.7072 −1.47907
\(490\) 0 0
\(491\) −9.12714 −0.411902 −0.205951 0.978562i \(-0.566029\pi\)
−0.205951 + 0.978562i \(0.566029\pi\)
\(492\) 0 0
\(493\) −14.9865 −0.674957
\(494\) 0 0
\(495\) −61.9228 −2.78322
\(496\) 0 0
\(497\) 46.1257 2.06902
\(498\) 0 0
\(499\) 19.8928 0.890524 0.445262 0.895400i \(-0.353111\pi\)
0.445262 + 0.895400i \(0.353111\pi\)
\(500\) 0 0
\(501\) 23.2125 1.03706
\(502\) 0 0
\(503\) −18.9716 −0.845904 −0.422952 0.906152i \(-0.639006\pi\)
−0.422952 + 0.906152i \(0.639006\pi\)
\(504\) 0 0
\(505\) −35.1960 −1.56620
\(506\) 0 0
\(507\) −11.1191 −0.493818
\(508\) 0 0
\(509\) 24.2981 1.07700 0.538498 0.842627i \(-0.318992\pi\)
0.538498 + 0.842627i \(0.318992\pi\)
\(510\) 0 0
\(511\) −36.0830 −1.59622
\(512\) 0 0
\(513\) −2.50625 −0.110654
\(514\) 0 0
\(515\) −26.3651 −1.16178
\(516\) 0 0
\(517\) 79.1642 3.48164
\(518\) 0 0
\(519\) −4.01270 −0.176138
\(520\) 0 0
\(521\) −10.1930 −0.446563 −0.223282 0.974754i \(-0.571677\pi\)
−0.223282 + 0.974754i \(0.571677\pi\)
\(522\) 0 0
\(523\) 1.42683 0.0623907 0.0311954 0.999513i \(-0.490069\pi\)
0.0311954 + 0.999513i \(0.490069\pi\)
\(524\) 0 0
\(525\) −19.8565 −0.866608
\(526\) 0 0
\(527\) 3.06501 0.133514
\(528\) 0 0
\(529\) −5.78247 −0.251412
\(530\) 0 0
\(531\) −48.2903 −2.09562
\(532\) 0 0
\(533\) −2.13363 −0.0924176
\(534\) 0 0
\(535\) 12.7801 0.552530
\(536\) 0 0
\(537\) 49.1950 2.12292
\(538\) 0 0
\(539\) −48.2075 −2.07644
\(540\) 0 0
\(541\) −28.1257 −1.20922 −0.604610 0.796522i \(-0.706671\pi\)
−0.604610 + 0.796522i \(0.706671\pi\)
\(542\) 0 0
\(543\) −29.8263 −1.27997
\(544\) 0 0
\(545\) 39.8261 1.70596
\(546\) 0 0
\(547\) 45.2937 1.93662 0.968310 0.249753i \(-0.0803493\pi\)
0.968310 + 0.249753i \(0.0803493\pi\)
\(548\) 0 0
\(549\) −23.6511 −1.00940
\(550\) 0 0
\(551\) −8.19638 −0.349177
\(552\) 0 0
\(553\) −9.40900 −0.400111
\(554\) 0 0
\(555\) 56.5587 2.40078
\(556\) 0 0
\(557\) −13.9151 −0.589601 −0.294801 0.955559i \(-0.595253\pi\)
−0.294801 + 0.955559i \(0.595253\pi\)
\(558\) 0 0
\(559\) −45.4904 −1.92404
\(560\) 0 0
\(561\) 28.6849 1.21108
\(562\) 0 0
\(563\) 10.3367 0.435641 0.217820 0.975989i \(-0.430105\pi\)
0.217820 + 0.975989i \(0.430105\pi\)
\(564\) 0 0
\(565\) −8.17013 −0.343720
\(566\) 0 0
\(567\) −20.3799 −0.855877
\(568\) 0 0
\(569\) 39.2369 1.64490 0.822448 0.568840i \(-0.192608\pi\)
0.822448 + 0.568840i \(0.192608\pi\)
\(570\) 0 0
\(571\) 33.4374 1.39931 0.699657 0.714479i \(-0.253336\pi\)
0.699657 + 0.714479i \(0.253336\pi\)
\(572\) 0 0
\(573\) −30.5725 −1.27718
\(574\) 0 0
\(575\) −8.04210 −0.335379
\(576\) 0 0
\(577\) −22.3203 −0.929206 −0.464603 0.885519i \(-0.653803\pi\)
−0.464603 + 0.885519i \(0.653803\pi\)
\(578\) 0 0
\(579\) 9.98409 0.414925
\(580\) 0 0
\(581\) −3.83747 −0.159205
\(582\) 0 0
\(583\) −9.66656 −0.400348
\(584\) 0 0
\(585\) 43.1791 1.78523
\(586\) 0 0
\(587\) 43.6753 1.80267 0.901337 0.433119i \(-0.142587\pi\)
0.901337 + 0.433119i \(0.142587\pi\)
\(588\) 0 0
\(589\) 1.67631 0.0690712
\(590\) 0 0
\(591\) 68.7581 2.82833
\(592\) 0 0
\(593\) −19.6978 −0.808890 −0.404445 0.914562i \(-0.632535\pi\)
−0.404445 + 0.914562i \(0.632535\pi\)
\(594\) 0 0
\(595\) 18.7157 0.767267
\(596\) 0 0
\(597\) 9.31842 0.381377
\(598\) 0 0
\(599\) −34.2570 −1.39970 −0.699851 0.714289i \(-0.746750\pi\)
−0.699851 + 0.714289i \(0.746750\pi\)
\(600\) 0 0
\(601\) −19.1801 −0.782373 −0.391186 0.920311i \(-0.627935\pi\)
−0.391186 + 0.920311i \(0.627935\pi\)
\(602\) 0 0
\(603\) 16.1711 0.658536
\(604\) 0 0
\(605\) 64.2968 2.61404
\(606\) 0 0
\(607\) −17.1796 −0.697297 −0.348649 0.937253i \(-0.613359\pi\)
−0.348649 + 0.937253i \(0.613359\pi\)
\(608\) 0 0
\(609\) 83.9731 3.40276
\(610\) 0 0
\(611\) −55.2016 −2.23322
\(612\) 0 0
\(613\) 13.4502 0.543247 0.271623 0.962404i \(-0.412439\pi\)
0.271623 + 0.962404i \(0.412439\pi\)
\(614\) 0 0
\(615\) 3.57081 0.143989
\(616\) 0 0
\(617\) −15.2065 −0.612191 −0.306095 0.952001i \(-0.599023\pi\)
−0.306095 + 0.952001i \(0.599023\pi\)
\(618\) 0 0
\(619\) 40.1696 1.61455 0.807277 0.590173i \(-0.200940\pi\)
0.807277 + 0.590173i \(0.200940\pi\)
\(620\) 0 0
\(621\) 10.3994 0.417315
\(622\) 0 0
\(623\) −8.75868 −0.350909
\(624\) 0 0
\(625\) −30.9343 −1.23737
\(626\) 0 0
\(627\) 15.6883 0.626529
\(628\) 0 0
\(629\) −14.8917 −0.593770
\(630\) 0 0
\(631\) 12.0318 0.478980 0.239490 0.970899i \(-0.423020\pi\)
0.239490 + 0.970899i \(0.423020\pi\)
\(632\) 0 0
\(633\) 7.51216 0.298581
\(634\) 0 0
\(635\) 9.46081 0.375441
\(636\) 0 0
\(637\) 33.6153 1.33189
\(638\) 0 0
\(639\) 46.8925 1.85504
\(640\) 0 0
\(641\) 28.1707 1.11267 0.556337 0.830957i \(-0.312206\pi\)
0.556337 + 0.830957i \(0.312206\pi\)
\(642\) 0 0
\(643\) 38.2190 1.50721 0.753605 0.657327i \(-0.228313\pi\)
0.753605 + 0.657327i \(0.228313\pi\)
\(644\) 0 0
\(645\) 76.1321 2.99770
\(646\) 0 0
\(647\) 7.26170 0.285487 0.142743 0.989760i \(-0.454408\pi\)
0.142743 + 0.989760i \(0.454408\pi\)
\(648\) 0 0
\(649\) 72.7372 2.85518
\(650\) 0 0
\(651\) −17.1741 −0.673105
\(652\) 0 0
\(653\) 16.0972 0.629932 0.314966 0.949103i \(-0.398007\pi\)
0.314966 + 0.949103i \(0.398007\pi\)
\(654\) 0 0
\(655\) 25.7320 1.00543
\(656\) 0 0
\(657\) −36.6828 −1.43113
\(658\) 0 0
\(659\) −17.8679 −0.696035 −0.348018 0.937488i \(-0.613145\pi\)
−0.348018 + 0.937488i \(0.613145\pi\)
\(660\) 0 0
\(661\) −21.6386 −0.841644 −0.420822 0.907143i \(-0.638258\pi\)
−0.420822 + 0.907143i \(0.638258\pi\)
\(662\) 0 0
\(663\) −20.0021 −0.776817
\(664\) 0 0
\(665\) 10.2359 0.396933
\(666\) 0 0
\(667\) 34.0101 1.31687
\(668\) 0 0
\(669\) 61.7928 2.38905
\(670\) 0 0
\(671\) 35.6244 1.37526
\(672\) 0 0
\(673\) −43.9881 −1.69561 −0.847807 0.530304i \(-0.822078\pi\)
−0.847807 + 0.530304i \(0.822078\pi\)
\(674\) 0 0
\(675\) −4.85745 −0.186963
\(676\) 0 0
\(677\) −29.6983 −1.14140 −0.570699 0.821160i \(-0.693328\pi\)
−0.570699 + 0.821160i \(0.693328\pi\)
\(678\) 0 0
\(679\) 36.0344 1.38287
\(680\) 0 0
\(681\) 19.5975 0.750977
\(682\) 0 0
\(683\) −6.20351 −0.237371 −0.118685 0.992932i \(-0.537868\pi\)
−0.118685 + 0.992932i \(0.537868\pi\)
\(684\) 0 0
\(685\) 11.2128 0.428420
\(686\) 0 0
\(687\) 50.7024 1.93442
\(688\) 0 0
\(689\) 6.74053 0.256794
\(690\) 0 0
\(691\) −28.1172 −1.06963 −0.534814 0.844970i \(-0.679618\pi\)
−0.534814 + 0.844970i \(0.679618\pi\)
\(692\) 0 0
\(693\) −91.3557 −3.47032
\(694\) 0 0
\(695\) 25.0464 0.950063
\(696\) 0 0
\(697\) −0.940179 −0.0356118
\(698\) 0 0
\(699\) 78.9093 2.98462
\(700\) 0 0
\(701\) −8.35403 −0.315527 −0.157764 0.987477i \(-0.550428\pi\)
−0.157764 + 0.987477i \(0.550428\pi\)
\(702\) 0 0
\(703\) −8.14452 −0.307177
\(704\) 0 0
\(705\) 92.3846 3.47940
\(706\) 0 0
\(707\) −51.9252 −1.95285
\(708\) 0 0
\(709\) 31.7187 1.19122 0.595610 0.803274i \(-0.296910\pi\)
0.595610 + 0.803274i \(0.296910\pi\)
\(710\) 0 0
\(711\) −9.56541 −0.358731
\(712\) 0 0
\(713\) −6.95569 −0.260493
\(714\) 0 0
\(715\) −65.0384 −2.43230
\(716\) 0 0
\(717\) −42.3088 −1.58005
\(718\) 0 0
\(719\) 33.7972 1.26042 0.630211 0.776424i \(-0.282968\pi\)
0.630211 + 0.776424i \(0.282968\pi\)
\(720\) 0 0
\(721\) −38.8968 −1.44859
\(722\) 0 0
\(723\) −63.0543 −2.34501
\(724\) 0 0
\(725\) −15.8857 −0.589979
\(726\) 0 0
\(727\) 32.8637 1.21885 0.609423 0.792845i \(-0.291401\pi\)
0.609423 + 0.792845i \(0.291401\pi\)
\(728\) 0 0
\(729\) −40.5412 −1.50153
\(730\) 0 0
\(731\) −20.0453 −0.741401
\(732\) 0 0
\(733\) 14.3118 0.528618 0.264309 0.964438i \(-0.414856\pi\)
0.264309 + 0.964438i \(0.414856\pi\)
\(734\) 0 0
\(735\) −56.2581 −2.07511
\(736\) 0 0
\(737\) −24.3576 −0.897225
\(738\) 0 0
\(739\) −22.6101 −0.831726 −0.415863 0.909427i \(-0.636520\pi\)
−0.415863 + 0.909427i \(0.636520\pi\)
\(740\) 0 0
\(741\) −10.9395 −0.401873
\(742\) 0 0
\(743\) 12.6222 0.463062 0.231531 0.972828i \(-0.425627\pi\)
0.231531 + 0.972828i \(0.425627\pi\)
\(744\) 0 0
\(745\) 13.5337 0.495838
\(746\) 0 0
\(747\) −3.90126 −0.142740
\(748\) 0 0
\(749\) 18.8546 0.688932
\(750\) 0 0
\(751\) −4.69943 −0.171485 −0.0857423 0.996317i \(-0.527326\pi\)
−0.0857423 + 0.996317i \(0.527326\pi\)
\(752\) 0 0
\(753\) 26.8783 0.979498
\(754\) 0 0
\(755\) 13.8763 0.505009
\(756\) 0 0
\(757\) −21.8920 −0.795679 −0.397839 0.917455i \(-0.630240\pi\)
−0.397839 + 0.917455i \(0.630240\pi\)
\(758\) 0 0
\(759\) −65.0969 −2.36287
\(760\) 0 0
\(761\) −0.164985 −0.00598068 −0.00299034 0.999996i \(-0.500952\pi\)
−0.00299034 + 0.999996i \(0.500952\pi\)
\(762\) 0 0
\(763\) 58.7560 2.12711
\(764\) 0 0
\(765\) 19.0268 0.687915
\(766\) 0 0
\(767\) −50.7200 −1.83139
\(768\) 0 0
\(769\) −8.55812 −0.308614 −0.154307 0.988023i \(-0.549314\pi\)
−0.154307 + 0.988023i \(0.549314\pi\)
\(770\) 0 0
\(771\) −53.2200 −1.91667
\(772\) 0 0
\(773\) −9.49557 −0.341532 −0.170766 0.985312i \(-0.554624\pi\)
−0.170766 + 0.985312i \(0.554624\pi\)
\(774\) 0 0
\(775\) 3.24892 0.116705
\(776\) 0 0
\(777\) 83.4419 2.99346
\(778\) 0 0
\(779\) −0.514201 −0.0184232
\(780\) 0 0
\(781\) −70.6317 −2.52740
\(782\) 0 0
\(783\) 20.5422 0.734117
\(784\) 0 0
\(785\) 18.7761 0.670149
\(786\) 0 0
\(787\) 8.59840 0.306500 0.153250 0.988187i \(-0.451026\pi\)
0.153250 + 0.988187i \(0.451026\pi\)
\(788\) 0 0
\(789\) −14.0430 −0.499945
\(790\) 0 0
\(791\) −12.0535 −0.428574
\(792\) 0 0
\(793\) −24.8410 −0.882131
\(794\) 0 0
\(795\) −11.2809 −0.400091
\(796\) 0 0
\(797\) −23.3568 −0.827340 −0.413670 0.910427i \(-0.635753\pi\)
−0.413670 + 0.910427i \(0.635753\pi\)
\(798\) 0 0
\(799\) −24.3245 −0.860538
\(800\) 0 0
\(801\) −8.90428 −0.314617
\(802\) 0 0
\(803\) 55.2534 1.94985
\(804\) 0 0
\(805\) −42.4730 −1.49698
\(806\) 0 0
\(807\) 37.0583 1.30451
\(808\) 0 0
\(809\) −11.0634 −0.388968 −0.194484 0.980906i \(-0.562303\pi\)
−0.194484 + 0.980906i \(0.562303\pi\)
\(810\) 0 0
\(811\) −12.9630 −0.455193 −0.227596 0.973756i \(-0.573087\pi\)
−0.227596 + 0.973756i \(0.573087\pi\)
\(812\) 0 0
\(813\) −22.5909 −0.792299
\(814\) 0 0
\(815\) 32.6778 1.14465
\(816\) 0 0
\(817\) −10.9631 −0.383551
\(818\) 0 0
\(819\) 63.7028 2.22595
\(820\) 0 0
\(821\) 14.8255 0.517415 0.258707 0.965956i \(-0.416703\pi\)
0.258707 + 0.965956i \(0.416703\pi\)
\(822\) 0 0
\(823\) 4.04240 0.140909 0.0704547 0.997515i \(-0.477555\pi\)
0.0704547 + 0.997515i \(0.477555\pi\)
\(824\) 0 0
\(825\) 30.4060 1.05860
\(826\) 0 0
\(827\) 33.6070 1.16863 0.584314 0.811527i \(-0.301364\pi\)
0.584314 + 0.811527i \(0.301364\pi\)
\(828\) 0 0
\(829\) 38.3399 1.33160 0.665800 0.746130i \(-0.268090\pi\)
0.665800 + 0.746130i \(0.268090\pi\)
\(830\) 0 0
\(831\) 59.8498 2.07617
\(832\) 0 0
\(833\) 14.8125 0.513223
\(834\) 0 0
\(835\) −23.1916 −0.802579
\(836\) 0 0
\(837\) −4.20125 −0.145217
\(838\) 0 0
\(839\) −37.6832 −1.30097 −0.650484 0.759520i \(-0.725434\pi\)
−0.650484 + 0.759520i \(0.725434\pi\)
\(840\) 0 0
\(841\) 38.1806 1.31657
\(842\) 0 0
\(843\) −21.8015 −0.750883
\(844\) 0 0
\(845\) 11.1091 0.382165
\(846\) 0 0
\(847\) 94.8581 3.25936
\(848\) 0 0
\(849\) 57.4580 1.97195
\(850\) 0 0
\(851\) 33.7949 1.15847
\(852\) 0 0
\(853\) 7.06647 0.241951 0.120976 0.992655i \(-0.461398\pi\)
0.120976 + 0.992655i \(0.461398\pi\)
\(854\) 0 0
\(855\) 10.4061 0.355881
\(856\) 0 0
\(857\) 7.95910 0.271878 0.135939 0.990717i \(-0.456595\pi\)
0.135939 + 0.990717i \(0.456595\pi\)
\(858\) 0 0
\(859\) −13.1466 −0.448558 −0.224279 0.974525i \(-0.572003\pi\)
−0.224279 + 0.974525i \(0.572003\pi\)
\(860\) 0 0
\(861\) 5.26807 0.179535
\(862\) 0 0
\(863\) 34.9886 1.19102 0.595512 0.803346i \(-0.296949\pi\)
0.595512 + 0.803346i \(0.296949\pi\)
\(864\) 0 0
\(865\) 4.00909 0.136313
\(866\) 0 0
\(867\) 36.0050 1.22279
\(868\) 0 0
\(869\) 14.4079 0.488754
\(870\) 0 0
\(871\) 16.9847 0.575504
\(872\) 0 0
\(873\) 36.6335 1.23985
\(874\) 0 0
\(875\) −31.3411 −1.05952
\(876\) 0 0
\(877\) 49.1816 1.66075 0.830373 0.557208i \(-0.188127\pi\)
0.830373 + 0.557208i \(0.188127\pi\)
\(878\) 0 0
\(879\) −22.7795 −0.768333
\(880\) 0 0
\(881\) −7.58658 −0.255598 −0.127799 0.991800i \(-0.540791\pi\)
−0.127799 + 0.991800i \(0.540791\pi\)
\(882\) 0 0
\(883\) −15.1875 −0.511101 −0.255551 0.966796i \(-0.582257\pi\)
−0.255551 + 0.966796i \(0.582257\pi\)
\(884\) 0 0
\(885\) 84.8842 2.85335
\(886\) 0 0
\(887\) 32.9000 1.10467 0.552337 0.833621i \(-0.313736\pi\)
0.552337 + 0.833621i \(0.313736\pi\)
\(888\) 0 0
\(889\) 13.9577 0.468125
\(890\) 0 0
\(891\) 31.2075 1.04549
\(892\) 0 0
\(893\) −13.3035 −0.445185
\(894\) 0 0
\(895\) −49.1507 −1.64293
\(896\) 0 0
\(897\) 45.3924 1.51561
\(898\) 0 0
\(899\) −13.7397 −0.458244
\(900\) 0 0
\(901\) 2.97021 0.0989519
\(902\) 0 0
\(903\) 112.319 3.73774
\(904\) 0 0
\(905\) 29.7995 0.990568
\(906\) 0 0
\(907\) 55.9392 1.85743 0.928715 0.370793i \(-0.120914\pi\)
0.928715 + 0.370793i \(0.120914\pi\)
\(908\) 0 0
\(909\) −52.7884 −1.75088
\(910\) 0 0
\(911\) −44.6551 −1.47949 −0.739745 0.672887i \(-0.765054\pi\)
−0.739745 + 0.672887i \(0.765054\pi\)
\(912\) 0 0
\(913\) 5.87627 0.194476
\(914\) 0 0
\(915\) 41.5736 1.37438
\(916\) 0 0
\(917\) 37.9629 1.25364
\(918\) 0 0
\(919\) 7.52507 0.248229 0.124115 0.992268i \(-0.460391\pi\)
0.124115 + 0.992268i \(0.460391\pi\)
\(920\) 0 0
\(921\) −30.0209 −0.989222
\(922\) 0 0
\(923\) 49.2518 1.62114
\(924\) 0 0
\(925\) −15.7852 −0.519014
\(926\) 0 0
\(927\) −39.5434 −1.29878
\(928\) 0 0
\(929\) −10.4360 −0.342395 −0.171198 0.985237i \(-0.554764\pi\)
−0.171198 + 0.985237i \(0.554764\pi\)
\(930\) 0 0
\(931\) 8.10124 0.265507
\(932\) 0 0
\(933\) 29.8439 0.977044
\(934\) 0 0
\(935\) −28.6591 −0.937252
\(936\) 0 0
\(937\) −19.1073 −0.624207 −0.312104 0.950048i \(-0.601034\pi\)
−0.312104 + 0.950048i \(0.601034\pi\)
\(938\) 0 0
\(939\) −48.2600 −1.57491
\(940\) 0 0
\(941\) 43.2146 1.40875 0.704377 0.709826i \(-0.251226\pi\)
0.704377 + 0.709826i \(0.251226\pi\)
\(942\) 0 0
\(943\) 2.13363 0.0694804
\(944\) 0 0
\(945\) −25.6538 −0.834518
\(946\) 0 0
\(947\) −1.25385 −0.0407446 −0.0203723 0.999792i \(-0.506485\pi\)
−0.0203723 + 0.999792i \(0.506485\pi\)
\(948\) 0 0
\(949\) −38.5285 −1.25069
\(950\) 0 0
\(951\) −68.8100 −2.23132
\(952\) 0 0
\(953\) −35.0739 −1.13616 −0.568078 0.822975i \(-0.692313\pi\)
−0.568078 + 0.822975i \(0.692313\pi\)
\(954\) 0 0
\(955\) 30.5450 0.988413
\(956\) 0 0
\(957\) −128.587 −4.15663
\(958\) 0 0
\(959\) 16.5425 0.534184
\(960\) 0 0
\(961\) −28.1900 −0.909354
\(962\) 0 0
\(963\) 19.1680 0.617681
\(964\) 0 0
\(965\) −9.97512 −0.321110
\(966\) 0 0
\(967\) 1.66764 0.0536275 0.0268138 0.999640i \(-0.491464\pi\)
0.0268138 + 0.999640i \(0.491464\pi\)
\(968\) 0 0
\(969\) −4.82047 −0.154856
\(970\) 0 0
\(971\) −16.1136 −0.517110 −0.258555 0.965996i \(-0.583246\pi\)
−0.258555 + 0.965996i \(0.583246\pi\)
\(972\) 0 0
\(973\) 36.9513 1.18460
\(974\) 0 0
\(975\) −21.2022 −0.679015
\(976\) 0 0
\(977\) 33.8857 1.08410 0.542051 0.840346i \(-0.317648\pi\)
0.542051 + 0.840346i \(0.317648\pi\)
\(978\) 0 0
\(979\) 13.4121 0.428651
\(980\) 0 0
\(981\) 59.7328 1.90712
\(982\) 0 0
\(983\) 9.89492 0.315599 0.157799 0.987471i \(-0.449560\pi\)
0.157799 + 0.987471i \(0.449560\pi\)
\(984\) 0 0
\(985\) −68.6963 −2.18885
\(986\) 0 0
\(987\) 136.296 4.33836
\(988\) 0 0
\(989\) 45.4904 1.44651
\(990\) 0 0
\(991\) −31.3693 −0.996478 −0.498239 0.867040i \(-0.666020\pi\)
−0.498239 + 0.867040i \(0.666020\pi\)
\(992\) 0 0
\(993\) −32.3747 −1.02738
\(994\) 0 0
\(995\) −9.31004 −0.295148
\(996\) 0 0
\(997\) 4.55606 0.144292 0.0721459 0.997394i \(-0.477015\pi\)
0.0721459 + 0.997394i \(0.477015\pi\)
\(998\) 0 0
\(999\) 20.4122 0.645813
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 4864.2.a.bm.1.2 8
4.3 odd 2 4864.2.a.br.1.8 8
8.3 odd 2 inner 4864.2.a.bm.1.1 8
8.5 even 2 4864.2.a.br.1.7 8
16.3 odd 4 2432.2.c.i.1217.15 yes 16
16.5 even 4 2432.2.c.i.1217.16 yes 16
16.11 odd 4 2432.2.c.i.1217.2 yes 16
16.13 even 4 2432.2.c.i.1217.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2432.2.c.i.1217.1 16 16.13 even 4
2432.2.c.i.1217.2 yes 16 16.11 odd 4
2432.2.c.i.1217.15 yes 16 16.3 odd 4
2432.2.c.i.1217.16 yes 16 16.5 even 4
4864.2.a.bm.1.1 8 8.3 odd 2 inner
4864.2.a.bm.1.2 8 1.1 even 1 trivial
4864.2.a.br.1.7 8 8.5 even 2
4864.2.a.br.1.8 8 4.3 odd 2