Properties

Label 8112.2.a.by.1.3
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 1.3
Root \(-1.24698\) of defining polynomial
Character \(\chi\) \(=\) 8112.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +0.246980 q^{5} +1.75302 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +0.246980 q^{5} +1.75302 q^{7} +1.00000 q^{9} +5.65279 q^{11} -0.246980 q^{15} -3.80194 q^{17} +5.58211 q^{19} -1.75302 q^{21} -8.34481 q^{23} -4.93900 q^{25} -1.00000 q^{27} -5.93900 q^{29} +5.26875 q^{31} -5.65279 q^{33} +0.432960 q^{35} -3.19806 q^{37} -0.445042 q^{41} -1.71379 q^{43} +0.246980 q^{45} -6.73556 q^{47} -3.92692 q^{49} +3.80194 q^{51} -1.06100 q^{53} +1.39612 q^{55} -5.58211 q^{57} -13.7017 q^{59} -8.51573 q^{61} +1.75302 q^{63} +5.96077 q^{67} +8.34481 q^{69} -5.71917 q^{71} -7.35690 q^{73} +4.93900 q^{75} +9.90946 q^{77} -4.45473 q^{79} +1.00000 q^{81} -10.1860 q^{83} -0.939001 q^{85} +5.93900 q^{87} -0.137063 q^{89} -5.26875 q^{93} +1.37867 q^{95} +13.6896 q^{97} +5.65279 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{3} - 4q^{5} + 10q^{7} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{3} - 4q^{5} + 10q^{7} + 3q^{9} - q^{11} + 4q^{15} - 7q^{17} + 11q^{19} - 10q^{21} - 2q^{23} - 5q^{25} - 3q^{27} - 8q^{29} + 8q^{31} + q^{33} - 18q^{35} - 14q^{37} - q^{41} + 3q^{43} - 4q^{45} - 9q^{47} + 17q^{49} + 7q^{51} - 13q^{53} + 13q^{55} - 11q^{57} - 14q^{59} - 13q^{61} + 10q^{63} + 5q^{67} + 2q^{69} - 6q^{71} - 18q^{73} + 5q^{75} - 15q^{77} + 9q^{79} + 3q^{81} - 16q^{83} + 7q^{85} + 8q^{87} + 5q^{89} - 8q^{93} - 3q^{95} - 5q^{97} - q^{99} + O(q^{100}) \)

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
\(5\) 0.246980 0.110453 0.0552263 0.998474i \(-0.482412\pi\)
0.0552263 + 0.998474i \(0.482412\pi\)
\(6\) 0 0
\(7\) 1.75302 0.662579 0.331290 0.943529i \(-0.392516\pi\)
0.331290 + 0.943529i \(0.392516\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 5.65279 1.70438 0.852191 0.523232i \(-0.175274\pi\)
0.852191 + 0.523232i \(0.175274\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) −0.246980 −0.0637699
\(16\) 0 0
\(17\) −3.80194 −0.922105 −0.461053 0.887373i \(-0.652528\pi\)
−0.461053 + 0.887373i \(0.652528\pi\)
\(18\) 0 0
\(19\) 5.58211 1.28062 0.640311 0.768115i \(-0.278805\pi\)
0.640311 + 0.768115i \(0.278805\pi\)
\(20\) 0 0
\(21\) −1.75302 −0.382540
\(22\) 0 0
\(23\) −8.34481 −1.74001 −0.870007 0.493039i \(-0.835886\pi\)
−0.870007 + 0.493039i \(0.835886\pi\)
\(24\) 0 0
\(25\) −4.93900 −0.987800
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −5.93900 −1.10284 −0.551422 0.834226i \(-0.685915\pi\)
−0.551422 + 0.834226i \(0.685915\pi\)
\(30\) 0 0
\(31\) 5.26875 0.946295 0.473148 0.880983i \(-0.343118\pi\)
0.473148 + 0.880983i \(0.343118\pi\)
\(32\) 0 0
\(33\) −5.65279 −0.984025
\(34\) 0 0
\(35\) 0.432960 0.0731836
\(36\) 0 0
\(37\) −3.19806 −0.525758 −0.262879 0.964829i \(-0.584672\pi\)
−0.262879 + 0.964829i \(0.584672\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −0.445042 −0.0695039 −0.0347519 0.999396i \(-0.511064\pi\)
−0.0347519 + 0.999396i \(0.511064\pi\)
\(42\) 0 0
\(43\) −1.71379 −0.261351 −0.130675 0.991425i \(-0.541715\pi\)
−0.130675 + 0.991425i \(0.541715\pi\)
\(44\) 0 0
\(45\) 0.246980 0.0368175
\(46\) 0 0
\(47\) −6.73556 −0.982483 −0.491241 0.871024i \(-0.663457\pi\)
−0.491241 + 0.871024i \(0.663457\pi\)
\(48\) 0 0
\(49\) −3.92692 −0.560988
\(50\) 0 0
\(51\) 3.80194 0.532378
\(52\) 0 0
\(53\) −1.06100 −0.145739 −0.0728697 0.997341i \(-0.523216\pi\)
−0.0728697 + 0.997341i \(0.523216\pi\)
\(54\) 0 0
\(55\) 1.39612 0.188253
\(56\) 0 0
\(57\) −5.58211 −0.739368
\(58\) 0 0
\(59\) −13.7017 −1.78381 −0.891905 0.452222i \(-0.850631\pi\)
−0.891905 + 0.452222i \(0.850631\pi\)
\(60\) 0 0
\(61\) −8.51573 −1.09033 −0.545164 0.838330i \(-0.683533\pi\)
−0.545164 + 0.838330i \(0.683533\pi\)
\(62\) 0 0
\(63\) 1.75302 0.220860
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 5.96077 0.728224 0.364112 0.931355i \(-0.381373\pi\)
0.364112 + 0.931355i \(0.381373\pi\)
\(68\) 0 0
\(69\) 8.34481 1.00460
\(70\) 0 0
\(71\) −5.71917 −0.678740 −0.339370 0.940653i \(-0.610214\pi\)
−0.339370 + 0.940653i \(0.610214\pi\)
\(72\) 0 0
\(73\) −7.35690 −0.861060 −0.430530 0.902576i \(-0.641673\pi\)
−0.430530 + 0.902576i \(0.641673\pi\)
\(74\) 0 0
\(75\) 4.93900 0.570307
\(76\) 0 0
\(77\) 9.90946 1.12929
\(78\) 0 0
\(79\) −4.45473 −0.501196 −0.250598 0.968091i \(-0.580627\pi\)
−0.250598 + 0.968091i \(0.580627\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −10.1860 −1.11806 −0.559028 0.829149i \(-0.688826\pi\)
−0.559028 + 0.829149i \(0.688826\pi\)
\(84\) 0 0
\(85\) −0.939001 −0.101849
\(86\) 0 0
\(87\) 5.93900 0.636728
\(88\) 0 0
\(89\) −0.137063 −0.0145287 −0.00726434 0.999974i \(-0.502312\pi\)
−0.00726434 + 0.999974i \(0.502312\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −5.26875 −0.546344
\(94\) 0 0
\(95\) 1.37867 0.141448
\(96\) 0 0
\(97\) 13.6896 1.38997 0.694986 0.719024i \(-0.255411\pi\)
0.694986 + 0.719024i \(0.255411\pi\)
\(98\) 0 0
\(99\) 5.65279 0.568127
\(100\) 0 0
\(101\) −5.41119 −0.538434 −0.269217 0.963080i \(-0.586765\pi\)
−0.269217 + 0.963080i \(0.586765\pi\)
\(102\) 0 0
\(103\) −13.7560 −1.35542 −0.677710 0.735330i \(-0.737027\pi\)
−0.677710 + 0.735330i \(0.737027\pi\)
\(104\) 0 0
\(105\) −0.432960 −0.0422526
\(106\) 0 0
\(107\) 12.8170 1.23907 0.619533 0.784970i \(-0.287322\pi\)
0.619533 + 0.784970i \(0.287322\pi\)
\(108\) 0 0
\(109\) −12.1468 −1.16345 −0.581724 0.813386i \(-0.697622\pi\)
−0.581724 + 0.813386i \(0.697622\pi\)
\(110\) 0 0
\(111\) 3.19806 0.303547
\(112\) 0 0
\(113\) −1.63773 −0.154064 −0.0770322 0.997029i \(-0.524544\pi\)
−0.0770322 + 0.997029i \(0.524544\pi\)
\(114\) 0 0
\(115\) −2.06100 −0.192189
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −6.66487 −0.610968
\(120\) 0 0
\(121\) 20.9541 1.90492
\(122\) 0 0
\(123\) 0.445042 0.0401281
\(124\) 0 0
\(125\) −2.45473 −0.219558
\(126\) 0 0
\(127\) −10.7995 −0.958305 −0.479152 0.877732i \(-0.659056\pi\)
−0.479152 + 0.877732i \(0.659056\pi\)
\(128\) 0 0
\(129\) 1.71379 0.150891
\(130\) 0 0
\(131\) −0.907542 −0.0792923 −0.0396462 0.999214i \(-0.512623\pi\)
−0.0396462 + 0.999214i \(0.512623\pi\)
\(132\) 0 0
\(133\) 9.78554 0.848514
\(134\) 0 0
\(135\) −0.246980 −0.0212566
\(136\) 0 0
\(137\) 9.54825 0.815762 0.407881 0.913035i \(-0.366268\pi\)
0.407881 + 0.913035i \(0.366268\pi\)
\(138\) 0 0
\(139\) 4.09246 0.347118 0.173559 0.984823i \(-0.444473\pi\)
0.173559 + 0.984823i \(0.444473\pi\)
\(140\) 0 0
\(141\) 6.73556 0.567237
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −1.46681 −0.121812
\(146\) 0 0
\(147\) 3.92692 0.323887
\(148\) 0 0
\(149\) 15.3884 1.26066 0.630332 0.776326i \(-0.282919\pi\)
0.630332 + 0.776326i \(0.282919\pi\)
\(150\) 0 0
\(151\) −3.67456 −0.299032 −0.149516 0.988759i \(-0.547772\pi\)
−0.149516 + 0.988759i \(0.547772\pi\)
\(152\) 0 0
\(153\) −3.80194 −0.307368
\(154\) 0 0
\(155\) 1.30127 0.104521
\(156\) 0 0
\(157\) −4.87800 −0.389307 −0.194653 0.980872i \(-0.562358\pi\)
−0.194653 + 0.980872i \(0.562358\pi\)
\(158\) 0 0
\(159\) 1.06100 0.0841427
\(160\) 0 0
\(161\) −14.6286 −1.15290
\(162\) 0 0
\(163\) 8.63102 0.676034 0.338017 0.941140i \(-0.390244\pi\)
0.338017 + 0.941140i \(0.390244\pi\)
\(164\) 0 0
\(165\) −1.39612 −0.108688
\(166\) 0 0
\(167\) 9.46980 0.732795 0.366397 0.930458i \(-0.380591\pi\)
0.366397 + 0.930458i \(0.380591\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 5.58211 0.426874
\(172\) 0 0
\(173\) 4.77479 0.363021 0.181510 0.983389i \(-0.441901\pi\)
0.181510 + 0.983389i \(0.441901\pi\)
\(174\) 0 0
\(175\) −8.65817 −0.654496
\(176\) 0 0
\(177\) 13.7017 1.02988
\(178\) 0 0
\(179\) −3.43535 −0.256770 −0.128385 0.991724i \(-0.540979\pi\)
−0.128385 + 0.991724i \(0.540979\pi\)
\(180\) 0 0
\(181\) −13.4862 −1.00242 −0.501210 0.865326i \(-0.667112\pi\)
−0.501210 + 0.865326i \(0.667112\pi\)
\(182\) 0 0
\(183\) 8.51573 0.629501
\(184\) 0 0
\(185\) −0.789856 −0.0580714
\(186\) 0 0
\(187\) −21.4916 −1.57162
\(188\) 0 0
\(189\) −1.75302 −0.127513
\(190\) 0 0
\(191\) 1.30127 0.0941569 0.0470784 0.998891i \(-0.485009\pi\)
0.0470784 + 0.998891i \(0.485009\pi\)
\(192\) 0 0
\(193\) 9.19567 0.661919 0.330959 0.943645i \(-0.392628\pi\)
0.330959 + 0.943645i \(0.392628\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −4.11231 −0.292990 −0.146495 0.989211i \(-0.546799\pi\)
−0.146495 + 0.989211i \(0.546799\pi\)
\(198\) 0 0
\(199\) 24.7724 1.75607 0.878034 0.478598i \(-0.158855\pi\)
0.878034 + 0.478598i \(0.158855\pi\)
\(200\) 0 0
\(201\) −5.96077 −0.420440
\(202\) 0 0
\(203\) −10.4112 −0.730722
\(204\) 0 0
\(205\) −0.109916 −0.00767688
\(206\) 0 0
\(207\) −8.34481 −0.580005
\(208\) 0 0
\(209\) 31.5545 2.18267
\(210\) 0 0
\(211\) 5.93900 0.408858 0.204429 0.978881i \(-0.434466\pi\)
0.204429 + 0.978881i \(0.434466\pi\)
\(212\) 0 0
\(213\) 5.71917 0.391871
\(214\) 0 0
\(215\) −0.423272 −0.0288669
\(216\) 0 0
\(217\) 9.23623 0.626996
\(218\) 0 0
\(219\) 7.35690 0.497133
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 14.2010 0.950972 0.475486 0.879723i \(-0.342272\pi\)
0.475486 + 0.879723i \(0.342272\pi\)
\(224\) 0 0
\(225\) −4.93900 −0.329267
\(226\) 0 0
\(227\) 16.0073 1.06244 0.531221 0.847233i \(-0.321733\pi\)
0.531221 + 0.847233i \(0.321733\pi\)
\(228\) 0 0
\(229\) −1.84117 −0.121668 −0.0608339 0.998148i \(-0.519376\pi\)
−0.0608339 + 0.998148i \(0.519376\pi\)
\(230\) 0 0
\(231\) −9.90946 −0.651995
\(232\) 0 0
\(233\) −23.4252 −1.53464 −0.767318 0.641267i \(-0.778409\pi\)
−0.767318 + 0.641267i \(0.778409\pi\)
\(234\) 0 0
\(235\) −1.66355 −0.108518
\(236\) 0 0
\(237\) 4.45473 0.289366
\(238\) 0 0
\(239\) 14.6015 0.944491 0.472246 0.881467i \(-0.343444\pi\)
0.472246 + 0.881467i \(0.343444\pi\)
\(240\) 0 0
\(241\) −8.63102 −0.555973 −0.277987 0.960585i \(-0.589667\pi\)
−0.277987 + 0.960585i \(0.589667\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −0.969869 −0.0619627
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 10.1860 0.645510
\(250\) 0 0
\(251\) 3.80194 0.239976 0.119988 0.992775i \(-0.461714\pi\)
0.119988 + 0.992775i \(0.461714\pi\)
\(252\) 0 0
\(253\) −47.1715 −2.96565
\(254\) 0 0
\(255\) 0.939001 0.0588025
\(256\) 0 0
\(257\) −20.5961 −1.28475 −0.642375 0.766391i \(-0.722051\pi\)
−0.642375 + 0.766391i \(0.722051\pi\)
\(258\) 0 0
\(259\) −5.60627 −0.348357
\(260\) 0 0
\(261\) −5.93900 −0.367615
\(262\) 0 0
\(263\) −0.332733 −0.0205172 −0.0102586 0.999947i \(-0.503265\pi\)
−0.0102586 + 0.999947i \(0.503265\pi\)
\(264\) 0 0
\(265\) −0.262045 −0.0160973
\(266\) 0 0
\(267\) 0.137063 0.00838814
\(268\) 0 0
\(269\) 27.3032 1.66471 0.832353 0.554247i \(-0.186994\pi\)
0.832353 + 0.554247i \(0.186994\pi\)
\(270\) 0 0
\(271\) 27.9855 1.70000 0.850000 0.526783i \(-0.176602\pi\)
0.850000 + 0.526783i \(0.176602\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −27.9191 −1.68359
\(276\) 0 0
\(277\) −2.10321 −0.126370 −0.0631849 0.998002i \(-0.520126\pi\)
−0.0631849 + 0.998002i \(0.520126\pi\)
\(278\) 0 0
\(279\) 5.26875 0.315432
\(280\) 0 0
\(281\) −27.2349 −1.62470 −0.812349 0.583172i \(-0.801811\pi\)
−0.812349 + 0.583172i \(0.801811\pi\)
\(282\) 0 0
\(283\) −5.28382 −0.314090 −0.157045 0.987591i \(-0.550197\pi\)
−0.157045 + 0.987591i \(0.550197\pi\)
\(284\) 0 0
\(285\) −1.37867 −0.0816651
\(286\) 0 0
\(287\) −0.780167 −0.0460518
\(288\) 0 0
\(289\) −2.54527 −0.149722
\(290\) 0 0
\(291\) −13.6896 −0.802500
\(292\) 0 0
\(293\) −32.6625 −1.90816 −0.954081 0.299548i \(-0.903164\pi\)
−0.954081 + 0.299548i \(0.903164\pi\)
\(294\) 0 0
\(295\) −3.38404 −0.197027
\(296\) 0 0
\(297\) −5.65279 −0.328008
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −3.00431 −0.173166
\(302\) 0 0
\(303\) 5.41119 0.310865
\(304\) 0 0
\(305\) −2.10321 −0.120430
\(306\) 0 0
\(307\) −20.7614 −1.18491 −0.592457 0.805602i \(-0.701842\pi\)
−0.592457 + 0.805602i \(0.701842\pi\)
\(308\) 0 0
\(309\) 13.7560 0.782552
\(310\) 0 0
\(311\) 11.3013 0.640836 0.320418 0.947276i \(-0.396177\pi\)
0.320418 + 0.947276i \(0.396177\pi\)
\(312\) 0 0
\(313\) −4.27173 −0.241453 −0.120726 0.992686i \(-0.538522\pi\)
−0.120726 + 0.992686i \(0.538522\pi\)
\(314\) 0 0
\(315\) 0.432960 0.0243945
\(316\) 0 0
\(317\) −15.4776 −0.869307 −0.434653 0.900598i \(-0.643129\pi\)
−0.434653 + 0.900598i \(0.643129\pi\)
\(318\) 0 0
\(319\) −33.5719 −1.87967
\(320\) 0 0
\(321\) −12.8170 −0.715375
\(322\) 0 0
\(323\) −21.2228 −1.18087
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 12.1468 0.671717
\(328\) 0 0
\(329\) −11.8076 −0.650973
\(330\) 0 0
\(331\) 6.06829 0.333544 0.166772 0.985996i \(-0.446666\pi\)
0.166772 + 0.985996i \(0.446666\pi\)
\(332\) 0 0
\(333\) −3.19806 −0.175253
\(334\) 0 0
\(335\) 1.47219 0.0804343
\(336\) 0 0
\(337\) 12.1239 0.660432 0.330216 0.943905i \(-0.392878\pi\)
0.330216 + 0.943905i \(0.392878\pi\)
\(338\) 0 0
\(339\) 1.63773 0.0889491
\(340\) 0 0
\(341\) 29.7832 1.61285
\(342\) 0 0
\(343\) −19.1551 −1.03428
\(344\) 0 0
\(345\) 2.06100 0.110960
\(346\) 0 0
\(347\) −23.1497 −1.24274 −0.621371 0.783516i \(-0.713424\pi\)
−0.621371 + 0.783516i \(0.713424\pi\)
\(348\) 0 0
\(349\) 22.1957 1.18811 0.594053 0.804426i \(-0.297527\pi\)
0.594053 + 0.804426i \(0.297527\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 5.07069 0.269885 0.134943 0.990853i \(-0.456915\pi\)
0.134943 + 0.990853i \(0.456915\pi\)
\(354\) 0 0
\(355\) −1.41252 −0.0749687
\(356\) 0 0
\(357\) 6.66487 0.352743
\(358\) 0 0
\(359\) −16.6746 −0.880050 −0.440025 0.897986i \(-0.645030\pi\)
−0.440025 + 0.897986i \(0.645030\pi\)
\(360\) 0 0
\(361\) 12.1599 0.639995
\(362\) 0 0
\(363\) −20.9541 −1.09980
\(364\) 0 0
\(365\) −1.81700 −0.0951063
\(366\) 0 0
\(367\) 1.17928 0.0615577 0.0307789 0.999526i \(-0.490201\pi\)
0.0307789 + 0.999526i \(0.490201\pi\)
\(368\) 0 0
\(369\) −0.445042 −0.0231680
\(370\) 0 0
\(371\) −1.85995 −0.0965639
\(372\) 0 0
\(373\) −30.0925 −1.55813 −0.779064 0.626944i \(-0.784305\pi\)
−0.779064 + 0.626944i \(0.784305\pi\)
\(374\) 0 0
\(375\) 2.45473 0.126762
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −19.1631 −0.984345 −0.492172 0.870498i \(-0.663797\pi\)
−0.492172 + 0.870498i \(0.663797\pi\)
\(380\) 0 0
\(381\) 10.7995 0.553277
\(382\) 0 0
\(383\) −15.3884 −0.786308 −0.393154 0.919473i \(-0.628616\pi\)
−0.393154 + 0.919473i \(0.628616\pi\)
\(384\) 0 0
\(385\) 2.44743 0.124733
\(386\) 0 0
\(387\) −1.71379 −0.0871169
\(388\) 0 0
\(389\) −24.0315 −1.21844 −0.609222 0.793000i \(-0.708518\pi\)
−0.609222 + 0.793000i \(0.708518\pi\)
\(390\) 0 0
\(391\) 31.7265 1.60448
\(392\) 0 0
\(393\) 0.907542 0.0457794
\(394\) 0 0
\(395\) −1.10023 −0.0553585
\(396\) 0 0
\(397\) −29.6015 −1.48566 −0.742828 0.669482i \(-0.766516\pi\)
−0.742828 + 0.669482i \(0.766516\pi\)
\(398\) 0 0
\(399\) −9.78554 −0.489890
\(400\) 0 0
\(401\) 21.1032 1.05384 0.526922 0.849914i \(-0.323346\pi\)
0.526922 + 0.849914i \(0.323346\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0.246980 0.0122725
\(406\) 0 0
\(407\) −18.0780 −0.896092
\(408\) 0 0
\(409\) 36.0224 1.78119 0.890596 0.454796i \(-0.150288\pi\)
0.890596 + 0.454796i \(0.150288\pi\)
\(410\) 0 0
\(411\) −9.54825 −0.470981
\(412\) 0 0
\(413\) −24.0194 −1.18192
\(414\) 0 0
\(415\) −2.51573 −0.123492
\(416\) 0 0
\(417\) −4.09246 −0.200409
\(418\) 0 0
\(419\) −5.96854 −0.291582 −0.145791 0.989315i \(-0.546573\pi\)
−0.145791 + 0.989315i \(0.546573\pi\)
\(420\) 0 0
\(421\) 2.09544 0.102126 0.0510628 0.998695i \(-0.483739\pi\)
0.0510628 + 0.998695i \(0.483739\pi\)
\(422\) 0 0
\(423\) −6.73556 −0.327494
\(424\) 0 0
\(425\) 18.7778 0.910856
\(426\) 0 0
\(427\) −14.9282 −0.722429
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −2.88577 −0.139003 −0.0695014 0.997582i \(-0.522141\pi\)
−0.0695014 + 0.997582i \(0.522141\pi\)
\(432\) 0 0
\(433\) −12.6485 −0.607847 −0.303924 0.952696i \(-0.598297\pi\)
−0.303924 + 0.952696i \(0.598297\pi\)
\(434\) 0 0
\(435\) 1.46681 0.0703283
\(436\) 0 0
\(437\) −46.5816 −2.22830
\(438\) 0 0
\(439\) 10.9269 0.521513 0.260757 0.965405i \(-0.416028\pi\)
0.260757 + 0.965405i \(0.416028\pi\)
\(440\) 0 0
\(441\) −3.92692 −0.186996
\(442\) 0 0
\(443\) 19.2403 0.914133 0.457067 0.889433i \(-0.348900\pi\)
0.457067 + 0.889433i \(0.348900\pi\)
\(444\) 0 0
\(445\) −0.0338518 −0.00160473
\(446\) 0 0
\(447\) −15.3884 −0.727844
\(448\) 0 0
\(449\) 28.8200 1.36010 0.680050 0.733166i \(-0.261958\pi\)
0.680050 + 0.733166i \(0.261958\pi\)
\(450\) 0 0
\(451\) −2.51573 −0.118461
\(452\) 0 0
\(453\) 3.67456 0.172646
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 18.0707 0.845311 0.422656 0.906290i \(-0.361098\pi\)
0.422656 + 0.906290i \(0.361098\pi\)
\(458\) 0 0
\(459\) 3.80194 0.177459
\(460\) 0 0
\(461\) 7.56763 0.352460 0.176230 0.984349i \(-0.443610\pi\)
0.176230 + 0.984349i \(0.443610\pi\)
\(462\) 0 0
\(463\) 35.3551 1.64309 0.821545 0.570143i \(-0.193112\pi\)
0.821545 + 0.570143i \(0.193112\pi\)
\(464\) 0 0
\(465\) −1.30127 −0.0603451
\(466\) 0 0
\(467\) −13.0000 −0.601568 −0.300784 0.953692i \(-0.597248\pi\)
−0.300784 + 0.953692i \(0.597248\pi\)
\(468\) 0 0
\(469\) 10.4494 0.482506
\(470\) 0 0
\(471\) 4.87800 0.224766
\(472\) 0 0
\(473\) −9.68771 −0.445441
\(474\) 0 0
\(475\) −27.5700 −1.26500
\(476\) 0 0
\(477\) −1.06100 −0.0485798
\(478\) 0 0
\(479\) −25.5265 −1.16633 −0.583167 0.812352i \(-0.698187\pi\)
−0.583167 + 0.812352i \(0.698187\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 14.6286 0.665626
\(484\) 0 0
\(485\) 3.38106 0.153526
\(486\) 0 0
\(487\) 16.0073 0.725360 0.362680 0.931914i \(-0.381862\pi\)
0.362680 + 0.931914i \(0.381862\pi\)
\(488\) 0 0
\(489\) −8.63102 −0.390308
\(490\) 0 0
\(491\) −20.7385 −0.935917 −0.467959 0.883750i \(-0.655010\pi\)
−0.467959 + 0.883750i \(0.655010\pi\)
\(492\) 0 0
\(493\) 22.5797 1.01694
\(494\) 0 0
\(495\) 1.39612 0.0627511
\(496\) 0 0
\(497\) −10.0258 −0.449719
\(498\) 0 0
\(499\) −8.06770 −0.361160 −0.180580 0.983560i \(-0.557797\pi\)
−0.180580 + 0.983560i \(0.557797\pi\)
\(500\) 0 0
\(501\) −9.46980 −0.423079
\(502\) 0 0
\(503\) 30.2422 1.34843 0.674216 0.738534i \(-0.264482\pi\)
0.674216 + 0.738534i \(0.264482\pi\)
\(504\) 0 0
\(505\) −1.33645 −0.0594714
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −16.0495 −0.711382 −0.355691 0.934604i \(-0.615754\pi\)
−0.355691 + 0.934604i \(0.615754\pi\)
\(510\) 0 0
\(511\) −12.8968 −0.570520
\(512\) 0 0
\(513\) −5.58211 −0.246456
\(514\) 0 0
\(515\) −3.39745 −0.149710
\(516\) 0 0
\(517\) −38.0747 −1.67452
\(518\) 0 0
\(519\) −4.77479 −0.209590
\(520\) 0 0
\(521\) −2.69309 −0.117986 −0.0589931 0.998258i \(-0.518789\pi\)
−0.0589931 + 0.998258i \(0.518789\pi\)
\(522\) 0 0
\(523\) −35.3957 −1.54774 −0.773872 0.633342i \(-0.781683\pi\)
−0.773872 + 0.633342i \(0.781683\pi\)
\(524\) 0 0
\(525\) 8.65817 0.377874
\(526\) 0 0
\(527\) −20.0315 −0.872584
\(528\) 0 0
\(529\) 46.6359 2.02765
\(530\) 0 0
\(531\) −13.7017 −0.594604
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 3.16554 0.136858
\(536\) 0 0
\(537\) 3.43535 0.148246
\(538\) 0 0
\(539\) −22.1981 −0.956138
\(540\) 0 0
\(541\) 34.7338 1.49332 0.746660 0.665205i \(-0.231656\pi\)
0.746660 + 0.665205i \(0.231656\pi\)
\(542\) 0 0
\(543\) 13.4862 0.578748
\(544\) 0 0
\(545\) −3.00000 −0.128506
\(546\) 0 0
\(547\) 26.1183 1.11674 0.558368 0.829593i \(-0.311428\pi\)
0.558368 + 0.829593i \(0.311428\pi\)
\(548\) 0 0
\(549\) −8.51573 −0.363442
\(550\) 0 0
\(551\) −33.1521 −1.41233
\(552\) 0 0
\(553\) −7.80923 −0.332082
\(554\) 0 0
\(555\) 0.789856 0.0335275
\(556\) 0 0
\(557\) −24.7748 −1.04974 −0.524871 0.851182i \(-0.675886\pi\)
−0.524871 + 0.851182i \(0.675886\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 21.4916 0.907375
\(562\) 0 0
\(563\) −5.26098 −0.221724 −0.110862 0.993836i \(-0.535361\pi\)
−0.110862 + 0.993836i \(0.535361\pi\)
\(564\) 0 0
\(565\) −0.404485 −0.0170168
\(566\) 0 0
\(567\) 1.75302 0.0736199
\(568\) 0 0
\(569\) −33.7458 −1.41470 −0.707350 0.706864i \(-0.750109\pi\)
−0.707350 + 0.706864i \(0.750109\pi\)
\(570\) 0 0
\(571\) −23.0887 −0.966234 −0.483117 0.875556i \(-0.660495\pi\)
−0.483117 + 0.875556i \(0.660495\pi\)
\(572\) 0 0
\(573\) −1.30127 −0.0543615
\(574\) 0 0
\(575\) 41.2150 1.71879
\(576\) 0 0
\(577\) −3.57002 −0.148622 −0.0743110 0.997235i \(-0.523676\pi\)
−0.0743110 + 0.997235i \(0.523676\pi\)
\(578\) 0 0
\(579\) −9.19567 −0.382159
\(580\) 0 0
\(581\) −17.8562 −0.740801
\(582\) 0 0
\(583\) −5.99761 −0.248396
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −11.4625 −0.473108 −0.236554 0.971618i \(-0.576018\pi\)
−0.236554 + 0.971618i \(0.576018\pi\)
\(588\) 0 0
\(589\) 29.4107 1.21185
\(590\) 0 0
\(591\) 4.11231 0.169158
\(592\) 0 0
\(593\) 21.8538 0.897430 0.448715 0.893675i \(-0.351882\pi\)
0.448715 + 0.893675i \(0.351882\pi\)
\(594\) 0 0
\(595\) −1.64609 −0.0674830
\(596\) 0 0
\(597\) −24.7724 −1.01387
\(598\) 0 0
\(599\) −27.0573 −1.10553 −0.552765 0.833337i \(-0.686427\pi\)
−0.552765 + 0.833337i \(0.686427\pi\)
\(600\) 0 0
\(601\) 10.8780 0.443723 0.221861 0.975078i \(-0.428787\pi\)
0.221861 + 0.975078i \(0.428787\pi\)
\(602\) 0 0
\(603\) 5.96077 0.242741
\(604\) 0 0
\(605\) 5.17523 0.210403
\(606\) 0 0
\(607\) 29.6359 1.20289 0.601443 0.798916i \(-0.294593\pi\)
0.601443 + 0.798916i \(0.294593\pi\)
\(608\) 0 0
\(609\) 10.4112 0.421883
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −10.2343 −0.413360 −0.206680 0.978409i \(-0.566266\pi\)
−0.206680 + 0.978409i \(0.566266\pi\)
\(614\) 0 0
\(615\) 0.109916 0.00443225
\(616\) 0 0
\(617\) −26.2828 −1.05810 −0.529052 0.848590i \(-0.677452\pi\)
−0.529052 + 0.848590i \(0.677452\pi\)
\(618\) 0 0
\(619\) 29.0834 1.16896 0.584479 0.811408i \(-0.301299\pi\)
0.584479 + 0.811408i \(0.301299\pi\)
\(620\) 0 0
\(621\) 8.34481 0.334866
\(622\) 0 0
\(623\) −0.240275 −0.00962641
\(624\) 0 0
\(625\) 24.0887 0.963549
\(626\) 0 0
\(627\) −31.5545 −1.26016
\(628\) 0 0
\(629\) 12.1588 0.484804
\(630\) 0 0
\(631\) 25.4480 1.01307 0.506535 0.862219i \(-0.330926\pi\)
0.506535 + 0.862219i \(0.330926\pi\)
\(632\) 0 0
\(633\) −5.93900 −0.236054
\(634\) 0 0
\(635\) −2.66727 −0.105847
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −5.71917 −0.226247
\(640\) 0 0
\(641\) −26.7409 −1.05620 −0.528102 0.849181i \(-0.677096\pi\)
−0.528102 + 0.849181i \(0.677096\pi\)
\(642\) 0 0
\(643\) −32.9614 −1.29987 −0.649935 0.759990i \(-0.725204\pi\)
−0.649935 + 0.759990i \(0.725204\pi\)
\(644\) 0 0
\(645\) 0.423272 0.0166663
\(646\) 0 0
\(647\) −34.4946 −1.35612 −0.678060 0.735006i \(-0.737179\pi\)
−0.678060 + 0.735006i \(0.737179\pi\)
\(648\) 0 0
\(649\) −77.4529 −3.04029
\(650\) 0 0
\(651\) −9.23623 −0.361996
\(652\) 0 0
\(653\) 36.1517 1.41472 0.707362 0.706852i \(-0.249885\pi\)
0.707362 + 0.706852i \(0.249885\pi\)
\(654\) 0 0
\(655\) −0.224144 −0.00875805
\(656\) 0 0
\(657\) −7.35690 −0.287020
\(658\) 0 0
\(659\) 6.81700 0.265553 0.132776 0.991146i \(-0.457611\pi\)
0.132776 + 0.991146i \(0.457611\pi\)
\(660\) 0 0
\(661\) 10.8944 0.423743 0.211871 0.977298i \(-0.432044\pi\)
0.211871 + 0.977298i \(0.432044\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 2.41683 0.0937206
\(666\) 0 0
\(667\) 49.5599 1.91897
\(668\) 0 0
\(669\) −14.2010 −0.549044
\(670\) 0 0
\(671\) −48.1377 −1.85833
\(672\) 0 0
\(673\) 20.7385 0.799412 0.399706 0.916643i \(-0.369112\pi\)
0.399706 + 0.916643i \(0.369112\pi\)
\(674\) 0 0
\(675\) 4.93900 0.190102
\(676\) 0 0
\(677\) −25.5786 −0.983067 −0.491534 0.870859i \(-0.663564\pi\)
−0.491534 + 0.870859i \(0.663564\pi\)
\(678\) 0 0
\(679\) 23.9982 0.920966
\(680\) 0 0
\(681\) −16.0073 −0.613401
\(682\) 0 0
\(683\) 21.6310 0.827688 0.413844 0.910348i \(-0.364186\pi\)
0.413844 + 0.910348i \(0.364186\pi\)
\(684\) 0 0
\(685\) 2.35822 0.0901031
\(686\) 0 0
\(687\) 1.84117 0.0702449
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 2.62996 0.100048 0.0500242 0.998748i \(-0.484070\pi\)
0.0500242 + 0.998748i \(0.484070\pi\)
\(692\) 0 0
\(693\) 9.90946 0.376429
\(694\) 0 0
\(695\) 1.01075 0.0383401
\(696\) 0 0
\(697\) 1.69202 0.0640899
\(698\) 0 0
\(699\) 23.4252 0.886022
\(700\) 0 0
\(701\) 40.0925 1.51427 0.757136 0.653258i \(-0.226598\pi\)
0.757136 + 0.653258i \(0.226598\pi\)
\(702\) 0 0
\(703\) −17.8519 −0.673298
\(704\) 0 0
\(705\) 1.66355 0.0626528
\(706\) 0 0
\(707\) −9.48593 −0.356755
\(708\) 0 0
\(709\) −23.2097 −0.871657 −0.435829 0.900030i \(-0.643545\pi\)
−0.435829 + 0.900030i \(0.643545\pi\)
\(710\) 0 0
\(711\) −4.45473 −0.167065
\(712\) 0 0
\(713\) −43.9667 −1.64657
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −14.6015 −0.545302
\(718\) 0 0
\(719\) 26.0146 0.970181 0.485090 0.874464i \(-0.338787\pi\)
0.485090 + 0.874464i \(0.338787\pi\)
\(720\) 0 0
\(721\) −24.1146 −0.898073
\(722\) 0 0
\(723\) 8.63102 0.320991
\(724\) 0 0
\(725\) 29.3327 1.08939
\(726\) 0 0
\(727\) 16.5472 0.613701 0.306851 0.951758i \(-0.400725\pi\)
0.306851 + 0.951758i \(0.400725\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 6.51573 0.240993
\(732\) 0 0
\(733\) −18.8750 −0.697165 −0.348582 0.937278i \(-0.613337\pi\)
−0.348582 + 0.937278i \(0.613337\pi\)
\(734\) 0 0
\(735\) 0.969869 0.0357742
\(736\) 0 0
\(737\) 33.6950 1.24117
\(738\) 0 0
\(739\) 47.3239 1.74084 0.870419 0.492312i \(-0.163848\pi\)
0.870419 + 0.492312i \(0.163848\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −8.88769 −0.326058 −0.163029 0.986621i \(-0.552126\pi\)
−0.163029 + 0.986621i \(0.552126\pi\)
\(744\) 0 0
\(745\) 3.80061 0.139244
\(746\) 0 0
\(747\) −10.1860 −0.372686
\(748\) 0 0
\(749\) 22.4685 0.820980
\(750\) 0 0
\(751\) −0.710808 −0.0259377 −0.0129689 0.999916i \(-0.504128\pi\)
−0.0129689 + 0.999916i \(0.504128\pi\)
\(752\) 0 0
\(753\) −3.80194 −0.138550
\(754\) 0 0
\(755\) −0.907542 −0.0330288
\(756\) 0 0
\(757\) 9.78554 0.355662 0.177831 0.984061i \(-0.443092\pi\)
0.177831 + 0.984061i \(0.443092\pi\)
\(758\) 0 0
\(759\) 47.1715 1.71222
\(760\) 0 0
\(761\) 18.8810 0.684435 0.342218 0.939621i \(-0.388822\pi\)
0.342218 + 0.939621i \(0.388822\pi\)
\(762\) 0 0
\(763\) −21.2935 −0.770877
\(764\) 0 0
\(765\) −0.939001 −0.0339497
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 12.4349 0.448413 0.224207 0.974542i \(-0.428021\pi\)
0.224207 + 0.974542i \(0.428021\pi\)
\(770\) 0 0
\(771\) 20.5961 0.741751
\(772\) 0 0
\(773\) −45.6746 −1.64280 −0.821400 0.570353i \(-0.806807\pi\)
−0.821400 + 0.570353i \(0.806807\pi\)
\(774\) 0 0
\(775\) −26.0224 −0.934751
\(776\) 0 0
\(777\) 5.60627 0.201124
\(778\) 0 0
\(779\) −2.48427 −0.0890082
\(780\) 0 0
\(781\) −32.3293 −1.15683
\(782\) 0 0
\(783\) 5.93900 0.212243
\(784\) 0 0
\(785\) −1.20477 −0.0430000
\(786\) 0 0
\(787\) −4.51871 −0.161075 −0.0805374 0.996752i \(-0.525664\pi\)
−0.0805374 + 0.996752i \(0.525664\pi\)
\(788\) 0 0
\(789\) 0.332733 0.0118456
\(790\) 0 0
\(791\) −2.87097 −0.102080
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0.262045 0.00929378
\(796\) 0 0
\(797\) −28.7391 −1.01799 −0.508996 0.860769i \(-0.669983\pi\)
−0.508996 + 0.860769i \(0.669983\pi\)
\(798\) 0 0
\(799\) 25.6082 0.905953
\(800\) 0 0
\(801\) −0.137063 −0.00484289
\(802\) 0 0
\(803\) −41.5870 −1.46757
\(804\) 0 0
\(805\) −3.61297 −0.127341
\(806\) 0 0
\(807\) −27.3032 −0.961118
\(808\) 0 0
\(809\) 5.42891 0.190870 0.0954352 0.995436i \(-0.469576\pi\)
0.0954352 + 0.995436i \(0.469576\pi\)
\(810\) 0 0
\(811\) 0.629104 0.0220908 0.0110454 0.999939i \(-0.496484\pi\)
0.0110454 + 0.999939i \(0.496484\pi\)
\(812\) 0 0
\(813\) −27.9855 −0.981495
\(814\) 0 0
\(815\) 2.13169 0.0746697
\(816\) 0 0
\(817\) −9.56657 −0.334692
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −36.2640 −1.26562 −0.632811 0.774307i \(-0.718099\pi\)
−0.632811 + 0.774307i \(0.718099\pi\)
\(822\) 0 0
\(823\) −41.7396 −1.45495 −0.727476 0.686133i \(-0.759307\pi\)
−0.727476 + 0.686133i \(0.759307\pi\)
\(824\) 0 0
\(825\) 27.9191 0.972020
\(826\) 0 0
\(827\) 38.1997 1.32833 0.664167 0.747584i \(-0.268786\pi\)
0.664167 + 0.747584i \(0.268786\pi\)
\(828\) 0 0
\(829\) 15.9788 0.554967 0.277484 0.960730i \(-0.410500\pi\)
0.277484 + 0.960730i \(0.410500\pi\)
\(830\) 0 0
\(831\) 2.10321 0.0729596
\(832\) 0 0
\(833\) 14.9299 0.517290
\(834\) 0 0
\(835\) 2.33885 0.0809391
\(836\) 0 0
\(837\) −5.26875 −0.182115
\(838\) 0 0
\(839\) −4.63879 −0.160149 −0.0800744 0.996789i \(-0.525516\pi\)
−0.0800744 + 0.996789i \(0.525516\pi\)
\(840\) 0 0
\(841\) 6.27173 0.216267
\(842\) 0 0
\(843\) 27.2349 0.938020
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 36.7329 1.26216
\(848\) 0 0
\(849\) 5.28382 0.181340
\(850\) 0 0
\(851\) 26.6872 0.914827
\(852\) 0 0
\(853\) −18.3884 −0.629605 −0.314803 0.949157i \(-0.601938\pi\)
−0.314803 + 0.949157i \(0.601938\pi\)
\(854\) 0 0
\(855\) 1.37867 0.0471494
\(856\) 0 0
\(857\) 28.1849 0.962778 0.481389 0.876507i \(-0.340132\pi\)
0.481389 + 0.876507i \(0.340132\pi\)
\(858\) 0 0
\(859\) −33.3957 −1.13944 −0.569722 0.821837i \(-0.692949\pi\)
−0.569722 + 0.821837i \(0.692949\pi\)
\(860\) 0 0
\(861\) 0.780167 0.0265880
\(862\) 0 0
\(863\) −43.0640 −1.46592 −0.732958 0.680274i \(-0.761861\pi\)
−0.732958 + 0.680274i \(0.761861\pi\)
\(864\) 0 0
\(865\) 1.17928 0.0400966
\(866\) 0 0
\(867\) 2.54527 0.0864419
\(868\) 0 0
\(869\) −25.1817 −0.854230
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 13.6896 0.463324
\(874\) 0 0
\(875\) −4.30319 −0.145474
\(876\) 0 0
\(877\) 30.1702 1.01877 0.509387 0.860537i \(-0.329872\pi\)
0.509387 + 0.860537i \(0.329872\pi\)
\(878\) 0 0
\(879\) 32.6625 1.10168
\(880\) 0 0
\(881\) −3.56273 −0.120031 −0.0600157 0.998197i \(-0.519115\pi\)
−0.0600157 + 0.998197i \(0.519115\pi\)
\(882\) 0 0
\(883\) 10.2088 0.343554 0.171777 0.985136i \(-0.445049\pi\)
0.171777 + 0.985136i \(0.445049\pi\)
\(884\) 0 0
\(885\) 3.38404 0.113753
\(886\) 0 0
\(887\) 9.33645 0.313487 0.156744 0.987639i \(-0.449900\pi\)
0.156744 + 0.987639i \(0.449900\pi\)
\(888\) 0 0
\(889\) −18.9318 −0.634953
\(890\) 0 0
\(891\) 5.65279 0.189376
\(892\) 0 0
\(893\) −37.5986 −1.25819
\(894\) 0 0
\(895\) −0.848462 −0.0283610
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −31.2911 −1.04362
\(900\) 0 0
\(901\) 4.03385 0.134387
\(902\) 0 0
\(903\) 3.00431 0.0999772
\(904\) 0 0
\(905\) −3.33081 −0.110720
\(906\) 0 0
\(907\) 41.0804 1.36405 0.682026 0.731328i \(-0.261099\pi\)
0.682026 + 0.731328i \(0.261099\pi\)
\(908\) 0 0
\(909\) −5.41119 −0.179478
\(910\) 0 0
\(911\) 18.9705 0.628519 0.314260 0.949337i \(-0.398244\pi\)
0.314260 + 0.949337i \(0.398244\pi\)
\(912\) 0 0
\(913\) −57.5792 −1.90559
\(914\) 0 0
\(915\) 2.10321 0.0695300
\(916\) 0 0
\(917\) −1.59094 −0.0525375
\(918\) 0 0
\(919\) −29.0019 −0.956685 −0.478343 0.878173i \(-0.658762\pi\)
−0.478343 + 0.878173i \(0.658762\pi\)
\(920\) 0 0
\(921\) 20.7614 0.684111
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 15.7952 0.519344
\(926\) 0 0
\(927\) −13.7560 −0.451806
\(928\) 0 0
\(929\) 50.7211 1.66410 0.832052 0.554697i \(-0.187166\pi\)
0.832052 + 0.554697i \(0.187166\pi\)
\(930\) 0 0
\(931\) −21.9205 −0.718415
\(932\) 0 0
\(933\) −11.3013 −0.369987
\(934\) 0 0
\(935\) −5.30798 −0.173589
\(936\) 0 0
\(937\) 51.3051 1.67606 0.838032 0.545620i \(-0.183706\pi\)
0.838032 + 0.545620i \(0.183706\pi\)
\(938\) 0 0
\(939\) 4.27173 0.139403
\(940\) 0 0
\(941\) 34.7036 1.13131 0.565653 0.824643i \(-0.308624\pi\)
0.565653 + 0.824643i \(0.308624\pi\)
\(942\) 0 0
\(943\) 3.71379 0.120938
\(944\) 0 0
\(945\) −0.432960 −0.0140842
\(946\) 0 0
\(947\) −13.0127 −0.422855 −0.211428 0.977394i \(-0.567811\pi\)
−0.211428 + 0.977394i \(0.567811\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 15.4776 0.501894
\(952\) 0 0
\(953\) 26.0151 0.842711 0.421355 0.906896i \(-0.361555\pi\)
0.421355 + 0.906896i \(0.361555\pi\)
\(954\) 0 0
\(955\) 0.321388 0.0103999
\(956\) 0 0
\(957\) 33.5719 1.08523
\(958\) 0 0
\(959\) 16.7383 0.540507
\(960\) 0 0
\(961\) −3.24027 −0.104525
\(962\) 0 0
\(963\) 12.8170 0.413022
\(964\) 0 0
\(965\) 2.27114 0.0731107
\(966\) 0 0
\(967\) 36.6644 1.17905 0.589524 0.807751i \(-0.299315\pi\)
0.589524 + 0.807751i \(0.299315\pi\)
\(968\) 0 0
\(969\) 21.2228 0.681775
\(970\) 0 0
\(971\) 37.8465 1.21455 0.607277 0.794490i \(-0.292262\pi\)
0.607277 + 0.794490i \(0.292262\pi\)
\(972\) 0 0
\(973\) 7.17416 0.229993
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −28.8998 −0.924586 −0.462293 0.886727i \(-0.652973\pi\)
−0.462293 + 0.886727i \(0.652973\pi\)
\(978\) 0 0
\(979\) −0.774791 −0.0247624
\(980\) 0 0
\(981\) −12.1468 −0.387816
\(982\) 0 0
\(983\) −19.3991 −0.618735 −0.309368 0.950942i \(-0.600117\pi\)
−0.309368 + 0.950942i \(0.600117\pi\)
\(984\) 0 0
\(985\) −1.01566 −0.0323615
\(986\) 0 0
\(987\) 11.8076 0.375839
\(988\) 0 0
\(989\) 14.3013 0.454754
\(990\) 0 0
\(991\) 5.18300 0.164643 0.0823217 0.996606i \(-0.473767\pi\)
0.0823217 + 0.996606i \(0.473767\pi\)
\(992\) 0 0
\(993\) −6.06829 −0.192572
\(994\) 0 0
\(995\) 6.11828 0.193962
\(996\) 0 0
\(997\) −49.3642 −1.56338 −0.781690 0.623667i \(-0.785642\pi\)
−0.781690 + 0.623667i \(0.785642\pi\)
\(998\) 0 0
\(999\) 3.19806 0.101182
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.1.3 3
4.3 odd 2 507.2.a.j.1.2 3
12.11 even 2 1521.2.a.q.1.2 3
13.12 even 2 8112.2.a.cf.1.1 3
52.3 odd 6 507.2.e.k.22.2 6
52.7 even 12 507.2.j.h.361.3 12
52.11 even 12 507.2.j.h.316.4 12
52.15 even 12 507.2.j.h.316.3 12
52.19 even 12 507.2.j.h.361.4 12
52.23 odd 6 507.2.e.j.22.2 6
52.31 even 4 507.2.b.g.337.4 6
52.35 odd 6 507.2.e.k.484.2 6
52.43 odd 6 507.2.e.j.484.2 6
52.47 even 4 507.2.b.g.337.3 6
52.51 odd 2 507.2.a.k.1.2 yes 3
156.47 odd 4 1521.2.b.m.1351.4 6
156.83 odd 4 1521.2.b.m.1351.3 6
156.155 even 2 1521.2.a.p.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.2.a.j.1.2 3 4.3 odd 2
507.2.a.k.1.2 yes 3 52.51 odd 2
507.2.b.g.337.3 6 52.47 even 4
507.2.b.g.337.4 6 52.31 even 4
507.2.e.j.22.2 6 52.23 odd 6
507.2.e.j.484.2 6 52.43 odd 6
507.2.e.k.22.2 6 52.3 odd 6
507.2.e.k.484.2 6 52.35 odd 6
507.2.j.h.316.3 12 52.15 even 12
507.2.j.h.316.4 12 52.11 even 12
507.2.j.h.361.3 12 52.7 even 12
507.2.j.h.361.4 12 52.19 even 12
1521.2.a.p.1.2 3 156.155 even 2
1521.2.a.q.1.2 3 12.11 even 2
1521.2.b.m.1351.3 6 156.83 odd 4
1521.2.b.m.1351.4 6 156.47 odd 4
8112.2.a.by.1.3 3 1.1 even 1 trivial
8112.2.a.cf.1.1 3 13.12 even 2