Properties

Label 8112.2.a.cp.1.3
Level $8112$
Weight $2$
Character 8112.1
Self dual yes
Analytic conductor $64.775$
Analytic rank $0$
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: \(0\)
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

\(f(q)\) \(=\) \(q+1.00000 q^{3} +3.69202 q^{5} +0.801938 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +3.69202 q^{5} +0.801938 q^{7} +1.00000 q^{9} +2.85086 q^{11} +3.69202 q^{15} +2.93900 q^{17} +2.44504 q^{19} +0.801938 q^{21} +7.78986 q^{23} +8.63102 q^{25} +1.00000 q^{27} +3.85086 q^{29} -2.34481 q^{31} +2.85086 q^{33} +2.96077 q^{35} +7.44504 q^{37} -0.850855 q^{41} +1.61596 q^{43} +3.69202 q^{45} -2.44504 q^{47} -6.35690 q^{49} +2.93900 q^{51} -9.96077 q^{53} +10.5254 q^{55} +2.44504 q^{57} +5.38404 q^{59} -13.2567 q^{61} +0.801938 q^{63} -14.3937 q^{67} +7.78986 q^{69} -8.12498 q^{71} -11.8877 q^{73} +8.63102 q^{75} +2.28621 q^{77} -5.40581 q^{79} +1.00000 q^{81} -7.04892 q^{83} +10.8509 q^{85} +3.85086 q^{87} +1.13169 q^{89} -2.34481 q^{93} +9.02715 q^{95} -5.94438 q^{97} +2.85086 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{3} + 6q^{5} - 2q^{7} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{3} + 6q^{5} - 2q^{7} + 3q^{9} - 5q^{11} + 6q^{15} - q^{17} + 7q^{19} - 2q^{21} + 11q^{25} + 3q^{27} - 2q^{29} + 16q^{31} - 5q^{33} - 4q^{35} + 22q^{37} + 11q^{41} + 15q^{43} + 6q^{45} - 7q^{47} - 15q^{49} - q^{51} - 17q^{53} - 3q^{55} + 7q^{57} + 6q^{59} - 13q^{61} - 2q^{63} - 11q^{67} + 6q^{73} + 11q^{75} + 15q^{77} - 3q^{79} + 3q^{81} - 12q^{83} + 19q^{85} - 2q^{87} + q^{89} + 16q^{93} + 21q^{95} - 5q^{97} - 5q^{99} + O(q^{100}) \)

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
\(5\) 3.69202 1.65112 0.825561 0.564313i \(-0.190859\pi\)
0.825561 + 0.564313i \(0.190859\pi\)
\(6\) 0 0
\(7\) 0.801938 0.303104 0.151552 0.988449i \(-0.451573\pi\)
0.151552 + 0.988449i \(0.451573\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.85086 0.859565 0.429783 0.902932i \(-0.358590\pi\)
0.429783 + 0.902932i \(0.358590\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) 3.69202 0.953276
\(16\) 0 0
\(17\) 2.93900 0.712812 0.356406 0.934331i \(-0.384002\pi\)
0.356406 + 0.934331i \(0.384002\pi\)
\(18\) 0 0
\(19\) 2.44504 0.560931 0.280466 0.959864i \(-0.409511\pi\)
0.280466 + 0.959864i \(0.409511\pi\)
\(20\) 0 0
\(21\) 0.801938 0.174997
\(22\) 0 0
\(23\) 7.78986 1.62430 0.812149 0.583451i \(-0.198298\pi\)
0.812149 + 0.583451i \(0.198298\pi\)
\(24\) 0 0
\(25\) 8.63102 1.72620
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 3.85086 0.715086 0.357543 0.933897i \(-0.383615\pi\)
0.357543 + 0.933897i \(0.383615\pi\)
\(30\) 0 0
\(31\) −2.34481 −0.421141 −0.210571 0.977579i \(-0.567532\pi\)
−0.210571 + 0.977579i \(0.567532\pi\)
\(32\) 0 0
\(33\) 2.85086 0.496270
\(34\) 0 0
\(35\) 2.96077 0.500462
\(36\) 0 0
\(37\) 7.44504 1.22396 0.611979 0.790874i \(-0.290374\pi\)
0.611979 + 0.790874i \(0.290374\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −0.850855 −0.132881 −0.0664406 0.997790i \(-0.521164\pi\)
−0.0664406 + 0.997790i \(0.521164\pi\)
\(42\) 0 0
\(43\) 1.61596 0.246431 0.123216 0.992380i \(-0.460679\pi\)
0.123216 + 0.992380i \(0.460679\pi\)
\(44\) 0 0
\(45\) 3.69202 0.550374
\(46\) 0 0
\(47\) −2.44504 −0.356646 −0.178323 0.983972i \(-0.557067\pi\)
−0.178323 + 0.983972i \(0.557067\pi\)
\(48\) 0 0
\(49\) −6.35690 −0.908128
\(50\) 0 0
\(51\) 2.93900 0.411542
\(52\) 0 0
\(53\) −9.96077 −1.36822 −0.684109 0.729380i \(-0.739809\pi\)
−0.684109 + 0.729380i \(0.739809\pi\)
\(54\) 0 0
\(55\) 10.5254 1.41925
\(56\) 0 0
\(57\) 2.44504 0.323854
\(58\) 0 0
\(59\) 5.38404 0.700943 0.350471 0.936573i \(-0.386021\pi\)
0.350471 + 0.936573i \(0.386021\pi\)
\(60\) 0 0
\(61\) −13.2567 −1.69734 −0.848671 0.528921i \(-0.822597\pi\)
−0.848671 + 0.528921i \(0.822597\pi\)
\(62\) 0 0
\(63\) 0.801938 0.101035
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −14.3937 −1.75847 −0.879237 0.476384i \(-0.841947\pi\)
−0.879237 + 0.476384i \(0.841947\pi\)
\(68\) 0 0
\(69\) 7.78986 0.937788
\(70\) 0 0
\(71\) −8.12498 −0.964258 −0.482129 0.876100i \(-0.660136\pi\)
−0.482129 + 0.876100i \(0.660136\pi\)
\(72\) 0 0
\(73\) −11.8877 −1.39135 −0.695674 0.718357i \(-0.744894\pi\)
−0.695674 + 0.718357i \(0.744894\pi\)
\(74\) 0 0
\(75\) 8.63102 0.996625
\(76\) 0 0
\(77\) 2.28621 0.260538
\(78\) 0 0
\(79\) −5.40581 −0.608202 −0.304101 0.952640i \(-0.598356\pi\)
−0.304101 + 0.952640i \(0.598356\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −7.04892 −0.773719 −0.386860 0.922139i \(-0.626440\pi\)
−0.386860 + 0.922139i \(0.626440\pi\)
\(84\) 0 0
\(85\) 10.8509 1.17694
\(86\) 0 0
\(87\) 3.85086 0.412855
\(88\) 0 0
\(89\) 1.13169 0.119959 0.0599793 0.998200i \(-0.480897\pi\)
0.0599793 + 0.998200i \(0.480897\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −2.34481 −0.243146
\(94\) 0 0
\(95\) 9.02715 0.926166
\(96\) 0 0
\(97\) −5.94438 −0.603560 −0.301780 0.953378i \(-0.597581\pi\)
−0.301780 + 0.953378i \(0.597581\pi\)
\(98\) 0 0
\(99\) 2.85086 0.286522
\(100\) 0 0
\(101\) 4.62565 0.460269 0.230134 0.973159i \(-0.426083\pi\)
0.230134 + 0.973159i \(0.426083\pi\)
\(102\) 0 0
\(103\) −1.20775 −0.119003 −0.0595016 0.998228i \(-0.518951\pi\)
−0.0595016 + 0.998228i \(0.518951\pi\)
\(104\) 0 0
\(105\) 2.96077 0.288942
\(106\) 0 0
\(107\) 9.52111 0.920440 0.460220 0.887805i \(-0.347771\pi\)
0.460220 + 0.887805i \(0.347771\pi\)
\(108\) 0 0
\(109\) −1.78448 −0.170922 −0.0854611 0.996342i \(-0.527236\pi\)
−0.0854611 + 0.996342i \(0.527236\pi\)
\(110\) 0 0
\(111\) 7.44504 0.706652
\(112\) 0 0
\(113\) 4.95108 0.465759 0.232879 0.972506i \(-0.425185\pi\)
0.232879 + 0.972506i \(0.425185\pi\)
\(114\) 0 0
\(115\) 28.7603 2.68191
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 2.35690 0.216056
\(120\) 0 0
\(121\) −2.87263 −0.261148
\(122\) 0 0
\(123\) −0.850855 −0.0767190
\(124\) 0 0
\(125\) 13.4058 1.19905
\(126\) 0 0
\(127\) −5.67025 −0.503153 −0.251577 0.967837i \(-0.580949\pi\)
−0.251577 + 0.967837i \(0.580949\pi\)
\(128\) 0 0
\(129\) 1.61596 0.142277
\(130\) 0 0
\(131\) −18.2228 −1.59213 −0.796067 0.605208i \(-0.793090\pi\)
−0.796067 + 0.605208i \(0.793090\pi\)
\(132\) 0 0
\(133\) 1.96077 0.170020
\(134\) 0 0
\(135\) 3.69202 0.317759
\(136\) 0 0
\(137\) 9.45042 0.807404 0.403702 0.914891i \(-0.367723\pi\)
0.403702 + 0.914891i \(0.367723\pi\)
\(138\) 0 0
\(139\) −4.01507 −0.340553 −0.170277 0.985396i \(-0.554466\pi\)
−0.170277 + 0.985396i \(0.554466\pi\)
\(140\) 0 0
\(141\) −2.44504 −0.205910
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 14.2174 1.18069
\(146\) 0 0
\(147\) −6.35690 −0.524308
\(148\) 0 0
\(149\) 19.4058 1.58979 0.794893 0.606750i \(-0.207527\pi\)
0.794893 + 0.606750i \(0.207527\pi\)
\(150\) 0 0
\(151\) 12.3623 1.00603 0.503014 0.864278i \(-0.332225\pi\)
0.503014 + 0.864278i \(0.332225\pi\)
\(152\) 0 0
\(153\) 2.93900 0.237604
\(154\) 0 0
\(155\) −8.65710 −0.695355
\(156\) 0 0
\(157\) −18.6775 −1.49063 −0.745315 0.666712i \(-0.767701\pi\)
−0.745315 + 0.666712i \(0.767701\pi\)
\(158\) 0 0
\(159\) −9.96077 −0.789941
\(160\) 0 0
\(161\) 6.24698 0.492331
\(162\) 0 0
\(163\) −12.3394 −0.966499 −0.483250 0.875483i \(-0.660544\pi\)
−0.483250 + 0.875483i \(0.660544\pi\)
\(164\) 0 0
\(165\) 10.5254 0.819403
\(166\) 0 0
\(167\) −11.4940 −0.889429 −0.444715 0.895672i \(-0.646695\pi\)
−0.444715 + 0.895672i \(0.646695\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 2.44504 0.186977
\(172\) 0 0
\(173\) 12.1142 0.921028 0.460514 0.887653i \(-0.347665\pi\)
0.460514 + 0.887653i \(0.347665\pi\)
\(174\) 0 0
\(175\) 6.92154 0.523219
\(176\) 0 0
\(177\) 5.38404 0.404689
\(178\) 0 0
\(179\) 0.538565 0.0402542 0.0201271 0.999797i \(-0.493593\pi\)
0.0201271 + 0.999797i \(0.493593\pi\)
\(180\) 0 0
\(181\) −23.2838 −1.73067 −0.865336 0.501192i \(-0.832895\pi\)
−0.865336 + 0.501192i \(0.832895\pi\)
\(182\) 0 0
\(183\) −13.2567 −0.979961
\(184\) 0 0
\(185\) 27.4873 2.02090
\(186\) 0 0
\(187\) 8.37867 0.612709
\(188\) 0 0
\(189\) 0.801938 0.0583324
\(190\) 0 0
\(191\) −16.7657 −1.21312 −0.606561 0.795037i \(-0.707452\pi\)
−0.606561 + 0.795037i \(0.707452\pi\)
\(192\) 0 0
\(193\) 25.7439 1.85309 0.926544 0.376186i \(-0.122765\pi\)
0.926544 + 0.376186i \(0.122765\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 21.4209 1.52617 0.763087 0.646296i \(-0.223683\pi\)
0.763087 + 0.646296i \(0.223683\pi\)
\(198\) 0 0
\(199\) −3.52781 −0.250080 −0.125040 0.992152i \(-0.539906\pi\)
−0.125040 + 0.992152i \(0.539906\pi\)
\(200\) 0 0
\(201\) −14.3937 −1.01526
\(202\) 0 0
\(203\) 3.08815 0.216745
\(204\) 0 0
\(205\) −3.14138 −0.219403
\(206\) 0 0
\(207\) 7.78986 0.541432
\(208\) 0 0
\(209\) 6.97046 0.482157
\(210\) 0 0
\(211\) −1.21552 −0.0836799 −0.0418399 0.999124i \(-0.513322\pi\)
−0.0418399 + 0.999124i \(0.513322\pi\)
\(212\) 0 0
\(213\) −8.12498 −0.556715
\(214\) 0 0
\(215\) 5.96615 0.406888
\(216\) 0 0
\(217\) −1.88040 −0.127650
\(218\) 0 0
\(219\) −11.8877 −0.803296
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 17.3884 1.16441 0.582205 0.813042i \(-0.302190\pi\)
0.582205 + 0.813042i \(0.302190\pi\)
\(224\) 0 0
\(225\) 8.63102 0.575402
\(226\) 0 0
\(227\) −17.4155 −1.15591 −0.577954 0.816070i \(-0.696149\pi\)
−0.577954 + 0.816070i \(0.696149\pi\)
\(228\) 0 0
\(229\) 18.7603 1.23972 0.619858 0.784714i \(-0.287190\pi\)
0.619858 + 0.784714i \(0.287190\pi\)
\(230\) 0 0
\(231\) 2.28621 0.150421
\(232\) 0 0
\(233\) 3.95108 0.258844 0.129422 0.991590i \(-0.458688\pi\)
0.129422 + 0.991590i \(0.458688\pi\)
\(234\) 0 0
\(235\) −9.02715 −0.588866
\(236\) 0 0
\(237\) −5.40581 −0.351145
\(238\) 0 0
\(239\) −0.818331 −0.0529334 −0.0264667 0.999650i \(-0.508426\pi\)
−0.0264667 + 0.999650i \(0.508426\pi\)
\(240\) 0 0
\(241\) 6.03252 0.388589 0.194295 0.980943i \(-0.437758\pi\)
0.194295 + 0.980943i \(0.437758\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −23.4698 −1.49943
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −7.04892 −0.446707
\(250\) 0 0
\(251\) 26.8799 1.69665 0.848323 0.529479i \(-0.177613\pi\)
0.848323 + 0.529479i \(0.177613\pi\)
\(252\) 0 0
\(253\) 22.2078 1.39619
\(254\) 0 0
\(255\) 10.8509 0.679507
\(256\) 0 0
\(257\) −9.05323 −0.564725 −0.282362 0.959308i \(-0.591118\pi\)
−0.282362 + 0.959308i \(0.591118\pi\)
\(258\) 0 0
\(259\) 5.97046 0.370986
\(260\) 0 0
\(261\) 3.85086 0.238362
\(262\) 0 0
\(263\) −23.1511 −1.42756 −0.713778 0.700372i \(-0.753017\pi\)
−0.713778 + 0.700372i \(0.753017\pi\)
\(264\) 0 0
\(265\) −36.7754 −2.25909
\(266\) 0 0
\(267\) 1.13169 0.0692581
\(268\) 0 0
\(269\) −2.42088 −0.147604 −0.0738018 0.997273i \(-0.523513\pi\)
−0.0738018 + 0.997273i \(0.523513\pi\)
\(270\) 0 0
\(271\) 21.4450 1.30269 0.651347 0.758780i \(-0.274204\pi\)
0.651347 + 0.758780i \(0.274204\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 24.6058 1.48379
\(276\) 0 0
\(277\) −14.8073 −0.889685 −0.444843 0.895609i \(-0.646740\pi\)
−0.444843 + 0.895609i \(0.646740\pi\)
\(278\) 0 0
\(279\) −2.34481 −0.140380
\(280\) 0 0
\(281\) 14.5036 0.865215 0.432608 0.901582i \(-0.357594\pi\)
0.432608 + 0.901582i \(0.357594\pi\)
\(282\) 0 0
\(283\) −25.6722 −1.52605 −0.763026 0.646368i \(-0.776287\pi\)
−0.763026 + 0.646368i \(0.776287\pi\)
\(284\) 0 0
\(285\) 9.02715 0.534722
\(286\) 0 0
\(287\) −0.682333 −0.0402768
\(288\) 0 0
\(289\) −8.36227 −0.491898
\(290\) 0 0
\(291\) −5.94438 −0.348466
\(292\) 0 0
\(293\) −26.5230 −1.54949 −0.774746 0.632273i \(-0.782122\pi\)
−0.774746 + 0.632273i \(0.782122\pi\)
\(294\) 0 0
\(295\) 19.8780 1.15734
\(296\) 0 0
\(297\) 2.85086 0.165423
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 1.29590 0.0746943
\(302\) 0 0
\(303\) 4.62565 0.265736
\(304\) 0 0
\(305\) −48.9439 −2.80252
\(306\) 0 0
\(307\) 8.24698 0.470680 0.235340 0.971913i \(-0.424380\pi\)
0.235340 + 0.971913i \(0.424380\pi\)
\(308\) 0 0
\(309\) −1.20775 −0.0687066
\(310\) 0 0
\(311\) 14.4179 0.817564 0.408782 0.912632i \(-0.365954\pi\)
0.408782 + 0.912632i \(0.365954\pi\)
\(312\) 0 0
\(313\) 14.2338 0.804544 0.402272 0.915520i \(-0.368221\pi\)
0.402272 + 0.915520i \(0.368221\pi\)
\(314\) 0 0
\(315\) 2.96077 0.166821
\(316\) 0 0
\(317\) −6.84415 −0.384406 −0.192203 0.981355i \(-0.561563\pi\)
−0.192203 + 0.981355i \(0.561563\pi\)
\(318\) 0 0
\(319\) 10.9782 0.614663
\(320\) 0 0
\(321\) 9.52111 0.531416
\(322\) 0 0
\(323\) 7.18598 0.399839
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −1.78448 −0.0986819
\(328\) 0 0
\(329\) −1.96077 −0.108101
\(330\) 0 0
\(331\) −9.44265 −0.519015 −0.259507 0.965741i \(-0.583560\pi\)
−0.259507 + 0.965741i \(0.583560\pi\)
\(332\) 0 0
\(333\) 7.44504 0.407986
\(334\) 0 0
\(335\) −53.1420 −2.90346
\(336\) 0 0
\(337\) 2.64310 0.143979 0.0719895 0.997405i \(-0.477065\pi\)
0.0719895 + 0.997405i \(0.477065\pi\)
\(338\) 0 0
\(339\) 4.95108 0.268906
\(340\) 0 0
\(341\) −6.68473 −0.361998
\(342\) 0 0
\(343\) −10.7114 −0.578361
\(344\) 0 0
\(345\) 28.7603 1.54840
\(346\) 0 0
\(347\) 10.1588 0.545355 0.272677 0.962106i \(-0.412091\pi\)
0.272677 + 0.962106i \(0.412091\pi\)
\(348\) 0 0
\(349\) 10.4397 0.558822 0.279411 0.960172i \(-0.409861\pi\)
0.279411 + 0.960172i \(0.409861\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 18.2911 0.973538 0.486769 0.873531i \(-0.338175\pi\)
0.486769 + 0.873531i \(0.338175\pi\)
\(354\) 0 0
\(355\) −29.9976 −1.59211
\(356\) 0 0
\(357\) 2.35690 0.124740
\(358\) 0 0
\(359\) 15.2731 0.806081 0.403041 0.915182i \(-0.367953\pi\)
0.403041 + 0.915182i \(0.367953\pi\)
\(360\) 0 0
\(361\) −13.0218 −0.685356
\(362\) 0 0
\(363\) −2.87263 −0.150774
\(364\) 0 0
\(365\) −43.8896 −2.29729
\(366\) 0 0
\(367\) 22.2717 1.16258 0.581288 0.813698i \(-0.302549\pi\)
0.581288 + 0.813698i \(0.302549\pi\)
\(368\) 0 0
\(369\) −0.850855 −0.0442937
\(370\) 0 0
\(371\) −7.98792 −0.414712
\(372\) 0 0
\(373\) 4.12631 0.213652 0.106826 0.994278i \(-0.465931\pi\)
0.106826 + 0.994278i \(0.465931\pi\)
\(374\) 0 0
\(375\) 13.4058 0.692273
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 10.7071 0.549986 0.274993 0.961446i \(-0.411324\pi\)
0.274993 + 0.961446i \(0.411324\pi\)
\(380\) 0 0
\(381\) −5.67025 −0.290496
\(382\) 0 0
\(383\) −6.52648 −0.333488 −0.166744 0.986000i \(-0.553325\pi\)
−0.166744 + 0.986000i \(0.553325\pi\)
\(384\) 0 0
\(385\) 8.44073 0.430179
\(386\) 0 0
\(387\) 1.61596 0.0821437
\(388\) 0 0
\(389\) −11.7922 −0.597891 −0.298945 0.954270i \(-0.596635\pi\)
−0.298945 + 0.954270i \(0.596635\pi\)
\(390\) 0 0
\(391\) 22.8944 1.15782
\(392\) 0 0
\(393\) −18.2228 −0.919219
\(394\) 0 0
\(395\) −19.9584 −1.00422
\(396\) 0 0
\(397\) −12.5429 −0.629509 −0.314754 0.949173i \(-0.601922\pi\)
−0.314754 + 0.949173i \(0.601922\pi\)
\(398\) 0 0
\(399\) 1.96077 0.0981613
\(400\) 0 0
\(401\) 17.8702 0.892397 0.446198 0.894934i \(-0.352778\pi\)
0.446198 + 0.894934i \(0.352778\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 3.69202 0.183458
\(406\) 0 0
\(407\) 21.2247 1.05207
\(408\) 0 0
\(409\) −27.9119 −1.38015 −0.690076 0.723737i \(-0.742423\pi\)
−0.690076 + 0.723737i \(0.742423\pi\)
\(410\) 0 0
\(411\) 9.45042 0.466155
\(412\) 0 0
\(413\) 4.31767 0.212459
\(414\) 0 0
\(415\) −26.0248 −1.27750
\(416\) 0 0
\(417\) −4.01507 −0.196619
\(418\) 0 0
\(419\) 16.4034 0.801360 0.400680 0.916218i \(-0.368774\pi\)
0.400680 + 0.916218i \(0.368774\pi\)
\(420\) 0 0
\(421\) −3.03684 −0.148006 −0.0740032 0.997258i \(-0.523577\pi\)
−0.0740032 + 0.997258i \(0.523577\pi\)
\(422\) 0 0
\(423\) −2.44504 −0.118882
\(424\) 0 0
\(425\) 25.3666 1.23046
\(426\) 0 0
\(427\) −10.6310 −0.514471
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 3.33811 0.160791 0.0803955 0.996763i \(-0.474382\pi\)
0.0803955 + 0.996763i \(0.474382\pi\)
\(432\) 0 0
\(433\) 11.9028 0.572010 0.286005 0.958228i \(-0.407673\pi\)
0.286005 + 0.958228i \(0.407673\pi\)
\(434\) 0 0
\(435\) 14.2174 0.681674
\(436\) 0 0
\(437\) 19.0465 0.911119
\(438\) 0 0
\(439\) −3.71810 −0.177455 −0.0887277 0.996056i \(-0.528280\pi\)
−0.0887277 + 0.996056i \(0.528280\pi\)
\(440\) 0 0
\(441\) −6.35690 −0.302709
\(442\) 0 0
\(443\) −1.45712 −0.0692300 −0.0346150 0.999401i \(-0.511021\pi\)
−0.0346150 + 0.999401i \(0.511021\pi\)
\(444\) 0 0
\(445\) 4.17821 0.198066
\(446\) 0 0
\(447\) 19.4058 0.917863
\(448\) 0 0
\(449\) −12.1274 −0.572326 −0.286163 0.958181i \(-0.592380\pi\)
−0.286163 + 0.958181i \(0.592380\pi\)
\(450\) 0 0
\(451\) −2.42566 −0.114220
\(452\) 0 0
\(453\) 12.3623 0.580830
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −3.44803 −0.161292 −0.0806459 0.996743i \(-0.525698\pi\)
−0.0806459 + 0.996743i \(0.525698\pi\)
\(458\) 0 0
\(459\) 2.93900 0.137181
\(460\) 0 0
\(461\) −6.75600 −0.314658 −0.157329 0.987546i \(-0.550288\pi\)
−0.157329 + 0.987546i \(0.550288\pi\)
\(462\) 0 0
\(463\) 7.45175 0.346312 0.173156 0.984894i \(-0.444604\pi\)
0.173156 + 0.984894i \(0.444604\pi\)
\(464\) 0 0
\(465\) −8.65710 −0.401464
\(466\) 0 0
\(467\) −32.6098 −1.50900 −0.754502 0.656298i \(-0.772121\pi\)
−0.754502 + 0.656298i \(0.772121\pi\)
\(468\) 0 0
\(469\) −11.5429 −0.533001
\(470\) 0 0
\(471\) −18.6775 −0.860616
\(472\) 0 0
\(473\) 4.60686 0.211824
\(474\) 0 0
\(475\) 21.1032 0.968282
\(476\) 0 0
\(477\) −9.96077 −0.456072
\(478\) 0 0
\(479\) −2.82908 −0.129264 −0.0646321 0.997909i \(-0.520587\pi\)
−0.0646321 + 0.997909i \(0.520587\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 6.24698 0.284247
\(484\) 0 0
\(485\) −21.9468 −0.996552
\(486\) 0 0
\(487\) 41.2935 1.87119 0.935594 0.353079i \(-0.114865\pi\)
0.935594 + 0.353079i \(0.114865\pi\)
\(488\) 0 0
\(489\) −12.3394 −0.558009
\(490\) 0 0
\(491\) 34.6698 1.56463 0.782313 0.622886i \(-0.214040\pi\)
0.782313 + 0.622886i \(0.214040\pi\)
\(492\) 0 0
\(493\) 11.3177 0.509722
\(494\) 0 0
\(495\) 10.5254 0.473082
\(496\) 0 0
\(497\) −6.51573 −0.292270
\(498\) 0 0
\(499\) −17.9409 −0.803146 −0.401573 0.915827i \(-0.631536\pi\)
−0.401573 + 0.915827i \(0.631536\pi\)
\(500\) 0 0
\(501\) −11.4940 −0.513512
\(502\) 0 0
\(503\) 26.1812 1.16736 0.583681 0.811983i \(-0.301612\pi\)
0.583681 + 0.811983i \(0.301612\pi\)
\(504\) 0 0
\(505\) 17.0780 0.759960
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −5.50604 −0.244051 −0.122025 0.992527i \(-0.538939\pi\)
−0.122025 + 0.992527i \(0.538939\pi\)
\(510\) 0 0
\(511\) −9.53319 −0.421723
\(512\) 0 0
\(513\) 2.44504 0.107951
\(514\) 0 0
\(515\) −4.45904 −0.196489
\(516\) 0 0
\(517\) −6.97046 −0.306560
\(518\) 0 0
\(519\) 12.1142 0.531756
\(520\) 0 0
\(521\) −26.7211 −1.17067 −0.585336 0.810791i \(-0.699037\pi\)
−0.585336 + 0.810791i \(0.699037\pi\)
\(522\) 0 0
\(523\) −36.5230 −1.59704 −0.798520 0.601968i \(-0.794383\pi\)
−0.798520 + 0.601968i \(0.794383\pi\)
\(524\) 0 0
\(525\) 6.92154 0.302081
\(526\) 0 0
\(527\) −6.89141 −0.300195
\(528\) 0 0
\(529\) 37.6819 1.63834
\(530\) 0 0
\(531\) 5.38404 0.233648
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 35.1521 1.51976
\(536\) 0 0
\(537\) 0.538565 0.0232408
\(538\) 0 0
\(539\) −18.1226 −0.780595
\(540\) 0 0
\(541\) 18.4655 0.793893 0.396947 0.917842i \(-0.370070\pi\)
0.396947 + 0.917842i \(0.370070\pi\)
\(542\) 0 0
\(543\) −23.2838 −0.999204
\(544\) 0 0
\(545\) −6.58834 −0.282213
\(546\) 0 0
\(547\) −39.8471 −1.70374 −0.851870 0.523753i \(-0.824532\pi\)
−0.851870 + 0.523753i \(0.824532\pi\)
\(548\) 0 0
\(549\) −13.2567 −0.565781
\(550\) 0 0
\(551\) 9.41550 0.401114
\(552\) 0 0
\(553\) −4.33513 −0.184348
\(554\) 0 0
\(555\) 27.4873 1.16677
\(556\) 0 0
\(557\) 9.20477 0.390018 0.195009 0.980801i \(-0.437526\pi\)
0.195009 + 0.980801i \(0.437526\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 8.37867 0.353748
\(562\) 0 0
\(563\) 0.975246 0.0411017 0.0205509 0.999789i \(-0.493458\pi\)
0.0205509 + 0.999789i \(0.493458\pi\)
\(564\) 0 0
\(565\) 18.2795 0.769024
\(566\) 0 0
\(567\) 0.801938 0.0336782
\(568\) 0 0
\(569\) −16.8944 −0.708250 −0.354125 0.935198i \(-0.615221\pi\)
−0.354125 + 0.935198i \(0.615221\pi\)
\(570\) 0 0
\(571\) 44.3226 1.85484 0.927421 0.374019i \(-0.122021\pi\)
0.927421 + 0.374019i \(0.122021\pi\)
\(572\) 0 0
\(573\) −16.7657 −0.700397
\(574\) 0 0
\(575\) 67.2344 2.80387
\(576\) 0 0
\(577\) 3.56704 0.148498 0.0742489 0.997240i \(-0.476344\pi\)
0.0742489 + 0.997240i \(0.476344\pi\)
\(578\) 0 0
\(579\) 25.7439 1.06988
\(580\) 0 0
\(581\) −5.65279 −0.234517
\(582\) 0 0
\(583\) −28.3967 −1.17607
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 16.1172 0.665229 0.332614 0.943063i \(-0.392069\pi\)
0.332614 + 0.943063i \(0.392069\pi\)
\(588\) 0 0
\(589\) −5.73317 −0.236231
\(590\) 0 0
\(591\) 21.4209 0.881137
\(592\) 0 0
\(593\) −42.8611 −1.76010 −0.880048 0.474885i \(-0.842490\pi\)
−0.880048 + 0.474885i \(0.842490\pi\)
\(594\) 0 0
\(595\) 8.70171 0.356735
\(596\) 0 0
\(597\) −3.52781 −0.144384
\(598\) 0 0
\(599\) −40.9420 −1.67284 −0.836422 0.548086i \(-0.815357\pi\)
−0.836422 + 0.548086i \(0.815357\pi\)
\(600\) 0 0
\(601\) 1.18705 0.0484206 0.0242103 0.999707i \(-0.492293\pi\)
0.0242103 + 0.999707i \(0.492293\pi\)
\(602\) 0 0
\(603\) −14.3937 −0.586158
\(604\) 0 0
\(605\) −10.6058 −0.431187
\(606\) 0 0
\(607\) −19.9922 −0.811460 −0.405730 0.913993i \(-0.632983\pi\)
−0.405730 + 0.913993i \(0.632983\pi\)
\(608\) 0 0
\(609\) 3.08815 0.125138
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −33.3618 −1.34747 −0.673735 0.738973i \(-0.735311\pi\)
−0.673735 + 0.738973i \(0.735311\pi\)
\(614\) 0 0
\(615\) −3.14138 −0.126672
\(616\) 0 0
\(617\) 11.6233 0.467935 0.233967 0.972244i \(-0.424829\pi\)
0.233967 + 0.972244i \(0.424829\pi\)
\(618\) 0 0
\(619\) 16.5381 0.664722 0.332361 0.943152i \(-0.392155\pi\)
0.332361 + 0.943152i \(0.392155\pi\)
\(620\) 0 0
\(621\) 7.78986 0.312596
\(622\) 0 0
\(623\) 0.907542 0.0363599
\(624\) 0 0
\(625\) 6.33944 0.253577
\(626\) 0 0
\(627\) 6.97046 0.278373
\(628\) 0 0
\(629\) 21.8810 0.872452
\(630\) 0 0
\(631\) 36.4416 1.45072 0.725358 0.688372i \(-0.241674\pi\)
0.725358 + 0.688372i \(0.241674\pi\)
\(632\) 0 0
\(633\) −1.21552 −0.0483126
\(634\) 0 0
\(635\) −20.9347 −0.830768
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −8.12498 −0.321419
\(640\) 0 0
\(641\) 27.2067 1.07460 0.537300 0.843391i \(-0.319444\pi\)
0.537300 + 0.843391i \(0.319444\pi\)
\(642\) 0 0
\(643\) 5.06962 0.199926 0.0999632 0.994991i \(-0.468127\pi\)
0.0999632 + 0.994991i \(0.468127\pi\)
\(644\) 0 0
\(645\) 5.96615 0.234917
\(646\) 0 0
\(647\) −19.3207 −0.759573 −0.379787 0.925074i \(-0.624003\pi\)
−0.379787 + 0.925074i \(0.624003\pi\)
\(648\) 0 0
\(649\) 15.3491 0.602506
\(650\) 0 0
\(651\) −1.88040 −0.0736985
\(652\) 0 0
\(653\) 35.2355 1.37887 0.689436 0.724347i \(-0.257859\pi\)
0.689436 + 0.724347i \(0.257859\pi\)
\(654\) 0 0
\(655\) −67.2790 −2.62881
\(656\) 0 0
\(657\) −11.8877 −0.463783
\(658\) 0 0
\(659\) 4.36168 0.169907 0.0849535 0.996385i \(-0.472926\pi\)
0.0849535 + 0.996385i \(0.472926\pi\)
\(660\) 0 0
\(661\) −15.4709 −0.601747 −0.300873 0.953664i \(-0.597278\pi\)
−0.300873 + 0.953664i \(0.597278\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 7.23921 0.280725
\(666\) 0 0
\(667\) 29.9976 1.16151
\(668\) 0 0
\(669\) 17.3884 0.672273
\(670\) 0 0
\(671\) −37.7928 −1.45898
\(672\) 0 0
\(673\) −11.7409 −0.452580 −0.226290 0.974060i \(-0.572660\pi\)
−0.226290 + 0.974060i \(0.572660\pi\)
\(674\) 0 0
\(675\) 8.63102 0.332208
\(676\) 0 0
\(677\) 3.44504 0.132404 0.0662019 0.997806i \(-0.478912\pi\)
0.0662019 + 0.997806i \(0.478912\pi\)
\(678\) 0 0
\(679\) −4.76702 −0.182941
\(680\) 0 0
\(681\) −17.4155 −0.667363
\(682\) 0 0
\(683\) −20.4058 −0.780807 −0.390403 0.920644i \(-0.627664\pi\)
−0.390403 + 0.920644i \(0.627664\pi\)
\(684\) 0 0
\(685\) 34.8911 1.33312
\(686\) 0 0
\(687\) 18.7603 0.715751
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −27.8039 −1.05771 −0.528854 0.848713i \(-0.677378\pi\)
−0.528854 + 0.848713i \(0.677378\pi\)
\(692\) 0 0
\(693\) 2.28621 0.0868459
\(694\) 0 0
\(695\) −14.8237 −0.562295
\(696\) 0 0
\(697\) −2.50066 −0.0947194
\(698\) 0 0
\(699\) 3.95108 0.149444
\(700\) 0 0
\(701\) 11.9715 0.452158 0.226079 0.974109i \(-0.427409\pi\)
0.226079 + 0.974109i \(0.427409\pi\)
\(702\) 0 0
\(703\) 18.2034 0.686556
\(704\) 0 0
\(705\) −9.02715 −0.339982
\(706\) 0 0
\(707\) 3.70948 0.139509
\(708\) 0 0
\(709\) 32.2664 1.21179 0.605894 0.795545i \(-0.292815\pi\)
0.605894 + 0.795545i \(0.292815\pi\)
\(710\) 0 0
\(711\) −5.40581 −0.202734
\(712\) 0 0
\(713\) −18.2658 −0.684058
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −0.818331 −0.0305611
\(718\) 0 0
\(719\) −12.1086 −0.451574 −0.225787 0.974177i \(-0.572495\pi\)
−0.225787 + 0.974177i \(0.572495\pi\)
\(720\) 0 0
\(721\) −0.968541 −0.0360704
\(722\) 0 0
\(723\) 6.03252 0.224352
\(724\) 0 0
\(725\) 33.2368 1.23438
\(726\) 0 0
\(727\) 16.6200 0.616402 0.308201 0.951321i \(-0.400273\pi\)
0.308201 + 0.951321i \(0.400273\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 4.74930 0.175659
\(732\) 0 0
\(733\) 17.7912 0.657132 0.328566 0.944481i \(-0.393435\pi\)
0.328566 + 0.944481i \(0.393435\pi\)
\(734\) 0 0
\(735\) −23.4698 −0.865696
\(736\) 0 0
\(737\) −41.0344 −1.51152
\(738\) 0 0
\(739\) 27.3618 1.00652 0.503260 0.864135i \(-0.332134\pi\)
0.503260 + 0.864135i \(0.332134\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −8.38596 −0.307651 −0.153826 0.988098i \(-0.549159\pi\)
−0.153826 + 0.988098i \(0.549159\pi\)
\(744\) 0 0
\(745\) 71.6467 2.62493
\(746\) 0 0
\(747\) −7.04892 −0.257906
\(748\) 0 0
\(749\) 7.63533 0.278989
\(750\) 0 0
\(751\) 38.7778 1.41502 0.707511 0.706703i \(-0.249818\pi\)
0.707511 + 0.706703i \(0.249818\pi\)
\(752\) 0 0
\(753\) 26.8799 0.979559
\(754\) 0 0
\(755\) 45.6418 1.66107
\(756\) 0 0
\(757\) 12.9729 0.471506 0.235753 0.971813i \(-0.424244\pi\)
0.235753 + 0.971813i \(0.424244\pi\)
\(758\) 0 0
\(759\) 22.2078 0.806090
\(760\) 0 0
\(761\) −5.15585 −0.186899 −0.0934497 0.995624i \(-0.529789\pi\)
−0.0934497 + 0.995624i \(0.529789\pi\)
\(762\) 0 0
\(763\) −1.43104 −0.0518072
\(764\) 0 0
\(765\) 10.8509 0.392313
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −35.5013 −1.28021 −0.640104 0.768288i \(-0.721109\pi\)
−0.640104 + 0.768288i \(0.721109\pi\)
\(770\) 0 0
\(771\) −9.05323 −0.326044
\(772\) 0 0
\(773\) −6.15585 −0.221411 −0.110705 0.993853i \(-0.535311\pi\)
−0.110705 + 0.993853i \(0.535311\pi\)
\(774\) 0 0
\(775\) −20.2381 −0.726976
\(776\) 0 0
\(777\) 5.97046 0.214189
\(778\) 0 0
\(779\) −2.08038 −0.0745372
\(780\) 0 0
\(781\) −23.1631 −0.828843
\(782\) 0 0
\(783\) 3.85086 0.137618
\(784\) 0 0
\(785\) −68.9579 −2.46121
\(786\) 0 0
\(787\) −14.2107 −0.506558 −0.253279 0.967393i \(-0.581509\pi\)
−0.253279 + 0.967393i \(0.581509\pi\)
\(788\) 0 0
\(789\) −23.1511 −0.824200
\(790\) 0 0
\(791\) 3.97046 0.141173
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −36.7754 −1.30429
\(796\) 0 0
\(797\) 11.9022 0.421596 0.210798 0.977530i \(-0.432394\pi\)
0.210798 + 0.977530i \(0.432394\pi\)
\(798\) 0 0
\(799\) −7.18598 −0.254222
\(800\) 0 0
\(801\) 1.13169 0.0399862
\(802\) 0 0
\(803\) −33.8901 −1.19596
\(804\) 0 0
\(805\) 23.0640 0.812899
\(806\) 0 0
\(807\) −2.42088 −0.0852190
\(808\) 0 0
\(809\) 1.61596 0.0568140 0.0284070 0.999596i \(-0.490957\pi\)
0.0284070 + 0.999596i \(0.490957\pi\)
\(810\) 0 0
\(811\) 51.9657 1.82476 0.912381 0.409342i \(-0.134242\pi\)
0.912381 + 0.409342i \(0.134242\pi\)
\(812\) 0 0
\(813\) 21.4450 0.752110
\(814\) 0 0
\(815\) −45.5575 −1.59581
\(816\) 0 0
\(817\) 3.95108 0.138231
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 54.8327 1.91367 0.956837 0.290627i \(-0.0938638\pi\)
0.956837 + 0.290627i \(0.0938638\pi\)
\(822\) 0 0
\(823\) 46.8514 1.63314 0.816569 0.577247i \(-0.195873\pi\)
0.816569 + 0.577247i \(0.195873\pi\)
\(824\) 0 0
\(825\) 24.6058 0.856664
\(826\) 0 0
\(827\) −21.2021 −0.737270 −0.368635 0.929574i \(-0.620175\pi\)
−0.368635 + 0.929574i \(0.620175\pi\)
\(828\) 0 0
\(829\) 18.2972 0.635489 0.317744 0.948176i \(-0.397075\pi\)
0.317744 + 0.948176i \(0.397075\pi\)
\(830\) 0 0
\(831\) −14.8073 −0.513660
\(832\) 0 0
\(833\) −18.6829 −0.647325
\(834\) 0 0
\(835\) −42.4359 −1.46856
\(836\) 0 0
\(837\) −2.34481 −0.0810486
\(838\) 0 0
\(839\) −21.1414 −0.729881 −0.364941 0.931031i \(-0.618911\pi\)
−0.364941 + 0.931031i \(0.618911\pi\)
\(840\) 0 0
\(841\) −14.1709 −0.488652
\(842\) 0 0
\(843\) 14.5036 0.499532
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −2.30367 −0.0791549
\(848\) 0 0
\(849\) −25.6722 −0.881067
\(850\) 0 0
\(851\) 57.9958 1.98807
\(852\) 0 0
\(853\) −7.13036 −0.244139 −0.122069 0.992522i \(-0.538953\pi\)
−0.122069 + 0.992522i \(0.538953\pi\)
\(854\) 0 0
\(855\) 9.02715 0.308722
\(856\) 0 0
\(857\) −44.7741 −1.52945 −0.764726 0.644355i \(-0.777126\pi\)
−0.764726 + 0.644355i \(0.777126\pi\)
\(858\) 0 0
\(859\) −57.3782 −1.95772 −0.978859 0.204535i \(-0.934432\pi\)
−0.978859 + 0.204535i \(0.934432\pi\)
\(860\) 0 0
\(861\) −0.682333 −0.0232538
\(862\) 0 0
\(863\) 6.67563 0.227241 0.113621 0.993524i \(-0.463755\pi\)
0.113621 + 0.993524i \(0.463755\pi\)
\(864\) 0 0
\(865\) 44.7260 1.52073
\(866\) 0 0
\(867\) −8.36227 −0.283998
\(868\) 0 0
\(869\) −15.4112 −0.522789
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −5.94438 −0.201187
\(874\) 0 0
\(875\) 10.7506 0.363438
\(876\) 0 0
\(877\) 25.2983 0.854263 0.427131 0.904190i \(-0.359524\pi\)
0.427131 + 0.904190i \(0.359524\pi\)
\(878\) 0 0
\(879\) −26.5230 −0.894599
\(880\) 0 0
\(881\) −35.3787 −1.19194 −0.595969 0.803008i \(-0.703232\pi\)
−0.595969 + 0.803008i \(0.703232\pi\)
\(882\) 0 0
\(883\) 11.5851 0.389869 0.194935 0.980816i \(-0.437551\pi\)
0.194935 + 0.980816i \(0.437551\pi\)
\(884\) 0 0
\(885\) 19.8780 0.668192
\(886\) 0 0
\(887\) 27.2892 0.916281 0.458141 0.888880i \(-0.348516\pi\)
0.458141 + 0.888880i \(0.348516\pi\)
\(888\) 0 0
\(889\) −4.54719 −0.152508
\(890\) 0 0
\(891\) 2.85086 0.0955072
\(892\) 0 0
\(893\) −5.97823 −0.200054
\(894\) 0 0
\(895\) 1.98839 0.0664646
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −9.02954 −0.301152
\(900\) 0 0
\(901\) −29.2747 −0.975282
\(902\) 0 0
\(903\) 1.29590 0.0431247
\(904\) 0 0
\(905\) −85.9643 −2.85755
\(906\) 0 0
\(907\) 30.4219 1.01014 0.505072 0.863077i \(-0.331466\pi\)
0.505072 + 0.863077i \(0.331466\pi\)
\(908\) 0 0
\(909\) 4.62565 0.153423
\(910\) 0 0
\(911\) −53.5719 −1.77492 −0.887459 0.460887i \(-0.847531\pi\)
−0.887459 + 0.460887i \(0.847531\pi\)
\(912\) 0 0
\(913\) −20.0954 −0.665062
\(914\) 0 0
\(915\) −48.9439 −1.61804
\(916\) 0 0
\(917\) −14.6136 −0.482582
\(918\) 0 0
\(919\) 36.1672 1.19305 0.596523 0.802596i \(-0.296549\pi\)
0.596523 + 0.802596i \(0.296549\pi\)
\(920\) 0 0
\(921\) 8.24698 0.271747
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 64.2583 2.11280
\(926\) 0 0
\(927\) −1.20775 −0.0396677
\(928\) 0 0
\(929\) −13.3478 −0.437927 −0.218964 0.975733i \(-0.570268\pi\)
−0.218964 + 0.975733i \(0.570268\pi\)
\(930\) 0 0
\(931\) −15.5429 −0.509397
\(932\) 0 0
\(933\) 14.4179 0.472021
\(934\) 0 0
\(935\) 30.9342 1.01166
\(936\) 0 0
\(937\) 38.6872 1.26386 0.631928 0.775027i \(-0.282264\pi\)
0.631928 + 0.775027i \(0.282264\pi\)
\(938\) 0 0
\(939\) 14.2338 0.464504
\(940\) 0 0
\(941\) 35.7275 1.16468 0.582342 0.812944i \(-0.302136\pi\)
0.582342 + 0.812944i \(0.302136\pi\)
\(942\) 0 0
\(943\) −6.62804 −0.215839
\(944\) 0 0
\(945\) 2.96077 0.0963139
\(946\) 0 0
\(947\) 17.8436 0.579838 0.289919 0.957051i \(-0.406372\pi\)
0.289919 + 0.957051i \(0.406372\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −6.84415 −0.221937
\(952\) 0 0
\(953\) 20.1691 0.653342 0.326671 0.945138i \(-0.394073\pi\)
0.326671 + 0.945138i \(0.394073\pi\)
\(954\) 0 0
\(955\) −61.8993 −2.00301
\(956\) 0 0
\(957\) 10.9782 0.354876
\(958\) 0 0
\(959\) 7.57865 0.244727
\(960\) 0 0
\(961\) −25.5018 −0.822640
\(962\) 0 0
\(963\) 9.52111 0.306813
\(964\) 0 0
\(965\) 95.0471 3.05967
\(966\) 0 0
\(967\) 32.7894 1.05444 0.527218 0.849730i \(-0.323235\pi\)
0.527218 + 0.849730i \(0.323235\pi\)
\(968\) 0 0
\(969\) 7.18598 0.230847
\(970\) 0 0
\(971\) 26.9845 0.865973 0.432986 0.901401i \(-0.357460\pi\)
0.432986 + 0.901401i \(0.357460\pi\)
\(972\) 0 0
\(973\) −3.21983 −0.103223
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −16.9571 −0.542504 −0.271252 0.962508i \(-0.587438\pi\)
−0.271252 + 0.962508i \(0.587438\pi\)
\(978\) 0 0
\(979\) 3.22627 0.103112
\(980\) 0 0
\(981\) −1.78448 −0.0569740
\(982\) 0 0
\(983\) 32.6631 1.04179 0.520895 0.853621i \(-0.325598\pi\)
0.520895 + 0.853621i \(0.325598\pi\)
\(984\) 0 0
\(985\) 79.0863 2.51990
\(986\) 0 0
\(987\) −1.96077 −0.0624120
\(988\) 0 0
\(989\) 12.5881 0.400277
\(990\) 0 0
\(991\) −7.64310 −0.242791 −0.121396 0.992604i \(-0.538737\pi\)
−0.121396 + 0.992604i \(0.538737\pi\)
\(992\) 0 0
\(993\) −9.44265 −0.299653
\(994\) 0 0
\(995\) −13.0248 −0.412912
\(996\) 0 0
\(997\) 36.1256 1.14411 0.572054 0.820216i \(-0.306147\pi\)
0.572054 + 0.820216i \(0.306147\pi\)
\(998\) 0 0
\(999\) 7.44504 0.235551
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.cp.1.3 3
4.3 odd 2 507.2.a.l.1.2 yes 3
12.11 even 2 1521.2.a.n.1.2 3
13.12 even 2 8112.2.a.cg.1.1 3
52.3 odd 6 507.2.e.i.22.2 6
52.7 even 12 507.2.j.i.361.5 12
52.11 even 12 507.2.j.i.316.2 12
52.15 even 12 507.2.j.i.316.5 12
52.19 even 12 507.2.j.i.361.2 12
52.23 odd 6 507.2.e.l.22.2 6
52.31 even 4 507.2.b.f.337.2 6
52.35 odd 6 507.2.e.i.484.2 6
52.43 odd 6 507.2.e.l.484.2 6
52.47 even 4 507.2.b.f.337.5 6
52.51 odd 2 507.2.a.i.1.2 3
156.47 odd 4 1521.2.b.k.1351.2 6
156.83 odd 4 1521.2.b.k.1351.5 6
156.155 even 2 1521.2.a.s.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.2.a.i.1.2 3 52.51 odd 2
507.2.a.l.1.2 yes 3 4.3 odd 2
507.2.b.f.337.2 6 52.31 even 4
507.2.b.f.337.5 6 52.47 even 4
507.2.e.i.22.2 6 52.3 odd 6
507.2.e.i.484.2 6 52.35 odd 6
507.2.e.l.22.2 6 52.23 odd 6
507.2.e.l.484.2 6 52.43 odd 6
507.2.j.i.316.2 12 52.11 even 12
507.2.j.i.316.5 12 52.15 even 12
507.2.j.i.361.2 12 52.19 even 12
507.2.j.i.361.5 12 52.7 even 12
1521.2.a.n.1.2 3 12.11 even 2
1521.2.a.s.1.2 3 156.155 even 2
1521.2.b.k.1351.2 6 156.47 odd 4
1521.2.b.k.1351.5 6 156.83 odd 4
8112.2.a.cg.1.1 3 13.12 even 2
8112.2.a.cp.1.3 3 1.1 even 1 trivial