Properties

Label 8008.2.a.n.1.9
Level 8008
Weight 2
Character 8008.1
Self dual yes
Analytic conductor 63.944
Analytic rank 0
Dimension 9
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8008.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.9442019386\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(-2.82774\)
Character \(\chi\) = 8008.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.82774 q^{3} +0.233208 q^{5} -1.00000 q^{7} +4.99612 q^{9} +O(q^{10})\) \(q+2.82774 q^{3} +0.233208 q^{5} -1.00000 q^{7} +4.99612 q^{9} +1.00000 q^{11} -1.00000 q^{13} +0.659452 q^{15} -1.37647 q^{17} -2.67708 q^{19} -2.82774 q^{21} -3.07457 q^{23} -4.94561 q^{25} +5.64452 q^{27} +9.22098 q^{29} +6.01875 q^{31} +2.82774 q^{33} -0.233208 q^{35} -4.26255 q^{37} -2.82774 q^{39} +8.07364 q^{41} +6.43767 q^{43} +1.16514 q^{45} +11.0424 q^{47} +1.00000 q^{49} -3.89232 q^{51} +3.87764 q^{53} +0.233208 q^{55} -7.57010 q^{57} -9.61739 q^{59} +10.7387 q^{61} -4.99612 q^{63} -0.233208 q^{65} +11.2232 q^{67} -8.69409 q^{69} +0.633784 q^{71} +6.61461 q^{73} -13.9849 q^{75} -1.00000 q^{77} +1.86994 q^{79} +0.972871 q^{81} +11.0882 q^{83} -0.321005 q^{85} +26.0746 q^{87} +12.4318 q^{89} +1.00000 q^{91} +17.0195 q^{93} -0.624318 q^{95} +5.54876 q^{97} +4.99612 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q - 3q^{3} + q^{5} - 9q^{7} + 12q^{9} + O(q^{10}) \) \( 9q - 3q^{3} + q^{5} - 9q^{7} + 12q^{9} + 9q^{11} - 9q^{13} + 11q^{15} + 7q^{17} - 17q^{19} + 3q^{21} + 11q^{23} + 18q^{25} - 9q^{27} + 9q^{29} - 8q^{31} - 3q^{33} - q^{35} + 2q^{37} + 3q^{39} + 18q^{41} + 7q^{43} + 5q^{45} + 15q^{47} + 9q^{49} - 7q^{51} - 4q^{53} + q^{55} + 22q^{57} - 23q^{59} + 12q^{61} - 12q^{63} - q^{65} - 16q^{67} - 32q^{69} - 6q^{71} + 4q^{73} - 14q^{75} - 9q^{77} + 21q^{79} + 5q^{81} - 16q^{83} + 53q^{85} + 41q^{87} + 5q^{89} + 9q^{91} + 29q^{93} + 19q^{95} + 18q^{97} + 12q^{99} + O(q^{100}) \)

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
\(3\) 2.82774 1.63260 0.816299 0.577630i \(-0.196022\pi\)
0.816299 + 0.577630i \(0.196022\pi\)
\(4\) 0 0
\(5\) 0.233208 0.104294 0.0521469 0.998639i \(-0.483394\pi\)
0.0521469 + 0.998639i \(0.483394\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 4.99612 1.66537
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 0.659452 0.170270
\(16\) 0 0
\(17\) −1.37647 −0.333844 −0.166922 0.985970i \(-0.553383\pi\)
−0.166922 + 0.985970i \(0.553383\pi\)
\(18\) 0 0
\(19\) −2.67708 −0.614165 −0.307083 0.951683i \(-0.599353\pi\)
−0.307083 + 0.951683i \(0.599353\pi\)
\(20\) 0 0
\(21\) −2.82774 −0.617064
\(22\) 0 0
\(23\) −3.07457 −0.641093 −0.320546 0.947233i \(-0.603866\pi\)
−0.320546 + 0.947233i \(0.603866\pi\)
\(24\) 0 0
\(25\) −4.94561 −0.989123
\(26\) 0 0
\(27\) 5.64452 1.08629
\(28\) 0 0
\(29\) 9.22098 1.71229 0.856147 0.516733i \(-0.172852\pi\)
0.856147 + 0.516733i \(0.172852\pi\)
\(30\) 0 0
\(31\) 6.01875 1.08100 0.540500 0.841344i \(-0.318235\pi\)
0.540500 + 0.841344i \(0.318235\pi\)
\(32\) 0 0
\(33\) 2.82774 0.492247
\(34\) 0 0
\(35\) −0.233208 −0.0394194
\(36\) 0 0
\(37\) −4.26255 −0.700759 −0.350379 0.936608i \(-0.613947\pi\)
−0.350379 + 0.936608i \(0.613947\pi\)
\(38\) 0 0
\(39\) −2.82774 −0.452801
\(40\) 0 0
\(41\) 8.07364 1.26089 0.630445 0.776234i \(-0.282872\pi\)
0.630445 + 0.776234i \(0.282872\pi\)
\(42\) 0 0
\(43\) 6.43767 0.981735 0.490868 0.871234i \(-0.336680\pi\)
0.490868 + 0.871234i \(0.336680\pi\)
\(44\) 0 0
\(45\) 1.16514 0.173688
\(46\) 0 0
\(47\) 11.0424 1.61070 0.805350 0.592799i \(-0.201977\pi\)
0.805350 + 0.592799i \(0.201977\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −3.89232 −0.545033
\(52\) 0 0
\(53\) 3.87764 0.532635 0.266318 0.963885i \(-0.414193\pi\)
0.266318 + 0.963885i \(0.414193\pi\)
\(54\) 0 0
\(55\) 0.233208 0.0314458
\(56\) 0 0
\(57\) −7.57010 −1.00268
\(58\) 0 0
\(59\) −9.61739 −1.25208 −0.626039 0.779792i \(-0.715325\pi\)
−0.626039 + 0.779792i \(0.715325\pi\)
\(60\) 0 0
\(61\) 10.7387 1.37495 0.687475 0.726208i \(-0.258719\pi\)
0.687475 + 0.726208i \(0.258719\pi\)
\(62\) 0 0
\(63\) −4.99612 −0.629452
\(64\) 0 0
\(65\) −0.233208 −0.0289259
\(66\) 0 0
\(67\) 11.2232 1.37113 0.685566 0.728011i \(-0.259555\pi\)
0.685566 + 0.728011i \(0.259555\pi\)
\(68\) 0 0
\(69\) −8.69409 −1.04665
\(70\) 0 0
\(71\) 0.633784 0.0752163 0.0376082 0.999293i \(-0.488026\pi\)
0.0376082 + 0.999293i \(0.488026\pi\)
\(72\) 0 0
\(73\) 6.61461 0.774182 0.387091 0.922042i \(-0.373480\pi\)
0.387091 + 0.922042i \(0.373480\pi\)
\(74\) 0 0
\(75\) −13.9849 −1.61484
\(76\) 0 0
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) 1.86994 0.210384 0.105192 0.994452i \(-0.466454\pi\)
0.105192 + 0.994452i \(0.466454\pi\)
\(80\) 0 0
\(81\) 0.972871 0.108097
\(82\) 0 0
\(83\) 11.0882 1.21709 0.608543 0.793521i \(-0.291754\pi\)
0.608543 + 0.793521i \(0.291754\pi\)
\(84\) 0 0
\(85\) −0.321005 −0.0348179
\(86\) 0 0
\(87\) 26.0746 2.79549
\(88\) 0 0
\(89\) 12.4318 1.31777 0.658886 0.752243i \(-0.271028\pi\)
0.658886 + 0.752243i \(0.271028\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 0 0
\(93\) 17.0195 1.76484
\(94\) 0 0
\(95\) −0.624318 −0.0640537
\(96\) 0 0
\(97\) 5.54876 0.563391 0.281695 0.959504i \(-0.409103\pi\)
0.281695 + 0.959504i \(0.409103\pi\)
\(98\) 0 0
\(99\) 4.99612 0.502129
\(100\) 0 0
\(101\) 3.29342 0.327707 0.163854 0.986485i \(-0.447608\pi\)
0.163854 + 0.986485i \(0.447608\pi\)
\(102\) 0 0
\(103\) 4.71506 0.464589 0.232294 0.972646i \(-0.425377\pi\)
0.232294 + 0.972646i \(0.425377\pi\)
\(104\) 0 0
\(105\) −0.659452 −0.0643560
\(106\) 0 0
\(107\) −6.23341 −0.602607 −0.301303 0.953528i \(-0.597422\pi\)
−0.301303 + 0.953528i \(0.597422\pi\)
\(108\) 0 0
\(109\) 17.1305 1.64080 0.820401 0.571789i \(-0.193750\pi\)
0.820401 + 0.571789i \(0.193750\pi\)
\(110\) 0 0
\(111\) −12.0534 −1.14406
\(112\) 0 0
\(113\) −13.0752 −1.23001 −0.615005 0.788523i \(-0.710846\pi\)
−0.615005 + 0.788523i \(0.710846\pi\)
\(114\) 0 0
\(115\) −0.717015 −0.0668620
\(116\) 0 0
\(117\) −4.99612 −0.461892
\(118\) 0 0
\(119\) 1.37647 0.126181
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 22.8302 2.05853
\(124\) 0 0
\(125\) −2.31940 −0.207453
\(126\) 0 0
\(127\) 9.93787 0.881844 0.440922 0.897546i \(-0.354652\pi\)
0.440922 + 0.897546i \(0.354652\pi\)
\(128\) 0 0
\(129\) 18.2041 1.60278
\(130\) 0 0
\(131\) −16.4134 −1.43405 −0.717023 0.697050i \(-0.754496\pi\)
−0.717023 + 0.697050i \(0.754496\pi\)
\(132\) 0 0
\(133\) 2.67708 0.232133
\(134\) 0 0
\(135\) 1.31635 0.113293
\(136\) 0 0
\(137\) −1.35339 −0.115628 −0.0578140 0.998327i \(-0.518413\pi\)
−0.0578140 + 0.998327i \(0.518413\pi\)
\(138\) 0 0
\(139\) 18.9706 1.60907 0.804533 0.593908i \(-0.202416\pi\)
0.804533 + 0.593908i \(0.202416\pi\)
\(140\) 0 0
\(141\) 31.2251 2.62963
\(142\) 0 0
\(143\) −1.00000 −0.0836242
\(144\) 0 0
\(145\) 2.15041 0.178582
\(146\) 0 0
\(147\) 2.82774 0.233228
\(148\) 0 0
\(149\) −11.4974 −0.941901 −0.470950 0.882160i \(-0.656089\pi\)
−0.470950 + 0.882160i \(0.656089\pi\)
\(150\) 0 0
\(151\) 5.60453 0.456090 0.228045 0.973651i \(-0.426767\pi\)
0.228045 + 0.973651i \(0.426767\pi\)
\(152\) 0 0
\(153\) −6.87704 −0.555975
\(154\) 0 0
\(155\) 1.40362 0.112742
\(156\) 0 0
\(157\) −1.67781 −0.133904 −0.0669521 0.997756i \(-0.521327\pi\)
−0.0669521 + 0.997756i \(0.521327\pi\)
\(158\) 0 0
\(159\) 10.9650 0.869579
\(160\) 0 0
\(161\) 3.07457 0.242310
\(162\) 0 0
\(163\) −12.6136 −0.987971 −0.493986 0.869470i \(-0.664460\pi\)
−0.493986 + 0.869470i \(0.664460\pi\)
\(164\) 0 0
\(165\) 0.659452 0.0513383
\(166\) 0 0
\(167\) −22.0678 −1.70765 −0.853827 0.520557i \(-0.825724\pi\)
−0.853827 + 0.520557i \(0.825724\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −13.3750 −1.02282
\(172\) 0 0
\(173\) 6.88776 0.523667 0.261833 0.965113i \(-0.415673\pi\)
0.261833 + 0.965113i \(0.415673\pi\)
\(174\) 0 0
\(175\) 4.94561 0.373853
\(176\) 0 0
\(177\) −27.1955 −2.04414
\(178\) 0 0
\(179\) −10.9773 −0.820482 −0.410241 0.911977i \(-0.634555\pi\)
−0.410241 + 0.911977i \(0.634555\pi\)
\(180\) 0 0
\(181\) 0.875823 0.0650994 0.0325497 0.999470i \(-0.489637\pi\)
0.0325497 + 0.999470i \(0.489637\pi\)
\(182\) 0 0
\(183\) 30.3663 2.24474
\(184\) 0 0
\(185\) −0.994061 −0.0730848
\(186\) 0 0
\(187\) −1.37647 −0.100658
\(188\) 0 0
\(189\) −5.64452 −0.410578
\(190\) 0 0
\(191\) −1.48144 −0.107193 −0.0535966 0.998563i \(-0.517068\pi\)
−0.0535966 + 0.998563i \(0.517068\pi\)
\(192\) 0 0
\(193\) −17.5835 −1.26569 −0.632845 0.774278i \(-0.718113\pi\)
−0.632845 + 0.774278i \(0.718113\pi\)
\(194\) 0 0
\(195\) −0.659452 −0.0472244
\(196\) 0 0
\(197\) −5.60141 −0.399084 −0.199542 0.979889i \(-0.563945\pi\)
−0.199542 + 0.979889i \(0.563945\pi\)
\(198\) 0 0
\(199\) −15.2498 −1.08103 −0.540513 0.841336i \(-0.681770\pi\)
−0.540513 + 0.841336i \(0.681770\pi\)
\(200\) 0 0
\(201\) 31.7363 2.23851
\(202\) 0 0
\(203\) −9.22098 −0.647186
\(204\) 0 0
\(205\) 1.88284 0.131503
\(206\) 0 0
\(207\) −15.3609 −1.06766
\(208\) 0 0
\(209\) −2.67708 −0.185178
\(210\) 0 0
\(211\) −7.62190 −0.524713 −0.262357 0.964971i \(-0.584500\pi\)
−0.262357 + 0.964971i \(0.584500\pi\)
\(212\) 0 0
\(213\) 1.79218 0.122798
\(214\) 0 0
\(215\) 1.50132 0.102389
\(216\) 0 0
\(217\) −6.01875 −0.408580
\(218\) 0 0
\(219\) 18.7044 1.26393
\(220\) 0 0
\(221\) 1.37647 0.0925917
\(222\) 0 0
\(223\) −5.72478 −0.383360 −0.191680 0.981457i \(-0.561394\pi\)
−0.191680 + 0.981457i \(0.561394\pi\)
\(224\) 0 0
\(225\) −24.7089 −1.64726
\(226\) 0 0
\(227\) 13.2484 0.879327 0.439663 0.898163i \(-0.355098\pi\)
0.439663 + 0.898163i \(0.355098\pi\)
\(228\) 0 0
\(229\) −5.53650 −0.365862 −0.182931 0.983126i \(-0.558559\pi\)
−0.182931 + 0.983126i \(0.558559\pi\)
\(230\) 0 0
\(231\) −2.82774 −0.186052
\(232\) 0 0
\(233\) −8.78101 −0.575263 −0.287632 0.957741i \(-0.592868\pi\)
−0.287632 + 0.957741i \(0.592868\pi\)
\(234\) 0 0
\(235\) 2.57518 0.167986
\(236\) 0 0
\(237\) 5.28770 0.343473
\(238\) 0 0
\(239\) 8.36273 0.540940 0.270470 0.962728i \(-0.412821\pi\)
0.270470 + 0.962728i \(0.412821\pi\)
\(240\) 0 0
\(241\) −30.5523 −1.96805 −0.984023 0.178040i \(-0.943024\pi\)
−0.984023 + 0.178040i \(0.943024\pi\)
\(242\) 0 0
\(243\) −14.1825 −0.909810
\(244\) 0 0
\(245\) 0.233208 0.0148991
\(246\) 0 0
\(247\) 2.67708 0.170339
\(248\) 0 0
\(249\) 31.3545 1.98701
\(250\) 0 0
\(251\) −17.0290 −1.07486 −0.537429 0.843309i \(-0.680605\pi\)
−0.537429 + 0.843309i \(0.680605\pi\)
\(252\) 0 0
\(253\) −3.07457 −0.193297
\(254\) 0 0
\(255\) −0.907720 −0.0568436
\(256\) 0 0
\(257\) −2.20347 −0.137449 −0.0687244 0.997636i \(-0.521893\pi\)
−0.0687244 + 0.997636i \(0.521893\pi\)
\(258\) 0 0
\(259\) 4.26255 0.264862
\(260\) 0 0
\(261\) 46.0692 2.85161
\(262\) 0 0
\(263\) 22.3070 1.37551 0.687755 0.725943i \(-0.258596\pi\)
0.687755 + 0.725943i \(0.258596\pi\)
\(264\) 0 0
\(265\) 0.904298 0.0555506
\(266\) 0 0
\(267\) 35.1540 2.15139
\(268\) 0 0
\(269\) −16.6954 −1.01794 −0.508968 0.860785i \(-0.669973\pi\)
−0.508968 + 0.860785i \(0.669973\pi\)
\(270\) 0 0
\(271\) 16.0899 0.977394 0.488697 0.872454i \(-0.337472\pi\)
0.488697 + 0.872454i \(0.337472\pi\)
\(272\) 0 0
\(273\) 2.82774 0.171143
\(274\) 0 0
\(275\) −4.94561 −0.298232
\(276\) 0 0
\(277\) 6.30137 0.378613 0.189306 0.981918i \(-0.439376\pi\)
0.189306 + 0.981918i \(0.439376\pi\)
\(278\) 0 0
\(279\) 30.0704 1.80027
\(280\) 0 0
\(281\) 18.7296 1.11731 0.558656 0.829400i \(-0.311317\pi\)
0.558656 + 0.829400i \(0.311317\pi\)
\(282\) 0 0
\(283\) −28.6955 −1.70577 −0.852886 0.522098i \(-0.825150\pi\)
−0.852886 + 0.522098i \(0.825150\pi\)
\(284\) 0 0
\(285\) −1.76541 −0.104574
\(286\) 0 0
\(287\) −8.07364 −0.476572
\(288\) 0 0
\(289\) −15.1053 −0.888548
\(290\) 0 0
\(291\) 15.6905 0.919791
\(292\) 0 0
\(293\) 3.45232 0.201687 0.100843 0.994902i \(-0.467846\pi\)
0.100843 + 0.994902i \(0.467846\pi\)
\(294\) 0 0
\(295\) −2.24285 −0.130584
\(296\) 0 0
\(297\) 5.64452 0.327528
\(298\) 0 0
\(299\) 3.07457 0.177807
\(300\) 0 0
\(301\) −6.43767 −0.371061
\(302\) 0 0
\(303\) 9.31294 0.535014
\(304\) 0 0
\(305\) 2.50435 0.143399
\(306\) 0 0
\(307\) 22.1992 1.26697 0.633487 0.773753i \(-0.281623\pi\)
0.633487 + 0.773753i \(0.281623\pi\)
\(308\) 0 0
\(309\) 13.3330 0.758486
\(310\) 0 0
\(311\) −24.4459 −1.38620 −0.693099 0.720843i \(-0.743755\pi\)
−0.693099 + 0.720843i \(0.743755\pi\)
\(312\) 0 0
\(313\) 6.66325 0.376629 0.188314 0.982109i \(-0.439698\pi\)
0.188314 + 0.982109i \(0.439698\pi\)
\(314\) 0 0
\(315\) −1.16514 −0.0656480
\(316\) 0 0
\(317\) 23.3545 1.31172 0.655860 0.754883i \(-0.272306\pi\)
0.655860 + 0.754883i \(0.272306\pi\)
\(318\) 0 0
\(319\) 9.22098 0.516276
\(320\) 0 0
\(321\) −17.6265 −0.983815
\(322\) 0 0
\(323\) 3.68494 0.205036
\(324\) 0 0
\(325\) 4.94561 0.274333
\(326\) 0 0
\(327\) 48.4405 2.67877
\(328\) 0 0
\(329\) −11.0424 −0.608788
\(330\) 0 0
\(331\) −12.5712 −0.690976 −0.345488 0.938423i \(-0.612287\pi\)
−0.345488 + 0.938423i \(0.612287\pi\)
\(332\) 0 0
\(333\) −21.2962 −1.16703
\(334\) 0 0
\(335\) 2.61734 0.143001
\(336\) 0 0
\(337\) −8.04391 −0.438179 −0.219090 0.975705i \(-0.570309\pi\)
−0.219090 + 0.975705i \(0.570309\pi\)
\(338\) 0 0
\(339\) −36.9732 −2.00811
\(340\) 0 0
\(341\) 6.01875 0.325934
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −2.02753 −0.109159
\(346\) 0 0
\(347\) 9.68601 0.519972 0.259986 0.965612i \(-0.416282\pi\)
0.259986 + 0.965612i \(0.416282\pi\)
\(348\) 0 0
\(349\) −22.8823 −1.22486 −0.612432 0.790523i \(-0.709809\pi\)
−0.612432 + 0.790523i \(0.709809\pi\)
\(350\) 0 0
\(351\) −5.64452 −0.301282
\(352\) 0 0
\(353\) −17.6747 −0.940729 −0.470364 0.882472i \(-0.655877\pi\)
−0.470364 + 0.882472i \(0.655877\pi\)
\(354\) 0 0
\(355\) 0.147804 0.00784460
\(356\) 0 0
\(357\) 3.89232 0.206003
\(358\) 0 0
\(359\) 11.8212 0.623901 0.311950 0.950098i \(-0.399018\pi\)
0.311950 + 0.950098i \(0.399018\pi\)
\(360\) 0 0
\(361\) −11.8332 −0.622801
\(362\) 0 0
\(363\) 2.82774 0.148418
\(364\) 0 0
\(365\) 1.54258 0.0807424
\(366\) 0 0
\(367\) −26.8779 −1.40301 −0.701507 0.712662i \(-0.747489\pi\)
−0.701507 + 0.712662i \(0.747489\pi\)
\(368\) 0 0
\(369\) 40.3369 2.09985
\(370\) 0 0
\(371\) −3.87764 −0.201317
\(372\) 0 0
\(373\) −17.9892 −0.931445 −0.465723 0.884931i \(-0.654206\pi\)
−0.465723 + 0.884931i \(0.654206\pi\)
\(374\) 0 0
\(375\) −6.55866 −0.338688
\(376\) 0 0
\(377\) −9.22098 −0.474905
\(378\) 0 0
\(379\) 0.474924 0.0243952 0.0121976 0.999926i \(-0.496117\pi\)
0.0121976 + 0.999926i \(0.496117\pi\)
\(380\) 0 0
\(381\) 28.1017 1.43970
\(382\) 0 0
\(383\) 10.4478 0.533857 0.266929 0.963716i \(-0.413991\pi\)
0.266929 + 0.963716i \(0.413991\pi\)
\(384\) 0 0
\(385\) −0.233208 −0.0118854
\(386\) 0 0
\(387\) 32.1634 1.63496
\(388\) 0 0
\(389\) −6.69516 −0.339458 −0.169729 0.985491i \(-0.554289\pi\)
−0.169729 + 0.985491i \(0.554289\pi\)
\(390\) 0 0
\(391\) 4.23207 0.214025
\(392\) 0 0
\(393\) −46.4129 −2.34122
\(394\) 0 0
\(395\) 0.436085 0.0219418
\(396\) 0 0
\(397\) 30.9425 1.55296 0.776480 0.630142i \(-0.217003\pi\)
0.776480 + 0.630142i \(0.217003\pi\)
\(398\) 0 0
\(399\) 7.57010 0.378979
\(400\) 0 0
\(401\) 28.2888 1.41267 0.706337 0.707875i \(-0.250346\pi\)
0.706337 + 0.707875i \(0.250346\pi\)
\(402\) 0 0
\(403\) −6.01875 −0.299815
\(404\) 0 0
\(405\) 0.226881 0.0112738
\(406\) 0 0
\(407\) −4.26255 −0.211287
\(408\) 0 0
\(409\) −15.5923 −0.770988 −0.385494 0.922710i \(-0.625969\pi\)
−0.385494 + 0.922710i \(0.625969\pi\)
\(410\) 0 0
\(411\) −3.82704 −0.188774
\(412\) 0 0
\(413\) 9.61739 0.473241
\(414\) 0 0
\(415\) 2.58586 0.126935
\(416\) 0 0
\(417\) 53.6440 2.62696
\(418\) 0 0
\(419\) −7.66727 −0.374571 −0.187285 0.982306i \(-0.559969\pi\)
−0.187285 + 0.982306i \(0.559969\pi\)
\(420\) 0 0
\(421\) 11.7765 0.573951 0.286976 0.957938i \(-0.407350\pi\)
0.286976 + 0.957938i \(0.407350\pi\)
\(422\) 0 0
\(423\) 55.1692 2.68242
\(424\) 0 0
\(425\) 6.80751 0.330213
\(426\) 0 0
\(427\) −10.7387 −0.519683
\(428\) 0 0
\(429\) −2.82774 −0.136525
\(430\) 0 0
\(431\) −2.78161 −0.133985 −0.0669926 0.997753i \(-0.521340\pi\)
−0.0669926 + 0.997753i \(0.521340\pi\)
\(432\) 0 0
\(433\) 37.5833 1.80614 0.903068 0.429497i \(-0.141309\pi\)
0.903068 + 0.429497i \(0.141309\pi\)
\(434\) 0 0
\(435\) 6.08080 0.291552
\(436\) 0 0
\(437\) 8.23089 0.393737
\(438\) 0 0
\(439\) 14.4168 0.688078 0.344039 0.938955i \(-0.388205\pi\)
0.344039 + 0.938955i \(0.388205\pi\)
\(440\) 0 0
\(441\) 4.99612 0.237911
\(442\) 0 0
\(443\) 36.1621 1.71811 0.859056 0.511882i \(-0.171052\pi\)
0.859056 + 0.511882i \(0.171052\pi\)
\(444\) 0 0
\(445\) 2.89921 0.137436
\(446\) 0 0
\(447\) −32.5116 −1.53774
\(448\) 0 0
\(449\) −29.9928 −1.41545 −0.707724 0.706489i \(-0.750278\pi\)
−0.707724 + 0.706489i \(0.750278\pi\)
\(450\) 0 0
\(451\) 8.07364 0.380173
\(452\) 0 0
\(453\) 15.8481 0.744611
\(454\) 0 0
\(455\) 0.233208 0.0109330
\(456\) 0 0
\(457\) −1.98949 −0.0930646 −0.0465323 0.998917i \(-0.514817\pi\)
−0.0465323 + 0.998917i \(0.514817\pi\)
\(458\) 0 0
\(459\) −7.76954 −0.362651
\(460\) 0 0
\(461\) −12.9126 −0.601401 −0.300700 0.953719i \(-0.597220\pi\)
−0.300700 + 0.953719i \(0.597220\pi\)
\(462\) 0 0
\(463\) 35.4548 1.64773 0.823863 0.566789i \(-0.191814\pi\)
0.823863 + 0.566789i \(0.191814\pi\)
\(464\) 0 0
\(465\) 3.96908 0.184062
\(466\) 0 0
\(467\) −36.1342 −1.67209 −0.836046 0.548659i \(-0.815138\pi\)
−0.836046 + 0.548659i \(0.815138\pi\)
\(468\) 0 0
\(469\) −11.2232 −0.518239
\(470\) 0 0
\(471\) −4.74442 −0.218612
\(472\) 0 0
\(473\) 6.43767 0.296004
\(474\) 0 0
\(475\) 13.2398 0.607485
\(476\) 0 0
\(477\) 19.3732 0.887037
\(478\) 0 0
\(479\) 28.2792 1.29211 0.646055 0.763291i \(-0.276418\pi\)
0.646055 + 0.763291i \(0.276418\pi\)
\(480\) 0 0
\(481\) 4.26255 0.194355
\(482\) 0 0
\(483\) 8.69409 0.395595
\(484\) 0 0
\(485\) 1.29402 0.0587582
\(486\) 0 0
\(487\) −20.9278 −0.948330 −0.474165 0.880436i \(-0.657250\pi\)
−0.474165 + 0.880436i \(0.657250\pi\)
\(488\) 0 0
\(489\) −35.6679 −1.61296
\(490\) 0 0
\(491\) −19.1424 −0.863884 −0.431942 0.901901i \(-0.642171\pi\)
−0.431942 + 0.901901i \(0.642171\pi\)
\(492\) 0 0
\(493\) −12.6925 −0.571639
\(494\) 0 0
\(495\) 1.16514 0.0523690
\(496\) 0 0
\(497\) −0.633784 −0.0284291
\(498\) 0 0
\(499\) −33.6064 −1.50443 −0.752214 0.658918i \(-0.771014\pi\)
−0.752214 + 0.658918i \(0.771014\pi\)
\(500\) 0 0
\(501\) −62.4019 −2.78791
\(502\) 0 0
\(503\) 14.5290 0.647814 0.323907 0.946089i \(-0.395003\pi\)
0.323907 + 0.946089i \(0.395003\pi\)
\(504\) 0 0
\(505\) 0.768052 0.0341779
\(506\) 0 0
\(507\) 2.82774 0.125584
\(508\) 0 0
\(509\) 27.6413 1.22518 0.612590 0.790401i \(-0.290128\pi\)
0.612590 + 0.790401i \(0.290128\pi\)
\(510\) 0 0
\(511\) −6.61461 −0.292613
\(512\) 0 0
\(513\) −15.1109 −0.667160
\(514\) 0 0
\(515\) 1.09959 0.0484537
\(516\) 0 0
\(517\) 11.0424 0.485645
\(518\) 0 0
\(519\) 19.4768 0.854937
\(520\) 0 0
\(521\) 2.07965 0.0911111 0.0455555 0.998962i \(-0.485494\pi\)
0.0455555 + 0.998962i \(0.485494\pi\)
\(522\) 0 0
\(523\) −18.3341 −0.801696 −0.400848 0.916145i \(-0.631284\pi\)
−0.400848 + 0.916145i \(0.631284\pi\)
\(524\) 0 0
\(525\) 13.9849 0.610352
\(526\) 0 0
\(527\) −8.28466 −0.360886
\(528\) 0 0
\(529\) −13.5470 −0.589000
\(530\) 0 0
\(531\) −48.0496 −2.08518
\(532\) 0 0
\(533\) −8.07364 −0.349708
\(534\) 0 0
\(535\) −1.45368 −0.0628482
\(536\) 0 0
\(537\) −31.0410 −1.33952
\(538\) 0 0
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) 16.1030 0.692321 0.346161 0.938175i \(-0.387485\pi\)
0.346161 + 0.938175i \(0.387485\pi\)
\(542\) 0 0
\(543\) 2.47660 0.106281
\(544\) 0 0
\(545\) 3.99497 0.171126
\(546\) 0 0
\(547\) −0.467630 −0.0199944 −0.00999720 0.999950i \(-0.503182\pi\)
−0.00999720 + 0.999950i \(0.503182\pi\)
\(548\) 0 0
\(549\) 53.6519 2.28981
\(550\) 0 0
\(551\) −24.6854 −1.05163
\(552\) 0 0
\(553\) −1.86994 −0.0795179
\(554\) 0 0
\(555\) −2.81095 −0.119318
\(556\) 0 0
\(557\) −42.4841 −1.80011 −0.900055 0.435777i \(-0.856474\pi\)
−0.900055 + 0.435777i \(0.856474\pi\)
\(558\) 0 0
\(559\) −6.43767 −0.272284
\(560\) 0 0
\(561\) −3.89232 −0.164334
\(562\) 0 0
\(563\) 8.27832 0.348890 0.174445 0.984667i \(-0.444187\pi\)
0.174445 + 0.984667i \(0.444187\pi\)
\(564\) 0 0
\(565\) −3.04924 −0.128282
\(566\) 0 0
\(567\) −0.972871 −0.0408567
\(568\) 0 0
\(569\) −5.10154 −0.213868 −0.106934 0.994266i \(-0.534103\pi\)
−0.106934 + 0.994266i \(0.534103\pi\)
\(570\) 0 0
\(571\) 15.3622 0.642887 0.321443 0.946929i \(-0.395832\pi\)
0.321443 + 0.946929i \(0.395832\pi\)
\(572\) 0 0
\(573\) −4.18913 −0.175003
\(574\) 0 0
\(575\) 15.2056 0.634119
\(576\) 0 0
\(577\) −7.19740 −0.299632 −0.149816 0.988714i \(-0.547868\pi\)
−0.149816 + 0.988714i \(0.547868\pi\)
\(578\) 0 0
\(579\) −49.7217 −2.06636
\(580\) 0 0
\(581\) −11.0882 −0.460015
\(582\) 0 0
\(583\) 3.87764 0.160596
\(584\) 0 0
\(585\) −1.16514 −0.0481725
\(586\) 0 0
\(587\) 22.6709 0.935729 0.467864 0.883800i \(-0.345024\pi\)
0.467864 + 0.883800i \(0.345024\pi\)
\(588\) 0 0
\(589\) −16.1127 −0.663913
\(590\) 0 0
\(591\) −15.8393 −0.651544
\(592\) 0 0
\(593\) 34.6355 1.42231 0.711154 0.703036i \(-0.248173\pi\)
0.711154 + 0.703036i \(0.248173\pi\)
\(594\) 0 0
\(595\) 0.321005 0.0131599
\(596\) 0 0
\(597\) −43.1224 −1.76488
\(598\) 0 0
\(599\) −18.3020 −0.747800 −0.373900 0.927469i \(-0.621980\pi\)
−0.373900 + 0.927469i \(0.621980\pi\)
\(600\) 0 0
\(601\) −5.23738 −0.213637 −0.106819 0.994279i \(-0.534066\pi\)
−0.106819 + 0.994279i \(0.534066\pi\)
\(602\) 0 0
\(603\) 56.0724 2.28345
\(604\) 0 0
\(605\) 0.233208 0.00948126
\(606\) 0 0
\(607\) −26.3505 −1.06954 −0.534768 0.844999i \(-0.679601\pi\)
−0.534768 + 0.844999i \(0.679601\pi\)
\(608\) 0 0
\(609\) −26.0746 −1.05659
\(610\) 0 0
\(611\) −11.0424 −0.446728
\(612\) 0 0
\(613\) 47.4507 1.91652 0.958259 0.285903i \(-0.0922936\pi\)
0.958259 + 0.285903i \(0.0922936\pi\)
\(614\) 0 0
\(615\) 5.32418 0.214692
\(616\) 0 0
\(617\) −15.2072 −0.612220 −0.306110 0.951996i \(-0.599027\pi\)
−0.306110 + 0.951996i \(0.599027\pi\)
\(618\) 0 0
\(619\) −30.5023 −1.22599 −0.612995 0.790086i \(-0.710036\pi\)
−0.612995 + 0.790086i \(0.710036\pi\)
\(620\) 0 0
\(621\) −17.3545 −0.696411
\(622\) 0 0
\(623\) −12.4318 −0.498071
\(624\) 0 0
\(625\) 24.1872 0.967487
\(626\) 0 0
\(627\) −7.57010 −0.302321
\(628\) 0 0
\(629\) 5.86729 0.233944
\(630\) 0 0
\(631\) 0.914236 0.0363952 0.0181976 0.999834i \(-0.494207\pi\)
0.0181976 + 0.999834i \(0.494207\pi\)
\(632\) 0 0
\(633\) −21.5528 −0.856645
\(634\) 0 0
\(635\) 2.31759 0.0919709
\(636\) 0 0
\(637\) −1.00000 −0.0396214
\(638\) 0 0
\(639\) 3.16646 0.125263
\(640\) 0 0
\(641\) −7.74078 −0.305742 −0.152871 0.988246i \(-0.548852\pi\)
−0.152871 + 0.988246i \(0.548852\pi\)
\(642\) 0 0
\(643\) −8.11887 −0.320177 −0.160089 0.987103i \(-0.551178\pi\)
−0.160089 + 0.987103i \(0.551178\pi\)
\(644\) 0 0
\(645\) 4.24534 0.167160
\(646\) 0 0
\(647\) −7.99447 −0.314295 −0.157148 0.987575i \(-0.550230\pi\)
−0.157148 + 0.987575i \(0.550230\pi\)
\(648\) 0 0
\(649\) −9.61739 −0.377516
\(650\) 0 0
\(651\) −17.0195 −0.667046
\(652\) 0 0
\(653\) 21.8267 0.854146 0.427073 0.904217i \(-0.359545\pi\)
0.427073 + 0.904217i \(0.359545\pi\)
\(654\) 0 0
\(655\) −3.82774 −0.149562
\(656\) 0 0
\(657\) 33.0474 1.28930
\(658\) 0 0
\(659\) 24.7032 0.962301 0.481150 0.876638i \(-0.340219\pi\)
0.481150 + 0.876638i \(0.340219\pi\)
\(660\) 0 0
\(661\) −40.6473 −1.58100 −0.790499 0.612463i \(-0.790179\pi\)
−0.790499 + 0.612463i \(0.790179\pi\)
\(662\) 0 0
\(663\) 3.89232 0.151165
\(664\) 0 0
\(665\) 0.624318 0.0242100
\(666\) 0 0
\(667\) −28.3506 −1.09774
\(668\) 0 0
\(669\) −16.1882 −0.625872
\(670\) 0 0
\(671\) 10.7387 0.414563
\(672\) 0 0
\(673\) 21.6599 0.834929 0.417464 0.908693i \(-0.362919\pi\)
0.417464 + 0.908693i \(0.362919\pi\)
\(674\) 0 0
\(675\) −27.9156 −1.07447
\(676\) 0 0
\(677\) −1.68620 −0.0648059 −0.0324029 0.999475i \(-0.510316\pi\)
−0.0324029 + 0.999475i \(0.510316\pi\)
\(678\) 0 0
\(679\) −5.54876 −0.212942
\(680\) 0 0
\(681\) 37.4630 1.43559
\(682\) 0 0
\(683\) 19.7272 0.754839 0.377420 0.926042i \(-0.376811\pi\)
0.377420 + 0.926042i \(0.376811\pi\)
\(684\) 0 0
\(685\) −0.315622 −0.0120593
\(686\) 0 0
\(687\) −15.6558 −0.597305
\(688\) 0 0
\(689\) −3.87764 −0.147726
\(690\) 0 0
\(691\) −1.36535 −0.0519405 −0.0259703 0.999663i \(-0.508268\pi\)
−0.0259703 + 0.999663i \(0.508268\pi\)
\(692\) 0 0
\(693\) −4.99612 −0.189787
\(694\) 0 0
\(695\) 4.42410 0.167816
\(696\) 0 0
\(697\) −11.1132 −0.420941
\(698\) 0 0
\(699\) −24.8304 −0.939173
\(700\) 0 0
\(701\) 23.7869 0.898418 0.449209 0.893427i \(-0.351706\pi\)
0.449209 + 0.893427i \(0.351706\pi\)
\(702\) 0 0
\(703\) 11.4112 0.430382
\(704\) 0 0
\(705\) 7.28194 0.274254
\(706\) 0 0
\(707\) −3.29342 −0.123862
\(708\) 0 0
\(709\) 20.1134 0.755373 0.377687 0.925934i \(-0.376720\pi\)
0.377687 + 0.925934i \(0.376720\pi\)
\(710\) 0 0
\(711\) 9.34244 0.350369
\(712\) 0 0
\(713\) −18.5051 −0.693021
\(714\) 0 0
\(715\) −0.233208 −0.00872149
\(716\) 0 0
\(717\) 23.6477 0.883138
\(718\) 0 0
\(719\) −0.628754 −0.0234485 −0.0117243 0.999931i \(-0.503732\pi\)
−0.0117243 + 0.999931i \(0.503732\pi\)
\(720\) 0 0
\(721\) −4.71506 −0.175598
\(722\) 0 0
\(723\) −86.3940 −3.21303
\(724\) 0 0
\(725\) −45.6034 −1.69367
\(726\) 0 0
\(727\) 7.70045 0.285594 0.142797 0.989752i \(-0.454390\pi\)
0.142797 + 0.989752i \(0.454390\pi\)
\(728\) 0 0
\(729\) −43.0231 −1.59345
\(730\) 0 0
\(731\) −8.86129 −0.327747
\(732\) 0 0
\(733\) 46.7135 1.72540 0.862702 0.505713i \(-0.168771\pi\)
0.862702 + 0.505713i \(0.168771\pi\)
\(734\) 0 0
\(735\) 0.659452 0.0243243
\(736\) 0 0
\(737\) 11.2232 0.413412
\(738\) 0 0
\(739\) −17.6767 −0.650247 −0.325123 0.945672i \(-0.605406\pi\)
−0.325123 + 0.945672i \(0.605406\pi\)
\(740\) 0 0
\(741\) 7.57010 0.278095
\(742\) 0 0
\(743\) −18.7809 −0.689004 −0.344502 0.938786i \(-0.611952\pi\)
−0.344502 + 0.938786i \(0.611952\pi\)
\(744\) 0 0
\(745\) −2.68128 −0.0982345
\(746\) 0 0
\(747\) 55.3979 2.02690
\(748\) 0 0
\(749\) 6.23341 0.227764
\(750\) 0 0
\(751\) −44.2451 −1.61453 −0.807264 0.590191i \(-0.799052\pi\)
−0.807264 + 0.590191i \(0.799052\pi\)
\(752\) 0 0
\(753\) −48.1535 −1.75481
\(754\) 0 0
\(755\) 1.30702 0.0475674
\(756\) 0 0
\(757\) −24.6289 −0.895153 −0.447576 0.894246i \(-0.647713\pi\)
−0.447576 + 0.894246i \(0.647713\pi\)
\(758\) 0 0
\(759\) −8.69409 −0.315576
\(760\) 0 0
\(761\) −0.473922 −0.0171796 −0.00858982 0.999963i \(-0.502734\pi\)
−0.00858982 + 0.999963i \(0.502734\pi\)
\(762\) 0 0
\(763\) −17.1305 −0.620165
\(764\) 0 0
\(765\) −1.60378 −0.0579848
\(766\) 0 0
\(767\) 9.61739 0.347264
\(768\) 0 0
\(769\) −22.5935 −0.814743 −0.407372 0.913262i \(-0.633555\pi\)
−0.407372 + 0.913262i \(0.633555\pi\)
\(770\) 0 0
\(771\) −6.23084 −0.224398
\(772\) 0 0
\(773\) 52.3031 1.88121 0.940605 0.339502i \(-0.110258\pi\)
0.940605 + 0.339502i \(0.110258\pi\)
\(774\) 0 0
\(775\) −29.7664 −1.06924
\(776\) 0 0
\(777\) 12.0534 0.432413
\(778\) 0 0
\(779\) −21.6138 −0.774395
\(780\) 0 0
\(781\) 0.633784 0.0226786
\(782\) 0 0
\(783\) 52.0480 1.86004
\(784\) 0 0
\(785\) −0.391280 −0.0139654
\(786\) 0 0
\(787\) −53.8006 −1.91778 −0.958892 0.283770i \(-0.908415\pi\)
−0.958892 + 0.283770i \(0.908415\pi\)
\(788\) 0 0
\(789\) 63.0785 2.24565
\(790\) 0 0
\(791\) 13.0752 0.464900
\(792\) 0 0
\(793\) −10.7387 −0.381343
\(794\) 0 0
\(795\) 2.55712 0.0906918
\(796\) 0 0
\(797\) 21.8313 0.773305 0.386653 0.922225i \(-0.373631\pi\)
0.386653 + 0.922225i \(0.373631\pi\)
\(798\) 0 0
\(799\) −15.1996 −0.537723
\(800\) 0 0
\(801\) 62.1110 2.19458
\(802\) 0 0
\(803\) 6.61461 0.233425
\(804\) 0 0
\(805\) 0.717015 0.0252715
\(806\) 0 0
\(807\) −47.2102 −1.66188
\(808\) 0 0
\(809\) −15.7343 −0.553189 −0.276595 0.960987i \(-0.589206\pi\)
−0.276595 + 0.960987i \(0.589206\pi\)
\(810\) 0 0
\(811\) −36.4679 −1.28056 −0.640280 0.768142i \(-0.721182\pi\)
−0.640280 + 0.768142i \(0.721182\pi\)
\(812\) 0 0
\(813\) 45.4982 1.59569
\(814\) 0 0
\(815\) −2.94159 −0.103039
\(816\) 0 0
\(817\) −17.2342 −0.602948
\(818\) 0 0
\(819\) 4.99612 0.174579
\(820\) 0 0
\(821\) 21.3255 0.744264 0.372132 0.928180i \(-0.378627\pi\)
0.372132 + 0.928180i \(0.378627\pi\)
\(822\) 0 0
\(823\) −5.13206 −0.178892 −0.0894462 0.995992i \(-0.528510\pi\)
−0.0894462 + 0.995992i \(0.528510\pi\)
\(824\) 0 0
\(825\) −13.9849 −0.486892
\(826\) 0 0
\(827\) 20.9148 0.727280 0.363640 0.931540i \(-0.381534\pi\)
0.363640 + 0.931540i \(0.381534\pi\)
\(828\) 0 0
\(829\) 42.1026 1.46228 0.731141 0.682226i \(-0.238988\pi\)
0.731141 + 0.682226i \(0.238988\pi\)
\(830\) 0 0
\(831\) 17.8186 0.618122
\(832\) 0 0
\(833\) −1.37647 −0.0476920
\(834\) 0 0
\(835\) −5.14638 −0.178098
\(836\) 0 0
\(837\) 33.9730 1.17428
\(838\) 0 0
\(839\) −9.91855 −0.342426 −0.171213 0.985234i \(-0.554769\pi\)
−0.171213 + 0.985234i \(0.554769\pi\)
\(840\) 0 0
\(841\) 56.0265 1.93195
\(842\) 0 0
\(843\) 52.9623 1.82412
\(844\) 0 0
\(845\) 0.233208 0.00802261
\(846\) 0 0
\(847\) −1.00000 −0.0343604
\(848\) 0 0
\(849\) −81.1435 −2.78484
\(850\) 0 0
\(851\) 13.1055 0.449251
\(852\) 0 0
\(853\) 15.4273 0.528221 0.264111 0.964492i \(-0.414922\pi\)
0.264111 + 0.964492i \(0.414922\pi\)
\(854\) 0 0
\(855\) −3.11917 −0.106673
\(856\) 0 0
\(857\) 32.0757 1.09568 0.547842 0.836582i \(-0.315449\pi\)
0.547842 + 0.836582i \(0.315449\pi\)
\(858\) 0 0
\(859\) 55.6842 1.89992 0.949961 0.312368i \(-0.101122\pi\)
0.949961 + 0.312368i \(0.101122\pi\)
\(860\) 0 0
\(861\) −22.8302 −0.778050
\(862\) 0 0
\(863\) −13.7556 −0.468245 −0.234122 0.972207i \(-0.575222\pi\)
−0.234122 + 0.972207i \(0.575222\pi\)
\(864\) 0 0
\(865\) 1.60628 0.0546152
\(866\) 0 0
\(867\) −42.7139 −1.45064
\(868\) 0 0
\(869\) 1.86994 0.0634333
\(870\) 0 0
\(871\) −11.2232 −0.380283
\(872\) 0 0
\(873\) 27.7223 0.938257
\(874\) 0 0
\(875\) 2.31940 0.0784100
\(876\) 0 0
\(877\) 31.4276 1.06123 0.530617 0.847612i \(-0.321960\pi\)
0.530617 + 0.847612i \(0.321960\pi\)
\(878\) 0 0
\(879\) 9.76227 0.329273
\(880\) 0 0
\(881\) −0.0935359 −0.00315131 −0.00157565 0.999999i \(-0.500502\pi\)
−0.00157565 + 0.999999i \(0.500502\pi\)
\(882\) 0 0
\(883\) 1.12055 0.0377094 0.0188547 0.999822i \(-0.493998\pi\)
0.0188547 + 0.999822i \(0.493998\pi\)
\(884\) 0 0
\(885\) −6.34221 −0.213191
\(886\) 0 0
\(887\) −7.49853 −0.251776 −0.125888 0.992044i \(-0.540178\pi\)
−0.125888 + 0.992044i \(0.540178\pi\)
\(888\) 0 0
\(889\) −9.93787 −0.333306
\(890\) 0 0
\(891\) 0.972871 0.0325924
\(892\) 0 0
\(893\) −29.5615 −0.989237
\(894\) 0 0
\(895\) −2.56000 −0.0855713
\(896\) 0 0
\(897\) 8.69409 0.290287
\(898\) 0 0
\(899\) 55.4988 1.85099
\(900\) 0 0
\(901\) −5.33748 −0.177817
\(902\) 0 0
\(903\) −18.2041 −0.605793
\(904\) 0 0
\(905\) 0.204249 0.00678947
\(906\) 0 0
\(907\) −55.2841 −1.83568 −0.917839 0.396953i \(-0.870068\pi\)
−0.917839 + 0.396953i \(0.870068\pi\)
\(908\) 0 0
\(909\) 16.4543 0.545755
\(910\) 0 0
\(911\) −32.9859 −1.09287 −0.546436 0.837501i \(-0.684016\pi\)
−0.546436 + 0.837501i \(0.684016\pi\)
\(912\) 0 0
\(913\) 11.0882 0.366965
\(914\) 0 0
\(915\) 7.08167 0.234113
\(916\) 0 0
\(917\) 16.4134 0.542018
\(918\) 0 0
\(919\) −32.9500 −1.08692 −0.543460 0.839435i \(-0.682886\pi\)
−0.543460 + 0.839435i \(0.682886\pi\)
\(920\) 0 0
\(921\) 62.7736 2.06846
\(922\) 0 0
\(923\) −0.633784 −0.0208613
\(924\) 0 0
\(925\) 21.0809 0.693136
\(926\) 0 0
\(927\) 23.5570 0.773714
\(928\) 0 0
\(929\) −42.5624 −1.39643 −0.698213 0.715890i \(-0.746021\pi\)
−0.698213 + 0.715890i \(0.746021\pi\)
\(930\) 0 0
\(931\) −2.67708 −0.0877379
\(932\) 0 0
\(933\) −69.1266 −2.26310
\(934\) 0 0
\(935\) −0.321005 −0.0104980
\(936\) 0 0
\(937\) 19.3789 0.633082 0.316541 0.948579i \(-0.397479\pi\)
0.316541 + 0.948579i \(0.397479\pi\)
\(938\) 0 0
\(939\) 18.8419 0.614883
\(940\) 0 0
\(941\) −37.8923 −1.23525 −0.617627 0.786471i \(-0.711906\pi\)
−0.617627 + 0.786471i \(0.711906\pi\)
\(942\) 0 0
\(943\) −24.8230 −0.808348
\(944\) 0 0
\(945\) −1.31635 −0.0428208
\(946\) 0 0
\(947\) −25.5405 −0.829956 −0.414978 0.909831i \(-0.636211\pi\)
−0.414978 + 0.909831i \(0.636211\pi\)
\(948\) 0 0
\(949\) −6.61461 −0.214719
\(950\) 0 0
\(951\) 66.0405 2.14151
\(952\) 0 0
\(953\) 29.0985 0.942592 0.471296 0.881975i \(-0.343786\pi\)
0.471296 + 0.881975i \(0.343786\pi\)
\(954\) 0 0
\(955\) −0.345484 −0.0111796
\(956\) 0 0
\(957\) 26.0746 0.842871
\(958\) 0 0
\(959\) 1.35339 0.0437033
\(960\) 0 0
\(961\) 5.22539 0.168561
\(962\) 0 0
\(963\) −31.1429 −1.00357
\(964\) 0 0
\(965\) −4.10062 −0.132004
\(966\) 0 0
\(967\) 46.9373 1.50940 0.754700 0.656070i \(-0.227782\pi\)
0.754700 + 0.656070i \(0.227782\pi\)
\(968\) 0 0
\(969\) 10.4201 0.334740
\(970\) 0 0
\(971\) −1.51495 −0.0486171 −0.0243085 0.999705i \(-0.507738\pi\)
−0.0243085 + 0.999705i \(0.507738\pi\)
\(972\) 0 0
\(973\) −18.9706 −0.608170
\(974\) 0 0
\(975\) 13.9849 0.447876
\(976\) 0 0
\(977\) −31.2958 −1.00124 −0.500621 0.865667i \(-0.666895\pi\)
−0.500621 + 0.865667i \(0.666895\pi\)
\(978\) 0 0
\(979\) 12.4318 0.397323
\(980\) 0 0
\(981\) 85.5859 2.73255
\(982\) 0 0
\(983\) 25.9289 0.827004 0.413502 0.910503i \(-0.364305\pi\)
0.413502 + 0.910503i \(0.364305\pi\)
\(984\) 0 0
\(985\) −1.30630 −0.0416220
\(986\) 0 0
\(987\) −31.2251 −0.993905
\(988\) 0 0
\(989\) −19.7931 −0.629383
\(990\) 0 0
\(991\) 24.6397 0.782707 0.391354 0.920240i \(-0.372007\pi\)
0.391354 + 0.920240i \(0.372007\pi\)
\(992\) 0 0
\(993\) −35.5481 −1.12809
\(994\) 0 0
\(995\) −3.55637 −0.112744
\(996\) 0 0
\(997\) 1.64604 0.0521305 0.0260653 0.999660i \(-0.491702\pi\)
0.0260653 + 0.999660i \(0.491702\pi\)
\(998\) 0 0
\(999\) −24.0600 −0.761226
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 8008.2.a.n.1.9 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8008.2.a.n.1.9 9 1.1 even 1 trivial