Properties

Label 9408.2.a.eb.1.2
Level 9408
Weight 2
Character 9408.1
Self dual yes
Analytic conductor 75.123
Analytic rank 0
Dimension 2
CM no
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 9408 = 2^{6} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9408.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(75.1232582216\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 4704)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 9408.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +2.82843 q^{5} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +2.82843 q^{5} +1.00000 q^{9} +2.82843 q^{11} +2.82843 q^{15} +2.82843 q^{17} +4.00000 q^{19} +8.48528 q^{23} +3.00000 q^{25} +1.00000 q^{27} -2.00000 q^{29} +2.82843 q^{33} +6.00000 q^{37} +8.48528 q^{41} -11.3137 q^{43} +2.82843 q^{45} -8.00000 q^{47} +2.82843 q^{51} -6.00000 q^{53} +8.00000 q^{55} +4.00000 q^{57} +12.0000 q^{59} -5.65685 q^{61} +5.65685 q^{67} +8.48528 q^{69} +2.82843 q^{71} -5.65685 q^{73} +3.00000 q^{75} +5.65685 q^{79} +1.00000 q^{81} -4.00000 q^{83} +8.00000 q^{85} -2.00000 q^{87} -2.82843 q^{89} +11.3137 q^{95} -16.9706 q^{97} +2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} + 2q^{9} + 8q^{19} + 6q^{25} + 2q^{27} - 4q^{29} + 12q^{37} - 16q^{47} - 12q^{53} + 16q^{55} + 8q^{57} + 24q^{59} + 6q^{75} + 2q^{81} - 8q^{83} + 16q^{85} - 4q^{87} + O(q^{100}) \)

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
\(5\) 2.82843 1.26491 0.632456 0.774597i \(-0.282047\pi\)
0.632456 + 0.774597i \(0.282047\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.82843 0.852803 0.426401 0.904534i \(-0.359781\pi\)
0.426401 + 0.904534i \(0.359781\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 2.82843 0.730297
\(16\) 0 0
\(17\) 2.82843 0.685994 0.342997 0.939336i \(-0.388558\pi\)
0.342997 + 0.939336i \(0.388558\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 8.48528 1.76930 0.884652 0.466252i \(-0.154396\pi\)
0.884652 + 0.466252i \(0.154396\pi\)
\(24\) 0 0
\(25\) 3.00000 0.600000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 2.82843 0.492366
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 8.48528 1.32518 0.662589 0.748983i \(-0.269458\pi\)
0.662589 + 0.748983i \(0.269458\pi\)
\(42\) 0 0
\(43\) −11.3137 −1.72532 −0.862662 0.505781i \(-0.831205\pi\)
−0.862662 + 0.505781i \(0.831205\pi\)
\(44\) 0 0
\(45\) 2.82843 0.421637
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 2.82843 0.396059
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 8.00000 1.07872
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) 0 0
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) −5.65685 −0.724286 −0.362143 0.932123i \(-0.617955\pi\)
−0.362143 + 0.932123i \(0.617955\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 5.65685 0.691095 0.345547 0.938401i \(-0.387693\pi\)
0.345547 + 0.938401i \(0.387693\pi\)
\(68\) 0 0
\(69\) 8.48528 1.02151
\(70\) 0 0
\(71\) 2.82843 0.335673 0.167836 0.985815i \(-0.446322\pi\)
0.167836 + 0.985815i \(0.446322\pi\)
\(72\) 0 0
\(73\) −5.65685 −0.662085 −0.331042 0.943616i \(-0.607400\pi\)
−0.331042 + 0.943616i \(0.607400\pi\)
\(74\) 0 0
\(75\) 3.00000 0.346410
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 5.65685 0.636446 0.318223 0.948016i \(-0.396914\pi\)
0.318223 + 0.948016i \(0.396914\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 8.00000 0.867722
\(86\) 0 0
\(87\) −2.00000 −0.214423
\(88\) 0 0
\(89\) −2.82843 −0.299813 −0.149906 0.988700i \(-0.547897\pi\)
−0.149906 + 0.988700i \(0.547897\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 11.3137 1.16076
\(96\) 0 0
\(97\) −16.9706 −1.72310 −0.861550 0.507673i \(-0.830506\pi\)
−0.861550 + 0.507673i \(0.830506\pi\)
\(98\) 0 0
\(99\) 2.82843 0.284268
\(100\) 0 0
\(101\) −14.1421 −1.40720 −0.703598 0.710599i \(-0.748424\pi\)
−0.703598 + 0.710599i \(0.748424\pi\)
\(102\) 0 0
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.82843 0.273434 0.136717 0.990610i \(-0.456345\pi\)
0.136717 + 0.990610i \(0.456345\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) 6.00000 0.569495
\(112\) 0 0
\(113\) 18.0000 1.69330 0.846649 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) 0 0
\(115\) 24.0000 2.23801
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −3.00000 −0.272727
\(122\) 0 0
\(123\) 8.48528 0.765092
\(124\) 0 0
\(125\) −5.65685 −0.505964
\(126\) 0 0
\(127\) −16.9706 −1.50589 −0.752947 0.658081i \(-0.771368\pi\)
−0.752947 + 0.658081i \(0.771368\pi\)
\(128\) 0 0
\(129\) −11.3137 −0.996116
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 2.82843 0.243432
\(136\) 0 0
\(137\) −10.0000 −0.854358 −0.427179 0.904167i \(-0.640493\pi\)
−0.427179 + 0.904167i \(0.640493\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −5.65685 −0.469776
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) 2.82843 0.228665
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 16.9706 1.35440 0.677199 0.735800i \(-0.263194\pi\)
0.677199 + 0.735800i \(0.263194\pi\)
\(158\) 0 0
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 16.9706 1.32924 0.664619 0.747183i \(-0.268594\pi\)
0.664619 + 0.747183i \(0.268594\pi\)
\(164\) 0 0
\(165\) 8.00000 0.622799
\(166\) 0 0
\(167\) 24.0000 1.85718 0.928588 0.371113i \(-0.121024\pi\)
0.928588 + 0.371113i \(0.121024\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 4.00000 0.305888
\(172\) 0 0
\(173\) 8.48528 0.645124 0.322562 0.946548i \(-0.395456\pi\)
0.322562 + 0.946548i \(0.395456\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 12.0000 0.901975
\(178\) 0 0
\(179\) −2.82843 −0.211407 −0.105703 0.994398i \(-0.533709\pi\)
−0.105703 + 0.994398i \(0.533709\pi\)
\(180\) 0 0
\(181\) −22.6274 −1.68188 −0.840941 0.541126i \(-0.817998\pi\)
−0.840941 + 0.541126i \(0.817998\pi\)
\(182\) 0 0
\(183\) −5.65685 −0.418167
\(184\) 0 0
\(185\) 16.9706 1.24770
\(186\) 0 0
\(187\) 8.00000 0.585018
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −19.7990 −1.43260 −0.716302 0.697790i \(-0.754167\pi\)
−0.716302 + 0.697790i \(0.754167\pi\)
\(192\) 0 0
\(193\) 10.0000 0.719816 0.359908 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 10.0000 0.712470 0.356235 0.934396i \(-0.384060\pi\)
0.356235 + 0.934396i \(0.384060\pi\)
\(198\) 0 0
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 0 0
\(201\) 5.65685 0.399004
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 24.0000 1.67623
\(206\) 0 0
\(207\) 8.48528 0.589768
\(208\) 0 0
\(209\) 11.3137 0.782586
\(210\) 0 0
\(211\) 22.6274 1.55774 0.778868 0.627188i \(-0.215794\pi\)
0.778868 + 0.627188i \(0.215794\pi\)
\(212\) 0 0
\(213\) 2.82843 0.193801
\(214\) 0 0
\(215\) −32.0000 −2.18238
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −5.65685 −0.382255
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 16.0000 1.07144 0.535720 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) 0 0
\(225\) 3.00000 0.200000
\(226\) 0 0
\(227\) 28.0000 1.85843 0.929213 0.369546i \(-0.120487\pi\)
0.929213 + 0.369546i \(0.120487\pi\)
\(228\) 0 0
\(229\) −11.3137 −0.747631 −0.373815 0.927503i \(-0.621951\pi\)
−0.373815 + 0.927503i \(0.621951\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) −22.6274 −1.47605
\(236\) 0 0
\(237\) 5.65685 0.367452
\(238\) 0 0
\(239\) −14.1421 −0.914779 −0.457389 0.889267i \(-0.651215\pi\)
−0.457389 + 0.889267i \(0.651215\pi\)
\(240\) 0 0
\(241\) 16.9706 1.09317 0.546585 0.837404i \(-0.315928\pi\)
0.546585 + 0.837404i \(0.315928\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) 20.0000 1.26239 0.631194 0.775625i \(-0.282565\pi\)
0.631194 + 0.775625i \(0.282565\pi\)
\(252\) 0 0
\(253\) 24.0000 1.50887
\(254\) 0 0
\(255\) 8.00000 0.500979
\(256\) 0 0
\(257\) 8.48528 0.529297 0.264649 0.964345i \(-0.414744\pi\)
0.264649 + 0.964345i \(0.414744\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) 0 0
\(263\) −14.1421 −0.872041 −0.436021 0.899937i \(-0.643613\pi\)
−0.436021 + 0.899937i \(0.643613\pi\)
\(264\) 0 0
\(265\) −16.9706 −1.04249
\(266\) 0 0
\(267\) −2.82843 −0.173097
\(268\) 0 0
\(269\) −8.48528 −0.517357 −0.258678 0.965964i \(-0.583287\pi\)
−0.258678 + 0.965964i \(0.583287\pi\)
\(270\) 0 0
\(271\) −24.0000 −1.45790 −0.728948 0.684569i \(-0.759990\pi\)
−0.728948 + 0.684569i \(0.759990\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 8.48528 0.511682
\(276\) 0 0
\(277\) 10.0000 0.600842 0.300421 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) 20.0000 1.18888 0.594438 0.804141i \(-0.297374\pi\)
0.594438 + 0.804141i \(0.297374\pi\)
\(284\) 0 0
\(285\) 11.3137 0.670166
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −9.00000 −0.529412
\(290\) 0 0
\(291\) −16.9706 −0.994832
\(292\) 0 0
\(293\) 8.48528 0.495715 0.247858 0.968796i \(-0.420273\pi\)
0.247858 + 0.968796i \(0.420273\pi\)
\(294\) 0 0
\(295\) 33.9411 1.97613
\(296\) 0 0
\(297\) 2.82843 0.164122
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −14.1421 −0.812444
\(304\) 0 0
\(305\) −16.0000 −0.916157
\(306\) 0 0
\(307\) 4.00000 0.228292 0.114146 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(308\) 0 0
\(309\) −16.0000 −0.910208
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −22.6274 −1.27898 −0.639489 0.768801i \(-0.720854\pi\)
−0.639489 + 0.768801i \(0.720854\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −30.0000 −1.68497 −0.842484 0.538721i \(-0.818908\pi\)
−0.842484 + 0.538721i \(0.818908\pi\)
\(318\) 0 0
\(319\) −5.65685 −0.316723
\(320\) 0 0
\(321\) 2.82843 0.157867
\(322\) 0 0
\(323\) 11.3137 0.629512
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −10.0000 −0.553001
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −22.6274 −1.24372 −0.621858 0.783130i \(-0.713622\pi\)
−0.621858 + 0.783130i \(0.713622\pi\)
\(332\) 0 0
\(333\) 6.00000 0.328798
\(334\) 0 0
\(335\) 16.0000 0.874173
\(336\) 0 0
\(337\) −34.0000 −1.85210 −0.926049 0.377403i \(-0.876817\pi\)
−0.926049 + 0.377403i \(0.876817\pi\)
\(338\) 0 0
\(339\) 18.0000 0.977626
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 24.0000 1.29212
\(346\) 0 0
\(347\) 19.7990 1.06287 0.531433 0.847100i \(-0.321654\pi\)
0.531433 + 0.847100i \(0.321654\pi\)
\(348\) 0 0
\(349\) 16.9706 0.908413 0.454207 0.890896i \(-0.349923\pi\)
0.454207 + 0.890896i \(0.349923\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −31.1127 −1.65596 −0.827981 0.560756i \(-0.810510\pi\)
−0.827981 + 0.560756i \(0.810510\pi\)
\(354\) 0 0
\(355\) 8.00000 0.424596
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 2.82843 0.149279 0.0746393 0.997211i \(-0.476219\pi\)
0.0746393 + 0.997211i \(0.476219\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) −3.00000 −0.157459
\(364\) 0 0
\(365\) −16.0000 −0.837478
\(366\) 0 0
\(367\) 32.0000 1.67039 0.835193 0.549957i \(-0.185356\pi\)
0.835193 + 0.549957i \(0.185356\pi\)
\(368\) 0 0
\(369\) 8.48528 0.441726
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 18.0000 0.932005 0.466002 0.884783i \(-0.345694\pi\)
0.466002 + 0.884783i \(0.345694\pi\)
\(374\) 0 0
\(375\) −5.65685 −0.292119
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) −16.9706 −0.869428
\(382\) 0 0
\(383\) 8.00000 0.408781 0.204390 0.978889i \(-0.434479\pi\)
0.204390 + 0.978889i \(0.434479\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −11.3137 −0.575108
\(388\) 0 0
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) 24.0000 1.21373
\(392\) 0 0
\(393\) 12.0000 0.605320
\(394\) 0 0
\(395\) 16.0000 0.805047
\(396\) 0 0
\(397\) −28.2843 −1.41955 −0.709773 0.704430i \(-0.751203\pi\)
−0.709773 + 0.704430i \(0.751203\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 2.82843 0.140546
\(406\) 0 0
\(407\) 16.9706 0.841200
\(408\) 0 0
\(409\) −5.65685 −0.279713 −0.139857 0.990172i \(-0.544664\pi\)
−0.139857 + 0.990172i \(0.544664\pi\)
\(410\) 0 0
\(411\) −10.0000 −0.493264
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −11.3137 −0.555368
\(416\) 0 0
\(417\) 4.00000 0.195881
\(418\) 0 0
\(419\) 4.00000 0.195413 0.0977064 0.995215i \(-0.468849\pi\)
0.0977064 + 0.995215i \(0.468849\pi\)
\(420\) 0 0
\(421\) 26.0000 1.26716 0.633581 0.773676i \(-0.281584\pi\)
0.633581 + 0.773676i \(0.281584\pi\)
\(422\) 0 0
\(423\) −8.00000 −0.388973
\(424\) 0 0
\(425\) 8.48528 0.411597
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −19.7990 −0.953684 −0.476842 0.878989i \(-0.658219\pi\)
−0.476842 + 0.878989i \(0.658219\pi\)
\(432\) 0 0
\(433\) 33.9411 1.63111 0.815553 0.578682i \(-0.196433\pi\)
0.815553 + 0.578682i \(0.196433\pi\)
\(434\) 0 0
\(435\) −5.65685 −0.271225
\(436\) 0 0
\(437\) 33.9411 1.62362
\(438\) 0 0
\(439\) −16.0000 −0.763638 −0.381819 0.924237i \(-0.624702\pi\)
−0.381819 + 0.924237i \(0.624702\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −36.7696 −1.74697 −0.873487 0.486847i \(-0.838147\pi\)
−0.873487 + 0.486847i \(0.838147\pi\)
\(444\) 0 0
\(445\) −8.00000 −0.379236
\(446\) 0 0
\(447\) −6.00000 −0.283790
\(448\) 0 0
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 0 0
\(451\) 24.0000 1.13012
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −6.00000 −0.280668 −0.140334 0.990104i \(-0.544818\pi\)
−0.140334 + 0.990104i \(0.544818\pi\)
\(458\) 0 0
\(459\) 2.82843 0.132020
\(460\) 0 0
\(461\) 19.7990 0.922131 0.461065 0.887366i \(-0.347467\pi\)
0.461065 + 0.887366i \(0.347467\pi\)
\(462\) 0 0
\(463\) −5.65685 −0.262896 −0.131448 0.991323i \(-0.541963\pi\)
−0.131448 + 0.991323i \(0.541963\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 16.9706 0.781962
\(472\) 0 0
\(473\) −32.0000 −1.47136
\(474\) 0 0
\(475\) 12.0000 0.550598
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) 0 0
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −48.0000 −2.17957
\(486\) 0 0
\(487\) −11.3137 −0.512673 −0.256337 0.966588i \(-0.582516\pi\)
−0.256337 + 0.966588i \(0.582516\pi\)
\(488\) 0 0
\(489\) 16.9706 0.767435
\(490\) 0 0
\(491\) −36.7696 −1.65939 −0.829693 0.558219i \(-0.811485\pi\)
−0.829693 + 0.558219i \(0.811485\pi\)
\(492\) 0 0
\(493\) −5.65685 −0.254772
\(494\) 0 0
\(495\) 8.00000 0.359573
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −22.6274 −1.01294 −0.506471 0.862257i \(-0.669050\pi\)
−0.506471 + 0.862257i \(0.669050\pi\)
\(500\) 0 0
\(501\) 24.0000 1.07224
\(502\) 0 0
\(503\) 16.0000 0.713405 0.356702 0.934218i \(-0.383901\pi\)
0.356702 + 0.934218i \(0.383901\pi\)
\(504\) 0 0
\(505\) −40.0000 −1.77998
\(506\) 0 0
\(507\) −13.0000 −0.577350
\(508\) 0 0
\(509\) 14.1421 0.626839 0.313420 0.949615i \(-0.398525\pi\)
0.313420 + 0.949615i \(0.398525\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 4.00000 0.176604
\(514\) 0 0
\(515\) −45.2548 −1.99417
\(516\) 0 0
\(517\) −22.6274 −0.995153
\(518\) 0 0
\(519\) 8.48528 0.372463
\(520\) 0 0
\(521\) −19.7990 −0.867409 −0.433705 0.901055i \(-0.642794\pi\)
−0.433705 + 0.901055i \(0.642794\pi\)
\(522\) 0 0
\(523\) −36.0000 −1.57417 −0.787085 0.616844i \(-0.788411\pi\)
−0.787085 + 0.616844i \(0.788411\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 49.0000 2.13043
\(530\) 0 0
\(531\) 12.0000 0.520756
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 8.00000 0.345870
\(536\) 0 0
\(537\) −2.82843 −0.122056
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 34.0000 1.46177 0.730887 0.682498i \(-0.239107\pi\)
0.730887 + 0.682498i \(0.239107\pi\)
\(542\) 0 0
\(543\) −22.6274 −0.971035
\(544\) 0 0
\(545\) −28.2843 −1.21157
\(546\) 0 0
\(547\) −16.9706 −0.725609 −0.362804 0.931865i \(-0.618181\pi\)
−0.362804 + 0.931865i \(0.618181\pi\)
\(548\) 0 0
\(549\) −5.65685 −0.241429
\(550\) 0 0
\(551\) −8.00000 −0.340811
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 16.9706 0.720360
\(556\) 0 0
\(557\) −30.0000 −1.27114 −0.635570 0.772043i \(-0.719235\pi\)
−0.635570 + 0.772043i \(0.719235\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) 0 0
\(563\) 20.0000 0.842900 0.421450 0.906852i \(-0.361521\pi\)
0.421450 + 0.906852i \(0.361521\pi\)
\(564\) 0 0
\(565\) 50.9117 2.14187
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −26.0000 −1.08998 −0.544988 0.838444i \(-0.683466\pi\)
−0.544988 + 0.838444i \(0.683466\pi\)
\(570\) 0 0
\(571\) 39.5980 1.65712 0.828562 0.559897i \(-0.189159\pi\)
0.828562 + 0.559897i \(0.189159\pi\)
\(572\) 0 0
\(573\) −19.7990 −0.827115
\(574\) 0 0
\(575\) 25.4558 1.06158
\(576\) 0 0
\(577\) 33.9411 1.41299 0.706494 0.707719i \(-0.250276\pi\)
0.706494 + 0.707719i \(0.250276\pi\)
\(578\) 0 0
\(579\) 10.0000 0.415586
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −16.9706 −0.702849
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 20.0000 0.825488 0.412744 0.910847i \(-0.364570\pi\)
0.412744 + 0.910847i \(0.364570\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 10.0000 0.411345
\(592\) 0 0
\(593\) 2.82843 0.116150 0.0580748 0.998312i \(-0.481504\pi\)
0.0580748 + 0.998312i \(0.481504\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −8.00000 −0.327418
\(598\) 0 0
\(599\) 14.1421 0.577832 0.288916 0.957354i \(-0.406705\pi\)
0.288916 + 0.957354i \(0.406705\pi\)
\(600\) 0 0
\(601\) −33.9411 −1.38449 −0.692244 0.721664i \(-0.743378\pi\)
−0.692244 + 0.721664i \(0.743378\pi\)
\(602\) 0 0
\(603\) 5.65685 0.230365
\(604\) 0 0
\(605\) −8.48528 −0.344976
\(606\) 0 0
\(607\) 32.0000 1.29884 0.649420 0.760430i \(-0.275012\pi\)
0.649420 + 0.760430i \(0.275012\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −2.00000 −0.0807792 −0.0403896 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) 0 0
\(615\) 24.0000 0.967773
\(616\) 0 0
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 0 0
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) 0 0
\(621\) 8.48528 0.340503
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −31.0000 −1.24000
\(626\) 0 0
\(627\) 11.3137 0.451826
\(628\) 0 0
\(629\) 16.9706 0.676661
\(630\) 0 0
\(631\) 5.65685 0.225196 0.112598 0.993641i \(-0.464083\pi\)
0.112598 + 0.993641i \(0.464083\pi\)
\(632\) 0 0
\(633\) 22.6274 0.899359
\(634\) 0 0
\(635\) −48.0000 −1.90482
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 2.82843 0.111891
\(640\) 0 0
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 0 0
\(643\) 12.0000 0.473234 0.236617 0.971603i \(-0.423961\pi\)
0.236617 + 0.971603i \(0.423961\pi\)
\(644\) 0 0
\(645\) −32.0000 −1.26000
\(646\) 0 0
\(647\) 24.0000 0.943537 0.471769 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) 0 0
\(649\) 33.9411 1.33231
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −18.0000 −0.704394 −0.352197 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) 0 0
\(655\) 33.9411 1.32619
\(656\) 0 0
\(657\) −5.65685 −0.220695
\(658\) 0 0
\(659\) 25.4558 0.991619 0.495809 0.868431i \(-0.334871\pi\)
0.495809 + 0.868431i \(0.334871\pi\)
\(660\) 0 0
\(661\) −28.2843 −1.10013 −0.550065 0.835122i \(-0.685397\pi\)
−0.550065 + 0.835122i \(0.685397\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −16.9706 −0.657103
\(668\) 0 0
\(669\) 16.0000 0.618596
\(670\) 0 0
\(671\) −16.0000 −0.617673
\(672\) 0 0
\(673\) −26.0000 −1.00223 −0.501113 0.865382i \(-0.667076\pi\)
−0.501113 + 0.865382i \(0.667076\pi\)
\(674\) 0 0
\(675\) 3.00000 0.115470
\(676\) 0 0
\(677\) −36.7696 −1.41317 −0.706584 0.707629i \(-0.749765\pi\)
−0.706584 + 0.707629i \(0.749765\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 28.0000 1.07296
\(682\) 0 0
\(683\) −25.4558 −0.974041 −0.487020 0.873391i \(-0.661916\pi\)
−0.487020 + 0.873391i \(0.661916\pi\)
\(684\) 0 0
\(685\) −28.2843 −1.08069
\(686\) 0 0
\(687\) −11.3137 −0.431645
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 12.0000 0.456502 0.228251 0.973602i \(-0.426699\pi\)
0.228251 + 0.973602i \(0.426699\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 11.3137 0.429153
\(696\) 0 0
\(697\) 24.0000 0.909065
\(698\) 0 0
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) −34.0000 −1.28416 −0.642081 0.766637i \(-0.721929\pi\)
−0.642081 + 0.766637i \(0.721929\pi\)
\(702\) 0 0
\(703\) 24.0000 0.905177
\(704\) 0 0
\(705\) −22.6274 −0.852198
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 14.0000 0.525781 0.262891 0.964826i \(-0.415324\pi\)
0.262891 + 0.964826i \(0.415324\pi\)
\(710\) 0 0
\(711\) 5.65685 0.212149
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −14.1421 −0.528148
\(718\) 0 0
\(719\) 24.0000 0.895049 0.447524 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 16.9706 0.631142
\(724\) 0 0
\(725\) −6.00000 −0.222834
\(726\) 0 0
\(727\) −16.0000 −0.593407 −0.296704 0.954970i \(-0.595887\pi\)
−0.296704 + 0.954970i \(0.595887\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −32.0000 −1.18356
\(732\) 0 0
\(733\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 16.0000 0.589368
\(738\) 0 0
\(739\) 16.9706 0.624272 0.312136 0.950037i \(-0.398955\pi\)
0.312136 + 0.950037i \(0.398955\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −31.1127 −1.14141 −0.570707 0.821154i \(-0.693331\pi\)
−0.570707 + 0.821154i \(0.693331\pi\)
\(744\) 0 0
\(745\) −16.9706 −0.621753
\(746\) 0 0
\(747\) −4.00000 −0.146352
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) 20.0000 0.728841
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −2.00000 −0.0726912 −0.0363456 0.999339i \(-0.511572\pi\)
−0.0363456 + 0.999339i \(0.511572\pi\)
\(758\) 0 0
\(759\) 24.0000 0.871145
\(760\) 0 0
\(761\) 19.7990 0.717713 0.358856 0.933393i \(-0.383167\pi\)
0.358856 + 0.933393i \(0.383167\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 8.00000 0.289241
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 33.9411 1.22395 0.611974 0.790878i \(-0.290376\pi\)
0.611974 + 0.790878i \(0.290376\pi\)
\(770\) 0 0
\(771\) 8.48528 0.305590
\(772\) 0 0
\(773\) 31.1127 1.11905 0.559523 0.828815i \(-0.310984\pi\)
0.559523 + 0.828815i \(0.310984\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 33.9411 1.21607
\(780\) 0 0
\(781\) 8.00000 0.286263
\(782\) 0 0
\(783\) −2.00000 −0.0714742
\(784\) 0 0
\(785\) 48.0000 1.71319
\(786\) 0 0
\(787\) −4.00000 −0.142585 −0.0712923 0.997455i \(-0.522712\pi\)
−0.0712923 + 0.997455i \(0.522712\pi\)
\(788\) 0 0
\(789\) −14.1421 −0.503473
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −16.9706 −0.601884
\(796\) 0 0
\(797\) −25.4558 −0.901692 −0.450846 0.892602i \(-0.648878\pi\)
−0.450846 + 0.892602i \(0.648878\pi\)
\(798\) 0 0
\(799\) −22.6274 −0.800500
\(800\) 0 0
\(801\) −2.82843 −0.0999376
\(802\) 0 0
\(803\) −16.0000 −0.564628
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −8.48528 −0.298696
\(808\) 0 0
\(809\) 10.0000 0.351581 0.175791 0.984428i \(-0.443752\pi\)
0.175791 + 0.984428i \(0.443752\pi\)
\(810\) 0 0
\(811\) 44.0000 1.54505 0.772524 0.634985i \(-0.218994\pi\)
0.772524 + 0.634985i \(0.218994\pi\)
\(812\) 0 0
\(813\) −24.0000 −0.841717
\(814\) 0 0
\(815\) 48.0000 1.68137
\(816\) 0 0
\(817\) −45.2548 −1.58327
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −6.00000 −0.209401 −0.104701 0.994504i \(-0.533388\pi\)
−0.104701 + 0.994504i \(0.533388\pi\)
\(822\) 0 0
\(823\) −5.65685 −0.197186 −0.0985928 0.995128i \(-0.531434\pi\)
−0.0985928 + 0.995128i \(0.531434\pi\)
\(824\) 0 0
\(825\) 8.48528 0.295420
\(826\) 0 0
\(827\) 36.7696 1.27860 0.639301 0.768956i \(-0.279224\pi\)
0.639301 + 0.768956i \(0.279224\pi\)
\(828\) 0 0
\(829\) 22.6274 0.785883 0.392941 0.919564i \(-0.371458\pi\)
0.392941 + 0.919564i \(0.371458\pi\)
\(830\) 0 0
\(831\) 10.0000 0.346896
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 67.8823 2.34916
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −8.00000 −0.276191 −0.138095 0.990419i \(-0.544098\pi\)
−0.138095 + 0.990419i \(0.544098\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) 6.00000 0.206651
\(844\) 0 0
\(845\) −36.7696 −1.26491
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 20.0000 0.686398
\(850\) 0 0
\(851\) 50.9117 1.74523
\(852\) 0 0
\(853\) 39.5980 1.35581 0.677905 0.735150i \(-0.262888\pi\)
0.677905 + 0.735150i \(0.262888\pi\)
\(854\) 0 0
\(855\) 11.3137 0.386921
\(856\) 0 0
\(857\) −2.82843 −0.0966172 −0.0483086 0.998832i \(-0.515383\pi\)
−0.0483086 + 0.998832i \(0.515383\pi\)
\(858\) 0 0
\(859\) −52.0000 −1.77422 −0.887109 0.461561i \(-0.847290\pi\)
−0.887109 + 0.461561i \(0.847290\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 31.1127 1.05909 0.529544 0.848282i \(-0.322363\pi\)
0.529544 + 0.848282i \(0.322363\pi\)
\(864\) 0 0
\(865\) 24.0000 0.816024
\(866\) 0 0
\(867\) −9.00000 −0.305656
\(868\) 0 0
\(869\) 16.0000 0.542763
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −16.9706 −0.574367
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 38.0000 1.28317 0.641584 0.767052i \(-0.278277\pi\)
0.641584 + 0.767052i \(0.278277\pi\)
\(878\) 0 0
\(879\) 8.48528 0.286201
\(880\) 0 0
\(881\) −31.1127 −1.04821 −0.524107 0.851653i \(-0.675601\pi\)
−0.524107 + 0.851653i \(0.675601\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(884\) 0 0
\(885\) 33.9411 1.14092
\(886\) 0 0
\(887\) −24.0000 −0.805841 −0.402921 0.915235i \(-0.632005\pi\)
−0.402921 + 0.915235i \(0.632005\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 2.82843 0.0947559
\(892\) 0 0
\(893\) −32.0000 −1.07084
\(894\) 0 0
\(895\) −8.00000 −0.267411
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −16.9706 −0.565371
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −64.0000 −2.12743
\(906\) 0 0
\(907\) −33.9411 −1.12700 −0.563498 0.826117i \(-0.690545\pi\)
−0.563498 + 0.826117i \(0.690545\pi\)
\(908\) 0 0
\(909\) −14.1421 −0.469065
\(910\) 0 0
\(911\) −42.4264 −1.40565 −0.702825 0.711363i \(-0.748078\pi\)
−0.702825 + 0.711363i \(0.748078\pi\)
\(912\) 0 0
\(913\) −11.3137 −0.374429
\(914\) 0 0
\(915\) −16.0000 −0.528944
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 11.3137 0.373205 0.186602 0.982436i \(-0.440252\pi\)
0.186602 + 0.982436i \(0.440252\pi\)
\(920\) 0 0
\(921\) 4.00000 0.131804
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 18.0000 0.591836
\(926\) 0 0
\(927\) −16.0000 −0.525509
\(928\) 0 0
\(929\) −36.7696 −1.20637 −0.603185 0.797601i \(-0.706102\pi\)
−0.603185 + 0.797601i \(0.706102\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 22.6274 0.739996
\(936\) 0 0
\(937\) 22.6274 0.739205 0.369603 0.929190i \(-0.379494\pi\)
0.369603 + 0.929190i \(0.379494\pi\)
\(938\) 0 0
\(939\) −22.6274 −0.738418
\(940\) 0 0
\(941\) 14.1421 0.461020 0.230510 0.973070i \(-0.425960\pi\)
0.230510 + 0.973070i \(0.425960\pi\)
\(942\) 0 0
\(943\) 72.0000 2.34464
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 14.1421 0.459558 0.229779 0.973243i \(-0.426200\pi\)
0.229779 + 0.973243i \(0.426200\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −30.0000 −0.972817
\(952\) 0 0
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) 0 0
\(955\) −56.0000 −1.81212
\(956\) 0 0
\(957\) −5.65685 −0.182860
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 2.82843 0.0911448
\(964\) 0 0
\(965\) 28.2843 0.910503
\(966\) 0 0
\(967\) −39.5980 −1.27339 −0.636693 0.771118i \(-0.719698\pi\)
−0.636693 + 0.771118i \(0.719698\pi\)
\(968\) 0 0
\(969\) 11.3137 0.363449
\(970\) 0 0
\(971\) −52.0000 −1.66876 −0.834380 0.551190i \(-0.814174\pi\)
−0.834380 + 0.551190i \(0.814174\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −34.0000 −1.08776 −0.543878 0.839164i \(-0.683045\pi\)
−0.543878 + 0.839164i \(0.683045\pi\)
\(978\) 0 0
\(979\) −8.00000 −0.255681
\(980\) 0 0
\(981\) −10.0000 −0.319275
\(982\) 0 0
\(983\) −24.0000 −0.765481 −0.382741 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) 0 0
\(985\) 28.2843 0.901212
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −96.0000 −3.05262
\(990\) 0 0
\(991\) 33.9411 1.07818 0.539088 0.842250i \(-0.318769\pi\)
0.539088 + 0.842250i \(0.318769\pi\)
\(992\) 0 0
\(993\) −22.6274 −0.718059
\(994\) 0 0
\(995\) −22.6274 −0.717337
\(996\) 0 0
\(997\) −50.9117 −1.61239 −0.806195 0.591650i \(-0.798477\pi\)
−0.806195 + 0.591650i \(0.798477\pi\)
\(998\) 0 0
\(999\) 6.00000 0.189832
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 9408.2.a.eb.1.2 2
4.3 odd 2 9408.2.a.dk.1.2 2
7.6 odd 2 9408.2.a.dk.1.1 2
8.3 odd 2 4704.2.a.br.1.1 yes 2
8.5 even 2 4704.2.a.bi.1.1 2
28.27 even 2 inner 9408.2.a.eb.1.1 2
56.13 odd 2 4704.2.a.br.1.2 yes 2
56.27 even 2 4704.2.a.bi.1.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4704.2.a.bi.1.1 2 8.5 even 2
4704.2.a.bi.1.2 yes 2 56.27 even 2
4704.2.a.br.1.1 yes 2 8.3 odd 2
4704.2.a.br.1.2 yes 2 56.13 odd 2
9408.2.a.dk.1.1 2 7.6 odd 2
9408.2.a.dk.1.2 2 4.3 odd 2
9408.2.a.eb.1.1 2 28.27 even 2 inner
9408.2.a.eb.1.2 2 1.1 even 1 trivial