Properties

Label 784.2.f.b.783.1
Level $784$
Weight $2$
Character 784.783
Analytic conductor $6.260$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 784.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.26027151847\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 112)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 783.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 784.783
Dual form 784.2.f.b.783.2

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.73205i q^{5} -2.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.73205i q^{5} -2.00000 q^{9} -1.73205i q^{11} -1.73205i q^{15} -5.19615i q^{17} +7.00000 q^{19} -8.66025i q^{23} +2.00000 q^{25} -5.00000 q^{27} -6.00000 q^{29} +5.00000 q^{31} -1.73205i q^{33} -5.00000 q^{37} +6.92820i q^{41} -3.46410i q^{43} +3.46410i q^{45} +3.00000 q^{47} -5.19615i q^{51} -9.00000 q^{53} -3.00000 q^{55} +7.00000 q^{57} +9.00000 q^{59} -8.66025i q^{61} +5.19615i q^{67} -8.66025i q^{69} +3.46410i q^{71} +1.73205i q^{73} +2.00000 q^{75} +5.19615i q^{79} +1.00000 q^{81} +12.0000 q^{83} -9.00000 q^{85} -6.00000 q^{87} +12.1244i q^{89} +5.00000 q^{93} -12.1244i q^{95} +6.92820i q^{97} +3.46410i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 4q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - 4q^{9} + 14q^{19} + 4q^{25} - 10q^{27} - 12q^{29} + 10q^{31} - 10q^{37} + 6q^{47} - 18q^{53} - 6q^{55} + 14q^{57} + 18q^{59} + 4q^{75} + 2q^{81} + 24q^{83} - 18q^{85} - 12q^{87} + 10q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/784\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(687\) \(689\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 0.288675 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) 0 0
\(5\) − 1.73205i − 0.774597i −0.921954 0.387298i \(-0.873408\pi\)
0.921954 0.387298i \(-0.126592\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) − 1.73205i − 0.522233i −0.965307 0.261116i \(-0.915909\pi\)
0.965307 0.261116i \(-0.0840907\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) − 1.73205i − 0.447214i
\(16\) 0 0
\(17\) − 5.19615i − 1.26025i −0.776493 0.630126i \(-0.783003\pi\)
0.776493 0.630126i \(-0.216997\pi\)
\(18\) 0 0
\(19\) 7.00000 1.60591 0.802955 0.596040i \(-0.203260\pi\)
0.802955 + 0.596040i \(0.203260\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 8.66025i − 1.80579i −0.429863 0.902894i \(-0.641438\pi\)
0.429863 0.902894i \(-0.358562\pi\)
\(24\) 0 0
\(25\) 2.00000 0.400000
\(26\) 0 0
\(27\) −5.00000 −0.962250
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 5.00000 0.898027 0.449013 0.893525i \(-0.351776\pi\)
0.449013 + 0.893525i \(0.351776\pi\)
\(32\) 0 0
\(33\) − 1.73205i − 0.301511i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −5.00000 −0.821995 −0.410997 0.911636i \(-0.634819\pi\)
−0.410997 + 0.911636i \(0.634819\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.92820i 1.08200i 0.841021 + 0.541002i \(0.181955\pi\)
−0.841021 + 0.541002i \(0.818045\pi\)
\(42\) 0 0
\(43\) − 3.46410i − 0.528271i −0.964486 0.264135i \(-0.914913\pi\)
0.964486 0.264135i \(-0.0850865\pi\)
\(44\) 0 0
\(45\) 3.46410i 0.516398i
\(46\) 0 0
\(47\) 3.00000 0.437595 0.218797 0.975770i \(-0.429787\pi\)
0.218797 + 0.975770i \(0.429787\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) − 5.19615i − 0.727607i
\(52\) 0 0
\(53\) −9.00000 −1.23625 −0.618123 0.786082i \(-0.712106\pi\)
−0.618123 + 0.786082i \(0.712106\pi\)
\(54\) 0 0
\(55\) −3.00000 −0.404520
\(56\) 0 0
\(57\) 7.00000 0.927173
\(58\) 0 0
\(59\) 9.00000 1.17170 0.585850 0.810419i \(-0.300761\pi\)
0.585850 + 0.810419i \(0.300761\pi\)
\(60\) 0 0
\(61\) − 8.66025i − 1.10883i −0.832240 0.554416i \(-0.812942\pi\)
0.832240 0.554416i \(-0.187058\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 5.19615i 0.634811i 0.948290 + 0.317406i \(0.102812\pi\)
−0.948290 + 0.317406i \(0.897188\pi\)
\(68\) 0 0
\(69\) − 8.66025i − 1.04257i
\(70\) 0 0
\(71\) 3.46410i 0.411113i 0.978645 + 0.205557i \(0.0659005\pi\)
−0.978645 + 0.205557i \(0.934100\pi\)
\(72\) 0 0
\(73\) 1.73205i 0.202721i 0.994850 + 0.101361i \(0.0323196\pi\)
−0.994850 + 0.101361i \(0.967680\pi\)
\(74\) 0 0
\(75\) 2.00000 0.230940
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 5.19615i 0.584613i 0.956325 + 0.292306i \(0.0944227\pi\)
−0.956325 + 0.292306i \(0.905577\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) −9.00000 −0.976187
\(86\) 0 0
\(87\) −6.00000 −0.643268
\(88\) 0 0
\(89\) 12.1244i 1.28518i 0.766211 + 0.642590i \(0.222140\pi\)
−0.766211 + 0.642590i \(0.777860\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 5.00000 0.518476
\(94\) 0 0
\(95\) − 12.1244i − 1.24393i
\(96\) 0 0
\(97\) 6.92820i 0.703452i 0.936103 + 0.351726i \(0.114405\pi\)
−0.936103 + 0.351726i \(0.885595\pi\)
\(98\) 0 0
\(99\) 3.46410i 0.348155i
\(100\) 0 0
\(101\) − 5.19615i − 0.517036i −0.966006 0.258518i \(-0.916766\pi\)
0.966006 0.258518i \(-0.0832342\pi\)
\(102\) 0 0
\(103\) −1.00000 −0.0985329 −0.0492665 0.998786i \(-0.515688\pi\)
−0.0492665 + 0.998786i \(0.515688\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 5.19615i 0.502331i 0.967944 + 0.251166i \(0.0808138\pi\)
−0.967944 + 0.251166i \(0.919186\pi\)
\(108\) 0 0
\(109\) 11.0000 1.05361 0.526804 0.849987i \(-0.323390\pi\)
0.526804 + 0.849987i \(0.323390\pi\)
\(110\) 0 0
\(111\) −5.00000 −0.474579
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) −15.0000 −1.39876
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 8.00000 0.727273
\(122\) 0 0
\(123\) 6.92820i 0.624695i
\(124\) 0 0
\(125\) − 12.1244i − 1.08444i
\(126\) 0 0
\(127\) 3.46410i 0.307389i 0.988118 + 0.153695i \(0.0491172\pi\)
−0.988118 + 0.153695i \(0.950883\pi\)
\(128\) 0 0
\(129\) − 3.46410i − 0.304997i
\(130\) 0 0
\(131\) −21.0000 −1.83478 −0.917389 0.397991i \(-0.869707\pi\)
−0.917389 + 0.397991i \(0.869707\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 8.66025i 0.745356i
\(136\) 0 0
\(137\) 3.00000 0.256307 0.128154 0.991754i \(-0.459095\pi\)
0.128154 + 0.991754i \(0.459095\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 3.00000 0.252646
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 10.3923i 0.863034i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −9.00000 −0.737309 −0.368654 0.929567i \(-0.620181\pi\)
−0.368654 + 0.929567i \(0.620181\pi\)
\(150\) 0 0
\(151\) 12.1244i 0.986666i 0.869841 + 0.493333i \(0.164222\pi\)
−0.869841 + 0.493333i \(0.835778\pi\)
\(152\) 0 0
\(153\) 10.3923i 0.840168i
\(154\) 0 0
\(155\) − 8.66025i − 0.695608i
\(156\) 0 0
\(157\) 15.5885i 1.24409i 0.782980 + 0.622047i \(0.213699\pi\)
−0.782980 + 0.622047i \(0.786301\pi\)
\(158\) 0 0
\(159\) −9.00000 −0.713746
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 12.1244i 0.949653i 0.880079 + 0.474826i \(0.157489\pi\)
−0.880079 + 0.474826i \(0.842511\pi\)
\(164\) 0 0
\(165\) −3.00000 −0.233550
\(166\) 0 0
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) −14.0000 −1.07061
\(172\) 0 0
\(173\) − 8.66025i − 0.658427i −0.944256 0.329213i \(-0.893216\pi\)
0.944256 0.329213i \(-0.106784\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 9.00000 0.676481
\(178\) 0 0
\(179\) − 15.5885i − 1.16514i −0.812782 0.582568i \(-0.802048\pi\)
0.812782 0.582568i \(-0.197952\pi\)
\(180\) 0 0
\(181\) − 6.92820i − 0.514969i −0.966282 0.257485i \(-0.917106\pi\)
0.966282 0.257485i \(-0.0828937\pi\)
\(182\) 0 0
\(183\) − 8.66025i − 0.640184i
\(184\) 0 0
\(185\) 8.66025i 0.636715i
\(186\) 0 0
\(187\) −9.00000 −0.658145
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 12.1244i 0.877288i 0.898661 + 0.438644i \(0.144541\pi\)
−0.898661 + 0.438644i \(0.855459\pi\)
\(192\) 0 0
\(193\) −5.00000 −0.359908 −0.179954 0.983675i \(-0.557595\pi\)
−0.179954 + 0.983675i \(0.557595\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 0 0
\(199\) −11.0000 −0.779769 −0.389885 0.920864i \(-0.627485\pi\)
−0.389885 + 0.920864i \(0.627485\pi\)
\(200\) 0 0
\(201\) 5.19615i 0.366508i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 12.0000 0.838116
\(206\) 0 0
\(207\) 17.3205i 1.20386i
\(208\) 0 0
\(209\) − 12.1244i − 0.838659i
\(210\) 0 0
\(211\) 24.2487i 1.66935i 0.550743 + 0.834675i \(0.314345\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) 0 0
\(213\) 3.46410i 0.237356i
\(214\) 0 0
\(215\) −6.00000 −0.409197
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 1.73205i 0.117041i
\(220\) 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
\(225\) −4.00000 −0.266667
\(226\) 0 0
\(227\) 21.0000 1.39382 0.696909 0.717159i \(-0.254558\pi\)
0.696909 + 0.717159i \(0.254558\pi\)
\(228\) 0 0
\(229\) 12.1244i 0.801200i 0.916253 + 0.400600i \(0.131198\pi\)
−0.916253 + 0.400600i \(0.868802\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 3.00000 0.196537 0.0982683 0.995160i \(-0.468670\pi\)
0.0982683 + 0.995160i \(0.468670\pi\)
\(234\) 0 0
\(235\) − 5.19615i − 0.338960i
\(236\) 0 0
\(237\) 5.19615i 0.337526i
\(238\) 0 0
\(239\) 10.3923i 0.672222i 0.941822 + 0.336111i \(0.109112\pi\)
−0.941822 + 0.336111i \(0.890888\pi\)
\(240\) 0 0
\(241\) − 12.1244i − 0.780998i −0.920603 0.390499i \(-0.872302\pi\)
0.920603 0.390499i \(-0.127698\pi\)
\(242\) 0 0
\(243\) 16.0000 1.02640
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 12.0000 0.760469
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) −15.0000 −0.943042
\(254\) 0 0
\(255\) −9.00000 −0.563602
\(256\) 0 0
\(257\) 19.0526i 1.18847i 0.804293 + 0.594233i \(0.202544\pi\)
−0.804293 + 0.594233i \(0.797456\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 12.0000 0.742781
\(262\) 0 0
\(263\) − 8.66025i − 0.534014i −0.963695 0.267007i \(-0.913965\pi\)
0.963695 0.267007i \(-0.0860347\pi\)
\(264\) 0 0
\(265\) 15.5885i 0.957591i
\(266\) 0 0
\(267\) 12.1244i 0.741999i
\(268\) 0 0
\(269\) 1.73205i 0.105605i 0.998605 + 0.0528025i \(0.0168154\pi\)
−0.998605 + 0.0528025i \(0.983185\pi\)
\(270\) 0 0
\(271\) −1.00000 −0.0607457 −0.0303728 0.999539i \(-0.509669\pi\)
−0.0303728 + 0.999539i \(0.509669\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 3.46410i − 0.208893i
\(276\) 0 0
\(277\) −17.0000 −1.02143 −0.510716 0.859750i \(-0.670619\pi\)
−0.510716 + 0.859750i \(0.670619\pi\)
\(278\) 0 0
\(279\) −10.0000 −0.598684
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) −11.0000 −0.653882 −0.326941 0.945045i \(-0.606018\pi\)
−0.326941 + 0.945045i \(0.606018\pi\)
\(284\) 0 0
\(285\) − 12.1244i − 0.718185i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −10.0000 −0.588235
\(290\) 0 0
\(291\) 6.92820i 0.406138i
\(292\) 0 0
\(293\) − 20.7846i − 1.21425i −0.794606 0.607125i \(-0.792323\pi\)
0.794606 0.607125i \(-0.207677\pi\)
\(294\) 0 0
\(295\) − 15.5885i − 0.907595i
\(296\) 0 0
\(297\) 8.66025i 0.502519i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) − 5.19615i − 0.298511i
\(304\) 0 0
\(305\) −15.0000 −0.858898
\(306\) 0 0
\(307\) −20.0000 −1.14146 −0.570730 0.821138i \(-0.693340\pi\)
−0.570730 + 0.821138i \(0.693340\pi\)
\(308\) 0 0
\(309\) −1.00000 −0.0568880
\(310\) 0 0
\(311\) 33.0000 1.87126 0.935629 0.352985i \(-0.114833\pi\)
0.935629 + 0.352985i \(0.114833\pi\)
\(312\) 0 0
\(313\) − 29.4449i − 1.66432i −0.554534 0.832161i \(-0.687103\pi\)
0.554534 0.832161i \(-0.312897\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −21.0000 −1.17948 −0.589739 0.807594i \(-0.700769\pi\)
−0.589739 + 0.807594i \(0.700769\pi\)
\(318\) 0 0
\(319\) 10.3923i 0.581857i
\(320\) 0 0
\(321\) 5.19615i 0.290021i
\(322\) 0 0
\(323\) − 36.3731i − 2.02385i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 11.0000 0.608301
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) − 29.4449i − 1.61844i −0.587508 0.809218i \(-0.699891\pi\)
0.587508 0.809218i \(-0.300109\pi\)
\(332\) 0 0
\(333\) 10.0000 0.547997
\(334\) 0 0
\(335\) 9.00000 0.491723
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 0 0
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) − 8.66025i − 0.468979i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −15.0000 −0.807573
\(346\) 0 0
\(347\) − 8.66025i − 0.464907i −0.972608 0.232453i \(-0.925325\pi\)
0.972608 0.232453i \(-0.0746753\pi\)
\(348\) 0 0
\(349\) 27.7128i 1.48343i 0.670714 + 0.741716i \(0.265988\pi\)
−0.670714 + 0.741716i \(0.734012\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 19.0526i − 1.01407i −0.861927 0.507033i \(-0.830742\pi\)
0.861927 0.507033i \(-0.169258\pi\)
\(354\) 0 0
\(355\) 6.00000 0.318447
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 15.5885i − 0.822727i −0.911471 0.411364i \(-0.865053\pi\)
0.911471 0.411364i \(-0.134947\pi\)
\(360\) 0 0
\(361\) 30.0000 1.57895
\(362\) 0 0
\(363\) 8.00000 0.419891
\(364\) 0 0
\(365\) 3.00000 0.157027
\(366\) 0 0
\(367\) 1.00000 0.0521996 0.0260998 0.999659i \(-0.491691\pi\)
0.0260998 + 0.999659i \(0.491691\pi\)
\(368\) 0 0
\(369\) − 13.8564i − 0.721336i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −13.0000 −0.673114 −0.336557 0.941663i \(-0.609263\pi\)
−0.336557 + 0.941663i \(0.609263\pi\)
\(374\) 0 0
\(375\) − 12.1244i − 0.626099i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) − 10.3923i − 0.533817i −0.963722 0.266908i \(-0.913998\pi\)
0.963722 0.266908i \(-0.0860021\pi\)
\(380\) 0 0
\(381\) 3.46410i 0.177471i
\(382\) 0 0
\(383\) 27.0000 1.37964 0.689818 0.723983i \(-0.257691\pi\)
0.689818 + 0.723983i \(0.257691\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 6.92820i 0.352180i
\(388\) 0 0
\(389\) −21.0000 −1.06474 −0.532371 0.846511i \(-0.678699\pi\)
−0.532371 + 0.846511i \(0.678699\pi\)
\(390\) 0 0
\(391\) −45.0000 −2.27575
\(392\) 0 0
\(393\) −21.0000 −1.05931
\(394\) 0 0
\(395\) 9.00000 0.452839
\(396\) 0 0
\(397\) 12.1244i 0.608504i 0.952592 + 0.304252i \(0.0984065\pi\)
−0.952592 + 0.304252i \(0.901594\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 3.00000 0.149813 0.0749064 0.997191i \(-0.476134\pi\)
0.0749064 + 0.997191i \(0.476134\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) − 1.73205i − 0.0860663i
\(406\) 0 0
\(407\) 8.66025i 0.429273i
\(408\) 0 0
\(409\) − 5.19615i − 0.256933i −0.991714 0.128467i \(-0.958994\pi\)
0.991714 0.128467i \(-0.0410055\pi\)
\(410\) 0 0
\(411\) 3.00000 0.147979
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) − 20.7846i − 1.02028i
\(416\) 0 0
\(417\) −4.00000 −0.195881
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) 0 0
\(423\) −6.00000 −0.291730
\(424\) 0 0
\(425\) − 10.3923i − 0.504101i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 29.4449i − 1.41831i −0.705053 0.709155i \(-0.749077\pi\)
0.705053 0.709155i \(-0.250923\pi\)
\(432\) 0 0
\(433\) − 34.6410i − 1.66474i −0.554220 0.832370i \(-0.686983\pi\)
0.554220 0.832370i \(-0.313017\pi\)
\(434\) 0 0
\(435\) 10.3923i 0.498273i
\(436\) 0 0
\(437\) − 60.6218i − 2.89993i
\(438\) 0 0
\(439\) 19.0000 0.906821 0.453410 0.891302i \(-0.350207\pi\)
0.453410 + 0.891302i \(0.350207\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 22.5167i − 1.06980i −0.844916 0.534899i \(-0.820349\pi\)
0.844916 0.534899i \(-0.179651\pi\)
\(444\) 0 0
\(445\) 21.0000 0.995495
\(446\) 0 0
\(447\) −9.00000 −0.425685
\(448\) 0 0
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) 12.0000 0.565058
\(452\) 0 0
\(453\) 12.1244i 0.569652i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −1.00000 −0.0467780 −0.0233890 0.999726i \(-0.507446\pi\)
−0.0233890 + 0.999726i \(0.507446\pi\)
\(458\) 0 0
\(459\) 25.9808i 1.21268i
\(460\) 0 0
\(461\) 27.7128i 1.29071i 0.763881 + 0.645357i \(0.223291\pi\)
−0.763881 + 0.645357i \(0.776709\pi\)
\(462\) 0 0
\(463\) 3.46410i 0.160990i 0.996755 + 0.0804952i \(0.0256502\pi\)
−0.996755 + 0.0804952i \(0.974350\pi\)
\(464\) 0 0
\(465\) − 8.66025i − 0.401610i
\(466\) 0 0
\(467\) 3.00000 0.138823 0.0694117 0.997588i \(-0.477888\pi\)
0.0694117 + 0.997588i \(0.477888\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 15.5885i 0.718278i
\(472\) 0 0
\(473\) −6.00000 −0.275880
\(474\) 0 0
\(475\) 14.0000 0.642364
\(476\) 0 0
\(477\) 18.0000 0.824163
\(478\) 0 0
\(479\) 9.00000 0.411220 0.205610 0.978634i \(-0.434082\pi\)
0.205610 + 0.978634i \(0.434082\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 12.0000 0.544892
\(486\) 0 0
\(487\) − 8.66025i − 0.392434i −0.980561 0.196217i \(-0.937134\pi\)
0.980561 0.196217i \(-0.0628656\pi\)
\(488\) 0 0
\(489\) 12.1244i 0.548282i
\(490\) 0 0
\(491\) 17.3205i 0.781664i 0.920462 + 0.390832i \(0.127813\pi\)
−0.920462 + 0.390832i \(0.872187\pi\)
\(492\) 0 0
\(493\) 31.1769i 1.40414i
\(494\) 0 0
\(495\) 6.00000 0.269680
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 12.1244i 0.542761i 0.962472 + 0.271380i \(0.0874801\pi\)
−0.962472 + 0.271380i \(0.912520\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) 0 0
\(503\) 24.0000 1.07011 0.535054 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) 0 0
\(505\) −9.00000 −0.400495
\(506\) 0 0
\(507\) 13.0000 0.577350
\(508\) 0 0
\(509\) − 15.5885i − 0.690946i −0.938429 0.345473i \(-0.887718\pi\)
0.938429 0.345473i \(-0.112282\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −35.0000 −1.54529
\(514\) 0 0
\(515\) 1.73205i 0.0763233i
\(516\) 0 0
\(517\) − 5.19615i − 0.228527i
\(518\) 0 0
\(519\) − 8.66025i − 0.380143i
\(520\) 0 0
\(521\) 29.4449i 1.29000i 0.764181 + 0.645001i \(0.223143\pi\)
−0.764181 + 0.645001i \(0.776857\pi\)
\(522\) 0 0
\(523\) 23.0000 1.00572 0.502860 0.864368i \(-0.332281\pi\)
0.502860 + 0.864368i \(0.332281\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 25.9808i − 1.13174i
\(528\) 0 0
\(529\) −52.0000 −2.26087
\(530\) 0 0
\(531\) −18.0000 −0.781133
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 9.00000 0.389104
\(536\) 0 0
\(537\) − 15.5885i − 0.672692i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 7.00000 0.300954 0.150477 0.988614i \(-0.451919\pi\)
0.150477 + 0.988614i \(0.451919\pi\)
\(542\) 0 0
\(543\) − 6.92820i − 0.297318i
\(544\) 0 0
\(545\) − 19.0526i − 0.816122i
\(546\) 0 0
\(547\) − 24.2487i − 1.03680i −0.855138 0.518400i \(-0.826528\pi\)
0.855138 0.518400i \(-0.173472\pi\)
\(548\) 0 0
\(549\) 17.3205i 0.739221i
\(550\) 0 0
\(551\) −42.0000 −1.78926
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 8.66025i 0.367607i
\(556\) 0 0
\(557\) 3.00000 0.127114 0.0635570 0.997978i \(-0.479756\pi\)
0.0635570 + 0.997978i \(0.479756\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −9.00000 −0.379980
\(562\) 0 0
\(563\) −15.0000 −0.632175 −0.316087 0.948730i \(-0.602369\pi\)
−0.316087 + 0.948730i \(0.602369\pi\)
\(564\) 0 0
\(565\) − 10.3923i − 0.437208i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 39.0000 1.63497 0.817483 0.575953i \(-0.195369\pi\)
0.817483 + 0.575953i \(0.195369\pi\)
\(570\) 0 0
\(571\) 12.1244i 0.507388i 0.967284 + 0.253694i \(0.0816457\pi\)
−0.967284 + 0.253694i \(0.918354\pi\)
\(572\) 0 0
\(573\) 12.1244i 0.506502i
\(574\) 0 0
\(575\) − 17.3205i − 0.722315i
\(576\) 0 0
\(577\) 15.5885i 0.648956i 0.945893 + 0.324478i \(0.105189\pi\)
−0.945893 + 0.324478i \(0.894811\pi\)
\(578\) 0 0
\(579\) −5.00000 −0.207793
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 15.5885i 0.645608i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 0 0
\(589\) 35.0000 1.44215
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) 0 0
\(593\) 19.0526i 0.782395i 0.920307 + 0.391197i \(0.127939\pi\)
−0.920307 + 0.391197i \(0.872061\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −11.0000 −0.450200
\(598\) 0 0
\(599\) 32.9090i 1.34462i 0.740268 + 0.672312i \(0.234699\pi\)
−0.740268 + 0.672312i \(0.765301\pi\)
\(600\) 0 0
\(601\) − 6.92820i − 0.282607i −0.989966 0.141304i \(-0.954871\pi\)
0.989966 0.141304i \(-0.0451294\pi\)
\(602\) 0 0
\(603\) − 10.3923i − 0.423207i
\(604\) 0 0
\(605\) − 13.8564i − 0.563343i
\(606\) 0 0
\(607\) −29.0000 −1.17707 −0.588537 0.808470i \(-0.700296\pi\)
−0.588537 + 0.808470i \(0.700296\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 11.0000 0.444286 0.222143 0.975014i \(-0.428695\pi\)
0.222143 + 0.975014i \(0.428695\pi\)
\(614\) 0 0
\(615\) 12.0000 0.483887
\(616\) 0 0
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) 0 0
\(619\) −7.00000 −0.281354 −0.140677 0.990056i \(-0.544928\pi\)
−0.140677 + 0.990056i \(0.544928\pi\)
\(620\) 0 0
\(621\) 43.3013i 1.73762i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) 0 0
\(627\) − 12.1244i − 0.484200i
\(628\) 0 0
\(629\) 25.9808i 1.03592i
\(630\) 0 0
\(631\) 45.0333i 1.79275i 0.443298 + 0.896374i \(0.353808\pi\)
−0.443298 + 0.896374i \(0.646192\pi\)
\(632\) 0 0
\(633\) 24.2487i 0.963800i
\(634\) 0 0
\(635\) 6.00000 0.238103
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) − 6.92820i − 0.274075i
\(640\) 0 0
\(641\) −9.00000 −0.355479 −0.177739 0.984078i \(-0.556878\pi\)
−0.177739 + 0.984078i \(0.556878\pi\)
\(642\) 0 0
\(643\) 4.00000 0.157745 0.0788723 0.996885i \(-0.474868\pi\)
0.0788723 + 0.996885i \(0.474868\pi\)
\(644\) 0 0
\(645\) −6.00000 −0.236250
\(646\) 0 0
\(647\) −15.0000 −0.589711 −0.294855 0.955542i \(-0.595271\pi\)
−0.294855 + 0.955542i \(0.595271\pi\)
\(648\) 0 0
\(649\) − 15.5885i − 0.611900i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 3.00000 0.117399 0.0586995 0.998276i \(-0.481305\pi\)
0.0586995 + 0.998276i \(0.481305\pi\)
\(654\) 0 0
\(655\) 36.3731i 1.42121i
\(656\) 0 0
\(657\) − 3.46410i − 0.135147i
\(658\) 0 0
\(659\) 24.2487i 0.944596i 0.881439 + 0.472298i \(0.156575\pi\)
−0.881439 + 0.472298i \(0.843425\pi\)
\(660\) 0 0
\(661\) − 39.8372i − 1.54949i −0.632276 0.774743i \(-0.717879\pi\)
0.632276 0.774743i \(-0.282121\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 51.9615i 2.01196i
\(668\) 0 0
\(669\) 16.0000 0.618596
\(670\) 0 0
\(671\) −15.0000 −0.579069
\(672\) 0 0
\(673\) −50.0000 −1.92736 −0.963679 0.267063i \(-0.913947\pi\)
−0.963679 + 0.267063i \(0.913947\pi\)
\(674\) 0 0
\(675\) −10.0000 −0.384900
\(676\) 0 0
\(677\) 39.8372i 1.53107i 0.643396 + 0.765533i \(0.277525\pi\)
−0.643396 + 0.765533i \(0.722475\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 21.0000 0.804722
\(682\) 0 0
\(683\) − 43.3013i − 1.65688i −0.560080 0.828439i \(-0.689230\pi\)
0.560080 0.828439i \(-0.310770\pi\)
\(684\) 0 0
\(685\) − 5.19615i − 0.198535i
\(686\) 0 0
\(687\) 12.1244i 0.462573i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −13.0000 −0.494543 −0.247272 0.968946i \(-0.579534\pi\)
−0.247272 + 0.968946i \(0.579534\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 6.92820i 0.262802i
\(696\) 0 0
\(697\) 36.0000 1.36360
\(698\) 0 0
\(699\) 3.00000 0.113470
\(700\) 0 0
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) 0 0
\(703\) −35.0000 −1.32005
\(704\) 0 0
\(705\) − 5.19615i − 0.195698i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 35.0000 1.31445 0.657226 0.753693i \(-0.271730\pi\)
0.657226 + 0.753693i \(0.271730\pi\)
\(710\) 0 0
\(711\) − 10.3923i − 0.389742i
\(712\) 0 0
\(713\) − 43.3013i − 1.62165i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 10.3923i 0.388108i
\(718\) 0 0
\(719\) −33.0000 −1.23069 −0.615346 0.788257i \(-0.710984\pi\)
−0.615346 + 0.788257i \(0.710984\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) − 12.1244i − 0.450910i
\(724\) 0 0
\(725\) −12.0000 −0.445669
\(726\) 0 0
\(727\) 32.0000 1.18681 0.593407 0.804902i \(-0.297782\pi\)
0.593407 + 0.804902i \(0.297782\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −18.0000 −0.665754
\(732\) 0 0
\(733\) 19.0526i 0.703722i 0.936052 + 0.351861i \(0.114451\pi\)
−0.936052 + 0.351861i \(0.885549\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 9.00000 0.331519
\(738\) 0 0
\(739\) − 1.73205i − 0.0637145i −0.999492 0.0318573i \(-0.989858\pi\)
0.999492 0.0318573i \(-0.0101422\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 38.1051i − 1.39794i −0.715150 0.698971i \(-0.753642\pi\)
0.715150 0.698971i \(-0.246358\pi\)
\(744\) 0 0
\(745\) 15.5885i 0.571117i
\(746\) 0 0
\(747\) −24.0000 −0.878114
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 39.8372i 1.45368i 0.686807 + 0.726839i \(0.259012\pi\)
−0.686807 + 0.726839i \(0.740988\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 21.0000 0.764268
\(756\) 0 0
\(757\) 26.0000 0.944986 0.472493 0.881334i \(-0.343354\pi\)
0.472493 + 0.881334i \(0.343354\pi\)
\(758\) 0 0
\(759\) −15.0000 −0.544466
\(760\) 0 0
\(761\) 5.19615i 0.188360i 0.995555 + 0.0941802i \(0.0300230\pi\)
−0.995555 + 0.0941802i \(0.969977\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 18.0000 0.650791
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) − 6.92820i − 0.249837i −0.992167 0.124919i \(-0.960133\pi\)
0.992167 0.124919i \(-0.0398670\pi\)
\(770\) 0 0
\(771\) 19.0526i 0.686161i
\(772\) 0 0
\(773\) 8.66025i 0.311488i 0.987797 + 0.155744i \(0.0497774\pi\)
−0.987797 + 0.155744i \(0.950223\pi\)
\(774\) 0 0
\(775\) 10.0000 0.359211
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 48.4974i 1.73760i
\(780\) 0 0
\(781\) 6.00000 0.214697
\(782\) 0 0
\(783\) 30.0000 1.07211
\(784\) 0 0
\(785\) 27.0000 0.963671
\(786\) 0 0
\(787\) −7.00000 −0.249523 −0.124762 0.992187i \(-0.539817\pi\)
−0.124762 + 0.992187i \(0.539817\pi\)
\(788\) 0 0
\(789\) − 8.66025i − 0.308313i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 15.5885i 0.552866i
\(796\) 0 0
\(797\) 13.8564i 0.490819i 0.969419 + 0.245410i \(0.0789224\pi\)
−0.969419 + 0.245410i \(0.921078\pi\)
\(798\) 0 0
\(799\) − 15.5885i − 0.551480i
\(800\) 0 0
\(801\) − 24.2487i − 0.856786i
\(802\) 0 0
\(803\) 3.00000 0.105868
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 1.73205i 0.0609711i
\(808\) 0 0
\(809\) 15.0000 0.527372 0.263686 0.964609i \(-0.415062\pi\)
0.263686 + 0.964609i \(0.415062\pi\)
\(810\) 0 0
\(811\) −4.00000 −0.140459 −0.0702295 0.997531i \(-0.522373\pi\)
−0.0702295 + 0.997531i \(0.522373\pi\)
\(812\) 0 0
\(813\) −1.00000 −0.0350715
\(814\) 0 0
\(815\) 21.0000 0.735598
\(816\) 0 0
\(817\) − 24.2487i − 0.848355i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −57.0000 −1.98931 −0.994657 0.103236i \(-0.967080\pi\)
−0.994657 + 0.103236i \(0.967080\pi\)
\(822\) 0 0
\(823\) − 22.5167i − 0.784881i −0.919777 0.392441i \(-0.871631\pi\)
0.919777 0.392441i \(-0.128369\pi\)
\(824\) 0 0
\(825\) − 3.46410i − 0.120605i
\(826\) 0 0
\(827\) 38.1051i 1.32504i 0.749042 + 0.662522i \(0.230514\pi\)
−0.749042 + 0.662522i \(0.769486\pi\)
\(828\) 0 0
\(829\) 8.66025i 0.300783i 0.988627 + 0.150392i \(0.0480534\pi\)
−0.988627 + 0.150392i \(0.951947\pi\)
\(830\) 0 0
\(831\) −17.0000 −0.589723
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) − 20.7846i − 0.719281i
\(836\) 0 0
\(837\) −25.0000 −0.864126
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) 6.00000 0.206651
\(844\) 0 0
\(845\) − 22.5167i − 0.774597i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −11.0000 −0.377519
\(850\) 0 0
\(851\) 43.3013i 1.48435i
\(852\) 0 0
\(853\) − 13.8564i − 0.474434i −0.971457 0.237217i \(-0.923765\pi\)
0.971457 0.237217i \(-0.0762353\pi\)
\(854\) 0 0
\(855\) 24.2487i 0.829288i
\(856\) 0 0
\(857\) 43.3013i 1.47914i 0.673078 + 0.739572i \(0.264972\pi\)
−0.673078 + 0.739572i \(0.735028\pi\)
\(858\) 0 0
\(859\) −41.0000 −1.39890 −0.699451 0.714681i \(-0.746572\pi\)
−0.699451 + 0.714681i \(0.746572\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 46.7654i 1.59191i 0.605355 + 0.795956i \(0.293031\pi\)
−0.605355 + 0.795956i \(0.706969\pi\)
\(864\) 0 0
\(865\) −15.0000 −0.510015
\(866\) 0 0
\(867\) −10.0000 −0.339618
\(868\) 0 0
\(869\) 9.00000 0.305304
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) − 13.8564i − 0.468968i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 47.0000 1.58708 0.793539 0.608520i \(-0.208236\pi\)
0.793539 + 0.608520i \(0.208236\pi\)
\(878\) 0 0
\(879\) − 20.7846i − 0.701047i
\(880\) 0 0
\(881\) − 13.8564i − 0.466834i −0.972377 0.233417i \(-0.925009\pi\)
0.972377 0.233417i \(-0.0749907\pi\)
\(882\) 0 0
\(883\) − 45.0333i − 1.51549i −0.652550 0.757746i \(-0.726301\pi\)
0.652550 0.757746i \(-0.273699\pi\)
\(884\) 0 0
\(885\) − 15.5885i − 0.524000i
\(886\) 0 0
\(887\) 3.00000 0.100730 0.0503651 0.998731i \(-0.483962\pi\)
0.0503651 + 0.998731i \(0.483962\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) − 1.73205i − 0.0580259i
\(892\) 0 0
\(893\) 21.0000 0.702738
\(894\) 0 0
\(895\) −27.0000 −0.902510
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −30.0000 −1.00056
\(900\) 0 0
\(901\) 46.7654i 1.55798i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −12.0000 −0.398893
\(906\) 0 0
\(907\) 5.19615i 0.172535i 0.996272 + 0.0862677i \(0.0274940\pi\)
−0.996272 + 0.0862677i \(0.972506\pi\)
\(908\) 0 0
\(909\) 10.3923i 0.344691i
\(910\) 0 0
\(911\) − 10.3923i − 0.344312i −0.985070 0.172156i \(-0.944927\pi\)
0.985070 0.172156i \(-0.0550734\pi\)
\(912\) 0 0
\(913\) − 20.7846i − 0.687870i
\(914\) 0 0
\(915\) −15.0000 −0.495885
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) − 43.3013i − 1.42838i −0.699953 0.714189i \(-0.746796\pi\)
0.699953 0.714189i \(-0.253204\pi\)
\(920\) 0 0
\(921\) −20.0000 −0.659022
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −10.0000 −0.328798
\(926\) 0 0
\(927\) 2.00000 0.0656886
\(928\) 0 0
\(929\) 25.9808i 0.852401i 0.904629 + 0.426201i \(0.140148\pi\)
−0.904629 + 0.426201i \(0.859852\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 33.0000 1.08037
\(934\) 0 0
\(935\) 15.5885i 0.509797i
\(936\) 0 0
\(937\) 41.5692i 1.35801i 0.734135 + 0.679004i \(0.237588\pi\)
−0.734135 + 0.679004i \(0.762412\pi\)
\(938\) 0 0
\(939\) − 29.4449i − 0.960897i
\(940\) 0 0
\(941\) 29.4449i 0.959875i 0.877303 + 0.479938i \(0.159341\pi\)
−0.877303 + 0.479938i \(0.840659\pi\)
\(942\) 0 0
\(943\) 60.0000 1.95387
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 29.4449i − 0.956830i −0.878134 0.478415i \(-0.841212\pi\)
0.878134 0.478415i \(-0.158788\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −21.0000 −0.680972
\(952\) 0 0
\(953\) −18.0000 −0.583077 −0.291539 0.956559i \(-0.594167\pi\)
−0.291539 + 0.956559i \(0.594167\pi\)
\(954\) 0 0
\(955\) 21.0000 0.679544
\(956\) 0 0
\(957\) 10.3923i 0.335936i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −6.00000 −0.193548
\(962\) 0 0
\(963\) − 10.3923i − 0.334887i
\(964\) 0 0
\(965\) 8.66025i 0.278783i
\(966\) 0 0
\(967\) 31.1769i 1.00258i 0.865279 + 0.501291i \(0.167141\pi\)
−0.865279 + 0.501291i \(0.832859\pi\)
\(968\) 0 0
\(969\) − 36.3731i − 1.16847i
\(970\) 0 0
\(971\) 3.00000 0.0962746 0.0481373 0.998841i \(-0.484672\pi\)
0.0481373 + 0.998841i \(0.484672\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 27.0000 0.863807 0.431903 0.901920i \(-0.357842\pi\)
0.431903 + 0.901920i \(0.357842\pi\)
\(978\) 0 0
\(979\) 21.0000 0.671163
\(980\) 0 0
\(981\) −22.0000 −0.702406
\(982\) 0 0
\(983\) −3.00000 −0.0956851 −0.0478426 0.998855i \(-0.515235\pi\)
−0.0478426 + 0.998855i \(0.515235\pi\)
\(984\) 0 0
\(985\) 10.3923i 0.331126i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −30.0000 −0.953945
\(990\) 0 0
\(991\) 5.19615i 0.165061i 0.996589 + 0.0825306i \(0.0263002\pi\)
−0.996589 + 0.0825306i \(0.973700\pi\)
\(992\) 0 0
\(993\) − 29.4449i − 0.934405i
\(994\) 0 0
\(995\) 19.0526i 0.604007i
\(996\) 0 0
\(997\) 1.73205i 0.0548546i 0.999624 + 0.0274273i \(0.00873148\pi\)
−0.999624 + 0.0274273i \(0.991269\pi\)
\(998\) 0 0
\(999\) 25.0000 0.790965
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 784.2.f.b.783.1 2
3.2 odd 2 7056.2.b.m.1567.2 2
4.3 odd 2 784.2.f.a.783.1 2
7.2 even 3 784.2.p.c.31.1 2
7.3 odd 6 784.2.p.d.607.1 2
7.4 even 3 112.2.p.a.47.1 yes 2
7.5 odd 6 112.2.p.b.31.1 yes 2
7.6 odd 2 784.2.f.a.783.2 2
8.3 odd 2 3136.2.f.b.3135.2 2
8.5 even 2 3136.2.f.a.3135.2 2
12.11 even 2 7056.2.b.b.1567.2 2
21.5 even 6 1008.2.cs.c.703.1 2
21.11 odd 6 1008.2.cs.f.271.1 2
21.20 even 2 7056.2.b.b.1567.1 2
28.3 even 6 784.2.p.c.607.1 2
28.11 odd 6 112.2.p.b.47.1 yes 2
28.19 even 6 112.2.p.a.31.1 2
28.23 odd 6 784.2.p.d.31.1 2
28.27 even 2 inner 784.2.f.b.783.2 2
56.5 odd 6 448.2.p.a.255.1 2
56.11 odd 6 448.2.p.a.383.1 2
56.13 odd 2 3136.2.f.b.3135.1 2
56.19 even 6 448.2.p.b.255.1 2
56.27 even 2 3136.2.f.a.3135.1 2
56.53 even 6 448.2.p.b.383.1 2
84.11 even 6 1008.2.cs.c.271.1 2
84.47 odd 6 1008.2.cs.f.703.1 2
84.83 odd 2 7056.2.b.m.1567.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.2.p.a.31.1 2 28.19 even 6
112.2.p.a.47.1 yes 2 7.4 even 3
112.2.p.b.31.1 yes 2 7.5 odd 6
112.2.p.b.47.1 yes 2 28.11 odd 6
448.2.p.a.255.1 2 56.5 odd 6
448.2.p.a.383.1 2 56.11 odd 6
448.2.p.b.255.1 2 56.19 even 6
448.2.p.b.383.1 2 56.53 even 6
784.2.f.a.783.1 2 4.3 odd 2
784.2.f.a.783.2 2 7.6 odd 2
784.2.f.b.783.1 2 1.1 even 1 trivial
784.2.f.b.783.2 2 28.27 even 2 inner
784.2.p.c.31.1 2 7.2 even 3
784.2.p.c.607.1 2 28.3 even 6
784.2.p.d.31.1 2 28.23 odd 6
784.2.p.d.607.1 2 7.3 odd 6
1008.2.cs.c.271.1 2 84.11 even 6
1008.2.cs.c.703.1 2 21.5 even 6
1008.2.cs.f.271.1 2 21.11 odd 6
1008.2.cs.f.703.1 2 84.47 odd 6
3136.2.f.a.3135.1 2 56.27 even 2
3136.2.f.a.3135.2 2 8.5 even 2
3136.2.f.b.3135.1 2 56.13 odd 2
3136.2.f.b.3135.2 2 8.3 odd 2
7056.2.b.b.1567.1 2 21.20 even 2
7056.2.b.b.1567.2 2 12.11 even 2
7056.2.b.m.1567.1 2 84.83 odd 2
7056.2.b.m.1567.2 2 3.2 odd 2