Properties

Label 7056.2.a.cy.1.4
Level 7056
Weight 2
Character 7056.1
Self dual yes
Analytic conductor 56.342
Analytic rank 0
Dimension 4
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 7056 = 2^{4} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7056.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(56.3424436662\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{7})\)
Defining polynomial: \(x^{4} - 8 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1764)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(2.57794\) of defining polynomial
Character \(\chi\) \(=\) 7056.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.74166 q^{5} +O(q^{10})\) \(q+3.74166 q^{5} +5.29150 q^{11} -4.24264 q^{13} +3.74166 q^{17} -2.82843 q^{19} -5.29150 q^{23} +9.00000 q^{25} +5.29150 q^{29} -8.48528 q^{31} +4.00000 q^{37} +3.74166 q^{41} -8.00000 q^{43} +7.48331 q^{47} +10.5830 q^{53} +19.7990 q^{55} +7.48331 q^{59} +9.89949 q^{61} -15.8745 q^{65} -12.0000 q^{67} +15.8745 q^{71} +1.41421 q^{73} +4.00000 q^{79} +14.9666 q^{83} +14.0000 q^{85} -3.74166 q^{89} -10.5830 q^{95} -9.89949 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 36q^{25} + 16q^{37} - 32q^{43} - 48q^{67} + 16q^{79} + 56q^{85} + O(q^{100}) \)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 3.74166 1.67332 0.836660 0.547723i \(-0.184505\pi\)
0.836660 + 0.547723i \(0.184505\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.29150 1.59545 0.797724 0.603023i \(-0.206037\pi\)
0.797724 + 0.603023i \(0.206037\pi\)
\(12\) 0 0
\(13\) −4.24264 −1.17670 −0.588348 0.808608i \(-0.700222\pi\)
−0.588348 + 0.808608i \(0.700222\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.74166 0.907485 0.453743 0.891133i \(-0.350089\pi\)
0.453743 + 0.891133i \(0.350089\pi\)
\(18\) 0 0
\(19\) −2.82843 −0.648886 −0.324443 0.945905i \(-0.605177\pi\)
−0.324443 + 0.945905i \(0.605177\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −5.29150 −1.10335 −0.551677 0.834058i \(-0.686012\pi\)
−0.551677 + 0.834058i \(0.686012\pi\)
\(24\) 0 0
\(25\) 9.00000 1.80000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 5.29150 0.982607 0.491304 0.870988i \(-0.336521\pi\)
0.491304 + 0.870988i \(0.336521\pi\)
\(30\) 0 0
\(31\) −8.48528 −1.52400 −0.762001 0.647576i \(-0.775783\pi\)
−0.762001 + 0.647576i \(0.775783\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 4.00000 0.657596 0.328798 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.74166 0.584349 0.292174 0.956365i \(-0.405621\pi\)
0.292174 + 0.956365i \(0.405621\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 7.48331 1.09155 0.545777 0.837931i \(-0.316235\pi\)
0.545777 + 0.837931i \(0.316235\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 10.5830 1.45369 0.726844 0.686803i \(-0.240986\pi\)
0.726844 + 0.686803i \(0.240986\pi\)
\(54\) 0 0
\(55\) 19.7990 2.66970
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 7.48331 0.974245 0.487122 0.873334i \(-0.338047\pi\)
0.487122 + 0.873334i \(0.338047\pi\)
\(60\) 0 0
\(61\) 9.89949 1.26750 0.633750 0.773538i \(-0.281515\pi\)
0.633750 + 0.773538i \(0.281515\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −15.8745 −1.96899
\(66\) 0 0
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 15.8745 1.88396 0.941979 0.335673i \(-0.108964\pi\)
0.941979 + 0.335673i \(0.108964\pi\)
\(72\) 0 0
\(73\) 1.41421 0.165521 0.0827606 0.996569i \(-0.473626\pi\)
0.0827606 + 0.996569i \(0.473626\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 14.9666 1.64280 0.821401 0.570352i \(-0.193193\pi\)
0.821401 + 0.570352i \(0.193193\pi\)
\(84\) 0 0
\(85\) 14.0000 1.51851
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −3.74166 −0.396615 −0.198307 0.980140i \(-0.563544\pi\)
−0.198307 + 0.980140i \(0.563544\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −10.5830 −1.08579
\(96\) 0 0
\(97\) −9.89949 −1.00514 −0.502571 0.864536i \(-0.667612\pi\)
−0.502571 + 0.864536i \(0.667612\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −11.2250 −1.11693 −0.558463 0.829529i \(-0.688609\pi\)
−0.558463 + 0.829529i \(0.688609\pi\)
\(102\) 0 0
\(103\) 2.82843 0.278693 0.139347 0.990244i \(-0.455500\pi\)
0.139347 + 0.990244i \(0.455500\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −5.29150 −0.511549 −0.255774 0.966736i \(-0.582330\pi\)
−0.255774 + 0.966736i \(0.582330\pi\)
\(108\) 0 0
\(109\) 16.0000 1.53252 0.766261 0.642529i \(-0.222115\pi\)
0.766261 + 0.642529i \(0.222115\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) −19.7990 −1.84627
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 17.0000 1.54545
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 14.9666 1.33866
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −15.8745 −1.35625 −0.678125 0.734946i \(-0.737207\pi\)
−0.678125 + 0.734946i \(0.737207\pi\)
\(138\) 0 0
\(139\) 5.65685 0.479808 0.239904 0.970797i \(-0.422884\pi\)
0.239904 + 0.970797i \(0.422884\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −22.4499 −1.87736
\(144\) 0 0
\(145\) 19.7990 1.64422
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 10.5830 0.866994 0.433497 0.901155i \(-0.357280\pi\)
0.433497 + 0.901155i \(0.357280\pi\)
\(150\) 0 0
\(151\) 4.00000 0.325515 0.162758 0.986666i \(-0.447961\pi\)
0.162758 + 0.986666i \(0.447961\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −31.7490 −2.55014
\(156\) 0 0
\(157\) −9.89949 −0.790066 −0.395033 0.918667i \(-0.629267\pi\)
−0.395033 + 0.918667i \(0.629267\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 20.0000 1.56652 0.783260 0.621694i \(-0.213555\pi\)
0.783260 + 0.621694i \(0.213555\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −7.48331 −0.579076 −0.289538 0.957166i \(-0.593502\pi\)
−0.289538 + 0.957166i \(0.593502\pi\)
\(168\) 0 0
\(169\) 5.00000 0.384615
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 11.2250 0.853419 0.426709 0.904389i \(-0.359673\pi\)
0.426709 + 0.904389i \(0.359673\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 15.8745 1.18652 0.593258 0.805012i \(-0.297841\pi\)
0.593258 + 0.805012i \(0.297841\pi\)
\(180\) 0 0
\(181\) 4.24264 0.315353 0.157676 0.987491i \(-0.449600\pi\)
0.157676 + 0.987491i \(0.449600\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 14.9666 1.10037
\(186\) 0 0
\(187\) 19.7990 1.44785
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −5.29150 −0.382880 −0.191440 0.981504i \(-0.561316\pi\)
−0.191440 + 0.981504i \(0.561316\pi\)
\(192\) 0 0
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −10.5830 −0.754008 −0.377004 0.926212i \(-0.623046\pi\)
−0.377004 + 0.926212i \(0.623046\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 14.0000 0.977802
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −14.9666 −1.03526
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −29.9333 −2.04143
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −15.8745 −1.06783
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 22.4499 1.49006 0.745028 0.667034i \(-0.232436\pi\)
0.745028 + 0.667034i \(0.232436\pi\)
\(228\) 0 0
\(229\) −12.7279 −0.841085 −0.420542 0.907273i \(-0.638160\pi\)
−0.420542 + 0.907273i \(0.638160\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 15.8745 1.03997 0.519987 0.854174i \(-0.325937\pi\)
0.519987 + 0.854174i \(0.325937\pi\)
\(234\) 0 0
\(235\) 28.0000 1.82652
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 15.8745 1.02684 0.513418 0.858138i \(-0.328379\pi\)
0.513418 + 0.858138i \(0.328379\pi\)
\(240\) 0 0
\(241\) −29.6985 −1.91305 −0.956524 0.291654i \(-0.905794\pi\)
−0.956524 + 0.291654i \(0.905794\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 12.0000 0.763542
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −7.48331 −0.472343 −0.236171 0.971711i \(-0.575893\pi\)
−0.236171 + 0.971711i \(0.575893\pi\)
\(252\) 0 0
\(253\) −28.0000 −1.76034
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −11.2250 −0.700195 −0.350097 0.936713i \(-0.613851\pi\)
−0.350097 + 0.936713i \(0.613851\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −15.8745 −0.978864 −0.489432 0.872041i \(-0.662796\pi\)
−0.489432 + 0.872041i \(0.662796\pi\)
\(264\) 0 0
\(265\) 39.5980 2.43248
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −3.74166 −0.228133 −0.114066 0.993473i \(-0.536388\pi\)
−0.114066 + 0.993473i \(0.536388\pi\)
\(270\) 0 0
\(271\) −19.7990 −1.20270 −0.601351 0.798985i \(-0.705371\pi\)
−0.601351 + 0.798985i \(0.705371\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 47.6235 2.87181
\(276\) 0 0
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −5.29150 −0.315665 −0.157832 0.987466i \(-0.550451\pi\)
−0.157832 + 0.987466i \(0.550451\pi\)
\(282\) 0 0
\(283\) −31.1127 −1.84946 −0.924729 0.380626i \(-0.875708\pi\)
−0.924729 + 0.380626i \(0.875708\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −3.00000 −0.176471
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −26.1916 −1.53013 −0.765065 0.643953i \(-0.777293\pi\)
−0.765065 + 0.643953i \(0.777293\pi\)
\(294\) 0 0
\(295\) 28.0000 1.63022
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 22.4499 1.29831
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 37.0405 2.12093
\(306\) 0 0
\(307\) −14.1421 −0.807134 −0.403567 0.914950i \(-0.632230\pi\)
−0.403567 + 0.914950i \(0.632230\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −7.48331 −0.424340 −0.212170 0.977233i \(-0.568053\pi\)
−0.212170 + 0.977233i \(0.568053\pi\)
\(312\) 0 0
\(313\) −12.7279 −0.719425 −0.359712 0.933063i \(-0.617125\pi\)
−0.359712 + 0.933063i \(0.617125\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 10.5830 0.594401 0.297200 0.954815i \(-0.403947\pi\)
0.297200 + 0.954815i \(0.403947\pi\)
\(318\) 0 0
\(319\) 28.0000 1.56770
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −10.5830 −0.588854
\(324\) 0 0
\(325\) −38.1838 −2.11805
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −44.8999 −2.45314
\(336\) 0 0
\(337\) 8.00000 0.435788 0.217894 0.975972i \(-0.430081\pi\)
0.217894 + 0.975972i \(0.430081\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −44.8999 −2.43147
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −15.8745 −0.852188 −0.426094 0.904679i \(-0.640111\pi\)
−0.426094 + 0.904679i \(0.640111\pi\)
\(348\) 0 0
\(349\) 29.6985 1.58972 0.794862 0.606791i \(-0.207543\pi\)
0.794862 + 0.606791i \(0.207543\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −18.7083 −0.995742 −0.497871 0.867251i \(-0.665885\pi\)
−0.497871 + 0.867251i \(0.665885\pi\)
\(354\) 0 0
\(355\) 59.3970 3.15246
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 5.29150 0.279275 0.139637 0.990203i \(-0.455406\pi\)
0.139637 + 0.990203i \(0.455406\pi\)
\(360\) 0 0
\(361\) −11.0000 −0.578947
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 5.29150 0.276970
\(366\) 0 0
\(367\) 11.3137 0.590571 0.295285 0.955409i \(-0.404585\pi\)
0.295285 + 0.955409i \(0.404585\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 10.0000 0.517780 0.258890 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −22.4499 −1.15623
\(378\) 0 0
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 14.9666 0.764759 0.382380 0.924005i \(-0.375105\pi\)
0.382380 + 0.924005i \(0.375105\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 5.29150 0.268290 0.134145 0.990962i \(-0.457171\pi\)
0.134145 + 0.990962i \(0.457171\pi\)
\(390\) 0 0
\(391\) −19.7990 −1.00128
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 14.9666 0.753053
\(396\) 0 0
\(397\) 32.5269 1.63248 0.816239 0.577714i \(-0.196055\pi\)
0.816239 + 0.577714i \(0.196055\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 15.8745 0.792735 0.396368 0.918092i \(-0.370271\pi\)
0.396368 + 0.918092i \(0.370271\pi\)
\(402\) 0 0
\(403\) 36.0000 1.79329
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 21.1660 1.04916
\(408\) 0 0
\(409\) −21.2132 −1.04893 −0.524463 0.851433i \(-0.675734\pi\)
−0.524463 + 0.851433i \(0.675734\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 56.0000 2.74893
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 7.48331 0.365584 0.182792 0.983152i \(-0.441487\pi\)
0.182792 + 0.983152i \(0.441487\pi\)
\(420\) 0 0
\(421\) 6.00000 0.292422 0.146211 0.989253i \(-0.453292\pi\)
0.146211 + 0.989253i \(0.453292\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 33.6749 1.63347
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 15.8745 0.764648 0.382324 0.924028i \(-0.375124\pi\)
0.382324 + 0.924028i \(0.375124\pi\)
\(432\) 0 0
\(433\) −4.24264 −0.203888 −0.101944 0.994790i \(-0.532506\pi\)
−0.101944 + 0.994790i \(0.532506\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 14.9666 0.715951
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −5.29150 −0.251407 −0.125703 0.992068i \(-0.540119\pi\)
−0.125703 + 0.992068i \(0.540119\pi\)
\(444\) 0 0
\(445\) −14.0000 −0.663664
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 19.7990 0.932298
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 18.0000 0.842004 0.421002 0.907060i \(-0.361678\pi\)
0.421002 + 0.907060i \(0.361678\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −3.74166 −0.174266 −0.0871332 0.996197i \(-0.527771\pi\)
−0.0871332 + 0.996197i \(0.527771\pi\)
\(462\) 0 0
\(463\) 8.00000 0.371792 0.185896 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −37.4166 −1.73143 −0.865716 0.500535i \(-0.833137\pi\)
−0.865716 + 0.500535i \(0.833137\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −42.3320 −1.94643
\(474\) 0 0
\(475\) −25.4558 −1.16799
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −37.4166 −1.70961 −0.854803 0.518952i \(-0.826322\pi\)
−0.854803 + 0.518952i \(0.826322\pi\)
\(480\) 0 0
\(481\) −16.9706 −0.773791
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −37.0405 −1.68192
\(486\) 0 0
\(487\) 12.0000 0.543772 0.271886 0.962329i \(-0.412353\pi\)
0.271886 + 0.962329i \(0.412353\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −15.8745 −0.716407 −0.358203 0.933644i \(-0.616611\pi\)
−0.358203 + 0.933644i \(0.616611\pi\)
\(492\) 0 0
\(493\) 19.7990 0.891702
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −32.0000 −1.43252 −0.716258 0.697835i \(-0.754147\pi\)
−0.716258 + 0.697835i \(0.754147\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 14.9666 0.667329 0.333665 0.942692i \(-0.391715\pi\)
0.333665 + 0.942692i \(0.391715\pi\)
\(504\) 0 0
\(505\) −42.0000 −1.86898
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 41.1582 1.82431 0.912153 0.409849i \(-0.134419\pi\)
0.912153 + 0.409849i \(0.134419\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 10.5830 0.466343
\(516\) 0 0
\(517\) 39.5980 1.74152
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −18.7083 −0.819625 −0.409812 0.912170i \(-0.634406\pi\)
−0.409812 + 0.912170i \(0.634406\pi\)
\(522\) 0 0
\(523\) 16.9706 0.742071 0.371035 0.928619i \(-0.379003\pi\)
0.371035 + 0.928619i \(0.379003\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −31.7490 −1.38301
\(528\) 0 0
\(529\) 5.00000 0.217391
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −15.8745 −0.687601
\(534\) 0 0
\(535\) −19.7990 −0.855985
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 10.0000 0.429934 0.214967 0.976621i \(-0.431036\pi\)
0.214967 + 0.976621i \(0.431036\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 59.8665 2.56440
\(546\) 0 0
\(547\) −16.0000 −0.684111 −0.342055 0.939680i \(-0.611123\pi\)
−0.342055 + 0.939680i \(0.611123\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −14.9666 −0.637600
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −31.7490 −1.34525 −0.672624 0.739984i \(-0.734833\pi\)
−0.672624 + 0.739984i \(0.734833\pi\)
\(558\) 0 0
\(559\) 33.9411 1.43556
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 7.48331 0.315384 0.157692 0.987488i \(-0.449595\pi\)
0.157692 + 0.987488i \(0.449595\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 26.4575 1.10916 0.554578 0.832132i \(-0.312880\pi\)
0.554578 + 0.832132i \(0.312880\pi\)
\(570\) 0 0
\(571\) −40.0000 −1.67395 −0.836974 0.547243i \(-0.815677\pi\)
−0.836974 + 0.547243i \(0.815677\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −47.6235 −1.98604
\(576\) 0 0
\(577\) −21.2132 −0.883117 −0.441559 0.897232i \(-0.645574\pi\)
−0.441559 + 0.897232i \(0.645574\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 56.0000 2.31928
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 22.4499 0.926608 0.463304 0.886199i \(-0.346664\pi\)
0.463304 + 0.886199i \(0.346664\pi\)
\(588\) 0 0
\(589\) 24.0000 0.988903
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −26.1916 −1.07556 −0.537780 0.843085i \(-0.680737\pi\)
−0.537780 + 0.843085i \(0.680737\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −5.29150 −0.216205 −0.108102 0.994140i \(-0.534477\pi\)
−0.108102 + 0.994140i \(0.534477\pi\)
\(600\) 0 0
\(601\) 4.24264 0.173061 0.0865305 0.996249i \(-0.472422\pi\)
0.0865305 + 0.996249i \(0.472422\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 63.6082 2.58604
\(606\) 0 0
\(607\) −16.9706 −0.688814 −0.344407 0.938820i \(-0.611920\pi\)
−0.344407 + 0.938820i \(0.611920\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −31.7490 −1.28443
\(612\) 0 0
\(613\) −24.0000 −0.969351 −0.484675 0.874694i \(-0.661062\pi\)
−0.484675 + 0.874694i \(0.661062\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 26.4575 1.06514 0.532570 0.846386i \(-0.321226\pi\)
0.532570 + 0.846386i \(0.321226\pi\)
\(618\) 0 0
\(619\) −11.3137 −0.454736 −0.227368 0.973809i \(-0.573012\pi\)
−0.227368 + 0.973809i \(0.573012\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 14.9666 0.596759
\(630\) 0 0
\(631\) 40.0000 1.59237 0.796187 0.605050i \(-0.206847\pi\)
0.796187 + 0.605050i \(0.206847\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 29.9333 1.18787
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 37.0405 1.46301 0.731506 0.681835i \(-0.238818\pi\)
0.731506 + 0.681835i \(0.238818\pi\)
\(642\) 0 0
\(643\) 14.1421 0.557711 0.278856 0.960333i \(-0.410045\pi\)
0.278856 + 0.960333i \(0.410045\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 22.4499 0.882598 0.441299 0.897360i \(-0.354518\pi\)
0.441299 + 0.897360i \(0.354518\pi\)
\(648\) 0 0
\(649\) 39.5980 1.55436
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −15.8745 −0.621217 −0.310609 0.950538i \(-0.600533\pi\)
−0.310609 + 0.950538i \(0.600533\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 47.6235 1.85515 0.927575 0.373638i \(-0.121890\pi\)
0.927575 + 0.373638i \(0.121890\pi\)
\(660\) 0 0
\(661\) −21.2132 −0.825098 −0.412549 0.910935i \(-0.635361\pi\)
−0.412549 + 0.910935i \(0.635361\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −28.0000 −1.08416
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 52.3832 2.02223
\(672\) 0 0
\(673\) 36.0000 1.38770 0.693849 0.720121i \(-0.255914\pi\)
0.693849 + 0.720121i \(0.255914\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −33.6749 −1.29423 −0.647116 0.762391i \(-0.724025\pi\)
−0.647116 + 0.762391i \(0.724025\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 15.8745 0.607421 0.303711 0.952764i \(-0.401774\pi\)
0.303711 + 0.952764i \(0.401774\pi\)
\(684\) 0 0
\(685\) −59.3970 −2.26944
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −44.8999 −1.71055
\(690\) 0 0
\(691\) 16.9706 0.645591 0.322795 0.946469i \(-0.395377\pi\)
0.322795 + 0.946469i \(0.395377\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 21.1660 0.802873
\(696\) 0 0
\(697\) 14.0000 0.530288
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 15.8745 0.599572 0.299786 0.954006i \(-0.403085\pi\)
0.299786 + 0.954006i \(0.403085\pi\)
\(702\) 0 0
\(703\) −11.3137 −0.426705
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −8.00000 −0.300446 −0.150223 0.988652i \(-0.547999\pi\)
−0.150223 + 0.988652i \(0.547999\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 44.8999 1.68151
\(714\) 0 0
\(715\) −84.0000 −3.14142
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 29.9333 1.11632 0.558161 0.829733i \(-0.311507\pi\)
0.558161 + 0.829733i \(0.311507\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 47.6235 1.76869
\(726\) 0 0
\(727\) 14.1421 0.524503 0.262251 0.965000i \(-0.415535\pi\)
0.262251 + 0.965000i \(0.415535\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −29.9333 −1.10712
\(732\) 0 0
\(733\) −46.6690 −1.72376 −0.861880 0.507112i \(-0.830713\pi\)
−0.861880 + 0.507112i \(0.830713\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −63.4980 −2.33898
\(738\) 0 0
\(739\) −12.0000 −0.441427 −0.220714 0.975339i \(-0.570839\pi\)
−0.220714 + 0.975339i \(0.570839\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 37.0405 1.35888 0.679442 0.733729i \(-0.262222\pi\)
0.679442 + 0.733729i \(0.262222\pi\)
\(744\) 0 0
\(745\) 39.5980 1.45076
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −32.0000 −1.16770 −0.583848 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 14.9666 0.544691
\(756\) 0 0
\(757\) −16.0000 −0.581530 −0.290765 0.956795i \(-0.593910\pi\)
−0.290765 + 0.956795i \(0.593910\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 41.1582 1.49198 0.745992 0.665955i \(-0.231976\pi\)
0.745992 + 0.665955i \(0.231976\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −31.7490 −1.14639
\(768\) 0 0
\(769\) −4.24264 −0.152994 −0.0764968 0.997070i \(-0.524373\pi\)
−0.0764968 + 0.997070i \(0.524373\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −48.6415 −1.74951 −0.874757 0.484561i \(-0.838979\pi\)
−0.874757 + 0.484561i \(0.838979\pi\)
\(774\) 0 0
\(775\) −76.3675 −2.74320
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −10.5830 −0.379176
\(780\) 0 0
\(781\) 84.0000 3.00576
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −37.0405 −1.32203
\(786\) 0 0
\(787\) −39.5980 −1.41152 −0.705758 0.708453i \(-0.749393\pi\)
−0.705758 + 0.708453i \(0.749393\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −42.0000 −1.49146
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −11.2250 −0.397609 −0.198804 0.980039i \(-0.563706\pi\)
−0.198804 + 0.980039i \(0.563706\pi\)
\(798\) 0 0
\(799\) 28.0000 0.990569
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 7.48331 0.264080
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 21.1660 0.744157 0.372079 0.928201i \(-0.378645\pi\)
0.372079 + 0.928201i \(0.378645\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 74.8331 2.62129
\(816\) 0 0
\(817\) 22.6274 0.791633
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −31.7490 −1.10805 −0.554024 0.832501i \(-0.686908\pi\)
−0.554024 + 0.832501i \(0.686908\pi\)
\(822\) 0 0
\(823\) −4.00000 −0.139431 −0.0697156 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 15.8745 0.552011 0.276005 0.961156i \(-0.410989\pi\)
0.276005 + 0.961156i \(0.410989\pi\)
\(828\) 0 0
\(829\) 18.3848 0.638530 0.319265 0.947666i \(-0.396564\pi\)
0.319265 + 0.947666i \(0.396564\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −28.0000 −0.968980
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 7.48331 0.258353 0.129176 0.991622i \(-0.458767\pi\)
0.129176 + 0.991622i \(0.458767\pi\)
\(840\) 0 0
\(841\) −1.00000 −0.0344828
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 18.7083 0.643585
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −21.1660 −0.725561
\(852\) 0 0
\(853\) −55.1543 −1.88845 −0.944224 0.329304i \(-0.893186\pi\)
−0.944224 + 0.329304i \(0.893186\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 41.1582 1.40594 0.702969 0.711220i \(-0.251857\pi\)
0.702969 + 0.711220i \(0.251857\pi\)
\(858\) 0 0
\(859\) 19.7990 0.675533 0.337766 0.941230i \(-0.390329\pi\)
0.337766 + 0.941230i \(0.390329\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −5.29150 −0.180125 −0.0900624 0.995936i \(-0.528707\pi\)
−0.0900624 + 0.995936i \(0.528707\pi\)
\(864\) 0 0
\(865\) 42.0000 1.42804
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 21.1660 0.718008
\(870\) 0 0
\(871\) 50.9117 1.72508
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 8.00000 0.270141 0.135070 0.990836i \(-0.456874\pi\)
0.135070 + 0.990836i \(0.456874\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −56.1249 −1.89089 −0.945447 0.325775i \(-0.894375\pi\)
−0.945447 + 0.325775i \(0.894375\pi\)
\(882\) 0 0
\(883\) −8.00000 −0.269221 −0.134611 0.990899i \(-0.542978\pi\)
−0.134611 + 0.990899i \(0.542978\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 37.4166 1.25633 0.628163 0.778082i \(-0.283807\pi\)
0.628163 + 0.778082i \(0.283807\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −21.1660 −0.708294
\(894\) 0 0
\(895\) 59.3970 1.98542
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −44.8999 −1.49750
\(900\) 0 0
\(901\) 39.5980 1.31920
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 15.8745 0.527686
\(906\) 0 0
\(907\) −16.0000 −0.531271 −0.265636 0.964073i \(-0.585582\pi\)
−0.265636 + 0.964073i \(0.585582\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −5.29150 −0.175315 −0.0876577 0.996151i \(-0.527938\pi\)
−0.0876577 + 0.996151i \(0.527938\pi\)
\(912\) 0 0
\(913\) 79.1960 2.62100
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −4.00000 −0.131948 −0.0659739 0.997821i \(-0.521015\pi\)
−0.0659739 + 0.997821i \(0.521015\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −67.3498 −2.21685
\(924\) 0 0
\(925\) 36.0000 1.18367
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −18.7083 −0.613799 −0.306899 0.951742i \(-0.599292\pi\)
−0.306899 + 0.951742i \(0.599292\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 74.0810 2.42271
\(936\) 0 0
\(937\) −35.3553 −1.15501 −0.577504 0.816388i \(-0.695973\pi\)
−0.577504 + 0.816388i \(0.695973\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 26.1916 0.853822 0.426911 0.904294i \(-0.359602\pi\)
0.426911 + 0.904294i \(0.359602\pi\)
\(942\) 0 0
\(943\) −19.7990 −0.644744
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −47.6235 −1.54756 −0.773778 0.633457i \(-0.781636\pi\)
−0.773778 + 0.633457i \(0.781636\pi\)
\(948\) 0 0
\(949\) −6.00000 −0.194768
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 42.3320 1.37127 0.685634 0.727946i \(-0.259525\pi\)
0.685634 + 0.727946i \(0.259525\pi\)
\(954\) 0 0
\(955\) −19.7990 −0.640680
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 41.0000 1.32258
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 7.48331 0.240896
\(966\) 0 0
\(967\) −48.0000 −1.54358 −0.771788 0.635880i \(-0.780637\pi\)
−0.771788 + 0.635880i \(0.780637\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −14.9666 −0.480302 −0.240151 0.970736i \(-0.577197\pi\)
−0.240151 + 0.970736i \(0.577197\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −15.8745 −0.507871 −0.253935 0.967221i \(-0.581725\pi\)
−0.253935 + 0.967221i \(0.581725\pi\)
\(978\) 0 0
\(979\) −19.7990 −0.632778
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 14.9666 0.477361 0.238681 0.971098i \(-0.423285\pi\)
0.238681 + 0.971098i \(0.423285\pi\)
\(984\) 0 0
\(985\) −39.5980 −1.26170
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 42.3320 1.34608
\(990\) 0 0
\(991\) −44.0000 −1.39771 −0.698853 0.715265i \(-0.746306\pi\)
−0.698853 + 0.715265i \(0.746306\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 1.41421 0.0447886 0.0223943 0.999749i \(-0.492871\pi\)
0.0223943 + 0.999749i \(0.492871\pi\)
\(998\) 0 0
\(999\) 0 0
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 7056.2.a.cy.1.4 4
3.2 odd 2 inner 7056.2.a.cy.1.1 4
4.3 odd 2 1764.2.a.m.1.3 yes 4
7.6 odd 2 inner 7056.2.a.cy.1.2 4
12.11 even 2 1764.2.a.m.1.2 yes 4
21.20 even 2 inner 7056.2.a.cy.1.3 4
28.3 even 6 1764.2.k.m.1549.4 8
28.11 odd 6 1764.2.k.m.1549.2 8
28.19 even 6 1764.2.k.m.361.4 8
28.23 odd 6 1764.2.k.m.361.2 8
28.27 even 2 1764.2.a.m.1.1 4
84.11 even 6 1764.2.k.m.1549.3 8
84.23 even 6 1764.2.k.m.361.3 8
84.47 odd 6 1764.2.k.m.361.1 8
84.59 odd 6 1764.2.k.m.1549.1 8
84.83 odd 2 1764.2.a.m.1.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1764.2.a.m.1.1 4 28.27 even 2
1764.2.a.m.1.2 yes 4 12.11 even 2
1764.2.a.m.1.3 yes 4 4.3 odd 2
1764.2.a.m.1.4 yes 4 84.83 odd 2
1764.2.k.m.361.1 8 84.47 odd 6
1764.2.k.m.361.2 8 28.23 odd 6
1764.2.k.m.361.3 8 84.23 even 6
1764.2.k.m.361.4 8 28.19 even 6
1764.2.k.m.1549.1 8 84.59 odd 6
1764.2.k.m.1549.2 8 28.11 odd 6
1764.2.k.m.1549.3 8 84.11 even 6
1764.2.k.m.1549.4 8 28.3 even 6
7056.2.a.cy.1.1 4 3.2 odd 2 inner
7056.2.a.cy.1.2 4 7.6 odd 2 inner
7056.2.a.cy.1.3 4 21.20 even 2 inner
7056.2.a.cy.1.4 4 1.1 even 1 trivial