Properties

Label 4900.2.a.bf.1.3
Level $4900$
Weight $2$
Character 4900.1
Self dual yes
Analytic conductor $39.127$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4900.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(39.1266969904\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{19})\)
Defining polynomial: \(x^{4} - 11 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 140)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(3.04547\) of defining polynomial
Character \(\chi\) \(=\) 4900.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.73205 q^{3} +O(q^{10})\) \(q+1.73205 q^{3} -5.27492 q^{11} +2.62685 q^{13} +0.418627 q^{17} -3.27492 q^{19} +7.82300 q^{23} -5.19615 q^{27} +4.27492 q^{29} -3.27492 q^{31} -9.13642 q^{33} +9.97368 q^{37} +4.54983 q^{39} +3.72508 q^{41} +2.15068 q^{43} +6.50958 q^{47} +0.725083 q^{51} +5.67232 q^{53} -5.67232 q^{57} +3.27492 q^{59} +13.5498 q^{61} -3.52165 q^{67} +13.5498 q^{69} -4.54983 q^{71} +6.50958 q^{73} +7.27492 q^{79} -9.00000 q^{81} -7.40437 q^{83} +7.40437 q^{87} +7.00000 q^{89} -5.67232 q^{93} -6.92820 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + O(q^{10}) \) \( 4 q - 6 q^{11} + 2 q^{19} + 2 q^{29} + 2 q^{31} - 12 q^{39} + 30 q^{41} + 18 q^{51} - 2 q^{59} + 24 q^{61} + 24 q^{69} + 12 q^{71} + 14 q^{79} - 36 q^{81} + 28 q^{89} + O(q^{100}) \)

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
\(3\) 1.73205 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −5.27492 −1.59045 −0.795224 0.606316i \(-0.792647\pi\)
−0.795224 + 0.606316i \(0.792647\pi\)
\(12\) 0 0
\(13\) 2.62685 0.728557 0.364278 0.931290i \(-0.381316\pi\)
0.364278 + 0.931290i \(0.381316\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.418627 0.101532 0.0507659 0.998711i \(-0.483834\pi\)
0.0507659 + 0.998711i \(0.483834\pi\)
\(18\) 0 0
\(19\) −3.27492 −0.751318 −0.375659 0.926758i \(-0.622584\pi\)
−0.375659 + 0.926758i \(0.622584\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 7.82300 1.63121 0.815604 0.578610i \(-0.196405\pi\)
0.815604 + 0.578610i \(0.196405\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −5.19615 −1.00000
\(28\) 0 0
\(29\) 4.27492 0.793832 0.396916 0.917855i \(-0.370080\pi\)
0.396916 + 0.917855i \(0.370080\pi\)
\(30\) 0 0
\(31\) −3.27492 −0.588192 −0.294096 0.955776i \(-0.595019\pi\)
−0.294096 + 0.955776i \(0.595019\pi\)
\(32\) 0 0
\(33\) −9.13642 −1.59045
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 9.97368 1.63966 0.819831 0.572605i \(-0.194067\pi\)
0.819831 + 0.572605i \(0.194067\pi\)
\(38\) 0 0
\(39\) 4.54983 0.728557
\(40\) 0 0
\(41\) 3.72508 0.581760 0.290880 0.956760i \(-0.406052\pi\)
0.290880 + 0.956760i \(0.406052\pi\)
\(42\) 0 0
\(43\) 2.15068 0.327975 0.163988 0.986462i \(-0.447564\pi\)
0.163988 + 0.986462i \(0.447564\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.50958 0.949519 0.474760 0.880115i \(-0.342535\pi\)
0.474760 + 0.880115i \(0.342535\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0.725083 0.101532
\(52\) 0 0
\(53\) 5.67232 0.779153 0.389577 0.920994i \(-0.372621\pi\)
0.389577 + 0.920994i \(0.372621\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −5.67232 −0.751318
\(58\) 0 0
\(59\) 3.27492 0.426358 0.213179 0.977013i \(-0.431618\pi\)
0.213179 + 0.977013i \(0.431618\pi\)
\(60\) 0 0
\(61\) 13.5498 1.73488 0.867439 0.497543i \(-0.165764\pi\)
0.867439 + 0.497543i \(0.165764\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −3.52165 −0.430237 −0.215119 0.976588i \(-0.569014\pi\)
−0.215119 + 0.976588i \(0.569014\pi\)
\(68\) 0 0
\(69\) 13.5498 1.63121
\(70\) 0 0
\(71\) −4.54983 −0.539966 −0.269983 0.962865i \(-0.587018\pi\)
−0.269983 + 0.962865i \(0.587018\pi\)
\(72\) 0 0
\(73\) 6.50958 0.761888 0.380944 0.924598i \(-0.375599\pi\)
0.380944 + 0.924598i \(0.375599\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 7.27492 0.818492 0.409246 0.912424i \(-0.365792\pi\)
0.409246 + 0.912424i \(0.365792\pi\)
\(80\) 0 0
\(81\) −9.00000 −1.00000
\(82\) 0 0
\(83\) −7.40437 −0.812736 −0.406368 0.913710i \(-0.633205\pi\)
−0.406368 + 0.913710i \(0.633205\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 7.40437 0.793832
\(88\) 0 0
\(89\) 7.00000 0.741999 0.370999 0.928633i \(-0.379015\pi\)
0.370999 + 0.928633i \(0.379015\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −5.67232 −0.588192
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −6.92820 −0.703452 −0.351726 0.936103i \(-0.614405\pi\)
−0.351726 + 0.936103i \(0.614405\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 13.5498 1.34826 0.674129 0.738613i \(-0.264519\pi\)
0.674129 + 0.738613i \(0.264519\pi\)
\(102\) 0 0
\(103\) 11.2871 1.11215 0.556076 0.831132i \(-0.312306\pi\)
0.556076 + 0.831132i \(0.312306\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −3.52165 −0.340450 −0.170225 0.985405i \(-0.554450\pi\)
−0.170225 + 0.985405i \(0.554450\pi\)
\(108\) 0 0
\(109\) −11.5498 −1.10627 −0.553137 0.833090i \(-0.686569\pi\)
−0.553137 + 0.833090i \(0.686569\pi\)
\(110\) 0 0
\(111\) 17.2749 1.63966
\(112\) 0 0
\(113\) −4.30136 −0.404637 −0.202319 0.979320i \(-0.564848\pi\)
−0.202319 + 0.979320i \(0.564848\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 16.8248 1.52952
\(122\) 0 0
\(123\) 6.45203 0.581760
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 15.6460 1.38836 0.694179 0.719802i \(-0.255768\pi\)
0.694179 + 0.719802i \(0.255768\pi\)
\(128\) 0 0
\(129\) 3.72508 0.327975
\(130\) 0 0
\(131\) 10.7251 0.937055 0.468527 0.883449i \(-0.344785\pi\)
0.468527 + 0.883449i \(0.344785\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 21.3183 1.82135 0.910674 0.413126i \(-0.135563\pi\)
0.910674 + 0.413126i \(0.135563\pi\)
\(138\) 0 0
\(139\) −13.0997 −1.11110 −0.555550 0.831483i \(-0.687492\pi\)
−0.555550 + 0.831483i \(0.687492\pi\)
\(140\) 0 0
\(141\) 11.2749 0.949519
\(142\) 0 0
\(143\) −13.8564 −1.15873
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −7.54983 −0.618507 −0.309253 0.950980i \(-0.600079\pi\)
−0.309253 + 0.950980i \(0.600079\pi\)
\(150\) 0 0
\(151\) −12.7251 −1.03555 −0.517776 0.855516i \(-0.673240\pi\)
−0.517776 + 0.855516i \(0.673240\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −2.20822 −0.176235 −0.0881176 0.996110i \(-0.528085\pi\)
−0.0881176 + 0.996110i \(0.528085\pi\)
\(158\) 0 0
\(159\) 9.82475 0.779153
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −5.67232 −0.444291 −0.222145 0.975014i \(-0.571306\pi\)
−0.222145 + 0.975014i \(0.571306\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −0.476171 −0.0368472 −0.0184236 0.999830i \(-0.505865\pi\)
−0.0184236 + 0.999830i \(0.505865\pi\)
\(168\) 0 0
\(169\) −6.09967 −0.469205
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 20.4811 1.55715 0.778573 0.627553i \(-0.215944\pi\)
0.778573 + 0.627553i \(0.215944\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 5.67232 0.426358
\(178\) 0 0
\(179\) −7.27492 −0.543753 −0.271876 0.962332i \(-0.587644\pi\)
−0.271876 + 0.962332i \(0.587644\pi\)
\(180\) 0 0
\(181\) 24.2749 1.80434 0.902170 0.431380i \(-0.141973\pi\)
0.902170 + 0.431380i \(0.141973\pi\)
\(182\) 0 0
\(183\) 23.4690 1.73488
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −2.20822 −0.161481
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0.175248 0.0126805 0.00634026 0.999980i \(-0.497982\pi\)
0.00634026 + 0.999980i \(0.497982\pi\)
\(192\) 0 0
\(193\) −21.3183 −1.53453 −0.767263 0.641332i \(-0.778382\pi\)
−0.767263 + 0.641332i \(0.778382\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −8.60271 −0.612918 −0.306459 0.951884i \(-0.599144\pi\)
−0.306459 + 0.951884i \(0.599144\pi\)
\(198\) 0 0
\(199\) −17.2749 −1.22459 −0.612293 0.790631i \(-0.709753\pi\)
−0.612293 + 0.790631i \(0.709753\pi\)
\(200\) 0 0
\(201\) −6.09967 −0.430237
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 17.2749 1.19493
\(210\) 0 0
\(211\) 25.6495 1.76578 0.882892 0.469576i \(-0.155593\pi\)
0.882892 + 0.469576i \(0.155593\pi\)
\(212\) 0 0
\(213\) −7.88054 −0.539966
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 11.2749 0.761888
\(220\) 0 0
\(221\) 1.09967 0.0739717
\(222\) 0 0
\(223\) −8.71780 −0.583787 −0.291893 0.956451i \(-0.594285\pi\)
−0.291893 + 0.956451i \(0.594285\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −19.5287 −1.29617 −0.648084 0.761569i \(-0.724429\pi\)
−0.648084 + 0.761569i \(0.724429\pi\)
\(228\) 0 0
\(229\) −3.27492 −0.216413 −0.108206 0.994128i \(-0.534511\pi\)
−0.108206 + 0.994128i \(0.534511\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −14.2750 −0.935189 −0.467594 0.883943i \(-0.654879\pi\)
−0.467594 + 0.883943i \(0.654879\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 12.6005 0.818492
\(238\) 0 0
\(239\) 0.549834 0.0355658 0.0177829 0.999842i \(-0.494339\pi\)
0.0177829 + 0.999842i \(0.494339\pi\)
\(240\) 0 0
\(241\) −9.82475 −0.632868 −0.316434 0.948615i \(-0.602486\pi\)
−0.316434 + 0.948615i \(0.602486\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −8.60271 −0.547377
\(248\) 0 0
\(249\) −12.8248 −0.812736
\(250\) 0 0
\(251\) 20.5498 1.29709 0.648547 0.761175i \(-0.275377\pi\)
0.648547 + 0.761175i \(0.275377\pi\)
\(252\) 0 0
\(253\) −41.2657 −2.59435
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −11.6482 −0.726594 −0.363297 0.931673i \(-0.618349\pi\)
−0.363297 + 0.931673i \(0.618349\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −0.779710 −0.0480790 −0.0240395 0.999711i \(-0.507653\pi\)
−0.0240395 + 0.999711i \(0.507653\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 12.1244 0.741999
\(268\) 0 0
\(269\) 14.4502 0.881042 0.440521 0.897742i \(-0.354794\pi\)
0.440521 + 0.897742i \(0.354794\pi\)
\(270\) 0 0
\(271\) −9.82475 −0.596811 −0.298406 0.954439i \(-0.596455\pi\)
−0.298406 + 0.954439i \(0.596455\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −14.2750 −0.857704 −0.428852 0.903375i \(-0.641082\pi\)
−0.428852 + 0.903375i \(0.641082\pi\)
\(278\) 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
\(283\) 10.9260 0.649484 0.324742 0.945803i \(-0.394722\pi\)
0.324742 + 0.945803i \(0.394722\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −16.8248 −0.989691
\(290\) 0 0
\(291\) −12.0000 −0.703452
\(292\) 0 0
\(293\) 6.92820 0.404750 0.202375 0.979308i \(-0.435134\pi\)
0.202375 + 0.979308i \(0.435134\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 27.4093 1.59045
\(298\) 0 0
\(299\) 20.5498 1.18843
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 23.4690 1.34826
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 26.5145 1.51326 0.756631 0.653843i \(-0.226844\pi\)
0.756631 + 0.653843i \(0.226844\pi\)
\(308\) 0 0
\(309\) 19.5498 1.11215
\(310\) 0 0
\(311\) 9.82475 0.557111 0.278555 0.960420i \(-0.410144\pi\)
0.278555 + 0.960420i \(0.410144\pi\)
\(312\) 0 0
\(313\) −33.5002 −1.89354 −0.946772 0.321904i \(-0.895677\pi\)
−0.946772 + 0.321904i \(0.895677\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −25.6197 −1.43894 −0.719472 0.694521i \(-0.755616\pi\)
−0.719472 + 0.694521i \(0.755616\pi\)
\(318\) 0 0
\(319\) −22.5498 −1.26255
\(320\) 0 0
\(321\) −6.09967 −0.340450
\(322\) 0 0
\(323\) −1.37097 −0.0762827
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −20.0049 −1.10627
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 17.8248 0.979737 0.489868 0.871796i \(-0.337045\pi\)
0.489868 + 0.871796i \(0.337045\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 4.30136 0.234310 0.117155 0.993114i \(-0.462623\pi\)
0.117155 + 0.993114i \(0.462623\pi\)
\(338\) 0 0
\(339\) −7.45017 −0.404637
\(340\) 0 0
\(341\) 17.2749 0.935489
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −12.1244 −0.650870 −0.325435 0.945564i \(-0.605511\pi\)
−0.325435 + 0.945564i \(0.605511\pi\)
\(348\) 0 0
\(349\) 3.72508 0.199399 0.0996996 0.995018i \(-0.468212\pi\)
0.0996996 + 0.995018i \(0.468212\pi\)
\(350\) 0 0
\(351\) −13.6495 −0.728557
\(352\) 0 0
\(353\) 8.18408 0.435595 0.217797 0.975994i \(-0.430113\pi\)
0.217797 + 0.975994i \(0.430113\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 36.3746 1.91978 0.959889 0.280382i \(-0.0904610\pi\)
0.959889 + 0.280382i \(0.0904610\pi\)
\(360\) 0 0
\(361\) −8.27492 −0.435522
\(362\) 0 0
\(363\) 29.1413 1.52952
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −6.03341 −0.314941 −0.157471 0.987524i \(-0.550334\pi\)
−0.157471 + 0.987524i \(0.550334\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −9.97368 −0.516417 −0.258209 0.966089i \(-0.583132\pi\)
−0.258209 + 0.966089i \(0.583132\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 11.2296 0.578352
\(378\) 0 0
\(379\) −21.6495 −1.11206 −0.556030 0.831162i \(-0.687676\pi\)
−0.556030 + 0.831162i \(0.687676\pi\)
\(380\) 0 0
\(381\) 27.0997 1.38836
\(382\) 0 0
\(383\) −6.14849 −0.314173 −0.157087 0.987585i \(-0.550210\pi\)
−0.157087 + 0.987585i \(0.550210\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 32.3746 1.64146 0.820728 0.571319i \(-0.193568\pi\)
0.820728 + 0.571319i \(0.193568\pi\)
\(390\) 0 0
\(391\) 3.27492 0.165620
\(392\) 0 0
\(393\) 18.5764 0.937055
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 10.8109 0.542585 0.271293 0.962497i \(-0.412549\pi\)
0.271293 + 0.962497i \(0.412549\pi\)
\(398\) 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
\(403\) −8.60271 −0.428532
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −52.6103 −2.60780
\(408\) 0 0
\(409\) 20.0997 0.993865 0.496932 0.867789i \(-0.334460\pi\)
0.496932 + 0.867789i \(0.334460\pi\)
\(410\) 0 0
\(411\) 36.9244 1.82135
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −22.6893 −1.11110
\(418\) 0 0
\(419\) 13.0997 0.639961 0.319980 0.947424i \(-0.396324\pi\)
0.319980 + 0.947424i \(0.396324\pi\)
\(420\) 0 0
\(421\) −4.27492 −0.208347 −0.104173 0.994559i \(-0.533220\pi\)
−0.104173 + 0.994559i \(0.533220\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −24.0000 −1.15873
\(430\) 0 0
\(431\) −18.3746 −0.885073 −0.442536 0.896751i \(-0.645921\pi\)
−0.442536 + 0.896751i \(0.645921\pi\)
\(432\) 0 0
\(433\) −18.1578 −0.872606 −0.436303 0.899800i \(-0.643712\pi\)
−0.436303 + 0.899800i \(0.643712\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −25.6197 −1.22556
\(438\) 0 0
\(439\) −23.8248 −1.13709 −0.568547 0.822651i \(-0.692494\pi\)
−0.568547 + 0.822651i \(0.692494\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 12.1244 0.576046 0.288023 0.957624i \(-0.407002\pi\)
0.288023 + 0.957624i \(0.407002\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −13.0767 −0.618507
\(448\) 0 0
\(449\) −3.17525 −0.149849 −0.0749246 0.997189i \(-0.523872\pi\)
−0.0749246 + 0.997189i \(0.523872\pi\)
\(450\) 0 0
\(451\) −19.6495 −0.925259
\(452\) 0 0
\(453\) −22.0405 −1.03555
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 1.37097 0.0641312 0.0320656 0.999486i \(-0.489791\pi\)
0.0320656 + 0.999486i \(0.489791\pi\)
\(458\) 0 0
\(459\) −2.17525 −0.101532
\(460\) 0 0
\(461\) 14.0000 0.652045 0.326023 0.945362i \(-0.394291\pi\)
0.326023 + 0.945362i \(0.394291\pi\)
\(462\) 0 0
\(463\) −2.15068 −0.0999505 −0.0499752 0.998750i \(-0.515914\pi\)
−0.0499752 + 0.998750i \(0.515914\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 15.7035 0.726673 0.363337 0.931658i \(-0.381637\pi\)
0.363337 + 0.931658i \(0.381637\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −3.82475 −0.176235
\(472\) 0 0
\(473\) −11.3446 −0.521627
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −9.82475 −0.448904 −0.224452 0.974485i \(-0.572059\pi\)
−0.224452 + 0.974485i \(0.572059\pi\)
\(480\) 0 0
\(481\) 26.1993 1.19459
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −2.93039 −0.132789 −0.0663943 0.997793i \(-0.521150\pi\)
−0.0663943 + 0.997793i \(0.521150\pi\)
\(488\) 0 0
\(489\) −9.82475 −0.444291
\(490\) 0 0
\(491\) −28.5498 −1.28844 −0.644218 0.764842i \(-0.722817\pi\)
−0.644218 + 0.764842i \(0.722817\pi\)
\(492\) 0 0
\(493\) 1.78959 0.0805993
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 1.62541 0.0727635 0.0363818 0.999338i \(-0.488417\pi\)
0.0363818 + 0.999338i \(0.488417\pi\)
\(500\) 0 0
\(501\) −0.824752 −0.0368472
\(502\) 0 0
\(503\) 31.7682 1.41647 0.708236 0.705975i \(-0.249491\pi\)
0.708236 + 0.705975i \(0.249491\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −10.5649 −0.469205
\(508\) 0 0
\(509\) 14.4502 0.640492 0.320246 0.947334i \(-0.396234\pi\)
0.320246 + 0.947334i \(0.396234\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 17.0170 0.751318
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −34.3375 −1.51016
\(518\) 0 0
\(519\) 35.4743 1.55715
\(520\) 0 0
\(521\) −9.82475 −0.430430 −0.215215 0.976567i \(-0.569045\pi\)
−0.215215 + 0.976567i \(0.569045\pi\)
\(522\) 0 0
\(523\) 7.34683 0.321254 0.160627 0.987015i \(-0.448648\pi\)
0.160627 + 0.987015i \(0.448648\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −1.37097 −0.0597203
\(528\) 0 0
\(529\) 38.1993 1.66084
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 9.78523 0.423845
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −12.6005 −0.543753
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −17.5498 −0.754526 −0.377263 0.926106i \(-0.623135\pi\)
−0.377263 + 0.926106i \(0.623135\pi\)
\(542\) 0 0
\(543\) 42.0454 1.80434
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 20.5386 0.878168 0.439084 0.898446i \(-0.355303\pi\)
0.439084 + 0.898446i \(0.355303\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −14.0000 −0.596420
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −9.97368 −0.422598 −0.211299 0.977421i \(-0.567769\pi\)
−0.211299 + 0.977421i \(0.567769\pi\)
\(558\) 0 0
\(559\) 5.64950 0.238949
\(560\) 0 0
\(561\) −3.82475 −0.161481
\(562\) 0 0
\(563\) −22.6317 −0.953814 −0.476907 0.878954i \(-0.658242\pi\)
−0.476907 + 0.878954i \(0.658242\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 8.37459 0.351081 0.175540 0.984472i \(-0.443833\pi\)
0.175540 + 0.984472i \(0.443833\pi\)
\(570\) 0 0
\(571\) 7.27492 0.304446 0.152223 0.988346i \(-0.451357\pi\)
0.152223 + 0.988346i \(0.451357\pi\)
\(572\) 0 0
\(573\) 0.303539 0.0126805
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −3.88273 −0.161640 −0.0808200 0.996729i \(-0.525754\pi\)
−0.0808200 + 0.996729i \(0.525754\pi\)
\(578\) 0 0
\(579\) −36.9244 −1.53453
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −29.9210 −1.23920
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 20.8997 0.862623 0.431311 0.902203i \(-0.358051\pi\)
0.431311 + 0.902203i \(0.358051\pi\)
\(588\) 0 0
\(589\) 10.7251 0.441919
\(590\) 0 0
\(591\) −14.9003 −0.612918
\(592\) 0 0
\(593\) 33.3851 1.37096 0.685482 0.728090i \(-0.259592\pi\)
0.685482 + 0.728090i \(0.259592\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −29.9210 −1.22459
\(598\) 0 0
\(599\) 5.27492 0.215527 0.107764 0.994177i \(-0.465631\pi\)
0.107764 + 0.994177i \(0.465631\pi\)
\(600\) 0 0
\(601\) −14.0000 −0.571072 −0.285536 0.958368i \(-0.592172\pi\)
−0.285536 + 0.958368i \(0.592172\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 11.4022 0.462801 0.231400 0.972859i \(-0.425669\pi\)
0.231400 + 0.972859i \(0.425669\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 17.0997 0.691779
\(612\) 0 0
\(613\) 28.3616 1.14551 0.572757 0.819725i \(-0.305874\pi\)
0.572757 + 0.819725i \(0.305874\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 31.2920 1.25977 0.629884 0.776689i \(-0.283102\pi\)
0.629884 + 0.776689i \(0.283102\pi\)
\(618\) 0 0
\(619\) 8.92442 0.358703 0.179351 0.983785i \(-0.442600\pi\)
0.179351 + 0.983785i \(0.442600\pi\)
\(620\) 0 0
\(621\) −40.6495 −1.63121
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 29.9210 1.19493
\(628\) 0 0
\(629\) 4.17525 0.166478
\(630\) 0 0
\(631\) 33.0997 1.31768 0.658839 0.752284i \(-0.271048\pi\)
0.658839 + 0.752284i \(0.271048\pi\)
\(632\) 0 0
\(633\) 44.4262 1.76578
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −2.09967 −0.0829319 −0.0414660 0.999140i \(-0.513203\pi\)
−0.0414660 + 0.999140i \(0.513203\pi\)
\(642\) 0 0
\(643\) −31.4071 −1.23857 −0.619287 0.785164i \(-0.712578\pi\)
−0.619287 + 0.785164i \(0.712578\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 26.9331 1.05885 0.529425 0.848357i \(-0.322408\pi\)
0.529425 + 0.848357i \(0.322408\pi\)
\(648\) 0 0
\(649\) −17.2749 −0.678100
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −28.3616 −1.10988 −0.554938 0.831892i \(-0.687258\pi\)
−0.554938 + 0.831892i \(0.687258\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −40.5498 −1.57960 −0.789799 0.613366i \(-0.789815\pi\)
−0.789799 + 0.613366i \(0.789815\pi\)
\(660\) 0 0
\(661\) 0.450166 0.0175094 0.00875471 0.999962i \(-0.497213\pi\)
0.00875471 + 0.999962i \(0.497213\pi\)
\(662\) 0 0
\(663\) 1.90468 0.0739717
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 33.4427 1.29491
\(668\) 0 0
\(669\) −15.0997 −0.583787
\(670\) 0 0
\(671\) −71.4743 −2.75923
\(672\) 0 0
\(673\) 31.2920 1.20622 0.603109 0.797659i \(-0.293928\pi\)
0.603109 + 0.797659i \(0.293928\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 46.4043 1.78346 0.891731 0.452566i \(-0.149491\pi\)
0.891731 + 0.452566i \(0.149491\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −33.8248 −1.29617
\(682\) 0 0
\(683\) −19.1676 −0.733430 −0.366715 0.930333i \(-0.619518\pi\)
−0.366715 + 0.930333i \(0.619518\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −5.67232 −0.216413
\(688\) 0 0
\(689\) 14.9003 0.567657
\(690\) 0 0
\(691\) −30.3746 −1.15550 −0.577752 0.816212i \(-0.696070\pi\)
−0.577752 + 0.816212i \(0.696070\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 1.55942 0.0590672
\(698\) 0 0
\(699\) −24.7251 −0.935189
\(700\) 0 0
\(701\) 8.82475 0.333306 0.166653 0.986016i \(-0.446704\pi\)
0.166653 + 0.986016i \(0.446704\pi\)
\(702\) 0 0
\(703\) −32.6630 −1.23191
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −10.4502 −0.392464 −0.196232 0.980557i \(-0.562871\pi\)
−0.196232 + 0.980557i \(0.562871\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −25.6197 −0.959465
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0.952341 0.0355658
\(718\) 0 0
\(719\) 30.3746 1.13278 0.566390 0.824137i \(-0.308339\pi\)
0.566390 + 0.824137i \(0.308339\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −17.0170 −0.632868
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 3.10302 0.115085 0.0575423 0.998343i \(-0.481674\pi\)
0.0575423 + 0.998343i \(0.481674\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) 0.900331 0.0332999
\(732\) 0 0
\(733\) 37.6865 1.39198 0.695991 0.718050i \(-0.254965\pi\)
0.695991 + 0.718050i \(0.254965\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 18.5764 0.684270
\(738\) 0 0
\(739\) −20.9244 −0.769717 −0.384859 0.922976i \(-0.625750\pi\)
−0.384859 + 0.922976i \(0.625750\pi\)
\(740\) 0 0
\(741\) −14.9003 −0.547377
\(742\) 0 0
\(743\) 6.45203 0.236702 0.118351 0.992972i \(-0.462239\pi\)
0.118351 + 0.992972i \(0.462239\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −14.7251 −0.537326 −0.268663 0.963234i \(-0.586582\pi\)
−0.268663 + 0.963234i \(0.586582\pi\)
\(752\) 0 0
\(753\) 35.5934 1.29709
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −35.5934 −1.29366 −0.646831 0.762633i \(-0.723906\pi\)
−0.646831 + 0.762633i \(0.723906\pi\)
\(758\) 0 0
\(759\) −71.4743 −2.59435
\(760\) 0 0
\(761\) −22.9244 −0.831010 −0.415505 0.909591i \(-0.636395\pi\)
−0.415505 + 0.909591i \(0.636395\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 8.60271 0.310626
\(768\) 0 0
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) −20.1752 −0.726594
\(772\) 0 0
\(773\) −40.3133 −1.44997 −0.724985 0.688765i \(-0.758153\pi\)
−0.724985 + 0.688765i \(0.758153\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −12.1993 −0.437087
\(780\) 0 0
\(781\) 24.0000 0.858788
\(782\) 0 0
\(783\) −22.2131 −0.793832
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −1.73205 −0.0617409 −0.0308705 0.999523i \(-0.509828\pi\)
−0.0308705 + 0.999523i \(0.509828\pi\)
\(788\) 0 0
\(789\) −1.35050 −0.0480790
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 35.5934 1.26396
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −46.8229 −1.65855 −0.829276 0.558839i \(-0.811247\pi\)
−0.829276 + 0.558839i \(0.811247\pi\)
\(798\) 0 0
\(799\) 2.72508 0.0964065
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −34.3375 −1.21174
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 25.0284 0.881042
\(808\) 0 0
\(809\) −17.1993 −0.604697 −0.302348 0.953198i \(-0.597771\pi\)
−0.302348 + 0.953198i \(0.597771\pi\)
\(810\) 0 0
\(811\) −7.45017 −0.261611 −0.130805 0.991408i \(-0.541756\pi\)
−0.130805 + 0.991408i \(0.541756\pi\)
\(812\) 0 0
\(813\) −17.0170 −0.596811
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −7.04329 −0.246414
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −20.3746 −0.711078 −0.355539 0.934661i \(-0.615703\pi\)
−0.355539 + 0.934661i \(0.615703\pi\)
\(822\) 0 0
\(823\) −46.1583 −1.60898 −0.804488 0.593968i \(-0.797560\pi\)
−0.804488 + 0.593968i \(0.797560\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 15.0547 0.523505 0.261752 0.965135i \(-0.415700\pi\)
0.261752 + 0.965135i \(0.415700\pi\)
\(828\) 0 0
\(829\) 50.9244 1.76868 0.884339 0.466845i \(-0.154609\pi\)
0.884339 + 0.466845i \(0.154609\pi\)
\(830\) 0 0
\(831\) −24.7251 −0.857704
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 17.0170 0.588192
\(838\) 0 0
\(839\) −41.0997 −1.41892 −0.709459 0.704747i \(-0.751061\pi\)
−0.709459 + 0.704747i \(0.751061\pi\)
\(840\) 0 0
\(841\) −10.7251 −0.369830
\(842\) 0 0
\(843\) 10.3923 0.357930
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 18.9244 0.649484
\(850\) 0 0
\(851\) 78.0241 2.67463
\(852\) 0 0
\(853\) 13.1342 0.449708 0.224854 0.974392i \(-0.427810\pi\)
0.224854 + 0.974392i \(0.427810\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 37.6865 1.28735 0.643673 0.765301i \(-0.277410\pi\)
0.643673 + 0.765301i \(0.277410\pi\)
\(858\) 0 0
\(859\) 2.37459 0.0810198 0.0405099 0.999179i \(-0.487102\pi\)
0.0405099 + 0.999179i \(0.487102\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 16.4257 0.559138 0.279569 0.960126i \(-0.409808\pi\)
0.279569 + 0.960126i \(0.409808\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −29.1413 −0.989691
\(868\) 0 0
\(869\) −38.3746 −1.30177
\(870\) 0 0
\(871\) −9.25083 −0.313452
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 22.8777 0.772527 0.386263 0.922389i \(-0.373766\pi\)
0.386263 + 0.922389i \(0.373766\pi\)
\(878\) 0 0
\(879\) 12.0000 0.404750
\(880\) 0 0
\(881\) 43.0241 1.44952 0.724759 0.689002i \(-0.241951\pi\)
0.724759 + 0.689002i \(0.241951\pi\)
\(882\) 0 0
\(883\) −55.5407 −1.86909 −0.934547 0.355840i \(-0.884195\pi\)
−0.934547 + 0.355840i \(0.884195\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −39.2301 −1.31722 −0.658609 0.752486i \(-0.728855\pi\)
−0.658609 + 0.752486i \(0.728855\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 47.4743 1.59045
\(892\) 0 0
\(893\) −21.3183 −0.713391
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 35.5934 1.18843
\(898\) 0 0
\(899\) −14.0000 −0.466926
\(900\) 0 0
\(901\) 2.37459 0.0791089
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 41.8569 1.38984 0.694918 0.719089i \(-0.255440\pi\)
0.694918 + 0.719089i \(0.255440\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −25.0997 −0.831589 −0.415795 0.909459i \(-0.636496\pi\)
−0.415795 + 0.909459i \(0.636496\pi\)
\(912\) 0 0
\(913\) 39.0575 1.29261
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −46.9244 −1.54789 −0.773947 0.633250i \(-0.781720\pi\)
−0.773947 + 0.633250i \(0.781720\pi\)
\(920\) 0 0
\(921\) 45.9244 1.51326
\(922\) 0 0
\(923\) −11.9517 −0.393396
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 48.0997 1.57810 0.789049 0.614330i \(-0.210573\pi\)
0.789049 + 0.614330i \(0.210573\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 17.0170 0.557111
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 24.3638 0.795931 0.397965 0.917400i \(-0.369716\pi\)
0.397965 + 0.917400i \(0.369716\pi\)
\(938\) 0 0
\(939\) −58.0241 −1.89354
\(940\) 0 0
\(941\) 3.27492 0.106759 0.0533796 0.998574i \(-0.483001\pi\)
0.0533796 + 0.998574i \(0.483001\pi\)
\(942\) 0 0
\(943\) 29.1413 0.948972
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 10.5649 0.343314 0.171657 0.985157i \(-0.445088\pi\)
0.171657 + 0.985157i \(0.445088\pi\)
\(948\) 0 0
\(949\) 17.0997 0.555079
\(950\) 0 0
\(951\) −44.3746 −1.43894
\(952\) 0 0
\(953\) −22.6893 −0.734978 −0.367489 0.930028i \(-0.619782\pi\)
−0.367489 + 0.930028i \(0.619782\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −39.0575 −1.26255
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −20.2749 −0.654030
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 2.15068 0.0691611 0.0345806 0.999402i \(-0.488990\pi\)
0.0345806 + 0.999402i \(0.488990\pi\)
\(968\) 0 0
\(969\) −2.37459 −0.0762827
\(970\) 0 0
\(971\) 36.9244 1.18496 0.592481 0.805585i \(-0.298149\pi\)
0.592481 + 0.805585i \(0.298149\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 29.9210 0.957259 0.478629 0.878017i \(-0.341134\pi\)
0.478629 + 0.878017i \(0.341134\pi\)
\(978\) 0 0
\(979\) −36.9244 −1.18011
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 45.9281 1.46488 0.732440 0.680832i \(-0.238382\pi\)
0.732440 + 0.680832i \(0.238382\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 16.8248 0.534996
\(990\) 0 0
\(991\) 33.4743 1.06334 0.531672 0.846950i \(-0.321564\pi\)
0.531672 + 0.846950i \(0.321564\pi\)
\(992\) 0 0
\(993\) 30.8734 0.979737
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0.418627 0.0132580 0.00662902 0.999978i \(-0.497890\pi\)
0.00662902 + 0.999978i \(0.497890\pi\)
\(998\) 0 0
\(999\) −51.8248 −1.63966
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 4900.2.a.bf.1.3 4
5.2 odd 4 980.2.e.c.589.2 4
5.3 odd 4 980.2.e.c.589.4 4
5.4 even 2 inner 4900.2.a.bf.1.1 4
7.3 odd 6 700.2.i.f.401.3 8
7.5 odd 6 700.2.i.f.501.3 8
7.6 odd 2 4900.2.a.be.1.1 4
35.2 odd 12 980.2.q.g.949.1 4
35.3 even 12 140.2.q.a.9.2 4
35.12 even 12 140.2.q.a.109.2 yes 4
35.13 even 4 980.2.e.f.589.1 4
35.17 even 12 140.2.q.b.9.1 yes 4
35.18 odd 12 980.2.q.g.569.1 4
35.19 odd 6 700.2.i.f.501.2 8
35.23 odd 12 980.2.q.b.949.1 4
35.24 odd 6 700.2.i.f.401.2 8
35.27 even 4 980.2.e.f.589.3 4
35.32 odd 12 980.2.q.b.569.2 4
35.33 even 12 140.2.q.b.109.2 yes 4
35.34 odd 2 4900.2.a.be.1.3 4
105.17 odd 12 1260.2.bm.b.289.2 4
105.38 odd 12 1260.2.bm.a.289.1 4
105.47 odd 12 1260.2.bm.a.109.1 4
105.68 odd 12 1260.2.bm.b.109.1 4
140.3 odd 12 560.2.bw.e.289.2 4
140.47 odd 12 560.2.bw.e.529.2 4
140.87 odd 12 560.2.bw.a.289.1 4
140.103 odd 12 560.2.bw.a.529.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
140.2.q.a.9.2 4 35.3 even 12
140.2.q.a.109.2 yes 4 35.12 even 12
140.2.q.b.9.1 yes 4 35.17 even 12
140.2.q.b.109.2 yes 4 35.33 even 12
560.2.bw.a.289.1 4 140.87 odd 12
560.2.bw.a.529.2 4 140.103 odd 12
560.2.bw.e.289.2 4 140.3 odd 12
560.2.bw.e.529.2 4 140.47 odd 12
700.2.i.f.401.2 8 35.24 odd 6
700.2.i.f.401.3 8 7.3 odd 6
700.2.i.f.501.2 8 35.19 odd 6
700.2.i.f.501.3 8 7.5 odd 6
980.2.e.c.589.2 4 5.2 odd 4
980.2.e.c.589.4 4 5.3 odd 4
980.2.e.f.589.1 4 35.13 even 4
980.2.e.f.589.3 4 35.27 even 4
980.2.q.b.569.2 4 35.32 odd 12
980.2.q.b.949.1 4 35.23 odd 12
980.2.q.g.569.1 4 35.18 odd 12
980.2.q.g.949.1 4 35.2 odd 12
1260.2.bm.a.109.1 4 105.47 odd 12
1260.2.bm.a.289.1 4 105.38 odd 12
1260.2.bm.b.109.1 4 105.68 odd 12
1260.2.bm.b.289.2 4 105.17 odd 12
4900.2.a.be.1.1 4 7.6 odd 2
4900.2.a.be.1.3 4 35.34 odd 2
4900.2.a.bf.1.1 4 5.4 even 2 inner
4900.2.a.bf.1.3 4 1.1 even 1 trivial