Properties

Label 5520.2.a.bk.1.2
Level $5520$
Weight $2$
Character 5520.1
Self dual yes
Analytic conductor $44.077$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5520.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(44.0774219157\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2760)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 5520.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +6.24264 q^{11} -2.58579 q^{13} -1.00000 q^{15} -1.58579 q^{17} -7.07107 q^{19} -1.00000 q^{21} +1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -7.24264 q^{29} -0.656854 q^{31} -6.24264 q^{33} +1.00000 q^{35} -3.00000 q^{37} +2.58579 q^{39} -4.41421 q^{41} -2.00000 q^{43} +1.00000 q^{45} -2.58579 q^{47} -6.00000 q^{49} +1.58579 q^{51} -3.58579 q^{53} +6.24264 q^{55} +7.07107 q^{57} -2.07107 q^{59} -13.0711 q^{61} +1.00000 q^{63} -2.58579 q^{65} +0.656854 q^{67} -1.00000 q^{69} +12.0711 q^{71} -15.5563 q^{73} -1.00000 q^{75} +6.24264 q^{77} +5.65685 q^{79} +1.00000 q^{81} -1.92893 q^{83} -1.58579 q^{85} +7.24264 q^{87} -5.17157 q^{89} -2.58579 q^{91} +0.656854 q^{93} -7.07107 q^{95} -6.82843 q^{97} +6.24264 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} + 4 q^{11} - 8 q^{13} - 2 q^{15} - 6 q^{17} - 2 q^{21} + 2 q^{23} + 2 q^{25} - 2 q^{27} - 6 q^{29} + 10 q^{31} - 4 q^{33} + 2 q^{35} - 6 q^{37} + 8 q^{39} - 6 q^{41} - 4 q^{43} + 2 q^{45} - 8 q^{47} - 12 q^{49} + 6 q^{51} - 10 q^{53} + 4 q^{55} + 10 q^{59} - 12 q^{61} + 2 q^{63} - 8 q^{65} - 10 q^{67} - 2 q^{69} + 10 q^{71} - 2 q^{75} + 4 q^{77} + 2 q^{81} - 18 q^{83} - 6 q^{85} + 6 q^{87} - 16 q^{89} - 8 q^{91} - 10 q^{93} - 8 q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964 0.188982 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 6.24264 1.88223 0.941113 0.338091i \(-0.109781\pi\)
0.941113 + 0.338091i \(0.109781\pi\)
\(12\) 0 0
\(13\) −2.58579 −0.717168 −0.358584 0.933497i \(-0.616740\pi\)
−0.358584 + 0.933497i \(0.616740\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) −1.58579 −0.384610 −0.192305 0.981335i \(-0.561596\pi\)
−0.192305 + 0.981335i \(0.561596\pi\)
\(18\) 0 0
\(19\) −7.07107 −1.62221 −0.811107 0.584898i \(-0.801135\pi\)
−0.811107 + 0.584898i \(0.801135\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −7.24264 −1.34492 −0.672462 0.740131i \(-0.734763\pi\)
−0.672462 + 0.740131i \(0.734763\pi\)
\(30\) 0 0
\(31\) −0.656854 −0.117975 −0.0589873 0.998259i \(-0.518787\pi\)
−0.0589873 + 0.998259i \(0.518787\pi\)
\(32\) 0 0
\(33\) −6.24264 −1.08670
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) −3.00000 −0.493197 −0.246598 0.969118i \(-0.579313\pi\)
−0.246598 + 0.969118i \(0.579313\pi\)
\(38\) 0 0
\(39\) 2.58579 0.414057
\(40\) 0 0
\(41\) −4.41421 −0.689384 −0.344692 0.938716i \(-0.612017\pi\)
−0.344692 + 0.938716i \(0.612017\pi\)
\(42\) 0 0
\(43\) −2.00000 −0.304997 −0.152499 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) −2.58579 −0.377176 −0.188588 0.982056i \(-0.560391\pi\)
−0.188588 + 0.982056i \(0.560391\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) 1.58579 0.222055
\(52\) 0 0
\(53\) −3.58579 −0.492546 −0.246273 0.969201i \(-0.579206\pi\)
−0.246273 + 0.969201i \(0.579206\pi\)
\(54\) 0 0
\(55\) 6.24264 0.841757
\(56\) 0 0
\(57\) 7.07107 0.936586
\(58\) 0 0
\(59\) −2.07107 −0.269630 −0.134815 0.990871i \(-0.543044\pi\)
−0.134815 + 0.990871i \(0.543044\pi\)
\(60\) 0 0
\(61\) −13.0711 −1.67358 −0.836789 0.547525i \(-0.815570\pi\)
−0.836789 + 0.547525i \(0.815570\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) −2.58579 −0.320727
\(66\) 0 0
\(67\) 0.656854 0.0802475 0.0401238 0.999195i \(-0.487225\pi\)
0.0401238 + 0.999195i \(0.487225\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 12.0711 1.43257 0.716286 0.697807i \(-0.245841\pi\)
0.716286 + 0.697807i \(0.245841\pi\)
\(72\) 0 0
\(73\) −15.5563 −1.82073 −0.910366 0.413803i \(-0.864200\pi\)
−0.910366 + 0.413803i \(0.864200\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 6.24264 0.711415
\(78\) 0 0
\(79\) 5.65685 0.636446 0.318223 0.948016i \(-0.396914\pi\)
0.318223 + 0.948016i \(0.396914\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −1.92893 −0.211728 −0.105864 0.994381i \(-0.533761\pi\)
−0.105864 + 0.994381i \(0.533761\pi\)
\(84\) 0 0
\(85\) −1.58579 −0.172003
\(86\) 0 0
\(87\) 7.24264 0.776493
\(88\) 0 0
\(89\) −5.17157 −0.548186 −0.274093 0.961703i \(-0.588378\pi\)
−0.274093 + 0.961703i \(0.588378\pi\)
\(90\) 0 0
\(91\) −2.58579 −0.271064
\(92\) 0 0
\(93\) 0.656854 0.0681126
\(94\) 0 0
\(95\) −7.07107 −0.725476
\(96\) 0 0
\(97\) −6.82843 −0.693322 −0.346661 0.937991i \(-0.612685\pi\)
−0.346661 + 0.937991i \(0.612685\pi\)
\(98\) 0 0
\(99\) 6.24264 0.627409
\(100\) 0 0
\(101\) 11.2426 1.11868 0.559342 0.828937i \(-0.311054\pi\)
0.559342 + 0.828937i \(0.311054\pi\)
\(102\) 0 0
\(103\) −6.48528 −0.639014 −0.319507 0.947584i \(-0.603517\pi\)
−0.319507 + 0.947584i \(0.603517\pi\)
\(104\) 0 0
\(105\) −1.00000 −0.0975900
\(106\) 0 0
\(107\) 5.24264 0.506825 0.253413 0.967358i \(-0.418447\pi\)
0.253413 + 0.967358i \(0.418447\pi\)
\(108\) 0 0
\(109\) 18.3848 1.76094 0.880471 0.474100i \(-0.157226\pi\)
0.880471 + 0.474100i \(0.157226\pi\)
\(110\) 0 0
\(111\) 3.00000 0.284747
\(112\) 0 0
\(113\) −9.24264 −0.869474 −0.434737 0.900557i \(-0.643159\pi\)
−0.434737 + 0.900557i \(0.643159\pi\)
\(114\) 0 0
\(115\) 1.00000 0.0932505
\(116\) 0 0
\(117\) −2.58579 −0.239056
\(118\) 0 0
\(119\) −1.58579 −0.145369
\(120\) 0 0
\(121\) 27.9706 2.54278
\(122\) 0 0
\(123\) 4.41421 0.398016
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −8.58579 −0.761865 −0.380933 0.924603i \(-0.624397\pi\)
−0.380933 + 0.924603i \(0.624397\pi\)
\(128\) 0 0
\(129\) 2.00000 0.176090
\(130\) 0 0
\(131\) 5.65685 0.494242 0.247121 0.968985i \(-0.420516\pi\)
0.247121 + 0.968985i \(0.420516\pi\)
\(132\) 0 0
\(133\) −7.07107 −0.613139
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 16.4853 1.40843 0.704216 0.709985i \(-0.251299\pi\)
0.704216 + 0.709985i \(0.251299\pi\)
\(138\) 0 0
\(139\) −19.4853 −1.65272 −0.826360 0.563142i \(-0.809592\pi\)
−0.826360 + 0.563142i \(0.809592\pi\)
\(140\) 0 0
\(141\) 2.58579 0.217763
\(142\) 0 0
\(143\) −16.1421 −1.34987
\(144\) 0 0
\(145\) −7.24264 −0.601469
\(146\) 0 0
\(147\) 6.00000 0.494872
\(148\) 0 0
\(149\) −13.4142 −1.09894 −0.549468 0.835515i \(-0.685169\pi\)
−0.549468 + 0.835515i \(0.685169\pi\)
\(150\) 0 0
\(151\) 11.6569 0.948621 0.474311 0.880358i \(-0.342697\pi\)
0.474311 + 0.880358i \(0.342697\pi\)
\(152\) 0 0
\(153\) −1.58579 −0.128203
\(154\) 0 0
\(155\) −0.656854 −0.0527598
\(156\) 0 0
\(157\) −15.4853 −1.23586 −0.617930 0.786233i \(-0.712028\pi\)
−0.617930 + 0.786233i \(0.712028\pi\)
\(158\) 0 0
\(159\) 3.58579 0.284371
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) 3.65685 0.286427 0.143213 0.989692i \(-0.454256\pi\)
0.143213 + 0.989692i \(0.454256\pi\)
\(164\) 0 0
\(165\) −6.24264 −0.485989
\(166\) 0 0
\(167\) −3.07107 −0.237646 −0.118823 0.992915i \(-0.537912\pi\)
−0.118823 + 0.992915i \(0.537912\pi\)
\(168\) 0 0
\(169\) −6.31371 −0.485670
\(170\) 0 0
\(171\) −7.07107 −0.540738
\(172\) 0 0
\(173\) 11.3137 0.860165 0.430083 0.902790i \(-0.358484\pi\)
0.430083 + 0.902790i \(0.358484\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 2.07107 0.155671
\(178\) 0 0
\(179\) 23.6569 1.76820 0.884098 0.467301i \(-0.154774\pi\)
0.884098 + 0.467301i \(0.154774\pi\)
\(180\) 0 0
\(181\) 22.1421 1.64581 0.822906 0.568178i \(-0.192351\pi\)
0.822906 + 0.568178i \(0.192351\pi\)
\(182\) 0 0
\(183\) 13.0711 0.966241
\(184\) 0 0
\(185\) −3.00000 −0.220564
\(186\) 0 0
\(187\) −9.89949 −0.723923
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −5.89949 −0.426872 −0.213436 0.976957i \(-0.568466\pi\)
−0.213436 + 0.976957i \(0.568466\pi\)
\(192\) 0 0
\(193\) 24.4853 1.76249 0.881245 0.472661i \(-0.156706\pi\)
0.881245 + 0.472661i \(0.156706\pi\)
\(194\) 0 0
\(195\) 2.58579 0.185172
\(196\) 0 0
\(197\) −25.7990 −1.83810 −0.919051 0.394139i \(-0.871043\pi\)
−0.919051 + 0.394139i \(0.871043\pi\)
\(198\) 0 0
\(199\) 27.4558 1.94629 0.973147 0.230186i \(-0.0739335\pi\)
0.973147 + 0.230186i \(0.0739335\pi\)
\(200\) 0 0
\(201\) −0.656854 −0.0463309
\(202\) 0 0
\(203\) −7.24264 −0.508334
\(204\) 0 0
\(205\) −4.41421 −0.308302
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) −44.1421 −3.05338
\(210\) 0 0
\(211\) −15.8284 −1.08967 −0.544837 0.838542i \(-0.683408\pi\)
−0.544837 + 0.838542i \(0.683408\pi\)
\(212\) 0 0
\(213\) −12.0711 −0.827096
\(214\) 0 0
\(215\) −2.00000 −0.136399
\(216\) 0 0
\(217\) −0.656854 −0.0445902
\(218\) 0 0
\(219\) 15.5563 1.05120
\(220\) 0 0
\(221\) 4.10051 0.275830
\(222\) 0 0
\(223\) −6.97056 −0.466783 −0.233392 0.972383i \(-0.574982\pi\)
−0.233392 + 0.972383i \(0.574982\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −20.0000 −1.32745 −0.663723 0.747978i \(-0.731025\pi\)
−0.663723 + 0.747978i \(0.731025\pi\)
\(228\) 0 0
\(229\) 1.17157 0.0774197 0.0387099 0.999250i \(-0.487675\pi\)
0.0387099 + 0.999250i \(0.487675\pi\)
\(230\) 0 0
\(231\) −6.24264 −0.410736
\(232\) 0 0
\(233\) −29.6569 −1.94289 −0.971443 0.237275i \(-0.923746\pi\)
−0.971443 + 0.237275i \(0.923746\pi\)
\(234\) 0 0
\(235\) −2.58579 −0.168678
\(236\) 0 0
\(237\) −5.65685 −0.367452
\(238\) 0 0
\(239\) −8.07107 −0.522074 −0.261037 0.965329i \(-0.584064\pi\)
−0.261037 + 0.965329i \(0.584064\pi\)
\(240\) 0 0
\(241\) 12.2426 0.788618 0.394309 0.918978i \(-0.370984\pi\)
0.394309 + 0.918978i \(0.370984\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −6.00000 −0.383326
\(246\) 0 0
\(247\) 18.2843 1.16340
\(248\) 0 0
\(249\) 1.92893 0.122241
\(250\) 0 0
\(251\) −15.6569 −0.988252 −0.494126 0.869390i \(-0.664512\pi\)
−0.494126 + 0.869390i \(0.664512\pi\)
\(252\) 0 0
\(253\) 6.24264 0.392471
\(254\) 0 0
\(255\) 1.58579 0.0993058
\(256\) 0 0
\(257\) 12.7279 0.793946 0.396973 0.917830i \(-0.370061\pi\)
0.396973 + 0.917830i \(0.370061\pi\)
\(258\) 0 0
\(259\) −3.00000 −0.186411
\(260\) 0 0
\(261\) −7.24264 −0.448308
\(262\) 0 0
\(263\) −4.27208 −0.263428 −0.131714 0.991288i \(-0.542048\pi\)
−0.131714 + 0.991288i \(0.542048\pi\)
\(264\) 0 0
\(265\) −3.58579 −0.220273
\(266\) 0 0
\(267\) 5.17157 0.316495
\(268\) 0 0
\(269\) −15.5858 −0.950282 −0.475141 0.879910i \(-0.657603\pi\)
−0.475141 + 0.879910i \(0.657603\pi\)
\(270\) 0 0
\(271\) 9.97056 0.605669 0.302834 0.953043i \(-0.402067\pi\)
0.302834 + 0.953043i \(0.402067\pi\)
\(272\) 0 0
\(273\) 2.58579 0.156499
\(274\) 0 0
\(275\) 6.24264 0.376445
\(276\) 0 0
\(277\) −0.343146 −0.0206176 −0.0103088 0.999947i \(-0.503281\pi\)
−0.0103088 + 0.999947i \(0.503281\pi\)
\(278\) 0 0
\(279\) −0.656854 −0.0393248
\(280\) 0 0
\(281\) −5.07107 −0.302515 −0.151257 0.988494i \(-0.548332\pi\)
−0.151257 + 0.988494i \(0.548332\pi\)
\(282\) 0 0
\(283\) −26.4558 −1.57264 −0.786318 0.617822i \(-0.788015\pi\)
−0.786318 + 0.617822i \(0.788015\pi\)
\(284\) 0 0
\(285\) 7.07107 0.418854
\(286\) 0 0
\(287\) −4.41421 −0.260563
\(288\) 0 0
\(289\) −14.4853 −0.852075
\(290\) 0 0
\(291\) 6.82843 0.400289
\(292\) 0 0
\(293\) 15.0416 0.878741 0.439371 0.898306i \(-0.355201\pi\)
0.439371 + 0.898306i \(0.355201\pi\)
\(294\) 0 0
\(295\) −2.07107 −0.120582
\(296\) 0 0
\(297\) −6.24264 −0.362235
\(298\) 0 0
\(299\) −2.58579 −0.149540
\(300\) 0 0
\(301\) −2.00000 −0.115278
\(302\) 0 0
\(303\) −11.2426 −0.645873
\(304\) 0 0
\(305\) −13.0711 −0.748447
\(306\) 0 0
\(307\) −14.3848 −0.820983 −0.410491 0.911865i \(-0.634643\pi\)
−0.410491 + 0.911865i \(0.634643\pi\)
\(308\) 0 0
\(309\) 6.48528 0.368935
\(310\) 0 0
\(311\) −12.9706 −0.735493 −0.367747 0.929926i \(-0.619871\pi\)
−0.367747 + 0.929926i \(0.619871\pi\)
\(312\) 0 0
\(313\) 19.0000 1.07394 0.536972 0.843600i \(-0.319568\pi\)
0.536972 + 0.843600i \(0.319568\pi\)
\(314\) 0 0
\(315\) 1.00000 0.0563436
\(316\) 0 0
\(317\) −27.5563 −1.54772 −0.773859 0.633357i \(-0.781676\pi\)
−0.773859 + 0.633357i \(0.781676\pi\)
\(318\) 0 0
\(319\) −45.2132 −2.53145
\(320\) 0 0
\(321\) −5.24264 −0.292616
\(322\) 0 0
\(323\) 11.2132 0.623919
\(324\) 0 0
\(325\) −2.58579 −0.143434
\(326\) 0 0
\(327\) −18.3848 −1.01668
\(328\) 0 0
\(329\) −2.58579 −0.142559
\(330\) 0 0
\(331\) 8.79899 0.483636 0.241818 0.970322i \(-0.422256\pi\)
0.241818 + 0.970322i \(0.422256\pi\)
\(332\) 0 0
\(333\) −3.00000 −0.164399
\(334\) 0 0
\(335\) 0.656854 0.0358878
\(336\) 0 0
\(337\) −3.65685 −0.199202 −0.0996008 0.995027i \(-0.531757\pi\)
−0.0996008 + 0.995027i \(0.531757\pi\)
\(338\) 0 0
\(339\) 9.24264 0.501991
\(340\) 0 0
\(341\) −4.10051 −0.222055
\(342\) 0 0
\(343\) −13.0000 −0.701934
\(344\) 0 0
\(345\) −1.00000 −0.0538382
\(346\) 0 0
\(347\) −18.8284 −1.01076 −0.505381 0.862896i \(-0.668648\pi\)
−0.505381 + 0.862896i \(0.668648\pi\)
\(348\) 0 0
\(349\) −6.17157 −0.330357 −0.165178 0.986264i \(-0.552820\pi\)
−0.165178 + 0.986264i \(0.552820\pi\)
\(350\) 0 0
\(351\) 2.58579 0.138019
\(352\) 0 0
\(353\) −24.2426 −1.29031 −0.645153 0.764054i \(-0.723206\pi\)
−0.645153 + 0.764054i \(0.723206\pi\)
\(354\) 0 0
\(355\) 12.0711 0.640666
\(356\) 0 0
\(357\) 1.58579 0.0839287
\(358\) 0 0
\(359\) 20.2426 1.06837 0.534183 0.845369i \(-0.320619\pi\)
0.534183 + 0.845369i \(0.320619\pi\)
\(360\) 0 0
\(361\) 31.0000 1.63158
\(362\) 0 0
\(363\) −27.9706 −1.46807
\(364\) 0 0
\(365\) −15.5563 −0.814257
\(366\) 0 0
\(367\) −9.14214 −0.477216 −0.238608 0.971116i \(-0.576691\pi\)
−0.238608 + 0.971116i \(0.576691\pi\)
\(368\) 0 0
\(369\) −4.41421 −0.229795
\(370\) 0 0
\(371\) −3.58579 −0.186165
\(372\) 0 0
\(373\) 8.97056 0.464478 0.232239 0.972659i \(-0.425395\pi\)
0.232239 + 0.972659i \(0.425395\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 18.7279 0.964537
\(378\) 0 0
\(379\) 6.34315 0.325826 0.162913 0.986640i \(-0.447911\pi\)
0.162913 + 0.986640i \(0.447911\pi\)
\(380\) 0 0
\(381\) 8.58579 0.439863
\(382\) 0 0
\(383\) −11.9289 −0.609540 −0.304770 0.952426i \(-0.598580\pi\)
−0.304770 + 0.952426i \(0.598580\pi\)
\(384\) 0 0
\(385\) 6.24264 0.318154
\(386\) 0 0
\(387\) −2.00000 −0.101666
\(388\) 0 0
\(389\) −7.85786 −0.398410 −0.199205 0.979958i \(-0.563836\pi\)
−0.199205 + 0.979958i \(0.563836\pi\)
\(390\) 0 0
\(391\) −1.58579 −0.0801967
\(392\) 0 0
\(393\) −5.65685 −0.285351
\(394\) 0 0
\(395\) 5.65685 0.284627
\(396\) 0 0
\(397\) −16.0000 −0.803017 −0.401508 0.915855i \(-0.631514\pi\)
−0.401508 + 0.915855i \(0.631514\pi\)
\(398\) 0 0
\(399\) 7.07107 0.353996
\(400\) 0 0
\(401\) 2.48528 0.124109 0.0620545 0.998073i \(-0.480235\pi\)
0.0620545 + 0.998073i \(0.480235\pi\)
\(402\) 0 0
\(403\) 1.69848 0.0846076
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) −18.7279 −0.928309
\(408\) 0 0
\(409\) 22.4558 1.11037 0.555185 0.831727i \(-0.312647\pi\)
0.555185 + 0.831727i \(0.312647\pi\)
\(410\) 0 0
\(411\) −16.4853 −0.813159
\(412\) 0 0
\(413\) −2.07107 −0.101911
\(414\) 0 0
\(415\) −1.92893 −0.0946876
\(416\) 0 0
\(417\) 19.4853 0.954198
\(418\) 0 0
\(419\) 0.384776 0.0187976 0.00939878 0.999956i \(-0.497008\pi\)
0.00939878 + 0.999956i \(0.497008\pi\)
\(420\) 0 0
\(421\) −38.8701 −1.89441 −0.947205 0.320628i \(-0.896106\pi\)
−0.947205 + 0.320628i \(0.896106\pi\)
\(422\) 0 0
\(423\) −2.58579 −0.125725
\(424\) 0 0
\(425\) −1.58579 −0.0769219
\(426\) 0 0
\(427\) −13.0711 −0.632553
\(428\) 0 0
\(429\) 16.1421 0.779350
\(430\) 0 0
\(431\) 10.8284 0.521587 0.260793 0.965395i \(-0.416016\pi\)
0.260793 + 0.965395i \(0.416016\pi\)
\(432\) 0 0
\(433\) 29.8284 1.43346 0.716731 0.697349i \(-0.245637\pi\)
0.716731 + 0.697349i \(0.245637\pi\)
\(434\) 0 0
\(435\) 7.24264 0.347258
\(436\) 0 0
\(437\) −7.07107 −0.338255
\(438\) 0 0
\(439\) 41.7990 1.99496 0.997478 0.0709697i \(-0.0226094\pi\)
0.997478 + 0.0709697i \(0.0226094\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) 0 0
\(443\) 0.928932 0.0441349 0.0220675 0.999756i \(-0.492975\pi\)
0.0220675 + 0.999756i \(0.492975\pi\)
\(444\) 0 0
\(445\) −5.17157 −0.245156
\(446\) 0 0
\(447\) 13.4142 0.634471
\(448\) 0 0
\(449\) −6.27208 −0.295998 −0.147999 0.988988i \(-0.547283\pi\)
−0.147999 + 0.988988i \(0.547283\pi\)
\(450\) 0 0
\(451\) −27.5563 −1.29758
\(452\) 0 0
\(453\) −11.6569 −0.547687
\(454\) 0 0
\(455\) −2.58579 −0.121224
\(456\) 0 0
\(457\) 19.0000 0.888783 0.444391 0.895833i \(-0.353420\pi\)
0.444391 + 0.895833i \(0.353420\pi\)
\(458\) 0 0
\(459\) 1.58579 0.0740182
\(460\) 0 0
\(461\) 1.51472 0.0705475 0.0352737 0.999378i \(-0.488770\pi\)
0.0352737 + 0.999378i \(0.488770\pi\)
\(462\) 0 0
\(463\) 20.0416 0.931414 0.465707 0.884939i \(-0.345800\pi\)
0.465707 + 0.884939i \(0.345800\pi\)
\(464\) 0 0
\(465\) 0.656854 0.0304609
\(466\) 0 0
\(467\) 7.58579 0.351028 0.175514 0.984477i \(-0.443841\pi\)
0.175514 + 0.984477i \(0.443841\pi\)
\(468\) 0 0
\(469\) 0.656854 0.0303307
\(470\) 0 0
\(471\) 15.4853 0.713524
\(472\) 0 0
\(473\) −12.4853 −0.574074
\(474\) 0 0
\(475\) −7.07107 −0.324443
\(476\) 0 0
\(477\) −3.58579 −0.164182
\(478\) 0 0
\(479\) −9.89949 −0.452319 −0.226160 0.974090i \(-0.572617\pi\)
−0.226160 + 0.974090i \(0.572617\pi\)
\(480\) 0 0
\(481\) 7.75736 0.353705
\(482\) 0 0
\(483\) −1.00000 −0.0455016
\(484\) 0 0
\(485\) −6.82843 −0.310063
\(486\) 0 0
\(487\) 9.07107 0.411049 0.205525 0.978652i \(-0.434110\pi\)
0.205525 + 0.978652i \(0.434110\pi\)
\(488\) 0 0
\(489\) −3.65685 −0.165369
\(490\) 0 0
\(491\) −5.10051 −0.230183 −0.115091 0.993355i \(-0.536716\pi\)
−0.115091 + 0.993355i \(0.536716\pi\)
\(492\) 0 0
\(493\) 11.4853 0.517271
\(494\) 0 0
\(495\) 6.24264 0.280586
\(496\) 0 0
\(497\) 12.0711 0.541461
\(498\) 0 0
\(499\) 4.17157 0.186745 0.0933726 0.995631i \(-0.470235\pi\)
0.0933726 + 0.995631i \(0.470235\pi\)
\(500\) 0 0
\(501\) 3.07107 0.137205
\(502\) 0 0
\(503\) 34.6985 1.54713 0.773564 0.633718i \(-0.218472\pi\)
0.773564 + 0.633718i \(0.218472\pi\)
\(504\) 0 0
\(505\) 11.2426 0.500291
\(506\) 0 0
\(507\) 6.31371 0.280402
\(508\) 0 0
\(509\) −23.7990 −1.05487 −0.527436 0.849595i \(-0.676846\pi\)
−0.527436 + 0.849595i \(0.676846\pi\)
\(510\) 0 0
\(511\) −15.5563 −0.688172
\(512\) 0 0
\(513\) 7.07107 0.312195
\(514\) 0 0
\(515\) −6.48528 −0.285776
\(516\) 0 0
\(517\) −16.1421 −0.709930
\(518\) 0 0
\(519\) −11.3137 −0.496617
\(520\) 0 0
\(521\) 9.07107 0.397411 0.198705 0.980059i \(-0.436326\pi\)
0.198705 + 0.980059i \(0.436326\pi\)
\(522\) 0 0
\(523\) −41.6569 −1.82153 −0.910764 0.412928i \(-0.864506\pi\)
−0.910764 + 0.412928i \(0.864506\pi\)
\(524\) 0 0
\(525\) −1.00000 −0.0436436
\(526\) 0 0
\(527\) 1.04163 0.0453741
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −2.07107 −0.0898767
\(532\) 0 0
\(533\) 11.4142 0.494404
\(534\) 0 0
\(535\) 5.24264 0.226659
\(536\) 0 0
\(537\) −23.6569 −1.02087
\(538\) 0 0
\(539\) −37.4558 −1.61334
\(540\) 0 0
\(541\) 27.5147 1.18295 0.591475 0.806323i \(-0.298546\pi\)
0.591475 + 0.806323i \(0.298546\pi\)
\(542\) 0 0
\(543\) −22.1421 −0.950210
\(544\) 0 0
\(545\) 18.3848 0.787517
\(546\) 0 0
\(547\) 31.3137 1.33888 0.669439 0.742867i \(-0.266535\pi\)
0.669439 + 0.742867i \(0.266535\pi\)
\(548\) 0 0
\(549\) −13.0711 −0.557860
\(550\) 0 0
\(551\) 51.2132 2.18176
\(552\) 0 0
\(553\) 5.65685 0.240554
\(554\) 0 0
\(555\) 3.00000 0.127343
\(556\) 0 0
\(557\) 28.4142 1.20395 0.601974 0.798515i \(-0.294381\pi\)
0.601974 + 0.798515i \(0.294381\pi\)
\(558\) 0 0
\(559\) 5.17157 0.218734
\(560\) 0 0
\(561\) 9.89949 0.417957
\(562\) 0 0
\(563\) −5.24264 −0.220951 −0.110475 0.993879i \(-0.535237\pi\)
−0.110475 + 0.993879i \(0.535237\pi\)
\(564\) 0 0
\(565\) −9.24264 −0.388841
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −24.1421 −1.01209 −0.506045 0.862507i \(-0.668893\pi\)
−0.506045 + 0.862507i \(0.668893\pi\)
\(570\) 0 0
\(571\) −5.27208 −0.220630 −0.110315 0.993897i \(-0.535186\pi\)
−0.110315 + 0.993897i \(0.535186\pi\)
\(572\) 0 0
\(573\) 5.89949 0.246455
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) 45.9411 1.91255 0.956277 0.292462i \(-0.0944746\pi\)
0.956277 + 0.292462i \(0.0944746\pi\)
\(578\) 0 0
\(579\) −24.4853 −1.01757
\(580\) 0 0
\(581\) −1.92893 −0.0800256
\(582\) 0 0
\(583\) −22.3848 −0.927083
\(584\) 0 0
\(585\) −2.58579 −0.106909
\(586\) 0 0
\(587\) −14.0000 −0.577842 −0.288921 0.957353i \(-0.593296\pi\)
−0.288921 + 0.957353i \(0.593296\pi\)
\(588\) 0 0
\(589\) 4.64466 0.191380
\(590\) 0 0
\(591\) 25.7990 1.06123
\(592\) 0 0
\(593\) 26.8701 1.10342 0.551711 0.834036i \(-0.313975\pi\)
0.551711 + 0.834036i \(0.313975\pi\)
\(594\) 0 0
\(595\) −1.58579 −0.0650109
\(596\) 0 0
\(597\) −27.4558 −1.12369
\(598\) 0 0
\(599\) −8.48528 −0.346699 −0.173350 0.984860i \(-0.555459\pi\)
−0.173350 + 0.984860i \(0.555459\pi\)
\(600\) 0 0
\(601\) −22.3137 −0.910195 −0.455098 0.890442i \(-0.650396\pi\)
−0.455098 + 0.890442i \(0.650396\pi\)
\(602\) 0 0
\(603\) 0.656854 0.0267492
\(604\) 0 0
\(605\) 27.9706 1.13717
\(606\) 0 0
\(607\) −34.0416 −1.38171 −0.690854 0.722995i \(-0.742765\pi\)
−0.690854 + 0.722995i \(0.742765\pi\)
\(608\) 0 0
\(609\) 7.24264 0.293487
\(610\) 0 0
\(611\) 6.68629 0.270498
\(612\) 0 0
\(613\) 3.37258 0.136217 0.0681087 0.997678i \(-0.478304\pi\)
0.0681087 + 0.997678i \(0.478304\pi\)
\(614\) 0 0
\(615\) 4.41421 0.177998
\(616\) 0 0
\(617\) 9.10051 0.366373 0.183186 0.983078i \(-0.441359\pi\)
0.183186 + 0.983078i \(0.441359\pi\)
\(618\) 0 0
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) −5.17157 −0.207195
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 44.1421 1.76287
\(628\) 0 0
\(629\) 4.75736 0.189688
\(630\) 0 0
\(631\) 12.1005 0.481714 0.240857 0.970561i \(-0.422572\pi\)
0.240857 + 0.970561i \(0.422572\pi\)
\(632\) 0 0
\(633\) 15.8284 0.629123
\(634\) 0 0
\(635\) −8.58579 −0.340717
\(636\) 0 0
\(637\) 15.5147 0.614716
\(638\) 0 0
\(639\) 12.0711 0.477524
\(640\) 0 0
\(641\) −6.92893 −0.273676 −0.136838 0.990593i \(-0.543694\pi\)
−0.136838 + 0.990593i \(0.543694\pi\)
\(642\) 0 0
\(643\) −26.1716 −1.03211 −0.516053 0.856557i \(-0.672599\pi\)
−0.516053 + 0.856557i \(0.672599\pi\)
\(644\) 0 0
\(645\) 2.00000 0.0787499
\(646\) 0 0
\(647\) 17.2721 0.679035 0.339518 0.940600i \(-0.389736\pi\)
0.339518 + 0.940600i \(0.389736\pi\)
\(648\) 0 0
\(649\) −12.9289 −0.507505
\(650\) 0 0
\(651\) 0.656854 0.0257441
\(652\) 0 0
\(653\) −37.8406 −1.48082 −0.740409 0.672157i \(-0.765368\pi\)
−0.740409 + 0.672157i \(0.765368\pi\)
\(654\) 0 0
\(655\) 5.65685 0.221032
\(656\) 0 0
\(657\) −15.5563 −0.606911
\(658\) 0 0
\(659\) −23.6985 −0.923162 −0.461581 0.887098i \(-0.652718\pi\)
−0.461581 + 0.887098i \(0.652718\pi\)
\(660\) 0 0
\(661\) 1.51472 0.0589157 0.0294579 0.999566i \(-0.490622\pi\)
0.0294579 + 0.999566i \(0.490622\pi\)
\(662\) 0 0
\(663\) −4.10051 −0.159250
\(664\) 0 0
\(665\) −7.07107 −0.274204
\(666\) 0 0
\(667\) −7.24264 −0.280436
\(668\) 0 0
\(669\) 6.97056 0.269498
\(670\) 0 0
\(671\) −81.5980 −3.15006
\(672\) 0 0
\(673\) −20.8701 −0.804482 −0.402241 0.915534i \(-0.631769\pi\)
−0.402241 + 0.915534i \(0.631769\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 10.8995 0.418902 0.209451 0.977819i \(-0.432832\pi\)
0.209451 + 0.977819i \(0.432832\pi\)
\(678\) 0 0
\(679\) −6.82843 −0.262051
\(680\) 0 0
\(681\) 20.0000 0.766402
\(682\) 0 0
\(683\) −19.2721 −0.737426 −0.368713 0.929543i \(-0.620201\pi\)
−0.368713 + 0.929543i \(0.620201\pi\)
\(684\) 0 0
\(685\) 16.4853 0.629870
\(686\) 0 0
\(687\) −1.17157 −0.0446983
\(688\) 0 0
\(689\) 9.27208 0.353238
\(690\) 0 0
\(691\) −44.6274 −1.69771 −0.848853 0.528628i \(-0.822707\pi\)
−0.848853 + 0.528628i \(0.822707\pi\)
\(692\) 0 0
\(693\) 6.24264 0.237138
\(694\) 0 0
\(695\) −19.4853 −0.739119
\(696\) 0 0
\(697\) 7.00000 0.265144
\(698\) 0 0
\(699\) 29.6569 1.12173
\(700\) 0 0
\(701\) 29.6985 1.12170 0.560848 0.827919i \(-0.310475\pi\)
0.560848 + 0.827919i \(0.310475\pi\)
\(702\) 0 0
\(703\) 21.2132 0.800071
\(704\) 0 0
\(705\) 2.58579 0.0973864
\(706\) 0 0
\(707\) 11.2426 0.422823
\(708\) 0 0
\(709\) 22.9289 0.861114 0.430557 0.902563i \(-0.358317\pi\)
0.430557 + 0.902563i \(0.358317\pi\)
\(710\) 0 0
\(711\) 5.65685 0.212149
\(712\) 0 0
\(713\) −0.656854 −0.0245994
\(714\) 0 0
\(715\) −16.1421 −0.603682
\(716\) 0 0
\(717\) 8.07107 0.301420
\(718\) 0 0
\(719\) 2.41421 0.0900350 0.0450175 0.998986i \(-0.485666\pi\)
0.0450175 + 0.998986i \(0.485666\pi\)
\(720\) 0 0
\(721\) −6.48528 −0.241524
\(722\) 0 0
\(723\) −12.2426 −0.455309
\(724\) 0 0
\(725\) −7.24264 −0.268985
\(726\) 0 0
\(727\) −19.1421 −0.709943 −0.354971 0.934877i \(-0.615509\pi\)
−0.354971 + 0.934877i \(0.615509\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 3.17157 0.117305
\(732\) 0 0
\(733\) −29.8284 −1.10174 −0.550869 0.834592i \(-0.685704\pi\)
−0.550869 + 0.834592i \(0.685704\pi\)
\(734\) 0 0
\(735\) 6.00000 0.221313
\(736\) 0 0
\(737\) 4.10051 0.151044
\(738\) 0 0
\(739\) −25.1421 −0.924868 −0.462434 0.886654i \(-0.653024\pi\)
−0.462434 + 0.886654i \(0.653024\pi\)
\(740\) 0 0
\(741\) −18.2843 −0.671689
\(742\) 0 0
\(743\) −18.6274 −0.683374 −0.341687 0.939814i \(-0.610998\pi\)
−0.341687 + 0.939814i \(0.610998\pi\)
\(744\) 0 0
\(745\) −13.4142 −0.491459
\(746\) 0 0
\(747\) −1.92893 −0.0705759
\(748\) 0 0
\(749\) 5.24264 0.191562
\(750\) 0 0
\(751\) 11.2721 0.411324 0.205662 0.978623i \(-0.434065\pi\)
0.205662 + 0.978623i \(0.434065\pi\)
\(752\) 0 0
\(753\) 15.6569 0.570567
\(754\) 0 0
\(755\) 11.6569 0.424236
\(756\) 0 0
\(757\) −47.0000 −1.70824 −0.854122 0.520073i \(-0.825905\pi\)
−0.854122 + 0.520073i \(0.825905\pi\)
\(758\) 0 0
\(759\) −6.24264 −0.226594
\(760\) 0 0
\(761\) −30.8995 −1.12011 −0.560053 0.828457i \(-0.689219\pi\)
−0.560053 + 0.828457i \(0.689219\pi\)
\(762\) 0 0
\(763\) 18.3848 0.665574
\(764\) 0 0
\(765\) −1.58579 −0.0573342
\(766\) 0 0
\(767\) 5.35534 0.193370
\(768\) 0 0
\(769\) −19.4142 −0.700094 −0.350047 0.936732i \(-0.613834\pi\)
−0.350047 + 0.936732i \(0.613834\pi\)
\(770\) 0 0
\(771\) −12.7279 −0.458385
\(772\) 0 0
\(773\) 25.1127 0.903241 0.451620 0.892210i \(-0.350846\pi\)
0.451620 + 0.892210i \(0.350846\pi\)
\(774\) 0 0
\(775\) −0.656854 −0.0235949
\(776\) 0 0
\(777\) 3.00000 0.107624
\(778\) 0 0
\(779\) 31.2132 1.11833
\(780\) 0 0
\(781\) 75.3553 2.69643
\(782\) 0 0
\(783\) 7.24264 0.258831
\(784\) 0 0
\(785\) −15.4853 −0.552693
\(786\) 0 0
\(787\) 27.7696 0.989878 0.494939 0.868928i \(-0.335190\pi\)
0.494939 + 0.868928i \(0.335190\pi\)
\(788\) 0 0
\(789\) 4.27208 0.152090
\(790\) 0 0
\(791\) −9.24264 −0.328630
\(792\) 0 0
\(793\) 33.7990 1.20024
\(794\) 0 0
\(795\) 3.58579 0.127175
\(796\) 0 0
\(797\) 37.5269 1.32927 0.664636 0.747168i \(-0.268587\pi\)
0.664636 + 0.747168i \(0.268587\pi\)
\(798\) 0 0
\(799\) 4.10051 0.145065
\(800\) 0 0
\(801\) −5.17157 −0.182729
\(802\) 0 0
\(803\) −97.1127 −3.42703
\(804\) 0 0
\(805\) 1.00000 0.0352454
\(806\) 0 0
\(807\) 15.5858 0.548646
\(808\) 0 0
\(809\) 8.41421 0.295828 0.147914 0.989000i \(-0.452744\pi\)
0.147914 + 0.989000i \(0.452744\pi\)
\(810\) 0 0
\(811\) 29.8284 1.04742 0.523709 0.851897i \(-0.324548\pi\)
0.523709 + 0.851897i \(0.324548\pi\)
\(812\) 0 0
\(813\) −9.97056 −0.349683
\(814\) 0 0
\(815\) 3.65685 0.128094
\(816\) 0 0
\(817\) 14.1421 0.494771
\(818\) 0 0
\(819\) −2.58579 −0.0903547
\(820\) 0 0
\(821\) −23.1127 −0.806639 −0.403319 0.915059i \(-0.632144\pi\)
−0.403319 + 0.915059i \(0.632144\pi\)
\(822\) 0 0
\(823\) −6.62742 −0.231017 −0.115509 0.993306i \(-0.536850\pi\)
−0.115509 + 0.993306i \(0.536850\pi\)
\(824\) 0 0
\(825\) −6.24264 −0.217341
\(826\) 0 0
\(827\) −17.0416 −0.592596 −0.296298 0.955096i \(-0.595752\pi\)
−0.296298 + 0.955096i \(0.595752\pi\)
\(828\) 0 0
\(829\) −40.7990 −1.41701 −0.708504 0.705707i \(-0.750629\pi\)
−0.708504 + 0.705707i \(0.750629\pi\)
\(830\) 0 0
\(831\) 0.343146 0.0119036
\(832\) 0 0
\(833\) 9.51472 0.329665
\(834\) 0 0
\(835\) −3.07107 −0.106279
\(836\) 0 0
\(837\) 0.656854 0.0227042
\(838\) 0 0
\(839\) 51.6569 1.78339 0.891696 0.452634i \(-0.149516\pi\)
0.891696 + 0.452634i \(0.149516\pi\)
\(840\) 0 0
\(841\) 23.4558 0.808822
\(842\) 0 0
\(843\) 5.07107 0.174657
\(844\) 0 0
\(845\) −6.31371 −0.217198
\(846\) 0 0
\(847\) 27.9706 0.961080
\(848\) 0 0
\(849\) 26.4558 0.907962
\(850\) 0 0
\(851\) −3.00000 −0.102839
\(852\) 0 0
\(853\) 10.4853 0.359009 0.179505 0.983757i \(-0.442551\pi\)
0.179505 + 0.983757i \(0.442551\pi\)
\(854\) 0 0
\(855\) −7.07107 −0.241825
\(856\) 0 0
\(857\) −31.6569 −1.08138 −0.540689 0.841223i \(-0.681836\pi\)
−0.540689 + 0.841223i \(0.681836\pi\)
\(858\) 0 0
\(859\) 16.4558 0.561466 0.280733 0.959786i \(-0.409422\pi\)
0.280733 + 0.959786i \(0.409422\pi\)
\(860\) 0 0
\(861\) 4.41421 0.150436
\(862\) 0 0
\(863\) 36.1421 1.23029 0.615146 0.788413i \(-0.289097\pi\)
0.615146 + 0.788413i \(0.289097\pi\)
\(864\) 0 0
\(865\) 11.3137 0.384678
\(866\) 0 0
\(867\) 14.4853 0.491946
\(868\) 0 0
\(869\) 35.3137 1.19794
\(870\) 0 0
\(871\) −1.69848 −0.0575510
\(872\) 0 0
\(873\) −6.82843 −0.231107
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) 7.17157 0.242167 0.121083 0.992642i \(-0.461363\pi\)
0.121083 + 0.992642i \(0.461363\pi\)
\(878\) 0 0
\(879\) −15.0416 −0.507342
\(880\) 0 0
\(881\) 22.5858 0.760934 0.380467 0.924794i \(-0.375763\pi\)
0.380467 + 0.924794i \(0.375763\pi\)
\(882\) 0 0
\(883\) 20.3848 0.686002 0.343001 0.939335i \(-0.388557\pi\)
0.343001 + 0.939335i \(0.388557\pi\)
\(884\) 0 0
\(885\) 2.07107 0.0696182
\(886\) 0 0
\(887\) 14.4853 0.486368 0.243184 0.969980i \(-0.421808\pi\)
0.243184 + 0.969980i \(0.421808\pi\)
\(888\) 0 0
\(889\) −8.58579 −0.287958
\(890\) 0 0
\(891\) 6.24264 0.209136
\(892\) 0 0
\(893\) 18.2843 0.611860
\(894\) 0 0
\(895\) 23.6569 0.790761
\(896\) 0 0
\(897\) 2.58579 0.0863369
\(898\) 0 0
\(899\) 4.75736 0.158667
\(900\) 0 0
\(901\) 5.68629 0.189438
\(902\) 0 0
\(903\) 2.00000 0.0665558
\(904\) 0 0
\(905\) 22.1421 0.736029
\(906\) 0 0
\(907\) 31.9706 1.06157 0.530783 0.847508i \(-0.321898\pi\)
0.530783 + 0.847508i \(0.321898\pi\)
\(908\) 0 0
\(909\) 11.2426 0.372895
\(910\) 0 0
\(911\) 28.9706 0.959838 0.479919 0.877313i \(-0.340666\pi\)
0.479919 + 0.877313i \(0.340666\pi\)
\(912\) 0 0
\(913\) −12.0416 −0.398520
\(914\) 0 0
\(915\) 13.0711 0.432116
\(916\) 0 0
\(917\) 5.65685 0.186806
\(918\) 0 0
\(919\) 37.9411 1.25156 0.625781 0.779999i \(-0.284780\pi\)
0.625781 + 0.779999i \(0.284780\pi\)
\(920\) 0 0
\(921\) 14.3848 0.473994
\(922\) 0 0
\(923\) −31.2132 −1.02740
\(924\) 0 0
\(925\) −3.00000 −0.0986394
\(926\) 0 0
\(927\) −6.48528 −0.213005
\(928\) 0 0
\(929\) −57.3848 −1.88273 −0.941367 0.337385i \(-0.890458\pi\)
−0.941367 + 0.337385i \(0.890458\pi\)
\(930\) 0 0
\(931\) 42.4264 1.39047
\(932\) 0 0
\(933\) 12.9706 0.424637
\(934\) 0 0
\(935\) −9.89949 −0.323748
\(936\) 0 0
\(937\) 50.8284 1.66049 0.830246 0.557397i \(-0.188200\pi\)
0.830246 + 0.557397i \(0.188200\pi\)
\(938\) 0 0
\(939\) −19.0000 −0.620042
\(940\) 0 0
\(941\) 4.44365 0.144859 0.0724294 0.997374i \(-0.476925\pi\)
0.0724294 + 0.997374i \(0.476925\pi\)
\(942\) 0 0
\(943\) −4.41421 −0.143747
\(944\) 0 0
\(945\) −1.00000 −0.0325300
\(946\) 0 0
\(947\) −49.4558 −1.60710 −0.803549 0.595238i \(-0.797058\pi\)
−0.803549 + 0.595238i \(0.797058\pi\)
\(948\) 0 0
\(949\) 40.2254 1.30577
\(950\) 0 0
\(951\) 27.5563 0.893576
\(952\) 0 0
\(953\) −34.6863 −1.12360 −0.561800 0.827273i \(-0.689891\pi\)
−0.561800 + 0.827273i \(0.689891\pi\)
\(954\) 0 0
\(955\) −5.89949 −0.190903
\(956\) 0 0
\(957\) 45.2132 1.46154
\(958\) 0 0
\(959\) 16.4853 0.532337
\(960\) 0 0
\(961\) −30.5685 −0.986082
\(962\) 0 0
\(963\) 5.24264 0.168942
\(964\) 0 0
\(965\) 24.4853 0.788209
\(966\) 0 0
\(967\) −30.5269 −0.981679 −0.490840 0.871250i \(-0.663310\pi\)
−0.490840 + 0.871250i \(0.663310\pi\)
\(968\) 0 0
\(969\) −11.2132 −0.360220
\(970\) 0 0
\(971\) −2.54416 −0.0816458 −0.0408229 0.999166i \(-0.512998\pi\)
−0.0408229 + 0.999166i \(0.512998\pi\)
\(972\) 0 0
\(973\) −19.4853 −0.624669
\(974\) 0 0
\(975\) 2.58579 0.0828114
\(976\) 0 0
\(977\) −18.6985 −0.598218 −0.299109 0.954219i \(-0.596689\pi\)
−0.299109 + 0.954219i \(0.596689\pi\)
\(978\) 0 0
\(979\) −32.2843 −1.03181
\(980\) 0 0
\(981\) 18.3848 0.586981
\(982\) 0 0
\(983\) −9.24264 −0.294794 −0.147397 0.989077i \(-0.547090\pi\)
−0.147397 + 0.989077i \(0.547090\pi\)
\(984\) 0 0
\(985\) −25.7990 −0.822024
\(986\) 0 0
\(987\) 2.58579 0.0823065
\(988\) 0 0
\(989\) −2.00000 −0.0635963
\(990\) 0 0
\(991\) 29.8284 0.947531 0.473766 0.880651i \(-0.342894\pi\)
0.473766 + 0.880651i \(0.342894\pi\)
\(992\) 0 0
\(993\) −8.79899 −0.279228
\(994\) 0 0
\(995\) 27.4558 0.870409
\(996\) 0 0
\(997\) 56.0833 1.77617 0.888087 0.459675i \(-0.152034\pi\)
0.888087 + 0.459675i \(0.152034\pi\)
\(998\) 0 0
\(999\) 3.00000 0.0949158
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 5520.2.a.bk.1.2 2
4.3 odd 2 2760.2.a.q.1.1 2
12.11 even 2 8280.2.a.y.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2760.2.a.q.1.1 2 4.3 odd 2
5520.2.a.bk.1.2 2 1.1 even 1 trivial
8280.2.a.y.1.2 2 12.11 even 2