Properties

Label 5520.2.a.bt.1.2
Level $5520$
Weight $2$
Character 5520.1
Self dual yes
Analytic conductor $44.077$
Analytic rank $0$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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

\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.00000 q^{5} +3.82843 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.00000 q^{5} +3.82843 q^{7} +1.00000 q^{9} -1.41421 q^{11} +3.41421 q^{13} +1.00000 q^{15} +0.414214 q^{17} +4.58579 q^{19} +3.82843 q^{21} +1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -6.41421 q^{29} +1.00000 q^{31} -1.41421 q^{33} +3.82843 q^{35} +9.48528 q^{37} +3.41421 q^{39} -1.24264 q^{41} -3.65685 q^{43} +1.00000 q^{45} -2.24264 q^{47} +7.65685 q^{49} +0.414214 q^{51} -3.24264 q^{53} -1.41421 q^{55} +4.58579 q^{57} +11.7279 q^{59} -1.41421 q^{61} +3.82843 q^{63} +3.41421 q^{65} -1.48528 q^{67} +1.00000 q^{69} -0.0710678 q^{71} -6.24264 q^{73} +1.00000 q^{75} -5.41421 q^{77} +16.9706 q^{79} +1.00000 q^{81} -11.2426 q^{83} +0.414214 q^{85} -6.41421 q^{87} -14.8284 q^{89} +13.0711 q^{91} +1.00000 q^{93} +4.58579 q^{95} -10.1421 q^{97} -1.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} + 4 q^{13} + 2 q^{15} - 2 q^{17} + 12 q^{19} + 2 q^{21} + 2 q^{23} + 2 q^{25} + 2 q^{27} - 10 q^{29} + 2 q^{31} + 2 q^{35} + 2 q^{37} + 4 q^{39} + 6 q^{41} + 4 q^{43} + 2 q^{45} + 4 q^{47} + 4 q^{49} - 2 q^{51} + 2 q^{53} + 12 q^{57} - 2 q^{59} + 2 q^{63} + 4 q^{65} + 14 q^{67} + 2 q^{69} + 14 q^{71} - 4 q^{73} + 2 q^{75} - 8 q^{77} + 2 q^{81} - 14 q^{83} - 2 q^{85} - 10 q^{87} - 24 q^{89} + 12 q^{91} + 2 q^{93} + 12 q^{95} + 8 q^{97}+O(q^{100}) \) 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
\(7\) 3.82843 1.44701 0.723505 0.690319i \(-0.242530\pi\)
0.723505 + 0.690319i \(0.242530\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.41421 −0.426401 −0.213201 0.977008i \(-0.568389\pi\)
−0.213201 + 0.977008i \(0.568389\pi\)
\(12\) 0 0
\(13\) 3.41421 0.946932 0.473466 0.880812i \(-0.343003\pi\)
0.473466 + 0.880812i \(0.343003\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 0.414214 0.100462 0.0502308 0.998738i \(-0.484004\pi\)
0.0502308 + 0.998738i \(0.484004\pi\)
\(18\) 0 0
\(19\) 4.58579 1.05205 0.526026 0.850469i \(-0.323682\pi\)
0.526026 + 0.850469i \(0.323682\pi\)
\(20\) 0 0
\(21\) 3.82843 0.835431
\(22\) 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
\(29\) −6.41421 −1.19109 −0.595545 0.803322i \(-0.703064\pi\)
−0.595545 + 0.803322i \(0.703064\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605 0.0898027 0.995960i \(-0.471376\pi\)
0.0898027 + 0.995960i \(0.471376\pi\)
\(32\) 0 0
\(33\) −1.41421 −0.246183
\(34\) 0 0
\(35\) 3.82843 0.647122
\(36\) 0 0
\(37\) 9.48528 1.55937 0.779685 0.626172i \(-0.215379\pi\)
0.779685 + 0.626172i \(0.215379\pi\)
\(38\) 0 0
\(39\) 3.41421 0.546712
\(40\) 0 0
\(41\) −1.24264 −0.194068 −0.0970339 0.995281i \(-0.530936\pi\)
−0.0970339 + 0.995281i \(0.530936\pi\)
\(42\) 0 0
\(43\) −3.65685 −0.557665 −0.278833 0.960340i \(-0.589947\pi\)
−0.278833 + 0.960340i \(0.589947\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) −2.24264 −0.327123 −0.163561 0.986533i \(-0.552298\pi\)
−0.163561 + 0.986533i \(0.552298\pi\)
\(48\) 0 0
\(49\) 7.65685 1.09384
\(50\) 0 0
\(51\) 0.414214 0.0580015
\(52\) 0 0
\(53\) −3.24264 −0.445411 −0.222705 0.974886i \(-0.571489\pi\)
−0.222705 + 0.974886i \(0.571489\pi\)
\(54\) 0 0
\(55\) −1.41421 −0.190693
\(56\) 0 0
\(57\) 4.58579 0.607402
\(58\) 0 0
\(59\) 11.7279 1.52685 0.763423 0.645899i \(-0.223517\pi\)
0.763423 + 0.645899i \(0.223517\pi\)
\(60\) 0 0
\(61\) −1.41421 −0.181071 −0.0905357 0.995893i \(-0.528858\pi\)
−0.0905357 + 0.995893i \(0.528858\pi\)
\(62\) 0 0
\(63\) 3.82843 0.482336
\(64\) 0 0
\(65\) 3.41421 0.423481
\(66\) 0 0
\(67\) −1.48528 −0.181456 −0.0907280 0.995876i \(-0.528919\pi\)
−0.0907280 + 0.995876i \(0.528919\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −0.0710678 −0.00843420 −0.00421710 0.999991i \(-0.501342\pi\)
−0.00421710 + 0.999991i \(0.501342\pi\)
\(72\) 0 0
\(73\) −6.24264 −0.730646 −0.365323 0.930881i \(-0.619041\pi\)
−0.365323 + 0.930881i \(0.619041\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) −5.41421 −0.617007
\(78\) 0 0
\(79\) 16.9706 1.90934 0.954669 0.297670i \(-0.0962096\pi\)
0.954669 + 0.297670i \(0.0962096\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −11.2426 −1.23404 −0.617020 0.786947i \(-0.711660\pi\)
−0.617020 + 0.786947i \(0.711660\pi\)
\(84\) 0 0
\(85\) 0.414214 0.0449278
\(86\) 0 0
\(87\) −6.41421 −0.687676
\(88\) 0 0
\(89\) −14.8284 −1.57181 −0.785905 0.618347i \(-0.787803\pi\)
−0.785905 + 0.618347i \(0.787803\pi\)
\(90\) 0 0
\(91\) 13.0711 1.37022
\(92\) 0 0
\(93\) 1.00000 0.103695
\(94\) 0 0
\(95\) 4.58579 0.470492
\(96\) 0 0
\(97\) −10.1421 −1.02978 −0.514889 0.857257i \(-0.672167\pi\)
−0.514889 + 0.857257i \(0.672167\pi\)
\(98\) 0 0
\(99\) −1.41421 −0.142134
\(100\) 0 0
\(101\) 4.07107 0.405086 0.202543 0.979273i \(-0.435079\pi\)
0.202543 + 0.979273i \(0.435079\pi\)
\(102\) 0 0
\(103\) 18.4853 1.82141 0.910704 0.413059i \(-0.135540\pi\)
0.910704 + 0.413059i \(0.135540\pi\)
\(104\) 0 0
\(105\) 3.82843 0.373616
\(106\) 0 0
\(107\) −6.41421 −0.620085 −0.310043 0.950723i \(-0.600343\pi\)
−0.310043 + 0.950723i \(0.600343\pi\)
\(108\) 0 0
\(109\) −4.58579 −0.439239 −0.219619 0.975586i \(-0.570482\pi\)
−0.219619 + 0.975586i \(0.570482\pi\)
\(110\) 0 0
\(111\) 9.48528 0.900303
\(112\) 0 0
\(113\) −0.899495 −0.0846174 −0.0423087 0.999105i \(-0.513471\pi\)
−0.0423087 + 0.999105i \(0.513471\pi\)
\(114\) 0 0
\(115\) 1.00000 0.0932505
\(116\) 0 0
\(117\) 3.41421 0.315644
\(118\) 0 0
\(119\) 1.58579 0.145369
\(120\) 0 0
\(121\) −9.00000 −0.818182
\(122\) 0 0
\(123\) −1.24264 −0.112045
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −11.5563 −1.02546 −0.512730 0.858550i \(-0.671366\pi\)
−0.512730 + 0.858550i \(0.671366\pi\)
\(128\) 0 0
\(129\) −3.65685 −0.321968
\(130\) 0 0
\(131\) 16.0000 1.39793 0.698963 0.715158i \(-0.253645\pi\)
0.698963 + 0.715158i \(0.253645\pi\)
\(132\) 0 0
\(133\) 17.5563 1.52233
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) −2.82843 −0.241649 −0.120824 0.992674i \(-0.538554\pi\)
−0.120824 + 0.992674i \(0.538554\pi\)
\(138\) 0 0
\(139\) 0.514719 0.0436579 0.0218289 0.999762i \(-0.493051\pi\)
0.0218289 + 0.999762i \(0.493051\pi\)
\(140\) 0 0
\(141\) −2.24264 −0.188864
\(142\) 0 0
\(143\) −4.82843 −0.403773
\(144\) 0 0
\(145\) −6.41421 −0.532671
\(146\) 0 0
\(147\) 7.65685 0.631527
\(148\) 0 0
\(149\) 6.24264 0.511417 0.255709 0.966754i \(-0.417691\pi\)
0.255709 + 0.966754i \(0.417691\pi\)
\(150\) 0 0
\(151\) −1.31371 −0.106908 −0.0534540 0.998570i \(-0.517023\pi\)
−0.0534540 + 0.998570i \(0.517023\pi\)
\(152\) 0 0
\(153\) 0.414214 0.0334872
\(154\) 0 0
\(155\) 1.00000 0.0803219
\(156\) 0 0
\(157\) −3.00000 −0.239426 −0.119713 0.992809i \(-0.538197\pi\)
−0.119713 + 0.992809i \(0.538197\pi\)
\(158\) 0 0
\(159\) −3.24264 −0.257158
\(160\) 0 0
\(161\) 3.82843 0.301722
\(162\) 0 0
\(163\) 14.9706 1.17258 0.586292 0.810099i \(-0.300587\pi\)
0.586292 + 0.810099i \(0.300587\pi\)
\(164\) 0 0
\(165\) −1.41421 −0.110096
\(166\) 0 0
\(167\) 10.2426 0.792599 0.396300 0.918121i \(-0.370294\pi\)
0.396300 + 0.918121i \(0.370294\pi\)
\(168\) 0 0
\(169\) −1.34315 −0.103319
\(170\) 0 0
\(171\) 4.58579 0.350684
\(172\) 0 0
\(173\) 10.3431 0.786375 0.393187 0.919458i \(-0.371372\pi\)
0.393187 + 0.919458i \(0.371372\pi\)
\(174\) 0 0
\(175\) 3.82843 0.289402
\(176\) 0 0
\(177\) 11.7279 0.881525
\(178\) 0 0
\(179\) 8.34315 0.623596 0.311798 0.950148i \(-0.399069\pi\)
0.311798 + 0.950148i \(0.399069\pi\)
\(180\) 0 0
\(181\) −20.4853 −1.52266 −0.761329 0.648365i \(-0.775453\pi\)
−0.761329 + 0.648365i \(0.775453\pi\)
\(182\) 0 0
\(183\) −1.41421 −0.104542
\(184\) 0 0
\(185\) 9.48528 0.697372
\(186\) 0 0
\(187\) −0.585786 −0.0428369
\(188\) 0 0
\(189\) 3.82843 0.278477
\(190\) 0 0
\(191\) −14.2426 −1.03056 −0.515281 0.857021i \(-0.672312\pi\)
−0.515281 + 0.857021i \(0.672312\pi\)
\(192\) 0 0
\(193\) −10.1421 −0.730047 −0.365023 0.930998i \(-0.618939\pi\)
−0.365023 + 0.930998i \(0.618939\pi\)
\(194\) 0 0
\(195\) 3.41421 0.244497
\(196\) 0 0
\(197\) −12.8284 −0.913988 −0.456994 0.889470i \(-0.651074\pi\)
−0.456994 + 0.889470i \(0.651074\pi\)
\(198\) 0 0
\(199\) −2.48528 −0.176177 −0.0880885 0.996113i \(-0.528076\pi\)
−0.0880885 + 0.996113i \(0.528076\pi\)
\(200\) 0 0
\(201\) −1.48528 −0.104764
\(202\) 0 0
\(203\) −24.5563 −1.72352
\(204\) 0 0
\(205\) −1.24264 −0.0867898
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) −6.48528 −0.448596
\(210\) 0 0
\(211\) 18.7990 1.29418 0.647088 0.762415i \(-0.275987\pi\)
0.647088 + 0.762415i \(0.275987\pi\)
\(212\) 0 0
\(213\) −0.0710678 −0.00486949
\(214\) 0 0
\(215\) −3.65685 −0.249395
\(216\) 0 0
\(217\) 3.82843 0.259891
\(218\) 0 0
\(219\) −6.24264 −0.421839
\(220\) 0 0
\(221\) 1.41421 0.0951303
\(222\) 0 0
\(223\) −14.9706 −1.00250 −0.501252 0.865302i \(-0.667127\pi\)
−0.501252 + 0.865302i \(0.667127\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 23.3137 1.54739 0.773693 0.633561i \(-0.218408\pi\)
0.773693 + 0.633561i \(0.218408\pi\)
\(228\) 0 0
\(229\) 0.485281 0.0320683 0.0160341 0.999871i \(-0.494896\pi\)
0.0160341 + 0.999871i \(0.494896\pi\)
\(230\) 0 0
\(231\) −5.41421 −0.356229
\(232\) 0 0
\(233\) 5.65685 0.370593 0.185296 0.982683i \(-0.440675\pi\)
0.185296 + 0.982683i \(0.440675\pi\)
\(234\) 0 0
\(235\) −2.24264 −0.146294
\(236\) 0 0
\(237\) 16.9706 1.10236
\(238\) 0 0
\(239\) −1.58579 −0.102576 −0.0512880 0.998684i \(-0.516333\pi\)
−0.0512880 + 0.998684i \(0.516333\pi\)
\(240\) 0 0
\(241\) 27.2132 1.75296 0.876478 0.481441i \(-0.159887\pi\)
0.876478 + 0.481441i \(0.159887\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 7.65685 0.489178
\(246\) 0 0
\(247\) 15.6569 0.996222
\(248\) 0 0
\(249\) −11.2426 −0.712473
\(250\) 0 0
\(251\) −18.9706 −1.19741 −0.598706 0.800969i \(-0.704318\pi\)
−0.598706 + 0.800969i \(0.704318\pi\)
\(252\) 0 0
\(253\) −1.41421 −0.0889108
\(254\) 0 0
\(255\) 0.414214 0.0259391
\(256\) 0 0
\(257\) −8.58579 −0.535567 −0.267783 0.963479i \(-0.586291\pi\)
−0.267783 + 0.963479i \(0.586291\pi\)
\(258\) 0 0
\(259\) 36.3137 2.25642
\(260\) 0 0
\(261\) −6.41421 −0.397030
\(262\) 0 0
\(263\) −9.58579 −0.591085 −0.295542 0.955330i \(-0.595500\pi\)
−0.295542 + 0.955330i \(0.595500\pi\)
\(264\) 0 0
\(265\) −3.24264 −0.199194
\(266\) 0 0
\(267\) −14.8284 −0.907485
\(268\) 0 0
\(269\) −2.07107 −0.126275 −0.0631376 0.998005i \(-0.520111\pi\)
−0.0631376 + 0.998005i \(0.520111\pi\)
\(270\) 0 0
\(271\) 4.31371 0.262039 0.131020 0.991380i \(-0.458175\pi\)
0.131020 + 0.991380i \(0.458175\pi\)
\(272\) 0 0
\(273\) 13.0711 0.791097
\(274\) 0 0
\(275\) −1.41421 −0.0852803
\(276\) 0 0
\(277\) −14.9706 −0.899494 −0.449747 0.893156i \(-0.648486\pi\)
−0.449747 + 0.893156i \(0.648486\pi\)
\(278\) 0 0
\(279\) 1.00000 0.0598684
\(280\) 0 0
\(281\) 12.2426 0.730335 0.365167 0.930942i \(-0.381012\pi\)
0.365167 + 0.930942i \(0.381012\pi\)
\(282\) 0 0
\(283\) −19.6274 −1.16673 −0.583364 0.812211i \(-0.698264\pi\)
−0.583364 + 0.812211i \(0.698264\pi\)
\(284\) 0 0
\(285\) 4.58579 0.271639
\(286\) 0 0
\(287\) −4.75736 −0.280818
\(288\) 0 0
\(289\) −16.8284 −0.989907
\(290\) 0 0
\(291\) −10.1421 −0.594543
\(292\) 0 0
\(293\) −8.89949 −0.519914 −0.259957 0.965620i \(-0.583708\pi\)
−0.259957 + 0.965620i \(0.583708\pi\)
\(294\) 0 0
\(295\) 11.7279 0.682826
\(296\) 0 0
\(297\) −1.41421 −0.0820610
\(298\) 0 0
\(299\) 3.41421 0.197449
\(300\) 0 0
\(301\) −14.0000 −0.806947
\(302\) 0 0
\(303\) 4.07107 0.233877
\(304\) 0 0
\(305\) −1.41421 −0.0809776
\(306\) 0 0
\(307\) 15.8995 0.907432 0.453716 0.891146i \(-0.350098\pi\)
0.453716 + 0.891146i \(0.350098\pi\)
\(308\) 0 0
\(309\) 18.4853 1.05159
\(310\) 0 0
\(311\) −8.68629 −0.492554 −0.246277 0.969199i \(-0.579207\pi\)
−0.246277 + 0.969199i \(0.579207\pi\)
\(312\) 0 0
\(313\) −7.14214 −0.403697 −0.201849 0.979417i \(-0.564695\pi\)
−0.201849 + 0.979417i \(0.564695\pi\)
\(314\) 0 0
\(315\) 3.82843 0.215707
\(316\) 0 0
\(317\) −4.58579 −0.257563 −0.128782 0.991673i \(-0.541107\pi\)
−0.128782 + 0.991673i \(0.541107\pi\)
\(318\) 0 0
\(319\) 9.07107 0.507882
\(320\) 0 0
\(321\) −6.41421 −0.358006
\(322\) 0 0
\(323\) 1.89949 0.105691
\(324\) 0 0
\(325\) 3.41421 0.189386
\(326\) 0 0
\(327\) −4.58579 −0.253595
\(328\) 0 0
\(329\) −8.58579 −0.473350
\(330\) 0 0
\(331\) 18.1716 0.998800 0.499400 0.866372i \(-0.333554\pi\)
0.499400 + 0.866372i \(0.333554\pi\)
\(332\) 0 0
\(333\) 9.48528 0.519790
\(334\) 0 0
\(335\) −1.48528 −0.0811496
\(336\) 0 0
\(337\) −19.6569 −1.07078 −0.535389 0.844606i \(-0.679835\pi\)
−0.535389 + 0.844606i \(0.679835\pi\)
\(338\) 0 0
\(339\) −0.899495 −0.0488539
\(340\) 0 0
\(341\) −1.41421 −0.0765840
\(342\) 0 0
\(343\) 2.51472 0.135782
\(344\) 0 0
\(345\) 1.00000 0.0538382
\(346\) 0 0
\(347\) 12.4853 0.670245 0.335123 0.942175i \(-0.391222\pi\)
0.335123 + 0.942175i \(0.391222\pi\)
\(348\) 0 0
\(349\) −10.1716 −0.544472 −0.272236 0.962231i \(-0.587763\pi\)
−0.272236 + 0.962231i \(0.587763\pi\)
\(350\) 0 0
\(351\) 3.41421 0.182237
\(352\) 0 0
\(353\) 26.0416 1.38606 0.693028 0.720911i \(-0.256276\pi\)
0.693028 + 0.720911i \(0.256276\pi\)
\(354\) 0 0
\(355\) −0.0710678 −0.00377189
\(356\) 0 0
\(357\) 1.58579 0.0839287
\(358\) 0 0
\(359\) −9.07107 −0.478753 −0.239376 0.970927i \(-0.576943\pi\)
−0.239376 + 0.970927i \(0.576943\pi\)
\(360\) 0 0
\(361\) 2.02944 0.106812
\(362\) 0 0
\(363\) −9.00000 −0.472377
\(364\) 0 0
\(365\) −6.24264 −0.326755
\(366\) 0 0
\(367\) −3.00000 −0.156599 −0.0782994 0.996930i \(-0.524949\pi\)
−0.0782994 + 0.996930i \(0.524949\pi\)
\(368\) 0 0
\(369\) −1.24264 −0.0646893
\(370\) 0 0
\(371\) −12.4142 −0.644514
\(372\) 0 0
\(373\) 6.34315 0.328436 0.164218 0.986424i \(-0.447490\pi\)
0.164218 + 0.986424i \(0.447490\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −21.8995 −1.12788
\(378\) 0 0
\(379\) 25.6569 1.31790 0.658952 0.752185i \(-0.271000\pi\)
0.658952 + 0.752185i \(0.271000\pi\)
\(380\) 0 0
\(381\) −11.5563 −0.592050
\(382\) 0 0
\(383\) 11.7279 0.599269 0.299634 0.954054i \(-0.403135\pi\)
0.299634 + 0.954054i \(0.403135\pi\)
\(384\) 0 0
\(385\) −5.41421 −0.275934
\(386\) 0 0
\(387\) −3.65685 −0.185888
\(388\) 0 0
\(389\) 31.4558 1.59487 0.797437 0.603402i \(-0.206188\pi\)
0.797437 + 0.603402i \(0.206188\pi\)
\(390\) 0 0
\(391\) 0.414214 0.0209477
\(392\) 0 0
\(393\) 16.0000 0.807093
\(394\) 0 0
\(395\) 16.9706 0.853882
\(396\) 0 0
\(397\) −10.3431 −0.519108 −0.259554 0.965729i \(-0.583575\pi\)
−0.259554 + 0.965729i \(0.583575\pi\)
\(398\) 0 0
\(399\) 17.5563 0.878917
\(400\) 0 0
\(401\) 21.5147 1.07439 0.537197 0.843457i \(-0.319483\pi\)
0.537197 + 0.843457i \(0.319483\pi\)
\(402\) 0 0
\(403\) 3.41421 0.170074
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) −13.4142 −0.664918
\(408\) 0 0
\(409\) 14.1716 0.700739 0.350370 0.936612i \(-0.386056\pi\)
0.350370 + 0.936612i \(0.386056\pi\)
\(410\) 0 0
\(411\) −2.82843 −0.139516
\(412\) 0 0
\(413\) 44.8995 2.20936
\(414\) 0 0
\(415\) −11.2426 −0.551880
\(416\) 0 0
\(417\) 0.514719 0.0252059
\(418\) 0 0
\(419\) 35.3553 1.72722 0.863611 0.504159i \(-0.168198\pi\)
0.863611 + 0.504159i \(0.168198\pi\)
\(420\) 0 0
\(421\) −20.8701 −1.01714 −0.508572 0.861019i \(-0.669827\pi\)
−0.508572 + 0.861019i \(0.669827\pi\)
\(422\) 0 0
\(423\) −2.24264 −0.109041
\(424\) 0 0
\(425\) 0.414214 0.0200923
\(426\) 0 0
\(427\) −5.41421 −0.262012
\(428\) 0 0
\(429\) −4.82843 −0.233119
\(430\) 0 0
\(431\) −11.5147 −0.554644 −0.277322 0.960777i \(-0.589447\pi\)
−0.277322 + 0.960777i \(0.589447\pi\)
\(432\) 0 0
\(433\) 26.3137 1.26456 0.632278 0.774742i \(-0.282120\pi\)
0.632278 + 0.774742i \(0.282120\pi\)
\(434\) 0 0
\(435\) −6.41421 −0.307538
\(436\) 0 0
\(437\) 4.58579 0.219368
\(438\) 0 0
\(439\) −5.79899 −0.276771 −0.138385 0.990378i \(-0.544191\pi\)
−0.138385 + 0.990378i \(0.544191\pi\)
\(440\) 0 0
\(441\) 7.65685 0.364612
\(442\) 0 0
\(443\) 3.89949 0.185271 0.0926353 0.995700i \(-0.470471\pi\)
0.0926353 + 0.995700i \(0.470471\pi\)
\(444\) 0 0
\(445\) −14.8284 −0.702935
\(446\) 0 0
\(447\) 6.24264 0.295267
\(448\) 0 0
\(449\) 20.8995 0.986308 0.493154 0.869942i \(-0.335844\pi\)
0.493154 + 0.869942i \(0.335844\pi\)
\(450\) 0 0
\(451\) 1.75736 0.0827508
\(452\) 0 0
\(453\) −1.31371 −0.0617234
\(454\) 0 0
\(455\) 13.0711 0.612781
\(456\) 0 0
\(457\) 40.4558 1.89244 0.946222 0.323517i \(-0.104865\pi\)
0.946222 + 0.323517i \(0.104865\pi\)
\(458\) 0 0
\(459\) 0.414214 0.0193338
\(460\) 0 0
\(461\) 23.4558 1.09245 0.546224 0.837639i \(-0.316065\pi\)
0.546224 + 0.837639i \(0.316065\pi\)
\(462\) 0 0
\(463\) 26.0416 1.21026 0.605129 0.796128i \(-0.293122\pi\)
0.605129 + 0.796128i \(0.293122\pi\)
\(464\) 0 0
\(465\) 1.00000 0.0463739
\(466\) 0 0
\(467\) −14.4142 −0.667010 −0.333505 0.942748i \(-0.608231\pi\)
−0.333505 + 0.942748i \(0.608231\pi\)
\(468\) 0 0
\(469\) −5.68629 −0.262569
\(470\) 0 0
\(471\) −3.00000 −0.138233
\(472\) 0 0
\(473\) 5.17157 0.237789
\(474\) 0 0
\(475\) 4.58579 0.210410
\(476\) 0 0
\(477\) −3.24264 −0.148470
\(478\) 0 0
\(479\) −39.8995 −1.82305 −0.911527 0.411240i \(-0.865096\pi\)
−0.911527 + 0.411240i \(0.865096\pi\)
\(480\) 0 0
\(481\) 32.3848 1.47662
\(482\) 0 0
\(483\) 3.82843 0.174199
\(484\) 0 0
\(485\) −10.1421 −0.460531
\(486\) 0 0
\(487\) −16.9289 −0.767123 −0.383562 0.923515i \(-0.625303\pi\)
−0.383562 + 0.923515i \(0.625303\pi\)
\(488\) 0 0
\(489\) 14.9706 0.676992
\(490\) 0 0
\(491\) −42.8995 −1.93603 −0.968014 0.250898i \(-0.919274\pi\)
−0.968014 + 0.250898i \(0.919274\pi\)
\(492\) 0 0
\(493\) −2.65685 −0.119659
\(494\) 0 0
\(495\) −1.41421 −0.0635642
\(496\) 0 0
\(497\) −0.272078 −0.0122044
\(498\) 0 0
\(499\) −26.4558 −1.18433 −0.592163 0.805818i \(-0.701726\pi\)
−0.592163 + 0.805818i \(0.701726\pi\)
\(500\) 0 0
\(501\) 10.2426 0.457607
\(502\) 0 0
\(503\) −29.2426 −1.30386 −0.651932 0.758277i \(-0.726041\pi\)
−0.651932 + 0.758277i \(0.726041\pi\)
\(504\) 0 0
\(505\) 4.07107 0.181160
\(506\) 0 0
\(507\) −1.34315 −0.0596512
\(508\) 0 0
\(509\) −3.51472 −0.155787 −0.0778936 0.996962i \(-0.524819\pi\)
−0.0778936 + 0.996962i \(0.524819\pi\)
\(510\) 0 0
\(511\) −23.8995 −1.05725
\(512\) 0 0
\(513\) 4.58579 0.202467
\(514\) 0 0
\(515\) 18.4853 0.814559
\(516\) 0 0
\(517\) 3.17157 0.139486
\(518\) 0 0
\(519\) 10.3431 0.454014
\(520\) 0 0
\(521\) 0.727922 0.0318908 0.0159454 0.999873i \(-0.494924\pi\)
0.0159454 + 0.999873i \(0.494924\pi\)
\(522\) 0 0
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 0 0
\(525\) 3.82843 0.167086
\(526\) 0 0
\(527\) 0.414214 0.0180434
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 11.7279 0.508948
\(532\) 0 0
\(533\) −4.24264 −0.183769
\(534\) 0 0
\(535\) −6.41421 −0.277311
\(536\) 0 0
\(537\) 8.34315 0.360033
\(538\) 0 0
\(539\) −10.8284 −0.466413
\(540\) 0 0
\(541\) 19.1127 0.821719 0.410860 0.911699i \(-0.365229\pi\)
0.410860 + 0.911699i \(0.365229\pi\)
\(542\) 0 0
\(543\) −20.4853 −0.879108
\(544\) 0 0
\(545\) −4.58579 −0.196434
\(546\) 0 0
\(547\) −17.6569 −0.754953 −0.377476 0.926019i \(-0.623208\pi\)
−0.377476 + 0.926019i \(0.623208\pi\)
\(548\) 0 0
\(549\) −1.41421 −0.0603572
\(550\) 0 0
\(551\) −29.4142 −1.25309
\(552\) 0 0
\(553\) 64.9706 2.76283
\(554\) 0 0
\(555\) 9.48528 0.402628
\(556\) 0 0
\(557\) −31.5269 −1.33584 −0.667919 0.744234i \(-0.732815\pi\)
−0.667919 + 0.744234i \(0.732815\pi\)
\(558\) 0 0
\(559\) −12.4853 −0.528071
\(560\) 0 0
\(561\) −0.585786 −0.0247319
\(562\) 0 0
\(563\) 46.4142 1.95613 0.978063 0.208310i \(-0.0667962\pi\)
0.978063 + 0.208310i \(0.0667962\pi\)
\(564\) 0 0
\(565\) −0.899495 −0.0378420
\(566\) 0 0
\(567\) 3.82843 0.160779
\(568\) 0 0
\(569\) 1.51472 0.0635003 0.0317502 0.999496i \(-0.489892\pi\)
0.0317502 + 0.999496i \(0.489892\pi\)
\(570\) 0 0
\(571\) −17.8995 −0.749071 −0.374535 0.927213i \(-0.622198\pi\)
−0.374535 + 0.927213i \(0.622198\pi\)
\(572\) 0 0
\(573\) −14.2426 −0.594995
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) −32.2843 −1.34401 −0.672006 0.740546i \(-0.734567\pi\)
−0.672006 + 0.740546i \(0.734567\pi\)
\(578\) 0 0
\(579\) −10.1421 −0.421493
\(580\) 0 0
\(581\) −43.0416 −1.78567
\(582\) 0 0
\(583\) 4.58579 0.189924
\(584\) 0 0
\(585\) 3.41421 0.141160
\(586\) 0 0
\(587\) 8.62742 0.356092 0.178046 0.984022i \(-0.443022\pi\)
0.178046 + 0.984022i \(0.443022\pi\)
\(588\) 0 0
\(589\) 4.58579 0.188954
\(590\) 0 0
\(591\) −12.8284 −0.527691
\(592\) 0 0
\(593\) −21.0711 −0.865285 −0.432643 0.901566i \(-0.642419\pi\)
−0.432643 + 0.901566i \(0.642419\pi\)
\(594\) 0 0
\(595\) 1.58579 0.0650109
\(596\) 0 0
\(597\) −2.48528 −0.101716
\(598\) 0 0
\(599\) −10.1421 −0.414396 −0.207198 0.978299i \(-0.566435\pi\)
−0.207198 + 0.978299i \(0.566435\pi\)
\(600\) 0 0
\(601\) 11.3431 0.462697 0.231348 0.972871i \(-0.425686\pi\)
0.231348 + 0.972871i \(0.425686\pi\)
\(602\) 0 0
\(603\) −1.48528 −0.0604853
\(604\) 0 0
\(605\) −9.00000 −0.365902
\(606\) 0 0
\(607\) −12.7279 −0.516610 −0.258305 0.966063i \(-0.583164\pi\)
−0.258305 + 0.966063i \(0.583164\pi\)
\(608\) 0 0
\(609\) −24.5563 −0.995073
\(610\) 0 0
\(611\) −7.65685 −0.309763
\(612\) 0 0
\(613\) 40.6274 1.64093 0.820463 0.571700i \(-0.193716\pi\)
0.820463 + 0.571700i \(0.193716\pi\)
\(614\) 0 0
\(615\) −1.24264 −0.0501081
\(616\) 0 0
\(617\) 8.75736 0.352558 0.176279 0.984340i \(-0.443594\pi\)
0.176279 + 0.984340i \(0.443594\pi\)
\(618\) 0 0
\(619\) 40.2843 1.61916 0.809581 0.587008i \(-0.199694\pi\)
0.809581 + 0.587008i \(0.199694\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) −56.7696 −2.27442
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −6.48528 −0.258997
\(628\) 0 0
\(629\) 3.92893 0.156657
\(630\) 0 0
\(631\) 1.41421 0.0562990 0.0281495 0.999604i \(-0.491039\pi\)
0.0281495 + 0.999604i \(0.491039\pi\)
\(632\) 0 0
\(633\) 18.7990 0.747193
\(634\) 0 0
\(635\) −11.5563 −0.458600
\(636\) 0 0
\(637\) 26.1421 1.03579
\(638\) 0 0
\(639\) −0.0710678 −0.00281140
\(640\) 0 0
\(641\) −18.5858 −0.734094 −0.367047 0.930202i \(-0.619631\pi\)
−0.367047 + 0.930202i \(0.619631\pi\)
\(642\) 0 0
\(643\) −42.6569 −1.68222 −0.841111 0.540862i \(-0.818098\pi\)
−0.841111 + 0.540862i \(0.818098\pi\)
\(644\) 0 0
\(645\) −3.65685 −0.143988
\(646\) 0 0
\(647\) −34.3848 −1.35181 −0.675903 0.736991i \(-0.736246\pi\)
−0.675903 + 0.736991i \(0.736246\pi\)
\(648\) 0 0
\(649\) −16.5858 −0.651049
\(650\) 0 0
\(651\) 3.82843 0.150048
\(652\) 0 0
\(653\) 8.72792 0.341550 0.170775 0.985310i \(-0.445373\pi\)
0.170775 + 0.985310i \(0.445373\pi\)
\(654\) 0 0
\(655\) 16.0000 0.625172
\(656\) 0 0
\(657\) −6.24264 −0.243549
\(658\) 0 0
\(659\) −30.3848 −1.18362 −0.591811 0.806076i \(-0.701587\pi\)
−0.591811 + 0.806076i \(0.701587\pi\)
\(660\) 0 0
\(661\) −49.1127 −1.91026 −0.955131 0.296183i \(-0.904286\pi\)
−0.955131 + 0.296183i \(0.904286\pi\)
\(662\) 0 0
\(663\) 1.41421 0.0549235
\(664\) 0 0
\(665\) 17.5563 0.680806
\(666\) 0 0
\(667\) −6.41421 −0.248359
\(668\) 0 0
\(669\) −14.9706 −0.578795
\(670\) 0 0
\(671\) 2.00000 0.0772091
\(672\) 0 0
\(673\) 32.7279 1.26157 0.630784 0.775958i \(-0.282733\pi\)
0.630784 + 0.775958i \(0.282733\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 33.1838 1.27536 0.637678 0.770303i \(-0.279895\pi\)
0.637678 + 0.770303i \(0.279895\pi\)
\(678\) 0 0
\(679\) −38.8284 −1.49010
\(680\) 0 0
\(681\) 23.3137 0.893383
\(682\) 0 0
\(683\) 39.0122 1.49276 0.746380 0.665520i \(-0.231790\pi\)
0.746380 + 0.665520i \(0.231790\pi\)
\(684\) 0 0
\(685\) −2.82843 −0.108069
\(686\) 0 0
\(687\) 0.485281 0.0185146
\(688\) 0 0
\(689\) −11.0711 −0.421774
\(690\) 0 0
\(691\) −18.9706 −0.721674 −0.360837 0.932629i \(-0.617509\pi\)
−0.360837 + 0.932629i \(0.617509\pi\)
\(692\) 0 0
\(693\) −5.41421 −0.205669
\(694\) 0 0
\(695\) 0.514719 0.0195244
\(696\) 0 0
\(697\) −0.514719 −0.0194964
\(698\) 0 0
\(699\) 5.65685 0.213962
\(700\) 0 0
\(701\) 18.4437 0.696607 0.348304 0.937382i \(-0.386758\pi\)
0.348304 + 0.937382i \(0.386758\pi\)
\(702\) 0 0
\(703\) 43.4975 1.64054
\(704\) 0 0
\(705\) −2.24264 −0.0844627
\(706\) 0 0
\(707\) 15.5858 0.586164
\(708\) 0 0
\(709\) −41.6985 −1.56602 −0.783010 0.622009i \(-0.786317\pi\)
−0.783010 + 0.622009i \(0.786317\pi\)
\(710\) 0 0
\(711\) 16.9706 0.636446
\(712\) 0 0
\(713\) 1.00000 0.0374503
\(714\) 0 0
\(715\) −4.82843 −0.180573
\(716\) 0 0
\(717\) −1.58579 −0.0592223
\(718\) 0 0
\(719\) −1.44365 −0.0538391 −0.0269195 0.999638i \(-0.508570\pi\)
−0.0269195 + 0.999638i \(0.508570\pi\)
\(720\) 0 0
\(721\) 70.7696 2.63560
\(722\) 0 0
\(723\) 27.2132 1.01207
\(724\) 0 0
\(725\) −6.41421 −0.238218
\(726\) 0 0
\(727\) −9.00000 −0.333792 −0.166896 0.985975i \(-0.553374\pi\)
−0.166896 + 0.985975i \(0.553374\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −1.51472 −0.0560239
\(732\) 0 0
\(733\) −6.31371 −0.233202 −0.116601 0.993179i \(-0.537200\pi\)
−0.116601 + 0.993179i \(0.537200\pi\)
\(734\) 0 0
\(735\) 7.65685 0.282427
\(736\) 0 0
\(737\) 2.10051 0.0773731
\(738\) 0 0
\(739\) 11.1421 0.409870 0.204935 0.978776i \(-0.434302\pi\)
0.204935 + 0.978776i \(0.434302\pi\)
\(740\) 0 0
\(741\) 15.6569 0.575169
\(742\) 0 0
\(743\) −4.00000 −0.146746 −0.0733729 0.997305i \(-0.523376\pi\)
−0.0733729 + 0.997305i \(0.523376\pi\)
\(744\) 0 0
\(745\) 6.24264 0.228713
\(746\) 0 0
\(747\) −11.2426 −0.411347
\(748\) 0 0
\(749\) −24.5563 −0.897269
\(750\) 0 0
\(751\) 30.9289 1.12861 0.564306 0.825565i \(-0.309144\pi\)
0.564306 + 0.825565i \(0.309144\pi\)
\(752\) 0 0
\(753\) −18.9706 −0.691326
\(754\) 0 0
\(755\) −1.31371 −0.0478107
\(756\) 0 0
\(757\) 3.82843 0.139147 0.0695733 0.997577i \(-0.477836\pi\)
0.0695733 + 0.997577i \(0.477836\pi\)
\(758\) 0 0
\(759\) −1.41421 −0.0513327
\(760\) 0 0
\(761\) −38.0711 −1.38008 −0.690038 0.723774i \(-0.742406\pi\)
−0.690038 + 0.723774i \(0.742406\pi\)
\(762\) 0 0
\(763\) −17.5563 −0.635583
\(764\) 0 0
\(765\) 0.414214 0.0149759
\(766\) 0 0
\(767\) 40.0416 1.44582
\(768\) 0 0
\(769\) −40.7279 −1.46869 −0.734343 0.678778i \(-0.762510\pi\)
−0.734343 + 0.678778i \(0.762510\pi\)
\(770\) 0 0
\(771\) −8.58579 −0.309210
\(772\) 0 0
\(773\) 25.7990 0.927925 0.463963 0.885855i \(-0.346427\pi\)
0.463963 + 0.885855i \(0.346427\pi\)
\(774\) 0 0
\(775\) 1.00000 0.0359211
\(776\) 0 0
\(777\) 36.3137 1.30275
\(778\) 0 0
\(779\) −5.69848 −0.204169
\(780\) 0 0
\(781\) 0.100505 0.00359635
\(782\) 0 0
\(783\) −6.41421 −0.229225
\(784\) 0 0
\(785\) −3.00000 −0.107075
\(786\) 0 0
\(787\) −37.2843 −1.32904 −0.664520 0.747270i \(-0.731364\pi\)
−0.664520 + 0.747270i \(0.731364\pi\)
\(788\) 0 0
\(789\) −9.58579 −0.341263
\(790\) 0 0
\(791\) −3.44365 −0.122442
\(792\) 0 0
\(793\) −4.82843 −0.171462
\(794\) 0 0
\(795\) −3.24264 −0.115005
\(796\) 0 0
\(797\) −15.1005 −0.534887 −0.267444 0.963573i \(-0.586179\pi\)
−0.267444 + 0.963573i \(0.586179\pi\)
\(798\) 0 0
\(799\) −0.928932 −0.0328633
\(800\) 0 0
\(801\) −14.8284 −0.523937
\(802\) 0 0
\(803\) 8.82843 0.311548
\(804\) 0 0
\(805\) 3.82843 0.134934
\(806\) 0 0
\(807\) −2.07107 −0.0729050
\(808\) 0 0
\(809\) 4.27208 0.150198 0.0750991 0.997176i \(-0.476073\pi\)
0.0750991 + 0.997176i \(0.476073\pi\)
\(810\) 0 0
\(811\) −23.1421 −0.812630 −0.406315 0.913733i \(-0.633186\pi\)
−0.406315 + 0.913733i \(0.633186\pi\)
\(812\) 0 0
\(813\) 4.31371 0.151288
\(814\) 0 0
\(815\) 14.9706 0.524396
\(816\) 0 0
\(817\) −16.7696 −0.586692
\(818\) 0 0
\(819\) 13.0711 0.456740
\(820\) 0 0
\(821\) −17.4558 −0.609213 −0.304607 0.952478i \(-0.598525\pi\)
−0.304607 + 0.952478i \(0.598525\pi\)
\(822\) 0 0
\(823\) 42.3431 1.47599 0.737995 0.674807i \(-0.235773\pi\)
0.737995 + 0.674807i \(0.235773\pi\)
\(824\) 0 0
\(825\) −1.41421 −0.0492366
\(826\) 0 0
\(827\) −27.7279 −0.964194 −0.482097 0.876118i \(-0.660125\pi\)
−0.482097 + 0.876118i \(0.660125\pi\)
\(828\) 0 0
\(829\) −2.17157 −0.0754218 −0.0377109 0.999289i \(-0.512007\pi\)
−0.0377109 + 0.999289i \(0.512007\pi\)
\(830\) 0 0
\(831\) −14.9706 −0.519323
\(832\) 0 0
\(833\) 3.17157 0.109888
\(834\) 0 0
\(835\) 10.2426 0.354461
\(836\) 0 0
\(837\) 1.00000 0.0345651
\(838\) 0 0
\(839\) −39.6569 −1.36911 −0.684553 0.728963i \(-0.740003\pi\)
−0.684553 + 0.728963i \(0.740003\pi\)
\(840\) 0 0
\(841\) 12.1421 0.418694
\(842\) 0 0
\(843\) 12.2426 0.421659
\(844\) 0 0
\(845\) −1.34315 −0.0462056
\(846\) 0 0
\(847\) −34.4558 −1.18392
\(848\) 0 0
\(849\) −19.6274 −0.673611
\(850\) 0 0
\(851\) 9.48528 0.325151
\(852\) 0 0
\(853\) −10.4853 −0.359009 −0.179505 0.983757i \(-0.557449\pi\)
−0.179505 + 0.983757i \(0.557449\pi\)
\(854\) 0 0
\(855\) 4.58579 0.156831
\(856\) 0 0
\(857\) −53.3137 −1.82116 −0.910581 0.413331i \(-0.864365\pi\)
−0.910581 + 0.413331i \(0.864365\pi\)
\(858\) 0 0
\(859\) 48.1716 1.64359 0.821796 0.569781i \(-0.192972\pi\)
0.821796 + 0.569781i \(0.192972\pi\)
\(860\) 0 0
\(861\) −4.75736 −0.162130
\(862\) 0 0
\(863\) −29.5147 −1.00469 −0.502346 0.864666i \(-0.667530\pi\)
−0.502346 + 0.864666i \(0.667530\pi\)
\(864\) 0 0
\(865\) 10.3431 0.351678
\(866\) 0 0
\(867\) −16.8284 −0.571523
\(868\) 0 0
\(869\) −24.0000 −0.814144
\(870\) 0 0
\(871\) −5.07107 −0.171827
\(872\) 0 0
\(873\) −10.1421 −0.343259
\(874\) 0 0
\(875\) 3.82843 0.129424
\(876\) 0 0
\(877\) −12.8284 −0.433185 −0.216593 0.976262i \(-0.569494\pi\)
−0.216593 + 0.976262i \(0.569494\pi\)
\(878\) 0 0
\(879\) −8.89949 −0.300173
\(880\) 0 0
\(881\) −25.0711 −0.844666 −0.422333 0.906441i \(-0.638789\pi\)
−0.422333 + 0.906441i \(0.638789\pi\)
\(882\) 0 0
\(883\) 47.0711 1.58407 0.792034 0.610477i \(-0.209022\pi\)
0.792034 + 0.610477i \(0.209022\pi\)
\(884\) 0 0
\(885\) 11.7279 0.394230
\(886\) 0 0
\(887\) −0.828427 −0.0278159 −0.0139079 0.999903i \(-0.504427\pi\)
−0.0139079 + 0.999903i \(0.504427\pi\)
\(888\) 0 0
\(889\) −44.2426 −1.48385
\(890\) 0 0
\(891\) −1.41421 −0.0473779
\(892\) 0 0
\(893\) −10.2843 −0.344150
\(894\) 0 0
\(895\) 8.34315 0.278881
\(896\) 0 0
\(897\) 3.41421 0.113997
\(898\) 0 0
\(899\) −6.41421 −0.213926
\(900\) 0 0
\(901\) −1.34315 −0.0447467
\(902\) 0 0
\(903\) −14.0000 −0.465891
\(904\) 0 0
\(905\) −20.4853 −0.680954
\(906\) 0 0
\(907\) −23.8284 −0.791210 −0.395605 0.918421i \(-0.629465\pi\)
−0.395605 + 0.918421i \(0.629465\pi\)
\(908\) 0 0
\(909\) 4.07107 0.135029
\(910\) 0 0
\(911\) 4.97056 0.164682 0.0823410 0.996604i \(-0.473760\pi\)
0.0823410 + 0.996604i \(0.473760\pi\)
\(912\) 0 0
\(913\) 15.8995 0.526196
\(914\) 0 0
\(915\) −1.41421 −0.0467525
\(916\) 0 0
\(917\) 61.2548 2.02281
\(918\) 0 0
\(919\) 32.2843 1.06496 0.532480 0.846443i \(-0.321260\pi\)
0.532480 + 0.846443i \(0.321260\pi\)
\(920\) 0 0
\(921\) 15.8995 0.523906
\(922\) 0 0
\(923\) −0.242641 −0.00798662
\(924\) 0 0
\(925\) 9.48528 0.311874
\(926\) 0 0
\(927\) 18.4853 0.607136
\(928\) 0 0
\(929\) −41.2426 −1.35313 −0.676564 0.736384i \(-0.736532\pi\)
−0.676564 + 0.736384i \(0.736532\pi\)
\(930\) 0 0
\(931\) 35.1127 1.15077
\(932\) 0 0
\(933\) −8.68629 −0.284376
\(934\) 0 0
\(935\) −0.585786 −0.0191573
\(936\) 0 0
\(937\) −0.485281 −0.0158535 −0.00792673 0.999969i \(-0.502523\pi\)
−0.00792673 + 0.999969i \(0.502523\pi\)
\(938\) 0 0
\(939\) −7.14214 −0.233075
\(940\) 0 0
\(941\) −58.5269 −1.90792 −0.953961 0.299929i \(-0.903037\pi\)
−0.953961 + 0.299929i \(0.903037\pi\)
\(942\) 0 0
\(943\) −1.24264 −0.0404659
\(944\) 0 0
\(945\) 3.82843 0.124539
\(946\) 0 0
\(947\) 11.5147 0.374178 0.187089 0.982343i \(-0.440095\pi\)
0.187089 + 0.982343i \(0.440095\pi\)
\(948\) 0 0
\(949\) −21.3137 −0.691872
\(950\) 0 0
\(951\) −4.58579 −0.148704
\(952\) 0 0
\(953\) 40.6274 1.31605 0.658026 0.752996i \(-0.271392\pi\)
0.658026 + 0.752996i \(0.271392\pi\)
\(954\) 0 0
\(955\) −14.2426 −0.460881
\(956\) 0 0
\(957\) 9.07107 0.293226
\(958\) 0 0
\(959\) −10.8284 −0.349668
\(960\) 0 0
\(961\) −30.0000 −0.967742
\(962\) 0 0
\(963\) −6.41421 −0.206695
\(964\) 0 0
\(965\) −10.1421 −0.326487
\(966\) 0 0
\(967\) −5.61522 −0.180573 −0.0902867 0.995916i \(-0.528778\pi\)
−0.0902867 + 0.995916i \(0.528778\pi\)
\(968\) 0 0
\(969\) 1.89949 0.0610206
\(970\) 0 0
\(971\) −50.4264 −1.61826 −0.809130 0.587629i \(-0.800061\pi\)
−0.809130 + 0.587629i \(0.800061\pi\)
\(972\) 0 0
\(973\) 1.97056 0.0631733
\(974\) 0 0
\(975\) 3.41421 0.109342
\(976\) 0 0
\(977\) 3.87006 0.123814 0.0619071 0.998082i \(-0.480282\pi\)
0.0619071 + 0.998082i \(0.480282\pi\)
\(978\) 0 0
\(979\) 20.9706 0.670222
\(980\) 0 0
\(981\) −4.58579 −0.146413
\(982\) 0 0
\(983\) −27.5269 −0.877972 −0.438986 0.898494i \(-0.644662\pi\)
−0.438986 + 0.898494i \(0.644662\pi\)
\(984\) 0 0
\(985\) −12.8284 −0.408748
\(986\) 0 0
\(987\) −8.58579 −0.273289
\(988\) 0 0
\(989\) −3.65685 −0.116281
\(990\) 0 0
\(991\) 0.455844 0.0144804 0.00724018 0.999974i \(-0.497695\pi\)
0.00724018 + 0.999974i \(0.497695\pi\)
\(992\) 0 0
\(993\) 18.1716 0.576657
\(994\) 0 0
\(995\) −2.48528 −0.0787887
\(996\) 0 0
\(997\) 36.4853 1.15550 0.577750 0.816214i \(-0.303931\pi\)
0.577750 + 0.816214i \(0.303931\pi\)
\(998\) 0 0
\(999\) 9.48528 0.300101
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.bt.1.2 2
4.3 odd 2 2760.2.a.n.1.1 2
12.11 even 2 8280.2.a.x.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2760.2.a.n.1.1 2 4.3 odd 2
5520.2.a.bt.1.2 2 1.1 even 1 trivial
8280.2.a.x.1.1 2 12.11 even 2