Properties

Label 8550.2.a.bw.1.2
Level $8550$
Weight $2$
Character 8550.1
Self dual yes
Analytic conductor $68.272$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8550.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(68.2720937282\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1710)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 8550.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{7} +1.00000 q^{8} +3.46410 q^{11} -5.46410 q^{13} -2.00000 q^{14} +1.00000 q^{16} -3.46410 q^{17} +1.00000 q^{19} +3.46410 q^{22} +6.92820 q^{23} -5.46410 q^{26} -2.00000 q^{28} -3.46410 q^{29} -1.46410 q^{31} +1.00000 q^{32} -3.46410 q^{34} +1.46410 q^{37} +1.00000 q^{38} -5.46410 q^{43} +3.46410 q^{44} +6.92820 q^{46} +6.92820 q^{47} -3.00000 q^{49} -5.46410 q^{52} -12.9282 q^{53} -2.00000 q^{56} -3.46410 q^{58} +3.46410 q^{59} +2.00000 q^{61} -1.46410 q^{62} +1.00000 q^{64} -14.9282 q^{67} -3.46410 q^{68} +6.92820 q^{71} +4.92820 q^{73} +1.46410 q^{74} +1.00000 q^{76} -6.92820 q^{77} -1.46410 q^{79} -2.53590 q^{83} -5.46410 q^{86} +3.46410 q^{88} -6.92820 q^{89} +10.9282 q^{91} +6.92820 q^{92} +6.92820 q^{94} -18.3923 q^{97} -3.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} - 4q^{7} + 2q^{8} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} - 4q^{7} + 2q^{8} - 4q^{13} - 4q^{14} + 2q^{16} + 2q^{19} - 4q^{26} - 4q^{28} + 4q^{31} + 2q^{32} - 4q^{37} + 2q^{38} - 4q^{43} - 6q^{49} - 4q^{52} - 12q^{53} - 4q^{56} + 4q^{61} + 4q^{62} + 2q^{64} - 16q^{67} - 4q^{73} - 4q^{74} + 2q^{76} + 4q^{79} - 12q^{83} - 4q^{86} + 8q^{91} - 16q^{97} - 6q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 3.46410 1.04447 0.522233 0.852803i \(-0.325099\pi\)
0.522233 + 0.852803i \(0.325099\pi\)
\(12\) 0 0
\(13\) −5.46410 −1.51547 −0.757735 0.652563i \(-0.773694\pi\)
−0.757735 + 0.652563i \(0.773694\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.46410 −0.840168 −0.420084 0.907485i \(-0.637999\pi\)
−0.420084 + 0.907485i \(0.637999\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 3.46410 0.738549
\(23\) 6.92820 1.44463 0.722315 0.691564i \(-0.243078\pi\)
0.722315 + 0.691564i \(0.243078\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −5.46410 −1.07160
\(27\) 0 0
\(28\) −2.00000 −0.377964
\(29\) −3.46410 −0.643268 −0.321634 0.946864i \(-0.604232\pi\)
−0.321634 + 0.946864i \(0.604232\pi\)
\(30\) 0 0
\(31\) −1.46410 −0.262960 −0.131480 0.991319i \(-0.541973\pi\)
−0.131480 + 0.991319i \(0.541973\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −3.46410 −0.594089
\(35\) 0 0
\(36\) 0 0
\(37\) 1.46410 0.240697 0.120348 0.992732i \(-0.461599\pi\)
0.120348 + 0.992732i \(0.461599\pi\)
\(38\) 1.00000 0.162221
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −5.46410 −0.833268 −0.416634 0.909074i \(-0.636790\pi\)
−0.416634 + 0.909074i \(0.636790\pi\)
\(44\) 3.46410 0.522233
\(45\) 0 0
\(46\) 6.92820 1.02151
\(47\) 6.92820 1.01058 0.505291 0.862949i \(-0.331385\pi\)
0.505291 + 0.862949i \(0.331385\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) −5.46410 −0.757735
\(53\) −12.9282 −1.77583 −0.887913 0.460012i \(-0.847845\pi\)
−0.887913 + 0.460012i \(0.847845\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −2.00000 −0.267261
\(57\) 0 0
\(58\) −3.46410 −0.454859
\(59\) 3.46410 0.450988 0.225494 0.974245i \(-0.427600\pi\)
0.225494 + 0.974245i \(0.427600\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −1.46410 −0.185941
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −14.9282 −1.82377 −0.911885 0.410445i \(-0.865373\pi\)
−0.911885 + 0.410445i \(0.865373\pi\)
\(68\) −3.46410 −0.420084
\(69\) 0 0
\(70\) 0 0
\(71\) 6.92820 0.822226 0.411113 0.911584i \(-0.365140\pi\)
0.411113 + 0.911584i \(0.365140\pi\)
\(72\) 0 0
\(73\) 4.92820 0.576803 0.288401 0.957510i \(-0.406876\pi\)
0.288401 + 0.957510i \(0.406876\pi\)
\(74\) 1.46410 0.170198
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) −6.92820 −0.789542
\(78\) 0 0
\(79\) −1.46410 −0.164724 −0.0823622 0.996602i \(-0.526246\pi\)
−0.0823622 + 0.996602i \(0.526246\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −2.53590 −0.278351 −0.139176 0.990268i \(-0.544445\pi\)
−0.139176 + 0.990268i \(0.544445\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −5.46410 −0.589209
\(87\) 0 0
\(88\) 3.46410 0.369274
\(89\) −6.92820 −0.734388 −0.367194 0.930144i \(-0.619682\pi\)
−0.367194 + 0.930144i \(0.619682\pi\)
\(90\) 0 0
\(91\) 10.9282 1.14559
\(92\) 6.92820 0.722315
\(93\) 0 0
\(94\) 6.92820 0.714590
\(95\) 0 0
\(96\) 0 0
\(97\) −18.3923 −1.86746 −0.933728 0.357984i \(-0.883464\pi\)
−0.933728 + 0.357984i \(0.883464\pi\)
\(98\) −3.00000 −0.303046
\(99\) 0 0
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) 4.92820 0.485590 0.242795 0.970078i \(-0.421936\pi\)
0.242795 + 0.970078i \(0.421936\pi\)
\(104\) −5.46410 −0.535799
\(105\) 0 0
\(106\) −12.9282 −1.25570
\(107\) 6.92820 0.669775 0.334887 0.942258i \(-0.391302\pi\)
0.334887 + 0.942258i \(0.391302\pi\)
\(108\) 0 0
\(109\) −14.3923 −1.37853 −0.689266 0.724508i \(-0.742067\pi\)
−0.689266 + 0.724508i \(0.742067\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −2.00000 −0.188982
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −3.46410 −0.321634
\(117\) 0 0
\(118\) 3.46410 0.318896
\(119\) 6.92820 0.635107
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 2.00000 0.181071
\(123\) 0 0
\(124\) −1.46410 −0.131480
\(125\) 0 0
\(126\) 0 0
\(127\) −14.0000 −1.24230 −0.621150 0.783692i \(-0.713334\pi\)
−0.621150 + 0.783692i \(0.713334\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −3.46410 −0.302660 −0.151330 0.988483i \(-0.548356\pi\)
−0.151330 + 0.988483i \(0.548356\pi\)
\(132\) 0 0
\(133\) −2.00000 −0.173422
\(134\) −14.9282 −1.28960
\(135\) 0 0
\(136\) −3.46410 −0.297044
\(137\) −15.4641 −1.32119 −0.660594 0.750744i \(-0.729695\pi\)
−0.660594 + 0.750744i \(0.729695\pi\)
\(138\) 0 0
\(139\) 9.85641 0.836009 0.418005 0.908445i \(-0.362730\pi\)
0.418005 + 0.908445i \(0.362730\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 6.92820 0.581402
\(143\) −18.9282 −1.58286
\(144\) 0 0
\(145\) 0 0
\(146\) 4.92820 0.407861
\(147\) 0 0
\(148\) 1.46410 0.120348
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) −8.39230 −0.682956 −0.341478 0.939890i \(-0.610927\pi\)
−0.341478 + 0.939890i \(0.610927\pi\)
\(152\) 1.00000 0.0811107
\(153\) 0 0
\(154\) −6.92820 −0.558291
\(155\) 0 0
\(156\) 0 0
\(157\) 15.3205 1.22271 0.611355 0.791357i \(-0.290625\pi\)
0.611355 + 0.791357i \(0.290625\pi\)
\(158\) −1.46410 −0.116478
\(159\) 0 0
\(160\) 0 0
\(161\) −13.8564 −1.09204
\(162\) 0 0
\(163\) 6.53590 0.511931 0.255966 0.966686i \(-0.417607\pi\)
0.255966 + 0.966686i \(0.417607\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) −2.53590 −0.196824
\(167\) −18.9282 −1.46471 −0.732354 0.680924i \(-0.761578\pi\)
−0.732354 + 0.680924i \(0.761578\pi\)
\(168\) 0 0
\(169\) 16.8564 1.29665
\(170\) 0 0
\(171\) 0 0
\(172\) −5.46410 −0.416634
\(173\) −18.0000 −1.36851 −0.684257 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 3.46410 0.261116
\(177\) 0 0
\(178\) −6.92820 −0.519291
\(179\) −1.60770 −0.120165 −0.0600824 0.998193i \(-0.519136\pi\)
−0.0600824 + 0.998193i \(0.519136\pi\)
\(180\) 0 0
\(181\) 18.3923 1.36709 0.683545 0.729909i \(-0.260437\pi\)
0.683545 + 0.729909i \(0.260437\pi\)
\(182\) 10.9282 0.810052
\(183\) 0 0
\(184\) 6.92820 0.510754
\(185\) 0 0
\(186\) 0 0
\(187\) −12.0000 −0.877527
\(188\) 6.92820 0.505291
\(189\) 0 0
\(190\) 0 0
\(191\) −11.3205 −0.819123 −0.409562 0.912282i \(-0.634318\pi\)
−0.409562 + 0.912282i \(0.634318\pi\)
\(192\) 0 0
\(193\) −6.39230 −0.460128 −0.230064 0.973175i \(-0.573894\pi\)
−0.230064 + 0.973175i \(0.573894\pi\)
\(194\) −18.3923 −1.32049
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 0.928203 0.0661317 0.0330659 0.999453i \(-0.489473\pi\)
0.0330659 + 0.999453i \(0.489473\pi\)
\(198\) 0 0
\(199\) 2.92820 0.207575 0.103787 0.994600i \(-0.466904\pi\)
0.103787 + 0.994600i \(0.466904\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 6.00000 0.422159
\(203\) 6.92820 0.486265
\(204\) 0 0
\(205\) 0 0
\(206\) 4.92820 0.343364
\(207\) 0 0
\(208\) −5.46410 −0.378867
\(209\) 3.46410 0.239617
\(210\) 0 0
\(211\) −17.8564 −1.22929 −0.614643 0.788806i \(-0.710700\pi\)
−0.614643 + 0.788806i \(0.710700\pi\)
\(212\) −12.9282 −0.887913
\(213\) 0 0
\(214\) 6.92820 0.473602
\(215\) 0 0
\(216\) 0 0
\(217\) 2.92820 0.198779
\(218\) −14.3923 −0.974770
\(219\) 0 0
\(220\) 0 0
\(221\) 18.9282 1.27325
\(222\) 0 0
\(223\) 23.8564 1.59754 0.798772 0.601634i \(-0.205484\pi\)
0.798772 + 0.601634i \(0.205484\pi\)
\(224\) −2.00000 −0.133631
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) −28.9282 −1.91163 −0.955815 0.293970i \(-0.905024\pi\)
−0.955815 + 0.293970i \(0.905024\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −3.46410 −0.227429
\(233\) −8.53590 −0.559205 −0.279603 0.960116i \(-0.590203\pi\)
−0.279603 + 0.960116i \(0.590203\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 3.46410 0.225494
\(237\) 0 0
\(238\) 6.92820 0.449089
\(239\) −2.53590 −0.164034 −0.0820168 0.996631i \(-0.526136\pi\)
−0.0820168 + 0.996631i \(0.526136\pi\)
\(240\) 0 0
\(241\) 0.143594 0.00924967 0.00462484 0.999989i \(-0.498528\pi\)
0.00462484 + 0.999989i \(0.498528\pi\)
\(242\) 1.00000 0.0642824
\(243\) 0 0
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) 0 0
\(247\) −5.46410 −0.347672
\(248\) −1.46410 −0.0929705
\(249\) 0 0
\(250\) 0 0
\(251\) 10.3923 0.655956 0.327978 0.944685i \(-0.393633\pi\)
0.327978 + 0.944685i \(0.393633\pi\)
\(252\) 0 0
\(253\) 24.0000 1.50887
\(254\) −14.0000 −0.878438
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −24.9282 −1.55498 −0.777489 0.628896i \(-0.783507\pi\)
−0.777489 + 0.628896i \(0.783507\pi\)
\(258\) 0 0
\(259\) −2.92820 −0.181950
\(260\) 0 0
\(261\) 0 0
\(262\) −3.46410 −0.214013
\(263\) −25.8564 −1.59437 −0.797187 0.603732i \(-0.793680\pi\)
−0.797187 + 0.603732i \(0.793680\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −2.00000 −0.122628
\(267\) 0 0
\(268\) −14.9282 −0.911885
\(269\) −22.3923 −1.36528 −0.682641 0.730753i \(-0.739169\pi\)
−0.682641 + 0.730753i \(0.739169\pi\)
\(270\) 0 0
\(271\) 16.7846 1.01959 0.509796 0.860295i \(-0.329721\pi\)
0.509796 + 0.860295i \(0.329721\pi\)
\(272\) −3.46410 −0.210042
\(273\) 0 0
\(274\) −15.4641 −0.934221
\(275\) 0 0
\(276\) 0 0
\(277\) −0.392305 −0.0235713 −0.0117857 0.999931i \(-0.503752\pi\)
−0.0117857 + 0.999931i \(0.503752\pi\)
\(278\) 9.85641 0.591148
\(279\) 0 0
\(280\) 0 0
\(281\) 17.0718 1.01842 0.509209 0.860643i \(-0.329938\pi\)
0.509209 + 0.860643i \(0.329938\pi\)
\(282\) 0 0
\(283\) 20.3923 1.21220 0.606098 0.795390i \(-0.292734\pi\)
0.606098 + 0.795390i \(0.292734\pi\)
\(284\) 6.92820 0.411113
\(285\) 0 0
\(286\) −18.9282 −1.11925
\(287\) 0 0
\(288\) 0 0
\(289\) −5.00000 −0.294118
\(290\) 0 0
\(291\) 0 0
\(292\) 4.92820 0.288401
\(293\) 12.9282 0.755274 0.377637 0.925954i \(-0.376737\pi\)
0.377637 + 0.925954i \(0.376737\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 1.46410 0.0850992
\(297\) 0 0
\(298\) −6.00000 −0.347571
\(299\) −37.8564 −2.18929
\(300\) 0 0
\(301\) 10.9282 0.629891
\(302\) −8.39230 −0.482923
\(303\) 0 0
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) 0 0
\(307\) 24.7846 1.41453 0.707266 0.706947i \(-0.249928\pi\)
0.707266 + 0.706947i \(0.249928\pi\)
\(308\) −6.92820 −0.394771
\(309\) 0 0
\(310\) 0 0
\(311\) 16.3923 0.929522 0.464761 0.885436i \(-0.346140\pi\)
0.464761 + 0.885436i \(0.346140\pi\)
\(312\) 0 0
\(313\) −22.7846 −1.28786 −0.643931 0.765083i \(-0.722698\pi\)
−0.643931 + 0.765083i \(0.722698\pi\)
\(314\) 15.3205 0.864586
\(315\) 0 0
\(316\) −1.46410 −0.0823622
\(317\) −19.8564 −1.11525 −0.557623 0.830094i \(-0.688287\pi\)
−0.557623 + 0.830094i \(0.688287\pi\)
\(318\) 0 0
\(319\) −12.0000 −0.671871
\(320\) 0 0
\(321\) 0 0
\(322\) −13.8564 −0.772187
\(323\) −3.46410 −0.192748
\(324\) 0 0
\(325\) 0 0
\(326\) 6.53590 0.361990
\(327\) 0 0
\(328\) 0 0
\(329\) −13.8564 −0.763928
\(330\) 0 0
\(331\) 14.9282 0.820528 0.410264 0.911967i \(-0.365437\pi\)
0.410264 + 0.911967i \(0.365437\pi\)
\(332\) −2.53590 −0.139176
\(333\) 0 0
\(334\) −18.9282 −1.03571
\(335\) 0 0
\(336\) 0 0
\(337\) −1.32051 −0.0719327 −0.0359663 0.999353i \(-0.511451\pi\)
−0.0359663 + 0.999353i \(0.511451\pi\)
\(338\) 16.8564 0.916868
\(339\) 0 0
\(340\) 0 0
\(341\) −5.07180 −0.274653
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) −5.46410 −0.294605
\(345\) 0 0
\(346\) −18.0000 −0.967686
\(347\) 37.1769 1.99576 0.997881 0.0650705i \(-0.0207272\pi\)
0.997881 + 0.0650705i \(0.0207272\pi\)
\(348\) 0 0
\(349\) −11.8564 −0.634659 −0.317329 0.948315i \(-0.602786\pi\)
−0.317329 + 0.948315i \(0.602786\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 3.46410 0.184637
\(353\) −13.6077 −0.724265 −0.362132 0.932127i \(-0.617951\pi\)
−0.362132 + 0.932127i \(0.617951\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −6.92820 −0.367194
\(357\) 0 0
\(358\) −1.60770 −0.0849693
\(359\) −33.4641 −1.76617 −0.883084 0.469215i \(-0.844537\pi\)
−0.883084 + 0.469215i \(0.844537\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 18.3923 0.966678
\(363\) 0 0
\(364\) 10.9282 0.572793
\(365\) 0 0
\(366\) 0 0
\(367\) 25.7128 1.34220 0.671099 0.741368i \(-0.265823\pi\)
0.671099 + 0.741368i \(0.265823\pi\)
\(368\) 6.92820 0.361158
\(369\) 0 0
\(370\) 0 0
\(371\) 25.8564 1.34240
\(372\) 0 0
\(373\) −0.392305 −0.0203128 −0.0101564 0.999948i \(-0.503233\pi\)
−0.0101564 + 0.999948i \(0.503233\pi\)
\(374\) −12.0000 −0.620505
\(375\) 0 0
\(376\) 6.92820 0.357295
\(377\) 18.9282 0.974852
\(378\) 0 0
\(379\) −9.07180 −0.465987 −0.232993 0.972478i \(-0.574852\pi\)
−0.232993 + 0.972478i \(0.574852\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −11.3205 −0.579208
\(383\) 18.9282 0.967186 0.483593 0.875293i \(-0.339331\pi\)
0.483593 + 0.875293i \(0.339331\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −6.39230 −0.325360
\(387\) 0 0
\(388\) −18.3923 −0.933728
\(389\) 26.7846 1.35803 0.679017 0.734123i \(-0.262406\pi\)
0.679017 + 0.734123i \(0.262406\pi\)
\(390\) 0 0
\(391\) −24.0000 −1.21373
\(392\) −3.00000 −0.151523
\(393\) 0 0
\(394\) 0.928203 0.0467622
\(395\) 0 0
\(396\) 0 0
\(397\) −19.3205 −0.969669 −0.484834 0.874606i \(-0.661120\pi\)
−0.484834 + 0.874606i \(0.661120\pi\)
\(398\) 2.92820 0.146778
\(399\) 0 0
\(400\) 0 0
\(401\) 12.0000 0.599251 0.299626 0.954057i \(-0.403138\pi\)
0.299626 + 0.954057i \(0.403138\pi\)
\(402\) 0 0
\(403\) 8.00000 0.398508
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) 6.92820 0.343841
\(407\) 5.07180 0.251400
\(408\) 0 0
\(409\) −30.7846 −1.52220 −0.761100 0.648634i \(-0.775341\pi\)
−0.761100 + 0.648634i \(0.775341\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 4.92820 0.242795
\(413\) −6.92820 −0.340915
\(414\) 0 0
\(415\) 0 0
\(416\) −5.46410 −0.267900
\(417\) 0 0
\(418\) 3.46410 0.169435
\(419\) −13.6077 −0.664779 −0.332390 0.943142i \(-0.607855\pi\)
−0.332390 + 0.943142i \(0.607855\pi\)
\(420\) 0 0
\(421\) 39.1769 1.90937 0.954683 0.297625i \(-0.0961944\pi\)
0.954683 + 0.297625i \(0.0961944\pi\)
\(422\) −17.8564 −0.869236
\(423\) 0 0
\(424\) −12.9282 −0.627849
\(425\) 0 0
\(426\) 0 0
\(427\) −4.00000 −0.193574
\(428\) 6.92820 0.334887
\(429\) 0 0
\(430\) 0 0
\(431\) −18.9282 −0.911739 −0.455870 0.890047i \(-0.650672\pi\)
−0.455870 + 0.890047i \(0.650672\pi\)
\(432\) 0 0
\(433\) 21.3205 1.02460 0.512299 0.858807i \(-0.328794\pi\)
0.512299 + 0.858807i \(0.328794\pi\)
\(434\) 2.92820 0.140558
\(435\) 0 0
\(436\) −14.3923 −0.689266
\(437\) 6.92820 0.331421
\(438\) 0 0
\(439\) −3.32051 −0.158479 −0.0792396 0.996856i \(-0.525249\pi\)
−0.0792396 + 0.996856i \(0.525249\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 18.9282 0.900323
\(443\) −4.39230 −0.208685 −0.104342 0.994541i \(-0.533274\pi\)
−0.104342 + 0.994541i \(0.533274\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 23.8564 1.12963
\(447\) 0 0
\(448\) −2.00000 −0.0944911
\(449\) 1.85641 0.0876092 0.0438046 0.999040i \(-0.486052\pi\)
0.0438046 + 0.999040i \(0.486052\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −6.00000 −0.282216
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 22.0000 1.02912 0.514558 0.857455i \(-0.327956\pi\)
0.514558 + 0.857455i \(0.327956\pi\)
\(458\) −28.9282 −1.35173
\(459\) 0 0
\(460\) 0 0
\(461\) 36.9282 1.71992 0.859959 0.510363i \(-0.170489\pi\)
0.859959 + 0.510363i \(0.170489\pi\)
\(462\) 0 0
\(463\) 15.0718 0.700446 0.350223 0.936666i \(-0.386106\pi\)
0.350223 + 0.936666i \(0.386106\pi\)
\(464\) −3.46410 −0.160817
\(465\) 0 0
\(466\) −8.53590 −0.395418
\(467\) 21.4641 0.993240 0.496620 0.867968i \(-0.334574\pi\)
0.496620 + 0.867968i \(0.334574\pi\)
\(468\) 0 0
\(469\) 29.8564 1.37864
\(470\) 0 0
\(471\) 0 0
\(472\) 3.46410 0.159448
\(473\) −18.9282 −0.870320
\(474\) 0 0
\(475\) 0 0
\(476\) 6.92820 0.317554
\(477\) 0 0
\(478\) −2.53590 −0.115989
\(479\) −7.60770 −0.347604 −0.173802 0.984781i \(-0.555605\pi\)
−0.173802 + 0.984781i \(0.555605\pi\)
\(480\) 0 0
\(481\) −8.00000 −0.364769
\(482\) 0.143594 0.00654051
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 0 0
\(486\) 0 0
\(487\) −34.7846 −1.57624 −0.788121 0.615521i \(-0.788946\pi\)
−0.788121 + 0.615521i \(0.788946\pi\)
\(488\) 2.00000 0.0905357
\(489\) 0 0
\(490\) 0 0
\(491\) −41.3205 −1.86477 −0.932384 0.361469i \(-0.882275\pi\)
−0.932384 + 0.361469i \(0.882275\pi\)
\(492\) 0 0
\(493\) 12.0000 0.540453
\(494\) −5.46410 −0.245842
\(495\) 0 0
\(496\) −1.46410 −0.0657401
\(497\) −13.8564 −0.621545
\(498\) 0 0
\(499\) −22.9282 −1.02641 −0.513204 0.858267i \(-0.671541\pi\)
−0.513204 + 0.858267i \(0.671541\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 10.3923 0.463831
\(503\) −6.92820 −0.308913 −0.154457 0.988000i \(-0.549363\pi\)
−0.154457 + 0.988000i \(0.549363\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 24.0000 1.06693
\(507\) 0 0
\(508\) −14.0000 −0.621150
\(509\) −36.2487 −1.60670 −0.803348 0.595510i \(-0.796950\pi\)
−0.803348 + 0.595510i \(0.796950\pi\)
\(510\) 0 0
\(511\) −9.85641 −0.436022
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −24.9282 −1.09954
\(515\) 0 0
\(516\) 0 0
\(517\) 24.0000 1.05552
\(518\) −2.92820 −0.128658
\(519\) 0 0
\(520\) 0 0
\(521\) −12.0000 −0.525730 −0.262865 0.964833i \(-0.584667\pi\)
−0.262865 + 0.964833i \(0.584667\pi\)
\(522\) 0 0
\(523\) 28.0000 1.22435 0.612177 0.790721i \(-0.290294\pi\)
0.612177 + 0.790721i \(0.290294\pi\)
\(524\) −3.46410 −0.151330
\(525\) 0 0
\(526\) −25.8564 −1.12739
\(527\) 5.07180 0.220931
\(528\) 0 0
\(529\) 25.0000 1.08696
\(530\) 0 0
\(531\) 0 0
\(532\) −2.00000 −0.0867110
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) −14.9282 −0.644800
\(537\) 0 0
\(538\) −22.3923 −0.965401
\(539\) −10.3923 −0.447628
\(540\) 0 0
\(541\) −39.5692 −1.70121 −0.850607 0.525802i \(-0.823765\pi\)
−0.850607 + 0.525802i \(0.823765\pi\)
\(542\) 16.7846 0.720961
\(543\) 0 0
\(544\) −3.46410 −0.148522
\(545\) 0 0
\(546\) 0 0
\(547\) −26.9282 −1.15137 −0.575683 0.817673i \(-0.695264\pi\)
−0.575683 + 0.817673i \(0.695264\pi\)
\(548\) −15.4641 −0.660594
\(549\) 0 0
\(550\) 0 0
\(551\) −3.46410 −0.147576
\(552\) 0 0
\(553\) 2.92820 0.124520
\(554\) −0.392305 −0.0166674
\(555\) 0 0
\(556\) 9.85641 0.418005
\(557\) 12.9282 0.547786 0.273893 0.961760i \(-0.411689\pi\)
0.273893 + 0.961760i \(0.411689\pi\)
\(558\) 0 0
\(559\) 29.8564 1.26279
\(560\) 0 0
\(561\) 0 0
\(562\) 17.0718 0.720130
\(563\) −8.78461 −0.370227 −0.185114 0.982717i \(-0.559265\pi\)
−0.185114 + 0.982717i \(0.559265\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 20.3923 0.857153
\(567\) 0 0
\(568\) 6.92820 0.290701
\(569\) −25.8564 −1.08396 −0.541978 0.840392i \(-0.682325\pi\)
−0.541978 + 0.840392i \(0.682325\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) −18.9282 −0.791428
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 6.78461 0.282447 0.141223 0.989978i \(-0.454896\pi\)
0.141223 + 0.989978i \(0.454896\pi\)
\(578\) −5.00000 −0.207973
\(579\) 0 0
\(580\) 0 0
\(581\) 5.07180 0.210414
\(582\) 0 0
\(583\) −44.7846 −1.85479
\(584\) 4.92820 0.203931
\(585\) 0 0
\(586\) 12.9282 0.534059
\(587\) 40.3923 1.66717 0.833584 0.552392i \(-0.186285\pi\)
0.833584 + 0.552392i \(0.186285\pi\)
\(588\) 0 0
\(589\) −1.46410 −0.0603273
\(590\) 0 0
\(591\) 0 0
\(592\) 1.46410 0.0601742
\(593\) 12.2487 0.502994 0.251497 0.967858i \(-0.419077\pi\)
0.251497 + 0.967858i \(0.419077\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) 0 0
\(598\) −37.8564 −1.54806
\(599\) −8.78461 −0.358929 −0.179465 0.983764i \(-0.557437\pi\)
−0.179465 + 0.983764i \(0.557437\pi\)
\(600\) 0 0
\(601\) 19.0718 0.777955 0.388977 0.921247i \(-0.372828\pi\)
0.388977 + 0.921247i \(0.372828\pi\)
\(602\) 10.9282 0.445400
\(603\) 0 0
\(604\) −8.39230 −0.341478
\(605\) 0 0
\(606\) 0 0
\(607\) −24.6410 −1.00015 −0.500074 0.865983i \(-0.666694\pi\)
−0.500074 + 0.865983i \(0.666694\pi\)
\(608\) 1.00000 0.0405554
\(609\) 0 0
\(610\) 0 0
\(611\) −37.8564 −1.53151
\(612\) 0 0
\(613\) 8.39230 0.338962 0.169481 0.985533i \(-0.445791\pi\)
0.169481 + 0.985533i \(0.445791\pi\)
\(614\) 24.7846 1.00023
\(615\) 0 0
\(616\) −6.92820 −0.279145
\(617\) −22.3923 −0.901480 −0.450740 0.892655i \(-0.648840\pi\)
−0.450740 + 0.892655i \(0.648840\pi\)
\(618\) 0 0
\(619\) 28.7846 1.15695 0.578476 0.815700i \(-0.303648\pi\)
0.578476 + 0.815700i \(0.303648\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 16.3923 0.657272
\(623\) 13.8564 0.555145
\(624\) 0 0
\(625\) 0 0
\(626\) −22.7846 −0.910656
\(627\) 0 0
\(628\) 15.3205 0.611355
\(629\) −5.07180 −0.202226
\(630\) 0 0
\(631\) −22.9282 −0.912757 −0.456379 0.889786i \(-0.650854\pi\)
−0.456379 + 0.889786i \(0.650854\pi\)
\(632\) −1.46410 −0.0582388
\(633\) 0 0
\(634\) −19.8564 −0.788599
\(635\) 0 0
\(636\) 0 0
\(637\) 16.3923 0.649487
\(638\) −12.0000 −0.475085
\(639\) 0 0
\(640\) 0 0
\(641\) 10.1436 0.400648 0.200324 0.979730i \(-0.435801\pi\)
0.200324 + 0.979730i \(0.435801\pi\)
\(642\) 0 0
\(643\) −45.1769 −1.78160 −0.890802 0.454392i \(-0.849857\pi\)
−0.890802 + 0.454392i \(0.849857\pi\)
\(644\) −13.8564 −0.546019
\(645\) 0 0
\(646\) −3.46410 −0.136293
\(647\) 17.0718 0.671162 0.335581 0.942011i \(-0.391067\pi\)
0.335581 + 0.942011i \(0.391067\pi\)
\(648\) 0 0
\(649\) 12.0000 0.471041
\(650\) 0 0
\(651\) 0 0
\(652\) 6.53590 0.255966
\(653\) −45.7128 −1.78888 −0.894440 0.447187i \(-0.852426\pi\)
−0.894440 + 0.447187i \(0.852426\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) −13.8564 −0.540179
\(659\) −22.3923 −0.872280 −0.436140 0.899879i \(-0.643655\pi\)
−0.436140 + 0.899879i \(0.643655\pi\)
\(660\) 0 0
\(661\) 42.3923 1.64887 0.824435 0.565957i \(-0.191493\pi\)
0.824435 + 0.565957i \(0.191493\pi\)
\(662\) 14.9282 0.580201
\(663\) 0 0
\(664\) −2.53590 −0.0984119
\(665\) 0 0
\(666\) 0 0
\(667\) −24.0000 −0.929284
\(668\) −18.9282 −0.732354
\(669\) 0 0
\(670\) 0 0
\(671\) 6.92820 0.267460
\(672\) 0 0
\(673\) −2.67949 −0.103287 −0.0516434 0.998666i \(-0.516446\pi\)
−0.0516434 + 0.998666i \(0.516446\pi\)
\(674\) −1.32051 −0.0508641
\(675\) 0 0
\(676\) 16.8564 0.648323
\(677\) 18.0000 0.691796 0.345898 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) 0 0
\(679\) 36.7846 1.41166
\(680\) 0 0
\(681\) 0 0
\(682\) −5.07180 −0.194209
\(683\) −25.8564 −0.989368 −0.494684 0.869073i \(-0.664716\pi\)
−0.494684 + 0.869073i \(0.664716\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 20.0000 0.763604
\(687\) 0 0
\(688\) −5.46410 −0.208317
\(689\) 70.6410 2.69121
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) −18.0000 −0.684257
\(693\) 0 0
\(694\) 37.1769 1.41122
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) −11.8564 −0.448772
\(699\) 0 0
\(700\) 0 0
\(701\) −23.0718 −0.871410 −0.435705 0.900090i \(-0.643501\pi\)
−0.435705 + 0.900090i \(0.643501\pi\)
\(702\) 0 0
\(703\) 1.46410 0.0552196
\(704\) 3.46410 0.130558
\(705\) 0 0
\(706\) −13.6077 −0.512132
\(707\) −12.0000 −0.451306
\(708\) 0 0
\(709\) 46.7846 1.75703 0.878516 0.477712i \(-0.158534\pi\)
0.878516 + 0.477712i \(0.158534\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −6.92820 −0.259645
\(713\) −10.1436 −0.379881
\(714\) 0 0
\(715\) 0 0
\(716\) −1.60770 −0.0600824
\(717\) 0 0
\(718\) −33.4641 −1.24887
\(719\) −14.5359 −0.542098 −0.271049 0.962566i \(-0.587370\pi\)
−0.271049 + 0.962566i \(0.587370\pi\)
\(720\) 0 0
\(721\) −9.85641 −0.367072
\(722\) 1.00000 0.0372161
\(723\) 0 0
\(724\) 18.3923 0.683545
\(725\) 0 0
\(726\) 0 0
\(727\) −0.143594 −0.00532559 −0.00266279 0.999996i \(-0.500848\pi\)
−0.00266279 + 0.999996i \(0.500848\pi\)
\(728\) 10.9282 0.405026
\(729\) 0 0
\(730\) 0 0
\(731\) 18.9282 0.700085
\(732\) 0 0
\(733\) 41.1769 1.52090 0.760452 0.649394i \(-0.224977\pi\)
0.760452 + 0.649394i \(0.224977\pi\)
\(734\) 25.7128 0.949077
\(735\) 0 0
\(736\) 6.92820 0.255377
\(737\) −51.7128 −1.90487
\(738\) 0 0
\(739\) −12.7846 −0.470289 −0.235145 0.971960i \(-0.575556\pi\)
−0.235145 + 0.971960i \(0.575556\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 25.8564 0.949219
\(743\) −10.1436 −0.372132 −0.186066 0.982537i \(-0.559574\pi\)
−0.186066 + 0.982537i \(0.559574\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −0.392305 −0.0143633
\(747\) 0 0
\(748\) −12.0000 −0.438763
\(749\) −13.8564 −0.506302
\(750\) 0 0
\(751\) −1.46410 −0.0534258 −0.0267129 0.999643i \(-0.508504\pi\)
−0.0267129 + 0.999643i \(0.508504\pi\)
\(752\) 6.92820 0.252646
\(753\) 0 0
\(754\) 18.9282 0.689325
\(755\) 0 0
\(756\) 0 0
\(757\) 41.1769 1.49660 0.748300 0.663360i \(-0.230870\pi\)
0.748300 + 0.663360i \(0.230870\pi\)
\(758\) −9.07180 −0.329502
\(759\) 0 0
\(760\) 0 0
\(761\) 53.5692 1.94188 0.970941 0.239318i \(-0.0769237\pi\)
0.970941 + 0.239318i \(0.0769237\pi\)
\(762\) 0 0
\(763\) 28.7846 1.04207
\(764\) −11.3205 −0.409562
\(765\) 0 0
\(766\) 18.9282 0.683904
\(767\) −18.9282 −0.683458
\(768\) 0 0
\(769\) 27.8564 1.00453 0.502264 0.864714i \(-0.332501\pi\)
0.502264 + 0.864714i \(0.332501\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −6.39230 −0.230064
\(773\) 7.85641 0.282575 0.141288 0.989969i \(-0.454876\pi\)
0.141288 + 0.989969i \(0.454876\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −18.3923 −0.660245
\(777\) 0 0
\(778\) 26.7846 0.960275
\(779\) 0 0
\(780\) 0 0
\(781\) 24.0000 0.858788
\(782\) −24.0000 −0.858238
\(783\) 0 0
\(784\) −3.00000 −0.107143
\(785\) 0 0
\(786\) 0 0
\(787\) −20.0000 −0.712923 −0.356462 0.934310i \(-0.616017\pi\)
−0.356462 + 0.934310i \(0.616017\pi\)
\(788\) 0.928203 0.0330659
\(789\) 0 0
\(790\) 0 0
\(791\) 12.0000 0.426671
\(792\) 0 0
\(793\) −10.9282 −0.388072
\(794\) −19.3205 −0.685659
\(795\) 0 0
\(796\) 2.92820 0.103787
\(797\) 33.7128 1.19417 0.597085 0.802178i \(-0.296326\pi\)
0.597085 + 0.802178i \(0.296326\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) 0 0
\(801\) 0 0
\(802\) 12.0000 0.423735
\(803\) 17.0718 0.602451
\(804\) 0 0
\(805\) 0 0
\(806\) 8.00000 0.281788
\(807\) 0 0
\(808\) 6.00000 0.211079
\(809\) 46.6410 1.63981 0.819905 0.572499i \(-0.194026\pi\)
0.819905 + 0.572499i \(0.194026\pi\)
\(810\) 0 0
\(811\) 44.0000 1.54505 0.772524 0.634985i \(-0.218994\pi\)
0.772524 + 0.634985i \(0.218994\pi\)
\(812\) 6.92820 0.243132
\(813\) 0 0
\(814\) 5.07180 0.177766
\(815\) 0 0
\(816\) 0 0
\(817\) −5.46410 −0.191165
\(818\) −30.7846 −1.07636
\(819\) 0 0
\(820\) 0 0
\(821\) −7.85641 −0.274190 −0.137095 0.990558i \(-0.543777\pi\)
−0.137095 + 0.990558i \(0.543777\pi\)
\(822\) 0 0
\(823\) 42.7846 1.49138 0.745689 0.666294i \(-0.232121\pi\)
0.745689 + 0.666294i \(0.232121\pi\)
\(824\) 4.92820 0.171682
\(825\) 0 0
\(826\) −6.92820 −0.241063
\(827\) −5.07180 −0.176364 −0.0881818 0.996104i \(-0.528106\pi\)
−0.0881818 + 0.996104i \(0.528106\pi\)
\(828\) 0 0
\(829\) 47.4641 1.64850 0.824248 0.566229i \(-0.191598\pi\)
0.824248 + 0.566229i \(0.191598\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −5.46410 −0.189434
\(833\) 10.3923 0.360072
\(834\) 0 0
\(835\) 0 0
\(836\) 3.46410 0.119808
\(837\) 0 0
\(838\) −13.6077 −0.470070
\(839\) 13.8564 0.478376 0.239188 0.970973i \(-0.423119\pi\)
0.239188 + 0.970973i \(0.423119\pi\)
\(840\) 0 0
\(841\) −17.0000 −0.586207
\(842\) 39.1769 1.35013
\(843\) 0 0
\(844\) −17.8564 −0.614643
\(845\) 0 0
\(846\) 0 0
\(847\) −2.00000 −0.0687208
\(848\) −12.9282 −0.443956
\(849\) 0 0
\(850\) 0 0
\(851\) 10.1436 0.347718
\(852\) 0 0
\(853\) −19.3205 −0.661522 −0.330761 0.943715i \(-0.607305\pi\)
−0.330761 + 0.943715i \(0.607305\pi\)
\(854\) −4.00000 −0.136877
\(855\) 0 0
\(856\) 6.92820 0.236801
\(857\) 14.7846 0.505033 0.252516 0.967593i \(-0.418742\pi\)
0.252516 + 0.967593i \(0.418742\pi\)
\(858\) 0 0
\(859\) −41.8564 −1.42812 −0.714061 0.700083i \(-0.753146\pi\)
−0.714061 + 0.700083i \(0.753146\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −18.9282 −0.644697
\(863\) 42.9282 1.46129 0.730647 0.682756i \(-0.239219\pi\)
0.730647 + 0.682756i \(0.239219\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 21.3205 0.724500
\(867\) 0 0
\(868\) 2.92820 0.0993897
\(869\) −5.07180 −0.172049
\(870\) 0 0
\(871\) 81.5692 2.76387
\(872\) −14.3923 −0.487385
\(873\) 0 0
\(874\) 6.92820 0.234350
\(875\) 0 0
\(876\) 0 0
\(877\) 23.6077 0.797175 0.398588 0.917130i \(-0.369500\pi\)
0.398588 + 0.917130i \(0.369500\pi\)
\(878\) −3.32051 −0.112062
\(879\) 0 0
\(880\) 0 0
\(881\) −6.92820 −0.233417 −0.116709 0.993166i \(-0.537234\pi\)
−0.116709 + 0.993166i \(0.537234\pi\)
\(882\) 0 0
\(883\) −3.60770 −0.121409 −0.0607043 0.998156i \(-0.519335\pi\)
−0.0607043 + 0.998156i \(0.519335\pi\)
\(884\) 18.9282 0.636624
\(885\) 0 0
\(886\) −4.39230 −0.147562
\(887\) −41.5692 −1.39576 −0.697879 0.716216i \(-0.745873\pi\)
−0.697879 + 0.716216i \(0.745873\pi\)
\(888\) 0 0
\(889\) 28.0000 0.939090
\(890\) 0 0
\(891\) 0 0
\(892\) 23.8564 0.798772
\(893\) 6.92820 0.231843
\(894\) 0 0
\(895\) 0 0
\(896\) −2.00000 −0.0668153
\(897\) 0 0
\(898\) 1.85641 0.0619491
\(899\) 5.07180 0.169154
\(900\) 0 0
\(901\) 44.7846 1.49199
\(902\) 0 0
\(903\) 0 0
\(904\) −6.00000 −0.199557
\(905\) 0 0
\(906\) 0 0
\(907\) 16.0000 0.531271 0.265636 0.964073i \(-0.414418\pi\)
0.265636 + 0.964073i \(0.414418\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −10.1436 −0.336072 −0.168036 0.985781i \(-0.553743\pi\)
−0.168036 + 0.985781i \(0.553743\pi\)
\(912\) 0 0
\(913\) −8.78461 −0.290728
\(914\) 22.0000 0.727695
\(915\) 0 0
\(916\) −28.9282 −0.955815
\(917\) 6.92820 0.228789
\(918\) 0 0
\(919\) 38.9282 1.28412 0.642061 0.766653i \(-0.278079\pi\)
0.642061 + 0.766653i \(0.278079\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 36.9282 1.21617
\(923\) −37.8564 −1.24606
\(924\) 0 0
\(925\) 0 0
\(926\) 15.0718 0.495290
\(927\) 0 0
\(928\) −3.46410 −0.113715
\(929\) −12.0000 −0.393707 −0.196854 0.980433i \(-0.563072\pi\)
−0.196854 + 0.980433i \(0.563072\pi\)
\(930\) 0 0
\(931\) −3.00000 −0.0983210
\(932\) −8.53590 −0.279603
\(933\) 0 0
\(934\) 21.4641 0.702327
\(935\) 0 0
\(936\) 0 0
\(937\) −43.5692 −1.42334 −0.711672 0.702512i \(-0.752062\pi\)
−0.711672 + 0.702512i \(0.752062\pi\)
\(938\) 29.8564 0.974846
\(939\) 0 0
\(940\) 0 0
\(941\) −48.2487 −1.57286 −0.786432 0.617677i \(-0.788074\pi\)
−0.786432 + 0.617677i \(0.788074\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 3.46410 0.112747
\(945\) 0 0
\(946\) −18.9282 −0.615409
\(947\) 0.679492 0.0220805 0.0110403 0.999939i \(-0.496486\pi\)
0.0110403 + 0.999939i \(0.496486\pi\)
\(948\) 0 0
\(949\) −26.9282 −0.874126
\(950\) 0 0
\(951\) 0 0
\(952\) 6.92820 0.224544
\(953\) 18.0000 0.583077 0.291539 0.956559i \(-0.405833\pi\)
0.291539 + 0.956559i \(0.405833\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −2.53590 −0.0820168
\(957\) 0 0
\(958\) −7.60770 −0.245793
\(959\) 30.9282 0.998724
\(960\) 0 0
\(961\) −28.8564 −0.930852
\(962\) −8.00000 −0.257930
\(963\) 0 0
\(964\) 0.143594 0.00462484
\(965\) 0 0
\(966\) 0 0
\(967\) −7.07180 −0.227414 −0.113707 0.993514i \(-0.536272\pi\)
−0.113707 + 0.993514i \(0.536272\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) 0 0
\(971\) −18.6795 −0.599453 −0.299727 0.954025i \(-0.596895\pi\)
−0.299727 + 0.954025i \(0.596895\pi\)
\(972\) 0 0
\(973\) −19.7128 −0.631964
\(974\) −34.7846 −1.11457
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) 23.5692 0.754046 0.377023 0.926204i \(-0.376948\pi\)
0.377023 + 0.926204i \(0.376948\pi\)
\(978\) 0 0
\(979\) −24.0000 −0.767043
\(980\) 0 0
\(981\) 0 0
\(982\) −41.3205 −1.31859
\(983\) 60.4974 1.92957 0.964784 0.263043i \(-0.0847262\pi\)
0.964784 + 0.263043i \(0.0847262\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 12.0000 0.382158
\(987\) 0 0
\(988\) −5.46410 −0.173836
\(989\) −37.8564 −1.20376
\(990\) 0 0
\(991\) −8.39230 −0.266590 −0.133295 0.991076i \(-0.542556\pi\)
−0.133295 + 0.991076i \(0.542556\pi\)
\(992\) −1.46410 −0.0464853
\(993\) 0 0
\(994\) −13.8564 −0.439499
\(995\) 0 0
\(996\) 0 0
\(997\) −12.3923 −0.392468 −0.196234 0.980557i \(-0.562871\pi\)
−0.196234 + 0.980557i \(0.562871\pi\)
\(998\) −22.9282 −0.725780
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 8550.2.a.bw.1.2 2
3.2 odd 2 8550.2.a.bo.1.1 2
5.4 even 2 1710.2.a.u.1.2 2
15.14 odd 2 1710.2.a.y.1.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1710.2.a.u.1.2 2 5.4 even 2
1710.2.a.y.1.1 yes 2 15.14 odd 2
8550.2.a.bo.1.1 2 3.2 odd 2
8550.2.a.bw.1.2 2 1.1 even 1 trivial