Properties

Label 8550.2.a.bw.1.1
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 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 8550.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{7} +1.00000 q^{8} -3.46410 q^{11} +1.46410 q^{13} -2.00000 q^{14} +1.00000 q^{16} +3.46410 q^{17} +1.00000 q^{19} -3.46410 q^{22} -6.92820 q^{23} +1.46410 q^{26} -2.00000 q^{28} +3.46410 q^{29} +5.46410 q^{31} +1.00000 q^{32} +3.46410 q^{34} -5.46410 q^{37} +1.00000 q^{38} +1.46410 q^{43} -3.46410 q^{44} -6.92820 q^{46} -6.92820 q^{47} -3.00000 q^{49} +1.46410 q^{52} +0.928203 q^{53} -2.00000 q^{56} +3.46410 q^{58} -3.46410 q^{59} +2.00000 q^{61} +5.46410 q^{62} +1.00000 q^{64} -1.07180 q^{67} +3.46410 q^{68} -6.92820 q^{71} -8.92820 q^{73} -5.46410 q^{74} +1.00000 q^{76} +6.92820 q^{77} +5.46410 q^{79} -9.46410 q^{83} +1.46410 q^{86} -3.46410 q^{88} +6.92820 q^{89} -2.92820 q^{91} -6.92820 q^{92} -6.92820 q^{94} +2.39230 q^{97} -3.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} - 4 q^{7} + 2 q^{8} + O(q^{10}) \) \( 2 q + 2 q^{2} + 2 q^{4} - 4 q^{7} + 2 q^{8} - 4 q^{13} - 4 q^{14} + 2 q^{16} + 2 q^{19} - 4 q^{26} - 4 q^{28} + 4 q^{31} + 2 q^{32} - 4 q^{37} + 2 q^{38} - 4 q^{43} - 6 q^{49} - 4 q^{52} - 12 q^{53} - 4 q^{56} + 4 q^{61} + 4 q^{62} + 2 q^{64} - 16 q^{67} - 4 q^{73} - 4 q^{74} + 2 q^{76} + 4 q^{79} - 12 q^{83} - 4 q^{86} + 8 q^{91} - 16 q^{97} - 6 q^{98} + O(q^{100}) \)

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 \(-0.674901\pi\)
−0.522233 + 0.852803i \(0.674901\pi\)
\(12\) 0 0
\(13\) 1.46410 0.406069 0.203034 0.979172i \(-0.434920\pi\)
0.203034 + 0.979172i \(0.434920\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.46410 0.840168 0.420084 0.907485i \(-0.362001\pi\)
0.420084 + 0.907485i \(0.362001\pi\)
\(18\) 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 \(-0.756922\pi\)
−0.722315 + 0.691564i \(0.756922\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 1.46410 0.287134
\(27\) 0 0
\(28\) −2.00000 −0.377964
\(29\) 3.46410 0.643268 0.321634 0.946864i \(-0.395768\pi\)
0.321634 + 0.946864i \(0.395768\pi\)
\(30\) 0 0
\(31\) 5.46410 0.981382 0.490691 0.871334i \(-0.336744\pi\)
0.490691 + 0.871334i \(0.336744\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 3.46410 0.594089
\(35\) 0 0
\(36\) 0 0
\(37\) −5.46410 −0.898293 −0.449146 0.893458i \(-0.648272\pi\)
−0.449146 + 0.893458i \(0.648272\pi\)
\(38\) 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
\(43\) 1.46410 0.223273 0.111637 0.993749i \(-0.464391\pi\)
0.111637 + 0.993749i \(0.464391\pi\)
\(44\) −3.46410 −0.522233
\(45\) 0 0
\(46\) −6.92820 −1.02151
\(47\) −6.92820 −1.01058 −0.505291 0.862949i \(-0.668615\pi\)
−0.505291 + 0.862949i \(0.668615\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 1.46410 0.203034
\(53\) 0.928203 0.127499 0.0637493 0.997966i \(-0.479694\pi\)
0.0637493 + 0.997966i \(0.479694\pi\)
\(54\) 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 \(-0.572400\pi\)
−0.225494 + 0.974245i \(0.572400\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 5.46410 0.693942
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −1.07180 −0.130941 −0.0654704 0.997855i \(-0.520855\pi\)
−0.0654704 + 0.997855i \(0.520855\pi\)
\(68\) 3.46410 0.420084
\(69\) 0 0
\(70\) 0 0
\(71\) −6.92820 −0.822226 −0.411113 0.911584i \(-0.634860\pi\)
−0.411113 + 0.911584i \(0.634860\pi\)
\(72\) 0 0
\(73\) −8.92820 −1.04497 −0.522484 0.852649i \(-0.674994\pi\)
−0.522484 + 0.852649i \(0.674994\pi\)
\(74\) −5.46410 −0.635189
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) 6.92820 0.789542
\(78\) 0 0
\(79\) 5.46410 0.614759 0.307380 0.951587i \(-0.400548\pi\)
0.307380 + 0.951587i \(0.400548\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −9.46410 −1.03882 −0.519410 0.854525i \(-0.673848\pi\)
−0.519410 + 0.854525i \(0.673848\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 1.46410 0.157878
\(87\) 0 0
\(88\) −3.46410 −0.369274
\(89\) 6.92820 0.734388 0.367194 0.930144i \(-0.380318\pi\)
0.367194 + 0.930144i \(0.380318\pi\)
\(90\) 0 0
\(91\) −2.92820 −0.306959
\(92\) −6.92820 −0.722315
\(93\) 0 0
\(94\) −6.92820 −0.714590
\(95\) 0 0
\(96\) 0 0
\(97\) 2.39230 0.242902 0.121451 0.992597i \(-0.461245\pi\)
0.121451 + 0.992597i \(0.461245\pi\)
\(98\) −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
\(103\) −8.92820 −0.879722 −0.439861 0.898066i \(-0.644972\pi\)
−0.439861 + 0.898066i \(0.644972\pi\)
\(104\) 1.46410 0.143567
\(105\) 0 0
\(106\) 0.928203 0.0901551
\(107\) −6.92820 −0.669775 −0.334887 0.942258i \(-0.608698\pi\)
−0.334887 + 0.942258i \(0.608698\pi\)
\(108\) 0 0
\(109\) 6.39230 0.612272 0.306136 0.951988i \(-0.400964\pi\)
0.306136 + 0.951988i \(0.400964\pi\)
\(110\) 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
\(124\) 5.46410 0.490691
\(125\) 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 \(-0.451644\pi\)
0.151330 + 0.988483i \(0.451644\pi\)
\(132\) 0 0
\(133\) −2.00000 −0.173422
\(134\) −1.07180 −0.0925891
\(135\) 0 0
\(136\) 3.46410 0.297044
\(137\) −8.53590 −0.729271 −0.364636 0.931150i \(-0.618806\pi\)
−0.364636 + 0.931150i \(0.618806\pi\)
\(138\) 0 0
\(139\) −17.8564 −1.51456 −0.757280 0.653090i \(-0.773472\pi\)
−0.757280 + 0.653090i \(0.773472\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −6.92820 −0.581402
\(143\) −5.07180 −0.424125
\(144\) 0 0
\(145\) 0 0
\(146\) −8.92820 −0.738903
\(147\) 0 0
\(148\) −5.46410 −0.449146
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) 12.3923 1.00847 0.504236 0.863566i \(-0.331774\pi\)
0.504236 + 0.863566i \(0.331774\pi\)
\(152\) 1.00000 0.0811107
\(153\) 0 0
\(154\) 6.92820 0.558291
\(155\) 0 0
\(156\) 0 0
\(157\) −19.3205 −1.54194 −0.770972 0.636869i \(-0.780229\pi\)
−0.770972 + 0.636869i \(0.780229\pi\)
\(158\) 5.46410 0.434701
\(159\) 0 0
\(160\) 0 0
\(161\) 13.8564 1.09204
\(162\) 0 0
\(163\) 13.4641 1.05459 0.527295 0.849682i \(-0.323206\pi\)
0.527295 + 0.849682i \(0.323206\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) −9.46410 −0.734557
\(167\) −5.07180 −0.392467 −0.196234 0.980557i \(-0.562871\pi\)
−0.196234 + 0.980557i \(0.562871\pi\)
\(168\) 0 0
\(169\) −10.8564 −0.835108
\(170\) 0 0
\(171\) 0 0
\(172\) 1.46410 0.111637
\(173\) −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
\(179\) −22.3923 −1.67368 −0.836840 0.547448i \(-0.815599\pi\)
−0.836840 + 0.547448i \(0.815599\pi\)
\(180\) 0 0
\(181\) −2.39230 −0.177819 −0.0889093 0.996040i \(-0.528338\pi\)
−0.0889093 + 0.996040i \(0.528338\pi\)
\(182\) −2.92820 −0.217053
\(183\) 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
\(191\) 23.3205 1.68741 0.843706 0.536805i \(-0.180369\pi\)
0.843706 + 0.536805i \(0.180369\pi\)
\(192\) 0 0
\(193\) 14.3923 1.03598 0.517990 0.855386i \(-0.326680\pi\)
0.517990 + 0.855386i \(0.326680\pi\)
\(194\) 2.39230 0.171757
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) −12.9282 −0.921096 −0.460548 0.887635i \(-0.652347\pi\)
−0.460548 + 0.887635i \(0.652347\pi\)
\(198\) 0 0
\(199\) −10.9282 −0.774680 −0.387340 0.921937i \(-0.626606\pi\)
−0.387340 + 0.921937i \(0.626606\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 6.00000 0.422159
\(203\) −6.92820 −0.486265
\(204\) 0 0
\(205\) 0 0
\(206\) −8.92820 −0.622057
\(207\) 0 0
\(208\) 1.46410 0.101517
\(209\) −3.46410 −0.239617
\(210\) 0 0
\(211\) 9.85641 0.678543 0.339272 0.940688i \(-0.389819\pi\)
0.339272 + 0.940688i \(0.389819\pi\)
\(212\) 0.928203 0.0637493
\(213\) 0 0
\(214\) −6.92820 −0.473602
\(215\) 0 0
\(216\) 0 0
\(217\) −10.9282 −0.741855
\(218\) 6.39230 0.432942
\(219\) 0 0
\(220\) 0 0
\(221\) 5.07180 0.341166
\(222\) 0 0
\(223\) −3.85641 −0.258244 −0.129122 0.991629i \(-0.541216\pi\)
−0.129122 + 0.991629i \(0.541216\pi\)
\(224\) −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
\(229\) −15.0718 −0.995972 −0.497986 0.867185i \(-0.665927\pi\)
−0.497986 + 0.867185i \(0.665927\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 3.46410 0.227429
\(233\) −15.4641 −1.01309 −0.506543 0.862214i \(-0.669077\pi\)
−0.506543 + 0.862214i \(0.669077\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −3.46410 −0.225494
\(237\) 0 0
\(238\) −6.92820 −0.449089
\(239\) −9.46410 −0.612182 −0.306091 0.952002i \(-0.599021\pi\)
−0.306091 + 0.952002i \(0.599021\pi\)
\(240\) 0 0
\(241\) 27.8564 1.79439 0.897194 0.441636i \(-0.145602\pi\)
0.897194 + 0.441636i \(0.145602\pi\)
\(242\) 1.00000 0.0642824
\(243\) 0 0
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) 0 0
\(247\) 1.46410 0.0931586
\(248\) 5.46410 0.346971
\(249\) 0 0
\(250\) 0 0
\(251\) −10.3923 −0.655956 −0.327978 0.944685i \(-0.606367\pi\)
−0.327978 + 0.944685i \(0.606367\pi\)
\(252\) 0 0
\(253\) 24.0000 1.50887
\(254\) −14.0000 −0.878438
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −11.0718 −0.690640 −0.345320 0.938485i \(-0.612230\pi\)
−0.345320 + 0.938485i \(0.612230\pi\)
\(258\) 0 0
\(259\) 10.9282 0.679046
\(260\) 0 0
\(261\) 0 0
\(262\) 3.46410 0.214013
\(263\) 1.85641 0.114471 0.0572355 0.998361i \(-0.481771\pi\)
0.0572355 + 0.998361i \(0.481771\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −2.00000 −0.122628
\(267\) 0 0
\(268\) −1.07180 −0.0654704
\(269\) −1.60770 −0.0980229 −0.0490115 0.998798i \(-0.515607\pi\)
−0.0490115 + 0.998798i \(0.515607\pi\)
\(270\) 0 0
\(271\) −24.7846 −1.50556 −0.752779 0.658273i \(-0.771287\pi\)
−0.752779 + 0.658273i \(0.771287\pi\)
\(272\) 3.46410 0.210042
\(273\) 0 0
\(274\) −8.53590 −0.515672
\(275\) 0 0
\(276\) 0 0
\(277\) 20.3923 1.22525 0.612627 0.790372i \(-0.290113\pi\)
0.612627 + 0.790372i \(0.290113\pi\)
\(278\) −17.8564 −1.07096
\(279\) 0 0
\(280\) 0 0
\(281\) 30.9282 1.84502 0.922511 0.385971i \(-0.126133\pi\)
0.922511 + 0.385971i \(0.126133\pi\)
\(282\) 0 0
\(283\) −0.392305 −0.0233201 −0.0116601 0.999932i \(-0.503712\pi\)
−0.0116601 + 0.999932i \(0.503712\pi\)
\(284\) −6.92820 −0.411113
\(285\) 0 0
\(286\) −5.07180 −0.299902
\(287\) 0 0
\(288\) 0 0
\(289\) −5.00000 −0.294118
\(290\) 0 0
\(291\) 0 0
\(292\) −8.92820 −0.522484
\(293\) −0.928203 −0.0542262 −0.0271131 0.999632i \(-0.508631\pi\)
−0.0271131 + 0.999632i \(0.508631\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −5.46410 −0.317594
\(297\) 0 0
\(298\) −6.00000 −0.347571
\(299\) −10.1436 −0.586619
\(300\) 0 0
\(301\) −2.92820 −0.168779
\(302\) 12.3923 0.713097
\(303\) 0 0
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) 0 0
\(307\) −16.7846 −0.957948 −0.478974 0.877829i \(-0.658991\pi\)
−0.478974 + 0.877829i \(0.658991\pi\)
\(308\) 6.92820 0.394771
\(309\) 0 0
\(310\) 0 0
\(311\) −4.39230 −0.249065 −0.124532 0.992216i \(-0.539743\pi\)
−0.124532 + 0.992216i \(0.539743\pi\)
\(312\) 0 0
\(313\) 18.7846 1.06177 0.530884 0.847444i \(-0.321860\pi\)
0.530884 + 0.847444i \(0.321860\pi\)
\(314\) −19.3205 −1.09032
\(315\) 0 0
\(316\) 5.46410 0.307380
\(317\) 7.85641 0.441260 0.220630 0.975358i \(-0.429189\pi\)
0.220630 + 0.975358i \(0.429189\pi\)
\(318\) 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
\(326\) 13.4641 0.745708
\(327\) 0 0
\(328\) 0 0
\(329\) 13.8564 0.763928
\(330\) 0 0
\(331\) 1.07180 0.0589113 0.0294556 0.999566i \(-0.490623\pi\)
0.0294556 + 0.999566i \(0.490623\pi\)
\(332\) −9.46410 −0.519410
\(333\) 0 0
\(334\) −5.07180 −0.277516
\(335\) 0 0
\(336\) 0 0
\(337\) 33.3205 1.81508 0.907542 0.419962i \(-0.137956\pi\)
0.907542 + 0.419962i \(0.137956\pi\)
\(338\) −10.8564 −0.590511
\(339\) 0 0
\(340\) 0 0
\(341\) −18.9282 −1.02502
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 1.46410 0.0789391
\(345\) 0 0
\(346\) −18.0000 −0.967686
\(347\) −25.1769 −1.35157 −0.675784 0.737100i \(-0.736195\pi\)
−0.675784 + 0.737100i \(0.736195\pi\)
\(348\) 0 0
\(349\) 15.8564 0.848774 0.424387 0.905481i \(-0.360490\pi\)
0.424387 + 0.905481i \(0.360490\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −3.46410 −0.184637
\(353\) −34.3923 −1.83052 −0.915259 0.402866i \(-0.868014\pi\)
−0.915259 + 0.402866i \(0.868014\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 6.92820 0.367194
\(357\) 0 0
\(358\) −22.3923 −1.18347
\(359\) −26.5359 −1.40051 −0.700256 0.713892i \(-0.746931\pi\)
−0.700256 + 0.713892i \(0.746931\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −2.39230 −0.125737
\(363\) 0 0
\(364\) −2.92820 −0.153480
\(365\) 0 0
\(366\) 0 0
\(367\) −29.7128 −1.55100 −0.775498 0.631350i \(-0.782501\pi\)
−0.775498 + 0.631350i \(0.782501\pi\)
\(368\) −6.92820 −0.361158
\(369\) 0 0
\(370\) 0 0
\(371\) −1.85641 −0.0963798
\(372\) 0 0
\(373\) 20.3923 1.05587 0.527937 0.849284i \(-0.322966\pi\)
0.527937 + 0.849284i \(0.322966\pi\)
\(374\) −12.0000 −0.620505
\(375\) 0 0
\(376\) −6.92820 −0.357295
\(377\) 5.07180 0.261211
\(378\) 0 0
\(379\) −22.9282 −1.17774 −0.588871 0.808227i \(-0.700428\pi\)
−0.588871 + 0.808227i \(0.700428\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 23.3205 1.19318
\(383\) 5.07180 0.259157 0.129578 0.991569i \(-0.458638\pi\)
0.129578 + 0.991569i \(0.458638\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 14.3923 0.732549
\(387\) 0 0
\(388\) 2.39230 0.121451
\(389\) −14.7846 −0.749609 −0.374805 0.927104i \(-0.622290\pi\)
−0.374805 + 0.927104i \(0.622290\pi\)
\(390\) 0 0
\(391\) −24.0000 −1.21373
\(392\) −3.00000 −0.151523
\(393\) 0 0
\(394\) −12.9282 −0.651313
\(395\) 0 0
\(396\) 0 0
\(397\) 15.3205 0.768914 0.384457 0.923143i \(-0.374389\pi\)
0.384457 + 0.923143i \(0.374389\pi\)
\(398\) −10.9282 −0.547781
\(399\) 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
\(407\) 18.9282 0.938236
\(408\) 0 0
\(409\) 10.7846 0.533265 0.266632 0.963798i \(-0.414089\pi\)
0.266632 + 0.963798i \(0.414089\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −8.92820 −0.439861
\(413\) 6.92820 0.340915
\(414\) 0 0
\(415\) 0 0
\(416\) 1.46410 0.0717835
\(417\) 0 0
\(418\) −3.46410 −0.169435
\(419\) −34.3923 −1.68017 −0.840087 0.542452i \(-0.817496\pi\)
−0.840087 + 0.542452i \(0.817496\pi\)
\(420\) 0 0
\(421\) −23.1769 −1.12957 −0.564787 0.825237i \(-0.691042\pi\)
−0.564787 + 0.825237i \(0.691042\pi\)
\(422\) 9.85641 0.479802
\(423\) 0 0
\(424\) 0.928203 0.0450775
\(425\) 0 0
\(426\) 0 0
\(427\) −4.00000 −0.193574
\(428\) −6.92820 −0.334887
\(429\) 0 0
\(430\) 0 0
\(431\) −5.07180 −0.244300 −0.122150 0.992512i \(-0.538979\pi\)
−0.122150 + 0.992512i \(0.538979\pi\)
\(432\) 0 0
\(433\) −13.3205 −0.640143 −0.320071 0.947393i \(-0.603707\pi\)
−0.320071 + 0.947393i \(0.603707\pi\)
\(434\) −10.9282 −0.524571
\(435\) 0 0
\(436\) 6.39230 0.306136
\(437\) −6.92820 −0.331421
\(438\) 0 0
\(439\) 31.3205 1.49485 0.747423 0.664348i \(-0.231291\pi\)
0.747423 + 0.664348i \(0.231291\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 5.07180 0.241241
\(443\) 16.3923 0.778822 0.389411 0.921064i \(-0.372679\pi\)
0.389411 + 0.921064i \(0.372679\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −3.85641 −0.182606
\(447\) 0 0
\(448\) −2.00000 −0.0944911
\(449\) −25.8564 −1.22024 −0.610120 0.792309i \(-0.708879\pi\)
−0.610120 + 0.792309i \(0.708879\pi\)
\(450\) 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 \(0.327956\pi\)
\(458\) −15.0718 −0.704259
\(459\) 0 0
\(460\) 0 0
\(461\) 23.0718 1.07456 0.537280 0.843404i \(-0.319452\pi\)
0.537280 + 0.843404i \(0.319452\pi\)
\(462\) 0 0
\(463\) 28.9282 1.34441 0.672204 0.740366i \(-0.265348\pi\)
0.672204 + 0.740366i \(0.265348\pi\)
\(464\) 3.46410 0.160817
\(465\) 0 0
\(466\) −15.4641 −0.716361
\(467\) 14.5359 0.672641 0.336321 0.941748i \(-0.390817\pi\)
0.336321 + 0.941748i \(0.390817\pi\)
\(468\) 0 0
\(469\) 2.14359 0.0989820
\(470\) 0 0
\(471\) 0 0
\(472\) −3.46410 −0.159448
\(473\) −5.07180 −0.233201
\(474\) 0 0
\(475\) 0 0
\(476\) −6.92820 −0.317554
\(477\) 0 0
\(478\) −9.46410 −0.432878
\(479\) −28.3923 −1.29728 −0.648639 0.761096i \(-0.724661\pi\)
−0.648639 + 0.761096i \(0.724661\pi\)
\(480\) 0 0
\(481\) −8.00000 −0.364769
\(482\) 27.8564 1.26882
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 0 0
\(486\) 0 0
\(487\) 6.78461 0.307440 0.153720 0.988114i \(-0.450875\pi\)
0.153720 + 0.988114i \(0.450875\pi\)
\(488\) 2.00000 0.0905357
\(489\) 0 0
\(490\) 0 0
\(491\) −6.67949 −0.301441 −0.150721 0.988576i \(-0.548159\pi\)
−0.150721 + 0.988576i \(0.548159\pi\)
\(492\) 0 0
\(493\) 12.0000 0.540453
\(494\) 1.46410 0.0658730
\(495\) 0 0
\(496\) 5.46410 0.245345
\(497\) 13.8564 0.621545
\(498\) 0 0
\(499\) −9.07180 −0.406109 −0.203055 0.979167i \(-0.565087\pi\)
−0.203055 + 0.979167i \(0.565087\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −10.3923 −0.463831
\(503\) 6.92820 0.308913 0.154457 0.988000i \(-0.450637\pi\)
0.154457 + 0.988000i \(0.450637\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 24.0000 1.06693
\(507\) 0 0
\(508\) −14.0000 −0.621150
\(509\) 12.2487 0.542915 0.271457 0.962450i \(-0.412494\pi\)
0.271457 + 0.962450i \(0.412494\pi\)
\(510\) 0 0
\(511\) 17.8564 0.789921
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −11.0718 −0.488356
\(515\) 0 0
\(516\) 0 0
\(517\) 24.0000 1.05552
\(518\) 10.9282 0.480158
\(519\) 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
\(526\) 1.85641 0.0809432
\(527\) 18.9282 0.824525
\(528\) 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
\(536\) −1.07180 −0.0462946
\(537\) 0 0
\(538\) −1.60770 −0.0693127
\(539\) 10.3923 0.447628
\(540\) 0 0
\(541\) 43.5692 1.87319 0.936594 0.350418i \(-0.113960\pi\)
0.936594 + 0.350418i \(0.113960\pi\)
\(542\) −24.7846 −1.06459
\(543\) 0 0
\(544\) 3.46410 0.148522
\(545\) 0 0
\(546\) 0 0
\(547\) −13.0718 −0.558910 −0.279455 0.960159i \(-0.590154\pi\)
−0.279455 + 0.960159i \(0.590154\pi\)
\(548\) −8.53590 −0.364636
\(549\) 0 0
\(550\) 0 0
\(551\) 3.46410 0.147576
\(552\) 0 0
\(553\) −10.9282 −0.464714
\(554\) 20.3923 0.866386
\(555\) 0 0
\(556\) −17.8564 −0.757280
\(557\) −0.928203 −0.0393292 −0.0196646 0.999807i \(-0.506260\pi\)
−0.0196646 + 0.999807i \(0.506260\pi\)
\(558\) 0 0
\(559\) 2.14359 0.0906643
\(560\) 0 0
\(561\) 0 0
\(562\) 30.9282 1.30463
\(563\) 32.7846 1.38171 0.690853 0.722995i \(-0.257235\pi\)
0.690853 + 0.722995i \(0.257235\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −0.392305 −0.0164898
\(567\) 0 0
\(568\) −6.92820 −0.290701
\(569\) 1.85641 0.0778246 0.0389123 0.999243i \(-0.487611\pi\)
0.0389123 + 0.999243i \(0.487611\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) −5.07180 −0.212062
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −34.7846 −1.44810 −0.724051 0.689746i \(-0.757722\pi\)
−0.724051 + 0.689746i \(0.757722\pi\)
\(578\) −5.00000 −0.207973
\(579\) 0 0
\(580\) 0 0
\(581\) 18.9282 0.785274
\(582\) 0 0
\(583\) −3.21539 −0.133168
\(584\) −8.92820 −0.369452
\(585\) 0 0
\(586\) −0.928203 −0.0383437
\(587\) 19.6077 0.809296 0.404648 0.914472i \(-0.367394\pi\)
0.404648 + 0.914472i \(0.367394\pi\)
\(588\) 0 0
\(589\) 5.46410 0.225144
\(590\) 0 0
\(591\) 0 0
\(592\) −5.46410 −0.224573
\(593\) −36.2487 −1.48856 −0.744278 0.667870i \(-0.767206\pi\)
−0.744278 + 0.667870i \(0.767206\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) 0 0
\(598\) −10.1436 −0.414802
\(599\) 32.7846 1.33954 0.669771 0.742567i \(-0.266392\pi\)
0.669771 + 0.742567i \(0.266392\pi\)
\(600\) 0 0
\(601\) 32.9282 1.34317 0.671585 0.740928i \(-0.265614\pi\)
0.671585 + 0.740928i \(0.265614\pi\)
\(602\) −2.92820 −0.119345
\(603\) 0 0
\(604\) 12.3923 0.504236
\(605\) 0 0
\(606\) 0 0
\(607\) 44.6410 1.81192 0.905961 0.423360i \(-0.139149\pi\)
0.905961 + 0.423360i \(0.139149\pi\)
\(608\) 1.00000 0.0405554
\(609\) 0 0
\(610\) 0 0
\(611\) −10.1436 −0.410366
\(612\) 0 0
\(613\) −12.3923 −0.500520 −0.250260 0.968179i \(-0.580516\pi\)
−0.250260 + 0.968179i \(0.580516\pi\)
\(614\) −16.7846 −0.677372
\(615\) 0 0
\(616\) 6.92820 0.279145
\(617\) −1.60770 −0.0647234 −0.0323617 0.999476i \(-0.510303\pi\)
−0.0323617 + 0.999476i \(0.510303\pi\)
\(618\) 0 0
\(619\) −12.7846 −0.513857 −0.256928 0.966430i \(-0.582710\pi\)
−0.256928 + 0.966430i \(0.582710\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −4.39230 −0.176115
\(623\) −13.8564 −0.555145
\(624\) 0 0
\(625\) 0 0
\(626\) 18.7846 0.750784
\(627\) 0 0
\(628\) −19.3205 −0.770972
\(629\) −18.9282 −0.754717
\(630\) 0 0
\(631\) −9.07180 −0.361143 −0.180571 0.983562i \(-0.557795\pi\)
−0.180571 + 0.983562i \(0.557795\pi\)
\(632\) 5.46410 0.217350
\(633\) 0 0
\(634\) 7.85641 0.312018
\(635\) 0 0
\(636\) 0 0
\(637\) −4.39230 −0.174029
\(638\) −12.0000 −0.475085
\(639\) 0 0
\(640\) 0 0
\(641\) 37.8564 1.49524 0.747619 0.664128i \(-0.231197\pi\)
0.747619 + 0.664128i \(0.231197\pi\)
\(642\) 0 0
\(643\) 17.1769 0.677391 0.338696 0.940896i \(-0.390014\pi\)
0.338696 + 0.940896i \(0.390014\pi\)
\(644\) 13.8564 0.546019
\(645\) 0 0
\(646\) 3.46410 0.136293
\(647\) 30.9282 1.21591 0.607957 0.793970i \(-0.291989\pi\)
0.607957 + 0.793970i \(0.291989\pi\)
\(648\) 0 0
\(649\) 12.0000 0.471041
\(650\) 0 0
\(651\) 0 0
\(652\) 13.4641 0.527295
\(653\) 9.71281 0.380092 0.190046 0.981775i \(-0.439136\pi\)
0.190046 + 0.981775i \(0.439136\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 13.8564 0.540179
\(659\) −1.60770 −0.0626269 −0.0313135 0.999510i \(-0.509969\pi\)
−0.0313135 + 0.999510i \(0.509969\pi\)
\(660\) 0 0
\(661\) 21.6077 0.840442 0.420221 0.907422i \(-0.361953\pi\)
0.420221 + 0.907422i \(0.361953\pi\)
\(662\) 1.07180 0.0416566
\(663\) 0 0
\(664\) −9.46410 −0.367278
\(665\) 0 0
\(666\) 0 0
\(667\) −24.0000 −0.929284
\(668\) −5.07180 −0.196234
\(669\) 0 0
\(670\) 0 0
\(671\) −6.92820 −0.267460
\(672\) 0 0
\(673\) −37.3205 −1.43860 −0.719300 0.694700i \(-0.755537\pi\)
−0.719300 + 0.694700i \(0.755537\pi\)
\(674\) 33.3205 1.28346
\(675\) 0 0
\(676\) −10.8564 −0.417554
\(677\) 18.0000 0.691796 0.345898 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) 0 0
\(679\) −4.78461 −0.183616
\(680\) 0 0
\(681\) 0 0
\(682\) −18.9282 −0.724798
\(683\) 1.85641 0.0710334 0.0355167 0.999369i \(-0.488692\pi\)
0.0355167 + 0.999369i \(0.488692\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 20.0000 0.763604
\(687\) 0 0
\(688\) 1.46410 0.0558184
\(689\) 1.35898 0.0517732
\(690\) 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
\(694\) −25.1769 −0.955703
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 15.8564 0.600174
\(699\) 0 0
\(700\) 0 0
\(701\) −36.9282 −1.39476 −0.697379 0.716702i \(-0.745651\pi\)
−0.697379 + 0.716702i \(0.745651\pi\)
\(702\) 0 0
\(703\) −5.46410 −0.206082
\(704\) −3.46410 −0.130558
\(705\) 0 0
\(706\) −34.3923 −1.29437
\(707\) −12.0000 −0.451306
\(708\) 0 0
\(709\) 5.21539 0.195868 0.0979340 0.995193i \(-0.468777\pi\)
0.0979340 + 0.995193i \(0.468777\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 6.92820 0.259645
\(713\) −37.8564 −1.41773
\(714\) 0 0
\(715\) 0 0
\(716\) −22.3923 −0.836840
\(717\) 0 0
\(718\) −26.5359 −0.990311
\(719\) −21.4641 −0.800476 −0.400238 0.916411i \(-0.631073\pi\)
−0.400238 + 0.916411i \(0.631073\pi\)
\(720\) 0 0
\(721\) 17.8564 0.665007
\(722\) 1.00000 0.0372161
\(723\) 0 0
\(724\) −2.39230 −0.0889093
\(725\) 0 0
\(726\) 0 0
\(727\) −27.8564 −1.03314 −0.516568 0.856246i \(-0.672791\pi\)
−0.516568 + 0.856246i \(0.672791\pi\)
\(728\) −2.92820 −0.108526
\(729\) 0 0
\(730\) 0 0
\(731\) 5.07180 0.187587
\(732\) 0 0
\(733\) −21.1769 −0.782187 −0.391094 0.920351i \(-0.627903\pi\)
−0.391094 + 0.920351i \(0.627903\pi\)
\(734\) −29.7128 −1.09672
\(735\) 0 0
\(736\) −6.92820 −0.255377
\(737\) 3.71281 0.136763
\(738\) 0 0
\(739\) 28.7846 1.05886 0.529429 0.848354i \(-0.322406\pi\)
0.529429 + 0.848354i \(0.322406\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −1.85641 −0.0681508
\(743\) −37.8564 −1.38882 −0.694408 0.719581i \(-0.744334\pi\)
−0.694408 + 0.719581i \(0.744334\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 20.3923 0.746615
\(747\) 0 0
\(748\) −12.0000 −0.438763
\(749\) 13.8564 0.506302
\(750\) 0 0
\(751\) 5.46410 0.199388 0.0996939 0.995018i \(-0.468214\pi\)
0.0996939 + 0.995018i \(0.468214\pi\)
\(752\) −6.92820 −0.252646
\(753\) 0 0
\(754\) 5.07180 0.184704
\(755\) 0 0
\(756\) 0 0
\(757\) −21.1769 −0.769688 −0.384844 0.922982i \(-0.625745\pi\)
−0.384844 + 0.922982i \(0.625745\pi\)
\(758\) −22.9282 −0.832790
\(759\) 0 0
\(760\) 0 0
\(761\) −29.5692 −1.07188 −0.535942 0.844255i \(-0.680043\pi\)
−0.535942 + 0.844255i \(0.680043\pi\)
\(762\) 0 0
\(763\) −12.7846 −0.462834
\(764\) 23.3205 0.843706
\(765\) 0 0
\(766\) 5.07180 0.183251
\(767\) −5.07180 −0.183132
\(768\) 0 0
\(769\) 0.143594 0.00517812 0.00258906 0.999997i \(-0.499176\pi\)
0.00258906 + 0.999997i \(0.499176\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 14.3923 0.517990
\(773\) −19.8564 −0.714185 −0.357093 0.934069i \(-0.616232\pi\)
−0.357093 + 0.934069i \(0.616232\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 2.39230 0.0858787
\(777\) 0 0
\(778\) −14.7846 −0.530054
\(779\) 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 \(0.616017\pi\)
\(788\) −12.9282 −0.460548
\(789\) 0 0
\(790\) 0 0
\(791\) 12.0000 0.426671
\(792\) 0 0
\(793\) 2.92820 0.103984
\(794\) 15.3205 0.543704
\(795\) 0 0
\(796\) −10.9282 −0.387340
\(797\) −21.7128 −0.769107 −0.384554 0.923103i \(-0.625645\pi\)
−0.384554 + 0.923103i \(0.625645\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) 0 0
\(801\) 0 0
\(802\) 12.0000 0.423735
\(803\) 30.9282 1.09143
\(804\) 0 0
\(805\) 0 0
\(806\) 8.00000 0.281788
\(807\) 0 0
\(808\) 6.00000 0.211079
\(809\) −22.6410 −0.796016 −0.398008 0.917382i \(-0.630298\pi\)
−0.398008 + 0.917382i \(0.630298\pi\)
\(810\) 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
\(814\) 18.9282 0.663433
\(815\) 0 0
\(816\) 0 0
\(817\) 1.46410 0.0512224
\(818\) 10.7846 0.377075
\(819\) 0 0
\(820\) 0 0
\(821\) 19.8564 0.692993 0.346497 0.938051i \(-0.387371\pi\)
0.346497 + 0.938051i \(0.387371\pi\)
\(822\) 0 0
\(823\) 1.21539 0.0423658 0.0211829 0.999776i \(-0.493257\pi\)
0.0211829 + 0.999776i \(0.493257\pi\)
\(824\) −8.92820 −0.311029
\(825\) 0 0
\(826\) 6.92820 0.241063
\(827\) −18.9282 −0.658198 −0.329099 0.944295i \(-0.606745\pi\)
−0.329099 + 0.944295i \(0.606745\pi\)
\(828\) 0 0
\(829\) 40.5359 1.40787 0.703935 0.710264i \(-0.251425\pi\)
0.703935 + 0.710264i \(0.251425\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 1.46410 0.0507586
\(833\) −10.3923 −0.360072
\(834\) 0 0
\(835\) 0 0
\(836\) −3.46410 −0.119808
\(837\) 0 0
\(838\) −34.3923 −1.18806
\(839\) −13.8564 −0.478376 −0.239188 0.970973i \(-0.576881\pi\)
−0.239188 + 0.970973i \(0.576881\pi\)
\(840\) 0 0
\(841\) −17.0000 −0.586207
\(842\) −23.1769 −0.798729
\(843\) 0 0
\(844\) 9.85641 0.339272
\(845\) 0 0
\(846\) 0 0
\(847\) −2.00000 −0.0687208
\(848\) 0.928203 0.0318746
\(849\) 0 0
\(850\) 0 0
\(851\) 37.8564 1.29770
\(852\) 0 0
\(853\) 15.3205 0.524564 0.262282 0.964991i \(-0.415525\pi\)
0.262282 + 0.964991i \(0.415525\pi\)
\(854\) −4.00000 −0.136877
\(855\) 0 0
\(856\) −6.92820 −0.236801
\(857\) −26.7846 −0.914945 −0.457472 0.889224i \(-0.651245\pi\)
−0.457472 + 0.889224i \(0.651245\pi\)
\(858\) 0 0
\(859\) −14.1436 −0.482573 −0.241287 0.970454i \(-0.577569\pi\)
−0.241287 + 0.970454i \(0.577569\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −5.07180 −0.172746
\(863\) 29.0718 0.989615 0.494808 0.869002i \(-0.335238\pi\)
0.494808 + 0.869002i \(0.335238\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −13.3205 −0.452649
\(867\) 0 0
\(868\) −10.9282 −0.370927
\(869\) −18.9282 −0.642095
\(870\) 0 0
\(871\) −1.56922 −0.0531710
\(872\) 6.39230 0.216471
\(873\) 0 0
\(874\) −6.92820 −0.234350
\(875\) 0 0
\(876\) 0 0
\(877\) 44.3923 1.49902 0.749511 0.661992i \(-0.230289\pi\)
0.749511 + 0.661992i \(0.230289\pi\)
\(878\) 31.3205 1.05702
\(879\) 0 0
\(880\) 0 0
\(881\) 6.92820 0.233417 0.116709 0.993166i \(-0.462766\pi\)
0.116709 + 0.993166i \(0.462766\pi\)
\(882\) 0 0
\(883\) −24.3923 −0.820866 −0.410433 0.911891i \(-0.634623\pi\)
−0.410433 + 0.911891i \(0.634623\pi\)
\(884\) 5.07180 0.170583
\(885\) 0 0
\(886\) 16.3923 0.550710
\(887\) 41.5692 1.39576 0.697879 0.716216i \(-0.254127\pi\)
0.697879 + 0.716216i \(0.254127\pi\)
\(888\) 0 0
\(889\) 28.0000 0.939090
\(890\) 0 0
\(891\) 0 0
\(892\) −3.85641 −0.129122
\(893\) −6.92820 −0.231843
\(894\) 0 0
\(895\) 0 0
\(896\) −2.00000 −0.0668153
\(897\) 0 0
\(898\) −25.8564 −0.862839
\(899\) 18.9282 0.631291
\(900\) 0 0
\(901\) 3.21539 0.107120
\(902\) 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
\(911\) −37.8564 −1.25424 −0.627119 0.778923i \(-0.715766\pi\)
−0.627119 + 0.778923i \(0.715766\pi\)
\(912\) 0 0
\(913\) 32.7846 1.08501
\(914\) 22.0000 0.727695
\(915\) 0 0
\(916\) −15.0718 −0.497986
\(917\) −6.92820 −0.228789
\(918\) 0 0
\(919\) 25.0718 0.827042 0.413521 0.910495i \(-0.364299\pi\)
0.413521 + 0.910495i \(0.364299\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 23.0718 0.759829
\(923\) −10.1436 −0.333880
\(924\) 0 0
\(925\) 0 0
\(926\) 28.9282 0.950640
\(927\) 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
\(932\) −15.4641 −0.506543
\(933\) 0 0
\(934\) 14.5359 0.475629
\(935\) 0 0
\(936\) 0 0
\(937\) 39.5692 1.29267 0.646335 0.763054i \(-0.276301\pi\)
0.646335 + 0.763054i \(0.276301\pi\)
\(938\) 2.14359 0.0699908
\(939\) 0 0
\(940\) 0 0
\(941\) 0.248711 0.00810776 0.00405388 0.999992i \(-0.498710\pi\)
0.00405388 + 0.999992i \(0.498710\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −3.46410 −0.112747
\(945\) 0 0
\(946\) −5.07180 −0.164898
\(947\) 35.3205 1.14776 0.573881 0.818939i \(-0.305437\pi\)
0.573881 + 0.818939i \(0.305437\pi\)
\(948\) 0 0
\(949\) −13.0718 −0.424328
\(950\) 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
\(956\) −9.46410 −0.306091
\(957\) 0 0
\(958\) −28.3923 −0.917314
\(959\) 17.0718 0.551277
\(960\) 0 0
\(961\) −1.14359 −0.0368901
\(962\) −8.00000 −0.257930
\(963\) 0 0
\(964\) 27.8564 0.897194
\(965\) 0 0
\(966\) 0 0
\(967\) −20.9282 −0.673006 −0.336503 0.941682i \(-0.609244\pi\)
−0.336503 + 0.941682i \(0.609244\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) 0 0
\(971\) −53.3205 −1.71114 −0.855568 0.517690i \(-0.826792\pi\)
−0.855568 + 0.517690i \(0.826792\pi\)
\(972\) 0 0
\(973\) 35.7128 1.14490
\(974\) 6.78461 0.217393
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) −59.5692 −1.90579 −0.952894 0.303303i \(-0.901910\pi\)
−0.952894 + 0.303303i \(0.901910\pi\)
\(978\) 0 0
\(979\) −24.0000 −0.767043
\(980\) 0 0
\(981\) 0 0
\(982\) −6.67949 −0.213151
\(983\) −36.4974 −1.16409 −0.582043 0.813158i \(-0.697747\pi\)
−0.582043 + 0.813158i \(0.697747\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 12.0000 0.382158
\(987\) 0 0
\(988\) 1.46410 0.0465793
\(989\) −10.1436 −0.322548
\(990\) 0 0
\(991\) 12.3923 0.393655 0.196827 0.980438i \(-0.436936\pi\)
0.196827 + 0.980438i \(0.436936\pi\)
\(992\) 5.46410 0.173485
\(993\) 0 0
\(994\) 13.8564 0.439499
\(995\) 0 0
\(996\) 0 0
\(997\) 8.39230 0.265787 0.132893 0.991130i \(-0.457573\pi\)
0.132893 + 0.991130i \(0.457573\pi\)
\(998\) −9.07180 −0.287163
\(999\) 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.1.1 2
3.2 odd 2 8550.2.a.bo.1.2 2
5.4 even 2 1710.2.a.u.1.1 2
15.14 odd 2 1710.2.a.y.1.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1710.2.a.u.1.1 2 5.4 even 2
1710.2.a.y.1.2 yes 2 15.14 odd 2
8550.2.a.bo.1.2 2 3.2 odd 2
8550.2.a.bw.1.1 2 1.1 even 1 trivial