Properties

Label 8550.2.a.x.1.1
Level $8550$
Weight $2$
Character 8550.1
Self dual yes
Analytic conductor $68.272$
Analytic rank $1$
Dimension $1$
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: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 570)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 8550.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{8} -4.00000 q^{11} -2.00000 q^{13} +1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{19} -4.00000 q^{22} +4.00000 q^{23} -2.00000 q^{26} -6.00000 q^{29} +4.00000 q^{31} +1.00000 q^{32} +2.00000 q^{34} +6.00000 q^{37} -1.00000 q^{38} -10.0000 q^{41} +4.00000 q^{43} -4.00000 q^{44} +4.00000 q^{46} -12.0000 q^{47} -7.00000 q^{49} -2.00000 q^{52} +6.00000 q^{53} -6.00000 q^{58} +12.0000 q^{59} -2.00000 q^{61} +4.00000 q^{62} +1.00000 q^{64} -4.00000 q^{67} +2.00000 q^{68} -8.00000 q^{71} +6.00000 q^{73} +6.00000 q^{74} -1.00000 q^{76} -4.00000 q^{79} -10.0000 q^{82} -12.0000 q^{83} +4.00000 q^{86} -4.00000 q^{88} -10.0000 q^{89} +4.00000 q^{92} -12.0000 q^{94} -2.00000 q^{97} -7.00000 q^{98} +O(q^{100})\)

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
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) −4.00000 −0.852803
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) 0 0
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) −1.00000 −0.162221
\(39\) 0 0
\(40\) 0 0
\(41\) −10.0000 −1.56174 −0.780869 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) −12.0000 −1.75038 −0.875190 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) −2.00000 −0.277350
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 4.00000 0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 2.00000 0.242536
\(69\) 0 0
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 6.00000 0.697486
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) 0 0
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −10.0000 −1.10432
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) −4.00000 −0.426401
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 4.00000 0.417029
\(93\) 0 0
\(94\) −12.0000 −1.23771
\(95\) 0 0
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) −7.00000 −0.707107
\(99\) 0 0
\(100\) 0 0
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 0 0
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 0 0
\(109\) −6.00000 −0.574696 −0.287348 0.957826i \(-0.592774\pi\)
−0.287348 + 0.957826i \(0.592774\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) 0 0
\(118\) 12.0000 1.10469
\(119\) 0 0
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) −2.00000 −0.181071
\(123\) 0 0
\(124\) 4.00000 0.359211
\(125\) 0 0
\(126\) 0 0
\(127\) 4.00000 0.354943 0.177471 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) 2.00000 0.171499
\(137\) 10.0000 0.854358 0.427179 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) 0 0
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −8.00000 −0.671345
\(143\) 8.00000 0.668994
\(144\) 0 0
\(145\) 0 0
\(146\) 6.00000 0.496564
\(147\) 0 0
\(148\) 6.00000 0.493197
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 0 0
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) −4.00000 −0.318223
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) −10.0000 −0.780869
\(165\) 0 0
\(166\) −12.0000 −0.931381
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) 4.00000 0.304997
\(173\) −10.0000 −0.760286 −0.380143 0.924928i \(-0.624125\pi\)
−0.380143 + 0.924928i \(0.624125\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) 0 0
\(178\) −10.0000 −0.749532
\(179\) −20.0000 −1.49487 −0.747435 0.664335i \(-0.768715\pi\)
−0.747435 + 0.664335i \(0.768715\pi\)
\(180\) 0 0
\(181\) 18.0000 1.33793 0.668965 0.743294i \(-0.266738\pi\)
0.668965 + 0.743294i \(0.266738\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 4.00000 0.294884
\(185\) 0 0
\(186\) 0 0
\(187\) −8.00000 −0.585018
\(188\) −12.0000 −0.875190
\(189\) 0 0
\(190\) 0 0
\(191\) −20.0000 −1.44715 −0.723575 0.690246i \(-0.757502\pi\)
−0.723575 + 0.690246i \(0.757502\pi\)
\(192\) 0 0
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) −7.00000 −0.500000
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −10.0000 −0.703598
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 4.00000 0.278693
\(207\) 0 0
\(208\) −2.00000 −0.138675
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 6.00000 0.412082
\(213\) 0 0
\(214\) −4.00000 −0.273434
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −6.00000 −0.406371
\(219\) 0 0
\(220\) 0 0
\(221\) −4.00000 −0.269069
\(222\) 0 0
\(223\) −12.0000 −0.803579 −0.401790 0.915732i \(-0.631612\pi\)
−0.401790 + 0.915732i \(0.631612\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −14.0000 −0.931266
\(227\) −20.0000 −1.32745 −0.663723 0.747978i \(-0.731025\pi\)
−0.663723 + 0.747978i \(0.731025\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 12.0000 0.781133
\(237\) 0 0
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) 26.0000 1.67481 0.837404 0.546585i \(-0.184072\pi\)
0.837404 + 0.546585i \(0.184072\pi\)
\(242\) 5.00000 0.321412
\(243\) 0 0
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) 0 0
\(247\) 2.00000 0.127257
\(248\) 4.00000 0.254000
\(249\) 0 0
\(250\) 0 0
\(251\) 20.0000 1.26239 0.631194 0.775625i \(-0.282565\pi\)
0.631194 + 0.775625i \(0.282565\pi\)
\(252\) 0 0
\(253\) −16.0000 −1.00591
\(254\) 4.00000 0.250982
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 2.00000 0.124757 0.0623783 0.998053i \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 12.0000 0.741362
\(263\) −4.00000 −0.246651 −0.123325 0.992366i \(-0.539356\pi\)
−0.123325 + 0.992366i \(0.539356\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −4.00000 −0.244339
\(269\) 2.00000 0.121942 0.0609711 0.998140i \(-0.480580\pi\)
0.0609711 + 0.998140i \(0.480580\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) 10.0000 0.604122
\(275\) 0 0
\(276\) 0 0
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) −20.0000 −1.19952
\(279\) 0 0
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) 28.0000 1.66443 0.832214 0.554455i \(-0.187073\pi\)
0.832214 + 0.554455i \(0.187073\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) 8.00000 0.473050
\(287\) 0 0
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) 6.00000 0.351123
\(293\) −2.00000 −0.116841 −0.0584206 0.998292i \(-0.518606\pi\)
−0.0584206 + 0.998292i \(0.518606\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 6.00000 0.348743
\(297\) 0 0
\(298\) −18.0000 −1.04271
\(299\) −8.00000 −0.462652
\(300\) 0 0
\(301\) 0 0
\(302\) −12.0000 −0.690522
\(303\) 0 0
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) 0 0
\(307\) −4.00000 −0.228292 −0.114146 0.993464i \(-0.536413\pi\)
−0.114146 + 0.993464i \(0.536413\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 28.0000 1.58773 0.793867 0.608091i \(-0.208065\pi\)
0.793867 + 0.608091i \(0.208065\pi\)
\(312\) 0 0
\(313\) 6.00000 0.339140 0.169570 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) 2.00000 0.112867
\(315\) 0 0
\(316\) −4.00000 −0.225018
\(317\) −26.0000 −1.46031 −0.730153 0.683284i \(-0.760551\pi\)
−0.730153 + 0.683284i \(0.760551\pi\)
\(318\) 0 0
\(319\) 24.0000 1.34374
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −2.00000 −0.111283
\(324\) 0 0
\(325\) 0 0
\(326\) 4.00000 0.221540
\(327\) 0 0
\(328\) −10.0000 −0.552158
\(329\) 0 0
\(330\) 0 0
\(331\) 28.0000 1.53902 0.769510 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) −12.0000 −0.658586
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −26.0000 −1.41631 −0.708155 0.706057i \(-0.750472\pi\)
−0.708155 + 0.706057i \(0.750472\pi\)
\(338\) −9.00000 −0.489535
\(339\) 0 0
\(340\) 0 0
\(341\) −16.0000 −0.866449
\(342\) 0 0
\(343\) 0 0
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) −10.0000 −0.537603
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 0 0
\(349\) −10.0000 −0.535288 −0.267644 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −4.00000 −0.213201
\(353\) 26.0000 1.38384 0.691920 0.721974i \(-0.256765\pi\)
0.691920 + 0.721974i \(0.256765\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −10.0000 −0.529999
\(357\) 0 0
\(358\) −20.0000 −1.05703
\(359\) −12.0000 −0.633336 −0.316668 0.948536i \(-0.602564\pi\)
−0.316668 + 0.948536i \(0.602564\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 18.0000 0.946059
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −24.0000 −1.25279 −0.626395 0.779506i \(-0.715470\pi\)
−0.626395 + 0.779506i \(0.715470\pi\)
\(368\) 4.00000 0.208514
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 22.0000 1.13912 0.569558 0.821951i \(-0.307114\pi\)
0.569558 + 0.821951i \(0.307114\pi\)
\(374\) −8.00000 −0.413670
\(375\) 0 0
\(376\) −12.0000 −0.618853
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −20.0000 −1.02329
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −2.00000 −0.101797
\(387\) 0 0
\(388\) −2.00000 −0.101535
\(389\) 22.0000 1.11544 0.557722 0.830028i \(-0.311675\pi\)
0.557722 + 0.830028i \(0.311675\pi\)
\(390\) 0 0
\(391\) 8.00000 0.404577
\(392\) −7.00000 −0.353553
\(393\) 0 0
\(394\) 2.00000 0.100759
\(395\) 0 0
\(396\) 0 0
\(397\) 18.0000 0.903394 0.451697 0.892171i \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 14.0000 0.699127 0.349563 0.936913i \(-0.386330\pi\)
0.349563 + 0.936913i \(0.386330\pi\)
\(402\) 0 0
\(403\) −8.00000 −0.398508
\(404\) −10.0000 −0.497519
\(405\) 0 0
\(406\) 0 0
\(407\) −24.0000 −1.18964
\(408\) 0 0
\(409\) 18.0000 0.890043 0.445021 0.895520i \(-0.353196\pi\)
0.445021 + 0.895520i \(0.353196\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 4.00000 0.197066
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) 0 0
\(418\) 4.00000 0.195646
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) 26.0000 1.26716 0.633581 0.773676i \(-0.281584\pi\)
0.633581 + 0.773676i \(0.281584\pi\)
\(422\) −4.00000 −0.194717
\(423\) 0 0
\(424\) 6.00000 0.291386
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −4.00000 −0.193347
\(429\) 0 0
\(430\) 0 0
\(431\) −8.00000 −0.385346 −0.192673 0.981263i \(-0.561716\pi\)
−0.192673 + 0.981263i \(0.561716\pi\)
\(432\) 0 0
\(433\) −34.0000 −1.63394 −0.816968 0.576683i \(-0.804347\pi\)
−0.816968 + 0.576683i \(0.804347\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −6.00000 −0.287348
\(437\) −4.00000 −0.191346
\(438\) 0 0
\(439\) 12.0000 0.572729 0.286364 0.958121i \(-0.407553\pi\)
0.286364 + 0.958121i \(0.407553\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −4.00000 −0.190261
\(443\) 4.00000 0.190046 0.0950229 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −12.0000 −0.568216
\(447\) 0 0
\(448\) 0 0
\(449\) 30.0000 1.41579 0.707894 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) 0 0
\(451\) 40.0000 1.88353
\(452\) −14.0000 −0.658505
\(453\) 0 0
\(454\) −20.0000 −0.938647
\(455\) 0 0
\(456\) 0 0
\(457\) 6.00000 0.280668 0.140334 0.990104i \(-0.455182\pi\)
0.140334 + 0.990104i \(0.455182\pi\)
\(458\) −10.0000 −0.467269
\(459\) 0 0
\(460\) 0 0
\(461\) 38.0000 1.76984 0.884918 0.465746i \(-0.154214\pi\)
0.884918 + 0.465746i \(0.154214\pi\)
\(462\) 0 0
\(463\) −8.00000 −0.371792 −0.185896 0.982569i \(-0.559519\pi\)
−0.185896 + 0.982569i \(0.559519\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 18.0000 0.833834
\(467\) −28.0000 −1.29569 −0.647843 0.761774i \(-0.724329\pi\)
−0.647843 + 0.761774i \(0.724329\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 12.0000 0.552345
\(473\) −16.0000 −0.735681
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) −12.0000 −0.548867
\(479\) −20.0000 −0.913823 −0.456912 0.889512i \(-0.651044\pi\)
−0.456912 + 0.889512i \(0.651044\pi\)
\(480\) 0 0
\(481\) −12.0000 −0.547153
\(482\) 26.0000 1.18427
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) 0 0
\(486\) 0 0
\(487\) 20.0000 0.906287 0.453143 0.891438i \(-0.350303\pi\)
0.453143 + 0.891438i \(0.350303\pi\)
\(488\) −2.00000 −0.0905357
\(489\) 0 0
\(490\) 0 0
\(491\) 36.0000 1.62466 0.812329 0.583200i \(-0.198200\pi\)
0.812329 + 0.583200i \(0.198200\pi\)
\(492\) 0 0
\(493\) −12.0000 −0.540453
\(494\) 2.00000 0.0899843
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) 0 0
\(498\) 0 0
\(499\) −36.0000 −1.61158 −0.805791 0.592200i \(-0.798259\pi\)
−0.805791 + 0.592200i \(0.798259\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 20.0000 0.892644
\(503\) −36.0000 −1.60516 −0.802580 0.596544i \(-0.796540\pi\)
−0.802580 + 0.596544i \(0.796540\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −16.0000 −0.711287
\(507\) 0 0
\(508\) 4.00000 0.177471
\(509\) 18.0000 0.797836 0.398918 0.916987i \(-0.369386\pi\)
0.398918 + 0.916987i \(0.369386\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 2.00000 0.0882162
\(515\) 0 0
\(516\) 0 0
\(517\) 48.0000 2.11104
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 0 0
\(523\) −28.0000 −1.22435 −0.612177 0.790721i \(-0.709706\pi\)
−0.612177 + 0.790721i \(0.709706\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) −4.00000 −0.174408
\(527\) 8.00000 0.348485
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 20.0000 0.866296
\(534\) 0 0
\(535\) 0 0
\(536\) −4.00000 −0.172774
\(537\) 0 0
\(538\) 2.00000 0.0862261
\(539\) 28.0000 1.20605
\(540\) 0 0
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) 8.00000 0.343629
\(543\) 0 0
\(544\) 2.00000 0.0857493
\(545\) 0 0
\(546\) 0 0
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) 10.0000 0.427179
\(549\) 0 0
\(550\) 0 0
\(551\) 6.00000 0.255609
\(552\) 0 0
\(553\) 0 0
\(554\) 2.00000 0.0849719
\(555\) 0 0
\(556\) −20.0000 −0.848189
\(557\) 42.0000 1.77960 0.889799 0.456354i \(-0.150845\pi\)
0.889799 + 0.456354i \(0.150845\pi\)
\(558\) 0 0
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) 6.00000 0.253095
\(563\) 4.00000 0.168580 0.0842900 0.996441i \(-0.473138\pi\)
0.0842900 + 0.996441i \(0.473138\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 28.0000 1.17693
\(567\) 0 0
\(568\) −8.00000 −0.335673
\(569\) 38.0000 1.59304 0.796521 0.604610i \(-0.206671\pi\)
0.796521 + 0.604610i \(0.206671\pi\)
\(570\) 0 0
\(571\) −44.0000 −1.84134 −0.920671 0.390339i \(-0.872358\pi\)
−0.920671 + 0.390339i \(0.872358\pi\)
\(572\) 8.00000 0.334497
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 30.0000 1.24892 0.624458 0.781058i \(-0.285320\pi\)
0.624458 + 0.781058i \(0.285320\pi\)
\(578\) −13.0000 −0.540729
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −24.0000 −0.993978
\(584\) 6.00000 0.248282
\(585\) 0 0
\(586\) −2.00000 −0.0826192
\(587\) −28.0000 −1.15568 −0.577842 0.816149i \(-0.696105\pi\)
−0.577842 + 0.816149i \(0.696105\pi\)
\(588\) 0 0
\(589\) −4.00000 −0.164817
\(590\) 0 0
\(591\) 0 0
\(592\) 6.00000 0.246598
\(593\) 10.0000 0.410651 0.205325 0.978694i \(-0.434175\pi\)
0.205325 + 0.978694i \(0.434175\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −18.0000 −0.737309
\(597\) 0 0
\(598\) −8.00000 −0.327144
\(599\) −16.0000 −0.653742 −0.326871 0.945069i \(-0.605994\pi\)
−0.326871 + 0.945069i \(0.605994\pi\)
\(600\) 0 0
\(601\) −38.0000 −1.55005 −0.775026 0.631929i \(-0.782263\pi\)
−0.775026 + 0.631929i \(0.782263\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −12.0000 −0.488273
\(605\) 0 0
\(606\) 0 0
\(607\) −4.00000 −0.162355 −0.0811775 0.996700i \(-0.525868\pi\)
−0.0811775 + 0.996700i \(0.525868\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 0 0
\(610\) 0 0
\(611\) 24.0000 0.970936
\(612\) 0 0
\(613\) −22.0000 −0.888572 −0.444286 0.895885i \(-0.646543\pi\)
−0.444286 + 0.895885i \(0.646543\pi\)
\(614\) −4.00000 −0.161427
\(615\) 0 0
\(616\) 0 0
\(617\) −14.0000 −0.563619 −0.281809 0.959470i \(-0.590935\pi\)
−0.281809 + 0.959470i \(0.590935\pi\)
\(618\) 0 0
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 28.0000 1.12270
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 6.00000 0.239808
\(627\) 0 0
\(628\) 2.00000 0.0798087
\(629\) 12.0000 0.478471
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) −4.00000 −0.159111
\(633\) 0 0
\(634\) −26.0000 −1.03259
\(635\) 0 0
\(636\) 0 0
\(637\) 14.0000 0.554700
\(638\) 24.0000 0.950169
\(639\) 0 0
\(640\) 0 0
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 0 0
\(643\) 4.00000 0.157745 0.0788723 0.996885i \(-0.474868\pi\)
0.0788723 + 0.996885i \(0.474868\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −2.00000 −0.0786889
\(647\) 36.0000 1.41531 0.707653 0.706560i \(-0.249754\pi\)
0.707653 + 0.706560i \(0.249754\pi\)
\(648\) 0 0
\(649\) −48.0000 −1.88416
\(650\) 0 0
\(651\) 0 0
\(652\) 4.00000 0.156652
\(653\) −6.00000 −0.234798 −0.117399 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −10.0000 −0.390434
\(657\) 0 0
\(658\) 0 0
\(659\) 12.0000 0.467454 0.233727 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 0 0
\(661\) 10.0000 0.388955 0.194477 0.980907i \(-0.437699\pi\)
0.194477 + 0.980907i \(0.437699\pi\)
\(662\) 28.0000 1.08825
\(663\) 0 0
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) 0 0
\(667\) −24.0000 −0.929284
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 8.00000 0.308837
\(672\) 0 0
\(673\) 14.0000 0.539660 0.269830 0.962908i \(-0.413032\pi\)
0.269830 + 0.962908i \(0.413032\pi\)
\(674\) −26.0000 −1.00148
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) −42.0000 −1.61419 −0.807096 0.590421i \(-0.798962\pi\)
−0.807096 + 0.590421i \(0.798962\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) −16.0000 −0.612672
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 4.00000 0.152499
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) −4.00000 −0.152167 −0.0760836 0.997101i \(-0.524242\pi\)
−0.0760836 + 0.997101i \(0.524242\pi\)
\(692\) −10.0000 −0.380143
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) 0 0
\(696\) 0 0
\(697\) −20.0000 −0.757554
\(698\) −10.0000 −0.378506
\(699\) 0 0
\(700\) 0 0
\(701\) −18.0000 −0.679851 −0.339925 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) 0 0
\(703\) −6.00000 −0.226294
\(704\) −4.00000 −0.150756
\(705\) 0 0
\(706\) 26.0000 0.978523
\(707\) 0 0
\(708\) 0 0
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −10.0000 −0.374766
\(713\) 16.0000 0.599205
\(714\) 0 0
\(715\) 0 0
\(716\) −20.0000 −0.747435
\(717\) 0 0
\(718\) −12.0000 −0.447836
\(719\) −28.0000 −1.04422 −0.522112 0.852877i \(-0.674856\pi\)
−0.522112 + 0.852877i \(0.674856\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 1.00000 0.0372161
\(723\) 0 0
\(724\) 18.0000 0.668965
\(725\) 0 0
\(726\) 0 0
\(727\) −40.0000 −1.48352 −0.741759 0.670667i \(-0.766008\pi\)
−0.741759 + 0.670667i \(0.766008\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 8.00000 0.295891
\(732\) 0 0
\(733\) −46.0000 −1.69905 −0.849524 0.527549i \(-0.823111\pi\)
−0.849524 + 0.527549i \(0.823111\pi\)
\(734\) −24.0000 −0.885856
\(735\) 0 0
\(736\) 4.00000 0.147442
\(737\) 16.0000 0.589368
\(738\) 0 0
\(739\) 52.0000 1.91285 0.956425 0.291977i \(-0.0943129\pi\)
0.956425 + 0.291977i \(0.0943129\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 16.0000 0.586983 0.293492 0.955962i \(-0.405183\pi\)
0.293492 + 0.955962i \(0.405183\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 22.0000 0.805477
\(747\) 0 0
\(748\) −8.00000 −0.292509
\(749\) 0 0
\(750\) 0 0
\(751\) −4.00000 −0.145962 −0.0729810 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) −12.0000 −0.437595
\(753\) 0 0
\(754\) 12.0000 0.437014
\(755\) 0 0
\(756\) 0 0
\(757\) −38.0000 −1.38113 −0.690567 0.723269i \(-0.742639\pi\)
−0.690567 + 0.723269i \(0.742639\pi\)
\(758\) −20.0000 −0.726433
\(759\) 0 0
\(760\) 0 0
\(761\) −18.0000 −0.652499 −0.326250 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −20.0000 −0.723575
\(765\) 0 0
\(766\) 0 0
\(767\) −24.0000 −0.866590
\(768\) 0 0
\(769\) −46.0000 −1.65880 −0.829401 0.558653i \(-0.811318\pi\)
−0.829401 + 0.558653i \(0.811318\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −2.00000 −0.0719816
\(773\) 30.0000 1.07903 0.539513 0.841978i \(-0.318609\pi\)
0.539513 + 0.841978i \(0.318609\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −2.00000 −0.0717958
\(777\) 0 0
\(778\) 22.0000 0.788738
\(779\) 10.0000 0.358287
\(780\) 0 0
\(781\) 32.0000 1.14505
\(782\) 8.00000 0.286079
\(783\) 0 0
\(784\) −7.00000 −0.250000
\(785\) 0 0
\(786\) 0 0
\(787\) 52.0000 1.85360 0.926800 0.375555i \(-0.122548\pi\)
0.926800 + 0.375555i \(0.122548\pi\)
\(788\) 2.00000 0.0712470
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 4.00000 0.142044
\(794\) 18.0000 0.638796
\(795\) 0 0
\(796\) 0 0
\(797\) 54.0000 1.91278 0.956389 0.292096i \(-0.0943526\pi\)
0.956389 + 0.292096i \(0.0943526\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) 0 0
\(801\) 0 0
\(802\) 14.0000 0.494357
\(803\) −24.0000 −0.846942
\(804\) 0 0
\(805\) 0 0
\(806\) −8.00000 −0.281788
\(807\) 0 0
\(808\) −10.0000 −0.351799
\(809\) −34.0000 −1.19538 −0.597688 0.801729i \(-0.703914\pi\)
−0.597688 + 0.801729i \(0.703914\pi\)
\(810\) 0 0
\(811\) −20.0000 −0.702295 −0.351147 0.936320i \(-0.614208\pi\)
−0.351147 + 0.936320i \(0.614208\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −24.0000 −0.841200
\(815\) 0 0
\(816\) 0 0
\(817\) −4.00000 −0.139942
\(818\) 18.0000 0.629355
\(819\) 0 0
\(820\) 0 0
\(821\) 22.0000 0.767805 0.383903 0.923374i \(-0.374580\pi\)
0.383903 + 0.923374i \(0.374580\pi\)
\(822\) 0 0
\(823\) −8.00000 −0.278862 −0.139431 0.990232i \(-0.544527\pi\)
−0.139431 + 0.990232i \(0.544527\pi\)
\(824\) 4.00000 0.139347
\(825\) 0 0
\(826\) 0 0
\(827\) 36.0000 1.25184 0.625921 0.779886i \(-0.284723\pi\)
0.625921 + 0.779886i \(0.284723\pi\)
\(828\) 0 0
\(829\) 10.0000 0.347314 0.173657 0.984806i \(-0.444442\pi\)
0.173657 + 0.984806i \(0.444442\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −2.00000 −0.0693375
\(833\) −14.0000 −0.485071
\(834\) 0 0
\(835\) 0 0
\(836\) 4.00000 0.138343
\(837\) 0 0
\(838\) −12.0000 −0.414533
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 26.0000 0.896019
\(843\) 0 0
\(844\) −4.00000 −0.137686
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 6.00000 0.206041
\(849\) 0 0
\(850\) 0 0
\(851\) 24.0000 0.822709
\(852\) 0 0
\(853\) −14.0000 −0.479351 −0.239675 0.970853i \(-0.577041\pi\)
−0.239675 + 0.970853i \(0.577041\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −4.00000 −0.136717
\(857\) 26.0000 0.888143 0.444072 0.895991i \(-0.353534\pi\)
0.444072 + 0.895991i \(0.353534\pi\)
\(858\) 0 0
\(859\) 36.0000 1.22830 0.614152 0.789188i \(-0.289498\pi\)
0.614152 + 0.789188i \(0.289498\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −8.00000 −0.272481
\(863\) 8.00000 0.272323 0.136162 0.990687i \(-0.456523\pi\)
0.136162 + 0.990687i \(0.456523\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −34.0000 −1.15537
\(867\) 0 0
\(868\) 0 0
\(869\) 16.0000 0.542763
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) −6.00000 −0.203186
\(873\) 0 0
\(874\) −4.00000 −0.135302
\(875\) 0 0
\(876\) 0 0
\(877\) 46.0000 1.55331 0.776655 0.629926i \(-0.216915\pi\)
0.776655 + 0.629926i \(0.216915\pi\)
\(878\) 12.0000 0.404980
\(879\) 0 0
\(880\) 0 0
\(881\) 30.0000 1.01073 0.505363 0.862907i \(-0.331359\pi\)
0.505363 + 0.862907i \(0.331359\pi\)
\(882\) 0 0
\(883\) −20.0000 −0.673054 −0.336527 0.941674i \(-0.609252\pi\)
−0.336527 + 0.941674i \(0.609252\pi\)
\(884\) −4.00000 −0.134535
\(885\) 0 0
\(886\) 4.00000 0.134383
\(887\) −48.0000 −1.61168 −0.805841 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) −12.0000 −0.401790
\(893\) 12.0000 0.401565
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 30.0000 1.00111
\(899\) −24.0000 −0.800445
\(900\) 0 0
\(901\) 12.0000 0.399778
\(902\) 40.0000 1.33185
\(903\) 0 0
\(904\) −14.0000 −0.465633
\(905\) 0 0
\(906\) 0 0
\(907\) −28.0000 −0.929725 −0.464862 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) −20.0000 −0.663723
\(909\) 0 0
\(910\) 0 0
\(911\) −24.0000 −0.795155 −0.397578 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) 0 0
\(913\) 48.0000 1.58857
\(914\) 6.00000 0.198462
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) 0 0
\(918\) 0 0
\(919\) 8.00000 0.263896 0.131948 0.991257i \(-0.457877\pi\)
0.131948 + 0.991257i \(0.457877\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 38.0000 1.25146
\(923\) 16.0000 0.526646
\(924\) 0 0
\(925\) 0 0
\(926\) −8.00000 −0.262896
\(927\) 0 0
\(928\) −6.00000 −0.196960
\(929\) 6.00000 0.196854 0.0984268 0.995144i \(-0.468619\pi\)
0.0984268 + 0.995144i \(0.468619\pi\)
\(930\) 0 0
\(931\) 7.00000 0.229416
\(932\) 18.0000 0.589610
\(933\) 0 0
\(934\) −28.0000 −0.916188
\(935\) 0 0
\(936\) 0 0
\(937\) −10.0000 −0.326686 −0.163343 0.986569i \(-0.552228\pi\)
−0.163343 + 0.986569i \(0.552228\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −30.0000 −0.977972 −0.488986 0.872292i \(-0.662633\pi\)
−0.488986 + 0.872292i \(0.662633\pi\)
\(942\) 0 0
\(943\) −40.0000 −1.30258
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) −16.0000 −0.520205
\(947\) 52.0000 1.68977 0.844886 0.534946i \(-0.179668\pi\)
0.844886 + 0.534946i \(0.179668\pi\)
\(948\) 0 0
\(949\) −12.0000 −0.389536
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 42.0000 1.36051 0.680257 0.732974i \(-0.261868\pi\)
0.680257 + 0.732974i \(0.261868\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −12.0000 −0.388108
\(957\) 0 0
\(958\) −20.0000 −0.646171
\(959\) 0 0
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) −12.0000 −0.386896
\(963\) 0 0
\(964\) 26.0000 0.837404
\(965\) 0 0
\(966\) 0 0
\(967\) 32.0000 1.02905 0.514525 0.857475i \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(968\) 5.00000 0.160706
\(969\) 0 0
\(970\) 0 0
\(971\) 36.0000 1.15529 0.577647 0.816286i \(-0.303971\pi\)
0.577647 + 0.816286i \(0.303971\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 20.0000 0.640841
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) 2.00000 0.0639857 0.0319928 0.999488i \(-0.489815\pi\)
0.0319928 + 0.999488i \(0.489815\pi\)
\(978\) 0 0
\(979\) 40.0000 1.27841
\(980\) 0 0
\(981\) 0 0
\(982\) 36.0000 1.14881
\(983\) 56.0000 1.78612 0.893061 0.449935i \(-0.148553\pi\)
0.893061 + 0.449935i \(0.148553\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −12.0000 −0.382158
\(987\) 0 0
\(988\) 2.00000 0.0636285
\(989\) 16.0000 0.508770
\(990\) 0 0
\(991\) 4.00000 0.127064 0.0635321 0.997980i \(-0.479763\pi\)
0.0635321 + 0.997980i \(0.479763\pi\)
\(992\) 4.00000 0.127000
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −38.0000 −1.20347 −0.601736 0.798695i \(-0.705524\pi\)
−0.601736 + 0.798695i \(0.705524\pi\)
\(998\) −36.0000 −1.13956
\(999\) 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.x.1.1 1
3.2 odd 2 2850.2.a.m.1.1 1
5.4 even 2 1710.2.a.i.1.1 1
15.2 even 4 2850.2.d.i.799.1 2
15.8 even 4 2850.2.d.i.799.2 2
15.14 odd 2 570.2.a.g.1.1 1
60.59 even 2 4560.2.a.s.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
570.2.a.g.1.1 1 15.14 odd 2
1710.2.a.i.1.1 1 5.4 even 2
2850.2.a.m.1.1 1 3.2 odd 2
2850.2.d.i.799.1 2 15.2 even 4
2850.2.d.i.799.2 2 15.8 even 4
4560.2.a.s.1.1 1 60.59 even 2
8550.2.a.x.1.1 1 1.1 even 1 trivial