Properties

Label 8550.2.a.cm.1.3
Level $8550$
Weight $2$
Character 8550.1
Self dual yes
Analytic conductor $68.272$
Analytic rank $1$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.788.1
Defining polynomial: \(x^{3} - x^{2} - 7 x - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(3.35386\) of defining polynomial
Character \(\chi\) \(=\) 8550.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +2.35386 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +2.35386 q^{7} +1.00000 q^{8} -4.89449 q^{11} +1.89449 q^{13} +2.35386 q^{14} +1.00000 q^{16} -2.35386 q^{17} -1.00000 q^{19} -4.89449 q^{22} -6.16707 q^{23} +1.89449 q^{26} +2.35386 q^{28} -4.27258 q^{29} +7.70771 q^{31} +1.00000 q^{32} -2.35386 q^{34} -7.78899 q^{37} -1.00000 q^{38} -3.54064 q^{41} +1.73194 q^{43} -4.89449 q^{44} -6.16707 q^{46} -5.32962 q^{47} -1.45936 q^{49} +1.89449 q^{52} -1.97577 q^{53} +2.35386 q^{56} -4.27258 q^{58} -6.81322 q^{59} +3.97577 q^{61} +7.70771 q^{62} +1.00000 q^{64} -5.19131 q^{67} -2.35386 q^{68} +4.14284 q^{71} -4.37809 q^{73} -7.78899 q^{74} -1.00000 q^{76} -11.5209 q^{77} +9.49670 q^{79} -3.54064 q^{82} -4.43513 q^{83} +1.73194 q^{86} -4.89449 q^{88} -3.62191 q^{89} +4.45936 q^{91} -6.16707 q^{92} -5.32962 q^{94} -7.06157 q^{97} -1.45936 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} + 3q^{4} - 2q^{7} + 3q^{8} + O(q^{10}) \) \( 3q + 3q^{2} + 3q^{4} - 2q^{7} + 3q^{8} - 5q^{11} - 4q^{13} - 2q^{14} + 3q^{16} + 2q^{17} - 3q^{19} - 5q^{22} - q^{23} - 4q^{26} - 2q^{28} - 5q^{29} + 5q^{31} + 3q^{32} + 2q^{34} - 4q^{37} - 3q^{38} - 10q^{41} - 2q^{43} - 5q^{44} - q^{46} + 4q^{47} - 5q^{49} - 4q^{52} + 5q^{53} - 2q^{56} - 5q^{58} - 12q^{59} + q^{61} + 5q^{62} + 3q^{64} - 9q^{67} + 2q^{68} - 16q^{71} - 15q^{73} - 4q^{74} - 3q^{76} - 8q^{77} - 9q^{79} - 10q^{82} - 3q^{83} - 2q^{86} - 5q^{88} - 9q^{89} + 14q^{91} - q^{92} + 4q^{94} + 6q^{97} - 5q^{98} + O(q^{100}) \)

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
\(7\) 2.35386 0.889674 0.444837 0.895612i \(-0.353262\pi\)
0.444837 + 0.895612i \(0.353262\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) −4.89449 −1.47575 −0.737873 0.674940i \(-0.764169\pi\)
−0.737873 + 0.674940i \(0.764169\pi\)
\(12\) 0 0
\(13\) 1.89449 0.525438 0.262719 0.964872i \(-0.415381\pi\)
0.262719 + 0.964872i \(0.415381\pi\)
\(14\) 2.35386 0.629094
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −2.35386 −0.570894 −0.285447 0.958395i \(-0.592142\pi\)
−0.285447 + 0.958395i \(0.592142\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) −4.89449 −1.04351
\(23\) −6.16707 −1.28592 −0.642962 0.765898i \(-0.722295\pi\)
−0.642962 + 0.765898i \(0.722295\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 1.89449 0.371541
\(27\) 0 0
\(28\) 2.35386 0.444837
\(29\) −4.27258 −0.793398 −0.396699 0.917949i \(-0.629844\pi\)
−0.396699 + 0.917949i \(0.629844\pi\)
\(30\) 0 0
\(31\) 7.70771 1.38435 0.692173 0.721732i \(-0.256654\pi\)
0.692173 + 0.721732i \(0.256654\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −2.35386 −0.403683
\(35\) 0 0
\(36\) 0 0
\(37\) −7.78899 −1.28050 −0.640251 0.768166i \(-0.721170\pi\)
−0.640251 + 0.768166i \(0.721170\pi\)
\(38\) −1.00000 −0.162221
\(39\) 0 0
\(40\) 0 0
\(41\) −3.54064 −0.552955 −0.276477 0.961020i \(-0.589167\pi\)
−0.276477 + 0.961020i \(0.589167\pi\)
\(42\) 0 0
\(43\) 1.73194 0.264119 0.132060 0.991242i \(-0.457841\pi\)
0.132060 + 0.991242i \(0.457841\pi\)
\(44\) −4.89449 −0.737873
\(45\) 0 0
\(46\) −6.16707 −0.909286
\(47\) −5.32962 −0.777405 −0.388703 0.921363i \(-0.627077\pi\)
−0.388703 + 0.921363i \(0.627077\pi\)
\(48\) 0 0
\(49\) −1.45936 −0.208480
\(50\) 0 0
\(51\) 0 0
\(52\) 1.89449 0.262719
\(53\) −1.97577 −0.271392 −0.135696 0.990750i \(-0.543327\pi\)
−0.135696 + 0.990750i \(0.543327\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 2.35386 0.314547
\(57\) 0 0
\(58\) −4.27258 −0.561017
\(59\) −6.81322 −0.887006 −0.443503 0.896273i \(-0.646264\pi\)
−0.443503 + 0.896273i \(0.646264\pi\)
\(60\) 0 0
\(61\) 3.97577 0.509045 0.254522 0.967067i \(-0.418082\pi\)
0.254522 + 0.967067i \(0.418082\pi\)
\(62\) 7.70771 0.978880
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −5.19131 −0.634219 −0.317110 0.948389i \(-0.602712\pi\)
−0.317110 + 0.948389i \(0.602712\pi\)
\(68\) −2.35386 −0.285447
\(69\) 0 0
\(70\) 0 0
\(71\) 4.14284 0.491665 0.245832 0.969312i \(-0.420939\pi\)
0.245832 + 0.969312i \(0.420939\pi\)
\(72\) 0 0
\(73\) −4.37809 −0.512417 −0.256208 0.966622i \(-0.582473\pi\)
−0.256208 + 0.966622i \(0.582473\pi\)
\(74\) −7.78899 −0.905451
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) −11.5209 −1.31293
\(78\) 0 0
\(79\) 9.49670 1.06846 0.534231 0.845339i \(-0.320601\pi\)
0.534231 + 0.845339i \(0.320601\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −3.54064 −0.390998
\(83\) −4.43513 −0.486819 −0.243409 0.969924i \(-0.578266\pi\)
−0.243409 + 0.969924i \(0.578266\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 1.73194 0.186760
\(87\) 0 0
\(88\) −4.89449 −0.521755
\(89\) −3.62191 −0.383922 −0.191961 0.981403i \(-0.561485\pi\)
−0.191961 + 0.981403i \(0.561485\pi\)
\(90\) 0 0
\(91\) 4.45936 0.467468
\(92\) −6.16707 −0.642962
\(93\) 0 0
\(94\) −5.32962 −0.549709
\(95\) 0 0
\(96\) 0 0
\(97\) −7.06157 −0.716994 −0.358497 0.933531i \(-0.616711\pi\)
−0.358497 + 0.933531i \(0.616711\pi\)
\(98\) −1.45936 −0.147418
\(99\) 0 0
\(100\) 0 0
\(101\) −6.10551 −0.607521 −0.303760 0.952748i \(-0.598242\pi\)
−0.303760 + 0.952748i \(0.598242\pi\)
\(102\) 0 0
\(103\) −11.2483 −1.10833 −0.554166 0.832406i \(-0.686963\pi\)
−0.554166 + 0.832406i \(0.686963\pi\)
\(104\) 1.89449 0.185770
\(105\) 0 0
\(106\) −1.97577 −0.191903
\(107\) 15.3099 1.48007 0.740033 0.672571i \(-0.234810\pi\)
0.740033 + 0.672571i \(0.234810\pi\)
\(108\) 0 0
\(109\) −14.7077 −1.40874 −0.704372 0.709831i \(-0.748771\pi\)
−0.704372 + 0.709831i \(0.748771\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 2.35386 0.222418
\(113\) −3.29681 −0.310138 −0.155069 0.987904i \(-0.549560\pi\)
−0.155069 + 0.987904i \(0.549560\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −4.27258 −0.396699
\(117\) 0 0
\(118\) −6.81322 −0.627208
\(119\) −5.54064 −0.507909
\(120\) 0 0
\(121\) 12.9561 1.17782
\(122\) 3.97577 0.359949
\(123\) 0 0
\(124\) 7.70771 0.692173
\(125\) 0 0
\(126\) 0 0
\(127\) 6.62644 0.588001 0.294001 0.955805i \(-0.405013\pi\)
0.294001 + 0.955805i \(0.405013\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −16.9803 −1.48358 −0.741788 0.670635i \(-0.766022\pi\)
−0.741788 + 0.670635i \(0.766022\pi\)
\(132\) 0 0
\(133\) −2.35386 −0.204105
\(134\) −5.19131 −0.448461
\(135\) 0 0
\(136\) −2.35386 −0.201841
\(137\) 12.8703 1.09958 0.549790 0.835303i \(-0.314708\pi\)
0.549790 + 0.835303i \(0.314708\pi\)
\(138\) 0 0
\(139\) 9.30992 0.789657 0.394828 0.918755i \(-0.370804\pi\)
0.394828 + 0.918755i \(0.370804\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 4.14284 0.347660
\(143\) −9.27258 −0.775412
\(144\) 0 0
\(145\) 0 0
\(146\) −4.37809 −0.362333
\(147\) 0 0
\(148\) −7.78899 −0.640251
\(149\) 16.1231 1.32086 0.660429 0.750888i \(-0.270374\pi\)
0.660429 + 0.750888i \(0.270374\pi\)
\(150\) 0 0
\(151\) −15.6638 −1.27470 −0.637350 0.770575i \(-0.719969\pi\)
−0.637350 + 0.770575i \(0.719969\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 0 0
\(154\) −11.5209 −0.928383
\(155\) 0 0
\(156\) 0 0
\(157\) 14.2483 1.13714 0.568571 0.822634i \(-0.307496\pi\)
0.568571 + 0.822634i \(0.307496\pi\)
\(158\) 9.49670 0.755517
\(159\) 0 0
\(160\) 0 0
\(161\) −14.5164 −1.14405
\(162\) 0 0
\(163\) −6.00000 −0.469956 −0.234978 0.972001i \(-0.575502\pi\)
−0.234978 + 0.972001i \(0.575502\pi\)
\(164\) −3.54064 −0.276477
\(165\) 0 0
\(166\) −4.43513 −0.344233
\(167\) 0.756178 0.0585148 0.0292574 0.999572i \(-0.490686\pi\)
0.0292574 + 0.999572i \(0.490686\pi\)
\(168\) 0 0
\(169\) −9.41090 −0.723915
\(170\) 0 0
\(171\) 0 0
\(172\) 1.73194 0.132060
\(173\) 13.6880 1.04068 0.520340 0.853959i \(-0.325805\pi\)
0.520340 + 0.853959i \(0.325805\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −4.89449 −0.368936
\(177\) 0 0
\(178\) −3.62191 −0.271474
\(179\) −23.2044 −1.73438 −0.867189 0.497978i \(-0.834076\pi\)
−0.867189 + 0.497978i \(0.834076\pi\)
\(180\) 0 0
\(181\) 6.65067 0.494340 0.247170 0.968972i \(-0.420499\pi\)
0.247170 + 0.968972i \(0.420499\pi\)
\(182\) 4.45936 0.330550
\(183\) 0 0
\(184\) −6.16707 −0.454643
\(185\) 0 0
\(186\) 0 0
\(187\) 11.5209 0.842494
\(188\) −5.32962 −0.388703
\(189\) 0 0
\(190\) 0 0
\(191\) −14.7890 −1.07009 −0.535047 0.844822i \(-0.679706\pi\)
−0.535047 + 0.844822i \(0.679706\pi\)
\(192\) 0 0
\(193\) −11.5977 −0.834819 −0.417410 0.908718i \(-0.637062\pi\)
−0.417410 + 0.908718i \(0.637062\pi\)
\(194\) −7.06157 −0.506991
\(195\) 0 0
\(196\) −1.45936 −0.104240
\(197\) 12.6022 0.897870 0.448935 0.893564i \(-0.351803\pi\)
0.448935 + 0.893564i \(0.351803\pi\)
\(198\) 0 0
\(199\) −23.7405 −1.68292 −0.841460 0.540319i \(-0.818304\pi\)
−0.841460 + 0.540319i \(0.818304\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −6.10551 −0.429582
\(203\) −10.0570 −0.705866
\(204\) 0 0
\(205\) 0 0
\(206\) −11.2483 −0.783710
\(207\) 0 0
\(208\) 1.89449 0.131359
\(209\) 4.89449 0.338559
\(210\) 0 0
\(211\) 19.9848 1.37581 0.687906 0.725800i \(-0.258530\pi\)
0.687906 + 0.725800i \(0.258530\pi\)
\(212\) −1.97577 −0.135696
\(213\) 0 0
\(214\) 15.3099 1.04656
\(215\) 0 0
\(216\) 0 0
\(217\) 18.1428 1.23162
\(218\) −14.7077 −0.996132
\(219\) 0 0
\(220\) 0 0
\(221\) −4.45936 −0.299969
\(222\) 0 0
\(223\) −18.7122 −1.25306 −0.626532 0.779396i \(-0.715526\pi\)
−0.626532 + 0.779396i \(0.715526\pi\)
\(224\) 2.35386 0.157274
\(225\) 0 0
\(226\) −3.29681 −0.219301
\(227\) 13.0813 0.868235 0.434117 0.900856i \(-0.357060\pi\)
0.434117 + 0.900856i \(0.357060\pi\)
\(228\) 0 0
\(229\) −9.89902 −0.654146 −0.327073 0.944999i \(-0.606062\pi\)
−0.327073 + 0.944999i \(0.606062\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −4.27258 −0.280509
\(233\) 12.6395 0.828044 0.414022 0.910267i \(-0.364124\pi\)
0.414022 + 0.910267i \(0.364124\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −6.81322 −0.443503
\(237\) 0 0
\(238\) −5.54064 −0.359146
\(239\) 20.1605 1.30407 0.652036 0.758188i \(-0.273915\pi\)
0.652036 + 0.758188i \(0.273915\pi\)
\(240\) 0 0
\(241\) −11.7693 −0.758126 −0.379063 0.925371i \(-0.623754\pi\)
−0.379063 + 0.925371i \(0.623754\pi\)
\(242\) 12.9561 0.832847
\(243\) 0 0
\(244\) 3.97577 0.254522
\(245\) 0 0
\(246\) 0 0
\(247\) −1.89449 −0.120544
\(248\) 7.70771 0.489440
\(249\) 0 0
\(250\) 0 0
\(251\) 18.3715 1.15960 0.579799 0.814760i \(-0.303131\pi\)
0.579799 + 0.814760i \(0.303131\pi\)
\(252\) 0 0
\(253\) 30.1847 1.89770
\(254\) 6.62644 0.415780
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 3.83293 0.239091 0.119546 0.992829i \(-0.461856\pi\)
0.119546 + 0.992829i \(0.461856\pi\)
\(258\) 0 0
\(259\) −18.3341 −1.13923
\(260\) 0 0
\(261\) 0 0
\(262\) −16.9803 −1.04905
\(263\) −21.2483 −1.31023 −0.655115 0.755529i \(-0.727380\pi\)
−0.655115 + 0.755529i \(0.727380\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −2.35386 −0.144324
\(267\) 0 0
\(268\) −5.19131 −0.317110
\(269\) −5.78899 −0.352961 −0.176480 0.984304i \(-0.556471\pi\)
−0.176480 + 0.984304i \(0.556471\pi\)
\(270\) 0 0
\(271\) 4.01971 0.244180 0.122090 0.992519i \(-0.461040\pi\)
0.122090 + 0.992519i \(0.461040\pi\)
\(272\) −2.35386 −0.142723
\(273\) 0 0
\(274\) 12.8703 0.777521
\(275\) 0 0
\(276\) 0 0
\(277\) −12.0989 −0.726953 −0.363476 0.931603i \(-0.618410\pi\)
−0.363476 + 0.931603i \(0.618410\pi\)
\(278\) 9.30992 0.558372
\(279\) 0 0
\(280\) 0 0
\(281\) −9.21101 −0.549483 −0.274742 0.961518i \(-0.588592\pi\)
−0.274742 + 0.961518i \(0.588592\pi\)
\(282\) 0 0
\(283\) 8.49670 0.505076 0.252538 0.967587i \(-0.418735\pi\)
0.252538 + 0.967587i \(0.418735\pi\)
\(284\) 4.14284 0.245832
\(285\) 0 0
\(286\) −9.27258 −0.548299
\(287\) −8.33415 −0.491949
\(288\) 0 0
\(289\) −11.4594 −0.674080
\(290\) 0 0
\(291\) 0 0
\(292\) −4.37809 −0.256208
\(293\) 12.0989 0.706825 0.353413 0.935468i \(-0.385021\pi\)
0.353413 + 0.935468i \(0.385021\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −7.78899 −0.452726
\(297\) 0 0
\(298\) 16.1231 0.933988
\(299\) −11.6835 −0.675673
\(300\) 0 0
\(301\) 4.07675 0.234980
\(302\) −15.6638 −0.901349
\(303\) 0 0
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) 0 0
\(307\) 20.9318 1.19464 0.597321 0.802002i \(-0.296232\pi\)
0.597321 + 0.802002i \(0.296232\pi\)
\(308\) −11.5209 −0.656466
\(309\) 0 0
\(310\) 0 0
\(311\) −14.9561 −0.848080 −0.424040 0.905643i \(-0.639388\pi\)
−0.424040 + 0.905643i \(0.639388\pi\)
\(312\) 0 0
\(313\) 32.2902 1.82515 0.912575 0.408909i \(-0.134091\pi\)
0.912575 + 0.408909i \(0.134091\pi\)
\(314\) 14.2483 0.804081
\(315\) 0 0
\(316\) 9.49670 0.534231
\(317\) 24.2726 1.36328 0.681642 0.731686i \(-0.261266\pi\)
0.681642 + 0.731686i \(0.261266\pi\)
\(318\) 0 0
\(319\) 20.9121 1.17085
\(320\) 0 0
\(321\) 0 0
\(322\) −14.5164 −0.808968
\(323\) 2.35386 0.130972
\(324\) 0 0
\(325\) 0 0
\(326\) −6.00000 −0.332309
\(327\) 0 0
\(328\) −3.54064 −0.195499
\(329\) −12.5452 −0.691637
\(330\) 0 0
\(331\) 6.80869 0.374240 0.187120 0.982337i \(-0.440085\pi\)
0.187120 + 0.982337i \(0.440085\pi\)
\(332\) −4.43513 −0.243409
\(333\) 0 0
\(334\) 0.756178 0.0413762
\(335\) 0 0
\(336\) 0 0
\(337\) −16.1231 −0.878283 −0.439142 0.898418i \(-0.644717\pi\)
−0.439142 + 0.898418i \(0.644717\pi\)
\(338\) −9.41090 −0.511885
\(339\) 0 0
\(340\) 0 0
\(341\) −37.7253 −2.04294
\(342\) 0 0
\(343\) −19.9121 −1.07515
\(344\) 1.73194 0.0933802
\(345\) 0 0
\(346\) 13.6880 0.735872
\(347\) 23.3670 1.25440 0.627202 0.778857i \(-0.284200\pi\)
0.627202 + 0.778857i \(0.284200\pi\)
\(348\) 0 0
\(349\) 16.5977 0.888453 0.444227 0.895914i \(-0.353478\pi\)
0.444227 + 0.895914i \(0.353478\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −4.89449 −0.260877
\(353\) −5.32105 −0.283211 −0.141605 0.989923i \(-0.545226\pi\)
−0.141605 + 0.989923i \(0.545226\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −3.62191 −0.191961
\(357\) 0 0
\(358\) −23.2044 −1.22639
\(359\) −6.67038 −0.352049 −0.176025 0.984386i \(-0.556324\pi\)
−0.176025 + 0.984386i \(0.556324\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 6.65067 0.349551
\(363\) 0 0
\(364\) 4.45936 0.233734
\(365\) 0 0
\(366\) 0 0
\(367\) −8.14284 −0.425053 −0.212526 0.977155i \(-0.568169\pi\)
−0.212526 + 0.977155i \(0.568169\pi\)
\(368\) −6.16707 −0.321481
\(369\) 0 0
\(370\) 0 0
\(371\) −4.65067 −0.241451
\(372\) 0 0
\(373\) −3.94296 −0.204159 −0.102079 0.994776i \(-0.532550\pi\)
−0.102079 + 0.994776i \(0.532550\pi\)
\(374\) 11.5209 0.595733
\(375\) 0 0
\(376\) −5.32962 −0.274854
\(377\) −8.09438 −0.416882
\(378\) 0 0
\(379\) −33.2044 −1.70560 −0.852798 0.522241i \(-0.825096\pi\)
−0.852798 + 0.522241i \(0.825096\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −14.7890 −0.756670
\(383\) −19.3867 −0.990612 −0.495306 0.868719i \(-0.664944\pi\)
−0.495306 + 0.868719i \(0.664944\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −11.5977 −0.590306
\(387\) 0 0
\(388\) −7.06157 −0.358497
\(389\) 6.87026 0.348336 0.174168 0.984716i \(-0.444276\pi\)
0.174168 + 0.984716i \(0.444276\pi\)
\(390\) 0 0
\(391\) 14.5164 0.734126
\(392\) −1.45936 −0.0737090
\(393\) 0 0
\(394\) 12.6022 0.634890
\(395\) 0 0
\(396\) 0 0
\(397\) −21.8879 −1.09852 −0.549261 0.835651i \(-0.685091\pi\)
−0.549261 + 0.835651i \(0.685091\pi\)
\(398\) −23.7405 −1.19000
\(399\) 0 0
\(400\) 0 0
\(401\) −7.49670 −0.374367 −0.187184 0.982325i \(-0.559936\pi\)
−0.187184 + 0.982325i \(0.559936\pi\)
\(402\) 0 0
\(403\) 14.6022 0.727388
\(404\) −6.10551 −0.303760
\(405\) 0 0
\(406\) −10.0570 −0.499123
\(407\) 38.1231 1.88969
\(408\) 0 0
\(409\) 4.21101 0.208221 0.104111 0.994566i \(-0.466800\pi\)
0.104111 + 0.994566i \(0.466800\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −11.2483 −0.554166
\(413\) −16.0373 −0.789146
\(414\) 0 0
\(415\) 0 0
\(416\) 1.89449 0.0928852
\(417\) 0 0
\(418\) 4.89449 0.239397
\(419\) −10.8703 −0.531047 −0.265523 0.964104i \(-0.585545\pi\)
−0.265523 + 0.964104i \(0.585545\pi\)
\(420\) 0 0
\(421\) 33.7496 1.64485 0.822427 0.568871i \(-0.192620\pi\)
0.822427 + 0.568871i \(0.192620\pi\)
\(422\) 19.9848 0.972846
\(423\) 0 0
\(424\) −1.97577 −0.0959517
\(425\) 0 0
\(426\) 0 0
\(427\) 9.35838 0.452884
\(428\) 15.3099 0.740033
\(429\) 0 0
\(430\) 0 0
\(431\) 15.7011 0.756296 0.378148 0.925745i \(-0.376561\pi\)
0.378148 + 0.925745i \(0.376561\pi\)
\(432\) 0 0
\(433\) −15.1847 −0.729730 −0.364865 0.931060i \(-0.618885\pi\)
−0.364865 + 0.931060i \(0.618885\pi\)
\(434\) 18.1428 0.870884
\(435\) 0 0
\(436\) −14.7077 −0.704372
\(437\) 6.16707 0.295011
\(438\) 0 0
\(439\) −25.4482 −1.21458 −0.607289 0.794481i \(-0.707743\pi\)
−0.607289 + 0.794481i \(0.707743\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −4.45936 −0.212110
\(443\) −24.4351 −1.16095 −0.580474 0.814279i \(-0.697133\pi\)
−0.580474 + 0.814279i \(0.697133\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −18.7122 −0.886050
\(447\) 0 0
\(448\) 2.35386 0.111209
\(449\) −26.5012 −1.25067 −0.625335 0.780356i \(-0.715038\pi\)
−0.625335 + 0.780356i \(0.715038\pi\)
\(450\) 0 0
\(451\) 17.3296 0.816020
\(452\) −3.29681 −0.155069
\(453\) 0 0
\(454\) 13.0813 0.613935
\(455\) 0 0
\(456\) 0 0
\(457\) 36.3230 1.69912 0.849560 0.527493i \(-0.176868\pi\)
0.849560 + 0.527493i \(0.176868\pi\)
\(458\) −9.89902 −0.462551
\(459\) 0 0
\(460\) 0 0
\(461\) −23.0813 −1.07500 −0.537501 0.843263i \(-0.680632\pi\)
−0.537501 + 0.843263i \(0.680632\pi\)
\(462\) 0 0
\(463\) −31.0419 −1.44264 −0.721319 0.692603i \(-0.756464\pi\)
−0.721319 + 0.692603i \(0.756464\pi\)
\(464\) −4.27258 −0.198350
\(465\) 0 0
\(466\) 12.6395 0.585515
\(467\) −16.6925 −0.772438 −0.386219 0.922407i \(-0.626219\pi\)
−0.386219 + 0.922407i \(0.626219\pi\)
\(468\) 0 0
\(469\) −12.2196 −0.564248
\(470\) 0 0
\(471\) 0 0
\(472\) −6.81322 −0.313604
\(473\) −8.47699 −0.389772
\(474\) 0 0
\(475\) 0 0
\(476\) −5.54064 −0.253955
\(477\) 0 0
\(478\) 20.1605 0.922118
\(479\) 12.5891 0.575211 0.287605 0.957749i \(-0.407141\pi\)
0.287605 + 0.957749i \(0.407141\pi\)
\(480\) 0 0
\(481\) −14.7562 −0.672824
\(482\) −11.7693 −0.536076
\(483\) 0 0
\(484\) 12.9561 0.588912
\(485\) 0 0
\(486\) 0 0
\(487\) −20.7077 −0.938356 −0.469178 0.883104i \(-0.655450\pi\)
−0.469178 + 0.883104i \(0.655450\pi\)
\(488\) 3.97577 0.179975
\(489\) 0 0
\(490\) 0 0
\(491\) −20.0747 −0.905957 −0.452979 0.891521i \(-0.649639\pi\)
−0.452979 + 0.891521i \(0.649639\pi\)
\(492\) 0 0
\(493\) 10.0570 0.452946
\(494\) −1.89449 −0.0852373
\(495\) 0 0
\(496\) 7.70771 0.346086
\(497\) 9.75165 0.437421
\(498\) 0 0
\(499\) 41.5956 1.86207 0.931037 0.364924i \(-0.118905\pi\)
0.931037 + 0.364924i \(0.118905\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 18.3715 0.819959
\(503\) −16.0000 −0.713405 −0.356702 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 30.1847 1.34187
\(507\) 0 0
\(508\) 6.62644 0.294001
\(509\) −25.8879 −1.14746 −0.573730 0.819044i \(-0.694504\pi\)
−0.573730 + 0.819044i \(0.694504\pi\)
\(510\) 0 0
\(511\) −10.3054 −0.455884
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 3.83293 0.169063
\(515\) 0 0
\(516\) 0 0
\(517\) 26.0858 1.14725
\(518\) −18.3341 −0.805556
\(519\) 0 0
\(520\) 0 0
\(521\) 24.5386 1.07505 0.537527 0.843247i \(-0.319359\pi\)
0.537527 + 0.843247i \(0.319359\pi\)
\(522\) 0 0
\(523\) 4.21101 0.184135 0.0920674 0.995753i \(-0.470652\pi\)
0.0920674 + 0.995753i \(0.470652\pi\)
\(524\) −16.9803 −0.741788
\(525\) 0 0
\(526\) −21.2483 −0.926472
\(527\) −18.1428 −0.790315
\(528\) 0 0
\(529\) 15.0328 0.653600
\(530\) 0 0
\(531\) 0 0
\(532\) −2.35386 −0.102053
\(533\) −6.70771 −0.290543
\(534\) 0 0
\(535\) 0 0
\(536\) −5.19131 −0.224230
\(537\) 0 0
\(538\) −5.78899 −0.249581
\(539\) 7.14284 0.307664
\(540\) 0 0
\(541\) −12.7208 −0.546910 −0.273455 0.961885i \(-0.588167\pi\)
−0.273455 + 0.961885i \(0.588167\pi\)
\(542\) 4.01971 0.172661
\(543\) 0 0
\(544\) −2.35386 −0.100921
\(545\) 0 0
\(546\) 0 0
\(547\) 5.97577 0.255505 0.127753 0.991806i \(-0.459224\pi\)
0.127753 + 0.991806i \(0.459224\pi\)
\(548\) 12.8703 0.549790
\(549\) 0 0
\(550\) 0 0
\(551\) 4.27258 0.182018
\(552\) 0 0
\(553\) 22.3539 0.950583
\(554\) −12.0989 −0.514033
\(555\) 0 0
\(556\) 9.30992 0.394828
\(557\) 29.6441 1.25606 0.628030 0.778189i \(-0.283862\pi\)
0.628030 + 0.778189i \(0.283862\pi\)
\(558\) 0 0
\(559\) 3.28116 0.138778
\(560\) 0 0
\(561\) 0 0
\(562\) −9.21101 −0.388543
\(563\) 24.3427 1.02592 0.512962 0.858411i \(-0.328548\pi\)
0.512962 + 0.858411i \(0.328548\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 8.49670 0.357143
\(567\) 0 0
\(568\) 4.14284 0.173830
\(569\) 23.6264 0.990472 0.495236 0.868759i \(-0.335082\pi\)
0.495236 + 0.868759i \(0.335082\pi\)
\(570\) 0 0
\(571\) 8.81322 0.368822 0.184411 0.982849i \(-0.440962\pi\)
0.184411 + 0.982849i \(0.440962\pi\)
\(572\) −9.27258 −0.387706
\(573\) 0 0
\(574\) −8.33415 −0.347861
\(575\) 0 0
\(576\) 0 0
\(577\) 36.9121 1.53667 0.768336 0.640047i \(-0.221085\pi\)
0.768336 + 0.640047i \(0.221085\pi\)
\(578\) −11.4594 −0.476647
\(579\) 0 0
\(580\) 0 0
\(581\) −10.4397 −0.433110
\(582\) 0 0
\(583\) 9.67038 0.400506
\(584\) −4.37809 −0.181167
\(585\) 0 0
\(586\) 12.0989 0.499801
\(587\) −15.9737 −0.659305 −0.329652 0.944102i \(-0.606932\pi\)
−0.329652 + 0.944102i \(0.606932\pi\)
\(588\) 0 0
\(589\) −7.70771 −0.317591
\(590\) 0 0
\(591\) 0 0
\(592\) −7.78899 −0.320125
\(593\) 4.22007 0.173297 0.0866487 0.996239i \(-0.472384\pi\)
0.0866487 + 0.996239i \(0.472384\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 16.1231 0.660429
\(597\) 0 0
\(598\) −11.6835 −0.477773
\(599\) 21.6552 0.884807 0.442404 0.896816i \(-0.354126\pi\)
0.442404 + 0.896816i \(0.354126\pi\)
\(600\) 0 0
\(601\) −30.0944 −1.22758 −0.613788 0.789471i \(-0.710355\pi\)
−0.613788 + 0.789471i \(0.710355\pi\)
\(602\) 4.07675 0.166156
\(603\) 0 0
\(604\) −15.6638 −0.637350
\(605\) 0 0
\(606\) 0 0
\(607\) 14.9979 0.608747 0.304373 0.952553i \(-0.401553\pi\)
0.304373 + 0.952553i \(0.401553\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 0 0
\(610\) 0 0
\(611\) −10.0969 −0.408478
\(612\) 0 0
\(613\) 16.7562 0.676776 0.338388 0.941007i \(-0.390118\pi\)
0.338388 + 0.941007i \(0.390118\pi\)
\(614\) 20.9318 0.844740
\(615\) 0 0
\(616\) −11.5209 −0.464192
\(617\) 18.1141 0.729245 0.364623 0.931155i \(-0.381198\pi\)
0.364623 + 0.931155i \(0.381198\pi\)
\(618\) 0 0
\(619\) −36.0176 −1.44767 −0.723835 0.689973i \(-0.757622\pi\)
−0.723835 + 0.689973i \(0.757622\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −14.9561 −0.599683
\(623\) −8.52546 −0.341565
\(624\) 0 0
\(625\) 0 0
\(626\) 32.2902 1.29058
\(627\) 0 0
\(628\) 14.2483 0.568571
\(629\) 18.3341 0.731030
\(630\) 0 0
\(631\) 17.6749 0.703627 0.351813 0.936070i \(-0.385565\pi\)
0.351813 + 0.936070i \(0.385565\pi\)
\(632\) 9.49670 0.377758
\(633\) 0 0
\(634\) 24.2726 0.963987
\(635\) 0 0
\(636\) 0 0
\(637\) −2.76475 −0.109543
\(638\) 20.9121 0.827919
\(639\) 0 0
\(640\) 0 0
\(641\) 2.08580 0.0823842 0.0411921 0.999151i \(-0.486884\pi\)
0.0411921 + 0.999151i \(0.486884\pi\)
\(642\) 0 0
\(643\) −37.5295 −1.48002 −0.740010 0.672596i \(-0.765179\pi\)
−0.740010 + 0.672596i \(0.765179\pi\)
\(644\) −14.5164 −0.572026
\(645\) 0 0
\(646\) 2.35386 0.0926112
\(647\) −9.12521 −0.358749 −0.179375 0.983781i \(-0.557407\pi\)
−0.179375 + 0.983781i \(0.557407\pi\)
\(648\) 0 0
\(649\) 33.3473 1.30899
\(650\) 0 0
\(651\) 0 0
\(652\) −6.00000 −0.234978
\(653\) −13.2529 −0.518625 −0.259313 0.965793i \(-0.583496\pi\)
−0.259313 + 0.965793i \(0.583496\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −3.54064 −0.138239
\(657\) 0 0
\(658\) −12.5452 −0.489061
\(659\) −24.9364 −0.971382 −0.485691 0.874130i \(-0.661432\pi\)
−0.485691 + 0.874130i \(0.661432\pi\)
\(660\) 0 0
\(661\) −29.9606 −1.16533 −0.582666 0.812712i \(-0.697990\pi\)
−0.582666 + 0.812712i \(0.697990\pi\)
\(662\) 6.80869 0.264627
\(663\) 0 0
\(664\) −4.43513 −0.172116
\(665\) 0 0
\(666\) 0 0
\(667\) 26.3493 1.02025
\(668\) 0.756178 0.0292574
\(669\) 0 0
\(670\) 0 0
\(671\) −19.4594 −0.751220
\(672\) 0 0
\(673\) −2.85055 −0.109881 −0.0549404 0.998490i \(-0.517497\pi\)
−0.0549404 + 0.998490i \(0.517497\pi\)
\(674\) −16.1231 −0.621040
\(675\) 0 0
\(676\) −9.41090 −0.361958
\(677\) 42.5562 1.63557 0.817784 0.575526i \(-0.195203\pi\)
0.817784 + 0.575526i \(0.195203\pi\)
\(678\) 0 0
\(679\) −16.6219 −0.637890
\(680\) 0 0
\(681\) 0 0
\(682\) −37.7253 −1.44458
\(683\) 12.2286 0.467916 0.233958 0.972247i \(-0.424832\pi\)
0.233958 + 0.972247i \(0.424832\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −19.9121 −0.760248
\(687\) 0 0
\(688\) 1.73194 0.0660298
\(689\) −3.74308 −0.142600
\(690\) 0 0
\(691\) 21.7581 0.827719 0.413859 0.910341i \(-0.364180\pi\)
0.413859 + 0.910341i \(0.364180\pi\)
\(692\) 13.6880 0.520340
\(693\) 0 0
\(694\) 23.3670 0.886998
\(695\) 0 0
\(696\) 0 0
\(697\) 8.33415 0.315678
\(698\) 16.5977 0.628231
\(699\) 0 0
\(700\) 0 0
\(701\) −28.6592 −1.08244 −0.541222 0.840880i \(-0.682038\pi\)
−0.541222 + 0.840880i \(0.682038\pi\)
\(702\) 0 0
\(703\) 7.78899 0.293767
\(704\) −4.89449 −0.184468
\(705\) 0 0
\(706\) −5.32105 −0.200260
\(707\) −14.3715 −0.540495
\(708\) 0 0
\(709\) 6.69461 0.251421 0.125711 0.992067i \(-0.459879\pi\)
0.125711 + 0.992067i \(0.459879\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −3.62191 −0.135737
\(713\) −47.5340 −1.78016
\(714\) 0 0
\(715\) 0 0
\(716\) −23.2044 −0.867189
\(717\) 0 0
\(718\) −6.67038 −0.248936
\(719\) −24.7980 −0.924811 −0.462405 0.886669i \(-0.653014\pi\)
−0.462405 + 0.886669i \(0.653014\pi\)
\(720\) 0 0
\(721\) −26.4770 −0.986055
\(722\) 1.00000 0.0372161
\(723\) 0 0
\(724\) 6.65067 0.247170
\(725\) 0 0
\(726\) 0 0
\(727\) 33.0328 1.22512 0.612560 0.790424i \(-0.290140\pi\)
0.612560 + 0.790424i \(0.290140\pi\)
\(728\) 4.45936 0.165275
\(729\) 0 0
\(730\) 0 0
\(731\) −4.07675 −0.150784
\(732\) 0 0
\(733\) 42.5098 1.57014 0.785068 0.619410i \(-0.212628\pi\)
0.785068 + 0.619410i \(0.212628\pi\)
\(734\) −8.14284 −0.300558
\(735\) 0 0
\(736\) −6.16707 −0.227321
\(737\) 25.4088 0.935946
\(738\) 0 0
\(739\) 1.20441 0.0443049 0.0221524 0.999755i \(-0.492948\pi\)
0.0221524 + 0.999755i \(0.492948\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −4.65067 −0.170731
\(743\) −21.5870 −0.791951 −0.395976 0.918261i \(-0.629594\pi\)
−0.395976 + 0.918261i \(0.629594\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −3.94296 −0.144362
\(747\) 0 0
\(748\) 11.5209 0.421247
\(749\) 36.0373 1.31678
\(750\) 0 0
\(751\) −42.2089 −1.54023 −0.770113 0.637908i \(-0.779800\pi\)
−0.770113 + 0.637908i \(0.779800\pi\)
\(752\) −5.32962 −0.194351
\(753\) 0 0
\(754\) −8.09438 −0.294780
\(755\) 0 0
\(756\) 0 0
\(757\) 32.3867 1.17711 0.588557 0.808456i \(-0.299696\pi\)
0.588557 + 0.808456i \(0.299696\pi\)
\(758\) −33.2044 −1.20604
\(759\) 0 0
\(760\) 0 0
\(761\) 43.1362 1.56369 0.781844 0.623475i \(-0.214279\pi\)
0.781844 + 0.623475i \(0.214279\pi\)
\(762\) 0 0
\(763\) −34.6198 −1.25332
\(764\) −14.7890 −0.535047
\(765\) 0 0
\(766\) −19.3867 −0.700469
\(767\) −12.9076 −0.466066
\(768\) 0 0
\(769\) −33.7541 −1.21720 −0.608602 0.793476i \(-0.708269\pi\)
−0.608602 + 0.793476i \(0.708269\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −11.5977 −0.417410
\(773\) −44.9055 −1.61514 −0.807570 0.589772i \(-0.799217\pi\)
−0.807570 + 0.589772i \(0.799217\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −7.06157 −0.253495
\(777\) 0 0
\(778\) 6.87026 0.246311
\(779\) 3.54064 0.126856
\(780\) 0 0
\(781\) −20.2771 −0.725572
\(782\) 14.5164 0.519106
\(783\) 0 0
\(784\) −1.45936 −0.0521201
\(785\) 0 0
\(786\) 0 0
\(787\) −23.0550 −0.821821 −0.410910 0.911676i \(-0.634789\pi\)
−0.410910 + 0.911676i \(0.634789\pi\)
\(788\) 12.6022 0.448935
\(789\) 0 0
\(790\) 0 0
\(791\) −7.76023 −0.275922
\(792\) 0 0
\(793\) 7.53206 0.267471
\(794\) −21.8879 −0.776772
\(795\) 0 0
\(796\) −23.7405 −0.841460
\(797\) −35.7384 −1.26592 −0.632960 0.774184i \(-0.718160\pi\)
−0.632960 + 0.774184i \(0.718160\pi\)
\(798\) 0 0
\(799\) 12.5452 0.443816
\(800\) 0 0
\(801\) 0 0
\(802\) −7.49670 −0.264718
\(803\) 21.4285 0.756196
\(804\) 0 0
\(805\) 0 0
\(806\) 14.6022 0.514341
\(807\) 0 0
\(808\) −6.10551 −0.214791
\(809\) −21.0525 −0.740167 −0.370084 0.928998i \(-0.620671\pi\)
−0.370084 + 0.928998i \(0.620671\pi\)
\(810\) 0 0
\(811\) 17.3912 0.610687 0.305344 0.952242i \(-0.401229\pi\)
0.305344 + 0.952242i \(0.401229\pi\)
\(812\) −10.0570 −0.352933
\(813\) 0 0
\(814\) 38.1231 1.33622
\(815\) 0 0
\(816\) 0 0
\(817\) −1.73194 −0.0605931
\(818\) 4.21101 0.147235
\(819\) 0 0
\(820\) 0 0
\(821\) −6.43966 −0.224746 −0.112373 0.993666i \(-0.535845\pi\)
−0.112373 + 0.993666i \(0.535845\pi\)
\(822\) 0 0
\(823\) 22.8308 0.795833 0.397917 0.917422i \(-0.369733\pi\)
0.397917 + 0.917422i \(0.369733\pi\)
\(824\) −11.2483 −0.391855
\(825\) 0 0
\(826\) −16.0373 −0.558010
\(827\) 29.2705 1.01784 0.508918 0.860815i \(-0.330046\pi\)
0.508918 + 0.860815i \(0.330046\pi\)
\(828\) 0 0
\(829\) −22.5053 −0.781640 −0.390820 0.920467i \(-0.627809\pi\)
−0.390820 + 0.920467i \(0.627809\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 1.89449 0.0656797
\(833\) 3.43513 0.119020
\(834\) 0 0
\(835\) 0 0
\(836\) 4.89449 0.169280
\(837\) 0 0
\(838\) −10.8703 −0.375507
\(839\) 25.2241 0.870833 0.435417 0.900229i \(-0.356601\pi\)
0.435417 + 0.900229i \(0.356601\pi\)
\(840\) 0 0
\(841\) −10.7450 −0.370519
\(842\) 33.7496 1.16309
\(843\) 0 0
\(844\) 19.9848 0.687906
\(845\) 0 0
\(846\) 0 0
\(847\) 30.4967 1.04788
\(848\) −1.97577 −0.0678481
\(849\) 0 0
\(850\) 0 0
\(851\) 48.0353 1.64663
\(852\) 0 0
\(853\) 19.4548 0.666121 0.333060 0.942905i \(-0.391919\pi\)
0.333060 + 0.942905i \(0.391919\pi\)
\(854\) 9.35838 0.320237
\(855\) 0 0
\(856\) 15.3099 0.523282
\(857\) −50.2351 −1.71600 −0.858000 0.513650i \(-0.828293\pi\)
−0.858000 + 0.513650i \(0.828293\pi\)
\(858\) 0 0
\(859\) −5.82840 −0.198862 −0.0994312 0.995044i \(-0.531702\pi\)
−0.0994312 + 0.995044i \(0.531702\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 15.7011 0.534782
\(863\) −8.37356 −0.285039 −0.142520 0.989792i \(-0.545520\pi\)
−0.142520 + 0.989792i \(0.545520\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −15.1847 −0.515997
\(867\) 0 0
\(868\) 18.1428 0.615808
\(869\) −46.4815 −1.57678
\(870\) 0 0
\(871\) −9.83490 −0.333243
\(872\) −14.7077 −0.498066
\(873\) 0 0
\(874\) 6.16707 0.208604
\(875\) 0 0
\(876\) 0 0
\(877\) 23.9031 0.807149 0.403575 0.914947i \(-0.367768\pi\)
0.403575 + 0.914947i \(0.367768\pi\)
\(878\) −25.4482 −0.858836
\(879\) 0 0
\(880\) 0 0
\(881\) −25.0419 −0.843682 −0.421841 0.906670i \(-0.638616\pi\)
−0.421841 + 0.906670i \(0.638616\pi\)
\(882\) 0 0
\(883\) 9.51188 0.320100 0.160050 0.987109i \(-0.448834\pi\)
0.160050 + 0.987109i \(0.448834\pi\)
\(884\) −4.45936 −0.149985
\(885\) 0 0
\(886\) −24.4351 −0.820914
\(887\) −40.7824 −1.36934 −0.684669 0.728854i \(-0.740053\pi\)
−0.684669 + 0.728854i \(0.740053\pi\)
\(888\) 0 0
\(889\) 15.5977 0.523129
\(890\) 0 0
\(891\) 0 0
\(892\) −18.7122 −0.626532
\(893\) 5.32962 0.178349
\(894\) 0 0
\(895\) 0 0
\(896\) 2.35386 0.0786368
\(897\) 0 0
\(898\) −26.5012 −0.884357
\(899\) −32.9318 −1.09834
\(900\) 0 0
\(901\) 4.65067 0.154936
\(902\) 17.3296 0.577013
\(903\) 0 0
\(904\) −3.29681 −0.109650
\(905\) 0 0
\(906\) 0 0
\(907\) 9.05299 0.300600 0.150300 0.988640i \(-0.451976\pi\)
0.150300 + 0.988640i \(0.451976\pi\)
\(908\) 13.0813 0.434117
\(909\) 0 0
\(910\) 0 0
\(911\) −28.2947 −0.937446 −0.468723 0.883345i \(-0.655286\pi\)
−0.468723 + 0.883345i \(0.655286\pi\)
\(912\) 0 0
\(913\) 21.7077 0.718420
\(914\) 36.3230 1.20146
\(915\) 0 0
\(916\) −9.89902 −0.327073
\(917\) −39.9692 −1.31990
\(918\) 0 0
\(919\) 42.3801 1.39799 0.698995 0.715127i \(-0.253631\pi\)
0.698995 + 0.715127i \(0.253631\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −23.0813 −0.760141
\(923\) 7.84858 0.258339
\(924\) 0 0
\(925\) 0 0
\(926\) −31.0419 −1.02010
\(927\) 0 0
\(928\) −4.27258 −0.140254
\(929\) 38.7430 1.27112 0.635558 0.772053i \(-0.280770\pi\)
0.635558 + 0.772053i \(0.280770\pi\)
\(930\) 0 0
\(931\) 1.45936 0.0478287
\(932\) 12.6395 0.414022
\(933\) 0 0
\(934\) −16.6925 −0.546196
\(935\) 0 0
\(936\) 0 0
\(937\) −3.20441 −0.104683 −0.0523417 0.998629i \(-0.516669\pi\)
−0.0523417 + 0.998629i \(0.516669\pi\)
\(938\) −12.2196 −0.398984
\(939\) 0 0
\(940\) 0 0
\(941\) 3.14492 0.102521 0.0512607 0.998685i \(-0.483676\pi\)
0.0512607 + 0.998685i \(0.483676\pi\)
\(942\) 0 0
\(943\) 21.8354 0.711058
\(944\) −6.81322 −0.221751
\(945\) 0 0
\(946\) −8.47699 −0.275611
\(947\) −29.0509 −0.944028 −0.472014 0.881591i \(-0.656473\pi\)
−0.472014 + 0.881591i \(0.656473\pi\)
\(948\) 0 0
\(949\) −8.29426 −0.269243
\(950\) 0 0
\(951\) 0 0
\(952\) −5.54064 −0.179573
\(953\) −11.5452 −0.373985 −0.186992 0.982361i \(-0.559874\pi\)
−0.186992 + 0.982361i \(0.559874\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 20.1605 0.652036
\(957\) 0 0
\(958\) 12.5891 0.406735
\(959\) 30.2947 0.978268
\(960\) 0 0
\(961\) 28.4088 0.916413
\(962\) −14.7562 −0.475758
\(963\) 0 0
\(964\) −11.7693 −0.379063
\(965\) 0 0
\(966\) 0 0
\(967\) 17.2044 0.553256 0.276628 0.960977i \(-0.410783\pi\)
0.276628 + 0.960977i \(0.410783\pi\)
\(968\) 12.9561 0.416424
\(969\) 0 0
\(970\) 0 0
\(971\) 38.2771 1.22837 0.614185 0.789162i \(-0.289485\pi\)
0.614185 + 0.789162i \(0.289485\pi\)
\(972\) 0 0
\(973\) 21.9142 0.702537
\(974\) −20.7077 −0.663518
\(975\) 0 0
\(976\) 3.97577 0.127261
\(977\) 16.5340 0.528971 0.264485 0.964390i \(-0.414798\pi\)
0.264485 + 0.964390i \(0.414798\pi\)
\(978\) 0 0
\(979\) 17.7274 0.566571
\(980\) 0 0
\(981\) 0 0
\(982\) −20.0747 −0.640608
\(983\) 41.7783 1.33252 0.666261 0.745719i \(-0.267894\pi\)
0.666261 + 0.745719i \(0.267894\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 10.0570 0.320281
\(987\) 0 0
\(988\) −1.89449 −0.0602718
\(989\) −10.6810 −0.339637
\(990\) 0 0
\(991\) 27.7845 0.882602 0.441301 0.897359i \(-0.354517\pi\)
0.441301 + 0.897359i \(0.354517\pi\)
\(992\) 7.70771 0.244720
\(993\) 0 0
\(994\) 9.75165 0.309304
\(995\) 0 0
\(996\) 0 0
\(997\) −16.9297 −0.536170 −0.268085 0.963395i \(-0.586391\pi\)
−0.268085 + 0.963395i \(0.586391\pi\)
\(998\) 41.5956 1.31669
\(999\) 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.cm.1.3 yes 3
3.2 odd 2 8550.2.a.cc.1.3 3
5.4 even 2 8550.2.a.ch.1.1 yes 3
15.14 odd 2 8550.2.a.ct.1.1 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8550.2.a.cc.1.3 3 3.2 odd 2
8550.2.a.ch.1.1 yes 3 5.4 even 2
8550.2.a.cm.1.3 yes 3 1.1 even 1 trivial
8550.2.a.ct.1.1 yes 3 15.14 odd 2