Properties

Label 8550.2.a.cj.1.2
Level $8550$
Weight $2$
Character 8550.1
Self dual yes
Analytic conductor $68.272$
Analytic rank $0$
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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.257.1
Defining polynomial: \(x^{3} - x^{2} - 4 x + 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 950)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.19869\) of defining polynomial
Character \(\chi\) \(=\) 8550.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +2.46980 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +2.46980 q^{7} -1.00000 q^{8} -0.728896 q^{11} -6.23163 q^{13} -2.46980 q^{14} +1.00000 q^{16} -0.563139 q^{17} -1.00000 q^{19} +0.728896 q^{22} -4.63555 q^{23} +6.23163 q^{26} +2.46980 q^{28} -10.2316 q^{29} +6.06587 q^{31} -1.00000 q^{32} +0.563139 q^{34} -5.72890 q^{37} +1.00000 q^{38} -4.79476 q^{41} +8.06587 q^{43} -0.728896 q^{44} +4.63555 q^{46} -8.12628 q^{47} -0.900112 q^{49} -6.23163 q^{52} +1.53020 q^{53} -2.46980 q^{56} +10.2316 q^{58} +5.76183 q^{59} +10.9396 q^{61} -6.06587 q^{62} +1.00000 q^{64} +12.9330 q^{67} -0.563139 q^{68} +4.39738 q^{71} -4.09334 q^{73} +5.72890 q^{74} -1.00000 q^{76} -1.80022 q^{77} +15.3370 q^{79} +4.79476 q^{82} -7.85517 q^{83} -8.06587 q^{86} +0.728896 q^{88} +10.0000 q^{89} -15.3908 q^{91} -4.63555 q^{92} +8.12628 q^{94} +11.0055 q^{97} +0.900112 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{2} + 3q^{4} + 2q^{7} - 3q^{8} + O(q^{10}) \) \( 3q - 3q^{2} + 3q^{4} + 2q^{7} - 3q^{8} - 2q^{11} - 2q^{13} - 2q^{14} + 3q^{16} + 4q^{17} - 3q^{19} + 2q^{22} - 14q^{23} + 2q^{26} + 2q^{28} - 14q^{29} - 4q^{31} - 3q^{32} - 4q^{34} - 17q^{37} + 3q^{38} + 8q^{41} + 2q^{43} - 2q^{44} + 14q^{46} - 13q^{47} + 25q^{49} - 2q^{52} + 10q^{53} - 2q^{56} + 14q^{58} + 6q^{59} + 22q^{61} + 4q^{62} + 3q^{64} + 4q^{68} + 2q^{71} - 12q^{73} + 17q^{74} - 3q^{76} + 50q^{77} + 24q^{79} - 8q^{82} - 12q^{83} - 2q^{86} + 2q^{88} + 30q^{89} - 7q^{91} - 14q^{92} + 13q^{94} - 25q^{98} + O(q^{100}) \)

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
\(7\) 2.46980 0.933495 0.466747 0.884391i \(-0.345426\pi\)
0.466747 + 0.884391i \(0.345426\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) −0.728896 −0.219770 −0.109885 0.993944i \(-0.535048\pi\)
−0.109885 + 0.993944i \(0.535048\pi\)
\(12\) 0 0
\(13\) −6.23163 −1.72834 −0.864171 0.503198i \(-0.832157\pi\)
−0.864171 + 0.503198i \(0.832157\pi\)
\(14\) −2.46980 −0.660081
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −0.563139 −0.136581 −0.0682907 0.997665i \(-0.521755\pi\)
−0.0682907 + 0.997665i \(0.521755\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0.728896 0.155401
\(23\) −4.63555 −0.966579 −0.483290 0.875460i \(-0.660558\pi\)
−0.483290 + 0.875460i \(0.660558\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 6.23163 1.22212
\(27\) 0 0
\(28\) 2.46980 0.466747
\(29\) −10.2316 −1.89997 −0.949983 0.312303i \(-0.898900\pi\)
−0.949983 + 0.312303i \(0.898900\pi\)
\(30\) 0 0
\(31\) 6.06587 1.08946 0.544731 0.838611i \(-0.316632\pi\)
0.544731 + 0.838611i \(0.316632\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 0.563139 0.0965776
\(35\) 0 0
\(36\) 0 0
\(37\) −5.72890 −0.941825 −0.470912 0.882180i \(-0.656075\pi\)
−0.470912 + 0.882180i \(0.656075\pi\)
\(38\) 1.00000 0.162221
\(39\) 0 0
\(40\) 0 0
\(41\) −4.79476 −0.748816 −0.374408 0.927264i \(-0.622154\pi\)
−0.374408 + 0.927264i \(0.622154\pi\)
\(42\) 0 0
\(43\) 8.06587 1.23003 0.615017 0.788514i \(-0.289149\pi\)
0.615017 + 0.788514i \(0.289149\pi\)
\(44\) −0.728896 −0.109885
\(45\) 0 0
\(46\) 4.63555 0.683475
\(47\) −8.12628 −1.18534 −0.592670 0.805446i \(-0.701926\pi\)
−0.592670 + 0.805446i \(0.701926\pi\)
\(48\) 0 0
\(49\) −0.900112 −0.128587
\(50\) 0 0
\(51\) 0 0
\(52\) −6.23163 −0.864171
\(53\) 1.53020 0.210190 0.105095 0.994462i \(-0.466485\pi\)
0.105095 + 0.994462i \(0.466485\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −2.46980 −0.330040
\(57\) 0 0
\(58\) 10.2316 1.34348
\(59\) 5.76183 0.750126 0.375063 0.926999i \(-0.377621\pi\)
0.375063 + 0.926999i \(0.377621\pi\)
\(60\) 0 0
\(61\) 10.9396 1.40067 0.700336 0.713814i \(-0.253034\pi\)
0.700336 + 0.713814i \(0.253034\pi\)
\(62\) −6.06587 −0.770366
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 12.9330 1.58002 0.790012 0.613092i \(-0.210074\pi\)
0.790012 + 0.613092i \(0.210074\pi\)
\(68\) −0.563139 −0.0682907
\(69\) 0 0
\(70\) 0 0
\(71\) 4.39738 0.521873 0.260937 0.965356i \(-0.415969\pi\)
0.260937 + 0.965356i \(0.415969\pi\)
\(72\) 0 0
\(73\) −4.09334 −0.479090 −0.239545 0.970885i \(-0.576998\pi\)
−0.239545 + 0.970885i \(0.576998\pi\)
\(74\) 5.72890 0.665971
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) −1.80022 −0.205155
\(78\) 0 0
\(79\) 15.3370 1.72554 0.862772 0.505593i \(-0.168726\pi\)
0.862772 + 0.505593i \(0.168726\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 4.79476 0.529493
\(83\) −7.85517 −0.862217 −0.431109 0.902300i \(-0.641877\pi\)
−0.431109 + 0.902300i \(0.641877\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −8.06587 −0.869765
\(87\) 0 0
\(88\) 0.728896 0.0777006
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) −15.3908 −1.61340
\(92\) −4.63555 −0.483290
\(93\) 0 0
\(94\) 8.12628 0.838162
\(95\) 0 0
\(96\) 0 0
\(97\) 11.0055 1.11744 0.558718 0.829358i \(-0.311294\pi\)
0.558718 + 0.829358i \(0.311294\pi\)
\(98\) 0.900112 0.0909250
\(99\) 0 0
\(100\) 0 0
\(101\) 9.73436 0.968605 0.484302 0.874901i \(-0.339074\pi\)
0.484302 + 0.874901i \(0.339074\pi\)
\(102\) 0 0
\(103\) −8.33151 −0.820928 −0.410464 0.911877i \(-0.634633\pi\)
−0.410464 + 0.911877i \(0.634633\pi\)
\(104\) 6.23163 0.611061
\(105\) 0 0
\(106\) −1.53020 −0.148627
\(107\) 15.0264 1.45266 0.726328 0.687348i \(-0.241225\pi\)
0.726328 + 0.687348i \(0.241225\pi\)
\(108\) 0 0
\(109\) −13.6619 −1.30858 −0.654288 0.756245i \(-0.727032\pi\)
−0.654288 + 0.756245i \(0.727032\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 2.46980 0.233374
\(113\) −1.20524 −0.113379 −0.0566895 0.998392i \(-0.518055\pi\)
−0.0566895 + 0.998392i \(0.518055\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −10.2316 −0.949983
\(117\) 0 0
\(118\) −5.76183 −0.530419
\(119\) −1.39084 −0.127498
\(120\) 0 0
\(121\) −10.4687 −0.951701
\(122\) −10.9396 −0.990424
\(123\) 0 0
\(124\) 6.06587 0.544731
\(125\) 0 0
\(126\) 0 0
\(127\) 9.52366 0.845088 0.422544 0.906342i \(-0.361137\pi\)
0.422544 + 0.906342i \(0.361137\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) 17.4028 1.52049 0.760247 0.649635i \(-0.225078\pi\)
0.760247 + 0.649635i \(0.225078\pi\)
\(132\) 0 0
\(133\) −2.46980 −0.214158
\(134\) −12.9330 −1.11725
\(135\) 0 0
\(136\) 0.563139 0.0482888
\(137\) −2.25910 −0.193008 −0.0965040 0.995333i \(-0.530766\pi\)
−0.0965040 + 0.995333i \(0.530766\pi\)
\(138\) 0 0
\(139\) 16.8606 1.43010 0.715050 0.699073i \(-0.246404\pi\)
0.715050 + 0.699073i \(0.246404\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −4.39738 −0.369020
\(143\) 4.54221 0.379838
\(144\) 0 0
\(145\) 0 0
\(146\) 4.09334 0.338768
\(147\) 0 0
\(148\) −5.72890 −0.470912
\(149\) 13.0055 1.06545 0.532724 0.846289i \(-0.321168\pi\)
0.532724 + 0.846289i \(0.321168\pi\)
\(150\) 0 0
\(151\) 17.5895 1.43142 0.715708 0.698400i \(-0.246104\pi\)
0.715708 + 0.698400i \(0.246104\pi\)
\(152\) 1.00000 0.0811107
\(153\) 0 0
\(154\) 1.80022 0.145066
\(155\) 0 0
\(156\) 0 0
\(157\) 9.52366 0.760071 0.380035 0.924972i \(-0.375912\pi\)
0.380035 + 0.924972i \(0.375912\pi\)
\(158\) −15.3370 −1.22014
\(159\) 0 0
\(160\) 0 0
\(161\) −11.4489 −0.902297
\(162\) 0 0
\(163\) 13.1921 1.03329 0.516644 0.856200i \(-0.327181\pi\)
0.516644 + 0.856200i \(0.327181\pi\)
\(164\) −4.79476 −0.374408
\(165\) 0 0
\(166\) 7.85517 0.609680
\(167\) −1.81331 −0.140318 −0.0701591 0.997536i \(-0.522351\pi\)
−0.0701591 + 0.997536i \(0.522351\pi\)
\(168\) 0 0
\(169\) 25.8332 1.98717
\(170\) 0 0
\(171\) 0 0
\(172\) 8.06587 0.615017
\(173\) 12.5237 0.952156 0.476078 0.879403i \(-0.342058\pi\)
0.476078 + 0.879403i \(0.342058\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −0.728896 −0.0549426
\(177\) 0 0
\(178\) −10.0000 −0.749532
\(179\) −1.39738 −0.104445 −0.0522226 0.998635i \(-0.516631\pi\)
−0.0522226 + 0.998635i \(0.516631\pi\)
\(180\) 0 0
\(181\) 5.72890 0.425825 0.212913 0.977071i \(-0.431705\pi\)
0.212913 + 0.977071i \(0.431705\pi\)
\(182\) 15.3908 1.14084
\(183\) 0 0
\(184\) 4.63555 0.341737
\(185\) 0 0
\(186\) 0 0
\(187\) 0.410470 0.0300165
\(188\) −8.12628 −0.592670
\(189\) 0 0
\(190\) 0 0
\(191\) 27.0198 1.95509 0.977544 0.210733i \(-0.0675849\pi\)
0.977544 + 0.210733i \(0.0675849\pi\)
\(192\) 0 0
\(193\) −23.0713 −1.66071 −0.830355 0.557234i \(-0.811862\pi\)
−0.830355 + 0.557234i \(0.811862\pi\)
\(194\) −11.0055 −0.790146
\(195\) 0 0
\(196\) −0.900112 −0.0642937
\(197\) −0.794765 −0.0566247 −0.0283123 0.999599i \(-0.509013\pi\)
−0.0283123 + 0.999599i \(0.509013\pi\)
\(198\) 0 0
\(199\) 8.07241 0.572238 0.286119 0.958194i \(-0.407635\pi\)
0.286119 + 0.958194i \(0.407635\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −9.73436 −0.684907
\(203\) −25.2700 −1.77361
\(204\) 0 0
\(205\) 0 0
\(206\) 8.33151 0.580484
\(207\) 0 0
\(208\) −6.23163 −0.432085
\(209\) 0.728896 0.0504188
\(210\) 0 0
\(211\) −11.6410 −0.801400 −0.400700 0.916209i \(-0.631233\pi\)
−0.400700 + 0.916209i \(0.631233\pi\)
\(212\) 1.53020 0.105095
\(213\) 0 0
\(214\) −15.0264 −1.02718
\(215\) 0 0
\(216\) 0 0
\(217\) 14.9815 1.01701
\(218\) 13.6619 0.925304
\(219\) 0 0
\(220\) 0 0
\(221\) 3.50927 0.236059
\(222\) 0 0
\(223\) −15.6554 −1.04836 −0.524182 0.851607i \(-0.675629\pi\)
−0.524182 + 0.851607i \(0.675629\pi\)
\(224\) −2.46980 −0.165020
\(225\) 0 0
\(226\) 1.20524 0.0801710
\(227\) −4.80131 −0.318674 −0.159337 0.987224i \(-0.550936\pi\)
−0.159337 + 0.987224i \(0.550936\pi\)
\(228\) 0 0
\(229\) 2.79476 0.184683 0.0923416 0.995727i \(-0.470565\pi\)
0.0923416 + 0.995727i \(0.470565\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 10.2316 0.671739
\(233\) −11.5422 −0.756155 −0.378078 0.925774i \(-0.623415\pi\)
−0.378078 + 0.925774i \(0.623415\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 5.76183 0.375063
\(237\) 0 0
\(238\) 1.39084 0.0901547
\(239\) 26.2251 1.69636 0.848180 0.529708i \(-0.177699\pi\)
0.848180 + 0.529708i \(0.177699\pi\)
\(240\) 0 0
\(241\) −12.0659 −0.777231 −0.388615 0.921400i \(-0.627047\pi\)
−0.388615 + 0.921400i \(0.627047\pi\)
\(242\) 10.4687 0.672954
\(243\) 0 0
\(244\) 10.9396 0.700336
\(245\) 0 0
\(246\) 0 0
\(247\) 6.23163 0.396509
\(248\) −6.06587 −0.385183
\(249\) 0 0
\(250\) 0 0
\(251\) −13.5237 −0.853606 −0.426803 0.904345i \(-0.640360\pi\)
−0.426803 + 0.904345i \(0.640360\pi\)
\(252\) 0 0
\(253\) 3.37884 0.212426
\(254\) −9.52366 −0.597568
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −22.7398 −1.41847 −0.709235 0.704972i \(-0.750960\pi\)
−0.709235 + 0.704972i \(0.750960\pi\)
\(258\) 0 0
\(259\) −14.1492 −0.879188
\(260\) 0 0
\(261\) 0 0
\(262\) −17.4028 −1.07515
\(263\) 6.67395 0.411533 0.205767 0.978601i \(-0.434031\pi\)
0.205767 + 0.978601i \(0.434031\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 2.46980 0.151433
\(267\) 0 0
\(268\) 12.9330 0.790012
\(269\) 29.7398 1.81327 0.906634 0.421917i \(-0.138643\pi\)
0.906634 + 0.421917i \(0.138643\pi\)
\(270\) 0 0
\(271\) 1.11189 0.0675426 0.0337713 0.999430i \(-0.489248\pi\)
0.0337713 + 0.999430i \(0.489248\pi\)
\(272\) −0.563139 −0.0341453
\(273\) 0 0
\(274\) 2.25910 0.136477
\(275\) 0 0
\(276\) 0 0
\(277\) 13.1263 0.788682 0.394341 0.918964i \(-0.370973\pi\)
0.394341 + 0.918964i \(0.370973\pi\)
\(278\) −16.8606 −1.01123
\(279\) 0 0
\(280\) 0 0
\(281\) −22.9265 −1.36768 −0.683840 0.729632i \(-0.739691\pi\)
−0.683840 + 0.729632i \(0.739691\pi\)
\(282\) 0 0
\(283\) 0.860634 0.0511594 0.0255797 0.999673i \(-0.491857\pi\)
0.0255797 + 0.999673i \(0.491857\pi\)
\(284\) 4.39738 0.260937
\(285\) 0 0
\(286\) −4.54221 −0.268586
\(287\) −11.8421 −0.699016
\(288\) 0 0
\(289\) −16.6829 −0.981346
\(290\) 0 0
\(291\) 0 0
\(292\) −4.09334 −0.239545
\(293\) −19.8212 −1.15796 −0.578982 0.815340i \(-0.696550\pi\)
−0.578982 + 0.815340i \(0.696550\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 5.72890 0.332985
\(297\) 0 0
\(298\) −13.0055 −0.753386
\(299\) 28.8870 1.67058
\(300\) 0 0
\(301\) 19.9210 1.14823
\(302\) −17.5895 −1.01216
\(303\) 0 0
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) 0 0
\(307\) −16.8661 −0.962599 −0.481299 0.876556i \(-0.659835\pi\)
−0.481299 + 0.876556i \(0.659835\pi\)
\(308\) −1.80022 −0.102577
\(309\) 0 0
\(310\) 0 0
\(311\) −10.3250 −0.585475 −0.292738 0.956193i \(-0.594566\pi\)
−0.292738 + 0.956193i \(0.594566\pi\)
\(312\) 0 0
\(313\) −15.7684 −0.891281 −0.445641 0.895212i \(-0.647024\pi\)
−0.445641 + 0.895212i \(0.647024\pi\)
\(314\) −9.52366 −0.537451
\(315\) 0 0
\(316\) 15.3370 0.862772
\(317\) 2.17230 0.122009 0.0610043 0.998138i \(-0.480570\pi\)
0.0610043 + 0.998138i \(0.480570\pi\)
\(318\) 0 0
\(319\) 7.45779 0.417556
\(320\) 0 0
\(321\) 0 0
\(322\) 11.4489 0.638020
\(323\) 0.563139 0.0313339
\(324\) 0 0
\(325\) 0 0
\(326\) −13.1921 −0.730645
\(327\) 0 0
\(328\) 4.79476 0.264747
\(329\) −20.0702 −1.10651
\(330\) 0 0
\(331\) 11.9791 0.658429 0.329215 0.944255i \(-0.393216\pi\)
0.329215 + 0.944255i \(0.393216\pi\)
\(332\) −7.85517 −0.431109
\(333\) 0 0
\(334\) 1.81331 0.0992200
\(335\) 0 0
\(336\) 0 0
\(337\) −11.1921 −0.609675 −0.304838 0.952404i \(-0.598602\pi\)
−0.304838 + 0.952404i \(0.598602\pi\)
\(338\) −25.8332 −1.40514
\(339\) 0 0
\(340\) 0 0
\(341\) −4.42139 −0.239432
\(342\) 0 0
\(343\) −19.5117 −1.05353
\(344\) −8.06587 −0.434883
\(345\) 0 0
\(346\) −12.5237 −0.673276
\(347\) 20.3843 1.09429 0.547143 0.837039i \(-0.315715\pi\)
0.547143 + 0.837039i \(0.315715\pi\)
\(348\) 0 0
\(349\) −0.252557 −0.0135191 −0.00675954 0.999977i \(-0.502152\pi\)
−0.00675954 + 0.999977i \(0.502152\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0.728896 0.0388503
\(353\) 28.6434 1.52453 0.762267 0.647263i \(-0.224086\pi\)
0.762267 + 0.647263i \(0.224086\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 10.0000 0.529999
\(357\) 0 0
\(358\) 1.39738 0.0738540
\(359\) 18.5741 0.980301 0.490151 0.871638i \(-0.336942\pi\)
0.490151 + 0.871638i \(0.336942\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −5.72890 −0.301104
\(363\) 0 0
\(364\) −15.3908 −0.806699
\(365\) 0 0
\(366\) 0 0
\(367\) −4.79476 −0.250285 −0.125142 0.992139i \(-0.539939\pi\)
−0.125142 + 0.992139i \(0.539939\pi\)
\(368\) −4.63555 −0.241645
\(369\) 0 0
\(370\) 0 0
\(371\) 3.77929 0.196211
\(372\) 0 0
\(373\) 15.9485 0.825783 0.412892 0.910780i \(-0.364519\pi\)
0.412892 + 0.910780i \(0.364519\pi\)
\(374\) −0.410470 −0.0212249
\(375\) 0 0
\(376\) 8.12628 0.419081
\(377\) 63.7597 3.28379
\(378\) 0 0
\(379\) −16.8013 −0.863025 −0.431513 0.902107i \(-0.642020\pi\)
−0.431513 + 0.902107i \(0.642020\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −27.0198 −1.38246
\(383\) −26.7948 −1.36915 −0.684574 0.728943i \(-0.740012\pi\)
−0.684574 + 0.728943i \(0.740012\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 23.0713 1.17430
\(387\) 0 0
\(388\) 11.0055 0.558718
\(389\) −18.3424 −0.929998 −0.464999 0.885311i \(-0.653945\pi\)
−0.464999 + 0.885311i \(0.653945\pi\)
\(390\) 0 0
\(391\) 2.61046 0.132017
\(392\) 0.900112 0.0454625
\(393\) 0 0
\(394\) 0.794765 0.0400397
\(395\) 0 0
\(396\) 0 0
\(397\) 21.2162 1.06481 0.532404 0.846490i \(-0.321289\pi\)
0.532404 + 0.846490i \(0.321289\pi\)
\(398\) −8.07241 −0.404633
\(399\) 0 0
\(400\) 0 0
\(401\) 27.8661 1.39157 0.695783 0.718252i \(-0.255057\pi\)
0.695783 + 0.718252i \(0.255057\pi\)
\(402\) 0 0
\(403\) −37.8002 −1.88296
\(404\) 9.73436 0.484302
\(405\) 0 0
\(406\) 25.2700 1.25413
\(407\) 4.17577 0.206985
\(408\) 0 0
\(409\) −18.0899 −0.894487 −0.447243 0.894412i \(-0.647594\pi\)
−0.447243 + 0.894412i \(0.647594\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −8.33151 −0.410464
\(413\) 14.2305 0.700239
\(414\) 0 0
\(415\) 0 0
\(416\) 6.23163 0.305531
\(417\) 0 0
\(418\) −0.728896 −0.0356515
\(419\) 15.3370 0.749260 0.374630 0.927174i \(-0.377770\pi\)
0.374630 + 0.927174i \(0.377770\pi\)
\(420\) 0 0
\(421\) −8.42486 −0.410602 −0.205301 0.978699i \(-0.565817\pi\)
−0.205301 + 0.978699i \(0.565817\pi\)
\(422\) 11.6410 0.566676
\(423\) 0 0
\(424\) −1.53020 −0.0743133
\(425\) 0 0
\(426\) 0 0
\(427\) 27.0185 1.30752
\(428\) 15.0264 0.726328
\(429\) 0 0
\(430\) 0 0
\(431\) −16.2766 −0.784014 −0.392007 0.919962i \(-0.628219\pi\)
−0.392007 + 0.919962i \(0.628219\pi\)
\(432\) 0 0
\(433\) −18.1976 −0.874521 −0.437261 0.899335i \(-0.644051\pi\)
−0.437261 + 0.899335i \(0.644051\pi\)
\(434\) −14.9815 −0.719133
\(435\) 0 0
\(436\) −13.6619 −0.654288
\(437\) 4.63555 0.221749
\(438\) 0 0
\(439\) 20.4214 0.974660 0.487330 0.873218i \(-0.337971\pi\)
0.487330 + 0.873218i \(0.337971\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −3.50927 −0.166919
\(443\) −21.6685 −1.02950 −0.514750 0.857340i \(-0.672115\pi\)
−0.514750 + 0.857340i \(0.672115\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 15.6554 0.741305
\(447\) 0 0
\(448\) 2.46980 0.116687
\(449\) −23.8133 −1.12382 −0.561910 0.827198i \(-0.689933\pi\)
−0.561910 + 0.827198i \(0.689933\pi\)
\(450\) 0 0
\(451\) 3.49489 0.164568
\(452\) −1.20524 −0.0566895
\(453\) 0 0
\(454\) 4.80131 0.225337
\(455\) 0 0
\(456\) 0 0
\(457\) 9.15813 0.428399 0.214200 0.976790i \(-0.431286\pi\)
0.214200 + 0.976790i \(0.431286\pi\)
\(458\) −2.79476 −0.130591
\(459\) 0 0
\(460\) 0 0
\(461\) 7.78931 0.362784 0.181392 0.983411i \(-0.441940\pi\)
0.181392 + 0.983411i \(0.441940\pi\)
\(462\) 0 0
\(463\) 0.405011 0.0188225 0.00941123 0.999956i \(-0.497004\pi\)
0.00941123 + 0.999956i \(0.497004\pi\)
\(464\) −10.2316 −0.474991
\(465\) 0 0
\(466\) 11.5422 0.534682
\(467\) 1.95814 0.0906118 0.0453059 0.998973i \(-0.485574\pi\)
0.0453059 + 0.998973i \(0.485574\pi\)
\(468\) 0 0
\(469\) 31.9420 1.47494
\(470\) 0 0
\(471\) 0 0
\(472\) −5.76183 −0.265210
\(473\) −5.87918 −0.270325
\(474\) 0 0
\(475\) 0 0
\(476\) −1.39084 −0.0637490
\(477\) 0 0
\(478\) −26.2251 −1.19951
\(479\) 22.9210 1.04729 0.523645 0.851937i \(-0.324572\pi\)
0.523645 + 0.851937i \(0.324572\pi\)
\(480\) 0 0
\(481\) 35.7003 1.62780
\(482\) 12.0659 0.549585
\(483\) 0 0
\(484\) −10.4687 −0.475850
\(485\) 0 0
\(486\) 0 0
\(487\) −23.9450 −1.08505 −0.542527 0.840038i \(-0.682532\pi\)
−0.542527 + 0.840038i \(0.682532\pi\)
\(488\) −10.9396 −0.495212
\(489\) 0 0
\(490\) 0 0
\(491\) −3.86826 −0.174572 −0.0872861 0.996183i \(-0.527819\pi\)
−0.0872861 + 0.996183i \(0.527819\pi\)
\(492\) 0 0
\(493\) 5.76183 0.259500
\(494\) −6.23163 −0.280374
\(495\) 0 0
\(496\) 6.06587 0.272366
\(497\) 10.8606 0.487166
\(498\) 0 0
\(499\) −7.92104 −0.354595 −0.177297 0.984157i \(-0.556735\pi\)
−0.177297 + 0.984157i \(0.556735\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 13.5237 0.603591
\(503\) 16.6949 0.744388 0.372194 0.928155i \(-0.378606\pi\)
0.372194 + 0.928155i \(0.378606\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −3.37884 −0.150208
\(507\) 0 0
\(508\) 9.52366 0.422544
\(509\) −24.4028 −1.08164 −0.540818 0.841139i \(-0.681885\pi\)
−0.540818 + 0.841139i \(0.681885\pi\)
\(510\) 0 0
\(511\) −10.1097 −0.447228
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 22.7398 1.00301
\(515\) 0 0
\(516\) 0 0
\(517\) 5.92321 0.260503
\(518\) 14.1492 0.621680
\(519\) 0 0
\(520\) 0 0
\(521\) −26.0659 −1.14197 −0.570983 0.820962i \(-0.693438\pi\)
−0.570983 + 0.820962i \(0.693438\pi\)
\(522\) 0 0
\(523\) 17.8726 0.781516 0.390758 0.920493i \(-0.372213\pi\)
0.390758 + 0.920493i \(0.372213\pi\)
\(524\) 17.4028 0.760247
\(525\) 0 0
\(526\) −6.67395 −0.290998
\(527\) −3.41593 −0.148800
\(528\) 0 0
\(529\) −1.51166 −0.0657243
\(530\) 0 0
\(531\) 0 0
\(532\) −2.46980 −0.107079
\(533\) 29.8792 1.29421
\(534\) 0 0
\(535\) 0 0
\(536\) −12.9330 −0.558623
\(537\) 0 0
\(538\) −29.7398 −1.28217
\(539\) 0.656088 0.0282597
\(540\) 0 0
\(541\) 23.0604 0.991444 0.495722 0.868481i \(-0.334903\pi\)
0.495722 + 0.868481i \(0.334903\pi\)
\(542\) −1.11189 −0.0477598
\(543\) 0 0
\(544\) 0.563139 0.0241444
\(545\) 0 0
\(546\) 0 0
\(547\) −27.7584 −1.18686 −0.593431 0.804885i \(-0.702227\pi\)
−0.593431 + 0.804885i \(0.702227\pi\)
\(548\) −2.25910 −0.0965040
\(549\) 0 0
\(550\) 0 0
\(551\) 10.2316 0.435882
\(552\) 0 0
\(553\) 37.8792 1.61079
\(554\) −13.1263 −0.557682
\(555\) 0 0
\(556\) 16.8606 0.715050
\(557\) 8.60808 0.364736 0.182368 0.983230i \(-0.441624\pi\)
0.182368 + 0.983230i \(0.441624\pi\)
\(558\) 0 0
\(559\) −50.2635 −2.12592
\(560\) 0 0
\(561\) 0 0
\(562\) 22.9265 0.967096
\(563\) −7.93959 −0.334614 −0.167307 0.985905i \(-0.553507\pi\)
−0.167307 + 0.985905i \(0.553507\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −0.860634 −0.0361751
\(567\) 0 0
\(568\) −4.39738 −0.184510
\(569\) 1.65757 0.0694889 0.0347444 0.999396i \(-0.488938\pi\)
0.0347444 + 0.999396i \(0.488938\pi\)
\(570\) 0 0
\(571\) −14.0528 −0.588091 −0.294045 0.955791i \(-0.595002\pi\)
−0.294045 + 0.955791i \(0.595002\pi\)
\(572\) 4.54221 0.189919
\(573\) 0 0
\(574\) 11.8421 0.494279
\(575\) 0 0
\(576\) 0 0
\(577\) −1.36445 −0.0568027 −0.0284014 0.999597i \(-0.509042\pi\)
−0.0284014 + 0.999597i \(0.509042\pi\)
\(578\) 16.6829 0.693916
\(579\) 0 0
\(580\) 0 0
\(581\) −19.4007 −0.804876
\(582\) 0 0
\(583\) −1.11536 −0.0461935
\(584\) 4.09334 0.169384
\(585\) 0 0
\(586\) 19.8212 0.818804
\(587\) 24.6609 1.01786 0.508931 0.860807i \(-0.330041\pi\)
0.508931 + 0.860807i \(0.330041\pi\)
\(588\) 0 0
\(589\) −6.06587 −0.249940
\(590\) 0 0
\(591\) 0 0
\(592\) −5.72890 −0.235456
\(593\) 0.747443 0.0306938 0.0153469 0.999882i \(-0.495115\pi\)
0.0153469 + 0.999882i \(0.495115\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 13.0055 0.532724
\(597\) 0 0
\(598\) −28.8870 −1.18128
\(599\) −1.40501 −0.0574072 −0.0287036 0.999588i \(-0.509138\pi\)
−0.0287036 + 0.999588i \(0.509138\pi\)
\(600\) 0 0
\(601\) 29.7453 1.21334 0.606668 0.794956i \(-0.292506\pi\)
0.606668 + 0.794956i \(0.292506\pi\)
\(602\) −19.9210 −0.811921
\(603\) 0 0
\(604\) 17.5895 0.715708
\(605\) 0 0
\(606\) 0 0
\(607\) 5.66849 0.230077 0.115038 0.993361i \(-0.463301\pi\)
0.115038 + 0.993361i \(0.463301\pi\)
\(608\) 1.00000 0.0405554
\(609\) 0 0
\(610\) 0 0
\(611\) 50.6399 2.04867
\(612\) 0 0
\(613\) −2.99454 −0.120948 −0.0604742 0.998170i \(-0.519261\pi\)
−0.0604742 + 0.998170i \(0.519261\pi\)
\(614\) 16.8661 0.680660
\(615\) 0 0
\(616\) 1.80022 0.0725331
\(617\) 32.3370 1.30184 0.650919 0.759147i \(-0.274384\pi\)
0.650919 + 0.759147i \(0.274384\pi\)
\(618\) 0 0
\(619\) −20.9265 −0.841107 −0.420554 0.907268i \(-0.638164\pi\)
−0.420554 + 0.907268i \(0.638164\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 10.3250 0.413994
\(623\) 24.6980 0.989503
\(624\) 0 0
\(625\) 0 0
\(626\) 15.7684 0.630231
\(627\) 0 0
\(628\) 9.52366 0.380035
\(629\) 3.22617 0.128636
\(630\) 0 0
\(631\) 22.8002 0.907663 0.453831 0.891088i \(-0.350057\pi\)
0.453831 + 0.891088i \(0.350057\pi\)
\(632\) −15.3370 −0.610072
\(633\) 0 0
\(634\) −2.17230 −0.0862731
\(635\) 0 0
\(636\) 0 0
\(637\) 5.60916 0.222243
\(638\) −7.45779 −0.295257
\(639\) 0 0
\(640\) 0 0
\(641\) −36.3184 −1.43449 −0.717246 0.696820i \(-0.754598\pi\)
−0.717246 + 0.696820i \(0.754598\pi\)
\(642\) 0 0
\(643\) 13.6135 0.536865 0.268433 0.963298i \(-0.413494\pi\)
0.268433 + 0.963298i \(0.413494\pi\)
\(644\) −11.4489 −0.451148
\(645\) 0 0
\(646\) −0.563139 −0.0221564
\(647\) 0.0724126 0.00284683 0.00142342 0.999999i \(-0.499547\pi\)
0.00142342 + 0.999999i \(0.499547\pi\)
\(648\) 0 0
\(649\) −4.19978 −0.164856
\(650\) 0 0
\(651\) 0 0
\(652\) 13.1921 0.516644
\(653\) 43.5346 1.70364 0.851820 0.523835i \(-0.175499\pi\)
0.851820 + 0.523835i \(0.175499\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −4.79476 −0.187204
\(657\) 0 0
\(658\) 20.0702 0.782420
\(659\) 33.4512 1.30308 0.651538 0.758616i \(-0.274124\pi\)
0.651538 + 0.758616i \(0.274124\pi\)
\(660\) 0 0
\(661\) −2.89465 −0.112589 −0.0562945 0.998414i \(-0.517929\pi\)
−0.0562945 + 0.998414i \(0.517929\pi\)
\(662\) −11.9791 −0.465580
\(663\) 0 0
\(664\) 7.85517 0.304840
\(665\) 0 0
\(666\) 0 0
\(667\) 47.4292 1.83647
\(668\) −1.81331 −0.0701591
\(669\) 0 0
\(670\) 0 0
\(671\) −7.97382 −0.307826
\(672\) 0 0
\(673\) 42.8475 1.65165 0.825826 0.563925i \(-0.190709\pi\)
0.825826 + 0.563925i \(0.190709\pi\)
\(674\) 11.1921 0.431105
\(675\) 0 0
\(676\) 25.8332 0.993583
\(677\) −34.4567 −1.32428 −0.662139 0.749381i \(-0.730351\pi\)
−0.662139 + 0.749381i \(0.730351\pi\)
\(678\) 0 0
\(679\) 27.1812 1.04312
\(680\) 0 0
\(681\) 0 0
\(682\) 4.42139 0.169304
\(683\) 2.73436 0.104627 0.0523136 0.998631i \(-0.483340\pi\)
0.0523136 + 0.998631i \(0.483340\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 19.5117 0.744959
\(687\) 0 0
\(688\) 8.06587 0.307508
\(689\) −9.53566 −0.363280
\(690\) 0 0
\(691\) −4.74198 −0.180394 −0.0901968 0.995924i \(-0.528750\pi\)
−0.0901968 + 0.995924i \(0.528750\pi\)
\(692\) 12.5237 0.476078
\(693\) 0 0
\(694\) −20.3843 −0.773777
\(695\) 0 0
\(696\) 0 0
\(697\) 2.70012 0.102274
\(698\) 0.252557 0.00955943
\(699\) 0 0
\(700\) 0 0
\(701\) −33.1372 −1.25157 −0.625787 0.779994i \(-0.715222\pi\)
−0.625787 + 0.779994i \(0.715222\pi\)
\(702\) 0 0
\(703\) 5.72890 0.216069
\(704\) −0.728896 −0.0274713
\(705\) 0 0
\(706\) −28.6434 −1.07801
\(707\) 24.0419 0.904187
\(708\) 0 0
\(709\) 5.37884 0.202006 0.101003 0.994886i \(-0.467795\pi\)
0.101003 + 0.994886i \(0.467795\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −10.0000 −0.374766
\(713\) −28.1187 −1.05305
\(714\) 0 0
\(715\) 0 0
\(716\) −1.39738 −0.0522226
\(717\) 0 0
\(718\) −18.5741 −0.693178
\(719\) −33.0857 −1.23389 −0.616944 0.787007i \(-0.711630\pi\)
−0.616944 + 0.787007i \(0.711630\pi\)
\(720\) 0 0
\(721\) −20.5771 −0.766332
\(722\) −1.00000 −0.0372161
\(723\) 0 0
\(724\) 5.72890 0.212913
\(725\) 0 0
\(726\) 0 0
\(727\) 2.62463 0.0973423 0.0486711 0.998815i \(-0.484501\pi\)
0.0486711 + 0.998815i \(0.484501\pi\)
\(728\) 15.3908 0.570422
\(729\) 0 0
\(730\) 0 0
\(731\) −4.54221 −0.168000
\(732\) 0 0
\(733\) 0.608077 0.0224598 0.0112299 0.999937i \(-0.496425\pi\)
0.0112299 + 0.999937i \(0.496425\pi\)
\(734\) 4.79476 0.176978
\(735\) 0 0
\(736\) 4.63555 0.170869
\(737\) −9.42685 −0.347242
\(738\) 0 0
\(739\) 36.3974 1.33890 0.669450 0.742857i \(-0.266530\pi\)
0.669450 + 0.742857i \(0.266530\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −3.77929 −0.138742
\(743\) 23.0713 0.846405 0.423202 0.906035i \(-0.360906\pi\)
0.423202 + 0.906035i \(0.360906\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −15.9485 −0.583917
\(747\) 0 0
\(748\) 0.410470 0.0150083
\(749\) 37.1121 1.35605
\(750\) 0 0
\(751\) 4.75290 0.173436 0.0867179 0.996233i \(-0.472362\pi\)
0.0867179 + 0.996233i \(0.472362\pi\)
\(752\) −8.12628 −0.296335
\(753\) 0 0
\(754\) −63.7597 −2.32199
\(755\) 0 0
\(756\) 0 0
\(757\) 26.8057 0.974269 0.487135 0.873327i \(-0.338042\pi\)
0.487135 + 0.873327i \(0.338042\pi\)
\(758\) 16.8013 0.610251
\(759\) 0 0
\(760\) 0 0
\(761\) −25.4478 −0.922481 −0.461241 0.887275i \(-0.652596\pi\)
−0.461241 + 0.887275i \(0.652596\pi\)
\(762\) 0 0
\(763\) −33.7422 −1.22155
\(764\) 27.0198 0.977544
\(765\) 0 0
\(766\) 26.7948 0.968134
\(767\) −35.9056 −1.29648
\(768\) 0 0
\(769\) −14.6421 −0.528007 −0.264004 0.964522i \(-0.585043\pi\)
−0.264004 + 0.964522i \(0.585043\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −23.0713 −0.830355
\(773\) 16.8462 0.605917 0.302959 0.953004i \(-0.402026\pi\)
0.302959 + 0.953004i \(0.402026\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −11.0055 −0.395073
\(777\) 0 0
\(778\) 18.3424 0.657608
\(779\) 4.79476 0.171790
\(780\) 0 0
\(781\) −3.20524 −0.114692
\(782\) −2.61046 −0.0933499
\(783\) 0 0
\(784\) −0.900112 −0.0321469
\(785\) 0 0
\(786\) 0 0
\(787\) 18.7367 0.667893 0.333946 0.942592i \(-0.391620\pi\)
0.333946 + 0.942592i \(0.391620\pi\)
\(788\) −0.794765 −0.0283123
\(789\) 0 0
\(790\) 0 0
\(791\) −2.97668 −0.105839
\(792\) 0 0
\(793\) −68.1714 −2.42084
\(794\) −21.2162 −0.752933
\(795\) 0 0
\(796\) 8.07241 0.286119
\(797\) 37.9900 1.34567 0.672837 0.739791i \(-0.265075\pi\)
0.672837 + 0.739791i \(0.265075\pi\)
\(798\) 0 0
\(799\) 4.57623 0.161895
\(800\) 0 0
\(801\) 0 0
\(802\) −27.8661 −0.983986
\(803\) 2.98362 0.105290
\(804\) 0 0
\(805\) 0 0
\(806\) 37.8002 1.33146
\(807\) 0 0
\(808\) −9.73436 −0.342453
\(809\) −21.8857 −0.769461 −0.384731 0.923029i \(-0.625706\pi\)
−0.384731 + 0.923029i \(0.625706\pi\)
\(810\) 0 0
\(811\) 37.8595 1.32943 0.664714 0.747098i \(-0.268553\pi\)
0.664714 + 0.747098i \(0.268553\pi\)
\(812\) −25.2700 −0.886804
\(813\) 0 0
\(814\) −4.17577 −0.146361
\(815\) 0 0
\(816\) 0 0
\(817\) −8.06587 −0.282189
\(818\) 18.0899 0.632498
\(819\) 0 0
\(820\) 0 0
\(821\) 20.1976 0.704901 0.352451 0.935830i \(-0.385348\pi\)
0.352451 + 0.935830i \(0.385348\pi\)
\(822\) 0 0
\(823\) −4.34590 −0.151489 −0.0757443 0.997127i \(-0.524133\pi\)
−0.0757443 + 0.997127i \(0.524133\pi\)
\(824\) 8.33151 0.290242
\(825\) 0 0
\(826\) −14.2305 −0.495144
\(827\) 56.8375 1.97643 0.988217 0.153057i \(-0.0489119\pi\)
0.988217 + 0.153057i \(0.0489119\pi\)
\(828\) 0 0
\(829\) −17.4543 −0.606214 −0.303107 0.952957i \(-0.598024\pi\)
−0.303107 + 0.952957i \(0.598024\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −6.23163 −0.216043
\(833\) 0.506888 0.0175626
\(834\) 0 0
\(835\) 0 0
\(836\) 0.728896 0.0252094
\(837\) 0 0
\(838\) −15.3370 −0.529807
\(839\) −1.77622 −0.0613219 −0.0306609 0.999530i \(-0.509761\pi\)
−0.0306609 + 0.999530i \(0.509761\pi\)
\(840\) 0 0
\(841\) 75.6862 2.60987
\(842\) 8.42486 0.290340
\(843\) 0 0
\(844\) −11.6410 −0.400700
\(845\) 0 0
\(846\) 0 0
\(847\) −25.8556 −0.888408
\(848\) 1.53020 0.0525475
\(849\) 0 0
\(850\) 0 0
\(851\) 26.5566 0.910348
\(852\) 0 0
\(853\) −16.1187 −0.551892 −0.275946 0.961173i \(-0.588991\pi\)
−0.275946 + 0.961173i \(0.588991\pi\)
\(854\) −27.0185 −0.924556
\(855\) 0 0
\(856\) −15.0264 −0.513591
\(857\) 47.2031 1.61243 0.806213 0.591625i \(-0.201514\pi\)
0.806213 + 0.591625i \(0.201514\pi\)
\(858\) 0 0
\(859\) 37.3239 1.27347 0.636737 0.771081i \(-0.280284\pi\)
0.636737 + 0.771081i \(0.280284\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 16.2766 0.554382
\(863\) 1.33697 0.0455111 0.0227555 0.999741i \(-0.492756\pi\)
0.0227555 + 0.999741i \(0.492756\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 18.1976 0.618380
\(867\) 0 0
\(868\) 14.9815 0.508504
\(869\) −11.1791 −0.379224
\(870\) 0 0
\(871\) −80.5939 −2.73082
\(872\) 13.6619 0.462652
\(873\) 0 0
\(874\) −4.63555 −0.156800
\(875\) 0 0
\(876\) 0 0
\(877\) 13.1647 0.444539 0.222270 0.974985i \(-0.428653\pi\)
0.222270 + 0.974985i \(0.428653\pi\)
\(878\) −20.4214 −0.689188
\(879\) 0 0
\(880\) 0 0
\(881\) 10.2052 0.343823 0.171912 0.985112i \(-0.445006\pi\)
0.171912 + 0.985112i \(0.445006\pi\)
\(882\) 0 0
\(883\) 1.49966 0.0504674 0.0252337 0.999682i \(-0.491967\pi\)
0.0252337 + 0.999682i \(0.491967\pi\)
\(884\) 3.50927 0.118030
\(885\) 0 0
\(886\) 21.6685 0.727967
\(887\) 46.9505 1.57644 0.788222 0.615391i \(-0.211002\pi\)
0.788222 + 0.615391i \(0.211002\pi\)
\(888\) 0 0
\(889\) 23.5215 0.788886
\(890\) 0 0
\(891\) 0 0
\(892\) −15.6554 −0.524182
\(893\) 8.12628 0.271936
\(894\) 0 0
\(895\) 0 0
\(896\) −2.46980 −0.0825101
\(897\) 0 0
\(898\) 23.8133 0.794661
\(899\) −62.0637 −2.06994
\(900\) 0 0
\(901\) −0.861719 −0.0287080
\(902\) −3.49489 −0.116367
\(903\) 0 0
\(904\) 1.20524 0.0400855
\(905\) 0 0
\(906\) 0 0
\(907\) 16.3668 0.543452 0.271726 0.962375i \(-0.412406\pi\)
0.271726 + 0.962375i \(0.412406\pi\)
\(908\) −4.80131 −0.159337
\(909\) 0 0
\(910\) 0 0
\(911\) −32.3293 −1.07112 −0.535559 0.844498i \(-0.679899\pi\)
−0.535559 + 0.844498i \(0.679899\pi\)
\(912\) 0 0
\(913\) 5.72561 0.189490
\(914\) −9.15813 −0.302924
\(915\) 0 0
\(916\) 2.79476 0.0923416
\(917\) 42.9815 1.41937
\(918\) 0 0
\(919\) −22.8157 −0.752620 −0.376310 0.926494i \(-0.622807\pi\)
−0.376310 + 0.926494i \(0.622807\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −7.78931 −0.256527
\(923\) −27.4028 −0.901976
\(924\) 0 0
\(925\) 0 0
\(926\) −0.405011 −0.0133095
\(927\) 0 0
\(928\) 10.2316 0.335870
\(929\) 27.2436 0.893834 0.446917 0.894575i \(-0.352522\pi\)
0.446917 + 0.894575i \(0.352522\pi\)
\(930\) 0 0
\(931\) 0.900112 0.0295000
\(932\) −11.5422 −0.378078
\(933\) 0 0
\(934\) −1.95814 −0.0640722
\(935\) 0 0
\(936\) 0 0
\(937\) −51.5006 −1.68245 −0.841225 0.540685i \(-0.818165\pi\)
−0.841225 + 0.540685i \(0.818165\pi\)
\(938\) −31.9420 −1.04294
\(939\) 0 0
\(940\) 0 0
\(941\) 40.8541 1.33181 0.665903 0.746039i \(-0.268047\pi\)
0.665903 + 0.746039i \(0.268047\pi\)
\(942\) 0 0
\(943\) 22.2264 0.723791
\(944\) 5.76183 0.187532
\(945\) 0 0
\(946\) 5.87918 0.191149
\(947\) −38.2526 −1.24304 −0.621521 0.783398i \(-0.713485\pi\)
−0.621521 + 0.783398i \(0.713485\pi\)
\(948\) 0 0
\(949\) 25.5082 0.828031
\(950\) 0 0
\(951\) 0 0
\(952\) 1.39084 0.0450773
\(953\) −54.1187 −1.75308 −0.876538 0.481334i \(-0.840153\pi\)
−0.876538 + 0.481334i \(0.840153\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 26.2251 0.848180
\(957\) 0 0
\(958\) −22.9210 −0.740545
\(959\) −5.57952 −0.180172
\(960\) 0 0
\(961\) 5.79476 0.186928
\(962\) −35.7003 −1.15103
\(963\) 0 0
\(964\) −12.0659 −0.388615
\(965\) 0 0
\(966\) 0 0
\(967\) 28.4214 0.913970 0.456985 0.889474i \(-0.348929\pi\)
0.456985 + 0.889474i \(0.348929\pi\)
\(968\) 10.4687 0.336477
\(969\) 0 0
\(970\) 0 0
\(971\) −30.8057 −0.988601 −0.494301 0.869291i \(-0.664576\pi\)
−0.494301 + 0.869291i \(0.664576\pi\)
\(972\) 0 0
\(973\) 41.6423 1.33499
\(974\) 23.9450 0.767249
\(975\) 0 0
\(976\) 10.9396 0.350168
\(977\) −37.6554 −1.20470 −0.602351 0.798231i \(-0.705769\pi\)
−0.602351 + 0.798231i \(0.705769\pi\)
\(978\) 0 0
\(979\) −7.28896 −0.232956
\(980\) 0 0
\(981\) 0 0
\(982\) 3.86826 0.123441
\(983\) 38.7948 1.23736 0.618680 0.785643i \(-0.287668\pi\)
0.618680 + 0.785643i \(0.287668\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −5.76183 −0.183494
\(987\) 0 0
\(988\) 6.23163 0.198254
\(989\) −37.3898 −1.18893
\(990\) 0 0
\(991\) −15.0295 −0.477427 −0.238713 0.971090i \(-0.576726\pi\)
−0.238713 + 0.971090i \(0.576726\pi\)
\(992\) −6.06587 −0.192592
\(993\) 0 0
\(994\) −10.8606 −0.344478
\(995\) 0 0
\(996\) 0 0
\(997\) 1.18123 0.0374099 0.0187050 0.999825i \(-0.494046\pi\)
0.0187050 + 0.999825i \(0.494046\pi\)
\(998\) 7.92104 0.250736
\(999\) 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.cj.1.2 3
3.2 odd 2 950.2.a.m.1.1 yes 3
5.4 even 2 8550.2.a.co.1.2 3
12.11 even 2 7600.2.a.bm.1.3 3
15.2 even 4 950.2.b.g.799.6 6
15.8 even 4 950.2.b.g.799.1 6
15.14 odd 2 950.2.a.k.1.3 3
60.59 even 2 7600.2.a.cb.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
950.2.a.k.1.3 3 15.14 odd 2
950.2.a.m.1.1 yes 3 3.2 odd 2
950.2.b.g.799.1 6 15.8 even 4
950.2.b.g.799.6 6 15.2 even 4
7600.2.a.bm.1.3 3 12.11 even 2
7600.2.a.cb.1.1 3 60.59 even 2
8550.2.a.cj.1.2 3 1.1 even 1 trivial
8550.2.a.co.1.2 3 5.4 even 2