Properties

Label 8550.2.a.cp.1.3
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: 3.3.993.1
Defining polynomial: \(x^{3} - x^{2} - 6 x + 3\)
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 1.3
Root \(2.77339\) of defining polynomial
Character \(\chi\) \(=\) 8550.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +2.69168 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +2.69168 q^{7} +1.00000 q^{8} -5.54677 q^{11} +2.91829 q^{13} +2.69168 q^{14} +1.00000 q^{16} +4.91829 q^{17} +1.00000 q^{19} -5.54677 q^{22} +3.60997 q^{23} +2.91829 q^{26} +2.69168 q^{28} -1.08171 q^{29} +7.54677 q^{31} +1.00000 q^{32} +4.91829 q^{34} -4.54677 q^{37} +1.00000 q^{38} +9.54677 q^{43} -5.54677 q^{44} +3.60997 q^{46} -0.836581 q^{47} +0.245129 q^{49} +2.91829 q^{52} +9.78523 q^{53} +2.69168 q^{56} -1.08171 q^{58} -12.9933 q^{59} -7.38336 q^{61} +7.54677 q^{62} +1.00000 q^{64} -2.85510 q^{67} +4.91829 q^{68} -14.4769 q^{71} -5.15674 q^{73} -4.54677 q^{74} +1.00000 q^{76} -14.9301 q^{77} -3.09355 q^{79} +1.71019 q^{83} +9.54677 q^{86} -5.54677 q^{88} +5.09355 q^{89} +7.85510 q^{91} +3.60997 q^{92} -0.836581 q^{94} +17.2570 q^{97} +0.245129 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} + 3q^{4} - 2q^{7} + 3q^{8} + O(q^{10}) \) \( 3q + 3q^{2} + 3q^{4} - 2q^{7} + 3q^{8} - 2q^{11} + 6q^{13} - 2q^{14} + 3q^{16} + 12q^{17} + 3q^{19} - 2q^{22} - 2q^{23} + 6q^{26} - 2q^{28} - 6q^{29} + 8q^{31} + 3q^{32} + 12q^{34} + q^{37} + 3q^{38} + 14q^{43} - 2q^{44} - 2q^{46} + 3q^{47} + 9q^{49} + 6q^{52} - 10q^{53} - 2q^{56} - 6q^{58} - 6q^{59} - 2q^{61} + 8q^{62} + 3q^{64} - 4q^{67} + 12q^{68} + 6q^{71} + 12q^{73} + q^{74} + 3q^{76} - 10q^{77} + 20q^{79} - 4q^{83} + 14q^{86} - 2q^{88} - 14q^{89} + 19q^{91} - 2q^{92} + 3q^{94} + 28q^{97} + 9q^{98} + O(q^{100}) \)

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
\(7\) 2.69168 1.01736 0.508679 0.860956i \(-0.330134\pi\)
0.508679 + 0.860956i \(0.330134\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) −5.54677 −1.67242 −0.836208 0.548413i \(-0.815232\pi\)
−0.836208 + 0.548413i \(0.815232\pi\)
\(12\) 0 0
\(13\) 2.91829 0.809388 0.404694 0.914452i \(-0.367378\pi\)
0.404694 + 0.914452i \(0.367378\pi\)
\(14\) 2.69168 0.719381
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.91829 1.19286 0.596430 0.802665i \(-0.296585\pi\)
0.596430 + 0.802665i \(0.296585\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) −5.54677 −1.18258
\(23\) 3.60997 0.752730 0.376365 0.926471i \(-0.377174\pi\)
0.376365 + 0.926471i \(0.377174\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 2.91829 0.572324
\(27\) 0 0
\(28\) 2.69168 0.508679
\(29\) −1.08171 −0.200868 −0.100434 0.994944i \(-0.532023\pi\)
−0.100434 + 0.994944i \(0.532023\pi\)
\(30\) 0 0
\(31\) 7.54677 1.35544 0.677720 0.735320i \(-0.262968\pi\)
0.677720 + 0.735320i \(0.262968\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 4.91829 0.843480
\(35\) 0 0
\(36\) 0 0
\(37\) −4.54677 −0.747485 −0.373743 0.927532i \(-0.621926\pi\)
−0.373743 + 0.927532i \(0.621926\pi\)
\(38\) 1.00000 0.162221
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 9.54677 1.45587 0.727935 0.685646i \(-0.240480\pi\)
0.727935 + 0.685646i \(0.240480\pi\)
\(44\) −5.54677 −0.836208
\(45\) 0 0
\(46\) 3.60997 0.532261
\(47\) −0.836581 −0.122028 −0.0610139 0.998137i \(-0.519433\pi\)
−0.0610139 + 0.998137i \(0.519433\pi\)
\(48\) 0 0
\(49\) 0.245129 0.0350184
\(50\) 0 0
\(51\) 0 0
\(52\) 2.91829 0.404694
\(53\) 9.78523 1.34410 0.672052 0.740504i \(-0.265413\pi\)
0.672052 + 0.740504i \(0.265413\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 2.69168 0.359691
\(57\) 0 0
\(58\) −1.08171 −0.142035
\(59\) −12.9933 −1.69159 −0.845793 0.533511i \(-0.820872\pi\)
−0.845793 + 0.533511i \(0.820872\pi\)
\(60\) 0 0
\(61\) −7.38336 −0.945342 −0.472671 0.881239i \(-0.656710\pi\)
−0.472671 + 0.881239i \(0.656710\pi\)
\(62\) 7.54677 0.958441
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −2.85510 −0.348806 −0.174403 0.984674i \(-0.555799\pi\)
−0.174403 + 0.984674i \(0.555799\pi\)
\(68\) 4.91829 0.596430
\(69\) 0 0
\(70\) 0 0
\(71\) −14.4769 −1.71809 −0.859046 0.511898i \(-0.828943\pi\)
−0.859046 + 0.511898i \(0.828943\pi\)
\(72\) 0 0
\(73\) −5.15674 −0.603551 −0.301776 0.953379i \(-0.597579\pi\)
−0.301776 + 0.953379i \(0.597579\pi\)
\(74\) −4.54677 −0.528552
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) −14.9301 −1.70145
\(78\) 0 0
\(79\) −3.09355 −0.348052 −0.174026 0.984741i \(-0.555678\pi\)
−0.174026 + 0.984741i \(0.555678\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1.71019 0.187718 0.0938591 0.995585i \(-0.470080\pi\)
0.0938591 + 0.995585i \(0.470080\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 9.54677 1.02946
\(87\) 0 0
\(88\) −5.54677 −0.591288
\(89\) 5.09355 0.539915 0.269958 0.962872i \(-0.412990\pi\)
0.269958 + 0.962872i \(0.412990\pi\)
\(90\) 0 0
\(91\) 7.85510 0.823438
\(92\) 3.60997 0.376365
\(93\) 0 0
\(94\) −0.836581 −0.0862867
\(95\) 0 0
\(96\) 0 0
\(97\) 17.2570 1.75218 0.876090 0.482148i \(-0.160143\pi\)
0.876090 + 0.482148i \(0.160143\pi\)
\(98\) 0.245129 0.0247618
\(99\) 0 0
\(100\) 0 0
\(101\) 9.38336 0.933679 0.466839 0.884342i \(-0.345393\pi\)
0.466839 + 0.884342i \(0.345393\pi\)
\(102\) 0 0
\(103\) 12.9301 1.27404 0.637022 0.770846i \(-0.280166\pi\)
0.637022 + 0.770846i \(0.280166\pi\)
\(104\) 2.91829 0.286162
\(105\) 0 0
\(106\) 9.78523 0.950425
\(107\) 18.9420 1.83119 0.915595 0.402102i \(-0.131720\pi\)
0.915595 + 0.402102i \(0.131720\pi\)
\(108\) 0 0
\(109\) −2.69168 −0.257816 −0.128908 0.991657i \(-0.541147\pi\)
−0.128908 + 0.991657i \(0.541147\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 2.69168 0.254340
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −1.08171 −0.100434
\(117\) 0 0
\(118\) −12.9933 −1.19613
\(119\) 13.2385 1.21357
\(120\) 0 0
\(121\) 19.7667 1.79697
\(122\) −7.38336 −0.668458
\(123\) 0 0
\(124\) 7.54677 0.677720
\(125\) 0 0
\(126\) 0 0
\(127\) 1.87361 0.166256 0.0831282 0.996539i \(-0.473509\pi\)
0.0831282 + 0.996539i \(0.473509\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) 11.5468 1.00885 0.504423 0.863457i \(-0.331705\pi\)
0.504423 + 0.863457i \(0.331705\pi\)
\(132\) 0 0
\(133\) 2.69168 0.233398
\(134\) −2.85510 −0.246643
\(135\) 0 0
\(136\) 4.91829 0.421740
\(137\) 4.23845 0.362115 0.181058 0.983472i \(-0.442048\pi\)
0.181058 + 0.983472i \(0.442048\pi\)
\(138\) 0 0
\(139\) 9.21994 0.782025 0.391012 0.920385i \(-0.372125\pi\)
0.391012 + 0.920385i \(0.372125\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −14.4769 −1.21487
\(143\) −16.1871 −1.35363
\(144\) 0 0
\(145\) 0 0
\(146\) −5.15674 −0.426775
\(147\) 0 0
\(148\) −4.54677 −0.373743
\(149\) 7.25697 0.594514 0.297257 0.954797i \(-0.403928\pi\)
0.297257 + 0.954797i \(0.403928\pi\)
\(150\) 0 0
\(151\) 20.0000 1.62758 0.813788 0.581161i \(-0.197401\pi\)
0.813788 + 0.581161i \(0.197401\pi\)
\(152\) 1.00000 0.0811107
\(153\) 0 0
\(154\) −14.9301 −1.20310
\(155\) 0 0
\(156\) 0 0
\(157\) 9.54677 0.761916 0.380958 0.924592i \(-0.375594\pi\)
0.380958 + 0.924592i \(0.375594\pi\)
\(158\) −3.09355 −0.246110
\(159\) 0 0
\(160\) 0 0
\(161\) 9.71687 0.765797
\(162\) 0 0
\(163\) −15.7102 −1.23052 −0.615259 0.788325i \(-0.710948\pi\)
−0.615259 + 0.788325i \(0.710948\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 1.71019 0.132737
\(167\) 6.45323 0.499366 0.249683 0.968328i \(-0.419674\pi\)
0.249683 + 0.968328i \(0.419674\pi\)
\(168\) 0 0
\(169\) −4.48358 −0.344891
\(170\) 0 0
\(171\) 0 0
\(172\) 9.54677 0.727935
\(173\) −0.873614 −0.0664196 −0.0332098 0.999448i \(-0.510573\pi\)
−0.0332098 + 0.999448i \(0.510573\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −5.54677 −0.418104
\(177\) 0 0
\(178\) 5.09355 0.381778
\(179\) 12.3834 0.925575 0.462788 0.886469i \(-0.346849\pi\)
0.462788 + 0.886469i \(0.346849\pi\)
\(180\) 0 0
\(181\) −17.6403 −1.31119 −0.655597 0.755111i \(-0.727583\pi\)
−0.655597 + 0.755111i \(0.727583\pi\)
\(182\) 7.85510 0.582259
\(183\) 0 0
\(184\) 3.60997 0.266130
\(185\) 0 0
\(186\) 0 0
\(187\) −27.2806 −1.99496
\(188\) −0.836581 −0.0610139
\(189\) 0 0
\(190\) 0 0
\(191\) 15.1567 1.09670 0.548352 0.836248i \(-0.315256\pi\)
0.548352 + 0.836248i \(0.315256\pi\)
\(192\) 0 0
\(193\) −23.3834 −1.68317 −0.841585 0.540124i \(-0.818377\pi\)
−0.841585 + 0.540124i \(0.818377\pi\)
\(194\) 17.2570 1.23898
\(195\) 0 0
\(196\) 0.245129 0.0175092
\(197\) −23.0935 −1.64535 −0.822674 0.568514i \(-0.807519\pi\)
−0.822674 + 0.568514i \(0.807519\pi\)
\(198\) 0 0
\(199\) −25.7852 −1.82787 −0.913933 0.405865i \(-0.866970\pi\)
−0.913933 + 0.405865i \(0.866970\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 9.38336 0.660211
\(203\) −2.91161 −0.204355
\(204\) 0 0
\(205\) 0 0
\(206\) 12.9301 0.900885
\(207\) 0 0
\(208\) 2.91829 0.202347
\(209\) −5.54677 −0.383678
\(210\) 0 0
\(211\) 14.2266 0.979400 0.489700 0.871891i \(-0.337106\pi\)
0.489700 + 0.871891i \(0.337106\pi\)
\(212\) 9.78523 0.672052
\(213\) 0 0
\(214\) 18.9420 1.29485
\(215\) 0 0
\(216\) 0 0
\(217\) 20.3135 1.37897
\(218\) −2.69168 −0.182303
\(219\) 0 0
\(220\) 0 0
\(221\) 14.3530 0.965487
\(222\) 0 0
\(223\) 4.45323 0.298210 0.149105 0.988821i \(-0.452361\pi\)
0.149105 + 0.988821i \(0.452361\pi\)
\(224\) 2.69168 0.179845
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) 5.94865 0.394826 0.197413 0.980320i \(-0.436746\pi\)
0.197413 + 0.980320i \(0.436746\pi\)
\(228\) 0 0
\(229\) 1.09355 0.0722638 0.0361319 0.999347i \(-0.488496\pi\)
0.0361319 + 0.999347i \(0.488496\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −1.08171 −0.0710177
\(233\) 14.0935 0.923299 0.461650 0.887062i \(-0.347258\pi\)
0.461650 + 0.887062i \(0.347258\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −12.9933 −0.845793
\(237\) 0 0
\(238\) 13.2385 0.858121
\(239\) −15.6100 −1.00972 −0.504862 0.863200i \(-0.668457\pi\)
−0.504862 + 0.863200i \(0.668457\pi\)
\(240\) 0 0
\(241\) −5.21994 −0.336246 −0.168123 0.985766i \(-0.553771\pi\)
−0.168123 + 0.985766i \(0.553771\pi\)
\(242\) 19.7667 1.27065
\(243\) 0 0
\(244\) −7.38336 −0.472671
\(245\) 0 0
\(246\) 0 0
\(247\) 2.91829 0.185686
\(248\) 7.54677 0.479221
\(249\) 0 0
\(250\) 0 0
\(251\) 28.6403 1.80776 0.903881 0.427785i \(-0.140706\pi\)
0.903881 + 0.427785i \(0.140706\pi\)
\(252\) 0 0
\(253\) −20.0237 −1.25888
\(254\) 1.87361 0.117561
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 6.45323 0.402541 0.201271 0.979536i \(-0.435493\pi\)
0.201271 + 0.979536i \(0.435493\pi\)
\(258\) 0 0
\(259\) −12.2385 −0.760460
\(260\) 0 0
\(261\) 0 0
\(262\) 11.5468 0.713362
\(263\) 22.1871 1.36812 0.684058 0.729428i \(-0.260214\pi\)
0.684058 + 0.729428i \(0.260214\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 2.69168 0.165037
\(267\) 0 0
\(268\) −2.85510 −0.174403
\(269\) 20.4070 1.24424 0.622119 0.782922i \(-0.286272\pi\)
0.622119 + 0.782922i \(0.286272\pi\)
\(270\) 0 0
\(271\) 15.9368 0.968092 0.484046 0.875043i \(-0.339167\pi\)
0.484046 + 0.875043i \(0.339167\pi\)
\(272\) 4.91829 0.298215
\(273\) 0 0
\(274\) 4.23845 0.256054
\(275\) 0 0
\(276\) 0 0
\(277\) −4.02368 −0.241759 −0.120880 0.992667i \(-0.538572\pi\)
−0.120880 + 0.992667i \(0.538572\pi\)
\(278\) 9.21994 0.552975
\(279\) 0 0
\(280\) 0 0
\(281\) −1.67316 −0.0998124 −0.0499062 0.998754i \(-0.515892\pi\)
−0.0499062 + 0.998754i \(0.515892\pi\)
\(282\) 0 0
\(283\) −16.8931 −1.00419 −0.502095 0.864812i \(-0.667437\pi\)
−0.502095 + 0.864812i \(0.667437\pi\)
\(284\) −14.4769 −0.859046
\(285\) 0 0
\(286\) −16.1871 −0.957163
\(287\) 0 0
\(288\) 0 0
\(289\) 7.18958 0.422916
\(290\) 0 0
\(291\) 0 0
\(292\) −5.15674 −0.301776
\(293\) −1.08171 −0.0631942 −0.0315971 0.999501i \(-0.510059\pi\)
−0.0315971 + 0.999501i \(0.510059\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −4.54677 −0.264276
\(297\) 0 0
\(298\) 7.25697 0.420385
\(299\) 10.5349 0.609251
\(300\) 0 0
\(301\) 25.6968 1.48114
\(302\) 20.0000 1.15087
\(303\) 0 0
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) 0 0
\(307\) −2.38336 −0.136025 −0.0680126 0.997684i \(-0.521666\pi\)
−0.0680126 + 0.997684i \(0.521666\pi\)
\(308\) −14.9301 −0.850723
\(309\) 0 0
\(310\) 0 0
\(311\) 29.4584 1.67043 0.835216 0.549922i \(-0.185343\pi\)
0.835216 + 0.549922i \(0.185343\pi\)
\(312\) 0 0
\(313\) −32.0355 −1.81075 −0.905377 0.424608i \(-0.860412\pi\)
−0.905377 + 0.424608i \(0.860412\pi\)
\(314\) 9.54677 0.538756
\(315\) 0 0
\(316\) −3.09355 −0.174026
\(317\) −26.6665 −1.49774 −0.748870 0.662717i \(-0.769403\pi\)
−0.748870 + 0.662717i \(0.769403\pi\)
\(318\) 0 0
\(319\) 6.00000 0.335936
\(320\) 0 0
\(321\) 0 0
\(322\) 9.71687 0.541500
\(323\) 4.91829 0.273661
\(324\) 0 0
\(325\) 0 0
\(326\) −15.7102 −0.870107
\(327\) 0 0
\(328\) 0 0
\(329\) −2.25181 −0.124146
\(330\) 0 0
\(331\) −19.8721 −1.09227 −0.546135 0.837697i \(-0.683901\pi\)
−0.546135 + 0.837697i \(0.683901\pi\)
\(332\) 1.71019 0.0938591
\(333\) 0 0
\(334\) 6.45323 0.353105
\(335\) 0 0
\(336\) 0 0
\(337\) −27.5705 −1.50186 −0.750929 0.660383i \(-0.770394\pi\)
−0.750929 + 0.660383i \(0.770394\pi\)
\(338\) −4.48358 −0.243875
\(339\) 0 0
\(340\) 0 0
\(341\) −41.8603 −2.26686
\(342\) 0 0
\(343\) −18.1819 −0.981732
\(344\) 9.54677 0.514728
\(345\) 0 0
\(346\) −0.873614 −0.0469658
\(347\) −27.2806 −1.46450 −0.732251 0.681035i \(-0.761530\pi\)
−0.732251 + 0.681035i \(0.761530\pi\)
\(348\) 0 0
\(349\) 30.9538 1.65692 0.828460 0.560049i \(-0.189218\pi\)
0.828460 + 0.560049i \(0.189218\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −5.54677 −0.295644
\(353\) 11.1819 0.595154 0.297577 0.954698i \(-0.403821\pi\)
0.297577 + 0.954698i \(0.403821\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 5.09355 0.269958
\(357\) 0 0
\(358\) 12.3834 0.654481
\(359\) −20.3387 −1.07343 −0.536717 0.843762i \(-0.680336\pi\)
−0.536717 + 0.843762i \(0.680336\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −17.6403 −0.927155
\(363\) 0 0
\(364\) 7.85510 0.411719
\(365\) 0 0
\(366\) 0 0
\(367\) 32.9538 1.72017 0.860087 0.510147i \(-0.170409\pi\)
0.860087 + 0.510147i \(0.170409\pi\)
\(368\) 3.60997 0.188183
\(369\) 0 0
\(370\) 0 0
\(371\) 26.3387 1.36744
\(372\) 0 0
\(373\) 10.0869 0.522278 0.261139 0.965301i \(-0.415902\pi\)
0.261139 + 0.965301i \(0.415902\pi\)
\(374\) −27.2806 −1.41065
\(375\) 0 0
\(376\) −0.836581 −0.0431434
\(377\) −3.15674 −0.162581
\(378\) 0 0
\(379\) −18.4256 −0.946457 −0.473229 0.880940i \(-0.656912\pi\)
−0.473229 + 0.880940i \(0.656912\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 15.1567 0.775486
\(383\) 28.1871 1.44029 0.720147 0.693822i \(-0.244074\pi\)
0.720147 + 0.693822i \(0.244074\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −23.3834 −1.19018
\(387\) 0 0
\(388\) 17.2570 0.876090
\(389\) −20.0237 −1.01524 −0.507620 0.861581i \(-0.669475\pi\)
−0.507620 + 0.861581i \(0.669475\pi\)
\(390\) 0 0
\(391\) 17.7549 0.897902
\(392\) 0.245129 0.0123809
\(393\) 0 0
\(394\) −23.0935 −1.16344
\(395\) 0 0
\(396\) 0 0
\(397\) 8.32684 0.417912 0.208956 0.977925i \(-0.432993\pi\)
0.208956 + 0.977925i \(0.432993\pi\)
\(398\) −25.7852 −1.29250
\(399\) 0 0
\(400\) 0 0
\(401\) 22.1501 1.10612 0.553061 0.833141i \(-0.313460\pi\)
0.553061 + 0.833141i \(0.313460\pi\)
\(402\) 0 0
\(403\) 22.0237 1.09708
\(404\) 9.38336 0.466839
\(405\) 0 0
\(406\) −2.91161 −0.144501
\(407\) 25.2199 1.25011
\(408\) 0 0
\(409\) 34.0237 1.68236 0.841181 0.540753i \(-0.181861\pi\)
0.841181 + 0.540753i \(0.181861\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 12.9301 0.637022
\(413\) −34.9738 −1.72095
\(414\) 0 0
\(415\) 0 0
\(416\) 2.91829 0.143081
\(417\) 0 0
\(418\) −5.54677 −0.271302
\(419\) −37.5334 −1.83363 −0.916814 0.399315i \(-0.869248\pi\)
−0.916814 + 0.399315i \(0.869248\pi\)
\(420\) 0 0
\(421\) 33.8341 1.64897 0.824487 0.565882i \(-0.191464\pi\)
0.824487 + 0.565882i \(0.191464\pi\)
\(422\) 14.2266 0.692541
\(423\) 0 0
\(424\) 9.78523 0.475213
\(425\) 0 0
\(426\) 0 0
\(427\) −19.8736 −0.961752
\(428\) 18.9420 0.915595
\(429\) 0 0
\(430\) 0 0
\(431\) −24.8037 −1.19475 −0.597377 0.801960i \(-0.703790\pi\)
−0.597377 + 0.801960i \(0.703790\pi\)
\(432\) 0 0
\(433\) −1.68651 −0.0810487 −0.0405244 0.999179i \(-0.512903\pi\)
−0.0405244 + 0.999179i \(0.512903\pi\)
\(434\) 20.3135 0.975079
\(435\) 0 0
\(436\) −2.69168 −0.128908
\(437\) 3.60997 0.172688
\(438\) 0 0
\(439\) 0.326839 0.0155992 0.00779958 0.999970i \(-0.497517\pi\)
0.00779958 + 0.999970i \(0.497517\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 14.3530 0.682703
\(443\) 0.490258 0.0232929 0.0116464 0.999932i \(-0.496293\pi\)
0.0116464 + 0.999932i \(0.496293\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 4.45323 0.210866
\(447\) 0 0
\(448\) 2.69168 0.127170
\(449\) −23.5468 −1.11124 −0.555621 0.831436i \(-0.687519\pi\)
−0.555621 + 0.831436i \(0.687519\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −6.00000 −0.282216
\(453\) 0 0
\(454\) 5.94865 0.279184
\(455\) 0 0
\(456\) 0 0
\(457\) 8.01184 0.374778 0.187389 0.982286i \(-0.439997\pi\)
0.187389 + 0.982286i \(0.439997\pi\)
\(458\) 1.09355 0.0510982
\(459\) 0 0
\(460\) 0 0
\(461\) −3.83658 −0.178687 −0.0893437 0.996001i \(-0.528477\pi\)
−0.0893437 + 0.996001i \(0.528477\pi\)
\(462\) 0 0
\(463\) 6.58381 0.305975 0.152988 0.988228i \(-0.451110\pi\)
0.152988 + 0.988228i \(0.451110\pi\)
\(464\) −1.08171 −0.0502171
\(465\) 0 0
\(466\) 14.0935 0.652871
\(467\) 22.6033 1.04596 0.522978 0.852346i \(-0.324821\pi\)
0.522978 + 0.852346i \(0.324821\pi\)
\(468\) 0 0
\(469\) −7.68500 −0.354860
\(470\) 0 0
\(471\) 0 0
\(472\) −12.9933 −0.598066
\(473\) −52.9538 −2.43482
\(474\) 0 0
\(475\) 0 0
\(476\) 13.2385 0.606783
\(477\) 0 0
\(478\) −15.6100 −0.713983
\(479\) 35.7904 1.63530 0.817652 0.575712i \(-0.195275\pi\)
0.817652 + 0.575712i \(0.195275\pi\)
\(480\) 0 0
\(481\) −13.2688 −0.605006
\(482\) −5.21994 −0.237762
\(483\) 0 0
\(484\) 19.7667 0.898487
\(485\) 0 0
\(486\) 0 0
\(487\) 21.4070 0.970045 0.485023 0.874502i \(-0.338811\pi\)
0.485023 + 0.874502i \(0.338811\pi\)
\(488\) −7.38336 −0.334229
\(489\) 0 0
\(490\) 0 0
\(491\) 4.32684 0.195267 0.0976337 0.995222i \(-0.468873\pi\)
0.0976337 + 0.995222i \(0.468873\pi\)
\(492\) 0 0
\(493\) −5.32016 −0.239608
\(494\) 2.91829 0.131300
\(495\) 0 0
\(496\) 7.54677 0.338860
\(497\) −38.9672 −1.74792
\(498\) 0 0
\(499\) −37.6968 −1.68754 −0.843771 0.536703i \(-0.819670\pi\)
−0.843771 + 0.536703i \(0.819670\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 28.6403 1.27828
\(503\) −16.9183 −0.754349 −0.377175 0.926142i \(-0.623104\pi\)
−0.377175 + 0.926142i \(0.623104\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −20.0237 −0.890161
\(507\) 0 0
\(508\) 1.87361 0.0831282
\(509\) 6.79955 0.301385 0.150692 0.988581i \(-0.451850\pi\)
0.150692 + 0.988581i \(0.451850\pi\)
\(510\) 0 0
\(511\) −13.8803 −0.614028
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 6.45323 0.284640
\(515\) 0 0
\(516\) 0 0
\(517\) 4.64032 0.204081
\(518\) −12.2385 −0.537727
\(519\) 0 0
\(520\) 0 0
\(521\) 1.35968 0.0595685 0.0297842 0.999556i \(-0.490518\pi\)
0.0297842 + 0.999556i \(0.490518\pi\)
\(522\) 0 0
\(523\) 21.9486 0.959747 0.479874 0.877338i \(-0.340682\pi\)
0.479874 + 0.877338i \(0.340682\pi\)
\(524\) 11.5468 0.504423
\(525\) 0 0
\(526\) 22.1871 0.967404
\(527\) 37.1172 1.61685
\(528\) 0 0
\(529\) −9.96813 −0.433397
\(530\) 0 0
\(531\) 0 0
\(532\) 2.69168 0.116699
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) −2.85510 −0.123321
\(537\) 0 0
\(538\) 20.4070 0.879810
\(539\) −1.35968 −0.0585654
\(540\) 0 0
\(541\) −39.8973 −1.71532 −0.857659 0.514218i \(-0.828082\pi\)
−0.857659 + 0.514218i \(0.828082\pi\)
\(542\) 15.9368 0.684544
\(543\) 0 0
\(544\) 4.91829 0.210870
\(545\) 0 0
\(546\) 0 0
\(547\) 7.34632 0.314106 0.157053 0.987590i \(-0.449801\pi\)
0.157053 + 0.987590i \(0.449801\pi\)
\(548\) 4.23845 0.181058
\(549\) 0 0
\(550\) 0 0
\(551\) −1.08171 −0.0460824
\(552\) 0 0
\(553\) −8.32684 −0.354093
\(554\) −4.02368 −0.170950
\(555\) 0 0
\(556\) 9.21994 0.391012
\(557\) −26.9672 −1.14264 −0.571318 0.820729i \(-0.693568\pi\)
−0.571318 + 0.820729i \(0.693568\pi\)
\(558\) 0 0
\(559\) 27.8603 1.17836
\(560\) 0 0
\(561\) 0 0
\(562\) −1.67316 −0.0705780
\(563\) −44.8973 −1.89220 −0.946098 0.323881i \(-0.895012\pi\)
−0.946098 + 0.323881i \(0.895012\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −16.8931 −0.710070
\(567\) 0 0
\(568\) −14.4769 −0.607437
\(569\) −31.1172 −1.30450 −0.652251 0.758003i \(-0.726175\pi\)
−0.652251 + 0.758003i \(0.726175\pi\)
\(570\) 0 0
\(571\) 33.2570 1.39176 0.695880 0.718158i \(-0.255014\pi\)
0.695880 + 0.718158i \(0.255014\pi\)
\(572\) −16.1871 −0.676817
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −23.4702 −0.977078 −0.488539 0.872542i \(-0.662470\pi\)
−0.488539 + 0.872542i \(0.662470\pi\)
\(578\) 7.18958 0.299047
\(579\) 0 0
\(580\) 0 0
\(581\) 4.60329 0.190977
\(582\) 0 0
\(583\) −54.2765 −2.24790
\(584\) −5.15674 −0.213388
\(585\) 0 0
\(586\) −1.08171 −0.0446850
\(587\) −10.1501 −0.418938 −0.209469 0.977815i \(-0.567174\pi\)
−0.209469 + 0.977815i \(0.567174\pi\)
\(588\) 0 0
\(589\) 7.54677 0.310959
\(590\) 0 0
\(591\) 0 0
\(592\) −4.54677 −0.186871
\(593\) −16.6732 −0.684685 −0.342342 0.939575i \(-0.611220\pi\)
−0.342342 + 0.939575i \(0.611220\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 7.25697 0.297257
\(597\) 0 0
\(598\) 10.5349 0.430806
\(599\) −18.2765 −0.746756 −0.373378 0.927679i \(-0.621800\pi\)
−0.373378 + 0.927679i \(0.621800\pi\)
\(600\) 0 0
\(601\) 7.96297 0.324816 0.162408 0.986724i \(-0.448074\pi\)
0.162408 + 0.986724i \(0.448074\pi\)
\(602\) 25.6968 1.04733
\(603\) 0 0
\(604\) 20.0000 0.813788
\(605\) 0 0
\(606\) 0 0
\(607\) 6.93013 0.281285 0.140643 0.990060i \(-0.455083\pi\)
0.140643 + 0.990060i \(0.455083\pi\)
\(608\) 1.00000 0.0405554
\(609\) 0 0
\(610\) 0 0
\(611\) −2.44139 −0.0987679
\(612\) 0 0
\(613\) 34.2765 1.38441 0.692206 0.721700i \(-0.256639\pi\)
0.692206 + 0.721700i \(0.256639\pi\)
\(614\) −2.38336 −0.0961844
\(615\) 0 0
\(616\) −14.9301 −0.601552
\(617\) 35.5139 1.42974 0.714869 0.699259i \(-0.246486\pi\)
0.714869 + 0.699259i \(0.246486\pi\)
\(618\) 0 0
\(619\) −29.5334 −1.18705 −0.593524 0.804816i \(-0.702264\pi\)
−0.593524 + 0.804816i \(0.702264\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 29.4584 1.18117
\(623\) 13.7102 0.549287
\(624\) 0 0
\(625\) 0 0
\(626\) −32.0355 −1.28040
\(627\) 0 0
\(628\) 9.54677 0.380958
\(629\) −22.3624 −0.891646
\(630\) 0 0
\(631\) −15.1634 −0.603646 −0.301823 0.953364i \(-0.597595\pi\)
−0.301823 + 0.953364i \(0.597595\pi\)
\(632\) −3.09355 −0.123055
\(633\) 0 0
\(634\) −26.6665 −1.05906
\(635\) 0 0
\(636\) 0 0
\(637\) 0.715358 0.0283435
\(638\) 6.00000 0.237542
\(639\) 0 0
\(640\) 0 0
\(641\) 25.0802 0.990608 0.495304 0.868720i \(-0.335057\pi\)
0.495304 + 0.868720i \(0.335057\pi\)
\(642\) 0 0
\(643\) −19.1306 −0.754437 −0.377218 0.926124i \(-0.623119\pi\)
−0.377218 + 0.926124i \(0.623119\pi\)
\(644\) 9.71687 0.382898
\(645\) 0 0
\(646\) 4.91829 0.193508
\(647\) 12.4019 0.487568 0.243784 0.969830i \(-0.421611\pi\)
0.243784 + 0.969830i \(0.421611\pi\)
\(648\) 0 0
\(649\) 72.0710 2.82904
\(650\) 0 0
\(651\) 0 0
\(652\) −15.7102 −0.615259
\(653\) −10.7801 −0.421856 −0.210928 0.977502i \(-0.567649\pi\)
−0.210928 + 0.977502i \(0.567649\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) −2.25181 −0.0877845
\(659\) −8.56529 −0.333656 −0.166828 0.985986i \(-0.553353\pi\)
−0.166828 + 0.985986i \(0.553353\pi\)
\(660\) 0 0
\(661\) −26.2883 −1.02250 −0.511248 0.859433i \(-0.670817\pi\)
−0.511248 + 0.859433i \(0.670817\pi\)
\(662\) −19.8721 −0.772351
\(663\) 0 0
\(664\) 1.71019 0.0663684
\(665\) 0 0
\(666\) 0 0
\(667\) −3.90494 −0.151200
\(668\) 6.45323 0.249683
\(669\) 0 0
\(670\) 0 0
\(671\) 40.9538 1.58100
\(672\) 0 0
\(673\) 12.9301 0.498420 0.249210 0.968449i \(-0.419829\pi\)
0.249210 + 0.968449i \(0.419829\pi\)
\(674\) −27.5705 −1.06197
\(675\) 0 0
\(676\) −4.48358 −0.172445
\(677\) −15.9250 −0.612046 −0.306023 0.952024i \(-0.598998\pi\)
−0.306023 + 0.952024i \(0.598998\pi\)
\(678\) 0 0
\(679\) 46.4502 1.78260
\(680\) 0 0
\(681\) 0 0
\(682\) −41.8603 −1.60291
\(683\) 13.2898 0.508520 0.254260 0.967136i \(-0.418168\pi\)
0.254260 + 0.967136i \(0.418168\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −18.1819 −0.694190
\(687\) 0 0
\(688\) 9.54677 0.363967
\(689\) 28.5561 1.08790
\(690\) 0 0
\(691\) 27.2570 1.03690 0.518452 0.855107i \(-0.326508\pi\)
0.518452 + 0.855107i \(0.326508\pi\)
\(692\) −0.873614 −0.0332098
\(693\) 0 0
\(694\) −27.2806 −1.03556
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 30.9538 1.17162
\(699\) 0 0
\(700\) 0 0
\(701\) 41.4441 1.56532 0.782660 0.622449i \(-0.213862\pi\)
0.782660 + 0.622449i \(0.213862\pi\)
\(702\) 0 0
\(703\) −4.54677 −0.171485
\(704\) −5.54677 −0.209052
\(705\) 0 0
\(706\) 11.1819 0.420838
\(707\) 25.2570 0.949886
\(708\) 0 0
\(709\) −36.7904 −1.38169 −0.690846 0.723002i \(-0.742762\pi\)
−0.690846 + 0.723002i \(0.742762\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 5.09355 0.190889
\(713\) 27.2436 1.02028
\(714\) 0 0
\(715\) 0 0
\(716\) 12.3834 0.462788
\(717\) 0 0
\(718\) −20.3387 −0.759033
\(719\) 4.13726 0.154294 0.0771469 0.997020i \(-0.475419\pi\)
0.0771469 + 0.997020i \(0.475419\pi\)
\(720\) 0 0
\(721\) 34.8037 1.29616
\(722\) 1.00000 0.0372161
\(723\) 0 0
\(724\) −17.6403 −0.655597
\(725\) 0 0
\(726\) 0 0
\(727\) 40.4374 1.49974 0.749870 0.661585i \(-0.230116\pi\)
0.749870 + 0.661585i \(0.230116\pi\)
\(728\) 7.85510 0.291129
\(729\) 0 0
\(730\) 0 0
\(731\) 46.9538 1.73665
\(732\) 0 0
\(733\) −28.1264 −1.03887 −0.519436 0.854509i \(-0.673858\pi\)
−0.519436 + 0.854509i \(0.673858\pi\)
\(734\) 32.9538 1.21635
\(735\) 0 0
\(736\) 3.60997 0.133065
\(737\) 15.8366 0.583348
\(738\) 0 0
\(739\) −38.1501 −1.40337 −0.701686 0.712486i \(-0.747569\pi\)
−0.701686 + 0.712486i \(0.747569\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 26.3387 0.966923
\(743\) −17.8232 −0.653871 −0.326935 0.945047i \(-0.606016\pi\)
−0.326935 + 0.945047i \(0.606016\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 10.0869 0.369307
\(747\) 0 0
\(748\) −27.2806 −0.997479
\(749\) 50.9857 1.86298
\(750\) 0 0
\(751\) 23.6968 0.864710 0.432355 0.901703i \(-0.357683\pi\)
0.432355 + 0.901703i \(0.357683\pi\)
\(752\) −0.836581 −0.0305070
\(753\) 0 0
\(754\) −3.15674 −0.114962
\(755\) 0 0
\(756\) 0 0
\(757\) 6.51394 0.236753 0.118377 0.992969i \(-0.462231\pi\)
0.118377 + 0.992969i \(0.462231\pi\)
\(758\) −18.4256 −0.669246
\(759\) 0 0
\(760\) 0 0
\(761\) −34.7786 −1.26072 −0.630361 0.776302i \(-0.717093\pi\)
−0.630361 + 0.776302i \(0.717093\pi\)
\(762\) 0 0
\(763\) −7.24513 −0.262291
\(764\) 15.1567 0.548352
\(765\) 0 0
\(766\) 28.1871 1.01844
\(767\) −37.9183 −1.36915
\(768\) 0 0
\(769\) −27.6850 −0.998347 −0.499173 0.866502i \(-0.666363\pi\)
−0.499173 + 0.866502i \(0.666363\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −23.3834 −0.841585
\(773\) −19.9738 −0.718409 −0.359205 0.933259i \(-0.616952\pi\)
−0.359205 + 0.933259i \(0.616952\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 17.2570 0.619489
\(777\) 0 0
\(778\) −20.0237 −0.717884
\(779\) 0 0
\(780\) 0 0
\(781\) 80.3001 2.87336
\(782\) 17.7549 0.634913
\(783\) 0 0
\(784\) 0.245129 0.00875461
\(785\) 0 0
\(786\) 0 0
\(787\) −42.0118 −1.49756 −0.748780 0.662818i \(-0.769360\pi\)
−0.748780 + 0.662818i \(0.769360\pi\)
\(788\) −23.0935 −0.822674
\(789\) 0 0
\(790\) 0 0
\(791\) −16.1501 −0.574230
\(792\) 0 0
\(793\) −21.5468 −0.765148
\(794\) 8.32684 0.295508
\(795\) 0 0
\(796\) −25.7852 −0.913933
\(797\) −45.1054 −1.59771 −0.798857 0.601520i \(-0.794562\pi\)
−0.798857 + 0.601520i \(0.794562\pi\)
\(798\) 0 0
\(799\) −4.11455 −0.145562
\(800\) 0 0
\(801\) 0 0
\(802\) 22.1501 0.782146
\(803\) 28.6033 1.00939
\(804\) 0 0
\(805\) 0 0
\(806\) 22.0237 0.775751
\(807\) 0 0
\(808\) 9.38336 0.330105
\(809\) −6.85510 −0.241012 −0.120506 0.992713i \(-0.538452\pi\)
−0.120506 + 0.992713i \(0.538452\pi\)
\(810\) 0 0
\(811\) −15.1819 −0.533110 −0.266555 0.963820i \(-0.585885\pi\)
−0.266555 + 0.963820i \(0.585885\pi\)
\(812\) −2.91161 −0.102178
\(813\) 0 0
\(814\) 25.2199 0.883958
\(815\) 0 0
\(816\) 0 0
\(817\) 9.54677 0.333999
\(818\) 34.0237 1.18961
\(819\) 0 0
\(820\) 0 0
\(821\) −22.5006 −0.785276 −0.392638 0.919693i \(-0.628437\pi\)
−0.392638 + 0.919693i \(0.628437\pi\)
\(822\) 0 0
\(823\) 16.1896 0.564333 0.282167 0.959365i \(-0.408947\pi\)
0.282167 + 0.959365i \(0.408947\pi\)
\(824\) 12.9301 0.450442
\(825\) 0 0
\(826\) −34.9738 −1.21690
\(827\) −25.0817 −0.872177 −0.436088 0.899904i \(-0.643636\pi\)
−0.436088 + 0.899904i \(0.643636\pi\)
\(828\) 0 0
\(829\) 21.3068 0.740016 0.370008 0.929029i \(-0.379355\pi\)
0.370008 + 0.929029i \(0.379355\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 2.91829 0.101174
\(833\) 1.20562 0.0417721
\(834\) 0 0
\(835\) 0 0
\(836\) −5.54677 −0.191839
\(837\) 0 0
\(838\) −37.5334 −1.29657
\(839\) 11.4727 0.396082 0.198041 0.980194i \(-0.436542\pi\)
0.198041 + 0.980194i \(0.436542\pi\)
\(840\) 0 0
\(841\) −27.8299 −0.959652
\(842\) 33.8341 1.16600
\(843\) 0 0
\(844\) 14.2266 0.489700
\(845\) 0 0
\(846\) 0 0
\(847\) 53.2056 1.82817
\(848\) 9.78523 0.336026
\(849\) 0 0
\(850\) 0 0
\(851\) −16.4137 −0.562655
\(852\) 0 0
\(853\) 38.9908 1.33502 0.667511 0.744600i \(-0.267360\pi\)
0.667511 + 0.744600i \(0.267360\pi\)
\(854\) −19.8736 −0.680061
\(855\) 0 0
\(856\) 18.9420 0.647423
\(857\) −29.8973 −1.02127 −0.510636 0.859797i \(-0.670590\pi\)
−0.510636 + 0.859797i \(0.670590\pi\)
\(858\) 0 0
\(859\) −6.47691 −0.220989 −0.110495 0.993877i \(-0.535243\pi\)
−0.110495 + 0.993877i \(0.535243\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −24.8037 −0.844819
\(863\) −5.09355 −0.173386 −0.0866932 0.996235i \(-0.527630\pi\)
−0.0866932 + 0.996235i \(0.527630\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −1.68651 −0.0573101
\(867\) 0 0
\(868\) 20.3135 0.689485
\(869\) 17.1592 0.582087
\(870\) 0 0
\(871\) −8.33200 −0.282319
\(872\) −2.69168 −0.0911517
\(873\) 0 0
\(874\) 3.60997 0.122109
\(875\) 0 0
\(876\) 0 0
\(877\) 29.8341 1.00743 0.503713 0.863871i \(-0.331967\pi\)
0.503713 + 0.863871i \(0.331967\pi\)
\(878\) 0.326839 0.0110303
\(879\) 0 0
\(880\) 0 0
\(881\) 11.5796 0.390127 0.195064 0.980791i \(-0.437509\pi\)
0.195064 + 0.980791i \(0.437509\pi\)
\(882\) 0 0
\(883\) 48.0237 1.61613 0.808063 0.589096i \(-0.200516\pi\)
0.808063 + 0.589096i \(0.200516\pi\)
\(884\) 14.3530 0.482744
\(885\) 0 0
\(886\) 0.490258 0.0164705
\(887\) 9.38336 0.315062 0.157531 0.987514i \(-0.449647\pi\)
0.157531 + 0.987514i \(0.449647\pi\)
\(888\) 0 0
\(889\) 5.04316 0.169142
\(890\) 0 0
\(891\) 0 0
\(892\) 4.45323 0.149105
\(893\) −0.836581 −0.0279951
\(894\) 0 0
\(895\) 0 0
\(896\) 2.69168 0.0899226
\(897\) 0 0
\(898\) −23.5468 −0.785766
\(899\) −8.16342 −0.272265
\(900\) 0 0
\(901\) 48.1266 1.60333
\(902\) 0 0
\(903\) 0 0
\(904\) −6.00000 −0.199557
\(905\) 0 0
\(906\) 0 0
\(907\) 19.4347 0.645319 0.322659 0.946515i \(-0.395423\pi\)
0.322659 + 0.946515i \(0.395423\pi\)
\(908\) 5.94865 0.197413
\(909\) 0 0
\(910\) 0 0
\(911\) 19.9866 0.662187 0.331094 0.943598i \(-0.392582\pi\)
0.331094 + 0.943598i \(0.392582\pi\)
\(912\) 0 0
\(913\) −9.48606 −0.313943
\(914\) 8.01184 0.265008
\(915\) 0 0
\(916\) 1.09355 0.0361319
\(917\) 31.0802 1.02636
\(918\) 0 0
\(919\) 6.15158 0.202922 0.101461 0.994840i \(-0.467648\pi\)
0.101461 + 0.994840i \(0.467648\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −3.83658 −0.126351
\(923\) −42.2478 −1.39060
\(924\) 0 0
\(925\) 0 0
\(926\) 6.58381 0.216357
\(927\) 0 0
\(928\) −1.08171 −0.0355089
\(929\) 7.17010 0.235243 0.117622 0.993058i \(-0.462473\pi\)
0.117622 + 0.993058i \(0.462473\pi\)
\(930\) 0 0
\(931\) 0.245129 0.00803378
\(932\) 14.0935 0.461650
\(933\) 0 0
\(934\) 22.6033 0.739602
\(935\) 0 0
\(936\) 0 0
\(937\) 24.2646 0.792690 0.396345 0.918102i \(-0.370278\pi\)
0.396345 + 0.918102i \(0.370278\pi\)
\(938\) −7.68500 −0.250924
\(939\) 0 0
\(940\) 0 0
\(941\) 7.44806 0.242800 0.121400 0.992604i \(-0.461262\pi\)
0.121400 + 0.992604i \(0.461262\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −12.9933 −0.422897
\(945\) 0 0
\(946\) −52.9538 −1.72168
\(947\) −52.9538 −1.72077 −0.860384 0.509647i \(-0.829776\pi\)
−0.860384 + 0.509647i \(0.829776\pi\)
\(948\) 0 0
\(949\) −15.0489 −0.488507
\(950\) 0 0
\(951\) 0 0
\(952\) 13.2385 0.429061
\(953\) −12.8694 −0.416881 −0.208441 0.978035i \(-0.566839\pi\)
−0.208441 + 0.978035i \(0.566839\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −15.6100 −0.504862
\(957\) 0 0
\(958\) 35.7904 1.15634
\(959\) 11.4085 0.368401
\(960\) 0 0
\(961\) 25.9538 0.837220
\(962\) −13.2688 −0.427804
\(963\) 0 0
\(964\) −5.21994 −0.168123
\(965\) 0 0
\(966\) 0 0
\(967\) −25.8603 −0.831610 −0.415805 0.909454i \(-0.636500\pi\)
−0.415805 + 0.909454i \(0.636500\pi\)
\(968\) 19.7667 0.635326
\(969\) 0 0
\(970\) 0 0
\(971\) −35.0935 −1.12621 −0.563103 0.826387i \(-0.690392\pi\)
−0.563103 + 0.826387i \(0.690392\pi\)
\(972\) 0 0
\(973\) 24.8171 0.795600
\(974\) 21.4070 0.685926
\(975\) 0 0
\(976\) −7.38336 −0.236335
\(977\) −42.1737 −1.34926 −0.674629 0.738157i \(-0.735696\pi\)
−0.674629 + 0.738157i \(0.735696\pi\)
\(978\) 0 0
\(979\) −28.2528 −0.902963
\(980\) 0 0
\(981\) 0 0
\(982\) 4.32684 0.138075
\(983\) −42.6270 −1.35959 −0.679795 0.733403i \(-0.737931\pi\)
−0.679795 + 0.733403i \(0.737931\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −5.32016 −0.169428
\(987\) 0 0
\(988\) 2.91829 0.0928432
\(989\) 34.4636 1.09588
\(990\) 0 0
\(991\) −34.1737 −1.08556 −0.542782 0.839873i \(-0.682629\pi\)
−0.542782 + 0.839873i \(0.682629\pi\)
\(992\) 7.54677 0.239610
\(993\) 0 0
\(994\) −38.9672 −1.23596
\(995\) 0 0
\(996\) 0 0
\(997\) −29.3834 −0.930580 −0.465290 0.885158i \(-0.654050\pi\)
−0.465290 + 0.885158i \(0.654050\pi\)
\(998\) −37.6968 −1.19327
\(999\) 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.cp.1.3 3
3.2 odd 2 950.2.a.j.1.3 3
5.4 even 2 8550.2.a.ci.1.1 3
12.11 even 2 7600.2.a.bz.1.1 3
15.2 even 4 950.2.b.h.799.1 6
15.8 even 4 950.2.b.h.799.6 6
15.14 odd 2 950.2.a.l.1.1 yes 3
60.59 even 2 7600.2.a.bk.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
950.2.a.j.1.3 3 3.2 odd 2
950.2.a.l.1.1 yes 3 15.14 odd 2
950.2.b.h.799.1 6 15.2 even 4
950.2.b.h.799.6 6 15.8 even 4
7600.2.a.bk.1.3 3 60.59 even 2
7600.2.a.bz.1.1 3 12.11 even 2
8550.2.a.ci.1.1 3 5.4 even 2
8550.2.a.cp.1.3 3 1.1 even 1 trivial