Properties

Label 8550.2.a.cq.1.1
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: 3.3.148.1
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 570)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.48119\) of defining polynomial
Character \(\chi\) \(=\) 8550.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -3.35026 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -3.35026 q^{7} +1.00000 q^{8} +0.962389 q^{11} -1.61213 q^{13} -3.35026 q^{14} +1.00000 q^{16} +0.387873 q^{17} +1.00000 q^{19} +0.962389 q^{22} +0.962389 q^{23} -1.61213 q^{26} -3.35026 q^{28} -6.96239 q^{29} +3.35026 q^{31} +1.00000 q^{32} +0.387873 q^{34} -1.61213 q^{37} +1.00000 q^{38} +9.27504 q^{41} +6.18664 q^{43} +0.962389 q^{44} +0.962389 q^{46} -0.962389 q^{47} +4.22425 q^{49} -1.61213 q^{52} -6.00000 q^{53} -3.35026 q^{56} -6.96239 q^{58} -10.3127 q^{59} +11.9248 q^{61} +3.35026 q^{62} +1.00000 q^{64} -7.22425 q^{67} +0.387873 q^{68} -7.22425 q^{71} -3.22425 q^{73} -1.61213 q^{74} +1.00000 q^{76} -3.22425 q^{77} -3.35026 q^{79} +9.27504 q^{82} -15.0132 q^{83} +6.18664 q^{86} +0.962389 q^{88} -4.64974 q^{89} +5.40105 q^{91} +0.962389 q^{92} -0.962389 q^{94} -10.9624 q^{97} +4.22425 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} + 3q^{4} + 3q^{8} + O(q^{10}) \) \( 3q + 3q^{2} + 3q^{4} + 3q^{8} - 8q^{11} - 4q^{13} + 3q^{16} + 2q^{17} + 3q^{19} - 8q^{22} - 8q^{23} - 4q^{26} - 10q^{29} + 3q^{32} + 2q^{34} - 4q^{37} + 3q^{38} - 4q^{41} + 6q^{43} - 8q^{44} - 8q^{46} + 8q^{47} + 11q^{49} - 4q^{52} - 18q^{53} - 10q^{58} - 10q^{59} + 14q^{61} + 3q^{64} - 20q^{67} + 2q^{68} - 20q^{71} - 8q^{73} - 4q^{74} + 3q^{76} - 8q^{77} - 4q^{82} - 4q^{83} + 6q^{86} - 8q^{88} - 24q^{89} - 24q^{91} - 8q^{92} + 8q^{94} - 22q^{97} + 11q^{98} + O(q^{100}) \)

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
\(7\) −3.35026 −1.26628 −0.633140 0.774037i \(-0.718234\pi\)
−0.633140 + 0.774037i \(0.718234\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 0.962389 0.290171 0.145086 0.989419i \(-0.453654\pi\)
0.145086 + 0.989419i \(0.453654\pi\)
\(12\) 0 0
\(13\) −1.61213 −0.447124 −0.223562 0.974690i \(-0.571768\pi\)
−0.223562 + 0.974690i \(0.571768\pi\)
\(14\) −3.35026 −0.895395
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0.387873 0.0940731 0.0470365 0.998893i \(-0.485022\pi\)
0.0470365 + 0.998893i \(0.485022\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0.962389 0.205182
\(23\) 0.962389 0.200672 0.100336 0.994954i \(-0.468008\pi\)
0.100336 + 0.994954i \(0.468008\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −1.61213 −0.316164
\(27\) 0 0
\(28\) −3.35026 −0.633140
\(29\) −6.96239 −1.29288 −0.646442 0.762964i \(-0.723744\pi\)
−0.646442 + 0.762964i \(0.723744\pi\)
\(30\) 0 0
\(31\) 3.35026 0.601725 0.300862 0.953668i \(-0.402726\pi\)
0.300862 + 0.953668i \(0.402726\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 0.387873 0.0665197
\(35\) 0 0
\(36\) 0 0
\(37\) −1.61213 −0.265032 −0.132516 0.991181i \(-0.542306\pi\)
−0.132516 + 0.991181i \(0.542306\pi\)
\(38\) 1.00000 0.162221
\(39\) 0 0
\(40\) 0 0
\(41\) 9.27504 1.44852 0.724259 0.689528i \(-0.242182\pi\)
0.724259 + 0.689528i \(0.242182\pi\)
\(42\) 0 0
\(43\) 6.18664 0.943454 0.471727 0.881745i \(-0.343631\pi\)
0.471727 + 0.881745i \(0.343631\pi\)
\(44\) 0.962389 0.145086
\(45\) 0 0
\(46\) 0.962389 0.141896
\(47\) −0.962389 −0.140379 −0.0701894 0.997534i \(-0.522360\pi\)
−0.0701894 + 0.997534i \(0.522360\pi\)
\(48\) 0 0
\(49\) 4.22425 0.603465
\(50\) 0 0
\(51\) 0 0
\(52\) −1.61213 −0.223562
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −3.35026 −0.447698
\(57\) 0 0
\(58\) −6.96239 −0.914206
\(59\) −10.3127 −1.34259 −0.671296 0.741189i \(-0.734262\pi\)
−0.671296 + 0.741189i \(0.734262\pi\)
\(60\) 0 0
\(61\) 11.9248 1.52681 0.763406 0.645919i \(-0.223526\pi\)
0.763406 + 0.645919i \(0.223526\pi\)
\(62\) 3.35026 0.425484
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −7.22425 −0.882583 −0.441292 0.897364i \(-0.645480\pi\)
−0.441292 + 0.897364i \(0.645480\pi\)
\(68\) 0.387873 0.0470365
\(69\) 0 0
\(70\) 0 0
\(71\) −7.22425 −0.857361 −0.428681 0.903456i \(-0.641021\pi\)
−0.428681 + 0.903456i \(0.641021\pi\)
\(72\) 0 0
\(73\) −3.22425 −0.377370 −0.188685 0.982038i \(-0.560423\pi\)
−0.188685 + 0.982038i \(0.560423\pi\)
\(74\) −1.61213 −0.187406
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) −3.22425 −0.367438
\(78\) 0 0
\(79\) −3.35026 −0.376934 −0.188467 0.982080i \(-0.560352\pi\)
−0.188467 + 0.982080i \(0.560352\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 9.27504 1.02426
\(83\) −15.0132 −1.64791 −0.823955 0.566655i \(-0.808237\pi\)
−0.823955 + 0.566655i \(0.808237\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 6.18664 0.667123
\(87\) 0 0
\(88\) 0.962389 0.102591
\(89\) −4.64974 −0.492871 −0.246436 0.969159i \(-0.579259\pi\)
−0.246436 + 0.969159i \(0.579259\pi\)
\(90\) 0 0
\(91\) 5.40105 0.566184
\(92\) 0.962389 0.100336
\(93\) 0 0
\(94\) −0.962389 −0.0992628
\(95\) 0 0
\(96\) 0 0
\(97\) −10.9624 −1.11306 −0.556531 0.830827i \(-0.687868\pi\)
−0.556531 + 0.830827i \(0.687868\pi\)
\(98\) 4.22425 0.426714
\(99\) 0 0
\(100\) 0 0
\(101\) −2.72496 −0.271144 −0.135572 0.990768i \(-0.543287\pi\)
−0.135572 + 0.990768i \(0.543287\pi\)
\(102\) 0 0
\(103\) 0.574515 0.0566087 0.0283043 0.999599i \(-0.490989\pi\)
0.0283043 + 0.999599i \(0.490989\pi\)
\(104\) −1.61213 −0.158082
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) −10.7005 −1.03446 −0.517229 0.855847i \(-0.673037\pi\)
−0.517229 + 0.855847i \(0.673037\pi\)
\(108\) 0 0
\(109\) 10.1260 0.969896 0.484948 0.874543i \(-0.338839\pi\)
0.484948 + 0.874543i \(0.338839\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −3.35026 −0.316570
\(113\) −20.5501 −1.93319 −0.966594 0.256311i \(-0.917493\pi\)
−0.966594 + 0.256311i \(0.917493\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −6.96239 −0.646442
\(117\) 0 0
\(118\) −10.3127 −0.949356
\(119\) −1.29948 −0.119123
\(120\) 0 0
\(121\) −10.0738 −0.915801
\(122\) 11.9248 1.07962
\(123\) 0 0
\(124\) 3.35026 0.300862
\(125\) 0 0
\(126\) 0 0
\(127\) 1.35026 0.119816 0.0599082 0.998204i \(-0.480919\pi\)
0.0599082 + 0.998204i \(0.480919\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −12.4387 −1.08677 −0.543385 0.839483i \(-0.682858\pi\)
−0.543385 + 0.839483i \(0.682858\pi\)
\(132\) 0 0
\(133\) −3.35026 −0.290505
\(134\) −7.22425 −0.624080
\(135\) 0 0
\(136\) 0.387873 0.0332598
\(137\) 18.1622 1.55170 0.775851 0.630916i \(-0.217321\pi\)
0.775851 + 0.630916i \(0.217321\pi\)
\(138\) 0 0
\(139\) 8.77575 0.744349 0.372175 0.928163i \(-0.378612\pi\)
0.372175 + 0.928163i \(0.378612\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −7.22425 −0.606246
\(143\) −1.55149 −0.129742
\(144\) 0 0
\(145\) 0 0
\(146\) −3.22425 −0.266841
\(147\) 0 0
\(148\) −1.61213 −0.132516
\(149\) −15.9756 −1.30877 −0.654385 0.756162i \(-0.727072\pi\)
−0.654385 + 0.756162i \(0.727072\pi\)
\(150\) 0 0
\(151\) 18.4241 1.49933 0.749665 0.661818i \(-0.230215\pi\)
0.749665 + 0.661818i \(0.230215\pi\)
\(152\) 1.00000 0.0811107
\(153\) 0 0
\(154\) −3.22425 −0.259818
\(155\) 0 0
\(156\) 0 0
\(157\) −13.7889 −1.10048 −0.550238 0.835008i \(-0.685463\pi\)
−0.550238 + 0.835008i \(0.685463\pi\)
\(158\) −3.35026 −0.266533
\(159\) 0 0
\(160\) 0 0
\(161\) −3.22425 −0.254107
\(162\) 0 0
\(163\) −3.73813 −0.292793 −0.146397 0.989226i \(-0.546768\pi\)
−0.146397 + 0.989226i \(0.546768\pi\)
\(164\) 9.27504 0.724259
\(165\) 0 0
\(166\) −15.0132 −1.16525
\(167\) −15.4763 −1.19759 −0.598795 0.800902i \(-0.704354\pi\)
−0.598795 + 0.800902i \(0.704354\pi\)
\(168\) 0 0
\(169\) −10.4010 −0.800081
\(170\) 0 0
\(171\) 0 0
\(172\) 6.18664 0.471727
\(173\) 1.47627 0.112239 0.0561194 0.998424i \(-0.482127\pi\)
0.0561194 + 0.998424i \(0.482127\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0.962389 0.0725428
\(177\) 0 0
\(178\) −4.64974 −0.348513
\(179\) 14.3127 1.06978 0.534889 0.844922i \(-0.320353\pi\)
0.534889 + 0.844922i \(0.320353\pi\)
\(180\) 0 0
\(181\) −8.82653 −0.656071 −0.328035 0.944665i \(-0.606387\pi\)
−0.328035 + 0.944665i \(0.606387\pi\)
\(182\) 5.40105 0.400352
\(183\) 0 0
\(184\) 0.962389 0.0709482
\(185\) 0 0
\(186\) 0 0
\(187\) 0.373285 0.0272973
\(188\) −0.962389 −0.0701894
\(189\) 0 0
\(190\) 0 0
\(191\) −2.31265 −0.167338 −0.0836688 0.996494i \(-0.526664\pi\)
−0.0836688 + 0.996494i \(0.526664\pi\)
\(192\) 0 0
\(193\) 7.58769 0.546174 0.273087 0.961989i \(-0.411955\pi\)
0.273087 + 0.961989i \(0.411955\pi\)
\(194\) −10.9624 −0.787054
\(195\) 0 0
\(196\) 4.22425 0.301732
\(197\) 18.8119 1.34030 0.670148 0.742228i \(-0.266231\pi\)
0.670148 + 0.742228i \(0.266231\pi\)
\(198\) 0 0
\(199\) −9.40105 −0.666423 −0.333211 0.942852i \(-0.608132\pi\)
−0.333211 + 0.942852i \(0.608132\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −2.72496 −0.191728
\(203\) 23.3258 1.63715
\(204\) 0 0
\(205\) 0 0
\(206\) 0.574515 0.0400284
\(207\) 0 0
\(208\) −1.61213 −0.111781
\(209\) 0.962389 0.0665698
\(210\) 0 0
\(211\) 4.77575 0.328776 0.164388 0.986396i \(-0.447435\pi\)
0.164388 + 0.986396i \(0.447435\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) −10.7005 −0.731473
\(215\) 0 0
\(216\) 0 0
\(217\) −11.2243 −0.761952
\(218\) 10.1260 0.685820
\(219\) 0 0
\(220\) 0 0
\(221\) −0.625301 −0.0420623
\(222\) 0 0
\(223\) −24.6761 −1.65243 −0.826216 0.563353i \(-0.809511\pi\)
−0.826216 + 0.563353i \(0.809511\pi\)
\(224\) −3.35026 −0.223849
\(225\) 0 0
\(226\) −20.5501 −1.36697
\(227\) −15.4763 −1.02720 −0.513598 0.858031i \(-0.671688\pi\)
−0.513598 + 0.858031i \(0.671688\pi\)
\(228\) 0 0
\(229\) 21.3258 1.40925 0.704625 0.709580i \(-0.251115\pi\)
0.704625 + 0.709580i \(0.251115\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −6.96239 −0.457103
\(233\) 9.01317 0.590473 0.295236 0.955424i \(-0.404602\pi\)
0.295236 + 0.955424i \(0.404602\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −10.3127 −0.671296
\(237\) 0 0
\(238\) −1.29948 −0.0842326
\(239\) 0.135857 0.00878787 0.00439393 0.999990i \(-0.498601\pi\)
0.00439393 + 0.999990i \(0.498601\pi\)
\(240\) 0 0
\(241\) −25.8496 −1.66512 −0.832558 0.553938i \(-0.813125\pi\)
−0.832558 + 0.553938i \(0.813125\pi\)
\(242\) −10.0738 −0.647569
\(243\) 0 0
\(244\) 11.9248 0.763406
\(245\) 0 0
\(246\) 0 0
\(247\) −1.61213 −0.102577
\(248\) 3.35026 0.212742
\(249\) 0 0
\(250\) 0 0
\(251\) 10.1114 0.638227 0.319114 0.947716i \(-0.396615\pi\)
0.319114 + 0.947716i \(0.396615\pi\)
\(252\) 0 0
\(253\) 0.926192 0.0582292
\(254\) 1.35026 0.0847230
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −26.9986 −1.68413 −0.842063 0.539380i \(-0.818659\pi\)
−0.842063 + 0.539380i \(0.818659\pi\)
\(258\) 0 0
\(259\) 5.40105 0.335605
\(260\) 0 0
\(261\) 0 0
\(262\) −12.4387 −0.768463
\(263\) −15.0376 −0.927259 −0.463629 0.886029i \(-0.653453\pi\)
−0.463629 + 0.886029i \(0.653453\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −3.35026 −0.205418
\(267\) 0 0
\(268\) −7.22425 −0.441292
\(269\) 4.51388 0.275216 0.137608 0.990487i \(-0.456059\pi\)
0.137608 + 0.990487i \(0.456059\pi\)
\(270\) 0 0
\(271\) 15.8496 0.962792 0.481396 0.876503i \(-0.340130\pi\)
0.481396 + 0.876503i \(0.340130\pi\)
\(272\) 0.387873 0.0235183
\(273\) 0 0
\(274\) 18.1622 1.09722
\(275\) 0 0
\(276\) 0 0
\(277\) 10.3127 0.619627 0.309814 0.950797i \(-0.399733\pi\)
0.309814 + 0.950797i \(0.399733\pi\)
\(278\) 8.77575 0.526334
\(279\) 0 0
\(280\) 0 0
\(281\) −24.3488 −1.45253 −0.726265 0.687415i \(-0.758746\pi\)
−0.726265 + 0.687415i \(0.758746\pi\)
\(282\) 0 0
\(283\) 26.2882 1.56267 0.781336 0.624111i \(-0.214539\pi\)
0.781336 + 0.624111i \(0.214539\pi\)
\(284\) −7.22425 −0.428681
\(285\) 0 0
\(286\) −1.55149 −0.0917417
\(287\) −31.0738 −1.83423
\(288\) 0 0
\(289\) −16.8496 −0.991150
\(290\) 0 0
\(291\) 0 0
\(292\) −3.22425 −0.188685
\(293\) 13.0738 0.763780 0.381890 0.924208i \(-0.375273\pi\)
0.381890 + 0.924208i \(0.375273\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −1.61213 −0.0937030
\(297\) 0 0
\(298\) −15.9756 −0.925439
\(299\) −1.55149 −0.0897251
\(300\) 0 0
\(301\) −20.7269 −1.19468
\(302\) 18.4241 1.06019
\(303\) 0 0
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) 0 0
\(307\) −7.07381 −0.403724 −0.201862 0.979414i \(-0.564699\pi\)
−0.201862 + 0.979414i \(0.564699\pi\)
\(308\) −3.22425 −0.183719
\(309\) 0 0
\(310\) 0 0
\(311\) −26.3127 −1.49205 −0.746027 0.665916i \(-0.768041\pi\)
−0.746027 + 0.665916i \(0.768041\pi\)
\(312\) 0 0
\(313\) −18.7005 −1.05702 −0.528508 0.848928i \(-0.677248\pi\)
−0.528508 + 0.848928i \(0.677248\pi\)
\(314\) −13.7889 −0.778154
\(315\) 0 0
\(316\) −3.35026 −0.188467
\(317\) −26.4749 −1.48698 −0.743488 0.668749i \(-0.766830\pi\)
−0.743488 + 0.668749i \(0.766830\pi\)
\(318\) 0 0
\(319\) −6.70052 −0.375157
\(320\) 0 0
\(321\) 0 0
\(322\) −3.22425 −0.179681
\(323\) 0.387873 0.0215818
\(324\) 0 0
\(325\) 0 0
\(326\) −3.73813 −0.207036
\(327\) 0 0
\(328\) 9.27504 0.512128
\(329\) 3.22425 0.177759
\(330\) 0 0
\(331\) −30.7005 −1.68745 −0.843727 0.536773i \(-0.819643\pi\)
−0.843727 + 0.536773i \(0.819643\pi\)
\(332\) −15.0132 −0.823955
\(333\) 0 0
\(334\) −15.4763 −0.846824
\(335\) 0 0
\(336\) 0 0
\(337\) 6.81194 0.371070 0.185535 0.982638i \(-0.440598\pi\)
0.185535 + 0.982638i \(0.440598\pi\)
\(338\) −10.4010 −0.565742
\(339\) 0 0
\(340\) 0 0
\(341\) 3.22425 0.174603
\(342\) 0 0
\(343\) 9.29948 0.502125
\(344\) 6.18664 0.333561
\(345\) 0 0
\(346\) 1.47627 0.0793648
\(347\) −18.3879 −0.987113 −0.493556 0.869714i \(-0.664303\pi\)
−0.493556 + 0.869714i \(0.664303\pi\)
\(348\) 0 0
\(349\) 31.1490 1.66737 0.833685 0.552241i \(-0.186227\pi\)
0.833685 + 0.552241i \(0.186227\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0.962389 0.0512955
\(353\) −7.61213 −0.405153 −0.202576 0.979266i \(-0.564931\pi\)
−0.202576 + 0.979266i \(0.564931\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −4.64974 −0.246436
\(357\) 0 0
\(358\) 14.3127 0.756447
\(359\) 35.3112 1.86366 0.931828 0.362901i \(-0.118214\pi\)
0.931828 + 0.362901i \(0.118214\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −8.82653 −0.463912
\(363\) 0 0
\(364\) 5.40105 0.283092
\(365\) 0 0
\(366\) 0 0
\(367\) 31.9756 1.66911 0.834555 0.550924i \(-0.185725\pi\)
0.834555 + 0.550924i \(0.185725\pi\)
\(368\) 0.962389 0.0501680
\(369\) 0 0
\(370\) 0 0
\(371\) 20.1016 1.04362
\(372\) 0 0
\(373\) 26.4894 1.37157 0.685786 0.727804i \(-0.259459\pi\)
0.685786 + 0.727804i \(0.259459\pi\)
\(374\) 0.373285 0.0193021
\(375\) 0 0
\(376\) −0.962389 −0.0496314
\(377\) 11.2243 0.578078
\(378\) 0 0
\(379\) −19.3258 −0.992701 −0.496350 0.868122i \(-0.665327\pi\)
−0.496350 + 0.868122i \(0.665327\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −2.31265 −0.118325
\(383\) −3.37470 −0.172439 −0.0862195 0.996276i \(-0.527479\pi\)
−0.0862195 + 0.996276i \(0.527479\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 7.58769 0.386203
\(387\) 0 0
\(388\) −10.9624 −0.556531
\(389\) 11.3503 0.575481 0.287741 0.957708i \(-0.407096\pi\)
0.287741 + 0.957708i \(0.407096\pi\)
\(390\) 0 0
\(391\) 0.373285 0.0188778
\(392\) 4.22425 0.213357
\(393\) 0 0
\(394\) 18.8119 0.947732
\(395\) 0 0
\(396\) 0 0
\(397\) −18.8364 −0.945371 −0.472685 0.881231i \(-0.656715\pi\)
−0.472685 + 0.881231i \(0.656715\pi\)
\(398\) −9.40105 −0.471232
\(399\) 0 0
\(400\) 0 0
\(401\) 4.12601 0.206043 0.103022 0.994679i \(-0.467149\pi\)
0.103022 + 0.994679i \(0.467149\pi\)
\(402\) 0 0
\(403\) −5.40105 −0.269045
\(404\) −2.72496 −0.135572
\(405\) 0 0
\(406\) 23.3258 1.15764
\(407\) −1.55149 −0.0769046
\(408\) 0 0
\(409\) 2.52373 0.124790 0.0623952 0.998052i \(-0.480126\pi\)
0.0623952 + 0.998052i \(0.480126\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0.574515 0.0283043
\(413\) 34.5501 1.70010
\(414\) 0 0
\(415\) 0 0
\(416\) −1.61213 −0.0790410
\(417\) 0 0
\(418\) 0.962389 0.0470720
\(419\) −7.51247 −0.367008 −0.183504 0.983019i \(-0.558744\pi\)
−0.183504 + 0.983019i \(0.558744\pi\)
\(420\) 0 0
\(421\) 3.67750 0.179230 0.0896152 0.995976i \(-0.471436\pi\)
0.0896152 + 0.995976i \(0.471436\pi\)
\(422\) 4.77575 0.232480
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) 0 0
\(427\) −39.9511 −1.93337
\(428\) −10.7005 −0.517229
\(429\) 0 0
\(430\) 0 0
\(431\) 10.3272 0.497446 0.248723 0.968575i \(-0.419989\pi\)
0.248723 + 0.968575i \(0.419989\pi\)
\(432\) 0 0
\(433\) 31.5877 1.51801 0.759004 0.651086i \(-0.225686\pi\)
0.759004 + 0.651086i \(0.225686\pi\)
\(434\) −11.2243 −0.538781
\(435\) 0 0
\(436\) 10.1260 0.484948
\(437\) 0.962389 0.0460373
\(438\) 0 0
\(439\) −38.1524 −1.82091 −0.910456 0.413605i \(-0.864269\pi\)
−0.910456 + 0.413605i \(0.864269\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −0.625301 −0.0297425
\(443\) −16.3127 −0.775037 −0.387519 0.921862i \(-0.626668\pi\)
−0.387519 + 0.921862i \(0.626668\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −24.6761 −1.16845
\(447\) 0 0
\(448\) −3.35026 −0.158285
\(449\) 37.5271 1.77101 0.885506 0.464629i \(-0.153812\pi\)
0.885506 + 0.464629i \(0.153812\pi\)
\(450\) 0 0
\(451\) 8.92619 0.420318
\(452\) −20.5501 −0.966594
\(453\) 0 0
\(454\) −15.4763 −0.726337
\(455\) 0 0
\(456\) 0 0
\(457\) −8.00000 −0.374224 −0.187112 0.982339i \(-0.559913\pi\)
−0.187112 + 0.982339i \(0.559913\pi\)
\(458\) 21.3258 0.996490
\(459\) 0 0
\(460\) 0 0
\(461\) −12.3780 −0.576502 −0.288251 0.957555i \(-0.593074\pi\)
−0.288251 + 0.957555i \(0.593074\pi\)
\(462\) 0 0
\(463\) 32.4504 1.50810 0.754049 0.656818i \(-0.228098\pi\)
0.754049 + 0.656818i \(0.228098\pi\)
\(464\) −6.96239 −0.323221
\(465\) 0 0
\(466\) 9.01317 0.417527
\(467\) −7.53690 −0.348766 −0.174383 0.984678i \(-0.555793\pi\)
−0.174383 + 0.984678i \(0.555793\pi\)
\(468\) 0 0
\(469\) 24.2031 1.11760
\(470\) 0 0
\(471\) 0 0
\(472\) −10.3127 −0.474678
\(473\) 5.95395 0.273763
\(474\) 0 0
\(475\) 0 0
\(476\) −1.29948 −0.0595614
\(477\) 0 0
\(478\) 0.135857 0.00621396
\(479\) −15.2097 −0.694947 −0.347474 0.937690i \(-0.612960\pi\)
−0.347474 + 0.937690i \(0.612960\pi\)
\(480\) 0 0
\(481\) 2.59895 0.118502
\(482\) −25.8496 −1.17741
\(483\) 0 0
\(484\) −10.0738 −0.457900
\(485\) 0 0
\(486\) 0 0
\(487\) −1.19982 −0.0543689 −0.0271844 0.999630i \(-0.508654\pi\)
−0.0271844 + 0.999630i \(0.508654\pi\)
\(488\) 11.9248 0.539809
\(489\) 0 0
\(490\) 0 0
\(491\) 5.11283 0.230739 0.115369 0.993323i \(-0.463195\pi\)
0.115369 + 0.993323i \(0.463195\pi\)
\(492\) 0 0
\(493\) −2.70052 −0.121625
\(494\) −1.61213 −0.0725330
\(495\) 0 0
\(496\) 3.35026 0.150431
\(497\) 24.2031 1.08566
\(498\) 0 0
\(499\) 14.2981 0.640069 0.320035 0.947406i \(-0.396305\pi\)
0.320035 + 0.947406i \(0.396305\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 10.1114 0.451295
\(503\) −11.5125 −0.513316 −0.256658 0.966502i \(-0.582621\pi\)
−0.256658 + 0.966502i \(0.582621\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0.926192 0.0411742
\(507\) 0 0
\(508\) 1.35026 0.0599082
\(509\) 31.9610 1.41665 0.708323 0.705889i \(-0.249452\pi\)
0.708323 + 0.705889i \(0.249452\pi\)
\(510\) 0 0
\(511\) 10.8021 0.477857
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −26.9986 −1.19086
\(515\) 0 0
\(516\) 0 0
\(517\) −0.926192 −0.0407339
\(518\) 5.40105 0.237308
\(519\) 0 0
\(520\) 0 0
\(521\) 33.2750 1.45781 0.728903 0.684617i \(-0.240031\pi\)
0.728903 + 0.684617i \(0.240031\pi\)
\(522\) 0 0
\(523\) −9.29948 −0.406638 −0.203319 0.979113i \(-0.565173\pi\)
−0.203319 + 0.979113i \(0.565173\pi\)
\(524\) −12.4387 −0.543385
\(525\) 0 0
\(526\) −15.0376 −0.655671
\(527\) 1.29948 0.0566061
\(528\) 0 0
\(529\) −22.0738 −0.959731
\(530\) 0 0
\(531\) 0 0
\(532\) −3.35026 −0.145252
\(533\) −14.9525 −0.647666
\(534\) 0 0
\(535\) 0 0
\(536\) −7.22425 −0.312040
\(537\) 0 0
\(538\) 4.51388 0.194607
\(539\) 4.06537 0.175108
\(540\) 0 0
\(541\) 28.5501 1.22746 0.613732 0.789515i \(-0.289668\pi\)
0.613732 + 0.789515i \(0.289668\pi\)
\(542\) 15.8496 0.680797
\(543\) 0 0
\(544\) 0.387873 0.0166299
\(545\) 0 0
\(546\) 0 0
\(547\) −28.4749 −1.21750 −0.608748 0.793363i \(-0.708328\pi\)
−0.608748 + 0.793363i \(0.708328\pi\)
\(548\) 18.1622 0.775851
\(549\) 0 0
\(550\) 0 0
\(551\) −6.96239 −0.296608
\(552\) 0 0
\(553\) 11.2243 0.477304
\(554\) 10.3127 0.438143
\(555\) 0 0
\(556\) 8.77575 0.372175
\(557\) 4.88717 0.207076 0.103538 0.994626i \(-0.466984\pi\)
0.103538 + 0.994626i \(0.466984\pi\)
\(558\) 0 0
\(559\) −9.97365 −0.421841
\(560\) 0 0
\(561\) 0 0
\(562\) −24.3488 −1.02709
\(563\) −30.8021 −1.29815 −0.649077 0.760723i \(-0.724845\pi\)
−0.649077 + 0.760723i \(0.724845\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 26.2882 1.10498
\(567\) 0 0
\(568\) −7.22425 −0.303123
\(569\) 24.1260 1.01141 0.505707 0.862705i \(-0.331232\pi\)
0.505707 + 0.862705i \(0.331232\pi\)
\(570\) 0 0
\(571\) −5.67276 −0.237398 −0.118699 0.992930i \(-0.537872\pi\)
−0.118699 + 0.992930i \(0.537872\pi\)
\(572\) −1.55149 −0.0648712
\(573\) 0 0
\(574\) −31.0738 −1.29700
\(575\) 0 0
\(576\) 0 0
\(577\) −32.0000 −1.33218 −0.666089 0.745873i \(-0.732033\pi\)
−0.666089 + 0.745873i \(0.732033\pi\)
\(578\) −16.8496 −0.700849
\(579\) 0 0
\(580\) 0 0
\(581\) 50.2981 2.08672
\(582\) 0 0
\(583\) −5.77433 −0.239148
\(584\) −3.22425 −0.133421
\(585\) 0 0
\(586\) 13.0738 0.540074
\(587\) 13.6121 0.561833 0.280916 0.959732i \(-0.409362\pi\)
0.280916 + 0.959732i \(0.409362\pi\)
\(588\) 0 0
\(589\) 3.35026 0.138045
\(590\) 0 0
\(591\) 0 0
\(592\) −1.61213 −0.0662580
\(593\) −23.4617 −0.963456 −0.481728 0.876321i \(-0.659991\pi\)
−0.481728 + 0.876321i \(0.659991\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −15.9756 −0.654385
\(597\) 0 0
\(598\) −1.55149 −0.0634452
\(599\) −10.0263 −0.409665 −0.204833 0.978797i \(-0.565665\pi\)
−0.204833 + 0.978797i \(0.565665\pi\)
\(600\) 0 0
\(601\) 11.7743 0.480285 0.240143 0.970738i \(-0.422806\pi\)
0.240143 + 0.970738i \(0.422806\pi\)
\(602\) −20.7269 −0.844764
\(603\) 0 0
\(604\) 18.4241 0.749665
\(605\) 0 0
\(606\) 0 0
\(607\) 38.4993 1.56264 0.781319 0.624132i \(-0.214547\pi\)
0.781319 + 0.624132i \(0.214547\pi\)
\(608\) 1.00000 0.0405554
\(609\) 0 0
\(610\) 0 0
\(611\) 1.55149 0.0627667
\(612\) 0 0
\(613\) −7.61213 −0.307451 −0.153725 0.988114i \(-0.549127\pi\)
−0.153725 + 0.988114i \(0.549127\pi\)
\(614\) −7.07381 −0.285476
\(615\) 0 0
\(616\) −3.22425 −0.129909
\(617\) 29.5369 1.18911 0.594555 0.804055i \(-0.297328\pi\)
0.594555 + 0.804055i \(0.297328\pi\)
\(618\) 0 0
\(619\) −22.5501 −0.906364 −0.453182 0.891418i \(-0.649711\pi\)
−0.453182 + 0.891418i \(0.649711\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −26.3127 −1.05504
\(623\) 15.5778 0.624113
\(624\) 0 0
\(625\) 0 0
\(626\) −18.7005 −0.747423
\(627\) 0 0
\(628\) −13.7889 −0.550238
\(629\) −0.625301 −0.0249324
\(630\) 0 0
\(631\) −37.9248 −1.50976 −0.754881 0.655862i \(-0.772305\pi\)
−0.754881 + 0.655862i \(0.772305\pi\)
\(632\) −3.35026 −0.133266
\(633\) 0 0
\(634\) −26.4749 −1.05145
\(635\) 0 0
\(636\) 0 0
\(637\) −6.81003 −0.269823
\(638\) −6.70052 −0.265276
\(639\) 0 0
\(640\) 0 0
\(641\) −6.67609 −0.263690 −0.131845 0.991270i \(-0.542090\pi\)
−0.131845 + 0.991270i \(0.542090\pi\)
\(642\) 0 0
\(643\) −25.2605 −0.996175 −0.498087 0.867127i \(-0.665964\pi\)
−0.498087 + 0.867127i \(0.665964\pi\)
\(644\) −3.22425 −0.127053
\(645\) 0 0
\(646\) 0.387873 0.0152607
\(647\) 16.9135 0.664939 0.332469 0.943114i \(-0.392118\pi\)
0.332469 + 0.943114i \(0.392118\pi\)
\(648\) 0 0
\(649\) −9.92478 −0.389582
\(650\) 0 0
\(651\) 0 0
\(652\) −3.73813 −0.146397
\(653\) 24.1114 0.943553 0.471776 0.881718i \(-0.343613\pi\)
0.471776 + 0.881718i \(0.343613\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 9.27504 0.362129
\(657\) 0 0
\(658\) 3.22425 0.125694
\(659\) 20.3879 0.794199 0.397099 0.917776i \(-0.370017\pi\)
0.397099 + 0.917776i \(0.370017\pi\)
\(660\) 0 0
\(661\) 17.6023 0.684649 0.342325 0.939582i \(-0.388786\pi\)
0.342325 + 0.939582i \(0.388786\pi\)
\(662\) −30.7005 −1.19321
\(663\) 0 0
\(664\) −15.0132 −0.582624
\(665\) 0 0
\(666\) 0 0
\(667\) −6.70052 −0.259445
\(668\) −15.4763 −0.598795
\(669\) 0 0
\(670\) 0 0
\(671\) 11.4763 0.443036
\(672\) 0 0
\(673\) −44.3634 −1.71008 −0.855042 0.518558i \(-0.826469\pi\)
−0.855042 + 0.518558i \(0.826469\pi\)
\(674\) 6.81194 0.262386
\(675\) 0 0
\(676\) −10.4010 −0.400040
\(677\) 8.70052 0.334388 0.167194 0.985924i \(-0.446529\pi\)
0.167194 + 0.985924i \(0.446529\pi\)
\(678\) 0 0
\(679\) 36.7269 1.40945
\(680\) 0 0
\(681\) 0 0
\(682\) 3.22425 0.123463
\(683\) −37.8759 −1.44928 −0.724641 0.689127i \(-0.757994\pi\)
−0.724641 + 0.689127i \(0.757994\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 9.29948 0.355056
\(687\) 0 0
\(688\) 6.18664 0.235864
\(689\) 9.67276 0.368503
\(690\) 0 0
\(691\) −0.775746 −0.0295108 −0.0147554 0.999891i \(-0.504697\pi\)
−0.0147554 + 0.999891i \(0.504697\pi\)
\(692\) 1.47627 0.0561194
\(693\) 0 0
\(694\) −18.3879 −0.697994
\(695\) 0 0
\(696\) 0 0
\(697\) 3.59754 0.136266
\(698\) 31.1490 1.17901
\(699\) 0 0
\(700\) 0 0
\(701\) −42.3752 −1.60049 −0.800245 0.599674i \(-0.795297\pi\)
−0.800245 + 0.599674i \(0.795297\pi\)
\(702\) 0 0
\(703\) −1.61213 −0.0608025
\(704\) 0.962389 0.0362714
\(705\) 0 0
\(706\) −7.61213 −0.286486
\(707\) 9.12933 0.343344
\(708\) 0 0
\(709\) 36.2784 1.36246 0.681231 0.732068i \(-0.261445\pi\)
0.681231 + 0.732068i \(0.261445\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −4.64974 −0.174256
\(713\) 3.22425 0.120749
\(714\) 0 0
\(715\) 0 0
\(716\) 14.3127 0.534889
\(717\) 0 0
\(718\) 35.3112 1.31780
\(719\) 42.5355 1.58631 0.793153 0.609022i \(-0.208438\pi\)
0.793153 + 0.609022i \(0.208438\pi\)
\(720\) 0 0
\(721\) −1.92478 −0.0716824
\(722\) 1.00000 0.0372161
\(723\) 0 0
\(724\) −8.82653 −0.328035
\(725\) 0 0
\(726\) 0 0
\(727\) 0.378024 0.0140201 0.00701007 0.999975i \(-0.497769\pi\)
0.00701007 + 0.999975i \(0.497769\pi\)
\(728\) 5.40105 0.200176
\(729\) 0 0
\(730\) 0 0
\(731\) 2.39963 0.0887536
\(732\) 0 0
\(733\) −26.0118 −0.960766 −0.480383 0.877059i \(-0.659502\pi\)
−0.480383 + 0.877059i \(0.659502\pi\)
\(734\) 31.9756 1.18024
\(735\) 0 0
\(736\) 0.962389 0.0354741
\(737\) −6.95254 −0.256100
\(738\) 0 0
\(739\) −44.8773 −1.65084 −0.825419 0.564520i \(-0.809061\pi\)
−0.825419 + 0.564520i \(0.809061\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 20.1016 0.737952
\(743\) −4.67418 −0.171479 −0.0857394 0.996318i \(-0.527325\pi\)
−0.0857394 + 0.996318i \(0.527325\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 26.4894 0.969847
\(747\) 0 0
\(748\) 0.373285 0.0136486
\(749\) 35.8496 1.30991
\(750\) 0 0
\(751\) −6.57452 −0.239907 −0.119954 0.992779i \(-0.538275\pi\)
−0.119954 + 0.992779i \(0.538275\pi\)
\(752\) −0.962389 −0.0350947
\(753\) 0 0
\(754\) 11.2243 0.408763
\(755\) 0 0
\(756\) 0 0
\(757\) 15.5633 0.565656 0.282828 0.959171i \(-0.408727\pi\)
0.282828 + 0.959171i \(0.408727\pi\)
\(758\) −19.3258 −0.701946
\(759\) 0 0
\(760\) 0 0
\(761\) 23.8759 0.865501 0.432750 0.901514i \(-0.357543\pi\)
0.432750 + 0.901514i \(0.357543\pi\)
\(762\) 0 0
\(763\) −33.9248 −1.22816
\(764\) −2.31265 −0.0836688
\(765\) 0 0
\(766\) −3.37470 −0.121933
\(767\) 16.6253 0.600305
\(768\) 0 0
\(769\) 30.4749 1.09895 0.549476 0.835510i \(-0.314827\pi\)
0.549476 + 0.835510i \(0.314827\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 7.58769 0.273087
\(773\) 16.3272 0.587250 0.293625 0.955921i \(-0.405138\pi\)
0.293625 + 0.955921i \(0.405138\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −10.9624 −0.393527
\(777\) 0 0
\(778\) 11.3503 0.406927
\(779\) 9.27504 0.332313
\(780\) 0 0
\(781\) −6.95254 −0.248781
\(782\) 0.373285 0.0133486
\(783\) 0 0
\(784\) 4.22425 0.150866
\(785\) 0 0
\(786\) 0 0
\(787\) −30.9525 −1.10334 −0.551669 0.834063i \(-0.686009\pi\)
−0.551669 + 0.834063i \(0.686009\pi\)
\(788\) 18.8119 0.670148
\(789\) 0 0
\(790\) 0 0
\(791\) 68.8481 2.44796
\(792\) 0 0
\(793\) −19.2243 −0.682673
\(794\) −18.8364 −0.668478
\(795\) 0 0
\(796\) −9.40105 −0.333211
\(797\) 14.8773 0.526982 0.263491 0.964662i \(-0.415126\pi\)
0.263491 + 0.964662i \(0.415126\pi\)
\(798\) 0 0
\(799\) −0.373285 −0.0132059
\(800\) 0 0
\(801\) 0 0
\(802\) 4.12601 0.145694
\(803\) −3.10299 −0.109502
\(804\) 0 0
\(805\) 0 0
\(806\) −5.40105 −0.190244
\(807\) 0 0
\(808\) −2.72496 −0.0958638
\(809\) −25.4471 −0.894672 −0.447336 0.894366i \(-0.647627\pi\)
−0.447336 + 0.894366i \(0.647627\pi\)
\(810\) 0 0
\(811\) −53.3522 −1.87345 −0.936724 0.350069i \(-0.886158\pi\)
−0.936724 + 0.350069i \(0.886158\pi\)
\(812\) 23.3258 0.818576
\(813\) 0 0
\(814\) −1.55149 −0.0543798
\(815\) 0 0
\(816\) 0 0
\(817\) 6.18664 0.216443
\(818\) 2.52373 0.0882402
\(819\) 0 0
\(820\) 0 0
\(821\) 27.9756 0.976354 0.488177 0.872745i \(-0.337662\pi\)
0.488177 + 0.872745i \(0.337662\pi\)
\(822\) 0 0
\(823\) 7.87399 0.274470 0.137235 0.990539i \(-0.456178\pi\)
0.137235 + 0.990539i \(0.456178\pi\)
\(824\) 0.574515 0.0200142
\(825\) 0 0
\(826\) 34.5501 1.20215
\(827\) 40.1016 1.39447 0.697234 0.716843i \(-0.254414\pi\)
0.697234 + 0.716843i \(0.254414\pi\)
\(828\) 0 0
\(829\) 47.2262 1.64023 0.820116 0.572197i \(-0.193909\pi\)
0.820116 + 0.572197i \(0.193909\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −1.61213 −0.0558904
\(833\) 1.63847 0.0567698
\(834\) 0 0
\(835\) 0 0
\(836\) 0.962389 0.0332849
\(837\) 0 0
\(838\) −7.51247 −0.259514
\(839\) 3.89843 0.134589 0.0672944 0.997733i \(-0.478563\pi\)
0.0672944 + 0.997733i \(0.478563\pi\)
\(840\) 0 0
\(841\) 19.4749 0.671547
\(842\) 3.67750 0.126735
\(843\) 0 0
\(844\) 4.77575 0.164388
\(845\) 0 0
\(846\) 0 0
\(847\) 33.7499 1.15966
\(848\) −6.00000 −0.206041
\(849\) 0 0
\(850\) 0 0
\(851\) −1.55149 −0.0531845
\(852\) 0 0
\(853\) −35.1900 −1.20488 −0.602441 0.798164i \(-0.705805\pi\)
−0.602441 + 0.798164i \(0.705805\pi\)
\(854\) −39.9511 −1.36710
\(855\) 0 0
\(856\) −10.7005 −0.365736
\(857\) −12.1768 −0.415951 −0.207976 0.978134i \(-0.566687\pi\)
−0.207976 + 0.978134i \(0.566687\pi\)
\(858\) 0 0
\(859\) 57.3522 1.95683 0.978415 0.206648i \(-0.0662554\pi\)
0.978415 + 0.206648i \(0.0662554\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 10.3272 0.351747
\(863\) 14.0752 0.479126 0.239563 0.970881i \(-0.422996\pi\)
0.239563 + 0.970881i \(0.422996\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 31.5877 1.07339
\(867\) 0 0
\(868\) −11.2243 −0.380976
\(869\) −3.22425 −0.109375
\(870\) 0 0
\(871\) 11.6464 0.394624
\(872\) 10.1260 0.342910
\(873\) 0 0
\(874\) 0.962389 0.0325533
\(875\) 0 0
\(876\) 0 0
\(877\) 18.8627 0.636949 0.318475 0.947931i \(-0.396829\pi\)
0.318475 + 0.947931i \(0.396829\pi\)
\(878\) −38.1524 −1.28758
\(879\) 0 0
\(880\) 0 0
\(881\) 19.1490 0.645147 0.322574 0.946544i \(-0.395452\pi\)
0.322574 + 0.946544i \(0.395452\pi\)
\(882\) 0 0
\(883\) 42.9135 1.44415 0.722077 0.691812i \(-0.243187\pi\)
0.722077 + 0.691812i \(0.243187\pi\)
\(884\) −0.625301 −0.0210311
\(885\) 0 0
\(886\) −16.3127 −0.548034
\(887\) −22.7005 −0.762209 −0.381104 0.924532i \(-0.624456\pi\)
−0.381104 + 0.924532i \(0.624456\pi\)
\(888\) 0 0
\(889\) −4.52373 −0.151721
\(890\) 0 0
\(891\) 0 0
\(892\) −24.6761 −0.826216
\(893\) −0.962389 −0.0322051
\(894\) 0 0
\(895\) 0 0
\(896\) −3.35026 −0.111924
\(897\) 0 0
\(898\) 37.5271 1.25229
\(899\) −23.3258 −0.777960
\(900\) 0 0
\(901\) −2.32724 −0.0775316
\(902\) 8.92619 0.297210
\(903\) 0 0
\(904\) −20.5501 −0.683485
\(905\) 0 0
\(906\) 0 0
\(907\) 14.5501 0.483127 0.241564 0.970385i \(-0.422340\pi\)
0.241564 + 0.970385i \(0.422340\pi\)
\(908\) −15.4763 −0.513598
\(909\) 0 0
\(910\) 0 0
\(911\) 0.998585 0.0330846 0.0165423 0.999863i \(-0.494734\pi\)
0.0165423 + 0.999863i \(0.494734\pi\)
\(912\) 0 0
\(913\) −14.4485 −0.478176
\(914\) −8.00000 −0.264616
\(915\) 0 0
\(916\) 21.3258 0.704625
\(917\) 41.6728 1.37616
\(918\) 0 0
\(919\) −33.7743 −1.11411 −0.557056 0.830475i \(-0.688069\pi\)
−0.557056 + 0.830475i \(0.688069\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −12.3780 −0.407649
\(923\) 11.6464 0.383346
\(924\) 0 0
\(925\) 0 0
\(926\) 32.4504 1.06639
\(927\) 0 0
\(928\) −6.96239 −0.228552
\(929\) −14.8773 −0.488109 −0.244054 0.969762i \(-0.578478\pi\)
−0.244054 + 0.969762i \(0.578478\pi\)
\(930\) 0 0
\(931\) 4.22425 0.138444
\(932\) 9.01317 0.295236
\(933\) 0 0
\(934\) −7.53690 −0.246615
\(935\) 0 0
\(936\) 0 0
\(937\) 22.2981 0.728446 0.364223 0.931312i \(-0.381335\pi\)
0.364223 + 0.931312i \(0.381335\pi\)
\(938\) 24.2031 0.790261
\(939\) 0 0
\(940\) 0 0
\(941\) 36.3634 1.18541 0.592707 0.805418i \(-0.298059\pi\)
0.592707 + 0.805418i \(0.298059\pi\)
\(942\) 0 0
\(943\) 8.92619 0.290677
\(944\) −10.3127 −0.335648
\(945\) 0 0
\(946\) 5.95395 0.193580
\(947\) −50.3390 −1.63580 −0.817899 0.575362i \(-0.804861\pi\)
−0.817899 + 0.575362i \(0.804861\pi\)
\(948\) 0 0
\(949\) 5.19791 0.168731
\(950\) 0 0
\(951\) 0 0
\(952\) −1.29948 −0.0421163
\(953\) 16.3272 0.528891 0.264446 0.964401i \(-0.414811\pi\)
0.264446 + 0.964401i \(0.414811\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0.135857 0.00439393
\(957\) 0 0
\(958\) −15.2097 −0.491402
\(959\) −60.8481 −1.96489
\(960\) 0 0
\(961\) −19.7757 −0.637927
\(962\) 2.59895 0.0837936
\(963\) 0 0
\(964\) −25.8496 −0.832558
\(965\) 0 0
\(966\) 0 0
\(967\) 0.276454 0.00889015 0.00444507 0.999990i \(-0.498585\pi\)
0.00444507 + 0.999990i \(0.498585\pi\)
\(968\) −10.0738 −0.323784
\(969\) 0 0
\(970\) 0 0
\(971\) −37.9102 −1.21660 −0.608298 0.793709i \(-0.708147\pi\)
−0.608298 + 0.793709i \(0.708147\pi\)
\(972\) 0 0
\(973\) −29.4010 −0.942554
\(974\) −1.19982 −0.0384446
\(975\) 0 0
\(976\) 11.9248 0.381703
\(977\) −28.3996 −0.908585 −0.454292 0.890853i \(-0.650108\pi\)
−0.454292 + 0.890853i \(0.650108\pi\)
\(978\) 0 0
\(979\) −4.47486 −0.143017
\(980\) 0 0
\(981\) 0 0
\(982\) 5.11283 0.163157
\(983\) 0.926192 0.0295409 0.0147705 0.999891i \(-0.495298\pi\)
0.0147705 + 0.999891i \(0.495298\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −2.70052 −0.0860022
\(987\) 0 0
\(988\) −1.61213 −0.0512886
\(989\) 5.95395 0.189325
\(990\) 0 0
\(991\) 30.5256 0.969679 0.484839 0.874603i \(-0.338878\pi\)
0.484839 + 0.874603i \(0.338878\pi\)
\(992\) 3.35026 0.106371
\(993\) 0 0
\(994\) 24.2031 0.767677
\(995\) 0 0
\(996\) 0 0
\(997\) 21.0132 0.665494 0.332747 0.943016i \(-0.392025\pi\)
0.332747 + 0.943016i \(0.392025\pi\)
\(998\) 14.2981 0.452597
\(999\) 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.cq.1.1 3
3.2 odd 2 2850.2.a.bl.1.1 3
5.2 odd 4 1710.2.d.f.1369.6 6
5.3 odd 4 1710.2.d.f.1369.3 6
5.4 even 2 8550.2.a.ce.1.3 3
15.2 even 4 570.2.d.c.229.1 6
15.8 even 4 570.2.d.c.229.4 yes 6
15.14 odd 2 2850.2.a.bm.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
570.2.d.c.229.1 6 15.2 even 4
570.2.d.c.229.4 yes 6 15.8 even 4
1710.2.d.f.1369.3 6 5.3 odd 4
1710.2.d.f.1369.6 6 5.2 odd 4
2850.2.a.bl.1.1 3 3.2 odd 2
2850.2.a.bm.1.3 3 15.14 odd 2
8550.2.a.ce.1.3 3 5.4 even 2
8550.2.a.cq.1.1 3 1.1 even 1 trivial