Properties

Label 5445.2.a.bk.1.3
Level $5445$
Weight $2$
Character 5445.1
Self dual yes
Analytic conductor $43.479$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5445.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(43.4785439006\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.5725.1
Defining polynomial: \(x^{4} - x^{3} - 8 x^{2} + 6 x + 11\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.55157\) of defining polynomial
Character \(\chi\) \(=\) 5445.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.933531 q^{2} -1.12852 q^{4} +1.00000 q^{5} -2.04108 q^{7} -2.92057 q^{8} +O(q^{10})\) \(q+0.933531 q^{2} -1.12852 q^{4} +1.00000 q^{5} -2.04108 q^{7} -2.92057 q^{8} +0.933531 q^{10} -1.44843 q^{13} -1.90541 q^{14} -0.469405 q^{16} -0.867063 q^{17} +3.12852 q^{19} -1.12852 q^{20} +4.70547 q^{23} +1.00000 q^{25} -1.35216 q^{26} +2.30340 q^{28} -2.03835 q^{29} +10.6136 q^{31} +5.40294 q^{32} -0.809430 q^{34} -2.04108 q^{35} +4.15664 q^{37} +2.92057 q^{38} -2.92057 q^{40} +0.805012 q^{41} -2.34089 q^{43} +4.39271 q^{46} -10.3803 q^{47} -2.83399 q^{49} +0.933531 q^{50} +1.63459 q^{52} -7.21596 q^{53} +5.96112 q^{56} -1.90286 q^{58} +8.32351 q^{59} -8.76752 q^{61} +9.90814 q^{62} +5.98262 q^{64} -1.44843 q^{65} -3.15664 q^{67} +0.978497 q^{68} -1.90541 q^{70} -12.8707 q^{71} -14.7184 q^{73} +3.88035 q^{74} -3.53059 q^{76} -16.9992 q^{79} -0.469405 q^{80} +0.751504 q^{82} -6.14148 q^{83} -0.867063 q^{85} -2.18529 q^{86} -3.77194 q^{89} +2.95637 q^{91} -5.31022 q^{92} -9.69031 q^{94} +3.12852 q^{95} -1.93795 q^{97} -2.64562 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - q^{2} + 9q^{4} + 4q^{5} - 8q^{7} - 3q^{8} + O(q^{10}) \) \( 4q - q^{2} + 9q^{4} + 4q^{5} - 8q^{7} - 3q^{8} - q^{10} - 15q^{13} - 7q^{14} + 7q^{16} + 6q^{17} - q^{19} + 9q^{20} + q^{23} + 4q^{25} + 18q^{26} - 31q^{28} - 17q^{29} + 15q^{31} + 8q^{32} - 35q^{34} - 8q^{35} - q^{37} + 3q^{38} - 3q^{40} + 12q^{41} - 14q^{43} - 9q^{46} - 14q^{47} + 20q^{49} - q^{50} - 39q^{52} - 2q^{53} + 12q^{56} - 11q^{58} + 11q^{59} + q^{61} + 30q^{62} - 3q^{64} - 15q^{65} + 5q^{67} + 19q^{68} - 7q^{70} + 3q^{71} - 45q^{73} - 29q^{74} - 23q^{76} + 7q^{80} + 11q^{82} - 15q^{83} + 6q^{85} + 10q^{86} - 2q^{89} + 16q^{91} - 34q^{92} - 29q^{94} - q^{95} - 26q^{97} + q^{98} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.933531 0.660106 0.330053 0.943962i \(-0.392933\pi\)
0.330053 + 0.943962i \(0.392933\pi\)
\(3\) 0 0
\(4\) −1.12852 −0.564260
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −2.04108 −0.771456 −0.385728 0.922613i \(-0.626050\pi\)
−0.385728 + 0.922613i \(0.626050\pi\)
\(8\) −2.92057 −1.03258
\(9\) 0 0
\(10\) 0.933531 0.295209
\(11\) 0 0
\(12\) 0 0
\(13\) −1.44843 −0.401724 −0.200862 0.979620i \(-0.564374\pi\)
−0.200862 + 0.979620i \(0.564374\pi\)
\(14\) −1.90541 −0.509243
\(15\) 0 0
\(16\) −0.469405 −0.117351
\(17\) −0.867063 −0.210294 −0.105147 0.994457i \(-0.533531\pi\)
−0.105147 + 0.994457i \(0.533531\pi\)
\(18\) 0 0
\(19\) 3.12852 0.717732 0.358866 0.933389i \(-0.383164\pi\)
0.358866 + 0.933389i \(0.383164\pi\)
\(20\) −1.12852 −0.252345
\(21\) 0 0
\(22\) 0 0
\(23\) 4.70547 0.981159 0.490580 0.871396i \(-0.336785\pi\)
0.490580 + 0.871396i \(0.336785\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −1.35216 −0.265180
\(27\) 0 0
\(28\) 2.30340 0.435301
\(29\) −2.03835 −0.378512 −0.189256 0.981928i \(-0.560608\pi\)
−0.189256 + 0.981928i \(0.560608\pi\)
\(30\) 0 0
\(31\) 10.6136 1.90626 0.953131 0.302558i \(-0.0978407\pi\)
0.953131 + 0.302558i \(0.0978407\pi\)
\(32\) 5.40294 0.955113
\(33\) 0 0
\(34\) −0.809430 −0.138816
\(35\) −2.04108 −0.345005
\(36\) 0 0
\(37\) 4.15664 0.683347 0.341674 0.939819i \(-0.389006\pi\)
0.341674 + 0.939819i \(0.389006\pi\)
\(38\) 2.92057 0.473779
\(39\) 0 0
\(40\) −2.92057 −0.461783
\(41\) 0.805012 0.125722 0.0628609 0.998022i \(-0.479978\pi\)
0.0628609 + 0.998022i \(0.479978\pi\)
\(42\) 0 0
\(43\) −2.34089 −0.356982 −0.178491 0.983942i \(-0.557122\pi\)
−0.178491 + 0.983942i \(0.557122\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 4.39271 0.647669
\(47\) −10.3803 −1.51412 −0.757060 0.653346i \(-0.773365\pi\)
−0.757060 + 0.653346i \(0.773365\pi\)
\(48\) 0 0
\(49\) −2.83399 −0.404856
\(50\) 0.933531 0.132021
\(51\) 0 0
\(52\) 1.63459 0.226676
\(53\) −7.21596 −0.991188 −0.495594 0.868554i \(-0.665050\pi\)
−0.495594 + 0.868554i \(0.665050\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 5.96112 0.796588
\(57\) 0 0
\(58\) −1.90286 −0.249858
\(59\) 8.32351 1.08363 0.541814 0.840498i \(-0.317738\pi\)
0.541814 + 0.840498i \(0.317738\pi\)
\(60\) 0 0
\(61\) −8.76752 −1.12257 −0.561283 0.827624i \(-0.689692\pi\)
−0.561283 + 0.827624i \(0.689692\pi\)
\(62\) 9.90814 1.25834
\(63\) 0 0
\(64\) 5.98262 0.747828
\(65\) −1.44843 −0.179656
\(66\) 0 0
\(67\) −3.15664 −0.385645 −0.192822 0.981234i \(-0.561764\pi\)
−0.192822 + 0.981234i \(0.561764\pi\)
\(68\) 0.978497 0.118660
\(69\) 0 0
\(70\) −1.90541 −0.227740
\(71\) −12.8707 −1.52747 −0.763733 0.645532i \(-0.776636\pi\)
−0.763733 + 0.645532i \(0.776636\pi\)
\(72\) 0 0
\(73\) −14.7184 −1.72266 −0.861331 0.508044i \(-0.830369\pi\)
−0.861331 + 0.508044i \(0.830369\pi\)
\(74\) 3.88035 0.451082
\(75\) 0 0
\(76\) −3.53059 −0.404987
\(77\) 0 0
\(78\) 0 0
\(79\) −16.9992 −1.91256 −0.956278 0.292458i \(-0.905527\pi\)
−0.956278 + 0.292458i \(0.905527\pi\)
\(80\) −0.469405 −0.0524811
\(81\) 0 0
\(82\) 0.751504 0.0829897
\(83\) −6.14148 −0.674115 −0.337058 0.941484i \(-0.609432\pi\)
−0.337058 + 0.941484i \(0.609432\pi\)
\(84\) 0 0
\(85\) −0.867063 −0.0940461
\(86\) −2.18529 −0.235646
\(87\) 0 0
\(88\) 0 0
\(89\) −3.77194 −0.399825 −0.199913 0.979814i \(-0.564066\pi\)
−0.199913 + 0.979814i \(0.564066\pi\)
\(90\) 0 0
\(91\) 2.95637 0.309912
\(92\) −5.31022 −0.553628
\(93\) 0 0
\(94\) −9.69031 −0.999480
\(95\) 3.12852 0.320979
\(96\) 0 0
\(97\) −1.93795 −0.196769 −0.0983845 0.995148i \(-0.531367\pi\)
−0.0983845 + 0.995148i \(0.531367\pi\)
\(98\) −2.64562 −0.267248
\(99\) 0 0
\(100\) −1.12852 −0.112852
\(101\) 5.25176 0.522570 0.261285 0.965262i \(-0.415854\pi\)
0.261285 + 0.965262i \(0.415854\pi\)
\(102\) 0 0
\(103\) −13.6007 −1.34011 −0.670056 0.742310i \(-0.733730\pi\)
−0.670056 + 0.742310i \(0.733730\pi\)
\(104\) 4.23026 0.414811
\(105\) 0 0
\(106\) −6.73632 −0.654290
\(107\) 9.99063 0.965831 0.482915 0.875667i \(-0.339578\pi\)
0.482915 + 0.875667i \(0.339578\pi\)
\(108\) 0 0
\(109\) −6.75183 −0.646708 −0.323354 0.946278i \(-0.604811\pi\)
−0.323354 + 0.946278i \(0.604811\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0.958094 0.0905314
\(113\) 5.24461 0.493371 0.246686 0.969096i \(-0.420658\pi\)
0.246686 + 0.969096i \(0.420658\pi\)
\(114\) 0 0
\(115\) 4.70547 0.438788
\(116\) 2.30032 0.213579
\(117\) 0 0
\(118\) 7.77025 0.715310
\(119\) 1.76974 0.162232
\(120\) 0 0
\(121\) 0 0
\(122\) −8.18476 −0.741013
\(123\) 0 0
\(124\) −11.9777 −1.07563
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −20.6751 −1.83462 −0.917311 0.398172i \(-0.869645\pi\)
−0.917311 + 0.398172i \(0.869645\pi\)
\(128\) −5.22091 −0.461468
\(129\) 0 0
\(130\) −1.35216 −0.118592
\(131\) −20.3089 −1.77439 −0.887197 0.461392i \(-0.847350\pi\)
−0.887197 + 0.461392i \(0.847350\pi\)
\(132\) 0 0
\(133\) −6.38556 −0.553698
\(134\) −2.94682 −0.254567
\(135\) 0 0
\(136\) 2.53232 0.217144
\(137\) 3.47982 0.297301 0.148650 0.988890i \(-0.452507\pi\)
0.148650 + 0.988890i \(0.452507\pi\)
\(138\) 0 0
\(139\) 2.73086 0.231629 0.115814 0.993271i \(-0.463052\pi\)
0.115814 + 0.993271i \(0.463052\pi\)
\(140\) 2.30340 0.194673
\(141\) 0 0
\(142\) −12.0152 −1.00829
\(143\) 0 0
\(144\) 0 0
\(145\) −2.03835 −0.169276
\(146\) −13.7401 −1.13714
\(147\) 0 0
\(148\) −4.69085 −0.385585
\(149\) 9.10841 0.746190 0.373095 0.927793i \(-0.378297\pi\)
0.373095 + 0.927793i \(0.378297\pi\)
\(150\) 0 0
\(151\) 21.5744 1.75570 0.877852 0.478933i \(-0.158976\pi\)
0.877852 + 0.478933i \(0.158976\pi\)
\(152\) −9.13706 −0.741114
\(153\) 0 0
\(154\) 0 0
\(155\) 10.6136 0.852506
\(156\) 0 0
\(157\) −23.7940 −1.89896 −0.949482 0.313821i \(-0.898391\pi\)
−0.949482 + 0.313821i \(0.898391\pi\)
\(158\) −15.8693 −1.26249
\(159\) 0 0
\(160\) 5.40294 0.427140
\(161\) −9.60425 −0.756921
\(162\) 0 0
\(163\) −1.80332 −0.141247 −0.0706236 0.997503i \(-0.522499\pi\)
−0.0706236 + 0.997503i \(0.522499\pi\)
\(164\) −0.908472 −0.0709397
\(165\) 0 0
\(166\) −5.73326 −0.444988
\(167\) 24.5874 1.90263 0.951315 0.308220i \(-0.0997333\pi\)
0.951315 + 0.308220i \(0.0997333\pi\)
\(168\) 0 0
\(169\) −10.9020 −0.838618
\(170\) −0.809430 −0.0620804
\(171\) 0 0
\(172\) 2.64174 0.201430
\(173\) 16.2115 1.23254 0.616270 0.787535i \(-0.288643\pi\)
0.616270 + 0.787535i \(0.288643\pi\)
\(174\) 0 0
\(175\) −2.04108 −0.154291
\(176\) 0 0
\(177\) 0 0
\(178\) −3.52123 −0.263927
\(179\) −16.9697 −1.26837 −0.634186 0.773181i \(-0.718665\pi\)
−0.634186 + 0.773181i \(0.718665\pi\)
\(180\) 0 0
\(181\) 18.3508 1.36400 0.682002 0.731350i \(-0.261109\pi\)
0.682002 + 0.731350i \(0.261109\pi\)
\(182\) 2.75986 0.204575
\(183\) 0 0
\(184\) −13.7427 −1.01312
\(185\) 4.15664 0.305602
\(186\) 0 0
\(187\) 0 0
\(188\) 11.7143 0.854356
\(189\) 0 0
\(190\) 2.92057 0.211880
\(191\) 15.3985 1.11420 0.557099 0.830446i \(-0.311914\pi\)
0.557099 + 0.830446i \(0.311914\pi\)
\(192\) 0 0
\(193\) −7.37942 −0.531182 −0.265591 0.964086i \(-0.585567\pi\)
−0.265591 + 0.964086i \(0.585567\pi\)
\(194\) −1.80914 −0.129888
\(195\) 0 0
\(196\) 3.19822 0.228444
\(197\) −3.14676 −0.224197 −0.112099 0.993697i \(-0.535757\pi\)
−0.112099 + 0.993697i \(0.535757\pi\)
\(198\) 0 0
\(199\) −3.25757 −0.230923 −0.115462 0.993312i \(-0.536835\pi\)
−0.115462 + 0.993312i \(0.536835\pi\)
\(200\) −2.92057 −0.206516
\(201\) 0 0
\(202\) 4.90268 0.344951
\(203\) 4.16043 0.292005
\(204\) 0 0
\(205\) 0.805012 0.0562245
\(206\) −12.6966 −0.884617
\(207\) 0 0
\(208\) 0.679903 0.0471428
\(209\) 0 0
\(210\) 0 0
\(211\) 19.9242 1.37164 0.685818 0.727773i \(-0.259445\pi\)
0.685818 + 0.727773i \(0.259445\pi\)
\(212\) 8.14335 0.559288
\(213\) 0 0
\(214\) 9.32657 0.637551
\(215\) −2.34089 −0.159647
\(216\) 0 0
\(217\) −21.6632 −1.47060
\(218\) −6.30305 −0.426896
\(219\) 0 0
\(220\) 0 0
\(221\) 1.25588 0.0844799
\(222\) 0 0
\(223\) 10.4801 0.701802 0.350901 0.936412i \(-0.385875\pi\)
0.350901 + 0.936412i \(0.385875\pi\)
\(224\) −11.0278 −0.736828
\(225\) 0 0
\(226\) 4.89601 0.325678
\(227\) −17.6731 −1.17301 −0.586503 0.809947i \(-0.699496\pi\)
−0.586503 + 0.809947i \(0.699496\pi\)
\(228\) 0 0
\(229\) 4.12967 0.272897 0.136448 0.990647i \(-0.456431\pi\)
0.136448 + 0.990647i \(0.456431\pi\)
\(230\) 4.39271 0.289646
\(231\) 0 0
\(232\) 5.95314 0.390843
\(233\) −13.5990 −0.890898 −0.445449 0.895307i \(-0.646956\pi\)
−0.445449 + 0.895307i \(0.646956\pi\)
\(234\) 0 0
\(235\) −10.3803 −0.677135
\(236\) −9.39324 −0.611448
\(237\) 0 0
\(238\) 1.65211 0.107090
\(239\) −8.00632 −0.517886 −0.258943 0.965893i \(-0.583374\pi\)
−0.258943 + 0.965893i \(0.583374\pi\)
\(240\) 0 0
\(241\) −0.501943 −0.0323330 −0.0161665 0.999869i \(-0.505146\pi\)
−0.0161665 + 0.999869i \(0.505146\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 9.89432 0.633419
\(245\) −2.83399 −0.181057
\(246\) 0 0
\(247\) −4.53146 −0.288330
\(248\) −30.9978 −1.96836
\(249\) 0 0
\(250\) 0.933531 0.0590417
\(251\) 18.5405 1.17027 0.585133 0.810937i \(-0.301042\pi\)
0.585133 + 0.810937i \(0.301042\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −19.3009 −1.21105
\(255\) 0 0
\(256\) −16.8391 −1.05245
\(257\) −7.38010 −0.460358 −0.230179 0.973148i \(-0.573931\pi\)
−0.230179 + 0.973148i \(0.573931\pi\)
\(258\) 0 0
\(259\) −8.48403 −0.527172
\(260\) 1.63459 0.101373
\(261\) 0 0
\(262\) −18.9590 −1.17129
\(263\) 3.05950 0.188657 0.0943285 0.995541i \(-0.469930\pi\)
0.0943285 + 0.995541i \(0.469930\pi\)
\(264\) 0 0
\(265\) −7.21596 −0.443273
\(266\) −5.96112 −0.365500
\(267\) 0 0
\(268\) 3.56233 0.217604
\(269\) 17.6801 1.07797 0.538987 0.842314i \(-0.318807\pi\)
0.538987 + 0.842314i \(0.318807\pi\)
\(270\) 0 0
\(271\) −15.2992 −0.929358 −0.464679 0.885479i \(-0.653830\pi\)
−0.464679 + 0.885479i \(0.653830\pi\)
\(272\) 0.407004 0.0246782
\(273\) 0 0
\(274\) 3.24852 0.196250
\(275\) 0 0
\(276\) 0 0
\(277\) −17.8238 −1.07093 −0.535463 0.844559i \(-0.679863\pi\)
−0.535463 + 0.844559i \(0.679863\pi\)
\(278\) 2.54935 0.152900
\(279\) 0 0
\(280\) 5.96112 0.356245
\(281\) 17.2990 1.03197 0.515985 0.856597i \(-0.327426\pi\)
0.515985 + 0.856597i \(0.327426\pi\)
\(282\) 0 0
\(283\) −25.7764 −1.53225 −0.766124 0.642693i \(-0.777817\pi\)
−0.766124 + 0.642693i \(0.777817\pi\)
\(284\) 14.5248 0.861887
\(285\) 0 0
\(286\) 0 0
\(287\) −1.64309 −0.0969888
\(288\) 0 0
\(289\) −16.2482 −0.955777
\(290\) −1.90286 −0.111740
\(291\) 0 0
\(292\) 16.6100 0.972029
\(293\) −13.5306 −0.790468 −0.395234 0.918581i \(-0.629336\pi\)
−0.395234 + 0.918581i \(0.629336\pi\)
\(294\) 0 0
\(295\) 8.32351 0.484613
\(296\) −12.1398 −0.705609
\(297\) 0 0
\(298\) 8.50299 0.492565
\(299\) −6.81557 −0.394155
\(300\) 0 0
\(301\) 4.77794 0.275396
\(302\) 20.1404 1.15895
\(303\) 0 0
\(304\) −1.46854 −0.0842268
\(305\) −8.76752 −0.502027
\(306\) 0 0
\(307\) −13.0268 −0.743478 −0.371739 0.928337i \(-0.621238\pi\)
−0.371739 + 0.928337i \(0.621238\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 9.90814 0.562745
\(311\) −1.90319 −0.107920 −0.0539601 0.998543i \(-0.517184\pi\)
−0.0539601 + 0.998543i \(0.517184\pi\)
\(312\) 0 0
\(313\) 10.9324 0.617934 0.308967 0.951073i \(-0.400017\pi\)
0.308967 + 0.951073i \(0.400017\pi\)
\(314\) −22.2124 −1.25352
\(315\) 0 0
\(316\) 19.1839 1.07918
\(317\) 16.0723 0.902708 0.451354 0.892345i \(-0.350941\pi\)
0.451354 + 0.892345i \(0.350941\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 5.98262 0.334439
\(321\) 0 0
\(322\) −8.96587 −0.499648
\(323\) −2.71262 −0.150934
\(324\) 0 0
\(325\) −1.44843 −0.0803447
\(326\) −1.68346 −0.0932382
\(327\) 0 0
\(328\) −2.35109 −0.129817
\(329\) 21.1870 1.16808
\(330\) 0 0
\(331\) 1.39579 0.0767195 0.0383597 0.999264i \(-0.487787\pi\)
0.0383597 + 0.999264i \(0.487787\pi\)
\(332\) 6.93078 0.380376
\(333\) 0 0
\(334\) 22.9531 1.25594
\(335\) −3.15664 −0.172466
\(336\) 0 0
\(337\) −2.76340 −0.150532 −0.0752660 0.997163i \(-0.523981\pi\)
−0.0752660 + 0.997163i \(0.523981\pi\)
\(338\) −10.1774 −0.553577
\(339\) 0 0
\(340\) 0.978497 0.0530664
\(341\) 0 0
\(342\) 0 0
\(343\) 20.0720 1.08378
\(344\) 6.83672 0.368611
\(345\) 0 0
\(346\) 15.1340 0.813608
\(347\) 3.12683 0.167857 0.0839286 0.996472i \(-0.473253\pi\)
0.0839286 + 0.996472i \(0.473253\pi\)
\(348\) 0 0
\(349\) 13.1277 0.702709 0.351355 0.936242i \(-0.385721\pi\)
0.351355 + 0.936242i \(0.385721\pi\)
\(350\) −1.90541 −0.101849
\(351\) 0 0
\(352\) 0 0
\(353\) −10.7984 −0.574739 −0.287370 0.957820i \(-0.592781\pi\)
−0.287370 + 0.957820i \(0.592781\pi\)
\(354\) 0 0
\(355\) −12.8707 −0.683103
\(356\) 4.25671 0.225605
\(357\) 0 0
\(358\) −15.8417 −0.837260
\(359\) −23.9716 −1.26517 −0.632585 0.774491i \(-0.718006\pi\)
−0.632585 + 0.774491i \(0.718006\pi\)
\(360\) 0 0
\(361\) −9.21237 −0.484861
\(362\) 17.1310 0.900388
\(363\) 0 0
\(364\) −3.33632 −0.174871
\(365\) −14.7184 −0.770398
\(366\) 0 0
\(367\) 8.87287 0.463160 0.231580 0.972816i \(-0.425610\pi\)
0.231580 + 0.972816i \(0.425610\pi\)
\(368\) −2.20877 −0.115140
\(369\) 0 0
\(370\) 3.88035 0.201730
\(371\) 14.7283 0.764658
\(372\) 0 0
\(373\) −26.3389 −1.36378 −0.681888 0.731456i \(-0.738841\pi\)
−0.681888 + 0.731456i \(0.738841\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 30.3163 1.56345
\(377\) 2.95242 0.152057
\(378\) 0 0
\(379\) 4.69610 0.241223 0.120611 0.992700i \(-0.461515\pi\)
0.120611 + 0.992700i \(0.461515\pi\)
\(380\) −3.53059 −0.181116
\(381\) 0 0
\(382\) 14.3750 0.735489
\(383\) 6.89931 0.352538 0.176269 0.984342i \(-0.443597\pi\)
0.176269 + 0.984342i \(0.443597\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −6.88892 −0.350637
\(387\) 0 0
\(388\) 2.18701 0.111029
\(389\) −2.40462 −0.121919 −0.0609596 0.998140i \(-0.519416\pi\)
−0.0609596 + 0.998140i \(0.519416\pi\)
\(390\) 0 0
\(391\) −4.07994 −0.206331
\(392\) 8.27688 0.418045
\(393\) 0 0
\(394\) −2.93760 −0.147994
\(395\) −16.9992 −0.855321
\(396\) 0 0
\(397\) 37.6715 1.89068 0.945338 0.326091i \(-0.105732\pi\)
0.945338 + 0.326091i \(0.105732\pi\)
\(398\) −3.04104 −0.152434
\(399\) 0 0
\(400\) −0.469405 −0.0234703
\(401\) 12.5168 0.625060 0.312530 0.949908i \(-0.398824\pi\)
0.312530 + 0.949908i \(0.398824\pi\)
\(402\) 0 0
\(403\) −15.3731 −0.765790
\(404\) −5.92671 −0.294865
\(405\) 0 0
\(406\) 3.88390 0.192754
\(407\) 0 0
\(408\) 0 0
\(409\) −26.4622 −1.30847 −0.654237 0.756290i \(-0.727010\pi\)
−0.654237 + 0.756290i \(0.727010\pi\)
\(410\) 0.751504 0.0371141
\(411\) 0 0
\(412\) 15.3486 0.756171
\(413\) −16.9889 −0.835971
\(414\) 0 0
\(415\) −6.14148 −0.301473
\(416\) −7.82580 −0.383691
\(417\) 0 0
\(418\) 0 0
\(419\) −30.9711 −1.51304 −0.756518 0.653973i \(-0.773101\pi\)
−0.756518 + 0.653973i \(0.773101\pi\)
\(420\) 0 0
\(421\) 18.2435 0.889132 0.444566 0.895746i \(-0.353358\pi\)
0.444566 + 0.895746i \(0.353358\pi\)
\(422\) 18.5998 0.905426
\(423\) 0 0
\(424\) 21.0747 1.02348
\(425\) −0.867063 −0.0420587
\(426\) 0 0
\(427\) 17.8952 0.866010
\(428\) −11.2746 −0.544979
\(429\) 0 0
\(430\) −2.18529 −0.105384
\(431\) −30.3488 −1.46185 −0.730925 0.682458i \(-0.760911\pi\)
−0.730925 + 0.682458i \(0.760911\pi\)
\(432\) 0 0
\(433\) 12.7012 0.610382 0.305191 0.952291i \(-0.401280\pi\)
0.305191 + 0.952291i \(0.401280\pi\)
\(434\) −20.2233 −0.970750
\(435\) 0 0
\(436\) 7.61957 0.364911
\(437\) 14.7212 0.704209
\(438\) 0 0
\(439\) −25.6564 −1.22451 −0.612257 0.790659i \(-0.709738\pi\)
−0.612257 + 0.790659i \(0.709738\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 1.17241 0.0557657
\(443\) 26.2148 1.24550 0.622751 0.782420i \(-0.286015\pi\)
0.622751 + 0.782420i \(0.286015\pi\)
\(444\) 0 0
\(445\) −3.77194 −0.178807
\(446\) 9.78354 0.463264
\(447\) 0 0
\(448\) −12.2110 −0.576916
\(449\) 12.0061 0.566605 0.283302 0.959031i \(-0.408570\pi\)
0.283302 + 0.959031i \(0.408570\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −5.91864 −0.278390
\(453\) 0 0
\(454\) −16.4984 −0.774309
\(455\) 2.95637 0.138597
\(456\) 0 0
\(457\) 13.7890 0.645022 0.322511 0.946566i \(-0.395473\pi\)
0.322511 + 0.946566i \(0.395473\pi\)
\(458\) 3.85518 0.180141
\(459\) 0 0
\(460\) −5.31022 −0.247590
\(461\) −11.3262 −0.527515 −0.263758 0.964589i \(-0.584962\pi\)
−0.263758 + 0.964589i \(0.584962\pi\)
\(462\) 0 0
\(463\) 6.18784 0.287573 0.143787 0.989609i \(-0.454072\pi\)
0.143787 + 0.989609i \(0.454072\pi\)
\(464\) 0.956812 0.0444189
\(465\) 0 0
\(466\) −12.6951 −0.588087
\(467\) −8.48288 −0.392541 −0.196270 0.980550i \(-0.562883\pi\)
−0.196270 + 0.980550i \(0.562883\pi\)
\(468\) 0 0
\(469\) 6.44295 0.297508
\(470\) −9.69031 −0.446981
\(471\) 0 0
\(472\) −24.3094 −1.11893
\(473\) 0 0
\(474\) 0 0
\(475\) 3.12852 0.143546
\(476\) −1.99719 −0.0915411
\(477\) 0 0
\(478\) −7.47415 −0.341860
\(479\) −9.23554 −0.421982 −0.210991 0.977488i \(-0.567669\pi\)
−0.210991 + 0.977488i \(0.567669\pi\)
\(480\) 0 0
\(481\) −6.02062 −0.274517
\(482\) −0.468579 −0.0213432
\(483\) 0 0
\(484\) 0 0
\(485\) −1.93795 −0.0879977
\(486\) 0 0
\(487\) 20.1708 0.914024 0.457012 0.889460i \(-0.348920\pi\)
0.457012 + 0.889460i \(0.348920\pi\)
\(488\) 25.6062 1.15914
\(489\) 0 0
\(490\) −2.64562 −0.119517
\(491\) 5.33594 0.240807 0.120404 0.992725i \(-0.461581\pi\)
0.120404 + 0.992725i \(0.461581\pi\)
\(492\) 0 0
\(493\) 1.76738 0.0795986
\(494\) −4.23026 −0.190328
\(495\) 0 0
\(496\) −4.98209 −0.223702
\(497\) 26.2700 1.17837
\(498\) 0 0
\(499\) −39.2816 −1.75849 −0.879243 0.476374i \(-0.841951\pi\)
−0.879243 + 0.476374i \(0.841951\pi\)
\(500\) −1.12852 −0.0504689
\(501\) 0 0
\(502\) 17.3081 0.772500
\(503\) −32.4181 −1.44545 −0.722727 0.691134i \(-0.757111\pi\)
−0.722727 + 0.691134i \(0.757111\pi\)
\(504\) 0 0
\(505\) 5.25176 0.233700
\(506\) 0 0
\(507\) 0 0
\(508\) 23.3323 1.03520
\(509\) 17.0656 0.756421 0.378211 0.925720i \(-0.376539\pi\)
0.378211 + 0.925720i \(0.376539\pi\)
\(510\) 0 0
\(511\) 30.0415 1.32896
\(512\) −5.27803 −0.233258
\(513\) 0 0
\(514\) −6.88955 −0.303885
\(515\) −13.6007 −0.599316
\(516\) 0 0
\(517\) 0 0
\(518\) −7.92011 −0.347990
\(519\) 0 0
\(520\) 4.23026 0.185509
\(521\) 15.0471 0.659224 0.329612 0.944116i \(-0.393082\pi\)
0.329612 + 0.944116i \(0.393082\pi\)
\(522\) 0 0
\(523\) −37.0416 −1.61972 −0.809859 0.586624i \(-0.800456\pi\)
−0.809859 + 0.586624i \(0.800456\pi\)
\(524\) 22.9189 1.00122
\(525\) 0 0
\(526\) 2.85614 0.124534
\(527\) −9.20267 −0.400875
\(528\) 0 0
\(529\) −0.858520 −0.0373270
\(530\) −6.73632 −0.292607
\(531\) 0 0
\(532\) 7.20623 0.312430
\(533\) −1.16601 −0.0505054
\(534\) 0 0
\(535\) 9.99063 0.431933
\(536\) 9.21919 0.398208
\(537\) 0 0
\(538\) 16.5049 0.711577
\(539\) 0 0
\(540\) 0 0
\(541\) 0.177099 0.00761407 0.00380704 0.999993i \(-0.498788\pi\)
0.00380704 + 0.999993i \(0.498788\pi\)
\(542\) −14.2822 −0.613475
\(543\) 0 0
\(544\) −4.68468 −0.200854
\(545\) −6.75183 −0.289217
\(546\) 0 0
\(547\) 1.21270 0.0518511 0.0259256 0.999664i \(-0.491747\pi\)
0.0259256 + 0.999664i \(0.491747\pi\)
\(548\) −3.92704 −0.167755
\(549\) 0 0
\(550\) 0 0
\(551\) −6.37702 −0.271670
\(552\) 0 0
\(553\) 34.6967 1.47545
\(554\) −16.6390 −0.706925
\(555\) 0 0
\(556\) −3.08183 −0.130699
\(557\) −1.93846 −0.0821352 −0.0410676 0.999156i \(-0.513076\pi\)
−0.0410676 + 0.999156i \(0.513076\pi\)
\(558\) 0 0
\(559\) 3.39062 0.143408
\(560\) 0.958094 0.0404869
\(561\) 0 0
\(562\) 16.1491 0.681210
\(563\) 20.9958 0.884866 0.442433 0.896802i \(-0.354115\pi\)
0.442433 + 0.896802i \(0.354115\pi\)
\(564\) 0 0
\(565\) 5.24461 0.220642
\(566\) −24.0631 −1.01145
\(567\) 0 0
\(568\) 37.5897 1.57723
\(569\) 39.3757 1.65071 0.825357 0.564612i \(-0.190974\pi\)
0.825357 + 0.564612i \(0.190974\pi\)
\(570\) 0 0
\(571\) 30.6796 1.28390 0.641950 0.766746i \(-0.278126\pi\)
0.641950 + 0.766746i \(0.278126\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −1.53388 −0.0640229
\(575\) 4.70547 0.196232
\(576\) 0 0
\(577\) −34.7038 −1.44474 −0.722369 0.691508i \(-0.756947\pi\)
−0.722369 + 0.691508i \(0.756947\pi\)
\(578\) −15.1682 −0.630914
\(579\) 0 0
\(580\) 2.30032 0.0955155
\(581\) 12.5353 0.520050
\(582\) 0 0
\(583\) 0 0
\(584\) 42.9862 1.77878
\(585\) 0 0
\(586\) −12.6313 −0.521793
\(587\) −8.05710 −0.332552 −0.166276 0.986079i \(-0.553174\pi\)
−0.166276 + 0.986079i \(0.553174\pi\)
\(588\) 0 0
\(589\) 33.2049 1.36818
\(590\) 7.77025 0.319896
\(591\) 0 0
\(592\) −1.95115 −0.0801917
\(593\) 28.1409 1.15561 0.577805 0.816175i \(-0.303909\pi\)
0.577805 + 0.816175i \(0.303909\pi\)
\(594\) 0 0
\(595\) 1.76974 0.0725524
\(596\) −10.2790 −0.421045
\(597\) 0 0
\(598\) −6.36255 −0.260184
\(599\) 28.5988 1.16851 0.584257 0.811568i \(-0.301386\pi\)
0.584257 + 0.811568i \(0.301386\pi\)
\(600\) 0 0
\(601\) −7.27610 −0.296799 −0.148399 0.988928i \(-0.547412\pi\)
−0.148399 + 0.988928i \(0.547412\pi\)
\(602\) 4.46035 0.181790
\(603\) 0 0
\(604\) −24.3472 −0.990672
\(605\) 0 0
\(606\) 0 0
\(607\) −5.34785 −0.217063 −0.108531 0.994093i \(-0.534615\pi\)
−0.108531 + 0.994093i \(0.534615\pi\)
\(608\) 16.9032 0.685515
\(609\) 0 0
\(610\) −8.18476 −0.331391
\(611\) 15.0352 0.608257
\(612\) 0 0
\(613\) −4.64099 −0.187448 −0.0937238 0.995598i \(-0.529877\pi\)
−0.0937238 + 0.995598i \(0.529877\pi\)
\(614\) −12.1609 −0.490774
\(615\) 0 0
\(616\) 0 0
\(617\) −24.5843 −0.989727 −0.494864 0.868971i \(-0.664782\pi\)
−0.494864 + 0.868971i \(0.664782\pi\)
\(618\) 0 0
\(619\) −6.13789 −0.246703 −0.123351 0.992363i \(-0.539364\pi\)
−0.123351 + 0.992363i \(0.539364\pi\)
\(620\) −11.9777 −0.481035
\(621\) 0 0
\(622\) −1.77669 −0.0712387
\(623\) 7.69884 0.308447
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 10.2057 0.407902
\(627\) 0 0
\(628\) 26.8519 1.07151
\(629\) −3.60407 −0.143704
\(630\) 0 0
\(631\) −17.7085 −0.704966 −0.352483 0.935818i \(-0.614662\pi\)
−0.352483 + 0.935818i \(0.614662\pi\)
\(632\) 49.6473 1.97486
\(633\) 0 0
\(634\) 15.0040 0.595883
\(635\) −20.6751 −0.820468
\(636\) 0 0
\(637\) 4.10485 0.162640
\(638\) 0 0
\(639\) 0 0
\(640\) −5.22091 −0.206375
\(641\) −39.8621 −1.57446 −0.787230 0.616659i \(-0.788486\pi\)
−0.787230 + 0.616659i \(0.788486\pi\)
\(642\) 0 0
\(643\) −17.8662 −0.704576 −0.352288 0.935892i \(-0.614596\pi\)
−0.352288 + 0.935892i \(0.614596\pi\)
\(644\) 10.8386 0.427100
\(645\) 0 0
\(646\) −2.53232 −0.0996327
\(647\) −5.04075 −0.198172 −0.0990862 0.995079i \(-0.531592\pi\)
−0.0990862 + 0.995079i \(0.531592\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −1.35216 −0.0530360
\(651\) 0 0
\(652\) 2.03509 0.0797001
\(653\) −40.7212 −1.59354 −0.796771 0.604281i \(-0.793460\pi\)
−0.796771 + 0.604281i \(0.793460\pi\)
\(654\) 0 0
\(655\) −20.3089 −0.793533
\(656\) −0.377877 −0.0147536
\(657\) 0 0
\(658\) 19.7787 0.771054
\(659\) −48.7556 −1.89925 −0.949624 0.313390i \(-0.898535\pi\)
−0.949624 + 0.313390i \(0.898535\pi\)
\(660\) 0 0
\(661\) −41.0061 −1.59495 −0.797477 0.603350i \(-0.793832\pi\)
−0.797477 + 0.603350i \(0.793832\pi\)
\(662\) 1.30301 0.0506430
\(663\) 0 0
\(664\) 17.9366 0.696076
\(665\) −6.38556 −0.247621
\(666\) 0 0
\(667\) −9.59140 −0.371380
\(668\) −27.7474 −1.07358
\(669\) 0 0
\(670\) −2.94682 −0.113846
\(671\) 0 0
\(672\) 0 0
\(673\) −30.4936 −1.17544 −0.587722 0.809063i \(-0.699975\pi\)
−0.587722 + 0.809063i \(0.699975\pi\)
\(674\) −2.57972 −0.0993671
\(675\) 0 0
\(676\) 12.3032 0.473198
\(677\) 43.4424 1.66963 0.834813 0.550534i \(-0.185576\pi\)
0.834813 + 0.550534i \(0.185576\pi\)
\(678\) 0 0
\(679\) 3.95551 0.151799
\(680\) 2.53232 0.0971099
\(681\) 0 0
\(682\) 0 0
\(683\) 43.5970 1.66819 0.834096 0.551619i \(-0.185990\pi\)
0.834096 + 0.551619i \(0.185990\pi\)
\(684\) 0 0
\(685\) 3.47982 0.132957
\(686\) 18.7378 0.715413
\(687\) 0 0
\(688\) 1.09882 0.0418923
\(689\) 10.4518 0.398184
\(690\) 0 0
\(691\) −8.30897 −0.316088 −0.158044 0.987432i \(-0.550519\pi\)
−0.158044 + 0.987432i \(0.550519\pi\)
\(692\) −18.2950 −0.695473
\(693\) 0 0
\(694\) 2.91900 0.110804
\(695\) 2.73086 0.103588
\(696\) 0 0
\(697\) −0.697996 −0.0264385
\(698\) 12.2551 0.463863
\(699\) 0 0
\(700\) 2.30340 0.0870603
\(701\) −9.08744 −0.343228 −0.171614 0.985164i \(-0.554898\pi\)
−0.171614 + 0.985164i \(0.554898\pi\)
\(702\) 0 0
\(703\) 13.0041 0.490460
\(704\) 0 0
\(705\) 0 0
\(706\) −10.0806 −0.379389
\(707\) −10.7193 −0.403139
\(708\) 0 0
\(709\) −4.85568 −0.182359 −0.0911794 0.995834i \(-0.529064\pi\)
−0.0911794 + 0.995834i \(0.529064\pi\)
\(710\) −12.0152 −0.450921
\(711\) 0 0
\(712\) 11.0162 0.412850
\(713\) 49.9421 1.87035
\(714\) 0 0
\(715\) 0 0
\(716\) 19.1506 0.715691
\(717\) 0 0
\(718\) −22.3782 −0.835147
\(719\) −23.9034 −0.891448 −0.445724 0.895171i \(-0.647054\pi\)
−0.445724 + 0.895171i \(0.647054\pi\)
\(720\) 0 0
\(721\) 27.7600 1.03384
\(722\) −8.60003 −0.320060
\(723\) 0 0
\(724\) −20.7092 −0.769653
\(725\) −2.03835 −0.0757024
\(726\) 0 0
\(727\) −11.3674 −0.421592 −0.210796 0.977530i \(-0.567606\pi\)
−0.210796 + 0.977530i \(0.567606\pi\)
\(728\) −8.63429 −0.320008
\(729\) 0 0
\(730\) −13.7401 −0.508545
\(731\) 2.02969 0.0750710
\(732\) 0 0
\(733\) −34.1178 −1.26017 −0.630085 0.776526i \(-0.716980\pi\)
−0.630085 + 0.776526i \(0.716980\pi\)
\(734\) 8.28311 0.305735
\(735\) 0 0
\(736\) 25.4234 0.937118
\(737\) 0 0
\(738\) 0 0
\(739\) 2.33848 0.0860225 0.0430113 0.999075i \(-0.486305\pi\)
0.0430113 + 0.999075i \(0.486305\pi\)
\(740\) −4.69085 −0.172439
\(741\) 0 0
\(742\) 13.7494 0.504755
\(743\) −14.4885 −0.531533 −0.265767 0.964037i \(-0.585625\pi\)
−0.265767 + 0.964037i \(0.585625\pi\)
\(744\) 0 0
\(745\) 9.10841 0.333706
\(746\) −24.5882 −0.900238
\(747\) 0 0
\(748\) 0 0
\(749\) −20.3917 −0.745096
\(750\) 0 0
\(751\) 33.3203 1.21587 0.607937 0.793985i \(-0.291997\pi\)
0.607937 + 0.793985i \(0.291997\pi\)
\(752\) 4.87256 0.177684
\(753\) 0 0
\(754\) 2.75617 0.100374
\(755\) 21.5744 0.785174
\(756\) 0 0
\(757\) 40.3778 1.46755 0.733777 0.679390i \(-0.237756\pi\)
0.733777 + 0.679390i \(0.237756\pi\)
\(758\) 4.38396 0.159233
\(759\) 0 0
\(760\) −9.13706 −0.331436
\(761\) 49.8971 1.80877 0.904384 0.426720i \(-0.140331\pi\)
0.904384 + 0.426720i \(0.140331\pi\)
\(762\) 0 0
\(763\) 13.7810 0.498907
\(764\) −17.3775 −0.628697
\(765\) 0 0
\(766\) 6.44072 0.232713
\(767\) −12.0561 −0.435319
\(768\) 0 0
\(769\) 4.93932 0.178116 0.0890582 0.996026i \(-0.471614\pi\)
0.0890582 + 0.996026i \(0.471614\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 8.32781 0.299725
\(773\) 33.4710 1.20387 0.601934 0.798546i \(-0.294397\pi\)
0.601934 + 0.798546i \(0.294397\pi\)
\(774\) 0 0
\(775\) 10.6136 0.381252
\(776\) 5.65992 0.203179
\(777\) 0 0
\(778\) −2.24479 −0.0804797
\(779\) 2.51850 0.0902345
\(780\) 0 0
\(781\) 0 0
\(782\) −3.80875 −0.136201
\(783\) 0 0
\(784\) 1.33029 0.0475104
\(785\) −23.7940 −0.849243
\(786\) 0 0
\(787\) 25.9400 0.924661 0.462331 0.886708i \(-0.347013\pi\)
0.462331 + 0.886708i \(0.347013\pi\)
\(788\) 3.55118 0.126506
\(789\) 0 0
\(790\) −15.8693 −0.564603
\(791\) −10.7047 −0.380614
\(792\) 0 0
\(793\) 12.6992 0.450961
\(794\) 35.1675 1.24805
\(795\) 0 0
\(796\) 3.67623 0.130301
\(797\) −9.86264 −0.349353 −0.174676 0.984626i \(-0.555888\pi\)
−0.174676 + 0.984626i \(0.555888\pi\)
\(798\) 0 0
\(799\) 9.00035 0.318410
\(800\) 5.40294 0.191023
\(801\) 0 0
\(802\) 11.6848 0.412606
\(803\) 0 0
\(804\) 0 0
\(805\) −9.60425 −0.338505
\(806\) −14.3513 −0.505503
\(807\) 0 0
\(808\) −15.3381 −0.539594
\(809\) 33.0590 1.16229 0.581146 0.813799i \(-0.302604\pi\)
0.581146 + 0.813799i \(0.302604\pi\)
\(810\) 0 0
\(811\) 5.21312 0.183057 0.0915286 0.995802i \(-0.470825\pi\)
0.0915286 + 0.995802i \(0.470825\pi\)
\(812\) −4.69513 −0.164767
\(813\) 0 0
\(814\) 0 0
\(815\) −1.80332 −0.0631677
\(816\) 0 0
\(817\) −7.32351 −0.256217
\(818\) −24.7033 −0.863731
\(819\) 0 0
\(820\) −0.908472 −0.0317252
\(821\) 1.63933 0.0572131 0.0286066 0.999591i \(-0.490893\pi\)
0.0286066 + 0.999591i \(0.490893\pi\)
\(822\) 0 0
\(823\) −17.8894 −0.623586 −0.311793 0.950150i \(-0.600930\pi\)
−0.311793 + 0.950150i \(0.600930\pi\)
\(824\) 39.7217 1.38377
\(825\) 0 0
\(826\) −15.8597 −0.551830
\(827\) −35.5449 −1.23602 −0.618009 0.786171i \(-0.712061\pi\)
−0.618009 + 0.786171i \(0.712061\pi\)
\(828\) 0 0
\(829\) −26.9512 −0.936053 −0.468026 0.883715i \(-0.655035\pi\)
−0.468026 + 0.883715i \(0.655035\pi\)
\(830\) −5.73326 −0.199004
\(831\) 0 0
\(832\) −8.66544 −0.300420
\(833\) 2.45725 0.0851386
\(834\) 0 0
\(835\) 24.5874 0.850882
\(836\) 0 0
\(837\) 0 0
\(838\) −28.9124 −0.998764
\(839\) 11.8373 0.408667 0.204334 0.978901i \(-0.434497\pi\)
0.204334 + 0.978901i \(0.434497\pi\)
\(840\) 0 0
\(841\) −24.8451 −0.856729
\(842\) 17.0308 0.586921
\(843\) 0 0
\(844\) −22.4848 −0.773959
\(845\) −10.9020 −0.375041
\(846\) 0 0
\(847\) 0 0
\(848\) 3.38721 0.116317
\(849\) 0 0
\(850\) −0.809430 −0.0277632
\(851\) 19.5590 0.670472
\(852\) 0 0
\(853\) −30.3043 −1.03760 −0.518799 0.854896i \(-0.673621\pi\)
−0.518799 + 0.854896i \(0.673621\pi\)
\(854\) 16.7057 0.571659
\(855\) 0 0
\(856\) −29.1783 −0.997295
\(857\) 29.2318 0.998540 0.499270 0.866446i \(-0.333602\pi\)
0.499270 + 0.866446i \(0.333602\pi\)
\(858\) 0 0
\(859\) −36.7151 −1.25270 −0.626351 0.779541i \(-0.715452\pi\)
−0.626351 + 0.779541i \(0.715452\pi\)
\(860\) 2.64174 0.0900824
\(861\) 0 0
\(862\) −28.3315 −0.964976
\(863\) −43.5758 −1.48334 −0.741669 0.670766i \(-0.765966\pi\)
−0.741669 + 0.670766i \(0.765966\pi\)
\(864\) 0 0
\(865\) 16.2115 0.551209
\(866\) 11.8570 0.402917
\(867\) 0 0
\(868\) 24.4474 0.829798
\(869\) 0 0
\(870\) 0 0
\(871\) 4.57219 0.154923
\(872\) 19.7192 0.667777
\(873\) 0 0
\(874\) 13.7427 0.464853
\(875\) −2.04108 −0.0690011
\(876\) 0 0
\(877\) −11.1173 −0.375404 −0.187702 0.982226i \(-0.560104\pi\)
−0.187702 + 0.982226i \(0.560104\pi\)
\(878\) −23.9511 −0.808309
\(879\) 0 0
\(880\) 0 0
\(881\) 39.1155 1.31783 0.658917 0.752216i \(-0.271015\pi\)
0.658917 + 0.752216i \(0.271015\pi\)
\(882\) 0 0
\(883\) 19.5187 0.656857 0.328428 0.944529i \(-0.393481\pi\)
0.328428 + 0.944529i \(0.393481\pi\)
\(884\) −1.41729 −0.0476686
\(885\) 0 0
\(886\) 24.4723 0.822164
\(887\) 5.19108 0.174299 0.0871497 0.996195i \(-0.472224\pi\)
0.0871497 + 0.996195i \(0.472224\pi\)
\(888\) 0 0
\(889\) 42.1996 1.41533
\(890\) −3.52123 −0.118032
\(891\) 0 0
\(892\) −11.8270 −0.395999
\(893\) −32.4749 −1.08673
\(894\) 0 0
\(895\) −16.9697 −0.567233
\(896\) 10.6563 0.356002
\(897\) 0 0
\(898\) 11.2081 0.374019
\(899\) −21.6343 −0.721543
\(900\) 0 0
\(901\) 6.25669 0.208440
\(902\) 0 0
\(903\) 0 0
\(904\) −15.3173 −0.509444
\(905\) 18.3508 0.610001
\(906\) 0 0
\(907\) −42.4379 −1.40913 −0.704563 0.709641i \(-0.748857\pi\)
−0.704563 + 0.709641i \(0.748857\pi\)
\(908\) 19.9445 0.661880
\(909\) 0 0
\(910\) 2.75986 0.0914886
\(911\) 19.7499 0.654344 0.327172 0.944965i \(-0.393904\pi\)
0.327172 + 0.944965i \(0.393904\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 12.8725 0.425783
\(915\) 0 0
\(916\) −4.66042 −0.153985
\(917\) 41.4520 1.36887
\(918\) 0 0
\(919\) −24.4853 −0.807696 −0.403848 0.914826i \(-0.632328\pi\)
−0.403848 + 0.914826i \(0.632328\pi\)
\(920\) −13.7427 −0.453082
\(921\) 0 0
\(922\) −10.5734 −0.348216
\(923\) 18.6423 0.613619
\(924\) 0 0
\(925\) 4.15664 0.136669
\(926\) 5.77654 0.189829
\(927\) 0 0
\(928\) −11.0131 −0.361522
\(929\) 6.05289 0.198589 0.0992944 0.995058i \(-0.468341\pi\)
0.0992944 + 0.995058i \(0.468341\pi\)
\(930\) 0 0
\(931\) −8.86620 −0.290578
\(932\) 15.3467 0.502698
\(933\) 0 0
\(934\) −7.91903 −0.259119
\(935\) 0 0
\(936\) 0 0
\(937\) 6.83865 0.223409 0.111704 0.993741i \(-0.464369\pi\)
0.111704 + 0.993741i \(0.464369\pi\)
\(938\) 6.01470 0.196387
\(939\) 0 0
\(940\) 11.7143 0.382080
\(941\) 3.95417 0.128902 0.0644512 0.997921i \(-0.479470\pi\)
0.0644512 + 0.997921i \(0.479470\pi\)
\(942\) 0 0
\(943\) 3.78796 0.123353
\(944\) −3.90710 −0.127165
\(945\) 0 0
\(946\) 0 0
\(947\) 40.1742 1.30549 0.652743 0.757579i \(-0.273618\pi\)
0.652743 + 0.757579i \(0.273618\pi\)
\(948\) 0 0
\(949\) 21.3187 0.692034
\(950\) 2.92057 0.0947558
\(951\) 0 0
\(952\) −5.16866 −0.167517
\(953\) 41.2470 1.33612 0.668061 0.744106i \(-0.267124\pi\)
0.668061 + 0.744106i \(0.267124\pi\)
\(954\) 0 0
\(955\) 15.3985 0.498284
\(956\) 9.03529 0.292222
\(957\) 0 0
\(958\) −8.62166 −0.278553
\(959\) −7.10258 −0.229354
\(960\) 0 0
\(961\) 81.6488 2.63383
\(962\) −5.62044 −0.181210
\(963\) 0 0
\(964\) 0.566452 0.0182442
\(965\) −7.37942 −0.237552
\(966\) 0 0
\(967\) 9.16826 0.294831 0.147416 0.989075i \(-0.452904\pi\)
0.147416 + 0.989075i \(0.452904\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −1.80914 −0.0580879
\(971\) 3.71264 0.119144 0.0595722 0.998224i \(-0.481026\pi\)
0.0595722 + 0.998224i \(0.481026\pi\)
\(972\) 0 0
\(973\) −5.57391 −0.178691
\(974\) 18.8300 0.603353
\(975\) 0 0
\(976\) 4.11552 0.131735
\(977\) −12.3932 −0.396495 −0.198247 0.980152i \(-0.563525\pi\)
−0.198247 + 0.980152i \(0.563525\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 3.19822 0.102163
\(981\) 0 0
\(982\) 4.98126 0.158958
\(983\) −23.2949 −0.742992 −0.371496 0.928435i \(-0.621155\pi\)
−0.371496 + 0.928435i \(0.621155\pi\)
\(984\) 0 0
\(985\) −3.14676 −0.100264
\(986\) 1.64990 0.0525436
\(987\) 0 0
\(988\) 5.11384 0.162693
\(989\) −11.0150 −0.350256
\(990\) 0 0
\(991\) −37.7826 −1.20020 −0.600101 0.799924i \(-0.704873\pi\)
−0.600101 + 0.799924i \(0.704873\pi\)
\(992\) 57.3447 1.82070
\(993\) 0 0
\(994\) 24.5239 0.777851
\(995\) −3.25757 −0.103272
\(996\) 0 0
\(997\) 7.87590 0.249432 0.124716 0.992192i \(-0.460198\pi\)
0.124716 + 0.992192i \(0.460198\pi\)
\(998\) −36.6706 −1.16079
\(999\) 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 5445.2.a.bk.1.3 4
3.2 odd 2 1815.2.a.v.1.2 4
11.3 even 5 495.2.n.b.361.2 8
11.4 even 5 495.2.n.b.181.2 8
11.10 odd 2 5445.2.a.br.1.2 4
15.14 odd 2 9075.2.a.cq.1.3 4
33.14 odd 10 165.2.m.b.31.1 yes 8
33.26 odd 10 165.2.m.b.16.1 8
33.32 even 2 1815.2.a.r.1.3 4
165.14 odd 10 825.2.n.i.526.2 8
165.47 even 20 825.2.bx.g.724.2 16
165.59 odd 10 825.2.n.i.676.2 8
165.92 even 20 825.2.bx.g.49.3 16
165.113 even 20 825.2.bx.g.724.3 16
165.158 even 20 825.2.bx.g.49.2 16
165.164 even 2 9075.2.a.dg.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.m.b.16.1 8 33.26 odd 10
165.2.m.b.31.1 yes 8 33.14 odd 10
495.2.n.b.181.2 8 11.4 even 5
495.2.n.b.361.2 8 11.3 even 5
825.2.n.i.526.2 8 165.14 odd 10
825.2.n.i.676.2 8 165.59 odd 10
825.2.bx.g.49.2 16 165.158 even 20
825.2.bx.g.49.3 16 165.92 even 20
825.2.bx.g.724.2 16 165.47 even 20
825.2.bx.g.724.3 16 165.113 even 20
1815.2.a.r.1.3 4 33.32 even 2
1815.2.a.v.1.2 4 3.2 odd 2
5445.2.a.bk.1.3 4 1.1 even 1 trivial
5445.2.a.br.1.2 4 11.10 odd 2
9075.2.a.cq.1.3 4 15.14 odd 2
9075.2.a.dg.1.2 4 165.164 even 2