Properties

Label 585.2.a.n.1.2
Level $585$
Weight $2$
Character 585.1
Self dual yes
Analytic conductor $4.671$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.67124851824\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 195)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.81361\) of defining polynomial
Character \(\chi\) \(=\) 585.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.289169 q^{2} -1.91638 q^{4} +1.00000 q^{5} +4.91638 q^{7} -1.13249 q^{8} +O(q^{10})\) \(q+0.289169 q^{2} -1.91638 q^{4} +1.00000 q^{5} +4.91638 q^{7} -1.13249 q^{8} +0.289169 q^{10} -4.91638 q^{11} +1.00000 q^{13} +1.42166 q^{14} +3.50528 q^{16} +4.33804 q^{17} +2.57834 q^{19} -1.91638 q^{20} -1.42166 q^{22} +6.33804 q^{23} +1.00000 q^{25} +0.289169 q^{26} -9.42166 q^{28} -6.00000 q^{29} +1.42166 q^{31} +3.27861 q^{32} +1.25443 q^{34} +4.91638 q^{35} +9.49472 q^{37} +0.745574 q^{38} -1.13249 q^{40} -4.33804 q^{41} -1.15667 q^{43} +9.42166 q^{44} +1.83276 q^{46} +5.42166 q^{47} +17.1708 q^{49} +0.289169 q^{50} -1.91638 q^{52} +0.338044 q^{53} -4.91638 q^{55} -5.56777 q^{56} -1.73501 q^{58} +11.2544 q^{59} -10.1708 q^{61} +0.411100 q^{62} -6.06249 q^{64} +1.00000 q^{65} -7.25443 q^{67} -8.31335 q^{68} +1.42166 q^{70} -0.916382 q^{71} -3.15667 q^{73} +2.74557 q^{74} -4.94108 q^{76} -24.1708 q^{77} -3.49472 q^{79} +3.50528 q^{80} -1.25443 q^{82} -11.2544 q^{83} +4.33804 q^{85} -0.334474 q^{86} +5.56777 q^{88} +0.338044 q^{89} +4.91638 q^{91} -12.1461 q^{92} +1.56777 q^{94} +2.57834 q^{95} -12.3380 q^{97} +4.96526 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 8 q^{4} + 3 q^{5} + q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 8 q^{4} + 3 q^{5} + q^{7} - 6 q^{8} - q^{11} + 3 q^{13} + 6 q^{14} + 26 q^{16} + q^{17} + 6 q^{19} + 8 q^{20} - 6 q^{22} + 7 q^{23} + 3 q^{25} - 30 q^{28} - 18 q^{29} + 6 q^{31} - 22 q^{32} - 22 q^{34} + q^{35} + 13 q^{37} + 28 q^{38} - 6 q^{40} - q^{41} + 30 q^{44} - 22 q^{46} + 18 q^{47} + 12 q^{49} + 8 q^{52} - 11 q^{53} - q^{55} + 16 q^{56} + 8 q^{59} + 9 q^{61} - 28 q^{62} + 30 q^{64} + 3 q^{65} + 4 q^{67} - 18 q^{68} + 6 q^{70} + 11 q^{71} - 6 q^{73} + 34 q^{74} + 4 q^{76} - 33 q^{77} + 5 q^{79} + 26 q^{80} + 22 q^{82} - 8 q^{83} + q^{85} - 56 q^{86} - 16 q^{88} - 11 q^{89} + q^{91} - 2 q^{92} - 28 q^{94} + 6 q^{95} - 25 q^{97} - 10 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.289169 0.204473 0.102237 0.994760i \(-0.467400\pi\)
0.102237 + 0.994760i \(0.467400\pi\)
\(3\) 0 0
\(4\) −1.91638 −0.958191
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 4.91638 1.85822 0.929109 0.369807i \(-0.120576\pi\)
0.929109 + 0.369807i \(0.120576\pi\)
\(8\) −1.13249 −0.400397
\(9\) 0 0
\(10\) 0.289169 0.0914431
\(11\) −4.91638 −1.48234 −0.741172 0.671315i \(-0.765730\pi\)
−0.741172 + 0.671315i \(0.765730\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 1.42166 0.379955
\(15\) 0 0
\(16\) 3.50528 0.876320
\(17\) 4.33804 1.05213 0.526065 0.850444i \(-0.323667\pi\)
0.526065 + 0.850444i \(0.323667\pi\)
\(18\) 0 0
\(19\) 2.57834 0.591511 0.295756 0.955264i \(-0.404429\pi\)
0.295756 + 0.955264i \(0.404429\pi\)
\(20\) −1.91638 −0.428516
\(21\) 0 0
\(22\) −1.42166 −0.303100
\(23\) 6.33804 1.32157 0.660787 0.750574i \(-0.270223\pi\)
0.660787 + 0.750574i \(0.270223\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0.289169 0.0567106
\(27\) 0 0
\(28\) −9.42166 −1.78053
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 1.42166 0.255338 0.127669 0.991817i \(-0.459250\pi\)
0.127669 + 0.991817i \(0.459250\pi\)
\(32\) 3.27861 0.579581
\(33\) 0 0
\(34\) 1.25443 0.215132
\(35\) 4.91638 0.831020
\(36\) 0 0
\(37\) 9.49472 1.56092 0.780461 0.625204i \(-0.214984\pi\)
0.780461 + 0.625204i \(0.214984\pi\)
\(38\) 0.745574 0.120948
\(39\) 0 0
\(40\) −1.13249 −0.179063
\(41\) −4.33804 −0.677489 −0.338744 0.940878i \(-0.610002\pi\)
−0.338744 + 0.940878i \(0.610002\pi\)
\(42\) 0 0
\(43\) −1.15667 −0.176391 −0.0881956 0.996103i \(-0.528110\pi\)
−0.0881956 + 0.996103i \(0.528110\pi\)
\(44\) 9.42166 1.42037
\(45\) 0 0
\(46\) 1.83276 0.270226
\(47\) 5.42166 0.790831 0.395415 0.918502i \(-0.370601\pi\)
0.395415 + 0.918502i \(0.370601\pi\)
\(48\) 0 0
\(49\) 17.1708 2.45297
\(50\) 0.289169 0.0408946
\(51\) 0 0
\(52\) −1.91638 −0.265754
\(53\) 0.338044 0.0464340 0.0232170 0.999730i \(-0.492609\pi\)
0.0232170 + 0.999730i \(0.492609\pi\)
\(54\) 0 0
\(55\) −4.91638 −0.662925
\(56\) −5.56777 −0.744025
\(57\) 0 0
\(58\) −1.73501 −0.227818
\(59\) 11.2544 1.46520 0.732601 0.680659i \(-0.238306\pi\)
0.732601 + 0.680659i \(0.238306\pi\)
\(60\) 0 0
\(61\) −10.1708 −1.30224 −0.651119 0.758975i \(-0.725700\pi\)
−0.651119 + 0.758975i \(0.725700\pi\)
\(62\) 0.411100 0.0522098
\(63\) 0 0
\(64\) −6.06249 −0.757812
\(65\) 1.00000 0.124035
\(66\) 0 0
\(67\) −7.25443 −0.886269 −0.443135 0.896455i \(-0.646134\pi\)
−0.443135 + 0.896455i \(0.646134\pi\)
\(68\) −8.31335 −1.00814
\(69\) 0 0
\(70\) 1.42166 0.169921
\(71\) −0.916382 −0.108754 −0.0543772 0.998520i \(-0.517317\pi\)
−0.0543772 + 0.998520i \(0.517317\pi\)
\(72\) 0 0
\(73\) −3.15667 −0.369461 −0.184730 0.982789i \(-0.559141\pi\)
−0.184730 + 0.982789i \(0.559141\pi\)
\(74\) 2.74557 0.319166
\(75\) 0 0
\(76\) −4.94108 −0.566780
\(77\) −24.1708 −2.75452
\(78\) 0 0
\(79\) −3.49472 −0.393187 −0.196593 0.980485i \(-0.562988\pi\)
−0.196593 + 0.980485i \(0.562988\pi\)
\(80\) 3.50528 0.391902
\(81\) 0 0
\(82\) −1.25443 −0.138528
\(83\) −11.2544 −1.23533 −0.617667 0.786440i \(-0.711922\pi\)
−0.617667 + 0.786440i \(0.711922\pi\)
\(84\) 0 0
\(85\) 4.33804 0.470527
\(86\) −0.334474 −0.0360672
\(87\) 0 0
\(88\) 5.56777 0.593527
\(89\) 0.338044 0.0358326 0.0179163 0.999839i \(-0.494297\pi\)
0.0179163 + 0.999839i \(0.494297\pi\)
\(90\) 0 0
\(91\) 4.91638 0.515377
\(92\) −12.1461 −1.26632
\(93\) 0 0
\(94\) 1.56777 0.161704
\(95\) 2.57834 0.264532
\(96\) 0 0
\(97\) −12.3380 −1.25274 −0.626369 0.779526i \(-0.715460\pi\)
−0.626369 + 0.779526i \(0.715460\pi\)
\(98\) 4.96526 0.501567
\(99\) 0 0
\(100\) −1.91638 −0.191638
\(101\) −10.6761 −1.06231 −0.531155 0.847274i \(-0.678242\pi\)
−0.531155 + 0.847274i \(0.678242\pi\)
\(102\) 0 0
\(103\) −14.5089 −1.42960 −0.714800 0.699329i \(-0.753482\pi\)
−0.714800 + 0.699329i \(0.753482\pi\)
\(104\) −1.13249 −0.111050
\(105\) 0 0
\(106\) 0.0977518 0.00949450
\(107\) −4.17081 −0.403207 −0.201604 0.979467i \(-0.564615\pi\)
−0.201604 + 0.979467i \(0.564615\pi\)
\(108\) 0 0
\(109\) −3.83276 −0.367112 −0.183556 0.983009i \(-0.558761\pi\)
−0.183556 + 0.983009i \(0.558761\pi\)
\(110\) −1.42166 −0.135550
\(111\) 0 0
\(112\) 17.2333 1.62839
\(113\) 0.843326 0.0793334 0.0396667 0.999213i \(-0.487370\pi\)
0.0396667 + 0.999213i \(0.487370\pi\)
\(114\) 0 0
\(115\) 6.33804 0.591026
\(116\) 11.4983 1.06759
\(117\) 0 0
\(118\) 3.25443 0.299594
\(119\) 21.3275 1.95509
\(120\) 0 0
\(121\) 13.1708 1.19735
\(122\) −2.94108 −0.266273
\(123\) 0 0
\(124\) −2.72445 −0.244663
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 1.83276 0.162631 0.0813157 0.996688i \(-0.474088\pi\)
0.0813157 + 0.996688i \(0.474088\pi\)
\(128\) −8.31029 −0.734533
\(129\) 0 0
\(130\) 0.289169 0.0253618
\(131\) −5.83276 −0.509611 −0.254805 0.966992i \(-0.582011\pi\)
−0.254805 + 0.966992i \(0.582011\pi\)
\(132\) 0 0
\(133\) 12.6761 1.09916
\(134\) −2.09775 −0.181218
\(135\) 0 0
\(136\) −4.91281 −0.421270
\(137\) 16.5089 1.41045 0.705223 0.708985i \(-0.250847\pi\)
0.705223 + 0.708985i \(0.250847\pi\)
\(138\) 0 0
\(139\) 7.49472 0.635694 0.317847 0.948142i \(-0.397040\pi\)
0.317847 + 0.948142i \(0.397040\pi\)
\(140\) −9.42166 −0.796276
\(141\) 0 0
\(142\) −0.264989 −0.0222374
\(143\) −4.91638 −0.411128
\(144\) 0 0
\(145\) −6.00000 −0.498273
\(146\) −0.912811 −0.0755448
\(147\) 0 0
\(148\) −18.1955 −1.49566
\(149\) −20.4842 −1.67813 −0.839064 0.544033i \(-0.816897\pi\)
−0.839064 + 0.544033i \(0.816897\pi\)
\(150\) 0 0
\(151\) −16.4111 −1.33552 −0.667758 0.744378i \(-0.732746\pi\)
−0.667758 + 0.744378i \(0.732746\pi\)
\(152\) −2.91995 −0.236839
\(153\) 0 0
\(154\) −6.98944 −0.563225
\(155\) 1.42166 0.114191
\(156\) 0 0
\(157\) −21.6655 −1.72910 −0.864549 0.502549i \(-0.832396\pi\)
−0.864549 + 0.502549i \(0.832396\pi\)
\(158\) −1.01056 −0.0803961
\(159\) 0 0
\(160\) 3.27861 0.259197
\(161\) 31.1602 2.45577
\(162\) 0 0
\(163\) 6.07306 0.475678 0.237839 0.971305i \(-0.423561\pi\)
0.237839 + 0.971305i \(0.423561\pi\)
\(164\) 8.31335 0.649163
\(165\) 0 0
\(166\) −3.25443 −0.252592
\(167\) 0.745574 0.0576942 0.0288471 0.999584i \(-0.490816\pi\)
0.0288471 + 0.999584i \(0.490816\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 1.25443 0.0962101
\(171\) 0 0
\(172\) 2.21663 0.169016
\(173\) −0.843326 −0.0641169 −0.0320584 0.999486i \(-0.510206\pi\)
−0.0320584 + 0.999486i \(0.510206\pi\)
\(174\) 0 0
\(175\) 4.91638 0.371644
\(176\) −17.2333 −1.29901
\(177\) 0 0
\(178\) 0.0977518 0.00732681
\(179\) 18.9894 1.41934 0.709669 0.704536i \(-0.248845\pi\)
0.709669 + 0.704536i \(0.248845\pi\)
\(180\) 0 0
\(181\) 17.4947 1.30037 0.650186 0.759775i \(-0.274691\pi\)
0.650186 + 0.759775i \(0.274691\pi\)
\(182\) 1.42166 0.105381
\(183\) 0 0
\(184\) −7.17780 −0.529154
\(185\) 9.49472 0.698066
\(186\) 0 0
\(187\) −21.3275 −1.55962
\(188\) −10.3900 −0.757767
\(189\) 0 0
\(190\) 0.745574 0.0540896
\(191\) 22.5089 1.62868 0.814342 0.580386i \(-0.197098\pi\)
0.814342 + 0.580386i \(0.197098\pi\)
\(192\) 0 0
\(193\) 2.65139 0.190851 0.0954257 0.995437i \(-0.469579\pi\)
0.0954257 + 0.995437i \(0.469579\pi\)
\(194\) −3.56777 −0.256151
\(195\) 0 0
\(196\) −32.9058 −2.35042
\(197\) 12.9894 0.925459 0.462730 0.886500i \(-0.346870\pi\)
0.462730 + 0.886500i \(0.346870\pi\)
\(198\) 0 0
\(199\) −2.84333 −0.201558 −0.100779 0.994909i \(-0.532134\pi\)
−0.100779 + 0.994909i \(0.532134\pi\)
\(200\) −1.13249 −0.0800794
\(201\) 0 0
\(202\) −3.08719 −0.217214
\(203\) −29.4983 −2.07037
\(204\) 0 0
\(205\) −4.33804 −0.302982
\(206\) −4.19550 −0.292315
\(207\) 0 0
\(208\) 3.50528 0.243048
\(209\) −12.6761 −0.876823
\(210\) 0 0
\(211\) 6.31335 0.434629 0.217314 0.976102i \(-0.430270\pi\)
0.217314 + 0.976102i \(0.430270\pi\)
\(212\) −0.647822 −0.0444926
\(213\) 0 0
\(214\) −1.20607 −0.0824450
\(215\) −1.15667 −0.0788845
\(216\) 0 0
\(217\) 6.98944 0.474474
\(218\) −1.10831 −0.0750645
\(219\) 0 0
\(220\) 9.42166 0.635208
\(221\) 4.33804 0.291808
\(222\) 0 0
\(223\) 19.2544 1.28937 0.644686 0.764448i \(-0.276988\pi\)
0.644686 + 0.764448i \(0.276988\pi\)
\(224\) 16.1189 1.07699
\(225\) 0 0
\(226\) 0.243863 0.0162215
\(227\) 13.0872 0.868627 0.434314 0.900762i \(-0.356991\pi\)
0.434314 + 0.900762i \(0.356991\pi\)
\(228\) 0 0
\(229\) −24.5089 −1.61959 −0.809795 0.586713i \(-0.800422\pi\)
−0.809795 + 0.586713i \(0.800422\pi\)
\(230\) 1.83276 0.120849
\(231\) 0 0
\(232\) 6.79497 0.446111
\(233\) −8.33804 −0.546243 −0.273122 0.961979i \(-0.588056\pi\)
−0.273122 + 0.961979i \(0.588056\pi\)
\(234\) 0 0
\(235\) 5.42166 0.353670
\(236\) −21.5678 −1.40394
\(237\) 0 0
\(238\) 6.16724 0.399763
\(239\) −8.91638 −0.576753 −0.288376 0.957517i \(-0.593115\pi\)
−0.288376 + 0.957517i \(0.593115\pi\)
\(240\) 0 0
\(241\) −6.00000 −0.386494 −0.193247 0.981150i \(-0.561902\pi\)
−0.193247 + 0.981150i \(0.561902\pi\)
\(242\) 3.80858 0.244825
\(243\) 0 0
\(244\) 19.4911 1.24779
\(245\) 17.1708 1.09700
\(246\) 0 0
\(247\) 2.57834 0.164056
\(248\) −1.61003 −0.102237
\(249\) 0 0
\(250\) 0.289169 0.0182886
\(251\) 6.31335 0.398495 0.199248 0.979949i \(-0.436150\pi\)
0.199248 + 0.979949i \(0.436150\pi\)
\(252\) 0 0
\(253\) −31.1602 −1.95903
\(254\) 0.529977 0.0332537
\(255\) 0 0
\(256\) 9.72191 0.607619
\(257\) 11.1567 0.695934 0.347967 0.937507i \(-0.386872\pi\)
0.347967 + 0.937507i \(0.386872\pi\)
\(258\) 0 0
\(259\) 46.6797 2.90053
\(260\) −1.91638 −0.118849
\(261\) 0 0
\(262\) −1.68665 −0.104202
\(263\) −8.00000 −0.493301 −0.246651 0.969104i \(-0.579330\pi\)
−0.246651 + 0.969104i \(0.579330\pi\)
\(264\) 0 0
\(265\) 0.338044 0.0207659
\(266\) 3.66553 0.224748
\(267\) 0 0
\(268\) 13.9022 0.849215
\(269\) −18.6761 −1.13870 −0.569351 0.822095i \(-0.692805\pi\)
−0.569351 + 0.822095i \(0.692805\pi\)
\(270\) 0 0
\(271\) 6.57834 0.399606 0.199803 0.979836i \(-0.435970\pi\)
0.199803 + 0.979836i \(0.435970\pi\)
\(272\) 15.2061 0.922003
\(273\) 0 0
\(274\) 4.77384 0.288398
\(275\) −4.91638 −0.296469
\(276\) 0 0
\(277\) 25.6655 1.54209 0.771046 0.636779i \(-0.219734\pi\)
0.771046 + 0.636779i \(0.219734\pi\)
\(278\) 2.16724 0.129982
\(279\) 0 0
\(280\) −5.56777 −0.332738
\(281\) −3.15667 −0.188311 −0.0941557 0.995557i \(-0.530015\pi\)
−0.0941557 + 0.995557i \(0.530015\pi\)
\(282\) 0 0
\(283\) −3.47002 −0.206271 −0.103136 0.994667i \(-0.532888\pi\)
−0.103136 + 0.994667i \(0.532888\pi\)
\(284\) 1.75614 0.104208
\(285\) 0 0
\(286\) −1.42166 −0.0840647
\(287\) −21.3275 −1.25892
\(288\) 0 0
\(289\) 1.81863 0.106978
\(290\) −1.73501 −0.101883
\(291\) 0 0
\(292\) 6.04939 0.354014
\(293\) −28.6550 −1.67404 −0.837020 0.547172i \(-0.815704\pi\)
−0.837020 + 0.547172i \(0.815704\pi\)
\(294\) 0 0
\(295\) 11.2544 0.655258
\(296\) −10.7527 −0.624989
\(297\) 0 0
\(298\) −5.92337 −0.343132
\(299\) 6.33804 0.366539
\(300\) 0 0
\(301\) −5.68665 −0.327773
\(302\) −4.74557 −0.273077
\(303\) 0 0
\(304\) 9.03780 0.518353
\(305\) −10.1708 −0.582379
\(306\) 0 0
\(307\) −1.92694 −0.109977 −0.0549883 0.998487i \(-0.517512\pi\)
−0.0549883 + 0.998487i \(0.517512\pi\)
\(308\) 46.3205 2.63935
\(309\) 0 0
\(310\) 0.411100 0.0233489
\(311\) 29.9789 1.69995 0.849973 0.526826i \(-0.176618\pi\)
0.849973 + 0.526826i \(0.176618\pi\)
\(312\) 0 0
\(313\) −16.3133 −0.922085 −0.461042 0.887378i \(-0.652524\pi\)
−0.461042 + 0.887378i \(0.652524\pi\)
\(314\) −6.26499 −0.353554
\(315\) 0 0
\(316\) 6.69721 0.376748
\(317\) −30.6761 −1.72294 −0.861470 0.507808i \(-0.830456\pi\)
−0.861470 + 0.507808i \(0.830456\pi\)
\(318\) 0 0
\(319\) 29.4983 1.65159
\(320\) −6.06249 −0.338904
\(321\) 0 0
\(322\) 9.01056 0.502139
\(323\) 11.1849 0.622347
\(324\) 0 0
\(325\) 1.00000 0.0554700
\(326\) 1.75614 0.0972634
\(327\) 0 0
\(328\) 4.91281 0.271265
\(329\) 26.6550 1.46954
\(330\) 0 0
\(331\) −10.0978 −0.555023 −0.277511 0.960722i \(-0.589510\pi\)
−0.277511 + 0.960722i \(0.589510\pi\)
\(332\) 21.5678 1.18369
\(333\) 0 0
\(334\) 0.215597 0.0117969
\(335\) −7.25443 −0.396352
\(336\) 0 0
\(337\) −1.32391 −0.0721180 −0.0360590 0.999350i \(-0.511480\pi\)
−0.0360590 + 0.999350i \(0.511480\pi\)
\(338\) 0.289169 0.0157287
\(339\) 0 0
\(340\) −8.31335 −0.450855
\(341\) −6.98944 −0.378499
\(342\) 0 0
\(343\) 50.0036 2.69994
\(344\) 1.30993 0.0706265
\(345\) 0 0
\(346\) −0.243863 −0.0131102
\(347\) 7.49472 0.402338 0.201169 0.979557i \(-0.435526\pi\)
0.201169 + 0.979557i \(0.435526\pi\)
\(348\) 0 0
\(349\) −22.1461 −1.18545 −0.592727 0.805403i \(-0.701949\pi\)
−0.592727 + 0.805403i \(0.701949\pi\)
\(350\) 1.42166 0.0759911
\(351\) 0 0
\(352\) −16.1189 −0.859139
\(353\) −4.50885 −0.239982 −0.119991 0.992775i \(-0.538287\pi\)
−0.119991 + 0.992775i \(0.538287\pi\)
\(354\) 0 0
\(355\) −0.916382 −0.0486365
\(356\) −0.647822 −0.0343345
\(357\) 0 0
\(358\) 5.49115 0.290216
\(359\) 20.4111 1.07726 0.538628 0.842543i \(-0.318943\pi\)
0.538628 + 0.842543i \(0.318943\pi\)
\(360\) 0 0
\(361\) −12.3522 −0.650115
\(362\) 5.05892 0.265891
\(363\) 0 0
\(364\) −9.42166 −0.493829
\(365\) −3.15667 −0.165228
\(366\) 0 0
\(367\) −10.3133 −0.538352 −0.269176 0.963091i \(-0.586751\pi\)
−0.269176 + 0.963091i \(0.586751\pi\)
\(368\) 22.2166 1.15812
\(369\) 0 0
\(370\) 2.74557 0.142736
\(371\) 1.66196 0.0862844
\(372\) 0 0
\(373\) 18.6761 0.967011 0.483506 0.875341i \(-0.339363\pi\)
0.483506 + 0.875341i \(0.339363\pi\)
\(374\) −6.16724 −0.318900
\(375\) 0 0
\(376\) −6.14000 −0.316646
\(377\) −6.00000 −0.309016
\(378\) 0 0
\(379\) −28.7527 −1.47693 −0.738464 0.674293i \(-0.764448\pi\)
−0.738464 + 0.674293i \(0.764448\pi\)
\(380\) −4.94108 −0.253472
\(381\) 0 0
\(382\) 6.50885 0.333022
\(383\) 14.2439 0.727827 0.363914 0.931433i \(-0.381440\pi\)
0.363914 + 0.931433i \(0.381440\pi\)
\(384\) 0 0
\(385\) −24.1708 −1.23186
\(386\) 0.766699 0.0390240
\(387\) 0 0
\(388\) 23.6444 1.20036
\(389\) −34.6761 −1.75815 −0.879074 0.476686i \(-0.841838\pi\)
−0.879074 + 0.476686i \(0.841838\pi\)
\(390\) 0 0
\(391\) 27.4947 1.39047
\(392\) −19.4458 −0.982163
\(393\) 0 0
\(394\) 3.75614 0.189231
\(395\) −3.49472 −0.175838
\(396\) 0 0
\(397\) 7.18137 0.360423 0.180211 0.983628i \(-0.442322\pi\)
0.180211 + 0.983628i \(0.442322\pi\)
\(398\) −0.822200 −0.0412132
\(399\) 0 0
\(400\) 3.50528 0.175264
\(401\) −37.8610 −1.89069 −0.945345 0.326072i \(-0.894275\pi\)
−0.945345 + 0.326072i \(0.894275\pi\)
\(402\) 0 0
\(403\) 1.42166 0.0708181
\(404\) 20.4595 1.01790
\(405\) 0 0
\(406\) −8.52998 −0.423336
\(407\) −46.6797 −2.31382
\(408\) 0 0
\(409\) −14.0000 −0.692255 −0.346128 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) −1.25443 −0.0619517
\(411\) 0 0
\(412\) 27.8045 1.36983
\(413\) 55.3311 2.72266
\(414\) 0 0
\(415\) −11.2544 −0.552458
\(416\) 3.27861 0.160747
\(417\) 0 0
\(418\) −3.66553 −0.179287
\(419\) 33.4983 1.63650 0.818249 0.574864i \(-0.194945\pi\)
0.818249 + 0.574864i \(0.194945\pi\)
\(420\) 0 0
\(421\) 13.5194 0.658896 0.329448 0.944174i \(-0.393137\pi\)
0.329448 + 0.944174i \(0.393137\pi\)
\(422\) 1.82562 0.0888699
\(423\) 0 0
\(424\) −0.382833 −0.0185920
\(425\) 4.33804 0.210426
\(426\) 0 0
\(427\) −50.0036 −2.41984
\(428\) 7.99286 0.386349
\(429\) 0 0
\(430\) −0.334474 −0.0161298
\(431\) −12.4111 −0.597822 −0.298911 0.954281i \(-0.596623\pi\)
−0.298911 + 0.954281i \(0.596623\pi\)
\(432\) 0 0
\(433\) −17.3239 −0.832534 −0.416267 0.909242i \(-0.636662\pi\)
−0.416267 + 0.909242i \(0.636662\pi\)
\(434\) 2.02113 0.0970171
\(435\) 0 0
\(436\) 7.34504 0.351763
\(437\) 16.3416 0.781725
\(438\) 0 0
\(439\) 0.651393 0.0310893 0.0155446 0.999879i \(-0.495052\pi\)
0.0155446 + 0.999879i \(0.495052\pi\)
\(440\) 5.56777 0.265433
\(441\) 0 0
\(442\) 1.25443 0.0596670
\(443\) 8.84690 0.420329 0.210164 0.977666i \(-0.432600\pi\)
0.210164 + 0.977666i \(0.432600\pi\)
\(444\) 0 0
\(445\) 0.338044 0.0160248
\(446\) 5.56777 0.263642
\(447\) 0 0
\(448\) −29.8055 −1.40818
\(449\) −4.33804 −0.204725 −0.102362 0.994747i \(-0.532640\pi\)
−0.102362 + 0.994747i \(0.532640\pi\)
\(450\) 0 0
\(451\) 21.3275 1.00427
\(452\) −1.61613 −0.0760166
\(453\) 0 0
\(454\) 3.78440 0.177611
\(455\) 4.91638 0.230484
\(456\) 0 0
\(457\) 15.3275 0.716989 0.358495 0.933532i \(-0.383290\pi\)
0.358495 + 0.933532i \(0.383290\pi\)
\(458\) −7.08719 −0.331163
\(459\) 0 0
\(460\) −12.1461 −0.566315
\(461\) 11.8575 0.552257 0.276128 0.961121i \(-0.410948\pi\)
0.276128 + 0.961121i \(0.410948\pi\)
\(462\) 0 0
\(463\) −26.4147 −1.22759 −0.613797 0.789464i \(-0.710359\pi\)
−0.613797 + 0.789464i \(0.710359\pi\)
\(464\) −21.0317 −0.976372
\(465\) 0 0
\(466\) −2.41110 −0.111692
\(467\) 33.6691 1.55802 0.779010 0.627012i \(-0.215722\pi\)
0.779010 + 0.627012i \(0.215722\pi\)
\(468\) 0 0
\(469\) −35.6655 −1.64688
\(470\) 1.56777 0.0723160
\(471\) 0 0
\(472\) −12.7456 −0.586663
\(473\) 5.68665 0.261473
\(474\) 0 0
\(475\) 2.57834 0.118302
\(476\) −40.8716 −1.87335
\(477\) 0 0
\(478\) −2.57834 −0.117930
\(479\) −10.7491 −0.491141 −0.245570 0.969379i \(-0.578975\pi\)
−0.245570 + 0.969379i \(0.578975\pi\)
\(480\) 0 0
\(481\) 9.49472 0.432922
\(482\) −1.73501 −0.0790276
\(483\) 0 0
\(484\) −25.2403 −1.14729
\(485\) −12.3380 −0.560242
\(486\) 0 0
\(487\) 22.7491 1.03086 0.515431 0.856931i \(-0.327632\pi\)
0.515431 + 0.856931i \(0.327632\pi\)
\(488\) 11.5184 0.521413
\(489\) 0 0
\(490\) 4.96526 0.224307
\(491\) 17.6867 0.798187 0.399094 0.916910i \(-0.369325\pi\)
0.399094 + 0.916910i \(0.369325\pi\)
\(492\) 0 0
\(493\) −26.0283 −1.17225
\(494\) 0.745574 0.0335450
\(495\) 0 0
\(496\) 4.98333 0.223758
\(497\) −4.50528 −0.202089
\(498\) 0 0
\(499\) 19.9305 0.892212 0.446106 0.894980i \(-0.352810\pi\)
0.446106 + 0.894980i \(0.352810\pi\)
\(500\) −1.91638 −0.0857032
\(501\) 0 0
\(502\) 1.82562 0.0814815
\(503\) −33.3522 −1.48710 −0.743550 0.668680i \(-0.766860\pi\)
−0.743550 + 0.668680i \(0.766860\pi\)
\(504\) 0 0
\(505\) −10.6761 −0.475080
\(506\) −9.01056 −0.400568
\(507\) 0 0
\(508\) −3.51227 −0.155832
\(509\) −13.8363 −0.613285 −0.306642 0.951825i \(-0.599206\pi\)
−0.306642 + 0.951825i \(0.599206\pi\)
\(510\) 0 0
\(511\) −15.5194 −0.686538
\(512\) 19.4319 0.858775
\(513\) 0 0
\(514\) 3.22616 0.142300
\(515\) −14.5089 −0.639336
\(516\) 0 0
\(517\) −26.6550 −1.17228
\(518\) 13.4983 0.593081
\(519\) 0 0
\(520\) −1.13249 −0.0496632
\(521\) 23.3522 1.02308 0.511539 0.859260i \(-0.329076\pi\)
0.511539 + 0.859260i \(0.329076\pi\)
\(522\) 0 0
\(523\) −4.00000 −0.174908 −0.0874539 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) 11.1778 0.488304
\(525\) 0 0
\(526\) −2.31335 −0.100867
\(527\) 6.16724 0.268649
\(528\) 0 0
\(529\) 17.1708 0.746557
\(530\) 0.0977518 0.00424607
\(531\) 0 0
\(532\) −24.2922 −1.05320
\(533\) −4.33804 −0.187902
\(534\) 0 0
\(535\) −4.17081 −0.180320
\(536\) 8.21560 0.354860
\(537\) 0 0
\(538\) −5.40054 −0.232834
\(539\) −84.4182 −3.63615
\(540\) 0 0
\(541\) 16.1744 0.695391 0.347695 0.937608i \(-0.386964\pi\)
0.347695 + 0.937608i \(0.386964\pi\)
\(542\) 1.90225 0.0817086
\(543\) 0 0
\(544\) 14.2227 0.609795
\(545\) −3.83276 −0.164178
\(546\) 0 0
\(547\) 9.68665 0.414171 0.207086 0.978323i \(-0.433602\pi\)
0.207086 + 0.978323i \(0.433602\pi\)
\(548\) −31.6373 −1.35148
\(549\) 0 0
\(550\) −1.42166 −0.0606199
\(551\) −15.4700 −0.659045
\(552\) 0 0
\(553\) −17.1814 −0.730626
\(554\) 7.42166 0.315316
\(555\) 0 0
\(556\) −14.3627 −0.609116
\(557\) −0.647822 −0.0274491 −0.0137246 0.999906i \(-0.504369\pi\)
−0.0137246 + 0.999906i \(0.504369\pi\)
\(558\) 0 0
\(559\) −1.15667 −0.0489221
\(560\) 17.2333 0.728240
\(561\) 0 0
\(562\) −0.912811 −0.0385046
\(563\) 16.3169 0.687676 0.343838 0.939029i \(-0.388273\pi\)
0.343838 + 0.939029i \(0.388273\pi\)
\(564\) 0 0
\(565\) 0.843326 0.0354790
\(566\) −1.00342 −0.0421769
\(567\) 0 0
\(568\) 1.03780 0.0435450
\(569\) −44.6550 −1.87203 −0.936017 0.351956i \(-0.885517\pi\)
−0.936017 + 0.351956i \(0.885517\pi\)
\(570\) 0 0
\(571\) −6.67252 −0.279236 −0.139618 0.990205i \(-0.544587\pi\)
−0.139618 + 0.990205i \(0.544587\pi\)
\(572\) 9.42166 0.393940
\(573\) 0 0
\(574\) −6.16724 −0.257415
\(575\) 6.33804 0.264315
\(576\) 0 0
\(577\) 15.3275 0.638091 0.319046 0.947739i \(-0.396638\pi\)
0.319046 + 0.947739i \(0.396638\pi\)
\(578\) 0.525891 0.0218742
\(579\) 0 0
\(580\) 11.4983 0.477440
\(581\) −55.3311 −2.29552
\(582\) 0 0
\(583\) −1.66196 −0.0688312
\(584\) 3.57492 0.147931
\(585\) 0 0
\(586\) −8.28611 −0.342296
\(587\) −12.2650 −0.506230 −0.253115 0.967436i \(-0.581455\pi\)
−0.253115 + 0.967436i \(0.581455\pi\)
\(588\) 0 0
\(589\) 3.66553 0.151035
\(590\) 3.25443 0.133983
\(591\) 0 0
\(592\) 33.2817 1.36787
\(593\) 5.85389 0.240390 0.120195 0.992750i \(-0.461648\pi\)
0.120195 + 0.992750i \(0.461648\pi\)
\(594\) 0 0
\(595\) 21.3275 0.874342
\(596\) 39.2555 1.60797
\(597\) 0 0
\(598\) 1.83276 0.0749473
\(599\) −27.1355 −1.10873 −0.554364 0.832274i \(-0.687039\pi\)
−0.554364 + 0.832274i \(0.687039\pi\)
\(600\) 0 0
\(601\) 34.1708 1.39386 0.696928 0.717141i \(-0.254550\pi\)
0.696928 + 0.717141i \(0.254550\pi\)
\(602\) −1.64440 −0.0670208
\(603\) 0 0
\(604\) 31.4499 1.27968
\(605\) 13.1708 0.535469
\(606\) 0 0
\(607\) 10.3133 0.418606 0.209303 0.977851i \(-0.432881\pi\)
0.209303 + 0.977851i \(0.432881\pi\)
\(608\) 8.45335 0.342829
\(609\) 0 0
\(610\) −2.94108 −0.119081
\(611\) 5.42166 0.219337
\(612\) 0 0
\(613\) 0.484156 0.0195549 0.00977744 0.999952i \(-0.496888\pi\)
0.00977744 + 0.999952i \(0.496888\pi\)
\(614\) −0.557212 −0.0224872
\(615\) 0 0
\(616\) 27.3733 1.10290
\(617\) 7.15667 0.288117 0.144058 0.989569i \(-0.453985\pi\)
0.144058 + 0.989569i \(0.453985\pi\)
\(618\) 0 0
\(619\) 5.42166 0.217915 0.108958 0.994046i \(-0.465249\pi\)
0.108958 + 0.994046i \(0.465249\pi\)
\(620\) −2.72445 −0.109416
\(621\) 0 0
\(622\) 8.66895 0.347593
\(623\) 1.66196 0.0665848
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −4.71731 −0.188542
\(627\) 0 0
\(628\) 41.5194 1.65681
\(629\) 41.1885 1.64229
\(630\) 0 0
\(631\) −10.7244 −0.426934 −0.213467 0.976950i \(-0.568476\pi\)
−0.213467 + 0.976950i \(0.568476\pi\)
\(632\) 3.95775 0.157431
\(633\) 0 0
\(634\) −8.87056 −0.352295
\(635\) 1.83276 0.0727310
\(636\) 0 0
\(637\) 17.1708 0.680332
\(638\) 8.52998 0.337705
\(639\) 0 0
\(640\) −8.31029 −0.328493
\(641\) 0.362741 0.0143274 0.00716370 0.999974i \(-0.497720\pi\)
0.00716370 + 0.999974i \(0.497720\pi\)
\(642\) 0 0
\(643\) −9.39697 −0.370580 −0.185290 0.982684i \(-0.559322\pi\)
−0.185290 + 0.982684i \(0.559322\pi\)
\(644\) −59.7149 −2.35310
\(645\) 0 0
\(646\) 3.23433 0.127253
\(647\) −18.0036 −0.707793 −0.353897 0.935285i \(-0.615144\pi\)
−0.353897 + 0.935285i \(0.615144\pi\)
\(648\) 0 0
\(649\) −55.3311 −2.17193
\(650\) 0.289169 0.0113421
\(651\) 0 0
\(652\) −11.6383 −0.455791
\(653\) 34.8222 1.36270 0.681349 0.731959i \(-0.261394\pi\)
0.681349 + 0.731959i \(0.261394\pi\)
\(654\) 0 0
\(655\) −5.83276 −0.227905
\(656\) −15.2061 −0.593697
\(657\) 0 0
\(658\) 7.70778 0.300480
\(659\) −11.4700 −0.446809 −0.223404 0.974726i \(-0.571717\pi\)
−0.223404 + 0.974726i \(0.571717\pi\)
\(660\) 0 0
\(661\) 12.1672 0.473251 0.236625 0.971601i \(-0.423959\pi\)
0.236625 + 0.971601i \(0.423959\pi\)
\(662\) −2.91995 −0.113487
\(663\) 0 0
\(664\) 12.7456 0.494624
\(665\) 12.6761 0.491558
\(666\) 0 0
\(667\) −38.0283 −1.47246
\(668\) −1.42880 −0.0552821
\(669\) 0 0
\(670\) −2.09775 −0.0810432
\(671\) 50.0036 1.93037
\(672\) 0 0
\(673\) −27.9789 −1.07851 −0.539253 0.842144i \(-0.681293\pi\)
−0.539253 + 0.842144i \(0.681293\pi\)
\(674\) −0.382833 −0.0147462
\(675\) 0 0
\(676\) −1.91638 −0.0737070
\(677\) −22.9930 −0.883693 −0.441847 0.897091i \(-0.645676\pi\)
−0.441847 + 0.897091i \(0.645676\pi\)
\(678\) 0 0
\(679\) −60.6585 −2.32786
\(680\) −4.91281 −0.188398
\(681\) 0 0
\(682\) −2.02113 −0.0773929
\(683\) 28.6066 1.09460 0.547301 0.836936i \(-0.315655\pi\)
0.547301 + 0.836936i \(0.315655\pi\)
\(684\) 0 0
\(685\) 16.5089 0.630771
\(686\) 14.4595 0.552065
\(687\) 0 0
\(688\) −4.05447 −0.154575
\(689\) 0.338044 0.0128785
\(690\) 0 0
\(691\) −19.4005 −0.738031 −0.369016 0.929423i \(-0.620305\pi\)
−0.369016 + 0.929423i \(0.620305\pi\)
\(692\) 1.61613 0.0614362
\(693\) 0 0
\(694\) 2.16724 0.0822672
\(695\) 7.49472 0.284291
\(696\) 0 0
\(697\) −18.8186 −0.712806
\(698\) −6.40396 −0.242393
\(699\) 0 0
\(700\) −9.42166 −0.356105
\(701\) 38.9683 1.47181 0.735906 0.677083i \(-0.236756\pi\)
0.735906 + 0.677083i \(0.236756\pi\)
\(702\) 0 0
\(703\) 24.4806 0.923303
\(704\) 29.8055 1.12334
\(705\) 0 0
\(706\) −1.30382 −0.0490698
\(707\) −52.4877 −1.97400
\(708\) 0 0
\(709\) −17.5194 −0.657955 −0.328978 0.944338i \(-0.606704\pi\)
−0.328978 + 0.944338i \(0.606704\pi\)
\(710\) −0.264989 −0.00994485
\(711\) 0 0
\(712\) −0.382833 −0.0143473
\(713\) 9.01056 0.337448
\(714\) 0 0
\(715\) −4.91638 −0.183862
\(716\) −36.3910 −1.36000
\(717\) 0 0
\(718\) 5.90225 0.220270
\(719\) 4.33447 0.161649 0.0808243 0.996728i \(-0.474245\pi\)
0.0808243 + 0.996728i \(0.474245\pi\)
\(720\) 0 0
\(721\) −71.3311 −2.65651
\(722\) −3.57186 −0.132931
\(723\) 0 0
\(724\) −33.5266 −1.24600
\(725\) −6.00000 −0.222834
\(726\) 0 0
\(727\) 22.1672 0.822137 0.411069 0.911604i \(-0.365156\pi\)
0.411069 + 0.911604i \(0.365156\pi\)
\(728\) −5.56777 −0.206355
\(729\) 0 0
\(730\) −0.912811 −0.0337846
\(731\) −5.01770 −0.185586
\(732\) 0 0
\(733\) −2.83976 −0.104889 −0.0524444 0.998624i \(-0.516701\pi\)
−0.0524444 + 0.998624i \(0.516701\pi\)
\(734\) −2.98230 −0.110079
\(735\) 0 0
\(736\) 20.7799 0.765959
\(737\) 35.6655 1.31376
\(738\) 0 0
\(739\) −43.9305 −1.61601 −0.808005 0.589176i \(-0.799453\pi\)
−0.808005 + 0.589176i \(0.799453\pi\)
\(740\) −18.1955 −0.668880
\(741\) 0 0
\(742\) 0.480585 0.0176428
\(743\) −41.0872 −1.50734 −0.753671 0.657251i \(-0.771719\pi\)
−0.753671 + 0.657251i \(0.771719\pi\)
\(744\) 0 0
\(745\) −20.4842 −0.750481
\(746\) 5.40054 0.197728
\(747\) 0 0
\(748\) 40.8716 1.49441
\(749\) −20.5053 −0.749247
\(750\) 0 0
\(751\) 23.6902 0.864468 0.432234 0.901761i \(-0.357725\pi\)
0.432234 + 0.901761i \(0.357725\pi\)
\(752\) 19.0045 0.693021
\(753\) 0 0
\(754\) −1.73501 −0.0631854
\(755\) −16.4111 −0.597261
\(756\) 0 0
\(757\) 9.32391 0.338883 0.169442 0.985540i \(-0.445804\pi\)
0.169442 + 0.985540i \(0.445804\pi\)
\(758\) −8.31438 −0.301992
\(759\) 0 0
\(760\) −2.91995 −0.105918
\(761\) 42.8222 1.55230 0.776152 0.630546i \(-0.217169\pi\)
0.776152 + 0.630546i \(0.217169\pi\)
\(762\) 0 0
\(763\) −18.8433 −0.682174
\(764\) −43.1355 −1.56059
\(765\) 0 0
\(766\) 4.11888 0.148821
\(767\) 11.2544 0.406374
\(768\) 0 0
\(769\) −17.3239 −0.624716 −0.312358 0.949964i \(-0.601119\pi\)
−0.312358 + 0.949964i \(0.601119\pi\)
\(770\) −6.98944 −0.251882
\(771\) 0 0
\(772\) −5.08108 −0.182872
\(773\) 11.6373 0.418563 0.209282 0.977855i \(-0.432887\pi\)
0.209282 + 0.977855i \(0.432887\pi\)
\(774\) 0 0
\(775\) 1.42166 0.0510676
\(776\) 13.9728 0.501593
\(777\) 0 0
\(778\) −10.0272 −0.359494
\(779\) −11.1849 −0.400742
\(780\) 0 0
\(781\) 4.50528 0.161212
\(782\) 7.95061 0.284313
\(783\) 0 0
\(784\) 60.1885 2.14959
\(785\) −21.6655 −0.773276
\(786\) 0 0
\(787\) 20.9411 0.746469 0.373234 0.927737i \(-0.378249\pi\)
0.373234 + 0.927737i \(0.378249\pi\)
\(788\) −24.8927 −0.886766
\(789\) 0 0
\(790\) −1.01056 −0.0359542
\(791\) 4.14611 0.147419
\(792\) 0 0
\(793\) −10.1708 −0.361176
\(794\) 2.07663 0.0736967
\(795\) 0 0
\(796\) 5.44890 0.193131
\(797\) −20.3380 −0.720410 −0.360205 0.932873i \(-0.617293\pi\)
−0.360205 + 0.932873i \(0.617293\pi\)
\(798\) 0 0
\(799\) 23.5194 0.832057
\(800\) 3.27861 0.115916
\(801\) 0 0
\(802\) −10.9482 −0.386595
\(803\) 15.5194 0.547668
\(804\) 0 0
\(805\) 31.1602 1.09825
\(806\) 0.411100 0.0144804
\(807\) 0 0
\(808\) 12.0906 0.425346
\(809\) −7.68665 −0.270248 −0.135124 0.990829i \(-0.543143\pi\)
−0.135124 + 0.990829i \(0.543143\pi\)
\(810\) 0 0
\(811\) −44.4111 −1.55948 −0.779742 0.626101i \(-0.784650\pi\)
−0.779742 + 0.626101i \(0.784650\pi\)
\(812\) 56.5300 1.98381
\(813\) 0 0
\(814\) −13.4983 −0.473115
\(815\) 6.07306 0.212730
\(816\) 0 0
\(817\) −2.98230 −0.104337
\(818\) −4.04836 −0.141548
\(819\) 0 0
\(820\) 8.31335 0.290315
\(821\) 46.4630 1.62157 0.810785 0.585343i \(-0.199040\pi\)
0.810785 + 0.585343i \(0.199040\pi\)
\(822\) 0 0
\(823\) 46.5089 1.62120 0.810598 0.585603i \(-0.199142\pi\)
0.810598 + 0.585603i \(0.199142\pi\)
\(824\) 16.4312 0.572408
\(825\) 0 0
\(826\) 16.0000 0.556711
\(827\) −39.4005 −1.37009 −0.685045 0.728500i \(-0.740218\pi\)
−0.685045 + 0.728500i \(0.740218\pi\)
\(828\) 0 0
\(829\) 47.6444 1.65476 0.827379 0.561644i \(-0.189831\pi\)
0.827379 + 0.561644i \(0.189831\pi\)
\(830\) −3.25443 −0.112963
\(831\) 0 0
\(832\) −6.06249 −0.210179
\(833\) 74.4877 2.58085
\(834\) 0 0
\(835\) 0.745574 0.0258017
\(836\) 24.2922 0.840164
\(837\) 0 0
\(838\) 9.68665 0.334620
\(839\) 39.9058 1.37770 0.688851 0.724903i \(-0.258115\pi\)
0.688851 + 0.724903i \(0.258115\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 3.90939 0.134726
\(843\) 0 0
\(844\) −12.0988 −0.416457
\(845\) 1.00000 0.0344010
\(846\) 0 0
\(847\) 64.7527 2.22493
\(848\) 1.18494 0.0406910
\(849\) 0 0
\(850\) 1.25443 0.0430265
\(851\) 60.1779 2.06287
\(852\) 0 0
\(853\) 29.5019 1.01012 0.505062 0.863083i \(-0.331470\pi\)
0.505062 + 0.863083i \(0.331470\pi\)
\(854\) −14.4595 −0.494793
\(855\) 0 0
\(856\) 4.72342 0.161443
\(857\) −8.33804 −0.284822 −0.142411 0.989808i \(-0.545485\pi\)
−0.142411 + 0.989808i \(0.545485\pi\)
\(858\) 0 0
\(859\) −4.17081 −0.142306 −0.0711531 0.997465i \(-0.522668\pi\)
−0.0711531 + 0.997465i \(0.522668\pi\)
\(860\) 2.21663 0.0755864
\(861\) 0 0
\(862\) −3.58890 −0.122238
\(863\) 3.93051 0.133796 0.0668981 0.997760i \(-0.478690\pi\)
0.0668981 + 0.997760i \(0.478690\pi\)
\(864\) 0 0
\(865\) −0.843326 −0.0286739
\(866\) −5.00953 −0.170231
\(867\) 0 0
\(868\) −13.3944 −0.454637
\(869\) 17.1814 0.582838
\(870\) 0 0
\(871\) −7.25443 −0.245807
\(872\) 4.34058 0.146991
\(873\) 0 0
\(874\) 4.72548 0.159842
\(875\) 4.91638 0.166204
\(876\) 0 0
\(877\) −23.0177 −0.777253 −0.388626 0.921395i \(-0.627050\pi\)
−0.388626 + 0.921395i \(0.627050\pi\)
\(878\) 0.188362 0.00635692
\(879\) 0 0
\(880\) −17.2333 −0.580934
\(881\) 15.3522 0.517228 0.258614 0.965981i \(-0.416734\pi\)
0.258614 + 0.965981i \(0.416734\pi\)
\(882\) 0 0
\(883\) 42.8011 1.44037 0.720185 0.693782i \(-0.244057\pi\)
0.720185 + 0.693782i \(0.244057\pi\)
\(884\) −8.31335 −0.279608
\(885\) 0 0
\(886\) 2.55824 0.0859459
\(887\) 53.1885 1.78590 0.892948 0.450160i \(-0.148633\pi\)
0.892948 + 0.450160i \(0.148633\pi\)
\(888\) 0 0
\(889\) 9.01056 0.302205
\(890\) 0.0977518 0.00327665
\(891\) 0 0
\(892\) −36.8988 −1.23546
\(893\) 13.9789 0.467785
\(894\) 0 0
\(895\) 18.9894 0.634747
\(896\) −40.8566 −1.36492
\(897\) 0 0
\(898\) −1.25443 −0.0418607
\(899\) −8.52998 −0.284491
\(900\) 0 0
\(901\) 1.46645 0.0488546
\(902\) 6.16724 0.205347
\(903\) 0 0
\(904\) −0.955062 −0.0317649
\(905\) 17.4947 0.581544
\(906\) 0 0
\(907\) 11.8116 0.392199 0.196099 0.980584i \(-0.437172\pi\)
0.196099 + 0.980584i \(0.437172\pi\)
\(908\) −25.0800 −0.832311
\(909\) 0 0
\(910\) 1.42166 0.0471277
\(911\) −44.1955 −1.46426 −0.732131 0.681164i \(-0.761474\pi\)
−0.732131 + 0.681164i \(0.761474\pi\)
\(912\) 0 0
\(913\) 55.3311 1.83119
\(914\) 4.43223 0.146605
\(915\) 0 0
\(916\) 46.9683 1.55188
\(917\) −28.6761 −0.946968
\(918\) 0 0
\(919\) 55.2096 1.82120 0.910599 0.413291i \(-0.135621\pi\)
0.910599 + 0.413291i \(0.135621\pi\)
\(920\) −7.17780 −0.236645
\(921\) 0 0
\(922\) 3.42880 0.112922
\(923\) −0.916382 −0.0301631
\(924\) 0 0
\(925\) 9.49472 0.312184
\(926\) −7.63829 −0.251010
\(927\) 0 0
\(928\) −19.6716 −0.645753
\(929\) −22.9930 −0.754376 −0.377188 0.926137i \(-0.623109\pi\)
−0.377188 + 0.926137i \(0.623109\pi\)
\(930\) 0 0
\(931\) 44.2721 1.45096
\(932\) 15.9789 0.523405
\(933\) 0 0
\(934\) 9.73604 0.318573
\(935\) −21.3275 −0.697483
\(936\) 0 0
\(937\) 7.97887 0.260658 0.130329 0.991471i \(-0.458397\pi\)
0.130329 + 0.991471i \(0.458397\pi\)
\(938\) −10.3133 −0.336743
\(939\) 0 0
\(940\) −10.3900 −0.338884
\(941\) −41.5019 −1.35292 −0.676461 0.736478i \(-0.736487\pi\)
−0.676461 + 0.736478i \(0.736487\pi\)
\(942\) 0 0
\(943\) −27.4947 −0.895351
\(944\) 39.4499 1.28399
\(945\) 0 0
\(946\) 1.64440 0.0534641
\(947\) −47.4499 −1.54192 −0.770958 0.636886i \(-0.780222\pi\)
−0.770958 + 0.636886i \(0.780222\pi\)
\(948\) 0 0
\(949\) −3.15667 −0.102470
\(950\) 0.745574 0.0241896
\(951\) 0 0
\(952\) −24.1533 −0.782811
\(953\) −30.3663 −0.983661 −0.491831 0.870691i \(-0.663672\pi\)
−0.491831 + 0.870691i \(0.663672\pi\)
\(954\) 0 0
\(955\) 22.5089 0.728369
\(956\) 17.0872 0.552639
\(957\) 0 0
\(958\) −3.10831 −0.100425
\(959\) 81.1638 2.62092
\(960\) 0 0
\(961\) −28.9789 −0.934802
\(962\) 2.74557 0.0885209
\(963\) 0 0
\(964\) 11.4983 0.370335
\(965\) 2.65139 0.0853514
\(966\) 0 0
\(967\) −41.0943 −1.32150 −0.660752 0.750604i \(-0.729763\pi\)
−0.660752 + 0.750604i \(0.729763\pi\)
\(968\) −14.9159 −0.479414
\(969\) 0 0
\(970\) −3.56777 −0.114554
\(971\) −38.1744 −1.22507 −0.612537 0.790442i \(-0.709851\pi\)
−0.612537 + 0.790442i \(0.709851\pi\)
\(972\) 0 0
\(973\) 36.8469 1.18126
\(974\) 6.57834 0.210784
\(975\) 0 0
\(976\) −35.6515 −1.14118
\(977\) 10.4806 0.335304 0.167652 0.985846i \(-0.446382\pi\)
0.167652 + 0.985846i \(0.446382\pi\)
\(978\) 0 0
\(979\) −1.66196 −0.0531163
\(980\) −32.9058 −1.05114
\(981\) 0 0
\(982\) 5.11442 0.163208
\(983\) 0.0766264 0.00244400 0.00122200 0.999999i \(-0.499611\pi\)
0.00122200 + 0.999999i \(0.499611\pi\)
\(984\) 0 0
\(985\) 12.9894 0.413878
\(986\) −7.52656 −0.239694
\(987\) 0 0
\(988\) −4.94108 −0.157197
\(989\) −7.33105 −0.233114
\(990\) 0 0
\(991\) −13.8575 −0.440197 −0.220098 0.975478i \(-0.570638\pi\)
−0.220098 + 0.975478i \(0.570638\pi\)
\(992\) 4.66107 0.147989
\(993\) 0 0
\(994\) −1.30279 −0.0413219
\(995\) −2.84333 −0.0901395
\(996\) 0 0
\(997\) −10.3416 −0.327522 −0.163761 0.986500i \(-0.552363\pi\)
−0.163761 + 0.986500i \(0.552363\pi\)
\(998\) 5.76328 0.182433
\(999\) 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 585.2.a.n.1.2 3
3.2 odd 2 195.2.a.e.1.2 3
4.3 odd 2 9360.2.a.dd.1.1 3
5.2 odd 4 2925.2.c.w.2224.4 6
5.3 odd 4 2925.2.c.w.2224.3 6
5.4 even 2 2925.2.a.bh.1.2 3
12.11 even 2 3120.2.a.bj.1.1 3
13.12 even 2 7605.2.a.bx.1.2 3
15.2 even 4 975.2.c.i.274.3 6
15.8 even 4 975.2.c.i.274.4 6
15.14 odd 2 975.2.a.o.1.2 3
21.20 even 2 9555.2.a.bq.1.2 3
39.38 odd 2 2535.2.a.bc.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
195.2.a.e.1.2 3 3.2 odd 2
585.2.a.n.1.2 3 1.1 even 1 trivial
975.2.a.o.1.2 3 15.14 odd 2
975.2.c.i.274.3 6 15.2 even 4
975.2.c.i.274.4 6 15.8 even 4
2535.2.a.bc.1.2 3 39.38 odd 2
2925.2.a.bh.1.2 3 5.4 even 2
2925.2.c.w.2224.3 6 5.3 odd 4
2925.2.c.w.2224.4 6 5.2 odd 4
3120.2.a.bj.1.1 3 12.11 even 2
7605.2.a.bx.1.2 3 13.12 even 2
9360.2.a.dd.1.1 3 4.3 odd 2
9555.2.a.bq.1.2 3 21.20 even 2