Properties

Label 5054.2.a.bf.1.8
Level $5054$
Weight $2$
Character 5054.1
Self dual yes
Analytic conductor $40.356$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 5054 = 2 \cdot 7 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5054.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.3563931816\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: 8.8.19520000000.1
Defining polynomial: \( x^{8} - 2x^{7} - 12x^{6} + 16x^{5} + 50x^{4} - 24x^{3} - 72x^{2} - 32x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(-0.761439\) of defining polynomial
Character \(\chi\) \(=\) 5054.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +2.13415 q^{3} +1.00000 q^{4} -1.04552 q^{5} -2.13415 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.55458 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +2.13415 q^{3} +1.00000 q^{4} -1.04552 q^{5} -2.13415 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.55458 q^{9} +1.04552 q^{10} -2.42021 q^{11} +2.13415 q^{12} +1.15517 q^{13} +1.00000 q^{14} -2.23129 q^{15} +1.00000 q^{16} +3.05251 q^{17} -1.55458 q^{18} -1.04552 q^{20} -2.13415 q^{21} +2.42021 q^{22} -0.514528 q^{23} -2.13415 q^{24} -3.90689 q^{25} -1.15517 q^{26} -3.08473 q^{27} -1.00000 q^{28} +5.57522 q^{29} +2.23129 q^{30} +2.17892 q^{31} -1.00000 q^{32} -5.16509 q^{33} -3.05251 q^{34} +1.04552 q^{35} +1.55458 q^{36} -1.13825 q^{37} +2.46531 q^{39} +1.04552 q^{40} +1.73850 q^{41} +2.13415 q^{42} -6.37863 q^{43} -2.42021 q^{44} -1.62534 q^{45} +0.514528 q^{46} -0.273517 q^{47} +2.13415 q^{48} +1.00000 q^{49} +3.90689 q^{50} +6.51450 q^{51} +1.15517 q^{52} -6.18481 q^{53} +3.08473 q^{54} +2.53037 q^{55} +1.00000 q^{56} -5.57522 q^{58} -10.2131 q^{59} -2.23129 q^{60} -9.38737 q^{61} -2.17892 q^{62} -1.55458 q^{63} +1.00000 q^{64} -1.20775 q^{65} +5.16509 q^{66} +4.12745 q^{67} +3.05251 q^{68} -1.09808 q^{69} -1.04552 q^{70} -11.0844 q^{71} -1.55458 q^{72} -1.05367 q^{73} +1.13825 q^{74} -8.33788 q^{75} +2.42021 q^{77} -2.46531 q^{78} +13.4572 q^{79} -1.04552 q^{80} -11.2470 q^{81} -1.73850 q^{82} +15.1827 q^{83} -2.13415 q^{84} -3.19145 q^{85} +6.37863 q^{86} +11.8983 q^{87} +2.42021 q^{88} +9.44592 q^{89} +1.62534 q^{90} -1.15517 q^{91} -0.514528 q^{92} +4.65013 q^{93} +0.273517 q^{94} -2.13415 q^{96} -12.6176 q^{97} -1.00000 q^{98} -3.76242 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} - 4 q^{3} + 8 q^{4} + 6 q^{5} + 4 q^{6} - 8 q^{7} - 8 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{2} - 4 q^{3} + 8 q^{4} + 6 q^{5} + 4 q^{6} - 8 q^{7} - 8 q^{8} + 8 q^{9} - 6 q^{10} + 4 q^{11} - 4 q^{12} - 6 q^{13} + 8 q^{14} - 8 q^{15} + 8 q^{16} + 2 q^{17} - 8 q^{18} + 6 q^{20} + 4 q^{21} - 4 q^{22} + 20 q^{23} + 4 q^{24} - 8 q^{25} + 6 q^{26} - 22 q^{27} - 8 q^{28} - 16 q^{29} + 8 q^{30} - 22 q^{31} - 8 q^{32} + 8 q^{33} - 2 q^{34} - 6 q^{35} + 8 q^{36} + 12 q^{37} + 8 q^{39} - 6 q^{40} - 36 q^{41} - 4 q^{42} + 4 q^{44} - 4 q^{45} - 20 q^{46} + 6 q^{47} - 4 q^{48} + 8 q^{49} + 8 q^{50} + 4 q^{51} - 6 q^{52} - 16 q^{53} + 22 q^{54} + 18 q^{55} + 8 q^{56} + 16 q^{58} - 14 q^{59} - 8 q^{60} + 10 q^{61} + 22 q^{62} - 8 q^{63} + 8 q^{64} - 32 q^{65} - 8 q^{66} + 20 q^{67} + 2 q^{68} - 40 q^{69} + 6 q^{70} - 8 q^{72} - 18 q^{73} - 12 q^{74} - 16 q^{75} - 4 q^{77} - 8 q^{78} - 4 q^{79} + 6 q^{80} + 12 q^{81} + 36 q^{82} + 36 q^{83} + 4 q^{84} - 16 q^{85} + 28 q^{87} - 4 q^{88} - 18 q^{89} + 4 q^{90} + 6 q^{91} + 20 q^{92} + 16 q^{93} - 6 q^{94} + 4 q^{96} - 26 q^{97} - 8 q^{98} + 4 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 2.13415 1.23215 0.616075 0.787687i \(-0.288722\pi\)
0.616075 + 0.787687i \(0.288722\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.04552 −0.467570 −0.233785 0.972288i \(-0.575111\pi\)
−0.233785 + 0.972288i \(0.575111\pi\)
\(6\) −2.13415 −0.871262
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.55458 0.518194
\(10\) 1.04552 0.330622
\(11\) −2.42021 −0.729721 −0.364861 0.931062i \(-0.618883\pi\)
−0.364861 + 0.931062i \(0.618883\pi\)
\(12\) 2.13415 0.616075
\(13\) 1.15517 0.320387 0.160194 0.987086i \(-0.448788\pi\)
0.160194 + 0.987086i \(0.448788\pi\)
\(14\) 1.00000 0.267261
\(15\) −2.23129 −0.576116
\(16\) 1.00000 0.250000
\(17\) 3.05251 0.740341 0.370171 0.928964i \(-0.379299\pi\)
0.370171 + 0.928964i \(0.379299\pi\)
\(18\) −1.55458 −0.366419
\(19\) 0 0
\(20\) −1.04552 −0.233785
\(21\) −2.13415 −0.465709
\(22\) 2.42021 0.515991
\(23\) −0.514528 −0.107287 −0.0536433 0.998560i \(-0.517083\pi\)
−0.0536433 + 0.998560i \(0.517083\pi\)
\(24\) −2.13415 −0.435631
\(25\) −3.90689 −0.781379
\(26\) −1.15517 −0.226548
\(27\) −3.08473 −0.593657
\(28\) −1.00000 −0.188982
\(29\) 5.57522 1.03529 0.517646 0.855595i \(-0.326808\pi\)
0.517646 + 0.855595i \(0.326808\pi\)
\(30\) 2.23129 0.407376
\(31\) 2.17892 0.391345 0.195673 0.980669i \(-0.437311\pi\)
0.195673 + 0.980669i \(0.437311\pi\)
\(32\) −1.00000 −0.176777
\(33\) −5.16509 −0.899126
\(34\) −3.05251 −0.523500
\(35\) 1.04552 0.176725
\(36\) 1.55458 0.259097
\(37\) −1.13825 −0.187127 −0.0935633 0.995613i \(-0.529826\pi\)
−0.0935633 + 0.995613i \(0.529826\pi\)
\(38\) 0 0
\(39\) 2.46531 0.394765
\(40\) 1.04552 0.165311
\(41\) 1.73850 0.271508 0.135754 0.990743i \(-0.456654\pi\)
0.135754 + 0.990743i \(0.456654\pi\)
\(42\) 2.13415 0.329306
\(43\) −6.37863 −0.972732 −0.486366 0.873755i \(-0.661678\pi\)
−0.486366 + 0.873755i \(0.661678\pi\)
\(44\) −2.42021 −0.364861
\(45\) −1.62534 −0.242292
\(46\) 0.514528 0.0758631
\(47\) −0.273517 −0.0398966 −0.0199483 0.999801i \(-0.506350\pi\)
−0.0199483 + 0.999801i \(0.506350\pi\)
\(48\) 2.13415 0.308038
\(49\) 1.00000 0.142857
\(50\) 3.90689 0.552518
\(51\) 6.51450 0.912212
\(52\) 1.15517 0.160194
\(53\) −6.18481 −0.849549 −0.424774 0.905299i \(-0.639647\pi\)
−0.424774 + 0.905299i \(0.639647\pi\)
\(54\) 3.08473 0.419779
\(55\) 2.53037 0.341196
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) −5.57522 −0.732062
\(59\) −10.2131 −1.32964 −0.664818 0.747005i \(-0.731491\pi\)
−0.664818 + 0.747005i \(0.731491\pi\)
\(60\) −2.23129 −0.288058
\(61\) −9.38737 −1.20193 −0.600965 0.799275i \(-0.705217\pi\)
−0.600965 + 0.799275i \(0.705217\pi\)
\(62\) −2.17892 −0.276723
\(63\) −1.55458 −0.195859
\(64\) 1.00000 0.125000
\(65\) −1.20775 −0.149803
\(66\) 5.16509 0.635778
\(67\) 4.12745 0.504249 0.252124 0.967695i \(-0.418871\pi\)
0.252124 + 0.967695i \(0.418871\pi\)
\(68\) 3.05251 0.370171
\(69\) −1.09808 −0.132193
\(70\) −1.04552 −0.124963
\(71\) −11.0844 −1.31547 −0.657737 0.753248i \(-0.728486\pi\)
−0.657737 + 0.753248i \(0.728486\pi\)
\(72\) −1.55458 −0.183209
\(73\) −1.05367 −0.123323 −0.0616615 0.998097i \(-0.519640\pi\)
−0.0616615 + 0.998097i \(0.519640\pi\)
\(74\) 1.13825 0.132318
\(75\) −8.33788 −0.962776
\(76\) 0 0
\(77\) 2.42021 0.275809
\(78\) −2.46531 −0.279141
\(79\) 13.4572 1.51406 0.757029 0.653381i \(-0.226650\pi\)
0.757029 + 0.653381i \(0.226650\pi\)
\(80\) −1.04552 −0.116892
\(81\) −11.2470 −1.24967
\(82\) −1.73850 −0.191985
\(83\) 15.1827 1.66652 0.833261 0.552880i \(-0.186471\pi\)
0.833261 + 0.552880i \(0.186471\pi\)
\(84\) −2.13415 −0.232855
\(85\) −3.19145 −0.346161
\(86\) 6.37863 0.687825
\(87\) 11.8983 1.27563
\(88\) 2.42021 0.257995
\(89\) 9.44592 1.00127 0.500633 0.865660i \(-0.333101\pi\)
0.500633 + 0.865660i \(0.333101\pi\)
\(90\) 1.62534 0.171326
\(91\) −1.15517 −0.121095
\(92\) −0.514528 −0.0536433
\(93\) 4.65013 0.482196
\(94\) 0.273517 0.0282112
\(95\) 0 0
\(96\) −2.13415 −0.217815
\(97\) −12.6176 −1.28112 −0.640559 0.767909i \(-0.721297\pi\)
−0.640559 + 0.767909i \(0.721297\pi\)
\(98\) −1.00000 −0.101015
\(99\) −3.76242 −0.378137
\(100\) −3.90689 −0.390689
\(101\) −1.49415 −0.148673 −0.0743366 0.997233i \(-0.523684\pi\)
−0.0743366 + 0.997233i \(0.523684\pi\)
\(102\) −6.51450 −0.645031
\(103\) −19.3707 −1.90865 −0.954325 0.298772i \(-0.903423\pi\)
−0.954325 + 0.298772i \(0.903423\pi\)
\(104\) −1.15517 −0.113274
\(105\) 2.23129 0.217751
\(106\) 6.18481 0.600722
\(107\) −8.80175 −0.850897 −0.425449 0.904983i \(-0.639884\pi\)
−0.425449 + 0.904983i \(0.639884\pi\)
\(108\) −3.08473 −0.296829
\(109\) −16.6094 −1.59089 −0.795447 0.606023i \(-0.792764\pi\)
−0.795447 + 0.606023i \(0.792764\pi\)
\(110\) −2.53037 −0.241262
\(111\) −2.42919 −0.230568
\(112\) −1.00000 −0.0944911
\(113\) 6.81605 0.641200 0.320600 0.947215i \(-0.396115\pi\)
0.320600 + 0.947215i \(0.396115\pi\)
\(114\) 0 0
\(115\) 0.537949 0.0501640
\(116\) 5.57522 0.517646
\(117\) 1.79581 0.166023
\(118\) 10.2131 0.940195
\(119\) −3.05251 −0.279823
\(120\) 2.23129 0.203688
\(121\) −5.14258 −0.467507
\(122\) 9.38737 0.849893
\(123\) 3.71022 0.334539
\(124\) 2.17892 0.195673
\(125\) 9.31231 0.832919
\(126\) 1.55458 0.138493
\(127\) 2.87127 0.254784 0.127392 0.991852i \(-0.459339\pi\)
0.127392 + 0.991852i \(0.459339\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −13.6129 −1.19855
\(130\) 1.20775 0.105927
\(131\) 3.59135 0.313778 0.156889 0.987616i \(-0.449854\pi\)
0.156889 + 0.987616i \(0.449854\pi\)
\(132\) −5.16509 −0.449563
\(133\) 0 0
\(134\) −4.12745 −0.356558
\(135\) 3.22514 0.277576
\(136\) −3.05251 −0.261750
\(137\) −3.24610 −0.277333 −0.138666 0.990339i \(-0.544282\pi\)
−0.138666 + 0.990339i \(0.544282\pi\)
\(138\) 1.09808 0.0934747
\(139\) 6.21267 0.526951 0.263476 0.964666i \(-0.415131\pi\)
0.263476 + 0.964666i \(0.415131\pi\)
\(140\) 1.04552 0.0883624
\(141\) −0.583726 −0.0491586
\(142\) 11.0844 0.930181
\(143\) −2.79576 −0.233793
\(144\) 1.55458 0.129549
\(145\) −5.82899 −0.484071
\(146\) 1.05367 0.0872026
\(147\) 2.13415 0.176021
\(148\) −1.13825 −0.0935633
\(149\) −15.3029 −1.25366 −0.626831 0.779155i \(-0.715648\pi\)
−0.626831 + 0.779155i \(0.715648\pi\)
\(150\) 8.33788 0.680785
\(151\) −8.44026 −0.686859 −0.343429 0.939179i \(-0.611589\pi\)
−0.343429 + 0.939179i \(0.611589\pi\)
\(152\) 0 0
\(153\) 4.74537 0.383641
\(154\) −2.42021 −0.195026
\(155\) −2.27810 −0.182981
\(156\) 2.46531 0.197383
\(157\) −1.70931 −0.136418 −0.0682089 0.997671i \(-0.521728\pi\)
−0.0682089 + 0.997671i \(0.521728\pi\)
\(158\) −13.4572 −1.07060
\(159\) −13.1993 −1.04677
\(160\) 1.04552 0.0826554
\(161\) 0.514528 0.0405505
\(162\) 11.2470 0.883649
\(163\) 19.0388 1.49123 0.745616 0.666376i \(-0.232155\pi\)
0.745616 + 0.666376i \(0.232155\pi\)
\(164\) 1.73850 0.135754
\(165\) 5.40019 0.420404
\(166\) −15.1827 −1.17841
\(167\) −7.85185 −0.607594 −0.303797 0.952737i \(-0.598255\pi\)
−0.303797 + 0.952737i \(0.598255\pi\)
\(168\) 2.13415 0.164653
\(169\) −11.6656 −0.897352
\(170\) 3.19145 0.244773
\(171\) 0 0
\(172\) −6.37863 −0.486366
\(173\) −8.12917 −0.618049 −0.309025 0.951054i \(-0.600003\pi\)
−0.309025 + 0.951054i \(0.600003\pi\)
\(174\) −11.8983 −0.902010
\(175\) 3.90689 0.295333
\(176\) −2.42021 −0.182430
\(177\) −21.7963 −1.63831
\(178\) −9.44592 −0.708002
\(179\) −14.2499 −1.06509 −0.532544 0.846402i \(-0.678764\pi\)
−0.532544 + 0.846402i \(0.678764\pi\)
\(180\) −1.62534 −0.121146
\(181\) −16.9244 −1.25798 −0.628992 0.777412i \(-0.716532\pi\)
−0.628992 + 0.777412i \(0.716532\pi\)
\(182\) 1.15517 0.0856271
\(183\) −20.0340 −1.48096
\(184\) 0.514528 0.0379315
\(185\) 1.19006 0.0874947
\(186\) −4.65013 −0.340964
\(187\) −7.38771 −0.540243
\(188\) −0.273517 −0.0199483
\(189\) 3.08473 0.224381
\(190\) 0 0
\(191\) −13.4517 −0.973328 −0.486664 0.873589i \(-0.661786\pi\)
−0.486664 + 0.873589i \(0.661786\pi\)
\(192\) 2.13415 0.154019
\(193\) −23.6578 −1.70293 −0.851463 0.524415i \(-0.824284\pi\)
−0.851463 + 0.524415i \(0.824284\pi\)
\(194\) 12.6176 0.905888
\(195\) −2.57752 −0.184580
\(196\) 1.00000 0.0714286
\(197\) 16.5603 1.17988 0.589938 0.807449i \(-0.299152\pi\)
0.589938 + 0.807449i \(0.299152\pi\)
\(198\) 3.76242 0.267383
\(199\) 7.68305 0.544637 0.272318 0.962207i \(-0.412210\pi\)
0.272318 + 0.962207i \(0.412210\pi\)
\(200\) 3.90689 0.276259
\(201\) 8.80859 0.621310
\(202\) 1.49415 0.105128
\(203\) −5.57522 −0.391303
\(204\) 6.51450 0.456106
\(205\) −1.81763 −0.126949
\(206\) 19.3707 1.34962
\(207\) −0.799877 −0.0555953
\(208\) 1.15517 0.0800968
\(209\) 0 0
\(210\) −2.23129 −0.153974
\(211\) −19.3109 −1.32942 −0.664708 0.747104i \(-0.731444\pi\)
−0.664708 + 0.747104i \(0.731444\pi\)
\(212\) −6.18481 −0.424774
\(213\) −23.6557 −1.62086
\(214\) 8.80175 0.601675
\(215\) 6.66897 0.454820
\(216\) 3.08473 0.209889
\(217\) −2.17892 −0.147915
\(218\) 16.6094 1.12493
\(219\) −2.24869 −0.151953
\(220\) 2.53037 0.170598
\(221\) 3.52617 0.237196
\(222\) 2.42919 0.163036
\(223\) 9.51413 0.637113 0.318557 0.947904i \(-0.396802\pi\)
0.318557 + 0.947904i \(0.396802\pi\)
\(224\) 1.00000 0.0668153
\(225\) −6.07359 −0.404906
\(226\) −6.81605 −0.453397
\(227\) 25.2920 1.67869 0.839343 0.543602i \(-0.182940\pi\)
0.839343 + 0.543602i \(0.182940\pi\)
\(228\) 0 0
\(229\) 21.2960 1.40728 0.703639 0.710558i \(-0.251557\pi\)
0.703639 + 0.710558i \(0.251557\pi\)
\(230\) −0.537949 −0.0354713
\(231\) 5.16509 0.339838
\(232\) −5.57522 −0.366031
\(233\) −12.7292 −0.833915 −0.416958 0.908926i \(-0.636904\pi\)
−0.416958 + 0.908926i \(0.636904\pi\)
\(234\) −1.79581 −0.117396
\(235\) 0.285967 0.0186544
\(236\) −10.2131 −0.664818
\(237\) 28.7197 1.86555
\(238\) 3.05251 0.197865
\(239\) 7.53887 0.487649 0.243824 0.969819i \(-0.421598\pi\)
0.243824 + 0.969819i \(0.421598\pi\)
\(240\) −2.23129 −0.144029
\(241\) −24.0728 −1.55067 −0.775333 0.631552i \(-0.782418\pi\)
−0.775333 + 0.631552i \(0.782418\pi\)
\(242\) 5.14258 0.330577
\(243\) −14.7486 −0.946123
\(244\) −9.38737 −0.600965
\(245\) −1.04552 −0.0667957
\(246\) −3.71022 −0.236555
\(247\) 0 0
\(248\) −2.17892 −0.138362
\(249\) 32.4022 2.05341
\(250\) −9.31231 −0.588962
\(251\) 0.365805 0.0230894 0.0115447 0.999933i \(-0.496325\pi\)
0.0115447 + 0.999933i \(0.496325\pi\)
\(252\) −1.55458 −0.0979295
\(253\) 1.24527 0.0782893
\(254\) −2.87127 −0.180160
\(255\) −6.81102 −0.426523
\(256\) 1.00000 0.0625000
\(257\) −8.33390 −0.519855 −0.259927 0.965628i \(-0.583699\pi\)
−0.259927 + 0.965628i \(0.583699\pi\)
\(258\) 13.6129 0.847504
\(259\) 1.13825 0.0707272
\(260\) −1.20775 −0.0749017
\(261\) 8.66713 0.536482
\(262\) −3.59135 −0.221874
\(263\) −22.5679 −1.39160 −0.695799 0.718237i \(-0.744950\pi\)
−0.695799 + 0.718237i \(0.744950\pi\)
\(264\) 5.16509 0.317889
\(265\) 6.46633 0.397223
\(266\) 0 0
\(267\) 20.1590 1.23371
\(268\) 4.12745 0.252124
\(269\) −20.4358 −1.24599 −0.622996 0.782225i \(-0.714085\pi\)
−0.622996 + 0.782225i \(0.714085\pi\)
\(270\) −3.22514 −0.196276
\(271\) 17.2844 1.04995 0.524976 0.851117i \(-0.324074\pi\)
0.524976 + 0.851117i \(0.324074\pi\)
\(272\) 3.05251 0.185085
\(273\) −2.46531 −0.149207
\(274\) 3.24610 0.196104
\(275\) 9.45550 0.570188
\(276\) −1.09808 −0.0660966
\(277\) −21.1338 −1.26981 −0.634904 0.772591i \(-0.718960\pi\)
−0.634904 + 0.772591i \(0.718960\pi\)
\(278\) −6.21267 −0.372611
\(279\) 3.38731 0.202793
\(280\) −1.04552 −0.0624816
\(281\) 19.0891 1.13876 0.569380 0.822074i \(-0.307183\pi\)
0.569380 + 0.822074i \(0.307183\pi\)
\(282\) 0.583726 0.0347604
\(283\) −21.5165 −1.27902 −0.639512 0.768781i \(-0.720864\pi\)
−0.639512 + 0.768781i \(0.720864\pi\)
\(284\) −11.0844 −0.657737
\(285\) 0 0
\(286\) 2.79576 0.165317
\(287\) −1.73850 −0.102620
\(288\) −1.55458 −0.0916047
\(289\) −7.68221 −0.451895
\(290\) 5.82899 0.342290
\(291\) −26.9277 −1.57853
\(292\) −1.05367 −0.0616615
\(293\) 11.3698 0.664231 0.332116 0.943239i \(-0.392238\pi\)
0.332116 + 0.943239i \(0.392238\pi\)
\(294\) −2.13415 −0.124466
\(295\) 10.6780 0.621698
\(296\) 1.13825 0.0661592
\(297\) 7.46570 0.433204
\(298\) 15.3029 0.886473
\(299\) −0.594369 −0.0343732
\(300\) −8.33788 −0.481388
\(301\) 6.37863 0.367658
\(302\) 8.44026 0.485682
\(303\) −3.18873 −0.183188
\(304\) 0 0
\(305\) 9.81466 0.561986
\(306\) −4.74537 −0.271275
\(307\) 9.12185 0.520612 0.260306 0.965526i \(-0.416177\pi\)
0.260306 + 0.965526i \(0.416177\pi\)
\(308\) 2.42021 0.137904
\(309\) −41.3399 −2.35174
\(310\) 2.27810 0.129387
\(311\) 17.3492 0.983782 0.491891 0.870657i \(-0.336306\pi\)
0.491891 + 0.870657i \(0.336306\pi\)
\(312\) −2.46531 −0.139571
\(313\) −2.42580 −0.137114 −0.0685571 0.997647i \(-0.521840\pi\)
−0.0685571 + 0.997647i \(0.521840\pi\)
\(314\) 1.70931 0.0964619
\(315\) 1.62534 0.0915777
\(316\) 13.4572 0.757029
\(317\) 8.88675 0.499130 0.249565 0.968358i \(-0.419712\pi\)
0.249565 + 0.968358i \(0.419712\pi\)
\(318\) 13.1993 0.740179
\(319\) −13.4932 −0.755474
\(320\) −1.04552 −0.0584462
\(321\) −18.7842 −1.04843
\(322\) −0.514528 −0.0286735
\(323\) 0 0
\(324\) −11.2470 −0.624834
\(325\) −4.51313 −0.250344
\(326\) −19.0388 −1.05446
\(327\) −35.4469 −1.96022
\(328\) −1.73850 −0.0959927
\(329\) 0.273517 0.0150795
\(330\) −5.40019 −0.297271
\(331\) −19.5033 −1.07200 −0.536000 0.844218i \(-0.680065\pi\)
−0.536000 + 0.844218i \(0.680065\pi\)
\(332\) 15.1827 0.833261
\(333\) −1.76950 −0.0969679
\(334\) 7.85185 0.429634
\(335\) −4.31533 −0.235771
\(336\) −2.13415 −0.116427
\(337\) 5.72554 0.311890 0.155945 0.987766i \(-0.450158\pi\)
0.155945 + 0.987766i \(0.450158\pi\)
\(338\) 11.6656 0.634524
\(339\) 14.5465 0.790055
\(340\) −3.19145 −0.173081
\(341\) −5.27344 −0.285573
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 6.37863 0.343913
\(345\) 1.14806 0.0618095
\(346\) 8.12917 0.437027
\(347\) 34.0907 1.83009 0.915044 0.403355i \(-0.132156\pi\)
0.915044 + 0.403355i \(0.132156\pi\)
\(348\) 11.8983 0.637817
\(349\) 31.1329 1.66651 0.833253 0.552892i \(-0.186476\pi\)
0.833253 + 0.552892i \(0.186476\pi\)
\(350\) −3.90689 −0.208832
\(351\) −3.56340 −0.190200
\(352\) 2.42021 0.128998
\(353\) 7.00294 0.372729 0.186364 0.982481i \(-0.440330\pi\)
0.186364 + 0.982481i \(0.440330\pi\)
\(354\) 21.7963 1.15846
\(355\) 11.5889 0.615076
\(356\) 9.44592 0.500633
\(357\) −6.51450 −0.344784
\(358\) 14.2499 0.753131
\(359\) 1.04073 0.0549274 0.0274637 0.999623i \(-0.491257\pi\)
0.0274637 + 0.999623i \(0.491257\pi\)
\(360\) 1.62534 0.0856631
\(361\) 0 0
\(362\) 16.9244 0.889529
\(363\) −10.9750 −0.576039
\(364\) −1.15517 −0.0605475
\(365\) 1.10163 0.0576621
\(366\) 20.0340 1.04720
\(367\) −2.88851 −0.150779 −0.0753896 0.997154i \(-0.524020\pi\)
−0.0753896 + 0.997154i \(0.524020\pi\)
\(368\) −0.514528 −0.0268216
\(369\) 2.70264 0.140694
\(370\) −1.19006 −0.0618681
\(371\) 6.18481 0.321099
\(372\) 4.65013 0.241098
\(373\) 7.01824 0.363391 0.181695 0.983355i \(-0.441842\pi\)
0.181695 + 0.983355i \(0.441842\pi\)
\(374\) 7.38771 0.382009
\(375\) 19.8738 1.02628
\(376\) 0.273517 0.0141056
\(377\) 6.44034 0.331694
\(378\) −3.08473 −0.158662
\(379\) −14.5650 −0.748151 −0.374076 0.927398i \(-0.622040\pi\)
−0.374076 + 0.927398i \(0.622040\pi\)
\(380\) 0 0
\(381\) 6.12772 0.313933
\(382\) 13.4517 0.688247
\(383\) −12.1342 −0.620027 −0.310014 0.950732i \(-0.600334\pi\)
−0.310014 + 0.950732i \(0.600334\pi\)
\(384\) −2.13415 −0.108908
\(385\) −2.53037 −0.128960
\(386\) 23.6578 1.20415
\(387\) −9.91610 −0.504064
\(388\) −12.6176 −0.640559
\(389\) 24.2664 1.23036 0.615179 0.788388i \(-0.289084\pi\)
0.615179 + 0.788388i \(0.289084\pi\)
\(390\) 2.57752 0.130518
\(391\) −1.57060 −0.0794287
\(392\) −1.00000 −0.0505076
\(393\) 7.66447 0.386622
\(394\) −16.5603 −0.834298
\(395\) −14.0698 −0.707928
\(396\) −3.76242 −0.189069
\(397\) −14.1299 −0.709161 −0.354581 0.935025i \(-0.615376\pi\)
−0.354581 + 0.935025i \(0.615376\pi\)
\(398\) −7.68305 −0.385116
\(399\) 0 0
\(400\) −3.90689 −0.195345
\(401\) 13.7977 0.689025 0.344512 0.938782i \(-0.388044\pi\)
0.344512 + 0.938782i \(0.388044\pi\)
\(402\) −8.80859 −0.439333
\(403\) 2.51703 0.125382
\(404\) −1.49415 −0.0743366
\(405\) 11.7590 0.584307
\(406\) 5.57522 0.276693
\(407\) 2.75480 0.136550
\(408\) −6.51450 −0.322516
\(409\) −12.7520 −0.630544 −0.315272 0.949001i \(-0.602096\pi\)
−0.315272 + 0.949001i \(0.602096\pi\)
\(410\) 1.81763 0.0897665
\(411\) −6.92765 −0.341716
\(412\) −19.3707 −0.954325
\(413\) 10.2131 0.502555
\(414\) 0.799877 0.0393118
\(415\) −15.8738 −0.779215
\(416\) −1.15517 −0.0566370
\(417\) 13.2587 0.649283
\(418\) 0 0
\(419\) 25.8240 1.26159 0.630793 0.775951i \(-0.282730\pi\)
0.630793 + 0.775951i \(0.282730\pi\)
\(420\) 2.23129 0.108876
\(421\) −5.42077 −0.264192 −0.132096 0.991237i \(-0.542171\pi\)
−0.132096 + 0.991237i \(0.542171\pi\)
\(422\) 19.3109 0.940039
\(423\) −0.425205 −0.0206742
\(424\) 6.18481 0.300361
\(425\) −11.9258 −0.578487
\(426\) 23.6557 1.14612
\(427\) 9.38737 0.454287
\(428\) −8.80175 −0.425449
\(429\) −5.96656 −0.288068
\(430\) −6.66897 −0.321606
\(431\) 34.0208 1.63872 0.819362 0.573276i \(-0.194328\pi\)
0.819362 + 0.573276i \(0.194328\pi\)
\(432\) −3.08473 −0.148414
\(433\) 13.3211 0.640172 0.320086 0.947389i \(-0.396288\pi\)
0.320086 + 0.947389i \(0.396288\pi\)
\(434\) 2.17892 0.104591
\(435\) −12.4399 −0.596448
\(436\) −16.6094 −0.795447
\(437\) 0 0
\(438\) 2.24869 0.107447
\(439\) −22.4500 −1.07148 −0.535739 0.844384i \(-0.679967\pi\)
−0.535739 + 0.844384i \(0.679967\pi\)
\(440\) −2.53037 −0.120631
\(441\) 1.55458 0.0740277
\(442\) −3.52617 −0.167723
\(443\) −7.75275 −0.368344 −0.184172 0.982894i \(-0.558960\pi\)
−0.184172 + 0.982894i \(0.558960\pi\)
\(444\) −2.42919 −0.115284
\(445\) −9.87588 −0.468162
\(446\) −9.51413 −0.450507
\(447\) −32.6586 −1.54470
\(448\) −1.00000 −0.0472456
\(449\) −22.0007 −1.03828 −0.519139 0.854690i \(-0.673747\pi\)
−0.519139 + 0.854690i \(0.673747\pi\)
\(450\) 6.07359 0.286312
\(451\) −4.20754 −0.198125
\(452\) 6.81605 0.320600
\(453\) −18.0128 −0.846313
\(454\) −25.2920 −1.18701
\(455\) 1.20775 0.0566203
\(456\) 0 0
\(457\) −23.2272 −1.08652 −0.543262 0.839563i \(-0.682811\pi\)
−0.543262 + 0.839563i \(0.682811\pi\)
\(458\) −21.2960 −0.995096
\(459\) −9.41617 −0.439509
\(460\) 0.537949 0.0250820
\(461\) 7.71392 0.359273 0.179637 0.983733i \(-0.442508\pi\)
0.179637 + 0.983733i \(0.442508\pi\)
\(462\) −5.16509 −0.240302
\(463\) 24.4450 1.13605 0.568027 0.823010i \(-0.307707\pi\)
0.568027 + 0.823010i \(0.307707\pi\)
\(464\) 5.57522 0.258823
\(465\) −4.86180 −0.225460
\(466\) 12.7292 0.589667
\(467\) −33.8146 −1.56475 −0.782377 0.622806i \(-0.785993\pi\)
−0.782377 + 0.622806i \(0.785993\pi\)
\(468\) 1.79581 0.0830114
\(469\) −4.12745 −0.190588
\(470\) −0.285967 −0.0131907
\(471\) −3.64792 −0.168087
\(472\) 10.2131 0.470097
\(473\) 15.4376 0.709823
\(474\) −28.7197 −1.31914
\(475\) 0 0
\(476\) −3.05251 −0.139911
\(477\) −9.61479 −0.440231
\(478\) −7.53887 −0.344820
\(479\) 19.3169 0.882610 0.441305 0.897357i \(-0.354516\pi\)
0.441305 + 0.897357i \(0.354516\pi\)
\(480\) 2.23129 0.101844
\(481\) −1.31487 −0.0599530
\(482\) 24.0728 1.09649
\(483\) 1.09808 0.0499643
\(484\) −5.14258 −0.233754
\(485\) 13.1919 0.599012
\(486\) 14.7486 0.669010
\(487\) −1.36447 −0.0618301 −0.0309150 0.999522i \(-0.509842\pi\)
−0.0309150 + 0.999522i \(0.509842\pi\)
\(488\) 9.38737 0.424946
\(489\) 40.6315 1.83742
\(490\) 1.04552 0.0472317
\(491\) 19.9180 0.898889 0.449444 0.893308i \(-0.351622\pi\)
0.449444 + 0.893308i \(0.351622\pi\)
\(492\) 3.71022 0.167269
\(493\) 17.0184 0.766469
\(494\) 0 0
\(495\) 3.93367 0.176806
\(496\) 2.17892 0.0978364
\(497\) 11.0844 0.497202
\(498\) −32.4022 −1.45198
\(499\) 19.6662 0.880381 0.440190 0.897905i \(-0.354911\pi\)
0.440190 + 0.897905i \(0.354911\pi\)
\(500\) 9.31231 0.416459
\(501\) −16.7570 −0.748648
\(502\) −0.365805 −0.0163267
\(503\) 32.8861 1.46632 0.733158 0.680058i \(-0.238045\pi\)
0.733158 + 0.680058i \(0.238045\pi\)
\(504\) 1.55458 0.0692466
\(505\) 1.56216 0.0695151
\(506\) −1.24527 −0.0553589
\(507\) −24.8961 −1.10567
\(508\) 2.87127 0.127392
\(509\) 26.7240 1.18452 0.592259 0.805748i \(-0.298236\pi\)
0.592259 + 0.805748i \(0.298236\pi\)
\(510\) 6.81102 0.301597
\(511\) 1.05367 0.0466117
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 8.33390 0.367593
\(515\) 20.2524 0.892427
\(516\) −13.6129 −0.599276
\(517\) 0.661970 0.0291134
\(518\) −1.13825 −0.0500117
\(519\) −17.3488 −0.761530
\(520\) 1.20775 0.0529635
\(521\) 23.0195 1.00850 0.504250 0.863558i \(-0.331769\pi\)
0.504250 + 0.863558i \(0.331769\pi\)
\(522\) −8.66713 −0.379350
\(523\) −10.1838 −0.445306 −0.222653 0.974898i \(-0.571472\pi\)
−0.222653 + 0.974898i \(0.571472\pi\)
\(524\) 3.59135 0.156889
\(525\) 8.33788 0.363895
\(526\) 22.5679 0.984008
\(527\) 6.65116 0.289729
\(528\) −5.16509 −0.224782
\(529\) −22.7353 −0.988490
\(530\) −6.46633 −0.280879
\(531\) −15.8772 −0.689010
\(532\) 0 0
\(533\) 2.00827 0.0869878
\(534\) −20.1590 −0.872365
\(535\) 9.20239 0.397854
\(536\) −4.12745 −0.178279
\(537\) −30.4114 −1.31235
\(538\) 20.4358 0.881050
\(539\) −2.42021 −0.104246
\(540\) 3.22514 0.138788
\(541\) 10.7486 0.462117 0.231059 0.972940i \(-0.425781\pi\)
0.231059 + 0.972940i \(0.425781\pi\)
\(542\) −17.2844 −0.742429
\(543\) −36.1193 −1.55003
\(544\) −3.05251 −0.130875
\(545\) 17.3654 0.743854
\(546\) 2.46531 0.105505
\(547\) −0.512859 −0.0219283 −0.0109641 0.999940i \(-0.503490\pi\)
−0.0109641 + 0.999940i \(0.503490\pi\)
\(548\) −3.24610 −0.138666
\(549\) −14.5934 −0.622833
\(550\) −9.45550 −0.403184
\(551\) 0 0
\(552\) 1.09808 0.0467373
\(553\) −13.4572 −0.572260
\(554\) 21.1338 0.897890
\(555\) 2.53976 0.107807
\(556\) 6.21267 0.263476
\(557\) −44.1525 −1.87080 −0.935401 0.353590i \(-0.884961\pi\)
−0.935401 + 0.353590i \(0.884961\pi\)
\(558\) −3.38731 −0.143396
\(559\) −7.36841 −0.311651
\(560\) 1.04552 0.0441812
\(561\) −15.7665 −0.665660
\(562\) −19.0891 −0.805225
\(563\) −10.1263 −0.426772 −0.213386 0.976968i \(-0.568449\pi\)
−0.213386 + 0.976968i \(0.568449\pi\)
\(564\) −0.583726 −0.0245793
\(565\) −7.12630 −0.299806
\(566\) 21.5165 0.904406
\(567\) 11.2470 0.472330
\(568\) 11.0844 0.465090
\(569\) 23.0647 0.966922 0.483461 0.875366i \(-0.339379\pi\)
0.483461 + 0.875366i \(0.339379\pi\)
\(570\) 0 0
\(571\) −9.68457 −0.405287 −0.202643 0.979253i \(-0.564953\pi\)
−0.202643 + 0.979253i \(0.564953\pi\)
\(572\) −2.79576 −0.116897
\(573\) −28.7078 −1.19929
\(574\) 1.73850 0.0725636
\(575\) 2.01021 0.0838314
\(576\) 1.55458 0.0647743
\(577\) 31.4649 1.30990 0.654950 0.755672i \(-0.272690\pi\)
0.654950 + 0.755672i \(0.272690\pi\)
\(578\) 7.68221 0.319538
\(579\) −50.4892 −2.09826
\(580\) −5.82899 −0.242036
\(581\) −15.1827 −0.629886
\(582\) 26.9277 1.11619
\(583\) 14.9685 0.619934
\(584\) 1.05367 0.0436013
\(585\) −1.87755 −0.0776272
\(586\) −11.3698 −0.469683
\(587\) −32.2780 −1.33226 −0.666128 0.745838i \(-0.732049\pi\)
−0.666128 + 0.745838i \(0.732049\pi\)
\(588\) 2.13415 0.0880107
\(589\) 0 0
\(590\) −10.6780 −0.439607
\(591\) 35.3422 1.45378
\(592\) −1.13825 −0.0467817
\(593\) −38.8819 −1.59669 −0.798345 0.602200i \(-0.794291\pi\)
−0.798345 + 0.602200i \(0.794291\pi\)
\(594\) −7.46570 −0.306322
\(595\) 3.19145 0.130837
\(596\) −15.3029 −0.626831
\(597\) 16.3968 0.671074
\(598\) 0.594369 0.0243056
\(599\) 37.9488 1.55055 0.775273 0.631626i \(-0.217612\pi\)
0.775273 + 0.631626i \(0.217612\pi\)
\(600\) 8.33788 0.340393
\(601\) 25.7005 1.04835 0.524173 0.851612i \(-0.324375\pi\)
0.524173 + 0.851612i \(0.324375\pi\)
\(602\) −6.37863 −0.259973
\(603\) 6.41647 0.261299
\(604\) −8.44026 −0.343429
\(605\) 5.37666 0.218592
\(606\) 3.18873 0.129533
\(607\) −26.8607 −1.09024 −0.545121 0.838357i \(-0.683516\pi\)
−0.545121 + 0.838357i \(0.683516\pi\)
\(608\) 0 0
\(609\) −11.8983 −0.482145
\(610\) −9.81466 −0.397384
\(611\) −0.315960 −0.0127824
\(612\) 4.74537 0.191820
\(613\) 31.4009 1.26827 0.634135 0.773223i \(-0.281357\pi\)
0.634135 + 0.773223i \(0.281357\pi\)
\(614\) −9.12185 −0.368128
\(615\) −3.87910 −0.156420
\(616\) −2.42021 −0.0975131
\(617\) −11.9350 −0.480484 −0.240242 0.970713i \(-0.577227\pi\)
−0.240242 + 0.970713i \(0.577227\pi\)
\(618\) 41.3399 1.66293
\(619\) −2.76014 −0.110940 −0.0554698 0.998460i \(-0.517666\pi\)
−0.0554698 + 0.998460i \(0.517666\pi\)
\(620\) −2.27810 −0.0914907
\(621\) 1.58718 0.0636914
\(622\) −17.3492 −0.695639
\(623\) −9.44592 −0.378443
\(624\) 2.46531 0.0986913
\(625\) 9.79827 0.391931
\(626\) 2.42580 0.0969543
\(627\) 0 0
\(628\) −1.70931 −0.0682089
\(629\) −3.47450 −0.138538
\(630\) −1.62534 −0.0647552
\(631\) 26.0991 1.03899 0.519495 0.854473i \(-0.326120\pi\)
0.519495 + 0.854473i \(0.326120\pi\)
\(632\) −13.4572 −0.535300
\(633\) −41.2122 −1.63804
\(634\) −8.88675 −0.352938
\(635\) −3.00197 −0.119129
\(636\) −13.1993 −0.523386
\(637\) 1.15517 0.0457696
\(638\) 13.4932 0.534201
\(639\) −17.2316 −0.681671
\(640\) 1.04552 0.0413277
\(641\) −30.2320 −1.19409 −0.597046 0.802207i \(-0.703659\pi\)
−0.597046 + 0.802207i \(0.703659\pi\)
\(642\) 18.7842 0.741354
\(643\) −5.69885 −0.224741 −0.112370 0.993666i \(-0.535844\pi\)
−0.112370 + 0.993666i \(0.535844\pi\)
\(644\) 0.514528 0.0202753
\(645\) 14.2326 0.560406
\(646\) 0 0
\(647\) −38.9990 −1.53321 −0.766605 0.642119i \(-0.778055\pi\)
−0.766605 + 0.642119i \(0.778055\pi\)
\(648\) 11.2470 0.441825
\(649\) 24.7179 0.970263
\(650\) 4.51313 0.177020
\(651\) −4.65013 −0.182253
\(652\) 19.0388 0.745616
\(653\) 9.49438 0.371544 0.185772 0.982593i \(-0.440521\pi\)
0.185772 + 0.982593i \(0.440521\pi\)
\(654\) 35.4469 1.38608
\(655\) −3.75482 −0.146713
\(656\) 1.73850 0.0678771
\(657\) −1.63802 −0.0639053
\(658\) −0.273517 −0.0106628
\(659\) −34.7488 −1.35362 −0.676811 0.736157i \(-0.736638\pi\)
−0.676811 + 0.736157i \(0.736638\pi\)
\(660\) 5.40019 0.210202
\(661\) 42.2506 1.64336 0.821679 0.569950i \(-0.193038\pi\)
0.821679 + 0.569950i \(0.193038\pi\)
\(662\) 19.5033 0.758019
\(663\) 7.52537 0.292261
\(664\) −15.1827 −0.589204
\(665\) 0 0
\(666\) 1.76950 0.0685667
\(667\) −2.86861 −0.111073
\(668\) −7.85185 −0.303797
\(669\) 20.3045 0.785019
\(670\) 4.31533 0.166716
\(671\) 22.7194 0.877074
\(672\) 2.13415 0.0823265
\(673\) −27.7918 −1.07130 −0.535648 0.844441i \(-0.679933\pi\)
−0.535648 + 0.844441i \(0.679933\pi\)
\(674\) −5.72554 −0.220540
\(675\) 12.0517 0.463871
\(676\) −11.6656 −0.448676
\(677\) −6.95181 −0.267180 −0.133590 0.991037i \(-0.542650\pi\)
−0.133590 + 0.991037i \(0.542650\pi\)
\(678\) −14.5465 −0.558653
\(679\) 12.6176 0.484217
\(680\) 3.19145 0.122386
\(681\) 53.9768 2.06839
\(682\) 5.27344 0.201931
\(683\) 27.6511 1.05804 0.529020 0.848609i \(-0.322560\pi\)
0.529020 + 0.848609i \(0.322560\pi\)
\(684\) 0 0
\(685\) 3.39385 0.129672
\(686\) 1.00000 0.0381802
\(687\) 45.4487 1.73398
\(688\) −6.37863 −0.243183
\(689\) −7.14452 −0.272184
\(690\) −1.14806 −0.0437059
\(691\) −28.9707 −1.10210 −0.551050 0.834472i \(-0.685773\pi\)
−0.551050 + 0.834472i \(0.685773\pi\)
\(692\) −8.12917 −0.309025
\(693\) 3.76242 0.142922
\(694\) −34.0907 −1.29407
\(695\) −6.49545 −0.246387
\(696\) −11.8983 −0.451005
\(697\) 5.30678 0.201009
\(698\) −31.1329 −1.17840
\(699\) −27.1659 −1.02751
\(700\) 3.90689 0.147667
\(701\) 36.1561 1.36560 0.682798 0.730607i \(-0.260763\pi\)
0.682798 + 0.730607i \(0.260763\pi\)
\(702\) 3.56340 0.134492
\(703\) 0 0
\(704\) −2.42021 −0.0912151
\(705\) 0.610296 0.0229851
\(706\) −7.00294 −0.263559
\(707\) 1.49415 0.0561932
\(708\) −21.7963 −0.819156
\(709\) −38.9545 −1.46297 −0.731483 0.681860i \(-0.761171\pi\)
−0.731483 + 0.681860i \(0.761171\pi\)
\(710\) −11.5889 −0.434924
\(711\) 20.9204 0.784576
\(712\) −9.44592 −0.354001
\(713\) −1.12112 −0.0419861
\(714\) 6.51450 0.243799
\(715\) 2.92302 0.109315
\(716\) −14.2499 −0.532544
\(717\) 16.0890 0.600856
\(718\) −1.04073 −0.0388395
\(719\) −32.1104 −1.19751 −0.598757 0.800931i \(-0.704338\pi\)
−0.598757 + 0.800931i \(0.704338\pi\)
\(720\) −1.62534 −0.0605730
\(721\) 19.3707 0.721402
\(722\) 0 0
\(723\) −51.3749 −1.91065
\(724\) −16.9244 −0.628992
\(725\) −21.7818 −0.808955
\(726\) 10.9750 0.407321
\(727\) 30.2622 1.12236 0.561181 0.827693i \(-0.310347\pi\)
0.561181 + 0.827693i \(0.310347\pi\)
\(728\) 1.15517 0.0428135
\(729\) 2.26540 0.0839035
\(730\) −1.10163 −0.0407733
\(731\) −19.4708 −0.720154
\(732\) −20.0340 −0.740479
\(733\) −2.13269 −0.0787727 −0.0393864 0.999224i \(-0.512540\pi\)
−0.0393864 + 0.999224i \(0.512540\pi\)
\(734\) 2.88851 0.106617
\(735\) −2.23129 −0.0823023
\(736\) 0.514528 0.0189658
\(737\) −9.98931 −0.367961
\(738\) −2.70264 −0.0994857
\(739\) 10.6328 0.391133 0.195567 0.980690i \(-0.437345\pi\)
0.195567 + 0.980690i \(0.437345\pi\)
\(740\) 1.19006 0.0437474
\(741\) 0 0
\(742\) −6.18481 −0.227051
\(743\) −30.3286 −1.11265 −0.556324 0.830966i \(-0.687789\pi\)
−0.556324 + 0.830966i \(0.687789\pi\)
\(744\) −4.65013 −0.170482
\(745\) 15.9995 0.586175
\(746\) −7.01824 −0.256956
\(747\) 23.6028 0.863582
\(748\) −7.38771 −0.270121
\(749\) 8.80175 0.321609
\(750\) −19.8738 −0.725690
\(751\) 11.9117 0.434664 0.217332 0.976098i \(-0.430265\pi\)
0.217332 + 0.976098i \(0.430265\pi\)
\(752\) −0.273517 −0.00997415
\(753\) 0.780681 0.0284496
\(754\) −6.44034 −0.234543
\(755\) 8.82444 0.321154
\(756\) 3.08473 0.112191
\(757\) 2.55529 0.0928738 0.0464369 0.998921i \(-0.485213\pi\)
0.0464369 + 0.998921i \(0.485213\pi\)
\(758\) 14.5650 0.529023
\(759\) 2.65758 0.0964642
\(760\) 0 0
\(761\) −17.1391 −0.621293 −0.310647 0.950525i \(-0.600546\pi\)
−0.310647 + 0.950525i \(0.600546\pi\)
\(762\) −6.12772 −0.221984
\(763\) 16.6094 0.601301
\(764\) −13.4517 −0.486664
\(765\) −4.96137 −0.179379
\(766\) 12.1342 0.438425
\(767\) −11.7979 −0.425998
\(768\) 2.13415 0.0770094
\(769\) −30.9530 −1.11620 −0.558098 0.829775i \(-0.688469\pi\)
−0.558098 + 0.829775i \(0.688469\pi\)
\(770\) 2.53037 0.0911883
\(771\) −17.7858 −0.640539
\(772\) −23.6578 −0.851463
\(773\) −49.9764 −1.79753 −0.898763 0.438435i \(-0.855533\pi\)
−0.898763 + 0.438435i \(0.855533\pi\)
\(774\) 9.91610 0.356427
\(775\) −8.51280 −0.305789
\(776\) 12.6176 0.452944
\(777\) 2.42919 0.0871465
\(778\) −24.2664 −0.869994
\(779\) 0 0
\(780\) −2.57752 −0.0922901
\(781\) 26.8265 0.959929
\(782\) 1.57060 0.0561646
\(783\) −17.1981 −0.614608
\(784\) 1.00000 0.0357143
\(785\) 1.78711 0.0637848
\(786\) −7.66447 −0.273383
\(787\) −33.0457 −1.17795 −0.588976 0.808150i \(-0.700469\pi\)
−0.588976 + 0.808150i \(0.700469\pi\)
\(788\) 16.5603 0.589938
\(789\) −48.1633 −1.71466
\(790\) 14.0698 0.500581
\(791\) −6.81605 −0.242351
\(792\) 3.76242 0.133692
\(793\) −10.8440 −0.385083
\(794\) 14.1299 0.501453
\(795\) 13.8001 0.489439
\(796\) 7.68305 0.272318
\(797\) 53.7061 1.90237 0.951184 0.308624i \(-0.0998685\pi\)
0.951184 + 0.308624i \(0.0998685\pi\)
\(798\) 0 0
\(799\) −0.834913 −0.0295371
\(800\) 3.90689 0.138130
\(801\) 14.6845 0.518850
\(802\) −13.7977 −0.487214
\(803\) 2.55011 0.0899915
\(804\) 8.80859 0.310655
\(805\) −0.537949 −0.0189602
\(806\) −2.51703 −0.0886585
\(807\) −43.6130 −1.53525
\(808\) 1.49415 0.0525639
\(809\) −38.4512 −1.35187 −0.675936 0.736960i \(-0.736260\pi\)
−0.675936 + 0.736960i \(0.736260\pi\)
\(810\) −11.7590 −0.413168
\(811\) −34.6028 −1.21507 −0.607535 0.794293i \(-0.707842\pi\)
−0.607535 + 0.794293i \(0.707842\pi\)
\(812\) −5.57522 −0.195652
\(813\) 36.8875 1.29370
\(814\) −2.75480 −0.0965556
\(815\) −19.9054 −0.697255
\(816\) 6.51450 0.228053
\(817\) 0 0
\(818\) 12.7520 0.445862
\(819\) −1.79581 −0.0627507
\(820\) −1.81763 −0.0634745
\(821\) 5.21981 0.182173 0.0910864 0.995843i \(-0.470966\pi\)
0.0910864 + 0.995843i \(0.470966\pi\)
\(822\) 6.92765 0.241630
\(823\) −5.08619 −0.177294 −0.0886468 0.996063i \(-0.528254\pi\)
−0.0886468 + 0.996063i \(0.528254\pi\)
\(824\) 19.3707 0.674809
\(825\) 20.1794 0.702558
\(826\) −10.2131 −0.355360
\(827\) 18.5906 0.646459 0.323229 0.946321i \(-0.395231\pi\)
0.323229 + 0.946321i \(0.395231\pi\)
\(828\) −0.799877 −0.0277976
\(829\) −0.766359 −0.0266168 −0.0133084 0.999911i \(-0.504236\pi\)
−0.0133084 + 0.999911i \(0.504236\pi\)
\(830\) 15.8738 0.550988
\(831\) −45.1027 −1.56459
\(832\) 1.15517 0.0400484
\(833\) 3.05251 0.105763
\(834\) −13.2587 −0.459113
\(835\) 8.20925 0.284093
\(836\) 0 0
\(837\) −6.72138 −0.232325
\(838\) −25.8240 −0.892076
\(839\) 41.0401 1.41686 0.708431 0.705780i \(-0.249403\pi\)
0.708431 + 0.705780i \(0.249403\pi\)
\(840\) −2.23129 −0.0769868
\(841\) 2.08304 0.0718290
\(842\) 5.42077 0.186812
\(843\) 40.7390 1.40312
\(844\) −19.3109 −0.664708
\(845\) 12.1966 0.419575
\(846\) 0.425205 0.0146189
\(847\) 5.14258 0.176701
\(848\) −6.18481 −0.212387
\(849\) −45.9194 −1.57595
\(850\) 11.9258 0.409052
\(851\) 0.585660 0.0200762
\(852\) −23.6557 −0.810431
\(853\) 18.3849 0.629488 0.314744 0.949177i \(-0.398081\pi\)
0.314744 + 0.949177i \(0.398081\pi\)
\(854\) −9.38737 −0.321229
\(855\) 0 0
\(856\) 8.80175 0.300838
\(857\) 9.11999 0.311533 0.155766 0.987794i \(-0.450215\pi\)
0.155766 + 0.987794i \(0.450215\pi\)
\(858\) 5.96656 0.203695
\(859\) −34.9079 −1.19104 −0.595521 0.803340i \(-0.703054\pi\)
−0.595521 + 0.803340i \(0.703054\pi\)
\(860\) 6.66897 0.227410
\(861\) −3.71022 −0.126444
\(862\) −34.0208 −1.15875
\(863\) −9.40882 −0.320280 −0.160140 0.987094i \(-0.551195\pi\)
−0.160140 + 0.987094i \(0.551195\pi\)
\(864\) 3.08473 0.104945
\(865\) 8.49919 0.288981
\(866\) −13.3211 −0.452670
\(867\) −16.3950 −0.556802
\(868\) −2.17892 −0.0739573
\(869\) −32.5694 −1.10484
\(870\) 12.4399 0.421753
\(871\) 4.76792 0.161555
\(872\) 16.6094 0.562466
\(873\) −19.6150 −0.663868
\(874\) 0 0
\(875\) −9.31231 −0.314814
\(876\) −2.24869 −0.0759763
\(877\) 53.5586 1.80854 0.904272 0.426957i \(-0.140414\pi\)
0.904272 + 0.426957i \(0.140414\pi\)
\(878\) 22.4500 0.757649
\(879\) 24.2648 0.818433
\(880\) 2.53037 0.0852989
\(881\) −27.8500 −0.938289 −0.469144 0.883121i \(-0.655438\pi\)
−0.469144 + 0.883121i \(0.655438\pi\)
\(882\) −1.55458 −0.0523455
\(883\) −53.9182 −1.81449 −0.907246 0.420601i \(-0.861819\pi\)
−0.907246 + 0.420601i \(0.861819\pi\)
\(884\) 3.52617 0.118598
\(885\) 22.7884 0.766025
\(886\) 7.75275 0.260459
\(887\) 39.8816 1.33909 0.669547 0.742770i \(-0.266488\pi\)
0.669547 + 0.742770i \(0.266488\pi\)
\(888\) 2.42919 0.0815181
\(889\) −2.87127 −0.0962995
\(890\) 9.87588 0.331040
\(891\) 27.2202 0.911910
\(892\) 9.51413 0.318557
\(893\) 0 0
\(894\) 32.6586 1.09227
\(895\) 14.8985 0.498003
\(896\) 1.00000 0.0334077
\(897\) −1.26847 −0.0423530
\(898\) 22.0007 0.734174
\(899\) 12.1479 0.405157
\(900\) −6.07359 −0.202453
\(901\) −18.8792 −0.628956
\(902\) 4.20754 0.140096
\(903\) 13.6129 0.453010
\(904\) −6.81605 −0.226698
\(905\) 17.6948 0.588195
\(906\) 18.0128 0.598434
\(907\) 38.8333 1.28944 0.644719 0.764420i \(-0.276974\pi\)
0.644719 + 0.764420i \(0.276974\pi\)
\(908\) 25.2920 0.839343
\(909\) −2.32277 −0.0770416
\(910\) −1.20775 −0.0400366
\(911\) −24.5969 −0.814931 −0.407466 0.913221i \(-0.633587\pi\)
−0.407466 + 0.913221i \(0.633587\pi\)
\(912\) 0 0
\(913\) −36.7454 −1.21610
\(914\) 23.2272 0.768288
\(915\) 20.9459 0.692451
\(916\) 21.2960 0.703639
\(917\) −3.59135 −0.118597
\(918\) 9.41617 0.310780
\(919\) −22.9886 −0.758323 −0.379161 0.925331i \(-0.623787\pi\)
−0.379161 + 0.925331i \(0.623787\pi\)
\(920\) −0.537949 −0.0177356
\(921\) 19.4674 0.641472
\(922\) −7.71392 −0.254044
\(923\) −12.8044 −0.421461
\(924\) 5.16509 0.169919
\(925\) 4.44701 0.146217
\(926\) −24.4450 −0.803311
\(927\) −30.1133 −0.989051
\(928\) −5.57522 −0.183015
\(929\) 34.9140 1.14549 0.572746 0.819733i \(-0.305878\pi\)
0.572746 + 0.819733i \(0.305878\pi\)
\(930\) 4.86180 0.159425
\(931\) 0 0
\(932\) −12.7292 −0.416958
\(933\) 37.0257 1.21217
\(934\) 33.8146 1.10645
\(935\) 7.72398 0.252601
\(936\) −1.79581 −0.0586979
\(937\) 33.9944 1.11055 0.555274 0.831667i \(-0.312613\pi\)
0.555274 + 0.831667i \(0.312613\pi\)
\(938\) 4.12745 0.134766
\(939\) −5.17701 −0.168945
\(940\) 0.285967 0.00932722
\(941\) 25.6708 0.836843 0.418421 0.908253i \(-0.362584\pi\)
0.418421 + 0.908253i \(0.362584\pi\)
\(942\) 3.64792 0.118856
\(943\) −0.894508 −0.0291292
\(944\) −10.2131 −0.332409
\(945\) −3.22514 −0.104914
\(946\) −15.4376 −0.501920
\(947\) 42.6574 1.38618 0.693089 0.720852i \(-0.256249\pi\)
0.693089 + 0.720852i \(0.256249\pi\)
\(948\) 28.7197 0.932774
\(949\) −1.21717 −0.0395111
\(950\) 0 0
\(951\) 18.9656 0.615003
\(952\) 3.05251 0.0989323
\(953\) −21.5394 −0.697731 −0.348865 0.937173i \(-0.613433\pi\)
−0.348865 + 0.937173i \(0.613433\pi\)
\(954\) 9.61479 0.311290
\(955\) 14.0639 0.455099
\(956\) 7.53887 0.243824
\(957\) −28.7965 −0.930858
\(958\) −19.3169 −0.624100
\(959\) 3.24610 0.104822
\(960\) −2.23129 −0.0720145
\(961\) −26.2523 −0.846849
\(962\) 1.31487 0.0423931
\(963\) −13.6831 −0.440930
\(964\) −24.0728 −0.775333
\(965\) 24.7346 0.796237
\(966\) −1.09808 −0.0353301
\(967\) 8.45314 0.271835 0.135917 0.990720i \(-0.456602\pi\)
0.135917 + 0.990720i \(0.456602\pi\)
\(968\) 5.14258 0.165289
\(969\) 0 0
\(970\) −13.1919 −0.423566
\(971\) −33.4395 −1.07313 −0.536563 0.843860i \(-0.680278\pi\)
−0.536563 + 0.843860i \(0.680278\pi\)
\(972\) −14.7486 −0.473061
\(973\) −6.21267 −0.199169
\(974\) 1.36447 0.0437205
\(975\) −9.63169 −0.308461
\(976\) −9.38737 −0.300482
\(977\) 38.5495 1.23331 0.616654 0.787234i \(-0.288488\pi\)
0.616654 + 0.787234i \(0.288488\pi\)
\(978\) −40.6315 −1.29925
\(979\) −22.8611 −0.730645
\(980\) −1.04552 −0.0333978
\(981\) −25.8207 −0.824392
\(982\) −19.9180 −0.635610
\(983\) 4.75637 0.151705 0.0758524 0.997119i \(-0.475832\pi\)
0.0758524 + 0.997119i \(0.475832\pi\)
\(984\) −3.71022 −0.118277
\(985\) −17.3141 −0.551674
\(986\) −17.0184 −0.541976
\(987\) 0.583726 0.0185802
\(988\) 0 0
\(989\) 3.28198 0.104361
\(990\) −3.93367 −0.125020
\(991\) 11.1093 0.352897 0.176449 0.984310i \(-0.443539\pi\)
0.176449 + 0.984310i \(0.443539\pi\)
\(992\) −2.17892 −0.0691808
\(993\) −41.6230 −1.32087
\(994\) −11.0844 −0.351575
\(995\) −8.03276 −0.254656
\(996\) 32.4022 1.02670
\(997\) 2.17801 0.0689784 0.0344892 0.999405i \(-0.489020\pi\)
0.0344892 + 0.999405i \(0.489020\pi\)
\(998\) −19.6662 −0.622523
\(999\) 3.51119 0.111089
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 5054.2.a.bf.1.8 8
19.18 odd 2 5054.2.a.bi.1.1 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5054.2.a.bf.1.8 8 1.1 even 1 trivial
5054.2.a.bi.1.1 yes 8 19.18 odd 2