Properties

Label 4029.2.a.k.1.12
Level 4029
Weight 2
Character 4029.1
Self dual yes
Analytic conductor 32.172
Analytic rank 0
Dimension 31
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4029 = 3 \cdot 17 \cdot 79 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4029.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.1717269744\)
Analytic rank: \(0\)
Dimension: \(31\)
Coefficient ring index: multiple of None
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.12
Character \(\chi\) = 4029.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.639237 q^{2} +1.00000 q^{3} -1.59138 q^{4} -1.53158 q^{5} -0.639237 q^{6} -2.74819 q^{7} +2.29574 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.639237 q^{2} +1.00000 q^{3} -1.59138 q^{4} -1.53158 q^{5} -0.639237 q^{6} -2.74819 q^{7} +2.29574 q^{8} +1.00000 q^{9} +0.979040 q^{10} +4.38312 q^{11} -1.59138 q^{12} +2.19700 q^{13} +1.75675 q^{14} -1.53158 q^{15} +1.71523 q^{16} +1.00000 q^{17} -0.639237 q^{18} +4.55900 q^{19} +2.43731 q^{20} -2.74819 q^{21} -2.80185 q^{22} -7.35985 q^{23} +2.29574 q^{24} -2.65427 q^{25} -1.40440 q^{26} +1.00000 q^{27} +4.37341 q^{28} -1.31023 q^{29} +0.979040 q^{30} -5.83625 q^{31} -5.68792 q^{32} +4.38312 q^{33} -0.639237 q^{34} +4.20907 q^{35} -1.59138 q^{36} +4.21370 q^{37} -2.91428 q^{38} +2.19700 q^{39} -3.51610 q^{40} +3.78873 q^{41} +1.75675 q^{42} +3.27995 q^{43} -6.97519 q^{44} -1.53158 q^{45} +4.70469 q^{46} -5.56863 q^{47} +1.71523 q^{48} +0.552565 q^{49} +1.69671 q^{50} +1.00000 q^{51} -3.49625 q^{52} -1.40021 q^{53} -0.639237 q^{54} -6.71308 q^{55} -6.30914 q^{56} +4.55900 q^{57} +0.837548 q^{58} +14.2575 q^{59} +2.43731 q^{60} -2.49345 q^{61} +3.73075 q^{62} -2.74819 q^{63} +0.205472 q^{64} -3.36487 q^{65} -2.80185 q^{66} -0.208432 q^{67} -1.59138 q^{68} -7.35985 q^{69} -2.69059 q^{70} +5.71350 q^{71} +2.29574 q^{72} -1.55512 q^{73} -2.69355 q^{74} -2.65427 q^{75} -7.25508 q^{76} -12.0457 q^{77} -1.40440 q^{78} +1.00000 q^{79} -2.62700 q^{80} +1.00000 q^{81} -2.42190 q^{82} -14.1845 q^{83} +4.37341 q^{84} -1.53158 q^{85} -2.09666 q^{86} -1.31023 q^{87} +10.0625 q^{88} +8.42513 q^{89} +0.979040 q^{90} -6.03777 q^{91} +11.7123 q^{92} -5.83625 q^{93} +3.55967 q^{94} -6.98245 q^{95} -5.68792 q^{96} +1.90697 q^{97} -0.353220 q^{98} +4.38312 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 31q + 4q^{2} + 31q^{3} + 34q^{4} + 11q^{5} + 4q^{6} + 4q^{7} + 12q^{8} + 31q^{9} + O(q^{10}) \) \( 31q + 4q^{2} + 31q^{3} + 34q^{4} + 11q^{5} + 4q^{6} + 4q^{7} + 12q^{8} + 31q^{9} + 5q^{10} + 26q^{11} + 34q^{12} + 7q^{13} + 19q^{14} + 11q^{15} + 40q^{16} + 31q^{17} + 4q^{18} + 32q^{19} + 23q^{20} + 4q^{21} + 2q^{22} + 29q^{23} + 12q^{24} + 32q^{25} + 13q^{26} + 31q^{27} - 13q^{28} + 25q^{29} + 5q^{30} + 22q^{31} + 28q^{32} + 26q^{33} + 4q^{34} + 20q^{35} + 34q^{36} - 4q^{37} + 19q^{38} + 7q^{39} - 3q^{40} + 33q^{41} + 19q^{42} + 6q^{43} + 30q^{44} + 11q^{45} - 11q^{46} + 23q^{47} + 40q^{48} + 31q^{49} + 6q^{50} + 31q^{51} - 7q^{52} + 12q^{53} + 4q^{54} + 40q^{56} + 32q^{57} + 9q^{58} + 27q^{59} + 23q^{60} - 4q^{61} + 25q^{62} + 4q^{63} + 10q^{64} + 54q^{65} + 2q^{66} + 34q^{68} + 29q^{69} - 59q^{70} + 35q^{71} + 12q^{72} + 5q^{73} + 48q^{74} + 32q^{75} + 32q^{76} + 42q^{77} + 13q^{78} + 31q^{79} + 24q^{80} + 31q^{81} + 5q^{82} + 67q^{83} - 13q^{84} + 11q^{85} - 20q^{86} + 25q^{87} - 7q^{88} + 22q^{89} + 5q^{90} + 16q^{91} + 57q^{92} + 22q^{93} + 45q^{94} + 73q^{95} + 28q^{96} - 13q^{97} - 19q^{98} + 26q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.639237 −0.452009 −0.226004 0.974126i \(-0.572566\pi\)
−0.226004 + 0.974126i \(0.572566\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.59138 −0.795688
\(5\) −1.53158 −0.684942 −0.342471 0.939528i \(-0.611264\pi\)
−0.342471 + 0.939528i \(0.611264\pi\)
\(6\) −0.639237 −0.260967
\(7\) −2.74819 −1.03872 −0.519360 0.854556i \(-0.673829\pi\)
−0.519360 + 0.854556i \(0.673829\pi\)
\(8\) 2.29574 0.811667
\(9\) 1.00000 0.333333
\(10\) 0.979040 0.309600
\(11\) 4.38312 1.32156 0.660780 0.750580i \(-0.270226\pi\)
0.660780 + 0.750580i \(0.270226\pi\)
\(12\) −1.59138 −0.459391
\(13\) 2.19700 0.609337 0.304668 0.952458i \(-0.401454\pi\)
0.304668 + 0.952458i \(0.401454\pi\)
\(14\) 1.75675 0.469510
\(15\) −1.53158 −0.395451
\(16\) 1.71523 0.428807
\(17\) 1.00000 0.242536
\(18\) −0.639237 −0.150670
\(19\) 4.55900 1.04591 0.522953 0.852362i \(-0.324830\pi\)
0.522953 + 0.852362i \(0.324830\pi\)
\(20\) 2.43731 0.545000
\(21\) −2.74819 −0.599705
\(22\) −2.80185 −0.597357
\(23\) −7.35985 −1.53463 −0.767317 0.641267i \(-0.778409\pi\)
−0.767317 + 0.641267i \(0.778409\pi\)
\(24\) 2.29574 0.468616
\(25\) −2.65427 −0.530855
\(26\) −1.40440 −0.275426
\(27\) 1.00000 0.192450
\(28\) 4.37341 0.826496
\(29\) −1.31023 −0.243304 −0.121652 0.992573i \(-0.538819\pi\)
−0.121652 + 0.992573i \(0.538819\pi\)
\(30\) 0.979040 0.178748
\(31\) −5.83625 −1.04822 −0.524110 0.851650i \(-0.675602\pi\)
−0.524110 + 0.851650i \(0.675602\pi\)
\(32\) −5.68792 −1.00549
\(33\) 4.38312 0.763003
\(34\) −0.639237 −0.109628
\(35\) 4.20907 0.711462
\(36\) −1.59138 −0.265229
\(37\) 4.21370 0.692727 0.346364 0.938100i \(-0.387416\pi\)
0.346364 + 0.938100i \(0.387416\pi\)
\(38\) −2.91428 −0.472759
\(39\) 2.19700 0.351801
\(40\) −3.51610 −0.555945
\(41\) 3.78873 0.591700 0.295850 0.955234i \(-0.404397\pi\)
0.295850 + 0.955234i \(0.404397\pi\)
\(42\) 1.75675 0.271072
\(43\) 3.27995 0.500187 0.250094 0.968222i \(-0.419539\pi\)
0.250094 + 0.968222i \(0.419539\pi\)
\(44\) −6.97519 −1.05155
\(45\) −1.53158 −0.228314
\(46\) 4.70469 0.693669
\(47\) −5.56863 −0.812268 −0.406134 0.913814i \(-0.633123\pi\)
−0.406134 + 0.913814i \(0.633123\pi\)
\(48\) 1.71523 0.247572
\(49\) 0.552565 0.0789379
\(50\) 1.69671 0.239951
\(51\) 1.00000 0.140028
\(52\) −3.49625 −0.484842
\(53\) −1.40021 −0.192333 −0.0961667 0.995365i \(-0.530658\pi\)
−0.0961667 + 0.995365i \(0.530658\pi\)
\(54\) −0.639237 −0.0869892
\(55\) −6.71308 −0.905191
\(56\) −6.30914 −0.843094
\(57\) 4.55900 0.603854
\(58\) 0.837548 0.109975
\(59\) 14.2575 1.85616 0.928081 0.372378i \(-0.121458\pi\)
0.928081 + 0.372378i \(0.121458\pi\)
\(60\) 2.43731 0.314656
\(61\) −2.49345 −0.319253 −0.159626 0.987177i \(-0.551029\pi\)
−0.159626 + 0.987177i \(0.551029\pi\)
\(62\) 3.73075 0.473805
\(63\) −2.74819 −0.346240
\(64\) 0.205472 0.0256840
\(65\) −3.36487 −0.417360
\(66\) −2.80185 −0.344884
\(67\) −0.208432 −0.0254640 −0.0127320 0.999919i \(-0.504053\pi\)
−0.0127320 + 0.999919i \(0.504053\pi\)
\(68\) −1.59138 −0.192983
\(69\) −7.35985 −0.886022
\(70\) −2.69059 −0.321587
\(71\) 5.71350 0.678068 0.339034 0.940774i \(-0.389900\pi\)
0.339034 + 0.940774i \(0.389900\pi\)
\(72\) 2.29574 0.270556
\(73\) −1.55512 −0.182013 −0.0910067 0.995850i \(-0.529008\pi\)
−0.0910067 + 0.995850i \(0.529008\pi\)
\(74\) −2.69355 −0.313119
\(75\) −2.65427 −0.306489
\(76\) −7.25508 −0.832214
\(77\) −12.0457 −1.37273
\(78\) −1.40440 −0.159017
\(79\) 1.00000 0.112509
\(80\) −2.62700 −0.293708
\(81\) 1.00000 0.111111
\(82\) −2.42190 −0.267454
\(83\) −14.1845 −1.55695 −0.778476 0.627674i \(-0.784007\pi\)
−0.778476 + 0.627674i \(0.784007\pi\)
\(84\) 4.37341 0.477178
\(85\) −1.53158 −0.166123
\(86\) −2.09666 −0.226089
\(87\) −1.31023 −0.140471
\(88\) 10.0625 1.07267
\(89\) 8.42513 0.893062 0.446531 0.894768i \(-0.352659\pi\)
0.446531 + 0.894768i \(0.352659\pi\)
\(90\) 0.979040 0.103200
\(91\) −6.03777 −0.632930
\(92\) 11.7123 1.22109
\(93\) −5.83625 −0.605191
\(94\) 3.55967 0.367152
\(95\) −6.98245 −0.716384
\(96\) −5.68792 −0.580521
\(97\) 1.90697 0.193623 0.0968116 0.995303i \(-0.469136\pi\)
0.0968116 + 0.995303i \(0.469136\pi\)
\(98\) −0.353220 −0.0356806
\(99\) 4.38312 0.440520
\(100\) 4.22395 0.422395
\(101\) −12.6867 −1.26238 −0.631188 0.775630i \(-0.717432\pi\)
−0.631188 + 0.775630i \(0.717432\pi\)
\(102\) −0.639237 −0.0632939
\(103\) 18.8984 1.86211 0.931057 0.364873i \(-0.118888\pi\)
0.931057 + 0.364873i \(0.118888\pi\)
\(104\) 5.04373 0.494579
\(105\) 4.20907 0.410763
\(106\) 0.895065 0.0869364
\(107\) −1.99970 −0.193318 −0.0966590 0.995318i \(-0.530816\pi\)
−0.0966590 + 0.995318i \(0.530816\pi\)
\(108\) −1.59138 −0.153130
\(109\) 18.5527 1.77702 0.888511 0.458856i \(-0.151741\pi\)
0.888511 + 0.458856i \(0.151741\pi\)
\(110\) 4.29125 0.409155
\(111\) 4.21370 0.399946
\(112\) −4.71378 −0.445410
\(113\) −2.42445 −0.228073 −0.114037 0.993477i \(-0.536378\pi\)
−0.114037 + 0.993477i \(0.536378\pi\)
\(114\) −2.91428 −0.272947
\(115\) 11.2722 1.05114
\(116\) 2.08507 0.193594
\(117\) 2.19700 0.203112
\(118\) −9.11389 −0.839002
\(119\) −2.74819 −0.251926
\(120\) −3.51610 −0.320975
\(121\) 8.21172 0.746520
\(122\) 1.59390 0.144305
\(123\) 3.78873 0.341618
\(124\) 9.28766 0.834057
\(125\) 11.7231 1.04855
\(126\) 1.75675 0.156503
\(127\) −3.87578 −0.343920 −0.171960 0.985104i \(-0.555010\pi\)
−0.171960 + 0.985104i \(0.555010\pi\)
\(128\) 11.2445 0.993882
\(129\) 3.27995 0.288783
\(130\) 2.15095 0.188651
\(131\) −2.04395 −0.178581 −0.0892906 0.996006i \(-0.528460\pi\)
−0.0892906 + 0.996006i \(0.528460\pi\)
\(132\) −6.97519 −0.607112
\(133\) −12.5290 −1.08640
\(134\) 0.133237 0.0115099
\(135\) −1.53158 −0.131817
\(136\) 2.29574 0.196858
\(137\) 11.9562 1.02149 0.510743 0.859733i \(-0.329370\pi\)
0.510743 + 0.859733i \(0.329370\pi\)
\(138\) 4.70469 0.400490
\(139\) 15.8345 1.34307 0.671533 0.740974i \(-0.265636\pi\)
0.671533 + 0.740974i \(0.265636\pi\)
\(140\) −6.69821 −0.566102
\(141\) −5.56863 −0.468963
\(142\) −3.65228 −0.306493
\(143\) 9.62969 0.805275
\(144\) 1.71523 0.142936
\(145\) 2.00672 0.166649
\(146\) 0.994093 0.0822717
\(147\) 0.552565 0.0455748
\(148\) −6.70557 −0.551195
\(149\) −2.92417 −0.239557 −0.119779 0.992801i \(-0.538219\pi\)
−0.119779 + 0.992801i \(0.538219\pi\)
\(150\) 1.69671 0.138536
\(151\) 5.51386 0.448712 0.224356 0.974507i \(-0.427972\pi\)
0.224356 + 0.974507i \(0.427972\pi\)
\(152\) 10.4663 0.848927
\(153\) 1.00000 0.0808452
\(154\) 7.70003 0.620486
\(155\) 8.93866 0.717970
\(156\) −3.49625 −0.279924
\(157\) −18.8903 −1.50761 −0.753803 0.657100i \(-0.771783\pi\)
−0.753803 + 0.657100i \(0.771783\pi\)
\(158\) −0.639237 −0.0508550
\(159\) −1.40021 −0.111044
\(160\) 8.71148 0.688703
\(161\) 20.2263 1.59406
\(162\) −0.639237 −0.0502232
\(163\) −0.132491 −0.0103775 −0.00518875 0.999987i \(-0.501652\pi\)
−0.00518875 + 0.999987i \(0.501652\pi\)
\(164\) −6.02929 −0.470809
\(165\) −6.71308 −0.522612
\(166\) 9.06727 0.703756
\(167\) 15.4476 1.19537 0.597685 0.801731i \(-0.296087\pi\)
0.597685 + 0.801731i \(0.296087\pi\)
\(168\) −6.30914 −0.486761
\(169\) −8.17321 −0.628708
\(170\) 0.979040 0.0750890
\(171\) 4.55900 0.348635
\(172\) −5.21963 −0.397993
\(173\) 23.8426 1.81272 0.906359 0.422508i \(-0.138850\pi\)
0.906359 + 0.422508i \(0.138850\pi\)
\(174\) 0.837548 0.0634943
\(175\) 7.29446 0.551409
\(176\) 7.51805 0.566694
\(177\) 14.2575 1.07166
\(178\) −5.38566 −0.403672
\(179\) 11.4316 0.854436 0.427218 0.904149i \(-0.359494\pi\)
0.427218 + 0.904149i \(0.359494\pi\)
\(180\) 2.43731 0.181667
\(181\) 8.05109 0.598433 0.299216 0.954185i \(-0.403275\pi\)
0.299216 + 0.954185i \(0.403275\pi\)
\(182\) 3.85957 0.286090
\(183\) −2.49345 −0.184321
\(184\) −16.8963 −1.24561
\(185\) −6.45360 −0.474478
\(186\) 3.73075 0.273552
\(187\) 4.38312 0.320525
\(188\) 8.86178 0.646312
\(189\) −2.74819 −0.199902
\(190\) 4.46344 0.323812
\(191\) 9.52470 0.689183 0.344592 0.938753i \(-0.388017\pi\)
0.344592 + 0.938753i \(0.388017\pi\)
\(192\) 0.205472 0.0148287
\(193\) 6.09563 0.438773 0.219387 0.975638i \(-0.429594\pi\)
0.219387 + 0.975638i \(0.429594\pi\)
\(194\) −1.21900 −0.0875195
\(195\) −3.36487 −0.240963
\(196\) −0.879339 −0.0628099
\(197\) 22.4157 1.59705 0.798527 0.601959i \(-0.205613\pi\)
0.798527 + 0.601959i \(0.205613\pi\)
\(198\) −2.80185 −0.199119
\(199\) −1.68609 −0.119524 −0.0597620 0.998213i \(-0.519034\pi\)
−0.0597620 + 0.998213i \(0.519034\pi\)
\(200\) −6.09353 −0.430877
\(201\) −0.208432 −0.0147016
\(202\) 8.10983 0.570605
\(203\) 3.60077 0.252724
\(204\) −1.59138 −0.111419
\(205\) −5.80273 −0.405280
\(206\) −12.0806 −0.841692
\(207\) −7.35985 −0.511545
\(208\) 3.76835 0.261288
\(209\) 19.9826 1.38223
\(210\) −2.69059 −0.185668
\(211\) −10.3572 −0.713022 −0.356511 0.934291i \(-0.616034\pi\)
−0.356511 + 0.934291i \(0.616034\pi\)
\(212\) 2.22826 0.153037
\(213\) 5.71350 0.391483
\(214\) 1.27828 0.0873814
\(215\) −5.02349 −0.342599
\(216\) 2.29574 0.156205
\(217\) 16.0391 1.08881
\(218\) −11.8595 −0.803230
\(219\) −1.55512 −0.105086
\(220\) 10.6830 0.720250
\(221\) 2.19700 0.147786
\(222\) −2.69355 −0.180779
\(223\) 2.90854 0.194771 0.0973853 0.995247i \(-0.468952\pi\)
0.0973853 + 0.995247i \(0.468952\pi\)
\(224\) 15.6315 1.04442
\(225\) −2.65427 −0.176952
\(226\) 1.54980 0.103091
\(227\) 11.1090 0.737332 0.368666 0.929562i \(-0.379815\pi\)
0.368666 + 0.929562i \(0.379815\pi\)
\(228\) −7.25508 −0.480479
\(229\) −10.3041 −0.680916 −0.340458 0.940260i \(-0.610582\pi\)
−0.340458 + 0.940260i \(0.610582\pi\)
\(230\) −7.20559 −0.475123
\(231\) −12.0457 −0.792546
\(232\) −3.00795 −0.197482
\(233\) 6.98827 0.457817 0.228908 0.973448i \(-0.426484\pi\)
0.228908 + 0.973448i \(0.426484\pi\)
\(234\) −1.40440 −0.0918086
\(235\) 8.52878 0.556356
\(236\) −22.6890 −1.47693
\(237\) 1.00000 0.0649570
\(238\) 1.75675 0.113873
\(239\) 15.5426 1.00537 0.502685 0.864470i \(-0.332346\pi\)
0.502685 + 0.864470i \(0.332346\pi\)
\(240\) −2.62700 −0.169572
\(241\) −6.67896 −0.430229 −0.215115 0.976589i \(-0.569013\pi\)
−0.215115 + 0.976589i \(0.569013\pi\)
\(242\) −5.24923 −0.337434
\(243\) 1.00000 0.0641500
\(244\) 3.96801 0.254026
\(245\) −0.846296 −0.0540678
\(246\) −2.42190 −0.154415
\(247\) 10.0161 0.637309
\(248\) −13.3985 −0.850806
\(249\) −14.1845 −0.898907
\(250\) −7.49384 −0.473952
\(251\) 28.7096 1.81213 0.906067 0.423135i \(-0.139070\pi\)
0.906067 + 0.423135i \(0.139070\pi\)
\(252\) 4.37341 0.275499
\(253\) −32.2591 −2.02811
\(254\) 2.47755 0.155455
\(255\) −1.53158 −0.0959110
\(256\) −7.59884 −0.474928
\(257\) −3.46899 −0.216389 −0.108195 0.994130i \(-0.534507\pi\)
−0.108195 + 0.994130i \(0.534507\pi\)
\(258\) −2.09666 −0.130533
\(259\) −11.5801 −0.719549
\(260\) 5.35477 0.332089
\(261\) −1.31023 −0.0811012
\(262\) 1.30657 0.0807203
\(263\) −26.4058 −1.62825 −0.814124 0.580690i \(-0.802783\pi\)
−0.814124 + 0.580690i \(0.802783\pi\)
\(264\) 10.0625 0.619304
\(265\) 2.14453 0.131737
\(266\) 8.00900 0.491064
\(267\) 8.42513 0.515610
\(268\) 0.331693 0.0202614
\(269\) 28.2967 1.72528 0.862639 0.505820i \(-0.168810\pi\)
0.862639 + 0.505820i \(0.168810\pi\)
\(270\) 0.979040 0.0595825
\(271\) −6.19456 −0.376292 −0.188146 0.982141i \(-0.560248\pi\)
−0.188146 + 0.982141i \(0.560248\pi\)
\(272\) 1.71523 0.104001
\(273\) −6.03777 −0.365422
\(274\) −7.64284 −0.461721
\(275\) −11.6340 −0.701556
\(276\) 11.7123 0.704997
\(277\) −8.05092 −0.483733 −0.241866 0.970310i \(-0.577760\pi\)
−0.241866 + 0.970310i \(0.577760\pi\)
\(278\) −10.1220 −0.607078
\(279\) −5.83625 −0.349407
\(280\) 9.66293 0.577470
\(281\) 20.6325 1.23083 0.615415 0.788203i \(-0.288988\pi\)
0.615415 + 0.788203i \(0.288988\pi\)
\(282\) 3.55967 0.211975
\(283\) 29.1318 1.73170 0.865852 0.500301i \(-0.166777\pi\)
0.865852 + 0.500301i \(0.166777\pi\)
\(284\) −9.09233 −0.539531
\(285\) −6.98245 −0.413605
\(286\) −6.15566 −0.363992
\(287\) −10.4122 −0.614611
\(288\) −5.68792 −0.335164
\(289\) 1.00000 0.0588235
\(290\) −1.28277 −0.0753268
\(291\) 1.90697 0.111788
\(292\) 2.47479 0.144826
\(293\) −13.6054 −0.794833 −0.397417 0.917638i \(-0.630093\pi\)
−0.397417 + 0.917638i \(0.630093\pi\)
\(294\) −0.353220 −0.0206002
\(295\) −21.8364 −1.27136
\(296\) 9.67356 0.562264
\(297\) 4.38312 0.254334
\(298\) 1.86924 0.108282
\(299\) −16.1696 −0.935110
\(300\) 4.22395 0.243870
\(301\) −9.01392 −0.519554
\(302\) −3.52466 −0.202822
\(303\) −12.6867 −0.728833
\(304\) 7.81972 0.448492
\(305\) 3.81890 0.218670
\(306\) −0.639237 −0.0365428
\(307\) −2.41280 −0.137706 −0.0688530 0.997627i \(-0.521934\pi\)
−0.0688530 + 0.997627i \(0.521934\pi\)
\(308\) 19.1692 1.09226
\(309\) 18.8984 1.07509
\(310\) −5.71392 −0.324529
\(311\) 9.07562 0.514631 0.257316 0.966327i \(-0.417162\pi\)
0.257316 + 0.966327i \(0.417162\pi\)
\(312\) 5.04373 0.285545
\(313\) −19.7211 −1.11470 −0.557351 0.830277i \(-0.688182\pi\)
−0.557351 + 0.830277i \(0.688182\pi\)
\(314\) 12.0754 0.681452
\(315\) 4.20907 0.237154
\(316\) −1.59138 −0.0895219
\(317\) 5.46386 0.306881 0.153441 0.988158i \(-0.450965\pi\)
0.153441 + 0.988158i \(0.450965\pi\)
\(318\) 0.895065 0.0501928
\(319\) −5.74289 −0.321540
\(320\) −0.314696 −0.0175921
\(321\) −1.99970 −0.111612
\(322\) −12.9294 −0.720527
\(323\) 4.55900 0.253669
\(324\) −1.59138 −0.0884098
\(325\) −5.83143 −0.323470
\(326\) 0.0846933 0.00469073
\(327\) 18.5527 1.02596
\(328\) 8.69794 0.480264
\(329\) 15.3037 0.843718
\(330\) 4.29125 0.236225
\(331\) −6.21216 −0.341451 −0.170726 0.985319i \(-0.554611\pi\)
−0.170726 + 0.985319i \(0.554611\pi\)
\(332\) 22.5729 1.23885
\(333\) 4.21370 0.230909
\(334\) −9.87467 −0.540318
\(335\) 0.319229 0.0174413
\(336\) −4.71378 −0.257158
\(337\) 28.9127 1.57498 0.787489 0.616329i \(-0.211381\pi\)
0.787489 + 0.616329i \(0.211381\pi\)
\(338\) 5.22462 0.284182
\(339\) −2.42445 −0.131678
\(340\) 2.43731 0.132182
\(341\) −25.5810 −1.38529
\(342\) −2.91428 −0.157586
\(343\) 17.7188 0.956725
\(344\) 7.52991 0.405985
\(345\) 11.2722 0.606873
\(346\) −15.2411 −0.819365
\(347\) −7.99135 −0.428998 −0.214499 0.976724i \(-0.568812\pi\)
−0.214499 + 0.976724i \(0.568812\pi\)
\(348\) 2.08507 0.111771
\(349\) −6.34237 −0.339499 −0.169750 0.985487i \(-0.554296\pi\)
−0.169750 + 0.985487i \(0.554296\pi\)
\(350\) −4.66289 −0.249242
\(351\) 2.19700 0.117267
\(352\) −24.9308 −1.32882
\(353\) −10.2365 −0.544836 −0.272418 0.962179i \(-0.587823\pi\)
−0.272418 + 0.962179i \(0.587823\pi\)
\(354\) −9.11389 −0.484398
\(355\) −8.75066 −0.464437
\(356\) −13.4076 −0.710599
\(357\) −2.74819 −0.145450
\(358\) −7.30749 −0.386213
\(359\) 21.7170 1.14618 0.573089 0.819493i \(-0.305745\pi\)
0.573089 + 0.819493i \(0.305745\pi\)
\(360\) −3.51610 −0.185315
\(361\) 1.78445 0.0939183
\(362\) −5.14656 −0.270497
\(363\) 8.21172 0.431003
\(364\) 9.60836 0.503615
\(365\) 2.38179 0.124669
\(366\) 1.59390 0.0833146
\(367\) −29.6223 −1.54627 −0.773135 0.634242i \(-0.781312\pi\)
−0.773135 + 0.634242i \(0.781312\pi\)
\(368\) −12.6238 −0.658063
\(369\) 3.78873 0.197233
\(370\) 4.12538 0.214468
\(371\) 3.84804 0.199780
\(372\) 9.28766 0.481543
\(373\) −27.0448 −1.40032 −0.700162 0.713984i \(-0.746889\pi\)
−0.700162 + 0.713984i \(0.746889\pi\)
\(374\) −2.80185 −0.144880
\(375\) 11.7231 0.605379
\(376\) −12.7841 −0.659291
\(377\) −2.87857 −0.148254
\(378\) 1.75675 0.0903573
\(379\) −3.69238 −0.189665 −0.0948325 0.995493i \(-0.530232\pi\)
−0.0948325 + 0.995493i \(0.530232\pi\)
\(380\) 11.1117 0.570018
\(381\) −3.87578 −0.198562
\(382\) −6.08854 −0.311517
\(383\) −38.5338 −1.96899 −0.984493 0.175425i \(-0.943870\pi\)
−0.984493 + 0.175425i \(0.943870\pi\)
\(384\) 11.2445 0.573818
\(385\) 18.4488 0.940240
\(386\) −3.89655 −0.198329
\(387\) 3.27995 0.166729
\(388\) −3.03470 −0.154064
\(389\) 14.4916 0.734755 0.367378 0.930072i \(-0.380256\pi\)
0.367378 + 0.930072i \(0.380256\pi\)
\(390\) 2.15095 0.108917
\(391\) −7.35985 −0.372204
\(392\) 1.26855 0.0640713
\(393\) −2.04395 −0.103104
\(394\) −14.3290 −0.721883
\(395\) −1.53158 −0.0770620
\(396\) −6.97519 −0.350516
\(397\) −8.04317 −0.403675 −0.201837 0.979419i \(-0.564691\pi\)
−0.201837 + 0.979419i \(0.564691\pi\)
\(398\) 1.07781 0.0540260
\(399\) −12.5290 −0.627235
\(400\) −4.55269 −0.227634
\(401\) 33.2791 1.66188 0.830939 0.556363i \(-0.187804\pi\)
0.830939 + 0.556363i \(0.187804\pi\)
\(402\) 0.133237 0.00664527
\(403\) −12.8222 −0.638720
\(404\) 20.1893 1.00446
\(405\) −1.53158 −0.0761046
\(406\) −2.30174 −0.114234
\(407\) 18.4691 0.915481
\(408\) 2.29574 0.113656
\(409\) 21.9818 1.08693 0.543466 0.839431i \(-0.317112\pi\)
0.543466 + 0.839431i \(0.317112\pi\)
\(410\) 3.70932 0.183190
\(411\) 11.9562 0.589755
\(412\) −30.0745 −1.48166
\(413\) −39.1822 −1.92803
\(414\) 4.70469 0.231223
\(415\) 21.7247 1.06642
\(416\) −12.4963 −0.612683
\(417\) 15.8345 0.775420
\(418\) −12.7736 −0.624779
\(419\) −0.856858 −0.0418602 −0.0209301 0.999781i \(-0.506663\pi\)
−0.0209301 + 0.999781i \(0.506663\pi\)
\(420\) −6.69821 −0.326839
\(421\) −30.9844 −1.51009 −0.755043 0.655675i \(-0.772384\pi\)
−0.755043 + 0.655675i \(0.772384\pi\)
\(422\) 6.62074 0.322292
\(423\) −5.56863 −0.270756
\(424\) −3.21451 −0.156111
\(425\) −2.65427 −0.128751
\(426\) −3.65228 −0.176954
\(427\) 6.85247 0.331614
\(428\) 3.18227 0.153821
\(429\) 9.62969 0.464926
\(430\) 3.21120 0.154858
\(431\) 32.3595 1.55870 0.779351 0.626587i \(-0.215549\pi\)
0.779351 + 0.626587i \(0.215549\pi\)
\(432\) 1.71523 0.0825240
\(433\) 2.15428 0.103528 0.0517640 0.998659i \(-0.483516\pi\)
0.0517640 + 0.998659i \(0.483516\pi\)
\(434\) −10.2528 −0.492151
\(435\) 2.00672 0.0962147
\(436\) −29.5242 −1.41395
\(437\) −33.5535 −1.60508
\(438\) 0.994093 0.0474996
\(439\) −31.0399 −1.48145 −0.740727 0.671806i \(-0.765519\pi\)
−0.740727 + 0.671806i \(0.765519\pi\)
\(440\) −15.4115 −0.734714
\(441\) 0.552565 0.0263126
\(442\) −1.40440 −0.0668006
\(443\) −12.8171 −0.608960 −0.304480 0.952519i \(-0.598483\pi\)
−0.304480 + 0.952519i \(0.598483\pi\)
\(444\) −6.70557 −0.318232
\(445\) −12.9037 −0.611696
\(446\) −1.85925 −0.0880380
\(447\) −2.92417 −0.138309
\(448\) −0.564677 −0.0266785
\(449\) 17.0630 0.805251 0.402626 0.915365i \(-0.368098\pi\)
0.402626 + 0.915365i \(0.368098\pi\)
\(450\) 1.69671 0.0799837
\(451\) 16.6064 0.781967
\(452\) 3.85821 0.181475
\(453\) 5.51386 0.259064
\(454\) −7.10130 −0.333281
\(455\) 9.24730 0.433520
\(456\) 10.4663 0.490128
\(457\) 21.2560 0.994313 0.497157 0.867661i \(-0.334377\pi\)
0.497157 + 0.867661i \(0.334377\pi\)
\(458\) 6.58678 0.307780
\(459\) 1.00000 0.0466760
\(460\) −17.9383 −0.836376
\(461\) −3.89040 −0.181194 −0.0905970 0.995888i \(-0.528878\pi\)
−0.0905970 + 0.995888i \(0.528878\pi\)
\(462\) 7.70003 0.358238
\(463\) 21.5731 1.00259 0.501294 0.865277i \(-0.332857\pi\)
0.501294 + 0.865277i \(0.332857\pi\)
\(464\) −2.24734 −0.104330
\(465\) 8.93866 0.414520
\(466\) −4.46716 −0.206937
\(467\) −8.01654 −0.370961 −0.185481 0.982648i \(-0.559384\pi\)
−0.185481 + 0.982648i \(0.559384\pi\)
\(468\) −3.49625 −0.161614
\(469\) 0.572810 0.0264499
\(470\) −5.45191 −0.251478
\(471\) −18.8903 −0.870417
\(472\) 32.7314 1.50659
\(473\) 14.3764 0.661027
\(474\) −0.639237 −0.0293611
\(475\) −12.1008 −0.555224
\(476\) 4.37341 0.200455
\(477\) −1.40021 −0.0641111
\(478\) −9.93544 −0.454436
\(479\) 12.0584 0.550963 0.275482 0.961306i \(-0.411163\pi\)
0.275482 + 0.961306i \(0.411163\pi\)
\(480\) 8.71148 0.397623
\(481\) 9.25747 0.422104
\(482\) 4.26944 0.194468
\(483\) 20.2263 0.920328
\(484\) −13.0679 −0.593997
\(485\) −2.92067 −0.132621
\(486\) −0.639237 −0.0289964
\(487\) −8.21823 −0.372404 −0.186202 0.982512i \(-0.559618\pi\)
−0.186202 + 0.982512i \(0.559618\pi\)
\(488\) −5.72430 −0.259127
\(489\) −0.132491 −0.00599146
\(490\) 0.540984 0.0244391
\(491\) −11.6895 −0.527539 −0.263770 0.964586i \(-0.584966\pi\)
−0.263770 + 0.964586i \(0.584966\pi\)
\(492\) −6.02929 −0.271822
\(493\) −1.31023 −0.0590098
\(494\) −6.40266 −0.288069
\(495\) −6.71308 −0.301730
\(496\) −10.0105 −0.449485
\(497\) −15.7018 −0.704322
\(498\) 9.06727 0.406314
\(499\) 21.1973 0.948920 0.474460 0.880277i \(-0.342643\pi\)
0.474460 + 0.880277i \(0.342643\pi\)
\(500\) −18.6559 −0.834316
\(501\) 15.4476 0.690147
\(502\) −18.3522 −0.819100
\(503\) 14.8819 0.663553 0.331776 0.943358i \(-0.392352\pi\)
0.331776 + 0.943358i \(0.392352\pi\)
\(504\) −6.30914 −0.281031
\(505\) 19.4307 0.864654
\(506\) 20.6212 0.916725
\(507\) −8.17321 −0.362985
\(508\) 6.16783 0.273653
\(509\) 25.5199 1.13115 0.565575 0.824697i \(-0.308654\pi\)
0.565575 + 0.824697i \(0.308654\pi\)
\(510\) 0.979040 0.0433526
\(511\) 4.27378 0.189061
\(512\) −17.6315 −0.779211
\(513\) 4.55900 0.201285
\(514\) 2.21751 0.0978100
\(515\) −28.9443 −1.27544
\(516\) −5.21963 −0.229781
\(517\) −24.4079 −1.07346
\(518\) 7.40240 0.325243
\(519\) 23.8426 1.04657
\(520\) −7.72486 −0.338758
\(521\) −18.9658 −0.830908 −0.415454 0.909614i \(-0.636377\pi\)
−0.415454 + 0.909614i \(0.636377\pi\)
\(522\) 0.837548 0.0366585
\(523\) −29.6697 −1.29736 −0.648682 0.761060i \(-0.724679\pi\)
−0.648682 + 0.761060i \(0.724679\pi\)
\(524\) 3.25270 0.142095
\(525\) 7.29446 0.318356
\(526\) 16.8795 0.735983
\(527\) −5.83625 −0.254231
\(528\) 7.51805 0.327181
\(529\) 31.1674 1.35510
\(530\) −1.37086 −0.0595464
\(531\) 14.2575 0.618721
\(532\) 19.9384 0.864437
\(533\) 8.32382 0.360545
\(534\) −5.38566 −0.233060
\(535\) 3.06269 0.132412
\(536\) −0.478505 −0.0206683
\(537\) 11.4316 0.493309
\(538\) −18.0883 −0.779841
\(539\) 2.42196 0.104321
\(540\) 2.43731 0.104885
\(541\) −11.5696 −0.497418 −0.248709 0.968578i \(-0.580006\pi\)
−0.248709 + 0.968578i \(0.580006\pi\)
\(542\) 3.95979 0.170088
\(543\) 8.05109 0.345505
\(544\) −5.68792 −0.243868
\(545\) −28.4148 −1.21716
\(546\) 3.85957 0.165174
\(547\) −0.0317545 −0.00135772 −0.000678862 1.00000i \(-0.500216\pi\)
−0.000678862 1.00000i \(0.500216\pi\)
\(548\) −19.0268 −0.812784
\(549\) −2.49345 −0.106418
\(550\) 7.43688 0.317110
\(551\) −5.97334 −0.254473
\(552\) −16.8963 −0.719155
\(553\) −2.74819 −0.116865
\(554\) 5.14645 0.218652
\(555\) −6.45360 −0.273940
\(556\) −25.1987 −1.06866
\(557\) −31.4076 −1.33078 −0.665391 0.746495i \(-0.731735\pi\)
−0.665391 + 0.746495i \(0.731735\pi\)
\(558\) 3.73075 0.157935
\(559\) 7.20603 0.304782
\(560\) 7.21951 0.305080
\(561\) 4.38312 0.185055
\(562\) −13.1890 −0.556346
\(563\) −29.5588 −1.24576 −0.622878 0.782319i \(-0.714037\pi\)
−0.622878 + 0.782319i \(0.714037\pi\)
\(564\) 8.86178 0.373148
\(565\) 3.71323 0.156217
\(566\) −18.6221 −0.782745
\(567\) −2.74819 −0.115413
\(568\) 13.1167 0.550365
\(569\) 27.3442 1.14633 0.573163 0.819441i \(-0.305716\pi\)
0.573163 + 0.819441i \(0.305716\pi\)
\(570\) 4.46344 0.186953
\(571\) −31.5959 −1.32225 −0.661124 0.750277i \(-0.729920\pi\)
−0.661124 + 0.750277i \(0.729920\pi\)
\(572\) −15.3245 −0.640748
\(573\) 9.52470 0.397900
\(574\) 6.65584 0.277809
\(575\) 19.5351 0.814668
\(576\) 0.205472 0.00856134
\(577\) 25.2670 1.05188 0.525940 0.850522i \(-0.323714\pi\)
0.525940 + 0.850522i \(0.323714\pi\)
\(578\) −0.639237 −0.0265888
\(579\) 6.09563 0.253326
\(580\) −3.19344 −0.132600
\(581\) 38.9818 1.61724
\(582\) −1.21900 −0.0505294
\(583\) −6.13728 −0.254180
\(584\) −3.57016 −0.147734
\(585\) −3.36487 −0.139120
\(586\) 8.69705 0.359272
\(587\) 44.1293 1.82141 0.910705 0.413057i \(-0.135539\pi\)
0.910705 + 0.413057i \(0.135539\pi\)
\(588\) −0.879339 −0.0362633
\(589\) −26.6074 −1.09634
\(590\) 13.9586 0.574667
\(591\) 22.4157 0.922060
\(592\) 7.22745 0.297047
\(593\) −0.859655 −0.0353018 −0.0176509 0.999844i \(-0.505619\pi\)
−0.0176509 + 0.999844i \(0.505619\pi\)
\(594\) −2.80185 −0.114961
\(595\) 4.20907 0.172555
\(596\) 4.65346 0.190613
\(597\) −1.68609 −0.0690073
\(598\) 10.3362 0.422678
\(599\) 8.77920 0.358708 0.179354 0.983785i \(-0.442599\pi\)
0.179354 + 0.983785i \(0.442599\pi\)
\(600\) −6.09353 −0.248767
\(601\) −28.3246 −1.15539 −0.577693 0.816254i \(-0.696047\pi\)
−0.577693 + 0.816254i \(0.696047\pi\)
\(602\) 5.76204 0.234843
\(603\) −0.208432 −0.00848799
\(604\) −8.77462 −0.357034
\(605\) −12.5769 −0.511322
\(606\) 8.10983 0.329439
\(607\) −0.208548 −0.00846471 −0.00423236 0.999991i \(-0.501347\pi\)
−0.00423236 + 0.999991i \(0.501347\pi\)
\(608\) −25.9312 −1.05165
\(609\) 3.60077 0.145910
\(610\) −2.44118 −0.0988406
\(611\) −12.2342 −0.494945
\(612\) −1.59138 −0.0643276
\(613\) −18.9206 −0.764197 −0.382098 0.924122i \(-0.624798\pi\)
−0.382098 + 0.924122i \(0.624798\pi\)
\(614\) 1.54235 0.0622443
\(615\) −5.80273 −0.233989
\(616\) −27.6537 −1.11420
\(617\) 1.74435 0.0702249 0.0351125 0.999383i \(-0.488821\pi\)
0.0351125 + 0.999383i \(0.488821\pi\)
\(618\) −12.0806 −0.485951
\(619\) 35.0971 1.41067 0.705335 0.708874i \(-0.250796\pi\)
0.705335 + 0.708874i \(0.250796\pi\)
\(620\) −14.2248 −0.571280
\(621\) −7.35985 −0.295341
\(622\) −5.80147 −0.232618
\(623\) −23.1539 −0.927641
\(624\) 3.76835 0.150855
\(625\) −4.68345 −0.187338
\(626\) 12.6065 0.503855
\(627\) 19.9826 0.798029
\(628\) 30.0615 1.19958
\(629\) 4.21370 0.168011
\(630\) −2.69059 −0.107196
\(631\) 13.6340 0.542762 0.271381 0.962472i \(-0.412520\pi\)
0.271381 + 0.962472i \(0.412520\pi\)
\(632\) 2.29574 0.0913197
\(633\) −10.3572 −0.411664
\(634\) −3.49270 −0.138713
\(635\) 5.93606 0.235565
\(636\) 2.22826 0.0883561
\(637\) 1.21398 0.0480998
\(638\) 3.67107 0.145339
\(639\) 5.71350 0.226023
\(640\) −17.2218 −0.680751
\(641\) 38.0087 1.50125 0.750627 0.660726i \(-0.229751\pi\)
0.750627 + 0.660726i \(0.229751\pi\)
\(642\) 1.27828 0.0504497
\(643\) 34.8705 1.37516 0.687579 0.726110i \(-0.258673\pi\)
0.687579 + 0.726110i \(0.258673\pi\)
\(644\) −32.1876 −1.26837
\(645\) −5.02349 −0.197800
\(646\) −2.91428 −0.114661
\(647\) 8.31044 0.326717 0.163359 0.986567i \(-0.447767\pi\)
0.163359 + 0.986567i \(0.447767\pi\)
\(648\) 2.29574 0.0901852
\(649\) 62.4921 2.45303
\(650\) 3.72767 0.146211
\(651\) 16.0391 0.628623
\(652\) 0.210843 0.00825726
\(653\) 34.1786 1.33751 0.668757 0.743481i \(-0.266827\pi\)
0.668757 + 0.743481i \(0.266827\pi\)
\(654\) −11.8595 −0.463745
\(655\) 3.13047 0.122318
\(656\) 6.49854 0.253725
\(657\) −1.55512 −0.0606712
\(658\) −9.78267 −0.381368
\(659\) −35.6527 −1.38883 −0.694416 0.719574i \(-0.744337\pi\)
−0.694416 + 0.719574i \(0.744337\pi\)
\(660\) 10.6830 0.415836
\(661\) 15.7411 0.612256 0.306128 0.951990i \(-0.400966\pi\)
0.306128 + 0.951990i \(0.400966\pi\)
\(662\) 3.97104 0.154339
\(663\) 2.19700 0.0853242
\(664\) −32.5640 −1.26373
\(665\) 19.1891 0.744122
\(666\) −2.69355 −0.104373
\(667\) 9.64310 0.373382
\(668\) −24.5829 −0.951141
\(669\) 2.90854 0.112451
\(670\) −0.204063 −0.00788364
\(671\) −10.9291 −0.421912
\(672\) 15.6315 0.602998
\(673\) 44.9580 1.73301 0.866503 0.499173i \(-0.166363\pi\)
0.866503 + 0.499173i \(0.166363\pi\)
\(674\) −18.4821 −0.711904
\(675\) −2.65427 −0.102163
\(676\) 13.0066 0.500256
\(677\) 11.5958 0.445662 0.222831 0.974857i \(-0.428470\pi\)
0.222831 + 0.974857i \(0.428470\pi\)
\(678\) 1.54980 0.0595197
\(679\) −5.24072 −0.201120
\(680\) −3.51610 −0.134836
\(681\) 11.1090 0.425699
\(682\) 16.3523 0.626162
\(683\) 12.6798 0.485177 0.242589 0.970129i \(-0.422003\pi\)
0.242589 + 0.970129i \(0.422003\pi\)
\(684\) −7.25508 −0.277405
\(685\) −18.3118 −0.699658
\(686\) −11.3265 −0.432448
\(687\) −10.3041 −0.393127
\(688\) 5.62586 0.214484
\(689\) −3.07625 −0.117196
\(690\) −7.20559 −0.274312
\(691\) −3.53616 −0.134522 −0.0672609 0.997735i \(-0.521426\pi\)
−0.0672609 + 0.997735i \(0.521426\pi\)
\(692\) −37.9425 −1.44236
\(693\) −12.0457 −0.457576
\(694\) 5.10837 0.193911
\(695\) −24.2518 −0.919922
\(696\) −3.00795 −0.114016
\(697\) 3.78873 0.143508
\(698\) 4.05428 0.153457
\(699\) 6.98827 0.264321
\(700\) −11.6082 −0.438750
\(701\) −0.126135 −0.00476404 −0.00238202 0.999997i \(-0.500758\pi\)
−0.00238202 + 0.999997i \(0.500758\pi\)
\(702\) −1.40440 −0.0530057
\(703\) 19.2102 0.724527
\(704\) 0.900609 0.0339430
\(705\) 8.52878 0.321212
\(706\) 6.54358 0.246271
\(707\) 34.8656 1.31125
\(708\) −22.6890 −0.852703
\(709\) −10.2108 −0.383474 −0.191737 0.981446i \(-0.561412\pi\)
−0.191737 + 0.981446i \(0.561412\pi\)
\(710\) 5.59375 0.209930
\(711\) 1.00000 0.0375029
\(712\) 19.3419 0.724869
\(713\) 42.9539 1.60864
\(714\) 1.75675 0.0657446
\(715\) −14.7486 −0.551566
\(716\) −18.1919 −0.679864
\(717\) 15.5426 0.580451
\(718\) −13.8823 −0.518083
\(719\) 43.3429 1.61642 0.808209 0.588896i \(-0.200437\pi\)
0.808209 + 0.588896i \(0.200437\pi\)
\(720\) −2.62700 −0.0979026
\(721\) −51.9365 −1.93421
\(722\) −1.14069 −0.0424519
\(723\) −6.67896 −0.248393
\(724\) −12.8123 −0.476166
\(725\) 3.47771 0.129159
\(726\) −5.24923 −0.194817
\(727\) 43.0937 1.59826 0.799129 0.601160i \(-0.205295\pi\)
0.799129 + 0.601160i \(0.205295\pi\)
\(728\) −13.8612 −0.513728
\(729\) 1.00000 0.0370370
\(730\) −1.52253 −0.0563513
\(731\) 3.27995 0.121313
\(732\) 3.96801 0.146662
\(733\) −26.2300 −0.968826 −0.484413 0.874840i \(-0.660967\pi\)
−0.484413 + 0.874840i \(0.660967\pi\)
\(734\) 18.9356 0.698927
\(735\) −0.846296 −0.0312161
\(736\) 41.8622 1.54306
\(737\) −0.913580 −0.0336522
\(738\) −2.42190 −0.0891513
\(739\) −13.9075 −0.511595 −0.255797 0.966730i \(-0.582338\pi\)
−0.255797 + 0.966730i \(0.582338\pi\)
\(740\) 10.2701 0.377536
\(741\) 10.0161 0.367950
\(742\) −2.45981 −0.0903025
\(743\) 3.68086 0.135038 0.0675189 0.997718i \(-0.478492\pi\)
0.0675189 + 0.997718i \(0.478492\pi\)
\(744\) −13.3985 −0.491213
\(745\) 4.47859 0.164083
\(746\) 17.2880 0.632959
\(747\) −14.1845 −0.518984
\(748\) −6.97519 −0.255038
\(749\) 5.49555 0.200803
\(750\) −7.49384 −0.273636
\(751\) −9.88190 −0.360596 −0.180298 0.983612i \(-0.557706\pi\)
−0.180298 + 0.983612i \(0.557706\pi\)
\(752\) −9.55147 −0.348306
\(753\) 28.7096 1.04624
\(754\) 1.84009 0.0670121
\(755\) −8.44490 −0.307341
\(756\) 4.37341 0.159059
\(757\) −8.05261 −0.292677 −0.146339 0.989235i \(-0.546749\pi\)
−0.146339 + 0.989235i \(0.546749\pi\)
\(758\) 2.36031 0.0857303
\(759\) −32.2591 −1.17093
\(760\) −16.0299 −0.581465
\(761\) −33.5101 −1.21474 −0.607369 0.794420i \(-0.707775\pi\)
−0.607369 + 0.794420i \(0.707775\pi\)
\(762\) 2.47755 0.0897520
\(763\) −50.9863 −1.84583
\(764\) −15.1574 −0.548375
\(765\) −1.53158 −0.0553743
\(766\) 24.6322 0.889999
\(767\) 31.3236 1.13103
\(768\) −7.59884 −0.274200
\(769\) −3.96696 −0.143052 −0.0715261 0.997439i \(-0.522787\pi\)
−0.0715261 + 0.997439i \(0.522787\pi\)
\(770\) −11.7932 −0.424997
\(771\) −3.46899 −0.124933
\(772\) −9.70044 −0.349127
\(773\) 7.08867 0.254962 0.127481 0.991841i \(-0.459311\pi\)
0.127481 + 0.991841i \(0.459311\pi\)
\(774\) −2.09666 −0.0753630
\(775\) 15.4910 0.556453
\(776\) 4.37790 0.157158
\(777\) −11.5801 −0.415432
\(778\) −9.26359 −0.332116
\(779\) 17.2728 0.618863
\(780\) 5.35477 0.191731
\(781\) 25.0430 0.896107
\(782\) 4.70469 0.168239
\(783\) −1.31023 −0.0468238
\(784\) 0.947776 0.0338491
\(785\) 28.9319 1.03262
\(786\) 1.30657 0.0466039
\(787\) −10.1096 −0.360369 −0.180185 0.983633i \(-0.557670\pi\)
−0.180185 + 0.983633i \(0.557670\pi\)
\(788\) −35.6718 −1.27076
\(789\) −26.4058 −0.940070
\(790\) 0.979040 0.0348327
\(791\) 6.66286 0.236904
\(792\) 10.0625 0.357555
\(793\) −5.47809 −0.194533
\(794\) 5.14149 0.182465
\(795\) 2.14453 0.0760585
\(796\) 2.68321 0.0951039
\(797\) −13.3690 −0.473554 −0.236777 0.971564i \(-0.576091\pi\)
−0.236777 + 0.971564i \(0.576091\pi\)
\(798\) 8.00900 0.283516
\(799\) −5.56863 −0.197004
\(800\) 15.0973 0.533770
\(801\) 8.42513 0.297687
\(802\) −21.2732 −0.751184
\(803\) −6.81629 −0.240542
\(804\) 0.331693 0.0116979
\(805\) −30.9781 −1.09183
\(806\) 8.19643 0.288707
\(807\) 28.2967 0.996090
\(808\) −29.1254 −1.02463
\(809\) −23.4715 −0.825213 −0.412607 0.910909i \(-0.635382\pi\)
−0.412607 + 0.910909i \(0.635382\pi\)
\(810\) 0.979040 0.0344000
\(811\) −12.6488 −0.444161 −0.222081 0.975028i \(-0.571285\pi\)
−0.222081 + 0.975028i \(0.571285\pi\)
\(812\) −5.73017 −0.201090
\(813\) −6.19456 −0.217253
\(814\) −11.8062 −0.413805
\(815\) 0.202920 0.00710799
\(816\) 1.71523 0.0600450
\(817\) 14.9533 0.523148
\(818\) −14.0516 −0.491303
\(819\) −6.03777 −0.210977
\(820\) 9.23432 0.322477
\(821\) −40.6010 −1.41699 −0.708493 0.705718i \(-0.750624\pi\)
−0.708493 + 0.705718i \(0.750624\pi\)
\(822\) −7.64284 −0.266575
\(823\) 10.5219 0.366769 0.183385 0.983041i \(-0.441295\pi\)
0.183385 + 0.983041i \(0.441295\pi\)
\(824\) 43.3858 1.51142
\(825\) −11.6340 −0.405044
\(826\) 25.0467 0.871487
\(827\) 48.7409 1.69489 0.847444 0.530885i \(-0.178140\pi\)
0.847444 + 0.530885i \(0.178140\pi\)
\(828\) 11.7123 0.407030
\(829\) −30.0911 −1.04511 −0.522554 0.852606i \(-0.675021\pi\)
−0.522554 + 0.852606i \(0.675021\pi\)
\(830\) −13.8872 −0.482032
\(831\) −8.05092 −0.279283
\(832\) 0.451422 0.0156502
\(833\) 0.552565 0.0191452
\(834\) −10.1220 −0.350497
\(835\) −23.6591 −0.818758
\(836\) −31.7999 −1.09982
\(837\) −5.83625 −0.201730
\(838\) 0.547735 0.0189212
\(839\) −26.6346 −0.919529 −0.459765 0.888041i \(-0.652066\pi\)
−0.459765 + 0.888041i \(0.652066\pi\)
\(840\) 9.66293 0.333403
\(841\) −27.2833 −0.940803
\(842\) 19.8064 0.682572
\(843\) 20.6325 0.710620
\(844\) 16.4823 0.567343
\(845\) 12.5179 0.430629
\(846\) 3.55967 0.122384
\(847\) −22.5674 −0.775424
\(848\) −2.40168 −0.0824739
\(849\) 29.1318 0.999800
\(850\) 1.69671 0.0581967
\(851\) −31.0122 −1.06308
\(852\) −9.09233 −0.311498
\(853\) 8.27726 0.283408 0.141704 0.989909i \(-0.454742\pi\)
0.141704 + 0.989909i \(0.454742\pi\)
\(854\) −4.38035 −0.149893
\(855\) −6.98245 −0.238795
\(856\) −4.59078 −0.156910
\(857\) −3.76957 −0.128766 −0.0643831 0.997925i \(-0.520508\pi\)
−0.0643831 + 0.997925i \(0.520508\pi\)
\(858\) −6.15566 −0.210151
\(859\) 17.1490 0.585115 0.292557 0.956248i \(-0.405494\pi\)
0.292557 + 0.956248i \(0.405494\pi\)
\(860\) 7.99426 0.272602
\(861\) −10.4122 −0.354846
\(862\) −20.6854 −0.704547
\(863\) −18.0469 −0.614323 −0.307161 0.951657i \(-0.599379\pi\)
−0.307161 + 0.951657i \(0.599379\pi\)
\(864\) −5.68792 −0.193507
\(865\) −36.5167 −1.24161
\(866\) −1.37710 −0.0467956
\(867\) 1.00000 0.0339618
\(868\) −25.5243 −0.866351
\(869\) 4.38312 0.148687
\(870\) −1.28277 −0.0434899
\(871\) −0.457923 −0.0155161
\(872\) 42.5921 1.44235
\(873\) 1.90697 0.0645411
\(874\) 21.4487 0.725512
\(875\) −32.2174 −1.08915
\(876\) 2.47479 0.0836153
\(877\) 43.5591 1.47089 0.735443 0.677586i \(-0.236974\pi\)
0.735443 + 0.677586i \(0.236974\pi\)
\(878\) 19.8419 0.669631
\(879\) −13.6054 −0.458897
\(880\) −11.5145 −0.388153
\(881\) 1.71058 0.0576308 0.0288154 0.999585i \(-0.490826\pi\)
0.0288154 + 0.999585i \(0.490826\pi\)
\(882\) −0.353220 −0.0118935
\(883\) 13.0727 0.439931 0.219966 0.975508i \(-0.429405\pi\)
0.219966 + 0.975508i \(0.429405\pi\)
\(884\) −3.49625 −0.117591
\(885\) −21.8364 −0.734022
\(886\) 8.19318 0.275255
\(887\) −31.7780 −1.06700 −0.533501 0.845800i \(-0.679124\pi\)
−0.533501 + 0.845800i \(0.679124\pi\)
\(888\) 9.67356 0.324623
\(889\) 10.6514 0.357237
\(890\) 8.24854 0.276492
\(891\) 4.38312 0.146840
\(892\) −4.62859 −0.154977
\(893\) −25.3873 −0.849555
\(894\) 1.86924 0.0625167
\(895\) −17.5083 −0.585239
\(896\) −30.9020 −1.03236
\(897\) −16.1696 −0.539886
\(898\) −10.9073 −0.363981
\(899\) 7.64683 0.255036
\(900\) 4.22395 0.140798
\(901\) −1.40021 −0.0466477
\(902\) −10.6155 −0.353456
\(903\) −9.01392 −0.299965
\(904\) −5.56591 −0.185119
\(905\) −12.3309 −0.409892
\(906\) −3.52466 −0.117099
\(907\) 20.7824 0.690069 0.345035 0.938590i \(-0.387867\pi\)
0.345035 + 0.938590i \(0.387867\pi\)
\(908\) −17.6786 −0.586686
\(909\) −12.6867 −0.420792
\(910\) −5.91122 −0.195955
\(911\) 13.0279 0.431633 0.215816 0.976434i \(-0.430759\pi\)
0.215816 + 0.976434i \(0.430759\pi\)
\(912\) 7.81972 0.258937
\(913\) −62.1724 −2.05761
\(914\) −13.5876 −0.449438
\(915\) 3.81890 0.126249
\(916\) 16.3977 0.541796
\(917\) 5.61718 0.185496
\(918\) −0.639237 −0.0210980
\(919\) −20.8694 −0.688417 −0.344208 0.938893i \(-0.611853\pi\)
−0.344208 + 0.938893i \(0.611853\pi\)
\(920\) 25.8780 0.853172
\(921\) −2.41280 −0.0795046
\(922\) 2.48689 0.0819013
\(923\) 12.5525 0.413172
\(924\) 19.1692 0.630619
\(925\) −11.1843 −0.367738
\(926\) −13.7903 −0.453179
\(927\) 18.8984 0.620705
\(928\) 7.45248 0.244640
\(929\) −32.2722 −1.05882 −0.529408 0.848367i \(-0.677586\pi\)
−0.529408 + 0.848367i \(0.677586\pi\)
\(930\) −5.71392 −0.187367
\(931\) 2.51914 0.0825616
\(932\) −11.1210 −0.364279
\(933\) 9.07562 0.297123
\(934\) 5.12447 0.167678
\(935\) −6.71308 −0.219541
\(936\) 5.04373 0.164860
\(937\) −9.16845 −0.299520 −0.149760 0.988722i \(-0.547850\pi\)
−0.149760 + 0.988722i \(0.547850\pi\)
\(938\) −0.366162 −0.0119556
\(939\) −19.7211 −0.643574
\(940\) −13.5725 −0.442686
\(941\) −34.3260 −1.11900 −0.559498 0.828832i \(-0.689006\pi\)
−0.559498 + 0.828832i \(0.689006\pi\)
\(942\) 12.0754 0.393436
\(943\) −27.8845 −0.908044
\(944\) 24.4548 0.795936
\(945\) 4.20907 0.136921
\(946\) −9.18992 −0.298790
\(947\) 53.4886 1.73815 0.869074 0.494683i \(-0.164716\pi\)
0.869074 + 0.494683i \(0.164716\pi\)
\(948\) −1.59138 −0.0516855
\(949\) −3.41660 −0.110908
\(950\) 7.73530 0.250966
\(951\) 5.46386 0.177178
\(952\) −6.30914 −0.204480
\(953\) −31.6375 −1.02484 −0.512420 0.858735i \(-0.671251\pi\)
−0.512420 + 0.858735i \(0.671251\pi\)
\(954\) 0.895065 0.0289788
\(955\) −14.5878 −0.472050
\(956\) −24.7342 −0.799961
\(957\) −5.74289 −0.185641
\(958\) −7.70819 −0.249040
\(959\) −32.8579 −1.06104
\(960\) −0.314696 −0.0101568
\(961\) 3.06178 0.0987670
\(962\) −5.91772 −0.190795
\(963\) −1.99970 −0.0644393
\(964\) 10.6287 0.342328
\(965\) −9.33593 −0.300534
\(966\) −12.9294 −0.415997
\(967\) 47.3885 1.52391 0.761955 0.647630i \(-0.224240\pi\)
0.761955 + 0.647630i \(0.224240\pi\)
\(968\) 18.8520 0.605925
\(969\) 4.55900 0.146456
\(970\) 1.86700 0.0599457
\(971\) 5.60608 0.179908 0.0899538 0.995946i \(-0.471328\pi\)
0.0899538 + 0.995946i \(0.471328\pi\)
\(972\) −1.59138 −0.0510434
\(973\) −43.5163 −1.39507
\(974\) 5.25340 0.168330
\(975\) −5.83143 −0.186755
\(976\) −4.27683 −0.136898
\(977\) −28.3396 −0.906663 −0.453332 0.891342i \(-0.649765\pi\)
−0.453332 + 0.891342i \(0.649765\pi\)
\(978\) 0.0846933 0.00270819
\(979\) 36.9283 1.18023
\(980\) 1.34677 0.0430211
\(981\) 18.5527 0.592341
\(982\) 7.47236 0.238453
\(983\) −39.7618 −1.26821 −0.634103 0.773249i \(-0.718630\pi\)
−0.634103 + 0.773249i \(0.718630\pi\)
\(984\) 8.69794 0.277280
\(985\) −34.3314 −1.09389
\(986\) 0.837548 0.0266730
\(987\) 15.3037 0.487121
\(988\) −15.9394 −0.507099
\(989\) −24.1399 −0.767604
\(990\) 4.29125 0.136385
\(991\) 10.0089 0.317943 0.158971 0.987283i \(-0.449182\pi\)
0.158971 + 0.987283i \(0.449182\pi\)
\(992\) 33.1961 1.05398
\(993\) −6.21216 −0.197137
\(994\) 10.0372 0.318360
\(995\) 2.58238 0.0818670
\(996\) 22.5729 0.715249
\(997\) −3.49583 −0.110714 −0.0553571 0.998467i \(-0.517630\pi\)
−0.0553571 + 0.998467i \(0.517630\pi\)
\(998\) −13.5501 −0.428920
\(999\) 4.21370 0.133315
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 4029.2.a.k.1.12 31
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4029.2.a.k.1.12 31 1.1 even 1 trivial