Properties

Label 4029.2.a.l.1.18
Level 4029
Weight 2
Character 4029.1
Self dual yes
Analytic conductor 32.172
Analytic rank 0
Dimension 32
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: \(32\)
Coefficient ring index: multiple of None
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.18
Character \(\chi\) = 4029.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.403182 q^{2} -1.00000 q^{3} -1.83744 q^{4} +2.23661 q^{5} -0.403182 q^{6} -0.778796 q^{7} -1.54719 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.403182 q^{2} -1.00000 q^{3} -1.83744 q^{4} +2.23661 q^{5} -0.403182 q^{6} -0.778796 q^{7} -1.54719 q^{8} +1.00000 q^{9} +0.901761 q^{10} +0.498396 q^{11} +1.83744 q^{12} -0.674164 q^{13} -0.313997 q^{14} -2.23661 q^{15} +3.05109 q^{16} -1.00000 q^{17} +0.403182 q^{18} -7.73513 q^{19} -4.10965 q^{20} +0.778796 q^{21} +0.200944 q^{22} +2.53354 q^{23} +1.54719 q^{24} +0.00242448 q^{25} -0.271811 q^{26} -1.00000 q^{27} +1.43099 q^{28} -1.10688 q^{29} -0.901761 q^{30} +8.74229 q^{31} +4.32452 q^{32} -0.498396 q^{33} -0.403182 q^{34} -1.74186 q^{35} -1.83744 q^{36} +2.84423 q^{37} -3.11867 q^{38} +0.674164 q^{39} -3.46046 q^{40} +5.58140 q^{41} +0.313997 q^{42} -1.12704 q^{43} -0.915774 q^{44} +2.23661 q^{45} +1.02148 q^{46} +10.1996 q^{47} -3.05109 q^{48} -6.39348 q^{49} +0.000977508 q^{50} +1.00000 q^{51} +1.23874 q^{52} -1.21499 q^{53} -0.403182 q^{54} +1.11472 q^{55} +1.20494 q^{56} +7.73513 q^{57} -0.446275 q^{58} -12.2640 q^{59} +4.10965 q^{60} -8.15571 q^{61} +3.52473 q^{62} -0.778796 q^{63} -4.35861 q^{64} -1.50784 q^{65} -0.200944 q^{66} +1.40853 q^{67} +1.83744 q^{68} -2.53354 q^{69} -0.702288 q^{70} +10.1181 q^{71} -1.54719 q^{72} +8.98549 q^{73} +1.14674 q^{74} -0.00242448 q^{75} +14.2129 q^{76} -0.388148 q^{77} +0.271811 q^{78} +1.00000 q^{79} +6.82410 q^{80} +1.00000 q^{81} +2.25032 q^{82} +12.9061 q^{83} -1.43099 q^{84} -2.23661 q^{85} -0.454401 q^{86} +1.10688 q^{87} -0.771112 q^{88} -7.67819 q^{89} +0.901761 q^{90} +0.525036 q^{91} -4.65524 q^{92} -8.74229 q^{93} +4.11232 q^{94} -17.3005 q^{95} -4.32452 q^{96} -3.48632 q^{97} -2.57774 q^{98} +0.498396 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32q - q^{2} - 32q^{3} + 41q^{4} - q^{5} + q^{6} + 4q^{7} - 3q^{8} + 32q^{9} + O(q^{10}) \) \( 32q - q^{2} - 32q^{3} + 41q^{4} - q^{5} + q^{6} + 4q^{7} - 3q^{8} + 32q^{9} + 17q^{10} + 8q^{11} - 41q^{12} + 17q^{13} + q^{14} + q^{15} + 55q^{16} - 32q^{17} - q^{18} + 48q^{19} - 7q^{20} - 4q^{21} - 4q^{22} - 19q^{23} + 3q^{24} + 63q^{25} + 27q^{26} - 32q^{27} + 17q^{28} - 15q^{29} - 17q^{30} + 20q^{31} + 13q^{32} - 8q^{33} + q^{34} + 22q^{35} + 41q^{36} + 6q^{37} + 11q^{38} - 17q^{39} + 47q^{40} + q^{41} - q^{42} + 40q^{43} + 22q^{44} - q^{45} + 5q^{46} - 5q^{47} - 55q^{48} + 88q^{49} + 17q^{50} + 32q^{51} + 23q^{52} - 34q^{53} + q^{54} + 48q^{55} - 48q^{57} - 9q^{58} + 41q^{59} + 7q^{60} + 20q^{61} + 15q^{62} + 4q^{63} + 93q^{64} - 58q^{65} + 4q^{66} + 52q^{67} - 41q^{68} + 19q^{69} + 25q^{70} + q^{71} - 3q^{72} + 19q^{73} + 12q^{74} - 63q^{75} + 128q^{76} - 20q^{77} - 27q^{78} + 32q^{79} - 16q^{80} + 32q^{81} - 5q^{82} + 31q^{83} - 17q^{84} + q^{85} - 62q^{86} + 15q^{87} + 35q^{88} + 18q^{89} + 17q^{90} + 48q^{91} - 75q^{92} - 20q^{93} + 29q^{94} + 5q^{95} - 13q^{96} + 17q^{97} + 30q^{98} + 8q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.403182 0.285093 0.142546 0.989788i \(-0.454471\pi\)
0.142546 + 0.989788i \(0.454471\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.83744 −0.918722
\(5\) 2.23661 1.00024 0.500121 0.865955i \(-0.333289\pi\)
0.500121 + 0.865955i \(0.333289\pi\)
\(6\) −0.403182 −0.164598
\(7\) −0.778796 −0.294357 −0.147179 0.989110i \(-0.547019\pi\)
−0.147179 + 0.989110i \(0.547019\pi\)
\(8\) −1.54719 −0.547014
\(9\) 1.00000 0.333333
\(10\) 0.901761 0.285162
\(11\) 0.498396 0.150272 0.0751360 0.997173i \(-0.476061\pi\)
0.0751360 + 0.997173i \(0.476061\pi\)
\(12\) 1.83744 0.530424
\(13\) −0.674164 −0.186980 −0.0934898 0.995620i \(-0.529802\pi\)
−0.0934898 + 0.995620i \(0.529802\pi\)
\(14\) −0.313997 −0.0839191
\(15\) −2.23661 −0.577490
\(16\) 3.05109 0.762772
\(17\) −1.00000 −0.242536
\(18\) 0.403182 0.0950310
\(19\) −7.73513 −1.77456 −0.887280 0.461231i \(-0.847408\pi\)
−0.887280 + 0.461231i \(0.847408\pi\)
\(20\) −4.10965 −0.918945
\(21\) 0.778796 0.169947
\(22\) 0.200944 0.0428415
\(23\) 2.53354 0.528280 0.264140 0.964484i \(-0.414912\pi\)
0.264140 + 0.964484i \(0.414912\pi\)
\(24\) 1.54719 0.315819
\(25\) 0.00242448 0.000484897 0
\(26\) −0.271811 −0.0533065
\(27\) −1.00000 −0.192450
\(28\) 1.43099 0.270432
\(29\) −1.10688 −0.205543 −0.102771 0.994705i \(-0.532771\pi\)
−0.102771 + 0.994705i \(0.532771\pi\)
\(30\) −0.901761 −0.164638
\(31\) 8.74229 1.57016 0.785081 0.619394i \(-0.212622\pi\)
0.785081 + 0.619394i \(0.212622\pi\)
\(32\) 4.32452 0.764475
\(33\) −0.498396 −0.0867595
\(34\) −0.403182 −0.0691452
\(35\) −1.74186 −0.294428
\(36\) −1.83744 −0.306241
\(37\) 2.84423 0.467588 0.233794 0.972286i \(-0.424886\pi\)
0.233794 + 0.972286i \(0.424886\pi\)
\(38\) −3.11867 −0.505915
\(39\) 0.674164 0.107953
\(40\) −3.46046 −0.547147
\(41\) 5.58140 0.871668 0.435834 0.900027i \(-0.356453\pi\)
0.435834 + 0.900027i \(0.356453\pi\)
\(42\) 0.313997 0.0484507
\(43\) −1.12704 −0.171871 −0.0859356 0.996301i \(-0.527388\pi\)
−0.0859356 + 0.996301i \(0.527388\pi\)
\(44\) −0.915774 −0.138058
\(45\) 2.23661 0.333414
\(46\) 1.02148 0.150609
\(47\) 10.1996 1.48777 0.743886 0.668307i \(-0.232981\pi\)
0.743886 + 0.668307i \(0.232981\pi\)
\(48\) −3.05109 −0.440387
\(49\) −6.39348 −0.913354
\(50\) 0.000977508 0 0.000138241 0
\(51\) 1.00000 0.140028
\(52\) 1.23874 0.171782
\(53\) −1.21499 −0.166892 −0.0834458 0.996512i \(-0.526593\pi\)
−0.0834458 + 0.996512i \(0.526593\pi\)
\(54\) −0.403182 −0.0548661
\(55\) 1.11472 0.150308
\(56\) 1.20494 0.161017
\(57\) 7.73513 1.02454
\(58\) −0.446275 −0.0585987
\(59\) −12.2640 −1.59664 −0.798319 0.602235i \(-0.794277\pi\)
−0.798319 + 0.602235i \(0.794277\pi\)
\(60\) 4.10965 0.530553
\(61\) −8.15571 −1.04423 −0.522116 0.852875i \(-0.674857\pi\)
−0.522116 + 0.852875i \(0.674857\pi\)
\(62\) 3.52473 0.447642
\(63\) −0.778796 −0.0981190
\(64\) −4.35861 −0.544826
\(65\) −1.50784 −0.187025
\(66\) −0.200944 −0.0247345
\(67\) 1.40853 0.172079 0.0860394 0.996292i \(-0.472579\pi\)
0.0860394 + 0.996292i \(0.472579\pi\)
\(68\) 1.83744 0.222823
\(69\) −2.53354 −0.305003
\(70\) −0.702288 −0.0839394
\(71\) 10.1181 1.20079 0.600397 0.799702i \(-0.295009\pi\)
0.600397 + 0.799702i \(0.295009\pi\)
\(72\) −1.54719 −0.182338
\(73\) 8.98549 1.05167 0.525836 0.850586i \(-0.323752\pi\)
0.525836 + 0.850586i \(0.323752\pi\)
\(74\) 1.14674 0.133306
\(75\) −0.00242448 −0.000279955 0
\(76\) 14.2129 1.63033
\(77\) −0.388148 −0.0442336
\(78\) 0.271811 0.0307765
\(79\) 1.00000 0.112509
\(80\) 6.82410 0.762957
\(81\) 1.00000 0.111111
\(82\) 2.25032 0.248506
\(83\) 12.9061 1.41663 0.708316 0.705896i \(-0.249455\pi\)
0.708316 + 0.705896i \(0.249455\pi\)
\(84\) −1.43099 −0.156134
\(85\) −2.23661 −0.242594
\(86\) −0.454401 −0.0489993
\(87\) 1.10688 0.118670
\(88\) −0.771112 −0.0822009
\(89\) −7.67819 −0.813886 −0.406943 0.913453i \(-0.633405\pi\)
−0.406943 + 0.913453i \(0.633405\pi\)
\(90\) 0.901761 0.0950540
\(91\) 0.525036 0.0550388
\(92\) −4.65524 −0.485343
\(93\) −8.74229 −0.906533
\(94\) 4.11232 0.424153
\(95\) −17.3005 −1.77499
\(96\) −4.32452 −0.441370
\(97\) −3.48632 −0.353982 −0.176991 0.984212i \(-0.556636\pi\)
−0.176991 + 0.984212i \(0.556636\pi\)
\(98\) −2.57774 −0.260391
\(99\) 0.498396 0.0500906
\(100\) −0.00445485 −0.000445485 0
\(101\) 2.51669 0.250420 0.125210 0.992130i \(-0.460040\pi\)
0.125210 + 0.992130i \(0.460040\pi\)
\(102\) 0.403182 0.0399210
\(103\) −14.6148 −1.44004 −0.720020 0.693953i \(-0.755868\pi\)
−0.720020 + 0.693953i \(0.755868\pi\)
\(104\) 1.04306 0.102280
\(105\) 1.74186 0.169988
\(106\) −0.489862 −0.0475796
\(107\) 18.9401 1.83101 0.915505 0.402308i \(-0.131792\pi\)
0.915505 + 0.402308i \(0.131792\pi\)
\(108\) 1.83744 0.176808
\(109\) −6.45006 −0.617804 −0.308902 0.951094i \(-0.599962\pi\)
−0.308902 + 0.951094i \(0.599962\pi\)
\(110\) 0.449434 0.0428518
\(111\) −2.84423 −0.269962
\(112\) −2.37617 −0.224527
\(113\) 12.8246 1.20643 0.603216 0.797578i \(-0.293886\pi\)
0.603216 + 0.797578i \(0.293886\pi\)
\(114\) 3.11867 0.292090
\(115\) 5.66655 0.528408
\(116\) 2.03383 0.188837
\(117\) −0.674164 −0.0623265
\(118\) −4.94463 −0.455190
\(119\) 0.778796 0.0713921
\(120\) 3.46046 0.315895
\(121\) −10.7516 −0.977418
\(122\) −3.28824 −0.297703
\(123\) −5.58140 −0.503258
\(124\) −16.0635 −1.44254
\(125\) −11.1776 −0.999757
\(126\) −0.313997 −0.0279730
\(127\) 13.7539 1.22046 0.610231 0.792223i \(-0.291077\pi\)
0.610231 + 0.792223i \(0.291077\pi\)
\(128\) −10.4064 −0.919801
\(129\) 1.12704 0.0992299
\(130\) −0.607935 −0.0533195
\(131\) 20.6934 1.80799 0.903997 0.427539i \(-0.140619\pi\)
0.903997 + 0.427539i \(0.140619\pi\)
\(132\) 0.915774 0.0797079
\(133\) 6.02409 0.522354
\(134\) 0.567893 0.0490584
\(135\) −2.23661 −0.192497
\(136\) 1.54719 0.132670
\(137\) −11.2179 −0.958410 −0.479205 0.877703i \(-0.659075\pi\)
−0.479205 + 0.877703i \(0.659075\pi\)
\(138\) −1.02148 −0.0869541
\(139\) 13.9759 1.18542 0.592709 0.805416i \(-0.298058\pi\)
0.592709 + 0.805416i \(0.298058\pi\)
\(140\) 3.20057 0.270498
\(141\) −10.1996 −0.858965
\(142\) 4.07943 0.342338
\(143\) −0.336001 −0.0280978
\(144\) 3.05109 0.254257
\(145\) −2.47566 −0.205592
\(146\) 3.62279 0.299824
\(147\) 6.39348 0.527325
\(148\) −5.22611 −0.429583
\(149\) 15.0359 1.23179 0.615894 0.787829i \(-0.288795\pi\)
0.615894 + 0.787829i \(0.288795\pi\)
\(150\) −0.000977508 0 −7.98132e−5 0
\(151\) −0.348781 −0.0283834 −0.0141917 0.999899i \(-0.504518\pi\)
−0.0141917 + 0.999899i \(0.504518\pi\)
\(152\) 11.9677 0.970709
\(153\) −1.00000 −0.0808452
\(154\) −0.156495 −0.0126107
\(155\) 19.5531 1.57054
\(156\) −1.23874 −0.0991785
\(157\) 13.6435 1.08887 0.544437 0.838802i \(-0.316744\pi\)
0.544437 + 0.838802i \(0.316744\pi\)
\(158\) 0.403182 0.0320755
\(159\) 1.21499 0.0963549
\(160\) 9.67227 0.764660
\(161\) −1.97311 −0.155503
\(162\) 0.403182 0.0316770
\(163\) 9.10119 0.712860 0.356430 0.934322i \(-0.383994\pi\)
0.356430 + 0.934322i \(0.383994\pi\)
\(164\) −10.2555 −0.800821
\(165\) −1.11472 −0.0867806
\(166\) 5.20352 0.403872
\(167\) −4.27495 −0.330805 −0.165403 0.986226i \(-0.552892\pi\)
−0.165403 + 0.986226i \(0.552892\pi\)
\(168\) −1.20494 −0.0929635
\(169\) −12.5455 −0.965039
\(170\) −0.901761 −0.0691619
\(171\) −7.73513 −0.591520
\(172\) 2.07086 0.157902
\(173\) 12.5207 0.951933 0.475967 0.879463i \(-0.342098\pi\)
0.475967 + 0.879463i \(0.342098\pi\)
\(174\) 0.446275 0.0338320
\(175\) −0.00188818 −0.000142733 0
\(176\) 1.52065 0.114623
\(177\) 12.2640 0.921819
\(178\) −3.09571 −0.232033
\(179\) 20.9004 1.56217 0.781084 0.624425i \(-0.214667\pi\)
0.781084 + 0.624425i \(0.214667\pi\)
\(180\) −4.10965 −0.306315
\(181\) 18.4174 1.36895 0.684477 0.729035i \(-0.260031\pi\)
0.684477 + 0.729035i \(0.260031\pi\)
\(182\) 0.211685 0.0156912
\(183\) 8.15571 0.602887
\(184\) −3.91987 −0.288977
\(185\) 6.36142 0.467701
\(186\) −3.52473 −0.258446
\(187\) −0.498396 −0.0364463
\(188\) −18.7413 −1.36685
\(189\) 0.778796 0.0566490
\(190\) −6.97524 −0.506037
\(191\) 6.19282 0.448097 0.224049 0.974578i \(-0.428073\pi\)
0.224049 + 0.974578i \(0.428073\pi\)
\(192\) 4.35861 0.314555
\(193\) −7.33927 −0.528292 −0.264146 0.964483i \(-0.585090\pi\)
−0.264146 + 0.964483i \(0.585090\pi\)
\(194\) −1.40562 −0.100918
\(195\) 1.50784 0.107979
\(196\) 11.7477 0.839118
\(197\) 13.3425 0.950611 0.475305 0.879821i \(-0.342338\pi\)
0.475305 + 0.879821i \(0.342338\pi\)
\(198\) 0.200944 0.0142805
\(199\) 5.24795 0.372017 0.186009 0.982548i \(-0.440445\pi\)
0.186009 + 0.982548i \(0.440445\pi\)
\(200\) −0.00375113 −0.000265245 0
\(201\) −1.40853 −0.0993497
\(202\) 1.01468 0.0713928
\(203\) 0.862034 0.0605029
\(204\) −1.83744 −0.128647
\(205\) 12.4834 0.871879
\(206\) −5.89243 −0.410545
\(207\) 2.53354 0.176093
\(208\) −2.05694 −0.142623
\(209\) −3.85515 −0.266667
\(210\) 0.702288 0.0484625
\(211\) −1.52927 −0.105279 −0.0526397 0.998614i \(-0.516763\pi\)
−0.0526397 + 0.998614i \(0.516763\pi\)
\(212\) 2.23248 0.153327
\(213\) −10.1181 −0.693279
\(214\) 7.63631 0.522008
\(215\) −2.52074 −0.171913
\(216\) 1.54719 0.105273
\(217\) −6.80845 −0.462188
\(218\) −2.60055 −0.176132
\(219\) −8.98549 −0.607183
\(220\) −2.04823 −0.138092
\(221\) 0.674164 0.0453492
\(222\) −1.14674 −0.0769642
\(223\) 6.63445 0.444276 0.222138 0.975015i \(-0.428696\pi\)
0.222138 + 0.975015i \(0.428696\pi\)
\(224\) −3.36792 −0.225029
\(225\) 0.00242448 0.000161632 0
\(226\) 5.17063 0.343945
\(227\) 1.69278 0.112354 0.0561768 0.998421i \(-0.482109\pi\)
0.0561768 + 0.998421i \(0.482109\pi\)
\(228\) −14.2129 −0.941270
\(229\) 13.9117 0.919312 0.459656 0.888097i \(-0.347973\pi\)
0.459656 + 0.888097i \(0.347973\pi\)
\(230\) 2.28465 0.150645
\(231\) 0.388148 0.0255383
\(232\) 1.71255 0.112435
\(233\) 9.20029 0.602731 0.301366 0.953509i \(-0.402558\pi\)
0.301366 + 0.953509i \(0.402558\pi\)
\(234\) −0.271811 −0.0177688
\(235\) 22.8126 1.48813
\(236\) 22.5344 1.46687
\(237\) −1.00000 −0.0649570
\(238\) 0.313997 0.0203534
\(239\) −0.640793 −0.0414494 −0.0207247 0.999785i \(-0.506597\pi\)
−0.0207247 + 0.999785i \(0.506597\pi\)
\(240\) −6.82410 −0.440494
\(241\) 19.4410 1.25231 0.626153 0.779700i \(-0.284629\pi\)
0.626153 + 0.779700i \(0.284629\pi\)
\(242\) −4.33485 −0.278655
\(243\) −1.00000 −0.0641500
\(244\) 14.9857 0.959358
\(245\) −14.2997 −0.913575
\(246\) −2.25032 −0.143475
\(247\) 5.21475 0.331807
\(248\) −13.5260 −0.858900
\(249\) −12.9061 −0.817893
\(250\) −4.50662 −0.285024
\(251\) 3.20740 0.202449 0.101225 0.994864i \(-0.467724\pi\)
0.101225 + 0.994864i \(0.467724\pi\)
\(252\) 1.43099 0.0901441
\(253\) 1.26271 0.0793857
\(254\) 5.54533 0.347945
\(255\) 2.23661 0.140062
\(256\) 4.52156 0.282597
\(257\) 0.710998 0.0443508 0.0221754 0.999754i \(-0.492941\pi\)
0.0221754 + 0.999754i \(0.492941\pi\)
\(258\) 0.454401 0.0282897
\(259\) −2.21507 −0.137638
\(260\) 2.77058 0.171824
\(261\) −1.10688 −0.0685142
\(262\) 8.34323 0.515446
\(263\) 14.3661 0.885855 0.442927 0.896558i \(-0.353940\pi\)
0.442927 + 0.896558i \(0.353940\pi\)
\(264\) 0.771112 0.0474587
\(265\) −2.71746 −0.166932
\(266\) 2.42880 0.148920
\(267\) 7.67819 0.469898
\(268\) −2.58809 −0.158093
\(269\) 24.3387 1.48396 0.741980 0.670422i \(-0.233887\pi\)
0.741980 + 0.670422i \(0.233887\pi\)
\(270\) −0.901761 −0.0548794
\(271\) 15.8359 0.961960 0.480980 0.876732i \(-0.340281\pi\)
0.480980 + 0.876732i \(0.340281\pi\)
\(272\) −3.05109 −0.184999
\(273\) −0.525036 −0.0317766
\(274\) −4.52286 −0.273236
\(275\) 0.00120835 7.28663e−5 0
\(276\) 4.65524 0.280213
\(277\) −20.6484 −1.24064 −0.620322 0.784347i \(-0.712998\pi\)
−0.620322 + 0.784347i \(0.712998\pi\)
\(278\) 5.63483 0.337954
\(279\) 8.74229 0.523387
\(280\) 2.69499 0.161056
\(281\) −4.66208 −0.278116 −0.139058 0.990284i \(-0.544408\pi\)
−0.139058 + 0.990284i \(0.544408\pi\)
\(282\) −4.11232 −0.244885
\(283\) 31.6649 1.88228 0.941141 0.338013i \(-0.109755\pi\)
0.941141 + 0.338013i \(0.109755\pi\)
\(284\) −18.5914 −1.10320
\(285\) 17.3005 1.02479
\(286\) −0.135469 −0.00801048
\(287\) −4.34677 −0.256582
\(288\) 4.32452 0.254825
\(289\) 1.00000 0.0588235
\(290\) −0.998142 −0.0586129
\(291\) 3.48632 0.204372
\(292\) −16.5103 −0.966195
\(293\) −22.8762 −1.33644 −0.668221 0.743963i \(-0.732944\pi\)
−0.668221 + 0.743963i \(0.732944\pi\)
\(294\) 2.57774 0.150337
\(295\) −27.4298 −1.59702
\(296\) −4.40056 −0.255777
\(297\) −0.498396 −0.0289198
\(298\) 6.06220 0.351174
\(299\) −1.70802 −0.0987776
\(300\) 0.00445485 0.000257201 0
\(301\) 0.877730 0.0505915
\(302\) −0.140622 −0.00809190
\(303\) −2.51669 −0.144580
\(304\) −23.6006 −1.35359
\(305\) −18.2411 −1.04448
\(306\) −0.403182 −0.0230484
\(307\) −17.8390 −1.01813 −0.509063 0.860729i \(-0.670008\pi\)
−0.509063 + 0.860729i \(0.670008\pi\)
\(308\) 0.713201 0.0406384
\(309\) 14.6148 0.831408
\(310\) 7.88346 0.447750
\(311\) 5.52688 0.313401 0.156700 0.987646i \(-0.449914\pi\)
0.156700 + 0.987646i \(0.449914\pi\)
\(312\) −1.04306 −0.0590516
\(313\) −17.5669 −0.992940 −0.496470 0.868054i \(-0.665371\pi\)
−0.496470 + 0.868054i \(0.665371\pi\)
\(314\) 5.50083 0.310430
\(315\) −1.74186 −0.0981428
\(316\) −1.83744 −0.103364
\(317\) 17.7766 0.998435 0.499217 0.866477i \(-0.333621\pi\)
0.499217 + 0.866477i \(0.333621\pi\)
\(318\) 0.489862 0.0274701
\(319\) −0.551665 −0.0308873
\(320\) −9.74850 −0.544958
\(321\) −18.9401 −1.05713
\(322\) −0.795524 −0.0443328
\(323\) 7.73513 0.430394
\(324\) −1.83744 −0.102080
\(325\) −0.00163450 −9.06657e−5 0
\(326\) 3.66944 0.203231
\(327\) 6.45006 0.356689
\(328\) −8.63548 −0.476815
\(329\) −7.94344 −0.437936
\(330\) −0.449434 −0.0247405
\(331\) −31.5772 −1.73564 −0.867819 0.496880i \(-0.834479\pi\)
−0.867819 + 0.496880i \(0.834479\pi\)
\(332\) −23.7143 −1.30149
\(333\) 2.84423 0.155863
\(334\) −1.72358 −0.0943102
\(335\) 3.15032 0.172121
\(336\) 2.37617 0.129631
\(337\) −25.5349 −1.39097 −0.695486 0.718539i \(-0.744811\pi\)
−0.695486 + 0.718539i \(0.744811\pi\)
\(338\) −5.05812 −0.275126
\(339\) −12.8246 −0.696534
\(340\) 4.10965 0.222877
\(341\) 4.35712 0.235951
\(342\) −3.11867 −0.168638
\(343\) 10.4308 0.563209
\(344\) 1.74374 0.0940160
\(345\) −5.66655 −0.305077
\(346\) 5.04813 0.271389
\(347\) −28.7493 −1.54334 −0.771672 0.636020i \(-0.780579\pi\)
−0.771672 + 0.636020i \(0.780579\pi\)
\(348\) −2.03383 −0.109025
\(349\) −17.4935 −0.936404 −0.468202 0.883621i \(-0.655098\pi\)
−0.468202 + 0.883621i \(0.655098\pi\)
\(350\) −0.000761279 0 −4.06921e−5 0
\(351\) 0.674164 0.0359842
\(352\) 2.15532 0.114879
\(353\) −15.6665 −0.833844 −0.416922 0.908942i \(-0.636891\pi\)
−0.416922 + 0.908942i \(0.636891\pi\)
\(354\) 4.94463 0.262804
\(355\) 22.6302 1.20109
\(356\) 14.1082 0.747735
\(357\) −0.778796 −0.0412182
\(358\) 8.42667 0.445363
\(359\) −9.66108 −0.509892 −0.254946 0.966955i \(-0.582058\pi\)
−0.254946 + 0.966955i \(0.582058\pi\)
\(360\) −3.46046 −0.182382
\(361\) 40.8322 2.14906
\(362\) 7.42556 0.390279
\(363\) 10.7516 0.564313
\(364\) −0.964725 −0.0505653
\(365\) 20.0970 1.05193
\(366\) 3.28824 0.171879
\(367\) −8.52917 −0.445219 −0.222609 0.974908i \(-0.571457\pi\)
−0.222609 + 0.974908i \(0.571457\pi\)
\(368\) 7.73007 0.402958
\(369\) 5.58140 0.290556
\(370\) 2.56481 0.133338
\(371\) 0.946229 0.0491257
\(372\) 16.0635 0.832852
\(373\) 34.2075 1.77120 0.885599 0.464450i \(-0.153748\pi\)
0.885599 + 0.464450i \(0.153748\pi\)
\(374\) −0.200944 −0.0103906
\(375\) 11.1776 0.577210
\(376\) −15.7808 −0.813832
\(377\) 0.746220 0.0384323
\(378\) 0.313997 0.0161502
\(379\) −10.9891 −0.564472 −0.282236 0.959345i \(-0.591076\pi\)
−0.282236 + 0.959345i \(0.591076\pi\)
\(380\) 31.7886 1.63072
\(381\) −13.7539 −0.704634
\(382\) 2.49684 0.127749
\(383\) 12.0453 0.615487 0.307743 0.951469i \(-0.400426\pi\)
0.307743 + 0.951469i \(0.400426\pi\)
\(384\) 10.4064 0.531047
\(385\) −0.868136 −0.0442443
\(386\) −2.95906 −0.150612
\(387\) −1.12704 −0.0572904
\(388\) 6.40592 0.325211
\(389\) −3.81884 −0.193623 −0.0968115 0.995303i \(-0.530864\pi\)
−0.0968115 + 0.995303i \(0.530864\pi\)
\(390\) 0.607935 0.0307840
\(391\) −2.53354 −0.128127
\(392\) 9.89192 0.499617
\(393\) −20.6934 −1.04385
\(394\) 5.37944 0.271012
\(395\) 2.23661 0.112536
\(396\) −0.915774 −0.0460194
\(397\) 36.9438 1.85415 0.927077 0.374870i \(-0.122313\pi\)
0.927077 + 0.374870i \(0.122313\pi\)
\(398\) 2.11588 0.106060
\(399\) −6.02409 −0.301581
\(400\) 0.00739731 0.000369866 0
\(401\) −38.1603 −1.90563 −0.952816 0.303547i \(-0.901829\pi\)
−0.952816 + 0.303547i \(0.901829\pi\)
\(402\) −0.567893 −0.0283239
\(403\) −5.89374 −0.293588
\(404\) −4.62427 −0.230066
\(405\) 2.23661 0.111138
\(406\) 0.347557 0.0172490
\(407\) 1.41755 0.0702653
\(408\) −1.54719 −0.0765973
\(409\) −16.0003 −0.791163 −0.395582 0.918431i \(-0.629457\pi\)
−0.395582 + 0.918431i \(0.629457\pi\)
\(410\) 5.03309 0.248567
\(411\) 11.2179 0.553338
\(412\) 26.8539 1.32300
\(413\) 9.55115 0.469981
\(414\) 1.02148 0.0502030
\(415\) 28.8660 1.41698
\(416\) −2.91544 −0.142941
\(417\) −13.9759 −0.684402
\(418\) −1.55433 −0.0760248
\(419\) 25.5435 1.24788 0.623941 0.781472i \(-0.285531\pi\)
0.623941 + 0.781472i \(0.285531\pi\)
\(420\) −3.20057 −0.156172
\(421\) 18.4572 0.899548 0.449774 0.893143i \(-0.351505\pi\)
0.449774 + 0.893143i \(0.351505\pi\)
\(422\) −0.616575 −0.0300144
\(423\) 10.1996 0.495924
\(424\) 1.87982 0.0912921
\(425\) −0.00242448 −0.000117605 0
\(426\) −4.07943 −0.197649
\(427\) 6.35163 0.307377
\(428\) −34.8014 −1.68219
\(429\) 0.336001 0.0162223
\(430\) −1.01632 −0.0490112
\(431\) −11.8175 −0.569230 −0.284615 0.958642i \(-0.591866\pi\)
−0.284615 + 0.958642i \(0.591866\pi\)
\(432\) −3.05109 −0.146796
\(433\) −30.0818 −1.44564 −0.722821 0.691036i \(-0.757155\pi\)
−0.722821 + 0.691036i \(0.757155\pi\)
\(434\) −2.74505 −0.131767
\(435\) 2.47566 0.118699
\(436\) 11.8516 0.567590
\(437\) −19.5973 −0.937465
\(438\) −3.62279 −0.173104
\(439\) 25.0422 1.19520 0.597599 0.801795i \(-0.296121\pi\)
0.597599 + 0.801795i \(0.296121\pi\)
\(440\) −1.72468 −0.0822208
\(441\) −6.39348 −0.304451
\(442\) 0.271811 0.0129287
\(443\) −2.64132 −0.125493 −0.0627464 0.998030i \(-0.519986\pi\)
−0.0627464 + 0.998030i \(0.519986\pi\)
\(444\) 5.22611 0.248020
\(445\) −17.1731 −0.814084
\(446\) 2.67489 0.126660
\(447\) −15.0359 −0.711173
\(448\) 3.39446 0.160373
\(449\) −12.2723 −0.579167 −0.289583 0.957153i \(-0.593517\pi\)
−0.289583 + 0.957153i \(0.593517\pi\)
\(450\) 0.000977508 0 4.60802e−5 0
\(451\) 2.78175 0.130987
\(452\) −23.5644 −1.10838
\(453\) 0.348781 0.0163872
\(454\) 0.682498 0.0320312
\(455\) 1.17430 0.0550521
\(456\) −11.9677 −0.560439
\(457\) −6.62566 −0.309935 −0.154968 0.987920i \(-0.549527\pi\)
−0.154968 + 0.987920i \(0.549527\pi\)
\(458\) 5.60896 0.262089
\(459\) 1.00000 0.0466760
\(460\) −10.4120 −0.485460
\(461\) −7.53578 −0.350976 −0.175488 0.984482i \(-0.556150\pi\)
−0.175488 + 0.984482i \(0.556150\pi\)
\(462\) 0.156495 0.00728078
\(463\) −6.78894 −0.315509 −0.157754 0.987478i \(-0.550425\pi\)
−0.157754 + 0.987478i \(0.550425\pi\)
\(464\) −3.37719 −0.156782
\(465\) −19.5531 −0.906753
\(466\) 3.70939 0.171834
\(467\) 1.29759 0.0600453 0.0300227 0.999549i \(-0.490442\pi\)
0.0300227 + 0.999549i \(0.490442\pi\)
\(468\) 1.23874 0.0572607
\(469\) −1.09695 −0.0506526
\(470\) 9.19765 0.424256
\(471\) −13.6435 −0.628661
\(472\) 18.9747 0.873383
\(473\) −0.561709 −0.0258274
\(474\) −0.403182 −0.0185188
\(475\) −0.0187537 −0.000860478 0
\(476\) −1.43099 −0.0655895
\(477\) −1.21499 −0.0556306
\(478\) −0.258356 −0.0118169
\(479\) 25.0686 1.14541 0.572707 0.819760i \(-0.305893\pi\)
0.572707 + 0.819760i \(0.305893\pi\)
\(480\) −9.67227 −0.441477
\(481\) −1.91748 −0.0874294
\(482\) 7.83827 0.357023
\(483\) 1.97311 0.0897797
\(484\) 19.7555 0.897976
\(485\) −7.79754 −0.354068
\(486\) −0.403182 −0.0182887
\(487\) 20.4671 0.927454 0.463727 0.885978i \(-0.346512\pi\)
0.463727 + 0.885978i \(0.346512\pi\)
\(488\) 12.6184 0.571209
\(489\) −9.10119 −0.411570
\(490\) −5.76539 −0.260454
\(491\) −8.85107 −0.399443 −0.199722 0.979853i \(-0.564004\pi\)
−0.199722 + 0.979853i \(0.564004\pi\)
\(492\) 10.2555 0.462354
\(493\) 1.10688 0.0498514
\(494\) 2.10249 0.0945957
\(495\) 1.11472 0.0501028
\(496\) 26.6735 1.19768
\(497\) −7.87991 −0.353462
\(498\) −5.20352 −0.233175
\(499\) 36.7177 1.64371 0.821855 0.569696i \(-0.192939\pi\)
0.821855 + 0.569696i \(0.192939\pi\)
\(500\) 20.5383 0.918499
\(501\) 4.27495 0.190991
\(502\) 1.29317 0.0577168
\(503\) −43.1318 −1.92315 −0.961576 0.274538i \(-0.911475\pi\)
−0.961576 + 0.274538i \(0.911475\pi\)
\(504\) 1.20494 0.0536725
\(505\) 5.62884 0.250480
\(506\) 0.509101 0.0226323
\(507\) 12.5455 0.557165
\(508\) −25.2720 −1.12127
\(509\) 40.7051 1.80422 0.902110 0.431505i \(-0.142017\pi\)
0.902110 + 0.431505i \(0.142017\pi\)
\(510\) 0.901761 0.0399307
\(511\) −6.99786 −0.309567
\(512\) 22.6357 1.00037
\(513\) 7.73513 0.341514
\(514\) 0.286662 0.0126441
\(515\) −32.6876 −1.44039
\(516\) −2.07086 −0.0911647
\(517\) 5.08346 0.223570
\(518\) −0.893077 −0.0392396
\(519\) −12.5207 −0.549599
\(520\) 2.33292 0.102305
\(521\) −28.1518 −1.23335 −0.616675 0.787218i \(-0.711521\pi\)
−0.616675 + 0.787218i \(0.711521\pi\)
\(522\) −0.446275 −0.0195329
\(523\) −22.4523 −0.981772 −0.490886 0.871224i \(-0.663327\pi\)
−0.490886 + 0.871224i \(0.663327\pi\)
\(524\) −38.0230 −1.66104
\(525\) 0.00188818 8.24068e−5 0
\(526\) 5.79218 0.252551
\(527\) −8.74229 −0.380820
\(528\) −1.52065 −0.0661778
\(529\) −16.5812 −0.720920
\(530\) −1.09563 −0.0475912
\(531\) −12.2640 −0.532212
\(532\) −11.0689 −0.479899
\(533\) −3.76278 −0.162984
\(534\) 3.09571 0.133964
\(535\) 42.3616 1.83145
\(536\) −2.17926 −0.0941295
\(537\) −20.9004 −0.901919
\(538\) 9.81295 0.423066
\(539\) −3.18648 −0.137251
\(540\) 4.10965 0.176851
\(541\) 25.0818 1.07835 0.539176 0.842193i \(-0.318736\pi\)
0.539176 + 0.842193i \(0.318736\pi\)
\(542\) 6.38474 0.274248
\(543\) −18.4174 −0.790365
\(544\) −4.32452 −0.185412
\(545\) −14.4263 −0.617954
\(546\) −0.211685 −0.00905929
\(547\) −5.18321 −0.221618 −0.110809 0.993842i \(-0.535344\pi\)
−0.110809 + 0.993842i \(0.535344\pi\)
\(548\) 20.6123 0.880512
\(549\) −8.15571 −0.348077
\(550\) 0.000487186 0 2.07737e−5 0
\(551\) 8.56187 0.364748
\(552\) 3.91987 0.166841
\(553\) −0.778796 −0.0331178
\(554\) −8.32508 −0.353699
\(555\) −6.36142 −0.270027
\(556\) −25.6799 −1.08907
\(557\) −24.9582 −1.05751 −0.528756 0.848774i \(-0.677341\pi\)
−0.528756 + 0.848774i \(0.677341\pi\)
\(558\) 3.52473 0.149214
\(559\) 0.759807 0.0321364
\(560\) −5.31458 −0.224582
\(561\) 0.498396 0.0210423
\(562\) −1.87967 −0.0792890
\(563\) 27.5026 1.15909 0.579547 0.814939i \(-0.303229\pi\)
0.579547 + 0.814939i \(0.303229\pi\)
\(564\) 18.7413 0.789150
\(565\) 28.6835 1.20673
\(566\) 12.7667 0.536625
\(567\) −0.778796 −0.0327063
\(568\) −15.6546 −0.656852
\(569\) 27.8853 1.16901 0.584506 0.811389i \(-0.301288\pi\)
0.584506 + 0.811389i \(0.301288\pi\)
\(570\) 6.97524 0.292161
\(571\) 32.5538 1.36233 0.681167 0.732128i \(-0.261473\pi\)
0.681167 + 0.732128i \(0.261473\pi\)
\(572\) 0.617382 0.0258141
\(573\) −6.19282 −0.258709
\(574\) −1.75254 −0.0731496
\(575\) 0.00614253 0.000256161 0
\(576\) −4.35861 −0.181609
\(577\) 1.88444 0.0784504 0.0392252 0.999230i \(-0.487511\pi\)
0.0392252 + 0.999230i \(0.487511\pi\)
\(578\) 0.403182 0.0167702
\(579\) 7.33927 0.305010
\(580\) 4.54889 0.188882
\(581\) −10.0512 −0.416996
\(582\) 1.40562 0.0582649
\(583\) −0.605546 −0.0250791
\(584\) −13.9023 −0.575280
\(585\) −1.50784 −0.0623416
\(586\) −9.22328 −0.381010
\(587\) −34.9913 −1.44425 −0.722123 0.691765i \(-0.756833\pi\)
−0.722123 + 0.691765i \(0.756833\pi\)
\(588\) −11.7477 −0.484465
\(589\) −67.6227 −2.78635
\(590\) −11.0592 −0.455300
\(591\) −13.3425 −0.548835
\(592\) 8.67799 0.356663
\(593\) −29.1167 −1.19568 −0.597840 0.801615i \(-0.703974\pi\)
−0.597840 + 0.801615i \(0.703974\pi\)
\(594\) −0.200944 −0.00824484
\(595\) 1.74186 0.0714094
\(596\) −27.6276 −1.13167
\(597\) −5.24795 −0.214784
\(598\) −0.688645 −0.0281608
\(599\) 30.3609 1.24051 0.620255 0.784400i \(-0.287029\pi\)
0.620255 + 0.784400i \(0.287029\pi\)
\(600\) 0.00375113 0.000153139 0
\(601\) 37.5860 1.53316 0.766582 0.642146i \(-0.221956\pi\)
0.766582 + 0.642146i \(0.221956\pi\)
\(602\) 0.353885 0.0144233
\(603\) 1.40853 0.0573596
\(604\) 0.640865 0.0260764
\(605\) −24.0471 −0.977655
\(606\) −1.01468 −0.0412187
\(607\) 20.4958 0.831898 0.415949 0.909388i \(-0.363449\pi\)
0.415949 + 0.909388i \(0.363449\pi\)
\(608\) −33.4507 −1.35661
\(609\) −0.862034 −0.0349314
\(610\) −7.35450 −0.297775
\(611\) −6.87624 −0.278183
\(612\) 1.83744 0.0742743
\(613\) 32.1035 1.29665 0.648324 0.761365i \(-0.275470\pi\)
0.648324 + 0.761365i \(0.275470\pi\)
\(614\) −7.19237 −0.290260
\(615\) −12.4834 −0.503380
\(616\) 0.600539 0.0241964
\(617\) −45.1098 −1.81605 −0.908025 0.418915i \(-0.862411\pi\)
−0.908025 + 0.418915i \(0.862411\pi\)
\(618\) 5.89243 0.237028
\(619\) −2.03300 −0.0817133 −0.0408567 0.999165i \(-0.513009\pi\)
−0.0408567 + 0.999165i \(0.513009\pi\)
\(620\) −35.9277 −1.44289
\(621\) −2.53354 −0.101668
\(622\) 2.22834 0.0893483
\(623\) 5.97974 0.239573
\(624\) 2.05694 0.0823433
\(625\) −25.0121 −1.00048
\(626\) −7.08266 −0.283080
\(627\) 3.85515 0.153960
\(628\) −25.0692 −1.00037
\(629\) −2.84423 −0.113407
\(630\) −0.702288 −0.0279798
\(631\) −13.5352 −0.538828 −0.269414 0.963024i \(-0.586830\pi\)
−0.269414 + 0.963024i \(0.586830\pi\)
\(632\) −1.54719 −0.0615439
\(633\) 1.52927 0.0607830
\(634\) 7.16722 0.284647
\(635\) 30.7621 1.22076
\(636\) −2.23248 −0.0885234
\(637\) 4.31025 0.170779
\(638\) −0.222421 −0.00880575
\(639\) 10.1181 0.400265
\(640\) −23.2750 −0.920024
\(641\) −9.93730 −0.392500 −0.196250 0.980554i \(-0.562876\pi\)
−0.196250 + 0.980554i \(0.562876\pi\)
\(642\) −7.63631 −0.301381
\(643\) 21.6200 0.852609 0.426305 0.904580i \(-0.359815\pi\)
0.426305 + 0.904580i \(0.359815\pi\)
\(644\) 3.62548 0.142864
\(645\) 2.52074 0.0992540
\(646\) 3.11867 0.122702
\(647\) −20.7064 −0.814052 −0.407026 0.913417i \(-0.633434\pi\)
−0.407026 + 0.913417i \(0.633434\pi\)
\(648\) −1.54719 −0.0607793
\(649\) −6.11233 −0.239930
\(650\) −0.000659001 0 −2.58482e−5 0
\(651\) 6.80845 0.266844
\(652\) −16.7229 −0.654920
\(653\) −29.1487 −1.14068 −0.570339 0.821409i \(-0.693188\pi\)
−0.570339 + 0.821409i \(0.693188\pi\)
\(654\) 2.60055 0.101690
\(655\) 46.2832 1.80843
\(656\) 17.0293 0.664884
\(657\) 8.98549 0.350558
\(658\) −3.20265 −0.124852
\(659\) −24.7434 −0.963867 −0.481934 0.876208i \(-0.660065\pi\)
−0.481934 + 0.876208i \(0.660065\pi\)
\(660\) 2.04823 0.0797272
\(661\) 45.1631 1.75664 0.878321 0.478072i \(-0.158664\pi\)
0.878321 + 0.478072i \(0.158664\pi\)
\(662\) −12.7314 −0.494818
\(663\) −0.674164 −0.0261824
\(664\) −19.9682 −0.774917
\(665\) 13.4735 0.522481
\(666\) 1.14674 0.0444353
\(667\) −2.80433 −0.108584
\(668\) 7.85498 0.303918
\(669\) −6.63445 −0.256503
\(670\) 1.27015 0.0490703
\(671\) −4.06477 −0.156919
\(672\) 3.36792 0.129920
\(673\) 41.6152 1.60415 0.802073 0.597226i \(-0.203730\pi\)
0.802073 + 0.597226i \(0.203730\pi\)
\(674\) −10.2952 −0.396556
\(675\) −0.00242448 −9.33184e−5 0
\(676\) 23.0517 0.886602
\(677\) 23.5954 0.906845 0.453422 0.891296i \(-0.350203\pi\)
0.453422 + 0.891296i \(0.350203\pi\)
\(678\) −5.17063 −0.198577
\(679\) 2.71513 0.104197
\(680\) 3.46046 0.132703
\(681\) −1.69278 −0.0648674
\(682\) 1.75671 0.0672680
\(683\) −24.8814 −0.952061 −0.476031 0.879429i \(-0.657925\pi\)
−0.476031 + 0.879429i \(0.657925\pi\)
\(684\) 14.2129 0.543443
\(685\) −25.0901 −0.958642
\(686\) 4.20551 0.160567
\(687\) −13.9117 −0.530765
\(688\) −3.43868 −0.131099
\(689\) 0.819103 0.0312053
\(690\) −2.28465 −0.0869752
\(691\) −30.8808 −1.17476 −0.587380 0.809311i \(-0.699841\pi\)
−0.587380 + 0.809311i \(0.699841\pi\)
\(692\) −23.0061 −0.874562
\(693\) −0.388148 −0.0147445
\(694\) −11.5912 −0.439997
\(695\) 31.2586 1.18571
\(696\) −1.71255 −0.0649142
\(697\) −5.58140 −0.211411
\(698\) −7.05306 −0.266962
\(699\) −9.20029 −0.347987
\(700\) 0.00346942 0.000131132 0
\(701\) −20.2883 −0.766278 −0.383139 0.923691i \(-0.625157\pi\)
−0.383139 + 0.923691i \(0.625157\pi\)
\(702\) 0.271811 0.0102588
\(703\) −22.0005 −0.829763
\(704\) −2.17231 −0.0818720
\(705\) −22.8126 −0.859174
\(706\) −6.31645 −0.237723
\(707\) −1.95998 −0.0737128
\(708\) −22.5344 −0.846895
\(709\) 41.7807 1.56911 0.784554 0.620061i \(-0.212892\pi\)
0.784554 + 0.620061i \(0.212892\pi\)
\(710\) 9.12409 0.342421
\(711\) 1.00000 0.0375029
\(712\) 11.8796 0.445207
\(713\) 22.1490 0.829485
\(714\) −0.313997 −0.0117510
\(715\) −0.751502 −0.0281046
\(716\) −38.4033 −1.43520
\(717\) 0.640793 0.0239308
\(718\) −3.89518 −0.145367
\(719\) −52.3091 −1.95080 −0.975400 0.220442i \(-0.929250\pi\)
−0.975400 + 0.220442i \(0.929250\pi\)
\(720\) 6.82410 0.254319
\(721\) 11.3820 0.423886
\(722\) 16.4628 0.612683
\(723\) −19.4410 −0.723019
\(724\) −33.8409 −1.25769
\(725\) −0.00268361 −9.96669e−5 0
\(726\) 4.33485 0.160882
\(727\) −19.7947 −0.734145 −0.367072 0.930192i \(-0.619640\pi\)
−0.367072 + 0.930192i \(0.619640\pi\)
\(728\) −0.812330 −0.0301070
\(729\) 1.00000 0.0370370
\(730\) 8.10277 0.299897
\(731\) 1.12704 0.0416849
\(732\) −14.9857 −0.553886
\(733\) −36.7643 −1.35792 −0.678960 0.734176i \(-0.737569\pi\)
−0.678960 + 0.734176i \(0.737569\pi\)
\(734\) −3.43881 −0.126929
\(735\) 14.2997 0.527453
\(736\) 10.9564 0.403857
\(737\) 0.702003 0.0258586
\(738\) 2.25032 0.0828355
\(739\) −37.5974 −1.38304 −0.691522 0.722356i \(-0.743059\pi\)
−0.691522 + 0.722356i \(0.743059\pi\)
\(740\) −11.6888 −0.429687
\(741\) −5.21475 −0.191569
\(742\) 0.381503 0.0140054
\(743\) −1.86258 −0.0683315 −0.0341657 0.999416i \(-0.510877\pi\)
−0.0341657 + 0.999416i \(0.510877\pi\)
\(744\) 13.5260 0.495886
\(745\) 33.6294 1.23209
\(746\) 13.7919 0.504956
\(747\) 12.9061 0.472211
\(748\) 0.915774 0.0334840
\(749\) −14.7505 −0.538970
\(750\) 4.50662 0.164559
\(751\) 1.28506 0.0468925 0.0234463 0.999725i \(-0.492536\pi\)
0.0234463 + 0.999725i \(0.492536\pi\)
\(752\) 31.1200 1.13483
\(753\) −3.20740 −0.116884
\(754\) 0.300862 0.0109568
\(755\) −0.780087 −0.0283903
\(756\) −1.43099 −0.0520447
\(757\) 8.09018 0.294042 0.147021 0.989133i \(-0.453031\pi\)
0.147021 + 0.989133i \(0.453031\pi\)
\(758\) −4.43061 −0.160927
\(759\) −1.26271 −0.0458334
\(760\) 26.7671 0.970945
\(761\) −29.5147 −1.06991 −0.534953 0.844882i \(-0.679671\pi\)
−0.534953 + 0.844882i \(0.679671\pi\)
\(762\) −5.54533 −0.200886
\(763\) 5.02328 0.181855
\(764\) −11.3790 −0.411677
\(765\) −2.23661 −0.0808648
\(766\) 4.85646 0.175471
\(767\) 8.26795 0.298539
\(768\) −4.52156 −0.163158
\(769\) −20.2893 −0.731651 −0.365826 0.930683i \(-0.619213\pi\)
−0.365826 + 0.930683i \(0.619213\pi\)
\(770\) −0.350017 −0.0126137
\(771\) −0.710998 −0.0256060
\(772\) 13.4855 0.485354
\(773\) −1.00188 −0.0360351 −0.0180176 0.999838i \(-0.505735\pi\)
−0.0180176 + 0.999838i \(0.505735\pi\)
\(774\) −0.454401 −0.0163331
\(775\) 0.0211955 0.000761366 0
\(776\) 5.39399 0.193633
\(777\) 2.21507 0.0794652
\(778\) −1.53969 −0.0552006
\(779\) −43.1728 −1.54683
\(780\) −2.77058 −0.0992026
\(781\) 5.04281 0.180446
\(782\) −1.02148 −0.0365280
\(783\) 1.10688 0.0395567
\(784\) −19.5071 −0.696681
\(785\) 30.5153 1.08914
\(786\) −8.34323 −0.297593
\(787\) −39.3642 −1.40318 −0.701592 0.712579i \(-0.747527\pi\)
−0.701592 + 0.712579i \(0.747527\pi\)
\(788\) −24.5160 −0.873347
\(789\) −14.3661 −0.511448
\(790\) 0.901761 0.0320832
\(791\) −9.98771 −0.355122
\(792\) −0.771112 −0.0274003
\(793\) 5.49829 0.195250
\(794\) 14.8951 0.528606
\(795\) 2.71746 0.0963783
\(796\) −9.64282 −0.341781
\(797\) 23.6370 0.837267 0.418633 0.908155i \(-0.362509\pi\)
0.418633 + 0.908155i \(0.362509\pi\)
\(798\) −2.42880 −0.0859787
\(799\) −10.1996 −0.360838
\(800\) 0.0104847 0.000370691 0
\(801\) −7.67819 −0.271295
\(802\) −15.3855 −0.543282
\(803\) 4.47833 0.158037
\(804\) 2.58809 0.0912748
\(805\) −4.41308 −0.155541
\(806\) −2.37625 −0.0836999
\(807\) −24.3387 −0.856764
\(808\) −3.89379 −0.136983
\(809\) 15.6980 0.551914 0.275957 0.961170i \(-0.411005\pi\)
0.275957 + 0.961170i \(0.411005\pi\)
\(810\) 0.901761 0.0316847
\(811\) −8.98332 −0.315447 −0.157723 0.987483i \(-0.550415\pi\)
−0.157723 + 0.987483i \(0.550415\pi\)
\(812\) −1.58394 −0.0555854
\(813\) −15.8359 −0.555388
\(814\) 0.571531 0.0200321
\(815\) 20.3558 0.713033
\(816\) 3.05109 0.106809
\(817\) 8.71776 0.304996
\(818\) −6.45103 −0.225555
\(819\) 0.525036 0.0183463
\(820\) −22.9376 −0.801015
\(821\) 9.56013 0.333651 0.166825 0.985986i \(-0.446648\pi\)
0.166825 + 0.985986i \(0.446648\pi\)
\(822\) 4.52286 0.157753
\(823\) −9.91028 −0.345451 −0.172725 0.984970i \(-0.555257\pi\)
−0.172725 + 0.984970i \(0.555257\pi\)
\(824\) 22.6119 0.787722
\(825\) −0.00120835 −4.20694e−5 0
\(826\) 3.85085 0.133988
\(827\) 5.23444 0.182019 0.0910096 0.995850i \(-0.470991\pi\)
0.0910096 + 0.995850i \(0.470991\pi\)
\(828\) −4.65524 −0.161781
\(829\) −24.6873 −0.857425 −0.428713 0.903441i \(-0.641033\pi\)
−0.428713 + 0.903441i \(0.641033\pi\)
\(830\) 11.6382 0.403970
\(831\) 20.6484 0.716286
\(832\) 2.93842 0.101871
\(833\) 6.39348 0.221521
\(834\) −5.63483 −0.195118
\(835\) −9.56139 −0.330886
\(836\) 7.08363 0.244993
\(837\) −8.74229 −0.302178
\(838\) 10.2987 0.355762
\(839\) 36.3784 1.25592 0.627961 0.778245i \(-0.283890\pi\)
0.627961 + 0.778245i \(0.283890\pi\)
\(840\) −2.69499 −0.0929860
\(841\) −27.7748 −0.957752
\(842\) 7.44160 0.256455
\(843\) 4.66208 0.160571
\(844\) 2.80995 0.0967224
\(845\) −28.0594 −0.965273
\(846\) 4.11232 0.141384
\(847\) 8.37330 0.287710
\(848\) −3.70704 −0.127300
\(849\) −31.6649 −1.08674
\(850\) −0.000977508 0 −3.35283e−5 0
\(851\) 7.20597 0.247018
\(852\) 18.5914 0.636931
\(853\) 31.6583 1.08396 0.541979 0.840392i \(-0.317675\pi\)
0.541979 + 0.840392i \(0.317675\pi\)
\(854\) 2.56086 0.0876309
\(855\) −17.3005 −0.591664
\(856\) −29.3039 −1.00159
\(857\) 3.81736 0.130399 0.0651993 0.997872i \(-0.479232\pi\)
0.0651993 + 0.997872i \(0.479232\pi\)
\(858\) 0.135469 0.00462485
\(859\) −10.8678 −0.370805 −0.185403 0.982663i \(-0.559359\pi\)
−0.185403 + 0.982663i \(0.559359\pi\)
\(860\) 4.63172 0.157940
\(861\) 4.34677 0.148138
\(862\) −4.76462 −0.162283
\(863\) 22.9695 0.781890 0.390945 0.920414i \(-0.372148\pi\)
0.390945 + 0.920414i \(0.372148\pi\)
\(864\) −4.32452 −0.147123
\(865\) 28.0040 0.952164
\(866\) −12.1285 −0.412142
\(867\) −1.00000 −0.0339618
\(868\) 12.5102 0.424622
\(869\) 0.498396 0.0169069
\(870\) 0.998142 0.0338402
\(871\) −0.949578 −0.0321752
\(872\) 9.97947 0.337947
\(873\) −3.48632 −0.117994
\(874\) −7.90128 −0.267265
\(875\) 8.70509 0.294286
\(876\) 16.5103 0.557833
\(877\) −55.0667 −1.85947 −0.929735 0.368230i \(-0.879964\pi\)
−0.929735 + 0.368230i \(0.879964\pi\)
\(878\) 10.0966 0.340742
\(879\) 22.8762 0.771596
\(880\) 3.40110 0.114651
\(881\) −46.0944 −1.55296 −0.776479 0.630143i \(-0.782996\pi\)
−0.776479 + 0.630143i \(0.782996\pi\)
\(882\) −2.57774 −0.0867969
\(883\) 24.6246 0.828683 0.414341 0.910122i \(-0.364012\pi\)
0.414341 + 0.910122i \(0.364012\pi\)
\(884\) −1.23874 −0.0416633
\(885\) 27.4298 0.922042
\(886\) −1.06493 −0.0357771
\(887\) 33.1461 1.11294 0.556468 0.830869i \(-0.312156\pi\)
0.556468 + 0.830869i \(0.312156\pi\)
\(888\) 4.40056 0.147673
\(889\) −10.7115 −0.359252
\(890\) −6.92389 −0.232089
\(891\) 0.498396 0.0166969
\(892\) −12.1904 −0.408166
\(893\) −78.8956 −2.64014
\(894\) −6.06220 −0.202750
\(895\) 46.7460 1.56255
\(896\) 8.10443 0.270750
\(897\) 1.70802 0.0570293
\(898\) −4.94798 −0.165116
\(899\) −9.67667 −0.322735
\(900\) −0.00445485 −0.000148495 0
\(901\) 1.21499 0.0404772
\(902\) 1.12155 0.0373435
\(903\) −0.877730 −0.0292090
\(904\) −19.8420 −0.659935
\(905\) 41.1925 1.36928
\(906\) 0.140622 0.00467186
\(907\) −36.7207 −1.21929 −0.609646 0.792674i \(-0.708688\pi\)
−0.609646 + 0.792674i \(0.708688\pi\)
\(908\) −3.11039 −0.103222
\(909\) 2.51669 0.0834732
\(910\) 0.473457 0.0156950
\(911\) −25.2975 −0.838142 −0.419071 0.907953i \(-0.637644\pi\)
−0.419071 + 0.907953i \(0.637644\pi\)
\(912\) 23.6006 0.781493
\(913\) 6.43236 0.212880
\(914\) −2.67135 −0.0883603
\(915\) 18.2411 0.603033
\(916\) −25.5620 −0.844592
\(917\) −16.1160 −0.532196
\(918\) 0.403182 0.0133070
\(919\) 52.6510 1.73680 0.868398 0.495868i \(-0.165150\pi\)
0.868398 + 0.495868i \(0.165150\pi\)
\(920\) −8.76722 −0.289047
\(921\) 17.8390 0.587815
\(922\) −3.03829 −0.100061
\(923\) −6.82125 −0.224524
\(924\) −0.713201 −0.0234626
\(925\) 0.00689578 0.000226732 0
\(926\) −2.73718 −0.0899493
\(927\) −14.6148 −0.480014
\(928\) −4.78673 −0.157132
\(929\) −33.9435 −1.11365 −0.556825 0.830630i \(-0.687981\pi\)
−0.556825 + 0.830630i \(0.687981\pi\)
\(930\) −7.88346 −0.258509
\(931\) 49.4544 1.62080
\(932\) −16.9050 −0.553743
\(933\) −5.52688 −0.180942
\(934\) 0.523165 0.0171185
\(935\) −1.11472 −0.0364551
\(936\) 1.04306 0.0340935
\(937\) 19.7797 0.646175 0.323088 0.946369i \(-0.395279\pi\)
0.323088 + 0.946369i \(0.395279\pi\)
\(938\) −0.442272 −0.0144407
\(939\) 17.5669 0.573274
\(940\) −41.9169 −1.36718
\(941\) 14.2250 0.463722 0.231861 0.972749i \(-0.425518\pi\)
0.231861 + 0.972749i \(0.425518\pi\)
\(942\) −5.50083 −0.179227
\(943\) 14.1407 0.460485
\(944\) −37.4186 −1.21787
\(945\) 1.74186 0.0566628
\(946\) −0.226471 −0.00736322
\(947\) −7.28486 −0.236726 −0.118363 0.992970i \(-0.537765\pi\)
−0.118363 + 0.992970i \(0.537765\pi\)
\(948\) 1.83744 0.0596774
\(949\) −6.05770 −0.196641
\(950\) −0.00756115 −0.000245316 0
\(951\) −17.7766 −0.576446
\(952\) −1.20494 −0.0390525
\(953\) −55.7114 −1.80467 −0.902335 0.431036i \(-0.858148\pi\)
−0.902335 + 0.431036i \(0.858148\pi\)
\(954\) −0.489862 −0.0158599
\(955\) 13.8509 0.448206
\(956\) 1.17742 0.0380805
\(957\) 0.551665 0.0178328
\(958\) 10.1072 0.326549
\(959\) 8.73645 0.282115
\(960\) 9.74850 0.314632
\(961\) 45.4276 1.46541
\(962\) −0.773092 −0.0249255
\(963\) 18.9401 0.610336
\(964\) −35.7218 −1.15052
\(965\) −16.4151 −0.528421
\(966\) 0.795524 0.0255956
\(967\) 32.5130 1.04555 0.522774 0.852471i \(-0.324897\pi\)
0.522774 + 0.852471i \(0.324897\pi\)
\(968\) 16.6348 0.534662
\(969\) −7.73513 −0.248488
\(970\) −3.14383 −0.100942
\(971\) −42.0401 −1.34913 −0.674565 0.738215i \(-0.735669\pi\)
−0.674565 + 0.738215i \(0.735669\pi\)
\(972\) 1.83744 0.0589360
\(973\) −10.8844 −0.348936
\(974\) 8.25198 0.264410
\(975\) 0.00163450 5.23459e−5 0
\(976\) −24.8838 −0.796511
\(977\) −19.4564 −0.622465 −0.311233 0.950334i \(-0.600742\pi\)
−0.311233 + 0.950334i \(0.600742\pi\)
\(978\) −3.66944 −0.117336
\(979\) −3.82678 −0.122304
\(980\) 26.2749 0.839322
\(981\) −6.45006 −0.205935
\(982\) −3.56859 −0.113878
\(983\) 19.2125 0.612783 0.306392 0.951906i \(-0.400878\pi\)
0.306392 + 0.951906i \(0.400878\pi\)
\(984\) 8.63548 0.275289
\(985\) 29.8419 0.950841
\(986\) 0.446275 0.0142123
\(987\) 7.94344 0.252843
\(988\) −9.58181 −0.304838
\(989\) −2.85539 −0.0907962
\(990\) 0.449434 0.0142839
\(991\) 22.4509 0.713176 0.356588 0.934262i \(-0.383940\pi\)
0.356588 + 0.934262i \(0.383940\pi\)
\(992\) 37.8062 1.20035
\(993\) 31.5772 1.00207
\(994\) −3.17704 −0.100770
\(995\) 11.7376 0.372108
\(996\) 23.7143 0.751416
\(997\) −11.4473 −0.362540 −0.181270 0.983433i \(-0.558021\pi\)
−0.181270 + 0.983433i \(0.558021\pi\)
\(998\) 14.8039 0.468610
\(999\) −2.84423 −0.0899873
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.l.1.18 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4029.2.a.l.1.18 32 1.1 even 1 trivial