Properties

Label 4033.2.a.f.1.20
Level 4033
Weight 2
Character 4033.1
Self dual yes
Analytic conductor 32.204
Analytic rank 0
Dimension 85
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4033 = 37 \cdot 109 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4033.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.2036671352\)
Analytic rank: \(0\)
Dimension: \(85\)
Coefficient ring index: multiple of None
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.20
Character \(\chi\) = 4033.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.66354 q^{2} +1.93600 q^{3} +0.767368 q^{4} -2.27919 q^{5} -3.22061 q^{6} +0.280246 q^{7} +2.05053 q^{8} +0.748091 q^{9} +O(q^{10})\) \(q-1.66354 q^{2} +1.93600 q^{3} +0.767368 q^{4} -2.27919 q^{5} -3.22061 q^{6} +0.280246 q^{7} +2.05053 q^{8} +0.748091 q^{9} +3.79153 q^{10} -0.838736 q^{11} +1.48562 q^{12} -6.44849 q^{13} -0.466201 q^{14} -4.41251 q^{15} -4.94588 q^{16} -5.62782 q^{17} -1.24448 q^{18} +7.99234 q^{19} -1.74898 q^{20} +0.542556 q^{21} +1.39527 q^{22} +1.07829 q^{23} +3.96983 q^{24} +0.194721 q^{25} +10.7273 q^{26} -4.35969 q^{27} +0.215052 q^{28} +7.63317 q^{29} +7.34040 q^{30} -6.25208 q^{31} +4.12661 q^{32} -1.62379 q^{33} +9.36210 q^{34} -0.638735 q^{35} +0.574061 q^{36} +1.00000 q^{37} -13.2956 q^{38} -12.4843 q^{39} -4.67356 q^{40} -0.298149 q^{41} -0.902565 q^{42} -2.25887 q^{43} -0.643619 q^{44} -1.70504 q^{45} -1.79378 q^{46} +4.42223 q^{47} -9.57522 q^{48} -6.92146 q^{49} -0.323927 q^{50} -10.8954 q^{51} -4.94836 q^{52} -10.6911 q^{53} +7.25253 q^{54} +1.91164 q^{55} +0.574654 q^{56} +15.4732 q^{57} -12.6981 q^{58} +3.13251 q^{59} -3.38602 q^{60} +12.4654 q^{61} +10.4006 q^{62} +0.209650 q^{63} +3.02698 q^{64} +14.6973 q^{65} +2.70124 q^{66} +12.7046 q^{67} -4.31861 q^{68} +2.08756 q^{69} +1.06256 q^{70} +11.3068 q^{71} +1.53399 q^{72} +11.0100 q^{73} -1.66354 q^{74} +0.376980 q^{75} +6.13306 q^{76} -0.235053 q^{77} +20.7681 q^{78} -16.2743 q^{79} +11.2726 q^{80} -10.6846 q^{81} +0.495983 q^{82} +7.64634 q^{83} +0.416340 q^{84} +12.8269 q^{85} +3.75772 q^{86} +14.7778 q^{87} -1.71986 q^{88} -14.3218 q^{89} +2.83641 q^{90} -1.80716 q^{91} +0.827443 q^{92} -12.1040 q^{93} -7.35655 q^{94} -18.2161 q^{95} +7.98911 q^{96} +0.322581 q^{97} +11.5141 q^{98} -0.627450 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 85q + 11q^{2} + 21q^{3} + 93q^{4} + 12q^{5} + 4q^{6} + 17q^{7} + 30q^{8} + 98q^{9} + O(q^{10}) \) \( 85q + 11q^{2} + 21q^{3} + 93q^{4} + 12q^{5} + 4q^{6} + 17q^{7} + 30q^{8} + 98q^{9} + 9q^{10} + 37q^{11} + 44q^{12} + 14q^{13} + 26q^{14} + 27q^{15} + 85q^{16} + 34q^{17} + 3q^{18} + 15q^{19} + 15q^{20} + 17q^{21} + q^{22} + 72q^{23} + 15q^{24} + 85q^{25} + 33q^{26} + 69q^{27} + 7q^{28} + 19q^{29} - 9q^{30} + 23q^{31} + 51q^{32} + 32q^{33} + 49q^{34} + 40q^{35} + 121q^{36} + 85q^{37} + 84q^{38} + 39q^{39} + 22q^{40} + 55q^{41} - 28q^{42} + 78q^{44} + 28q^{45} + 17q^{46} + 184q^{47} + 97q^{48} + 88q^{49} + 26q^{50} + 27q^{51} + 73q^{52} + 64q^{53} + 31q^{54} + 39q^{55} + 68q^{56} - 33q^{57} + 28q^{58} + 60q^{59} - 22q^{60} + 7q^{61} + 70q^{62} + 28q^{63} + 102q^{64} + 17q^{65} - 15q^{66} + 82q^{67} + 92q^{68} + 22q^{69} - 41q^{70} + 113q^{71} - 19q^{73} + 11q^{74} + 45q^{75} + 34q^{76} + 64q^{77} + 29q^{78} + 23q^{79} + 54q^{80} + 149q^{81} + 4q^{82} + 100q^{83} - 49q^{84} - 5q^{85} - 24q^{86} + 65q^{87} + 14q^{88} + 84q^{89} - 21q^{90} + 32q^{91} + 95q^{92} + 19q^{93} - 47q^{94} + 102q^{95} + 29q^{96} + 7q^{97} + 26q^{98} + 107q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.66354 −1.17630 −0.588150 0.808752i \(-0.700144\pi\)
−0.588150 + 0.808752i \(0.700144\pi\)
\(3\) 1.93600 1.11775 0.558875 0.829252i \(-0.311233\pi\)
0.558875 + 0.829252i \(0.311233\pi\)
\(4\) 0.767368 0.383684
\(5\) −2.27919 −1.01929 −0.509643 0.860386i \(-0.670223\pi\)
−0.509643 + 0.860386i \(0.670223\pi\)
\(6\) −3.22061 −1.31481
\(7\) 0.280246 0.105923 0.0529616 0.998597i \(-0.483134\pi\)
0.0529616 + 0.998597i \(0.483134\pi\)
\(8\) 2.05053 0.724973
\(9\) 0.748091 0.249364
\(10\) 3.79153 1.19899
\(11\) −0.838736 −0.252888 −0.126444 0.991974i \(-0.540356\pi\)
−0.126444 + 0.991974i \(0.540356\pi\)
\(12\) 1.48562 0.428862
\(13\) −6.44849 −1.78849 −0.894244 0.447580i \(-0.852286\pi\)
−0.894244 + 0.447580i \(0.852286\pi\)
\(14\) −0.466201 −0.124597
\(15\) −4.41251 −1.13931
\(16\) −4.94588 −1.23647
\(17\) −5.62782 −1.36495 −0.682473 0.730911i \(-0.739096\pi\)
−0.682473 + 0.730911i \(0.739096\pi\)
\(18\) −1.24448 −0.293327
\(19\) 7.99234 1.83357 0.916784 0.399383i \(-0.130776\pi\)
0.916784 + 0.399383i \(0.130776\pi\)
\(20\) −1.74898 −0.391084
\(21\) 0.542556 0.118396
\(22\) 1.39527 0.297473
\(23\) 1.07829 0.224839 0.112419 0.993661i \(-0.464140\pi\)
0.112419 + 0.993661i \(0.464140\pi\)
\(24\) 3.96983 0.810338
\(25\) 0.194721 0.0389442
\(26\) 10.7273 2.10380
\(27\) −4.35969 −0.839023
\(28\) 0.215052 0.0406410
\(29\) 7.63317 1.41744 0.708722 0.705488i \(-0.249272\pi\)
0.708722 + 0.705488i \(0.249272\pi\)
\(30\) 7.34040 1.34017
\(31\) −6.25208 −1.12291 −0.561454 0.827508i \(-0.689758\pi\)
−0.561454 + 0.827508i \(0.689758\pi\)
\(32\) 4.12661 0.729488
\(33\) −1.62379 −0.282666
\(34\) 9.36210 1.60559
\(35\) −0.638735 −0.107966
\(36\) 0.574061 0.0956768
\(37\) 1.00000 0.164399
\(38\) −13.2956 −2.15683
\(39\) −12.4843 −1.99908
\(40\) −4.67356 −0.738955
\(41\) −0.298149 −0.0465630 −0.0232815 0.999729i \(-0.507411\pi\)
−0.0232815 + 0.999729i \(0.507411\pi\)
\(42\) −0.902565 −0.139269
\(43\) −2.25887 −0.344474 −0.172237 0.985056i \(-0.555099\pi\)
−0.172237 + 0.985056i \(0.555099\pi\)
\(44\) −0.643619 −0.0970292
\(45\) −1.70504 −0.254173
\(46\) −1.79378 −0.264478
\(47\) 4.42223 0.645048 0.322524 0.946561i \(-0.395469\pi\)
0.322524 + 0.946561i \(0.395469\pi\)
\(48\) −9.57522 −1.38206
\(49\) −6.92146 −0.988780
\(50\) −0.323927 −0.0458102
\(51\) −10.8954 −1.52567
\(52\) −4.94836 −0.686214
\(53\) −10.6911 −1.46854 −0.734268 0.678859i \(-0.762475\pi\)
−0.734268 + 0.678859i \(0.762475\pi\)
\(54\) 7.25253 0.986944
\(55\) 1.91164 0.257766
\(56\) 0.574654 0.0767914
\(57\) 15.4732 2.04947
\(58\) −12.6981 −1.66734
\(59\) 3.13251 0.407818 0.203909 0.978990i \(-0.434635\pi\)
0.203909 + 0.978990i \(0.434635\pi\)
\(60\) −3.38602 −0.437134
\(61\) 12.4654 1.59604 0.798018 0.602633i \(-0.205882\pi\)
0.798018 + 0.602633i \(0.205882\pi\)
\(62\) 10.4006 1.32088
\(63\) 0.209650 0.0264134
\(64\) 3.02698 0.378373
\(65\) 14.6973 1.82298
\(66\) 2.70124 0.332500
\(67\) 12.7046 1.55211 0.776057 0.630663i \(-0.217217\pi\)
0.776057 + 0.630663i \(0.217217\pi\)
\(68\) −4.31861 −0.523708
\(69\) 2.08756 0.251313
\(70\) 1.06256 0.127000
\(71\) 11.3068 1.34187 0.670933 0.741518i \(-0.265894\pi\)
0.670933 + 0.741518i \(0.265894\pi\)
\(72\) 1.53399 0.180782
\(73\) 11.0100 1.28863 0.644313 0.764762i \(-0.277144\pi\)
0.644313 + 0.764762i \(0.277144\pi\)
\(74\) −1.66354 −0.193383
\(75\) 0.376980 0.0435299
\(76\) 6.13306 0.703511
\(77\) −0.235053 −0.0267867
\(78\) 20.7681 2.35152
\(79\) −16.2743 −1.83101 −0.915503 0.402312i \(-0.868207\pi\)
−0.915503 + 0.402312i \(0.868207\pi\)
\(80\) 11.2726 1.26032
\(81\) −10.6846 −1.18718
\(82\) 0.495983 0.0547721
\(83\) 7.64634 0.839295 0.419647 0.907687i \(-0.362154\pi\)
0.419647 + 0.907687i \(0.362154\pi\)
\(84\) 0.416340 0.0454264
\(85\) 12.8269 1.39127
\(86\) 3.75772 0.405205
\(87\) 14.7778 1.58435
\(88\) −1.71986 −0.183337
\(89\) −14.3218 −1.51811 −0.759055 0.651026i \(-0.774339\pi\)
−0.759055 + 0.651026i \(0.774339\pi\)
\(90\) 2.83641 0.298984
\(91\) −1.80716 −0.189442
\(92\) 0.827443 0.0862669
\(93\) −12.1040 −1.25513
\(94\) −7.35655 −0.758771
\(95\) −18.2161 −1.86893
\(96\) 7.98911 0.815385
\(97\) 0.322581 0.0327531 0.0163765 0.999866i \(-0.494787\pi\)
0.0163765 + 0.999866i \(0.494787\pi\)
\(98\) 11.5141 1.16310
\(99\) −0.627450 −0.0630611
\(100\) 0.149423 0.0149423
\(101\) 2.90172 0.288732 0.144366 0.989524i \(-0.453886\pi\)
0.144366 + 0.989524i \(0.453886\pi\)
\(102\) 18.1250 1.79464
\(103\) 17.9316 1.76685 0.883426 0.468571i \(-0.155231\pi\)
0.883426 + 0.468571i \(0.155231\pi\)
\(104\) −13.2228 −1.29661
\(105\) −1.23659 −0.120679
\(106\) 17.7851 1.72744
\(107\) 15.8842 1.53559 0.767793 0.640698i \(-0.221355\pi\)
0.767793 + 0.640698i \(0.221355\pi\)
\(108\) −3.34549 −0.321920
\(109\) −1.00000 −0.0957826
\(110\) −3.18009 −0.303210
\(111\) 1.93600 0.183757
\(112\) −1.38606 −0.130971
\(113\) −18.6792 −1.75719 −0.878596 0.477565i \(-0.841519\pi\)
−0.878596 + 0.477565i \(0.841519\pi\)
\(114\) −25.7402 −2.41079
\(115\) −2.45763 −0.229175
\(116\) 5.85744 0.543850
\(117\) −4.82405 −0.445984
\(118\) −5.21106 −0.479717
\(119\) −1.57717 −0.144579
\(120\) −9.04801 −0.825967
\(121\) −10.2965 −0.936047
\(122\) −20.7368 −1.87742
\(123\) −0.577216 −0.0520458
\(124\) −4.79765 −0.430842
\(125\) 10.9522 0.979591
\(126\) −0.348761 −0.0310701
\(127\) 5.55628 0.493040 0.246520 0.969138i \(-0.420713\pi\)
0.246520 + 0.969138i \(0.420713\pi\)
\(128\) −13.2887 −1.17457
\(129\) −4.37316 −0.385036
\(130\) −24.4496 −2.14437
\(131\) 18.1781 1.58822 0.794112 0.607772i \(-0.207936\pi\)
0.794112 + 0.607772i \(0.207936\pi\)
\(132\) −1.24605 −0.108454
\(133\) 2.23982 0.194217
\(134\) −21.1346 −1.82575
\(135\) 9.93658 0.855205
\(136\) −11.5400 −0.989549
\(137\) 6.57044 0.561351 0.280675 0.959803i \(-0.409442\pi\)
0.280675 + 0.959803i \(0.409442\pi\)
\(138\) −3.47275 −0.295620
\(139\) −14.8864 −1.26264 −0.631322 0.775521i \(-0.717487\pi\)
−0.631322 + 0.775521i \(0.717487\pi\)
\(140\) −0.490145 −0.0414248
\(141\) 8.56142 0.721002
\(142\) −18.8093 −1.57844
\(143\) 5.40858 0.452288
\(144\) −3.69997 −0.308331
\(145\) −17.3975 −1.44478
\(146\) −18.3156 −1.51581
\(147\) −13.3999 −1.10521
\(148\) 0.767368 0.0630772
\(149\) 19.0624 1.56165 0.780827 0.624748i \(-0.214798\pi\)
0.780827 + 0.624748i \(0.214798\pi\)
\(150\) −0.627122 −0.0512043
\(151\) −2.24853 −0.182983 −0.0914914 0.995806i \(-0.529163\pi\)
−0.0914914 + 0.995806i \(0.529163\pi\)
\(152\) 16.3886 1.32929
\(153\) −4.21012 −0.340368
\(154\) 0.391020 0.0315092
\(155\) 14.2497 1.14456
\(156\) −9.58002 −0.767015
\(157\) −11.6346 −0.928546 −0.464273 0.885692i \(-0.653684\pi\)
−0.464273 + 0.885692i \(0.653684\pi\)
\(158\) 27.0730 2.15381
\(159\) −20.6980 −1.64146
\(160\) −9.40534 −0.743557
\(161\) 0.302186 0.0238156
\(162\) 17.7743 1.39648
\(163\) −3.04285 −0.238335 −0.119167 0.992874i \(-0.538023\pi\)
−0.119167 + 0.992874i \(0.538023\pi\)
\(164\) −0.228790 −0.0178655
\(165\) 3.70093 0.288117
\(166\) −12.7200 −0.987263
\(167\) 7.23086 0.559540 0.279770 0.960067i \(-0.409742\pi\)
0.279770 + 0.960067i \(0.409742\pi\)
\(168\) 1.11253 0.0858336
\(169\) 28.5830 2.19869
\(170\) −21.3380 −1.63655
\(171\) 5.97899 0.457225
\(172\) −1.73338 −0.132169
\(173\) 21.0233 1.59837 0.799187 0.601083i \(-0.205264\pi\)
0.799187 + 0.601083i \(0.205264\pi\)
\(174\) −24.5835 −1.86367
\(175\) 0.0545699 0.00412510
\(176\) 4.14829 0.312689
\(177\) 6.06454 0.455838
\(178\) 23.8249 1.78575
\(179\) −11.3923 −0.851501 −0.425751 0.904841i \(-0.639990\pi\)
−0.425751 + 0.904841i \(0.639990\pi\)
\(180\) −1.30840 −0.0975220
\(181\) 8.47716 0.630102 0.315051 0.949075i \(-0.397978\pi\)
0.315051 + 0.949075i \(0.397978\pi\)
\(182\) 3.00629 0.222841
\(183\) 24.1331 1.78397
\(184\) 2.21107 0.163002
\(185\) −2.27919 −0.167570
\(186\) 20.1355 1.47641
\(187\) 4.72025 0.345179
\(188\) 3.39347 0.247495
\(189\) −1.22179 −0.0888720
\(190\) 30.3032 2.19843
\(191\) 11.8817 0.859727 0.429864 0.902894i \(-0.358562\pi\)
0.429864 + 0.902894i \(0.358562\pi\)
\(192\) 5.86023 0.422926
\(193\) 6.04742 0.435303 0.217651 0.976027i \(-0.430160\pi\)
0.217651 + 0.976027i \(0.430160\pi\)
\(194\) −0.536626 −0.0385275
\(195\) 28.4540 2.03764
\(196\) −5.31131 −0.379379
\(197\) 17.0340 1.21363 0.606813 0.794845i \(-0.292448\pi\)
0.606813 + 0.794845i \(0.292448\pi\)
\(198\) 1.04379 0.0741789
\(199\) −18.3588 −1.30142 −0.650712 0.759325i \(-0.725529\pi\)
−0.650712 + 0.759325i \(0.725529\pi\)
\(200\) 0.399283 0.0282335
\(201\) 24.5961 1.73487
\(202\) −4.82713 −0.339636
\(203\) 2.13917 0.150140
\(204\) −8.36081 −0.585374
\(205\) 0.679539 0.0474611
\(206\) −29.8299 −2.07835
\(207\) 0.806657 0.0560665
\(208\) 31.8934 2.21141
\(209\) −6.70346 −0.463688
\(210\) 2.05712 0.141955
\(211\) −22.1723 −1.52640 −0.763202 0.646160i \(-0.776374\pi\)
−0.763202 + 0.646160i \(0.776374\pi\)
\(212\) −8.20401 −0.563454
\(213\) 21.8899 1.49987
\(214\) −26.4241 −1.80631
\(215\) 5.14839 0.351118
\(216\) −8.93970 −0.608269
\(217\) −1.75212 −0.118942
\(218\) 1.66354 0.112669
\(219\) 21.3154 1.44036
\(220\) 1.46693 0.0989005
\(221\) 36.2909 2.44119
\(222\) −3.22061 −0.216153
\(223\) 4.62064 0.309421 0.154711 0.987960i \(-0.450555\pi\)
0.154711 + 0.987960i \(0.450555\pi\)
\(224\) 1.15647 0.0772697
\(225\) 0.145669 0.00971128
\(226\) 31.0736 2.06699
\(227\) −3.67592 −0.243979 −0.121990 0.992531i \(-0.538927\pi\)
−0.121990 + 0.992531i \(0.538927\pi\)
\(228\) 11.8736 0.786349
\(229\) −15.1612 −1.00188 −0.500940 0.865482i \(-0.667012\pi\)
−0.500940 + 0.865482i \(0.667012\pi\)
\(230\) 4.08836 0.269579
\(231\) −0.455061 −0.0299408
\(232\) 15.6521 1.02761
\(233\) 11.8018 0.773164 0.386582 0.922255i \(-0.373656\pi\)
0.386582 + 0.922255i \(0.373656\pi\)
\(234\) 8.02501 0.524611
\(235\) −10.0791 −0.657489
\(236\) 2.40379 0.156473
\(237\) −31.5071 −2.04661
\(238\) 2.62369 0.170069
\(239\) −6.27461 −0.405871 −0.202935 0.979192i \(-0.565048\pi\)
−0.202935 + 0.979192i \(0.565048\pi\)
\(240\) 21.8238 1.40872
\(241\) −12.3768 −0.797257 −0.398628 0.917113i \(-0.630514\pi\)
−0.398628 + 0.917113i \(0.630514\pi\)
\(242\) 17.1287 1.10107
\(243\) −7.60635 −0.487948
\(244\) 9.56558 0.612373
\(245\) 15.7753 1.00785
\(246\) 0.960222 0.0612215
\(247\) −51.5385 −3.27932
\(248\) −12.8201 −0.814078
\(249\) 14.8033 0.938121
\(250\) −18.2194 −1.15229
\(251\) −13.6941 −0.864365 −0.432183 0.901786i \(-0.642256\pi\)
−0.432183 + 0.901786i \(0.642256\pi\)
\(252\) 0.160878 0.0101344
\(253\) −0.904399 −0.0568591
\(254\) −9.24310 −0.579964
\(255\) 24.8328 1.55509
\(256\) 16.0524 1.00327
\(257\) −4.62370 −0.288418 −0.144209 0.989547i \(-0.546064\pi\)
−0.144209 + 0.989547i \(0.546064\pi\)
\(258\) 7.27494 0.452918
\(259\) 0.280246 0.0174137
\(260\) 11.2783 0.699448
\(261\) 5.71030 0.353459
\(262\) −30.2399 −1.86823
\(263\) 11.8481 0.730587 0.365293 0.930892i \(-0.380969\pi\)
0.365293 + 0.930892i \(0.380969\pi\)
\(264\) −3.32964 −0.204925
\(265\) 24.3671 1.49686
\(266\) −3.72604 −0.228458
\(267\) −27.7270 −1.69687
\(268\) 9.74910 0.595521
\(269\) 25.5005 1.55480 0.777398 0.629010i \(-0.216539\pi\)
0.777398 + 0.629010i \(0.216539\pi\)
\(270\) −16.5299 −1.00598
\(271\) −20.1831 −1.22603 −0.613017 0.790069i \(-0.710044\pi\)
−0.613017 + 0.790069i \(0.710044\pi\)
\(272\) 27.8345 1.68772
\(273\) −3.49867 −0.211749
\(274\) −10.9302 −0.660317
\(275\) −0.163320 −0.00984855
\(276\) 1.60193 0.0964248
\(277\) 8.63800 0.519007 0.259504 0.965742i \(-0.416441\pi\)
0.259504 + 0.965742i \(0.416441\pi\)
\(278\) 24.7641 1.48525
\(279\) −4.67713 −0.280012
\(280\) −1.30975 −0.0782724
\(281\) 4.95819 0.295781 0.147890 0.989004i \(-0.452752\pi\)
0.147890 + 0.989004i \(0.452752\pi\)
\(282\) −14.2423 −0.848115
\(283\) 28.2083 1.67681 0.838405 0.545047i \(-0.183488\pi\)
0.838405 + 0.545047i \(0.183488\pi\)
\(284\) 8.67645 0.514853
\(285\) −35.2663 −2.08900
\(286\) −8.99739 −0.532027
\(287\) −0.0835551 −0.00493210
\(288\) 3.08708 0.181908
\(289\) 14.6723 0.863078
\(290\) 28.9414 1.69950
\(291\) 0.624516 0.0366098
\(292\) 8.44873 0.494425
\(293\) −6.78686 −0.396493 −0.198246 0.980152i \(-0.563525\pi\)
−0.198246 + 0.980152i \(0.563525\pi\)
\(294\) 22.2913 1.30006
\(295\) −7.13960 −0.415683
\(296\) 2.05053 0.119185
\(297\) 3.65663 0.212179
\(298\) −31.7111 −1.83697
\(299\) −6.95332 −0.402121
\(300\) 0.289282 0.0167017
\(301\) −0.633039 −0.0364878
\(302\) 3.74052 0.215243
\(303\) 5.61773 0.322730
\(304\) −39.5292 −2.26715
\(305\) −28.4112 −1.62682
\(306\) 7.00370 0.400375
\(307\) 23.6448 1.34948 0.674739 0.738057i \(-0.264256\pi\)
0.674739 + 0.738057i \(0.264256\pi\)
\(308\) −0.180372 −0.0102776
\(309\) 34.7155 1.97490
\(310\) −23.7050 −1.34635
\(311\) 18.4730 1.04751 0.523755 0.851869i \(-0.324531\pi\)
0.523755 + 0.851869i \(0.324531\pi\)
\(312\) −25.5994 −1.44928
\(313\) 10.9631 0.619671 0.309836 0.950790i \(-0.399726\pi\)
0.309836 + 0.950790i \(0.399726\pi\)
\(314\) 19.3547 1.09225
\(315\) −0.477832 −0.0269228
\(316\) −12.4884 −0.702527
\(317\) 18.3449 1.03035 0.515176 0.857085i \(-0.327727\pi\)
0.515176 + 0.857085i \(0.327727\pi\)
\(318\) 34.4319 1.93085
\(319\) −6.40221 −0.358455
\(320\) −6.89908 −0.385670
\(321\) 30.7518 1.71640
\(322\) −0.502699 −0.0280143
\(323\) −44.9794 −2.50272
\(324\) −8.19904 −0.455502
\(325\) −1.25566 −0.0696513
\(326\) 5.06191 0.280353
\(327\) −1.93600 −0.107061
\(328\) −0.611364 −0.0337569
\(329\) 1.23931 0.0683255
\(330\) −6.15665 −0.338913
\(331\) −9.73247 −0.534945 −0.267473 0.963565i \(-0.586188\pi\)
−0.267473 + 0.963565i \(0.586188\pi\)
\(332\) 5.86755 0.322024
\(333\) 0.748091 0.0409951
\(334\) −12.0288 −0.658188
\(335\) −28.9562 −1.58205
\(336\) −2.68342 −0.146393
\(337\) −4.65652 −0.253657 −0.126828 0.991925i \(-0.540480\pi\)
−0.126828 + 0.991925i \(0.540480\pi\)
\(338\) −47.5489 −2.58632
\(339\) −36.1629 −1.96410
\(340\) 9.84294 0.533808
\(341\) 5.24385 0.283970
\(342\) −9.94630 −0.537834
\(343\) −3.90144 −0.210658
\(344\) −4.63188 −0.249734
\(345\) −4.75796 −0.256160
\(346\) −34.9731 −1.88017
\(347\) −16.7576 −0.899595 −0.449797 0.893131i \(-0.648504\pi\)
−0.449797 + 0.893131i \(0.648504\pi\)
\(348\) 11.3400 0.607888
\(349\) 8.47731 0.453780 0.226890 0.973920i \(-0.427144\pi\)
0.226890 + 0.973920i \(0.427144\pi\)
\(350\) −0.0907793 −0.00485236
\(351\) 28.1134 1.50058
\(352\) −3.46113 −0.184479
\(353\) −1.59783 −0.0850438 −0.0425219 0.999096i \(-0.513539\pi\)
−0.0425219 + 0.999096i \(0.513539\pi\)
\(354\) −10.0886 −0.536203
\(355\) −25.7703 −1.36775
\(356\) −10.9901 −0.582474
\(357\) −3.05341 −0.161603
\(358\) 18.9516 1.00162
\(359\) 13.0006 0.686145 0.343073 0.939309i \(-0.388532\pi\)
0.343073 + 0.939309i \(0.388532\pi\)
\(360\) −3.49625 −0.184268
\(361\) 44.8775 2.36197
\(362\) −14.1021 −0.741190
\(363\) −19.9341 −1.04627
\(364\) −1.38676 −0.0726859
\(365\) −25.0940 −1.31348
\(366\) −40.1464 −2.09848
\(367\) 20.5598 1.07322 0.536608 0.843832i \(-0.319706\pi\)
0.536608 + 0.843832i \(0.319706\pi\)
\(368\) −5.33308 −0.278006
\(369\) −0.223042 −0.0116111
\(370\) 3.79153 0.197112
\(371\) −2.99614 −0.155552
\(372\) −9.28824 −0.481573
\(373\) 29.3613 1.52027 0.760134 0.649766i \(-0.225133\pi\)
0.760134 + 0.649766i \(0.225133\pi\)
\(374\) −7.85233 −0.406034
\(375\) 21.2034 1.09494
\(376\) 9.06793 0.467643
\(377\) −49.2224 −2.53508
\(378\) 2.03249 0.104540
\(379\) 19.6145 1.00753 0.503765 0.863841i \(-0.331948\pi\)
0.503765 + 0.863841i \(0.331948\pi\)
\(380\) −13.9784 −0.717079
\(381\) 10.7570 0.551095
\(382\) −19.7656 −1.01130
\(383\) 2.75677 0.140864 0.0704321 0.997517i \(-0.477562\pi\)
0.0704321 + 0.997517i \(0.477562\pi\)
\(384\) −25.7270 −1.31287
\(385\) 0.535730 0.0273033
\(386\) −10.0601 −0.512047
\(387\) −1.68984 −0.0858993
\(388\) 0.247538 0.0125668
\(389\) 34.9851 1.77381 0.886907 0.461949i \(-0.152850\pi\)
0.886907 + 0.461949i \(0.152850\pi\)
\(390\) −47.3344 −2.39687
\(391\) −6.06841 −0.306893
\(392\) −14.1927 −0.716839
\(393\) 35.1927 1.77524
\(394\) −28.3368 −1.42759
\(395\) 37.0923 1.86632
\(396\) −0.481485 −0.0241955
\(397\) −4.70489 −0.236132 −0.118066 0.993006i \(-0.537669\pi\)
−0.118066 + 0.993006i \(0.537669\pi\)
\(398\) 30.5407 1.53087
\(399\) 4.33629 0.217086
\(400\) −0.963068 −0.0481534
\(401\) 17.4936 0.873590 0.436795 0.899561i \(-0.356114\pi\)
0.436795 + 0.899561i \(0.356114\pi\)
\(402\) −40.9166 −2.04073
\(403\) 40.3165 2.00831
\(404\) 2.22669 0.110782
\(405\) 24.3523 1.21008
\(406\) −3.55859 −0.176610
\(407\) −0.838736 −0.0415746
\(408\) −22.3415 −1.10607
\(409\) 16.3741 0.809649 0.404824 0.914394i \(-0.367333\pi\)
0.404824 + 0.914394i \(0.367333\pi\)
\(410\) −1.13044 −0.0558285
\(411\) 12.7204 0.627449
\(412\) 13.7601 0.677912
\(413\) 0.877874 0.0431974
\(414\) −1.34191 −0.0659511
\(415\) −17.4275 −0.855481
\(416\) −26.6104 −1.30468
\(417\) −28.8200 −1.41132
\(418\) 11.1515 0.545437
\(419\) −12.1757 −0.594821 −0.297411 0.954750i \(-0.596123\pi\)
−0.297411 + 0.954750i \(0.596123\pi\)
\(420\) −0.948920 −0.0463025
\(421\) 0.0221456 0.00107931 0.000539655 1.00000i \(-0.499828\pi\)
0.000539655 1.00000i \(0.499828\pi\)
\(422\) 36.8845 1.79551
\(423\) 3.30823 0.160851
\(424\) −21.9225 −1.06465
\(425\) −1.09586 −0.0531568
\(426\) −36.4147 −1.76430
\(427\) 3.49339 0.169057
\(428\) 12.1890 0.589180
\(429\) 10.4710 0.505544
\(430\) −8.56456 −0.413020
\(431\) −13.2466 −0.638067 −0.319034 0.947743i \(-0.603358\pi\)
−0.319034 + 0.947743i \(0.603358\pi\)
\(432\) 21.5625 1.03743
\(433\) −8.04356 −0.386549 −0.193274 0.981145i \(-0.561911\pi\)
−0.193274 + 0.981145i \(0.561911\pi\)
\(434\) 2.91473 0.139911
\(435\) −33.6815 −1.61490
\(436\) −0.767368 −0.0367503
\(437\) 8.61804 0.412257
\(438\) −35.4590 −1.69430
\(439\) −15.7492 −0.751666 −0.375833 0.926687i \(-0.622643\pi\)
−0.375833 + 0.926687i \(0.622643\pi\)
\(440\) 3.91988 0.186873
\(441\) −5.17788 −0.246566
\(442\) −60.3714 −2.87157
\(443\) 7.49477 0.356087 0.178044 0.984023i \(-0.443023\pi\)
0.178044 + 0.984023i \(0.443023\pi\)
\(444\) 1.48562 0.0705045
\(445\) 32.6422 1.54739
\(446\) −7.68663 −0.363972
\(447\) 36.9048 1.74554
\(448\) 0.848301 0.0400784
\(449\) 11.2197 0.529489 0.264744 0.964319i \(-0.414712\pi\)
0.264744 + 0.964319i \(0.414712\pi\)
\(450\) −0.242327 −0.0114234
\(451\) 0.250068 0.0117752
\(452\) −14.3338 −0.674206
\(453\) −4.35315 −0.204529
\(454\) 6.11504 0.286993
\(455\) 4.11888 0.193096
\(456\) 31.7282 1.48581
\(457\) −13.7712 −0.644189 −0.322095 0.946707i \(-0.604387\pi\)
−0.322095 + 0.946707i \(0.604387\pi\)
\(458\) 25.2212 1.17851
\(459\) 24.5356 1.14522
\(460\) −1.88590 −0.0879307
\(461\) −36.5579 −1.70267 −0.851335 0.524622i \(-0.824207\pi\)
−0.851335 + 0.524622i \(0.824207\pi\)
\(462\) 0.757013 0.0352194
\(463\) −21.5275 −1.00047 −0.500233 0.865891i \(-0.666752\pi\)
−0.500233 + 0.865891i \(0.666752\pi\)
\(464\) −37.7527 −1.75263
\(465\) 27.5874 1.27934
\(466\) −19.6328 −0.909473
\(467\) −30.2345 −1.39909 −0.699543 0.714591i \(-0.746613\pi\)
−0.699543 + 0.714591i \(0.746613\pi\)
\(468\) −3.70182 −0.171117
\(469\) 3.56042 0.164405
\(470\) 16.7670 0.773404
\(471\) −22.5247 −1.03788
\(472\) 6.42332 0.295657
\(473\) 1.89459 0.0871135
\(474\) 52.4133 2.40742
\(475\) 1.55628 0.0714069
\(476\) −1.21027 −0.0554728
\(477\) −7.99792 −0.366200
\(478\) 10.4381 0.477426
\(479\) 17.9662 0.820895 0.410447 0.911884i \(-0.365373\pi\)
0.410447 + 0.911884i \(0.365373\pi\)
\(480\) −18.2087 −0.831111
\(481\) −6.44849 −0.294026
\(482\) 20.5892 0.937814
\(483\) 0.585032 0.0266199
\(484\) −7.90122 −0.359146
\(485\) −0.735224 −0.0333848
\(486\) 12.6535 0.573973
\(487\) 5.06597 0.229561 0.114781 0.993391i \(-0.463384\pi\)
0.114781 + 0.993391i \(0.463384\pi\)
\(488\) 25.5608 1.15708
\(489\) −5.89096 −0.266398
\(490\) −26.2429 −1.18553
\(491\) −12.8341 −0.579194 −0.289597 0.957149i \(-0.593521\pi\)
−0.289597 + 0.957149i \(0.593521\pi\)
\(492\) −0.442937 −0.0199691
\(493\) −42.9581 −1.93473
\(494\) 85.7364 3.85746
\(495\) 1.43008 0.0642773
\(496\) 30.9221 1.38844
\(497\) 3.16868 0.142135
\(498\) −24.6259 −1.10351
\(499\) −38.1852 −1.70940 −0.854702 0.519119i \(-0.826260\pi\)
−0.854702 + 0.519119i \(0.826260\pi\)
\(500\) 8.40433 0.375853
\(501\) 13.9989 0.625426
\(502\) 22.7807 1.01675
\(503\) 15.9093 0.709358 0.354679 0.934988i \(-0.384590\pi\)
0.354679 + 0.934988i \(0.384590\pi\)
\(504\) 0.429894 0.0191490
\(505\) −6.61359 −0.294301
\(506\) 1.50450 0.0668834
\(507\) 55.3366 2.45758
\(508\) 4.26371 0.189172
\(509\) 36.1965 1.60438 0.802191 0.597067i \(-0.203667\pi\)
0.802191 + 0.597067i \(0.203667\pi\)
\(510\) −41.3104 −1.82926
\(511\) 3.08552 0.136495
\(512\) −0.126323 −0.00558272
\(513\) −34.8441 −1.53841
\(514\) 7.69171 0.339267
\(515\) −40.8695 −1.80093
\(516\) −3.35582 −0.147732
\(517\) −3.70908 −0.163125
\(518\) −0.466201 −0.0204837
\(519\) 40.7011 1.78658
\(520\) 30.1374 1.32161
\(521\) −7.59295 −0.332653 −0.166327 0.986071i \(-0.553191\pi\)
−0.166327 + 0.986071i \(0.553191\pi\)
\(522\) −9.49932 −0.415774
\(523\) −5.89319 −0.257691 −0.128846 0.991665i \(-0.541127\pi\)
−0.128846 + 0.991665i \(0.541127\pi\)
\(524\) 13.9493 0.609376
\(525\) 0.105647 0.00461082
\(526\) −19.7098 −0.859390
\(527\) 35.1856 1.53271
\(528\) 8.03108 0.349508
\(529\) −21.8373 −0.949448
\(530\) −40.5357 −1.76076
\(531\) 2.34340 0.101695
\(532\) 1.71877 0.0745180
\(533\) 1.92261 0.0832774
\(534\) 46.1250 1.99603
\(535\) −36.2032 −1.56520
\(536\) 26.0512 1.12524
\(537\) −22.0555 −0.951765
\(538\) −42.4212 −1.82891
\(539\) 5.80528 0.250051
\(540\) 7.62501 0.328128
\(541\) −24.5930 −1.05733 −0.528667 0.848829i \(-0.677308\pi\)
−0.528667 + 0.848829i \(0.677308\pi\)
\(542\) 33.5754 1.44219
\(543\) 16.4118 0.704297
\(544\) −23.2238 −0.995712
\(545\) 2.27919 0.0976299
\(546\) 5.82017 0.249080
\(547\) −14.2064 −0.607423 −0.303711 0.952764i \(-0.598226\pi\)
−0.303711 + 0.952764i \(0.598226\pi\)
\(548\) 5.04195 0.215381
\(549\) 9.32528 0.397993
\(550\) 0.271689 0.0115849
\(551\) 61.0068 2.59898
\(552\) 4.28062 0.182195
\(553\) −4.56082 −0.193946
\(554\) −14.3697 −0.610509
\(555\) −4.41251 −0.187301
\(556\) −11.4233 −0.484456
\(557\) 29.8773 1.26594 0.632972 0.774175i \(-0.281835\pi\)
0.632972 + 0.774175i \(0.281835\pi\)
\(558\) 7.78059 0.329379
\(559\) 14.5663 0.616088
\(560\) 3.15911 0.133497
\(561\) 9.13840 0.385824
\(562\) −8.24814 −0.347927
\(563\) 32.9464 1.38852 0.694262 0.719722i \(-0.255731\pi\)
0.694262 + 0.719722i \(0.255731\pi\)
\(564\) 6.56976 0.276637
\(565\) 42.5735 1.79108
\(566\) −46.9257 −1.97243
\(567\) −2.99433 −0.125750
\(568\) 23.1849 0.972817
\(569\) −15.8391 −0.664012 −0.332006 0.943277i \(-0.607725\pi\)
−0.332006 + 0.943277i \(0.607725\pi\)
\(570\) 58.6670 2.45729
\(571\) 24.5961 1.02931 0.514656 0.857397i \(-0.327920\pi\)
0.514656 + 0.857397i \(0.327920\pi\)
\(572\) 4.15037 0.173536
\(573\) 23.0029 0.960960
\(574\) 0.138997 0.00580164
\(575\) 0.209966 0.00875617
\(576\) 2.26446 0.0943524
\(577\) −12.3499 −0.514133 −0.257066 0.966394i \(-0.582756\pi\)
−0.257066 + 0.966394i \(0.582756\pi\)
\(578\) −24.4080 −1.01524
\(579\) 11.7078 0.486559
\(580\) −13.3502 −0.554339
\(581\) 2.14286 0.0889007
\(582\) −1.03891 −0.0430641
\(583\) 8.96702 0.371376
\(584\) 22.5764 0.934219
\(585\) 10.9949 0.454585
\(586\) 11.2902 0.466395
\(587\) 3.61984 0.149407 0.0747034 0.997206i \(-0.476199\pi\)
0.0747034 + 0.997206i \(0.476199\pi\)
\(588\) −10.2827 −0.424051
\(589\) −49.9688 −2.05893
\(590\) 11.8770 0.488969
\(591\) 32.9779 1.35653
\(592\) −4.94588 −0.203275
\(593\) 7.64434 0.313915 0.156958 0.987605i \(-0.449831\pi\)
0.156958 + 0.987605i \(0.449831\pi\)
\(594\) −6.08295 −0.249587
\(595\) 3.59469 0.147368
\(596\) 14.6279 0.599181
\(597\) −35.5427 −1.45467
\(598\) 11.5671 0.473015
\(599\) −16.9673 −0.693263 −0.346632 0.938001i \(-0.612675\pi\)
−0.346632 + 0.938001i \(0.612675\pi\)
\(600\) 0.773010 0.0315580
\(601\) 25.8925 1.05618 0.528089 0.849189i \(-0.322909\pi\)
0.528089 + 0.849189i \(0.322909\pi\)
\(602\) 1.05309 0.0429206
\(603\) 9.50419 0.387041
\(604\) −1.72545 −0.0702075
\(605\) 23.4678 0.954100
\(606\) −9.34532 −0.379628
\(607\) 11.0903 0.450140 0.225070 0.974343i \(-0.427739\pi\)
0.225070 + 0.974343i \(0.427739\pi\)
\(608\) 32.9813 1.33757
\(609\) 4.14142 0.167819
\(610\) 47.2631 1.91363
\(611\) −28.5167 −1.15366
\(612\) −3.23071 −0.130594
\(613\) 27.3252 1.10365 0.551827 0.833958i \(-0.313931\pi\)
0.551827 + 0.833958i \(0.313931\pi\)
\(614\) −39.3340 −1.58739
\(615\) 1.31559 0.0530496
\(616\) −0.481983 −0.0194197
\(617\) 17.2199 0.693249 0.346624 0.938004i \(-0.387328\pi\)
0.346624 + 0.938004i \(0.387328\pi\)
\(618\) −57.7507 −2.32307
\(619\) 11.9350 0.479709 0.239855 0.970809i \(-0.422900\pi\)
0.239855 + 0.970809i \(0.422900\pi\)
\(620\) 10.9348 0.439151
\(621\) −4.70100 −0.188645
\(622\) −30.7306 −1.23219
\(623\) −4.01364 −0.160803
\(624\) 61.7457 2.47181
\(625\) −25.9357 −1.03743
\(626\) −18.2376 −0.728920
\(627\) −12.9779 −0.518287
\(628\) −8.92805 −0.356268
\(629\) −5.62782 −0.224396
\(630\) 0.794893 0.0316693
\(631\) −10.1591 −0.404427 −0.202213 0.979341i \(-0.564813\pi\)
−0.202213 + 0.979341i \(0.564813\pi\)
\(632\) −33.3711 −1.32743
\(633\) −42.9255 −1.70614
\(634\) −30.5175 −1.21200
\(635\) −12.6638 −0.502549
\(636\) −15.8830 −0.629800
\(637\) 44.6329 1.76842
\(638\) 10.6503 0.421651
\(639\) 8.45849 0.334613
\(640\) 30.2876 1.19722
\(641\) 40.5582 1.60195 0.800977 0.598695i \(-0.204314\pi\)
0.800977 + 0.598695i \(0.204314\pi\)
\(642\) −51.1569 −2.01900
\(643\) −29.2228 −1.15243 −0.576217 0.817297i \(-0.695472\pi\)
−0.576217 + 0.817297i \(0.695472\pi\)
\(644\) 0.231888 0.00913766
\(645\) 9.96728 0.392461
\(646\) 74.8251 2.94395
\(647\) 21.9980 0.864830 0.432415 0.901675i \(-0.357662\pi\)
0.432415 + 0.901675i \(0.357662\pi\)
\(648\) −21.9092 −0.860675
\(649\) −2.62735 −0.103132
\(650\) 2.08884 0.0819309
\(651\) −3.39211 −0.132947
\(652\) −2.33499 −0.0914452
\(653\) 42.7225 1.67186 0.835930 0.548836i \(-0.184929\pi\)
0.835930 + 0.548836i \(0.184929\pi\)
\(654\) 3.22061 0.125936
\(655\) −41.4313 −1.61885
\(656\) 1.47461 0.0575738
\(657\) 8.23649 0.321336
\(658\) −2.06165 −0.0803714
\(659\) −6.12509 −0.238600 −0.119300 0.992858i \(-0.538065\pi\)
−0.119300 + 0.992858i \(0.538065\pi\)
\(660\) 2.83998 0.110546
\(661\) 11.5308 0.448496 0.224248 0.974532i \(-0.428007\pi\)
0.224248 + 0.974532i \(0.428007\pi\)
\(662\) 16.1904 0.629256
\(663\) 70.2591 2.72864
\(664\) 15.6791 0.608466
\(665\) −5.10499 −0.197963
\(666\) −1.24448 −0.0482226
\(667\) 8.23075 0.318696
\(668\) 5.54873 0.214687
\(669\) 8.94556 0.345855
\(670\) 48.1699 1.86096
\(671\) −10.4552 −0.403619
\(672\) 2.23892 0.0863681
\(673\) 14.3009 0.551258 0.275629 0.961264i \(-0.411114\pi\)
0.275629 + 0.961264i \(0.411114\pi\)
\(674\) 7.74631 0.298377
\(675\) −0.848925 −0.0326751
\(676\) 21.9336 0.843602
\(677\) −9.51420 −0.365660 −0.182830 0.983145i \(-0.558526\pi\)
−0.182830 + 0.983145i \(0.558526\pi\)
\(678\) 60.1585 2.31037
\(679\) 0.0904020 0.00346931
\(680\) 26.3020 1.00863
\(681\) −7.11657 −0.272707
\(682\) −8.72335 −0.334034
\(683\) 33.5549 1.28394 0.641971 0.766729i \(-0.278117\pi\)
0.641971 + 0.766729i \(0.278117\pi\)
\(684\) 4.58809 0.175430
\(685\) −14.9753 −0.572177
\(686\) 6.49020 0.247797
\(687\) −29.3520 −1.11985
\(688\) 11.1721 0.425932
\(689\) 68.9415 2.62646
\(690\) 7.91506 0.301321
\(691\) 43.7734 1.66522 0.832610 0.553860i \(-0.186846\pi\)
0.832610 + 0.553860i \(0.186846\pi\)
\(692\) 16.1326 0.613270
\(693\) −0.175841 −0.00667963
\(694\) 27.8769 1.05819
\(695\) 33.9289 1.28700
\(696\) 30.3024 1.14861
\(697\) 1.67793 0.0635560
\(698\) −14.1023 −0.533782
\(699\) 22.8483 0.864203
\(700\) 0.0418752 0.00158273
\(701\) −30.7993 −1.16327 −0.581637 0.813448i \(-0.697588\pi\)
−0.581637 + 0.813448i \(0.697588\pi\)
\(702\) −46.7678 −1.76514
\(703\) 7.99234 0.301437
\(704\) −2.53884 −0.0956861
\(705\) −19.5131 −0.734907
\(706\) 2.65805 0.100037
\(707\) 0.813197 0.0305834
\(708\) 4.65373 0.174898
\(709\) −11.5682 −0.434451 −0.217226 0.976121i \(-0.569701\pi\)
−0.217226 + 0.976121i \(0.569701\pi\)
\(710\) 42.8700 1.60888
\(711\) −12.1747 −0.456586
\(712\) −29.3674 −1.10059
\(713\) −6.74155 −0.252473
\(714\) 5.07947 0.190094
\(715\) −12.3272 −0.461011
\(716\) −8.74209 −0.326707
\(717\) −12.1476 −0.453662
\(718\) −21.6270 −0.807113
\(719\) −44.7252 −1.66797 −0.833985 0.551787i \(-0.813946\pi\)
−0.833985 + 0.551787i \(0.813946\pi\)
\(720\) 8.43294 0.314277
\(721\) 5.02526 0.187150
\(722\) −74.6555 −2.77839
\(723\) −23.9614 −0.891133
\(724\) 6.50510 0.241760
\(725\) 1.48634 0.0552013
\(726\) 33.1611 1.23072
\(727\) −34.3694 −1.27469 −0.637345 0.770579i \(-0.719967\pi\)
−0.637345 + 0.770579i \(0.719967\pi\)
\(728\) −3.70565 −0.137341
\(729\) 17.3280 0.641778
\(730\) 41.7448 1.54505
\(731\) 12.7125 0.470189
\(732\) 18.5189 0.684480
\(733\) 21.6862 0.801000 0.400500 0.916297i \(-0.368836\pi\)
0.400500 + 0.916297i \(0.368836\pi\)
\(734\) −34.2021 −1.26242
\(735\) 30.5411 1.12652
\(736\) 4.44967 0.164017
\(737\) −10.6558 −0.392512
\(738\) 0.371040 0.0136582
\(739\) 49.7361 1.82957 0.914787 0.403937i \(-0.132358\pi\)
0.914787 + 0.403937i \(0.132358\pi\)
\(740\) −1.74898 −0.0642938
\(741\) −99.7784 −3.66545
\(742\) 4.98421 0.182976
\(743\) −16.4574 −0.603762 −0.301881 0.953346i \(-0.597614\pi\)
−0.301881 + 0.953346i \(0.597614\pi\)
\(744\) −24.8197 −0.909935
\(745\) −43.4469 −1.59177
\(746\) −48.8436 −1.78829
\(747\) 5.72015 0.209289
\(748\) 3.62217 0.132440
\(749\) 4.45149 0.162654
\(750\) −35.2727 −1.28798
\(751\) −15.0153 −0.547917 −0.273958 0.961742i \(-0.588333\pi\)
−0.273958 + 0.961742i \(0.588333\pi\)
\(752\) −21.8718 −0.797583
\(753\) −26.5118 −0.966144
\(754\) 81.8834 2.98202
\(755\) 5.12483 0.186512
\(756\) −0.937561 −0.0340987
\(757\) −3.76004 −0.136661 −0.0683305 0.997663i \(-0.521767\pi\)
−0.0683305 + 0.997663i \(0.521767\pi\)
\(758\) −32.6295 −1.18516
\(759\) −1.75091 −0.0635542
\(760\) −37.3527 −1.35492
\(761\) −13.2940 −0.481908 −0.240954 0.970537i \(-0.577460\pi\)
−0.240954 + 0.970537i \(0.577460\pi\)
\(762\) −17.8946 −0.648254
\(763\) −0.280246 −0.0101456
\(764\) 9.11761 0.329864
\(765\) 9.59567 0.346932
\(766\) −4.58599 −0.165699
\(767\) −20.2000 −0.729378
\(768\) 31.0774 1.12141
\(769\) −20.6511 −0.744697 −0.372349 0.928093i \(-0.621447\pi\)
−0.372349 + 0.928093i \(0.621447\pi\)
\(770\) −0.891209 −0.0321169
\(771\) −8.95147 −0.322379
\(772\) 4.64059 0.167019
\(773\) −51.4157 −1.84929 −0.924647 0.380826i \(-0.875640\pi\)
−0.924647 + 0.380826i \(0.875640\pi\)
\(774\) 2.81111 0.101043
\(775\) −1.21741 −0.0437308
\(776\) 0.661462 0.0237451
\(777\) 0.542556 0.0194641
\(778\) −58.1991 −2.08654
\(779\) −2.38291 −0.0853765
\(780\) 21.8347 0.781808
\(781\) −9.48339 −0.339342
\(782\) 10.0950 0.360998
\(783\) −33.2783 −1.18927
\(784\) 34.2327 1.22260
\(785\) 26.5176 0.946454
\(786\) −58.5445 −2.08821
\(787\) −35.5400 −1.26686 −0.633432 0.773799i \(-0.718354\pi\)
−0.633432 + 0.773799i \(0.718354\pi\)
\(788\) 13.0714 0.465648
\(789\) 22.9380 0.816613
\(790\) −61.7046 −2.19535
\(791\) −5.23478 −0.186127
\(792\) −1.28661 −0.0457176
\(793\) −80.3832 −2.85449
\(794\) 7.82678 0.277762
\(795\) 47.1747 1.67311
\(796\) −14.0880 −0.499335
\(797\) −13.9000 −0.492362 −0.246181 0.969224i \(-0.579176\pi\)
−0.246181 + 0.969224i \(0.579176\pi\)
\(798\) −7.21360 −0.255359
\(799\) −24.8875 −0.880456
\(800\) 0.803538 0.0284094
\(801\) −10.7140 −0.378561
\(802\) −29.1013 −1.02760
\(803\) −9.23450 −0.325878
\(804\) 18.8742 0.665643
\(805\) −0.688741 −0.0242749
\(806\) −67.0681 −2.36237
\(807\) 49.3690 1.73787
\(808\) 5.95008 0.209323
\(809\) −22.2891 −0.783642 −0.391821 0.920042i \(-0.628155\pi\)
−0.391821 + 0.920042i \(0.628155\pi\)
\(810\) −40.5111 −1.42342
\(811\) −6.47918 −0.227515 −0.113757 0.993509i \(-0.536289\pi\)
−0.113757 + 0.993509i \(0.536289\pi\)
\(812\) 1.64153 0.0576063
\(813\) −39.0744 −1.37040
\(814\) 1.39527 0.0489042
\(815\) 6.93525 0.242931
\(816\) 53.8876 1.88644
\(817\) −18.0536 −0.631617
\(818\) −27.2390 −0.952391
\(819\) −1.35192 −0.0472400
\(820\) 0.521456 0.0182100
\(821\) 23.1854 0.809175 0.404587 0.914499i \(-0.367415\pi\)
0.404587 + 0.914499i \(0.367415\pi\)
\(822\) −21.1608 −0.738069
\(823\) −19.3137 −0.673233 −0.336617 0.941642i \(-0.609283\pi\)
−0.336617 + 0.941642i \(0.609283\pi\)
\(824\) 36.7693 1.28092
\(825\) −0.316187 −0.0110082
\(826\) −1.46038 −0.0508131
\(827\) −18.4629 −0.642019 −0.321009 0.947076i \(-0.604022\pi\)
−0.321009 + 0.947076i \(0.604022\pi\)
\(828\) 0.619003 0.0215118
\(829\) 51.5371 1.78996 0.894979 0.446108i \(-0.147190\pi\)
0.894979 + 0.446108i \(0.147190\pi\)
\(830\) 28.9913 1.00630
\(831\) 16.7232 0.580120
\(832\) −19.5195 −0.676715
\(833\) 38.9527 1.34963
\(834\) 47.9432 1.66014
\(835\) −16.4805 −0.570332
\(836\) −5.14402 −0.177910
\(837\) 27.2572 0.942146
\(838\) 20.2548 0.699689
\(839\) −26.3690 −0.910358 −0.455179 0.890400i \(-0.650425\pi\)
−0.455179 + 0.890400i \(0.650425\pi\)
\(840\) −2.53567 −0.0874890
\(841\) 29.2652 1.00915
\(842\) −0.0368401 −0.00126959
\(843\) 9.59904 0.330609
\(844\) −17.0143 −0.585656
\(845\) −65.1461 −2.24109
\(846\) −5.50337 −0.189210
\(847\) −2.88556 −0.0991491
\(848\) 52.8770 1.81580
\(849\) 54.6113 1.87425
\(850\) 1.82300 0.0625284
\(851\) 1.07829 0.0369632
\(852\) 16.7976 0.575476
\(853\) 38.9511 1.33366 0.666830 0.745210i \(-0.267651\pi\)
0.666830 + 0.745210i \(0.267651\pi\)
\(854\) −5.81140 −0.198862
\(855\) −13.6273 −0.466043
\(856\) 32.5711 1.11326
\(857\) −35.1996 −1.20239 −0.601197 0.799101i \(-0.705309\pi\)
−0.601197 + 0.799101i \(0.705309\pi\)
\(858\) −17.4189 −0.594672
\(859\) −40.2126 −1.37204 −0.686018 0.727584i \(-0.740643\pi\)
−0.686018 + 0.727584i \(0.740643\pi\)
\(860\) 3.95071 0.134718
\(861\) −0.161763 −0.00551285
\(862\) 22.0363 0.750559
\(863\) −28.3165 −0.963905 −0.481953 0.876197i \(-0.660072\pi\)
−0.481953 + 0.876197i \(0.660072\pi\)
\(864\) −17.9907 −0.612058
\(865\) −47.9162 −1.62920
\(866\) 13.3808 0.454698
\(867\) 28.4056 0.964705
\(868\) −1.34452 −0.0456361
\(869\) 13.6499 0.463040
\(870\) 56.0305 1.89961
\(871\) −81.9254 −2.77594
\(872\) −2.05053 −0.0694398
\(873\) 0.241320 0.00816743
\(874\) −14.3365 −0.484938
\(875\) 3.06930 0.103761
\(876\) 16.3567 0.552643
\(877\) −20.1023 −0.678807 −0.339404 0.940641i \(-0.610225\pi\)
−0.339404 + 0.940641i \(0.610225\pi\)
\(878\) 26.1994 0.884186
\(879\) −13.1393 −0.443179
\(880\) −9.45475 −0.318720
\(881\) −0.269412 −0.00907671 −0.00453836 0.999990i \(-0.501445\pi\)
−0.00453836 + 0.999990i \(0.501445\pi\)
\(882\) 8.61362 0.290036
\(883\) −27.8467 −0.937115 −0.468557 0.883433i \(-0.655226\pi\)
−0.468557 + 0.883433i \(0.655226\pi\)
\(884\) 27.8485 0.936645
\(885\) −13.8222 −0.464630
\(886\) −12.4679 −0.418866
\(887\) 31.6191 1.06166 0.530832 0.847477i \(-0.321880\pi\)
0.530832 + 0.847477i \(0.321880\pi\)
\(888\) 3.96983 0.133219
\(889\) 1.55713 0.0522244
\(890\) −54.3016 −1.82019
\(891\) 8.96158 0.300224
\(892\) 3.54573 0.118720
\(893\) 35.3439 1.18274
\(894\) −61.3926 −2.05328
\(895\) 25.9653 0.867924
\(896\) −3.72412 −0.124414
\(897\) −13.4616 −0.449471
\(898\) −18.6644 −0.622838
\(899\) −47.7232 −1.59166
\(900\) 0.111782 0.00372606
\(901\) 60.1676 2.00447
\(902\) −0.415998 −0.0138512
\(903\) −1.22556 −0.0407842
\(904\) −38.3024 −1.27392
\(905\) −19.3211 −0.642255
\(906\) 7.24164 0.240587
\(907\) 25.3006 0.840093 0.420047 0.907503i \(-0.362014\pi\)
0.420047 + 0.907503i \(0.362014\pi\)
\(908\) −2.82078 −0.0936109
\(909\) 2.17075 0.0719993
\(910\) −6.85192 −0.227139
\(911\) 34.9013 1.15633 0.578166 0.815919i \(-0.303769\pi\)
0.578166 + 0.815919i \(0.303769\pi\)
\(912\) −76.5284 −2.53411
\(913\) −6.41326 −0.212248
\(914\) 22.9089 0.757761
\(915\) −55.0040 −1.81837
\(916\) −11.6342 −0.384405
\(917\) 5.09433 0.168230
\(918\) −40.8159 −1.34713
\(919\) −23.2761 −0.767807 −0.383903 0.923373i \(-0.625420\pi\)
−0.383903 + 0.923373i \(0.625420\pi\)
\(920\) −5.03945 −0.166146
\(921\) 45.7762 1.50838
\(922\) 60.8155 2.00285
\(923\) −72.9115 −2.39991
\(924\) −0.349199 −0.0114878
\(925\) 0.194721 0.00640240
\(926\) 35.8118 1.17685
\(927\) 13.4144 0.440588
\(928\) 31.4991 1.03401
\(929\) −8.95774 −0.293894 −0.146947 0.989144i \(-0.546945\pi\)
−0.146947 + 0.989144i \(0.546945\pi\)
\(930\) −45.8928 −1.50488
\(931\) −55.3187 −1.81300
\(932\) 9.05635 0.296650
\(933\) 35.7638 1.17085
\(934\) 50.2963 1.64575
\(935\) −10.7584 −0.351836
\(936\) −9.89188 −0.323326
\(937\) −1.88070 −0.0614398 −0.0307199 0.999528i \(-0.509780\pi\)
−0.0307199 + 0.999528i \(0.509780\pi\)
\(938\) −5.92290 −0.193389
\(939\) 21.2245 0.692637
\(940\) −7.73438 −0.252268
\(941\) 19.2503 0.627543 0.313772 0.949498i \(-0.398407\pi\)
0.313772 + 0.949498i \(0.398407\pi\)
\(942\) 37.4707 1.22086
\(943\) −0.321490 −0.0104692
\(944\) −15.4930 −0.504255
\(945\) 2.78469 0.0905860
\(946\) −3.15173 −0.102472
\(947\) 26.0077 0.845137 0.422569 0.906331i \(-0.361129\pi\)
0.422569 + 0.906331i \(0.361129\pi\)
\(948\) −24.1775 −0.785249
\(949\) −70.9979 −2.30469
\(950\) −2.58893 −0.0839961
\(951\) 35.5157 1.15167
\(952\) −3.23405 −0.104816
\(953\) 2.68314 0.0869153 0.0434576 0.999055i \(-0.486163\pi\)
0.0434576 + 0.999055i \(0.486163\pi\)
\(954\) 13.3049 0.430761
\(955\) −27.0806 −0.876308
\(956\) −4.81493 −0.155726
\(957\) −12.3947 −0.400663
\(958\) −29.8874 −0.965619
\(959\) 1.84134 0.0594600
\(960\) −13.3566 −0.431083
\(961\) 8.08856 0.260921
\(962\) 10.7273 0.345863
\(963\) 11.8828 0.382919
\(964\) −9.49752 −0.305895
\(965\) −13.7832 −0.443698
\(966\) −0.973224 −0.0313130
\(967\) 9.28493 0.298583 0.149292 0.988793i \(-0.452301\pi\)
0.149292 + 0.988793i \(0.452301\pi\)
\(968\) −21.1134 −0.678609
\(969\) −87.0801 −2.79742
\(970\) 1.22307 0.0392705
\(971\) 55.7109 1.78785 0.893924 0.448219i \(-0.147942\pi\)
0.893924 + 0.448219i \(0.147942\pi\)
\(972\) −5.83687 −0.187218
\(973\) −4.17185 −0.133743
\(974\) −8.42745 −0.270033
\(975\) −2.43095 −0.0778527
\(976\) −61.6526 −1.97345
\(977\) 8.36084 0.267487 0.133743 0.991016i \(-0.457300\pi\)
0.133743 + 0.991016i \(0.457300\pi\)
\(978\) 9.79985 0.313365
\(979\) 12.0122 0.383912
\(980\) 12.1055 0.386696
\(981\) −0.748091 −0.0238847
\(982\) 21.3500 0.681307
\(983\) −1.96092 −0.0625438 −0.0312719 0.999511i \(-0.509956\pi\)
−0.0312719 + 0.999511i \(0.509956\pi\)
\(984\) −1.18360 −0.0377318
\(985\) −38.8239 −1.23703
\(986\) 71.4625 2.27583
\(987\) 2.39931 0.0763708
\(988\) −39.5490 −1.25822
\(989\) −2.43571 −0.0774510
\(990\) −2.37900 −0.0756095
\(991\) −30.7573 −0.977039 −0.488519 0.872553i \(-0.662463\pi\)
−0.488519 + 0.872553i \(0.662463\pi\)
\(992\) −25.7999 −0.819148
\(993\) −18.8421 −0.597934
\(994\) −5.27123 −0.167193
\(995\) 41.8433 1.32652
\(996\) 11.3596 0.359942
\(997\) 15.8530 0.502069 0.251035 0.967978i \(-0.419229\pi\)
0.251035 + 0.967978i \(0.419229\pi\)
\(998\) 63.5226 2.01077
\(999\) −4.35969 −0.137935
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 4033.2.a.f.1.20 85
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4033.2.a.f.1.20 85 1.1 even 1 trivial