Properties

Label 418.2.a.h.1.2
Level $418$
Weight $2$
Character 418.1
Self dual yes
Analytic conductor $3.338$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 418 = 2 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 418.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.33774680449\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.469.1
Defining polynomial: \( x^{3} - x^{2} - 5x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.772866\) of defining polynomial
Character \(\chi\) \(=\) 418.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +0.772866 q^{3} +1.00000 q^{4} +1.22713 q^{5} +0.772866 q^{6} +3.40268 q^{7} +1.00000 q^{8} -2.40268 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +0.772866 q^{3} +1.00000 q^{4} +1.22713 q^{5} +0.772866 q^{6} +3.40268 q^{7} +1.00000 q^{8} -2.40268 q^{9} +1.22713 q^{10} -1.00000 q^{11} +0.772866 q^{12} -1.40268 q^{13} +3.40268 q^{14} +0.948410 q^{15} +1.00000 q^{16} -4.80536 q^{17} -2.40268 q^{18} -1.00000 q^{19} +1.22713 q^{20} +2.62981 q^{21} -1.00000 q^{22} +6.80536 q^{23} +0.772866 q^{24} -3.49414 q^{25} -1.40268 q^{26} -4.17554 q^{27} +3.40268 q^{28} +8.03249 q^{29} +0.948410 q^{30} -4.94841 q^{31} +1.00000 q^{32} -0.772866 q^{33} -4.80536 q^{34} +4.17554 q^{35} -2.40268 q^{36} +2.35109 q^{37} -1.00000 q^{38} -1.08408 q^{39} +1.22713 q^{40} +2.94841 q^{41} +2.62981 q^{42} -9.12395 q^{43} -1.00000 q^{44} -2.94841 q^{45} +6.80536 q^{46} +5.25963 q^{47} +0.772866 q^{48} +4.57822 q^{49} -3.49414 q^{50} -3.71390 q^{51} -1.40268 q^{52} -4.45427 q^{53} -4.17554 q^{54} -1.22713 q^{55} +3.40268 q^{56} -0.772866 q^{57} +8.03249 q^{58} -14.5193 q^{59} +0.948410 q^{60} +2.00000 q^{61} -4.94841 q^{62} -8.17554 q^{63} +1.00000 q^{64} -1.72128 q^{65} -0.772866 q^{66} +3.40268 q^{67} -4.80536 q^{68} +5.25963 q^{69} +4.17554 q^{70} -7.57822 q^{71} -2.40268 q^{72} -10.0650 q^{73} +2.35109 q^{74} -2.70050 q^{75} -1.00000 q^{76} -3.40268 q^{77} -1.08408 q^{78} -3.71390 q^{79} +1.22713 q^{80} +3.98090 q^{81} +2.94841 q^{82} +10.6697 q^{83} +2.62981 q^{84} -5.89682 q^{85} -9.12395 q^{86} +6.20804 q^{87} -1.00000 q^{88} +14.9883 q^{89} -2.94841 q^{90} -4.77287 q^{91} +6.80536 q^{92} -3.82446 q^{93} +5.25963 q^{94} -1.22713 q^{95} +0.772866 q^{96} +2.35109 q^{97} +4.57822 q^{98} +2.40268 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + q^{3} + 3 q^{4} + 5 q^{5} + q^{6} + q^{7} + 3 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} + q^{3} + 3 q^{4} + 5 q^{5} + q^{6} + q^{7} + 3 q^{8} + 2 q^{9} + 5 q^{10} - 3 q^{11} + q^{12} + 5 q^{13} + q^{14} - 9 q^{15} + 3 q^{16} + 4 q^{17} + 2 q^{18} - 3 q^{19} + 5 q^{20} - 3 q^{22} + 2 q^{23} + q^{24} + 4 q^{25} + 5 q^{26} - 2 q^{27} + q^{28} + 7 q^{29} - 9 q^{30} - 3 q^{31} + 3 q^{32} - q^{33} + 4 q^{34} + 2 q^{35} + 2 q^{36} - 14 q^{37} - 3 q^{38} + 2 q^{39} + 5 q^{40} - 3 q^{41} - 5 q^{43} - 3 q^{44} + 3 q^{45} + 2 q^{46} + q^{48} - 6 q^{49} + 4 q^{50} + 2 q^{51} + 5 q^{52} - 16 q^{53} - 2 q^{54} - 5 q^{55} + q^{56} - q^{57} + 7 q^{58} - 12 q^{59} - 9 q^{60} + 6 q^{61} - 3 q^{62} - 14 q^{63} + 3 q^{64} + 8 q^{65} - q^{66} + q^{67} + 4 q^{68} + 2 q^{70} - 3 q^{71} + 2 q^{72} + 4 q^{73} - 14 q^{74} - 41 q^{75} - 3 q^{76} - q^{77} + 2 q^{78} + 2 q^{79} + 5 q^{80} - 17 q^{81} - 3 q^{82} + 7 q^{83} + 6 q^{85} - 5 q^{86} - 9 q^{87} - 3 q^{88} + 16 q^{89} + 3 q^{90} - 13 q^{91} + 2 q^{92} - 22 q^{93} - 5 q^{95} + q^{96} - 14 q^{97} - 6 q^{98} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0.772866 0.446214 0.223107 0.974794i \(-0.428380\pi\)
0.223107 + 0.974794i \(0.428380\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.22713 0.548791 0.274396 0.961617i \(-0.411522\pi\)
0.274396 + 0.961617i \(0.411522\pi\)
\(6\) 0.772866 0.315521
\(7\) 3.40268 1.28609 0.643046 0.765828i \(-0.277670\pi\)
0.643046 + 0.765828i \(0.277670\pi\)
\(8\) 1.00000 0.353553
\(9\) −2.40268 −0.800893
\(10\) 1.22713 0.388054
\(11\) −1.00000 −0.301511
\(12\) 0.772866 0.223107
\(13\) −1.40268 −0.389033 −0.194517 0.980899i \(-0.562314\pi\)
−0.194517 + 0.980899i \(0.562314\pi\)
\(14\) 3.40268 0.909404
\(15\) 0.948410 0.244878
\(16\) 1.00000 0.250000
\(17\) −4.80536 −1.16547 −0.582735 0.812662i \(-0.698018\pi\)
−0.582735 + 0.812662i \(0.698018\pi\)
\(18\) −2.40268 −0.566317
\(19\) −1.00000 −0.229416
\(20\) 1.22713 0.274396
\(21\) 2.62981 0.573872
\(22\) −1.00000 −0.213201
\(23\) 6.80536 1.41902 0.709508 0.704698i \(-0.248917\pi\)
0.709508 + 0.704698i \(0.248917\pi\)
\(24\) 0.772866 0.157761
\(25\) −3.49414 −0.698828
\(26\) −1.40268 −0.275088
\(27\) −4.17554 −0.803584
\(28\) 3.40268 0.643046
\(29\) 8.03249 1.49160 0.745798 0.666172i \(-0.232068\pi\)
0.745798 + 0.666172i \(0.232068\pi\)
\(30\) 0.948410 0.173155
\(31\) −4.94841 −0.888761 −0.444380 0.895838i \(-0.646576\pi\)
−0.444380 + 0.895838i \(0.646576\pi\)
\(32\) 1.00000 0.176777
\(33\) −0.772866 −0.134539
\(34\) −4.80536 −0.824112
\(35\) 4.17554 0.705796
\(36\) −2.40268 −0.400446
\(37\) 2.35109 0.386517 0.193258 0.981148i \(-0.438094\pi\)
0.193258 + 0.981148i \(0.438094\pi\)
\(38\) −1.00000 −0.162221
\(39\) −1.08408 −0.173592
\(40\) 1.22713 0.194027
\(41\) 2.94841 0.460464 0.230232 0.973136i \(-0.426051\pi\)
0.230232 + 0.973136i \(0.426051\pi\)
\(42\) 2.62981 0.405789
\(43\) −9.12395 −1.39139 −0.695695 0.718337i \(-0.744903\pi\)
−0.695695 + 0.718337i \(0.744903\pi\)
\(44\) −1.00000 −0.150756
\(45\) −2.94841 −0.439523
\(46\) 6.80536 1.00340
\(47\) 5.25963 0.767195 0.383598 0.923500i \(-0.374685\pi\)
0.383598 + 0.923500i \(0.374685\pi\)
\(48\) 0.772866 0.111554
\(49\) 4.57822 0.654032
\(50\) −3.49414 −0.494146
\(51\) −3.71390 −0.520049
\(52\) −1.40268 −0.194517
\(53\) −4.45427 −0.611841 −0.305920 0.952057i \(-0.598964\pi\)
−0.305920 + 0.952057i \(0.598964\pi\)
\(54\) −4.17554 −0.568220
\(55\) −1.22713 −0.165467
\(56\) 3.40268 0.454702
\(57\) −0.772866 −0.102369
\(58\) 8.03249 1.05472
\(59\) −14.5193 −1.89025 −0.945123 0.326715i \(-0.894058\pi\)
−0.945123 + 0.326715i \(0.894058\pi\)
\(60\) 0.948410 0.122439
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −4.94841 −0.628449
\(63\) −8.17554 −1.03002
\(64\) 1.00000 0.125000
\(65\) −1.72128 −0.213498
\(66\) −0.772866 −0.0951332
\(67\) 3.40268 0.415703 0.207852 0.978160i \(-0.433353\pi\)
0.207852 + 0.978160i \(0.433353\pi\)
\(68\) −4.80536 −0.582735
\(69\) 5.25963 0.633185
\(70\) 4.17554 0.499073
\(71\) −7.57822 −0.899370 −0.449685 0.893187i \(-0.648464\pi\)
−0.449685 + 0.893187i \(0.648464\pi\)
\(72\) −2.40268 −0.283158
\(73\) −10.0650 −1.17802 −0.589009 0.808127i \(-0.700482\pi\)
−0.589009 + 0.808127i \(0.700482\pi\)
\(74\) 2.35109 0.273309
\(75\) −2.70050 −0.311827
\(76\) −1.00000 −0.114708
\(77\) −3.40268 −0.387771
\(78\) −1.08408 −0.122748
\(79\) −3.71390 −0.417846 −0.208923 0.977932i \(-0.566996\pi\)
−0.208923 + 0.977932i \(0.566996\pi\)
\(80\) 1.22713 0.137198
\(81\) 3.98090 0.442322
\(82\) 2.94841 0.325597
\(83\) 10.6697 1.17115 0.585575 0.810618i \(-0.300869\pi\)
0.585575 + 0.810618i \(0.300869\pi\)
\(84\) 2.62981 0.286936
\(85\) −5.89682 −0.639600
\(86\) −9.12395 −0.983861
\(87\) 6.20804 0.665571
\(88\) −1.00000 −0.106600
\(89\) 14.9883 1.58875 0.794377 0.607425i \(-0.207797\pi\)
0.794377 + 0.607425i \(0.207797\pi\)
\(90\) −2.94841 −0.310790
\(91\) −4.77287 −0.500332
\(92\) 6.80536 0.709508
\(93\) −3.82446 −0.396578
\(94\) 5.25963 0.542489
\(95\) −1.22713 −0.125901
\(96\) 0.772866 0.0788803
\(97\) 2.35109 0.238717 0.119358 0.992851i \(-0.461916\pi\)
0.119358 + 0.992851i \(0.461916\pi\)
\(98\) 4.57822 0.462470
\(99\) 2.40268 0.241478
\(100\) −3.49414 −0.349414
\(101\) 14.0650 1.39952 0.699759 0.714379i \(-0.253291\pi\)
0.699759 + 0.714379i \(0.253291\pi\)
\(102\) −3.71390 −0.367730
\(103\) −15.5782 −1.53497 −0.767484 0.641068i \(-0.778492\pi\)
−0.767484 + 0.641068i \(0.778492\pi\)
\(104\) −1.40268 −0.137544
\(105\) 3.22713 0.314936
\(106\) −4.45427 −0.432637
\(107\) −4.35109 −0.420636 −0.210318 0.977633i \(-0.567450\pi\)
−0.210318 + 0.977633i \(0.567450\pi\)
\(108\) −4.17554 −0.401792
\(109\) 6.00000 0.574696 0.287348 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) −1.22713 −0.117003
\(111\) 1.81708 0.172469
\(112\) 3.40268 0.321523
\(113\) −19.9618 −1.87785 −0.938924 0.344124i \(-0.888176\pi\)
−0.938924 + 0.344124i \(0.888176\pi\)
\(114\) −0.772866 −0.0723855
\(115\) 8.35109 0.778743
\(116\) 8.03249 0.745798
\(117\) 3.37019 0.311574
\(118\) −14.5193 −1.33661
\(119\) −16.3511 −1.49890
\(120\) 0.948410 0.0865776
\(121\) 1.00000 0.0909091
\(122\) 2.00000 0.181071
\(123\) 2.27872 0.205466
\(124\) −4.94841 −0.444380
\(125\) −10.4235 −0.932302
\(126\) −8.17554 −0.728335
\(127\) 11.7139 1.03944 0.519720 0.854337i \(-0.326036\pi\)
0.519720 + 0.854337i \(0.326036\pi\)
\(128\) 1.00000 0.0883883
\(129\) −7.05159 −0.620858
\(130\) −1.72128 −0.150966
\(131\) 1.85695 0.162242 0.0811211 0.996704i \(-0.474150\pi\)
0.0811211 + 0.996704i \(0.474150\pi\)
\(132\) −0.772866 −0.0672693
\(133\) −3.40268 −0.295050
\(134\) 3.40268 0.293947
\(135\) −5.12395 −0.441000
\(136\) −4.80536 −0.412056
\(137\) 10.5973 0.905390 0.452695 0.891665i \(-0.350463\pi\)
0.452695 + 0.891665i \(0.350463\pi\)
\(138\) 5.25963 0.447729
\(139\) 14.7347 1.24978 0.624889 0.780713i \(-0.285144\pi\)
0.624889 + 0.780713i \(0.285144\pi\)
\(140\) 4.17554 0.352898
\(141\) 4.06498 0.342333
\(142\) −7.57822 −0.635950
\(143\) 1.40268 0.117298
\(144\) −2.40268 −0.200223
\(145\) 9.85695 0.818575
\(146\) −10.0650 −0.832984
\(147\) 3.53835 0.291838
\(148\) 2.35109 0.193258
\(149\) 18.3511 1.50338 0.751690 0.659517i \(-0.229239\pi\)
0.751690 + 0.659517i \(0.229239\pi\)
\(150\) −2.70050 −0.220495
\(151\) −7.64891 −0.622460 −0.311230 0.950335i \(-0.600741\pi\)
−0.311230 + 0.950335i \(0.600741\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 11.5457 0.933417
\(154\) −3.40268 −0.274196
\(155\) −6.07236 −0.487744
\(156\) −1.08408 −0.0867960
\(157\) 8.38358 0.669083 0.334541 0.942381i \(-0.391419\pi\)
0.334541 + 0.942381i \(0.391419\pi\)
\(158\) −3.71390 −0.295462
\(159\) −3.44255 −0.273012
\(160\) 1.22713 0.0970135
\(161\) 23.1564 1.82498
\(162\) 3.98090 0.312769
\(163\) −14.2479 −1.11598 −0.557991 0.829847i \(-0.688428\pi\)
−0.557991 + 0.829847i \(0.688428\pi\)
\(164\) 2.94841 0.230232
\(165\) −0.948410 −0.0738336
\(166\) 10.6697 0.828128
\(167\) −10.8703 −0.841172 −0.420586 0.907253i \(-0.638176\pi\)
−0.420586 + 0.907253i \(0.638176\pi\)
\(168\) 2.62981 0.202894
\(169\) −11.0325 −0.848653
\(170\) −5.89682 −0.452265
\(171\) 2.40268 0.183737
\(172\) −9.12395 −0.695695
\(173\) 3.39530 0.258140 0.129070 0.991635i \(-0.458801\pi\)
0.129070 + 0.991635i \(0.458801\pi\)
\(174\) 6.20804 0.470630
\(175\) −11.8894 −0.898757
\(176\) −1.00000 −0.0753778
\(177\) −11.2214 −0.843454
\(178\) 14.9883 1.12342
\(179\) −0.311217 −0.0232614 −0.0116307 0.999932i \(-0.503702\pi\)
−0.0116307 + 0.999932i \(0.503702\pi\)
\(180\) −2.94841 −0.219762
\(181\) −15.3246 −1.13907 −0.569535 0.821967i \(-0.692877\pi\)
−0.569535 + 0.821967i \(0.692877\pi\)
\(182\) −4.77287 −0.353788
\(183\) 1.54573 0.114264
\(184\) 6.80536 0.501698
\(185\) 2.88510 0.212117
\(186\) −3.82446 −0.280423
\(187\) 4.80536 0.351403
\(188\) 5.25963 0.383598
\(189\) −14.2080 −1.03348
\(190\) −1.22713 −0.0890257
\(191\) 26.8703 1.94427 0.972135 0.234422i \(-0.0753199\pi\)
0.972135 + 0.234422i \(0.0753199\pi\)
\(192\) 0.772866 0.0557768
\(193\) 18.5665 1.33645 0.668223 0.743961i \(-0.267055\pi\)
0.668223 + 0.743961i \(0.267055\pi\)
\(194\) 2.35109 0.168798
\(195\) −1.33031 −0.0952658
\(196\) 4.57822 0.327016
\(197\) 14.9085 1.06219 0.531095 0.847312i \(-0.321781\pi\)
0.531095 + 0.847312i \(0.321781\pi\)
\(198\) 2.40268 0.170751
\(199\) 15.1564 1.07441 0.537206 0.843451i \(-0.319480\pi\)
0.537206 + 0.843451i \(0.319480\pi\)
\(200\) −3.49414 −0.247073
\(201\) 2.62981 0.185493
\(202\) 14.0650 0.989609
\(203\) 27.3320 1.91833
\(204\) −3.71390 −0.260025
\(205\) 3.61810 0.252699
\(206\) −15.5782 −1.08539
\(207\) −16.3511 −1.13648
\(208\) −1.40268 −0.0972583
\(209\) 1.00000 0.0691714
\(210\) 3.22713 0.222693
\(211\) 11.1564 0.768041 0.384021 0.923324i \(-0.374539\pi\)
0.384021 + 0.923324i \(0.374539\pi\)
\(212\) −4.45427 −0.305920
\(213\) −5.85695 −0.401311
\(214\) −4.35109 −0.297434
\(215\) −11.1963 −0.763583
\(216\) −4.17554 −0.284110
\(217\) −16.8378 −1.14303
\(218\) 6.00000 0.406371
\(219\) −7.77888 −0.525648
\(220\) −1.22713 −0.0827334
\(221\) 6.74037 0.453407
\(222\) 1.81708 0.121954
\(223\) 1.61072 0.107861 0.0539307 0.998545i \(-0.482825\pi\)
0.0539307 + 0.998545i \(0.482825\pi\)
\(224\) 3.40268 0.227351
\(225\) 8.39530 0.559687
\(226\) −19.9618 −1.32784
\(227\) −16.7672 −1.11288 −0.556438 0.830889i \(-0.687832\pi\)
−0.556438 + 0.830889i \(0.687832\pi\)
\(228\) −0.772866 −0.0511843
\(229\) −8.55913 −0.565603 −0.282801 0.959178i \(-0.591264\pi\)
−0.282801 + 0.959178i \(0.591264\pi\)
\(230\) 8.35109 0.550654
\(231\) −2.62981 −0.173029
\(232\) 8.03249 0.527359
\(233\) 5.44255 0.356553 0.178277 0.983980i \(-0.442948\pi\)
0.178277 + 0.983980i \(0.442948\pi\)
\(234\) 3.37019 0.220316
\(235\) 6.45427 0.421030
\(236\) −14.5193 −0.945123
\(237\) −2.87034 −0.186449
\(238\) −16.3511 −1.05988
\(239\) 11.2921 0.730426 0.365213 0.930924i \(-0.380996\pi\)
0.365213 + 0.930924i \(0.380996\pi\)
\(240\) 0.948410 0.0612196
\(241\) 11.4101 0.734987 0.367493 0.930026i \(-0.380216\pi\)
0.367493 + 0.930026i \(0.380216\pi\)
\(242\) 1.00000 0.0642824
\(243\) 15.6033 1.00095
\(244\) 2.00000 0.128037
\(245\) 5.61810 0.358927
\(246\) 2.27872 0.145286
\(247\) 1.40268 0.0892503
\(248\) −4.94841 −0.314224
\(249\) 8.24623 0.522584
\(250\) −10.4235 −0.659237
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) −8.17554 −0.515011
\(253\) −6.80536 −0.427849
\(254\) 11.7139 0.734995
\(255\) −4.55745 −0.285399
\(256\) 1.00000 0.0625000
\(257\) 24.2331 1.51162 0.755811 0.654790i \(-0.227243\pi\)
0.755811 + 0.654790i \(0.227243\pi\)
\(258\) −7.05159 −0.439013
\(259\) 8.00000 0.497096
\(260\) −1.72128 −0.106749
\(261\) −19.2995 −1.19461
\(262\) 1.85695 0.114723
\(263\) 20.5665 1.26819 0.634093 0.773257i \(-0.281374\pi\)
0.634093 + 0.773257i \(0.281374\pi\)
\(264\) −0.772866 −0.0475666
\(265\) −5.46599 −0.335773
\(266\) −3.40268 −0.208632
\(267\) 11.5839 0.708925
\(268\) 3.40268 0.207852
\(269\) −1.36281 −0.0830918 −0.0415459 0.999137i \(-0.513228\pi\)
−0.0415459 + 0.999137i \(0.513228\pi\)
\(270\) −5.12395 −0.311834
\(271\) −23.1963 −1.40908 −0.704538 0.709666i \(-0.748846\pi\)
−0.704538 + 0.709666i \(0.748846\pi\)
\(272\) −4.80536 −0.291368
\(273\) −3.68878 −0.223255
\(274\) 10.5973 0.640208
\(275\) 3.49414 0.210705
\(276\) 5.25963 0.316592
\(277\) 31.0385 1.86492 0.932462 0.361269i \(-0.117656\pi\)
0.932462 + 0.361269i \(0.117656\pi\)
\(278\) 14.7347 0.883727
\(279\) 11.8894 0.711802
\(280\) 4.17554 0.249537
\(281\) −2.77287 −0.165415 −0.0827076 0.996574i \(-0.526357\pi\)
−0.0827076 + 0.996574i \(0.526357\pi\)
\(282\) 4.06498 0.242066
\(283\) −9.23451 −0.548935 −0.274467 0.961596i \(-0.588502\pi\)
−0.274467 + 0.961596i \(0.588502\pi\)
\(284\) −7.57822 −0.449685
\(285\) −0.948410 −0.0561790
\(286\) 1.40268 0.0829421
\(287\) 10.0325 0.592199
\(288\) −2.40268 −0.141579
\(289\) 6.09146 0.358321
\(290\) 9.85695 0.578820
\(291\) 1.81708 0.106519
\(292\) −10.0650 −0.589009
\(293\) 17.7390 1.03632 0.518162 0.855283i \(-0.326616\pi\)
0.518162 + 0.855283i \(0.326616\pi\)
\(294\) 3.53835 0.206361
\(295\) −17.8171 −1.03735
\(296\) 2.35109 0.136654
\(297\) 4.17554 0.242290
\(298\) 18.3511 1.06305
\(299\) −9.54573 −0.552044
\(300\) −2.70050 −0.155914
\(301\) −31.0459 −1.78946
\(302\) −7.64891 −0.440145
\(303\) 10.8703 0.624485
\(304\) −1.00000 −0.0573539
\(305\) 2.45427 0.140531
\(306\) 11.5457 0.660026
\(307\) 20.9735 1.19702 0.598511 0.801115i \(-0.295759\pi\)
0.598511 + 0.801115i \(0.295759\pi\)
\(308\) −3.40268 −0.193886
\(309\) −12.0399 −0.684924
\(310\) −6.07236 −0.344887
\(311\) −3.36281 −0.190687 −0.0953436 0.995444i \(-0.530395\pi\)
−0.0953436 + 0.995444i \(0.530395\pi\)
\(312\) −1.08408 −0.0613741
\(313\) −4.94103 −0.279284 −0.139642 0.990202i \(-0.544595\pi\)
−0.139642 + 0.990202i \(0.544595\pi\)
\(314\) 8.38358 0.473113
\(315\) −10.0325 −0.565267
\(316\) −3.71390 −0.208923
\(317\) 14.2861 0.802388 0.401194 0.915993i \(-0.368595\pi\)
0.401194 + 0.915993i \(0.368595\pi\)
\(318\) −3.44255 −0.193049
\(319\) −8.03249 −0.449733
\(320\) 1.22713 0.0685989
\(321\) −3.36281 −0.187694
\(322\) 23.1564 1.29046
\(323\) 4.80536 0.267377
\(324\) 3.98090 0.221161
\(325\) 4.90116 0.271867
\(326\) −14.2479 −0.789119
\(327\) 4.63719 0.256437
\(328\) 2.94841 0.162799
\(329\) 17.8968 0.986684
\(330\) −0.948410 −0.0522082
\(331\) −31.9293 −1.75499 −0.877497 0.479582i \(-0.840788\pi\)
−0.877497 + 0.479582i \(0.840788\pi\)
\(332\) 10.6697 0.585575
\(333\) −5.64891 −0.309558
\(334\) −10.8703 −0.594799
\(335\) 4.17554 0.228134
\(336\) 2.62981 0.143468
\(337\) −8.84523 −0.481830 −0.240915 0.970546i \(-0.577448\pi\)
−0.240915 + 0.970546i \(0.577448\pi\)
\(338\) −11.0325 −0.600088
\(339\) −15.4278 −0.837923
\(340\) −5.89682 −0.319800
\(341\) 4.94841 0.267971
\(342\) 2.40268 0.129922
\(343\) −8.24053 −0.444947
\(344\) −9.12395 −0.491931
\(345\) 6.45427 0.347486
\(346\) 3.39530 0.182532
\(347\) 5.27439 0.283144 0.141572 0.989928i \(-0.454784\pi\)
0.141572 + 0.989928i \(0.454784\pi\)
\(348\) 6.20804 0.332786
\(349\) −10.9085 −0.583921 −0.291960 0.956430i \(-0.594308\pi\)
−0.291960 + 0.956430i \(0.594308\pi\)
\(350\) −11.8894 −0.635517
\(351\) 5.85695 0.312621
\(352\) −1.00000 −0.0533002
\(353\) −22.1300 −1.17786 −0.588930 0.808184i \(-0.700451\pi\)
−0.588930 + 0.808184i \(0.700451\pi\)
\(354\) −11.2214 −0.596412
\(355\) −9.29950 −0.493566
\(356\) 14.9883 0.794377
\(357\) −12.6372 −0.668831
\(358\) −0.311217 −0.0164483
\(359\) −16.3910 −0.865082 −0.432541 0.901614i \(-0.642383\pi\)
−0.432541 + 0.901614i \(0.642383\pi\)
\(360\) −2.94841 −0.155395
\(361\) 1.00000 0.0526316
\(362\) −15.3246 −0.805444
\(363\) 0.772866 0.0405649
\(364\) −4.77287 −0.250166
\(365\) −12.3511 −0.646486
\(366\) 1.54573 0.0807967
\(367\) −32.3511 −1.68871 −0.844357 0.535782i \(-0.820017\pi\)
−0.844357 + 0.535782i \(0.820017\pi\)
\(368\) 6.80536 0.354754
\(369\) −7.08408 −0.368783
\(370\) 2.88510 0.149989
\(371\) −15.1564 −0.786883
\(372\) −3.82446 −0.198289
\(373\) 17.5782 0.910166 0.455083 0.890449i \(-0.349610\pi\)
0.455083 + 0.890449i \(0.349610\pi\)
\(374\) 4.80536 0.248479
\(375\) −8.05593 −0.416006
\(376\) 5.25963 0.271245
\(377\) −11.2670 −0.580280
\(378\) −14.2080 −0.730783
\(379\) 8.93365 0.458891 0.229445 0.973322i \(-0.426309\pi\)
0.229445 + 0.973322i \(0.426309\pi\)
\(380\) −1.22713 −0.0629507
\(381\) 9.05327 0.463813
\(382\) 26.8703 1.37481
\(383\) −16.0399 −0.819599 −0.409800 0.912176i \(-0.634401\pi\)
−0.409800 + 0.912176i \(0.634401\pi\)
\(384\) 0.772866 0.0394401
\(385\) −4.17554 −0.212805
\(386\) 18.5665 0.945010
\(387\) 21.9219 1.11435
\(388\) 2.35109 0.119358
\(389\) 17.2271 0.873450 0.436725 0.899595i \(-0.356138\pi\)
0.436725 + 0.899595i \(0.356138\pi\)
\(390\) −1.33031 −0.0673631
\(391\) −32.7022 −1.65382
\(392\) 4.57822 0.231235
\(393\) 1.43517 0.0723948
\(394\) 14.9085 0.751081
\(395\) −4.55745 −0.229310
\(396\) 2.40268 0.120739
\(397\) 15.0784 0.756762 0.378381 0.925650i \(-0.376481\pi\)
0.378381 + 0.925650i \(0.376481\pi\)
\(398\) 15.1564 0.759724
\(399\) −2.62981 −0.131655
\(400\) −3.49414 −0.174707
\(401\) −13.1564 −0.657002 −0.328501 0.944504i \(-0.606543\pi\)
−0.328501 + 0.944504i \(0.606543\pi\)
\(402\) 2.62981 0.131163
\(403\) 6.94103 0.345757
\(404\) 14.0650 0.699759
\(405\) 4.88510 0.242743
\(406\) 27.3320 1.35646
\(407\) −2.35109 −0.116539
\(408\) −3.71390 −0.183865
\(409\) −16.3836 −0.810116 −0.405058 0.914291i \(-0.632749\pi\)
−0.405058 + 0.914291i \(0.632749\pi\)
\(410\) 3.61810 0.178685
\(411\) 8.19030 0.403998
\(412\) −15.5782 −0.767484
\(413\) −49.4044 −2.43103
\(414\) −16.3511 −0.803612
\(415\) 13.0931 0.642717
\(416\) −1.40268 −0.0687720
\(417\) 11.3879 0.557669
\(418\) 1.00000 0.0489116
\(419\) 1.12966 0.0551874 0.0275937 0.999619i \(-0.491216\pi\)
0.0275937 + 0.999619i \(0.491216\pi\)
\(420\) 3.22713 0.157468
\(421\) −2.63719 −0.128529 −0.0642645 0.997933i \(-0.520470\pi\)
−0.0642645 + 0.997933i \(0.520470\pi\)
\(422\) 11.1564 0.543087
\(423\) −12.6372 −0.614441
\(424\) −4.45427 −0.216318
\(425\) 16.7906 0.814464
\(426\) −5.85695 −0.283770
\(427\) 6.80536 0.329334
\(428\) −4.35109 −0.210318
\(429\) 1.08408 0.0523400
\(430\) −11.1963 −0.539934
\(431\) −28.4958 −1.37260 −0.686298 0.727321i \(-0.740765\pi\)
−0.686298 + 0.727321i \(0.740765\pi\)
\(432\) −4.17554 −0.200896
\(433\) −37.5075 −1.80250 −0.901249 0.433302i \(-0.857348\pi\)
−0.901249 + 0.433302i \(0.857348\pi\)
\(434\) −16.8378 −0.808243
\(435\) 7.61810 0.365260
\(436\) 6.00000 0.287348
\(437\) −6.80536 −0.325544
\(438\) −7.77888 −0.371689
\(439\) −32.1300 −1.53348 −0.766740 0.641958i \(-0.778122\pi\)
−0.766740 + 0.641958i \(0.778122\pi\)
\(440\) −1.22713 −0.0585013
\(441\) −11.0000 −0.523810
\(442\) 6.74037 0.320607
\(443\) 33.0385 1.56971 0.784853 0.619681i \(-0.212738\pi\)
0.784853 + 0.619681i \(0.212738\pi\)
\(444\) 1.81708 0.0862346
\(445\) 18.3926 0.871895
\(446\) 1.61072 0.0762696
\(447\) 14.1829 0.670829
\(448\) 3.40268 0.160761
\(449\) 7.88206 0.371977 0.185989 0.982552i \(-0.440451\pi\)
0.185989 + 0.982552i \(0.440451\pi\)
\(450\) 8.39530 0.395758
\(451\) −2.94841 −0.138835
\(452\) −19.9618 −0.938924
\(453\) −5.91158 −0.277750
\(454\) −16.7672 −0.786922
\(455\) −5.85695 −0.274578
\(456\) −0.772866 −0.0361927
\(457\) 24.9353 1.16643 0.583213 0.812320i \(-0.301795\pi\)
0.583213 + 0.812320i \(0.301795\pi\)
\(458\) −8.55913 −0.399942
\(459\) 20.0650 0.936553
\(460\) 8.35109 0.389372
\(461\) −26.6874 −1.24296 −0.621478 0.783431i \(-0.713468\pi\)
−0.621478 + 0.783431i \(0.713468\pi\)
\(462\) −2.62981 −0.122350
\(463\) −32.7022 −1.51980 −0.759900 0.650041i \(-0.774752\pi\)
−0.759900 + 0.650041i \(0.774752\pi\)
\(464\) 8.03249 0.372899
\(465\) −4.69312 −0.217638
\(466\) 5.44255 0.252121
\(467\) 11.3776 0.526491 0.263246 0.964729i \(-0.415207\pi\)
0.263246 + 0.964729i \(0.415207\pi\)
\(468\) 3.37019 0.155787
\(469\) 11.5782 0.534633
\(470\) 6.45427 0.297713
\(471\) 6.47938 0.298554
\(472\) −14.5193 −0.668303
\(473\) 9.12395 0.419520
\(474\) −2.87034 −0.131839
\(475\) 3.49414 0.160322
\(476\) −16.3511 −0.749451
\(477\) 10.7022 0.490019
\(478\) 11.2921 0.516489
\(479\) −12.5973 −0.575586 −0.287793 0.957693i \(-0.592922\pi\)
−0.287793 + 0.957693i \(0.592922\pi\)
\(480\) 0.948410 0.0432888
\(481\) −3.29782 −0.150368
\(482\) 11.4101 0.519714
\(483\) 17.8968 0.814334
\(484\) 1.00000 0.0454545
\(485\) 2.88510 0.131006
\(486\) 15.6033 0.707782
\(487\) 9.69616 0.439375 0.219688 0.975570i \(-0.429496\pi\)
0.219688 + 0.975570i \(0.429496\pi\)
\(488\) 2.00000 0.0905357
\(489\) −11.0117 −0.497967
\(490\) 5.61810 0.253800
\(491\) −11.7538 −0.530440 −0.265220 0.964188i \(-0.585445\pi\)
−0.265220 + 0.964188i \(0.585445\pi\)
\(492\) 2.27872 0.102733
\(493\) −38.5990 −1.73841
\(494\) 1.40268 0.0631095
\(495\) 2.94841 0.132521
\(496\) −4.94841 −0.222190
\(497\) −25.7863 −1.15667
\(498\) 8.24623 0.369523
\(499\) 26.6640 1.19364 0.596822 0.802374i \(-0.296430\pi\)
0.596822 + 0.802374i \(0.296430\pi\)
\(500\) −10.4235 −0.466151
\(501\) −8.40131 −0.375343
\(502\) −12.0000 −0.535586
\(503\) 11.1622 0.497696 0.248848 0.968543i \(-0.419948\pi\)
0.248848 + 0.968543i \(0.419948\pi\)
\(504\) −8.17554 −0.364168
\(505\) 17.2596 0.768043
\(506\) −6.80536 −0.302535
\(507\) −8.52663 −0.378681
\(508\) 11.7139 0.519720
\(509\) −35.9618 −1.59398 −0.796989 0.603993i \(-0.793575\pi\)
−0.796989 + 0.603993i \(0.793575\pi\)
\(510\) −4.55745 −0.201807
\(511\) −34.2479 −1.51504
\(512\) 1.00000 0.0441942
\(513\) 4.17554 0.184355
\(514\) 24.2331 1.06888
\(515\) −19.1166 −0.842377
\(516\) −7.05159 −0.310429
\(517\) −5.25963 −0.231318
\(518\) 8.00000 0.351500
\(519\) 2.62411 0.115186
\(520\) −1.72128 −0.0754829
\(521\) 10.1300 0.443802 0.221901 0.975069i \(-0.428774\pi\)
0.221901 + 0.975069i \(0.428774\pi\)
\(522\) −19.2995 −0.844716
\(523\) −7.93502 −0.346974 −0.173487 0.984836i \(-0.555503\pi\)
−0.173487 + 0.984836i \(0.555503\pi\)
\(524\) 1.85695 0.0811211
\(525\) −9.18894 −0.401038
\(526\) 20.5665 0.896742
\(527\) 23.7789 1.03582
\(528\) −0.772866 −0.0336347
\(529\) 23.3129 1.01360
\(530\) −5.46599 −0.237427
\(531\) 34.8851 1.51388
\(532\) −3.40268 −0.147525
\(533\) −4.13567 −0.179136
\(534\) 11.5839 0.501286
\(535\) −5.33937 −0.230841
\(536\) 3.40268 0.146973
\(537\) −0.240529 −0.0103796
\(538\) −1.36281 −0.0587548
\(539\) −4.57822 −0.197198
\(540\) −5.12395 −0.220500
\(541\) 10.8436 0.466201 0.233100 0.972453i \(-0.425113\pi\)
0.233100 + 0.972453i \(0.425113\pi\)
\(542\) −23.1963 −0.996367
\(543\) −11.8439 −0.508269
\(544\) −4.80536 −0.206028
\(545\) 7.36281 0.315388
\(546\) −3.68878 −0.157865
\(547\) −10.3129 −0.440947 −0.220474 0.975393i \(-0.570760\pi\)
−0.220474 + 0.975393i \(0.570760\pi\)
\(548\) 10.5973 0.452695
\(549\) −4.80536 −0.205088
\(550\) 3.49414 0.148991
\(551\) −8.03249 −0.342196
\(552\) 5.25963 0.223865
\(553\) −12.6372 −0.537388
\(554\) 31.0385 1.31870
\(555\) 2.22980 0.0946496
\(556\) 14.7347 0.624889
\(557\) −14.4811 −0.613582 −0.306791 0.951777i \(-0.599255\pi\)
−0.306791 + 0.951777i \(0.599255\pi\)
\(558\) 11.8894 0.503320
\(559\) 12.7980 0.541297
\(560\) 4.17554 0.176449
\(561\) 3.71390 0.156801
\(562\) −2.77287 −0.116966
\(563\) 10.5340 0.443956 0.221978 0.975052i \(-0.428749\pi\)
0.221978 + 0.975052i \(0.428749\pi\)
\(564\) 4.06498 0.171167
\(565\) −24.4958 −1.03055
\(566\) −9.23451 −0.388156
\(567\) 13.5457 0.568867
\(568\) −7.57822 −0.317975
\(569\) 10.3762 0.434993 0.217496 0.976061i \(-0.430211\pi\)
0.217496 + 0.976061i \(0.430211\pi\)
\(570\) −0.948410 −0.0397245
\(571\) 33.2847 1.39292 0.696461 0.717594i \(-0.254757\pi\)
0.696461 + 0.717594i \(0.254757\pi\)
\(572\) 1.40268 0.0586489
\(573\) 20.7672 0.867561
\(574\) 10.0325 0.418748
\(575\) −23.7789 −0.991648
\(576\) −2.40268 −0.100112
\(577\) −2.93365 −0.122129 −0.0610647 0.998134i \(-0.519450\pi\)
−0.0610647 + 0.998134i \(0.519450\pi\)
\(578\) 6.09146 0.253371
\(579\) 14.3494 0.596341
\(580\) 9.85695 0.409287
\(581\) 36.3055 1.50621
\(582\) 1.81708 0.0753202
\(583\) 4.45427 0.184477
\(584\) −10.0650 −0.416492
\(585\) 4.13567 0.170989
\(586\) 17.7390 0.732792
\(587\) 8.90854 0.367695 0.183847 0.982955i \(-0.441145\pi\)
0.183847 + 0.982955i \(0.441145\pi\)
\(588\) 3.53835 0.145919
\(589\) 4.94841 0.203896
\(590\) −17.8171 −0.733517
\(591\) 11.5223 0.473964
\(592\) 2.35109 0.0966292
\(593\) 16.3129 0.669890 0.334945 0.942238i \(-0.391282\pi\)
0.334945 + 0.942238i \(0.391282\pi\)
\(594\) 4.17554 0.171325
\(595\) −20.0650 −0.822584
\(596\) 18.3511 0.751690
\(597\) 11.7139 0.479418
\(598\) −9.54573 −0.390354
\(599\) 40.9826 1.67450 0.837251 0.546818i \(-0.184161\pi\)
0.837251 + 0.546818i \(0.184161\pi\)
\(600\) −2.70050 −0.110248
\(601\) 0.349413 0.0142528 0.00712642 0.999975i \(-0.497732\pi\)
0.00712642 + 0.999975i \(0.497732\pi\)
\(602\) −31.0459 −1.26534
\(603\) −8.17554 −0.332934
\(604\) −7.64891 −0.311230
\(605\) 1.22713 0.0498901
\(606\) 10.8703 0.441577
\(607\) 5.56049 0.225693 0.112847 0.993612i \(-0.464003\pi\)
0.112847 + 0.993612i \(0.464003\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 21.1240 0.855986
\(610\) 2.45427 0.0993704
\(611\) −7.37757 −0.298464
\(612\) 11.5457 0.466709
\(613\) −19.7407 −0.797319 −0.398659 0.917099i \(-0.630524\pi\)
−0.398659 + 0.917099i \(0.630524\pi\)
\(614\) 20.9735 0.846422
\(615\) 2.79630 0.112758
\(616\) −3.40268 −0.137098
\(617\) 10.5973 0.426632 0.213316 0.976983i \(-0.431574\pi\)
0.213316 + 0.976983i \(0.431574\pi\)
\(618\) −12.0399 −0.484315
\(619\) 4.22112 0.169661 0.0848306 0.996395i \(-0.472965\pi\)
0.0848306 + 0.996395i \(0.472965\pi\)
\(620\) −6.07236 −0.243872
\(621\) −28.4161 −1.14030
\(622\) −3.36281 −0.134836
\(623\) 51.0003 2.04328
\(624\) −1.08408 −0.0433980
\(625\) 4.67973 0.187189
\(626\) −4.94103 −0.197483
\(627\) 0.772866 0.0308653
\(628\) 8.38358 0.334541
\(629\) −11.2978 −0.450474
\(630\) −10.0325 −0.399704
\(631\) 13.6905 0.545009 0.272504 0.962155i \(-0.412148\pi\)
0.272504 + 0.962155i \(0.412148\pi\)
\(632\) −3.71390 −0.147731
\(633\) 8.62243 0.342711
\(634\) 14.2861 0.567374
\(635\) 14.3745 0.570436
\(636\) −3.44255 −0.136506
\(637\) −6.42178 −0.254440
\(638\) −8.03249 −0.318009
\(639\) 18.2080 0.720299
\(640\) 1.22713 0.0485067
\(641\) −42.9883 −1.69794 −0.848968 0.528445i \(-0.822775\pi\)
−0.848968 + 0.528445i \(0.822775\pi\)
\(642\) −3.36281 −0.132719
\(643\) 16.3658 0.645406 0.322703 0.946500i \(-0.395408\pi\)
0.322703 + 0.946500i \(0.395408\pi\)
\(644\) 23.1564 0.912492
\(645\) −8.65325 −0.340721
\(646\) 4.80536 0.189064
\(647\) 39.9886 1.57211 0.786057 0.618154i \(-0.212119\pi\)
0.786057 + 0.618154i \(0.212119\pi\)
\(648\) 3.98090 0.156385
\(649\) 14.5193 0.569931
\(650\) 4.90116 0.192239
\(651\) −13.0134 −0.510035
\(652\) −14.2479 −0.557991
\(653\) −37.5976 −1.47131 −0.735655 0.677357i \(-0.763125\pi\)
−0.735655 + 0.677357i \(0.763125\pi\)
\(654\) 4.63719 0.181329
\(655\) 2.27872 0.0890371
\(656\) 2.94841 0.115116
\(657\) 24.1829 0.943466
\(658\) 17.8968 0.697691
\(659\) −11.5075 −0.448270 −0.224135 0.974558i \(-0.571956\pi\)
−0.224135 + 0.974558i \(0.571956\pi\)
\(660\) −0.948410 −0.0369168
\(661\) −29.2362 −1.13716 −0.568578 0.822629i \(-0.692506\pi\)
−0.568578 + 0.822629i \(0.692506\pi\)
\(662\) −31.9293 −1.24097
\(663\) 5.20940 0.202316
\(664\) 10.6697 0.414064
\(665\) −4.17554 −0.161921
\(666\) −5.64891 −0.218891
\(667\) 54.6640 2.11660
\(668\) −10.8703 −0.420586
\(669\) 1.24487 0.0481293
\(670\) 4.17554 0.161315
\(671\) −2.00000 −0.0772091
\(672\) 2.62981 0.101447
\(673\) −47.0784 −1.81474 −0.907369 0.420335i \(-0.861913\pi\)
−0.907369 + 0.420335i \(0.861913\pi\)
\(674\) −8.84523 −0.340706
\(675\) 14.5899 0.561567
\(676\) −11.0325 −0.424327
\(677\) 43.2539 1.66238 0.831192 0.555986i \(-0.187659\pi\)
0.831192 + 0.555986i \(0.187659\pi\)
\(678\) −15.4278 −0.592501
\(679\) 8.00000 0.307012
\(680\) −5.89682 −0.226133
\(681\) −12.9588 −0.496581
\(682\) 4.94841 0.189484
\(683\) 34.8556 1.33371 0.666856 0.745187i \(-0.267640\pi\)
0.666856 + 0.745187i \(0.267640\pi\)
\(684\) 2.40268 0.0918687
\(685\) 13.0043 0.496870
\(686\) −8.24053 −0.314625
\(687\) −6.61505 −0.252380
\(688\) −9.12395 −0.347847
\(689\) 6.24791 0.238026
\(690\) 6.45427 0.245710
\(691\) 28.4811 1.08347 0.541735 0.840549i \(-0.317768\pi\)
0.541735 + 0.840549i \(0.317768\pi\)
\(692\) 3.39530 0.129070
\(693\) 8.17554 0.310563
\(694\) 5.27439 0.200213
\(695\) 18.0814 0.685867
\(696\) 6.20804 0.235315
\(697\) −14.1682 −0.536657
\(698\) −10.9085 −0.412894
\(699\) 4.20636 0.159099
\(700\) −11.8894 −0.449379
\(701\) 12.8703 0.486106 0.243053 0.970013i \(-0.421851\pi\)
0.243053 + 0.970013i \(0.421851\pi\)
\(702\) 5.85695 0.221056
\(703\) −2.35109 −0.0886730
\(704\) −1.00000 −0.0376889
\(705\) 4.98828 0.187870
\(706\) −22.1300 −0.832872
\(707\) 47.8586 1.79991
\(708\) −11.2214 −0.421727
\(709\) 1.41744 0.0532330 0.0266165 0.999646i \(-0.491527\pi\)
0.0266165 + 0.999646i \(0.491527\pi\)
\(710\) −9.29950 −0.349004
\(711\) 8.92330 0.334650
\(712\) 14.9883 0.561710
\(713\) −33.6757 −1.26116
\(714\) −12.6372 −0.472935
\(715\) 1.72128 0.0643721
\(716\) −0.311217 −0.0116307
\(717\) 8.72729 0.325927
\(718\) −16.3910 −0.611705
\(719\) −31.1564 −1.16194 −0.580970 0.813925i \(-0.697327\pi\)
−0.580970 + 0.813925i \(0.697327\pi\)
\(720\) −2.94841 −0.109881
\(721\) −53.0077 −1.97411
\(722\) 1.00000 0.0372161
\(723\) 8.81844 0.327961
\(724\) −15.3246 −0.569535
\(725\) −28.0667 −1.04237
\(726\) 0.772866 0.0286837
\(727\) −8.57221 −0.317926 −0.158963 0.987285i \(-0.550815\pi\)
−0.158963 + 0.987285i \(0.550815\pi\)
\(728\) −4.77287 −0.176894
\(729\) 0.116574 0.00431757
\(730\) −12.3511 −0.457134
\(731\) 43.8439 1.62162
\(732\) 1.54573 0.0571319
\(733\) 44.6787 1.65025 0.825123 0.564952i \(-0.191105\pi\)
0.825123 + 0.564952i \(0.191105\pi\)
\(734\) −32.3511 −1.19410
\(735\) 4.34203 0.160158
\(736\) 6.80536 0.250849
\(737\) −3.40268 −0.125339
\(738\) −7.08408 −0.260769
\(739\) 29.4898 1.08480 0.542400 0.840120i \(-0.317516\pi\)
0.542400 + 0.840120i \(0.317516\pi\)
\(740\) 2.88510 0.106058
\(741\) 1.08408 0.0398248
\(742\) −15.1564 −0.556411
\(743\) −13.2596 −0.486449 −0.243224 0.969970i \(-0.578205\pi\)
−0.243224 + 0.969970i \(0.578205\pi\)
\(744\) −3.82446 −0.140211
\(745\) 22.5193 0.825042
\(746\) 17.5782 0.643584
\(747\) −25.6358 −0.937966
\(748\) 4.80536 0.175701
\(749\) −14.8054 −0.540976
\(750\) −8.05593 −0.294161
\(751\) −10.1829 −0.371580 −0.185790 0.982589i \(-0.559484\pi\)
−0.185790 + 0.982589i \(0.559484\pi\)
\(752\) 5.25963 0.191799
\(753\) −9.27439 −0.337977
\(754\) −11.2670 −0.410320
\(755\) −9.38624 −0.341600
\(756\) −14.2080 −0.516741
\(757\) 40.1122 1.45790 0.728952 0.684565i \(-0.240008\pi\)
0.728952 + 0.684565i \(0.240008\pi\)
\(758\) 8.93365 0.324485
\(759\) −5.25963 −0.190912
\(760\) −1.22713 −0.0445128
\(761\) −38.7819 −1.40584 −0.702922 0.711267i \(-0.748122\pi\)
−0.702922 + 0.711267i \(0.748122\pi\)
\(762\) 9.05327 0.327965
\(763\) 20.4161 0.739111
\(764\) 26.8703 0.972135
\(765\) 14.1682 0.512251
\(766\) −16.0399 −0.579544
\(767\) 20.3658 0.735368
\(768\) 0.772866 0.0278884
\(769\) 8.15340 0.294019 0.147010 0.989135i \(-0.453035\pi\)
0.147010 + 0.989135i \(0.453035\pi\)
\(770\) −4.17554 −0.150476
\(771\) 18.7290 0.674507
\(772\) 18.5665 0.668223
\(773\) −24.3893 −0.877222 −0.438611 0.898677i \(-0.644529\pi\)
−0.438611 + 0.898677i \(0.644529\pi\)
\(774\) 21.9219 0.787968
\(775\) 17.2904 0.621091
\(776\) 2.35109 0.0843992
\(777\) 6.18292 0.221811
\(778\) 17.2271 0.617623
\(779\) −2.94841 −0.105638
\(780\) −1.33031 −0.0476329
\(781\) 7.57822 0.271170
\(782\) −32.7022 −1.16943
\(783\) −33.5400 −1.19862
\(784\) 4.57822 0.163508
\(785\) 10.2878 0.367187
\(786\) 1.43517 0.0511909
\(787\) −46.3779 −1.65319 −0.826596 0.562795i \(-0.809726\pi\)
−0.826596 + 0.562795i \(0.809726\pi\)
\(788\) 14.9085 0.531095
\(789\) 15.8951 0.565882
\(790\) −4.55745 −0.162147
\(791\) −67.9236 −2.41509
\(792\) 2.40268 0.0853755
\(793\) −2.80536 −0.0996212
\(794\) 15.0784 0.535112
\(795\) −4.22447 −0.149827
\(796\) 15.1564 0.537206
\(797\) 3.89682 0.138032 0.0690162 0.997616i \(-0.478014\pi\)
0.0690162 + 0.997616i \(0.478014\pi\)
\(798\) −2.62981 −0.0930944
\(799\) −25.2744 −0.894144
\(800\) −3.49414 −0.123537
\(801\) −36.0120 −1.27242
\(802\) −13.1564 −0.464570
\(803\) 10.0650 0.355186
\(804\) 2.62981 0.0927464
\(805\) 28.4161 1.00153
\(806\) 6.94103 0.244487
\(807\) −1.05327 −0.0370767
\(808\) 14.0650 0.494804
\(809\) −43.3896 −1.52550 −0.762748 0.646695i \(-0.776151\pi\)
−0.762748 + 0.646695i \(0.776151\pi\)
\(810\) 4.88510 0.171645
\(811\) −12.9233 −0.453798 −0.226899 0.973918i \(-0.572859\pi\)
−0.226899 + 0.973918i \(0.572859\pi\)
\(812\) 27.3320 0.959165
\(813\) −17.9276 −0.628750
\(814\) −2.35109 −0.0824056
\(815\) −17.4841 −0.612441
\(816\) −3.71390 −0.130012
\(817\) 9.12395 0.319207
\(818\) −16.3836 −0.572838
\(819\) 11.4677 0.400713
\(820\) 3.61810 0.126349
\(821\) −36.5842 −1.27680 −0.638399 0.769705i \(-0.720403\pi\)
−0.638399 + 0.769705i \(0.720403\pi\)
\(822\) 8.19030 0.285670
\(823\) −21.6107 −0.753302 −0.376651 0.926355i \(-0.622924\pi\)
−0.376651 + 0.926355i \(0.622924\pi\)
\(824\) −15.5782 −0.542693
\(825\) 2.70050 0.0940194
\(826\) −49.4044 −1.71900
\(827\) −34.8851 −1.21307 −0.606537 0.795055i \(-0.707442\pi\)
−0.606537 + 0.795055i \(0.707442\pi\)
\(828\) −16.3511 −0.568240
\(829\) 31.7407 1.10240 0.551200 0.834373i \(-0.314170\pi\)
0.551200 + 0.834373i \(0.314170\pi\)
\(830\) 13.0931 0.454469
\(831\) 23.9886 0.832155
\(832\) −1.40268 −0.0486291
\(833\) −22.0000 −0.762255
\(834\) 11.3879 0.394331
\(835\) −13.3394 −0.461628
\(836\) 1.00000 0.0345857
\(837\) 20.6623 0.714194
\(838\) 1.12966 0.0390234
\(839\) 13.4603 0.464701 0.232350 0.972632i \(-0.425358\pi\)
0.232350 + 0.972632i \(0.425358\pi\)
\(840\) 3.22713 0.111347
\(841\) 35.5209 1.22486
\(842\) −2.63719 −0.0908837
\(843\) −2.14305 −0.0738106
\(844\) 11.1564 0.384021
\(845\) −13.5384 −0.465733
\(846\) −12.6372 −0.434476
\(847\) 3.40268 0.116917
\(848\) −4.45427 −0.152960
\(849\) −7.13704 −0.244943
\(850\) 16.7906 0.575913
\(851\) 16.0000 0.548473
\(852\) −5.85695 −0.200656
\(853\) 36.2981 1.24282 0.621412 0.783484i \(-0.286559\pi\)
0.621412 + 0.783484i \(0.286559\pi\)
\(854\) 6.80536 0.232875
\(855\) 2.94841 0.100833
\(856\) −4.35109 −0.148717
\(857\) −43.8569 −1.49812 −0.749062 0.662499i \(-0.769496\pi\)
−0.749062 + 0.662499i \(0.769496\pi\)
\(858\) 1.08408 0.0370100
\(859\) −41.1980 −1.40566 −0.702829 0.711359i \(-0.748080\pi\)
−0.702829 + 0.711359i \(0.748080\pi\)
\(860\) −11.1963 −0.381791
\(861\) 7.75377 0.264248
\(862\) −28.4958 −0.970571
\(863\) −24.8720 −0.846653 −0.423327 0.905977i \(-0.639138\pi\)
−0.423327 + 0.905977i \(0.639138\pi\)
\(864\) −4.17554 −0.142055
\(865\) 4.16649 0.141665
\(866\) −37.5075 −1.27456
\(867\) 4.70788 0.159888
\(868\) −16.8378 −0.571514
\(869\) 3.71390 0.125985
\(870\) 7.61810 0.258278
\(871\) −4.77287 −0.161722
\(872\) 6.00000 0.203186
\(873\) −5.64891 −0.191187
\(874\) −6.80536 −0.230195
\(875\) −35.4677 −1.19903
\(876\) −7.77888 −0.262824
\(877\) 39.8911 1.34703 0.673514 0.739175i \(-0.264784\pi\)
0.673514 + 0.739175i \(0.264784\pi\)
\(878\) −32.1300 −1.08433
\(879\) 13.7099 0.462422
\(880\) −1.22713 −0.0413667
\(881\) −2.88343 −0.0971451 −0.0485725 0.998820i \(-0.515467\pi\)
−0.0485725 + 0.998820i \(0.515467\pi\)
\(882\) −11.0000 −0.370389
\(883\) −4.27134 −0.143742 −0.0718711 0.997414i \(-0.522897\pi\)
−0.0718711 + 0.997414i \(0.522897\pi\)
\(884\) 6.74037 0.226703
\(885\) −13.7702 −0.462880
\(886\) 33.0385 1.10995
\(887\) 22.3129 0.749194 0.374597 0.927188i \(-0.377781\pi\)
0.374597 + 0.927188i \(0.377781\pi\)
\(888\) 1.81708 0.0609771
\(889\) 39.8586 1.33682
\(890\) 18.3926 0.616523
\(891\) −3.98090 −0.133365
\(892\) 1.61072 0.0539307
\(893\) −5.25963 −0.176007
\(894\) 14.1829 0.474348
\(895\) −0.381905 −0.0127657
\(896\) 3.40268 0.113676
\(897\) −7.37757 −0.246330
\(898\) 7.88206 0.263028
\(899\) −39.7481 −1.32567
\(900\) 8.39530 0.279843
\(901\) 21.4044 0.713082
\(902\) −2.94841 −0.0981713
\(903\) −23.9943 −0.798480
\(904\) −19.9618 −0.663920
\(905\) −18.8054 −0.625111
\(906\) −5.91158 −0.196399
\(907\) −35.5578 −1.18068 −0.590338 0.807156i \(-0.701006\pi\)
−0.590338 + 0.807156i \(0.701006\pi\)
\(908\) −16.7672 −0.556438
\(909\) −33.7936 −1.12086
\(910\) −5.85695 −0.194156
\(911\) −49.1980 −1.63000 −0.815001 0.579459i \(-0.803264\pi\)
−0.815001 + 0.579459i \(0.803264\pi\)
\(912\) −0.772866 −0.0255921
\(913\) −10.6697 −0.353115
\(914\) 24.9353 0.824787
\(915\) 1.89682 0.0627069
\(916\) −8.55913 −0.282801
\(917\) 6.31860 0.208658
\(918\) 20.0650 0.662243
\(919\) −8.47200 −0.279466 −0.139733 0.990189i \(-0.544624\pi\)
−0.139733 + 0.990189i \(0.544624\pi\)
\(920\) 8.35109 0.275327
\(921\) 16.2097 0.534128
\(922\) −26.6874 −0.878903
\(923\) 10.6298 0.349885
\(924\) −2.62981 −0.0865145
\(925\) −8.21504 −0.270109
\(926\) −32.7022 −1.07466
\(927\) 37.4295 1.22934
\(928\) 8.03249 0.263679
\(929\) −16.0017 −0.524998 −0.262499 0.964932i \(-0.584547\pi\)
−0.262499 + 0.964932i \(0.584547\pi\)
\(930\) −4.69312 −0.153894
\(931\) −4.57822 −0.150045
\(932\) 5.44255 0.178277
\(933\) −2.59900 −0.0850874
\(934\) 11.3776 0.372285
\(935\) 5.89682 0.192847
\(936\) 3.37019 0.110158
\(937\) −15.9766 −0.521932 −0.260966 0.965348i \(-0.584041\pi\)
−0.260966 + 0.965348i \(0.584041\pi\)
\(938\) 11.5782 0.378042
\(939\) −3.81875 −0.124620
\(940\) 6.45427 0.210515
\(941\) −21.0915 −0.687562 −0.343781 0.939050i \(-0.611708\pi\)
−0.343781 + 0.939050i \(0.611708\pi\)
\(942\) 6.47938 0.211110
\(943\) 20.0650 0.653406
\(944\) −14.5193 −0.472561
\(945\) −17.4352 −0.567166
\(946\) 9.12395 0.296645
\(947\) 20.9735 0.681548 0.340774 0.940145i \(-0.389311\pi\)
0.340774 + 0.940145i \(0.389311\pi\)
\(948\) −2.87034 −0.0932244
\(949\) 14.1179 0.458288
\(950\) 3.49414 0.113365
\(951\) 11.0412 0.358037
\(952\) −16.3511 −0.529942
\(953\) 3.27439 0.106068 0.0530339 0.998593i \(-0.483111\pi\)
0.0530339 + 0.998593i \(0.483111\pi\)
\(954\) 10.7022 0.346496
\(955\) 32.9735 1.06700
\(956\) 11.2921 0.365213
\(957\) −6.20804 −0.200677
\(958\) −12.5973 −0.407001
\(959\) 36.0593 1.16441
\(960\) 0.948410 0.0306098
\(961\) −6.51324 −0.210104
\(962\) −3.29782 −0.106326
\(963\) 10.4543 0.336884
\(964\) 11.4101 0.367493
\(965\) 22.7836 0.733430
\(966\) 17.8968 0.575821
\(967\) −29.2744 −0.941401 −0.470700 0.882293i \(-0.655999\pi\)
−0.470700 + 0.882293i \(0.655999\pi\)
\(968\) 1.00000 0.0321412
\(969\) 3.71390 0.119308
\(970\) 2.88510 0.0926350
\(971\) −38.3836 −1.23179 −0.615894 0.787829i \(-0.711205\pi\)
−0.615894 + 0.787829i \(0.711205\pi\)
\(972\) 15.6033 0.500477
\(973\) 50.1373 1.60733
\(974\) 9.69616 0.310685
\(975\) 3.78794 0.121311
\(976\) 2.00000 0.0640184
\(977\) 10.9233 0.349467 0.174734 0.984616i \(-0.444094\pi\)
0.174734 + 0.984616i \(0.444094\pi\)
\(978\) −11.0117 −0.352116
\(979\) −14.9883 −0.479028
\(980\) 5.61810 0.179463
\(981\) −14.4161 −0.460270
\(982\) −11.7538 −0.375078
\(983\) −32.1196 −1.02446 −0.512228 0.858849i \(-0.671180\pi\)
−0.512228 + 0.858849i \(0.671180\pi\)
\(984\) 2.27872 0.0726431
\(985\) 18.2948 0.582920
\(986\) −38.5990 −1.22924
\(987\) 13.8318 0.440272
\(988\) 1.40268 0.0446252
\(989\) −62.0918 −1.97440
\(990\) 2.94841 0.0937066
\(991\) 4.37620 0.139015 0.0695073 0.997581i \(-0.477857\pi\)
0.0695073 + 0.997581i \(0.477857\pi\)
\(992\) −4.94841 −0.157112
\(993\) −24.6771 −0.783103
\(994\) −25.7863 −0.817890
\(995\) 18.5990 0.589628
\(996\) 8.24623 0.261292
\(997\) −8.37788 −0.265330 −0.132665 0.991161i \(-0.542353\pi\)
−0.132665 + 0.991161i \(0.542353\pi\)
\(998\) 26.6640 0.844034
\(999\) −9.81708 −0.310599
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 418.2.a.h.1.2 3
3.2 odd 2 3762.2.a.bd.1.2 3
4.3 odd 2 3344.2.a.p.1.2 3
11.10 odd 2 4598.2.a.bm.1.2 3
19.18 odd 2 7942.2.a.bc.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.a.h.1.2 3 1.1 even 1 trivial
3344.2.a.p.1.2 3 4.3 odd 2
3762.2.a.bd.1.2 3 3.2 odd 2
4598.2.a.bm.1.2 3 11.10 odd 2
7942.2.a.bc.1.2 3 19.18 odd 2