Properties

Label 7942.2.a.bc.1.2
Level $7942$
Weight $2$
Character 7942.1
Self dual yes
Analytic conductor $63.417$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7942 = 2 \cdot 11 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7942.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.4171892853\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.469.1
Defining polynomial: \(x^{3} - x^{2} - 5 x + 4\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 418)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.772866\) of defining polynomial
Character \(\chi\) \(=\) 7942.1

$q$-expansion

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −0.772866 −0.446214 −0.223107 0.974794i \(-0.571620\pi\)
−0.223107 + 0.974794i \(0.571620\pi\)
\(4\) 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 \(-0.437686\pi\)
0.194517 + 0.980899i \(0.437686\pi\)
\(14\) −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
\(19\) 0 0
\(20\) 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 \(-0.767932\pi\)
−0.745798 + 0.666172i \(0.767932\pi\)
\(30\) 0.948410 0.173155
\(31\) 4.94841 0.888761 0.444380 0.895838i \(-0.353424\pi\)
0.444380 + 0.895838i \(0.353424\pi\)
\(32\) −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 \(-0.561906\pi\)
−0.193258 + 0.981148i \(0.561906\pi\)
\(38\) 0 0
\(39\) −1.08408 −0.173592
\(40\) −1.22713 −0.194027
\(41\) −2.94841 −0.460464 −0.230232 0.973136i \(-0.573949\pi\)
−0.230232 + 0.973136i \(0.573949\pi\)
\(42\) 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 \(-0.401036\pi\)
0.305920 + 0.952057i \(0.401036\pi\)
\(54\) −4.17554 −0.568220
\(55\) −1.22713 −0.165467
\(56\) −3.40268 −0.454702
\(57\) 0 0
\(58\) 8.03249 1.05472
\(59\) 14.5193 1.89025 0.945123 0.326715i \(-0.105942\pi\)
0.945123 + 0.326715i \(0.105942\pi\)
\(60\) −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 \(-0.566647\pi\)
−0.207852 + 0.978160i \(0.566647\pi\)
\(68\) −4.80536 −0.582735
\(69\) −5.25963 −0.633185
\(70\) −4.17554 −0.499073
\(71\) 7.57822 0.899370 0.449685 0.893187i \(-0.351536\pi\)
0.449685 + 0.893187i \(0.351536\pi\)
\(72\) 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
\(76\) 0 0
\(77\) −3.40268 −0.387771
\(78\) 1.08408 0.122748
\(79\) 3.71390 0.417846 0.208923 0.977932i \(-0.433004\pi\)
0.208923 + 0.977932i \(0.433004\pi\)
\(80\) 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 \(-0.792203\pi\)
−0.794377 + 0.607425i \(0.792203\pi\)
\(90\) 2.94841 0.310790
\(91\) 4.77287 0.500332
\(92\) 6.80536 0.709508
\(93\) −3.82446 −0.396578
\(94\) −5.25963 −0.542489
\(95\) 0 0
\(96\) 0.772866 0.0788803
\(97\) −2.35109 −0.238717 −0.119358 0.992851i \(-0.538084\pi\)
−0.119358 + 0.992851i \(0.538084\pi\)
\(98\) −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 \(-0.221508\pi\)
0.767484 + 0.641068i \(0.221508\pi\)
\(104\) −1.40268 −0.137544
\(105\) −3.22713 −0.314936
\(106\) −4.45427 −0.432637
\(107\) 4.35109 0.420636 0.210318 0.977633i \(-0.432550\pi\)
0.210318 + 0.977633i \(0.432550\pi\)
\(108\) 4.17554 0.401792
\(109\) −6.00000 −0.574696 −0.287348 0.957826i \(-0.592774\pi\)
−0.287348 + 0.957826i \(0.592774\pi\)
\(110\) 1.22713 0.117003
\(111\) 1.81708 0.172469
\(112\) 3.40268 0.321523
\(113\) 19.9618 1.87785 0.938924 0.344124i \(-0.111824\pi\)
0.938924 + 0.344124i \(0.111824\pi\)
\(114\) 0 0
\(115\) 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 \(-0.673964\pi\)
−0.519720 + 0.854337i \(0.673964\pi\)
\(128\) −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
\(133\) 0 0
\(134\) 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 \(-0.399259\pi\)
0.311230 + 0.950335i \(0.399259\pi\)
\(152\) 0 0
\(153\) 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 \(-0.361824\pi\)
0.420586 + 0.907253i \(0.361824\pi\)
\(168\) 2.62981 0.202894
\(169\) −11.0325 −0.848653
\(170\) 5.89682 0.452265
\(171\) 0 0
\(172\) −9.12395 −0.695695
\(173\) −3.39530 −0.258140 −0.129070 0.991635i \(-0.541199\pi\)
−0.129070 + 0.991635i \(0.541199\pi\)
\(174\) −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 \(-0.496298\pi\)
0.0116307 + 0.999932i \(0.496298\pi\)
\(180\) −2.94841 −0.219762
\(181\) 15.3246 1.13907 0.569535 0.821967i \(-0.307123\pi\)
0.569535 + 0.821967i \(0.307123\pi\)
\(182\) −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
\(190\) 0 0
\(191\) 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 \(-0.732945\pi\)
−0.668223 + 0.743961i \(0.732945\pi\)
\(194\) 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
\(209\) 0 0
\(210\) 3.22713 0.222693
\(211\) −11.1564 −0.768041 −0.384021 0.923324i \(-0.625461\pi\)
−0.384021 + 0.923324i \(0.625461\pi\)
\(212\) 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 \(-0.517175\pi\)
−0.0539307 + 0.998545i \(0.517175\pi\)
\(224\) −3.40268 −0.227351
\(225\) 8.39530 0.559687
\(226\) −19.9618 −1.32784
\(227\) 16.7672 1.11288 0.556438 0.830889i \(-0.312168\pi\)
0.556438 + 0.830889i \(0.312168\pi\)
\(228\) 0 0
\(229\) −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 \(-0.619784\pi\)
−0.367493 + 0.930026i \(0.619784\pi\)
\(242\) −1.00000 −0.0642824
\(243\) −15.6033 −1.00095
\(244\) 2.00000 0.128037
\(245\) 5.61810 0.358927
\(246\) −2.27872 −0.145286
\(247\) 0 0
\(248\) −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 \(-0.772757\pi\)
−0.755811 + 0.654790i \(0.772757\pi\)
\(258\) −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
\(266\) 0 0
\(267\) 11.5839 0.708925
\(268\) −3.40268 −0.207852
\(269\) 1.36281 0.0830918 0.0415459 0.999137i \(-0.486772\pi\)
0.0415459 + 0.999137i \(0.486772\pi\)
\(270\) −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 \(-0.473643\pi\)
0.0827076 + 0.996574i \(0.473643\pi\)
\(282\) 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 0
\(286\) 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 \(-0.673384\pi\)
−0.518162 + 0.855283i \(0.673384\pi\)
\(294\) 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
\(304\) 0 0
\(305\) 2.45427 0.140531
\(306\) −11.5457 −0.660026
\(307\) −20.9735 −1.19702 −0.598511 0.801115i \(-0.704241\pi\)
−0.598511 + 0.801115i \(0.704241\pi\)
\(308\) −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 \(-0.631405\pi\)
−0.401194 + 0.915993i \(0.631405\pi\)
\(318\) 3.44255 0.193049
\(319\) 8.03249 0.449733
\(320\) 1.22713 0.0685989
\(321\) −3.36281 −0.187694
\(322\) −23.1564 −1.29046
\(323\) 0 0
\(324\) 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 \(-0.159212\pi\)
0.877497 + 0.479582i \(0.159212\pi\)
\(332\) 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 \(-0.422552\pi\)
0.240915 + 0.970546i \(0.422552\pi\)
\(338\) 11.0325 0.600088
\(339\) −15.4278 −0.837923
\(340\) −5.89682 −0.319800
\(341\) −4.94841 −0.267971
\(342\) 0 0
\(343\) −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
\(361\) 0 0
\(362\) −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 \(-0.650390\pi\)
−0.455083 + 0.890449i \(0.650390\pi\)
\(374\) −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 \(-0.573691\pi\)
−0.229445 + 0.973322i \(0.573691\pi\)
\(380\) 0 0
\(381\) 9.05327 0.463813
\(382\) −26.8703 −1.37481
\(383\) 16.0399 0.819599 0.409800 0.912176i \(-0.365599\pi\)
0.409800 + 0.912176i \(0.365599\pi\)
\(384\) 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
\(399\) 0 0
\(400\) −3.49414 −0.174707
\(401\) 13.1564 0.657002 0.328501 0.944504i \(-0.393457\pi\)
0.328501 + 0.944504i \(0.393457\pi\)
\(402\) −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 \(-0.367251\pi\)
0.405058 + 0.914291i \(0.367251\pi\)
\(410\) 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
\(418\) 0 0
\(419\) 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 \(-0.479530\pi\)
0.0642645 + 0.997933i \(0.479530\pi\)
\(422\) 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 \(-0.259235\pi\)
0.686298 + 0.727321i \(0.259235\pi\)
\(432\) 4.17554 0.200896
\(433\) 37.5075 1.80250 0.901249 0.433302i \(-0.142652\pi\)
0.901249 + 0.433302i \(0.142652\pi\)
\(434\) −16.8378 −0.808243
\(435\) 7.61810 0.365260
\(436\) −6.00000 −0.287348
\(437\) 0 0
\(438\) −7.77888 −0.371689
\(439\) 32.1300 1.53348 0.766740 0.641958i \(-0.221878\pi\)
0.766740 + 0.641958i \(0.221878\pi\)
\(440\) 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 \(-0.559549\pi\)
−0.185989 + 0.982552i \(0.559549\pi\)
\(450\) −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 0
\(457\) 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
\(475\) 0 0
\(476\) −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 \(-0.570504\pi\)
−0.219688 + 0.975570i \(0.570504\pi\)
\(488\) −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
\(494\) 0 0
\(495\) 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 \(-0.206425\pi\)
0.796989 + 0.603993i \(0.206425\pi\)
\(510\) −4.55745 −0.201807
\(511\) −34.2479 −1.51504
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 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 \(-0.571226\pi\)
−0.221901 + 0.975069i \(0.571226\pi\)
\(522\) −19.2995 −0.844716
\(523\) 7.93502 0.346974 0.173487 0.984836i \(-0.444497\pi\)
0.173487 + 0.984836i \(0.444497\pi\)
\(524\) 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
\(532\) 0 0
\(533\) −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 \(-0.429240\pi\)
0.220474 + 0.975393i \(0.429240\pi\)
\(548\) 10.5973 0.452695
\(549\) −4.80536 −0.205088
\(550\) −3.49414 −0.148991
\(551\) 0 0
\(552\) 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 \(-0.571251\pi\)
−0.221978 + 0.975052i \(0.571251\pi\)
\(564\) −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 \(-0.569789\pi\)
−0.217496 + 0.976061i \(0.569789\pi\)
\(570\) 0 0
\(571\) 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
\(589\) 0 0
\(590\) −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 \(-0.815839\pi\)
−0.837251 + 0.546818i \(0.815839\pi\)
\(600\) −2.70050 −0.110248
\(601\) −0.349413 −0.0142528 −0.00712642 0.999975i \(-0.502268\pi\)
−0.00712642 + 0.999975i \(0.502268\pi\)
\(602\) 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 \(-0.535997\pi\)
−0.112847 + 0.993612i \(0.535997\pi\)
\(608\) 0 0
\(609\) 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 0
\(628\) 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 \(-0.177225\pi\)
0.848968 + 0.528445i \(0.177225\pi\)
\(642\) 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
\(646\) 0 0
\(647\) 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 \(-0.428044\pi\)
0.224135 + 0.974558i \(0.428044\pi\)
\(660\) 0.948410 0.0369168
\(661\) 29.2362 1.13716 0.568578 0.822629i \(-0.307494\pi\)
0.568578 + 0.822629i \(0.307494\pi\)
\(662\) −31.9293 −1.24097
\(663\) 5.20940 0.202316
\(664\) −10.6697 −0.414064
\(665\) 0 0
\(666\) −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 \(-0.138087\pi\)
0.907369 + 0.420335i \(0.138087\pi\)
\(674\) −8.84523 −0.340706
\(675\) −14.5899 −0.561567
\(676\) −11.0325 −0.424327
\(677\) −43.2539 −1.66238 −0.831192 0.555986i \(-0.812341\pi\)
−0.831192 + 0.555986i \(0.812341\pi\)
\(678\) 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 \(-0.732360\pi\)
−0.666856 + 0.745187i \(0.732360\pi\)
\(684\) 0 0
\(685\) 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
\(703\) 0 0
\(704\) −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
\(722\) 0 0
\(723\) 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
\(741\) 0 0
\(742\) −15.1564 −0.556411
\(743\) 13.2596 0.486449 0.243224 0.969970i \(-0.421795\pi\)
0.243224 + 0.969970i \(0.421795\pi\)
\(744\) 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 \(-0.440516\pi\)
0.185790 + 0.982589i \(0.440516\pi\)
\(752\) 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
\(760\) 0 0
\(761\) −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 \(-0.355471\pi\)
0.438611 + 0.898677i \(0.355471\pi\)
\(774\) −21.9219 −0.787968
\(775\) −17.2904 −0.621091
\(776\) 2.35109 0.0843992
\(777\) 6.18292 0.221811
\(778\) −17.2271 −0.617623
\(779\) 0 0
\(780\) −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 \(-0.190274\pi\)
0.826596 + 0.562795i \(0.190274\pi\)
\(788\) 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 \(-0.521986\pi\)
−0.0690162 + 0.997616i \(0.521986\pi\)
\(798\) 0 0
\(799\) −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 \(-0.427141\pi\)
0.226899 + 0.973918i \(0.427141\pi\)
\(812\) −27.3320 −0.959165
\(813\) 17.9276 0.628750
\(814\) −2.35109 −0.0824056
\(815\) −17.4841 −0.612441
\(816\) 3.71390 0.130012
\(817\) 0 0
\(818\) −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 \(-0.292558\pi\)
0.606537 + 0.795055i \(0.292558\pi\)
\(828\) −16.3511 −0.568240
\(829\) −31.7407 −1.10240 −0.551200 0.834373i \(-0.685830\pi\)
−0.551200 + 0.834373i \(0.685830\pi\)
\(830\) −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
\(836\) 0 0
\(837\) 20.6623 0.714194
\(838\) −1.12966 −0.0390234
\(839\) −13.4603 −0.464701 −0.232350 0.972632i \(-0.574642\pi\)
−0.232350 + 0.972632i \(0.574642\pi\)
\(840\) 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
\(855\) 0 0
\(856\) −4.35109 −0.148717
\(857\) 43.8569 1.49812 0.749062 0.662499i \(-0.230504\pi\)
0.749062 + 0.662499i \(0.230504\pi\)
\(858\) −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 \(-0.360862\pi\)
0.423327 + 0.905977i \(0.360862\pi\)
\(864\) −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
\(874\) 0 0
\(875\) −35.4677 −1.19903
\(876\) 7.77888 0.262824
\(877\) −39.8911 −1.34703 −0.673514 0.739175i \(-0.735216\pi\)
−0.673514 + 0.739175i \(0.735216\pi\)
\(878\) −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 \(-0.622219\pi\)
−0.374597 + 0.927188i \(0.622219\pi\)
\(888\) −1.81708 −0.0609771
\(889\) −39.8586 −1.33682
\(890\) 18.3926 0.616523
\(891\) −3.98090 −0.133365
\(892\) −1.61072 −0.0539307
\(893\) 0 0
\(894\) 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 \(-0.298994\pi\)
0.590338 + 0.807156i \(0.298994\pi\)
\(908\) 16.7672 0.556438
\(909\) −33.7936 −1.12086
\(910\) −5.85695 −0.194156
\(911\) 49.1980 1.63000 0.815001 0.579459i \(-0.196736\pi\)
0.815001 + 0.579459i \(0.196736\pi\)
\(912\) 0 0
\(913\) −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
\(931\) 0 0
\(932\) 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 \(-0.388292\pi\)
0.343781 + 0.939050i \(0.388292\pi\)
\(942\) 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
\(950\) 0 0
\(951\) 11.0412 0.358037
\(952\) 16.3511 0.529942
\(953\) −3.27439 −0.106068 −0.0530339 0.998593i \(-0.516889\pi\)
−0.0530339 + 0.998593i \(0.516889\pi\)
\(954\) 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
\(969\) 0 0
\(970\) 2.88510 0.0926350
\(971\) 38.3836 1.23179 0.615894 0.787829i \(-0.288795\pi\)
0.615894 + 0.787829i \(0.288795\pi\)
\(972\) −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 \(-0.555906\pi\)
−0.174734 + 0.984616i \(0.555906\pi\)
\(978\) −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 \(-0.328820\pi\)
0.512228 + 0.858849i \(0.328820\pi\)
\(984\) −2.27872 −0.0726431
\(985\) 18.2948 0.582920
\(986\) −38.5990 −1.22924
\(987\) −13.8318 −0.440272
\(988\) 0 0
\(989\) −62.0918 −1.97440
\(990\) −2.94841 −0.0937066
\(991\) −4.37620 −0.139015 −0.0695073 0.997581i \(-0.522143\pi\)
−0.0695073 + 0.997581i \(0.522143\pi\)
\(992\) −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 7942.2.a.bc.1.2 3
19.18 odd 2 418.2.a.h.1.2 3
57.56 even 2 3762.2.a.bd.1.2 3
76.75 even 2 3344.2.a.p.1.2 3
209.208 even 2 4598.2.a.bm.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.a.h.1.2 3 19.18 odd 2
3344.2.a.p.1.2 3 76.75 even 2
3762.2.a.bd.1.2 3 57.56 even 2
4598.2.a.bm.1.2 3 209.208 even 2
7942.2.a.bc.1.2 3 1.1 even 1 trivial