Properties

Label 8001.2.a.z.1.32
Level $8001$
Weight $2$
Character 8001.1
Self dual yes
Analytic conductor $63.888$
Analytic rank $1$
Dimension $32$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8001.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.8883066572\)
Analytic rank: \(1\)
Dimension: \(32\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.32
Character \(\chi\) \(=\) 8001.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.69601 q^{2} +5.26849 q^{4} -3.35934 q^{5} -1.00000 q^{7} +8.81189 q^{8} +O(q^{10})\) \(q+2.69601 q^{2} +5.26849 q^{4} -3.35934 q^{5} -1.00000 q^{7} +8.81189 q^{8} -9.05682 q^{10} +2.89389 q^{11} -2.10825 q^{13} -2.69601 q^{14} +13.2200 q^{16} -6.29317 q^{17} -0.167183 q^{19} -17.6986 q^{20} +7.80197 q^{22} -4.03441 q^{23} +6.28516 q^{25} -5.68387 q^{26} -5.26849 q^{28} +0.392019 q^{29} -7.54513 q^{31} +18.0175 q^{32} -16.9665 q^{34} +3.35934 q^{35} -2.42977 q^{37} -0.450728 q^{38} -29.6021 q^{40} -0.160937 q^{41} -6.15714 q^{43} +15.2464 q^{44} -10.8768 q^{46} +8.81645 q^{47} +1.00000 q^{49} +16.9449 q^{50} -11.1073 q^{52} -2.87949 q^{53} -9.72156 q^{55} -8.81189 q^{56} +1.05689 q^{58} -3.45891 q^{59} -12.0314 q^{61} -20.3418 q^{62} +22.1354 q^{64} +7.08233 q^{65} +9.36114 q^{67} -33.1555 q^{68} +9.05682 q^{70} -12.0277 q^{71} -11.2562 q^{73} -6.55069 q^{74} -0.880801 q^{76} -2.89389 q^{77} +3.36380 q^{79} -44.4104 q^{80} -0.433889 q^{82} +17.3020 q^{83} +21.1409 q^{85} -16.5997 q^{86} +25.5006 q^{88} +14.6779 q^{89} +2.10825 q^{91} -21.2552 q^{92} +23.7693 q^{94} +0.561624 q^{95} -1.23267 q^{97} +2.69601 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32q + 30q^{4} - 32q^{7} + O(q^{10}) \) \( 32q + 30q^{4} - 32q^{7} - 16q^{10} - 14q^{13} + 18q^{16} - 30q^{19} - 10q^{22} + 36q^{25} - 30q^{28} - 58q^{31} - 34q^{34} + 8q^{37} - 34q^{40} + 6q^{43} - 36q^{46} + 32q^{49} - 56q^{52} - 88q^{55} - 22q^{58} - 46q^{61} + 20q^{64} - 8q^{67} + 16q^{70} - 60q^{73} - 128q^{76} - 74q^{79} - 52q^{82} - 16q^{85} - 64q^{88} + 14q^{91} - 58q^{94} - 44q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.69601 1.90637 0.953185 0.302389i \(-0.0977841\pi\)
0.953185 + 0.302389i \(0.0977841\pi\)
\(3\) 0 0
\(4\) 5.26849 2.63424
\(5\) −3.35934 −1.50234 −0.751171 0.660107i \(-0.770511\pi\)
−0.751171 + 0.660107i \(0.770511\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 8.81189 3.11547
\(9\) 0 0
\(10\) −9.05682 −2.86402
\(11\) 2.89389 0.872541 0.436271 0.899816i \(-0.356299\pi\)
0.436271 + 0.899816i \(0.356299\pi\)
\(12\) 0 0
\(13\) −2.10825 −0.584723 −0.292362 0.956308i \(-0.594441\pi\)
−0.292362 + 0.956308i \(0.594441\pi\)
\(14\) −2.69601 −0.720540
\(15\) 0 0
\(16\) 13.2200 3.30500
\(17\) −6.29317 −1.52632 −0.763160 0.646210i \(-0.776353\pi\)
−0.763160 + 0.646210i \(0.776353\pi\)
\(18\) 0 0
\(19\) −0.167183 −0.0383544 −0.0191772 0.999816i \(-0.506105\pi\)
−0.0191772 + 0.999816i \(0.506105\pi\)
\(20\) −17.6986 −3.95754
\(21\) 0 0
\(22\) 7.80197 1.66339
\(23\) −4.03441 −0.841233 −0.420616 0.907239i \(-0.638186\pi\)
−0.420616 + 0.907239i \(0.638186\pi\)
\(24\) 0 0
\(25\) 6.28516 1.25703
\(26\) −5.68387 −1.11470
\(27\) 0 0
\(28\) −5.26849 −0.995651
\(29\) 0.392019 0.0727961 0.0363981 0.999337i \(-0.488412\pi\)
0.0363981 + 0.999337i \(0.488412\pi\)
\(30\) 0 0
\(31\) −7.54513 −1.35514 −0.677572 0.735456i \(-0.736968\pi\)
−0.677572 + 0.735456i \(0.736968\pi\)
\(32\) 18.0175 3.18507
\(33\) 0 0
\(34\) −16.9665 −2.90973
\(35\) 3.35934 0.567832
\(36\) 0 0
\(37\) −2.42977 −0.399452 −0.199726 0.979852i \(-0.564005\pi\)
−0.199726 + 0.979852i \(0.564005\pi\)
\(38\) −0.450728 −0.0731177
\(39\) 0 0
\(40\) −29.6021 −4.68051
\(41\) −0.160937 −0.0251342 −0.0125671 0.999921i \(-0.504000\pi\)
−0.0125671 + 0.999921i \(0.504000\pi\)
\(42\) 0 0
\(43\) −6.15714 −0.938955 −0.469478 0.882944i \(-0.655558\pi\)
−0.469478 + 0.882944i \(0.655558\pi\)
\(44\) 15.2464 2.29849
\(45\) 0 0
\(46\) −10.8768 −1.60370
\(47\) 8.81645 1.28601 0.643006 0.765861i \(-0.277687\pi\)
0.643006 + 0.765861i \(0.277687\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 16.9449 2.39637
\(51\) 0 0
\(52\) −11.1073 −1.54030
\(53\) −2.87949 −0.395528 −0.197764 0.980250i \(-0.563368\pi\)
−0.197764 + 0.980250i \(0.563368\pi\)
\(54\) 0 0
\(55\) −9.72156 −1.31086
\(56\) −8.81189 −1.17754
\(57\) 0 0
\(58\) 1.05689 0.138776
\(59\) −3.45891 −0.450312 −0.225156 0.974323i \(-0.572289\pi\)
−0.225156 + 0.974323i \(0.572289\pi\)
\(60\) 0 0
\(61\) −12.0314 −1.54046 −0.770232 0.637763i \(-0.779860\pi\)
−0.770232 + 0.637763i \(0.779860\pi\)
\(62\) −20.3418 −2.58341
\(63\) 0 0
\(64\) 22.1354 2.76693
\(65\) 7.08233 0.878455
\(66\) 0 0
\(67\) 9.36114 1.14364 0.571822 0.820377i \(-0.306237\pi\)
0.571822 + 0.820377i \(0.306237\pi\)
\(68\) −33.1555 −4.02070
\(69\) 0 0
\(70\) 9.05682 1.08250
\(71\) −12.0277 −1.42742 −0.713710 0.700441i \(-0.752987\pi\)
−0.713710 + 0.700441i \(0.752987\pi\)
\(72\) 0 0
\(73\) −11.2562 −1.31743 −0.658717 0.752390i \(-0.728901\pi\)
−0.658717 + 0.752390i \(0.728901\pi\)
\(74\) −6.55069 −0.761503
\(75\) 0 0
\(76\) −0.880801 −0.101035
\(77\) −2.89389 −0.329790
\(78\) 0 0
\(79\) 3.36380 0.378457 0.189228 0.981933i \(-0.439401\pi\)
0.189228 + 0.981933i \(0.439401\pi\)
\(80\) −44.4104 −4.96524
\(81\) 0 0
\(82\) −0.433889 −0.0479150
\(83\) 17.3020 1.89914 0.949569 0.313559i \(-0.101521\pi\)
0.949569 + 0.313559i \(0.101521\pi\)
\(84\) 0 0
\(85\) 21.1409 2.29305
\(86\) −16.5997 −1.79000
\(87\) 0 0
\(88\) 25.5006 2.71838
\(89\) 14.6779 1.55585 0.777927 0.628355i \(-0.216271\pi\)
0.777927 + 0.628355i \(0.216271\pi\)
\(90\) 0 0
\(91\) 2.10825 0.221005
\(92\) −21.2552 −2.21601
\(93\) 0 0
\(94\) 23.7693 2.45161
\(95\) 0.561624 0.0576215
\(96\) 0 0
\(97\) −1.23267 −0.125159 −0.0625793 0.998040i \(-0.519933\pi\)
−0.0625793 + 0.998040i \(0.519933\pi\)
\(98\) 2.69601 0.272338
\(99\) 0 0
\(100\) 33.1133 3.31133
\(101\) −12.2730 −1.22121 −0.610603 0.791937i \(-0.709073\pi\)
−0.610603 + 0.791937i \(0.709073\pi\)
\(102\) 0 0
\(103\) −3.50168 −0.345030 −0.172515 0.985007i \(-0.555189\pi\)
−0.172515 + 0.985007i \(0.555189\pi\)
\(104\) −18.5777 −1.82169
\(105\) 0 0
\(106\) −7.76314 −0.754023
\(107\) −11.7913 −1.13991 −0.569953 0.821677i \(-0.693039\pi\)
−0.569953 + 0.821677i \(0.693039\pi\)
\(108\) 0 0
\(109\) −8.25140 −0.790341 −0.395170 0.918608i \(-0.629314\pi\)
−0.395170 + 0.918608i \(0.629314\pi\)
\(110\) −26.2095 −2.49897
\(111\) 0 0
\(112\) −13.2200 −1.24917
\(113\) −13.2805 −1.24932 −0.624662 0.780895i \(-0.714763\pi\)
−0.624662 + 0.780895i \(0.714763\pi\)
\(114\) 0 0
\(115\) 13.5530 1.26382
\(116\) 2.06535 0.191763
\(117\) 0 0
\(118\) −9.32528 −0.858462
\(119\) 6.29317 0.576894
\(120\) 0 0
\(121\) −2.62539 −0.238672
\(122\) −32.4368 −2.93669
\(123\) 0 0
\(124\) −39.7514 −3.56978
\(125\) −4.31730 −0.386151
\(126\) 0 0
\(127\) −1.00000 −0.0887357
\(128\) 23.6424 2.08971
\(129\) 0 0
\(130\) 19.0940 1.67466
\(131\) 19.3595 1.69145 0.845726 0.533618i \(-0.179168\pi\)
0.845726 + 0.533618i \(0.179168\pi\)
\(132\) 0 0
\(133\) 0.167183 0.0144966
\(134\) 25.2378 2.18021
\(135\) 0 0
\(136\) −55.4547 −4.75520
\(137\) 15.6936 1.34079 0.670397 0.742003i \(-0.266124\pi\)
0.670397 + 0.742003i \(0.266124\pi\)
\(138\) 0 0
\(139\) −7.82773 −0.663939 −0.331970 0.943290i \(-0.607713\pi\)
−0.331970 + 0.943290i \(0.607713\pi\)
\(140\) 17.6986 1.49581
\(141\) 0 0
\(142\) −32.4267 −2.72119
\(143\) −6.10105 −0.510195
\(144\) 0 0
\(145\) −1.31693 −0.109365
\(146\) −30.3468 −2.51152
\(147\) 0 0
\(148\) −12.8012 −1.05225
\(149\) −15.7545 −1.29066 −0.645328 0.763905i \(-0.723279\pi\)
−0.645328 + 0.763905i \(0.723279\pi\)
\(150\) 0 0
\(151\) −1.99541 −0.162384 −0.0811922 0.996698i \(-0.525873\pi\)
−0.0811922 + 0.996698i \(0.525873\pi\)
\(152\) −1.47320 −0.119492
\(153\) 0 0
\(154\) −7.80197 −0.628701
\(155\) 25.3466 2.03589
\(156\) 0 0
\(157\) 4.40954 0.351919 0.175960 0.984397i \(-0.443697\pi\)
0.175960 + 0.984397i \(0.443697\pi\)
\(158\) 9.06884 0.721478
\(159\) 0 0
\(160\) −60.5269 −4.78507
\(161\) 4.03441 0.317956
\(162\) 0 0
\(163\) −9.81692 −0.768921 −0.384460 0.923142i \(-0.625612\pi\)
−0.384460 + 0.923142i \(0.625612\pi\)
\(164\) −0.847896 −0.0662095
\(165\) 0 0
\(166\) 46.6463 3.62046
\(167\) −15.3402 −1.18706 −0.593531 0.804811i \(-0.702266\pi\)
−0.593531 + 0.804811i \(0.702266\pi\)
\(168\) 0 0
\(169\) −8.55528 −0.658099
\(170\) 56.9962 4.37141
\(171\) 0 0
\(172\) −32.4388 −2.47344
\(173\) 12.7598 0.970110 0.485055 0.874484i \(-0.338800\pi\)
0.485055 + 0.874484i \(0.338800\pi\)
\(174\) 0 0
\(175\) −6.28516 −0.475114
\(176\) 38.2572 2.88375
\(177\) 0 0
\(178\) 39.5718 2.96603
\(179\) 1.20349 0.0899531 0.0449766 0.998988i \(-0.485679\pi\)
0.0449766 + 0.998988i \(0.485679\pi\)
\(180\) 0 0
\(181\) −8.27628 −0.615171 −0.307585 0.951520i \(-0.599521\pi\)
−0.307585 + 0.951520i \(0.599521\pi\)
\(182\) 5.68387 0.421316
\(183\) 0 0
\(184\) −35.5508 −2.62084
\(185\) 8.16243 0.600113
\(186\) 0 0
\(187\) −18.2118 −1.33178
\(188\) 46.4494 3.38767
\(189\) 0 0
\(190\) 1.51415 0.109848
\(191\) −16.1586 −1.16920 −0.584598 0.811323i \(-0.698748\pi\)
−0.584598 + 0.811323i \(0.698748\pi\)
\(192\) 0 0
\(193\) 10.3243 0.743159 0.371579 0.928401i \(-0.378816\pi\)
0.371579 + 0.928401i \(0.378816\pi\)
\(194\) −3.32329 −0.238598
\(195\) 0 0
\(196\) 5.26849 0.376321
\(197\) −19.7255 −1.40539 −0.702693 0.711493i \(-0.748019\pi\)
−0.702693 + 0.711493i \(0.748019\pi\)
\(198\) 0 0
\(199\) 1.25357 0.0888634 0.0444317 0.999012i \(-0.485852\pi\)
0.0444317 + 0.999012i \(0.485852\pi\)
\(200\) 55.3841 3.91625
\(201\) 0 0
\(202\) −33.0881 −2.32807
\(203\) −0.392019 −0.0275144
\(204\) 0 0
\(205\) 0.540643 0.0377601
\(206\) −9.44056 −0.657755
\(207\) 0 0
\(208\) −27.8710 −1.93251
\(209\) −0.483809 −0.0334658
\(210\) 0 0
\(211\) 1.81620 0.125032 0.0625162 0.998044i \(-0.480088\pi\)
0.0625162 + 0.998044i \(0.480088\pi\)
\(212\) −15.1706 −1.04192
\(213\) 0 0
\(214\) −31.7894 −2.17308
\(215\) 20.6839 1.41063
\(216\) 0 0
\(217\) 7.54513 0.512197
\(218\) −22.2459 −1.50668
\(219\) 0 0
\(220\) −51.2179 −3.45311
\(221\) 13.2676 0.892474
\(222\) 0 0
\(223\) 0.622163 0.0416631 0.0208316 0.999783i \(-0.493369\pi\)
0.0208316 + 0.999783i \(0.493369\pi\)
\(224\) −18.0175 −1.20384
\(225\) 0 0
\(226\) −35.8044 −2.38167
\(227\) 25.3816 1.68464 0.842319 0.538980i \(-0.181190\pi\)
0.842319 + 0.538980i \(0.181190\pi\)
\(228\) 0 0
\(229\) −0.702452 −0.0464193 −0.0232097 0.999731i \(-0.507389\pi\)
−0.0232097 + 0.999731i \(0.507389\pi\)
\(230\) 36.5389 2.40931
\(231\) 0 0
\(232\) 3.45443 0.226794
\(233\) 16.1341 1.05698 0.528490 0.848939i \(-0.322758\pi\)
0.528490 + 0.848939i \(0.322758\pi\)
\(234\) 0 0
\(235\) −29.6175 −1.93203
\(236\) −18.2232 −1.18623
\(237\) 0 0
\(238\) 16.9665 1.09977
\(239\) 8.34664 0.539899 0.269949 0.962874i \(-0.412993\pi\)
0.269949 + 0.962874i \(0.412993\pi\)
\(240\) 0 0
\(241\) 11.7059 0.754043 0.377021 0.926205i \(-0.376948\pi\)
0.377021 + 0.926205i \(0.376948\pi\)
\(242\) −7.07809 −0.454997
\(243\) 0 0
\(244\) −63.3873 −4.05796
\(245\) −3.35934 −0.214620
\(246\) 0 0
\(247\) 0.352463 0.0224267
\(248\) −66.4868 −4.22192
\(249\) 0 0
\(250\) −11.6395 −0.736146
\(251\) 17.8668 1.12774 0.563871 0.825863i \(-0.309311\pi\)
0.563871 + 0.825863i \(0.309311\pi\)
\(252\) 0 0
\(253\) −11.6751 −0.734010
\(254\) −2.69601 −0.169163
\(255\) 0 0
\(256\) 19.4694 1.21683
\(257\) −9.30841 −0.580643 −0.290321 0.956929i \(-0.593762\pi\)
−0.290321 + 0.956929i \(0.593762\pi\)
\(258\) 0 0
\(259\) 2.42977 0.150979
\(260\) 37.3131 2.31406
\(261\) 0 0
\(262\) 52.1936 3.22453
\(263\) 12.0727 0.744435 0.372217 0.928146i \(-0.378598\pi\)
0.372217 + 0.928146i \(0.378598\pi\)
\(264\) 0 0
\(265\) 9.67318 0.594219
\(266\) 0.450728 0.0276359
\(267\) 0 0
\(268\) 49.3190 3.01264
\(269\) −27.9575 −1.70460 −0.852299 0.523056i \(-0.824792\pi\)
−0.852299 + 0.523056i \(0.824792\pi\)
\(270\) 0 0
\(271\) −11.6402 −0.707089 −0.353545 0.935418i \(-0.615024\pi\)
−0.353545 + 0.935418i \(0.615024\pi\)
\(272\) −83.1957 −5.04448
\(273\) 0 0
\(274\) 42.3101 2.55605
\(275\) 18.1886 1.09681
\(276\) 0 0
\(277\) 26.0313 1.56407 0.782036 0.623234i \(-0.214181\pi\)
0.782036 + 0.623234i \(0.214181\pi\)
\(278\) −21.1037 −1.26571
\(279\) 0 0
\(280\) 29.6021 1.76906
\(281\) 17.6693 1.05406 0.527031 0.849846i \(-0.323305\pi\)
0.527031 + 0.849846i \(0.323305\pi\)
\(282\) 0 0
\(283\) −9.81022 −0.583157 −0.291579 0.956547i \(-0.594180\pi\)
−0.291579 + 0.956547i \(0.594180\pi\)
\(284\) −63.3676 −3.76017
\(285\) 0 0
\(286\) −16.4485 −0.972620
\(287\) 0.160937 0.00949982
\(288\) 0 0
\(289\) 22.6041 1.32965
\(290\) −3.55045 −0.208490
\(291\) 0 0
\(292\) −59.3030 −3.47044
\(293\) 18.0827 1.05641 0.528203 0.849118i \(-0.322866\pi\)
0.528203 + 0.849118i \(0.322866\pi\)
\(294\) 0 0
\(295\) 11.6197 0.676523
\(296\) −21.4109 −1.24448
\(297\) 0 0
\(298\) −42.4743 −2.46047
\(299\) 8.50554 0.491888
\(300\) 0 0
\(301\) 6.15714 0.354892
\(302\) −5.37966 −0.309565
\(303\) 0 0
\(304\) −2.21016 −0.126761
\(305\) 40.4176 2.31431
\(306\) 0 0
\(307\) 4.05704 0.231547 0.115774 0.993276i \(-0.463065\pi\)
0.115774 + 0.993276i \(0.463065\pi\)
\(308\) −15.2464 −0.868746
\(309\) 0 0
\(310\) 68.3349 3.88116
\(311\) 13.5710 0.769543 0.384772 0.923012i \(-0.374280\pi\)
0.384772 + 0.923012i \(0.374280\pi\)
\(312\) 0 0
\(313\) 23.3786 1.32144 0.660719 0.750633i \(-0.270251\pi\)
0.660719 + 0.750633i \(0.270251\pi\)
\(314\) 11.8882 0.670888
\(315\) 0 0
\(316\) 17.7221 0.996947
\(317\) −31.8794 −1.79053 −0.895263 0.445537i \(-0.853013\pi\)
−0.895263 + 0.445537i \(0.853013\pi\)
\(318\) 0 0
\(319\) 1.13446 0.0635176
\(320\) −74.3603 −4.15687
\(321\) 0 0
\(322\) 10.8768 0.606142
\(323\) 1.05211 0.0585411
\(324\) 0 0
\(325\) −13.2507 −0.735016
\(326\) −26.4666 −1.46585
\(327\) 0 0
\(328\) −1.41816 −0.0783048
\(329\) −8.81645 −0.486067
\(330\) 0 0
\(331\) −32.8969 −1.80818 −0.904090 0.427343i \(-0.859450\pi\)
−0.904090 + 0.427343i \(0.859450\pi\)
\(332\) 91.1552 5.00279
\(333\) 0 0
\(334\) −41.3574 −2.26298
\(335\) −31.4472 −1.71815
\(336\) 0 0
\(337\) 12.2361 0.666544 0.333272 0.942831i \(-0.391847\pi\)
0.333272 + 0.942831i \(0.391847\pi\)
\(338\) −23.0652 −1.25458
\(339\) 0 0
\(340\) 111.381 6.04046
\(341\) −21.8348 −1.18242
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −54.2560 −2.92529
\(345\) 0 0
\(346\) 34.4006 1.84939
\(347\) −4.98147 −0.267419 −0.133710 0.991021i \(-0.542689\pi\)
−0.133710 + 0.991021i \(0.542689\pi\)
\(348\) 0 0
\(349\) −7.29876 −0.390693 −0.195347 0.980734i \(-0.562583\pi\)
−0.195347 + 0.980734i \(0.562583\pi\)
\(350\) −16.9449 −0.905742
\(351\) 0 0
\(352\) 52.1406 2.77911
\(353\) −12.2191 −0.650356 −0.325178 0.945653i \(-0.605424\pi\)
−0.325178 + 0.945653i \(0.605424\pi\)
\(354\) 0 0
\(355\) 40.4050 2.14447
\(356\) 77.3303 4.09850
\(357\) 0 0
\(358\) 3.24463 0.171484
\(359\) 36.0746 1.90395 0.951973 0.306181i \(-0.0990514\pi\)
0.951973 + 0.306181i \(0.0990514\pi\)
\(360\) 0 0
\(361\) −18.9720 −0.998529
\(362\) −22.3130 −1.17274
\(363\) 0 0
\(364\) 11.1073 0.582180
\(365\) 37.8133 1.97924
\(366\) 0 0
\(367\) −25.0223 −1.30616 −0.653078 0.757291i \(-0.726523\pi\)
−0.653078 + 0.757291i \(0.726523\pi\)
\(368\) −53.3348 −2.78027
\(369\) 0 0
\(370\) 22.0060 1.14404
\(371\) 2.87949 0.149496
\(372\) 0 0
\(373\) −12.9187 −0.668905 −0.334452 0.942413i \(-0.608551\pi\)
−0.334452 + 0.942413i \(0.608551\pi\)
\(374\) −49.0992 −2.53886
\(375\) 0 0
\(376\) 77.6896 4.00653
\(377\) −0.826474 −0.0425656
\(378\) 0 0
\(379\) 1.07672 0.0553076 0.0276538 0.999618i \(-0.491196\pi\)
0.0276538 + 0.999618i \(0.491196\pi\)
\(380\) 2.95891 0.151789
\(381\) 0 0
\(382\) −43.5638 −2.22892
\(383\) 14.0723 0.719060 0.359530 0.933133i \(-0.382937\pi\)
0.359530 + 0.933133i \(0.382937\pi\)
\(384\) 0 0
\(385\) 9.72156 0.495457
\(386\) 27.8344 1.41674
\(387\) 0 0
\(388\) −6.49430 −0.329698
\(389\) 0.896424 0.0454505 0.0227253 0.999742i \(-0.492766\pi\)
0.0227253 + 0.999742i \(0.492766\pi\)
\(390\) 0 0
\(391\) 25.3892 1.28399
\(392\) 8.81189 0.445067
\(393\) 0 0
\(394\) −53.1803 −2.67918
\(395\) −11.3001 −0.568571
\(396\) 0 0
\(397\) 15.5645 0.781158 0.390579 0.920569i \(-0.372275\pi\)
0.390579 + 0.920569i \(0.372275\pi\)
\(398\) 3.37965 0.169406
\(399\) 0 0
\(400\) 83.0898 4.15449
\(401\) −21.5171 −1.07451 −0.537257 0.843419i \(-0.680539\pi\)
−0.537257 + 0.843419i \(0.680539\pi\)
\(402\) 0 0
\(403\) 15.9070 0.792385
\(404\) −64.6600 −3.21696
\(405\) 0 0
\(406\) −1.05689 −0.0524525
\(407\) −7.03149 −0.348538
\(408\) 0 0
\(409\) −0.413177 −0.0204303 −0.0102151 0.999948i \(-0.503252\pi\)
−0.0102151 + 0.999948i \(0.503252\pi\)
\(410\) 1.45758 0.0719848
\(411\) 0 0
\(412\) −18.4485 −0.908894
\(413\) 3.45891 0.170202
\(414\) 0 0
\(415\) −58.1232 −2.85316
\(416\) −37.9854 −1.86239
\(417\) 0 0
\(418\) −1.30436 −0.0637982
\(419\) 36.4195 1.77921 0.889605 0.456730i \(-0.150980\pi\)
0.889605 + 0.456730i \(0.150980\pi\)
\(420\) 0 0
\(421\) −13.2124 −0.643934 −0.321967 0.946751i \(-0.604344\pi\)
−0.321967 + 0.946751i \(0.604344\pi\)
\(422\) 4.89650 0.238358
\(423\) 0 0
\(424\) −25.3737 −1.23226
\(425\) −39.5536 −1.91863
\(426\) 0 0
\(427\) 12.0314 0.582241
\(428\) −62.1222 −3.00279
\(429\) 0 0
\(430\) 55.7642 2.68919
\(431\) 33.8851 1.63219 0.816095 0.577918i \(-0.196135\pi\)
0.816095 + 0.577918i \(0.196135\pi\)
\(432\) 0 0
\(433\) −4.57030 −0.219635 −0.109817 0.993952i \(-0.535027\pi\)
−0.109817 + 0.993952i \(0.535027\pi\)
\(434\) 20.3418 0.976436
\(435\) 0 0
\(436\) −43.4724 −2.08195
\(437\) 0.674485 0.0322650
\(438\) 0 0
\(439\) −15.1264 −0.721945 −0.360973 0.932576i \(-0.617555\pi\)
−0.360973 + 0.932576i \(0.617555\pi\)
\(440\) −85.6653 −4.08393
\(441\) 0 0
\(442\) 35.7696 1.70139
\(443\) −21.4687 −1.02001 −0.510005 0.860171i \(-0.670357\pi\)
−0.510005 + 0.860171i \(0.670357\pi\)
\(444\) 0 0
\(445\) −49.3080 −2.33743
\(446\) 1.67736 0.0794253
\(447\) 0 0
\(448\) −22.1354 −1.04580
\(449\) 27.3231 1.28945 0.644727 0.764413i \(-0.276971\pi\)
0.644727 + 0.764413i \(0.276971\pi\)
\(450\) 0 0
\(451\) −0.465735 −0.0219306
\(452\) −69.9681 −3.29102
\(453\) 0 0
\(454\) 68.4292 3.21154
\(455\) −7.08233 −0.332025
\(456\) 0 0
\(457\) 17.8679 0.835826 0.417913 0.908487i \(-0.362762\pi\)
0.417913 + 0.908487i \(0.362762\pi\)
\(458\) −1.89382 −0.0884924
\(459\) 0 0
\(460\) 71.4036 3.32921
\(461\) −5.37461 −0.250320 −0.125160 0.992137i \(-0.539944\pi\)
−0.125160 + 0.992137i \(0.539944\pi\)
\(462\) 0 0
\(463\) 21.8586 1.01586 0.507928 0.861399i \(-0.330411\pi\)
0.507928 + 0.861399i \(0.330411\pi\)
\(464\) 5.18249 0.240591
\(465\) 0 0
\(466\) 43.4978 2.01500
\(467\) −32.8761 −1.52133 −0.760663 0.649147i \(-0.775126\pi\)
−0.760663 + 0.649147i \(0.775126\pi\)
\(468\) 0 0
\(469\) −9.36114 −0.432257
\(470\) −79.8491 −3.68316
\(471\) 0 0
\(472\) −30.4796 −1.40294
\(473\) −17.8181 −0.819277
\(474\) 0 0
\(475\) −1.05077 −0.0482127
\(476\) 33.1555 1.51968
\(477\) 0 0
\(478\) 22.5026 1.02925
\(479\) 15.8690 0.725073 0.362536 0.931970i \(-0.381911\pi\)
0.362536 + 0.931970i \(0.381911\pi\)
\(480\) 0 0
\(481\) 5.12256 0.233569
\(482\) 31.5592 1.43748
\(483\) 0 0
\(484\) −13.8318 −0.628720
\(485\) 4.14095 0.188031
\(486\) 0 0
\(487\) 8.98925 0.407342 0.203671 0.979039i \(-0.434713\pi\)
0.203671 + 0.979039i \(0.434713\pi\)
\(488\) −106.019 −4.79927
\(489\) 0 0
\(490\) −9.05682 −0.409146
\(491\) 1.39367 0.0628953 0.0314476 0.999505i \(-0.489988\pi\)
0.0314476 + 0.999505i \(0.489988\pi\)
\(492\) 0 0
\(493\) −2.46705 −0.111110
\(494\) 0.950246 0.0427536
\(495\) 0 0
\(496\) −99.7465 −4.47875
\(497\) 12.0277 0.539514
\(498\) 0 0
\(499\) −5.96722 −0.267129 −0.133565 0.991040i \(-0.542642\pi\)
−0.133565 + 0.991040i \(0.542642\pi\)
\(500\) −22.7456 −1.01722
\(501\) 0 0
\(502\) 48.1691 2.14989
\(503\) 28.1974 1.25726 0.628630 0.777704i \(-0.283616\pi\)
0.628630 + 0.777704i \(0.283616\pi\)
\(504\) 0 0
\(505\) 41.2291 1.83467
\(506\) −31.4763 −1.39929
\(507\) 0 0
\(508\) −5.26849 −0.233751
\(509\) 19.4653 0.862786 0.431393 0.902164i \(-0.358022\pi\)
0.431393 + 0.902164i \(0.358022\pi\)
\(510\) 0 0
\(511\) 11.2562 0.497944
\(512\) 5.20488 0.230026
\(513\) 0 0
\(514\) −25.0956 −1.10692
\(515\) 11.7633 0.518354
\(516\) 0 0
\(517\) 25.5139 1.12210
\(518\) 6.55069 0.287821
\(519\) 0 0
\(520\) 62.4087 2.73680
\(521\) −18.0600 −0.791221 −0.395611 0.918418i \(-0.629467\pi\)
−0.395611 + 0.918418i \(0.629467\pi\)
\(522\) 0 0
\(523\) −0.634346 −0.0277380 −0.0138690 0.999904i \(-0.504415\pi\)
−0.0138690 + 0.999904i \(0.504415\pi\)
\(524\) 101.996 4.45570
\(525\) 0 0
\(526\) 32.5482 1.41917
\(527\) 47.4828 2.06838
\(528\) 0 0
\(529\) −6.72353 −0.292328
\(530\) 26.0790 1.13280
\(531\) 0 0
\(532\) 0.880801 0.0381876
\(533\) 0.339296 0.0146965
\(534\) 0 0
\(535\) 39.6109 1.71253
\(536\) 82.4893 3.56299
\(537\) 0 0
\(538\) −75.3737 −3.24959
\(539\) 2.89389 0.124649
\(540\) 0 0
\(541\) 12.8668 0.553186 0.276593 0.960987i \(-0.410795\pi\)
0.276593 + 0.960987i \(0.410795\pi\)
\(542\) −31.3820 −1.34797
\(543\) 0 0
\(544\) −113.387 −4.86144
\(545\) 27.7193 1.18736
\(546\) 0 0
\(547\) 25.5679 1.09320 0.546602 0.837393i \(-0.315921\pi\)
0.546602 + 0.837393i \(0.315921\pi\)
\(548\) 82.6814 3.53198
\(549\) 0 0
\(550\) 49.0367 2.09093
\(551\) −0.0655389 −0.00279205
\(552\) 0 0
\(553\) −3.36380 −0.143043
\(554\) 70.1808 2.98170
\(555\) 0 0
\(556\) −41.2403 −1.74898
\(557\) 40.7860 1.72816 0.864079 0.503357i \(-0.167902\pi\)
0.864079 + 0.503357i \(0.167902\pi\)
\(558\) 0 0
\(559\) 12.9808 0.549029
\(560\) 44.4104 1.87668
\(561\) 0 0
\(562\) 47.6367 2.00943
\(563\) 10.0286 0.422657 0.211329 0.977415i \(-0.432221\pi\)
0.211329 + 0.977415i \(0.432221\pi\)
\(564\) 0 0
\(565\) 44.6137 1.87691
\(566\) −26.4485 −1.11171
\(567\) 0 0
\(568\) −105.986 −4.44709
\(569\) −29.2372 −1.22569 −0.612844 0.790204i \(-0.709975\pi\)
−0.612844 + 0.790204i \(0.709975\pi\)
\(570\) 0 0
\(571\) 10.8647 0.454672 0.227336 0.973816i \(-0.426998\pi\)
0.227336 + 0.973816i \(0.426998\pi\)
\(572\) −32.1433 −1.34398
\(573\) 0 0
\(574\) 0.433889 0.0181102
\(575\) −25.3569 −1.05746
\(576\) 0 0
\(577\) −3.53035 −0.146971 −0.0734853 0.997296i \(-0.523412\pi\)
−0.0734853 + 0.997296i \(0.523412\pi\)
\(578\) 60.9408 2.53480
\(579\) 0 0
\(580\) −6.93821 −0.288093
\(581\) −17.3020 −0.717807
\(582\) 0 0
\(583\) −8.33293 −0.345115
\(584\) −99.1881 −4.10443
\(585\) 0 0
\(586\) 48.7513 2.01390
\(587\) 30.2595 1.24894 0.624472 0.781048i \(-0.285314\pi\)
0.624472 + 0.781048i \(0.285314\pi\)
\(588\) 0 0
\(589\) 1.26142 0.0519758
\(590\) 31.3268 1.28970
\(591\) 0 0
\(592\) −32.1215 −1.32019
\(593\) −31.0070 −1.27330 −0.636652 0.771151i \(-0.719681\pi\)
−0.636652 + 0.771151i \(0.719681\pi\)
\(594\) 0 0
\(595\) −21.1409 −0.866693
\(596\) −83.0022 −3.39990
\(597\) 0 0
\(598\) 22.9311 0.937721
\(599\) 35.9127 1.46735 0.733676 0.679499i \(-0.237803\pi\)
0.733676 + 0.679499i \(0.237803\pi\)
\(600\) 0 0
\(601\) −46.4045 −1.89288 −0.946440 0.322879i \(-0.895349\pi\)
−0.946440 + 0.322879i \(0.895349\pi\)
\(602\) 16.5997 0.676555
\(603\) 0 0
\(604\) −10.5128 −0.427760
\(605\) 8.81958 0.358567
\(606\) 0 0
\(607\) −13.7737 −0.559058 −0.279529 0.960137i \(-0.590178\pi\)
−0.279529 + 0.960137i \(0.590178\pi\)
\(608\) −3.01222 −0.122162
\(609\) 0 0
\(610\) 108.966 4.41192
\(611\) −18.5873 −0.751961
\(612\) 0 0
\(613\) −33.4089 −1.34937 −0.674686 0.738105i \(-0.735721\pi\)
−0.674686 + 0.738105i \(0.735721\pi\)
\(614\) 10.9378 0.441415
\(615\) 0 0
\(616\) −25.5006 −1.02745
\(617\) −26.6250 −1.07188 −0.535941 0.844256i \(-0.680043\pi\)
−0.535941 + 0.844256i \(0.680043\pi\)
\(618\) 0 0
\(619\) −3.87967 −0.155937 −0.0779686 0.996956i \(-0.524843\pi\)
−0.0779686 + 0.996956i \(0.524843\pi\)
\(620\) 133.538 5.36303
\(621\) 0 0
\(622\) 36.5877 1.46703
\(623\) −14.6779 −0.588058
\(624\) 0 0
\(625\) −16.9225 −0.676902
\(626\) 63.0291 2.51915
\(627\) 0 0
\(628\) 23.2316 0.927041
\(629\) 15.2910 0.609691
\(630\) 0 0
\(631\) 24.1518 0.961467 0.480733 0.876867i \(-0.340370\pi\)
0.480733 + 0.876867i \(0.340370\pi\)
\(632\) 29.6414 1.17907
\(633\) 0 0
\(634\) −85.9473 −3.41340
\(635\) 3.35934 0.133311
\(636\) 0 0
\(637\) −2.10825 −0.0835319
\(638\) 3.05852 0.121088
\(639\) 0 0
\(640\) −79.4228 −3.13946
\(641\) −3.52432 −0.139202 −0.0696012 0.997575i \(-0.522173\pi\)
−0.0696012 + 0.997575i \(0.522173\pi\)
\(642\) 0 0
\(643\) 19.9282 0.785890 0.392945 0.919562i \(-0.371456\pi\)
0.392945 + 0.919562i \(0.371456\pi\)
\(644\) 21.2552 0.837574
\(645\) 0 0
\(646\) 2.83651 0.111601
\(647\) 38.7661 1.52405 0.762027 0.647546i \(-0.224205\pi\)
0.762027 + 0.647546i \(0.224205\pi\)
\(648\) 0 0
\(649\) −10.0097 −0.392916
\(650\) −35.7240 −1.40121
\(651\) 0 0
\(652\) −51.7203 −2.02552
\(653\) −23.5705 −0.922386 −0.461193 0.887300i \(-0.652579\pi\)
−0.461193 + 0.887300i \(0.652579\pi\)
\(654\) 0 0
\(655\) −65.0353 −2.54114
\(656\) −2.12759 −0.0830683
\(657\) 0 0
\(658\) −23.7693 −0.926623
\(659\) 7.43377 0.289579 0.144789 0.989463i \(-0.453750\pi\)
0.144789 + 0.989463i \(0.453750\pi\)
\(660\) 0 0
\(661\) 3.67368 0.142890 0.0714449 0.997445i \(-0.477239\pi\)
0.0714449 + 0.997445i \(0.477239\pi\)
\(662\) −88.6906 −3.44706
\(663\) 0 0
\(664\) 152.463 5.91671
\(665\) −0.561624 −0.0217789
\(666\) 0 0
\(667\) −1.58157 −0.0612385
\(668\) −80.8197 −3.12701
\(669\) 0 0
\(670\) −84.7822 −3.27542
\(671\) −34.8176 −1.34412
\(672\) 0 0
\(673\) 45.7468 1.76341 0.881706 0.471800i \(-0.156396\pi\)
0.881706 + 0.471800i \(0.156396\pi\)
\(674\) 32.9888 1.27068
\(675\) 0 0
\(676\) −45.0734 −1.73359
\(677\) −49.9400 −1.91935 −0.959674 0.281114i \(-0.909296\pi\)
−0.959674 + 0.281114i \(0.909296\pi\)
\(678\) 0 0
\(679\) 1.23267 0.0473055
\(680\) 186.291 7.14395
\(681\) 0 0
\(682\) −58.8668 −2.25413
\(683\) −31.8312 −1.21799 −0.608993 0.793176i \(-0.708426\pi\)
−0.608993 + 0.793176i \(0.708426\pi\)
\(684\) 0 0
\(685\) −52.7201 −2.01433
\(686\) −2.69601 −0.102934
\(687\) 0 0
\(688\) −81.3973 −3.10324
\(689\) 6.07068 0.231275
\(690\) 0 0
\(691\) 15.3903 0.585475 0.292737 0.956193i \(-0.405434\pi\)
0.292737 + 0.956193i \(0.405434\pi\)
\(692\) 67.2249 2.55551
\(693\) 0 0
\(694\) −13.4301 −0.509800
\(695\) 26.2960 0.997464
\(696\) 0 0
\(697\) 1.01281 0.0383628
\(698\) −19.6775 −0.744806
\(699\) 0 0
\(700\) −33.1133 −1.25157
\(701\) 16.6857 0.630210 0.315105 0.949057i \(-0.397960\pi\)
0.315105 + 0.949057i \(0.397960\pi\)
\(702\) 0 0
\(703\) 0.406216 0.0153207
\(704\) 64.0575 2.41426
\(705\) 0 0
\(706\) −32.9428 −1.23982
\(707\) 12.2730 0.461573
\(708\) 0 0
\(709\) 13.3407 0.501022 0.250511 0.968114i \(-0.419401\pi\)
0.250511 + 0.968114i \(0.419401\pi\)
\(710\) 108.932 4.08816
\(711\) 0 0
\(712\) 129.340 4.84722
\(713\) 30.4401 1.13999
\(714\) 0 0
\(715\) 20.4955 0.766488
\(716\) 6.34058 0.236958
\(717\) 0 0
\(718\) 97.2577 3.62962
\(719\) −20.3493 −0.758903 −0.379451 0.925212i \(-0.623887\pi\)
−0.379451 + 0.925212i \(0.623887\pi\)
\(720\) 0 0
\(721\) 3.50168 0.130409
\(722\) −51.1489 −1.90356
\(723\) 0 0
\(724\) −43.6035 −1.62051
\(725\) 2.46390 0.0915071
\(726\) 0 0
\(727\) 6.19761 0.229856 0.114928 0.993374i \(-0.463336\pi\)
0.114928 + 0.993374i \(0.463336\pi\)
\(728\) 18.5777 0.688534
\(729\) 0 0
\(730\) 101.945 3.77316
\(731\) 38.7480 1.43315
\(732\) 0 0
\(733\) −20.4599 −0.755703 −0.377851 0.925866i \(-0.623337\pi\)
−0.377851 + 0.925866i \(0.623337\pi\)
\(734\) −67.4606 −2.49001
\(735\) 0 0
\(736\) −72.6899 −2.67939
\(737\) 27.0901 0.997877
\(738\) 0 0
\(739\) −40.7478 −1.49893 −0.749466 0.662042i \(-0.769690\pi\)
−0.749466 + 0.662042i \(0.769690\pi\)
\(740\) 43.0036 1.58085
\(741\) 0 0
\(742\) 7.76314 0.284994
\(743\) 4.75496 0.174443 0.0872213 0.996189i \(-0.472201\pi\)
0.0872213 + 0.996189i \(0.472201\pi\)
\(744\) 0 0
\(745\) 52.9246 1.93901
\(746\) −34.8290 −1.27518
\(747\) 0 0
\(748\) −95.9485 −3.50822
\(749\) 11.7913 0.430844
\(750\) 0 0
\(751\) 30.3761 1.10844 0.554220 0.832370i \(-0.313017\pi\)
0.554220 + 0.832370i \(0.313017\pi\)
\(752\) 116.553 4.25026
\(753\) 0 0
\(754\) −2.22819 −0.0811457
\(755\) 6.70327 0.243957
\(756\) 0 0
\(757\) 2.83567 0.103064 0.0515321 0.998671i \(-0.483590\pi\)
0.0515321 + 0.998671i \(0.483590\pi\)
\(758\) 2.90286 0.105437
\(759\) 0 0
\(760\) 4.94897 0.179518
\(761\) −10.8152 −0.392052 −0.196026 0.980599i \(-0.562804\pi\)
−0.196026 + 0.980599i \(0.562804\pi\)
\(762\) 0 0
\(763\) 8.25140 0.298721
\(764\) −85.1314 −3.07994
\(765\) 0 0
\(766\) 37.9391 1.37079
\(767\) 7.29226 0.263308
\(768\) 0 0
\(769\) −12.8275 −0.462571 −0.231286 0.972886i \(-0.574293\pi\)
−0.231286 + 0.972886i \(0.574293\pi\)
\(770\) 26.2095 0.944524
\(771\) 0 0
\(772\) 54.3934 1.95766
\(773\) −34.8329 −1.25285 −0.626427 0.779480i \(-0.715483\pi\)
−0.626427 + 0.779480i \(0.715483\pi\)
\(774\) 0 0
\(775\) −47.4223 −1.70346
\(776\) −10.8621 −0.389928
\(777\) 0 0
\(778\) 2.41677 0.0866455
\(779\) 0.0269060 0.000964006 0
\(780\) 0 0
\(781\) −34.8067 −1.24548
\(782\) 68.4498 2.44776
\(783\) 0 0
\(784\) 13.2200 0.472142
\(785\) −14.8131 −0.528703
\(786\) 0 0
\(787\) 7.06256 0.251753 0.125876 0.992046i \(-0.459826\pi\)
0.125876 + 0.992046i \(0.459826\pi\)
\(788\) −103.924 −3.70213
\(789\) 0 0
\(790\) −30.4653 −1.08391
\(791\) 13.2805 0.472200
\(792\) 0 0
\(793\) 25.3652 0.900745
\(794\) 41.9620 1.48918
\(795\) 0 0
\(796\) 6.60443 0.234088
\(797\) 4.18736 0.148324 0.0741619 0.997246i \(-0.476372\pi\)
0.0741619 + 0.997246i \(0.476372\pi\)
\(798\) 0 0
\(799\) −55.4835 −1.96286
\(800\) 113.243 4.00374
\(801\) 0 0
\(802\) −58.0104 −2.04842
\(803\) −32.5741 −1.14952
\(804\) 0 0
\(805\) −13.5530 −0.477679
\(806\) 42.8855 1.51058
\(807\) 0 0
\(808\) −108.148 −3.80463
\(809\) 5.20778 0.183096 0.0915479 0.995801i \(-0.470819\pi\)
0.0915479 + 0.995801i \(0.470819\pi\)
\(810\) 0 0
\(811\) 40.9326 1.43734 0.718668 0.695353i \(-0.244752\pi\)
0.718668 + 0.695353i \(0.244752\pi\)
\(812\) −2.06535 −0.0724795
\(813\) 0 0
\(814\) −18.9570 −0.664442
\(815\) 32.9784 1.15518
\(816\) 0 0
\(817\) 1.02937 0.0360131
\(818\) −1.11393 −0.0389476
\(819\) 0 0
\(820\) 2.84837 0.0994694
\(821\) 9.27823 0.323813 0.161906 0.986806i \(-0.448236\pi\)
0.161906 + 0.986806i \(0.448236\pi\)
\(822\) 0 0
\(823\) −33.4733 −1.16681 −0.583404 0.812182i \(-0.698279\pi\)
−0.583404 + 0.812182i \(0.698279\pi\)
\(824\) −30.8564 −1.07493
\(825\) 0 0
\(826\) 9.32528 0.324468
\(827\) 11.0635 0.384716 0.192358 0.981325i \(-0.438387\pi\)
0.192358 + 0.981325i \(0.438387\pi\)
\(828\) 0 0
\(829\) −18.5732 −0.645075 −0.322538 0.946557i \(-0.604536\pi\)
−0.322538 + 0.946557i \(0.604536\pi\)
\(830\) −156.701 −5.43917
\(831\) 0 0
\(832\) −46.6670 −1.61789
\(833\) −6.29317 −0.218046
\(834\) 0 0
\(835\) 51.5330 1.78337
\(836\) −2.54894 −0.0881571
\(837\) 0 0
\(838\) 98.1875 3.39183
\(839\) 43.0577 1.48652 0.743258 0.669004i \(-0.233279\pi\)
0.743258 + 0.669004i \(0.233279\pi\)
\(840\) 0 0
\(841\) −28.8463 −0.994701
\(842\) −35.6209 −1.22758
\(843\) 0 0
\(844\) 9.56862 0.329366
\(845\) 28.7401 0.988690
\(846\) 0 0
\(847\) 2.62539 0.0902095
\(848\) −38.0668 −1.30722
\(849\) 0 0
\(850\) −106.637 −3.65762
\(851\) 9.80269 0.336032
\(852\) 0 0
\(853\) 17.7530 0.607850 0.303925 0.952696i \(-0.401703\pi\)
0.303925 + 0.952696i \(0.401703\pi\)
\(854\) 32.4368 1.10997
\(855\) 0 0
\(856\) −103.903 −3.55134
\(857\) −2.85537 −0.0975377 −0.0487688 0.998810i \(-0.515530\pi\)
−0.0487688 + 0.998810i \(0.515530\pi\)
\(858\) 0 0
\(859\) 19.0355 0.649485 0.324742 0.945803i \(-0.394722\pi\)
0.324742 + 0.945803i \(0.394722\pi\)
\(860\) 108.973 3.71595
\(861\) 0 0
\(862\) 91.3548 3.11156
\(863\) −24.5348 −0.835173 −0.417586 0.908637i \(-0.637124\pi\)
−0.417586 + 0.908637i \(0.637124\pi\)
\(864\) 0 0
\(865\) −42.8645 −1.45744
\(866\) −12.3216 −0.418705
\(867\) 0 0
\(868\) 39.7514 1.34925
\(869\) 9.73446 0.330219
\(870\) 0 0
\(871\) −19.7356 −0.668716
\(872\) −72.7104 −2.46229
\(873\) 0 0
\(874\) 1.81842 0.0615090
\(875\) 4.31730 0.145951
\(876\) 0 0
\(877\) 0.731536 0.0247022 0.0123511 0.999924i \(-0.496068\pi\)
0.0123511 + 0.999924i \(0.496068\pi\)
\(878\) −40.7810 −1.37629
\(879\) 0 0
\(880\) −128.519 −4.33237
\(881\) −24.2382 −0.816605 −0.408302 0.912847i \(-0.633879\pi\)
−0.408302 + 0.912847i \(0.633879\pi\)
\(882\) 0 0
\(883\) −24.7304 −0.832245 −0.416122 0.909309i \(-0.636611\pi\)
−0.416122 + 0.909309i \(0.636611\pi\)
\(884\) 69.9001 2.35099
\(885\) 0 0
\(886\) −57.8800 −1.94452
\(887\) −2.92462 −0.0981992 −0.0490996 0.998794i \(-0.515635\pi\)
−0.0490996 + 0.998794i \(0.515635\pi\)
\(888\) 0 0
\(889\) 1.00000 0.0335389
\(890\) −132.935 −4.45600
\(891\) 0 0
\(892\) 3.27786 0.109751
\(893\) −1.47396 −0.0493242
\(894\) 0 0
\(895\) −4.04293 −0.135140
\(896\) −23.6424 −0.789836
\(897\) 0 0
\(898\) 73.6633 2.45818
\(899\) −2.95783 −0.0986493
\(900\) 0 0
\(901\) 18.1211 0.603702
\(902\) −1.25563 −0.0418078
\(903\) 0 0
\(904\) −117.026 −3.89223
\(905\) 27.8028 0.924197
\(906\) 0 0
\(907\) 22.1789 0.736440 0.368220 0.929739i \(-0.379967\pi\)
0.368220 + 0.929739i \(0.379967\pi\)
\(908\) 133.723 4.43774
\(909\) 0 0
\(910\) −19.0940 −0.632961
\(911\) 4.10731 0.136081 0.0680406 0.997683i \(-0.478325\pi\)
0.0680406 + 0.997683i \(0.478325\pi\)
\(912\) 0 0
\(913\) 50.0700 1.65708
\(914\) 48.1721 1.59339
\(915\) 0 0
\(916\) −3.70086 −0.122280
\(917\) −19.3595 −0.639309
\(918\) 0 0
\(919\) 18.2568 0.602238 0.301119 0.953587i \(-0.402640\pi\)
0.301119 + 0.953587i \(0.402640\pi\)
\(920\) 119.427 3.93739
\(921\) 0 0
\(922\) −14.4900 −0.477203
\(923\) 25.3573 0.834646
\(924\) 0 0
\(925\) −15.2715 −0.502124
\(926\) 58.9312 1.93660
\(927\) 0 0
\(928\) 7.06320 0.231861
\(929\) 48.7368 1.59900 0.799501 0.600665i \(-0.205097\pi\)
0.799501 + 0.600665i \(0.205097\pi\)
\(930\) 0 0
\(931\) −0.167183 −0.00547920
\(932\) 85.0023 2.78434
\(933\) 0 0
\(934\) −88.6345 −2.90021
\(935\) 61.1795 2.00078
\(936\) 0 0
\(937\) −18.6686 −0.609875 −0.304938 0.952372i \(-0.598636\pi\)
−0.304938 + 0.952372i \(0.598636\pi\)
\(938\) −25.2378 −0.824042
\(939\) 0 0
\(940\) −156.039 −5.08944
\(941\) 51.5302 1.67984 0.839918 0.542713i \(-0.182603\pi\)
0.839918 + 0.542713i \(0.182603\pi\)
\(942\) 0 0
\(943\) 0.649287 0.0211437
\(944\) −45.7268 −1.48828
\(945\) 0 0
\(946\) −48.0378 −1.56184
\(947\) 27.8818 0.906036 0.453018 0.891501i \(-0.350347\pi\)
0.453018 + 0.891501i \(0.350347\pi\)
\(948\) 0 0
\(949\) 23.7308 0.770335
\(950\) −2.83290 −0.0919113
\(951\) 0 0
\(952\) 55.4547 1.79730
\(953\) 39.2006 1.26983 0.634916 0.772582i \(-0.281035\pi\)
0.634916 + 0.772582i \(0.281035\pi\)
\(954\) 0 0
\(955\) 54.2822 1.75653
\(956\) 43.9741 1.42223
\(957\) 0 0
\(958\) 42.7830 1.38226
\(959\) −15.6936 −0.506772
\(960\) 0 0
\(961\) 25.9289 0.836417
\(962\) 13.8105 0.445268
\(963\) 0 0
\(964\) 61.6724 1.98633
\(965\) −34.6828 −1.11648
\(966\) 0 0
\(967\) 31.0201 0.997539 0.498770 0.866735i \(-0.333785\pi\)
0.498770 + 0.866735i \(0.333785\pi\)
\(968\) −23.1347 −0.743576
\(969\) 0 0
\(970\) 11.1641 0.358456
\(971\) 7.85796 0.252174 0.126087 0.992019i \(-0.459758\pi\)
0.126087 + 0.992019i \(0.459758\pi\)
\(972\) 0 0
\(973\) 7.82773 0.250945
\(974\) 24.2351 0.776544
\(975\) 0 0
\(976\) −159.055 −5.09123
\(977\) 33.1002 1.05897 0.529484 0.848320i \(-0.322385\pi\)
0.529484 + 0.848320i \(0.322385\pi\)
\(978\) 0 0
\(979\) 42.4762 1.35755
\(980\) −17.6986 −0.565362
\(981\) 0 0
\(982\) 3.75734 0.119902
\(983\) 5.35215 0.170707 0.0853535 0.996351i \(-0.472798\pi\)
0.0853535 + 0.996351i \(0.472798\pi\)
\(984\) 0 0
\(985\) 66.2647 2.11137
\(986\) −6.65119 −0.211817
\(987\) 0 0
\(988\) 1.85695 0.0590774
\(989\) 24.8404 0.789880
\(990\) 0 0
\(991\) 31.1999 0.991097 0.495549 0.868580i \(-0.334967\pi\)
0.495549 + 0.868580i \(0.334967\pi\)
\(992\) −135.944 −4.31623
\(993\) 0 0
\(994\) 32.4267 1.02851
\(995\) −4.21118 −0.133503
\(996\) 0 0
\(997\) −62.0455 −1.96500 −0.982501 0.186257i \(-0.940364\pi\)
−0.982501 + 0.186257i \(0.940364\pi\)
\(998\) −16.0877 −0.509247
\(999\) 0 0
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 8001.2.a.z.1.32 yes 32
3.2 odd 2 inner 8001.2.a.z.1.1 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8001.2.a.z.1.1 32 3.2 odd 2 inner
8001.2.a.z.1.32 yes 32 1.1 even 1 trivial