Properties

Label 69.2.c.a.68.3
Level $69$
Weight $2$
Character 69.68
Analytic conductor $0.551$
Analytic rank $0$
Dimension $6$
CM discriminant -23
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 69.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.550967773947\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.8869743.1
Defining polynomial: \(x^{6} - 3 x^{3} + 8\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 68.3
Root \(1.33454 - 0.467979i\) of defining polynomial
Character \(\chi\) \(=\) 69.68
Dual form 69.2.c.a.68.4

$q$-expansion

\(f(q)\) \(=\) \(q-0.935958i q^{2} +(-0.227452 + 1.71705i) q^{3} +1.12398 q^{4} +(1.60709 + 0.212885i) q^{6} -2.92392i q^{8} +(-2.89653 - 0.781094i) q^{9} +O(q^{10})\) \(q-0.935958i q^{2} +(-0.227452 + 1.71705i) q^{3} +1.12398 q^{4} +(1.60709 + 0.212885i) q^{6} -2.92392i q^{8} +(-2.89653 - 0.781094i) q^{9} +(-0.255652 + 1.92994i) q^{12} -4.88325 q^{13} -0.488695 q^{16} +(-0.731071 + 2.71103i) q^{18} +4.79583i q^{23} +(5.02051 + 0.665051i) q^{24} -5.00000 q^{25} +4.57052i q^{26} +(2.00000 - 4.79583i) q^{27} -8.43039i q^{29} +11.1312 q^{31} -5.39043i q^{32} +(-3.25565 - 0.877936i) q^{36} +(1.11071 - 8.38480i) q^{39} +12.1742i q^{41} +4.48870 q^{46} +6.55848i q^{47} +(0.111155 - 0.839115i) q^{48} +7.00000 q^{49} +4.67979i q^{50} -5.48870 q^{52} +(-4.48870 - 1.87192i) q^{54} -7.89049 q^{58} +9.59166i q^{59} -10.4184i q^{62} -6.02261 q^{64} +(-8.23469 - 1.09082i) q^{69} -14.0461i q^{71} +(-2.28385 + 8.46921i) q^{72} -7.61268 q^{73} +(1.13726 - 8.58526i) q^{75} +(-7.84782 - 1.03957i) q^{78} +(7.77979 + 4.52492i) q^{81} +11.3946 q^{82} +(14.4754 + 1.91751i) q^{87} +5.39043i q^{92} +(-2.53182 + 19.1129i) q^{93} +6.13846 q^{94} +(9.25565 + 1.22607i) q^{96} -6.55170i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 12q^{4} + 3q^{6} + O(q^{10}) \) \( 6q - 12q^{4} + 3q^{6} - 15q^{12} + 24q^{16} + 21q^{18} - 6q^{24} - 30q^{25} + 12q^{27} - 33q^{36} - 24q^{39} + 69q^{48} + 42q^{49} - 6q^{52} + 30q^{58} - 90q^{64} - 42q^{72} - 51q^{78} + 66q^{82} + 48q^{87} - 6q^{93} - 78q^{94} + 69q^{96} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/69\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(47\)
\(\chi(n)\) \(-1\) \(-1\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.935958i 0.661822i −0.943662 0.330911i \(-0.892644\pi\)
0.943662 0.330911i \(-0.107356\pi\)
\(3\) −0.227452 + 1.71705i −0.131319 + 0.991340i
\(4\) 1.12398 0.561992
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 1.60709 + 0.212885i 0.656091 + 0.0869101i
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 2.92392i 1.03376i
\(9\) −2.89653 0.781094i −0.965510 0.260365i
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) −0.255652 + 1.92994i −0.0738005 + 0.557125i
\(13\) −4.88325 −1.35437 −0.677185 0.735812i \(-0.736801\pi\)
−0.677185 + 0.735812i \(0.736801\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −0.488695 −0.122174
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −0.731071 + 2.71103i −0.172315 + 0.638996i
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.79583i 1.00000i
\(24\) 5.02051 + 0.665051i 1.02481 + 0.135753i
\(25\) −5.00000 −1.00000
\(26\) 4.57052i 0.896353i
\(27\) 2.00000 4.79583i 0.384900 0.922958i
\(28\) 0 0
\(29\) 8.43039i 1.56548i −0.622346 0.782742i \(-0.713820\pi\)
0.622346 0.782742i \(-0.286180\pi\)
\(30\) 0 0
\(31\) 11.1312 1.99923 0.999613 0.0278144i \(-0.00885474\pi\)
0.999613 + 0.0278144i \(0.00885474\pi\)
\(32\) 5.39043i 0.952903i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −3.25565 0.877936i −0.542609 0.146323i
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 1.11071 8.38480i 0.177855 1.34264i
\(40\) 0 0
\(41\) 12.1742i 1.90129i 0.310274 + 0.950647i \(0.399579\pi\)
−0.310274 + 0.950647i \(0.600421\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 4.48870 0.661822
\(47\) 6.55848i 0.956652i 0.878182 + 0.478326i \(0.158756\pi\)
−0.878182 + 0.478326i \(0.841244\pi\)
\(48\) 0.111155 0.839115i 0.0160438 0.121116i
\(49\) 7.00000 1.00000
\(50\) 4.67979i 0.661822i
\(51\) 0 0
\(52\) −5.48870 −0.761145
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) −4.48870 1.87192i −0.610834 0.254735i
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −7.89049 −1.03607
\(59\) 9.59166i 1.24873i 0.781133 + 0.624364i \(0.214642\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 10.4184i 1.32313i
\(63\) 0 0
\(64\) −6.02261 −0.752826
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) −8.23469 1.09082i −0.991340 0.131319i
\(70\) 0 0
\(71\) 14.0461i 1.66697i −0.552542 0.833485i \(-0.686342\pi\)
0.552542 0.833485i \(-0.313658\pi\)
\(72\) −2.28385 + 8.46921i −0.269155 + 0.998106i
\(73\) −7.61268 −0.890997 −0.445498 0.895283i \(-0.646973\pi\)
−0.445498 + 0.895283i \(0.646973\pi\)
\(74\) 0 0
\(75\) 1.13726 8.58526i 0.131319 0.991340i
\(76\) 0 0
\(77\) 0 0
\(78\) −7.84782 1.03957i −0.888590 0.117709i
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 7.77979 + 4.52492i 0.864421 + 0.502769i
\(82\) 11.3946 1.25832
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 14.4754 + 1.91751i 1.55193 + 0.205579i
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 5.39043i 0.561992i
\(93\) −2.53182 + 19.1129i −0.262537 + 1.98191i
\(94\) 6.13846 0.633134
\(95\) 0 0
\(96\) 9.25565 + 1.22607i 0.944651 + 0.125135i
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 6.55170i 0.661822i
\(99\) 0 0
\(100\) −5.61992 −0.561992
\(101\) 19.1833i 1.90881i −0.298511 0.954406i \(-0.596490\pi\)
0.298511 0.954406i \(-0.403510\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 14.2782i 1.40010i
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 2.24797 5.39043i 0.216311 0.518695i
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 9.47562i 0.879789i
\(117\) 14.1445 + 3.81428i 1.30766 + 0.352630i
\(118\) 8.97739 0.826436
\(119\) 0 0
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) −20.9038 2.76905i −1.88483 0.249677i
\(124\) 12.5113 1.12355
\(125\) 0 0
\(126\) 0 0
\(127\) 8.40180 0.745539 0.372769 0.927924i \(-0.378408\pi\)
0.372769 + 0.927924i \(0.378408\pi\)
\(128\) 5.14396i 0.454666i
\(129\) 0 0
\(130\) 0 0
\(131\) 2.81465i 0.245917i −0.992412 0.122958i \(-0.960762\pi\)
0.992412 0.122958i \(-0.0392382\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) −1.02096 + 7.70732i −0.0869101 + 0.656091i
\(139\) −20.8977 −1.77252 −0.886261 0.463186i \(-0.846706\pi\)
−0.886261 + 0.463186i \(0.846706\pi\)
\(140\) 0 0
\(141\) −11.2612 1.49174i −0.948368 0.125627i
\(142\) −13.1466 −1.10324
\(143\) 0 0
\(144\) 1.41552 + 0.381717i 0.117960 + 0.0318097i
\(145\) 0 0
\(146\) 7.12515i 0.589681i
\(147\) −1.59216 + 12.0194i −0.131319 + 0.991340i
\(148\) 0 0
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) −8.03544 1.06443i −0.656091 0.0869101i
\(151\) 13.8606 1.12796 0.563982 0.825787i \(-0.309269\pi\)
0.563982 + 0.825787i \(0.309269\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 1.24842 9.42437i 0.0999532 0.754554i
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 4.23514 7.28155i 0.332744 0.572093i
\(163\) −23.6272 −1.85062 −0.925311 0.379210i \(-0.876196\pi\)
−0.925311 + 0.379210i \(0.876196\pi\)
\(164\) 13.6836i 1.06851i
\(165\) 0 0
\(166\) 0 0
\(167\) 9.59166i 0.742225i 0.928588 + 0.371113i \(0.121024\pi\)
−0.928588 + 0.371113i \(0.878976\pi\)
\(168\) 0 0
\(169\) 10.8462 0.834321
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 19.1833i 1.45848i −0.684257 0.729241i \(-0.739873\pi\)
0.684257 0.729241i \(-0.260127\pi\)
\(174\) 1.79471 13.5484i 0.136056 1.02710i
\(175\) 0 0
\(176\) 0 0
\(177\) −16.4694 2.18164i −1.23791 0.163982i
\(178\) 0 0
\(179\) 17.7900i 1.32968i 0.746984 + 0.664842i \(0.231501\pi\)
−0.746984 + 0.664842i \(0.768499\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 14.0226 1.03376
\(185\) 0 0
\(186\) 17.8888 + 2.36968i 1.31167 + 0.173753i
\(187\) 0 0
\(188\) 7.37162i 0.537631i
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 1.36985 10.3411i 0.0988608 0.746307i
\(193\) 27.1457 1.95399 0.976995 0.213262i \(-0.0684089\pi\)
0.976995 + 0.213262i \(0.0684089\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 7.86788 0.561992
\(197\) 0.942731i 0.0671668i 0.999436 + 0.0335834i \(0.0106919\pi\)
−0.999436 + 0.0335834i \(0.989308\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 14.6196i 1.03376i
\(201\) 0 0
\(202\) −17.9548 −1.26329
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 3.74599 13.8913i 0.260365 0.965510i
\(208\) 2.38642 0.165469
\(209\) 0 0
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 0 0
\(213\) 24.1179 + 3.19482i 1.65253 + 0.218906i
\(214\) 0 0
\(215\) 0 0
\(216\) −14.0226 5.84783i −0.954118 0.397895i
\(217\) 0 0
\(218\) 0 0
\(219\) 1.73152 13.0714i 0.117005 0.883281i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 8.00000 0.535720 0.267860 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 0 0
\(225\) 14.4827 + 3.90547i 0.965510 + 0.260365i
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −24.6498 −1.61834
\(233\) 29.0350i 1.90215i −0.308965 0.951073i \(-0.599983\pi\)
0.308965 0.951073i \(-0.400017\pi\)
\(234\) 3.57000 13.2387i 0.233378 0.865438i
\(235\) 0 0
\(236\) 10.7809i 0.701775i
\(237\) 0 0
\(238\) 0 0
\(239\) 27.1631i 1.75703i 0.477711 + 0.878517i \(0.341467\pi\)
−0.477711 + 0.878517i \(0.658533\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 10.2955i 0.661822i
\(243\) −9.53906 + 12.3291i −0.611931 + 0.790911i
\(244\) 0 0
\(245\) 0 0
\(246\) −2.59172 + 19.5650i −0.165242 + 1.24742i
\(247\) 0 0
\(248\) 32.5468i 2.06672i
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 7.86373i 0.493414i
\(255\) 0 0
\(256\) −16.8597 −1.05373
\(257\) 19.6619i 1.22647i −0.789899 0.613237i \(-0.789867\pi\)
0.789899 0.613237i \(-0.210133\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −6.58493 + 24.4189i −0.407597 + 1.51149i
\(262\) −2.63439 −0.162753
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 32.7788i 1.99856i 0.0379247 + 0.999281i \(0.487925\pi\)
−0.0379247 + 0.999281i \(0.512075\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) −9.25565 1.22607i −0.557125 0.0738005i
\(277\) 29.8751 1.79502 0.897511 0.440992i \(-0.145373\pi\)
0.897511 + 0.440992i \(0.145373\pi\)
\(278\) 19.5594i 1.17309i
\(279\) −32.2419 8.69453i −1.93027 0.520528i
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) −1.39620 + 10.5400i −0.0831428 + 0.627651i
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 15.7876i 0.936823i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −4.21043 + 15.6136i −0.248102 + 0.920038i
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) −8.55652 −0.500733
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 11.2496 + 1.49020i 0.656091 + 0.0869101i
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 23.4193i 1.35437i
\(300\) 1.27826 9.64968i 0.0738005 0.557125i
\(301\) 0 0
\(302\) 12.9730i 0.746511i
\(303\) 32.9388 + 4.36329i 1.89228 + 0.250664i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 20.0000 1.14146 0.570730 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 34.6508i 1.96486i −0.186621 0.982432i \(-0.559754\pi\)
0.186621 0.982432i \(-0.440246\pi\)
\(312\) −24.5164 3.24761i −1.38797 0.183860i
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 19.1833i 1.07744i −0.842484 0.538721i \(-0.818908\pi\)
0.842484 0.538721i \(-0.181092\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 8.74435 + 5.08594i 0.485797 + 0.282552i
\(325\) 24.4163 1.35437
\(326\) 22.1140i 1.22478i
\(327\) 0 0
\(328\) 35.5964 1.96548
\(329\) 0 0
\(330\) 0 0
\(331\) −18.1683 −0.998620 −0.499310 0.866423i \(-0.666413\pi\)
−0.499310 + 0.866423i \(0.666413\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 8.97739 0.491221
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 10.1516i 0.552172i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −17.9548 −0.965255
\(347\) 9.59166i 0.514907i 0.966291 + 0.257454i \(0.0828835\pi\)
−0.966291 + 0.257454i \(0.917117\pi\)
\(348\) 16.2701 + 2.15525i 0.872170 + 0.115533i
\(349\) −10.3421 −0.553600 −0.276800 0.960928i \(-0.589274\pi\)
−0.276800 + 0.960928i \(0.589274\pi\)
\(350\) 0 0
\(351\) −9.76651 + 23.4193i −0.521298 + 1.25003i
\(352\) 0 0
\(353\) 21.5473i 1.14685i 0.819258 + 0.573425i \(0.194386\pi\)
−0.819258 + 0.573425i \(0.805614\pi\)
\(354\) −2.04193 + 15.4146i −0.108527 + 0.819279i
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 16.6507 0.880015
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 19.0000 1.00000
\(362\) 0 0
\(363\) 2.50197 18.8876i 0.131319 0.991340i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 2.34370i 0.122174i
\(369\) 9.50921 35.2630i 0.495030 1.83572i
\(370\) 0 0
\(371\) 0 0
\(372\) −2.84572 + 21.4826i −0.147544 + 1.11382i
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 19.1764 0.988949
\(377\) 41.1678i 2.12025i
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) −1.91101 + 14.4263i −0.0979038 + 0.739083i
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 8.83244 + 1.17000i 0.450729 + 0.0597065i
\(385\) 0 0
\(386\) 25.4072i 1.29319i
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 20.4674i 1.03376i
\(393\) 4.83289 + 0.640197i 0.243787 + 0.0322937i
\(394\) 0.882357 0.0444525
\(395\) 0 0
\(396\) 0 0
\(397\) −39.6416 −1.98956 −0.994778 0.102061i \(-0.967456\pi\)
−0.994778 + 0.102061i \(0.967456\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 2.44348 0.122174
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) −54.3566 −2.70769
\(404\) 21.5617i 1.07274i
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −36.9122 −1.82519 −0.912595 0.408864i \(-0.865925\pi\)
−0.912595 + 0.408864i \(0.865925\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) −13.0016 3.50609i −0.638996 0.172315i
\(415\) 0 0
\(416\) 26.3229i 1.29058i
\(417\) 4.75323 35.8825i 0.232767 1.75717i
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 3.74383i 0.182247i
\(423\) 5.12279 18.9968i 0.249078 0.923658i
\(424\) 0 0
\(425\) 0 0
\(426\) 2.99022 22.5734i 0.144877 1.09368i
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) −0.977391 + 2.34370i −0.0470247 + 0.112761i
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) −12.2342 1.62063i −0.584575 0.0774366i
\(439\) 5.67237 0.270728 0.135364 0.990796i \(-0.456780\pi\)
0.135364 + 0.990796i \(0.456780\pi\)
\(440\) 0 0
\(441\) −20.2757 5.46766i −0.965510 0.260365i
\(442\) 0 0
\(443\) 38.3946i 1.82418i 0.409988 + 0.912091i \(0.365533\pi\)
−0.409988 + 0.912091i \(0.634467\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 7.48766i 0.354551i
\(447\) 0 0
\(448\) 0 0
\(449\) 38.3667i 1.81063i 0.424736 + 0.905317i \(0.360367\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 3.65535 13.5552i 0.172315 0.638996i
\(451\) 0 0
\(452\) 0 0
\(453\) −3.15263 + 23.7994i −0.148124 + 1.11820i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 23.4057i 1.09011i 0.838399 + 0.545056i \(0.183492\pi\)
−0.838399 + 0.545056i \(0.816508\pi\)
\(462\) 0 0
\(463\) 32.0000 1.48717 0.743583 0.668644i \(-0.233125\pi\)
0.743583 + 0.668644i \(0.233125\pi\)
\(464\) 4.11989i 0.191261i
\(465\) 0 0
\(466\) −27.1755 −1.25888
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 15.8982 + 4.28719i 0.734894 + 0.198175i
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 28.0452 1.29089
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 25.4235 1.16284
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −12.3638 −0.561992
\(485\) 0 0
\(486\) 11.5395 + 8.92815i 0.523443 + 0.404989i
\(487\) 16.5901 0.751768 0.375884 0.926667i \(-0.377339\pi\)
0.375884 + 0.926667i \(0.377339\pi\)
\(488\) 0 0
\(489\) 5.37404 40.5690i 0.243023 1.83460i
\(490\) 0 0
\(491\) 4.67301i 0.210890i −0.994425 0.105445i \(-0.966373\pi\)
0.994425 0.105445i \(-0.0336267\pi\)
\(492\) −23.4955 3.11237i −1.05926 0.140316i
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −5.43978 −0.244253
\(497\) 0 0
\(498\) 0 0
\(499\) 43.1602 1.93211 0.966057 0.258328i \(-0.0831715\pi\)
0.966057 + 0.258328i \(0.0831715\pi\)
\(500\) 0 0
\(501\) −16.4694 2.18164i −0.735798 0.0974686i
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −2.46698 + 18.6234i −0.109563 + 0.827096i
\(508\) 9.44348 0.418987
\(509\) 17.8035i 0.789127i −0.918869 0.394564i \(-0.870896\pi\)
0.918869 0.394564i \(-0.129104\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 5.49209i 0.242718i
\(513\) 0 0
\(514\) −18.4027 −0.811708
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 32.9388 + 4.36329i 1.44585 + 0.191527i
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 22.8551 + 6.16321i 1.00034 + 0.269757i
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 3.16362i 0.138203i
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 7.49199 27.7826i 0.325125 1.20566i
\(532\) 0 0
\(533\) 59.4498i 2.57506i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −30.5463 4.04637i −1.31817 0.174614i
\(538\) 30.6796 1.32269
\(539\) 0 0
\(540\) 0 0
\(541\) 0.575595 0.0247468 0.0123734 0.999923i \(-0.496061\pi\)
0.0123734 + 0.999923i \(0.496061\pi\)
\(542\) 14.9753i 0.643245i
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 45.8896 1.96210 0.981049 0.193761i \(-0.0620688\pi\)
0.981049 + 0.193761i \(0.0620688\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) −3.18947 + 24.0775i −0.135753 + 1.02481i
\(553\) 0 0
\(554\) 27.9618i 1.18799i
\(555\) 0 0
\(556\) −23.4887 −0.996143
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) −8.13771 + 30.1771i −0.344497 + 1.27750i
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) −12.6574 1.67669i −0.532975 0.0706014i
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) −41.0697 −1.72325
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 23.9792i 1.00000i
\(576\) 17.4447 + 4.70422i 0.726861 + 0.196009i
\(577\) 32.6045 1.35734 0.678672 0.734441i \(-0.262556\pi\)
0.678672 + 0.734441i \(0.262556\pi\)
\(578\) 15.9113i 0.661822i
\(579\) −6.17434 + 46.6106i −0.256597 + 1.93707i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 22.2588i 0.921077i
\(585\) 0 0
\(586\) 0 0
\(587\) 25.2776i 1.04332i −0.853154 0.521660i \(-0.825313\pi\)
0.853154 0.521660i \(-0.174687\pi\)
\(588\) −1.78957 + 13.5096i −0.0738005 + 0.557125i
\(589\) 0 0
\(590\) 0 0
\(591\) −1.61872 0.214426i −0.0665852 0.00882032i
\(592\) 0 0
\(593\) 38.3667i 1.57553i 0.615976 + 0.787765i \(0.288762\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) −21.9194 −0.896353
\(599\) 9.59166i 0.391905i 0.980613 + 0.195952i \(0.0627798\pi\)
−0.980613 + 0.195952i \(0.937220\pi\)
\(600\) −25.1026 3.32525i −1.02481 0.135753i
\(601\) −42.3711 −1.72835 −0.864176 0.503190i \(-0.832159\pi\)
−0.864176 + 0.503190i \(0.832159\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 15.5791 0.633906
\(605\) 0 0
\(606\) 4.08385 30.8293i 0.165895 1.25235i
\(607\) −40.0000 −1.62355 −0.811775 0.583970i \(-0.801498\pi\)
−0.811775 + 0.583970i \(0.801498\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 32.0267i 1.29566i
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 18.7192i 0.755444i
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 23.0000 + 9.59166i 0.922958 + 0.384900i
\(622\) −32.4316 −1.30039
\(623\) 0 0
\(624\) −0.542797 + 4.09761i −0.0217293 + 0.164036i
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0.909808 6.86821i 0.0361616 0.272987i
\(634\) −17.9548 −0.713075
\(635\) 0 0
\(636\) 0 0
\(637\) −34.1828 −1.35437
\(638\) 0 0
\(639\) −10.9714 + 40.6851i −0.434020 + 1.60948i
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 47.7677i 1.87794i 0.343996 + 0.938971i \(0.388219\pi\)
−0.343996 + 0.938971i \(0.611781\pi\)
\(648\) 13.2305 22.7474i 0.519743 0.893604i
\(649\) 0 0
\(650\) 22.8526i 0.896353i
\(651\) 0 0
\(652\) −26.5565 −1.04003
\(653\) 49.6396i 1.94255i −0.237962 0.971274i \(-0.576480\pi\)
0.237962 0.971274i \(-0.423520\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 5.94949i 0.232288i
\(657\) 22.0504 + 5.94622i 0.860267 + 0.231984i
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 17.0048i 0.660909i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 40.4307 1.56548
\(668\) 10.7809i 0.417124i
\(669\) −1.81962 + 13.7364i −0.0703504 + 0.531080i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 21.6868 0.835966 0.417983 0.908455i \(-0.362737\pi\)
0.417983 + 0.908455i \(0.362737\pi\)
\(674\) 0 0
\(675\) −10.0000 + 23.9792i −0.384900 + 0.922958i
\(676\) 12.1909 0.468881
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 44.0239i 1.68453i −0.539066 0.842263i \(-0.681223\pi\)
0.539066 0.842263i \(-0.318777\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 44.0000 1.67384 0.836919 0.547326i \(-0.184354\pi\)
0.836919 + 0.547326i \(0.184354\pi\)
\(692\) 21.5617i 0.819654i
\(693\) 0 0
\(694\) 8.97739 0.340777
\(695\) 0 0
\(696\) 5.60664 42.3249i 0.212519 1.60432i
\(697\) 0 0
\(698\) 9.67977i 0.366385i
\(699\) 49.8546 + 6.60407i 1.88567 + 0.249789i
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 21.9194 + 9.14104i 0.827296 + 0.345006i
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 20.1674 0.759010
\(707\) 0 0
\(708\) −18.5113 2.45213i −0.695697 0.0921567i
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 53.3835i 1.99923i
\(714\) 0 0
\(715\) 0 0
\(716\) 19.9956i 0.747272i
\(717\) −46.6404 6.17830i −1.74182 0.230733i
\(718\) 0 0
\(719\) 47.9583i 1.78854i −0.447524 0.894272i \(-0.647694\pi\)
0.447524 0.894272i \(-0.352306\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 17.7832i 0.661822i
\(723\) 0 0
\(724\) 0 0
\(725\) 42.1520i 1.56548i
\(726\) −17.6780 2.34174i −0.656091 0.0869101i
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) −19.0000 19.1833i −0.703704 0.710494i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 25.8516 0.952903
\(737\) 0 0
\(738\) −33.0047 8.90022i −1.21492 0.327622i
\(739\) −15.4389 −0.567928 −0.283964 0.958835i \(-0.591650\pi\)
−0.283964 + 0.958835i \(0.591650\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 55.8845 + 7.40283i 2.04882 + 0.271401i
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 3.20510i 0.116878i
\(753\) 0 0
\(754\) 38.5313 1.40323
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 44.0103i 1.59537i 0.603072 + 0.797687i \(0.293943\pi\)
−0.603072 + 0.797687i \(0.706057\pi\)
\(762\) 13.5024 + 1.78862i 0.489141 + 0.0647949i
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 46.8385i 1.69124i
\(768\) 3.83478 28.9491i 0.138376 1.04461i
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 33.7605 + 4.47214i 1.21585 + 0.161060i
\(772\) 30.5113 1.09813
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) −55.6561 −1.99923
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −40.4307 16.8608i −1.44488 0.602555i
\(784\) −3.42087 −0.122174
\(785\) 0 0
\(786\) 0.599197 4.52338i 0.0213727 0.161344i
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 1.05961i 0.0377472i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 37.1029i 1.31673i
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 26.9522i 0.952903i
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 50.8755i 1.79201i
\(807\) −56.2830