Properties

Label 729.2.e.f.163.1
Level $729$
Weight $2$
Character 729.163
Analytic conductor $5.821$
Analytic rank $0$
Dimension $6$
CM discriminant -3
Inner twists $12$

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Newspace parameters

Level: \( N \) \(=\) \( 729 = 3^{6} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 729.e (of order \(9\), degree \(6\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.82109430735\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Defining polynomial: \(x^{6} - x^{3} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 27)
Sato-Tate group: $\mathrm{U}(1)[D_{9}]$

Embedding invariants

Embedding label 163.1
Root \(-0.173648 + 0.984808i\) of defining polynomial
Character \(\chi\) \(=\) 729.163
Dual form 729.2.e.f.568.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-1.53209 + 1.28558i) q^{4} +(-0.766044 - 0.642788i) q^{7} +O(q^{10})\) \(q+(-1.53209 + 1.28558i) q^{4} +(-0.766044 - 0.642788i) q^{7} +(-4.69846 - 1.71010i) q^{13} +(0.694593 - 3.93923i) q^{16} +(3.50000 - 6.06218i) q^{19} +(4.69846 - 1.71010i) q^{25} +2.00000 q^{28} +(-3.06418 + 2.57115i) q^{31} +(-5.50000 - 9.52628i) q^{37} +(1.38919 - 7.87846i) q^{43} +(-1.04189 - 5.90885i) q^{49} +(9.39693 - 3.42020i) q^{52} +(-0.766044 - 0.642788i) q^{61} +(4.00000 + 6.92820i) q^{64} +(-4.69846 - 1.71010i) q^{67} +(3.50000 - 6.06218i) q^{73} +(2.43107 + 13.7873i) q^{76} +(-15.9748 + 5.81434i) q^{79} +(2.50000 + 4.33013i) q^{91} +(-3.29932 + 18.7113i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + O(q^{10}) \) \( 6q + 21q^{19} + 12q^{28} - 33q^{37} + 24q^{64} + 21q^{73} + 15q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/729\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{8}{9}\right)\)

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 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(3\) 0 0
\(4\) −1.53209 + 1.28558i −0.766044 + 0.642788i
\(5\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(6\) 0 0
\(7\) −0.766044 0.642788i −0.289538 0.242951i 0.486436 0.873716i \(-0.338297\pi\)
−0.775974 + 0.630765i \(0.782741\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(12\) 0 0
\(13\) −4.69846 1.71010i −1.30312 0.474297i −0.405108 0.914269i \(-0.632766\pi\)
−0.898011 + 0.439972i \(0.854988\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0.694593 3.93923i 0.173648 0.984808i
\(17\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(18\) 0 0
\(19\) 3.50000 6.06218i 0.802955 1.39076i −0.114708 0.993399i \(-0.536593\pi\)
0.917663 0.397360i \(-0.130073\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(24\) 0 0
\(25\) 4.69846 1.71010i 0.939693 0.342020i
\(26\) 0 0
\(27\) 0 0
\(28\) 2.00000 0.377964
\(29\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(30\) 0 0
\(31\) −3.06418 + 2.57115i −0.550343 + 0.461792i −0.875057 0.484020i \(-0.839176\pi\)
0.324714 + 0.945812i \(0.394732\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −5.50000 9.52628i −0.904194 1.56611i −0.821995 0.569495i \(-0.807139\pi\)
−0.0821995 0.996616i \(-0.526194\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(42\) 0 0
\(43\) 1.38919 7.87846i 0.211849 1.20145i −0.674443 0.738327i \(-0.735616\pi\)
0.886292 0.463127i \(-0.153273\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(48\) 0 0
\(49\) −1.04189 5.90885i −0.148841 0.844121i
\(50\) 0 0
\(51\) 0 0
\(52\) 9.39693 3.42020i 1.30312 0.474297i
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(60\) 0 0
\(61\) −0.766044 0.642788i −0.0980819 0.0823005i 0.592428 0.805623i \(-0.298169\pi\)
−0.690510 + 0.723323i \(0.742614\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 4.00000 + 6.92820i 0.500000 + 0.866025i
\(65\) 0 0
\(66\) 0 0
\(67\) −4.69846 1.71010i −0.574009 0.208922i 0.0386729 0.999252i \(-0.487687\pi\)
−0.612682 + 0.790330i \(0.709909\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(72\) 0 0
\(73\) 3.50000 6.06218i 0.409644 0.709524i −0.585206 0.810885i \(-0.698986\pi\)
0.994850 + 0.101361i \(0.0323196\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 2.43107 + 13.7873i 0.278863 + 1.58151i
\(77\) 0 0
\(78\) 0 0
\(79\) −15.9748 + 5.81434i −1.79730 + 0.654165i −0.798677 + 0.601760i \(0.794466\pi\)
−0.998626 + 0.0524041i \(0.983312\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(90\) 0 0
\(91\) 2.50000 + 4.33013i 0.262071 + 0.453921i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −3.29932 + 18.7113i −0.334995 + 1.89985i 0.0922897 + 0.995732i \(0.470581\pi\)
−0.427284 + 0.904117i \(0.640530\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −5.00000 + 8.66025i −0.500000 + 0.866025i
\(101\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(102\) 0 0
\(103\) −2.25743 12.8025i −0.222431 1.26147i −0.867536 0.497374i \(-0.834298\pi\)
0.645105 0.764094i \(-0.276813\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −3.06418 + 2.57115i −0.289538 + 0.242951i
\(113\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 10.3366 + 3.76222i 0.939693 + 0.342020i
\(122\) 0 0
\(123\) 0 0
\(124\) 1.38919 7.87846i 0.124753 0.707507i
\(125\) 0 0
\(126\) 0 0
\(127\) −10.0000 + 17.3205i −0.887357 + 1.53695i −0.0443678 + 0.999015i \(0.514127\pi\)
−0.842989 + 0.537931i \(0.819206\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(132\) 0 0
\(133\) −6.57785 + 2.39414i −0.570372 + 0.207598i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(138\) 0 0
\(139\) 17.6190 14.7841i 1.49443 1.25397i 0.605564 0.795796i \(-0.292947\pi\)
0.888861 0.458176i \(-0.151497\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 20.6732 + 7.52444i 1.69933 + 0.618505i
\(149\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(150\) 0 0
\(151\) −3.29932 + 18.7113i −0.268494 + 1.52271i 0.490402 + 0.871496i \(0.336850\pi\)
−0.758896 + 0.651211i \(0.774261\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 2.43107 + 13.7873i 0.194021 + 1.10035i 0.913806 + 0.406150i \(0.133129\pi\)
−0.719785 + 0.694197i \(0.755760\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −25.0000 −1.95815 −0.979076 0.203497i \(-0.934769\pi\)
−0.979076 + 0.203497i \(0.934769\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(168\) 0 0
\(169\) 9.19253 + 7.71345i 0.707118 + 0.593342i
\(170\) 0 0
\(171\) 0 0
\(172\) 8.00000 + 13.8564i 0.609994 + 1.05654i
\(173\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(174\) 0 0
\(175\) −4.69846 1.71010i −0.355170 0.129271i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(180\) 0 0
\(181\) 3.50000 6.06218i 0.260153 0.450598i −0.706129 0.708083i \(-0.749560\pi\)
0.966282 + 0.257485i \(0.0828937\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(192\) 0 0
\(193\) 17.6190 14.7841i 1.26824 1.06418i 0.273492 0.961874i \(-0.411821\pi\)
0.994753 0.102310i \(-0.0326233\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 9.19253 + 7.71345i 0.656610 + 0.550961i
\(197\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(198\) 0 0
\(199\) −5.50000 9.52628i −0.389885 0.675300i 0.602549 0.798082i \(-0.294152\pi\)
−0.992434 + 0.122782i \(0.960818\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −10.0000 + 17.3205i −0.693375 + 1.20096i
\(209\) 0 0
\(210\) 0 0
\(211\) −2.25743 12.8025i −0.155408 0.881361i −0.958412 0.285388i \(-0.907878\pi\)
0.803005 0.595973i \(-0.203233\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 4.00000 0.271538
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −21.4492 17.9981i −1.43635 1.20524i −0.941838 0.336066i \(-0.890903\pi\)
−0.494509 0.869172i \(-0.664652\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(228\) 0 0
\(229\) 20.6732 + 7.52444i 1.36613 + 0.497229i 0.917943 0.396713i \(-0.129849\pi\)
0.448183 + 0.893942i \(0.352071\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(240\) 0 0
\(241\) −15.9748 + 5.81434i −1.02903 + 0.374535i −0.800710 0.599052i \(-0.795544\pi\)
−0.228316 + 0.973587i \(0.573322\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) 0 0
\(247\) −26.8116 + 22.4976i −1.70598 + 1.43149i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −15.0351 5.47232i −0.939693 0.342020i
\(257\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(258\) 0 0
\(259\) −1.91013 + 10.8329i −0.118690 + 0.673123i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 9.39693 3.42020i 0.574009 0.208922i
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 29.0000 1.76162 0.880812 0.473466i \(-0.156997\pi\)
0.880812 + 0.473466i \(0.156997\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 19.9172 + 16.7125i 1.19671 + 1.00416i 0.999718 + 0.0237496i \(0.00756043\pi\)
0.196988 + 0.980406i \(0.436884\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(282\) 0 0
\(283\) −30.0702 10.9446i −1.78749 0.650592i −0.999386 0.0350443i \(-0.988843\pi\)
−0.788100 0.615547i \(-0.788935\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 8.50000 14.7224i 0.500000 0.866025i
\(290\) 0 0
\(291\) 0 0
\(292\) 2.43107 + 13.7873i 0.142268 + 0.806841i
\(293\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −6.12836 + 5.14230i −0.353233 + 0.296397i
\(302\) 0 0
\(303\) 0 0
\(304\) −21.4492 17.9981i −1.23020 1.03226i
\(305\) 0 0
\(306\) 0 0
\(307\) 8.00000 + 13.8564i 0.456584 + 0.790827i 0.998778 0.0494267i \(-0.0157394\pi\)
−0.542194 + 0.840254i \(0.682406\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(312\) 0 0
\(313\) 6.07769 34.4683i 0.343531 1.94826i 0.0271446 0.999632i \(-0.491359\pi\)
0.316387 0.948630i \(-0.397530\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 17.0000 29.4449i 0.956325 1.65640i
\(317\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −25.0000 −1.38675
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −0.766044 0.642788i −0.0421056 0.0353308i 0.621492 0.783420i \(-0.286527\pi\)
−0.663598 + 0.748090i \(0.730971\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −4.69846 1.71010i −0.255942 0.0931551i 0.210863 0.977516i \(-0.432373\pi\)
−0.466805 + 0.884361i \(0.654595\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −6.50000 + 11.2583i −0.350967 + 0.607893i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(348\) 0 0
\(349\) 34.7686 12.6547i 1.86112 0.677393i 0.882996 0.469381i \(-0.155523\pi\)
0.978126 0.208012i \(-0.0666992\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(360\) 0 0
\(361\) −15.0000 25.9808i −0.789474 1.36741i
\(362\) 0 0
\(363\) 0 0
\(364\) −9.39693 3.42020i −0.492533 0.179267i
\(365\) 0 0
\(366\) 0 0
\(367\) 6.07769 34.4683i 0.317253 1.79923i −0.242048 0.970264i \(-0.577819\pi\)
0.559301 0.828965i \(-0.311070\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −2.25743 12.8025i −0.116885 0.662888i −0.985800 0.167926i \(-0.946293\pi\)
0.868914 0.494962i \(-0.164818\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 29.0000 1.48963 0.744815 0.667271i \(-0.232538\pi\)
0.744815 + 0.667271i \(0.232538\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) −19.0000 32.9090i −0.964579 1.67070i
\(389\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 17.0000 29.4449i 0.853206 1.47780i −0.0250943 0.999685i \(-0.507989\pi\)
0.878300 0.478110i \(-0.158678\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −3.47296 19.6962i −0.173648 0.984808i
\(401\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(402\) 0 0
\(403\) 18.7939 6.84040i 0.936188 0.340745i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −23.7474 + 19.9264i −1.17423 + 0.985298i −0.174232 + 0.984705i \(0.555744\pi\)
−1.00000 0.000593299i \(0.999811\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 19.9172 + 16.7125i 0.981248 + 0.823365i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(420\) 0 0
\(421\) −3.29932 + 18.7113i −0.160799 + 0.911935i 0.792492 + 0.609882i \(0.208783\pi\)
−0.953291 + 0.302053i \(0.902328\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0.173648 + 0.984808i 0.00840342 + 0.0476582i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −3.06418 + 2.57115i −0.146748 + 0.123136i
\(437\) 0 0
\(438\) 0 0
\(439\) −21.4492 17.9981i −1.02372 0.859000i −0.0336266 0.999434i \(-0.510706\pi\)
−0.990090 + 0.140434i \(0.955150\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 1.38919 7.87846i 0.0656328 0.372222i
\(449\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 9.39693 3.42020i 0.439570 0.159990i −0.112749 0.993624i \(-0.535966\pi\)
0.552318 + 0.833633i \(0.313743\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(462\) 0 0
\(463\) 17.6190 14.7841i 0.818825 0.687076i −0.133871 0.990999i \(-0.542741\pi\)
0.952697 + 0.303923i \(0.0982964\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(468\) 0 0
\(469\) 2.50000 + 4.33013i 0.115439 + 0.199947i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 6.07769 34.4683i 0.278863 1.58151i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(480\) 0 0
\(481\) 9.55065 + 54.1644i 0.435472 + 2.46969i
\(482\) 0 0
\(483\) 0 0
\(484\) −20.6732 + 7.52444i −0.939693 + 0.342020i
\(485\) 0 0
\(486\) 0 0
\(487\) −25.0000 −1.13286 −0.566429 0.824110i \(-0.691675\pi\)
−0.566429 + 0.824110i \(0.691675\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 8.00000 + 13.8564i 0.359211 + 0.622171i
\(497\) 0 0
\(498\) 0 0
\(499\) −30.0702 10.9446i −1.34613 0.489950i −0.434389 0.900725i \(-0.643036\pi\)
−0.911736 + 0.410776i \(0.865258\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) −6.94593 39.3923i −0.308176 1.74775i
\(509\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(510\) 0 0
\(511\) −6.57785 + 2.39414i −0.290987 + 0.105911i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(522\) 0 0
\(523\) 21.5000 + 37.2391i 0.940129 + 1.62835i 0.765222 + 0.643767i \(0.222629\pi\)
0.174908 + 0.984585i \(0.444037\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −3.99391 + 22.6506i −0.173648 + 0.984808i
\(530\) 0 0
\(531\) 0 0
\(532\) 7.00000 12.1244i 0.303488 0.525657i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 29.0000 1.24681 0.623404 0.781900i \(-0.285749\pi\)
0.623404 + 0.781900i \(0.285749\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −0.766044 0.642788i −0.0327537 0.0274836i 0.626264 0.779611i \(-0.284583\pi\)
−0.659018 + 0.752128i \(0.729028\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 15.9748 + 5.81434i 0.679317 + 0.247251i
\(554\) 0 0
\(555\) 0 0
\(556\) −7.98782 + 45.3012i −0.338759 + 1.92120i
\(557\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(558\) 0 0
\(559\) −20.0000 + 34.6410i −0.845910 + 1.46516i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(570\) 0 0
\(571\) −23.7474 + 19.9264i −0.993797 + 0.833895i −0.986113 0.166076i \(-0.946890\pi\)
−0.00768386 + 0.999970i \(0.502446\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −5.50000 9.52628i −0.228968 0.396584i 0.728535 0.685009i \(-0.240202\pi\)
−0.957503 + 0.288425i \(0.906868\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(588\) 0 0
\(589\) 4.86215 + 27.5746i 0.200341 + 1.13619i
\(590\) 0 0
\(591\) 0 0
\(592\) −41.3465 + 15.0489i −1.69933 + 0.618505i
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(600\) 0 0
\(601\) 19.9172 + 16.7125i 0.812438 + 0.681716i 0.951188 0.308611i \(-0.0998642\pi\)
−0.138751 + 0.990327i \(0.544309\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −19.0000 32.9090i −0.773099 1.33905i
\(605\) 0 0
\(606\) 0 0
\(607\) 46.0449 + 16.7590i 1.86891 + 0.680226i 0.970520 + 0.241020i \(0.0774820\pi\)
0.898386 + 0.439206i \(0.144740\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −23.5000 + 40.7032i −0.949156 + 1.64399i −0.201948 + 0.979396i \(0.564727\pi\)
−0.747208 + 0.664590i \(0.768606\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(618\) 0 0
\(619\) −15.9748 + 5.81434i −0.642080 + 0.233698i −0.642481 0.766302i \(-0.722095\pi\)
0.000400419 1.00000i \(0.499873\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 19.1511 16.0697i 0.766044 0.642788i
\(626\) 0 0
\(627\) 0 0
\(628\) −21.4492 17.9981i −0.855918 0.718201i
\(629\) 0 0
\(630\) 0 0
\(631\) 21.5000 + 37.2391i 0.855901 + 1.48246i 0.875806 + 0.482663i \(0.160330\pi\)
−0.0199047 + 0.999802i \(0.506336\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −5.20945 + 29.5442i −0.206406 + 1.17059i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(642\) 0 0
\(643\) −6.94593 39.3923i −0.273921 1.55348i −0.742370 0.669991i \(-0.766298\pi\)
0.468449 0.883491i \(-0.344813\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 38.3022 32.1394i 1.50003 1.25868i
\(653\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(660\) 0 0
\(661\) 46.0449 + 16.7590i 1.79094 + 0.651849i 0.999157 + 0.0410470i \(0.0130693\pi\)
0.791783 + 0.610802i \(0.209153\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 34.7686 12.6547i 1.34023 0.487805i 0.430346 0.902664i \(-0.358391\pi\)
0.909886 + 0.414859i \(0.136169\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −24.0000 −0.923077
\(677\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(678\) 0 0
\(679\) 14.5548 12.2130i 0.558564 0.468691i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −30.0702 10.9446i −1.14641 0.417261i
\(689\) 0 0
\(690\) 0 0
\(691\) 1.38919 7.87846i 0.0528471 0.299711i −0.946916 0.321481i \(-0.895819\pi\)
0.999763 + 0.0217706i \(0.00693034\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 9.39693 3.42020i 0.355170 0.129271i
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) −77.0000 −2.90411
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 40.6004 + 34.0677i 1.52478 + 1.27944i 0.825108 + 0.564975i \(0.191114\pi\)
0.699671 + 0.714466i \(0.253330\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(720\) 0 0
\(721\) −6.50000 + 11.2583i −0.242073 + 0.419282i
\(722\) 0 0
\(723\) 0 0
\(724\) 2.43107 + 13.7873i 0.0903502 + 0.512401i
\(725\) 0 0
\(726\) 0 0
\(727\) −41.3465 + 15.0489i −1.53346 + 0.558132i −0.964465 0.264211i \(-0.914888\pi\)
−0.568991 + 0.822344i \(0.692666\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 38.3022 32.1394i 1.41472 1.18710i 0.460629 0.887593i \(-0.347624\pi\)
0.954096 0.299503i \(-0.0968207\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 8.00000 + 13.8564i 0.294285 + 0.509716i 0.974818 0.223001i \(-0.0715853\pi\)
−0.680534 + 0.732717i \(0.738252\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 7.11958 + 40.3771i 0.259797 + 1.47338i 0.783452 + 0.621452i \(0.213457\pi\)
−0.523655 + 0.851930i \(0.675432\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 29.0000 1.05402 0.527011 0.849858i \(-0.323312\pi\)
0.527011 + 0.849858i \(0.323312\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(762\) 0 0
\(763\) −1.53209 1.28558i −0.0554653 0.0465409i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 46.0449 + 16.7590i 1.66042 + 0.604345i 0.990429 0.138022i \(-0.0440745\pi\)
0.669994 + 0.742367i \(0.266297\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −7.98782 + 45.3012i −0.287488 + 1.63042i
\(773\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(774\) 0 0
\(775\) −10.0000 + 17.3205i −0.359211 + 0.622171i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −24.0000 −0.857143
\(785\) 0 0
\(786\) 0 0
\(787\) −23.7474 + 19.9264i −0.846503 + 0.710300i −0.959017 0.283350i \(-0.908554\pi\)
0.112514 + 0.993650i \(0.464110\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 2.50000 + 4.33013i 0.0887776 + 0.153767i
\(794\) 0 0
\(795\) 0 0
\(796\) 20.6732 + 7.52444i 0.732743 + 0.266697i
\(797\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)