Properties

Label 729.2.e.f.82.1
Level $729$
Weight $2$
Character 729.82
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: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
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 82.1
Root \(-0.766044 + 0.642788i\) of defining polynomial
Character \(\chi\) \(=\) 729.82
Dual form 729.2.e.f.649.1

$q$-expansion

\(f(q)\) \(=\) \(q+(1.87939 - 0.684040i) q^{4} +(0.939693 + 0.342020i) q^{7} +O(q^{10})\) \(q+(1.87939 - 0.684040i) q^{4} +(0.939693 + 0.342020i) q^{7} +(0.868241 - 4.92404i) q^{13} +(3.06418 - 2.57115i) q^{16} +(3.50000 + 6.06218i) q^{19} +(-0.868241 - 4.92404i) q^{25} +2.00000 q^{28} +(3.75877 - 1.36808i) q^{31} +(-5.50000 + 9.52628i) q^{37} +(6.12836 - 5.14230i) q^{43} +(-4.59627 - 3.85673i) q^{49} +(-1.73648 - 9.84808i) q^{52} +(0.939693 + 0.342020i) q^{61} +(4.00000 - 6.92820i) q^{64} +(0.868241 - 4.92404i) q^{67} +(3.50000 + 6.06218i) q^{73} +(10.7246 + 8.99903i) q^{76} +(2.95202 + 16.7417i) q^{79} +(2.50000 - 4.33013i) q^{91} +(-14.5548 + 12.2130i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 21 q^{19} + 12 q^{28} - 33 q^{37} + 24 q^{64} + 21 q^{73} + 15 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{4}{9}\right)\)

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 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(3\) 0 0
\(4\) 1.87939 0.684040i 0.939693 0.342020i
\(5\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(6\) 0 0
\(7\) 0.939693 + 0.342020i 0.355170 + 0.129271i 0.513442 0.858124i \(-0.328370\pi\)
−0.158272 + 0.987396i \(0.550592\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(12\) 0 0
\(13\) 0.868241 4.92404i 0.240807 1.36568i −0.589226 0.807968i \(-0.700567\pi\)
0.830033 0.557714i \(-0.188322\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 3.06418 2.57115i 0.766044 0.642788i
\(17\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(18\) 0 0
\(19\) 3.50000 + 6.06218i 0.802955 + 1.39076i 0.917663 + 0.397360i \(0.130073\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(24\) 0 0
\(25\) −0.868241 4.92404i −0.173648 0.984808i
\(26\) 0 0
\(27\) 0 0
\(28\) 2.00000 0.377964
\(29\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(30\) 0 0
\(31\) 3.75877 1.36808i 0.675095 0.245715i 0.0183550 0.999832i \(-0.494157\pi\)
0.656740 + 0.754117i \(0.271935\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −5.50000 + 9.52628i −0.904194 + 1.56611i −0.0821995 + 0.996616i \(0.526194\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(42\) 0 0
\(43\) 6.12836 5.14230i 0.934565 0.784194i −0.0420659 0.999115i \(-0.513394\pi\)
0.976631 + 0.214921i \(0.0689495\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(48\) 0 0
\(49\) −4.59627 3.85673i −0.656610 0.550961i
\(50\) 0 0
\(51\) 0 0
\(52\) −1.73648 9.84808i −0.240807 1.36568i
\(53\) 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 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(60\) 0 0
\(61\) 0.939693 + 0.342020i 0.120315 + 0.0437912i 0.401476 0.915869i \(-0.368497\pi\)
−0.281161 + 0.959661i \(0.590719\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 4.00000 6.92820i 0.500000 0.866025i
\(65\) 0 0
\(66\) 0 0
\(67\) 0.868241 4.92404i 0.106073 0.601567i −0.884714 0.466134i \(-0.845646\pi\)
0.990787 0.135433i \(-0.0432425\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(72\) 0 0
\(73\) 3.50000 + 6.06218i 0.409644 + 0.709524i 0.994850 0.101361i \(-0.0323196\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 10.7246 + 8.99903i 1.23020 + 1.03226i
\(77\) 0 0
\(78\) 0 0
\(79\) 2.95202 + 16.7417i 0.332128 + 1.88359i 0.453930 + 0.891038i \(0.350022\pi\)
−0.121802 + 0.992554i \(0.538867\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(90\) 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
\(97\) −14.5548 + 12.2130i −1.47782 + 1.24004i −0.569346 + 0.822098i \(0.692804\pi\)
−0.908474 + 0.417941i \(0.862752\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −5.00000 8.66025i −0.500000 0.866025i
\(101\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(102\) 0 0
\(103\) −9.95858 8.35624i −0.981248 0.823365i 0.00302937 0.999995i \(-0.499036\pi\)
−0.984277 + 0.176631i \(0.943480\pi\)
\(104\) 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
\(112\) 3.75877 1.36808i 0.355170 0.129271i
\(113\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −1.91013 + 10.8329i −0.173648 + 0.984808i
\(122\) 0 0
\(123\) 0 0
\(124\) 6.12836 5.14230i 0.550343 0.461792i
\(125\) 0 0
\(126\) 0 0
\(127\) −10.0000 17.3205i −0.887357 1.53695i −0.842989 0.537931i \(-0.819206\pi\)
−0.0443678 0.999015i \(-0.514127\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(132\) 0 0
\(133\) 1.21554 + 6.89365i 0.105400 + 0.597756i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(138\) 0 0
\(139\) −21.6129 + 7.86646i −1.83318 + 0.667225i −0.841223 + 0.540689i \(0.818164\pi\)
−0.991962 + 0.126536i \(0.959614\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) −3.82026 + 21.6658i −0.314023 + 1.78092i
\(149\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(150\) 0 0
\(151\) −14.5548 + 12.2130i −1.18446 + 0.993877i −0.184517 + 0.982829i \(0.559072\pi\)
−0.999939 + 0.0110477i \(0.996483\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 10.7246 + 8.99903i 0.855918 + 0.718201i 0.961085 0.276254i \(-0.0890931\pi\)
−0.105167 + 0.994455i \(0.533538\pi\)
\(158\) 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 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(168\) 0 0
\(169\) −11.2763 4.10424i −0.867409 0.315711i
\(170\) 0 0
\(171\) 0 0
\(172\) 8.00000 13.8564i 0.609994 1.05654i
\(173\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(174\) 0 0
\(175\) 0.868241 4.92404i 0.0656328 0.372222i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(180\) 0 0
\(181\) 3.50000 + 6.06218i 0.260153 + 0.450598i 0.966282 0.257485i \(-0.0828937\pi\)
−0.706129 + 0.708083i \(0.749560\pi\)
\(182\) 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 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(192\) 0 0
\(193\) −21.6129 + 7.86646i −1.55573 + 0.566240i −0.969754 0.244086i \(-0.921512\pi\)
−0.585979 + 0.810326i \(0.699290\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −11.2763 4.10424i −0.805451 0.293160i
\(197\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(198\) 0 0
\(199\) −5.50000 + 9.52628i −0.389885 + 0.675300i −0.992434 0.122782i \(-0.960818\pi\)
0.602549 + 0.798082i \(0.294152\pi\)
\(200\) 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
\(211\) −9.95858 8.35624i −0.685577 0.575267i 0.232053 0.972703i \(-0.425456\pi\)
−0.917630 + 0.397436i \(0.869900\pi\)
\(212\) 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
\(223\) 26.3114 + 9.57656i 1.76194 + 0.641294i 0.999980 0.00632846i \(-0.00201443\pi\)
0.761961 + 0.647623i \(0.224237\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(228\) 0 0
\(229\) −3.82026 + 21.6658i −0.252450 + 1.43171i 0.550085 + 0.835109i \(0.314595\pi\)
−0.802535 + 0.596606i \(0.796516\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(240\) 0 0
\(241\) 2.95202 + 16.7417i 0.190156 + 1.07843i 0.919150 + 0.393909i \(0.128877\pi\)
−0.728993 + 0.684521i \(0.760011\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) 0 0
\(247\) 32.8892 11.9707i 2.09269 0.761678i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 2.77837 15.7569i 0.173648 0.984808i
\(257\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(258\) 0 0
\(259\) −8.42649 + 7.07066i −0.523597 + 0.439350i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −1.73648 9.84808i −0.106073 0.601567i
\(269\) 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
\(277\) −24.4320 8.89252i −1.46798 0.534300i −0.520427 0.853906i \(-0.674227\pi\)
−0.947550 + 0.319606i \(0.896449\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(282\) 0 0
\(283\) 5.55674 31.5138i 0.330314 1.87330i −0.139030 0.990288i \(-0.544398\pi\)
0.469344 0.883016i \(-0.344491\pi\)
\(284\) 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
\(292\) 10.7246 + 8.99903i 0.627611 + 0.526628i
\(293\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 7.51754 2.73616i 0.433304 0.157710i
\(302\) 0 0
\(303\) 0 0
\(304\) 26.3114 + 9.57656i 1.50906 + 0.549254i
\(305\) 0 0
\(306\) 0 0
\(307\) 8.00000 13.8564i 0.456584 0.790827i −0.542194 0.840254i \(-0.682406\pi\)
0.998778 + 0.0494267i \(0.0157394\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(312\) 0 0
\(313\) 26.8116 22.4976i 1.51548 1.27164i 0.663345 0.748314i \(-0.269136\pi\)
0.852134 0.523324i \(-0.175308\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 17.0000 + 29.4449i 0.956325 + 1.65640i
\(317\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(318\) 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
\(331\) 0.939693 + 0.342020i 0.0516502 + 0.0187991i 0.367716 0.929938i \(-0.380140\pi\)
−0.316066 + 0.948737i \(0.602362\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0.868241 4.92404i 0.0472961 0.268229i −0.951985 0.306145i \(-0.900961\pi\)
0.999281 + 0.0379157i \(0.0120718\pi\)
\(338\) 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 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(348\) 0 0
\(349\) −6.42498 36.4379i −0.343921 1.95048i −0.308920 0.951088i \(-0.599967\pi\)
−0.0350017 0.999387i \(-0.511144\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(360\) 0 0
\(361\) −15.0000 + 25.9808i −0.789474 + 1.36741i
\(362\) 0 0
\(363\) 0 0
\(364\) 1.73648 9.84808i 0.0910164 0.516180i
\(365\) 0 0
\(366\) 0 0
\(367\) 26.8116 22.4976i 1.39955 1.17436i 0.438254 0.898851i \(-0.355597\pi\)
0.961298 0.275512i \(-0.0888475\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −9.95858 8.35624i −0.515636 0.432670i 0.347472 0.937691i \(-0.387040\pi\)
−0.863107 + 0.505021i \(0.831485\pi\)
\(374\) 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 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) −19.0000 + 32.9090i −0.964579 + 1.67070i
\(389\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(390\) 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 0.878300 + 0.478110i \(0.158678\pi\)
−0.0250943 + 0.999685i \(0.507989\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −15.3209 12.8558i −0.766044 0.642788i
\(401\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(402\) 0 0
\(403\) −3.47296 19.6962i −0.173001 0.981135i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 29.1305 10.6026i 1.44041 0.524266i 0.500514 0.865729i \(-0.333144\pi\)
0.939895 + 0.341463i \(0.110922\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −24.4320 8.89252i −1.20368 0.438103i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(420\) 0 0
\(421\) −14.5548 + 12.2130i −0.709360 + 0.595223i −0.924419 0.381377i \(-0.875450\pi\)
0.215060 + 0.976601i \(0.431005\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0.766044 + 0.642788i 0.0370715 + 0.0311067i
\(428\) 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
\(436\) 3.75877 1.36808i 0.180012 0.0655192i
\(437\) 0 0
\(438\) 0 0
\(439\) 26.3114 + 9.57656i 1.25577 + 0.457064i 0.882349 0.470596i \(-0.155961\pi\)
0.373425 + 0.927660i \(0.378183\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 6.12836 5.14230i 0.289538 0.242951i
\(449\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −1.73648 9.84808i −0.0812292 0.460674i −0.998107 0.0615051i \(-0.980410\pi\)
0.916878 0.399169i \(-0.130701\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(462\) 0 0
\(463\) −21.6129 + 7.86646i −1.00444 + 0.365586i −0.791294 0.611435i \(-0.790592\pi\)
−0.213144 + 0.977021i \(0.568370\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(468\) 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
\(475\) 26.8116 22.4976i 1.23020 1.03226i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(480\) 0 0
\(481\) 42.1324 + 35.3533i 1.92107 + 1.61197i
\(482\) 0 0
\(483\) 0 0
\(484\) 3.82026 + 21.6658i 0.173648 + 0.984808i
\(485\) 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 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(492\) 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
\(499\) 5.55674 31.5138i 0.248754 1.41075i −0.562857 0.826555i \(-0.690298\pi\)
0.811610 0.584199i \(-0.198591\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) −30.6418 25.7115i −1.35951 1.14076i
\(509\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(510\) 0 0
\(511\) 1.21554 + 6.89365i 0.0537722 + 0.304957i
\(512\) 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 \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(522\) 0 0
\(523\) 21.5000 37.2391i 0.940129 1.62835i 0.174908 0.984585i \(-0.444037\pi\)
0.765222 0.643767i \(-0.222629\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −17.6190 + 14.7841i −0.766044 + 0.642788i
\(530\) 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
\(547\) 0.939693 + 0.342020i 0.0401784 + 0.0146237i 0.362031 0.932166i \(-0.382083\pi\)
−0.321853 + 0.946790i \(0.604306\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −2.95202 + 16.7417i −0.125533 + 0.711931i
\(554\) 0 0
\(555\) 0 0
\(556\) −35.2380 + 29.5682i −1.49443 + 1.25397i
\(557\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(558\) 0 0
\(559\) −20.0000 34.6410i −0.845910 1.46516i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(570\) 0 0
\(571\) 29.1305 10.6026i 1.21907 0.443706i 0.349231 0.937037i \(-0.386443\pi\)
0.869842 + 0.493331i \(0.164221\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −5.50000 + 9.52628i −0.228968 + 0.396584i −0.957503 0.288425i \(-0.906868\pi\)
0.728535 + 0.685009i \(0.240202\pi\)
\(578\) 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 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(588\) 0 0
\(589\) 21.4492 + 17.9981i 0.883801 + 0.741597i
\(590\) 0 0
\(591\) 0 0
\(592\) 7.64052 + 43.3315i 0.314023 + 1.78092i
\(593\) 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 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(600\) 0 0
\(601\) −24.4320 8.89252i −0.996602 0.362734i −0.208329 0.978059i \(-0.566802\pi\)
−0.788273 + 0.615325i \(0.789025\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −19.0000 + 32.9090i −0.773099 + 1.33905i
\(605\) 0 0
\(606\) 0 0
\(607\) −8.50876 + 48.2556i −0.345360 + 1.95863i −0.0688294 + 0.997628i \(0.521926\pi\)
−0.276531 + 0.961005i \(0.589185\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −23.5000 40.7032i −0.949156 1.64399i −0.747208 0.664590i \(-0.768606\pi\)
−0.201948 0.979396i \(-0.564727\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(618\) 0 0
\(619\) 2.95202 + 16.7417i 0.118652 + 0.672907i 0.984877 + 0.173254i \(0.0554281\pi\)
−0.866226 + 0.499653i \(0.833461\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −23.4923 + 8.55050i −0.939693 + 0.342020i
\(626\) 0 0
\(627\) 0 0
\(628\) 26.3114 + 9.57656i 1.04994 + 0.382147i
\(629\) 0 0
\(630\) 0 0
\(631\) 21.5000 37.2391i 0.855901 1.48246i −0.0199047 0.999802i \(-0.506336\pi\)
0.875806 0.482663i \(-0.160330\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −22.9813 + 19.2836i −0.910554 + 0.764045i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(642\) 0 0
\(643\) −30.6418 25.7115i −1.20839 1.01396i −0.999350 0.0360565i \(-0.988520\pi\)
−0.209044 0.977906i \(-0.567035\pi\)
\(644\) 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
\(652\) −46.9846 + 17.1010i −1.84006 + 0.669727i
\(653\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(660\) 0 0
\(661\) −8.50876 + 48.2556i −0.330952 + 1.87692i 0.133078 + 0.991106i \(0.457514\pi\)
−0.464031 + 0.885819i \(0.653597\pi\)
\(662\) 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
\(673\) −6.42498 36.4379i −0.247665 1.40458i −0.814221 0.580554i \(-0.802836\pi\)
0.566557 0.824023i \(-0.308275\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −24.0000 −0.923077
\(677\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(678\) 0 0
\(679\) −17.8542 + 6.49838i −0.685180 + 0.249385i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 5.55674 31.5138i 0.211849 1.20145i
\(689\) 0 0
\(690\) 0 0
\(691\) 6.12836 5.14230i 0.233134 0.195622i −0.518735 0.854935i \(-0.673597\pi\)
0.751869 + 0.659313i \(0.229153\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −1.73648 9.84808i −0.0656328 0.372222i
\(701\) 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
\(709\) −49.8037 18.1271i −1.87042 0.680776i −0.968581 0.248700i \(-0.919997\pi\)
−0.901837 0.432077i \(-0.857781\pi\)
\(710\) 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 \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(720\) 0 0
\(721\) −6.50000 11.2583i −0.242073 0.419282i
\(722\) 0 0
\(723\) 0 0
\(724\) 10.7246 + 8.99903i 0.398577 + 0.334446i
\(725\) 0 0
\(726\) 0 0
\(727\) 7.64052 + 43.3315i 0.283371 + 1.60708i 0.711046 + 0.703145i \(0.248222\pi\)
−0.427675 + 0.903933i \(0.640667\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −46.9846 + 17.1010i −1.73542 + 0.631640i −0.998992 0.0448796i \(-0.985710\pi\)
−0.736425 + 0.676520i \(0.763487\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 8.00000 13.8564i 0.294285 0.509716i −0.680534 0.732717i \(-0.738252\pi\)
0.974818 + 0.223001i \(0.0715853\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 31.4078 + 26.3543i 1.14609 + 0.961682i 0.999621 0.0275338i \(-0.00876539\pi\)
0.146467 + 0.989216i \(0.453210\pi\)
\(752\) 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 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(762\) 0 0
\(763\) 1.87939 + 0.684040i 0.0680383 + 0.0247639i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −8.50876 + 48.2556i −0.306834 + 1.74014i 0.307912 + 0.951415i \(0.400370\pi\)
−0.614745 + 0.788726i \(0.710741\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −35.2380 + 29.5682i −1.26824 + 1.06418i
\(773\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(774\) 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
\(787\) 29.1305 10.6026i 1.03839 0.377943i 0.234120 0.972208i \(-0.424779\pi\)
0.804269 + 0.594265i \(0.202557\pi\)
\(788\) 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
\(796\) −3.82026 + 21.6658i −0.135406 + 0.767923i
\(797\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(798\) 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