Properties

Label 729.2.e.f.325.1
Level $729$
Weight $2$
Character 729.325
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 325.1
Root \(0.939693 + 0.342020i\) of defining polynomial
Character \(\chi\) \(=\) 729.325
Dual form 729.2.e.f.406.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-0.347296 + 1.96962i) q^{4} +(-0.173648 - 0.984808i) q^{7} +O(q^{10})\) \(q+(-0.347296 + 1.96962i) q^{4} +(-0.173648 - 0.984808i) q^{7} +(3.83022 + 3.21394i) q^{13} +(-3.75877 - 1.36808i) q^{16} +(3.50000 + 6.06218i) q^{19} +(-3.83022 + 3.21394i) q^{25} +2.00000 q^{28} +(-0.694593 + 3.93923i) q^{31} +(-5.50000 + 9.52628i) q^{37} +(-7.51754 - 2.73616i) q^{43} +(5.63816 - 2.05212i) q^{49} +(-7.66044 + 6.42788i) q^{52} +(-0.173648 - 0.984808i) q^{61} +(4.00000 - 6.92820i) q^{64} +(3.83022 + 3.21394i) q^{67} +(3.50000 + 6.06218i) q^{73} +(-13.1557 + 4.78828i) q^{76} +(13.0228 - 10.9274i) q^{79} +(2.50000 - 4.33013i) q^{91} +(17.8542 + 6.49838i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + O(q^{10}) \) \( 6q + 21q^{19} + 12q^{28} - 33q^{37} + 24q^{64} + 21q^{73} + 15q^{91} + O(q^{100}) \)

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{7}{9}\right)\)

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 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(3\) 0 0
\(4\) −0.347296 + 1.96962i −0.173648 + 0.984808i
\(5\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(6\) 0 0
\(7\) −0.173648 0.984808i −0.0656328 0.372222i −0.999878 0.0155920i \(-0.995037\pi\)
0.934246 0.356630i \(-0.116074\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(12\) 0 0
\(13\) 3.83022 + 3.21394i 1.06231 + 0.891386i 0.994334 0.106301i \(-0.0339006\pi\)
0.0679785 + 0.997687i \(0.478345\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −3.75877 1.36808i −0.939693 0.342020i
\(17\) 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 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(24\) 0 0
\(25\) −3.83022 + 3.21394i −0.766044 + 0.642788i
\(26\) 0 0
\(27\) 0 0
\(28\) 2.00000 0.377964
\(29\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(30\) 0 0
\(31\) −0.694593 + 3.93923i −0.124753 + 0.707507i 0.856702 + 0.515812i \(0.172510\pi\)
−0.981455 + 0.191695i \(0.938602\pi\)
\(32\) 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 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(42\) 0 0
\(43\) −7.51754 2.73616i −1.14641 0.417261i −0.302188 0.953248i \(-0.597717\pi\)
−0.844226 + 0.535988i \(0.819939\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(48\) 0 0
\(49\) 5.63816 2.05212i 0.805451 0.293160i
\(50\) 0 0
\(51\) 0 0
\(52\) −7.66044 + 6.42788i −1.06231 + 0.891386i
\(53\) 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 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(60\) 0 0
\(61\) −0.173648 0.984808i −0.0222334 0.126092i 0.971671 0.236338i \(-0.0759472\pi\)
−0.993904 + 0.110246i \(0.964836\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 4.00000 6.92820i 0.500000 0.866025i
\(65\) 0 0
\(66\) 0 0
\(67\) 3.83022 + 3.21394i 0.467936 + 0.392645i 0.846041 0.533118i \(-0.178980\pi\)
−0.378105 + 0.925763i \(0.623424\pi\)
\(68\) 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
\(76\) −13.1557 + 4.78828i −1.50906 + 0.549254i
\(77\) 0 0
\(78\) 0 0
\(79\) 13.0228 10.9274i 1.46517 1.22943i 0.544696 0.838633i \(-0.316645\pi\)
0.920478 0.390794i \(-0.127800\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(84\) 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
\(97\) 17.8542 + 6.49838i 1.81282 + 0.659811i 0.996631 + 0.0820195i \(0.0261370\pi\)
0.816185 + 0.577791i \(0.196085\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −5.00000 8.66025i −0.500000 0.866025i
\(101\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(102\) 0 0
\(103\) 12.2160 4.44626i 1.20368 0.438103i 0.339172 0.940724i \(-0.389853\pi\)
0.864507 + 0.502621i \(0.167631\pi\)
\(104\) 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
\(112\) −0.694593 + 3.93923i −0.0656328 + 0.372222i
\(113\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −8.42649 7.07066i −0.766044 0.642788i
\(122\) 0 0
\(123\) 0 0
\(124\) −7.51754 2.73616i −0.675095 0.245715i
\(125\) 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 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(132\) 0 0
\(133\) 5.36231 4.49951i 0.464971 0.390157i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(138\) 0 0
\(139\) 3.99391 22.6506i 0.338759 1.92120i −0.0476387 0.998865i \(-0.515170\pi\)
0.386398 0.922332i \(-0.373719\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) −16.8530 14.1413i −1.38531 1.16241i
\(149\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(150\) 0 0
\(151\) 17.8542 + 6.49838i 1.45295 + 0.528831i 0.943414 0.331618i \(-0.107594\pi\)
0.509537 + 0.860449i \(0.329817\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −13.1557 + 4.78828i −1.04994 + 0.382147i −0.808640 0.588304i \(-0.799796\pi\)
−0.241299 + 0.970451i \(0.577574\pi\)
\(158\) 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 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(168\) 0 0
\(169\) 2.08378 + 11.8177i 0.160291 + 0.909053i
\(170\) 0 0
\(171\) 0 0
\(172\) 8.00000 13.8564i 0.609994 1.05654i
\(173\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(174\) 0 0
\(175\) 3.83022 + 3.21394i 0.289538 + 0.242951i
\(176\) 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 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(192\) 0 0
\(193\) 3.99391 22.6506i 0.287488 1.63042i −0.408773 0.912636i \(-0.634043\pi\)
0.696261 0.717788i \(-0.254845\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 2.08378 + 11.8177i 0.148841 + 0.844121i
\(197\) 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
\(211\) 12.2160 4.44626i 0.840984 0.306093i 0.114625 0.993409i \(-0.463433\pi\)
0.726359 + 0.687315i \(0.241211\pi\)
\(212\) 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
\(223\) −4.86215 27.5746i −0.325594 1.84653i −0.505471 0.862844i \(-0.668681\pi\)
0.179877 0.983689i \(-0.442430\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(228\) 0 0
\(229\) −16.8530 14.1413i −1.11368 0.934485i −0.115408 0.993318i \(-0.536818\pi\)
−0.998268 + 0.0588329i \(0.981262\pi\)
\(230\) 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 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(240\) 0 0
\(241\) 13.0228 10.9274i 0.838869 0.703895i −0.118440 0.992961i \(-0.537789\pi\)
0.957309 + 0.289066i \(0.0933448\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) 0 0
\(247\) −6.07769 + 34.4683i −0.386714 + 2.19316i
\(248\) 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
\(256\) 12.2567 + 10.2846i 0.766044 + 0.642788i
\(257\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(258\) 0 0
\(259\) 10.3366 + 3.76222i 0.642286 + 0.233773i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −7.66044 + 6.42788i −0.467936 + 0.392645i
\(269\) 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
\(277\) 4.51485 + 25.6050i 0.271271 + 1.53846i 0.750562 + 0.660800i \(0.229783\pi\)
−0.479291 + 0.877656i \(0.659106\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(282\) 0 0
\(283\) 24.5134 + 20.5692i 1.45717 + 1.22271i 0.927130 + 0.374741i \(0.122268\pi\)
0.530042 + 0.847971i \(0.322176\pi\)
\(284\) 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
\(292\) −13.1557 + 4.78828i −0.769879 + 0.280213i
\(293\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −1.38919 + 7.87846i −0.0800713 + 0.454107i
\(302\) 0 0
\(303\) 0 0
\(304\) −4.86215 27.5746i −0.278863 1.58151i
\(305\) 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 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(312\) 0 0
\(313\) −32.8892 11.9707i −1.85901 0.676624i −0.979731 0.200316i \(-0.935803\pi\)
−0.879279 0.476308i \(-0.841975\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 17.0000 + 29.4449i 0.956325 + 1.65640i
\(317\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(318\) 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
\(331\) −0.173648 0.984808i −0.00954457 0.0541299i 0.979663 0.200648i \(-0.0643046\pi\)
−0.989208 + 0.146518i \(0.953193\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 3.83022 + 3.21394i 0.208645 + 0.175074i 0.741122 0.671370i \(-0.234294\pi\)
−0.532476 + 0.846445i \(0.678738\pi\)
\(338\) 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 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(348\) 0 0
\(349\) −28.3436 + 23.7831i −1.51720 + 1.27308i −0.669207 + 0.743076i \(0.733366\pi\)
−0.847994 + 0.530006i \(0.822190\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(354\) 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
\(364\) 7.66044 + 6.42788i 0.401516 + 0.336912i
\(365\) 0 0
\(366\) 0 0
\(367\) −32.8892 11.9707i −1.71680 0.624866i −0.719249 0.694752i \(-0.755514\pi\)
−0.997555 + 0.0698862i \(0.977736\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 12.2160 4.44626i 0.632521 0.230219i −0.00580736 0.999983i \(-0.501849\pi\)
0.638328 + 0.769764i \(0.279626\pi\)
\(374\) 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 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) −19.0000 + 32.9090i −0.964579 + 1.67070i
\(389\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(390\) 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
\(400\) 18.7939 6.84040i 0.939693 0.342020i
\(401\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(402\) 0 0
\(403\) −15.3209 + 12.8558i −0.763188 + 0.640391i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −5.38309 + 30.5290i −0.266177 + 1.50956i 0.499486 + 0.866322i \(0.333522\pi\)
−0.765663 + 0.643242i \(0.777589\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 4.51485 + 25.6050i 0.222431 + 1.26147i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(420\) 0 0
\(421\) 17.8542 + 6.49838i 0.870159 + 0.316712i 0.738231 0.674548i \(-0.235661\pi\)
0.131927 + 0.991259i \(0.457883\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −0.939693 + 0.342020i −0.0454749 + 0.0165515i
\(428\) 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
\(436\) −0.694593 + 3.93923i −0.0332650 + 0.188655i
\(437\) 0 0
\(438\) 0 0
\(439\) −4.86215 27.5746i −0.232058 1.31606i −0.848722 0.528839i \(-0.822628\pi\)
0.616665 0.787226i \(-0.288483\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −7.51754 2.73616i −0.355170 0.129271i
\(449\) 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
\(457\) −7.66044 + 6.42788i −0.358340 + 0.300683i −0.804129 0.594455i \(-0.797368\pi\)
0.445788 + 0.895138i \(0.352923\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(462\) 0 0
\(463\) 3.99391 22.6506i 0.185613 1.05266i −0.739553 0.673098i \(-0.764963\pi\)
0.925166 0.379563i \(-0.123926\pi\)
\(464\) 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
\(475\) −32.8892 11.9707i −1.50906 0.549254i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(480\) 0 0
\(481\) −51.6831 + 18.8111i −2.35655 + 0.857713i
\(482\) 0 0
\(483\) 0 0
\(484\) 16.8530 14.1413i 0.766044 0.642788i
\(485\) 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 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(492\) 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
\(499\) 24.5134 + 20.5692i 1.09737 + 0.920804i 0.997246 0.0741708i \(-0.0236310\pi\)
0.100126 + 0.994975i \(0.468075\pi\)
\(500\) 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
\(508\) 37.5877 13.6808i 1.66768 0.606988i
\(509\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(510\) 0 0
\(511\) 5.36231 4.49951i 0.237215 0.199047i
\(512\) 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
\(529\) 21.6129 + 7.86646i 0.939693 + 0.342020i
\(530\) 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
\(547\) −0.173648 0.984808i −0.00742466 0.0421073i 0.980870 0.194662i \(-0.0623610\pi\)
−0.988295 + 0.152555i \(0.951250\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −13.0228 10.9274i −0.553784 0.464680i
\(554\) 0 0
\(555\) 0 0
\(556\) 43.2259 + 15.7329i 1.83318 + 0.667225i
\(557\) 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 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(570\) 0 0
\(571\) −5.38309 + 30.5290i −0.225275 + 1.27760i 0.636882 + 0.770961i \(0.280224\pi\)
−0.862158 + 0.506640i \(0.830887\pi\)
\(572\) 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 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(588\) 0 0
\(589\) −26.3114 + 9.57656i −1.08414 + 0.394595i
\(590\) 0 0
\(591\) 0 0
\(592\) 33.7060 28.2827i 1.38531 1.16241i
\(593\) 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 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(600\) 0 0
\(601\) 4.51485 + 25.6050i 0.184165 + 1.04445i 0.927024 + 0.375002i \(0.122358\pi\)
−0.742859 + 0.669448i \(0.766531\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −19.0000 + 32.9090i −0.773099 + 1.33905i
\(605\) 0 0
\(606\) 0 0
\(607\) −37.5362 31.4966i −1.52355 1.27841i −0.829557 0.558422i \(-0.811407\pi\)
−0.693990 0.719985i \(-0.744149\pi\)
\(608\) 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 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(618\) 0 0
\(619\) 13.0228 10.9274i 0.523429 0.439209i −0.342396 0.939556i \(-0.611239\pi\)
0.865825 + 0.500347i \(0.166794\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 4.34120 24.6202i 0.173648 0.984808i
\(626\) 0 0
\(627\) 0 0
\(628\) −4.86215 27.5746i −0.194021 1.10035i
\(629\) 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
\(637\) 28.1908 + 10.2606i 1.11696 + 0.406540i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(642\) 0 0
\(643\) 37.5877 13.6808i 1.48231 0.539518i 0.530901 0.847434i \(-0.321854\pi\)
0.951414 + 0.307916i \(0.0996315\pi\)
\(644\) 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
\(652\) 8.68241 49.2404i 0.340029 1.92840i
\(653\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(660\) 0 0
\(661\) −37.5362 31.4966i −1.45999 1.22508i −0.924862 0.380303i \(-0.875820\pi\)
−0.535126 0.844772i \(-0.679736\pi\)
\(662\) 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
\(673\) −28.3436 + 23.7831i −1.09257 + 0.916773i −0.996903 0.0786409i \(-0.974942\pi\)
−0.0956642 + 0.995414i \(0.530497\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −24.0000 −0.923077
\(677\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(678\) 0 0
\(679\) 3.29932 18.7113i 0.126616 0.718076i
\(680\) 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
\(688\) 24.5134 + 20.5692i 0.934565 + 0.784194i
\(689\) 0 0
\(690\) 0 0
\(691\) −7.51754 2.73616i −0.285981 0.104088i 0.195047 0.980794i \(-0.437514\pi\)
−0.481028 + 0.876705i \(0.659736\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −7.66044 + 6.42788i −0.289538 + 0.242951i
\(701\) 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
\(709\) 9.20335 + 52.1948i 0.345639 + 1.96022i 0.268910 + 0.963165i \(0.413337\pi\)
0.0767291 + 0.997052i \(0.475552\pi\)
\(710\) 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
\(724\) −13.1557 + 4.78828i −0.488928 + 0.177955i
\(725\) 0 0
\(726\) 0 0
\(727\) 33.7060 28.2827i 1.25008 1.04895i 0.253419 0.967357i \(-0.418445\pi\)
0.996666 0.0815889i \(-0.0259995\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 8.68241 49.2404i 0.320692 1.81874i −0.217671 0.976022i \(-0.569846\pi\)
0.538363 0.842713i \(-0.319043\pi\)
\(734\) 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 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −38.5274 + 14.0228i −1.40588 + 0.511700i −0.929919 0.367764i \(-0.880123\pi\)
−0.475965 + 0.879464i \(0.657901\pi\)
\(752\) 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 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(762\) 0 0
\(763\) −0.347296 1.96962i −0.0125730 0.0713049i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −37.5362 31.4966i −1.35359 1.13580i −0.977905 0.209048i \(-0.932963\pi\)
−0.375684 0.926748i \(-0.622592\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 43.2259 + 15.7329i 1.55573 + 0.566240i
\(773\) 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
\(787\) −5.38309 + 30.5290i −0.191887 + 1.08824i 0.724897 + 0.688858i \(0.241887\pi\)
−0.916783 + 0.399385i \(0.869224\pi\)
\(788\) 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
\(796\) −16.8530 14.1413i −0.597338 0.501226i
\(797\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(798\) 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 \(0.5\pi\)