Properties

Label 588.3.p.c.569.1
Level $588$
Weight $3$
Character 588.569
Analytic conductor $16.022$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 588.p (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(16.0218395444\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 12)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label 569.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 588.569
Dual form 588.3.p.c.557.1

$q$-expansion

\(f(q)\) \(=\) \(q+(1.50000 + 2.59808i) q^{3} +(-4.50000 + 7.79423i) q^{9} +O(q^{10})\) \(q+(1.50000 + 2.59808i) q^{3} +(-4.50000 + 7.79423i) q^{9} -22.0000 q^{13} +(-13.0000 + 22.5167i) q^{19} +(-12.5000 - 21.6506i) q^{25} -27.0000 q^{27} +(23.0000 + 39.8372i) q^{31} +(-13.0000 + 22.5167i) q^{37} +(-33.0000 - 57.1577i) q^{39} -22.0000 q^{43} -78.0000 q^{57} +(-37.0000 + 64.0859i) q^{61} +(-61.0000 - 105.655i) q^{67} +(23.0000 + 39.8372i) q^{73} +(37.5000 - 64.9519i) q^{75} +(71.0000 - 122.976i) q^{79} +(-40.5000 - 70.1481i) q^{81} +(-69.0000 + 119.512i) q^{93} +2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 3q^{3} - 9q^{9} + O(q^{10}) \) \( 2q + 3q^{3} - 9q^{9} - 44q^{13} - 26q^{19} - 25q^{25} - 54q^{27} + 46q^{31} - 26q^{37} - 66q^{39} - 44q^{43} - 156q^{57} - 74q^{61} - 122q^{67} + 46q^{73} + 75q^{75} + 142q^{79} - 81q^{81} - 138q^{93} + 4q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/588\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(295\) \(493\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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
\(3\) 1.50000 + 2.59808i 0.500000 + 0.866025i
\(4\) 0 0
\(5\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −4.50000 + 7.79423i −0.500000 + 0.866025i
\(10\) 0 0
\(11\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(12\) 0 0
\(13\) −22.0000 −1.69231 −0.846154 0.532939i \(-0.821088\pi\)
−0.846154 + 0.532939i \(0.821088\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(18\) 0 0
\(19\) −13.0000 + 22.5167i −0.684211 + 1.18509i 0.289474 + 0.957186i \(0.406520\pi\)
−0.973684 + 0.227901i \(0.926814\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(24\) 0 0
\(25\) −12.5000 21.6506i −0.500000 0.866025i
\(26\) 0 0
\(27\) −27.0000 −1.00000
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 23.0000 + 39.8372i 0.741935 + 1.28507i 0.951613 + 0.307299i \(0.0994253\pi\)
−0.209677 + 0.977771i \(0.567241\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −13.0000 + 22.5167i −0.351351 + 0.608558i −0.986486 0.163843i \(-0.947611\pi\)
0.635135 + 0.772401i \(0.280944\pi\)
\(38\) 0 0
\(39\) −33.0000 57.1577i −0.846154 1.46558i
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −22.0000 −0.511628 −0.255814 0.966726i \(-0.582343\pi\)
−0.255814 + 0.966726i \(0.582343\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −78.0000 −1.36842
\(58\) 0 0
\(59\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(60\) 0 0
\(61\) −37.0000 + 64.0859i −0.606557 + 1.05059i 0.385246 + 0.922814i \(0.374117\pi\)
−0.991803 + 0.127774i \(0.959217\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −61.0000 105.655i −0.910448 1.57694i −0.813433 0.581659i \(-0.802404\pi\)
−0.0970149 0.995283i \(-0.530929\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 23.0000 + 39.8372i 0.315068 + 0.545715i 0.979452 0.201677i \(-0.0646392\pi\)
−0.664384 + 0.747392i \(0.731306\pi\)
\(74\) 0 0
\(75\) 37.5000 64.9519i 0.500000 0.866025i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 71.0000 122.976i 0.898734 1.55665i 0.0696203 0.997574i \(-0.477821\pi\)
0.829114 0.559080i \(-0.188845\pi\)
\(80\) 0 0
\(81\) −40.5000 70.1481i −0.500000 0.866025i
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −69.0000 + 119.512i −0.741935 + 1.28507i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 2.00000 0.0206186 0.0103093 0.999947i \(-0.496718\pi\)
0.0103093 + 0.999947i \(0.496718\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(102\) 0 0
\(103\) −97.0000 + 168.009i −0.941748 + 1.63115i −0.179612 + 0.983738i \(0.557484\pi\)
−0.762136 + 0.647417i \(0.775849\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(108\) 0 0
\(109\) 107.000 + 185.329i 0.981651 + 1.70027i 0.655963 + 0.754793i \(0.272263\pi\)
0.325688 + 0.945477i \(0.394404\pi\)
\(110\) 0 0
\(111\) −78.0000 −0.702703
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 99.0000 171.473i 0.846154 1.46558i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −60.5000 + 104.789i −0.500000 + 0.866025i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 146.000 1.14961 0.574803 0.818292i \(-0.305079\pi\)
0.574803 + 0.818292i \(0.305079\pi\)
\(128\) 0 0
\(129\) −33.0000 57.1577i −0.255814 0.443083i
\(130\) 0 0
\(131\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(138\) 0 0
\(139\) −22.0000 −0.158273 −0.0791367 0.996864i \(-0.525216\pi\)
−0.0791367 + 0.996864i \(0.525216\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(150\) 0 0
\(151\) 143.000 + 247.683i 0.947020 + 1.64029i 0.751656 + 0.659556i \(0.229256\pi\)
0.195364 + 0.980731i \(0.437411\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 59.0000 + 102.191i 0.375796 + 0.650898i 0.990446 0.137902i \(-0.0440359\pi\)
−0.614650 + 0.788800i \(0.710703\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 131.000 226.899i 0.803681 1.39202i −0.113497 0.993538i \(-0.536205\pi\)
0.917178 0.398478i \(-0.130461\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 315.000 1.86391
\(170\) 0 0
\(171\) −117.000 202.650i −0.684211 1.18509i
\(172\) 0 0
\(173\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(180\) 0 0
\(181\) 314.000 1.73481 0.867403 0.497606i \(-0.165787\pi\)
0.867403 + 0.497606i \(0.165787\pi\)
\(182\) 0 0
\(183\) −222.000 −1.21311
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(192\) 0 0
\(193\) 191.000 + 330.822i 0.989637 + 1.71410i 0.619171 + 0.785256i \(0.287469\pi\)
0.370466 + 0.928846i \(0.379198\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) −193.000 334.286i −0.969849 1.67983i −0.695980 0.718061i \(-0.745030\pi\)
−0.273869 0.961767i \(-0.588304\pi\)
\(200\) 0 0
\(201\) 183.000 316.965i 0.910448 1.57694i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −166.000 −0.786730 −0.393365 0.919382i \(-0.628689\pi\)
−0.393365 + 0.919382i \(0.628689\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −69.0000 + 119.512i −0.315068 + 0.545715i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 338.000 1.51570 0.757848 0.652432i \(-0.226251\pi\)
0.757848 + 0.652432i \(0.226251\pi\)
\(224\) 0 0
\(225\) 225.000 1.00000
\(226\) 0 0
\(227\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(228\) 0 0
\(229\) −13.0000 + 22.5167i −0.0567686 + 0.0983260i −0.893013 0.450031i \(-0.851413\pi\)
0.836245 + 0.548357i \(0.184746\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 426.000 1.79747
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 143.000 + 247.683i 0.593361 + 1.02773i 0.993776 + 0.111397i \(0.0355327\pi\)
−0.400415 + 0.916334i \(0.631134\pi\)
\(242\) 0 0
\(243\) 121.500 210.444i 0.500000 0.866025i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 286.000 495.367i 1.15789 2.00553i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(270\) 0 0
\(271\) −121.000 + 209.578i −0.446494 + 0.773351i −0.998155 0.0607176i \(-0.980661\pi\)
0.551661 + 0.834069i \(0.313994\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −61.0000 105.655i −0.220217 0.381426i 0.734657 0.678439i \(-0.237343\pi\)
−0.954874 + 0.297012i \(0.904010\pi\)
\(278\) 0 0
\(279\) −414.000 −1.48387
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) −229.000 396.640i −0.809187 1.40155i −0.913428 0.407001i \(-0.866574\pi\)
0.104240 0.994552i \(-0.466759\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −144.500 + 250.281i −0.500000 + 0.866025i
\(290\) 0 0
\(291\) 3.00000 + 5.19615i 0.0103093 + 0.0178562i
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −358.000 −1.16612 −0.583062 0.812428i \(-0.698145\pi\)
−0.583062 + 0.812428i \(0.698145\pi\)
\(308\) 0 0
\(309\) −582.000 −1.88350
\(310\) 0 0
\(311\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(312\) 0 0
\(313\) 71.0000 122.976i 0.226837 0.392893i −0.730032 0.683413i \(-0.760495\pi\)
0.956869 + 0.290520i \(0.0938282\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 275.000 + 476.314i 0.846154 + 1.46558i
\(326\) 0 0
\(327\) −321.000 + 555.988i −0.981651 + 1.70027i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −181.000 + 313.501i −0.546828 + 0.947134i 0.451662 + 0.892189i \(0.350831\pi\)
−0.998489 + 0.0549442i \(0.982502\pi\)
\(332\) 0 0
\(333\) −117.000 202.650i −0.351351 0.608558i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 482.000 1.43027 0.715134 0.698988i \(-0.246366\pi\)
0.715134 + 0.698988i \(0.246366\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(348\) 0 0
\(349\) −502.000 −1.43840 −0.719198 0.694805i \(-0.755490\pi\)
−0.719198 + 0.694805i \(0.755490\pi\)
\(350\) 0 0
\(351\) 594.000 1.69231
\(352\) 0 0
\(353\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(360\) 0 0
\(361\) −157.500 272.798i −0.436288 0.755673i
\(362\) 0 0
\(363\) −363.000 −1.00000
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 359.000 + 621.806i 0.978202 + 1.69429i 0.668937 + 0.743319i \(0.266749\pi\)
0.309264 + 0.950976i \(0.399917\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −349.000 + 604.486i −0.935657 + 1.62061i −0.162198 + 0.986758i \(0.551858\pi\)
−0.773458 + 0.633847i \(0.781475\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −694.000 −1.83113 −0.915567 0.402165i \(-0.868258\pi\)
−0.915567 + 0.402165i \(0.868258\pi\)
\(380\) 0 0
\(381\) 219.000 + 379.319i 0.574803 + 0.995588i
\(382\) 0 0
\(383\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 99.0000 171.473i 0.255814 0.443083i
\(388\) 0 0
\(389\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −181.000 + 313.501i −0.455919 + 0.789676i −0.998741 0.0501728i \(-0.984023\pi\)
0.542821 + 0.839848i \(0.317356\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(402\) 0 0
\(403\) −506.000 876.418i −1.25558 2.17473i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −313.000 542.132i −0.765281 1.32551i −0.940098 0.340905i \(-0.889267\pi\)
0.174817 0.984601i \(-0.444067\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −33.0000 57.1577i −0.0791367 0.137069i
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) −358.000 −0.850356 −0.425178 0.905110i \(-0.639789\pi\)
−0.425178 + 0.905110i \(0.639789\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(432\) 0 0
\(433\) −862.000 −1.99076 −0.995381 0.0960028i \(-0.969394\pi\)
−0.995381 + 0.0960028i \(0.969394\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 47.0000 81.4064i 0.107062 0.185436i −0.807517 0.589844i \(-0.799189\pi\)
0.914579 + 0.404408i \(0.132522\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −429.000 + 743.050i −0.947020 + 1.64029i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 407.000 704.945i 0.890591 1.54255i 0.0514223 0.998677i \(-0.483625\pi\)
0.839168 0.543872i \(-0.183042\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −526.000 −1.13607 −0.568035 0.823005i \(-0.692296\pi\)
−0.568035 + 0.823005i \(0.692296\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −177.000 + 306.573i −0.375796 + 0.650898i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 650.000 1.36842
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(480\) 0 0
\(481\) 286.000 495.367i 0.594595 1.02987i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −481.000 833.116i −0.987680 1.71071i −0.629363 0.777111i \(-0.716684\pi\)
−0.358316 0.933600i \(-0.616649\pi\)
\(488\) 0 0
\(489\) 786.000 1.60736
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −13.0000 + 22.5167i −0.0260521 + 0.0451236i −0.878758 0.477269i \(-0.841627\pi\)
0.852705 + 0.522392i \(0.174960\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 472.500 + 818.394i 0.931953 + 1.61419i
\(508\) 0 0
\(509\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 351.000 607.950i 0.684211 1.18509i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(522\) 0 0
\(523\) 491.000 850.437i 0.938815 1.62607i 0.171128 0.985249i \(-0.445259\pi\)
0.767686 0.640826i \(-0.221408\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −264.500 458.127i −0.500000 0.866025i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −517.000 + 895.470i −0.955638 + 1.65521i −0.222736 + 0.974879i \(0.571499\pi\)
−0.732902 + 0.680334i \(0.761835\pi\)
\(542\) 0 0
\(543\) 471.000 + 815.796i 0.867403 + 1.50239i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 506.000 0.925046 0.462523 0.886607i \(-0.346944\pi\)
0.462523 + 0.886607i \(0.346944\pi\)
\(548\) 0 0
\(549\) −333.000 576.773i −0.606557 1.05059i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(558\) 0 0
\(559\) 484.000 0.865832
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(570\) 0 0
\(571\) 443.000 + 767.299i 0.775832 + 1.34378i 0.934326 + 0.356420i \(0.116003\pi\)
−0.158494 + 0.987360i \(0.550664\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −481.000 833.116i −0.833622 1.44388i −0.895147 0.445770i \(-0.852930\pi\)
0.0615251 0.998106i \(-0.480404\pi\)
\(578\) 0 0
\(579\) −573.000 + 992.465i −0.989637 + 1.71410i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) −1196.00 −2.03056
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 579.000 1002.86i 0.969849 1.67983i
\(598\) 0 0
\(599\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(600\) 0 0
\(601\) −526.000 −0.875208 −0.437604 0.899168i \(-0.644173\pi\)
−0.437604 + 0.899168i \(0.644173\pi\)
\(602\) 0 0
\(603\) 1098.00 1.82090
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 407.000 704.945i 0.670511 1.16136i −0.307249 0.951629i \(-0.599408\pi\)
0.977759 0.209729i \(-0.0672583\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 563.000 + 975.145i 0.918434 + 1.59077i 0.801794 + 0.597600i \(0.203879\pi\)
0.116639 + 0.993174i \(0.462788\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 107.000 + 185.329i 0.172859 + 0.299401i 0.939418 0.342773i \(-0.111366\pi\)
−0.766559 + 0.642174i \(0.778033\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −312.500 + 541.266i −0.500000 + 0.866025i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 674.000 1.06815 0.534073 0.845438i \(-0.320661\pi\)
0.534073 + 0.845438i \(0.320661\pi\)
\(632\) 0 0
\(633\) −249.000 431.281i −0.393365 0.681328i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(642\) 0 0
\(643\) 314.000 0.488336 0.244168 0.969733i \(-0.421485\pi\)
0.244168 + 0.969733i \(0.421485\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −414.000 −0.630137
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −61.0000 105.655i −0.0922844 0.159841i 0.816188 0.577787i \(-0.196084\pi\)
−0.908472 + 0.417946i \(0.862750\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 507.000 + 878.150i 0.757848 + 1.31263i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 1154.00 1.71471 0.857355 0.514725i \(-0.172106\pi\)
0.857355 + 0.514725i \(0.172106\pi\)
\(674\) 0 0
\(675\) 337.500 + 584.567i 0.500000 + 0.866025i
\(676\) 0 0
\(677\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −78.0000 −0.113537
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 659.000 1141.42i 0.953690 1.65184i 0.216353 0.976315i \(-0.430584\pi\)
0.737337 0.675525i \(-0.236083\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) −338.000 585.433i −0.480797 0.832764i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 467.000 808.868i 0.658674 1.14086i −0.322285 0.946643i \(-0.604451\pi\)
0.980959 0.194214i \(-0.0622158\pi\)
\(710\) 0 0
\(711\) 639.000 + 1106.78i 0.898734 + 1.55665i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −429.000 + 743.050i −0.593361 + 1.02773i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 482.000 0.662999 0.331499 0.943455i \(-0.392446\pi\)
0.331499 + 0.943455i \(0.392446\pi\)
\(728\) 0 0
\(729\) 729.000 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −517.000 + 895.470i −0.705321 + 1.22165i 0.261255 + 0.965270i \(0.415864\pi\)
−0.966576 + 0.256381i \(0.917470\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 611.000 + 1058.28i 0.826793 + 1.43205i 0.900541 + 0.434771i \(0.143171\pi\)
−0.0737483 + 0.997277i \(0.523496\pi\)
\(740\) 0 0
\(741\) 1716.00 2.31579
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −601.000 + 1040.96i −0.800266 + 1.38610i 0.119174 + 0.992873i \(0.461975\pi\)
−0.919441 + 0.393229i \(0.871358\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −838.000 −1.10700 −0.553501 0.832849i \(-0.686708\pi\)
−0.553501 + 0.832849i \(0.686708\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −1534.00 −1.99480 −0.997399 0.0720749i \(-0.977038\pi\)
−0.997399 + 0.0720749i \(0.977038\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(774\) 0 0
\(775\) 575.000 995.929i 0.741935 1.28507i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −781.000 1352.73i −0.992376 1.71885i −0.602922 0.797800i \(-0.705997\pi\)
−0.389454 0.921046i \(-0.627336\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 814.000 1409.89i 1.02648 1.77792i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 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 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)