Properties

Label 588.4.k.a.521.1
Level $588$
Weight $4$
Character 588.521
Analytic conductor $34.693$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 588.k (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(34.6931230834\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label 521.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 588.521
Dual form 588.4.k.a.509.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-4.50000 - 2.59808i) q^{3} +(13.5000 + 23.3827i) q^{9} +O(q^{10})\) \(q+(-4.50000 - 2.59808i) q^{3} +(13.5000 + 23.3827i) q^{9} -29.4449i q^{13} +(-25.5000 + 14.7224i) q^{19} +(62.5000 - 108.253i) q^{25} -140.296i q^{27} +(298.500 + 172.339i) q^{31} +(-216.500 - 374.989i) q^{37} +(-76.5000 + 132.502i) q^{39} -449.000 q^{43} +153.000 q^{57} +(-810.000 + 467.654i) q^{61} +(-503.500 + 872.088i) q^{67} +(-730.500 - 421.754i) q^{73} +(-562.500 + 324.760i) q^{75} +(-251.500 - 435.611i) q^{79} +(-364.500 + 631.333i) q^{81} +(-895.500 - 1551.05i) q^{93} -1371.78i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 9 q^{3} + 27 q^{9} + O(q^{10}) \) \( 2 q - 9 q^{3} + 27 q^{9} - 51 q^{19} + 125 q^{25} + 597 q^{31} - 433 q^{37} - 153 q^{39} - 898 q^{43} + 306 q^{57} - 1620 q^{61} - 1007 q^{67} - 1461 q^{73} - 1125 q^{75} - 503 q^{79} - 729 q^{81} - 1791 q^{93} + O(q^{100}) \)

Character values

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

\(n\) \(197\) \(295\) \(493\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{1}{6}\right)\)

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
\(3\) −4.50000 2.59808i −0.866025 0.500000i
\(4\) 0 0
\(5\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 13.5000 + 23.3827i 0.500000 + 0.866025i
\(10\) 0 0
\(11\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(12\) 0 0
\(13\) 29.4449i 0.628195i −0.949391 0.314098i \(-0.898298\pi\)
0.949391 0.314098i \(-0.101702\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(18\) 0 0
\(19\) −25.5000 + 14.7224i −0.307900 + 0.177766i −0.645986 0.763349i \(-0.723554\pi\)
0.338086 + 0.941115i \(0.390220\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(24\) 0 0
\(25\) 62.5000 108.253i 0.500000 0.866025i
\(26\) 0 0
\(27\) 140.296i 1.00000i
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 298.500 + 172.339i 1.72943 + 0.998484i 0.892233 + 0.451576i \(0.149138\pi\)
0.837192 + 0.546908i \(0.184195\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −216.500 374.989i −0.961956 1.66616i −0.717579 0.696477i \(-0.754750\pi\)
−0.244377 0.969680i \(-0.578583\pi\)
\(38\) 0 0
\(39\) −76.5000 + 132.502i −0.314098 + 0.544033i
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −449.000 −1.59237 −0.796184 0.605054i \(-0.793151\pi\)
−0.796184 + 0.605054i \(0.793151\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 153.000 0.355532
\(58\) 0 0
\(59\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(60\) 0 0
\(61\) −810.000 + 467.654i −1.70016 + 0.981589i −0.754578 + 0.656210i \(0.772158\pi\)
−0.945584 + 0.325379i \(0.894508\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −503.500 + 872.088i −0.918094 + 1.59019i −0.115787 + 0.993274i \(0.536939\pi\)
−0.802307 + 0.596912i \(0.796394\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −730.500 421.754i −1.17121 0.676200i −0.217248 0.976117i \(-0.569708\pi\)
−0.953966 + 0.299916i \(0.903041\pi\)
\(74\) 0 0
\(75\) −562.500 + 324.760i −0.866025 + 0.500000i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −251.500 435.611i −0.358177 0.620380i 0.629480 0.777017i \(-0.283268\pi\)
−0.987656 + 0.156637i \(0.949935\pi\)
\(80\) 0 0
\(81\) −364.500 + 631.333i −0.500000 + 0.866025i
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 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
\(91\) 0 0
\(92\) 0 0
\(93\) −895.500 1551.05i −0.998484 1.72943i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1371.78i 1.43591i −0.696088 0.717957i \(-0.745078\pi\)
0.696088 0.717957i \(-0.254922\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(102\) 0 0
\(103\) −1810.50 + 1045.29i −1.73198 + 0.999959i −0.861446 + 0.507850i \(0.830440\pi\)
−0.870534 + 0.492109i \(0.836226\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(108\) 0 0
\(109\) −783.500 + 1357.06i −0.688493 + 1.19250i 0.283833 + 0.958874i \(0.408394\pi\)
−0.972325 + 0.233630i \(0.924939\pi\)
\(110\) 0 0
\(111\) 2249.93i 1.92391i
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 688.500 397.506i 0.544033 0.314098i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −665.500 1152.68i −0.500000 0.866025i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −2647.00 −1.84947 −0.924737 0.380606i \(-0.875715\pi\)
−0.924737 + 0.380606i \(0.875715\pi\)
\(128\) 0 0
\(129\) 2020.50 + 1166.54i 1.37903 + 0.796184i
\(130\) 0 0
\(131\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(138\) 0 0
\(139\) 1217.63i 0.743008i −0.928431 0.371504i \(-0.878842\pi\)
0.928431 0.371504i \(-0.121158\pi\)
\(140\) 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 \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(150\) 0 0
\(151\) 874.000 1513.81i 0.471027 0.815843i −0.528424 0.848981i \(-0.677217\pi\)
0.999451 + 0.0331378i \(0.0105500\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −702.000 405.300i −0.356852 0.206028i 0.310847 0.950460i \(-0.399387\pi\)
−0.667699 + 0.744432i \(0.732721\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 1700.00 + 2944.49i 0.816897 + 1.41491i 0.907957 + 0.419062i \(0.137641\pi\)
−0.0910600 + 0.995845i \(0.529026\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 1330.00 0.605371
\(170\) 0 0
\(171\) −688.500 397.506i −0.307900 0.177766i
\(172\) 0 0
\(173\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(180\) 0 0
\(181\) 4709.45i 1.93398i −0.254814 0.966990i \(-0.582014\pi\)
0.254814 0.966990i \(-0.417986\pi\)
\(182\) 0 0
\(183\) 4860.00 1.96318
\(184\) 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 \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(192\) 0 0
\(193\) 2555.50 4426.26i 0.953103 1.65082i 0.214453 0.976734i \(-0.431203\pi\)
0.738650 0.674089i \(-0.235464\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 1755.00 + 1013.25i 0.625169 + 0.360942i 0.778879 0.627175i \(-0.215789\pi\)
−0.153710 + 0.988116i \(0.549122\pi\)
\(200\) 0 0
\(201\) 4531.50 2616.26i 1.59019 0.918094i
\(202\) 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
\(211\) −6032.00 −1.96806 −0.984028 0.178011i \(-0.943034\pi\)
−0.984028 + 0.178011i \(0.943034\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 2191.50 + 3795.79i 0.676200 + 1.17121i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 5830.08i 1.75072i 0.483469 + 0.875362i \(0.339377\pi\)
−0.483469 + 0.875362i \(0.660623\pi\)
\(224\) 0 0
\(225\) 3375.00 1.00000
\(226\) 0 0
\(227\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(228\) 0 0
\(229\) −1054.50 + 608.816i −0.304294 + 0.175684i −0.644370 0.764714i \(-0.722880\pi\)
0.340076 + 0.940398i \(0.389547\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 2613.66i 0.716353i
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −1080.00 623.538i −0.288668 0.166662i 0.348673 0.937244i \(-0.386632\pi\)
−0.637341 + 0.770582i \(0.719966\pi\)
\(242\) 0 0
\(243\) 3280.50 1894.00i 0.866025 0.500000i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 433.500 + 750.844i 0.111672 + 0.193421i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(270\) 0 0
\(271\) 7695.00 4442.71i 1.72486 0.995850i 0.816928 0.576739i \(-0.195675\pi\)
0.907935 0.419111i \(-0.137658\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 2583.50 4474.75i 0.560388 0.970620i −0.437074 0.899425i \(-0.643985\pi\)
0.997462 0.0711951i \(-0.0226813\pi\)
\(278\) 0 0
\(279\) 9306.31i 1.99697i
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) −7534.50 4350.05i −1.58261 0.913722i −0.994476 0.104961i \(-0.966528\pi\)
−0.588137 0.808761i \(-0.700138\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 2456.50 + 4254.78i 0.500000 + 0.866025i
\(290\) 0 0
\(291\) −3564.00 + 6173.03i −0.717957 + 1.24354i
\(292\) 0 0
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 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
\(307\) 8419.50i 1.56523i −0.622505 0.782616i \(-0.713885\pi\)
0.622505 0.782616i \(-0.286115\pi\)
\(308\) 0 0
\(309\) 10863.0 1.99992
\(310\) 0 0
\(311\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(312\) 0 0
\(313\) −9559.50 + 5519.18i −1.72631 + 0.996685i −0.822478 + 0.568797i \(0.807409\pi\)
−0.903832 + 0.427888i \(0.859258\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −3187.50 1840.30i −0.544033 0.314098i
\(326\) 0 0
\(327\) 7051.50 4071.19i 1.19250 0.688493i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 4949.50 + 8572.79i 0.821901 + 1.42357i 0.904265 + 0.426971i \(0.140420\pi\)
−0.0823644 + 0.996602i \(0.526247\pi\)
\(332\) 0 0
\(333\) 5845.50 10124.7i 0.961956 1.66616i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 7363.00 1.19017 0.595086 0.803662i \(-0.297118\pi\)
0.595086 + 0.803662i \(0.297118\pi\)
\(338\) 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 \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(348\) 0 0
\(349\) 5300.08i 0.812913i −0.913670 0.406456i \(-0.866764\pi\)
0.913670 0.406456i \(-0.133236\pi\)
\(350\) 0 0
\(351\) −4131.00 −0.628195
\(352\) 0 0
\(353\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(360\) 0 0
\(361\) −2996.00 + 5189.22i −0.436798 + 0.756557i
\(362\) 0 0
\(363\) 6916.08i 1.00000i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −2536.50 1464.45i −0.360774 0.208293i 0.308646 0.951177i \(-0.400124\pi\)
−0.669420 + 0.742884i \(0.733457\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −125.500 217.372i −0.0174213 0.0301746i 0.857183 0.515011i \(-0.172212\pi\)
−0.874605 + 0.484837i \(0.838879\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −14687.0 −1.99056 −0.995278 0.0970683i \(-0.969053\pi\)
−0.995278 + 0.0970683i \(0.969053\pi\)
\(380\) 0 0
\(381\) 11911.5 + 6877.11i 1.60169 + 0.924737i
\(382\) 0 0
\(383\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −6061.50 10498.8i −0.796184 1.37903i
\(388\) 0 0
\(389\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 7723.50 4459.16i 0.976401 0.563726i 0.0752196 0.997167i \(-0.476034\pi\)
0.901182 + 0.433441i \(0.142701\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(402\) 0 0
\(403\) 5074.50 8789.29i 0.627243 1.08642i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 12394.5 + 7155.97i 1.49846 + 0.865134i 0.999998 0.00177990i \(-0.000566559\pi\)
0.498458 + 0.866914i \(0.333900\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −3163.50 + 5479.34i −0.371504 + 0.643464i
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −10459.0 −1.21078 −0.605392 0.795927i \(-0.706984\pi\)
−0.605392 + 0.795927i \(0.706984\pi\)
\(422\) 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 \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(432\) 0 0
\(433\) 11159.6i 1.23856i −0.785170 0.619280i \(-0.787425\pi\)
0.785170 0.619280i \(-0.212575\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −9315.00 + 5378.02i −1.01271 + 0.584690i −0.911985 0.410224i \(-0.865450\pi\)
−0.100728 + 0.994914i \(0.532117\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(444\) 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
\(453\) −7866.00 + 4541.44i −0.815843 + 0.471027i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −9603.50 16633.7i −0.983004 1.70261i −0.650491 0.759514i \(-0.725437\pi\)
−0.332513 0.943099i \(-0.607897\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 7811.00 0.784034 0.392017 0.919958i \(-0.371777\pi\)
0.392017 + 0.919958i \(0.371777\pi\)
\(464\) 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
\(469\) 0 0
\(470\) 0 0
\(471\) 2106.00 + 3647.70i 0.206028 + 0.356852i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 3680.61i 0.355532i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(480\) 0 0
\(481\) −11041.5 + 6374.81i −1.04667 + 0.604296i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −7398.50 + 12814.6i −0.688415 + 1.19237i 0.283936 + 0.958843i \(0.408360\pi\)
−0.972351 + 0.233526i \(0.924974\pi\)
\(488\) 0 0
\(489\) 17666.9i 1.63379i
\(490\) 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
\(499\) −3303.50 5721.83i −0.296363 0.513315i 0.678938 0.734195i \(-0.262440\pi\)
−0.975301 + 0.220880i \(0.929107\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −5985.00 3455.44i −0.524267 0.302685i
\(508\) 0 0
\(509\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 2065.50 + 3577.55i 0.177766 + 0.307900i
\(514\) 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
\(523\) −17980.5 + 10381.0i −1.50331 + 0.867938i −0.503320 + 0.864100i \(0.667888\pi\)
−0.999993 + 0.00383755i \(0.998778\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −6083.50 + 10536.9i −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
\(541\) −10394.5 18003.8i −0.826053 1.43077i −0.901112 0.433586i \(-0.857248\pi\)
0.0750596 0.997179i \(-0.476085\pi\)
\(542\) 0 0
\(543\) −12235.5 + 21192.5i −0.966990 + 1.67488i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 1640.00 0.128193 0.0640963 0.997944i \(-0.479584\pi\)
0.0640963 + 0.997944i \(0.479584\pi\)
\(548\) 0 0
\(549\) −21870.0 12626.7i −1.70016 0.981589i
\(550\) 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 \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(558\) 0 0
\(559\) 13220.7i 1.00032i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(570\) 0 0
\(571\) −314.500 + 544.730i −0.0230498 + 0.0399234i −0.877320 0.479905i \(-0.840671\pi\)
0.854270 + 0.519829i \(0.174004\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −4048.50 2337.40i −0.292099 0.168644i 0.346789 0.937943i \(-0.387272\pi\)
−0.638888 + 0.769300i \(0.720605\pi\)
\(578\) 0 0
\(579\) −22999.5 + 13278.8i −1.65082 + 0.953103i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) −10149.0 −0.709987
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −5265.00 9119.25i −0.360942 0.625169i
\(598\) 0 0
\(599\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(600\) 0 0
\(601\) 26935.1i 1.82813i 0.405567 + 0.914065i \(0.367074\pi\)
−0.405567 + 0.914065i \(0.632926\pi\)
\(602\) 0 0
\(603\) −27189.0 −1.83619
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 17278.5 9975.75i 1.15538 0.667056i 0.205184 0.978723i \(-0.434221\pi\)
0.950191 + 0.311667i \(0.100887\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −8695.00 + 15060.2i −0.572900 + 0.992292i 0.423366 + 0.905959i \(0.360848\pi\)
−0.996266 + 0.0863334i \(0.972485\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 26674.5 + 15400.5i 1.73205 + 0.999999i 0.866625 + 0.498959i \(0.166284\pi\)
0.865424 + 0.501040i \(0.167049\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −7812.50 13531.6i −0.500000 0.866025i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 1892.00 0.119365 0.0596825 0.998217i \(-0.480991\pi\)
0.0596825 + 0.998217i \(0.480991\pi\)
\(632\) 0 0
\(633\) 27144.0 + 15671.6i 1.70439 + 0.984028i
\(634\) 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 \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(642\) 0 0
\(643\) 3521.26i 0.215964i −0.994153 0.107982i \(-0.965561\pi\)
0.994153 0.107982i \(-0.0344389\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 22774.7i 1.35240i
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −27106.5 15649.9i −1.59504 0.920896i −0.992423 0.122864i \(-0.960792\pi\)
−0.602615 0.798032i \(-0.705875\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 15147.0 26235.4i 0.875362 1.51617i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 33949.0 1.94448 0.972242 0.233977i \(-0.0751742\pi\)
0.972242 + 0.233977i \(0.0751742\pi\)
\(674\) 0 0
\(675\) −15187.5 8768.51i −0.866025 0.500000i
\(676\) 0 0
\(677\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 6327.00 0.351368
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 2053.50 1185.59i 0.113052 0.0652705i −0.442408 0.896814i \(-0.645876\pi\)
0.555460 + 0.831543i \(0.312542\pi\)
\(692\) 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
\(703\) 11041.5 + 6374.81i 0.592373 + 0.342007i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 18073.0 + 31303.4i 0.957328 + 1.65814i 0.728948 + 0.684569i \(0.240010\pi\)
0.228381 + 0.973572i \(0.426657\pi\)
\(710\) 0 0
\(711\) 6790.50 11761.5i 0.358177 0.620380i
\(712\) 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
\(721\) 0 0
\(722\) 0 0
\(723\) 3240.00 + 5611.84i 0.166662 + 0.288668i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 28182.2i 1.43772i −0.695157 0.718858i \(-0.744665\pi\)
0.695157 0.718858i \(-0.255335\pi\)
\(728\) 0 0
\(729\) −19683.0 −1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 27190.5 15698.4i 1.37013 0.791044i 0.379184 0.925321i \(-0.376205\pi\)
0.990944 + 0.134277i \(0.0428712\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 18711.5 32409.3i 0.931412 1.61325i 0.150502 0.988610i \(-0.451911\pi\)
0.780910 0.624644i \(-0.214756\pi\)
\(740\) 0 0
\(741\) 4505.06i 0.223344i
\(742\) 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
\(751\) 20510.5 + 35525.2i 0.996590 + 1.72614i 0.569757 + 0.821813i \(0.307037\pi\)
0.426832 + 0.904331i \(0.359630\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 41470.0 1.99109 0.995543 0.0943039i \(-0.0300625\pi\)
0.995543 + 0.0943039i \(0.0300625\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 17211.4i 0.807098i 0.914958 + 0.403549i \(0.132224\pi\)
−0.914958 + 0.403549i \(0.867776\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(774\) 0 0
\(775\) 37312.5 21542.4i 1.72943 0.998484i
\(776\) 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
\(787\) 7047.00 + 4068.59i 0.319185 + 0.184281i 0.651029 0.759053i \(-0.274338\pi\)
−0.331844 + 0.943334i \(0.607671\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 13770.0 + 23850.3i 0.616629 + 1.06803i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0