Properties

Label 441.4.e.j.226.1
Level $441$
Weight $4$
Character 441.226
Analytic conductor $26.020$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 441.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(26.0198423125\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 9)
Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

Embedding invariants

Embedding label 226.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 441.226
Dual form 441.4.e.j.361.1

$q$-expansion

\(f(q)\) \(=\) \(q+(4.00000 + 6.92820i) q^{4} +O(q^{10})\) \(q+(4.00000 + 6.92820i) q^{4} +70.0000 q^{13} +(-32.0000 + 55.4256i) q^{16} +(28.0000 - 48.4974i) q^{19} +(62.5000 + 108.253i) q^{25} +(154.000 + 266.736i) q^{31} +(-55.0000 + 95.2628i) q^{37} -520.000 q^{43} +(280.000 + 484.974i) q^{52} +(91.0000 - 157.617i) q^{61} -512.000 q^{64} +(440.000 + 762.102i) q^{67} +(595.000 + 1030.57i) q^{73} +448.000 q^{76} +(-442.000 + 765.566i) q^{79} +1330.00 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 8q^{4} + O(q^{10}) \) \( 2q + 8q^{4} + 140q^{13} - 64q^{16} + 56q^{19} + 125q^{25} + 308q^{31} - 110q^{37} - 1040q^{43} + 560q^{52} + 182q^{61} - 1024q^{64} + 880q^{67} + 1190q^{73} + 896q^{76} - 884q^{79} + 2660q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(344\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\)

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 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(3\) 0 0
\(4\) 4.00000 + 6.92820i 0.500000 + 0.866025i
\(5\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(12\) 0 0
\(13\) 70.0000 1.49342 0.746712 0.665148i \(-0.231631\pi\)
0.746712 + 0.665148i \(0.231631\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −32.0000 + 55.4256i −0.500000 + 0.866025i
\(17\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(18\) 0 0
\(19\) 28.0000 48.4974i 0.338086 0.585583i −0.645986 0.763349i \(-0.723554\pi\)
0.984073 + 0.177766i \(0.0568871\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(24\) 0 0
\(25\) 62.5000 + 108.253i 0.500000 + 0.866025i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 154.000 + 266.736i 0.892233 + 1.54539i 0.837192 + 0.546908i \(0.184195\pi\)
0.0550403 + 0.998484i \(0.482471\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −55.0000 + 95.2628i −0.244377 + 0.423273i −0.961956 0.273204i \(-0.911917\pi\)
0.717579 + 0.696477i \(0.245250\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −520.000 −1.84417 −0.922084 0.386989i \(-0.873515\pi\)
−0.922084 + 0.386989i \(0.873515\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 280.000 + 484.974i 0.746712 + 1.29334i
\(53\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(60\) 0 0
\(61\) 91.0000 157.617i 0.191006 0.330832i −0.754578 0.656210i \(-0.772158\pi\)
0.945584 + 0.325379i \(0.105492\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −512.000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 440.000 + 762.102i 0.802307 + 1.38964i 0.918094 + 0.396362i \(0.129728\pi\)
−0.115787 + 0.993274i \(0.536939\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 595.000 + 1030.57i 0.953966 + 1.65232i 0.736718 + 0.676200i \(0.236375\pi\)
0.217248 + 0.976117i \(0.430292\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 448.000 0.676173
\(77\) 0 0
\(78\) 0 0
\(79\) −442.000 + 765.566i −0.629480 + 1.09029i 0.358177 + 0.933654i \(0.383399\pi\)
−0.987656 + 0.156637i \(0.949935\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 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 \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1330.00 1.39218 0.696088 0.717957i \(-0.254922\pi\)
0.696088 + 0.717957i \(0.254922\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −500.000 + 866.025i −0.500000 + 0.866025i
\(101\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(102\) 0 0
\(103\) 910.000 1576.17i 0.870534 1.50781i 0.00908799 0.999959i \(-0.497107\pi\)
0.861446 0.507850i \(-0.169560\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(108\) 0 0
\(109\) 323.000 + 559.452i 0.283833 + 0.491613i 0.972325 0.233630i \(-0.0750606\pi\)
−0.688493 + 0.725243i \(0.741727\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 665.500 1152.68i 0.500000 0.866025i
\(122\) 0 0
\(123\) 0 0
\(124\) −1232.00 + 2133.89i −0.892233 + 1.54539i
\(125\) 0 0
\(126\) 0 0
\(127\) 380.000 0.265508 0.132754 0.991149i \(-0.457618\pi\)
0.132754 + 0.991149i \(0.457618\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(138\) 0 0
\(139\) −2576.00 −1.57190 −0.785948 0.618293i \(-0.787825\pi\)
−0.785948 + 0.618293i \(0.787825\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) −880.000 −0.488754
\(149\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(150\) 0 0
\(151\) −874.000 1513.81i −0.471027 0.815843i 0.528424 0.848981i \(-0.322783\pi\)
−0.999451 + 0.0331378i \(0.989450\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −1925.00 3334.20i −0.978546 1.69489i −0.667699 0.744432i \(-0.732721\pi\)
−0.310847 0.950460i \(-0.600613\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 1700.00 2944.49i 0.816897 1.41491i −0.0910600 0.995845i \(-0.529026\pi\)
0.907957 0.419062i \(-0.137641\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 2703.00 1.23031
\(170\) 0 0
\(171\) 0 0
\(172\) −2080.00 3602.67i −0.922084 1.59710i
\(173\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(180\) 0 0
\(181\) −3458.00 −1.42006 −0.710031 0.704171i \(-0.751319\pi\)
−0.710031 + 0.704171i \(0.751319\pi\)
\(182\) 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 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(192\) 0 0
\(193\) 575.000 + 995.929i 0.214453 + 0.371443i 0.953103 0.302646i \(-0.0978698\pi\)
−0.738650 + 0.674089i \(0.764536\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) −2618.00 4534.51i −0.932588 1.61529i −0.778879 0.627175i \(-0.784211\pi\)
−0.153710 0.988116i \(-0.549122\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −2240.00 + 3879.79i −0.746712 + 1.29334i
\(209\) 0 0
\(210\) 0 0
\(211\) 6032.00 1.96806 0.984028 0.178011i \(-0.0569664\pi\)
0.984028 + 0.178011i \(0.0569664\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 3220.00 0.966938 0.483469 0.875362i \(-0.339377\pi\)
0.483469 + 0.875362i \(0.339377\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(228\) 0 0
\(229\) 2233.00 3867.67i 0.644370 1.11608i −0.340076 0.940398i \(-0.610453\pi\)
0.984447 0.175684i \(-0.0562138\pi\)
\(230\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −3689.00 6389.54i −0.986014 1.70783i −0.637341 0.770582i \(-0.719966\pi\)
−0.348673 0.937244i \(-0.613368\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 1456.00 0.382012
\(245\) 0 0
\(246\) 0 0
\(247\) 1960.00 3394.82i 0.504906 0.874523i
\(248\) 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
\(256\) −2048.00 3547.24i −0.500000 0.866025i
\(257\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −3520.00 + 6096.82i −0.802307 + 1.38964i
\(269\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(270\) 0 0
\(271\) 406.000 703.213i 0.0910064 0.157628i −0.816928 0.576739i \(-0.804325\pi\)
0.907935 + 0.419111i \(0.137658\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 2015.00 + 3490.08i 0.437074 + 0.757035i 0.997462 0.0711951i \(-0.0226813\pi\)
−0.560388 + 0.828230i \(0.689348\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 2800.00 + 4849.74i 0.588137 + 1.01868i 0.994476 + 0.104961i \(0.0334717\pi\)
−0.406340 + 0.913722i \(0.633195\pi\)
\(284\) 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
\(291\) 0 0
\(292\) −4760.00 + 8244.56i −0.953966 + 1.65232i
\(293\) 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
\(304\) 1792.00 + 3103.84i 0.338086 + 0.585583i
\(305\) 0 0
\(306\) 0 0
\(307\) −10640.0 −1.97804 −0.989018 0.147797i \(-0.952782\pi\)
−0.989018 + 0.147797i \(0.952782\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(312\) 0 0
\(313\) 5005.00 8668.91i 0.903832 1.56548i 0.0813539 0.996685i \(-0.474076\pi\)
0.822478 0.568797i \(-0.192591\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −7072.00 −1.25896
\(317\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 4375.00 + 7577.72i 0.746712 + 1.29334i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −496.000 + 859.097i −0.0823644 + 0.142659i −0.904265 0.426971i \(-0.859580\pi\)
0.821901 + 0.569631i \(0.192914\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −4930.00 −0.796897 −0.398448 0.917191i \(-0.630451\pi\)
−0.398448 + 0.917191i \(0.630451\pi\)
\(338\) 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 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(348\) 0 0
\(349\) 11914.0 1.82734 0.913670 0.406456i \(-0.133236\pi\)
0.913670 + 0.406456i \(0.133236\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(360\) 0 0
\(361\) 1861.50 + 3224.21i 0.271395 + 0.470070i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 2170.00 + 3758.55i 0.308646 + 0.534591i 0.978066 0.208293i \(-0.0667908\pi\)
−0.669420 + 0.742884i \(0.733457\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −6175.00 + 10695.4i −0.857183 + 1.48469i 0.0174213 + 0.999848i \(0.494454\pi\)
−0.874605 + 0.484837i \(0.838879\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −8584.00 −1.16340 −0.581702 0.813402i \(-0.697613\pi\)
−0.581702 + 0.813402i \(0.697613\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 5320.00 + 9214.51i 0.696088 + 1.20566i
\(389\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 595.000 1030.57i 0.0752196 0.130284i −0.825962 0.563726i \(-0.809368\pi\)
0.901182 + 0.433441i \(0.142701\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −8000.00 −1.00000
\(401\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(402\) 0 0
\(403\) 10780.0 + 18671.5i 1.33248 + 2.30793i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 4123.00 + 7141.25i 0.498458 + 0.863354i 0.999998 0.00177990i \(-0.000566559\pi\)
−0.501541 + 0.865134i \(0.667233\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 14560.0 1.74107
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 17138.0 1.98398 0.991989 0.126322i \(-0.0403172\pi\)
0.991989 + 0.126322i \(0.0403172\pi\)
\(422\) 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 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(432\) 0 0
\(433\) 2590.00 0.287454 0.143727 0.989617i \(-0.454091\pi\)
0.143727 + 0.989617i \(0.454091\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −2584.00 + 4475.62i −0.283833 + 0.491613i
\(437\) 0 0
\(438\) 0 0
\(439\) 7462.00 12924.6i 0.811257 1.40514i −0.100728 0.994914i \(-0.532117\pi\)
0.911985 0.410224i \(-0.134550\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −6355.00 + 11007.2i −0.650491 + 1.12668i 0.332513 + 0.943099i \(0.392103\pi\)
−0.983004 + 0.183585i \(0.941230\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) −19780.0 −1.98543 −0.992716 0.120482i \(-0.961556\pi\)
−0.992716 + 0.120482i \(0.961556\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 7000.00 0.676173
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(480\) 0 0
\(481\) −3850.00 + 6668.40i −0.364958 + 0.632126i
\(482\) 0 0
\(483\) 0 0
\(484\) 10648.0 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) −10450.0 18099.9i −0.972351 1.68416i −0.688415 0.725317i \(-0.741693\pi\)
−0.283936 0.958843i \(-0.591640\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −19712.0 −1.78447
\(497\) 0 0
\(498\) 0 0
\(499\) 7568.00 13108.2i 0.678938 1.17596i −0.296363 0.955075i \(-0.595774\pi\)
0.975301 0.220880i \(-0.0708930\pi\)
\(500\) 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
\(507\) 0 0
\(508\) 1520.00 + 2632.72i 0.132754 + 0.229937i
\(509\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 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
\(523\) −6020.00 + 10426.9i −0.503320 + 0.871775i 0.496673 + 0.867938i \(0.334555\pi\)
−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
\(541\) 11339.0 19639.7i 0.901112 1.56077i 0.0750596 0.997179i \(-0.476085\pi\)
0.826053 0.563593i \(-0.190581\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 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
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −10304.0 17847.1i −0.785948 1.36130i
\(557\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(558\) 0 0
\(559\) −36400.0 −2.75413
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(570\) 0 0
\(571\) −11656.0 20188.8i −0.854270 1.47964i −0.877320 0.479905i \(-0.840671\pi\)
0.0230498 0.999734i \(-0.492662\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −8855.00 15337.3i −0.638888 1.10659i −0.985677 0.168644i \(-0.946061\pi\)
0.346789 0.937943i \(-0.387272\pi\)
\(578\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 17248.0 1.20661
\(590\) 0 0
\(591\) 0 0
\(592\) −3520.00 6096.82i −0.244377 0.423273i
\(593\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(600\) 0 0
\(601\) 29302.0 1.98877 0.994387 0.105801i \(-0.0337408\pi\)
0.994387 + 0.105801i \(0.0337408\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 6992.00 12110.5i 0.471027 0.815843i
\(605\) 0 0
\(606\) 0 0
\(607\) −14210.0 + 24612.4i −0.950191 + 1.64578i −0.205184 + 0.978723i \(0.565779\pi\)
−0.745007 + 0.667056i \(0.767554\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −8695.00 15060.2i −0.572900 0.992292i −0.996266 0.0863334i \(-0.972485\pi\)
0.423366 0.905959i \(-0.360848\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) −13328.0 23084.8i −0.865424 1.49896i −0.866625 0.498959i \(-0.833716\pi\)
0.00120126 0.999999i \(-0.499618\pi\)
\(620\) 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
\(628\) 15400.0 26673.6i 0.978546 1.69489i
\(629\) 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
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(642\) 0 0
\(643\) −13160.0 −0.807122 −0.403561 0.914953i \(-0.632228\pi\)
−0.403561 + 0.914953i \(0.632228\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 27200.0 1.63379
\(653\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −10241.0 17737.9i −0.602615 1.04376i −0.992423 0.122864i \(-0.960792\pi\)
0.389808 0.920896i \(-0.372541\pi\)
\(662\) 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
\(673\) 24050.0 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 10812.0 + 18726.9i 0.615157 + 1.06548i
\(677\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 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
\(688\) 16640.0 28821.3i 0.922084 1.59710i
\(689\) 0 0
\(690\) 0 0
\(691\) −8036.00 + 13918.8i −0.442408 + 0.766273i −0.997868 0.0652705i \(-0.979209\pi\)
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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 3080.00 + 5334.72i 0.165241 + 0.286206i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −18073.0 + 31303.4i −0.957328 + 1.65814i −0.228381 + 0.973572i \(0.573343\pi\)
−0.728948 + 0.684569i \(0.759990\pi\)
\(710\) 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
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) −13832.0 23957.7i −0.710031 1.22981i
\(725\) 0 0
\(726\) 0 0
\(727\) 10780.0 0.549942 0.274971 0.961452i \(-0.411332\pi\)
0.274971 + 0.961452i \(0.411332\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 7525.00 13033.7i 0.379184 0.656767i −0.611759 0.791044i \(-0.709538\pi\)
0.990944 + 0.134277i \(0.0428712\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −15688.0 27172.4i −0.780910 1.35258i −0.931412 0.363966i \(-0.881422\pi\)
0.150502 0.988610i \(-0.451911\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 11726.0 20310.0i 0.569757 0.986849i −0.426832 0.904331i \(-0.640370\pi\)
0.996590 0.0825179i \(-0.0262962\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −41470.0 −1.99109 −0.995543 0.0943039i \(-0.969937\pi\)
−0.995543 + 0.0943039i \(0.969937\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 4606.00 0.215990 0.107995 0.994151i \(-0.465557\pi\)
0.107995 + 0.994151i \(0.465557\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −4600.00 + 7967.43i −0.214453 + 0.371443i
\(773\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(774\) 0 0
\(775\) −19250.0 + 33342.0i −0.892233 + 1.54539i
\(776\) 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
\(787\) 21700.0 + 37585.5i 0.982874 + 1.70239i 0.651029 + 0.759053i \(0.274338\pi\)
0.331844 + 0.943334i \(0.392329\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 6370.00 11033.2i 0.285253 0.494072i
\(794\) 0 0
\(795\) 0 0
\(796\) 20944.0 36276.1i 0.932588 1.61529i
\(797\) 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 0
\(807\) 0 0