Properties

Label 637.2.r.b.324.1
Level $637$
Weight $2$
Character 637.324
Analytic conductor $5.086$
Analytic rank $0$
Dimension $4$
CM discriminant -91
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.r (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.08647060876\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-13})\)
Defining polynomial: \(x^{4} - 13 x^{2} + 169\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label 324.1
Root \(3.12250 + 1.80278i\) of defining polynomial
Character \(\chi\) \(=\) 637.324
Dual form 637.2.r.b.116.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-1.00000 - 1.73205i) q^{4} +(-3.12250 - 1.80278i) q^{5} +(1.50000 - 2.59808i) q^{9} +O(q^{10})\) \(q+(-1.00000 - 1.73205i) q^{4} +(-3.12250 - 1.80278i) q^{5} +(1.50000 - 2.59808i) q^{9} +3.60555i q^{13} +(-2.00000 + 3.46410i) q^{16} +(-3.12250 - 1.80278i) q^{19} +7.21110i q^{20} +(-0.500000 + 0.866025i) q^{23} +(4.00000 + 6.92820i) q^{25} -5.00000 q^{29} +(-9.36750 + 5.40833i) q^{31} -6.00000 q^{36} +7.21110i q^{41} +9.00000 q^{43} +(-9.36750 + 5.40833i) q^{45} +(-3.12250 - 1.80278i) q^{47} +(6.24500 - 3.60555i) q^{52} +(-5.50000 - 9.52628i) q^{53} +(12.4900 - 7.21110i) q^{59} +8.00000 q^{64} +(6.50000 - 11.2583i) q^{65} +(-9.36750 + 5.40833i) q^{73} +7.21110i q^{76} +(-7.50000 + 12.9904i) q^{79} +(12.4900 - 7.21110i) q^{80} +(-4.50000 - 7.79423i) q^{81} -18.0278i q^{83} +(-3.12250 - 1.80278i) q^{89} +2.00000 q^{92} +(6.50000 + 11.2583i) q^{95} -18.0278i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 6 q^{9} + O(q^{10}) \) \( 4 q - 4 q^{4} + 6 q^{9} - 8 q^{16} - 2 q^{23} + 16 q^{25} - 20 q^{29} - 24 q^{36} + 36 q^{43} - 22 q^{53} + 32 q^{64} + 26 q^{65} - 30 q^{79} - 18 q^{81} + 8 q^{92} + 26 q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/637\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(248\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{3}\right)\)

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 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(3\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(4\) −1.00000 1.73205i −0.500000 0.866025i
\(5\) −3.12250 1.80278i −1.39642 0.806226i −0.402408 0.915460i \(-0.631827\pi\)
−0.994016 + 0.109235i \(0.965160\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 1.50000 2.59808i 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
\(13\) 3.60555i 1.00000i
\(14\) 0 0
\(15\) 0 0
\(16\) −2.00000 + 3.46410i −0.500000 + 0.866025i
\(17\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(18\) 0 0
\(19\) −3.12250 1.80278i −0.716350 0.413585i 0.0970575 0.995279i \(-0.469057\pi\)
−0.813408 + 0.581694i \(0.802390\pi\)
\(20\) 7.21110i 1.61245i
\(21\) 0 0
\(22\) 0 0
\(23\) −0.500000 + 0.866025i −0.104257 + 0.180579i −0.913434 0.406986i \(-0.866580\pi\)
0.809177 + 0.587565i \(0.199913\pi\)
\(24\) 0 0
\(25\) 4.00000 + 6.92820i 0.800000 + 1.38564i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −5.00000 −0.928477 −0.464238 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) 0 0
\(31\) −9.36750 + 5.40833i −1.68245 + 0.971364i −0.722428 + 0.691446i \(0.756974\pi\)
−0.960024 + 0.279918i \(0.909693\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −6.00000 −1.00000
\(37\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 7.21110i 1.12619i 0.826394 + 0.563093i \(0.190389\pi\)
−0.826394 + 0.563093i \(0.809611\pi\)
\(42\) 0 0
\(43\) 9.00000 1.37249 0.686244 0.727372i \(-0.259258\pi\)
0.686244 + 0.727372i \(0.259258\pi\)
\(44\) 0 0
\(45\) −9.36750 + 5.40833i −1.39642 + 0.806226i
\(46\) 0 0
\(47\) −3.12250 1.80278i −0.455463 0.262962i 0.254671 0.967028i \(-0.418033\pi\)
−0.710135 + 0.704066i \(0.751366\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 6.24500 3.60555i 0.866025 0.500000i
\(53\) −5.50000 9.52628i −0.755483 1.30854i −0.945134 0.326683i \(-0.894069\pi\)
0.189651 0.981852i \(-0.439264\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 12.4900 7.21110i 1.62606 0.938806i 0.640806 0.767703i \(-0.278600\pi\)
0.985253 0.171103i \(-0.0547329\pi\)
\(60\) 0 0
\(61\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 8.00000 1.00000
\(65\) 6.50000 11.2583i 0.806226 1.39642i
\(66\) 0 0
\(67\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −9.36750 + 5.40833i −1.09638 + 0.632997i −0.935269 0.353939i \(-0.884842\pi\)
−0.161114 + 0.986936i \(0.551509\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 7.21110i 0.827170i
\(77\) 0 0
\(78\) 0 0
\(79\) −7.50000 + 12.9904i −0.843816 + 1.46153i 0.0428296 + 0.999082i \(0.486363\pi\)
−0.886646 + 0.462450i \(0.846971\pi\)
\(80\) 12.4900 7.21110i 1.39642 0.806226i
\(81\) −4.50000 7.79423i −0.500000 0.866025i
\(82\) 0 0
\(83\) 18.0278i 1.97880i −0.145204 0.989402i \(-0.546384\pi\)
0.145204 0.989402i \(-0.453616\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −3.12250 1.80278i −0.330984 0.191094i 0.325294 0.945613i \(-0.394537\pi\)
−0.656278 + 0.754519i \(0.727870\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 2.00000 0.208514
\(93\) 0 0
\(94\) 0 0
\(95\) 6.50000 + 11.2583i 0.666886 + 1.15508i
\(96\) 0 0
\(97\) 18.0278i 1.83044i −0.402953 0.915221i \(-0.632016\pi\)
0.402953 0.915221i \(-0.367984\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 8.00000 13.8564i 0.800000 1.38564i
\(101\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(102\) 0 0
\(103\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −4.00000 + 6.92820i −0.386695 + 0.669775i −0.992003 0.126217i \(-0.959717\pi\)
0.605308 + 0.795991i \(0.293050\pi\)
\(108\) 0 0
\(109\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −19.0000 −1.78737 −0.893685 0.448695i \(-0.851889\pi\)
−0.893685 + 0.448695i \(0.851889\pi\)
\(114\) 0 0
\(115\) 3.12250 1.80278i 0.291175 0.168110i
\(116\) 5.00000 + 8.66025i 0.464238 + 0.804084i
\(117\) 9.36750 + 5.40833i 0.866025 + 0.500000i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −5.50000 + 9.52628i −0.500000 + 0.866025i
\(122\) 0 0
\(123\) 0 0
\(124\) 18.7350 + 10.8167i 1.68245 + 0.971364i
\(125\) 10.8167i 0.967471i
\(126\) 0 0
\(127\) −12.0000 −1.06483 −0.532414 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) 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 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 6.00000 + 10.3923i 0.500000 + 0.866025i
\(145\) 15.6125 + 9.01388i 1.29655 + 0.748562i
\(146\) 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
\(151\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 39.0000 3.13256
\(156\) 0 0
\(157\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(164\) 12.4900 7.21110i 0.975305 0.563093i
\(165\) 0 0
\(166\) 0 0
\(167\) 18.0278i 1.39503i −0.716570 0.697515i \(-0.754289\pi\)
0.716570 0.697515i \(-0.245711\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) −9.36750 + 5.40833i −0.716350 + 0.413585i
\(172\) −9.00000 15.5885i −0.686244 1.18861i
\(173\) 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
\(179\) −12.5000 21.6506i −0.934294 1.61824i −0.775888 0.630870i \(-0.782698\pi\)
−0.158406 0.987374i \(-0.550635\pi\)
\(180\) 18.7350 + 10.8167i 1.39642 + 0.806226i
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 7.21110i 0.525924i
\(189\) 0 0
\(190\) 0 0
\(191\) 10.0000 17.3205i 0.723575 1.25327i −0.235983 0.971757i \(-0.575831\pi\)
0.959558 0.281511i \(-0.0908356\pi\)
\(192\) 0 0
\(193\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 13.0000 22.5167i 0.907959 1.57263i
\(206\) 0 0
\(207\) 1.50000 + 2.59808i 0.104257 + 0.180579i
\(208\) −12.4900 7.21110i −0.866025 0.500000i
\(209\) 0 0
\(210\) 0 0
\(211\) −5.00000 −0.344214 −0.172107 0.985078i \(-0.555058\pi\)
−0.172107 + 0.985078i \(0.555058\pi\)
\(212\) −11.0000 + 19.0526i −0.755483 + 1.30854i
\(213\) 0 0
\(214\) 0 0
\(215\) −28.1025 16.2250i −1.91657 1.10653i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 18.0278i 1.20723i −0.797277 0.603614i \(-0.793727\pi\)
0.797277 0.603614i \(-0.206273\pi\)
\(224\) 0 0
\(225\) 24.0000 1.60000
\(226\) 0 0
\(227\) 12.4900 7.21110i 0.828990 0.478618i −0.0245166 0.999699i \(-0.507805\pi\)
0.853507 + 0.521082i \(0.174471\pi\)
\(228\) 0 0
\(229\) 18.7350 + 10.8167i 1.23804 + 0.714785i 0.968694 0.248258i \(-0.0798581\pi\)
0.269349 + 0.963043i \(0.413191\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −14.5000 + 25.1147i −0.949927 + 1.64532i −0.204354 + 0.978897i \(0.565509\pi\)
−0.745573 + 0.666424i \(0.767824\pi\)
\(234\) 0 0
\(235\) 6.50000 + 11.2583i 0.424013 + 0.734412i
\(236\) −24.9800 14.4222i −1.62606 0.938806i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −9.36750 + 5.40833i −0.603414 + 0.348381i −0.770383 0.637581i \(-0.779935\pi\)
0.166970 + 0.985962i \(0.446602\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 6.50000 11.2583i 0.413585 0.716350i
\(248\) 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
\(256\) −8.00000 13.8564i −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
\(260\) −26.0000 −1.61245
\(261\) −7.50000 + 12.9904i −0.464238 + 0.804084i
\(262\) 0 0
\(263\) 15.5000 + 26.8468i 0.955771 + 1.65544i 0.732594 + 0.680666i \(0.238309\pi\)
0.223177 + 0.974778i \(0.428357\pi\)
\(264\) 0 0
\(265\) 39.6611i 2.43636i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(270\) 0 0
\(271\) −24.9800 14.4222i −1.51743 0.876087i −0.999790 0.0204858i \(-0.993479\pi\)
−0.517636 0.855601i \(-0.673188\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 8.50000 + 14.7224i 0.510716 + 0.884585i 0.999923 + 0.0124177i \(0.00395278\pi\)
−0.489207 + 0.872167i \(0.662714\pi\)
\(278\) 0 0
\(279\) 32.4500i 1.94273i
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(284\) 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
\(292\) 18.7350 + 10.8167i 1.09638 + 0.632997i
\(293\) 18.0278i 1.05319i −0.850115 0.526596i \(-0.823468\pi\)
0.850115 0.526596i \(-0.176532\pi\)
\(294\) 0 0
\(295\) −52.0000 −3.02756
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −3.12250 1.80278i −0.180579 0.104257i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 12.4900 7.21110i 0.716350 0.413585i
\(305\) 0 0
\(306\) 0 0
\(307\) 32.4500i 1.85202i 0.377503 + 0.926009i \(0.376783\pi\)
−0.377503 + 0.926009i \(0.623217\pi\)
\(308\) 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
\(313\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 30.0000 1.68763
\(317\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −24.9800 14.4222i −1.39642 0.806226i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −9.00000 + 15.5885i −0.500000 + 0.866025i
\(325\) −24.9800 + 14.4222i −1.38564 + 0.800000i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(332\) −31.2250 + 18.0278i −1.71369 + 0.989402i
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 23.0000 1.25289 0.626445 0.779466i \(-0.284509\pi\)
0.626445 + 0.779466i \(0.284509\pi\)
\(338\) 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
\(347\) −16.0000 27.7128i −0.858925 1.48770i −0.872955 0.487800i \(-0.837799\pi\)
0.0140303 0.999902i \(-0.495534\pi\)
\(348\) 0 0
\(349\) 32.4500i 1.73701i 0.495683 + 0.868503i \(0.334918\pi\)
−0.495683 + 0.868503i \(0.665082\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −31.2250 + 18.0278i −1.66194 + 0.959521i −0.690150 + 0.723666i \(0.742456\pi\)
−0.971788 + 0.235854i \(0.924211\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 7.21110i 0.382188i
\(357\) 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
\(361\) −3.00000 5.19615i −0.157895 0.273482i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 39.0000 2.04135
\(366\) 0 0
\(367\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(368\) −2.00000 3.46410i −0.104257 0.180579i
\(369\) 18.7350 + 10.8167i 0.975305 + 0.563093i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 3.00000 5.19615i 0.155334 0.269047i −0.777847 0.628454i \(-0.783688\pi\)
0.933181 + 0.359408i \(0.117021\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 18.0278i 0.928477i
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 13.0000 22.5167i 0.666886 1.15508i
\(381\) 0 0
\(382\) 0 0
\(383\) −24.9800 14.4222i −1.27642 0.736940i −0.300230 0.953867i \(-0.597063\pi\)
−0.976188 + 0.216927i \(0.930397\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 13.5000 23.3827i 0.686244 1.18861i
\(388\) −31.2250 + 18.0278i −1.58521 + 0.915221i
\(389\) 5.00000 + 8.66025i 0.253510 + 0.439092i 0.964490 0.264120i \(-0.0850816\pi\)
−0.710980 + 0.703213i \(0.751748\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 46.8375 27.0416i 2.35665 1.36061i
\(396\) 0 0
\(397\) −3.12250 1.80278i −0.156714 0.0904787i 0.419592 0.907713i \(-0.362173\pi\)
−0.576306 + 0.817234i \(0.695506\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −32.0000 −1.60000
\(401\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(402\) 0 0
\(403\) −19.5000 33.7750i −0.971364 1.68245i
\(404\) 0 0
\(405\) 32.4500i 1.61245i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 34.3475 19.8305i 1.69837 0.980557i 0.751069 0.660224i \(-0.229539\pi\)
0.947305 0.320333i \(-0.103795\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −32.5000 + 56.2917i −1.59536 + 2.76325i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) −9.36750 + 5.40833i −0.455463 + 0.262962i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 16.0000 0.773389
\(429\) 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
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3.12250 1.80278i 0.149369 0.0862385i
\(438\) 0 0
\(439\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 20.5000 35.5070i 0.973984 1.68699i 0.290738 0.956803i \(-0.406099\pi\)
0.683247 0.730188i \(-0.260567\pi\)
\(444\) 0 0
\(445\) 6.50000 + 11.2583i 0.308130 + 0.533696i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 19.0000 + 32.9090i 0.893685 + 1.54791i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) −6.24500 3.60555i −0.291175 0.168110i
\(461\) 7.21110i 0.335855i 0.985799 + 0.167927i \(0.0537074\pi\)
−0.985799 + 0.167927i \(0.946293\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 10.0000 17.3205i 0.464238 0.804084i
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(468\) 21.6333i 1.00000i
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 28.8444i 1.32347i
\(476\) 0 0
\(477\) −33.0000 −1.51097
\(478\) 0 0
\(479\) 34.3475 19.8305i 1.56938 0.906080i 0.573135 0.819461i \(-0.305727\pi\)
0.996241 0.0866194i \(-0.0276064\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 22.0000 1.00000
\(485\) −32.5000 + 56.2917i −1.47575 + 2.55607i
\(486\) 0 0
\(487\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −40.0000 −1.80517 −0.902587 0.430507i \(-0.858335\pi\)
−0.902587 + 0.430507i \(0.858335\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 43.2666i 1.94273i
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(500\) −18.7350 + 10.8167i −0.837854 + 0.483735i
\(501\) 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
\(508\) 12.0000 + 20.7846i 0.532414 + 0.922168i
\(509\) −3.12250 1.80278i −0.138402 0.0799066i 0.429200 0.903209i \(-0.358796\pi\)
−0.567602 + 0.823303i \(0.692129\pi\)
\(510\) 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
\(523\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 11.0000 + 19.0526i 0.478261 + 0.828372i
\(530\) 0 0
\(531\) 43.2666i 1.87761i
\(532\) 0 0
\(533\) −26.0000 −1.12619
\(534\) 0 0
\(535\) 24.9800 14.4222i 1.07998 0.623526i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 37.0000 1.58201 0.791003 0.611812i \(-0.209559\pi\)
0.791003 + 0.611812i \(0.209559\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 15.6125 + 9.01388i 0.665115 + 0.384004i
\(552\) 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
\(559\) 32.4500i 1.37249i
\(560\) 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
\(565\) 59.3275 + 34.2527i 2.49593 + 1.44102i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −0.500000 + 0.866025i −0.0209611 + 0.0363057i −0.876316 0.481737i \(-0.840006\pi\)
0.855355 + 0.518043i \(0.173339\pi\)
\(570\) 0 0
\(571\) 1.50000 + 2.59808i 0.0627730 + 0.108726i 0.895704 0.444651i \(-0.146672\pi\)
−0.832931 + 0.553377i \(0.813339\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −8.00000 −0.333623
\(576\) 12.0000 20.7846i 0.500000 0.866025i
\(577\) −31.2250 + 18.0278i −1.29991 + 0.750505i −0.980389 0.197073i \(-0.936856\pi\)
−0.319524 + 0.947578i \(0.603523\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 36.0555i 1.49712i
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −19.5000 33.7750i −0.806226 1.39642i
\(586\) 0 0
\(587\) 18.0278i 0.744085i −0.928216 0.372043i \(-0.878658\pi\)
0.928216 0.372043i \(-0.121342\pi\)
\(588\) 0 0
\(589\) 39.0000 1.60697
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 40.5925 + 23.4361i 1.66693 + 0.962405i 0.969277 + 0.245973i \(0.0791073\pi\)
0.697657 + 0.716432i \(0.254226\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −5.50000 9.52628i −0.224724 0.389233i 0.731513 0.681828i \(-0.238815\pi\)
−0.956237 + 0.292595i \(0.905481\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 34.3475 19.8305i 1.39642 0.806226i
\(606\) 0 0
\(607\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 6.50000 11.2583i 0.262962 0.455463i
\(612\) 0 0
\(613\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 12.4900 7.21110i 0.502015 0.289839i −0.227530 0.973771i \(-0.573065\pi\)
0.729545 + 0.683932i \(0.239732\pi\)
\(620\) −39.0000 67.5500i −1.56628 2.71287i
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0.500000 0.866025i 0.0200000 0.0346410i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 37.4700 + 21.6333i 1.48695 + 0.858492i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 8.50000 + 14.7224i 0.335730 + 0.581501i 0.983625 0.180229i \(-0.0576838\pi\)
−0.647895 + 0.761730i \(0.724350\pi\)
\(642\) 0 0
\(643\) 43.2666i 1.70627i −0.521691 0.853134i \(-0.674699\pi\)
0.521691 0.853134i \(-0.325301\pi\)
\(644\) 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
\(652\) 0 0
\(653\) 17.0000 29.4449i 0.665261 1.15227i −0.313953 0.949439i \(-0.601653\pi\)
0.979214 0.202828i \(-0.0650132\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −24.9800 14.4222i −0.975305 0.563093i
\(657\) 32.4500i 1.26599i
\(658\) 0 0
\(659\) −19.0000 −0.740135 −0.370067 0.929005i \(-0.620665\pi\)
−0.370067 + 0.929005i \(0.620665\pi\)
\(660\) 0 0
\(661\) −9.36750 + 5.40833i −0.364353 + 0.210360i −0.670989 0.741468i \(-0.734130\pi\)
0.306635 + 0.951827i \(0.400797\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 2.50000 4.33013i 0.0968004 0.167663i
\(668\) −31.2250 + 18.0278i −1.20813 + 0.697515i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 51.0000 1.96591 0.982953 0.183858i \(-0.0588587\pi\)
0.982953 + 0.183858i \(0.0588587\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 13.0000 + 22.5167i 0.500000 + 0.866025i
\(677\) 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 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(684\) 18.7350 + 10.8167i 0.716350 + 0.413585i
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −18.0000 + 31.1769i −0.686244 + 1.18861i
\(689\) 34.3475 19.8305i 1.30854 0.755483i
\(690\) 0 0
\(691\) 40.5925 + 23.4361i 1.54421 + 0.891551i 0.998566 + 0.0535342i \(0.0170486\pi\)
0.545645 + 0.838016i \(0.316285\pi\)
\(692\) 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
\(701\) 23.0000 0.868698 0.434349 0.900745i \(-0.356978\pi\)
0.434349 + 0.900745i \(0.356978\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(710\) 0 0
\(711\) 22.5000 + 38.9711i 0.843816 + 1.46153i
\(712\) 0 0
\(713\) 10.8167i 0.405087i
\(714\) 0 0
\(715\) 0 0
\(716\) −25.0000 + 43.3013i −0.934294 + 1.61824i
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(720\) 43.2666i 1.61245i
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −20.0000 34.6410i −0.742781 1.28654i
\(726\) 0 0
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −46.8375 27.0416i −1.72998 0.998806i −0.889422 0.457087i \(-0.848893\pi\)
−0.840560 0.541718i \(-0.817774\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −46.8375 27.0416i −1.71369 0.989402i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 13.5000 23.3827i 0.492622 0.853246i −0.507342 0.861745i \(-0.669372\pi\)
0.999964 + 0.00849853i \(0.00270520\pi\)
\(752\) 12.4900 7.21110i 0.455463 0.262962i
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −47.0000 −1.70824 −0.854122 0.520073i \(-0.825905\pi\)
−0.854122 + 0.520073i \(0.825905\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 40.5925 + 23.4361i 1.47148 + 0.849557i 0.999486 0.0320465i \(-0.0102025\pi\)
0.471990 + 0.881604i \(0.343536\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −40.0000 −1.44715
\(765\) 0 0
\(766\) 0 0
\(767\) 26.0000 + 45.0333i 0.938806 + 1.62606i
\(768\) 0 0
\(769\) 32.4500i 1.17018i 0.810970 + 0.585088i \(0.198940\pi\)
−0.810970 + 0.585088i \(0.801060\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −31.2250 + 18.0278i −1.12308 + 0.648413i −0.942187 0.335089i \(-0.891234\pi\)
−0.180898 + 0.983502i \(0.557900\pi\)
\(774\) 0 0
\(775\) −74.9400 43.2666i −2.69192 1.55418i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 13.0000 22.5167i 0.465773 0.806743i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 34.3475 19.8305i 1.22436 0.706882i 0.258512 0.966008i \(-0.416768\pi\)
0.965844 + 0.259126i \(0.0834344\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 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
\(801\) −9.36750 + 5.40833i −0.330984 + 0.191094i
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0