Properties

Label 384.3.h.b.65.2
Level $384$
Weight $3$
Character 384.65
Analytic conductor $10.463$
Analytic rank $0$
Dimension $2$
CM discriminant -8
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 384.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.4632421514\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 65.2
Root \(1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 384.65
Dual form 384.3.h.b.65.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-1.00000 + 2.82843i) q^{3} +(-7.00000 - 5.65685i) q^{9} +O(q^{10})\) \(q+(-1.00000 + 2.82843i) q^{3} +(-7.00000 - 5.65685i) q^{9} -14.0000 q^{11} -33.9411i q^{17} +16.9706i q^{19} -25.0000 q^{25} +(23.0000 - 14.1421i) q^{27} +(14.0000 - 39.5980i) q^{33} -67.8823i q^{41} -84.8528i q^{43} -49.0000 q^{49} +(96.0000 + 33.9411i) q^{51} +(-48.0000 - 16.9706i) q^{57} -82.0000 q^{59} +118.794i q^{67} +142.000 q^{73} +(25.0000 - 70.7107i) q^{75} +(17.0000 + 79.1960i) q^{81} -158.000 q^{83} +101.823i q^{89} -94.0000 q^{97} +(98.0000 + 79.1960i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} - 14q^{9} + O(q^{10}) \) \( 2q - 2q^{3} - 14q^{9} - 28q^{11} - 50q^{25} + 46q^{27} + 28q^{33} - 98q^{49} + 192q^{51} - 96q^{57} - 164q^{59} + 284q^{73} + 50q^{75} + 34q^{81} - 316q^{83} - 188q^{97} + 196q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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
\(3\) −1.00000 + 2.82843i −0.333333 + 0.942809i
\(4\) 0 0
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) −7.00000 5.65685i −0.777778 0.628539i
\(10\) 0 0
\(11\) −14.0000 −1.27273 −0.636364 0.771389i \(-0.719562\pi\)
−0.636364 + 0.771389i \(0.719562\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 33.9411i 1.99654i −0.0588235 0.998268i \(-0.518735\pi\)
0.0588235 0.998268i \(-0.481265\pi\)
\(18\) 0 0
\(19\) 16.9706i 0.893188i 0.894737 + 0.446594i \(0.147363\pi\)
−0.894737 + 0.446594i \(0.852637\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) −25.0000 −1.00000
\(26\) 0 0
\(27\) 23.0000 14.1421i 0.851852 0.523783i
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 14.0000 39.5980i 0.424242 1.19994i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 67.8823i 1.65566i −0.560976 0.827832i \(-0.689574\pi\)
0.560976 0.827832i \(-0.310426\pi\)
\(42\) 0 0
\(43\) 84.8528i 1.97332i −0.162791 0.986661i \(-0.552050\pi\)
0.162791 0.986661i \(-0.447950\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) −49.0000 −1.00000
\(50\) 0 0
\(51\) 96.0000 + 33.9411i 1.88235 + 0.665512i
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −48.0000 16.9706i −0.842105 0.297729i
\(58\) 0 0
\(59\) −82.0000 −1.38983 −0.694915 0.719092i \(-0.744558\pi\)
−0.694915 + 0.719092i \(0.744558\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 118.794i 1.77304i 0.462687 + 0.886522i \(0.346886\pi\)
−0.462687 + 0.886522i \(0.653114\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 142.000 1.94521 0.972603 0.232473i \(-0.0746819\pi\)
0.972603 + 0.232473i \(0.0746819\pi\)
\(74\) 0 0
\(75\) 25.0000 70.7107i 0.333333 0.942809i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 17.0000 + 79.1960i 0.209877 + 0.977728i
\(82\) 0 0
\(83\) −158.000 −1.90361 −0.951807 0.306697i \(-0.900776\pi\)
−0.951807 + 0.306697i \(0.900776\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 101.823i 1.14408i 0.820225 + 0.572041i \(0.193848\pi\)
−0.820225 + 0.572041i \(0.806152\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −94.0000 −0.969072 −0.484536 0.874771i \(-0.661012\pi\)
−0.484536 + 0.874771i \(0.661012\pi\)
\(98\) 0 0
\(99\) 98.0000 + 79.1960i 0.989899 + 0.799959i
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −178.000 −1.66355 −0.831776 0.555112i \(-0.812675\pi\)
−0.831776 + 0.555112i \(0.812675\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 203.647i 1.80218i 0.433628 + 0.901092i \(0.357233\pi\)
−0.433628 + 0.901092i \(0.642767\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 75.0000 0.619835
\(122\) 0 0
\(123\) 192.000 + 67.8823i 1.56098 + 0.551888i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) 240.000 + 84.8528i 1.86047 + 0.657774i
\(130\) 0 0
\(131\) 62.0000 0.473282 0.236641 0.971597i \(-0.423953\pi\)
0.236641 + 0.971597i \(0.423953\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 135.765i 0.990982i 0.868613 + 0.495491i \(0.165012\pi\)
−0.868613 + 0.495491i \(0.834988\pi\)
\(138\) 0 0
\(139\) 186.676i 1.34299i −0.741007 0.671497i \(-0.765652\pi\)
0.741007 0.671497i \(-0.234348\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 49.0000 138.593i 0.333333 0.942809i
\(148\) 0 0
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) −192.000 + 237.588i −1.25490 + 1.55286i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 50.9117i 0.312342i −0.987730 0.156171i \(-0.950085\pi\)
0.987730 0.156171i \(-0.0499150\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 169.000 1.00000
\(170\) 0 0
\(171\) 96.0000 118.794i 0.561404 0.694701i
\(172\) 0 0
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 82.0000 231.931i 0.463277 1.31034i
\(178\) 0 0
\(179\) −34.0000 −0.189944 −0.0949721 0.995480i \(-0.530276\pi\)
−0.0949721 + 0.995480i \(0.530276\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 475.176i 2.54105i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) −98.0000 −0.507772 −0.253886 0.967234i \(-0.581709\pi\)
−0.253886 + 0.967234i \(0.581709\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) −336.000 118.794i −1.67164 0.591015i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 237.588i 1.13678i
\(210\) 0 0
\(211\) 356.382i 1.68901i −0.535545 0.844507i \(-0.679894\pi\)
0.535545 0.844507i \(-0.320106\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −142.000 + 401.637i −0.648402 + 1.83396i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 175.000 + 141.421i 0.777778 + 0.628539i
\(226\) 0 0
\(227\) −446.000 −1.96476 −0.982379 0.186900i \(-0.940156\pi\)
−0.982379 + 0.186900i \(0.940156\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 169.706i 0.728350i −0.931330 0.364175i \(-0.881351\pi\)
0.931330 0.364175i \(-0.118649\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −194.000 −0.804979 −0.402490 0.915425i \(-0.631855\pi\)
−0.402490 + 0.915425i \(0.631855\pi\)
\(242\) 0 0
\(243\) −241.000 31.1127i −0.991770 0.128036i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 158.000 446.891i 0.634538 1.79474i
\(250\) 0 0
\(251\) 466.000 1.85657 0.928287 0.371865i \(-0.121282\pi\)
0.928287 + 0.371865i \(0.121282\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 339.411i 1.32067i −0.750973 0.660333i \(-0.770415\pi\)
0.750973 0.660333i \(-0.229585\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −288.000 101.823i −1.07865 0.381361i
\(268\) 0 0
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 350.000 1.27273
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 509.117i 1.81180i −0.423488 0.905902i \(-0.639194\pi\)
0.423488 0.905902i \(-0.360806\pi\)
\(282\) 0 0
\(283\) 560.029i 1.97890i 0.144876 + 0.989450i \(0.453722\pi\)
−0.144876 + 0.989450i \(0.546278\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −863.000 −2.98616
\(290\) 0 0
\(291\) 94.0000 265.872i 0.323024 0.913650i
\(292\) 0 0
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −322.000 + 197.990i −1.08418 + 0.666633i
\(298\) 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
\(307\) 288.500i 0.939738i −0.882736 0.469869i \(-0.844301\pi\)
0.882736 0.469869i \(-0.155699\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 526.000 1.68051 0.840256 0.542191i \(-0.182405\pi\)
0.840256 + 0.542191i \(0.182405\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 178.000 503.460i 0.554517 1.56841i
\(322\) 0 0
\(323\) 576.000 1.78328
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 661.852i 1.99955i −0.0211480 0.999776i \(-0.506732\pi\)
0.0211480 0.999776i \(-0.493268\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 478.000 1.41840 0.709199 0.705009i \(-0.249057\pi\)
0.709199 + 0.705009i \(0.249057\pi\)
\(338\) 0 0
\(339\) −576.000 203.647i −1.69912 0.600728i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 658.000 1.89625 0.948127 0.317892i \(-0.102975\pi\)
0.948127 + 0.317892i \(0.102975\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 678.823i 1.92301i 0.274788 + 0.961505i \(0.411392\pi\)
−0.274788 + 0.961505i \(0.588608\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 73.0000 0.202216
\(362\) 0 0
\(363\) −75.0000 + 212.132i −0.206612 + 0.584386i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) −384.000 + 475.176i −1.04065 + 1.28774i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 322.441i 0.850767i 0.905013 + 0.425383i \(0.139861\pi\)
−0.905013 + 0.425383i \(0.860139\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −480.000 + 593.970i −1.24031 + 1.53481i
\(388\) 0 0
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −62.0000 + 175.362i −0.157761 + 0.446215i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 237.588i 0.592488i −0.955112 0.296244i \(-0.904266\pi\)
0.955112 0.296244i \(-0.0957342\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −334.000 −0.816626 −0.408313 0.912842i \(-0.633883\pi\)
−0.408313 + 0.912842i \(0.633883\pi\)
\(410\) 0 0
\(411\) −384.000 135.765i −0.934307 0.330327i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 528.000 + 186.676i 1.26619 + 0.447665i
\(418\) 0 0
\(419\) 514.000 1.22673 0.613365 0.789799i \(-0.289815\pi\)
0.613365 + 0.789799i \(0.289815\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 848.528i 1.99654i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 578.000 1.33487 0.667436 0.744667i \(-0.267392\pi\)
0.667436 + 0.744667i \(0.267392\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 343.000 + 277.186i 0.777778 + 0.628539i
\(442\) 0 0
\(443\) −878.000 −1.98194 −0.990971 0.134079i \(-0.957192\pi\)
−0.990971 + 0.134079i \(0.957192\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 237.588i 0.529149i 0.964365 + 0.264574i \(0.0852315\pi\)
−0.964365 + 0.264574i \(0.914769\pi\)
\(450\) 0 0
\(451\) 950.352i 2.10721i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −238.000 −0.520788 −0.260394 0.965502i \(-0.583852\pi\)
−0.260394 + 0.965502i \(0.583852\pi\)
\(458\) 0 0
\(459\) −480.000 780.646i −1.04575 1.70075i
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 34.0000 0.0728051 0.0364026 0.999337i \(-0.488410\pi\)
0.0364026 + 0.999337i \(0.488410\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1187.94i 2.51150i
\(474\) 0 0
\(475\) 424.264i 0.893188i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) 0 0
\(489\) 144.000 + 50.9117i 0.294479 + 0.104114i
\(490\) 0 0
\(491\) 782.000 1.59267 0.796334 0.604857i \(-0.206770\pi\)
0.796334 + 0.604857i \(0.206770\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 593.970i 1.19032i 0.803607 + 0.595160i \(0.202911\pi\)
−0.803607 + 0.595160i \(0.797089\pi\)
\(500\) 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
\(507\) −169.000 + 478.004i −0.333333 + 0.942809i
\(508\) 0 0
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 240.000 + 390.323i 0.467836 + 0.760863i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 271.529i 0.521169i −0.965451 0.260584i \(-0.916085\pi\)
0.965451 0.260584i \(-0.0839152\pi\)
\(522\) 0 0
\(523\) 967.322i 1.84956i −0.380497 0.924782i \(-0.624247\pi\)
0.380497 0.924782i \(-0.375753\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 529.000 1.00000
\(530\) 0 0
\(531\) 574.000 + 463.862i 1.08098 + 0.873563i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 34.0000 96.1665i 0.0633147 0.179081i
\(538\) 0 0
\(539\) 686.000 1.27273
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 390.323i 0.713570i −0.934186 0.356785i \(-0.883873\pi\)
0.934186 0.356785i \(-0.116127\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −1344.00 475.176i −2.39572 0.847016i
\(562\) 0 0
\(563\) 226.000 0.401421 0.200710 0.979651i \(-0.435675\pi\)
0.200710 + 0.979651i \(0.435675\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 950.352i 1.67021i −0.550088 0.835107i \(-0.685406\pi\)
0.550088 0.835107i \(-0.314594\pi\)
\(570\) 0 0
\(571\) 933.381i 1.63464i −0.576182 0.817321i \(-0.695458\pi\)
0.576182 0.817321i \(-0.304542\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −2.00000 −0.00346620 −0.00173310 0.999998i \(-0.500552\pi\)
−0.00173310 + 0.999998i \(0.500552\pi\)
\(578\) 0 0
\(579\) 98.0000 277.186i 0.169257 0.478732i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −1138.00 −1.93867 −0.969336 0.245741i \(-0.920969\pi\)
−0.969336 + 0.245741i \(0.920969\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 814.587i 1.37367i 0.726813 + 0.686836i \(0.241001\pi\)
−0.726813 + 0.686836i \(0.758999\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) −914.000 −1.52080 −0.760399 0.649456i \(-0.774997\pi\)
−0.760399 + 0.649456i \(0.774997\pi\)
\(602\) 0 0
\(603\) 672.000 831.558i 1.11443 1.37903i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1187.94i 1.92535i −0.270665 0.962674i \(-0.587243\pi\)
0.270665 0.962674i \(-0.412757\pi\)
\(618\) 0 0
\(619\) 1103.09i 1.78205i 0.453958 + 0.891023i \(0.350012\pi\)
−0.453958 + 0.891023i \(0.649988\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 625.000 1.00000
\(626\) 0 0
\(627\) 672.000 + 237.588i 1.07177 + 0.378928i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) 1008.00 + 356.382i 1.59242 + 0.563004i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1187.94i 1.85326i −0.375975 0.926630i \(-0.622692\pi\)
0.375975 0.926630i \(-0.377308\pi\)
\(642\) 0 0
\(643\) 424.264i 0.659820i 0.944012 + 0.329910i \(0.107018\pi\)
−0.944012 + 0.329910i \(0.892982\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 1148.00 1.76888
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −994.000 803.273i −1.51294 1.22264i
\(658\) 0 0
\(659\) −994.000 −1.50835 −0.754173 0.656676i \(-0.771962\pi\)
−0.754173 + 0.656676i \(0.771962\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 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
\(673\) −1246.00 −1.85141 −0.925706 0.378244i \(-0.876528\pi\)
−0.925706 + 0.378244i \(0.876528\pi\)
\(674\) 0 0
\(675\) −575.000 + 353.553i −0.851852 + 0.523783i
\(676\) 0 0
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 446.000 1261.48i 0.654919 1.85239i
\(682\) 0 0
\(683\) −398.000 −0.582723 −0.291362 0.956613i \(-0.594108\pi\)
−0.291362 + 0.956613i \(0.594108\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 1170.97i 1.69460i 0.531114 + 0.847300i \(0.321773\pi\)
−0.531114 + 0.847300i \(0.678227\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −2304.00 −3.30560
\(698\) 0 0
\(699\) 480.000 + 169.706i 0.686695 + 0.242783i
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 194.000 548.715i 0.268326 0.758942i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) 329.000 650.538i 0.451303 0.892371i
\(730\) 0 0
\(731\) −2880.00 −3.93981
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1663.12i 2.25660i
\(738\) 0 0
\(739\) 1442.50i 1.95196i −0.217862 0.975980i \(-0.569908\pi\)
0.217862 0.975980i \(-0.430092\pi\)
\(740\) 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
\(747\) 1106.00 + 893.783i 1.48059 + 1.19650i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) −466.000 + 1318.05i −0.618858 + 1.75039i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 610.940i 0.802812i −0.915900 0.401406i \(-0.868522\pi\)
0.915900 0.401406i \(-0.131478\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 1054.00 1.37061 0.685306 0.728256i \(-0.259669\pi\)
0.685306 + 0.728256i \(0.259669\pi\)
\(770\) 0 0
\(771\) 960.000 + 339.411i 1.24514 + 0.440222i
\(772\) 0 0
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1152.00 1.47882
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 1272.79i 1.61727i −0.588310 0.808635i \(-0.700207\pi\)
0.588310 0.808635i \(-0.299793\pi\)
\(788\) 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
\(801\) 576.000 712.764i 0.719101 0.889842i
\(802\) 0 0
\(803\) −1988.00 −2.47572
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 339.411i 0.419544i 0.977750 + 0.209772i \(0.0672722\pi\)
−0.977750 + 0.209772i \(0.932728\pi\)
\(810\) 0 0
\(811\) 1612.20i