Properties

Label 384.6.f.b.191.4
Level $384$
Weight $6$
Character 384.191
Analytic conductor $61.587$
Analytic rank $0$
Dimension $4$
CM discriminant -8
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 384.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(61.5873868082\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3}\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 191.4
Root \(-0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 384.191
Dual form 384.6.f.b.191.3

$q$-expansion

\(f(q)\) \(=\) \(q+(15.5563 + 1.00000i) q^{3} +(241.000 + 31.1127i) q^{9} +O(q^{10})\) \(q+(15.5563 + 1.00000i) q^{3} +(241.000 + 31.1127i) q^{9} -474.000i q^{11} -1419.87i q^{17} -1264.31 q^{19} -3125.00 q^{25} +(3717.97 + 725.000i) q^{27} +(474.000 - 7373.71i) q^{33} -16416.2i q^{41} -8918.03 q^{43} +16807.0 q^{49} +(1419.87 - 22088.0i) q^{51} +(-19668.0 - 1264.31i) q^{57} -48486.0i q^{59} +29774.9 q^{67} -50402.0 q^{73} +(-48613.6 - 3125.00i) q^{75} +(57113.0 + 14996.3i) q^{81} -89298.0i q^{83} +149279. i q^{89} +85450.0 q^{97} +(14747.4 - 114234. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 964q^{9} + O(q^{10}) \) \( 4q + 964q^{9} - 12500q^{25} + 1896q^{33} + 67228q^{49} - 78672q^{57} - 201608q^{73} + 228452q^{81} + 341800q^{97} + O(q^{100}) \)

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
\(3\) 15.5563 + 1.00000i 0.997940 + 0.0641500i
\(4\) 0 0
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 241.000 + 31.1127i 0.991770 + 0.128036i
\(10\) 0 0
\(11\) 474.000i 1.18113i −0.806991 0.590564i \(-0.798906\pi\)
0.806991 0.590564i \(-0.201094\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1419.87i 1.19159i −0.803137 0.595794i \(-0.796837\pi\)
0.803137 0.595794i \(-0.203163\pi\)
\(18\) 0 0
\(19\) −1264.31 −0.803468 −0.401734 0.915756i \(-0.631592\pi\)
−0.401734 + 0.915756i \(0.631592\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −3125.00 −1.00000
\(26\) 0 0
\(27\) 3717.97 + 725.000i 0.981513 + 0.191394i
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 474.000 7373.71i 0.0757693 1.17869i
\(34\) 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
\(41\) 16416.2i 1.52515i −0.646899 0.762575i \(-0.723935\pi\)
0.646899 0.762575i \(-0.276065\pi\)
\(42\) 0 0
\(43\) −8918.03 −0.735526 −0.367763 0.929920i \(-0.619876\pi\)
−0.367763 + 0.929920i \(0.619876\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 16807.0 1.00000
\(50\) 0 0
\(51\) 1419.87 22088.0i 0.0764405 1.18913i
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −19668.0 1264.31i −0.801813 0.0515425i
\(58\) 0 0
\(59\) 48486.0i 1.81337i −0.421809 0.906685i \(-0.638605\pi\)
0.421809 0.906685i \(-0.361395\pi\)
\(60\) 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
\(67\) 29774.9 0.810331 0.405166 0.914243i \(-0.367214\pi\)
0.405166 + 0.914243i \(0.367214\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −50402.0 −1.10698 −0.553491 0.832855i \(-0.686705\pi\)
−0.553491 + 0.832855i \(0.686705\pi\)
\(74\) 0 0
\(75\) −48613.6 3125.00i −0.997940 0.0641500i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 57113.0 + 14996.3i 0.967214 + 0.253964i
\(82\) 0 0
\(83\) 89298.0i 1.42281i −0.702783 0.711404i \(-0.748059\pi\)
0.702783 0.711404i \(-0.251941\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 149279.i 1.99767i 0.0482961 + 0.998833i \(0.484621\pi\)
−0.0482961 + 0.998833i \(0.515379\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 85450.0 0.922110 0.461055 0.887372i \(-0.347471\pi\)
0.461055 + 0.887372i \(0.347471\pi\)
\(98\) 0 0
\(99\) 14747.4 114234.i 0.151227 1.17141i
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 235686.i 1.99010i −0.0993883 0.995049i \(-0.531689\pi\)
0.0993883 0.995049i \(-0.468311\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 89480.1i 0.659220i 0.944117 + 0.329610i \(0.106917\pi\)
−0.944117 + 0.329610i \(0.893083\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −63625.0 −0.395061
\(122\) 0 0
\(123\) 16416.2 255376.i 0.0978385 1.52201i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) −138732. 8918.03i −0.734011 0.0471840i
\(130\) 0 0
\(131\) 385902.i 1.96471i −0.187022 0.982356i \(-0.559884\pi\)
0.187022 0.982356i \(-0.440116\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 119224.i 0.542702i −0.962480 0.271351i \(-0.912530\pi\)
0.962480 0.271351i \(-0.0874705\pi\)
\(138\) 0 0
\(139\) −439105. −1.92766 −0.963832 0.266512i \(-0.914129\pi\)
−0.963832 + 0.266512i \(0.914129\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 261456. + 16807.0i 0.997940 + 0.0641500i
\(148\) 0 0
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 44176.0 342189.i 0.152566 1.18178i
\(154\) 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
\(163\) 626952. 1.84827 0.924135 0.382067i \(-0.124788\pi\)
0.924135 + 0.382067i \(0.124788\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 371293. 1.00000
\(170\) 0 0
\(171\) −304698. 39336.0i −0.796855 0.102873i
\(172\) 0 0
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 48486.0 754265.i 0.116328 1.80963i
\(178\) 0 0
\(179\) 446382.i 1.04130i −0.853772 0.520648i \(-0.825690\pi\)
0.853772 0.520648i \(-0.174310\pi\)
\(180\) 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
\(187\) −673019. −1.40742
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) −1.02430e6 −1.97940 −0.989699 0.143165i \(-0.954272\pi\)
−0.989699 + 0.143165i \(0.954272\pi\)
\(194\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 463188. + 29774.9i 0.808662 + 0.0519828i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 599281.i 0.948998i
\(210\) 0 0
\(211\) 1.04702e6 1.61901 0.809507 0.587110i \(-0.199734\pi\)
0.809507 + 0.587110i \(0.199734\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −784071. 50402.0i −1.10470 0.0710129i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) −753125. 97227.2i −0.991770 0.128036i
\(226\) 0 0
\(227\) 1.38435e6i 1.78312i 0.452900 + 0.891561i \(0.350390\pi\)
−0.452900 + 0.891561i \(0.649610\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1.33043e6i 1.60547i 0.596336 + 0.802735i \(0.296623\pi\)
−0.596336 + 0.802735i \(0.703377\pi\)
\(234\) 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
\(241\) 1.74717e6 1.93773 0.968866 0.247587i \(-0.0796376\pi\)
0.968866 + 0.247587i \(0.0796376\pi\)
\(242\) 0 0
\(243\) 873473. + 290401.i 0.948930 + 0.315488i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 89298.0 1.38915e6i 0.0912732 1.41988i
\(250\) 0 0
\(251\) 1.62673e6i 1.62978i 0.579613 + 0.814892i \(0.303204\pi\)
−0.579613 + 0.814892i \(0.696796\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2.06078e6i 1.94625i 0.230274 + 0.973126i \(0.426038\pi\)
−0.230274 + 0.973126i \(0.573962\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −149279. + 2.32223e6i −0.128150 + 1.99355i
\(268\) 0 0
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.48125e6i 1.18113i
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 2.52282e6i 1.90599i 0.302992 + 0.952993i \(0.402014\pi\)
−0.302992 + 0.952993i \(0.597986\pi\)
\(282\) 0 0
\(283\) −2.45830e6 −1.82460 −0.912301 0.409520i \(-0.865696\pi\)
−0.912301 + 0.409520i \(0.865696\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −596175. −0.419884
\(290\) 0 0
\(291\) 1.32929e6 + 85450.0i 0.920211 + 0.0591534i
\(292\) 0 0
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 343650. 1.76232e6i 0.226061 1.15929i
\(298\) 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
\(307\) 3.10484e6 1.88015 0.940077 0.340962i \(-0.110753\pi\)
0.940077 + 0.340962i \(0.110753\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −3.43345e6 −1.98093 −0.990467 0.137752i \(-0.956012\pi\)
−0.990467 + 0.137752i \(0.956012\pi\)
\(314\) 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
\(321\) 235686. 3.66641e6i 0.127665 1.98600i
\(322\) 0 0
\(323\) 1.79515e6i 0.957403i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −2.66602e6 −1.33750 −0.668748 0.743489i \(-0.733170\pi\)
−0.668748 + 0.743489i \(0.733170\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 3.86401e6 1.85338 0.926689 0.375829i \(-0.122642\pi\)
0.926689 + 0.375829i \(0.122642\pi\)
\(338\) 0 0
\(339\) −89480.1 + 1.39198e6i −0.0422890 + 0.657862i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 3.24529e6i 1.44687i 0.690393 + 0.723435i \(0.257438\pi\)
−0.690393 + 0.723435i \(0.742562\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 418404.i 0.178714i 0.996000 + 0.0893570i \(0.0284812\pi\)
−0.996000 + 0.0893570i \(0.971519\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −877627. −0.354439
\(362\) 0 0
\(363\) −989773. 63625.0i −0.394247 0.0253432i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) 510752. 3.95630e6i 0.195274 1.51260i
\(370\) 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
\(379\) 4.98043e6 1.78102 0.890510 0.454964i \(-0.150348\pi\)
0.890510 + 0.454964i \(0.150348\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −2.14925e6 277464.i −0.729472 0.0941736i
\(388\) 0 0
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 385902. 6.00323e6i 0.126036 1.96066i
\(394\) 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
\(401\) 4.70380e6i 1.46079i 0.683024 + 0.730396i \(0.260664\pi\)
−0.683024 + 0.730396i \(0.739336\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −1.77938e6 −0.525970 −0.262985 0.964800i \(-0.584707\pi\)
−0.262985 + 0.964800i \(0.584707\pi\)
\(410\) 0 0
\(411\) 119224. 1.85469e6i 0.0348144 0.541585i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −6.83087e6 439105.i −1.92369 0.123660i
\(418\) 0 0
\(419\) 5.47202e6i 1.52269i −0.648345 0.761347i \(-0.724539\pi\)
0.648345 0.761347i \(-0.275461\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 4.43710e6i 1.19159i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 3.93980e6 1.00984 0.504922 0.863165i \(-0.331521\pi\)
0.504922 + 0.863165i \(0.331521\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 4.05049e6 + 522911.i 0.991770 + 0.128036i
\(442\) 0 0
\(443\) 7.79860e6i 1.88802i −0.329912 0.944012i \(-0.607019\pi\)
0.329912 0.944012i \(-0.392981\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 5.30154e6i 1.24104i 0.784190 + 0.620521i \(0.213079\pi\)
−0.784190 + 0.620521i \(0.786921\pi\)
\(450\) 0 0
\(451\) −7.78127e6 −1.80140
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −1.12941e6 −0.252966 −0.126483 0.991969i \(-0.540369\pi\)
−0.126483 + 0.991969i \(0.540369\pi\)
\(458\) 0 0
\(459\) 1.02941e6 5.27903e6i 0.228063 1.16956i
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 6.03165e6i 1.27981i −0.768456 0.639903i \(-0.778975\pi\)
0.768456 0.639903i \(-0.221025\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 4.22715e6i 0.868749i
\(474\) 0 0
\(475\) 3.95096e6 0.803468
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 9.75308e6 + 626952.i 1.84446 + 0.118567i
\(490\) 0 0
\(491\) 567798.i 0.106289i −0.998587 0.0531447i \(-0.983076\pi\)
0.998587 0.0531447i \(-0.0169245\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −254346. −0.0457271 −0.0228636 0.999739i \(-0.507278\pi\)
−0.0228636 + 0.999739i \(0.507278\pi\)
\(500\) 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
\(507\) 5.77596e6 + 371293.i 0.997940 + 0.0641500i
\(508\) 0 0
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −4.70065e6 916623.i −0.788614 0.153779i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 9.79627e6i 1.58113i 0.612381 + 0.790563i \(0.290212\pi\)
−0.612381 + 0.790563i \(0.709788\pi\)
\(522\) 0 0
\(523\) −2.36813e6 −0.378575 −0.189287 0.981922i \(-0.560618\pi\)
−0.189287 + 0.981922i \(0.560618\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −6.43634e6 −1.00000
\(530\) 0 0
\(531\) 1.50853e6 1.16851e7i 0.232176 1.79844i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 446382. 6.94407e6i 0.0667992 1.03915i
\(538\) 0 0
\(539\) 7.96652e6i 1.18113i
\(540\) 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
\(547\) −1.10674e7 −1.58153 −0.790767 0.612118i \(-0.790318\pi\)
−0.790767 + 0.612118i \(0.790318\pi\)
\(548\) 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
\(561\) −1.04697e7 673019.i −1.40452 0.0902859i
\(562\) 0 0
\(563\) 4.15965e6i 0.553077i −0.961003 0.276539i \(-0.910813\pi\)
0.961003 0.276539i \(-0.0891874\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 9.60102e6i 1.24319i −0.783340 0.621594i \(-0.786486\pi\)
0.783340 0.621594i \(-0.213514\pi\)
\(570\) 0 0
\(571\) 1.14040e7 1.46375 0.731875 0.681438i \(-0.238645\pi\)
0.731875 + 0.681438i \(0.238645\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 1.13587e7 1.42033 0.710164 0.704036i \(-0.248621\pi\)
0.710164 + 0.704036i \(0.248621\pi\)
\(578\) 0 0
\(579\) −1.59343e7 1.02430e6i −1.97532 0.126978i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 9.71089e6i 1.16322i −0.813466 0.581612i \(-0.802422\pi\)
0.813466 0.581612i \(-0.197578\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 5.42022e6i 0.632965i −0.948598 0.316483i \(-0.897498\pi\)
0.948598 0.316483i \(-0.102502\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) −3.45547e6 −0.390231 −0.195115 0.980780i \(-0.562508\pi\)
−0.195115 + 0.980780i \(0.562508\pi\)
\(602\) 0 0
\(603\) 7.17574e6 + 926376.i 0.803662 + 0.103751i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 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
\(617\) 1.88174e7i 1.98997i −0.100023 0.994985i \(-0.531891\pi\)
0.100023 0.994985i \(-0.468109\pi\)
\(618\) 0 0
\(619\) 1.76152e7 1.84783 0.923914 0.382599i \(-0.124971\pi\)
0.923914 + 0.382599i \(0.124971\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 9.76562e6 1.00000
\(626\) 0 0
\(627\) −599281. + 9.32263e6i −0.0608782 + 0.947043i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 1.62879e7 + 1.04702e6i 1.61568 + 0.103860i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 3.68841e6i 0.354564i −0.984160 0.177282i \(-0.943270\pi\)
0.984160 0.177282i \(-0.0567304\pi\)
\(642\) 0 0
\(643\) −1.56210e7 −1.48998 −0.744990 0.667075i \(-0.767546\pi\)
−0.744990 + 0.667075i \(0.767546\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) −2.29824e7 −2.14182
\(650\) 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
\(657\) −1.21469e7 1.56814e6i −1.09787 0.141733i
\(658\) 0 0
\(659\) 2.17584e7i 1.95170i 0.218435 + 0.975852i \(0.429905\pi\)
−0.218435 + 0.975852i \(0.570095\pi\)
\(660\) 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
\(673\) 1.93814e7 1.64949 0.824743 0.565508i \(-0.191320\pi\)
0.824743 + 0.565508i \(0.191320\pi\)
\(674\) 0 0
\(675\) −1.16186e7 2.26562e6i −0.981513 0.191394i
\(676\) 0 0
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −1.38435e6 + 2.15354e7i −0.114387 + 1.77945i
\(682\) 0 0
\(683\) 2.43566e7i 1.99786i 0.0462568 + 0.998930i \(0.485271\pi\)
−0.0462568 + 0.998930i \(0.514729\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 1.44709e7 1.15292 0.576460 0.817125i \(-0.304434\pi\)
0.576460 + 0.817125i \(0.304434\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −2.33089e7 −1.81735
\(698\) 0 0
\(699\) −1.33043e6 + 2.06966e7i −0.102991 + 1.60216i
\(700\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 2.71796e7 + 1.74717e6i 1.93374 + 0.124306i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 1.32977e7 + 5.39105e6i 0.926737 + 0.375712i
\(730\) 0 0
\(731\) 1.26624e7i 0.876444i
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.41133e7i 0.957104i
\(738\) 0 0
\(739\) −2.88667e7 −1.94440 −0.972201 0.234146i \(-0.924771\pi\)
−0.972201 + 0.234146i \(0.924771\pi\)
\(740\) 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
\(747\) 2.77830e6 2.15208e7i 0.182170 1.41110i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) −1.62673e6 + 2.53059e7i −0.104551 + 1.62643i
\(754\) 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
\(761\) 2.74352e7i 1.71730i 0.512563 + 0.858650i \(0.328696\pi\)
−0.512563 + 0.858650i \(0.671304\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 2.92618e7 1.78437 0.892185 0.451670i \(-0.149172\pi\)
0.892185 + 0.451670i \(0.149172\pi\)
\(770\) 0 0
\(771\) −2.06078e6 + 3.20582e7i −0.124852 + 1.94224i
\(772\) 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
\(779\) 2.07551e7i 1.22541i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 2.46125e7 1.41651 0.708253 0.705959i \(-0.249484\pi\)
0.708253 + 0.705959i \(0.249484\pi\)
\(788\) 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
\(801\) −4.64446e6 + 3.59762e7i −0.255773 + 1.98122i
\(802\) 0 0
\(803\) 2.38905e7i 1.30749i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0