Properties

Label 585.1.o.a.298.2
Level $585$
Weight $1$
Character 585.298
Analytic conductor $0.292$
Analytic rank $0$
Dimension $8$
Projective image $D_{8}$
CM discriminant -39
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 585.o (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.291953032390\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{16})\)
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{8}\)
Projective field: Galois closure of 8.2.12049171875.2

Embedding invariants

Embedding label 298.2
Root \(-0.923880 - 0.382683i\) of defining polynomial
Character \(\chi\) \(=\) 585.298
Dual form 585.1.o.a.532.2

$q$-expansion

\(f(q)\) \(=\) \(q+(-0.541196 - 0.541196i) q^{2} -0.414214i q^{4} +(-0.382683 - 0.923880i) q^{5} +(-0.765367 + 0.765367i) q^{8} +O(q^{10})\) \(q+(-0.541196 - 0.541196i) q^{2} -0.414214i q^{4} +(-0.382683 - 0.923880i) q^{5} +(-0.765367 + 0.765367i) q^{8} +(-0.292893 + 0.707107i) q^{10} -0.765367i q^{11} +(-0.707107 - 0.707107i) q^{13} +0.414214 q^{16} +(-0.382683 + 0.158513i) q^{20} +(-0.414214 + 0.414214i) q^{22} +(-0.707107 + 0.707107i) q^{25} +0.765367i q^{26} +(0.541196 + 0.541196i) q^{32} +(1.00000 + 0.414214i) q^{40} -1.84776i q^{41} +(-1.00000 - 1.00000i) q^{43} -0.317025 q^{44} +(1.30656 + 1.30656i) q^{47} -1.00000i q^{49} +0.765367 q^{50} +(-0.292893 + 0.292893i) q^{52} +(-0.707107 + 0.292893i) q^{55} +1.84776 q^{59} +1.41421 q^{61} -1.00000i q^{64} +(-0.382683 + 0.923880i) q^{65} +1.84776i q^{71} +1.41421i q^{79} +(-0.158513 - 0.382683i) q^{80} +(-1.00000 + 1.00000i) q^{82} +(1.30656 - 1.30656i) q^{83} +1.08239i q^{86} +(0.585786 + 0.585786i) q^{88} -0.765367 q^{89} -1.41421i q^{94} +(-0.541196 + 0.541196i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{10} - 8 q^{16} + 8 q^{22} + 8 q^{40} - 8 q^{43} - 8 q^{52} - 8 q^{82} + 16 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/585\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\) \(496\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{4}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.541196 0.541196i −0.541196 0.541196i 0.382683 0.923880i \(-0.375000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(3\) 0 0
\(4\) 0.414214i 0.414214i
\(5\) −0.382683 0.923880i −0.382683 0.923880i
\(6\) 0 0
\(7\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(8\) −0.765367 + 0.765367i −0.765367 + 0.765367i
\(9\) 0 0
\(10\) −0.292893 + 0.707107i −0.292893 + 0.707107i
\(11\) 0.765367i 0.765367i −0.923880 0.382683i \(-0.875000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(12\) 0 0
\(13\) −0.707107 0.707107i −0.707107 0.707107i
\(14\) 0 0
\(15\) 0 0
\(16\) 0.414214 0.414214
\(17\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −0.382683 + 0.158513i −0.382683 + 0.158513i
\(21\) 0 0
\(22\) −0.414214 + 0.414214i −0.414214 + 0.414214i
\(23\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(24\) 0 0
\(25\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(26\) 0.765367i 0.765367i
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0.541196 + 0.541196i 0.541196 + 0.541196i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 1.00000 + 0.414214i 1.00000 + 0.414214i
\(41\) 1.84776i 1.84776i −0.382683 0.923880i \(-0.625000\pi\)
0.382683 0.923880i \(-0.375000\pi\)
\(42\) 0 0
\(43\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(44\) −0.317025 −0.317025
\(45\) 0 0
\(46\) 0 0
\(47\) 1.30656 + 1.30656i 1.30656 + 1.30656i 0.923880 + 0.382683i \(0.125000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(48\) 0 0
\(49\) 1.00000i 1.00000i
\(50\) 0.765367 0.765367
\(51\) 0 0
\(52\) −0.292893 + 0.292893i −0.292893 + 0.292893i
\(53\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(54\) 0 0
\(55\) −0.707107 + 0.292893i −0.707107 + 0.292893i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.84776 1.84776 0.923880 0.382683i \(-0.125000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(60\) 0 0
\(61\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000i 1.00000i
\(65\) −0.382683 + 0.923880i −0.382683 + 0.923880i
\(66\) 0 0
\(67\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 1.84776i 1.84776i 0.382683 + 0.923880i \(0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(72\) 0 0
\(73\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(80\) −0.158513 0.382683i −0.158513 0.382683i
\(81\) 0 0
\(82\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(83\) 1.30656 1.30656i 1.30656 1.30656i 0.382683 0.923880i \(-0.375000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 1.08239i 1.08239i
\(87\) 0 0
\(88\) 0.585786 + 0.585786i 0.585786 + 0.585786i
\(89\) −0.765367 −0.765367 −0.382683 0.923880i \(-0.625000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 1.41421i 1.41421i
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(98\) −0.541196 + 0.541196i −0.541196 + 0.541196i
\(99\) 0 0
\(100\) 0.292893 + 0.292893i 0.292893 + 0.292893i
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(104\) 1.08239 1.08239
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0.541196 + 0.224171i 0.541196 + 0.224171i
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) −1.00000 1.00000i −1.00000 1.00000i
\(119\) 0 0
\(120\) 0 0
\(121\) 0.414214 0.414214
\(122\) −0.765367 0.765367i −0.765367 0.765367i
\(123\) 0 0
\(124\) 0 0
\(125\) 0.923880 + 0.382683i 0.923880 + 0.382683i
\(126\) 0 0
\(127\) 1.41421 1.41421i 1.41421 1.41421i 0.707107 0.707107i \(-0.250000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0.707107 0.292893i 0.707107 0.292893i
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −0.541196 0.541196i −0.541196 0.541196i 0.382683 0.923880i \(-0.375000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 1.00000 1.00000i 1.00000 1.00000i
\(143\) −0.541196 + 0.541196i −0.541196 + 0.541196i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1.84776 −1.84776 −0.923880 0.382683i \(-0.875000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(158\) 0.765367 0.765367i 0.765367 0.765367i
\(159\) 0 0
\(160\) 0.292893 0.707107i 0.292893 0.707107i
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(164\) −0.765367 −0.765367
\(165\) 0 0
\(166\) −1.41421 −1.41421
\(167\) −0.541196 0.541196i −0.541196 0.541196i 0.382683 0.923880i \(-0.375000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(168\) 0 0
\(169\) 1.00000i 1.00000i
\(170\) 0 0
\(171\) 0 0
\(172\) −0.414214 + 0.414214i −0.414214 + 0.414214i
\(173\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0.317025i 0.317025i
\(177\) 0 0
\(178\) 0.414214 + 0.414214i 0.414214 + 0.414214i
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) 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
\(188\) 0.541196 0.541196i 0.541196 0.541196i
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −0.414214 −0.414214
\(197\) −1.30656 1.30656i −1.30656 1.30656i −0.923880 0.382683i \(-0.875000\pi\)
−0.382683 0.923880i \(-0.625000\pi\)
\(198\) 0 0
\(199\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(200\) 1.08239i 1.08239i
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −1.70711 + 0.707107i −1.70711 + 0.707107i
\(206\) 1.08239i 1.08239i
\(207\) 0 0
\(208\) −0.292893 0.292893i −0.292893 0.292893i
\(209\) 0 0
\(210\) 0 0
\(211\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −0.541196 + 1.30656i −0.541196 + 1.30656i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0.121320 + 0.292893i 0.121320 + 0.292893i
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0.541196 + 0.541196i 0.541196 + 0.541196i 0.923880 0.382683i \(-0.125000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(234\) 0 0
\(235\) 0.707107 1.70711i 0.707107 1.70711i
\(236\) 0.765367i 0.765367i
\(237\) 0 0
\(238\) 0 0
\(239\) −1.84776 −1.84776 −0.923880 0.382683i \(-0.875000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) −0.224171 0.224171i −0.224171 0.224171i
\(243\) 0 0
\(244\) 0.585786i 0.585786i
\(245\) −0.923880 + 0.382683i −0.923880 + 0.382683i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) −0.292893 0.707107i −0.292893 0.707107i
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −1.53073 −1.53073
\(255\) 0 0
\(256\) −1.00000 −1.00000
\(257\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0.382683 + 0.158513i 0.382683 + 0.158513i
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0.585786i 0.585786i
\(275\) 0.541196 + 0.541196i 0.541196 + 0.541196i
\(276\) 0 0
\(277\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0.765367i 0.765367i 0.923880 + 0.382683i \(0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(282\) 0 0
\(283\) 1.41421 + 1.41421i 1.41421 + 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(284\) 0.765367 0.765367
\(285\) 0 0
\(286\) 0.585786 0.585786
\(287\) 0 0
\(288\) 0 0
\(289\) 1.00000i 1.00000i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0.541196 0.541196i 0.541196 0.541196i −0.382683 0.923880i \(-0.625000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(294\) 0 0
\(295\) −0.707107 1.70711i −0.707107 1.70711i
\(296\) 0 0
\(297\) 0 0
\(298\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −0.541196 1.30656i −0.541196 1.30656i
\(306\) 0 0
\(307\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −1.41421 1.41421i −1.41421 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 0.707107i \(-0.750000\pi\)
\(314\) 1.08239 1.08239
\(315\) 0 0
\(316\) 0.585786 0.585786
\(317\) 1.30656 + 1.30656i 1.30656 + 1.30656i 0.923880 + 0.382683i \(0.125000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −0.923880 + 0.382683i −0.923880 + 0.382683i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 1.00000 1.00000
\(326\) 0 0
\(327\) 0 0
\(328\) 1.41421 + 1.41421i 1.41421 + 1.41421i
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) −0.541196 0.541196i −0.541196 0.541196i
\(333\) 0 0
\(334\) 0.585786i 0.585786i
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(338\) 0.541196 0.541196i 0.541196 0.541196i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 1.53073 1.53073
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0.414214 0.414214i 0.414214 0.414214i
\(353\) −0.541196 + 0.541196i −0.541196 + 0.541196i −0.923880 0.382683i \(-0.875000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(354\) 0 0
\(355\) 1.70711 0.707107i 1.70711 0.707107i
\(356\) 0.317025i 0.317025i
\(357\) 0 0
\(358\) 0 0
\(359\) 0.765367 0.765367 0.382683 0.923880i \(-0.375000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(360\) 0 0
\(361\) −1.00000 −1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −2.00000 −2.00000
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −1.30656 + 1.30656i −1.30656 + 1.30656i −0.382683 + 0.923880i \(0.625000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0.765367 + 0.765367i 0.765367 + 0.765367i
\(393\) 0 0
\(394\) 1.41421i 1.41421i
\(395\) 1.30656 0.541196i 1.30656 0.541196i
\(396\) 0 0
\(397\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(398\) 0.765367 0.765367i 0.765367 0.765367i
\(399\) 0 0
\(400\) −0.292893 + 0.292893i −0.292893 + 0.292893i
\(401\) 1.84776i 1.84776i 0.382683 + 0.923880i \(0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) 1.30656 + 0.541196i 1.30656 + 0.541196i
\(411\) 0 0
\(412\) 0.414214 0.414214i 0.414214 0.414214i
\(413\) 0 0
\(414\) 0 0
\(415\) −1.70711 0.707107i −1.70711 0.707107i
\(416\) 0.765367i 0.765367i
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0.765367 + 0.765367i 0.765367 + 0.765367i
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 1.00000 0.414214i 1.00000 0.414214i
\(431\) 0.765367i 0.765367i 0.923880 + 0.382683i \(0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(432\) 0 0
\(433\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0.317025 0.765367i 0.317025 0.765367i
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(444\) 0 0
\(445\) 0.292893 + 0.707107i 0.292893 + 0.707107i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 1.84776 1.84776 0.923880 0.382683i \(-0.125000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(450\) 0 0
\(451\) −1.41421 −1.41421
\(452\) 0 0
\(453\) 0 0
\(454\) 0.585786i 0.585786i
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0.765367i 0.765367i −0.923880 0.382683i \(-0.875000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(462\) 0 0
\(463\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −1.30656 + 0.541196i −1.30656 + 0.541196i
\(471\) 0 0
\(472\) −1.41421 + 1.41421i −1.41421 + 1.41421i
\(473\) −0.765367 + 0.765367i −0.765367 + 0.765367i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(479\) 0.765367 0.765367 0.382683 0.923880i \(-0.375000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0.171573i 0.171573i
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(488\) −1.08239 + 1.08239i −1.08239 + 1.08239i
\(489\) 0 0
\(490\) 0.707107 + 0.292893i 0.707107 + 0.292893i
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) 0.158513 0.382683i 0.158513 0.382683i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) −0.585786 0.585786i −0.585786 0.585786i
\(509\) 0.765367 0.765367 0.382683 0.923880i \(-0.375000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0.541196 + 0.541196i 0.541196 + 0.541196i
\(513\) 0 0
\(514\) 0 0
\(515\) 0.541196 1.30656i 0.541196 1.30656i
\(516\) 0 0
\(517\) 1.00000 1.00000i 1.00000 1.00000i
\(518\) 0 0
\(519\) 0 0
\(520\) −0.414214 1.00000i −0.414214 1.00000i
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 1.00000i 1.00000i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −1.30656 + 1.30656i −1.30656 + 1.30656i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −0.765367 −0.765367
\(540\) 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
\(547\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(548\) −0.224171 + 0.224171i −0.224171 + 0.224171i
\(549\) 0 0
\(550\) 0.585786i 0.585786i
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) −1.08239 −1.08239
\(555\) 0 0
\(556\) 0 0
\(557\) 0.541196 + 0.541196i 0.541196 + 0.541196i 0.923880 0.382683i \(-0.125000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(558\) 0 0
\(559\) 1.41421i 1.41421i
\(560\) 0 0
\(561\) 0 0
\(562\) 0.414214 0.414214i 0.414214 0.414214i
\(563\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 1.53073i 1.53073i
\(567\) 0 0
\(568\) −1.41421 1.41421i −1.41421 1.41421i
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(572\) 0.224171 + 0.224171i 0.224171 + 0.224171i
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(578\) 0.541196 0.541196i 0.541196 0.541196i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) −0.585786 −0.585786
\(587\) 0.541196 + 0.541196i 0.541196 + 0.541196i 0.923880 0.382683i \(-0.125000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) −0.541196 + 1.30656i −0.541196 + 1.30656i
\(591\) 0 0
\(592\) 0 0
\(593\) 1.30656 1.30656i 1.30656 1.30656i 0.382683 0.923880i \(-0.375000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0.765367i 0.765367i
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −0.158513 0.382683i −0.158513 0.382683i
\(606\) 0 0
\(607\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) −0.414214 + 1.00000i −0.414214 + 1.00000i
\(611\) 1.84776i 1.84776i
\(612\) 0 0
\(613\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −1.30656 1.30656i −1.30656 1.30656i −0.923880 0.382683i \(-0.875000\pi\)
−0.382683 0.923880i \(-0.625000\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000i 1.00000i
\(626\) 1.53073i 1.53073i
\(627\) 0 0
\(628\) 0.414214 + 0.414214i 0.414214 + 0.414214i
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) −1.08239 1.08239i −1.08239 1.08239i
\(633\) 0 0
\(634\) 1.41421i 1.41421i
\(635\) −1.84776 0.765367i −1.84776 0.765367i
\(636\) 0 0
\(637\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(648\) 0 0
\(649\) 1.41421i 1.41421i
\(650\) −0.541196 0.541196i −0.541196 0.541196i
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0.765367i 0.765367i
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 2.00000i 2.00000i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) −0.224171 + 0.224171i −0.224171 + 0.224171i
\(669\) 0 0
\(670\) 0 0
\(671\) 1.08239i 1.08239i
\(672\) 0 0
\(673\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0.414214 0.414214
\(677\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0.541196 0.541196i 0.541196 0.541196i −0.382683 0.923880i \(-0.625000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(684\) 0 0
\(685\) −0.292893 + 0.707107i −0.292893 + 0.707107i
\(686\) 0 0
\(687\) 0 0
\(688\) −0.414214 0.414214i −0.414214 0.414214i
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 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
\(703\) 0 0
\(704\) −0.765367 −0.765367
\(705\) 0 0
\(706\) 0.585786 0.585786
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(710\) −1.30656 0.541196i −1.30656 0.541196i
\(711\) 0 0
\(712\) 0.585786 0.585786i 0.585786 0.585786i
\(713\) 0 0
\(714\) 0 0
\(715\) 0.707107 + 0.292893i 0.707107 + 0.292893i
\(716\) 0 0
\(717\) 0 0
\(718\) −0.414214 0.414214i −0.414214 0.414214i
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0.541196 + 0.541196i 0.541196 + 0.541196i
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 1.41421 1.41421i 1.41421 1.41421i 0.707107 0.707107i \(-0.250000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(734\) −1.08239 −1.08239
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −1.30656 + 1.30656i −1.30656 + 1.30656i −0.382683 + 0.923880i \(0.625000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(744\) 0 0
\(745\) 0.707107 + 1.70711i 0.707107 + 1.70711i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0.541196 + 0.541196i 0.541196 + 0.541196i
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0.765367i 0.765367i −0.923880 0.382683i \(-0.875000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 1.41421 1.41421
\(767\) −1.30656 1.30656i −1.30656 1.30656i
\(768\) 0 0
\(769\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 1.30656 1.30656i 1.30656 1.30656i 0.382683 0.923880i \(-0.375000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 1.41421 1.41421
\(782\) 0 0
\(783\) 0 0
\(784\) 0.414214i 0.414214i
\(785\) 1.30656 + 0.541196i 1.30656 + 0.541196i
\(786\) 0 0
\(787\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(788\) −0.541196 + 0.541196i −0.541196 + 0.541196i
\(789\) 0 0
\(790\) −1.00000 0.414214i −1.00000 0.414214i
\(791\) 0 0
\(792\) 0 0
\(793\) −1.00000 1.00000i −1.00000 1.00000i
\(794\) 0 0
\(795\) 0 0
\(796\) 0.585786 0.585786
\(797\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −0.765367 −0.765367
\(801\) 0 0
\(802\) 1.00000 1.00000i 1.00000 1.00000i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0