Properties

Label 799.1.e.b.140.1
Level $799$
Weight $1$
Character 799.140
Analytic conductor $0.399$
Analytic rank $0$
Dimension $8$
Projective image $D_{20}$
CM discriminant -47
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 799 = 17 \cdot 47 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 799.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.398752945094\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{20})\)
Defining polynomial: \(x^{8} - x^{6} + x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{20}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{20} - \cdots)\)

Embedding invariants

Embedding label 140.1
Root \(0.587785 - 0.809017i\) of defining polynomial
Character \(\chi\) \(=\) 799.140
Dual form 799.1.e.b.234.3

$q$-expansion

\(f(q)\) \(=\) \(q-0.618034i q^{2} +(-1.39680 + 1.39680i) q^{3} +0.618034 q^{4} +(0.863271 + 0.863271i) q^{6} +(-1.26007 - 1.26007i) q^{7} -1.00000i q^{8} -2.90211i q^{9} +O(q^{10})\) \(q-0.618034i q^{2} +(-1.39680 + 1.39680i) q^{3} +0.618034 q^{4} +(0.863271 + 0.863271i) q^{6} +(-1.26007 - 1.26007i) q^{7} -1.00000i q^{8} -2.90211i q^{9} +(-0.863271 + 0.863271i) q^{12} +(-0.778768 + 0.778768i) q^{14} +(-0.809017 - 0.587785i) q^{17} -1.79360 q^{18} +3.52015 q^{21} +(1.39680 + 1.39680i) q^{24} -1.00000i q^{25} +(2.65688 + 2.65688i) q^{27} +(-0.778768 - 0.778768i) q^{28} -1.00000i q^{32} +(-0.363271 + 0.500000i) q^{34} -1.79360i q^{36} +(1.39680 - 1.39680i) q^{37} -2.17557i q^{42} -1.00000 q^{47} +2.17557i q^{49} -0.618034 q^{50} +(1.95106 - 0.309017i) q^{51} +1.90211i q^{53} +(1.64204 - 1.64204i) q^{54} +(-1.26007 + 1.26007i) q^{56} -0.618034i q^{59} +(-0.642040 - 0.642040i) q^{61} +(-3.65688 + 3.65688i) q^{63} -0.618034 q^{64} +(-0.500000 - 0.363271i) q^{68} +(-0.221232 + 0.221232i) q^{71} -2.90211 q^{72} +(-0.863271 - 0.863271i) q^{74} +(1.39680 + 1.39680i) q^{75} +(-0.221232 - 0.221232i) q^{79} -4.52015 q^{81} +2.17557 q^{84} +0.618034 q^{89} +0.618034i q^{94} +(1.39680 + 1.39680i) q^{96} +(0.642040 - 0.642040i) q^{97} +1.34458 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{3} - 4 q^{4} + 4 q^{6} + 2 q^{7} + O(q^{10}) \) \( 8 q - 2 q^{3} - 4 q^{4} + 4 q^{6} + 2 q^{7} - 4 q^{12} - 6 q^{14} - 2 q^{17} + 4 q^{18} + 4 q^{21} + 2 q^{24} - 6 q^{28} + 2 q^{37} - 8 q^{47} + 4 q^{50} + 8 q^{51} + 10 q^{54} + 2 q^{56} - 2 q^{61} - 8 q^{63} + 4 q^{64} - 4 q^{68} - 2 q^{71} - 8 q^{72} - 4 q^{74} + 2 q^{75} - 2 q^{79} - 12 q^{81} + 8 q^{84} - 4 q^{89} + 2 q^{96} + 2 q^{97} - 4 q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/799\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(377\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.618034i 0.618034i −0.951057 0.309017i \(-0.900000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(3\) −1.39680 + 1.39680i −1.39680 + 1.39680i −0.587785 + 0.809017i \(0.700000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(4\) 0.618034 0.618034
\(5\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(6\) 0.863271 + 0.863271i 0.863271 + 0.863271i
\(7\) −1.26007 1.26007i −1.26007 1.26007i −0.951057 0.309017i \(-0.900000\pi\)
−0.309017 0.951057i \(-0.600000\pi\)
\(8\) 1.00000i 1.00000i
\(9\) 2.90211i 2.90211i
\(10\) 0 0
\(11\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(12\) −0.863271 + 0.863271i −0.863271 + 0.863271i
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) −0.778768 + 0.778768i −0.778768 + 0.778768i
\(15\) 0 0
\(16\) 0 0
\(17\) −0.809017 0.587785i −0.809017 0.587785i
\(18\) −1.79360 −1.79360
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 3.52015 3.52015
\(22\) 0 0
\(23\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(24\) 1.39680 + 1.39680i 1.39680 + 1.39680i
\(25\) 1.00000i 1.00000i
\(26\) 0 0
\(27\) 2.65688 + 2.65688i 2.65688 + 2.65688i
\(28\) −0.778768 0.778768i −0.778768 0.778768i
\(29\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(30\) 0 0
\(31\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(32\) 1.00000i 1.00000i
\(33\) 0 0
\(34\) −0.363271 + 0.500000i −0.363271 + 0.500000i
\(35\) 0 0
\(36\) 1.79360i 1.79360i
\(37\) 1.39680 1.39680i 1.39680 1.39680i 0.587785 0.809017i \(-0.300000\pi\)
0.809017 0.587785i \(-0.200000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(42\) 2.17557i 2.17557i
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.00000 −1.00000
\(48\) 0 0
\(49\) 2.17557i 2.17557i
\(50\) −0.618034 −0.618034
\(51\) 1.95106 0.309017i 1.95106 0.309017i
\(52\) 0 0
\(53\) 1.90211i 1.90211i 0.309017 + 0.951057i \(0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(54\) 1.64204 1.64204i 1.64204 1.64204i
\(55\) 0 0
\(56\) −1.26007 + 1.26007i −1.26007 + 1.26007i
\(57\) 0 0
\(58\) 0 0
\(59\) 0.618034i 0.618034i −0.951057 0.309017i \(-0.900000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(60\) 0 0
\(61\) −0.642040 0.642040i −0.642040 0.642040i 0.309017 0.951057i \(-0.400000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(62\) 0 0
\(63\) −3.65688 + 3.65688i −3.65688 + 3.65688i
\(64\) −0.618034 −0.618034
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) −0.500000 0.363271i −0.500000 0.363271i
\(69\) 0 0
\(70\) 0 0
\(71\) −0.221232 + 0.221232i −0.221232 + 0.221232i −0.809017 0.587785i \(-0.800000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(72\) −2.90211 −2.90211
\(73\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(74\) −0.863271 0.863271i −0.863271 0.863271i
\(75\) 1.39680 + 1.39680i 1.39680 + 1.39680i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −0.221232 0.221232i −0.221232 0.221232i 0.587785 0.809017i \(-0.300000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(80\) 0 0
\(81\) −4.52015 −4.52015
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 2.17557 2.17557
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0.618034i 0.618034i
\(95\) 0 0
\(96\) 1.39680 + 1.39680i 1.39680 + 1.39680i
\(97\) 0.642040 0.642040i 0.642040 0.642040i −0.309017 0.951057i \(-0.600000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(98\) 1.34458 1.34458
\(99\) 0 0
\(100\) 0.618034i 0.618034i
\(101\) −1.17557 −1.17557 −0.587785 0.809017i \(-0.700000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(102\) −0.190983 1.20582i −0.190983 1.20582i
\(103\) 1.17557 1.17557 0.587785 0.809017i \(-0.300000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 1.17557 1.17557
\(107\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(108\) 1.64204 + 1.64204i 1.64204 + 1.64204i
\(109\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(110\) 0 0
\(111\) 3.90211i 3.90211i
\(112\) 0 0
\(113\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) −0.381966 −0.381966
\(119\) 0.278768 + 1.76007i 0.278768 + 1.76007i
\(120\) 0 0
\(121\) 1.00000i 1.00000i
\(122\) −0.396802 + 0.396802i −0.396802 + 0.396802i
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 2.26007 + 2.26007i 2.26007 + 2.26007i
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0.618034i 0.618034i
\(129\) 0 0
\(130\) 0 0
\(131\) 0.642040 0.642040i 0.642040 0.642040i −0.309017 0.951057i \(-0.600000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) −0.587785 + 0.809017i −0.587785 + 0.809017i
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(140\) 0 0
\(141\) 1.39680 1.39680i 1.39680 1.39680i
\(142\) 0.136729 + 0.136729i 0.136729 + 0.136729i
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −3.03884 3.03884i −3.03884 3.03884i
\(148\) 0.863271 0.863271i 0.863271 0.863271i
\(149\) −1.17557 −1.17557 −0.587785 0.809017i \(-0.700000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(150\) 0.863271 0.863271i 0.863271 0.863271i
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) −1.70582 + 2.34786i −1.70582 + 2.34786i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 1.90211 1.90211 0.951057 0.309017i \(-0.100000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(158\) −0.136729 + 0.136729i −0.136729 + 0.136729i
\(159\) −2.65688 2.65688i −2.65688 2.65688i
\(160\) 0 0
\(161\) 0 0
\(162\) 2.79360i 2.79360i
\(163\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(168\) 3.52015i 3.52015i
\(169\) 1.00000 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −1.26007 + 1.26007i −1.26007 + 1.26007i −0.309017 + 0.951057i \(0.600000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(174\) 0 0
\(175\) −1.26007 + 1.26007i −1.26007 + 1.26007i
\(176\) 0 0
\(177\) 0.863271 + 0.863271i 0.863271 + 0.863271i
\(178\) 0.381966i 0.381966i
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(182\) 0 0
\(183\) 1.79360 1.79360
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) −0.618034 −0.618034
\(189\) 6.69572i 6.69572i
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0.863271 0.863271i 0.863271 0.863271i
\(193\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(194\) −0.396802 0.396802i −0.396802 0.396802i
\(195\) 0 0
\(196\) 1.34458i 1.34458i
\(197\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(200\) −1.00000 −1.00000
\(201\) 0 0
\(202\) 0.726543i 0.726543i
\(203\) 0 0
\(204\) 1.20582 0.190983i 1.20582 0.190983i
\(205\) 0 0
\(206\) 0.726543i 0.726543i
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(212\) 1.17557i 1.17557i
\(213\) 0.618034i 0.618034i
\(214\) 0 0
\(215\) 0 0
\(216\) 2.65688 2.65688i 2.65688 2.65688i
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 2.41164 2.41164
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) −1.26007 + 1.26007i −1.26007 + 1.26007i
\(225\) −2.90211 −2.90211
\(226\) 0 0
\(227\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(228\) 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 0
\(236\) 0.381966i 0.381966i
\(237\) 0.618034 0.618034
\(238\) 1.08779 0.172288i 1.08779 0.172288i
\(239\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(240\) 0 0
\(241\) 1.39680 1.39680i 1.39680 1.39680i 0.587785 0.809017i \(-0.300000\pi\)
0.809017 0.587785i \(-0.200000\pi\)
\(242\) 0.618034 0.618034
\(243\) 3.65688 3.65688i 3.65688 3.65688i
\(244\) −0.396802 0.396802i −0.396802 0.396802i
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 1.17557 1.17557 0.587785 0.809017i \(-0.300000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(252\) −2.26007 + 2.26007i −2.26007 + 2.26007i
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −1.00000 −1.00000
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) −3.52015 −3.52015
\(260\) 0 0
\(261\) 0 0
\(262\) −0.396802 0.396802i −0.396802 0.396802i
\(263\) 0.618034i 0.618034i −0.951057 0.309017i \(-0.900000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −0.863271 + 0.863271i −0.863271 + 0.863271i
\(268\) 0 0
\(269\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(270\) 0 0
\(271\) −1.17557 −1.17557 −0.587785 0.809017i \(-0.700000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0.221232 0.221232i 0.221232 0.221232i −0.587785 0.809017i \(-0.700000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) −0.863271 0.863271i −0.863271 0.863271i
\(283\) 1.26007 + 1.26007i 1.26007 + 1.26007i 0.951057 + 0.309017i \(0.100000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(284\) −0.136729 + 0.136729i −0.136729 + 0.136729i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −2.90211 −2.90211
\(289\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(290\) 0 0
\(291\) 1.79360i 1.79360i
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) −1.87811 + 1.87811i −1.87811 + 1.87811i
\(295\) 0 0
\(296\) −1.39680 1.39680i −1.39680 1.39680i
\(297\) 0 0
\(298\) 0.726543i 0.726543i
\(299\) 0 0
\(300\) 0.863271 + 0.863271i 0.863271 + 0.863271i
\(301\) 0 0
\(302\) 0 0
\(303\) 1.64204 1.64204i 1.64204 1.64204i
\(304\) 0 0
\(305\) 0 0
\(306\) 1.45106 + 1.05425i 1.45106 + 1.05425i
\(307\) −1.17557 −1.17557 −0.587785 0.809017i \(-0.700000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(308\) 0 0
\(309\) −1.64204 + 1.64204i −1.64204 + 1.64204i
\(310\) 0 0
\(311\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(312\) 0 0
\(313\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(314\) 1.17557i 1.17557i
\(315\) 0 0
\(316\) −0.136729 0.136729i −0.136729 0.136729i
\(317\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(318\) −1.64204 + 1.64204i −1.64204 + 1.64204i
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −2.79360 −2.79360
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 1.26007 + 1.26007i 1.26007 + 1.26007i
\(330\) 0 0
\(331\) 0.618034i 0.618034i 0.951057 + 0.309017i \(0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(332\) 0 0
\(333\) −4.05368 4.05368i −4.05368 4.05368i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −1.26007 + 1.26007i −1.26007 + 1.26007i −0.309017 + 0.951057i \(0.600000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(338\) 0.618034i 0.618034i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1.48131 1.48131i 1.48131 1.48131i
\(344\) 0 0
\(345\) 0 0
\(346\) 0.778768 + 0.778768i 0.778768 + 0.778768i
\(347\) 1.39680 + 1.39680i 1.39680 + 1.39680i 0.809017 + 0.587785i \(0.200000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0.778768 + 0.778768i 0.778768 + 0.778768i
\(351\) 0 0
\(352\) 0 0
\(353\) 1.90211 1.90211 0.951057 0.309017i \(-0.100000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(354\) 0.533531 0.533531i 0.533531 0.533531i
\(355\) 0 0
\(356\) 0.381966 0.381966
\(357\) −2.84786 2.06909i −2.84786 2.06909i
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −1.00000 −1.00000
\(362\) 0 0
\(363\) −1.39680 1.39680i −1.39680 1.39680i
\(364\) 0 0
\(365\) 0 0
\(366\) 1.10851i 1.10851i
\(367\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 2.39680 2.39680i 2.39680 2.39680i
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 1.00000i 1.00000i
\(377\) 0 0
\(378\) −4.13818 −4.13818
\(379\) 1.26007 1.26007i 1.26007 1.26007i 0.309017 0.951057i \(-0.400000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 1.17557i 1.17557i −0.809017 0.587785i \(-0.800000\pi\)
0.809017 0.587785i \(-0.200000\pi\)
\(384\) 0.863271 + 0.863271i 0.863271 + 0.863271i
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0.396802 0.396802i 0.396802 0.396802i
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 2.17557 2.17557
\(393\) 1.79360i 1.79360i
\(394\) 0.618034 0.618034i 0.618034 0.618034i
\(395\) 0 0
\(396\) 0 0
\(397\) 0.221232 + 0.221232i 0.221232 + 0.221232i 0.809017 0.587785i \(-0.200000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1.39680 + 1.39680i 1.39680 + 1.39680i 0.809017 + 0.587785i \(0.200000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −0.726543 −0.726543
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) −0.309017 1.95106i −0.309017 1.95106i
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0.726543 0.726543
\(413\) −0.778768 + 0.778768i −0.778768 + 0.778768i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 2.90211i 2.90211i
\(424\) 1.90211 1.90211
\(425\) −0.587785 + 0.809017i −0.587785 + 0.809017i
\(426\) −0.381966 −0.381966
\(427\) 1.61803i 1.61803i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −0.642040 0.642040i −0.642040 0.642040i 0.309017 0.951057i \(-0.400000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 6.31375 6.31375
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 2.41164i 2.41164i
\(445\) 0 0
\(446\) 0 0
\(447\) 1.64204 1.64204i 1.64204 1.64204i
\(448\) 0.778768 + 0.778768i 0.778768 + 0.778768i
\(449\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(450\) 1.79360i 1.79360i
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 1.17557i 1.17557i −0.809017 0.587785i \(-0.800000\pi\)
0.809017 0.587785i \(-0.200000\pi\)
\(458\) 0 0
\(459\) −0.587785 3.71113i −0.587785 3.71113i
\(460\) 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
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −2.65688 + 2.65688i −2.65688 + 2.65688i
\(472\) −0.618034 −0.618034
\(473\) 0 0
\(474\) 0.381966i 0.381966i
\(475\) 0 0
\(476\) 0.172288 + 1.08779i 0.172288 + 1.08779i
\(477\) 5.52015 5.52015
\(478\) 1.00000i 1.00000i
\(479\) 1.26007 1.26007i 1.26007 1.26007i 0.309017 0.951057i \(-0.400000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −0.863271 0.863271i −0.863271 0.863271i
\(483\) 0 0
\(484\) 0.618034i 0.618034i
\(485\) 0 0
\(486\) −2.26007 2.26007i −2.26007 2.26007i
\(487\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(488\) −0.642040 + 0.642040i −0.642040 + 0.642040i
\(489\) 0 0
\(490\) 0 0
\(491\) 1.90211i 1.90211i −0.309017 0.951057i \(-0.600000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0.557537 0.557537
\(498\) 0 0
\(499\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0.726543i 0.726543i
\(503\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(504\) 3.65688 + 3.65688i 3.65688 + 3.65688i
\(505\) 0 0
\(506\) 0 0
\(507\) −1.39680 + 1.39680i −1.39680 + 1.39680i
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 2.17557i 2.17557i
\(519\) 3.52015i 3.52015i
\(520\) 0 0
\(521\) −1.26007 1.26007i −1.26007 1.26007i −0.951057 0.309017i \(-0.900000\pi\)
−0.309017 0.951057i \(-0.600000\pi\)
\(522\) 0 0
\(523\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(524\) 0.396802 0.396802i 0.396802 0.396802i
\(525\) 3.52015i 3.52015i
\(526\) −0.381966 −0.381966
\(527\) 0 0
\(528\) 0 0
\(529\) 1.00000i 1.00000i
\(530\) 0 0
\(531\) −1.79360 −1.79360
\(532\) 0 0
\(533\) 0 0
\(534\) 0.533531 + 0.533531i 0.533531 + 0.533531i
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) −0.618034 0.618034i −0.618034 0.618034i
\(539\) 0 0
\(540\) 0 0
\(541\) −1.26007 + 1.26007i −1.26007 + 1.26007i −0.309017 + 0.951057i \(0.600000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(542\) 0.726543i 0.726543i
\(543\) 0 0
\(544\) −0.587785 + 0.809017i −0.587785 + 0.809017i
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(548\) 0 0
\(549\) −1.86327 + 1.86327i −1.86327 + 1.86327i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0.557537i 0.557537i
\(554\) −0.136729 0.136729i −0.136729 0.136729i
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0.863271 0.863271i 0.863271 0.863271i
\(565\) 0 0
\(566\) 0.778768 0.778768i 0.778768 0.778768i
\(567\) 5.69572 + 5.69572i 5.69572 + 5.69572i
\(568\) 0.221232 + 0.221232i 0.221232 + 0.221232i
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) −0.221232 0.221232i −0.221232 0.221232i 0.587785 0.809017i \(-0.300000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 1.79360i 1.79360i
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 0.587785 0.190983i 0.587785 0.190983i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 1.10851 1.10851
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) −1.87811 1.87811i −1.87811 1.87811i
\(589\) 0 0
\(590\) 0 0
\(591\) −2.79360 −2.79360
\(592\) 0 0
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −0.726543 −0.726543
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 1.39680 1.39680i 1.39680 1.39680i
\(601\) 0.642040 + 0.642040i 0.642040 + 0.642040i 0.951057 0.309017i \(-0.100000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) −1.01484 1.01484i −1.01484 1.01484i
\(607\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) −1.05425 + 1.45106i −1.05425 + 1.45106i
\(613\) −0.618034 −0.618034 −0.309017 0.951057i \(-0.600000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(614\) 0.726543i 0.726543i
\(615\) 0 0
\(616\) 0 0
\(617\) −0.221232 + 0.221232i −0.221232 + 0.221232i −0.809017 0.587785i \(-0.800000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(618\) 1.01484 + 1.01484i 1.01484 + 1.01484i
\(619\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −0.778768 0.778768i −0.778768 0.778768i
\(624\) 0 0
\(625\) −1.00000 −1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 1.17557 1.17557
\(629\) −1.95106 + 0.309017i −1.95106 + 0.309017i
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) −0.221232 + 0.221232i −0.221232 + 0.221232i
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) −1.64204 1.64204i −1.64204 1.64204i
\(637\) 0 0
\(638\) 0 0
\(639\) 0.642040 + 0.642040i 0.642040 + 0.642040i
\(640\) 0 0
\(641\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(642\) 0 0
\(643\) 0.642040 0.642040i 0.642040 0.642040i −0.309017 0.951057i \(-0.600000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −1.90211 −1.90211 −0.951057 0.309017i \(-0.900000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(648\) 4.52015i 4.52015i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −1.39680 1.39680i −1.39680 1.39680i −0.809017 0.587785i \(-0.800000\pi\)
−0.587785 0.809017i \(-0.700000\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0.778768 0.778768i 0.778768 0.778768i
\(659\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(660\) 0 0
\(661\) 1.61803i 1.61803i 0.587785 + 0.809017i \(0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(662\) 0.381966 0.381966
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) −2.50531 + 2.50531i −2.50531 + 2.50531i
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 3.52015i 3.52015i
\(673\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(674\) 0.778768 + 0.778768i 0.778768 + 0.778768i
\(675\) 2.65688 2.65688i 2.65688 2.65688i
\(676\) 0.618034 0.618034
\(677\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(678\) 0 0
\(679\) −1.61803 −1.61803
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0.642040 0.642040i 0.642040 0.642040i −0.309017 0.951057i \(-0.600000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −0.915497 0.915497i −0.915497 0.915497i
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(692\) −0.778768 + 0.778768i −0.778768 + 0.778768i
\(693\) 0 0
\(694\) 0.863271 0.863271i 0.863271 0.863271i
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −0.778768 + 0.778768i −0.778768 + 0.778768i
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 1.17557i 1.17557i
\(707\) 1.48131 + 1.48131i 1.48131 + 1.48131i
\(708\) 0.533531 + 0.533531i 0.533531 + 0.533531i
\(709\) −1.26007 + 1.26007i −1.26007 + 1.26007i −0.309017 + 0.951057i \(0.600000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(710\) 0 0
\(711\) −0.642040 + 0.642040i −0.642040 + 0.642040i
\(712\) 0.618034i 0.618034i
\(713\) 0 0
\(714\) −1.27877 + 1.76007i −1.27877 + 1.76007i
\(715\) 0 0
\(716\) 0 0
\(717\) −2.26007 + 2.26007i −2.26007 + 2.26007i
\(718\) 0 0
\(719\) −0.642040 + 0.642040i −0.642040 + 0.642040i −0.951057 0.309017i \(-0.900000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(720\) 0 0
\(721\) −1.48131 1.48131i −1.48131 1.48131i
\(722\) 0.618034i 0.618034i
\(723\) 3.90211i 3.90211i
\(724\) 0 0
\(725\) 0 0
\(726\) −0.863271 + 0.863271i −0.863271 + 0.863271i
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 5.69572i 5.69572i
\(730\) 0 0
\(731\) 0 0
\(732\) 1.10851 1.10851
\(733\) 1.61803i 1.61803i 0.587785 + 0.809017i \(0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 1.90211i 1.90211i 0.309017 + 0.951057i \(0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −1.48131 1.48131i −1.48131 1.48131i
\(743\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(752\) 0 0
\(753\) −1.64204 + 1.64204i −1.64204 + 1.64204i
\(754\) 0 0
\(755\) 0 0
\(756\) 4.13818i 4.13818i
\(757\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(758\) −0.778768 0.778768i −0.778768 0.778768i
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) −0.726543 −0.726543
\(767\) 0 0
\(768\) 1.39680 1.39680i 1.39680 1.39680i
\(769\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 1.61803i 1.61803i −0.587785 0.809017i \(-0.700000\pi\)
0.587785 0.809017i \(-0.300000\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −0.642040 0.642040i −0.642040 0.642040i
\(777\) 4.91695 4.91695i 4.91695 4.91695i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 1.10851 1.10851
\(787\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(788\) 0.618034 + 0.618034i 0.618034 + 0.618034i
\(789\) 0.863271 + 0.863271i 0.863271 + 0.863271i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0.136729 0.136729i 0.136729 0.136729i
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(800\) −1.00000 −1.00000
\(801\) 1.79360i 1.79360i
\(802\) 0.863271 0.863271i 0.863271 0.863271i
\(803\) 0 0
\(804\) 0 0
\(805\) 0