Properties

Label 1089.1.s.a.94.1
Level $1089$
Weight $1$
Character 1089.94
Analytic conductor $0.543$
Analytic rank $0$
Dimension $8$
Projective image $D_{3}$
CM discriminant -11
Inner twists $16$

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Newspace parameters

Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1089.s (of order \(30\), degree \(8\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.543481798757\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{15})\)
Defining polynomial: \(x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 99)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.891.1
Artin image: $C_{15}\times S_3$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{30} + \cdots)\)

Embedding invariants

Embedding label 94.1
Root \(0.669131 + 0.743145i\) of defining polynomial
Character \(\chi\) \(=\) 1089.94
Dual form 1089.1.s.a.475.1

$q$-expansion

\(f(q)\) \(=\) \(q+(0.913545 - 0.406737i) q^{3} +(-0.104528 - 0.994522i) q^{4} +(-0.669131 - 0.743145i) q^{5} +(0.669131 - 0.743145i) q^{9} +O(q^{10})\) \(q+(0.913545 - 0.406737i) q^{3} +(-0.104528 - 0.994522i) q^{4} +(-0.669131 - 0.743145i) q^{5} +(0.669131 - 0.743145i) q^{9} +(-0.500000 - 0.866025i) q^{12} +(-0.913545 - 0.406737i) q^{15} +(-0.978148 + 0.207912i) q^{16} +(-0.669131 + 0.743145i) q^{20} +(-1.00000 + 1.73205i) q^{23} +(0.309017 - 0.951057i) q^{27} +(0.978148 + 0.207912i) q^{31} +(-0.809017 - 0.587785i) q^{36} +(0.809017 - 0.587785i) q^{37} -1.00000 q^{45} +(0.104528 - 0.994522i) q^{47} +(-0.809017 + 0.587785i) q^{48} +(0.669131 + 0.743145i) q^{49} +(-0.309017 - 0.951057i) q^{53} +(0.104528 + 0.994522i) q^{59} +(-0.309017 + 0.951057i) q^{60} +(0.309017 + 0.951057i) q^{64} +(0.500000 - 0.866025i) q^{67} +(-0.209057 + 1.98904i) q^{69} +(-0.309017 + 0.951057i) q^{71} +(0.809017 + 0.587785i) q^{80} +(-0.104528 - 0.994522i) q^{81} +2.00000 q^{89} +(1.82709 + 0.813473i) q^{92} +(0.978148 - 0.207912i) q^{93} +(-0.669131 + 0.743145i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + q^{3} + q^{4} - q^{5} + q^{9} + O(q^{10}) \) \( 8q + q^{3} + q^{4} - q^{5} + q^{9} - 4q^{12} - q^{15} + q^{16} - q^{20} - 8q^{23} - 2q^{27} - q^{31} - 2q^{36} + 2q^{37} - 8q^{45} - q^{47} - 2q^{48} + q^{49} + 2q^{53} - q^{59} + 2q^{60} - 2q^{64} + 4q^{67} + 2q^{69} + 2q^{71} + 2q^{80} + q^{81} + 16q^{89} + 2q^{92} - q^{93} - q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(848\)
\(\chi(n)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{1}{3}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 0.669131 0.743145i \(-0.266667\pi\)
−0.669131 + 0.743145i \(0.733333\pi\)
\(3\) 0.913545 0.406737i 0.913545 0.406737i
\(4\) −0.104528 0.994522i −0.104528 0.994522i
\(5\) −0.669131 0.743145i −0.669131 0.743145i 0.309017 0.951057i \(-0.400000\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(6\) 0 0
\(7\) 0 0 −0.913545 0.406737i \(-0.866667\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(8\) 0 0
\(9\) 0.669131 0.743145i 0.669131 0.743145i
\(10\) 0 0
\(11\) 0 0
\(12\) −0.500000 0.866025i −0.500000 0.866025i
\(13\) 0 0 −0.978148 0.207912i \(-0.933333\pi\)
0.978148 + 0.207912i \(0.0666667\pi\)
\(14\) 0 0
\(15\) −0.913545 0.406737i −0.913545 0.406737i
\(16\) −0.978148 + 0.207912i −0.978148 + 0.207912i
\(17\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(18\) 0 0
\(19\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(20\) −0.669131 + 0.743145i −0.669131 + 0.743145i
\(21\) 0 0
\(22\) 0 0
\(23\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0.309017 0.951057i 0.309017 0.951057i
\(28\) 0 0
\(29\) 0 0 −0.913545 0.406737i \(-0.866667\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(30\) 0 0
\(31\) 0.978148 + 0.207912i 0.978148 + 0.207912i 0.669131 0.743145i \(-0.266667\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −0.809017 0.587785i −0.809017 0.587785i
\(37\) 0.809017 0.587785i 0.809017 0.587785i −0.104528 0.994522i \(-0.533333\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 0.913545 0.406737i \(-0.133333\pi\)
−0.913545 + 0.406737i \(0.866667\pi\)
\(42\) 0 0
\(43\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(44\) 0 0
\(45\) −1.00000 −1.00000
\(46\) 0 0
\(47\) 0.104528 0.994522i 0.104528 0.994522i −0.809017 0.587785i \(-0.800000\pi\)
0.913545 0.406737i \(-0.133333\pi\)
\(48\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(49\) 0.669131 + 0.743145i 0.669131 + 0.743145i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −0.309017 0.951057i −0.309017 0.951057i −0.978148 0.207912i \(-0.933333\pi\)
0.669131 0.743145i \(-0.266667\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0.104528 + 0.994522i 0.104528 + 0.994522i 0.913545 + 0.406737i \(0.133333\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(60\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(61\) 0 0 0.978148 0.207912i \(-0.0666667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(65\) 0 0
\(66\) 0 0
\(67\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(68\) 0 0
\(69\) −0.209057 + 1.98904i −0.209057 + 1.98904i
\(70\) 0 0
\(71\) −0.309017 + 0.951057i −0.309017 + 0.951057i 0.669131 + 0.743145i \(0.266667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(72\) 0 0
\(73\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.669131 0.743145i \(-0.266667\pi\)
−0.669131 + 0.743145i \(0.733333\pi\)
\(80\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(81\) −0.104528 0.994522i −0.104528 0.994522i
\(82\) 0 0
\(83\) 0 0 0.978148 0.207912i \(-0.0666667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1.82709 + 0.813473i 1.82709 + 0.813473i
\(93\) 0.978148 0.207912i 0.978148 0.207912i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −0.669131 + 0.743145i −0.669131 + 0.743145i −0.978148 0.207912i \(-0.933333\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 0.669131 0.743145i \(-0.266667\pi\)
−0.669131 + 0.743145i \(0.733333\pi\)
\(102\) 0 0
\(103\) 0.104528 + 0.994522i 0.104528 + 0.994522i 0.913545 + 0.406737i \(0.133333\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(108\) −0.978148 0.207912i −0.978148 0.207912i
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0.500000 0.866025i 0.500000 0.866025i
\(112\) 0 0
\(113\) −0.913545 + 0.406737i −0.913545 + 0.406737i −0.809017 0.587785i \(-0.800000\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(114\) 0 0
\(115\) 1.95630 0.415823i 1.95630 0.415823i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 0 0
\(124\) 0.104528 0.994522i 0.104528 0.994522i
\(125\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(126\) 0 0
\(127\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −0.913545 + 0.406737i −0.913545 + 0.406737i
\(136\) 0 0
\(137\) 0.978148 0.207912i 0.978148 0.207912i 0.309017 0.951057i \(-0.400000\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(138\) 0 0
\(139\) 0 0 −0.104528 0.994522i \(-0.533333\pi\)
0.104528 + 0.994522i \(0.466667\pi\)
\(140\) 0 0
\(141\) −0.309017 0.951057i −0.309017 0.951057i
\(142\) 0 0
\(143\) 0 0
\(144\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(145\) 0 0
\(146\) 0 0
\(147\) 0.913545 + 0.406737i 0.913545 + 0.406737i
\(148\) −0.669131 0.743145i −0.669131 0.743145i
\(149\) 0 0 −0.669131 0.743145i \(-0.733333\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(150\) 0 0
\(151\) 0 0 0.104528 0.994522i \(-0.466667\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −0.500000 0.866025i −0.500000 0.866025i
\(156\) 0 0
\(157\) −0.913545 + 0.406737i −0.913545 + 0.406737i −0.809017 0.587785i \(-0.800000\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(158\) 0 0
\(159\) −0.669131 0.743145i −0.669131 0.743145i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −0.309017 0.951057i −0.309017 0.951057i −0.978148 0.207912i \(-0.933333\pi\)
0.669131 0.743145i \(-0.266667\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.978148 0.207912i \(-0.933333\pi\)
0.978148 + 0.207912i \(0.0666667\pi\)
\(168\) 0 0
\(169\) 0.913545 + 0.406737i 0.913545 + 0.406737i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 0.104528 0.994522i \(-0.466667\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(178\) 0 0
\(179\) 0.809017 + 0.587785i 0.809017 + 0.587785i 0.913545 0.406737i \(-0.133333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(180\) 0.104528 + 0.994522i 0.104528 + 0.994522i
\(181\) −0.309017 + 0.951057i −0.309017 + 0.951057i 0.669131 + 0.743145i \(0.266667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −0.978148 0.207912i −0.978148 0.207912i
\(186\) 0 0
\(187\) 0 0
\(188\) −1.00000 −1.00000
\(189\) 0 0
\(190\) 0 0
\(191\) −0.913545 0.406737i −0.913545 0.406737i −0.104528 0.994522i \(-0.533333\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(192\) 0.669131 + 0.743145i 0.669131 + 0.743145i
\(193\) 0 0 −0.669131 0.743145i \(-0.733333\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0.669131 0.743145i 0.669131 0.743145i
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(200\) 0 0
\(201\) 0.104528 0.994522i 0.104528 0.994522i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0.618034 + 1.90211i 0.618034 + 1.90211i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 −0.978148 0.207912i \(-0.933333\pi\)
0.978148 + 0.207912i \(0.0666667\pi\)
\(212\) −0.913545 + 0.406737i −0.913545 + 0.406737i
\(213\) 0.104528 + 0.994522i 0.104528 + 0.994522i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −0.209057 + 1.98904i −0.209057 + 1.98904i −0.104528 + 0.994522i \(0.533333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 −0.913545 0.406737i \(-0.866667\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(228\) 0 0
\(229\) −1.95630 0.415823i −1.95630 0.415823i −0.978148 0.207912i \(-0.933333\pi\)
−0.978148 0.207912i \(-0.933333\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(234\) 0 0
\(235\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(236\) 0.978148 0.207912i 0.978148 0.207912i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 0.913545 0.406737i \(-0.133333\pi\)
−0.913545 + 0.406737i \(0.866667\pi\)
\(240\) 0.978148 + 0.207912i 0.978148 + 0.207912i
\(241\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(242\) 0 0
\(243\) −0.500000 0.866025i −0.500000 0.866025i
\(244\) 0 0
\(245\) 0.104528 0.994522i 0.104528 0.994522i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0.618034 + 1.90211i 0.618034 + 1.90211i 0.309017 + 0.951057i \(0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0.913545 0.406737i 0.913545 0.406737i
\(257\) −0.209057 1.98904i −0.209057 1.98904i −0.104528 0.994522i \(-0.533333\pi\)
−0.104528 0.994522i \(-0.533333\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(264\) 0 0
\(265\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(266\) 0 0
\(267\) 1.82709 0.813473i 1.82709 0.813473i
\(268\) −0.913545 0.406737i −0.913545 0.406737i
\(269\) −0.309017 + 0.951057i −0.309017 + 0.951057i 0.669131 + 0.743145i \(0.266667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(270\) 0 0
\(271\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 2.00000 2.00000
\(277\) 0 0 0.669131 0.743145i \(-0.266667\pi\)
−0.669131 + 0.743145i \(0.733333\pi\)
\(278\) 0 0
\(279\) 0.809017 0.587785i 0.809017 0.587785i
\(280\) 0 0
\(281\) 0 0 0.978148 0.207912i \(-0.0666667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(282\) 0 0
\(283\) 0 0 0.913545 0.406737i \(-0.133333\pi\)
−0.913545 + 0.406737i \(0.866667\pi\)
\(284\) 0.978148 + 0.207912i 0.978148 + 0.207912i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −0.809017 0.587785i −0.809017 0.587785i
\(290\) 0 0
\(291\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(292\) 0 0
\(293\) 0 0 −0.104528 0.994522i \(-0.533333\pi\)
0.104528 + 0.994522i \(0.466667\pi\)
\(294\) 0 0
\(295\) 0.669131 0.743145i 0.669131 0.743145i
\(296\) 0 0
\(297\) 0 0
\(298\) 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
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(310\) 0 0
\(311\) −0.913545 + 0.406737i −0.913545 + 0.406737i −0.809017 0.587785i \(-0.800000\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(312\) 0 0
\(313\) −1.95630 + 0.415823i −1.95630 + 0.415823i −0.978148 + 0.207912i \(0.933333\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.33826 1.48629i 1.33826 1.48629i 0.669131 0.743145i \(-0.266667\pi\)
0.669131 0.743145i \(-0.266667\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0.500000 0.866025i 0.500000 0.866025i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −0.978148 + 0.207912i −0.978148 + 0.207912i
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(332\) 0 0
\(333\) 0.104528 0.994522i 0.104528 0.994522i
\(334\) 0 0
\(335\) −0.978148 + 0.207912i −0.978148 + 0.207912i
\(336\) 0 0
\(337\) 0 0 −0.104528 0.994522i \(-0.533333\pi\)
0.104528 + 0.994522i \(0.466667\pi\)
\(338\) 0 0
\(339\) −0.669131 + 0.743145i −0.669131 + 0.743145i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 1.61803 1.17557i 1.61803 1.17557i
\(346\) 0 0
\(347\) 0 0 −0.669131 0.743145i \(-0.733333\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(348\) 0 0
\(349\) 0 0 0.104528 0.994522i \(-0.466667\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(354\) 0 0
\(355\) 0.913545 0.406737i 0.913545 0.406737i
\(356\) −0.209057 1.98904i −0.209057 1.98904i
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(360\) 0 0
\(361\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −0.913545 0.406737i −0.913545 0.406737i −0.104528 0.994522i \(-0.533333\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(368\) 0.618034 1.90211i 0.618034 1.90211i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −0.309017 0.951057i −0.309017 0.951057i
\(373\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(374\) 0 0
\(375\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0.618034 1.90211i 0.618034 1.90211i 0.309017 0.951057i \(-0.400000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0.978148 + 0.207912i 0.978148 + 0.207912i 0.669131 0.743145i \(-0.266667\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(389\) −0.913545 0.406737i −0.913545 0.406737i −0.104528 0.994522i \(-0.533333\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −0.669131 0.743145i −0.669131 0.743145i 0.309017 0.951057i \(-0.400000\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −0.669131 + 0.743145i −0.669131 + 0.743145i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 −0.978148 0.207912i \(-0.933333\pi\)
0.978148 + 0.207912i \(0.0666667\pi\)
\(410\) 0 0
\(411\) 0.809017 0.587785i 0.809017 0.587785i
\(412\) 0.978148 0.207912i 0.978148 0.207912i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(420\) 0 0
\(421\) 0.104528 0.994522i 0.104528 0.994522i −0.809017 0.587785i \(-0.800000\pi\)
0.913545 0.406737i \(-0.133333\pi\)
\(422\) 0 0
\(423\) −0.669131 0.743145i −0.669131 0.743145i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(432\) −0.104528 + 0.994522i −0.104528 + 0.994522i
\(433\) −1.61803 + 1.17557i −1.61803 + 1.17557i −0.809017 + 0.587785i \(0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(440\) 0 0
\(441\) 1.00000 1.00000
\(442\) 0 0
\(443\) 0.104528 0.994522i 0.104528 0.994522i −0.809017 0.587785i \(-0.800000\pi\)
0.913545 0.406737i \(-0.133333\pi\)
\(444\) −0.913545 0.406737i −0.913545 0.406737i
\(445\) −1.33826 1.48629i −1.33826 1.48629i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −0.309017 0.951057i −0.309017 0.951057i −0.978148 0.207912i \(-0.933333\pi\)
0.669131 0.743145i \(-0.266667\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 0.978148 0.207912i \(-0.0666667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) −0.618034 1.90211i −0.618034 1.90211i
\(461\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(462\) 0 0
\(463\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(464\) 0 0
\(465\) −0.809017 0.587785i −0.809017 0.587785i
\(466\) 0 0
\(467\) −0.309017 + 0.951057i −0.309017 + 0.951057i 0.669131 + 0.743145i \(0.266667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −0.669131 + 0.743145i −0.669131 + 0.743145i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −0.913545 0.406737i −0.913545 0.406737i
\(478\) 0 0
\(479\) 0 0 0.978148 0.207912i \(-0.0666667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 1.00000 1.00000
\(486\) 0 0
\(487\) 0.809017 + 0.587785i 0.809017 + 0.587785i 0.913545 0.406737i \(-0.133333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(488\) 0 0
\(489\) −0.669131 0.743145i −0.669131 0.743145i
\(490\) 0 0
\(491\) 0 0 −0.104528 0.994522i \(-0.533333\pi\)
0.104528 + 0.994522i \(0.466667\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −1.00000 −1.00000
\(497\) 0 0
\(498\) 0 0
\(499\) 0.104528 + 0.994522i 0.104528 + 0.994522i 0.913545 + 0.406737i \(0.133333\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(500\) 0.669131 + 0.743145i 0.669131 + 0.743145i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 1.00000 1.00000
\(508\) 0 0
\(509\) 1.82709 0.813473i 1.82709 0.813473i 0.913545 0.406737i \(-0.133333\pi\)
0.913545 0.406737i \(-0.133333\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0.669131 0.743145i 0.669131 0.743145i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0.809017 0.587785i 0.809017 0.587785i −0.104528 0.994522i \(-0.533333\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(522\) 0 0
\(523\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −1.50000 2.59808i −1.50000 2.59808i
\(530\) 0 0
\(531\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0.978148 + 0.207912i 0.978148 + 0.207912i
\(538\) 0 0
\(539\) 0 0
\(540\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(541\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(542\) 0 0
\(543\) 0.104528 + 0.994522i 0.104528 + 0.994522i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 0.104528 0.994522i \(-0.466667\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(548\) −0.309017 0.951057i −0.309017 0.951057i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −0.978148 + 0.207912i −0.978148 + 0.207912i
\(556\) 0 0
\(557\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.978148 0.207912i \(-0.933333\pi\)
0.978148 + 0.207912i \(0.0666667\pi\)
\(564\) −0.913545 + 0.406737i −0.913545 + 0.406737i
\(565\) 0.913545 + 0.406737i 0.913545 + 0.406737i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 0.104528 0.994522i \(-0.466667\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(570\) 0 0
\(571\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(572\) 0 0
\(573\) −1.00000 −1.00000
\(574\) 0 0
\(575\) 0 0
\(576\) 0.913545 + 0.406737i 0.913545 + 0.406737i
\(577\) −0.309017 + 0.951057i −0.309017 + 0.951057i 0.669131 + 0.743145i \(0.266667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −0.913545 0.406737i −0.913545 0.406737i −0.104528 0.994522i \(-0.533333\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(588\) 0.309017 0.951057i 0.309017 0.951057i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −0.669131 + 0.743145i −0.669131 + 0.743145i
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −0.913545 + 0.406737i −0.913545 + 0.406737i
\(598\) 0 0
\(599\) 1.33826 + 1.48629i 1.33826 + 1.48629i 0.669131 + 0.743145i \(0.266667\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(600\) 0 0
\(601\) 0 0 −0.913545 0.406737i \(-0.866667\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(602\) 0 0
\(603\) −0.309017 0.951057i −0.309017 0.951057i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 −0.978148 0.207912i \(-0.933333\pi\)
0.978148 + 0.207912i \(0.0666667\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(618\) 0 0
\(619\) 0.104528 0.994522i 0.104528 0.994522i −0.809017 0.587785i \(-0.800000\pi\)
0.913545 0.406737i \(-0.133333\pi\)
\(620\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(621\) 1.33826 + 1.48629i 1.33826 + 1.48629i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0.978148 + 0.207912i 0.978148 + 0.207912i
\(626\) 0 0
\(627\) 0 0
\(628\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(629\) 0 0
\(630\) 0 0
\(631\) 0.809017 0.587785i 0.809017 0.587785i −0.104528 0.994522i \(-0.533333\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) −0.669131 + 0.743145i −0.669131 + 0.743145i
\(637\) 0 0
\(638\) 0 0
\(639\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(640\) 0 0
\(641\) −0.209057 + 1.98904i −0.209057 + 1.98904i −0.104528 + 0.994522i \(0.533333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(642\) 0 0
\(643\) 1.33826 + 1.48629i 1.33826 + 1.48629i 0.669131 + 0.743145i \(0.266667\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0.618034 + 1.90211i 0.618034 + 1.90211i 0.309017 + 0.951057i \(0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −0.913545 + 0.406737i −0.913545 + 0.406737i
\(653\) 0.104528 + 0.994522i 0.104528 + 0.994522i 0.913545 + 0.406737i \(0.133333\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(660\) 0 0
\(661\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0.618034 + 1.90211i 0.618034 + 1.90211i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 0.669131 0.743145i \(-0.266667\pi\)
−0.669131 + 0.743145i \(0.733333\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0.309017 0.951057i 0.309017 0.951057i
\(677\) 0 0 0.978148 0.207912i \(-0.0666667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(684\) 0 0
\(685\) −0.809017 0.587785i −0.809017 0.587785i
\(686\) 0 0
\(687\) −1.95630 + 0.415823i −1.95630 + 0.415823i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −0.669131 + 0.743145i −0.669131 + 0.743145i −0.978148 0.207912i \(-0.933333\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(692\) 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 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(706\) 0 0
\(707\) 0 0
\(708\) 0.809017 0.587785i 0.809017 0.587785i
\(709\) 0.978148 0.207912i 0.978148 0.207912i 0.309017 0.951057i \(-0.400000\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −1.33826 + 1.48629i −1.33826 + 1.48629i
\(714\) 0 0
\(715\) 0 0
\(716\) 0.500000 0.866025i 0.500000 0.866025i
\(717\) 0 0
\(718\) 0 0
\(719\) 0.809017 0.587785i 0.809017 0.587785i −0.104528 0.994522i \(-0.533333\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(720\) 0.978148 0.207912i 0.978148 0.207912i
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0.978148 + 0.207912i 0.978148 + 0.207912i
\(725\) 0 0
\(726\) 0 0
\(727\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(728\) 0 0
\(729\) −0.809017 0.587785i −0.809017 0.587785i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 −0.104528 0.994522i \(-0.533333\pi\)
0.104528 + 0.994522i \(0.466667\pi\)
\(734\) 0 0
\(735\) −0.309017 0.951057i −0.309017 0.951057i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(740\) −0.104528 + 0.994522i −0.104528 + 0.994522i
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.669131 0.743145i \(-0.733333\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −0.913545 + 0.406737i −0.913545 + 0.406737i −0.809017 0.587785i \(-0.800000\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(752\) 0.104528 + 0.994522i 0.104528 + 0.994522i
\(753\) 1.33826 + 1.48629i 1.33826 + 1.48629i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −0.309017 0.951057i −0.309017 0.951057i −0.978148 0.207912i \(-0.933333\pi\)
0.669131 0.743145i \(-0.266667\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.978148 0.207912i \(-0.933333\pi\)
0.978148 + 0.207912i \(0.0666667\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0.669131 0.743145i 0.669131 0.743145i
\(769\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(770\) 0 0
\(771\) −1.00000 1.73205i −1.00000 1.73205i
\(772\) 0 0
\(773\) −1.61803 1.17557i −1.61803 1.17557i −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 0.587785i \(-0.800000\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −0.809017 0.587785i −0.809017 0.587785i
\(785\) 0.913545 + 0.406737i 0.913545 + 0.406737i
\(786\) 0 0
\(787\) 0 0 −0.669131 0.743145i \(-0.733333\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −0.104528 + 0.994522i −0.104528 + 0.994522i
\(796\) 0.104528 + 0.994522i 0.104528 + 0.994522i
\(797\) −0.669131 0.743145i −0.669131 0.743145i 0.309017 0.951057i \(-0.400000\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 1.33826 1.48629i 1.33826 1.48629i
\(802\) 0 0
\(803\) 0 0
\(804\) −1.00000 −1.00000
\(805\) 0