Properties

Label 111.1.i.a.47.1
Level 111
Weight 1
Character 111.47
Analytic conductor 0.055
Analytic rank 0
Dimension 2
Projective image \(D_{3}\)
CM disc. -3
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 111 = 3 \cdot 37 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 111.i (of order \(6\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.0553962164023\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.4107.1
Artin image size \(18\)
Artin image $C_3\times S_3$
Artin field Galois closure of 6.0.36963.1

Embedding invariants

Embedding label 47.1
Root \(0.500000 + 0.866025i\)
Character \(\chi\) = 111.47
Dual form 111.1.i.a.26.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{4} +(0.500000 + 0.866025i) q^{7} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})\) \(q+(-0.500000 - 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{4} +(0.500000 + 0.866025i) q^{7} +(-0.500000 + 0.866025i) q^{9} +(-0.500000 + 0.866025i) q^{12} +(0.500000 + 0.866025i) q^{13} +(-0.500000 + 0.866025i) q^{16} +(-1.00000 - 1.73205i) q^{19} +(0.500000 - 0.866025i) q^{21} +(-0.500000 + 0.866025i) q^{25} +1.00000 q^{27} +(0.500000 - 0.866025i) q^{28} -1.00000 q^{31} +1.00000 q^{36} +1.00000 q^{37} +(0.500000 - 0.866025i) q^{39} -1.00000 q^{43} +1.00000 q^{48} +(0.500000 - 0.866025i) q^{52} +(-1.00000 + 1.73205i) q^{57} +(-1.00000 - 1.73205i) q^{61} -1.00000 q^{63} +1.00000 q^{64} +(0.500000 + 0.866025i) q^{67} -1.00000 q^{73} +1.00000 q^{75} +(-1.00000 + 1.73205i) q^{76} +(0.500000 + 0.866025i) q^{79} +(-0.500000 - 0.866025i) q^{81} -1.00000 q^{84} +(-0.500000 + 0.866025i) q^{91} +(0.500000 + 0.866025i) q^{93} -1.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{3} - q^{4} + q^{7} - q^{9} + O(q^{10}) \) \( 2q - q^{3} - q^{4} + q^{7} - q^{9} - q^{12} + q^{13} - q^{16} - 2q^{19} + q^{21} - q^{25} + 2q^{27} + q^{28} - 2q^{31} + 2q^{36} + 2q^{37} + q^{39} - 2q^{43} + 2q^{48} + q^{52} - 2q^{57} - 2q^{61} - 2q^{63} + 2q^{64} + q^{67} - 2q^{73} + 2q^{75} - 2q^{76} + q^{79} - q^{81} - 2q^{84} - q^{91} + q^{93} - 2q^{97} + O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/111\mathbb{Z}\right)^\times\).

\(n\) \(38\) \(76\)
\(\chi(n)\) \(-1\) \(e\left(\frac{2}{3}\right)\)

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 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(3\) −0.500000 0.866025i −0.500000 0.866025i
\(4\) −0.500000 0.866025i −0.500000 0.866025i
\(5\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(6\) 0 0
\(7\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(8\) 0 0
\(9\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(13\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(17\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(18\) 0 0
\(19\) −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(20\) 0 0
\(21\) 0.500000 0.866025i 0.500000 0.866025i
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(26\) 0 0
\(27\) 1.00000 1.00000
\(28\) 0.500000 0.866025i 0.500000 0.866025i
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 1.00000
\(37\) 1.00000 1.00000
\(38\) 0 0
\(39\) 0.500000 0.866025i 0.500000 0.866025i
\(40\) 0 0
\(41\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(42\) 0 0
\(43\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 1.00000 1.00000
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0.500000 0.866025i 0.500000 0.866025i
\(53\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(58\) 0 0
\(59\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(60\) 0 0
\(61\) −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(62\) 0 0
\(63\) −1.00000 −1.00000
\(64\) 1.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(72\) 0 0
\(73\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(74\) 0 0
\(75\) 1.00000 1.00000
\(76\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(77\) 0 0
\(78\) 0 0
\(79\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(80\) 0 0
\(81\) −0.500000 0.866025i −0.500000 0.866025i
\(82\) 0 0
\(83\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(84\) −1.00000 −1.00000
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(90\) 0 0
\(91\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(92\) 0 0
\(93\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 1.00000 1.00000
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(108\) −0.500000 0.866025i −0.500000 0.866025i
\(109\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(110\) 0 0
\(111\) −0.500000 0.866025i −0.500000 0.866025i
\(112\) −1.00000 −1.00000
\(113\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −1.00000 −1.00000
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(125\) 0 0
\(126\) 0 0
\(127\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(128\) 0 0
\(129\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(130\) 0 0
\(131\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(132\) 0 0
\(133\) 1.00000 1.73205i 1.00000 1.73205i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −0.500000 0.866025i −0.500000 0.866025i
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) −0.500000 0.866025i −0.500000 0.866025i
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) −1.00000 −1.00000
\(157\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 2.00000 2.00000
\(172\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(173\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(174\) 0 0
\(175\) −1.00000 −1.00000
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(182\) 0 0
\(183\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) −0.500000 0.866025i −0.500000 0.866025i
\(193\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(198\) 0 0
\(199\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(200\) 0 0
\(201\) 0.500000 0.866025i 0.500000 0.866025i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −1.00000 −1.00000
\(209\) 0 0
\(210\) 0 0
\(211\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −0.500000 0.866025i −0.500000 0.866025i
\(218\) 0 0
\(219\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(224\) 0 0
\(225\) −0.500000 0.866025i −0.500000 0.866025i
\(226\) 0 0
\(227\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(228\) 2.00000 2.00000
\(229\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0.500000 0.866025i 0.500000 0.866025i
\(238\) 0 0
\(239\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(240\) 0 0
\(241\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(242\) 0 0
\(243\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(244\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(245\) 0 0
\(246\) 0 0
\(247\) 1.00000 1.73205i 1.00000 1.73205i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −0.500000 0.866025i −0.500000 0.866025i
\(257\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(258\) 0 0
\(259\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(260\) 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 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0.500000 0.866025i 0.500000 0.866025i
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(272\) 0 0
\(273\) 1.00000 1.00000
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(278\) 0 0
\(279\) 0.500000 0.866025i 0.500000 0.866025i
\(280\) 0 0
\(281\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(282\) 0 0
\(283\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −0.500000 0.866025i −0.500000 0.866025i
\(290\) 0 0
\(291\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(292\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(293\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) −0.500000 0.866025i −0.500000 0.866025i
\(301\) −0.500000 0.866025i −0.500000 0.866025i
\(302\) 0 0
\(303\) 0 0
\(304\) 2.00000 2.00000
\(305\) 0 0
\(306\) 0 0
\(307\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(308\) 0 0
\(309\) −1.00000 1.73205i −1.00000 1.73205i
\(310\) 0 0
\(311\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(312\) 0 0
\(313\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0.500000 0.866025i 0.500000 0.866025i
\(317\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(325\) −1.00000 −1.00000
\(326\) 0 0
\(327\) −1.00000 −1.00000
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(332\) 0 0
\(333\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(334\) 0 0
\(335\) 0 0
\(336\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(337\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 1.00000
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(350\) 0 0
\(351\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(352\) 0 0
\(353\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) −1.50000 + 2.59808i −1.50000 + 2.59808i
\(362\) 0 0
\(363\) −0.500000 0.866025i −0.500000 0.866025i
\(364\) 1.00000 1.00000
\(365\) 0 0
\(366\) 0 0
\(367\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0.500000 0.866025i 0.500000 0.866025i
\(373\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(380\) 0 0
\(381\) −1.00000 −1.00000
\(382\) 0 0
\(383\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0.500000 0.866025i 0.500000 0.866025i
\(388\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(389\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(390\) 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
\(399\) −2.00000 −2.00000
\(400\) −0.500000 0.866025i −0.500000 0.866025i
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) −0.500000 0.866025i −0.500000 0.866025i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(410\) 0 0
\(411\) 0 0
\(412\) −1.00000 1.73205i −1.00000 1.73205i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −1.00000 −1.00000
\(418\) 0 0
\(419\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(420\) 0 0
\(421\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 1.00000 1.73205i 1.00000 1.73205i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(432\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(433\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(434\) 0 0
\(435\) 0 0
\(436\) −1.00000 −1.00000
\(437\) 0 0
\(438\) 0 0
\(439\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(449\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0.500000 0.866025i 0.500000 0.866025i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(462\) 0 0
\(463\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(469\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(470\) 0 0
\(471\) −1.00000 −1.00000
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 2.00000 2.00000
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(480\) 0 0
\(481\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(482\) 0 0
\(483\) 0 0
\(484\) −0.500000 0.866025i −0.500000 0.866025i
\(485\) 0 0
\(486\) 0 0
\(487\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(488\) 0 0
\(489\) 2.00000 2.00000
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0.500000 0.866025i 0.500000 0.866025i
\(497\) 0 0
\(498\) 0 0
\(499\) −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) −1.00000 −1.00000
\(509\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(510\) 0 0
\(511\) −0.500000 0.866025i −0.500000 0.866025i
\(512\) 0 0
\(513\) −1.00000 1.73205i −1.00000 1.73205i
\(514\) 0 0
\(515\) 0 0
\(516\) 0.500000 0.866025i 0.500000 0.866025i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(522\) 0 0
\(523\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(524\) 0 0
\(525\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 1.00000 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) −2.00000 −2.00000
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(542\) 0 0
\(543\) 0.500000 0.866025i 0.500000 0.866025i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(548\) 0 0
\(549\) 2.00000 2.00000
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(554\) 0 0
\(555\) 0 0
\(556\) −1.00000 −1.00000
\(557\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(558\) 0 0
\(559\) −0.500000 0.866025i −0.500000 0.866025i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0.500000 0.866025i 0.500000 0.866025i
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(577\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(578\) 0 0
\(579\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(588\) 0 0
\(589\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(590\) 0 0
\(591\) 0 0
\(592\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(598\) 0 0
\(599\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(600\) 0 0
\(601\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(602\) 0 0
\(603\) −1.00000 −1.00000
\(604\) 0.500000 0.866025i 0.500000 0.866025i
\(605\) 0 0
\(606\) 0 0
\(607\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(618\) 0 0
\(619\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(625\) −0.500000 0.866025i −0.500000 0.866025i
\(626\) 0 0
\(627\) 0 0
\(628\) −1.00000 −1.00000
\(629\) 0 0
\(630\) 0 0
\(631\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(632\) 0 0
\(633\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(642\) 0 0
\(643\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(652\) 2.00000 2.00000
\(653\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0.500000 0.866025i 0.500000 0.866025i
\(658\) 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 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(674\) 0 0
\(675\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) −0.500000 0.866025i −0.500000 0.866025i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(684\) −1.00000 1.73205i −1.00000 1.73205i
\(685\) 0 0
\(686\) 0 0
\(687\) 0.500000 0.866025i 0.500000 0.866025i
\(688\) 0.500000 0.866025i 0.500000 0.866025i
\(689\) 0 0
\(690\) 0 0
\(691\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(701\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(702\) 0 0
\(703\) −1.00000 1.73205i −1.00000 1.73205i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(710\) 0 0
\(711\) −1.00000 −1.00000
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(720\) 0 0
\(721\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(722\) 0 0
\(723\) 0.500000 0.866025i 0.500000 0.866025i
\(724\) 0.500000 0.866025i 0.500000 0.866025i
\(725\) 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
\(729\) 1.00000 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 2.00000 2.00000
\(733\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(740\) 0 0
\(741\) −2.00000 −2.00000
\(742\) 0 0
\(743\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0.500000 0.866025i 0.500000 0.866025i
\(757\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(762\) 0 0
\(763\) 1.00000 1.00000
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(769\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(773\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(774\) 0 0
\(775\) 0.500000 0.866025i 0.500000 0.866025i
\(776\) 0 0
\(777\) 0.500000 0.866025i 0.500000 0.866025i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 1.00000 1.73205i 1.00000 1.73205i
\(794\) 0 0
\(795\) 0 0
\(796\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(797\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) −1.00000 −1.00000
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(810\) 0 0
\(811\) −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(812\) 0 0
\(813\) −1.00000 −1.00000
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(818\) 0 0
\(819\) −0.500000 0.866025i −0.500000 0.866025i
\(820\) 0 0
\(821\)