Properties

Label 819.1.bj.a.107.1
Level $819$
Weight $1$
Character 819.107
Analytic conductor $0.409$
Analytic rank $0$
Dimension $4$
Projective image $S_{4}$
CM/RM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 819.bj (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.408734245346\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} - 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(S_{4}\)
Projective field: Galois closure of 4.2.223587.1

Embedding invariants

Embedding label 107.1
Root \(-1.22474 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 819.107
Dual form 819.1.bj.a.620.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.41421i q^{2} -1.00000 q^{4} +(0.500000 - 0.866025i) q^{7} +O(q^{10})\) \(q-1.41421i q^{2} -1.00000 q^{4} +(0.500000 - 0.866025i) q^{7} +(0.500000 + 0.866025i) q^{13} +(-1.22474 - 0.707107i) q^{14} -1.00000 q^{16} -1.41421i q^{17} +(0.500000 + 0.866025i) q^{19} +(-0.500000 - 0.866025i) q^{25} +(1.22474 - 0.707107i) q^{26} +(-0.500000 + 0.866025i) q^{28} +(-1.22474 + 0.707107i) q^{29} +1.41421i q^{32} -2.00000 q^{34} -1.00000 q^{37} +(1.22474 - 0.707107i) q^{38} +(0.500000 - 0.866025i) q^{43} +(1.22474 + 0.707107i) q^{47} +(-0.500000 - 0.866025i) q^{49} +(-1.22474 + 0.707107i) q^{50} +(-0.500000 - 0.866025i) q^{52} +(1.00000 + 1.73205i) q^{58} +1.41421i q^{59} +(0.500000 + 0.866025i) q^{61} +1.00000 q^{64} +1.41421i q^{68} +(1.22474 + 0.707107i) q^{71} +(0.500000 + 0.866025i) q^{73} +1.41421i q^{74} +(-0.500000 - 0.866025i) q^{76} +(-1.22474 - 0.707107i) q^{86} +1.00000 q^{91} +(1.00000 - 1.73205i) q^{94} +(-0.500000 + 0.866025i) q^{97} +(-1.22474 + 0.707107i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} + 2q^{7} + O(q^{10}) \) \( 4q - 4q^{4} + 2q^{7} + 2q^{13} - 4q^{16} + 2q^{19} - 2q^{25} - 2q^{28} - 8q^{34} - 4q^{37} + 2q^{43} - 2q^{49} - 2q^{52} + 4q^{58} + 2q^{61} + 4q^{64} + 2q^{73} - 2q^{76} + 4q^{91} + 4q^{94} - 2q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/819\mathbb{Z}\right)^\times\).

\(n\) \(92\) \(379\) \(703\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421i 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(3\) 0 0
\(4\) −1.00000 −1.00000
\(5\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(6\) 0 0
\(7\) 0.500000 0.866025i 0.500000 0.866025i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(12\) 0 0
\(13\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(14\) −1.22474 0.707107i −1.22474 0.707107i
\(15\) 0 0
\(16\) −1.00000 −1.00000
\(17\) 1.41421i 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(18\) 0 0
\(19\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) −0.500000 0.866025i −0.500000 0.866025i
\(26\) 1.22474 0.707107i 1.22474 0.707107i
\(27\) 0 0
\(28\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(29\) −1.22474 + 0.707107i −1.22474 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(30\) 0 0
\(31\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(32\) 1.41421i 1.41421i
\(33\) 0 0
\(34\) −2.00000 −2.00000
\(35\) 0 0
\(36\) 0 0
\(37\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(38\) 1.22474 0.707107i 1.22474 0.707107i
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(42\) 0 0
\(43\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.22474 + 0.707107i 1.22474 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(48\) 0 0
\(49\) −0.500000 0.866025i −0.500000 0.866025i
\(50\) −1.22474 + 0.707107i −1.22474 + 0.707107i
\(51\) 0 0
\(52\) −0.500000 0.866025i −0.500000 0.866025i
\(53\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(59\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(60\) 0 0
\(61\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(68\) 1.41421i 1.41421i
\(69\) 0 0
\(70\) 0 0
\(71\) 1.22474 + 0.707107i 1.22474 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(72\) 0 0
\(73\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(74\) 1.41421i 1.41421i
\(75\) 0 0
\(76\) −0.500000 0.866025i −0.500000 0.866025i
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1.22474 0.707107i −1.22474 0.707107i
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 1.00000 1.00000
\(92\) 0 0
\(93\) 0 0
\(94\) 1.00000 1.73205i 1.00000 1.73205i
\(95\) 0 0
\(96\) 0 0
\(97\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(98\) −1.22474 + 0.707107i −1.22474 + 0.707107i
\(99\) 0 0
\(100\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(101\) 1.22474 + 0.707107i 1.22474 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(102\) 0 0
\(103\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(108\) 0 0
\(109\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(113\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 1.22474 0.707107i 1.22474 0.707107i
\(117\) 0 0
\(118\) 2.00000 2.00000
\(119\) −1.22474 0.707107i −1.22474 0.707107i
\(120\) 0 0
\(121\) −0.500000 0.866025i −0.500000 0.866025i
\(122\) 1.22474 0.707107i 1.22474 0.707107i
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −1.22474 0.707107i −1.22474 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(132\) 0 0
\(133\) 1.00000 1.00000
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1.41421i 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(138\) 0 0
\(139\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 1.00000 1.73205i 1.00000 1.73205i
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 1.22474 0.707107i 1.22474 0.707107i
\(147\) 0 0
\(148\) 1.00000 1.00000
\(149\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(150\) 0 0
\(151\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.22474 0.707107i 1.22474 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(168\) 0 0
\(169\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(170\) 0 0
\(171\) 0 0
\(172\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(173\) −1.22474 + 0.707107i −1.22474 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(174\) 0 0
\(175\) −1.00000 −1.00000
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −1.22474 0.707107i −1.22474 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(180\) 0 0
\(181\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(182\) 1.41421i 1.41421i
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) −1.22474 0.707107i −1.22474 0.707107i
\(189\) 0 0
\(190\) 0 0
\(191\) −1.22474 + 0.707107i −1.22474 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(192\) 0 0
\(193\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(194\) 1.22474 + 0.707107i 1.22474 + 0.707107i
\(195\) 0 0
\(196\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(197\) 1.22474 0.707107i 1.22474 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(198\) 0 0
\(199\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 1.00000 1.73205i 1.00000 1.73205i
\(203\) 1.41421i 1.41421i
\(204\) 0 0
\(205\) 0 0
\(206\) −1.22474 0.707107i −1.22474 0.707107i
\(207\) 0 0
\(208\) −0.500000 0.866025i −0.500000 0.866025i
\(209\) 0 0
\(210\) 0 0
\(211\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 2.00000 2.00000
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −1.22474 + 0.707107i −1.22474 + 0.707107i
\(219\) 0 0
\(220\) 0 0
\(221\) 1.22474 0.707107i 1.22474 0.707107i
\(222\) 0 0
\(223\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(224\) 1.22474 + 0.707107i 1.22474 + 0.707107i
\(225\) 0 0
\(226\) 0 0
\(227\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(228\) 0 0
\(229\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 1.41421i 1.41421i
\(237\) 0 0
\(238\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(239\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(240\) 0 0
\(241\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(242\) −1.22474 + 0.707107i −1.22474 + 0.707107i
\(243\) 0 0
\(244\) −0.500000 0.866025i −0.500000 0.866025i
\(245\) 0 0
\(246\) 0 0
\(247\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 1.22474 + 0.707107i 1.22474 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 1.22474 0.707107i 1.22474 0.707107i
\(255\) 0 0
\(256\) 1.00000 1.00000
\(257\) 1.41421i 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(258\) 0 0
\(259\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(260\) 0 0
\(261\) 0 0
\(262\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(263\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 1.41421i 1.41421i
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(272\) 1.41421i 1.41421i
\(273\) 0 0
\(274\) −2.00000 −2.00000
\(275\) 0 0
\(276\) 0 0
\(277\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(284\) −1.22474 0.707107i −1.22474 0.707107i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −1.00000 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) −0.500000 0.866025i −0.500000 0.866025i
\(293\) −1.22474 0.707107i −1.22474 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −0.500000 0.866025i −0.500000 0.866025i
\(302\) 0 0
\(303\) 0 0
\(304\) −0.500000 0.866025i −0.500000 0.866025i
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(312\) 0 0
\(313\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(314\) −1.22474 + 0.707107i −1.22474 + 0.707107i
\(315\) 0 0
\(316\) 0 0
\(317\) 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
\(323\) 1.22474 0.707107i 1.22474 0.707107i
\(324\) 0 0
\(325\) 0.500000 0.866025i 0.500000 0.866025i
\(326\) 1.22474 0.707107i 1.22474 0.707107i
\(327\) 0 0
\(328\) 0 0
\(329\) 1.22474 0.707107i 1.22474 0.707107i
\(330\) 0 0
\(331\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) −1.00000 1.73205i −1.00000 1.73205i
\(335\) 0 0
\(336\) 0 0
\(337\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(338\) 1.22474 + 0.707107i 1.22474 + 0.707107i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −1.00000
\(344\) 0 0
\(345\) 0 0
\(346\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(347\) 1.41421i 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(348\) 0 0
\(349\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(350\) 1.41421i 1.41421i
\(351\) 0 0
\(352\) 0 0
\(353\) −1.22474 0.707107i −1.22474 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(359\) −1.22474 0.707107i −1.22474 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(360\) 0 0
\(361\) 0 0
\(362\) 1.41421i 1.41421i
\(363\) 0 0
\(364\) −1.00000 −1.00000
\(365\) 0 0
\(366\) 0 0
\(367\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −1.22474 0.707107i −1.22474 0.707107i
\(378\) 0 0
\(379\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(383\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 1.22474 + 0.707107i 1.22474 + 0.707107i
\(387\) 0 0
\(388\) 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
\(394\) −1.00000 1.73205i −1.00000 1.73205i
\(395\) 0 0
\(396\) 0 0
\(397\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(398\) 1.41421i 1.41421i
\(399\) 0 0
\(400\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(401\) 1.41421i 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −1.22474 0.707107i −1.22474 0.707107i
\(405\) 0 0
\(406\) 2.00000 2.00000
\(407\) 0 0
\(408\) 0 0
\(409\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(413\) 1.22474 + 0.707107i 1.22474 + 0.707107i
\(414\) 0 0
\(415\) 0 0
\(416\) −1.22474 + 0.707107i −1.22474 + 0.707107i
\(417\) 0 0
\(418\) 0 0
\(419\) 1.22474 0.707107i 1.22474 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(422\) −1.22474 + 0.707107i −1.22474 + 0.707107i
\(423\) 0 0
\(424\) 0 0
\(425\) −1.22474 + 0.707107i −1.22474 + 0.707107i
\(426\) 0 0
\(427\) 1.00000 1.00000
\(428\) 1.41421i 1.41421i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(432\) 0 0
\(433\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(437\) 0 0
\(438\) 0 0
\(439\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −1.00000 1.73205i −1.00000 1.73205i
\(443\) 1.22474 + 0.707107i 1.22474 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0.500000 0.866025i 0.500000 0.866025i
\(449\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 2.00000 2.00000
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(458\) −1.22474 0.707107i −1.22474 0.707107i
\(459\) 0 0
\(460\) 0 0
\(461\) −1.22474 0.707107i −1.22474 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(462\) 0 0
\(463\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(464\) 1.22474 0.707107i 1.22474 0.707107i
\(465\) 0 0
\(466\) 0 0
\(467\) −1.22474 0.707107i −1.22474 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0.500000 0.866025i 0.500000 0.866025i
\(476\) 1.22474 + 0.707107i 1.22474 + 0.707107i
\(477\) 0 0
\(478\) 2.00000 2.00000
\(479\) 1.22474 + 0.707107i 1.22474 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(480\) 0 0
\(481\) −0.500000 0.866025i −0.500000 0.866025i
\(482\) 2.82843i 2.82843i
\(483\) 0 0
\(484\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(485\) 0 0
\(486\) 0 0
\(487\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(492\) 0 0
\(493\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(494\) 1.22474 + 0.707107i 1.22474 + 0.707107i
\(495\) 0 0
\(496\) 0 0
\(497\) 1.22474 0.707107i 1.22474 0.707107i
\(498\) 0 0
\(499\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(500\) 0 0
\(501\) 0 0
\(502\) 1.00000 1.73205i 1.00000 1.73205i
\(503\) −1.22474 0.707107i −1.22474 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) −0.500000 0.866025i −0.500000 0.866025i
\(509\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(510\) 0 0
\(511\) 1.00000 1.00000
\(512\) 1.41421i 1.41421i
\(513\) 0 0
\(514\) −2.00000 −2.00000
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 1.22474 + 0.707107i 1.22474 + 0.707107i
\(519\) 0 0
\(520\) 0 0
\(521\) −1.22474 + 0.707107i −1.22474 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(522\) 0 0
\(523\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(524\) 1.22474 + 0.707107i 1.22474 + 0.707107i
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 1.00000 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) −1.00000 −1.00000
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(542\) 1.41421i 1.41421i
\(543\) 0 0
\(544\) 2.00000 2.00000
\(545\) 0 0
\(546\) 0 0
\(547\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(548\) 1.41421i 1.41421i
\(549\) 0 0
\(550\) 0 0
\(551\) −1.22474 0.707107i −1.22474 0.707107i
\(552\) 0 0
\(553\) 0 0
\(554\) 1.41421i 1.41421i
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(558\) 0 0
\(559\) 1.00000 1.00000
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 1.41421i 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 1.22474 + 0.707107i 1.22474 + 0.707107i
\(567\) 0 0
\(568\) 0 0
\(569\) 1.41421i 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(570\) 0 0
\(571\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(578\) 1.41421i 1.41421i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(587\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 1.00000 1.00000
\(593\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 1.22474 0.707107i 1.22474 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(600\) 0 0
\(601\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(602\) −1.22474 + 0.707107i −1.22474 + 0.707107i
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(608\) −1.22474 + 0.707107i −1.22474 + 0.707107i
\(609\) 0 0
\(610\) 0 0
\(611\) 1.41421i 1.41421i
\(612\) 0 0
\(613\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(614\) 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
\(619\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(626\) 1.22474 + 0.707107i 1.22474 + 0.707107i
\(627\) 0 0
\(628\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(629\) 1.41421i 1.41421i
\(630\) 0 0
\(631\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0.500000 0.866025i 0.500000 0.866025i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(644\) 0 0
\(645\) 0 0
\(646\) −1.00000 1.73205i −1.00000 1.73205i
\(647\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −1.22474 0.707107i −1.22474 0.707107i
\(651\) 0 0
\(652\) −0.500000 0.866025i −0.500000 0.866025i
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) −1.00000 1.73205i −1.00000 1.73205i
\(659\) −1.22474 0.707107i −1.22474 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(660\) 0 0
\(661\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(662\) −1.22474 + 0.707107i −1.22474 + 0.707107i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) −1.22474 + 0.707107i −1.22474 + 0.707107i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(674\) 1.41421i 1.41421i
\(675\) 0 0
\(676\) 0.500000 0.866025i 0.500000 0.866025i
\(677\) 1.22474 + 0.707107i 1.22474 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(678\) 0 0
\(679\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 1.41421i 1.41421i
\(687\) 0 0
\(688\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(689\) 0 0
\(690\) 0 0
\(691\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(692\) 1.22474 0.707107i 1.22474 0.707107i
\(693\) 0 0
\(694\) −2.00000 −2.00000
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 1.22474 0.707107i 1.22474 0.707107i
\(699\) 0 0
\(700\) 1.00000 1.00000
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) −0.500000 0.866025i −0.500000 0.866025i
\(704\) 0 0
\(705\) 0 0
\(706\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(707\) 1.22474 0.707107i 1.22474 0.707107i
\(708\) 0 0
\(709\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 1.22474 + 0.707107i 1.22474 + 0.707107i
\(717\) 0 0
\(718\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(719\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(720\) 0 0
\(721\) −0.500000 0.866025i −0.500000 0.866025i
\(722\) 0 0
\(723\) 0 0
\(724\) 1.00000 1.00000
\(725\) 1.22474 + 0.707107i 1.22474 + 0.707107i
\(726\) 0 0
\(727\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −1.22474 0.707107i −1.22474 0.707107i
\(732\) 0 0
\(733\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(734\) −1.22474 0.707107i −1.22474 0.707107i
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 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
\(749\) 1.22474 + 0.707107i 1.22474 + 0.707107i
\(750\) 0 0
\(751\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(752\) −1.22474 0.707107i −1.22474 0.707107i
\(753\) 0 0
\(754\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(762\) 0 0
\(763\) −1.00000 −1.00000
\(764\) 1.22474 0.707107i 1.22474 0.707107i
\(765\) 0 0
\(766\) 0 0
\(767\) −1.22474 + 0.707107i −1.22474 + 0.707107i
\(768\) 0 0
\(769\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0.500000 0.866025i 0.500000 0.866025i
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 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
\(784\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(785\) 0 0
\(786\) 0 0
\(787\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(788\) −1.22474 + 0.707107i −1.22474 + 0.707107i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(794\) 1.22474 0.707107i 1.22474 0.707107i
\(795\) 0 0
\(796\) −1.00000 −1.00000
\(797\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(798\) 0 0
\(799\) 1.00000 1.73205i 1.00000 1.73205i
\(800\) 1.22474 0.707107i 1.22474 0.707107i
\(801\) 0 0
\(802\) −2.00000 −2.00000
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0