Properties

Label 563.1.b.b.562.2
Level 563
Weight 1
Character 563.562
Analytic conductor 0.281
Analytic rank 0
Dimension 2
Projective image \(S_{4}\)
CM/RM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 563 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 563.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.280973602112\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(S_{4}\)
Projective field Galois closure of 4.2.563.1
Artin image $\GL(2,3)$
Artin field Galois closure of 8.2.178453547.2

Embedding invariants

Embedding label 562.2
Root \(-1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 563.562
Dual form 563.1.b.b.562.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.41421i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.41421i q^{5} -1.41421i q^{6} +1.00000 q^{7} +O(q^{10})\) \(q+1.41421i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.41421i q^{5} -1.41421i q^{6} +1.00000 q^{7} -2.00000 q^{10} +1.00000 q^{12} +1.41421i q^{14} -1.41421i q^{15} -1.00000 q^{16} -1.00000 q^{17} +1.00000 q^{19} -1.41421i q^{20} -1.00000 q^{21} -1.00000 q^{23} -1.00000 q^{25} +1.00000 q^{27} -1.00000 q^{28} +2.00000 q^{30} -1.41421i q^{31} -1.41421i q^{32} -1.41421i q^{34} +1.41421i q^{35} +1.41421i q^{38} -1.41421i q^{42} +1.41421i q^{43} -1.41421i q^{46} +1.00000 q^{47} +1.00000 q^{48} -1.41421i q^{50} +1.00000 q^{51} +1.41421i q^{53} +1.41421i q^{54} -1.00000 q^{57} +1.00000 q^{59} +1.41421i q^{60} +1.00000 q^{61} +2.00000 q^{62} +1.00000 q^{64} +1.00000 q^{67} +1.00000 q^{68} +1.00000 q^{69} -2.00000 q^{70} +1.00000 q^{71} +1.00000 q^{75} -1.00000 q^{76} +1.41421i q^{79} -1.41421i q^{80} -1.00000 q^{81} -1.41421i q^{83} +1.00000 q^{84} -1.41421i q^{85} -2.00000 q^{86} -1.41421i q^{89} +1.00000 q^{92} +1.41421i q^{93} +1.41421i q^{94} +1.41421i q^{95} +1.41421i q^{96} +1.41421i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} - 2q^{4} + 2q^{7} + O(q^{10}) \) \( 2q - 2q^{3} - 2q^{4} + 2q^{7} - 4q^{10} + 2q^{12} - 2q^{16} - 2q^{17} + 2q^{19} - 2q^{21} - 2q^{23} - 2q^{25} + 2q^{27} - 2q^{28} + 4q^{30} + 2q^{47} + 2q^{48} + 2q^{51} - 2q^{57} + 2q^{59} + 2q^{61} + 4q^{62} + 2q^{64} + 2q^{67} + 2q^{68} + 2q^{69} - 4q^{70} + 2q^{71} + 2q^{75} - 2q^{76} - 2q^{81} + 2q^{84} - 4q^{86} + 2q^{92} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/563\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(4\) −1.00000 −1.00000
\(5\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(6\) − 1.41421i − 1.41421i
\(7\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) −2.00000 −2.00000
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 1.00000 1.00000
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 1.41421i 1.41421i
\(15\) − 1.41421i − 1.41421i
\(16\) −1.00000 −1.00000
\(17\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(18\) 0 0
\(19\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(20\) − 1.41421i − 1.41421i
\(21\) −1.00000 −1.00000
\(22\) 0 0
\(23\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(24\) 0 0
\(25\) −1.00000 −1.00000
\(26\) 0 0
\(27\) 1.00000 1.00000
\(28\) −1.00000 −1.00000
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 2.00000 2.00000
\(31\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(32\) − 1.41421i − 1.41421i
\(33\) 0 0
\(34\) − 1.41421i − 1.41421i
\(35\) 1.41421i 1.41421i
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 1.41421i 1.41421i
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) − 1.41421i − 1.41421i
\(43\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) − 1.41421i − 1.41421i
\(47\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(48\) 1.00000 1.00000
\(49\) 0 0
\(50\) − 1.41421i − 1.41421i
\(51\) 1.00000 1.00000
\(52\) 0 0
\(53\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(54\) 1.41421i 1.41421i
\(55\) 0 0
\(56\) 0 0
\(57\) −1.00000 −1.00000
\(58\) 0 0
\(59\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(60\) 1.41421i 1.41421i
\(61\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(62\) 2.00000 2.00000
\(63\) 0 0
\(64\) 1.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(68\) 1.00000 1.00000
\(69\) 1.00000 1.00000
\(70\) −2.00000 −2.00000
\(71\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 1.00000 1.00000
\(76\) −1.00000 −1.00000
\(77\) 0 0
\(78\) 0 0
\(79\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(80\) − 1.41421i − 1.41421i
\(81\) −1.00000 −1.00000
\(82\) 0 0
\(83\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(84\) 1.00000 1.00000
\(85\) − 1.41421i − 1.41421i
\(86\) −2.00000 −2.00000
\(87\) 0 0
\(88\) 0 0
\(89\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1.00000 1.00000
\(93\) 1.41421i 1.41421i
\(94\) 1.41421i 1.41421i
\(95\) 1.41421i 1.41421i
\(96\) 1.41421i 1.41421i
\(97\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 1.00000 1.00000
\(101\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(102\) 1.41421i 1.41421i
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) − 1.41421i − 1.41421i
\(106\) −2.00000 −2.00000
\(107\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(108\) −1.00000 −1.00000
\(109\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −1.00000 −1.00000
\(113\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(114\) − 1.41421i − 1.41421i
\(115\) − 1.41421i − 1.41421i
\(116\) 0 0
\(117\) 0 0
\(118\) 1.41421i 1.41421i
\(119\) −1.00000 −1.00000
\(120\) 0 0
\(121\) −1.00000 −1.00000
\(122\) 1.41421i 1.41421i
\(123\) 0 0
\(124\) 1.41421i 1.41421i
\(125\) 0 0
\(126\) 0 0
\(127\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(128\) 0 0
\(129\) − 1.41421i − 1.41421i
\(130\) 0 0
\(131\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(132\) 0 0
\(133\) 1.00000 1.00000
\(134\) 1.41421i 1.41421i
\(135\) 1.41421i 1.41421i
\(136\) 0 0
\(137\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(138\) 1.41421i 1.41421i
\(139\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(140\) − 1.41421i − 1.41421i
\(141\) −1.00000 −1.00000
\(142\) 1.41421i 1.41421i
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(150\) 1.41421i 1.41421i
\(151\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 2.00000 2.00000
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) −2.00000 −2.00000
\(159\) − 1.41421i − 1.41421i
\(160\) 2.00000 2.00000
\(161\) −1.00000 −1.00000
\(162\) − 1.41421i − 1.41421i
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 2.00000 2.00000
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) −1.00000 −1.00000
\(170\) 2.00000 2.00000
\(171\) 0 0
\(172\) − 1.41421i − 1.41421i
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) −1.00000 −1.00000
\(176\) 0 0
\(177\) −1.00000 −1.00000
\(178\) 2.00000 2.00000
\(179\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(182\) 0 0
\(183\) −1.00000 −1.00000
\(184\) 0 0
\(185\) 0 0
\(186\) −2.00000 −2.00000
\(187\) 0 0
\(188\) −1.00000 −1.00000
\(189\) 1.00000 1.00000
\(190\) −2.00000 −2.00000
\(191\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(192\) −1.00000 −1.00000
\(193\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(194\) −2.00000 −2.00000
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(200\) 0 0
\(201\) −1.00000 −1.00000
\(202\) − 1.41421i − 1.41421i
\(203\) 0 0
\(204\) −1.00000 −1.00000
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 2.00000 2.00000
\(211\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(212\) − 1.41421i − 1.41421i
\(213\) −1.00000 −1.00000
\(214\) 1.41421i 1.41421i
\(215\) −2.00000 −2.00000
\(216\) 0 0
\(217\) − 1.41421i − 1.41421i
\(218\) 2.00000 2.00000
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(224\) − 1.41421i − 1.41421i
\(225\) 0 0
\(226\) − 1.41421i − 1.41421i
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 1.00000 1.00000
\(229\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(230\) 2.00000 2.00000
\(231\) 0 0
\(232\) 0 0
\(233\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(234\) 0 0
\(235\) 1.41421i 1.41421i
\(236\) −1.00000 −1.00000
\(237\) − 1.41421i − 1.41421i
\(238\) − 1.41421i − 1.41421i
\(239\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(240\) 1.41421i 1.41421i
\(241\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(242\) − 1.41421i − 1.41421i
\(243\) 0 0
\(244\) −1.00000 −1.00000
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 1.41421i 1.41421i
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) − 1.41421i − 1.41421i
\(255\) 1.41421i 1.41421i
\(256\) 1.00000 1.00000
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 2.00000 2.00000
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 2.00000 2.00000
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) −2.00000 −2.00000
\(266\) 1.41421i 1.41421i
\(267\) 1.41421i 1.41421i
\(268\) −1.00000 −1.00000
\(269\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(270\) −2.00000 −2.00000
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 1.00000 1.00000
\(273\) 0 0
\(274\) 2.82843i 2.82843i
\(275\) 0 0
\(276\) −1.00000 −1.00000
\(277\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(278\) 2.00000 2.00000
\(279\) 0 0
\(280\) 0 0
\(281\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(282\) − 1.41421i − 1.41421i
\(283\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(284\) −1.00000 −1.00000
\(285\) − 1.41421i − 1.41421i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 0 0
\(290\) 0 0
\(291\) − 1.41421i − 1.41421i
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 1.41421i 1.41421i
\(296\) 0 0
\(297\) 0 0
\(298\) 1.41421i 1.41421i
\(299\) 0 0
\(300\) −1.00000 −1.00000
\(301\) 1.41421i 1.41421i
\(302\) −2.00000 −2.00000
\(303\) 1.00000 1.00000
\(304\) −1.00000 −1.00000
\(305\) 1.41421i 1.41421i
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 2.82843i 2.82843i
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) − 1.41421i − 1.41421i
\(317\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(318\) 2.00000 2.00000
\(319\) 0 0
\(320\) 1.41421i 1.41421i
\(321\) −1.00000 −1.00000
\(322\) − 1.41421i − 1.41421i
\(323\) −1.00000 −1.00000
\(324\) 1.00000 1.00000
\(325\) 0 0
\(326\) 0 0
\(327\) 1.41421i 1.41421i
\(328\) 0 0
\(329\) 1.00000 1.00000
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 1.41421i 1.41421i
\(333\) 0 0
\(334\) 0 0
\(335\) 1.41421i 1.41421i
\(336\) 1.00000 1.00000
\(337\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(338\) − 1.41421i − 1.41421i
\(339\) 1.00000 1.00000
\(340\) 1.41421i 1.41421i
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −1.00000
\(344\) 0 0
\(345\) 1.41421i 1.41421i
\(346\) 0 0
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(350\) − 1.41421i − 1.41421i
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) − 1.41421i − 1.41421i
\(355\) 1.41421i 1.41421i
\(356\) 1.41421i 1.41421i
\(357\) 1.00000 1.00000
\(358\) − 2.82843i − 2.82843i
\(359\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(360\) 0 0
\(361\) 0 0
\(362\) − 1.41421i − 1.41421i
\(363\) 1.00000 1.00000
\(364\) 0 0
\(365\) 0 0
\(366\) − 1.41421i − 1.41421i
\(367\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(368\) 1.00000 1.00000
\(369\) 0 0
\(370\) 0 0
\(371\) 1.41421i 1.41421i
\(372\) − 1.41421i − 1.41421i
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 1.41421i 1.41421i
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) − 1.41421i − 1.41421i
\(381\) 1.00000 1.00000
\(382\) 1.41421i 1.41421i
\(383\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 1.41421i 1.41421i
\(387\) 0 0
\(388\) − 1.41421i − 1.41421i
\(389\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(390\) 0 0
\(391\) 1.00000 1.00000
\(392\) 0 0
\(393\) 1.41421i 1.41421i
\(394\) 0 0
\(395\) −2.00000 −2.00000
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 2.00000 2.00000
\(399\) −1.00000 −1.00000
\(400\) 1.00000 1.00000
\(401\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(402\) − 1.41421i − 1.41421i
\(403\) 0 0
\(404\) 1.00000 1.00000
\(405\) − 1.41421i − 1.41421i
\(406\) 0 0
\(407\) 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
\(411\) −2.00000 −2.00000
\(412\) 0 0
\(413\) 1.00000 1.00000
\(414\) 0 0
\(415\) 2.00000 2.00000
\(416\) 0 0
\(417\) 1.41421i 1.41421i
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 1.41421i 1.41421i
\(421\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(422\) 1.41421i 1.41421i
\(423\) 0 0
\(424\) 0 0
\(425\) 1.00000 1.00000
\(426\) − 1.41421i − 1.41421i
\(427\) 1.00000 1.00000
\(428\) −1.00000 −1.00000
\(429\) 0 0
\(430\) − 2.82843i − 2.82843i
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) −1.00000 −1.00000
\(433\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(434\) 2.00000 2.00000
\(435\) 0 0
\(436\) 1.41421i 1.41421i
\(437\) −1.00000 −1.00000
\(438\) 0 0
\(439\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(444\) 0 0
\(445\) 2.00000 2.00000
\(446\) − 1.41421i − 1.41421i
\(447\) −1.00000 −1.00000
\(448\) 1.00000 1.00000
\(449\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 1.00000 1.00000
\(453\) − 1.41421i − 1.41421i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) −2.00000 −2.00000
\(459\) −1.00000 −1.00000
\(460\) 1.41421i 1.41421i
\(461\) 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
\(465\) −2.00000 −2.00000
\(466\) 2.00000 2.00000
\(467\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(468\) 0 0
\(469\) 1.00000 1.00000
\(470\) −2.00000 −2.00000
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 2.00000 2.00000
\(475\) −1.00000 −1.00000
\(476\) 1.00000 1.00000
\(477\) 0 0
\(478\) 2.00000 2.00000
\(479\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(480\) −2.00000 −2.00000
\(481\) 0 0
\(482\) 1.41421i 1.41421i
\(483\) 1.00000 1.00000
\(484\) 1.00000 1.00000
\(485\) −2.00000 −2.00000
\(486\) 0 0
\(487\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 1.41421i 1.41421i
\(497\) 1.00000 1.00000
\(498\) −2.00000 −2.00000
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(504\) 0 0
\(505\) − 1.41421i − 1.41421i
\(506\) 0 0
\(507\) 1.00000 1.00000
\(508\) 1.00000 1.00000
\(509\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(510\) −2.00000 −2.00000
\(511\) 0 0
\(512\) 1.41421i 1.41421i
\(513\) 1.00000 1.00000
\(514\) 0 0
\(515\) 0 0
\(516\) 1.41421i 1.41421i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 1.41421i 1.41421i
\(525\) 1.00000 1.00000
\(526\) 0 0
\(527\) 1.41421i 1.41421i
\(528\) 0 0
\(529\) 0 0
\(530\) − 2.82843i − 2.82843i
\(531\) 0 0
\(532\) −1.00000 −1.00000
\(533\) 0 0
\(534\) −2.00000 −2.00000
\(535\) 1.41421i 1.41421i
\(536\) 0 0
\(537\) 2.00000 2.00000
\(538\) 1.41421i 1.41421i
\(539\) 0 0
\(540\) − 1.41421i − 1.41421i
\(541\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(542\) 0 0
\(543\) 1.00000 1.00000
\(544\) 1.41421i 1.41421i
\(545\) 2.00000 2.00000
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) −2.00000 −2.00000
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 1.41421i 1.41421i
\(554\) − 1.41421i − 1.41421i
\(555\) 0 0
\(556\) 1.41421i 1.41421i
\(557\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) − 1.41421i − 1.41421i
\(561\) 0 0
\(562\) − 1.41421i − 1.41421i
\(563\) −1.00000 −1.00000
\(564\) 1.00000 1.00000
\(565\) − 1.41421i − 1.41421i
\(566\) −2.00000 −2.00000
\(567\) −1.00000 −1.00000
\(568\) 0 0
\(569\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(570\) 2.00000 2.00000
\(571\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(572\) 0 0
\(573\) −1.00000 −1.00000
\(574\) 0 0
\(575\) 1.00000 1.00000
\(576\) 0 0
\(577\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(578\) 0 0
\(579\) −1.00000 −1.00000
\(580\) 0 0
\(581\) − 1.41421i − 1.41421i
\(582\) 2.00000 2.00000
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) − 1.41421i − 1.41421i
\(590\) −2.00000 −2.00000
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) − 1.41421i − 1.41421i
\(596\) −1.00000 −1.00000
\(597\) 1.41421i 1.41421i
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(602\) −2.00000 −2.00000
\(603\) 0 0
\(604\) − 1.41421i − 1.41421i
\(605\) − 1.41421i − 1.41421i
\(606\) 1.41421i 1.41421i
\(607\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(608\) − 1.41421i − 1.41421i
\(609\) 0 0
\(610\) −2.00000 −2.00000
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) −2.00000 −2.00000
\(621\) −1.00000 −1.00000
\(622\) 0 0
\(623\) − 1.41421i − 1.41421i
\(624\) 0 0
\(625\) −1.00000 −1.00000
\(626\) 2.00000 2.00000
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) −1.00000 −1.00000
\(634\) −2.00000 −2.00000
\(635\) − 1.41421i − 1.41421i
\(636\) 1.41421i 1.41421i
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(642\) − 1.41421i − 1.41421i
\(643\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(644\) 1.00000 1.00000
\(645\) 2.00000 2.00000
\(646\) − 1.41421i − 1.41421i
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 1.41421i 1.41421i
\(652\) 0 0
\(653\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(654\) −2.00000 −2.00000
\(655\) 2.00000 2.00000
\(656\) 0 0
\(657\) 0 0
\(658\) 1.41421i 1.41421i
\(659\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1.41421i 1.41421i
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 1.00000 1.00000
\(670\) −2.00000 −2.00000
\(671\) 0 0
\(672\) 1.41421i 1.41421i
\(673\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(674\) − 1.41421i − 1.41421i
\(675\) −1.00000 −1.00000
\(676\) 1.00000 1.00000
\(677\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(678\) 1.41421i 1.41421i
\(679\) 1.41421i 1.41421i
\(680\) 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
\(685\) 2.82843i 2.82843i
\(686\) − 1.41421i − 1.41421i
\(687\) − 1.41421i − 1.41421i
\(688\) − 1.41421i − 1.41421i
\(689\) 0 0
\(690\) −2.00000 −2.00000
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 2.00000 2.00000
\(696\) 0 0
\(697\) 0 0
\(698\) 1.41421i 1.41421i
\(699\) 1.41421i 1.41421i
\(700\) 1.00000 1.00000
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) − 1.41421i − 1.41421i
\(706\) 0 0
\(707\) −1.00000 −1.00000
\(708\) 1.00000 1.00000
\(709\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(710\) −2.00000 −2.00000
\(711\) 0 0
\(712\) 0 0
\(713\) 1.41421i 1.41421i
\(714\) 1.41421i 1.41421i
\(715\) 0 0
\(716\) 2.00000 2.00000
\(717\) 1.41421i 1.41421i
\(718\) 2.00000 2.00000
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −1.00000 −1.00000
\(724\) 1.00000 1.00000
\(725\) 0 0
\(726\) 1.41421i 1.41421i
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 1.00000 1.00000
\(730\) 0 0
\(731\) − 1.41421i − 1.41421i
\(732\) 1.00000 1.00000
\(733\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(734\) 2.00000 2.00000
\(735\) 0 0
\(736\) 1.41421i 1.41421i
\(737\) 0 0
\(738\) 0 0
\(739\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −2.00000 −2.00000
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 1.41421i 1.41421i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 1.00000 1.00000
\(750\) 0 0
\(751\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(752\) −1.00000 −1.00000
\(753\) 0 0
\(754\) 0 0
\(755\) −2.00000 −2.00000
\(756\) −1.00000 −1.00000
\(757\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 1.41421i 1.41421i
\(763\) − 1.41421i − 1.41421i
\(764\) −1.00000 −1.00000
\(765\) 0 0
\(766\) − 1.41421i − 1.41421i
\(767\) 0 0
\(768\) −1.00000 −1.00000
\(769\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −1.00000 −1.00000
\(773\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(774\) 0 0
\(775\) 1.41421i 1.41421i
\(776\) 0 0
\(777\) 0 0
\(778\) −2.00000 −2.00000
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 1.41421i 1.41421i
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) −2.00000 −2.00000
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) − 2.82843i − 2.82843i
\(791\) −1.00000 −1.00000
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 2.00000 2.00000
\(796\) 1.41421i 1.41421i
\(797\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(798\) − 1.41421i − 1.41421i
\(799\) −1.00000 −1.00000
\(800\) 1.41421i 1.41421i
\(801\) 0 0
\(802\) 1.41421i 1.41421i
\(803\) 0 0
\(804\) 1.00000 1.00000
\(805\) − 1.41421i − 1.41421i
\(806\) 0 0
\(807\) −1.00000 −1.00000
\(808\) 0 0
\(809\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(810\) 2.00000 2.00000
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) −1.00000 −1.00000
\(817\) 1.41421i 1.41421i
\(818\) − 1.41421i − 1.41421i
\(819\) 0 0
\(820\) 0 0
\(821\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(822\) − 2.82843i − 2.82843i
\(823\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 1.41421i 1.41421i
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 2.82843i 2.82843i
\(831\) 1.00000 1.00000
\(832\) 0 0
\(833\) 0 0
\(834\) −2.00000 −2.00000
\(835\) 0 0
\(836\) 0 0
\(837\) − 1.41421i − 1.41421i
\(838\) 0 0
\(839\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(840\) 0 0
\(841\) 1.00000 1.00000
\(842\) 1.41421i 1.41421i
\(843\) 1.00000 1.00000
\(844\) −1.00000 −1.00000
\(845\) − 1.41421i − 1.41421i
\(846\) 0 0
\(847\) −1.00000 −1.00000
\(848\) − 1.41421i − 1.41421i
\(849\) − 1.41421i − 1.41421i
\(850\) 1.41421i 1.41421i
\(851\) 0 0
\(852\) 1.00000 1.00000
\(853\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(854\) 1.41421i 1.41421i
\(855\) 0 0
\(856\) 0 0
\(857\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(858\) 0 0
\(859\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(860\) 2.00000 2.00000
\(861\) 0 0
\(862\) 0 0
\(863\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(864\) − 1.41421i − 1.41421i
\(865\) 0 0
\(866\) 2.00000 2.00000
\(867\) 0 0
\(868\) 1.41421i 1.41421i
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) − 1.41421i − 1.41421i
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(878\) 1.41421i 1.41421i
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) − 1.41421i − 1.41421i
\(886\) 2.00000 2.00000
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) −1.00000 −1.00000
\(890\) 2.82843i 2.82843i
\(891\) 0 0
\(892\) 1.00000 1.00000
\(893\) 1.00000 1.00000
\(894\) − 1.41421i − 1.41421i
\(895\) − 2.82843i − 2.82843i
\(896\) 0 0
\(897\) 0 0
\(898\) − 1.41421i − 1.41421i
\(899\) 0 0
\(900\) 0 0
\(901\) − 1.41421i − 1.41421i
\(902\) 0 0
\(903\) − 1.41421i − 1.41421i
\(904\) 0 0
\(905\) − 1.41421i − 1.41421i
\(906\) 2.00000 2.00000
\(907\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 1.00000 1.00000
\(913\) 0 0
\(914\) 0 0
\(915\) − 1.41421i − 1.41421i
\(916\) − 1.41421i − 1.41421i
\(917\) − 1.41421i − 1.41421i
\(918\) − 1.41421i − 1.41421i
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(930\) − 2.82843i − 2.82843i
\(931\) 0 0
\(932\) 1.41421i 1.41421i
\(933\) 0 0
\(934\) − 1.41421i − 1.41421i
\(935\) 0 0
\(936\) 0 0
\(937\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(938\) 1.41421i 1.41421i
\(939\) 1.41421i 1.41421i
\(940\) − 1.41421i − 1.41421i
\(941\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −1.00000 −1.00000
\(945\) 1.41421i 1.41421i
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 1.41421i 1.41421i
\(949\) 0 0
\(950\) − 1.41421i − 1.41421i
\(951\) − 1.41421i − 1.41421i
\(952\) 0 0
\(953\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 0 0
\(955\) 1.41421i 1.41421i
\(956\) 1.41421i 1.41421i
\(957\) 0 0
\(958\) 2.00000 2.00000
\(959\) 2.00000 2.00000
\(960\) − 1.41421i − 1.41421i
\(961\) −1.00000 −1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) −1.00000 −1.00000
\(965\) 1.41421i 1.41421i
\(966\) 1.41421i 1.41421i
\(967\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(968\) 0 0
\(969\) 1.00000 1.00000
\(970\) − 2.82843i − 2.82843i
\(971\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(972\) 0 0
\(973\) − 1.41421i − 1.41421i
\(974\) 2.00000 2.00000
\(975\) 0 0
\(976\) −1.00000 −1.00000
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) − 1.41421i − 1.41421i
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −1.00000 −1.00000
\(988\) 0 0
\(989\) − 1.41421i − 1.41421i
\(990\) 0 0
\(991\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(992\) −2.00000 −2.00000
\(993\) 0 0
\(994\) 1.41421i 1.41421i
\(995\) 2.00000 2.00000
\(996\) − 1.41421i − 1.41421i
\(997\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(998\) 0 0
\(999\) 0 0
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 \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 563.1.b.b.562.2 yes 2
563.562 odd 2 inner 563.1.b.b.562.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
563.1.b.b.562.1 2 563.562 odd 2 inner
563.1.b.b.562.2 yes 2 1.1 even 1 trivial