Properties

Label 643.1.b.b.642.2
Level $643$
Weight $1$
Character 643.642
Analytic conductor $0.321$
Analytic rank $0$
Dimension $2$
Projective image $S_{4}$
CM/RM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 643 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 643.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.320898803123\)
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 4.2.643.1
Artin image $\GL(2,3)$
Artin field Galois closure of 8.2.265847707.1

Embedding invariants

Embedding label 642.2
Root \(-1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 643.642
Dual form 643.1.b.b.642.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.41421i q^{2} +1.41421i q^{3} -1.00000 q^{4} -2.00000 q^{6} +1.00000 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.41421i q^{2} +1.41421i q^{3} -1.00000 q^{4} -2.00000 q^{6} +1.00000 q^{7} -1.00000 q^{9} +1.41421i q^{11} -1.41421i q^{12} -1.41421i q^{13} +1.41421i q^{14} -1.00000 q^{16} -1.41421i q^{17} -1.41421i q^{18} -1.41421i q^{19} +1.41421i q^{21} -2.00000 q^{22} -1.00000 q^{23} +1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{28} +1.00000 q^{29} -1.00000 q^{31} -1.41421i q^{32} -2.00000 q^{33} +2.00000 q^{34} +1.00000 q^{36} +2.00000 q^{38} +2.00000 q^{39} -1.41421i q^{41} -2.00000 q^{42} -1.41421i q^{44} -1.41421i q^{46} -1.41421i q^{48} +1.41421i q^{50} +2.00000 q^{51} +1.41421i q^{52} -1.00000 q^{53} +2.00000 q^{57} +1.41421i q^{58} +1.41421i q^{59} -1.41421i q^{62} -1.00000 q^{63} +1.00000 q^{64} -2.82843i q^{66} +1.41421i q^{67} +1.41421i q^{68} -1.41421i q^{69} +1.41421i q^{73} +1.41421i q^{75} +1.41421i q^{76} +1.41421i q^{77} +2.82843i q^{78} -1.41421i q^{79} -1.00000 q^{81} +2.00000 q^{82} -1.00000 q^{83} -1.41421i q^{84} +1.41421i q^{87} +1.00000 q^{89} -1.41421i q^{91} +1.00000 q^{92} -1.41421i q^{93} +2.00000 q^{96} -1.00000 q^{97} -1.41421i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} - 4q^{6} + 2q^{7} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} - 4q^{6} + 2q^{7} - 2q^{9} - 2q^{16} - 4q^{22} - 2q^{23} + 2q^{25} + 4q^{26} - 2q^{28} + 2q^{29} - 2q^{31} - 4q^{33} + 4q^{34} + 2q^{36} + 4q^{38} + 4q^{39} - 4q^{42} + 4q^{51} - 2q^{53} + 4q^{57} - 2q^{63} + 2q^{64} - 2q^{81} + 4q^{82} - 2q^{83} + 2q^{89} + 2q^{92} + 4q^{96} - 2q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/643\mathbb{Z}\right)^\times\).

\(n\) \(11\)
\(\chi(n)\) \(-1\)

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 \(0.750000\pi\)
\(3\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(4\) −1.00000 −1.00000
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) −2.00000 −2.00000
\(7\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(8\) 0 0
\(9\) −1.00000 −1.00000
\(10\) 0 0
\(11\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(12\) − 1.41421i − 1.41421i
\(13\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(14\) 1.41421i 1.41421i
\(15\) 0 0
\(16\) −1.00000 −1.00000
\(17\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(18\) − 1.41421i − 1.41421i
\(19\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(20\) 0 0
\(21\) 1.41421i 1.41421i
\(22\) −2.00000 −2.00000
\(23\) −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
\(26\) 2.00000 2.00000
\(27\) 0 0
\(28\) −1.00000 −1.00000
\(29\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(30\) 0 0
\(31\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(32\) − 1.41421i − 1.41421i
\(33\) −2.00000 −2.00000
\(34\) 2.00000 2.00000
\(35\) 0 0
\(36\) 1.00000 1.00000
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 2.00000 2.00000
\(39\) 2.00000 2.00000
\(40\) 0 0
\(41\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(42\) −2.00000 −2.00000
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) − 1.41421i − 1.41421i
\(45\) 0 0
\(46\) − 1.41421i − 1.41421i
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) − 1.41421i − 1.41421i
\(49\) 0 0
\(50\) 1.41421i 1.41421i
\(51\) 2.00000 2.00000
\(52\) 1.41421i 1.41421i
\(53\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.00000 2.00000
\(58\) 1.41421i 1.41421i
\(59\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) − 1.41421i − 1.41421i
\(63\) −1.00000 −1.00000
\(64\) 1.00000 1.00000
\(65\) 0 0
\(66\) − 2.82843i − 2.82843i
\(67\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(68\) 1.41421i 1.41421i
\(69\) − 1.41421i − 1.41421i
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(74\) 0 0
\(75\) 1.41421i 1.41421i
\(76\) 1.41421i 1.41421i
\(77\) 1.41421i 1.41421i
\(78\) 2.82843i 2.82843i
\(79\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(80\) 0 0
\(81\) −1.00000 −1.00000
\(82\) 2.00000 2.00000
\(83\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(84\) − 1.41421i − 1.41421i
\(85\) 0 0
\(86\) 0 0
\(87\) 1.41421i 1.41421i
\(88\) 0 0
\(89\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(90\) 0 0
\(91\) − 1.41421i − 1.41421i
\(92\) 1.00000 1.00000
\(93\) − 1.41421i − 1.41421i
\(94\) 0 0
\(95\) 0 0
\(96\) 2.00000 2.00000
\(97\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(98\) 0 0
\(99\) − 1.41421i − 1.41421i
\(100\) −1.00000 −1.00000
\(101\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(102\) 2.82843i 2.82843i
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) − 1.41421i − 1.41421i
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −1.00000 −1.00000
\(113\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(114\) 2.82843i 2.82843i
\(115\) 0 0
\(116\) −1.00000 −1.00000
\(117\) 1.41421i 1.41421i
\(118\) −2.00000 −2.00000
\(119\) − 1.41421i − 1.41421i
\(120\) 0 0
\(121\) −1.00000 −1.00000
\(122\) 0 0
\(123\) 2.00000 2.00000
\(124\) 1.00000 1.00000
\(125\) 0 0
\(126\) − 1.41421i − 1.41421i
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(132\) 2.00000 2.00000
\(133\) − 1.41421i − 1.41421i
\(134\) −2.00000 −2.00000
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 2.00000 2.00000
\(139\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 2.00000 2.00000
\(144\) 1.00000 1.00000
\(145\) 0 0
\(146\) −2.00000 −2.00000
\(147\) 0 0
\(148\) 0 0
\(149\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(150\) −2.00000 −2.00000
\(151\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(152\) 0 0
\(153\) 1.41421i 1.41421i
\(154\) −2.00000 −2.00000
\(155\) 0 0
\(156\) −2.00000 −2.00000
\(157\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(158\) 2.00000 2.00000
\(159\) − 1.41421i − 1.41421i
\(160\) 0 0
\(161\) −1.00000 −1.00000
\(162\) − 1.41421i − 1.41421i
\(163\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(164\) 1.41421i 1.41421i
\(165\) 0 0
\(166\) − 1.41421i − 1.41421i
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −1.00000 −1.00000
\(170\) 0 0
\(171\) 1.41421i 1.41421i
\(172\) 0 0
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) −2.00000 −2.00000
\(175\) 1.00000 1.00000
\(176\) − 1.41421i − 1.41421i
\(177\) −2.00000 −2.00000
\(178\) 1.41421i 1.41421i
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 2.00000 2.00000
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 2.00000 2.00000
\(187\) 2.00000 2.00000
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 1.41421i 1.41421i
\(193\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(194\) − 1.41421i − 1.41421i
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 2.00000 2.00000
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) −2.00000 −2.00000
\(202\) − 1.41421i − 1.41421i
\(203\) 1.00000 1.00000
\(204\) −2.00000 −2.00000
\(205\) 0 0
\(206\) 0 0
\(207\) 1.00000 1.00000
\(208\) 1.41421i 1.41421i
\(209\) 2.00000 2.00000
\(210\) 0 0
\(211\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(212\) 1.00000 1.00000
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −1.00000 −1.00000
\(218\) −2.00000 −2.00000
\(219\) −2.00000 −2.00000
\(220\) 0 0
\(221\) −2.00000 −2.00000
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) − 1.41421i − 1.41421i
\(225\) −1.00000 −1.00000
\(226\) −2.00000 −2.00000
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) −2.00000 −2.00000
\(229\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(230\) 0 0
\(231\) −2.00000 −2.00000
\(232\) 0 0
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) −2.00000 −2.00000
\(235\) 0 0
\(236\) − 1.41421i − 1.41421i
\(237\) 2.00000 2.00000
\(238\) 2.00000 2.00000
\(239\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(240\) 0 0
\(241\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(242\) − 1.41421i − 1.41421i
\(243\) − 1.41421i − 1.41421i
\(244\) 0 0
\(245\) 0 0
\(246\) 2.82843i 2.82843i
\(247\) −2.00000 −2.00000
\(248\) 0 0
\(249\) − 1.41421i − 1.41421i
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 1.00000 1.00000
\(253\) − 1.41421i − 1.41421i
\(254\) 0 0
\(255\) 0 0
\(256\) 1.00000 1.00000
\(257\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −1.00000 −1.00000
\(262\) 1.41421i 1.41421i
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 2.00000 2.00000
\(267\) 1.41421i 1.41421i
\(268\) − 1.41421i − 1.41421i
\(269\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 1.41421i 1.41421i
\(273\) 2.00000 2.00000
\(274\) 0 0
\(275\) 1.41421i 1.41421i
\(276\) 1.41421i 1.41421i
\(277\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(278\) 1.41421i 1.41421i
\(279\) 1.00000 1.00000
\(280\) 0 0
\(281\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(282\) 0 0
\(283\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 2.82843i 2.82843i
\(287\) − 1.41421i − 1.41421i
\(288\) 1.41421i 1.41421i
\(289\) −1.00000 −1.00000
\(290\) 0 0
\(291\) − 1.41421i − 1.41421i
\(292\) − 1.41421i − 1.41421i
\(293\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 1.41421i 1.41421i
\(299\) 1.41421i 1.41421i
\(300\) − 1.41421i − 1.41421i
\(301\) 0 0
\(302\) 2.00000 2.00000
\(303\) − 1.41421i − 1.41421i
\(304\) 1.41421i 1.41421i
\(305\) 0 0
\(306\) −2.00000 −2.00000
\(307\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(308\) − 1.41421i − 1.41421i
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(314\) 2.00000 2.00000
\(315\) 0 0
\(316\) 1.41421i 1.41421i
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 2.00000 2.00000
\(319\) 1.41421i 1.41421i
\(320\) 0 0
\(321\) 0 0
\(322\) − 1.41421i − 1.41421i
\(323\) −2.00000 −2.00000
\(324\) 1.00000 1.00000
\(325\) − 1.41421i − 1.41421i
\(326\) 0 0
\(327\) −2.00000 −2.00000
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(332\) 1.00000 1.00000
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) − 1.41421i − 1.41421i
\(337\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(338\) − 1.41421i − 1.41421i
\(339\) −2.00000 −2.00000
\(340\) 0 0
\(341\) − 1.41421i − 1.41421i
\(342\) −2.00000 −2.00000
\(343\) −1.00000 −1.00000
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) − 1.41421i − 1.41421i
\(349\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(350\) 1.41421i 1.41421i
\(351\) 0 0
\(352\) 2.00000 2.00000
\(353\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(354\) − 2.82843i − 2.82843i
\(355\) 0 0
\(356\) −1.00000 −1.00000
\(357\) 2.00000 2.00000
\(358\) 0 0
\(359\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(360\) 0 0
\(361\) −1.00000 −1.00000
\(362\) 0 0
\(363\) − 1.41421i − 1.41421i
\(364\) 1.41421i 1.41421i
\(365\) 0 0
\(366\) 0 0
\(367\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(368\) 1.00000 1.00000
\(369\) 1.41421i 1.41421i
\(370\) 0 0
\(371\) −1.00000 −1.00000
\(372\) 1.41421i 1.41421i
\(373\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(374\) 2.82843i 2.82843i
\(375\) 0 0
\(376\) 0 0
\(377\) − 1.41421i − 1.41421i
\(378\) 0 0
\(379\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) − 1.41421i − 1.41421i
\(387\) 0 0
\(388\) 1.00000 1.00000
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 1.41421i 1.41421i
\(392\) 0 0
\(393\) 1.41421i 1.41421i
\(394\) 0 0
\(395\) 0 0
\(396\) 1.41421i 1.41421i
\(397\) −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
\(400\) −1.00000 −1.00000
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) − 2.82843i − 2.82843i
\(403\) 1.41421i 1.41421i
\(404\) 1.00000 1.00000
\(405\) 0 0
\(406\) 1.41421i 1.41421i
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 1.41421i 1.41421i
\(414\) 1.41421i 1.41421i
\(415\) 0 0
\(416\) −2.00000 −2.00000
\(417\) 1.41421i 1.41421i
\(418\) 2.82843i 2.82843i
\(419\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(420\) 0 0
\(421\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(422\) − 1.41421i − 1.41421i
\(423\) 0 0
\(424\) 0 0
\(425\) − 1.41421i − 1.41421i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 2.82843i 2.82843i
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) − 1.41421i − 1.41421i
\(435\) 0 0
\(436\) − 1.41421i − 1.41421i
\(437\) 1.41421i 1.41421i
\(438\) − 2.82843i − 2.82843i
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) − 2.82843i − 2.82843i
\(443\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 1.41421i 1.41421i
\(448\) 1.00000 1.00000
\(449\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(450\) − 1.41421i − 1.41421i
\(451\) 2.00000 2.00000
\(452\) − 1.41421i − 1.41421i
\(453\) 2.00000 2.00000
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 1.41421i 1.41421i
\(459\) 0 0
\(460\) 0 0
\(461\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(462\) − 2.82843i − 2.82843i
\(463\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(464\) −1.00000 −1.00000
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) − 1.41421i − 1.41421i
\(469\) 1.41421i 1.41421i
\(470\) 0 0
\(471\) 2.00000 2.00000
\(472\) 0 0
\(473\) 0 0
\(474\) 2.82843i 2.82843i
\(475\) − 1.41421i − 1.41421i
\(476\) 1.41421i 1.41421i
\(477\) 1.00000 1.00000
\(478\) 2.00000 2.00000
\(479\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 1.41421i 1.41421i
\(483\) − 1.41421i − 1.41421i
\(484\) 1.00000 1.00000
\(485\) 0 0
\(486\) 2.00000 2.00000
\(487\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) −2.00000 −2.00000
\(493\) − 1.41421i − 1.41421i
\(494\) − 2.82843i − 2.82843i
\(495\) 0 0
\(496\) 1.00000 1.00000
\(497\) 0 0
\(498\) 2.00000 2.00000
\(499\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 2.00000 2.00000
\(507\) − 1.41421i − 1.41421i
\(508\) 0 0
\(509\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(510\) 0 0
\(511\) 1.41421i 1.41421i
\(512\) 1.41421i 1.41421i
\(513\) 0 0
\(514\) − 1.41421i − 1.41421i
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) − 1.41421i − 1.41421i
\(523\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(524\) −1.00000 −1.00000
\(525\) 1.41421i 1.41421i
\(526\) 0 0
\(527\) 1.41421i 1.41421i
\(528\) 2.00000 2.00000
\(529\) 0 0
\(530\) 0 0
\(531\) − 1.41421i − 1.41421i
\(532\) 1.41421i 1.41421i
\(533\) −2.00000 −2.00000
\(534\) −2.00000 −2.00000
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 1.41421i 1.41421i
\(539\) 0 0
\(540\) 0 0
\(541\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(542\) 0 0
\(543\) 0 0
\(544\) −2.00000 −2.00000
\(545\) 0 0
\(546\) 2.82843i 2.82843i
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) −2.00000 −2.00000
\(551\) − 1.41421i − 1.41421i
\(552\) 0 0
\(553\) − 1.41421i − 1.41421i
\(554\) 1.41421i 1.41421i
\(555\) 0 0
\(556\) −1.00000 −1.00000
\(557\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(558\) 1.41421i 1.41421i
\(559\) 0 0
\(560\) 0 0
\(561\) 2.82843i 2.82843i
\(562\) − 1.41421i − 1.41421i
\(563\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −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 \(0.750000\pi\)
\(570\) 0 0
\(571\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(572\) −2.00000 −2.00000
\(573\) 0 0
\(574\) 2.00000 2.00000
\(575\) −1.00000 −1.00000
\(576\) −1.00000 −1.00000
\(577\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(578\) − 1.41421i − 1.41421i
\(579\) − 1.41421i − 1.41421i
\(580\) 0 0
\(581\) −1.00000 −1.00000
\(582\) 2.00000 2.00000
\(583\) − 1.41421i − 1.41421i
\(584\) 0 0
\(585\) 0 0
\(586\) − 2.82843i − 2.82843i
\(587\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(588\) 0 0
\(589\) 1.41421i 1.41421i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −1.00000 −1.00000
\(597\) 0 0
\(598\) −2.00000 −2.00000
\(599\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) − 1.41421i − 1.41421i
\(604\) 1.41421i 1.41421i
\(605\) 0 0
\(606\) 2.00000 2.00000
\(607\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(608\) −2.00000 −2.00000
\(609\) 1.41421i 1.41421i
\(610\) 0 0
\(611\) 0 0
\(612\) − 1.41421i − 1.41421i
\(613\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(614\) − 1.41421i − 1.41421i
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 1.00000 1.00000
\(624\) −2.00000 −2.00000
\(625\) 1.00000 1.00000
\(626\) −2.00000 −2.00000
\(627\) 2.82843i 2.82843i
\(628\) 1.41421i 1.41421i
\(629\) 0 0
\(630\) 0 0
\(631\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(632\) 0 0
\(633\) − 1.41421i − 1.41421i
\(634\) 0 0
\(635\) 0 0
\(636\) 1.41421i 1.41421i
\(637\) 0 0
\(638\) −2.00000 −2.00000
\(639\) 0 0
\(640\) 0 0
\(641\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(642\) 0 0
\(643\) −1.00000 −1.00000
\(644\) 1.00000 1.00000
\(645\) 0 0
\(646\) − 2.82843i − 2.82843i
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) −2.00000 −2.00000
\(650\) 2.00000 2.00000
\(651\) − 1.41421i − 1.41421i
\(652\) 0 0
\(653\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(654\) − 2.82843i − 2.82843i
\(655\) 0 0
\(656\) 1.41421i 1.41421i
\(657\) − 1.41421i − 1.41421i
\(658\) 0 0
\(659\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(662\) 1.41421i 1.41421i
\(663\) − 2.82843i − 2.82843i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −1.00000 −1.00000
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 2.00000 2.00000
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 2.00000 2.00000
\(675\) 0 0
\(676\) 1.00000 1.00000
\(677\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(678\) − 2.82843i − 2.82843i
\(679\) −1.00000 −1.00000
\(680\) 0 0
\(681\) 0 0
\(682\) 2.00000 2.00000
\(683\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(684\) − 1.41421i − 1.41421i
\(685\) 0 0
\(686\) − 1.41421i − 1.41421i
\(687\) 1.41421i 1.41421i
\(688\) 0 0
\(689\) 1.41421i 1.41421i
\(690\) 0 0
\(691\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(692\) 0 0
\(693\) − 1.41421i − 1.41421i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −2.00000 −2.00000
\(698\) −2.00000 −2.00000
\(699\) 0 0
\(700\) −1.00000 −1.00000
\(701\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 1.41421i 1.41421i
\(705\) 0 0
\(706\) 2.00000 2.00000
\(707\) −1.00000 −1.00000
\(708\) 2.00000 2.00000
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 1.41421i 1.41421i
\(712\) 0 0
\(713\) 1.00000 1.00000
\(714\) 2.82843i 2.82843i
\(715\) 0 0
\(716\) 0 0
\(717\) 2.00000 2.00000
\(718\) − 1.41421i − 1.41421i
\(719\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) − 1.41421i − 1.41421i
\(723\) 1.41421i 1.41421i
\(724\) 0 0
\(725\) 1.00000 1.00000
\(726\) 2.00000 2.00000
\(727\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(728\) 0 0
\(729\) 1.00000 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(734\) 1.41421i 1.41421i
\(735\) 0 0
\(736\) 1.41421i 1.41421i
\(737\) −2.00000 −2.00000
\(738\) −2.00000 −2.00000
\(739\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(740\) 0 0
\(741\) − 2.82843i − 2.82843i
\(742\) − 1.41421i − 1.41421i
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 1.41421i 1.41421i
\(747\) 1.00000 1.00000
\(748\) −2.00000 −2.00000
\(749\) 0 0
\(750\) 0 0
\(751\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 2.00000 2.00000
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 1.41421i 1.41421i
\(759\) 2.00000 2.00000
\(760\) 0 0
\(761\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(762\) 0 0
\(763\) 1.41421i 1.41421i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 2.00000 2.00000
\(768\) 1.41421i 1.41421i
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) − 1.41421i − 1.41421i
\(772\) 1.00000 1.00000
\(773\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(774\) 0 0
\(775\) −1.00000 −1.00000
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −2.00000 −2.00000
\(780\) 0 0
\(781\) 0 0
\(782\) −2.00000 −2.00000
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) −2.00000 −2.00000
\(787\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 1.41421i 1.41421i
\(792\) 0 0
\(793\) 0 0
\(794\) − 1.41421i − 1.41421i
\(795\) 0 0
\(796\) 0 0
\(797\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(798\) 2.82843i 2.82843i
\(799\) 0 0
\(800\) − 1.41421i − 1.41421i
\(801\) −1.00000 −1.00000
\(802\) 0 0
\(803\) −2.00000 −2.00000
\(804\) 2.00000 2.00000
\(805\) 0 0
\(806\) −2.00000 −2.00000
\(807\) 1.41421i 1.41421i
\(808\) 0 0
\(809\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) −1.00000 −1.00000
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) −2.00000 −2.00000
\(817\) 0 0
\(818\) 0 0
\(819\) 1.41421i 1.41421i
\(820\) 0 0
\(821\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(822\) 0 0
\(823\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(824\) 0 0
\(825\) −2.00000 −2.00000
\(826\) −2.00000 −2.00000
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) −1.00000 −1.00000
\(829\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(830\) 0 0
\(831\) 1.41421i 1.41421i
\(832\) − 1.41421i − 1.41421i
\(833\) 0 0
\(834\) −2.00000 −2.00000
\(835\) 0 0
\(836\) −2.00000 −2.00000
\(837\) 0 0
\(838\) 1.41421i 1.41421i
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 0 0
\(842\) − 1.41421i − 1.41421i
\(843\) − 1.41421i − 1.41421i
\(844\) 1.00000 1.00000
\(845\) 0 0
\(846\) 0 0
\(847\) −1.00000 −1.00000
\(848\) 1.00000 1.00000
\(849\) −2.00000 −2.00000
\(850\) 2.00000 2.00000
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(858\) −4.00000 −4.00000
\(859\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(860\) 0 0
\(861\) 2.00000 2.00000
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 1.41421i − 1.41421i
\(868\) 1.00000 1.00000
\(869\) 2.00000 2.00000
\(870\) 0 0
\(871\) 2.00000 2.00000
\(872\) 0 0
\(873\) 1.00000 1.00000
\(874\) −2.00000 −2.00000
\(875\) 0 0
\(876\) 2.00000 2.00000
\(877\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(878\) 0 0
\(879\) − 2.82843i − 2.82843i
\(880\) 0 0
\(881\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(882\) 0 0
\(883\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(884\) 2.00000 2.00000
\(885\) 0 0
\(886\) − 1.41421i − 1.41421i
\(887\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) − 1.41421i − 1.41421i
\(892\) 0 0
\(893\) 0 0
\(894\) −2.00000 −2.00000
\(895\) 0 0
\(896\) 0 0
\(897\) −2.00000 −2.00000
\(898\) 1.41421i 1.41421i
\(899\) −1.00000 −1.00000
\(900\) 1.00000 1.00000
\(901\) 1.41421i 1.41421i
\(902\) 2.82843i 2.82843i
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 2.82843i 2.82843i
\(907\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(908\) 0 0
\(909\) 1.00000 1.00000
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) −2.00000 −2.00000
\(913\) − 1.41421i − 1.41421i
\(914\) 0 0
\(915\) 0 0
\(916\) −1.00000 −1.00000
\(917\) 1.00000 1.00000
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) − 1.41421i − 1.41421i
\(922\) −2.00000 −2.00000
\(923\) 0 0
\(924\) 2.00000 2.00000
\(925\) 0 0
\(926\) 1.41421i 1.41421i
\(927\) 0 0
\(928\) − 1.41421i − 1.41421i
\(929\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(938\) −2.00000 −2.00000
\(939\) −2.00000 −2.00000
\(940\) 0 0
\(941\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(942\) 2.82843i 2.82843i
\(943\) 1.41421i 1.41421i
\(944\) − 1.41421i − 1.41421i
\(945\) 0 0
\(946\) 0 0
\(947\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(948\) −2.00000 −2.00000
\(949\) 2.00000 2.00000
\(950\) 2.00000 2.00000
\(951\) 0 0
\(952\) 0 0
\(953\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(954\) 1.41421i 1.41421i
\(955\) 0 0
\(956\) 1.41421i 1.41421i
\(957\) −2.00000 −2.00000
\(958\) 1.41421i 1.41421i
\(959\) 0 0
\(960\) 0 0
\(961\) 0 0
\(962\) 0 0
\(963\) 0 0
\(964\) −1.00000 −1.00000
\(965\) 0 0
\(966\) 2.00000 2.00000
\(967\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) 0 0
\(969\) − 2.82843i − 2.82843i
\(970\) 0 0
\(971\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(972\) 1.41421i 1.41421i
\(973\) 1.00000 1.00000
\(974\) −2.00000 −2.00000
\(975\) 2.00000 2.00000
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 1.41421i 1.41421i
\(980\) 0 0
\(981\) − 1.41421i − 1.41421i
\(982\) 0 0
\(983\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 2.00000 2.00000
\(987\) 0 0
\(988\) 2.00000 2.00000
\(989\) 0 0
\(990\) 0 0
\(991\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(992\) 1.41421i 1.41421i
\(993\) 1.41421i 1.41421i
\(994\) 0 0
\(995\) 0 0
\(996\) 1.41421i 1.41421i
\(997\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(998\) 2.00000 2.00000
\(999\) 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 643.1.b.b.642.2 yes 2
643.642 odd 2 inner 643.1.b.b.642.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
643.1.b.b.642.1 2 643.642 odd 2 inner
643.1.b.b.642.2 yes 2 1.1 even 1 trivial