Properties

Label 1931.1.b.b.1930.1
Level 1931
Weight 1
Character 1931.1930
Analytic conductor 0.964
Analytic rank 0
Dimension 2
Projective image \(S_{4}\)
CM/RM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 1931 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1931.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.963694539394\)
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 4.2.1931.1
Artin image $\GL(2,3)$
Artin field Galois closure of 8.2.7200237491.1

Embedding invariants

Embedding label 1930.1
Root \(1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 1931.1930
Dual form 1931.1.b.b.1930.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.41421i q^{2} -1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.41421i q^{2} -1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{9} +1.41421i q^{10} +1.00000 q^{11} -1.41421i q^{14} -1.00000 q^{16} -1.41421i q^{17} +1.41421i q^{18} +1.00000 q^{20} -1.41421i q^{22} -1.00000 q^{23} -1.00000 q^{28} -1.41421i q^{29} -1.41421i q^{31} +1.41421i q^{32} -2.00000 q^{34} -1.00000 q^{35} +1.00000 q^{36} +1.00000 q^{37} +1.00000 q^{41} -1.00000 q^{43} -1.00000 q^{44} +1.00000 q^{45} +1.41421i q^{46} -1.41421i q^{53} -1.00000 q^{55} -2.00000 q^{58} -1.00000 q^{59} +1.41421i q^{61} -2.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} +1.41421i q^{67} +1.41421i q^{68} +1.41421i q^{70} -1.41421i q^{74} +1.00000 q^{77} +1.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -1.41421i q^{82} +1.00000 q^{83} +1.41421i q^{85} +1.41421i q^{86} +1.41421i q^{89} -1.41421i q^{90} +1.00000 q^{92} -1.00000 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} - 2q^{5} + 2q^{7} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} - 2q^{5} + 2q^{7} - 2q^{9} + 2q^{11} - 2q^{16} + 2q^{20} - 2q^{23} - 2q^{28} - 4q^{34} - 2q^{35} + 2q^{36} + 2q^{37} + 2q^{41} - 2q^{43} - 2q^{44} + 2q^{45} - 2q^{55} - 4q^{58} - 2q^{59} - 4q^{62} - 2q^{63} + 2q^{64} + 2q^{77} + 2q^{79} + 2q^{80} + 2q^{81} + 2q^{83} + 2q^{92} - 2q^{97} - 2q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1931\mathbb{Z}\right)^\times\).

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

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 \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(3\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) −1.00000 −1.00000
\(5\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(6\) 0 0
\(7\) 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
\(10\) 1.41421i 1.41421i
\(11\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) − 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
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 1.00000 1.00000
\(21\) 0 0
\(22\) − 1.41421i − 1.41421i
\(23\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) −1.00000 −1.00000
\(29\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(30\) 0 0
\(31\) − 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
\(34\) −2.00000 −2.00000
\(35\) −1.00000 −1.00000
\(36\) 1.00000 1.00000
\(37\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(42\) 0 0
\(43\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(44\) −1.00000 −1.00000
\(45\) 1.00000 1.00000
\(46\) 1.41421i 1.41421i
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(54\) 0 0
\(55\) −1.00000 −1.00000
\(56\) 0 0
\(57\) 0 0
\(58\) −2.00000 −2.00000
\(59\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(60\) 0 0
\(61\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(62\) −2.00000 −2.00000
\(63\) −1.00000 −1.00000
\(64\) 1.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(68\) 1.41421i 1.41421i
\(69\) 0 0
\(70\) 1.41421i 1.41421i
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) − 1.41421i − 1.41421i
\(75\) 0 0
\(76\) 0 0
\(77\) 1.00000 1.00000
\(78\) 0 0
\(79\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(80\) 1.00000 1.00000
\(81\) 1.00000 1.00000
\(82\) − 1.41421i − 1.41421i
\(83\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(84\) 0 0
\(85\) 1.41421i 1.41421i
\(86\) 1.41421i 1.41421i
\(87\) 0 0
\(88\) 0 0
\(89\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(90\) − 1.41421i − 1.41421i
\(91\) 0 0
\(92\) 1.00000 1.00000
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(98\) 0 0
\(99\) −1.00000 −1.00000
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −2.00000 −2.00000
\(107\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(108\) 0 0
\(109\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(110\) 1.41421i 1.41421i
\(111\) 0 0
\(112\) −1.00000 −1.00000
\(113\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(114\) 0 0
\(115\) 1.00000 1.00000
\(116\) 1.41421i 1.41421i
\(117\) 0 0
\(118\) 1.41421i 1.41421i
\(119\) − 1.41421i − 1.41421i
\(120\) 0 0
\(121\) 0 0
\(122\) 2.00000 2.00000
\(123\) 0 0
\(124\) 1.41421i 1.41421i
\(125\) 1.00000 1.00000
\(126\) 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 \(0.333333\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 2.00000 2.00000
\(135\) 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
\(139\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(140\) 1.00000 1.00000
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 1.00000 1.00000
\(145\) 1.41421i 1.41421i
\(146\) 0 0
\(147\) 0 0
\(148\) −1.00000 −1.00000
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(152\) 0 0
\(153\) 1.41421i 1.41421i
\(154\) − 1.41421i − 1.41421i
\(155\) 1.41421i 1.41421i
\(156\) 0 0
\(157\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) − 1.41421i − 1.41421i
\(159\) 0 0
\(160\) − 1.41421i − 1.41421i
\(161\) −1.00000 −1.00000
\(162\) − 1.41421i − 1.41421i
\(163\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(164\) −1.00000 −1.00000
\(165\) 0 0
\(166\) − 1.41421i − 1.41421i
\(167\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(168\) 0 0
\(169\) 1.00000 1.00000
\(170\) 2.00000 2.00000
\(171\) 0 0
\(172\) 1.00000 1.00000
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1.00000 −1.00000
\(177\) 0 0
\(178\) 2.00000 2.00000
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) −1.00000 −1.00000
\(181\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −1.00000 −1.00000
\(186\) 0 0
\(187\) − 1.41421i − 1.41421i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) −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
\(197\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(198\) 1.41421i 1.41421i
\(199\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 1.41421i − 1.41421i
\(204\) 0 0
\(205\) −1.00000 −1.00000
\(206\) 1.41421i 1.41421i
\(207\) 1.00000 1.00000
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(212\) 1.41421i 1.41421i
\(213\) 0 0
\(214\) 1.41421i 1.41421i
\(215\) 1.00000 1.00000
\(216\) 0 0
\(217\) − 1.41421i − 1.41421i
\(218\) − 1.41421i − 1.41421i
\(219\) 0 0
\(220\) 1.00000 1.00000
\(221\) 0 0
\(222\) 0 0
\(223\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(224\) 1.41421i 1.41421i
\(225\) 0 0
\(226\) −2.00000 −2.00000
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(230\) − 1.41421i − 1.41421i
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 1.00000 1.00000
\(237\) 0 0
\(238\) −2.00000 −2.00000
\(239\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) − 1.41421i − 1.41421i
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) − 1.41421i − 1.41421i
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 1.00000 1.00000
\(253\) −1.00000 −1.00000
\(254\) 0 0
\(255\) 0 0
\(256\) 1.00000 1.00000
\(257\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(258\) 0 0
\(259\) 1.00000 1.00000
\(260\) 0 0
\(261\) 1.41421i 1.41421i
\(262\) − 1.41421i − 1.41421i
\(263\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(264\) 0 0
\(265\) 1.41421i 1.41421i
\(266\) 0 0
\(267\) 0 0
\(268\) − 1.41421i − 1.41421i
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(272\) 1.41421i 1.41421i
\(273\) 0 0
\(274\) −2.00000 −2.00000
\(275\) 0 0
\(276\) 0 0
\(277\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(278\) −2.00000 −2.00000
\(279\) 1.41421i 1.41421i
\(280\) 0 0
\(281\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.00000 1.00000
\(288\) − 1.41421i − 1.41421i
\(289\) −1.00000 −1.00000
\(290\) 2.00000 2.00000
\(291\) 0 0
\(292\) 0 0
\(293\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(294\) 0 0
\(295\) 1.00000 1.00000
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −1.00000 −1.00000
\(302\) − 1.41421i − 1.41421i
\(303\) 0 0
\(304\) 0 0
\(305\) − 1.41421i − 1.41421i
\(306\) 2.00000 2.00000
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) −1.00000 −1.00000
\(309\) 0 0
\(310\) 2.00000 2.00000
\(311\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(312\) 0 0
\(313\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(314\) 0 0
\(315\) 1.00000 1.00000
\(316\) −1.00000 −1.00000
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) − 1.41421i − 1.41421i
\(320\) −1.00000 −1.00000
\(321\) 0 0
\(322\) 1.41421i 1.41421i
\(323\) 0 0
\(324\) −1.00000 −1.00000
\(325\) 0 0
\(326\) − 2.82843i − 2.82843i
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(332\) −1.00000 −1.00000
\(333\) −1.00000 −1.00000
\(334\) 2.00000 2.00000
\(335\) − 1.41421i − 1.41421i
\(336\) 0 0
\(337\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(338\) − 1.41421i − 1.41421i
\(339\) 0 0
\(340\) − 1.41421i − 1.41421i
\(341\) − 1.41421i − 1.41421i
\(342\) 0 0
\(343\) −1.00000 −1.00000
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.41421i 1.41421i
\(353\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) − 1.41421i − 1.41421i
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) −1.00000 −1.00000
\(362\) − 1.41421i − 1.41421i
\(363\) 0 0
\(364\) 0 0
\(365\) 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
\(369\) −1.00000 −1.00000
\(370\) 1.41421i 1.41421i
\(371\) − 1.41421i − 1.41421i
\(372\) 0 0
\(373\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(374\) −2.00000 −2.00000
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) −1.00000 −1.00000
\(386\) 1.41421i 1.41421i
\(387\) 1.00000 1.00000
\(388\) 1.00000 1.00000
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 1.41421i 1.41421i
\(392\) 0 0
\(393\) 0 0
\(394\) − 1.41421i − 1.41421i
\(395\) −1.00000 −1.00000
\(396\) 1.00000 1.00000
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 2.00000 2.00000
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −1.00000 −1.00000
\(406\) −2.00000 −2.00000
\(407\) 1.00000 1.00000
\(408\) 0 0
\(409\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(410\) 1.41421i 1.41421i
\(411\) 0 0
\(412\) 1.00000 1.00000
\(413\) −1.00000 −1.00000
\(414\) − 1.41421i − 1.41421i
\(415\) −1.00000 −1.00000
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(420\) 0 0
\(421\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(422\) − 1.41421i − 1.41421i
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 1.41421i 1.41421i
\(428\) 1.00000 1.00000
\(429\) 0 0
\(430\) − 1.41421i − 1.41421i
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(434\) −2.00000 −2.00000
\(435\) 0 0
\(436\) −1.00000 −1.00000
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(444\) 0 0
\(445\) − 1.41421i − 1.41421i
\(446\) − 1.41421i − 1.41421i
\(447\) 0 0
\(448\) 1.00000 1.00000
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 1.00000 1.00000
\(452\) 1.41421i 1.41421i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) −1.00000 −1.00000
\(461\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 1.41421i 1.41421i
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) 1.41421i 1.41421i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −1.00000 −1.00000
\(474\) 0 0
\(475\) 0 0
\(476\) 1.41421i 1.41421i
\(477\) 1.41421i 1.41421i
\(478\) 1.41421i 1.41421i
\(479\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 1.00000 1.00000
\(486\) 0 0
\(487\) − 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
\(493\) −2.00000 −2.00000
\(494\) 0 0
\(495\) 1.00000 1.00000
\(496\) 1.41421i 1.41421i
\(497\) 0 0
\(498\) 0 0
\(499\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(500\) −1.00000 −1.00000
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 1.41421i 1.41421i
\(507\) 0 0
\(508\) 0 0
\(509\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) − 1.41421i − 1.41421i
\(513\) 0 0
\(514\) 2.00000 2.00000
\(515\) 1.00000 1.00000
\(516\) 0 0
\(517\) 0 0
\(518\) − 1.41421i − 1.41421i
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 2.00000 2.00000
\(523\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(524\) −1.00000 −1.00000
\(525\) 0 0
\(526\) − 2.82843i − 2.82843i
\(527\) −2.00000 −2.00000
\(528\) 0 0
\(529\) 0 0
\(530\) 2.00000 2.00000
\(531\) 1.00000 1.00000
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 1.00000 1.00000
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(542\) 2.00000 2.00000
\(543\) 0 0
\(544\) 2.00000 2.00000
\(545\) −1.00000 −1.00000
\(546\) 0 0
\(547\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(548\) 1.41421i 1.41421i
\(549\) − 1.41421i − 1.41421i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 1.00000 1.00000
\(554\) −2.00000 −2.00000
\(555\) 0 0
\(556\) 1.41421i 1.41421i
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 2.00000 2.00000
\(559\) 0 0
\(560\) 1.00000 1.00000
\(561\) 0 0
\(562\) 1.41421i 1.41421i
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 1.41421i 1.41421i
\(566\) 0 0
\(567\) 1.00000 1.00000
\(568\) 0 0
\(569\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(570\) 0 0
\(571\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) − 1.41421i − 1.41421i
\(575\) 0 0
\(576\) −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
\(579\) 0 0
\(580\) − 1.41421i − 1.41421i
\(581\) 1.00000 1.00000
\(582\) 0 0
\(583\) − 1.41421i − 1.41421i
\(584\) 0 0
\(585\) 0 0
\(586\) 2.00000 2.00000
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) − 1.41421i − 1.41421i
\(591\) 0 0
\(592\) −1.00000 −1.00000
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 1.41421i 1.41421i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(602\) 1.41421i 1.41421i
\(603\) − 1.41421i − 1.41421i
\(604\) −1.00000 −1.00000
\(605\) 0 0
\(606\) 0 0
\(607\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) −2.00000 −2.00000
\(611\) 0 0
\(612\) − 1.41421i − 1.41421i
\(613\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(620\) − 1.41421i − 1.41421i
\(621\) 0 0
\(622\) −2.00000 −2.00000
\(623\) 1.41421i 1.41421i
\(624\) 0 0
\(625\) −1.00000 −1.00000
\(626\) −2.00000 −2.00000
\(627\) 0 0
\(628\) 0 0
\(629\) − 1.41421i − 1.41421i
\(630\) − 1.41421i − 1.41421i
\(631\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) −2.00000 −2.00000
\(639\) 0 0
\(640\) 0 0
\(641\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(644\) 1.00000 1.00000
\(645\) 0 0
\(646\) 0 0
\(647\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(648\) 0 0
\(649\) −1.00000 −1.00000
\(650\) 0 0
\(651\) 0 0
\(652\) −2.00000 −2.00000
\(653\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(654\) 0 0
\(655\) −1.00000 −1.00000
\(656\) −1.00000 −1.00000
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 2.00000 2.00000
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 1.41421i 1.41421i
\(667\) 1.41421i 1.41421i
\(668\) − 1.41421i − 1.41421i
\(669\) 0 0
\(670\) −2.00000 −2.00000
\(671\) 1.41421i 1.41421i
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −1.00000 −1.00000
\(677\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(678\) 0 0
\(679\) −1.00000 −1.00000
\(680\) 0 0
\(681\) 0 0
\(682\) −2.00000 −2.00000
\(683\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(684\) 0 0
\(685\) 1.41421i 1.41421i
\(686\) 1.41421i 1.41421i
\(687\) 0 0
\(688\) 1.00000 1.00000
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) −1.00000 −1.00000
\(694\) 0 0
\(695\) 1.41421i 1.41421i
\(696\) 0 0
\(697\) − 1.41421i − 1.41421i
\(698\) 2.00000 2.00000
\(699\) 0 0
\(700\) 0 0
\(701\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 1.00000 1.00000
\(705\) 0 0
\(706\) 1.41421i 1.41421i
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(710\) 0 0
\(711\) −1.00000 −1.00000
\(712\) 0 0
\(713\) 1.41421i 1.41421i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(720\) −1.00000 −1.00000
\(721\) −1.00000 −1.00000
\(722\) 1.41421i 1.41421i
\(723\) 0 0
\(724\) −1.00000 −1.00000
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) −1.00000 −1.00000
\(730\) 0 0
\(731\) 1.41421i 1.41421i
\(732\) 0 0
\(733\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(734\) − 1.41421i − 1.41421i
\(735\) 0 0
\(736\) − 1.41421i − 1.41421i
\(737\) 1.41421i 1.41421i
\(738\) 1.41421i 1.41421i
\(739\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(740\) 1.00000 1.00000
\(741\) 0 0
\(742\) −2.00000 −2.00000
\(743\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 1.41421i 1.41421i
\(747\) −1.00000 −1.00000
\(748\) 1.41421i 1.41421i
\(749\) −1.00000 −1.00000
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −1.00000 −1.00000
\(756\) 0 0
\(757\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(758\) −2.00000 −2.00000
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 1.00000 1.00000
\(764\) 0 0
\(765\) − 1.41421i − 1.41421i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(770\) 1.41421i 1.41421i
\(771\) 0 0
\(772\) 1.00000 1.00000
\(773\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(774\) − 1.41421i − 1.41421i
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 2.00000 2.00000
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(788\) −1.00000 −1.00000
\(789\) 0 0
\(790\) 1.41421i 1.41421i
\(791\) − 1.41421i − 1.41421i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) − 1.41421i − 1.41421i
\(797\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) − 1.41421i − 1.41421i
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 1.00000 1.00000
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(810\) 1.41421i 1.41421i
\(811\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(812\) 1.41421i 1.41421i
\(813\) 0 0
\(814\) − 1.41421i − 1.41421i
\(815\) −2.00000 −2.00000
\(816\) 0 0
\(817\) 0 0
\(818\) 2.00000 2.00000
\(819\) 0 0
\(820\) 1.00000 1.00000
\(821\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 1.41421i 1.41421i
\(827\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(828\) −1.00000 −1.00000
\(829\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(830\) 1.41421i 1.41421i
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) − 1.41421i − 1.41421i
\(836\) 0 0
\(837\) 0 0
\(838\) 1.41421i 1.41421i
\(839\) 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
\(842\) − 2.82843i − 2.82843i
\(843\) 0 0
\(844\) −1.00000 −1.00000
\(845\) −1.00000 −1.00000
\(846\) 0 0
\(847\) 0 0
\(848\) 1.41421i 1.41421i
\(849\) 0 0
\(850\) 0 0
\(851\) −1.00000 −1.00000
\(852\) 0 0
\(853\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(854\) 2.00000 2.00000
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(860\) −1.00000 −1.00000
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 1.41421i 1.41421i
\(867\) 0 0
\(868\) 1.41421i 1.41421i
\(869\) 1.00000 1.00000
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 1.00000 1.00000
\(874\) 0 0
\(875\) 1.00000 1.00000
\(876\) 0 0
\(877\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 1.00000 1.00000
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(884\) 0 0
\(885\) 0 0
\(886\) − 1.41421i − 1.41421i
\(887\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −2.00000 −2.00000
\(891\) 1.00000 1.00000
\(892\) −1.00000 −1.00000
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −2.00000 −2.00000
\(900\) 0 0
\(901\) −2.00000 −2.00000
\(902\) − 1.41421i − 1.41421i
\(903\) 0 0
\(904\) 0 0
\(905\) −1.00000 −1.00000
\(906\) 0 0
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(912\) 0 0
\(913\) 1.00000 1.00000
\(914\) −2.00000 −2.00000
\(915\) 0 0
\(916\) 0 0
\(917\) 1.00000 1.00000
\(918\) 0 0
\(919\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 1.41421i − 1.41421i
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 1.00000 1.00000
\(928\) 2.00000 2.00000
\(929\) 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
\(935\) 1.41421i 1.41421i
\(936\) 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
\(939\) 0 0
\(940\) 0 0
\(941\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(942\) 0 0
\(943\) −1.00000 −1.00000
\(944\) 1.00000 1.00000
\(945\) 0 0
\(946\) 1.41421i 1.41421i
\(947\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(954\) 2.00000 2.00000
\(955\) 0 0
\(956\) 1.00000 1.00000
\(957\) 0 0
\(958\) 2.00000 2.00000
\(959\) − 1.41421i − 1.41421i
\(960\) 0 0
\(961\) −1.00000 −1.00000
\(962\) 0 0
\(963\) 1.00000 1.00000
\(964\) 0 0
\(965\) 1.00000 1.00000
\(966\) 0 0
\(967\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) − 1.41421i − 1.41421i
\(971\) 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
\(976\) − 1.41421i − 1.41421i
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 1.41421i 1.41421i
\(980\) 0 0
\(981\) −1.00000 −1.00000
\(982\) 1.41421i 1.41421i
\(983\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(984\) 0 0
\(985\) −1.00000 −1.00000
\(986\) 2.82843i 2.82843i
\(987\) 0 0
\(988\) 0 0
\(989\) 1.00000 1.00000
\(990\) − 1.41421i − 1.41421i
\(991\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(992\) 2.00000 2.00000
\(993\) 0 0
\(994\) 0 0
\(995\) − 1.41421i − 1.41421i
\(996\) 0 0
\(997\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(998\) 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 1931.1.b.b.1930.1 2
1931.1930 odd 2 inner 1931.1.b.b.1930.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1931.1.b.b.1930.1 2 1.1 even 1 trivial
1931.1.b.b.1930.2 yes 2 1931.1930 odd 2 inner