Properties

Label 331.1.b.b.330.1
Level 331
Weight 1
Character 331.330
Analytic conductor 0.165
Analytic rank 0
Dimension 2
Projective image \(S_{4}\)
CM/RM no
Inner twists 2

Related objects

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Newspace parameters

Level: \( N \) \(=\) \( 331 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 331.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.165190519182\)
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 4.2.331.1
Artin image $\GL(2,3)$
Artin field Galois closure of 8.2.36264691.2

Embedding invariants

Embedding label 330.1
Root \(1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 331.330
Dual form 331.1.b.b.330.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.41421i q^{2} -1.41421i q^{3} -1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{6} +1.41421i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.41421i q^{2} -1.41421i q^{3} -1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{6} +1.41421i q^{7} -1.00000 q^{9} +1.41421i q^{10} -1.41421i q^{11} +1.41421i q^{12} +2.00000 q^{14} +1.41421i q^{15} -1.00000 q^{16} +1.00000 q^{17} +1.41421i q^{18} +1.00000 q^{19} +1.00000 q^{20} +2.00000 q^{21} -2.00000 q^{22} -1.41421i q^{28} +1.41421i q^{29} +2.00000 q^{30} +1.00000 q^{31} +1.41421i q^{32} -2.00000 q^{33} -1.41421i q^{34} -1.41421i q^{35} +1.00000 q^{36} -1.41421i q^{37} -1.41421i q^{38} +1.41421i q^{41} -2.82843i q^{42} -1.00000 q^{43} +1.41421i q^{44} +1.00000 q^{45} +1.41421i q^{48} -1.00000 q^{49} -1.41421i q^{51} -1.00000 q^{53} +1.41421i q^{55} -1.41421i q^{57} +2.00000 q^{58} -1.41421i q^{60} +1.41421i q^{61} -1.41421i q^{62} -1.41421i q^{63} +1.00000 q^{64} +2.82843i q^{66} +1.00000 q^{67} -1.00000 q^{68} -2.00000 q^{70} +1.00000 q^{71} -2.00000 q^{74} -1.00000 q^{76} +2.00000 q^{77} +1.00000 q^{79} +1.00000 q^{80} -1.00000 q^{81} +2.00000 q^{82} -2.00000 q^{84} -1.00000 q^{85} +1.41421i q^{86} +2.00000 q^{87} -1.41421i q^{90} -1.41421i q^{93} -1.00000 q^{95} +2.00000 q^{96} +1.41421i q^{97} +1.41421i q^{98} +1.41421i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} - 2q^{5} - 4q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} - 2q^{5} - 4q^{6} - 2q^{9} + 4q^{14} - 2q^{16} + 2q^{17} + 2q^{19} + 2q^{20} + 4q^{21} - 4q^{22} + 4q^{30} + 2q^{31} - 4q^{33} + 2q^{36} - 2q^{43} + 2q^{45} - 2q^{49} - 2q^{53} + 4q^{58} + 2q^{64} + 2q^{67} - 2q^{68} - 4q^{70} + 2q^{71} - 4q^{74} - 2q^{76} + 4q^{77} + 2q^{79} + 2q^{80} - 2q^{81} + 4q^{82} - 4q^{84} - 2q^{85} + 4q^{87} - 2q^{95} + 4q^{96} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/331\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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 \(-0.250000\pi\)
\(3\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(4\) −1.00000 −1.00000
\(5\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(6\) −2.00000 −2.00000
\(7\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(8\) 0 0
\(9\) −1.00000 −1.00000
\(10\) 1.41421i 1.41421i
\(11\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(12\) 1.41421i 1.41421i
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 2.00000 2.00000
\(15\) 1.41421i 1.41421i
\(16\) −1.00000 −1.00000
\(17\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(18\) 1.41421i 1.41421i
\(19\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(20\) 1.00000 1.00000
\(21\) 2.00000 2.00000
\(22\) −2.00000 −2.00000
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) − 1.41421i − 1.41421i
\(29\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(30\) 2.00000 2.00000
\(31\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(32\) 1.41421i 1.41421i
\(33\) −2.00000 −2.00000
\(34\) − 1.41421i − 1.41421i
\(35\) − 1.41421i − 1.41421i
\(36\) 1.00000 1.00000
\(37\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(38\) − 1.41421i − 1.41421i
\(39\) 0 0
\(40\) 0 0
\(41\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(42\) − 2.82843i − 2.82843i
\(43\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(44\) 1.41421i 1.41421i
\(45\) 1.00000 1.00000
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 1.41421i 1.41421i
\(49\) −1.00000 −1.00000
\(50\) 0 0
\(51\) − 1.41421i − 1.41421i
\(52\) 0 0
\(53\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(54\) 0 0
\(55\) 1.41421i 1.41421i
\(56\) 0 0
\(57\) − 1.41421i − 1.41421i
\(58\) 2.00000 2.00000
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) − 1.41421i − 1.41421i
\(61\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(62\) − 1.41421i − 1.41421i
\(63\) − 1.41421i − 1.41421i
\(64\) 1.00000 1.00000
\(65\) 0 0
\(66\) 2.82843i 2.82843i
\(67\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(68\) −1.00000 −1.00000
\(69\) 0 0
\(70\) −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 \(\pi\)
\(74\) −2.00000 −2.00000
\(75\) 0 0
\(76\) −1.00000 −1.00000
\(77\) 2.00000 2.00000
\(78\) 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
\(82\) 2.00000 2.00000
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) −2.00000 −2.00000
\(85\) −1.00000 −1.00000
\(86\) 1.41421i 1.41421i
\(87\) 2.00000 2.00000
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) − 1.41421i − 1.41421i
\(91\) 0 0
\(92\) 0 0
\(93\) − 1.41421i − 1.41421i
\(94\) 0 0
\(95\) −1.00000 −1.00000
\(96\) 2.00000 2.00000
\(97\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(98\) 1.41421i 1.41421i
\(99\) 1.41421i 1.41421i
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) −2.00000 −2.00000
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) −2.00000 −2.00000
\(106\) 1.41421i 1.41421i
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(110\) 2.00000 2.00000
\(111\) −2.00000 −2.00000
\(112\) − 1.41421i − 1.41421i
\(113\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(114\) −2.00000 −2.00000
\(115\) 0 0
\(116\) − 1.41421i − 1.41421i
\(117\) 0 0
\(118\) 0 0
\(119\) 1.41421i 1.41421i
\(120\) 0 0
\(121\) −1.00000 −1.00000
\(122\) 2.00000 2.00000
\(123\) 2.00000 2.00000
\(124\) −1.00000 −1.00000
\(125\) 1.00000 1.00000
\(126\) −2.00000 −2.00000
\(127\) −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
\(131\) 1.00000 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
\(134\) − 1.41421i − 1.41421i
\(135\) 0 0
\(136\) 0 0
\(137\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 1.41421i 1.41421i
\(141\) 0 0
\(142\) − 1.41421i − 1.41421i
\(143\) 0 0
\(144\) 1.00000 1.00000
\(145\) − 1.41421i − 1.41421i
\(146\) 0 0
\(147\) 1.41421i 1.41421i
\(148\) 1.41421i 1.41421i
\(149\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) −1.00000 −1.00000
\(154\) − 2.82843i − 2.82843i
\(155\) −1.00000 −1.00000
\(156\) 0 0
\(157\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(158\) − 1.41421i − 1.41421i
\(159\) 1.41421i 1.41421i
\(160\) − 1.41421i − 1.41421i
\(161\) 0 0
\(162\) 1.41421i 1.41421i
\(163\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(164\) − 1.41421i − 1.41421i
\(165\) 2.00000 2.00000
\(166\) 0 0
\(167\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(168\) 0 0
\(169\) 1.00000 1.00000
\(170\) 1.41421i 1.41421i
\(171\) −1.00000 −1.00000
\(172\) 1.00000 1.00000
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) − 2.82843i − 2.82843i
\(175\) 0 0
\(176\) 1.41421i 1.41421i
\(177\) 0 0
\(178\) 0 0
\(179\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(180\) −1.00000 −1.00000
\(181\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(182\) 0 0
\(183\) 2.00000 2.00000
\(184\) 0 0
\(185\) 1.41421i 1.41421i
\(186\) −2.00000 −2.00000
\(187\) − 1.41421i − 1.41421i
\(188\) 0 0
\(189\) 0 0
\(190\) 1.41421i 1.41421i
\(191\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(192\) − 1.41421i − 1.41421i
\(193\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(194\) 2.00000 2.00000
\(195\) 0 0
\(196\) 1.00000 1.00000
\(197\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(198\) 2.00000 2.00000
\(199\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(200\) 0 0
\(201\) − 1.41421i − 1.41421i
\(202\) 0 0
\(203\) −2.00000 −2.00000
\(204\) 1.41421i 1.41421i
\(205\) − 1.41421i − 1.41421i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 1.41421i − 1.41421i
\(210\) 2.82843i 2.82843i
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 1.00000 1.00000
\(213\) − 1.41421i − 1.41421i
\(214\) 0 0
\(215\) 1.00000 1.00000
\(216\) 0 0
\(217\) 1.41421i 1.41421i
\(218\) 1.41421i 1.41421i
\(219\) 0 0
\(220\) − 1.41421i − 1.41421i
\(221\) 0 0
\(222\) 2.82843i 2.82843i
\(223\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(224\) −2.00000 −2.00000
\(225\) 0 0
\(226\) 2.82843i 2.82843i
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 1.41421i 1.41421i
\(229\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(230\) 0 0
\(231\) − 2.82843i − 2.82843i
\(232\) 0 0
\(233\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 1.41421i − 1.41421i
\(238\) 2.00000 2.00000
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) − 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
\(243\) 1.41421i 1.41421i
\(244\) − 1.41421i − 1.41421i
\(245\) 1.00000 1.00000
\(246\) − 2.82843i − 2.82843i
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) − 1.41421i − 1.41421i
\(251\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(252\) 1.41421i 1.41421i
\(253\) 0 0
\(254\) 1.41421i 1.41421i
\(255\) 1.41421i 1.41421i
\(256\) 1.00000 1.00000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 2.00000 2.00000
\(259\) 2.00000 2.00000
\(260\) 0 0
\(261\) − 1.41421i − 1.41421i
\(262\) − 1.41421i − 1.41421i
\(263\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(264\) 0 0
\(265\) 1.00000 1.00000
\(266\) 2.00000 2.00000
\(267\) 0 0
\(268\) −1.00000 −1.00000
\(269\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(270\) 0 0
\(271\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(272\) −1.00000 −1.00000
\(273\) 0 0
\(274\) −2.00000 −2.00000
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) −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
\(283\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(284\) −1.00000 −1.00000
\(285\) 1.41421i 1.41421i
\(286\) 0 0
\(287\) −2.00000 −2.00000
\(288\) − 1.41421i − 1.41421i
\(289\) 0 0
\(290\) −2.00000 −2.00000
\(291\) 2.00000 2.00000
\(292\) 0 0
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 2.00000 2.00000
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 1.41421i 1.41421i
\(299\) 0 0
\(300\) 0 0
\(301\) − 1.41421i − 1.41421i
\(302\) 0 0
\(303\) 0 0
\(304\) −1.00000 −1.00000
\(305\) − 1.41421i − 1.41421i
\(306\) 1.41421i 1.41421i
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) −2.00000 −2.00000
\(309\) 0 0
\(310\) 1.41421i 1.41421i
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(314\) 1.41421i 1.41421i
\(315\) 1.41421i 1.41421i
\(316\) −1.00000 −1.00000
\(317\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(318\) 2.00000 2.00000
\(319\) 2.00000 2.00000
\(320\) −1.00000 −1.00000
\(321\) 0 0
\(322\) 0 0
\(323\) 1.00000 1.00000
\(324\) 1.00000 1.00000
\(325\) 0 0
\(326\) − 1.41421i − 1.41421i
\(327\) 1.41421i 1.41421i
\(328\) 0 0
\(329\) 0 0
\(330\) − 2.82843i − 2.82843i
\(331\) −1.00000 −1.00000
\(332\) 0 0
\(333\) 1.41421i 1.41421i
\(334\) 1.41421i 1.41421i
\(335\) −1.00000 −1.00000
\(336\) −2.00000 −2.00000
\(337\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(338\) − 1.41421i − 1.41421i
\(339\) 2.82843i 2.82843i
\(340\) 1.00000 1.00000
\(341\) − 1.41421i − 1.41421i
\(342\) 1.41421i 1.41421i
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) −2.00000 −2.00000
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 2.00000 2.00000
\(353\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(354\) 0 0
\(355\) −1.00000 −1.00000
\(356\) 0 0
\(357\) 2.00000 2.00000
\(358\) 1.41421i 1.41421i
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 0 0
\(362\) 2.00000 2.00000
\(363\) 1.41421i 1.41421i
\(364\) 0 0
\(365\) 0 0
\(366\) − 2.82843i − 2.82843i
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) − 1.41421i − 1.41421i
\(370\) 2.00000 2.00000
\(371\) − 1.41421i − 1.41421i
\(372\) 1.41421i 1.41421i
\(373\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(374\) −2.00000 −2.00000
\(375\) − 1.41421i − 1.41421i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 1.00000 1.00000
\(381\) 1.41421i 1.41421i
\(382\) 1.41421i 1.41421i
\(383\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(384\) 0 0
\(385\) −2.00000 −2.00000
\(386\) 0 0
\(387\) 1.00000 1.00000
\(388\) − 1.41421i − 1.41421i
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) − 1.41421i − 1.41421i
\(394\) − 1.41421i − 1.41421i
\(395\) −1.00000 −1.00000
\(396\) − 1.41421i − 1.41421i
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) −2.00000 −2.00000
\(399\) 2.00000 2.00000
\(400\) 0 0
\(401\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(402\) −2.00000 −2.00000
\(403\) 0 0
\(404\) 0 0
\(405\) 1.00000 1.00000
\(406\) 2.82843i 2.82843i
\(407\) −2.00000 −2.00000
\(408\) 0 0
\(409\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(410\) −2.00000 −2.00000
\(411\) −2.00000 −2.00000
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) −2.00000 −2.00000
\(419\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(420\) 2.00000 2.00000
\(421\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) −2.00000 −2.00000
\(427\) −2.00000 −2.00000
\(428\) 0 0
\(429\) 0 0
\(430\) − 1.41421i − 1.41421i
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(434\) 2.00000 2.00000
\(435\) −2.00000 −2.00000
\(436\) 1.00000 1.00000
\(437\) 0 0
\(438\) 0 0
\(439\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(440\) 0 0
\(441\) 1.00000 1.00000
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 2.00000 2.00000
\(445\) 0 0
\(446\) 1.41421i 1.41421i
\(447\) 1.41421i 1.41421i
\(448\) 1.41421i 1.41421i
\(449\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(450\) 0 0
\(451\) 2.00000 2.00000
\(452\) 2.00000 2.00000
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(458\) −2.00000 −2.00000
\(459\) 0 0
\(460\) 0 0
\(461\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(462\) −4.00000 −4.00000
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) − 1.41421i − 1.41421i
\(465\) 1.41421i 1.41421i
\(466\) − 1.41421i − 1.41421i
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) 1.41421i 1.41421i
\(470\) 0 0
\(471\) 1.41421i 1.41421i
\(472\) 0 0
\(473\) 1.41421i 1.41421i
\(474\) −2.00000 −2.00000
\(475\) 0 0
\(476\) − 1.41421i − 1.41421i
\(477\) 1.00000 1.00000
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) −2.00000 −2.00000
\(481\) 0 0
\(482\) − 1.41421i − 1.41421i
\(483\) 0 0
\(484\) 1.00000 1.00000
\(485\) − 1.41421i − 1.41421i
\(486\) 2.00000 2.00000
\(487\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(488\) 0 0
\(489\) − 1.41421i − 1.41421i
\(490\) − 1.41421i − 1.41421i
\(491\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(492\) −2.00000 −2.00000
\(493\) 1.41421i 1.41421i
\(494\) 0 0
\(495\) − 1.41421i − 1.41421i
\(496\) −1.00000 −1.00000
\(497\) 1.41421i 1.41421i
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) −1.00000 −1.00000
\(501\) 1.41421i 1.41421i
\(502\) 2.00000 2.00000
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 1.41421i − 1.41421i
\(508\) 1.00000 1.00000
\(509\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(510\) 2.00000 2.00000
\(511\) 0 0
\(512\) − 1.41421i − 1.41421i
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) − 1.41421i − 1.41421i
\(517\) 0 0
\(518\) − 2.82843i − 2.82843i
\(519\) 0 0
\(520\) 0 0
\(521\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(522\) −2.00000 −2.00000
\(523\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(524\) −1.00000 −1.00000
\(525\) 0 0
\(526\) −2.00000 −2.00000
\(527\) 1.00000 1.00000
\(528\) 2.00000 2.00000
\(529\) 1.00000 1.00000
\(530\) − 1.41421i − 1.41421i
\(531\) 0 0
\(532\) − 1.41421i − 1.41421i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 1.41421i 1.41421i
\(538\) 1.41421i 1.41421i
\(539\) 1.41421i 1.41421i
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) − 1.41421i − 1.41421i
\(543\) 2.00000 2.00000
\(544\) 1.41421i 1.41421i
\(545\) 1.00000 1.00000
\(546\) 0 0
\(547\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(548\) 1.41421i 1.41421i
\(549\) − 1.41421i − 1.41421i
\(550\) 0 0
\(551\) 1.41421i 1.41421i
\(552\) 0 0
\(553\) 1.41421i 1.41421i
\(554\) 0 0
\(555\) 2.00000 2.00000
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 1.41421i 1.41421i
\(559\) 0 0
\(560\) 1.41421i 1.41421i
\(561\) −2.00000 −2.00000
\(562\) 1.41421i 1.41421i
\(563\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(564\) 0 0
\(565\) 2.00000 2.00000
\(566\) 1.41421i 1.41421i
\(567\) − 1.41421i − 1.41421i
\(568\) 0 0
\(569\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(570\) 2.00000 2.00000
\(571\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(572\) 0 0
\(573\) 1.41421i 1.41421i
\(574\) 2.82843i 2.82843i
\(575\) 0 0
\(576\) −1.00000 −1.00000
\(577\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 1.41421i 1.41421i
\(581\) 0 0
\(582\) − 2.82843i − 2.82843i
\(583\) 1.41421i 1.41421i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) − 1.41421i − 1.41421i
\(589\) 1.00000 1.00000
\(590\) 0 0
\(591\) − 1.41421i − 1.41421i
\(592\) 1.41421i 1.41421i
\(593\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(594\) 0 0
\(595\) − 1.41421i − 1.41421i
\(596\) 1.00000 1.00000
\(597\) −2.00000 −2.00000
\(598\) 0 0
\(599\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(600\) 0 0
\(601\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(602\) −2.00000 −2.00000
\(603\) −1.00000 −1.00000
\(604\) 0 0
\(605\) 1.00000 1.00000
\(606\) 0 0
\(607\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(608\) 1.41421i 1.41421i
\(609\) 2.82843i 2.82843i
\(610\) −2.00000 −2.00000
\(611\) 0 0
\(612\) 1.00000 1.00000
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) −2.00000 −2.00000
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 1.00000 1.00000
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −1.00000 −1.00000
\(626\) 1.41421i 1.41421i
\(627\) −2.00000 −2.00000
\(628\) 1.00000 1.00000
\(629\) − 1.41421i − 1.41421i
\(630\) 2.00000 2.00000
\(631\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 2.00000 2.00000
\(635\) 1.00000 1.00000
\(636\) − 1.41421i − 1.41421i
\(637\) 0 0
\(638\) − 2.82843i − 2.82843i
\(639\) −1.00000 −1.00000
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) − 1.41421i − 1.41421i
\(646\) − 1.41421i − 1.41421i
\(647\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 2.00000 2.00000
\(652\) −1.00000 −1.00000
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 2.00000 2.00000
\(655\) −1.00000 −1.00000
\(656\) − 1.41421i − 1.41421i
\(657\) 0 0
\(658\) 0 0
\(659\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(660\) −2.00000 −2.00000
\(661\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(662\) 1.41421i 1.41421i
\(663\) 0 0
\(664\) 0 0
\(665\) − 1.41421i − 1.41421i
\(666\) 2.00000 2.00000
\(667\) 0 0
\(668\) 1.00000 1.00000
\(669\) 1.41421i 1.41421i
\(670\) 1.41421i 1.41421i
\(671\) 2.00000 2.00000
\(672\) 2.82843i 2.82843i
\(673\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(674\) 1.41421i 1.41421i
\(675\) 0 0
\(676\) −1.00000 −1.00000
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 4.00000 4.00000
\(679\) −2.00000 −2.00000
\(680\) 0 0
\(681\) 0 0
\(682\) −2.00000 −2.00000
\(683\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(684\) 1.00000 1.00000
\(685\) 1.41421i 1.41421i
\(686\) 0 0
\(687\) −2.00000 −2.00000
\(688\) 1.00000 1.00000
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) −2.00000 −2.00000
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 1.41421i 1.41421i
\(698\) 0 0
\(699\) − 1.41421i − 1.41421i
\(700\) 0 0
\(701\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(702\) 0 0
\(703\) − 1.41421i − 1.41421i
\(704\) − 1.41421i − 1.41421i
\(705\) 0 0
\(706\) − 1.41421i − 1.41421i
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 1.41421i 1.41421i
\(711\) −1.00000 −1.00000
\(712\) 0 0
\(713\) 0 0
\(714\) − 2.82843i − 2.82843i
\(715\) 0 0
\(716\) 1.00000 1.00000
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) −1.00000 −1.00000
\(721\) 0 0
\(722\) 0 0
\(723\) − 1.41421i − 1.41421i
\(724\) − 1.41421i − 1.41421i
\(725\) 0 0
\(726\) 2.00000 2.00000
\(727\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(728\) 0 0
\(729\) 1.00000 1.00000
\(730\) 0 0
\(731\) −1.00000 −1.00000
\(732\) −2.00000 −2.00000
\(733\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) 0 0
\(735\) − 1.41421i − 1.41421i
\(736\) 0 0
\(737\) − 1.41421i − 1.41421i
\(738\) −2.00000 −2.00000
\(739\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(740\) − 1.41421i − 1.41421i
\(741\) 0 0
\(742\) −2.00000 −2.00000
\(743\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(744\) 0 0
\(745\) 1.00000 1.00000
\(746\) 2.00000 2.00000
\(747\) 0 0
\(748\) 1.41421i 1.41421i
\(749\) 0 0
\(750\) −2.00000 −2.00000
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) 2.00000 2.00000
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 2.00000 2.00000
\(763\) − 1.41421i − 1.41421i
\(764\) 1.00000 1.00000
\(765\) 1.00000 1.00000
\(766\) −2.00000 −2.00000
\(767\) 0 0
\(768\) − 1.41421i − 1.41421i
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 2.82843i 2.82843i
\(771\) 0 0
\(772\) 0 0
\(773\) −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
\(777\) − 2.82843i − 2.82843i
\(778\) 0 0
\(779\) 1.41421i 1.41421i
\(780\) 0 0
\(781\) − 1.41421i − 1.41421i
\(782\) 0 0
\(783\) 0 0
\(784\) 1.00000 1.00000
\(785\) 1.00000 1.00000
\(786\) −2.00000 −2.00000
\(787\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) −1.00000 −1.00000
\(789\) −2.00000 −2.00000
\(790\) 1.41421i 1.41421i
\(791\) − 2.82843i − 2.82843i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) − 1.41421i − 1.41421i
\(796\) 1.41421i 1.41421i
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) − 2.82843i − 2.82843i
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) − 1.41421i − 1.41421i
\(803\) 0 0
\(804\) 1.41421i 1.41421i
\(805\) 0 0
\(806\) 0 0
\(807\) 1.41421i 1.41421i
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) − 1.41421i − 1.41421i
\(811\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(812\) 2.00000 2.00000
\(813\) − 1.41421i − 1.41421i
\(814\) 2.82843i 2.82843i
\(815\) −1.00000 −1.00000
\(816\) 1.41421i 1.41421i
\(817\) −1.00000 −1.00000
\(818\) −2.00000 −2.00000
\(819\) 0 0
\(820\) 1.41421i 1.41421i
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 2.82843i 2.82843i
\(823\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(828\) 0 0
\(829\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −1.00000 −1.00000
\(834\) 0 0
\(835\) 1.00000 1.00000
\(836\) 1.41421i 1.41421i
\(837\) 0 0
\(838\) 1.41421i 1.41421i
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −1.00000 −1.00000
\(842\) −2.00000 −2.00000
\(843\) 1.41421i 1.41421i
\(844\) 0 0
\(845\) −1.00000 −1.00000
\(846\) 0 0
\(847\) − 1.41421i − 1.41421i
\(848\) 1.00000 1.00000
\(849\) 1.41421i 1.41421i
\(850\) 0 0
\(851\) 0 0
\(852\) 1.41421i 1.41421i
\(853\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(854\) 2.82843i 2.82843i
\(855\) 1.00000 1.00000
\(856\) 0 0
\(857\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(858\) 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
\(861\) 2.82843i 2.82843i
\(862\) 0 0
\(863\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) − 1.41421i − 1.41421i
\(867\) 0 0
\(868\) − 1.41421i − 1.41421i
\(869\) − 1.41421i − 1.41421i
\(870\) 2.82843i 2.82843i
\(871\) 0 0
\(872\) 0 0
\(873\) − 1.41421i − 1.41421i
\(874\) 0 0
\(875\) 1.41421i 1.41421i
\(876\) 0 0
\(877\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(878\) −2.00000 −2.00000
\(879\) 0 0
\(880\) − 1.41421i − 1.41421i
\(881\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(882\) − 1.41421i − 1.41421i
\(883\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(888\) 0 0
\(889\) − 1.41421i − 1.41421i
\(890\) 0 0
\(891\) 1.41421i 1.41421i
\(892\) 1.00000 1.00000
\(893\) 0 0
\(894\) 2.00000 2.00000
\(895\) 1.00000 1.00000
\(896\) 0 0
\(897\) 0 0
\(898\) − 1.41421i − 1.41421i
\(899\) 1.41421i 1.41421i
\(900\) 0 0
\(901\) −1.00000 −1.00000
\(902\) − 2.82843i − 2.82843i
\(903\) −2.00000 −2.00000
\(904\) 0 0
\(905\) − 1.41421i − 1.41421i
\(906\) 0 0
\(907\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(912\) 1.41421i 1.41421i
\(913\) 0 0
\(914\) 1.41421i 1.41421i
\(915\) −2.00000 −2.00000
\(916\) 1.41421i 1.41421i
\(917\) 1.41421i 1.41421i
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 1.41421i − 1.41421i
\(923\) 0 0
\(924\) 2.82843i 2.82843i
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) −2.00000 −2.00000
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 2.00000 2.00000
\(931\) −1.00000 −1.00000
\(932\) −1.00000 −1.00000
\(933\) 0 0
\(934\) 0 0
\(935\) 1.41421i 1.41421i
\(936\) 0 0
\(937\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(938\) 2.00000 2.00000
\(939\) 1.41421i 1.41421i
\(940\) 0 0
\(941\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(942\) 2.00000 2.00000
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 2.00000 2.00000
\(947\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(948\) 1.41421i 1.41421i
\(949\) 0 0
\(950\) 0 0
\(951\) 2.00000 2.00000
\(952\) 0 0
\(953\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) − 1.41421i − 1.41421i
\(955\) 1.00000 1.00000
\(956\) 0 0
\(957\) − 2.82843i − 2.82843i
\(958\) 0 0
\(959\) 2.00000 2.00000
\(960\) 1.41421i 1.41421i
\(961\) 0 0
\(962\) 0 0
\(963\) 0 0
\(964\) −1.00000 −1.00000
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 0 0
\(969\) − 1.41421i − 1.41421i
\(970\) −2.00000 −2.00000
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) − 1.41421i − 1.41421i
\(973\) 0 0
\(974\) − 1.41421i − 1.41421i
\(975\) 0 0
\(976\) − 1.41421i − 1.41421i
\(977\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(978\) −2.00000 −2.00000
\(979\) 0 0
\(980\) −1.00000 −1.00000
\(981\) 1.00000 1.00000
\(982\) −2.00000 −2.00000
\(983\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(984\) 0 0
\(985\) −1.00000 −1.00000
\(986\) 2.00000 2.00000
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) −2.00000 −2.00000
\(991\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(992\) 1.41421i 1.41421i
\(993\) 1.41421i 1.41421i
\(994\) 2.00000 2.00000
\(995\) 1.41421i 1.41421i
\(996\) 0 0
\(997\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(998\) 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 331.1.b.b.330.1 2
3.2 odd 2 2979.1.c.d.2647.2 2
331.330 odd 2 inner 331.1.b.b.330.2 yes 2
993.992 even 2 2979.1.c.d.2647.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
331.1.b.b.330.1 2 1.1 even 1 trivial
331.1.b.b.330.2 yes 2 331.330 odd 2 inner
2979.1.c.d.2647.1 2 993.992 even 2
2979.1.c.d.2647.2 2 3.2 odd 2