Properties

Label 119.1.d.a.118.2
Level 119
Weight 1
Character 119.118
Self dual yes
Analytic conductor 0.059
Analytic rank 0
Dimension 2
Projective image \(D_{5}\)
CM discriminant -119
Inner twists 2

Related objects

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Newspace parameters

Level: \( N \) \(=\) \( 119 = 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 119.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.0593887365033\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{5}\)
Projective field Galois closure of 5.1.14161.1
Artin image $D_5$
Artin field Galois closure of 5.1.14161.1

Embedding invariants

Embedding label 118.2
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 119.118

$q$-expansion

\(f(q)\) \(=\) \(q+0.618034 q^{2} +0.618034 q^{3} -0.618034 q^{4} -1.61803 q^{5} +0.381966 q^{6} +1.00000 q^{7} -1.00000 q^{8} -0.618034 q^{9} +O(q^{10})\) \(q+0.618034 q^{2} +0.618034 q^{3} -0.618034 q^{4} -1.61803 q^{5} +0.381966 q^{6} +1.00000 q^{7} -1.00000 q^{8} -0.618034 q^{9} -1.00000 q^{10} -0.381966 q^{12} +0.618034 q^{14} -1.00000 q^{15} +1.00000 q^{17} -0.381966 q^{18} +1.00000 q^{20} +0.618034 q^{21} -0.618034 q^{24} +1.61803 q^{25} -1.00000 q^{27} -0.618034 q^{28} -0.618034 q^{30} -1.61803 q^{31} +1.00000 q^{32} +0.618034 q^{34} -1.61803 q^{35} +0.381966 q^{36} +1.61803 q^{40} +0.618034 q^{41} +0.381966 q^{42} +0.618034 q^{43} +1.00000 q^{45} +1.00000 q^{49} +1.00000 q^{50} +0.618034 q^{51} -1.61803 q^{53} -0.618034 q^{54} -1.00000 q^{56} +0.618034 q^{60} +0.618034 q^{61} -1.00000 q^{62} -0.618034 q^{63} +0.618034 q^{64} -1.61803 q^{67} -0.618034 q^{68} -1.00000 q^{70} +0.618034 q^{72} +0.618034 q^{73} +1.00000 q^{75} +0.381966 q^{82} -0.381966 q^{84} -1.61803 q^{85} +0.381966 q^{86} +0.618034 q^{90} -1.00000 q^{93} +0.618034 q^{96} -1.61803 q^{97} +0.618034 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} - q^{3} + q^{4} - q^{5} + 3q^{6} + 2q^{7} - 2q^{8} + q^{9} + O(q^{10}) \) \( 2q - q^{2} - q^{3} + q^{4} - q^{5} + 3q^{6} + 2q^{7} - 2q^{8} + q^{9} - 2q^{10} - 3q^{12} - q^{14} - 2q^{15} + 2q^{17} - 3q^{18} + 2q^{20} - q^{21} + q^{24} + q^{25} - 2q^{27} + q^{28} + q^{30} - q^{31} + 2q^{32} - q^{34} - q^{35} + 3q^{36} + q^{40} - q^{41} + 3q^{42} - q^{43} + 2q^{45} + 2q^{49} + 2q^{50} - q^{51} - q^{53} + q^{54} - 2q^{56} - q^{60} - q^{61} - 2q^{62} + q^{63} - q^{64} - q^{67} + q^{68} - 2q^{70} - q^{72} - q^{73} + 2q^{75} + 3q^{82} - 3q^{84} - q^{85} + 3q^{86} - q^{90} - 2q^{93} - q^{96} - q^{97} - q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/119\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(71\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(3\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(4\) −0.618034 −0.618034
\(5\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(6\) 0.381966 0.381966
\(7\) 1.00000 1.00000
\(8\) −1.00000 −1.00000
\(9\) −0.618034 −0.618034
\(10\) −1.00000 −1.00000
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) −0.381966 −0.381966
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0.618034 0.618034
\(15\) −1.00000 −1.00000
\(16\) 0 0
\(17\) 1.00000 1.00000
\(18\) −0.381966 −0.381966
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 1.00000 1.00000
\(21\) 0.618034 0.618034
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) −0.618034 −0.618034
\(25\) 1.61803 1.61803
\(26\) 0 0
\(27\) −1.00000 −1.00000
\(28\) −0.618034 −0.618034
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) −0.618034 −0.618034
\(31\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(32\) 1.00000 1.00000
\(33\) 0 0
\(34\) 0.618034 0.618034
\(35\) −1.61803 −1.61803
\(36\) 0.381966 0.381966
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 1.61803 1.61803
\(41\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(42\) 0.381966 0.381966
\(43\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(44\) 0 0
\(45\) 1.00000 1.00000
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 1.00000 1.00000
\(50\) 1.00000 1.00000
\(51\) 0.618034 0.618034
\(52\) 0 0
\(53\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(54\) −0.618034 −0.618034
\(55\) 0 0
\(56\) −1.00000 −1.00000
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0.618034 0.618034
\(61\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(62\) −1.00000 −1.00000
\(63\) −0.618034 −0.618034
\(64\) 0.618034 0.618034
\(65\) 0 0
\(66\) 0 0
\(67\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(68\) −0.618034 −0.618034
\(69\) 0 0
\(70\) −1.00000 −1.00000
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0.618034 0.618034
\(73\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(74\) 0 0
\(75\) 1.00000 1.00000
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0.381966 0.381966
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) −0.381966 −0.381966
\(85\) −1.61803 −1.61803
\(86\) 0.381966 0.381966
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0.618034 0.618034
\(91\) 0 0
\(92\) 0 0
\(93\) −1.00000 −1.00000
\(94\) 0 0
\(95\) 0 0
\(96\) 0.618034 0.618034
\(97\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(98\) 0.618034 0.618034
\(99\) 0 0
\(100\) −1.00000 −1.00000
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0.381966 0.381966
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) −1.00000 −1.00000
\(106\) −1.00000 −1.00000
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0.618034 0.618034
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 1.00000 1.00000
\(120\) 1.00000 1.00000
\(121\) 1.00000 1.00000
\(122\) 0.381966 0.381966
\(123\) 0.381966 0.381966
\(124\) 1.00000 1.00000
\(125\) −1.00000 −1.00000
\(126\) −0.381966 −0.381966
\(127\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(128\) −0.618034 −0.618034
\(129\) 0.381966 0.381966
\(130\) 0 0
\(131\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(132\) 0 0
\(133\) 0 0
\(134\) −1.00000 −1.00000
\(135\) 1.61803 1.61803
\(136\) −1.00000 −1.00000
\(137\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(138\) 0 0
\(139\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(140\) 1.00000 1.00000
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0.381966 0.381966
\(147\) 0.618034 0.618034
\(148\) 0 0
\(149\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(150\) 0.618034 0.618034
\(151\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(152\) 0 0
\(153\) −0.618034 −0.618034
\(154\) 0 0
\(155\) 2.61803 2.61803
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) −1.00000 −1.00000
\(160\) −1.61803 −1.61803
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) −0.381966 −0.381966
\(165\) 0 0
\(166\) 0 0
\(167\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(168\) −0.618034 −0.618034
\(169\) 1.00000 1.00000
\(170\) −1.00000 −1.00000
\(171\) 0 0
\(172\) −0.381966 −0.381966
\(173\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(174\) 0 0
\(175\) 1.61803 1.61803
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(180\) −0.618034 −0.618034
\(181\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(182\) 0 0
\(183\) 0.381966 0.381966
\(184\) 0 0
\(185\) 0 0
\(186\) −0.618034 −0.618034
\(187\) 0 0
\(188\) 0 0
\(189\) −1.00000 −1.00000
\(190\) 0 0
\(191\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(192\) 0.381966 0.381966
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) −1.00000 −1.00000
\(195\) 0 0
\(196\) −0.618034 −0.618034
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(200\) −1.61803 −1.61803
\(201\) −1.00000 −1.00000
\(202\) 0 0
\(203\) 0 0
\(204\) −0.381966 −0.381966
\(205\) −1.00000 −1.00000
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) −0.618034 −0.618034
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 1.00000 1.00000
\(213\) 0 0
\(214\) 0 0
\(215\) −1.00000 −1.00000
\(216\) 1.00000 1.00000
\(217\) −1.61803 −1.61803
\(218\) 0 0
\(219\) 0.381966 0.381966
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 1.00000 1.00000
\(225\) −1.00000 −1.00000
\(226\) 0 0
\(227\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0.618034 0.618034
\(239\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(240\) 0 0
\(241\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(242\) 0.618034 0.618034
\(243\) 1.00000 1.00000
\(244\) −0.381966 −0.381966
\(245\) −1.61803 −1.61803
\(246\) 0.236068 0.236068
\(247\) 0 0
\(248\) 1.61803 1.61803
\(249\) 0 0
\(250\) −0.618034 −0.618034
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0.381966 0.381966
\(253\) 0 0
\(254\) −1.00000 −1.00000
\(255\) −1.00000 −1.00000
\(256\) −1.00000 −1.00000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0.236068 0.236068
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 1.23607 1.23607
\(263\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(264\) 0 0
\(265\) 2.61803 2.61803
\(266\) 0 0
\(267\) 0 0
\(268\) 1.00000 1.00000
\(269\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(270\) 1.00000 1.00000
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0.381966 0.381966
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) −1.00000 −1.00000
\(279\) 1.00000 1.00000
\(280\) 1.61803 1.61803
\(281\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(282\) 0 0
\(283\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0.618034 0.618034
\(288\) −0.618034 −0.618034
\(289\) 1.00000 1.00000
\(290\) 0 0
\(291\) −1.00000 −1.00000
\(292\) −0.381966 −0.381966
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0.381966 0.381966
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0.381966 0.381966
\(299\) 0 0
\(300\) −0.618034 −0.618034
\(301\) 0.618034 0.618034
\(302\) 0.381966 0.381966
\(303\) 0 0
\(304\) 0 0
\(305\) −1.00000 −1.00000
\(306\) −0.381966 −0.381966
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 1.61803 1.61803
\(311\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(312\) 0 0
\(313\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(314\) 0 0
\(315\) 1.00000 1.00000
\(316\) 0 0
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) −0.618034 −0.618034
\(319\) 0 0
\(320\) −1.00000 −1.00000
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) −0.618034 −0.618034
\(329\) 0 0
\(330\) 0 0
\(331\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0.381966 0.381966
\(335\) 2.61803 2.61803
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0.618034 0.618034
\(339\) 0 0
\(340\) 1.00000 1.00000
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 1.00000
\(344\) −0.618034 −0.618034
\(345\) 0 0
\(346\) 0.381966 0.381966
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 1.00000 1.00000
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0.618034 0.618034
\(358\) 0.381966 0.381966
\(359\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(360\) −1.00000 −1.00000
\(361\) 1.00000 1.00000
\(362\) 1.23607 1.23607
\(363\) 0.618034 0.618034
\(364\) 0 0
\(365\) −1.00000 −1.00000
\(366\) 0.236068 0.236068
\(367\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(368\) 0 0
\(369\) −0.381966 −0.381966
\(370\) 0 0
\(371\) −1.61803 −1.61803
\(372\) 0.618034 0.618034
\(373\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(374\) 0 0
\(375\) −0.618034 −0.618034
\(376\) 0 0
\(377\) 0 0
\(378\) −0.618034 −0.618034
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) −1.00000 −1.00000
\(382\) −1.00000 −1.00000
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) −0.381966 −0.381966
\(385\) 0 0
\(386\) 0 0
\(387\) −0.381966 −0.381966
\(388\) 1.00000 1.00000
\(389\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −1.00000 −1.00000
\(393\) 1.23607 1.23607
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(398\) 0.381966 0.381966
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) −0.618034 −0.618034
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) −0.618034 −0.618034
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) −0.618034 −0.618034
\(411\) 0.381966 0.381966
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −1.00000 −1.00000
\(418\) 0 0
\(419\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(420\) 0.618034 0.618034
\(421\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 1.61803 1.61803
\(425\) 1.61803 1.61803
\(426\) 0 0
\(427\) 0.618034 0.618034
\(428\) 0 0
\(429\) 0 0
\(430\) −0.618034 −0.618034
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) −1.00000 −1.00000
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0.236068 0.236068
\(439\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(440\) 0 0
\(441\) −0.618034 −0.618034
\(442\) 0 0
\(443\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0.381966 0.381966
\(448\) 0.618034 0.618034
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) −0.618034 −0.618034
\(451\) 0 0
\(452\) 0 0
\(453\) 0.381966 0.381966
\(454\) −1.00000 −1.00000
\(455\) 0 0
\(456\) 0 0
\(457\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(458\) 0 0
\(459\) −1.00000 −1.00000
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(464\) 0 0
\(465\) 1.61803 1.61803
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) −1.61803 −1.61803
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) −0.618034 −0.618034
\(477\) 1.00000 1.00000
\(478\) 0.381966 0.381966
\(479\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(480\) −1.00000 −1.00000
\(481\) 0 0
\(482\) −1.00000 −1.00000
\(483\) 0 0
\(484\) −0.618034 −0.618034
\(485\) 2.61803 2.61803
\(486\) 0.618034 0.618034
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) −0.618034 −0.618034
\(489\) 0 0
\(490\) −1.00000 −1.00000
\(491\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(492\) −0.236068 −0.236068
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 0.618034 0.618034
\(501\) 0.381966 0.381966
\(502\) 0 0
\(503\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(504\) 0.618034 0.618034
\(505\) 0 0
\(506\) 0 0
\(507\) 0.618034 0.618034
\(508\) 1.00000 1.00000
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) −0.618034 −0.618034
\(511\) 0.618034 0.618034
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) −0.236068 −0.236068
\(517\) 0 0
\(518\) 0 0
\(519\) 0.381966 0.381966
\(520\) 0 0
\(521\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) −1.23607 −1.23607
\(525\) 1.00000 1.00000
\(526\) 1.23607 1.23607
\(527\) −1.61803 −1.61803
\(528\) 0 0
\(529\) 1.00000 1.00000
\(530\) 1.61803 1.61803
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 1.61803 1.61803
\(537\) 0.381966 0.381966
\(538\) 1.23607 1.23607
\(539\) 0 0
\(540\) −1.00000 −1.00000
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 1.23607 1.23607
\(544\) 1.00000 1.00000
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) −0.381966 −0.381966
\(549\) −0.381966 −0.381966
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 1.00000 1.00000
\(557\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(558\) 0.618034 0.618034
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −1.00000 −1.00000
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −1.00000 −1.00000
\(567\) 0 0
\(568\) 0 0
\(569\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) −1.00000 −1.00000
\(574\) 0.381966 0.381966
\(575\) 0 0
\(576\) −0.381966 −0.381966
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 0.618034 0.618034
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) −0.618034 −0.618034
\(583\) 0 0
\(584\) −0.618034 −0.618034
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) −0.381966 −0.381966
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) −1.61803 −1.61803
\(596\) −0.381966 −0.381966
\(597\) 0.381966 0.381966
\(598\) 0 0
\(599\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(600\) −1.00000 −1.00000
\(601\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(602\) 0.381966 0.381966
\(603\) 1.00000 1.00000
\(604\) −0.381966 −0.381966
\(605\) −1.61803 −1.61803
\(606\) 0 0
\(607\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) −0.618034 −0.618034
\(611\) 0 0
\(612\) 0.381966 0.381966
\(613\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(614\) 0 0
\(615\) −0.618034 −0.618034
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(620\) −1.61803 −1.61803
\(621\) 0 0
\(622\) −1.00000 −1.00000
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) −1.00000 −1.00000
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0.618034 0.618034
\(631\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 2.61803 2.61803
\(636\) 0.618034 0.618034
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 1.00000 1.00000
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(644\) 0 0
\(645\) −0.618034 −0.618034
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −1.00000 −1.00000
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) −3.23607 −3.23607
\(656\) 0 0
\(657\) −0.381966 −0.381966
\(658\) 0 0
\(659\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) −1.00000 −1.00000
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) −0.381966 −0.381966
\(669\) 0 0
\(670\) 1.61803 1.61803
\(671\) 0 0
\(672\) 0.618034 0.618034
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) −1.61803 −1.61803
\(676\) −0.618034 −0.618034
\(677\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(678\) 0 0
\(679\) −1.61803 −1.61803
\(680\) 1.61803 1.61803
\(681\) −1.00000 −1.00000
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) −1.00000 −1.00000
\(686\) 0.618034 0.618034
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(692\) −0.381966 −0.381966
\(693\) 0 0
\(694\) 0 0
\(695\) 2.61803 2.61803
\(696\) 0 0
\(697\) 0.618034 0.618034
\(698\) 0 0
\(699\) 0 0
\(700\) −1.00000 −1.00000
\(701\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0.381966 0.381966
\(715\) 0 0
\(716\) −0.381966 −0.381966
\(717\) 0.381966 0.381966
\(718\) −1.00000 −1.00000
\(719\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0.618034 0.618034
\(723\) −1.00000 −1.00000
\(724\) −1.23607 −1.23607
\(725\) 0 0
\(726\) 0.381966 0.381966
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 0.618034 0.618034
\(730\) −0.618034 −0.618034
\(731\) 0.618034 0.618034
\(732\) −0.236068 −0.236068
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0.381966 0.381966
\(735\) −1.00000 −1.00000
\(736\) 0 0
\(737\) 0 0
\(738\) −0.236068 −0.236068
\(739\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −1.00000 −1.00000
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 1.00000 1.00000
\(745\) −1.00000 −1.00000
\(746\) 0.381966 0.381966
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) −0.381966 −0.381966
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −1.00000 −1.00000
\(756\) 0.618034 0.618034
\(757\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) −0.618034 −0.618034
\(763\) 0 0
\(764\) 1.00000 1.00000
\(765\) 1.00000 1.00000
\(766\) 0 0
\(767\) 0 0
\(768\) −0.618034 −0.618034
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) −0.236068 −0.236068
\(775\) −2.61803 −2.61803
\(776\) 1.61803 1.61803
\(777\) 0 0
\(778\) −1.00000 −1.00000
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0.763932 0.763932
\(787\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(788\) 0 0
\(789\) 1.23607 1.23607
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) −1.00000 −1.00000
\(795\) 1.61803 1.61803
\(796\) −0.381966 −0.381966
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 1.61803 1.61803
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0.618034 0.618034
\(805\) 0 0
\(806\) 0 0
\(807\) 1.23607 1.23607
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0.618034 0.618034
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0.236068 0.236068
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 1.00000 1.00000
\(834\) −0.618034 −0.618034
\(835\) −1.00000 −1.00000
\(836\) 0 0
\(837\) 1.61803 1.61803
\(838\) −1.00000 −1.00000
\(839\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(840\) 1.00000 1.00000
\(841\) 1.00000 1.00000
\(842\) 0.381966 0.381966
\(843\) −1.00000 −1.00000
\(844\) 0 0
\(845\) −1.61803 −1.61803
\(846\) 0 0
\(847\) 1.00000 1.00000
\(848\) 0 0
\(849\) −1.00000 −1.00000
\(850\) 1.00000 1.00000
\(851\) 0 0
\(852\) 0 0
\(853\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(854\) 0.381966 0.381966
\(855\) 0 0
\(856\) 0 0
\(857\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0.618034 0.618034
\(861\) 0.381966 0.381966
\(862\) 0 0
\(863\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(864\) −1.00000 −1.00000
\(865\) −1.00000 −1.00000
\(866\) 0 0
\(867\) 0.618034 0.618034
\(868\) 1.00000 1.00000
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 1.00000 1.00000
\(874\) 0 0
\(875\) −1.00000 −1.00000
\(876\) −0.236068 −0.236068
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0.381966 0.381966
\(879\) 0 0
\(880\) 0 0
\(881\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(882\) −0.381966 −0.381966
\(883\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 1.23607 1.23607
\(887\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(888\) 0 0
\(889\) −1.61803 −1.61803
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0.236068 0.236068
\(895\) −1.00000 −1.00000
\(896\) −0.618034 −0.618034
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0.618034 0.618034
\(901\) −1.61803 −1.61803
\(902\) 0 0
\(903\) 0.381966 0.381966
\(904\) 0 0
\(905\) −3.23607 −3.23607
\(906\) 0.236068 0.236068
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 1.00000 1.00000
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −1.00000 −1.00000
\(915\) −0.618034 −0.618034
\(916\) 0 0
\(917\) 2.00000 2.00000
\(918\) −0.618034 −0.618034
\(919\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0.381966 0.381966
\(927\) 0 0
\(928\) 0 0
\(929\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(930\) 1.00000 1.00000
\(931\) 0 0
\(932\) 0 0
\(933\) −1.00000 −1.00000
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) −1.00000 −1.00000
\(939\) −1.00000 −1.00000
\(940\) 0 0
\(941\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 1.61803 1.61803
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) −1.00000 −1.00000
\(953\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(954\) 0.618034 0.618034
\(955\) 2.61803 2.61803
\(956\) −0.381966 −0.381966
\(957\) 0 0
\(958\) 0.381966 0.381966
\(959\) 0.618034 0.618034
\(960\) −0.618034 −0.618034
\(961\) 1.61803 1.61803
\(962\) 0 0
\(963\) 0 0
\(964\) 1.00000 1.00000
\(965\) 0 0
\(966\) 0 0
\(967\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(968\) −1.00000 −1.00000
\(969\) 0 0
\(970\) 1.61803 1.61803
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) −0.618034 −0.618034
\(973\) −1.61803 −1.61803
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 1.00000 1.00000
\(981\) 0 0
\(982\) −1.00000 −1.00000
\(983\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(984\) −0.381966 −0.381966
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) −1.61803 −1.61803
\(993\) −1.00000 −1.00000
\(994\) 0 0
\(995\) −1.00000 −1.00000
\(996\) 0 0
\(997\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(998\) 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 119.1.d.a.118.2 2
3.2 odd 2 1071.1.h.b.118.1 2
4.3 odd 2 1904.1.n.b.1665.1 2
5.2 odd 4 2975.1.b.b.2974.3 4
5.3 odd 4 2975.1.b.b.2974.2 4
5.4 even 2 2975.1.h.d.951.1 2
7.2 even 3 833.1.h.b.815.1 4
7.3 odd 6 833.1.h.a.509.1 4
7.4 even 3 833.1.h.b.509.1 4
7.5 odd 6 833.1.h.a.815.1 4
7.6 odd 2 119.1.d.b.118.2 yes 2
17.2 even 8 2023.1.f.b.251.4 8
17.3 odd 16 2023.1.l.b.1266.4 16
17.4 even 4 2023.1.c.e.1735.1 4
17.5 odd 16 2023.1.l.b.468.1 16
17.6 odd 16 2023.1.l.b.1868.4 16
17.7 odd 16 2023.1.l.b.1889.1 16
17.8 even 8 2023.1.f.b.1483.1 8
17.9 even 8 2023.1.f.b.1483.2 8
17.10 odd 16 2023.1.l.b.1889.2 16
17.11 odd 16 2023.1.l.b.1868.3 16
17.12 odd 16 2023.1.l.b.468.2 16
17.13 even 4 2023.1.c.e.1735.2 4
17.14 odd 16 2023.1.l.b.1266.3 16
17.15 even 8 2023.1.f.b.251.3 8
17.16 even 2 119.1.d.b.118.2 yes 2
21.20 even 2 1071.1.h.a.118.1 2
28.27 even 2 1904.1.n.a.1665.2 2
35.13 even 4 2975.1.b.a.2974.2 4
35.27 even 4 2975.1.b.a.2974.3 4
35.34 odd 2 2975.1.h.c.951.1 2
51.50 odd 2 1071.1.h.a.118.1 2
68.67 odd 2 1904.1.n.a.1665.2 2
85.33 odd 4 2975.1.b.a.2974.2 4
85.67 odd 4 2975.1.b.a.2974.3 4
85.84 even 2 2975.1.h.c.951.1 2
119.6 even 16 2023.1.l.b.1868.3 16
119.13 odd 4 2023.1.c.e.1735.1 4
119.16 even 6 833.1.h.a.815.1 4
119.20 even 16 2023.1.l.b.1266.3 16
119.27 even 16 2023.1.l.b.1889.1 16
119.33 odd 6 833.1.h.b.815.1 4
119.41 even 16 2023.1.l.b.1889.2 16
119.48 even 16 2023.1.l.b.1266.4 16
119.55 odd 4 2023.1.c.e.1735.2 4
119.62 even 16 2023.1.l.b.1868.4 16
119.67 even 6 833.1.h.a.509.1 4
119.76 odd 8 2023.1.f.b.1483.2 8
119.83 odd 8 2023.1.f.b.251.4 8
119.90 even 16 2023.1.l.b.468.2 16
119.97 even 16 2023.1.l.b.468.1 16
119.101 odd 6 833.1.h.b.509.1 4
119.104 odd 8 2023.1.f.b.251.3 8
119.111 odd 8 2023.1.f.b.1483.1 8
119.118 odd 2 CM 119.1.d.a.118.2 2
357.356 even 2 1071.1.h.b.118.1 2
476.475 even 2 1904.1.n.b.1665.1 2
595.118 even 4 2975.1.b.b.2974.2 4
595.237 even 4 2975.1.b.b.2974.3 4
595.594 odd 2 2975.1.h.d.951.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
119.1.d.a.118.2 2 1.1 even 1 trivial
119.1.d.a.118.2 2 119.118 odd 2 CM
119.1.d.b.118.2 yes 2 7.6 odd 2
119.1.d.b.118.2 yes 2 17.16 even 2
833.1.h.a.509.1 4 7.3 odd 6
833.1.h.a.509.1 4 119.67 even 6
833.1.h.a.815.1 4 7.5 odd 6
833.1.h.a.815.1 4 119.16 even 6
833.1.h.b.509.1 4 7.4 even 3
833.1.h.b.509.1 4 119.101 odd 6
833.1.h.b.815.1 4 7.2 even 3
833.1.h.b.815.1 4 119.33 odd 6
1071.1.h.a.118.1 2 21.20 even 2
1071.1.h.a.118.1 2 51.50 odd 2
1071.1.h.b.118.1 2 3.2 odd 2
1071.1.h.b.118.1 2 357.356 even 2
1904.1.n.a.1665.2 2 28.27 even 2
1904.1.n.a.1665.2 2 68.67 odd 2
1904.1.n.b.1665.1 2 4.3 odd 2
1904.1.n.b.1665.1 2 476.475 even 2
2023.1.c.e.1735.1 4 17.4 even 4
2023.1.c.e.1735.1 4 119.13 odd 4
2023.1.c.e.1735.2 4 17.13 even 4
2023.1.c.e.1735.2 4 119.55 odd 4
2023.1.f.b.251.3 8 17.15 even 8
2023.1.f.b.251.3 8 119.104 odd 8
2023.1.f.b.251.4 8 17.2 even 8
2023.1.f.b.251.4 8 119.83 odd 8
2023.1.f.b.1483.1 8 17.8 even 8
2023.1.f.b.1483.1 8 119.111 odd 8
2023.1.f.b.1483.2 8 17.9 even 8
2023.1.f.b.1483.2 8 119.76 odd 8
2023.1.l.b.468.1 16 17.5 odd 16
2023.1.l.b.468.1 16 119.97 even 16
2023.1.l.b.468.2 16 17.12 odd 16
2023.1.l.b.468.2 16 119.90 even 16
2023.1.l.b.1266.3 16 17.14 odd 16
2023.1.l.b.1266.3 16 119.20 even 16
2023.1.l.b.1266.4 16 17.3 odd 16
2023.1.l.b.1266.4 16 119.48 even 16
2023.1.l.b.1868.3 16 17.11 odd 16
2023.1.l.b.1868.3 16 119.6 even 16
2023.1.l.b.1868.4 16 17.6 odd 16
2023.1.l.b.1868.4 16 119.62 even 16
2023.1.l.b.1889.1 16 17.7 odd 16
2023.1.l.b.1889.1 16 119.27 even 16
2023.1.l.b.1889.2 16 17.10 odd 16
2023.1.l.b.1889.2 16 119.41 even 16
2975.1.b.a.2974.2 4 35.13 even 4
2975.1.b.a.2974.2 4 85.33 odd 4
2975.1.b.a.2974.3 4 35.27 even 4
2975.1.b.a.2974.3 4 85.67 odd 4
2975.1.b.b.2974.2 4 5.3 odd 4
2975.1.b.b.2974.2 4 595.118 even 4
2975.1.b.b.2974.3 4 5.2 odd 4
2975.1.b.b.2974.3 4 595.237 even 4
2975.1.h.c.951.1 2 35.34 odd 2
2975.1.h.c.951.1 2 85.84 even 2
2975.1.h.d.951.1 2 5.4 even 2
2975.1.h.d.951.1 2 595.594 odd 2