Properties

Label 116.1.d.a.115.1
Level 116
Weight 1
Character 116.115
Self dual Yes
Analytic conductor 0.058
Analytic rank 0
Dimension 1
Projective image \(D_{3}\)
CM disc. -116
Inner twists 2

Related objects

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Newspace parameters

Level: \( N \) = \( 116 = 2^{2} \cdot 29 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 116.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(0.0578915414654\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.116.1
Artin image size \(12\)
Artin image $D_6$
Artin field Galois closure of 6.0.53824.1

Embedding invariants

Embedding label 115.1
Root \(0\)
Character \(\chi\) = 116.115

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{10} +1.00000 q^{11} +1.00000 q^{12} -1.00000 q^{13} -1.00000 q^{15} +1.00000 q^{16} -2.00000 q^{19} -1.00000 q^{20} -1.00000 q^{22} -1.00000 q^{24} +1.00000 q^{26} -1.00000 q^{27} +1.00000 q^{29} +1.00000 q^{30} +1.00000 q^{31} -1.00000 q^{32} +1.00000 q^{33} +2.00000 q^{38} -1.00000 q^{39} +1.00000 q^{40} +1.00000 q^{43} +1.00000 q^{44} +1.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{52} -1.00000 q^{53} +1.00000 q^{54} -1.00000 q^{55} -2.00000 q^{57} -1.00000 q^{58} -1.00000 q^{60} -1.00000 q^{62} +1.00000 q^{64} +1.00000 q^{65} -1.00000 q^{66} -2.00000 q^{76} +1.00000 q^{78} +1.00000 q^{79} -1.00000 q^{80} -1.00000 q^{81} -1.00000 q^{86} +1.00000 q^{87} -1.00000 q^{88} +1.00000 q^{93} -1.00000 q^{94} +2.00000 q^{95} -1.00000 q^{96} -1.00000 q^{98} +O(q^{100})\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/116\mathbb{Z}\right)^\times\).

\(n\) \(59\) \(89\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −1.00000
\(3\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(4\) 1.00000 1.00000
\(5\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(6\) −1.00000 −1.00000
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) −1.00000 −1.00000
\(9\) 0 0
\(10\) 1.00000 1.00000
\(11\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(12\) 1.00000 1.00000
\(13\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(14\) 0 0
\(15\) −1.00000 −1.00000
\(16\) 1.00000 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(20\) −1.00000 −1.00000
\(21\) 0 0
\(22\) −1.00000 −1.00000
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) −1.00000 −1.00000
\(25\) 0 0
\(26\) 1.00000 1.00000
\(27\) −1.00000 −1.00000
\(28\) 0 0
\(29\) 1.00000 1.00000
\(30\) 1.00000 1.00000
\(31\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(32\) −1.00000 −1.00000
\(33\) 1.00000 1.00000
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 2.00000 2.00000
\(39\) −1.00000 −1.00000
\(40\) 1.00000 1.00000
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(44\) 1.00000 1.00000
\(45\) 0 0
\(46\) 0 0
\(47\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(48\) 1.00000 1.00000
\(49\) 1.00000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) −1.00000 −1.00000
\(53\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(54\) 1.00000 1.00000
\(55\) −1.00000 −1.00000
\(56\) 0 0
\(57\) −2.00000 −2.00000
\(58\) −1.00000 −1.00000
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) −1.00000 −1.00000
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) −1.00000 −1.00000
\(63\) 0 0
\(64\) 1.00000 1.00000
\(65\) 1.00000 1.00000
\(66\) −1.00000 −1.00000
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −2.00000 −2.00000
\(77\) 0 0
\(78\) 1.00000 1.00000
\(79\) 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
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1.00000 −1.00000
\(87\) 1.00000 1.00000
\(88\) −1.00000 −1.00000
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 1.00000 1.00000
\(94\) −1.00000 −1.00000
\(95\) 2.00000 2.00000
\(96\) −1.00000 −1.00000
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) −1.00000 −1.00000
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 1.00000 1.00000
\(105\) 0 0
\(106\) 1.00000 1.00000
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) −1.00000 −1.00000
\(109\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(110\) 1.00000 1.00000
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 2.00000 2.00000
\(115\) 0 0
\(116\) 1.00000 1.00000
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 1.00000 1.00000
\(121\) 0 0
\(122\) 0 0
\(123\) 0 0
\(124\) 1.00000 1.00000
\(125\) 1.00000 1.00000
\(126\) 0 0
\(127\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(128\) −1.00000 −1.00000
\(129\) 1.00000 1.00000
\(130\) −1.00000 −1.00000
\(131\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(132\) 1.00000 1.00000
\(133\) 0 0
\(134\) 0 0
\(135\) 1.00000 1.00000
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 1.00000 1.00000
\(142\) 0 0
\(143\) −1.00000 −1.00000
\(144\) 0 0
\(145\) −1.00000 −1.00000
\(146\) 0 0
\(147\) 1.00000 1.00000
\(148\) 0 0
\(149\) −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 \(\pi\)
\(152\) 2.00000 2.00000
\(153\) 0 0
\(154\) 0 0
\(155\) −1.00000 −1.00000
\(156\) −1.00000 −1.00000
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) −1.00000 −1.00000
\(159\) −1.00000 −1.00000
\(160\) 1.00000 1.00000
\(161\) 0 0
\(162\) 1.00000 1.00000
\(163\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(164\) 0 0
\(165\) −1.00000 −1.00000
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 0 0
\(172\) 1.00000 1.00000
\(173\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(174\) −1.00000 −1.00000
\(175\) 0 0
\(176\) 1.00000 1.00000
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) −1.00000 −1.00000
\(187\) 0 0
\(188\) 1.00000 1.00000
\(189\) 0 0
\(190\) −2.00000 −2.00000
\(191\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(192\) 1.00000 1.00000
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 1.00000 1.00000
\(196\) 1.00000 1.00000
\(197\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −1.00000 −1.00000
\(209\) −2.00000 −2.00000
\(210\) 0 0
\(211\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(212\) −1.00000 −1.00000
\(213\) 0 0
\(214\) 0 0
\(215\) −1.00000 −1.00000
\(216\) 1.00000 1.00000
\(217\) 0 0
\(218\) 1.00000 1.00000
\(219\) 0 0
\(220\) −1.00000 −1.00000
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) −2.00000 −2.00000
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −1.00000 −1.00000
\(233\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(234\) 0 0
\(235\) −1.00000 −1.00000
\(236\) 0 0
\(237\) 1.00000 1.00000
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) −1.00000 −1.00000
\(241\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −1.00000 −1.00000
\(246\) 0 0
\(247\) 2.00000 2.00000
\(248\) −1.00000 −1.00000
\(249\) 0 0
\(250\) −1.00000 −1.00000
\(251\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 2.00000 2.00000
\(255\) 0 0
\(256\) 1.00000 1.00000
\(257\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(258\) −1.00000 −1.00000
\(259\) 0 0
\(260\) 1.00000 1.00000
\(261\) 0 0
\(262\) 2.00000 2.00000
\(263\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(264\) −1.00000 −1.00000
\(265\) 1.00000 1.00000
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) −1.00000 −1.00000
\(271\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(282\) −1.00000 −1.00000
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 2.00000 2.00000
\(286\) 1.00000 1.00000
\(287\) 0 0
\(288\) 0 0
\(289\) 1.00000 1.00000
\(290\) 1.00000 1.00000
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) −1.00000 −1.00000
\(295\) 0 0
\(296\) 0 0
\(297\) −1.00000 −1.00000
\(298\) 1.00000 1.00000
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) −2.00000 −2.00000
\(305\) 0 0
\(306\) 0 0
\(307\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 1.00000 1.00000
\(311\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(312\) 1.00000 1.00000
\(313\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 1.00000 1.00000
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 1.00000 1.00000
\(319\) 1.00000 1.00000
\(320\) −1.00000 −1.00000
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −1.00000 −1.00000
\(325\) 0 0
\(326\) −1.00000 −1.00000
\(327\) −1.00000 −1.00000
\(328\) 0 0
\(329\) 0 0
\(330\) 1.00000 1.00000
\(331\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 1.00000 1.00000
\(342\) 0 0
\(343\) 0 0
\(344\) −1.00000 −1.00000
\(345\) 0 0
\(346\) −2.00000 −2.00000
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 1.00000 1.00000
\(349\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(350\) 0 0
\(351\) 1.00000 1.00000
\(352\) −1.00000 −1.00000
\(353\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(360\) 0 0
\(361\) 3.00000 3.00000
\(362\) 1.00000 1.00000
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 1.00000 1.00000
\(373\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(374\) 0 0
\(375\) 1.00000 1.00000
\(376\) −1.00000 −1.00000
\(377\) −1.00000 −1.00000
\(378\) 0 0
\(379\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(380\) 2.00000 2.00000
\(381\) −2.00000 −2.00000
\(382\) 2.00000 2.00000
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) −1.00000 −1.00000
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) −1.00000 −1.00000
\(391\) 0 0
\(392\) −1.00000 −1.00000
\(393\) −2.00000 −2.00000
\(394\) −2.00000 −2.00000
\(395\) −1.00000 −1.00000
\(396\) 0 0
\(397\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(402\) 0 0
\(403\) −1.00000 −1.00000
\(404\) 0 0
\(405\) 1.00000 1.00000
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 1.00000 1.00000
\(417\) 0 0
\(418\) 2.00000 2.00000
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) −1.00000 −1.00000
\(423\) 0 0
\(424\) 1.00000 1.00000
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −1.00000 −1.00000
\(430\) 1.00000 1.00000
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) −1.00000 −1.00000
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) −1.00000 −1.00000
\(436\) −1.00000 −1.00000
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 1.00000 1.00000
\(441\) 0 0
\(442\) 0 0
\(443\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −1.00000 −1.00000
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 2.00000 2.00000
\(457\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 1.00000 1.00000
\(465\) −1.00000 −1.00000
\(466\) 1.00000 1.00000
\(467\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 1.00000 1.00000
\(471\) 0 0
\(472\) 0 0
\(473\) 1.00000 1.00000
\(474\) −1.00000 −1.00000
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(480\) 1.00000 1.00000
\(481\) 0 0
\(482\) 1.00000 1.00000
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 1.00000 1.00000
\(490\) 1.00000 1.00000
\(491\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) −2.00000 −2.00000
\(495\) 0 0
\(496\) 1.00000 1.00000
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 1.00000 1.00000
\(501\) 0 0
\(502\) −1.00000 −1.00000
\(503\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) −2.00000 −2.00000
\(509\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −1.00000
\(513\) 2.00000 2.00000
\(514\) 1.00000 1.00000
\(515\) 0 0
\(516\) 1.00000 1.00000
\(517\) 1.00000 1.00000
\(518\) 0 0
\(519\) 2.00000 2.00000
\(520\) −1.00000 −1.00000
\(521\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) −2.00000 −2.00000
\(525\) 0 0
\(526\) −1.00000 −1.00000
\(527\) 0 0
\(528\) 1.00000 1.00000
\(529\) 1.00000 1.00000
\(530\) −1.00000 −1.00000
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 1.00000 1.00000
\(540\) 1.00000 1.00000
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) −1.00000 −1.00000
\(543\) −1.00000 −1.00000
\(544\) 0 0
\(545\) 1.00000 1.00000
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −2.00000 −2.00000
\(552\) 0 0
\(553\) 0 0
\(554\) −2.00000 −2.00000
\(555\) 0 0
\(556\) 0 0
\(557\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(558\) 0 0
\(559\) −1.00000 −1.00000
\(560\) 0 0
\(561\) 0 0
\(562\) 1.00000 1.00000
\(563\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(564\) 1.00000 1.00000
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) −2.00000 −2.00000
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) −1.00000 −1.00000
\(573\) −2.00000 −2.00000
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) −1.00000 −1.00000
\(579\) 0 0
\(580\) −1.00000 −1.00000
\(581\) 0 0
\(582\) 0 0
\(583\) −1.00000 −1.00000
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 1.00000 1.00000
\(589\) −2.00000 −2.00000
\(590\) 0 0
\(591\) 2.00000 2.00000
\(592\) 0 0
\(593\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(594\) 1.00000 1.00000
\(595\) 0 0
\(596\) −1.00000 −1.00000
\(597\) 0 0
\(598\) 0 0
\(599\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(608\) 2.00000 2.00000
\(609\) 0 0
\(610\) 0 0
\(611\) −1.00000 −1.00000
\(612\) 0 0
\(613\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(614\) −1.00000 −1.00000
\(615\) 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 \(0.333333\pi\)
\(620\) −1.00000 −1.00000
\(621\) 0 0
\(622\) 2.00000 2.00000
\(623\) 0 0
\(624\) −1.00000 −1.00000
\(625\) −1.00000 −1.00000
\(626\) 1.00000 1.00000
\(627\) −2.00000 −2.00000
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) −1.00000 −1.00000
\(633\) 1.00000 1.00000
\(634\) 0 0
\(635\) 2.00000 2.00000
\(636\) −1.00000 −1.00000
\(637\) −1.00000 −1.00000
\(638\) −1.00000 −1.00000
\(639\) 0 0
\(640\) 1.00000 1.00000
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) −1.00000 −1.00000
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 1.00000 1.00000
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 1.00000 1.00000
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 1.00000 1.00000
\(655\) 2.00000 2.00000
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(660\) −1.00000 −1.00000
\(661\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(662\) −1.00000 −1.00000
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) −1.00000 −1.00000
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 1.00000 1.00000
\(689\) 1.00000 1.00000
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 2.00000 2.00000
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) −1.00000 −1.00000
\(697\) 0 0
\(698\) 1.00000 1.00000
\(699\) −1.00000 −1.00000
\(700\) 0 0
\(701\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(702\) −1.00000 −1.00000
\(703\) 0 0
\(704\) 1.00000 1.00000
\(705\) −1.00000 −1.00000
\(706\) −2.00000 −2.00000
\(707\) 0 0
\(708\) 0 0
\(709\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 1.00000 1.00000
\(716\) 0 0
\(717\) 0 0
\(718\) −1.00000 −1.00000
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −3.00000 −3.00000
\(723\) −1.00000 −1.00000
\(724\) −1.00000 −1.00000
\(725\) 0 0
\(726\) 0 0
\(727\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 1.00000 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 2.00000 2.00000
\(735\) −1.00000 −1.00000
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(740\) 0 0
\(741\) 2.00000 2.00000
\(742\) 0 0
\(743\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(744\) −1.00000 −1.00000
\(745\) 1.00000 1.00000
\(746\) 1.00000 1.00000
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) −1.00000 −1.00000
\(751\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(752\) 1.00000 1.00000
\(753\) 1.00000 1.00000
\(754\) 1.00000 1.00000
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 2.00000 2.00000
\(759\) 0 0
\(760\) −2.00000 −2.00000
\(761\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(762\) 2.00000 2.00000
\(763\) 0 0
\(764\) −2.00000 −2.00000
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 1.00000 1.00000
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) −1.00000 −1.00000
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 1.00000 1.00000
\(781\) 0 0
\(782\) 0 0
\(783\) −1.00000 −1.00000
\(784\) 1.00000 1.00000
\(785\) 0 0
\(786\) 2.00000 2.00000
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 2.00000 2.00000
\(789\) 1.00000 1.00000
\(790\) 1.00000 1.00000
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 1.00000 1.00000
\(795\) 1.00000 1.00000
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 1.00000 1.00000
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 1.00000 1.00000
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) −1.00000 −1.00000
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 1.00000 1.00000
\(814\) 0 0
\(815\) −1.00000 −1.00000
\(816\) 0 0
\(817\) −2.00000 −2.00000
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(822\) 0 0
\(823\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(824\) 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
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 2.00000 2.00000
\(832\) −1.00000 −1.00000
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) −2.00000 −2.00000
\(837\) −1.00000 −1.00000
\(838\) 0 0
\(839\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(840\) 0 0
\(841\) 1.00000 1.00000
\(842\) 0 0
\(843\) −1.00000 −1.00000
\(844\) 1.00000 1.00000
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) −1.00000 −1.00000
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(858\) 1.00000 1.00000
\(859\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(860\) −1.00000 −1.00000
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 1.00000 1.00000
\(865\) −2.00000 −2.00000
\(866\) 0 0
\(867\) 1.00000 1.00000
\(868\) 0 0
\(869\) 1.00000 1.00000
\(870\) 1.00000 1.00000
\(871\) 0 0
\(872\) 1.00000 1.00000
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) −1.00000 −1.00000
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 2.00000 2.00000
\(887\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −1.00000 −1.00000
\(892\) 0 0
\(893\) −2.00000 −2.00000
\(894\) 1.00000 1.00000
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 1.00000 1.00000
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 1.00000 1.00000
\(906\) 0 0
\(907\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(912\) −2.00000 −2.00000
\(913\) 0 0
\(914\) −2.00000 −2.00000
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 1.00000 1.00000
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) −1.00000 −1.00000
\(929\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(930\) 1.00000 1.00000
\(931\) −2.00000 −2.00000
\(932\) −1.00000 −1.00000
\(933\) −2.00000 −2.00000
\(934\) −1.00000 −1.00000
\(935\) 0 0
\(936\) 0 0
\(937\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(938\) 0 0
\(939\) −1.00000 −1.00000
\(940\) −1.00000 −1.00000
\(941\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) −1.00000 −1.00000
\(947\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(948\) 1.00000 1.00000
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(954\) 0 0
\(955\) 2.00000 2.00000
\(956\) 0 0
\(957\) 1.00000 1.00000
\(958\) −1.00000 −1.00000
\(959\) 0 0
\(960\) −1.00000 −1.00000
\(961\) 0 0
\(962\) 0 0
\(963\) 0 0
\(964\) −1.00000 −1.00000
\(965\) 0 0
\(966\) 0 0
\(967\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(978\) −1.00000 −1.00000
\(979\) 0 0
\(980\) −1.00000 −1.00000
\(981\) 0 0
\(982\) −1.00000 −1.00000
\(983\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(984\) 0 0
\(985\) −2.00000 −2.00000
\(986\) 0 0
\(987\) 0 0
\(988\) 2.00000 2.00000
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) −1.00000 −1.00000
\(993\) 1.00000 1.00000
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 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 \(a_p\))