Properties

Label 356.1.d.b.355.1
Level 356
Weight 1
Character 356.355
Self dual Yes
Analytic conductor 0.178
Analytic rank 0
Dimension 1
Projective image \(D_{3}\)
CM disc. -356
Inner twists 2

Related objects

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Newspace parameters

Level: \( N \) = \( 356 = 2^{2} \cdot 89 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 356.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(0.177667144497\)
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 3.1.356.1
Artin image size \(6\)
Artin image $S_3$
Artin field Galois closure of 3.1.356.1

Embedding invariants

Embedding label 355.1
Root \(0\)
Character \(\chi\) = 356.355

$q$-expansion

\(f(q)\) \(=\) \(q\)\(+1.00000 q^{2}\) \(-1.00000 q^{3}\) \(+1.00000 q^{4}\) \(-1.00000 q^{5}\) \(-1.00000 q^{6}\) \(+2.00000 q^{7}\) \(+1.00000 q^{8}\) \(+O(q^{10})\) \(q\)\(+1.00000 q^{2}\) \(-1.00000 q^{3}\) \(+1.00000 q^{4}\) \(-1.00000 q^{5}\) \(-1.00000 q^{6}\) \(+2.00000 q^{7}\) \(+1.00000 q^{8}\) \(-1.00000 q^{10}\) \(-1.00000 q^{12}\) \(+2.00000 q^{14}\) \(+1.00000 q^{15}\) \(+1.00000 q^{16}\) \(-1.00000 q^{17}\) \(-1.00000 q^{19}\) \(-1.00000 q^{20}\) \(-2.00000 q^{21}\) \(-1.00000 q^{23}\) \(-1.00000 q^{24}\) \(+1.00000 q^{27}\) \(+2.00000 q^{28}\) \(+1.00000 q^{30}\) \(-1.00000 q^{31}\) \(+1.00000 q^{32}\) \(-1.00000 q^{34}\) \(-2.00000 q^{35}\) \(-1.00000 q^{38}\) \(-1.00000 q^{40}\) \(-2.00000 q^{42}\) \(-1.00000 q^{43}\) \(-1.00000 q^{46}\) \(-1.00000 q^{48}\) \(+3.00000 q^{49}\) \(+1.00000 q^{51}\) \(-1.00000 q^{53}\) \(+1.00000 q^{54}\) \(+2.00000 q^{56}\) \(+1.00000 q^{57}\) \(+2.00000 q^{59}\) \(+1.00000 q^{60}\) \(-1.00000 q^{62}\) \(+1.00000 q^{64}\) \(-1.00000 q^{68}\) \(+1.00000 q^{69}\) \(-2.00000 q^{70}\) \(-1.00000 q^{73}\) \(-1.00000 q^{76}\) \(-1.00000 q^{80}\) \(-1.00000 q^{81}\) \(+2.00000 q^{83}\) \(-2.00000 q^{84}\) \(+1.00000 q^{85}\) \(-1.00000 q^{86}\) \(+1.00000 q^{89}\) \(-1.00000 q^{92}\) \(+1.00000 q^{93}\) \(+1.00000 q^{95}\) \(-1.00000 q^{96}\) \(-1.00000 q^{97}\) \(+3.00000 q^{98}\) \(+O(q^{100})\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/356\mathbb{Z}\right)^\times\).

\(n\) \(179\) \(181\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(4\) 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
\(7\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(8\) 1.00000 1.00000
\(9\) 0 0
\(10\) −1.00000 −1.00000
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) −1.00000 −1.00000
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 2.00000 2.00000
\(15\) 1.00000 1.00000
\(16\) 1.00000 1.00000
\(17\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(18\) 0 0
\(19\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(20\) −1.00000 −1.00000
\(21\) −2.00000 −2.00000
\(22\) 0 0
\(23\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(24\) −1.00000 −1.00000
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000 1.00000
\(28\) 2.00000 2.00000
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 1.00000 1.00000
\(31\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(32\) 1.00000 1.00000
\(33\) 0 0
\(34\) −1.00000 −1.00000
\(35\) −2.00000 −2.00000
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) −1.00000 −1.00000
\(39\) 0 0
\(40\) −1.00000 −1.00000
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) −2.00000 −2.00000
\(43\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −1.00000 −1.00000
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) −1.00000 −1.00000
\(49\) 3.00000 3.00000
\(50\) 0 0
\(51\) 1.00000 1.00000
\(52\) 0 0
\(53\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(54\) 1.00000 1.00000
\(55\) 0 0
\(56\) 2.00000 2.00000
\(57\) 1.00000 1.00000
\(58\) 0 0
\(59\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(60\) 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
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) −1.00000 −1.00000
\(69\) 1.00000 1.00000
\(70\) −2.00000 −2.00000
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −1.00000 −1.00000
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) −1.00000 −1.00000
\(81\) −1.00000 −1.00000
\(82\) 0 0
\(83\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(84\) −2.00000 −2.00000
\(85\) 1.00000 1.00000
\(86\) −1.00000 −1.00000
\(87\) 0 0
\(88\) 0 0
\(89\) 1.00000 1.00000
\(90\) 0 0
\(91\) 0 0
\(92\) −1.00000 −1.00000
\(93\) 1.00000 1.00000
\(94\) 0 0
\(95\) 1.00000 1.00000
\(96\) −1.00000 −1.00000
\(97\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(98\) 3.00000 3.00000
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 1.00000 1.00000
\(103\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(104\) 0 0
\(105\) 2.00000 2.00000
\(106\) −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 \(0.666667\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 2.00000 2.00000
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 1.00000 1.00000
\(115\) 1.00000 1.00000
\(116\) 0 0
\(117\) 0 0
\(118\) 2.00000 2.00000
\(119\) −2.00000 −2.00000
\(120\) 1.00000 1.00000
\(121\) 1.00000 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) −1.00000 −1.00000
\(125\) 1.00000 1.00000
\(126\) 0 0
\(127\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(128\) 1.00000 1.00000
\(129\) 1.00000 1.00000
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) −2.00000 −2.00000
\(134\) 0 0
\(135\) −1.00000 −1.00000
\(136\) −1.00000 −1.00000
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 1.00000 1.00000
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) −2.00000 −2.00000
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) −1.00000 −1.00000
\(147\) −3.00000 −3.00000
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(152\) −1.00000 −1.00000
\(153\) 0 0
\(154\) 0 0
\(155\) 1.00000 1.00000
\(156\) 0 0
\(157\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(158\) 0 0
\(159\) 1.00000 1.00000
\(160\) −1.00000 −1.00000
\(161\) −2.00000 −2.00000
\(162\) −1.00000 −1.00000
\(163\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 2.00000 2.00000
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) −2.00000 −2.00000
\(169\) 1.00000 1.00000
\(170\) 1.00000 1.00000
\(171\) 0 0
\(172\) −1.00000 −1.00000
\(173\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −2.00000 −2.00000
\(178\) 1.00000 1.00000
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −1.00000 −1.00000
\(185\) 0 0
\(186\) 1.00000 1.00000
\(187\) 0 0
\(188\) 0 0
\(189\) 2.00000 2.00000
\(190\) 1.00000 1.00000
\(191\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(192\) −1.00000 −1.00000
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) −1.00000 −1.00000
\(195\) 0 0
\(196\) 3.00000 3.00000
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 1.00000 1.00000
\(205\) 0 0
\(206\) −1.00000 −1.00000
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 2.00000 2.00000
\(211\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(212\) −1.00000 −1.00000
\(213\) 0 0
\(214\) 0 0
\(215\) 1.00000 1.00000
\(216\) 1.00000 1.00000
\(217\) −2.00000 −2.00000
\(218\) −1.00000 −1.00000
\(219\) 1.00000 1.00000
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 2.00000 2.00000
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 1.00000 1.00000
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 1.00000 1.00000
\(231\) 0 0
\(232\) 0 0
\(233\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(234\) 0 0
\(235\) 0 0
\(236\) 2.00000 2.00000
\(237\) 0 0
\(238\) −2.00000 −2.00000
\(239\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(240\) 1.00000 1.00000
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 1.00000 1.00000
\(243\) 0 0
\(244\) 0 0
\(245\) −3.00000 −3.00000
\(246\) 0 0
\(247\) 0 0
\(248\) −1.00000 −1.00000
\(249\) −2.00000 −2.00000
\(250\) 1.00000 1.00000
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −1.00000 −1.00000
\(255\) −1.00000 −1.00000
\(256\) 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
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 1.00000 1.00000
\(266\) −2.00000 −2.00000
\(267\) −1.00000 −1.00000
\(268\) 0 0
\(269\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(270\) −1.00000 −1.00000
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) −1.00000 −1.00000
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 1.00000 1.00000
\(277\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) −2.00000 −2.00000
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) −1.00000 −1.00000
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 0 0
\(290\) 0 0
\(291\) 1.00000 1.00000
\(292\) −1.00000 −1.00000
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) −3.00000 −3.00000
\(295\) −2.00000 −2.00000
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −2.00000 −2.00000
\(302\) 2.00000 2.00000
\(303\) 0 0
\(304\) −1.00000 −1.00000
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 1.00000 1.00000
\(310\) 1.00000 1.00000
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 2.00000 2.00000
\(315\) 0 0
\(316\) 0 0
\(317\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(318\) 1.00000 1.00000
\(319\) 0 0
\(320\) −1.00000 −1.00000
\(321\) 0 0
\(322\) −2.00000 −2.00000
\(323\) 1.00000 1.00000
\(324\) −1.00000 −1.00000
\(325\) 0 0
\(326\) −1.00000 −1.00000
\(327\) 1.00000 1.00000
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 2.00000 2.00000
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) −2.00000 −2.00000
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 1.00000 1.00000
\(339\) 0 0
\(340\) 1.00000 1.00000
\(341\) 0 0
\(342\) 0 0
\(343\) 4.00000 4.00000
\(344\) −1.00000 −1.00000
\(345\) −1.00000 −1.00000
\(346\) −1.00000 −1.00000
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) −2.00000 −2.00000
\(355\) 0 0
\(356\) 1.00000 1.00000
\(357\) 2.00000 2.00000
\(358\) 0 0
\(359\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(360\) 0 0
\(361\) 0 0
\(362\) 0 0
\(363\) −1.00000 −1.00000
\(364\) 0 0
\(365\) 1.00000 1.00000
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) −1.00000 −1.00000
\(369\) 0 0
\(370\) 0 0
\(371\) −2.00000 −2.00000
\(372\) 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
\(376\) 0 0
\(377\) 0 0
\(378\) 2.00000 2.00000
\(379\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(380\) 1.00000 1.00000
\(381\) 1.00000 1.00000
\(382\) −1.00000 −1.00000
\(383\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(384\) −1.00000 −1.00000
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) −1.00000 −1.00000
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 1.00000 1.00000
\(392\) 3.00000 3.00000
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 2.00000 2.00000
\(400\) 0 0
\(401\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 1.00000 1.00000
\(406\) 0 0
\(407\) 0 0
\(408\) 1.00000 1.00000
\(409\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(410\) 0 0
\(411\) 0 0
\(412\) −1.00000 −1.00000
\(413\) 4.00000 4.00000
\(414\) 0 0
\(415\) −2.00000 −2.00000
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(420\) 2.00000 2.00000
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 2.00000 2.00000
\(423\) 0 0
\(424\) −1.00000 −1.00000
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 1.00000 1.00000
\(431\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(432\) 1.00000 1.00000
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) −2.00000 −2.00000
\(435\) 0 0
\(436\) −1.00000 −1.00000
\(437\) 1.00000 1.00000
\(438\) 1.00000 1.00000
\(439\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) −1.00000 −1.00000
\(446\) 0 0
\(447\) 0 0
\(448\) 2.00000 2.00000
\(449\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −2.00000 −2.00000
\(454\) 0 0
\(455\) 0 0
\(456\) 1.00000 1.00000
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) −1.00000 −1.00000
\(460\) 1.00000 1.00000
\(461\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) −1.00000 −1.00000
\(466\) 2.00000 2.00000
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −2.00000 −2.00000
\(472\) 2.00000 2.00000
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) −2.00000 −2.00000
\(477\) 0 0
\(478\) −1.00000 −1.00000
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 1.00000 1.00000
\(481\) 0 0
\(482\) 0 0
\(483\) 2.00000 2.00000
\(484\) 1.00000 1.00000
\(485\) 1.00000 1.00000
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 1.00000 1.00000
\(490\) −3.00000 −3.00000
\(491\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −1.00000 −1.00000
\(497\) 0 0
\(498\) −2.00000 −2.00000
\(499\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(500\) 1.00000 1.00000
\(501\) 0 0
\(502\) 0 0
\(503\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −1.00000 −1.00000
\(508\) −1.00000 −1.00000
\(509\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(510\) −1.00000 −1.00000
\(511\) −2.00000 −2.00000
\(512\) 1.00000 1.00000
\(513\) −1.00000 −1.00000
\(514\) −1.00000 −1.00000
\(515\) 1.00000 1.00000
\(516\) 1.00000 1.00000
\(517\) 0 0
\(518\) 0 0
\(519\) 1.00000 1.00000
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1.00000 1.00000
\(528\) 0 0
\(529\) 0 0
\(530\) 1.00000 1.00000
\(531\) 0 0
\(532\) −2.00000 −2.00000
\(533\) 0 0
\(534\) −1.00000 −1.00000
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) −1.00000 −1.00000
\(539\) 0 0
\(540\) −1.00000 −1.00000
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) −1.00000 −1.00000
\(545\) 1.00000 1.00000
\(546\) 0 0
\(547\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 1.00000 1.00000
\(553\) 0 0
\(554\) −1.00000 −1.00000
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −2.00000 −2.00000
\(561\) 0 0
\(562\) 0 0
\(563\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −2.00000 −2.00000
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) −1.00000 −1.00000
\(571\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(572\) 0 0
\(573\) 1.00000 1.00000
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 4.00000 4.00000
\(582\) 1.00000 1.00000
\(583\) 0 0
\(584\) −1.00000 −1.00000
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) −3.00000 −3.00000
\(589\) 1.00000 1.00000
\(590\) −2.00000 −2.00000
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 2.00000 2.00000
\(596\) 0 0
\(597\) 0 0
\(598\) 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 \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(602\) −2.00000 −2.00000
\(603\) 0 0
\(604\) 2.00000 2.00000
\(605\) −1.00000 −1.00000
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) −1.00000 −1.00000
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 1.00000 1.00000
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 1.00000 1.00000
\(621\) −1.00000 −1.00000
\(622\) 0 0
\(623\) 2.00000 2.00000
\(624\) 0 0
\(625\) −1.00000 −1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 2.00000 2.00000
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) −2.00000 −2.00000
\(634\) −1.00000 −1.00000
\(635\) 1.00000 1.00000
\(636\) 1.00000 1.00000
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) −1.00000 −1.00000
\(641\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) −2.00000 −2.00000
\(645\) −1.00000 −1.00000
\(646\) 1.00000 1.00000
\(647\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(648\) −1.00000 −1.00000
\(649\) 0 0
\(650\) 0 0
\(651\) 2.00000 2.00000
\(652\) −1.00000 −1.00000
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 1.00000 1.00000
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 2.00000 2.00000
\(665\) 2.00000 2.00000
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) −2.00000 −2.00000
\(673\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 1.00000 1.00000
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) −2.00000 −2.00000
\(680\) 1.00000 1.00000
\(681\) 0 0
\(682\) 0 0
\(683\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 4.00000 4.00000
\(687\) 0 0
\(688\) −1.00000 −1.00000
\(689\) 0 0
\(690\) −1.00000 −1.00000
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) −1.00000 −1.00000
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −2.00000 −2.00000
\(700\) 0 0
\(701\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) −2.00000 −2.00000
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 1.00000 1.00000
\(713\) 1.00000 1.00000
\(714\) 2.00000 2.00000
\(715\) 0 0
\(716\) 0 0
\(717\) 1.00000 1.00000
\(718\) 2.00000 2.00000
\(719\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(720\) 0 0
\(721\) −2.00000 −2.00000
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) −1.00000 −1.00000
\(727\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(728\) 0 0
\(729\) 1.00000 1.00000
\(730\) 1.00000 1.00000
\(731\) 1.00000 1.00000
\(732\) 0 0
\(733\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(734\) 0 0
\(735\) 3.00000 3.00000
\(736\) −1.00000 −1.00000
\(737\) 0 0
\(738\) 0 0
\(739\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −2.00000 −2.00000
\(743\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(744\) 1.00000 1.00000
\(745\) 0 0
\(746\) −1.00000 −1.00000
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) −1.00000 −1.00000
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −2.00000 −2.00000
\(756\) 2.00000 2.00000
\(757\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(758\) −1.00000 −1.00000
\(759\) 0 0
\(760\) 1.00000 1.00000
\(761\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(762\) 1.00000 1.00000
\(763\) −2.00000 −2.00000
\(764\) −1.00000 −1.00000
\(765\) 0 0
\(766\) −1.00000 −1.00000
\(767\) 0 0
\(768\) −1.00000 −1.00000
\(769\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(770\) 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
\(776\) −1.00000 −1.00000
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 1.00000 1.00000
\(783\) 0 0
\(784\) 3.00000 3.00000
\(785\) −2.00000 −2.00000
\(786\) 0 0
\(787\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −1.00000 −1.00000
\(796\) 0 0
\(797\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(798\) 2.00000 2.00000
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 2.00000 2.00000
\(803\) 0 0
\(804\) 0 0
\(805\) 2.00000 2.00000
\(806\) 0 0
\(807\) 1.00000 1.00000
\(808\) 0 0
\(809\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(810\) 1.00000 1.00000
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 1.00000 1.00000
\(816\) 1.00000 1.00000
\(817\) 1.00000 1.00000
\(818\) 2.00000 2.00000
\(819\) 0 0
\(820\) 0 0
\(821\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) −1.00000 −1.00000
\(825\) 0 0
\(826\) 4.00000 4.00000
\(827\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) −2.00000 −2.00000
\(831\) 1.00000 1.00000
\(832\) 0 0
\(833\) −3.00000 −3.00000
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −1.00000 −1.00000
\(838\) −1.00000 −1.00000
\(839\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(840\) 2.00000 2.00000
\(841\) 1.00000 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 2.00000 2.00000
\(845\) −1.00000 −1.00000
\(846\) 0 0
\(847\) 2.00000 2.00000
\(848\) −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
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(860\) 1.00000 1.00000
\(861\) 0 0
\(862\) −1.00000 −1.00000
\(863\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(864\) 1.00000 1.00000
\(865\) 1.00000 1.00000
\(866\) 0 0
\(867\) 0 0
\(868\) −2.00000 −2.00000
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) −1.00000 −1.00000
\(873\) 0 0
\(874\) 1.00000 1.00000
\(875\) 2.00000 2.00000
\(876\) 1.00000 1.00000
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) −1.00000 −1.00000
\(879\) 0 0
\(880\) 0 0
\(881\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(882\) 0 0
\(883\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(884\) 0 0
\(885\) 2.00000 2.00000
\(886\) 0 0
\(887\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(888\) 0 0
\(889\) −2.00000 −2.00000
\(890\) −1.00000 −1.00000
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 2.00000 2.00000
\(897\) 0 0
\(898\) −1.00000 −1.00000
\(899\) 0 0
\(900\) 0 0
\(901\) 1.00000 1.00000
\(902\) 0 0
\(903\) 2.00000 2.00000
\(904\) 0 0
\(905\) 0 0
\(906\) −2.00000 −2.00000
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 1.00000 1.00000
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) −1.00000 −1.00000
\(919\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(920\) 1.00000 1.00000
\(921\) 0 0
\(922\) 2.00000 2.00000
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(930\) −1.00000 −1.00000
\(931\) −3.00000 −3.00000
\(932\) 2.00000 2.00000
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) −2.00000 −2.00000
\(943\) 0 0
\(944\) 2.00000 2.00000
\(945\) −2.00000 −2.00000
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 1.00000 1.00000
\(952\) −2.00000 −2.00000
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 1.00000 1.00000
\(956\) −1.00000 −1.00000
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 1.00000 1.00000
\(961\) 0 0
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 2.00000 2.00000
\(967\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(968\) 1.00000 1.00000
\(969\) −1.00000 −1.00000
\(970\) 1.00000 1.00000
\(971\) 0 0 1.00000 \(0\)
−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
\(980\) −3.00000 −3.00000
\(981\) 0 0
\(982\) −1.00000 −1.00000
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 1.00000 1.00000
\(990\) 0 0
\(991\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(992\) −1.00000 −1.00000
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) −2.00000 −2.00000
\(997\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(998\) 2.00000 2.00000
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))