Properties

Label 104.1.h.a.51.1
Level 104
Weight 1
Character 104.51
Self dual Yes
Analytic conductor 0.052
Analytic rank 0
Dimension 1
Projective image \(D_{3}\)
CM disc. -104
Inner twists 2

Related objects

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Newspace parameters

Level: \( N \) = \( 104 = 2^{3} \cdot 13 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 104.h (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(0.0519027613138\)
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 3.1.104.1
Artin image size \(12\)
Artin image $D_6$
Artin field Galois closure of 6.0.86528.1

Embedding invariants

Embedding label 51.1
Root \(0\)
Character \(\chi\) = 104.51

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} -1.00000 q^{10} -1.00000 q^{12} -1.00000 q^{13} -1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} +1.00000 q^{20} -1.00000 q^{21} +1.00000 q^{24} +1.00000 q^{26} +1.00000 q^{27} +1.00000 q^{28} +1.00000 q^{30} -2.00000 q^{31} -1.00000 q^{32} +1.00000 q^{34} +1.00000 q^{35} +1.00000 q^{37} +1.00000 q^{39} -1.00000 q^{40} +1.00000 q^{42} -1.00000 q^{43} +1.00000 q^{47} -1.00000 q^{48} +1.00000 q^{51} -1.00000 q^{52} -1.00000 q^{54} -1.00000 q^{56} -1.00000 q^{60} +2.00000 q^{62} +1.00000 q^{64} -1.00000 q^{65} -1.00000 q^{68} -1.00000 q^{70} +1.00000 q^{71} -1.00000 q^{74} -1.00000 q^{78} +1.00000 q^{80} -1.00000 q^{81} -1.00000 q^{84} -1.00000 q^{85} +1.00000 q^{86} -1.00000 q^{91} +2.00000 q^{93} -1.00000 q^{94} +1.00000 q^{96} +O(q^{100})\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/104\mathbb{Z}\right)^\times\).

\(n\) \(41\) \(53\) \(79\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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 \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(6\) 1.00000 1.00000
\(7\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(8\) −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
\(13\) −1.00000 −1.00000
\(14\) −1.00000 −1.00000
\(15\) −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
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 1.00000 1.00000
\(21\) −1.00000 −1.00000
\(22\) 0 0
\(23\) 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
\(28\) 1.00000 1.00000
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 1.00000 1.00000
\(31\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(32\) −1.00000 −1.00000
\(33\) 0 0
\(34\) 1.00000 1.00000
\(35\) 1.00000 1.00000
\(36\) 0 0
\(37\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(38\) 0 0
\(39\) 1.00000 1.00000
\(40\) −1.00000 −1.00000
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 1.00000 1.00000
\(43\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 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
\(49\) 0 0
\(50\) 0 0
\(51\) 1.00000 1.00000
\(52\) −1.00000 −1.00000
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) −1.00000 −1.00000
\(55\) 0 0
\(56\) −1.00000 −1.00000
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) −1.00000 −1.00000
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 2.00000 2.00000
\(63\) 0 0
\(64\) 1.00000 1.00000
\(65\) −1.00000 −1.00000
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) −1.00000 −1.00000
\(69\) 0 0
\(70\) −1.00000 −1.00000
\(71\) 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 \(\pi\)
\(74\) −1.00000 −1.00000
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) −1.00000 −1.00000
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 1.00000 1.00000
\(81\) −1.00000 −1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) −1.00000 −1.00000
\(85\) −1.00000 −1.00000
\(86\) 1.00000 1.00000
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) −1.00000 −1.00000
\(92\) 0 0
\(93\) 2.00000 2.00000
\(94\) −1.00000 −1.00000
\(95\) 0 0
\(96\) 1.00000 1.00000
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) −1.00000 −1.00000
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 1.00000 1.00000
\(105\) −1.00000 −1.00000
\(106\) 0 0
\(107\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(108\) 1.00000 1.00000
\(109\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(110\) 0 0
\(111\) −1.00000 −1.00000
\(112\) 1.00000 1.00000
\(113\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(114\) 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 0
\(123\) 0 0
\(124\) −2.00000 −2.00000
\(125\) −1.00000 −1.00000
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) −1.00000 −1.00000
\(129\) 1.00000 1.00000
\(130\) 1.00000 1.00000
\(131\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 1.00000 1.00000
\(136\) 1.00000 1.00000
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(140\) 1.00000 1.00000
\(141\) −1.00000 −1.00000
\(142\) −1.00000 −1.00000
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 1.00000 1.00000
\(149\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −2.00000 −2.00000
\(156\) 1.00000 1.00000
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) −1.00000 −1.00000
\(161\) 0 0
\(162\) 1.00000 1.00000
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(168\) 1.00000 1.00000
\(169\) 1.00000 1.00000
\(170\) 1.00000 1.00000
\(171\) 0 0
\(172\) −1.00000 −1.00000
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 1.00000 1.00000
\(183\) 0 0
\(184\) 0 0
\(185\) 1.00000 1.00000
\(186\) −2.00000 −2.00000
\(187\) 0 0
\(188\) 1.00000 1.00000
\(189\) 1.00000 1.00000
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−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
\(196\) 0 0
\(197\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(198\) 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
\(206\) 0 0
\(207\) 0 0
\(208\) −1.00000 −1.00000
\(209\) 0 0
\(210\) 1.00000 1.00000
\(211\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(212\) 0 0
\(213\) −1.00000 −1.00000
\(214\) −2.00000 −2.00000
\(215\) −1.00000 −1.00000
\(216\) −1.00000 −1.00000
\(217\) −2.00000 −2.00000
\(218\) −1.00000 −1.00000
\(219\) 0 0
\(220\) 0 0
\(221\) 1.00000 1.00000
\(222\) 1.00000 1.00000
\(223\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(224\) −1.00000 −1.00000
\(225\) 0 0
\(226\) −2.00000 −2.00000
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −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
\(237\) 0 0
\(238\) 1.00000 1.00000
\(239\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(240\) −1.00000 −1.00000
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) −1.00000 −1.00000
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 2.00000 2.00000
\(249\) 0 0
\(250\) 1.00000 1.00000
\(251\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 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
\(259\) 1.00000 1.00000
\(260\) −1.00000 −1.00000
\(261\) 0 0
\(262\) 1.00000 1.00000
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 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 \(0.333333\pi\)
\(272\) −1.00000 −1.00000
\(273\) 1.00000 1.00000
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 1.00000 1.00000
\(279\) 0 0
\(280\) −1.00000 −1.00000
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 1.00000 1.00000
\(283\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(284\) 1.00000 1.00000
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 0 0
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −1.00000 −1.00000
\(297\) 0 0
\(298\) 2.00000 2.00000
\(299\) 0 0
\(300\) 0 0
\(301\) −1.00000 −1.00000
\(302\) −1.00000 −1.00000
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 2.00000 2.00000
\(311\) 0 0 1.00000 \(0\)
−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
\(316\) 0 0
\(317\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 1.00000 1.00000
\(321\) −2.00000 −2.00000
\(322\) 0 0
\(323\) 0 0
\(324\) −1.00000 −1.00000
\(325\) 0 0
\(326\) 0 0
\(327\) −1.00000 −1.00000
\(328\) 0 0
\(329\) 1.00000 1.00000
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 2.00000 2.00000
\(335\) 0 0
\(336\) −1.00000 −1.00000
\(337\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(338\) −1.00000 −1.00000
\(339\) −2.00000 −2.00000
\(340\) −1.00000 −1.00000
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −1.00000
\(344\) 1.00000 1.00000
\(345\) 0 0
\(346\) 0 0
\(347\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(348\) 0 0
\(349\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(350\) 0 0
\(351\) −1.00000 −1.00000
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 1.00000 1.00000
\(356\) 0 0
\(357\) 1.00000 1.00000
\(358\) 1.00000 1.00000
\(359\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 1.00000 1.00000
\(362\) 0 0
\(363\) −1.00000 −1.00000
\(364\) −1.00000 −1.00000
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) −1.00000 −1.00000
\(371\) 0 0
\(372\) 2.00000 2.00000
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 1.00000 1.00000
\(376\) −1.00000 −1.00000
\(377\) 0 0
\(378\) −1.00000 −1.00000
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(384\) 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
\(392\) 0 0
\(393\) 1.00000 1.00000
\(394\) −1.00000 −1.00000
\(395\) 0 0
\(396\) 0 0
\(397\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 2.00000 2.00000
\(404\) 0 0
\(405\) −1.00000 −1.00000
\(406\) 0 0
\(407\) 0 0
\(408\) −1.00000 −1.00000
\(409\) 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
\(417\) 1.00000 1.00000
\(418\) 0 0
\(419\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(420\) −1.00000 −1.00000
\(421\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(422\) 1.00000 1.00000
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 1.00000 1.00000
\(427\) 0 0
\(428\) 2.00000 2.00000
\(429\) 0 0
\(430\) 1.00000 1.00000
\(431\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(432\) 1.00000 1.00000
\(433\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(434\) 2.00000 2.00000
\(435\) 0 0
\(436\) 1.00000 1.00000
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −1.00000 −1.00000
\(443\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(444\) −1.00000 −1.00000
\(445\) 0 0
\(446\) −1.00000 −1.00000
\(447\) 2.00000 2.00000
\(448\) 1.00000 1.00000
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 2.00000 2.00000
\(453\) −1.00000 −1.00000
\(454\) 0 0
\(455\) −1.00000 −1.00000
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) −1.00000 −1.00000
\(459\) −1.00000 −1.00000
\(460\) 0 0
\(461\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(462\) 0 0
\(463\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 2.00000 2.00000
\(466\) 1.00000 1.00000
\(467\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(468\) 0 0
\(469\) 0 0
\(470\) −1.00000 −1.00000
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) −1.00000 −1.00000
\(477\) 0 0
\(478\) −1.00000 −1.00000
\(479\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(480\) 1.00000 1.00000
\(481\) −1.00000 −1.00000
\(482\) 0 0
\(483\) 0 0
\(484\) 1.00000 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −2.00000 −2.00000
\(497\) 1.00000 1.00000
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) −1.00000 −1.00000
\(501\) 2.00000 2.00000
\(502\) −2.00000 −2.00000
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −1.00000 −1.00000
\(508\) 0 0
\(509\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(510\) −1.00000 −1.00000
\(511\) 0 0
\(512\) −1.00000 −1.00000
\(513\) 0 0
\(514\) 1.00000 1.00000
\(515\) 0 0
\(516\) 1.00000 1.00000
\(517\) 0 0
\(518\) −1.00000 −1.00000
\(519\) 0 0
\(520\) 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
\(523\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(524\) −1.00000 −1.00000
\(525\) 0 0
\(526\) 0 0
\(527\) 2.00000 2.00000
\(528\) 0 0
\(529\) 1.00000 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 2.00000 2.00000
\(536\) 0 0
\(537\) 1.00000 1.00000
\(538\) 0 0
\(539\) 0 0
\(540\) 1.00000 1.00000
\(541\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(542\) −1.00000 −1.00000
\(543\) 0 0
\(544\) 1.00000 1.00000
\(545\) 1.00000 1.00000
\(546\) −1.00000 −1.00000
\(547\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −1.00000 −1.00000
\(556\) −1.00000 −1.00000
\(557\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(558\) 0 0
\(559\) 1.00000 1.00000
\(560\) 1.00000 1.00000
\(561\) 0 0
\(562\) 0 0
\(563\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(564\) −1.00000 −1.00000
\(565\) 2.00000 2.00000
\(566\) −2.00000 −2.00000
\(567\) −1.00000 −1.00000
\(568\) −1.00000 −1.00000
\(569\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(570\) 0 0
\(571\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 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
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) −1.00000 −1.00000
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −1.00000 −1.00000
\(592\) 1.00000 1.00000
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) −1.00000 −1.00000
\(596\) −2.00000 −2.00000
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(602\) 1.00000 1.00000
\(603\) 0 0
\(604\) 1.00000 1.00000
\(605\) 1.00000 1.00000
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −1.00000 −1.00000
\(612\) 0 0
\(613\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) −2.00000 −2.00000
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 1.00000 1.00000
\(625\) −1.00000 −1.00000
\(626\) 1.00000 1.00000
\(627\) 0 0
\(628\) 0 0
\(629\) −1.00000 −1.00000
\(630\) 0 0
\(631\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(632\) 0 0
\(633\) 1.00000 1.00000
\(634\) 2.00000 2.00000
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) −1.00000 −1.00000
\(641\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(642\) 2.00000 2.00000
\(643\) 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
\(651\) 2.00000 2.00000
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 1.00000 1.00000
\(655\) −1.00000 −1.00000
\(656\) 0 0
\(657\) 0 0
\(658\) −1.00000 −1.00000
\(659\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(660\) 0 0
\(661\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) −1.00000 −1.00000
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) −2.00000 −2.00000
\(669\) −1.00000 −1.00000
\(670\) 0 0
\(671\) 0 0
\(672\) 1.00000 1.00000
\(673\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(674\) 1.00000 1.00000
\(675\) 0 0
\(676\) 1.00000 1.00000
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 2.00000 2.00000
\(679\) 0 0
\(680\) 1.00000 1.00000
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 1.00000 1.00000
\(687\) −1.00000 −1.00000
\(688\) −1.00000 −1.00000
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 1.00000 1.00000
\(695\) −1.00000 −1.00000
\(696\) 0 0
\(697\) 0 0
\(698\) −1.00000 −1.00000
\(699\) 1.00000 1.00000
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 1.00000 1.00000
\(703\) 0 0
\(704\) 0 0
\(705\) −1.00000 −1.00000
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(710\) −1.00000 −1.00000
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) −1.00000 −1.00000
\(715\) 0 0
\(716\) −1.00000 −1.00000
\(717\) −1.00000 −1.00000
\(718\) 2.00000 2.00000
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −1.00000 −1.00000
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 1.00000 1.00000
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 1.00000 1.00000
\(729\) 1.00000 1.00000
\(730\) 0 0
\(731\) 1.00000 1.00000
\(732\) 0 0
\(733\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 1.00000 1.00000
\(741\) 0 0
\(742\) 0 0
\(743\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(744\) −2.00000 −2.00000
\(745\) −2.00000 −2.00000
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 2.00000 2.00000
\(750\) −1.00000 −1.00000
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 1.00000 1.00000
\(753\) −2.00000 −2.00000
\(754\) 0 0
\(755\) 1.00000 1.00000
\(756\) 1.00000 1.00000
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 1.00000 1.00000
\(764\) 0 0
\(765\) 0 0
\(766\) −1.00000 −1.00000
\(767\) 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
\(773\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −1.00000 −1.00000
\(778\) 0 0
\(779\) 0 0
\(780\) 1.00000 1.00000
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) −1.00000 −1.00000
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 1.00000 1.00000
\(789\) 0 0
\(790\) 0 0
\(791\) 2.00000 2.00000
\(792\) 0 0
\(793\) 0 0
\(794\) 2.00000 2.00000
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) −1.00000 −1.00000
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) −2.00000 −2.00000
\(807\) 0 0
\(808\) 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
\(813\) −1.00000 −1.00000
\(814\) 0 0
\(815\) 0 0
\(816\) 1.00000 1.00000
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(822\) 0 0
\(823\) 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
\(832\) −1.00000 −1.00000
\(833\) 0 0
\(834\) −1.00000 −1.00000
\(835\) −2.00000 −2.00000
\(836\) 0 0
\(837\) −2.00000 −2.00000
\(838\) 1.00000 1.00000
\(839\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(840\) 1.00000 1.00000
\(841\) 1.00000 1.00000
\(842\) −1.00000 −1.00000
\(843\) 0 0
\(844\) −1.00000 −1.00000
\(845\) 1.00000 1.00000
\(846\) 0 0
\(847\) 1.00000 1.00000
\(848\) 0 0
\(849\) −2.00000 −2.00000
\(850\) 0 0
\(851\) 0 0
\(852\) −1.00000 −1.00000
\(853\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −2.00000 −2.00000
\(857\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(858\) 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 \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(864\) −1.00000 −1.00000
\(865\) 0 0
\(866\) 1.00000 1.00000
\(867\) 0 0
\(868\) −2.00000 −2.00000
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) −1.00000 −1.00000
\(873\) 0 0
\(874\) 0 0
\(875\) −1.00000 −1.00000
\(876\) 0 0
\(877\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(878\) 0 0
\(879\) −1.00000 −1.00000
\(880\) 0 0
\(881\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(882\) 0 0
\(883\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(884\) 1.00000 1.00000
\(885\) 0 0
\(886\) 1.00000 1.00000
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 1.00000 1.00000
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 1.00000 1.00000
\(893\) 0 0
\(894\) −2.00000 −2.00000
\(895\) −1.00000 −1.00000
\(896\) −1.00000 −1.00000
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 1.00000 1.00000
\(904\) −2.00000 −2.00000
\(905\) 0 0
\(906\) 1.00000 1.00000
\(907\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 1.00000 1.00000
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 1.00000 1.00000
\(917\) −1.00000 −1.00000
\(918\) 1.00000 1.00000
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −1.00000 −1.00000
\(923\) −1.00000 −1.00000
\(924\) 0 0
\(925\) 0 0
\(926\) 2.00000 2.00000
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) −2.00000 −2.00000
\(931\) 0 0
\(932\) −1.00000 −1.00000
\(933\) 0 0
\(934\) −2.00000 −2.00000
\(935\) 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 \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 1.00000 1.00000
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 2.00000 2.00000
\(952\) 1.00000 1.00000
\(953\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 1.00000 1.00000
\(957\) 0 0
\(958\) −1.00000 −1.00000
\(959\) 0 0
\(960\) −1.00000 −1.00000
\(961\) 3.00000 3.00000
\(962\) 1.00000 1.00000
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(968\) −1.00000 −1.00000
\(969\) 0 0
\(970\) 0 0
\(971\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(972\) 0 0
\(973\) −1.00000 −1.00000
\(974\) 2.00000 2.00000
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 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
\(985\) 1.00000 1.00000
\(986\) 0 0
\(987\) −1.00000 −1.00000
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 2.00000 2.00000
\(993\) 0 0
\(994\) −1.00000 −1.00000
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 0 0
\(999\) 1.00000 1.00000
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 \(a_p\))