Properties

Label 152.1.g.a.37.1
Level 152
Weight 1
Character 152.37
Self dual Yes
Analytic conductor 0.076
Analytic rank 0
Dimension 1
Projective image \(D_{3}\)
CM disc. -152
Inner twists 2

Related objects

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Newspace parameters

Level: \( N \) = \( 152 = 2^{3} \cdot 19 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 152.g (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(0.0758578819202\)
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 3.1.152.1
Artin image size \(12\)
Artin image $D_6$
Artin field Galois closure of 6.2.184832.1

Embedding invariants

Embedding label 37.1
Root \(0\)
Character \(\chi\) = 152.37

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{12} +1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -1.00000 q^{17} -1.00000 q^{19} -1.00000 q^{21} -1.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{26} -1.00000 q^{27} -1.00000 q^{28} +1.00000 q^{29} -1.00000 q^{32} +1.00000 q^{34} -2.00000 q^{37} +1.00000 q^{38} +1.00000 q^{39} +1.00000 q^{42} +1.00000 q^{46} +2.00000 q^{47} +1.00000 q^{48} -1.00000 q^{50} -1.00000 q^{51} +1.00000 q^{52} +1.00000 q^{53} +1.00000 q^{54} +1.00000 q^{56} -1.00000 q^{57} -1.00000 q^{58} +1.00000 q^{59} +1.00000 q^{64} +1.00000 q^{67} -1.00000 q^{68} -1.00000 q^{69} -1.00000 q^{73} +2.00000 q^{74} +1.00000 q^{75} -1.00000 q^{76} -1.00000 q^{78} -1.00000 q^{81} -1.00000 q^{84} +1.00000 q^{87} -1.00000 q^{91} -1.00000 q^{92} -2.00000 q^{94} -1.00000 q^{96} +O(q^{100})\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/152\mathbb{Z}\right)^\times\).

\(n\) \(39\) \(77\) \(97\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(4\) 1.00000 1.00000
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) −1.00000 −1.00000
\(7\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(8\) −1.00000 −1.00000
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 1.00000 1.00000
\(13\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(14\) 1.00000 1.00000
\(15\) 0 0
\(16\) 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
\(20\) 0 0
\(21\) −1.00000 −1.00000
\(22\) 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
\(25\) 1.00000 1.00000
\(26\) −1.00000 −1.00000
\(27\) −1.00000 −1.00000
\(28\) −1.00000 −1.00000
\(29\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −1.00000 −1.00000
\(33\) 0 0
\(34\) 1.00000 1.00000
\(35\) 0 0
\(36\) 0 0
\(37\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(38\) 1.00000 1.00000
\(39\) 1.00000 1.00000
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 1.00000 1.00000
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 1.00000 1.00000
\(47\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(48\) 1.00000 1.00000
\(49\) 0 0
\(50\) −1.00000 −1.00000
\(51\) −1.00000 −1.00000
\(52\) 1.00000 1.00000
\(53\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(54\) 1.00000 1.00000
\(55\) 0 0
\(56\) 1.00000 1.00000
\(57\) −1.00000 −1.00000
\(58\) −1.00000 −1.00000
\(59\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(68\) −1.00000 −1.00000
\(69\) −1.00000 −1.00000
\(70\) 0 0
\(71\) 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 \(0.666667\pi\)
\(74\) 2.00000 2.00000
\(75\) 1.00000 1.00000
\(76\) −1.00000 −1.00000
\(77\) 0 0
\(78\) −1.00000 −1.00000
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) −1.00000 −1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) −1.00000 −1.00000
\(85\) 0 0
\(86\) 0 0
\(87\) 1.00000 1.00000
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) −1.00000 −1.00000
\(92\) −1.00000 −1.00000
\(93\) 0 0
\(94\) −2.00000 −2.00000
\(95\) 0 0
\(96\) −1.00000 −1.00000
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 1.00000 1.00000
\(101\) 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
\(105\) 0 0
\(106\) −1.00000 −1.00000
\(107\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(108\) −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
\(111\) −2.00000 −2.00000
\(112\) −1.00000 −1.00000
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 1.00000 1.00000
\(115\) 0 0
\(116\) 1.00000 1.00000
\(117\) 0 0
\(118\) −1.00000 −1.00000
\(119\) 1.00000 1.00000
\(120\) 0 0
\(121\) 1.00000 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) −1.00000 −1.00000
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 1.00000 1.00000
\(134\) −1.00000 −1.00000
\(135\) 0 0
\(136\) 1.00000 1.00000
\(137\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(138\) 1.00000 1.00000
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 2.00000 2.00000
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 1.00000 1.00000
\(147\) 0 0
\(148\) −2.00000 −2.00000
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) −1.00000 −1.00000
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 1.00000 1.00000
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 1.00000 1.00000
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 1.00000 1.00000
\(160\) 0 0
\(161\) 1.00000 1.00000
\(162\) 1.00000 1.00000
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 1.00000 1.00000
\(169\) 0 0
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(174\) −1.00000 −1.00000
\(175\) −1.00000 −1.00000
\(176\) 0 0
\(177\) 1.00000 1.00000
\(178\) 0 0
\(179\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(182\) 1.00000 1.00000
\(183\) 0 0
\(184\) 1.00000 1.00000
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 2.00000 2.00000
\(189\) 1.00000 1.00000
\(190\) 0 0
\(191\) −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 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(200\) −1.00000 −1.00000
\(201\) 1.00000 1.00000
\(202\) 0 0
\(203\) −1.00000 −1.00000
\(204\) −1.00000 −1.00000
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 1.00000 1.00000
\(209\) 0 0
\(210\) 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
\(214\) −1.00000 −1.00000
\(215\) 0 0
\(216\) 1.00000 1.00000
\(217\) 0 0
\(218\) −1.00000 −1.00000
\(219\) −1.00000 −1.00000
\(220\) 0 0
\(221\) −1.00000 −1.00000
\(222\) 2.00000 2.00000
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 1.00000 1.00000
\(225\) 0 0
\(226\) 0 0
\(227\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(228\) −1.00000 −1.00000
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −1.00000 −1.00000
\(233\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(234\) 0 0
\(235\) 0 0
\(236\) 1.00000 1.00000
\(237\) 0 0
\(238\) −1.00000 −1.00000
\(239\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(240\) 0 0
\(241\) 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
\(247\) −1.00000 −1.00000
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 1.00000 1.00000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 2.00000 2.00000
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(264\) 0 0
\(265\) 0 0
\(266\) −1.00000 −1.00000
\(267\) 0 0
\(268\) 1.00000 1.00000
\(269\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(272\) −1.00000 −1.00000
\(273\) −1.00000 −1.00000
\(274\) 1.00000 1.00000
\(275\) 0 0
\(276\) −1.00000 −1.00000
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) −2.00000 −2.00000
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 0 0
\(290\) 0 0
\(291\) 0 0
\(292\) −1.00000 −1.00000
\(293\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 2.00000 2.00000
\(297\) 0 0
\(298\) 0 0
\(299\) −1.00000 −1.00000
\(300\) 1.00000 1.00000
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) −1.00000 −1.00000
\(305\) 0 0
\(306\) 0 0
\(307\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(312\) −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
\(317\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(318\) −1.00000 −1.00000
\(319\) 0 0
\(320\) 0 0
\(321\) 1.00000 1.00000
\(322\) −1.00000 −1.00000
\(323\) 1.00000 1.00000
\(324\) −1.00000 −1.00000
\(325\) 1.00000 1.00000
\(326\) 0 0
\(327\) 1.00000 1.00000
\(328\) 0 0
\(329\) −2.00000 −2.00000
\(330\) 0 0
\(331\) 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
\(336\) −1.00000 −1.00000
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 1.00000
\(344\) 0 0
\(345\) 0 0
\(346\) 2.00000 2.00000
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 1.00000 1.00000
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 1.00000 1.00000
\(351\) −1.00000 −1.00000
\(352\) 0 0
\(353\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(354\) −1.00000 −1.00000
\(355\) 0 0
\(356\) 0 0
\(357\) 1.00000 1.00000
\(358\) 2.00000 2.00000
\(359\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(360\) 0 0
\(361\) 1.00000 1.00000
\(362\) 2.00000 2.00000
\(363\) 1.00000 1.00000
\(364\) −1.00000 −1.00000
\(365\) 0 0
\(366\) 0 0
\(367\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(368\) −1.00000 −1.00000
\(369\) 0 0
\(370\) 0 0
\(371\) −1.00000 −1.00000
\(372\) 0 0
\(373\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −2.00000 −2.00000
\(377\) 1.00000 1.00000
\(378\) −1.00000 −1.00000
\(379\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 1.00000 1.00000
\(383\) 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 \(\pi\)
\(390\) 0 0
\(391\) 1.00000 1.00000
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 1.00000 1.00000
\(399\) 1.00000 1.00000
\(400\) 1.00000 1.00000
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) −1.00000 −1.00000
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 1.00000 1.00000
\(407\) 0 0
\(408\) 1.00000 1.00000
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) −1.00000 −1.00000
\(412\) 0 0
\(413\) −1.00000 −1.00000
\(414\) 0 0
\(415\) 0 0
\(416\) −1.00000 −1.00000
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 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
\(424\) −1.00000 −1.00000
\(425\) −1.00000 −1.00000
\(426\) 0 0
\(427\) 0 0
\(428\) 1.00000 1.00000
\(429\) 0 0
\(430\) 0 0
\(431\) 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
\(435\) 0 0
\(436\) 1.00000 1.00000
\(437\) 1.00000 1.00000
\(438\) 1.00000 1.00000
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 1.00000 1.00000
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) −2.00000 −2.00000
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −1.00000 −1.00000
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) −1.00000 −1.00000
\(455\) 0 0
\(456\) 1.00000 1.00000
\(457\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(458\) 0 0
\(459\) 1.00000 1.00000
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(464\) 1.00000 1.00000
\(465\) 0 0
\(466\) −2.00000 −2.00000
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) −1.00000 −1.00000
\(470\) 0 0
\(471\) 0 0
\(472\) −1.00000 −1.00000
\(473\) 0 0
\(474\) 0 0
\(475\) −1.00000 −1.00000
\(476\) 1.00000 1.00000
\(477\) 0 0
\(478\) 1.00000 1.00000
\(479\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(480\) 0 0
\(481\) −2.00000 −2.00000
\(482\) 0 0
\(483\) 1.00000 1.00000
\(484\) 1.00000 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) −1.00000 −1.00000
\(494\) 1.00000 1.00000
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 1.00000 1.00000
\(512\) −1.00000 −1.00000
\(513\) 1.00000 1.00000
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) −2.00000 −2.00000
\(519\) −2.00000 −2.00000
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(524\) 0 0
\(525\) −1.00000 −1.00000
\(526\) −2.00000 −2.00000
\(527\) 0 0
\(528\) 0 0
\(529\) 0 0
\(530\) 0 0
\(531\) 0 0
\(532\) 1.00000 1.00000
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) −1.00000 −1.00000
\(537\) −2.00000 −2.00000
\(538\) 2.00000 2.00000
\(539\) 0 0
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 1.00000 1.00000
\(543\) −2.00000 −2.00000
\(544\) 1.00000 1.00000
\(545\) 0 0
\(546\) 1.00000 1.00000
\(547\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(548\) −1.00000 −1.00000
\(549\) 0 0
\(550\) 0 0
\(551\) −1.00000 −1.00000
\(552\) 1.00000 1.00000
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(564\) 2.00000 2.00000
\(565\) 0 0
\(566\) 0 0
\(567\) 1.00000 1.00000
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) −1.00000 −1.00000
\(574\) 0 0
\(575\) −1.00000 −1.00000
\(576\) 0 0
\(577\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 1.00000 1.00000
\(585\) 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
\(591\) 0 0
\(592\) −2.00000 −2.00000
\(593\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −1.00000 −1.00000
\(598\) 1.00000 1.00000
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) −1.00000 −1.00000
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 1.00000 1.00000
\(609\) −1.00000 −1.00000
\(610\) 0 0
\(611\) 2.00000 2.00000
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 2.00000 2.00000
\(615\) 0 0
\(616\) 0 0
\(617\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 1.00000 1.00000
\(622\) 1.00000 1.00000
\(623\) 0 0
\(624\) 1.00000 1.00000
\(625\) 1.00000 1.00000
\(626\) 1.00000 1.00000
\(627\) 0 0
\(628\) 0 0
\(629\) 2.00000 2.00000
\(630\) 0 0
\(631\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(632\) 0 0
\(633\) 1.00000 1.00000
\(634\) −1.00000 −1.00000
\(635\) 0 0
\(636\) 1.00000 1.00000
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) −1.00000 −1.00000
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 1.00000 1.00000
\(645\) 0 0
\(646\) −1.00000 −1.00000
\(647\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(648\) 1.00000 1.00000
\(649\) 0 0
\(650\) −1.00000 −1.00000
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) −1.00000 −1.00000
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 2.00000 2.00000
\(659\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(660\) 0 0
\(661\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(662\) −1.00000 −1.00000
\(663\) −1.00000 −1.00000
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −1.00000 −1.00000
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 1.00000 1.00000
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) −1.00000 −1.00000
\(676\) 0 0
\(677\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 1.00000 1.00000
\(682\) 0 0
\(683\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −1.00000 −1.00000
\(687\) 0 0
\(688\) 0 0
\(689\) 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
\(698\) 0 0
\(699\) 2.00000 2.00000
\(700\) −1.00000 −1.00000
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 1.00000 1.00000
\(703\) 2.00000 2.00000
\(704\) 0 0
\(705\) 0 0
\(706\) 1.00000 1.00000
\(707\) 0 0
\(708\) 1.00000 1.00000
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) −1.00000 −1.00000
\(715\) 0 0
\(716\) −2.00000 −2.00000
\(717\) −1.00000 −1.00000
\(718\) 1.00000 1.00000
\(719\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −1.00000 −1.00000
\(723\) 0 0
\(724\) −2.00000 −2.00000
\(725\) 1.00000 1.00000
\(726\) −1.00000 −1.00000
\(727\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(728\) 1.00000 1.00000
\(729\) 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
\(735\) 0 0
\(736\) 1.00000 1.00000
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) −1.00000 −1.00000
\(742\) 1.00000 1.00000
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −1.00000 −1.00000
\(747\) 0 0
\(748\) 0 0
\(749\) −1.00000 −1.00000
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 2.00000 2.00000
\(753\) 0 0
\(754\) −1.00000 −1.00000
\(755\) 0 0
\(756\) 1.00000 1.00000
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) −1.00000 −1.00000
\(759\) 0 0
\(760\) 0 0
\(761\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(762\) 0 0
\(763\) −1.00000 −1.00000
\(764\) −1.00000 −1.00000
\(765\) 0 0
\(766\) 0 0
\(767\) 1.00000 1.00000
\(768\) 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
\(771\) 0 0
\(772\) 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
\(777\) 2.00000 2.00000
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) −1.00000 −1.00000
\(783\) −1.00000 −1.00000
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(788\) 0 0
\(789\) 2.00000 2.00000
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) −1.00000 −1.00000
\(797\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(798\) −1.00000 −1.00000
\(799\) −2.00000 −2.00000
\(800\) −1.00000 −1.00000
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 1.00000 1.00000
\(805\) 0 0
\(806\) 0 0
\(807\) −2.00000 −2.00000
\(808\) 0 0
\(809\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(810\) 0 0
\(811\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(812\) −1.00000 −1.00000
\(813\) −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
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 1.00000 1.00000
\(823\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 1.00000 1.00000
\(827\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(828\) 0 0
\(829\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 1.00000 1.00000
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 0 0
\(842\) −1.00000 −1.00000
\(843\) 0 0
\(844\) 1.00000 1.00000
\(845\) 0 0
\(846\) 0 0
\(847\) −1.00000 −1.00000
\(848\) 1.00000 1.00000
\(849\) 0 0
\(850\) 1.00000 1.00000
\(851\) 2.00000 2.00000
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −1.00000 −1.00000
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 1.00000 1.00000
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 1.00000 1.00000
\(872\) −1.00000 −1.00000
\(873\) 0 0
\(874\) −1.00000 −1.00000
\(875\) 0 0
\(876\) −1.00000 −1.00000
\(877\) 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
\(881\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) −1.00000 −1.00000
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 2.00000 2.00000
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −2.00000 −2.00000
\(894\) 0 0
\(895\) 0 0
\(896\) 1.00000 1.00000
\(897\) −1.00000 −1.00000
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −1.00000 −1.00000
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(908\) 1.00000 1.00000
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) −1.00000 −1.00000
\(913\) 0 0
\(914\) 1.00000 1.00000
\(915\) 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 \(0.666667\pi\)
\(920\) 0 0
\(921\) −2.00000 −2.00000
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −2.00000 −2.00000
\(926\) −2.00000 −2.00000
\(927\) 0 0
\(928\) −1.00000 −1.00000
\(929\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 2.00000 2.00000
\(933\) −1.00000 −1.00000
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(938\) 1.00000 1.00000
\(939\) −1.00000 −1.00000
\(940\) 0 0
\(941\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 1.00000 1.00000
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) −1.00000 −1.00000
\(950\) 1.00000 1.00000
\(951\) 1.00000 1.00000
\(952\) −1.00000 −1.00000
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −1.00000 −1.00000
\(957\) 0 0
\(958\) −2.00000 −2.00000
\(959\) 1.00000 1.00000
\(960\) 0 0
\(961\) 1.00000 1.00000
\(962\) 2.00000 2.00000
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) −1.00000 −1.00000
\(967\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(968\) −1.00000 −1.00000
\(969\) 1.00000 1.00000
\(970\) 0 0
\(971\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 1.00000 1.00000
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 1.00000 1.00000
\(987\) −2.00000 −2.00000
\(988\) −1.00000 −1.00000
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 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
\(999\) 2.00000 2.00000
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 \(a_p\))