Properties

Label 2004.1.g.c.2003.2
Level 2004
Weight 1
Character 2004.2003
Self dual Yes
Analytic conductor 1.000
Analytic rank 0
Dimension 2
Projective image \(D_{4}\)
CM disc. -2004
Inner twists 4

Related objects

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Newspace parameters

Level: \( N \) = \( 2004 = 2^{2} \cdot 3 \cdot 167 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 2004.g (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(1.00012628532\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{4}\)
Projective field Galois closure of 4.2.24048.2
Artin image size \(16\)
Artin image $D_8$
Artin field Galois closure of 8.2.96577152768.2

Embedding invariants

Embedding label 2003.2
Root \(-1.41421\)
Character \(\chi\) = 2004.2003

$q$-expansion

\(f(q)\) \(=\) \(q\)\(-1.00000 q^{2}\) \(+1.00000 q^{3}\) \(+1.00000 q^{4}\) \(+1.41421 q^{5}\) \(-1.00000 q^{6}\) \(-1.00000 q^{8}\) \(+1.00000 q^{9}\) \(+O(q^{10})\) \(q\)\(-1.00000 q^{2}\) \(+1.00000 q^{3}\) \(+1.00000 q^{4}\) \(+1.41421 q^{5}\) \(-1.00000 q^{6}\) \(-1.00000 q^{8}\) \(+1.00000 q^{9}\) \(-1.41421 q^{10}\) \(+1.00000 q^{12}\) \(+1.41421 q^{15}\) \(+1.00000 q^{16}\) \(-1.41421 q^{17}\) \(-1.00000 q^{18}\) \(+1.41421 q^{20}\) \(-1.00000 q^{24}\) \(+1.00000 q^{25}\) \(+1.00000 q^{27}\) \(-1.41421 q^{30}\) \(-1.00000 q^{32}\) \(+1.41421 q^{34}\) \(+1.00000 q^{36}\) \(-1.41421 q^{40}\) \(-1.41421 q^{41}\) \(+1.41421 q^{43}\) \(+1.41421 q^{45}\) \(+1.00000 q^{48}\) \(+1.00000 q^{49}\) \(-1.00000 q^{50}\) \(-1.41421 q^{51}\) \(-1.41421 q^{53}\) \(-1.00000 q^{54}\) \(+1.41421 q^{60}\) \(+1.00000 q^{64}\) \(-1.41421 q^{67}\) \(-1.41421 q^{68}\) \(-1.00000 q^{72}\) \(+1.00000 q^{75}\) \(-1.41421 q^{79}\) \(+1.41421 q^{80}\) \(+1.00000 q^{81}\) \(+1.41421 q^{82}\) \(-2.00000 q^{85}\) \(-1.41421 q^{86}\) \(-1.41421 q^{90}\) \(-1.00000 q^{96}\) \(-2.00000 q^{97}\) \(-1.00000 q^{98}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 2q^{3} \) \(\mathstrut +\mathstrut 2q^{4} \) \(\mathstrut -\mathstrut 2q^{6} \) \(\mathstrut -\mathstrut 2q^{8} \) \(\mathstrut +\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 2q^{3} \) \(\mathstrut +\mathstrut 2q^{4} \) \(\mathstrut -\mathstrut 2q^{6} \) \(\mathstrut -\mathstrut 2q^{8} \) \(\mathstrut +\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut 2q^{12} \) \(\mathstrut +\mathstrut 2q^{16} \) \(\mathstrut -\mathstrut 2q^{18} \) \(\mathstrut -\mathstrut 2q^{24} \) \(\mathstrut +\mathstrut 2q^{25} \) \(\mathstrut +\mathstrut 2q^{27} \) \(\mathstrut -\mathstrut 2q^{32} \) \(\mathstrut +\mathstrut 2q^{36} \) \(\mathstrut +\mathstrut 2q^{48} \) \(\mathstrut +\mathstrut 2q^{49} \) \(\mathstrut -\mathstrut 2q^{50} \) \(\mathstrut -\mathstrut 2q^{54} \) \(\mathstrut +\mathstrut 2q^{64} \) \(\mathstrut -\mathstrut 2q^{72} \) \(\mathstrut +\mathstrut 2q^{75} \) \(\mathstrut +\mathstrut 2q^{81} \) \(\mathstrut -\mathstrut 4q^{85} \) \(\mathstrut -\mathstrut 2q^{96} \) \(\mathstrut -\mathstrut 4q^{97} \) \(\mathstrut -\mathstrut 2q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2004\mathbb{Z}\right)^\times\).

\(n\) \(673\) \(1003\) \(1337\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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
\(4\) 1.00000 1.00000
\(5\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(6\) −1.00000 −1.00000
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) −1.00000 −1.00000
\(9\) 1.00000 1.00000
\(10\) −1.41421 −1.41421
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 1.00000 1.00000
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 1.41421 1.41421
\(16\) 1.00000 1.00000
\(17\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(18\) −1.00000 −1.00000
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 1.41421 1.41421
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) −1.00000 −1.00000
\(25\) 1.00000 1.00000
\(26\) 0 0
\(27\) 1.00000 1.00000
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) −1.41421 −1.41421
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −1.00000 −1.00000
\(33\) 0 0
\(34\) 1.41421 1.41421
\(35\) 0 0
\(36\) 1.00000 1.00000
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −1.41421 −1.41421
\(41\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(42\) 0 0
\(43\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(44\) 0 0
\(45\) 1.41421 1.41421
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.00000 1.00000
\(49\) 1.00000 1.00000
\(50\) −1.00000 −1.00000
\(51\) −1.41421 −1.41421
\(52\) 0 0
\(53\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(54\) −1.00000 −1.00000
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 1.41421 1.41421
\(61\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(68\) −1.41421 −1.41421
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) −1.00000 −1.00000
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 1.00000 1.00000
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(80\) 1.41421 1.41421
\(81\) 1.00000 1.00000
\(82\) 1.41421 1.41421
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) −2.00000 −2.00000
\(86\) −1.41421 −1.41421
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) −1.41421 −1.41421
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) −1.00000 −1.00000
\(97\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(98\) −1.00000 −1.00000
\(99\) 0 0
\(100\) 1.00000 1.00000
\(101\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(102\) 1.41421 1.41421
\(103\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 1.41421 1.41421
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 1.00000 1.00000
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) −1.41421 −1.41421
\(121\) −1.00000 −1.00000
\(122\) 0 0
\(123\) −1.41421 −1.41421
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) −1.00000 −1.00000
\(129\) 1.41421 1.41421
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 1.41421 1.41421
\(135\) 1.41421 1.41421
\(136\) 1.41421 1.41421
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 1.00000 1.00000
\(145\) 0 0
\(146\) 0 0
\(147\) 1.00000 1.00000
\(148\) 0 0
\(149\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(150\) −1.00000 −1.00000
\(151\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(152\) 0 0
\(153\) −1.41421 −1.41421
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(158\) 1.41421 1.41421
\(159\) −1.41421 −1.41421
\(160\) −1.41421 −1.41421
\(161\) 0 0
\(162\) −1.00000 −1.00000
\(163\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(164\) −1.41421 −1.41421
\(165\) 0 0
\(166\) 0 0
\(167\) 1.00000 1.00000
\(168\) 0 0
\(169\) 1.00000 1.00000
\(170\) 2.00000 2.00000
\(171\) 0 0
\(172\) 1.41421 1.41421
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(180\) 1.41421 1.41421
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 1.00000 1.00000
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 2.00000 2.00000
\(195\) 0 0
\(196\) 1.00000 1.00000
\(197\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) −1.00000 −1.00000
\(201\) −1.41421 −1.41421
\(202\) −1.41421 −1.41421
\(203\) 0 0
\(204\) −1.41421 −1.41421
\(205\) −2.00000 −2.00000
\(206\) 1.41421 1.41421
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) −1.41421 −1.41421
\(213\) 0 0
\(214\) 0 0
\(215\) 2.00000 2.00000
\(216\) −1.00000 −1.00000
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 1.00000 1.00000
\(226\) −1.41421 −1.41421
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −1.41421 −1.41421
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 1.41421 1.41421
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 1.00000 1.00000
\(243\) 1.00000 1.00000
\(244\) 0 0
\(245\) 1.41421 1.41421
\(246\) 1.41421 1.41421
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −2.00000 −2.00000
\(256\) 1.00000 1.00000
\(257\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(258\) −1.41421 −1.41421
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) −2.00000 −2.00000
\(266\) 0 0
\(267\) 0 0
\(268\) −1.41421 −1.41421
\(269\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(270\) −1.41421 −1.41421
\(271\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(272\) −1.41421 −1.41421
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) −1.41421 −1.41421
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −1.00000 −1.00000
\(289\) 1.00000 1.00000
\(290\) 0 0
\(291\) −2.00000 −2.00000
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) −1.00000 −1.00000
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 1.41421 1.41421
\(299\) 0 0
\(300\) 1.00000 1.00000
\(301\) 0 0
\(302\) −1.41421 −1.41421
\(303\) 1.41421 1.41421
\(304\) 0 0
\(305\) 0 0
\(306\) 1.41421 1.41421
\(307\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(308\) 0 0
\(309\) −1.41421 −1.41421
\(310\) 0 0
\(311\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 2.00000 2.00000
\(315\) 0 0
\(316\) −1.41421 −1.41421
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 1.41421 1.41421
\(319\) 0 0
\(320\) 1.41421 1.41421
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 1.00000 1.00000
\(325\) 0 0
\(326\) −1.41421 −1.41421
\(327\) 0 0
\(328\) 1.41421 1.41421
\(329\) 0 0
\(330\) 0 0
\(331\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) −1.00000 −1.00000
\(335\) −2.00000 −2.00000
\(336\) 0 0
\(337\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(338\) −1.00000 −1.00000
\(339\) 1.41421 1.41421
\(340\) −2.00000 −2.00000
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) −1.41421 −1.41421
\(345\) 0 0
\(346\) 0 0
\(347\) 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 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) −2.00000 −2.00000
\(359\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(360\) −1.41421 −1.41421
\(361\) 1.00000 1.00000
\(362\) 0 0
\(363\) −1.00000 −1.00000
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) −1.41421 −1.41421
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) −1.00000 −1.00000
\(385\) 0 0
\(386\) 0 0
\(387\) 1.41421 1.41421
\(388\) −2.00000 −2.00000
\(389\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −1.00000 −1.00000
\(393\) 0 0
\(394\) −1.41421 −1.41421
\(395\) −2.00000 −2.00000
\(396\) 0 0
\(397\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 1.00000 1.00000
\(401\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(402\) 1.41421 1.41421
\(403\) 0 0
\(404\) 1.41421 1.41421
\(405\) 1.41421 1.41421
\(406\) 0 0
\(407\) 0 0
\(408\) 1.41421 1.41421
\(409\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) 2.00000 2.00000
\(411\) 0 0
\(412\) −1.41421 −1.41421
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 1.41421 1.41421
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 1.41421 1.41421
\(425\) −1.41421 −1.41421
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) −2.00000 −2.00000
\(431\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(432\) 1.00000 1.00000
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(440\) 0 0
\(441\) 1.00000 1.00000
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −1.41421 −1.41421
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) −1.00000 −1.00000
\(451\) 0 0
\(452\) 1.41421 1.41421
\(453\) 1.41421 1.41421
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) −1.41421 −1.41421
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −2.00000 −2.00000
\(472\) 0 0
\(473\) 0 0
\(474\) 1.41421 1.41421
\(475\) 0 0
\(476\) 0 0
\(477\) −1.41421 −1.41421
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) −1.41421 −1.41421
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −1.00000 −1.00000
\(485\) −2.82843 −2.82843
\(486\) −1.00000 −1.00000
\(487\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(488\) 0 0
\(489\) 1.41421 1.41421
\(490\) −1.41421 −1.41421
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) −1.41421 −1.41421
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(500\) 0 0
\(501\) 1.00000 1.00000
\(502\) 2.00000 2.00000
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 2.00000 2.00000
\(506\) 0 0
\(507\) 1.00000 1.00000
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 2.00000 2.00000
\(511\) 0 0
\(512\) −1.00000 −1.00000
\(513\) 0 0
\(514\) −1.41421 −1.41421
\(515\) −2.00000 −2.00000
\(516\) 1.41421 1.41421
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 2.00000 2.00000
\(527\) 0 0
\(528\) 0 0
\(529\) 1.00000 1.00000
\(530\) 2.00000 2.00000
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 1.41421 1.41421
\(537\) 2.00000 2.00000
\(538\) 1.41421 1.41421
\(539\) 0 0
\(540\) 1.41421 1.41421
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 1.41421 1.41421
\(543\) 0 0
\(544\) 1.41421 1.41421
\(545\) 0 0
\(546\) 0 0
\(547\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 1.41421 1.41421
\(557\) 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 \(\pi\)
\(564\) 0 0
\(565\) 2.00000 2.00000
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(570\) 0 0
\(571\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 1.00000
\(577\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(578\) −1.00000 −1.00000
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 2.00000 2.00000
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 1.00000 1.00000
\(589\) 0 0
\(590\) 0 0
\(591\) 1.41421 1.41421
\(592\) 0 0
\(593\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −1.41421 −1.41421
\(597\) 0 0
\(598\) 0 0
\(599\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(600\) −1.00000 −1.00000
\(601\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(602\) 0 0
\(603\) −1.41421 −1.41421
\(604\) 1.41421 1.41421
\(605\) −1.41421 −1.41421
\(606\) −1.41421 −1.41421
\(607\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) −1.41421 −1.41421
\(613\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(614\) −1.41421 −1.41421
\(615\) −2.00000 −2.00000
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 1.41421 1.41421
\(619\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 2.00000 2.00000
\(623\) 0 0
\(624\) 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 \(\pi\)
\(632\) 1.41421 1.41421
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) −1.41421 −1.41421
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) −1.41421 −1.41421
\(641\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(642\) 0 0
\(643\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(644\) 0 0
\(645\) 2.00000 2.00000
\(646\) 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
\(652\) 1.41421 1.41421
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −1.41421 −1.41421
\(657\) 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 \(\pi\)
\(662\) 1.41421 1.41421
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 1.00000 1.00000
\(669\) 0 0
\(670\) 2.00000 2.00000
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 2.00000 2.00000
\(675\) 1.00000 1.00000
\(676\) 1.00000 1.00000
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) −1.41421 −1.41421
\(679\) 0 0
\(680\) 2.00000 2.00000
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 1.41421 1.41421
\(689\) 0 0
\(690\) 0 0
\(691\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 2.00000 2.00000
\(696\) 0 0
\(697\) 2.00000 2.00000
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) −1.41421 −1.41421
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 2.00000 2.00000
\(717\) 0 0
\(718\) −2.00000 −2.00000
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 1.41421 1.41421
\(721\) 0 0
\(722\) −1.00000 −1.00000
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 1.00000 1.00000
\(727\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(728\) 0 0
\(729\) 1.00000 1.00000
\(730\) 0 0
\(731\) −2.00000 −2.00000
\(732\) 0 0
\(733\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) 0 0
\(735\) 1.41421 1.41421
\(736\) 0 0
\(737\) 0 0
\(738\) 1.41421 1.41421
\(739\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(744\) 0 0
\(745\) −2.00000 −2.00000
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(752\) 0 0
\(753\) −2.00000 −2.00000
\(754\) 0 0
\(755\) 2.00000 2.00000
\(756\) 0 0
\(757\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(758\) −1.41421 −1.41421
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −2.00000 −2.00000
\(766\) 0 0
\(767\) 0 0
\(768\) 1.00000 1.00000
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 1.41421 1.41421
\(772\) 0 0
\(773\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(774\) −1.41421 −1.41421
\(775\) 0 0
\(776\) 2.00000 2.00000
\(777\) 0 0
\(778\) −1.41421 −1.41421
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 1.00000 1.00000
\(785\) −2.82843 −2.82843
\(786\) 0 0
\(787\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(788\) 1.41421 1.41421
\(789\) −2.00000 −2.00000
\(790\) 2.00000 2.00000
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −2.00000 −2.00000
\(796\) 0 0
\(797\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −1.00000 −1.00000
\(801\) 0 0
\(802\) −1.41421 −1.41421
\(803\) 0 0
\(804\) −1.41421 −1.41421
\(805\) 0 0
\(806\) 0 0
\(807\) −1.41421 −1.41421
\(808\) −1.41421 −1.41421
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) −1.41421 −1.41421
\(811\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(812\) 0 0
\(813\) −1.41421 −1.41421
\(814\) 0 0
\(815\) 2.00000 2.00000
\(816\) −1.41421 −1.41421
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) −2.00000 −2.00000
\(821\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(822\) 0 0
\(823\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(824\) 1.41421 1.41421
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −1.41421 −1.41421
\(834\) −1.41421 −1.41421
\(835\) 1.41421 1.41421
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 1.00000 1.00000
\(842\) 2.00000 2.00000
\(843\) 0 0
\(844\) 0 0
\(845\) 1.41421 1.41421
\(846\) 0 0
\(847\) 0 0
\(848\) −1.41421 −1.41421
\(849\) 0 0
\(850\) 1.41421 1.41421
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 2.00000 2.00000
\(861\) 0 0
\(862\) 2.00000 2.00000
\(863\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(864\) −1.00000 −1.00000
\(865\) 0 0
\(866\) 0 0
\(867\) 1.00000 1.00000
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −2.00000 −2.00000
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(878\) −1.41421 −1.41421
\(879\) 0 0
\(880\) 0 0
\(881\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(882\) −1.00000 −1.00000
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 1.41421 1.41421
\(895\) 2.82843 2.82843
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 1.00000 1.00000
\(901\) 2.00000 2.00000
\(902\) 0 0
\(903\) 0 0
\(904\) −1.41421 −1.41421
\(905\) 0 0
\(906\) −1.41421 −1.41421
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) 1.41421 1.41421
\(910\) 0 0
\(911\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 1.41421 1.41421
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 1.41421 1.41421
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 1.41421 1.41421
\(927\) −1.41421 −1.41421
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −2.00000 −2.00000
\(934\) −2.00000 −2.00000
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(942\) 2.00000 2.00000
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(948\) −1.41421 −1.41421
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(954\) 1.41421 1.41421
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 1.41421 1.41421
\(961\) 1.00000 1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 1.00000 1.00000
\(969\) 0 0
\(970\) 2.82843 2.82843
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 1.00000 1.00000
\(973\) 0 0
\(974\) 1.41421 1.41421
\(975\) 0 0
\(976\) 0 0
\(977\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(978\) −1.41421 −1.41421
\(979\) 0 0
\(980\) 1.41421 1.41421
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 1.41421 1.41421
\(985\) 2.00000 2.00000
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(992\) 0 0
\(993\) −1.41421 −1.41421
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(998\) 1.41421 1.41421
\(999\) 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 \(a_p\))