Properties

Label 20.5.d.b.19.1
Level 20
Weight 5
Character 20.19
Self dual Yes
Analytic conductor 2.067
Analytic rank 0
Dimension 1
CM disc. -20
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) = \( 5 \)
Character orbit: \([\chi]\) = 20.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(2.06739926168\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 19.1
Root \(0\)
Character \(\chi\) = 20.19

$q$-expansion

\(f(q)\) \(=\) \(q+4.00000 q^{2} -2.00000 q^{3} +16.0000 q^{4} +25.0000 q^{5} -8.00000 q^{6} -82.0000 q^{7} +64.0000 q^{8} -77.0000 q^{9} +O(q^{10})\) \(q+4.00000 q^{2} -2.00000 q^{3} +16.0000 q^{4} +25.0000 q^{5} -8.00000 q^{6} -82.0000 q^{7} +64.0000 q^{8} -77.0000 q^{9} +100.000 q^{10} -32.0000 q^{12} -328.000 q^{14} -50.0000 q^{15} +256.000 q^{16} -308.000 q^{18} +400.000 q^{20} +164.000 q^{21} +878.000 q^{23} -128.000 q^{24} +625.000 q^{25} +316.000 q^{27} -1312.00 q^{28} -1198.00 q^{29} -200.000 q^{30} +1024.00 q^{32} -2050.00 q^{35} -1232.00 q^{36} +1600.00 q^{40} +482.000 q^{41} +656.000 q^{42} +2078.00 q^{43} -1925.00 q^{45} +3512.00 q^{46} -4402.00 q^{47} -512.000 q^{48} +4323.00 q^{49} +2500.00 q^{50} +1264.00 q^{54} -5248.00 q^{56} -4792.00 q^{58} -800.000 q^{60} -4078.00 q^{61} +6314.00 q^{63} +4096.00 q^{64} +4478.00 q^{67} -1756.00 q^{69} -8200.00 q^{70} -4928.00 q^{72} -1250.00 q^{75} +6400.00 q^{80} +5605.00 q^{81} +1928.00 q^{82} -8002.00 q^{83} +2624.00 q^{84} +8312.00 q^{86} +2396.00 q^{87} +4322.00 q^{89} -7700.00 q^{90} +14048.0 q^{92} -17608.0 q^{94} -2048.00 q^{96} +17292.0 q^{98} +O(q^{100})\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/20\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(17\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.00000 1.00000
\(3\) −2.00000 −0.222222 −0.111111 0.993808i \(-0.535441\pi\)
−0.111111 + 0.993808i \(0.535441\pi\)
\(4\) 16.0000 1.00000
\(5\) 25.0000 1.00000
\(6\) −8.00000 −0.222222
\(7\) −82.0000 −1.67347 −0.836735 0.547608i \(-0.815538\pi\)
−0.836735 + 0.547608i \(0.815538\pi\)
\(8\) 64.0000 1.00000
\(9\) −77.0000 −0.950617
\(10\) 100.000 1.00000
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) −32.0000 −0.222222
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) −328.000 −1.67347
\(15\) −50.0000 −0.222222
\(16\) 256.000 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) −308.000 −0.950617
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 400.000 1.00000
\(21\) 164.000 0.371882
\(22\) 0 0
\(23\) 878.000 1.65974 0.829868 0.557960i \(-0.188416\pi\)
0.829868 + 0.557960i \(0.188416\pi\)
\(24\) −128.000 −0.222222
\(25\) 625.000 1.00000
\(26\) 0 0
\(27\) 316.000 0.433471
\(28\) −1312.00 −1.67347
\(29\) −1198.00 −1.42449 −0.712247 0.701929i \(-0.752323\pi\)
−0.712247 + 0.701929i \(0.752323\pi\)
\(30\) −200.000 −0.222222
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 1024.00 1.00000
\(33\) 0 0
\(34\) 0 0
\(35\) −2050.00 −1.67347
\(36\) −1232.00 −0.950617
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 1600.00 1.00000
\(41\) 482.000 0.286734 0.143367 0.989670i \(-0.454207\pi\)
0.143367 + 0.989670i \(0.454207\pi\)
\(42\) 656.000 0.371882
\(43\) 2078.00 1.12385 0.561925 0.827188i \(-0.310061\pi\)
0.561925 + 0.827188i \(0.310061\pi\)
\(44\) 0 0
\(45\) −1925.00 −0.950617
\(46\) 3512.00 1.65974
\(47\) −4402.00 −1.99276 −0.996378 0.0850293i \(-0.972902\pi\)
−0.996378 + 0.0850293i \(0.972902\pi\)
\(48\) −512.000 −0.222222
\(49\) 4323.00 1.80050
\(50\) 2500.00 1.00000
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 1264.00 0.433471
\(55\) 0 0
\(56\) −5248.00 −1.67347
\(57\) 0 0
\(58\) −4792.00 −1.42449
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) −800.000 −0.222222
\(61\) −4078.00 −1.09594 −0.547971 0.836497i \(-0.684600\pi\)
−0.547971 + 0.836497i \(0.684600\pi\)
\(62\) 0 0
\(63\) 6314.00 1.59083
\(64\) 4096.00 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 4478.00 0.997550 0.498775 0.866732i \(-0.333783\pi\)
0.498775 + 0.866732i \(0.333783\pi\)
\(68\) 0 0
\(69\) −1756.00 −0.368830
\(70\) −8200.00 −1.67347
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) −4928.00 −0.950617
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) −1250.00 −0.222222
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 6400.00 1.00000
\(81\) 5605.00 0.854291
\(82\) 1928.00 0.286734
\(83\) −8002.00 −1.16156 −0.580781 0.814060i \(-0.697253\pi\)
−0.580781 + 0.814060i \(0.697253\pi\)
\(84\) 2624.00 0.371882
\(85\) 0 0
\(86\) 8312.00 1.12385
\(87\) 2396.00 0.316554
\(88\) 0 0
\(89\) 4322.00 0.545638 0.272819 0.962065i \(-0.412044\pi\)
0.272819 + 0.962065i \(0.412044\pi\)
\(90\) −7700.00 −0.950617
\(91\) 0 0
\(92\) 14048.0 1.65974
\(93\) 0 0
\(94\) −17608.0 −1.99276
\(95\) 0 0
\(96\) −2048.00 −0.222222
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 17292.0 1.80050
\(99\) 0 0
\(100\) 10000.0 1.00000
\(101\) −5518.00 −0.540927 −0.270464 0.962730i \(-0.587177\pi\)
−0.270464 + 0.962730i \(0.587177\pi\)
\(102\) 0 0
\(103\) −19282.0 −1.81751 −0.908757 0.417326i \(-0.862967\pi\)
−0.908757 + 0.417326i \(0.862967\pi\)
\(104\) 0 0
\(105\) 4100.00 0.371882
\(106\) 0 0
\(107\) −7522.00 −0.657001 −0.328500 0.944504i \(-0.606543\pi\)
−0.328500 + 0.944504i \(0.606543\pi\)
\(108\) 5056.00 0.433471
\(109\) −22318.0 −1.87846 −0.939231 0.343287i \(-0.888460\pi\)
−0.939231 + 0.343287i \(0.888460\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −20992.0 −1.67347
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 21950.0 1.65974
\(116\) −19168.0 −1.42449
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) −3200.00 −0.222222
\(121\) 14641.0 1.00000
\(122\) −16312.0 −1.09594
\(123\) −964.000 −0.0637187
\(124\) 0 0
\(125\) 15625.0 1.00000
\(126\) 25256.0 1.59083
\(127\) 23438.0 1.45316 0.726579 0.687082i \(-0.241109\pi\)
0.726579 + 0.687082i \(0.241109\pi\)
\(128\) 16384.0 1.00000
\(129\) −4156.00 −0.249745
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 17912.0 0.997550
\(135\) 7900.00 0.433471
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) −7024.00 −0.368830
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) −32800.0 −1.67347
\(141\) 8804.00 0.442835
\(142\) 0 0
\(143\) 0 0
\(144\) −19712.0 −0.950617
\(145\) −29950.0 −1.42449
\(146\) 0 0
\(147\) −8646.00 −0.400111
\(148\) 0 0
\(149\) 32882.0 1.48110 0.740552 0.671999i \(-0.234564\pi\)
0.740552 + 0.671999i \(0.234564\pi\)
\(150\) −5000.00 −0.222222
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 25600.0 1.00000
\(161\) −71996.0 −2.77752
\(162\) 22420.0 0.854291
\(163\) −26242.0 −0.987692 −0.493846 0.869549i \(-0.664409\pi\)
−0.493846 + 0.869549i \(0.664409\pi\)
\(164\) 7712.00 0.286734
\(165\) 0 0
\(166\) −32008.0 −1.16156
\(167\) 3758.00 0.134748 0.0673742 0.997728i \(-0.478538\pi\)
0.0673742 + 0.997728i \(0.478538\pi\)
\(168\) 10496.0 0.371882
\(169\) 28561.0 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 33248.0 1.12385
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 9584.00 0.316554
\(175\) −51250.0 −1.67347
\(176\) 0 0
\(177\) 0 0
\(178\) 17288.0 0.545638
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) −30800.0 −0.950617
\(181\) 62642.0 1.91209 0.956045 0.293219i \(-0.0947265\pi\)
0.956045 + 0.293219i \(0.0947265\pi\)
\(182\) 0 0
\(183\) 8156.00 0.243543
\(184\) 56192.0 1.65974
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) −70432.0 −1.99276
\(189\) −25912.0 −0.725400
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) −8192.00 −0.222222
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 69168.0 1.80050
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 40000.0 1.00000
\(201\) −8956.00 −0.221678
\(202\) −22072.0 −0.540927
\(203\) 98236.0 2.38385
\(204\) 0 0
\(205\) 12050.0 0.286734
\(206\) −77128.0 −1.81751
\(207\) −67606.0 −1.57777
\(208\) 0 0
\(209\) 0 0
\(210\) 16400.0 0.371882
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) −30088.0 −0.657001
\(215\) 51950.0 1.12385
\(216\) 20224.0 0.433471
\(217\) 0 0
\(218\) −89272.0 −1.87846
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 90638.0 1.82264 0.911319 0.411700i \(-0.135065\pi\)
0.911319 + 0.411700i \(0.135065\pi\)
\(224\) −83968.0 −1.67347
\(225\) −48125.0 −0.950617
\(226\) 0 0
\(227\) 23678.0 0.459508 0.229754 0.973249i \(-0.426208\pi\)
0.229754 + 0.973249i \(0.426208\pi\)
\(228\) 0 0
\(229\) −36238.0 −0.691024 −0.345512 0.938414i \(-0.612295\pi\)
−0.345512 + 0.938414i \(0.612295\pi\)
\(230\) 87800.0 1.65974
\(231\) 0 0
\(232\) −76672.0 −1.42449
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) −110050. −1.99276
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) −12800.0 −0.222222
\(241\) −24958.0 −0.429710 −0.214855 0.976646i \(-0.568928\pi\)
−0.214855 + 0.976646i \(0.568928\pi\)
\(242\) 58564.0 1.00000
\(243\) −36806.0 −0.623313
\(244\) −65248.0 −1.09594
\(245\) 108075. 1.80050
\(246\) −3856.00 −0.0637187
\(247\) 0 0
\(248\) 0 0
\(249\) 16004.0 0.258125
\(250\) 62500.0 1.00000
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 101024. 1.59083
\(253\) 0 0
\(254\) 93752.0 1.45316
\(255\) 0 0
\(256\) 65536.0 1.00000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) −16624.0 −0.249745
\(259\) 0 0
\(260\) 0 0
\(261\) 92246.0 1.35415
\(262\) 0 0
\(263\) −57682.0 −0.833928 −0.416964 0.908923i \(-0.636906\pi\)
−0.416964 + 0.908923i \(0.636906\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −8644.00 −0.121253
\(268\) 71648.0 0.997550
\(269\) −143278. −1.98004 −0.990022 0.140911i \(-0.954997\pi\)
−0.990022 + 0.140911i \(0.954997\pi\)
\(270\) 31600.0 0.433471
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) −28096.0 −0.368830
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) −131200. −1.67347
\(281\) 16802.0 0.212789 0.106394 0.994324i \(-0.466069\pi\)
0.106394 + 0.994324i \(0.466069\pi\)
\(282\) 35216.0 0.442835
\(283\) −60322.0 −0.753187 −0.376594 0.926379i \(-0.622905\pi\)
−0.376594 + 0.926379i \(0.622905\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −39524.0 −0.479841
\(288\) −78848.0 −0.950617
\(289\) 83521.0 1.00000
\(290\) −119800. −1.42449
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) −34584.0 −0.400111
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 131528. 1.48110
\(299\) 0 0
\(300\) −20000.0 −0.222222
\(301\) −170396. −1.88073
\(302\) 0 0
\(303\) 11036.0 0.120206
\(304\) 0 0
\(305\) −101950. −1.09594
\(306\) 0 0
\(307\) 166718. 1.76891 0.884455 0.466625i \(-0.154530\pi\)
0.884455 + 0.466625i \(0.154530\pi\)
\(308\) 0 0
\(309\) 38564.0 0.403892
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 157850. 1.59083
\(316\) 0 0
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 102400. 1.00000
\(321\) 15044.0 0.146000
\(322\) −287984. −2.77752
\(323\) 0 0
\(324\) 89680.0 0.854291
\(325\) 0 0
\(326\) −104968. −0.987692
\(327\) 44636.0 0.417436
\(328\) 30848.0 0.286734
\(329\) 360964. 3.33482
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) −128032. −1.16156
\(333\) 0 0
\(334\) 15032.0 0.134748
\(335\) 111950. 0.997550
\(336\) 41984.0 0.371882
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 114244. 1.00000
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −157604. −1.33961
\(344\) 132992. 1.12385
\(345\) −43900.0 −0.368830
\(346\) 0 0
\(347\) −227362. −1.88825 −0.944124 0.329591i \(-0.893089\pi\)
−0.944124 + 0.329591i \(0.893089\pi\)
\(348\) 38336.0 0.316554
\(349\) −243118. −1.99603 −0.998013 0.0630059i \(-0.979931\pi\)
−0.998013 + 0.0630059i \(0.979931\pi\)
\(350\) −205000. −1.67347
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 69152.0 0.545638
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) −123200. −0.950617
\(361\) 130321. 1.00000
\(362\) 250568. 1.91209
\(363\) −29282.0 −0.222222
\(364\) 0 0
\(365\) 0 0
\(366\) 32624.0 0.243543
\(367\) 254798. 1.89175 0.945875 0.324530i \(-0.105206\pi\)
0.945875 + 0.324530i \(0.105206\pi\)
\(368\) 224768. 1.65974
\(369\) −37114.0 −0.272574
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) −31250.0 −0.222222
\(376\) −281728. −1.99276
\(377\) 0 0
\(378\) −103648. −0.725400
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) −46876.0 −0.322924
\(382\) 0 0
\(383\) −291442. −1.98680 −0.993401 0.114693i \(-0.963412\pi\)
−0.993401 + 0.114693i \(0.963412\pi\)
\(384\) −32768.0 −0.222222
\(385\) 0 0
\(386\) 0 0
\(387\) −160006. −1.06835
\(388\) 0 0
\(389\) −261838. −1.73035 −0.865174 0.501472i \(-0.832792\pi\)
−0.865174 + 0.501472i \(0.832792\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 276672. 1.80050
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 160000. 1.00000
\(401\) −93118.0 −0.579088 −0.289544 0.957165i \(-0.593504\pi\)
−0.289544 + 0.957165i \(0.593504\pi\)
\(402\) −35824.0 −0.221678
\(403\) 0 0
\(404\) −88288.0 −0.540927
\(405\) 140125. 0.854291
\(406\) 392944. 2.38385
\(407\) 0 0
\(408\) 0 0
\(409\) 308642. 1.84505 0.922526 0.385936i \(-0.126121\pi\)
0.922526 + 0.385936i \(0.126121\pi\)
\(410\) 48200.0 0.286734
\(411\) 0 0
\(412\) −308512. −1.81751
\(413\) 0 0
\(414\) −270424. −1.57777
\(415\) −200050. −1.16156
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 65600.0 0.371882
\(421\) 250802. 1.41503 0.707517 0.706696i \(-0.249815\pi\)
0.707517 + 0.706696i \(0.249815\pi\)
\(422\) 0 0
\(423\) 338954. 1.89435
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 334396. 1.83403
\(428\) −120352. −0.657001
\(429\) 0 0
\(430\) 207800. 1.12385
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 80896.0 0.433471
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 59900.0 0.316554
\(436\) −357088. −1.87846
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) −332871. −1.71159
\(442\) 0 0
\(443\) 241118. 1.22863 0.614317 0.789060i \(-0.289432\pi\)
0.614317 + 0.789060i \(0.289432\pi\)
\(444\) 0 0
\(445\) 108050. 0.545638
\(446\) 362552. 1.82264
\(447\) −65764.0 −0.329134
\(448\) −335872. −1.67347
\(449\) −244798. −1.21427 −0.607135 0.794599i \(-0.707681\pi\)
−0.607135 + 0.794599i \(0.707681\pi\)
\(450\) −192500. −0.950617
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 94712.0 0.459508
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) −144952. −0.691024
\(459\) 0 0
\(460\) 351200. 1.65974
\(461\) 283922. 1.33597 0.667986 0.744174i \(-0.267157\pi\)
0.667986 + 0.744174i \(0.267157\pi\)
\(462\) 0 0
\(463\) 154958. 0.722856 0.361428 0.932400i \(-0.382289\pi\)
0.361428 + 0.932400i \(0.382289\pi\)
\(464\) −306688. −1.42449
\(465\) 0 0
\(466\) 0 0
\(467\) −420802. −1.92950 −0.964748 0.263174i \(-0.915231\pi\)
−0.964748 + 0.263174i \(0.915231\pi\)
\(468\) 0 0
\(469\) −367196. −1.66937
\(470\) −440200. −1.99276
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) −51200.0 −0.222222
\(481\) 0 0
\(482\) −99832.0 −0.429710
\(483\) 143992. 0.617226
\(484\) 234256. 1.00000
\(485\) 0 0
\(486\) −147224. −0.623313
\(487\) −240082. −1.01228 −0.506141 0.862451i \(-0.668929\pi\)
−0.506141 + 0.862451i \(0.668929\pi\)
\(488\) −260992. −1.09594
\(489\) 52484.0 0.219487
\(490\) 432300. 1.80050
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) −15424.0 −0.0637187
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 64016.0 0.258125
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 250000. 1.00000
\(501\) −7516.00 −0.0299441
\(502\) 0 0
\(503\) 333038. 1.31631 0.658154 0.752883i \(-0.271337\pi\)
0.658154 + 0.752883i \(0.271337\pi\)
\(504\) 404096. 1.59083
\(505\) −137950. −0.540927
\(506\) 0 0
\(507\) −57122.0 −0.222222
\(508\) 375008. 1.45316
\(509\) 446162. 1.72209 0.861047 0.508525i \(-0.169809\pi\)
0.861047 + 0.508525i \(0.169809\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 262144. 1.00000
\(513\) 0 0
\(514\) 0 0
\(515\) −482050. −1.81751
\(516\) −66496.0 −0.249745
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −21598.0 −0.0795679 −0.0397840 0.999208i \(-0.512667\pi\)
−0.0397840 + 0.999208i \(0.512667\pi\)
\(522\) 368984. 1.35415
\(523\) −520162. −1.90167 −0.950835 0.309697i \(-0.899772\pi\)
−0.950835 + 0.309697i \(0.899772\pi\)
\(524\) 0 0
\(525\) 102500. 0.371882
\(526\) −230728. −0.833928
\(527\) 0 0
\(528\) 0 0
\(529\) 491043. 1.75472
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) −34576.0 −0.121253
\(535\) −188050. −0.657001
\(536\) 286592. 0.997550
\(537\) 0 0
\(538\) −573112. −1.98004
\(539\) 0 0
\(540\) 126400. 0.433471
\(541\) −454318. −1.55226 −0.776132 0.630571i \(-0.782821\pi\)
−0.776132 + 0.630571i \(0.782821\pi\)
\(542\) 0 0
\(543\) −125284. −0.424909
\(544\) 0 0
\(545\) −557950. −1.87846
\(546\) 0 0
\(547\) 576638. 1.92721 0.963604 0.267334i \(-0.0861426\pi\)
0.963604 + 0.267334i \(0.0861426\pi\)
\(548\) 0 0
\(549\) 314006. 1.04182
\(550\) 0 0
\(551\) 0 0
\(552\) −112384. −0.368830
\(553\) 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
\(560\) −524800. −1.67347
\(561\) 0 0
\(562\) 67208.0 0.212789
\(563\) 629438. 1.98580 0.992902 0.118939i \(-0.0379494\pi\)
0.992902 + 0.118939i \(0.0379494\pi\)
\(564\) 140864. 0.442835
\(565\) 0 0
\(566\) −241288. −0.753187
\(567\) −459610. −1.42963
\(568\) 0 0
\(569\) −622558. −1.92289 −0.961447 0.274991i \(-0.911325\pi\)
−0.961447 + 0.274991i \(0.911325\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −158096. −0.479841
\(575\) 548750. 1.65974
\(576\) −315392. −0.950617
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 334084. 1.00000
\(579\) 0 0
\(580\) −479200. −1.42449
\(581\) 656164. 1.94384
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 468638. 1.36007 0.680035 0.733180i \(-0.261965\pi\)
0.680035 + 0.733180i \(0.261965\pi\)
\(588\) −138336. −0.400111
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 526112. 1.48110
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) −80000.0 −0.222222
\(601\) −547678. −1.51627 −0.758135 0.652098i \(-0.773889\pi\)
−0.758135 + 0.652098i \(0.773889\pi\)
\(602\) −681584. −1.88073
\(603\) −344806. −0.948288
\(604\) 0 0
\(605\) 366025. 1.00000
\(606\) 44144.0 0.120206
\(607\) 192398. 0.522184 0.261092 0.965314i \(-0.415917\pi\)
0.261092 + 0.965314i \(0.415917\pi\)
\(608\) 0 0
\(609\) −196472. −0.529744
\(610\) −407800. −1.09594
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 666872. 1.76891
\(615\) −24100.0 −0.0637187
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 154256. 0.403892
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 277448. 0.719446
\(622\) 0 0
\(623\) −354404. −0.913109
\(624\) 0 0
\(625\) 390625. 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 631400. 1.59083
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 585950. 1.45316
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 409600. 1.00000
\(641\) 473282. 1.15187 0.575936 0.817495i \(-0.304638\pi\)
0.575936 + 0.817495i \(0.304638\pi\)
\(642\) 60176.0 0.146000
\(643\) −663682. −1.60523 −0.802617 0.596495i \(-0.796559\pi\)
−0.802617 + 0.596495i \(0.796559\pi\)
\(644\) −1.15194e6 −2.77752
\(645\) −103900. −0.249745
\(646\) 0 0
\(647\) 76718.0 0.183269 0.0916344 0.995793i \(-0.470791\pi\)
0.0916344 + 0.995793i \(0.470791\pi\)
\(648\) 358720. 0.854291
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −419872. −0.987692
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 178544. 0.417436
\(655\) 0 0
\(656\) 123392. 0.286734
\(657\) 0 0
\(658\) 1.44386e6 3.33482
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −785038. −1.79675 −0.898375 0.439229i \(-0.855252\pi\)
−0.898375 + 0.439229i \(0.855252\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) −512128. −1.16156
\(665\) 0 0
\(666\) 0 0
\(667\) −1.05184e6 −2.36428
\(668\) 60128.0 0.134748
\(669\) −181276. −0.405031
\(670\) 447800. 0.997550
\(671\) 0 0
\(672\) 167936. 0.371882
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 197500. 0.433471
\(676\) 456976. 1.00000
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −47356.0 −0.102113
\(682\) 0 0
\(683\) −623842. −1.33731 −0.668657 0.743571i \(-0.733130\pi\)
−0.668657 + 0.743571i \(0.733130\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −630416. −1.33961
\(687\) 72476.0 0.153561
\(688\) 531968. 1.12385
\(689\) 0 0
\(690\) −175600. −0.368830
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −909448. −1.88825
\(695\) 0 0
\(696\) 153344. 0.316554
\(697\) 0 0
\(698\) −972472. −1.99603
\(699\) 0 0
\(700\) −820000. −1.67347
\(701\) −169198. −0.344318 −0.172159 0.985069i \(-0.555074\pi\)
−0.172159 + 0.985069i \(0.555074\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 220100. 0.442835
\(706\) 0 0
\(707\) 452476. 0.905225
\(708\) 0 0
\(709\) −518158. −1.03079 −0.515394 0.856953i \(-0.672355\pi\)
−0.515394 + 0.856953i \(0.672355\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 276608. 0.545638
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) −492800. −0.950617
\(721\) 1.58112e6 3.04155
\(722\) 521284. 1.00000
\(723\) 49916.0 0.0954912
\(724\) 1.00227e6 1.91209
\(725\) −748750. −1.42449
\(726\) −117128. −0.222222
\(727\) 1.00504e6 1.90158 0.950788 0.309842i \(-0.100276\pi\)
0.950788 + 0.309842i \(0.100276\pi\)
\(728\) 0 0
\(729\) −380393. −0.715777
\(730\) 0 0
\(731\) 0 0
\(732\) 130496. 0.243543
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 1.01919e6 1.89175
\(735\) −216150. −0.400111
\(736\) 899072. 1.65974
\(737\) 0 0
\(738\) −148456. −0.272574
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −520402. −0.942674 −0.471337 0.881953i \(-0.656228\pi\)
−0.471337 + 0.881953i \(0.656228\pi\)
\(744\) 0 0
\(745\) 822050. 1.48110
\(746\) 0 0
\(747\) 616154. 1.10420
\(748\) 0 0
\(749\) 616804. 1.09947
\(750\) −125000. −0.222222
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) −1.12691e6 −1.99276
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) −414592. −0.725400
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1.09968e6 −1.89887 −0.949437 0.313957i \(-0.898345\pi\)
−0.949437 + 0.313957i \(0.898345\pi\)
\(762\) −187504. −0.322924
\(763\) 1.83008e6 3.14355
\(764\) 0 0
\(765\) 0 0
\(766\) −1.16577e6 −1.98680
\(767\) 0 0
\(768\) −131072. −0.222222
\(769\) 618242. 1.04546 0.522728 0.852499i \(-0.324914\pi\)
0.522728 + 0.852499i \(0.324914\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) −640024. −1.06835
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) −1.04735e6 −1.73035
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −378568. −0.617476
\(784\) 1.10669e6 1.80050
\(785\) 0 0
\(786\) 0 0
\(787\) −1.22528e6 −1.97827 −0.989137 0.146994i \(-0.953040\pi\)
−0.989137 + 0.146994i \(0.953040\pi\)
\(788\) 0 0
\(789\) 115364. 0.185317
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 640000. 1.00000
\(801\) −332794. −0.518693
\(802\) −372472. −0.579088
\(803\) 0 0
\(804\) −143296. −0.221678
\(805\) −1.79990e6 −2.77752
\(806\) 0 0
\(807\) 286556. 0.440010
\(808\) −353152. −0.540927
\(809\) 375842. 0.574260 0.287130 0.957892i \(-0.407299\pi\)
0.287130 + 0.957892i \(0.407299\pi\)
\(810\) 560500. 0.854291
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 1.57178e6 2.38385
\(813\) 0 0
\(814\) 0 0
\(815\) −656050. −0.987692
\(816\) 0 0
\(817\) 0 0
\(818\) 1.23457e6 1.84505
\(819\) 0 0
\(820\) 192800. 0.286734
\(821\) −909838. −1.34983 −0.674913 0.737897i \(-0.735819\pi\)
−0.674913 + 0.737897i \(0.735819\pi\)
\(822\) 0 0
\(823\) 594158. 0.877207 0.438604 0.898681i \(-0.355473\pi\)
0.438604 + 0.898681i \(0.355473\pi\)
\(824\) −1.23405e6 −1.81751
\(825\) 0 0
\(826\) 0 0
\(827\) −1.01264e6 −1.48062 −0.740312 0.672263i \(-0.765322\pi\)
−0.740312 + 0.672263i \(0.765322\pi\)
\(828\) −1.08170e6 −1.57777
\(829\) 810002. 1.17863 0.589314 0.807904i \(-0.299398\pi\)
0.589314 + 0.807904i \(0.299398\pi\)
\(830\) −800200. −1.16156
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 93950.0 0.134748
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 262400. 0.371882
\(841\) 727923. 1.02919
\(842\) 1.00321e6 1.41503
\(843\) −33604.0 −0.0472864
\(844\) 0 0
\(845\) 714025. 1.00000
\(846\) 1.35582e6 1.89435
\(847\) −1.20056e6 −1.67347
\(848\) 0 0
\(849\) 120644. 0.167375
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 1.33758e6 1.83403
\(855\) 0 0
\(856\) −481408. −0.657001
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 831200. 1.12385
\(861\) 79048.0 0.106631
\(862\) 0 0
\(863\) 1.18696e6 1.59373 0.796863 0.604160i \(-0.206491\pi\)
0.796863 + 0.604160i \(0.206491\pi\)
\(864\) 323584. 0.433471
\(865\) 0 0
\(866\) 0 0
\(867\) −167042. −0.222222
\(868\) 0 0
\(869\) 0 0
\(870\) 239600. 0.316554
\(871\) 0 0
\(872\) −1.42835e6 −1.87846
\(873\) 0 0
\(874\) 0 0
\(875\) −1.28125e6 −1.67347
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1.06560e6 1.37291 0.686457 0.727171i \(-0.259165\pi\)
0.686457 + 0.727171i \(0.259165\pi\)
\(882\) −1.33148e6 −1.71159
\(883\) 68798.0 0.0882377 0.0441189 0.999026i \(-0.485952\pi\)
0.0441189 + 0.999026i \(0.485952\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 964472. 1.22863
\(887\) −644242. −0.818845 −0.409423 0.912345i \(-0.634270\pi\)
−0.409423 + 0.912345i \(0.634270\pi\)
\(888\) 0 0
\(889\) −1.92192e6 −2.43182
\(890\) 432200. 0.545638
\(891\) 0 0
\(892\) 1.45021e6 1.82264
\(893\) 0 0
\(894\) −263056. −0.329134
\(895\) 0 0
\(896\) −1.34349e6 −1.67347
\(897\) 0 0
\(898\) −979192. −1.21427
\(899\) 0 0
\(900\) −770000. −0.950617
\(901\) 0 0
\(902\) 0 0
\(903\) 340792. 0.417940
\(904\) 0 0
\(905\) 1.56605e6 1.91209
\(906\) 0 0
\(907\) 1.58032e6 1.92101 0.960506 0.278261i \(-0.0897579\pi\)
0.960506 + 0.278261i \(0.0897579\pi\)
\(908\) 378848. 0.459508
\(909\) 424886. 0.514215
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 203900. 0.243543
\(916\) −579808. −0.691024
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 1.40480e6 1.65974
\(921\) −333436. −0.393091
\(922\) 1.13569e6 1.33597
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 619832. 0.722856
\(927\) 1.48471e6 1.72776
\(928\) −1.22675e6 −1.42449
\(929\) −1.41024e6 −1.63403 −0.817017 0.576614i \(-0.804374\pi\)
−0.817017 + 0.576614i \(0.804374\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) −1.68321e6 −1.92950
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) −1.46878e6 −1.66937
\(939\) 0 0
\(940\) −1.76080e6 −1.99276
\(941\) −1.75704e6 −1.98428 −0.992138 0.125152i \(-0.960058\pi\)
−0.992138 + 0.125152i \(0.960058\pi\)
\(942\) 0 0
\(943\) 423196. 0.475903
\(944\) 0 0
\(945\) −647800. −0.725400
\(946\) 0 0
\(947\) 1.46080e6 1.62888 0.814442 0.580245i \(-0.197043\pi\)
0.814442 + 0.580245i \(0.197043\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) −204800. −0.222222
\(961\) 923521. 1.00000
\(962\) 0 0
\(963\) 579194. 0.624556
\(964\) −399328. −0.429710
\(965\) 0 0
\(966\) 575968. 0.617226
\(967\) −1.81064e6 −1.93633 −0.968166 0.250311i \(-0.919467\pi\)
−0.968166 + 0.250311i \(0.919467\pi\)
\(968\) 937024. 1.00000
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) −588896. −0.623313
\(973\) 0 0
\(974\) −960328. −1.01228
\(975\) 0 0
\(976\) −1.04397e6 −1.09594
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 209936. 0.219487
\(979\) 0 0
\(980\) 1.72920e6 1.80050
\(981\) 1.71849e6 1.78570
\(982\) 0 0
\(983\) −1.85192e6 −1.91653 −0.958265 0.285881i \(-0.907714\pi\)
−0.958265 + 0.285881i \(0.907714\pi\)
\(984\) −61696.0 −0.0637187
\(985\) 0 0
\(986\) 0 0
\(987\) −721928. −0.741071
\(988\) 0 0
\(989\) 1.82448e6 1.86529
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 256064. 0.258125
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))