Properties

Label 71.1.b.a.70.1
Level 71
Weight 1
Character 71.70
Self dual Yes
Analytic conductor 0.035
Analytic rank 0
Dimension 3
Projective image \(D_{7}\)
CM disc. -71
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 71 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 71.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(0.0354336158969\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{7}\)
Projective field Galois closure of 7.1.357911.1
Artin image size \(14\)
Artin image $D_7$
Artin field Galois closure of 7.1.357911.1

Embedding invariants

Embedding label 70.1
Root \(1.80194\)
Character \(\chi\) = 71.70

$q$-expansion

\(f(q)\) \(=\) \(q-1.80194 q^{2} -0.445042 q^{3} +2.24698 q^{4} +1.24698 q^{5} +0.801938 q^{6} -2.24698 q^{8} -0.801938 q^{9} +O(q^{10})\) \(q-1.80194 q^{2} -0.445042 q^{3} +2.24698 q^{4} +1.24698 q^{5} +0.801938 q^{6} -2.24698 q^{8} -0.801938 q^{9} -2.24698 q^{10} -1.00000 q^{12} -0.554958 q^{15} +1.80194 q^{16} +1.44504 q^{18} -1.80194 q^{19} +2.80194 q^{20} +1.00000 q^{24} +0.554958 q^{25} +0.801938 q^{27} -0.445042 q^{29} +1.00000 q^{30} -1.00000 q^{32} -1.80194 q^{36} -1.80194 q^{37} +3.24698 q^{38} -2.80194 q^{40} +1.24698 q^{43} -1.00000 q^{45} -0.801938 q^{48} +1.00000 q^{49} -1.00000 q^{50} -1.44504 q^{54} +0.801938 q^{57} +0.801938 q^{58} -1.24698 q^{60} +1.00000 q^{71} +1.80194 q^{72} +1.24698 q^{73} +3.24698 q^{74} -0.246980 q^{75} -4.04892 q^{76} +1.24698 q^{79} +2.24698 q^{80} +0.445042 q^{81} -1.80194 q^{83} -2.24698 q^{86} +0.198062 q^{87} -0.445042 q^{89} +1.80194 q^{90} -2.24698 q^{95} +0.445042 q^{96} -1.80194 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - q^{2} - q^{3} + 2q^{4} - q^{5} - 2q^{6} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 3q - q^{2} - q^{3} + 2q^{4} - q^{5} - 2q^{6} - 2q^{8} + 2q^{9} - 2q^{10} - 3q^{12} - 2q^{15} + q^{16} + 4q^{18} - q^{19} + 4q^{20} + 3q^{24} + 2q^{25} - 2q^{27} - q^{29} + 3q^{30} - 3q^{32} - q^{36} - q^{37} + 5q^{38} - 4q^{40} - q^{43} - 3q^{45} + 2q^{48} + 3q^{49} - 3q^{50} - 4q^{54} - 2q^{57} - 2q^{58} + q^{60} + 3q^{71} + q^{72} - q^{73} + 5q^{74} + 4q^{75} - 3q^{76} - q^{79} + 2q^{80} + q^{81} - q^{83} - 2q^{86} + 5q^{87} - q^{89} + q^{90} - 2q^{95} + q^{96} - q^{98} + O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/71\mathbb{Z}\right)^\times\).

\(n\) \(7\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(3\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(4\) 2.24698 2.24698
\(5\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(6\) 0.801938 0.801938
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) −2.24698 −2.24698
\(9\) −0.801938 −0.801938
\(10\) −2.24698 −2.24698
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) −1.00000 −1.00000
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) −0.554958 −0.554958
\(16\) 1.80194 1.80194
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 1.44504 1.44504
\(19\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(20\) 2.80194 2.80194
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 1.00000 1.00000
\(25\) 0.554958 0.554958
\(26\) 0 0
\(27\) 0.801938 0.801938
\(28\) 0 0
\(29\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(30\) 1.00000 1.00000
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −1.00000 −1.00000
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −1.80194 −1.80194
\(37\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(38\) 3.24698 3.24698
\(39\) 0 0
\(40\) −2.80194 −2.80194
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(44\) 0 0
\(45\) −1.00000 −1.00000
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) −0.801938 −0.801938
\(49\) 1.00000 1.00000
\(50\) −1.00000 −1.00000
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) −1.44504 −1.44504
\(55\) 0 0
\(56\) 0 0
\(57\) 0.801938 0.801938
\(58\) 0.801938 0.801938
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) −1.24698 −1.24698
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 1.00000 1.00000
\(72\) 1.80194 1.80194
\(73\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(74\) 3.24698 3.24698
\(75\) −0.246980 −0.246980
\(76\) −4.04892 −4.04892
\(77\) 0 0
\(78\) 0 0
\(79\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(80\) 2.24698 2.24698
\(81\) 0.445042 0.445042
\(82\) 0 0
\(83\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −2.24698 −2.24698
\(87\) 0.198062 0.198062
\(88\) 0 0
\(89\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(90\) 1.80194 1.80194
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −2.24698 −2.24698
\(96\) 0.445042 0.445042
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) −1.80194 −1.80194
\(99\) 0 0
\(100\) 1.24698 1.24698
\(101\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(102\) 0 0
\(103\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(108\) 1.80194 1.80194
\(109\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(110\) 0 0
\(111\) 0.801938 0.801938
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) −1.44504 −1.44504
\(115\) 0 0
\(116\) −1.00000 −1.00000
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 1.24698 1.24698
\(121\) 1.00000 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −0.554958 −0.554958
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 1.00000 1.00000
\(129\) −0.554958 −0.554958
\(130\) 0 0
\(131\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 1.00000 1.00000
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −1.80194 −1.80194
\(143\) 0 0
\(144\) −1.44504 −1.44504
\(145\) −0.554958 −0.554958
\(146\) −2.24698 −2.24698
\(147\) −0.445042 −0.445042
\(148\) −4.04892 −4.04892
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0.445042 0.445042
\(151\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(152\) 4.04892 4.04892
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(158\) −2.24698 −2.24698
\(159\) 0 0
\(160\) −1.24698 −1.24698
\(161\) 0 0
\(162\) −0.801938 −0.801938
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 3.24698 3.24698
\(167\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(168\) 0 0
\(169\) 1.00000 1.00000
\(170\) 0 0
\(171\) 1.44504 1.44504
\(172\) 2.80194 2.80194
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) −0.356896 −0.356896
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0.801938 0.801938
\(179\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(180\) −2.24698 −2.24698
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −2.24698 −2.24698
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 4.04892 4.04892
\(191\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 2.24698 2.24698
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(200\) −1.24698 −1.24698
\(201\) 0 0
\(202\) 3.24698 3.24698
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0.801938 0.801938
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) −0.445042 −0.445042
\(214\) −3.60388 −3.60388
\(215\) 1.55496 1.55496
\(216\) −1.80194 −1.80194
\(217\) 0 0
\(218\) 0.801938 0.801938
\(219\) −0.554958 −0.554958
\(220\) 0 0
\(221\) 0 0
\(222\) −1.44504 −1.44504
\(223\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(224\) 0 0
\(225\) −0.445042 −0.445042
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 1.80194 1.80194
\(229\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 1.00000 1.00000
\(233\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −0.554958 −0.554958
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) −1.00000 −1.00000
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) −1.80194 −1.80194
\(243\) −1.00000 −1.00000
\(244\) 0 0
\(245\) 1.24698 1.24698
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0.801938 0.801938
\(250\) 1.00000 1.00000
\(251\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −1.80194 −1.80194
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 1.00000 1.00000
\(259\) 0 0
\(260\) 0 0
\(261\) 0.356896 0.356896
\(262\) −2.24698 −2.24698
\(263\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0.198062 0.198062
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) −1.80194 −1.80194
\(271\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(278\) 0 0
\(279\) 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 \(\pi\)
\(284\) 2.24698 2.24698
\(285\) 1.00000 1.00000
\(286\) 0 0
\(287\) 0 0
\(288\) 0.801938 0.801938
\(289\) 1.00000 1.00000
\(290\) 1.00000 1.00000
\(291\) 0 0
\(292\) 2.80194 2.80194
\(293\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(294\) 0.801938 0.801938
\(295\) 0 0
\(296\) 4.04892 4.04892
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) −0.554958 −0.554958
\(301\) 0 0
\(302\) 0.801938 0.801938
\(303\) 0.801938 0.801938
\(304\) −3.24698 −3.24698
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0.198062 0.198062
\(310\) 0 0
\(311\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(312\) 0 0
\(313\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(314\) 0.801938 0.801938
\(315\) 0 0
\(316\) 2.80194 2.80194
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −0.890084 −0.890084
\(322\) 0 0
\(323\) 0 0
\(324\) 1.00000 1.00000
\(325\) 0 0
\(326\) 0 0
\(327\) 0.198062 0.198062
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) −4.04892 −4.04892
\(333\) 1.44504 1.44504
\(334\) −2.24698 −2.24698
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) −1.80194 −1.80194
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) −2.60388 −2.60388
\(343\) 0 0
\(344\) −2.80194 −2.80194
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0.445042 0.445042
\(349\) 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
\(355\) 1.24698 1.24698
\(356\) −1.00000 −1.00000
\(357\) 0 0
\(358\) −2.24698 −2.24698
\(359\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(360\) 2.24698 2.24698
\(361\) 2.24698 2.24698
\(362\) 0 0
\(363\) −0.445042 −0.445042
\(364\) 0 0
\(365\) 1.55496 1.55496
\(366\) 0 0
\(367\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 4.04892 4.04892
\(371\) 0 0
\(372\) 0 0
\(373\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(374\) 0 0
\(375\) 0.246980 0.246980
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(380\) −5.04892 −5.04892
\(381\) 0 0
\(382\) 0.801938 0.801938
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) −0.445042 −0.445042
\(385\) 0 0
\(386\) 0 0
\(387\) −1.00000 −1.00000
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −2.24698 −2.24698
\(393\) −0.554958 −0.554958
\(394\) 0 0
\(395\) 1.55496 1.55496
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 3.24698 3.24698
\(399\) 0 0
\(400\) 1.00000 1.00000
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −4.04892 −4.04892
\(405\) 0.554958 0.554958
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −1.00000 −1.00000
\(413\) 0 0
\(414\) 0 0
\(415\) −2.24698 −2.24698
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0.801938 0.801938
\(427\) 0 0
\(428\) 4.49396 4.49396
\(429\) 0 0
\(430\) −2.80194 −2.80194
\(431\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(432\) 1.44504 1.44504
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0.246980 0.246980
\(436\) −1.00000 −1.00000
\(437\) 0 0
\(438\) 1.00000 1.00000
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) −0.801938 −0.801938
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 1.80194 1.80194
\(445\) −0.554958 −0.554958
\(446\) 3.24698 3.24698
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0.801938 0.801938
\(451\) 0 0
\(452\) 0 0
\(453\) 0.198062 0.198062
\(454\) 0 0
\(455\) 0 0
\(456\) −1.80194 −1.80194
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) −2.24698 −2.24698
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(464\) −0.801938 −0.801938
\(465\) 0 0
\(466\) 3.24698 3.24698
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0.198062 0.198062
\(472\) 0 0
\(473\) 0 0
\(474\) 1.00000 1.00000
\(475\) −1.00000 −1.00000
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0.554958 0.554958
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 2.24698 2.24698
\(485\) 0 0
\(486\) 1.80194 1.80194
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) −2.24698 −2.24698
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) −1.44504 −1.44504
\(499\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(500\) −1.24698 −1.24698
\(501\) −0.554958 −0.554958
\(502\) 3.24698 3.24698
\(503\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(504\) 0 0
\(505\) −2.24698 −2.24698
\(506\) 0 0
\(507\) −0.445042 −0.445042
\(508\) 0 0
\(509\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(510\) 0 0
\(511\) 0 0
\(512\) 2.24698 2.24698
\(513\) −1.44504 −1.44504
\(514\) 0 0
\(515\) −0.554958 −0.554958
\(516\) −1.24698 −1.24698
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(522\) −0.643104 −0.643104
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 2.80194 2.80194
\(525\) 0 0
\(526\) −2.24698 −2.24698
\(527\) 0 0
\(528\) 0 0
\(529\) 1.00000 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) −0.356896 −0.356896
\(535\) 2.49396 2.49396
\(536\) 0 0
\(537\) −0.554958 −0.554958
\(538\) 0 0
\(539\) 0 0
\(540\) 2.24698 2.24698
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0.801938 0.801938
\(543\) 0 0
\(544\) 0 0
\(545\) −0.554958 −0.554958
\(546\) 0 0
\(547\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0.801938 0.801938
\(552\) 0 0
\(553\) 0 0
\(554\) 3.24698 3.24698
\(555\) 1.00000 1.00000
\(556\) 0 0
\(557\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) −2.24698 −2.24698
\(569\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(570\) −1.80194 −1.80194
\(571\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(572\) 0 0
\(573\) 0.198062 0.198062
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(578\) −1.80194 −1.80194
\(579\) 0 0
\(580\) −1.24698 −1.24698
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) −2.80194 −2.80194
\(585\) 0 0
\(586\) −3.60388 −3.60388
\(587\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(588\) −1.00000 −1.00000
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −3.24698 −3.24698
\(593\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0.801938 0.801938
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0.554958 0.554958
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −1.00000 −1.00000
\(605\) 1.24698 1.24698
\(606\) −1.44504 −1.44504
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 1.80194 1.80194
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(618\) −0.356896 −0.356896
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 3.24698 3.24698
\(623\) 0 0
\(624\) 0 0
\(625\) −1.24698 −1.24698
\(626\) 3.24698 3.24698
\(627\) 0 0
\(628\) −1.00000 −1.00000
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) −2.80194 −2.80194
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −0.801938 −0.801938
\(640\) 1.24698 1.24698
\(641\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(642\) 1.60388 1.60388
\(643\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(644\) 0 0
\(645\) −0.692021 −0.692021
\(646\) 0 0
\(647\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(648\) −1.00000 −1.00000
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) −0.356896 −0.356896
\(655\) 1.55496 1.55496
\(656\) 0 0
\(657\) −1.00000 −1.00000
\(658\) 0 0
\(659\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 4.04892 4.04892
\(665\) 0 0
\(666\) −2.60388 −2.60388
\(667\) 0 0
\(668\) 2.80194 2.80194
\(669\) 0.801938 0.801938
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 0.445042 0.445042
\(676\) 2.24698 2.24698
\(677\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 3.24698 3.24698
\(685\) 0 0
\(686\) 0 0
\(687\) −0.554958 −0.554958
\(688\) 2.24698 2.24698
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) −0.445042 −0.445042
\(697\) 0 0
\(698\) 0 0
\(699\) 0.801938 0.801938
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 3.24698 3.24698
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) −2.24698 −2.24698
\(711\) −1.00000 −1.00000
\(712\) 1.00000 1.00000
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 2.80194 2.80194
\(717\) 0 0
\(718\) −2.24698 −2.24698
\(719\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(720\) −1.80194 −1.80194
\(721\) 0 0
\(722\) −4.04892 −4.04892
\(723\) 0 0
\(724\) 0 0
\(725\) −0.246980 −0.246980
\(726\) 0.801938 0.801938
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −2.80194 −2.80194
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) −2.24698 −2.24698
\(735\) −0.554958 −0.554958
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(740\) −5.04892 −5.04892
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −2.24698 −2.24698
\(747\) 1.44504 1.44504
\(748\) 0 0
\(749\) 0 0
\(750\) −0.445042 −0.445042
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 0.801938 0.801938
\(754\) 0 0
\(755\) −0.554958 −0.554958
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0.801938 0.801938
\(759\) 0 0
\(760\) 5.04892 5.04892
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −1.00000 −1.00000
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0.801938 0.801938
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 1.80194 1.80194
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −0.356896 −0.356896
\(784\) 1.80194 1.80194
\(785\) −0.554958 −0.554958
\(786\) 1.00000 1.00000
\(787\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(788\) 0 0
\(789\) −0.554958 −0.554958
\(790\) −2.80194 −2.80194
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) −4.04892 −4.04892
\(797\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −0.554958 −0.554958
\(801\) 0.356896 0.356896
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 4.04892 4.04892
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) −1.00000 −1.00000
\(811\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(812\) 0 0
\(813\) 0.198062 0.198062
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −2.24698 −2.24698
\(818\) 0.801938 0.801938
\(819\) 0 0
\(820\) 0 0
\(821\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 1.00000 1.00000
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(830\) 4.04892 4.04892
\(831\) 0.801938 0.801938
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 1.55496 1.55496
\(836\) 0 0
\(837\) 0 0
\(838\) 0.801938 0.801938
\(839\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(840\) 0 0
\(841\) −0.801938 −0.801938
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 1.24698 1.24698
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) −1.00000 −1.00000
\(853\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(854\) 0 0
\(855\) 1.80194 1.80194
\(856\) −4.49396 −4.49396
\(857\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 3.49396 3.49396
\(861\) 0 0
\(862\) 3.24698 3.24698
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) −0.801938 −0.801938
\(865\) 0 0
\(866\) 0 0
\(867\) −0.445042 −0.445042
\(868\) 0 0
\(869\) 0 0
\(870\) −0.445042 −0.445042
\(871\) 0 0
\(872\) 1.00000 1.00000
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) −1.24698 −1.24698
\(877\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(878\) 0 0
\(879\) −0.890084 −0.890084
\(880\) 0 0
\(881\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(882\) 1.44504 1.44504
\(883\) 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 \(\pi\)
\(888\) −1.80194 −1.80194
\(889\) 0 0
\(890\) 1.00000 1.00000
\(891\) 0 0
\(892\) −4.04892 −4.04892
\(893\) 0 0
\(894\) 0 0
\(895\) 1.55496 1.55496
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −1.00000 −1.00000
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) −0.356896 −0.356896
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) 1.44504 1.44504
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 1.44504 1.44504
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 2.80194 2.80194
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −1.00000 −1.00000
\(926\) 3.24698 3.24698
\(927\) 0.356896 0.356896
\(928\) 0.445042 0.445042
\(929\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(930\) 0 0
\(931\) −1.80194 −1.80194
\(932\) −4.04892 −4.04892
\(933\) 0.801938 0.801938
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 0.801938 0.801938
\(940\) 0 0
\(941\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(942\) −0.356896 −0.356896
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(948\) −1.24698 −1.24698
\(949\) 0 0
\(950\) 1.80194 1.80194
\(951\) 0 0
\(952\) 0 0
\(953\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(954\) 0 0
\(955\) −0.554958 −0.554958
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 1.00000 1.00000
\(962\) 0 0
\(963\) −1.60388 −1.60388
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) −2.24698 −2.24698
\(969\) 0 0
\(970\) 0 0
\(971\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(972\) −2.24698 −2.24698
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 2.80194 2.80194
\(981\) 0.356896 0.356896
\(982\) 0 0
\(983\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −2.24698 −2.24698
\(996\) 1.80194 1.80194
\(997\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(998\) 0.801938 0.801938
\(999\) −1.44504 −1.44504
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 \(a_p\))