Properties

Label 71.1.b.a.70.2
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 70.2
Root \(0.445042\)
Character \(\chi\) = 71.70

$q$-expansion

\(f(q)\) \(=\) \(q-0.445042 q^{2} +1.24698 q^{3} -0.801938 q^{4} -1.80194 q^{5} -0.554958 q^{6} +0.801938 q^{8} +0.554958 q^{9} +O(q^{10})\) \(q-0.445042 q^{2} +1.24698 q^{3} -0.801938 q^{4} -1.80194 q^{5} -0.554958 q^{6} +0.801938 q^{8} +0.554958 q^{9} +0.801938 q^{10} -1.00000 q^{12} -2.24698 q^{15} +0.445042 q^{16} -0.246980 q^{18} -0.445042 q^{19} +1.44504 q^{20} +1.00000 q^{24} +2.24698 q^{25} -0.554958 q^{27} +1.24698 q^{29} +1.00000 q^{30} -1.00000 q^{32} -0.445042 q^{36} -0.445042 q^{37} +0.198062 q^{38} -1.44504 q^{40} -1.80194 q^{43} -1.00000 q^{45} +0.554958 q^{48} +1.00000 q^{49} -1.00000 q^{50} +0.246980 q^{54} -0.554958 q^{57} -0.554958 q^{58} +1.80194 q^{60} +1.00000 q^{71} +0.445042 q^{72} -1.80194 q^{73} +0.198062 q^{74} +2.80194 q^{75} +0.356896 q^{76} -1.80194 q^{79} -0.801938 q^{80} -1.24698 q^{81} -0.445042 q^{83} +0.801938 q^{86} +1.55496 q^{87} +1.24698 q^{89} +0.445042 q^{90} +0.801938 q^{95} -1.24698 q^{96} -0.445042 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - q^{2} - q^{3} + 2q^{4} - q^{5} - 2q^{6} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 3q - q^{2} - q^{3} + 2q^{4} - q^{5} - 2q^{6} - 2q^{8} + 2q^{9} - 2q^{10} - 3q^{12} - 2q^{15} + q^{16} + 4q^{18} - q^{19} + 4q^{20} + 3q^{24} + 2q^{25} - 2q^{27} - q^{29} + 3q^{30} - 3q^{32} - q^{36} - q^{37} + 5q^{38} - 4q^{40} - q^{43} - 3q^{45} + 2q^{48} + 3q^{49} - 3q^{50} - 4q^{54} - 2q^{57} - 2q^{58} + q^{60} + 3q^{71} + q^{72} - q^{73} + 5q^{74} + 4q^{75} - 3q^{76} - q^{79} + 2q^{80} + q^{81} - q^{83} - 2q^{86} + 5q^{87} - q^{89} + q^{90} - 2q^{95} + q^{96} - q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(3\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(4\) −0.801938 −0.801938
\(5\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(6\) −0.554958 −0.554958
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0.801938 0.801938
\(9\) 0.554958 0.554958
\(10\) 0.801938 0.801938
\(11\) 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
\(15\) −2.24698 −2.24698
\(16\) 0.445042 0.445042
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) −0.246980 −0.246980
\(19\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(20\) 1.44504 1.44504
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 1.00000 1.00000
\(25\) 2.24698 2.24698
\(26\) 0 0
\(27\) −0.554958 −0.554958
\(28\) 0 0
\(29\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(30\) 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
\(36\) −0.445042 −0.445042
\(37\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(38\) 0.198062 0.198062
\(39\) 0 0
\(40\) −1.44504 −1.44504
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(44\) 0 0
\(45\) −1.00000 −1.00000
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0.554958 0.554958
\(49\) 1.00000 1.00000
\(50\) −1.00000 −1.00000
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0.246980 0.246980
\(55\) 0 0
\(56\) 0 0
\(57\) −0.554958 −0.554958
\(58\) −0.554958 −0.554958
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 1.80194 1.80194
\(61\) 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
\(72\) 0.445042 0.445042
\(73\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(74\) 0.198062 0.198062
\(75\) 2.80194 2.80194
\(76\) 0.356896 0.356896
\(77\) 0 0
\(78\) 0 0
\(79\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(80\) −0.801938 −0.801938
\(81\) −1.24698 −1.24698
\(82\) 0 0
\(83\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0.801938 0.801938
\(87\) 1.55496 1.55496
\(88\) 0 0
\(89\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(90\) 0.445042 0.445042
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0.801938 0.801938
\(96\) −1.24698 −1.24698
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) −0.445042 −0.445042
\(99\) 0 0
\(100\) −1.80194 −1.80194
\(101\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(102\) 0 0
\(103\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(108\) 0.445042 0.445042
\(109\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(110\) 0 0
\(111\) −0.554958 −0.554958
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0.246980 0.246980
\(115\) 0 0
\(116\) −1.00000 −1.00000
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) −1.80194 −1.80194
\(121\) 1.00000 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −2.24698 −2.24698
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 1.00000 1.00000
\(129\) −2.24698 −2.24698
\(130\) 0 0
\(131\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(132\) 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
\(142\) −0.445042 −0.445042
\(143\) 0 0
\(144\) 0.246980 0.246980
\(145\) −2.24698 −2.24698
\(146\) 0.801938 0.801938
\(147\) 1.24698 1.24698
\(148\) 0.356896 0.356896
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) −1.24698 −1.24698
\(151\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(152\) −0.356896 −0.356896
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(158\) 0.801938 0.801938
\(159\) 0 0
\(160\) 1.80194 1.80194
\(161\) 0 0
\(162\) 0.554958 0.554958
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0.198062 0.198062
\(167\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(168\) 0 0
\(169\) 1.00000 1.00000
\(170\) 0 0
\(171\) −0.246980 −0.246980
\(172\) 1.44504 1.44504
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) −0.692021 −0.692021
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) −0.554958 −0.554958
\(179\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(180\) 0.801938 0.801938
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0.801938 0.801938
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) −0.356896 −0.356896
\(191\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −0.801938 −0.801938
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(200\) 1.80194 1.80194
\(201\) 0 0
\(202\) 0.198062 0.198062
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) −0.554958 −0.554958
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 1.24698 1.24698
\(214\) −0.890084 −0.890084
\(215\) 3.24698 3.24698
\(216\) −0.445042 −0.445042
\(217\) 0 0
\(218\) −0.554958 −0.554958
\(219\) −2.24698 −2.24698
\(220\) 0 0
\(221\) 0 0
\(222\) 0.246980 0.246980
\(223\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(224\) 0 0
\(225\) 1.24698 1.24698
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0.445042 0.445042
\(229\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 1.00000 1.00000
\(233\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −2.24698 −2.24698
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) −1.00000 −1.00000
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) −0.445042 −0.445042
\(243\) −1.00000 −1.00000
\(244\) 0 0
\(245\) −1.80194 −1.80194
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −0.554958 −0.554958
\(250\) 1.00000 1.00000
\(251\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −0.445042 −0.445042
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 1.00000 1.00000
\(259\) 0 0
\(260\) 0 0
\(261\) 0.692021 0.692021
\(262\) 0.801938 0.801938
\(263\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 1.55496 1.55496
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) −0.445042 −0.445042
\(271\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(278\) 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 \(\pi\)
\(284\) −0.801938 −0.801938
\(285\) 1.00000 1.00000
\(286\) 0 0
\(287\) 0 0
\(288\) −0.554958 −0.554958
\(289\) 1.00000 1.00000
\(290\) 1.00000 1.00000
\(291\) 0 0
\(292\) 1.44504 1.44504
\(293\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(294\) −0.554958 −0.554958
\(295\) 0 0
\(296\) −0.356896 −0.356896
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) −2.24698 −2.24698
\(301\) 0 0
\(302\) −0.554958 −0.554958
\(303\) −0.554958 −0.554958
\(304\) −0.198062 −0.198062
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 1.55496 1.55496
\(310\) 0 0
\(311\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(312\) 0 0
\(313\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(314\) −0.554958 −0.554958
\(315\) 0 0
\(316\) 1.44504 1.44504
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 2.49396 2.49396
\(322\) 0 0
\(323\) 0 0
\(324\) 1.00000 1.00000
\(325\) 0 0
\(326\) 0 0
\(327\) 1.55496 1.55496
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0.356896 0.356896
\(333\) −0.246980 −0.246980
\(334\) 0.801938 0.801938
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) −0.445042 −0.445042
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0.109916 0.109916
\(343\) 0 0
\(344\) −1.44504 −1.44504
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) −1.24698 −1.24698
\(349\) 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
\(355\) −1.80194 −1.80194
\(356\) −1.00000 −1.00000
\(357\) 0 0
\(358\) 0.801938 0.801938
\(359\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(360\) −0.801938 −0.801938
\(361\) −0.801938 −0.801938
\(362\) 0 0
\(363\) 1.24698 1.24698
\(364\) 0 0
\(365\) 3.24698 3.24698
\(366\) 0 0
\(367\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) −0.356896 −0.356896
\(371\) 0 0
\(372\) 0 0
\(373\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(374\) 0 0
\(375\) −2.80194 −2.80194
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(380\) −0.643104 −0.643104
\(381\) 0 0
\(382\) −0.554958 −0.554958
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 1.24698 1.24698
\(385\) 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
\(392\) 0.801938 0.801938
\(393\) −2.24698 −2.24698
\(394\) 0 0
\(395\) 3.24698 3.24698
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0.198062 0.198062
\(399\) 0 0
\(400\) 1.00000 1.00000
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0.356896 0.356896
\(405\) 2.24698 2.24698
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −1.00000 −1.00000
\(413\) 0 0
\(414\) 0 0
\(415\) 0.801938 0.801938
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) −0.554958 −0.554958
\(427\) 0 0
\(428\) −1.60388 −1.60388
\(429\) 0 0
\(430\) −1.44504 −1.44504
\(431\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(432\) −0.246980 −0.246980
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) −2.80194 −2.80194
\(436\) −1.00000 −1.00000
\(437\) 0 0
\(438\) 1.00000 1.00000
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 0.554958 0.554958
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0.445042 0.445042
\(445\) −2.24698 −2.24698
\(446\) 0.198062 0.198062
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) −0.554958 −0.554958
\(451\) 0 0
\(452\) 0 0
\(453\) 1.55496 1.55496
\(454\) 0 0
\(455\) 0 0
\(456\) −0.445042 −0.445042
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0.801938 0.801938
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(464\) 0.554958 0.554958
\(465\) 0 0
\(466\) 0.198062 0.198062
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 1.55496 1.55496
\(472\) 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 \(\pi\)
\(480\) 2.24698 2.24698
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −0.801938 −0.801938
\(485\) 0 0
\(486\) 0.445042 0.445042
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0.801938 0.801938
\(491\) 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
\(498\) 0.246980 0.246980
\(499\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(500\) 1.80194 1.80194
\(501\) −2.24698 −2.24698
\(502\) 0.198062 0.198062
\(503\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(504\) 0 0
\(505\) 0.801938 0.801938
\(506\) 0 0
\(507\) 1.24698 1.24698
\(508\) 0 0
\(509\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(510\) 0 0
\(511\) 0 0
\(512\) −0.801938 −0.801938
\(513\) 0.246980 0.246980
\(514\) 0 0
\(515\) −2.24698 −2.24698
\(516\) 1.80194 1.80194
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(522\) −0.307979 −0.307979
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 1.44504 1.44504
\(525\) 0 0
\(526\) 0.801938 0.801938
\(527\) 0 0
\(528\) 0 0
\(529\) 1.00000 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) −0.692021 −0.692021
\(535\) −3.60388 −3.60388
\(536\) 0 0
\(537\) −2.24698 −2.24698
\(538\) 0 0
\(539\) 0 0
\(540\) −0.801938 −0.801938
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) −0.554958 −0.554958
\(543\) 0 0
\(544\) 0 0
\(545\) −2.24698 −2.24698
\(546\) 0 0
\(547\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −0.554958 −0.554958
\(552\) 0 0
\(553\) 0 0
\(554\) 0.198062 0.198062
\(555\) 1.00000 1.00000
\(556\) 0 0
\(557\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(558\) 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
\(568\) 0.801938 0.801938
\(569\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(570\) −0.445042 −0.445042
\(571\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(572\) 0 0
\(573\) 1.55496 1.55496
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(578\) −0.445042 −0.445042
\(579\) 0 0
\(580\) 1.80194 1.80194
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) −1.44504 −1.44504
\(585\) 0 0
\(586\) −0.890084 −0.890084
\(587\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(588\) −1.00000 −1.00000
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −0.198062 −0.198062
\(593\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −0.554958 −0.554958
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 2.24698 2.24698
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −1.00000 −1.00000
\(605\) −1.80194 −1.80194
\(606\) 0.246980 0.246980
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0.445042 0.445042
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(618\) −0.692021 −0.692021
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0.198062 0.198062
\(623\) 0 0
\(624\) 0 0
\(625\) 1.80194 1.80194
\(626\) 0.198062 0.198062
\(627\) 0 0
\(628\) −1.00000 −1.00000
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) −1.44504 −1.44504
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0.554958 0.554958
\(640\) −1.80194 −1.80194
\(641\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(642\) −1.10992 −1.10992
\(643\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(644\) 0 0
\(645\) 4.04892 4.04892
\(646\) 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 \(\pi\)
\(654\) −0.692021 −0.692021
\(655\) 3.24698 3.24698
\(656\) 0 0
\(657\) −1.00000 −1.00000
\(658\) 0 0
\(659\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) −0.356896 −0.356896
\(665\) 0 0
\(666\) 0.109916 0.109916
\(667\) 0 0
\(668\) 1.44504 1.44504
\(669\) −0.554958 −0.554958
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) −1.24698 −1.24698
\(676\) −0.801938 −0.801938
\(677\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0.198062 0.198062
\(685\) 0 0
\(686\) 0 0
\(687\) −2.24698 −2.24698
\(688\) −0.801938 −0.801938
\(689\) 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
\(696\) 1.24698 1.24698
\(697\) 0 0
\(698\) 0 0
\(699\) −0.554958 −0.554958
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0.198062 0.198062
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0.801938 0.801938
\(711\) −1.00000 −1.00000
\(712\) 1.00000 1.00000
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 1.44504 1.44504
\(717\) 0 0
\(718\) 0.801938 0.801938
\(719\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(720\) −0.445042 −0.445042
\(721\) 0 0
\(722\) 0.356896 0.356896
\(723\) 0 0
\(724\) 0 0
\(725\) 2.80194 2.80194
\(726\) −0.554958 −0.554958
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −1.44504 −1.44504
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0.801938 0.801938
\(735\) −2.24698 −2.24698
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(740\) −0.643104 −0.643104
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0.801938 0.801938
\(747\) −0.246980 −0.246980
\(748\) 0 0
\(749\) 0 0
\(750\) 1.24698 1.24698
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) −0.554958 −0.554958
\(754\) 0 0
\(755\) −2.24698 −2.24698
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) −0.554958 −0.554958
\(759\) 0 0
\(760\) 0.643104 0.643104
\(761\) 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
\(768\) −0.554958 −0.554958
\(769\) 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 \(\pi\)
\(774\) 0.445042 0.445042
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −0.692021 −0.692021
\(784\) 0.445042 0.445042
\(785\) −2.24698 −2.24698
\(786\) 1.00000 1.00000
\(787\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(788\) 0 0
\(789\) −2.24698 −2.24698
\(790\) −1.44504 −1.44504
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0.356896 0.356896
\(797\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −2.24698 −2.24698
\(801\) 0.692021 0.692021
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) −0.356896 −0.356896
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) −1.00000 −1.00000
\(811\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(812\) 0 0
\(813\) 1.55496 1.55496
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0.801938 0.801938
\(818\) −0.554958 −0.554958
\(819\) 0 0
\(820\) 0 0
\(821\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(822\) 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
\(829\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(830\) −0.356896 −0.356896
\(831\) −0.554958 −0.554958
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 3.24698 3.24698
\(836\) 0 0
\(837\) 0 0
\(838\) −0.554958 −0.554958
\(839\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(840\) 0 0
\(841\) 0.554958 0.554958
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −1.80194 −1.80194
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) −1.00000 −1.00000
\(853\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(854\) 0 0
\(855\) 0.445042 0.445042
\(856\) 1.60388 1.60388
\(857\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) −2.60388 −2.60388
\(861\) 0 0
\(862\) 0.198062 0.198062
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0.554958 0.554958
\(865\) 0 0
\(866\) 0 0
\(867\) 1.24698 1.24698
\(868\) 0 0
\(869\) 0 0
\(870\) 1.24698 1.24698
\(871\) 0 0
\(872\) 1.00000 1.00000
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 1.80194 1.80194
\(877\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(878\) 0 0
\(879\) 2.49396 2.49396
\(880\) 0 0
\(881\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(882\) −0.246980 −0.246980
\(883\) 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 \(\pi\)
\(888\) −0.445042 −0.445042
\(889\) 0 0
\(890\) 1.00000 1.00000
\(891\) 0 0
\(892\) 0.356896 0.356896
\(893\) 0 0
\(894\) 0 0
\(895\) 3.24698 3.24698
\(896\) 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
\(906\) −0.692021 −0.692021
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) −0.246980 −0.246980
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) −0.246980 −0.246980
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 1.44504 1.44504
\(917\) 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
\(926\) 0.198062 0.198062
\(927\) 0.692021 0.692021
\(928\) −1.24698 −1.24698
\(929\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(930\) 0 0
\(931\) −0.445042 −0.445042
\(932\) 0.356896 0.356896
\(933\) −0.554958 −0.554958
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) −0.554958 −0.554958
\(940\) 0 0
\(941\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(942\) −0.692021 −0.692021
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(948\) 1.80194 1.80194
\(949\) 0 0
\(950\) 0.445042 0.445042
\(951\) 0 0
\(952\) 0 0
\(953\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(954\) 0 0
\(955\) −2.24698 −2.24698
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 1.00000 1.00000
\(962\) 0 0
\(963\) 1.10992 1.10992
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 0.801938 0.801938
\(969\) 0 0
\(970\) 0 0
\(971\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(972\) 0.801938 0.801938
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 1.44504 1.44504
\(981\) 0.692021 0.692021
\(982\) 0 0
\(983\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(984\) 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
\(995\) 0.801938 0.801938
\(996\) 0.445042 0.445042
\(997\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(998\) −0.554958 −0.554958
\(999\) 0.246980 0.246980
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 \(a_p\))