Properties

Label 15.3.d.b.14.1
Level 15
Weight 3
Character 15.14
Self dual Yes
Analytic conductor 0.409
Analytic rank 0
Dimension 1
CM disc. -15
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 15 = 3 \cdot 5 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 15.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(0.40872039654\)
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 14.1
Root \(0\)
Character \(\chi\) = 15.14

$q$-expansion

\(f(q)\) \(=\) \(q\)\(+1.00000 q^{2}\) \(-3.00000 q^{3}\) \(-3.00000 q^{4}\) \(+5.00000 q^{5}\) \(-3.00000 q^{6}\) \(-7.00000 q^{8}\) \(+9.00000 q^{9}\) \(+O(q^{10})\) \(q\)\(+1.00000 q^{2}\) \(-3.00000 q^{3}\) \(-3.00000 q^{4}\) \(+5.00000 q^{5}\) \(-3.00000 q^{6}\) \(-7.00000 q^{8}\) \(+9.00000 q^{9}\) \(+5.00000 q^{10}\) \(+9.00000 q^{12}\) \(-15.0000 q^{15}\) \(+5.00000 q^{16}\) \(-14.0000 q^{17}\) \(+9.00000 q^{18}\) \(-22.0000 q^{19}\) \(-15.0000 q^{20}\) \(+34.0000 q^{23}\) \(+21.0000 q^{24}\) \(+25.0000 q^{25}\) \(-27.0000 q^{27}\) \(-15.0000 q^{30}\) \(+2.00000 q^{31}\) \(+33.0000 q^{32}\) \(-14.0000 q^{34}\) \(-27.0000 q^{36}\) \(-22.0000 q^{38}\) \(-35.0000 q^{40}\) \(+45.0000 q^{45}\) \(+34.0000 q^{46}\) \(-14.0000 q^{47}\) \(-15.0000 q^{48}\) \(+49.0000 q^{49}\) \(+25.0000 q^{50}\) \(+42.0000 q^{51}\) \(-86.0000 q^{53}\) \(-27.0000 q^{54}\) \(+66.0000 q^{57}\) \(+45.0000 q^{60}\) \(-118.000 q^{61}\) \(+2.00000 q^{62}\) \(+13.0000 q^{64}\) \(+42.0000 q^{68}\) \(-102.000 q^{69}\) \(-63.0000 q^{72}\) \(-75.0000 q^{75}\) \(+66.0000 q^{76}\) \(+98.0000 q^{79}\) \(+25.0000 q^{80}\) \(+81.0000 q^{81}\) \(+154.000 q^{83}\) \(-70.0000 q^{85}\) \(+45.0000 q^{90}\) \(-102.000 q^{92}\) \(-6.00000 q^{93}\) \(-14.0000 q^{94}\) \(-110.000 q^{95}\) \(-99.0000 q^{96}\) \(+49.0000 q^{98}\) \(+O(q^{100})\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/15\mathbb{Z}\right)^\times\).

\(n\) \(7\) \(11\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.500000 0.250000 0.968246i \(-0.419569\pi\)
0.250000 + 0.968246i \(0.419569\pi\)
\(3\) −3.00000 −1.00000
\(4\) −3.00000 −0.750000
\(5\) 5.00000 1.00000
\(6\) −3.00000 −0.500000
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) −7.00000 −0.875000
\(9\) 9.00000 1.00000
\(10\) 5.00000 0.500000
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 9.00000 0.750000
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) −15.0000 −1.00000
\(16\) 5.00000 0.312500
\(17\) −14.0000 −0.823529 −0.411765 0.911290i \(-0.635087\pi\)
−0.411765 + 0.911290i \(0.635087\pi\)
\(18\) 9.00000 0.500000
\(19\) −22.0000 −1.15789 −0.578947 0.815365i \(-0.696536\pi\)
−0.578947 + 0.815365i \(0.696536\pi\)
\(20\) −15.0000 −0.750000
\(21\) 0 0
\(22\) 0 0
\(23\) 34.0000 1.47826 0.739130 0.673562i \(-0.235237\pi\)
0.739130 + 0.673562i \(0.235237\pi\)
\(24\) 21.0000 0.875000
\(25\) 25.0000 1.00000
\(26\) 0 0
\(27\) −27.0000 −1.00000
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) −15.0000 −0.500000
\(31\) 2.00000 0.0645161 0.0322581 0.999480i \(-0.489730\pi\)
0.0322581 + 0.999480i \(0.489730\pi\)
\(32\) 33.0000 1.03125
\(33\) 0 0
\(34\) −14.0000 −0.411765
\(35\) 0 0
\(36\) −27.0000 −0.750000
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) −22.0000 −0.578947
\(39\) 0 0
\(40\) −35.0000 −0.875000
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 45.0000 1.00000
\(46\) 34.0000 0.739130
\(47\) −14.0000 −0.297872 −0.148936 0.988847i \(-0.547585\pi\)
−0.148936 + 0.988847i \(0.547585\pi\)
\(48\) −15.0000 −0.312500
\(49\) 49.0000 1.00000
\(50\) 25.0000 0.500000
\(51\) 42.0000 0.823529
\(52\) 0 0
\(53\) −86.0000 −1.62264 −0.811321 0.584601i \(-0.801251\pi\)
−0.811321 + 0.584601i \(0.801251\pi\)
\(54\) −27.0000 −0.500000
\(55\) 0 0
\(56\) 0 0
\(57\) 66.0000 1.15789
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 45.0000 0.750000
\(61\) −118.000 −1.93443 −0.967213 0.253966i \(-0.918265\pi\)
−0.967213 + 0.253966i \(0.918265\pi\)
\(62\) 2.00000 0.0322581
\(63\) 0 0
\(64\) 13.0000 0.203125
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 42.0000 0.617647
\(69\) −102.000 −1.47826
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) −63.0000 −0.875000
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) −75.0000 −1.00000
\(76\) 66.0000 0.868421
\(77\) 0 0
\(78\) 0 0
\(79\) 98.0000 1.24051 0.620253 0.784402i \(-0.287030\pi\)
0.620253 + 0.784402i \(0.287030\pi\)
\(80\) 25.0000 0.312500
\(81\) 81.0000 1.00000
\(82\) 0 0
\(83\) 154.000 1.85542 0.927711 0.373300i \(-0.121774\pi\)
0.927711 + 0.373300i \(0.121774\pi\)
\(84\) 0 0
\(85\) −70.0000 −0.823529
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 45.0000 0.500000
\(91\) 0 0
\(92\) −102.000 −1.10870
\(93\) −6.00000 −0.0645161
\(94\) −14.0000 −0.148936
\(95\) −110.000 −1.15789
\(96\) −99.0000 −1.03125
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 49.0000 0.500000
\(99\) 0 0
\(100\) −75.0000 −0.750000
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 42.0000 0.411765
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −86.0000 −0.811321
\(107\) 106.000 0.990654 0.495327 0.868707i \(-0.335048\pi\)
0.495327 + 0.868707i \(0.335048\pi\)
\(108\) 81.0000 0.750000
\(109\) −22.0000 −0.201835 −0.100917 0.994895i \(-0.532178\pi\)
−0.100917 + 0.994895i \(0.532178\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −206.000 −1.82301 −0.911504 0.411290i \(-0.865078\pi\)
−0.911504 + 0.411290i \(0.865078\pi\)
\(114\) 66.0000 0.578947
\(115\) 170.000 1.47826
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 105.000 0.875000
\(121\) 121.000 1.00000
\(122\) −118.000 −0.967213
\(123\) 0 0
\(124\) −6.00000 −0.0483871
\(125\) 125.000 1.00000
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) −119.000 −0.929688
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −135.000 −1.00000
\(136\) 98.0000 0.720588
\(137\) 226.000 1.64964 0.824818 0.565399i \(-0.191278\pi\)
0.824818 + 0.565399i \(0.191278\pi\)
\(138\) −102.000 −0.739130
\(139\) −262.000 −1.88489 −0.942446 0.334358i \(-0.891480\pi\)
−0.942446 + 0.334358i \(0.891480\pi\)
\(140\) 0 0
\(141\) 42.0000 0.297872
\(142\) 0 0
\(143\) 0 0
\(144\) 45.0000 0.312500
\(145\) 0 0
\(146\) 0 0
\(147\) −147.000 −1.00000
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) −75.0000 −0.500000
\(151\) −238.000 −1.57616 −0.788079 0.615574i \(-0.788924\pi\)
−0.788079 + 0.615574i \(0.788924\pi\)
\(152\) 154.000 1.01316
\(153\) −126.000 −0.823529
\(154\) 0 0
\(155\) 10.0000 0.0645161
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 98.0000 0.620253
\(159\) 258.000 1.62264
\(160\) 165.000 1.03125
\(161\) 0 0
\(162\) 81.0000 0.500000
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 154.000 0.927711
\(167\) −254.000 −1.52096 −0.760479 0.649362i \(-0.775036\pi\)
−0.760479 + 0.649362i \(0.775036\pi\)
\(168\) 0 0
\(169\) 169.000 1.00000
\(170\) −70.0000 −0.411765
\(171\) −198.000 −1.15789
\(172\) 0 0
\(173\) 154.000 0.890173 0.445087 0.895487i \(-0.353173\pi\)
0.445087 + 0.895487i \(0.353173\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) −135.000 −0.750000
\(181\) 122.000 0.674033 0.337017 0.941499i \(-0.390582\pi\)
0.337017 + 0.941499i \(0.390582\pi\)
\(182\) 0 0
\(183\) 354.000 1.93443
\(184\) −238.000 −1.29348
\(185\) 0 0
\(186\) −6.00000 −0.0322581
\(187\) 0 0
\(188\) 42.0000 0.223404
\(189\) 0 0
\(190\) −110.000 −0.578947
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) −39.0000 −0.203125
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −147.000 −0.750000
\(197\) −374.000 −1.89848 −0.949239 0.314557i \(-0.898144\pi\)
−0.949239 + 0.314557i \(0.898144\pi\)
\(198\) 0 0
\(199\) −142.000 −0.713568 −0.356784 0.934187i \(-0.616127\pi\)
−0.356784 + 0.934187i \(0.616127\pi\)
\(200\) −175.000 −0.875000
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) −126.000 −0.617647
\(205\) 0 0
\(206\) 0 0
\(207\) 306.000 1.47826
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 362.000 1.71564 0.857820 0.513950i \(-0.171818\pi\)
0.857820 + 0.513950i \(0.171818\pi\)
\(212\) 258.000 1.21698
\(213\) 0 0
\(214\) 106.000 0.495327
\(215\) 0 0
\(216\) 189.000 0.875000
\(217\) 0 0
\(218\) −22.0000 −0.100917
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 225.000 1.00000
\(226\) −206.000 −0.911504
\(227\) −134.000 −0.590308 −0.295154 0.955450i \(-0.595371\pi\)
−0.295154 + 0.955450i \(0.595371\pi\)
\(228\) −198.000 −0.868421
\(229\) 218.000 0.951965 0.475983 0.879455i \(-0.342093\pi\)
0.475983 + 0.879455i \(0.342093\pi\)
\(230\) 170.000 0.739130
\(231\) 0 0
\(232\) 0 0
\(233\) 34.0000 0.145923 0.0729614 0.997335i \(-0.476755\pi\)
0.0729614 + 0.997335i \(0.476755\pi\)
\(234\) 0 0
\(235\) −70.0000 −0.297872
\(236\) 0 0
\(237\) −294.000 −1.24051
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) −75.0000 −0.312500
\(241\) −478.000 −1.98340 −0.991701 0.128564i \(-0.958963\pi\)
−0.991701 + 0.128564i \(0.958963\pi\)
\(242\) 121.000 0.500000
\(243\) −243.000 −1.00000
\(244\) 354.000 1.45082
\(245\) 245.000 1.00000
\(246\) 0 0
\(247\) 0 0
\(248\) −14.0000 −0.0564516
\(249\) −462.000 −1.85542
\(250\) 125.000 0.500000
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 210.000 0.823529
\(256\) −171.000 −0.667969
\(257\) 466.000 1.81323 0.906615 0.421959i \(-0.138657\pi\)
0.906615 + 0.421959i \(0.138657\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −446.000 −1.69582 −0.847909 0.530142i \(-0.822139\pi\)
−0.847909 + 0.530142i \(0.822139\pi\)
\(264\) 0 0
\(265\) −430.000 −1.62264
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) −135.000 −0.500000
\(271\) 482.000 1.77860 0.889299 0.457326i \(-0.151193\pi\)
0.889299 + 0.457326i \(0.151193\pi\)
\(272\) −70.0000 −0.257353
\(273\) 0 0
\(274\) 226.000 0.824818
\(275\) 0 0
\(276\) 306.000 1.10870
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) −262.000 −0.942446
\(279\) 18.0000 0.0645161
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 42.0000 0.148936
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 330.000 1.15789
\(286\) 0 0
\(287\) 0 0
\(288\) 297.000 1.03125
\(289\) −93.0000 −0.321799
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 394.000 1.34471 0.672355 0.740229i \(-0.265283\pi\)
0.672355 + 0.740229i \(0.265283\pi\)
\(294\) −147.000 −0.500000
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 225.000 0.750000
\(301\) 0 0
\(302\) −238.000 −0.788079
\(303\) 0 0
\(304\) −110.000 −0.361842
\(305\) −590.000 −1.93443
\(306\) −126.000 −0.411765
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 10.0000 0.0322581
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −294.000 −0.930380
\(317\) −134.000 −0.422713 −0.211356 0.977409i \(-0.567788\pi\)
−0.211356 + 0.977409i \(0.567788\pi\)
\(318\) 258.000 0.811321
\(319\) 0 0
\(320\) 65.0000 0.203125
\(321\) −318.000 −0.990654
\(322\) 0 0
\(323\) 308.000 0.953560
\(324\) −243.000 −0.750000
\(325\) 0 0
\(326\) 0 0
\(327\) 66.0000 0.201835
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 122.000 0.368580 0.184290 0.982872i \(-0.441001\pi\)
0.184290 + 0.982872i \(0.441001\pi\)
\(332\) −462.000 −1.39157
\(333\) 0 0
\(334\) −254.000 −0.760479
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 169.000 0.500000
\(339\) 618.000 1.82301
\(340\) 210.000 0.617647
\(341\) 0 0
\(342\) −198.000 −0.578947
\(343\) 0 0
\(344\) 0 0
\(345\) −510.000 −1.47826
\(346\) 154.000 0.445087
\(347\) 586.000 1.68876 0.844380 0.535744i \(-0.179969\pi\)
0.844380 + 0.535744i \(0.179969\pi\)
\(348\) 0 0
\(349\) 458.000 1.31232 0.656160 0.754621i \(-0.272179\pi\)
0.656160 + 0.754621i \(0.272179\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 274.000 0.776204 0.388102 0.921616i \(-0.373131\pi\)
0.388102 + 0.921616i \(0.373131\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) −315.000 −0.875000
\(361\) 123.000 0.340720
\(362\) 122.000 0.337017
\(363\) −363.000 −1.00000
\(364\) 0 0
\(365\) 0 0
\(366\) 354.000 0.967213
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 170.000 0.461957
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 18.0000 0.0483871
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) −375.000 −1.00000
\(376\) 98.0000 0.260638
\(377\) 0 0
\(378\) 0 0
\(379\) −742.000 −1.95778 −0.978892 0.204379i \(-0.934482\pi\)
−0.978892 + 0.204379i \(0.934482\pi\)
\(380\) 330.000 0.868421
\(381\) 0 0
\(382\) 0 0
\(383\) −686.000 −1.79112 −0.895561 0.444938i \(-0.853226\pi\)
−0.895561 + 0.444938i \(0.853226\pi\)
\(384\) 357.000 0.929688
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) −476.000 −1.21739
\(392\) −343.000 −0.875000
\(393\) 0 0
\(394\) −374.000 −0.949239
\(395\) 490.000 1.24051
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) −142.000 −0.356784
\(399\) 0 0
\(400\) 125.000 0.312500
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 405.000 1.00000
\(406\) 0 0
\(407\) 0 0
\(408\) −294.000 −0.720588
\(409\) −142.000 −0.347188 −0.173594 0.984817i \(-0.555538\pi\)
−0.173594 + 0.984817i \(0.555538\pi\)
\(410\) 0 0
\(411\) −678.000 −1.64964
\(412\) 0 0
\(413\) 0 0
\(414\) 306.000 0.739130
\(415\) 770.000 1.85542
\(416\) 0 0
\(417\) 786.000 1.88489
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 602.000 1.42993 0.714964 0.699161i \(-0.246443\pi\)
0.714964 + 0.699161i \(0.246443\pi\)
\(422\) 362.000 0.857820
\(423\) −126.000 −0.297872
\(424\) 602.000 1.41981
\(425\) −350.000 −0.823529
\(426\) 0 0
\(427\) 0 0
\(428\) −318.000 −0.742991
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) −135.000 −0.312500
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 66.0000 0.151376
\(437\) −748.000 −1.71167
\(438\) 0 0
\(439\) −622.000 −1.41686 −0.708428 0.705783i \(-0.750595\pi\)
−0.708428 + 0.705783i \(0.750595\pi\)
\(440\) 0 0
\(441\) 441.000 1.00000
\(442\) 0 0
\(443\) −566.000 −1.27765 −0.638826 0.769351i \(-0.720580\pi\)
−0.638826 + 0.769351i \(0.720580\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 225.000 0.500000
\(451\) 0 0
\(452\) 618.000 1.36726
\(453\) 714.000 1.57616
\(454\) −134.000 −0.295154
\(455\) 0 0
\(456\) −462.000 −1.01316
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 218.000 0.475983
\(459\) 378.000 0.823529
\(460\) −510.000 −1.10870
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) −30.0000 −0.0645161
\(466\) 34.0000 0.0729614
\(467\) 346.000 0.740899 0.370450 0.928853i \(-0.379204\pi\)
0.370450 + 0.928853i \(0.379204\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −70.0000 −0.148936
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) −294.000 −0.620253
\(475\) −550.000 −1.15789
\(476\) 0 0
\(477\) −774.000 −1.62264
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) −495.000 −1.03125
\(481\) 0 0
\(482\) −478.000 −0.991701
\(483\) 0 0
\(484\) −363.000 −0.750000
\(485\) 0 0
\(486\) −243.000 −0.500000
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 826.000 1.69262
\(489\) 0 0
\(490\) 245.000 0.500000
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 10.0000 0.0201613
\(497\) 0 0
\(498\) −462.000 −0.927711
\(499\) 938.000 1.87976 0.939880 0.341506i \(-0.110937\pi\)
0.939880 + 0.341506i \(0.110937\pi\)
\(500\) −375.000 −0.750000
\(501\) 762.000 1.52096
\(502\) 0 0
\(503\) 994.000 1.97614 0.988072 0.153995i \(-0.0492141\pi\)
0.988072 + 0.153995i \(0.0492141\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −507.000 −1.00000
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 210.000 0.411765
\(511\) 0 0
\(512\) 305.000 0.595703
\(513\) 594.000 1.15789
\(514\) 466.000 0.906615
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −462.000 −0.890173
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −446.000 −0.847909
\(527\) −28.0000 −0.0531309
\(528\) 0 0
\(529\) 627.000 1.18526
\(530\) −430.000 −0.811321
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 530.000 0.990654
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 405.000 0.750000
\(541\) −1078.00 −1.99261 −0.996303 0.0859072i \(-0.972621\pi\)
−0.996303 + 0.0859072i \(0.972621\pi\)
\(542\) 482.000 0.889299
\(543\) −366.000 −0.674033
\(544\) −462.000 −0.849265
\(545\) −110.000 −0.201835
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) −678.000 −1.23723
\(549\) −1062.00 −1.93443
\(550\) 0 0
\(551\) 0 0
\(552\) 714.000 1.29348
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 786.000 1.41367
\(557\) −614.000 −1.10233 −0.551167 0.834395i \(-0.685817\pi\)
−0.551167 + 0.834395i \(0.685817\pi\)
\(558\) 18.0000 0.0322581
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 154.000 0.273535 0.136767 0.990603i \(-0.456329\pi\)
0.136767 + 0.990603i \(0.456329\pi\)
\(564\) −126.000 −0.223404
\(565\) −1030.00 −1.82301
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 330.000 0.578947
\(571\) −358.000 −0.626970 −0.313485 0.949593i \(-0.601497\pi\)
−0.313485 + 0.949593i \(0.601497\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 850.000 1.47826
\(576\) 117.000 0.203125
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) −93.0000 −0.160900
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 394.000 0.672355
\(587\) −854.000 −1.45486 −0.727428 0.686184i \(-0.759284\pi\)
−0.727428 + 0.686184i \(0.759284\pi\)
\(588\) 441.000 0.750000
\(589\) −44.0000 −0.0747029
\(590\) 0 0
\(591\) 1122.00 1.89848
\(592\) 0 0
\(593\) −1166.00 −1.96627 −0.983137 0.182873i \(-0.941460\pi\)
−0.983137 + 0.182873i \(0.941460\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 426.000 0.713568
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 525.000 0.875000
\(601\) 242.000 0.402662 0.201331 0.979523i \(-0.435473\pi\)
0.201331 + 0.979523i \(0.435473\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 714.000 1.18212
\(605\) 605.000 1.00000
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) −726.000 −1.19408
\(609\) 0 0
\(610\) −590.000 −0.967213
\(611\) 0 0
\(612\) 378.000 0.617647
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1186.00 1.92220 0.961102 0.276193i \(-0.0890730\pi\)
0.961102 + 0.276193i \(0.0890730\pi\)
\(618\) 0 0
\(619\) 698.000 1.12763 0.563813 0.825903i \(-0.309334\pi\)
0.563813 + 0.825903i \(0.309334\pi\)
\(620\) −30.0000 −0.0483871
\(621\) −918.000 −1.47826
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 625.000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −238.000 −0.377179 −0.188590 0.982056i \(-0.560392\pi\)
−0.188590 + 0.982056i \(0.560392\pi\)
\(632\) −686.000 −1.08544
\(633\) −1086.00 −1.71564
\(634\) −134.000 −0.211356
\(635\) 0 0
\(636\) −774.000 −1.21698
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) −595.000 −0.929688
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) −318.000 −0.495327
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 308.000 0.476780
\(647\) 706.000 1.09119 0.545595 0.838049i \(-0.316304\pi\)
0.545595 + 0.838049i \(0.316304\pi\)
\(648\) −567.000 −0.875000
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1114.00 1.70597 0.852986 0.521933i \(-0.174789\pi\)
0.852986 + 0.521933i \(0.174789\pi\)
\(654\) 66.0000 0.100917
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −838.000 −1.26778 −0.633888 0.773425i \(-0.718542\pi\)
−0.633888 + 0.773425i \(0.718542\pi\)
\(662\) 122.000 0.184290
\(663\) 0 0
\(664\) −1078.00 −1.62349
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 762.000 1.14072
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) −675.000 −1.00000
\(676\) −507.000 −0.750000
\(677\) −374.000 −0.552437 −0.276219 0.961095i \(-0.589081\pi\)
−0.276219 + 0.961095i \(0.589081\pi\)
\(678\) 618.000 0.911504
\(679\) 0 0
\(680\) 490.000 0.720588
\(681\) 402.000 0.590308
\(682\) 0 0
\(683\) −86.0000 −0.125915 −0.0629575 0.998016i \(-0.520053\pi\)
−0.0629575 + 0.998016i \(0.520053\pi\)
\(684\) 594.000 0.868421
\(685\) 1130.00 1.64964
\(686\) 0 0
\(687\) −654.000 −0.951965
\(688\) 0 0
\(689\) 0 0
\(690\) −510.000 −0.739130
\(691\) 1322.00 1.91317 0.956585 0.291455i \(-0.0941392\pi\)
0.956585 + 0.291455i \(0.0941392\pi\)
\(692\) −462.000 −0.667630
\(693\) 0 0
\(694\) 586.000 0.844380
\(695\) −1310.00 −1.88489
\(696\) 0 0
\(697\) 0 0
\(698\) 458.000 0.656160
\(699\) −102.000 −0.145923
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 210.000 0.297872
\(706\) 274.000 0.388102
\(707\) 0 0
\(708\) 0 0
\(709\) −742.000 −1.04654 −0.523272 0.852166i \(-0.675289\pi\)
−0.523272 + 0.852166i \(0.675289\pi\)
\(710\) 0 0
\(711\) 882.000 1.24051
\(712\) 0 0
\(713\) 68.0000 0.0953717
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 225.000 0.312500
\(721\) 0 0
\(722\) 123.000 0.170360
\(723\) 1434.00 1.98340
\(724\) −366.000 −0.505525
\(725\) 0 0
\(726\) −363.000 −0.500000
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 729.000 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) −1062.00 −1.45082
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) −735.000 −1.00000
\(736\) 1122.00 1.52446
\(737\) 0 0
\(738\) 0 0
\(739\) −1462.00 −1.97835 −0.989175 0.146744i \(-0.953121\pi\)
−0.989175 + 0.146744i \(0.953121\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 514.000 0.691790 0.345895 0.938273i \(-0.387575\pi\)
0.345895 + 0.938273i \(0.387575\pi\)
\(744\) 42.0000 0.0564516
\(745\) 0 0
\(746\) 0 0
\(747\) 1386.00 1.85542
\(748\) 0 0
\(749\) 0 0
\(750\) −375.000 −0.500000
\(751\) −1438.00 −1.91478 −0.957390 0.288798i \(-0.906745\pi\)
−0.957390 + 0.288798i \(0.906745\pi\)
\(752\) −70.0000 −0.0930851
\(753\) 0 0
\(754\) 0 0
\(755\) −1190.00 −1.57616
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) −742.000 −0.978892
\(759\) 0 0
\(760\) 770.000 1.01316
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −630.000 −0.823529
\(766\) −686.000 −0.895561
\(767\) 0 0
\(768\) 513.000 0.667969
\(769\) 578.000 0.751625 0.375813 0.926696i \(-0.377364\pi\)
0.375813 + 0.926696i \(0.377364\pi\)
\(770\) 0 0
\(771\) −1398.00 −1.81323
\(772\) 0 0
\(773\) −1526.00 −1.97413 −0.987063 0.160330i \(-0.948744\pi\)
−0.987063 + 0.160330i \(0.948744\pi\)
\(774\) 0 0
\(775\) 50.0000 0.0645161
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) −476.000 −0.608696
\(783\) 0 0
\(784\) 245.000 0.312500
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 1122.00 1.42386
\(789\) 1338.00 1.69582
\(790\) 490.000 0.620253
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 1290.00 1.62264
\(796\) 426.000 0.535176
\(797\) 826.000 1.03639 0.518193 0.855264i \(-0.326605\pi\)
0.518193 + 0.855264i \(0.326605\pi\)
\(798\) 0 0
\(799\) 196.000 0.245307
\(800\) 825.000 1.03125
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 405.000 0.500000
\(811\) 1082.00 1.33416 0.667078 0.744988i \(-0.267545\pi\)
0.667078 + 0.744988i \(0.267545\pi\)
\(812\) 0 0
\(813\) −1446.00 −1.77860
\(814\) 0 0
\(815\) 0 0
\(816\) 210.000 0.257353
\(817\) 0 0
\(818\) −142.000 −0.173594
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) −678.000 −0.824818
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −374.000 −0.452237 −0.226119 0.974100i \(-0.572604\pi\)
−0.226119 + 0.974100i \(0.572604\pi\)
\(828\) −918.000 −1.10870
\(829\) −502.000 −0.605549 −0.302774 0.953062i \(-0.597913\pi\)
−0.302774 + 0.953062i \(0.597913\pi\)
\(830\) 770.000 0.927711
\(831\) 0 0
\(832\) 0 0
\(833\) −686.000 −0.823529
\(834\) 786.000 0.942446
\(835\) −1270.00 −1.52096
\(836\) 0 0
\(837\) −54.0000 −0.0645161
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 841.000 1.00000
\(842\) 602.000 0.714964
\(843\) 0 0
\(844\) −1086.00 −1.28673
\(845\) 845.000 1.00000
\(846\) −126.000 −0.148936
\(847\) 0 0
\(848\) −430.000 −0.507075
\(849\) 0 0
\(850\) −350.000 −0.411765
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) −990.000 −1.15789
\(856\) −742.000 −0.866822
\(857\) 1666.00 1.94399 0.971995 0.235000i \(-0.0755091\pi\)
0.971995 + 0.235000i \(0.0755091\pi\)
\(858\) 0 0
\(859\) 218.000 0.253783 0.126892 0.991917i \(-0.459500\pi\)
0.126892 + 0.991917i \(0.459500\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 274.000 0.317497 0.158749 0.987319i \(-0.449254\pi\)
0.158749 + 0.987319i \(0.449254\pi\)
\(864\) −891.000 −1.03125
\(865\) 770.000 0.890173
\(866\) 0 0
\(867\) 279.000 0.321799
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 154.000 0.176606
\(873\) 0 0
\(874\) −748.000 −0.855835
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) −622.000 −0.708428
\(879\) −1182.00 −1.34471
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 441.000 0.500000
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −566.000 −0.638826
\(887\) −1694.00 −1.90981 −0.954904 0.296914i \(-0.904042\pi\)
−0.954904 + 0.296914i \(0.904042\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 308.000 0.344905
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −675.000 −0.750000
\(901\) 1204.00 1.33629
\(902\) 0 0
\(903\) 0 0
\(904\) 1442.00 1.59513
\(905\) 610.000 0.674033
\(906\) 714.000 0.788079
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 402.000 0.442731
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 330.000 0.361842
\(913\) 0 0
\(914\) 0 0
\(915\) 1770.00 1.93443
\(916\) −654.000 −0.713974
\(917\) 0 0
\(918\) 378.000 0.411765
\(919\) 1298.00 1.41240 0.706202 0.708010i \(-0.250407\pi\)
0.706202 + 0.708010i \(0.250407\pi\)
\(920\) −1190.00 −1.29348
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) −30.0000 −0.0322581
\(931\) −1078.00 −1.15789
\(932\) −102.000 −0.109442
\(933\) 0 0
\(934\) 346.000 0.370450
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 210.000 0.223404
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −1574.00 −1.66209 −0.831045 0.556205i \(-0.812257\pi\)
−0.831045 + 0.556205i \(0.812257\pi\)
\(948\) 882.000 0.930380
\(949\) 0 0
\(950\) −550.000 −0.578947
\(951\) 402.000 0.422713
\(952\) 0 0
\(953\) 1474.00 1.54669 0.773347 0.633983i \(-0.218581\pi\)
0.773347 + 0.633983i \(0.218581\pi\)
\(954\) −774.000 −0.811321
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) −195.000 −0.203125
\(961\) −957.000 −0.995838
\(962\) 0 0
\(963\) 954.000 0.990654
\(964\) 1434.00 1.48755
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) −847.000 −0.875000
\(969\) −924.000 −0.953560
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 729.000 0.750000
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) −590.000 −0.604508
\(977\) −1934.00 −1.97953 −0.989765 0.142710i \(-0.954418\pi\)
−0.989765 + 0.142710i \(0.954418\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −735.000 −0.750000
\(981\) −198.000 −0.201835
\(982\) 0 0
\(983\) 1954.00 1.98779 0.993896 0.110319i \(-0.0351872\pi\)
0.993896 + 0.110319i \(0.0351872\pi\)
\(984\) 0 0
\(985\) −1870.00 −1.89848
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −958.000 −0.966700 −0.483350 0.875427i \(-0.660580\pi\)
−0.483350 + 0.875427i \(0.660580\pi\)
\(992\) 66.0000 0.0665323
\(993\) −366.000 −0.368580
\(994\) 0 0
\(995\) −710.000 −0.713568
\(996\) 1386.00 1.39157
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 938.000 0.939880
\(999\) 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 \(a_p\))