Properties

Label 3.43.b.a.2.1
Level 3
Weight 43
Character 3.2
Self dual Yes
Analytic conductor 33.518
Analytic rank 0
Dimension 1
CM disc. -3
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 43 \)
Character orbit: \([\chi]\) = 3.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(33.5183121516\)
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 2.1
Root \(0\)
Character \(\chi\) = 3.2

$q$-expansion

\(f(q)\) \(=\) \(q\)\(-1.04604e10 q^{3}\) \(+4.39805e12 q^{4}\) \(+1.46246e17 q^{7}\) \(+1.09419e20 q^{9}\) \(+O(q^{10})\) \(q\)\(-1.04604e10 q^{3}\) \(+4.39805e12 q^{4}\) \(+1.46246e17 q^{7}\) \(+1.09419e20 q^{9}\) \(-4.60051e22 q^{12}\) \(-3.57172e23 q^{13}\) \(+1.93428e25 q^{16}\) \(-1.65282e26 q^{19}\) \(-1.52979e27 q^{21}\) \(+2.27374e29 q^{25}\) \(-1.14456e30 q^{27}\) \(+6.43197e29 q^{28}\) \(+4.00719e31 q^{31}\) \(+4.81230e32 q^{36}\) \(+1.62704e33 q^{37}\) \(+3.73615e33 q^{39}\) \(+3.01952e34 q^{43}\) \(-2.02333e35 q^{48}\) \(-2.90586e35 q^{49}\) \(-1.57086e36 q^{52}\) \(+1.72891e36 q^{57}\) \(+5.58739e37 q^{61}\) \(+1.60021e37 q^{63}\) \(+8.50706e37 q^{64}\) \(-3.97023e38 q^{67}\) \(-1.16786e39 q^{73}\) \(-2.37841e39 q^{75}\) \(-7.26919e38 q^{76}\) \(+1.40825e40 q^{79}\) \(+1.19725e40 q^{81}\) \(-6.72807e39 q^{84}\) \(-5.22350e40 q^{91}\) \(-4.19166e41 q^{93}\) \(+2.26618e41 q^{97}\) \(+O(q^{100})\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(3\) −1.04604e10 −1.00000
\(4\) 4.39805e12 1.00000
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 1.46246e17 0.261834 0.130917 0.991393i \(-0.458208\pi\)
0.130917 + 0.991393i \(0.458208\pi\)
\(8\) 0 0
\(9\) 1.09419e20 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) −4.60051e22 −1.00000
\(13\) −3.57172e23 −1.44566 −0.722832 0.691024i \(-0.757160\pi\)
−0.722832 + 0.691024i \(0.757160\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.93428e25 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) −1.65282e26 −0.231420 −0.115710 0.993283i \(-0.536914\pi\)
−0.115710 + 0.993283i \(0.536914\pi\)
\(20\) 0 0
\(21\) −1.52979e27 −0.261834
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 2.27374e29 1.00000
\(26\) 0 0
\(27\) −1.14456e30 −1.00000
\(28\) 6.43197e29 0.261834
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 4.00719e31 1.92417 0.962087 0.272743i \(-0.0879310\pi\)
0.962087 + 0.272743i \(0.0879310\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 4.81230e32 1.00000
\(37\) 1.62704e33 1.90179 0.950894 0.309518i \(-0.100168\pi\)
0.950894 + 0.309518i \(0.100168\pi\)
\(38\) 0 0
\(39\) 3.73615e33 1.44566
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 3.01952e34 1.50349 0.751745 0.659454i \(-0.229212\pi\)
0.751745 + 0.659454i \(0.229212\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) −2.02333e35 −1.00000
\(49\) −2.90586e35 −0.931443
\(50\) 0 0
\(51\) 0 0
\(52\) −1.57086e36 −1.44566
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.72891e36 0.231420
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 5.58739e37 1.80004 0.900021 0.435846i \(-0.143551\pi\)
0.900021 + 0.435846i \(0.143551\pi\)
\(62\) 0 0
\(63\) 1.60021e37 0.261834
\(64\) 8.50706e37 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −3.97023e38 −1.78338 −0.891692 0.452642i \(-0.850481\pi\)
−0.891692 + 0.452642i \(0.850481\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −1.16786e39 −0.866183 −0.433092 0.901350i \(-0.642577\pi\)
−0.433092 + 0.901350i \(0.642577\pi\)
\(74\) 0 0
\(75\) −2.37841e39 −1.00000
\(76\) −7.26919e38 −0.231420
\(77\) 0 0
\(78\) 0 0
\(79\) 1.40825e40 1.98844 0.994218 0.107380i \(-0.0342461\pi\)
0.994218 + 0.107380i \(0.0342461\pi\)
\(80\) 0 0
\(81\) 1.19725e40 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) −6.72807e39 −0.261834
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) −5.22350e40 −0.378523
\(92\) 0 0
\(93\) −4.19166e41 −1.92417
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 2.26618e41 0.429623 0.214811 0.976656i \(-0.431086\pi\)
0.214811 + 0.976656i \(0.431086\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 1.00000e42 1.00000
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 2.25411e42 1.21170 0.605849 0.795580i \(-0.292834\pi\)
0.605849 + 0.795580i \(0.292834\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) −5.03383e42 −1.00000
\(109\) 7.71406e42 1.26278 0.631388 0.775467i \(-0.282486\pi\)
0.631388 + 0.775467i \(0.282486\pi\)
\(110\) 0 0
\(111\) −1.70194e43 −1.90179
\(112\) 2.82881e42 0.261834
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −3.90814e43 −1.44566
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 5.47637e43 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 1.76238e44 1.92417
\(125\) 0 0
\(126\) 0 0
\(127\) 8.74831e43 0.578157 0.289079 0.957305i \(-0.406651\pi\)
0.289079 + 0.957305i \(0.406651\pi\)
\(128\) 0 0
\(129\) −3.15853e44 −1.50349
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) −2.41719e43 −0.0605935
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) −2.00663e45 −1.99139 −0.995696 0.0926805i \(-0.970456\pi\)
−0.995696 + 0.0926805i \(0.970456\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 2.11647e45 1.00000
\(145\) 0 0
\(146\) 0 0
\(147\) 3.03963e45 0.931443
\(148\) 7.15580e45 1.90179
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) −9.57378e45 −1.66943 −0.834714 0.550684i \(-0.814367\pi\)
−0.834714 + 0.550684i \(0.814367\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 1.64317e46 1.44566
\(157\) −2.52740e46 −1.94438 −0.972190 0.234194i \(-0.924755\pi\)
−0.972190 + 0.234194i \(0.924755\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −3.90772e46 −1.36768 −0.683838 0.729634i \(-0.739691\pi\)
−0.683838 + 0.729634i \(0.739691\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 6.65311e46 1.08994
\(170\) 0 0
\(171\) −1.80850e46 −0.231420
\(172\) 1.32800e47 1.50349
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 3.32525e46 0.261834
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) −2.98915e46 −0.115957 −0.0579786 0.998318i \(-0.518466\pi\)
−0.0579786 + 0.998318i \(0.518466\pi\)
\(182\) 0 0
\(183\) −5.84461e47 −1.80004
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −1.67388e47 −0.261834
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) −8.89868e47 −1.00000
\(193\) 1.97159e48 1.98662 0.993308 0.115491i \(-0.0368443\pi\)
0.993308 + 0.115491i \(0.0368443\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −1.27801e48 −0.931443
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 1.66753e48 0.883401 0.441701 0.897163i \(-0.354375\pi\)
0.441701 + 0.897163i \(0.354375\pi\)
\(200\) 0 0
\(201\) 4.15300e48 1.78338
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −6.90872e48 −1.44566
\(209\) 0 0
\(210\) 0 0
\(211\) −1.03860e49 −1.60886 −0.804430 0.594048i \(-0.797529\pi\)
−0.804430 + 0.594048i \(0.797529\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 5.86036e48 0.503813
\(218\) 0 0
\(219\) 1.22162e49 0.866183
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −2.92851e49 −1.41986 −0.709928 0.704274i \(-0.751273\pi\)
−0.709928 + 0.704274i \(0.751273\pi\)
\(224\) 0 0
\(225\) 2.48790e49 1.00000
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 7.60383e48 0.231420
\(229\) 6.69541e49 1.85880 0.929399 0.369077i \(-0.120326\pi\)
0.929399 + 0.369077i \(0.120326\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −1.47308e50 −1.98844
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −1.47098e50 −1.39717 −0.698583 0.715529i \(-0.746186\pi\)
−0.698583 + 0.715529i \(0.746186\pi\)
\(242\) 0 0
\(243\) −1.25237e50 −1.00000
\(244\) 2.45736e50 1.80004
\(245\) 0 0
\(246\) 0 0
\(247\) 5.90342e49 0.334555
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 7.03780e49 0.261834
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 3.74144e50 1.00000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 2.37948e50 0.497952
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −1.74613e51 −1.78338
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) −7.19126e50 −0.581367 −0.290683 0.956819i \(-0.593883\pi\)
−0.290683 + 0.956819i \(0.593883\pi\)
\(272\) 0 0
\(273\) 5.46397e50 0.378523
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −3.91354e51 −1.99754 −0.998769 0.0496016i \(-0.984205\pi\)
−0.998769 + 0.0496016i \(0.984205\pi\)
\(278\) 0 0
\(279\) 4.38463e51 1.92417
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 4.99356e51 1.62516 0.812579 0.582852i \(-0.198063\pi\)
0.812579 + 0.582852i \(0.198063\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 4.77370e51 1.00000
\(290\) 0 0
\(291\) −2.37050e51 −0.429623
\(292\) −5.13629e51 −0.866183
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) −1.04604e52 −1.00000
\(301\) 4.41593e51 0.393664
\(302\) 0 0
\(303\) 0 0
\(304\) −3.19702e51 −0.231420
\(305\) 0 0
\(306\) 0 0
\(307\) −1.64577e52 −0.969313 −0.484657 0.874705i \(-0.661055\pi\)
−0.484657 + 0.874705i \(0.661055\pi\)
\(308\) 0 0
\(309\) −2.35788e52 −1.21170
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 5.07668e52 1.99135 0.995677 0.0928829i \(-0.0296082\pi\)
0.995677 + 0.0928829i \(0.0296082\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 6.19356e52 1.98844
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 5.26557e52 1.00000
\(325\) −8.12116e52 −1.44566
\(326\) 0 0
\(327\) −8.06918e52 −1.26278
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −6.67255e52 −0.808917 −0.404458 0.914556i \(-0.632540\pi\)
−0.404458 + 0.914556i \(0.632540\pi\)
\(332\) 0 0
\(333\) 1.78029e53 1.90179
\(334\) 0 0
\(335\) 0 0
\(336\) −2.95904e52 −0.261834
\(337\) −2.05646e53 −1.70959 −0.854794 0.518967i \(-0.826317\pi\)
−0.854794 + 0.518967i \(0.826317\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −8.81219e52 −0.505717
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) 4.57361e53 1.82358 0.911790 0.410657i \(-0.134701\pi\)
0.911790 + 0.410657i \(0.134701\pi\)
\(350\) 0 0
\(351\) 4.08805e53 1.44566
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) −4.82777e53 −0.946445
\(362\) 0 0
\(363\) −5.72848e53 −1.00000
\(364\) −2.29732e53 −0.378523
\(365\) 0 0
\(366\) 0 0
\(367\) 4.53140e53 0.628412 0.314206 0.949355i \(-0.398262\pi\)
0.314206 + 0.949355i \(0.398262\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −1.84351e54 −1.92417
\(373\) 5.59645e53 0.552112 0.276056 0.961142i \(-0.410972\pi\)
0.276056 + 0.961142i \(0.410972\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 2.10584e54 1.48594 0.742972 0.669322i \(-0.233415\pi\)
0.742972 + 0.669322i \(0.233415\pi\)
\(380\) 0 0
\(381\) −9.15104e53 −0.578157
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 3.30393e54 1.50349
\(388\) 9.96675e53 0.429623
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −3.71013e54 −0.988072 −0.494036 0.869441i \(-0.664479\pi\)
−0.494036 + 0.869441i \(0.664479\pi\)
\(398\) 0 0
\(399\) 2.52846e53 0.0605935
\(400\) 4.39805e54 1.00000
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) −1.43126e55 −2.78171
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −7.31829e54 −1.04285 −0.521423 0.853299i \(-0.674598\pi\)
−0.521423 + 0.853299i \(0.674598\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 9.91370e54 1.21170
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 2.09901e55 1.99139
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) −5.17954e54 −0.402130 −0.201065 0.979578i \(-0.564440\pi\)
−0.201065 + 0.979578i \(0.564440\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 8.17135e54 0.471312
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) −2.21390e55 −1.00000
\(433\) 2.01495e55 0.866999 0.433500 0.901154i \(-0.357279\pi\)
0.433500 + 0.901154i \(0.357279\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 3.39268e55 1.26278
\(437\) 0 0
\(438\) 0 0
\(439\) −4.81824e55 −1.55286 −0.776432 0.630201i \(-0.782972\pi\)
−0.776432 + 0.630201i \(0.782972\pi\)
\(440\) 0 0
\(441\) −3.17956e55 −0.931443
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) −7.48522e55 −1.90179
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 1.24412e55 0.261834
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 1.00145e56 1.66943
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 1.27291e56 1.76424 0.882120 0.471024i \(-0.156115\pi\)
0.882120 + 0.471024i \(0.156115\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −2.27293e55 −0.239544 −0.119772 0.992801i \(-0.538216\pi\)
−0.119772 + 0.992801i \(0.538216\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) −1.71882e56 −1.44566
\(469\) −5.80631e55 −0.466950
\(470\) 0 0
\(471\) 2.64375e56 1.94438
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −3.75808e55 −0.231420
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) −5.81133e56 −2.74934
\(482\) 0 0
\(483\) 0 0
\(484\) 2.40853e56 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) −5.41599e56 −1.97501 −0.987506 0.157584i \(-0.949630\pi\)
−0.987506 + 0.157584i \(0.949630\pi\)
\(488\) 0 0
\(489\) 4.08761e56 1.36768
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 7.75103e56 1.92417
\(497\) 0 0
\(498\) 0 0
\(499\) 4.75732e56 1.04052 0.520261 0.854008i \(-0.325835\pi\)
0.520261 + 0.854008i \(0.325835\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −6.95939e56 −1.08994
\(508\) 3.84755e56 0.578157
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) −1.70795e56 −0.226796
\(512\) 0 0
\(513\) 1.89176e56 0.231420
\(514\) 0 0
\(515\) 0 0
\(516\) −1.38913e57 −1.50349
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) −1.11031e57 −0.905546 −0.452773 0.891626i \(-0.649565\pi\)
−0.452773 + 0.891626i \(0.649565\pi\)
\(524\) 0 0
\(525\) −3.47833e56 −0.261834
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 1.55801e57 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) −1.06309e56 −0.0605935
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 4.99077e57 1.99998 0.999989 0.00462563i \(-0.00147239\pi\)
0.999989 + 0.00462563i \(0.00147239\pi\)
\(542\) 0 0
\(543\) 3.12676e56 0.115957
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −3.92097e57 −1.24641 −0.623204 0.782059i \(-0.714169\pi\)
−0.623204 + 0.782059i \(0.714169\pi\)
\(548\) 0 0
\(549\) 6.11367e57 1.80004
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 2.05951e57 0.520639
\(554\) 0 0
\(555\) 0 0
\(556\) −8.82526e57 −1.99139
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) −1.07849e58 −2.17354
\(560\) 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
\(567\) 1.75093e57 0.261834
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 3.09305e57 0.399051 0.199525 0.979893i \(-0.436060\pi\)
0.199525 + 0.979893i \(0.436060\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 9.30834e57 1.00000
\(577\) 1.85663e58 1.92324 0.961619 0.274387i \(-0.0884749\pi\)
0.961619 + 0.274387i \(0.0884749\pi\)
\(578\) 0 0
\(579\) −2.06236e58 −1.98662
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 1.33684e58 0.931443
\(589\) −6.62317e57 −0.445292
\(590\) 0 0
\(591\) 0 0
\(592\) 3.14715e58 1.90179
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −1.74429e58 −0.883401
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 3.93880e57 0.173381 0.0866903 0.996235i \(-0.472371\pi\)
0.0866903 + 0.996235i \(0.472371\pi\)
\(602\) 0 0
\(603\) −4.34419e58 −1.78338
\(604\) −4.21059e58 −1.66943
\(605\) 0 0
\(606\) 0 0
\(607\) −1.52066e58 −0.543339 −0.271669 0.962391i \(-0.587576\pi\)
−0.271669 + 0.962391i \(0.587576\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 4.36230e58 1.26778 0.633892 0.773421i \(-0.281456\pi\)
0.633892 + 0.773421i \(0.281456\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 4.09829e58 0.970731 0.485366 0.874311i \(-0.338687\pi\)
0.485366 + 0.874311i \(0.338687\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 7.22676e58 1.44566
\(625\) 5.16988e58 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) −1.11156e59 −1.94438
\(629\) 0 0
\(630\) 0 0
\(631\) −8.47186e58 −1.34079 −0.670397 0.742002i \(-0.733876\pi\)
−0.670397 + 0.742002i \(0.733876\pi\)
\(632\) 0 0
\(633\) 1.08642e59 1.60886
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 1.03789e59 1.34655
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) −1.67557e59 −1.78539 −0.892696 0.450660i \(-0.851189\pi\)
−0.892696 + 0.450660i \(0.851189\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −6.13014e58 −0.503813
\(652\) −1.71863e59 −1.36768
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −1.27786e59 −0.866183
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 3.12492e59 1.86470 0.932350 0.361557i \(-0.117755\pi\)
0.932350 + 0.361557i \(0.117755\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 3.06332e59 1.41986
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 1.71754e59 0.702411 0.351206 0.936298i \(-0.385772\pi\)
0.351206 + 0.936298i \(0.385772\pi\)
\(674\) 0 0
\(675\) −2.60243e59 −1.00000
\(676\) 2.92607e59 1.08994
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 3.31419e58 0.112490
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) −7.95387e58 −0.231420
\(685\) 0 0
\(686\) 0 0
\(687\) −7.00364e59 −1.85880
\(688\) 5.84060e59 1.50349
\(689\) 0 0
\(690\) 0 0
\(691\) −8.43782e59 −1.98240 −0.991200 0.132371i \(-0.957741\pi\)
−0.991200 + 0.132371i \(0.957741\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 1.46246e59 0.261834
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) −2.68921e59 −0.440111
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −8.34229e59 −1.14213 −0.571064 0.820906i \(-0.693469\pi\)
−0.571064 + 0.820906i \(0.693469\pi\)
\(710\) 0 0
\(711\) 1.54090e60 1.98844
\(712\) 0 0
\(713\) 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 \(\pi\)
\(720\) 0 0
\(721\) 3.29655e59 0.317263
\(722\) 0 0
\(723\) 1.53869e60 1.39717
\(724\) −1.31464e59 −0.115957
\(725\) 0 0
\(726\) 0 0
\(727\) 1.79563e60 1.45209 0.726045 0.687647i \(-0.241356\pi\)
0.726045 + 0.687647i \(0.241356\pi\)
\(728\) 0 0
\(729\) 1.31002e60 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) −2.57049e60 −1.80004
\(733\) −1.96528e60 −1.33734 −0.668670 0.743560i \(-0.733136\pi\)
−0.668670 + 0.743560i \(0.733136\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −3.48715e60 −1.99957 −0.999786 0.0206731i \(-0.993419\pi\)
−0.999786 + 0.0206731i \(0.993419\pi\)
\(740\) 0 0
\(741\) −6.17519e59 −0.334555
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 2.89369e60 1.18308 0.591540 0.806276i \(-0.298520\pi\)
0.591540 + 0.806276i \(0.298520\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) −7.36179e59 −0.261834
\(757\) 1.42240e60 0.492049 0.246024 0.969264i \(-0.420876\pi\)
0.246024 + 0.969264i \(0.420876\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 1.12815e60 0.330637
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −3.91368e60 −1.00000
\(769\) −4.49510e59 −0.111760 −0.0558800 0.998437i \(-0.517796\pi\)
−0.0558800 + 0.998437i \(0.517796\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 8.67116e60 1.98662
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 9.11129e60 1.92417
\(776\) 0 0
\(777\) −2.48902e60 −0.497952
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −5.62074e60 −0.931443
\(785\) 0 0
\(786\) 0 0
\(787\) −1.11699e61 −1.70836 −0.854181 0.519976i \(-0.825941\pi\)
−0.854181 + 0.519976i \(0.825941\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −1.99566e61 −2.60226
\(794\) 0 0
\(795\) 0 0
\(796\) 7.33386e60 0.883401
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 1.82651e61 1.78338
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) 3.60960e60 0.293779 0.146890 0.989153i \(-0.453074\pi\)
0.146890 + 0.989153i \(0.453074\pi\)
\(812\) 0 0
\(813\) 7.52232e60 0.581367
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −4.99073e60 −0.347937
\(818\) 0 0
\(819\) −5.71551e60 −0.378523
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 2.89785e61 1.73252 0.866262 0.499590i \(-0.166516\pi\)
0.866262 + 0.499590i \(0.166516\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) −1.53813e61 −0.789490 −0.394745 0.918791i \(-0.629167\pi\)
−0.394745 + 0.918791i \(0.629167\pi\)
\(830\) 0 0
\(831\) 4.09371e61 1.99754
\(832\) −3.03849e61 −1.44566
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −4.58647e61 −1.92417
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 2.63461e61 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) −4.56782e61 −1.60886
\(845\) 0 0
\(846\) 0 0
\(847\) 8.00898e60 0.261834
\(848\) 0 0
\(849\) −5.22344e61 −1.62516
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 1.96709e61 0.554491 0.277246 0.960799i \(-0.410578\pi\)
0.277246 + 0.960799i \(0.410578\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 8.21606e61 1.99897 0.999487 0.0320345i \(-0.0101987\pi\)
0.999487 + 0.0320345i \(0.0101987\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −4.99345e61 −1.00000
\(868\) 2.57741e61 0.503813
\(869\) 0 0
\(870\) 0 0
\(871\) 1.41806e62 2.57817
\(872\) 0 0
\(873\) 2.47963e61 0.429623
\(874\) 0 0
\(875\) 0 0
\(876\) 5.37274e61 0.866183
\(877\) −1.08468e62 −1.70730 −0.853649 0.520848i \(-0.825616\pi\)
−0.853649 + 0.520848i \(0.825616\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) −1.19985e62 −1.63663 −0.818317 0.574768i \(-0.805092\pi\)
−0.818317 + 0.574768i \(0.805092\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 1.27941e61 0.151381
\(890\) 0 0
\(891\) 0 0
\(892\) −1.28797e62 −1.41986
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 1.09419e62 1.00000
\(901\) 0 0
\(902\) 0 0
\(903\) −4.61922e61 −0.393664
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −1.57556e62 −1.22372 −0.611862 0.790965i \(-0.709579\pi\)
−0.611862 + 0.790965i \(0.709579\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 3.34420e61 0.231420
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 2.94467e62 1.85880
\(917\) 0 0
\(918\) 0 0
\(919\) −2.61130e62 −1.53898 −0.769488 0.638661i \(-0.779488\pi\)
−0.769488 + 0.638661i \(0.779488\pi\)
\(920\) 0 0
\(921\) 1.72154e62 0.969313
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 3.69946e62 1.90179
\(926\) 0 0
\(927\) 2.46643e62 1.21170
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 4.80286e61 0.215554
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −4.64100e62 −1.82003 −0.910017 0.414570i \(-0.863932\pi\)
−0.910017 + 0.414570i \(0.863932\pi\)
\(938\) 0 0
\(939\) −5.31038e62 −1.99135
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) −6.47868e62 −1.98844
\(949\) 4.17126e62 1.25221
\(950\) 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
\(960\) 0 0
\(961\) 1.17205e63 2.70244
\(962\) 0 0
\(963\) 0 0
\(964\) −6.46943e62 −1.39717
\(965\) 0 0
\(966\) 0 0
\(967\) −3.72659e62 −0.753972 −0.376986 0.926219i \(-0.623040\pi\)
−0.376986 + 0.926219i \(0.623040\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) −5.50797e62 −1.00000
\(973\) −2.93462e62 −0.521413
\(974\) 0 0
\(975\) 8.49502e62 1.44566
\(976\) 1.08076e63 1.80004
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 8.44064e62 1.26278
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 2.59635e62 0.334555
\(989\) 0 0
\(990\) 0 0
\(991\) −7.74736e62 −0.936714 −0.468357 0.883539i \(-0.655154\pi\)
−0.468357 + 0.883539i \(0.655154\pi\)
\(992\) 0 0
\(993\) 6.97972e62 0.808917
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −1.73262e63 −1.84546 −0.922729 0.385449i \(-0.874046\pi\)
−0.922729 + 0.385449i \(0.874046\pi\)
\(998\) 0 0
\(999\) −1.86225e63 −1.90179
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 \(a_p\))