Properties

Label 7.17.b.a.6.1
Level 7
Weight 17
Character 7.6
Self dual Yes
Analytic conductor 11.363
Analytic rank 0
Dimension 1
CM disc. -7
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 7 \)
Weight: \( k \) = \( 17 \)
Character orbit: \([\chi]\) = 7.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(11.36271807\)
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 6.1
Root \(0\)
Character \(\chi\) = 7.6

$q$-expansion

\(f(q)\) \(=\) \(q+449.000 q^{2} +136065. q^{4} +5.76480e6 q^{7} +3.16675e7 q^{8} +4.30467e7 q^{9} +O(q^{10})\) \(q+449.000 q^{2} +136065. q^{4} +5.76480e6 q^{7} +3.16675e7 q^{8} +4.30467e7 q^{9} -2.55690e8 q^{11} +2.58840e9 q^{14} +5.30156e9 q^{16} +1.93280e10 q^{18} -1.14805e11 q^{22} -1.56184e11 q^{23} +1.52588e11 q^{25} +7.84388e11 q^{28} -9.88787e11 q^{29} +3.05038e11 q^{32} +5.85715e12 q^{36} -2.72396e12 q^{37} +2.18633e13 q^{43} -3.47905e13 q^{44} -7.01266e13 q^{46} +3.32329e13 q^{49} +6.85120e13 q^{50} +1.11956e14 q^{53} +1.82557e14 q^{56} -4.43965e14 q^{58} +2.48156e14 q^{63} -2.10481e14 q^{64} -5.61251e14 q^{67} +5.00488e14 q^{71} +1.36318e15 q^{72} -1.22306e15 q^{74} -1.47400e15 q^{77} +1.13983e15 q^{79} +1.85302e15 q^{81} +9.81662e15 q^{86} -8.09707e15 q^{88} -2.12512e16 q^{92} +1.49216e16 q^{98} -1.10066e16 q^{99} +O(q^{100})\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 449.000 1.75391 0.876953 0.480576i \(-0.159572\pi\)
0.876953 + 0.480576i \(0.159572\pi\)
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 136065. 2.07619
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 5.76480e6 1.00000
\(8\) 3.16675e7 1.88753
\(9\) 4.30467e7 1.00000
\(10\) 0 0
\(11\) −2.55690e8 −1.19281 −0.596406 0.802683i \(-0.703405\pi\)
−0.596406 + 0.802683i \(0.703405\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 2.58840e9 1.75391
\(15\) 0 0
\(16\) 5.30156e9 1.23437
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 1.93280e10 1.75391
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −1.14805e11 −2.09208
\(23\) −1.56184e11 −1.99441 −0.997204 0.0747259i \(-0.976192\pi\)
−0.997204 + 0.0747259i \(0.976192\pi\)
\(24\) 0 0
\(25\) 1.52588e11 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 7.84388e11 2.07619
\(29\) −9.88787e11 −1.97660 −0.988300 0.152524i \(-0.951260\pi\)
−0.988300 + 0.152524i \(0.951260\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 3.05038e11 0.277431
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 5.85715e12 2.07619
\(37\) −2.72396e12 −0.775508 −0.387754 0.921763i \(-0.626749\pi\)
−0.387754 + 0.921763i \(0.626749\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 2.18633e13 1.87054 0.935272 0.353930i \(-0.115155\pi\)
0.935272 + 0.353930i \(0.115155\pi\)
\(44\) −3.47905e13 −2.47650
\(45\) 0 0
\(46\) −7.01266e13 −3.49801
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 3.32329e13 1.00000
\(50\) 6.85120e13 1.75391
\(51\) 0 0
\(52\) 0 0
\(53\) 1.11956e14 1.79821 0.899106 0.437730i \(-0.144217\pi\)
0.899106 + 0.437730i \(0.144217\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.82557e14 1.88753
\(57\) 0 0
\(58\) −4.43965e14 −3.46677
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 2.48156e14 1.00000
\(64\) −2.10481e14 −0.747778
\(65\) 0 0
\(66\) 0 0
\(67\) −5.61251e14 −1.38216 −0.691081 0.722778i \(-0.742865\pi\)
−0.691081 + 0.722778i \(0.742865\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 5.00488e14 0.775045 0.387523 0.921860i \(-0.373331\pi\)
0.387523 + 0.921860i \(0.373331\pi\)
\(72\) 1.36318e15 1.88753
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) −1.22306e15 −1.36017
\(75\) 0 0
\(76\) 0 0
\(77\) −1.47400e15 −1.19281
\(78\) 0 0
\(79\) 1.13983e15 0.751315 0.375658 0.926759i \(-0.377417\pi\)
0.375658 + 0.926759i \(0.377417\pi\)
\(80\) 0 0
\(81\) 1.85302e15 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 9.81662e15 3.28076
\(87\) 0 0
\(88\) −8.09707e15 −2.25147
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −2.12512e16 −4.14076
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 1.49216e16 1.75391
\(99\) −1.10066e16 −1.19281
\(100\) 2.07619e16 2.07619
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 5.02683e16 3.15390
\(107\) −1.86596e16 −1.08601 −0.543004 0.839730i \(-0.682713\pi\)
−0.543004 + 0.839730i \(0.682713\pi\)
\(108\) 0 0
\(109\) −2.41222e16 −1.21061 −0.605305 0.795994i \(-0.706949\pi\)
−0.605305 + 0.795994i \(0.706949\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 3.05624e16 1.23437
\(113\) 3.36940e15 0.126743 0.0633716 0.997990i \(-0.479815\pi\)
0.0633716 + 0.997990i \(0.479815\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −1.34539e17 −4.10379
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.94277e16 0.422803
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 1.11422e17 1.75391
\(127\) 1.35082e17 1.99603 0.998017 0.0629511i \(-0.0200512\pi\)
0.998017 + 0.0629511i \(0.0200512\pi\)
\(128\) −1.14497e17 −1.58896
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −2.52002e17 −2.42418
\(135\) 0 0
\(136\) 0 0
\(137\) 1.76666e17 1.42360 0.711800 0.702383i \(-0.247880\pi\)
0.711800 + 0.702383i \(0.247880\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 2.24719e17 1.35936
\(143\) 0 0
\(144\) 2.28215e17 1.23437
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) −3.70635e17 −1.61010
\(149\) 3.05538e17 1.25769 0.628847 0.777529i \(-0.283527\pi\)
0.628847 + 0.777529i \(0.283527\pi\)
\(150\) 0 0
\(151\) −5.11302e17 −1.89174 −0.945872 0.324540i \(-0.894790\pi\)
−0.945872 + 0.324540i \(0.894790\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −6.61827e17 −2.09208
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 5.11782e17 1.31774
\(159\) 0 0
\(160\) 0 0
\(161\) −9.00370e17 −1.99441
\(162\) 8.32006e17 1.75391
\(163\) −2.57293e17 −0.516330 −0.258165 0.966101i \(-0.583118\pi\)
−0.258165 + 0.966101i \(0.583118\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 6.65417e17 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 2.97483e18 3.88360
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 8.79639e17 1.00000
\(176\) −1.35556e18 −1.47237
\(177\) 0 0
\(178\) 0 0
\(179\) −1.55549e18 −1.47586 −0.737928 0.674880i \(-0.764196\pi\)
−0.737928 + 0.674880i \(0.764196\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −4.94596e18 −3.76451
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −5.06222e16 −0.0285808 −0.0142904 0.999898i \(-0.504549\pi\)
−0.0142904 + 0.999898i \(0.504549\pi\)
\(192\) 0 0
\(193\) 1.06490e18 0.553157 0.276578 0.960991i \(-0.410799\pi\)
0.276578 + 0.960991i \(0.410799\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 4.52184e18 2.07619
\(197\) 4.51503e18 1.99036 0.995179 0.0980766i \(-0.0312690\pi\)
0.995179 + 0.0980766i \(0.0312690\pi\)
\(198\) −4.94197e18 −2.09208
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 4.83208e18 1.88753
\(201\) 0 0
\(202\) 0 0
\(203\) −5.70016e18 −1.97660
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −6.72321e18 −1.99441
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 6.78059e18 1.72587 0.862935 0.505315i \(-0.168624\pi\)
0.862935 + 0.505315i \(0.168624\pi\)
\(212\) 1.52333e19 3.73343
\(213\) 0 0
\(214\) −8.37817e18 −1.90475
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −1.08309e19 −2.12330
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 1.75848e18 0.277431
\(225\) 6.56841e18 1.00000
\(226\) 1.51286e18 0.222296
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −3.13124e19 −3.73089
\(233\) 1.65498e19 1.90522 0.952611 0.304192i \(-0.0983863\pi\)
0.952611 + 0.304192i \(0.0983863\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −1.59739e19 −1.50047 −0.750234 0.661173i \(-0.770059\pi\)
−0.750234 + 0.661173i \(0.770059\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 8.72302e18 0.741556
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 3.37653e19 2.07619
\(253\) 3.99347e19 2.37896
\(254\) 6.06518e19 3.50086
\(255\) 0 0
\(256\) −3.76150e19 −2.03912
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) −1.57031e19 −0.775508
\(260\) 0 0
\(261\) −4.25640e19 −1.97660
\(262\) 0 0
\(263\) −3.97386e19 −1.73607 −0.868034 0.496505i \(-0.834616\pi\)
−0.868034 + 0.496505i \(0.834616\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −7.63666e19 −2.86963
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 7.93229e19 2.49686
\(275\) −3.90152e19 −1.19281
\(276\) 0 0
\(277\) 1.17111e19 0.337878 0.168939 0.985626i \(-0.445966\pi\)
0.168939 + 0.985626i \(0.445966\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −4.91957e18 −0.126554 −0.0632771 0.997996i \(-0.520155\pi\)
−0.0632771 + 0.997996i \(0.520155\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 6.80989e19 1.60914
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 1.31309e19 0.277431
\(289\) 4.86612e19 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −8.62609e19 −1.46380
\(297\) 0 0
\(298\) 1.37187e20 2.20588
\(299\) 0 0
\(300\) 0 0
\(301\) 1.26038e20 1.87054
\(302\) −2.29575e20 −3.31794
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) −2.00560e20 −2.47650
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 1.55091e20 1.55987
\(317\) 2.35552e19 0.231000 0.115500 0.993307i \(-0.463153\pi\)
0.115500 + 0.993307i \(0.463153\pi\)
\(318\) 0 0
\(319\) 2.52823e20 2.35771
\(320\) 0 0
\(321\) 0 0
\(322\) −4.04266e20 −3.49801
\(323\) 0 0
\(324\) 2.52131e20 2.07619
\(325\) 0 0
\(326\) −1.15525e20 −0.905595
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −1.99417e20 −1.38401 −0.692005 0.721893i \(-0.743272\pi\)
−0.692005 + 0.721893i \(0.743272\pi\)
\(332\) 0 0
\(333\) −1.17257e20 −0.775508
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −3.05652e20 −1.83733 −0.918666 0.395034i \(-0.870733\pi\)
−0.918666 + 0.395034i \(0.870733\pi\)
\(338\) 2.98772e20 1.75391
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1.91581e20 1.00000
\(344\) 6.92356e20 3.53071
\(345\) 0 0
\(346\) 0 0
\(347\) −6.29786e19 −0.299611 −0.149805 0.988716i \(-0.547865\pi\)
−0.149805 + 0.988716i \(0.547865\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 3.94958e20 1.75391
\(351\) 0 0
\(352\) −7.79952e19 −0.330923
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) −6.98416e20 −2.58851
\(359\) −5.33002e20 −1.93185 −0.965924 0.258825i \(-0.916665\pi\)
−0.965924 + 0.258825i \(0.916665\pi\)
\(360\) 0 0
\(361\) 2.88441e20 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) −8.28019e20 −2.46183
\(369\) 0 0
\(370\) 0 0
\(371\) 6.45405e20 1.79821
\(372\) 0 0
\(373\) −7.22084e20 −1.92716 −0.963579 0.267424i \(-0.913828\pi\)
−0.963579 + 0.267424i \(0.913828\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 2.51941e20 0.591814 0.295907 0.955217i \(-0.404378\pi\)
0.295907 + 0.955217i \(0.404378\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −2.27294e19 −0.0501280
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 4.78138e20 0.970185
\(387\) 9.41143e20 1.87054
\(388\) 0 0
\(389\) −3.75792e20 −0.716722 −0.358361 0.933583i \(-0.616664\pi\)
−0.358361 + 0.933583i \(0.616664\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 1.05240e21 1.88753
\(393\) 0 0
\(394\) 2.02725e21 3.49090
\(395\) 0 0
\(396\) −1.49762e21 −2.47650
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 8.08954e20 1.23437
\(401\) −1.16330e21 −1.73995 −0.869976 0.493094i \(-0.835866\pi\)
−0.869976 + 0.493094i \(0.835866\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) −2.55937e21 −3.46677
\(407\) 6.96488e20 0.925036
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) −3.01872e21 −3.49801
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) −3.31587e19 −0.0336001 −0.0168001 0.999859i \(-0.505348\pi\)
−0.0168001 + 0.999859i \(0.505348\pi\)
\(422\) 3.04449e21 3.02701
\(423\) 0 0
\(424\) 3.54537e21 3.39418
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −2.53892e21 −2.25475
\(429\) 0 0
\(430\) 0 0
\(431\) 1.38814e20 0.116578 0.0582889 0.998300i \(-0.481436\pi\)
0.0582889 + 0.998300i \(0.481436\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −3.28218e21 −2.51345
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 1.43057e21 1.00000
\(442\) 0 0
\(443\) −2.00159e20 −0.134941 −0.0674706 0.997721i \(-0.521493\pi\)
−0.0674706 + 0.997721i \(0.521493\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −1.21338e21 −0.747778
\(449\) 2.40516e21 1.45604 0.728020 0.685556i \(-0.240441\pi\)
0.728020 + 0.685556i \(0.240441\pi\)
\(450\) 2.94922e21 1.75391
\(451\) 0 0
\(452\) 4.58457e20 0.263142
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −2.39845e21 −1.26067 −0.630335 0.776323i \(-0.717083\pi\)
−0.630335 + 0.776323i \(0.717083\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 4.10410e21 1.94343 0.971716 0.236151i \(-0.0758861\pi\)
0.971716 + 0.236151i \(0.0758861\pi\)
\(464\) −5.24211e21 −2.43985
\(465\) 0 0
\(466\) 7.43086e21 3.34158
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) −3.23550e21 −1.38216
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −5.59023e21 −2.23121
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 4.81935e21 1.79821
\(478\) −7.17226e21 −2.63168
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 2.64343e21 0.877818
\(485\) 0 0
\(486\) 0 0
\(487\) −6.16705e21 −1.94915 −0.974576 0.224059i \(-0.928069\pi\)
−0.974576 + 0.224059i \(0.928069\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −2.27966e21 −0.674868 −0.337434 0.941349i \(-0.609559\pi\)
−0.337434 + 0.941349i \(0.609559\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 2.88522e21 0.775045
\(498\) 0 0
\(499\) −5.80134e21 −1.50912 −0.754560 0.656231i \(-0.772150\pi\)
−0.754560 + 0.656231i \(0.772150\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 7.85848e21 1.88753
\(505\) 0 0
\(506\) 1.79307e22 4.17247
\(507\) 0 0
\(508\) 1.83799e22 4.14414
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −9.38548e21 −1.98745
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) −7.05068e21 −1.36017
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) −1.91113e22 −3.46677
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −1.78426e22 −3.04490
\(527\) 0 0
\(528\) 0 0
\(529\) 1.82609e22 2.97766
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) −1.77734e22 −2.60887
\(537\) 0 0
\(538\) 0 0
\(539\) −8.49733e21 −1.19281
\(540\) 0 0
\(541\) 1.25303e22 1.70759 0.853796 0.520608i \(-0.174295\pi\)
0.853796 + 0.520608i \(0.174295\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 1.18030e22 1.47263 0.736314 0.676640i \(-0.236565\pi\)
0.736314 + 0.676640i \(0.236565\pi\)
\(548\) 2.40380e22 2.95566
\(549\) 0 0
\(550\) −1.75178e22 −2.09208
\(551\) 0 0
\(552\) 0 0
\(553\) 6.57088e21 0.751315
\(554\) 5.25829e21 0.592607
\(555\) 0 0
\(556\) 0 0
\(557\) −1.64658e22 −1.77722 −0.888608 0.458667i \(-0.848327\pi\)
−0.888608 + 0.458667i \(0.848327\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −2.20889e21 −0.221964
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 1.06823e22 1.00000
\(568\) 1.58492e22 1.46292
\(569\) 4.05166e21 0.368753 0.184376 0.982856i \(-0.440973\pi\)
0.184376 + 0.982856i \(0.440973\pi\)
\(570\) 0 0
\(571\) 5.15395e21 0.456090 0.228045 0.973651i \(-0.426767\pi\)
0.228045 + 0.973651i \(0.426767\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −2.38318e22 −1.99441
\(576\) −9.06051e21 −0.747778
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 2.18489e22 1.75391
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −2.86261e22 −2.14493
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −1.44412e22 −0.957261
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 4.15730e22 2.61121
\(597\) 0 0
\(598\) 0 0
\(599\) 1.30525e22 0.787551 0.393775 0.919207i \(-0.371169\pi\)
0.393775 + 0.919207i \(0.371169\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 5.65909e22 3.28076
\(603\) −2.41600e22 −1.38216
\(604\) −6.95704e22 −3.92761
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 5.59127e21 0.280431 0.140216 0.990121i \(-0.455220\pi\)
0.140216 + 0.990121i \(0.455220\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) −4.66780e22 −2.25147
\(617\) −3.43785e22 −1.63684 −0.818419 0.574623i \(-0.805149\pi\)
−0.818419 + 0.574623i \(0.805149\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 2.32831e22 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 2.29226e22 0.912070 0.456035 0.889962i \(-0.349269\pi\)
0.456035 + 0.889962i \(0.349269\pi\)
\(632\) 3.60955e22 1.41813
\(633\) 0 0
\(634\) 1.05763e22 0.405153
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 1.13518e23 4.13521
\(639\) 2.15444e22 0.775045
\(640\) 0 0
\(641\) 4.65002e22 1.63151 0.815757 0.578395i \(-0.196321\pi\)
0.815757 + 0.578395i \(0.196321\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) −1.22509e23 −4.14076
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 5.86806e22 1.88753
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −3.50086e22 −1.07200
\(653\) −6.48581e22 −1.96182 −0.980909 0.194467i \(-0.937702\pi\)
−0.980909 + 0.194467i \(0.937702\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −5.14324e22 −1.44595 −0.722975 0.690874i \(-0.757226\pi\)
−0.722975 + 0.690874i \(0.757226\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) −8.95384e22 −2.42742
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) −5.26486e22 −1.36017
\(667\) 1.54433e23 3.94215
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −7.61624e22 −1.80976 −0.904879 0.425670i \(-0.860039\pi\)
−0.904879 + 0.425670i \(0.860039\pi\)
\(674\) −1.37238e23 −3.22251
\(675\) 0 0
\(676\) 9.05399e22 2.07619
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −4.90124e21 −0.103500 −0.0517500 0.998660i \(-0.516480\pi\)
−0.0517500 + 0.998660i \(0.516480\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 8.60200e22 1.75391
\(687\) 0 0
\(688\) 1.15910e23 2.30894
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) −6.34510e22 −1.19281
\(694\) −2.82774e22 −0.525489
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 1.19688e23 2.07619
\(701\) 9.59565e22 1.64562 0.822812 0.568314i \(-0.192404\pi\)
0.822812 + 0.568314i \(0.192404\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 5.38179e22 0.891960
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −2.98237e22 −0.467081 −0.233541 0.972347i \(-0.575031\pi\)
−0.233541 + 0.972347i \(0.575031\pi\)
\(710\) 0 0
\(711\) 4.90658e22 0.751315
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −2.11648e23 −3.06415
\(717\) 0 0
\(718\) −2.39318e23 −3.38828
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 1.29510e23 1.75391
\(723\) 0 0
\(724\) 0 0
\(725\) −1.50877e23 −1.97660
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 7.97664e22 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) −4.76421e22 −0.553310
\(737\) 1.43506e23 1.64866
\(738\) 0 0
\(739\) 5.03340e21 0.0565856 0.0282928 0.999600i \(-0.490993\pi\)
0.0282928 + 0.999600i \(0.490993\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 2.89787e23 3.15390
\(743\) 1.51791e23 1.63432 0.817159 0.576413i \(-0.195548\pi\)
0.817159 + 0.576413i \(0.195548\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −3.24216e23 −3.38005
\(747\) 0 0
\(748\) 0 0
\(749\) −1.07569e23 −1.08601
\(750\) 0 0
\(751\) −4.14381e22 −0.409525 −0.204762 0.978812i \(-0.565642\pi\)
−0.204762 + 0.978812i \(0.565642\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −2.15454e23 −1.99796 −0.998982 0.0451177i \(-0.985634\pi\)
−0.998982 + 0.0451177i \(0.985634\pi\)
\(758\) 1.13121e23 1.03799
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) −1.39060e23 −1.21061
\(764\) −6.88791e21 −0.0593390
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1.44895e23 1.14846
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 4.22573e23 3.28076
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) −1.68731e23 −1.25706
\(779\) 0 0
\(780\) 0 0
\(781\) −1.27970e23 −0.924484
\(782\) 0 0
\(783\) 0 0
\(784\) 1.76186e23 1.23437
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 6.14338e23 4.13236
\(789\) 0 0
\(790\) 0 0
\(791\) 1.94239e22 0.126743
\(792\) −3.48552e23 −2.25147
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 4.65451e22 0.277431
\(801\) 0 0
\(802\) −5.22322e23 −3.05171
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 2.38865e23 1.30186 0.650929 0.759138i \(-0.274379\pi\)
0.650929 + 0.759138i \(0.274379\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) −7.75592e23 −4.10379
\(813\) 0 0
\(814\) 3.12723e23 1.62243
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 4.12742e23 1.99955 0.999777 0.0211189i \(-0.00672286\pi\)
0.999777 + 0.0211189i \(0.00672286\pi\)
\(822\) 0 0
\(823\) −4.20586e23 −1.99828 −0.999139 0.0414872i \(-0.986790\pi\)
−0.999139 + 0.0414872i \(0.986790\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 8.69945e22 0.397601 0.198801 0.980040i \(-0.436295\pi\)
0.198801 + 0.980040i \(0.436295\pi\)
\(828\) −9.14794e23 −4.14076
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 7.27453e23 2.90695
\(842\) −1.48883e22 −0.0589315
\(843\) 0 0
\(844\) 9.22601e23 3.58323
\(845\) 0 0
\(846\) 0 0
\(847\) 1.11997e23 0.422803
\(848\) 5.93543e23 2.21965
\(849\) 0 0
\(850\) 0 0
\(851\) 4.25439e23 1.54668
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −5.90904e23 −2.04987
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 6.23276e22 0.204467
\(863\) −3.54814e23 −1.15323 −0.576613 0.817017i \(-0.695626\pi\)
−0.576613 + 0.817017i \(0.695626\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −2.91442e23 −0.896179
\(870\) 0 0
\(871\) 0 0
\(872\) −7.63889e23 −2.28506
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −6.51792e23 −1.86257 −0.931284 0.364293i \(-0.881311\pi\)
−0.931284 + 0.364293i \(0.881311\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 6.42325e23 1.75391
\(883\) −7.34537e23 −1.98760 −0.993798 0.111199i \(-0.964531\pi\)
−0.993798 + 0.111199i \(0.964531\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −8.98712e22 −0.236674
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 7.78721e23 1.99603
\(890\) 0 0
\(891\) −4.73799e23 −1.19281
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) −6.60052e23 −1.58896
\(897\) 0 0
\(898\) 1.07992e24 2.55376
\(899\) 0 0
\(900\) 8.93730e23 2.07619
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 1.06700e23 0.239232
\(905\) 0 0
\(906\) 0 0
\(907\) 1.48980e23 0.325289 0.162645 0.986685i \(-0.447998\pi\)
0.162645 + 0.986685i \(0.447998\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −3.87116e23 −0.816007 −0.408004 0.912980i \(-0.633775\pi\)
−0.408004 + 0.912980i \(0.633775\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −1.07691e24 −2.21110
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −4.69772e23 −0.923344 −0.461672 0.887051i \(-0.652750\pi\)
−0.461672 + 0.887051i \(0.652750\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −4.15643e23 −0.775508
\(926\) 1.84274e24 3.40860
\(927\) 0 0
\(928\) −3.01618e23 −0.548369
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 2.25185e24 3.95560
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) −1.45274e24 −2.42418
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) −2.51001e24 −3.91333
\(947\) 7.12280e23 1.10116 0.550581 0.834782i \(-0.314406\pi\)
0.550581 + 0.834782i \(0.314406\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1.05416e24 1.54940 0.774698 0.632331i \(-0.217902\pi\)
0.774698 + 0.632331i \(0.217902\pi\)
\(954\) 2.16389e24 3.15390
\(955\) 0 0
\(956\) −2.17348e24 −3.11525
\(957\) 0 0
\(958\) 0 0
\(959\) 1.01844e24 1.42360
\(960\) 0 0
\(961\) 7.27423e23 1.00000
\(962\) 0 0
\(963\) −8.03236e23 −1.08601
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 2.77130e23 0.362470 0.181235 0.983440i \(-0.441990\pi\)
0.181235 + 0.983440i \(0.441990\pi\)
\(968\) 6.15226e23 0.798053
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −2.76900e24 −3.41863
\(975\) 0 0
\(976\) 0 0
\(977\) 1.30752e24 1.57505 0.787523 0.616285i \(-0.211363\pi\)
0.787523 + 0.616285i \(0.211363\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −1.03838e24 −1.21061
\(982\) −1.02357e24 −1.18366
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −3.41470e24 −3.73063
\(990\) 0 0
\(991\) 1.85984e24 1.99933 0.999667 0.0258078i \(-0.00821579\pi\)
0.999667 + 0.0258078i \(0.00821579\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 1.29546e24 1.35936
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) −2.60480e24 −2.64685
\(999\) 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 \(a_p\))