Properties

Label 7.9.b.a.6.1
Level 7
Weight 9
Character 7.6
Self dual Yes
Analytic conductor 2.852
Analytic rank 0
Dimension 1
CM disc. -7
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 7 \)
Weight: \( k \) = \( 9 \)
Character orbit: \([\chi]\) = 7.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(2.85165027043\)
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

\(f(q)\) \(=\) \(q\)\(-31.0000 q^{2}\) \(+705.000 q^{4}\) \(+2401.00 q^{7}\) \(-13919.0 q^{8}\) \(+6561.00 q^{9}\) \(+O(q^{10})\) \(q\)\(-31.0000 q^{2}\) \(+705.000 q^{4}\) \(+2401.00 q^{7}\) \(-13919.0 q^{8}\) \(+6561.00 q^{9}\) \(+13154.0 q^{11}\) \(-74431.0 q^{14}\) \(+251009. q^{16}\) \(-203391. q^{18}\) \(-407774. q^{22}\) \(-20926.0 q^{23}\) \(+390625. q^{25}\) \(+1.69270e6 q^{28}\) \(+108194. q^{29}\) \(-4.21802e6 q^{32}\) \(+4.62550e6 q^{36}\) \(-2.07389e6 q^{37}\) \(-6.72605e6 q^{43}\) \(+9.27357e6 q^{44}\) \(+648706. q^{46}\) \(+5.76480e6 q^{49}\) \(-1.21094e7 q^{50}\) \(+1.53778e7 q^{53}\) \(-3.34195e7 q^{56}\) \(-3.35401e6 q^{58}\) \(+1.57530e7 q^{63}\) \(+6.65002e7 q^{64}\) \(-1.58393e7 q^{67}\) \(-4.23320e7 q^{71}\) \(-9.13226e7 q^{72}\) \(+6.42905e7 q^{74}\) \(+3.15828e7 q^{77}\) \(-6.46068e7 q^{79}\) \(+4.30467e7 q^{81}\) \(+2.08507e8 q^{86}\) \(-1.83091e8 q^{88}\) \(-1.47528e7 q^{92}\) \(-1.78709e8 q^{98}\) \(+8.63034e7 q^{99}\) \(+O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −31.0000 −1.93750 −0.968750 0.248039i \(-0.920214\pi\)
−0.968750 + 0.248039i \(0.920214\pi\)
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 705.000 2.75391
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 2401.00 1.00000
\(8\) −13919.0 −3.39819
\(9\) 6561.00 1.00000
\(10\) 0 0
\(11\) 13154.0 0.898436 0.449218 0.893422i \(-0.351703\pi\)
0.449218 + 0.893422i \(0.351703\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) −74431.0 −1.93750
\(15\) 0 0
\(16\) 251009. 3.83009
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) −203391. −1.93750
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −407774. −1.74072
\(23\) −20926.0 −0.0747782 −0.0373891 0.999301i \(-0.511904\pi\)
−0.0373891 + 0.999301i \(0.511904\pi\)
\(24\) 0 0
\(25\) 390625. 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 1.69270e6 2.75391
\(29\) 108194. 0.152972 0.0764859 0.997071i \(-0.475630\pi\)
0.0764859 + 0.997071i \(0.475630\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −4.21802e6 −4.02261
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 4.62550e6 2.75391
\(37\) −2.07389e6 −1.10657 −0.553284 0.832993i \(-0.686626\pi\)
−0.553284 + 0.832993i \(0.686626\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −6.72605e6 −1.96737 −0.983685 0.179900i \(-0.942423\pi\)
−0.983685 + 0.179900i \(0.942423\pi\)
\(44\) 9.27357e6 2.47421
\(45\) 0 0
\(46\) 648706. 0.144883
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 5.76480e6 1.00000
\(50\) −1.21094e7 −1.93750
\(51\) 0 0
\(52\) 0 0
\(53\) 1.53778e7 1.94890 0.974450 0.224604i \(-0.0721087\pi\)
0.974450 + 0.224604i \(0.0721087\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −3.34195e7 −3.39819
\(57\) 0 0
\(58\) −3.35401e6 −0.296383
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 1.57530e7 1.00000
\(64\) 6.65002e7 3.96372
\(65\) 0 0
\(66\) 0 0
\(67\) −1.58393e7 −0.786027 −0.393014 0.919533i \(-0.628567\pi\)
−0.393014 + 0.919533i \(0.628567\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −4.23320e7 −1.66585 −0.832923 0.553388i \(-0.813335\pi\)
−0.832923 + 0.553388i \(0.813335\pi\)
\(72\) −9.13226e7 −3.39819
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 6.42905e7 2.14397
\(75\) 0 0
\(76\) 0 0
\(77\) 3.15828e7 0.898436
\(78\) 0 0
\(79\) −6.46068e7 −1.65871 −0.829354 0.558723i \(-0.811292\pi\)
−0.829354 + 0.558723i \(0.811292\pi\)
\(80\) 0 0
\(81\) 4.30467e7 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 2.08507e8 3.81178
\(87\) 0 0
\(88\) −1.83091e8 −3.05306
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −1.47528e7 −0.205932
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) −1.78709e8 −1.93750
\(99\) 8.63034e7 0.898436
\(100\) 2.75391e8 2.75391
\(101\) 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
\(106\) −4.76711e8 −3.77599
\(107\) −1.25316e8 −0.956030 −0.478015 0.878352i \(-0.658644\pi\)
−0.478015 + 0.878352i \(0.658644\pi\)
\(108\) 0 0
\(109\) −1.25416e8 −0.888476 −0.444238 0.895909i \(-0.646526\pi\)
−0.444238 + 0.895909i \(0.646526\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 6.02673e8 3.83009
\(113\) 2.37778e8 1.45834 0.729168 0.684335i \(-0.239907\pi\)
0.729168 + 0.684335i \(0.239907\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 7.62768e7 0.421270
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −4.13312e7 −0.192813
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) −4.88342e8 −1.93750
\(127\) 5.20031e8 1.99901 0.999504 0.0314912i \(-0.0100256\pi\)
0.999504 + 0.0314912i \(0.0100256\pi\)
\(128\) −9.81693e8 −3.65709
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 4.91019e8 1.52293
\(135\) 0 0
\(136\) 0 0
\(137\) −6.51814e8 −1.85030 −0.925149 0.379605i \(-0.876060\pi\)
−0.925149 + 0.379605i \(0.876060\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 1.31229e9 3.22758
\(143\) 0 0
\(144\) 1.64687e9 3.83009
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) −1.46209e9 −3.04738
\(149\) −8.89611e8 −1.80491 −0.902454 0.430786i \(-0.858236\pi\)
−0.902454 + 0.430786i \(0.858236\pi\)
\(150\) 0 0
\(151\) −1.71055e8 −0.329023 −0.164512 0.986375i \(-0.552605\pi\)
−0.164512 + 0.986375i \(0.552605\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −9.79065e8 −1.74072
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 2.00281e9 3.21375
\(159\) 0 0
\(160\) 0 0
\(161\) −5.02433e7 −0.0747782
\(162\) −1.33445e9 −1.93750
\(163\) 8.59843e8 1.21806 0.609030 0.793147i \(-0.291559\pi\)
0.609030 + 0.793147i \(0.291559\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 8.15731e8 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) −4.74186e9 −5.41795
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 9.37891e8 1.00000
\(176\) 3.30177e9 3.44109
\(177\) 0 0
\(178\) 0 0
\(179\) 7.43255e8 0.723978 0.361989 0.932182i \(-0.382098\pi\)
0.361989 + 0.932182i \(0.382098\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 2.91269e8 0.254111
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −1.86863e9 −1.40407 −0.702036 0.712141i \(-0.747726\pi\)
−0.702036 + 0.712141i \(0.747726\pi\)
\(192\) 0 0
\(193\) 2.21701e9 1.59786 0.798930 0.601424i \(-0.205400\pi\)
0.798930 + 0.601424i \(0.205400\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 4.06418e9 2.75391
\(197\) −3.00864e9 −1.99759 −0.998794 0.0490975i \(-0.984365\pi\)
−0.998794 + 0.0490975i \(0.984365\pi\)
\(198\) −2.67541e9 −1.74072
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) −5.43711e9 −3.39819
\(201\) 0 0
\(202\) 0 0
\(203\) 2.59774e8 0.152972
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −1.37295e8 −0.0747782
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −3.82599e9 −1.93025 −0.965126 0.261787i \(-0.915688\pi\)
−0.965126 + 0.261787i \(0.915688\pi\)
\(212\) 1.08413e10 5.36709
\(213\) 0 0
\(214\) 3.88480e9 1.85231
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 3.88789e9 1.72142
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) −1.01275e10 −4.02261
\(225\) 2.56289e9 1.00000
\(226\) −7.37111e9 −2.82553
\(227\) 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
\(232\) −1.50595e9 −0.519828
\(233\) 5.82434e9 1.97616 0.988082 0.153930i \(-0.0491932\pi\)
0.988082 + 0.153930i \(0.0491932\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −2.30608e9 −0.706776 −0.353388 0.935477i \(-0.614970\pi\)
−0.353388 + 0.935477i \(0.614970\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 1.28127e9 0.373575
\(243\) 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 \(\pi\)
\(252\) 1.11058e10 2.75391
\(253\) −2.75261e8 −0.0671834
\(254\) −1.61210e10 −3.87308
\(255\) 0 0
\(256\) 1.34084e10 3.12190
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) −4.97940e9 −1.10657
\(260\) 0 0
\(261\) 7.09861e8 0.152972
\(262\) 0 0
\(263\) 2.45793e9 0.513744 0.256872 0.966445i \(-0.417308\pi\)
0.256872 + 0.966445i \(0.417308\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −1.11667e10 −2.16464
\(269\) 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
\(274\) 2.02062e10 3.58495
\(275\) 5.13828e9 0.898436
\(276\) 0 0
\(277\) −9.00181e9 −1.52901 −0.764506 0.644616i \(-0.777017\pi\)
−0.764506 + 0.644616i \(0.777017\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 8.53387e9 1.36874 0.684369 0.729135i \(-0.260078\pi\)
0.684369 + 0.729135i \(0.260078\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) −2.98440e10 −4.58759
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −2.76744e10 −4.02261
\(289\) 6.97576e9 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
\(296\) 2.88664e10 3.76033
\(297\) 0 0
\(298\) 2.75779e10 3.49701
\(299\) 0 0
\(300\) 0 0
\(301\) −1.61492e10 −1.96737
\(302\) 5.30269e9 0.637483
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 2.22658e10 2.47421
\(309\) 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
\(316\) −4.55478e10 −4.56793
\(317\) −1.50830e10 −1.49365 −0.746827 0.665019i \(-0.768424\pi\)
−0.746827 + 0.665019i \(0.768424\pi\)
\(318\) 0 0
\(319\) 1.42318e9 0.137435
\(320\) 0 0
\(321\) 0 0
\(322\) 1.55754e9 0.144883
\(323\) 0 0
\(324\) 3.03479e10 2.75391
\(325\) 0 0
\(326\) −2.66551e10 −2.35999
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 9.42105e9 0.784851 0.392425 0.919784i \(-0.371636\pi\)
0.392425 + 0.919784i \(0.371636\pi\)
\(332\) 0 0
\(333\) −1.36068e10 −1.10657
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 5.20199e9 0.403320 0.201660 0.979456i \(-0.435366\pi\)
0.201660 + 0.979456i \(0.435366\pi\)
\(338\) −2.52877e10 −1.93750
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1.38413e10 1.00000
\(344\) 9.36198e10 6.68550
\(345\) 0 0
\(346\) 0 0
\(347\) 1.89057e10 1.30399 0.651995 0.758223i \(-0.273932\pi\)
0.651995 + 0.758223i \(0.273932\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) −2.90746e10 −1.93750
\(351\) 0 0
\(352\) −5.54838e10 −3.61406
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) −2.30409e10 −1.40271
\(359\) −4.33626e9 −0.261058 −0.130529 0.991444i \(-0.541668\pi\)
−0.130529 + 0.991444i \(0.541668\pi\)
\(360\) 0 0
\(361\) 1.69836e10 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 \(\pi\)
\(368\) −5.25261e9 −0.286407
\(369\) 0 0
\(370\) 0 0
\(371\) 3.69220e10 1.94890
\(372\) 0 0
\(373\) 5.22427e9 0.269892 0.134946 0.990853i \(-0.456914\pi\)
0.134946 + 0.990853i \(0.456914\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −3.32169e10 −1.60991 −0.804956 0.593335i \(-0.797811\pi\)
−0.804956 + 0.593335i \(0.797811\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 5.79275e10 2.72039
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −6.87274e10 −3.09585
\(387\) −4.41296e10 −1.96737
\(388\) 0 0
\(389\) −2.59393e10 −1.13282 −0.566409 0.824124i \(-0.691668\pi\)
−0.566409 + 0.824124i \(0.691668\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −8.02403e10 −3.39819
\(393\) 0 0
\(394\) 9.32680e10 3.87033
\(395\) 0 0
\(396\) 6.08439e10 2.47421
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 9.80504e10 3.83009
\(401\) −1.31857e10 −0.509949 −0.254974 0.966948i \(-0.582067\pi\)
−0.254974 + 0.966948i \(0.582067\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) −8.05299e9 −0.296383
\(407\) −2.72799e10 −0.994180
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 4.25616e9 0.144883
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 4.40519e10 1.40228 0.701142 0.713022i \(-0.252674\pi\)
0.701142 + 0.713022i \(0.252674\pi\)
\(422\) 1.18606e11 3.73986
\(423\) 0 0
\(424\) −2.14043e11 −6.62274
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −8.83478e10 −2.63282
\(429\) 0 0
\(430\) 0 0
\(431\) 5.02026e10 1.45485 0.727423 0.686189i \(-0.240718\pi\)
0.727423 + 0.686189i \(0.240718\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −8.84180e10 −2.44678
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 3.78229e10 1.00000
\(442\) 0 0
\(443\) −5.25970e10 −1.36567 −0.682836 0.730572i \(-0.739254\pi\)
−0.682836 + 0.730572i \(0.739254\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 1.59667e11 3.96372
\(449\) 7.55570e10 1.85904 0.929521 0.368768i \(-0.120220\pi\)
0.929521 + 0.368768i \(0.120220\pi\)
\(450\) −7.94496e10 −1.93750
\(451\) 0 0
\(452\) 1.67633e11 4.01612
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 3.75045e10 0.859843 0.429921 0.902866i \(-0.358541\pi\)
0.429921 + 0.902866i \(0.358541\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −9.12559e10 −1.98581 −0.992904 0.118920i \(-0.962057\pi\)
−0.992904 + 0.118920i \(0.962057\pi\)
\(464\) 2.71577e10 0.585896
\(465\) 0 0
\(466\) −1.80554e11 −3.82882
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) −3.80302e10 −0.786027
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −8.84744e10 −1.76756
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 1.00893e11 1.94890
\(478\) 7.14883e10 1.36938
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −2.91385e10 −0.530989
\(485\) 0 0
\(486\) 0 0
\(487\) −1.26840e10 −0.225496 −0.112748 0.993624i \(-0.535965\pi\)
−0.112748 + 0.993624i \(0.535965\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 6.69045e10 1.15114 0.575572 0.817751i \(-0.304780\pi\)
0.575572 + 0.817751i \(0.304780\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −1.01639e11 −1.66585
\(498\) 0 0
\(499\) 4.34400e10 0.700628 0.350314 0.936632i \(-0.386075\pi\)
0.350314 + 0.936632i \(0.386075\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) −2.19265e11 −3.39819
\(505\) 0 0
\(506\) 8.53308e9 0.130168
\(507\) 0 0
\(508\) 3.66622e11 5.50508
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.64348e11 −2.39158
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 1.54361e11 2.14397
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) −2.20057e10 −0.296383
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −7.61958e10 −0.995379
\(527\) 0 0
\(528\) 0 0
\(529\) −7.78731e10 −0.994408
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 2.20468e11 2.67107
\(537\) 0 0
\(538\) 0 0
\(539\) 7.58302e10 0.898436
\(540\) 0 0
\(541\) 1.64943e11 1.92551 0.962755 0.270374i \(-0.0871473\pi\)
0.962755 + 0.270374i \(0.0871473\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −1.66832e11 −1.86350 −0.931749 0.363102i \(-0.881718\pi\)
−0.931749 + 0.363102i \(0.881718\pi\)
\(548\) −4.59529e11 −5.09554
\(549\) 0 0
\(550\) −1.59287e11 −1.74072
\(551\) 0 0
\(552\) 0 0
\(553\) −1.55121e11 −1.65871
\(554\) 2.79056e11 2.96246
\(555\) 0 0
\(556\) 0 0
\(557\) −4.54320e10 −0.471999 −0.236000 0.971753i \(-0.575836\pi\)
−0.236000 + 0.971753i \(0.575836\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −2.64550e11 −2.65193
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 1.03355e11 1.00000
\(568\) 5.89219e11 5.66087
\(569\) −1.61328e11 −1.53908 −0.769538 0.638601i \(-0.779513\pi\)
−0.769538 + 0.638601i \(0.779513\pi\)
\(570\) 0 0
\(571\) 1.66597e11 1.56719 0.783596 0.621271i \(-0.213383\pi\)
0.783596 + 0.621271i \(0.213383\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −8.17422e9 −0.0747782
\(576\) 4.36308e11 3.96372
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) −2.16248e11 −1.93750
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 2.02279e11 1.75096
\(584\) 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
\(592\) −5.20564e11 −4.23826
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −6.27176e11 −4.97055
\(597\) 0 0
\(598\) 0 0
\(599\) 2.14941e11 1.66960 0.834798 0.550556i \(-0.185584\pi\)
0.834798 + 0.550556i \(0.185584\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 5.00626e11 3.81178
\(603\) −1.03922e11 −0.786027
\(604\) −1.20593e11 −0.906100
\(605\) 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
\(613\) 2.13231e11 1.51011 0.755055 0.655662i \(-0.227610\pi\)
0.755055 + 0.655662i \(0.227610\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) −4.39600e11 −3.05306
\(617\) −8.73356e10 −0.602630 −0.301315 0.953525i \(-0.597426\pi\)
−0.301315 + 0.953525i \(0.597426\pi\)
\(618\) 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
\(625\) 1.52588e11 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −2.70532e11 −1.70648 −0.853239 0.521519i \(-0.825365\pi\)
−0.853239 + 0.521519i \(0.825365\pi\)
\(632\) 8.99263e11 5.63661
\(633\) 0 0
\(634\) 4.67572e11 2.89395
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) −4.41187e10 −0.266281
\(639\) −2.77740e11 −1.66585
\(640\) 0 0
\(641\) −3.21718e11 −1.90565 −0.952827 0.303515i \(-0.901840\pi\)
−0.952827 + 0.303515i \(0.901840\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) −3.54215e10 −0.205932
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) −5.99167e11 −3.39819
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 6.06189e11 3.35442
\(653\) −3.55288e10 −0.195402 −0.0977008 0.995216i \(-0.531149\pi\)
−0.0977008 + 0.995216i \(0.531149\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −1.40383e11 −0.744345 −0.372172 0.928164i \(-0.621387\pi\)
−0.372172 + 0.928164i \(0.621387\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) −2.92052e11 −1.52065
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 4.21810e11 2.14397
\(667\) −2.26407e9 −0.0114389
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 8.94776e10 0.436168 0.218084 0.975930i \(-0.430019\pi\)
0.218084 + 0.975930i \(0.430019\pi\)
\(674\) −1.61262e11 −0.781433
\(675\) 0 0
\(676\) 5.75090e11 2.75391
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 2.99681e11 1.37713 0.688567 0.725172i \(-0.258240\pi\)
0.688567 + 0.725172i \(0.258240\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −4.29080e11 −1.93750
\(687\) 0 0
\(688\) −1.68830e12 −7.53521
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 2.07214e11 0.898436
\(694\) −5.86076e11 −2.52648
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 6.61213e11 2.75391
\(701\) 4.61061e11 1.90935 0.954676 0.297648i \(-0.0962021\pi\)
0.954676 + 0.297648i \(0.0962021\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 8.74743e11 3.56115
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 3.12856e11 1.23811 0.619056 0.785347i \(-0.287516\pi\)
0.619056 + 0.785347i \(0.287516\pi\)
\(710\) 0 0
\(711\) −4.23886e11 −1.65871
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 5.23994e11 1.99377
\(717\) 0 0
\(718\) 1.34424e11 0.505801
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −5.26490e11 −1.93750
\(723\) 0 0
\(724\) 0 0
\(725\) 4.22633e10 0.152972
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 2.82430e11 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
\(736\) 8.82662e10 0.300804
\(737\) −2.08350e11 −0.706195
\(738\) 0 0
\(739\) −4.27712e11 −1.43408 −0.717040 0.697032i \(-0.754504\pi\)
−0.717040 + 0.697032i \(0.754504\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −1.14458e12 −3.77599
\(743\) 5.80987e11 1.90639 0.953194 0.302358i \(-0.0977739\pi\)
0.953194 + 0.302358i \(0.0977739\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −1.61952e11 −0.522916
\(747\) 0 0
\(748\) 0 0
\(749\) −3.00884e11 −0.956030
\(750\) 0 0
\(751\) −4.01165e11 −1.26114 −0.630570 0.776132i \(-0.717179\pi\)
−0.630570 + 0.776132i \(0.717179\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 1.48198e10 0.0451292 0.0225646 0.999745i \(-0.492817\pi\)
0.0225646 + 0.999745i \(0.492817\pi\)
\(758\) 1.02972e12 3.11920
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) −3.01123e11 −0.888476
\(764\) −1.31738e12 −3.86668
\(765\) 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
\(772\) 1.56299e12 4.40036
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 1.36802e12 3.81178
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 8.04119e11 2.19484
\(779\) 0 0
\(780\) 0 0
\(781\) −5.56835e11 −1.49666
\(782\) 0 0
\(783\) 0 0
\(784\) 1.44702e12 3.83009
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) −2.12109e12 −5.50117
\(789\) 0 0
\(790\) 0 0
\(791\) 5.70904e11 1.45834
\(792\) −1.20126e12 −3.05306
\(793\) 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
\(800\) −1.64766e12 −4.02261
\(801\) 0 0
\(802\) 4.08758e11 0.988026
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 7.78347e11 1.81710 0.908551 0.417774i \(-0.137190\pi\)
0.908551 + 0.417774i \(0.137190\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 1.83141e11 0.421270
\(813\) 0 0
\(814\) 8.45677e11 1.92622
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −9.08612e11 −1.99989 −0.999944 0.0105600i \(-0.996639\pi\)
−0.999944 + 0.0105600i \(0.996639\pi\)
\(822\) 0 0
\(823\) 1.90374e10 0.0414961 0.0207481 0.999785i \(-0.493395\pi\)
0.0207481 + 0.999785i \(0.493395\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 7.24287e11 1.54842 0.774210 0.632929i \(-0.218148\pi\)
0.774210 + 0.632929i \(0.218148\pi\)
\(828\) −9.67933e10 −0.205932
\(829\) 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
\(841\) −4.88540e11 −0.976600
\(842\) −1.36561e12 −2.71692
\(843\) 0 0
\(844\) −2.69732e12 −5.31573
\(845\) 0 0
\(846\) 0 0
\(847\) −9.92361e10 −0.192813
\(848\) 3.85996e12 7.46447
\(849\) 0 0
\(850\) 0 0
\(851\) 4.33981e10 0.0827471
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 1.74427e12 3.24877
\(857\) 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
\(862\) −1.55628e12 −2.81876
\(863\) 5.10419e11 0.920203 0.460101 0.887866i \(-0.347813\pi\)
0.460101 + 0.887866i \(0.347813\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −8.49838e11 −1.49024
\(870\) 0 0
\(871\) 0 0
\(872\) 1.74566e12 3.01921
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −2.19301e11 −0.370718 −0.185359 0.982671i \(-0.559345\pi\)
−0.185359 + 0.982671i \(0.559345\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) −1.17251e12 −1.93750
\(883\) −6.77048e10 −0.111372 −0.0556861 0.998448i \(-0.517735\pi\)
−0.0556861 + 0.998448i \(0.517735\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 1.63051e12 2.64599
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 1.24859e12 1.99901
\(890\) 0 0
\(891\) 5.66237e11 0.898436
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) −2.35705e12 −3.65709
\(897\) 0 0
\(898\) −2.34227e12 −3.60190
\(899\) 0 0
\(900\) 1.80684e12 2.75391
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) −3.30963e12 −4.95571
\(905\) 0 0
\(906\) 0 0
\(907\) 1.03197e12 1.52489 0.762445 0.647053i \(-0.223999\pi\)
0.762445 + 0.647053i \(0.223999\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −7.49459e11 −1.08811 −0.544057 0.839048i \(-0.683113\pi\)
−0.544057 + 0.839048i \(0.683113\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −1.16264e12 −1.66595
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 7.40117e11 1.03762 0.518810 0.854889i \(-0.326375\pi\)
0.518810 + 0.854889i \(0.326375\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −8.10112e11 −1.10657
\(926\) 2.82893e12 3.84750
\(927\) 0 0
\(928\) −4.56364e11 −0.615346
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 4.10616e12 5.44217
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 1.17894e12 1.52293
\(939\) 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
\(946\) 2.74271e12 3.42464
\(947\) −1.41632e12 −1.76101 −0.880506 0.474035i \(-0.842797\pi\)
−0.880506 + 0.474035i \(0.842797\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1.55399e12 −1.88398 −0.941992 0.335635i \(-0.891049\pi\)
−0.941992 + 0.335635i \(0.891049\pi\)
\(954\) −3.12770e12 −3.77599
\(955\) 0 0
\(956\) −1.62578e12 −1.94640
\(957\) 0 0
\(958\) 0 0
\(959\) −1.56501e12 −1.85030
\(960\) 0 0
\(961\) 8.52891e11 1.00000
\(962\) 0 0
\(963\) −8.22198e11 −0.956030
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 1.34397e12 1.53703 0.768516 0.639830i \(-0.220995\pi\)
0.768516 + 0.639830i \(0.220995\pi\)
\(968\) 5.75288e11 0.655216
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 3.93203e11 0.436899
\(975\) 0 0
\(976\) 0 0
\(977\) 1.72274e12 1.89078 0.945390 0.325943i \(-0.105682\pi\)
0.945390 + 0.325943i \(0.105682\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −8.22852e11 −0.888476
\(982\) −2.07404e12 −2.23034
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 1.40749e11 0.147116
\(990\) 0 0
\(991\) −1.92881e12 −1.99983 −0.999917 0.0129050i \(-0.995892\pi\)
−0.999917 + 0.0129050i \(0.995892\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 3.15081e12 3.22758
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) −1.34664e12 −1.35747
\(999\) 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 \(a_p\))