Properties

Label 12.5.c.a.5.1
Level 12
Weight 5
Character 12.5
Self dual Yes
Analytic conductor 1.240
Analytic rank 0
Dimension 1
CM disc. -3
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) = \( 5 \)
Character orbit: \([\chi]\) = 12.c (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(1.24043955701\)
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 5.1
Root \(0\)
Character \(\chi\) = 12.5

$q$-expansion

\(f(q)\) \(=\) \(q+9.00000 q^{3} -94.0000 q^{7} +81.0000 q^{9} +O(q^{10})\) \(q+9.00000 q^{3} -94.0000 q^{7} +81.0000 q^{9} +146.000 q^{13} -46.0000 q^{19} -846.000 q^{21} +625.000 q^{25} +729.000 q^{27} +194.000 q^{31} -2062.00 q^{37} +1314.00 q^{39} -3214.00 q^{43} +6435.00 q^{49} -414.000 q^{57} -1966.00 q^{61} -7614.00 q^{63} +5906.00 q^{67} -8542.00 q^{73} +5625.00 q^{75} +7682.00 q^{79} +6561.00 q^{81} -13724.0 q^{91} +1746.00 q^{93} -18814.0 q^{97} +O(q^{100})\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/12\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(7\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 9.00000 1.00000
\(4\) 0 0
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) −94.0000 −1.91837 −0.959184 0.282784i \(-0.908742\pi\)
−0.959184 + 0.282784i \(0.908742\pi\)
\(8\) 0 0
\(9\) 81.0000 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 146.000 0.863905 0.431953 0.901896i \(-0.357825\pi\)
0.431953 + 0.901896i \(0.357825\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) −46.0000 −0.127424 −0.0637119 0.997968i \(-0.520294\pi\)
−0.0637119 + 0.997968i \(0.520294\pi\)
\(20\) 0 0
\(21\) −846.000 −1.91837
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 625.000 1.00000
\(26\) 0 0
\(27\) 729.000 1.00000
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 194.000 0.201873 0.100937 0.994893i \(-0.467816\pi\)
0.100937 + 0.994893i \(0.467816\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2062.00 −1.50621 −0.753104 0.657901i \(-0.771445\pi\)
−0.753104 + 0.657901i \(0.771445\pi\)
\(38\) 0 0
\(39\) 1314.00 0.863905
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −3214.00 −1.73824 −0.869118 0.494604i \(-0.835313\pi\)
−0.869118 + 0.494604i \(0.835313\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 6435.00 2.68013
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −414.000 −0.127424
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) −1966.00 −0.528353 −0.264176 0.964474i \(-0.585100\pi\)
−0.264176 + 0.964474i \(0.585100\pi\)
\(62\) 0 0
\(63\) −7614.00 −1.91837
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 5906.00 1.31566 0.657830 0.753166i \(-0.271474\pi\)
0.657830 + 0.753166i \(0.271474\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −8542.00 −1.60293 −0.801464 0.598043i \(-0.795945\pi\)
−0.801464 + 0.598043i \(0.795945\pi\)
\(74\) 0 0
\(75\) 5625.00 1.00000
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 7682.00 1.23089 0.615446 0.788179i \(-0.288976\pi\)
0.615446 + 0.788179i \(0.288976\pi\)
\(80\) 0 0
\(81\) 6561.00 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) −13724.0 −1.65729
\(92\) 0 0
\(93\) 1746.00 0.201873
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −18814.0 −1.99957 −0.999787 0.0206175i \(-0.993437\pi\)
−0.999787 + 0.0206175i \(0.993437\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 16418.0 1.54755 0.773777 0.633458i \(-0.218365\pi\)
0.773777 + 0.633458i \(0.218365\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 22034.0 1.85456 0.927279 0.374371i \(-0.122141\pi\)
0.927279 + 0.374371i \(0.122141\pi\)
\(110\) 0 0
\(111\) −18558.0 −1.50621
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 11826.0 0.863905
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 14641.0 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −10942.0 −0.678405 −0.339203 0.940713i \(-0.610157\pi\)
−0.339203 + 0.940713i \(0.610157\pi\)
\(128\) 0 0
\(129\) −28926.0 −1.73824
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 4324.00 0.244446
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) −38158.0 −1.97495 −0.987475 0.157777i \(-0.949567\pi\)
−0.987475 + 0.157777i \(0.949567\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 57915.0 2.68013
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 36194.0 1.58739 0.793693 0.608318i \(-0.208156\pi\)
0.793693 + 0.608318i \(0.208156\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −35374.0 −1.43511 −0.717554 0.696502i \(-0.754739\pi\)
−0.717554 + 0.696502i \(0.754739\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 15506.0 0.583612 0.291806 0.956477i \(-0.405744\pi\)
0.291806 + 0.956477i \(0.405744\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) −7245.00 −0.253668
\(170\) 0 0
\(171\) −3726.00 −0.127424
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) −58750.0 −1.91837
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) 33074.0 1.00955 0.504777 0.863250i \(-0.331575\pi\)
0.504777 + 0.863250i \(0.331575\pi\)
\(182\) 0 0
\(183\) −17694.0 −0.528353
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −68526.0 −1.91837
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 71426.0 1.91753 0.958764 0.284203i \(-0.0917291\pi\)
0.958764 + 0.284203i \(0.0917291\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 69794.0 1.76243 0.881215 0.472715i \(-0.156726\pi\)
0.881215 + 0.472715i \(0.156726\pi\)
\(200\) 0 0
\(201\) 53154.0 1.31566
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −61486.0 −1.38106 −0.690528 0.723306i \(-0.742622\pi\)
−0.690528 + 0.723306i \(0.742622\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −18236.0 −0.387267
\(218\) 0 0
\(219\) −76878.0 −1.60293
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 14786.0 0.297332 0.148666 0.988888i \(-0.452502\pi\)
0.148666 + 0.988888i \(0.452502\pi\)
\(224\) 0 0
\(225\) 50625.0 1.00000
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) −104206. −1.98711 −0.993555 0.113354i \(-0.963841\pi\)
−0.993555 + 0.113354i \(0.963841\pi\)
\(230\) 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
\(237\) 69138.0 1.23089
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −34366.0 −0.591691 −0.295845 0.955236i \(-0.595601\pi\)
−0.295845 + 0.955236i \(0.595601\pi\)
\(242\) 0 0
\(243\) 59049.0 1.00000
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −6716.00 −0.110082
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 193828. 2.88946
\(260\) 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
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) −88318.0 −1.20257 −0.601285 0.799034i \(-0.705345\pi\)
−0.601285 + 0.799034i \(0.705345\pi\)
\(272\) 0 0
\(273\) −123516. −1.65729
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −138574. −1.80602 −0.903009 0.429621i \(-0.858647\pi\)
−0.903009 + 0.429621i \(0.858647\pi\)
\(278\) 0 0
\(279\) 15714.0 0.201873
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 49586.0 0.619136 0.309568 0.950877i \(-0.399816\pi\)
0.309568 + 0.950877i \(0.399816\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 83521.0 1.00000
\(290\) 0 0
\(291\) −169326. −1.99957
\(292\) 0 0
\(293\) 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
\(300\) 0 0
\(301\) 302116. 3.33458
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −60334.0 −0.640155 −0.320078 0.947391i \(-0.603709\pi\)
−0.320078 + 0.947391i \(0.603709\pi\)
\(308\) 0 0
\(309\) 147762. 1.54755
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) −175774. −1.79418 −0.897090 0.441848i \(-0.854323\pi\)
−0.897090 + 0.441848i \(0.854323\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 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
\(324\) 0 0
\(325\) 91250.0 0.863905
\(326\) 0 0
\(327\) 198306. 1.85456
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −88078.0 −0.803917 −0.401959 0.915658i \(-0.631670\pi\)
−0.401959 + 0.915658i \(0.631670\pi\)
\(332\) 0 0
\(333\) −167022. −1.50621
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 5186.00 0.0456639 0.0228319 0.999739i \(-0.492732\pi\)
0.0228319 + 0.999739i \(0.492732\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −379196. −3.22311
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) 8402.00 0.0689814 0.0344907 0.999405i \(-0.489019\pi\)
0.0344907 + 0.999405i \(0.489019\pi\)
\(350\) 0 0
\(351\) 106434. 0.863905
\(352\) 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
\(361\) −128205. −0.983763
\(362\) 0 0
\(363\) 131769. 1.00000
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 246146. 1.82751 0.913757 0.406261i \(-0.133168\pi\)
0.913757 + 0.406261i \(0.133168\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 208946. 1.50181 0.750907 0.660407i \(-0.229616\pi\)
0.750907 + 0.660407i \(0.229616\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 194354. 1.35305 0.676527 0.736418i \(-0.263484\pi\)
0.676527 + 0.736418i \(0.263484\pi\)
\(380\) 0 0
\(381\) −98478.0 −0.678405
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −260334. −1.73824
\(388\) 0 0
\(389\) 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
\(397\) −184174. −1.16855 −0.584275 0.811556i \(-0.698621\pi\)
−0.584275 + 0.811556i \(0.698621\pi\)
\(398\) 0 0
\(399\) 38916.0 0.244446
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 28324.0 0.174399
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 57314.0 0.342621 0.171311 0.985217i \(-0.445200\pi\)
0.171311 + 0.985217i \(0.445200\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −343422. −1.97495
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) −226318. −1.27689 −0.638447 0.769666i \(-0.720423\pi\)
−0.638447 + 0.769666i \(0.720423\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 184804. 1.01357
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 368066. 1.96313 0.981567 0.191119i \(-0.0612116\pi\)
0.981567 + 0.191119i \(0.0612116\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −376606. −1.95415 −0.977076 0.212892i \(-0.931712\pi\)
−0.977076 + 0.212892i \(0.931712\pi\)
\(440\) 0 0
\(441\) 521235. 2.68013
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 325746. 1.58739
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 244898. 1.17261 0.586304 0.810091i \(-0.300582\pi\)
0.586304 + 0.810091i \(0.300582\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −152062. −0.709347 −0.354673 0.934990i \(-0.615408\pi\)
−0.354673 + 0.934990i \(0.615408\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) −555164. −2.52392
\(470\) 0 0
\(471\) −318366. −1.43511
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −28750.0 −0.127424
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) −301052. −1.30122
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 451106. 1.90204 0.951022 0.309122i \(-0.100035\pi\)
0.951022 + 0.309122i \(0.100035\pi\)
\(488\) 0 0
\(489\) 139554. 0.583612
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −497326. −1.99729 −0.998643 0.0520865i \(-0.983413\pi\)
−0.998643 + 0.0520865i \(0.983413\pi\)
\(500\) 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
\(507\) −65205.0 −0.253668
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 802948. 3.07500
\(512\) 0 0
\(513\) −33534.0 −0.127424
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 417266. 1.52549 0.762745 0.646699i \(-0.223851\pi\)
0.762745 + 0.646699i \(0.223851\pi\)
\(524\) 0 0
\(525\) −528750. −1.91837
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 279841. 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 483794. 1.65297 0.826487 0.562956i \(-0.190336\pi\)
0.826487 + 0.562956i \(0.190336\pi\)
\(542\) 0 0
\(543\) 297666. 1.00955
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −342382. −1.14429 −0.572145 0.820152i \(-0.693889\pi\)
−0.572145 + 0.820152i \(0.693889\pi\)
\(548\) 0 0
\(549\) −159246. −0.528353
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −722108. −2.36130
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) −469244. −1.50167
\(560\) 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
\(567\) −616734. −1.91837
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 132914. 0.407660 0.203830 0.979006i \(-0.434661\pi\)
0.203830 + 0.979006i \(0.434661\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 259586. 0.779704 0.389852 0.920878i \(-0.372526\pi\)
0.389852 + 0.920878i \(0.372526\pi\)
\(578\) 0 0
\(579\) 642834. 1.91753
\(580\) 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 \(\pi\)
\(588\) 0 0
\(589\) −8924.00 −0.0257234
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 628146. 1.76243
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) −445726. −1.23401 −0.617005 0.786959i \(-0.711654\pi\)
−0.617005 + 0.786959i \(0.711654\pi\)
\(602\) 0 0
\(603\) 478386. 1.31566
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −74302.0 −0.201662 −0.100831 0.994904i \(-0.532150\pi\)
−0.100831 + 0.994904i \(0.532150\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 516338. 1.37408 0.687042 0.726618i \(-0.258909\pi\)
0.687042 + 0.726618i \(0.258909\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) −720526. −1.88048 −0.940239 0.340515i \(-0.889399\pi\)
−0.940239 + 0.340515i \(0.889399\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 390625. 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −342046. −0.859065 −0.429532 0.903052i \(-0.641322\pi\)
−0.429532 + 0.903052i \(0.641322\pi\)
\(632\) 0 0
\(633\) −553374. −1.38106
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 939510. 2.31538
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) −728302. −1.76153 −0.880764 0.473555i \(-0.842970\pi\)
−0.880764 + 0.473555i \(0.842970\pi\)
\(644\) 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
\(651\) −164124. −0.387267
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −691902. −1.60293
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −858958. −1.96593 −0.982967 0.183781i \(-0.941166\pi\)
−0.982967 + 0.183781i \(0.941166\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 133074. 0.297332
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 425858. 0.940231 0.470116 0.882605i \(-0.344212\pi\)
0.470116 + 0.882605i \(0.344212\pi\)
\(674\) 0 0
\(675\) 455625. 1.00000
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 1.76852e6 3.83592
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −937854. −1.98711
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 782162. 1.63810 0.819050 0.573722i \(-0.194501\pi\)
0.819050 + 0.573722i \(0.194501\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 94852.0 0.191927
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −133006. −0.264593 −0.132297 0.991210i \(-0.542235\pi\)
−0.132297 + 0.991210i \(0.542235\pi\)
\(710\) 0 0
\(711\) 622242. 1.23089
\(712\) 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
\(721\) −1.54329e6 −2.96878
\(722\) 0 0
\(723\) −309294. −0.591691
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −824734. −1.56043 −0.780216 0.625510i \(-0.784891\pi\)
−0.780216 + 0.625510i \(0.784891\pi\)
\(728\) 0 0
\(729\) 531441. 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −5422.00 −0.0100914 −0.00504570 0.999987i \(-0.501606\pi\)
−0.00504570 + 0.999987i \(0.501606\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 401042. 0.734346 0.367173 0.930153i \(-0.380326\pi\)
0.367173 + 0.930153i \(0.380326\pi\)
\(740\) 0 0
\(741\) −60444.0 −0.110082
\(742\) 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
\(751\) 316802. 0.561705 0.280852 0.959751i \(-0.409383\pi\)
0.280852 + 0.959751i \(0.409383\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −443854. −0.774548 −0.387274 0.921965i \(-0.626583\pi\)
−0.387274 + 0.921965i \(0.626583\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) −2.07120e6 −3.55772
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 1.17043e6 1.97922 0.989610 0.143775i \(-0.0459241\pi\)
0.989610 + 0.143775i \(0.0459241\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 121250. 0.201873
\(776\) 0 0
\(777\) 1.74445e6 2.88946
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 1.20111e6 1.93924 0.969621 0.244613i \(-0.0786610\pi\)
0.969621 + 0.244613i \(0.0786610\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −287036. −0.456447
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 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 \(\pi\)
\(810\) 0 0
\(811\) 976754. 1.48506 0.742529 0.669814i \(-0.233626\pi\)
0.742529 + 0.669814i \(0.233626\pi\)
\(812\) 0 0
\(813\) −794862. −1.20257
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 147844. 0.221493
\(818\) 0 0
\(819\) −1.11164e6 −1.65729
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) −235294. −0.347385 −0.173693 0.984800i \(-0.555570\pi\)
−0.173693 + 0.984800i \(0.555570\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) −1.16472e6 −1.69477 −0.847387 0.530976i \(-0.821825\pi\)
−0.847387 + 0.530976i \(0.821825\pi\)
\(830\) 0 0
\(831\) −1.24717e6 −1.80602
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 141426. 0.201873
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 707281. 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −1.37625e6 −1.91837
\(848\) 0 0
\(849\) 446274. 0.619136
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 1.29375e6 1.77808 0.889039 0.457831i \(-0.151373\pi\)
0.889039 + 0.457831i \(0.151373\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 534962. 0.724998 0.362499 0.931984i \(-0.381924\pi\)
0.362499 + 0.931984i \(0.381924\pi\)
\(860\) 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
\(867\) 751689. 1.00000
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 862276. 1.13661
\(872\) 0 0
\(873\) −1.52393e6 −1.99957
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −1.18065e6 −1.53505 −0.767527 0.641017i \(-0.778513\pi\)
−0.767527 + 0.641017i \(0.778513\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 1.33743e6 1.71533 0.857666 0.514207i \(-0.171914\pi\)
0.857666 + 0.514207i \(0.171914\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 1.02855e6 1.30143
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 2.71904e6 3.33458
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −1.59950e6 −1.94433 −0.972166 0.234295i \(-0.924722\pi\)
−0.972166 + 0.234295i \(0.924722\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −939166. −1.11202 −0.556008 0.831177i \(-0.687668\pi\)
−0.556008 + 0.831177i \(0.687668\pi\)
\(920\) 0 0
\(921\) −543006. −0.640155
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −1.28875e6 −1.50621
\(926\) 0 0
\(927\) 1.32986e6 1.54755
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) −296010. −0.341513
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −320734. −0.365314 −0.182657 0.983177i \(-0.558470\pi\)
−0.182657 + 0.983177i \(0.558470\pi\)
\(938\) 0 0
\(939\) −1.58197e6 −1.79418
\(940\) 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 \(\pi\)
\(948\) 0 0
\(949\) −1.24713e6 −1.38478
\(950\) 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
\(961\) −885885. −0.959247
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 482978. 0.516505 0.258252 0.966077i \(-0.416853\pi\)
0.258252 + 0.966077i \(0.416853\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 3.58685e6 3.78868
\(974\) 0 0
\(975\) 821250. 0.863905
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 1.78475e6 1.85456
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −1.96205e6 −1.99785 −0.998923 0.0464053i \(-0.985223\pi\)
−0.998923 + 0.0464053i \(0.985223\pi\)
\(992\) 0 0
\(993\) −792702. −0.803917
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 1.59922e6 1.60886 0.804428 0.594050i \(-0.202472\pi\)
0.804428 + 0.594050i \(0.202472\pi\)
\(998\) 0 0
\(999\) −1.50320e6 −1.50621
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 \(a_p\))