Properties

Label 7524.2.l.b.2089.1
Level 7524
Weight 2
Character 7524.2089
Analytic conductor 60.079
Analytic rank 0
Dimension 8
CM disc. -627
Inner twists 8

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Newspace parameters

Level: \( N \) = \( 7524 = 2^{2} \cdot 3^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 7524.l (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(60.0794424808\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.488455618816.6
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{4} \)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 2089.1
Root \(-1.67945 - 2.15831i\)
Character \(\chi\) = 7524.2089
Dual form 7524.2.l.b.2089.5

$q$-expansion

\(f(q)\) \(=\) \(q+O(q^{10})\) \(q-3.31662i q^{11} -7.15439 q^{13} -8.19666i q^{17} -4.35890 q^{19} -5.00000 q^{25} +7.00000 q^{49} +5.72842i q^{53} +11.7284i q^{59} -0.271584i q^{71} +12.7454 q^{79} +14.8299i q^{83} +17.7284i q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 40q^{25} + 56q^{49} + O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7524\mathbb{Z}\right)^\times\).

\(n\) \(2377\) \(3763\) \(4105\) \(6689\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 3.31662i − 1.00000i
\(12\) 0 0
\(13\) −7.15439 −1.98427 −0.992135 0.125173i \(-0.960051\pi\)
−0.992135 + 0.125173i \(0.960051\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 8.19666i − 1.98798i −0.109461 0.993991i \(-0.534912\pi\)
0.109461 0.993991i \(-0.465088\pi\)
\(18\) 0 0
\(19\) −4.35890 −1.00000
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 5.72842i 0.786858i 0.919355 + 0.393429i \(0.128711\pi\)
−0.919355 + 0.393429i \(0.871289\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 11.7284i 1.52691i 0.645861 + 0.763455i \(0.276498\pi\)
−0.645861 + 0.763455i \(0.723502\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 0.271584i − 0.0322311i −0.999870 0.0161155i \(-0.994870\pi\)
0.999870 0.0161155i \(-0.00512996\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 12.7454 1.43397 0.716983 0.697091i \(-0.245523\pi\)
0.716983 + 0.697091i \(0.245523\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 14.8299i 1.62779i 0.581009 + 0.813897i \(0.302658\pi\)
−0.581009 + 0.813897i \(0.697342\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 17.7284i 1.87921i 0.342263 + 0.939604i \(0.388807\pi\)
−0.342263 + 0.939604i \(0.611193\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 13.2665i 1.32007i 0.751237 + 0.660033i \(0.229458\pi\)
−0.751237 + 0.660033i \(0.770542\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 17.4356 1.67003 0.835014 0.550229i \(-0.185460\pi\)
0.835014 + 0.550229i \(0.185460\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 6.27158i − 0.589981i −0.955500 0.294990i \(-0.904684\pi\)
0.955500 0.294990i \(-0.0953165\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 15.8722 1.40843 0.704214 0.709987i \(-0.251299\pi\)
0.704214 + 0.709987i \(0.251299\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 17.9567i 1.56889i 0.620200 + 0.784444i \(0.287051\pi\)
−0.620200 + 0.784444i \(0.712949\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 23.7284i 1.98427i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 5.06984i − 0.415337i −0.978199 0.207669i \(-0.933412\pi\)
0.978199 0.207669i \(-0.0665876\pi\)
\(150\) 0 0
\(151\) −8.71780 −0.709444 −0.354722 0.934972i \(-0.615425\pi\)
−0.354722 + 0.934972i \(0.615425\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −21.1852 −1.69077 −0.845383 0.534160i \(-0.820628\pi\)
−0.845383 + 0.534160i \(0.820628\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 19.1852 1.50270 0.751352 0.659901i \(-0.229402\pi\)
0.751352 + 0.659901i \(0.229402\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 38.1852 2.93733
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 12.2716i − 0.917221i −0.888637 0.458611i \(-0.848347\pi\)
0.888637 0.458611i \(-0.151653\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −27.1852 −1.98798
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) −4.02756 −0.289910 −0.144955 0.989438i \(-0.546304\pi\)
−0.144955 + 0.989438i \(0.546304\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 11.3235i − 0.806765i −0.915032 0.403382i \(-0.867834\pi\)
0.915032 0.403382i \(-0.132166\pi\)
\(198\) 0 0
\(199\) −15.1852 −1.07645 −0.538227 0.842800i \(-0.680906\pi\)
−0.538227 + 0.842800i \(0.680906\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 14.4568i 1.00000i
\(210\) 0 0
\(211\) −27.0541 −1.86248 −0.931242 0.364402i \(-0.881273\pi\)
−0.931242 + 0.364402i \(0.881273\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 58.6421i 3.94469i
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 13.1852 0.871306 0.435653 0.900115i \(-0.356518\pi\)
0.435653 + 0.900115i \(0.356518\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 26.5330i − 1.73823i −0.494606 0.869117i \(-0.664688\pi\)
0.494606 0.869117i \(-0.335312\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 6.63325i − 0.429069i −0.976716 0.214535i \(-0.931177\pi\)
0.976716 0.214535i \(-0.0688235\pi\)
\(240\) 0 0
\(241\) −10.2812 −0.662271 −0.331135 0.943583i \(-0.607432\pi\)
−0.331135 + 0.943583i \(0.607432\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 31.1852 1.98427
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 28.9137i 1.80358i 0.432169 + 0.901792i \(0.357748\pi\)
−0.432169 + 0.901792i \(0.642252\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 31.2232i − 1.92531i −0.270737 0.962653i \(-0.587267\pi\)
0.270737 0.962653i \(-0.412733\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 18.2716i − 1.11404i −0.830500 0.557019i \(-0.811945\pi\)
0.830500 0.557019i \(-0.188055\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 16.5831i 1.00000i
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −50.1852 −2.95207
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −30.1810 −1.72252 −0.861259 0.508166i \(-0.830324\pi\)
−0.861259 + 0.508166i \(0.830324\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −9.18525 −0.519181 −0.259590 0.965719i \(-0.583588\pi\)
−0.259590 + 0.965719i \(0.583588\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 28.9137i 1.62395i 0.583690 + 0.811977i \(0.301608\pi\)
−0.583690 + 0.811977i \(0.698392\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 35.7284i 1.98798i
\(324\) 0 0
\(325\) 35.7719 1.98427
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −34.8712 −1.89955 −0.949777 0.312926i \(-0.898691\pi\)
−0.949777 + 0.312926i \(0.898691\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 11.7031i 0.628255i 0.949381 + 0.314127i \(0.101712\pi\)
−0.949381 + 0.314127i \(0.898288\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 34.3501i − 1.81293i −0.422285 0.906463i \(-0.638772\pi\)
0.422285 0.906463i \(-0.361228\pi\)
\(360\) 0 0
\(361\) 19.0000 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 7.18525 0.375067 0.187533 0.982258i \(-0.439951\pi\)
0.187533 + 0.982258i \(0.439951\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 17.4356 0.902781 0.451390 0.892327i \(-0.350928\pi\)
0.451390 + 0.892327i \(0.350928\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 24.2716i − 1.24022i −0.784515 0.620110i \(-0.787088\pi\)
0.784515 0.620110i \(-0.212912\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 37.1852 1.86627 0.933137 0.359521i \(-0.117060\pi\)
0.933137 + 0.359521i \(0.117060\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 28.9137i 1.44388i 0.691956 + 0.721940i \(0.256749\pi\)
−0.691956 + 0.721940i \(0.743251\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 32.6451 1.61420 0.807098 0.590417i \(-0.201037\pi\)
0.807098 + 0.590417i \(0.201037\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 40.9833i 1.98798i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 9.61854 0.459068 0.229534 0.973301i \(-0.426280\pi\)
0.229534 + 0.973301i \(0.426280\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 41.7284i 1.96929i 0.174581 + 0.984643i \(0.444143\pi\)
−0.174581 + 0.984643i \(0.555857\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 37.8565i 1.76315i 0.472042 + 0.881576i \(0.343517\pi\)
−0.472042 + 0.881576i \(0.656483\pi\)
\(462\) 0 0
\(463\) −39.1852 −1.82109 −0.910546 0.413407i \(-0.864339\pi\)
−0.910546 + 0.413407i \(0.864339\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 21.7945 1.00000
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) − 28.0964i − 1.28376i −0.766806 0.641879i \(-0.778155\pi\)
0.766806 0.641879i \(-0.221845\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 21.0836i 0.951488i 0.879584 + 0.475744i \(0.157821\pi\)
−0.879584 + 0.475744i \(0.842179\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −3.18525 −0.142591 −0.0712957 0.997455i \(-0.522713\pi\)
−0.0712957 + 0.997455i \(0.522713\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 33.1662i 1.47881i 0.673261 + 0.739405i \(0.264893\pi\)
−0.673261 + 0.739405i \(0.735107\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 28.9137i 1.28158i 0.767718 + 0.640788i \(0.221392\pi\)
−0.767718 + 0.640788i \(0.778608\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 30.2716i − 1.32622i −0.748521 0.663111i \(-0.769236\pi\)
0.748521 0.663111i \(-0.230764\pi\)
\(522\) 0 0
\(523\) −23.9273 −1.04627 −0.523134 0.852250i \(-0.675237\pi\)
−0.523134 + 0.852250i \(0.675237\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −23.0000 −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
\(539\) − 23.2164i − 1.00000i
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −8.71780 −0.372746 −0.186373 0.982479i \(-0.559673\pi\)
−0.186373 + 0.982479i \(0.559673\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 13.2665i 0.562120i 0.959690 + 0.281060i \(0.0906859\pi\)
−0.959690 + 0.281060i \(0.909314\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 1.18525 0.0493425 0.0246713 0.999696i \(-0.492146\pi\)
0.0246713 + 0.999696i \(0.492146\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 18.9990 0.786858
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 1.94302i − 0.0797901i −0.999204 0.0398950i \(-0.987298\pi\)
0.999204 0.0398950i \(-0.0127024\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 47.7284i 1.95013i 0.221919 + 0.975065i \(0.428768\pi\)
−0.221919 + 0.975065i \(0.571232\pi\)
\(600\) 0 0
\(601\) −34.8712 −1.42243 −0.711213 0.702977i \(-0.751854\pi\)
−0.711213 + 0.702977i \(0.751854\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 43.5890 1.76922 0.884611 0.466329i \(-0.154424\pi\)
0.884611 + 0.466329i \(0.154424\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) 43.1852 1.73576 0.867881 0.496772i \(-0.165482\pi\)
0.867881 + 0.496772i \(0.165482\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −4.00000 −0.159237 −0.0796187 0.996825i \(-0.525370\pi\)
−0.0796187 + 0.996825i \(0.525370\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −50.0807 −1.98427
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 28.9137i 1.14202i 0.820943 + 0.571011i \(0.193448\pi\)
−0.820943 + 0.571011i \(0.806552\pi\)
\(642\) 0 0
\(643\) 8.00000 0.315489 0.157745 0.987480i \(-0.449578\pi\)
0.157745 + 0.987480i \(0.449578\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 38.8988 1.52691
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −46.9539 −1.80994 −0.904970 0.425476i \(-0.860107\pi\)
−0.904970 + 0.425476i \(0.860107\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 36.2716i − 1.38789i −0.720026 0.693947i \(-0.755870\pi\)
0.720026 0.693947i \(-0.244130\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 40.9833i − 1.56134i
\(690\) 0 0
\(691\) −16.0000 −0.608669 −0.304334 0.952565i \(-0.598434\pi\)
−0.304334 + 0.952565i \(0.598434\pi\)
\(692\) 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
\(701\) 44.1101i 1.66602i 0.553261 + 0.833008i \(0.313383\pi\)
−0.553261 + 0.833008i \(0.686617\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −45.1852 −1.69697 −0.848484 0.529221i \(-0.822484\pi\)
−0.848484 + 0.529221i \(0.822484\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 20.0000 0.741759 0.370879 0.928681i \(-0.379056\pi\)
0.370879 + 0.928681i \(0.379056\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 2.81475 0.102304 0.0511519 0.998691i \(-0.483711\pi\)
0.0511519 + 0.998691i \(0.483711\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 53.0660i 1.92364i 0.273681 + 0.961820i \(0.411759\pi\)
−0.273681 + 0.961820i \(0.588241\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 83.9096i − 3.02980i
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 53.7284i 1.93248i 0.257650 + 0.966238i \(0.417052\pi\)
−0.257650 + 0.966238i \(0.582948\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −0.900742 −0.0322311
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 55.6717 1.98448 0.992241 0.124333i \(-0.0396790\pi\)
0.992241 + 0.124333i \(0.0396790\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 28.9137i 1.02417i 0.858933 + 0.512087i \(0.171128\pi\)
−0.858933 + 0.512087i \(0.828872\pi\)
\(798\) 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
\(809\) 34.7297i 1.22103i 0.792005 + 0.610515i \(0.209037\pi\)
−0.792005 + 0.610515i \(0.790963\pi\)
\(810\) 0 0
\(811\) −33.3078 −1.16959 −0.584797 0.811180i \(-0.698826\pi\)
−0.584797 + 0.811180i \(0.698826\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 54.2498i − 1.89333i −0.322218 0.946666i \(-0.604428\pi\)
0.322218 0.946666i \(-0.395572\pi\)
\(822\) 0 0
\(823\) −28.0000 −0.976019 −0.488009 0.872838i \(-0.662277\pi\)
−0.488009 + 0.872838i \(0.662277\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 57.3766i − 1.98798i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) − 57.8273i − 1.99642i −0.0597970 0.998211i \(-0.519045\pi\)
0.0597970 0.998211i \(-0.480955\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) −4.81475 −0.164277 −0.0821386 0.996621i \(-0.526175\pi\)
−0.0821386 + 0.996621i \(0.526175\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 57.8273i − 1.96847i −0.176879 0.984233i \(-0.556600\pi\)
0.176879 0.984233i \(-0.443400\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 42.2716i − 1.43397i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −13.4080 −0.452757 −0.226379 0.974039i \(-0.572689\pi\)
−0.226379 + 0.974039i \(0.572689\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 32.0000 1.07689 0.538443 0.842662i \(-0.319013\pi\)
0.538443 + 0.842662i \(0.319013\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 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
\(901\) 46.9539 1.56426
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 57.8273i − 1.91590i −0.286927 0.957952i \(-0.592634\pi\)
0.286927 0.957952i \(-0.407366\pi\)
\(912\) 0 0
\(913\) 49.1852 1.62779
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 1.94302i 0.0639551i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 0 0
\(931\) −30.5123 −1.00000
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 31.0000 1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 59.7284i 1.91678i 0.285469 + 0.958388i \(0.407851\pi\)
−0.285469 + 0.958388i \(0.592149\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 28.9137i 0.925030i 0.886611 + 0.462515i \(0.153053\pi\)
−0.886611 + 0.462515i \(0.846947\pi\)
\(978\) 0 0
\(979\) 58.7985 1.87921
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) − 57.8273i − 1.84441i −0.386707 0.922203i \(-0.626387\pi\)
0.386707 0.922203i \(-0.373613\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 0 0
\(999\) 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 \(a_p\))