Properties

Label 7524.2.l.b.2089.2
Level 7524
Weight 2
Character 7524.2089
Analytic conductor 60.079
Analytic rank 0
Dimension 8
CM disc. -627
Inner twists 8

Related objects

Downloads

Learn more about

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 2089.2
Root \(2.67945 - 2.15831i\)
Character \(\chi\) = 7524.2089
Dual form 7524.2.l.b.2089.6

$q$-expansion

\(f(q)\) \(=\) \(q\)\(+O(q^{10})\) \(q\)\(-3.31662i q^{11}\) \(-2.79549 q^{13}\) \(+4.88004i q^{17}\) \(+4.35890 q^{19}\) \(-5.00000 q^{25}\) \(+7.00000 q^{49}\) \(-8.72842i q^{53}\) \(-2.72842i q^{59}\) \(-14.7284i q^{71}\) \(+17.1043 q^{79}\) \(+1.75321i q^{83}\) \(+3.27158i q^{89}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(8q \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut -\mathstrut 40q^{25} \) \(\mathstrut +\mathstrut 56q^{49} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
\(13\) −2.79549 −0.775329 −0.387664 0.921801i \(-0.626718\pi\)
−0.387664 + 0.921801i \(0.626718\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.88004i 1.18358i 0.806091 + 0.591791i \(0.201579\pi\)
−0.806091 + 0.591791i \(0.798421\pi\)
\(18\) 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
\(53\) − 8.72842i − 1.19894i −0.800397 0.599470i \(-0.795378\pi\)
0.800397 0.599470i \(-0.204622\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 2.72842i − 0.355210i −0.984102 0.177605i \(-0.943165\pi\)
0.984102 0.177605i \(-0.0568349\pi\)
\(60\) 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
\(71\) − 14.7284i − 1.74794i −0.485979 0.873971i \(-0.661537\pi\)
0.485979 0.873971i \(-0.338463\pi\)
\(72\) 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
\(79\) 17.1043 1.92438 0.962190 0.272380i \(-0.0878107\pi\)
0.962190 + 0.272380i \(0.0878107\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1.75321i 0.192440i 0.995360 + 0.0962201i \(0.0306753\pi\)
−0.995360 + 0.0962201i \(0.969325\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.27158i 0.346787i 0.984853 + 0.173394i \(0.0554733\pi\)
−0.984853 + 0.173394i \(0.944527\pi\)
\(90\) 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 \(-0.814540\pi\)
−0.835014 + 0.550229i \(0.814540\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 20.7284i − 1.94997i −0.222281 0.974983i \(-0.571350\pi\)
0.222281 0.974983i \(-0.428650\pi\)
\(114\) 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
\(127\) −5.92231 −0.525520 −0.262760 0.964861i \(-0.584633\pi\)
−0.262760 + 0.964861i \(0.584633\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 21.2734i − 1.85866i −0.369248 0.929331i \(-0.620385\pi\)
0.369248 0.929331i \(-0.379615\pi\)
\(132\) 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
\(143\) 9.27158i 0.775329i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 18.1465i − 1.48662i −0.668946 0.743311i \(-0.733254\pi\)
0.668946 0.743311i \(-0.266746\pi\)
\(150\) 0 0
\(151\) 8.71780 0.709444 0.354722 0.934972i \(-0.384575\pi\)
0.354722 + 0.934972i \(0.384575\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 22.1852 1.77058 0.885288 0.465044i \(-0.153961\pi\)
0.885288 + 0.465044i \(0.153961\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −24.1852 −1.89433 −0.947167 0.320740i \(-0.896069\pi\)
−0.947167 + 0.320740i \(0.896069\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −5.18525 −0.398865
\(170\) 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
\(179\) − 26.7284i − 1.99778i −0.0471502 0.998888i \(-0.515014\pi\)
0.0471502 0.998888i \(-0.484986\pi\)
\(180\) 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
\(187\) 16.1852 1.18358
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) −25.8221 −1.85871 −0.929356 0.369184i \(-0.879637\pi\)
−0.929356 + 0.369184i \(0.879637\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 27.9066i 1.98826i 0.108176 + 0.994132i \(0.465499\pi\)
−0.108176 + 0.994132i \(0.534501\pi\)
\(198\) 0 0
\(199\) 28.1852 1.99800 0.999000 0.0447181i \(-0.0142390\pi\)
0.999000 + 0.0447181i \(0.0142390\pi\)
\(200\) 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
\(211\) −22.6952 −1.56240 −0.781202 0.624278i \(-0.785393\pi\)
−0.781202 + 0.624278i \(0.785393\pi\)
\(212\) 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
\(221\) − 13.6421i − 0.917666i
\(222\) 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
\(229\) −30.1852 −1.99470 −0.997349 0.0727709i \(-0.976816\pi\)
−0.997349 + 0.0727709i \(0.976816\pi\)
\(230\) 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
\(241\) 20.2311 1.30320 0.651599 0.758563i \(-0.274098\pi\)
0.651599 + 0.758563i \(0.274098\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −12.1852 −0.775329
\(248\) 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 \(-0.642252\pi\)
0.432169 0.901792i \(-0.357748\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 8.00686i 0.493724i 0.969051 + 0.246862i \(0.0793994\pi\)
−0.969051 + 0.246862i \(0.920601\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 32.7284i − 1.99549i −0.0671428 0.997743i \(-0.521388\pi\)
0.0671428 0.997743i \(-0.478612\pi\)
\(270\) 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
\(289\) −6.81475 −0.400868
\(290\) 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
\(307\) 0.331335 0.0189103 0.00945514 0.999955i \(-0.496990\pi\)
0.00945514 + 0.999955i \(0.496990\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 34.1852 1.93226 0.966132 0.258047i \(-0.0830791\pi\)
0.966132 + 0.258047i \(0.0830791\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 28.9137i − 1.62395i −0.583690 0.811977i \(-0.698392\pi\)
0.583690 0.811977i \(-0.301608\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 21.2716i 1.18358i
\(324\) 0 0
\(325\) 13.9774 0.775329
\(326\) 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 \(-0.101309\pi\)
0.949777 + 0.312926i \(0.101309\pi\)
\(338\) 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
\(347\) 24.7798i 1.33025i 0.746733 + 0.665124i \(0.231621\pi\)
−0.746733 + 0.665124i \(0.768379\pi\)
\(348\) 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
\(359\) 31.0334i 1.63788i 0.573878 + 0.818941i \(0.305438\pi\)
−0.573878 + 0.818941i \(0.694562\pi\)
\(360\) 0 0
\(361\) 19.0000 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −36.1852 −1.88885 −0.944427 0.328720i \(-0.893383\pi\)
−0.944427 + 0.328720i \(0.893383\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −17.4356 −0.902781 −0.451390 0.892327i \(-0.649072\pi\)
−0.451390 + 0.892327i \(0.649072\pi\)
\(374\) 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
\(383\) − 38.7284i − 1.97893i −0.144774 0.989465i \(-0.546245\pi\)
0.144774 0.989465i \(-0.453755\pi\)
\(384\) 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
\(397\) −6.18525 −0.310429 −0.155214 0.987881i \(-0.549607\pi\)
−0.155214 + 0.987881i \(0.549607\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 28.9137i − 1.44388i −0.691956 0.721940i \(-0.743251\pi\)
0.691956 0.721940i \(-0.256749\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 37.0040 1.82973 0.914865 0.403759i \(-0.132297\pi\)
0.914865 + 0.403759i \(0.132297\pi\)
\(410\) 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
\(425\) − 24.4002i − 1.18358i
\(426\) 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
\(439\) 40.1308 1.91534 0.957670 0.287868i \(-0.0929465\pi\)
0.957670 + 0.287868i \(0.0929465\pi\)
\(440\) 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
\(449\) 27.2716i 1.28703i 0.765435 + 0.643513i \(0.222524\pi\)
−0.765435 + 0.643513i \(0.777476\pi\)
\(450\) 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
\(461\) − 1.37361i − 0.0639753i −0.999488 0.0319877i \(-0.989816\pi\)
0.999488 0.0319877i \(-0.0101837\pi\)
\(462\) 0 0
\(463\) 4.18525 0.194505 0.0972525 0.995260i \(-0.468995\pi\)
0.0972525 + 0.995260i \(0.468995\pi\)
\(464\) 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
\(479\) − 15.0197i − 0.686268i −0.939286 0.343134i \(-0.888511\pi\)
0.939286 0.343134i \(-0.111489\pi\)
\(480\) 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
\(491\) − 44.2999i − 1.99923i −0.0277858 0.999614i \(-0.508846\pi\)
0.0277858 0.999614i \(-0.491154\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 40.1852 1.79894 0.899469 0.436984i \(-0.143953\pi\)
0.899469 + 0.436984i \(0.143953\pi\)
\(500\) 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 \(-0.778608\pi\)
0.767718 0.640788i \(-0.221392\pi\)
\(510\) 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
\(521\) − 44.7284i − 1.95959i −0.200011 0.979794i \(-0.564098\pi\)
0.200011 0.979794i \(-0.435902\pi\)
\(522\) 0 0
\(523\) −45.7218 −1.99928 −0.999638 0.0269223i \(-0.991429\pi\)
−0.999638 + 0.0269223i \(0.991429\pi\)
\(524\) 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 \(-0.440327\pi\)
0.186373 + 0.982479i \(0.440327\pi\)
\(548\) 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
\(577\) −42.1852 −1.75619 −0.878097 0.478482i \(-0.841187\pi\)
−0.878097 + 0.478482i \(0.841187\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −28.9489 −1.19894
\(584\) 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
\(593\) − 41.1731i − 1.69078i −0.534152 0.845388i \(-0.679369\pi\)
0.534152 0.845388i \(-0.320631\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 33.2716i 1.35944i 0.733472 + 0.679720i \(0.237899\pi\)
−0.733472 + 0.679720i \(0.762101\pi\)
\(600\) 0 0
\(601\) 34.8712 1.42243 0.711213 0.702977i \(-0.248146\pi\)
0.711213 + 0.702977i \(0.248146\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −43.5890 −1.76922 −0.884611 0.466329i \(-0.845576\pi\)
−0.884611 + 0.466329i \(0.845576\pi\)
\(608\) 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
\(619\) −0.185248 −0.00744576 −0.00372288 0.999993i \(-0.501185\pi\)
−0.00372288 + 0.999993i \(0.501185\pi\)
\(620\) 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
\(637\) −19.5684 −0.775329
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 28.9137i − 1.14202i −0.820943 0.571011i \(-0.806552\pi\)
0.820943 0.571011i \(-0.193448\pi\)
\(642\) 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
\(649\) −9.04913 −0.355210
\(650\) 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
\(673\) −42.5950 −1.64192 −0.820958 0.570989i \(-0.806560\pi\)
−0.820958 + 0.570989i \(0.806560\pi\)
\(674\) 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
\(683\) − 50.7284i − 1.94107i −0.240962 0.970534i \(-0.577463\pi\)
0.240962 0.970534i \(-0.422537\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 24.4002i 0.929573i
\(690\) 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
\(701\) − 47.4268i − 1.79128i −0.444776 0.895642i \(-0.646717\pi\)
0.444776 0.895642i \(-0.353283\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −1.81475 −0.0681544 −0.0340772 0.999419i \(-0.510849\pi\)
−0.0340772 + 0.999419i \(0.510849\pi\)
\(710\) 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
\(757\) 46.1852 1.67863 0.839316 0.543644i \(-0.182956\pi\)
0.839316 + 0.543644i \(0.182956\pi\)
\(758\) 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
\(767\) 7.62725i 0.275404i
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 39.2716i 1.41250i 0.707962 + 0.706250i \(0.249615\pi\)
−0.707962 + 0.706250i \(0.750385\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −48.8486 −1.74794
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 33.8772 1.20759 0.603796 0.797139i \(-0.293654\pi\)
0.603796 + 0.797139i \(0.293654\pi\)
\(788\) 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 \(-0.828872\pi\)
0.858933 0.512087i \(-0.171128\pi\)
\(798\) 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
\(809\) 21.6530i 0.761278i 0.924724 + 0.380639i \(0.124296\pi\)
−0.924724 + 0.380639i \(0.875704\pi\)
\(810\) 0 0
\(811\) 23.3579 0.820207 0.410104 0.912039i \(-0.365493\pi\)
0.410104 + 0.912039i \(0.365493\pi\)
\(812\) 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
\(821\) 11.1337i 0.388568i 0.980945 + 0.194284i \(0.0622384\pi\)
−0.980945 + 0.194284i \(0.937762\pi\)
\(822\) 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
\(833\) 34.1603i 1.18358i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 57.8273i 1.99642i 0.0597970 + 0.998211i \(0.480955\pi\)
−0.0597970 + 0.998211i \(0.519045\pi\)
\(840\) 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
\(859\) −48.1852 −1.64406 −0.822030 0.569445i \(-0.807158\pi\)
−0.822030 + 0.569445i \(0.807158\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 57.8273i 1.96847i 0.176879 + 0.984233i \(0.443400\pi\)
−0.176879 + 0.984233i \(0.556600\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 56.7284i − 1.92438i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 43.2577 1.46071 0.730354 0.683069i \(-0.239355\pi\)
0.730354 + 0.683069i \(0.239355\pi\)
\(878\) 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
\(901\) 42.5950 1.41904
\(902\) 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 \(0.407366\pi\)
−0.286927 + 0.957952i \(0.592634\pi\)
\(912\) 0 0
\(913\) 5.81475 0.192440
\(914\) 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
\(923\) 41.1731i 1.35523i
\(924\) 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
\(971\) 45.2716i 1.45283i 0.687254 + 0.726417i \(0.258816\pi\)
−0.687254 + 0.726417i \(0.741184\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 28.9137i − 0.925030i −0.886611 0.462515i \(-0.846947\pi\)
0.886611 0.462515i \(-0.153053\pi\)
\(978\) 0 0
\(979\) 10.8506 0.346787
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 57.8273i 1.84441i 0.386707 + 0.922203i \(0.373613\pi\)
−0.386707 + 0.922203i \(0.626387\pi\)
\(984\) 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 \(a_p\))