Properties

Label 23.3.b.a.22.2
Level 23
Weight 3
Character 23.22
Self dual Yes
Analytic conductor 0.627
Analytic rank 0
Dimension 3
CM disc. -23
Inner twists 2

Related objects

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Newspace parameters

Level: \( N \) = \( 23 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 23.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(0.626704608029\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.621.1
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 22.2
Root \(-2.14510\)
Character \(\chi\) = 23.22

$q$-expansion

\(f(q)\) \(=\) \(q+0.601466 q^{2} +1.54364 q^{3} -3.63824 q^{4} +0.928445 q^{6} -4.59414 q^{8} -6.61718 q^{9} +O(q^{10})\) \(q+0.601466 q^{2} +1.54364 q^{3} -3.63824 q^{4} +0.928445 q^{6} -4.59414 q^{8} -6.61718 q^{9} -5.61612 q^{12} +23.5162 q^{13} +11.7897 q^{16} -3.98001 q^{18} -23.0000 q^{23} -7.09168 q^{24} +25.0000 q^{25} +14.1442 q^{26} -24.1073 q^{27} -42.4015 q^{29} -27.9663 q^{31} +25.4677 q^{32} +24.0749 q^{36} +36.3005 q^{39} +74.9986 q^{41} -13.8337 q^{46} -93.8839 q^{47} +18.1991 q^{48} +49.0000 q^{49} +15.0366 q^{50} -85.5575 q^{52} -14.4997 q^{54} -25.5030 q^{58} +26.0000 q^{59} -16.8208 q^{62} -31.8410 q^{64} -35.5037 q^{69} +140.916 q^{71} +30.4003 q^{72} -56.8366 q^{73} +38.5909 q^{75} +21.8335 q^{78} +22.3418 q^{81} +45.1091 q^{82} -65.4525 q^{87} +83.6795 q^{92} -43.1698 q^{93} -56.4679 q^{94} +39.3128 q^{96} +29.4718 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 12q^{4} - 33q^{6} - 21q^{8} + 27q^{9} + O(q^{10}) \) \( 3q + 12q^{4} - 33q^{6} - 21q^{8} + 27q^{9} + 3q^{12} + 48q^{16} + 39q^{18} - 69q^{23} - 132q^{24} + 75q^{25} + 87q^{26} - 114q^{27} - 84q^{32} + 255q^{36} - 42q^{39} + 231q^{48} + 147q^{49} - 309q^{52} - 297q^{54} - 273q^{58} + 78q^{59} + 303q^{62} - 45q^{64} - 33q^{72} + 399q^{78} + 243q^{81} - 129q^{82} + 246q^{87} - 276q^{92} - 546q^{93} - 57q^{94} - 21q^{96} + O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/23\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.601466 0.300733 0.150366 0.988630i \(-0.451955\pi\)
0.150366 + 0.988630i \(0.451955\pi\)
\(3\) 1.54364 0.514546 0.257273 0.966339i \(-0.417176\pi\)
0.257273 + 0.966339i \(0.417176\pi\)
\(4\) −3.63824 −0.909560
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0.928445 0.154741
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) −4.59414 −0.574267
\(9\) −6.61718 −0.735243
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) −5.61612 −0.468010
\(13\) 23.5162 1.80894 0.904469 0.426540i \(-0.140268\pi\)
0.904469 + 0.426540i \(0.140268\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 11.7897 0.736859
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) −3.98001 −0.221112
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −23.0000 −1.00000
\(24\) −7.09168 −0.295487
\(25\) 25.0000 1.00000
\(26\) 14.1442 0.544007
\(27\) −24.1073 −0.892862
\(28\) 0 0
\(29\) −42.4015 −1.46212 −0.731060 0.682314i \(-0.760974\pi\)
−0.731060 + 0.682314i \(0.760974\pi\)
\(30\) 0 0
\(31\) −27.9663 −0.902138 −0.451069 0.892489i \(-0.648957\pi\)
−0.451069 + 0.892489i \(0.648957\pi\)
\(32\) 25.4677 0.795865
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 24.0749 0.668747
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 36.3005 0.930781
\(40\) 0 0
\(41\) 74.9986 1.82924 0.914618 0.404320i \(-0.132492\pi\)
0.914618 + 0.404320i \(0.132492\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −13.8337 −0.300733
\(47\) −93.8839 −1.99753 −0.998765 0.0496817i \(-0.984179\pi\)
−0.998765 + 0.0496817i \(0.984179\pi\)
\(48\) 18.1991 0.379148
\(49\) 49.0000 1.00000
\(50\) 15.0366 0.300733
\(51\) 0 0
\(52\) −85.5575 −1.64534
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) −14.4997 −0.268513
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −25.5030 −0.439707
\(59\) 26.0000 0.440678 0.220339 0.975423i \(-0.429284\pi\)
0.220339 + 0.975423i \(0.429284\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) −16.8208 −0.271303
\(63\) 0 0
\(64\) −31.8410 −0.497516
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) −35.5037 −0.514546
\(70\) 0 0
\(71\) 140.916 1.98474 0.992368 0.123310i \(-0.0393509\pi\)
0.992368 + 0.123310i \(0.0393509\pi\)
\(72\) 30.4003 0.422226
\(73\) −56.8366 −0.778584 −0.389292 0.921114i \(-0.627280\pi\)
−0.389292 + 0.921114i \(0.627280\pi\)
\(74\) 0 0
\(75\) 38.5909 0.514546
\(76\) 0 0
\(77\) 0 0
\(78\) 21.8335 0.279916
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 22.3418 0.275825
\(82\) 45.1091 0.550111
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −65.4525 −0.752327
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 83.6795 0.909560
\(93\) −43.1698 −0.464191
\(94\) −56.4679 −0.600723
\(95\) 0 0
\(96\) 39.3128 0.409509
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 29.4718 0.300733
\(99\) 0 0
\(100\) −90.9560 −0.909560
\(101\) −166.000 −1.64356 −0.821782 0.569802i \(-0.807020\pi\)
−0.821782 + 0.569802i \(0.807020\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) −108.037 −1.03881
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 87.7080 0.812111
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 154.267 1.32988
\(117\) −155.611 −1.33001
\(118\) 15.6381 0.132526
\(119\) 0 0
\(120\) 0 0
\(121\) 121.000 1.00000
\(122\) 0 0
\(123\) 115.771 0.941225
\(124\) 101.748 0.820549
\(125\) 0 0
\(126\) 0 0
\(127\) 243.881 1.92032 0.960162 0.279443i \(-0.0901498\pi\)
0.960162 + 0.279443i \(0.0901498\pi\)
\(128\) −121.022 −0.945484
\(129\) 0 0
\(130\) 0 0
\(131\) −182.414 −1.39247 −0.696235 0.717814i \(-0.745143\pi\)
−0.696235 + 0.717814i \(0.745143\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) −21.3542 −0.154741
\(139\) 118.304 0.851109 0.425555 0.904933i \(-0.360079\pi\)
0.425555 + 0.904933i \(0.360079\pi\)
\(140\) 0 0
\(141\) −144.923 −1.02782
\(142\) 84.7563 0.596875
\(143\) 0 0
\(144\) −78.0149 −0.541770
\(145\) 0 0
\(146\) −34.1853 −0.234146
\(147\) 75.6382 0.514546
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 23.2111 0.154741
\(151\) −188.672 −1.24948 −0.624741 0.780832i \(-0.714796\pi\)
−0.624741 + 0.780832i \(0.714796\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) −132.070 −0.846601
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 13.4378 0.0829495
\(163\) −314.249 −1.92791 −0.963954 0.266070i \(-0.914275\pi\)
−0.963954 + 0.266070i \(0.914275\pi\)
\(164\) −272.863 −1.66380
\(165\) 0 0
\(166\) 0 0
\(167\) 242.000 1.44910 0.724551 0.689221i \(-0.242047\pi\)
0.724551 + 0.689221i \(0.242047\pi\)
\(168\) 0 0
\(169\) 384.011 2.27225
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −22.0000 −0.127168 −0.0635838 0.997977i \(-0.520253\pi\)
−0.0635838 + 0.997977i \(0.520253\pi\)
\(174\) −39.3674 −0.226249
\(175\) 0 0
\(176\) 0 0
\(177\) 40.1346 0.226749
\(178\) 0 0
\(179\) 287.187 1.60440 0.802198 0.597059i \(-0.203664\pi\)
0.802198 + 0.597059i \(0.203664\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 105.665 0.574267
\(185\) 0 0
\(186\) −25.9651 −0.139598
\(187\) 0 0
\(188\) 341.572 1.81687
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) −49.1510 −0.255995
\(193\) −36.1432 −0.187271 −0.0936353 0.995607i \(-0.529849\pi\)
−0.0936353 + 0.995607i \(0.529849\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −178.274 −0.909560
\(197\) −219.461 −1.11402 −0.557008 0.830507i \(-0.688051\pi\)
−0.557008 + 0.830507i \(0.688051\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) −114.853 −0.574267
\(201\) 0 0
\(202\) −99.8433 −0.494274
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 152.195 0.735243
\(208\) 277.250 1.33293
\(209\) 0 0
\(210\) 0 0
\(211\) −406.000 −1.92417 −0.962085 0.272749i \(-0.912067\pi\)
−0.962085 + 0.272749i \(0.912067\pi\)
\(212\) 0 0
\(213\) 217.524 1.02124
\(214\) 0 0
\(215\) 0 0
\(216\) 110.752 0.512741
\(217\) 0 0
\(218\) 0 0
\(219\) −87.7351 −0.400617
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −382.000 −1.71300 −0.856502 0.516143i \(-0.827367\pi\)
−0.856502 + 0.516143i \(0.827367\pi\)
\(224\) 0 0
\(225\) −165.430 −0.735243
\(226\) 0 0
\(227\) 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
\(232\) 194.798 0.839647
\(233\) −48.6597 −0.208840 −0.104420 0.994533i \(-0.533299\pi\)
−0.104420 + 0.994533i \(0.533299\pi\)
\(234\) −93.5946 −0.399977
\(235\) 0 0
\(236\) −94.5942 −0.400823
\(237\) 0 0
\(238\) 0 0
\(239\) −217.542 −0.910219 −0.455109 0.890436i \(-0.650400\pi\)
−0.455109 + 0.890436i \(0.650400\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 72.7773 0.300733
\(243\) 251.453 1.03479
\(244\) 0 0
\(245\) 0 0
\(246\) 69.6321 0.283057
\(247\) 0 0
\(248\) 128.481 0.518068
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 146.686 0.577505
\(255\) 0 0
\(256\) 54.5736 0.213178
\(257\) 367.540 1.43011 0.715057 0.699066i \(-0.246400\pi\)
0.715057 + 0.699066i \(0.246400\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 280.578 1.07501
\(262\) −109.716 −0.418762
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 303.541 1.12840 0.564202 0.825637i \(-0.309184\pi\)
0.564202 + 0.825637i \(0.309184\pi\)
\(270\) 0 0
\(271\) −286.000 −1.05535 −0.527675 0.849446i \(-0.676936\pi\)
−0.527675 + 0.849446i \(0.676936\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 129.171 0.468010
\(277\) −549.049 −1.98213 −0.991063 0.133391i \(-0.957413\pi\)
−0.991063 + 0.133391i \(0.957413\pi\)
\(278\) 71.1559 0.255956
\(279\) 185.058 0.663290
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) −87.1660 −0.309099
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) −512.687 −1.80524
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −168.524 −0.585154
\(289\) 289.000 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 206.785 0.708169
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 45.4938 0.154741
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −540.872 −1.80894
\(300\) −140.403 −0.468010
\(301\) 0 0
\(302\) −113.480 −0.375760
\(303\) −256.244 −0.845689
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −214.000 −0.697068 −0.348534 0.937296i \(-0.613320\pi\)
−0.348534 + 0.937296i \(0.613320\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 486.858 1.56546 0.782730 0.622361i \(-0.213826\pi\)
0.782730 + 0.622361i \(0.213826\pi\)
\(312\) −166.769 −0.534517
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 266.000 0.839117 0.419558 0.907728i \(-0.362185\pi\)
0.419558 + 0.907728i \(0.362185\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −81.2848 −0.250879
\(325\) 587.905 1.80894
\(326\) −189.010 −0.579785
\(327\) 0 0
\(328\) −344.554 −1.05047
\(329\) 0 0
\(330\) 0 0
\(331\) 661.999 2.00000 0.999999 0.00159277i \(-0.000506994\pi\)
0.999999 + 0.00159277i \(0.000506994\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 145.555 0.435792
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 230.969 0.683341
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −13.2322 −0.0382435
\(347\) 602.000 1.73487 0.867435 0.497550i \(-0.165767\pi\)
0.867435 + 0.497550i \(0.165767\pi\)
\(348\) 238.132 0.684287
\(349\) −26.0476 −0.0746349 −0.0373174 0.999303i \(-0.511881\pi\)
−0.0373174 + 0.999303i \(0.511881\pi\)
\(350\) 0 0
\(351\) −566.911 −1.61513
\(352\) 0 0
\(353\) −695.320 −1.96974 −0.984872 0.173284i \(-0.944562\pi\)
−0.984872 + 0.173284i \(0.944562\pi\)
\(354\) 24.1396 0.0681908
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 172.733 0.482494
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 361.000 1.00000
\(362\) 0 0
\(363\) 186.780 0.514546
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) −271.164 −0.736859
\(369\) −496.280 −1.34493
\(370\) 0 0
\(371\) 0 0
\(372\) 157.062 0.422210
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 431.316 1.14712
\(377\) −997.120 −2.64488
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 376.464 0.988095
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) −186.814 −0.486495
\(385\) 0 0
\(386\) −21.7389 −0.0563184
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −225.113 −0.574267
\(393\) −281.581 −0.716490
\(394\) −131.998 −0.335021
\(395\) 0 0
\(396\) 0 0
\(397\) −528.356 −1.33087 −0.665435 0.746455i \(-0.731754\pi\)
−0.665435 + 0.746455i \(0.731754\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 294.744 0.736859
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) −657.660 −1.63191
\(404\) 603.948 1.49492
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 256.398 0.626889 0.313445 0.949606i \(-0.398517\pi\)
0.313445 + 0.949606i \(0.398517\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 91.5402 0.221112
\(415\) 0 0
\(416\) 598.902 1.43967
\(417\) 182.619 0.437935
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) −244.195 −0.578661
\(423\) 621.247 1.46867
\(424\) 0 0
\(425\) 0 0
\(426\) 130.833 0.307120
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) −284.218 −0.657913
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) −52.7697 −0.120479
\(439\) 626.871 1.42795 0.713976 0.700171i \(-0.246893\pi\)
0.713976 + 0.700171i \(0.246893\pi\)
\(440\) 0 0
\(441\) −324.242 −0.735243
\(442\) 0 0
\(443\) 867.929 1.95921 0.979604 0.200938i \(-0.0643991\pi\)
0.979604 + 0.200938i \(0.0643991\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −229.760 −0.515157
\(447\) 0 0
\(448\) 0 0
\(449\) −574.000 −1.27840 −0.639198 0.769042i \(-0.720734\pi\)
−0.639198 + 0.769042i \(0.720734\pi\)
\(450\) −99.5002 −0.221112
\(451\) 0 0
\(452\) 0 0
\(453\) −291.241 −0.642916
\(454\) 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
\(461\) 542.680 1.17718 0.588590 0.808431i \(-0.299683\pi\)
0.588590 + 0.808431i \(0.299683\pi\)
\(462\) 0 0
\(463\) 98.0000 0.211663 0.105832 0.994384i \(-0.466250\pi\)
0.105832 + 0.994384i \(0.466250\pi\)
\(464\) −499.902 −1.07738
\(465\) 0 0
\(466\) −29.2671 −0.0628050
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 566.150 1.20972
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) −119.448 −0.253067
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) −130.844 −0.273733
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −440.227 −0.909560
\(485\) 0 0
\(486\) 151.240 0.311194
\(487\) −238.236 −0.489190 −0.244595 0.969625i \(-0.578655\pi\)
−0.244595 + 0.969625i \(0.578655\pi\)
\(488\) 0 0
\(489\) −485.086 −0.991997
\(490\) 0 0
\(491\) −301.732 −0.614526 −0.307263 0.951625i \(-0.599413\pi\)
−0.307263 + 0.951625i \(0.599413\pi\)
\(492\) −421.201 −0.856101
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −329.715 −0.664748
\(497\) 0 0
\(498\) 0 0
\(499\) −1.01465 −0.00203337 −0.00101668 0.999999i \(-0.500324\pi\)
−0.00101668 + 0.999999i \(0.500324\pi\)
\(500\) 0 0
\(501\) 373.560 0.745629
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 592.773 1.16918
\(508\) −887.298 −1.74665
\(509\) −989.779 −1.94456 −0.972278 0.233827i \(-0.924875\pi\)
−0.972278 + 0.233827i \(0.924875\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 516.912 1.00959
\(513\) 0 0
\(514\) 221.062 0.430082
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −33.9600 −0.0654336
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 168.758 0.323291
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 663.665 1.26654
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 529.000 1.00000
\(530\) 0 0
\(531\) −172.047 −0.324005
\(532\) 0 0
\(533\) 1763.68 3.30897
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 443.312 0.825535
\(538\) 182.569 0.339348
\(539\) 0 0
\(540\) 0 0
\(541\) 517.647 0.956834 0.478417 0.878133i \(-0.341211\pi\)
0.478417 + 0.878133i \(0.341211\pi\)
\(542\) −172.019 −0.317379
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −866.121 −1.58340 −0.791701 0.610909i \(-0.790804\pi\)
−0.791701 + 0.610909i \(0.790804\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 163.109 0.295487
\(553\) 0 0
\(554\) −330.234 −0.596091
\(555\) 0 0
\(556\) −430.419 −0.774135
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 111.306 0.199473
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 527.263 0.934864
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) −647.389 −1.13977
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −575.000 −1.00000
\(576\) 210.698 0.365795
\(577\) −950.813 −1.64786 −0.823928 0.566694i \(-0.808222\pi\)
−0.823928 + 0.566694i \(0.808222\pi\)
\(578\) 173.824 0.300733
\(579\) −55.7920 −0.0963593
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 261.115 0.447115
\(585\) 0 0
\(586\) 0 0
\(587\) 637.468 1.08598 0.542988 0.839740i \(-0.317293\pi\)
0.542988 + 0.839740i \(0.317293\pi\)
\(588\) −275.190 −0.468010
\(589\) 0 0
\(590\) 0 0
\(591\) −338.768 −0.573212
\(592\) 0 0
\(593\) −286.000 −0.482293 −0.241147 0.970489i \(-0.577524\pi\)
−0.241147 + 0.970489i \(0.577524\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) −325.316 −0.544007
\(599\) 1106.00 1.84641 0.923205 0.384307i \(-0.125560\pi\)
0.923205 + 0.384307i \(0.125560\pi\)
\(600\) −177.292 −0.295487
\(601\) −1201.97 −1.99995 −0.999973 0.00737547i \(-0.997652\pi\)
−0.999973 + 0.00737547i \(0.997652\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 686.434 1.13648
\(605\) 0 0
\(606\) −154.122 −0.254326
\(607\) 386.000 0.635914 0.317957 0.948105i \(-0.397003\pi\)
0.317957 + 0.948105i \(0.397003\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −2207.79 −3.61341
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) −128.714 −0.209631
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 554.467 0.892862
\(622\) 292.829 0.470785
\(623\) 0 0
\(624\) 427.973 0.685854
\(625\) 625.000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) −626.717 −0.990074
\(634\) 159.990 0.252350
\(635\) 0 0
\(636\) 0 0
\(637\) 1152.29 1.80894
\(638\) 0 0
\(639\) −932.469 −1.45926
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −230.059 −0.355578 −0.177789 0.984069i \(-0.556894\pi\)
−0.177789 + 0.984069i \(0.556894\pi\)
\(648\) −102.641 −0.158397
\(649\) 0 0
\(650\) 353.604 0.544007
\(651\) 0 0
\(652\) 1143.31 1.75355
\(653\) 56.2240 0.0861010 0.0430505 0.999073i \(-0.486292\pi\)
0.0430505 + 0.999073i \(0.486292\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 884.215 1.34789
\(657\) 376.098 0.572448
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 398.170 0.601465
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 975.233 1.46212
\(668\) −880.454 −1.31804
\(669\) −589.669 −0.881419
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 1323.09 1.96596 0.982982 0.183700i \(-0.0588075\pi\)
0.982982 + 0.183700i \(0.0588075\pi\)
\(674\) 0 0
\(675\) −602.682 −0.892862
\(676\) −1397.12 −2.06675
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −788.189 −1.15401 −0.577005 0.816741i \(-0.695779\pi\)
−0.577005 + 0.816741i \(0.695779\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 554.000 0.801737 0.400868 0.916136i \(-0.368708\pi\)
0.400868 + 0.916136i \(0.368708\pi\)
\(692\) 80.0413 0.115667
\(693\) 0 0
\(694\) 362.082 0.521732
\(695\) 0 0
\(696\) 300.698 0.432037
\(697\) 0 0
\(698\) −15.6667 −0.0224451
\(699\) −75.1129 −0.107458
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) −340.977 −0.485723
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) −418.211 −0.592367
\(707\) 0 0
\(708\) −146.019 −0.206242
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 643.224 0.902138
\(714\) 0 0
\(715\) 0 0
\(716\) −1044.85 −1.45929
\(717\) −335.806 −0.468349
\(718\) 0 0
\(719\) −862.000 −1.19889 −0.599444 0.800417i \(-0.704611\pi\)
−0.599444 + 0.800417i \(0.704611\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 217.129 0.300733
\(723\) 0 0
\(724\) 0 0
\(725\) −1060.04 −1.46212
\(726\) 112.342 0.154741
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 187.076 0.256620
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) −585.757 −0.795865
\(737\) 0 0
\(738\) −298.495 −0.404465
\(739\) 1316.84 1.78192 0.890958 0.454086i \(-0.150034\pi\)
0.890958 + 0.454086i \(0.150034\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 198.328 0.266570
\(745\) 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 \(\pi\)
\(752\) −1106.87 −1.47190
\(753\) 0 0
\(754\) −599.734 −0.795403
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1475.62 1.93906 0.969529 0.244977i \(-0.0787804\pi\)
0.969529 + 0.244977i \(0.0787804\pi\)
\(762\) 226.430 0.297152
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 611.421 0.797159
\(768\) 84.2418 0.109690
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 567.348 0.735860
\(772\) 131.498 0.170334
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) −699.157 −0.902138
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 1022.18 1.30547
\(784\) 577.697 0.736859
\(785\) 0 0
\(786\) −169.361 −0.215472
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 798.451 1.01326
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) −317.788 −0.400237
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 636.692 0.795865
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) −395.560 −0.490769
\(807\) 468.557 0.580615
\(808\) 762.627 0.943845
\(809\) 146.000 0.180470 0.0902349 0.995921i \(-0.471238\pi\)
0.0902349 + 0.995921i \(0.471238\pi\)
\(810\) 0 0
\(811\) −1620.09 −1.99764 −0.998820 0.0485755i \(-0.984532\pi\)
−0.998820 + 0.0485755i \(0.984532\pi\)
\(812\) 0 0
\(813\) −441.480 −0.543026
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 154.214 0.188526
\(819\) 0 0
\(820\) 0 0
\(821\) 1274.00 1.55177 0.775883 0.630877i \(-0.217305\pi\)
0.775883 + 0.630877i \(0.217305\pi\)
\(822\) 0 0
\(823\) 437.797 0.531952 0.265976 0.963980i \(-0.414306\pi\)
0.265976 + 0.963980i \(0.414306\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) −553.723 −0.668747
\(829\) −1654.00 −1.99517 −0.997587 0.0694210i \(-0.977885\pi\)
−0.997587 + 0.0694210i \(0.977885\pi\)
\(830\) 0 0
\(831\) −847.533 −1.01989
\(832\) −748.780 −0.899976
\(833\) 0 0
\(834\) 109.839 0.131701
\(835\) 0 0
\(836\) 0 0
\(837\) 674.191 0.805485
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 956.883 1.13779
\(842\) 0 0
\(843\) 0 0
\(844\) 1477.13 1.75015
\(845\) 0 0
\(846\) 373.659 0.441677
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) −791.403 −0.928877
\(853\) −1606.00 −1.88277 −0.941383 0.337339i \(-0.890473\pi\)
−0.941383 + 0.337339i \(0.890473\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −1589.30 −1.85449 −0.927244 0.374458i \(-0.877829\pi\)
−0.927244 + 0.374458i \(0.877829\pi\)
\(858\) 0 0
\(859\) −845.930 −0.984784 −0.492392 0.870374i \(-0.663877\pi\)
−0.492392 + 0.870374i \(0.663877\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −1395.38 −1.61690 −0.808448 0.588568i \(-0.799692\pi\)
−0.808448 + 0.588568i \(0.799692\pi\)
\(864\) −613.956 −0.710597
\(865\) 0 0
\(866\) 0 0
\(867\) 446.111 0.514546
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 319.201 0.364385
\(877\) −1558.00 −1.77651 −0.888255 0.459350i \(-0.848082\pi\)
−0.888255 + 0.459350i \(0.848082\pi\)
\(878\) 377.041 0.429432
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) −195.020 −0.221112
\(883\) 938.000 1.06229 0.531144 0.847282i \(-0.321762\pi\)
0.531144 + 0.847282i \(0.321762\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 522.029 0.589198
\(887\) 841.479 0.948680 0.474340 0.880342i \(-0.342687\pi\)
0.474340 + 0.880342i \(0.342687\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 1389.81 1.55808
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −834.910 −0.930781
\(898\) −345.241 −0.384456
\(899\) 1185.81 1.31903
\(900\) 601.872 0.668747
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) −175.171 −0.193346
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) 1098.45 1.20842
\(910\) 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
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) −330.338 −0.358674
\(922\) 326.404 0.354017
\(923\) 3313.81 3.59026
\(924\) 0 0
\(925\) 0 0
\(926\) 58.9436 0.0636540
\(927\) 0 0
\(928\) −1079.87 −1.16365
\(929\) −340.699 −0.366737 −0.183368 0.983044i \(-0.558700\pi\)
−0.183368 + 0.983044i \(0.558700\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 177.036 0.189952
\(933\) 751.533 0.805501
\(934\) 0 0
\(935\) 0 0
\(936\) 714.898 0.763780
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) −1724.97 −1.82924
\(944\) 306.533 0.324717
\(945\) 0 0
\(946\) 0 0
\(947\) −1846.71 −1.95006 −0.975031 0.222069i \(-0.928719\pi\)
−0.975031 + 0.222069i \(0.928719\pi\)
\(948\) 0 0
\(949\) −1336.58 −1.40841
\(950\) 0 0
\(951\) 410.607 0.431764
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 791.471 0.827898
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −178.887 −0.186147
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 1121.00 1.15926 0.579629 0.814881i \(-0.303198\pi\)
0.579629 + 0.814881i \(0.303198\pi\)
\(968\) −555.891 −0.574267
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) −914.846 −0.941200
\(973\) 0 0
\(974\) −143.291 −0.147116
\(975\) 907.511 0.930781
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) −291.763 −0.298326
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) −181.482 −0.184808
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) −531.867 −0.540515
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 1154.00 1.16448 0.582240 0.813017i \(-0.302176\pi\)
0.582240 + 0.813017i \(0.302176\pi\)
\(992\) −712.236 −0.717980
\(993\) 1021.89 1.02909
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −1318.00 −1.32197 −0.660983 0.750401i \(-0.729860\pi\)
−0.660983 + 0.750401i \(0.729860\pi\)
\(998\) −0.610277 −0.000611500 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 \(a_p\))