Properties

Label 23.3.b.a.22.3
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 22.3
Root \(2.66908\)
Character \(\chi\) = 23.22

$q$-expansion

\(f(q)\) \(=\) \(q+3.12398 q^{2} -5.79306 q^{3} +5.75927 q^{4} -18.0974 q^{6} +5.49593 q^{8} +24.5596 q^{9} +O(q^{10})\) \(q+3.12398 q^{2} -5.79306 q^{3} +5.75927 q^{4} -18.0974 q^{6} +5.49593 q^{8} +24.5596 q^{9} -33.3638 q^{12} -2.15383 q^{13} -5.86788 q^{16} +76.7237 q^{18} -23.0000 q^{23} -31.8383 q^{24} +25.0000 q^{25} -6.72853 q^{26} -90.1376 q^{27} -13.0715 q^{29} +61.9041 q^{31} -40.3149 q^{32} +141.445 q^{36} +12.4773 q^{39} -66.2117 q^{41} -71.8516 q^{46} +50.9864 q^{47} +33.9930 q^{48} +49.0000 q^{49} +78.0996 q^{50} -12.4045 q^{52} -281.588 q^{54} -40.8352 q^{58} +26.0000 q^{59} +193.387 q^{62} -102.472 q^{64} +133.240 q^{69} -55.2940 q^{71} +134.978 q^{72} -88.0471 q^{73} -144.827 q^{75} +38.9788 q^{78} +301.136 q^{81} -206.844 q^{82} +75.7242 q^{87} -132.463 q^{92} -358.614 q^{93} +159.281 q^{94} +233.547 q^{96} +153.075 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 12q^{4} - 33q^{6} - 21q^{8} + 27q^{9} + O(q^{10}) \) \( 3q + 12q^{4} - 33q^{6} - 21q^{8} + 27q^{9} + 3q^{12} + 48q^{16} + 39q^{18} - 69q^{23} - 132q^{24} + 75q^{25} + 87q^{26} - 114q^{27} - 84q^{32} + 255q^{36} - 42q^{39} + 231q^{48} + 147q^{49} - 309q^{52} - 297q^{54} - 273q^{58} + 78q^{59} + 303q^{62} - 45q^{64} - 33q^{72} + 399q^{78} + 243q^{81} - 129q^{82} + 246q^{87} - 276q^{92} - 546q^{93} - 57q^{94} - 21q^{96} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.12398 1.56199 0.780996 0.624536i \(-0.214712\pi\)
0.780996 + 0.624536i \(0.214712\pi\)
\(3\) −5.79306 −1.93102 −0.965510 0.260365i \(-0.916157\pi\)
−0.965510 + 0.260365i \(0.916157\pi\)
\(4\) 5.75927 1.43982
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) −18.0974 −3.01624
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 5.49593 0.686992
\(9\) 24.5596 2.72884
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) −33.3638 −2.78032
\(13\) −2.15383 −0.165679 −0.0828396 0.996563i \(-0.526399\pi\)
−0.0828396 + 0.996563i \(0.526399\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −5.86788 −0.366743
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 76.7237 4.26243
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −23.0000 −1.00000
\(24\) −31.8383 −1.32660
\(25\) 25.0000 1.00000
\(26\) −6.72853 −0.258790
\(27\) −90.1376 −3.33843
\(28\) 0 0
\(29\) −13.0715 −0.450742 −0.225371 0.974273i \(-0.572359\pi\)
−0.225371 + 0.974273i \(0.572359\pi\)
\(30\) 0 0
\(31\) 61.9041 1.99691 0.998453 0.0556073i \(-0.0177095\pi\)
0.998453 + 0.0556073i \(0.0177095\pi\)
\(32\) −40.3149 −1.25984
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 141.445 3.92903
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 12.4773 0.319930
\(40\) 0 0
\(41\) −66.2117 −1.61492 −0.807460 0.589922i \(-0.799158\pi\)
−0.807460 + 0.589922i \(0.799158\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −71.8516 −1.56199
\(47\) 50.9864 1.08482 0.542408 0.840115i \(-0.317513\pi\)
0.542408 + 0.840115i \(0.317513\pi\)
\(48\) 33.9930 0.708188
\(49\) 49.0000 1.00000
\(50\) 78.0996 1.56199
\(51\) 0 0
\(52\) −12.4045 −0.238548
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) −281.588 −5.21460
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −40.8352 −0.704056
\(59\) 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 \(\pi\)
\(62\) 193.387 3.11915
\(63\) 0 0
\(64\) −102.472 −1.60112
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 133.240 1.93102
\(70\) 0 0
\(71\) −55.2940 −0.778789 −0.389395 0.921071i \(-0.627316\pi\)
−0.389395 + 0.921071i \(0.627316\pi\)
\(72\) 134.978 1.87469
\(73\) −88.0471 −1.20612 −0.603062 0.797694i \(-0.706053\pi\)
−0.603062 + 0.797694i \(0.706053\pi\)
\(74\) 0 0
\(75\) −144.827 −1.93102
\(76\) 0 0
\(77\) 0 0
\(78\) 38.9788 0.499728
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 301.136 3.71773
\(82\) −206.844 −2.52249
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 75.7242 0.870393
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −132.463 −1.43982
\(93\) −358.614 −3.85607
\(94\) 159.281 1.69447
\(95\) 0 0
\(96\) 233.547 2.43278
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 153.075 1.56199
\(99\) 0 0
\(100\) 143.982 1.43982
\(101\) −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 \(\pi\)
\(104\) −11.8373 −0.113820
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) −519.127 −4.80673
\(109\) 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
\(116\) −75.2825 −0.648987
\(117\) −52.8971 −0.452112
\(118\) 81.2236 0.688335
\(119\) 0 0
\(120\) 0 0
\(121\) 121.000 1.00000
\(122\) 0 0
\(123\) 383.569 3.11844
\(124\) 356.522 2.87518
\(125\) 0 0
\(126\) 0 0
\(127\) −183.410 −1.44417 −0.722086 0.691803i \(-0.756816\pi\)
−0.722086 + 0.691803i \(0.756816\pi\)
\(128\) −158.860 −1.24109
\(129\) 0 0
\(130\) 0 0
\(131\) 254.078 1.93952 0.969762 0.244051i \(-0.0784764\pi\)
0.969762 + 0.244051i \(0.0784764\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 416.241 3.01624
\(139\) 158.715 1.14183 0.570917 0.821007i \(-0.306588\pi\)
0.570917 + 0.821007i \(0.306588\pi\)
\(140\) 0 0
\(141\) −295.367 −2.09480
\(142\) −172.738 −1.21646
\(143\) 0 0
\(144\) −144.113 −1.00078
\(145\) 0 0
\(146\) −275.058 −1.88396
\(147\) −283.860 −1.93102
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) −452.436 −3.01624
\(151\) −109.883 −0.727699 −0.363849 0.931458i \(-0.618538\pi\)
−0.363849 + 0.931458i \(0.618538\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 71.8600 0.460641
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 940.745 5.80707
\(163\) 232.242 1.42480 0.712400 0.701774i \(-0.247608\pi\)
0.712400 + 0.701774i \(0.247608\pi\)
\(164\) −381.331 −2.32519
\(165\) 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
\(169\) −164.361 −0.972550
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −22.0000 −0.127168 −0.0635838 0.997977i \(-0.520253\pi\)
−0.0635838 + 0.997977i \(0.520253\pi\)
\(174\) 236.561 1.35955
\(175\) 0 0
\(176\) 0 0
\(177\) −150.620 −0.850958
\(178\) 0 0
\(179\) 41.5170 0.231938 0.115969 0.993253i \(-0.463003\pi\)
0.115969 + 0.993253i \(0.463003\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −126.406 −0.686992
\(185\) 0 0
\(186\) −1120.30 −6.02314
\(187\) 0 0
\(188\) 293.644 1.56194
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 593.624 3.09179
\(193\) 350.889 1.81808 0.909038 0.416713i \(-0.136818\pi\)
0.909038 + 0.416713i \(0.136818\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 282.204 1.43982
\(197\) 393.111 1.99549 0.997744 0.0671289i \(-0.0213839\pi\)
0.997744 + 0.0671289i \(0.0213839\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 137.398 0.686992
\(201\) 0 0
\(202\) −518.581 −2.56723
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −564.870 −2.72884
\(208\) 12.6384 0.0607616
\(209\) 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
\(213\) 320.322 1.50386
\(214\) 0 0
\(215\) 0 0
\(216\) −495.390 −2.29347
\(217\) 0 0
\(218\) 0 0
\(219\) 510.062 2.32905
\(220\) 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
\(225\) 613.989 2.72884
\(226\) 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
\(232\) −71.8402 −0.309656
\(233\) −377.032 −1.61816 −0.809081 0.587697i \(-0.800035\pi\)
−0.809081 + 0.587697i \(0.800035\pi\)
\(234\) −165.250 −0.706195
\(235\) 0 0
\(236\) 149.741 0.634496
\(237\) 0 0
\(238\) 0 0
\(239\) −259.834 −1.08717 −0.543585 0.839354i \(-0.682934\pi\)
−0.543585 + 0.839354i \(0.682934\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 378.002 1.56199
\(243\) −933.264 −3.84059
\(244\) 0 0
\(245\) 0 0
\(246\) 1198.26 4.87098
\(247\) 0 0
\(248\) 340.221 1.37186
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −572.969 −2.25578
\(255\) 0 0
\(256\) −86.3891 −0.337457
\(257\) 127.410 0.495760 0.247880 0.968791i \(-0.420266\pi\)
0.247880 + 0.968791i \(0.420266\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −321.031 −1.23000
\(262\) 793.735 3.02952
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −536.452 −1.99425 −0.997123 0.0757948i \(-0.975851\pi\)
−0.997123 + 0.0757948i \(0.975851\pi\)
\(270\) 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
\(276\) 767.368 2.78032
\(277\) 338.523 1.22210 0.611052 0.791591i \(-0.290747\pi\)
0.611052 + 0.791591i \(0.290747\pi\)
\(278\) 495.823 1.78354
\(279\) 1520.34 5.44924
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) −922.722 −3.27206
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) −318.453 −1.12131
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −990.117 −3.43790
\(289\) 289.000 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) −507.087 −1.73660
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) −886.774 −3.01624
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 49.5381 0.165679
\(300\) −834.095 −2.78032
\(301\) 0 0
\(302\) −343.271 −1.13666
\(303\) 961.648 3.17376
\(304\) 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
\(311\) −578.675 −1.86069 −0.930346 0.366684i \(-0.880493\pi\)
−0.930346 + 0.366684i \(0.880493\pi\)
\(312\) 68.5742 0.219789
\(313\) 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
\(324\) 1734.33 5.35286
\(325\) −53.8457 −0.165679
\(326\) 725.521 2.22553
\(327\) 0 0
\(328\) −363.895 −1.10944
\(329\) 0 0
\(330\) 0 0
\(331\) −331.913 −1.00276 −0.501379 0.865228i \(-0.667174\pi\)
−0.501379 + 0.865228i \(0.667174\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 756.004 2.26348
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) −513.461 −1.51912
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −68.7276 −0.198635
\(347\) 602.000 1.73487 0.867435 0.497550i \(-0.165767\pi\)
0.867435 + 0.497550i \(0.165767\pi\)
\(348\) 436.116 1.25321
\(349\) −591.041 −1.69353 −0.846763 0.531970i \(-0.821452\pi\)
−0.846763 + 0.531970i \(0.821452\pi\)
\(350\) 0 0
\(351\) 194.141 0.553108
\(352\) 0 0
\(353\) 241.712 0.684736 0.342368 0.939566i \(-0.388771\pi\)
0.342368 + 0.939566i \(0.388771\pi\)
\(354\) −470.533 −1.32919
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 129.698 0.362286
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 361.000 1.00000
\(362\) 0 0
\(363\) −700.961 −1.93102
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 134.961 0.366743
\(369\) −1626.13 −4.40686
\(370\) 0 0
\(371\) 0 0
\(372\) −2065.36 −5.55203
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 280.218 0.745260
\(377\) 28.1538 0.0746786
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 1062.50 2.78873
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 920.284 2.39657
\(385\) 0 0
\(386\) 1096.17 2.83982
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 269.301 0.686992
\(393\) −1471.89 −3.74526
\(394\) 1228.07 3.11694
\(395\) 0 0
\(396\) 0 0
\(397\) 777.459 1.95833 0.979167 0.203056i \(-0.0650875\pi\)
0.979167 + 0.203056i \(0.0650875\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −146.697 −0.366743
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) −133.331 −0.330846
\(404\) −956.039 −2.36643
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 544.511 1.33132 0.665661 0.746254i \(-0.268150\pi\)
0.665661 + 0.746254i \(0.268150\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) −1764.64 −4.26243
\(415\) 0 0
\(416\) 86.8314 0.208729
\(417\) −919.446 −2.20491
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) −1268.34 −3.00554
\(423\) 1252.20 2.96029
\(424\) 0 0
\(425\) 0 0
\(426\) 1000.68 2.34901
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 528.917 1.22434
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 1593.43 3.63796
\(439\) −845.824 −1.92671 −0.963353 0.268236i \(-0.913559\pi\)
−0.963353 + 0.268236i \(0.913559\pi\)
\(440\) 0 0
\(441\) 1203.42 2.72884
\(442\) 0 0
\(443\) −588.144 −1.32764 −0.663820 0.747893i \(-0.731066\pi\)
−0.663820 + 0.747893i \(0.731066\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −1193.36 −2.67570
\(447\) 0 0
\(448\) 0 0
\(449\) −574.000 −1.27840 −0.639198 0.769042i \(-0.720734\pi\)
−0.639198 + 0.769042i \(0.720734\pi\)
\(450\) 1918.09 4.26243
\(451\) 0 0
\(452\) 0 0
\(453\) 636.556 1.40520
\(454\) 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
\(461\) 374.172 0.811654 0.405827 0.913950i \(-0.366984\pi\)
0.405827 + 0.913950i \(0.366984\pi\)
\(462\) 0 0
\(463\) 98.0000 0.211663 0.105832 0.994384i \(-0.466250\pi\)
0.105832 + 0.994384i \(0.466250\pi\)
\(464\) 76.7022 0.165306
\(465\) 0 0
\(466\) −1177.84 −2.52756
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) −304.649 −0.650959
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 142.894 0.302742
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) −811.716 −1.69815
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 696.872 1.43982
\(485\) 0 0
\(486\) −2915.50 −5.99897
\(487\) −698.770 −1.43485 −0.717423 0.696638i \(-0.754678\pi\)
−0.717423 + 0.696638i \(0.754678\pi\)
\(488\) 0 0
\(489\) −1345.39 −2.75132
\(490\) 0 0
\(491\) 960.163 1.95553 0.977763 0.209715i \(-0.0672535\pi\)
0.977763 + 0.209715i \(0.0672535\pi\)
\(492\) 2209.08 4.48999
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −363.246 −0.732350
\(497\) 0 0
\(498\) 0 0
\(499\) 864.800 1.73307 0.866533 0.499119i \(-0.166343\pi\)
0.866533 + 0.499119i \(0.166343\pi\)
\(500\) 0 0
\(501\) −1401.92 −2.79825
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 952.154 1.87802
\(508\) −1056.31 −2.07934
\(509\) 701.035 1.37728 0.688639 0.725104i \(-0.258208\pi\)
0.688639 + 0.725104i \(0.258208\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 365.561 0.713986
\(513\) 0 0
\(514\) 398.028 0.774373
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 127.447 0.245563
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) −1002.90 −1.92126
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 1463.30 2.79256
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 529.000 1.00000
\(530\) 0 0
\(531\) 638.549 1.20254
\(532\) 0 0
\(533\) 142.609 0.267559
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −240.510 −0.447878
\(538\) −1675.87 −3.11500
\(539\) 0 0
\(540\) 0 0
\(541\) −1081.67 −1.99939 −0.999694 0.0247449i \(-0.992123\pi\)
−0.999694 + 0.0247449i \(0.992123\pi\)
\(542\) −893.459 −1.64845
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 1011.85 1.84983 0.924913 0.380179i \(-0.124138\pi\)
0.924913 + 0.380179i \(0.124138\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 732.280 1.32660
\(553\) 0 0
\(554\) 1057.54 1.90892
\(555\) 0 0
\(556\) 914.083 1.64403
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 4749.51 8.51166
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) −1701.10 −3.01613
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) −303.892 −0.535022
\(569\) 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
\(576\) −2516.66 −4.36920
\(577\) −90.9437 −0.157615 −0.0788074 0.996890i \(-0.525111\pi\)
−0.0788074 + 0.996890i \(0.525111\pi\)
\(578\) 902.831 1.56199
\(579\) −2032.72 −3.51074
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) −483.901 −0.828598
\(585\) 0 0
\(586\) 0 0
\(587\) 535.041 0.911484 0.455742 0.890112i \(-0.349374\pi\)
0.455742 + 0.890112i \(0.349374\pi\)
\(588\) −1634.83 −2.78032
\(589\) 0 0
\(590\) 0 0
\(591\) −2277.32 −3.85333
\(592\) 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
\(598\) 154.756 0.258790
\(599\) 1106.00 1.84641 0.923205 0.384307i \(-0.125560\pi\)
0.923205 + 0.384307i \(0.125560\pi\)
\(600\) −795.957 −1.32660
\(601\) 593.306 0.987198 0.493599 0.869690i \(-0.335681\pi\)
0.493599 + 0.869690i \(0.335681\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −632.843 −1.04775
\(605\) 0 0
\(606\) 3004.17 4.95738
\(607\) 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
\(611\) −109.816 −0.179732
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) −668.532 −1.08881
\(615\) 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
\(621\) 2073.16 3.33843
\(622\) −1807.77 −2.90638
\(623\) 0 0
\(624\) −73.2151 −0.117332
\(625\) 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
\(633\) 2351.98 3.71561
\(634\) 830.980 1.31069
\(635\) 0 0
\(636\) 0 0
\(637\) −105.538 −0.165679
\(638\) 0 0
\(639\) −1358.00 −2.12519
\(640\) 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
\(647\) −987.754 −1.52667 −0.763334 0.646004i \(-0.776439\pi\)
−0.763334 + 0.646004i \(0.776439\pi\)
\(648\) 1655.03 2.55405
\(649\) 0 0
\(650\) −168.213 −0.258790
\(651\) 0 0
\(652\) 1337.55 2.05145
\(653\) −1158.09 −1.77350 −0.886748 0.462254i \(-0.847041\pi\)
−0.886748 + 0.462254i \(0.847041\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 388.523 0.592260
\(657\) −2162.40 −3.29132
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) −1036.89 −1.56630
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 300.645 0.450742
\(668\) 1393.74 2.08644
\(669\) 2212.95 3.30785
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −875.681 −1.30116 −0.650580 0.759438i \(-0.725474\pi\)
−0.650580 + 0.759438i \(0.725474\pi\)
\(674\) 0 0
\(675\) −2253.44 −3.33843
\(676\) −946.600 −1.40030
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −572.102 −0.837631 −0.418816 0.908071i \(-0.637555\pi\)
−0.418816 + 0.908071i \(0.637555\pi\)
\(684\) 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 \(0.368708\pi\)
\(692\) −126.704 −0.183098
\(693\) 0 0
\(694\) 1880.64 2.70985
\(695\) 0 0
\(696\) 416.175 0.597953
\(697\) 0 0
\(698\) −1846.40 −2.64527
\(699\) 2184.17 3.12470
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 606.493 0.863950
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 755.104 1.06955
\(707\) 0 0
\(708\) −867.459 −1.22522
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −1423.79 −1.99691
\(714\) 0 0
\(715\) 0 0
\(716\) 239.107 0.333949
\(717\) 1505.23 2.09935
\(718\) 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
\(722\) 1127.76 1.56199
\(723\) 0 0
\(724\) 0 0
\(725\) −326.788 −0.450742
\(726\) −2189.79 −3.01624
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 2696.23 3.69853
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 927.243 1.25984
\(737\) 0 0
\(738\) −5080.01 −6.88348
\(739\) −1239.64 −1.67746 −0.838729 0.544550i \(-0.816701\pi\)
−0.838729 + 0.544550i \(0.816701\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) −1970.92 −2.64908
\(745\) 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 \(\pi\)
\(752\) −299.182 −0.397848
\(753\) 0 0
\(754\) 87.9521 0.116647
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −414.909 −0.545216 −0.272608 0.962125i \(-0.587886\pi\)
−0.272608 + 0.962125i \(0.587886\pi\)
\(762\) 3319.25 4.35597
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −55.9996 −0.0730112
\(768\) 500.457 0.651637
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) −738.096 −0.957322
\(772\) 2020.86 2.61770
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 1547.60 1.99691
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 1178.24 1.50477
\(784\) −287.526 −0.366743
\(785\) 0 0
\(786\) −4598.15 −5.85007
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 2264.03 2.87314
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 2428.77 3.05890
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −1007.87 −1.25984
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) −416.523 −0.516778
\(807\) 3107.70 3.85093
\(808\) −912.325 −1.12911
\(809\) 146.000 0.180470 0.0902349 0.995921i \(-0.471238\pi\)
0.0902349 + 0.995921i \(0.471238\pi\)
\(810\) 0 0
\(811\) 741.809 0.914684 0.457342 0.889291i \(-0.348801\pi\)
0.457342 + 0.889291i \(0.348801\pi\)
\(812\) 0 0
\(813\) 1656.82 2.03790
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 1701.04 2.07951
\(819\) 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
\(823\) 1155.23 1.40369 0.701843 0.712332i \(-0.252361\pi\)
0.701843 + 0.712332i \(0.252361\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) −3253.24 −3.92903
\(829\) −1654.00 −1.99517 −0.997587 0.0694210i \(-0.977885\pi\)
−0.997587 + 0.0694210i \(0.977885\pi\)
\(830\) 0 0
\(831\) −1961.08 −2.35991
\(832\) 220.706 0.265272
\(833\) 0 0
\(834\) −2872.33 −3.44405
\(835\) 0 0
\(836\) 0 0
\(837\) −5579.88 −6.66653
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) −670.135 −0.796831
\(842\) 0 0
\(843\) 0 0
\(844\) −2338.26 −2.77045
\(845\) 0 0
\(846\) 3911.86 4.62395
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 1844.82 2.16528
\(853\) −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
\(857\) 238.815 0.278664 0.139332 0.990246i \(-0.455504\pi\)
0.139332 + 0.990246i \(0.455504\pi\)
\(858\) 0 0
\(859\) −872.004 −1.01514 −0.507570 0.861611i \(-0.669456\pi\)
−0.507570 + 0.861611i \(0.669456\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1577.46 1.82788 0.913939 0.405852i \(-0.133025\pi\)
0.913939 + 0.405852i \(0.133025\pi\)
\(864\) 3633.89 4.20589
\(865\) 0 0
\(866\) 0 0
\(867\) −1674.19 −1.93102
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 2937.59 3.35341
\(877\) −1558.00 −1.77651 −0.888255 0.459350i \(-0.848082\pi\)
−0.888255 + 0.459350i \(0.848082\pi\)
\(878\) −2642.34 −3.00950
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 3759.46 4.26243
\(883\) 938.000 1.06229 0.531144 0.847282i \(-0.321762\pi\)
0.531144 + 0.847282i \(0.321762\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −1837.35 −2.07376
\(887\) 931.755 1.05046 0.525228 0.850961i \(-0.323980\pi\)
0.525228 + 0.850961i \(0.323980\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) −2200.04 −2.46641
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −286.977 −0.319930
\(898\) −1793.17 −1.99684
\(899\) −809.181 −0.900090
\(900\) 3536.13 3.92903
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 1988.59 2.19491
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) −4076.89 −4.48503
\(910\) 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
\(921\) 1239.72 1.34605
\(922\) 1168.91 1.26780
\(923\) 119.094 0.129029
\(924\) 0 0
\(925\) 0 0
\(926\) 306.150 0.330616
\(927\) 0 0
\(928\) 526.977 0.567863
\(929\) 1752.14 1.88605 0.943026 0.332720i \(-0.107967\pi\)
0.943026 + 0.332720i \(0.107967\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −2171.43 −2.32986
\(933\) 3352.30 3.59303
\(934\) 0 0
\(935\) 0 0
\(936\) −290.719 −0.310597
\(937\) 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
\(943\) 1522.87 1.61492
\(944\) −152.565 −0.161615
\(945\) 0 0
\(946\) 0 0
\(947\) 559.105 0.590396 0.295198 0.955436i \(-0.404614\pi\)
0.295198 + 0.955436i \(0.404614\pi\)
\(948\) 0 0
\(949\) 189.638 0.199830
\(950\) 0 0
\(951\) −1540.95 −1.62035
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −1496.45 −1.56533
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 2871.11 2.98763
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −1925.34 −1.99104 −0.995522 0.0945328i \(-0.969864\pi\)
−0.995522 + 0.0945328i \(0.969864\pi\)
\(968\) 665.008 0.686992
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) −5374.92 −5.52975
\(973\) 0 0
\(974\) −2182.94 −2.24122
\(975\) 311.932 0.319930
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) −4202.99 −4.29754
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 2999.53 3.05451
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 2108.07 2.14235
\(985\) 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 \(0.302176\pi\)
\(992\) −2495.66 −2.51578
\(993\) 1922.79 1.93635
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −1318.00 −1.32197 −0.660983 0.750401i \(-0.729860\pi\)
−0.660983 + 0.750401i \(0.729860\pi\)
\(998\) 2701.62 2.70704
\(999\) 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 \(a_p\))