Properties

Label 800.4.a.o.1.1
Level $800$
Weight $4$
Character 800.1
Self dual yes
Analytic conductor $47.202$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 800.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(47.2015280046\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 160)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 800.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.47214 q^{3} +31.3050 q^{7} -7.00000 q^{9} +O(q^{10})\) \(q-4.47214 q^{3} +31.3050 q^{7} -7.00000 q^{9} -8.94427 q^{11} +62.0000 q^{13} +46.0000 q^{17} +107.331 q^{19} -140.000 q^{21} -192.302 q^{23} +152.053 q^{27} -90.0000 q^{29} -152.053 q^{31} +40.0000 q^{33} +214.000 q^{37} -277.272 q^{39} -10.0000 q^{41} +67.0820 q^{43} -398.020 q^{47} +637.000 q^{49} -205.718 q^{51} +678.000 q^{53} -480.000 q^{57} -411.437 q^{59} +250.000 q^{61} -219.135 q^{63} -49.1935 q^{67} +860.000 q^{69} -366.715 q^{71} -522.000 q^{73} -280.000 q^{77} +876.539 q^{79} -491.000 q^{81} -380.132 q^{83} +402.492 q^{87} +970.000 q^{89} +1940.91 q^{91} +680.000 q^{93} +934.000 q^{97} +62.6099 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 14 q^{9} + O(q^{10}) \) \( 2 q - 14 q^{9} + 124 q^{13} + 92 q^{17} - 280 q^{21} - 180 q^{29} + 80 q^{33} + 428 q^{37} - 20 q^{41} + 1274 q^{49} + 1356 q^{53} - 960 q^{57} + 500 q^{61} + 1720 q^{69} - 1044 q^{73} - 560 q^{77} - 982 q^{81} + 1940 q^{89} + 1360 q^{93} + 1868 q^{97} + O(q^{100}) \)

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
\(3\) −4.47214 −0.860663 −0.430331 0.902671i \(-0.641603\pi\)
−0.430331 + 0.902671i \(0.641603\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 31.3050 1.69031 0.845154 0.534522i \(-0.179509\pi\)
0.845154 + 0.534522i \(0.179509\pi\)
\(8\) 0 0
\(9\) −7.00000 −0.259259
\(10\) 0 0
\(11\) −8.94427 −0.245164 −0.122582 0.992458i \(-0.539117\pi\)
−0.122582 + 0.992458i \(0.539117\pi\)
\(12\) 0 0
\(13\) 62.0000 1.32275 0.661373 0.750057i \(-0.269974\pi\)
0.661373 + 0.750057i \(0.269974\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 46.0000 0.656273 0.328136 0.944630i \(-0.393579\pi\)
0.328136 + 0.944630i \(0.393579\pi\)
\(18\) 0 0
\(19\) 107.331 1.29597 0.647986 0.761652i \(-0.275611\pi\)
0.647986 + 0.761652i \(0.275611\pi\)
\(20\) 0 0
\(21\) −140.000 −1.45479
\(22\) 0 0
\(23\) −192.302 −1.74338 −0.871689 0.490059i \(-0.836975\pi\)
−0.871689 + 0.490059i \(0.836975\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 152.053 1.08380
\(28\) 0 0
\(29\) −90.0000 −0.576296 −0.288148 0.957586i \(-0.593039\pi\)
−0.288148 + 0.957586i \(0.593039\pi\)
\(30\) 0 0
\(31\) −152.053 −0.880950 −0.440475 0.897765i \(-0.645190\pi\)
−0.440475 + 0.897765i \(0.645190\pi\)
\(32\) 0 0
\(33\) 40.0000 0.211003
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 214.000 0.950848 0.475424 0.879757i \(-0.342295\pi\)
0.475424 + 0.879757i \(0.342295\pi\)
\(38\) 0 0
\(39\) −277.272 −1.13844
\(40\) 0 0
\(41\) −10.0000 −0.0380912 −0.0190456 0.999819i \(-0.506063\pi\)
−0.0190456 + 0.999819i \(0.506063\pi\)
\(42\) 0 0
\(43\) 67.0820 0.237905 0.118953 0.992900i \(-0.462046\pi\)
0.118953 + 0.992900i \(0.462046\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −398.020 −1.23526 −0.617630 0.786469i \(-0.711907\pi\)
−0.617630 + 0.786469i \(0.711907\pi\)
\(48\) 0 0
\(49\) 637.000 1.85714
\(50\) 0 0
\(51\) −205.718 −0.564830
\(52\) 0 0
\(53\) 678.000 1.75718 0.878589 0.477578i \(-0.158485\pi\)
0.878589 + 0.477578i \(0.158485\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −480.000 −1.11540
\(58\) 0 0
\(59\) −411.437 −0.907872 −0.453936 0.891034i \(-0.649981\pi\)
−0.453936 + 0.891034i \(0.649981\pi\)
\(60\) 0 0
\(61\) 250.000 0.524741 0.262371 0.964967i \(-0.415496\pi\)
0.262371 + 0.964967i \(0.415496\pi\)
\(62\) 0 0
\(63\) −219.135 −0.438228
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −49.1935 −0.0897006 −0.0448503 0.998994i \(-0.514281\pi\)
−0.0448503 + 0.998994i \(0.514281\pi\)
\(68\) 0 0
\(69\) 860.000 1.50046
\(70\) 0 0
\(71\) −366.715 −0.612973 −0.306486 0.951875i \(-0.599153\pi\)
−0.306486 + 0.951875i \(0.599153\pi\)
\(72\) 0 0
\(73\) −522.000 −0.836924 −0.418462 0.908234i \(-0.637431\pi\)
−0.418462 + 0.908234i \(0.637431\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −280.000 −0.414402
\(78\) 0 0
\(79\) 876.539 1.24833 0.624166 0.781291i \(-0.285439\pi\)
0.624166 + 0.781291i \(0.285439\pi\)
\(80\) 0 0
\(81\) −491.000 −0.673525
\(82\) 0 0
\(83\) −380.132 −0.502709 −0.251355 0.967895i \(-0.580876\pi\)
−0.251355 + 0.967895i \(0.580876\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 402.492 0.495997
\(88\) 0 0
\(89\) 970.000 1.15528 0.577639 0.816292i \(-0.303974\pi\)
0.577639 + 0.816292i \(0.303974\pi\)
\(90\) 0 0
\(91\) 1940.91 2.23585
\(92\) 0 0
\(93\) 680.000 0.758201
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 934.000 0.977663 0.488832 0.872378i \(-0.337423\pi\)
0.488832 + 0.872378i \(0.337423\pi\)
\(98\) 0 0
\(99\) 62.6099 0.0635609
\(100\) 0 0
\(101\) −602.000 −0.593082 −0.296541 0.955020i \(-0.595833\pi\)
−0.296541 + 0.955020i \(0.595833\pi\)
\(102\) 0 0
\(103\) 1829.10 1.74978 0.874888 0.484325i \(-0.160935\pi\)
0.874888 + 0.484325i \(0.160935\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1525.00 1.37782 0.688912 0.724845i \(-0.258089\pi\)
0.688912 + 0.724845i \(0.258089\pi\)
\(108\) 0 0
\(109\) 2154.00 1.89281 0.946403 0.322989i \(-0.104688\pi\)
0.946403 + 0.322989i \(0.104688\pi\)
\(110\) 0 0
\(111\) −957.037 −0.818360
\(112\) 0 0
\(113\) 2182.00 1.81651 0.908254 0.418420i \(-0.137416\pi\)
0.908254 + 0.418420i \(0.137416\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −434.000 −0.342934
\(118\) 0 0
\(119\) 1440.03 1.10930
\(120\) 0 0
\(121\) −1251.00 −0.939895
\(122\) 0 0
\(123\) 44.7214 0.0327837
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −1310.34 −0.915539 −0.457770 0.889071i \(-0.651352\pi\)
−0.457770 + 0.889071i \(0.651352\pi\)
\(128\) 0 0
\(129\) −300.000 −0.204756
\(130\) 0 0
\(131\) −205.718 −0.137204 −0.0686019 0.997644i \(-0.521854\pi\)
−0.0686019 + 0.997644i \(0.521854\pi\)
\(132\) 0 0
\(133\) 3360.00 2.19059
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2094.00 1.30586 0.652929 0.757419i \(-0.273540\pi\)
0.652929 + 0.757419i \(0.273540\pi\)
\(138\) 0 0
\(139\) −1377.42 −0.840511 −0.420256 0.907406i \(-0.638060\pi\)
−0.420256 + 0.907406i \(0.638060\pi\)
\(140\) 0 0
\(141\) 1780.00 1.06314
\(142\) 0 0
\(143\) −554.545 −0.324289
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −2848.75 −1.59837
\(148\) 0 0
\(149\) −334.000 −0.183640 −0.0918200 0.995776i \(-0.529268\pi\)
−0.0918200 + 0.995776i \(0.529268\pi\)
\(150\) 0 0
\(151\) 3139.44 1.69195 0.845973 0.533225i \(-0.179020\pi\)
0.845973 + 0.533225i \(0.179020\pi\)
\(152\) 0 0
\(153\) −322.000 −0.170145
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −834.000 −0.423952 −0.211976 0.977275i \(-0.567990\pi\)
−0.211976 + 0.977275i \(0.567990\pi\)
\(158\) 0 0
\(159\) −3032.11 −1.51234
\(160\) 0 0
\(161\) −6020.00 −2.94685
\(162\) 0 0
\(163\) 3090.25 1.48495 0.742475 0.669874i \(-0.233652\pi\)
0.742475 + 0.669874i \(0.233652\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −4.47214 −0.00207224 −0.00103612 0.999999i \(-0.500330\pi\)
−0.00103612 + 0.999999i \(0.500330\pi\)
\(168\) 0 0
\(169\) 1647.00 0.749659
\(170\) 0 0
\(171\) −751.319 −0.335993
\(172\) 0 0
\(173\) 1838.00 0.807749 0.403874 0.914814i \(-0.367663\pi\)
0.403874 + 0.914814i \(0.367663\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 1840.00 0.781372
\(178\) 0 0
\(179\) 1842.52 0.769365 0.384683 0.923049i \(-0.374311\pi\)
0.384683 + 0.923049i \(0.374311\pi\)
\(180\) 0 0
\(181\) 1862.00 0.764648 0.382324 0.924028i \(-0.375124\pi\)
0.382324 + 0.924028i \(0.375124\pi\)
\(182\) 0 0
\(183\) −1118.03 −0.451625
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −411.437 −0.160894
\(188\) 0 0
\(189\) 4760.00 1.83195
\(190\) 0 0
\(191\) −2066.13 −0.782721 −0.391360 0.920237i \(-0.627995\pi\)
−0.391360 + 0.920237i \(0.627995\pi\)
\(192\) 0 0
\(193\) −3378.00 −1.25986 −0.629932 0.776650i \(-0.716917\pi\)
−0.629932 + 0.776650i \(0.716917\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −66.0000 −0.0238696 −0.0119348 0.999929i \(-0.503799\pi\)
−0.0119348 + 0.999929i \(0.503799\pi\)
\(198\) 0 0
\(199\) −1216.42 −0.433316 −0.216658 0.976248i \(-0.569516\pi\)
−0.216658 + 0.976248i \(0.569516\pi\)
\(200\) 0 0
\(201\) 220.000 0.0772020
\(202\) 0 0
\(203\) −2817.45 −0.974118
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 1346.11 0.451987
\(208\) 0 0
\(209\) −960.000 −0.317725
\(210\) 0 0
\(211\) −5286.06 −1.72468 −0.862341 0.506329i \(-0.831002\pi\)
−0.862341 + 0.506329i \(0.831002\pi\)
\(212\) 0 0
\(213\) 1640.00 0.527563
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −4760.00 −1.48908
\(218\) 0 0
\(219\) 2334.45 0.720310
\(220\) 0 0
\(221\) 2852.00 0.868083
\(222\) 0 0
\(223\) −2965.03 −0.890371 −0.445186 0.895438i \(-0.646862\pi\)
−0.445186 + 0.895438i \(0.646862\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 4369.28 1.27753 0.638765 0.769402i \(-0.279446\pi\)
0.638765 + 0.769402i \(0.279446\pi\)
\(228\) 0 0
\(229\) −3250.00 −0.937843 −0.468921 0.883240i \(-0.655357\pi\)
−0.468921 + 0.883240i \(0.655357\pi\)
\(230\) 0 0
\(231\) 1252.20 0.356661
\(232\) 0 0
\(233\) −3298.00 −0.927293 −0.463646 0.886020i \(-0.653459\pi\)
−0.463646 + 0.886020i \(0.653459\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −3920.00 −1.07439
\(238\) 0 0
\(239\) −554.545 −0.150086 −0.0750429 0.997180i \(-0.523909\pi\)
−0.0750429 + 0.997180i \(0.523909\pi\)
\(240\) 0 0
\(241\) 5150.00 1.37652 0.688259 0.725465i \(-0.258375\pi\)
0.688259 + 0.725465i \(0.258375\pi\)
\(242\) 0 0
\(243\) −1909.60 −0.504119
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 6654.54 1.71424
\(248\) 0 0
\(249\) 1700.00 0.432663
\(250\) 0 0
\(251\) 1386.36 0.348631 0.174316 0.984690i \(-0.444229\pi\)
0.174316 + 0.984690i \(0.444229\pi\)
\(252\) 0 0
\(253\) 1720.00 0.427413
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 4166.00 1.01116 0.505580 0.862780i \(-0.331279\pi\)
0.505580 + 0.862780i \(0.331279\pi\)
\(258\) 0 0
\(259\) 6699.26 1.60723
\(260\) 0 0
\(261\) 630.000 0.149410
\(262\) 0 0
\(263\) −961.509 −0.225434 −0.112717 0.993627i \(-0.535955\pi\)
−0.112717 + 0.993627i \(0.535955\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −4337.97 −0.994305
\(268\) 0 0
\(269\) −1494.00 −0.338627 −0.169314 0.985562i \(-0.554155\pi\)
−0.169314 + 0.985562i \(0.554155\pi\)
\(270\) 0 0
\(271\) 5017.74 1.12474 0.562372 0.826884i \(-0.309889\pi\)
0.562372 + 0.826884i \(0.309889\pi\)
\(272\) 0 0
\(273\) −8680.00 −1.92431
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 1006.00 0.218212 0.109106 0.994030i \(-0.465201\pi\)
0.109106 + 0.994030i \(0.465201\pi\)
\(278\) 0 0
\(279\) 1064.37 0.228395
\(280\) 0 0
\(281\) −3210.00 −0.681468 −0.340734 0.940160i \(-0.610676\pi\)
−0.340734 + 0.940160i \(0.610676\pi\)
\(282\) 0 0
\(283\) −3635.85 −0.763705 −0.381853 0.924223i \(-0.624714\pi\)
−0.381853 + 0.924223i \(0.624714\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −313.050 −0.0643858
\(288\) 0 0
\(289\) −2797.00 −0.569306
\(290\) 0 0
\(291\) −4176.97 −0.841439
\(292\) 0 0
\(293\) 3622.00 0.722183 0.361091 0.932530i \(-0.382404\pi\)
0.361091 + 0.932530i \(0.382404\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −1360.00 −0.265708
\(298\) 0 0
\(299\) −11922.7 −2.30605
\(300\) 0 0
\(301\) 2100.00 0.402133
\(302\) 0 0
\(303\) 2692.23 0.510443
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −2088.49 −0.388261 −0.194131 0.980976i \(-0.562189\pi\)
−0.194131 + 0.980976i \(0.562189\pi\)
\(308\) 0 0
\(309\) −8180.00 −1.50597
\(310\) 0 0
\(311\) −8899.55 −1.62266 −0.811330 0.584589i \(-0.801256\pi\)
−0.811330 + 0.584589i \(0.801256\pi\)
\(312\) 0 0
\(313\) −8778.00 −1.58518 −0.792591 0.609754i \(-0.791268\pi\)
−0.792591 + 0.609754i \(0.791268\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 5046.00 0.894043 0.447021 0.894523i \(-0.352485\pi\)
0.447021 + 0.894523i \(0.352485\pi\)
\(318\) 0 0
\(319\) 804.984 0.141287
\(320\) 0 0
\(321\) −6820.00 −1.18584
\(322\) 0 0
\(323\) 4937.24 0.850512
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −9632.98 −1.62907
\(328\) 0 0
\(329\) −12460.0 −2.08797
\(330\) 0 0
\(331\) −313.050 −0.0519842 −0.0259921 0.999662i \(-0.508274\pi\)
−0.0259921 + 0.999662i \(0.508274\pi\)
\(332\) 0 0
\(333\) −1498.00 −0.246516
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 2574.00 0.416067 0.208034 0.978122i \(-0.433294\pi\)
0.208034 + 0.978122i \(0.433294\pi\)
\(338\) 0 0
\(339\) −9758.20 −1.56340
\(340\) 0 0
\(341\) 1360.00 0.215977
\(342\) 0 0
\(343\) 9203.66 1.44884
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −2643.03 −0.408892 −0.204446 0.978878i \(-0.565539\pi\)
−0.204446 + 0.978878i \(0.565539\pi\)
\(348\) 0 0
\(349\) −10170.0 −1.55985 −0.779925 0.625873i \(-0.784743\pi\)
−0.779925 + 0.625873i \(0.784743\pi\)
\(350\) 0 0
\(351\) 9427.26 1.43359
\(352\) 0 0
\(353\) 318.000 0.0479474 0.0239737 0.999713i \(-0.492368\pi\)
0.0239737 + 0.999713i \(0.492368\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −6440.00 −0.954737
\(358\) 0 0
\(359\) 12378.9 1.81987 0.909933 0.414755i \(-0.136133\pi\)
0.909933 + 0.414755i \(0.136133\pi\)
\(360\) 0 0
\(361\) 4661.00 0.679545
\(362\) 0 0
\(363\) 5594.64 0.808933
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 3072.36 0.436991 0.218496 0.975838i \(-0.429885\pi\)
0.218496 + 0.975838i \(0.429885\pi\)
\(368\) 0 0
\(369\) 70.0000 0.00987549
\(370\) 0 0
\(371\) 21224.8 2.97017
\(372\) 0 0
\(373\) 3278.00 0.455036 0.227518 0.973774i \(-0.426939\pi\)
0.227518 + 0.973774i \(0.426939\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −5580.00 −0.762293
\(378\) 0 0
\(379\) 5116.12 0.693397 0.346699 0.937977i \(-0.387303\pi\)
0.346699 + 0.937977i \(0.387303\pi\)
\(380\) 0 0
\(381\) 5860.00 0.787971
\(382\) 0 0
\(383\) 1149.34 0.153338 0.0766690 0.997057i \(-0.475572\pi\)
0.0766690 + 0.997057i \(0.475572\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −469.574 −0.0616791
\(388\) 0 0
\(389\) 834.000 0.108703 0.0543515 0.998522i \(-0.482691\pi\)
0.0543515 + 0.998522i \(0.482691\pi\)
\(390\) 0 0
\(391\) −8845.88 −1.14413
\(392\) 0 0
\(393\) 920.000 0.118086
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 8734.00 1.10415 0.552074 0.833795i \(-0.313837\pi\)
0.552074 + 0.833795i \(0.313837\pi\)
\(398\) 0 0
\(399\) −15026.4 −1.88536
\(400\) 0 0
\(401\) 242.000 0.0301369 0.0150685 0.999886i \(-0.495203\pi\)
0.0150685 + 0.999886i \(0.495203\pi\)
\(402\) 0 0
\(403\) −9427.26 −1.16527
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −1914.07 −0.233113
\(408\) 0 0
\(409\) −6514.00 −0.787522 −0.393761 0.919213i \(-0.628826\pi\)
−0.393761 + 0.919213i \(0.628826\pi\)
\(410\) 0 0
\(411\) −9364.65 −1.12390
\(412\) 0 0
\(413\) −12880.0 −1.53458
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 6160.00 0.723397
\(418\) 0 0
\(419\) 16081.8 1.87505 0.937527 0.347913i \(-0.113110\pi\)
0.937527 + 0.347913i \(0.113110\pi\)
\(420\) 0 0
\(421\) 7250.00 0.839295 0.419648 0.907687i \(-0.362154\pi\)
0.419648 + 0.907687i \(0.362154\pi\)
\(422\) 0 0
\(423\) 2786.14 0.320252
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 7826.24 0.886975
\(428\) 0 0
\(429\) 2480.00 0.279104
\(430\) 0 0
\(431\) 4981.96 0.556781 0.278390 0.960468i \(-0.410199\pi\)
0.278390 + 0.960468i \(0.410199\pi\)
\(432\) 0 0
\(433\) −11482.0 −1.27434 −0.637171 0.770723i \(-0.719895\pi\)
−0.637171 + 0.770723i \(0.719895\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −20640.0 −2.25937
\(438\) 0 0
\(439\) 3792.37 0.412301 0.206150 0.978520i \(-0.433906\pi\)
0.206150 + 0.978520i \(0.433906\pi\)
\(440\) 0 0
\(441\) −4459.00 −0.481481
\(442\) 0 0
\(443\) 746.847 0.0800988 0.0400494 0.999198i \(-0.487248\pi\)
0.0400494 + 0.999198i \(0.487248\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 1493.69 0.158052
\(448\) 0 0
\(449\) −1306.00 −0.137269 −0.0686347 0.997642i \(-0.521864\pi\)
−0.0686347 + 0.997642i \(0.521864\pi\)
\(450\) 0 0
\(451\) 89.4427 0.00933857
\(452\) 0 0
\(453\) −14040.0 −1.45620
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 9526.00 0.975071 0.487536 0.873103i \(-0.337896\pi\)
0.487536 + 0.873103i \(0.337896\pi\)
\(458\) 0 0
\(459\) 6994.42 0.711267
\(460\) 0 0
\(461\) 1518.00 0.153363 0.0766815 0.997056i \(-0.475568\pi\)
0.0766815 + 0.997056i \(0.475568\pi\)
\(462\) 0 0
\(463\) −17293.7 −1.73587 −0.867936 0.496676i \(-0.834554\pi\)
−0.867936 + 0.496676i \(0.834554\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 16980.7 1.68260 0.841299 0.540570i \(-0.181792\pi\)
0.841299 + 0.540570i \(0.181792\pi\)
\(468\) 0 0
\(469\) −1540.00 −0.151622
\(470\) 0 0
\(471\) 3729.76 0.364880
\(472\) 0 0
\(473\) −600.000 −0.0583256
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −4746.00 −0.455565
\(478\) 0 0
\(479\) 3810.26 0.363456 0.181728 0.983349i \(-0.441831\pi\)
0.181728 + 0.983349i \(0.441831\pi\)
\(480\) 0 0
\(481\) 13268.0 1.25773
\(482\) 0 0
\(483\) 26922.3 2.53624
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −1310.34 −0.121924 −0.0609620 0.998140i \(-0.519417\pi\)
−0.0609620 + 0.998140i \(0.519417\pi\)
\(488\) 0 0
\(489\) −13820.0 −1.27804
\(490\) 0 0
\(491\) 2960.55 0.272114 0.136057 0.990701i \(-0.456557\pi\)
0.136057 + 0.990701i \(0.456557\pi\)
\(492\) 0 0
\(493\) −4140.00 −0.378207
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −11480.0 −1.03611
\(498\) 0 0
\(499\) −19319.6 −1.73320 −0.866598 0.499006i \(-0.833699\pi\)
−0.866598 + 0.499006i \(0.833699\pi\)
\(500\) 0 0
\(501\) 20.0000 0.00178350
\(502\) 0 0
\(503\) −3072.36 −0.272345 −0.136173 0.990685i \(-0.543480\pi\)
−0.136173 + 0.990685i \(0.543480\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −7365.61 −0.645203
\(508\) 0 0
\(509\) 18550.0 1.61535 0.807676 0.589626i \(-0.200725\pi\)
0.807676 + 0.589626i \(0.200725\pi\)
\(510\) 0 0
\(511\) −16341.2 −1.41466
\(512\) 0 0
\(513\) 16320.0 1.40457
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 3560.00 0.302841
\(518\) 0 0
\(519\) −8219.79 −0.695200
\(520\) 0 0
\(521\) −2102.00 −0.176757 −0.0883784 0.996087i \(-0.528168\pi\)
−0.0883784 + 0.996087i \(0.528168\pi\)
\(522\) 0 0
\(523\) −17696.2 −1.47955 −0.739773 0.672856i \(-0.765067\pi\)
−0.739773 + 0.672856i \(0.765067\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −6994.42 −0.578144
\(528\) 0 0
\(529\) 24813.0 2.03937
\(530\) 0 0
\(531\) 2880.06 0.235374
\(532\) 0 0
\(533\) −620.000 −0.0503850
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −8240.00 −0.662164
\(538\) 0 0
\(539\) −5697.50 −0.455304
\(540\) 0 0
\(541\) −9922.00 −0.788503 −0.394251 0.919003i \(-0.628996\pi\)
−0.394251 + 0.919003i \(0.628996\pi\)
\(542\) 0 0
\(543\) −8327.12 −0.658105
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −3716.34 −0.290493 −0.145246 0.989396i \(-0.546397\pi\)
−0.145246 + 0.989396i \(0.546397\pi\)
\(548\) 0 0
\(549\) −1750.00 −0.136044
\(550\) 0 0
\(551\) −9659.81 −0.746864
\(552\) 0 0
\(553\) 27440.0 2.11007
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 15094.0 1.14821 0.574105 0.818781i \(-0.305350\pi\)
0.574105 + 0.818781i \(0.305350\pi\)
\(558\) 0 0
\(559\) 4159.09 0.314688
\(560\) 0 0
\(561\) 1840.00 0.138476
\(562\) 0 0
\(563\) −5657.25 −0.423490 −0.211745 0.977325i \(-0.567915\pi\)
−0.211745 + 0.977325i \(0.567915\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −15370.7 −1.13847
\(568\) 0 0
\(569\) −5906.00 −0.435136 −0.217568 0.976045i \(-0.569812\pi\)
−0.217568 + 0.976045i \(0.569812\pi\)
\(570\) 0 0
\(571\) −4892.52 −0.358573 −0.179287 0.983797i \(-0.557379\pi\)
−0.179287 + 0.983797i \(0.557379\pi\)
\(572\) 0 0
\(573\) 9240.00 0.673659
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 13286.0 0.958585 0.479292 0.877655i \(-0.340893\pi\)
0.479292 + 0.877655i \(0.340893\pi\)
\(578\) 0 0
\(579\) 15106.9 1.08432
\(580\) 0 0
\(581\) −11900.0 −0.849734
\(582\) 0 0
\(583\) −6064.22 −0.430796
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −9029.24 −0.634884 −0.317442 0.948278i \(-0.602824\pi\)
−0.317442 + 0.948278i \(0.602824\pi\)
\(588\) 0 0
\(589\) −16320.0 −1.14169
\(590\) 0 0
\(591\) 295.161 0.0205437
\(592\) 0 0
\(593\) −11442.0 −0.792355 −0.396178 0.918174i \(-0.629664\pi\)
−0.396178 + 0.918174i \(0.629664\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 5440.00 0.372939
\(598\) 0 0
\(599\) −14149.8 −0.965187 −0.482593 0.875845i \(-0.660305\pi\)
−0.482593 + 0.875845i \(0.660305\pi\)
\(600\) 0 0
\(601\) 3110.00 0.211081 0.105540 0.994415i \(-0.466343\pi\)
0.105540 + 0.994415i \(0.466343\pi\)
\(602\) 0 0
\(603\) 344.354 0.0232557
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 11193.8 0.748502 0.374251 0.927327i \(-0.377900\pi\)
0.374251 + 0.927327i \(0.377900\pi\)
\(608\) 0 0
\(609\) 12600.0 0.838387
\(610\) 0 0
\(611\) −24677.2 −1.63394
\(612\) 0 0
\(613\) 5342.00 0.351976 0.175988 0.984392i \(-0.443688\pi\)
0.175988 + 0.984392i \(0.443688\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −19714.0 −1.28631 −0.643157 0.765734i \(-0.722376\pi\)
−0.643157 + 0.765734i \(0.722376\pi\)
\(618\) 0 0
\(619\) −13166.0 −0.854903 −0.427451 0.904038i \(-0.640589\pi\)
−0.427451 + 0.904038i \(0.640589\pi\)
\(620\) 0 0
\(621\) −29240.0 −1.88947
\(622\) 0 0
\(623\) 30365.8 1.95278
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 4293.25 0.273454
\(628\) 0 0
\(629\) 9844.00 0.624016
\(630\) 0 0
\(631\) 12262.6 0.773639 0.386820 0.922155i \(-0.373574\pi\)
0.386820 + 0.922155i \(0.373574\pi\)
\(632\) 0 0
\(633\) 23640.0 1.48437
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 39494.0 2.45653
\(638\) 0 0
\(639\) 2567.01 0.158919
\(640\) 0 0
\(641\) −2690.00 −0.165754 −0.0828772 0.996560i \(-0.526411\pi\)
−0.0828772 + 0.996560i \(0.526411\pi\)
\(642\) 0 0
\(643\) −12240.2 −0.750712 −0.375356 0.926881i \(-0.622480\pi\)
−0.375356 + 0.926881i \(0.622480\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 17973.5 1.09214 0.546068 0.837741i \(-0.316124\pi\)
0.546068 + 0.837741i \(0.316124\pi\)
\(648\) 0 0
\(649\) 3680.00 0.222577
\(650\) 0 0
\(651\) 21287.4 1.28159
\(652\) 0 0
\(653\) 3478.00 0.208430 0.104215 0.994555i \(-0.466767\pi\)
0.104215 + 0.994555i \(0.466767\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 3654.00 0.216980
\(658\) 0 0
\(659\) −10572.1 −0.624934 −0.312467 0.949929i \(-0.601155\pi\)
−0.312467 + 0.949929i \(0.601155\pi\)
\(660\) 0 0
\(661\) −110.000 −0.00647277 −0.00323639 0.999995i \(-0.501030\pi\)
−0.00323639 + 0.999995i \(0.501030\pi\)
\(662\) 0 0
\(663\) −12754.5 −0.747127
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 17307.2 1.00470
\(668\) 0 0
\(669\) 13260.0 0.766310
\(670\) 0 0
\(671\) −2236.07 −0.128647
\(672\) 0 0
\(673\) 14278.0 0.817796 0.408898 0.912580i \(-0.365913\pi\)
0.408898 + 0.912580i \(0.365913\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −18386.0 −1.04377 −0.521884 0.853016i \(-0.674771\pi\)
−0.521884 + 0.853016i \(0.674771\pi\)
\(678\) 0 0
\(679\) 29238.8 1.65255
\(680\) 0 0
\(681\) −19540.0 −1.09952
\(682\) 0 0
\(683\) −15317.1 −0.858113 −0.429057 0.903278i \(-0.641154\pi\)
−0.429057 + 0.903278i \(0.641154\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 14534.4 0.807167
\(688\) 0 0
\(689\) 42036.0 2.32430
\(690\) 0 0
\(691\) 9507.76 0.523433 0.261717 0.965145i \(-0.415711\pi\)
0.261717 + 0.965145i \(0.415711\pi\)
\(692\) 0 0
\(693\) 1960.00 0.107438
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −460.000 −0.0249982
\(698\) 0 0
\(699\) 14749.1 0.798086
\(700\) 0 0
\(701\) −15830.0 −0.852911 −0.426456 0.904508i \(-0.640238\pi\)
−0.426456 + 0.904508i \(0.640238\pi\)
\(702\) 0 0
\(703\) 22968.9 1.23227
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −18845.6 −1.00249
\(708\) 0 0
\(709\) −20050.0 −1.06205 −0.531025 0.847356i \(-0.678193\pi\)
−0.531025 + 0.847356i \(0.678193\pi\)
\(710\) 0 0
\(711\) −6135.77 −0.323642
\(712\) 0 0
\(713\) 29240.0 1.53583
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 2480.00 0.129173
\(718\) 0 0
\(719\) 21126.4 1.09580 0.547900 0.836544i \(-0.315427\pi\)
0.547900 + 0.836544i \(0.315427\pi\)
\(720\) 0 0
\(721\) 57260.0 2.95766
\(722\) 0 0
\(723\) −23031.5 −1.18472
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 11336.9 0.578351 0.289175 0.957276i \(-0.406619\pi\)
0.289175 + 0.957276i \(0.406619\pi\)
\(728\) 0 0
\(729\) 21797.0 1.10740
\(730\) 0 0
\(731\) 3085.77 0.156131
\(732\) 0 0
\(733\) 17198.0 0.866607 0.433303 0.901248i \(-0.357348\pi\)
0.433303 + 0.901248i \(0.357348\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 440.000 0.0219913
\(738\) 0 0
\(739\) 4597.36 0.228845 0.114423 0.993432i \(-0.463498\pi\)
0.114423 + 0.993432i \(0.463498\pi\)
\(740\) 0 0
\(741\) −29760.0 −1.47539
\(742\) 0 0
\(743\) 2419.43 0.119462 0.0597309 0.998215i \(-0.480976\pi\)
0.0597309 + 0.998215i \(0.480976\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 2660.92 0.130332
\(748\) 0 0
\(749\) 47740.0 2.32895
\(750\) 0 0
\(751\) 7432.69 0.361149 0.180574 0.983561i \(-0.442204\pi\)
0.180574 + 0.983561i \(0.442204\pi\)
\(752\) 0 0
\(753\) −6200.00 −0.300054
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −11474.0 −0.550898 −0.275449 0.961316i \(-0.588826\pi\)
−0.275449 + 0.961316i \(0.588826\pi\)
\(758\) 0 0
\(759\) −7692.07 −0.367858
\(760\) 0 0
\(761\) 31802.0 1.51488 0.757439 0.652906i \(-0.226450\pi\)
0.757439 + 0.652906i \(0.226450\pi\)
\(762\) 0 0
\(763\) 67430.9 3.19942
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −25509.1 −1.20089
\(768\) 0 0
\(769\) −5310.00 −0.249003 −0.124502 0.992219i \(-0.539733\pi\)
−0.124502 + 0.992219i \(0.539733\pi\)
\(770\) 0 0
\(771\) −18630.9 −0.870267
\(772\) 0 0
\(773\) −37938.0 −1.76525 −0.882623 0.470082i \(-0.844224\pi\)
−0.882623 + 0.470082i \(0.844224\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −29960.0 −1.38328
\(778\) 0 0
\(779\) −1073.31 −0.0493651
\(780\) 0 0
\(781\) 3280.00 0.150279
\(782\) 0 0
\(783\) −13684.7 −0.624588
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −37633.0 −1.70454 −0.852270 0.523103i \(-0.824774\pi\)
−0.852270 + 0.523103i \(0.824774\pi\)
\(788\) 0 0
\(789\) 4300.00 0.194023
\(790\) 0 0
\(791\) 68307.4 3.07046
\(792\) 0 0
\(793\) 15500.0 0.694100
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 17526.0 0.778924 0.389462 0.921042i \(-0.372661\pi\)
0.389462 + 0.921042i \(0.372661\pi\)
\(798\) 0 0
\(799\) −18308.9 −0.810667
\(800\) 0 0
\(801\) −6790.00 −0.299517
\(802\) 0 0
\(803\) 4668.91 0.205183
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 6681.37 0.291444
\(808\) 0 0
\(809\) 8970.00 0.389825 0.194912 0.980821i \(-0.437558\pi\)
0.194912 + 0.980821i \(0.437558\pi\)
\(810\) 0 0
\(811\) −3550.88 −0.153746 −0.0768731 0.997041i \(-0.524494\pi\)
−0.0768731 + 0.997041i \(0.524494\pi\)
\(812\) 0 0
\(813\) −22440.0 −0.968026
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 7200.00 0.308318
\(818\) 0 0
\(819\) −13586.3 −0.579665
\(820\) 0 0
\(821\) −15550.0 −0.661022 −0.330511 0.943802i \(-0.607221\pi\)
−0.330511 + 0.943802i \(0.607221\pi\)
\(822\) 0 0
\(823\) 26712.1 1.13138 0.565689 0.824619i \(-0.308610\pi\)
0.565689 + 0.824619i \(0.308610\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 863.122 0.0362923 0.0181461 0.999835i \(-0.494224\pi\)
0.0181461 + 0.999835i \(0.494224\pi\)
\(828\) 0 0
\(829\) 19066.0 0.798781 0.399391 0.916781i \(-0.369222\pi\)
0.399391 + 0.916781i \(0.369222\pi\)
\(830\) 0 0
\(831\) −4498.97 −0.187807
\(832\) 0 0
\(833\) 29302.0 1.21879
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −23120.0 −0.954772
\(838\) 0 0
\(839\) −47744.5 −1.96463 −0.982315 0.187238i \(-0.940047\pi\)
−0.982315 + 0.187238i \(0.940047\pi\)
\(840\) 0 0
\(841\) −16289.0 −0.667883
\(842\) 0 0
\(843\) 14355.6 0.586514
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −39162.5 −1.58871
\(848\) 0 0
\(849\) 16260.0 0.657293
\(850\) 0 0
\(851\) −41152.6 −1.65769
\(852\) 0 0
\(853\) 14462.0 0.580503 0.290252 0.956950i \(-0.406261\pi\)
0.290252 + 0.956950i \(0.406261\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −29346.0 −1.16971 −0.584854 0.811138i \(-0.698848\pi\)
−0.584854 + 0.811138i \(0.698848\pi\)
\(858\) 0 0
\(859\) 22807.9 0.905932 0.452966 0.891528i \(-0.350366\pi\)
0.452966 + 0.891528i \(0.350366\pi\)
\(860\) 0 0
\(861\) 1400.00 0.0554145
\(862\) 0 0
\(863\) −24753.3 −0.976375 −0.488187 0.872739i \(-0.662342\pi\)
−0.488187 + 0.872739i \(0.662342\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 12508.6 0.489981
\(868\) 0 0
\(869\) −7840.00 −0.306046
\(870\) 0 0
\(871\) −3050.00 −0.118651
\(872\) 0 0
\(873\) −6538.00 −0.253468
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 32126.0 1.23696 0.618482 0.785799i \(-0.287748\pi\)
0.618482 + 0.785799i \(0.287748\pi\)
\(878\) 0 0
\(879\) −16198.1 −0.621556
\(880\) 0 0
\(881\) −33570.0 −1.28377 −0.641885 0.766801i \(-0.721848\pi\)
−0.641885 + 0.766801i \(0.721848\pi\)
\(882\) 0 0
\(883\) 6435.40 0.245265 0.122632 0.992452i \(-0.460866\pi\)
0.122632 + 0.992452i \(0.460866\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 46827.7 1.77263 0.886314 0.463084i \(-0.153257\pi\)
0.886314 + 0.463084i \(0.153257\pi\)
\(888\) 0 0
\(889\) −41020.0 −1.54754
\(890\) 0 0
\(891\) 4391.64 0.165124
\(892\) 0 0
\(893\) −42720.0 −1.60086
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 53320.0 1.98473
\(898\) 0 0
\(899\) 13684.7 0.507688
\(900\) 0 0
\(901\) 31188.0 1.15319
\(902\) 0 0
\(903\) −9391.49 −0.346101
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −11980.9 −0.438608 −0.219304 0.975657i \(-0.570379\pi\)
−0.219304 + 0.975657i \(0.570379\pi\)
\(908\) 0 0
\(909\) 4214.00 0.153762
\(910\) 0 0
\(911\) −24194.3 −0.879903 −0.439951 0.898022i \(-0.645004\pi\)
−0.439951 + 0.898022i \(0.645004\pi\)
\(912\) 0 0
\(913\) 3400.00 0.123246
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −6440.00 −0.231917
\(918\) 0 0
\(919\) −37512.3 −1.34648 −0.673240 0.739424i \(-0.735098\pi\)
−0.673240 + 0.739424i \(0.735098\pi\)
\(920\) 0 0
\(921\) 9340.00 0.334162
\(922\) 0 0
\(923\) −22736.3 −0.810808
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −12803.7 −0.453646
\(928\) 0 0
\(929\) −21994.0 −0.776749 −0.388374 0.921502i \(-0.626963\pi\)
−0.388374 + 0.921502i \(0.626963\pi\)
\(930\) 0 0
\(931\) 68370.0 2.40681
\(932\) 0 0
\(933\) 39800.0 1.39656
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 16286.0 0.567813 0.283906 0.958852i \(-0.408370\pi\)
0.283906 + 0.958852i \(0.408370\pi\)
\(938\) 0 0
\(939\) 39256.4 1.36431
\(940\) 0 0
\(941\) 24302.0 0.841894 0.420947 0.907085i \(-0.361698\pi\)
0.420947 + 0.907085i \(0.361698\pi\)
\(942\) 0 0
\(943\) 1923.02 0.0664073
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −19869.7 −0.681815 −0.340907 0.940097i \(-0.610734\pi\)
−0.340907 + 0.940097i \(0.610734\pi\)
\(948\) 0 0
\(949\) −32364.0 −1.10704
\(950\) 0 0
\(951\) −22566.4 −0.769470
\(952\) 0 0
\(953\) 22422.0 0.762140 0.381070 0.924546i \(-0.375556\pi\)
0.381070 + 0.924546i \(0.375556\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −3600.00 −0.121600
\(958\) 0 0
\(959\) 65552.6 2.20730
\(960\) 0 0
\(961\) −6671.00 −0.223927
\(962\) 0 0
\(963\) −10675.0 −0.357214
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −43777.7 −1.45584 −0.727920 0.685662i \(-0.759513\pi\)
−0.727920 + 0.685662i \(0.759513\pi\)
\(968\) 0 0
\(969\) −22080.0 −0.732004
\(970\) 0 0
\(971\) −25714.8 −0.849873 −0.424936 0.905223i \(-0.639704\pi\)
−0.424936 + 0.905223i \(0.639704\pi\)
\(972\) 0 0
\(973\) −43120.0 −1.42072
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −28986.0 −0.949175 −0.474588 0.880208i \(-0.657403\pi\)
−0.474588 + 0.880208i \(0.657403\pi\)
\(978\) 0 0
\(979\) −8675.94 −0.283232
\(980\) 0 0
\(981\) −15078.0 −0.490727
\(982\) 0 0
\(983\) −32123.4 −1.04229 −0.521147 0.853467i \(-0.674496\pi\)
−0.521147 + 0.853467i \(0.674496\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 55722.8 1.79704
\(988\) 0 0
\(989\) −12900.0 −0.414758
\(990\) 0 0
\(991\) −11994.3 −0.384471 −0.192235 0.981349i \(-0.561574\pi\)
−0.192235 + 0.981349i \(0.561574\pi\)
\(992\) 0 0
\(993\) 1400.00 0.0447408
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 406.000 0.0128968 0.00644842 0.999979i \(-0.497947\pi\)
0.00644842 + 0.999979i \(0.497947\pi\)
\(998\) 0 0
\(999\) 32539.3 1.03053
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 \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 800.4.a.o.1.1 2
4.3 odd 2 inner 800.4.a.o.1.2 2
5.2 odd 4 800.4.c.l.449.3 4
5.3 odd 4 800.4.c.l.449.1 4
5.4 even 2 160.4.a.d.1.2 yes 2
8.3 odd 2 1600.4.a.cg.1.1 2
8.5 even 2 1600.4.a.cg.1.2 2
15.14 odd 2 1440.4.a.bb.1.1 2
20.3 even 4 800.4.c.l.449.4 4
20.7 even 4 800.4.c.l.449.2 4
20.19 odd 2 160.4.a.d.1.1 2
40.19 odd 2 320.4.a.q.1.2 2
40.29 even 2 320.4.a.q.1.1 2
60.59 even 2 1440.4.a.bb.1.2 2
80.19 odd 4 1280.4.d.v.641.2 4
80.29 even 4 1280.4.d.v.641.4 4
80.59 odd 4 1280.4.d.v.641.3 4
80.69 even 4 1280.4.d.v.641.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
160.4.a.d.1.1 2 20.19 odd 2
160.4.a.d.1.2 yes 2 5.4 even 2
320.4.a.q.1.1 2 40.29 even 2
320.4.a.q.1.2 2 40.19 odd 2
800.4.a.o.1.1 2 1.1 even 1 trivial
800.4.a.o.1.2 2 4.3 odd 2 inner
800.4.c.l.449.1 4 5.3 odd 4
800.4.c.l.449.2 4 20.7 even 4
800.4.c.l.449.3 4 5.2 odd 4
800.4.c.l.449.4 4 20.3 even 4
1280.4.d.v.641.1 4 80.69 even 4
1280.4.d.v.641.2 4 80.19 odd 4
1280.4.d.v.641.3 4 80.59 odd 4
1280.4.d.v.641.4 4 80.29 even 4
1440.4.a.bb.1.1 2 15.14 odd 2
1440.4.a.bb.1.2 2 60.59 even 2
1600.4.a.cg.1.1 2 8.3 odd 2
1600.4.a.cg.1.2 2 8.5 even 2