Properties

Label 800.4.a.o.1.2
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 1.2
Root \(-0.618034\) 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 \(-0.358397\pi\)
0.430331 + 0.902671i \(0.358397\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −31.3050 −1.69031 −0.845154 0.534522i \(-0.820491\pi\)
−0.845154 + 0.534522i \(0.820491\pi\)
\(8\) 0 0
\(9\) −7.00000 −0.259259
\(10\) 0 0
\(11\) 8.94427 0.245164 0.122582 0.992458i \(-0.460883\pi\)
0.122582 + 0.992458i \(0.460883\pi\)
\(12\) 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 \(-0.724389\pi\)
−0.647986 + 0.761652i \(0.724389\pi\)
\(20\) 0 0
\(21\) −140.000 −1.45479
\(22\) 0 0
\(23\) 192.302 1.74338 0.871689 0.490059i \(-0.163025\pi\)
0.871689 + 0.490059i \(0.163025\pi\)
\(24\) 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 \(-0.354810\pi\)
0.440475 + 0.897765i \(0.354810\pi\)
\(32\) 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 \(-0.537954\pi\)
−0.118953 + 0.992900i \(0.537954\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 398.020 1.23526 0.617630 0.786469i \(-0.288093\pi\)
0.617630 + 0.786469i \(0.288093\pi\)
\(48\) 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 \(-0.350019\pi\)
0.453936 + 0.891034i \(0.350019\pi\)
\(60\) 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 \(-0.485719\pi\)
0.0448503 + 0.998994i \(0.485719\pi\)
\(68\) 0 0
\(69\) 860.000 1.50046
\(70\) 0 0
\(71\) 366.715 0.612973 0.306486 0.951875i \(-0.400847\pi\)
0.306486 + 0.951875i \(0.400847\pi\)
\(72\) 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 \(-0.714561\pi\)
−0.624166 + 0.781291i \(0.714561\pi\)
\(80\) 0 0
\(81\) −491.000 −0.673525
\(82\) 0 0
\(83\) 380.132 0.502709 0.251355 0.967895i \(-0.419124\pi\)
0.251355 + 0.967895i \(0.419124\pi\)
\(84\) 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 \(-0.839065\pi\)
−0.874888 + 0.484325i \(0.839065\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1525.00 −1.37782 −0.688912 0.724845i \(-0.741911\pi\)
−0.688912 + 0.724845i \(0.741911\pi\)
\(108\) 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 \(-0.348648\pi\)
0.457770 + 0.889071i \(0.348648\pi\)
\(128\) 0 0
\(129\) −300.000 −0.204756
\(130\) 0 0
\(131\) 205.718 0.137204 0.0686019 0.997644i \(-0.478146\pi\)
0.0686019 + 0.997644i \(0.478146\pi\)
\(132\) 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 \(-0.361940\pi\)
0.420256 + 0.907406i \(0.361940\pi\)
\(140\) 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 \(-0.820980\pi\)
−0.845973 + 0.533225i \(0.820980\pi\)
\(152\) 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 \(-0.766348\pi\)
−0.742475 + 0.669874i \(0.766348\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 4.47214 0.00207224 0.00103612 0.999999i \(-0.499670\pi\)
0.00103612 + 0.999999i \(0.499670\pi\)
\(168\) 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 \(-0.625689\pi\)
−0.384683 + 0.923049i \(0.625689\pi\)
\(180\) 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 \(-0.372005\pi\)
0.391360 + 0.920237i \(0.372005\pi\)
\(192\) 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 \(-0.430484\pi\)
0.216658 + 0.976248i \(0.430484\pi\)
\(200\) 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 \(-0.168998\pi\)
0.862341 + 0.506329i \(0.168998\pi\)
\(212\) 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 \(-0.353138\pi\)
0.445186 + 0.895438i \(0.353138\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −4369.28 −1.27753 −0.638765 0.769402i \(-0.720554\pi\)
−0.638765 + 0.769402i \(0.720554\pi\)
\(228\) 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 \(-0.476091\pi\)
0.0750429 + 0.997180i \(0.476091\pi\)
\(240\) 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 \(-0.555771\pi\)
−0.174316 + 0.984690i \(0.555771\pi\)
\(252\) 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 \(-0.464045\pi\)
0.112717 + 0.993627i \(0.464045\pi\)
\(264\) 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 \(-0.690111\pi\)
−0.562372 + 0.826884i \(0.690111\pi\)
\(272\) 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 \(-0.375286\pi\)
0.381853 + 0.924223i \(0.375286\pi\)
\(284\) 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 \(-0.437811\pi\)
0.194131 + 0.980976i \(0.437811\pi\)
\(308\) 0 0
\(309\) −8180.00 −1.50597
\(310\) 0 0
\(311\) 8899.55 1.62266 0.811330 0.584589i \(-0.198744\pi\)
0.811330 + 0.584589i \(0.198744\pi\)
\(312\) 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 \(-0.491726\pi\)
0.0259921 + 0.999662i \(0.491726\pi\)
\(332\) 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 \(-0.434461\pi\)
0.204446 + 0.978878i \(0.434461\pi\)
\(348\) 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 \(-0.863867\pi\)
−0.909933 + 0.414755i \(0.863867\pi\)
\(360\) 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 \(-0.570115\pi\)
−0.218496 + 0.975838i \(0.570115\pi\)
\(368\) 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 \(-0.612697\pi\)
−0.346699 + 0.937977i \(0.612697\pi\)
\(380\) 0 0
\(381\) 5860.00 0.787971
\(382\) 0 0
\(383\) −1149.34 −0.153338 −0.0766690 0.997057i \(-0.524428\pi\)
−0.0766690 + 0.997057i \(0.524428\pi\)
\(384\) 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 \(-0.886890\pi\)
−0.937527 + 0.347913i \(0.886890\pi\)
\(420\) 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 \(-0.589801\pi\)
−0.278390 + 0.960468i \(0.589801\pi\)
\(432\) 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 \(-0.566094\pi\)
−0.206150 + 0.978520i \(0.566094\pi\)
\(440\) 0 0
\(441\) −4459.00 −0.481481
\(442\) 0 0
\(443\) −746.847 −0.0800988 −0.0400494 0.999198i \(-0.512752\pi\)
−0.0400494 + 0.999198i \(0.512752\pi\)
\(444\) 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 \(-0.165446\pi\)
0.867936 + 0.496676i \(0.165446\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −16980.7 −1.68260 −0.841299 0.540570i \(-0.818208\pi\)
−0.841299 + 0.540570i \(0.818208\pi\)
\(468\) 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 \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(480\) 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 \(-0.480583\pi\)
0.0609620 + 0.998140i \(0.480583\pi\)
\(488\) 0 0
\(489\) −13820.0 −1.27804
\(490\) 0 0
\(491\) −2960.55 −0.272114 −0.136057 0.990701i \(-0.543443\pi\)
−0.136057 + 0.990701i \(0.543443\pi\)
\(492\) 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 \(-0.166301\pi\)
0.866598 + 0.499006i \(0.166301\pi\)
\(500\) 0 0
\(501\) 20.0000 0.00178350
\(502\) 0 0
\(503\) 3072.36 0.272345 0.136173 0.990685i \(-0.456520\pi\)
0.136173 + 0.990685i \(0.456520\pi\)
\(504\) 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 \(-0.234933\pi\)
0.739773 + 0.672856i \(0.234933\pi\)
\(524\) 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 \(-0.453603\pi\)
0.145246 + 0.989396i \(0.453603\pi\)
\(548\) 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 \(-0.432085\pi\)
0.211745 + 0.977325i \(0.432085\pi\)
\(564\) 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 \(-0.442621\pi\)
0.179287 + 0.983797i \(0.442621\pi\)
\(572\) 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 \(-0.397176\pi\)
0.317442 + 0.948278i \(0.397176\pi\)
\(588\) 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 \(-0.339695\pi\)
0.482593 + 0.875845i \(0.339695\pi\)
\(600\) 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 \(-0.622100\pi\)
−0.374251 + 0.927327i \(0.622100\pi\)
\(608\) 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 \(-0.359411\pi\)
0.427451 + 0.904038i \(0.359411\pi\)
\(620\) 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 \(-0.626426\pi\)
−0.386820 + 0.922155i \(0.626426\pi\)
\(632\) 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 \(-0.377520\pi\)
0.375356 + 0.926881i \(0.377520\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −17973.5 −1.09214 −0.546068 0.837741i \(-0.683876\pi\)
−0.546068 + 0.837741i \(0.683876\pi\)
\(648\) 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 \(-0.398845\pi\)
0.312467 + 0.949929i \(0.398845\pi\)
\(660\) 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 \(-0.358846\pi\)
0.429057 + 0.903278i \(0.358846\pi\)
\(684\) 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 \(-0.584289\pi\)
−0.261717 + 0.965145i \(0.584289\pi\)
\(692\) 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 \(-0.684573\pi\)
−0.547900 + 0.836544i \(0.684573\pi\)
\(720\) 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 \(-0.593381\pi\)
−0.289175 + 0.957276i \(0.593381\pi\)
\(728\) 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 \(-0.536502\pi\)
−0.114423 + 0.993432i \(0.536502\pi\)
\(740\) 0 0
\(741\) −29760.0 −1.47539
\(742\) 0 0
\(743\) −2419.43 −0.119462 −0.0597309 0.998215i \(-0.519024\pi\)
−0.0597309 + 0.998215i \(0.519024\pi\)
\(744\) 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 \(-0.557796\pi\)
−0.180574 + 0.983561i \(0.557796\pi\)
\(752\) 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 \(-0.175226\pi\)
0.852270 + 0.523103i \(0.175226\pi\)
\(788\) 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 \(-0.475506\pi\)
0.0768731 + 0.997041i \(0.475506\pi\)
\(812\) 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 \(-0.691390\pi\)
−0.565689 + 0.824619i \(0.691390\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −863.122 −0.0362923 −0.0181461 0.999835i \(-0.505776\pi\)
−0.0181461 + 0.999835i \(0.505776\pi\)
\(828\) 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 \(-0.0599534\pi\)
0.982315 + 0.187238i \(0.0599534\pi\)
\(840\) 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 \(-0.649634\pi\)
−0.452966 + 0.891528i \(0.649634\pi\)
\(860\) 0 0
\(861\) 1400.00 0.0554145
\(862\) 0 0
\(863\) 24753.3 0.976375 0.488187 0.872739i \(-0.337658\pi\)
0.488187 + 0.872739i \(0.337658\pi\)
\(864\) 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 \(-0.539134\pi\)
−0.122632 + 0.992452i \(0.539134\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −46827.7 −1.77263 −0.886314 0.463084i \(-0.846743\pi\)
−0.886314 + 0.463084i \(0.846743\pi\)
\(888\) 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 \(-0.429621\pi\)
0.219304 + 0.975657i \(0.429621\pi\)
\(908\) 0 0
\(909\) 4214.00 0.153762
\(910\) 0 0
\(911\) 24194.3 0.879903 0.439951 0.898022i \(-0.354996\pi\)
0.439951 + 0.898022i \(0.354996\pi\)
\(912\) 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 \(-0.264902\pi\)
0.673240 + 0.739424i \(0.264902\pi\)
\(920\) 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 \(-0.389266\pi\)
0.340907 + 0.940097i \(0.389266\pi\)
\(948\) 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 \(-0.240487\pi\)
0.727920 + 0.685662i \(0.240487\pi\)
\(968\) 0 0
\(969\) −22080.0 −0.732004
\(970\) 0 0
\(971\) 25714.8 0.849873 0.424936 0.905223i \(-0.360296\pi\)
0.424936 + 0.905223i \(0.360296\pi\)
\(972\) 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 \(-0.325504\pi\)
0.521147 + 0.853467i \(0.325504\pi\)
\(984\) 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 \(-0.438426\pi\)
0.192235 + 0.981349i \(0.438426\pi\)
\(992\) 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.1.2 2
4.3 odd 2 inner 800.4.a.o.1.1 2
5.2 odd 4 800.4.c.l.449.2 4
5.3 odd 4 800.4.c.l.449.4 4
5.4 even 2 160.4.a.d.1.1 2
8.3 odd 2 1600.4.a.cg.1.2 2
8.5 even 2 1600.4.a.cg.1.1 2
15.14 odd 2 1440.4.a.bb.1.2 2
20.3 even 4 800.4.c.l.449.1 4
20.7 even 4 800.4.c.l.449.3 4
20.19 odd 2 160.4.a.d.1.2 yes 2
40.19 odd 2 320.4.a.q.1.1 2
40.29 even 2 320.4.a.q.1.2 2
60.59 even 2 1440.4.a.bb.1.1 2
80.19 odd 4 1280.4.d.v.641.4 4
80.29 even 4 1280.4.d.v.641.2 4
80.59 odd 4 1280.4.d.v.641.1 4
80.69 even 4 1280.4.d.v.641.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
160.4.a.d.1.1 2 5.4 even 2
160.4.a.d.1.2 yes 2 20.19 odd 2
320.4.a.q.1.1 2 40.19 odd 2
320.4.a.q.1.2 2 40.29 even 2
800.4.a.o.1.1 2 4.3 odd 2 inner
800.4.a.o.1.2 2 1.1 even 1 trivial
800.4.c.l.449.1 4 20.3 even 4
800.4.c.l.449.2 4 5.2 odd 4
800.4.c.l.449.3 4 20.7 even 4
800.4.c.l.449.4 4 5.3 odd 4
1280.4.d.v.641.1 4 80.59 odd 4
1280.4.d.v.641.2 4 80.29 even 4
1280.4.d.v.641.3 4 80.69 even 4
1280.4.d.v.641.4 4 80.19 odd 4
1440.4.a.bb.1.1 2 60.59 even 2
1440.4.a.bb.1.2 2 15.14 odd 2
1600.4.a.cg.1.1 2 8.5 even 2
1600.4.a.cg.1.2 2 8.3 odd 2