Properties

Label 800.4.c.l.449.2
Level $800$
Weight $4$
Character 800.449
Analytic conductor $47.202$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 800.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(47.2015280046\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 160)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.2
Root \(-1.61803i\) of defining polynomial
Character \(\chi\) \(=\) 800.449
Dual form 800.4.c.l.449.4

$q$-expansion

\(f(q)\) \(=\) \(q-4.47214i q^{3} -31.3050i q^{7} +7.00000 q^{9} +O(q^{10})\) \(q-4.47214i q^{3} -31.3050i q^{7} +7.00000 q^{9} +8.94427 q^{11} -62.0000i q^{13} +46.0000i q^{17} +107.331 q^{19} -140.000 q^{21} -192.302i q^{23} -152.053i q^{27} +90.0000 q^{29} +152.053 q^{31} -40.0000i q^{33} +214.000i q^{37} -277.272 q^{39} -10.0000 q^{41} +67.0820i q^{43} +398.020i q^{47} -637.000 q^{49} +205.718 q^{51} -678.000i q^{53} -480.000i q^{57} -411.437 q^{59} +250.000 q^{61} -219.135i q^{63} +49.1935i q^{67} -860.000 q^{69} +366.715 q^{71} +522.000i q^{73} -280.000i q^{77} +876.539 q^{79} -491.000 q^{81} -380.132i q^{83} -402.492i q^{87} -970.000 q^{89} -1940.91 q^{91} -680.000i q^{93} +934.000i q^{97} +62.6099 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 28 q^{9} + O(q^{10}) \) \( 4 q + 28 q^{9} - 560 q^{21} + 360 q^{29} - 40 q^{41} - 2548 q^{49} + 1000 q^{61} - 3440 q^{69} - 1964 q^{81} - 3880 q^{89} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/800\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 4.47214i − 0.860663i −0.902671 0.430331i \(-0.858397\pi\)
0.902671 0.430331i \(-0.141603\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 31.3050i − 1.69031i −0.534522 0.845154i \(-0.679509\pi\)
0.534522 0.845154i \(-0.320491\pi\)
\(8\) 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
\(13\) − 62.0000i − 1.32275i −0.750057 0.661373i \(-0.769974\pi\)
0.750057 0.661373i \(-0.230026\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 46.0000i 0.656273i 0.944630 + 0.328136i \(0.106421\pi\)
−0.944630 + 0.328136i \(0.893579\pi\)
\(18\) 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
\(23\) − 192.302i − 1.74338i −0.490059 0.871689i \(-0.663025\pi\)
0.490059 0.871689i \(-0.336975\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 152.053i − 1.08380i
\(28\) 0 0
\(29\) 90.0000 0.576296 0.288148 0.957586i \(-0.406961\pi\)
0.288148 + 0.957586i \(0.406961\pi\)
\(30\) 0 0
\(31\) 152.053 0.880950 0.440475 0.897765i \(-0.354810\pi\)
0.440475 + 0.897765i \(0.354810\pi\)
\(32\) 0 0
\(33\) − 40.0000i − 0.211003i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 214.000i 0.950848i 0.879757 + 0.475424i \(0.157705\pi\)
−0.879757 + 0.475424i \(0.842295\pi\)
\(38\) 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
\(43\) 67.0820i 0.237905i 0.992900 + 0.118953i \(0.0379536\pi\)
−0.992900 + 0.118953i \(0.962046\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 398.020i 1.23526i 0.786469 + 0.617630i \(0.211907\pi\)
−0.786469 + 0.617630i \(0.788093\pi\)
\(48\) 0 0
\(49\) −637.000 −1.85714
\(50\) 0 0
\(51\) 205.718 0.564830
\(52\) 0 0
\(53\) − 678.000i − 1.75718i −0.477578 0.878589i \(-0.658485\pi\)
0.477578 0.878589i \(-0.341515\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 480.000i − 1.11540i
\(58\) 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
\(63\) − 219.135i − 0.438228i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 49.1935i 0.0897006i 0.998994 + 0.0448503i \(0.0142811\pi\)
−0.998994 + 0.0448503i \(0.985719\pi\)
\(68\) 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
\(73\) 522.000i 0.836924i 0.908234 + 0.418462i \(0.137431\pi\)
−0.908234 + 0.418462i \(0.862569\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 280.000i − 0.414402i
\(78\) 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
\(83\) − 380.132i − 0.502709i −0.967895 0.251355i \(-0.919124\pi\)
0.967895 0.251355i \(-0.0808760\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 402.492i − 0.495997i
\(88\) 0 0
\(89\) −970.000 −1.15528 −0.577639 0.816292i \(-0.696026\pi\)
−0.577639 + 0.816292i \(0.696026\pi\)
\(90\) 0 0
\(91\) −1940.91 −2.23585
\(92\) 0 0
\(93\) − 680.000i − 0.758201i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 934.000i 0.977663i 0.872378 + 0.488832i \(0.162577\pi\)
−0.872378 + 0.488832i \(0.837423\pi\)
\(98\) 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
\(103\) 1829.10i 1.74978i 0.484325 + 0.874888i \(0.339065\pi\)
−0.484325 + 0.874888i \(0.660935\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 1525.00i − 1.37782i −0.724845 0.688912i \(-0.758089\pi\)
0.724845 0.688912i \(-0.241911\pi\)
\(108\) 0 0
\(109\) −2154.00 −1.89281 −0.946403 0.322989i \(-0.895312\pi\)
−0.946403 + 0.322989i \(0.895312\pi\)
\(110\) 0 0
\(111\) 957.037 0.818360
\(112\) 0 0
\(113\) − 2182.00i − 1.81651i −0.418420 0.908254i \(-0.637416\pi\)
0.418420 0.908254i \(-0.362584\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 434.000i − 0.342934i
\(118\) 0 0
\(119\) 1440.03 1.10930
\(120\) 0 0
\(121\) −1251.00 −0.939895
\(122\) 0 0
\(123\) 44.7214i 0.0327837i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 1310.34i 0.915539i 0.889071 + 0.457770i \(0.151352\pi\)
−0.889071 + 0.457770i \(0.848648\pi\)
\(128\) 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
\(133\) − 3360.00i − 2.19059i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2094.00i 1.30586i 0.757419 + 0.652929i \(0.226460\pi\)
−0.757419 + 0.652929i \(0.773540\pi\)
\(138\) 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
\(143\) − 554.545i − 0.324289i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 2848.75i 1.59837i
\(148\) 0 0
\(149\) 334.000 0.183640 0.0918200 0.995776i \(-0.470732\pi\)
0.0918200 + 0.995776i \(0.470732\pi\)
\(150\) 0 0
\(151\) −3139.44 −1.69195 −0.845973 0.533225i \(-0.820980\pi\)
−0.845973 + 0.533225i \(0.820980\pi\)
\(152\) 0 0
\(153\) 322.000i 0.170145i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 834.000i − 0.423952i −0.977275 0.211976i \(-0.932010\pi\)
0.977275 0.211976i \(-0.0679898\pi\)
\(158\) 0 0
\(159\) −3032.11 −1.51234
\(160\) 0 0
\(161\) −6020.00 −2.94685
\(162\) 0 0
\(163\) 3090.25i 1.48495i 0.669874 + 0.742475i \(0.266348\pi\)
−0.669874 + 0.742475i \(0.733652\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 4.47214i 0.00207224i 0.999999 + 0.00103612i \(0.000329807\pi\)
−0.999999 + 0.00103612i \(0.999670\pi\)
\(168\) 0 0
\(169\) −1647.00 −0.749659
\(170\) 0 0
\(171\) 751.319 0.335993
\(172\) 0 0
\(173\) − 1838.00i − 0.807749i −0.914814 0.403874i \(-0.867663\pi\)
0.914814 0.403874i \(-0.132337\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 1840.00i 0.781372i
\(178\) 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
\(183\) − 1118.03i − 0.451625i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 411.437i 0.160894i
\(188\) 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
\(193\) 3378.00i 1.25986i 0.776650 + 0.629932i \(0.216917\pi\)
−0.776650 + 0.629932i \(0.783083\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 66.0000i − 0.0238696i −0.999929 0.0119348i \(-0.996201\pi\)
0.999929 0.0119348i \(-0.00379905\pi\)
\(198\) 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
\(203\) − 2817.45i − 0.974118i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 1346.11i − 0.451987i
\(208\) 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
\(213\) − 1640.00i − 0.527563i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 4760.00i − 1.48908i
\(218\) 0 0
\(219\) 2334.45 0.720310
\(220\) 0 0
\(221\) 2852.00 0.868083
\(222\) 0 0
\(223\) − 2965.03i − 0.890371i −0.895438 0.445186i \(-0.853138\pi\)
0.895438 0.445186i \(-0.146862\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 4369.28i − 1.27753i −0.769402 0.638765i \(-0.779446\pi\)
0.769402 0.638765i \(-0.220554\pi\)
\(228\) 0 0
\(229\) 3250.00 0.937843 0.468921 0.883240i \(-0.344643\pi\)
0.468921 + 0.883240i \(0.344643\pi\)
\(230\) 0 0
\(231\) −1252.20 −0.356661
\(232\) 0 0
\(233\) 3298.00i 0.927293i 0.886020 + 0.463646i \(0.153459\pi\)
−0.886020 + 0.463646i \(0.846541\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 3920.00i − 1.07439i
\(238\) 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
\(243\) − 1909.60i − 0.504119i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 6654.54i − 1.71424i
\(248\) 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
\(253\) − 1720.00i − 0.427413i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 4166.00i 1.01116i 0.862780 + 0.505580i \(0.168721\pi\)
−0.862780 + 0.505580i \(0.831279\pi\)
\(258\) 0 0
\(259\) 6699.26 1.60723
\(260\) 0 0
\(261\) 630.000 0.149410
\(262\) 0 0
\(263\) − 961.509i − 0.225434i −0.993627 0.112717i \(-0.964045\pi\)
0.993627 0.112717i \(-0.0359554\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 4337.97i 0.994305i
\(268\) 0 0
\(269\) 1494.00 0.338627 0.169314 0.985562i \(-0.445845\pi\)
0.169314 + 0.985562i \(0.445845\pi\)
\(270\) 0 0
\(271\) −5017.74 −1.12474 −0.562372 0.826884i \(-0.690111\pi\)
−0.562372 + 0.826884i \(0.690111\pi\)
\(272\) 0 0
\(273\) 8680.00i 1.92431i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 1006.00i 0.218212i 0.994030 + 0.109106i \(0.0347988\pi\)
−0.994030 + 0.109106i \(0.965201\pi\)
\(278\) 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
\(283\) − 3635.85i − 0.763705i −0.924223 0.381853i \(-0.875286\pi\)
0.924223 0.381853i \(-0.124714\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 313.050i 0.0643858i
\(288\) 0 0
\(289\) 2797.00 0.569306
\(290\) 0 0
\(291\) 4176.97 0.841439
\(292\) 0 0
\(293\) − 3622.00i − 0.722183i −0.932530 0.361091i \(-0.882404\pi\)
0.932530 0.361091i \(-0.117596\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 1360.00i − 0.265708i
\(298\) 0 0
\(299\) −11922.7 −2.30605
\(300\) 0 0
\(301\) 2100.00 0.402133
\(302\) 0 0
\(303\) 2692.23i 0.510443i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 2088.49i 0.388261i 0.980976 + 0.194131i \(0.0621886\pi\)
−0.980976 + 0.194131i \(0.937811\pi\)
\(308\) 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
\(313\) 8778.00i 1.58518i 0.609754 + 0.792591i \(0.291268\pi\)
−0.609754 + 0.792591i \(0.708732\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 5046.00i 0.894043i 0.894523 + 0.447021i \(0.147515\pi\)
−0.894523 + 0.447021i \(0.852485\pi\)
\(318\) 0 0
\(319\) 804.984 0.141287
\(320\) 0 0
\(321\) −6820.00 −1.18584
\(322\) 0 0
\(323\) 4937.24i 0.850512i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 9632.98i 1.62907i
\(328\) 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
\(333\) 1498.00i 0.246516i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 2574.00i 0.416067i 0.978122 + 0.208034i \(0.0667064\pi\)
−0.978122 + 0.208034i \(0.933294\pi\)
\(338\) 0 0
\(339\) −9758.20 −1.56340
\(340\) 0 0
\(341\) 1360.00 0.215977
\(342\) 0 0
\(343\) 9203.66i 1.44884i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 2643.03i 0.408892i 0.978878 + 0.204446i \(0.0655392\pi\)
−0.978878 + 0.204446i \(0.934461\pi\)
\(348\) 0 0
\(349\) 10170.0 1.55985 0.779925 0.625873i \(-0.215257\pi\)
0.779925 + 0.625873i \(0.215257\pi\)
\(350\) 0 0
\(351\) −9427.26 −1.43359
\(352\) 0 0
\(353\) − 318.000i − 0.0479474i −0.999713 0.0239737i \(-0.992368\pi\)
0.999713 0.0239737i \(-0.00763180\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 6440.00i − 0.954737i
\(358\) 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
\(363\) 5594.64i 0.808933i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 3072.36i − 0.436991i −0.975838 0.218496i \(-0.929885\pi\)
0.975838 0.218496i \(-0.0701149\pi\)
\(368\) 0 0
\(369\) −70.0000 −0.00987549
\(370\) 0 0
\(371\) −21224.8 −2.97017
\(372\) 0 0
\(373\) − 3278.00i − 0.455036i −0.973774 0.227518i \(-0.926939\pi\)
0.973774 0.227518i \(-0.0730610\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 5580.00i − 0.762293i
\(378\) 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
\(383\) 1149.34i 0.153338i 0.997057 + 0.0766690i \(0.0244285\pi\)
−0.997057 + 0.0766690i \(0.975572\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 469.574i 0.0616791i
\(388\) 0 0
\(389\) −834.000 −0.108703 −0.0543515 0.998522i \(-0.517309\pi\)
−0.0543515 + 0.998522i \(0.517309\pi\)
\(390\) 0 0
\(391\) 8845.88 1.14413
\(392\) 0 0
\(393\) − 920.000i − 0.118086i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 8734.00i 1.10415i 0.833795 + 0.552074i \(0.186163\pi\)
−0.833795 + 0.552074i \(0.813837\pi\)
\(398\) 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
\(403\) − 9427.26i − 1.16527i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1914.07i 0.233113i
\(408\) 0 0
\(409\) 6514.00 0.787522 0.393761 0.919213i \(-0.371174\pi\)
0.393761 + 0.919213i \(0.371174\pi\)
\(410\) 0 0
\(411\) 9364.65 1.12390
\(412\) 0 0
\(413\) 12880.0i 1.53458i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 6160.00i 0.723397i
\(418\) 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
\(423\) 2786.14i 0.320252i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 7826.24i − 0.886975i
\(428\) 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
\(433\) 11482.0i 1.27434i 0.770723 + 0.637171i \(0.219895\pi\)
−0.770723 + 0.637171i \(0.780105\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 20640.0i − 2.25937i
\(438\) 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
\(443\) 746.847i 0.0800988i 0.999198 + 0.0400494i \(0.0127515\pi\)
−0.999198 + 0.0400494i \(0.987248\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 1493.69i − 0.158052i
\(448\) 0 0
\(449\) 1306.00 0.137269 0.0686347 0.997642i \(-0.478136\pi\)
0.0686347 + 0.997642i \(0.478136\pi\)
\(450\) 0 0
\(451\) −89.4427 −0.00933857
\(452\) 0 0
\(453\) 14040.0i 1.45620i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 9526.00i 0.975071i 0.873103 + 0.487536i \(0.162104\pi\)
−0.873103 + 0.487536i \(0.837896\pi\)
\(458\) 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
\(463\) − 17293.7i − 1.73587i −0.496676 0.867936i \(-0.665446\pi\)
0.496676 0.867936i \(-0.334554\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 16980.7i − 1.68260i −0.540570 0.841299i \(-0.681792\pi\)
0.540570 0.841299i \(-0.318208\pi\)
\(468\) 0 0
\(469\) 1540.00 0.151622
\(470\) 0 0
\(471\) −3729.76 −0.364880
\(472\) 0 0
\(473\) 600.000i 0.0583256i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 4746.00i − 0.455565i
\(478\) 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
\(483\) 26922.3i 2.53624i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 1310.34i 0.121924i 0.998140 + 0.0609620i \(0.0194168\pi\)
−0.998140 + 0.0609620i \(0.980583\pi\)
\(488\) 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
\(493\) 4140.00i 0.378207i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 11480.0i − 1.03611i
\(498\) 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
\(503\) − 3072.36i − 0.272345i −0.990685 0.136173i \(-0.956520\pi\)
0.990685 0.136173i \(-0.0434802\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 7365.61i 0.645203i
\(508\) 0 0
\(509\) −18550.0 −1.61535 −0.807676 0.589626i \(-0.799275\pi\)
−0.807676 + 0.589626i \(0.799275\pi\)
\(510\) 0 0
\(511\) 16341.2 1.41466
\(512\) 0 0
\(513\) − 16320.0i − 1.40457i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 3560.00i 0.302841i
\(518\) 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
\(523\) − 17696.2i − 1.47955i −0.672856 0.739773i \(-0.734933\pi\)
0.672856 0.739773i \(-0.265067\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 6994.42i 0.578144i
\(528\) 0 0
\(529\) −24813.0 −2.03937
\(530\) 0 0
\(531\) −2880.06 −0.235374
\(532\) 0 0
\(533\) 620.000i 0.0503850i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 8240.00i − 0.662164i
\(538\) 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
\(543\) − 8327.12i − 0.658105i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 3716.34i 0.290493i 0.989396 + 0.145246i \(0.0463975\pi\)
−0.989396 + 0.145246i \(0.953603\pi\)
\(548\) 0 0
\(549\) 1750.00 0.136044
\(550\) 0 0
\(551\) 9659.81 0.746864
\(552\) 0 0
\(553\) − 27440.0i − 2.11007i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 15094.0i 1.14821i 0.818781 + 0.574105i \(0.194650\pi\)
−0.818781 + 0.574105i \(0.805350\pi\)
\(558\) 0 0
\(559\) 4159.09 0.314688
\(560\) 0 0
\(561\) 1840.00 0.138476
\(562\) 0 0
\(563\) − 5657.25i − 0.423490i −0.977325 0.211745i \(-0.932085\pi\)
0.977325 0.211745i \(-0.0679146\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 15370.7i 1.13847i
\(568\) 0 0
\(569\) 5906.00 0.435136 0.217568 0.976045i \(-0.430188\pi\)
0.217568 + 0.976045i \(0.430188\pi\)
\(570\) 0 0
\(571\) 4892.52 0.358573 0.179287 0.983797i \(-0.442621\pi\)
0.179287 + 0.983797i \(0.442621\pi\)
\(572\) 0 0
\(573\) − 9240.00i − 0.673659i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 13286.0i 0.958585i 0.877655 + 0.479292i \(0.159107\pi\)
−0.877655 + 0.479292i \(0.840893\pi\)
\(578\) 0 0
\(579\) 15106.9 1.08432
\(580\) 0 0
\(581\) −11900.0 −0.849734
\(582\) 0 0
\(583\) − 6064.22i − 0.430796i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 9029.24i 0.634884i 0.948278 + 0.317442i \(0.102824\pi\)
−0.948278 + 0.317442i \(0.897176\pi\)
\(588\) 0 0
\(589\) 16320.0 1.14169
\(590\) 0 0
\(591\) −295.161 −0.0205437
\(592\) 0 0
\(593\) 11442.0i 0.792355i 0.918174 + 0.396178i \(0.129664\pi\)
−0.918174 + 0.396178i \(0.870336\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 5440.00i 0.372939i
\(598\) 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
\(603\) 344.354i 0.0232557i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 11193.8i − 0.748502i −0.927327 0.374251i \(-0.877900\pi\)
0.927327 0.374251i \(-0.122100\pi\)
\(608\) 0 0
\(609\) −12600.0 −0.838387
\(610\) 0 0
\(611\) 24677.2 1.63394
\(612\) 0 0
\(613\) − 5342.00i − 0.351976i −0.984392 0.175988i \(-0.943688\pi\)
0.984392 0.175988i \(-0.0563120\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 19714.0i − 1.28631i −0.765734 0.643157i \(-0.777624\pi\)
0.765734 0.643157i \(-0.222376\pi\)
\(618\) 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
\(623\) 30365.8i 1.95278i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 4293.25i − 0.273454i
\(628\) 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
\(633\) − 23640.0i − 1.48437i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 39494.0i 2.45653i
\(638\) 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
\(643\) − 12240.2i − 0.750712i −0.926881 0.375356i \(-0.877520\pi\)
0.926881 0.375356i \(-0.122480\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 17973.5i − 1.09214i −0.837741 0.546068i \(-0.816124\pi\)
0.837741 0.546068i \(-0.183876\pi\)
\(648\) 0 0
\(649\) −3680.00 −0.222577
\(650\) 0 0
\(651\) −21287.4 −1.28159
\(652\) 0 0
\(653\) − 3478.00i − 0.208430i −0.994555 0.104215i \(-0.966767\pi\)
0.994555 0.104215i \(-0.0332330\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 3654.00i 0.216980i
\(658\) 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
\(663\) − 12754.5i − 0.747127i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 17307.2i − 1.00470i
\(668\) 0 0
\(669\) −13260.0 −0.766310
\(670\) 0 0
\(671\) 2236.07 0.128647
\(672\) 0 0
\(673\) − 14278.0i − 0.817796i −0.912580 0.408898i \(-0.865913\pi\)
0.912580 0.408898i \(-0.134087\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 18386.0i − 1.04377i −0.853016 0.521884i \(-0.825229\pi\)
0.853016 0.521884i \(-0.174771\pi\)
\(678\) 0 0
\(679\) 29238.8 1.65255
\(680\) 0 0
\(681\) −19540.0 −1.09952
\(682\) 0 0
\(683\) − 15317.1i − 0.858113i −0.903278 0.429057i \(-0.858846\pi\)
0.903278 0.429057i \(-0.141154\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 14534.4i − 0.807167i
\(688\) 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
\(693\) − 1960.00i − 0.107438i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 460.000i − 0.0249982i
\(698\) 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
\(703\) 22968.9i 1.23227i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 18845.6i 1.00249i
\(708\) 0 0
\(709\) 20050.0 1.06205 0.531025 0.847356i \(-0.321807\pi\)
0.531025 + 0.847356i \(0.321807\pi\)
\(710\) 0 0
\(711\) 6135.77 0.323642
\(712\) 0 0
\(713\) − 29240.0i − 1.53583i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 2480.00i 0.129173i
\(718\) 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
\(723\) − 23031.5i − 1.18472i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 11336.9i − 0.578351i −0.957276 0.289175i \(-0.906619\pi\)
0.957276 0.289175i \(-0.0933811\pi\)
\(728\) 0 0
\(729\) −21797.0 −1.10740
\(730\) 0 0
\(731\) −3085.77 −0.156131
\(732\) 0 0
\(733\) − 17198.0i − 0.866607i −0.901248 0.433303i \(-0.857348\pi\)
0.901248 0.433303i \(-0.142652\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 440.000i 0.0219913i
\(738\) 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
\(743\) 2419.43i 0.119462i 0.998215 + 0.0597309i \(0.0190243\pi\)
−0.998215 + 0.0597309i \(0.980976\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 2660.92i − 0.130332i
\(748\) 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
\(753\) 6200.00i 0.300054i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 11474.0i − 0.550898i −0.961316 0.275449i \(-0.911174\pi\)
0.961316 0.275449i \(-0.0888265\pi\)
\(758\) 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
\(763\) 67430.9i 3.19942i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 25509.1i 1.20089i
\(768\) 0 0
\(769\) 5310.00 0.249003 0.124502 0.992219i \(-0.460267\pi\)
0.124502 + 0.992219i \(0.460267\pi\)
\(770\) 0 0
\(771\) 18630.9 0.870267
\(772\) 0 0
\(773\) 37938.0i 1.76525i 0.470082 + 0.882623i \(0.344224\pi\)
−0.470082 + 0.882623i \(0.655776\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 29960.0i − 1.38328i
\(778\) 0 0
\(779\) −1073.31 −0.0493651
\(780\) 0 0
\(781\) 3280.00 0.150279
\(782\) 0 0
\(783\) − 13684.7i − 0.624588i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 37633.0i 1.70454i 0.523103 + 0.852270i \(0.324774\pi\)
−0.523103 + 0.852270i \(0.675226\pi\)
\(788\) 0 0
\(789\) −4300.00 −0.194023
\(790\) 0 0
\(791\) −68307.4 −3.07046
\(792\) 0 0
\(793\) − 15500.0i − 0.694100i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 17526.0i 0.778924i 0.921042 + 0.389462i \(0.127339\pi\)
−0.921042 + 0.389462i \(0.872661\pi\)
\(798\) 0 0
\(799\) −18308.9 −0.810667
\(800\) 0 0
\(801\) −6790.00 −0.299517
\(802\) 0 0
\(803\) 4668.91i 0.205183i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 6681.37i − 0.291444i
\(808\) 0 0
\(809\) −8970.00 −0.389825 −0.194912 0.980821i \(-0.562442\pi\)
−0.194912 + 0.980821i \(0.562442\pi\)
\(810\) 0 0
\(811\) 3550.88 0.153746 0.0768731 0.997041i \(-0.475506\pi\)
0.0768731 + 0.997041i \(0.475506\pi\)
\(812\) 0 0
\(813\) 22440.0i 0.968026i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 7200.00i 0.308318i
\(818\) 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
\(823\) 26712.1i 1.13138i 0.824619 + 0.565689i \(0.191390\pi\)
−0.824619 + 0.565689i \(0.808610\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 863.122i − 0.0362923i −0.999835 0.0181461i \(-0.994224\pi\)
0.999835 0.0181461i \(-0.00577641\pi\)
\(828\) 0 0
\(829\) −19066.0 −0.798781 −0.399391 0.916781i \(-0.630778\pi\)
−0.399391 + 0.916781i \(0.630778\pi\)
\(830\) 0 0
\(831\) 4498.97 0.187807
\(832\) 0 0
\(833\) − 29302.0i − 1.21879i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 23120.0i − 0.954772i
\(838\) 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
\(843\) 14355.6i 0.586514i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 39162.5i 1.58871i
\(848\) 0 0
\(849\) −16260.0 −0.657293
\(850\) 0 0
\(851\) 41152.6 1.65769
\(852\) 0 0
\(853\) − 14462.0i − 0.580503i −0.956950 0.290252i \(-0.906261\pi\)
0.956950 0.290252i \(-0.0937390\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 29346.0i − 1.16971i −0.811138 0.584854i \(-0.801152\pi\)
0.811138 0.584854i \(-0.198848\pi\)
\(858\) 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
\(863\) − 24753.3i − 0.976375i −0.872739 0.488187i \(-0.837658\pi\)
0.872739 0.488187i \(-0.162342\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 12508.6i − 0.489981i
\(868\) 0 0
\(869\) 7840.00 0.306046
\(870\) 0 0
\(871\) 3050.00 0.118651
\(872\) 0 0
\(873\) 6538.00i 0.253468i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 32126.0i 1.23696i 0.785799 + 0.618482i \(0.212252\pi\)
−0.785799 + 0.618482i \(0.787748\pi\)
\(878\) 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
\(883\) 6435.40i 0.245265i 0.992452 + 0.122632i \(0.0391336\pi\)
−0.992452 + 0.122632i \(0.960866\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 46827.7i − 1.77263i −0.463084 0.886314i \(-0.653257\pi\)
0.463084 0.886314i \(-0.346743\pi\)
\(888\) 0 0
\(889\) 41020.0 1.54754
\(890\) 0 0
\(891\) −4391.64 −0.165124
\(892\) 0 0
\(893\) 42720.0i 1.60086i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 53320.0i 1.98473i
\(898\) 0 0
\(899\) 13684.7 0.507688
\(900\) 0 0
\(901\) 31188.0 1.15319
\(902\) 0 0
\(903\) − 9391.49i − 0.346101i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 11980.9i 0.438608i 0.975657 + 0.219304i \(0.0703787\pi\)
−0.975657 + 0.219304i \(0.929621\pi\)
\(908\) 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
\(913\) − 3400.00i − 0.123246i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 6440.00i − 0.231917i
\(918\) 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
\(923\) − 22736.3i − 0.810808i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 12803.7i 0.453646i
\(928\) 0 0
\(929\) 21994.0 0.776749 0.388374 0.921502i \(-0.373037\pi\)
0.388374 + 0.921502i \(0.373037\pi\)
\(930\) 0 0
\(931\) −68370.0 −2.40681
\(932\) 0 0
\(933\) − 39800.0i − 1.39656i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 16286.0i 0.567813i 0.958852 + 0.283906i \(0.0916305\pi\)
−0.958852 + 0.283906i \(0.908370\pi\)
\(938\) 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
\(943\) 1923.02i 0.0664073i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 19869.7i 0.681815i 0.940097 + 0.340907i \(0.110734\pi\)
−0.940097 + 0.340907i \(0.889266\pi\)
\(948\) 0 0
\(949\) 32364.0 1.10704
\(950\) 0 0
\(951\) 22566.4 0.769470
\(952\) 0 0
\(953\) − 22422.0i − 0.762140i −0.924546 0.381070i \(-0.875556\pi\)
0.924546 0.381070i \(-0.124444\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 3600.00i − 0.121600i
\(958\) 0 0
\(959\) 65552.6 2.20730
\(960\) 0 0
\(961\) −6671.00 −0.223927
\(962\) 0 0
\(963\) − 10675.0i − 0.357214i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 43777.7i 1.45584i 0.685662 + 0.727920i \(0.259513\pi\)
−0.685662 + 0.727920i \(0.740487\pi\)
\(968\) 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
\(973\) 43120.0i 1.42072i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 28986.0i − 0.949175i −0.880208 0.474588i \(-0.842597\pi\)
0.880208 0.474588i \(-0.157403\pi\)
\(978\) 0 0
\(979\) −8675.94 −0.283232
\(980\) 0 0
\(981\) −15078.0 −0.490727
\(982\) 0 0
\(983\) − 32123.4i − 1.04229i −0.853467 0.521147i \(-0.825504\pi\)
0.853467 0.521147i \(-0.174496\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 55722.8i − 1.79704i
\(988\) 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
\(993\) − 1400.00i − 0.0447408i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 406.000i 0.0128968i 0.999979 + 0.00644842i \(0.00205261\pi\)
−0.999979 + 0.00644842i \(0.997947\pi\)
\(998\) 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.c.l.449.2 4
4.3 odd 2 inner 800.4.c.l.449.3 4
5.2 odd 4 160.4.a.d.1.1 2
5.3 odd 4 800.4.a.o.1.2 2
5.4 even 2 inner 800.4.c.l.449.4 4
15.2 even 4 1440.4.a.bb.1.2 2
20.3 even 4 800.4.a.o.1.1 2
20.7 even 4 160.4.a.d.1.2 yes 2
20.19 odd 2 inner 800.4.c.l.449.1 4
40.3 even 4 1600.4.a.cg.1.2 2
40.13 odd 4 1600.4.a.cg.1.1 2
40.27 even 4 320.4.a.q.1.1 2
40.37 odd 4 320.4.a.q.1.2 2
60.47 odd 4 1440.4.a.bb.1.1 2
80.27 even 4 1280.4.d.v.641.1 4
80.37 odd 4 1280.4.d.v.641.3 4
80.67 even 4 1280.4.d.v.641.4 4
80.77 odd 4 1280.4.d.v.641.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
160.4.a.d.1.1 2 5.2 odd 4
160.4.a.d.1.2 yes 2 20.7 even 4
320.4.a.q.1.1 2 40.27 even 4
320.4.a.q.1.2 2 40.37 odd 4
800.4.a.o.1.1 2 20.3 even 4
800.4.a.o.1.2 2 5.3 odd 4
800.4.c.l.449.1 4 20.19 odd 2 inner
800.4.c.l.449.2 4 1.1 even 1 trivial
800.4.c.l.449.3 4 4.3 odd 2 inner
800.4.c.l.449.4 4 5.4 even 2 inner
1280.4.d.v.641.1 4 80.27 even 4
1280.4.d.v.641.2 4 80.77 odd 4
1280.4.d.v.641.3 4 80.37 odd 4
1280.4.d.v.641.4 4 80.67 even 4
1440.4.a.bb.1.1 2 60.47 odd 4
1440.4.a.bb.1.2 2 15.2 even 4
1600.4.a.cg.1.1 2 40.13 odd 4
1600.4.a.cg.1.2 2 40.3 even 4