Properties

Label 800.3.b.i.351.4
Level $800$
Weight $3$
Character 800.351
Analytic conductor $21.798$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 800.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(21.7984211488\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.1827904.1
Defining polynomial: \( x^{6} + 9x^{4} + 14x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: no (minimal twist has level 160)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 351.4
Root \(0.273891i\) of defining polynomial
Character \(\chi\) \(=\) 800.351
Dual form 800.3.b.i.351.3

$q$-expansion

\(f(q)\) \(=\) \(q+0.547781i q^{3} +10.0566i q^{7} +8.69994 q^{9} +O(q^{10})\) \(q+0.547781i q^{3} +10.0566i q^{7} +8.69994 q^{9} -17.2087i q^{11} -4.41325 q^{13} +27.0176 q^{17} +4.82650i q^{19} -5.50881 q^{21} -15.2653i q^{23} +9.69569i q^{27} +2.38225 q^{29} +38.0352i q^{31} +9.42663 q^{33} +16.5691 q^{37} -2.41749i q^{39} -13.3177 q^{41} +59.7918i q^{43} +62.4388i q^{47} -52.1351 q^{49} +14.7997i q^{51} +71.5952 q^{53} -2.64386 q^{57} -68.8265i q^{59} +40.9439 q^{61} +87.4917i q^{63} +51.0080i q^{67} +8.36206 q^{69} -40.4527i q^{71} +35.8441 q^{73} +173.061 q^{77} +126.800i q^{79} +72.9883 q^{81} -75.1490i q^{83} +1.30495i q^{87} +106.523 q^{89} -44.3822i q^{91} -20.8350 q^{93} +85.4351 q^{97} -149.715i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 18 q^{9} + 80 q^{17} + 8 q^{21} - 44 q^{29} + 144 q^{33} + 208 q^{37} - 68 q^{41} + 62 q^{49} + 64 q^{53} + 400 q^{57} - 100 q^{61} + 184 q^{69} + 80 q^{73} + 400 q^{77} + 238 q^{81} - 76 q^{89} + 320 q^{93} + 208 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 0.547781i 0.182594i 0.995824 + 0.0912969i \(0.0291012\pi\)
−0.995824 + 0.0912969i \(0.970899\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 10.0566i 1.43666i 0.695705 + 0.718328i \(0.255092\pi\)
−0.695705 + 0.718328i \(0.744908\pi\)
\(8\) 0 0
\(9\) 8.69994 0.966660
\(10\) 0 0
\(11\) − 17.2087i − 1.56443i −0.623008 0.782216i \(-0.714089\pi\)
0.623008 0.782216i \(-0.285911\pi\)
\(12\) 0 0
\(13\) −4.41325 −0.339481 −0.169740 0.985489i \(-0.554293\pi\)
−0.169740 + 0.985489i \(0.554293\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 27.0176 1.58927 0.794636 0.607086i \(-0.207662\pi\)
0.794636 + 0.607086i \(0.207662\pi\)
\(18\) 0 0
\(19\) 4.82650i 0.254026i 0.991901 + 0.127013i \(0.0405390\pi\)
−0.991901 + 0.127013i \(0.959461\pi\)
\(20\) 0 0
\(21\) −5.50881 −0.262324
\(22\) 0 0
\(23\) − 15.2653i − 0.663710i −0.943330 0.331855i \(-0.892325\pi\)
0.943330 0.331855i \(-0.107675\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 9.69569i 0.359100i
\(28\) 0 0
\(29\) 2.38225 0.0821465 0.0410733 0.999156i \(-0.486922\pi\)
0.0410733 + 0.999156i \(0.486922\pi\)
\(30\) 0 0
\(31\) 38.0352i 1.22694i 0.789717 + 0.613472i \(0.210228\pi\)
−0.789717 + 0.613472i \(0.789772\pi\)
\(32\) 0 0
\(33\) 9.42663 0.285655
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 16.5691 0.447814 0.223907 0.974610i \(-0.428119\pi\)
0.223907 + 0.974610i \(0.428119\pi\)
\(38\) 0 0
\(39\) − 2.41749i − 0.0619870i
\(40\) 0 0
\(41\) −13.3177 −0.324822 −0.162411 0.986723i \(-0.551927\pi\)
−0.162411 + 0.986723i \(0.551927\pi\)
\(42\) 0 0
\(43\) 59.7918i 1.39051i 0.718765 + 0.695253i \(0.244708\pi\)
−0.718765 + 0.695253i \(0.755292\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 62.4388i 1.32849i 0.747517 + 0.664243i \(0.231246\pi\)
−0.747517 + 0.664243i \(0.768754\pi\)
\(48\) 0 0
\(49\) −52.1351 −1.06398
\(50\) 0 0
\(51\) 14.7997i 0.290191i
\(52\) 0 0
\(53\) 71.5952 1.35085 0.675427 0.737427i \(-0.263959\pi\)
0.675427 + 0.737427i \(0.263959\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −2.64386 −0.0463836
\(58\) 0 0
\(59\) − 68.8265i − 1.16655i −0.812274 0.583275i \(-0.801771\pi\)
0.812274 0.583275i \(-0.198229\pi\)
\(60\) 0 0
\(61\) 40.9439 0.671212 0.335606 0.942002i \(-0.391059\pi\)
0.335606 + 0.942002i \(0.391059\pi\)
\(62\) 0 0
\(63\) 87.4917i 1.38876i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 51.0080i 0.761313i 0.924717 + 0.380656i \(0.124302\pi\)
−0.924717 + 0.380656i \(0.875698\pi\)
\(68\) 0 0
\(69\) 8.36206 0.121189
\(70\) 0 0
\(71\) − 40.4527i − 0.569757i −0.958564 0.284878i \(-0.908047\pi\)
0.958564 0.284878i \(-0.0919532\pi\)
\(72\) 0 0
\(73\) 35.8441 0.491015 0.245508 0.969395i \(-0.421045\pi\)
0.245508 + 0.969395i \(0.421045\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 173.061 2.24755
\(78\) 0 0
\(79\) 126.800i 1.60506i 0.596612 + 0.802530i \(0.296513\pi\)
−0.596612 + 0.802530i \(0.703487\pi\)
\(80\) 0 0
\(81\) 72.9883 0.901090
\(82\) 0 0
\(83\) − 75.1490i − 0.905409i −0.891661 0.452705i \(-0.850459\pi\)
0.891661 0.452705i \(-0.149541\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 1.30495i 0.0149994i
\(88\) 0 0
\(89\) 106.523 1.19689 0.598445 0.801164i \(-0.295785\pi\)
0.598445 + 0.801164i \(0.295785\pi\)
\(90\) 0 0
\(91\) − 44.3822i − 0.487717i
\(92\) 0 0
\(93\) −20.8350 −0.224032
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 85.4351 0.880774 0.440387 0.897808i \(-0.354841\pi\)
0.440387 + 0.897808i \(0.354841\pi\)
\(98\) 0 0
\(99\) − 149.715i − 1.51227i
\(100\) 0 0
\(101\) −83.2877 −0.824631 −0.412315 0.911041i \(-0.635280\pi\)
−0.412315 + 0.911041i \(0.635280\pi\)
\(102\) 0 0
\(103\) − 47.2653i − 0.458887i −0.973322 0.229443i \(-0.926309\pi\)
0.973322 0.229443i \(-0.0736906\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 189.628i 1.77223i 0.463467 + 0.886114i \(0.346605\pi\)
−0.463467 + 0.886114i \(0.653395\pi\)
\(108\) 0 0
\(109\) −84.5617 −0.775795 −0.387898 0.921702i \(-0.626799\pi\)
−0.387898 + 0.921702i \(0.626799\pi\)
\(110\) 0 0
\(111\) 9.07626i 0.0817681i
\(112\) 0 0
\(113\) −114.558 −1.01379 −0.506896 0.862007i \(-0.669207\pi\)
−0.506896 + 0.862007i \(0.669207\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −38.3950 −0.328162
\(118\) 0 0
\(119\) 271.705i 2.28324i
\(120\) 0 0
\(121\) −175.141 −1.44745
\(122\) 0 0
\(123\) − 7.29518i − 0.0593104i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 44.4121i 0.349701i 0.984595 + 0.174851i \(0.0559443\pi\)
−0.984595 + 0.174851i \(0.944056\pi\)
\(128\) 0 0
\(129\) −32.7528 −0.253898
\(130\) 0 0
\(131\) − 47.1200i − 0.359695i −0.983695 0.179847i \(-0.942440\pi\)
0.983695 0.179847i \(-0.0575604\pi\)
\(132\) 0 0
\(133\) −48.5381 −0.364948
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 62.8617 0.458845 0.229422 0.973327i \(-0.426316\pi\)
0.229422 + 0.973327i \(0.426316\pi\)
\(138\) 0 0
\(139\) − 128.320i − 0.923167i −0.887097 0.461584i \(-0.847281\pi\)
0.887097 0.461584i \(-0.152719\pi\)
\(140\) 0 0
\(141\) −34.2028 −0.242573
\(142\) 0 0
\(143\) 75.9465i 0.531094i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 28.5586i − 0.194276i
\(148\) 0 0
\(149\) 165.561 1.11115 0.555574 0.831467i \(-0.312499\pi\)
0.555574 + 0.831467i \(0.312499\pi\)
\(150\) 0 0
\(151\) 218.558i 1.44741i 0.690111 + 0.723704i \(0.257562\pi\)
−0.690111 + 0.723704i \(0.742438\pi\)
\(152\) 0 0
\(153\) 235.052 1.53628
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −174.293 −1.11014 −0.555072 0.831802i \(-0.687309\pi\)
−0.555072 + 0.831802i \(0.687309\pi\)
\(158\) 0 0
\(159\) 39.2185i 0.246657i
\(160\) 0 0
\(161\) 153.517 0.953524
\(162\) 0 0
\(163\) 52.6353i 0.322916i 0.986880 + 0.161458i \(0.0516196\pi\)
−0.986880 + 0.161458i \(0.948380\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 76.6812i − 0.459169i −0.973289 0.229584i \(-0.926263\pi\)
0.973289 0.229584i \(-0.0737367\pi\)
\(168\) 0 0
\(169\) −149.523 −0.884753
\(170\) 0 0
\(171\) 41.9902i 0.245557i
\(172\) 0 0
\(173\) −9.72437 −0.0562102 −0.0281051 0.999605i \(-0.508947\pi\)
−0.0281051 + 0.999605i \(0.508947\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 37.7019 0.213005
\(178\) 0 0
\(179\) − 155.502i − 0.868728i −0.900738 0.434364i \(-0.856973\pi\)
0.900738 0.434364i \(-0.143027\pi\)
\(180\) 0 0
\(181\) 250.346 1.38313 0.691563 0.722316i \(-0.256923\pi\)
0.691563 + 0.722316i \(0.256923\pi\)
\(182\) 0 0
\(183\) 22.4283i 0.122559i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 464.939i − 2.48631i
\(188\) 0 0
\(189\) −97.5056 −0.515903
\(190\) 0 0
\(191\) − 111.128i − 0.581825i −0.956750 0.290912i \(-0.906041\pi\)
0.956750 0.290912i \(-0.0939588\pi\)
\(192\) 0 0
\(193\) 10.2701 0.0532130 0.0266065 0.999646i \(-0.491530\pi\)
0.0266065 + 0.999646i \(0.491530\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −163.213 −0.828492 −0.414246 0.910165i \(-0.635955\pi\)
−0.414246 + 0.910165i \(0.635955\pi\)
\(198\) 0 0
\(199\) − 195.028i − 0.980041i −0.871711 0.490021i \(-0.836989\pi\)
0.871711 0.490021i \(-0.163011\pi\)
\(200\) 0 0
\(201\) −27.9412 −0.139011
\(202\) 0 0
\(203\) 23.9573i 0.118016i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 132.807i − 0.641582i
\(208\) 0 0
\(209\) 83.0580 0.397407
\(210\) 0 0
\(211\) − 297.038i − 1.40776i −0.710317 0.703881i \(-0.751449\pi\)
0.710317 0.703881i \(-0.248551\pi\)
\(212\) 0 0
\(213\) 22.1592 0.104034
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −382.505 −1.76270
\(218\) 0 0
\(219\) 19.6347i 0.0896563i
\(220\) 0 0
\(221\) −119.236 −0.539527
\(222\) 0 0
\(223\) − 405.335i − 1.81764i −0.417184 0.908822i \(-0.636983\pi\)
0.417184 0.908822i \(-0.363017\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 36.6013i − 0.161239i −0.996745 0.0806196i \(-0.974310\pi\)
0.996745 0.0806196i \(-0.0256899\pi\)
\(228\) 0 0
\(229\) 91.1467 0.398021 0.199010 0.979997i \(-0.436227\pi\)
0.199010 + 0.979997i \(0.436227\pi\)
\(230\) 0 0
\(231\) 94.7997i 0.410388i
\(232\) 0 0
\(233\) −338.802 −1.45409 −0.727043 0.686592i \(-0.759106\pi\)
−0.727043 + 0.686592i \(0.759106\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −69.4585 −0.293074
\(238\) 0 0
\(239\) 30.7997i 0.128869i 0.997922 + 0.0644346i \(0.0205244\pi\)
−0.997922 + 0.0644346i \(0.979476\pi\)
\(240\) 0 0
\(241\) 240.282 0.997020 0.498510 0.866884i \(-0.333881\pi\)
0.498510 + 0.866884i \(0.333881\pi\)
\(242\) 0 0
\(243\) 127.243i 0.523633i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 21.3005i − 0.0862370i
\(248\) 0 0
\(249\) 41.1652 0.165322
\(250\) 0 0
\(251\) 86.0268i 0.342736i 0.985207 + 0.171368i \(0.0548187\pi\)
−0.985207 + 0.171368i \(0.945181\pi\)
\(252\) 0 0
\(253\) −262.697 −1.03833
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −355.963 −1.38507 −0.692536 0.721383i \(-0.743507\pi\)
−0.692536 + 0.721383i \(0.743507\pi\)
\(258\) 0 0
\(259\) 166.629i 0.643355i
\(260\) 0 0
\(261\) 20.7254 0.0794077
\(262\) 0 0
\(263\) 73.1254i 0.278043i 0.990289 + 0.139022i \(0.0443958\pi\)
−0.990289 + 0.139022i \(0.955604\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 58.3514i 0.218545i
\(268\) 0 0
\(269\) −268.002 −0.996290 −0.498145 0.867094i \(-0.665985\pi\)
−0.498145 + 0.867094i \(0.665985\pi\)
\(270\) 0 0
\(271\) 229.412i 0.846538i 0.906004 + 0.423269i \(0.139117\pi\)
−0.906004 + 0.423269i \(0.860883\pi\)
\(272\) 0 0
\(273\) 24.3118 0.0890541
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 126.327 0.456053 0.228026 0.973655i \(-0.426773\pi\)
0.228026 + 0.973655i \(0.426773\pi\)
\(278\) 0 0
\(279\) 330.904i 1.18604i
\(280\) 0 0
\(281\) −458.746 −1.63255 −0.816275 0.577664i \(-0.803964\pi\)
−0.816275 + 0.577664i \(0.803964\pi\)
\(282\) 0 0
\(283\) − 465.558i − 1.64508i −0.568706 0.822541i \(-0.692556\pi\)
0.568706 0.822541i \(-0.307444\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 133.931i − 0.466657i
\(288\) 0 0
\(289\) 440.952 1.52579
\(290\) 0 0
\(291\) 46.7997i 0.160824i
\(292\) 0 0
\(293\) 165.218 0.563884 0.281942 0.959431i \(-0.409021\pi\)
0.281942 + 0.959431i \(0.409021\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 166.851 0.561787
\(298\) 0 0
\(299\) 67.3697i 0.225317i
\(300\) 0 0
\(301\) −601.302 −1.99768
\(302\) 0 0
\(303\) − 45.6234i − 0.150572i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 235.339i − 0.766577i −0.923629 0.383288i \(-0.874792\pi\)
0.923629 0.383288i \(-0.125208\pi\)
\(308\) 0 0
\(309\) 25.8911 0.0837898
\(310\) 0 0
\(311\) − 210.665i − 0.677381i −0.940898 0.338691i \(-0.890016\pi\)
0.940898 0.338691i \(-0.109984\pi\)
\(312\) 0 0
\(313\) −318.738 −1.01833 −0.509166 0.860668i \(-0.670046\pi\)
−0.509166 + 0.860668i \(0.670046\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 70.3950 0.222066 0.111033 0.993817i \(-0.464584\pi\)
0.111033 + 0.993817i \(0.464584\pi\)
\(318\) 0 0
\(319\) − 40.9955i − 0.128513i
\(320\) 0 0
\(321\) −103.875 −0.323598
\(322\) 0 0
\(323\) 130.401i 0.403717i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 46.3213i − 0.141655i
\(328\) 0 0
\(329\) −627.922 −1.90858
\(330\) 0 0
\(331\) − 280.807i − 0.848359i −0.905578 0.424180i \(-0.860562\pi\)
0.905578 0.424180i \(-0.139438\pi\)
\(332\) 0 0
\(333\) 144.150 0.432884
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −120.969 −0.358959 −0.179480 0.983762i \(-0.557441\pi\)
−0.179480 + 0.983762i \(0.557441\pi\)
\(338\) 0 0
\(339\) − 62.7530i − 0.185112i
\(340\) 0 0
\(341\) 654.539 1.91947
\(342\) 0 0
\(343\) − 31.5280i − 0.0919182i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 214.333i − 0.617675i −0.951115 0.308838i \(-0.900060\pi\)
0.951115 0.308838i \(-0.0999399\pi\)
\(348\) 0 0
\(349\) 418.041 1.19782 0.598912 0.800815i \(-0.295600\pi\)
0.598912 + 0.800815i \(0.295600\pi\)
\(350\) 0 0
\(351\) − 42.7895i − 0.121907i
\(352\) 0 0
\(353\) 364.929 1.03379 0.516897 0.856048i \(-0.327087\pi\)
0.516897 + 0.856048i \(0.327087\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −148.835 −0.416905
\(358\) 0 0
\(359\) − 242.169i − 0.674567i −0.941403 0.337283i \(-0.890492\pi\)
0.941403 0.337283i \(-0.109508\pi\)
\(360\) 0 0
\(361\) 337.705 0.935471
\(362\) 0 0
\(363\) − 95.9389i − 0.264295i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 224.564i − 0.611891i −0.952049 0.305945i \(-0.901028\pi\)
0.952049 0.305945i \(-0.0989725\pi\)
\(368\) 0 0
\(369\) −115.863 −0.313992
\(370\) 0 0
\(371\) 720.004i 1.94071i
\(372\) 0 0
\(373\) −572.174 −1.53398 −0.766989 0.641660i \(-0.778246\pi\)
−0.766989 + 0.641660i \(0.778246\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −10.5135 −0.0278872
\(378\) 0 0
\(379\) − 122.896i − 0.324263i −0.986769 0.162132i \(-0.948163\pi\)
0.986769 0.162132i \(-0.0518369\pi\)
\(380\) 0 0
\(381\) −24.3281 −0.0638533
\(382\) 0 0
\(383\) 289.717i 0.756442i 0.925715 + 0.378221i \(0.123464\pi\)
−0.925715 + 0.378221i \(0.876536\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 520.185i 1.34415i
\(388\) 0 0
\(389\) −111.590 −0.286863 −0.143432 0.989660i \(-0.545814\pi\)
−0.143432 + 0.989660i \(0.545814\pi\)
\(390\) 0 0
\(391\) − 412.433i − 1.05482i
\(392\) 0 0
\(393\) 25.8114 0.0656780
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −705.314 −1.77661 −0.888305 0.459253i \(-0.848117\pi\)
−0.888305 + 0.459253i \(0.848117\pi\)
\(398\) 0 0
\(399\) − 26.5883i − 0.0666373i
\(400\) 0 0
\(401\) 432.052 1.07744 0.538718 0.842486i \(-0.318909\pi\)
0.538718 + 0.842486i \(0.318909\pi\)
\(402\) 0 0
\(403\) − 167.859i − 0.416524i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 285.134i − 0.700575i
\(408\) 0 0
\(409\) 98.8102 0.241590 0.120795 0.992677i \(-0.461456\pi\)
0.120795 + 0.992677i \(0.461456\pi\)
\(410\) 0 0
\(411\) 34.4345i 0.0837822i
\(412\) 0 0
\(413\) 692.160 1.67593
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 70.2914 0.168565
\(418\) 0 0
\(419\) 204.980i 0.489213i 0.969622 + 0.244607i \(0.0786588\pi\)
−0.969622 + 0.244607i \(0.921341\pi\)
\(420\) 0 0
\(421\) 449.956 1.06878 0.534390 0.845238i \(-0.320541\pi\)
0.534390 + 0.845238i \(0.320541\pi\)
\(422\) 0 0
\(423\) 543.214i 1.28419i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 411.756i 0.964301i
\(428\) 0 0
\(429\) −41.6021 −0.0969745
\(430\) 0 0
\(431\) − 314.588i − 0.729903i −0.931026 0.364952i \(-0.881086\pi\)
0.931026 0.364952i \(-0.118914\pi\)
\(432\) 0 0
\(433\) −330.441 −0.763143 −0.381571 0.924339i \(-0.624617\pi\)
−0.381571 + 0.924339i \(0.624617\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 73.6781 0.168600
\(438\) 0 0
\(439\) − 664.815i − 1.51439i −0.653191 0.757193i \(-0.726570\pi\)
0.653191 0.757193i \(-0.273430\pi\)
\(440\) 0 0
\(441\) −453.572 −1.02851
\(442\) 0 0
\(443\) 312.860i 0.706229i 0.935580 + 0.353115i \(0.114877\pi\)
−0.935580 + 0.353115i \(0.885123\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 90.6912i 0.202889i
\(448\) 0 0
\(449\) 246.330 0.548620 0.274310 0.961641i \(-0.411551\pi\)
0.274310 + 0.961641i \(0.411551\pi\)
\(450\) 0 0
\(451\) 229.181i 0.508161i
\(452\) 0 0
\(453\) −119.722 −0.264287
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 707.094 1.54725 0.773626 0.633642i \(-0.218441\pi\)
0.773626 + 0.633642i \(0.218441\pi\)
\(458\) 0 0
\(459\) 261.955i 0.570707i
\(460\) 0 0
\(461\) −611.288 −1.32600 −0.663002 0.748618i \(-0.730718\pi\)
−0.663002 + 0.748618i \(0.730718\pi\)
\(462\) 0 0
\(463\) − 334.063i − 0.721519i −0.932659 0.360760i \(-0.882518\pi\)
0.932659 0.360760i \(-0.117482\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 128.864i − 0.275939i −0.990436 0.137970i \(-0.955942\pi\)
0.990436 0.137970i \(-0.0440577\pi\)
\(468\) 0 0
\(469\) −512.966 −1.09374
\(470\) 0 0
\(471\) − 95.4742i − 0.202705i
\(472\) 0 0
\(473\) 1028.94 2.17535
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 622.874 1.30582
\(478\) 0 0
\(479\) − 510.257i − 1.06525i −0.846350 0.532627i \(-0.821205\pi\)
0.846350 0.532627i \(-0.178795\pi\)
\(480\) 0 0
\(481\) −73.1237 −0.152024
\(482\) 0 0
\(483\) 84.0939i 0.174107i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 616.472i 1.26586i 0.774211 + 0.632928i \(0.218147\pi\)
−0.774211 + 0.632928i \(0.781853\pi\)
\(488\) 0 0
\(489\) −28.8326 −0.0589624
\(490\) 0 0
\(491\) 603.190i 1.22849i 0.789114 + 0.614247i \(0.210540\pi\)
−0.789114 + 0.614247i \(0.789460\pi\)
\(492\) 0 0
\(493\) 64.3627 0.130553
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 406.817 0.818545
\(498\) 0 0
\(499\) 867.976i 1.73943i 0.493553 + 0.869716i \(0.335698\pi\)
−0.493553 + 0.869716i \(0.664302\pi\)
\(500\) 0 0
\(501\) 42.0045 0.0838413
\(502\) 0 0
\(503\) − 57.8408i − 0.114992i −0.998346 0.0574958i \(-0.981688\pi\)
0.998346 0.0574958i \(-0.0183116\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 81.9060i − 0.161550i
\(508\) 0 0
\(509\) 168.498 0.331038 0.165519 0.986207i \(-0.447070\pi\)
0.165519 + 0.986207i \(0.447070\pi\)
\(510\) 0 0
\(511\) 360.470i 0.705420i
\(512\) 0 0
\(513\) −46.7962 −0.0912207
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 1074.49 2.07833
\(518\) 0 0
\(519\) − 5.32682i − 0.0102636i
\(520\) 0 0
\(521\) −87.1060 −0.167190 −0.0835951 0.996500i \(-0.526640\pi\)
−0.0835951 + 0.996500i \(0.526640\pi\)
\(522\) 0 0
\(523\) 243.469i 0.465524i 0.972534 + 0.232762i \(0.0747763\pi\)
−0.972534 + 0.232762i \(0.925224\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1027.62i 1.94995i
\(528\) 0 0
\(529\) 295.969 0.559488
\(530\) 0 0
\(531\) − 598.786i − 1.12766i
\(532\) 0 0
\(533\) 58.7743 0.110271
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 85.1812 0.158624
\(538\) 0 0
\(539\) 897.179i 1.66452i
\(540\) 0 0
\(541\) −812.616 −1.50206 −0.751032 0.660266i \(-0.770443\pi\)
−0.751032 + 0.660266i \(0.770443\pi\)
\(542\) 0 0
\(543\) 137.135i 0.252550i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 104.713i 0.191431i 0.995409 + 0.0957155i \(0.0305139\pi\)
−0.995409 + 0.0957155i \(0.969486\pi\)
\(548\) 0 0
\(549\) 356.210 0.648833
\(550\) 0 0
\(551\) 11.4979i 0.0208674i
\(552\) 0 0
\(553\) −1275.17 −2.30592
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 260.018 0.466818 0.233409 0.972379i \(-0.425012\pi\)
0.233409 + 0.972379i \(0.425012\pi\)
\(558\) 0 0
\(559\) − 263.876i − 0.472050i
\(560\) 0 0
\(561\) 254.685 0.453984
\(562\) 0 0
\(563\) 39.9073i 0.0708833i 0.999372 + 0.0354416i \(0.0112838\pi\)
−0.999372 + 0.0354416i \(0.988716\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 734.014i 1.29456i
\(568\) 0 0
\(569\) 211.588 0.371860 0.185930 0.982563i \(-0.440470\pi\)
0.185930 + 0.982563i \(0.440470\pi\)
\(570\) 0 0
\(571\) − 556.938i − 0.975372i −0.873019 0.487686i \(-0.837841\pi\)
0.873019 0.487686i \(-0.162159\pi\)
\(572\) 0 0
\(573\) 60.8741 0.106237
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −516.233 −0.894684 −0.447342 0.894363i \(-0.647629\pi\)
−0.447342 + 0.894363i \(0.647629\pi\)
\(578\) 0 0
\(579\) 5.62577i 0.00971636i
\(580\) 0 0
\(581\) 755.742 1.30076
\(582\) 0 0
\(583\) − 1232.06i − 2.11332i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 934.677i 1.59229i 0.605103 + 0.796147i \(0.293132\pi\)
−0.605103 + 0.796147i \(0.706868\pi\)
\(588\) 0 0
\(589\) −183.577 −0.311676
\(590\) 0 0
\(591\) − 89.4050i − 0.151277i
\(592\) 0 0
\(593\) 634.665 1.07026 0.535131 0.844769i \(-0.320262\pi\)
0.535131 + 0.844769i \(0.320262\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 106.833 0.178949
\(598\) 0 0
\(599\) − 914.029i − 1.52593i −0.646443 0.762963i \(-0.723744\pi\)
0.646443 0.762963i \(-0.276256\pi\)
\(600\) 0 0
\(601\) −598.569 −0.995955 −0.497977 0.867190i \(-0.665924\pi\)
−0.497977 + 0.867190i \(0.665924\pi\)
\(602\) 0 0
\(603\) 443.766i 0.735930i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 201.830i − 0.332504i −0.986083 0.166252i \(-0.946834\pi\)
0.986083 0.166252i \(-0.0531664\pi\)
\(608\) 0 0
\(609\) −13.1234 −0.0215490
\(610\) 0 0
\(611\) − 275.558i − 0.450995i
\(612\) 0 0
\(613\) 71.9205 0.117325 0.0586627 0.998278i \(-0.481316\pi\)
0.0586627 + 0.998278i \(0.481316\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 204.485 0.331418 0.165709 0.986175i \(-0.447009\pi\)
0.165709 + 0.986175i \(0.447009\pi\)
\(618\) 0 0
\(619\) 287.512i 0.464479i 0.972659 + 0.232239i \(0.0746053\pi\)
−0.972659 + 0.232239i \(0.925395\pi\)
\(620\) 0 0
\(621\) 148.008 0.238338
\(622\) 0 0
\(623\) 1071.26i 1.71952i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 45.4976i 0.0725640i
\(628\) 0 0
\(629\) 447.658 0.711699
\(630\) 0 0
\(631\) 274.203i 0.434554i 0.976110 + 0.217277i \(0.0697175\pi\)
−0.976110 + 0.217277i \(0.930283\pi\)
\(632\) 0 0
\(633\) 162.712 0.257049
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 230.085 0.361201
\(638\) 0 0
\(639\) − 351.936i − 0.550761i
\(640\) 0 0
\(641\) 681.328 1.06291 0.531457 0.847085i \(-0.321645\pi\)
0.531457 + 0.847085i \(0.321645\pi\)
\(642\) 0 0
\(643\) − 527.270i − 0.820016i −0.912082 0.410008i \(-0.865526\pi\)
0.912082 0.410008i \(-0.134474\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 551.627i − 0.852592i −0.904584 0.426296i \(-0.859818\pi\)
0.904584 0.426296i \(-0.140182\pi\)
\(648\) 0 0
\(649\) −1184.42 −1.82499
\(650\) 0 0
\(651\) − 209.529i − 0.321857i
\(652\) 0 0
\(653\) −699.422 −1.07109 −0.535545 0.844507i \(-0.679894\pi\)
−0.535545 + 0.844507i \(0.679894\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 311.842 0.474645
\(658\) 0 0
\(659\) − 38.3257i − 0.0581574i −0.999577 0.0290787i \(-0.990743\pi\)
0.999577 0.0290787i \(-0.00925734\pi\)
\(660\) 0 0
\(661\) 392.364 0.593591 0.296796 0.954941i \(-0.404082\pi\)
0.296796 + 0.954941i \(0.404082\pi\)
\(662\) 0 0
\(663\) − 65.3150i − 0.0985143i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 36.3658i − 0.0545215i
\(668\) 0 0
\(669\) 222.035 0.331890
\(670\) 0 0
\(671\) − 704.594i − 1.05007i
\(672\) 0 0
\(673\) 427.290 0.634904 0.317452 0.948274i \(-0.397173\pi\)
0.317452 + 0.948274i \(0.397173\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −131.234 −0.193847 −0.0969233 0.995292i \(-0.530900\pi\)
−0.0969233 + 0.995292i \(0.530900\pi\)
\(678\) 0 0
\(679\) 859.186i 1.26537i
\(680\) 0 0
\(681\) 20.0495 0.0294413
\(682\) 0 0
\(683\) 896.229i 1.31219i 0.754676 + 0.656097i \(0.227794\pi\)
−0.754676 + 0.656097i \(0.772206\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 49.9285i 0.0726761i
\(688\) 0 0
\(689\) −315.968 −0.458589
\(690\) 0 0
\(691\) 520.465i 0.753206i 0.926375 + 0.376603i \(0.122908\pi\)
−0.926375 + 0.376603i \(0.877092\pi\)
\(692\) 0 0
\(693\) 1505.62 2.17262
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −359.812 −0.516230
\(698\) 0 0
\(699\) − 185.589i − 0.265507i
\(700\) 0 0
\(701\) 80.7527 0.115196 0.0575982 0.998340i \(-0.481656\pi\)
0.0575982 + 0.998340i \(0.481656\pi\)
\(702\) 0 0
\(703\) 79.9709i 0.113757i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 837.591i − 1.18471i
\(708\) 0 0
\(709\) 174.828 0.246584 0.123292 0.992370i \(-0.460655\pi\)
0.123292 + 0.992370i \(0.460655\pi\)
\(710\) 0 0
\(711\) 1103.15i 1.55155i
\(712\) 0 0
\(713\) 580.621 0.814335
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −16.8715 −0.0235307
\(718\) 0 0
\(719\) − 889.905i − 1.23770i −0.785510 0.618849i \(-0.787599\pi\)
0.785510 0.618849i \(-0.212401\pi\)
\(720\) 0 0
\(721\) 475.328 0.659263
\(722\) 0 0
\(723\) 131.622i 0.182050i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 408.818i − 0.562335i −0.959659 0.281168i \(-0.909278\pi\)
0.959659 0.281168i \(-0.0907217\pi\)
\(728\) 0 0
\(729\) 587.194 0.805478
\(730\) 0 0
\(731\) 1615.43i 2.20989i
\(732\) 0 0
\(733\) −388.369 −0.529835 −0.264917 0.964271i \(-0.585345\pi\)
−0.264917 + 0.964271i \(0.585345\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 877.783 1.19102
\(738\) 0 0
\(739\) − 109.561i − 0.148256i −0.997249 0.0741280i \(-0.976383\pi\)
0.997249 0.0741280i \(-0.0236173\pi\)
\(740\) 0 0
\(741\) 11.6680 0.0157463
\(742\) 0 0
\(743\) 732.717i 0.986160i 0.869984 + 0.493080i \(0.164129\pi\)
−0.869984 + 0.493080i \(0.835871\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 653.791i − 0.875222i
\(748\) 0 0
\(749\) −1907.02 −2.54608
\(750\) 0 0
\(751\) − 420.809i − 0.560332i −0.959952 0.280166i \(-0.909610\pi\)
0.959952 0.280166i \(-0.0903895\pi\)
\(752\) 0 0
\(753\) −47.1238 −0.0625814
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −1306.99 −1.72654 −0.863269 0.504744i \(-0.831587\pi\)
−0.863269 + 0.504744i \(0.831587\pi\)
\(758\) 0 0
\(759\) − 143.901i − 0.189592i
\(760\) 0 0
\(761\) −1402.25 −1.84264 −0.921320 0.388804i \(-0.872888\pi\)
−0.921320 + 0.388804i \(0.872888\pi\)
\(762\) 0 0
\(763\) − 850.402i − 1.11455i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 303.748i 0.396021i
\(768\) 0 0
\(769\) 359.952 0.468078 0.234039 0.972227i \(-0.424806\pi\)
0.234039 + 0.972227i \(0.424806\pi\)
\(770\) 0 0
\(771\) − 194.990i − 0.252905i
\(772\) 0 0
\(773\) −437.539 −0.566027 −0.283013 0.959116i \(-0.591334\pi\)
−0.283013 + 0.959116i \(0.591334\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −91.2762 −0.117473
\(778\) 0 0
\(779\) − 64.2778i − 0.0825132i
\(780\) 0 0
\(781\) −696.141 −0.891346
\(782\) 0 0
\(783\) 23.0975i 0.0294988i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 438.946i 0.557746i 0.960328 + 0.278873i \(0.0899608\pi\)
−0.960328 + 0.278873i \(0.910039\pi\)
\(788\) 0 0
\(789\) −40.0567 −0.0507690
\(790\) 0 0
\(791\) − 1152.07i − 1.45647i
\(792\) 0 0
\(793\) −180.696 −0.227864
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −982.414 −1.23264 −0.616320 0.787496i \(-0.711377\pi\)
−0.616320 + 0.787496i \(0.711377\pi\)
\(798\) 0 0
\(799\) 1686.95i 2.11133i
\(800\) 0 0
\(801\) 926.745 1.15699
\(802\) 0 0
\(803\) − 616.832i − 0.768160i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 146.806i − 0.181916i
\(808\) 0 0
\(809\) 449.408 0.555510 0.277755 0.960652i \(-0.410410\pi\)
0.277755 + 0.960652i \(0.410410\pi\)
\(810\) 0 0
\(811\) 610.106i 0.752288i 0.926561 + 0.376144i \(0.122750\pi\)
−0.926561 + 0.376144i \(0.877250\pi\)
\(812\) 0 0
\(813\) −125.667 −0.154572
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −288.585 −0.353225
\(818\) 0 0
\(819\) − 386.123i − 0.471456i
\(820\) 0 0
\(821\) −1099.76 −1.33954 −0.669770 0.742568i \(-0.733607\pi\)
−0.669770 + 0.742568i \(0.733607\pi\)
\(822\) 0 0
\(823\) 236.092i 0.286867i 0.989660 + 0.143434i \(0.0458143\pi\)
−0.989660 + 0.143434i \(0.954186\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1266.81i 1.53182i 0.642950 + 0.765908i \(0.277710\pi\)
−0.642950 + 0.765908i \(0.722290\pi\)
\(828\) 0 0
\(829\) 41.7642 0.0503790 0.0251895 0.999683i \(-0.491981\pi\)
0.0251895 + 0.999683i \(0.491981\pi\)
\(830\) 0 0
\(831\) 69.1993i 0.0832723i
\(832\) 0 0
\(833\) −1408.57 −1.69095
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −368.778 −0.440595
\(838\) 0 0
\(839\) 816.273i 0.972912i 0.873705 + 0.486456i \(0.161711\pi\)
−0.873705 + 0.486456i \(0.838289\pi\)
\(840\) 0 0
\(841\) −835.325 −0.993252
\(842\) 0 0
\(843\) − 251.293i − 0.298093i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 1761.32i − 2.07948i
\(848\) 0 0
\(849\) 255.024 0.300381
\(850\) 0 0
\(851\) − 252.933i − 0.297219i
\(852\) 0 0
\(853\) −1215.58 −1.42507 −0.712533 0.701639i \(-0.752452\pi\)
−0.712533 + 0.701639i \(0.752452\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −997.998 −1.16452 −0.582262 0.813001i \(-0.697832\pi\)
−0.582262 + 0.813001i \(0.697832\pi\)
\(858\) 0 0
\(859\) − 1110.42i − 1.29269i −0.763044 0.646347i \(-0.776296\pi\)
0.763044 0.646347i \(-0.223704\pi\)
\(860\) 0 0
\(861\) 73.3646 0.0852086
\(862\) 0 0
\(863\) − 1372.33i − 1.59019i −0.606486 0.795094i \(-0.707422\pi\)
0.606486 0.795094i \(-0.292578\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 241.545i 0.278599i
\(868\) 0 0
\(869\) 2182.06 2.51101
\(870\) 0 0
\(871\) − 225.111i − 0.258451i
\(872\) 0 0
\(873\) 743.280 0.851409
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 776.337 0.885219 0.442609 0.896714i \(-0.354053\pi\)
0.442609 + 0.896714i \(0.354053\pi\)
\(878\) 0 0
\(879\) 90.5034i 0.102962i
\(880\) 0 0
\(881\) 1047.38 1.18885 0.594427 0.804150i \(-0.297379\pi\)
0.594427 + 0.804150i \(0.297379\pi\)
\(882\) 0 0
\(883\) − 1175.41i − 1.33116i −0.746327 0.665579i \(-0.768185\pi\)
0.746327 0.665579i \(-0.231815\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 109.768i − 0.123752i −0.998084 0.0618762i \(-0.980292\pi\)
0.998084 0.0618762i \(-0.0197084\pi\)
\(888\) 0 0
\(889\) −446.634 −0.502401
\(890\) 0 0
\(891\) − 1256.04i − 1.40969i
\(892\) 0 0
\(893\) −301.361 −0.337470
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −36.9039 −0.0411414
\(898\) 0 0
\(899\) 90.6094i 0.100789i
\(900\) 0 0
\(901\) 1934.33 2.14687
\(902\) 0 0
\(903\) − 329.382i − 0.364764i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 1639.15i − 1.80722i −0.428358 0.903609i \(-0.640908\pi\)
0.428358 0.903609i \(-0.359092\pi\)
\(908\) 0 0
\(909\) −724.598 −0.797137
\(910\) 0 0
\(911\) 6.82023i 0.00748654i 0.999993 + 0.00374327i \(0.00119152\pi\)
−0.999993 + 0.00374327i \(0.998808\pi\)
\(912\) 0 0
\(913\) −1293.22 −1.41645
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 473.867 0.516757
\(918\) 0 0
\(919\) 72.4188i 0.0788017i 0.999223 + 0.0394009i \(0.0125449\pi\)
−0.999223 + 0.0394009i \(0.987455\pi\)
\(920\) 0 0
\(921\) 128.914 0.139972
\(922\) 0 0
\(923\) 178.528i 0.193421i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 411.205i − 0.443587i
\(928\) 0 0
\(929\) 368.537 0.396703 0.198352 0.980131i \(-0.436441\pi\)
0.198352 + 0.980131i \(0.436441\pi\)
\(930\) 0 0
\(931\) − 251.630i − 0.270279i
\(932\) 0 0
\(933\) 115.399 0.123686
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −590.172 −0.629852 −0.314926 0.949116i \(-0.601980\pi\)
−0.314926 + 0.949116i \(0.601980\pi\)
\(938\) 0 0
\(939\) − 174.599i − 0.185941i
\(940\) 0 0
\(941\) −189.017 −0.200869 −0.100434 0.994944i \(-0.532023\pi\)
−0.100434 + 0.994944i \(0.532023\pi\)
\(942\) 0 0
\(943\) 203.299i 0.215588i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 664.319i − 0.701498i −0.936470 0.350749i \(-0.885927\pi\)
0.936470 0.350749i \(-0.114073\pi\)
\(948\) 0 0
\(949\) −158.189 −0.166690
\(950\) 0 0
\(951\) 38.5610i 0.0405479i
\(952\) 0 0
\(953\) 173.943 0.182522 0.0912610 0.995827i \(-0.470910\pi\)
0.0912610 + 0.995827i \(0.470910\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 22.4566 0.0234656
\(958\) 0 0
\(959\) 632.175i 0.659202i
\(960\) 0 0
\(961\) −485.680 −0.505390
\(962\) 0 0
\(963\) 1649.76i 1.71314i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 255.364i − 0.264078i −0.991245 0.132039i \(-0.957848\pi\)
0.991245 0.132039i \(-0.0421524\pi\)
\(968\) 0 0
\(969\) −71.4309 −0.0737161
\(970\) 0 0
\(971\) − 1623.60i − 1.67209i −0.548659 0.836046i \(-0.684861\pi\)
0.548659 0.836046i \(-0.315139\pi\)
\(972\) 0 0
\(973\) 1290.46 1.32627
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 104.946 0.107416 0.0537082 0.998557i \(-0.482896\pi\)
0.0537082 + 0.998557i \(0.482896\pi\)
\(978\) 0 0
\(979\) − 1833.13i − 1.87245i
\(980\) 0 0
\(981\) −735.681 −0.749930
\(982\) 0 0
\(983\) − 889.933i − 0.905323i −0.891682 0.452662i \(-0.850475\pi\)
0.891682 0.452662i \(-0.149525\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 343.964i − 0.348494i
\(988\) 0 0
\(989\) 912.742 0.922894
\(990\) 0 0
\(991\) 1509.57i 1.52328i 0.648001 + 0.761639i \(0.275605\pi\)
−0.648001 + 0.761639i \(0.724395\pi\)
\(992\) 0 0
\(993\) 153.821 0.154905
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −1599.93 −1.60474 −0.802371 0.596825i \(-0.796429\pi\)
−0.802371 + 0.596825i \(0.796429\pi\)
\(998\) 0 0
\(999\) 160.649i 0.160810i
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 800.3.b.i.351.4 6
4.3 odd 2 inner 800.3.b.i.351.3 6
5.2 odd 4 160.3.h.a.159.4 yes 6
5.3 odd 4 160.3.h.b.159.3 yes 6
5.4 even 2 800.3.b.h.351.3 6
8.3 odd 2 1600.3.b.w.1151.4 6
8.5 even 2 1600.3.b.w.1151.3 6
15.2 even 4 1440.3.j.a.1279.5 6
15.8 even 4 1440.3.j.b.1279.6 6
20.3 even 4 160.3.h.a.159.3 6
20.7 even 4 160.3.h.b.159.4 yes 6
20.19 odd 2 800.3.b.h.351.4 6
40.3 even 4 320.3.h.g.319.4 6
40.13 odd 4 320.3.h.f.319.4 6
40.19 odd 2 1600.3.b.v.1151.3 6
40.27 even 4 320.3.h.f.319.3 6
40.29 even 2 1600.3.b.v.1151.4 6
40.37 odd 4 320.3.h.g.319.3 6
60.23 odd 4 1440.3.j.a.1279.6 6
60.47 odd 4 1440.3.j.b.1279.5 6
80.3 even 4 1280.3.e.i.639.4 6
80.13 odd 4 1280.3.e.h.639.3 6
80.27 even 4 1280.3.e.h.639.4 6
80.37 odd 4 1280.3.e.i.639.3 6
80.43 even 4 1280.3.e.g.639.3 6
80.53 odd 4 1280.3.e.f.639.4 6
80.67 even 4 1280.3.e.f.639.3 6
80.77 odd 4 1280.3.e.g.639.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
160.3.h.a.159.3 6 20.3 even 4
160.3.h.a.159.4 yes 6 5.2 odd 4
160.3.h.b.159.3 yes 6 5.3 odd 4
160.3.h.b.159.4 yes 6 20.7 even 4
320.3.h.f.319.3 6 40.27 even 4
320.3.h.f.319.4 6 40.13 odd 4
320.3.h.g.319.3 6 40.37 odd 4
320.3.h.g.319.4 6 40.3 even 4
800.3.b.h.351.3 6 5.4 even 2
800.3.b.h.351.4 6 20.19 odd 2
800.3.b.i.351.3 6 4.3 odd 2 inner
800.3.b.i.351.4 6 1.1 even 1 trivial
1280.3.e.f.639.3 6 80.67 even 4
1280.3.e.f.639.4 6 80.53 odd 4
1280.3.e.g.639.3 6 80.43 even 4
1280.3.e.g.639.4 6 80.77 odd 4
1280.3.e.h.639.3 6 80.13 odd 4
1280.3.e.h.639.4 6 80.27 even 4
1280.3.e.i.639.3 6 80.37 odd 4
1280.3.e.i.639.4 6 80.3 even 4
1440.3.j.a.1279.5 6 15.2 even 4
1440.3.j.a.1279.6 6 60.23 odd 4
1440.3.j.b.1279.5 6 60.47 odd 4
1440.3.j.b.1279.6 6 15.8 even 4
1600.3.b.v.1151.3 6 40.19 odd 2
1600.3.b.v.1151.4 6 40.29 even 2
1600.3.b.w.1151.3 6 8.5 even 2
1600.3.b.w.1151.4 6 8.3 odd 2