Properties

Label 882.4.a.y.1.1
Level $882$
Weight $4$
Character 882.1
Self dual yes
Analytic conductor $52.040$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 882.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(52.0396846251\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 882.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{2} +4.00000 q^{4} -7.07107 q^{5} -8.00000 q^{8} +O(q^{10})\) \(q-2.00000 q^{2} +4.00000 q^{4} -7.07107 q^{5} -8.00000 q^{8} +14.1421 q^{10} +40.0000 q^{11} -63.6396 q^{13} +16.0000 q^{16} +1.41421 q^{17} +11.3137 q^{19} -28.2843 q^{20} -80.0000 q^{22} +68.0000 q^{23} -75.0000 q^{25} +127.279 q^{26} +110.000 q^{29} -118.794 q^{31} -32.0000 q^{32} -2.82843 q^{34} -20.0000 q^{37} -22.6274 q^{38} +56.5685 q^{40} -49.4975 q^{41} -340.000 q^{43} +160.000 q^{44} -136.000 q^{46} +90.5097 q^{47} +150.000 q^{50} -254.558 q^{52} +628.000 q^{53} -282.843 q^{55} -220.000 q^{58} -876.812 q^{59} +917.825 q^{61} +237.588 q^{62} +64.0000 q^{64} +450.000 q^{65} +540.000 q^{67} +5.65685 q^{68} -420.000 q^{71} +289.914 q^{73} +40.0000 q^{74} +45.2548 q^{76} -760.000 q^{79} -113.137 q^{80} +98.9949 q^{82} +944.695 q^{83} -10.0000 q^{85} +680.000 q^{86} -320.000 q^{88} +1152.58 q^{89} +272.000 q^{92} -181.019 q^{94} -80.0000 q^{95} -502.046 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{2} + 8q^{4} - 16q^{8} + O(q^{10}) \) \( 2q - 4q^{2} + 8q^{4} - 16q^{8} + 80q^{11} + 32q^{16} - 160q^{22} + 136q^{23} - 150q^{25} + 220q^{29} - 64q^{32} - 40q^{37} - 680q^{43} + 320q^{44} - 272q^{46} + 300q^{50} + 1256q^{53} - 440q^{58} + 128q^{64} + 900q^{65} + 1080q^{67} - 840q^{71} + 80q^{74} - 1520q^{79} - 20q^{85} + 1360q^{86} - 640q^{88} + 544q^{92} - 160q^{95} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) 0 0
\(4\) 4.00000 0.500000
\(5\) −7.07107 −0.632456 −0.316228 0.948683i \(-0.602416\pi\)
−0.316228 + 0.948683i \(0.602416\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −8.00000 −0.353553
\(9\) 0 0
\(10\) 14.1421 0.447214
\(11\) 40.0000 1.09640 0.548202 0.836346i \(-0.315312\pi\)
0.548202 + 0.836346i \(0.315312\pi\)
\(12\) 0 0
\(13\) −63.6396 −1.35773 −0.678864 0.734264i \(-0.737527\pi\)
−0.678864 + 0.734264i \(0.737527\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 1.41421 0.0201763 0.0100882 0.999949i \(-0.496789\pi\)
0.0100882 + 0.999949i \(0.496789\pi\)
\(18\) 0 0
\(19\) 11.3137 0.136608 0.0683038 0.997665i \(-0.478241\pi\)
0.0683038 + 0.997665i \(0.478241\pi\)
\(20\) −28.2843 −0.316228
\(21\) 0 0
\(22\) −80.0000 −0.775275
\(23\) 68.0000 0.616477 0.308239 0.951309i \(-0.400260\pi\)
0.308239 + 0.951309i \(0.400260\pi\)
\(24\) 0 0
\(25\) −75.0000 −0.600000
\(26\) 127.279 0.960058
\(27\) 0 0
\(28\) 0 0
\(29\) 110.000 0.704362 0.352181 0.935932i \(-0.385440\pi\)
0.352181 + 0.935932i \(0.385440\pi\)
\(30\) 0 0
\(31\) −118.794 −0.688259 −0.344129 0.938922i \(-0.611826\pi\)
−0.344129 + 0.938922i \(0.611826\pi\)
\(32\) −32.0000 −0.176777
\(33\) 0 0
\(34\) −2.82843 −0.0142668
\(35\) 0 0
\(36\) 0 0
\(37\) −20.0000 −0.0888643 −0.0444322 0.999012i \(-0.514148\pi\)
−0.0444322 + 0.999012i \(0.514148\pi\)
\(38\) −22.6274 −0.0965961
\(39\) 0 0
\(40\) 56.5685 0.223607
\(41\) −49.4975 −0.188542 −0.0942708 0.995547i \(-0.530052\pi\)
−0.0942708 + 0.995547i \(0.530052\pi\)
\(42\) 0 0
\(43\) −340.000 −1.20580 −0.602901 0.797816i \(-0.705989\pi\)
−0.602901 + 0.797816i \(0.705989\pi\)
\(44\) 160.000 0.548202
\(45\) 0 0
\(46\) −136.000 −0.435915
\(47\) 90.5097 0.280898 0.140449 0.990088i \(-0.455145\pi\)
0.140449 + 0.990088i \(0.455145\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 150.000 0.424264
\(51\) 0 0
\(52\) −254.558 −0.678864
\(53\) 628.000 1.62759 0.813797 0.581150i \(-0.197397\pi\)
0.813797 + 0.581150i \(0.197397\pi\)
\(54\) 0 0
\(55\) −282.843 −0.693427
\(56\) 0 0
\(57\) 0 0
\(58\) −220.000 −0.498059
\(59\) −876.812 −1.93477 −0.967383 0.253316i \(-0.918479\pi\)
−0.967383 + 0.253316i \(0.918479\pi\)
\(60\) 0 0
\(61\) 917.825 1.92648 0.963241 0.268639i \(-0.0865738\pi\)
0.963241 + 0.268639i \(0.0865738\pi\)
\(62\) 237.588 0.486672
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) 450.000 0.858702
\(66\) 0 0
\(67\) 540.000 0.984649 0.492325 0.870412i \(-0.336147\pi\)
0.492325 + 0.870412i \(0.336147\pi\)
\(68\) 5.65685 0.0100882
\(69\) 0 0
\(70\) 0 0
\(71\) −420.000 −0.702040 −0.351020 0.936368i \(-0.614165\pi\)
−0.351020 + 0.936368i \(0.614165\pi\)
\(72\) 0 0
\(73\) 289.914 0.464820 0.232410 0.972618i \(-0.425339\pi\)
0.232410 + 0.972618i \(0.425339\pi\)
\(74\) 40.0000 0.0628366
\(75\) 0 0
\(76\) 45.2548 0.0683038
\(77\) 0 0
\(78\) 0 0
\(79\) −760.000 −1.08236 −0.541182 0.840906i \(-0.682023\pi\)
−0.541182 + 0.840906i \(0.682023\pi\)
\(80\) −113.137 −0.158114
\(81\) 0 0
\(82\) 98.9949 0.133319
\(83\) 944.695 1.24932 0.624661 0.780896i \(-0.285237\pi\)
0.624661 + 0.780896i \(0.285237\pi\)
\(84\) 0 0
\(85\) −10.0000 −0.0127606
\(86\) 680.000 0.852631
\(87\) 0 0
\(88\) −320.000 −0.387638
\(89\) 1152.58 1.37274 0.686369 0.727254i \(-0.259204\pi\)
0.686369 + 0.727254i \(0.259204\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 272.000 0.308239
\(93\) 0 0
\(94\) −181.019 −0.198625
\(95\) −80.0000 −0.0863982
\(96\) 0 0
\(97\) −502.046 −0.525516 −0.262758 0.964862i \(-0.584632\pi\)
−0.262758 + 0.964862i \(0.584632\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −300.000 −0.300000
\(101\) −1760.70 −1.73461 −0.867306 0.497776i \(-0.834150\pi\)
−0.867306 + 0.497776i \(0.834150\pi\)
\(102\) 0 0
\(103\) 226.274 0.216461 0.108230 0.994126i \(-0.465482\pi\)
0.108230 + 0.994126i \(0.465482\pi\)
\(104\) 509.117 0.480029
\(105\) 0 0
\(106\) −1256.00 −1.15088
\(107\) 2024.00 1.82867 0.914334 0.404961i \(-0.132715\pi\)
0.914334 + 0.404961i \(0.132715\pi\)
\(108\) 0 0
\(109\) −404.000 −0.355011 −0.177505 0.984120i \(-0.556803\pi\)
−0.177505 + 0.984120i \(0.556803\pi\)
\(110\) 565.685 0.490327
\(111\) 0 0
\(112\) 0 0
\(113\) 1008.00 0.839156 0.419578 0.907719i \(-0.362178\pi\)
0.419578 + 0.907719i \(0.362178\pi\)
\(114\) 0 0
\(115\) −480.833 −0.389895
\(116\) 440.000 0.352181
\(117\) 0 0
\(118\) 1753.62 1.36809
\(119\) 0 0
\(120\) 0 0
\(121\) 269.000 0.202104
\(122\) −1835.65 −1.36223
\(123\) 0 0
\(124\) −475.176 −0.344129
\(125\) 1414.21 1.01193
\(126\) 0 0
\(127\) 1000.00 0.698706 0.349353 0.936991i \(-0.386401\pi\)
0.349353 + 0.936991i \(0.386401\pi\)
\(128\) −128.000 −0.0883883
\(129\) 0 0
\(130\) −900.000 −0.607194
\(131\) 84.8528 0.0565926 0.0282963 0.999600i \(-0.490992\pi\)
0.0282963 + 0.999600i \(0.490992\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −1080.00 −0.696252
\(135\) 0 0
\(136\) −11.3137 −0.00713340
\(137\) 2034.00 1.26844 0.634220 0.773152i \(-0.281321\pi\)
0.634220 + 0.773152i \(0.281321\pi\)
\(138\) 0 0
\(139\) −1736.65 −1.05972 −0.529860 0.848085i \(-0.677756\pi\)
−0.529860 + 0.848085i \(0.677756\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 840.000 0.496417
\(143\) −2545.58 −1.48862
\(144\) 0 0
\(145\) −777.817 −0.445477
\(146\) −579.828 −0.328677
\(147\) 0 0
\(148\) −80.0000 −0.0444322
\(149\) 2140.00 1.17661 0.588307 0.808637i \(-0.299794\pi\)
0.588307 + 0.808637i \(0.299794\pi\)
\(150\) 0 0
\(151\) 2120.00 1.14254 0.571269 0.820763i \(-0.306451\pi\)
0.571269 + 0.820763i \(0.306451\pi\)
\(152\) −90.5097 −0.0482980
\(153\) 0 0
\(154\) 0 0
\(155\) 840.000 0.435293
\(156\) 0 0
\(157\) 1746.55 0.887835 0.443918 0.896068i \(-0.353588\pi\)
0.443918 + 0.896068i \(0.353588\pi\)
\(158\) 1520.00 0.765346
\(159\) 0 0
\(160\) 226.274 0.111803
\(161\) 0 0
\(162\) 0 0
\(163\) 3340.00 1.60496 0.802482 0.596677i \(-0.203513\pi\)
0.802482 + 0.596677i \(0.203513\pi\)
\(164\) −197.990 −0.0942708
\(165\) 0 0
\(166\) −1889.39 −0.883404
\(167\) 367.696 0.170378 0.0851890 0.996365i \(-0.472851\pi\)
0.0851890 + 0.996365i \(0.472851\pi\)
\(168\) 0 0
\(169\) 1853.00 0.843423
\(170\) 20.0000 0.00902312
\(171\) 0 0
\(172\) −1360.00 −0.602901
\(173\) 3389.87 1.48975 0.744876 0.667203i \(-0.232509\pi\)
0.744876 + 0.667203i \(0.232509\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 640.000 0.274101
\(177\) 0 0
\(178\) −2305.17 −0.970672
\(179\) 720.000 0.300644 0.150322 0.988637i \(-0.451969\pi\)
0.150322 + 0.988637i \(0.451969\pi\)
\(180\) 0 0
\(181\) −1854.03 −0.761377 −0.380689 0.924703i \(-0.624313\pi\)
−0.380689 + 0.924703i \(0.624313\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −544.000 −0.217958
\(185\) 141.421 0.0562027
\(186\) 0 0
\(187\) 56.5685 0.0221214
\(188\) 362.039 0.140449
\(189\) 0 0
\(190\) 160.000 0.0610927
\(191\) 3980.00 1.50776 0.753881 0.657011i \(-0.228179\pi\)
0.753881 + 0.657011i \(0.228179\pi\)
\(192\) 0 0
\(193\) 3710.00 1.38369 0.691844 0.722047i \(-0.256799\pi\)
0.691844 + 0.722047i \(0.256799\pi\)
\(194\) 1004.09 0.371596
\(195\) 0 0
\(196\) 0 0
\(197\) 956.000 0.345747 0.172874 0.984944i \(-0.444695\pi\)
0.172874 + 0.984944i \(0.444695\pi\)
\(198\) 0 0
\(199\) 4089.91 1.45691 0.728457 0.685092i \(-0.240238\pi\)
0.728457 + 0.685092i \(0.240238\pi\)
\(200\) 600.000 0.212132
\(201\) 0 0
\(202\) 3521.39 1.22656
\(203\) 0 0
\(204\) 0 0
\(205\) 350.000 0.119244
\(206\) −452.548 −0.153061
\(207\) 0 0
\(208\) −1018.23 −0.339432
\(209\) 452.548 0.149777
\(210\) 0 0
\(211\) 2868.00 0.935741 0.467870 0.883797i \(-0.345021\pi\)
0.467870 + 0.883797i \(0.345021\pi\)
\(212\) 2512.00 0.813797
\(213\) 0 0
\(214\) −4048.00 −1.29306
\(215\) 2404.16 0.762617
\(216\) 0 0
\(217\) 0 0
\(218\) 808.000 0.251031
\(219\) 0 0
\(220\) −1131.37 −0.346714
\(221\) −90.0000 −0.0273939
\(222\) 0 0
\(223\) 2630.44 0.789897 0.394949 0.918703i \(-0.370762\pi\)
0.394949 + 0.918703i \(0.370762\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −2016.00 −0.593373
\(227\) −169.706 −0.0496201 −0.0248100 0.999692i \(-0.507898\pi\)
−0.0248100 + 0.999692i \(0.507898\pi\)
\(228\) 0 0
\(229\) −3143.80 −0.907196 −0.453598 0.891206i \(-0.649860\pi\)
−0.453598 + 0.891206i \(0.649860\pi\)
\(230\) 961.665 0.275697
\(231\) 0 0
\(232\) −880.000 −0.249029
\(233\) −4482.00 −1.26020 −0.630098 0.776516i \(-0.716985\pi\)
−0.630098 + 0.776516i \(0.716985\pi\)
\(234\) 0 0
\(235\) −640.000 −0.177655
\(236\) −3507.25 −0.967383
\(237\) 0 0
\(238\) 0 0
\(239\) −1740.00 −0.470926 −0.235463 0.971883i \(-0.575661\pi\)
−0.235463 + 0.971883i \(0.575661\pi\)
\(240\) 0 0
\(241\) 1260.06 0.336796 0.168398 0.985719i \(-0.446141\pi\)
0.168398 + 0.985719i \(0.446141\pi\)
\(242\) −538.000 −0.142909
\(243\) 0 0
\(244\) 3671.30 0.963241
\(245\) 0 0
\(246\) 0 0
\(247\) −720.000 −0.185476
\(248\) 950.352 0.243336
\(249\) 0 0
\(250\) −2828.43 −0.715542
\(251\) −5826.56 −1.46522 −0.732608 0.680651i \(-0.761697\pi\)
−0.732608 + 0.680651i \(0.761697\pi\)
\(252\) 0 0
\(253\) 2720.00 0.675909
\(254\) −2000.00 −0.494060
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) −4688.12 −1.13789 −0.568943 0.822377i \(-0.692648\pi\)
−0.568943 + 0.822377i \(0.692648\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 1800.00 0.429351
\(261\) 0 0
\(262\) −169.706 −0.0400170
\(263\) 2172.00 0.509244 0.254622 0.967041i \(-0.418049\pi\)
0.254622 + 0.967041i \(0.418049\pi\)
\(264\) 0 0
\(265\) −4440.63 −1.02938
\(266\) 0 0
\(267\) 0 0
\(268\) 2160.00 0.492325
\(269\) −2708.22 −0.613840 −0.306920 0.951735i \(-0.599298\pi\)
−0.306920 + 0.951735i \(0.599298\pi\)
\(270\) 0 0
\(271\) 6188.60 1.38720 0.693599 0.720361i \(-0.256024\pi\)
0.693599 + 0.720361i \(0.256024\pi\)
\(272\) 22.6274 0.00504408
\(273\) 0 0
\(274\) −4068.00 −0.896923
\(275\) −3000.00 −0.657843
\(276\) 0 0
\(277\) 6130.00 1.32966 0.664830 0.746994i \(-0.268504\pi\)
0.664830 + 0.746994i \(0.268504\pi\)
\(278\) 3473.31 0.749335
\(279\) 0 0
\(280\) 0 0
\(281\) −1970.00 −0.418222 −0.209111 0.977892i \(-0.567057\pi\)
−0.209111 + 0.977892i \(0.567057\pi\)
\(282\) 0 0
\(283\) −1555.63 −0.326759 −0.163380 0.986563i \(-0.552240\pi\)
−0.163380 + 0.986563i \(0.552240\pi\)
\(284\) −1680.00 −0.351020
\(285\) 0 0
\(286\) 5091.17 1.05261
\(287\) 0 0
\(288\) 0 0
\(289\) −4911.00 −0.999593
\(290\) 1555.63 0.315000
\(291\) 0 0
\(292\) 1159.66 0.232410
\(293\) 7686.25 1.53254 0.766272 0.642516i \(-0.222109\pi\)
0.766272 + 0.642516i \(0.222109\pi\)
\(294\) 0 0
\(295\) 6200.00 1.22365
\(296\) 160.000 0.0314183
\(297\) 0 0
\(298\) −4280.00 −0.831992
\(299\) −4327.49 −0.837008
\(300\) 0 0
\(301\) 0 0
\(302\) −4240.00 −0.807896
\(303\) 0 0
\(304\) 181.019 0.0341519
\(305\) −6490.00 −1.21841
\(306\) 0 0
\(307\) −8598.42 −1.59849 −0.799247 0.601003i \(-0.794768\pi\)
−0.799247 + 0.601003i \(0.794768\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −1680.00 −0.307799
\(311\) 7580.18 1.38210 0.691050 0.722807i \(-0.257149\pi\)
0.691050 + 0.722807i \(0.257149\pi\)
\(312\) 0 0
\(313\) 3075.91 0.555466 0.277733 0.960658i \(-0.410417\pi\)
0.277733 + 0.960658i \(0.410417\pi\)
\(314\) −3493.11 −0.627794
\(315\) 0 0
\(316\) −3040.00 −0.541182
\(317\) −2324.00 −0.411763 −0.205881 0.978577i \(-0.566006\pi\)
−0.205881 + 0.978577i \(0.566006\pi\)
\(318\) 0 0
\(319\) 4400.00 0.772266
\(320\) −452.548 −0.0790569
\(321\) 0 0
\(322\) 0 0
\(323\) 16.0000 0.00275623
\(324\) 0 0
\(325\) 4772.97 0.814636
\(326\) −6680.00 −1.13488
\(327\) 0 0
\(328\) 395.980 0.0666595
\(329\) 0 0
\(330\) 0 0
\(331\) 9508.00 1.57887 0.789436 0.613832i \(-0.210373\pi\)
0.789436 + 0.613832i \(0.210373\pi\)
\(332\) 3778.78 0.624661
\(333\) 0 0
\(334\) −735.391 −0.120475
\(335\) −3818.38 −0.622747
\(336\) 0 0
\(337\) −4720.00 −0.762952 −0.381476 0.924379i \(-0.624584\pi\)
−0.381476 + 0.924379i \(0.624584\pi\)
\(338\) −3706.00 −0.596390
\(339\) 0 0
\(340\) −40.0000 −0.00638031
\(341\) −4751.76 −0.754610
\(342\) 0 0
\(343\) 0 0
\(344\) 2720.00 0.426316
\(345\) 0 0
\(346\) −6779.74 −1.05341
\(347\) −6504.00 −1.00620 −0.503102 0.864227i \(-0.667808\pi\)
−0.503102 + 0.864227i \(0.667808\pi\)
\(348\) 0 0
\(349\) −5256.63 −0.806249 −0.403125 0.915145i \(-0.632076\pi\)
−0.403125 + 0.915145i \(0.632076\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1280.00 −0.193819
\(353\) 12211.7 1.84126 0.920630 0.390435i \(-0.127676\pi\)
0.920630 + 0.390435i \(0.127676\pi\)
\(354\) 0 0
\(355\) 2969.85 0.444009
\(356\) 4610.34 0.686369
\(357\) 0 0
\(358\) −1440.00 −0.212588
\(359\) −7340.00 −1.07908 −0.539541 0.841959i \(-0.681402\pi\)
−0.539541 + 0.841959i \(0.681402\pi\)
\(360\) 0 0
\(361\) −6731.00 −0.981338
\(362\) 3708.07 0.538375
\(363\) 0 0
\(364\) 0 0
\(365\) −2050.00 −0.293978
\(366\) 0 0
\(367\) −7665.04 −1.09022 −0.545111 0.838364i \(-0.683513\pi\)
−0.545111 + 0.838364i \(0.683513\pi\)
\(368\) 1088.00 0.154119
\(369\) 0 0
\(370\) −282.843 −0.0397413
\(371\) 0 0
\(372\) 0 0
\(373\) −2990.00 −0.415057 −0.207529 0.978229i \(-0.566542\pi\)
−0.207529 + 0.978229i \(0.566542\pi\)
\(374\) −113.137 −0.0156422
\(375\) 0 0
\(376\) −724.077 −0.0993123
\(377\) −7000.36 −0.956331
\(378\) 0 0
\(379\) −11900.0 −1.61283 −0.806414 0.591351i \(-0.798595\pi\)
−0.806414 + 0.591351i \(0.798595\pi\)
\(380\) −320.000 −0.0431991
\(381\) 0 0
\(382\) −7960.00 −1.06615
\(383\) −9712.82 −1.29583 −0.647914 0.761714i \(-0.724358\pi\)
−0.647914 + 0.761714i \(0.724358\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −7420.00 −0.978415
\(387\) 0 0
\(388\) −2008.18 −0.262758
\(389\) 2150.00 0.280230 0.140115 0.990135i \(-0.455253\pi\)
0.140115 + 0.990135i \(0.455253\pi\)
\(390\) 0 0
\(391\) 96.1665 0.0124382
\(392\) 0 0
\(393\) 0 0
\(394\) −1912.00 −0.244480
\(395\) 5374.01 0.684546
\(396\) 0 0
\(397\) 3401.18 0.429976 0.214988 0.976617i \(-0.431029\pi\)
0.214988 + 0.976617i \(0.431029\pi\)
\(398\) −8179.81 −1.03019
\(399\) 0 0
\(400\) −1200.00 −0.150000
\(401\) −12090.0 −1.50560 −0.752800 0.658249i \(-0.771297\pi\)
−0.752800 + 0.658249i \(0.771297\pi\)
\(402\) 0 0
\(403\) 7560.00 0.934468
\(404\) −7042.78 −0.867306
\(405\) 0 0
\(406\) 0 0
\(407\) −800.000 −0.0974313
\(408\) 0 0
\(409\) 8192.54 0.990452 0.495226 0.868764i \(-0.335085\pi\)
0.495226 + 0.868764i \(0.335085\pi\)
\(410\) −700.000 −0.0843184
\(411\) 0 0
\(412\) 905.097 0.108230
\(413\) 0 0
\(414\) 0 0
\(415\) −6680.00 −0.790140
\(416\) 2036.47 0.240015
\(417\) 0 0
\(418\) −905.097 −0.105908
\(419\) −1046.52 −0.122019 −0.0610093 0.998137i \(-0.519432\pi\)
−0.0610093 + 0.998137i \(0.519432\pi\)
\(420\) 0 0
\(421\) −3870.00 −0.448010 −0.224005 0.974588i \(-0.571913\pi\)
−0.224005 + 0.974588i \(0.571913\pi\)
\(422\) −5736.00 −0.661669
\(423\) 0 0
\(424\) −5024.00 −0.575441
\(425\) −106.066 −0.0121058
\(426\) 0 0
\(427\) 0 0
\(428\) 8096.00 0.914334
\(429\) 0 0
\(430\) −4808.33 −0.539251
\(431\) −2700.00 −0.301750 −0.150875 0.988553i \(-0.548209\pi\)
−0.150875 + 0.988553i \(0.548209\pi\)
\(432\) 0 0
\(433\) −5876.06 −0.652160 −0.326080 0.945342i \(-0.605728\pi\)
−0.326080 + 0.945342i \(0.605728\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −1616.00 −0.177505
\(437\) 769.332 0.0842154
\(438\) 0 0
\(439\) 6346.99 0.690035 0.345017 0.938596i \(-0.387873\pi\)
0.345017 + 0.938596i \(0.387873\pi\)
\(440\) 2262.74 0.245164
\(441\) 0 0
\(442\) 180.000 0.0193704
\(443\) −6928.00 −0.743023 −0.371512 0.928428i \(-0.621160\pi\)
−0.371512 + 0.928428i \(0.621160\pi\)
\(444\) 0 0
\(445\) −8150.00 −0.868196
\(446\) −5260.87 −0.558542
\(447\) 0 0
\(448\) 0 0
\(449\) −1320.00 −0.138741 −0.0693704 0.997591i \(-0.522099\pi\)
−0.0693704 + 0.997591i \(0.522099\pi\)
\(450\) 0 0
\(451\) −1979.90 −0.206718
\(452\) 4032.00 0.419578
\(453\) 0 0
\(454\) 339.411 0.0350867
\(455\) 0 0
\(456\) 0 0
\(457\) 1290.00 0.132043 0.0660215 0.997818i \(-0.478969\pi\)
0.0660215 + 0.997818i \(0.478969\pi\)
\(458\) 6287.59 0.641485
\(459\) 0 0
\(460\) −1923.33 −0.194947
\(461\) −17642.3 −1.78240 −0.891198 0.453615i \(-0.850134\pi\)
−0.891198 + 0.453615i \(0.850134\pi\)
\(462\) 0 0
\(463\) 5680.00 0.570134 0.285067 0.958508i \(-0.407984\pi\)
0.285067 + 0.958508i \(0.407984\pi\)
\(464\) 1760.00 0.176090
\(465\) 0 0
\(466\) 8964.00 0.891093
\(467\) 7693.32 0.762322 0.381161 0.924509i \(-0.375524\pi\)
0.381161 + 0.924509i \(0.375524\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 1280.00 0.125621
\(471\) 0 0
\(472\) 7014.50 0.684043
\(473\) −13600.0 −1.32205
\(474\) 0 0
\(475\) −848.528 −0.0819645
\(476\) 0 0
\(477\) 0 0
\(478\) 3480.00 0.332995
\(479\) 16518.0 1.57563 0.787816 0.615911i \(-0.211212\pi\)
0.787816 + 0.615911i \(0.211212\pi\)
\(480\) 0 0
\(481\) 1272.79 0.120653
\(482\) −2520.13 −0.238151
\(483\) 0 0
\(484\) 1076.00 0.101052
\(485\) 3550.00 0.332365
\(486\) 0 0
\(487\) 13680.0 1.27290 0.636448 0.771320i \(-0.280403\pi\)
0.636448 + 0.771320i \(0.280403\pi\)
\(488\) −7342.60 −0.681114
\(489\) 0 0
\(490\) 0 0
\(491\) 2280.00 0.209562 0.104781 0.994495i \(-0.466586\pi\)
0.104781 + 0.994495i \(0.466586\pi\)
\(492\) 0 0
\(493\) 155.563 0.0142114
\(494\) 1440.00 0.131151
\(495\) 0 0
\(496\) −1900.70 −0.172065
\(497\) 0 0
\(498\) 0 0
\(499\) 860.000 0.0771521 0.0385760 0.999256i \(-0.487718\pi\)
0.0385760 + 0.999256i \(0.487718\pi\)
\(500\) 5656.85 0.505964
\(501\) 0 0
\(502\) 11653.1 1.03606
\(503\) 5730.39 0.507963 0.253982 0.967209i \(-0.418260\pi\)
0.253982 + 0.967209i \(0.418260\pi\)
\(504\) 0 0
\(505\) 12450.0 1.09706
\(506\) −5440.00 −0.477940
\(507\) 0 0
\(508\) 4000.00 0.349353
\(509\) 4589.12 0.399625 0.199813 0.979834i \(-0.435967\pi\)
0.199813 + 0.979834i \(0.435967\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −512.000 −0.0441942
\(513\) 0 0
\(514\) 9376.24 0.804607
\(515\) −1600.00 −0.136902
\(516\) 0 0
\(517\) 3620.39 0.307978
\(518\) 0 0
\(519\) 0 0
\(520\) −3600.00 −0.303597
\(521\) 14587.6 1.22667 0.613335 0.789823i \(-0.289828\pi\)
0.613335 + 0.789823i \(0.289828\pi\)
\(522\) 0 0
\(523\) −6109.40 −0.510794 −0.255397 0.966836i \(-0.582206\pi\)
−0.255397 + 0.966836i \(0.582206\pi\)
\(524\) 339.411 0.0282963
\(525\) 0 0
\(526\) −4344.00 −0.360090
\(527\) −168.000 −0.0138865
\(528\) 0 0
\(529\) −7543.00 −0.619956
\(530\) 8881.26 0.727882
\(531\) 0 0
\(532\) 0 0
\(533\) 3150.00 0.255988
\(534\) 0 0
\(535\) −14311.8 −1.15655
\(536\) −4320.00 −0.348126
\(537\) 0 0
\(538\) 5416.44 0.434051
\(539\) 0 0
\(540\) 0 0
\(541\) −17210.0 −1.36768 −0.683841 0.729631i \(-0.739692\pi\)
−0.683841 + 0.729631i \(0.739692\pi\)
\(542\) −12377.2 −0.980897
\(543\) 0 0
\(544\) −45.2548 −0.00356670
\(545\) 2856.71 0.224529
\(546\) 0 0
\(547\) 4060.00 0.317355 0.158677 0.987330i \(-0.449277\pi\)
0.158677 + 0.987330i \(0.449277\pi\)
\(548\) 8136.00 0.634220
\(549\) 0 0
\(550\) 6000.00 0.465165
\(551\) 1244.51 0.0962211
\(552\) 0 0
\(553\) 0 0
\(554\) −12260.0 −0.940212
\(555\) 0 0
\(556\) −6946.62 −0.529860
\(557\) 10356.0 0.787788 0.393894 0.919156i \(-0.371128\pi\)
0.393894 + 0.919156i \(0.371128\pi\)
\(558\) 0 0
\(559\) 21637.5 1.63715
\(560\) 0 0
\(561\) 0 0
\(562\) 3940.00 0.295728
\(563\) −23210.1 −1.73746 −0.868728 0.495289i \(-0.835062\pi\)
−0.868728 + 0.495289i \(0.835062\pi\)
\(564\) 0 0
\(565\) −7127.64 −0.530729
\(566\) 3111.27 0.231054
\(567\) 0 0
\(568\) 3360.00 0.248209
\(569\) 5890.00 0.433957 0.216979 0.976176i \(-0.430380\pi\)
0.216979 + 0.976176i \(0.430380\pi\)
\(570\) 0 0
\(571\) −5612.00 −0.411305 −0.205652 0.978625i \(-0.565932\pi\)
−0.205652 + 0.978625i \(0.565932\pi\)
\(572\) −10182.3 −0.744309
\(573\) 0 0
\(574\) 0 0
\(575\) −5100.00 −0.369886
\(576\) 0 0
\(577\) 17797.9 1.28412 0.642058 0.766656i \(-0.278081\pi\)
0.642058 + 0.766656i \(0.278081\pi\)
\(578\) 9822.00 0.706819
\(579\) 0 0
\(580\) −3111.27 −0.222739
\(581\) 0 0
\(582\) 0 0
\(583\) 25120.0 1.78450
\(584\) −2319.31 −0.164339
\(585\) 0 0
\(586\) −15372.5 −1.08367
\(587\) 7942.22 0.558451 0.279225 0.960226i \(-0.409922\pi\)
0.279225 + 0.960226i \(0.409922\pi\)
\(588\) 0 0
\(589\) −1344.00 −0.0940213
\(590\) −12400.0 −0.865254
\(591\) 0 0
\(592\) −320.000 −0.0222161
\(593\) −14078.5 −0.974932 −0.487466 0.873142i \(-0.662079\pi\)
−0.487466 + 0.873142i \(0.662079\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 8560.00 0.588307
\(597\) 0 0
\(598\) 8654.99 0.591854
\(599\) −7300.00 −0.497946 −0.248973 0.968510i \(-0.580093\pi\)
−0.248973 + 0.968510i \(0.580093\pi\)
\(600\) 0 0
\(601\) 8727.11 0.592323 0.296162 0.955138i \(-0.404293\pi\)
0.296162 + 0.955138i \(0.404293\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 8480.00 0.571269
\(605\) −1902.12 −0.127822
\(606\) 0 0
\(607\) −10606.6 −0.709240 −0.354620 0.935011i \(-0.615390\pi\)
−0.354620 + 0.935011i \(0.615390\pi\)
\(608\) −362.039 −0.0241490
\(609\) 0 0
\(610\) 12980.0 0.861549
\(611\) −5760.00 −0.381382
\(612\) 0 0
\(613\) −13980.0 −0.921121 −0.460560 0.887628i \(-0.652351\pi\)
−0.460560 + 0.887628i \(0.652351\pi\)
\(614\) 17196.8 1.13031
\(615\) 0 0
\(616\) 0 0
\(617\) −2654.00 −0.173170 −0.0865851 0.996244i \(-0.527595\pi\)
−0.0865851 + 0.996244i \(0.527595\pi\)
\(618\) 0 0
\(619\) −23883.2 −1.55081 −0.775403 0.631467i \(-0.782453\pi\)
−0.775403 + 0.631467i \(0.782453\pi\)
\(620\) 3360.00 0.217647
\(621\) 0 0
\(622\) −15160.4 −0.977292
\(623\) 0 0
\(624\) 0 0
\(625\) −625.000 −0.0400000
\(626\) −6151.83 −0.392774
\(627\) 0 0
\(628\) 6986.21 0.443918
\(629\) −28.2843 −0.00179295
\(630\) 0 0
\(631\) −6400.00 −0.403772 −0.201886 0.979409i \(-0.564707\pi\)
−0.201886 + 0.979409i \(0.564707\pi\)
\(632\) 6080.00 0.382673
\(633\) 0 0
\(634\) 4648.00 0.291160
\(635\) −7071.07 −0.441900
\(636\) 0 0
\(637\) 0 0
\(638\) −8800.00 −0.546074
\(639\) 0 0
\(640\) 905.097 0.0559017
\(641\) 15350.0 0.945848 0.472924 0.881103i \(-0.343199\pi\)
0.472924 + 0.881103i \(0.343199\pi\)
\(642\) 0 0
\(643\) 17847.4 1.09461 0.547303 0.836934i \(-0.315655\pi\)
0.547303 + 0.836934i \(0.315655\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −32.0000 −0.00194895
\(647\) −14000.7 −0.850734 −0.425367 0.905021i \(-0.639855\pi\)
−0.425367 + 0.905021i \(0.639855\pi\)
\(648\) 0 0
\(649\) −35072.5 −2.12129
\(650\) −9545.94 −0.576035
\(651\) 0 0
\(652\) 13360.0 0.802482
\(653\) 26382.0 1.58102 0.790511 0.612448i \(-0.209815\pi\)
0.790511 + 0.612448i \(0.209815\pi\)
\(654\) 0 0
\(655\) −600.000 −0.0357923
\(656\) −791.960 −0.0471354
\(657\) 0 0
\(658\) 0 0
\(659\) 14400.0 0.851205 0.425603 0.904910i \(-0.360062\pi\)
0.425603 + 0.904910i \(0.360062\pi\)
\(660\) 0 0
\(661\) 28582.7 1.68190 0.840951 0.541112i \(-0.181996\pi\)
0.840951 + 0.541112i \(0.181996\pi\)
\(662\) −19016.0 −1.11643
\(663\) 0 0
\(664\) −7557.56 −0.441702
\(665\) 0 0
\(666\) 0 0
\(667\) 7480.00 0.434223
\(668\) 1470.78 0.0851890
\(669\) 0 0
\(670\) 7636.75 0.440349
\(671\) 36713.0 2.11220
\(672\) 0 0
\(673\) −18120.0 −1.03785 −0.518926 0.854819i \(-0.673668\pi\)
−0.518926 + 0.854819i \(0.673668\pi\)
\(674\) 9440.00 0.539488
\(675\) 0 0
\(676\) 7412.00 0.421711
\(677\) 17797.9 1.01038 0.505191 0.863008i \(-0.331422\pi\)
0.505191 + 0.863008i \(0.331422\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 80.0000 0.00451156
\(681\) 0 0
\(682\) 9503.52 0.533590
\(683\) 1728.00 0.0968083 0.0484042 0.998828i \(-0.484586\pi\)
0.0484042 + 0.998828i \(0.484586\pi\)
\(684\) 0 0
\(685\) −14382.6 −0.802232
\(686\) 0 0
\(687\) 0 0
\(688\) −5440.00 −0.301451
\(689\) −39965.7 −2.20983
\(690\) 0 0
\(691\) 17089.4 0.940825 0.470412 0.882447i \(-0.344105\pi\)
0.470412 + 0.882447i \(0.344105\pi\)
\(692\) 13559.5 0.744876
\(693\) 0 0
\(694\) 13008.0 0.711494
\(695\) 12280.0 0.670226
\(696\) 0 0
\(697\) −70.0000 −0.00380407
\(698\) 10513.3 0.570104
\(699\) 0 0
\(700\) 0 0
\(701\) 13410.0 0.722523 0.361262 0.932465i \(-0.382346\pi\)
0.361262 + 0.932465i \(0.382346\pi\)
\(702\) 0 0
\(703\) −226.274 −0.0121395
\(704\) 2560.00 0.137051
\(705\) 0 0
\(706\) −24423.5 −1.30197
\(707\) 0 0
\(708\) 0 0
\(709\) −140.000 −0.00741581 −0.00370791 0.999993i \(-0.501180\pi\)
−0.00370791 + 0.999993i \(0.501180\pi\)
\(710\) −5939.70 −0.313962
\(711\) 0 0
\(712\) −9220.67 −0.485336
\(713\) −8077.99 −0.424296
\(714\) 0 0
\(715\) 18000.0 0.941485
\(716\) 2880.00 0.150322
\(717\) 0 0
\(718\) 14680.0 0.763026
\(719\) 25936.7 1.34531 0.672653 0.739958i \(-0.265155\pi\)
0.672653 + 0.739958i \(0.265155\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 13462.0 0.693911
\(723\) 0 0
\(724\) −7416.14 −0.380689
\(725\) −8250.00 −0.422617
\(726\) 0 0
\(727\) 9277.24 0.473279 0.236639 0.971598i \(-0.423954\pi\)
0.236639 + 0.971598i \(0.423954\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 4100.00 0.207874
\(731\) −480.833 −0.0243286
\(732\) 0 0
\(733\) −25477.1 −1.28379 −0.641894 0.766793i \(-0.721851\pi\)
−0.641894 + 0.766793i \(0.721851\pi\)
\(734\) 15330.1 0.770904
\(735\) 0 0
\(736\) −2176.00 −0.108979
\(737\) 21600.0 1.07957
\(738\) 0 0
\(739\) −6924.00 −0.344660 −0.172330 0.985039i \(-0.555129\pi\)
−0.172330 + 0.985039i \(0.555129\pi\)
\(740\) 565.685 0.0281014
\(741\) 0 0
\(742\) 0 0
\(743\) −29108.0 −1.43724 −0.718620 0.695403i \(-0.755226\pi\)
−0.718620 + 0.695403i \(0.755226\pi\)
\(744\) 0 0
\(745\) −15132.1 −0.744157
\(746\) 5980.00 0.293490
\(747\) 0 0
\(748\) 226.274 0.0110607
\(749\) 0 0
\(750\) 0 0
\(751\) 31448.0 1.52803 0.764017 0.645196i \(-0.223224\pi\)
0.764017 + 0.645196i \(0.223224\pi\)
\(752\) 1448.15 0.0702244
\(753\) 0 0
\(754\) 14000.7 0.676228
\(755\) −14990.7 −0.722604
\(756\) 0 0
\(757\) −13300.0 −0.638569 −0.319284 0.947659i \(-0.603443\pi\)
−0.319284 + 0.947659i \(0.603443\pi\)
\(758\) 23800.0 1.14044
\(759\) 0 0
\(760\) 640.000 0.0305464
\(761\) 4801.26 0.228706 0.114353 0.993440i \(-0.463520\pi\)
0.114353 + 0.993440i \(0.463520\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 15920.0 0.753881
\(765\) 0 0
\(766\) 19425.6 0.916288
\(767\) 55800.0 2.62689
\(768\) 0 0
\(769\) 16932.4 0.794015 0.397007 0.917815i \(-0.370049\pi\)
0.397007 + 0.917815i \(0.370049\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 14840.0 0.691844
\(773\) −19441.2 −0.904594 −0.452297 0.891867i \(-0.649395\pi\)
−0.452297 + 0.891867i \(0.649395\pi\)
\(774\) 0 0
\(775\) 8909.55 0.412955
\(776\) 4016.37 0.185798
\(777\) 0 0
\(778\) −4300.00 −0.198152
\(779\) −560.000 −0.0257562
\(780\) 0 0
\(781\) −16800.0 −0.769720
\(782\) −192.333 −0.00879516
\(783\) 0 0
\(784\) 0 0
\(785\) −12350.0 −0.561516
\(786\) 0 0
\(787\) −20732.4 −0.939046 −0.469523 0.882920i \(-0.655574\pi\)
−0.469523 + 0.882920i \(0.655574\pi\)
\(788\) 3824.00 0.172874
\(789\) 0 0
\(790\) −10748.0 −0.484047
\(791\) 0 0
\(792\) 0 0
\(793\) −58410.0 −2.61564
\(794\) −6802.37 −0.304039
\(795\) 0 0
\(796\) 16359.6 0.728457
\(797\) 28582.7 1.27033 0.635163 0.772378i \(-0.280933\pi\)
0.635163 + 0.772378i \(0.280933\pi\)
\(798\) 0 0
\(799\) 128.000 0.00566748
\(800\) 2400.00 0.106066
\(801\) 0 0
\(802\) 24180.0 1.06462
\(803\) 11596.6 0.509631
\(804\) 0 0
\(805\) 0 0
\(806\) −15120.0 −0.660768
\(807\) 0 0
\(808\) 14085.6 0.613278
\(809\) 22280.0 0.968261 0.484130 0.874996i \(-0.339136\pi\)
0.484130 + 0.874996i \(0.339136\pi\)
\(810\) 0 0
\(811\) 26100.7 1.13011 0.565056 0.825053i \(-0.308855\pi\)
0.565056 + 0.825053i \(0.308855\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 1600.00 0.0688943
\(815\) −23617.4 −1.01507
\(816\) 0 0
\(817\) −3846.66 −0.164722
\(818\) −16385.1 −0.700356
\(819\) 0 0
\(820\) 1400.00 0.0596221
\(821\) −3540.00 −0.150483 −0.0752417 0.997165i \(-0.523973\pi\)
−0.0752417 + 0.997165i \(0.523973\pi\)
\(822\) 0 0
\(823\) −28400.0 −1.20287 −0.601435 0.798922i \(-0.705404\pi\)
−0.601435 + 0.798922i \(0.705404\pi\)
\(824\) −1810.19 −0.0765304
\(825\) 0 0
\(826\) 0 0
\(827\) 38736.0 1.62876 0.814379 0.580334i \(-0.197078\pi\)
0.814379 + 0.580334i \(0.197078\pi\)
\(828\) 0 0
\(829\) −23409.5 −0.980754 −0.490377 0.871511i \(-0.663141\pi\)
−0.490377 + 0.871511i \(0.663141\pi\)
\(830\) 13360.0 0.558714
\(831\) 0 0
\(832\) −4072.94 −0.169716
\(833\) 0 0
\(834\) 0 0
\(835\) −2600.00 −0.107757
\(836\) 1810.19 0.0748886
\(837\) 0 0
\(838\) 2093.04 0.0862801
\(839\) −29387.4 −1.20925 −0.604627 0.796509i \(-0.706678\pi\)
−0.604627 + 0.796509i \(0.706678\pi\)
\(840\) 0 0
\(841\) −12289.0 −0.503875
\(842\) 7740.00 0.316791
\(843\) 0 0
\(844\) 11472.0 0.467870
\(845\) −13102.7 −0.533427
\(846\) 0 0
\(847\) 0 0
\(848\) 10048.0 0.406898
\(849\) 0 0
\(850\) 212.132 0.00856008
\(851\) −1360.00 −0.0547828
\(852\) 0 0
\(853\) 22252.7 0.893220 0.446610 0.894729i \(-0.352631\pi\)
0.446610 + 0.894729i \(0.352631\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −16192.0 −0.646532
\(857\) −13832.4 −0.551350 −0.275675 0.961251i \(-0.588901\pi\)
−0.275675 + 0.961251i \(0.588901\pi\)
\(858\) 0 0
\(859\) 39665.9 1.57553 0.787766 0.615975i \(-0.211238\pi\)
0.787766 + 0.615975i \(0.211238\pi\)
\(860\) 9616.65 0.381308
\(861\) 0 0
\(862\) 5400.00 0.213370
\(863\) 31988.0 1.26174 0.630871 0.775887i \(-0.282698\pi\)
0.630871 + 0.775887i \(0.282698\pi\)
\(864\) 0 0
\(865\) −23970.0 −0.942202
\(866\) 11752.1 0.461147
\(867\) 0 0
\(868\) 0 0
\(869\) −30400.0 −1.18671
\(870\) 0 0
\(871\) −34365.4 −1.33688
\(872\) 3232.00 0.125515
\(873\) 0 0
\(874\) −1538.66 −0.0595493
\(875\) 0 0
\(876\) 0 0
\(877\) −33460.0 −1.28833 −0.644164 0.764887i \(-0.722795\pi\)
−0.644164 + 0.764887i \(0.722795\pi\)
\(878\) −12694.0 −0.487928
\(879\) 0 0
\(880\) −4525.48 −0.173357
\(881\) 12791.6 0.489170 0.244585 0.969628i \(-0.421348\pi\)
0.244585 + 0.969628i \(0.421348\pi\)
\(882\) 0 0
\(883\) −35260.0 −1.34382 −0.671910 0.740633i \(-0.734526\pi\)
−0.671910 + 0.740633i \(0.734526\pi\)
\(884\) −360.000 −0.0136970
\(885\) 0 0
\(886\) 13856.0 0.525397
\(887\) 15437.6 0.584377 0.292188 0.956361i \(-0.405617\pi\)
0.292188 + 0.956361i \(0.405617\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 16300.0 0.613907
\(891\) 0 0
\(892\) 10521.7 0.394949
\(893\) 1024.00 0.0383727
\(894\) 0 0
\(895\) −5091.17 −0.190144
\(896\) 0 0
\(897\) 0 0
\(898\) 2640.00 0.0981046
\(899\) −13067.3 −0.484783
\(900\) 0 0
\(901\) 888.126 0.0328388
\(902\) 3959.80 0.146172
\(903\) 0 0
\(904\) −8064.00 −0.296687
\(905\) 13110.0 0.481537
\(906\) 0 0
\(907\) 33100.0 1.21176 0.605881 0.795556i \(-0.292821\pi\)
0.605881 + 0.795556i \(0.292821\pi\)
\(908\) −678.823 −0.0248100
\(909\) 0 0
\(910\) 0 0
\(911\) 39620.0 1.44091 0.720455 0.693502i \(-0.243933\pi\)
0.720455 + 0.693502i \(0.243933\pi\)
\(912\) 0 0
\(913\) 37787.8 1.36976
\(914\) −2580.00 −0.0933685
\(915\) 0 0
\(916\) −12575.2 −0.453598
\(917\) 0 0
\(918\) 0 0
\(919\) 20944.0 0.751772 0.375886 0.926666i \(-0.377338\pi\)
0.375886 + 0.926666i \(0.377338\pi\)
\(920\) 3846.66 0.137849
\(921\) 0 0
\(922\) 35284.6 1.26034
\(923\) 26728.6 0.953179
\(924\) 0 0
\(925\) 1500.00 0.0533186
\(926\) −11360.0 −0.403146
\(927\) 0 0
\(928\) −3520.00 −0.124515
\(929\) 46237.7 1.63295 0.816475 0.577381i \(-0.195925\pi\)
0.816475 + 0.577381i \(0.195925\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −17928.0 −0.630098
\(933\) 0 0
\(934\) −15386.6 −0.539043
\(935\) −400.000 −0.0139908
\(936\) 0 0
\(937\) −50522.8 −1.76148 −0.880740 0.473600i \(-0.842954\pi\)
−0.880740 + 0.473600i \(0.842954\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −2560.00 −0.0888277
\(941\) 9878.28 0.342213 0.171107 0.985253i \(-0.445266\pi\)
0.171107 + 0.985253i \(0.445266\pi\)
\(942\) 0 0
\(943\) −3365.83 −0.116232
\(944\) −14029.0 −0.483692
\(945\) 0 0
\(946\) 27200.0 0.934829
\(947\) 30216.0 1.03684 0.518420 0.855126i \(-0.326520\pi\)
0.518420 + 0.855126i \(0.326520\pi\)
\(948\) 0 0
\(949\) −18450.0 −0.631098
\(950\) 1697.06 0.0579577
\(951\) 0 0
\(952\) 0 0
\(953\) 35512.0 1.20708 0.603540 0.797333i \(-0.293757\pi\)
0.603540 + 0.797333i \(0.293757\pi\)
\(954\) 0 0
\(955\) −28142.8 −0.953593
\(956\) −6960.00 −0.235463
\(957\) 0 0
\(958\) −33036.0 −1.11414
\(959\) 0 0
\(960\) 0 0
\(961\) −15679.0 −0.526300
\(962\) −2545.58 −0.0853149
\(963\) 0 0
\(964\) 5040.26 0.168398
\(965\) −26233.7 −0.875121
\(966\) 0 0
\(967\) −720.000 −0.0239438 −0.0119719 0.999928i \(-0.503811\pi\)
−0.0119719 + 0.999928i \(0.503811\pi\)
\(968\) −2152.00 −0.0714544
\(969\) 0 0
\(970\) −7100.00 −0.235018
\(971\) −15386.6 −0.508528 −0.254264 0.967135i \(-0.581833\pi\)
−0.254264 + 0.967135i \(0.581833\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −27360.0 −0.900073
\(975\) 0 0
\(976\) 14685.2 0.481620
\(977\) −41574.0 −1.36138 −0.680691 0.732571i \(-0.738320\pi\)
−0.680691 + 0.732571i \(0.738320\pi\)
\(978\) 0 0
\(979\) 46103.4 1.50508
\(980\) 0 0
\(981\) 0 0
\(982\) −4560.00 −0.148183
\(983\) 44236.6 1.43533 0.717665 0.696389i \(-0.245211\pi\)
0.717665 + 0.696389i \(0.245211\pi\)
\(984\) 0 0
\(985\) −6759.94 −0.218670
\(986\) −311.127 −0.0100490
\(987\) 0 0
\(988\) −2880.00 −0.0927379
\(989\) −23120.0 −0.743350
\(990\) 0 0
\(991\) 12272.0 0.393373 0.196687 0.980466i \(-0.436982\pi\)
0.196687 + 0.980466i \(0.436982\pi\)
\(992\) 3801.41 0.121668
\(993\) 0 0
\(994\) 0 0
\(995\) −28920.0 −0.921433
\(996\) 0 0
\(997\) 57692.8 1.83265 0.916324 0.400437i \(-0.131142\pi\)
0.916324 + 0.400437i \(0.131142\pi\)
\(998\) −1720.00 −0.0545548
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 882.4.a.y.1.1 2
3.2 odd 2 882.4.a.be.1.2 yes 2
7.2 even 3 882.4.g.bg.361.2 4
7.3 odd 6 882.4.g.bg.667.1 4
7.4 even 3 882.4.g.bg.667.2 4
7.5 odd 6 882.4.g.bg.361.1 4
7.6 odd 2 inner 882.4.a.y.1.2 yes 2
21.2 odd 6 882.4.g.bc.361.1 4
21.5 even 6 882.4.g.bc.361.2 4
21.11 odd 6 882.4.g.bc.667.1 4
21.17 even 6 882.4.g.bc.667.2 4
21.20 even 2 882.4.a.be.1.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
882.4.a.y.1.1 2 1.1 even 1 trivial
882.4.a.y.1.2 yes 2 7.6 odd 2 inner
882.4.a.be.1.1 yes 2 21.20 even 2
882.4.a.be.1.2 yes 2 3.2 odd 2
882.4.g.bc.361.1 4 21.2 odd 6
882.4.g.bc.361.2 4 21.5 even 6
882.4.g.bc.667.1 4 21.11 odd 6
882.4.g.bc.667.2 4 21.17 even 6
882.4.g.bg.361.1 4 7.5 odd 6
882.4.g.bg.361.2 4 7.2 even 3
882.4.g.bg.667.1 4 7.3 odd 6
882.4.g.bg.667.2 4 7.4 even 3