Properties

Label 882.4.a.bg.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: no (minimal twist has level 98)
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

\(f(q)\) \(=\) \(q+2.00000 q^{2} +4.00000 q^{4} -19.7990 q^{5} +8.00000 q^{8} +O(q^{10})\) \(q+2.00000 q^{2} +4.00000 q^{4} -19.7990 q^{5} +8.00000 q^{8} -39.5980 q^{10} +14.0000 q^{11} -50.9117 q^{13} +16.0000 q^{16} +1.41421 q^{17} +1.41421 q^{19} -79.1960 q^{20} +28.0000 q^{22} -140.000 q^{23} +267.000 q^{25} -101.823 q^{26} +286.000 q^{29} +93.3381 q^{31} +32.0000 q^{32} +2.82843 q^{34} -38.0000 q^{37} +2.82843 q^{38} -158.392 q^{40} -125.865 q^{41} -34.0000 q^{43} +56.0000 q^{44} -280.000 q^{46} +523.259 q^{47} +534.000 q^{50} -203.647 q^{52} +74.0000 q^{53} -277.186 q^{55} +572.000 q^{58} +434.164 q^{59} -14.1421 q^{61} +186.676 q^{62} +64.0000 q^{64} +1008.00 q^{65} +684.000 q^{67} +5.65685 q^{68} -588.000 q^{71} +270.115 q^{73} -76.0000 q^{74} +5.65685 q^{76} +1220.00 q^{79} -316.784 q^{80} -251.730 q^{82} +422.850 q^{83} -28.0000 q^{85} -68.0000 q^{86} +112.000 q^{88} +618.011 q^{89} -560.000 q^{92} +1046.52 q^{94} -28.0000 q^{95} -1483.51 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{2} + 8q^{4} + 16q^{8} + O(q^{10}) \) \( 2q + 4q^{2} + 8q^{4} + 16q^{8} + 28q^{11} + 32q^{16} + 56q^{22} - 280q^{23} + 534q^{25} + 572q^{29} + 64q^{32} - 76q^{37} - 68q^{43} + 112q^{44} - 560q^{46} + 1068q^{50} + 148q^{53} + 1144q^{58} + 128q^{64} + 2016q^{65} + 1368q^{67} - 1176q^{71} - 152q^{74} + 2440q^{79} - 56q^{85} - 136q^{86} + 224q^{88} - 1120q^{92} - 56q^{95} + O(q^{100}) \)

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
\(5\) −19.7990 −1.77088 −0.885438 0.464758i \(-0.846141\pi\)
−0.885438 + 0.464758i \(0.846141\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 8.00000 0.353553
\(9\) 0 0
\(10\) −39.5980 −1.25220
\(11\) 14.0000 0.383742 0.191871 0.981420i \(-0.438545\pi\)
0.191871 + 0.981420i \(0.438545\pi\)
\(12\) 0 0
\(13\) −50.9117 −1.08618 −0.543091 0.839674i \(-0.682746\pi\)
−0.543091 + 0.839674i \(0.682746\pi\)
\(14\) 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
\(19\) 1.41421 0.0170759 0.00853797 0.999964i \(-0.497282\pi\)
0.00853797 + 0.999964i \(0.497282\pi\)
\(20\) −79.1960 −0.885438
\(21\) 0 0
\(22\) 28.0000 0.271346
\(23\) −140.000 −1.26922 −0.634609 0.772833i \(-0.718839\pi\)
−0.634609 + 0.772833i \(0.718839\pi\)
\(24\) 0 0
\(25\) 267.000 2.13600
\(26\) −101.823 −0.768046
\(27\) 0 0
\(28\) 0 0
\(29\) 286.000 1.83134 0.915670 0.401931i \(-0.131661\pi\)
0.915670 + 0.401931i \(0.131661\pi\)
\(30\) 0 0
\(31\) 93.3381 0.540775 0.270387 0.962752i \(-0.412848\pi\)
0.270387 + 0.962752i \(0.412848\pi\)
\(32\) 32.0000 0.176777
\(33\) 0 0
\(34\) 2.82843 0.0142668
\(35\) 0 0
\(36\) 0 0
\(37\) −38.0000 −0.168842 −0.0844211 0.996430i \(-0.526904\pi\)
−0.0844211 + 0.996430i \(0.526904\pi\)
\(38\) 2.82843 0.0120745
\(39\) 0 0
\(40\) −158.392 −0.626099
\(41\) −125.865 −0.479434 −0.239717 0.970843i \(-0.577055\pi\)
−0.239717 + 0.970843i \(0.577055\pi\)
\(42\) 0 0
\(43\) −34.0000 −0.120580 −0.0602901 0.998181i \(-0.519203\pi\)
−0.0602901 + 0.998181i \(0.519203\pi\)
\(44\) 56.0000 0.191871
\(45\) 0 0
\(46\) −280.000 −0.897473
\(47\) 523.259 1.62394 0.811970 0.583699i \(-0.198395\pi\)
0.811970 + 0.583699i \(0.198395\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 534.000 1.51038
\(51\) 0 0
\(52\) −203.647 −0.543091
\(53\) 74.0000 0.191786 0.0958932 0.995392i \(-0.469429\pi\)
0.0958932 + 0.995392i \(0.469429\pi\)
\(54\) 0 0
\(55\) −277.186 −0.679559
\(56\) 0 0
\(57\) 0 0
\(58\) 572.000 1.29495
\(59\) 434.164 0.958022 0.479011 0.877809i \(-0.340995\pi\)
0.479011 + 0.877809i \(0.340995\pi\)
\(60\) 0 0
\(61\) −14.1421 −0.0296839 −0.0148419 0.999890i \(-0.504725\pi\)
−0.0148419 + 0.999890i \(0.504725\pi\)
\(62\) 186.676 0.382385
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) 1008.00 1.92349
\(66\) 0 0
\(67\) 684.000 1.24722 0.623611 0.781735i \(-0.285665\pi\)
0.623611 + 0.781735i \(0.285665\pi\)
\(68\) 5.65685 0.0100882
\(69\) 0 0
\(70\) 0 0
\(71\) −588.000 −0.982856 −0.491428 0.870918i \(-0.663525\pi\)
−0.491428 + 0.870918i \(0.663525\pi\)
\(72\) 0 0
\(73\) 270.115 0.433076 0.216538 0.976274i \(-0.430523\pi\)
0.216538 + 0.976274i \(0.430523\pi\)
\(74\) −76.0000 −0.119389
\(75\) 0 0
\(76\) 5.65685 0.00853797
\(77\) 0 0
\(78\) 0 0
\(79\) 1220.00 1.73748 0.868739 0.495271i \(-0.164931\pi\)
0.868739 + 0.495271i \(0.164931\pi\)
\(80\) −316.784 −0.442719
\(81\) 0 0
\(82\) −251.730 −0.339011
\(83\) 422.850 0.559202 0.279601 0.960116i \(-0.409798\pi\)
0.279601 + 0.960116i \(0.409798\pi\)
\(84\) 0 0
\(85\) −28.0000 −0.0357297
\(86\) −68.0000 −0.0852631
\(87\) 0 0
\(88\) 112.000 0.135673
\(89\) 618.011 0.736057 0.368028 0.929815i \(-0.380033\pi\)
0.368028 + 0.929815i \(0.380033\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −560.000 −0.634609
\(93\) 0 0
\(94\) 1046.52 1.14830
\(95\) −28.0000 −0.0302394
\(96\) 0 0
\(97\) −1483.51 −1.55286 −0.776431 0.630202i \(-0.782972\pi\)
−0.776431 + 0.630202i \(0.782972\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 1068.00 1.06800
\(101\) 1128.54 1.11182 0.555912 0.831241i \(-0.312369\pi\)
0.555912 + 0.831241i \(0.312369\pi\)
\(102\) 0 0
\(103\) 868.327 0.830668 0.415334 0.909669i \(-0.363665\pi\)
0.415334 + 0.909669i \(0.363665\pi\)
\(104\) −407.294 −0.384023
\(105\) 0 0
\(106\) 148.000 0.135613
\(107\) 1684.00 1.52148 0.760740 0.649056i \(-0.224836\pi\)
0.760740 + 0.649056i \(0.224836\pi\)
\(108\) 0 0
\(109\) −818.000 −0.718809 −0.359405 0.933182i \(-0.617020\pi\)
−0.359405 + 0.933182i \(0.617020\pi\)
\(110\) −554.372 −0.480521
\(111\) 0 0
\(112\) 0 0
\(113\) 540.000 0.449548 0.224774 0.974411i \(-0.427836\pi\)
0.224774 + 0.974411i \(0.427836\pi\)
\(114\) 0 0
\(115\) 2771.86 2.24763
\(116\) 1144.00 0.915670
\(117\) 0 0
\(118\) 868.327 0.677424
\(119\) 0 0
\(120\) 0 0
\(121\) −1135.00 −0.852742
\(122\) −28.2843 −0.0209897
\(123\) 0 0
\(124\) 373.352 0.270387
\(125\) −2811.46 −2.01171
\(126\) 0 0
\(127\) 1720.00 1.20177 0.600887 0.799334i \(-0.294814\pi\)
0.600887 + 0.799334i \(0.294814\pi\)
\(128\) 128.000 0.0883883
\(129\) 0 0
\(130\) 2016.00 1.36011
\(131\) −1735.24 −1.15732 −0.578659 0.815570i \(-0.696424\pi\)
−0.578659 + 0.815570i \(0.696424\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 1368.00 0.881919
\(135\) 0 0
\(136\) 11.3137 0.00713340
\(137\) −828.000 −0.516356 −0.258178 0.966097i \(-0.583122\pi\)
−0.258178 + 0.966097i \(0.583122\pi\)
\(138\) 0 0
\(139\) 425.678 0.259752 0.129876 0.991530i \(-0.458542\pi\)
0.129876 + 0.991530i \(0.458542\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −1176.00 −0.694984
\(143\) −712.764 −0.416813
\(144\) 0 0
\(145\) −5662.51 −3.24308
\(146\) 540.230 0.306231
\(147\) 0 0
\(148\) −152.000 −0.0844211
\(149\) −2050.00 −1.12713 −0.563566 0.826071i \(-0.690571\pi\)
−0.563566 + 0.826071i \(0.690571\pi\)
\(150\) 0 0
\(151\) −472.000 −0.254376 −0.127188 0.991879i \(-0.540595\pi\)
−0.127188 + 0.991879i \(0.540595\pi\)
\(152\) 11.3137 0.00603726
\(153\) 0 0
\(154\) 0 0
\(155\) −1848.00 −0.957645
\(156\) 0 0
\(157\) 2211.83 1.12435 0.562176 0.827018i \(-0.309964\pi\)
0.562176 + 0.827018i \(0.309964\pi\)
\(158\) 2440.00 1.22858
\(159\) 0 0
\(160\) −633.568 −0.313050
\(161\) 0 0
\(162\) 0 0
\(163\) 3286.00 1.57901 0.789507 0.613741i \(-0.210336\pi\)
0.789507 + 0.613741i \(0.210336\pi\)
\(164\) −503.460 −0.239717
\(165\) 0 0
\(166\) 845.700 0.395416
\(167\) −1490.58 −0.690686 −0.345343 0.938476i \(-0.612237\pi\)
−0.345343 + 0.938476i \(0.612237\pi\)
\(168\) 0 0
\(169\) 395.000 0.179791
\(170\) −56.0000 −0.0252647
\(171\) 0 0
\(172\) −136.000 −0.0602901
\(173\) −2070.41 −0.909886 −0.454943 0.890521i \(-0.650340\pi\)
−0.454943 + 0.890521i \(0.650340\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 224.000 0.0959354
\(177\) 0 0
\(178\) 1236.02 0.520471
\(179\) −540.000 −0.225483 −0.112742 0.993624i \(-0.535963\pi\)
−0.112742 + 0.993624i \(0.535963\pi\)
\(180\) 0 0
\(181\) −3784.44 −1.55412 −0.777058 0.629429i \(-0.783289\pi\)
−0.777058 + 0.629429i \(0.783289\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −1120.00 −0.448736
\(185\) 752.362 0.298999
\(186\) 0 0
\(187\) 19.7990 0.00774249
\(188\) 2093.04 0.811970
\(189\) 0 0
\(190\) −56.0000 −0.0213825
\(191\) −1028.00 −0.389442 −0.194721 0.980859i \(-0.562380\pi\)
−0.194721 + 0.980859i \(0.562380\pi\)
\(192\) 0 0
\(193\) 4592.00 1.71264 0.856320 0.516446i \(-0.172745\pi\)
0.856320 + 0.516446i \(0.172745\pi\)
\(194\) −2967.02 −1.09804
\(195\) 0 0
\(196\) 0 0
\(197\) −794.000 −0.287158 −0.143579 0.989639i \(-0.545861\pi\)
−0.143579 + 0.989639i \(0.545861\pi\)
\(198\) 0 0
\(199\) −2486.19 −0.885634 −0.442817 0.896612i \(-0.646021\pi\)
−0.442817 + 0.896612i \(0.646021\pi\)
\(200\) 2136.00 0.755190
\(201\) 0 0
\(202\) 2257.08 0.786178
\(203\) 0 0
\(204\) 0 0
\(205\) 2492.00 0.849019
\(206\) 1736.65 0.587371
\(207\) 0 0
\(208\) −814.587 −0.271545
\(209\) 19.7990 0.00655275
\(210\) 0 0
\(211\) −2748.00 −0.896588 −0.448294 0.893886i \(-0.647968\pi\)
−0.448294 + 0.893886i \(0.647968\pi\)
\(212\) 296.000 0.0958932
\(213\) 0 0
\(214\) 3368.00 1.07585
\(215\) 673.166 0.213533
\(216\) 0 0
\(217\) 0 0
\(218\) −1636.00 −0.508275
\(219\) 0 0
\(220\) −1108.74 −0.339779
\(221\) −72.0000 −0.0219151
\(222\) 0 0
\(223\) 3428.05 1.02941 0.514707 0.857366i \(-0.327901\pi\)
0.514707 + 0.857366i \(0.327901\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 1080.00 0.317878
\(227\) 5290.57 1.54691 0.773453 0.633854i \(-0.218528\pi\)
0.773453 + 0.633854i \(0.218528\pi\)
\(228\) 0 0
\(229\) 2749.23 0.793338 0.396669 0.917962i \(-0.370166\pi\)
0.396669 + 0.917962i \(0.370166\pi\)
\(230\) 5543.72 1.58931
\(231\) 0 0
\(232\) 2288.00 0.647477
\(233\) −72.0000 −0.0202441 −0.0101221 0.999949i \(-0.503222\pi\)
−0.0101221 + 0.999949i \(0.503222\pi\)
\(234\) 0 0
\(235\) −10360.0 −2.87580
\(236\) 1736.65 0.479011
\(237\) 0 0
\(238\) 0 0
\(239\) −4308.00 −1.16595 −0.582974 0.812491i \(-0.698111\pi\)
−0.582974 + 0.812491i \(0.698111\pi\)
\(240\) 0 0
\(241\) 1540.08 0.411640 0.205820 0.978590i \(-0.434014\pi\)
0.205820 + 0.978590i \(0.434014\pi\)
\(242\) −2270.00 −0.602980
\(243\) 0 0
\(244\) −56.5685 −0.0148419
\(245\) 0 0
\(246\) 0 0
\(247\) −72.0000 −0.0185476
\(248\) 746.705 0.191193
\(249\) 0 0
\(250\) −5622.91 −1.42250
\(251\) 931.967 0.234363 0.117182 0.993110i \(-0.462614\pi\)
0.117182 + 0.993110i \(0.462614\pi\)
\(252\) 0 0
\(253\) −1960.00 −0.487052
\(254\) 3440.00 0.849783
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) 937.624 0.227577 0.113789 0.993505i \(-0.463701\pi\)
0.113789 + 0.993505i \(0.463701\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 4032.00 0.961746
\(261\) 0 0
\(262\) −3470.48 −0.818347
\(263\) −7140.00 −1.67404 −0.837018 0.547176i \(-0.815703\pi\)
−0.837018 + 0.547176i \(0.815703\pi\)
\(264\) 0 0
\(265\) −1465.13 −0.339630
\(266\) 0 0
\(267\) 0 0
\(268\) 2736.00 0.623611
\(269\) 4610.34 1.04497 0.522485 0.852648i \(-0.325005\pi\)
0.522485 + 0.852648i \(0.325005\pi\)
\(270\) 0 0
\(271\) 2364.57 0.530026 0.265013 0.964245i \(-0.414624\pi\)
0.265013 + 0.964245i \(0.414624\pi\)
\(272\) 22.6274 0.00504408
\(273\) 0 0
\(274\) −1656.00 −0.365119
\(275\) 3738.00 0.819672
\(276\) 0 0
\(277\) 4006.00 0.868943 0.434472 0.900686i \(-0.356935\pi\)
0.434472 + 0.900686i \(0.356935\pi\)
\(278\) 851.357 0.183673
\(279\) 0 0
\(280\) 0 0
\(281\) 5984.00 1.27038 0.635188 0.772358i \(-0.280923\pi\)
0.635188 + 0.772358i \(0.280923\pi\)
\(282\) 0 0
\(283\) 4928.53 1.03523 0.517617 0.855613i \(-0.326819\pi\)
0.517617 + 0.855613i \(0.326819\pi\)
\(284\) −2352.00 −0.491428
\(285\) 0 0
\(286\) −1425.53 −0.294731
\(287\) 0 0
\(288\) 0 0
\(289\) −4911.00 −0.999593
\(290\) −11325.0 −2.29320
\(291\) 0 0
\(292\) 1080.46 0.216538
\(293\) 1971.41 0.393076 0.196538 0.980496i \(-0.437030\pi\)
0.196538 + 0.980496i \(0.437030\pi\)
\(294\) 0 0
\(295\) −8596.00 −1.69654
\(296\) −304.000 −0.0596947
\(297\) 0 0
\(298\) −4100.00 −0.797002
\(299\) 7127.64 1.37860
\(300\) 0 0
\(301\) 0 0
\(302\) −944.000 −0.179871
\(303\) 0 0
\(304\) 22.6274 0.00426898
\(305\) 280.000 0.0525664
\(306\) 0 0
\(307\) 4767.31 0.886270 0.443135 0.896455i \(-0.353866\pi\)
0.443135 + 0.896455i \(0.353866\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −3696.00 −0.677157
\(311\) −6776.91 −1.23564 −0.617819 0.786320i \(-0.711984\pi\)
−0.617819 + 0.786320i \(0.711984\pi\)
\(312\) 0 0
\(313\) 6190.01 1.11783 0.558914 0.829226i \(-0.311218\pi\)
0.558914 + 0.829226i \(0.311218\pi\)
\(314\) 4423.66 0.795037
\(315\) 0 0
\(316\) 4880.00 0.868739
\(317\) −9826.00 −1.74096 −0.870478 0.492207i \(-0.836190\pi\)
−0.870478 + 0.492207i \(0.836190\pi\)
\(318\) 0 0
\(319\) 4004.00 0.702762
\(320\) −1267.14 −0.221359
\(321\) 0 0
\(322\) 0 0
\(323\) 2.00000 0.000344529 0
\(324\) 0 0
\(325\) −13593.4 −2.32008
\(326\) 6572.00 1.11653
\(327\) 0 0
\(328\) −1006.92 −0.169506
\(329\) 0 0
\(330\) 0 0
\(331\) −5738.00 −0.952837 −0.476418 0.879219i \(-0.658065\pi\)
−0.476418 + 0.879219i \(0.658065\pi\)
\(332\) 1691.40 0.279601
\(333\) 0 0
\(334\) −2981.16 −0.488389
\(335\) −13542.5 −2.20868
\(336\) 0 0
\(337\) −2254.00 −0.364342 −0.182171 0.983267i \(-0.558312\pi\)
−0.182171 + 0.983267i \(0.558312\pi\)
\(338\) 790.000 0.127131
\(339\) 0 0
\(340\) −112.000 −0.0178649
\(341\) 1306.73 0.207518
\(342\) 0 0
\(343\) 0 0
\(344\) −272.000 −0.0426316
\(345\) 0 0
\(346\) −4140.82 −0.643386
\(347\) 1986.00 0.307245 0.153623 0.988130i \(-0.450906\pi\)
0.153623 + 0.988130i \(0.450906\pi\)
\(348\) 0 0
\(349\) −6771.25 −1.03856 −0.519279 0.854605i \(-0.673800\pi\)
−0.519279 + 0.854605i \(0.673800\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 448.000 0.0678366
\(353\) 6993.29 1.05443 0.527217 0.849731i \(-0.323236\pi\)
0.527217 + 0.849731i \(0.323236\pi\)
\(354\) 0 0
\(355\) 11641.8 1.74052
\(356\) 2472.05 0.368028
\(357\) 0 0
\(358\) −1080.00 −0.159441
\(359\) −5944.00 −0.873850 −0.436925 0.899498i \(-0.643933\pi\)
−0.436925 + 0.899498i \(0.643933\pi\)
\(360\) 0 0
\(361\) −6857.00 −0.999708
\(362\) −7568.87 −1.09893
\(363\) 0 0
\(364\) 0 0
\(365\) −5348.00 −0.766924
\(366\) 0 0
\(367\) 842.871 0.119884 0.0599421 0.998202i \(-0.480908\pi\)
0.0599421 + 0.998202i \(0.480908\pi\)
\(368\) −2240.00 −0.317305
\(369\) 0 0
\(370\) 1504.72 0.211424
\(371\) 0 0
\(372\) 0 0
\(373\) −5726.00 −0.794855 −0.397428 0.917634i \(-0.630097\pi\)
−0.397428 + 0.917634i \(0.630097\pi\)
\(374\) 39.5980 0.00547477
\(375\) 0 0
\(376\) 4186.07 0.574149
\(377\) −14560.7 −1.98917
\(378\) 0 0
\(379\) 10330.0 1.40004 0.700022 0.714122i \(-0.253174\pi\)
0.700022 + 0.714122i \(0.253174\pi\)
\(380\) −112.000 −0.0151197
\(381\) 0 0
\(382\) −2056.00 −0.275377
\(383\) 1004.09 0.133960 0.0669800 0.997754i \(-0.478664\pi\)
0.0669800 + 0.997754i \(0.478664\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 9184.00 1.21102
\(387\) 0 0
\(388\) −5934.04 −0.776431
\(389\) −5210.00 −0.679068 −0.339534 0.940594i \(-0.610269\pi\)
−0.339534 + 0.940594i \(0.610269\pi\)
\(390\) 0 0
\(391\) −197.990 −0.0256081
\(392\) 0 0
\(393\) 0 0
\(394\) −1588.00 −0.203051
\(395\) −24154.8 −3.07686
\(396\) 0 0
\(397\) 73.5391 0.00929678 0.00464839 0.999989i \(-0.498520\pi\)
0.00464839 + 0.999989i \(0.498520\pi\)
\(398\) −4972.37 −0.626238
\(399\) 0 0
\(400\) 4272.00 0.534000
\(401\) 498.000 0.0620173 0.0310086 0.999519i \(-0.490128\pi\)
0.0310086 + 0.999519i \(0.490128\pi\)
\(402\) 0 0
\(403\) −4752.00 −0.587380
\(404\) 4514.17 0.555912
\(405\) 0 0
\(406\) 0 0
\(407\) −532.000 −0.0647918
\(408\) 0 0
\(409\) −3355.93 −0.405721 −0.202861 0.979208i \(-0.565024\pi\)
−0.202861 + 0.979208i \(0.565024\pi\)
\(410\) 4984.00 0.600347
\(411\) 0 0
\(412\) 3473.31 0.415334
\(413\) 0 0
\(414\) 0 0
\(415\) −8372.00 −0.990278
\(416\) −1629.17 −0.192012
\(417\) 0 0
\(418\) 39.5980 0.00463349
\(419\) 14545.2 1.69589 0.847946 0.530082i \(-0.177839\pi\)
0.847946 + 0.530082i \(0.177839\pi\)
\(420\) 0 0
\(421\) 10854.0 1.25651 0.628256 0.778007i \(-0.283769\pi\)
0.628256 + 0.778007i \(0.283769\pi\)
\(422\) −5496.00 −0.633984
\(423\) 0 0
\(424\) 592.000 0.0678067
\(425\) 377.595 0.0430966
\(426\) 0 0
\(427\) 0 0
\(428\) 6736.00 0.760740
\(429\) 0 0
\(430\) 1346.33 0.150990
\(431\) 5364.00 0.599477 0.299739 0.954021i \(-0.403100\pi\)
0.299739 + 0.954021i \(0.403100\pi\)
\(432\) 0 0
\(433\) 6487.00 0.719966 0.359983 0.932959i \(-0.382783\pi\)
0.359983 + 0.932959i \(0.382783\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −3272.00 −0.359405
\(437\) −197.990 −0.0216731
\(438\) 0 0
\(439\) −13932.8 −1.51476 −0.757378 0.652977i \(-0.773520\pi\)
−0.757378 + 0.652977i \(0.773520\pi\)
\(440\) −2217.49 −0.240260
\(441\) 0 0
\(442\) −144.000 −0.0154963
\(443\) −5996.00 −0.643067 −0.321533 0.946898i \(-0.604198\pi\)
−0.321533 + 0.946898i \(0.604198\pi\)
\(444\) 0 0
\(445\) −12236.0 −1.30347
\(446\) 6856.11 0.727906
\(447\) 0 0
\(448\) 0 0
\(449\) −2622.00 −0.275590 −0.137795 0.990461i \(-0.544001\pi\)
−0.137795 + 0.990461i \(0.544001\pi\)
\(450\) 0 0
\(451\) −1762.11 −0.183979
\(452\) 2160.00 0.224774
\(453\) 0 0
\(454\) 10581.1 1.09383
\(455\) 0 0
\(456\) 0 0
\(457\) 11208.0 1.14724 0.573619 0.819122i \(-0.305539\pi\)
0.573619 + 0.819122i \(0.305539\pi\)
\(458\) 5498.46 0.560974
\(459\) 0 0
\(460\) 11087.4 1.12381
\(461\) 9786.36 0.988712 0.494356 0.869260i \(-0.335404\pi\)
0.494356 + 0.869260i \(0.335404\pi\)
\(462\) 0 0
\(463\) 3952.00 0.396685 0.198342 0.980133i \(-0.436444\pi\)
0.198342 + 0.980133i \(0.436444\pi\)
\(464\) 4576.00 0.457835
\(465\) 0 0
\(466\) −144.000 −0.0143147
\(467\) 17506.5 1.73470 0.867352 0.497696i \(-0.165820\pi\)
0.867352 + 0.497696i \(0.165820\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −20720.0 −2.03349
\(471\) 0 0
\(472\) 3473.31 0.338712
\(473\) −476.000 −0.0462717
\(474\) 0 0
\(475\) 377.595 0.0364742
\(476\) 0 0
\(477\) 0 0
\(478\) −8616.00 −0.824449
\(479\) 2288.20 0.218268 0.109134 0.994027i \(-0.465192\pi\)
0.109134 + 0.994027i \(0.465192\pi\)
\(480\) 0 0
\(481\) 1934.64 0.183393
\(482\) 3080.16 0.291073
\(483\) 0 0
\(484\) −4540.00 −0.426371
\(485\) 29372.0 2.74993
\(486\) 0 0
\(487\) 972.000 0.0904426 0.0452213 0.998977i \(-0.485601\pi\)
0.0452213 + 0.998977i \(0.485601\pi\)
\(488\) −113.137 −0.0104948
\(489\) 0 0
\(490\) 0 0
\(491\) 7404.00 0.680525 0.340263 0.940330i \(-0.389484\pi\)
0.340263 + 0.940330i \(0.389484\pi\)
\(492\) 0 0
\(493\) 404.465 0.0369497
\(494\) −144.000 −0.0131151
\(495\) 0 0
\(496\) 1493.41 0.135194
\(497\) 0 0
\(498\) 0 0
\(499\) −12244.0 −1.09843 −0.549215 0.835681i \(-0.685073\pi\)
−0.549215 + 0.835681i \(0.685073\pi\)
\(500\) −11245.8 −1.00586
\(501\) 0 0
\(502\) 1863.93 0.165720
\(503\) −2415.48 −0.214117 −0.107058 0.994253i \(-0.534143\pi\)
−0.107058 + 0.994253i \(0.534143\pi\)
\(504\) 0 0
\(505\) −22344.0 −1.96890
\(506\) −3920.00 −0.344398
\(507\) 0 0
\(508\) 6880.00 0.600887
\(509\) −5707.77 −0.497038 −0.248519 0.968627i \(-0.579944\pi\)
−0.248519 + 0.968627i \(0.579944\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 512.000 0.0441942
\(513\) 0 0
\(514\) 1875.25 0.160921
\(515\) −17192.0 −1.47101
\(516\) 0 0
\(517\) 7325.63 0.623173
\(518\) 0 0
\(519\) 0 0
\(520\) 8064.00 0.680057
\(521\) 1.41421 0.000118921 0 5.94605e−5 1.00000i \(-0.499981\pi\)
5.94605e−5 1.00000i \(0.499981\pi\)
\(522\) 0 0
\(523\) 12257.0 1.02478 0.512391 0.858752i \(-0.328760\pi\)
0.512391 + 0.858752i \(0.328760\pi\)
\(524\) −6940.96 −0.578659
\(525\) 0 0
\(526\) −14280.0 −1.18372
\(527\) 132.000 0.0109108
\(528\) 0 0
\(529\) 7433.00 0.610915
\(530\) −2930.25 −0.240155
\(531\) 0 0
\(532\) 0 0
\(533\) 6408.00 0.520753
\(534\) 0 0
\(535\) −33341.5 −2.69435
\(536\) 5472.00 0.440960
\(537\) 0 0
\(538\) 9220.67 0.738906
\(539\) 0 0
\(540\) 0 0
\(541\) 2050.00 0.162914 0.0814569 0.996677i \(-0.474043\pi\)
0.0814569 + 0.996677i \(0.474043\pi\)
\(542\) 4729.13 0.374785
\(543\) 0 0
\(544\) 45.2548 0.00356670
\(545\) 16195.6 1.27292
\(546\) 0 0
\(547\) 14554.0 1.13763 0.568815 0.822465i \(-0.307402\pi\)
0.568815 + 0.822465i \(0.307402\pi\)
\(548\) −3312.00 −0.258178
\(549\) 0 0
\(550\) 7476.00 0.579596
\(551\) 404.465 0.0312719
\(552\) 0 0
\(553\) 0 0
\(554\) 8012.00 0.614435
\(555\) 0 0
\(556\) 1702.71 0.129876
\(557\) −6954.00 −0.528995 −0.264498 0.964386i \(-0.585206\pi\)
−0.264498 + 0.964386i \(0.585206\pi\)
\(558\) 0 0
\(559\) 1731.00 0.130972
\(560\) 0 0
\(561\) 0 0
\(562\) 11968.0 0.898291
\(563\) −1636.25 −0.122486 −0.0612429 0.998123i \(-0.519506\pi\)
−0.0612429 + 0.998123i \(0.519506\pi\)
\(564\) 0 0
\(565\) −10691.5 −0.796094
\(566\) 9857.07 0.732020
\(567\) 0 0
\(568\) −4704.00 −0.347492
\(569\) 7142.00 0.526201 0.263100 0.964768i \(-0.415255\pi\)
0.263100 + 0.964768i \(0.415255\pi\)
\(570\) 0 0
\(571\) −20606.0 −1.51022 −0.755109 0.655599i \(-0.772416\pi\)
−0.755109 + 0.655599i \(0.772416\pi\)
\(572\) −2851.05 −0.208407
\(573\) 0 0
\(574\) 0 0
\(575\) −37380.0 −2.71105
\(576\) 0 0
\(577\) 8803.48 0.635171 0.317585 0.948230i \(-0.397128\pi\)
0.317585 + 0.948230i \(0.397128\pi\)
\(578\) −9822.00 −0.706819
\(579\) 0 0
\(580\) −22650.0 −1.62154
\(581\) 0 0
\(582\) 0 0
\(583\) 1036.00 0.0735965
\(584\) 2160.92 0.153115
\(585\) 0 0
\(586\) 3942.83 0.277947
\(587\) 6503.97 0.457321 0.228661 0.973506i \(-0.426565\pi\)
0.228661 + 0.973506i \(0.426565\pi\)
\(588\) 0 0
\(589\) 132.000 0.00923424
\(590\) −17192.0 −1.19963
\(591\) 0 0
\(592\) −608.000 −0.0422106
\(593\) −23140.8 −1.60249 −0.801246 0.598335i \(-0.795829\pi\)
−0.801246 + 0.598335i \(0.795829\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −8200.00 −0.563566
\(597\) 0 0
\(598\) 14255.3 0.974818
\(599\) 11296.0 0.770521 0.385260 0.922808i \(-0.374112\pi\)
0.385260 + 0.922808i \(0.374112\pi\)
\(600\) 0 0
\(601\) −8727.11 −0.592323 −0.296162 0.955138i \(-0.595707\pi\)
−0.296162 + 0.955138i \(0.595707\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −1888.00 −0.127188
\(605\) 22471.9 1.51010
\(606\) 0 0
\(607\) −19736.8 −1.31975 −0.659877 0.751374i \(-0.729392\pi\)
−0.659877 + 0.751374i \(0.729392\pi\)
\(608\) 45.2548 0.00301863
\(609\) 0 0
\(610\) 560.000 0.0371701
\(611\) −26640.0 −1.76389
\(612\) 0 0
\(613\) 16962.0 1.11760 0.558800 0.829302i \(-0.311262\pi\)
0.558800 + 0.829302i \(0.311262\pi\)
\(614\) 9534.63 0.626688
\(615\) 0 0
\(616\) 0 0
\(617\) 19034.0 1.24194 0.620972 0.783832i \(-0.286738\pi\)
0.620972 + 0.783832i \(0.286738\pi\)
\(618\) 0 0
\(619\) 18677.5 1.21278 0.606392 0.795166i \(-0.292616\pi\)
0.606392 + 0.795166i \(0.292616\pi\)
\(620\) −7392.00 −0.478822
\(621\) 0 0
\(622\) −13553.8 −0.873728
\(623\) 0 0
\(624\) 0 0
\(625\) 22289.0 1.42650
\(626\) 12380.0 0.790424
\(627\) 0 0
\(628\) 8847.32 0.562176
\(629\) −53.7401 −0.00340661
\(630\) 0 0
\(631\) −14716.0 −0.928423 −0.464211 0.885724i \(-0.653662\pi\)
−0.464211 + 0.885724i \(0.653662\pi\)
\(632\) 9760.00 0.614291
\(633\) 0 0
\(634\) −19652.0 −1.23104
\(635\) −34054.3 −2.12819
\(636\) 0 0
\(637\) 0 0
\(638\) 8008.00 0.496928
\(639\) 0 0
\(640\) −2534.27 −0.156525
\(641\) −4730.00 −0.291457 −0.145728 0.989325i \(-0.546553\pi\)
−0.145728 + 0.989325i \(0.546553\pi\)
\(642\) 0 0
\(643\) −19056.5 −1.16877 −0.584383 0.811478i \(-0.698663\pi\)
−0.584383 + 0.811478i \(0.698663\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 4.00000 0.000243619 0
\(647\) 9342.29 0.567672 0.283836 0.958873i \(-0.408393\pi\)
0.283836 + 0.958873i \(0.408393\pi\)
\(648\) 0 0
\(649\) 6078.29 0.367633
\(650\) −27186.8 −1.64055
\(651\) 0 0
\(652\) 13144.0 0.789507
\(653\) −3774.00 −0.226169 −0.113084 0.993585i \(-0.536073\pi\)
−0.113084 + 0.993585i \(0.536073\pi\)
\(654\) 0 0
\(655\) 34356.0 2.04947
\(656\) −2013.84 −0.119859
\(657\) 0 0
\(658\) 0 0
\(659\) 21150.0 1.25021 0.625104 0.780541i \(-0.285057\pi\)
0.625104 + 0.780541i \(0.285057\pi\)
\(660\) 0 0
\(661\) 10377.5 0.610647 0.305324 0.952249i \(-0.401235\pi\)
0.305324 + 0.952249i \(0.401235\pi\)
\(662\) −11476.0 −0.673757
\(663\) 0 0
\(664\) 3382.80 0.197708
\(665\) 0 0
\(666\) 0 0
\(667\) −40040.0 −2.32437
\(668\) −5962.32 −0.345343
\(669\) 0 0
\(670\) −27085.0 −1.56177
\(671\) −197.990 −0.0113909
\(672\) 0 0
\(673\) −1164.00 −0.0666700 −0.0333350 0.999444i \(-0.510613\pi\)
−0.0333350 + 0.999444i \(0.510613\pi\)
\(674\) −4508.00 −0.257629
\(675\) 0 0
\(676\) 1580.00 0.0898953
\(677\) 27152.9 1.54146 0.770732 0.637160i \(-0.219891\pi\)
0.770732 + 0.637160i \(0.219891\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −224.000 −0.0126324
\(681\) 0 0
\(682\) 2613.47 0.146737
\(683\) 16596.0 0.929763 0.464882 0.885373i \(-0.346097\pi\)
0.464882 + 0.885373i \(0.346097\pi\)
\(684\) 0 0
\(685\) 16393.6 0.914403
\(686\) 0 0
\(687\) 0 0
\(688\) −544.000 −0.0301451
\(689\) −3767.46 −0.208315
\(690\) 0 0
\(691\) −11298.2 −0.622000 −0.311000 0.950410i \(-0.600664\pi\)
−0.311000 + 0.950410i \(0.600664\pi\)
\(692\) −8281.63 −0.454943
\(693\) 0 0
\(694\) 3972.00 0.217255
\(695\) −8428.00 −0.459989
\(696\) 0 0
\(697\) −178.000 −0.00967321
\(698\) −13542.5 −0.734372
\(699\) 0 0
\(700\) 0 0
\(701\) 2754.00 0.148384 0.0741920 0.997244i \(-0.476362\pi\)
0.0741920 + 0.997244i \(0.476362\pi\)
\(702\) 0 0
\(703\) −53.7401 −0.00288314
\(704\) 896.000 0.0479677
\(705\) 0 0
\(706\) 13986.6 0.745597
\(707\) 0 0
\(708\) 0 0
\(709\) 29434.0 1.55912 0.779561 0.626327i \(-0.215442\pi\)
0.779561 + 0.626327i \(0.215442\pi\)
\(710\) 23283.6 1.23073
\(711\) 0 0
\(712\) 4944.09 0.260235
\(713\) −13067.3 −0.686361
\(714\) 0 0
\(715\) 14112.0 0.738124
\(716\) −2160.00 −0.112742
\(717\) 0 0
\(718\) −11888.0 −0.617906
\(719\) −17669.2 −0.916480 −0.458240 0.888828i \(-0.651520\pi\)
−0.458240 + 0.888828i \(0.651520\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −13714.0 −0.706901
\(723\) 0 0
\(724\) −15137.7 −0.777058
\(725\) 76362.0 3.91174
\(726\) 0 0
\(727\) −28445.5 −1.45115 −0.725574 0.688144i \(-0.758426\pi\)
−0.725574 + 0.688144i \(0.758426\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −10696.0 −0.542297
\(731\) −48.0833 −0.00243286
\(732\) 0 0
\(733\) −22341.7 −1.12580 −0.562900 0.826525i \(-0.690314\pi\)
−0.562900 + 0.826525i \(0.690314\pi\)
\(734\) 1685.74 0.0847710
\(735\) 0 0
\(736\) −4480.00 −0.224368
\(737\) 9576.00 0.478611
\(738\) 0 0
\(739\) 20670.0 1.02890 0.514451 0.857520i \(-0.327996\pi\)
0.514451 + 0.857520i \(0.327996\pi\)
\(740\) 3009.45 0.149499
\(741\) 0 0
\(742\) 0 0
\(743\) 25400.0 1.25415 0.627076 0.778958i \(-0.284251\pi\)
0.627076 + 0.778958i \(0.284251\pi\)
\(744\) 0 0
\(745\) 40587.9 1.99601
\(746\) −11452.0 −0.562048
\(747\) 0 0
\(748\) 79.1960 0.00387124
\(749\) 0 0
\(750\) 0 0
\(751\) 29180.0 1.41783 0.708917 0.705292i \(-0.249184\pi\)
0.708917 + 0.705292i \(0.249184\pi\)
\(752\) 8372.14 0.405985
\(753\) 0 0
\(754\) −29121.5 −1.40655
\(755\) 9345.12 0.450469
\(756\) 0 0
\(757\) −26206.0 −1.25822 −0.629110 0.777316i \(-0.716581\pi\)
−0.629110 + 0.777316i \(0.716581\pi\)
\(758\) 20660.0 0.989980
\(759\) 0 0
\(760\) −224.000 −0.0106912
\(761\) 6863.18 0.326925 0.163463 0.986550i \(-0.447734\pi\)
0.163463 + 0.986550i \(0.447734\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −4112.00 −0.194721
\(765\) 0 0
\(766\) 2008.18 0.0947240
\(767\) −22104.0 −1.04059
\(768\) 0 0
\(769\) 9058.04 0.424761 0.212380 0.977187i \(-0.431878\pi\)
0.212380 + 0.977187i \(0.431878\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 18368.0 0.856320
\(773\) −132.936 −0.00618548 −0.00309274 0.999995i \(-0.500984\pi\)
−0.00309274 + 0.999995i \(0.500984\pi\)
\(774\) 0 0
\(775\) 24921.3 1.15509
\(776\) −11868.1 −0.549020
\(777\) 0 0
\(778\) −10420.0 −0.480174
\(779\) −178.000 −0.00818679
\(780\) 0 0
\(781\) −8232.00 −0.377163
\(782\) −395.980 −0.0181077
\(783\) 0 0
\(784\) 0 0
\(785\) −43792.0 −1.99109
\(786\) 0 0
\(787\) 8729.94 0.395411 0.197706 0.980261i \(-0.436651\pi\)
0.197706 + 0.980261i \(0.436651\pi\)
\(788\) −3176.00 −0.143579
\(789\) 0 0
\(790\) −48309.5 −2.17567
\(791\) 0 0
\(792\) 0 0
\(793\) 720.000 0.0322421
\(794\) 147.078 0.00657382
\(795\) 0 0
\(796\) −9944.75 −0.442817
\(797\) 7517.96 0.334128 0.167064 0.985946i \(-0.446571\pi\)
0.167064 + 0.985946i \(0.446571\pi\)
\(798\) 0 0
\(799\) 740.000 0.0327651
\(800\) 8544.00 0.377595
\(801\) 0 0
\(802\) 996.000 0.0438528
\(803\) 3781.61 0.166189
\(804\) 0 0
\(805\) 0 0
\(806\) −9504.00 −0.415340
\(807\) 0 0
\(808\) 9028.34 0.393089
\(809\) −3776.00 −0.164100 −0.0820501 0.996628i \(-0.526147\pi\)
−0.0820501 + 0.996628i \(0.526147\pi\)
\(810\) 0 0
\(811\) 36227.9 1.56860 0.784300 0.620382i \(-0.213023\pi\)
0.784300 + 0.620382i \(0.213023\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −1064.00 −0.0458147
\(815\) −65059.5 −2.79624
\(816\) 0 0
\(817\) −48.0833 −0.00205902
\(818\) −6711.86 −0.286888
\(819\) 0 0
\(820\) 9968.00 0.424509
\(821\) 16410.0 0.697580 0.348790 0.937201i \(-0.386593\pi\)
0.348790 + 0.937201i \(0.386593\pi\)
\(822\) 0 0
\(823\) 22072.0 0.934850 0.467425 0.884033i \(-0.345182\pi\)
0.467425 + 0.884033i \(0.345182\pi\)
\(824\) 6946.62 0.293686
\(825\) 0 0
\(826\) 0 0
\(827\) 11628.0 0.488930 0.244465 0.969658i \(-0.421388\pi\)
0.244465 + 0.969658i \(0.421388\pi\)
\(828\) 0 0
\(829\) 30906.2 1.29483 0.647417 0.762136i \(-0.275849\pi\)
0.647417 + 0.762136i \(0.275849\pi\)
\(830\) −16744.0 −0.700232
\(831\) 0 0
\(832\) −3258.35 −0.135773
\(833\) 0 0
\(834\) 0 0
\(835\) 29512.0 1.22312
\(836\) 79.1960 0.00327638
\(837\) 0 0
\(838\) 29090.4 1.19918
\(839\) 17884.1 0.735911 0.367955 0.929843i \(-0.380058\pi\)
0.367955 + 0.929843i \(0.380058\pi\)
\(840\) 0 0
\(841\) 57407.0 2.35381
\(842\) 21708.0 0.888488
\(843\) 0 0
\(844\) −10992.0 −0.448294
\(845\) −7820.60 −0.318387
\(846\) 0 0
\(847\) 0 0
\(848\) 1184.00 0.0479466
\(849\) 0 0
\(850\) 755.190 0.0304739
\(851\) 5320.00 0.214298
\(852\) 0 0
\(853\) 20755.0 0.833104 0.416552 0.909112i \(-0.363238\pi\)
0.416552 + 0.909112i \(0.363238\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 13472.0 0.537925
\(857\) 44919.7 1.79046 0.895231 0.445602i \(-0.147010\pi\)
0.895231 + 0.445602i \(0.147010\pi\)
\(858\) 0 0
\(859\) −69.2965 −0.00275246 −0.00137623 0.999999i \(-0.500438\pi\)
−0.00137623 + 0.999999i \(0.500438\pi\)
\(860\) 2692.66 0.106766
\(861\) 0 0
\(862\) 10728.0 0.423895
\(863\) 5452.00 0.215050 0.107525 0.994202i \(-0.465707\pi\)
0.107525 + 0.994202i \(0.465707\pi\)
\(864\) 0 0
\(865\) 40992.0 1.61129
\(866\) 12974.0 0.509093
\(867\) 0 0
\(868\) 0 0
\(869\) 17080.0 0.666743
\(870\) 0 0
\(871\) −34823.6 −1.35471
\(872\) −6544.00 −0.254137
\(873\) 0 0
\(874\) −395.980 −0.0153252
\(875\) 0 0
\(876\) 0 0
\(877\) 31106.0 1.19769 0.598845 0.800865i \(-0.295626\pi\)
0.598845 + 0.800865i \(0.295626\pi\)
\(878\) −27865.7 −1.07109
\(879\) 0 0
\(880\) −4434.97 −0.169890
\(881\) 5943.94 0.227306 0.113653 0.993521i \(-0.463745\pi\)
0.113653 + 0.993521i \(0.463745\pi\)
\(882\) 0 0
\(883\) 34796.0 1.32614 0.663068 0.748559i \(-0.269254\pi\)
0.663068 + 0.748559i \(0.269254\pi\)
\(884\) −288.000 −0.0109576
\(885\) 0 0
\(886\) −11992.0 −0.454717
\(887\) 9964.55 0.377200 0.188600 0.982054i \(-0.439605\pi\)
0.188600 + 0.982054i \(0.439605\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −24472.0 −0.921689
\(891\) 0 0
\(892\) 13712.2 0.514707
\(893\) 740.000 0.0277303
\(894\) 0 0
\(895\) 10691.5 0.399303
\(896\) 0 0
\(897\) 0 0
\(898\) −5244.00 −0.194871
\(899\) 26694.7 0.990343
\(900\) 0 0
\(901\) 104.652 0.00386954
\(902\) −3524.22 −0.130093
\(903\) 0 0
\(904\) 4320.00 0.158939
\(905\) 74928.0 2.75214
\(906\) 0 0
\(907\) −29756.0 −1.08934 −0.544670 0.838650i \(-0.683345\pi\)
−0.544670 + 0.838650i \(0.683345\pi\)
\(908\) 21162.3 0.773453
\(909\) 0 0
\(910\) 0 0
\(911\) −21440.0 −0.779735 −0.389868 0.920871i \(-0.627479\pi\)
−0.389868 + 0.920871i \(0.627479\pi\)
\(912\) 0 0
\(913\) 5919.90 0.214589
\(914\) 22416.0 0.811220
\(915\) 0 0
\(916\) 10996.9 0.396669
\(917\) 0 0
\(918\) 0 0
\(919\) −8288.00 −0.297493 −0.148746 0.988875i \(-0.547524\pi\)
−0.148746 + 0.988875i \(0.547524\pi\)
\(920\) 22174.9 0.794656
\(921\) 0 0
\(922\) 19572.7 0.699125
\(923\) 29936.1 1.06756
\(924\) 0 0
\(925\) −10146.0 −0.360647
\(926\) 7904.00 0.280498
\(927\) 0 0
\(928\) 9152.00 0.323738
\(929\) −45581.5 −1.60978 −0.804888 0.593427i \(-0.797774\pi\)
−0.804888 + 0.593427i \(0.797774\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −288.000 −0.0101221
\(933\) 0 0
\(934\) 35013.1 1.22662
\(935\) −392.000 −0.0137110
\(936\) 0 0
\(937\) −11665.8 −0.406731 −0.203365 0.979103i \(-0.565188\pi\)
−0.203365 + 0.979103i \(0.565188\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −41440.0 −1.43790
\(941\) 14.1421 0.000489926 0 0.000244963 1.00000i \(-0.499922\pi\)
0.000244963 1.00000i \(0.499922\pi\)
\(942\) 0 0
\(943\) 17621.1 0.608507
\(944\) 6946.62 0.239505
\(945\) 0 0
\(946\) −952.000 −0.0327190
\(947\) −14034.0 −0.481567 −0.240783 0.970579i \(-0.577404\pi\)
−0.240783 + 0.970579i \(0.577404\pi\)
\(948\) 0 0
\(949\) −13752.0 −0.470399
\(950\) 755.190 0.0257912
\(951\) 0 0
\(952\) 0 0
\(953\) 42698.0 1.45134 0.725668 0.688045i \(-0.241531\pi\)
0.725668 + 0.688045i \(0.241531\pi\)
\(954\) 0 0
\(955\) 20353.4 0.689654
\(956\) −17232.0 −0.582974
\(957\) 0 0
\(958\) 4576.40 0.154339
\(959\) 0 0
\(960\) 0 0
\(961\) −21079.0 −0.707563
\(962\) 3869.29 0.129679
\(963\) 0 0
\(964\) 6160.31 0.205820
\(965\) −90917.0 −3.03287
\(966\) 0 0
\(967\) −48492.0 −1.61261 −0.806307 0.591497i \(-0.798537\pi\)
−0.806307 + 0.591497i \(0.798537\pi\)
\(968\) −9080.00 −0.301490
\(969\) 0 0
\(970\) 58744.0 1.94449
\(971\) 52669.6 1.74073 0.870364 0.492409i \(-0.163884\pi\)
0.870364 + 0.492409i \(0.163884\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 1944.00 0.0639525
\(975\) 0 0
\(976\) −226.274 −0.00742096
\(977\) 55380.0 1.81347 0.906737 0.421698i \(-0.138566\pi\)
0.906737 + 0.421698i \(0.138566\pi\)
\(978\) 0 0
\(979\) 8652.16 0.282456
\(980\) 0 0
\(981\) 0 0
\(982\) 14808.0 0.481204
\(983\) −50535.5 −1.63971 −0.819854 0.572573i \(-0.805945\pi\)
−0.819854 + 0.572573i \(0.805945\pi\)
\(984\) 0 0
\(985\) 15720.4 0.508521
\(986\) 808.930 0.0261274
\(987\) 0 0
\(988\) −288.000 −0.00927379
\(989\) 4760.00 0.153043
\(990\) 0 0
\(991\) −39712.0 −1.27295 −0.636475 0.771297i \(-0.719608\pi\)
−0.636475 + 0.771297i \(0.719608\pi\)
\(992\) 2986.82 0.0955964
\(993\) 0 0
\(994\) 0 0
\(995\) 49224.0 1.56835
\(996\) 0 0
\(997\) −2186.37 −0.0694515 −0.0347258 0.999397i \(-0.511056\pi\)
−0.0347258 + 0.999397i \(0.511056\pi\)
\(998\) −24488.0 −0.776708
\(999\) 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.bg.1.1 2
3.2 odd 2 98.4.a.g.1.2 yes 2
7.2 even 3 882.4.g.ba.361.2 4
7.3 odd 6 882.4.g.ba.667.1 4
7.4 even 3 882.4.g.ba.667.2 4
7.5 odd 6 882.4.g.ba.361.1 4
7.6 odd 2 inner 882.4.a.bg.1.2 2
12.11 even 2 784.4.a.y.1.1 2
15.14 odd 2 2450.4.a.bx.1.1 2
21.2 odd 6 98.4.c.h.67.1 4
21.5 even 6 98.4.c.h.67.2 4
21.11 odd 6 98.4.c.h.79.1 4
21.17 even 6 98.4.c.h.79.2 4
21.20 even 2 98.4.a.g.1.1 2
84.83 odd 2 784.4.a.y.1.2 2
105.104 even 2 2450.4.a.bx.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
98.4.a.g.1.1 2 21.20 even 2
98.4.a.g.1.2 yes 2 3.2 odd 2
98.4.c.h.67.1 4 21.2 odd 6
98.4.c.h.67.2 4 21.5 even 6
98.4.c.h.79.1 4 21.11 odd 6
98.4.c.h.79.2 4 21.17 even 6
784.4.a.y.1.1 2 12.11 even 2
784.4.a.y.1.2 2 84.83 odd 2
882.4.a.bg.1.1 2 1.1 even 1 trivial
882.4.a.bg.1.2 2 7.6 odd 2 inner
882.4.g.ba.361.1 4 7.5 odd 6
882.4.g.ba.361.2 4 7.2 even 3
882.4.g.ba.667.1 4 7.3 odd 6
882.4.g.ba.667.2 4 7.4 even 3
2450.4.a.bx.1.1 2 15.14 odd 2
2450.4.a.bx.1.2 2 105.104 even 2