Properties

Label 1900.2.c.e.1749.3
Level $1900$
Weight $2$
Character 1900.1749
Analytic conductor $15.172$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1900.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(15.1715763840\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 380)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1749.3
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1900.1749
Dual form 1900.2.c.e.1749.2

$q$-expansion

\(f(q)\) \(=\) \(q+0.732051i q^{3} +2.00000i q^{7} +2.46410 q^{9} +O(q^{10})\) \(q+0.732051i q^{3} +2.00000i q^{7} +2.46410 q^{9} +3.46410 q^{11} -0.732051i q^{13} +3.46410i q^{17} -1.00000 q^{19} -1.46410 q^{21} -3.46410i q^{23} +4.00000i q^{27} +3.46410 q^{29} +5.46410 q^{31} +2.53590i q^{33} +3.26795i q^{37} +0.535898 q^{39} -6.00000 q^{41} -8.92820i q^{43} -0.928203i q^{47} +3.00000 q^{49} -2.53590 q^{51} +7.26795i q^{53} -0.732051i q^{57} +6.92820 q^{59} -8.39230 q^{61} +4.92820i q^{63} +3.26795i q^{67} +2.53590 q^{69} -9.46410 q^{71} +7.46410i q^{73} +6.92820i q^{77} +10.9282 q^{79} +4.46410 q^{81} +3.46410i q^{83} +2.53590i q^{87} +8.53590 q^{89} +1.46410 q^{91} +4.00000i q^{93} +14.5885i q^{97} +8.53590 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{9} - 4q^{19} + 8q^{21} + 8q^{31} + 16q^{39} - 24q^{41} + 12q^{49} - 24q^{51} + 8q^{61} + 24q^{69} - 24q^{71} + 16q^{79} + 4q^{81} + 48q^{89} - 8q^{91} + 48q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\) \(951\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) 0.732051i 0.422650i 0.977416 + 0.211325i \(0.0677778\pi\)
−0.977416 + 0.211325i \(0.932222\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.00000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) 0 0
\(9\) 2.46410 0.821367
\(10\) 0 0
\(11\) 3.46410 1.04447 0.522233 0.852803i \(-0.325099\pi\)
0.522233 + 0.852803i \(0.325099\pi\)
\(12\) 0 0
\(13\) − 0.732051i − 0.203034i −0.994834 0.101517i \(-0.967630\pi\)
0.994834 0.101517i \(-0.0323697\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.46410i 0.840168i 0.907485 + 0.420084i \(0.137999\pi\)
−0.907485 + 0.420084i \(0.862001\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −1.46410 −0.319493
\(22\) 0 0
\(23\) − 3.46410i − 0.722315i −0.932505 0.361158i \(-0.882382\pi\)
0.932505 0.361158i \(-0.117618\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 4.00000i 0.769800i
\(28\) 0 0
\(29\) 3.46410 0.643268 0.321634 0.946864i \(-0.395768\pi\)
0.321634 + 0.946864i \(0.395768\pi\)
\(30\) 0 0
\(31\) 5.46410 0.981382 0.490691 0.871334i \(-0.336744\pi\)
0.490691 + 0.871334i \(0.336744\pi\)
\(32\) 0 0
\(33\) 2.53590i 0.441443i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 3.26795i 0.537248i 0.963245 + 0.268624i \(0.0865688\pi\)
−0.963245 + 0.268624i \(0.913431\pi\)
\(38\) 0 0
\(39\) 0.535898 0.0858124
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) − 8.92820i − 1.36154i −0.732498 0.680769i \(-0.761646\pi\)
0.732498 0.680769i \(-0.238354\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 0.928203i − 0.135392i −0.997706 0.0676962i \(-0.978435\pi\)
0.997706 0.0676962i \(-0.0215649\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) −2.53590 −0.355097
\(52\) 0 0
\(53\) 7.26795i 0.998330i 0.866507 + 0.499165i \(0.166360\pi\)
−0.866507 + 0.499165i \(0.833640\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 0.732051i − 0.0969625i
\(58\) 0 0
\(59\) 6.92820 0.901975 0.450988 0.892530i \(-0.351072\pi\)
0.450988 + 0.892530i \(0.351072\pi\)
\(60\) 0 0
\(61\) −8.39230 −1.07452 −0.537262 0.843415i \(-0.680541\pi\)
−0.537262 + 0.843415i \(0.680541\pi\)
\(62\) 0 0
\(63\) 4.92820i 0.620895i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 3.26795i 0.399244i 0.979873 + 0.199622i \(0.0639713\pi\)
−0.979873 + 0.199622i \(0.936029\pi\)
\(68\) 0 0
\(69\) 2.53590 0.305286
\(70\) 0 0
\(71\) −9.46410 −1.12318 −0.561591 0.827415i \(-0.689811\pi\)
−0.561591 + 0.827415i \(0.689811\pi\)
\(72\) 0 0
\(73\) 7.46410i 0.873607i 0.899557 + 0.436804i \(0.143889\pi\)
−0.899557 + 0.436804i \(0.856111\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.92820i 0.789542i
\(78\) 0 0
\(79\) 10.9282 1.22952 0.614759 0.788715i \(-0.289253\pi\)
0.614759 + 0.788715i \(0.289253\pi\)
\(80\) 0 0
\(81\) 4.46410 0.496011
\(82\) 0 0
\(83\) 3.46410i 0.380235i 0.981761 + 0.190117i \(0.0608868\pi\)
−0.981761 + 0.190117i \(0.939113\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 2.53590i 0.271877i
\(88\) 0 0
\(89\) 8.53590 0.904803 0.452402 0.891814i \(-0.350567\pi\)
0.452402 + 0.891814i \(0.350567\pi\)
\(90\) 0 0
\(91\) 1.46410 0.153480
\(92\) 0 0
\(93\) 4.00000i 0.414781i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 14.5885i 1.48123i 0.671928 + 0.740617i \(0.265467\pi\)
−0.671928 + 0.740617i \(0.734533\pi\)
\(98\) 0 0
\(99\) 8.53590 0.857890
\(100\) 0 0
\(101\) 4.39230 0.437051 0.218525 0.975831i \(-0.429875\pi\)
0.218525 + 0.975831i \(0.429875\pi\)
\(102\) 0 0
\(103\) 3.66025i 0.360656i 0.983607 + 0.180328i \(0.0577158\pi\)
−0.983607 + 0.180328i \(0.942284\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 11.6603i 1.12724i 0.826034 + 0.563620i \(0.190592\pi\)
−0.826034 + 0.563620i \(0.809408\pi\)
\(108\) 0 0
\(109\) −6.39230 −0.612272 −0.306136 0.951988i \(-0.599036\pi\)
−0.306136 + 0.951988i \(0.599036\pi\)
\(110\) 0 0
\(111\) −2.39230 −0.227068
\(112\) 0 0
\(113\) 18.5885i 1.74865i 0.485336 + 0.874327i \(0.338697\pi\)
−0.485336 + 0.874327i \(0.661303\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 1.80385i − 0.166766i
\(118\) 0 0
\(119\) −6.92820 −0.635107
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) − 4.39230i − 0.396041i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 10.5885i − 0.939574i −0.882780 0.469787i \(-0.844331\pi\)
0.882780 0.469787i \(-0.155669\pi\)
\(128\) 0 0
\(129\) 6.53590 0.575454
\(130\) 0 0
\(131\) −5.07180 −0.443125 −0.221562 0.975146i \(-0.571116\pi\)
−0.221562 + 0.975146i \(0.571116\pi\)
\(132\) 0 0
\(133\) − 2.00000i − 0.173422i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 17.3205i − 1.47979i −0.672722 0.739895i \(-0.734875\pi\)
0.672722 0.739895i \(-0.265125\pi\)
\(138\) 0 0
\(139\) −4.53590 −0.384730 −0.192365 0.981323i \(-0.561616\pi\)
−0.192365 + 0.981323i \(0.561616\pi\)
\(140\) 0 0
\(141\) 0.679492 0.0572235
\(142\) 0 0
\(143\) − 2.53590i − 0.212062i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 2.19615i 0.181136i
\(148\) 0 0
\(149\) −16.3923 −1.34291 −0.671455 0.741045i \(-0.734330\pi\)
−0.671455 + 0.741045i \(0.734330\pi\)
\(150\) 0 0
\(151\) 12.3923 1.00847 0.504236 0.863566i \(-0.331774\pi\)
0.504236 + 0.863566i \(0.331774\pi\)
\(152\) 0 0
\(153\) 8.53590i 0.690086i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 16.5359i 1.31971i 0.751394 + 0.659854i \(0.229382\pi\)
−0.751394 + 0.659854i \(0.770618\pi\)
\(158\) 0 0
\(159\) −5.32051 −0.421944
\(160\) 0 0
\(161\) 6.92820 0.546019
\(162\) 0 0
\(163\) − 20.9282i − 1.63922i −0.572919 0.819612i \(-0.694189\pi\)
0.572919 0.819612i \(-0.305811\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 7.26795i − 0.562411i −0.959648 0.281205i \(-0.909266\pi\)
0.959648 0.281205i \(-0.0907342\pi\)
\(168\) 0 0
\(169\) 12.4641 0.958777
\(170\) 0 0
\(171\) −2.46410 −0.188435
\(172\) 0 0
\(173\) − 14.1962i − 1.07931i −0.841885 0.539657i \(-0.818554\pi\)
0.841885 0.539657i \(-0.181446\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 5.07180i 0.381220i
\(178\) 0 0
\(179\) 20.7846 1.55351 0.776757 0.629800i \(-0.216863\pi\)
0.776757 + 0.629800i \(0.216863\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 0 0
\(183\) − 6.14359i − 0.454148i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 12.0000i 0.877527i
\(188\) 0 0
\(189\) −8.00000 −0.581914
\(190\) 0 0
\(191\) −6.92820 −0.501307 −0.250654 0.968077i \(-0.580646\pi\)
−0.250654 + 0.968077i \(0.580646\pi\)
\(192\) 0 0
\(193\) 18.1962i 1.30979i 0.755721 + 0.654894i \(0.227287\pi\)
−0.755721 + 0.654894i \(0.772713\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 12.9282i − 0.921096i −0.887635 0.460548i \(-0.847653\pi\)
0.887635 0.460548i \(-0.152347\pi\)
\(198\) 0 0
\(199\) −13.0718 −0.926635 −0.463318 0.886192i \(-0.653341\pi\)
−0.463318 + 0.886192i \(0.653341\pi\)
\(200\) 0 0
\(201\) −2.39230 −0.168740
\(202\) 0 0
\(203\) 6.92820i 0.486265i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 8.53590i − 0.593286i
\(208\) 0 0
\(209\) −3.46410 −0.239617
\(210\) 0 0
\(211\) −15.3205 −1.05471 −0.527354 0.849646i \(-0.676816\pi\)
−0.527354 + 0.849646i \(0.676816\pi\)
\(212\) 0 0
\(213\) − 6.92820i − 0.474713i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 10.9282i 0.741855i
\(218\) 0 0
\(219\) −5.46410 −0.369230
\(220\) 0 0
\(221\) 2.53590 0.170583
\(222\) 0 0
\(223\) − 7.66025i − 0.512969i −0.966548 0.256484i \(-0.917436\pi\)
0.966548 0.256484i \(-0.0825642\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 2.19615i − 0.145764i −0.997341 0.0728819i \(-0.976780\pi\)
0.997341 0.0728819i \(-0.0232196\pi\)
\(228\) 0 0
\(229\) −10.5359 −0.696232 −0.348116 0.937452i \(-0.613178\pi\)
−0.348116 + 0.937452i \(0.613178\pi\)
\(230\) 0 0
\(231\) −5.07180 −0.333700
\(232\) 0 0
\(233\) − 12.9282i − 0.846955i −0.905907 0.423477i \(-0.860809\pi\)
0.905907 0.423477i \(-0.139191\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 8.00000i 0.519656i
\(238\) 0 0
\(239\) −13.8564 −0.896296 −0.448148 0.893959i \(-0.647916\pi\)
−0.448148 + 0.893959i \(0.647916\pi\)
\(240\) 0 0
\(241\) 15.8564 1.02140 0.510700 0.859759i \(-0.329386\pi\)
0.510700 + 0.859759i \(0.329386\pi\)
\(242\) 0 0
\(243\) 15.2679i 0.979439i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0.732051i 0.0465793i
\(248\) 0 0
\(249\) −2.53590 −0.160706
\(250\) 0 0
\(251\) −24.0000 −1.51487 −0.757433 0.652913i \(-0.773547\pi\)
−0.757433 + 0.652913i \(0.773547\pi\)
\(252\) 0 0
\(253\) − 12.0000i − 0.754434i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 9.12436i 0.569162i 0.958652 + 0.284581i \(0.0918544\pi\)
−0.958652 + 0.284581i \(0.908146\pi\)
\(258\) 0 0
\(259\) −6.53590 −0.406121
\(260\) 0 0
\(261\) 8.53590 0.528359
\(262\) 0 0
\(263\) − 15.4641i − 0.953557i −0.879023 0.476779i \(-0.841804\pi\)
0.879023 0.476779i \(-0.158196\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 6.24871i 0.382415i
\(268\) 0 0
\(269\) 10.3923 0.633630 0.316815 0.948487i \(-0.397387\pi\)
0.316815 + 0.948487i \(0.397387\pi\)
\(270\) 0 0
\(271\) 18.3923 1.11725 0.558626 0.829419i \(-0.311329\pi\)
0.558626 + 0.829419i \(0.311329\pi\)
\(272\) 0 0
\(273\) 1.07180i 0.0648681i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 2.39230i − 0.143740i −0.997414 0.0718698i \(-0.977103\pi\)
0.997414 0.0718698i \(-0.0228966\pi\)
\(278\) 0 0
\(279\) 13.4641 0.806075
\(280\) 0 0
\(281\) −12.9282 −0.771232 −0.385616 0.922659i \(-0.626011\pi\)
−0.385616 + 0.922659i \(0.626011\pi\)
\(282\) 0 0
\(283\) 2.39230i 0.142208i 0.997469 + 0.0711039i \(0.0226522\pi\)
−0.997469 + 0.0711039i \(0.977348\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 12.0000i − 0.708338i
\(288\) 0 0
\(289\) 5.00000 0.294118
\(290\) 0 0
\(291\) −10.6795 −0.626043
\(292\) 0 0
\(293\) − 11.6603i − 0.681199i −0.940208 0.340600i \(-0.889370\pi\)
0.940208 0.340600i \(-0.110630\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 13.8564i 0.804030i
\(298\) 0 0
\(299\) −2.53590 −0.146655
\(300\) 0 0
\(301\) 17.8564 1.02923
\(302\) 0 0
\(303\) 3.21539i 0.184719i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 18.1962i − 1.03851i −0.854620 0.519255i \(-0.826210\pi\)
0.854620 0.519255i \(-0.173790\pi\)
\(308\) 0 0
\(309\) −2.67949 −0.152431
\(310\) 0 0
\(311\) −22.3923 −1.26975 −0.634876 0.772614i \(-0.718949\pi\)
−0.634876 + 0.772614i \(0.718949\pi\)
\(312\) 0 0
\(313\) 30.7846i 1.74005i 0.493008 + 0.870025i \(0.335897\pi\)
−0.493008 + 0.870025i \(0.664103\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 21.1244i − 1.18646i −0.805032 0.593231i \(-0.797852\pi\)
0.805032 0.593231i \(-0.202148\pi\)
\(318\) 0 0
\(319\) 12.0000 0.671871
\(320\) 0 0
\(321\) −8.53590 −0.476427
\(322\) 0 0
\(323\) − 3.46410i − 0.192748i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 4.67949i − 0.258776i
\(328\) 0 0
\(329\) 1.85641 0.102347
\(330\) 0 0
\(331\) 26.2487 1.44276 0.721380 0.692540i \(-0.243508\pi\)
0.721380 + 0.692540i \(0.243508\pi\)
\(332\) 0 0
\(333\) 8.05256i 0.441278i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 20.7321i − 1.12935i −0.825314 0.564673i \(-0.809002\pi\)
0.825314 0.564673i \(-0.190998\pi\)
\(338\) 0 0
\(339\) −13.6077 −0.739069
\(340\) 0 0
\(341\) 18.9282 1.02502
\(342\) 0 0
\(343\) 20.0000i 1.07990i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 31.1769i − 1.67366i −0.547459 0.836832i \(-0.684405\pi\)
0.547459 0.836832i \(-0.315595\pi\)
\(348\) 0 0
\(349\) −7.07180 −0.378545 −0.189272 0.981925i \(-0.560613\pi\)
−0.189272 + 0.981925i \(0.560613\pi\)
\(350\) 0 0
\(351\) 2.92820 0.156296
\(352\) 0 0
\(353\) − 22.3923i − 1.19182i −0.803050 0.595911i \(-0.796791\pi\)
0.803050 0.595911i \(-0.203209\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 5.07180i − 0.268428i
\(358\) 0 0
\(359\) 15.4641 0.816164 0.408082 0.912945i \(-0.366198\pi\)
0.408082 + 0.912945i \(0.366198\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 0.732051i 0.0384227i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 23.8564i − 1.24529i −0.782503 0.622647i \(-0.786057\pi\)
0.782503 0.622647i \(-0.213943\pi\)
\(368\) 0 0
\(369\) −14.7846 −0.769656
\(370\) 0 0
\(371\) −14.5359 −0.754666
\(372\) 0 0
\(373\) − 14.5885i − 0.755362i −0.925936 0.377681i \(-0.876722\pi\)
0.925936 0.377681i \(-0.123278\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 2.53590i − 0.130605i
\(378\) 0 0
\(379\) −1.07180 −0.0550545 −0.0275273 0.999621i \(-0.508763\pi\)
−0.0275273 + 0.999621i \(0.508763\pi\)
\(380\) 0 0
\(381\) 7.75129 0.397111
\(382\) 0 0
\(383\) − 23.6603i − 1.20898i −0.796612 0.604491i \(-0.793376\pi\)
0.796612 0.604491i \(-0.206624\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 22.0000i − 1.11832i
\(388\) 0 0
\(389\) −18.0000 −0.912636 −0.456318 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) 0 0
\(391\) 12.0000 0.606866
\(392\) 0 0
\(393\) − 3.71281i − 0.187287i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 12.5359i − 0.629159i −0.949231 0.314579i \(-0.898137\pi\)
0.949231 0.314579i \(-0.101863\pi\)
\(398\) 0 0
\(399\) 1.46410 0.0732968
\(400\) 0 0
\(401\) −2.78461 −0.139057 −0.0695284 0.997580i \(-0.522149\pi\)
−0.0695284 + 0.997580i \(0.522149\pi\)
\(402\) 0 0
\(403\) − 4.00000i − 0.199254i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 11.3205i 0.561137i
\(408\) 0 0
\(409\) −6.39230 −0.316079 −0.158040 0.987433i \(-0.550517\pi\)
−0.158040 + 0.987433i \(0.550517\pi\)
\(410\) 0 0
\(411\) 12.6795 0.625433
\(412\) 0 0
\(413\) 13.8564i 0.681829i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 3.32051i − 0.162606i
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) −34.0000 −1.65706 −0.828529 0.559946i \(-0.810822\pi\)
−0.828529 + 0.559946i \(0.810822\pi\)
\(422\) 0 0
\(423\) − 2.28719i − 0.111207i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 16.7846i − 0.812264i
\(428\) 0 0
\(429\) 1.85641 0.0896281
\(430\) 0 0
\(431\) −26.5359 −1.27819 −0.639095 0.769128i \(-0.720691\pi\)
−0.639095 + 0.769128i \(0.720691\pi\)
\(432\) 0 0
\(433\) 39.6603i 1.90595i 0.303049 + 0.952975i \(0.401996\pi\)
−0.303049 + 0.952975i \(0.598004\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3.46410i 0.165710i
\(438\) 0 0
\(439\) −23.7128 −1.13175 −0.565875 0.824491i \(-0.691462\pi\)
−0.565875 + 0.824491i \(0.691462\pi\)
\(440\) 0 0
\(441\) 7.39230 0.352015
\(442\) 0 0
\(443\) − 34.3923i − 1.63403i −0.576618 0.817014i \(-0.695628\pi\)
0.576618 0.817014i \(-0.304372\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 12.0000i − 0.567581i
\(448\) 0 0
\(449\) −3.46410 −0.163481 −0.0817405 0.996654i \(-0.526048\pi\)
−0.0817405 + 0.996654i \(0.526048\pi\)
\(450\) 0 0
\(451\) −20.7846 −0.978709
\(452\) 0 0
\(453\) 9.07180i 0.426230i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 6.78461i − 0.317371i −0.987329 0.158685i \(-0.949274\pi\)
0.987329 0.158685i \(-0.0507255\pi\)
\(458\) 0 0
\(459\) −13.8564 −0.646762
\(460\) 0 0
\(461\) 33.7128 1.57016 0.785081 0.619393i \(-0.212621\pi\)
0.785081 + 0.619393i \(0.212621\pi\)
\(462\) 0 0
\(463\) − 11.4641i − 0.532782i −0.963865 0.266391i \(-0.914169\pi\)
0.963865 0.266391i \(-0.0858312\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 5.32051i 0.246204i 0.992394 + 0.123102i \(0.0392842\pi\)
−0.992394 + 0.123102i \(0.960716\pi\)
\(468\) 0 0
\(469\) −6.53590 −0.301800
\(470\) 0 0
\(471\) −12.1051 −0.557774
\(472\) 0 0
\(473\) − 30.9282i − 1.42208i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 17.9090i 0.819995i
\(478\) 0 0
\(479\) 1.60770 0.0734575 0.0367287 0.999325i \(-0.488306\pi\)
0.0367287 + 0.999325i \(0.488306\pi\)
\(480\) 0 0
\(481\) 2.39230 0.109080
\(482\) 0 0
\(483\) 5.07180i 0.230775i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 1.80385i − 0.0817401i −0.999164 0.0408701i \(-0.986987\pi\)
0.999164 0.0408701i \(-0.0130130\pi\)
\(488\) 0 0
\(489\) 15.3205 0.692817
\(490\) 0 0
\(491\) 5.07180 0.228887 0.114443 0.993430i \(-0.463492\pi\)
0.114443 + 0.993430i \(0.463492\pi\)
\(492\) 0 0
\(493\) 12.0000i 0.540453i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 18.9282i − 0.849046i
\(498\) 0 0
\(499\) −16.5359 −0.740248 −0.370124 0.928982i \(-0.620685\pi\)
−0.370124 + 0.928982i \(0.620685\pi\)
\(500\) 0 0
\(501\) 5.32051 0.237703
\(502\) 0 0
\(503\) − 8.53590i − 0.380597i −0.981726 0.190298i \(-0.939054\pi\)
0.981726 0.190298i \(-0.0609456\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 9.12436i 0.405227i
\(508\) 0 0
\(509\) −29.3205 −1.29961 −0.649804 0.760102i \(-0.725149\pi\)
−0.649804 + 0.760102i \(0.725149\pi\)
\(510\) 0 0
\(511\) −14.9282 −0.660385
\(512\) 0 0
\(513\) − 4.00000i − 0.176604i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 3.21539i − 0.141413i
\(518\) 0 0
\(519\) 10.3923 0.456172
\(520\) 0 0
\(521\) 26.7846 1.17346 0.586728 0.809784i \(-0.300416\pi\)
0.586728 + 0.809784i \(0.300416\pi\)
\(522\) 0 0
\(523\) − 35.3731i − 1.54676i −0.633945 0.773378i \(-0.718565\pi\)
0.633945 0.773378i \(-0.281435\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 18.9282i 0.824525i
\(528\) 0 0
\(529\) 11.0000 0.478261
\(530\) 0 0
\(531\) 17.0718 0.740853
\(532\) 0 0
\(533\) 4.39230i 0.190252i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 15.2154i 0.656593i
\(538\) 0 0
\(539\) 10.3923 0.447628
\(540\) 0 0
\(541\) 33.1769 1.42639 0.713193 0.700967i \(-0.247248\pi\)
0.713193 + 0.700967i \(0.247248\pi\)
\(542\) 0 0
\(543\) 10.2487i 0.439814i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 3.26795i 0.139727i 0.997557 + 0.0698637i \(0.0222564\pi\)
−0.997557 + 0.0698637i \(0.977744\pi\)
\(548\) 0 0
\(549\) −20.6795 −0.882579
\(550\) 0 0
\(551\) −3.46410 −0.147576
\(552\) 0 0
\(553\) 21.8564i 0.929429i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 29.3205i − 1.24235i −0.783672 0.621175i \(-0.786656\pi\)
0.783672 0.621175i \(-0.213344\pi\)
\(558\) 0 0
\(559\) −6.53590 −0.276439
\(560\) 0 0
\(561\) −8.78461 −0.370887
\(562\) 0 0
\(563\) − 33.8038i − 1.42466i −0.701844 0.712331i \(-0.747639\pi\)
0.701844 0.712331i \(-0.252361\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 8.92820i 0.374949i
\(568\) 0 0
\(569\) −5.32051 −0.223047 −0.111524 0.993762i \(-0.535573\pi\)
−0.111524 + 0.993762i \(0.535573\pi\)
\(570\) 0 0
\(571\) −2.39230 −0.100115 −0.0500574 0.998746i \(-0.515940\pi\)
−0.0500574 + 0.998746i \(0.515940\pi\)
\(572\) 0 0
\(573\) − 5.07180i − 0.211877i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 34.7846i 1.44810i 0.689746 + 0.724051i \(0.257722\pi\)
−0.689746 + 0.724051i \(0.742278\pi\)
\(578\) 0 0
\(579\) −13.3205 −0.553581
\(580\) 0 0
\(581\) −6.92820 −0.287430
\(582\) 0 0
\(583\) 25.1769i 1.04272i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 19.1769i − 0.791516i −0.918355 0.395758i \(-0.870482\pi\)
0.918355 0.395758i \(-0.129518\pi\)
\(588\) 0 0
\(589\) −5.46410 −0.225144
\(590\) 0 0
\(591\) 9.46410 0.389301
\(592\) 0 0
\(593\) − 4.14359i − 0.170157i −0.996374 0.0850785i \(-0.972886\pi\)
0.996374 0.0850785i \(-0.0271141\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 9.56922i − 0.391642i
\(598\) 0 0
\(599\) −20.7846 −0.849236 −0.424618 0.905373i \(-0.639592\pi\)
−0.424618 + 0.905373i \(0.639592\pi\)
\(600\) 0 0
\(601\) −44.6410 −1.82095 −0.910473 0.413570i \(-0.864282\pi\)
−0.910473 + 0.413570i \(0.864282\pi\)
\(602\) 0 0
\(603\) 8.05256i 0.327926i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 25.9090i 1.05161i 0.850604 + 0.525806i \(0.176236\pi\)
−0.850604 + 0.525806i \(0.823764\pi\)
\(608\) 0 0
\(609\) −5.07180 −0.205520
\(610\) 0 0
\(611\) −0.679492 −0.0274893
\(612\) 0 0
\(613\) 14.3923i 0.581300i 0.956829 + 0.290650i \(0.0938715\pi\)
−0.956829 + 0.290650i \(0.906129\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 22.3923i 0.901480i 0.892655 + 0.450740i \(0.148840\pi\)
−0.892655 + 0.450740i \(0.851160\pi\)
\(618\) 0 0
\(619\) −18.3923 −0.739249 −0.369625 0.929181i \(-0.620514\pi\)
−0.369625 + 0.929181i \(0.620514\pi\)
\(620\) 0 0
\(621\) 13.8564 0.556038
\(622\) 0 0
\(623\) 17.0718i 0.683967i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 2.53590i − 0.101274i
\(628\) 0 0
\(629\) −11.3205 −0.451378
\(630\) 0 0
\(631\) 4.53590 0.180571 0.0902856 0.995916i \(-0.471222\pi\)
0.0902856 + 0.995916i \(0.471222\pi\)
\(632\) 0 0
\(633\) − 11.2154i − 0.445772i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 2.19615i − 0.0870147i
\(638\) 0 0
\(639\) −23.3205 −0.922545
\(640\) 0 0
\(641\) −11.0718 −0.437310 −0.218655 0.975802i \(-0.570167\pi\)
−0.218655 + 0.975802i \(0.570167\pi\)
\(642\) 0 0
\(643\) − 6.39230i − 0.252088i −0.992025 0.126044i \(-0.959772\pi\)
0.992025 0.126044i \(-0.0402280\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 8.53590i 0.335581i 0.985823 + 0.167790i \(0.0536632\pi\)
−0.985823 + 0.167790i \(0.946337\pi\)
\(648\) 0 0
\(649\) 24.0000 0.942082
\(650\) 0 0
\(651\) −8.00000 −0.313545
\(652\) 0 0
\(653\) − 20.5359i − 0.803632i −0.915720 0.401816i \(-0.868379\pi\)
0.915720 0.401816i \(-0.131621\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 18.3923i 0.717552i
\(658\) 0 0
\(659\) 32.7846 1.27711 0.638554 0.769577i \(-0.279533\pi\)
0.638554 + 0.769577i \(0.279533\pi\)
\(660\) 0 0
\(661\) 46.7846 1.81971 0.909855 0.414926i \(-0.136193\pi\)
0.909855 + 0.414926i \(0.136193\pi\)
\(662\) 0 0
\(663\) 1.85641i 0.0720969i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 12.0000i − 0.464642i
\(668\) 0 0
\(669\) 5.60770 0.216806
\(670\) 0 0
\(671\) −29.0718 −1.12230
\(672\) 0 0
\(673\) 47.2679i 1.82205i 0.412356 + 0.911023i \(0.364706\pi\)
−0.412356 + 0.911023i \(0.635294\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 34.9808i − 1.34442i −0.740361 0.672210i \(-0.765345\pi\)
0.740361 0.672210i \(-0.234655\pi\)
\(678\) 0 0
\(679\) −29.1769 −1.11971
\(680\) 0 0
\(681\) 1.60770 0.0616070
\(682\) 0 0
\(683\) 0.339746i 0.0130000i 0.999979 + 0.00650001i \(0.00206903\pi\)
−0.999979 + 0.00650001i \(0.997931\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 7.71281i − 0.294262i
\(688\) 0 0
\(689\) 5.32051 0.202695
\(690\) 0 0
\(691\) −11.1769 −0.425190 −0.212595 0.977140i \(-0.568191\pi\)
−0.212595 + 0.977140i \(0.568191\pi\)
\(692\) 0 0
\(693\) 17.0718i 0.648504i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 20.7846i − 0.787273i
\(698\) 0 0
\(699\) 9.46410 0.357965
\(700\) 0 0
\(701\) −33.4641 −1.26392 −0.631961 0.775000i \(-0.717750\pi\)
−0.631961 + 0.775000i \(0.717750\pi\)
\(702\) 0 0
\(703\) − 3.26795i − 0.123253i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 8.78461i 0.330379i
\(708\) 0 0
\(709\) −10.7846 −0.405025 −0.202512 0.979280i \(-0.564911\pi\)
−0.202512 + 0.979280i \(0.564911\pi\)
\(710\) 0 0
\(711\) 26.9282 1.00989
\(712\) 0 0
\(713\) − 18.9282i − 0.708867i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 10.1436i − 0.378819i
\(718\) 0 0
\(719\) 17.3205 0.645946 0.322973 0.946408i \(-0.395318\pi\)
0.322973 + 0.946408i \(0.395318\pi\)
\(720\) 0 0
\(721\) −7.32051 −0.272630
\(722\) 0 0
\(723\) 11.6077i 0.431695i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 27.8564i 1.03314i 0.856246 + 0.516568i \(0.172791\pi\)
−0.856246 + 0.516568i \(0.827209\pi\)
\(728\) 0 0
\(729\) 2.21539 0.0820515
\(730\) 0 0
\(731\) 30.9282 1.14392
\(732\) 0 0
\(733\) 30.7846i 1.13706i 0.822664 + 0.568528i \(0.192487\pi\)
−0.822664 + 0.568528i \(0.807513\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 11.3205i 0.416996i
\(738\) 0 0
\(739\) 28.0000 1.03000 0.514998 0.857191i \(-0.327793\pi\)
0.514998 + 0.857191i \(0.327793\pi\)
\(740\) 0 0
\(741\) −0.535898 −0.0196867
\(742\) 0 0
\(743\) − 45.1244i − 1.65545i −0.561132 0.827726i \(-0.689634\pi\)
0.561132 0.827726i \(-0.310366\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 8.53590i 0.312312i
\(748\) 0 0
\(749\) −23.3205 −0.852113
\(750\) 0 0
\(751\) −18.5359 −0.676385 −0.338192 0.941077i \(-0.609815\pi\)
−0.338192 + 0.941077i \(0.609815\pi\)
\(752\) 0 0
\(753\) − 17.5692i − 0.640258i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 10.0000i − 0.363456i −0.983349 0.181728i \(-0.941831\pi\)
0.983349 0.181728i \(-0.0581691\pi\)
\(758\) 0 0
\(759\) 8.78461 0.318861
\(760\) 0 0
\(761\) 40.3923 1.46422 0.732110 0.681186i \(-0.238536\pi\)
0.732110 + 0.681186i \(0.238536\pi\)
\(762\) 0 0
\(763\) − 12.7846i − 0.462834i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 5.07180i − 0.183132i
\(768\) 0 0
\(769\) 37.4641 1.35099 0.675495 0.737365i \(-0.263930\pi\)
0.675495 + 0.737365i \(0.263930\pi\)
\(770\) 0 0
\(771\) −6.67949 −0.240556
\(772\) 0 0
\(773\) 16.0526i 0.577370i 0.957424 + 0.288685i \(0.0932181\pi\)
−0.957424 + 0.288685i \(0.906782\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 4.78461i − 0.171647i
\(778\) 0 0
\(779\) 6.00000 0.214972
\(780\) 0 0
\(781\) −32.7846 −1.17313
\(782\) 0 0
\(783\) 13.8564i 0.495188i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 1.80385i − 0.0643002i −0.999483 0.0321501i \(-0.989765\pi\)
0.999483 0.0321501i \(-0.0102355\pi\)
\(788\) 0 0
\(789\) 11.3205 0.403021
\(790\) 0 0
\(791\) −37.1769 −1.32186
\(792\) 0 0
\(793\) 6.14359i 0.218165i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 11.6603i 0.413027i 0.978444 + 0.206514i \(0.0662118\pi\)
−0.978444 + 0.206514i \(0.933788\pi\)
\(798\) 0 0
\(799\) 3.21539 0.113752
\(800\) 0 0
\(801\) 21.0333 0.743176
\(802\) 0 0
\(803\) 25.8564i 0.912453i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 7.60770i 0.267804i
\(808\) 0 0
\(809\) 45.7128 1.60718 0.803588 0.595185i \(-0.202921\pi\)
0.803588 + 0.595185i \(0.202921\pi\)
\(810\) 0 0
\(811\) 36.3923 1.27791 0.638953 0.769246i \(-0.279368\pi\)
0.638953 + 0.769246i \(0.279368\pi\)
\(812\) 0 0
\(813\) 13.4641i 0.472207i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 8.92820i 0.312358i
\(818\) 0 0
\(819\) 3.60770 0.126063
\(820\) 0 0
\(821\) −43.8564 −1.53060 −0.765300 0.643674i \(-0.777409\pi\)
−0.765300 + 0.643674i \(0.777409\pi\)
\(822\) 0 0
\(823\) − 43.5692i − 1.51873i −0.650666 0.759364i \(-0.725510\pi\)
0.650666 0.759364i \(-0.274490\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 12.3397i − 0.429095i −0.976714 0.214548i \(-0.931172\pi\)
0.976714 0.214548i \(-0.0688277\pi\)
\(828\) 0 0
\(829\) 40.2487 1.39790 0.698948 0.715173i \(-0.253652\pi\)
0.698948 + 0.715173i \(0.253652\pi\)
\(830\) 0 0
\(831\) 1.75129 0.0607515
\(832\) 0 0
\(833\) 10.3923i 0.360072i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 21.8564i 0.755468i
\(838\) 0 0
\(839\) −5.07180 −0.175098 −0.0875489 0.996160i \(-0.527903\pi\)
−0.0875489 + 0.996160i \(0.527903\pi\)
\(840\) 0 0
\(841\) −17.0000 −0.586207
\(842\) 0 0
\(843\) − 9.46410i − 0.325961i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 2.00000i 0.0687208i
\(848\) 0 0
\(849\) −1.75129 −0.0601041
\(850\) 0 0
\(851\) 11.3205 0.388062
\(852\) 0 0
\(853\) − 24.6410i − 0.843692i −0.906667 0.421846i \(-0.861382\pi\)
0.906667 0.421846i \(-0.138618\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 14.1962i 0.484931i 0.970160 + 0.242466i \(0.0779562\pi\)
−0.970160 + 0.242466i \(0.922044\pi\)
\(858\) 0 0
\(859\) 7.71281 0.263158 0.131579 0.991306i \(-0.457995\pi\)
0.131579 + 0.991306i \(0.457995\pi\)
\(860\) 0 0
\(861\) 8.78461 0.299379
\(862\) 0 0
\(863\) 40.7321i 1.38654i 0.720680 + 0.693268i \(0.243830\pi\)
−0.720680 + 0.693268i \(0.756170\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 3.66025i 0.124309i
\(868\) 0 0
\(869\) 37.8564 1.28419
\(870\) 0 0
\(871\) 2.39230 0.0810602
\(872\) 0 0
\(873\) 35.9474i 1.21664i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 50.9808i − 1.72150i −0.509030 0.860749i \(-0.669996\pi\)
0.509030 0.860749i \(-0.330004\pi\)
\(878\) 0 0
\(879\) 8.53590 0.287909
\(880\) 0 0
\(881\) −35.3205 −1.18998 −0.594989 0.803734i \(-0.702844\pi\)
−0.594989 + 0.803734i \(0.702844\pi\)
\(882\) 0 0
\(883\) 22.0000i 0.740359i 0.928960 + 0.370179i \(0.120704\pi\)
−0.928960 + 0.370179i \(0.879296\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 54.5885i 1.83290i 0.400148 + 0.916451i \(0.368959\pi\)
−0.400148 + 0.916451i \(0.631041\pi\)
\(888\) 0 0
\(889\) 21.1769 0.710251
\(890\) 0 0
\(891\) 15.4641 0.518067
\(892\) 0 0
\(893\) 0.928203i 0.0310611i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 1.85641i − 0.0619836i
\(898\) 0 0
\(899\) 18.9282 0.631291
\(900\) 0 0
\(901\) −25.1769 −0.838765
\(902\) 0 0
\(903\) 13.0718i 0.435002i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 38.5885i 1.28131i 0.767829 + 0.640654i \(0.221337\pi\)
−0.767829 + 0.640654i \(0.778663\pi\)
\(908\) 0 0
\(909\) 10.8231 0.358979
\(910\) 0 0
\(911\) 21.4641 0.711137 0.355569 0.934650i \(-0.384287\pi\)
0.355569 + 0.934650i \(0.384287\pi\)
\(912\) 0 0
\(913\) 12.0000i 0.397142i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 10.1436i − 0.334971i
\(918\) 0 0
\(919\) 52.4974 1.73173 0.865865 0.500278i \(-0.166769\pi\)
0.865865 + 0.500278i \(0.166769\pi\)
\(920\) 0 0
\(921\) 13.3205 0.438926
\(922\) 0 0
\(923\) 6.92820i 0.228045i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 9.01924i 0.296231i
\(928\) 0 0
\(929\) −35.5692 −1.16699 −0.583494 0.812117i \(-0.698315\pi\)
−0.583494 + 0.812117i \(0.698315\pi\)
\(930\) 0 0
\(931\) −3.00000 −0.0983210
\(932\) 0 0
\(933\) − 16.3923i − 0.536660i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 11.1769i − 0.365134i −0.983193 0.182567i \(-0.941559\pi\)
0.983193 0.182567i \(-0.0584406\pi\)
\(938\) 0 0
\(939\) −22.5359 −0.735431
\(940\) 0 0
\(941\) −16.6410 −0.542482 −0.271241 0.962512i \(-0.587434\pi\)
−0.271241 + 0.962512i \(0.587434\pi\)
\(942\) 0 0
\(943\) 20.7846i 0.676840i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 41.3205i 1.34274i 0.741124 + 0.671368i \(0.234293\pi\)
−0.741124 + 0.671368i \(0.765707\pi\)
\(948\) 0 0
\(949\) 5.46410 0.177372
\(950\) 0 0
\(951\) 15.4641 0.501458
\(952\) 0 0
\(953\) − 29.4115i − 0.952733i −0.879247 0.476367i \(-0.841953\pi\)
0.879247 0.476367i \(-0.158047\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 8.78461i 0.283966i
\(958\) 0 0
\(959\) 34.6410 1.11862
\(960\) 0 0
\(961\) −1.14359 −0.0368901
\(962\) 0 0
\(963\) 28.7321i 0.925877i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 10.6795i − 0.343429i −0.985147 0.171715i \(-0.945069\pi\)
0.985147 0.171715i \(-0.0549307\pi\)
\(968\) 0 0
\(969\) 2.53590 0.0814648
\(970\) 0 0
\(971\) −21.4641 −0.688816 −0.344408 0.938820i \(-0.611920\pi\)
−0.344408 + 0.938820i \(0.611920\pi\)
\(972\) 0 0
\(973\) − 9.07180i − 0.290828i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 21.8038i 0.697567i 0.937203 + 0.348783i \(0.113405\pi\)
−0.937203 + 0.348783i \(0.886595\pi\)
\(978\) 0 0
\(979\) 29.5692 0.945036
\(980\) 0 0
\(981\) −15.7513 −0.502900
\(982\) 0 0
\(983\) − 56.4449i − 1.80031i −0.435568 0.900156i \(-0.643452\pi\)
0.435568 0.900156i \(-0.356548\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 1.35898i 0.0432569i
\(988\) 0 0
\(989\) −30.9282 −0.983460
\(990\) 0 0
\(991\) 12.3923 0.393655 0.196827 0.980438i \(-0.436936\pi\)
0.196827 + 0.980438i \(0.436936\pi\)
\(992\) 0 0
\(993\) 19.2154i 0.609782i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 9.60770i 0.304279i 0.988359 + 0.152139i \(0.0486163\pi\)
−0.988359 + 0.152139i \(0.951384\pi\)
\(998\) 0 0
\(999\) −13.0718 −0.413573
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 1900.2.c.e.1749.3 4
5.2 odd 4 1900.2.a.d.1.2 2
5.3 odd 4 380.2.a.d.1.1 2
5.4 even 2 inner 1900.2.c.e.1749.2 4
15.8 even 4 3420.2.a.h.1.1 2
20.3 even 4 1520.2.a.l.1.2 2
20.7 even 4 7600.2.a.bf.1.1 2
40.3 even 4 6080.2.a.bj.1.1 2
40.13 odd 4 6080.2.a.z.1.2 2
95.18 even 4 7220.2.a.h.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.a.d.1.1 2 5.3 odd 4
1520.2.a.l.1.2 2 20.3 even 4
1900.2.a.d.1.2 2 5.2 odd 4
1900.2.c.e.1749.2 4 5.4 even 2 inner
1900.2.c.e.1749.3 4 1.1 even 1 trivial
3420.2.a.h.1.1 2 15.8 even 4
6080.2.a.z.1.2 2 40.13 odd 4
6080.2.a.bj.1.1 2 40.3 even 4
7220.2.a.h.1.2 2 95.18 even 4
7600.2.a.bf.1.1 2 20.7 even 4