Properties

Label 1600.2.d.b.801.1
Level $1600$
Weight $2$
Character 1600.801
Analytic conductor $12.776$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.7760643234\)
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: \( 2^{3} \)
Twist minimal: no (minimal twist has level 320)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 801.1
Root \(-0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1600.801
Dual form 1600.2.d.b.801.4

$q$-expansion

\(f(q)\) \(=\) \(q-2.73205i q^{3} -4.73205 q^{7} -4.46410 q^{9} +O(q^{10})\) \(q-2.73205i q^{3} -4.73205 q^{7} -4.46410 q^{9} +3.46410i q^{11} +3.46410i q^{13} +3.46410 q^{17} -2.00000i q^{19} +12.9282i q^{21} +2.19615 q^{23} +4.00000i q^{27} -2.53590 q^{31} +9.46410 q^{33} +6.00000i q^{37} +9.46410 q^{39} +9.46410 q^{41} +0.196152i q^{43} -2.19615 q^{47} +15.3923 q^{49} -9.46410i q^{51} +10.3923i q^{53} -5.46410 q^{57} +6.00000i q^{59} +0.928203i q^{61} +21.1244 q^{63} +0.196152i q^{67} -6.00000i q^{69} +16.3923 q^{71} -6.39230 q^{73} -16.3923i q^{77} -12.0000 q^{79} -2.46410 q^{81} -1.26795i q^{83} -12.9282 q^{89} -16.3923i q^{91} +6.92820i q^{93} -14.3923 q^{97} -15.4641i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 12q^{7} - 4q^{9} + O(q^{10}) \) \( 4q - 12q^{7} - 4q^{9} - 12q^{23} - 24q^{31} + 24q^{33} + 24q^{39} + 24q^{41} + 12q^{47} + 20q^{49} - 8q^{57} + 36q^{63} + 24q^{71} + 16q^{73} - 48q^{79} + 4q^{81} - 24q^{89} - 16q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1151\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) − 2.73205i − 1.57735i −0.614810 0.788675i \(-0.710767\pi\)
0.614810 0.788675i \(-0.289233\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −4.73205 −1.78855 −0.894274 0.447521i \(-0.852307\pi\)
−0.894274 + 0.447521i \(0.852307\pi\)
\(8\) 0 0
\(9\) −4.46410 −1.48803
\(10\) 0 0
\(11\) 3.46410i 1.04447i 0.852803 + 0.522233i \(0.174901\pi\)
−0.852803 + 0.522233i \(0.825099\pi\)
\(12\) 0 0
\(13\) 3.46410i 0.960769i 0.877058 + 0.480384i \(0.159503\pi\)
−0.877058 + 0.480384i \(0.840497\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.46410 0.840168 0.420084 0.907485i \(-0.362001\pi\)
0.420084 + 0.907485i \(0.362001\pi\)
\(18\) 0 0
\(19\) − 2.00000i − 0.458831i −0.973329 0.229416i \(-0.926318\pi\)
0.973329 0.229416i \(-0.0736815\pi\)
\(20\) 0 0
\(21\) 12.9282i 2.82117i
\(22\) 0 0
\(23\) 2.19615 0.457929 0.228965 0.973435i \(-0.426466\pi\)
0.228965 + 0.973435i \(0.426466\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 4.00000i 0.769800i
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −2.53590 −0.455461 −0.227730 0.973724i \(-0.573130\pi\)
−0.227730 + 0.973724i \(0.573130\pi\)
\(32\) 0 0
\(33\) 9.46410 1.64749
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 6.00000i 0.986394i 0.869918 + 0.493197i \(0.164172\pi\)
−0.869918 + 0.493197i \(0.835828\pi\)
\(38\) 0 0
\(39\) 9.46410 1.51547
\(40\) 0 0
\(41\) 9.46410 1.47804 0.739022 0.673681i \(-0.235288\pi\)
0.739022 + 0.673681i \(0.235288\pi\)
\(42\) 0 0
\(43\) 0.196152i 0.0299130i 0.999888 + 0.0149565i \(0.00476097\pi\)
−0.999888 + 0.0149565i \(0.995239\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.19615 −0.320342 −0.160171 0.987089i \(-0.551205\pi\)
−0.160171 + 0.987089i \(0.551205\pi\)
\(48\) 0 0
\(49\) 15.3923 2.19890
\(50\) 0 0
\(51\) − 9.46410i − 1.32524i
\(52\) 0 0
\(53\) 10.3923i 1.42749i 0.700404 + 0.713746i \(0.253003\pi\)
−0.700404 + 0.713746i \(0.746997\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −5.46410 −0.723738
\(58\) 0 0
\(59\) 6.00000i 0.781133i 0.920575 + 0.390567i \(0.127721\pi\)
−0.920575 + 0.390567i \(0.872279\pi\)
\(60\) 0 0
\(61\) 0.928203i 0.118844i 0.998233 + 0.0594221i \(0.0189258\pi\)
−0.998233 + 0.0594221i \(0.981074\pi\)
\(62\) 0 0
\(63\) 21.1244 2.66142
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0.196152i 0.0239638i 0.999928 + 0.0119819i \(0.00381405\pi\)
−0.999928 + 0.0119819i \(0.996186\pi\)
\(68\) 0 0
\(69\) − 6.00000i − 0.722315i
\(70\) 0 0
\(71\) 16.3923 1.94541 0.972704 0.232048i \(-0.0745426\pi\)
0.972704 + 0.232048i \(0.0745426\pi\)
\(72\) 0 0
\(73\) −6.39230 −0.748163 −0.374081 0.927396i \(-0.622042\pi\)
−0.374081 + 0.927396i \(0.622042\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 16.3923i − 1.86808i
\(78\) 0 0
\(79\) −12.0000 −1.35011 −0.675053 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) 0 0
\(81\) −2.46410 −0.273789
\(82\) 0 0
\(83\) − 1.26795i − 0.139176i −0.997576 0.0695878i \(-0.977832\pi\)
0.997576 0.0695878i \(-0.0221684\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −12.9282 −1.37039 −0.685193 0.728361i \(-0.740282\pi\)
−0.685193 + 0.728361i \(0.740282\pi\)
\(90\) 0 0
\(91\) − 16.3923i − 1.71838i
\(92\) 0 0
\(93\) 6.92820i 0.718421i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −14.3923 −1.46132 −0.730659 0.682743i \(-0.760787\pi\)
−0.730659 + 0.682743i \(0.760787\pi\)
\(98\) 0 0
\(99\) − 15.4641i − 1.55420i
\(100\) 0 0
\(101\) − 12.0000i − 1.19404i −0.802225 0.597022i \(-0.796350\pi\)
0.802225 0.597022i \(-0.203650\pi\)
\(102\) 0 0
\(103\) 2.19615 0.216393 0.108197 0.994130i \(-0.465492\pi\)
0.108197 + 0.994130i \(0.465492\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 13.2679i 1.28266i 0.767265 + 0.641331i \(0.221617\pi\)
−0.767265 + 0.641331i \(0.778383\pi\)
\(108\) 0 0
\(109\) 12.9282i 1.23830i 0.785274 + 0.619149i \(0.212522\pi\)
−0.785274 + 0.619149i \(0.787478\pi\)
\(110\) 0 0
\(111\) 16.3923 1.55589
\(112\) 0 0
\(113\) 12.9282 1.21618 0.608092 0.793867i \(-0.291935\pi\)
0.608092 + 0.793867i \(0.291935\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 15.4641i − 1.42966i
\(118\) 0 0
\(119\) −16.3923 −1.50268
\(120\) 0 0
\(121\) −1.00000 −0.0909091
\(122\) 0 0
\(123\) − 25.8564i − 2.33139i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 14.1962 1.25970 0.629852 0.776715i \(-0.283115\pi\)
0.629852 + 0.776715i \(0.283115\pi\)
\(128\) 0 0
\(129\) 0.535898 0.0471832
\(130\) 0 0
\(131\) 10.3923i 0.907980i 0.891007 + 0.453990i \(0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(132\) 0 0
\(133\) 9.46410i 0.820642i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0.928203 0.0793018 0.0396509 0.999214i \(-0.487375\pi\)
0.0396509 + 0.999214i \(0.487375\pi\)
\(138\) 0 0
\(139\) 10.0000i 0.848189i 0.905618 + 0.424094i \(0.139408\pi\)
−0.905618 + 0.424094i \(0.860592\pi\)
\(140\) 0 0
\(141\) 6.00000i 0.505291i
\(142\) 0 0
\(143\) −12.0000 −1.00349
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 42.0526i − 3.46844i
\(148\) 0 0
\(149\) 18.0000i 1.47462i 0.675556 + 0.737309i \(0.263904\pi\)
−0.675556 + 0.737309i \(0.736096\pi\)
\(150\) 0 0
\(151\) 9.46410 0.770178 0.385089 0.922880i \(-0.374171\pi\)
0.385089 + 0.922880i \(0.374171\pi\)
\(152\) 0 0
\(153\) −15.4641 −1.25020
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 12.9282i 1.03178i 0.856654 + 0.515891i \(0.172539\pi\)
−0.856654 + 0.515891i \(0.827461\pi\)
\(158\) 0 0
\(159\) 28.3923 2.25166
\(160\) 0 0
\(161\) −10.3923 −0.819028
\(162\) 0 0
\(163\) 16.1962i 1.26858i 0.773095 + 0.634290i \(0.218708\pi\)
−0.773095 + 0.634290i \(0.781292\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −2.19615 −0.169943 −0.0849717 0.996383i \(-0.527080\pi\)
−0.0849717 + 0.996383i \(0.527080\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 8.92820i 0.682757i
\(172\) 0 0
\(173\) − 6.00000i − 0.456172i −0.973641 0.228086i \(-0.926753\pi\)
0.973641 0.228086i \(-0.0732467\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 16.3923 1.23212
\(178\) 0 0
\(179\) 7.85641i 0.587215i 0.955926 + 0.293608i \(0.0948559\pi\)
−0.955926 + 0.293608i \(0.905144\pi\)
\(180\) 0 0
\(181\) − 6.92820i − 0.514969i −0.966282 0.257485i \(-0.917106\pi\)
0.966282 0.257485i \(-0.0828937\pi\)
\(182\) 0 0
\(183\) 2.53590 0.187459
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 12.0000i 0.877527i
\(188\) 0 0
\(189\) − 18.9282i − 1.37682i
\(190\) 0 0
\(191\) −4.39230 −0.317816 −0.158908 0.987293i \(-0.550797\pi\)
−0.158908 + 0.987293i \(0.550797\pi\)
\(192\) 0 0
\(193\) 14.3923 1.03598 0.517990 0.855386i \(-0.326680\pi\)
0.517990 + 0.855386i \(0.326680\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 10.3923i − 0.740421i −0.928948 0.370211i \(-0.879286\pi\)
0.928948 0.370211i \(-0.120714\pi\)
\(198\) 0 0
\(199\) 6.92820 0.491127 0.245564 0.969380i \(-0.421027\pi\)
0.245564 + 0.969380i \(0.421027\pi\)
\(200\) 0 0
\(201\) 0.535898 0.0377994
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −9.80385 −0.681415
\(208\) 0 0
\(209\) 6.92820 0.479234
\(210\) 0 0
\(211\) − 14.3923i − 0.990807i −0.868663 0.495404i \(-0.835020\pi\)
0.868663 0.495404i \(-0.164980\pi\)
\(212\) 0 0
\(213\) − 44.7846i − 3.06859i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 12.0000 0.814613
\(218\) 0 0
\(219\) 17.4641i 1.18011i
\(220\) 0 0
\(221\) 12.0000i 0.807207i
\(222\) 0 0
\(223\) −9.12436 −0.611012 −0.305506 0.952190i \(-0.598826\pi\)
−0.305506 + 0.952190i \(0.598826\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 12.5885i 0.835525i 0.908556 + 0.417763i \(0.137186\pi\)
−0.908556 + 0.417763i \(0.862814\pi\)
\(228\) 0 0
\(229\) − 5.07180i − 0.335154i −0.985859 0.167577i \(-0.946406\pi\)
0.985859 0.167577i \(-0.0535942\pi\)
\(230\) 0 0
\(231\) −44.7846 −2.94661
\(232\) 0 0
\(233\) 22.3923 1.46697 0.733484 0.679706i \(-0.237893\pi\)
0.733484 + 0.679706i \(0.237893\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 32.7846i 2.12959i
\(238\) 0 0
\(239\) −20.7846 −1.34444 −0.672222 0.740349i \(-0.734660\pi\)
−0.672222 + 0.740349i \(0.734660\pi\)
\(240\) 0 0
\(241\) 0.392305 0.0252706 0.0126353 0.999920i \(-0.495978\pi\)
0.0126353 + 0.999920i \(0.495978\pi\)
\(242\) 0 0
\(243\) 18.7321i 1.20166i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 6.92820 0.440831
\(248\) 0 0
\(249\) −3.46410 −0.219529
\(250\) 0 0
\(251\) 8.53590i 0.538781i 0.963031 + 0.269391i \(0.0868223\pi\)
−0.963031 + 0.269391i \(0.913178\pi\)
\(252\) 0 0
\(253\) 7.60770i 0.478292i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 0 0
\(259\) − 28.3923i − 1.76421i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 9.80385 0.604531 0.302266 0.953224i \(-0.402257\pi\)
0.302266 + 0.953224i \(0.402257\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 35.3205i 2.16158i
\(268\) 0 0
\(269\) − 26.7846i − 1.63309i −0.577284 0.816543i \(-0.695888\pi\)
0.577284 0.816543i \(-0.304112\pi\)
\(270\) 0 0
\(271\) −16.3923 −0.995762 −0.497881 0.867245i \(-0.665888\pi\)
−0.497881 + 0.867245i \(0.665888\pi\)
\(272\) 0 0
\(273\) −44.7846 −2.71049
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 7.85641i 0.472046i 0.971748 + 0.236023i \(0.0758440\pi\)
−0.971748 + 0.236023i \(0.924156\pi\)
\(278\) 0 0
\(279\) 11.3205 0.677741
\(280\) 0 0
\(281\) −7.60770 −0.453837 −0.226919 0.973914i \(-0.572865\pi\)
−0.226919 + 0.973914i \(0.572865\pi\)
\(282\) 0 0
\(283\) 20.5885i 1.22386i 0.790913 + 0.611928i \(0.209606\pi\)
−0.790913 + 0.611928i \(0.790394\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −44.7846 −2.64355
\(288\) 0 0
\(289\) −5.00000 −0.294118
\(290\) 0 0
\(291\) 39.3205i 2.30501i
\(292\) 0 0
\(293\) 30.0000i 1.75262i 0.481749 + 0.876309i \(0.340002\pi\)
−0.481749 + 0.876309i \(0.659998\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −13.8564 −0.804030
\(298\) 0 0
\(299\) 7.60770i 0.439964i
\(300\) 0 0
\(301\) − 0.928203i − 0.0535007i
\(302\) 0 0
\(303\) −32.7846 −1.88343
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 23.8038i − 1.35856i −0.733880 0.679279i \(-0.762293\pi\)
0.733880 0.679279i \(-0.237707\pi\)
\(308\) 0 0
\(309\) − 6.00000i − 0.341328i
\(310\) 0 0
\(311\) −28.3923 −1.60998 −0.804990 0.593288i \(-0.797829\pi\)
−0.804990 + 0.593288i \(0.797829\pi\)
\(312\) 0 0
\(313\) −22.0000 −1.24351 −0.621757 0.783210i \(-0.713581\pi\)
−0.621757 + 0.783210i \(0.713581\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.60770i 0.0902972i 0.998980 + 0.0451486i \(0.0143761\pi\)
−0.998980 + 0.0451486i \(0.985624\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 36.2487 2.02321
\(322\) 0 0
\(323\) − 6.92820i − 0.385496i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 35.3205 1.95323
\(328\) 0 0
\(329\) 10.3923 0.572946
\(330\) 0 0
\(331\) 26.3923i 1.45065i 0.688405 + 0.725326i \(0.258311\pi\)
−0.688405 + 0.725326i \(0.741689\pi\)
\(332\) 0 0
\(333\) − 26.7846i − 1.46779i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 0 0
\(339\) − 35.3205i − 1.91835i
\(340\) 0 0
\(341\) − 8.78461i − 0.475713i
\(342\) 0 0
\(343\) −39.7128 −2.14429
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 10.0526i − 0.539650i −0.962909 0.269825i \(-0.913034\pi\)
0.962909 0.269825i \(-0.0869658\pi\)
\(348\) 0 0
\(349\) 32.7846i 1.75492i 0.479650 + 0.877460i \(0.340764\pi\)
−0.479650 + 0.877460i \(0.659236\pi\)
\(350\) 0 0
\(351\) −13.8564 −0.739600
\(352\) 0 0
\(353\) −26.7846 −1.42560 −0.712800 0.701367i \(-0.752573\pi\)
−0.712800 + 0.701367i \(0.752573\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 44.7846i 2.37025i
\(358\) 0 0
\(359\) −32.7846 −1.73031 −0.865153 0.501508i \(-0.832779\pi\)
−0.865153 + 0.501508i \(0.832779\pi\)
\(360\) 0 0
\(361\) 15.0000 0.789474
\(362\) 0 0
\(363\) 2.73205i 0.143395i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 28.0526 1.46433 0.732166 0.681126i \(-0.238510\pi\)
0.732166 + 0.681126i \(0.238510\pi\)
\(368\) 0 0
\(369\) −42.2487 −2.19938
\(370\) 0 0
\(371\) − 49.1769i − 2.55314i
\(372\) 0 0
\(373\) − 19.8564i − 1.02813i −0.857753 0.514063i \(-0.828140\pi\)
0.857753 0.514063i \(-0.171860\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 2.00000i 0.102733i 0.998680 + 0.0513665i \(0.0163577\pi\)
−0.998680 + 0.0513665i \(0.983642\pi\)
\(380\) 0 0
\(381\) − 38.7846i − 1.98700i
\(382\) 0 0
\(383\) −14.1962 −0.725390 −0.362695 0.931908i \(-0.618143\pi\)
−0.362695 + 0.931908i \(0.618143\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 0.875644i − 0.0445115i
\(388\) 0 0
\(389\) − 14.7846i − 0.749609i −0.927104 0.374805i \(-0.877710\pi\)
0.927104 0.374805i \(-0.122290\pi\)
\(390\) 0 0
\(391\) 7.60770 0.384738
\(392\) 0 0
\(393\) 28.3923 1.43220
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 20.5359i − 1.03067i −0.856990 0.515334i \(-0.827668\pi\)
0.856990 0.515334i \(-0.172332\pi\)
\(398\) 0 0
\(399\) 25.8564 1.29444
\(400\) 0 0
\(401\) −31.8564 −1.59083 −0.795417 0.606063i \(-0.792748\pi\)
−0.795417 + 0.606063i \(0.792748\pi\)
\(402\) 0 0
\(403\) − 8.78461i − 0.437593i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −20.7846 −1.03025
\(408\) 0 0
\(409\) 24.3923 1.20612 0.603061 0.797695i \(-0.293948\pi\)
0.603061 + 0.797695i \(0.293948\pi\)
\(410\) 0 0
\(411\) − 2.53590i − 0.125087i
\(412\) 0 0
\(413\) − 28.3923i − 1.39709i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 27.3205 1.33789
\(418\) 0 0
\(419\) 12.9282i 0.631584i 0.948828 + 0.315792i \(0.102270\pi\)
−0.948828 + 0.315792i \(0.897730\pi\)
\(420\) 0 0
\(421\) − 6.00000i − 0.292422i −0.989253 0.146211i \(-0.953292\pi\)
0.989253 0.146211i \(-0.0467079\pi\)
\(422\) 0 0
\(423\) 9.80385 0.476679
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 4.39230i − 0.212559i
\(428\) 0 0
\(429\) 32.7846i 1.58286i
\(430\) 0 0
\(431\) 7.60770 0.366450 0.183225 0.983071i \(-0.441346\pi\)
0.183225 + 0.983071i \(0.441346\pi\)
\(432\) 0 0
\(433\) −5.60770 −0.269489 −0.134744 0.990880i \(-0.543021\pi\)
−0.134744 + 0.990880i \(0.543021\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 4.39230i − 0.210112i
\(438\) 0 0
\(439\) 5.07180 0.242064 0.121032 0.992649i \(-0.461380\pi\)
0.121032 + 0.992649i \(0.461380\pi\)
\(440\) 0 0
\(441\) −68.7128 −3.27204
\(442\) 0 0
\(443\) 17.6603i 0.839064i 0.907741 + 0.419532i \(0.137806\pi\)
−0.907741 + 0.419532i \(0.862194\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 49.1769 2.32599
\(448\) 0 0
\(449\) −9.46410 −0.446639 −0.223319 0.974745i \(-0.571689\pi\)
−0.223319 + 0.974745i \(0.571689\pi\)
\(450\) 0 0
\(451\) 32.7846i 1.54377i
\(452\) 0 0
\(453\) − 25.8564i − 1.21484i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 18.7846 0.878707 0.439353 0.898314i \(-0.355208\pi\)
0.439353 + 0.898314i \(0.355208\pi\)
\(458\) 0 0
\(459\) 13.8564i 0.646762i
\(460\) 0 0
\(461\) − 12.0000i − 0.558896i −0.960161 0.279448i \(-0.909849\pi\)
0.960161 0.279448i \(-0.0901514\pi\)
\(462\) 0 0
\(463\) 26.1962 1.21744 0.608719 0.793386i \(-0.291684\pi\)
0.608719 + 0.793386i \(0.291684\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 28.9808i − 1.34107i −0.741878 0.670535i \(-0.766065\pi\)
0.741878 0.670535i \(-0.233935\pi\)
\(468\) 0 0
\(469\) − 0.928203i − 0.0428604i
\(470\) 0 0
\(471\) 35.3205 1.62748
\(472\) 0 0
\(473\) −0.679492 −0.0312431
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 46.3923i − 2.12416i
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) −20.7846 −0.947697
\(482\) 0 0
\(483\) 28.3923i 1.29189i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −14.8756 −0.674080 −0.337040 0.941490i \(-0.609426\pi\)
−0.337040 + 0.941490i \(0.609426\pi\)
\(488\) 0 0
\(489\) 44.2487 2.00100
\(490\) 0 0
\(491\) 1.60770i 0.0725543i 0.999342 + 0.0362771i \(0.0115499\pi\)
−0.999342 + 0.0362771i \(0.988450\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −77.5692 −3.47946
\(498\) 0 0
\(499\) 43.5692i 1.95043i 0.221268 + 0.975213i \(0.428980\pi\)
−0.221268 + 0.975213i \(0.571020\pi\)
\(500\) 0 0
\(501\) 6.00000i 0.268060i
\(502\) 0 0
\(503\) 2.19615 0.0979216 0.0489608 0.998801i \(-0.484409\pi\)
0.0489608 + 0.998801i \(0.484409\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 2.73205i − 0.121335i
\(508\) 0 0
\(509\) − 32.7846i − 1.45315i −0.687086 0.726576i \(-0.741110\pi\)
0.687086 0.726576i \(-0.258890\pi\)
\(510\) 0 0
\(511\) 30.2487 1.33812
\(512\) 0 0
\(513\) 8.00000 0.353209
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 7.60770i − 0.334586i
\(518\) 0 0
\(519\) −16.3923 −0.719542
\(520\) 0 0
\(521\) 4.14359 0.181534 0.0907671 0.995872i \(-0.471068\pi\)
0.0907671 + 0.995872i \(0.471068\pi\)
\(522\) 0 0
\(523\) − 36.9808i − 1.61706i −0.588458 0.808528i \(-0.700265\pi\)
0.588458 0.808528i \(-0.299735\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −8.78461 −0.382664
\(528\) 0 0
\(529\) −18.1769 −0.790301
\(530\) 0 0
\(531\) − 26.7846i − 1.16235i
\(532\) 0 0
\(533\) 32.7846i 1.42006i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 21.4641 0.926244
\(538\) 0 0
\(539\) 53.3205i 2.29668i
\(540\) 0 0
\(541\) − 39.7128i − 1.70739i −0.520776 0.853694i \(-0.674357\pi\)
0.520776 0.853694i \(-0.325643\pi\)
\(542\) 0 0
\(543\) −18.9282 −0.812287
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 12.1962i − 0.521470i −0.965410 0.260735i \(-0.916035\pi\)
0.965410 0.260735i \(-0.0839649\pi\)
\(548\) 0 0
\(549\) − 4.14359i − 0.176844i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 56.7846 2.41473
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 26.7846i − 1.13490i −0.823408 0.567450i \(-0.807930\pi\)
0.823408 0.567450i \(-0.192070\pi\)
\(558\) 0 0
\(559\) −0.679492 −0.0287394
\(560\) 0 0
\(561\) 32.7846 1.38417
\(562\) 0 0
\(563\) − 15.8038i − 0.666053i −0.942918 0.333026i \(-0.891930\pi\)
0.942918 0.333026i \(-0.108070\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 11.6603 0.489685
\(568\) 0 0
\(569\) 6.24871 0.261960 0.130980 0.991385i \(-0.458188\pi\)
0.130980 + 0.991385i \(0.458188\pi\)
\(570\) 0 0
\(571\) − 9.60770i − 0.402070i −0.979584 0.201035i \(-0.935570\pi\)
0.979584 0.201035i \(-0.0644304\pi\)
\(572\) 0 0
\(573\) 12.0000i 0.501307i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 0 0
\(579\) − 39.3205i − 1.63410i
\(580\) 0 0
\(581\) 6.00000i 0.248922i
\(582\) 0 0
\(583\) −36.0000 −1.49097
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 31.5167i 1.30083i 0.759578 + 0.650416i \(0.225405\pi\)
−0.759578 + 0.650416i \(0.774595\pi\)
\(588\) 0 0
\(589\) 5.07180i 0.208980i
\(590\) 0 0
\(591\) −28.3923 −1.16790
\(592\) 0 0
\(593\) −12.9282 −0.530898 −0.265449 0.964125i \(-0.585520\pi\)
−0.265449 + 0.964125i \(0.585520\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 18.9282i − 0.774680i
\(598\) 0 0
\(599\) 12.0000 0.490307 0.245153 0.969484i \(-0.421162\pi\)
0.245153 + 0.969484i \(0.421162\pi\)
\(600\) 0 0
\(601\) 0.392305 0.0160024 0.00800122 0.999968i \(-0.497453\pi\)
0.00800122 + 0.999968i \(0.497453\pi\)
\(602\) 0 0
\(603\) − 0.875644i − 0.0356590i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −2.19615 −0.0891391 −0.0445695 0.999006i \(-0.514192\pi\)
−0.0445695 + 0.999006i \(0.514192\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 7.60770i − 0.307774i
\(612\) 0 0
\(613\) − 34.3923i − 1.38909i −0.719448 0.694546i \(-0.755605\pi\)
0.719448 0.694546i \(-0.244395\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −34.3923 −1.38458 −0.692291 0.721618i \(-0.743399\pi\)
−0.692291 + 0.721618i \(0.743399\pi\)
\(618\) 0 0
\(619\) − 34.7846i − 1.39811i −0.715067 0.699056i \(-0.753604\pi\)
0.715067 0.699056i \(-0.246396\pi\)
\(620\) 0 0
\(621\) 8.78461i 0.352514i
\(622\) 0 0
\(623\) 61.1769 2.45100
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 18.9282i − 0.755920i
\(628\) 0 0
\(629\) 20.7846i 0.828737i
\(630\) 0 0
\(631\) 14.5359 0.578665 0.289332 0.957229i \(-0.406567\pi\)
0.289332 + 0.957229i \(0.406567\pi\)
\(632\) 0 0
\(633\) −39.3205 −1.56285
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 53.3205i 2.11264i
\(638\) 0 0
\(639\) −73.1769 −2.89483
\(640\) 0 0
\(641\) 16.3923 0.647457 0.323729 0.946150i \(-0.395064\pi\)
0.323729 + 0.946150i \(0.395064\pi\)
\(642\) 0 0
\(643\) 20.5885i 0.811929i 0.913889 + 0.405965i \(0.133064\pi\)
−0.913889 + 0.405965i \(0.866936\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 5.41154 0.212750 0.106375 0.994326i \(-0.466076\pi\)
0.106375 + 0.994326i \(0.466076\pi\)
\(648\) 0 0
\(649\) −20.7846 −0.815867
\(650\) 0 0
\(651\) − 32.7846i − 1.28493i
\(652\) 0 0
\(653\) 43.1769i 1.68964i 0.535048 + 0.844822i \(0.320293\pi\)
−0.535048 + 0.844822i \(0.679707\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 28.5359 1.11329
\(658\) 0 0
\(659\) − 28.6410i − 1.11570i −0.829943 0.557848i \(-0.811627\pi\)
0.829943 0.557848i \(-0.188373\pi\)
\(660\) 0 0
\(661\) 47.5692i 1.85023i 0.379689 + 0.925114i \(0.376031\pi\)
−0.379689 + 0.925114i \(0.623969\pi\)
\(662\) 0 0
\(663\) 32.7846 1.27325
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 24.9282i 0.963780i
\(670\) 0 0
\(671\) −3.21539 −0.124129
\(672\) 0 0
\(673\) −23.1769 −0.893404 −0.446702 0.894683i \(-0.647402\pi\)
−0.446702 + 0.894683i \(0.647402\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 19.1769i − 0.737029i −0.929622 0.368514i \(-0.879867\pi\)
0.929622 0.368514i \(-0.120133\pi\)
\(678\) 0 0
\(679\) 68.1051 2.61363
\(680\) 0 0
\(681\) 34.3923 1.31792
\(682\) 0 0
\(683\) − 5.66025i − 0.216584i −0.994119 0.108292i \(-0.965462\pi\)
0.994119 0.108292i \(-0.0345381\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −13.8564 −0.528655
\(688\) 0 0
\(689\) −36.0000 −1.37149
\(690\) 0 0
\(691\) 26.3923i 1.00401i 0.864865 + 0.502005i \(0.167404\pi\)
−0.864865 + 0.502005i \(0.832596\pi\)
\(692\) 0 0
\(693\) 73.1769i 2.77976i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 32.7846 1.24181
\(698\) 0 0
\(699\) − 61.1769i − 2.31392i
\(700\) 0 0
\(701\) 26.7846i 1.01164i 0.862639 + 0.505820i \(0.168810\pi\)
−0.862639 + 0.505820i \(0.831190\pi\)
\(702\) 0 0
\(703\) 12.0000 0.452589
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 56.7846i 2.13561i
\(708\) 0 0
\(709\) 37.8564i 1.42173i 0.703330 + 0.710864i \(0.251696\pi\)
−0.703330 + 0.710864i \(0.748304\pi\)
\(710\) 0 0
\(711\) 53.5692 2.00900
\(712\) 0 0
\(713\) −5.56922 −0.208569
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 56.7846i 2.12066i
\(718\) 0 0
\(719\) 3.21539 0.119914 0.0599569 0.998201i \(-0.480904\pi\)
0.0599569 + 0.998201i \(0.480904\pi\)
\(720\) 0 0
\(721\) −10.3923 −0.387030
\(722\) 0 0
\(723\) − 1.07180i − 0.0398606i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 4.05256 0.150301 0.0751505 0.997172i \(-0.476056\pi\)
0.0751505 + 0.997172i \(0.476056\pi\)
\(728\) 0 0
\(729\) 43.7846 1.62165
\(730\) 0 0
\(731\) 0.679492i 0.0251319i
\(732\) 0 0
\(733\) 2.78461i 0.102852i 0.998677 + 0.0514260i \(0.0163766\pi\)
−0.998677 + 0.0514260i \(0.983623\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −0.679492 −0.0250294
\(738\) 0 0
\(739\) 2.00000i 0.0735712i 0.999323 + 0.0367856i \(0.0117119\pi\)
−0.999323 + 0.0367856i \(0.988288\pi\)
\(740\) 0 0
\(741\) − 18.9282i − 0.695345i
\(742\) 0 0
\(743\) −5.41154 −0.198530 −0.0992651 0.995061i \(-0.531649\pi\)
−0.0992651 + 0.995061i \(0.531649\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 5.66025i 0.207098i
\(748\) 0 0
\(749\) − 62.7846i − 2.29410i
\(750\) 0 0
\(751\) −19.6077 −0.715495 −0.357747 0.933818i \(-0.616455\pi\)
−0.357747 + 0.933818i \(0.616455\pi\)
\(752\) 0 0
\(753\) 23.3205 0.849847
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 36.9282i 1.34218i 0.741377 + 0.671089i \(0.234173\pi\)
−0.741377 + 0.671089i \(0.765827\pi\)
\(758\) 0 0
\(759\) 20.7846 0.754434
\(760\) 0 0
\(761\) 33.7128 1.22209 0.611044 0.791596i \(-0.290750\pi\)
0.611044 + 0.791596i \(0.290750\pi\)
\(762\) 0 0
\(763\) − 61.1769i − 2.21475i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −20.7846 −0.750489
\(768\) 0 0
\(769\) 34.7846 1.25437 0.627183 0.778872i \(-0.284208\pi\)
0.627183 + 0.778872i \(0.284208\pi\)
\(770\) 0 0
\(771\) 16.3923i 0.590354i
\(772\) 0 0
\(773\) − 25.6077i − 0.921045i −0.887648 0.460522i \(-0.847662\pi\)
0.887648 0.460522i \(-0.152338\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −77.5692 −2.78278
\(778\) 0 0
\(779\) − 18.9282i − 0.678173i
\(780\) 0 0
\(781\) 56.7846i 2.03191i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 16.1962i 0.577330i 0.957430 + 0.288665i \(0.0932115\pi\)
−0.957430 + 0.288665i \(0.906789\pi\)
\(788\) 0 0
\(789\) − 26.7846i − 0.953557i
\(790\) 0 0
\(791\) −61.1769 −2.17520
\(792\) 0 0
\(793\) −3.21539 −0.114182
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 22.3923i − 0.793176i −0.917997 0.396588i \(-0.870194\pi\)
0.917997 0.396588i \(-0.129806\pi\)
\(798\) 0 0
\(799\) −7.60770 −0.269141
\(800\) 0 0
\(801\) 57.7128 2.03918
\(802\) 0 0
\(803\) − 22.1436i − 0.781430i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −73.1769 −2.57595
\(808\) 0 0
\(809\) −47.5692 −1.67244 −0.836222 0.548391i \(-0.815241\pi\)
−0.836222 + 0.548391i \(0.815241\pi\)
\(810\) 0 0
\(811\) − 17.6077i − 0.618290i −0.951015 0.309145i \(-0.899957\pi\)
0.951015 0.309145i \(-0.100043\pi\)
\(812\) 0 0
\(813\) 44.7846i 1.57066i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0.392305 0.0137250
\(818\) 0 0
\(819\) 73.1769i 2.55701i
\(820\) 0 0
\(821\) − 9.21539i − 0.321619i −0.986985 0.160810i \(-0.948589\pi\)
0.986985 0.160810i \(-0.0514105\pi\)
\(822\) 0 0
\(823\) 48.8372 1.70236 0.851178 0.524877i \(-0.175889\pi\)
0.851178 + 0.524877i \(0.175889\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 1.26795i − 0.0440909i −0.999757 0.0220455i \(-0.992982\pi\)
0.999757 0.0220455i \(-0.00701786\pi\)
\(828\) 0 0
\(829\) − 9.21539i − 0.320064i −0.987112 0.160032i \(-0.948840\pi\)
0.987112 0.160032i \(-0.0511597\pi\)
\(830\) 0 0
\(831\) 21.4641 0.744581
\(832\) 0 0
\(833\) 53.3205 1.84745
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 10.1436i − 0.350614i
\(838\) 0 0
\(839\) −32.7846 −1.13185 −0.565925 0.824457i \(-0.691481\pi\)
−0.565925 + 0.824457i \(0.691481\pi\)
\(840\) 0 0
\(841\) 29.0000 1.00000
\(842\) 0 0
\(843\) 20.7846i 0.715860i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 4.73205 0.162595
\(848\) 0 0
\(849\) 56.2487 1.93045
\(850\) 0 0
\(851\) 13.1769i 0.451699i
\(852\) 0 0
\(853\) 3.46410i 0.118609i 0.998240 + 0.0593043i \(0.0188882\pi\)
−0.998240 + 0.0593043i \(0.981112\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 43.8564 1.49811 0.749053 0.662510i \(-0.230509\pi\)
0.749053 + 0.662510i \(0.230509\pi\)
\(858\) 0 0
\(859\) 31.5692i 1.07713i 0.842585 + 0.538564i \(0.181033\pi\)
−0.842585 + 0.538564i \(0.818967\pi\)
\(860\) 0 0
\(861\) 122.354i 4.16981i
\(862\) 0 0
\(863\) 10.9808 0.373789 0.186895 0.982380i \(-0.440158\pi\)
0.186895 + 0.982380i \(0.440158\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 13.6603i 0.463927i
\(868\) 0 0
\(869\) − 41.5692i − 1.41014i
\(870\) 0 0
\(871\) −0.679492 −0.0230237
\(872\) 0 0
\(873\) 64.2487 2.17449
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 19.8564i − 0.670503i −0.942129 0.335252i \(-0.891179\pi\)
0.942129 0.335252i \(-0.108821\pi\)
\(878\) 0 0
\(879\) 81.9615 2.76449
\(880\) 0 0
\(881\) −28.3923 −0.956561 −0.478281 0.878207i \(-0.658740\pi\)
−0.478281 + 0.878207i \(0.658740\pi\)
\(882\) 0 0
\(883\) 16.1962i 0.545044i 0.962150 + 0.272522i \(0.0878577\pi\)
−0.962150 + 0.272522i \(0.912142\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −51.3731 −1.72494 −0.862469 0.506109i \(-0.831083\pi\)
−0.862469 + 0.506109i \(0.831083\pi\)
\(888\) 0 0
\(889\) −67.1769 −2.25304
\(890\) 0 0
\(891\) − 8.53590i − 0.285963i
\(892\) 0 0
\(893\) 4.39230i 0.146983i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 20.7846 0.693978
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 36.0000i 1.19933i
\(902\) 0 0
\(903\) −2.53590 −0.0843894
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 32.5885i − 1.08208i −0.840996 0.541041i \(-0.818030\pi\)
0.840996 0.541041i \(-0.181970\pi\)
\(908\) 0 0
\(909\) 53.5692i 1.77678i
\(910\) 0 0
\(911\) 25.1769 0.834148 0.417074 0.908872i \(-0.363056\pi\)
0.417074 + 0.908872i \(0.363056\pi\)
\(912\) 0 0
\(913\) 4.39230 0.145364
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 49.1769i − 1.62396i
\(918\) 0 0
\(919\) −15.7128 −0.518318 −0.259159 0.965835i \(-0.583445\pi\)
−0.259159 + 0.965835i \(0.583445\pi\)
\(920\) 0 0
\(921\) −65.0333 −2.14292
\(922\) 0 0
\(923\) 56.7846i 1.86909i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −9.80385 −0.322001
\(928\) 0 0
\(929\) 28.3923 0.931521 0.465761 0.884911i \(-0.345781\pi\)
0.465761 + 0.884911i \(0.345781\pi\)
\(930\) 0 0
\(931\) − 30.7846i − 1.00892i
\(932\) 0 0
\(933\) 77.5692i 2.53950i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 30.3923 0.992873 0.496437 0.868073i \(-0.334641\pi\)
0.496437 + 0.868073i \(0.334641\pi\)
\(938\) 0 0
\(939\) 60.1051i 1.96146i
\(940\) 0 0
\(941\) − 20.7846i − 0.677559i −0.940866 0.338779i \(-0.889986\pi\)
0.940866 0.338779i \(-0.110014\pi\)
\(942\) 0 0
\(943\) 20.7846 0.676840
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 56.1962i 1.82613i 0.407815 + 0.913065i \(0.366291\pi\)
−0.407815 + 0.913065i \(0.633709\pi\)
\(948\) 0 0
\(949\) − 22.1436i − 0.718811i
\(950\) 0 0
\(951\) 4.39230 0.142430
\(952\) 0 0
\(953\) −11.0718 −0.358651 −0.179325 0.983790i \(-0.557391\pi\)
−0.179325 + 0.983790i \(0.557391\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −4.39230 −0.141835
\(960\) 0 0
\(961\) −24.5692 −0.792555
\(962\) 0 0
\(963\) − 59.2295i − 1.90864i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −2.19615 −0.0706235 −0.0353118 0.999376i \(-0.511242\pi\)
−0.0353118 + 0.999376i \(0.511242\pi\)
\(968\) 0 0
\(969\) −18.9282 −0.608061
\(970\) 0 0
\(971\) 57.0333i 1.83029i 0.403129 + 0.915143i \(0.367923\pi\)
−0.403129 + 0.915143i \(0.632077\pi\)
\(972\) 0 0
\(973\) − 47.3205i − 1.51703i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 29.3205 0.938046 0.469023 0.883186i \(-0.344606\pi\)
0.469023 + 0.883186i \(0.344606\pi\)
\(978\) 0 0
\(979\) − 44.7846i − 1.43132i
\(980\) 0 0
\(981\) − 57.7128i − 1.84263i
\(982\) 0 0
\(983\) 42.5885 1.35836 0.679180 0.733971i \(-0.262335\pi\)
0.679180 + 0.733971i \(0.262335\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 28.3923i − 0.903737i
\(988\) 0 0
\(989\) 0.430781i 0.0136980i
\(990\) 0 0
\(991\) −32.1051 −1.01985 −0.509926 0.860218i \(-0.670327\pi\)
−0.509926 + 0.860218i \(0.670327\pi\)
\(992\) 0 0
\(993\) 72.1051 2.28819
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 46.3923i − 1.46926i −0.678468 0.734630i \(-0.737356\pi\)
0.678468 0.734630i \(-0.262644\pi\)
\(998\) 0 0
\(999\) −24.0000 −0.759326
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 1600.2.d.b.801.1 4
4.3 odd 2 1600.2.d.h.801.4 4
5.2 odd 4 1600.2.f.d.1249.1 4
5.3 odd 4 1600.2.f.h.1249.4 4
5.4 even 2 320.2.d.b.161.4 yes 4
8.3 odd 2 1600.2.d.h.801.1 4
8.5 even 2 inner 1600.2.d.b.801.4 4
15.14 odd 2 2880.2.k.l.1441.2 4
16.3 odd 4 6400.2.a.y.1.1 2
16.5 even 4 6400.2.a.bf.1.1 2
16.11 odd 4 6400.2.a.cd.1.2 2
16.13 even 4 6400.2.a.ck.1.2 2
20.3 even 4 1600.2.f.e.1249.1 4
20.7 even 4 1600.2.f.i.1249.4 4
20.19 odd 2 320.2.d.a.161.1 4
40.3 even 4 1600.2.f.i.1249.3 4
40.13 odd 4 1600.2.f.d.1249.2 4
40.19 odd 2 320.2.d.a.161.4 yes 4
40.27 even 4 1600.2.f.e.1249.2 4
40.29 even 2 320.2.d.b.161.1 yes 4
40.37 odd 4 1600.2.f.h.1249.3 4
60.59 even 2 2880.2.k.e.1441.1 4
80.19 odd 4 1280.2.a.p.1.2 2
80.29 even 4 1280.2.a.c.1.1 2
80.59 odd 4 1280.2.a.b.1.1 2
80.69 even 4 1280.2.a.m.1.2 2
120.29 odd 2 2880.2.k.l.1441.4 4
120.59 even 2 2880.2.k.e.1441.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
320.2.d.a.161.1 4 20.19 odd 2
320.2.d.a.161.4 yes 4 40.19 odd 2
320.2.d.b.161.1 yes 4 40.29 even 2
320.2.d.b.161.4 yes 4 5.4 even 2
1280.2.a.b.1.1 2 80.59 odd 4
1280.2.a.c.1.1 2 80.29 even 4
1280.2.a.m.1.2 2 80.69 even 4
1280.2.a.p.1.2 2 80.19 odd 4
1600.2.d.b.801.1 4 1.1 even 1 trivial
1600.2.d.b.801.4 4 8.5 even 2 inner
1600.2.d.h.801.1 4 8.3 odd 2
1600.2.d.h.801.4 4 4.3 odd 2
1600.2.f.d.1249.1 4 5.2 odd 4
1600.2.f.d.1249.2 4 40.13 odd 4
1600.2.f.e.1249.1 4 20.3 even 4
1600.2.f.e.1249.2 4 40.27 even 4
1600.2.f.h.1249.3 4 40.37 odd 4
1600.2.f.h.1249.4 4 5.3 odd 4
1600.2.f.i.1249.3 4 40.3 even 4
1600.2.f.i.1249.4 4 20.7 even 4
2880.2.k.e.1441.1 4 60.59 even 2
2880.2.k.e.1441.3 4 120.59 even 2
2880.2.k.l.1441.2 4 15.14 odd 2
2880.2.k.l.1441.4 4 120.29 odd 2
6400.2.a.y.1.1 2 16.3 odd 4
6400.2.a.bf.1.1 2 16.5 even 4
6400.2.a.cd.1.2 2 16.11 odd 4
6400.2.a.ck.1.2 2 16.13 even 4