Properties

Label 1600.2.f.i.1249.4
Level $1600$
Weight $2$
Character 1600.1249
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.f (of order \(2\), degree \(1\), not 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: \(\Z[a_1, \ldots, a_{19}]\)
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 1249.4
Root \(-0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1600.1249
Dual form 1600.2.f.i.1249.3

$q$-expansion

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

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
\(3\) 2.73205 1.57735 0.788675 0.614810i \(-0.210767\pi\)
0.788675 + 0.614810i \(0.210767\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 4.73205i 1.78855i 0.447521 + 0.894274i \(0.352307\pi\)
−0.447521 + 0.894274i \(0.647693\pi\)
\(8\) 0 0
\(9\) 4.46410 1.48803
\(10\) 0 0
\(11\) − 3.46410i − 1.04447i −0.852803 0.522233i \(-0.825099\pi\)
0.852803 0.522233i \(-0.174901\pi\)
\(12\) 0 0
\(13\) 3.46410 0.960769 0.480384 0.877058i \(-0.340497\pi\)
0.480384 + 0.877058i \(0.340497\pi\)
\(14\) 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
\(19\) − 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
\(23\) 2.19615i 0.457929i 0.973435 + 0.228965i \(0.0735340\pi\)
−0.973435 + 0.228965i \(0.926466\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 4.00000 0.769800
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 2.53590 0.455461 0.227730 0.973724i \(-0.426870\pi\)
0.227730 + 0.973724i \(0.426870\pi\)
\(32\) 0 0
\(33\) − 9.46410i − 1.64749i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 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
\(43\) −0.196152 −0.0299130 −0.0149565 0.999888i \(-0.504761\pi\)
−0.0149565 + 0.999888i \(0.504761\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.19615i 0.320342i 0.987089 + 0.160171i \(0.0512045\pi\)
−0.987089 + 0.160171i \(0.948795\pi\)
\(48\) 0 0
\(49\) −15.3923 −2.19890
\(50\) 0 0
\(51\) 9.46410i 1.32524i
\(52\) 0 0
\(53\) 10.3923 1.42749 0.713746 0.700404i \(-0.246997\pi\)
0.713746 + 0.700404i \(0.246997\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 5.46410i − 0.723738i
\(58\) 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
\(63\) 21.1244i 2.66142i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0.196152 0.0239638 0.0119819 0.999928i \(-0.496186\pi\)
0.0119819 + 0.999928i \(0.496186\pi\)
\(68\) 0 0
\(69\) 6.00000i 0.722315i
\(70\) 0 0
\(71\) −16.3923 −1.94541 −0.972704 0.232048i \(-0.925457\pi\)
−0.972704 + 0.232048i \(0.925457\pi\)
\(72\) 0 0
\(73\) 6.39230i 0.748163i 0.927396 + 0.374081i \(0.122042\pi\)
−0.927396 + 0.374081i \(0.877958\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 16.3923 1.86808
\(78\) 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
\(83\) 1.26795 0.139176 0.0695878 0.997576i \(-0.477832\pi\)
0.0695878 + 0.997576i \(0.477832\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 12.9282 1.37039 0.685193 0.728361i \(-0.259718\pi\)
0.685193 + 0.728361i \(0.259718\pi\)
\(90\) 0 0
\(91\) 16.3923i 1.71838i
\(92\) 0 0
\(93\) 6.92820 0.718421
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 14.3923i − 1.46132i −0.682743 0.730659i \(-0.739213\pi\)
0.682743 0.730659i \(-0.260787\pi\)
\(98\) 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
\(103\) 2.19615i 0.216393i 0.994130 + 0.108197i \(0.0345076\pi\)
−0.994130 + 0.108197i \(0.965492\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 13.2679 1.28266 0.641331 0.767265i \(-0.278383\pi\)
0.641331 + 0.767265i \(0.278383\pi\)
\(108\) 0 0
\(109\) − 12.9282i − 1.23830i −0.785274 0.619149i \(-0.787478\pi\)
0.785274 0.619149i \(-0.212522\pi\)
\(110\) 0 0
\(111\) −16.3923 −1.55589
\(112\) 0 0
\(113\) − 12.9282i − 1.21618i −0.793867 0.608092i \(-0.791935\pi\)
0.793867 0.608092i \(-0.208065\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 15.4641 1.42966
\(118\) 0 0
\(119\) −16.3923 −1.50268
\(120\) 0 0
\(121\) −1.00000 −0.0909091
\(122\) 0 0
\(123\) 25.8564 2.33139
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 14.1962i − 1.25970i −0.776715 0.629852i \(-0.783115\pi\)
0.776715 0.629852i \(-0.216885\pi\)
\(128\) 0 0
\(129\) −0.535898 −0.0471832
\(130\) 0 0
\(131\) − 10.3923i − 0.907980i −0.891007 0.453990i \(-0.850000\pi\)
0.891007 0.453990i \(-0.150000\pi\)
\(132\) 0 0
\(133\) 9.46410 0.820642
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0.928203i 0.0793018i 0.999214 + 0.0396509i \(0.0126246\pi\)
−0.999214 + 0.0396509i \(0.987375\pi\)
\(138\) 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
\(143\) − 12.0000i − 1.00349i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −42.0526 −3.46844
\(148\) 0 0
\(149\) − 18.0000i − 1.47462i −0.675556 0.737309i \(-0.736096\pi\)
0.675556 0.737309i \(-0.263904\pi\)
\(150\) 0 0
\(151\) −9.46410 −0.770178 −0.385089 0.922880i \(-0.625829\pi\)
−0.385089 + 0.922880i \(0.625829\pi\)
\(152\) 0 0
\(153\) 15.4641i 1.25020i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −12.9282 −1.03178 −0.515891 0.856654i \(-0.672539\pi\)
−0.515891 + 0.856654i \(0.672539\pi\)
\(158\) 0 0
\(159\) 28.3923 2.25166
\(160\) 0 0
\(161\) −10.3923 −0.819028
\(162\) 0 0
\(163\) −16.1962 −1.26858 −0.634290 0.773095i \(-0.718708\pi\)
−0.634290 + 0.773095i \(0.718708\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2.19615i 0.169943i 0.996383 + 0.0849717i \(0.0270800\pi\)
−0.996383 + 0.0849717i \(0.972920\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) − 8.92820i − 0.682757i
\(172\) 0 0
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 16.3923i 1.23212i
\(178\) 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
\(183\) 2.53590i 0.187459i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 12.0000 0.877527
\(188\) 0 0
\(189\) 18.9282i 1.37682i
\(190\) 0 0
\(191\) 4.39230 0.317816 0.158908 0.987293i \(-0.449203\pi\)
0.158908 + 0.987293i \(0.449203\pi\)
\(192\) 0 0
\(193\) − 14.3923i − 1.03598i −0.855386 0.517990i \(-0.826680\pi\)
0.855386 0.517990i \(-0.173320\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 10.3923 0.740421 0.370211 0.928948i \(-0.379286\pi\)
0.370211 + 0.928948i \(0.379286\pi\)
\(198\) 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
\(207\) 9.80385i 0.681415i
\(208\) 0 0
\(209\) −6.92820 −0.479234
\(210\) 0 0
\(211\) 14.3923i 0.990807i 0.868663 + 0.495404i \(0.164980\pi\)
−0.868663 + 0.495404i \(0.835020\pi\)
\(212\) 0 0
\(213\) −44.7846 −3.06859
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 12.0000i 0.814613i
\(218\) 0 0
\(219\) 17.4641i 1.18011i
\(220\) 0 0
\(221\) 12.0000i 0.807207i
\(222\) 0 0
\(223\) − 9.12436i − 0.611012i −0.952190 0.305506i \(-0.901174\pi\)
0.952190 0.305506i \(-0.0988256\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 12.5885 0.835525 0.417763 0.908556i \(-0.362814\pi\)
0.417763 + 0.908556i \(0.362814\pi\)
\(228\) 0 0
\(229\) 5.07180i 0.335154i 0.985859 + 0.167577i \(0.0535942\pi\)
−0.985859 + 0.167577i \(0.946406\pi\)
\(230\) 0 0
\(231\) 44.7846 2.94661
\(232\) 0 0
\(233\) − 22.3923i − 1.46697i −0.679706 0.733484i \(-0.737893\pi\)
0.679706 0.733484i \(-0.262107\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −32.7846 −2.12959
\(238\) 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
\(243\) −18.7321 −1.20166
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 6.92820i − 0.440831i
\(248\) 0 0
\(249\) 3.46410 0.219529
\(250\) 0 0
\(251\) − 8.53590i − 0.538781i −0.963031 0.269391i \(-0.913178\pi\)
0.963031 0.269391i \(-0.0868223\pi\)
\(252\) 0 0
\(253\) 7.60770 0.478292
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 6.00000i − 0.374270i −0.982334 0.187135i \(-0.940080\pi\)
0.982334 0.187135i \(-0.0599201\pi\)
\(258\) 0 0
\(259\) − 28.3923i − 1.76421i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 9.80385i 0.604531i 0.953224 + 0.302266i \(0.0977429\pi\)
−0.953224 + 0.302266i \(0.902257\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 35.3205 2.16158
\(268\) 0 0
\(269\) 26.7846i 1.63309i 0.577284 + 0.816543i \(0.304112\pi\)
−0.577284 + 0.816543i \(0.695888\pi\)
\(270\) 0 0
\(271\) 16.3923 0.995762 0.497881 0.867245i \(-0.334112\pi\)
0.497881 + 0.867245i \(0.334112\pi\)
\(272\) 0 0
\(273\) 44.7846i 2.71049i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −7.85641 −0.472046 −0.236023 0.971748i \(-0.575844\pi\)
−0.236023 + 0.971748i \(0.575844\pi\)
\(278\) 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
\(283\) −20.5885 −1.22386 −0.611928 0.790913i \(-0.709606\pi\)
−0.611928 + 0.790913i \(0.709606\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 44.7846i 2.64355i
\(288\) 0 0
\(289\) 5.00000 0.294118
\(290\) 0 0
\(291\) − 39.3205i − 2.30501i
\(292\) 0 0
\(293\) 30.0000 1.75262 0.876309 0.481749i \(-0.159998\pi\)
0.876309 + 0.481749i \(0.159998\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 13.8564i − 0.804030i
\(298\) 0 0
\(299\) 7.60770i 0.439964i
\(300\) 0 0
\(301\) − 0.928203i − 0.0535007i
\(302\) 0 0
\(303\) − 32.7846i − 1.88343i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −23.8038 −1.35856 −0.679279 0.733880i \(-0.737707\pi\)
−0.679279 + 0.733880i \(0.737707\pi\)
\(308\) 0 0
\(309\) 6.00000i 0.341328i
\(310\) 0 0
\(311\) 28.3923 1.60998 0.804990 0.593288i \(-0.202171\pi\)
0.804990 + 0.593288i \(0.202171\pi\)
\(312\) 0 0
\(313\) 22.0000i 1.24351i 0.783210 + 0.621757i \(0.213581\pi\)
−0.783210 + 0.621757i \(0.786419\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −1.60770 −0.0902972 −0.0451486 0.998980i \(-0.514376\pi\)
−0.0451486 + 0.998980i \(0.514376\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 36.2487 2.02321
\(322\) 0 0
\(323\) 6.92820 0.385496
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 35.3205i − 1.95323i
\(328\) 0 0
\(329\) −10.3923 −0.572946
\(330\) 0 0
\(331\) − 26.3923i − 1.45065i −0.688405 0.725326i \(-0.741689\pi\)
0.688405 0.725326i \(-0.258311\pi\)
\(332\) 0 0
\(333\) −26.7846 −1.46779
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 14.0000i 0.762629i 0.924445 + 0.381314i \(0.124528\pi\)
−0.924445 + 0.381314i \(0.875472\pi\)
\(338\) 0 0
\(339\) − 35.3205i − 1.91835i
\(340\) 0 0
\(341\) − 8.78461i − 0.475713i
\(342\) 0 0
\(343\) − 39.7128i − 2.14429i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −10.0526 −0.539650 −0.269825 0.962909i \(-0.586966\pi\)
−0.269825 + 0.962909i \(0.586966\pi\)
\(348\) 0 0
\(349\) − 32.7846i − 1.75492i −0.479650 0.877460i \(-0.659236\pi\)
0.479650 0.877460i \(-0.340764\pi\)
\(350\) 0 0
\(351\) 13.8564 0.739600
\(352\) 0 0
\(353\) 26.7846i 1.42560i 0.701367 + 0.712800i \(0.252573\pi\)
−0.701367 + 0.712800i \(0.747427\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −44.7846 −2.37025
\(358\) 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
\(363\) −2.73205 −0.143395
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 28.0526i − 1.46433i −0.681126 0.732166i \(-0.738510\pi\)
0.681126 0.732166i \(-0.261490\pi\)
\(368\) 0 0
\(369\) 42.2487 2.19938
\(370\) 0 0
\(371\) 49.1769i 2.55314i
\(372\) 0 0
\(373\) −19.8564 −1.02813 −0.514063 0.857753i \(-0.671860\pi\)
−0.514063 + 0.857753i \(0.671860\pi\)
\(374\) 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
\(383\) − 14.1962i − 0.725390i −0.931908 0.362695i \(-0.881857\pi\)
0.931908 0.362695i \(-0.118143\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −0.875644 −0.0445115
\(388\) 0 0
\(389\) 14.7846i 0.749609i 0.927104 + 0.374805i \(0.122290\pi\)
−0.927104 + 0.374805i \(0.877710\pi\)
\(390\) 0 0
\(391\) −7.60770 −0.384738
\(392\) 0 0
\(393\) − 28.3923i − 1.43220i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 20.5359 1.03067 0.515334 0.856990i \(-0.327668\pi\)
0.515334 + 0.856990i \(0.327668\pi\)
\(398\) 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
\(403\) 8.78461 0.437593
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 20.7846i 1.03025i
\(408\) 0 0
\(409\) −24.3923 −1.20612 −0.603061 0.797695i \(-0.706052\pi\)
−0.603061 + 0.797695i \(0.706052\pi\)
\(410\) 0 0
\(411\) 2.53590i 0.125087i
\(412\) 0 0
\(413\) −28.3923 −1.39709
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 27.3205i 1.33789i
\(418\) 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
\(423\) 9.80385i 0.476679i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −4.39230 −0.212559
\(428\) 0 0
\(429\) − 32.7846i − 1.58286i
\(430\) 0 0
\(431\) −7.60770 −0.366450 −0.183225 0.983071i \(-0.558654\pi\)
−0.183225 + 0.983071i \(0.558654\pi\)
\(432\) 0 0
\(433\) 5.60770i 0.269489i 0.990880 + 0.134744i \(0.0430213\pi\)
−0.990880 + 0.134744i \(0.956979\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 4.39230 0.210112
\(438\) 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
\(443\) −17.6603 −0.839064 −0.419532 0.907741i \(-0.637806\pi\)
−0.419532 + 0.907741i \(0.637806\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 49.1769i − 2.32599i
\(448\) 0 0
\(449\) 9.46410 0.446639 0.223319 0.974745i \(-0.428311\pi\)
0.223319 + 0.974745i \(0.428311\pi\)
\(450\) 0 0
\(451\) − 32.7846i − 1.54377i
\(452\) 0 0
\(453\) −25.8564 −1.21484
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 18.7846i 0.878707i 0.898314 + 0.439353i \(0.144792\pi\)
−0.898314 + 0.439353i \(0.855208\pi\)
\(458\) 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
\(463\) 26.1962i 1.21744i 0.793386 + 0.608719i \(0.208316\pi\)
−0.793386 + 0.608719i \(0.791684\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −28.9808 −1.34107 −0.670535 0.741878i \(-0.733935\pi\)
−0.670535 + 0.741878i \(0.733935\pi\)
\(468\) 0 0
\(469\) 0.928203i 0.0428604i
\(470\) 0 0
\(471\) −35.3205 −1.62748
\(472\) 0 0
\(473\) 0.679492i 0.0312431i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 46.3923 2.12416
\(478\) 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
\(483\) −28.3923 −1.29189
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 14.8756i 0.674080i 0.941490 + 0.337040i \(0.109426\pi\)
−0.941490 + 0.337040i \(0.890574\pi\)
\(488\) 0 0
\(489\) −44.2487 −2.00100
\(490\) 0 0
\(491\) − 1.60770i − 0.0725543i −0.999342 0.0362771i \(-0.988450\pi\)
0.999342 0.0362771i \(-0.0115499\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 77.5692i − 3.47946i
\(498\) 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
\(503\) 2.19615i 0.0979216i 0.998801 + 0.0489608i \(0.0155909\pi\)
−0.998801 + 0.0489608i \(0.984409\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −2.73205 −0.121335
\(508\) 0 0
\(509\) 32.7846i 1.45315i 0.687086 + 0.726576i \(0.258890\pi\)
−0.687086 + 0.726576i \(0.741110\pi\)
\(510\) 0 0
\(511\) −30.2487 −1.33812
\(512\) 0 0
\(513\) − 8.00000i − 0.353209i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 7.60770 0.334586
\(518\) 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
\(523\) 36.9808 1.61706 0.808528 0.588458i \(-0.200265\pi\)
0.808528 + 0.588458i \(0.200265\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 8.78461i 0.382664i
\(528\) 0 0
\(529\) 18.1769 0.790301
\(530\) 0 0
\(531\) 26.7846i 1.16235i
\(532\) 0 0
\(533\) 32.7846 1.42006
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 21.4641i 0.926244i
\(538\) 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
\(543\) − 18.9282i − 0.812287i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −12.1962 −0.521470 −0.260735 0.965410i \(-0.583965\pi\)
−0.260735 + 0.965410i \(0.583965\pi\)
\(548\) 0 0
\(549\) 4.14359i 0.176844i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) − 56.7846i − 2.41473i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 26.7846 1.13490 0.567450 0.823408i \(-0.307930\pi\)
0.567450 + 0.823408i \(0.307930\pi\)
\(558\) 0 0
\(559\) −0.679492 −0.0287394
\(560\) 0 0
\(561\) 32.7846 1.38417
\(562\) 0 0
\(563\) 15.8038 0.666053 0.333026 0.942918i \(-0.391930\pi\)
0.333026 + 0.942918i \(0.391930\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 11.6603i − 0.489685i
\(568\) 0 0
\(569\) −6.24871 −0.261960 −0.130980 0.991385i \(-0.541812\pi\)
−0.130980 + 0.991385i \(0.541812\pi\)
\(570\) 0 0
\(571\) 9.60770i 0.402070i 0.979584 + 0.201035i \(0.0644304\pi\)
−0.979584 + 0.201035i \(0.935570\pi\)
\(572\) 0 0
\(573\) 12.0000 0.501307
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 2.00000i 0.0832611i 0.999133 + 0.0416305i \(0.0132552\pi\)
−0.999133 + 0.0416305i \(0.986745\pi\)
\(578\) 0 0
\(579\) − 39.3205i − 1.63410i
\(580\) 0 0
\(581\) 6.00000i 0.248922i
\(582\) 0 0
\(583\) − 36.0000i − 1.49097i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 31.5167 1.30083 0.650416 0.759578i \(-0.274595\pi\)
0.650416 + 0.759578i \(0.274595\pi\)
\(588\) 0 0
\(589\) − 5.07180i − 0.208980i
\(590\) 0 0
\(591\) 28.3923 1.16790
\(592\) 0 0
\(593\) 12.9282i 0.530898i 0.964125 + 0.265449i \(0.0855201\pi\)
−0.964125 + 0.265449i \(0.914480\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 18.9282 0.774680
\(598\) 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
\(603\) 0.875644 0.0356590
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 2.19615i 0.0891391i 0.999006 + 0.0445695i \(0.0141916\pi\)
−0.999006 + 0.0445695i \(0.985808\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 7.60770i 0.307774i
\(612\) 0 0
\(613\) −34.3923 −1.38909 −0.694546 0.719448i \(-0.744395\pi\)
−0.694546 + 0.719448i \(0.744395\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 34.3923i − 1.38458i −0.721618 0.692291i \(-0.756601\pi\)
0.721618 0.692291i \(-0.243399\pi\)
\(618\) 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
\(623\) 61.1769i 2.45100i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −18.9282 −0.755920
\(628\) 0 0
\(629\) − 20.7846i − 0.828737i
\(630\) 0 0
\(631\) −14.5359 −0.578665 −0.289332 0.957229i \(-0.593433\pi\)
−0.289332 + 0.957229i \(0.593433\pi\)
\(632\) 0 0
\(633\) 39.3205i 1.56285i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −53.3205 −2.11264
\(638\) 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
\(643\) −20.5885 −0.811929 −0.405965 0.913889i \(-0.633064\pi\)
−0.405965 + 0.913889i \(0.633064\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 5.41154i − 0.212750i −0.994326 0.106375i \(-0.966076\pi\)
0.994326 0.106375i \(-0.0339244\pi\)
\(648\) 0 0
\(649\) 20.7846 0.815867
\(650\) 0 0
\(651\) 32.7846i 1.28493i
\(652\) 0 0
\(653\) 43.1769 1.68964 0.844822 0.535048i \(-0.179707\pi\)
0.844822 + 0.535048i \(0.179707\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 28.5359i 1.11329i
\(658\) 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
\(663\) 32.7846i 1.27325i
\(664\) 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
\(673\) 23.1769i 0.893404i 0.894683 + 0.446702i \(0.147402\pi\)
−0.894683 + 0.446702i \(0.852598\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 19.1769 0.737029 0.368514 0.929622i \(-0.379867\pi\)
0.368514 + 0.929622i \(0.379867\pi\)
\(678\) 0 0
\(679\) 68.1051 2.61363
\(680\) 0 0
\(681\) 34.3923 1.31792
\(682\) 0 0
\(683\) 5.66025 0.216584 0.108292 0.994119i \(-0.465462\pi\)
0.108292 + 0.994119i \(0.465462\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 13.8564i 0.528655i
\(688\) 0 0
\(689\) 36.0000 1.37149
\(690\) 0 0
\(691\) − 26.3923i − 1.00401i −0.864865 0.502005i \(-0.832596\pi\)
0.864865 0.502005i \(-0.167404\pi\)
\(692\) 0 0
\(693\) 73.1769 2.77976
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 32.7846i 1.24181i
\(698\) 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
\(703\) 12.0000i 0.452589i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 56.7846 2.13561
\(708\) 0 0
\(709\) − 37.8564i − 1.42173i −0.703330 0.710864i \(-0.748304\pi\)
0.703330 0.710864i \(-0.251696\pi\)
\(710\) 0 0
\(711\) −53.5692 −2.00900
\(712\) 0 0
\(713\) 5.56922i 0.208569i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −56.7846 −2.12066
\(718\) 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
\(723\) 1.07180 0.0398606
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 4.05256i − 0.150301i −0.997172 0.0751505i \(-0.976056\pi\)
0.997172 0.0751505i \(-0.0239437\pi\)
\(728\) 0 0
\(729\) −43.7846 −1.62165
\(730\) 0 0
\(731\) − 0.679492i − 0.0251319i
\(732\) 0 0
\(733\) 2.78461 0.102852 0.0514260 0.998677i \(-0.483623\pi\)
0.0514260 + 0.998677i \(0.483623\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 0.679492i − 0.0250294i
\(738\) 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
\(743\) − 5.41154i − 0.198530i −0.995061 0.0992651i \(-0.968351\pi\)
0.995061 0.0992651i \(-0.0316492\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 5.66025 0.207098
\(748\) 0 0
\(749\) 62.7846i 2.29410i
\(750\) 0 0
\(751\) 19.6077 0.715495 0.357747 0.933818i \(-0.383545\pi\)
0.357747 + 0.933818i \(0.383545\pi\)
\(752\) 0 0
\(753\) − 23.3205i − 0.849847i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −36.9282 −1.34218 −0.671089 0.741377i \(-0.734173\pi\)
−0.671089 + 0.741377i \(0.734173\pi\)
\(758\) 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
\(763\) 61.1769 2.21475
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 20.7846i 0.750489i
\(768\) 0 0
\(769\) −34.7846 −1.25437 −0.627183 0.778872i \(-0.715792\pi\)
−0.627183 + 0.778872i \(0.715792\pi\)
\(770\) 0 0
\(771\) − 16.3923i − 0.590354i
\(772\) 0 0
\(773\) −25.6077 −0.921045 −0.460522 0.887648i \(-0.652338\pi\)
−0.460522 + 0.887648i \(0.652338\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 77.5692i − 2.78278i
\(778\) 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
\(787\) 16.1962 0.577330 0.288665 0.957430i \(-0.406789\pi\)
0.288665 + 0.957430i \(0.406789\pi\)
\(788\) 0 0
\(789\) 26.7846i 0.953557i
\(790\) 0 0
\(791\) 61.1769 2.17520
\(792\) 0 0
\(793\) 3.21539i 0.114182i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 22.3923 0.793176 0.396588 0.917997i \(-0.370194\pi\)
0.396588 + 0.917997i \(0.370194\pi\)
\(798\) 0 0
\(799\) −7.60770 −0.269141
\(800\) 0 0
\(801\) 57.7128 2.03918
\(802\) 0 0
\(803\) 22.1436 0.781430
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 73.1769i 2.57595i
\(808\) 0 0
\(809\) 47.5692 1.67244 0.836222 0.548391i \(-0.184759\pi\)
0.836222 + 0.548391i \(0.184759\pi\)
\(810\) 0 0
\(811\) 17.6077i 0.618290i 0.951015 + 0.309145i \(0.100043\pi\)
−0.951015 + 0.309145i \(0.899957\pi\)
\(812\) 0 0
\(813\) 44.7846 1.57066
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0.392305i 0.0137250i
\(818\) 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
\(823\) 48.8372i 1.70236i 0.524877 + 0.851178i \(0.324111\pi\)
−0.524877 + 0.851178i \(0.675889\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −1.26795 −0.0440909 −0.0220455 0.999757i \(-0.507018\pi\)
−0.0220455 + 0.999757i \(0.507018\pi\)
\(828\) 0 0
\(829\) 9.21539i 0.320064i 0.987112 + 0.160032i \(0.0511597\pi\)
−0.987112 + 0.160032i \(0.948840\pi\)
\(830\) 0 0
\(831\) −21.4641 −0.744581
\(832\) 0 0
\(833\) − 53.3205i − 1.84745i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 10.1436 0.350614
\(838\) 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
\(843\) −20.7846 −0.715860
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 4.73205i − 0.162595i
\(848\) 0 0
\(849\) −56.2487 −1.93045
\(850\) 0 0
\(851\) − 13.1769i − 0.451699i
\(852\) 0 0
\(853\) 3.46410 0.118609 0.0593043 0.998240i \(-0.481112\pi\)
0.0593043 + 0.998240i \(0.481112\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 43.8564i 1.49811i 0.662510 + 0.749053i \(0.269491\pi\)
−0.662510 + 0.749053i \(0.730509\pi\)
\(858\) 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
\(863\) 10.9808i 0.373789i 0.982380 + 0.186895i \(0.0598423\pi\)
−0.982380 + 0.186895i \(0.940158\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 13.6603 0.463927
\(868\) 0 0
\(869\) 41.5692i 1.41014i
\(870\) 0 0
\(871\) 0.679492 0.0230237
\(872\) 0 0
\(873\) − 64.2487i − 2.17449i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 19.8564 0.670503 0.335252 0.942129i \(-0.391179\pi\)
0.335252 + 0.942129i \(0.391179\pi\)
\(878\) 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
\(883\) −16.1962 −0.545044 −0.272522 0.962150i \(-0.587858\pi\)
−0.272522 + 0.962150i \(0.587858\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 51.3731i 1.72494i 0.506109 + 0.862469i \(0.331083\pi\)
−0.506109 + 0.862469i \(0.668917\pi\)
\(888\) 0 0
\(889\) 67.1769 2.25304
\(890\) 0 0
\(891\) 8.53590i 0.285963i
\(892\) 0 0
\(893\) 4.39230 0.146983
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 20.7846i 0.693978i
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 36.0000i 1.19933i
\(902\) 0 0
\(903\) − 2.53590i − 0.0843894i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −32.5885 −1.08208 −0.541041 0.840996i \(-0.681970\pi\)
−0.541041 + 0.840996i \(0.681970\pi\)
\(908\) 0 0
\(909\) − 53.5692i − 1.77678i
\(910\) 0 0
\(911\) −25.1769 −0.834148 −0.417074 0.908872i \(-0.636944\pi\)
−0.417074 + 0.908872i \(0.636944\pi\)
\(912\) 0 0
\(913\) − 4.39230i − 0.145364i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 49.1769 1.62396
\(918\) 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
\(923\) −56.7846 −1.86909
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 9.80385i 0.322001i
\(928\) 0 0
\(929\) −28.3923 −0.931521 −0.465761 0.884911i \(-0.654219\pi\)
−0.465761 + 0.884911i \(0.654219\pi\)
\(930\) 0 0
\(931\) 30.7846i 1.00892i
\(932\) 0 0
\(933\) 77.5692 2.53950
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 30.3923i 0.992873i 0.868073 + 0.496437i \(0.165359\pi\)
−0.868073 + 0.496437i \(0.834641\pi\)
\(938\) 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
\(943\) 20.7846i 0.676840i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 56.1962 1.82613 0.913065 0.407815i \(-0.133709\pi\)
0.913065 + 0.407815i \(0.133709\pi\)
\(948\) 0 0
\(949\) 22.1436i 0.718811i
\(950\) 0 0
\(951\) −4.39230 −0.142430
\(952\) 0 0
\(953\) 11.0718i 0.358651i 0.983790 + 0.179325i \(0.0573915\pi\)
−0.983790 + 0.179325i \(0.942609\pi\)
\(954\) 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
\(963\) 59.2295 1.90864
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 2.19615i 0.0706235i 0.999376 + 0.0353118i \(0.0112424\pi\)
−0.999376 + 0.0353118i \(0.988758\pi\)
\(968\) 0 0
\(969\) 18.9282 0.608061
\(970\) 0 0
\(971\) − 57.0333i − 1.83029i −0.403129 0.915143i \(-0.632077\pi\)
0.403129 0.915143i \(-0.367923\pi\)
\(972\) 0 0
\(973\) −47.3205 −1.51703
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 29.3205i 0.938046i 0.883186 + 0.469023i \(0.155394\pi\)
−0.883186 + 0.469023i \(0.844606\pi\)
\(978\) 0 0
\(979\) − 44.7846i − 1.43132i
\(980\) 0 0
\(981\) − 57.7128i − 1.84263i
\(982\) 0 0
\(983\) 42.5885i 1.35836i 0.733971 + 0.679180i \(0.237665\pi\)
−0.733971 + 0.679180i \(0.762335\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −28.3923 −0.903737
\(988\) 0 0
\(989\) − 0.430781i − 0.0136980i
\(990\) 0 0
\(991\) 32.1051 1.01985 0.509926 0.860218i \(-0.329673\pi\)
0.509926 + 0.860218i \(0.329673\pi\)
\(992\) 0 0
\(993\) − 72.1051i − 2.28819i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 46.3923 1.46926 0.734630 0.678468i \(-0.237356\pi\)
0.734630 + 0.678468i \(0.237356\pi\)
\(998\) 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.f.i.1249.4 4
4.3 odd 2 1600.2.f.d.1249.1 4
5.2 odd 4 320.2.d.a.161.1 4
5.3 odd 4 1600.2.d.h.801.4 4
5.4 even 2 1600.2.f.e.1249.1 4
8.3 odd 2 1600.2.f.h.1249.3 4
8.5 even 2 1600.2.f.e.1249.2 4
15.2 even 4 2880.2.k.e.1441.1 4
20.3 even 4 1600.2.d.b.801.1 4
20.7 even 4 320.2.d.b.161.4 yes 4
20.19 odd 2 1600.2.f.h.1249.4 4
40.3 even 4 1600.2.d.b.801.4 4
40.13 odd 4 1600.2.d.h.801.1 4
40.19 odd 2 1600.2.f.d.1249.2 4
40.27 even 4 320.2.d.b.161.1 yes 4
40.29 even 2 inner 1600.2.f.i.1249.3 4
40.37 odd 4 320.2.d.a.161.4 yes 4
60.47 odd 4 2880.2.k.l.1441.2 4
80.3 even 4 6400.2.a.ck.1.2 2
80.13 odd 4 6400.2.a.y.1.1 2
80.27 even 4 1280.2.a.m.1.2 2
80.37 odd 4 1280.2.a.b.1.1 2
80.43 even 4 6400.2.a.bf.1.1 2
80.53 odd 4 6400.2.a.cd.1.2 2
80.67 even 4 1280.2.a.c.1.1 2
80.77 odd 4 1280.2.a.p.1.2 2
120.77 even 4 2880.2.k.e.1441.3 4
120.107 odd 4 2880.2.k.l.1441.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
320.2.d.a.161.1 4 5.2 odd 4
320.2.d.a.161.4 yes 4 40.37 odd 4
320.2.d.b.161.1 yes 4 40.27 even 4
320.2.d.b.161.4 yes 4 20.7 even 4
1280.2.a.b.1.1 2 80.37 odd 4
1280.2.a.c.1.1 2 80.67 even 4
1280.2.a.m.1.2 2 80.27 even 4
1280.2.a.p.1.2 2 80.77 odd 4
1600.2.d.b.801.1 4 20.3 even 4
1600.2.d.b.801.4 4 40.3 even 4
1600.2.d.h.801.1 4 40.13 odd 4
1600.2.d.h.801.4 4 5.3 odd 4
1600.2.f.d.1249.1 4 4.3 odd 2
1600.2.f.d.1249.2 4 40.19 odd 2
1600.2.f.e.1249.1 4 5.4 even 2
1600.2.f.e.1249.2 4 8.5 even 2
1600.2.f.h.1249.3 4 8.3 odd 2
1600.2.f.h.1249.4 4 20.19 odd 2
1600.2.f.i.1249.3 4 40.29 even 2 inner
1600.2.f.i.1249.4 4 1.1 even 1 trivial
2880.2.k.e.1441.1 4 15.2 even 4
2880.2.k.e.1441.3 4 120.77 even 4
2880.2.k.l.1441.2 4 60.47 odd 4
2880.2.k.l.1441.4 4 120.107 odd 4
6400.2.a.y.1.1 2 80.13 odd 4
6400.2.a.bf.1.1 2 80.43 even 4
6400.2.a.cd.1.2 2 80.53 odd 4
6400.2.a.ck.1.2 2 80.3 even 4