Properties

Label 7728.2.a.bs.1.3
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7728.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(61.7083906820\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
Defining polynomial: \( x^{3} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3864)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.254102\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +2.93543 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +2.93543 q^{5} +1.00000 q^{7} +1.00000 q^{9} +3.68133 q^{11} -6.61676 q^{13} -2.93543 q^{15} -4.18953 q^{17} -1.17313 q^{19} -1.00000 q^{21} +1.00000 q^{23} +3.61676 q^{25} -1.00000 q^{27} +1.68133 q^{29} +1.17313 q^{31} -3.68133 q^{33} +2.93543 q^{35} -2.31867 q^{37} +6.61676 q^{39} -7.17313 q^{41} +2.74590 q^{43} +2.93543 q^{45} -12.2171 q^{47} +1.00000 q^{49} +4.18953 q^{51} -13.1854 q^{53} +10.8063 q^{55} +1.17313 q^{57} +3.06457 q^{59} -0.745898 q^{61} +1.00000 q^{63} -19.4231 q^{65} -8.46004 q^{67} -1.00000 q^{69} +11.5040 q^{71} -10.3187 q^{73} -3.61676 q^{75} +3.68133 q^{77} +6.18953 q^{79} +1.00000 q^{81} -1.33508 q^{83} -12.2981 q^{85} -1.68133 q^{87} +6.14137 q^{89} -6.61676 q^{91} -1.17313 q^{93} -3.44364 q^{95} -6.69774 q^{97} +3.68133 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{3} + q^{5} + 3 q^{7} + 3 q^{9} + 4 q^{11} - 5 q^{13} - q^{15} - 4 q^{17} + 2 q^{19} - 3 q^{21} + 3 q^{23} - 4 q^{25} - 3 q^{27} - 2 q^{29} - 2 q^{31} - 4 q^{33} + q^{35} - 14 q^{37} + 5 q^{39} - 16 q^{41} + 9 q^{43} + q^{45} - 10 q^{47} + 3 q^{49} + 4 q^{51} + q^{53} + 9 q^{55} - 2 q^{57} + 17 q^{59} - 3 q^{61} + 3 q^{63} - 20 q^{65} - 13 q^{67} - 3 q^{69} + q^{71} - 38 q^{73} + 4 q^{75} + 4 q^{77} + 10 q^{79} + 3 q^{81} - 8 q^{83} - 15 q^{85} + 2 q^{87} - q^{89} - 5 q^{91} + 2 q^{93} - q^{95} - 10 q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 2.93543 1.31277 0.656383 0.754428i \(-0.272086\pi\)
0.656383 + 0.754428i \(0.272086\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.68133 1.10996 0.554981 0.831863i \(-0.312725\pi\)
0.554981 + 0.831863i \(0.312725\pi\)
\(12\) 0 0
\(13\) −6.61676 −1.83516 −0.917580 0.397551i \(-0.869860\pi\)
−0.917580 + 0.397551i \(0.869860\pi\)
\(14\) 0 0
\(15\) −2.93543 −0.757925
\(16\) 0 0
\(17\) −4.18953 −1.01611 −0.508056 0.861324i \(-0.669636\pi\)
−0.508056 + 0.861324i \(0.669636\pi\)
\(18\) 0 0
\(19\) −1.17313 −0.269134 −0.134567 0.990905i \(-0.542964\pi\)
−0.134567 + 0.990905i \(0.542964\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 3.61676 0.723353
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.68133 0.312215 0.156108 0.987740i \(-0.450105\pi\)
0.156108 + 0.987740i \(0.450105\pi\)
\(30\) 0 0
\(31\) 1.17313 0.210700 0.105350 0.994435i \(-0.466404\pi\)
0.105350 + 0.994435i \(0.466404\pi\)
\(32\) 0 0
\(33\) −3.68133 −0.640837
\(34\) 0 0
\(35\) 2.93543 0.496179
\(36\) 0 0
\(37\) −2.31867 −0.381187 −0.190593 0.981669i \(-0.561041\pi\)
−0.190593 + 0.981669i \(0.561041\pi\)
\(38\) 0 0
\(39\) 6.61676 1.05953
\(40\) 0 0
\(41\) −7.17313 −1.12025 −0.560127 0.828407i \(-0.689248\pi\)
−0.560127 + 0.828407i \(0.689248\pi\)
\(42\) 0 0
\(43\) 2.74590 0.418746 0.209373 0.977836i \(-0.432858\pi\)
0.209373 + 0.977836i \(0.432858\pi\)
\(44\) 0 0
\(45\) 2.93543 0.437588
\(46\) 0 0
\(47\) −12.2171 −1.78205 −0.891025 0.453954i \(-0.850013\pi\)
−0.891025 + 0.453954i \(0.850013\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 4.18953 0.586652
\(52\) 0 0
\(53\) −13.1854 −1.81115 −0.905575 0.424187i \(-0.860560\pi\)
−0.905575 + 0.424187i \(0.860560\pi\)
\(54\) 0 0
\(55\) 10.8063 1.45712
\(56\) 0 0
\(57\) 1.17313 0.155385
\(58\) 0 0
\(59\) 3.06457 0.398973 0.199486 0.979901i \(-0.436073\pi\)
0.199486 + 0.979901i \(0.436073\pi\)
\(60\) 0 0
\(61\) −0.745898 −0.0955025 −0.0477512 0.998859i \(-0.515205\pi\)
−0.0477512 + 0.998859i \(0.515205\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) −19.4231 −2.40913
\(66\) 0 0
\(67\) −8.46004 −1.03356 −0.516779 0.856119i \(-0.672869\pi\)
−0.516779 + 0.856119i \(0.672869\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 11.5040 1.36528 0.682639 0.730756i \(-0.260832\pi\)
0.682639 + 0.730756i \(0.260832\pi\)
\(72\) 0 0
\(73\) −10.3187 −1.20771 −0.603854 0.797095i \(-0.706369\pi\)
−0.603854 + 0.797095i \(0.706369\pi\)
\(74\) 0 0
\(75\) −3.61676 −0.417628
\(76\) 0 0
\(77\) 3.68133 0.419527
\(78\) 0 0
\(79\) 6.18953 0.696377 0.348188 0.937425i \(-0.386797\pi\)
0.348188 + 0.937425i \(0.386797\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −1.33508 −0.146544 −0.0732718 0.997312i \(-0.523344\pi\)
−0.0732718 + 0.997312i \(0.523344\pi\)
\(84\) 0 0
\(85\) −12.2981 −1.33392
\(86\) 0 0
\(87\) −1.68133 −0.180258
\(88\) 0 0
\(89\) 6.14137 0.650984 0.325492 0.945545i \(-0.394470\pi\)
0.325492 + 0.945545i \(0.394470\pi\)
\(90\) 0 0
\(91\) −6.61676 −0.693625
\(92\) 0 0
\(93\) −1.17313 −0.121648
\(94\) 0 0
\(95\) −3.44364 −0.353310
\(96\) 0 0
\(97\) −6.69774 −0.680052 −0.340026 0.940416i \(-0.610436\pi\)
−0.340026 + 0.940416i \(0.610436\pi\)
\(98\) 0 0
\(99\) 3.68133 0.369988
\(100\) 0 0
\(101\) −17.6608 −1.75731 −0.878655 0.477456i \(-0.841559\pi\)
−0.878655 + 0.477456i \(0.841559\pi\)
\(102\) 0 0
\(103\) −5.01641 −0.494281 −0.247141 0.968980i \(-0.579491\pi\)
−0.247141 + 0.968980i \(0.579491\pi\)
\(104\) 0 0
\(105\) −2.93543 −0.286469
\(106\) 0 0
\(107\) −6.48763 −0.627183 −0.313591 0.949558i \(-0.601532\pi\)
−0.313591 + 0.949558i \(0.601532\pi\)
\(108\) 0 0
\(109\) 15.1526 1.45135 0.725676 0.688037i \(-0.241527\pi\)
0.725676 + 0.688037i \(0.241527\pi\)
\(110\) 0 0
\(111\) 2.31867 0.220078
\(112\) 0 0
\(113\) −4.90262 −0.461200 −0.230600 0.973049i \(-0.574069\pi\)
−0.230600 + 0.973049i \(0.574069\pi\)
\(114\) 0 0
\(115\) 2.93543 0.273730
\(116\) 0 0
\(117\) −6.61676 −0.611720
\(118\) 0 0
\(119\) −4.18953 −0.384054
\(120\) 0 0
\(121\) 2.55220 0.232018
\(122\) 0 0
\(123\) 7.17313 0.646779
\(124\) 0 0
\(125\) −4.06040 −0.363173
\(126\) 0 0
\(127\) 1.51237 0.134201 0.0671007 0.997746i \(-0.478625\pi\)
0.0671007 + 0.997746i \(0.478625\pi\)
\(128\) 0 0
\(129\) −2.74590 −0.241763
\(130\) 0 0
\(131\) −13.2939 −1.16150 −0.580748 0.814084i \(-0.697240\pi\)
−0.580748 + 0.814084i \(0.697240\pi\)
\(132\) 0 0
\(133\) −1.17313 −0.101723
\(134\) 0 0
\(135\) −2.93543 −0.252642
\(136\) 0 0
\(137\) −17.8021 −1.52094 −0.760469 0.649374i \(-0.775031\pi\)
−0.760469 + 0.649374i \(0.775031\pi\)
\(138\) 0 0
\(139\) −4.29809 −0.364560 −0.182280 0.983247i \(-0.558348\pi\)
−0.182280 + 0.983247i \(0.558348\pi\)
\(140\) 0 0
\(141\) 12.2171 1.02887
\(142\) 0 0
\(143\) −24.3585 −2.03696
\(144\) 0 0
\(145\) 4.93543 0.409865
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) 6.85446 0.561539 0.280770 0.959775i \(-0.409410\pi\)
0.280770 + 0.959775i \(0.409410\pi\)
\(150\) 0 0
\(151\) 11.2663 0.916842 0.458421 0.888735i \(-0.348415\pi\)
0.458421 + 0.888735i \(0.348415\pi\)
\(152\) 0 0
\(153\) −4.18953 −0.338704
\(154\) 0 0
\(155\) 3.44364 0.276599
\(156\) 0 0
\(157\) 13.2663 1.05877 0.529385 0.848382i \(-0.322423\pi\)
0.529385 + 0.848382i \(0.322423\pi\)
\(158\) 0 0
\(159\) 13.1854 1.04567
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) −11.1854 −0.876105 −0.438053 0.898949i \(-0.644332\pi\)
−0.438053 + 0.898949i \(0.644332\pi\)
\(164\) 0 0
\(165\) −10.8063 −0.841269
\(166\) 0 0
\(167\) −20.9149 −1.61844 −0.809220 0.587506i \(-0.800110\pi\)
−0.809220 + 0.587506i \(0.800110\pi\)
\(168\) 0 0
\(169\) 30.7816 2.36781
\(170\) 0 0
\(171\) −1.17313 −0.0897113
\(172\) 0 0
\(173\) 2.15672 0.163972 0.0819862 0.996633i \(-0.473874\pi\)
0.0819862 + 0.996633i \(0.473874\pi\)
\(174\) 0 0
\(175\) 3.61676 0.273402
\(176\) 0 0
\(177\) −3.06457 −0.230347
\(178\) 0 0
\(179\) 9.47645 0.708303 0.354152 0.935188i \(-0.384770\pi\)
0.354152 + 0.935188i \(0.384770\pi\)
\(180\) 0 0
\(181\) −2.95601 −0.219718 −0.109859 0.993947i \(-0.535040\pi\)
−0.109859 + 0.993947i \(0.535040\pi\)
\(182\) 0 0
\(183\) 0.745898 0.0551384
\(184\) 0 0
\(185\) −6.80630 −0.500409
\(186\) 0 0
\(187\) −15.4231 −1.12785
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −11.2663 −0.815204 −0.407602 0.913160i \(-0.633635\pi\)
−0.407602 + 0.913160i \(0.633635\pi\)
\(192\) 0 0
\(193\) −4.76647 −0.343098 −0.171549 0.985176i \(-0.554877\pi\)
−0.171549 + 0.985176i \(0.554877\pi\)
\(194\) 0 0
\(195\) 19.4231 1.39091
\(196\) 0 0
\(197\) 27.9711 1.99286 0.996429 0.0844385i \(-0.0269097\pi\)
0.996429 + 0.0844385i \(0.0269097\pi\)
\(198\) 0 0
\(199\) −13.3145 −0.943840 −0.471920 0.881641i \(-0.656439\pi\)
−0.471920 + 0.881641i \(0.656439\pi\)
\(200\) 0 0
\(201\) 8.46004 0.596725
\(202\) 0 0
\(203\) 1.68133 0.118006
\(204\) 0 0
\(205\) −21.0562 −1.47063
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) −4.31867 −0.298729
\(210\) 0 0
\(211\) 20.8185 1.43321 0.716604 0.697481i \(-0.245696\pi\)
0.716604 + 0.697481i \(0.245696\pi\)
\(212\) 0 0
\(213\) −11.5040 −0.788243
\(214\) 0 0
\(215\) 8.06040 0.549715
\(216\) 0 0
\(217\) 1.17313 0.0796371
\(218\) 0 0
\(219\) 10.3187 0.697271
\(220\) 0 0
\(221\) 27.7212 1.86473
\(222\) 0 0
\(223\) 16.8943 1.13132 0.565662 0.824637i \(-0.308621\pi\)
0.565662 + 0.824637i \(0.308621\pi\)
\(224\) 0 0
\(225\) 3.61676 0.241118
\(226\) 0 0
\(227\) 29.2733 1.94294 0.971470 0.237162i \(-0.0762171\pi\)
0.971470 + 0.237162i \(0.0762171\pi\)
\(228\) 0 0
\(229\) −11.7899 −0.779098 −0.389549 0.921006i \(-0.627369\pi\)
−0.389549 + 0.921006i \(0.627369\pi\)
\(230\) 0 0
\(231\) −3.68133 −0.242214
\(232\) 0 0
\(233\) 9.34731 0.612363 0.306181 0.951973i \(-0.400949\pi\)
0.306181 + 0.951973i \(0.400949\pi\)
\(234\) 0 0
\(235\) −35.8625 −2.33941
\(236\) 0 0
\(237\) −6.18953 −0.402053
\(238\) 0 0
\(239\) 13.6279 0.881518 0.440759 0.897625i \(-0.354709\pi\)
0.440759 + 0.897625i \(0.354709\pi\)
\(240\) 0 0
\(241\) −19.0768 −1.22885 −0.614423 0.788977i \(-0.710611\pi\)
−0.614423 + 0.788977i \(0.710611\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 2.93543 0.187538
\(246\) 0 0
\(247\) 7.76231 0.493904
\(248\) 0 0
\(249\) 1.33508 0.0846070
\(250\) 0 0
\(251\) 20.3379 1.28372 0.641859 0.766823i \(-0.278163\pi\)
0.641859 + 0.766823i \(0.278163\pi\)
\(252\) 0 0
\(253\) 3.68133 0.231443
\(254\) 0 0
\(255\) 12.2981 0.770136
\(256\) 0 0
\(257\) 19.5470 1.21931 0.609653 0.792668i \(-0.291309\pi\)
0.609653 + 0.792668i \(0.291309\pi\)
\(258\) 0 0
\(259\) −2.31867 −0.144075
\(260\) 0 0
\(261\) 1.68133 0.104072
\(262\) 0 0
\(263\) −30.3103 −1.86902 −0.934508 0.355943i \(-0.884160\pi\)
−0.934508 + 0.355943i \(0.884160\pi\)
\(264\) 0 0
\(265\) −38.7047 −2.37761
\(266\) 0 0
\(267\) −6.14137 −0.375846
\(268\) 0 0
\(269\) −18.3585 −1.11934 −0.559669 0.828717i \(-0.689072\pi\)
−0.559669 + 0.828717i \(0.689072\pi\)
\(270\) 0 0
\(271\) 17.1320 1.04069 0.520347 0.853955i \(-0.325803\pi\)
0.520347 + 0.853955i \(0.325803\pi\)
\(272\) 0 0
\(273\) 6.61676 0.400465
\(274\) 0 0
\(275\) 13.3145 0.802895
\(276\) 0 0
\(277\) −7.09215 −0.426126 −0.213063 0.977038i \(-0.568344\pi\)
−0.213063 + 0.977038i \(0.568344\pi\)
\(278\) 0 0
\(279\) 1.17313 0.0702333
\(280\) 0 0
\(281\) 22.2171 1.32536 0.662681 0.748902i \(-0.269418\pi\)
0.662681 + 0.748902i \(0.269418\pi\)
\(282\) 0 0
\(283\) 4.83911 0.287655 0.143828 0.989603i \(-0.454059\pi\)
0.143828 + 0.989603i \(0.454059\pi\)
\(284\) 0 0
\(285\) 3.44364 0.203983
\(286\) 0 0
\(287\) −7.17313 −0.423416
\(288\) 0 0
\(289\) 0.552195 0.0324821
\(290\) 0 0
\(291\) 6.69774 0.392628
\(292\) 0 0
\(293\) 22.3791 1.30740 0.653700 0.756754i \(-0.273216\pi\)
0.653700 + 0.756754i \(0.273216\pi\)
\(294\) 0 0
\(295\) 8.99583 0.523758
\(296\) 0 0
\(297\) −3.68133 −0.213612
\(298\) 0 0
\(299\) −6.61676 −0.382657
\(300\) 0 0
\(301\) 2.74590 0.158271
\(302\) 0 0
\(303\) 17.6608 1.01458
\(304\) 0 0
\(305\) −2.18953 −0.125372
\(306\) 0 0
\(307\) 8.60142 0.490909 0.245454 0.969408i \(-0.421063\pi\)
0.245454 + 0.969408i \(0.421063\pi\)
\(308\) 0 0
\(309\) 5.01641 0.285373
\(310\) 0 0
\(311\) −11.6660 −0.661517 −0.330759 0.943715i \(-0.607305\pi\)
−0.330759 + 0.943715i \(0.607305\pi\)
\(312\) 0 0
\(313\) 21.8625 1.23574 0.617872 0.786279i \(-0.287995\pi\)
0.617872 + 0.786279i \(0.287995\pi\)
\(314\) 0 0
\(315\) 2.93543 0.165393
\(316\) 0 0
\(317\) 0.685500 0.0385015 0.0192507 0.999815i \(-0.493872\pi\)
0.0192507 + 0.999815i \(0.493872\pi\)
\(318\) 0 0
\(319\) 6.18953 0.346547
\(320\) 0 0
\(321\) 6.48763 0.362104
\(322\) 0 0
\(323\) 4.91486 0.273470
\(324\) 0 0
\(325\) −23.9313 −1.32747
\(326\) 0 0
\(327\) −15.1526 −0.837938
\(328\) 0 0
\(329\) −12.2171 −0.673552
\(330\) 0 0
\(331\) −24.8133 −1.36386 −0.681931 0.731416i \(-0.738860\pi\)
−0.681931 + 0.731416i \(0.738860\pi\)
\(332\) 0 0
\(333\) −2.31867 −0.127062
\(334\) 0 0
\(335\) −24.8339 −1.35682
\(336\) 0 0
\(337\) −4.96302 −0.270353 −0.135176 0.990822i \(-0.543160\pi\)
−0.135176 + 0.990822i \(0.543160\pi\)
\(338\) 0 0
\(339\) 4.90262 0.266274
\(340\) 0 0
\(341\) 4.31867 0.233869
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) −2.93543 −0.158038
\(346\) 0 0
\(347\) −13.1731 −0.707171 −0.353585 0.935402i \(-0.615038\pi\)
−0.353585 + 0.935402i \(0.615038\pi\)
\(348\) 0 0
\(349\) −15.7899 −0.845213 −0.422607 0.906313i \(-0.638885\pi\)
−0.422607 + 0.906313i \(0.638885\pi\)
\(350\) 0 0
\(351\) 6.61676 0.353177
\(352\) 0 0
\(353\) 29.0028 1.54367 0.771833 0.635826i \(-0.219340\pi\)
0.771833 + 0.635826i \(0.219340\pi\)
\(354\) 0 0
\(355\) 33.7693 1.79229
\(356\) 0 0
\(357\) 4.18953 0.221734
\(358\) 0 0
\(359\) 5.19370 0.274113 0.137057 0.990563i \(-0.456236\pi\)
0.137057 + 0.990563i \(0.456236\pi\)
\(360\) 0 0
\(361\) −17.6238 −0.927567
\(362\) 0 0
\(363\) −2.55220 −0.133956
\(364\) 0 0
\(365\) −30.2898 −1.58544
\(366\) 0 0
\(367\) −10.5839 −0.552478 −0.276239 0.961089i \(-0.589088\pi\)
−0.276239 + 0.961089i \(0.589088\pi\)
\(368\) 0 0
\(369\) −7.17313 −0.373418
\(370\) 0 0
\(371\) −13.1854 −0.684550
\(372\) 0 0
\(373\) −23.0112 −1.19147 −0.595737 0.803180i \(-0.703140\pi\)
−0.595737 + 0.803180i \(0.703140\pi\)
\(374\) 0 0
\(375\) 4.06040 0.209678
\(376\) 0 0
\(377\) −11.1250 −0.572965
\(378\) 0 0
\(379\) −27.7417 −1.42500 −0.712498 0.701674i \(-0.752436\pi\)
−0.712498 + 0.701674i \(0.752436\pi\)
\(380\) 0 0
\(381\) −1.51237 −0.0774812
\(382\) 0 0
\(383\) 10.5275 0.537928 0.268964 0.963150i \(-0.413319\pi\)
0.268964 + 0.963150i \(0.413319\pi\)
\(384\) 0 0
\(385\) 10.8063 0.550740
\(386\) 0 0
\(387\) 2.74590 0.139582
\(388\) 0 0
\(389\) −25.0028 −1.26769 −0.633847 0.773458i \(-0.718525\pi\)
−0.633847 + 0.773458i \(0.718525\pi\)
\(390\) 0 0
\(391\) −4.18953 −0.211874
\(392\) 0 0
\(393\) 13.2939 0.670590
\(394\) 0 0
\(395\) 18.1690 0.914180
\(396\) 0 0
\(397\) 13.6454 0.684843 0.342422 0.939546i \(-0.388753\pi\)
0.342422 + 0.939546i \(0.388753\pi\)
\(398\) 0 0
\(399\) 1.17313 0.0587298
\(400\) 0 0
\(401\) −5.46421 −0.272870 −0.136435 0.990649i \(-0.543564\pi\)
−0.136435 + 0.990649i \(0.543564\pi\)
\(402\) 0 0
\(403\) −7.76231 −0.386668
\(404\) 0 0
\(405\) 2.93543 0.145863
\(406\) 0 0
\(407\) −8.53579 −0.423103
\(408\) 0 0
\(409\) −22.2223 −1.09882 −0.549412 0.835551i \(-0.685148\pi\)
−0.549412 + 0.835551i \(0.685148\pi\)
\(410\) 0 0
\(411\) 17.8021 0.878114
\(412\) 0 0
\(413\) 3.06457 0.150798
\(414\) 0 0
\(415\) −3.91903 −0.192377
\(416\) 0 0
\(417\) 4.29809 0.210479
\(418\) 0 0
\(419\) 5.00106 0.244318 0.122159 0.992511i \(-0.461018\pi\)
0.122159 + 0.992511i \(0.461018\pi\)
\(420\) 0 0
\(421\) −6.83388 −0.333063 −0.166532 0.986036i \(-0.553257\pi\)
−0.166532 + 0.986036i \(0.553257\pi\)
\(422\) 0 0
\(423\) −12.2171 −0.594017
\(424\) 0 0
\(425\) −15.1526 −0.735007
\(426\) 0 0
\(427\) −0.745898 −0.0360965
\(428\) 0 0
\(429\) 24.3585 1.17604
\(430\) 0 0
\(431\) 16.9547 0.816678 0.408339 0.912830i \(-0.366108\pi\)
0.408339 + 0.912830i \(0.366108\pi\)
\(432\) 0 0
\(433\) −15.5574 −0.747642 −0.373821 0.927501i \(-0.621953\pi\)
−0.373821 + 0.927501i \(0.621953\pi\)
\(434\) 0 0
\(435\) −4.93543 −0.236636
\(436\) 0 0
\(437\) −1.17313 −0.0561183
\(438\) 0 0
\(439\) 33.4147 1.59480 0.797399 0.603453i \(-0.206209\pi\)
0.797399 + 0.603453i \(0.206209\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −25.4506 −1.20920 −0.604598 0.796531i \(-0.706666\pi\)
−0.604598 + 0.796531i \(0.706666\pi\)
\(444\) 0 0
\(445\) 18.0276 0.854589
\(446\) 0 0
\(447\) −6.85446 −0.324205
\(448\) 0 0
\(449\) −39.0562 −1.84318 −0.921589 0.388168i \(-0.873108\pi\)
−0.921589 + 0.388168i \(0.873108\pi\)
\(450\) 0 0
\(451\) −26.4067 −1.24344
\(452\) 0 0
\(453\) −11.2663 −0.529339
\(454\) 0 0
\(455\) −19.4231 −0.910567
\(456\) 0 0
\(457\) −31.2182 −1.46032 −0.730162 0.683274i \(-0.760556\pi\)
−0.730162 + 0.683274i \(0.760556\pi\)
\(458\) 0 0
\(459\) 4.18953 0.195551
\(460\) 0 0
\(461\) 0.806297 0.0375530 0.0187765 0.999824i \(-0.494023\pi\)
0.0187765 + 0.999824i \(0.494023\pi\)
\(462\) 0 0
\(463\) −4.15672 −0.193179 −0.0965896 0.995324i \(-0.530793\pi\)
−0.0965896 + 0.995324i \(0.530793\pi\)
\(464\) 0 0
\(465\) −3.44364 −0.159695
\(466\) 0 0
\(467\) −8.69774 −0.402483 −0.201242 0.979542i \(-0.564498\pi\)
−0.201242 + 0.979542i \(0.564498\pi\)
\(468\) 0 0
\(469\) −8.46004 −0.390648
\(470\) 0 0
\(471\) −13.2663 −0.611281
\(472\) 0 0
\(473\) 10.1086 0.464792
\(474\) 0 0
\(475\) −4.24292 −0.194679
\(476\) 0 0
\(477\) −13.1854 −0.603716
\(478\) 0 0
\(479\) −21.8656 −0.999066 −0.499533 0.866295i \(-0.666495\pi\)
−0.499533 + 0.866295i \(0.666495\pi\)
\(480\) 0 0
\(481\) 15.3421 0.699539
\(482\) 0 0
\(483\) −1.00000 −0.0455016
\(484\) 0 0
\(485\) −19.6608 −0.892749
\(486\) 0 0
\(487\) −33.2939 −1.50869 −0.754346 0.656477i \(-0.772046\pi\)
−0.754346 + 0.656477i \(0.772046\pi\)
\(488\) 0 0
\(489\) 11.1854 0.505820
\(490\) 0 0
\(491\) −29.4405 −1.32863 −0.664316 0.747452i \(-0.731277\pi\)
−0.664316 + 0.747452i \(0.731277\pi\)
\(492\) 0 0
\(493\) −7.04399 −0.317245
\(494\) 0 0
\(495\) 10.8063 0.485707
\(496\) 0 0
\(497\) 11.5040 0.516026
\(498\) 0 0
\(499\) 39.0786 1.74940 0.874699 0.484667i \(-0.161059\pi\)
0.874699 + 0.484667i \(0.161059\pi\)
\(500\) 0 0
\(501\) 20.9149 0.934407
\(502\) 0 0
\(503\) −37.4353 −1.66916 −0.834579 0.550889i \(-0.814289\pi\)
−0.834579 + 0.550889i \(0.814289\pi\)
\(504\) 0 0
\(505\) −51.8420 −2.30694
\(506\) 0 0
\(507\) −30.7816 −1.36706
\(508\) 0 0
\(509\) 4.12914 0.183021 0.0915104 0.995804i \(-0.470831\pi\)
0.0915104 + 0.995804i \(0.470831\pi\)
\(510\) 0 0
\(511\) −10.3187 −0.456471
\(512\) 0 0
\(513\) 1.17313 0.0517948
\(514\) 0 0
\(515\) −14.7253 −0.648875
\(516\) 0 0
\(517\) −44.9753 −1.97801
\(518\) 0 0
\(519\) −2.15672 −0.0946695
\(520\) 0 0
\(521\) 31.0164 1.35885 0.679427 0.733743i \(-0.262229\pi\)
0.679427 + 0.733743i \(0.262229\pi\)
\(522\) 0 0
\(523\) 20.3791 0.891114 0.445557 0.895253i \(-0.353006\pi\)
0.445557 + 0.895253i \(0.353006\pi\)
\(524\) 0 0
\(525\) −3.61676 −0.157848
\(526\) 0 0
\(527\) −4.91486 −0.214095
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 3.06457 0.132991
\(532\) 0 0
\(533\) 47.4629 2.05585
\(534\) 0 0
\(535\) −19.0440 −0.823344
\(536\) 0 0
\(537\) −9.47645 −0.408939
\(538\) 0 0
\(539\) 3.68133 0.158566
\(540\) 0 0
\(541\) −8.12914 −0.349499 −0.174749 0.984613i \(-0.555912\pi\)
−0.174749 + 0.984613i \(0.555912\pi\)
\(542\) 0 0
\(543\) 2.95601 0.126854
\(544\) 0 0
\(545\) 44.4793 1.90528
\(546\) 0 0
\(547\) −30.2898 −1.29510 −0.647548 0.762024i \(-0.724206\pi\)
−0.647548 + 0.762024i \(0.724206\pi\)
\(548\) 0 0
\(549\) −0.745898 −0.0318342
\(550\) 0 0
\(551\) −1.97241 −0.0840277
\(552\) 0 0
\(553\) 6.18953 0.263206
\(554\) 0 0
\(555\) 6.80630 0.288911
\(556\) 0 0
\(557\) −2.97526 −0.126066 −0.0630328 0.998011i \(-0.520077\pi\)
−0.0630328 + 0.998011i \(0.520077\pi\)
\(558\) 0 0
\(559\) −18.1690 −0.768465
\(560\) 0 0
\(561\) 15.4231 0.651162
\(562\) 0 0
\(563\) 18.1882 0.766541 0.383271 0.923636i \(-0.374798\pi\)
0.383271 + 0.923636i \(0.374798\pi\)
\(564\) 0 0
\(565\) −14.3913 −0.605447
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) 13.8625 0.581147 0.290574 0.956853i \(-0.406154\pi\)
0.290574 + 0.956853i \(0.406154\pi\)
\(570\) 0 0
\(571\) 37.5522 1.57151 0.785755 0.618538i \(-0.212275\pi\)
0.785755 + 0.618538i \(0.212275\pi\)
\(572\) 0 0
\(573\) 11.2663 0.470658
\(574\) 0 0
\(575\) 3.61676 0.150829
\(576\) 0 0
\(577\) 9.80736 0.408286 0.204143 0.978941i \(-0.434559\pi\)
0.204143 + 0.978941i \(0.434559\pi\)
\(578\) 0 0
\(579\) 4.76647 0.198088
\(580\) 0 0
\(581\) −1.33508 −0.0553883
\(582\) 0 0
\(583\) −48.5397 −2.01031
\(584\) 0 0
\(585\) −19.4231 −0.803045
\(586\) 0 0
\(587\) 1.21295 0.0500638 0.0250319 0.999687i \(-0.492031\pi\)
0.0250319 + 0.999687i \(0.492031\pi\)
\(588\) 0 0
\(589\) −1.37623 −0.0567065
\(590\) 0 0
\(591\) −27.9711 −1.15058
\(592\) 0 0
\(593\) −32.9424 −1.35278 −0.676392 0.736542i \(-0.736457\pi\)
−0.676392 + 0.736542i \(0.736457\pi\)
\(594\) 0 0
\(595\) −12.2981 −0.504173
\(596\) 0 0
\(597\) 13.3145 0.544926
\(598\) 0 0
\(599\) −7.92425 −0.323776 −0.161888 0.986809i \(-0.551758\pi\)
−0.161888 + 0.986809i \(0.551758\pi\)
\(600\) 0 0
\(601\) −0.0893124 −0.00364313 −0.00182156 0.999998i \(-0.500580\pi\)
−0.00182156 + 0.999998i \(0.500580\pi\)
\(602\) 0 0
\(603\) −8.46004 −0.344520
\(604\) 0 0
\(605\) 7.49180 0.304585
\(606\) 0 0
\(607\) 39.5592 1.60566 0.802829 0.596209i \(-0.203327\pi\)
0.802829 + 0.596209i \(0.203327\pi\)
\(608\) 0 0
\(609\) −1.68133 −0.0681310
\(610\) 0 0
\(611\) 80.8378 3.27035
\(612\) 0 0
\(613\) −8.68940 −0.350962 −0.175481 0.984483i \(-0.556148\pi\)
−0.175481 + 0.984483i \(0.556148\pi\)
\(614\) 0 0
\(615\) 21.0562 0.849069
\(616\) 0 0
\(617\) −20.3668 −0.819938 −0.409969 0.912100i \(-0.634460\pi\)
−0.409969 + 0.912100i \(0.634460\pi\)
\(618\) 0 0
\(619\) −5.56754 −0.223778 −0.111889 0.993721i \(-0.535690\pi\)
−0.111889 + 0.993721i \(0.535690\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 6.14137 0.246049
\(624\) 0 0
\(625\) −30.0028 −1.20011
\(626\) 0 0
\(627\) 4.31867 0.172471
\(628\) 0 0
\(629\) 9.71414 0.387328
\(630\) 0 0
\(631\) −10.7306 −0.427176 −0.213588 0.976924i \(-0.568515\pi\)
−0.213588 + 0.976924i \(0.568515\pi\)
\(632\) 0 0
\(633\) −20.8185 −0.827462
\(634\) 0 0
\(635\) 4.43947 0.176175
\(636\) 0 0
\(637\) −6.61676 −0.262166
\(638\) 0 0
\(639\) 11.5040 0.455093
\(640\) 0 0
\(641\) 19.9466 0.787844 0.393922 0.919144i \(-0.371118\pi\)
0.393922 + 0.919144i \(0.371118\pi\)
\(642\) 0 0
\(643\) 14.6084 0.576100 0.288050 0.957615i \(-0.406993\pi\)
0.288050 + 0.957615i \(0.406993\pi\)
\(644\) 0 0
\(645\) −8.06040 −0.317378
\(646\) 0 0
\(647\) −28.0398 −1.10236 −0.551180 0.834387i \(-0.685822\pi\)
−0.551180 + 0.834387i \(0.685822\pi\)
\(648\) 0 0
\(649\) 11.2817 0.442845
\(650\) 0 0
\(651\) −1.17313 −0.0459785
\(652\) 0 0
\(653\) 22.7323 0.889585 0.444792 0.895634i \(-0.353277\pi\)
0.444792 + 0.895634i \(0.353277\pi\)
\(654\) 0 0
\(655\) −39.0234 −1.52477
\(656\) 0 0
\(657\) −10.3187 −0.402570
\(658\) 0 0
\(659\) 13.7141 0.534227 0.267114 0.963665i \(-0.413930\pi\)
0.267114 + 0.963665i \(0.413930\pi\)
\(660\) 0 0
\(661\) −11.7141 −0.455627 −0.227814 0.973705i \(-0.573158\pi\)
−0.227814 + 0.973705i \(0.573158\pi\)
\(662\) 0 0
\(663\) −27.7212 −1.07660
\(664\) 0 0
\(665\) −3.44364 −0.133538
\(666\) 0 0
\(667\) 1.68133 0.0651014
\(668\) 0 0
\(669\) −16.8943 −0.653171
\(670\) 0 0
\(671\) −2.74590 −0.106004
\(672\) 0 0
\(673\) −14.0192 −0.540402 −0.270201 0.962804i \(-0.587090\pi\)
−0.270201 + 0.962804i \(0.587090\pi\)
\(674\) 0 0
\(675\) −3.61676 −0.139209
\(676\) 0 0
\(677\) 6.30332 0.242256 0.121128 0.992637i \(-0.461349\pi\)
0.121128 + 0.992637i \(0.461349\pi\)
\(678\) 0 0
\(679\) −6.69774 −0.257036
\(680\) 0 0
\(681\) −29.2733 −1.12176
\(682\) 0 0
\(683\) 27.8984 1.06750 0.533752 0.845641i \(-0.320781\pi\)
0.533752 + 0.845641i \(0.320781\pi\)
\(684\) 0 0
\(685\) −52.2569 −1.99664
\(686\) 0 0
\(687\) 11.7899 0.449812
\(688\) 0 0
\(689\) 87.2444 3.32375
\(690\) 0 0
\(691\) −32.1414 −1.22272 −0.611358 0.791354i \(-0.709376\pi\)
−0.611358 + 0.791354i \(0.709376\pi\)
\(692\) 0 0
\(693\) 3.68133 0.139842
\(694\) 0 0
\(695\) −12.6168 −0.478581
\(696\) 0 0
\(697\) 30.0521 1.13830
\(698\) 0 0
\(699\) −9.34731 −0.353548
\(700\) 0 0
\(701\) −5.02342 −0.189732 −0.0948659 0.995490i \(-0.530242\pi\)
−0.0948659 + 0.995490i \(0.530242\pi\)
\(702\) 0 0
\(703\) 2.72009 0.102590
\(704\) 0 0
\(705\) 35.8625 1.35066
\(706\) 0 0
\(707\) −17.6608 −0.664201
\(708\) 0 0
\(709\) 6.26217 0.235181 0.117590 0.993062i \(-0.462483\pi\)
0.117590 + 0.993062i \(0.462483\pi\)
\(710\) 0 0
\(711\) 6.18953 0.232126
\(712\) 0 0
\(713\) 1.17313 0.0439340
\(714\) 0 0
\(715\) −71.5027 −2.67405
\(716\) 0 0
\(717\) −13.6279 −0.508945
\(718\) 0 0
\(719\) −34.2140 −1.27597 −0.637984 0.770050i \(-0.720231\pi\)
−0.637984 + 0.770050i \(0.720231\pi\)
\(720\) 0 0
\(721\) −5.01641 −0.186821
\(722\) 0 0
\(723\) 19.0768 0.709474
\(724\) 0 0
\(725\) 6.08097 0.225842
\(726\) 0 0
\(727\) −16.5358 −0.613278 −0.306639 0.951826i \(-0.599205\pi\)
−0.306639 + 0.951826i \(0.599205\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −11.5040 −0.425492
\(732\) 0 0
\(733\) 21.4590 0.792606 0.396303 0.918120i \(-0.370293\pi\)
0.396303 + 0.918120i \(0.370293\pi\)
\(734\) 0 0
\(735\) −2.93543 −0.108275
\(736\) 0 0
\(737\) −31.1442 −1.14721
\(738\) 0 0
\(739\) −2.23069 −0.0820571 −0.0410285 0.999158i \(-0.513063\pi\)
−0.0410285 + 0.999158i \(0.513063\pi\)
\(740\) 0 0
\(741\) −7.76231 −0.285155
\(742\) 0 0
\(743\) 33.5009 1.22903 0.614515 0.788905i \(-0.289352\pi\)
0.614515 + 0.788905i \(0.289352\pi\)
\(744\) 0 0
\(745\) 20.1208 0.737169
\(746\) 0 0
\(747\) −1.33508 −0.0488479
\(748\) 0 0
\(749\) −6.48763 −0.237053
\(750\) 0 0
\(751\) 22.8719 0.834608 0.417304 0.908767i \(-0.362975\pi\)
0.417304 + 0.908767i \(0.362975\pi\)
\(752\) 0 0
\(753\) −20.3379 −0.741155
\(754\) 0 0
\(755\) 33.0716 1.20360
\(756\) 0 0
\(757\) 16.4618 0.598315 0.299158 0.954204i \(-0.403294\pi\)
0.299158 + 0.954204i \(0.403294\pi\)
\(758\) 0 0
\(759\) −3.68133 −0.133624
\(760\) 0 0
\(761\) −30.5550 −1.10762 −0.553810 0.832643i \(-0.686826\pi\)
−0.553810 + 0.832643i \(0.686826\pi\)
\(762\) 0 0
\(763\) 15.1526 0.548559
\(764\) 0 0
\(765\) −12.2981 −0.444639
\(766\) 0 0
\(767\) −20.2775 −0.732179
\(768\) 0 0
\(769\) −0.508203 −0.0183263 −0.00916314 0.999958i \(-0.502917\pi\)
−0.00916314 + 0.999958i \(0.502917\pi\)
\(770\) 0 0
\(771\) −19.5470 −0.703967
\(772\) 0 0
\(773\) 28.5275 1.02606 0.513031 0.858370i \(-0.328523\pi\)
0.513031 + 0.858370i \(0.328523\pi\)
\(774\) 0 0
\(775\) 4.24292 0.152410
\(776\) 0 0
\(777\) 2.31867 0.0831818
\(778\) 0 0
\(779\) 8.41499 0.301498
\(780\) 0 0
\(781\) 42.3502 1.51541
\(782\) 0 0
\(783\) −1.68133 −0.0600859
\(784\) 0 0
\(785\) 38.9424 1.38992
\(786\) 0 0
\(787\) 33.3285 1.18803 0.594017 0.804453i \(-0.297541\pi\)
0.594017 + 0.804453i \(0.297541\pi\)
\(788\) 0 0
\(789\) 30.3103 1.07908
\(790\) 0 0
\(791\) −4.90262 −0.174317
\(792\) 0 0
\(793\) 4.93543 0.175262
\(794\) 0 0
\(795\) 38.7047 1.37272
\(796\) 0 0
\(797\) 55.0961 1.95160 0.975801 0.218660i \(-0.0701685\pi\)
0.975801 + 0.218660i \(0.0701685\pi\)
\(798\) 0 0
\(799\) 51.1840 1.81076
\(800\) 0 0
\(801\) 6.14137 0.216995
\(802\) 0 0
\(803\) −37.9864 −1.34051
\(804\) 0 0
\(805\) 2.93543 0.103460
\(806\) 0 0
\(807\) 18.3585 0.646250
\(808\) 0 0
\(809\) 13.6660 0.480470 0.240235 0.970715i \(-0.422775\pi\)
0.240235 + 0.970715i \(0.422775\pi\)
\(810\) 0 0
\(811\) −23.4887 −0.824799 −0.412400 0.911003i \(-0.635309\pi\)
−0.412400 + 0.911003i \(0.635309\pi\)
\(812\) 0 0
\(813\) −17.1320 −0.600845
\(814\) 0 0
\(815\) −32.8339 −1.15012
\(816\) 0 0
\(817\) −3.22129 −0.112699
\(818\) 0 0
\(819\) −6.61676 −0.231208
\(820\) 0 0
\(821\) −1.21117 −0.0422701 −0.0211350 0.999777i \(-0.506728\pi\)
−0.0211350 + 0.999777i \(0.506728\pi\)
\(822\) 0 0
\(823\) 12.5975 0.439122 0.219561 0.975599i \(-0.429538\pi\)
0.219561 + 0.975599i \(0.429538\pi\)
\(824\) 0 0
\(825\) −13.3145 −0.463551
\(826\) 0 0
\(827\) 14.0726 0.489354 0.244677 0.969605i \(-0.421318\pi\)
0.244677 + 0.969605i \(0.421318\pi\)
\(828\) 0 0
\(829\) −9.06847 −0.314961 −0.157480 0.987522i \(-0.550337\pi\)
−0.157480 + 0.987522i \(0.550337\pi\)
\(830\) 0 0
\(831\) 7.09215 0.246024
\(832\) 0 0
\(833\) −4.18953 −0.145159
\(834\) 0 0
\(835\) −61.3941 −2.12463
\(836\) 0 0
\(837\) −1.17313 −0.0405492
\(838\) 0 0
\(839\) −51.3009 −1.77110 −0.885552 0.464539i \(-0.846220\pi\)
−0.885552 + 0.464539i \(0.846220\pi\)
\(840\) 0 0
\(841\) −26.1731 −0.902522
\(842\) 0 0
\(843\) −22.2171 −0.765198
\(844\) 0 0
\(845\) 90.3572 3.10838
\(846\) 0 0
\(847\) 2.55220 0.0876945
\(848\) 0 0
\(849\) −4.83911 −0.166078
\(850\) 0 0
\(851\) −2.31867 −0.0794830
\(852\) 0 0
\(853\) −22.9753 −0.786658 −0.393329 0.919398i \(-0.628677\pi\)
−0.393329 + 0.919398i \(0.628677\pi\)
\(854\) 0 0
\(855\) −3.44364 −0.117770
\(856\) 0 0
\(857\) −22.4754 −0.767745 −0.383872 0.923386i \(-0.625410\pi\)
−0.383872 + 0.923386i \(0.625410\pi\)
\(858\) 0 0
\(859\) −14.3463 −0.489488 −0.244744 0.969588i \(-0.578704\pi\)
−0.244744 + 0.969588i \(0.578704\pi\)
\(860\) 0 0
\(861\) 7.17313 0.244460
\(862\) 0 0
\(863\) 38.8873 1.32374 0.661869 0.749619i \(-0.269763\pi\)
0.661869 + 0.749619i \(0.269763\pi\)
\(864\) 0 0
\(865\) 6.33091 0.215257
\(866\) 0 0
\(867\) −0.552195 −0.0187535
\(868\) 0 0
\(869\) 22.7857 0.772953
\(870\) 0 0
\(871\) 55.9781 1.89675
\(872\) 0 0
\(873\) −6.69774 −0.226684
\(874\) 0 0
\(875\) −4.06040 −0.137267
\(876\) 0 0
\(877\) 21.7969 0.736029 0.368015 0.929820i \(-0.380038\pi\)
0.368015 + 0.929820i \(0.380038\pi\)
\(878\) 0 0
\(879\) −22.3791 −0.754827
\(880\) 0 0
\(881\) −41.5079 −1.39844 −0.699219 0.714908i \(-0.746469\pi\)
−0.699219 + 0.714908i \(0.746469\pi\)
\(882\) 0 0
\(883\) 37.5920 1.26507 0.632536 0.774531i \(-0.282014\pi\)
0.632536 + 0.774531i \(0.282014\pi\)
\(884\) 0 0
\(885\) −8.99583 −0.302392
\(886\) 0 0
\(887\) 3.81748 0.128178 0.0640891 0.997944i \(-0.479586\pi\)
0.0640891 + 0.997944i \(0.479586\pi\)
\(888\) 0 0
\(889\) 1.51237 0.0507233
\(890\) 0 0
\(891\) 3.68133 0.123329
\(892\) 0 0
\(893\) 14.3322 0.479610
\(894\) 0 0
\(895\) 27.8175 0.929836
\(896\) 0 0
\(897\) 6.61676 0.220927
\(898\) 0 0
\(899\) 1.97241 0.0657837
\(900\) 0 0
\(901\) 55.2405 1.84033
\(902\) 0 0
\(903\) −2.74590 −0.0913778
\(904\) 0 0
\(905\) −8.67716 −0.288439
\(906\) 0 0
\(907\) 8.58084 0.284922 0.142461 0.989800i \(-0.454498\pi\)
0.142461 + 0.989800i \(0.454498\pi\)
\(908\) 0 0
\(909\) −17.6608 −0.585770
\(910\) 0 0
\(911\) 18.5082 0.613204 0.306602 0.951838i \(-0.400808\pi\)
0.306602 + 0.951838i \(0.400808\pi\)
\(912\) 0 0
\(913\) −4.91486 −0.162658
\(914\) 0 0
\(915\) 2.18953 0.0723838
\(916\) 0 0
\(917\) −13.2939 −0.439004
\(918\) 0 0
\(919\) −11.1648 −0.368292 −0.184146 0.982899i \(-0.558952\pi\)
−0.184146 + 0.982899i \(0.558952\pi\)
\(920\) 0 0
\(921\) −8.60142 −0.283426
\(922\) 0 0
\(923\) −76.1195 −2.50550
\(924\) 0 0
\(925\) −8.38608 −0.275733
\(926\) 0 0
\(927\) −5.01641 −0.164760
\(928\) 0 0
\(929\) −24.9271 −0.817831 −0.408916 0.912572i \(-0.634093\pi\)
−0.408916 + 0.912572i \(0.634093\pi\)
\(930\) 0 0
\(931\) −1.17313 −0.0384477
\(932\) 0 0
\(933\) 11.6660 0.381927
\(934\) 0 0
\(935\) −45.2733 −1.48060
\(936\) 0 0
\(937\) −14.5358 −0.474864 −0.237432 0.971404i \(-0.576306\pi\)
−0.237432 + 0.971404i \(0.576306\pi\)
\(938\) 0 0
\(939\) −21.8625 −0.713457
\(940\) 0 0
\(941\) 34.1208 1.11231 0.556153 0.831080i \(-0.312277\pi\)
0.556153 + 0.831080i \(0.312277\pi\)
\(942\) 0 0
\(943\) −7.17313 −0.233589
\(944\) 0 0
\(945\) −2.93543 −0.0954896
\(946\) 0 0
\(947\) 17.7641 0.577255 0.288628 0.957441i \(-0.406801\pi\)
0.288628 + 0.957441i \(0.406801\pi\)
\(948\) 0 0
\(949\) 68.2762 2.21634
\(950\) 0 0
\(951\) −0.685500 −0.0222288
\(952\) 0 0
\(953\) 53.7763 1.74199 0.870993 0.491295i \(-0.163476\pi\)
0.870993 + 0.491295i \(0.163476\pi\)
\(954\) 0 0
\(955\) −33.0716 −1.07017
\(956\) 0 0
\(957\) −6.18953 −0.200079
\(958\) 0 0
\(959\) −17.8021 −0.574861
\(960\) 0 0
\(961\) −29.6238 −0.955606
\(962\) 0 0
\(963\) −6.48763 −0.209061
\(964\) 0 0
\(965\) −13.9917 −0.450408
\(966\) 0 0
\(967\) −50.7805 −1.63299 −0.816495 0.577352i \(-0.804086\pi\)
−0.816495 + 0.577352i \(0.804086\pi\)
\(968\) 0 0
\(969\) −4.91486 −0.157888
\(970\) 0 0
\(971\) −13.6883 −0.439280 −0.219640 0.975581i \(-0.570488\pi\)
−0.219640 + 0.975581i \(0.570488\pi\)
\(972\) 0 0
\(973\) −4.29809 −0.137791
\(974\) 0 0
\(975\) 23.9313 0.766414
\(976\) 0 0
\(977\) −47.1473 −1.50838 −0.754188 0.656658i \(-0.771969\pi\)
−0.754188 + 0.656658i \(0.771969\pi\)
\(978\) 0 0
\(979\) 22.6084 0.722568
\(980\) 0 0
\(981\) 15.1526 0.483784
\(982\) 0 0
\(983\) −22.9888 −0.733230 −0.366615 0.930373i \(-0.619483\pi\)
−0.366615 + 0.930373i \(0.619483\pi\)
\(984\) 0 0
\(985\) 82.1072 2.61615
\(986\) 0 0
\(987\) 12.2171 0.388875
\(988\) 0 0
\(989\) 2.74590 0.0873145
\(990\) 0 0
\(991\) −48.9135 −1.55379 −0.776895 0.629631i \(-0.783206\pi\)
−0.776895 + 0.629631i \(0.783206\pi\)
\(992\) 0 0
\(993\) 24.8133 0.787426
\(994\) 0 0
\(995\) −39.0838 −1.23904
\(996\) 0 0
\(997\) −45.9536 −1.45537 −0.727683 0.685914i \(-0.759403\pi\)
−0.727683 + 0.685914i \(0.759403\pi\)
\(998\) 0 0
\(999\) 2.31867 0.0733595
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 7728.2.a.bs.1.3 3
4.3 odd 2 3864.2.a.q.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3864.2.a.q.1.3 3 4.3 odd 2
7728.2.a.bs.1.3 3 1.1 even 1 trivial