Properties

Label 1280.2.a.b.1.2
Level $1280$
Weight $2$
Character 1280.1
Self dual yes
Analytic conductor $10.221$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1280.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.2208514587\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 320)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 1280.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.732051 q^{3} -1.00000 q^{5} +1.26795 q^{7} -2.46410 q^{9} +O(q^{10})\) \(q+0.732051 q^{3} -1.00000 q^{5} +1.26795 q^{7} -2.46410 q^{9} -3.46410 q^{11} +3.46410 q^{13} -0.732051 q^{15} +3.46410 q^{17} +2.00000 q^{19} +0.928203 q^{21} +8.19615 q^{23} +1.00000 q^{25} -4.00000 q^{27} +9.46410 q^{31} -2.53590 q^{33} -1.26795 q^{35} +6.00000 q^{37} +2.53590 q^{39} -2.53590 q^{41} +10.1962 q^{43} +2.46410 q^{45} +8.19615 q^{47} -5.39230 q^{49} +2.53590 q^{51} -10.3923 q^{53} +3.46410 q^{55} +1.46410 q^{57} +6.00000 q^{59} -12.9282 q^{61} -3.12436 q^{63} -3.46410 q^{65} -10.1962 q^{67} +6.00000 q^{69} -4.39230 q^{71} +14.3923 q^{73} +0.732051 q^{75} -4.39230 q^{77} +12.0000 q^{79} +4.46410 q^{81} -4.73205 q^{83} -3.46410 q^{85} -0.928203 q^{89} +4.39230 q^{91} +6.92820 q^{93} -2.00000 q^{95} -6.39230 q^{97} +8.53590 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 2 q^{5} + 6 q^{7} + 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{3} - 2 q^{5} + 6 q^{7} + 2 q^{9} + 2 q^{15} + 4 q^{19} - 12 q^{21} + 6 q^{23} + 2 q^{25} - 8 q^{27} + 12 q^{31} - 12 q^{33} - 6 q^{35} + 12 q^{37} + 12 q^{39} - 12 q^{41} + 10 q^{43} - 2 q^{45} + 6 q^{47} + 10 q^{49} + 12 q^{51} - 4 q^{57} + 12 q^{59} - 12 q^{61} + 18 q^{63} - 10 q^{67} + 12 q^{69} + 12 q^{71} + 8 q^{73} - 2 q^{75} + 12 q^{77} + 24 q^{79} + 2 q^{81} - 6 q^{83} + 12 q^{89} - 12 q^{91} - 4 q^{95} + 8 q^{97} + 24 q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.732051 0.422650 0.211325 0.977416i \(-0.432222\pi\)
0.211325 + 0.977416i \(0.432222\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.26795 0.479240 0.239620 0.970867i \(-0.422977\pi\)
0.239620 + 0.970867i \(0.422977\pi\)
\(8\) 0 0
\(9\) −2.46410 −0.821367
\(10\) 0 0
\(11\) −3.46410 −1.04447 −0.522233 0.852803i \(-0.674901\pi\)
−0.522233 + 0.852803i \(0.674901\pi\)
\(12\) 0 0
\(13\) 3.46410 0.960769 0.480384 0.877058i \(-0.340497\pi\)
0.480384 + 0.877058i \(0.340497\pi\)
\(14\) 0 0
\(15\) −0.732051 −0.189015
\(16\) 0 0
\(17\) 3.46410 0.840168 0.420084 0.907485i \(-0.362001\pi\)
0.420084 + 0.907485i \(0.362001\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) 0.928203 0.202551
\(22\) 0 0
\(23\) 8.19615 1.70902 0.854508 0.519438i \(-0.173859\pi\)
0.854508 + 0.519438i \(0.173859\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −4.00000 −0.769800
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 9.46410 1.69980 0.849901 0.526942i \(-0.176661\pi\)
0.849901 + 0.526942i \(0.176661\pi\)
\(32\) 0 0
\(33\) −2.53590 −0.441443
\(34\) 0 0
\(35\) −1.26795 −0.214323
\(36\) 0 0
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 0 0
\(39\) 2.53590 0.406069
\(40\) 0 0
\(41\) −2.53590 −0.396041 −0.198020 0.980198i \(-0.563451\pi\)
−0.198020 + 0.980198i \(0.563451\pi\)
\(42\) 0 0
\(43\) 10.1962 1.55490 0.777449 0.628946i \(-0.216513\pi\)
0.777449 + 0.628946i \(0.216513\pi\)
\(44\) 0 0
\(45\) 2.46410 0.367327
\(46\) 0 0
\(47\) 8.19615 1.19553 0.597766 0.801671i \(-0.296055\pi\)
0.597766 + 0.801671i \(0.296055\pi\)
\(48\) 0 0
\(49\) −5.39230 −0.770329
\(50\) 0 0
\(51\) 2.53590 0.355097
\(52\) 0 0
\(53\) −10.3923 −1.42749 −0.713746 0.700404i \(-0.753003\pi\)
−0.713746 + 0.700404i \(0.753003\pi\)
\(54\) 0 0
\(55\) 3.46410 0.467099
\(56\) 0 0
\(57\) 1.46410 0.193925
\(58\) 0 0
\(59\) 6.00000 0.781133 0.390567 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) −12.9282 −1.65529 −0.827643 0.561254i \(-0.810319\pi\)
−0.827643 + 0.561254i \(0.810319\pi\)
\(62\) 0 0
\(63\) −3.12436 −0.393632
\(64\) 0 0
\(65\) −3.46410 −0.429669
\(66\) 0 0
\(67\) −10.1962 −1.24566 −0.622829 0.782358i \(-0.714017\pi\)
−0.622829 + 0.782358i \(0.714017\pi\)
\(68\) 0 0
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) −4.39230 −0.521271 −0.260635 0.965437i \(-0.583932\pi\)
−0.260635 + 0.965437i \(0.583932\pi\)
\(72\) 0 0
\(73\) 14.3923 1.68449 0.842246 0.539093i \(-0.181233\pi\)
0.842246 + 0.539093i \(0.181233\pi\)
\(74\) 0 0
\(75\) 0.732051 0.0845299
\(76\) 0 0
\(77\) −4.39230 −0.500550
\(78\) 0 0
\(79\) 12.0000 1.35011 0.675053 0.737769i \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) 0 0
\(81\) 4.46410 0.496011
\(82\) 0 0
\(83\) −4.73205 −0.519410 −0.259705 0.965688i \(-0.583625\pi\)
−0.259705 + 0.965688i \(0.583625\pi\)
\(84\) 0 0
\(85\) −3.46410 −0.375735
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −0.928203 −0.0983893 −0.0491947 0.998789i \(-0.515665\pi\)
−0.0491947 + 0.998789i \(0.515665\pi\)
\(90\) 0 0
\(91\) 4.39230 0.460439
\(92\) 0 0
\(93\) 6.92820 0.718421
\(94\) 0 0
\(95\) −2.00000 −0.205196
\(96\) 0 0
\(97\) −6.39230 −0.649040 −0.324520 0.945879i \(-0.605203\pi\)
−0.324520 + 0.945879i \(0.605203\pi\)
\(98\) 0 0
\(99\) 8.53590 0.857890
\(100\) 0 0
\(101\) 12.0000 1.19404 0.597022 0.802225i \(-0.296350\pi\)
0.597022 + 0.802225i \(0.296350\pi\)
\(102\) 0 0
\(103\) 8.19615 0.807591 0.403795 0.914849i \(-0.367691\pi\)
0.403795 + 0.914849i \(0.367691\pi\)
\(104\) 0 0
\(105\) −0.928203 −0.0905834
\(106\) 0 0
\(107\) −16.7321 −1.61755 −0.808774 0.588119i \(-0.799869\pi\)
−0.808774 + 0.588119i \(0.799869\pi\)
\(108\) 0 0
\(109\) −0.928203 −0.0889057 −0.0444529 0.999011i \(-0.514154\pi\)
−0.0444529 + 0.999011i \(0.514154\pi\)
\(110\) 0 0
\(111\) 4.39230 0.416899
\(112\) 0 0
\(113\) 0.928203 0.0873180 0.0436590 0.999046i \(-0.486098\pi\)
0.0436590 + 0.999046i \(0.486098\pi\)
\(114\) 0 0
\(115\) −8.19615 −0.764295
\(116\) 0 0
\(117\) −8.53590 −0.789144
\(118\) 0 0
\(119\) 4.39230 0.402642
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −1.85641 −0.167387
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 3.80385 0.337537 0.168768 0.985656i \(-0.446021\pi\)
0.168768 + 0.985656i \(0.446021\pi\)
\(128\) 0 0
\(129\) 7.46410 0.657178
\(130\) 0 0
\(131\) 10.3923 0.907980 0.453990 0.891007i \(-0.350000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(132\) 0 0
\(133\) 2.53590 0.219890
\(134\) 0 0
\(135\) 4.00000 0.344265
\(136\) 0 0
\(137\) −12.9282 −1.10453 −0.552265 0.833668i \(-0.686237\pi\)
−0.552265 + 0.833668i \(0.686237\pi\)
\(138\) 0 0
\(139\) 10.0000 0.848189 0.424094 0.905618i \(-0.360592\pi\)
0.424094 + 0.905618i \(0.360592\pi\)
\(140\) 0 0
\(141\) 6.00000 0.505291
\(142\) 0 0
\(143\) −12.0000 −1.00349
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −3.94744 −0.325579
\(148\) 0 0
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 0 0
\(151\) 2.53590 0.206368 0.103184 0.994662i \(-0.467097\pi\)
0.103184 + 0.994662i \(0.467097\pi\)
\(152\) 0 0
\(153\) −8.53590 −0.690086
\(154\) 0 0
\(155\) −9.46410 −0.760175
\(156\) 0 0
\(157\) 0.928203 0.0740787 0.0370393 0.999314i \(-0.488207\pi\)
0.0370393 + 0.999314i \(0.488207\pi\)
\(158\) 0 0
\(159\) −7.60770 −0.603329
\(160\) 0 0
\(161\) 10.3923 0.819028
\(162\) 0 0
\(163\) 5.80385 0.454592 0.227296 0.973826i \(-0.427011\pi\)
0.227296 + 0.973826i \(0.427011\pi\)
\(164\) 0 0
\(165\) 2.53590 0.197419
\(166\) 0 0
\(167\) −8.19615 −0.634237 −0.317119 0.948386i \(-0.602715\pi\)
−0.317119 + 0.948386i \(0.602715\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) −4.92820 −0.376869
\(172\) 0 0
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) 1.26795 0.0958479
\(176\) 0 0
\(177\) 4.39230 0.330146
\(178\) 0 0
\(179\) 19.8564 1.48414 0.742069 0.670324i \(-0.233845\pi\)
0.742069 + 0.670324i \(0.233845\pi\)
\(180\) 0 0
\(181\) −6.92820 −0.514969 −0.257485 0.966282i \(-0.582894\pi\)
−0.257485 + 0.966282i \(0.582894\pi\)
\(182\) 0 0
\(183\) −9.46410 −0.699607
\(184\) 0 0
\(185\) −6.00000 −0.441129
\(186\) 0 0
\(187\) −12.0000 −0.877527
\(188\) 0 0
\(189\) −5.07180 −0.368919
\(190\) 0 0
\(191\) −16.3923 −1.18611 −0.593053 0.805164i \(-0.702077\pi\)
−0.593053 + 0.805164i \(0.702077\pi\)
\(192\) 0 0
\(193\) 6.39230 0.460128 0.230064 0.973175i \(-0.426106\pi\)
0.230064 + 0.973175i \(0.426106\pi\)
\(194\) 0 0
\(195\) −2.53590 −0.181599
\(196\) 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 \(-0.578973\pi\)
−0.245564 + 0.969380i \(0.578973\pi\)
\(200\) 0 0
\(201\) −7.46410 −0.526477
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 2.53590 0.177115
\(206\) 0 0
\(207\) −20.1962 −1.40373
\(208\) 0 0
\(209\) −6.92820 −0.479234
\(210\) 0 0
\(211\) −6.39230 −0.440064 −0.220032 0.975493i \(-0.570616\pi\)
−0.220032 + 0.975493i \(0.570616\pi\)
\(212\) 0 0
\(213\) −3.21539 −0.220315
\(214\) 0 0
\(215\) −10.1962 −0.695372
\(216\) 0 0
\(217\) 12.0000 0.814613
\(218\) 0 0
\(219\) 10.5359 0.711950
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) 0 0
\(223\) 15.1244 1.01280 0.506401 0.862298i \(-0.330976\pi\)
0.506401 + 0.862298i \(0.330976\pi\)
\(224\) 0 0
\(225\) −2.46410 −0.164273
\(226\) 0 0
\(227\) −18.5885 −1.23376 −0.616880 0.787058i \(-0.711603\pi\)
−0.616880 + 0.787058i \(0.711603\pi\)
\(228\) 0 0
\(229\) 18.9282 1.25081 0.625405 0.780300i \(-0.284934\pi\)
0.625405 + 0.780300i \(0.284934\pi\)
\(230\) 0 0
\(231\) −3.21539 −0.211557
\(232\) 0 0
\(233\) 1.60770 0.105324 0.0526618 0.998612i \(-0.483229\pi\)
0.0526618 + 0.998612i \(0.483229\pi\)
\(234\) 0 0
\(235\) −8.19615 −0.534658
\(236\) 0 0
\(237\) 8.78461 0.570622
\(238\) 0 0
\(239\) −20.7846 −1.34444 −0.672222 0.740349i \(-0.734660\pi\)
−0.672222 + 0.740349i \(0.734660\pi\)
\(240\) 0 0
\(241\) −20.3923 −1.31358 −0.656792 0.754072i \(-0.728087\pi\)
−0.656792 + 0.754072i \(0.728087\pi\)
\(242\) 0 0
\(243\) 15.2679 0.979439
\(244\) 0 0
\(245\) 5.39230 0.344502
\(246\) 0 0
\(247\) 6.92820 0.440831
\(248\) 0 0
\(249\) −3.46410 −0.219529
\(250\) 0 0
\(251\) 15.4641 0.976085 0.488043 0.872820i \(-0.337711\pi\)
0.488043 + 0.872820i \(0.337711\pi\)
\(252\) 0 0
\(253\) −28.3923 −1.78501
\(254\) 0 0
\(255\) −2.53590 −0.158804
\(256\) 0 0
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 0 0
\(259\) 7.60770 0.472719
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −20.1962 −1.24535 −0.622674 0.782481i \(-0.713954\pi\)
−0.622674 + 0.782481i \(0.713954\pi\)
\(264\) 0 0
\(265\) 10.3923 0.638394
\(266\) 0 0
\(267\) −0.679492 −0.0415842
\(268\) 0 0
\(269\) 14.7846 0.901434 0.450717 0.892667i \(-0.351168\pi\)
0.450717 + 0.892667i \(0.351168\pi\)
\(270\) 0 0
\(271\) −4.39230 −0.266814 −0.133407 0.991061i \(-0.542592\pi\)
−0.133407 + 0.991061i \(0.542592\pi\)
\(272\) 0 0
\(273\) 3.21539 0.194604
\(274\) 0 0
\(275\) −3.46410 −0.208893
\(276\) 0 0
\(277\) −19.8564 −1.19306 −0.596528 0.802592i \(-0.703454\pi\)
−0.596528 + 0.802592i \(0.703454\pi\)
\(278\) 0 0
\(279\) −23.3205 −1.39616
\(280\) 0 0
\(281\) 28.3923 1.69374 0.846871 0.531798i \(-0.178483\pi\)
0.846871 + 0.531798i \(0.178483\pi\)
\(282\) 0 0
\(283\) 10.5885 0.629418 0.314709 0.949188i \(-0.398093\pi\)
0.314709 + 0.949188i \(0.398093\pi\)
\(284\) 0 0
\(285\) −1.46410 −0.0867259
\(286\) 0 0
\(287\) −3.21539 −0.189798
\(288\) 0 0
\(289\) −5.00000 −0.294118
\(290\) 0 0
\(291\) −4.67949 −0.274317
\(292\) 0 0
\(293\) 30.0000 1.75262 0.876309 0.481749i \(-0.159998\pi\)
0.876309 + 0.481749i \(0.159998\pi\)
\(294\) 0 0
\(295\) −6.00000 −0.349334
\(296\) 0 0
\(297\) 13.8564 0.804030
\(298\) 0 0
\(299\) 28.3923 1.64197
\(300\) 0 0
\(301\) 12.9282 0.745169
\(302\) 0 0
\(303\) 8.78461 0.504663
\(304\) 0 0
\(305\) 12.9282 0.740267
\(306\) 0 0
\(307\) −34.1962 −1.95168 −0.975839 0.218492i \(-0.929886\pi\)
−0.975839 + 0.218492i \(0.929886\pi\)
\(308\) 0 0
\(309\) 6.00000 0.341328
\(310\) 0 0
\(311\) −7.60770 −0.431393 −0.215696 0.976460i \(-0.569202\pi\)
−0.215696 + 0.976460i \(0.569202\pi\)
\(312\) 0 0
\(313\) −22.0000 −1.24351 −0.621757 0.783210i \(-0.713581\pi\)
−0.621757 + 0.783210i \(0.713581\pi\)
\(314\) 0 0
\(315\) 3.12436 0.176037
\(316\) 0 0
\(317\) −22.3923 −1.25768 −0.628839 0.777536i \(-0.716469\pi\)
−0.628839 + 0.777536i \(0.716469\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −12.2487 −0.683656
\(322\) 0 0
\(323\) 6.92820 0.385496
\(324\) 0 0
\(325\) 3.46410 0.192154
\(326\) 0 0
\(327\) −0.679492 −0.0375760
\(328\) 0 0
\(329\) 10.3923 0.572946
\(330\) 0 0
\(331\) 5.60770 0.308227 0.154113 0.988053i \(-0.450748\pi\)
0.154113 + 0.988053i \(0.450748\pi\)
\(332\) 0 0
\(333\) −14.7846 −0.810192
\(334\) 0 0
\(335\) 10.1962 0.557075
\(336\) 0 0
\(337\) −14.0000 −0.762629 −0.381314 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) 0 0
\(339\) 0.679492 0.0369049
\(340\) 0 0
\(341\) −32.7846 −1.77539
\(342\) 0 0
\(343\) −15.7128 −0.848412
\(344\) 0 0
\(345\) −6.00000 −0.323029
\(346\) 0 0
\(347\) −28.0526 −1.50594 −0.752970 0.658055i \(-0.771380\pi\)
−0.752970 + 0.658055i \(0.771380\pi\)
\(348\) 0 0
\(349\) −8.78461 −0.470229 −0.235115 0.971968i \(-0.575547\pi\)
−0.235115 + 0.971968i \(0.575547\pi\)
\(350\) 0 0
\(351\) −13.8564 −0.739600
\(352\) 0 0
\(353\) −14.7846 −0.786905 −0.393453 0.919345i \(-0.628719\pi\)
−0.393453 + 0.919345i \(0.628719\pi\)
\(354\) 0 0
\(355\) 4.39230 0.233119
\(356\) 0 0
\(357\) 3.21539 0.170177
\(358\) 0 0
\(359\) 8.78461 0.463634 0.231817 0.972759i \(-0.425533\pi\)
0.231817 + 0.972759i \(0.425533\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 0.732051 0.0384227
\(364\) 0 0
\(365\) −14.3923 −0.753328
\(366\) 0 0
\(367\) −10.0526 −0.524739 −0.262370 0.964967i \(-0.584504\pi\)
−0.262370 + 0.964967i \(0.584504\pi\)
\(368\) 0 0
\(369\) 6.24871 0.325295
\(370\) 0 0
\(371\) −13.1769 −0.684111
\(372\) 0 0
\(373\) 7.85641 0.406789 0.203395 0.979097i \(-0.434803\pi\)
0.203395 + 0.979097i \(0.434803\pi\)
\(374\) 0 0
\(375\) −0.732051 −0.0378029
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 2.00000 0.102733 0.0513665 0.998680i \(-0.483642\pi\)
0.0513665 + 0.998680i \(0.483642\pi\)
\(380\) 0 0
\(381\) 2.78461 0.142660
\(382\) 0 0
\(383\) −3.80385 −0.194368 −0.0971838 0.995266i \(-0.530983\pi\)
−0.0971838 + 0.995266i \(0.530983\pi\)
\(384\) 0 0
\(385\) 4.39230 0.223853
\(386\) 0 0
\(387\) −25.1244 −1.27714
\(388\) 0 0
\(389\) −26.7846 −1.35803 −0.679017 0.734123i \(-0.737594\pi\)
−0.679017 + 0.734123i \(0.737594\pi\)
\(390\) 0 0
\(391\) 28.3923 1.43586
\(392\) 0 0
\(393\) 7.60770 0.383757
\(394\) 0 0
\(395\) −12.0000 −0.603786
\(396\) 0 0
\(397\) 27.4641 1.37838 0.689192 0.724579i \(-0.257966\pi\)
0.689192 + 0.724579i \(0.257966\pi\)
\(398\) 0 0
\(399\) 1.85641 0.0929366
\(400\) 0 0
\(401\) −4.14359 −0.206921 −0.103461 0.994634i \(-0.532992\pi\)
−0.103461 + 0.994634i \(0.532992\pi\)
\(402\) 0 0
\(403\) 32.7846 1.63312
\(404\) 0 0
\(405\) −4.46410 −0.221823
\(406\) 0 0
\(407\) −20.7846 −1.03025
\(408\) 0 0
\(409\) −3.60770 −0.178389 −0.0891945 0.996014i \(-0.528429\pi\)
−0.0891945 + 0.996014i \(0.528429\pi\)
\(410\) 0 0
\(411\) −9.46410 −0.466830
\(412\) 0 0
\(413\) 7.60770 0.374350
\(414\) 0 0
\(415\) 4.73205 0.232287
\(416\) 0 0
\(417\) 7.32051 0.358487
\(418\) 0 0
\(419\) 0.928203 0.0453457 0.0226728 0.999743i \(-0.492782\pi\)
0.0226728 + 0.999743i \(0.492782\pi\)
\(420\) 0 0
\(421\) 6.00000 0.292422 0.146211 0.989253i \(-0.453292\pi\)
0.146211 + 0.989253i \(0.453292\pi\)
\(422\) 0 0
\(423\) −20.1962 −0.981971
\(424\) 0 0
\(425\) 3.46410 0.168034
\(426\) 0 0
\(427\) −16.3923 −0.793279
\(428\) 0 0
\(429\) −8.78461 −0.424125
\(430\) 0 0
\(431\) −28.3923 −1.36761 −0.683805 0.729665i \(-0.739676\pi\)
−0.683805 + 0.729665i \(0.739676\pi\)
\(432\) 0 0
\(433\) 26.3923 1.26833 0.634167 0.773196i \(-0.281343\pi\)
0.634167 + 0.773196i \(0.281343\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 16.3923 0.784150
\(438\) 0 0
\(439\) 18.9282 0.903394 0.451697 0.892171i \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(440\) 0 0
\(441\) 13.2872 0.632723
\(442\) 0 0
\(443\) −0.339746 −0.0161418 −0.00807091 0.999967i \(-0.502569\pi\)
−0.00807091 + 0.999967i \(0.502569\pi\)
\(444\) 0 0
\(445\) 0.928203 0.0440011
\(446\) 0 0
\(447\) −13.1769 −0.623247
\(448\) 0 0
\(449\) −2.53590 −0.119676 −0.0598382 0.998208i \(-0.519058\pi\)
−0.0598382 + 0.998208i \(0.519058\pi\)
\(450\) 0 0
\(451\) 8.78461 0.413651
\(452\) 0 0
\(453\) 1.85641 0.0872216
\(454\) 0 0
\(455\) −4.39230 −0.205914
\(456\) 0 0
\(457\) −22.7846 −1.06582 −0.532910 0.846172i \(-0.678901\pi\)
−0.532910 + 0.846172i \(0.678901\pi\)
\(458\) 0 0
\(459\) −13.8564 −0.646762
\(460\) 0 0
\(461\) −12.0000 −0.558896 −0.279448 0.960161i \(-0.590151\pi\)
−0.279448 + 0.960161i \(0.590151\pi\)
\(462\) 0 0
\(463\) 15.8038 0.734467 0.367234 0.930129i \(-0.380305\pi\)
0.367234 + 0.930129i \(0.380305\pi\)
\(464\) 0 0
\(465\) −6.92820 −0.321288
\(466\) 0 0
\(467\) 22.9808 1.06342 0.531711 0.846926i \(-0.321549\pi\)
0.531711 + 0.846926i \(0.321549\pi\)
\(468\) 0 0
\(469\) −12.9282 −0.596969
\(470\) 0 0
\(471\) 0.679492 0.0313093
\(472\) 0 0
\(473\) −35.3205 −1.62404
\(474\) 0 0
\(475\) 2.00000 0.0917663
\(476\) 0 0
\(477\) 25.6077 1.17250
\(478\) 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
\(483\) 7.60770 0.346162
\(484\) 0 0
\(485\) 6.39230 0.290260
\(486\) 0 0
\(487\) 39.1244 1.77289 0.886447 0.462830i \(-0.153166\pi\)
0.886447 + 0.462830i \(0.153166\pi\)
\(488\) 0 0
\(489\) 4.24871 0.192133
\(490\) 0 0
\(491\) 22.3923 1.01055 0.505275 0.862958i \(-0.331391\pi\)
0.505275 + 0.862958i \(0.331391\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −8.53590 −0.383660
\(496\) 0 0
\(497\) −5.56922 −0.249814
\(498\) 0 0
\(499\) 39.5692 1.77136 0.885681 0.464295i \(-0.153692\pi\)
0.885681 + 0.464295i \(0.153692\pi\)
\(500\) 0 0
\(501\) −6.00000 −0.268060
\(502\) 0 0
\(503\) 8.19615 0.365448 0.182724 0.983164i \(-0.441508\pi\)
0.182724 + 0.983164i \(0.441508\pi\)
\(504\) 0 0
\(505\) −12.0000 −0.533993
\(506\) 0 0
\(507\) −0.732051 −0.0325115
\(508\) 0 0
\(509\) 8.78461 0.389371 0.194685 0.980866i \(-0.437631\pi\)
0.194685 + 0.980866i \(0.437631\pi\)
\(510\) 0 0
\(511\) 18.2487 0.807275
\(512\) 0 0
\(513\) −8.00000 −0.353209
\(514\) 0 0
\(515\) −8.19615 −0.361166
\(516\) 0 0
\(517\) −28.3923 −1.24869
\(518\) 0 0
\(519\) 4.39230 0.192801
\(520\) 0 0
\(521\) −31.8564 −1.39565 −0.697827 0.716266i \(-0.745850\pi\)
−0.697827 + 0.716266i \(0.745850\pi\)
\(522\) 0 0
\(523\) −14.9808 −0.655063 −0.327531 0.944840i \(-0.606217\pi\)
−0.327531 + 0.944840i \(0.606217\pi\)
\(524\) 0 0
\(525\) 0.928203 0.0405101
\(526\) 0 0
\(527\) 32.7846 1.42812
\(528\) 0 0
\(529\) 44.1769 1.92074
\(530\) 0 0
\(531\) −14.7846 −0.641597
\(532\) 0 0
\(533\) −8.78461 −0.380504
\(534\) 0 0
\(535\) 16.7321 0.723390
\(536\) 0 0
\(537\) 14.5359 0.627270
\(538\) 0 0
\(539\) 18.6795 0.804583
\(540\) 0 0
\(541\) 15.7128 0.675547 0.337773 0.941227i \(-0.390326\pi\)
0.337773 + 0.941227i \(0.390326\pi\)
\(542\) 0 0
\(543\) −5.07180 −0.217652
\(544\) 0 0
\(545\) 0.928203 0.0397599
\(546\) 0 0
\(547\) −1.80385 −0.0771270 −0.0385635 0.999256i \(-0.512278\pi\)
−0.0385635 + 0.999256i \(0.512278\pi\)
\(548\) 0 0
\(549\) 31.8564 1.35960
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 15.2154 0.647024
\(554\) 0 0
\(555\) −4.39230 −0.186443
\(556\) 0 0
\(557\) −14.7846 −0.626444 −0.313222 0.949680i \(-0.601408\pi\)
−0.313222 + 0.949680i \(0.601408\pi\)
\(558\) 0 0
\(559\) 35.3205 1.49390
\(560\) 0 0
\(561\) −8.78461 −0.370887
\(562\) 0 0
\(563\) −26.1962 −1.10404 −0.552018 0.833832i \(-0.686142\pi\)
−0.552018 + 0.833832i \(0.686142\pi\)
\(564\) 0 0
\(565\) −0.928203 −0.0390498
\(566\) 0 0
\(567\) 5.66025 0.237708
\(568\) 0 0
\(569\) 42.2487 1.77116 0.885579 0.464489i \(-0.153762\pi\)
0.885579 + 0.464489i \(0.153762\pi\)
\(570\) 0 0
\(571\) −30.3923 −1.27188 −0.635939 0.771739i \(-0.719387\pi\)
−0.635939 + 0.771739i \(0.719387\pi\)
\(572\) 0 0
\(573\) −12.0000 −0.501307
\(574\) 0 0
\(575\) 8.19615 0.341803
\(576\) 0 0
\(577\) −2.00000 −0.0832611 −0.0416305 0.999133i \(-0.513255\pi\)
−0.0416305 + 0.999133i \(0.513255\pi\)
\(578\) 0 0
\(579\) 4.67949 0.194473
\(580\) 0 0
\(581\) −6.00000 −0.248922
\(582\) 0 0
\(583\) 36.0000 1.49097
\(584\) 0 0
\(585\) 8.53590 0.352916
\(586\) 0 0
\(587\) 13.5167 0.557892 0.278946 0.960307i \(-0.410015\pi\)
0.278946 + 0.960307i \(0.410015\pi\)
\(588\) 0 0
\(589\) 18.9282 0.779923
\(590\) 0 0
\(591\) 7.60770 0.312939
\(592\) 0 0
\(593\) −0.928203 −0.0381167 −0.0190584 0.999818i \(-0.506067\pi\)
−0.0190584 + 0.999818i \(0.506067\pi\)
\(594\) 0 0
\(595\) −4.39230 −0.180067
\(596\) 0 0
\(597\) −5.07180 −0.207575
\(598\) 0 0
\(599\) 12.0000 0.490307 0.245153 0.969484i \(-0.421162\pi\)
0.245153 + 0.969484i \(0.421162\pi\)
\(600\) 0 0
\(601\) 20.3923 0.831819 0.415910 0.909406i \(-0.363463\pi\)
0.415910 + 0.909406i \(0.363463\pi\)
\(602\) 0 0
\(603\) 25.1244 1.02314
\(604\) 0 0
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) 8.19615 0.332672 0.166336 0.986069i \(-0.446806\pi\)
0.166336 + 0.986069i \(0.446806\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 28.3923 1.14863
\(612\) 0 0
\(613\) −13.6077 −0.549610 −0.274805 0.961500i \(-0.588613\pi\)
−0.274805 + 0.961500i \(0.588613\pi\)
\(614\) 0 0
\(615\) 1.85641 0.0748575
\(616\) 0 0
\(617\) −13.6077 −0.547825 −0.273913 0.961755i \(-0.588318\pi\)
−0.273913 + 0.961755i \(0.588318\pi\)
\(618\) 0 0
\(619\) 6.78461 0.272696 0.136348 0.990661i \(-0.456463\pi\)
0.136348 + 0.990661i \(0.456463\pi\)
\(620\) 0 0
\(621\) −32.7846 −1.31560
\(622\) 0 0
\(623\) −1.17691 −0.0471521
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −5.07180 −0.202548
\(628\) 0 0
\(629\) 20.7846 0.828737
\(630\) 0 0
\(631\) 21.4641 0.854472 0.427236 0.904140i \(-0.359487\pi\)
0.427236 + 0.904140i \(0.359487\pi\)
\(632\) 0 0
\(633\) −4.67949 −0.185993
\(634\) 0 0
\(635\) −3.80385 −0.150951
\(636\) 0 0
\(637\) −18.6795 −0.740108
\(638\) 0 0
\(639\) 10.8231 0.428155
\(640\) 0 0
\(641\) −4.39230 −0.173486 −0.0867428 0.996231i \(-0.527646\pi\)
−0.0867428 + 0.996231i \(0.527646\pi\)
\(642\) 0 0
\(643\) −10.5885 −0.417568 −0.208784 0.977962i \(-0.566951\pi\)
−0.208784 + 0.977962i \(0.566951\pi\)
\(644\) 0 0
\(645\) −7.46410 −0.293899
\(646\) 0 0
\(647\) −36.5885 −1.43844 −0.719220 0.694782i \(-0.755501\pi\)
−0.719220 + 0.694782i \(0.755501\pi\)
\(648\) 0 0
\(649\) −20.7846 −0.815867
\(650\) 0 0
\(651\) 8.78461 0.344296
\(652\) 0 0
\(653\) 19.1769 0.750451 0.375225 0.926934i \(-0.377565\pi\)
0.375225 + 0.926934i \(0.377565\pi\)
\(654\) 0 0
\(655\) −10.3923 −0.406061
\(656\) 0 0
\(657\) −35.4641 −1.38359
\(658\) 0 0
\(659\) −40.6410 −1.58315 −0.791575 0.611073i \(-0.790738\pi\)
−0.791575 + 0.611073i \(0.790738\pi\)
\(660\) 0 0
\(661\) 35.5692 1.38348 0.691741 0.722146i \(-0.256844\pi\)
0.691741 + 0.722146i \(0.256844\pi\)
\(662\) 0 0
\(663\) 8.78461 0.341166
\(664\) 0 0
\(665\) −2.53590 −0.0983379
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 11.0718 0.428060
\(670\) 0 0
\(671\) 44.7846 1.72889
\(672\) 0 0
\(673\) −39.1769 −1.51016 −0.755080 0.655633i \(-0.772402\pi\)
−0.755080 + 0.655633i \(0.772402\pi\)
\(674\) 0 0
\(675\) −4.00000 −0.153960
\(676\) 0 0
\(677\) 43.1769 1.65942 0.829712 0.558192i \(-0.188505\pi\)
0.829712 + 0.558192i \(0.188505\pi\)
\(678\) 0 0
\(679\) −8.10512 −0.311046
\(680\) 0 0
\(681\) −13.6077 −0.521448
\(682\) 0 0
\(683\) −11.6603 −0.446167 −0.223084 0.974799i \(-0.571612\pi\)
−0.223084 + 0.974799i \(0.571612\pi\)
\(684\) 0 0
\(685\) 12.9282 0.493961
\(686\) 0 0
\(687\) 13.8564 0.528655
\(688\) 0 0
\(689\) −36.0000 −1.37149
\(690\) 0 0
\(691\) −5.60770 −0.213327 −0.106663 0.994295i \(-0.534017\pi\)
−0.106663 + 0.994295i \(0.534017\pi\)
\(692\) 0 0
\(693\) 10.8231 0.411135
\(694\) 0 0
\(695\) −10.0000 −0.379322
\(696\) 0 0
\(697\) −8.78461 −0.332741
\(698\) 0 0
\(699\) 1.17691 0.0445150
\(700\) 0 0
\(701\) −14.7846 −0.558407 −0.279204 0.960232i \(-0.590070\pi\)
−0.279204 + 0.960232i \(0.590070\pi\)
\(702\) 0 0
\(703\) 12.0000 0.452589
\(704\) 0 0
\(705\) −6.00000 −0.225973
\(706\) 0 0
\(707\) 15.2154 0.572234
\(708\) 0 0
\(709\) −10.1436 −0.380951 −0.190475 0.981692i \(-0.561003\pi\)
−0.190475 + 0.981692i \(0.561003\pi\)
\(710\) 0 0
\(711\) −29.5692 −1.10893
\(712\) 0 0
\(713\) 77.5692 2.90499
\(714\) 0 0
\(715\) 12.0000 0.448775
\(716\) 0 0
\(717\) −15.2154 −0.568229
\(718\) 0 0
\(719\) −44.7846 −1.67018 −0.835092 0.550110i \(-0.814586\pi\)
−0.835092 + 0.550110i \(0.814586\pi\)
\(720\) 0 0
\(721\) 10.3923 0.387030
\(722\) 0 0
\(723\) −14.9282 −0.555186
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 34.0526 1.26294 0.631470 0.775401i \(-0.282452\pi\)
0.631470 + 0.775401i \(0.282452\pi\)
\(728\) 0 0
\(729\) −2.21539 −0.0820515
\(730\) 0 0
\(731\) 35.3205 1.30638
\(732\) 0 0
\(733\) 38.7846 1.43254 0.716271 0.697822i \(-0.245847\pi\)
0.716271 + 0.697822i \(0.245847\pi\)
\(734\) 0 0
\(735\) 3.94744 0.145604
\(736\) 0 0
\(737\) 35.3205 1.30105
\(738\) 0 0
\(739\) −2.00000 −0.0735712 −0.0367856 0.999323i \(-0.511712\pi\)
−0.0367856 + 0.999323i \(0.511712\pi\)
\(740\) 0 0
\(741\) 5.07180 0.186317
\(742\) 0 0
\(743\) 36.5885 1.34230 0.671150 0.741321i \(-0.265801\pi\)
0.671150 + 0.741321i \(0.265801\pi\)
\(744\) 0 0
\(745\) 18.0000 0.659469
\(746\) 0 0
\(747\) 11.6603 0.426626
\(748\) 0 0
\(749\) −21.2154 −0.775193
\(750\) 0 0
\(751\) 40.3923 1.47394 0.736968 0.675928i \(-0.236257\pi\)
0.736968 + 0.675928i \(0.236257\pi\)
\(752\) 0 0
\(753\) 11.3205 0.412542
\(754\) 0 0
\(755\) −2.53590 −0.0922908
\(756\) 0 0
\(757\) 23.0718 0.838559 0.419279 0.907857i \(-0.362283\pi\)
0.419279 + 0.907857i \(0.362283\pi\)
\(758\) 0 0
\(759\) −20.7846 −0.754434
\(760\) 0 0
\(761\) 21.7128 0.787089 0.393544 0.919306i \(-0.371249\pi\)
0.393544 + 0.919306i \(0.371249\pi\)
\(762\) 0 0
\(763\) −1.17691 −0.0426072
\(764\) 0 0
\(765\) 8.53590 0.308616
\(766\) 0 0
\(767\) 20.7846 0.750489
\(768\) 0 0
\(769\) −6.78461 −0.244659 −0.122330 0.992490i \(-0.539037\pi\)
−0.122330 + 0.992490i \(0.539037\pi\)
\(770\) 0 0
\(771\) 4.39230 0.158185
\(772\) 0 0
\(773\) −46.3923 −1.66862 −0.834308 0.551299i \(-0.814132\pi\)
−0.834308 + 0.551299i \(0.814132\pi\)
\(774\) 0 0
\(775\) 9.46410 0.339961
\(776\) 0 0
\(777\) 5.56922 0.199795
\(778\) 0 0
\(779\) −5.07180 −0.181716
\(780\) 0 0
\(781\) 15.2154 0.544449
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −0.928203 −0.0331290
\(786\) 0 0
\(787\) 5.80385 0.206885 0.103442 0.994635i \(-0.467014\pi\)
0.103442 + 0.994635i \(0.467014\pi\)
\(788\) 0 0
\(789\) −14.7846 −0.526346
\(790\) 0 0
\(791\) 1.17691 0.0418463
\(792\) 0 0
\(793\) −44.7846 −1.59035
\(794\) 0 0
\(795\) 7.60770 0.269817
\(796\) 0 0
\(797\) 1.60770 0.0569475 0.0284737 0.999595i \(-0.490935\pi\)
0.0284737 + 0.999595i \(0.490935\pi\)
\(798\) 0 0
\(799\) 28.3923 1.00445
\(800\) 0 0
\(801\) 2.28719 0.0808138
\(802\) 0 0
\(803\) −49.8564 −1.75939
\(804\) 0 0
\(805\) −10.3923 −0.366281
\(806\) 0 0
\(807\) 10.8231 0.380991
\(808\) 0 0
\(809\) −35.5692 −1.25055 −0.625274 0.780406i \(-0.715013\pi\)
−0.625274 + 0.780406i \(0.715013\pi\)
\(810\) 0 0
\(811\) −38.3923 −1.34814 −0.674068 0.738669i \(-0.735455\pi\)
−0.674068 + 0.738669i \(0.735455\pi\)
\(812\) 0 0
\(813\) −3.21539 −0.112769
\(814\) 0 0
\(815\) −5.80385 −0.203300
\(816\) 0 0
\(817\) 20.3923 0.713436
\(818\) 0 0
\(819\) −10.8231 −0.378189
\(820\) 0 0
\(821\) 50.7846 1.77240 0.886198 0.463308i \(-0.153337\pi\)
0.886198 + 0.463308i \(0.153337\pi\)
\(822\) 0 0
\(823\) 30.8372 1.07492 0.537458 0.843290i \(-0.319385\pi\)
0.537458 + 0.843290i \(0.319385\pi\)
\(824\) 0 0
\(825\) −2.53590 −0.0882886
\(826\) 0 0
\(827\) 4.73205 0.164550 0.0822748 0.996610i \(-0.473781\pi\)
0.0822748 + 0.996610i \(0.473781\pi\)
\(828\) 0 0
\(829\) −50.7846 −1.76382 −0.881911 0.471416i \(-0.843743\pi\)
−0.881911 + 0.471416i \(0.843743\pi\)
\(830\) 0 0
\(831\) −14.5359 −0.504245
\(832\) 0 0
\(833\) −18.6795 −0.647206
\(834\) 0 0
\(835\) 8.19615 0.283640
\(836\) 0 0
\(837\) −37.8564 −1.30851
\(838\) 0 0
\(839\) 8.78461 0.303278 0.151639 0.988436i \(-0.451545\pi\)
0.151639 + 0.988436i \(0.451545\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 0 0
\(843\) 20.7846 0.715860
\(844\) 0 0
\(845\) 1.00000 0.0344010
\(846\) 0 0
\(847\) 1.26795 0.0435672
\(848\) 0 0
\(849\) 7.75129 0.266024
\(850\) 0 0
\(851\) 49.1769 1.68576
\(852\) 0 0
\(853\) −3.46410 −0.118609 −0.0593043 0.998240i \(-0.518888\pi\)
−0.0593043 + 0.998240i \(0.518888\pi\)
\(854\) 0 0
\(855\) 4.92820 0.168541
\(856\) 0 0
\(857\) 16.1436 0.551455 0.275727 0.961236i \(-0.411081\pi\)
0.275727 + 0.961236i \(0.411081\pi\)
\(858\) 0 0
\(859\) −51.5692 −1.75952 −0.879760 0.475419i \(-0.842296\pi\)
−0.879760 + 0.475419i \(0.842296\pi\)
\(860\) 0 0
\(861\) −2.35383 −0.0802183
\(862\) 0 0
\(863\) −40.9808 −1.39500 −0.697501 0.716584i \(-0.745705\pi\)
−0.697501 + 0.716584i \(0.745705\pi\)
\(864\) 0 0
\(865\) −6.00000 −0.204006
\(866\) 0 0
\(867\) −3.66025 −0.124309
\(868\) 0 0
\(869\) −41.5692 −1.41014
\(870\) 0 0
\(871\) −35.3205 −1.19679
\(872\) 0 0
\(873\) 15.7513 0.533100
\(874\) 0 0
\(875\) −1.26795 −0.0428645
\(876\) 0 0
\(877\) −7.85641 −0.265292 −0.132646 0.991163i \(-0.542347\pi\)
−0.132646 + 0.991163i \(0.542347\pi\)
\(878\) 0 0
\(879\) 21.9615 0.740744
\(880\) 0 0
\(881\) −7.60770 −0.256310 −0.128155 0.991754i \(-0.540905\pi\)
−0.128155 + 0.991754i \(0.540905\pi\)
\(882\) 0 0
\(883\) 5.80385 0.195315 0.0976575 0.995220i \(-0.468865\pi\)
0.0976575 + 0.995220i \(0.468865\pi\)
\(884\) 0 0
\(885\) −4.39230 −0.147646
\(886\) 0 0
\(887\) −21.3731 −0.717637 −0.358819 0.933407i \(-0.616820\pi\)
−0.358819 + 0.933407i \(0.616820\pi\)
\(888\) 0 0
\(889\) 4.82309 0.161761
\(890\) 0 0
\(891\) −15.4641 −0.518067
\(892\) 0 0
\(893\) 16.3923 0.548548
\(894\) 0 0
\(895\) −19.8564 −0.663726
\(896\) 0 0
\(897\) 20.7846 0.693978
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) 0 0
\(903\) 9.46410 0.314946
\(904\) 0 0
\(905\) 6.92820 0.230301
\(906\) 0 0
\(907\) 1.41154 0.0468695 0.0234348 0.999725i \(-0.492540\pi\)
0.0234348 + 0.999725i \(0.492540\pi\)
\(908\) 0 0
\(909\) −29.5692 −0.980749
\(910\) 0 0
\(911\) 37.1769 1.23173 0.615863 0.787853i \(-0.288807\pi\)
0.615863 + 0.787853i \(0.288807\pi\)
\(912\) 0 0
\(913\) 16.3923 0.542506
\(914\) 0 0
\(915\) 9.46410 0.312874
\(916\) 0 0
\(917\) 13.1769 0.435140
\(918\) 0 0
\(919\) 39.7128 1.31000 0.655002 0.755627i \(-0.272668\pi\)
0.655002 + 0.755627i \(0.272668\pi\)
\(920\) 0 0
\(921\) −25.0333 −0.824876
\(922\) 0 0
\(923\) −15.2154 −0.500821
\(924\) 0 0
\(925\) 6.00000 0.197279
\(926\) 0 0
\(927\) −20.1962 −0.663329
\(928\) 0 0
\(929\) 7.60770 0.249600 0.124800 0.992182i \(-0.460171\pi\)
0.124800 + 0.992182i \(0.460171\pi\)
\(930\) 0 0
\(931\) −10.7846 −0.353451
\(932\) 0 0
\(933\) −5.56922 −0.182328
\(934\) 0 0
\(935\) 12.0000 0.392442
\(936\) 0 0
\(937\) 9.60770 0.313870 0.156935 0.987609i \(-0.449839\pi\)
0.156935 + 0.987609i \(0.449839\pi\)
\(938\) 0 0
\(939\) −16.1051 −0.525571
\(940\) 0 0
\(941\) 20.7846 0.677559 0.338779 0.940866i \(-0.389986\pi\)
0.338779 + 0.940866i \(0.389986\pi\)
\(942\) 0 0
\(943\) −20.7846 −0.676840
\(944\) 0 0
\(945\) 5.07180 0.164986
\(946\) 0 0
\(947\) 45.8038 1.48843 0.744213 0.667943i \(-0.232825\pi\)
0.744213 + 0.667943i \(0.232825\pi\)
\(948\) 0 0
\(949\) 49.8564 1.61841
\(950\) 0 0
\(951\) −16.3923 −0.531557
\(952\) 0 0
\(953\) −24.9282 −0.807504 −0.403752 0.914869i \(-0.632294\pi\)
−0.403752 + 0.914869i \(0.632294\pi\)
\(954\) 0 0
\(955\) 16.3923 0.530443
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −16.3923 −0.529335
\(960\) 0 0
\(961\) 58.5692 1.88933
\(962\) 0 0
\(963\) 41.2295 1.32860
\(964\) 0 0
\(965\) −6.39230 −0.205776
\(966\) 0 0
\(967\) −8.19615 −0.263570 −0.131785 0.991278i \(-0.542071\pi\)
−0.131785 + 0.991278i \(0.542071\pi\)
\(968\) 0 0
\(969\) 5.07180 0.162930
\(970\) 0 0
\(971\) −33.0333 −1.06009 −0.530045 0.847970i \(-0.677825\pi\)
−0.530045 + 0.847970i \(0.677825\pi\)
\(972\) 0 0
\(973\) 12.6795 0.406486
\(974\) 0 0
\(975\) 2.53590 0.0812137
\(976\) 0 0
\(977\) 5.32051 0.170218 0.0851091 0.996372i \(-0.472876\pi\)
0.0851091 + 0.996372i \(0.472876\pi\)
\(978\) 0 0
\(979\) 3.21539 0.102764
\(980\) 0 0
\(981\) 2.28719 0.0730243
\(982\) 0 0
\(983\) −11.4115 −0.363972 −0.181986 0.983301i \(-0.558253\pi\)
−0.181986 + 0.983301i \(0.558253\pi\)
\(984\) 0 0
\(985\) −10.3923 −0.331126
\(986\) 0 0
\(987\) 7.60770 0.242156
\(988\) 0 0
\(989\) 83.5692 2.65735
\(990\) 0 0
\(991\) −44.1051 −1.40105 −0.700523 0.713630i \(-0.747050\pi\)
−0.700523 + 0.713630i \(0.747050\pi\)
\(992\) 0 0
\(993\) 4.10512 0.130272
\(994\) 0 0
\(995\) 6.92820 0.219639
\(996\) 0 0
\(997\) −25.6077 −0.811004 −0.405502 0.914094i \(-0.632903\pi\)
−0.405502 + 0.914094i \(0.632903\pi\)
\(998\) 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 1280.2.a.b.1.2 2
4.3 odd 2 1280.2.a.m.1.1 2
5.4 even 2 6400.2.a.cd.1.1 2
8.3 odd 2 1280.2.a.c.1.2 2
8.5 even 2 1280.2.a.p.1.1 2
16.3 odd 4 320.2.d.b.161.2 yes 4
16.5 even 4 320.2.d.a.161.2 4
16.11 odd 4 320.2.d.b.161.3 yes 4
16.13 even 4 320.2.d.a.161.3 yes 4
20.19 odd 2 6400.2.a.bf.1.2 2
40.19 odd 2 6400.2.a.ck.1.1 2
40.29 even 2 6400.2.a.y.1.2 2
48.5 odd 4 2880.2.k.e.1441.4 4
48.11 even 4 2880.2.k.l.1441.3 4
48.29 odd 4 2880.2.k.e.1441.2 4
48.35 even 4 2880.2.k.l.1441.1 4
80.3 even 4 1600.2.f.d.1249.3 4
80.13 odd 4 1600.2.f.i.1249.2 4
80.19 odd 4 1600.2.d.b.801.3 4
80.27 even 4 1600.2.f.d.1249.4 4
80.29 even 4 1600.2.d.h.801.2 4
80.37 odd 4 1600.2.f.i.1249.1 4
80.43 even 4 1600.2.f.h.1249.1 4
80.53 odd 4 1600.2.f.e.1249.4 4
80.59 odd 4 1600.2.d.b.801.2 4
80.67 even 4 1600.2.f.h.1249.2 4
80.69 even 4 1600.2.d.h.801.3 4
80.77 odd 4 1600.2.f.e.1249.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
320.2.d.a.161.2 4 16.5 even 4
320.2.d.a.161.3 yes 4 16.13 even 4
320.2.d.b.161.2 yes 4 16.3 odd 4
320.2.d.b.161.3 yes 4 16.11 odd 4
1280.2.a.b.1.2 2 1.1 even 1 trivial
1280.2.a.c.1.2 2 8.3 odd 2
1280.2.a.m.1.1 2 4.3 odd 2
1280.2.a.p.1.1 2 8.5 even 2
1600.2.d.b.801.2 4 80.59 odd 4
1600.2.d.b.801.3 4 80.19 odd 4
1600.2.d.h.801.2 4 80.29 even 4
1600.2.d.h.801.3 4 80.69 even 4
1600.2.f.d.1249.3 4 80.3 even 4
1600.2.f.d.1249.4 4 80.27 even 4
1600.2.f.e.1249.3 4 80.77 odd 4
1600.2.f.e.1249.4 4 80.53 odd 4
1600.2.f.h.1249.1 4 80.43 even 4
1600.2.f.h.1249.2 4 80.67 even 4
1600.2.f.i.1249.1 4 80.37 odd 4
1600.2.f.i.1249.2 4 80.13 odd 4
2880.2.k.e.1441.2 4 48.29 odd 4
2880.2.k.e.1441.4 4 48.5 odd 4
2880.2.k.l.1441.1 4 48.35 even 4
2880.2.k.l.1441.3 4 48.11 even 4
6400.2.a.y.1.2 2 40.29 even 2
6400.2.a.bf.1.2 2 20.19 odd 2
6400.2.a.cd.1.1 2 5.4 even 2
6400.2.a.ck.1.1 2 40.19 odd 2