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Properties

Label 8001.2.a.f.1.1
Level $8001$
Weight $2$
Character 8001.1
Self dual yes
Analytic conductor $63.888$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8001,2,Mod(1,8001)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8001, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8001.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8001.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(63.8883066572\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2667)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 8001.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} +2.00000 q^{4} -1.00000 q^{7} +O(q^{10})\) \(q+2.00000 q^{2} +2.00000 q^{4} -1.00000 q^{7} +2.00000 q^{13} -2.00000 q^{14} -4.00000 q^{16} -1.00000 q^{17} -2.00000 q^{19} -5.00000 q^{25} +4.00000 q^{26} -2.00000 q^{28} +9.00000 q^{29} -8.00000 q^{32} -2.00000 q^{34} -1.00000 q^{37} -4.00000 q^{38} +3.00000 q^{41} -6.00000 q^{43} -8.00000 q^{47} +1.00000 q^{49} -10.0000 q^{50} +4.00000 q^{52} -11.0000 q^{53} +18.0000 q^{58} +12.0000 q^{59} -10.0000 q^{61} -8.00000 q^{64} +4.00000 q^{67} -2.00000 q^{68} -8.00000 q^{71} -4.00000 q^{73} -2.00000 q^{74} -4.00000 q^{76} -13.0000 q^{79} +6.00000 q^{82} +6.00000 q^{83} -12.0000 q^{86} -12.0000 q^{89} -2.00000 q^{91} -16.0000 q^{94} -7.00000 q^{97} +2.00000 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 1.41421 0.707107 − 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(3\) 0 0
\(4\) 2.00000 1.00000
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) −1.00000 −0.242536 −0.121268 − 0.992620i \(-0.538696\pi\)
−0.121268 + 0.992620i \(0.538696\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 − 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 4.00000 0.784465
\(27\) 0 0
\(28\) −2.00000 −0.377964
\(29\) 9.00000 1.67126 0.835629 − 0.549294i \(-0.185103\pi\)
0.835629 + 0.549294i \(0.185103\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −8.00000 −1.41421
\(33\) 0 0
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) 0 0
\(37\) −1.00000 −0.164399 −0.0821995 − 0.996616i \(-0.526194\pi\)
−0.0821995 + 0.996616i \(0.526194\pi\)
\(38\) −4.00000 −0.648886
\(39\) 0 0
\(40\) 0 0
\(41\) 3.00000 0.468521 0.234261 − 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) 0 0
\(43\) −6.00000 −0.914991 −0.457496 − 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 − 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −10.0000 −1.41421
\(51\) 0 0
\(52\) 4.00000 0.554700
\(53\) −11.0000 −1.51097 −0.755483 − 0.655168i \(-0.772598\pi\)
−0.755483 + 0.655168i \(0.772598\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 18.0000 2.36352
\(59\) 12.0000 1.56227 0.781133 − 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 − 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 − 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) −2.00000 −0.242536
\(69\) 0 0
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) −4.00000 −0.468165 −0.234082 − 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) 0 0
\(79\) −13.0000 −1.46261 −0.731307 − 0.682048i \(-0.761089\pi\)
−0.731307 + 0.682048i \(0.761089\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 6.00000 0.662589
\(83\) 6.00000 0.658586 0.329293 − 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −12.0000 −1.29399
\(87\) 0 0
\(88\) 0 0
\(89\) −12.0000 −1.27200 −0.635999 − 0.771690i \(-0.719412\pi\)
−0.635999 + 0.771690i \(0.719412\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) 0 0
\(93\) 0 0
\(94\) −16.0000 −1.65027
\(95\) 0 0
\(96\) 0 0
\(97\) −7.00000 −0.710742 −0.355371 − 0.934725i \(-0.615646\pi\)
−0.355371 + 0.934725i \(0.615646\pi\)
\(98\) 2.00000 0.202031
\(99\) 0 0
\(100\) −10.0000 −1.00000
\(101\) −4.00000 −0.398015 −0.199007 − 0.979998i \(-0.563772\pi\)
−0.199007 + 0.979998i \(0.563772\pi\)
\(102\) 0 0
\(103\) −16.0000 −1.57653 −0.788263 − 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −22.0000 −2.13683
\(107\) −6.00000 −0.580042 −0.290021 − 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) 0 0
\(109\) 16.0000 1.53252 0.766261 − 0.642529i \(-0.222115\pi\)
0.766261 + 0.642529i \(0.222115\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 4.00000 0.377964
\(113\) 6.00000 0.564433 0.282216 − 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 18.0000 1.67126
\(117\) 0 0
\(118\) 24.0000 2.20938
\(119\) 1.00000 0.0916698
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) −20.0000 −1.81071
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 1.00000 0.0887357
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −1.00000 −0.0873704 −0.0436852 − 0.999045i \(-0.513910\pi\)
−0.0436852 + 0.999045i \(0.513910\pi\)
\(132\) 0 0
\(133\) 2.00000 0.173422
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) 0 0
\(137\) 6.00000 0.512615 0.256307 − 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) −7.00000 −0.593732 −0.296866 − 0.954919i \(-0.595942\pi\)
−0.296866 + 0.954919i \(0.595942\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −16.0000 −1.34269
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) −8.00000 −0.662085
\(147\) 0 0
\(148\) −2.00000 −0.164399
\(149\) 10.0000 0.819232 0.409616 − 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 − 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −12.0000 −0.957704 −0.478852 − 0.877896i \(-0.658947\pi\)
−0.478852 + 0.877896i \(0.658947\pi\)
\(158\) −26.0000 −2.06845
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −15.0000 −1.17489 −0.587445 − 0.809264i \(-0.699866\pi\)
−0.587445 + 0.809264i \(0.699866\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) 2.00000 0.154765 0.0773823 − 0.997001i \(-0.475344\pi\)
0.0773823 + 0.997001i \(0.475344\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) −12.0000 −0.914991
\(173\) −14.0000 −1.06440 −0.532200 − 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) 0 0
\(175\) 5.00000 0.377964
\(176\) 0 0
\(177\) 0 0
\(178\) −24.0000 −1.79888
\(179\) −18.0000 −1.34538 −0.672692 − 0.739923i \(-0.734862\pi\)
−0.672692 + 0.739923i \(0.734862\pi\)
\(180\) 0 0
\(181\) −21.0000 −1.56092 −0.780459 − 0.625207i \(-0.785014\pi\)
−0.780459 + 0.625207i \(0.785014\pi\)
\(182\) −4.00000 −0.296500
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) −16.0000 −1.16692
\(189\) 0 0
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 − 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 0 0
\(193\) 22.0000 1.58359 0.791797 − 0.610784i \(-0.209146\pi\)
0.791797 + 0.610784i \(0.209146\pi\)
\(194\) −14.0000 −1.00514
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) 16.0000 1.13995 0.569976 − 0.821661i \(-0.306952\pi\)
0.569976 + 0.821661i \(0.306952\pi\)
\(198\) 0 0
\(199\) 20.0000 1.41776 0.708881 − 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −8.00000 −0.562878
\(203\) −9.00000 −0.631676
\(204\) 0 0
\(205\) 0 0
\(206\) −32.0000 −2.22955
\(207\) 0 0
\(208\) −8.00000 −0.554700
\(209\) 0 0
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 − 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) −22.0000 −1.51097
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 32.0000 2.16731
\(219\) 0 0
\(220\) 0 0
\(221\) −2.00000 −0.134535
\(222\) 0 0
\(223\) −5.00000 −0.334825 −0.167412 − 0.985887i \(-0.553541\pi\)
−0.167412 + 0.985887i \(0.553541\pi\)
\(224\) 8.00000 0.534522
\(225\) 0 0
\(226\) 12.0000 0.798228
\(227\) 12.0000 0.796468 0.398234 − 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) −17.0000 −1.12339 −0.561696 − 0.827344i \(-0.689851\pi\)
−0.561696 + 0.827344i \(0.689851\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 11.0000 0.720634 0.360317 − 0.932830i \(-0.382669\pi\)
0.360317 + 0.932830i \(0.382669\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 24.0000 1.56227
\(237\) 0 0
\(238\) 2.00000 0.129641
\(239\) 21.0000 1.35838 0.679189 − 0.733964i \(-0.262332\pi\)
0.679189 + 0.733964i \(0.262332\pi\)
\(240\) 0 0
\(241\) 11.0000 0.708572 0.354286 − 0.935137i \(-0.384724\pi\)
0.354286 + 0.935137i \(0.384724\pi\)
\(242\) −22.0000 −1.41421
\(243\) 0 0
\(244\) −20.0000 −1.28037
\(245\) 0 0
\(246\) 0 0
\(247\) −4.00000 −0.254514
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −13.0000 −0.820553 −0.410276 − 0.911961i \(-0.634568\pi\)
−0.410276 + 0.911961i \(0.634568\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 2.00000 0.125491
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 22.0000 1.37232 0.686161 − 0.727450i \(-0.259294\pi\)
0.686161 + 0.727450i \(0.259294\pi\)
\(258\) 0 0
\(259\) 1.00000 0.0621370
\(260\) 0 0
\(261\) 0 0
\(262\) −2.00000 −0.123560
\(263\) −12.0000 −0.739952 −0.369976 − 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 4.00000 0.245256
\(267\) 0 0
\(268\) 8.00000 0.488678
\(269\) 1.00000 0.0609711 0.0304855 − 0.999535i \(-0.490295\pi\)
0.0304855 + 0.999535i \(0.490295\pi\)
\(270\) 0 0
\(271\) 4.00000 0.242983 0.121491 − 0.992592i \(-0.461232\pi\)
0.121491 + 0.992592i \(0.461232\pi\)
\(272\) 4.00000 0.242536
\(273\) 0 0
\(274\) 12.0000 0.724947
\(275\) 0 0
\(276\) 0 0
\(277\) −20.0000 −1.20168 −0.600842 − 0.799368i \(-0.705168\pi\)
−0.600842 + 0.799368i \(0.705168\pi\)
\(278\) −14.0000 −0.839664
\(279\) 0 0
\(280\) 0 0
\(281\) 3.00000 0.178965 0.0894825 − 0.995988i \(-0.471479\pi\)
0.0894825 + 0.995988i \(0.471479\pi\)
\(282\) 0 0
\(283\) −13.0000 −0.772770 −0.386385 − 0.922338i \(-0.626276\pi\)
−0.386385 + 0.922338i \(0.626276\pi\)
\(284\) −16.0000 −0.949425
\(285\) 0 0
\(286\) 0 0
\(287\) −3.00000 −0.177084
\(288\) 0 0
\(289\) −16.0000 −0.941176
\(290\) 0 0
\(291\) 0 0
\(292\) −8.00000 −0.468165
\(293\) 24.0000 1.40209 0.701047 − 0.713115i \(-0.252716\pi\)
0.701047 + 0.713115i \(0.252716\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 20.0000 1.15857
\(299\) 0 0
\(300\) 0 0
\(301\) 6.00000 0.345834
\(302\) 16.0000 0.920697
\(303\) 0 0
\(304\) 8.00000 0.458831
\(305\) 0 0
\(306\) 0 0
\(307\) −21.0000 −1.19853 −0.599267 − 0.800549i \(-0.704541\pi\)
−0.599267 + 0.800549i \(0.704541\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −4.00000 −0.226819 −0.113410 − 0.993548i \(-0.536177\pi\)
−0.113410 + 0.993548i \(0.536177\pi\)
\(312\) 0 0
\(313\) 27.0000 1.52613 0.763065 − 0.646322i \(-0.223694\pi\)
0.763065 + 0.646322i \(0.223694\pi\)
\(314\) −24.0000 −1.35440
\(315\) 0 0
\(316\) −26.0000 −1.46261
\(317\) 34.0000 1.90963 0.954815 − 0.297200i \(-0.0960529\pi\)
0.954815 + 0.297200i \(0.0960529\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 2.00000 0.111283
\(324\) 0 0
\(325\) −10.0000 −0.554700
\(326\) −30.0000 −1.66155
\(327\) 0 0
\(328\) 0 0
\(329\) 8.00000 0.441054
\(330\) 0 0
\(331\) 2.00000 0.109930 0.0549650 − 0.998488i \(-0.482495\pi\)
0.0549650 + 0.998488i \(0.482495\pi\)
\(332\) 12.0000 0.658586
\(333\) 0 0
\(334\) 4.00000 0.218870
\(335\) 0 0
\(336\) 0 0
\(337\) 10.0000 0.544735 0.272367 − 0.962193i \(-0.412193\pi\)
0.272367 + 0.962193i \(0.412193\pi\)
\(338\) −18.0000 −0.979071
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) −28.0000 −1.50529
\(347\) 15.0000 0.805242 0.402621 − 0.915367i \(-0.368099\pi\)
0.402621 + 0.915367i \(0.368099\pi\)
\(348\) 0 0
\(349\) −27.0000 −1.44528 −0.722638 − 0.691226i \(-0.757071\pi\)
−0.722638 + 0.691226i \(0.757071\pi\)
\(350\) 10.0000 0.534522
\(351\) 0 0
\(352\) 0 0
\(353\) −2.00000 −0.106449 −0.0532246 − 0.998583i \(-0.516950\pi\)
−0.0532246 + 0.998583i \(0.516950\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −24.0000 −1.27200
\(357\) 0 0
\(358\) −36.0000 −1.90266
\(359\) 27.0000 1.42501 0.712503 − 0.701669i \(-0.247562\pi\)
0.712503 + 0.701669i \(0.247562\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) −42.0000 −2.20747
\(363\) 0 0
\(364\) −4.00000 −0.209657
\(365\) 0 0
\(366\) 0 0
\(367\) 2.00000 0.104399 0.0521996 − 0.998637i \(-0.483377\pi\)
0.0521996 + 0.998637i \(0.483377\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 11.0000 0.571092
\(372\) 0 0
\(373\) 4.00000 0.207112 0.103556 − 0.994624i \(-0.466978\pi\)
0.103556 + 0.994624i \(0.466978\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 18.0000 0.927047
\(378\) 0 0
\(379\) −36.0000 −1.84920 −0.924598 − 0.380945i \(-0.875599\pi\)
−0.924598 + 0.380945i \(0.875599\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −16.0000 −0.818631
\(383\) 3.00000 0.153293 0.0766464 − 0.997058i \(-0.475579\pi\)
0.0766464 + 0.997058i \(0.475579\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 44.0000 2.23954
\(387\) 0 0
\(388\) −14.0000 −0.710742
\(389\) 10.0000 0.507020 0.253510 − 0.967333i \(-0.418415\pi\)
0.253510 + 0.967333i \(0.418415\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 32.0000 1.61214
\(395\) 0 0
\(396\) 0 0
\(397\) −16.0000 −0.803017 −0.401508 − 0.915855i \(-0.631514\pi\)
−0.401508 + 0.915855i \(0.631514\pi\)
\(398\) 40.0000 2.00502
\(399\) 0 0
\(400\) 20.0000 1.00000
\(401\) −9.00000 −0.449439 −0.224719 − 0.974424i \(-0.572147\pi\)
−0.224719 + 0.974424i \(0.572147\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −8.00000 −0.398015
\(405\) 0 0
\(406\) −18.0000 −0.893325
\(407\) 0 0
\(408\) 0 0
\(409\) 18.0000 0.890043 0.445021 − 0.895520i \(-0.353196\pi\)
0.445021 + 0.895520i \(0.353196\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −32.0000 −1.57653
\(413\) −12.0000 −0.590481
\(414\) 0 0
\(415\) 0 0
\(416\) −16.0000 −0.784465
\(417\) 0 0
\(418\) 0 0
\(419\) −28.0000 −1.36789 −0.683945 − 0.729534i \(-0.739737\pi\)
−0.683945 + 0.729534i \(0.739737\pi\)
\(420\) 0 0
\(421\) −2.00000 −0.0974740 −0.0487370 − 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) 8.00000 0.389434
\(423\) 0 0
\(424\) 0 0
\(425\) 5.00000 0.242536
\(426\) 0 0
\(427\) 10.0000 0.483934
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) 0 0
\(431\) 12.0000 0.578020 0.289010 − 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) 0 0
\(433\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 32.0000 1.53252
\(437\) 0 0
\(438\) 0 0
\(439\) −7.00000 −0.334092 −0.167046 − 0.985949i \(-0.553423\pi\)
−0.167046 + 0.985949i \(0.553423\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −4.00000 −0.190261
\(443\) −14.0000 −0.665160 −0.332580 − 0.943075i \(-0.607919\pi\)
−0.332580 + 0.943075i \(0.607919\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −10.0000 −0.473514
\(447\) 0 0
\(448\) 8.00000 0.377964
\(449\) −16.0000 −0.755087 −0.377543 − 0.925992i \(-0.623231\pi\)
−0.377543 + 0.925992i \(0.623231\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 12.0000 0.564433
\(453\) 0 0
\(454\) 24.0000 1.12638
\(455\) 0 0
\(456\) 0 0
\(457\) 35.0000 1.63723 0.818615 − 0.574342i \(-0.194742\pi\)
0.818615 + 0.574342i \(0.194742\pi\)
\(458\) −34.0000 −1.58872
\(459\) 0 0
\(460\) 0 0
\(461\) −40.0000 −1.86299 −0.931493 − 0.363760i \(-0.881493\pi\)
−0.931493 + 0.363760i \(0.881493\pi\)
\(462\) 0 0
\(463\) −36.0000 −1.67306 −0.836531 − 0.547920i \(-0.815420\pi\)
−0.836531 + 0.547920i \(0.815420\pi\)
\(464\) −36.0000 −1.67126
\(465\) 0 0
\(466\) 22.0000 1.01913
\(467\) −4.00000 −0.185098 −0.0925490 − 0.995708i \(-0.529501\pi\)
−0.0925490 + 0.995708i \(0.529501\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 10.0000 0.458831
\(476\) 2.00000 0.0916698
\(477\) 0 0
\(478\) 42.0000 1.92104
\(479\) 19.0000 0.868132 0.434066 − 0.900881i \(-0.357078\pi\)
0.434066 + 0.900881i \(0.357078\pi\)
\(480\) 0 0
\(481\) −2.00000 −0.0911922
\(482\) 22.0000 1.00207
\(483\) 0 0
\(484\) −22.0000 −1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) 30.0000 1.35943 0.679715 − 0.733476i \(-0.262104\pi\)
0.679715 + 0.733476i \(0.262104\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −44.0000 −1.98569 −0.992846 − 0.119401i \(-0.961903\pi\)
−0.992846 + 0.119401i \(0.961903\pi\)
\(492\) 0 0
\(493\) −9.00000 −0.405340
\(494\) −8.00000 −0.359937
\(495\) 0 0
\(496\) 0 0
\(497\) 8.00000 0.358849
\(498\) 0 0
\(499\) 10.0000 0.447661 0.223831 − 0.974628i \(-0.428144\pi\)
0.223831 + 0.974628i \(0.428144\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −26.0000 −1.16044
\(503\) −1.00000 −0.0445878 −0.0222939 − 0.999751i \(-0.507097\pi\)
−0.0222939 + 0.999751i \(0.507097\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 2.00000 0.0887357
\(509\) −41.0000 −1.81729 −0.908647 − 0.417566i \(-0.862883\pi\)
−0.908647 + 0.417566i \(0.862883\pi\)
\(510\) 0 0
\(511\) 4.00000 0.176950
\(512\) 32.0000 1.41421
\(513\) 0 0
\(514\) 44.0000 1.94076
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 2.00000 0.0878750
\(519\) 0 0
\(520\) 0 0
\(521\) 30.0000 1.31432 0.657162 − 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) 0 0
\(523\) 32.0000 1.39926 0.699631 − 0.714504i \(-0.253348\pi\)
0.699631 + 0.714504i \(0.253348\pi\)
\(524\) −2.00000 −0.0873704
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) 0 0
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 4.00000 0.173422
\(533\) 6.00000 0.259889
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 2.00000 0.0862261
\(539\) 0 0
\(540\) 0 0
\(541\) 24.0000 1.03184 0.515920 − 0.856637i \(-0.327450\pi\)
0.515920 + 0.856637i \(0.327450\pi\)
\(542\) 8.00000 0.343629
\(543\) 0 0
\(544\) 8.00000 0.342997
\(545\) 0 0
\(546\) 0 0
\(547\) −12.0000 −0.513083 −0.256541 − 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) 12.0000 0.512615
\(549\) 0 0
\(550\) 0 0
\(551\) −18.0000 −0.766826
\(552\) 0 0
\(553\) 13.0000 0.552816
\(554\) −40.0000 −1.69944
\(555\) 0 0
\(556\) −14.0000 −0.593732
\(557\) 40.0000 1.69485 0.847427 − 0.530912i \(-0.178150\pi\)
0.847427 + 0.530912i \(0.178150\pi\)
\(558\) 0 0
\(559\) −12.0000 −0.507546
\(560\) 0 0
\(561\) 0 0
\(562\) 6.00000 0.253095
\(563\) 32.0000 1.34864 0.674320 − 0.738440i \(-0.264437\pi\)
0.674320 + 0.738440i \(0.264437\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −26.0000 −1.09286
\(567\) 0 0
\(568\) 0 0
\(569\) 22.0000 0.922288 0.461144 − 0.887325i \(-0.347439\pi\)
0.461144 + 0.887325i \(0.347439\pi\)
\(570\) 0 0
\(571\) 14.0000 0.585882 0.292941 − 0.956131i \(-0.405366\pi\)
0.292941 + 0.956131i \(0.405366\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −6.00000 −0.250435
\(575\) 0 0
\(576\) 0 0
\(577\) 8.00000 0.333044 0.166522 − 0.986038i \(-0.446746\pi\)
0.166522 + 0.986038i \(0.446746\pi\)
\(578\) −32.0000 −1.33102
\(579\) 0 0
\(580\) 0 0
\(581\) −6.00000 −0.248922
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 48.0000 1.98286
\(587\) −17.0000 −0.701665 −0.350833 − 0.936438i \(-0.614101\pi\)
−0.350833 + 0.936438i \(0.614101\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 4.00000 0.164399
\(593\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 20.0000 0.819232
\(597\) 0 0
\(598\) 0 0
\(599\) −21.0000 −0.858037 −0.429018 − 0.903296i \(-0.641140\pi\)
−0.429018 + 0.903296i \(0.641140\pi\)
\(600\) 0 0
\(601\) 27.0000 1.10135 0.550676 − 0.834719i \(-0.314370\pi\)
0.550676 + 0.834719i \(0.314370\pi\)
\(602\) 12.0000 0.489083
\(603\) 0 0
\(604\) 16.0000 0.651031
\(605\) 0 0
\(606\) 0 0
\(607\) −10.0000 −0.405887 −0.202944 − 0.979190i \(-0.565051\pi\)
−0.202944 + 0.979190i \(0.565051\pi\)
\(608\) 16.0000 0.648886
\(609\) 0 0
\(610\) 0 0
\(611\) −16.0000 −0.647291
\(612\) 0 0
\(613\) 46.0000 1.85792 0.928961 − 0.370177i \(-0.120703\pi\)
0.928961 + 0.370177i \(0.120703\pi\)
\(614\) −42.0000 −1.69498
\(615\) 0 0
\(616\) 0 0
\(617\) 37.0000 1.48956 0.744782 − 0.667308i \(-0.232553\pi\)
0.744782 + 0.667308i \(0.232553\pi\)
\(618\) 0 0
\(619\) 1.00000 0.0401934 0.0200967 − 0.999798i \(-0.493603\pi\)
0.0200967 + 0.999798i \(0.493603\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −8.00000 −0.320771
\(623\) 12.0000 0.480770
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 54.0000 2.15827
\(627\) 0 0
\(628\) −24.0000 −0.957704
\(629\) 1.00000 0.0398726
\(630\) 0 0
\(631\) 10.0000 0.398094 0.199047 − 0.979990i \(-0.436215\pi\)
0.199047 + 0.979990i \(0.436215\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 68.0000 2.70063
\(635\) 0 0
\(636\) 0 0
\(637\) 2.00000 0.0792429
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −35.0000 −1.38242 −0.691208 − 0.722655i \(-0.742921\pi\)
−0.691208 + 0.722655i \(0.742921\pi\)
\(642\) 0 0
\(643\) 38.0000 1.49857 0.749287 − 0.662246i \(-0.230396\pi\)
0.749287 + 0.662246i \(0.230396\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 4.00000 0.157378
\(647\) −38.0000 −1.49393 −0.746967 − 0.664861i \(-0.768491\pi\)
−0.746967 + 0.664861i \(0.768491\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −20.0000 −0.784465
\(651\) 0 0
\(652\) −30.0000 −1.17489
\(653\) −14.0000 −0.547862 −0.273931 − 0.961749i \(-0.588324\pi\)
−0.273931 + 0.961749i \(0.588324\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −12.0000 −0.468521
\(657\) 0 0
\(658\) 16.0000 0.623745
\(659\) −4.00000 −0.155818 −0.0779089 − 0.996960i \(-0.524824\pi\)
−0.0779089 + 0.996960i \(0.524824\pi\)
\(660\) 0 0
\(661\) −28.0000 −1.08907 −0.544537 − 0.838737i \(-0.683295\pi\)
−0.544537 + 0.838737i \(0.683295\pi\)
\(662\) 4.00000 0.155464
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 4.00000 0.154765
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 25.0000 0.963679 0.481840 − 0.876259i \(-0.339969\pi\)
0.481840 + 0.876259i \(0.339969\pi\)
\(674\) 20.0000 0.770371
\(675\) 0 0
\(676\) −18.0000 −0.692308
\(677\) 21.0000 0.807096 0.403548 − 0.914959i \(-0.367777\pi\)
0.403548 + 0.914959i \(0.367777\pi\)
\(678\) 0 0
\(679\) 7.00000 0.268635
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −16.0000 −0.612223 −0.306111 − 0.951996i \(-0.599028\pi\)
−0.306111 + 0.951996i \(0.599028\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −2.00000 −0.0763604
\(687\) 0 0
\(688\) 24.0000 0.914991
\(689\) −22.0000 −0.838133
\(690\) 0 0
\(691\) −37.0000 −1.40755 −0.703773 − 0.710425i \(-0.748503\pi\)
−0.703773 + 0.710425i \(0.748503\pi\)
\(692\) −28.0000 −1.06440
\(693\) 0 0
\(694\) 30.0000 1.13878
\(695\) 0 0
\(696\) 0 0
\(697\) −3.00000 −0.113633
\(698\) −54.0000 −2.04393
\(699\) 0 0
\(700\) 10.0000 0.377964
\(701\) 22.0000 0.830929 0.415464 − 0.909610i \(-0.363619\pi\)
0.415464 + 0.909610i \(0.363619\pi\)
\(702\) 0 0
\(703\) 2.00000 0.0754314
\(704\) 0 0
\(705\) 0 0
\(706\) −4.00000 −0.150542
\(707\) 4.00000 0.150435
\(708\) 0 0
\(709\) 21.0000 0.788672 0.394336 − 0.918966i \(-0.370975\pi\)
0.394336 + 0.918966i \(0.370975\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −36.0000 −1.34538
\(717\) 0 0
\(718\) 54.0000 2.01526
\(719\) −21.0000 −0.783168 −0.391584 − 0.920142i \(-0.628073\pi\)
−0.391584 + 0.920142i \(0.628073\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) −30.0000 −1.11648
\(723\) 0 0
\(724\) −42.0000 −1.56092
\(725\) −45.0000 −1.67126
\(726\) 0 0
\(727\) −3.00000 −0.111264 −0.0556319 − 0.998451i \(-0.517717\pi\)
−0.0556319 + 0.998451i \(0.517717\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 6.00000 0.221918
\(732\) 0 0
\(733\) 14.0000 0.517102 0.258551 − 0.965998i \(-0.416755\pi\)
0.258551 + 0.965998i \(0.416755\pi\)
\(734\) 4.00000 0.147643
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 31.0000 1.14035 0.570177 − 0.821522i \(-0.306875\pi\)
0.570177 + 0.821522i \(0.306875\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 22.0000 0.807645
\(743\) 15.0000 0.550297 0.275148 − 0.961402i \(-0.411273\pi\)
0.275148 + 0.961402i \(0.411273\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 8.00000 0.292901
\(747\) 0 0
\(748\) 0 0
\(749\) 6.00000 0.219235
\(750\) 0 0
\(751\) 20.0000 0.729810 0.364905 − 0.931045i \(-0.381101\pi\)
0.364905 + 0.931045i \(0.381101\pi\)
\(752\) 32.0000 1.16692
\(753\) 0 0
\(754\) 36.0000 1.31104
\(755\) 0 0
\(756\) 0 0
\(757\) 23.0000 0.835949 0.417975 − 0.908459i \(-0.362740\pi\)
0.417975 + 0.908459i \(0.362740\pi\)
\(758\) −72.0000 −2.61516
\(759\) 0 0
\(760\) 0 0
\(761\) −4.00000 −0.145000 −0.0724999 − 0.997368i \(-0.523098\pi\)
−0.0724999 + 0.997368i \(0.523098\pi\)
\(762\) 0 0
\(763\) −16.0000 −0.579239
\(764\) −16.0000 −0.578860
\(765\) 0 0
\(766\) 6.00000 0.216789
\(767\) 24.0000 0.866590
\(768\) 0 0
\(769\) −41.0000 −1.47850 −0.739249 − 0.673432i \(-0.764819\pi\)
−0.739249 + 0.673432i \(0.764819\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 44.0000 1.58359
\(773\) −9.00000 −0.323708 −0.161854 − 0.986815i \(-0.551747\pi\)
−0.161854 + 0.986815i \(0.551747\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 20.0000 0.717035
\(779\) −6.00000 −0.214972
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −4.00000 −0.142857
\(785\) 0 0
\(786\) 0 0
\(787\) 14.0000 0.499046 0.249523 − 0.968369i \(-0.419726\pi\)
0.249523 + 0.968369i \(0.419726\pi\)
\(788\) 32.0000 1.13995
\(789\) 0 0
\(790\) 0 0
\(791\) −6.00000 −0.213335
\(792\) 0 0
\(793\) −20.0000 −0.710221
\(794\) −32.0000 −1.13564
\(795\) 0 0
\(796\) 40.0000 1.41776
\(797\) −34.0000 −1.20434 −0.602171 − 0.798367i \(-0.705697\pi\)
−0.602171 + 0.798367i \(0.705697\pi\)
\(798\) 0 0
\(799\) 8.00000 0.283020
\(800\) 40.0000 1.41421
\(801\) 0 0
\(802\) −18.0000 −0.635602
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 12.0000 0.421898 0.210949 − 0.977497i \(-0.432345\pi\)
0.210949 + 0.977497i \(0.432345\pi\)
\(810\) 0 0
\(811\) 36.0000 1.26413 0.632065 − 0.774915i \(-0.282207\pi\)
0.632065 + 0.774915i \(0.282207\pi\)
\(812\) −18.0000 −0.631676
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 12.0000 0.419827
\(818\) 36.0000 1.25871
\(819\) 0 0
\(820\) 0 0
\(821\) 37.0000 1.29131 0.645654 − 0.763630i \(-0.276585\pi\)
0.645654 + 0.763630i \(0.276585\pi\)
\(822\) 0 0
\(823\) 27.0000 0.941161 0.470580 − 0.882357i \(-0.344045\pi\)
0.470580 + 0.882357i \(0.344045\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) −24.0000 −0.835067
\(827\) 13.0000 0.452054 0.226027 − 0.974121i \(-0.427426\pi\)
0.226027 + 0.974121i \(0.427426\pi\)
\(828\) 0 0
\(829\) −6.00000 −0.208389 −0.104194 − 0.994557i \(-0.533226\pi\)
−0.104194 + 0.994557i \(0.533226\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −16.0000 −0.554700
\(833\) −1.00000 −0.0346479
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) −56.0000 −1.93449
\(839\) 54.0000 1.86429 0.932144 − 0.362089i \(-0.117936\pi\)
0.932144 + 0.362089i \(0.117936\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) −4.00000 −0.137849
\(843\) 0 0
\(844\) 8.00000 0.275371
\(845\) 0 0
\(846\) 0 0
\(847\) 11.0000 0.377964
\(848\) 44.0000 1.51097
\(849\) 0 0
\(850\) 10.0000 0.342997
\(851\) 0 0
\(852\) 0 0
\(853\) 5.00000 0.171197 0.0855984 − 0.996330i \(-0.472720\pi\)
0.0855984 + 0.996330i \(0.472720\pi\)
\(854\) 20.0000 0.684386
\(855\) 0 0
\(856\) 0 0
\(857\) −18.0000 −0.614868 −0.307434 − 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 0 0
\(859\) 5.00000 0.170598 0.0852989 − 0.996355i \(-0.472815\pi\)
0.0852989 + 0.996355i \(0.472815\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 24.0000 0.817443
\(863\) 35.0000 1.19141 0.595707 − 0.803202i \(-0.296872\pi\)
0.595707 + 0.803202i \(0.296872\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 3.00000 0.101303 0.0506514 − 0.998716i \(-0.483870\pi\)
0.0506514 + 0.998716i \(0.483870\pi\)
\(878\) −14.0000 −0.472477
\(879\) 0 0
\(880\) 0 0
\(881\) −42.0000 −1.41502 −0.707508 − 0.706705i \(-0.750181\pi\)
−0.707508 + 0.706705i \(0.750181\pi\)
\(882\) 0 0
\(883\) −12.0000 −0.403832 −0.201916 − 0.979403i \(-0.564717\pi\)
−0.201916 + 0.979403i \(0.564717\pi\)
\(884\) −4.00000 −0.134535
\(885\) 0 0
\(886\) −28.0000 −0.940678
\(887\) −20.0000 −0.671534 −0.335767 − 0.941945i \(-0.608996\pi\)
−0.335767 + 0.941945i \(0.608996\pi\)
\(888\) 0 0
\(889\) −1.00000 −0.0335389
\(890\) 0 0
\(891\) 0 0
\(892\) −10.0000 −0.334825
\(893\) 16.0000 0.535420
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −32.0000 −1.06785
\(899\) 0 0
\(900\) 0 0
\(901\) 11.0000 0.366463
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 47.0000 1.56061 0.780305 − 0.625400i \(-0.215064\pi\)
0.780305 + 0.625400i \(0.215064\pi\)
\(908\) 24.0000 0.796468
\(909\) 0 0
\(910\) 0 0
\(911\) −2.00000 −0.0662630 −0.0331315 − 0.999451i \(-0.510548\pi\)
−0.0331315 + 0.999451i \(0.510548\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 70.0000 2.31539
\(915\) 0 0
\(916\) −34.0000 −1.12339
\(917\) 1.00000 0.0330229
\(918\) 0 0
\(919\) 11.0000 0.362857 0.181428 − 0.983404i \(-0.441928\pi\)
0.181428 + 0.983404i \(0.441928\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −80.0000 −2.63466
\(923\) −16.0000 −0.526646
\(924\) 0 0
\(925\) 5.00000 0.164399
\(926\) −72.0000 −2.36607
\(927\) 0 0
\(928\) −72.0000 −2.36352
\(929\) 16.0000 0.524943 0.262471 − 0.964940i \(-0.415462\pi\)
0.262471 + 0.964940i \(0.415462\pi\)
\(930\) 0 0
\(931\) −2.00000 −0.0655474
\(932\) 22.0000 0.720634
\(933\) 0 0
\(934\) −8.00000 −0.261768
\(935\) 0 0
\(936\) 0 0
\(937\) −10.0000 −0.326686 −0.163343 − 0.986569i \(-0.552228\pi\)
−0.163343 + 0.986569i \(0.552228\pi\)
\(938\) −8.00000 −0.261209
\(939\) 0 0
\(940\) 0 0
\(941\) −1.00000 −0.0325991 −0.0162995 − 0.999867i \(-0.505189\pi\)
−0.0162995 + 0.999867i \(0.505189\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −48.0000 −1.56227
\(945\) 0 0
\(946\) 0 0
\(947\) −31.0000 −1.00736 −0.503682 − 0.863889i \(-0.668022\pi\)
−0.503682 + 0.863889i \(0.668022\pi\)
\(948\) 0 0
\(949\) −8.00000 −0.259691
\(950\) 20.0000 0.648886
\(951\) 0 0
\(952\) 0 0
\(953\) −32.0000 −1.03658 −0.518291 − 0.855204i \(-0.673432\pi\)
−0.518291 + 0.855204i \(0.673432\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 42.0000 1.35838
\(957\) 0 0
\(958\) 38.0000 1.22772
\(959\) −6.00000 −0.193750
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) −4.00000 −0.128965
\(963\) 0 0
\(964\) 22.0000 0.708572
\(965\) 0 0
\(966\) 0 0
\(967\) −30.0000 −0.964735 −0.482367 − 0.875969i \(-0.660223\pi\)
−0.482367 + 0.875969i \(0.660223\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −60.0000 −1.92549 −0.962746 − 0.270408i \(-0.912841\pi\)
−0.962746 + 0.270408i \(0.912841\pi\)
\(972\) 0 0
\(973\) 7.00000 0.224410
\(974\) 60.0000 1.92252
\(975\) 0 0
\(976\) 40.0000 1.28037
\(977\) 4.00000 0.127971 0.0639857 − 0.997951i \(-0.479619\pi\)
0.0639857 + 0.997951i \(0.479619\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) −88.0000 −2.80819
\(983\) −9.00000 −0.287055 −0.143528 − 0.989646i \(-0.545845\pi\)
−0.143528 + 0.989646i \(0.545845\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −18.0000 −0.573237
\(987\) 0 0
\(988\) −8.00000 −0.254514
\(989\) 0 0
\(990\) 0 0
\(991\) −14.0000 −0.444725 −0.222362 − 0.974964i \(-0.571377\pi\)
−0.222362 + 0.974964i \(0.571377\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 16.0000 0.507489
\(995\) 0 0
\(996\) 0 0
\(997\) −42.0000 −1.33015 −0.665077 − 0.746775i \(-0.731601\pi\)
−0.665077 + 0.746775i \(0.731601\pi\)
\(998\) 20.0000 0.633089
\(999\) 0 0
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 8001.2.a.f.1.1 1
3.2 odd 2 2667.2.a.a.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2667.2.a.a.1.1 ✓ 1 3.2 odd 2
8001.2.a.f.1.1 1 1.1 even 1 trivial