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Properties

Label 7744.2.a.o.1.1
Level $7744$
Weight $2$
Character 7744.1
Self dual yes
Analytic conductor $61.836$
Analytic rank $0$
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] = [7744,2,Mod(1,7744)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7744, 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("7744.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7744 = 2^{6} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7744.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: \(61.8361513253\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 352)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 7744.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +3.00000 q^{5} +4.00000 q^{7} -2.00000 q^{9} -2.00000 q^{13} -3.00000 q^{15} +8.00000 q^{17} +6.00000 q^{19} -4.00000 q^{21} +5.00000 q^{23} +4.00000 q^{25} +5.00000 q^{27} +4.00000 q^{29} +1.00000 q^{31} +12.0000 q^{35} -3.00000 q^{37} +2.00000 q^{39} +6.00000 q^{41} -6.00000 q^{43} -6.00000 q^{45} -12.0000 q^{47} +9.00000 q^{49} -8.00000 q^{51} +6.00000 q^{53} -6.00000 q^{57} -3.00000 q^{59} -8.00000 q^{63} -6.00000 q^{65} -11.0000 q^{67} -5.00000 q^{69} -5.00000 q^{71} +10.0000 q^{73} -4.00000 q^{75} +2.00000 q^{79} +1.00000 q^{81} -2.00000 q^{83} +24.0000 q^{85} -4.00000 q^{87} -5.00000 q^{89} -8.00000 q^{91} -1.00000 q^{93} +18.0000 q^{95} +13.0000 q^{97} +O(q^{100})\)

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
\(3\) −1.00000 −0.577350 −0.288675 − 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) 0 0
\(5\) 3.00000 1.34164 0.670820 − 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) 0 0
\(7\) 4.00000 1.51186 0.755929 − 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) −3.00000 −0.774597
\(16\) 0 0
\(17\) 8.00000 1.94029 0.970143 − 0.242536i \(-0.0779791\pi\)
0.970143 + 0.242536i \(0.0779791\pi\)
\(18\) 0 0
\(19\) 6.00000 1.37649 0.688247 − 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 0 0
\(21\) −4.00000 −0.872872
\(22\) 0 0
\(23\) 5.00000 1.04257 0.521286 − 0.853382i \(-0.325452\pi\)
0.521286 + 0.853382i \(0.325452\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) 5.00000 0.962250
\(28\) 0 0
\(29\) 4.00000 0.742781 0.371391 − 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605 0.0898027 − 0.995960i \(-0.471376\pi\)
0.0898027 + 0.995960i \(0.471376\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 12.0000 2.02837
\(36\) 0 0
\(37\) −3.00000 −0.493197 −0.246598 − 0.969118i \(-0.579313\pi\)
−0.246598 + 0.969118i \(0.579313\pi\)
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) −6.00000 −0.914991 −0.457496 − 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) 0 0
\(45\) −6.00000 −0.894427
\(46\) 0 0
\(47\) −12.0000 −1.75038 −0.875190 − 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) −8.00000 −1.12022
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −6.00000 −0.794719
\(58\) 0 0
\(59\) −3.00000 −0.390567 −0.195283 − 0.980747i \(-0.562563\pi\)
−0.195283 + 0.980747i \(0.562563\pi\)
\(60\) 0 0
\(61\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 0 0
\(63\) −8.00000 −1.00791
\(64\) 0 0
\(65\) −6.00000 −0.744208
\(66\) 0 0
\(67\) −11.0000 −1.34386 −0.671932 − 0.740613i \(-0.734535\pi\)
−0.671932 + 0.740613i \(0.734535\pi\)
\(68\) 0 0
\(69\) −5.00000 −0.601929
\(70\) 0 0
\(71\) −5.00000 −0.593391 −0.296695 − 0.954972i \(-0.595885\pi\)
−0.296695 + 0.954972i \(0.595885\pi\)
\(72\) 0 0
\(73\) 10.0000 1.17041 0.585206 − 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 0 0
\(75\) −4.00000 −0.461880
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 2.00000 0.225018 0.112509 − 0.993651i \(-0.464111\pi\)
0.112509 + 0.993651i \(0.464111\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −2.00000 −0.219529 −0.109764 − 0.993958i \(-0.535010\pi\)
−0.109764 + 0.993958i \(0.535010\pi\)
\(84\) 0 0
\(85\) 24.0000 2.60317
\(86\) 0 0
\(87\) −4.00000 −0.428845
\(88\) 0 0
\(89\) −5.00000 −0.529999 −0.264999 − 0.964249i \(-0.585372\pi\)
−0.264999 + 0.964249i \(0.585372\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) 0 0
\(93\) −1.00000 −0.103695
\(94\) 0 0
\(95\) 18.0000 1.84676
\(96\) 0 0
\(97\) 13.0000 1.31995 0.659975 − 0.751288i \(-0.270567\pi\)
0.659975 + 0.751288i \(0.270567\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −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 \(-0.210980\pi\)
0.788263 + 0.615338i \(0.210980\pi\)
\(104\) 0 0
\(105\) −12.0000 −1.17108
\(106\) 0 0
\(107\) −16.0000 −1.54678 −0.773389 − 0.633932i \(-0.781440\pi\)
−0.773389 + 0.633932i \(0.781440\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 − 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) 3.00000 0.284747
\(112\) 0 0
\(113\) 1.00000 0.0940721 0.0470360 − 0.998893i \(-0.485022\pi\)
0.0470360 + 0.998893i \(0.485022\pi\)
\(114\) 0 0
\(115\) 15.0000 1.39876
\(116\) 0 0
\(117\) 4.00000 0.369800
\(118\) 0 0
\(119\) 32.0000 2.93344
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) −6.00000 −0.541002
\(124\) 0 0
\(125\) −3.00000 −0.268328
\(126\) 0 0
\(127\) −2.00000 −0.177471 −0.0887357 − 0.996055i \(-0.528283\pi\)
−0.0887357 + 0.996055i \(0.528283\pi\)
\(128\) 0 0
\(129\) 6.00000 0.528271
\(130\) 0 0
\(131\) −8.00000 −0.698963 −0.349482 − 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) 0 0
\(133\) 24.0000 2.08106
\(134\) 0 0
\(135\) 15.0000 1.29099
\(136\) 0 0
\(137\) −3.00000 −0.256307 −0.128154 − 0.991754i \(-0.540905\pi\)
−0.128154 + 0.991754i \(0.540905\pi\)
\(138\) 0 0
\(139\) −8.00000 −0.678551 −0.339276 − 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) 0 0
\(141\) 12.0000 1.01058
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 12.0000 0.996546
\(146\) 0 0
\(147\) −9.00000 −0.742307
\(148\) 0 0
\(149\) 4.00000 0.327693 0.163846 − 0.986486i \(-0.447610\pi\)
0.163846 + 0.986486i \(0.447610\pi\)
\(150\) 0 0
\(151\) 10.0000 0.813788 0.406894 − 0.913475i \(-0.366612\pi\)
0.406894 + 0.913475i \(0.366612\pi\)
\(152\) 0 0
\(153\) −16.0000 −1.29352
\(154\) 0 0
\(155\) 3.00000 0.240966
\(156\) 0 0
\(157\) 23.0000 1.83560 0.917800 − 0.397043i \(-0.129964\pi\)
0.917800 + 0.397043i \(0.129964\pi\)
\(158\) 0 0
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) 20.0000 1.57622
\(162\) 0 0
\(163\) −20.0000 −1.56652 −0.783260 − 0.621694i \(-0.786445\pi\)
−0.783260 + 0.621694i \(0.786445\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −18.0000 −1.39288 −0.696441 − 0.717614i \(-0.745234\pi\)
−0.696441 + 0.717614i \(0.745234\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −12.0000 −0.917663
\(172\) 0 0
\(173\) −14.0000 −1.06440 −0.532200 − 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) 0 0
\(175\) 16.0000 1.20949
\(176\) 0 0
\(177\) 3.00000 0.225494
\(178\) 0 0
\(179\) 13.0000 0.971666 0.485833 − 0.874052i \(-0.338516\pi\)
0.485833 + 0.874052i \(0.338516\pi\)
\(180\) 0 0
\(181\) −7.00000 −0.520306 −0.260153 − 0.965567i \(-0.583773\pi\)
−0.260153 + 0.965567i \(0.583773\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −9.00000 −0.661693
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 20.0000 1.45479
\(190\) 0 0
\(191\) 11.0000 0.795932 0.397966 − 0.917400i \(-0.369716\pi\)
0.397966 + 0.917400i \(0.369716\pi\)
\(192\) 0 0
\(193\) 12.0000 0.863779 0.431889 − 0.901927i \(-0.357847\pi\)
0.431889 + 0.901927i \(0.357847\pi\)
\(194\) 0 0
\(195\) 6.00000 0.429669
\(196\) 0 0
\(197\) −18.0000 −1.28245 −0.641223 − 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) −24.0000 −1.70131 −0.850657 − 0.525720i \(-0.823796\pi\)
−0.850657 + 0.525720i \(0.823796\pi\)
\(200\) 0 0
\(201\) 11.0000 0.775880
\(202\) 0 0
\(203\) 16.0000 1.12298
\(204\) 0 0
\(205\) 18.0000 1.25717
\(206\) 0 0
\(207\) −10.0000 −0.695048
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 − 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 0 0
\(213\) 5.00000 0.342594
\(214\) 0 0
\(215\) −18.0000 −1.22759
\(216\) 0 0
\(217\) 4.00000 0.271538
\(218\) 0 0
\(219\) −10.0000 −0.675737
\(220\) 0 0
\(221\) −16.0000 −1.07628
\(222\) 0 0
\(223\) 13.0000 0.870544 0.435272 − 0.900299i \(-0.356652\pi\)
0.435272 + 0.900299i \(0.356652\pi\)
\(224\) 0 0
\(225\) −8.00000 −0.533333
\(226\) 0 0
\(227\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) −27.0000 −1.78421 −0.892105 − 0.451828i \(-0.850772\pi\)
−0.892105 + 0.451828i \(0.850772\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 12.0000 0.786146 0.393073 − 0.919507i \(-0.371412\pi\)
0.393073 + 0.919507i \(0.371412\pi\)
\(234\) 0 0
\(235\) −36.0000 −2.34838
\(236\) 0 0
\(237\) −2.00000 −0.129914
\(238\) 0 0
\(239\) −6.00000 −0.388108 −0.194054 − 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 0 0
\(241\) 20.0000 1.28831 0.644157 − 0.764894i \(-0.277208\pi\)
0.644157 + 0.764894i \(0.277208\pi\)
\(242\) 0 0
\(243\) −16.0000 −1.02640
\(244\) 0 0
\(245\) 27.0000 1.72497
\(246\) 0 0
\(247\) −12.0000 −0.763542
\(248\) 0 0
\(249\) 2.00000 0.126745
\(250\) 0 0
\(251\) −15.0000 −0.946792 −0.473396 − 0.880850i \(-0.656972\pi\)
−0.473396 + 0.880850i \(0.656972\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −24.0000 −1.50294
\(256\) 0 0
\(257\) −10.0000 −0.623783 −0.311891 − 0.950118i \(-0.600963\pi\)
−0.311891 + 0.950118i \(0.600963\pi\)
\(258\) 0 0
\(259\) −12.0000 −0.745644
\(260\) 0 0
\(261\) −8.00000 −0.495188
\(262\) 0 0
\(263\) 12.0000 0.739952 0.369976 − 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) 0 0
\(265\) 18.0000 1.10573
\(266\) 0 0
\(267\) 5.00000 0.305995
\(268\) 0 0
\(269\) 18.0000 1.09748 0.548740 − 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 0 0
\(271\) 22.0000 1.33640 0.668202 − 0.743980i \(-0.267064\pi\)
0.668202 + 0.743980i \(0.267064\pi\)
\(272\) 0 0
\(273\) 8.00000 0.484182
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 10.0000 0.600842 0.300421 − 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) 0 0
\(279\) −2.00000 −0.119737
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 − 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 0 0
\(283\) −8.00000 −0.475551 −0.237775 − 0.971320i \(-0.576418\pi\)
−0.237775 + 0.971320i \(0.576418\pi\)
\(284\) 0 0
\(285\) −18.0000 −1.06623
\(286\) 0 0
\(287\) 24.0000 1.41668
\(288\) 0 0
\(289\) 47.0000 2.76471
\(290\) 0 0
\(291\) −13.0000 −0.762073
\(292\) 0 0
\(293\) 22.0000 1.28525 0.642627 − 0.766179i \(-0.277845\pi\)
0.642627 + 0.766179i \(0.277845\pi\)
\(294\) 0 0
\(295\) −9.00000 −0.524000
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −10.0000 −0.578315
\(300\) 0 0
\(301\) −24.0000 −1.38334
\(302\) 0 0
\(303\) 4.00000 0.229794
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 22.0000 1.25561 0.627803 − 0.778372i \(-0.283954\pi\)
0.627803 + 0.778372i \(0.283954\pi\)
\(308\) 0 0
\(309\) −16.0000 −0.910208
\(310\) 0 0
\(311\) −8.00000 −0.453638 −0.226819 − 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) 0 0
\(313\) −25.0000 −1.41308 −0.706542 − 0.707671i \(-0.749746\pi\)
−0.706542 + 0.707671i \(0.749746\pi\)
\(314\) 0 0
\(315\) −24.0000 −1.35225
\(316\) 0 0
\(317\) 19.0000 1.06715 0.533573 − 0.845754i \(-0.320849\pi\)
0.533573 + 0.845754i \(0.320849\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 16.0000 0.893033
\(322\) 0 0
\(323\) 48.0000 2.67079
\(324\) 0 0
\(325\) −8.00000 −0.443760
\(326\) 0 0
\(327\) −2.00000 −0.110600
\(328\) 0 0
\(329\) −48.0000 −2.64633
\(330\) 0 0
\(331\) −13.0000 −0.714545 −0.357272 − 0.934000i \(-0.616293\pi\)
−0.357272 + 0.934000i \(0.616293\pi\)
\(332\) 0 0
\(333\) 6.00000 0.328798
\(334\) 0 0
\(335\) −33.0000 −1.80298
\(336\) 0 0
\(337\) −36.0000 −1.96104 −0.980522 − 0.196407i \(-0.937073\pi\)
−0.980522 + 0.196407i \(0.937073\pi\)
\(338\) 0 0
\(339\) −1.00000 −0.0543125
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 8.00000 0.431959
\(344\) 0 0
\(345\) −15.0000 −0.807573
\(346\) 0 0
\(347\) 30.0000 1.61048 0.805242 − 0.592946i \(-0.202035\pi\)
0.805242 + 0.592946i \(0.202035\pi\)
\(348\) 0 0
\(349\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) −10.0000 −0.533761
\(352\) 0 0
\(353\) 3.00000 0.159674 0.0798369 − 0.996808i \(-0.474560\pi\)
0.0798369 + 0.996808i \(0.474560\pi\)
\(354\) 0 0
\(355\) −15.0000 −0.796117
\(356\) 0 0
\(357\) −32.0000 −1.69362
\(358\) 0 0
\(359\) 6.00000 0.316668 0.158334 − 0.987386i \(-0.449388\pi\)
0.158334 + 0.987386i \(0.449388\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 30.0000 1.57027
\(366\) 0 0
\(367\) −7.00000 −0.365397 −0.182699 − 0.983169i \(-0.558483\pi\)
−0.182699 + 0.983169i \(0.558483\pi\)
\(368\) 0 0
\(369\) −12.0000 −0.624695
\(370\) 0 0
\(371\) 24.0000 1.24602
\(372\) 0 0
\(373\) −8.00000 −0.414224 −0.207112 − 0.978317i \(-0.566407\pi\)
−0.207112 + 0.978317i \(0.566407\pi\)
\(374\) 0 0
\(375\) 3.00000 0.154919
\(376\) 0 0
\(377\) −8.00000 −0.412021
\(378\) 0 0
\(379\) −17.0000 −0.873231 −0.436616 − 0.899648i \(-0.643823\pi\)
−0.436616 + 0.899648i \(0.643823\pi\)
\(380\) 0 0
\(381\) 2.00000 0.102463
\(382\) 0 0
\(383\) 13.0000 0.664269 0.332134 − 0.943232i \(-0.392231\pi\)
0.332134 + 0.943232i \(0.392231\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 12.0000 0.609994
\(388\) 0 0
\(389\) −17.0000 −0.861934 −0.430967 − 0.902368i \(-0.641828\pi\)
−0.430967 + 0.902368i \(0.641828\pi\)
\(390\) 0 0
\(391\) 40.0000 2.02289
\(392\) 0 0
\(393\) 8.00000 0.403547
\(394\) 0 0
\(395\) 6.00000 0.301893
\(396\) 0 0
\(397\) 10.0000 0.501886 0.250943 − 0.968002i \(-0.419259\pi\)
0.250943 + 0.968002i \(0.419259\pi\)
\(398\) 0 0
\(399\) −24.0000 −1.20150
\(400\) 0 0
\(401\) −14.0000 −0.699127 −0.349563 − 0.936913i \(-0.613670\pi\)
−0.349563 + 0.936913i \(0.613670\pi\)
\(402\) 0 0
\(403\) −2.00000 −0.0996271
\(404\) 0 0
\(405\) 3.00000 0.149071
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 14.0000 0.692255 0.346128 − 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 0 0
\(411\) 3.00000 0.147979
\(412\) 0 0
\(413\) −12.0000 −0.590481
\(414\) 0 0
\(415\) −6.00000 −0.294528
\(416\) 0 0
\(417\) 8.00000 0.391762
\(418\) 0 0
\(419\) −8.00000 −0.390826 −0.195413 − 0.980721i \(-0.562605\pi\)
−0.195413 + 0.980721i \(0.562605\pi\)
\(420\) 0 0
\(421\) 34.0000 1.65706 0.828529 − 0.559946i \(-0.189178\pi\)
0.828529 + 0.559946i \(0.189178\pi\)
\(422\) 0 0
\(423\) 24.0000 1.16692
\(424\) 0 0
\(425\) 32.0000 1.55223
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 16.0000 0.770693 0.385346 − 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) 0 0
\(433\) −19.0000 −0.913082 −0.456541 − 0.889702i \(-0.650912\pi\)
−0.456541 + 0.889702i \(0.650912\pi\)
\(434\) 0 0
\(435\) −12.0000 −0.575356
\(436\) 0 0
\(437\) 30.0000 1.43509
\(438\) 0 0
\(439\) −4.00000 −0.190910 −0.0954548 − 0.995434i \(-0.530431\pi\)
−0.0954548 + 0.995434i \(0.530431\pi\)
\(440\) 0 0
\(441\) −18.0000 −0.857143
\(442\) 0 0
\(443\) 21.0000 0.997740 0.498870 − 0.866677i \(-0.333748\pi\)
0.498870 + 0.866677i \(0.333748\pi\)
\(444\) 0 0
\(445\) −15.0000 −0.711068
\(446\) 0 0
\(447\) −4.00000 −0.189194
\(448\) 0 0
\(449\) 23.0000 1.08544 0.542719 − 0.839915i \(-0.317395\pi\)
0.542719 + 0.839915i \(0.317395\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −10.0000 −0.469841
\(454\) 0 0
\(455\) −24.0000 −1.12514
\(456\) 0 0
\(457\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(458\) 0 0
\(459\) 40.0000 1.86704
\(460\) 0 0
\(461\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 7.00000 0.325318 0.162659 − 0.986682i \(-0.447993\pi\)
0.162659 + 0.986682i \(0.447993\pi\)
\(464\) 0 0
\(465\) −3.00000 −0.139122
\(466\) 0 0
\(467\) 5.00000 0.231372 0.115686 − 0.993286i \(-0.463093\pi\)
0.115686 + 0.993286i \(0.463093\pi\)
\(468\) 0 0
\(469\) −44.0000 −2.03173
\(470\) 0 0
\(471\) −23.0000 −1.05978
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 24.0000 1.10120
\(476\) 0 0
\(477\) −12.0000 −0.549442
\(478\) 0 0
\(479\) −6.00000 −0.274147 −0.137073 − 0.990561i \(-0.543770\pi\)
−0.137073 + 0.990561i \(0.543770\pi\)
\(480\) 0 0
\(481\) 6.00000 0.273576
\(482\) 0 0
\(483\) −20.0000 −0.910032
\(484\) 0 0
\(485\) 39.0000 1.77090
\(486\) 0 0
\(487\) −11.0000 −0.498458 −0.249229 − 0.968445i \(-0.580177\pi\)
−0.249229 + 0.968445i \(0.580177\pi\)
\(488\) 0 0
\(489\) 20.0000 0.904431
\(490\) 0 0
\(491\) 24.0000 1.08310 0.541552 − 0.840667i \(-0.317837\pi\)
0.541552 + 0.840667i \(0.317837\pi\)
\(492\) 0 0
\(493\) 32.0000 1.44121
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −20.0000 −0.897123
\(498\) 0 0
\(499\) −20.0000 −0.895323 −0.447661 − 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) 0 0
\(501\) 18.0000 0.804181
\(502\) 0 0
\(503\) −14.0000 −0.624229 −0.312115 − 0.950044i \(-0.601037\pi\)
−0.312115 + 0.950044i \(0.601037\pi\)
\(504\) 0 0
\(505\) −12.0000 −0.533993
\(506\) 0 0
\(507\) 9.00000 0.399704
\(508\) 0 0
\(509\) −19.0000 −0.842160 −0.421080 − 0.907023i \(-0.638349\pi\)
−0.421080 + 0.907023i \(0.638349\pi\)
\(510\) 0 0
\(511\) 40.0000 1.76950
\(512\) 0 0
\(513\) 30.0000 1.32453
\(514\) 0 0
\(515\) 48.0000 2.11513
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 14.0000 0.614532
\(520\) 0 0
\(521\) −7.00000 −0.306676 −0.153338 − 0.988174i \(-0.549002\pi\)
−0.153338 + 0.988174i \(0.549002\pi\)
\(522\) 0 0
\(523\) 18.0000 0.787085 0.393543 − 0.919306i \(-0.371249\pi\)
0.393543 + 0.919306i \(0.371249\pi\)
\(524\) 0 0
\(525\) −16.0000 −0.698297
\(526\) 0 0
\(527\) 8.00000 0.348485
\(528\) 0 0
\(529\) 2.00000 0.0869565
\(530\) 0 0
\(531\) 6.00000 0.260378
\(532\) 0 0
\(533\) −12.0000 −0.519778
\(534\) 0 0
\(535\) −48.0000 −2.07522
\(536\) 0 0
\(537\) −13.0000 −0.560991
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −10.0000 −0.429934 −0.214967 − 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) 0 0
\(543\) 7.00000 0.300399
\(544\) 0 0
\(545\) 6.00000 0.257012
\(546\) 0 0
\(547\) −44.0000 −1.88130 −0.940652 − 0.339372i \(-0.889785\pi\)
−0.940652 + 0.339372i \(0.889785\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 24.0000 1.02243
\(552\) 0 0
\(553\) 8.00000 0.340195
\(554\) 0 0
\(555\) 9.00000 0.382029
\(556\) 0 0
\(557\) 36.0000 1.52537 0.762684 − 0.646771i \(-0.223881\pi\)
0.762684 + 0.646771i \(0.223881\pi\)
\(558\) 0 0
\(559\) 12.0000 0.507546
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −8.00000 −0.337160 −0.168580 − 0.985688i \(-0.553918\pi\)
−0.168580 + 0.985688i \(0.553918\pi\)
\(564\) 0 0
\(565\) 3.00000 0.126211
\(566\) 0 0
\(567\) 4.00000 0.167984
\(568\) 0 0
\(569\) −4.00000 −0.167689 −0.0838444 − 0.996479i \(-0.526720\pi\)
−0.0838444 + 0.996479i \(0.526720\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 − 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 0 0
\(573\) −11.0000 −0.459532
\(574\) 0 0
\(575\) 20.0000 0.834058
\(576\) 0 0
\(577\) −27.0000 −1.12402 −0.562012 − 0.827129i \(-0.689973\pi\)
−0.562012 + 0.827129i \(0.689973\pi\)
\(578\) 0 0
\(579\) −12.0000 −0.498703
\(580\) 0 0
\(581\) −8.00000 −0.331896
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 12.0000 0.496139
\(586\) 0 0
\(587\) 12.0000 0.495293 0.247647 − 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 0 0
\(589\) 6.00000 0.247226
\(590\) 0 0
\(591\) 18.0000 0.740421
\(592\) 0 0
\(593\) −4.00000 −0.164260 −0.0821302 − 0.996622i \(-0.526172\pi\)
−0.0821302 + 0.996622i \(0.526172\pi\)
\(594\) 0 0
\(595\) 96.0000 3.93562
\(596\) 0 0
\(597\) 24.0000 0.982255
\(598\) 0 0
\(599\) −16.0000 −0.653742 −0.326871 − 0.945069i \(-0.605994\pi\)
−0.326871 + 0.945069i \(0.605994\pi\)
\(600\) 0 0
\(601\) 32.0000 1.30531 0.652654 − 0.757656i \(-0.273656\pi\)
0.652654 + 0.757656i \(0.273656\pi\)
\(602\) 0 0
\(603\) 22.0000 0.895909
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 38.0000 1.54237 0.771186 − 0.636610i \(-0.219664\pi\)
0.771186 + 0.636610i \(0.219664\pi\)
\(608\) 0 0
\(609\) −16.0000 −0.648353
\(610\) 0 0
\(611\) 24.0000 0.970936
\(612\) 0 0
\(613\) 16.0000 0.646234 0.323117 − 0.946359i \(-0.395269\pi\)
0.323117 + 0.946359i \(0.395269\pi\)
\(614\) 0 0
\(615\) −18.0000 −0.725830
\(616\) 0 0
\(617\) −26.0000 −1.04672 −0.523360 − 0.852111i \(-0.675322\pi\)
−0.523360 + 0.852111i \(0.675322\pi\)
\(618\) 0 0
\(619\) −29.0000 −1.16561 −0.582804 − 0.812613i \(-0.698045\pi\)
−0.582804 + 0.812613i \(0.698045\pi\)
\(620\) 0 0
\(621\) 25.0000 1.00322
\(622\) 0 0
\(623\) −20.0000 −0.801283
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −24.0000 −0.956943
\(630\) 0 0
\(631\) −7.00000 −0.278666 −0.139333 − 0.990246i \(-0.544496\pi\)
−0.139333 + 0.990246i \(0.544496\pi\)
\(632\) 0 0
\(633\) −4.00000 −0.158986
\(634\) 0 0
\(635\) −6.00000 −0.238103
\(636\) 0 0
\(637\) −18.0000 −0.713186
\(638\) 0 0
\(639\) 10.0000 0.395594
\(640\) 0 0
\(641\) 35.0000 1.38242 0.691208 − 0.722655i \(-0.257079\pi\)
0.691208 + 0.722655i \(0.257079\pi\)
\(642\) 0 0
\(643\) −7.00000 −0.276053 −0.138027 − 0.990429i \(-0.544076\pi\)
−0.138027 + 0.990429i \(0.544076\pi\)
\(644\) 0 0
\(645\) 18.0000 0.708749
\(646\) 0 0
\(647\) 3.00000 0.117942 0.0589711 − 0.998260i \(-0.481218\pi\)
0.0589711 + 0.998260i \(0.481218\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −4.00000 −0.156772
\(652\) 0 0
\(653\) 13.0000 0.508729 0.254365 − 0.967108i \(-0.418134\pi\)
0.254365 + 0.967108i \(0.418134\pi\)
\(654\) 0 0
\(655\) −24.0000 −0.937758
\(656\) 0 0
\(657\) −20.0000 −0.780274
\(658\) 0 0
\(659\) 50.0000 1.94772 0.973862 − 0.227142i \(-0.0729380\pi\)
0.973862 + 0.227142i \(0.0729380\pi\)
\(660\) 0 0
\(661\) 19.0000 0.739014 0.369507 − 0.929228i \(-0.379527\pi\)
0.369507 + 0.929228i \(0.379527\pi\)
\(662\) 0 0
\(663\) 16.0000 0.621389
\(664\) 0 0
\(665\) 72.0000 2.79204
\(666\) 0 0
\(667\) 20.0000 0.774403
\(668\) 0 0
\(669\) −13.0000 −0.502609
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 10.0000 0.385472 0.192736 − 0.981251i \(-0.438264\pi\)
0.192736 + 0.981251i \(0.438264\pi\)
\(674\) 0 0
\(675\) 20.0000 0.769800
\(676\) 0 0
\(677\) 12.0000 0.461197 0.230599 − 0.973049i \(-0.425932\pi\)
0.230599 + 0.973049i \(0.425932\pi\)
\(678\) 0 0
\(679\) 52.0000 1.99558
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 20.0000 0.765279 0.382639 − 0.923898i \(-0.375015\pi\)
0.382639 + 0.923898i \(0.375015\pi\)
\(684\) 0 0
\(685\) −9.00000 −0.343872
\(686\) 0 0
\(687\) 27.0000 1.03011
\(688\) 0 0
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) −11.0000 −0.418460 −0.209230 − 0.977866i \(-0.567096\pi\)
−0.209230 + 0.977866i \(0.567096\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −24.0000 −0.910372
\(696\) 0 0
\(697\) 48.0000 1.81813
\(698\) 0 0
\(699\) −12.0000 −0.453882
\(700\) 0 0
\(701\) 36.0000 1.35970 0.679851 − 0.733351i \(-0.262045\pi\)
0.679851 + 0.733351i \(0.262045\pi\)
\(702\) 0 0
\(703\) −18.0000 −0.678883
\(704\) 0 0
\(705\) 36.0000 1.35584
\(706\) 0 0
\(707\) −16.0000 −0.601742
\(708\) 0 0
\(709\) 5.00000 0.187779 0.0938895 − 0.995583i \(-0.470070\pi\)
0.0938895 + 0.995583i \(0.470070\pi\)
\(710\) 0 0
\(711\) −4.00000 −0.150012
\(712\) 0 0
\(713\) 5.00000 0.187251
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 6.00000 0.224074
\(718\) 0 0
\(719\) 21.0000 0.783168 0.391584 − 0.920142i \(-0.371927\pi\)
0.391584 + 0.920142i \(0.371927\pi\)
\(720\) 0 0
\(721\) 64.0000 2.38348
\(722\) 0 0
\(723\) −20.0000 −0.743808
\(724\) 0 0
\(725\) 16.0000 0.594225
\(726\) 0 0
\(727\) −27.0000 −1.00137 −0.500687 − 0.865628i \(-0.666919\pi\)
−0.500687 + 0.865628i \(0.666919\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −48.0000 −1.77534
\(732\) 0 0
\(733\) 24.0000 0.886460 0.443230 − 0.896408i \(-0.353832\pi\)
0.443230 + 0.896408i \(0.353832\pi\)
\(734\) 0 0
\(735\) −27.0000 −0.995910
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −34.0000 −1.25071 −0.625355 − 0.780340i \(-0.715046\pi\)
−0.625355 + 0.780340i \(0.715046\pi\)
\(740\) 0 0
\(741\) 12.0000 0.440831
\(742\) 0 0
\(743\) 12.0000 0.440237 0.220119 − 0.975473i \(-0.429356\pi\)
0.220119 + 0.975473i \(0.429356\pi\)
\(744\) 0 0
\(745\) 12.0000 0.439646
\(746\) 0 0
\(747\) 4.00000 0.146352
\(748\) 0 0
\(749\) −64.0000 −2.33851
\(750\) 0 0
\(751\) 43.0000 1.56909 0.784546 − 0.620070i \(-0.212896\pi\)
0.784546 + 0.620070i \(0.212896\pi\)
\(752\) 0 0
\(753\) 15.0000 0.546630
\(754\) 0 0
\(755\) 30.0000 1.09181
\(756\) 0 0
\(757\) 26.0000 0.944986 0.472493 − 0.881334i \(-0.343354\pi\)
0.472493 + 0.881334i \(0.343354\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −42.0000 −1.52250 −0.761249 − 0.648459i \(-0.775414\pi\)
−0.761249 + 0.648459i \(0.775414\pi\)
\(762\) 0 0
\(763\) 8.00000 0.289619
\(764\) 0 0
\(765\) −48.0000 −1.73544
\(766\) 0 0
\(767\) 6.00000 0.216647
\(768\) 0 0
\(769\) −38.0000 −1.37032 −0.685158 − 0.728395i \(-0.740267\pi\)
−0.685158 + 0.728395i \(0.740267\pi\)
\(770\) 0 0
\(771\) 10.0000 0.360141
\(772\) 0 0
\(773\) −34.0000 −1.22290 −0.611448 − 0.791285i \(-0.709412\pi\)
−0.611448 + 0.791285i \(0.709412\pi\)
\(774\) 0 0
\(775\) 4.00000 0.143684
\(776\) 0 0
\(777\) 12.0000 0.430498
\(778\) 0 0
\(779\) 36.0000 1.28983
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 20.0000 0.714742
\(784\) 0 0
\(785\) 69.0000 2.46272
\(786\) 0 0
\(787\) −4.00000 −0.142585 −0.0712923 − 0.997455i \(-0.522712\pi\)
−0.0712923 + 0.997455i \(0.522712\pi\)
\(788\) 0 0
\(789\) −12.0000 −0.427211
\(790\) 0 0
\(791\) 4.00000 0.142224
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −18.0000 −0.638394
\(796\) 0 0
\(797\) 51.0000 1.80651 0.903256 − 0.429101i \(-0.141170\pi\)
0.903256 + 0.429101i \(0.141170\pi\)
\(798\) 0 0
\(799\) −96.0000 −3.39624
\(800\) 0 0
\(801\) 10.0000 0.353333
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 60.0000 2.11472
\(806\) 0 0
\(807\) −18.0000 −0.633630
\(808\) 0 0
\(809\) 6.00000 0.210949 0.105474 − 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) −14.0000 −0.491606 −0.245803 − 0.969320i \(-0.579052\pi\)
−0.245803 + 0.969320i \(0.579052\pi\)
\(812\) 0 0
\(813\) −22.0000 −0.771574
\(814\) 0 0
\(815\) −60.0000 −2.10171
\(816\) 0 0
\(817\) −36.0000 −1.25948
\(818\) 0 0
\(819\) 16.0000 0.559085
\(820\) 0 0
\(821\) −2.00000 −0.0698005 −0.0349002 − 0.999391i \(-0.511111\pi\)
−0.0349002 + 0.999391i \(0.511111\pi\)
\(822\) 0 0
\(823\) −15.0000 −0.522867 −0.261434 − 0.965221i \(-0.584195\pi\)
−0.261434 + 0.965221i \(0.584195\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −42.0000 −1.46048 −0.730242 − 0.683189i \(-0.760592\pi\)
−0.730242 + 0.683189i \(0.760592\pi\)
\(828\) 0 0
\(829\) 7.00000 0.243120 0.121560 − 0.992584i \(-0.461210\pi\)
0.121560 + 0.992584i \(0.461210\pi\)
\(830\) 0 0
\(831\) −10.0000 −0.346896
\(832\) 0 0
\(833\) 72.0000 2.49465
\(834\) 0 0
\(835\) −54.0000 −1.86875
\(836\) 0 0
\(837\) 5.00000 0.172825
\(838\) 0 0
\(839\) −55.0000 −1.89881 −0.949405 − 0.314053i \(-0.898313\pi\)
−0.949405 + 0.314053i \(0.898313\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 0 0
\(843\) 18.0000 0.619953
\(844\) 0 0
\(845\) −27.0000 −0.928828
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 8.00000 0.274559
\(850\) 0 0
\(851\) −15.0000 −0.514193
\(852\) 0 0
\(853\) 38.0000 1.30110 0.650548 − 0.759465i \(-0.274539\pi\)
0.650548 + 0.759465i \(0.274539\pi\)
\(854\) 0 0
\(855\) −36.0000 −1.23117
\(856\) 0 0
\(857\) 48.0000 1.63965 0.819824 − 0.572615i \(-0.194071\pi\)
0.819824 + 0.572615i \(0.194071\pi\)
\(858\) 0 0
\(859\) −15.0000 −0.511793 −0.255897 − 0.966704i \(-0.582371\pi\)
−0.255897 + 0.966704i \(0.582371\pi\)
\(860\) 0 0
\(861\) −24.0000 −0.817918
\(862\) 0 0
\(863\) 48.0000 1.63394 0.816970 − 0.576681i \(-0.195652\pi\)
0.816970 + 0.576681i \(0.195652\pi\)
\(864\) 0 0
\(865\) −42.0000 −1.42804
\(866\) 0 0
\(867\) −47.0000 −1.59620
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 22.0000 0.745442
\(872\) 0 0
\(873\) −26.0000 −0.879967
\(874\) 0 0
\(875\) −12.0000 −0.405674
\(876\) 0 0
\(877\) −6.00000 −0.202606 −0.101303 − 0.994856i \(-0.532301\pi\)
−0.101303 + 0.994856i \(0.532301\pi\)
\(878\) 0 0
\(879\) −22.0000 −0.742042
\(880\) 0 0
\(881\) 37.0000 1.24656 0.623281 − 0.781998i \(-0.285799\pi\)
0.623281 + 0.781998i \(0.285799\pi\)
\(882\) 0 0
\(883\) 28.0000 0.942275 0.471138 − 0.882060i \(-0.343844\pi\)
0.471138 + 0.882060i \(0.343844\pi\)
\(884\) 0 0
\(885\) 9.00000 0.302532
\(886\) 0 0
\(887\) 14.0000 0.470074 0.235037 − 0.971986i \(-0.424479\pi\)
0.235037 + 0.971986i \(0.424479\pi\)
\(888\) 0 0
\(889\) −8.00000 −0.268311
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −72.0000 −2.40939
\(894\) 0 0
\(895\) 39.0000 1.30363
\(896\) 0 0
\(897\) 10.0000 0.333890
\(898\) 0 0
\(899\) 4.00000 0.133407
\(900\) 0 0
\(901\) 48.0000 1.59911
\(902\) 0 0
\(903\) 24.0000 0.798670
\(904\) 0 0
\(905\) −21.0000 −0.698064
\(906\) 0 0
\(907\) −28.0000 −0.929725 −0.464862 − 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) 0 0
\(909\) 8.00000 0.265343
\(910\) 0 0
\(911\) −48.0000 −1.59031 −0.795155 − 0.606406i \(-0.792611\pi\)
−0.795155 + 0.606406i \(0.792611\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −32.0000 −1.05673
\(918\) 0 0
\(919\) −10.0000 −0.329870 −0.164935 − 0.986304i \(-0.552741\pi\)
−0.164935 + 0.986304i \(0.552741\pi\)
\(920\) 0 0
\(921\) −22.0000 −0.724925
\(922\) 0 0
\(923\) 10.0000 0.329154
\(924\) 0 0
\(925\) −12.0000 −0.394558
\(926\) 0 0
\(927\) −32.0000 −1.05102
\(928\) 0 0
\(929\) 2.00000 0.0656179 0.0328089 − 0.999462i \(-0.489555\pi\)
0.0328089 + 0.999462i \(0.489555\pi\)
\(930\) 0 0
\(931\) 54.0000 1.76978
\(932\) 0 0
\(933\) 8.00000 0.261908
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −10.0000 −0.326686 −0.163343 − 0.986569i \(-0.552228\pi\)
−0.163343 + 0.986569i \(0.552228\pi\)
\(938\) 0 0
\(939\) 25.0000 0.815844
\(940\) 0 0
\(941\) −42.0000 −1.36916 −0.684580 − 0.728937i \(-0.740015\pi\)
−0.684580 + 0.728937i \(0.740015\pi\)
\(942\) 0 0
\(943\) 30.0000 0.976934
\(944\) 0 0
\(945\) 60.0000 1.95180
\(946\) 0 0
\(947\) −27.0000 −0.877382 −0.438691 − 0.898638i \(-0.644558\pi\)
−0.438691 + 0.898638i \(0.644558\pi\)
\(948\) 0 0
\(949\) −20.0000 −0.649227
\(950\) 0 0
\(951\) −19.0000 −0.616117
\(952\) 0 0
\(953\) −4.00000 −0.129573 −0.0647864 − 0.997899i \(-0.520637\pi\)
−0.0647864 + 0.997899i \(0.520637\pi\)
\(954\) 0 0
\(955\) 33.0000 1.06785
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −12.0000 −0.387500
\(960\) 0 0
\(961\) −30.0000 −0.967742
\(962\) 0 0
\(963\) 32.0000 1.03119
\(964\) 0 0
\(965\) 36.0000 1.15888
\(966\) 0 0
\(967\) 4.00000 0.128631 0.0643157 − 0.997930i \(-0.479514\pi\)
0.0643157 + 0.997930i \(0.479514\pi\)
\(968\) 0 0
\(969\) −48.0000 −1.54198
\(970\) 0 0
\(971\) 51.0000 1.63667 0.818334 − 0.574743i \(-0.194898\pi\)
0.818334 + 0.574743i \(0.194898\pi\)
\(972\) 0 0
\(973\) −32.0000 −1.02587
\(974\) 0 0
\(975\) 8.00000 0.256205
\(976\) 0 0
\(977\) −7.00000 −0.223950 −0.111975 − 0.993711i \(-0.535718\pi\)
−0.111975 + 0.993711i \(0.535718\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −4.00000 −0.127710
\(982\) 0 0
\(983\) −59.0000 −1.88181 −0.940904 − 0.338674i \(-0.890022\pi\)
−0.940904 + 0.338674i \(0.890022\pi\)
\(984\) 0 0
\(985\) −54.0000 −1.72058
\(986\) 0 0
\(987\) 48.0000 1.52786
\(988\) 0 0
\(989\) −30.0000 −0.953945
\(990\) 0 0
\(991\) −56.0000 −1.77890 −0.889449 − 0.457034i \(-0.848912\pi\)
−0.889449 + 0.457034i \(0.848912\pi\)
\(992\) 0 0
\(993\) 13.0000 0.412543
\(994\) 0 0
\(995\) −72.0000 −2.28255
\(996\) 0 0
\(997\) −52.0000 −1.64686 −0.823428 − 0.567420i \(-0.807941\pi\)
−0.823428 + 0.567420i \(0.807941\pi\)
\(998\) 0 0
\(999\) −15.0000 −0.474579
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 7744.2.a.o.1.1 1
4.3 odd 2 7744.2.a.ba.1.1 1
8.3 odd 2 3872.2.a.d.1.1 1
8.5 even 2 3872.2.a.i.1.1 1
11.10 odd 2 704.2.a.e.1.1 1
33.32 even 2 6336.2.a.g.1.1 1
44.43 even 2 704.2.a.j.1.1 1
88.21 odd 2 352.2.a.d.1.1 yes 1
88.43 even 2 352.2.a.b.1.1 ✓ 1
132.131 odd 2 6336.2.a.l.1.1 1
176.21 odd 4 2816.2.c.l.1409.2 2
176.43 even 4 2816.2.c.b.1409.1 2
176.109 odd 4 2816.2.c.l.1409.1 2
176.131 even 4 2816.2.c.b.1409.2 2
264.131 odd 2 3168.2.a.z.1.1 1
264.197 even 2 3168.2.a.y.1.1 1
440.109 odd 2 8800.2.a.m.1.1 1
440.219 even 2 8800.2.a.p.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
352.2.a.b.1.1 ✓ 1 88.43 even 2
352.2.a.d.1.1 yes 1 88.21 odd 2
704.2.a.e.1.1 1 11.10 odd 2
704.2.a.j.1.1 1 44.43 even 2
2816.2.c.b.1409.1 2 176.43 even 4
2816.2.c.b.1409.2 2 176.131 even 4
2816.2.c.l.1409.1 2 176.109 odd 4
2816.2.c.l.1409.2 2 176.21 odd 4
3168.2.a.y.1.1 1 264.197 even 2
3168.2.a.z.1.1 1 264.131 odd 2
3872.2.a.d.1.1 1 8.3 odd 2
3872.2.a.i.1.1 1 8.5 even 2
6336.2.a.g.1.1 1 33.32 even 2
6336.2.a.l.1.1 1 132.131 odd 2
7744.2.a.o.1.1 1 1.1 even 1 trivial
7744.2.a.ba.1.1 1 4.3 odd 2
8800.2.a.m.1.1 1 440.109 odd 2
8800.2.a.p.1.1 1 440.219 even 2