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Properties

Label 6096.2.a.o.1.1
Level $6096$
Weight $2$
Character 6096.1
Self dual yes
Analytic conductor $48.677$
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] = [6096,2,Mod(1,6096)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6096, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("6096.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6096 = 2^{4} \cdot 3 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6096.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: \(48.6768050722\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3048)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 6096.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} -5.00000 q^{11} +4.00000 q^{13} +1.00000 q^{15} -3.00000 q^{17} -7.00000 q^{19} +1.00000 q^{21} -5.00000 q^{23} -4.00000 q^{25} +1.00000 q^{27} +6.00000 q^{29} +8.00000 q^{31} -5.00000 q^{33} +1.00000 q^{35} -4.00000 q^{37} +4.00000 q^{39} -9.00000 q^{41} -10.0000 q^{43} +1.00000 q^{45} +10.0000 q^{47} -6.00000 q^{49} -3.00000 q^{51} +11.0000 q^{53} -5.00000 q^{55} -7.00000 q^{57} +2.00000 q^{59} -8.00000 q^{61} +1.00000 q^{63} +4.00000 q^{65} -14.0000 q^{67} -5.00000 q^{69} -8.00000 q^{71} +6.00000 q^{73} -4.00000 q^{75} -5.00000 q^{77} -16.0000 q^{79} +1.00000 q^{81} -3.00000 q^{85} +6.00000 q^{87} +4.00000 q^{91} +8.00000 q^{93} -7.00000 q^{95} -10.0000 q^{97} -5.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 1.00000 0.447214 0.223607 − 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964 0.188982 − 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −5.00000 −1.50756 −0.753778 − 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 − 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) −3.00000 −0.727607 −0.363803 − 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) −7.00000 −1.60591 −0.802955 − 0.596040i \(-0.796740\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −5.00000 −1.04257 −0.521286 − 0.853382i \(-0.674548\pi\)
−0.521286 + 0.853382i \(0.674548\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 − 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 0 0
\(33\) −5.00000 −0.870388
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 − 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 0 0
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) −9.00000 −1.40556 −0.702782 − 0.711405i \(-0.748059\pi\)
−0.702782 + 0.711405i \(0.748059\pi\)
\(42\) 0 0
\(43\) −10.0000 −1.52499 −0.762493 − 0.646997i \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 10.0000 1.45865 0.729325 − 0.684167i \(-0.239834\pi\)
0.729325 + 0.684167i \(0.239834\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) −3.00000 −0.420084
\(52\) 0 0
\(53\) 11.0000 1.51097 0.755483 − 0.655168i \(-0.227402\pi\)
0.755483 + 0.655168i \(0.227402\pi\)
\(54\) 0 0
\(55\) −5.00000 −0.674200
\(56\) 0 0
\(57\) −7.00000 −0.927173
\(58\) 0 0
\(59\) 2.00000 0.260378 0.130189 − 0.991489i \(-0.458442\pi\)
0.130189 + 0.991489i \(0.458442\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 − 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) −14.0000 −1.71037 −0.855186 − 0.518321i \(-0.826557\pi\)
−0.855186 + 0.518321i \(0.826557\pi\)
\(68\) 0 0
\(69\) −5.00000 −0.601929
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 − 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 0 0
\(75\) −4.00000 −0.461880
\(76\) 0 0
\(77\) −5.00000 −0.569803
\(78\) 0 0
\(79\) −16.0000 −1.80014 −0.900070 − 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) −3.00000 −0.325396
\(86\) 0 0
\(87\) 6.00000 0.643268
\(88\) 0 0
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 0 0
\(93\) 8.00000 0.829561
\(94\) 0 0
\(95\) −7.00000 −0.718185
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 − 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 0 0
\(99\) −5.00000 −0.502519
\(100\) 0 0
\(101\) 3.00000 0.298511 0.149256 − 0.988799i \(-0.452312\pi\)
0.149256 + 0.988799i \(0.452312\pi\)
\(102\) 0 0
\(103\) −10.0000 −0.985329 −0.492665 − 0.870219i \(-0.663977\pi\)
−0.492665 + 0.870219i \(0.663977\pi\)
\(104\) 0 0
\(105\) 1.00000 0.0975900
\(106\) 0 0
\(107\) 4.00000 0.386695 0.193347 − 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) 0 0
\(109\) −6.00000 −0.574696 −0.287348 − 0.957826i \(-0.592774\pi\)
−0.287348 + 0.957826i \(0.592774\pi\)
\(110\) 0 0
\(111\) −4.00000 −0.379663
\(112\) 0 0
\(113\) 1.00000 0.0940721 0.0470360 − 0.998893i \(-0.485022\pi\)
0.0470360 + 0.998893i \(0.485022\pi\)
\(114\) 0 0
\(115\) −5.00000 −0.466252
\(116\) 0 0
\(117\) 4.00000 0.369800
\(118\) 0 0
\(119\) −3.00000 −0.275010
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 0 0
\(123\) −9.00000 −0.811503
\(124\) 0 0
\(125\) −9.00000 −0.804984
\(126\) 0 0
\(127\) 1.00000 0.0887357
\(128\) 0 0
\(129\) −10.0000 −0.880451
\(130\) 0 0
\(131\) 7.00000 0.611593 0.305796 − 0.952097i \(-0.401077\pi\)
0.305796 + 0.952097i \(0.401077\pi\)
\(132\) 0 0
\(133\) −7.00000 −0.606977
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 18.0000 1.53784 0.768922 − 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) 0 0
\(139\) 14.0000 1.18746 0.593732 − 0.804663i \(-0.297654\pi\)
0.593732 + 0.804663i \(0.297654\pi\)
\(140\) 0 0
\(141\) 10.0000 0.842152
\(142\) 0 0
\(143\) −20.0000 −1.67248
\(144\) 0 0
\(145\) 6.00000 0.498273
\(146\) 0 0
\(147\) −6.00000 −0.494872
\(148\) 0 0
\(149\) 6.00000 0.491539 0.245770 − 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) −1.00000 −0.0813788 −0.0406894 − 0.999172i \(-0.512955\pi\)
−0.0406894 + 0.999172i \(0.512955\pi\)
\(152\) 0 0
\(153\) −3.00000 −0.242536
\(154\) 0 0
\(155\) 8.00000 0.642575
\(156\) 0 0
\(157\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) 0 0
\(159\) 11.0000 0.872357
\(160\) 0 0
\(161\) −5.00000 −0.394055
\(162\) 0 0
\(163\) −17.0000 −1.33154 −0.665771 − 0.746156i \(-0.731897\pi\)
−0.665771 + 0.746156i \(0.731897\pi\)
\(164\) 0 0
\(165\) −5.00000 −0.389249
\(166\) 0 0
\(167\) −9.00000 −0.696441 −0.348220 − 0.937413i \(-0.613214\pi\)
−0.348220 + 0.937413i \(0.613214\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) −7.00000 −0.535303
\(172\) 0 0
\(173\) −6.00000 −0.456172 −0.228086 − 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) −4.00000 −0.302372
\(176\) 0 0
\(177\) 2.00000 0.150329
\(178\) 0 0
\(179\) 5.00000 0.373718 0.186859 − 0.982387i \(-0.440169\pi\)
0.186859 + 0.982387i \(0.440169\pi\)
\(180\) 0 0
\(181\) 17.0000 1.26360 0.631800 − 0.775131i \(-0.282316\pi\)
0.631800 + 0.775131i \(0.282316\pi\)
\(182\) 0 0
\(183\) −8.00000 −0.591377
\(184\) 0 0
\(185\) −4.00000 −0.294086
\(186\) 0 0
\(187\) 15.0000 1.09691
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −10.0000 −0.723575 −0.361787 − 0.932261i \(-0.617833\pi\)
−0.361787 + 0.932261i \(0.617833\pi\)
\(192\) 0 0
\(193\) 22.0000 1.58359 0.791797 − 0.610784i \(-0.209146\pi\)
0.791797 + 0.610784i \(0.209146\pi\)
\(194\) 0 0
\(195\) 4.00000 0.286446
\(196\) 0 0
\(197\) 12.0000 0.854965 0.427482 − 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 0 0
\(199\) −10.0000 −0.708881 −0.354441 − 0.935079i \(-0.615329\pi\)
−0.354441 + 0.935079i \(0.615329\pi\)
\(200\) 0 0
\(201\) −14.0000 −0.987484
\(202\) 0 0
\(203\) 6.00000 0.421117
\(204\) 0 0
\(205\) −9.00000 −0.628587
\(206\) 0 0
\(207\) −5.00000 −0.347524
\(208\) 0 0
\(209\) 35.0000 2.42100
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 − 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 0 0
\(213\) −8.00000 −0.548151
\(214\) 0 0
\(215\) −10.0000 −0.681994
\(216\) 0 0
\(217\) 8.00000 0.543075
\(218\) 0 0
\(219\) 6.00000 0.405442
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) 0 0
\(223\) −5.00000 −0.334825 −0.167412 − 0.985887i \(-0.553541\pi\)
−0.167412 + 0.985887i \(0.553541\pi\)
\(224\) 0 0
\(225\) −4.00000 −0.266667
\(226\) 0 0
\(227\) −23.0000 −1.52656 −0.763282 − 0.646066i \(-0.776413\pi\)
−0.763282 + 0.646066i \(0.776413\pi\)
\(228\) 0 0
\(229\) 10.0000 0.660819 0.330409 − 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 0 0
\(231\) −5.00000 −0.328976
\(232\) 0 0
\(233\) −26.0000 −1.70332 −0.851658 − 0.524097i \(-0.824403\pi\)
−0.851658 + 0.524097i \(0.824403\pi\)
\(234\) 0 0
\(235\) 10.0000 0.652328
\(236\) 0 0
\(237\) −16.0000 −1.03931
\(238\) 0 0
\(239\) 25.0000 1.61712 0.808558 − 0.588417i \(-0.200249\pi\)
0.808558 + 0.588417i \(0.200249\pi\)
\(240\) 0 0
\(241\) −20.0000 −1.28831 −0.644157 − 0.764894i \(-0.722792\pi\)
−0.644157 + 0.764894i \(0.722792\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −6.00000 −0.383326
\(246\) 0 0
\(247\) −28.0000 −1.78160
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 21.0000 1.32551 0.662754 − 0.748837i \(-0.269387\pi\)
0.662754 + 0.748837i \(0.269387\pi\)
\(252\) 0 0
\(253\) 25.0000 1.57174
\(254\) 0 0
\(255\) −3.00000 −0.187867
\(256\) 0 0
\(257\) −16.0000 −0.998053 −0.499026 − 0.866587i \(-0.666309\pi\)
−0.499026 + 0.866587i \(0.666309\pi\)
\(258\) 0 0
\(259\) −4.00000 −0.248548
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 0 0
\(263\) 24.0000 1.47990 0.739952 − 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 0 0
\(265\) 11.0000 0.675725
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −2.00000 −0.121942 −0.0609711 − 0.998140i \(-0.519420\pi\)
−0.0609711 + 0.998140i \(0.519420\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 − 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 0 0
\(273\) 4.00000 0.242091
\(274\) 0 0
\(275\) 20.0000 1.20605
\(276\) 0 0
\(277\) 19.0000 1.14160 0.570800 − 0.821089i \(-0.306633\pi\)
0.570800 + 0.821089i \(0.306633\pi\)
\(278\) 0 0
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) −16.0000 −0.954480 −0.477240 − 0.878773i \(-0.658363\pi\)
−0.477240 + 0.878773i \(0.658363\pi\)
\(282\) 0 0
\(283\) −32.0000 −1.90220 −0.951101 − 0.308879i \(-0.900046\pi\)
−0.951101 + 0.308879i \(0.900046\pi\)
\(284\) 0 0
\(285\) −7.00000 −0.414644
\(286\) 0 0
\(287\) −9.00000 −0.531253
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) −10.0000 −0.586210
\(292\) 0 0
\(293\) −3.00000 −0.175262 −0.0876309 − 0.996153i \(-0.527930\pi\)
−0.0876309 + 0.996153i \(0.527930\pi\)
\(294\) 0 0
\(295\) 2.00000 0.116445
\(296\) 0 0
\(297\) −5.00000 −0.290129
\(298\) 0 0
\(299\) −20.0000 −1.15663
\(300\) 0 0
\(301\) −10.0000 −0.576390
\(302\) 0 0
\(303\) 3.00000 0.172345
\(304\) 0 0
\(305\) −8.00000 −0.458079
\(306\) 0 0
\(307\) −10.0000 −0.570730 −0.285365 − 0.958419i \(-0.592115\pi\)
−0.285365 + 0.958419i \(0.592115\pi\)
\(308\) 0 0
\(309\) −10.0000 −0.568880
\(310\) 0 0
\(311\) −15.0000 −0.850572 −0.425286 − 0.905059i \(-0.639826\pi\)
−0.425286 + 0.905059i \(0.639826\pi\)
\(312\) 0 0
\(313\) 10.0000 0.565233 0.282617 − 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) 0 0
\(315\) 1.00000 0.0563436
\(316\) 0 0
\(317\) −3.00000 −0.168497 −0.0842484 − 0.996445i \(-0.526849\pi\)
−0.0842484 + 0.996445i \(0.526849\pi\)
\(318\) 0 0
\(319\) −30.0000 −1.67968
\(320\) 0 0
\(321\) 4.00000 0.223258
\(322\) 0 0
\(323\) 21.0000 1.16847
\(324\) 0 0
\(325\) −16.0000 −0.887520
\(326\) 0 0
\(327\) −6.00000 −0.331801
\(328\) 0 0
\(329\) 10.0000 0.551318
\(330\) 0 0
\(331\) −14.0000 −0.769510 −0.384755 − 0.923019i \(-0.625714\pi\)
−0.384755 + 0.923019i \(0.625714\pi\)
\(332\) 0 0
\(333\) −4.00000 −0.219199
\(334\) 0 0
\(335\) −14.0000 −0.764902
\(336\) 0 0
\(337\) −4.00000 −0.217894 −0.108947 − 0.994048i \(-0.534748\pi\)
−0.108947 + 0.994048i \(0.534748\pi\)
\(338\) 0 0
\(339\) 1.00000 0.0543125
\(340\) 0 0
\(341\) −40.0000 −2.16612
\(342\) 0 0
\(343\) −13.0000 −0.701934
\(344\) 0 0
\(345\) −5.00000 −0.269191
\(346\) 0 0
\(347\) 18.0000 0.966291 0.483145 − 0.875540i \(-0.339494\pi\)
0.483145 + 0.875540i \(0.339494\pi\)
\(348\) 0 0
\(349\) −7.00000 −0.374701 −0.187351 − 0.982293i \(-0.559990\pi\)
−0.187351 + 0.982293i \(0.559990\pi\)
\(350\) 0 0
\(351\) 4.00000 0.213504
\(352\) 0 0
\(353\) 6.00000 0.319348 0.159674 − 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) 0 0
\(355\) −8.00000 −0.424596
\(356\) 0 0
\(357\) −3.00000 −0.158777
\(358\) 0 0
\(359\) 31.0000 1.63612 0.818059 − 0.575135i \(-0.195050\pi\)
0.818059 + 0.575135i \(0.195050\pi\)
\(360\) 0 0
\(361\) 30.0000 1.57895
\(362\) 0 0
\(363\) 14.0000 0.734809
\(364\) 0 0
\(365\) 6.00000 0.314054
\(366\) 0 0
\(367\) −28.0000 −1.46159 −0.730794 − 0.682598i \(-0.760850\pi\)
−0.730794 + 0.682598i \(0.760850\pi\)
\(368\) 0 0
\(369\) −9.00000 −0.468521
\(370\) 0 0
\(371\) 11.0000 0.571092
\(372\) 0 0
\(373\) 31.0000 1.60512 0.802560 − 0.596572i \(-0.203471\pi\)
0.802560 + 0.596572i \(0.203471\pi\)
\(374\) 0 0
\(375\) −9.00000 −0.464758
\(376\) 0 0
\(377\) 24.0000 1.23606
\(378\) 0 0
\(379\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) 1.00000 0.0512316
\(382\) 0 0
\(383\) 4.00000 0.204390 0.102195 − 0.994764i \(-0.467413\pi\)
0.102195 + 0.994764i \(0.467413\pi\)
\(384\) 0 0
\(385\) −5.00000 −0.254824
\(386\) 0 0
\(387\) −10.0000 −0.508329
\(388\) 0 0
\(389\) −14.0000 −0.709828 −0.354914 − 0.934899i \(-0.615490\pi\)
−0.354914 + 0.934899i \(0.615490\pi\)
\(390\) 0 0
\(391\) 15.0000 0.758583
\(392\) 0 0
\(393\) 7.00000 0.353103
\(394\) 0 0
\(395\) −16.0000 −0.805047
\(396\) 0 0
\(397\) 18.0000 0.903394 0.451697 − 0.892171i \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(398\) 0 0
\(399\) −7.00000 −0.350438
\(400\) 0 0
\(401\) −18.0000 −0.898877 −0.449439 − 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 0 0
\(403\) 32.0000 1.59403
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 20.0000 0.991363
\(408\) 0 0
\(409\) 30.0000 1.48340 0.741702 − 0.670729i \(-0.234019\pi\)
0.741702 + 0.670729i \(0.234019\pi\)
\(410\) 0 0
\(411\) 18.0000 0.887875
\(412\) 0 0
\(413\) 2.00000 0.0984136
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 14.0000 0.685583
\(418\) 0 0
\(419\) −35.0000 −1.70986 −0.854931 − 0.518742i \(-0.826401\pi\)
−0.854931 + 0.518742i \(0.826401\pi\)
\(420\) 0 0
\(421\) −26.0000 −1.26716 −0.633581 − 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) 0 0
\(423\) 10.0000 0.486217
\(424\) 0 0
\(425\) 12.0000 0.582086
\(426\) 0 0
\(427\) −8.00000 −0.387147
\(428\) 0 0
\(429\) −20.0000 −0.965609
\(430\) 0 0
\(431\) −18.0000 −0.867029 −0.433515 − 0.901146i \(-0.642727\pi\)
−0.433515 + 0.901146i \(0.642727\pi\)
\(432\) 0 0
\(433\) −5.00000 −0.240285 −0.120142 − 0.992757i \(-0.538335\pi\)
−0.120142 + 0.992757i \(0.538335\pi\)
\(434\) 0 0
\(435\) 6.00000 0.287678
\(436\) 0 0
\(437\) 35.0000 1.67428
\(438\) 0 0
\(439\) 13.0000 0.620456 0.310228 − 0.950662i \(-0.399595\pi\)
0.310228 + 0.950662i \(0.399595\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) 0 0
\(443\) −19.0000 −0.902717 −0.451359 − 0.892343i \(-0.649060\pi\)
−0.451359 + 0.892343i \(0.649060\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 6.00000 0.283790
\(448\) 0 0
\(449\) 17.0000 0.802280 0.401140 − 0.916017i \(-0.368614\pi\)
0.401140 + 0.916017i \(0.368614\pi\)
\(450\) 0 0
\(451\) 45.0000 2.11897
\(452\) 0 0
\(453\) −1.00000 −0.0469841
\(454\) 0 0
\(455\) 4.00000 0.187523
\(456\) 0 0
\(457\) −13.0000 −0.608114 −0.304057 − 0.952654i \(-0.598341\pi\)
−0.304057 + 0.952654i \(0.598341\pi\)
\(458\) 0 0
\(459\) −3.00000 −0.140028
\(460\) 0 0
\(461\) 19.0000 0.884918 0.442459 − 0.896789i \(-0.354106\pi\)
0.442459 + 0.896789i \(0.354106\pi\)
\(462\) 0 0
\(463\) −22.0000 −1.02243 −0.511213 − 0.859454i \(-0.670804\pi\)
−0.511213 + 0.859454i \(0.670804\pi\)
\(464\) 0 0
\(465\) 8.00000 0.370991
\(466\) 0 0
\(467\) 6.00000 0.277647 0.138823 − 0.990317i \(-0.455668\pi\)
0.138823 + 0.990317i \(0.455668\pi\)
\(468\) 0 0
\(469\) −14.0000 −0.646460
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 50.0000 2.29900
\(474\) 0 0
\(475\) 28.0000 1.28473
\(476\) 0 0
\(477\) 11.0000 0.503655
\(478\) 0 0
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) −16.0000 −0.729537
\(482\) 0 0
\(483\) −5.00000 −0.227508
\(484\) 0 0
\(485\) −10.0000 −0.454077
\(486\) 0 0
\(487\) 8.00000 0.362515 0.181257 − 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) 0 0
\(489\) −17.0000 −0.768767
\(490\) 0 0
\(491\) −2.00000 −0.0902587 −0.0451294 − 0.998981i \(-0.514370\pi\)
−0.0451294 + 0.998981i \(0.514370\pi\)
\(492\) 0 0
\(493\) −18.0000 −0.810679
\(494\) 0 0
\(495\) −5.00000 −0.224733
\(496\) 0 0
\(497\) −8.00000 −0.358849
\(498\) 0 0
\(499\) 24.0000 1.07439 0.537194 − 0.843459i \(-0.319484\pi\)
0.537194 + 0.843459i \(0.319484\pi\)
\(500\) 0 0
\(501\) −9.00000 −0.402090
\(502\) 0 0
\(503\) 24.0000 1.07011 0.535054 − 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) 0 0
\(505\) 3.00000 0.133498
\(506\) 0 0
\(507\) 3.00000 0.133235
\(508\) 0 0
\(509\) −36.0000 −1.59567 −0.797836 − 0.602875i \(-0.794022\pi\)
−0.797836 + 0.602875i \(0.794022\pi\)
\(510\) 0 0
\(511\) 6.00000 0.265424
\(512\) 0 0
\(513\) −7.00000 −0.309058
\(514\) 0 0
\(515\) −10.0000 −0.440653
\(516\) 0 0
\(517\) −50.0000 −2.19900
\(518\) 0 0
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) −14.0000 −0.613351 −0.306676 − 0.951814i \(-0.599217\pi\)
−0.306676 + 0.951814i \(0.599217\pi\)
\(522\) 0 0
\(523\) 44.0000 1.92399 0.961993 − 0.273075i \(-0.0880406\pi\)
0.961993 + 0.273075i \(0.0880406\pi\)
\(524\) 0 0
\(525\) −4.00000 −0.174574
\(526\) 0 0
\(527\) −24.0000 −1.04546
\(528\) 0 0
\(529\) 2.00000 0.0869565
\(530\) 0 0
\(531\) 2.00000 0.0867926
\(532\) 0 0
\(533\) −36.0000 −1.55933
\(534\) 0 0
\(535\) 4.00000 0.172935
\(536\) 0 0
\(537\) 5.00000 0.215766
\(538\) 0 0
\(539\) 30.0000 1.29219
\(540\) 0 0
\(541\) 25.0000 1.07483 0.537417 − 0.843317i \(-0.319400\pi\)
0.537417 + 0.843317i \(0.319400\pi\)
\(542\) 0 0
\(543\) 17.0000 0.729540
\(544\) 0 0
\(545\) −6.00000 −0.257012
\(546\) 0 0
\(547\) −20.0000 −0.855138 −0.427569 − 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) 0 0
\(549\) −8.00000 −0.341432
\(550\) 0 0
\(551\) −42.0000 −1.78926
\(552\) 0 0
\(553\) −16.0000 −0.680389
\(554\) 0 0
\(555\) −4.00000 −0.169791
\(556\) 0 0
\(557\) −30.0000 −1.27114 −0.635570 − 0.772043i \(-0.719235\pi\)
−0.635570 + 0.772043i \(0.719235\pi\)
\(558\) 0 0
\(559\) −40.0000 −1.69182
\(560\) 0 0
\(561\) 15.0000 0.633300
\(562\) 0 0
\(563\) 12.0000 0.505740 0.252870 − 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 0 0
\(565\) 1.00000 0.0420703
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −15.0000 −0.628833 −0.314416 − 0.949285i \(-0.601809\pi\)
−0.314416 + 0.949285i \(0.601809\pi\)
\(570\) 0 0
\(571\) 42.0000 1.75765 0.878823 − 0.477149i \(-0.158330\pi\)
0.878823 + 0.477149i \(0.158330\pi\)
\(572\) 0 0
\(573\) −10.0000 −0.417756
\(574\) 0 0
\(575\) 20.0000 0.834058
\(576\) 0 0
\(577\) 25.0000 1.04076 0.520382 − 0.853934i \(-0.325790\pi\)
0.520382 + 0.853934i \(0.325790\pi\)
\(578\) 0 0
\(579\) 22.0000 0.914289
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −55.0000 −2.27787
\(584\) 0 0
\(585\) 4.00000 0.165380
\(586\) 0 0
\(587\) −12.0000 −0.495293 −0.247647 − 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 0 0
\(589\) −56.0000 −2.30744
\(590\) 0 0
\(591\) 12.0000 0.493614
\(592\) 0 0
\(593\) −30.0000 −1.23195 −0.615976 − 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) 0 0
\(595\) −3.00000 −0.122988
\(596\) 0 0
\(597\) −10.0000 −0.409273
\(598\) 0 0
\(599\) 33.0000 1.34834 0.674172 − 0.738575i \(-0.264501\pi\)
0.674172 + 0.738575i \(0.264501\pi\)
\(600\) 0 0
\(601\) −16.0000 −0.652654 −0.326327 − 0.945257i \(-0.605811\pi\)
−0.326327 + 0.945257i \(0.605811\pi\)
\(602\) 0 0
\(603\) −14.0000 −0.570124
\(604\) 0 0
\(605\) 14.0000 0.569181
\(606\) 0 0
\(607\) −34.0000 −1.38002 −0.690009 − 0.723801i \(-0.742393\pi\)
−0.690009 + 0.723801i \(0.742393\pi\)
\(608\) 0 0
\(609\) 6.00000 0.243132
\(610\) 0 0
\(611\) 40.0000 1.61823
\(612\) 0 0
\(613\) −9.00000 −0.363507 −0.181753 − 0.983344i \(-0.558177\pi\)
−0.181753 + 0.983344i \(0.558177\pi\)
\(614\) 0 0
\(615\) −9.00000 −0.362915
\(616\) 0 0
\(617\) −48.0000 −1.93241 −0.966204 − 0.257780i \(-0.917009\pi\)
−0.966204 + 0.257780i \(0.917009\pi\)
\(618\) 0 0
\(619\) −46.0000 −1.84890 −0.924448 − 0.381308i \(-0.875474\pi\)
−0.924448 + 0.381308i \(0.875474\pi\)
\(620\) 0 0
\(621\) −5.00000 −0.200643
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 0 0
\(627\) 35.0000 1.39777
\(628\) 0 0
\(629\) 12.0000 0.478471
\(630\) 0 0
\(631\) 48.0000 1.91085 0.955425 − 0.295234i \(-0.0953977\pi\)
0.955425 + 0.295234i \(0.0953977\pi\)
\(632\) 0 0
\(633\) −4.00000 −0.158986
\(634\) 0 0
\(635\) 1.00000 0.0396838
\(636\) 0 0
\(637\) −24.0000 −0.950915
\(638\) 0 0
\(639\) −8.00000 −0.316475
\(640\) 0 0
\(641\) −18.0000 −0.710957 −0.355479 − 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 0 0
\(643\) −13.0000 −0.512670 −0.256335 − 0.966588i \(-0.582515\pi\)
−0.256335 + 0.966588i \(0.582515\pi\)
\(644\) 0 0
\(645\) −10.0000 −0.393750
\(646\) 0 0
\(647\) 29.0000 1.14011 0.570054 − 0.821607i \(-0.306922\pi\)
0.570054 + 0.821607i \(0.306922\pi\)
\(648\) 0 0
\(649\) −10.0000 −0.392534
\(650\) 0 0
\(651\) 8.00000 0.313545
\(652\) 0 0
\(653\) −10.0000 −0.391330 −0.195665 − 0.980671i \(-0.562687\pi\)
−0.195665 + 0.980671i \(0.562687\pi\)
\(654\) 0 0
\(655\) 7.00000 0.273513
\(656\) 0 0
\(657\) 6.00000 0.234082
\(658\) 0 0
\(659\) 14.0000 0.545363 0.272681 − 0.962104i \(-0.412090\pi\)
0.272681 + 0.962104i \(0.412090\pi\)
\(660\) 0 0
\(661\) −16.0000 −0.622328 −0.311164 − 0.950356i \(-0.600719\pi\)
−0.311164 + 0.950356i \(0.600719\pi\)
\(662\) 0 0
\(663\) −12.0000 −0.466041
\(664\) 0 0
\(665\) −7.00000 −0.271448
\(666\) 0 0
\(667\) −30.0000 −1.16160
\(668\) 0 0
\(669\) −5.00000 −0.193311
\(670\) 0 0
\(671\) 40.0000 1.54418
\(672\) 0 0
\(673\) −46.0000 −1.77317 −0.886585 − 0.462566i \(-0.846929\pi\)
−0.886585 + 0.462566i \(0.846929\pi\)
\(674\) 0 0
\(675\) −4.00000 −0.153960
\(676\) 0 0
\(677\) 34.0000 1.30673 0.653363 − 0.757045i \(-0.273358\pi\)
0.653363 + 0.757045i \(0.273358\pi\)
\(678\) 0 0
\(679\) −10.0000 −0.383765
\(680\) 0 0
\(681\) −23.0000 −0.881362
\(682\) 0 0
\(683\) 4.00000 0.153056 0.0765279 − 0.997067i \(-0.475617\pi\)
0.0765279 + 0.997067i \(0.475617\pi\)
\(684\) 0 0
\(685\) 18.0000 0.687745
\(686\) 0 0
\(687\) 10.0000 0.381524
\(688\) 0 0
\(689\) 44.0000 1.67627
\(690\) 0 0
\(691\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(692\) 0 0
\(693\) −5.00000 −0.189934
\(694\) 0 0
\(695\) 14.0000 0.531050
\(696\) 0 0
\(697\) 27.0000 1.02270
\(698\) 0 0
\(699\) −26.0000 −0.983410
\(700\) 0 0
\(701\) 25.0000 0.944237 0.472118 − 0.881535i \(-0.343489\pi\)
0.472118 + 0.881535i \(0.343489\pi\)
\(702\) 0 0
\(703\) 28.0000 1.05604
\(704\) 0 0
\(705\) 10.0000 0.376622
\(706\) 0 0
\(707\) 3.00000 0.112827
\(708\) 0 0
\(709\) 50.0000 1.87779 0.938895 − 0.344204i \(-0.111851\pi\)
0.938895 + 0.344204i \(0.111851\pi\)
\(710\) 0 0
\(711\) −16.0000 −0.600047
\(712\) 0 0
\(713\) −40.0000 −1.49801
\(714\) 0 0
\(715\) −20.0000 −0.747958
\(716\) 0 0
\(717\) 25.0000 0.933642
\(718\) 0 0
\(719\) −34.0000 −1.26799 −0.633993 − 0.773339i \(-0.718585\pi\)
−0.633993 + 0.773339i \(0.718585\pi\)
\(720\) 0 0
\(721\) −10.0000 −0.372419
\(722\) 0 0
\(723\) −20.0000 −0.743808
\(724\) 0 0
\(725\) −24.0000 −0.891338
\(726\) 0 0
\(727\) 20.0000 0.741759 0.370879 − 0.928681i \(-0.379056\pi\)
0.370879 + 0.928681i \(0.379056\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 30.0000 1.10959
\(732\) 0 0
\(733\) −42.0000 −1.55131 −0.775653 − 0.631160i \(-0.782579\pi\)
−0.775653 + 0.631160i \(0.782579\pi\)
\(734\) 0 0
\(735\) −6.00000 −0.221313
\(736\) 0 0
\(737\) 70.0000 2.57848
\(738\) 0 0
\(739\) −44.0000 −1.61857 −0.809283 − 0.587419i \(-0.800144\pi\)
−0.809283 + 0.587419i \(0.800144\pi\)
\(740\) 0 0
\(741\) −28.0000 −1.02861
\(742\) 0 0
\(743\) 41.0000 1.50414 0.752072 − 0.659081i \(-0.229055\pi\)
0.752072 + 0.659081i \(0.229055\pi\)
\(744\) 0 0
\(745\) 6.00000 0.219823
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 4.00000 0.146157
\(750\) 0 0
\(751\) −24.0000 −0.875772 −0.437886 − 0.899030i \(-0.644273\pi\)
−0.437886 + 0.899030i \(0.644273\pi\)
\(752\) 0 0
\(753\) 21.0000 0.765283
\(754\) 0 0
\(755\) −1.00000 −0.0363937
\(756\) 0 0
\(757\) −40.0000 −1.45382 −0.726912 − 0.686730i \(-0.759045\pi\)
−0.726912 + 0.686730i \(0.759045\pi\)
\(758\) 0 0
\(759\) 25.0000 0.907443
\(760\) 0 0
\(761\) −30.0000 −1.08750 −0.543750 − 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) 0 0
\(763\) −6.00000 −0.217215
\(764\) 0 0
\(765\) −3.00000 −0.108465
\(766\) 0 0
\(767\) 8.00000 0.288863
\(768\) 0 0
\(769\) 4.00000 0.144244 0.0721218 − 0.997396i \(-0.477023\pi\)
0.0721218 + 0.997396i \(0.477023\pi\)
\(770\) 0 0
\(771\) −16.0000 −0.576226
\(772\) 0 0
\(773\) 48.0000 1.72644 0.863220 − 0.504828i \(-0.168444\pi\)
0.863220 + 0.504828i \(0.168444\pi\)
\(774\) 0 0
\(775\) −32.0000 −1.14947
\(776\) 0 0
\(777\) −4.00000 −0.143499
\(778\) 0 0
\(779\) 63.0000 2.25721
\(780\) 0 0
\(781\) 40.0000 1.43131
\(782\) 0 0
\(783\) 6.00000 0.214423
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 33.0000 1.17632 0.588161 − 0.808744i \(-0.299852\pi\)
0.588161 + 0.808744i \(0.299852\pi\)
\(788\) 0 0
\(789\) 24.0000 0.854423
\(790\) 0 0
\(791\) 1.00000 0.0355559
\(792\) 0 0
\(793\) −32.0000 −1.13635
\(794\) 0 0
\(795\) 11.0000 0.390130
\(796\) 0 0
\(797\) 38.0000 1.34603 0.673015 − 0.739629i \(-0.264999\pi\)
0.673015 + 0.739629i \(0.264999\pi\)
\(798\) 0 0
\(799\) −30.0000 −1.06132
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −30.0000 −1.05868
\(804\) 0 0
\(805\) −5.00000 −0.176227
\(806\) 0 0
\(807\) −2.00000 −0.0704033
\(808\) 0 0
\(809\) 21.0000 0.738321 0.369160 − 0.929366i \(-0.379645\pi\)
0.369160 + 0.929366i \(0.379645\pi\)
\(810\) 0 0
\(811\) −33.0000 −1.15879 −0.579393 − 0.815048i \(-0.696710\pi\)
−0.579393 + 0.815048i \(0.696710\pi\)
\(812\) 0 0
\(813\) 20.0000 0.701431
\(814\) 0 0
\(815\) −17.0000 −0.595484
\(816\) 0 0
\(817\) 70.0000 2.44899
\(818\) 0 0
\(819\) 4.00000 0.139771
\(820\) 0 0
\(821\) 43.0000 1.50071 0.750355 − 0.661035i \(-0.229882\pi\)
0.750355 + 0.661035i \(0.229882\pi\)
\(822\) 0 0
\(823\) 16.0000 0.557725 0.278862 − 0.960331i \(-0.410043\pi\)
0.278862 + 0.960331i \(0.410043\pi\)
\(824\) 0 0
\(825\) 20.0000 0.696311
\(826\) 0 0
\(827\) 28.0000 0.973655 0.486828 − 0.873498i \(-0.338154\pi\)
0.486828 + 0.873498i \(0.338154\pi\)
\(828\) 0 0
\(829\) 31.0000 1.07667 0.538337 − 0.842729i \(-0.319053\pi\)
0.538337 + 0.842729i \(0.319053\pi\)
\(830\) 0 0
\(831\) 19.0000 0.659103
\(832\) 0 0
\(833\) 18.0000 0.623663
\(834\) 0 0
\(835\) −9.00000 −0.311458
\(836\) 0 0
\(837\) 8.00000 0.276520
\(838\) 0 0
\(839\) 37.0000 1.27738 0.638691 − 0.769463i \(-0.279476\pi\)
0.638691 + 0.769463i \(0.279476\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) −16.0000 −0.551069
\(844\) 0 0
\(845\) 3.00000 0.103203
\(846\) 0 0
\(847\) 14.0000 0.481046
\(848\) 0 0
\(849\) −32.0000 −1.09824
\(850\) 0 0
\(851\) 20.0000 0.685591
\(852\) 0 0
\(853\) −31.0000 −1.06142 −0.530710 − 0.847554i \(-0.678075\pi\)
−0.530710 + 0.847554i \(0.678075\pi\)
\(854\) 0 0
\(855\) −7.00000 −0.239395
\(856\) 0 0
\(857\) −40.0000 −1.36637 −0.683187 − 0.730243i \(-0.739407\pi\)
−0.683187 + 0.730243i \(0.739407\pi\)
\(858\) 0 0
\(859\) 50.0000 1.70598 0.852989 − 0.521929i \(-0.174787\pi\)
0.852989 + 0.521929i \(0.174787\pi\)
\(860\) 0 0
\(861\) −9.00000 −0.306719
\(862\) 0 0
\(863\) −16.0000 −0.544646 −0.272323 − 0.962206i \(-0.587792\pi\)
−0.272323 + 0.962206i \(0.587792\pi\)
\(864\) 0 0
\(865\) −6.00000 −0.204006
\(866\) 0 0
\(867\) −8.00000 −0.271694
\(868\) 0 0
\(869\) 80.0000 2.71381
\(870\) 0 0
\(871\) −56.0000 −1.89749
\(872\) 0 0
\(873\) −10.0000 −0.338449
\(874\) 0 0
\(875\) −9.00000 −0.304256
\(876\) 0 0
\(877\) −22.0000 −0.742887 −0.371444 − 0.928456i \(-0.621137\pi\)
−0.371444 + 0.928456i \(0.621137\pi\)
\(878\) 0 0
\(879\) −3.00000 −0.101187
\(880\) 0 0
\(881\) −32.0000 −1.07811 −0.539054 − 0.842271i \(-0.681218\pi\)
−0.539054 + 0.842271i \(0.681218\pi\)
\(882\) 0 0
\(883\) 59.0000 1.98551 0.992754 − 0.120164i \(-0.0383421\pi\)
0.992754 + 0.120164i \(0.0383421\pi\)
\(884\) 0 0
\(885\) 2.00000 0.0672293
\(886\) 0 0
\(887\) 41.0000 1.37665 0.688323 − 0.725405i \(-0.258347\pi\)
0.688323 + 0.725405i \(0.258347\pi\)
\(888\) 0 0
\(889\) 1.00000 0.0335389
\(890\) 0 0
\(891\) −5.00000 −0.167506
\(892\) 0 0
\(893\) −70.0000 −2.34246
\(894\) 0 0
\(895\) 5.00000 0.167132
\(896\) 0 0
\(897\) −20.0000 −0.667781
\(898\) 0 0
\(899\) 48.0000 1.60089
\(900\) 0 0
\(901\) −33.0000 −1.09939
\(902\) 0 0
\(903\) −10.0000 −0.332779
\(904\) 0 0
\(905\) 17.0000 0.565099
\(906\) 0 0
\(907\) 29.0000 0.962929 0.481465 − 0.876466i \(-0.340105\pi\)
0.481465 + 0.876466i \(0.340105\pi\)
\(908\) 0 0
\(909\) 3.00000 0.0995037
\(910\) 0 0
\(911\) −38.0000 −1.25900 −0.629498 − 0.777002i \(-0.716739\pi\)
−0.629498 + 0.777002i \(0.716739\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −8.00000 −0.264472
\(916\) 0 0
\(917\) 7.00000 0.231160
\(918\) 0 0
\(919\) −28.0000 −0.923635 −0.461817 − 0.886975i \(-0.652802\pi\)
−0.461817 + 0.886975i \(0.652802\pi\)
\(920\) 0 0
\(921\) −10.0000 −0.329511
\(922\) 0 0
\(923\) −32.0000 −1.05329
\(924\) 0 0
\(925\) 16.0000 0.526077
\(926\) 0 0
\(927\) −10.0000 −0.328443
\(928\) 0 0
\(929\) 10.0000 0.328089 0.164045 − 0.986453i \(-0.447546\pi\)
0.164045 + 0.986453i \(0.447546\pi\)
\(930\) 0 0
\(931\) 42.0000 1.37649
\(932\) 0 0
\(933\) −15.0000 −0.491078
\(934\) 0 0
\(935\) 15.0000 0.490552
\(936\) 0 0
\(937\) −38.0000 −1.24141 −0.620703 − 0.784046i \(-0.713153\pi\)
−0.620703 + 0.784046i \(0.713153\pi\)
\(938\) 0 0
\(939\) 10.0000 0.326338
\(940\) 0 0
\(941\) 28.0000 0.912774 0.456387 − 0.889781i \(-0.349143\pi\)
0.456387 + 0.889781i \(0.349143\pi\)
\(942\) 0 0
\(943\) 45.0000 1.46540
\(944\) 0 0
\(945\) 1.00000 0.0325300
\(946\) 0 0
\(947\) −26.0000 −0.844886 −0.422443 − 0.906389i \(-0.638827\pi\)
−0.422443 + 0.906389i \(0.638827\pi\)
\(948\) 0 0
\(949\) 24.0000 0.779073
\(950\) 0 0
\(951\) −3.00000 −0.0972817
\(952\) 0 0
\(953\) −21.0000 −0.680257 −0.340128 − 0.940379i \(-0.610471\pi\)
−0.340128 + 0.940379i \(0.610471\pi\)
\(954\) 0 0
\(955\) −10.0000 −0.323592
\(956\) 0 0
\(957\) −30.0000 −0.969762
\(958\) 0 0
\(959\) 18.0000 0.581250
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) 4.00000 0.128898
\(964\) 0 0
\(965\) 22.0000 0.708205
\(966\) 0 0
\(967\) −32.0000 −1.02905 −0.514525 − 0.857475i \(-0.672032\pi\)
−0.514525 + 0.857475i \(0.672032\pi\)
\(968\) 0 0
\(969\) 21.0000 0.674617
\(970\) 0 0
\(971\) 36.0000 1.15529 0.577647 − 0.816286i \(-0.303971\pi\)
0.577647 + 0.816286i \(0.303971\pi\)
\(972\) 0 0
\(973\) 14.0000 0.448819
\(974\) 0 0
\(975\) −16.0000 −0.512410
\(976\) 0 0
\(977\) 18.0000 0.575871 0.287936 − 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −6.00000 −0.191565
\(982\) 0 0
\(983\) 14.0000 0.446531 0.223265 − 0.974758i \(-0.428328\pi\)
0.223265 + 0.974758i \(0.428328\pi\)
\(984\) 0 0
\(985\) 12.0000 0.382352
\(986\) 0 0
\(987\) 10.0000 0.318304
\(988\) 0 0
\(989\) 50.0000 1.58991
\(990\) 0 0
\(991\) −40.0000 −1.27064 −0.635321 − 0.772248i \(-0.719132\pi\)
−0.635321 + 0.772248i \(0.719132\pi\)
\(992\) 0 0
\(993\) −14.0000 −0.444277
\(994\) 0 0
\(995\) −10.0000 −0.317021
\(996\) 0 0
\(997\) −14.0000 −0.443384 −0.221692 − 0.975117i \(-0.571158\pi\)
−0.221692 + 0.975117i \(0.571158\pi\)
\(998\) 0 0
\(999\) −4.00000 −0.126554
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 6096.2.a.o.1.1 1
4.3 odd 2 3048.2.a.c.1.1 ✓ 1
12.11 even 2 9144.2.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3048.2.a.c.1.1 ✓ 1 4.3 odd 2
6096.2.a.o.1.1 1 1.1 even 1 trivial
9144.2.a.d.1.1 1 12.11 even 2