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Properties

Label 882.2.a.a.1.1
Level $882$
Weight $2$
Character 882.1
Self dual yes
Analytic conductor $7.043$
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] = [882,2,Mod(1,882)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(882, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("882.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 882.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: \(7.04280545828\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 126)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 882.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -3.00000 q^{5} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -3.00000 q^{5} -1.00000 q^{8} +3.00000 q^{10} +3.00000 q^{11} +2.00000 q^{13} +1.00000 q^{16} -6.00000 q^{17} +2.00000 q^{19} -3.00000 q^{20} -3.00000 q^{22} +6.00000 q^{23} +4.00000 q^{25} -2.00000 q^{26} -9.00000 q^{29} -7.00000 q^{31} -1.00000 q^{32} +6.00000 q^{34} -10.0000 q^{37} -2.00000 q^{38} +3.00000 q^{40} -4.00000 q^{43} +3.00000 q^{44} -6.00000 q^{46} -12.0000 q^{47} -4.00000 q^{50} +2.00000 q^{52} +3.00000 q^{53} -9.00000 q^{55} +9.00000 q^{58} +3.00000 q^{59} -4.00000 q^{61} +7.00000 q^{62} +1.00000 q^{64} -6.00000 q^{65} +2.00000 q^{67} -6.00000 q^{68} +2.00000 q^{73} +10.0000 q^{74} +2.00000 q^{76} +5.00000 q^{79} -3.00000 q^{80} -9.00000 q^{83} +18.0000 q^{85} +4.00000 q^{86} -3.00000 q^{88} +6.00000 q^{89} +6.00000 q^{92} +12.0000 q^{94} -6.00000 q^{95} -13.0000 q^{97} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −3.00000 −1.34164 −0.670820 − 0.741620i \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 3.00000 0.948683
\(11\) 3.00000 0.904534 0.452267 − 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −6.00000 −1.45521 −0.727607 − 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 − 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) −3.00000 −0.670820
\(21\) 0 0
\(22\) −3.00000 −0.639602
\(23\) 6.00000 1.25109 0.625543 − 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) 0 0
\(29\) −9.00000 −1.67126 −0.835629 − 0.549294i \(-0.814897\pi\)
−0.835629 + 0.549294i \(0.814897\pi\)
\(30\) 0 0
\(31\) −7.00000 −1.25724 −0.628619 − 0.777714i \(-0.716379\pi\)
−0.628619 + 0.777714i \(0.716379\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) 0 0
\(37\) −10.0000 −1.64399 −0.821995 − 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) −2.00000 −0.324443
\(39\) 0 0
\(40\) 3.00000 0.474342
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 3.00000 0.452267
\(45\) 0 0
\(46\) −6.00000 −0.884652
\(47\) −12.0000 −1.75038 −0.875190 − 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −4.00000 −0.565685
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) 3.00000 0.412082 0.206041 − 0.978543i \(-0.433942\pi\)
0.206041 + 0.978543i \(0.433942\pi\)
\(54\) 0 0
\(55\) −9.00000 −1.21356
\(56\) 0 0
\(57\) 0 0
\(58\) 9.00000 1.18176
\(59\) 3.00000 0.390567 0.195283 − 0.980747i \(-0.437437\pi\)
0.195283 + 0.980747i \(0.437437\pi\)
\(60\) 0 0
\(61\) −4.00000 −0.512148 −0.256074 − 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) 7.00000 0.889001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −6.00000 −0.744208
\(66\) 0 0
\(67\) 2.00000 0.244339 0.122169 − 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) −6.00000 −0.727607
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 − 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 10.0000 1.16248
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) 0 0
\(78\) 0 0
\(79\) 5.00000 0.562544 0.281272 − 0.959628i \(-0.409244\pi\)
0.281272 + 0.959628i \(0.409244\pi\)
\(80\) −3.00000 −0.335410
\(81\) 0 0
\(82\) 0 0
\(83\) −9.00000 −0.987878 −0.493939 − 0.869496i \(-0.664443\pi\)
−0.493939 + 0.869496i \(0.664443\pi\)
\(84\) 0 0
\(85\) 18.0000 1.95237
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) −3.00000 −0.319801
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 6.00000 0.625543
\(93\) 0 0
\(94\) 12.0000 1.23771
\(95\) −6.00000 −0.615587
\(96\) 0 0
\(97\) −13.0000 −1.31995 −0.659975 − 0.751288i \(-0.729433\pi\)
−0.659975 + 0.751288i \(0.729433\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 4.00000 0.400000
\(101\) −6.00000 −0.597022 −0.298511 − 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) −16.0000 −1.57653 −0.788263 − 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) −3.00000 −0.291386
\(107\) −3.00000 −0.290021 −0.145010 − 0.989430i \(-0.546322\pi\)
−0.145010 + 0.989430i \(0.546322\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.957826 −0.478913 − 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 9.00000 0.858116
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) −18.0000 −1.67851
\(116\) −9.00000 −0.835629
\(117\) 0 0
\(118\) −3.00000 −0.276172
\(119\) 0 0
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 4.00000 0.362143
\(123\) 0 0
\(124\) −7.00000 −0.628619
\(125\) 3.00000 0.268328
\(126\) 0 0
\(127\) −1.00000 −0.0887357 −0.0443678 − 0.999015i \(-0.514127\pi\)
−0.0443678 + 0.999015i \(0.514127\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 6.00000 0.526235
\(131\) 15.0000 1.31056 0.655278 − 0.755388i \(-0.272551\pi\)
0.655278 + 0.755388i \(0.272551\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −2.00000 −0.172774
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) −6.00000 −0.512615 −0.256307 − 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) 2.00000 0.169638 0.0848189 − 0.996396i \(-0.472969\pi\)
0.0848189 + 0.996396i \(0.472969\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 6.00000 0.501745
\(144\) 0 0
\(145\) 27.0000 2.24223
\(146\) −2.00000 −0.165521
\(147\) 0 0
\(148\) −10.0000 −0.821995
\(149\) 6.00000 0.491539 0.245770 − 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) 5.00000 0.406894 0.203447 − 0.979086i \(-0.434786\pi\)
0.203447 + 0.979086i \(0.434786\pi\)
\(152\) −2.00000 −0.162221
\(153\) 0 0
\(154\) 0 0
\(155\) 21.0000 1.68676
\(156\) 0 0
\(157\) −4.00000 −0.319235 −0.159617 − 0.987179i \(-0.551026\pi\)
−0.159617 + 0.987179i \(0.551026\pi\)
\(158\) −5.00000 −0.397779
\(159\) 0 0
\(160\) 3.00000 0.237171
\(161\) 0 0
\(162\) 0 0
\(163\) −10.0000 −0.783260 −0.391630 − 0.920123i \(-0.628089\pi\)
−0.391630 + 0.920123i \(0.628089\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 9.00000 0.698535
\(167\) 18.0000 1.39288 0.696441 − 0.717614i \(-0.254766\pi\)
0.696441 + 0.717614i \(0.254766\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −18.0000 −1.38054
\(171\) 0 0
\(172\) −4.00000 −0.304997
\(173\) 6.00000 0.456172 0.228086 − 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 3.00000 0.226134
\(177\) 0 0
\(178\) −6.00000 −0.449719
\(179\) 12.0000 0.896922 0.448461 − 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) 20.0000 1.48659 0.743294 − 0.668965i \(-0.233262\pi\)
0.743294 + 0.668965i \(0.233262\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −6.00000 −0.442326
\(185\) 30.0000 2.20564
\(186\) 0 0
\(187\) −18.0000 −1.31629
\(188\) −12.0000 −0.875190
\(189\) 0 0
\(190\) 6.00000 0.435286
\(191\) −12.0000 −0.868290 −0.434145 − 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 0 0
\(193\) −7.00000 −0.503871 −0.251936 − 0.967744i \(-0.581067\pi\)
−0.251936 + 0.967744i \(0.581067\pi\)
\(194\) 13.0000 0.933346
\(195\) 0 0
\(196\) 0 0
\(197\) 18.0000 1.28245 0.641223 − 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 0 0
\(199\) 8.00000 0.567105 0.283552 − 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) −4.00000 −0.282843
\(201\) 0 0
\(202\) 6.00000 0.422159
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 16.0000 1.11477
\(207\) 0 0
\(208\) 2.00000 0.138675
\(209\) 6.00000 0.415029
\(210\) 0 0
\(211\) −16.0000 −1.10149 −0.550743 − 0.834675i \(-0.685655\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) 3.00000 0.206041
\(213\) 0 0
\(214\) 3.00000 0.205076
\(215\) 12.0000 0.818393
\(216\) 0 0
\(217\) 0 0
\(218\) 10.0000 0.677285
\(219\) 0 0
\(220\) −9.00000 −0.606780
\(221\) −12.0000 −0.807207
\(222\) 0 0
\(223\) −1.00000 −0.0669650 −0.0334825 − 0.999439i \(-0.510660\pi\)
−0.0334825 + 0.999439i \(0.510660\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −15.0000 −0.995585 −0.497792 − 0.867296i \(-0.665856\pi\)
−0.497792 + 0.867296i \(0.665856\pi\)
\(228\) 0 0
\(229\) 20.0000 1.32164 0.660819 − 0.750546i \(-0.270209\pi\)
0.660819 + 0.750546i \(0.270209\pi\)
\(230\) 18.0000 1.18688
\(231\) 0 0
\(232\) 9.00000 0.590879
\(233\) 6.00000 0.393073 0.196537 − 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) 36.0000 2.34838
\(236\) 3.00000 0.195283
\(237\) 0 0
\(238\) 0 0
\(239\) −18.0000 −1.16432 −0.582162 − 0.813073i \(-0.697793\pi\)
−0.582162 + 0.813073i \(0.697793\pi\)
\(240\) 0 0
\(241\) 23.0000 1.48156 0.740780 − 0.671748i \(-0.234456\pi\)
0.740780 + 0.671748i \(0.234456\pi\)
\(242\) 2.00000 0.128565
\(243\) 0 0
\(244\) −4.00000 −0.256074
\(245\) 0 0
\(246\) 0 0
\(247\) 4.00000 0.254514
\(248\) 7.00000 0.444500
\(249\) 0 0
\(250\) −3.00000 −0.189737
\(251\) −9.00000 −0.568075 −0.284037 − 0.958813i \(-0.591674\pi\)
−0.284037 + 0.958813i \(0.591674\pi\)
\(252\) 0 0
\(253\) 18.0000 1.13165
\(254\) 1.00000 0.0627456
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 6.00000 0.374270 0.187135 − 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −6.00000 −0.372104
\(261\) 0 0
\(262\) −15.0000 −0.926703
\(263\) −24.0000 −1.47990 −0.739952 − 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 0 0
\(265\) −9.00000 −0.552866
\(266\) 0 0
\(267\) 0 0
\(268\) 2.00000 0.122169
\(269\) 3.00000 0.182913 0.0914566 − 0.995809i \(-0.470848\pi\)
0.0914566 + 0.995809i \(0.470848\pi\)
\(270\) 0 0
\(271\) −19.0000 −1.15417 −0.577084 − 0.816685i \(-0.695809\pi\)
−0.577084 + 0.816685i \(0.695809\pi\)
\(272\) −6.00000 −0.363803
\(273\) 0 0
\(274\) 6.00000 0.362473
\(275\) 12.0000 0.723627
\(276\) 0 0
\(277\) −4.00000 −0.240337 −0.120168 − 0.992754i \(-0.538343\pi\)
−0.120168 + 0.992754i \(0.538343\pi\)
\(278\) −2.00000 −0.119952
\(279\) 0 0
\(280\) 0 0
\(281\) 18.0000 1.07379 0.536895 − 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 0 0
\(283\) 20.0000 1.18888 0.594438 − 0.804141i \(-0.297374\pi\)
0.594438 + 0.804141i \(0.297374\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) −6.00000 −0.354787
\(287\) 0 0
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) −27.0000 −1.58549
\(291\) 0 0
\(292\) 2.00000 0.117041
\(293\) −9.00000 −0.525786 −0.262893 − 0.964825i \(-0.584677\pi\)
−0.262893 + 0.964825i \(0.584677\pi\)
\(294\) 0 0
\(295\) −9.00000 −0.524000
\(296\) 10.0000 0.581238
\(297\) 0 0
\(298\) −6.00000 −0.347571
\(299\) 12.0000 0.693978
\(300\) 0 0
\(301\) 0 0
\(302\) −5.00000 −0.287718
\(303\) 0 0
\(304\) 2.00000 0.114708
\(305\) 12.0000 0.687118
\(306\) 0 0
\(307\) 26.0000 1.48390 0.741949 − 0.670456i \(-0.233902\pi\)
0.741949 + 0.670456i \(0.233902\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −21.0000 −1.19272
\(311\) 12.0000 0.680458 0.340229 − 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 0 0
\(313\) 17.0000 0.960897 0.480448 − 0.877023i \(-0.340474\pi\)
0.480448 + 0.877023i \(0.340474\pi\)
\(314\) 4.00000 0.225733
\(315\) 0 0
\(316\) 5.00000 0.281272
\(317\) −21.0000 −1.17948 −0.589739 − 0.807594i \(-0.700769\pi\)
−0.589739 + 0.807594i \(0.700769\pi\)
\(318\) 0 0
\(319\) −27.0000 −1.51171
\(320\) −3.00000 −0.167705
\(321\) 0 0
\(322\) 0 0
\(323\) −12.0000 −0.667698
\(324\) 0 0
\(325\) 8.00000 0.443760
\(326\) 10.0000 0.553849
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 8.00000 0.439720 0.219860 − 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) −9.00000 −0.493939
\(333\) 0 0
\(334\) −18.0000 −0.984916
\(335\) −6.00000 −0.327815
\(336\) 0 0
\(337\) 5.00000 0.272367 0.136184 − 0.990684i \(-0.456516\pi\)
0.136184 + 0.990684i \(0.456516\pi\)
\(338\) 9.00000 0.489535
\(339\) 0 0
\(340\) 18.0000 0.976187
\(341\) −21.0000 −1.13721
\(342\) 0 0
\(343\) 0 0
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) 12.0000 0.644194 0.322097 − 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 0 0
\(349\) 14.0000 0.749403 0.374701 − 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −3.00000 −0.159901
\(353\) −24.0000 −1.27739 −0.638696 − 0.769460i \(-0.720526\pi\)
−0.638696 + 0.769460i \(0.720526\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) −30.0000 −1.58334 −0.791670 − 0.610949i \(-0.790788\pi\)
−0.791670 + 0.610949i \(0.790788\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) −20.0000 −1.05118
\(363\) 0 0
\(364\) 0 0
\(365\) −6.00000 −0.314054
\(366\) 0 0
\(367\) −37.0000 −1.93138 −0.965692 − 0.259690i \(-0.916380\pi\)
−0.965692 + 0.259690i \(0.916380\pi\)
\(368\) 6.00000 0.312772
\(369\) 0 0
\(370\) −30.0000 −1.55963
\(371\) 0 0
\(372\) 0 0
\(373\) −4.00000 −0.207112 −0.103556 − 0.994624i \(-0.533022\pi\)
−0.103556 + 0.994624i \(0.533022\pi\)
\(374\) 18.0000 0.930758
\(375\) 0 0
\(376\) 12.0000 0.618853
\(377\) −18.0000 −0.927047
\(378\) 0 0
\(379\) −28.0000 −1.43826 −0.719132 − 0.694874i \(-0.755460\pi\)
−0.719132 + 0.694874i \(0.755460\pi\)
\(380\) −6.00000 −0.307794
\(381\) 0 0
\(382\) 12.0000 0.613973
\(383\) −30.0000 −1.53293 −0.766464 − 0.642287i \(-0.777986\pi\)
−0.766464 + 0.642287i \(0.777986\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 7.00000 0.356291
\(387\) 0 0
\(388\) −13.0000 −0.659975
\(389\) 30.0000 1.52106 0.760530 − 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) 0 0
\(391\) −36.0000 −1.82060
\(392\) 0 0
\(393\) 0 0
\(394\) −18.0000 −0.906827
\(395\) −15.0000 −0.754732
\(396\) 0 0
\(397\) 8.00000 0.401508 0.200754 − 0.979642i \(-0.435661\pi\)
0.200754 + 0.979642i \(0.435661\pi\)
\(398\) −8.00000 −0.401004
\(399\) 0 0
\(400\) 4.00000 0.200000
\(401\) 24.0000 1.19850 0.599251 − 0.800561i \(-0.295465\pi\)
0.599251 + 0.800561i \(0.295465\pi\)
\(402\) 0 0
\(403\) −14.0000 −0.697390
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) 0 0
\(407\) −30.0000 −1.48704
\(408\) 0 0
\(409\) 11.0000 0.543915 0.271957 − 0.962309i \(-0.412329\pi\)
0.271957 + 0.962309i \(0.412329\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −16.0000 −0.788263
\(413\) 0 0
\(414\) 0 0
\(415\) 27.0000 1.32538
\(416\) −2.00000 −0.0980581
\(417\) 0 0
\(418\) −6.00000 −0.293470
\(419\) 36.0000 1.75872 0.879358 − 0.476162i \(-0.157972\pi\)
0.879358 + 0.476162i \(0.157972\pi\)
\(420\) 0 0
\(421\) 8.00000 0.389896 0.194948 − 0.980814i \(-0.437546\pi\)
0.194948 + 0.980814i \(0.437546\pi\)
\(422\) 16.0000 0.778868
\(423\) 0 0
\(424\) −3.00000 −0.145693
\(425\) −24.0000 −1.16417
\(426\) 0 0
\(427\) 0 0
\(428\) −3.00000 −0.145010
\(429\) 0 0
\(430\) −12.0000 −0.578691
\(431\) −24.0000 −1.15604 −0.578020 − 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 0 0
\(433\) 26.0000 1.24948 0.624740 − 0.780833i \(-0.285205\pi\)
0.624740 + 0.780833i \(0.285205\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −10.0000 −0.478913
\(437\) 12.0000 0.574038
\(438\) 0 0
\(439\) −19.0000 −0.906821 −0.453410 − 0.891302i \(-0.649793\pi\)
−0.453410 + 0.891302i \(0.649793\pi\)
\(440\) 9.00000 0.429058
\(441\) 0 0
\(442\) 12.0000 0.570782
\(443\) 15.0000 0.712672 0.356336 − 0.934358i \(-0.384026\pi\)
0.356336 + 0.934358i \(0.384026\pi\)
\(444\) 0 0
\(445\) −18.0000 −0.853282
\(446\) 1.00000 0.0473514
\(447\) 0 0
\(448\) 0 0
\(449\) −18.0000 −0.849473 −0.424736 − 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 15.0000 0.703985
\(455\) 0 0
\(456\) 0 0
\(457\) −13.0000 −0.608114 −0.304057 − 0.952654i \(-0.598341\pi\)
−0.304057 + 0.952654i \(0.598341\pi\)
\(458\) −20.0000 −0.934539
\(459\) 0 0
\(460\) −18.0000 −0.839254
\(461\) −18.0000 −0.838344 −0.419172 − 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) 0 0
\(463\) −4.00000 −0.185896 −0.0929479 − 0.995671i \(-0.529629\pi\)
−0.0929479 + 0.995671i \(0.529629\pi\)
\(464\) −9.00000 −0.417815
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) −12.0000 −0.555294 −0.277647 − 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −36.0000 −1.66056
\(471\) 0 0
\(472\) −3.00000 −0.138086
\(473\) −12.0000 −0.551761
\(474\) 0 0
\(475\) 8.00000 0.367065
\(476\) 0 0
\(477\) 0 0
\(478\) 18.0000 0.823301
\(479\) −24.0000 −1.09659 −0.548294 − 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) −20.0000 −0.911922
\(482\) −23.0000 −1.04762
\(483\) 0 0
\(484\) −2.00000 −0.0909091
\(485\) 39.0000 1.77090
\(486\) 0 0
\(487\) 11.0000 0.498458 0.249229 − 0.968445i \(-0.419823\pi\)
0.249229 + 0.968445i \(0.419823\pi\)
\(488\) 4.00000 0.181071
\(489\) 0 0
\(490\) 0 0
\(491\) −9.00000 −0.406164 −0.203082 − 0.979162i \(-0.565096\pi\)
−0.203082 + 0.979162i \(0.565096\pi\)
\(492\) 0 0
\(493\) 54.0000 2.43204
\(494\) −4.00000 −0.179969
\(495\) 0 0
\(496\) −7.00000 −0.314309
\(497\) 0 0
\(498\) 0 0
\(499\) 38.0000 1.70111 0.850557 − 0.525883i \(-0.176265\pi\)
0.850557 + 0.525883i \(0.176265\pi\)
\(500\) 3.00000 0.134164
\(501\) 0 0
\(502\) 9.00000 0.401690
\(503\) −18.0000 −0.802580 −0.401290 − 0.915951i \(-0.631438\pi\)
−0.401290 + 0.915951i \(0.631438\pi\)
\(504\) 0 0
\(505\) 18.0000 0.800989
\(506\) −18.0000 −0.800198
\(507\) 0 0
\(508\) −1.00000 −0.0443678
\(509\) 15.0000 0.664863 0.332432 − 0.943127i \(-0.392131\pi\)
0.332432 + 0.943127i \(0.392131\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −6.00000 −0.264649
\(515\) 48.0000 2.11513
\(516\) 0 0
\(517\) −36.0000 −1.58328
\(518\) 0 0
\(519\) 0 0
\(520\) 6.00000 0.263117
\(521\) −24.0000 −1.05146 −0.525730 − 0.850652i \(-0.676208\pi\)
−0.525730 + 0.850652i \(0.676208\pi\)
\(522\) 0 0
\(523\) 26.0000 1.13690 0.568450 − 0.822718i \(-0.307543\pi\)
0.568450 + 0.822718i \(0.307543\pi\)
\(524\) 15.0000 0.655278
\(525\) 0 0
\(526\) 24.0000 1.04645
\(527\) 42.0000 1.82955
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 9.00000 0.390935
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 9.00000 0.389104
\(536\) −2.00000 −0.0863868
\(537\) 0 0
\(538\) −3.00000 −0.129339
\(539\) 0 0
\(540\) 0 0
\(541\) −16.0000 −0.687894 −0.343947 − 0.938989i \(-0.611764\pi\)
−0.343947 + 0.938989i \(0.611764\pi\)
\(542\) 19.0000 0.816120
\(543\) 0 0
\(544\) 6.00000 0.257248
\(545\) 30.0000 1.28506
\(546\) 0 0
\(547\) 14.0000 0.598597 0.299298 − 0.954160i \(-0.403247\pi\)
0.299298 + 0.954160i \(0.403247\pi\)
\(548\) −6.00000 −0.256307
\(549\) 0 0
\(550\) −12.0000 −0.511682
\(551\) −18.0000 −0.766826
\(552\) 0 0
\(553\) 0 0
\(554\) 4.00000 0.169944
\(555\) 0 0
\(556\) 2.00000 0.0848189
\(557\) −15.0000 −0.635570 −0.317785 − 0.948163i \(-0.602939\pi\)
−0.317785 + 0.948163i \(0.602939\pi\)
\(558\) 0 0
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) −18.0000 −0.759284
\(563\) 3.00000 0.126435 0.0632175 − 0.998000i \(-0.479864\pi\)
0.0632175 + 0.998000i \(0.479864\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −20.0000 −0.840663
\(567\) 0 0
\(568\) 0 0
\(569\) −30.0000 −1.25767 −0.628833 − 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) 0 0
\(571\) 32.0000 1.33916 0.669579 − 0.742741i \(-0.266474\pi\)
0.669579 + 0.742741i \(0.266474\pi\)
\(572\) 6.00000 0.250873
\(573\) 0 0
\(574\) 0 0
\(575\) 24.0000 1.00087
\(576\) 0 0
\(577\) −1.00000 −0.0416305 −0.0208153 − 0.999783i \(-0.506626\pi\)
−0.0208153 + 0.999783i \(0.506626\pi\)
\(578\) −19.0000 −0.790296
\(579\) 0 0
\(580\) 27.0000 1.12111
\(581\) 0 0
\(582\) 0 0
\(583\) 9.00000 0.372742
\(584\) −2.00000 −0.0827606
\(585\) 0 0
\(586\) 9.00000 0.371787
\(587\) −27.0000 −1.11441 −0.557205 − 0.830375i \(-0.688126\pi\)
−0.557205 + 0.830375i \(0.688126\pi\)
\(588\) 0 0
\(589\) −14.0000 −0.576860
\(590\) 9.00000 0.370524
\(591\) 0 0
\(592\) −10.0000 −0.410997
\(593\) −30.0000 −1.23195 −0.615976 − 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 6.00000 0.245770
\(597\) 0 0
\(598\) −12.0000 −0.490716
\(599\) 30.0000 1.22577 0.612883 − 0.790173i \(-0.290010\pi\)
0.612883 + 0.790173i \(0.290010\pi\)
\(600\) 0 0
\(601\) 23.0000 0.938190 0.469095 − 0.883148i \(-0.344580\pi\)
0.469095 + 0.883148i \(0.344580\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 5.00000 0.203447
\(605\) 6.00000 0.243935
\(606\) 0 0
\(607\) −13.0000 −0.527654 −0.263827 − 0.964570i \(-0.584985\pi\)
−0.263827 + 0.964570i \(0.584985\pi\)
\(608\) −2.00000 −0.0811107
\(609\) 0 0
\(610\) −12.0000 −0.485866
\(611\) −24.0000 −0.970936
\(612\) 0 0
\(613\) 26.0000 1.05013 0.525065 − 0.851062i \(-0.324041\pi\)
0.525065 + 0.851062i \(0.324041\pi\)
\(614\) −26.0000 −1.04927
\(615\) 0 0
\(616\) 0 0
\(617\) −36.0000 −1.44931 −0.724653 − 0.689114i \(-0.758000\pi\)
−0.724653 + 0.689114i \(0.758000\pi\)
\(618\) 0 0
\(619\) −4.00000 −0.160774 −0.0803868 − 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) 21.0000 0.843380
\(621\) 0 0
\(622\) −12.0000 −0.481156
\(623\) 0 0
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) −17.0000 −0.679457
\(627\) 0 0
\(628\) −4.00000 −0.159617
\(629\) 60.0000 2.39236
\(630\) 0 0
\(631\) −37.0000 −1.47295 −0.736473 − 0.676467i \(-0.763510\pi\)
−0.736473 + 0.676467i \(0.763510\pi\)
\(632\) −5.00000 −0.198889
\(633\) 0 0
\(634\) 21.0000 0.834017
\(635\) 3.00000 0.119051
\(636\) 0 0
\(637\) 0 0
\(638\) 27.0000 1.06894
\(639\) 0 0
\(640\) 3.00000 0.118585
\(641\) 12.0000 0.473972 0.236986 − 0.971513i \(-0.423841\pi\)
0.236986 + 0.971513i \(0.423841\pi\)
\(642\) 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
\(646\) 12.0000 0.472134
\(647\) 12.0000 0.471769 0.235884 − 0.971781i \(-0.424201\pi\)
0.235884 + 0.971781i \(0.424201\pi\)
\(648\) 0 0
\(649\) 9.00000 0.353281
\(650\) −8.00000 −0.313786
\(651\) 0 0
\(652\) −10.0000 −0.391630
\(653\) −39.0000 −1.52619 −0.763094 − 0.646288i \(-0.776321\pi\)
−0.763094 + 0.646288i \(0.776321\pi\)
\(654\) 0 0
\(655\) −45.0000 −1.75830
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 36.0000 1.40236 0.701180 − 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) 0 0
\(661\) −40.0000 −1.55582 −0.777910 − 0.628376i \(-0.783720\pi\)
−0.777910 + 0.628376i \(0.783720\pi\)
\(662\) −8.00000 −0.310929
\(663\) 0 0
\(664\) 9.00000 0.349268
\(665\) 0 0
\(666\) 0 0
\(667\) −54.0000 −2.09089
\(668\) 18.0000 0.696441
\(669\) 0 0
\(670\) 6.00000 0.231800
\(671\) −12.0000 −0.463255
\(672\) 0 0
\(673\) 17.0000 0.655302 0.327651 − 0.944799i \(-0.393743\pi\)
0.327651 + 0.944799i \(0.393743\pi\)
\(674\) −5.00000 −0.192593
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 33.0000 1.26829 0.634147 − 0.773213i \(-0.281352\pi\)
0.634147 + 0.773213i \(0.281352\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −18.0000 −0.690268
\(681\) 0 0
\(682\) 21.0000 0.804132
\(683\) −51.0000 −1.95146 −0.975730 − 0.218975i \(-0.929729\pi\)
−0.975730 + 0.218975i \(0.929729\pi\)
\(684\) 0 0
\(685\) 18.0000 0.687745
\(686\) 0 0
\(687\) 0 0
\(688\) −4.00000 −0.152499
\(689\) 6.00000 0.228582
\(690\) 0 0
\(691\) 32.0000 1.21734 0.608669 − 0.793424i \(-0.291704\pi\)
0.608669 + 0.793424i \(0.291704\pi\)
\(692\) 6.00000 0.228086
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) −6.00000 −0.227593
\(696\) 0 0
\(697\) 0 0
\(698\) −14.0000 −0.529908
\(699\) 0 0
\(700\) 0 0
\(701\) −9.00000 −0.339925 −0.169963 − 0.985451i \(-0.554365\pi\)
−0.169963 + 0.985451i \(0.554365\pi\)
\(702\) 0 0
\(703\) −20.0000 −0.754314
\(704\) 3.00000 0.113067
\(705\) 0 0
\(706\) 24.0000 0.903252
\(707\) 0 0
\(708\) 0 0
\(709\) 14.0000 0.525781 0.262891 − 0.964826i \(-0.415324\pi\)
0.262891 + 0.964826i \(0.415324\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −6.00000 −0.224860
\(713\) −42.0000 −1.57291
\(714\) 0 0
\(715\) −18.0000 −0.673162
\(716\) 12.0000 0.448461
\(717\) 0 0
\(718\) 30.0000 1.11959
\(719\) 42.0000 1.56634 0.783168 − 0.621810i \(-0.213603\pi\)
0.783168 + 0.621810i \(0.213603\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 15.0000 0.558242
\(723\) 0 0
\(724\) 20.0000 0.743294
\(725\) −36.0000 −1.33701
\(726\) 0 0
\(727\) −31.0000 −1.14973 −0.574863 − 0.818250i \(-0.694945\pi\)
−0.574863 + 0.818250i \(0.694945\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 6.00000 0.222070
\(731\) 24.0000 0.887672
\(732\) 0 0
\(733\) 20.0000 0.738717 0.369358 − 0.929287i \(-0.379577\pi\)
0.369358 + 0.929287i \(0.379577\pi\)
\(734\) 37.0000 1.36569
\(735\) 0 0
\(736\) −6.00000 −0.221163
\(737\) 6.00000 0.221013
\(738\) 0 0
\(739\) 26.0000 0.956425 0.478213 − 0.878244i \(-0.341285\pi\)
0.478213 + 0.878244i \(0.341285\pi\)
\(740\) 30.0000 1.10282
\(741\) 0 0
\(742\) 0 0
\(743\) 54.0000 1.98107 0.990534 − 0.137268i \(-0.0438322\pi\)
0.990534 + 0.137268i \(0.0438322\pi\)
\(744\) 0 0
\(745\) −18.0000 −0.659469
\(746\) 4.00000 0.146450
\(747\) 0 0
\(748\) −18.0000 −0.658145
\(749\) 0 0
\(750\) 0 0
\(751\) 23.0000 0.839282 0.419641 − 0.907690i \(-0.362156\pi\)
0.419641 + 0.907690i \(0.362156\pi\)
\(752\) −12.0000 −0.437595
\(753\) 0 0
\(754\) 18.0000 0.655521
\(755\) −15.0000 −0.545906
\(756\) 0 0
\(757\) −28.0000 −1.01768 −0.508839 − 0.860862i \(-0.669925\pi\)
−0.508839 + 0.860862i \(0.669925\pi\)
\(758\) 28.0000 1.01701
\(759\) 0 0
\(760\) 6.00000 0.217643
\(761\) 42.0000 1.52250 0.761249 − 0.648459i \(-0.224586\pi\)
0.761249 + 0.648459i \(0.224586\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −12.0000 −0.434145
\(765\) 0 0
\(766\) 30.0000 1.08394
\(767\) 6.00000 0.216647
\(768\) 0 0
\(769\) 5.00000 0.180305 0.0901523 − 0.995928i \(-0.471265\pi\)
0.0901523 + 0.995928i \(0.471265\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −7.00000 −0.251936
\(773\) −6.00000 −0.215805 −0.107903 − 0.994161i \(-0.534413\pi\)
−0.107903 + 0.994161i \(0.534413\pi\)
\(774\) 0 0
\(775\) −28.0000 −1.00579
\(776\) 13.0000 0.466673
\(777\) 0 0
\(778\) −30.0000 −1.07555
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 36.0000 1.28736
\(783\) 0 0
\(784\) 0 0
\(785\) 12.0000 0.428298
\(786\) 0 0
\(787\) −16.0000 −0.570338 −0.285169 − 0.958477i \(-0.592050\pi\)
−0.285169 + 0.958477i \(0.592050\pi\)
\(788\) 18.0000 0.641223
\(789\) 0 0
\(790\) 15.0000 0.533676
\(791\) 0 0
\(792\) 0 0
\(793\) −8.00000 −0.284088
\(794\) −8.00000 −0.283909
\(795\) 0 0
\(796\) 8.00000 0.283552
\(797\) −27.0000 −0.956389 −0.478195 − 0.878254i \(-0.658709\pi\)
−0.478195 + 0.878254i \(0.658709\pi\)
\(798\) 0 0
\(799\) 72.0000 2.54718
\(800\) −4.00000 −0.141421
\(801\) 0 0
\(802\) −24.0000 −0.847469
\(803\) 6.00000 0.211735
\(804\) 0 0
\(805\) 0 0
\(806\) 14.0000 0.493129
\(807\) 0 0
\(808\) 6.00000 0.211079
\(809\) 12.0000 0.421898 0.210949 − 0.977497i \(-0.432345\pi\)
0.210949 + 0.977497i \(0.432345\pi\)
\(810\) 0 0
\(811\) 2.00000 0.0702295 0.0351147 − 0.999383i \(-0.488820\pi\)
0.0351147 + 0.999383i \(0.488820\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 30.0000 1.05150
\(815\) 30.0000 1.05085
\(816\) 0 0
\(817\) −8.00000 −0.279885
\(818\) −11.0000 −0.384606
\(819\) 0 0
\(820\) 0 0
\(821\) 15.0000 0.523504 0.261752 − 0.965135i \(-0.415700\pi\)
0.261752 + 0.965135i \(0.415700\pi\)
\(822\) 0 0
\(823\) −16.0000 −0.557725 −0.278862 − 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) 16.0000 0.557386
\(825\) 0 0
\(826\) 0 0
\(827\) −9.00000 −0.312961 −0.156480 − 0.987681i \(-0.550015\pi\)
−0.156480 + 0.987681i \(0.550015\pi\)
\(828\) 0 0
\(829\) 2.00000 0.0694629 0.0347314 − 0.999397i \(-0.488942\pi\)
0.0347314 + 0.999397i \(0.488942\pi\)
\(830\) −27.0000 −0.937184
\(831\) 0 0
\(832\) 2.00000 0.0693375
\(833\) 0 0
\(834\) 0 0
\(835\) −54.0000 −1.86875
\(836\) 6.00000 0.207514
\(837\) 0 0
\(838\) −36.0000 −1.24360
\(839\) 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
\(842\) −8.00000 −0.275698
\(843\) 0 0
\(844\) −16.0000 −0.550743
\(845\) 27.0000 0.928828
\(846\) 0 0
\(847\) 0 0
\(848\) 3.00000 0.103020
\(849\) 0 0
\(850\) 24.0000 0.823193
\(851\) −60.0000 −2.05677
\(852\) 0 0
\(853\) −46.0000 −1.57501 −0.787505 − 0.616308i \(-0.788628\pi\)
−0.787505 + 0.616308i \(0.788628\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 3.00000 0.102538
\(857\) 12.0000 0.409912 0.204956 − 0.978771i \(-0.434295\pi\)
0.204956 + 0.978771i \(0.434295\pi\)
\(858\) 0 0
\(859\) −16.0000 −0.545913 −0.272956 − 0.962026i \(-0.588002\pi\)
−0.272956 + 0.962026i \(0.588002\pi\)
\(860\) 12.0000 0.409197
\(861\) 0 0
\(862\) 24.0000 0.817443
\(863\) 6.00000 0.204242 0.102121 − 0.994772i \(-0.467437\pi\)
0.102121 + 0.994772i \(0.467437\pi\)
\(864\) 0 0
\(865\) −18.0000 −0.612018
\(866\) −26.0000 −0.883516
\(867\) 0 0
\(868\) 0 0
\(869\) 15.0000 0.508840
\(870\) 0 0
\(871\) 4.00000 0.135535
\(872\) 10.0000 0.338643
\(873\) 0 0
\(874\) −12.0000 −0.405906
\(875\) 0 0
\(876\) 0 0
\(877\) 56.0000 1.89099 0.945493 − 0.325643i \(-0.105581\pi\)
0.945493 + 0.325643i \(0.105581\pi\)
\(878\) 19.0000 0.641219
\(879\) 0 0
\(880\) −9.00000 −0.303390
\(881\) 18.0000 0.606435 0.303218 − 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) 0 0
\(883\) 20.0000 0.673054 0.336527 − 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) −15.0000 −0.503935
\(887\) 6.00000 0.201460 0.100730 − 0.994914i \(-0.467882\pi\)
0.100730 + 0.994914i \(0.467882\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 18.0000 0.603361
\(891\) 0 0
\(892\) −1.00000 −0.0334825
\(893\) −24.0000 −0.803129
\(894\) 0 0
\(895\) −36.0000 −1.20335
\(896\) 0 0
\(897\) 0 0
\(898\) 18.0000 0.600668
\(899\) 63.0000 2.10117
\(900\) 0 0
\(901\) −18.0000 −0.599667
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −60.0000 −1.99447
\(906\) 0 0
\(907\) 50.0000 1.66022 0.830111 − 0.557598i \(-0.188277\pi\)
0.830111 + 0.557598i \(0.188277\pi\)
\(908\) −15.0000 −0.497792
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) −27.0000 −0.893570
\(914\) 13.0000 0.430002
\(915\) 0 0
\(916\) 20.0000 0.660819
\(917\) 0 0
\(918\) 0 0
\(919\) −28.0000 −0.923635 −0.461817 − 0.886975i \(-0.652802\pi\)
−0.461817 + 0.886975i \(0.652802\pi\)
\(920\) 18.0000 0.593442
\(921\) 0 0
\(922\) 18.0000 0.592798
\(923\) 0 0
\(924\) 0 0
\(925\) −40.0000 −1.31519
\(926\) 4.00000 0.131448
\(927\) 0 0
\(928\) 9.00000 0.295439
\(929\) 24.0000 0.787414 0.393707 − 0.919236i \(-0.371192\pi\)
0.393707 + 0.919236i \(0.371192\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 6.00000 0.196537
\(933\) 0 0
\(934\) 12.0000 0.392652
\(935\) 54.0000 1.76599
\(936\) 0 0
\(937\) −37.0000 −1.20874 −0.604369 − 0.796705i \(-0.706575\pi\)
−0.604369 + 0.796705i \(0.706575\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 36.0000 1.17419
\(941\) −15.0000 −0.488986 −0.244493 − 0.969651i \(-0.578622\pi\)
−0.244493 + 0.969651i \(0.578622\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 3.00000 0.0976417
\(945\) 0 0
\(946\) 12.0000 0.390154
\(947\) −12.0000 −0.389948 −0.194974 − 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) 0 0
\(949\) 4.00000 0.129845
\(950\) −8.00000 −0.259554
\(951\) 0 0
\(952\) 0 0
\(953\) 36.0000 1.16615 0.583077 − 0.812417i \(-0.301849\pi\)
0.583077 + 0.812417i \(0.301849\pi\)
\(954\) 0 0
\(955\) 36.0000 1.16493
\(956\) −18.0000 −0.582162
\(957\) 0 0
\(958\) 24.0000 0.775405
\(959\) 0 0
\(960\) 0 0
\(961\) 18.0000 0.580645
\(962\) 20.0000 0.644826
\(963\) 0 0
\(964\) 23.0000 0.740780
\(965\) 21.0000 0.676014
\(966\) 0 0
\(967\) 29.0000 0.932577 0.466289 − 0.884633i \(-0.345591\pi\)
0.466289 + 0.884633i \(0.345591\pi\)
\(968\) 2.00000 0.0642824
\(969\) 0 0
\(970\) −39.0000 −1.25221
\(971\) −39.0000 −1.25157 −0.625785 − 0.779996i \(-0.715221\pi\)
−0.625785 + 0.779996i \(0.715221\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −11.0000 −0.352463
\(975\) 0 0
\(976\) −4.00000 −0.128037
\(977\) 12.0000 0.383914 0.191957 − 0.981403i \(-0.438517\pi\)
0.191957 + 0.981403i \(0.438517\pi\)
\(978\) 0 0
\(979\) 18.0000 0.575282
\(980\) 0 0
\(981\) 0 0
\(982\) 9.00000 0.287202
\(983\) −24.0000 −0.765481 −0.382741 − 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) 0 0
\(985\) −54.0000 −1.72058
\(986\) −54.0000 −1.71971
\(987\) 0 0
\(988\) 4.00000 0.127257
\(989\) −24.0000 −0.763156
\(990\) 0 0
\(991\) −19.0000 −0.603555 −0.301777 − 0.953378i \(-0.597580\pi\)
−0.301777 + 0.953378i \(0.597580\pi\)
\(992\) 7.00000 0.222250
\(993\) 0 0
\(994\) 0 0
\(995\) −24.0000 −0.760851
\(996\) 0 0
\(997\) −22.0000 −0.696747 −0.348373 − 0.937356i \(-0.613266\pi\)
−0.348373 + 0.937356i \(0.613266\pi\)
\(998\) −38.0000 −1.20287
\(999\) 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 882.2.a.a.1.1 1
3.2 odd 2 882.2.a.j.1.1 1
4.3 odd 2 7056.2.a.e.1.1 1
7.2 even 3 126.2.g.d.109.1 yes 2
7.3 odd 6 882.2.g.g.667.1 2
7.4 even 3 126.2.g.d.37.1 yes 2
7.5 odd 6 882.2.g.g.361.1 2
7.6 odd 2 882.2.a.e.1.1 1
12.11 even 2 7056.2.a.by.1.1 1
21.2 odd 6 126.2.g.a.109.1 yes 2
21.5 even 6 882.2.g.e.361.1 2
21.11 odd 6 126.2.g.a.37.1 ✓ 2
21.17 even 6 882.2.g.e.667.1 2
21.20 even 2 882.2.a.h.1.1 1
28.11 odd 6 1008.2.s.o.289.1 2
28.23 odd 6 1008.2.s.o.865.1 2
28.27 even 2 7056.2.a.bx.1.1 1
63.2 odd 6 1134.2.h.f.109.1 2
63.4 even 3 1134.2.h.j.541.1 2
63.11 odd 6 1134.2.e.k.919.1 2
63.16 even 3 1134.2.h.j.109.1 2
63.23 odd 6 1134.2.e.k.865.1 2
63.25 even 3 1134.2.e.g.919.1 2
63.32 odd 6 1134.2.h.f.541.1 2
63.58 even 3 1134.2.e.g.865.1 2
84.11 even 6 1008.2.s.b.289.1 2
84.23 even 6 1008.2.s.b.865.1 2
84.83 odd 2 7056.2.a.h.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.2.g.a.37.1 ✓ 2 21.11 odd 6
126.2.g.a.109.1 yes 2 21.2 odd 6
126.2.g.d.37.1 yes 2 7.4 even 3
126.2.g.d.109.1 yes 2 7.2 even 3
882.2.a.a.1.1 1 1.1 even 1 trivial
882.2.a.e.1.1 1 7.6 odd 2
882.2.a.h.1.1 1 21.20 even 2
882.2.a.j.1.1 1 3.2 odd 2
882.2.g.e.361.1 2 21.5 even 6
882.2.g.e.667.1 2 21.17 even 6
882.2.g.g.361.1 2 7.5 odd 6
882.2.g.g.667.1 2 7.3 odd 6
1008.2.s.b.289.1 2 84.11 even 6
1008.2.s.b.865.1 2 84.23 even 6
1008.2.s.o.289.1 2 28.11 odd 6
1008.2.s.o.865.1 2 28.23 odd 6
1134.2.e.g.865.1 2 63.58 even 3
1134.2.e.g.919.1 2 63.25 even 3
1134.2.e.k.865.1 2 63.23 odd 6
1134.2.e.k.919.1 2 63.11 odd 6
1134.2.h.f.109.1 2 63.2 odd 6
1134.2.h.f.541.1 2 63.32 odd 6
1134.2.h.j.109.1 2 63.16 even 3
1134.2.h.j.541.1 2 63.4 even 3
7056.2.a.e.1.1 1 4.3 odd 2
7056.2.a.h.1.1 1 84.83 odd 2
7056.2.a.bx.1.1 1 28.27 even 2
7056.2.a.by.1.1 1 12.11 even 2