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Properties

Label 882.2.a.j.1.1
Level $882$
Weight $2$
Character 882.1
Self dual yes
Analytic conductor $7.043$
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] = [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: \(0\)
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 \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) 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 \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 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 \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 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 \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 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 \(-0.185103\pi\)
0.835629 + 0.549294i \(0.185103\pi\)
\(30\) 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 \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) 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 \(-0.566058\pi\)
−0.206041 + 0.978543i \(0.566058\pi\)
\(54\) 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 \(-0.562563\pi\)
−0.195283 + 0.980747i \(0.562563\pi\)
\(60\) 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 \(-0.335557\pi\)
0.493939 + 0.869496i \(0.335557\pi\)
\(84\) 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 \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 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 \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 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 \(-0.453678\pi\)
0.145010 + 0.989430i \(0.453678\pi\)
\(108\) 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 \(-0.727449\pi\)
−0.655278 + 0.755388i \(0.727449\pi\)
\(132\) 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 \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 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 \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 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 \(-0.745234\pi\)
−0.696441 + 0.717614i \(0.745234\pi\)
\(168\) 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 \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 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 \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 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 \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 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 \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 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 \(-0.334144\pi\)
0.497792 + 0.867296i \(0.334144\pi\)
\(228\) 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 \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 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 \(-0.302207\pi\)
0.582162 + 0.813073i \(0.302207\pi\)
\(240\) 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 \(-0.408326\pi\)
0.284037 + 0.958813i \(0.408326\pi\)
\(252\) 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 \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 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 \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 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 \(-0.529152\pi\)
−0.0914566 + 0.995809i \(0.529152\pi\)
\(270\) 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 \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 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 \(-0.415323\pi\)
0.262893 + 0.964825i \(0.415323\pi\)
\(294\) 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 \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 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 \(-0.299231\pi\)
0.589739 + 0.807594i \(0.299231\pi\)
\(318\) 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 \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 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 \(-0.279474\pi\)
0.638696 + 0.769460i \(0.279474\pi\)
\(354\) 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 \(-0.209212\pi\)
0.791670 + 0.610949i \(0.209212\pi\)
\(360\) 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 \(-0.222014\pi\)
0.766464 + 0.642287i \(0.222014\pi\)
\(384\) 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 \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 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 \(-0.704535\pi\)
−0.599251 + 0.800561i \(0.704535\pi\)
\(402\) 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 \(-0.842028\pi\)
−0.879358 + 0.476162i \(0.842028\pi\)
\(420\) 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 \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) 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 \(-0.615974\pi\)
−0.356336 + 0.934358i \(0.615974\pi\)
\(444\) 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 \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 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 \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 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 \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 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 \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 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 \(-0.434904\pi\)
0.203082 + 0.979162i \(0.434904\pi\)
\(492\) 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 \(-0.368562\pi\)
0.401290 + 0.915951i \(0.368562\pi\)
\(504\) 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 \(-0.607869\pi\)
−0.332432 + 0.943127i \(0.607869\pi\)
\(510\) 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 \(-0.323792\pi\)
0.525730 + 0.850652i \(0.323792\pi\)
\(522\) 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 \(-0.397061\pi\)
0.317785 + 0.948163i \(0.397061\pi\)
\(558\) 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 \(-0.520136\pi\)
−0.0632175 + 0.998000i \(0.520136\pi\)
\(564\) 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 \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 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 \(-0.311874\pi\)
0.557205 + 0.830375i \(0.311874\pi\)
\(588\) 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 \(-0.288762\pi\)
0.615976 + 0.787765i \(0.288762\pi\)
\(594\) 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 \(-0.709990\pi\)
−0.612883 + 0.790173i \(0.709990\pi\)
\(600\) 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 \(-0.242000\pi\)
0.724653 + 0.689114i \(0.242000\pi\)
\(618\) 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 \(-0.576159\pi\)
−0.236986 + 0.971513i \(0.576159\pi\)
\(642\) 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 \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) 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 \(-0.223679\pi\)
0.763094 + 0.646288i \(0.223679\pi\)
\(654\) 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 \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 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 \(-0.718648\pi\)
−0.634147 + 0.773213i \(0.718648\pi\)
\(678\) 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 \(-0.0702714\pi\)
0.975730 + 0.218975i \(0.0702714\pi\)
\(684\) 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 \(-0.445635\pi\)
0.169963 + 0.985451i \(0.445635\pi\)
\(702\) 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 \(-0.786397\pi\)
−0.783168 + 0.621810i \(0.786397\pi\)
\(720\) 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 \(-0.956168\pi\)
−0.990534 + 0.137268i \(0.956168\pi\)
\(744\) 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 \(-0.775414\pi\)
−0.761249 + 0.648459i \(0.775414\pi\)
\(762\) 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 \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) 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 \(-0.341291\pi\)
0.478195 + 0.878254i \(0.341291\pi\)
\(798\) 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 \(-0.567655\pi\)
−0.210949 + 0.977497i \(0.567655\pi\)
\(810\) 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 \(-0.584300\pi\)
−0.261752 + 0.965135i \(0.584300\pi\)
\(822\) 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 \(-0.449985\pi\)
0.156480 + 0.987681i \(0.449985\pi\)
\(828\) 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 \(-0.882064\pi\)
−0.932144 + 0.362089i \(0.882064\pi\)
\(840\) 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 \(-0.565705\pi\)
−0.204956 + 0.978771i \(0.565705\pi\)
\(858\) 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 \(-0.532563\pi\)
−0.102121 + 0.994772i \(0.532563\pi\)
\(864\) 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 \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 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 \(-0.532118\pi\)
−0.100730 + 0.994914i \(0.532118\pi\)
\(888\) 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 \(-0.628808\pi\)
−0.393707 + 0.919236i \(0.628808\pi\)
\(930\) 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 \(-0.421378\pi\)
0.244493 + 0.969651i \(0.421378\pi\)
\(942\) 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 \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) 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 \(-0.698151\pi\)
−0.583077 + 0.812417i \(0.698151\pi\)
\(954\) 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 \(-0.284779\pi\)
0.625785 + 0.779996i \(0.284779\pi\)
\(972\) 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 \(-0.561483\pi\)
−0.191957 + 0.981403i \(0.561483\pi\)
\(978\) 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 \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) 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.j.1.1 1
3.2 odd 2 882.2.a.a.1.1 1
4.3 odd 2 7056.2.a.by.1.1 1
7.2 even 3 126.2.g.a.109.1 yes 2
7.3 odd 6 882.2.g.e.667.1 2
7.4 even 3 126.2.g.a.37.1 ✓ 2
7.5 odd 6 882.2.g.e.361.1 2
7.6 odd 2 882.2.a.h.1.1 1
12.11 even 2 7056.2.a.e.1.1 1
21.2 odd 6 126.2.g.d.109.1 yes 2
21.5 even 6 882.2.g.g.361.1 2
21.11 odd 6 126.2.g.d.37.1 yes 2
21.17 even 6 882.2.g.g.667.1 2
21.20 even 2 882.2.a.e.1.1 1
28.11 odd 6 1008.2.s.b.289.1 2
28.23 odd 6 1008.2.s.b.865.1 2
28.27 even 2 7056.2.a.h.1.1 1
63.2 odd 6 1134.2.h.j.109.1 2
63.4 even 3 1134.2.h.f.541.1 2
63.11 odd 6 1134.2.e.g.919.1 2
63.16 even 3 1134.2.h.f.109.1 2
63.23 odd 6 1134.2.e.g.865.1 2
63.25 even 3 1134.2.e.k.919.1 2
63.32 odd 6 1134.2.h.j.541.1 2
63.58 even 3 1134.2.e.k.865.1 2
84.11 even 6 1008.2.s.o.289.1 2
84.23 even 6 1008.2.s.o.865.1 2
84.83 odd 2 7056.2.a.bx.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.2.g.a.37.1 ✓ 2 7.4 even 3
126.2.g.a.109.1 yes 2 7.2 even 3
126.2.g.d.37.1 yes 2 21.11 odd 6
126.2.g.d.109.1 yes 2 21.2 odd 6
882.2.a.a.1.1 1 3.2 odd 2
882.2.a.e.1.1 1 21.20 even 2
882.2.a.h.1.1 1 7.6 odd 2
882.2.a.j.1.1 1 1.1 even 1 trivial
882.2.g.e.361.1 2 7.5 odd 6
882.2.g.e.667.1 2 7.3 odd 6
882.2.g.g.361.1 2 21.5 even 6
882.2.g.g.667.1 2 21.17 even 6
1008.2.s.b.289.1 2 28.11 odd 6
1008.2.s.b.865.1 2 28.23 odd 6
1008.2.s.o.289.1 2 84.11 even 6
1008.2.s.o.865.1 2 84.23 even 6
1134.2.e.g.865.1 2 63.23 odd 6
1134.2.e.g.919.1 2 63.11 odd 6
1134.2.e.k.865.1 2 63.58 even 3
1134.2.e.k.919.1 2 63.25 even 3
1134.2.h.f.109.1 2 63.16 even 3
1134.2.h.f.541.1 2 63.4 even 3
1134.2.h.j.109.1 2 63.2 odd 6
1134.2.h.j.541.1 2 63.32 odd 6
7056.2.a.e.1.1 1 12.11 even 2
7056.2.a.h.1.1 1 28.27 even 2
7056.2.a.bx.1.1 1 84.83 odd 2
7056.2.a.by.1.1 1 4.3 odd 2