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Properties

Label 2016.2.a.n.1.1
Level $2016$
Weight $2$
Character 2016.1
Self dual yes
Analytic conductor $16.098$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2016,2,Mod(1,2016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2016, 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("2016.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2016.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: \(16.0978410475\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 672)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2016.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.00000 q^{5} +1.00000 q^{7} +O(q^{10})\) \(q+4.00000 q^{5} +1.00000 q^{7} +2.00000 q^{11} -2.00000 q^{13} -4.00000 q^{19} +6.00000 q^{23} +11.0000 q^{25} +10.0000 q^{29} -8.00000 q^{31} +4.00000 q^{35} +10.0000 q^{37} +4.00000 q^{41} -8.00000 q^{43} +4.00000 q^{47} +1.00000 q^{49} -10.0000 q^{53} +8.00000 q^{55} -8.00000 q^{59} -6.00000 q^{61} -8.00000 q^{65} +4.00000 q^{67} -14.0000 q^{71} +6.00000 q^{73} +2.00000 q^{77} +4.00000 q^{79} +12.0000 q^{83} -4.00000 q^{89} -2.00000 q^{91} -16.0000 q^{95} -2.00000 q^{97} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 4.00000 1.78885 0.894427 − 0.447214i \(-0.147584\pi\)
0.894427 + 0.447214i \(0.147584\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 − 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.00000 1.25109 0.625543 − 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 0 0
\(25\) 11.0000 2.20000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 10.0000 1.85695 0.928477 − 0.371391i \(-0.121119\pi\)
0.928477 + 0.371391i \(0.121119\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 − 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 4.00000 0.676123
\(36\) 0 0
\(37\) 10.0000 1.64399 0.821995 − 0.569495i \(-0.192861\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.00000 0.624695 0.312348 − 0.949968i \(-0.398885\pi\)
0.312348 + 0.949968i \(0.398885\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 − 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.00000 0.583460 0.291730 − 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −10.0000 −1.37361 −0.686803 − 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) 0 0
\(55\) 8.00000 1.07872
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −8.00000 −1.04151 −0.520756 − 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 − 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −8.00000 −0.992278
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 − 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −14.0000 −1.66149 −0.830747 − 0.556650i \(-0.812086\pi\)
−0.830747 + 0.556650i \(0.812086\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 − 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.00000 0.227921
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 − 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 12.0000 1.31717 0.658586 − 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −4.00000 −0.423999 −0.212000 − 0.977270i \(-0.567998\pi\)
−0.212000 + 0.977270i \(0.567998\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −16.0000 −1.64157
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 10.0000 0.966736 0.483368 − 0.875417i \(-0.339413\pi\)
0.483368 + 0.875417i \(0.339413\pi\)
\(108\) 0 0
\(109\) 14.0000 1.34096 0.670478 − 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −2.00000 −0.188144 −0.0940721 − 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 0 0
\(115\) 24.0000 2.23801
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 24.0000 2.14663
\(126\) 0 0
\(127\) −20.0000 −1.77471 −0.887357 − 0.461084i \(-0.847461\pi\)
−0.887357 + 0.461084i \(0.847461\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 20.0000 1.74741 0.873704 − 0.486458i \(-0.161711\pi\)
0.873704 + 0.486458i \(0.161711\pi\)
\(132\) 0 0
\(133\) −4.00000 −0.346844
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −6.00000 −0.512615 −0.256307 − 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 − 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −4.00000 −0.334497
\(144\) 0 0
\(145\) 40.0000 3.32182
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 6.00000 0.491539 0.245770 − 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −32.0000 −2.57030
\(156\) 0 0
\(157\) −22.0000 −1.75579 −0.877896 − 0.478852i \(-0.841053\pi\)
−0.877896 + 0.478852i \(0.841053\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 6.00000 0.472866
\(162\) 0 0
\(163\) 20.0000 1.56652 0.783260 − 0.621694i \(-0.213555\pi\)
0.783260 + 0.621694i \(0.213555\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −20.0000 −1.54765 −0.773823 − 0.633402i \(-0.781658\pi\)
−0.773823 + 0.633402i \(0.781658\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −24.0000 −1.82469 −0.912343 − 0.409426i \(-0.865729\pi\)
−0.912343 + 0.409426i \(0.865729\pi\)
\(174\) 0 0
\(175\) 11.0000 0.831522
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 6.00000 0.448461 0.224231 − 0.974536i \(-0.428013\pi\)
0.224231 + 0.974536i \(0.428013\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 − 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 40.0000 2.94086
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −6.00000 −0.434145 −0.217072 − 0.976156i \(-0.569651\pi\)
−0.217072 + 0.976156i \(0.569651\pi\)
\(192\) 0 0
\(193\) 6.00000 0.431889 0.215945 − 0.976406i \(-0.430717\pi\)
0.215945 + 0.976406i \(0.430717\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −2.00000 −0.142494 −0.0712470 − 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 0 0
\(199\) −8.00000 −0.567105 −0.283552 − 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 10.0000 0.701862
\(204\) 0 0
\(205\) 16.0000 1.11749
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −8.00000 −0.553372
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 − 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −32.0000 −2.18238
\(216\) 0 0
\(217\) −8.00000 −0.543075
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −16.0000 −1.07144 −0.535720 − 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 14.0000 0.925146 0.462573 − 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 18.0000 1.17922 0.589610 − 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 0 0
\(235\) 16.0000 1.04372
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 6.00000 0.388108 0.194054 − 0.980991i \(-0.437836\pi\)
0.194054 + 0.980991i \(0.437836\pi\)
\(240\) 0 0
\(241\) −2.00000 −0.128831 −0.0644157 − 0.997923i \(-0.520518\pi\)
−0.0644157 + 0.997923i \(0.520518\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 4.00000 0.255551
\(246\) 0 0
\(247\) 8.00000 0.509028
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 24.0000 1.51487 0.757433 − 0.652913i \(-0.226453\pi\)
0.757433 + 0.652913i \(0.226453\pi\)
\(252\) 0 0
\(253\) 12.0000 0.754434
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 4.00000 0.249513 0.124757 − 0.992187i \(-0.460185\pi\)
0.124757 + 0.992187i \(0.460185\pi\)
\(258\) 0 0
\(259\) 10.0000 0.621370
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 30.0000 1.84988 0.924940 − 0.380114i \(-0.124115\pi\)
0.924940 + 0.380114i \(0.124115\pi\)
\(264\) 0 0
\(265\) −40.0000 −2.45718
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −12.0000 −0.731653 −0.365826 − 0.930683i \(-0.619214\pi\)
−0.365826 + 0.930683i \(0.619214\pi\)
\(270\) 0 0
\(271\) −24.0000 −1.45790 −0.728948 − 0.684569i \(-0.759990\pi\)
−0.728948 + 0.684569i \(0.759990\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 22.0000 1.32665
\(276\) 0 0
\(277\) 10.0000 0.600842 0.300421 − 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 − 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) −20.0000 −1.18888 −0.594438 − 0.804141i \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 4.00000 0.236113
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) −32.0000 −1.86311
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −12.0000 −0.693978
\(300\) 0 0
\(301\) −8.00000 −0.461112
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −24.0000 −1.37424
\(306\) 0 0
\(307\) −12.0000 −0.684876 −0.342438 − 0.939540i \(-0.611253\pi\)
−0.342438 + 0.939540i \(0.611253\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 12.0000 0.680458 0.340229 − 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 0 0
\(313\) −14.0000 −0.791327 −0.395663 − 0.918396i \(-0.629485\pi\)
−0.395663 + 0.918396i \(0.629485\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −18.0000 −1.01098 −0.505490 − 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 0 0
\(319\) 20.0000 1.11979
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −22.0000 −1.22034
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 4.00000 0.220527
\(330\) 0 0
\(331\) −32.0000 −1.75888 −0.879440 − 0.476011i \(-0.842082\pi\)
−0.879440 + 0.476011i \(0.842082\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 16.0000 0.874173
\(336\) 0 0
\(337\) −22.0000 −1.19842 −0.599208 − 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −16.0000 −0.866449
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −2.00000 −0.107366 −0.0536828 − 0.998558i \(-0.517096\pi\)
−0.0536828 + 0.998558i \(0.517096\pi\)
\(348\) 0 0
\(349\) −14.0000 −0.749403 −0.374701 − 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 8.00000 0.425797 0.212899 − 0.977074i \(-0.431710\pi\)
0.212899 + 0.977074i \(0.431710\pi\)
\(354\) 0 0
\(355\) −56.0000 −2.97217
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −22.0000 −1.16112 −0.580558 − 0.814219i \(-0.697165\pi\)
−0.580558 + 0.814219i \(0.697165\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 24.0000 1.25622
\(366\) 0 0
\(367\) 24.0000 1.25279 0.626395 − 0.779506i \(-0.284530\pi\)
0.626395 + 0.779506i \(0.284530\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −10.0000 −0.519174
\(372\) 0 0
\(373\) 14.0000 0.724893 0.362446 − 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −20.0000 −1.03005
\(378\) 0 0
\(379\) 32.0000 1.64373 0.821865 − 0.569683i \(-0.192934\pi\)
0.821865 + 0.569683i \(0.192934\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −8.00000 −0.408781 −0.204390 − 0.978889i \(-0.565521\pi\)
−0.204390 + 0.978889i \(0.565521\pi\)
\(384\) 0 0
\(385\) 8.00000 0.407718
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −30.0000 −1.52106 −0.760530 − 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 16.0000 0.805047
\(396\) 0 0
\(397\) −38.0000 −1.90717 −0.953583 − 0.301131i \(-0.902636\pi\)
−0.953583 + 0.301131i \(0.902636\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −22.0000 −1.09863 −0.549314 − 0.835616i \(-0.685111\pi\)
−0.549314 + 0.835616i \(0.685111\pi\)
\(402\) 0 0
\(403\) 16.0000 0.797017
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 20.0000 0.991363
\(408\) 0 0
\(409\) 22.0000 1.08783 0.543915 − 0.839140i \(-0.316941\pi\)
0.543915 + 0.839140i \(0.316941\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −8.00000 −0.393654
\(414\) 0 0
\(415\) 48.0000 2.35623
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −16.0000 −0.781651 −0.390826 − 0.920465i \(-0.627810\pi\)
−0.390826 + 0.920465i \(0.627810\pi\)
\(420\) 0 0
\(421\) −22.0000 −1.07221 −0.536107 − 0.844150i \(-0.680106\pi\)
−0.536107 + 0.844150i \(0.680106\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −6.00000 −0.290360
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 18.0000 0.867029 0.433515 − 0.901146i \(-0.357273\pi\)
0.433515 + 0.901146i \(0.357273\pi\)
\(432\) 0 0
\(433\) −14.0000 −0.672797 −0.336399 − 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −24.0000 −1.14808
\(438\) 0 0
\(439\) 8.00000 0.381819 0.190910 − 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 6.00000 0.285069 0.142534 − 0.989790i \(-0.454475\pi\)
0.142534 + 0.989790i \(0.454475\pi\)
\(444\) 0 0
\(445\) −16.0000 −0.758473
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −10.0000 −0.471929 −0.235965 − 0.971762i \(-0.575825\pi\)
−0.235965 + 0.971762i \(0.575825\pi\)
\(450\) 0 0
\(451\) 8.00000 0.376705
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −8.00000 −0.375046
\(456\) 0 0
\(457\) −22.0000 −1.02912 −0.514558 − 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) −20.0000 −0.929479 −0.464739 − 0.885448i \(-0.653852\pi\)
−0.464739 + 0.885448i \(0.653852\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 8.00000 0.370196 0.185098 − 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) 0 0
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −16.0000 −0.735681
\(474\) 0 0
\(475\) −44.0000 −2.01886
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 12.0000 0.548294 0.274147 − 0.961688i \(-0.411605\pi\)
0.274147 + 0.961688i \(0.411605\pi\)
\(480\) 0 0
\(481\) −20.0000 −0.911922
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −8.00000 −0.363261
\(486\) 0 0
\(487\) −40.0000 −1.81257 −0.906287 − 0.422664i \(-0.861095\pi\)
−0.906287 + 0.422664i \(0.861095\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −10.0000 −0.451294 −0.225647 − 0.974209i \(-0.572450\pi\)
−0.225647 + 0.974209i \(0.572450\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −14.0000 −0.627986
\(498\) 0 0
\(499\) 24.0000 1.07439 0.537194 − 0.843459i \(-0.319484\pi\)
0.537194 + 0.843459i \(0.319484\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −24.0000 −1.07011 −0.535054 − 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 36.0000 1.59567 0.797836 − 0.602875i \(-0.205978\pi\)
0.797836 + 0.602875i \(0.205978\pi\)
\(510\) 0 0
\(511\) 6.00000 0.265424
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 8.00000 0.351840
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) −36.0000 −1.57417 −0.787085 − 0.616844i \(-0.788411\pi\)
−0.787085 + 0.616844i \(0.788411\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −8.00000 −0.346518
\(534\) 0 0
\(535\) 40.0000 1.72935
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 2.00000 0.0861461
\(540\) 0 0
\(541\) −14.0000 −0.601907 −0.300954 − 0.953639i \(-0.597305\pi\)
−0.300954 + 0.953639i \(0.597305\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 56.0000 2.39878
\(546\) 0 0
\(547\) 12.0000 0.513083 0.256541 − 0.966533i \(-0.417417\pi\)
0.256541 + 0.966533i \(0.417417\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −40.0000 −1.70406
\(552\) 0 0
\(553\) 4.00000 0.170097
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 22.0000 0.932170 0.466085 − 0.884740i \(-0.345664\pi\)
0.466085 + 0.884740i \(0.345664\pi\)
\(558\) 0 0
\(559\) 16.0000 0.676728
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 24.0000 1.01148 0.505740 − 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) 0 0
\(565\) −8.00000 −0.336563
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −14.0000 −0.586911 −0.293455 − 0.955973i \(-0.594805\pi\)
−0.293455 + 0.955973i \(0.594805\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 − 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 66.0000 2.75239
\(576\) 0 0
\(577\) −6.00000 −0.249783 −0.124892 − 0.992170i \(-0.539858\pi\)
−0.124892 + 0.992170i \(0.539858\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 12.0000 0.497844
\(582\) 0 0
\(583\) −20.0000 −0.828315
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −48.0000 −1.98117 −0.990586 − 0.136892i \(-0.956289\pi\)
−0.990586 + 0.136892i \(0.956289\pi\)
\(588\) 0 0
\(589\) 32.0000 1.31854
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 24.0000 0.985562 0.492781 − 0.870153i \(-0.335980\pi\)
0.492781 + 0.870153i \(0.335980\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 10.0000 0.408589 0.204294 − 0.978909i \(-0.434510\pi\)
0.204294 + 0.978909i \(0.434510\pi\)
\(600\) 0 0
\(601\) 34.0000 1.38689 0.693444 − 0.720510i \(-0.256092\pi\)
0.693444 + 0.720510i \(0.256092\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −28.0000 −1.13836
\(606\) 0 0
\(607\) −8.00000 −0.324710 −0.162355 − 0.986732i \(-0.551909\pi\)
−0.162355 + 0.986732i \(0.551909\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −8.00000 −0.323645
\(612\) 0 0
\(613\) 6.00000 0.242338 0.121169 − 0.992632i \(-0.461336\pi\)
0.121169 + 0.992632i \(0.461336\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 18.0000 0.724653 0.362326 − 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) 0 0
\(619\) 36.0000 1.44696 0.723481 − 0.690344i \(-0.242541\pi\)
0.723481 + 0.690344i \(0.242541\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −4.00000 −0.160257
\(624\) 0 0
\(625\) 41.0000 1.64000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 4.00000 0.159237 0.0796187 − 0.996825i \(-0.474630\pi\)
0.0796187 + 0.996825i \(0.474630\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −80.0000 −3.17470
\(636\) 0 0
\(637\) −2.00000 −0.0792429
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 2.00000 0.0789953 0.0394976 − 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) 0 0
\(643\) −12.0000 −0.473234 −0.236617 − 0.971603i \(-0.576039\pi\)
−0.236617 + 0.971603i \(0.576039\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −12.0000 −0.471769 −0.235884 − 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) 0 0
\(649\) −16.0000 −0.628055
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 26.0000 1.01746 0.508729 − 0.860927i \(-0.330115\pi\)
0.508729 + 0.860927i \(0.330115\pi\)
\(654\) 0 0
\(655\) 80.0000 3.12586
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −6.00000 −0.233727 −0.116863 − 0.993148i \(-0.537284\pi\)
−0.116863 + 0.993148i \(0.537284\pi\)
\(660\) 0 0
\(661\) −22.0000 −0.855701 −0.427850 − 0.903850i \(-0.640729\pi\)
−0.427850 + 0.903850i \(0.640729\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −16.0000 −0.620453
\(666\) 0 0
\(667\) 60.0000 2.32321
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −12.0000 −0.463255
\(672\) 0 0
\(673\) −34.0000 −1.31060 −0.655302 − 0.755367i \(-0.727459\pi\)
−0.655302 + 0.755367i \(0.727459\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −16.0000 −0.614930 −0.307465 − 0.951559i \(-0.599481\pi\)
−0.307465 + 0.951559i \(0.599481\pi\)
\(678\) 0 0
\(679\) −2.00000 −0.0767530
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −18.0000 −0.688751 −0.344375 − 0.938832i \(-0.611909\pi\)
−0.344375 + 0.938832i \(0.611909\pi\)
\(684\) 0 0
\(685\) −24.0000 −0.916993
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 20.0000 0.761939
\(690\) 0 0
\(691\) −28.0000 −1.06517 −0.532585 − 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 16.0000 0.606915
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 18.0000 0.679851 0.339925 − 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) 0 0
\(703\) −40.0000 −1.50863
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −26.0000 −0.976450 −0.488225 − 0.872718i \(-0.662356\pi\)
−0.488225 + 0.872718i \(0.662356\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −48.0000 −1.79761
\(714\) 0 0
\(715\) −16.0000 −0.598366
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −32.0000 −1.19340 −0.596699 − 0.802465i \(-0.703521\pi\)
−0.596699 + 0.802465i \(0.703521\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 110.000 4.08530
\(726\) 0 0
\(727\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 38.0000 1.40356 0.701781 − 0.712393i \(-0.252388\pi\)
0.701781 + 0.712393i \(0.252388\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 8.00000 0.294684
\(738\) 0 0
\(739\) 20.0000 0.735712 0.367856 − 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −6.00000 −0.220119 −0.110059 − 0.993925i \(-0.535104\pi\)
−0.110059 + 0.993925i \(0.535104\pi\)
\(744\) 0 0
\(745\) 24.0000 0.879292
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 10.0000 0.365392
\(750\) 0 0
\(751\) 16.0000 0.583848 0.291924 − 0.956441i \(-0.405705\pi\)
0.291924 + 0.956441i \(0.405705\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −42.0000 −1.52652 −0.763258 − 0.646094i \(-0.776401\pi\)
−0.763258 + 0.646094i \(0.776401\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −12.0000 −0.435000 −0.217500 − 0.976060i \(-0.569790\pi\)
−0.217500 + 0.976060i \(0.569790\pi\)
\(762\) 0 0
\(763\) 14.0000 0.506834
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 16.0000 0.577727
\(768\) 0 0
\(769\) 34.0000 1.22607 0.613036 − 0.790055i \(-0.289948\pi\)
0.613036 + 0.790055i \(0.289948\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −24.0000 −0.863220 −0.431610 − 0.902060i \(-0.642054\pi\)
−0.431610 + 0.902060i \(0.642054\pi\)
\(774\) 0 0
\(775\) −88.0000 −3.16105
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −16.0000 −0.573259
\(780\) 0 0
\(781\) −28.0000 −1.00192
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −88.0000 −3.14085
\(786\) 0 0
\(787\) 36.0000 1.28326 0.641631 − 0.767014i \(-0.278258\pi\)
0.641631 + 0.767014i \(0.278258\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −2.00000 −0.0711118
\(792\) 0 0
\(793\) 12.0000 0.426132
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −24.0000 −0.850124 −0.425062 − 0.905164i \(-0.639748\pi\)
−0.425062 + 0.905164i \(0.639748\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 12.0000 0.423471
\(804\) 0 0
\(805\) 24.0000 0.845889
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 14.0000 0.492214 0.246107 − 0.969243i \(-0.420849\pi\)
0.246107 + 0.969243i \(0.420849\pi\)
\(810\) 0 0
\(811\) 20.0000 0.702295 0.351147 − 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 80.0000 2.80228
\(816\) 0 0
\(817\) 32.0000 1.11954
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −2.00000 −0.0698005 −0.0349002 − 0.999391i \(-0.511111\pi\)
−0.0349002 + 0.999391i \(0.511111\pi\)
\(822\) 0 0
\(823\) −4.00000 −0.139431 −0.0697156 − 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 26.0000 0.904109 0.452054 − 0.891990i \(-0.350691\pi\)
0.452054 + 0.891990i \(0.350691\pi\)
\(828\) 0 0
\(829\) 14.0000 0.486240 0.243120 − 0.969996i \(-0.421829\pi\)
0.243120 + 0.969996i \(0.421829\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −80.0000 −2.76851
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −44.0000 −1.51905 −0.759524 − 0.650479i \(-0.774568\pi\)
−0.759524 + 0.650479i \(0.774568\pi\)
\(840\) 0 0
\(841\) 71.0000 2.44828
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −36.0000 −1.23844
\(846\) 0 0
\(847\) −7.00000 −0.240523
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 60.0000 2.05677
\(852\) 0 0
\(853\) 42.0000 1.43805 0.719026 − 0.694983i \(-0.244588\pi\)
0.719026 + 0.694983i \(0.244588\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −28.0000 −0.956462 −0.478231 − 0.878234i \(-0.658722\pi\)
−0.478231 + 0.878234i \(0.658722\pi\)
\(858\) 0 0
\(859\) −20.0000 −0.682391 −0.341196 − 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 22.0000 0.748889 0.374444 − 0.927249i \(-0.377833\pi\)
0.374444 + 0.927249i \(0.377833\pi\)
\(864\) 0 0
\(865\) −96.0000 −3.26410
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 8.00000 0.271381
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 24.0000 0.811348
\(876\) 0 0
\(877\) −50.0000 −1.68838 −0.844190 − 0.536044i \(-0.819918\pi\)
−0.844190 + 0.536044i \(0.819918\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −48.0000 −1.61716 −0.808581 − 0.588386i \(-0.799764\pi\)
−0.808581 + 0.588386i \(0.799764\pi\)
\(882\) 0 0
\(883\) −8.00000 −0.269221 −0.134611 − 0.990899i \(-0.542978\pi\)
−0.134611 + 0.990899i \(0.542978\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 12.0000 0.402921 0.201460 − 0.979497i \(-0.435431\pi\)
0.201460 + 0.979497i \(0.435431\pi\)
\(888\) 0 0
\(889\) −20.0000 −0.670778
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −16.0000 −0.535420
\(894\) 0 0
\(895\) 24.0000 0.802232
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −80.0000 −2.66815
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 56.0000 1.86150
\(906\) 0 0
\(907\) 32.0000 1.06254 0.531271 − 0.847202i \(-0.321714\pi\)
0.531271 + 0.847202i \(0.321714\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 18.0000 0.596367 0.298183 − 0.954509i \(-0.403619\pi\)
0.298183 + 0.954509i \(0.403619\pi\)
\(912\) 0 0
\(913\) 24.0000 0.794284
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 20.0000 0.660458
\(918\) 0 0
\(919\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 28.0000 0.921631
\(924\) 0 0
\(925\) 110.000 3.61678
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 36.0000 1.18112 0.590561 − 0.806993i \(-0.298907\pi\)
0.590561 + 0.806993i \(0.298907\pi\)
\(930\) 0 0
\(931\) −4.00000 −0.131095
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 2.00000 0.0653372 0.0326686 − 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −4.00000 −0.130396 −0.0651981 − 0.997872i \(-0.520768\pi\)
−0.0651981 + 0.997872i \(0.520768\pi\)
\(942\) 0 0
\(943\) 24.0000 0.781548
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 18.0000 0.584921 0.292461 − 0.956278i \(-0.405526\pi\)
0.292461 + 0.956278i \(0.405526\pi\)
\(948\) 0 0
\(949\) −12.0000 −0.389536
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 6.00000 0.194359 0.0971795 − 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) 0 0
\(955\) −24.0000 −0.776622
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −6.00000 −0.193750
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 24.0000 0.772587
\(966\) 0 0
\(967\) 12.0000 0.385894 0.192947 − 0.981209i \(-0.438195\pi\)
0.192947 + 0.981209i \(0.438195\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 20.0000 0.641831 0.320915 − 0.947108i \(-0.396010\pi\)
0.320915 + 0.947108i \(0.396010\pi\)
\(972\) 0 0
\(973\) 4.00000 0.128234
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −14.0000 −0.447900 −0.223950 − 0.974601i \(-0.571895\pi\)
−0.223950 + 0.974601i \(0.571895\pi\)
\(978\) 0 0
\(979\) −8.00000 −0.255681
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) −8.00000 −0.254901
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −48.0000 −1.52631
\(990\) 0 0
\(991\) 16.0000 0.508257 0.254128 − 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −32.0000 −1.01447
\(996\) 0 0
\(997\) −6.00000 −0.190022 −0.0950110 − 0.995476i \(-0.530289\pi\)
−0.0950110 + 0.995476i \(0.530289\pi\)
\(998\) 0 0
\(999\) 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 2016.2.a.n.1.1 1
3.2 odd 2 672.2.a.e.1.1 yes 1
4.3 odd 2 2016.2.a.m.1.1 1
8.3 odd 2 4032.2.a.b.1.1 1
8.5 even 2 4032.2.a.c.1.1 1
12.11 even 2 672.2.a.a.1.1 ✓ 1
21.20 even 2 4704.2.a.q.1.1 1
24.5 odd 2 1344.2.a.j.1.1 1
24.11 even 2 1344.2.a.t.1.1 1
48.5 odd 4 5376.2.c.b.2689.1 2
48.11 even 4 5376.2.c.bf.2689.2 2
48.29 odd 4 5376.2.c.b.2689.2 2
48.35 even 4 5376.2.c.bf.2689.1 2
84.83 odd 2 4704.2.a.bg.1.1 1
168.83 odd 2 9408.2.a.c.1.1 1
168.125 even 2 9408.2.a.bq.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
672.2.a.a.1.1 ✓ 1 12.11 even 2
672.2.a.e.1.1 yes 1 3.2 odd 2
1344.2.a.j.1.1 1 24.5 odd 2
1344.2.a.t.1.1 1 24.11 even 2
2016.2.a.m.1.1 1 4.3 odd 2
2016.2.a.n.1.1 1 1.1 even 1 trivial
4032.2.a.b.1.1 1 8.3 odd 2
4032.2.a.c.1.1 1 8.5 even 2
4704.2.a.q.1.1 1 21.20 even 2
4704.2.a.bg.1.1 1 84.83 odd 2
5376.2.c.b.2689.1 2 48.5 odd 4
5376.2.c.b.2689.2 2 48.29 odd 4
5376.2.c.bf.2689.1 2 48.35 even 4
5376.2.c.bf.2689.2 2 48.11 even 4
9408.2.a.c.1.1 1 168.83 odd 2
9408.2.a.bq.1.1 1 168.125 even 2