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Properties

Label 3528.2.a.x.1.1
Level $3528$
Weight $2$
Character 3528.1
Self dual yes
Analytic conductor $28.171$
Analytic rank $1$
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] = [3528,2,Mod(1,3528)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3528, 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("3528.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3528.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: \(28.1712218331\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 56)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3528.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000 q^{5} +O(q^{10})\) \(q+2.00000 q^{5} +4.00000 q^{11} -2.00000 q^{13} -6.00000 q^{17} -8.00000 q^{19} -1.00000 q^{25} -6.00000 q^{29} -8.00000 q^{31} -2.00000 q^{37} +2.00000 q^{41} -4.00000 q^{43} -8.00000 q^{47} -6.00000 q^{53} +8.00000 q^{55} +6.00000 q^{61} -4.00000 q^{65} -4.00000 q^{67} +8.00000 q^{71} -10.0000 q^{73} +16.0000 q^{79} +8.00000 q^{83} -12.0000 q^{85} -6.00000 q^{89} -16.0000 q^{95} +6.00000 q^{97} +O(q^{100})\)

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
\(5\) 2.00000 0.894427 0.447214 − 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 − 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 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
\(17\) −6.00000 −1.45521 −0.727607 − 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) −8.00000 −1.83533 −0.917663 − 0.397360i \(-0.869927\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 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
\(35\) 0 0
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 − 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 − 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 − 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 8.00000 1.07872
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 − 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 − 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) −10.0000 −1.17041 −0.585206 − 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 16.0000 1.80014 0.900070 − 0.435745i \(-0.143515\pi\)
0.900070 + 0.435745i \(0.143515\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 8.00000 0.878114 0.439057 − 0.898459i \(-0.355313\pi\)
0.439057 + 0.898459i \(0.355313\pi\)
\(84\) 0 0
\(85\) −12.0000 −1.30158
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −16.0000 −1.64157
\(96\) 0 0
\(97\) 6.00000 0.609208 0.304604 − 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 2.00000 0.199007 0.0995037 − 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) 0 0
\(103\) 16.0000 1.57653 0.788263 − 0.615338i \(-0.210980\pi\)
0.788263 + 0.615338i \(0.210980\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 12.0000 1.16008 0.580042 − 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.957826 −0.478913 − 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 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
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 − 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 8.00000 0.698963 0.349482 − 0.936943i \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 6.00000 0.512615 0.256307 − 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) 8.00000 0.678551 0.339276 − 0.940687i \(-0.389818\pi\)
0.339276 + 0.940687i \(0.389818\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −8.00000 −0.668994
\(144\) 0 0
\(145\) −12.0000 −0.996546
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −6.00000 −0.491539 −0.245770 − 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −16.0000 −1.28515
\(156\) 0 0
\(157\) −18.0000 −1.43656 −0.718278 − 0.695756i \(-0.755069\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −12.0000 −0.939913 −0.469956 − 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 16.0000 1.23812 0.619059 − 0.785345i \(-0.287514\pi\)
0.619059 + 0.785345i \(0.287514\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 18.0000 1.36851 0.684257 − 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 4.00000 0.298974 0.149487 − 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 − 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −4.00000 −0.294086
\(186\) 0 0
\(187\) −24.0000 −1.75505
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 16.0000 1.15772 0.578860 − 0.815427i \(-0.303498\pi\)
0.578860 + 0.815427i \(0.303498\pi\)
\(192\) 0 0
\(193\) −14.0000 −1.00774 −0.503871 − 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −6.00000 −0.427482 −0.213741 − 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 0 0
\(199\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 4.00000 0.279372
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −32.0000 −2.21349
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 − 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −8.00000 −0.545595
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) 0 0
\(223\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −8.00000 −0.530979 −0.265489 − 0.964114i \(-0.585534\pi\)
−0.265489 + 0.964114i \(0.585534\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 − 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 22.0000 1.44127 0.720634 − 0.693316i \(-0.243851\pi\)
0.720634 + 0.693316i \(0.243851\pi\)
\(234\) 0 0
\(235\) −16.0000 −1.04372
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −8.00000 −0.517477 −0.258738 − 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 − 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 16.0000 1.01806
\(248\) 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
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2.00000 0.124757 0.0623783 − 0.998053i \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −24.0000 −1.47990 −0.739952 − 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 0 0
\(265\) −12.0000 −0.737154
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −14.0000 −0.853595 −0.426798 − 0.904347i \(-0.640358\pi\)
−0.426798 + 0.904347i \(0.640358\pi\)
\(270\) 0 0
\(271\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −4.00000 −0.241209
\(276\) 0 0
\(277\) 22.0000 1.32185 0.660926 − 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 − 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) −16.0000 −0.951101 −0.475551 − 0.879688i \(-0.657751\pi\)
−0.475551 + 0.879688i \(0.657751\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 18.0000 1.05157 0.525786 − 0.850617i \(-0.323771\pi\)
0.525786 + 0.850617i \(0.323771\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 12.0000 0.687118
\(306\) 0 0
\(307\) 8.00000 0.456584 0.228292 − 0.973593i \(-0.426686\pi\)
0.228292 + 0.973593i \(0.426686\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 8.00000 0.453638 0.226819 − 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) 0 0
\(313\) 14.0000 0.791327 0.395663 − 0.918396i \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −30.0000 −1.68497 −0.842484 − 0.538721i \(-0.818908\pi\)
−0.842484 + 0.538721i \(0.818908\pi\)
\(318\) 0 0
\(319\) −24.0000 −1.34374
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 48.0000 2.67079
\(324\) 0 0
\(325\) 2.00000 0.110940
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 − 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −8.00000 −0.437087
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 − 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −32.0000 −1.73290
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −12.0000 −0.644194 −0.322097 − 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 0 0
\(349\) 30.0000 1.60586 0.802932 − 0.596071i \(-0.203272\pi\)
0.802932 + 0.596071i \(0.203272\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −14.0000 −0.745145 −0.372572 − 0.928003i \(-0.621524\pi\)
−0.372572 + 0.928003i \(0.621524\pi\)
\(354\) 0 0
\(355\) 16.0000 0.849192
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −20.0000 −1.04685
\(366\) 0 0
\(367\) −16.0000 −0.835193 −0.417597 − 0.908633i \(-0.637127\pi\)
−0.417597 + 0.908633i \(0.637127\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −26.0000 −1.34623 −0.673114 − 0.739538i \(-0.735044\pi\)
−0.673114 + 0.739538i \(0.735044\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) −4.00000 −0.205466 −0.102733 − 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −24.0000 −1.22634 −0.613171 − 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 2.00000 0.101404 0.0507020 − 0.998714i \(-0.483854\pi\)
0.0507020 + 0.998714i \(0.483854\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 32.0000 1.61009
\(396\) 0 0
\(397\) 14.0000 0.702640 0.351320 − 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −18.0000 −0.898877 −0.449439 − 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 0 0
\(403\) 16.0000 0.797017
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −8.00000 −0.396545
\(408\) 0 0
\(409\) 14.0000 0.692255 0.346128 − 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 16.0000 0.785409
\(416\) 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
\(421\) −26.0000 −1.26716 −0.633581 − 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 6.00000 0.291043
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 8.00000 0.385346 0.192673 − 0.981263i \(-0.438284\pi\)
0.192673 + 0.981263i \(0.438284\pi\)
\(432\) 0 0
\(433\) −10.0000 −0.480569 −0.240285 − 0.970702i \(-0.577241\pi\)
−0.240285 + 0.970702i \(0.577241\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −24.0000 −1.14546 −0.572729 − 0.819745i \(-0.694115\pi\)
−0.572729 + 0.819745i \(0.694115\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −36.0000 −1.71041 −0.855206 − 0.518289i \(-0.826569\pi\)
−0.855206 + 0.518289i \(0.826569\pi\)
\(444\) 0 0
\(445\) −12.0000 −0.568855
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −34.0000 −1.60456 −0.802280 − 0.596948i \(-0.796380\pi\)
−0.802280 + 0.596948i \(0.796380\pi\)
\(450\) 0 0
\(451\) 8.00000 0.376705
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −38.0000 −1.77757 −0.888783 − 0.458329i \(-0.848448\pi\)
−0.888783 + 0.458329i \(0.848448\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −30.0000 −1.39724 −0.698620 − 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 0 0
\(463\) −16.0000 −0.743583 −0.371792 − 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −8.00000 −0.370196 −0.185098 − 0.982720i \(-0.559260\pi\)
−0.185098 + 0.982720i \(0.559260\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −16.0000 −0.735681
\(474\) 0 0
\(475\) 8.00000 0.367065
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −24.0000 −1.09659 −0.548294 − 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) 4.00000 0.182384
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 12.0000 0.544892
\(486\) 0 0
\(487\) −16.0000 −0.725029 −0.362515 − 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 − 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) 36.0000 1.62136
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −4.00000 −0.179065 −0.0895323 − 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 40.0000 1.78351 0.891756 − 0.452517i \(-0.149474\pi\)
0.891756 + 0.452517i \(0.149474\pi\)
\(504\) 0 0
\(505\) 4.00000 0.177998
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 34.0000 1.50702 0.753512 − 0.657434i \(-0.228358\pi\)
0.753512 + 0.657434i \(0.228358\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 32.0000 1.41009
\(516\) 0 0
\(517\) −32.0000 −1.40736
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 18.0000 0.788594 0.394297 − 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) 0 0
\(523\) 32.0000 1.39926 0.699631 − 0.714504i \(-0.253348\pi\)
0.699631 + 0.714504i \(0.253348\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 48.0000 2.09091
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −4.00000 −0.173259
\(534\) 0 0
\(535\) 24.0000 1.03761
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 14.0000 0.601907 0.300954 − 0.953639i \(-0.402695\pi\)
0.300954 + 0.953639i \(0.402695\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −20.0000 −0.856706
\(546\) 0 0
\(547\) 36.0000 1.53925 0.769624 − 0.638497i \(-0.220443\pi\)
0.769624 + 0.638497i \(0.220443\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 48.0000 2.04487
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −14.0000 −0.593199 −0.296600 − 0.955002i \(-0.595853\pi\)
−0.296600 + 0.955002i \(0.595853\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 32.0000 1.34864 0.674320 − 0.738440i \(-0.264437\pi\)
0.674320 + 0.738440i \(0.264437\pi\)
\(564\) 0 0
\(565\) −4.00000 −0.168281
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −26.0000 −1.08998 −0.544988 − 0.838444i \(-0.683466\pi\)
−0.544988 + 0.838444i \(0.683466\pi\)
\(570\) 0 0
\(571\) 28.0000 1.17176 0.585882 − 0.810397i \(-0.300748\pi\)
0.585882 + 0.810397i \(0.300748\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 14.0000 0.582828 0.291414 − 0.956597i \(-0.405874\pi\)
0.291414 + 0.956597i \(0.405874\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −24.0000 −0.993978
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −24.0000 −0.990586 −0.495293 − 0.868726i \(-0.664939\pi\)
−0.495293 + 0.868726i \(0.664939\pi\)
\(588\) 0 0
\(589\) 64.0000 2.63707
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 34.0000 1.39621 0.698106 − 0.715994i \(-0.254026\pi\)
0.698106 + 0.715994i \(0.254026\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 24.0000 0.980613 0.490307 − 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) −26.0000 −1.06056 −0.530281 − 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 10.0000 0.406558
\(606\) 0 0
\(607\) −32.0000 −1.29884 −0.649420 − 0.760430i \(-0.724988\pi\)
−0.649420 + 0.760430i \(0.724988\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 16.0000 0.647291
\(612\) 0 0
\(613\) −18.0000 −0.727013 −0.363507 − 0.931592i \(-0.618421\pi\)
−0.363507 + 0.931592i \(0.618421\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 38.0000 1.52982 0.764911 − 0.644136i \(-0.222783\pi\)
0.764911 + 0.644136i \(0.222783\pi\)
\(618\) 0 0
\(619\) 32.0000 1.28619 0.643094 − 0.765787i \(-0.277650\pi\)
0.643094 + 0.765787i \(0.277650\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 12.0000 0.478471
\(630\) 0 0
\(631\) 24.0000 0.955425 0.477712 − 0.878516i \(-0.341466\pi\)
0.477712 + 0.878516i \(0.341466\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −16.0000 −0.634941
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 30.0000 1.18493 0.592464 − 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) 0 0
\(643\) −16.0000 −0.630978 −0.315489 − 0.948929i \(-0.602169\pi\)
−0.315489 + 0.948929i \(0.602169\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −32.0000 −1.25805 −0.629025 − 0.777385i \(-0.716546\pi\)
−0.629025 + 0.777385i \(0.716546\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 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
\(655\) 16.0000 0.625172
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 12.0000 0.467454 0.233727 − 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 0 0
\(661\) −2.00000 −0.0777910 −0.0388955 − 0.999243i \(-0.512384\pi\)
−0.0388955 + 0.999243i \(0.512384\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 24.0000 0.926510
\(672\) 0 0
\(673\) −14.0000 −0.539660 −0.269830 − 0.962908i \(-0.586968\pi\)
−0.269830 + 0.962908i \(0.586968\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −6.00000 −0.230599 −0.115299 − 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −36.0000 −1.37750 −0.688751 − 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) 0 0
\(685\) 12.0000 0.458496
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) 8.00000 0.304334 0.152167 − 0.988355i \(-0.451375\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 16.0000 0.606915
\(696\) 0 0
\(697\) −12.0000 −0.454532
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 42.0000 1.58632 0.793159 − 0.609015i \(-0.208435\pi\)
0.793159 + 0.609015i \(0.208435\pi\)
\(702\) 0 0
\(703\) 16.0000 0.603451
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 30.0000 1.12667 0.563337 − 0.826227i \(-0.309517\pi\)
0.563337 + 0.826227i \(0.309517\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) −16.0000 −0.598366
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 8.00000 0.298350 0.149175 − 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 6.00000 0.222834
\(726\) 0 0
\(727\) −16.0000 −0.593407 −0.296704 − 0.954970i \(-0.595887\pi\)
−0.296704 + 0.954970i \(0.595887\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 24.0000 0.887672
\(732\) 0 0
\(733\) −26.0000 −0.960332 −0.480166 − 0.877178i \(-0.659424\pi\)
−0.480166 + 0.877178i \(0.659424\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −16.0000 −0.589368
\(738\) 0 0
\(739\) 52.0000 1.91285 0.956425 − 0.291977i \(-0.0943129\pi\)
0.956425 + 0.291977i \(0.0943129\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −32.0000 −1.17397 −0.586983 − 0.809599i \(-0.699684\pi\)
−0.586983 + 0.809599i \(0.699684\pi\)
\(744\) 0 0
\(745\) −12.0000 −0.439646
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −8.00000 −0.291924 −0.145962 − 0.989290i \(-0.546628\pi\)
−0.145962 + 0.989290i \(0.546628\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −2.00000 −0.0726912 −0.0363456 − 0.999339i \(-0.511572\pi\)
−0.0363456 + 0.999339i \(0.511572\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −30.0000 −1.08750 −0.543750 − 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 22.0000 0.793340 0.396670 − 0.917961i \(-0.370166\pi\)
0.396670 + 0.917961i \(0.370166\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 50.0000 1.79838 0.899188 − 0.437564i \(-0.144158\pi\)
0.899188 + 0.437564i \(0.144158\pi\)
\(774\) 0 0
\(775\) 8.00000 0.287368
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −16.0000 −0.573259
\(780\) 0 0
\(781\) 32.0000 1.14505
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −36.0000 −1.28490
\(786\) 0 0
\(787\) 40.0000 1.42585 0.712923 − 0.701242i \(-0.247371\pi\)
0.712923 + 0.701242i \(0.247371\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −12.0000 −0.426132
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −30.0000 −1.06265 −0.531327 − 0.847167i \(-0.678307\pi\)
−0.531327 + 0.847167i \(0.678307\pi\)
\(798\) 0 0
\(799\) 48.0000 1.69812
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −40.0000 −1.41157
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −26.0000 −0.914111 −0.457056 − 0.889438i \(-0.651096\pi\)
−0.457056 + 0.889438i \(0.651096\pi\)
\(810\) 0 0
\(811\) −16.0000 −0.561836 −0.280918 − 0.959732i \(-0.590639\pi\)
−0.280918 + 0.959732i \(0.590639\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −24.0000 −0.840683
\(816\) 0 0
\(817\) 32.0000 1.11954
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 10.0000 0.349002 0.174501 − 0.984657i \(-0.444169\pi\)
0.174501 + 0.984657i \(0.444169\pi\)
\(822\) 0 0
\(823\) 24.0000 0.836587 0.418294 − 0.908312i \(-0.362628\pi\)
0.418294 + 0.908312i \(0.362628\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 28.0000 0.973655 0.486828 − 0.873498i \(-0.338154\pi\)
0.486828 + 0.873498i \(0.338154\pi\)
\(828\) 0 0
\(829\) −26.0000 −0.903017 −0.451509 − 0.892267i \(-0.649114\pi\)
−0.451509 + 0.892267i \(0.649114\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 32.0000 1.10741
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −48.0000 −1.65714 −0.828572 − 0.559883i \(-0.810846\pi\)
−0.828572 + 0.559883i \(0.810846\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −18.0000 −0.619219
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −26.0000 −0.890223 −0.445112 − 0.895475i \(-0.646836\pi\)
−0.445112 + 0.895475i \(0.646836\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 18.0000 0.614868 0.307434 − 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 0 0
\(859\) 40.0000 1.36478 0.682391 − 0.730987i \(-0.260940\pi\)
0.682391 + 0.730987i \(0.260940\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −16.0000 −0.544646 −0.272323 − 0.962206i \(-0.587792\pi\)
−0.272323 + 0.962206i \(0.587792\pi\)
\(864\) 0 0
\(865\) 36.0000 1.22404
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 64.0000 2.17105
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 6.00000 0.202606 0.101303 − 0.994856i \(-0.467699\pi\)
0.101303 + 0.994856i \(0.467699\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 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 \(-0.609252\pi\)
−0.336527 + 0.941674i \(0.609252\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −16.0000 −0.537227 −0.268614 − 0.963248i \(-0.586566\pi\)
−0.268614 + 0.963248i \(0.586566\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 64.0000 2.14168
\(894\) 0 0
\(895\) 8.00000 0.267411
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 48.0000 1.60089
\(900\) 0 0
\(901\) 36.0000 1.19933
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −20.0000 −0.664822
\(906\) 0 0
\(907\) −28.0000 −0.929725 −0.464862 − 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −24.0000 −0.795155 −0.397578 − 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) 0 0
\(913\) 32.0000 1.05905
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −40.0000 −1.31948 −0.659739 − 0.751495i \(-0.729333\pi\)
−0.659739 + 0.751495i \(0.729333\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −16.0000 −0.526646
\(924\) 0 0
\(925\) 2.00000 0.0657596
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 10.0000 0.328089 0.164045 − 0.986453i \(-0.447546\pi\)
0.164045 + 0.986453i \(0.447546\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −48.0000 −1.56977
\(936\) 0 0
\(937\) −10.0000 −0.326686 −0.163343 − 0.986569i \(-0.552228\pi\)
−0.163343 + 0.986569i \(0.552228\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −54.0000 −1.76035 −0.880175 − 0.474650i \(-0.842575\pi\)
−0.880175 + 0.474650i \(0.842575\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −52.0000 −1.68977 −0.844886 − 0.534946i \(-0.820332\pi\)
−0.844886 + 0.534946i \(0.820332\pi\)
\(948\) 0 0
\(949\) 20.0000 0.649227
\(950\) 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
\(955\) 32.0000 1.03550
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −28.0000 −0.901352
\(966\) 0 0
\(967\) −16.0000 −0.514525 −0.257263 − 0.966342i \(-0.582821\pi\)
−0.257263 + 0.966342i \(0.582821\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 48.0000 1.54039 0.770197 − 0.637806i \(-0.220158\pi\)
0.770197 + 0.637806i \(0.220158\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −2.00000 −0.0639857 −0.0319928 − 0.999488i \(-0.510185\pi\)
−0.0319928 + 0.999488i \(0.510185\pi\)
\(978\) 0 0
\(979\) −24.0000 −0.767043
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) −12.0000 −0.382352
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 − 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 14.0000 0.443384 0.221692 − 0.975117i \(-0.428842\pi\)
0.221692 + 0.975117i \(0.428842\pi\)
\(998\) 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 3528.2.a.x.1.1 1
3.2 odd 2 392.2.a.d.1.1 1
4.3 odd 2 7056.2.a.bo.1.1 1
7.2 even 3 3528.2.s.e.361.1 2
7.3 odd 6 3528.2.s.t.3313.1 2
7.4 even 3 3528.2.s.e.3313.1 2
7.5 odd 6 3528.2.s.t.361.1 2
7.6 odd 2 504.2.a.c.1.1 1
12.11 even 2 784.2.a.e.1.1 1
15.14 odd 2 9800.2.a.u.1.1 1
21.2 odd 6 392.2.i.d.361.1 2
21.5 even 6 392.2.i.c.361.1 2
21.11 odd 6 392.2.i.d.177.1 2
21.17 even 6 392.2.i.c.177.1 2
21.20 even 2 56.2.a.a.1.1 ✓ 1
24.5 odd 2 3136.2.a.q.1.1 1
24.11 even 2 3136.2.a.p.1.1 1
28.27 even 2 1008.2.a.d.1.1 1
56.13 odd 2 4032.2.a.bb.1.1 1
56.27 even 2 4032.2.a.bk.1.1 1
84.11 even 6 784.2.i.g.177.1 2
84.23 even 6 784.2.i.g.753.1 2
84.47 odd 6 784.2.i.e.753.1 2
84.59 odd 6 784.2.i.e.177.1 2
84.83 odd 2 112.2.a.b.1.1 1
105.62 odd 4 1400.2.g.g.449.1 2
105.83 odd 4 1400.2.g.g.449.2 2
105.104 even 2 1400.2.a.g.1.1 1
168.83 odd 2 448.2.a.e.1.1 1
168.125 even 2 448.2.a.d.1.1 1
231.230 odd 2 6776.2.a.g.1.1 1
273.272 even 2 9464.2.a.c.1.1 1
336.83 odd 4 1792.2.b.d.897.1 2
336.125 even 4 1792.2.b.i.897.1 2
336.251 odd 4 1792.2.b.d.897.2 2
336.293 even 4 1792.2.b.i.897.2 2
420.83 even 4 2800.2.g.p.449.1 2
420.167 even 4 2800.2.g.p.449.2 2
420.419 odd 2 2800.2.a.p.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.2.a.a.1.1 ✓ 1 21.20 even 2
112.2.a.b.1.1 1 84.83 odd 2
392.2.a.d.1.1 1 3.2 odd 2
392.2.i.c.177.1 2 21.17 even 6
392.2.i.c.361.1 2 21.5 even 6
392.2.i.d.177.1 2 21.11 odd 6
392.2.i.d.361.1 2 21.2 odd 6
448.2.a.d.1.1 1 168.125 even 2
448.2.a.e.1.1 1 168.83 odd 2
504.2.a.c.1.1 1 7.6 odd 2
784.2.a.e.1.1 1 12.11 even 2
784.2.i.e.177.1 2 84.59 odd 6
784.2.i.e.753.1 2 84.47 odd 6
784.2.i.g.177.1 2 84.11 even 6
784.2.i.g.753.1 2 84.23 even 6
1008.2.a.d.1.1 1 28.27 even 2
1400.2.a.g.1.1 1 105.104 even 2
1400.2.g.g.449.1 2 105.62 odd 4
1400.2.g.g.449.2 2 105.83 odd 4
1792.2.b.d.897.1 2 336.83 odd 4
1792.2.b.d.897.2 2 336.251 odd 4
1792.2.b.i.897.1 2 336.125 even 4
1792.2.b.i.897.2 2 336.293 even 4
2800.2.a.p.1.1 1 420.419 odd 2
2800.2.g.p.449.1 2 420.83 even 4
2800.2.g.p.449.2 2 420.167 even 4
3136.2.a.p.1.1 1 24.11 even 2
3136.2.a.q.1.1 1 24.5 odd 2
3528.2.a.x.1.1 1 1.1 even 1 trivial
3528.2.s.e.361.1 2 7.2 even 3
3528.2.s.e.3313.1 2 7.4 even 3
3528.2.s.t.361.1 2 7.5 odd 6
3528.2.s.t.3313.1 2 7.3 odd 6
4032.2.a.bb.1.1 1 56.13 odd 2
4032.2.a.bk.1.1 1 56.27 even 2
6776.2.a.g.1.1 1 231.230 odd 2
7056.2.a.bo.1.1 1 4.3 odd 2
9464.2.a.c.1.1 1 273.272 even 2
9800.2.a.u.1.1 1 15.14 odd 2