Properties

Label 648.2.a.e.1.1
Level $648$
Weight $2$
Character 648.1
Self dual yes
Analytic conductor $5.174$
Analytic rank $1$
Dimension $2$
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] = [648,2,Mod(1,648)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(648, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("648.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 648 = 2^{3} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 648.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: \(5.17430605098\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 648.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.73205 q^{5} +3.46410 q^{7} +O(q^{10})\) \(q-3.73205 q^{5} +3.46410 q^{7} -2.00000 q^{11} -2.46410 q^{13} -2.26795 q^{17} -7.46410 q^{19} +4.92820 q^{23} +8.92820 q^{25} -4.26795 q^{29} -10.9282 q^{31} -12.9282 q^{35} -0.464102 q^{37} -6.92820 q^{41} -4.53590 q^{43} -6.92820 q^{47} +5.00000 q^{49} +10.9282 q^{53} +7.46410 q^{55} -8.00000 q^{59} +10.4641 q^{61} +9.19615 q^{65} -0.535898 q^{67} +2.00000 q^{71} +1.00000 q^{73} -6.92820 q^{77} +0.535898 q^{79} +2.92820 q^{83} +8.46410 q^{85} +5.19615 q^{89} -8.53590 q^{91} +27.8564 q^{95} -11.8564 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{5} - 4 q^{11} + 2 q^{13} - 8 q^{17} - 8 q^{19} - 4 q^{23} + 4 q^{25} - 12 q^{29} - 8 q^{31} - 12 q^{35} + 6 q^{37} - 16 q^{43} + 10 q^{49} + 8 q^{53} + 8 q^{55} - 16 q^{59} + 14 q^{61} + 8 q^{65} - 8 q^{67} + 4 q^{71} + 2 q^{73} + 8 q^{79} - 8 q^{83} + 10 q^{85} - 24 q^{91} + 28 q^{95} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(5\) −3.73205 −1.66902 −0.834512 0.550990i \(-0.814250\pi\)
−0.834512 + 0.550990i \(0.814250\pi\)
\(6\) 0 0
\(7\) 3.46410 1.30931 0.654654 0.755929i \(-0.272814\pi\)
0.654654 + 0.755929i \(0.272814\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) −2.46410 −0.683419 −0.341709 0.939806i \(-0.611006\pi\)
−0.341709 + 0.939806i \(0.611006\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.26795 −0.550058 −0.275029 0.961436i \(-0.588688\pi\)
−0.275029 + 0.961436i \(0.588688\pi\)
\(18\) 0 0
\(19\) −7.46410 −1.71238 −0.856191 0.516659i \(-0.827175\pi\)
−0.856191 + 0.516659i \(0.827175\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.92820 1.02760 0.513801 0.857910i \(-0.328237\pi\)
0.513801 + 0.857910i \(0.328237\pi\)
\(24\) 0 0
\(25\) 8.92820 1.78564
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −4.26795 −0.792538 −0.396269 0.918134i \(-0.629695\pi\)
−0.396269 + 0.918134i \(0.629695\pi\)
\(30\) 0 0
\(31\) −10.9282 −1.96276 −0.981382 0.192068i \(-0.938481\pi\)
−0.981382 + 0.192068i \(0.938481\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −12.9282 −2.18527
\(36\) 0 0
\(37\) −0.464102 −0.0762978 −0.0381489 0.999272i \(-0.512146\pi\)
−0.0381489 + 0.999272i \(0.512146\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.92820 −1.08200 −0.541002 0.841021i \(-0.681955\pi\)
−0.541002 + 0.841021i \(0.681955\pi\)
\(42\) 0 0
\(43\) −4.53590 −0.691718 −0.345859 0.938286i \(-0.612412\pi\)
−0.345859 + 0.938286i \(0.612412\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −6.92820 −1.01058 −0.505291 0.862949i \(-0.668615\pi\)
−0.505291 + 0.862949i \(0.668615\pi\)
\(48\) 0 0
\(49\) 5.00000 0.714286
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 10.9282 1.50110 0.750552 0.660811i \(-0.229788\pi\)
0.750552 + 0.660811i \(0.229788\pi\)
\(54\) 0 0
\(55\) 7.46410 1.00646
\(56\) 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
\(61\) 10.4641 1.33979 0.669895 0.742455i \(-0.266339\pi\)
0.669895 + 0.742455i \(0.266339\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 9.19615 1.14064
\(66\) 0 0
\(67\) −0.535898 −0.0654704 −0.0327352 0.999464i \(-0.510422\pi\)
−0.0327352 + 0.999464i \(0.510422\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) 0 0
\(73\) 1.00000 0.117041 0.0585206 0.998286i \(-0.481362\pi\)
0.0585206 + 0.998286i \(0.481362\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −6.92820 −0.789542
\(78\) 0 0
\(79\) 0.535898 0.0602933 0.0301466 0.999545i \(-0.490403\pi\)
0.0301466 + 0.999545i \(0.490403\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 2.92820 0.321412 0.160706 0.987002i \(-0.448623\pi\)
0.160706 + 0.987002i \(0.448623\pi\)
\(84\) 0 0
\(85\) 8.46410 0.918061
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 5.19615 0.550791 0.275396 0.961331i \(-0.411191\pi\)
0.275396 + 0.961331i \(0.411191\pi\)
\(90\) 0 0
\(91\) −8.53590 −0.894805
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 27.8564 2.85801
\(96\) 0 0
\(97\) −11.8564 −1.20384 −0.601918 0.798558i \(-0.705597\pi\)
−0.601918 + 0.798558i \(0.705597\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 12.0000 1.19404 0.597022 0.802225i \(-0.296350\pi\)
0.597022 + 0.802225i \(0.296350\pi\)
\(102\) 0 0
\(103\) 10.9282 1.07679 0.538394 0.842693i \(-0.319031\pi\)
0.538394 + 0.842693i \(0.319031\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 5.07180 0.490309 0.245155 0.969484i \(-0.421161\pi\)
0.245155 + 0.969484i \(0.421161\pi\)
\(108\) 0 0
\(109\) 4.46410 0.427583 0.213792 0.976879i \(-0.431419\pi\)
0.213792 + 0.976879i \(0.431419\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1.73205 0.162938 0.0814688 0.996676i \(-0.474039\pi\)
0.0814688 + 0.996676i \(0.474039\pi\)
\(114\) 0 0
\(115\) −18.3923 −1.71509
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −7.85641 −0.720196
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −14.6603 −1.31125
\(126\) 0 0
\(127\) 0.535898 0.0475533 0.0237766 0.999717i \(-0.492431\pi\)
0.0237766 + 0.999717i \(0.492431\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 7.07180 0.617866 0.308933 0.951084i \(-0.400028\pi\)
0.308933 + 0.951084i \(0.400028\pi\)
\(132\) 0 0
\(133\) −25.8564 −2.24203
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −20.6603 −1.76512 −0.882562 0.470195i \(-0.844183\pi\)
−0.882562 + 0.470195i \(0.844183\pi\)
\(138\) 0 0
\(139\) −5.07180 −0.430184 −0.215092 0.976594i \(-0.569005\pi\)
−0.215092 + 0.976594i \(0.569005\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 4.92820 0.412117
\(144\) 0 0
\(145\) 15.9282 1.32277
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 15.7321 1.28882 0.644410 0.764680i \(-0.277103\pi\)
0.644410 + 0.764680i \(0.277103\pi\)
\(150\) 0 0
\(151\) 1.07180 0.0872216 0.0436108 0.999049i \(-0.486114\pi\)
0.0436108 + 0.999049i \(0.486114\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 40.7846 3.27590
\(156\) 0 0
\(157\) 5.39230 0.430353 0.215176 0.976575i \(-0.430967\pi\)
0.215176 + 0.976575i \(0.430967\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 17.0718 1.34545
\(162\) 0 0
\(163\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −18.0000 −1.39288 −0.696441 0.717614i \(-0.745234\pi\)
−0.696441 + 0.717614i \(0.745234\pi\)
\(168\) 0 0
\(169\) −6.92820 −0.532939
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 7.73205 0.587857 0.293928 0.955827i \(-0.405037\pi\)
0.293928 + 0.955827i \(0.405037\pi\)
\(174\) 0 0
\(175\) 30.9282 2.33795
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 6.92820 0.517838 0.258919 0.965899i \(-0.416634\pi\)
0.258919 + 0.965899i \(0.416634\pi\)
\(180\) 0 0
\(181\) 7.85641 0.583962 0.291981 0.956424i \(-0.405686\pi\)
0.291981 + 0.956424i \(0.405686\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 1.73205 0.127343
\(186\) 0 0
\(187\) 4.53590 0.331698
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −19.8564 −1.43676 −0.718380 0.695651i \(-0.755116\pi\)
−0.718380 + 0.695651i \(0.755116\pi\)
\(192\) 0 0
\(193\) 17.9282 1.29050 0.645250 0.763971i \(-0.276753\pi\)
0.645250 + 0.763971i \(0.276753\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2.66025 0.189535 0.0947676 0.995499i \(-0.469789\pi\)
0.0947676 + 0.995499i \(0.469789\pi\)
\(198\) 0 0
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −14.7846 −1.03768
\(204\) 0 0
\(205\) 25.8564 1.80589
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 14.9282 1.03261
\(210\) 0 0
\(211\) −25.3205 −1.74314 −0.871568 0.490275i \(-0.836896\pi\)
−0.871568 + 0.490275i \(0.836896\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 16.9282 1.15449
\(216\) 0 0
\(217\) −37.8564 −2.56986
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 5.58846 0.375920
\(222\) 0 0
\(223\) 0.535898 0.0358864 0.0179432 0.999839i \(-0.494288\pi\)
0.0179432 + 0.999839i \(0.494288\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 14.7846 0.981289 0.490645 0.871360i \(-0.336761\pi\)
0.490645 + 0.871360i \(0.336761\pi\)
\(228\) 0 0
\(229\) −7.53590 −0.497986 −0.248993 0.968505i \(-0.580100\pi\)
−0.248993 + 0.968505i \(0.580100\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −9.73205 −0.637568 −0.318784 0.947827i \(-0.603274\pi\)
−0.318784 + 0.947827i \(0.603274\pi\)
\(234\) 0 0
\(235\) 25.8564 1.68669
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −10.9282 −0.706887 −0.353443 0.935456i \(-0.614989\pi\)
−0.353443 + 0.935456i \(0.614989\pi\)
\(240\) 0 0
\(241\) −8.07180 −0.519950 −0.259975 0.965615i \(-0.583714\pi\)
−0.259975 + 0.965615i \(0.583714\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −18.6603 −1.19216
\(246\) 0 0
\(247\) 18.3923 1.17027
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −10.0000 −0.631194 −0.315597 0.948893i \(-0.602205\pi\)
−0.315597 + 0.948893i \(0.602205\pi\)
\(252\) 0 0
\(253\) −9.85641 −0.619667
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 27.0526 1.68749 0.843746 0.536742i \(-0.180345\pi\)
0.843746 + 0.536742i \(0.180345\pi\)
\(258\) 0 0
\(259\) −1.60770 −0.0998973
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −28.7846 −1.77494 −0.887468 0.460870i \(-0.847537\pi\)
−0.887468 + 0.460870i \(0.847537\pi\)
\(264\) 0 0
\(265\) −40.7846 −2.50538
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −20.5167 −1.25092 −0.625461 0.780255i \(-0.715089\pi\)
−0.625461 + 0.780255i \(0.715089\pi\)
\(270\) 0 0
\(271\) 16.5359 1.00448 0.502242 0.864727i \(-0.332509\pi\)
0.502242 + 0.864727i \(0.332509\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −17.8564 −1.07678
\(276\) 0 0
\(277\) −14.0000 −0.841178 −0.420589 0.907251i \(-0.638177\pi\)
−0.420589 + 0.907251i \(0.638177\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −4.12436 −0.246038 −0.123019 0.992404i \(-0.539258\pi\)
−0.123019 + 0.992404i \(0.539258\pi\)
\(282\) 0 0
\(283\) −17.8564 −1.06145 −0.530727 0.847543i \(-0.678081\pi\)
−0.530727 + 0.847543i \(0.678081\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −24.0000 −1.41668
\(288\) 0 0
\(289\) −11.8564 −0.697436
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 6.12436 0.357789 0.178894 0.983868i \(-0.442748\pi\)
0.178894 + 0.983868i \(0.442748\pi\)
\(294\) 0 0
\(295\) 29.8564 1.73831
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −12.1436 −0.702282
\(300\) 0 0
\(301\) −15.7128 −0.905671
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −39.0526 −2.23614
\(306\) 0 0
\(307\) −18.9282 −1.08029 −0.540145 0.841572i \(-0.681631\pi\)
−0.540145 + 0.841572i \(0.681631\pi\)
\(308\) 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
\(313\) 23.7846 1.34439 0.672193 0.740376i \(-0.265353\pi\)
0.672193 + 0.740376i \(0.265353\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −7.73205 −0.434275 −0.217138 0.976141i \(-0.569672\pi\)
−0.217138 + 0.976141i \(0.569672\pi\)
\(318\) 0 0
\(319\) 8.53590 0.477919
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 16.9282 0.941910
\(324\) 0 0
\(325\) −22.0000 −1.22034
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −24.0000 −1.32316
\(330\) 0 0
\(331\) 20.2487 1.11297 0.556485 0.830858i \(-0.312150\pi\)
0.556485 + 0.830858i \(0.312150\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 2.00000 0.109272
\(336\) 0 0
\(337\) 7.85641 0.427966 0.213983 0.976837i \(-0.431356\pi\)
0.213983 + 0.976837i \(0.431356\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 21.8564 1.18359
\(342\) 0 0
\(343\) −6.92820 −0.374088
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −24.9282 −1.33822 −0.669108 0.743165i \(-0.733324\pi\)
−0.669108 + 0.743165i \(0.733324\pi\)
\(348\) 0 0
\(349\) −6.00000 −0.321173 −0.160586 0.987022i \(-0.551338\pi\)
−0.160586 + 0.987022i \(0.551338\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −2.14359 −0.114092 −0.0570460 0.998372i \(-0.518168\pi\)
−0.0570460 + 0.998372i \(0.518168\pi\)
\(354\) 0 0
\(355\) −7.46410 −0.396153
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 14.7846 0.780302 0.390151 0.920751i \(-0.372423\pi\)
0.390151 + 0.920751i \(0.372423\pi\)
\(360\) 0 0
\(361\) 36.7128 1.93225
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −3.73205 −0.195344
\(366\) 0 0
\(367\) 22.9282 1.19684 0.598421 0.801182i \(-0.295795\pi\)
0.598421 + 0.801182i \(0.295795\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 37.8564 1.96541
\(372\) 0 0
\(373\) −10.0000 −0.517780 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 10.5167 0.541636
\(378\) 0 0
\(379\) −24.7846 −1.27310 −0.636550 0.771235i \(-0.719639\pi\)
−0.636550 + 0.771235i \(0.719639\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −18.0000 −0.919757 −0.459879 0.887982i \(-0.652107\pi\)
−0.459879 + 0.887982i \(0.652107\pi\)
\(384\) 0 0
\(385\) 25.8564 1.31776
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −26.9282 −1.36531 −0.682657 0.730739i \(-0.739176\pi\)
−0.682657 + 0.730739i \(0.739176\pi\)
\(390\) 0 0
\(391\) −11.1769 −0.565241
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −2.00000 −0.100631
\(396\) 0 0
\(397\) 0.607695 0.0304993 0.0152497 0.999884i \(-0.495146\pi\)
0.0152497 + 0.999884i \(0.495146\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 17.1962 0.858735 0.429367 0.903130i \(-0.358737\pi\)
0.429367 + 0.903130i \(0.358737\pi\)
\(402\) 0 0
\(403\) 26.9282 1.34139
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0.928203 0.0460093
\(408\) 0 0
\(409\) 15.9282 0.787599 0.393799 0.919196i \(-0.371160\pi\)
0.393799 + 0.919196i \(0.371160\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −27.7128 −1.36366
\(414\) 0 0
\(415\) −10.9282 −0.536444
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −25.8564 −1.26317 −0.631584 0.775307i \(-0.717595\pi\)
−0.631584 + 0.775307i \(0.717595\pi\)
\(420\) 0 0
\(421\) 3.39230 0.165331 0.0826654 0.996577i \(-0.473657\pi\)
0.0826654 + 0.996577i \(0.473657\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −20.2487 −0.982207
\(426\) 0 0
\(427\) 36.2487 1.75420
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 26.9282 1.29709 0.648543 0.761178i \(-0.275379\pi\)
0.648543 + 0.761178i \(0.275379\pi\)
\(432\) 0 0
\(433\) 27.7846 1.33524 0.667622 0.744501i \(-0.267312\pi\)
0.667622 + 0.744501i \(0.267312\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −36.7846 −1.75965
\(438\) 0 0
\(439\) −12.0000 −0.572729 −0.286364 0.958121i \(-0.592447\pi\)
−0.286364 + 0.958121i \(0.592447\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 1.07180 0.0509226 0.0254613 0.999676i \(-0.491895\pi\)
0.0254613 + 0.999676i \(0.491895\pi\)
\(444\) 0 0
\(445\) −19.3923 −0.919283
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −28.7846 −1.35843 −0.679215 0.733939i \(-0.737680\pi\)
−0.679215 + 0.733939i \(0.737680\pi\)
\(450\) 0 0
\(451\) 13.8564 0.652473
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 31.8564 1.49345
\(456\) 0 0
\(457\) −37.9282 −1.77421 −0.887103 0.461571i \(-0.847286\pi\)
−0.887103 + 0.461571i \(0.847286\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 24.7846 1.15433 0.577167 0.816626i \(-0.304158\pi\)
0.577167 + 0.816626i \(0.304158\pi\)
\(462\) 0 0
\(463\) −40.0000 −1.85896 −0.929479 0.368875i \(-0.879743\pi\)
−0.929479 + 0.368875i \(0.879743\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −39.7128 −1.83769 −0.918845 0.394619i \(-0.870877\pi\)
−0.918845 + 0.394619i \(0.870877\pi\)
\(468\) 0 0
\(469\) −1.85641 −0.0857209
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 9.07180 0.417122
\(474\) 0 0
\(475\) −66.6410 −3.05770
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −11.8564 −0.541733 −0.270867 0.962617i \(-0.587310\pi\)
−0.270867 + 0.962617i \(0.587310\pi\)
\(480\) 0 0
\(481\) 1.14359 0.0521434
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 44.2487 2.00923
\(486\) 0 0
\(487\) 34.9282 1.58275 0.791374 0.611332i \(-0.209366\pi\)
0.791374 + 0.611332i \(0.209366\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 43.8564 1.97921 0.989606 0.143806i \(-0.0459340\pi\)
0.989606 + 0.143806i \(0.0459340\pi\)
\(492\) 0 0
\(493\) 9.67949 0.435942
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 6.92820 0.310772
\(498\) 0 0
\(499\) 24.2487 1.08552 0.542761 0.839887i \(-0.317379\pi\)
0.542761 + 0.839887i \(0.317379\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −25.8564 −1.15288 −0.576440 0.817139i \(-0.695559\pi\)
−0.576440 + 0.817139i \(0.695559\pi\)
\(504\) 0 0
\(505\) −44.7846 −1.99289
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 32.7846 1.45315 0.726576 0.687086i \(-0.241110\pi\)
0.726576 + 0.687086i \(0.241110\pi\)
\(510\) 0 0
\(511\) 3.46410 0.153243
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −40.7846 −1.79718
\(516\) 0 0
\(517\) 13.8564 0.609404
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 28.7846 1.26108 0.630538 0.776158i \(-0.282834\pi\)
0.630538 + 0.776158i \(0.282834\pi\)
\(522\) 0 0
\(523\) −35.4641 −1.55074 −0.775368 0.631509i \(-0.782436\pi\)
−0.775368 + 0.631509i \(0.782436\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 24.7846 1.07963
\(528\) 0 0
\(529\) 1.28719 0.0559647
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 17.0718 0.739462
\(534\) 0 0
\(535\) −18.9282 −0.818338
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −10.0000 −0.430730
\(540\) 0 0
\(541\) 7.39230 0.317820 0.158910 0.987293i \(-0.449202\pi\)
0.158910 + 0.987293i \(0.449202\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −16.6603 −0.713647
\(546\) 0 0
\(547\) −17.8564 −0.763485 −0.381742 0.924269i \(-0.624676\pi\)
−0.381742 + 0.924269i \(0.624676\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 31.8564 1.35713
\(552\) 0 0
\(553\) 1.85641 0.0789424
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 13.5885 0.575761 0.287881 0.957666i \(-0.407049\pi\)
0.287881 + 0.957666i \(0.407049\pi\)
\(558\) 0 0
\(559\) 11.1769 0.472733
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −1.07180 −0.0451708 −0.0225854 0.999745i \(-0.507190\pi\)
−0.0225854 + 0.999745i \(0.507190\pi\)
\(564\) 0 0
\(565\) −6.46410 −0.271947
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 43.5885 1.82732 0.913662 0.406476i \(-0.133242\pi\)
0.913662 + 0.406476i \(0.133242\pi\)
\(570\) 0 0
\(571\) −13.8564 −0.579873 −0.289936 0.957046i \(-0.593634\pi\)
−0.289936 + 0.957046i \(0.593634\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 44.0000 1.83493
\(576\) 0 0
\(577\) −2.85641 −0.118914 −0.0594569 0.998231i \(-0.518937\pi\)
−0.0594569 + 0.998231i \(0.518937\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 10.1436 0.420827
\(582\) 0 0
\(583\) −21.8564 −0.905200
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −26.0000 −1.07313 −0.536567 0.843857i \(-0.680279\pi\)
−0.536567 + 0.843857i \(0.680279\pi\)
\(588\) 0 0
\(589\) 81.5692 3.36100
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0.411543 0.0169000 0.00845002 0.999964i \(-0.497310\pi\)
0.00845002 + 0.999964i \(0.497310\pi\)
\(594\) 0 0
\(595\) 29.3205 1.20202
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 18.9282 0.773385 0.386693 0.922209i \(-0.373617\pi\)
0.386693 + 0.922209i \(0.373617\pi\)
\(600\) 0 0
\(601\) −6.85641 −0.279679 −0.139839 0.990174i \(-0.544659\pi\)
−0.139839 + 0.990174i \(0.544659\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 26.1244 1.06211
\(606\) 0 0
\(607\) 39.1769 1.59014 0.795071 0.606516i \(-0.207434\pi\)
0.795071 + 0.606516i \(0.207434\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 17.0718 0.690651
\(612\) 0 0
\(613\) 3.85641 0.155759 0.0778794 0.996963i \(-0.475185\pi\)
0.0778794 + 0.996963i \(0.475185\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −29.4449 −1.18541 −0.592703 0.805421i \(-0.701939\pi\)
−0.592703 + 0.805421i \(0.701939\pi\)
\(618\) 0 0
\(619\) −4.78461 −0.192310 −0.0961549 0.995366i \(-0.530654\pi\)
−0.0961549 + 0.995366i \(0.530654\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 18.0000 0.721155
\(624\) 0 0
\(625\) 10.0718 0.402872
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 1.05256 0.0419683
\(630\) 0 0
\(631\) 12.7846 0.508947 0.254474 0.967080i \(-0.418098\pi\)
0.254474 + 0.967080i \(0.418098\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −2.00000 −0.0793676
\(636\) 0 0
\(637\) −12.3205 −0.488156
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 29.9808 1.18417 0.592084 0.805876i \(-0.298305\pi\)
0.592084 + 0.805876i \(0.298305\pi\)
\(642\) 0 0
\(643\) 32.7846 1.29290 0.646449 0.762957i \(-0.276253\pi\)
0.646449 + 0.762957i \(0.276253\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −37.7128 −1.48264 −0.741322 0.671150i \(-0.765801\pi\)
−0.741322 + 0.671150i \(0.765801\pi\)
\(648\) 0 0
\(649\) 16.0000 0.628055
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −8.78461 −0.343768 −0.171884 0.985117i \(-0.554985\pi\)
−0.171884 + 0.985117i \(0.554985\pi\)
\(654\) 0 0
\(655\) −26.3923 −1.03123
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −2.78461 −0.108473 −0.0542365 0.998528i \(-0.517272\pi\)
−0.0542365 + 0.998528i \(0.517272\pi\)
\(660\) 0 0
\(661\) −30.3205 −1.17933 −0.589666 0.807648i \(-0.700740\pi\)
−0.589666 + 0.807648i \(0.700740\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 96.4974 3.74201
\(666\) 0 0
\(667\) −21.0333 −0.814413
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −20.9282 −0.807924
\(672\) 0 0
\(673\) 25.9282 0.999459 0.499729 0.866182i \(-0.333433\pi\)
0.499729 + 0.866182i \(0.333433\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −30.6410 −1.17763 −0.588815 0.808268i \(-0.700405\pi\)
−0.588815 + 0.808268i \(0.700405\pi\)
\(678\) 0 0
\(679\) −41.0718 −1.57619
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −1.07180 −0.0410112 −0.0205056 0.999790i \(-0.506528\pi\)
−0.0205056 + 0.999790i \(0.506528\pi\)
\(684\) 0 0
\(685\) 77.1051 2.94604
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −26.9282 −1.02588
\(690\) 0 0
\(691\) −17.6077 −0.669828 −0.334914 0.942249i \(-0.608707\pi\)
−0.334914 + 0.942249i \(0.608707\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 18.9282 0.717988
\(696\) 0 0
\(697\) 15.7128 0.595165
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −35.4449 −1.33873 −0.669367 0.742932i \(-0.733435\pi\)
−0.669367 + 0.742932i \(0.733435\pi\)
\(702\) 0 0
\(703\) 3.46410 0.130651
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 41.5692 1.56337
\(708\) 0 0
\(709\) 22.3205 0.838264 0.419132 0.907925i \(-0.362334\pi\)
0.419132 + 0.907925i \(0.362334\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −53.8564 −2.01694
\(714\) 0 0
\(715\) −18.3923 −0.687833
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −8.92820 −0.332966 −0.166483 0.986044i \(-0.553241\pi\)
−0.166483 + 0.986044i \(0.553241\pi\)
\(720\) 0 0
\(721\) 37.8564 1.40985
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −38.1051 −1.41519
\(726\) 0 0
\(727\) −31.1769 −1.15629 −0.578144 0.815935i \(-0.696223\pi\)
−0.578144 + 0.815935i \(0.696223\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 10.2872 0.380485
\(732\) 0 0
\(733\) 39.8564 1.47213 0.736065 0.676911i \(-0.236682\pi\)
0.736065 + 0.676911i \(0.236682\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.07180 0.0394801
\(738\) 0 0
\(739\) −28.0000 −1.03000 −0.514998 0.857191i \(-0.672207\pi\)
−0.514998 + 0.857191i \(0.672207\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 22.9282 0.841154 0.420577 0.907257i \(-0.361828\pi\)
0.420577 + 0.907257i \(0.361828\pi\)
\(744\) 0 0
\(745\) −58.7128 −2.15107
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 17.5692 0.641965
\(750\) 0 0
\(751\) −17.3205 −0.632034 −0.316017 0.948753i \(-0.602346\pi\)
−0.316017 + 0.948753i \(0.602346\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −4.00000 −0.145575
\(756\) 0 0
\(757\) −35.8564 −1.30322 −0.651612 0.758553i \(-0.725907\pi\)
−0.651612 + 0.758553i \(0.725907\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 49.1962 1.78336 0.891680 0.452667i \(-0.149527\pi\)
0.891680 + 0.452667i \(0.149527\pi\)
\(762\) 0 0
\(763\) 15.4641 0.559838
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 19.7128 0.711788
\(768\) 0 0
\(769\) −9.78461 −0.352842 −0.176421 0.984315i \(-0.556452\pi\)
−0.176421 + 0.984315i \(0.556452\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −28.2679 −1.01673 −0.508364 0.861142i \(-0.669749\pi\)
−0.508364 + 0.861142i \(0.669749\pi\)
\(774\) 0 0
\(775\) −97.5692 −3.50479
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 51.7128 1.85280
\(780\) 0 0
\(781\) −4.00000 −0.143131
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −20.1244 −0.718269
\(786\) 0 0
\(787\) 19.4641 0.693820 0.346910 0.937898i \(-0.387231\pi\)
0.346910 + 0.937898i \(0.387231\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 6.00000 0.213335
\(792\) 0 0
\(793\) −25.7846 −0.915638
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −3.48334 −0.123386 −0.0616931 0.998095i \(-0.519650\pi\)
−0.0616931 + 0.998095i \(0.519650\pi\)
\(798\) 0 0
\(799\) 15.7128 0.555879
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −2.00000 −0.0705785
\(804\) 0 0
\(805\) −63.7128 −2.24558
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −29.7321 −1.04532 −0.522662 0.852540i \(-0.675061\pi\)
−0.522662 + 0.852540i \(0.675061\pi\)
\(810\) 0 0
\(811\) 33.5692 1.17877 0.589387 0.807851i \(-0.299369\pi\)
0.589387 + 0.807851i \(0.299369\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 33.8564 1.18449
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 9.33975 0.325959 0.162980 0.986629i \(-0.447889\pi\)
0.162980 + 0.986629i \(0.447889\pi\)
\(822\) 0 0
\(823\) 20.0000 0.697156 0.348578 0.937280i \(-0.386665\pi\)
0.348578 + 0.937280i \(0.386665\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −11.8564 −0.412288 −0.206144 0.978522i \(-0.566091\pi\)
−0.206144 + 0.978522i \(0.566091\pi\)
\(828\) 0 0
\(829\) 23.8564 0.828567 0.414284 0.910148i \(-0.364032\pi\)
0.414284 + 0.910148i \(0.364032\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −11.3397 −0.392899
\(834\) 0 0
\(835\) 67.1769 2.32475
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −23.0718 −0.796527 −0.398263 0.917271i \(-0.630387\pi\)
−0.398263 + 0.917271i \(0.630387\pi\)
\(840\) 0 0
\(841\) −10.7846 −0.371883
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 25.8564 0.889487
\(846\) 0 0
\(847\) −24.2487 −0.833196
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −2.28719 −0.0784038
\(852\) 0 0
\(853\) 14.0000 0.479351 0.239675 0.970853i \(-0.422959\pi\)
0.239675 + 0.970853i \(0.422959\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −27.5885 −0.942404 −0.471202 0.882025i \(-0.656180\pi\)
−0.471202 + 0.882025i \(0.656180\pi\)
\(858\) 0 0
\(859\) −25.8564 −0.882209 −0.441105 0.897456i \(-0.645413\pi\)
−0.441105 + 0.897456i \(0.645413\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −32.9282 −1.12089 −0.560445 0.828192i \(-0.689370\pi\)
−0.560445 + 0.828192i \(0.689370\pi\)
\(864\) 0 0
\(865\) −28.8564 −0.981147
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −1.07180 −0.0363582
\(870\) 0 0
\(871\) 1.32051 0.0447437
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −50.7846 −1.71683
\(876\) 0 0
\(877\) 20.3205 0.686175 0.343087 0.939303i \(-0.388527\pi\)
0.343087 + 0.939303i \(0.388527\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −50.6410 −1.70614 −0.853070 0.521797i \(-0.825262\pi\)
−0.853070 + 0.521797i \(0.825262\pi\)
\(882\) 0 0
\(883\) 37.0718 1.24757 0.623783 0.781598i \(-0.285595\pi\)
0.623783 + 0.781598i \(0.285595\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −44.6410 −1.49890 −0.749449 0.662062i \(-0.769682\pi\)
−0.749449 + 0.662062i \(0.769682\pi\)
\(888\) 0 0
\(889\) 1.85641 0.0622619
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 51.7128 1.73050
\(894\) 0 0
\(895\) −25.8564 −0.864284
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 46.6410 1.55556
\(900\) 0 0
\(901\) −24.7846 −0.825695
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −29.3205 −0.974647
\(906\) 0 0
\(907\) 36.7846 1.22141 0.610706 0.791857i \(-0.290886\pi\)
0.610706 + 0.791857i \(0.290886\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −5.21539 −0.172794 −0.0863968 0.996261i \(-0.527535\pi\)
−0.0863968 + 0.996261i \(0.527535\pi\)
\(912\) 0 0
\(913\) −5.85641 −0.193819
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 24.4974 0.808976
\(918\) 0 0
\(919\) −27.1769 −0.896484 −0.448242 0.893912i \(-0.647950\pi\)
−0.448242 + 0.893912i \(0.647950\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −4.92820 −0.162214
\(924\) 0 0
\(925\) −4.14359 −0.136241
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 35.5885 1.16762 0.583810 0.811891i \(-0.301561\pi\)
0.583810 + 0.811891i \(0.301561\pi\)
\(930\) 0 0
\(931\) −37.3205 −1.22313
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −16.9282 −0.553611
\(936\) 0 0
\(937\) 32.5692 1.06399 0.531995 0.846747i \(-0.321443\pi\)
0.531995 + 0.846747i \(0.321443\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −6.12436 −0.199648 −0.0998241 0.995005i \(-0.531828\pi\)
−0.0998241 + 0.995005i \(0.531828\pi\)
\(942\) 0 0
\(943\) −34.1436 −1.11187
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −0.928203 −0.0301626 −0.0150813 0.999886i \(-0.504801\pi\)
−0.0150813 + 0.999886i \(0.504801\pi\)
\(948\) 0 0
\(949\) −2.46410 −0.0799881
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 37.9808 1.23032 0.615159 0.788403i \(-0.289092\pi\)
0.615159 + 0.788403i \(0.289092\pi\)
\(954\) 0 0
\(955\) 74.1051 2.39799
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −71.5692 −2.31109
\(960\) 0 0
\(961\) 88.4256 2.85244
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −66.9090 −2.15388
\(966\) 0 0
\(967\) 25.3205 0.814253 0.407126 0.913372i \(-0.366531\pi\)
0.407126 + 0.913372i \(0.366531\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 12.9282 0.414886 0.207443 0.978247i \(-0.433486\pi\)
0.207443 + 0.978247i \(0.433486\pi\)
\(972\) 0 0
\(973\) −17.5692 −0.563243
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −30.9282 −0.989481 −0.494740 0.869041i \(-0.664737\pi\)
−0.494740 + 0.869041i \(0.664737\pi\)
\(978\) 0 0
\(979\) −10.3923 −0.332140
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −36.0000 −1.14822 −0.574111 0.818778i \(-0.694652\pi\)
−0.574111 + 0.818778i \(0.694652\pi\)
\(984\) 0 0
\(985\) −9.92820 −0.316339
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −22.3538 −0.710810
\(990\) 0 0
\(991\) 30.1051 0.956321 0.478160 0.878273i \(-0.341304\pi\)
0.478160 + 0.878273i \(0.341304\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −14.9282 −0.473256
\(996\) 0 0
\(997\) 14.1769 0.448987 0.224494 0.974476i \(-0.427927\pi\)
0.224494 + 0.974476i \(0.427927\pi\)
\(998\) 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 648.2.a.e.1.1 2
3.2 odd 2 648.2.a.h.1.2 yes 2
4.3 odd 2 1296.2.a.m.1.1 2
8.3 odd 2 5184.2.a.bz.1.2 2
8.5 even 2 5184.2.a.cb.1.2 2
9.2 odd 6 648.2.i.i.433.1 4
9.4 even 3 648.2.i.j.217.2 4
9.5 odd 6 648.2.i.i.217.1 4
9.7 even 3 648.2.i.j.433.2 4
12.11 even 2 1296.2.a.q.1.2 2
24.5 odd 2 5184.2.a.bg.1.1 2
24.11 even 2 5184.2.a.bi.1.1 2
36.7 odd 6 1296.2.i.t.433.2 4
36.11 even 6 1296.2.i.r.433.1 4
36.23 even 6 1296.2.i.r.865.1 4
36.31 odd 6 1296.2.i.t.865.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
648.2.a.e.1.1 2 1.1 even 1 trivial
648.2.a.h.1.2 yes 2 3.2 odd 2
648.2.i.i.217.1 4 9.5 odd 6
648.2.i.i.433.1 4 9.2 odd 6
648.2.i.j.217.2 4 9.4 even 3
648.2.i.j.433.2 4 9.7 even 3
1296.2.a.m.1.1 2 4.3 odd 2
1296.2.a.q.1.2 2 12.11 even 2
1296.2.i.r.433.1 4 36.11 even 6
1296.2.i.r.865.1 4 36.23 even 6
1296.2.i.t.433.2 4 36.7 odd 6
1296.2.i.t.865.2 4 36.31 odd 6
5184.2.a.bg.1.1 2 24.5 odd 2
5184.2.a.bi.1.1 2 24.11 even 2
5184.2.a.bz.1.2 2 8.3 odd 2
5184.2.a.cb.1.2 2 8.5 even 2