Properties

Label 8036.2.a.m.1.6
Level $8036$
Weight $2$
Character 8036.1
Self dual yes
Analytic conductor $64.168$
Analytic rank $1$
Dimension $8$
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] = [8036,2,Mod(1,8036)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8036, 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("8036.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8036 = 2^{2} \cdot 7^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8036.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: \(64.1677830643\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 14x^{6} - 4x^{5} + 60x^{4} + 31x^{3} - 75x^{2} - 60x - 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1148)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-1.33338\) of defining polynomial
Character \(\chi\) \(=\) 8036.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.33338 q^{3} +1.76687 q^{5} -1.22209 q^{9} +O(q^{10})\) \(q+1.33338 q^{3} +1.76687 q^{5} -1.22209 q^{9} -2.48727 q^{11} -2.30214 q^{13} +2.35592 q^{15} -1.40413 q^{17} +3.95607 q^{19} -0.0405520 q^{23} -1.87817 q^{25} -5.62966 q^{27} +6.20359 q^{29} -2.98573 q^{31} -3.31648 q^{33} +0.195125 q^{37} -3.06963 q^{39} +1.00000 q^{41} -1.19665 q^{43} -2.15927 q^{45} -7.43927 q^{47} -1.87225 q^{51} -11.7999 q^{53} -4.39468 q^{55} +5.27497 q^{57} -5.50846 q^{59} +3.91456 q^{61} -4.06758 q^{65} +5.79845 q^{67} -0.0540714 q^{69} +10.2887 q^{71} -12.2984 q^{73} -2.50432 q^{75} -1.35940 q^{79} -3.84025 q^{81} +6.02722 q^{83} -2.48092 q^{85} +8.27178 q^{87} -0.226063 q^{89} -3.98113 q^{93} +6.98987 q^{95} +1.43170 q^{97} +3.03965 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{9} - 8 q^{11} - 7 q^{13} - q^{15} - q^{17} + 4 q^{19} - 3 q^{23} - 4 q^{25} - 12 q^{27} - 4 q^{29} + 4 q^{31} + 23 q^{33} - 31 q^{37} + 5 q^{39} + 8 q^{41} - 8 q^{43} + q^{45} + 24 q^{47} - 23 q^{51} - q^{53} + 2 q^{55} - 15 q^{57} + 4 q^{59} - 4 q^{61} - 24 q^{65} - 21 q^{69} + 8 q^{71} + 11 q^{73} - 15 q^{75} + 14 q^{79} - 28 q^{81} - 42 q^{83} - 20 q^{85} + 25 q^{87} - 11 q^{89} - 27 q^{93} - 15 q^{95} - 16 q^{97} - 8 q^{99}+O(q^{100}) \) 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
\(3\) 1.33338 0.769830 0.384915 0.922952i \(-0.374231\pi\)
0.384915 + 0.922952i \(0.374231\pi\)
\(4\) 0 0
\(5\) 1.76687 0.790168 0.395084 0.918645i \(-0.370715\pi\)
0.395084 + 0.918645i \(0.370715\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −1.22209 −0.407362
\(10\) 0 0
\(11\) −2.48727 −0.749939 −0.374970 0.927037i \(-0.622347\pi\)
−0.374970 + 0.927037i \(0.622347\pi\)
\(12\) 0 0
\(13\) −2.30214 −0.638498 −0.319249 0.947671i \(-0.603431\pi\)
−0.319249 + 0.947671i \(0.603431\pi\)
\(14\) 0 0
\(15\) 2.35592 0.608295
\(16\) 0 0
\(17\) −1.40413 −0.340552 −0.170276 0.985396i \(-0.554466\pi\)
−0.170276 + 0.985396i \(0.554466\pi\)
\(18\) 0 0
\(19\) 3.95607 0.907586 0.453793 0.891107i \(-0.350071\pi\)
0.453793 + 0.891107i \(0.350071\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −0.0405520 −0.00845568 −0.00422784 0.999991i \(-0.501346\pi\)
−0.00422784 + 0.999991i \(0.501346\pi\)
\(24\) 0 0
\(25\) −1.87817 −0.375634
\(26\) 0 0
\(27\) −5.62966 −1.08343
\(28\) 0 0
\(29\) 6.20359 1.15198 0.575989 0.817457i \(-0.304617\pi\)
0.575989 + 0.817457i \(0.304617\pi\)
\(30\) 0 0
\(31\) −2.98573 −0.536253 −0.268127 0.963384i \(-0.586405\pi\)
−0.268127 + 0.963384i \(0.586405\pi\)
\(32\) 0 0
\(33\) −3.31648 −0.577326
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0.195125 0.0320784 0.0160392 0.999871i \(-0.494894\pi\)
0.0160392 + 0.999871i \(0.494894\pi\)
\(38\) 0 0
\(39\) −3.06963 −0.491535
\(40\) 0 0
\(41\) 1.00000 0.156174
\(42\) 0 0
\(43\) −1.19665 −0.182488 −0.0912438 0.995829i \(-0.529084\pi\)
−0.0912438 + 0.995829i \(0.529084\pi\)
\(44\) 0 0
\(45\) −2.15927 −0.321884
\(46\) 0 0
\(47\) −7.43927 −1.08513 −0.542565 0.840014i \(-0.682547\pi\)
−0.542565 + 0.840014i \(0.682547\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −1.87225 −0.262167
\(52\) 0 0
\(53\) −11.7999 −1.62085 −0.810424 0.585844i \(-0.800763\pi\)
−0.810424 + 0.585844i \(0.800763\pi\)
\(54\) 0 0
\(55\) −4.39468 −0.592578
\(56\) 0 0
\(57\) 5.27497 0.698686
\(58\) 0 0
\(59\) −5.50846 −0.717140 −0.358570 0.933503i \(-0.616736\pi\)
−0.358570 + 0.933503i \(0.616736\pi\)
\(60\) 0 0
\(61\) 3.91456 0.501208 0.250604 0.968090i \(-0.419371\pi\)
0.250604 + 0.968090i \(0.419371\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −4.06758 −0.504521
\(66\) 0 0
\(67\) 5.79845 0.708394 0.354197 0.935171i \(-0.384754\pi\)
0.354197 + 0.935171i \(0.384754\pi\)
\(68\) 0 0
\(69\) −0.0540714 −0.00650943
\(70\) 0 0
\(71\) 10.2887 1.22105 0.610524 0.791997i \(-0.290959\pi\)
0.610524 + 0.791997i \(0.290959\pi\)
\(72\) 0 0
\(73\) −12.2984 −1.43942 −0.719710 0.694275i \(-0.755725\pi\)
−0.719710 + 0.694275i \(0.755725\pi\)
\(74\) 0 0
\(75\) −2.50432 −0.289174
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −1.35940 −0.152945 −0.0764723 0.997072i \(-0.524366\pi\)
−0.0764723 + 0.997072i \(0.524366\pi\)
\(80\) 0 0
\(81\) −3.84025 −0.426694
\(82\) 0 0
\(83\) 6.02722 0.661573 0.330787 0.943706i \(-0.392686\pi\)
0.330787 + 0.943706i \(0.392686\pi\)
\(84\) 0 0
\(85\) −2.48092 −0.269093
\(86\) 0 0
\(87\) 8.27178 0.886828
\(88\) 0 0
\(89\) −0.226063 −0.0239627 −0.0119813 0.999928i \(-0.503814\pi\)
−0.0119813 + 0.999928i \(0.503814\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −3.98113 −0.412824
\(94\) 0 0
\(95\) 6.98987 0.717145
\(96\) 0 0
\(97\) 1.43170 0.145367 0.0726836 0.997355i \(-0.476844\pi\)
0.0726836 + 0.997355i \(0.476844\pi\)
\(98\) 0 0
\(99\) 3.03965 0.305497
\(100\) 0 0
\(101\) −6.73578 −0.670235 −0.335117 0.942176i \(-0.608776\pi\)
−0.335117 + 0.942176i \(0.608776\pi\)
\(102\) 0 0
\(103\) −6.93264 −0.683093 −0.341547 0.939865i \(-0.610951\pi\)
−0.341547 + 0.939865i \(0.610951\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −11.6739 −1.12856 −0.564278 0.825585i \(-0.690845\pi\)
−0.564278 + 0.825585i \(0.690845\pi\)
\(108\) 0 0
\(109\) −2.20063 −0.210782 −0.105391 0.994431i \(-0.533609\pi\)
−0.105391 + 0.994431i \(0.533609\pi\)
\(110\) 0 0
\(111\) 0.260177 0.0246949
\(112\) 0 0
\(113\) 9.83479 0.925180 0.462590 0.886572i \(-0.346920\pi\)
0.462590 + 0.886572i \(0.346920\pi\)
\(114\) 0 0
\(115\) −0.0716501 −0.00668141
\(116\) 0 0
\(117\) 2.81341 0.260100
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −4.81350 −0.437591
\(122\) 0 0
\(123\) 1.33338 0.120227
\(124\) 0 0
\(125\) −12.1528 −1.08698
\(126\) 0 0
\(127\) −20.5602 −1.82443 −0.912213 0.409716i \(-0.865628\pi\)
−0.912213 + 0.409716i \(0.865628\pi\)
\(128\) 0 0
\(129\) −1.59560 −0.140484
\(130\) 0 0
\(131\) −8.26440 −0.722064 −0.361032 0.932553i \(-0.617575\pi\)
−0.361032 + 0.932553i \(0.617575\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −9.94689 −0.856092
\(136\) 0 0
\(137\) −0.681112 −0.0581913 −0.0290956 0.999577i \(-0.509263\pi\)
−0.0290956 + 0.999577i \(0.509263\pi\)
\(138\) 0 0
\(139\) −13.1700 −1.11706 −0.558531 0.829484i \(-0.688635\pi\)
−0.558531 + 0.829484i \(0.688635\pi\)
\(140\) 0 0
\(141\) −9.91941 −0.835365
\(142\) 0 0
\(143\) 5.72603 0.478835
\(144\) 0 0
\(145\) 10.9609 0.910257
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −15.2961 −1.25310 −0.626551 0.779380i \(-0.715534\pi\)
−0.626551 + 0.779380i \(0.715534\pi\)
\(150\) 0 0
\(151\) −3.98315 −0.324144 −0.162072 0.986779i \(-0.551818\pi\)
−0.162072 + 0.986779i \(0.551818\pi\)
\(152\) 0 0
\(153\) 1.71597 0.138728
\(154\) 0 0
\(155\) −5.27540 −0.423730
\(156\) 0 0
\(157\) 14.5140 1.15835 0.579173 0.815205i \(-0.303376\pi\)
0.579173 + 0.815205i \(0.303376\pi\)
\(158\) 0 0
\(159\) −15.7339 −1.24778
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 15.2886 1.19750 0.598749 0.800937i \(-0.295665\pi\)
0.598749 + 0.800937i \(0.295665\pi\)
\(164\) 0 0
\(165\) −5.85980 −0.456184
\(166\) 0 0
\(167\) −13.7533 −1.06426 −0.532130 0.846663i \(-0.678608\pi\)
−0.532130 + 0.846663i \(0.678608\pi\)
\(168\) 0 0
\(169\) −7.70017 −0.592321
\(170\) 0 0
\(171\) −4.83466 −0.369716
\(172\) 0 0
\(173\) 15.7093 1.19436 0.597179 0.802108i \(-0.296288\pi\)
0.597179 + 0.802108i \(0.296288\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −7.34489 −0.552076
\(178\) 0 0
\(179\) −6.21506 −0.464535 −0.232268 0.972652i \(-0.574615\pi\)
−0.232268 + 0.972652i \(0.574615\pi\)
\(180\) 0 0
\(181\) −15.6181 −1.16088 −0.580441 0.814302i \(-0.697120\pi\)
−0.580441 + 0.814302i \(0.697120\pi\)
\(182\) 0 0
\(183\) 5.21961 0.385845
\(184\) 0 0
\(185\) 0.344761 0.0253473
\(186\) 0 0
\(187\) 3.49245 0.255393
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −4.36348 −0.315730 −0.157865 0.987461i \(-0.550461\pi\)
−0.157865 + 0.987461i \(0.550461\pi\)
\(192\) 0 0
\(193\) −15.1674 −1.09177 −0.545887 0.837859i \(-0.683807\pi\)
−0.545887 + 0.837859i \(0.683807\pi\)
\(194\) 0 0
\(195\) −5.42364 −0.388395
\(196\) 0 0
\(197\) −4.34032 −0.309235 −0.154618 0.987974i \(-0.549415\pi\)
−0.154618 + 0.987974i \(0.549415\pi\)
\(198\) 0 0
\(199\) 3.07448 0.217944 0.108972 0.994045i \(-0.465244\pi\)
0.108972 + 0.994045i \(0.465244\pi\)
\(200\) 0 0
\(201\) 7.73157 0.545343
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 1.76687 0.123404
\(206\) 0 0
\(207\) 0.0495580 0.00344452
\(208\) 0 0
\(209\) −9.83981 −0.680634
\(210\) 0 0
\(211\) 0.480367 0.0330698 0.0165349 0.999863i \(-0.494737\pi\)
0.0165349 + 0.999863i \(0.494737\pi\)
\(212\) 0 0
\(213\) 13.7188 0.940000
\(214\) 0 0
\(215\) −2.11433 −0.144196
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −16.3985 −1.10811
\(220\) 0 0
\(221\) 3.23250 0.217442
\(222\) 0 0
\(223\) −7.20336 −0.482373 −0.241186 0.970479i \(-0.577537\pi\)
−0.241186 + 0.970479i \(0.577537\pi\)
\(224\) 0 0
\(225\) 2.29528 0.153019
\(226\) 0 0
\(227\) 7.92532 0.526022 0.263011 0.964793i \(-0.415284\pi\)
0.263011 + 0.964793i \(0.415284\pi\)
\(228\) 0 0
\(229\) 18.3980 1.21577 0.607886 0.794024i \(-0.292018\pi\)
0.607886 + 0.794024i \(0.292018\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 14.7877 0.968773 0.484386 0.874854i \(-0.339043\pi\)
0.484386 + 0.874854i \(0.339043\pi\)
\(234\) 0 0
\(235\) −13.1442 −0.857435
\(236\) 0 0
\(237\) −1.81260 −0.117741
\(238\) 0 0
\(239\) −25.7876 −1.66806 −0.834031 0.551717i \(-0.813973\pi\)
−0.834031 + 0.551717i \(0.813973\pi\)
\(240\) 0 0
\(241\) −16.1914 −1.04298 −0.521491 0.853257i \(-0.674624\pi\)
−0.521491 + 0.853257i \(0.674624\pi\)
\(242\) 0 0
\(243\) 11.7685 0.754947
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −9.10742 −0.579491
\(248\) 0 0
\(249\) 8.03660 0.509299
\(250\) 0 0
\(251\) 1.47665 0.0932054 0.0466027 0.998914i \(-0.485161\pi\)
0.0466027 + 0.998914i \(0.485161\pi\)
\(252\) 0 0
\(253\) 0.100864 0.00634124
\(254\) 0 0
\(255\) −3.30802 −0.207156
\(256\) 0 0
\(257\) −6.36748 −0.397193 −0.198596 0.980081i \(-0.563638\pi\)
−0.198596 + 0.980081i \(0.563638\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −7.58132 −0.469272
\(262\) 0 0
\(263\) 18.1131 1.11690 0.558450 0.829538i \(-0.311396\pi\)
0.558450 + 0.829538i \(0.311396\pi\)
\(264\) 0 0
\(265\) −20.8490 −1.28074
\(266\) 0 0
\(267\) −0.301430 −0.0184472
\(268\) 0 0
\(269\) −17.1746 −1.04715 −0.523577 0.851978i \(-0.675403\pi\)
−0.523577 + 0.851978i \(0.675403\pi\)
\(270\) 0 0
\(271\) −15.7499 −0.956739 −0.478369 0.878159i \(-0.658772\pi\)
−0.478369 + 0.878159i \(0.658772\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 4.67151 0.281703
\(276\) 0 0
\(277\) 19.6270 1.17927 0.589635 0.807670i \(-0.299272\pi\)
0.589635 + 0.807670i \(0.299272\pi\)
\(278\) 0 0
\(279\) 3.64882 0.218449
\(280\) 0 0
\(281\) 8.20950 0.489738 0.244869 0.969556i \(-0.421255\pi\)
0.244869 + 0.969556i \(0.421255\pi\)
\(282\) 0 0
\(283\) 17.9706 1.06824 0.534120 0.845409i \(-0.320643\pi\)
0.534120 + 0.845409i \(0.320643\pi\)
\(284\) 0 0
\(285\) 9.32018 0.552080
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −15.0284 −0.884024
\(290\) 0 0
\(291\) 1.90901 0.111908
\(292\) 0 0
\(293\) 7.45654 0.435616 0.217808 0.975992i \(-0.430109\pi\)
0.217808 + 0.975992i \(0.430109\pi\)
\(294\) 0 0
\(295\) −9.73273 −0.566662
\(296\) 0 0
\(297\) 14.0025 0.812506
\(298\) 0 0
\(299\) 0.0933562 0.00539893
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −8.98138 −0.515967
\(304\) 0 0
\(305\) 6.91652 0.396039
\(306\) 0 0
\(307\) 12.0440 0.687387 0.343693 0.939082i \(-0.388322\pi\)
0.343693 + 0.939082i \(0.388322\pi\)
\(308\) 0 0
\(309\) −9.24388 −0.525866
\(310\) 0 0
\(311\) 26.7692 1.51794 0.758970 0.651126i \(-0.225703\pi\)
0.758970 + 0.651126i \(0.225703\pi\)
\(312\) 0 0
\(313\) −2.24338 −0.126804 −0.0634018 0.997988i \(-0.520195\pi\)
−0.0634018 + 0.997988i \(0.520195\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −28.4626 −1.59862 −0.799309 0.600921i \(-0.794801\pi\)
−0.799309 + 0.600921i \(0.794801\pi\)
\(318\) 0 0
\(319\) −15.4300 −0.863914
\(320\) 0 0
\(321\) −15.5658 −0.868796
\(322\) 0 0
\(323\) −5.55485 −0.309080
\(324\) 0 0
\(325\) 4.32380 0.239841
\(326\) 0 0
\(327\) −2.93428 −0.162266
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 4.23741 0.232909 0.116455 0.993196i \(-0.462847\pi\)
0.116455 + 0.993196i \(0.462847\pi\)
\(332\) 0 0
\(333\) −0.238460 −0.0130675
\(334\) 0 0
\(335\) 10.2451 0.559751
\(336\) 0 0
\(337\) −9.10406 −0.495930 −0.247965 0.968769i \(-0.579762\pi\)
−0.247965 + 0.968769i \(0.579762\pi\)
\(338\) 0 0
\(339\) 13.1136 0.712231
\(340\) 0 0
\(341\) 7.42631 0.402157
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −0.0955372 −0.00514355
\(346\) 0 0
\(347\) 10.1193 0.543233 0.271617 0.962406i \(-0.412442\pi\)
0.271617 + 0.962406i \(0.412442\pi\)
\(348\) 0 0
\(349\) 7.60171 0.406910 0.203455 0.979084i \(-0.434783\pi\)
0.203455 + 0.979084i \(0.434783\pi\)
\(350\) 0 0
\(351\) 12.9603 0.691767
\(352\) 0 0
\(353\) 9.34227 0.497239 0.248619 0.968601i \(-0.420023\pi\)
0.248619 + 0.968601i \(0.420023\pi\)
\(354\) 0 0
\(355\) 18.1789 0.964834
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −4.57052 −0.241223 −0.120611 0.992700i \(-0.538485\pi\)
−0.120611 + 0.992700i \(0.538485\pi\)
\(360\) 0 0
\(361\) −3.34948 −0.176289
\(362\) 0 0
\(363\) −6.41825 −0.336871
\(364\) 0 0
\(365\) −21.7297 −1.13738
\(366\) 0 0
\(367\) 32.6165 1.70257 0.851283 0.524707i \(-0.175825\pi\)
0.851283 + 0.524707i \(0.175825\pi\)
\(368\) 0 0
\(369\) −1.22209 −0.0636192
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −25.5720 −1.32407 −0.662034 0.749474i \(-0.730306\pi\)
−0.662034 + 0.749474i \(0.730306\pi\)
\(374\) 0 0
\(375\) −16.2044 −0.836792
\(376\) 0 0
\(377\) −14.2815 −0.735536
\(378\) 0 0
\(379\) 32.0887 1.64828 0.824142 0.566383i \(-0.191658\pi\)
0.824142 + 0.566383i \(0.191658\pi\)
\(380\) 0 0
\(381\) −27.4147 −1.40450
\(382\) 0 0
\(383\) 0.470160 0.0240240 0.0120120 0.999928i \(-0.496176\pi\)
0.0120120 + 0.999928i \(0.496176\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 1.46241 0.0743385
\(388\) 0 0
\(389\) −15.4769 −0.784710 −0.392355 0.919814i \(-0.628340\pi\)
−0.392355 + 0.919814i \(0.628340\pi\)
\(390\) 0 0
\(391\) 0.0569403 0.00287960
\(392\) 0 0
\(393\) −11.0196 −0.555866
\(394\) 0 0
\(395\) −2.40189 −0.120852
\(396\) 0 0
\(397\) 28.3764 1.42417 0.712086 0.702092i \(-0.247750\pi\)
0.712086 + 0.702092i \(0.247750\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −25.0305 −1.24996 −0.624982 0.780639i \(-0.714894\pi\)
−0.624982 + 0.780639i \(0.714894\pi\)
\(402\) 0 0
\(403\) 6.87356 0.342396
\(404\) 0 0
\(405\) −6.78522 −0.337160
\(406\) 0 0
\(407\) −0.485328 −0.0240568
\(408\) 0 0
\(409\) 25.1559 1.24388 0.621940 0.783065i \(-0.286345\pi\)
0.621940 + 0.783065i \(0.286345\pi\)
\(410\) 0 0
\(411\) −0.908184 −0.0447974
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 10.6493 0.522754
\(416\) 0 0
\(417\) −17.5606 −0.859947
\(418\) 0 0
\(419\) −3.32306 −0.162342 −0.0811710 0.996700i \(-0.525866\pi\)
−0.0811710 + 0.996700i \(0.525866\pi\)
\(420\) 0 0
\(421\) −3.67892 −0.179300 −0.0896498 0.995973i \(-0.528575\pi\)
−0.0896498 + 0.995973i \(0.528575\pi\)
\(422\) 0 0
\(423\) 9.09143 0.442040
\(424\) 0 0
\(425\) 2.63720 0.127923
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 7.63500 0.368621
\(430\) 0 0
\(431\) 24.8602 1.19747 0.598737 0.800946i \(-0.295670\pi\)
0.598737 + 0.800946i \(0.295670\pi\)
\(432\) 0 0
\(433\) 38.5625 1.85320 0.926599 0.376052i \(-0.122719\pi\)
0.926599 + 0.376052i \(0.122719\pi\)
\(434\) 0 0
\(435\) 14.6152 0.700743
\(436\) 0 0
\(437\) −0.160427 −0.00767425
\(438\) 0 0
\(439\) 25.4440 1.21437 0.607187 0.794559i \(-0.292298\pi\)
0.607187 + 0.794559i \(0.292298\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 3.76272 0.178772 0.0893860 0.995997i \(-0.471510\pi\)
0.0893860 + 0.995997i \(0.471510\pi\)
\(444\) 0 0
\(445\) −0.399425 −0.0189346
\(446\) 0 0
\(447\) −20.3955 −0.964676
\(448\) 0 0
\(449\) 29.9012 1.41113 0.705563 0.708647i \(-0.250694\pi\)
0.705563 + 0.708647i \(0.250694\pi\)
\(450\) 0 0
\(451\) −2.48727 −0.117121
\(452\) 0 0
\(453\) −5.31107 −0.249536
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −12.1086 −0.566415 −0.283208 0.959059i \(-0.591399\pi\)
−0.283208 + 0.959059i \(0.591399\pi\)
\(458\) 0 0
\(459\) 7.90479 0.368964
\(460\) 0 0
\(461\) −3.92540 −0.182824 −0.0914120 0.995813i \(-0.529138\pi\)
−0.0914120 + 0.995813i \(0.529138\pi\)
\(462\) 0 0
\(463\) −13.6329 −0.633575 −0.316788 0.948496i \(-0.602604\pi\)
−0.316788 + 0.948496i \(0.602604\pi\)
\(464\) 0 0
\(465\) −7.03414 −0.326200
\(466\) 0 0
\(467\) 11.9213 0.551651 0.275826 0.961208i \(-0.411049\pi\)
0.275826 + 0.961208i \(0.411049\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 19.3528 0.891729
\(472\) 0 0
\(473\) 2.97639 0.136855
\(474\) 0 0
\(475\) −7.43018 −0.340920
\(476\) 0 0
\(477\) 14.4205 0.660272
\(478\) 0 0
\(479\) 2.55874 0.116912 0.0584558 0.998290i \(-0.481382\pi\)
0.0584558 + 0.998290i \(0.481382\pi\)
\(480\) 0 0
\(481\) −0.449205 −0.0204820
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 2.52963 0.114865
\(486\) 0 0
\(487\) 1.99748 0.0905143 0.0452572 0.998975i \(-0.485589\pi\)
0.0452572 + 0.998975i \(0.485589\pi\)
\(488\) 0 0
\(489\) 20.3856 0.921869
\(490\) 0 0
\(491\) 8.72991 0.393975 0.196988 0.980406i \(-0.436884\pi\)
0.196988 + 0.980406i \(0.436884\pi\)
\(492\) 0 0
\(493\) −8.71066 −0.392308
\(494\) 0 0
\(495\) 5.37067 0.241394
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −9.82275 −0.439727 −0.219863 0.975531i \(-0.570561\pi\)
−0.219863 + 0.975531i \(0.570561\pi\)
\(500\) 0 0
\(501\) −18.3384 −0.819299
\(502\) 0 0
\(503\) 16.7626 0.747407 0.373703 0.927548i \(-0.378088\pi\)
0.373703 + 0.927548i \(0.378088\pi\)
\(504\) 0 0
\(505\) −11.9012 −0.529598
\(506\) 0 0
\(507\) −10.2673 −0.455986
\(508\) 0 0
\(509\) 8.21673 0.364200 0.182100 0.983280i \(-0.441711\pi\)
0.182100 + 0.983280i \(0.441711\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −22.2714 −0.983305
\(514\) 0 0
\(515\) −12.2491 −0.539759
\(516\) 0 0
\(517\) 18.5035 0.813781
\(518\) 0 0
\(519\) 20.9466 0.919452
\(520\) 0 0
\(521\) −22.4558 −0.983805 −0.491903 0.870650i \(-0.663698\pi\)
−0.491903 + 0.870650i \(0.663698\pi\)
\(522\) 0 0
\(523\) −30.4841 −1.33298 −0.666488 0.745516i \(-0.732203\pi\)
−0.666488 + 0.745516i \(0.732203\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 4.19236 0.182622
\(528\) 0 0
\(529\) −22.9984 −0.999929
\(530\) 0 0
\(531\) 6.73181 0.292136
\(532\) 0 0
\(533\) −2.30214 −0.0997166
\(534\) 0 0
\(535\) −20.6262 −0.891749
\(536\) 0 0
\(537\) −8.28706 −0.357613
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 18.4708 0.794120 0.397060 0.917793i \(-0.370031\pi\)
0.397060 + 0.917793i \(0.370031\pi\)
\(542\) 0 0
\(543\) −20.8249 −0.893682
\(544\) 0 0
\(545\) −3.88823 −0.166553
\(546\) 0 0
\(547\) −26.1709 −1.11899 −0.559494 0.828834i \(-0.689005\pi\)
−0.559494 + 0.828834i \(0.689005\pi\)
\(548\) 0 0
\(549\) −4.78392 −0.204173
\(550\) 0 0
\(551\) 24.5419 1.04552
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0.459699 0.0195131
\(556\) 0 0
\(557\) 12.2203 0.517791 0.258896 0.965905i \(-0.416641\pi\)
0.258896 + 0.965905i \(0.416641\pi\)
\(558\) 0 0
\(559\) 2.75485 0.116518
\(560\) 0 0
\(561\) 4.65678 0.196609
\(562\) 0 0
\(563\) 29.2098 1.23105 0.615524 0.788118i \(-0.288945\pi\)
0.615524 + 0.788118i \(0.288945\pi\)
\(564\) 0 0
\(565\) 17.3768 0.731048
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 31.3875 1.31583 0.657916 0.753092i \(-0.271438\pi\)
0.657916 + 0.753092i \(0.271438\pi\)
\(570\) 0 0
\(571\) −38.7026 −1.61965 −0.809826 0.586670i \(-0.800439\pi\)
−0.809826 + 0.586670i \(0.800439\pi\)
\(572\) 0 0
\(573\) −5.81820 −0.243059
\(574\) 0 0
\(575\) 0.0761635 0.00317624
\(576\) 0 0
\(577\) 31.3521 1.30521 0.652603 0.757700i \(-0.273677\pi\)
0.652603 + 0.757700i \(0.273677\pi\)
\(578\) 0 0
\(579\) −20.2240 −0.840480
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 29.3496 1.21554
\(584\) 0 0
\(585\) 4.97093 0.205523
\(586\) 0 0
\(587\) −7.78318 −0.321246 −0.160623 0.987016i \(-0.551350\pi\)
−0.160623 + 0.987016i \(0.551350\pi\)
\(588\) 0 0
\(589\) −11.8118 −0.486696
\(590\) 0 0
\(591\) −5.78732 −0.238059
\(592\) 0 0
\(593\) −37.5107 −1.54038 −0.770190 0.637815i \(-0.779838\pi\)
−0.770190 + 0.637815i \(0.779838\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 4.09946 0.167780
\(598\) 0 0
\(599\) 21.9236 0.895775 0.447887 0.894090i \(-0.352177\pi\)
0.447887 + 0.894090i \(0.352177\pi\)
\(600\) 0 0
\(601\) −25.9765 −1.05960 −0.529802 0.848122i \(-0.677734\pi\)
−0.529802 + 0.848122i \(0.677734\pi\)
\(602\) 0 0
\(603\) −7.08621 −0.288573
\(604\) 0 0
\(605\) −8.50484 −0.345771
\(606\) 0 0
\(607\) −18.6351 −0.756375 −0.378188 0.925729i \(-0.623453\pi\)
−0.378188 + 0.925729i \(0.623453\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 17.1262 0.692853
\(612\) 0 0
\(613\) −26.7402 −1.08003 −0.540013 0.841657i \(-0.681581\pi\)
−0.540013 + 0.841657i \(0.681581\pi\)
\(614\) 0 0
\(615\) 2.35592 0.0949998
\(616\) 0 0
\(617\) 3.57592 0.143961 0.0719805 0.997406i \(-0.477068\pi\)
0.0719805 + 0.997406i \(0.477068\pi\)
\(618\) 0 0
\(619\) 3.31677 0.133312 0.0666561 0.997776i \(-0.478767\pi\)
0.0666561 + 0.997776i \(0.478767\pi\)
\(620\) 0 0
\(621\) 0.228294 0.00916113
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −12.0816 −0.483265
\(626\) 0 0
\(627\) −13.1203 −0.523972
\(628\) 0 0
\(629\) −0.273981 −0.0109244
\(630\) 0 0
\(631\) −41.9093 −1.66838 −0.834191 0.551475i \(-0.814065\pi\)
−0.834191 + 0.551475i \(0.814065\pi\)
\(632\) 0 0
\(633\) 0.640514 0.0254581
\(634\) 0 0
\(635\) −36.3273 −1.44160
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −12.5737 −0.497409
\(640\) 0 0
\(641\) −24.3984 −0.963681 −0.481840 0.876259i \(-0.660031\pi\)
−0.481840 + 0.876259i \(0.660031\pi\)
\(642\) 0 0
\(643\) −5.91248 −0.233166 −0.116583 0.993181i \(-0.537194\pi\)
−0.116583 + 0.993181i \(0.537194\pi\)
\(644\) 0 0
\(645\) −2.81921 −0.111006
\(646\) 0 0
\(647\) −19.0189 −0.747709 −0.373855 0.927487i \(-0.621964\pi\)
−0.373855 + 0.927487i \(0.621964\pi\)
\(648\) 0 0
\(649\) 13.7010 0.537812
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 13.9021 0.544032 0.272016 0.962293i \(-0.412310\pi\)
0.272016 + 0.962293i \(0.412310\pi\)
\(654\) 0 0
\(655\) −14.6021 −0.570552
\(656\) 0 0
\(657\) 15.0297 0.586365
\(658\) 0 0
\(659\) 20.2641 0.789379 0.394689 0.918815i \(-0.370852\pi\)
0.394689 + 0.918815i \(0.370852\pi\)
\(660\) 0 0
\(661\) −2.13328 −0.0829749 −0.0414875 0.999139i \(-0.513210\pi\)
−0.0414875 + 0.999139i \(0.513210\pi\)
\(662\) 0 0
\(663\) 4.31017 0.167393
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −0.251568 −0.00974076
\(668\) 0 0
\(669\) −9.60485 −0.371345
\(670\) 0 0
\(671\) −9.73655 −0.375875
\(672\) 0 0
\(673\) −2.81059 −0.108340 −0.0541702 0.998532i \(-0.517251\pi\)
−0.0541702 + 0.998532i \(0.517251\pi\)
\(674\) 0 0
\(675\) 10.5735 0.406973
\(676\) 0 0
\(677\) −11.0339 −0.424066 −0.212033 0.977262i \(-0.568008\pi\)
−0.212033 + 0.977262i \(0.568008\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 10.5675 0.404948
\(682\) 0 0
\(683\) 26.7222 1.02250 0.511248 0.859433i \(-0.329183\pi\)
0.511248 + 0.859433i \(0.329183\pi\)
\(684\) 0 0
\(685\) −1.20344 −0.0459809
\(686\) 0 0
\(687\) 24.5316 0.935938
\(688\) 0 0
\(689\) 27.1651 1.03491
\(690\) 0 0
\(691\) 22.7729 0.866320 0.433160 0.901317i \(-0.357398\pi\)
0.433160 + 0.901317i \(0.357398\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −23.2696 −0.882667
\(696\) 0 0
\(697\) −1.40413 −0.0531853
\(698\) 0 0
\(699\) 19.7177 0.745790
\(700\) 0 0
\(701\) −51.9521 −1.96220 −0.981102 0.193494i \(-0.938018\pi\)
−0.981102 + 0.193494i \(0.938018\pi\)
\(702\) 0 0
\(703\) 0.771929 0.0291139
\(704\) 0 0
\(705\) −17.5263 −0.660079
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −6.70478 −0.251803 −0.125902 0.992043i \(-0.540182\pi\)
−0.125902 + 0.992043i \(0.540182\pi\)
\(710\) 0 0
\(711\) 1.66131 0.0623038
\(712\) 0 0
\(713\) 0.121077 0.00453438
\(714\) 0 0
\(715\) 10.1171 0.378360
\(716\) 0 0
\(717\) −34.3848 −1.28412
\(718\) 0 0
\(719\) −29.1180 −1.08592 −0.542959 0.839759i \(-0.682696\pi\)
−0.542959 + 0.839759i \(0.682696\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −21.5894 −0.802919
\(724\) 0 0
\(725\) −11.6514 −0.432722
\(726\) 0 0
\(727\) −30.2981 −1.12369 −0.561846 0.827242i \(-0.689909\pi\)
−0.561846 + 0.827242i \(0.689909\pi\)
\(728\) 0 0
\(729\) 27.2126 1.00788
\(730\) 0 0
\(731\) 1.68026 0.0621465
\(732\) 0 0
\(733\) 20.3061 0.750021 0.375011 0.927020i \(-0.377639\pi\)
0.375011 + 0.927020i \(0.377639\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −14.4223 −0.531252
\(738\) 0 0
\(739\) −7.74185 −0.284788 −0.142394 0.989810i \(-0.545480\pi\)
−0.142394 + 0.989810i \(0.545480\pi\)
\(740\) 0 0
\(741\) −12.1437 −0.446110
\(742\) 0 0
\(743\) 31.5135 1.15612 0.578058 0.815995i \(-0.303811\pi\)
0.578058 + 0.815995i \(0.303811\pi\)
\(744\) 0 0
\(745\) −27.0262 −0.990162
\(746\) 0 0
\(747\) −7.36578 −0.269500
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 25.1877 0.919112 0.459556 0.888149i \(-0.348008\pi\)
0.459556 + 0.888149i \(0.348008\pi\)
\(752\) 0 0
\(753\) 1.96894 0.0717523
\(754\) 0 0
\(755\) −7.03771 −0.256128
\(756\) 0 0
\(757\) −28.2989 −1.02854 −0.514270 0.857628i \(-0.671937\pi\)
−0.514270 + 0.857628i \(0.671937\pi\)
\(758\) 0 0
\(759\) 0.134490 0.00488168
\(760\) 0 0
\(761\) −9.10751 −0.330147 −0.165074 0.986281i \(-0.552786\pi\)
−0.165074 + 0.986281i \(0.552786\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 3.03189 0.109618
\(766\) 0 0
\(767\) 12.6812 0.457893
\(768\) 0 0
\(769\) 16.0188 0.577653 0.288826 0.957381i \(-0.406735\pi\)
0.288826 + 0.957381i \(0.406735\pi\)
\(770\) 0 0
\(771\) −8.49031 −0.305771
\(772\) 0 0
\(773\) 30.3488 1.09157 0.545786 0.837925i \(-0.316231\pi\)
0.545786 + 0.837925i \(0.316231\pi\)
\(774\) 0 0
\(775\) 5.60771 0.201435
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 3.95607 0.141741
\(780\) 0 0
\(781\) −25.5908 −0.915712
\(782\) 0 0
\(783\) −34.9241 −1.24809
\(784\) 0 0
\(785\) 25.6444 0.915288
\(786\) 0 0
\(787\) 33.9886 1.21156 0.605781 0.795631i \(-0.292861\pi\)
0.605781 + 0.795631i \(0.292861\pi\)
\(788\) 0 0
\(789\) 24.1517 0.859823
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −9.01185 −0.320020
\(794\) 0 0
\(795\) −27.7997 −0.985954
\(796\) 0 0
\(797\) 44.5444 1.57784 0.788922 0.614494i \(-0.210640\pi\)
0.788922 + 0.614494i \(0.210640\pi\)
\(798\) 0 0
\(799\) 10.4457 0.369543
\(800\) 0 0
\(801\) 0.276269 0.00976148
\(802\) 0 0
\(803\) 30.5894 1.07948
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −22.9004 −0.806131
\(808\) 0 0
\(809\) −26.8608 −0.944376 −0.472188 0.881498i \(-0.656536\pi\)
−0.472188 + 0.881498i \(0.656536\pi\)
\(810\) 0 0
\(811\) 38.5647 1.35419 0.677095 0.735896i \(-0.263239\pi\)
0.677095 + 0.735896i \(0.263239\pi\)
\(812\) 0 0
\(813\) −21.0007 −0.736526
\(814\) 0 0
\(815\) 27.0130 0.946224
\(816\) 0 0
\(817\) −4.73404 −0.165623
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −27.8147 −0.970741 −0.485370 0.874309i \(-0.661315\pi\)
−0.485370 + 0.874309i \(0.661315\pi\)
\(822\) 0 0
\(823\) −22.7746 −0.793872 −0.396936 0.917846i \(-0.629927\pi\)
−0.396936 + 0.917846i \(0.629927\pi\)
\(824\) 0 0
\(825\) 6.22892 0.216863
\(826\) 0 0
\(827\) 36.6139 1.27319 0.636595 0.771198i \(-0.280342\pi\)
0.636595 + 0.771198i \(0.280342\pi\)
\(828\) 0 0
\(829\) 26.6456 0.925440 0.462720 0.886504i \(-0.346873\pi\)
0.462720 + 0.886504i \(0.346873\pi\)
\(830\) 0 0
\(831\) 26.1703 0.907837
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −24.3003 −0.840945
\(836\) 0 0
\(837\) 16.8087 0.580992
\(838\) 0 0
\(839\) 14.1618 0.488920 0.244460 0.969659i \(-0.421389\pi\)
0.244460 + 0.969659i \(0.421389\pi\)
\(840\) 0 0
\(841\) 9.48458 0.327054
\(842\) 0 0
\(843\) 10.9464 0.377015
\(844\) 0 0
\(845\) −13.6052 −0.468033
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 23.9617 0.822363
\(850\) 0 0
\(851\) −0.00791271 −0.000271244 0
\(852\) 0 0
\(853\) 14.6813 0.502677 0.251338 0.967899i \(-0.419129\pi\)
0.251338 + 0.967899i \(0.419129\pi\)
\(854\) 0 0
\(855\) −8.54222 −0.292138
\(856\) 0 0
\(857\) −3.60916 −0.123287 −0.0616433 0.998098i \(-0.519634\pi\)
−0.0616433 + 0.998098i \(0.519634\pi\)
\(858\) 0 0
\(859\) −44.9615 −1.53407 −0.767033 0.641607i \(-0.778268\pi\)
−0.767033 + 0.641607i \(0.778268\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −8.52948 −0.290347 −0.145173 0.989406i \(-0.546374\pi\)
−0.145173 + 0.989406i \(0.546374\pi\)
\(864\) 0 0
\(865\) 27.7563 0.943744
\(866\) 0 0
\(867\) −20.0387 −0.680548
\(868\) 0 0
\(869\) 3.38119 0.114699
\(870\) 0 0
\(871\) −13.3488 −0.452308
\(872\) 0 0
\(873\) −1.74966 −0.0592171
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −8.11893 −0.274157 −0.137078 0.990560i \(-0.543771\pi\)
−0.137078 + 0.990560i \(0.543771\pi\)
\(878\) 0 0
\(879\) 9.94244 0.335350
\(880\) 0 0
\(881\) 38.5733 1.29957 0.649784 0.760119i \(-0.274859\pi\)
0.649784 + 0.760119i \(0.274859\pi\)
\(882\) 0 0
\(883\) −37.4122 −1.25902 −0.629510 0.776992i \(-0.716744\pi\)
−0.629510 + 0.776992i \(0.716744\pi\)
\(884\) 0 0
\(885\) −12.9775 −0.436233
\(886\) 0 0
\(887\) 21.1258 0.709333 0.354667 0.934993i \(-0.384594\pi\)
0.354667 + 0.934993i \(0.384594\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 9.55173 0.319995
\(892\) 0 0
\(893\) −29.4303 −0.984848
\(894\) 0 0
\(895\) −10.9812 −0.367061
\(896\) 0 0
\(897\) 0.124480 0.00415626
\(898\) 0 0
\(899\) −18.5223 −0.617752
\(900\) 0 0
\(901\) 16.5687 0.551983
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −27.5951 −0.917292
\(906\) 0 0
\(907\) −19.6698 −0.653124 −0.326562 0.945176i \(-0.605890\pi\)
−0.326562 + 0.945176i \(0.605890\pi\)
\(908\) 0 0
\(909\) 8.23169 0.273028
\(910\) 0 0
\(911\) −32.5069 −1.07700 −0.538501 0.842625i \(-0.681009\pi\)
−0.538501 + 0.842625i \(0.681009\pi\)
\(912\) 0 0
\(913\) −14.9913 −0.496140
\(914\) 0 0
\(915\) 9.22238 0.304882
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 40.7307 1.34358 0.671791 0.740741i \(-0.265525\pi\)
0.671791 + 0.740741i \(0.265525\pi\)
\(920\) 0 0
\(921\) 16.0593 0.529171
\(922\) 0 0
\(923\) −23.6861 −0.779637
\(924\) 0 0
\(925\) −0.366478 −0.0120497
\(926\) 0 0
\(927\) 8.47228 0.278266
\(928\) 0 0
\(929\) 12.2903 0.403231 0.201616 0.979465i \(-0.435381\pi\)
0.201616 + 0.979465i \(0.435381\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 35.6936 1.16856
\(934\) 0 0
\(935\) 6.17071 0.201804
\(936\) 0 0
\(937\) −57.2280 −1.86956 −0.934779 0.355229i \(-0.884403\pi\)
−0.934779 + 0.355229i \(0.884403\pi\)
\(938\) 0 0
\(939\) −2.99129 −0.0976172
\(940\) 0 0
\(941\) 2.96786 0.0967496 0.0483748 0.998829i \(-0.484596\pi\)
0.0483748 + 0.998829i \(0.484596\pi\)
\(942\) 0 0
\(943\) −0.0405520 −0.00132055
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 9.66991 0.314230 0.157115 0.987580i \(-0.449781\pi\)
0.157115 + 0.987580i \(0.449781\pi\)
\(948\) 0 0
\(949\) 28.3126 0.919067
\(950\) 0 0
\(951\) −37.9515 −1.23066
\(952\) 0 0
\(953\) −14.5269 −0.470574 −0.235287 0.971926i \(-0.575603\pi\)
−0.235287 + 0.971926i \(0.575603\pi\)
\(954\) 0 0
\(955\) −7.70970 −0.249480
\(956\) 0 0
\(957\) −20.5741 −0.665067
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −22.0854 −0.712433
\(962\) 0 0
\(963\) 14.2665 0.459730
\(964\) 0 0
\(965\) −26.7988 −0.862685
\(966\) 0 0
\(967\) 43.5318 1.39989 0.699944 0.714198i \(-0.253208\pi\)
0.699944 + 0.714198i \(0.253208\pi\)
\(968\) 0 0
\(969\) −7.40675 −0.237939
\(970\) 0 0
\(971\) 21.9986 0.705969 0.352985 0.935629i \(-0.385167\pi\)
0.352985 + 0.935629i \(0.385167\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 5.76529 0.184637
\(976\) 0 0
\(977\) 14.7930 0.473271 0.236636 0.971598i \(-0.423955\pi\)
0.236636 + 0.971598i \(0.423955\pi\)
\(978\) 0 0
\(979\) 0.562280 0.0179706
\(980\) 0 0
\(981\) 2.68936 0.0858646
\(982\) 0 0
\(983\) −44.5499 −1.42092 −0.710460 0.703738i \(-0.751513\pi\)
−0.710460 + 0.703738i \(0.751513\pi\)
\(984\) 0 0
\(985\) −7.66879 −0.244348
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0.0485266 0.00154306
\(990\) 0 0
\(991\) −17.6446 −0.560500 −0.280250 0.959927i \(-0.590417\pi\)
−0.280250 + 0.959927i \(0.590417\pi\)
\(992\) 0 0
\(993\) 5.65010 0.179301
\(994\) 0 0
\(995\) 5.43220 0.172212
\(996\) 0 0
\(997\) 55.7934 1.76699 0.883497 0.468437i \(-0.155183\pi\)
0.883497 + 0.468437i \(0.155183\pi\)
\(998\) 0 0
\(999\) −1.09849 −0.0347546
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 8036.2.a.m.1.6 8
7.2 even 3 1148.2.i.d.165.3 16
7.4 even 3 1148.2.i.d.821.3 yes 16
7.6 odd 2 8036.2.a.n.1.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1148.2.i.d.165.3 16 7.2 even 3
1148.2.i.d.821.3 yes 16 7.4 even 3
8036.2.a.m.1.6 8 1.1 even 1 trivial
8036.2.a.n.1.3 8 7.6 odd 2