Properties

Label 6552.2.a.bb.1.2
Level $6552$
Weight $2$
Character 6552.1
Self dual yes
Analytic conductor $52.318$
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] = [6552,2,Mod(1,6552)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6552, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("6552.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6552 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6552.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: \(52.3179834043\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 8 \) 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 1.2
Root \(-2.37228\) of defining polynomial
Character \(\chi\) \(=\) 6552.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.37228 q^{5} -1.00000 q^{7} +O(q^{10})\) \(q+1.37228 q^{5} -1.00000 q^{7} -5.37228 q^{11} -1.00000 q^{13} +5.37228 q^{17} +7.37228 q^{19} -1.37228 q^{23} -3.11684 q^{25} -4.62772 q^{29} -10.7446 q^{31} -1.37228 q^{35} +5.37228 q^{37} +6.00000 q^{41} +7.37228 q^{43} -9.48913 q^{47} +1.00000 q^{49} +2.74456 q^{53} -7.37228 q^{55} +2.74456 q^{59} -12.1168 q^{61} -1.37228 q^{65} +12.0000 q^{67} -10.0000 q^{71} +9.37228 q^{73} +5.37228 q^{77} -14.7446 q^{79} +1.25544 q^{83} +7.37228 q^{85} -15.4891 q^{89} +1.00000 q^{91} +10.1168 q^{95} -15.4891 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{5} - 2 q^{7} - 5 q^{11} - 2 q^{13} + 5 q^{17} + 9 q^{19} + 3 q^{23} + 11 q^{25} - 15 q^{29} - 10 q^{31} + 3 q^{35} + 5 q^{37} + 12 q^{41} + 9 q^{43} + 4 q^{47} + 2 q^{49} - 6 q^{53} - 9 q^{55} - 6 q^{59} - 7 q^{61} + 3 q^{65} + 24 q^{67} - 20 q^{71} + 13 q^{73} + 5 q^{77} - 18 q^{79} + 14 q^{83} + 9 q^{85} - 8 q^{89} + 2 q^{91} + 3 q^{95} - 8 q^{97}+O(q^{100}) \) 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
\(5\) 1.37228 0.613703 0.306851 0.951757i \(-0.400725\pi\)
0.306851 + 0.951757i \(0.400725\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −5.37228 −1.61980 −0.809902 0.586565i \(-0.800480\pi\)
−0.809902 + 0.586565i \(0.800480\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.37228 1.30297 0.651485 0.758662i \(-0.274146\pi\)
0.651485 + 0.758662i \(0.274146\pi\)
\(18\) 0 0
\(19\) 7.37228 1.69132 0.845659 0.533724i \(-0.179208\pi\)
0.845659 + 0.533724i \(0.179208\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.37228 −0.286140 −0.143070 0.989713i \(-0.545697\pi\)
−0.143070 + 0.989713i \(0.545697\pi\)
\(24\) 0 0
\(25\) −3.11684 −0.623369
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −4.62772 −0.859346 −0.429673 0.902985i \(-0.641371\pi\)
−0.429673 + 0.902985i \(0.641371\pi\)
\(30\) 0 0
\(31\) −10.7446 −1.92978 −0.964890 0.262654i \(-0.915402\pi\)
−0.964890 + 0.262654i \(0.915402\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.37228 −0.231958
\(36\) 0 0
\(37\) 5.37228 0.883198 0.441599 0.897213i \(-0.354411\pi\)
0.441599 + 0.897213i \(0.354411\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 7.37228 1.12426 0.562131 0.827048i \(-0.309982\pi\)
0.562131 + 0.827048i \(0.309982\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −9.48913 −1.38413 −0.692066 0.721835i \(-0.743299\pi\)
−0.692066 + 0.721835i \(0.743299\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.74456 0.376995 0.188497 0.982074i \(-0.439638\pi\)
0.188497 + 0.982074i \(0.439638\pi\)
\(54\) 0 0
\(55\) −7.37228 −0.994078
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.74456 0.357312 0.178656 0.983912i \(-0.442825\pi\)
0.178656 + 0.983912i \(0.442825\pi\)
\(60\) 0 0
\(61\) −12.1168 −1.55140 −0.775701 0.631100i \(-0.782604\pi\)
−0.775701 + 0.631100i \(0.782604\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.37228 −0.170211
\(66\) 0 0
\(67\) 12.0000 1.46603 0.733017 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −10.0000 −1.18678 −0.593391 0.804914i \(-0.702211\pi\)
−0.593391 + 0.804914i \(0.702211\pi\)
\(72\) 0 0
\(73\) 9.37228 1.09694 0.548471 0.836169i \(-0.315210\pi\)
0.548471 + 0.836169i \(0.315210\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.37228 0.612228
\(78\) 0 0
\(79\) −14.7446 −1.65889 −0.829446 0.558586i \(-0.811344\pi\)
−0.829446 + 0.558586i \(0.811344\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1.25544 0.137802 0.0689011 0.997623i \(-0.478051\pi\)
0.0689011 + 0.997623i \(0.478051\pi\)
\(84\) 0 0
\(85\) 7.37228 0.799636
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −15.4891 −1.64184 −0.820922 0.571040i \(-0.806540\pi\)
−0.820922 + 0.571040i \(0.806540\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 10.1168 1.03797
\(96\) 0 0
\(97\) −15.4891 −1.57268 −0.786341 0.617792i \(-0.788027\pi\)
−0.786341 + 0.617792i \(0.788027\pi\)
\(98\) 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
\(103\) −6.11684 −0.602711 −0.301355 0.953512i \(-0.597439\pi\)
−0.301355 + 0.953512i \(0.597439\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −10.0000 −0.966736 −0.483368 0.875417i \(-0.660587\pi\)
−0.483368 + 0.875417i \(0.660587\pi\)
\(108\) 0 0
\(109\) 4.11684 0.394322 0.197161 0.980371i \(-0.436828\pi\)
0.197161 + 0.980371i \(0.436828\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 13.4891 1.26895 0.634475 0.772943i \(-0.281216\pi\)
0.634475 + 0.772943i \(0.281216\pi\)
\(114\) 0 0
\(115\) −1.88316 −0.175605
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −5.37228 −0.492476
\(120\) 0 0
\(121\) 17.8614 1.62376
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −11.1386 −0.996266
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −12.6277 −1.10329 −0.551644 0.834079i \(-0.685999\pi\)
−0.551644 + 0.834079i \(0.685999\pi\)
\(132\) 0 0
\(133\) −7.37228 −0.639258
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 12.8614 1.09882 0.549412 0.835552i \(-0.314852\pi\)
0.549412 + 0.835552i \(0.314852\pi\)
\(138\) 0 0
\(139\) −20.2337 −1.71620 −0.858100 0.513483i \(-0.828355\pi\)
−0.858100 + 0.513483i \(0.828355\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 5.37228 0.449253
\(144\) 0 0
\(145\) −6.35053 −0.527383
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −17.4891 −1.43276 −0.716382 0.697708i \(-0.754203\pi\)
−0.716382 + 0.697708i \(0.754203\pi\)
\(150\) 0 0
\(151\) −12.6277 −1.02763 −0.513815 0.857901i \(-0.671768\pi\)
−0.513815 + 0.857901i \(0.671768\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −14.7446 −1.18431
\(156\) 0 0
\(157\) 17.3723 1.38646 0.693229 0.720717i \(-0.256187\pi\)
0.693229 + 0.720717i \(0.256187\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 1.37228 0.108151
\(162\) 0 0
\(163\) −2.74456 −0.214971 −0.107485 0.994207i \(-0.534280\pi\)
−0.107485 + 0.994207i \(0.534280\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.88316 0.145723 0.0728615 0.997342i \(-0.476787\pi\)
0.0728615 + 0.997342i \(0.476787\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −22.0000 −1.67263 −0.836315 0.548250i \(-0.815294\pi\)
−0.836315 + 0.548250i \(0.815294\pi\)
\(174\) 0 0
\(175\) 3.11684 0.235611
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −26.2337 −1.96080 −0.980399 0.197023i \(-0.936873\pi\)
−0.980399 + 0.197023i \(0.936873\pi\)
\(180\) 0 0
\(181\) −0.510875 −0.0379730 −0.0189865 0.999820i \(-0.506044\pi\)
−0.0189865 + 0.999820i \(0.506044\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 7.37228 0.542021
\(186\) 0 0
\(187\) −28.8614 −2.11056
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 21.6060 1.56335 0.781677 0.623684i \(-0.214365\pi\)
0.781677 + 0.623684i \(0.214365\pi\)
\(192\) 0 0
\(193\) 3.48913 0.251153 0.125576 0.992084i \(-0.459922\pi\)
0.125576 + 0.992084i \(0.459922\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −18.7446 −1.33549 −0.667747 0.744388i \(-0.732741\pi\)
−0.667747 + 0.744388i \(0.732741\pi\)
\(198\) 0 0
\(199\) −11.3723 −0.806160 −0.403080 0.915165i \(-0.632060\pi\)
−0.403080 + 0.915165i \(0.632060\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 4.62772 0.324802
\(204\) 0 0
\(205\) 8.23369 0.575066
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −39.6060 −2.73960
\(210\) 0 0
\(211\) 6.11684 0.421101 0.210550 0.977583i \(-0.432474\pi\)
0.210550 + 0.977583i \(0.432474\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 10.1168 0.689963
\(216\) 0 0
\(217\) 10.7446 0.729388
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −5.37228 −0.361379
\(222\) 0 0
\(223\) 5.48913 0.367579 0.183790 0.982966i \(-0.441164\pi\)
0.183790 + 0.982966i \(0.441164\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −26.7446 −1.77510 −0.887549 0.460712i \(-0.847594\pi\)
−0.887549 + 0.460712i \(0.847594\pi\)
\(228\) 0 0
\(229\) −3.48913 −0.230568 −0.115284 0.993333i \(-0.536778\pi\)
−0.115284 + 0.993333i \(0.536778\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −12.0000 −0.786146 −0.393073 0.919507i \(-0.628588\pi\)
−0.393073 + 0.919507i \(0.628588\pi\)
\(234\) 0 0
\(235\) −13.0217 −0.849445
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 3.25544 0.210577 0.105288 0.994442i \(-0.466423\pi\)
0.105288 + 0.994442i \(0.466423\pi\)
\(240\) 0 0
\(241\) 19.4891 1.25540 0.627702 0.778453i \(-0.283995\pi\)
0.627702 + 0.778453i \(0.283995\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1.37228 0.0876718
\(246\) 0 0
\(247\) −7.37228 −0.469087
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 1.88316 0.118864 0.0594319 0.998232i \(-0.481071\pi\)
0.0594319 + 0.998232i \(0.481071\pi\)
\(252\) 0 0
\(253\) 7.37228 0.463491
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −23.4891 −1.46521 −0.732606 0.680653i \(-0.761696\pi\)
−0.732606 + 0.680653i \(0.761696\pi\)
\(258\) 0 0
\(259\) −5.37228 −0.333817
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −20.7446 −1.27916 −0.639582 0.768723i \(-0.720893\pi\)
−0.639582 + 0.768723i \(0.720893\pi\)
\(264\) 0 0
\(265\) 3.76631 0.231363
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −7.48913 −0.456620 −0.228310 0.973588i \(-0.573320\pi\)
−0.228310 + 0.973588i \(0.573320\pi\)
\(270\) 0 0
\(271\) 8.23369 0.500161 0.250080 0.968225i \(-0.419543\pi\)
0.250080 + 0.968225i \(0.419543\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 16.7446 1.00974
\(276\) 0 0
\(277\) 7.25544 0.435937 0.217968 0.975956i \(-0.430057\pi\)
0.217968 + 0.975956i \(0.430057\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −8.23369 −0.491181 −0.245590 0.969374i \(-0.578982\pi\)
−0.245590 + 0.969374i \(0.578982\pi\)
\(282\) 0 0
\(283\) 22.9783 1.36592 0.682958 0.730458i \(-0.260693\pi\)
0.682958 + 0.730458i \(0.260693\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −6.00000 −0.354169
\(288\) 0 0
\(289\) 11.8614 0.697730
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 15.4891 0.904884 0.452442 0.891794i \(-0.350553\pi\)
0.452442 + 0.891794i \(0.350553\pi\)
\(294\) 0 0
\(295\) 3.76631 0.219283
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 1.37228 0.0793611
\(300\) 0 0
\(301\) −7.37228 −0.424931
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −16.6277 −0.952100
\(306\) 0 0
\(307\) 4.00000 0.228292 0.114146 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −4.00000 −0.226819 −0.113410 0.993548i \(-0.536177\pi\)
−0.113410 + 0.993548i \(0.536177\pi\)
\(312\) 0 0
\(313\) 2.00000 0.113047 0.0565233 0.998401i \(-0.481998\pi\)
0.0565233 + 0.998401i \(0.481998\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 16.2337 0.911775 0.455887 0.890037i \(-0.349322\pi\)
0.455887 + 0.890037i \(0.349322\pi\)
\(318\) 0 0
\(319\) 24.8614 1.39197
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 39.6060 2.20374
\(324\) 0 0
\(325\) 3.11684 0.172891
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 9.48913 0.523152
\(330\) 0 0
\(331\) 24.2337 1.33200 0.666002 0.745950i \(-0.268004\pi\)
0.666002 + 0.745950i \(0.268004\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 16.4674 0.899709
\(336\) 0 0
\(337\) −6.62772 −0.361035 −0.180517 0.983572i \(-0.557777\pi\)
−0.180517 + 0.983572i \(0.557777\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 57.7228 3.12587
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −28.7446 −1.54309 −0.771544 0.636175i \(-0.780515\pi\)
−0.771544 + 0.636175i \(0.780515\pi\)
\(348\) 0 0
\(349\) −6.23369 −0.333682 −0.166841 0.985984i \(-0.553357\pi\)
−0.166841 + 0.985984i \(0.553357\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −11.4891 −0.611504 −0.305752 0.952111i \(-0.598908\pi\)
−0.305752 + 0.952111i \(0.598908\pi\)
\(354\) 0 0
\(355\) −13.7228 −0.728331
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 27.4891 1.45082 0.725410 0.688317i \(-0.241650\pi\)
0.725410 + 0.688317i \(0.241650\pi\)
\(360\) 0 0
\(361\) 35.3505 1.86055
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 12.8614 0.673197
\(366\) 0 0
\(367\) 26.9783 1.40825 0.704127 0.710074i \(-0.251339\pi\)
0.704127 + 0.710074i \(0.251339\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −2.74456 −0.142491
\(372\) 0 0
\(373\) −36.9783 −1.91466 −0.957331 0.288995i \(-0.906679\pi\)
−0.957331 + 0.288995i \(0.906679\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 4.62772 0.238340
\(378\) 0 0
\(379\) 24.2337 1.24480 0.622400 0.782699i \(-0.286158\pi\)
0.622400 + 0.782699i \(0.286158\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 8.86141 0.452797 0.226398 0.974035i \(-0.427305\pi\)
0.226398 + 0.974035i \(0.427305\pi\)
\(384\) 0 0
\(385\) 7.37228 0.375726
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −14.7446 −0.747579 −0.373790 0.927514i \(-0.621942\pi\)
−0.373790 + 0.927514i \(0.621942\pi\)
\(390\) 0 0
\(391\) −7.37228 −0.372832
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −20.2337 −1.01807
\(396\) 0 0
\(397\) 26.2337 1.31663 0.658316 0.752742i \(-0.271269\pi\)
0.658316 + 0.752742i \(0.271269\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −24.2337 −1.21017 −0.605086 0.796160i \(-0.706861\pi\)
−0.605086 + 0.796160i \(0.706861\pi\)
\(402\) 0 0
\(403\) 10.7446 0.535225
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −28.8614 −1.43061
\(408\) 0 0
\(409\) 2.62772 0.129932 0.0649662 0.997887i \(-0.479306\pi\)
0.0649662 + 0.997887i \(0.479306\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −2.74456 −0.135051
\(414\) 0 0
\(415\) 1.72281 0.0845696
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −23.3723 −1.14181 −0.570905 0.821016i \(-0.693408\pi\)
−0.570905 + 0.821016i \(0.693408\pi\)
\(420\) 0 0
\(421\) −12.9783 −0.632521 −0.316261 0.948672i \(-0.602427\pi\)
−0.316261 + 0.948672i \(0.602427\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −16.7446 −0.812231
\(426\) 0 0
\(427\) 12.1168 0.586375
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −31.4891 −1.51678 −0.758389 0.651802i \(-0.774013\pi\)
−0.758389 + 0.651802i \(0.774013\pi\)
\(432\) 0 0
\(433\) 20.9783 1.00815 0.504075 0.863660i \(-0.331833\pi\)
0.504075 + 0.863660i \(0.331833\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −10.1168 −0.483954
\(438\) 0 0
\(439\) −20.6277 −0.984507 −0.492254 0.870452i \(-0.663827\pi\)
−0.492254 + 0.870452i \(0.663827\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −30.0000 −1.42534 −0.712672 0.701498i \(-0.752515\pi\)
−0.712672 + 0.701498i \(0.752515\pi\)
\(444\) 0 0
\(445\) −21.2554 −1.00760
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −2.11684 −0.0999000 −0.0499500 0.998752i \(-0.515906\pi\)
−0.0499500 + 0.998752i \(0.515906\pi\)
\(450\) 0 0
\(451\) −32.2337 −1.51783
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 1.37228 0.0643335
\(456\) 0 0
\(457\) 12.7446 0.596165 0.298083 0.954540i \(-0.403653\pi\)
0.298083 + 0.954540i \(0.403653\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 12.1168 0.564338 0.282169 0.959365i \(-0.408946\pi\)
0.282169 + 0.959365i \(0.408946\pi\)
\(462\) 0 0
\(463\) −2.11684 −0.0983781 −0.0491890 0.998789i \(-0.515664\pi\)
−0.0491890 + 0.998789i \(0.515664\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 18.3505 0.849161 0.424581 0.905390i \(-0.360422\pi\)
0.424581 + 0.905390i \(0.360422\pi\)
\(468\) 0 0
\(469\) −12.0000 −0.554109
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −39.6060 −1.82108
\(474\) 0 0
\(475\) −22.9783 −1.05431
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 11.6060 0.530290 0.265145 0.964209i \(-0.414580\pi\)
0.265145 + 0.964209i \(0.414580\pi\)
\(480\) 0 0
\(481\) −5.37228 −0.244955
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −21.2554 −0.965160
\(486\) 0 0
\(487\) −8.00000 −0.362515 −0.181257 0.983436i \(-0.558017\pi\)
−0.181257 + 0.983436i \(0.558017\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 19.2554 0.868986 0.434493 0.900675i \(-0.356928\pi\)
0.434493 + 0.900675i \(0.356928\pi\)
\(492\) 0 0
\(493\) −24.8614 −1.11970
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 10.0000 0.448561
\(498\) 0 0
\(499\) −30.9783 −1.38678 −0.693388 0.720564i \(-0.743883\pi\)
−0.693388 + 0.720564i \(0.743883\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −26.9783 −1.20290 −0.601450 0.798910i \(-0.705410\pi\)
−0.601450 + 0.798910i \(0.705410\pi\)
\(504\) 0 0
\(505\) 2.74456 0.122131
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −5.37228 −0.238122 −0.119061 0.992887i \(-0.537988\pi\)
−0.119061 + 0.992887i \(0.537988\pi\)
\(510\) 0 0
\(511\) −9.37228 −0.414605
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −8.39403 −0.369885
\(516\) 0 0
\(517\) 50.9783 2.24202
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 44.1168 1.93279 0.966397 0.257054i \(-0.0827519\pi\)
0.966397 + 0.257054i \(0.0827519\pi\)
\(522\) 0 0
\(523\) −33.7228 −1.47460 −0.737298 0.675568i \(-0.763899\pi\)
−0.737298 + 0.675568i \(0.763899\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −57.7228 −2.51445
\(528\) 0 0
\(529\) −21.1168 −0.918124
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −6.00000 −0.259889
\(534\) 0 0
\(535\) −13.7228 −0.593289
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −5.37228 −0.231401
\(540\) 0 0
\(541\) −5.37228 −0.230972 −0.115486 0.993309i \(-0.536843\pi\)
−0.115486 + 0.993309i \(0.536843\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 5.64947 0.241997
\(546\) 0 0
\(547\) −30.9783 −1.32453 −0.662267 0.749268i \(-0.730406\pi\)
−0.662267 + 0.749268i \(0.730406\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −34.1168 −1.45343
\(552\) 0 0
\(553\) 14.7446 0.627003
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −2.74456 −0.116291 −0.0581454 0.998308i \(-0.518519\pi\)
−0.0581454 + 0.998308i \(0.518519\pi\)
\(558\) 0 0
\(559\) −7.37228 −0.311814
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −6.11684 −0.257794 −0.128897 0.991658i \(-0.541144\pi\)
−0.128897 + 0.991658i \(0.541144\pi\)
\(564\) 0 0
\(565\) 18.5109 0.778758
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −2.74456 −0.115058 −0.0575290 0.998344i \(-0.518322\pi\)
−0.0575290 + 0.998344i \(0.518322\pi\)
\(570\) 0 0
\(571\) −12.0000 −0.502184 −0.251092 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 4.27719 0.178371
\(576\) 0 0
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −1.25544 −0.0520843
\(582\) 0 0
\(583\) −14.7446 −0.610657
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 37.7228 1.55699 0.778494 0.627653i \(-0.215984\pi\)
0.778494 + 0.627653i \(0.215984\pi\)
\(588\) 0 0
\(589\) −79.2119 −3.26387
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 8.74456 0.359096 0.179548 0.983749i \(-0.442536\pi\)
0.179548 + 0.983749i \(0.442536\pi\)
\(594\) 0 0
\(595\) −7.37228 −0.302234
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 27.8832 1.13927 0.569637 0.821896i \(-0.307084\pi\)
0.569637 + 0.821896i \(0.307084\pi\)
\(600\) 0 0
\(601\) 39.4891 1.61080 0.805398 0.592735i \(-0.201952\pi\)
0.805398 + 0.592735i \(0.201952\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 24.5109 0.996509
\(606\) 0 0
\(607\) −10.1168 −0.410630 −0.205315 0.978696i \(-0.565822\pi\)
−0.205315 + 0.978696i \(0.565822\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 9.48913 0.383889
\(612\) 0 0
\(613\) −5.37228 −0.216984 −0.108492 0.994097i \(-0.534602\pi\)
−0.108492 + 0.994097i \(0.534602\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −12.8614 −0.517781 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(618\) 0 0
\(619\) 3.37228 0.135543 0.0677717 0.997701i \(-0.478411\pi\)
0.0677717 + 0.997701i \(0.478411\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 15.4891 0.620559
\(624\) 0 0
\(625\) 0.298936 0.0119574
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 28.8614 1.15078
\(630\) 0 0
\(631\) −23.6060 −0.939739 −0.469869 0.882736i \(-0.655699\pi\)
−0.469869 + 0.882736i \(0.655699\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 10.9783 0.435659
\(636\) 0 0
\(637\) −1.00000 −0.0396214
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −2.74456 −0.108404 −0.0542019 0.998530i \(-0.517261\pi\)
−0.0542019 + 0.998530i \(0.517261\pi\)
\(642\) 0 0
\(643\) −9.88316 −0.389754 −0.194877 0.980828i \(-0.562431\pi\)
−0.194877 + 0.980828i \(0.562431\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 28.4674 1.11917 0.559584 0.828774i \(-0.310961\pi\)
0.559584 + 0.828774i \(0.310961\pi\)
\(648\) 0 0
\(649\) −14.7446 −0.578775
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −35.8397 −1.40251 −0.701257 0.712908i \(-0.747377\pi\)
−0.701257 + 0.712908i \(0.747377\pi\)
\(654\) 0 0
\(655\) −17.3288 −0.677092
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 39.7228 1.54738 0.773691 0.633564i \(-0.218409\pi\)
0.773691 + 0.633564i \(0.218409\pi\)
\(660\) 0 0
\(661\) 10.2337 0.398044 0.199022 0.979995i \(-0.436223\pi\)
0.199022 + 0.979995i \(0.436223\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −10.1168 −0.392314
\(666\) 0 0
\(667\) 6.35053 0.245894
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 65.0951 2.51297
\(672\) 0 0
\(673\) 27.8832 1.07482 0.537408 0.843322i \(-0.319403\pi\)
0.537408 + 0.843322i \(0.319403\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 2.00000 0.0768662 0.0384331 0.999261i \(-0.487763\pi\)
0.0384331 + 0.999261i \(0.487763\pi\)
\(678\) 0 0
\(679\) 15.4891 0.594418
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −37.6060 −1.43895 −0.719476 0.694517i \(-0.755618\pi\)
−0.719476 + 0.694517i \(0.755618\pi\)
\(684\) 0 0
\(685\) 17.6495 0.674352
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −2.74456 −0.104560
\(690\) 0 0
\(691\) −34.9783 −1.33064 −0.665318 0.746560i \(-0.731704\pi\)
−0.665318 + 0.746560i \(0.731704\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −27.7663 −1.05324
\(696\) 0 0
\(697\) 32.2337 1.22094
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −25.2554 −0.953885 −0.476942 0.878935i \(-0.658255\pi\)
−0.476942 + 0.878935i \(0.658255\pi\)
\(702\) 0 0
\(703\) 39.6060 1.49377
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −2.00000 −0.0752177
\(708\) 0 0
\(709\) 12.5109 0.469856 0.234928 0.972013i \(-0.424515\pi\)
0.234928 + 0.972013i \(0.424515\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 14.7446 0.552188
\(714\) 0 0
\(715\) 7.37228 0.275708
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −10.9783 −0.409420 −0.204710 0.978823i \(-0.565625\pi\)
−0.204710 + 0.978823i \(0.565625\pi\)
\(720\) 0 0
\(721\) 6.11684 0.227803
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 14.4239 0.535689
\(726\) 0 0
\(727\) 2.35053 0.0871764 0.0435882 0.999050i \(-0.486121\pi\)
0.0435882 + 0.999050i \(0.486121\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 39.6060 1.46488
\(732\) 0 0
\(733\) 35.2554 1.30219 0.651095 0.758997i \(-0.274310\pi\)
0.651095 + 0.758997i \(0.274310\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −64.4674 −2.37469
\(738\) 0 0
\(739\) −5.25544 −0.193324 −0.0966622 0.995317i \(-0.530817\pi\)
−0.0966622 + 0.995317i \(0.530817\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 15.2554 0.559668 0.279834 0.960048i \(-0.409721\pi\)
0.279834 + 0.960048i \(0.409721\pi\)
\(744\) 0 0
\(745\) −24.0000 −0.879292
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 10.0000 0.365392
\(750\) 0 0
\(751\) −34.9783 −1.27637 −0.638187 0.769881i \(-0.720315\pi\)
−0.638187 + 0.769881i \(0.720315\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −17.3288 −0.630659
\(756\) 0 0
\(757\) 11.7228 0.426073 0.213036 0.977044i \(-0.431665\pi\)
0.213036 + 0.977044i \(0.431665\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −11.7228 −0.424952 −0.212476 0.977166i \(-0.568153\pi\)
−0.212476 + 0.977166i \(0.568153\pi\)
\(762\) 0 0
\(763\) −4.11684 −0.149040
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −2.74456 −0.0991004
\(768\) 0 0
\(769\) −41.8397 −1.50878 −0.754388 0.656428i \(-0.772066\pi\)
−0.754388 + 0.656428i \(0.772066\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −38.8614 −1.39775 −0.698874 0.715245i \(-0.746315\pi\)
−0.698874 + 0.715245i \(0.746315\pi\)
\(774\) 0 0
\(775\) 33.4891 1.20296
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 44.2337 1.58484
\(780\) 0 0
\(781\) 53.7228 1.92235
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 23.8397 0.850874
\(786\) 0 0
\(787\) −36.6277 −1.30564 −0.652819 0.757514i \(-0.726414\pi\)
−0.652819 + 0.757514i \(0.726414\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −13.4891 −0.479618
\(792\) 0 0
\(793\) 12.1168 0.430282
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 38.2337 1.35431 0.677153 0.735842i \(-0.263213\pi\)
0.677153 + 0.735842i \(0.263213\pi\)
\(798\) 0 0
\(799\) −50.9783 −1.80348
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −50.3505 −1.77683
\(804\) 0 0
\(805\) 1.88316 0.0663725
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 47.2119 1.65988 0.829942 0.557850i \(-0.188374\pi\)
0.829942 + 0.557850i \(0.188374\pi\)
\(810\) 0 0
\(811\) 38.3505 1.34667 0.673335 0.739338i \(-0.264861\pi\)
0.673335 + 0.739338i \(0.264861\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −3.76631 −0.131928
\(816\) 0 0
\(817\) 54.3505 1.90148
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 54.9783 1.91875 0.959377 0.282127i \(-0.0910399\pi\)
0.959377 + 0.282127i \(0.0910399\pi\)
\(822\) 0 0
\(823\) 33.7228 1.17550 0.587752 0.809041i \(-0.300013\pi\)
0.587752 + 0.809041i \(0.300013\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −24.3505 −0.846751 −0.423375 0.905954i \(-0.639155\pi\)
−0.423375 + 0.905954i \(0.639155\pi\)
\(828\) 0 0
\(829\) −50.6277 −1.75837 −0.879187 0.476477i \(-0.841913\pi\)
−0.879187 + 0.476477i \(0.841913\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 5.37228 0.186139
\(834\) 0 0
\(835\) 2.58422 0.0894306
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 9.48913 0.327601 0.163800 0.986493i \(-0.447625\pi\)
0.163800 + 0.986493i \(0.447625\pi\)
\(840\) 0 0
\(841\) −7.58422 −0.261525
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 1.37228 0.0472079
\(846\) 0 0
\(847\) −17.8614 −0.613725
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −7.37228 −0.252719
\(852\) 0 0
\(853\) 39.7228 1.36008 0.680042 0.733174i \(-0.261962\pi\)
0.680042 + 0.733174i \(0.261962\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −12.9783 −0.443329 −0.221664 0.975123i \(-0.571149\pi\)
−0.221664 + 0.975123i \(0.571149\pi\)
\(858\) 0 0
\(859\) −8.23369 −0.280930 −0.140465 0.990086i \(-0.544860\pi\)
−0.140465 + 0.990086i \(0.544860\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 41.2119 1.40287 0.701435 0.712733i \(-0.252543\pi\)
0.701435 + 0.712733i \(0.252543\pi\)
\(864\) 0 0
\(865\) −30.1902 −1.02650
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 79.2119 2.68708
\(870\) 0 0
\(871\) −12.0000 −0.406604
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 11.1386 0.376553
\(876\) 0 0
\(877\) 8.97825 0.303174 0.151587 0.988444i \(-0.451562\pi\)
0.151587 + 0.988444i \(0.451562\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −26.6277 −0.897111 −0.448555 0.893755i \(-0.648061\pi\)
−0.448555 + 0.893755i \(0.648061\pi\)
\(882\) 0 0
\(883\) 20.8614 0.702042 0.351021 0.936368i \(-0.385835\pi\)
0.351021 + 0.936368i \(0.385835\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −37.7228 −1.26661 −0.633304 0.773903i \(-0.718302\pi\)
−0.633304 + 0.773903i \(0.718302\pi\)
\(888\) 0 0
\(889\) −8.00000 −0.268311
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −69.9565 −2.34101
\(894\) 0 0
\(895\) −36.0000 −1.20335
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 49.7228 1.65835
\(900\) 0 0
\(901\) 14.7446 0.491213
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −0.701064 −0.0233041
\(906\) 0 0
\(907\) 20.0000 0.664089 0.332045 0.943264i \(-0.392262\pi\)
0.332045 + 0.943264i \(0.392262\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −14.6277 −0.484638 −0.242319 0.970197i \(-0.577908\pi\)
−0.242319 + 0.970197i \(0.577908\pi\)
\(912\) 0 0
\(913\) −6.74456 −0.223212
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 12.6277 0.417004
\(918\) 0 0
\(919\) 52.2337 1.72303 0.861515 0.507732i \(-0.169516\pi\)
0.861515 + 0.507732i \(0.169516\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 10.0000 0.329154
\(924\) 0 0
\(925\) −16.7446 −0.550558
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −7.48913 −0.245710 −0.122855 0.992425i \(-0.539205\pi\)
−0.122855 + 0.992425i \(0.539205\pi\)
\(930\) 0 0
\(931\) 7.37228 0.241617
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −39.6060 −1.29525
\(936\) 0 0
\(937\) −27.4891 −0.898031 −0.449015 0.893524i \(-0.648225\pi\)
−0.449015 + 0.893524i \(0.648225\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −18.0000 −0.586783 −0.293392 0.955992i \(-0.594784\pi\)
−0.293392 + 0.955992i \(0.594784\pi\)
\(942\) 0 0
\(943\) −8.23369 −0.268126
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 21.6060 0.702100 0.351050 0.936357i \(-0.385825\pi\)
0.351050 + 0.936357i \(0.385825\pi\)
\(948\) 0 0
\(949\) −9.37228 −0.304237
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 45.9565 1.48868 0.744339 0.667802i \(-0.232765\pi\)
0.744339 + 0.667802i \(0.232765\pi\)
\(954\) 0 0
\(955\) 29.6495 0.959434
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −12.8614 −0.415316
\(960\) 0 0
\(961\) 84.4456 2.72405
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 4.78806 0.154133
\(966\) 0 0
\(967\) −29.8832 −0.960978 −0.480489 0.877001i \(-0.659541\pi\)
−0.480489 + 0.877001i \(0.659541\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −48.0000 −1.54039 −0.770197 0.637806i \(-0.779842\pi\)
−0.770197 + 0.637806i \(0.779842\pi\)
\(972\) 0 0
\(973\) 20.2337 0.648662
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 14.1168 0.451638 0.225819 0.974169i \(-0.427494\pi\)
0.225819 + 0.974169i \(0.427494\pi\)
\(978\) 0 0
\(979\) 83.2119 2.65947
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −27.3723 −0.873040 −0.436520 0.899695i \(-0.643789\pi\)
−0.436520 + 0.899695i \(0.643789\pi\)
\(984\) 0 0
\(985\) −25.7228 −0.819597
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −10.1168 −0.321697
\(990\) 0 0
\(991\) 52.7011 1.67410 0.837052 0.547123i \(-0.184277\pi\)
0.837052 + 0.547123i \(0.184277\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −15.6060 −0.494742
\(996\) 0 0
\(997\) 30.4674 0.964911 0.482456 0.875920i \(-0.339745\pi\)
0.482456 + 0.875920i \(0.339745\pi\)
\(998\) 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 6552.2.a.bb.1.2 2
3.2 odd 2 6552.2.a.bm.1.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6552.2.a.bb.1.2 2 1.1 even 1 trivial
6552.2.a.bm.1.1 yes 2 3.2 odd 2