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Properties

Label 3328.2.a.l.1.1
Level $3328$
Weight $2$
Character 3328.1
Self dual yes
Analytic conductor $26.574$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3328,2,Mod(1,3328)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3328.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3328, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3328 = 2^{8} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3328.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,3,0,1,0,-3,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(26.5742137927\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1664)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3328.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} +1.00000 q^{5} -3.00000 q^{7} +6.00000 q^{9} +6.00000 q^{11} -1.00000 q^{13} +3.00000 q^{15} +1.00000 q^{17} -9.00000 q^{21} -6.00000 q^{23} -4.00000 q^{25} +9.00000 q^{27} +10.0000 q^{29} +18.0000 q^{33} -3.00000 q^{35} +9.00000 q^{37} -3.00000 q^{39} -4.00000 q^{41} +9.00000 q^{43} +6.00000 q^{45} +9.00000 q^{47} +2.00000 q^{49} +3.00000 q^{51} -2.00000 q^{53} +6.00000 q^{55} +12.0000 q^{61} -18.0000 q^{63} -1.00000 q^{65} +6.00000 q^{67} -18.0000 q^{69} -9.00000 q^{71} -12.0000 q^{75} -18.0000 q^{77} -6.00000 q^{79} +9.00000 q^{81} -6.00000 q^{83} +1.00000 q^{85} +30.0000 q^{87} -8.00000 q^{89} +3.00000 q^{91} -18.0000 q^{97} +36.0000 q^{99} +O(q^{100})\)

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
\(3\) 3.00000 1.73205 0.866025 − 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214 0.223607 − 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 0 0
\(7\) −3.00000 −1.13389 −0.566947 − 0.823754i \(-0.691875\pi\)
−0.566947 + 0.823754i \(0.691875\pi\)
\(8\) 0 0
\(9\) 6.00000 2.00000
\(10\) 0 0
\(11\) 6.00000 1.80907 0.904534 − 0.426401i \(-0.140219\pi\)
0.904534 + 0.426401i \(0.140219\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 3.00000 0.774597
\(16\) 0 0
\(17\) 1.00000 0.242536 0.121268 − 0.992620i \(-0.461304\pi\)
0.121268 + 0.992620i \(0.461304\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) −9.00000 −1.96396
\(22\) 0 0
\(23\) −6.00000 −1.25109 −0.625543 − 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) 9.00000 1.73205
\(28\) 0 0
\(29\) 10.0000 1.85695 0.928477 − 0.371391i \(-0.121119\pi\)
0.928477 + 0.371391i \(0.121119\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 18.0000 3.13340
\(34\) 0 0
\(35\) −3.00000 −0.507093
\(36\) 0 0
\(37\) 9.00000 1.47959 0.739795 − 0.672832i \(-0.234922\pi\)
0.739795 + 0.672832i \(0.234922\pi\)
\(38\) 0 0
\(39\) −3.00000 −0.480384
\(40\) 0 0
\(41\) −4.00000 −0.624695 −0.312348 − 0.949968i \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(42\) 0 0
\(43\) 9.00000 1.37249 0.686244 − 0.727372i \(-0.259258\pi\)
0.686244 + 0.727372i \(0.259258\pi\)
\(44\) 0 0
\(45\) 6.00000 0.894427
\(46\) 0 0
\(47\) 9.00000 1.31278 0.656392 − 0.754420i \(-0.272082\pi\)
0.656392 + 0.754420i \(0.272082\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) 3.00000 0.420084
\(52\) 0 0
\(53\) −2.00000 −0.274721 −0.137361 − 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) 6.00000 0.809040
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 12.0000 1.53644 0.768221 − 0.640184i \(-0.221142\pi\)
0.768221 + 0.640184i \(0.221142\pi\)
\(62\) 0 0
\(63\) −18.0000 −2.26779
\(64\) 0 0
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) 6.00000 0.733017 0.366508 − 0.930415i \(-0.380553\pi\)
0.366508 + 0.930415i \(0.380553\pi\)
\(68\) 0 0
\(69\) −18.0000 −2.16695
\(70\) 0 0
\(71\) −9.00000 −1.06810 −0.534052 − 0.845452i \(-0.679331\pi\)
−0.534052 + 0.845452i \(0.679331\pi\)
\(72\) 0 0
\(73\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 0 0
\(75\) −12.0000 −1.38564
\(76\) 0 0
\(77\) −18.0000 −2.05129
\(78\) 0 0
\(79\) −6.00000 −0.675053 −0.337526 − 0.941316i \(-0.609590\pi\)
−0.337526 + 0.941316i \(0.609590\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) −6.00000 −0.658586 −0.329293 − 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) 1.00000 0.108465
\(86\) 0 0
\(87\) 30.0000 3.21634
\(88\) 0 0
\(89\) −8.00000 −0.847998 −0.423999 − 0.905663i \(-0.639374\pi\)
−0.423999 + 0.905663i \(0.639374\pi\)
\(90\) 0 0
\(91\) 3.00000 0.314485
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −18.0000 −1.82762 −0.913812 − 0.406138i \(-0.866875\pi\)
−0.913812 + 0.406138i \(0.866875\pi\)
\(98\) 0 0
\(99\) 36.0000 3.61814
\(100\) 0 0
\(101\) 8.00000 0.796030 0.398015 − 0.917379i \(-0.369699\pi\)
0.398015 + 0.917379i \(0.369699\pi\)
\(102\) 0 0
\(103\) 6.00000 0.591198 0.295599 − 0.955312i \(-0.404481\pi\)
0.295599 + 0.955312i \(0.404481\pi\)
\(104\) 0 0
\(105\) −9.00000 −0.878310
\(106\) 0 0
\(107\) −12.0000 −1.16008 −0.580042 − 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) −7.00000 −0.670478 −0.335239 − 0.942133i \(-0.608817\pi\)
−0.335239 + 0.942133i \(0.608817\pi\)
\(110\) 0 0
\(111\) 27.0000 2.56273
\(112\) 0 0
\(113\) 10.0000 0.940721 0.470360 − 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) 0 0
\(115\) −6.00000 −0.559503
\(116\) 0 0
\(117\) −6.00000 −0.554700
\(118\) 0 0
\(119\) −3.00000 −0.275010
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) 0 0
\(123\) −12.0000 −1.08200
\(124\) 0 0
\(125\) −9.00000 −0.804984
\(126\) 0 0
\(127\) −6.00000 −0.532414 −0.266207 − 0.963916i \(-0.585770\pi\)
−0.266207 + 0.963916i \(0.585770\pi\)
\(128\) 0 0
\(129\) 27.0000 2.37722
\(130\) 0 0
\(131\) −15.0000 −1.31056 −0.655278 − 0.755388i \(-0.727449\pi\)
−0.655278 + 0.755388i \(0.727449\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 9.00000 0.774597
\(136\) 0 0
\(137\) 10.0000 0.854358 0.427179 − 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) 0 0
\(139\) 9.00000 0.763370 0.381685 − 0.924292i \(-0.375344\pi\)
0.381685 + 0.924292i \(0.375344\pi\)
\(140\) 0 0
\(141\) 27.0000 2.27381
\(142\) 0 0
\(143\) −6.00000 −0.501745
\(144\) 0 0
\(145\) 10.0000 0.830455
\(146\) 0 0
\(147\) 6.00000 0.494872
\(148\) 0 0
\(149\) −22.0000 −1.80231 −0.901155 − 0.433497i \(-0.857280\pi\)
−0.901155 + 0.433497i \(0.857280\pi\)
\(150\) 0 0
\(151\) −9.00000 −0.732410 −0.366205 − 0.930534i \(-0.619343\pi\)
−0.366205 + 0.930534i \(0.619343\pi\)
\(152\) 0 0
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 12.0000 0.957704 0.478852 − 0.877896i \(-0.341053\pi\)
0.478852 + 0.877896i \(0.341053\pi\)
\(158\) 0 0
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) 18.0000 1.41860
\(162\) 0 0
\(163\) 6.00000 0.469956 0.234978 − 0.972001i \(-0.424498\pi\)
0.234978 + 0.972001i \(0.424498\pi\)
\(164\) 0 0
\(165\) 18.0000 1.40130
\(166\) 0 0
\(167\) −24.0000 −1.85718 −0.928588 − 0.371113i \(-0.878976\pi\)
−0.928588 + 0.371113i \(0.878976\pi\)
\(168\) 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 \(-0.184706\pi\)
0.836315 + 0.548250i \(0.184706\pi\)
\(174\) 0 0
\(175\) 12.0000 0.907115
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 9.00000 0.672692 0.336346 − 0.941739i \(-0.390809\pi\)
0.336346 + 0.941739i \(0.390809\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 − 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) 36.0000 2.66120
\(184\) 0 0
\(185\) 9.00000 0.661693
\(186\) 0 0
\(187\) 6.00000 0.438763
\(188\) 0 0
\(189\) −27.0000 −1.96396
\(190\) 0 0
\(191\) −6.00000 −0.434145 −0.217072 − 0.976156i \(-0.569651\pi\)
−0.217072 + 0.976156i \(0.569651\pi\)
\(192\) 0 0
\(193\) 14.0000 1.00774 0.503871 − 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) 0 0
\(195\) −3.00000 −0.214834
\(196\) 0 0
\(197\) −1.00000 −0.0712470 −0.0356235 − 0.999365i \(-0.511342\pi\)
−0.0356235 + 0.999365i \(0.511342\pi\)
\(198\) 0 0
\(199\) −6.00000 −0.425329 −0.212664 − 0.977125i \(-0.568214\pi\)
−0.212664 + 0.977125i \(0.568214\pi\)
\(200\) 0 0
\(201\) 18.0000 1.26962
\(202\) 0 0
\(203\) −30.0000 −2.10559
\(204\) 0 0
\(205\) −4.00000 −0.279372
\(206\) 0 0
\(207\) −36.0000 −2.50217
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −27.0000 −1.85876 −0.929378 − 0.369129i \(-0.879656\pi\)
−0.929378 + 0.369129i \(0.879656\pi\)
\(212\) 0 0
\(213\) −27.0000 −1.85001
\(214\) 0 0
\(215\) 9.00000 0.613795
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −1.00000 −0.0672673
\(222\) 0 0
\(223\) 15.0000 1.00447 0.502237 − 0.864730i \(-0.332510\pi\)
0.502237 + 0.864730i \(0.332510\pi\)
\(224\) 0 0
\(225\) −24.0000 −1.60000
\(226\) 0 0
\(227\) −24.0000 −1.59294 −0.796468 − 0.604681i \(-0.793301\pi\)
−0.796468 + 0.604681i \(0.793301\pi\)
\(228\) 0 0
\(229\) −5.00000 −0.330409 −0.165205 − 0.986259i \(-0.552828\pi\)
−0.165205 + 0.986259i \(0.552828\pi\)
\(230\) 0 0
\(231\) −54.0000 −3.55294
\(232\) 0 0
\(233\) 17.0000 1.11371 0.556854 − 0.830611i \(-0.312008\pi\)
0.556854 + 0.830611i \(0.312008\pi\)
\(234\) 0 0
\(235\) 9.00000 0.587095
\(236\) 0 0
\(237\) −18.0000 −1.16923
\(238\) 0 0
\(239\) −15.0000 −0.970269 −0.485135 − 0.874439i \(-0.661229\pi\)
−0.485135 + 0.874439i \(0.661229\pi\)
\(240\) 0 0
\(241\) −12.0000 −0.772988 −0.386494 − 0.922292i \(-0.626314\pi\)
−0.386494 + 0.922292i \(0.626314\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 2.00000 0.127775
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −18.0000 −1.14070
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 − 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) −36.0000 −2.26330
\(254\) 0 0
\(255\) 3.00000 0.187867
\(256\) 0 0
\(257\) 17.0000 1.06043 0.530215 − 0.847863i \(-0.322111\pi\)
0.530215 + 0.847863i \(0.322111\pi\)
\(258\) 0 0
\(259\) −27.0000 −1.67770
\(260\) 0 0
\(261\) 60.0000 3.71391
\(262\) 0 0
\(263\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) −2.00000 −0.122859
\(266\) 0 0
\(267\) −24.0000 −1.46878
\(268\) 0 0
\(269\) 20.0000 1.21942 0.609711 − 0.792624i \(-0.291286\pi\)
0.609711 + 0.792624i \(0.291286\pi\)
\(270\) 0 0
\(271\) −15.0000 −0.911185 −0.455593 − 0.890188i \(-0.650573\pi\)
−0.455593 + 0.890188i \(0.650573\pi\)
\(272\) 0 0
\(273\) 9.00000 0.544705
\(274\) 0 0
\(275\) −24.0000 −1.44725
\(276\) 0 0
\(277\) 8.00000 0.480673 0.240337 − 0.970690i \(-0.422742\pi\)
0.240337 + 0.970690i \(0.422742\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 16.0000 0.954480 0.477240 − 0.878773i \(-0.341637\pi\)
0.477240 + 0.878773i \(0.341637\pi\)
\(282\) 0 0
\(283\) −24.0000 −1.42665 −0.713326 − 0.700832i \(-0.752812\pi\)
−0.713326 + 0.700832i \(0.752812\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 12.0000 0.708338
\(288\) 0 0
\(289\) −16.0000 −0.941176
\(290\) 0 0
\(291\) −54.0000 −3.16554
\(292\) 0 0
\(293\) 1.00000 0.0584206 0.0292103 − 0.999573i \(-0.490701\pi\)
0.0292103 + 0.999573i \(0.490701\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 54.0000 3.13340
\(298\) 0 0
\(299\) 6.00000 0.346989
\(300\) 0 0
\(301\) −27.0000 −1.55625
\(302\) 0 0
\(303\) 24.0000 1.37876
\(304\) 0 0
\(305\) 12.0000 0.687118
\(306\) 0 0
\(307\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 18.0000 1.02398
\(310\) 0 0
\(311\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 19.0000 1.07394 0.536972 − 0.843600i \(-0.319568\pi\)
0.536972 + 0.843600i \(0.319568\pi\)
\(314\) 0 0
\(315\) −18.0000 −1.01419
\(316\) 0 0
\(317\) −22.0000 −1.23564 −0.617822 − 0.786318i \(-0.711985\pi\)
−0.617822 + 0.786318i \(0.711985\pi\)
\(318\) 0 0
\(319\) 60.0000 3.35936
\(320\) 0 0
\(321\) −36.0000 −2.00932
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 4.00000 0.221880
\(326\) 0 0
\(327\) −21.0000 −1.16130
\(328\) 0 0
\(329\) −27.0000 −1.48856
\(330\) 0 0
\(331\) 6.00000 0.329790 0.164895 − 0.986311i \(-0.447272\pi\)
0.164895 + 0.986311i \(0.447272\pi\)
\(332\) 0 0
\(333\) 54.0000 2.95918
\(334\) 0 0
\(335\) 6.00000 0.327815
\(336\) 0 0
\(337\) −33.0000 −1.79762 −0.898812 − 0.438334i \(-0.855569\pi\)
−0.898812 + 0.438334i \(0.855569\pi\)
\(338\) 0 0
\(339\) 30.0000 1.62938
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 15.0000 0.809924
\(344\) 0 0
\(345\) −18.0000 −0.969087
\(346\) 0 0
\(347\) −3.00000 −0.161048 −0.0805242 − 0.996753i \(-0.525659\pi\)
−0.0805242 + 0.996753i \(0.525659\pi\)
\(348\) 0 0
\(349\) 9.00000 0.481759 0.240879 − 0.970555i \(-0.422564\pi\)
0.240879 + 0.970555i \(0.422564\pi\)
\(350\) 0 0
\(351\) −9.00000 −0.480384
\(352\) 0 0
\(353\) 22.0000 1.17094 0.585471 − 0.810693i \(-0.300910\pi\)
0.585471 + 0.810693i \(0.300910\pi\)
\(354\) 0 0
\(355\) −9.00000 −0.477670
\(356\) 0 0
\(357\) −9.00000 −0.476331
\(358\) 0 0
\(359\) −24.0000 −1.26667 −0.633336 − 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) 75.0000 3.93648
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 30.0000 1.56599 0.782994 − 0.622030i \(-0.213692\pi\)
0.782994 + 0.622030i \(0.213692\pi\)
\(368\) 0 0
\(369\) −24.0000 −1.24939
\(370\) 0 0
\(371\) 6.00000 0.311504
\(372\) 0 0
\(373\) 24.0000 1.24267 0.621336 − 0.783544i \(-0.286590\pi\)
0.621336 + 0.783544i \(0.286590\pi\)
\(374\) 0 0
\(375\) −27.0000 −1.39427
\(376\) 0 0
\(377\) −10.0000 −0.515026
\(378\) 0 0
\(379\) −30.0000 −1.54100 −0.770498 − 0.637442i \(-0.779993\pi\)
−0.770498 + 0.637442i \(0.779993\pi\)
\(380\) 0 0
\(381\) −18.0000 −0.922168
\(382\) 0 0
\(383\) −21.0000 −1.07305 −0.536525 − 0.843884i \(-0.680263\pi\)
−0.536525 + 0.843884i \(0.680263\pi\)
\(384\) 0 0
\(385\) −18.0000 −0.917365
\(386\) 0 0
\(387\) 54.0000 2.74497
\(388\) 0 0
\(389\) −4.00000 −0.202808 −0.101404 − 0.994845i \(-0.532333\pi\)
−0.101404 + 0.994845i \(0.532333\pi\)
\(390\) 0 0
\(391\) −6.00000 −0.303433
\(392\) 0 0
\(393\) −45.0000 −2.26995
\(394\) 0 0
\(395\) −6.00000 −0.301893
\(396\) 0 0
\(397\) 6.00000 0.301131 0.150566 − 0.988600i \(-0.451890\pi\)
0.150566 + 0.988600i \(0.451890\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −14.0000 −0.699127 −0.349563 − 0.936913i \(-0.613670\pi\)
−0.349563 + 0.936913i \(0.613670\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 9.00000 0.447214
\(406\) 0 0
\(407\) 54.0000 2.67668
\(408\) 0 0
\(409\) −18.0000 −0.890043 −0.445021 − 0.895520i \(-0.646804\pi\)
−0.445021 + 0.895520i \(0.646804\pi\)
\(410\) 0 0
\(411\) 30.0000 1.47979
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −6.00000 −0.294528
\(416\) 0 0
\(417\) 27.0000 1.32220
\(418\) 0 0
\(419\) 3.00000 0.146560 0.0732798 − 0.997311i \(-0.476653\pi\)
0.0732798 + 0.997311i \(0.476653\pi\)
\(420\) 0 0
\(421\) 17.0000 0.828529 0.414265 − 0.910156i \(-0.364039\pi\)
0.414265 + 0.910156i \(0.364039\pi\)
\(422\) 0 0
\(423\) 54.0000 2.62557
\(424\) 0 0
\(425\) −4.00000 −0.194029
\(426\) 0 0
\(427\) −36.0000 −1.74216
\(428\) 0 0
\(429\) −18.0000 −0.869048
\(430\) 0 0
\(431\) −3.00000 −0.144505 −0.0722525 − 0.997386i \(-0.523019\pi\)
−0.0722525 + 0.997386i \(0.523019\pi\)
\(432\) 0 0
\(433\) −25.0000 −1.20142 −0.600712 − 0.799466i \(-0.705116\pi\)
−0.600712 + 0.799466i \(0.705116\pi\)
\(434\) 0 0
\(435\) 30.0000 1.43839
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −36.0000 −1.71819 −0.859093 − 0.511819i \(-0.828972\pi\)
−0.859093 + 0.511819i \(0.828972\pi\)
\(440\) 0 0
\(441\) 12.0000 0.571429
\(442\) 0 0
\(443\) 15.0000 0.712672 0.356336 − 0.934358i \(-0.384026\pi\)
0.356336 + 0.934358i \(0.384026\pi\)
\(444\) 0 0
\(445\) −8.00000 −0.379236
\(446\) 0 0
\(447\) −66.0000 −3.12169
\(448\) 0 0
\(449\) −22.0000 −1.03824 −0.519122 − 0.854700i \(-0.673741\pi\)
−0.519122 + 0.854700i \(0.673741\pi\)
\(450\) 0 0
\(451\) −24.0000 −1.13012
\(452\) 0 0
\(453\) −27.0000 −1.26857
\(454\) 0 0
\(455\) 3.00000 0.140642
\(456\) 0 0
\(457\) 24.0000 1.12267 0.561336 − 0.827588i \(-0.310287\pi\)
0.561336 + 0.827588i \(0.310287\pi\)
\(458\) 0 0
\(459\) 9.00000 0.420084
\(460\) 0 0
\(461\) −19.0000 −0.884918 −0.442459 − 0.896789i \(-0.645894\pi\)
−0.442459 + 0.896789i \(0.645894\pi\)
\(462\) 0 0
\(463\) −24.0000 −1.11537 −0.557687 − 0.830051i \(-0.688311\pi\)
−0.557687 + 0.830051i \(0.688311\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 12.0000 0.555294 0.277647 − 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 0 0
\(469\) −18.0000 −0.831163
\(470\) 0 0
\(471\) 36.0000 1.65879
\(472\) 0 0
\(473\) 54.0000 2.48292
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −12.0000 −0.549442
\(478\) 0 0
\(479\) −15.0000 −0.685367 −0.342684 − 0.939451i \(-0.611336\pi\)
−0.342684 + 0.939451i \(0.611336\pi\)
\(480\) 0 0
\(481\) −9.00000 −0.410365
\(482\) 0 0
\(483\) 54.0000 2.45709
\(484\) 0 0
\(485\) −18.0000 −0.817338
\(486\) 0 0
\(487\) 24.0000 1.08754 0.543772 − 0.839233i \(-0.316996\pi\)
0.543772 + 0.839233i \(0.316996\pi\)
\(488\) 0 0
\(489\) 18.0000 0.813988
\(490\) 0 0
\(491\) −15.0000 −0.676941 −0.338470 − 0.940977i \(-0.609909\pi\)
−0.338470 + 0.940977i \(0.609909\pi\)
\(492\) 0 0
\(493\) 10.0000 0.450377
\(494\) 0 0
\(495\) 36.0000 1.61808
\(496\) 0 0
\(497\) 27.0000 1.21112
\(498\) 0 0
\(499\) 24.0000 1.07439 0.537194 − 0.843459i \(-0.319484\pi\)
0.537194 + 0.843459i \(0.319484\pi\)
\(500\) 0 0
\(501\) −72.0000 −3.21672
\(502\) 0 0
\(503\) −24.0000 −1.07011 −0.535054 − 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) 8.00000 0.355995
\(506\) 0 0
\(507\) 3.00000 0.133235
\(508\) 0 0
\(509\) −14.0000 −0.620539 −0.310270 − 0.950649i \(-0.600419\pi\)
−0.310270 + 0.950649i \(0.600419\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 6.00000 0.264392
\(516\) 0 0
\(517\) 54.0000 2.37492
\(518\) 0 0
\(519\) 66.0000 2.89708
\(520\) 0 0
\(521\) −29.0000 −1.27051 −0.635257 − 0.772301i \(-0.719106\pi\)
−0.635257 + 0.772301i \(0.719106\pi\)
\(522\) 0 0
\(523\) 12.0000 0.524723 0.262362 − 0.964970i \(-0.415499\pi\)
0.262362 + 0.964970i \(0.415499\pi\)
\(524\) 0 0
\(525\) 36.0000 1.57117
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 4.00000 0.173259
\(534\) 0 0
\(535\) −12.0000 −0.518805
\(536\) 0 0
\(537\) 27.0000 1.16514
\(538\) 0 0
\(539\) 12.0000 0.516877
\(540\) 0 0
\(541\) −25.0000 −1.07483 −0.537417 − 0.843317i \(-0.680600\pi\)
−0.537417 + 0.843317i \(0.680600\pi\)
\(542\) 0 0
\(543\) 6.00000 0.257485
\(544\) 0 0
\(545\) −7.00000 −0.299847
\(546\) 0 0
\(547\) 3.00000 0.128271 0.0641354 − 0.997941i \(-0.479571\pi\)
0.0641354 + 0.997941i \(0.479571\pi\)
\(548\) 0 0
\(549\) 72.0000 3.07289
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 18.0000 0.765438
\(554\) 0 0
\(555\) 27.0000 1.14609
\(556\) 0 0
\(557\) 23.0000 0.974541 0.487271 − 0.873251i \(-0.337993\pi\)
0.487271 + 0.873251i \(0.337993\pi\)
\(558\) 0 0
\(559\) −9.00000 −0.380659
\(560\) 0 0
\(561\) 18.0000 0.759961
\(562\) 0 0
\(563\) −3.00000 −0.126435 −0.0632175 − 0.998000i \(-0.520136\pi\)
−0.0632175 + 0.998000i \(0.520136\pi\)
\(564\) 0 0
\(565\) 10.0000 0.420703
\(566\) 0 0
\(567\) −27.0000 −1.13389
\(568\) 0 0
\(569\) −17.0000 −0.712677 −0.356339 − 0.934357i \(-0.615975\pi\)
−0.356339 + 0.934357i \(0.615975\pi\)
\(570\) 0 0
\(571\) −21.0000 −0.878823 −0.439411 − 0.898286i \(-0.644813\pi\)
−0.439411 + 0.898286i \(0.644813\pi\)
\(572\) 0 0
\(573\) −18.0000 −0.751961
\(574\) 0 0
\(575\) 24.0000 1.00087
\(576\) 0 0
\(577\) −16.0000 −0.666089 −0.333044 − 0.942911i \(-0.608076\pi\)
−0.333044 + 0.942911i \(0.608076\pi\)
\(578\) 0 0
\(579\) 42.0000 1.74546
\(580\) 0 0
\(581\) 18.0000 0.746766
\(582\) 0 0
\(583\) −12.0000 −0.496989
\(584\) 0 0
\(585\) −6.00000 −0.248069
\(586\) 0 0
\(587\) 30.0000 1.23823 0.619116 − 0.785299i \(-0.287491\pi\)
0.619116 + 0.785299i \(0.287491\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −3.00000 −0.123404
\(592\) 0 0
\(593\) 2.00000 0.0821302 0.0410651 − 0.999156i \(-0.486925\pi\)
0.0410651 + 0.999156i \(0.486925\pi\)
\(594\) 0 0
\(595\) −3.00000 −0.122988
\(596\) 0 0
\(597\) −18.0000 −0.736691
\(598\) 0 0
\(599\) 24.0000 0.980613 0.490307 − 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) −35.0000 −1.42768 −0.713840 − 0.700309i \(-0.753046\pi\)
−0.713840 + 0.700309i \(0.753046\pi\)
\(602\) 0 0
\(603\) 36.0000 1.46603
\(604\) 0 0
\(605\) 25.0000 1.01639
\(606\) 0 0
\(607\) −24.0000 −0.974130 −0.487065 − 0.873366i \(-0.661933\pi\)
−0.487065 + 0.873366i \(0.661933\pi\)
\(608\) 0 0
\(609\) −90.0000 −3.64698
\(610\) 0 0
\(611\) −9.00000 −0.364101
\(612\) 0 0
\(613\) 6.00000 0.242338 0.121169 − 0.992632i \(-0.461336\pi\)
0.121169 + 0.992632i \(0.461336\pi\)
\(614\) 0 0
\(615\) −12.0000 −0.483887
\(616\) 0 0
\(617\) 28.0000 1.12724 0.563619 − 0.826035i \(-0.309409\pi\)
0.563619 + 0.826035i \(0.309409\pi\)
\(618\) 0 0
\(619\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) −54.0000 −2.16695
\(622\) 0 0
\(623\) 24.0000 0.961540
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 9.00000 0.358854
\(630\) 0 0
\(631\) −21.0000 −0.835997 −0.417998 − 0.908448i \(-0.637268\pi\)
−0.417998 + 0.908448i \(0.637268\pi\)
\(632\) 0 0
\(633\) −81.0000 −3.21946
\(634\) 0 0
\(635\) −6.00000 −0.238103
\(636\) 0 0
\(637\) −2.00000 −0.0792429
\(638\) 0 0
\(639\) −54.0000 −2.13621
\(640\) 0 0
\(641\) 10.0000 0.394976 0.197488 − 0.980305i \(-0.436722\pi\)
0.197488 + 0.980305i \(0.436722\pi\)
\(642\) 0 0
\(643\) 6.00000 0.236617 0.118308 − 0.992977i \(-0.462253\pi\)
0.118308 + 0.992977i \(0.462253\pi\)
\(644\) 0 0
\(645\) 27.0000 1.06312
\(646\) 0 0
\(647\) −12.0000 −0.471769 −0.235884 − 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 28.0000 1.09572 0.547862 − 0.836569i \(-0.315442\pi\)
0.547862 + 0.836569i \(0.315442\pi\)
\(654\) 0 0
\(655\) −15.0000 −0.586098
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 36.0000 1.40236 0.701180 − 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) 0 0
\(661\) −42.0000 −1.63361 −0.816805 − 0.576913i \(-0.804257\pi\)
−0.816805 + 0.576913i \(0.804257\pi\)
\(662\) 0 0
\(663\) −3.00000 −0.116510
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −60.0000 −2.32321
\(668\) 0 0
\(669\) 45.0000 1.73980
\(670\) 0 0
\(671\) 72.0000 2.77953
\(672\) 0 0
\(673\) 1.00000 0.0385472 0.0192736 − 0.999814i \(-0.493865\pi\)
0.0192736 + 0.999814i \(0.493865\pi\)
\(674\) 0 0
\(675\) −36.0000 −1.38564
\(676\) 0 0
\(677\) −38.0000 −1.46046 −0.730229 − 0.683202i \(-0.760587\pi\)
−0.730229 + 0.683202i \(0.760587\pi\)
\(678\) 0 0
\(679\) 54.0000 2.07233
\(680\) 0 0
\(681\) −72.0000 −2.75905
\(682\) 0 0
\(683\) −30.0000 −1.14792 −0.573959 − 0.818884i \(-0.694593\pi\)
−0.573959 + 0.818884i \(0.694593\pi\)
\(684\) 0 0
\(685\) 10.0000 0.382080
\(686\) 0 0
\(687\) −15.0000 −0.572286
\(688\) 0 0
\(689\) 2.00000 0.0761939
\(690\) 0 0
\(691\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(692\) 0 0
\(693\) −108.000 −4.10258
\(694\) 0 0
\(695\) 9.00000 0.341389
\(696\) 0 0
\(697\) −4.00000 −0.151511
\(698\) 0 0
\(699\) 51.0000 1.92900
\(700\) 0 0
\(701\) 34.0000 1.28416 0.642081 − 0.766637i \(-0.278071\pi\)
0.642081 + 0.766637i \(0.278071\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 27.0000 1.01688
\(706\) 0 0
\(707\) −24.0000 −0.902613
\(708\) 0 0
\(709\) 10.0000 0.375558 0.187779 − 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 0 0
\(711\) −36.0000 −1.35011
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) −6.00000 −0.224387
\(716\) 0 0
\(717\) −45.0000 −1.68056
\(718\) 0 0
\(719\) −30.0000 −1.11881 −0.559406 − 0.828894i \(-0.688971\pi\)
−0.559406 + 0.828894i \(0.688971\pi\)
\(720\) 0 0
\(721\) −18.0000 −0.670355
\(722\) 0 0
\(723\) −36.0000 −1.33885
\(724\) 0 0
\(725\) −40.0000 −1.48556
\(726\) 0 0
\(727\) −30.0000 −1.11264 −0.556319 − 0.830969i \(-0.687787\pi\)
−0.556319 + 0.830969i \(0.687787\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 9.00000 0.332877
\(732\) 0 0
\(733\) 31.0000 1.14501 0.572506 − 0.819901i \(-0.305971\pi\)
0.572506 + 0.819901i \(0.305971\pi\)
\(734\) 0 0
\(735\) 6.00000 0.221313
\(736\) 0 0
\(737\) 36.0000 1.32608
\(738\) 0 0
\(739\) 12.0000 0.441427 0.220714 − 0.975339i \(-0.429161\pi\)
0.220714 + 0.975339i \(0.429161\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −15.0000 −0.550297 −0.275148 − 0.961402i \(-0.588727\pi\)
−0.275148 + 0.961402i \(0.588727\pi\)
\(744\) 0 0
\(745\) −22.0000 −0.806018
\(746\) 0 0
\(747\) −36.0000 −1.31717
\(748\) 0 0
\(749\) 36.0000 1.31541
\(750\) 0 0
\(751\) −36.0000 −1.31366 −0.656829 − 0.754039i \(-0.728103\pi\)
−0.656829 + 0.754039i \(0.728103\pi\)
\(752\) 0 0
\(753\) 36.0000 1.31191
\(754\) 0 0
\(755\) −9.00000 −0.327544
\(756\) 0 0
\(757\) 34.0000 1.23575 0.617876 − 0.786276i \(-0.287994\pi\)
0.617876 + 0.786276i \(0.287994\pi\)
\(758\) 0 0
\(759\) −108.000 −3.92015
\(760\) 0 0
\(761\) 20.0000 0.724999 0.362500 − 0.931984i \(-0.381923\pi\)
0.362500 + 0.931984i \(0.381923\pi\)
\(762\) 0 0
\(763\) 21.0000 0.760251
\(764\) 0 0
\(765\) 6.00000 0.216930
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −32.0000 −1.15395 −0.576975 − 0.816762i \(-0.695767\pi\)
−0.576975 + 0.816762i \(0.695767\pi\)
\(770\) 0 0
\(771\) 51.0000 1.83672
\(772\) 0 0
\(773\) 1.00000 0.0359675 0.0179838 − 0.999838i \(-0.494275\pi\)
0.0179838 + 0.999838i \(0.494275\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −81.0000 −2.90586
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −54.0000 −1.93227
\(782\) 0 0
\(783\) 90.0000 3.21634
\(784\) 0 0
\(785\) 12.0000 0.428298
\(786\) 0 0
\(787\) −24.0000 −0.855508 −0.427754 − 0.903895i \(-0.640695\pi\)
−0.427754 + 0.903895i \(0.640695\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −30.0000 −1.06668
\(792\) 0 0
\(793\) −12.0000 −0.426132
\(794\) 0 0
\(795\) −6.00000 −0.212798
\(796\) 0 0
\(797\) −20.0000 −0.708436 −0.354218 − 0.935163i \(-0.615253\pi\)
−0.354218 + 0.935163i \(0.615253\pi\)
\(798\) 0 0
\(799\) 9.00000 0.318397
\(800\) 0 0
\(801\) −48.0000 −1.69600
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 18.0000 0.634417
\(806\) 0 0
\(807\) 60.0000 2.11210
\(808\) 0 0
\(809\) 11.0000 0.386739 0.193370 − 0.981126i \(-0.438058\pi\)
0.193370 + 0.981126i \(0.438058\pi\)
\(810\) 0 0
\(811\) −12.0000 −0.421377 −0.210688 − 0.977553i \(-0.567571\pi\)
−0.210688 + 0.977553i \(0.567571\pi\)
\(812\) 0 0
\(813\) −45.0000 −1.57822
\(814\) 0 0
\(815\) 6.00000 0.210171
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 18.0000 0.628971
\(820\) 0 0
\(821\) −19.0000 −0.663105 −0.331552 − 0.943437i \(-0.607572\pi\)
−0.331552 + 0.943437i \(0.607572\pi\)
\(822\) 0 0
\(823\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(824\) 0 0
\(825\) −72.0000 −2.50672
\(826\) 0 0
\(827\) −12.0000 −0.417281 −0.208640 − 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 0 0
\(829\) 2.00000 0.0694629 0.0347314 − 0.999397i \(-0.488942\pi\)
0.0347314 + 0.999397i \(0.488942\pi\)
\(830\) 0 0
\(831\) 24.0000 0.832551
\(832\) 0 0
\(833\) 2.00000 0.0692959
\(834\) 0 0
\(835\) −24.0000 −0.830554
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 48.0000 1.65714 0.828572 − 0.559883i \(-0.189154\pi\)
0.828572 + 0.559883i \(0.189154\pi\)
\(840\) 0 0
\(841\) 71.0000 2.44828
\(842\) 0 0
\(843\) 48.0000 1.65321
\(844\) 0 0
\(845\) 1.00000 0.0344010
\(846\) 0 0
\(847\) −75.0000 −2.57703
\(848\) 0 0
\(849\) −72.0000 −2.47103
\(850\) 0 0
\(851\) −54.0000 −1.85110
\(852\) 0 0
\(853\) 15.0000 0.513590 0.256795 − 0.966466i \(-0.417333\pi\)
0.256795 + 0.966466i \(0.417333\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −34.0000 −1.16142 −0.580709 − 0.814111i \(-0.697225\pi\)
−0.580709 + 0.814111i \(0.697225\pi\)
\(858\) 0 0
\(859\) 36.0000 1.22830 0.614152 − 0.789188i \(-0.289498\pi\)
0.614152 + 0.789188i \(0.289498\pi\)
\(860\) 0 0
\(861\) 36.0000 1.22688
\(862\) 0 0
\(863\) 21.0000 0.714848 0.357424 − 0.933942i \(-0.383655\pi\)
0.357424 + 0.933942i \(0.383655\pi\)
\(864\) 0 0
\(865\) 22.0000 0.748022
\(866\) 0 0
\(867\) −48.0000 −1.63017
\(868\) 0 0
\(869\) −36.0000 −1.22122
\(870\) 0 0
\(871\) −6.00000 −0.203302
\(872\) 0 0
\(873\) −108.000 −3.65525
\(874\) 0 0
\(875\) 27.0000 0.912767
\(876\) 0 0
\(877\) 39.0000 1.31694 0.658468 − 0.752609i \(-0.271205\pi\)
0.658468 + 0.752609i \(0.271205\pi\)
\(878\) 0 0
\(879\) 3.00000 0.101187
\(880\) 0 0
\(881\) −19.0000 −0.640126 −0.320063 − 0.947396i \(-0.603704\pi\)
−0.320063 + 0.947396i \(0.603704\pi\)
\(882\) 0 0
\(883\) 27.0000 0.908622 0.454311 − 0.890843i \(-0.349885\pi\)
0.454311 + 0.890843i \(0.349885\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 24.0000 0.805841 0.402921 − 0.915235i \(-0.367995\pi\)
0.402921 + 0.915235i \(0.367995\pi\)
\(888\) 0 0
\(889\) 18.0000 0.603701
\(890\) 0 0
\(891\) 54.0000 1.80907
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 9.00000 0.300837
\(896\) 0 0
\(897\) 18.0000 0.601003
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −2.00000 −0.0666297
\(902\) 0 0
\(903\) −81.0000 −2.69551
\(904\) 0 0
\(905\) 2.00000 0.0664822
\(906\) 0 0
\(907\) −27.0000 −0.896520 −0.448260 − 0.893903i \(-0.647956\pi\)
−0.448260 + 0.893903i \(0.647956\pi\)
\(908\) 0 0
\(909\) 48.0000 1.59206
\(910\) 0 0
\(911\) −6.00000 −0.198789 −0.0993944 − 0.995048i \(-0.531691\pi\)
−0.0993944 + 0.995048i \(0.531691\pi\)
\(912\) 0 0
\(913\) −36.0000 −1.19143
\(914\) 0 0
\(915\) 36.0000 1.19012
\(916\) 0 0
\(917\) 45.0000 1.48603
\(918\) 0 0
\(919\) 12.0000 0.395843 0.197922 − 0.980218i \(-0.436581\pi\)
0.197922 + 0.980218i \(0.436581\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 9.00000 0.296239
\(924\) 0 0
\(925\) −36.0000 −1.18367
\(926\) 0 0
\(927\) 36.0000 1.18240
\(928\) 0 0
\(929\) 16.0000 0.524943 0.262471 − 0.964940i \(-0.415462\pi\)
0.262471 + 0.964940i \(0.415462\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 6.00000 0.196221
\(936\) 0 0
\(937\) −2.00000 −0.0653372 −0.0326686 − 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) 0 0
\(939\) 57.0000 1.86012
\(940\) 0 0
\(941\) 11.0000 0.358590 0.179295 − 0.983795i \(-0.442618\pi\)
0.179295 + 0.983795i \(0.442618\pi\)
\(942\) 0 0
\(943\) 24.0000 0.781548
\(944\) 0 0
\(945\) −27.0000 −0.878310
\(946\) 0 0
\(947\) −18.0000 −0.584921 −0.292461 − 0.956278i \(-0.594474\pi\)
−0.292461 + 0.956278i \(0.594474\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −66.0000 −2.14020
\(952\) 0 0
\(953\) 55.0000 1.78162 0.890812 − 0.454371i \(-0.150136\pi\)
0.890812 + 0.454371i \(0.150136\pi\)
\(954\) 0 0
\(955\) −6.00000 −0.194155
\(956\) 0 0
\(957\) 180.000 5.81857
\(958\) 0 0
\(959\) −30.0000 −0.968751
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) −72.0000 −2.32017
\(964\) 0 0
\(965\) 14.0000 0.450676
\(966\) 0 0
\(967\) 15.0000 0.482367 0.241184 − 0.970479i \(-0.422464\pi\)
0.241184 + 0.970479i \(0.422464\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −9.00000 −0.288824 −0.144412 − 0.989518i \(-0.546129\pi\)
−0.144412 + 0.989518i \(0.546129\pi\)
\(972\) 0 0
\(973\) −27.0000 −0.865580
\(974\) 0 0
\(975\) 12.0000 0.384308
\(976\) 0 0
\(977\) 38.0000 1.21573 0.607864 − 0.794041i \(-0.292027\pi\)
0.607864 + 0.794041i \(0.292027\pi\)
\(978\) 0 0
\(979\) −48.0000 −1.53409
\(980\) 0 0
\(981\) −42.0000 −1.34096
\(982\) 0 0
\(983\) 21.0000 0.669796 0.334898 − 0.942254i \(-0.391298\pi\)
0.334898 + 0.942254i \(0.391298\pi\)
\(984\) 0 0
\(985\) −1.00000 −0.0318626
\(986\) 0 0
\(987\) −81.0000 −2.57826
\(988\) 0 0
\(989\) −54.0000 −1.71710
\(990\) 0 0
\(991\) 48.0000 1.52477 0.762385 − 0.647124i \(-0.224028\pi\)
0.762385 + 0.647124i \(0.224028\pi\)
\(992\) 0 0
\(993\) 18.0000 0.571213
\(994\) 0 0
\(995\) −6.00000 −0.190213
\(996\) 0 0
\(997\) −18.0000 −0.570066 −0.285033 − 0.958518i \(-0.592005\pi\)
−0.285033 + 0.958518i \(0.592005\pi\)
\(998\) 0 0
\(999\) 81.0000 2.56273
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 3328.2.a.l.1.1 1
4.3 odd 2 3328.2.a.b.1.1 1
8.3 odd 2 3328.2.a.k.1.1 1
8.5 even 2 3328.2.a.a.1.1 1
16.3 odd 4 1664.2.b.a.833.1 ✓ 2
16.5 even 4 1664.2.b.d.833.1 yes 2
16.11 odd 4 1664.2.b.a.833.2 yes 2
16.13 even 4 1664.2.b.d.833.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1664.2.b.a.833.1 ✓ 2 16.3 odd 4
1664.2.b.a.833.2 yes 2 16.11 odd 4
1664.2.b.d.833.1 yes 2 16.5 even 4
1664.2.b.d.833.2 yes 2 16.13 even 4
3328.2.a.a.1.1 1 8.5 even 2
3328.2.a.b.1.1 1 4.3 odd 2
3328.2.a.k.1.1 1 8.3 odd 2
3328.2.a.l.1.1 1 1.1 even 1 trivial