Properties

Label 1456.2.a.c.1.1
Level $1456$
Weight $2$
Character 1456.1
Self dual yes
Analytic conductor $11.626$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1456,2,Mod(1,1456)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1456, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1456.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1456.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: \(11.6262185343\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 728)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1456.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{3} +3.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{3} +3.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{13} -6.00000 q^{15} -2.00000 q^{17} -5.00000 q^{19} +2.00000 q^{21} -1.00000 q^{23} +4.00000 q^{25} +4.00000 q^{27} -5.00000 q^{29} +3.00000 q^{31} -3.00000 q^{35} -6.00000 q^{37} +2.00000 q^{39} -2.00000 q^{41} -1.00000 q^{43} +3.00000 q^{45} -3.00000 q^{47} +1.00000 q^{49} +4.00000 q^{51} +11.0000 q^{53} +10.0000 q^{57} +8.00000 q^{59} -10.0000 q^{61} -1.00000 q^{63} -3.00000 q^{65} -2.00000 q^{67} +2.00000 q^{69} -14.0000 q^{71} -7.00000 q^{73} -8.00000 q^{75} -13.0000 q^{79} -11.0000 q^{81} +1.00000 q^{83} -6.00000 q^{85} +10.0000 q^{87} -11.0000 q^{89} +1.00000 q^{91} -6.00000 q^{93} -15.0000 q^{95} +5.00000 q^{97} +O(q^{100})\)

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
\(3\) −2.00000 −1.15470 −0.577350 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 0 0
\(5\) 3.00000 1.34164 0.670820 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) −6.00000 −1.54919
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) −5.00000 −1.14708 −0.573539 0.819178i \(-0.694430\pi\)
−0.573539 + 0.819178i \(0.694430\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 0 0
\(23\) −1.00000 −0.208514 −0.104257 0.994550i \(-0.533247\pi\)
−0.104257 + 0.994550i \(0.533247\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) 4.00000 0.769800
\(28\) 0 0
\(29\) −5.00000 −0.928477 −0.464238 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) 0 0
\(31\) 3.00000 0.538816 0.269408 0.963026i \(-0.413172\pi\)
0.269408 + 0.963026i \(0.413172\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.00000 −0.507093
\(36\) 0 0
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499 −0.0762493 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) 0 0
\(45\) 3.00000 0.447214
\(46\) 0 0
\(47\) −3.00000 −0.437595 −0.218797 0.975770i \(-0.570213\pi\)
−0.218797 + 0.975770i \(0.570213\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 4.00000 0.560112
\(52\) 0 0
\(53\) 11.0000 1.51097 0.755483 0.655168i \(-0.227402\pi\)
0.755483 + 0.655168i \(0.227402\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 10.0000 1.32453
\(58\) 0 0
\(59\) 8.00000 1.04151 0.520756 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) −3.00000 −0.372104
\(66\) 0 0
\(67\) −2.00000 −0.244339 −0.122169 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) 0 0
\(69\) 2.00000 0.240772
\(70\) 0 0
\(71\) −14.0000 −1.66149 −0.830747 0.556650i \(-0.812086\pi\)
−0.830747 + 0.556650i \(0.812086\pi\)
\(72\) 0 0
\(73\) −7.00000 −0.819288 −0.409644 0.912245i \(-0.634347\pi\)
−0.409644 + 0.912245i \(0.634347\pi\)
\(74\) 0 0
\(75\) −8.00000 −0.923760
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −13.0000 −1.46261 −0.731307 0.682048i \(-0.761089\pi\)
−0.731307 + 0.682048i \(0.761089\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) 1.00000 0.109764 0.0548821 0.998493i \(-0.482522\pi\)
0.0548821 + 0.998493i \(0.482522\pi\)
\(84\) 0 0
\(85\) −6.00000 −0.650791
\(86\) 0 0
\(87\) 10.0000 1.07211
\(88\) 0 0
\(89\) −11.0000 −1.16600 −0.582999 0.812473i \(-0.698121\pi\)
−0.582999 + 0.812473i \(0.698121\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 0 0
\(93\) −6.00000 −0.622171
\(94\) 0 0
\(95\) −15.0000 −1.53897
\(96\) 0 0
\(97\) 5.00000 0.507673 0.253837 0.967247i \(-0.418307\pi\)
0.253837 + 0.967247i \(0.418307\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 12.0000 1.19404 0.597022 0.802225i \(-0.296350\pi\)
0.597022 + 0.802225i \(0.296350\pi\)
\(102\) 0 0
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) 0 0
\(105\) 6.00000 0.585540
\(106\) 0 0
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) −4.00000 −0.383131 −0.191565 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) 0 0
\(111\) 12.0000 1.13899
\(112\) 0 0
\(113\) −15.0000 −1.41108 −0.705541 0.708669i \(-0.749296\pi\)
−0.705541 + 0.708669i \(0.749296\pi\)
\(114\) 0 0
\(115\) −3.00000 −0.279751
\(116\) 0 0
\(117\) −1.00000 −0.0924500
\(118\) 0 0
\(119\) 2.00000 0.183340
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 4.00000 0.360668
\(124\) 0 0
\(125\) −3.00000 −0.268328
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 0 0
\(129\) 2.00000 0.176090
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 5.00000 0.433555
\(134\) 0 0
\(135\) 12.0000 1.03280
\(136\) 0 0
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 6.00000 0.505291
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −15.0000 −1.24568
\(146\) 0 0
\(147\) −2.00000 −0.164957
\(148\) 0 0
\(149\) −16.0000 −1.31077 −0.655386 0.755295i \(-0.727494\pi\)
−0.655386 + 0.755295i \(0.727494\pi\)
\(150\) 0 0
\(151\) −2.00000 −0.162758 −0.0813788 0.996683i \(-0.525932\pi\)
−0.0813788 + 0.996683i \(0.525932\pi\)
\(152\) 0 0
\(153\) −2.00000 −0.161690
\(154\) 0 0
\(155\) 9.00000 0.722897
\(156\) 0 0
\(157\) 14.0000 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 0 0
\(159\) −22.0000 −1.74471
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) 12.0000 0.939913 0.469956 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 23.0000 1.77979 0.889897 0.456162i \(-0.150776\pi\)
0.889897 + 0.456162i \(0.150776\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −5.00000 −0.382360
\(172\) 0 0
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) −4.00000 −0.302372
\(176\) 0 0
\(177\) −16.0000 −1.20263
\(178\) 0 0
\(179\) −17.0000 −1.27064 −0.635320 0.772249i \(-0.719132\pi\)
−0.635320 + 0.772249i \(0.719132\pi\)
\(180\) 0 0
\(181\) 8.00000 0.594635 0.297318 0.954779i \(-0.403908\pi\)
0.297318 + 0.954779i \(0.403908\pi\)
\(182\) 0 0
\(183\) 20.0000 1.47844
\(184\) 0 0
\(185\) −18.0000 −1.32339
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −4.00000 −0.290957
\(190\) 0 0
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 0 0
\(193\) 10.0000 0.719816 0.359908 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) 0 0
\(195\) 6.00000 0.429669
\(196\) 0 0
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) −10.0000 −0.708881 −0.354441 0.935079i \(-0.615329\pi\)
−0.354441 + 0.935079i \(0.615329\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 0 0
\(203\) 5.00000 0.350931
\(204\) 0 0
\(205\) −6.00000 −0.419058
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 15.0000 1.03264 0.516321 0.856395i \(-0.327301\pi\)
0.516321 + 0.856395i \(0.327301\pi\)
\(212\) 0 0
\(213\) 28.0000 1.91853
\(214\) 0 0
\(215\) −3.00000 −0.204598
\(216\) 0 0
\(217\) −3.00000 −0.203653
\(218\) 0 0
\(219\) 14.0000 0.946032
\(220\) 0 0
\(221\) 2.00000 0.134535
\(222\) 0 0
\(223\) 17.0000 1.13840 0.569202 0.822198i \(-0.307252\pi\)
0.569202 + 0.822198i \(0.307252\pi\)
\(224\) 0 0
\(225\) 4.00000 0.266667
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 26.0000 1.71813 0.859064 0.511868i \(-0.171046\pi\)
0.859064 + 0.511868i \(0.171046\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −9.00000 −0.589610 −0.294805 0.955557i \(-0.595255\pi\)
−0.294805 + 0.955557i \(0.595255\pi\)
\(234\) 0 0
\(235\) −9.00000 −0.587095
\(236\) 0 0
\(237\) 26.0000 1.68888
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) 5.00000 0.322078 0.161039 0.986948i \(-0.448515\pi\)
0.161039 + 0.986948i \(0.448515\pi\)
\(242\) 0 0
\(243\) 10.0000 0.641500
\(244\) 0 0
\(245\) 3.00000 0.191663
\(246\) 0 0
\(247\) 5.00000 0.318142
\(248\) 0 0
\(249\) −2.00000 −0.126745
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 12.0000 0.751469
\(256\) 0 0
\(257\) −24.0000 −1.49708 −0.748539 0.663090i \(-0.769245\pi\)
−0.748539 + 0.663090i \(0.769245\pi\)
\(258\) 0 0
\(259\) 6.00000 0.372822
\(260\) 0 0
\(261\) −5.00000 −0.309492
\(262\) 0 0
\(263\) 21.0000 1.29492 0.647458 0.762101i \(-0.275832\pi\)
0.647458 + 0.762101i \(0.275832\pi\)
\(264\) 0 0
\(265\) 33.0000 2.02717
\(266\) 0 0
\(267\) 22.0000 1.34638
\(268\) 0 0
\(269\) 32.0000 1.95107 0.975537 0.219834i \(-0.0705517\pi\)
0.975537 + 0.219834i \(0.0705517\pi\)
\(270\) 0 0
\(271\) 24.0000 1.45790 0.728948 0.684569i \(-0.240010\pi\)
0.728948 + 0.684569i \(0.240010\pi\)
\(272\) 0 0
\(273\) −2.00000 −0.121046
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 1.00000 0.0600842 0.0300421 0.999549i \(-0.490436\pi\)
0.0300421 + 0.999549i \(0.490436\pi\)
\(278\) 0 0
\(279\) 3.00000 0.179605
\(280\) 0 0
\(281\) −20.0000 −1.19310 −0.596550 0.802576i \(-0.703462\pi\)
−0.596550 + 0.802576i \(0.703462\pi\)
\(282\) 0 0
\(283\) −32.0000 −1.90220 −0.951101 0.308879i \(-0.900046\pi\)
−0.951101 + 0.308879i \(0.900046\pi\)
\(284\) 0 0
\(285\) 30.0000 1.77705
\(286\) 0 0
\(287\) 2.00000 0.118056
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) −10.0000 −0.586210
\(292\) 0 0
\(293\) 23.0000 1.34367 0.671837 0.740699i \(-0.265505\pi\)
0.671837 + 0.740699i \(0.265505\pi\)
\(294\) 0 0
\(295\) 24.0000 1.39733
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 1.00000 0.0578315
\(300\) 0 0
\(301\) 1.00000 0.0576390
\(302\) 0 0
\(303\) −24.0000 −1.37876
\(304\) 0 0
\(305\) −30.0000 −1.71780
\(306\) 0 0
\(307\) 9.00000 0.513657 0.256829 0.966457i \(-0.417322\pi\)
0.256829 + 0.966457i \(0.417322\pi\)
\(308\) 0 0
\(309\) 32.0000 1.82042
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 0 0
\(313\) 28.0000 1.58265 0.791327 0.611393i \(-0.209391\pi\)
0.791327 + 0.611393i \(0.209391\pi\)
\(314\) 0 0
\(315\) −3.00000 −0.169031
\(316\) 0 0
\(317\) 20.0000 1.12331 0.561656 0.827371i \(-0.310164\pi\)
0.561656 + 0.827371i \(0.310164\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −24.0000 −1.33955
\(322\) 0 0
\(323\) 10.0000 0.556415
\(324\) 0 0
\(325\) −4.00000 −0.221880
\(326\) 0 0
\(327\) 8.00000 0.442401
\(328\) 0 0
\(329\) 3.00000 0.165395
\(330\) 0 0
\(331\) 6.00000 0.329790 0.164895 0.986311i \(-0.447272\pi\)
0.164895 + 0.986311i \(0.447272\pi\)
\(332\) 0 0
\(333\) −6.00000 −0.328798
\(334\) 0 0
\(335\) −6.00000 −0.327815
\(336\) 0 0
\(337\) 21.0000 1.14394 0.571971 0.820274i \(-0.306179\pi\)
0.571971 + 0.820274i \(0.306179\pi\)
\(338\) 0 0
\(339\) 30.0000 1.62938
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 6.00000 0.323029
\(346\) 0 0
\(347\) −32.0000 −1.71785 −0.858925 0.512101i \(-0.828867\pi\)
−0.858925 + 0.512101i \(0.828867\pi\)
\(348\) 0 0
\(349\) 1.00000 0.0535288 0.0267644 0.999642i \(-0.491480\pi\)
0.0267644 + 0.999642i \(0.491480\pi\)
\(350\) 0 0
\(351\) −4.00000 −0.213504
\(352\) 0 0
\(353\) 26.0000 1.38384 0.691920 0.721974i \(-0.256765\pi\)
0.691920 + 0.721974i \(0.256765\pi\)
\(354\) 0 0
\(355\) −42.0000 −2.22913
\(356\) 0 0
\(357\) −4.00000 −0.211702
\(358\) 0 0
\(359\) −14.0000 −0.738892 −0.369446 0.929252i \(-0.620452\pi\)
−0.369446 + 0.929252i \(0.620452\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) 0 0
\(363\) 22.0000 1.15470
\(364\) 0 0
\(365\) −21.0000 −1.09919
\(366\) 0 0
\(367\) −36.0000 −1.87918 −0.939592 0.342296i \(-0.888796\pi\)
−0.939592 + 0.342296i \(0.888796\pi\)
\(368\) 0 0
\(369\) −2.00000 −0.104116
\(370\) 0 0
\(371\) −11.0000 −0.571092
\(372\) 0 0
\(373\) 22.0000 1.13912 0.569558 0.821951i \(-0.307114\pi\)
0.569558 + 0.821951i \(0.307114\pi\)
\(374\) 0 0
\(375\) 6.00000 0.309839
\(376\) 0 0
\(377\) 5.00000 0.257513
\(378\) 0 0
\(379\) −28.0000 −1.43826 −0.719132 0.694874i \(-0.755460\pi\)
−0.719132 + 0.694874i \(0.755460\pi\)
\(380\) 0 0
\(381\) 16.0000 0.819705
\(382\) 0 0
\(383\) 16.0000 0.817562 0.408781 0.912633i \(-0.365954\pi\)
0.408781 + 0.912633i \(0.365954\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −1.00000 −0.0508329
\(388\) 0 0
\(389\) 10.0000 0.507020 0.253510 0.967333i \(-0.418415\pi\)
0.253510 + 0.967333i \(0.418415\pi\)
\(390\) 0 0
\(391\) 2.00000 0.101144
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −39.0000 −1.96230
\(396\) 0 0
\(397\) 13.0000 0.652451 0.326226 0.945292i \(-0.394223\pi\)
0.326226 + 0.945292i \(0.394223\pi\)
\(398\) 0 0
\(399\) −10.0000 −0.500626
\(400\) 0 0
\(401\) −12.0000 −0.599251 −0.299626 0.954057i \(-0.596862\pi\)
−0.299626 + 0.954057i \(0.596862\pi\)
\(402\) 0 0
\(403\) −3.00000 −0.149441
\(404\) 0 0
\(405\) −33.0000 −1.63978
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 13.0000 0.642809 0.321404 0.946942i \(-0.395845\pi\)
0.321404 + 0.946942i \(0.395845\pi\)
\(410\) 0 0
\(411\) −4.00000 −0.197305
\(412\) 0 0
\(413\) −8.00000 −0.393654
\(414\) 0 0
\(415\) 3.00000 0.147264
\(416\) 0 0
\(417\) 8.00000 0.391762
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) 0 0
\(423\) −3.00000 −0.145865
\(424\) 0 0
\(425\) −8.00000 −0.388057
\(426\) 0 0
\(427\) 10.0000 0.483934
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −14.0000 −0.674356 −0.337178 0.941441i \(-0.609472\pi\)
−0.337178 + 0.941441i \(0.609472\pi\)
\(432\) 0 0
\(433\) 16.0000 0.768911 0.384455 0.923144i \(-0.374389\pi\)
0.384455 + 0.923144i \(0.374389\pi\)
\(434\) 0 0
\(435\) 30.0000 1.43839
\(436\) 0 0
\(437\) 5.00000 0.239182
\(438\) 0 0
\(439\) −38.0000 −1.81364 −0.906821 0.421517i \(-0.861498\pi\)
−0.906821 + 0.421517i \(0.861498\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −17.0000 −0.807694 −0.403847 0.914826i \(-0.632327\pi\)
−0.403847 + 0.914826i \(0.632327\pi\)
\(444\) 0 0
\(445\) −33.0000 −1.56435
\(446\) 0 0
\(447\) 32.0000 1.51355
\(448\) 0 0
\(449\) −28.0000 −1.32140 −0.660701 0.750649i \(-0.729741\pi\)
−0.660701 + 0.750649i \(0.729741\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 4.00000 0.187936
\(454\) 0 0
\(455\) 3.00000 0.140642
\(456\) 0 0
\(457\) −6.00000 −0.280668 −0.140334 0.990104i \(-0.544818\pi\)
−0.140334 + 0.990104i \(0.544818\pi\)
\(458\) 0 0
\(459\) −8.00000 −0.373408
\(460\) 0 0
\(461\) −42.0000 −1.95614 −0.978068 0.208288i \(-0.933211\pi\)
−0.978068 + 0.208288i \(0.933211\pi\)
\(462\) 0 0
\(463\) −12.0000 −0.557687 −0.278844 0.960337i \(-0.589951\pi\)
−0.278844 + 0.960337i \(0.589951\pi\)
\(464\) 0 0
\(465\) −18.0000 −0.834730
\(466\) 0 0
\(467\) 8.00000 0.370196 0.185098 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) 0 0
\(469\) 2.00000 0.0923514
\(470\) 0 0
\(471\) −28.0000 −1.29017
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −20.0000 −0.917663
\(476\) 0 0
\(477\) 11.0000 0.503655
\(478\) 0 0
\(479\) −25.0000 −1.14228 −0.571140 0.820853i \(-0.693499\pi\)
−0.571140 + 0.820853i \(0.693499\pi\)
\(480\) 0 0
\(481\) 6.00000 0.273576
\(482\) 0 0
\(483\) −2.00000 −0.0910032
\(484\) 0 0
\(485\) 15.0000 0.681115
\(486\) 0 0
\(487\) 22.0000 0.996915 0.498458 0.866914i \(-0.333900\pi\)
0.498458 + 0.866914i \(0.333900\pi\)
\(488\) 0 0
\(489\) −24.0000 −1.08532
\(490\) 0 0
\(491\) 8.00000 0.361035 0.180517 0.983572i \(-0.442223\pi\)
0.180517 + 0.983572i \(0.442223\pi\)
\(492\) 0 0
\(493\) 10.0000 0.450377
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 14.0000 0.627986
\(498\) 0 0
\(499\) 10.0000 0.447661 0.223831 0.974628i \(-0.428144\pi\)
0.223831 + 0.974628i \(0.428144\pi\)
\(500\) 0 0
\(501\) −46.0000 −2.05513
\(502\) 0 0
\(503\) 18.0000 0.802580 0.401290 0.915951i \(-0.368562\pi\)
0.401290 + 0.915951i \(0.368562\pi\)
\(504\) 0 0
\(505\) 36.0000 1.60198
\(506\) 0 0
\(507\) −2.00000 −0.0888231
\(508\) 0 0
\(509\) 11.0000 0.487566 0.243783 0.969830i \(-0.421611\pi\)
0.243783 + 0.969830i \(0.421611\pi\)
\(510\) 0 0
\(511\) 7.00000 0.309662
\(512\) 0 0
\(513\) −20.0000 −0.883022
\(514\) 0 0
\(515\) −48.0000 −2.11513
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 12.0000 0.526742
\(520\) 0 0
\(521\) −4.00000 −0.175243 −0.0876216 0.996154i \(-0.527927\pi\)
−0.0876216 + 0.996154i \(0.527927\pi\)
\(522\) 0 0
\(523\) 6.00000 0.262362 0.131181 0.991358i \(-0.458123\pi\)
0.131181 + 0.991358i \(0.458123\pi\)
\(524\) 0 0
\(525\) 8.00000 0.349149
\(526\) 0 0
\(527\) −6.00000 −0.261364
\(528\) 0 0
\(529\) −22.0000 −0.956522
\(530\) 0 0
\(531\) 8.00000 0.347170
\(532\) 0 0
\(533\) 2.00000 0.0866296
\(534\) 0 0
\(535\) 36.0000 1.55642
\(536\) 0 0
\(537\) 34.0000 1.46721
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −6.00000 −0.257960 −0.128980 0.991647i \(-0.541170\pi\)
−0.128980 + 0.991647i \(0.541170\pi\)
\(542\) 0 0
\(543\) −16.0000 −0.686626
\(544\) 0 0
\(545\) −12.0000 −0.514024
\(546\) 0 0
\(547\) 1.00000 0.0427569 0.0213785 0.999771i \(-0.493195\pi\)
0.0213785 + 0.999771i \(0.493195\pi\)
\(548\) 0 0
\(549\) −10.0000 −0.426790
\(550\) 0 0
\(551\) 25.0000 1.06504
\(552\) 0 0
\(553\) 13.0000 0.552816
\(554\) 0 0
\(555\) 36.0000 1.52811
\(556\) 0 0
\(557\) −40.0000 −1.69485 −0.847427 0.530912i \(-0.821850\pi\)
−0.847427 + 0.530912i \(0.821850\pi\)
\(558\) 0 0
\(559\) 1.00000 0.0422955
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 8.00000 0.337160 0.168580 0.985688i \(-0.446082\pi\)
0.168580 + 0.985688i \(0.446082\pi\)
\(564\) 0 0
\(565\) −45.0000 −1.89316
\(566\) 0 0
\(567\) 11.0000 0.461957
\(568\) 0 0
\(569\) −21.0000 −0.880366 −0.440183 0.897908i \(-0.645086\pi\)
−0.440183 + 0.897908i \(0.645086\pi\)
\(570\) 0 0
\(571\) 23.0000 0.962520 0.481260 0.876578i \(-0.340179\pi\)
0.481260 + 0.876578i \(0.340179\pi\)
\(572\) 0 0
\(573\) −16.0000 −0.668410
\(574\) 0 0
\(575\) −4.00000 −0.166812
\(576\) 0 0
\(577\) 18.0000 0.749350 0.374675 0.927156i \(-0.377754\pi\)
0.374675 + 0.927156i \(0.377754\pi\)
\(578\) 0 0
\(579\) −20.0000 −0.831172
\(580\) 0 0
\(581\) −1.00000 −0.0414870
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −3.00000 −0.124035
\(586\) 0 0
\(587\) −3.00000 −0.123823 −0.0619116 0.998082i \(-0.519720\pi\)
−0.0619116 + 0.998082i \(0.519720\pi\)
\(588\) 0 0
\(589\) −15.0000 −0.618064
\(590\) 0 0
\(591\) 36.0000 1.48084
\(592\) 0 0
\(593\) −21.0000 −0.862367 −0.431183 0.902264i \(-0.641904\pi\)
−0.431183 + 0.902264i \(0.641904\pi\)
\(594\) 0 0
\(595\) 6.00000 0.245976
\(596\) 0 0
\(597\) 20.0000 0.818546
\(598\) 0 0
\(599\) −5.00000 −0.204294 −0.102147 0.994769i \(-0.532571\pi\)
−0.102147 + 0.994769i \(0.532571\pi\)
\(600\) 0 0
\(601\) 38.0000 1.55005 0.775026 0.631929i \(-0.217737\pi\)
0.775026 + 0.631929i \(0.217737\pi\)
\(602\) 0 0
\(603\) −2.00000 −0.0814463
\(604\) 0 0
\(605\) −33.0000 −1.34164
\(606\) 0 0
\(607\) 32.0000 1.29884 0.649420 0.760430i \(-0.275012\pi\)
0.649420 + 0.760430i \(0.275012\pi\)
\(608\) 0 0
\(609\) −10.0000 −0.405220
\(610\) 0 0
\(611\) 3.00000 0.121367
\(612\) 0 0
\(613\) −4.00000 −0.161558 −0.0807792 0.996732i \(-0.525741\pi\)
−0.0807792 + 0.996732i \(0.525741\pi\)
\(614\) 0 0
\(615\) 12.0000 0.483887
\(616\) 0 0
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 0 0
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) 0 0
\(623\) 11.0000 0.440706
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 12.0000 0.478471
\(630\) 0 0
\(631\) 38.0000 1.51276 0.756378 0.654135i \(-0.226967\pi\)
0.756378 + 0.654135i \(0.226967\pi\)
\(632\) 0 0
\(633\) −30.0000 −1.19239
\(634\) 0 0
\(635\) −24.0000 −0.952411
\(636\) 0 0
\(637\) −1.00000 −0.0396214
\(638\) 0 0
\(639\) −14.0000 −0.553831
\(640\) 0 0
\(641\) 33.0000 1.30342 0.651711 0.758468i \(-0.274052\pi\)
0.651711 + 0.758468i \(0.274052\pi\)
\(642\) 0 0
\(643\) −28.0000 −1.10421 −0.552106 0.833774i \(-0.686176\pi\)
−0.552106 + 0.833774i \(0.686176\pi\)
\(644\) 0 0
\(645\) 6.00000 0.236250
\(646\) 0 0
\(647\) 8.00000 0.314512 0.157256 0.987558i \(-0.449735\pi\)
0.157256 + 0.987558i \(0.449735\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 6.00000 0.235159
\(652\) 0 0
\(653\) 34.0000 1.33052 0.665261 0.746611i \(-0.268320\pi\)
0.665261 + 0.746611i \(0.268320\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −7.00000 −0.273096
\(658\) 0 0
\(659\) 5.00000 0.194772 0.0973862 0.995247i \(-0.468952\pi\)
0.0973862 + 0.995247i \(0.468952\pi\)
\(660\) 0 0
\(661\) 13.0000 0.505641 0.252821 0.967513i \(-0.418642\pi\)
0.252821 + 0.967513i \(0.418642\pi\)
\(662\) 0 0
\(663\) −4.00000 −0.155347
\(664\) 0 0
\(665\) 15.0000 0.581675
\(666\) 0 0
\(667\) 5.00000 0.193601
\(668\) 0 0
\(669\) −34.0000 −1.31452
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 45.0000 1.73462 0.867311 0.497766i \(-0.165846\pi\)
0.867311 + 0.497766i \(0.165846\pi\)
\(674\) 0 0
\(675\) 16.0000 0.615840
\(676\) 0 0
\(677\) 32.0000 1.22986 0.614930 0.788582i \(-0.289184\pi\)
0.614930 + 0.788582i \(0.289184\pi\)
\(678\) 0 0
\(679\) −5.00000 −0.191882
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −40.0000 −1.53056 −0.765279 0.643699i \(-0.777399\pi\)
−0.765279 + 0.643699i \(0.777399\pi\)
\(684\) 0 0
\(685\) 6.00000 0.229248
\(686\) 0 0
\(687\) −52.0000 −1.98392
\(688\) 0 0
\(689\) −11.0000 −0.419067
\(690\) 0 0
\(691\) 49.0000 1.86405 0.932024 0.362397i \(-0.118041\pi\)
0.932024 + 0.362397i \(0.118041\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −12.0000 −0.455186
\(696\) 0 0
\(697\) 4.00000 0.151511
\(698\) 0 0
\(699\) 18.0000 0.680823
\(700\) 0 0
\(701\) −11.0000 −0.415464 −0.207732 0.978186i \(-0.566608\pi\)
−0.207732 + 0.978186i \(0.566608\pi\)
\(702\) 0 0
\(703\) 30.0000 1.13147
\(704\) 0 0
\(705\) 18.0000 0.677919
\(706\) 0 0
\(707\) −12.0000 −0.451306
\(708\) 0 0
\(709\) 24.0000 0.901339 0.450669 0.892691i \(-0.351185\pi\)
0.450669 + 0.892691i \(0.351185\pi\)
\(710\) 0 0
\(711\) −13.0000 −0.487538
\(712\) 0 0
\(713\) −3.00000 −0.112351
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 24.0000 0.896296
\(718\) 0 0
\(719\) −8.00000 −0.298350 −0.149175 0.988811i \(-0.547662\pi\)
−0.149175 + 0.988811i \(0.547662\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) 0 0
\(723\) −10.0000 −0.371904
\(724\) 0 0
\(725\) −20.0000 −0.742781
\(726\) 0 0
\(727\) −6.00000 −0.222528 −0.111264 0.993791i \(-0.535490\pi\)
−0.111264 + 0.993791i \(0.535490\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 2.00000 0.0739727
\(732\) 0 0
\(733\) −39.0000 −1.44050 −0.720249 0.693716i \(-0.755972\pi\)
−0.720249 + 0.693716i \(0.755972\pi\)
\(734\) 0 0
\(735\) −6.00000 −0.221313
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 40.0000 1.47142 0.735712 0.677295i \(-0.236848\pi\)
0.735712 + 0.677295i \(0.236848\pi\)
\(740\) 0 0
\(741\) −10.0000 −0.367359
\(742\) 0 0
\(743\) −12.0000 −0.440237 −0.220119 0.975473i \(-0.570644\pi\)
−0.220119 + 0.975473i \(0.570644\pi\)
\(744\) 0 0
\(745\) −48.0000 −1.75858
\(746\) 0 0
\(747\) 1.00000 0.0365881
\(748\) 0 0
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) −25.0000 −0.912263 −0.456131 0.889912i \(-0.650765\pi\)
−0.456131 + 0.889912i \(0.650765\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −6.00000 −0.218362
\(756\) 0 0
\(757\) 17.0000 0.617876 0.308938 0.951082i \(-0.400027\pi\)
0.308938 + 0.951082i \(0.400027\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −49.0000 −1.77625 −0.888124 0.459603i \(-0.847992\pi\)
−0.888124 + 0.459603i \(0.847992\pi\)
\(762\) 0 0
\(763\) 4.00000 0.144810
\(764\) 0 0
\(765\) −6.00000 −0.216930
\(766\) 0 0
\(767\) −8.00000 −0.288863
\(768\) 0 0
\(769\) 23.0000 0.829401 0.414701 0.909958i \(-0.363886\pi\)
0.414701 + 0.909958i \(0.363886\pi\)
\(770\) 0 0
\(771\) 48.0000 1.72868
\(772\) 0 0
\(773\) 18.0000 0.647415 0.323708 0.946157i \(-0.395071\pi\)
0.323708 + 0.946157i \(0.395071\pi\)
\(774\) 0 0
\(775\) 12.0000 0.431053
\(776\) 0 0
\(777\) −12.0000 −0.430498
\(778\) 0 0
\(779\) 10.0000 0.358287
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −20.0000 −0.714742
\(784\) 0 0
\(785\) 42.0000 1.49904
\(786\) 0 0
\(787\) −1.00000 −0.0356462 −0.0178231 0.999841i \(-0.505674\pi\)
−0.0178231 + 0.999841i \(0.505674\pi\)
\(788\) 0 0
\(789\) −42.0000 −1.49524
\(790\) 0 0
\(791\) 15.0000 0.533339
\(792\) 0 0
\(793\) 10.0000 0.355110
\(794\) 0 0
\(795\) −66.0000 −2.34078
\(796\) 0 0
\(797\) 18.0000 0.637593 0.318796 0.947823i \(-0.396721\pi\)
0.318796 + 0.947823i \(0.396721\pi\)
\(798\) 0 0
\(799\) 6.00000 0.212265
\(800\) 0 0
\(801\) −11.0000 −0.388666
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 3.00000 0.105736
\(806\) 0 0
\(807\) −64.0000 −2.25291
\(808\) 0 0
\(809\) −43.0000 −1.51180 −0.755900 0.654687i \(-0.772800\pi\)
−0.755900 + 0.654687i \(0.772800\pi\)
\(810\) 0 0
\(811\) −44.0000 −1.54505 −0.772524 0.634985i \(-0.781006\pi\)
−0.772524 + 0.634985i \(0.781006\pi\)
\(812\) 0 0
\(813\) −48.0000 −1.68343
\(814\) 0 0
\(815\) 36.0000 1.26102
\(816\) 0 0
\(817\) 5.00000 0.174928
\(818\) 0 0
\(819\) 1.00000 0.0349428
\(820\) 0 0
\(821\) 6.00000 0.209401 0.104701 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) 0 0
\(823\) −8.00000 −0.278862 −0.139431 0.990232i \(-0.544527\pi\)
−0.139431 + 0.990232i \(0.544527\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −36.0000 −1.25184 −0.625921 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) 0 0
\(829\) −36.0000 −1.25033 −0.625166 0.780492i \(-0.714969\pi\)
−0.625166 + 0.780492i \(0.714969\pi\)
\(830\) 0 0
\(831\) −2.00000 −0.0693792
\(832\) 0 0
\(833\) −2.00000 −0.0692959
\(834\) 0 0
\(835\) 69.0000 2.38784
\(836\) 0 0
\(837\) 12.0000 0.414781
\(838\) 0 0
\(839\) −36.0000 −1.24286 −0.621429 0.783470i \(-0.713448\pi\)
−0.621429 + 0.783470i \(0.713448\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 0 0
\(843\) 40.0000 1.37767
\(844\) 0 0
\(845\) 3.00000 0.103203
\(846\) 0 0
\(847\) 11.0000 0.377964
\(848\) 0 0
\(849\) 64.0000 2.19647
\(850\) 0 0
\(851\) 6.00000 0.205677
\(852\) 0 0
\(853\) −25.0000 −0.855984 −0.427992 0.903783i \(-0.640779\pi\)
−0.427992 + 0.903783i \(0.640779\pi\)
\(854\) 0 0
\(855\) −15.0000 −0.512989
\(856\) 0 0
\(857\) −22.0000 −0.751506 −0.375753 0.926720i \(-0.622616\pi\)
−0.375753 + 0.926720i \(0.622616\pi\)
\(858\) 0 0
\(859\) −42.0000 −1.43302 −0.716511 0.697576i \(-0.754262\pi\)
−0.716511 + 0.697576i \(0.754262\pi\)
\(860\) 0 0
\(861\) −4.00000 −0.136320
\(862\) 0 0
\(863\) −36.0000 −1.22545 −0.612727 0.790295i \(-0.709928\pi\)
−0.612727 + 0.790295i \(0.709928\pi\)
\(864\) 0 0
\(865\) −18.0000 −0.612018
\(866\) 0 0
\(867\) 26.0000 0.883006
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 2.00000 0.0677674
\(872\) 0 0
\(873\) 5.00000 0.169224
\(874\) 0 0
\(875\) 3.00000 0.101419
\(876\) 0 0
\(877\) −26.0000 −0.877958 −0.438979 0.898497i \(-0.644660\pi\)
−0.438979 + 0.898497i \(0.644660\pi\)
\(878\) 0 0
\(879\) −46.0000 −1.55154
\(880\) 0 0
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 0 0
\(883\) 48.0000 1.61533 0.807664 0.589643i \(-0.200731\pi\)
0.807664 + 0.589643i \(0.200731\pi\)
\(884\) 0 0
\(885\) −48.0000 −1.61350
\(886\) 0 0
\(887\) −8.00000 −0.268614 −0.134307 0.990940i \(-0.542881\pi\)
−0.134307 + 0.990940i \(0.542881\pi\)
\(888\) 0 0
\(889\) 8.00000 0.268311
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 15.0000 0.501956
\(894\) 0 0
\(895\) −51.0000 −1.70474
\(896\) 0 0
\(897\) −2.00000 −0.0667781
\(898\) 0 0
\(899\) −15.0000 −0.500278
\(900\) 0 0
\(901\) −22.0000 −0.732926
\(902\) 0 0
\(903\) −2.00000 −0.0665558
\(904\) 0 0
\(905\) 24.0000 0.797787
\(906\) 0 0
\(907\) 35.0000 1.16216 0.581078 0.813848i \(-0.302631\pi\)
0.581078 + 0.813848i \(0.302631\pi\)
\(908\) 0 0
\(909\) 12.0000 0.398015
\(910\) 0 0
\(911\) 29.0000 0.960813 0.480406 0.877046i \(-0.340489\pi\)
0.480406 + 0.877046i \(0.340489\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 60.0000 1.98354
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) −18.0000 −0.593120
\(922\) 0 0
\(923\) 14.0000 0.460816
\(924\) 0 0
\(925\) −24.0000 −0.789115
\(926\) 0 0
\(927\) −16.0000 −0.525509
\(928\) 0 0
\(929\) −9.00000 −0.295280 −0.147640 0.989041i \(-0.547168\pi\)
−0.147640 + 0.989041i \(0.547168\pi\)
\(930\) 0 0
\(931\) −5.00000 −0.163868
\(932\) 0 0
\(933\) −48.0000 −1.57145
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 18.0000 0.588034 0.294017 0.955800i \(-0.405008\pi\)
0.294017 + 0.955800i \(0.405008\pi\)
\(938\) 0 0
\(939\) −56.0000 −1.82749
\(940\) 0 0
\(941\) 9.00000 0.293392 0.146696 0.989182i \(-0.453136\pi\)
0.146696 + 0.989182i \(0.453136\pi\)
\(942\) 0 0
\(943\) 2.00000 0.0651290
\(944\) 0 0
\(945\) −12.0000 −0.390360
\(946\) 0 0
\(947\) 58.0000 1.88475 0.942373 0.334563i \(-0.108589\pi\)
0.942373 + 0.334563i \(0.108589\pi\)
\(948\) 0 0
\(949\) 7.00000 0.227230
\(950\) 0 0
\(951\) −40.0000 −1.29709
\(952\) 0 0
\(953\) −11.0000 −0.356325 −0.178162 0.984001i \(-0.557015\pi\)
−0.178162 + 0.984001i \(0.557015\pi\)
\(954\) 0 0
\(955\) 24.0000 0.776622
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −2.00000 −0.0645834
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) 0 0
\(963\) 12.0000 0.386695
\(964\) 0 0
\(965\) 30.0000 0.965734
\(966\) 0 0
\(967\) −4.00000 −0.128631 −0.0643157 0.997930i \(-0.520486\pi\)
−0.0643157 + 0.997930i \(0.520486\pi\)
\(968\) 0 0
\(969\) −20.0000 −0.642493
\(970\) 0 0
\(971\) −30.0000 −0.962746 −0.481373 0.876516i \(-0.659862\pi\)
−0.481373 + 0.876516i \(0.659862\pi\)
\(972\) 0 0
\(973\) 4.00000 0.128234
\(974\) 0 0
\(975\) 8.00000 0.256205
\(976\) 0 0
\(977\) 44.0000 1.40768 0.703842 0.710356i \(-0.251466\pi\)
0.703842 + 0.710356i \(0.251466\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −4.00000 −0.127710
\(982\) 0 0
\(983\) −9.00000 −0.287055 −0.143528 0.989646i \(-0.545845\pi\)
−0.143528 + 0.989646i \(0.545845\pi\)
\(984\) 0 0
\(985\) −54.0000 −1.72058
\(986\) 0 0
\(987\) −6.00000 −0.190982
\(988\) 0 0
\(989\) 1.00000 0.0317982
\(990\) 0 0
\(991\) 48.0000 1.52477 0.762385 0.647124i \(-0.224028\pi\)
0.762385 + 0.647124i \(0.224028\pi\)
\(992\) 0 0
\(993\) −12.0000 −0.380808
\(994\) 0 0
\(995\) −30.0000 −0.951064
\(996\) 0 0
\(997\) 52.0000 1.64686 0.823428 0.567420i \(-0.192059\pi\)
0.823428 + 0.567420i \(0.192059\pi\)
\(998\) 0 0
\(999\) −24.0000 −0.759326
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 1456.2.a.c.1.1 1
4.3 odd 2 728.2.a.d.1.1 1
8.3 odd 2 5824.2.a.c.1.1 1
8.5 even 2 5824.2.a.ba.1.1 1
12.11 even 2 6552.2.a.c.1.1 1
28.27 even 2 5096.2.a.a.1.1 1
52.51 odd 2 9464.2.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
728.2.a.d.1.1 1 4.3 odd 2
1456.2.a.c.1.1 1 1.1 even 1 trivial
5096.2.a.a.1.1 1 28.27 even 2
5824.2.a.c.1.1 1 8.3 odd 2
5824.2.a.ba.1.1 1 8.5 even 2
6552.2.a.c.1.1 1 12.11 even 2
9464.2.a.e.1.1 1 52.51 odd 2