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Properties

Label 5408.2.a.b.1.1
Level $5408$
Weight $2$
Character 5408.1
Self dual yes
Analytic conductor $43.183$
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] = [5408,2,Mod(1,5408)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5408, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5408.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5408 = 2^{5} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5408.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: \(43.1830974131\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 416)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 5408.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} +3.00000 q^{5} +1.00000 q^{7} +6.00000 q^{9} -4.00000 q^{11} -9.00000 q^{15} +5.00000 q^{17} -6.00000 q^{19} -3.00000 q^{21} +6.00000 q^{23} +4.00000 q^{25} -9.00000 q^{27} -4.00000 q^{29} +12.0000 q^{33} +3.00000 q^{35} +3.00000 q^{37} -12.0000 q^{41} +3.00000 q^{43} +18.0000 q^{45} -7.00000 q^{47} -6.00000 q^{49} -15.0000 q^{51} -2.00000 q^{53} -12.0000 q^{55} +18.0000 q^{57} +2.00000 q^{59} -12.0000 q^{61} +6.00000 q^{63} -4.00000 q^{67} -18.0000 q^{69} +11.0000 q^{71} -6.00000 q^{73} -12.0000 q^{75} -4.00000 q^{77} +6.00000 q^{79} +9.00000 q^{81} -10.0000 q^{83} +15.0000 q^{85} +12.0000 q^{87} -6.00000 q^{89} -18.0000 q^{95} -24.0000 q^{99} +O(q^{100})\)

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 \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(4\) 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 0.188982 − 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) 0 0
\(9\) 6.00000 2.00000
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 − 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) −9.00000 −2.32379
\(16\) 0 0
\(17\) 5.00000 1.21268 0.606339 − 0.795206i \(-0.292637\pi\)
0.606339 + 0.795206i \(0.292637\pi\)
\(18\) 0 0
\(19\) −6.00000 −1.37649 −0.688247 − 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 0 0
\(21\) −3.00000 −0.654654
\(22\) 0 0
\(23\) 6.00000 1.25109 0.625543 − 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) −9.00000 −1.73205
\(28\) 0 0
\(29\) −4.00000 −0.742781 −0.371391 − 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 12.0000 2.08893
\(34\) 0 0
\(35\) 3.00000 0.507093
\(36\) 0 0
\(37\) 3.00000 0.493197 0.246598 − 0.969118i \(-0.420687\pi\)
0.246598 + 0.969118i \(0.420687\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −12.0000 −1.87409 −0.937043 − 0.349215i \(-0.886448\pi\)
−0.937043 + 0.349215i \(0.886448\pi\)
\(42\) 0 0
\(43\) 3.00000 0.457496 0.228748 − 0.973486i \(-0.426537\pi\)
0.228748 + 0.973486i \(0.426537\pi\)
\(44\) 0 0
\(45\) 18.0000 2.68328
\(46\) 0 0
\(47\) −7.00000 −1.02105 −0.510527 − 0.859861i \(-0.670550\pi\)
−0.510527 + 0.859861i \(0.670550\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) −15.0000 −2.10042
\(52\) 0 0
\(53\) −2.00000 −0.274721 −0.137361 − 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) −12.0000 −1.61808
\(56\) 0 0
\(57\) 18.0000 2.38416
\(58\) 0 0
\(59\) 2.00000 0.260378 0.130189 − 0.991489i \(-0.458442\pi\)
0.130189 + 0.991489i \(0.458442\pi\)
\(60\) 0 0
\(61\) −12.0000 −1.53644 −0.768221 − 0.640184i \(-0.778858\pi\)
−0.768221 + 0.640184i \(0.778858\pi\)
\(62\) 0 0
\(63\) 6.00000 0.755929
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) −18.0000 −2.16695
\(70\) 0 0
\(71\) 11.0000 1.30546 0.652730 − 0.757591i \(-0.273624\pi\)
0.652730 + 0.757591i \(0.273624\pi\)
\(72\) 0 0
\(73\) −6.00000 −0.702247 −0.351123 − 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 0 0
\(75\) −12.0000 −1.38564
\(76\) 0 0
\(77\) −4.00000 −0.455842
\(78\) 0 0
\(79\) 6.00000 0.675053 0.337526 − 0.941316i \(-0.390410\pi\)
0.337526 + 0.941316i \(0.390410\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) −10.0000 −1.09764 −0.548821 − 0.835940i \(-0.684923\pi\)
−0.548821 + 0.835940i \(0.684923\pi\)
\(84\) 0 0
\(85\) 15.0000 1.62698
\(86\) 0 0
\(87\) 12.0000 1.28654
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −18.0000 −1.84676
\(96\) 0 0
\(97\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 0 0
\(99\) −24.0000 −2.41209
\(100\) 0 0
\(101\) 4.00000 0.398015 0.199007 − 0.979998i \(-0.436228\pi\)
0.199007 + 0.979998i \(0.436228\pi\)
\(102\) 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 \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) 15.0000 1.43674 0.718370 − 0.695662i \(-0.244889\pi\)
0.718370 + 0.695662i \(0.244889\pi\)
\(110\) 0 0
\(111\) −9.00000 −0.854242
\(112\) 0 0
\(113\) −14.0000 −1.31701 −0.658505 − 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 0 0
\(115\) 18.0000 1.67851
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 5.00000 0.458349
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) 36.0000 3.24601
\(124\) 0 0
\(125\) −3.00000 −0.268328
\(126\) 0 0
\(127\) 18.0000 1.59724 0.798621 − 0.601834i \(-0.205563\pi\)
0.798621 + 0.601834i \(0.205563\pi\)
\(128\) 0 0
\(129\) −9.00000 −0.792406
\(130\) 0 0
\(131\) −9.00000 −0.786334 −0.393167 − 0.919467i \(-0.628621\pi\)
−0.393167 + 0.919467i \(0.628621\pi\)
\(132\) 0 0
\(133\) −6.00000 −0.520266
\(134\) 0 0
\(135\) −27.0000 −2.32379
\(136\) 0 0
\(137\) 6.00000 0.512615 0.256307 − 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) −9.00000 −0.763370 −0.381685 − 0.924292i \(-0.624656\pi\)
−0.381685 + 0.924292i \(0.624656\pi\)
\(140\) 0 0
\(141\) 21.0000 1.76852
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −12.0000 −0.996546
\(146\) 0 0
\(147\) 18.0000 1.48461
\(148\) 0 0
\(149\) 6.00000 0.491539 0.245770 − 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) 3.00000 0.244137 0.122068 − 0.992522i \(-0.461047\pi\)
0.122068 + 0.992522i \(0.461047\pi\)
\(152\) 0 0
\(153\) 30.0000 2.42536
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 14.0000 1.11732 0.558661 − 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 0 0
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 6.00000 0.472866
\(162\) 0 0
\(163\) −18.0000 −1.40987 −0.704934 − 0.709273i \(-0.749024\pi\)
−0.704934 + 0.709273i \(0.749024\pi\)
\(164\) 0 0
\(165\) 36.0000 2.80260
\(166\) 0 0
\(167\) 4.00000 0.309529 0.154765 − 0.987951i \(-0.450538\pi\)
0.154765 + 0.987951i \(0.450538\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) −36.0000 −2.75299
\(172\) 0 0
\(173\) 14.0000 1.06440 0.532200 − 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) 0 0
\(175\) 4.00000 0.302372
\(176\) 0 0
\(177\) −6.00000 −0.450988
\(178\) 0 0
\(179\) −21.0000 −1.56961 −0.784807 − 0.619740i \(-0.787238\pi\)
−0.784807 + 0.619740i \(0.787238\pi\)
\(180\) 0 0
\(181\) 6.00000 0.445976 0.222988 − 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) 0 0
\(183\) 36.0000 2.66120
\(184\) 0 0
\(185\) 9.00000 0.661693
\(186\) 0 0
\(187\) −20.0000 −1.46254
\(188\) 0 0
\(189\) −9.00000 −0.654654
\(190\) 0 0
\(191\) −24.0000 −1.73658 −0.868290 − 0.496058i \(-0.834780\pi\)
−0.868290 + 0.496058i \(0.834780\pi\)
\(192\) 0 0
\(193\) −18.0000 −1.29567 −0.647834 − 0.761781i \(-0.724325\pi\)
−0.647834 + 0.761781i \(0.724325\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −3.00000 −0.213741 −0.106871 − 0.994273i \(-0.534083\pi\)
−0.106871 + 0.994273i \(0.534083\pi\)
\(198\) 0 0
\(199\) 12.0000 0.850657 0.425329 − 0.905039i \(-0.360158\pi\)
0.425329 + 0.905039i \(0.360158\pi\)
\(200\) 0 0
\(201\) 12.0000 0.846415
\(202\) 0 0
\(203\) −4.00000 −0.280745
\(204\) 0 0
\(205\) −36.0000 −2.51435
\(206\) 0 0
\(207\) 36.0000 2.50217
\(208\) 0 0
\(209\) 24.0000 1.66011
\(210\) 0 0
\(211\) 3.00000 0.206529 0.103264 − 0.994654i \(-0.467071\pi\)
0.103264 + 0.994654i \(0.467071\pi\)
\(212\) 0 0
\(213\) −33.0000 −2.26112
\(214\) 0 0
\(215\) 9.00000 0.613795
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 18.0000 1.21633
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −21.0000 −1.40626 −0.703132 − 0.711059i \(-0.748216\pi\)
−0.703132 + 0.711059i \(0.748216\pi\)
\(224\) 0 0
\(225\) 24.0000 1.60000
\(226\) 0 0
\(227\) −8.00000 −0.530979 −0.265489 − 0.964114i \(-0.585534\pi\)
−0.265489 + 0.964114i \(0.585534\pi\)
\(228\) 0 0
\(229\) 9.00000 0.594737 0.297368 − 0.954763i \(-0.403891\pi\)
0.297368 + 0.954763i \(0.403891\pi\)
\(230\) 0 0
\(231\) 12.0000 0.789542
\(232\) 0 0
\(233\) −19.0000 −1.24473 −0.622366 − 0.782727i \(-0.713828\pi\)
−0.622366 + 0.782727i \(0.713828\pi\)
\(234\) 0 0
\(235\) −21.0000 −1.36989
\(236\) 0 0
\(237\) −18.0000 −1.16923
\(238\) 0 0
\(239\) 5.00000 0.323423 0.161712 − 0.986838i \(-0.448299\pi\)
0.161712 + 0.986838i \(0.448299\pi\)
\(240\) 0 0
\(241\) −6.00000 −0.386494 −0.193247 − 0.981150i \(-0.561902\pi\)
−0.193247 + 0.981150i \(0.561902\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −18.0000 −1.14998
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 30.0000 1.90117
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 − 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) −24.0000 −1.50887
\(254\) 0 0
\(255\) −45.0000 −2.81801
\(256\) 0 0
\(257\) 5.00000 0.311891 0.155946 − 0.987766i \(-0.450158\pi\)
0.155946 + 0.987766i \(0.450158\pi\)
\(258\) 0 0
\(259\) 3.00000 0.186411
\(260\) 0 0
\(261\) −24.0000 −1.48556
\(262\) 0 0
\(263\) 12.0000 0.739952 0.369976 − 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) 0 0
\(265\) −6.00000 −0.368577
\(266\) 0 0
\(267\) 18.0000 1.10158
\(268\) 0 0
\(269\) −4.00000 −0.243884 −0.121942 − 0.992537i \(-0.538912\pi\)
−0.121942 + 0.992537i \(0.538912\pi\)
\(270\) 0 0
\(271\) −11.0000 −0.668202 −0.334101 − 0.942537i \(-0.608433\pi\)
−0.334101 + 0.942537i \(0.608433\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −16.0000 −0.964836
\(276\) 0 0
\(277\) −12.0000 −0.721010 −0.360505 − 0.932757i \(-0.617396\pi\)
−0.360505 + 0.932757i \(0.617396\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 18.0000 1.07379 0.536895 − 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 0 0
\(283\) −12.0000 −0.713326 −0.356663 − 0.934233i \(-0.616086\pi\)
−0.356663 + 0.934233i \(0.616086\pi\)
\(284\) 0 0
\(285\) 54.0000 3.19868
\(286\) 0 0
\(287\) −12.0000 −0.708338
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −9.00000 −0.525786 −0.262893 − 0.964825i \(-0.584677\pi\)
−0.262893 + 0.964825i \(0.584677\pi\)
\(294\) 0 0
\(295\) 6.00000 0.349334
\(296\) 0 0
\(297\) 36.0000 2.08893
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 3.00000 0.172917
\(302\) 0 0
\(303\) −12.0000 −0.689382
\(304\) 0 0
\(305\) −36.0000 −2.06135
\(306\) 0 0
\(307\) 2.00000 0.114146 0.0570730 − 0.998370i \(-0.481823\pi\)
0.0570730 + 0.998370i \(0.481823\pi\)
\(308\) 0 0
\(309\) −18.0000 −1.02398
\(310\) 0 0
\(311\) 6.00000 0.340229 0.170114 − 0.985424i \(-0.445586\pi\)
0.170114 + 0.985424i \(0.445586\pi\)
\(312\) 0 0
\(313\) −21.0000 −1.18699 −0.593495 − 0.804838i \(-0.702252\pi\)
−0.593495 + 0.804838i \(0.702252\pi\)
\(314\) 0 0
\(315\) 18.0000 1.01419
\(316\) 0 0
\(317\) 6.00000 0.336994 0.168497 − 0.985702i \(-0.446109\pi\)
0.168497 + 0.985702i \(0.446109\pi\)
\(318\) 0 0
\(319\) 16.0000 0.895828
\(320\) 0 0
\(321\) −36.0000 −2.00932
\(322\) 0 0
\(323\) −30.0000 −1.66924
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −45.0000 −2.48851
\(328\) 0 0
\(329\) −7.00000 −0.385922
\(330\) 0 0
\(331\) −26.0000 −1.42909 −0.714545 − 0.699590i \(-0.753366\pi\)
−0.714545 + 0.699590i \(0.753366\pi\)
\(332\) 0 0
\(333\) 18.0000 0.986394
\(334\) 0 0
\(335\) −12.0000 −0.655630
\(336\) 0 0
\(337\) −13.0000 −0.708155 −0.354078 − 0.935216i \(-0.615205\pi\)
−0.354078 + 0.935216i \(0.615205\pi\)
\(338\) 0 0
\(339\) 42.0000 2.28113
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −13.0000 −0.701934
\(344\) 0 0
\(345\) −54.0000 −2.90726
\(346\) 0 0
\(347\) −21.0000 −1.12734 −0.563670 − 0.826000i \(-0.690611\pi\)
−0.563670 + 0.826000i \(0.690611\pi\)
\(348\) 0 0
\(349\) −21.0000 −1.12410 −0.562052 − 0.827102i \(-0.689988\pi\)
−0.562052 + 0.827102i \(0.689988\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 18.0000 0.958043 0.479022 − 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) 0 0
\(355\) 33.0000 1.75146
\(356\) 0 0
\(357\) −15.0000 −0.793884
\(358\) 0 0
\(359\) −32.0000 −1.68890 −0.844448 − 0.535638i \(-0.820071\pi\)
−0.844448 + 0.535638i \(0.820071\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 0 0
\(363\) −15.0000 −0.787296
\(364\) 0 0
\(365\) −18.0000 −0.942163
\(366\) 0 0
\(367\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) −72.0000 −3.74817
\(370\) 0 0
\(371\) −2.00000 −0.103835
\(372\) 0 0
\(373\) 12.0000 0.621336 0.310668 − 0.950518i \(-0.399447\pi\)
0.310668 + 0.950518i \(0.399447\pi\)
\(374\) 0 0
\(375\) 9.00000 0.464758
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 30.0000 1.54100 0.770498 − 0.637442i \(-0.220007\pi\)
0.770498 + 0.637442i \(0.220007\pi\)
\(380\) 0 0
\(381\) −54.0000 −2.76650
\(382\) 0 0
\(383\) 11.0000 0.562074 0.281037 − 0.959697i \(-0.409322\pi\)
0.281037 + 0.959697i \(0.409322\pi\)
\(384\) 0 0
\(385\) −12.0000 −0.611577
\(386\) 0 0
\(387\) 18.0000 0.914991
\(388\) 0 0
\(389\) 2.00000 0.101404 0.0507020 − 0.998714i \(-0.483854\pi\)
0.0507020 + 0.998714i \(0.483854\pi\)
\(390\) 0 0
\(391\) 30.0000 1.51717
\(392\) 0 0
\(393\) 27.0000 1.36197
\(394\) 0 0
\(395\) 18.0000 0.905678
\(396\) 0 0
\(397\) −30.0000 −1.50566 −0.752828 − 0.658217i \(-0.771311\pi\)
−0.752828 + 0.658217i \(0.771311\pi\)
\(398\) 0 0
\(399\) 18.0000 0.901127
\(400\) 0 0
\(401\) −6.00000 −0.299626 −0.149813 − 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 27.0000 1.34164
\(406\) 0 0
\(407\) −12.0000 −0.594818
\(408\) 0 0
\(409\) −6.00000 −0.296681 −0.148340 − 0.988936i \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) 0 0
\(411\) −18.0000 −0.887875
\(412\) 0 0
\(413\) 2.00000 0.0984136
\(414\) 0 0
\(415\) −30.0000 −1.47264
\(416\) 0 0
\(417\) 27.0000 1.32220
\(418\) 0 0
\(419\) −3.00000 −0.146560 −0.0732798 − 0.997311i \(-0.523347\pi\)
−0.0732798 + 0.997311i \(0.523347\pi\)
\(420\) 0 0
\(421\) −9.00000 −0.438633 −0.219317 − 0.975654i \(-0.570383\pi\)
−0.219317 + 0.975654i \(0.570383\pi\)
\(422\) 0 0
\(423\) −42.0000 −2.04211
\(424\) 0 0
\(425\) 20.0000 0.970143
\(426\) 0 0
\(427\) −12.0000 −0.580721
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −19.0000 −0.915198 −0.457599 − 0.889159i \(-0.651290\pi\)
−0.457599 + 0.889159i \(0.651290\pi\)
\(432\) 0 0
\(433\) 3.00000 0.144171 0.0720854 − 0.997398i \(-0.477035\pi\)
0.0720854 + 0.997398i \(0.477035\pi\)
\(434\) 0 0
\(435\) 36.0000 1.72607
\(436\) 0 0
\(437\) −36.0000 −1.72211
\(438\) 0 0
\(439\) −6.00000 −0.286364 −0.143182 − 0.989696i \(-0.545733\pi\)
−0.143182 + 0.989696i \(0.545733\pi\)
\(440\) 0 0
\(441\) −36.0000 −1.71429
\(442\) 0 0
\(443\) −39.0000 −1.85295 −0.926473 − 0.376361i \(-0.877175\pi\)
−0.926473 + 0.376361i \(0.877175\pi\)
\(444\) 0 0
\(445\) −18.0000 −0.853282
\(446\) 0 0
\(447\) −18.0000 −0.851371
\(448\) 0 0
\(449\) 12.0000 0.566315 0.283158 − 0.959073i \(-0.408618\pi\)
0.283158 + 0.959073i \(0.408618\pi\)
\(450\) 0 0
\(451\) 48.0000 2.26023
\(452\) 0 0
\(453\) −9.00000 −0.422857
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −18.0000 −0.842004 −0.421002 − 0.907060i \(-0.638322\pi\)
−0.421002 + 0.907060i \(0.638322\pi\)
\(458\) 0 0
\(459\) −45.0000 −2.10042
\(460\) 0 0
\(461\) −33.0000 −1.53696 −0.768482 − 0.639872i \(-0.778987\pi\)
−0.768482 + 0.639872i \(0.778987\pi\)
\(462\) 0 0
\(463\) 32.0000 1.48717 0.743583 − 0.668644i \(-0.233125\pi\)
0.743583 + 0.668644i \(0.233125\pi\)
\(464\) 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
\(469\) −4.00000 −0.184703
\(470\) 0 0
\(471\) −42.0000 −1.93526
\(472\) 0 0
\(473\) −12.0000 −0.551761
\(474\) 0 0
\(475\) −24.0000 −1.10120
\(476\) 0 0
\(477\) −12.0000 −0.549442
\(478\) 0 0
\(479\) 37.0000 1.69057 0.845287 − 0.534313i \(-0.179430\pi\)
0.845287 + 0.534313i \(0.179430\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) −18.0000 −0.819028
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 12.0000 0.543772 0.271886 − 0.962329i \(-0.412353\pi\)
0.271886 + 0.962329i \(0.412353\pi\)
\(488\) 0 0
\(489\) 54.0000 2.44196
\(490\) 0 0
\(491\) 15.0000 0.676941 0.338470 − 0.940977i \(-0.390091\pi\)
0.338470 + 0.940977i \(0.390091\pi\)
\(492\) 0 0
\(493\) −20.0000 −0.900755
\(494\) 0 0
\(495\) −72.0000 −3.23616
\(496\) 0 0
\(497\) 11.0000 0.493417
\(498\) 0 0
\(499\) −32.0000 −1.43252 −0.716258 − 0.697835i \(-0.754147\pi\)
−0.716258 + 0.697835i \(0.754147\pi\)
\(500\) 0 0
\(501\) −12.0000 −0.536120
\(502\) 0 0
\(503\) 6.00000 0.267527 0.133763 − 0.991013i \(-0.457294\pi\)
0.133763 + 0.991013i \(0.457294\pi\)
\(504\) 0 0
\(505\) 12.0000 0.533993
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −18.0000 −0.797836 −0.398918 − 0.916987i \(-0.630614\pi\)
−0.398918 + 0.916987i \(0.630614\pi\)
\(510\) 0 0
\(511\) −6.00000 −0.265424
\(512\) 0 0
\(513\) 54.0000 2.38416
\(514\) 0 0
\(515\) 18.0000 0.793175
\(516\) 0 0
\(517\) 28.0000 1.23144
\(518\) 0 0
\(519\) −42.0000 −1.84360
\(520\) 0 0
\(521\) 11.0000 0.481919 0.240959 − 0.970535i \(-0.422538\pi\)
0.240959 + 0.970535i \(0.422538\pi\)
\(522\) 0 0
\(523\) −24.0000 −1.04945 −0.524723 − 0.851273i \(-0.675831\pi\)
−0.524723 + 0.851273i \(0.675831\pi\)
\(524\) 0 0
\(525\) −12.0000 −0.523723
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 12.0000 0.520756
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 36.0000 1.55642
\(536\) 0 0
\(537\) 63.0000 2.71865
\(538\) 0 0
\(539\) 24.0000 1.03375
\(540\) 0 0
\(541\) 33.0000 1.41878 0.709390 − 0.704816i \(-0.248970\pi\)
0.709390 + 0.704816i \(0.248970\pi\)
\(542\) 0 0
\(543\) −18.0000 −0.772454
\(544\) 0 0
\(545\) 45.0000 1.92759
\(546\) 0 0
\(547\) 33.0000 1.41098 0.705489 − 0.708721i \(-0.250727\pi\)
0.705489 + 0.708721i \(0.250727\pi\)
\(548\) 0 0
\(549\) −72.0000 −3.07289
\(550\) 0 0
\(551\) 24.0000 1.02243
\(552\) 0 0
\(553\) 6.00000 0.255146
\(554\) 0 0
\(555\) −27.0000 −1.14609
\(556\) 0 0
\(557\) −3.00000 −0.127114 −0.0635570 − 0.997978i \(-0.520244\pi\)
−0.0635570 + 0.997978i \(0.520244\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 60.0000 2.53320
\(562\) 0 0
\(563\) 3.00000 0.126435 0.0632175 − 0.998000i \(-0.479864\pi\)
0.0632175 + 0.998000i \(0.479864\pi\)
\(564\) 0 0
\(565\) −42.0000 −1.76695
\(566\) 0 0
\(567\) 9.00000 0.377964
\(568\) 0 0
\(569\) 11.0000 0.461144 0.230572 − 0.973055i \(-0.425940\pi\)
0.230572 + 0.973055i \(0.425940\pi\)
\(570\) 0 0
\(571\) 21.0000 0.878823 0.439411 − 0.898286i \(-0.355187\pi\)
0.439411 + 0.898286i \(0.355187\pi\)
\(572\) 0 0
\(573\) 72.0000 3.00784
\(574\) 0 0
\(575\) 24.0000 1.00087
\(576\) 0 0
\(577\) 6.00000 0.249783 0.124892 − 0.992170i \(-0.460142\pi\)
0.124892 + 0.992170i \(0.460142\pi\)
\(578\) 0 0
\(579\) 54.0000 2.24416
\(580\) 0 0
\(581\) −10.0000 −0.414870
\(582\) 0 0
\(583\) 8.00000 0.331326
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −2.00000 −0.0825488 −0.0412744 − 0.999148i \(-0.513142\pi\)
−0.0412744 + 0.999148i \(0.513142\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 9.00000 0.370211
\(592\) 0 0
\(593\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 15.0000 0.614940
\(596\) 0 0
\(597\) −36.0000 −1.47338
\(598\) 0 0
\(599\) −30.0000 −1.22577 −0.612883 − 0.790173i \(-0.709990\pi\)
−0.612883 + 0.790173i \(0.709990\pi\)
\(600\) 0 0
\(601\) −3.00000 −0.122373 −0.0611863 − 0.998126i \(-0.519488\pi\)
−0.0611863 + 0.998126i \(0.519488\pi\)
\(602\) 0 0
\(603\) −24.0000 −0.977356
\(604\) 0 0
\(605\) 15.0000 0.609837
\(606\) 0 0
\(607\) −30.0000 −1.21766 −0.608831 − 0.793300i \(-0.708361\pi\)
−0.608831 + 0.793300i \(0.708361\pi\)
\(608\) 0 0
\(609\) 12.0000 0.486265
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −6.00000 −0.242338 −0.121169 − 0.992632i \(-0.538664\pi\)
−0.121169 + 0.992632i \(0.538664\pi\)
\(614\) 0 0
\(615\) 108.000 4.35498
\(616\) 0 0
\(617\) −36.0000 −1.44931 −0.724653 − 0.689114i \(-0.758000\pi\)
−0.724653 + 0.689114i \(0.758000\pi\)
\(618\) 0 0
\(619\) −44.0000 −1.76851 −0.884255 − 0.467005i \(-0.845333\pi\)
−0.884255 + 0.467005i \(0.845333\pi\)
\(620\) 0 0
\(621\) −54.0000 −2.16695
\(622\) 0 0
\(623\) −6.00000 −0.240385
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) −72.0000 −2.87540
\(628\) 0 0
\(629\) 15.0000 0.598089
\(630\) 0 0
\(631\) −21.0000 −0.835997 −0.417998 − 0.908448i \(-0.637268\pi\)
−0.417998 + 0.908448i \(0.637268\pi\)
\(632\) 0 0
\(633\) −9.00000 −0.357718
\(634\) 0 0
\(635\) 54.0000 2.14292
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 66.0000 2.61092
\(640\) 0 0
\(641\) 10.0000 0.394976 0.197488 − 0.980305i \(-0.436722\pi\)
0.197488 + 0.980305i \(0.436722\pi\)
\(642\) 0 0
\(643\) 4.00000 0.157745 0.0788723 − 0.996885i \(-0.474868\pi\)
0.0788723 + 0.996885i \(0.474868\pi\)
\(644\) 0 0
\(645\) −27.0000 −1.06312
\(646\) 0 0
\(647\) 18.0000 0.707653 0.353827 − 0.935311i \(-0.384880\pi\)
0.353827 + 0.935311i \(0.384880\pi\)
\(648\) 0 0
\(649\) −8.00000 −0.314027
\(650\) 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
\(655\) −27.0000 −1.05498
\(656\) 0 0
\(657\) −36.0000 −1.40449
\(658\) 0 0
\(659\) 24.0000 0.934907 0.467454 − 0.884018i \(-0.345171\pi\)
0.467454 + 0.884018i \(0.345171\pi\)
\(660\) 0 0
\(661\) −42.0000 −1.63361 −0.816805 − 0.576913i \(-0.804257\pi\)
−0.816805 + 0.576913i \(0.804257\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −18.0000 −0.698010
\(666\) 0 0
\(667\) −24.0000 −0.929284
\(668\) 0 0
\(669\) 63.0000 2.43572
\(670\) 0 0
\(671\) 48.0000 1.85302
\(672\) 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
\(677\) −14.0000 −0.538064 −0.269032 − 0.963131i \(-0.586704\pi\)
−0.269032 + 0.963131i \(0.586704\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 24.0000 0.919682
\(682\) 0 0
\(683\) −14.0000 −0.535695 −0.267848 − 0.963461i \(-0.586312\pi\)
−0.267848 + 0.963461i \(0.586312\pi\)
\(684\) 0 0
\(685\) 18.0000 0.687745
\(686\) 0 0
\(687\) −27.0000 −1.03011
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 8.00000 0.304334 0.152167 − 0.988355i \(-0.451375\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) 0 0
\(693\) −24.0000 −0.911685
\(694\) 0 0
\(695\) −27.0000 −1.02417
\(696\) 0 0
\(697\) −60.0000 −2.27266
\(698\) 0 0
\(699\) 57.0000 2.15594
\(700\) 0 0
\(701\) 26.0000 0.982006 0.491003 − 0.871158i \(-0.336630\pi\)
0.491003 + 0.871158i \(0.336630\pi\)
\(702\) 0 0
\(703\) −18.0000 −0.678883
\(704\) 0 0
\(705\) 63.0000 2.37272
\(706\) 0 0
\(707\) 4.00000 0.150435
\(708\) 0 0
\(709\) 42.0000 1.57734 0.788672 − 0.614815i \(-0.210769\pi\)
0.788672 + 0.614815i \(0.210769\pi\)
\(710\) 0 0
\(711\) 36.0000 1.35011
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −15.0000 −0.560185
\(718\) 0 0
\(719\) −48.0000 −1.79010 −0.895049 − 0.445968i \(-0.852860\pi\)
−0.895049 + 0.445968i \(0.852860\pi\)
\(720\) 0 0
\(721\) 6.00000 0.223452
\(722\) 0 0
\(723\) 18.0000 0.669427
\(724\) 0 0
\(725\) −16.0000 −0.594225
\(726\) 0 0
\(727\) −24.0000 −0.890111 −0.445055 − 0.895503i \(-0.646816\pi\)
−0.445055 + 0.895503i \(0.646816\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 15.0000 0.554795
\(732\) 0 0
\(733\) 33.0000 1.21888 0.609441 − 0.792831i \(-0.291394\pi\)
0.609441 + 0.792831i \(0.291394\pi\)
\(734\) 0 0
\(735\) 54.0000 1.99182
\(736\) 0 0
\(737\) 16.0000 0.589368
\(738\) 0 0
\(739\) −16.0000 −0.588570 −0.294285 − 0.955718i \(-0.595081\pi\)
−0.294285 + 0.955718i \(0.595081\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −11.0000 −0.403551 −0.201775 − 0.979432i \(-0.564671\pi\)
−0.201775 + 0.979432i \(0.564671\pi\)
\(744\) 0 0
\(745\) 18.0000 0.659469
\(746\) 0 0
\(747\) −60.0000 −2.19529
\(748\) 0 0
\(749\) 12.0000 0.438470
\(750\) 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
\(757\) 2.00000 0.0726912 0.0363456 − 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) 0 0
\(759\) 72.0000 2.61343
\(760\) 0 0
\(761\) 30.0000 1.08750 0.543750 − 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) 0 0
\(763\) 15.0000 0.543036
\(764\) 0 0
\(765\) 90.0000 3.25396
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −48.0000 −1.73092 −0.865462 − 0.500974i \(-0.832975\pi\)
−0.865462 + 0.500974i \(0.832975\pi\)
\(770\) 0 0
\(771\) −15.0000 −0.540212
\(772\) 0 0
\(773\) 27.0000 0.971123 0.485561 − 0.874203i \(-0.338615\pi\)
0.485561 + 0.874203i \(0.338615\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −9.00000 −0.322873
\(778\) 0 0
\(779\) 72.0000 2.57967
\(780\) 0 0
\(781\) −44.0000 −1.57444
\(782\) 0 0
\(783\) 36.0000 1.28654
\(784\) 0 0
\(785\) 42.0000 1.49904
\(786\) 0 0
\(787\) −32.0000 −1.14068 −0.570338 − 0.821410i \(-0.693188\pi\)
−0.570338 + 0.821410i \(0.693188\pi\)
\(788\) 0 0
\(789\) −36.0000 −1.28163
\(790\) 0 0
\(791\) −14.0000 −0.497783
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 18.0000 0.638394
\(796\) 0 0
\(797\) −50.0000 −1.77109 −0.885545 − 0.464553i \(-0.846215\pi\)
−0.885545 + 0.464553i \(0.846215\pi\)
\(798\) 0 0
\(799\) −35.0000 −1.23821
\(800\) 0 0
\(801\) −36.0000 −1.27200
\(802\) 0 0
\(803\) 24.0000 0.846942
\(804\) 0 0
\(805\) 18.0000 0.634417
\(806\) 0 0
\(807\) 12.0000 0.422420
\(808\) 0 0
\(809\) −25.0000 −0.878953 −0.439477 − 0.898254i \(-0.644836\pi\)
−0.439477 + 0.898254i \(0.644836\pi\)
\(810\) 0 0
\(811\) 24.0000 0.842754 0.421377 − 0.906886i \(-0.361547\pi\)
0.421377 + 0.906886i \(0.361547\pi\)
\(812\) 0 0
\(813\) 33.0000 1.15736
\(814\) 0 0
\(815\) −54.0000 −1.89154
\(816\) 0 0
\(817\) −18.0000 −0.629740
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −45.0000 −1.57051 −0.785255 − 0.619172i \(-0.787468\pi\)
−0.785255 + 0.619172i \(0.787468\pi\)
\(822\) 0 0
\(823\) −30.0000 −1.04573 −0.522867 − 0.852414i \(-0.675138\pi\)
−0.522867 + 0.852414i \(0.675138\pi\)
\(824\) 0 0
\(825\) 48.0000 1.67115
\(826\) 0 0
\(827\) 14.0000 0.486828 0.243414 − 0.969923i \(-0.421733\pi\)
0.243414 + 0.969923i \(0.421733\pi\)
\(828\) 0 0
\(829\) 52.0000 1.80603 0.903017 − 0.429604i \(-0.141347\pi\)
0.903017 + 0.429604i \(0.141347\pi\)
\(830\) 0 0
\(831\) 36.0000 1.24883
\(832\) 0 0
\(833\) −30.0000 −1.03944
\(834\) 0 0
\(835\) 12.0000 0.415277
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 56.0000 1.93333 0.966667 − 0.256036i \(-0.0824164\pi\)
0.966667 + 0.256036i \(0.0824164\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 0 0
\(843\) −54.0000 −1.85986
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 5.00000 0.171802
\(848\) 0 0
\(849\) 36.0000 1.23552
\(850\) 0 0
\(851\) 18.0000 0.617032
\(852\) 0 0
\(853\) 33.0000 1.12990 0.564949 − 0.825126i \(-0.308896\pi\)
0.564949 + 0.825126i \(0.308896\pi\)
\(854\) 0 0
\(855\) −108.000 −3.69352
\(856\) 0 0
\(857\) −10.0000 −0.341593 −0.170797 − 0.985306i \(-0.554634\pi\)
−0.170797 + 0.985306i \(0.554634\pi\)
\(858\) 0 0
\(859\) 48.0000 1.63774 0.818869 − 0.573980i \(-0.194601\pi\)
0.818869 + 0.573980i \(0.194601\pi\)
\(860\) 0 0
\(861\) 36.0000 1.22688
\(862\) 0 0
\(863\) 29.0000 0.987171 0.493586 − 0.869697i \(-0.335686\pi\)
0.493586 + 0.869697i \(0.335686\pi\)
\(864\) 0 0
\(865\) 42.0000 1.42804
\(866\) 0 0
\(867\) −24.0000 −0.815083
\(868\) 0 0
\(869\) −24.0000 −0.814144
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −3.00000 −0.101419
\(876\) 0 0
\(877\) −27.0000 −0.911725 −0.455863 − 0.890050i \(-0.650669\pi\)
−0.455863 + 0.890050i \(0.650669\pi\)
\(878\) 0 0
\(879\) 27.0000 0.910687
\(880\) 0 0
\(881\) 41.0000 1.38133 0.690663 − 0.723177i \(-0.257319\pi\)
0.690663 + 0.723177i \(0.257319\pi\)
\(882\) 0 0
\(883\) 9.00000 0.302874 0.151437 − 0.988467i \(-0.451610\pi\)
0.151437 + 0.988467i \(0.451610\pi\)
\(884\) 0 0
\(885\) −18.0000 −0.605063
\(886\) 0 0
\(887\) −24.0000 −0.805841 −0.402921 − 0.915235i \(-0.632005\pi\)
−0.402921 + 0.915235i \(0.632005\pi\)
\(888\) 0 0
\(889\) 18.0000 0.603701
\(890\) 0 0
\(891\) −36.0000 −1.20605
\(892\) 0 0
\(893\) 42.0000 1.40548
\(894\) 0 0
\(895\) −63.0000 −2.10586
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −10.0000 −0.333148
\(902\) 0 0
\(903\) −9.00000 −0.299501
\(904\) 0 0
\(905\) 18.0000 0.598340
\(906\) 0 0
\(907\) −21.0000 −0.697294 −0.348647 − 0.937254i \(-0.613359\pi\)
−0.348647 + 0.937254i \(0.613359\pi\)
\(908\) 0 0
\(909\) 24.0000 0.796030
\(910\) 0 0
\(911\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 40.0000 1.32381
\(914\) 0 0
\(915\) 108.000 3.57037
\(916\) 0 0
\(917\) −9.00000 −0.297206
\(918\) 0 0
\(919\) 36.0000 1.18753 0.593765 − 0.804638i \(-0.297641\pi\)
0.593765 + 0.804638i \(0.297641\pi\)
\(920\) 0 0
\(921\) −6.00000 −0.197707
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 12.0000 0.394558
\(926\) 0 0
\(927\) 36.0000 1.18240
\(928\) 0 0
\(929\) 36.0000 1.18112 0.590561 − 0.806993i \(-0.298907\pi\)
0.590561 + 0.806993i \(0.298907\pi\)
\(930\) 0 0
\(931\) 36.0000 1.17985
\(932\) 0 0
\(933\) −18.0000 −0.589294
\(934\) 0 0
\(935\) −60.0000 −1.96221
\(936\) 0 0
\(937\) 54.0000 1.76410 0.882052 − 0.471153i \(-0.156162\pi\)
0.882052 + 0.471153i \(0.156162\pi\)
\(938\) 0 0
\(939\) 63.0000 2.05593
\(940\) 0 0
\(941\) −3.00000 −0.0977972 −0.0488986 − 0.998804i \(-0.515571\pi\)
−0.0488986 + 0.998804i \(0.515571\pi\)
\(942\) 0 0
\(943\) −72.0000 −2.34464
\(944\) 0 0
\(945\) −27.0000 −0.878310
\(946\) 0 0
\(947\) −20.0000 −0.649913 −0.324956 − 0.945729i \(-0.605350\pi\)
−0.324956 + 0.945729i \(0.605350\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −18.0000 −0.583690
\(952\) 0 0
\(953\) −41.0000 −1.32812 −0.664060 − 0.747679i \(-0.731168\pi\)
−0.664060 + 0.747679i \(0.731168\pi\)
\(954\) 0 0
\(955\) −72.0000 −2.32987
\(956\) 0 0
\(957\) −48.0000 −1.55162
\(958\) 0 0
\(959\) 6.00000 0.193750
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 72.0000 2.32017
\(964\) 0 0
\(965\) −54.0000 −1.73832
\(966\) 0 0
\(967\) −33.0000 −1.06121 −0.530604 − 0.847620i \(-0.678035\pi\)
−0.530604 + 0.847620i \(0.678035\pi\)
\(968\) 0 0
\(969\) 90.0000 2.89122
\(970\) 0 0
\(971\) −15.0000 −0.481373 −0.240686 − 0.970603i \(-0.577373\pi\)
−0.240686 + 0.970603i \(0.577373\pi\)
\(972\) 0 0
\(973\) −9.00000 −0.288527
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −12.0000 −0.383914 −0.191957 − 0.981403i \(-0.561483\pi\)
−0.191957 + 0.981403i \(0.561483\pi\)
\(978\) 0 0
\(979\) 24.0000 0.767043
\(980\) 0 0
\(981\) 90.0000 2.87348
\(982\) 0 0
\(983\) −35.0000 −1.11633 −0.558163 − 0.829731i \(-0.688494\pi\)
−0.558163 + 0.829731i \(0.688494\pi\)
\(984\) 0 0
\(985\) −9.00000 −0.286764
\(986\) 0 0
\(987\) 21.0000 0.668437
\(988\) 0 0
\(989\) 18.0000 0.572367
\(990\) 0 0
\(991\) 54.0000 1.71537 0.857683 − 0.514178i \(-0.171903\pi\)
0.857683 + 0.514178i \(0.171903\pi\)
\(992\) 0 0
\(993\) 78.0000 2.47526
\(994\) 0 0
\(995\) 36.0000 1.14128
\(996\) 0 0
\(997\) 28.0000 0.886769 0.443384 − 0.896332i \(-0.353778\pi\)
0.443384 + 0.896332i \(0.353778\pi\)
\(998\) 0 0
\(999\) −27.0000 −0.854242
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 5408.2.a.b.1.1 1
4.3 odd 2 5408.2.a.m.1.1 1
13.5 odd 4 416.2.f.a.129.1 ✓ 2
13.8 odd 4 416.2.f.a.129.2 yes 2
13.12 even 2 5408.2.a.a.1.1 1
39.5 even 4 3744.2.c.c.3457.2 2
39.8 even 4 3744.2.c.c.3457.1 2
52.31 even 4 416.2.f.c.129.1 yes 2
52.47 even 4 416.2.f.c.129.2 yes 2
52.51 odd 2 5408.2.a.l.1.1 1
104.5 odd 4 832.2.f.e.129.2 2
104.21 odd 4 832.2.f.e.129.1 2
104.83 even 4 832.2.f.a.129.2 2
104.99 even 4 832.2.f.a.129.1 2
156.47 odd 4 3744.2.c.b.3457.1 2
156.83 odd 4 3744.2.c.b.3457.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
416.2.f.a.129.1 ✓ 2 13.5 odd 4
416.2.f.a.129.2 yes 2 13.8 odd 4
416.2.f.c.129.1 yes 2 52.31 even 4
416.2.f.c.129.2 yes 2 52.47 even 4
832.2.f.a.129.1 2 104.99 even 4
832.2.f.a.129.2 2 104.83 even 4
832.2.f.e.129.1 2 104.21 odd 4
832.2.f.e.129.2 2 104.5 odd 4
3744.2.c.b.3457.1 2 156.47 odd 4
3744.2.c.b.3457.2 2 156.83 odd 4
3744.2.c.c.3457.1 2 39.8 even 4
3744.2.c.c.3457.2 2 39.5 even 4
5408.2.a.a.1.1 1 13.12 even 2
5408.2.a.b.1.1 1 1.1 even 1 trivial
5408.2.a.l.1.1 1 52.51 odd 2
5408.2.a.m.1.1 1 4.3 odd 2