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Properties

Label 7644.2.a.a.1.1
Level $7644$
Weight $2$
Character 7644.1
Self dual yes
Analytic conductor $61.038$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7644,2,Mod(1,7644)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7644, 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("7644.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7644 = 2^{2} \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7644.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: \(61.0376473051\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 156)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 7644.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.00000 q^{9} -1.00000 q^{13} +6.00000 q^{17} -2.00000 q^{19} -5.00000 q^{25} -1.00000 q^{27} -6.00000 q^{29} -2.00000 q^{31} +2.00000 q^{37} +1.00000 q^{39} +12.0000 q^{41} -4.00000 q^{43} -6.00000 q^{51} +6.00000 q^{53} +2.00000 q^{57} -12.0000 q^{59} -2.00000 q^{61} -10.0000 q^{67} +12.0000 q^{71} -14.0000 q^{73} +5.00000 q^{75} +8.00000 q^{79} +1.00000 q^{81} -12.0000 q^{83} +6.00000 q^{87} +2.00000 q^{93} +10.0000 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 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
\(15\) 0 0
\(16\) 0 0
\(17\) 6.00000 1.45521 0.727607 − 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 − 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 − 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 12.0000 1.87409 0.937043 − 0.349215i \(-0.113552\pi\)
0.937043 + 0.349215i \(0.113552\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −6.00000 −0.840168
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.00000 0.264906
\(58\) 0 0
\(59\) −12.0000 −1.56227 −0.781133 − 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 − 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −10.0000 −1.22169 −0.610847 − 0.791748i \(-0.709171\pi\)
−0.610847 + 0.791748i \(0.709171\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 − 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) −14.0000 −1.63858 −0.819288 − 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) 0 0
\(75\) 5.00000 0.577350
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 − 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −12.0000 −1.31717 −0.658586 − 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 6.00000 0.643268
\(88\) 0 0
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 2.00000 0.207390
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 10.0000 1.01535 0.507673 − 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −18.0000 −1.79107 −0.895533 − 0.444994i \(-0.853206\pi\)
−0.895533 + 0.444994i \(0.853206\pi\)
\(102\) 0 0
\(103\) 16.0000 1.57653 0.788263 − 0.615338i \(-0.210980\pi\)
0.788263 + 0.615338i \(0.210980\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −12.0000 −1.16008 −0.580042 − 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) 14.0000 1.34096 0.670478 − 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 − 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −1.00000 −0.0924500
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) −12.0000 −1.08200
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −4.00000 −0.354943 −0.177471 − 0.984126i \(-0.556792\pi\)
−0.177471 + 0.984126i \(0.556792\pi\)
\(128\) 0 0
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 − 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −12.0000 −1.02523 −0.512615 − 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 0 0
\(139\) 16.0000 1.35710 0.678551 − 0.734553i \(-0.262608\pi\)
0.678551 + 0.734553i \(0.262608\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 12.0000 0.983078 0.491539 − 0.870855i \(-0.336434\pi\)
0.491539 + 0.870855i \(0.336434\pi\)
\(150\) 0 0
\(151\) −10.0000 −0.813788 −0.406894 − 0.913475i \(-0.633388\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(152\) 0 0
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −2.00000 −0.159617 −0.0798087 − 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 0 0
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 14.0000 1.09656 0.548282 − 0.836293i \(-0.315282\pi\)
0.548282 + 0.836293i \(0.315282\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 24.0000 1.85718 0.928588 − 0.371113i \(-0.121024\pi\)
0.928588 + 0.371113i \(0.121024\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −2.00000 −0.152944
\(172\) 0 0
\(173\) 6.00000 0.456172 0.228086 − 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 12.0000 0.901975
\(178\) 0 0
\(179\) 12.0000 0.896922 0.448461 − 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 − 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 2.00000 0.143963 0.0719816 − 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −24.0000 −1.70993 −0.854965 − 0.518686i \(-0.826421\pi\)
−0.854965 + 0.518686i \(0.826421\pi\)
\(198\) 0 0
\(199\) −20.0000 −1.41776 −0.708881 − 0.705328i \(-0.750800\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) 0 0
\(201\) 10.0000 0.705346
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 − 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 0 0
\(213\) −12.0000 −0.822226
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 14.0000 0.946032
\(220\) 0 0
\(221\) −6.00000 −0.403604
\(222\) 0 0
\(223\) −26.0000 −1.74109 −0.870544 − 0.492090i \(-0.836233\pi\)
−0.870544 + 0.492090i \(0.836233\pi\)
\(224\) 0 0
\(225\) −5.00000 −0.333333
\(226\) 0 0
\(227\) −24.0000 −1.59294 −0.796468 − 0.604681i \(-0.793301\pi\)
−0.796468 + 0.604681i \(0.793301\pi\)
\(228\) 0 0
\(229\) −14.0000 −0.925146 −0.462573 − 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 − 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −8.00000 −0.519656
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 − 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 − 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 2.00000 0.127257
\(248\) 0 0
\(249\) 12.0000 0.760469
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 − 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −6.00000 −0.374270 −0.187135 − 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) −24.0000 −1.47990 −0.739952 − 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −18.0000 −1.09748 −0.548740 − 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 0 0
\(271\) −26.0000 −1.57939 −0.789694 − 0.613501i \(-0.789761\pi\)
−0.789694 + 0.613501i \(0.789761\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 − 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 0 0
\(279\) −2.00000 −0.119737
\(280\) 0 0
\(281\) −12.0000 −0.715860 −0.357930 − 0.933748i \(-0.616517\pi\)
−0.357930 + 0.933748i \(0.616517\pi\)
\(282\) 0 0
\(283\) 16.0000 0.951101 0.475551 − 0.879688i \(-0.342249\pi\)
0.475551 + 0.879688i \(0.342249\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) −10.0000 −0.586210
\(292\) 0 0
\(293\) 12.0000 0.701047 0.350524 − 0.936554i \(-0.386004\pi\)
0.350524 + 0.936554i \(0.386004\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 18.0000 1.03407
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −2.00000 −0.114146 −0.0570730 − 0.998370i \(-0.518177\pi\)
−0.0570730 + 0.998370i \(0.518177\pi\)
\(308\) 0 0
\(309\) −16.0000 −0.910208
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 − 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 0 0
\(313\) 22.0000 1.24351 0.621757 − 0.783210i \(-0.286419\pi\)
0.621757 + 0.783210i \(0.286419\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −24.0000 −1.34797 −0.673987 − 0.738743i \(-0.735420\pi\)
−0.673987 + 0.738743i \(0.735420\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) −12.0000 −0.667698
\(324\) 0 0
\(325\) 5.00000 0.277350
\(326\) 0 0
\(327\) −14.0000 −0.774202
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −10.0000 −0.549650 −0.274825 − 0.961494i \(-0.588620\pi\)
−0.274825 + 0.961494i \(0.588620\pi\)
\(332\) 0 0
\(333\) 2.00000 0.109599
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 − 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 0 0
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −12.0000 −0.644194 −0.322097 − 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 0 0
\(349\) 22.0000 1.17763 0.588817 − 0.808267i \(-0.299594\pi\)
0.588817 + 0.808267i \(0.299594\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) −12.0000 −0.638696 −0.319348 − 0.947638i \(-0.603464\pi\)
−0.319348 + 0.947638i \(0.603464\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −24.0000 −1.26667 −0.633336 − 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 11.0000 0.577350
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −20.0000 −1.04399 −0.521996 − 0.852948i \(-0.674812\pi\)
−0.521996 + 0.852948i \(0.674812\pi\)
\(368\) 0 0
\(369\) 12.0000 0.624695
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 14.0000 0.724893 0.362446 − 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 6.00000 0.309016
\(378\) 0 0
\(379\) 26.0000 1.33553 0.667765 − 0.744372i \(-0.267251\pi\)
0.667765 + 0.744372i \(0.267251\pi\)
\(380\) 0 0
\(381\) 4.00000 0.204926
\(382\) 0 0
\(383\) −24.0000 −1.22634 −0.613171 − 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −4.00000 −0.203331
\(388\) 0 0
\(389\) 30.0000 1.52106 0.760530 − 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 12.0000 0.605320
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 22.0000 1.10415 0.552074 − 0.833795i \(-0.313837\pi\)
0.552074 + 0.833795i \(0.313837\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 24.0000 1.19850 0.599251 − 0.800561i \(-0.295465\pi\)
0.599251 + 0.800561i \(0.295465\pi\)
\(402\) 0 0
\(403\) 2.00000 0.0996271
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 10.0000 0.494468 0.247234 − 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 0 0
\(411\) 12.0000 0.591916
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −16.0000 −0.783523
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 − 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) −34.0000 −1.65706 −0.828529 − 0.559946i \(-0.810822\pi\)
−0.828529 + 0.559946i \(0.810822\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −30.0000 −1.45521
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 24.0000 1.15604 0.578020 − 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) 0 0
\(433\) −14.0000 −0.672797 −0.336399 − 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −8.00000 −0.381819 −0.190910 − 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −12.0000 −0.570137 −0.285069 − 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −12.0000 −0.567581
\(448\) 0 0
\(449\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 10.0000 0.469841
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 38.0000 1.77757 0.888783 − 0.458329i \(-0.151552\pi\)
0.888783 + 0.458329i \(0.151552\pi\)
\(458\) 0 0
\(459\) −6.00000 −0.280056
\(460\) 0 0
\(461\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) −22.0000 −1.02243 −0.511213 − 0.859454i \(-0.670804\pi\)
−0.511213 + 0.859454i \(0.670804\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −36.0000 −1.66588 −0.832941 − 0.553362i \(-0.813345\pi\)
−0.832941 + 0.553362i \(0.813345\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 2.00000 0.0921551
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 10.0000 0.458831
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) 0 0
\(479\) 12.0000 0.548294 0.274147 − 0.961688i \(-0.411605\pi\)
0.274147 + 0.961688i \(0.411605\pi\)
\(480\) 0 0
\(481\) −2.00000 −0.0911922
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 2.00000 0.0906287 0.0453143 − 0.998973i \(-0.485571\pi\)
0.0453143 + 0.998973i \(0.485571\pi\)
\(488\) 0 0
\(489\) −14.0000 −0.633102
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 − 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) −36.0000 −1.62136
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 14.0000 0.626726 0.313363 − 0.949633i \(-0.398544\pi\)
0.313363 + 0.949633i \(0.398544\pi\)
\(500\) 0 0
\(501\) −24.0000 −1.07224
\(502\) 0 0
\(503\) 24.0000 1.07011 0.535054 − 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −1.00000 −0.0444116
\(508\) 0 0
\(509\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 2.00000 0.0883022
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) −30.0000 −1.31432 −0.657162 − 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) 0 0
\(523\) 16.0000 0.699631 0.349816 − 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −12.0000 −0.522728
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) −12.0000 −0.520756
\(532\) 0 0
\(533\) −12.0000 −0.519778
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −12.0000 −0.517838
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −10.0000 −0.429934 −0.214967 − 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) 0 0
\(543\) 2.00000 0.0858282
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −16.0000 −0.684111 −0.342055 − 0.939680i \(-0.611123\pi\)
−0.342055 + 0.939680i \(0.611123\pi\)
\(548\) 0 0
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) 12.0000 0.511217
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −12.0000 −0.508456 −0.254228 − 0.967144i \(-0.581821\pi\)
−0.254228 + 0.967144i \(0.581821\pi\)
\(558\) 0 0
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −12.0000 −0.505740 −0.252870 − 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −18.0000 −0.754599 −0.377300 − 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) 0 0
\(571\) −40.0000 −1.67395 −0.836974 − 0.547243i \(-0.815677\pi\)
−0.836974 + 0.547243i \(0.815677\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −2.00000 −0.0832611 −0.0416305 − 0.999133i \(-0.513255\pi\)
−0.0416305 + 0.999133i \(0.513255\pi\)
\(578\) 0 0
\(579\) −2.00000 −0.0831172
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 12.0000 0.495293 0.247647 − 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 0 0
\(589\) 4.00000 0.164817
\(590\) 0 0
\(591\) 24.0000 0.987228
\(592\) 0 0
\(593\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 20.0000 0.818546
\(598\) 0 0
\(599\) −48.0000 −1.96123 −0.980613 − 0.195952i \(-0.937220\pi\)
−0.980613 + 0.195952i \(0.937220\pi\)
\(600\) 0 0
\(601\) −26.0000 −1.06056 −0.530281 − 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) 0 0
\(603\) −10.0000 −0.407231
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 4.00000 0.162355 0.0811775 − 0.996700i \(-0.474132\pi\)
0.0811775 + 0.996700i \(0.474132\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −22.0000 −0.888572 −0.444286 − 0.895885i \(-0.646543\pi\)
−0.444286 + 0.895885i \(0.646543\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −36.0000 −1.44931 −0.724653 − 0.689114i \(-0.758000\pi\)
−0.724653 + 0.689114i \(0.758000\pi\)
\(618\) 0 0
\(619\) 22.0000 0.884255 0.442127 − 0.896952i \(-0.354224\pi\)
0.442127 + 0.896952i \(0.354224\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 12.0000 0.478471
\(630\) 0 0
\(631\) −10.0000 −0.398094 −0.199047 − 0.979990i \(-0.563785\pi\)
−0.199047 + 0.979990i \(0.563785\pi\)
\(632\) 0 0
\(633\) 4.00000 0.158986
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 12.0000 0.474713
\(640\) 0 0
\(641\) 18.0000 0.710957 0.355479 − 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 0 0
\(643\) −14.0000 −0.552106 −0.276053 − 0.961142i \(-0.589027\pi\)
−0.276053 + 0.961142i \(0.589027\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 24.0000 0.943537 0.471769 − 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −30.0000 −1.17399 −0.586995 − 0.809590i \(-0.699689\pi\)
−0.586995 + 0.809590i \(0.699689\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −14.0000 −0.546192
\(658\) 0 0
\(659\) −36.0000 −1.40236 −0.701180 − 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 0 0
\(661\) −2.00000 −0.0777910 −0.0388955 − 0.999243i \(-0.512384\pi\)
−0.0388955 + 0.999243i \(0.512384\pi\)
\(662\) 0 0
\(663\) 6.00000 0.233021
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 26.0000 1.00522
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −10.0000 −0.385472 −0.192736 − 0.981251i \(-0.561736\pi\)
−0.192736 + 0.981251i \(0.561736\pi\)
\(674\) 0 0
\(675\) 5.00000 0.192450
\(676\) 0 0
\(677\) −6.00000 −0.230599 −0.115299 − 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 24.0000 0.919682
\(682\) 0 0
\(683\) −24.0000 −0.918334 −0.459167 − 0.888350i \(-0.651852\pi\)
−0.459167 + 0.888350i \(0.651852\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 14.0000 0.534133
\(688\) 0 0
\(689\) −6.00000 −0.228582
\(690\) 0 0
\(691\) −14.0000 −0.532585 −0.266293 − 0.963892i \(-0.585799\pi\)
−0.266293 + 0.963892i \(0.585799\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 72.0000 2.72719
\(698\) 0 0
\(699\) −6.00000 −0.226941
\(700\) 0 0
\(701\) −42.0000 −1.58632 −0.793159 − 0.609015i \(-0.791565\pi\)
−0.793159 + 0.609015i \(0.791565\pi\)
\(702\) 0 0
\(703\) −4.00000 −0.150863
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −10.0000 −0.375558 −0.187779 − 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 12.0000 0.448148
\(718\) 0 0
\(719\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −10.0000 −0.371904
\(724\) 0 0
\(725\) 30.0000 1.11417
\(726\) 0 0
\(727\) −44.0000 −1.63187 −0.815935 − 0.578144i \(-0.803777\pi\)
−0.815935 + 0.578144i \(0.803777\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −24.0000 −0.887672
\(732\) 0 0
\(733\) −38.0000 −1.40356 −0.701781 − 0.712393i \(-0.747612\pi\)
−0.701781 + 0.712393i \(0.747612\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 2.00000 0.0735712 0.0367856 − 0.999323i \(-0.488288\pi\)
0.0367856 + 0.999323i \(0.488288\pi\)
\(740\) 0 0
\(741\) −2.00000 −0.0734718
\(742\) 0 0
\(743\) 12.0000 0.440237 0.220119 − 0.975473i \(-0.429356\pi\)
0.220119 + 0.975473i \(0.429356\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −12.0000 −0.439057
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 32.0000 1.16770 0.583848 − 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) 0 0
\(753\) 12.0000 0.437304
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 2.00000 0.0726912 0.0363456 − 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 12.0000 0.435000 0.217500 − 0.976060i \(-0.430210\pi\)
0.217500 + 0.976060i \(0.430210\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 12.0000 0.433295
\(768\) 0 0
\(769\) 22.0000 0.793340 0.396670 − 0.917961i \(-0.370166\pi\)
0.396670 + 0.917961i \(0.370166\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) 0 0
\(773\) −12.0000 −0.431610 −0.215805 − 0.976436i \(-0.569238\pi\)
−0.215805 + 0.976436i \(0.569238\pi\)
\(774\) 0 0
\(775\) 10.0000 0.359211
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −24.0000 −0.859889
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 6.00000 0.214423
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −26.0000 −0.926800 −0.463400 − 0.886149i \(-0.653371\pi\)
−0.463400 + 0.886149i \(0.653371\pi\)
\(788\) 0 0
\(789\) 24.0000 0.854423
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 2.00000 0.0710221
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 18.0000 0.637593 0.318796 − 0.947823i \(-0.396721\pi\)
0.318796 + 0.947823i \(0.396721\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 18.0000 0.633630
\(808\) 0 0
\(809\) 6.00000 0.210949 0.105474 − 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) 22.0000 0.772524 0.386262 − 0.922389i \(-0.373766\pi\)
0.386262 + 0.922389i \(0.373766\pi\)
\(812\) 0 0
\(813\) 26.0000 0.911860
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 8.00000 0.279885
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −12.0000 −0.418803 −0.209401 − 0.977830i \(-0.567152\pi\)
−0.209401 + 0.977830i \(0.567152\pi\)
\(822\) 0 0
\(823\) −40.0000 −1.39431 −0.697156 − 0.716919i \(-0.745552\pi\)
−0.697156 + 0.716919i \(0.745552\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 48.0000 1.66912 0.834562 − 0.550914i \(-0.185721\pi\)
0.834562 + 0.550914i \(0.185721\pi\)
\(828\) 0 0
\(829\) −2.00000 −0.0694629 −0.0347314 − 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) 0 0
\(831\) 10.0000 0.346896
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 2.00000 0.0691301
\(838\) 0 0
\(839\) −36.0000 −1.24286 −0.621429 − 0.783470i \(-0.713448\pi\)
−0.621429 + 0.783470i \(0.713448\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) 12.0000 0.413302
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −16.0000 −0.549119
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 22.0000 0.753266 0.376633 − 0.926363i \(-0.377082\pi\)
0.376633 + 0.926363i \(0.377082\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 42.0000 1.43469 0.717346 − 0.696717i \(-0.245357\pi\)
0.717346 + 0.696717i \(0.245357\pi\)
\(858\) 0 0
\(859\) −44.0000 −1.50126 −0.750630 − 0.660722i \(-0.770250\pi\)
−0.750630 + 0.660722i \(0.770250\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −19.0000 −0.645274
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 10.0000 0.338837
\(872\) 0 0
\(873\) 10.0000 0.338449
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 50.0000 1.68838 0.844190 − 0.536044i \(-0.180082\pi\)
0.844190 + 0.536044i \(0.180082\pi\)
\(878\) 0 0
\(879\) −12.0000 −0.404750
\(880\) 0 0
\(881\) 42.0000 1.41502 0.707508 − 0.706705i \(-0.249819\pi\)
0.707508 + 0.706705i \(0.249819\pi\)
\(882\) 0 0
\(883\) −16.0000 −0.538443 −0.269221 − 0.963078i \(-0.586766\pi\)
−0.269221 + 0.963078i \(0.586766\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −24.0000 −0.805841 −0.402921 − 0.915235i \(-0.632005\pi\)
−0.402921 + 0.915235i \(0.632005\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 12.0000 0.400222
\(900\) 0 0
\(901\) 36.0000 1.19933
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 8.00000 0.265636 0.132818 − 0.991140i \(-0.457597\pi\)
0.132818 + 0.991140i \(0.457597\pi\)
\(908\) 0 0
\(909\) −18.0000 −0.597022
\(910\) 0 0
\(911\) 48.0000 1.59031 0.795155 − 0.606406i \(-0.207389\pi\)
0.795155 + 0.606406i \(0.207389\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −16.0000 −0.527791 −0.263896 − 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 0 0
\(921\) 2.00000 0.0659022
\(922\) 0 0
\(923\) −12.0000 −0.394985
\(924\) 0 0
\(925\) −10.0000 −0.328798
\(926\) 0 0
\(927\) 16.0000 0.525509
\(928\) 0 0
\(929\) 36.0000 1.18112 0.590561 − 0.806993i \(-0.298907\pi\)
0.590561 + 0.806993i \(0.298907\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 24.0000 0.785725
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 34.0000 1.11073 0.555366 − 0.831606i \(-0.312578\pi\)
0.555366 + 0.831606i \(0.312578\pi\)
\(938\) 0 0
\(939\) −22.0000 −0.717943
\(940\) 0 0
\(941\) 12.0000 0.391189 0.195594 − 0.980685i \(-0.437336\pi\)
0.195594 + 0.980685i \(0.437336\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −36.0000 −1.16984 −0.584921 − 0.811090i \(-0.698875\pi\)
−0.584921 + 0.811090i \(0.698875\pi\)
\(948\) 0 0
\(949\) 14.0000 0.454459
\(950\) 0 0
\(951\) 24.0000 0.778253
\(952\) 0 0
\(953\) −18.0000 −0.583077 −0.291539 − 0.956559i \(-0.594167\pi\)
−0.291539 + 0.956559i \(0.594167\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) −12.0000 −0.386695
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 50.0000 1.60789 0.803946 − 0.594703i \(-0.202730\pi\)
0.803946 + 0.594703i \(0.202730\pi\)
\(968\) 0 0
\(969\) 12.0000 0.385496
\(970\) 0 0
\(971\) −12.0000 −0.385098 −0.192549 − 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −5.00000 −0.160128
\(976\) 0 0
\(977\) 24.0000 0.767828 0.383914 − 0.923369i \(-0.374576\pi\)
0.383914 + 0.923369i \(0.374576\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 14.0000 0.446986
\(982\) 0 0
\(983\) −60.0000 −1.91370 −0.956851 − 0.290578i \(-0.906153\pi\)
−0.956851 + 0.290578i \(0.906153\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −52.0000 −1.65183 −0.825917 − 0.563791i \(-0.809342\pi\)
−0.825917 + 0.563791i \(0.809342\pi\)
\(992\) 0 0
\(993\) 10.0000 0.317340
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 10.0000 0.316703 0.158352 − 0.987383i \(-0.449382\pi\)
0.158352 + 0.987383i \(0.449382\pi\)
\(998\) 0 0
\(999\) −2.00000 −0.0632772
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 7644.2.a.a.1.1 1
7.6 odd 2 156.2.a.b.1.1 ✓ 1
21.20 even 2 468.2.a.c.1.1 1
28.27 even 2 624.2.a.b.1.1 1
35.13 even 4 3900.2.h.e.1249.2 2
35.27 even 4 3900.2.h.e.1249.1 2
35.34 odd 2 3900.2.a.a.1.1 1
56.13 odd 2 2496.2.a.h.1.1 1
56.27 even 2 2496.2.a.v.1.1 1
63.13 odd 6 4212.2.i.f.2809.1 2
63.20 even 6 4212.2.i.g.1405.1 2
63.34 odd 6 4212.2.i.f.1405.1 2
63.41 even 6 4212.2.i.g.2809.1 2
84.83 odd 2 1872.2.a.i.1.1 1
91.6 even 12 2028.2.q.d.361.1 4
91.20 even 12 2028.2.q.d.361.2 4
91.34 even 4 2028.2.b.d.337.1 2
91.41 even 12 2028.2.q.d.1837.2 4
91.48 odd 6 2028.2.i.b.2005.1 2
91.55 odd 6 2028.2.i.b.529.1 2
91.62 odd 6 2028.2.i.c.529.1 2
91.69 odd 6 2028.2.i.c.2005.1 2
91.76 even 12 2028.2.q.d.1837.1 4
91.83 even 4 2028.2.b.d.337.2 2
91.90 odd 2 2028.2.a.e.1.1 1
168.83 odd 2 7488.2.a.bb.1.1 1
168.125 even 2 7488.2.a.bf.1.1 1
273.83 odd 4 6084.2.b.a.4393.2 2
273.125 odd 4 6084.2.b.a.4393.1 2
273.272 even 2 6084.2.a.h.1.1 1
364.363 even 2 8112.2.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
156.2.a.b.1.1 ✓ 1 7.6 odd 2
468.2.a.c.1.1 1 21.20 even 2
624.2.a.b.1.1 1 28.27 even 2
1872.2.a.i.1.1 1 84.83 odd 2
2028.2.a.e.1.1 1 91.90 odd 2
2028.2.b.d.337.1 2 91.34 even 4
2028.2.b.d.337.2 2 91.83 even 4
2028.2.i.b.529.1 2 91.55 odd 6
2028.2.i.b.2005.1 2 91.48 odd 6
2028.2.i.c.529.1 2 91.62 odd 6
2028.2.i.c.2005.1 2 91.69 odd 6
2028.2.q.d.361.1 4 91.6 even 12
2028.2.q.d.361.2 4 91.20 even 12
2028.2.q.d.1837.1 4 91.76 even 12
2028.2.q.d.1837.2 4 91.41 even 12
2496.2.a.h.1.1 1 56.13 odd 2
2496.2.a.v.1.1 1 56.27 even 2
3900.2.a.a.1.1 1 35.34 odd 2
3900.2.h.e.1249.1 2 35.27 even 4
3900.2.h.e.1249.2 2 35.13 even 4
4212.2.i.f.1405.1 2 63.34 odd 6
4212.2.i.f.2809.1 2 63.13 odd 6
4212.2.i.g.1405.1 2 63.20 even 6
4212.2.i.g.2809.1 2 63.41 even 6
6084.2.a.h.1.1 1 273.272 even 2
6084.2.b.a.4393.1 2 273.125 odd 4
6084.2.b.a.4393.2 2 273.83 odd 4
7488.2.a.bb.1.1 1 168.83 odd 2
7488.2.a.bf.1.1 1 168.125 even 2
7644.2.a.a.1.1 1 1.1 even 1 trivial
8112.2.a.i.1.1 1 364.363 even 2