Properties

Label 5929.2.a.d.1.1
Level $5929$
Weight $2$
Character 5929.1
Self dual yes
Analytic conductor $47.343$
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] = [5929,2,Mod(1,5929)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5929, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("5929.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5929 = 7^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5929.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: \(47.3433033584\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 77)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 5929.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -2.00000 q^{4} -3.00000 q^{5} -2.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -2.00000 q^{4} -3.00000 q^{5} -2.00000 q^{9} +2.00000 q^{12} -4.00000 q^{13} +3.00000 q^{15} +4.00000 q^{16} -6.00000 q^{17} +2.00000 q^{19} +6.00000 q^{20} +3.00000 q^{23} +4.00000 q^{25} +5.00000 q^{27} +6.00000 q^{29} -5.00000 q^{31} +4.00000 q^{36} +11.0000 q^{37} +4.00000 q^{39} +6.00000 q^{41} -8.00000 q^{43} +6.00000 q^{45} -4.00000 q^{48} +6.00000 q^{51} +8.00000 q^{52} -6.00000 q^{53} -2.00000 q^{57} +9.00000 q^{59} -6.00000 q^{60} -10.0000 q^{61} -8.00000 q^{64} +12.0000 q^{65} +5.00000 q^{67} +12.0000 q^{68} -3.00000 q^{69} +9.00000 q^{71} +2.00000 q^{73} -4.00000 q^{75} -4.00000 q^{76} +10.0000 q^{79} -12.0000 q^{80} +1.00000 q^{81} +12.0000 q^{83} +18.0000 q^{85} -6.00000 q^{87} +3.00000 q^{89} -6.00000 q^{92} +5.00000 q^{93} -6.00000 q^{95} +1.00000 q^{97} +O(q^{100})\)

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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) −1.00000 −0.577350 −0.288675 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) −2.00000 −1.00000
\(5\) −3.00000 −1.34164 −0.670820 0.741620i \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) 0 0
\(12\) 2.00000 0.577350
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 3.00000 0.774597
\(16\) 4.00000 1.00000
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 6.00000 1.34164
\(21\) 0 0
\(22\) 0 0
\(23\) 3.00000 0.625543 0.312772 0.949828i \(-0.398743\pi\)
0.312772 + 0.949828i \(0.398743\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) 5.00000 0.962250
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −5.00000 −0.898027 −0.449013 0.893525i \(-0.648224\pi\)
−0.449013 + 0.893525i \(0.648224\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 4.00000 0.666667
\(37\) 11.0000 1.80839 0.904194 0.427121i \(-0.140472\pi\)
0.904194 + 0.427121i \(0.140472\pi\)
\(38\) 0 0
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) 6.00000 0.894427
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −4.00000 −0.577350
\(49\) 0 0
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) 8.00000 1.10940
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −2.00000 −0.264906
\(58\) 0 0
\(59\) 9.00000 1.17170 0.585850 0.810419i \(-0.300761\pi\)
0.585850 + 0.810419i \(0.300761\pi\)
\(60\) −6.00000 −0.774597
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 12.0000 1.48842
\(66\) 0 0
\(67\) 5.00000 0.610847 0.305424 0.952217i \(-0.401202\pi\)
0.305424 + 0.952217i \(0.401202\pi\)
\(68\) 12.0000 1.45521
\(69\) −3.00000 −0.361158
\(70\) 0 0
\(71\) 9.00000 1.06810 0.534052 0.845452i \(-0.320669\pi\)
0.534052 + 0.845452i \(0.320669\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 0 0
\(75\) −4.00000 −0.461880
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) 0 0
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) −12.0000 −1.34164
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) 18.0000 1.95237
\(86\) 0 0
\(87\) −6.00000 −0.643268
\(88\) 0 0
\(89\) 3.00000 0.317999 0.159000 0.987279i \(-0.449173\pi\)
0.159000 + 0.987279i \(0.449173\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −6.00000 −0.625543
\(93\) 5.00000 0.518476
\(94\) 0 0
\(95\) −6.00000 −0.615587
\(96\) 0 0
\(97\) 1.00000 0.101535 0.0507673 0.998711i \(-0.483833\pi\)
0.0507673 + 0.998711i \(0.483833\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −8.00000 −0.800000
\(101\) −12.0000 −1.19404 −0.597022 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) 0 0
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) −10.0000 −0.962250
\(109\) −20.0000 −1.91565 −0.957826 0.287348i \(-0.907226\pi\)
−0.957826 + 0.287348i \(0.907226\pi\)
\(110\) 0 0
\(111\) −11.0000 −1.04407
\(112\) 0 0
\(113\) −3.00000 −0.282216 −0.141108 0.989994i \(-0.545067\pi\)
−0.141108 + 0.989994i \(0.545067\pi\)
\(114\) 0 0
\(115\) −9.00000 −0.839254
\(116\) −12.0000 −1.11417
\(117\) 8.00000 0.739600
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) −6.00000 −0.541002
\(124\) 10.0000 0.898027
\(125\) 3.00000 0.268328
\(126\) 0 0
\(127\) −2.00000 −0.177471 −0.0887357 0.996055i \(-0.528283\pi\)
−0.0887357 + 0.996055i \(0.528283\pi\)
\(128\) 0 0
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −15.0000 −1.29099
\(136\) 0 0
\(137\) −3.00000 −0.256307 −0.128154 0.991754i \(-0.540905\pi\)
−0.128154 + 0.991754i \(0.540905\pi\)
\(138\) 0 0
\(139\) 14.0000 1.18746 0.593732 0.804663i \(-0.297654\pi\)
0.593732 + 0.804663i \(0.297654\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −8.00000 −0.666667
\(145\) −18.0000 −1.49482
\(146\) 0 0
\(147\) 0 0
\(148\) −22.0000 −1.80839
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) 10.0000 0.813788 0.406894 0.913475i \(-0.366612\pi\)
0.406894 + 0.913475i \(0.366612\pi\)
\(152\) 0 0
\(153\) 12.0000 0.970143
\(154\) 0 0
\(155\) 15.0000 1.20483
\(156\) −8.00000 −0.640513
\(157\) 13.0000 1.03751 0.518756 0.854922i \(-0.326395\pi\)
0.518756 + 0.854922i \(0.326395\pi\)
\(158\) 0 0
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 20.0000 1.56652 0.783260 0.621694i \(-0.213555\pi\)
0.783260 + 0.621694i \(0.213555\pi\)
\(164\) −12.0000 −0.937043
\(165\) 0 0
\(166\) 0 0
\(167\) 6.00000 0.464294 0.232147 0.972681i \(-0.425425\pi\)
0.232147 + 0.972681i \(0.425425\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) 16.0000 1.21999
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −9.00000 −0.676481
\(178\) 0 0
\(179\) −15.0000 −1.12115 −0.560576 0.828103i \(-0.689420\pi\)
−0.560576 + 0.828103i \(0.689420\pi\)
\(180\) −12.0000 −0.894427
\(181\) 7.00000 0.520306 0.260153 0.965567i \(-0.416227\pi\)
0.260153 + 0.965567i \(0.416227\pi\)
\(182\) 0 0
\(183\) 10.0000 0.739221
\(184\) 0 0
\(185\) −33.0000 −2.42621
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −27.0000 −1.95365 −0.976826 0.214036i \(-0.931339\pi\)
−0.976826 + 0.214036i \(0.931339\pi\)
\(192\) 8.00000 0.577350
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 0 0
\(195\) −12.0000 −0.859338
\(196\) 0 0
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) −5.00000 −0.352673
\(202\) 0 0
\(203\) 0 0
\(204\) −12.0000 −0.840168
\(205\) −18.0000 −1.25717
\(206\) 0 0
\(207\) −6.00000 −0.417029
\(208\) −16.0000 −1.10940
\(209\) 0 0
\(210\) 0 0
\(211\) −14.0000 −0.963800 −0.481900 0.876226i \(-0.660053\pi\)
−0.481900 + 0.876226i \(0.660053\pi\)
\(212\) 12.0000 0.824163
\(213\) −9.00000 −0.616670
\(214\) 0 0
\(215\) 24.0000 1.63679
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −2.00000 −0.135147
\(220\) 0 0
\(221\) 24.0000 1.61441
\(222\) 0 0
\(223\) 19.0000 1.27233 0.636167 0.771551i \(-0.280519\pi\)
0.636167 + 0.771551i \(0.280519\pi\)
\(224\) 0 0
\(225\) −8.00000 −0.533333
\(226\) 0 0
\(227\) −12.0000 −0.796468 −0.398234 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 4.00000 0.264906
\(229\) −5.00000 −0.330409 −0.165205 0.986259i \(-0.552828\pi\)
−0.165205 + 0.986259i \(0.552828\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −18.0000 −1.17170
\(237\) −10.0000 −0.649570
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 12.0000 0.774597
\(241\) −28.0000 −1.80364 −0.901819 0.432113i \(-0.857768\pi\)
−0.901819 + 0.432113i \(0.857768\pi\)
\(242\) 0 0
\(243\) −16.0000 −1.02640
\(244\) 20.0000 1.28037
\(245\) 0 0
\(246\) 0 0
\(247\) −8.00000 −0.509028
\(248\) 0 0
\(249\) −12.0000 −0.760469
\(250\) 0 0
\(251\) 9.00000 0.568075 0.284037 0.958813i \(-0.408326\pi\)
0.284037 + 0.958813i \(0.408326\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −18.0000 −1.12720
\(256\) 16.0000 1.00000
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −24.0000 −1.48842
\(261\) −12.0000 −0.742781
\(262\) 0 0
\(263\) 30.0000 1.84988 0.924940 0.380114i \(-0.124115\pi\)
0.924940 + 0.380114i \(0.124115\pi\)
\(264\) 0 0
\(265\) 18.0000 1.10573
\(266\) 0 0
\(267\) −3.00000 −0.183597
\(268\) −10.0000 −0.610847
\(269\) −30.0000 −1.82913 −0.914566 0.404436i \(-0.867468\pi\)
−0.914566 + 0.404436i \(0.867468\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) −24.0000 −1.45521
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 6.00000 0.361158
\(277\) −8.00000 −0.480673 −0.240337 0.970690i \(-0.577258\pi\)
−0.240337 + 0.970690i \(0.577258\pi\)
\(278\) 0 0
\(279\) 10.0000 0.598684
\(280\) 0 0
\(281\) −12.0000 −0.715860 −0.357930 0.933748i \(-0.616517\pi\)
−0.357930 + 0.933748i \(0.616517\pi\)
\(282\) 0 0
\(283\) 32.0000 1.90220 0.951101 0.308879i \(-0.0999539\pi\)
0.951101 + 0.308879i \(0.0999539\pi\)
\(284\) −18.0000 −1.06810
\(285\) 6.00000 0.355409
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) −1.00000 −0.0586210
\(292\) −4.00000 −0.234082
\(293\) −30.0000 −1.75262 −0.876309 0.481749i \(-0.840002\pi\)
−0.876309 + 0.481749i \(0.840002\pi\)
\(294\) 0 0
\(295\) −27.0000 −1.57200
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −12.0000 −0.693978
\(300\) 8.00000 0.461880
\(301\) 0 0
\(302\) 0 0
\(303\) 12.0000 0.689382
\(304\) 8.00000 0.458831
\(305\) 30.0000 1.71780
\(306\) 0 0
\(307\) 20.0000 1.14146 0.570730 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 0 0
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 19.0000 1.07394 0.536972 0.843600i \(-0.319568\pi\)
0.536972 + 0.843600i \(0.319568\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −20.0000 −1.12509
\(317\) 9.00000 0.505490 0.252745 0.967533i \(-0.418667\pi\)
0.252745 + 0.967533i \(0.418667\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 24.0000 1.34164
\(321\) 6.00000 0.334887
\(322\) 0 0
\(323\) −12.0000 −0.667698
\(324\) −2.00000 −0.111111
\(325\) −16.0000 −0.887520
\(326\) 0 0
\(327\) 20.0000 1.10600
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −1.00000 −0.0549650 −0.0274825 0.999622i \(-0.508749\pi\)
−0.0274825 + 0.999622i \(0.508749\pi\)
\(332\) −24.0000 −1.31717
\(333\) −22.0000 −1.20559
\(334\) 0 0
\(335\) −15.0000 −0.819538
\(336\) 0 0
\(337\) −14.0000 −0.762629 −0.381314 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) 0 0
\(339\) 3.00000 0.162938
\(340\) −36.0000 −1.95237
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 9.00000 0.484544
\(346\) 0 0
\(347\) 18.0000 0.966291 0.483145 0.875540i \(-0.339494\pi\)
0.483145 + 0.875540i \(0.339494\pi\)
\(348\) 12.0000 0.643268
\(349\) −10.0000 −0.535288 −0.267644 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 0 0
\(351\) −20.0000 −1.06752
\(352\) 0 0
\(353\) 3.00000 0.159674 0.0798369 0.996808i \(-0.474560\pi\)
0.0798369 + 0.996808i \(0.474560\pi\)
\(354\) 0 0
\(355\) −27.0000 −1.43301
\(356\) −6.00000 −0.317999
\(357\) 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
\(363\) 0 0
\(364\) 0 0
\(365\) −6.00000 −0.314054
\(366\) 0 0
\(367\) −17.0000 −0.887393 −0.443696 0.896177i \(-0.646333\pi\)
−0.443696 + 0.896177i \(0.646333\pi\)
\(368\) 12.0000 0.625543
\(369\) −12.0000 −0.624695
\(370\) 0 0
\(371\) 0 0
\(372\) −10.0000 −0.518476
\(373\) 4.00000 0.207112 0.103556 0.994624i \(-0.466978\pi\)
0.103556 + 0.994624i \(0.466978\pi\)
\(374\) 0 0
\(375\) −3.00000 −0.154919
\(376\) 0 0
\(377\) −24.0000 −1.23606
\(378\) 0 0
\(379\) 11.0000 0.565032 0.282516 0.959263i \(-0.408831\pi\)
0.282516 + 0.959263i \(0.408831\pi\)
\(380\) 12.0000 0.615587
\(381\) 2.00000 0.102463
\(382\) 0 0
\(383\) −21.0000 −1.07305 −0.536525 0.843884i \(-0.680263\pi\)
−0.536525 + 0.843884i \(0.680263\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 16.0000 0.813326
\(388\) −2.00000 −0.101535
\(389\) 33.0000 1.67317 0.836583 0.547840i \(-0.184550\pi\)
0.836583 + 0.547840i \(0.184550\pi\)
\(390\) 0 0
\(391\) −18.0000 −0.910299
\(392\) 0 0
\(393\) 6.00000 0.302660
\(394\) 0 0
\(395\) −30.0000 −1.50946
\(396\) 0 0
\(397\) −2.00000 −0.100377 −0.0501886 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 16.0000 0.800000
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 0 0
\(403\) 20.0000 0.996271
\(404\) 24.0000 1.19404
\(405\) −3.00000 −0.149071
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 0 0
\(411\) 3.00000 0.147979
\(412\) −8.00000 −0.394132
\(413\) 0 0
\(414\) 0 0
\(415\) −36.0000 −1.76717
\(416\) 0 0
\(417\) −14.0000 −0.685583
\(418\) 0 0
\(419\) −24.0000 −1.17248 −0.586238 0.810139i \(-0.699392\pi\)
−0.586238 + 0.810139i \(0.699392\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −24.0000 −1.16417
\(426\) 0 0
\(427\) 0 0
\(428\) 12.0000 0.580042
\(429\) 0 0
\(430\) 0 0
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 20.0000 0.962250
\(433\) −11.0000 −0.528626 −0.264313 0.964437i \(-0.585145\pi\)
−0.264313 + 0.964437i \(0.585145\pi\)
\(434\) 0 0
\(435\) 18.0000 0.863034
\(436\) 40.0000 1.91565
\(437\) 6.00000 0.287019
\(438\) 0 0
\(439\) 26.0000 1.24091 0.620456 0.784241i \(-0.286947\pi\)
0.620456 + 0.784241i \(0.286947\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 9.00000 0.427603 0.213801 0.976877i \(-0.431415\pi\)
0.213801 + 0.976877i \(0.431415\pi\)
\(444\) 22.0000 1.04407
\(445\) −9.00000 −0.426641
\(446\) 0 0
\(447\) −6.00000 −0.283790
\(448\) 0 0
\(449\) −9.00000 −0.424736 −0.212368 0.977190i \(-0.568118\pi\)
−0.212368 + 0.977190i \(0.568118\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 6.00000 0.282216
\(453\) −10.0000 −0.469841
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −8.00000 −0.374224 −0.187112 0.982339i \(-0.559913\pi\)
−0.187112 + 0.982339i \(0.559913\pi\)
\(458\) 0 0
\(459\) −30.0000 −1.40028
\(460\) 18.0000 0.839254
\(461\) −6.00000 −0.279448 −0.139724 0.990190i \(-0.544622\pi\)
−0.139724 + 0.990190i \(0.544622\pi\)
\(462\) 0 0
\(463\) 5.00000 0.232370 0.116185 0.993228i \(-0.462933\pi\)
0.116185 + 0.993228i \(0.462933\pi\)
\(464\) 24.0000 1.11417
\(465\) −15.0000 −0.695608
\(466\) 0 0
\(467\) −15.0000 −0.694117 −0.347059 0.937843i \(-0.612820\pi\)
−0.347059 + 0.937843i \(0.612820\pi\)
\(468\) −16.0000 −0.739600
\(469\) 0 0
\(470\) 0 0
\(471\) −13.0000 −0.599008
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 8.00000 0.367065
\(476\) 0 0
\(477\) 12.0000 0.549442
\(478\) 0 0
\(479\) −12.0000 −0.548294 −0.274147 0.961688i \(-0.588395\pi\)
−0.274147 + 0.961688i \(0.588395\pi\)
\(480\) 0 0
\(481\) −44.0000 −2.00623
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −3.00000 −0.136223
\(486\) 0 0
\(487\) 11.0000 0.498458 0.249229 0.968445i \(-0.419823\pi\)
0.249229 + 0.968445i \(0.419823\pi\)
\(488\) 0 0
\(489\) −20.0000 −0.904431
\(490\) 0 0
\(491\) 30.0000 1.35388 0.676941 0.736038i \(-0.263305\pi\)
0.676941 + 0.736038i \(0.263305\pi\)
\(492\) 12.0000 0.541002
\(493\) −36.0000 −1.62136
\(494\) 0 0
\(495\) 0 0
\(496\) −20.0000 −0.898027
\(497\) 0 0
\(498\) 0 0
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) −6.00000 −0.268328
\(501\) −6.00000 −0.268060
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 36.0000 1.60198
\(506\) 0 0
\(507\) −3.00000 −0.133235
\(508\) 4.00000 0.177471
\(509\) −21.0000 −0.930809 −0.465404 0.885098i \(-0.654091\pi\)
−0.465404 + 0.885098i \(0.654091\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 10.0000 0.441511
\(514\) 0 0
\(515\) −12.0000 −0.528783
\(516\) −16.0000 −0.704361
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −3.00000 −0.131432 −0.0657162 0.997838i \(-0.520933\pi\)
−0.0657162 + 0.997838i \(0.520933\pi\)
\(522\) 0 0
\(523\) −16.0000 −0.699631 −0.349816 0.936819i \(-0.613756\pi\)
−0.349816 + 0.936819i \(0.613756\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) 0 0
\(527\) 30.0000 1.30682
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) −18.0000 −0.781133
\(532\) 0 0
\(533\) −24.0000 −1.03956
\(534\) 0 0
\(535\) 18.0000 0.778208
\(536\) 0 0
\(537\) 15.0000 0.647298
\(538\) 0 0
\(539\) 0 0
\(540\) 30.0000 1.29099
\(541\) 16.0000 0.687894 0.343947 0.938989i \(-0.388236\pi\)
0.343947 + 0.938989i \(0.388236\pi\)
\(542\) 0 0
\(543\) −7.00000 −0.300399
\(544\) 0 0
\(545\) 60.0000 2.57012
\(546\) 0 0
\(547\) −8.00000 −0.342055 −0.171028 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) 6.00000 0.256307
\(549\) 20.0000 0.853579
\(550\) 0 0
\(551\) 12.0000 0.511217
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 33.0000 1.40077
\(556\) −28.0000 −1.18746
\(557\) −30.0000 −1.27114 −0.635570 0.772043i \(-0.719235\pi\)
−0.635570 + 0.772043i \(0.719235\pi\)
\(558\) 0 0
\(559\) 32.0000 1.35346
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 36.0000 1.51722 0.758610 0.651546i \(-0.225879\pi\)
0.758610 + 0.651546i \(0.225879\pi\)
\(564\) 0 0
\(565\) 9.00000 0.378633
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 18.0000 0.754599 0.377300 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) 0 0
\(573\) 27.0000 1.12794
\(574\) 0 0
\(575\) 12.0000 0.500435
\(576\) 16.0000 0.666667
\(577\) −11.0000 −0.457936 −0.228968 0.973434i \(-0.573535\pi\)
−0.228968 + 0.973434i \(0.573535\pi\)
\(578\) 0 0
\(579\) 14.0000 0.581820
\(580\) 36.0000 1.49482
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −24.0000 −0.992278
\(586\) 0 0
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 0 0
\(589\) −10.0000 −0.412043
\(590\) 0 0
\(591\) 18.0000 0.740421
\(592\) 44.0000 1.80839
\(593\) 6.00000 0.246390 0.123195 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −12.0000 −0.491539
\(597\) −16.0000 −0.654836
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 8.00000 0.326327 0.163163 0.986599i \(-0.447830\pi\)
0.163163 + 0.986599i \(0.447830\pi\)
\(602\) 0 0
\(603\) −10.0000 −0.407231
\(604\) −20.0000 −0.813788
\(605\) 0 0
\(606\) 0 0
\(607\) 14.0000 0.568242 0.284121 0.958788i \(-0.408298\pi\)
0.284121 + 0.958788i \(0.408298\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) −24.0000 −0.970143
\(613\) 16.0000 0.646234 0.323117 0.946359i \(-0.395269\pi\)
0.323117 + 0.946359i \(0.395269\pi\)
\(614\) 0 0
\(615\) 18.0000 0.725830
\(616\) 0 0
\(617\) −30.0000 −1.20775 −0.603877 0.797077i \(-0.706378\pi\)
−0.603877 + 0.797077i \(0.706378\pi\)
\(618\) 0 0
\(619\) 19.0000 0.763674 0.381837 0.924230i \(-0.375291\pi\)
0.381837 + 0.924230i \(0.375291\pi\)
\(620\) −30.0000 −1.20483
\(621\) 15.0000 0.601929
\(622\) 0 0
\(623\) 0 0
\(624\) 16.0000 0.640513
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) 0 0
\(628\) −26.0000 −1.03751
\(629\) −66.0000 −2.63159
\(630\) 0 0
\(631\) 11.0000 0.437903 0.218952 0.975736i \(-0.429736\pi\)
0.218952 + 0.975736i \(0.429736\pi\)
\(632\) 0 0
\(633\) 14.0000 0.556450
\(634\) 0 0
\(635\) 6.00000 0.238103
\(636\) −12.0000 −0.475831
\(637\) 0 0
\(638\) 0 0
\(639\) −18.0000 −0.712069
\(640\) 0 0
\(641\) 15.0000 0.592464 0.296232 0.955116i \(-0.404270\pi\)
0.296232 + 0.955116i \(0.404270\pi\)
\(642\) 0 0
\(643\) 49.0000 1.93237 0.966186 0.257847i \(-0.0830131\pi\)
0.966186 + 0.257847i \(0.0830131\pi\)
\(644\) 0 0
\(645\) −24.0000 −0.944999
\(646\) 0 0
\(647\) 33.0000 1.29736 0.648682 0.761060i \(-0.275321\pi\)
0.648682 + 0.761060i \(0.275321\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −40.0000 −1.56652
\(653\) 39.0000 1.52619 0.763094 0.646288i \(-0.223679\pi\)
0.763094 + 0.646288i \(0.223679\pi\)
\(654\) 0 0
\(655\) 18.0000 0.703318
\(656\) 24.0000 0.937043
\(657\) −4.00000 −0.156055
\(658\) 0 0
\(659\) −30.0000 −1.16863 −0.584317 0.811525i \(-0.698638\pi\)
−0.584317 + 0.811525i \(0.698638\pi\)
\(660\) 0 0
\(661\) 49.0000 1.90588 0.952940 0.303160i \(-0.0980418\pi\)
0.952940 + 0.303160i \(0.0980418\pi\)
\(662\) 0 0
\(663\) −24.0000 −0.932083
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 18.0000 0.696963
\(668\) −12.0000 −0.464294
\(669\) −19.0000 −0.734582
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 28.0000 1.07932 0.539660 0.841883i \(-0.318553\pi\)
0.539660 + 0.841883i \(0.318553\pi\)
\(674\) 0 0
\(675\) 20.0000 0.769800
\(676\) −6.00000 −0.230769
\(677\) −18.0000 −0.691796 −0.345898 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 12.0000 0.459841
\(682\) 0 0
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 8.00000 0.305888
\(685\) 9.00000 0.343872
\(686\) 0 0
\(687\) 5.00000 0.190762
\(688\) −32.0000 −1.21999
\(689\) 24.0000 0.914327
\(690\) 0 0
\(691\) −35.0000 −1.33146 −0.665731 0.746191i \(-0.731880\pi\)
−0.665731 + 0.746191i \(0.731880\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −42.0000 −1.59315
\(696\) 0 0
\(697\) −36.0000 −1.36360
\(698\) 0 0
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 22.0000 0.829746
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 18.0000 0.676481
\(709\) −1.00000 −0.0375558 −0.0187779 0.999824i \(-0.505978\pi\)
−0.0187779 + 0.999824i \(0.505978\pi\)
\(710\) 0 0
\(711\) −20.0000 −0.750059
\(712\) 0 0
\(713\) −15.0000 −0.561754
\(714\) 0 0
\(715\) 0 0
\(716\) 30.0000 1.12115
\(717\) −12.0000 −0.448148
\(718\) 0 0
\(719\) 39.0000 1.45445 0.727227 0.686397i \(-0.240809\pi\)
0.727227 + 0.686397i \(0.240809\pi\)
\(720\) 24.0000 0.894427
\(721\) 0 0
\(722\) 0 0
\(723\) 28.0000 1.04133
\(724\) −14.0000 −0.520306
\(725\) 24.0000 0.891338
\(726\) 0 0
\(727\) −17.0000 −0.630495 −0.315248 0.949009i \(-0.602088\pi\)
−0.315248 + 0.949009i \(0.602088\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 48.0000 1.77534
\(732\) −20.0000 −0.739221
\(733\) −4.00000 −0.147743 −0.0738717 0.997268i \(-0.523536\pi\)
−0.0738717 + 0.997268i \(0.523536\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 34.0000 1.25071 0.625355 0.780340i \(-0.284954\pi\)
0.625355 + 0.780340i \(0.284954\pi\)
\(740\) 66.0000 2.42621
\(741\) 8.00000 0.293887
\(742\) 0 0
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) −18.0000 −0.659469
\(746\) 0 0
\(747\) −24.0000 −0.878114
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −31.0000 −1.13121 −0.565603 0.824678i \(-0.691357\pi\)
−0.565603 + 0.824678i \(0.691357\pi\)
\(752\) 0 0
\(753\) −9.00000 −0.327978
\(754\) 0 0
\(755\) −30.0000 −1.09181
\(756\) 0 0
\(757\) 38.0000 1.38113 0.690567 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 48.0000 1.74000 0.869999 0.493053i \(-0.164119\pi\)
0.869999 + 0.493053i \(0.164119\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 54.0000 1.95365
\(765\) −36.0000 −1.30158
\(766\) 0 0
\(767\) −36.0000 −1.29988
\(768\) −16.0000 −0.577350
\(769\) −40.0000 −1.44244 −0.721218 0.692708i \(-0.756418\pi\)
−0.721218 + 0.692708i \(0.756418\pi\)
\(770\) 0 0
\(771\) −6.00000 −0.216085
\(772\) 28.0000 1.00774
\(773\) 6.00000 0.215805 0.107903 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) 0 0
\(775\) −20.0000 −0.718421
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 12.0000 0.429945
\(780\) 24.0000 0.859338
\(781\) 0 0
\(782\) 0 0
\(783\) 30.0000 1.07211
\(784\) 0 0
\(785\) −39.0000 −1.39197
\(786\) 0 0
\(787\) 50.0000 1.78231 0.891154 0.453701i \(-0.149897\pi\)
0.891154 + 0.453701i \(0.149897\pi\)
\(788\) 36.0000 1.28245
\(789\) −30.0000 −1.06803
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 40.0000 1.42044
\(794\) 0 0
\(795\) −18.0000 −0.638394
\(796\) −32.0000 −1.13421
\(797\) 21.0000 0.743858 0.371929 0.928261i \(-0.378696\pi\)
0.371929 + 0.928261i \(0.378696\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 0 0
\(803\) 0 0
\(804\) 10.0000 0.352673
\(805\) 0 0
\(806\) 0 0
\(807\) 30.0000 1.05605
\(808\) 0 0
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 0 0
\(811\) 2.00000 0.0702295 0.0351147 0.999383i \(-0.488820\pi\)
0.0351147 + 0.999383i \(0.488820\pi\)
\(812\) 0 0
\(813\) 16.0000 0.561144
\(814\) 0 0
\(815\) −60.0000 −2.10171
\(816\) 24.0000 0.840168
\(817\) −16.0000 −0.559769
\(818\) 0 0
\(819\) 0 0
\(820\) 36.0000 1.25717
\(821\) −18.0000 −0.628204 −0.314102 0.949389i \(-0.601703\pi\)
−0.314102 + 0.949389i \(0.601703\pi\)
\(822\) 0 0
\(823\) 23.0000 0.801730 0.400865 0.916137i \(-0.368710\pi\)
0.400865 + 0.916137i \(0.368710\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −36.0000 −1.25184 −0.625921 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) 12.0000 0.417029
\(829\) 25.0000 0.868286 0.434143 0.900844i \(-0.357051\pi\)
0.434143 + 0.900844i \(0.357051\pi\)
\(830\) 0 0
\(831\) 8.00000 0.277517
\(832\) 32.0000 1.10940
\(833\) 0 0
\(834\) 0 0
\(835\) −18.0000 −0.622916
\(836\) 0 0
\(837\) −25.0000 −0.864126
\(838\) 0 0
\(839\) 15.0000 0.517858 0.258929 0.965896i \(-0.416631\pi\)
0.258929 + 0.965896i \(0.416631\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) 12.0000 0.413302
\(844\) 28.0000 0.963800
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) 0 0
\(848\) −24.0000 −0.824163
\(849\) −32.0000 −1.09824
\(850\) 0 0
\(851\) 33.0000 1.13123
\(852\) 18.0000 0.616670
\(853\) −10.0000 −0.342393 −0.171197 0.985237i \(-0.554763\pi\)
−0.171197 + 0.985237i \(0.554763\pi\)
\(854\) 0 0
\(855\) 12.0000 0.410391
\(856\) 0 0
\(857\) −12.0000 −0.409912 −0.204956 0.978771i \(-0.565705\pi\)
−0.204956 + 0.978771i \(0.565705\pi\)
\(858\) 0 0
\(859\) 13.0000 0.443554 0.221777 0.975097i \(-0.428814\pi\)
0.221777 + 0.975097i \(0.428814\pi\)
\(860\) −48.0000 −1.63679
\(861\) 0 0
\(862\) 0 0
\(863\) −12.0000 −0.408485 −0.204242 0.978920i \(-0.565473\pi\)
−0.204242 + 0.978920i \(0.565473\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −19.0000 −0.645274
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −20.0000 −0.677674
\(872\) 0 0
\(873\) −2.00000 −0.0676897
\(874\) 0 0
\(875\) 0 0
\(876\) 4.00000 0.135147
\(877\) 22.0000 0.742887 0.371444 0.928456i \(-0.378863\pi\)
0.371444 + 0.928456i \(0.378863\pi\)
\(878\) 0 0
\(879\) 30.0000 1.01187
\(880\) 0 0
\(881\) 9.00000 0.303218 0.151609 0.988441i \(-0.451555\pi\)
0.151609 + 0.988441i \(0.451555\pi\)
\(882\) 0 0
\(883\) 20.0000 0.673054 0.336527 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) −48.0000 −1.61441
\(885\) 27.0000 0.907595
\(886\) 0 0
\(887\) −42.0000 −1.41022 −0.705111 0.709097i \(-0.749103\pi\)
−0.705111 + 0.709097i \(0.749103\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) −38.0000 −1.27233
\(893\) 0 0
\(894\) 0 0
\(895\) 45.0000 1.50418
\(896\) 0 0
\(897\) 12.0000 0.400668
\(898\) 0 0
\(899\) −30.0000 −1.00056
\(900\) 16.0000 0.533333
\(901\) 36.0000 1.19933
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −21.0000 −0.698064
\(906\) 0 0
\(907\) 8.00000 0.265636 0.132818 0.991140i \(-0.457597\pi\)
0.132818 + 0.991140i \(0.457597\pi\)
\(908\) 24.0000 0.796468
\(909\) 24.0000 0.796030
\(910\) 0 0
\(911\) 48.0000 1.59031 0.795155 0.606406i \(-0.207389\pi\)
0.795155 + 0.606406i \(0.207389\pi\)
\(912\) −8.00000 −0.264906
\(913\) 0 0
\(914\) 0 0
\(915\) −30.0000 −0.991769
\(916\) 10.0000 0.330409
\(917\) 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
\(921\) −20.0000 −0.659022
\(922\) 0 0
\(923\) −36.0000 −1.18495
\(924\) 0 0
\(925\) 44.0000 1.44671
\(926\) 0 0
\(927\) −8.00000 −0.262754
\(928\) 0 0
\(929\) −18.0000 −0.590561 −0.295280 0.955411i \(-0.595413\pi\)
−0.295280 + 0.955411i \(0.595413\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 12.0000 0.393073
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 20.0000 0.653372 0.326686 0.945133i \(-0.394068\pi\)
0.326686 + 0.945133i \(0.394068\pi\)
\(938\) 0 0
\(939\) −19.0000 −0.620042
\(940\) 0 0
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) 0 0
\(943\) 18.0000 0.586161
\(944\) 36.0000 1.17170
\(945\) 0 0
\(946\) 0 0
\(947\) −27.0000 −0.877382 −0.438691 0.898638i \(-0.644558\pi\)
−0.438691 + 0.898638i \(0.644558\pi\)
\(948\) 20.0000 0.649570
\(949\) −8.00000 −0.259691
\(950\) 0 0
\(951\) −9.00000 −0.291845
\(952\) 0 0
\(953\) 36.0000 1.16615 0.583077 0.812417i \(-0.301849\pi\)
0.583077 + 0.812417i \(0.301849\pi\)
\(954\) 0 0
\(955\) 81.0000 2.62110
\(956\) −24.0000 −0.776215
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) −24.0000 −0.774597
\(961\) −6.00000 −0.193548
\(962\) 0 0
\(963\) 12.0000 0.386695
\(964\) 56.0000 1.80364
\(965\) 42.0000 1.35203
\(966\) 0 0
\(967\) −14.0000 −0.450210 −0.225105 0.974335i \(-0.572272\pi\)
−0.225105 + 0.974335i \(0.572272\pi\)
\(968\) 0 0
\(969\) 12.0000 0.385496
\(970\) 0 0
\(971\) 39.0000 1.25157 0.625785 0.779996i \(-0.284779\pi\)
0.625785 + 0.779996i \(0.284779\pi\)
\(972\) 32.0000 1.02640
\(973\) 0 0
\(974\) 0 0
\(975\) 16.0000 0.512410
\(976\) −40.0000 −1.28037
\(977\) 9.00000 0.287936 0.143968 0.989582i \(-0.454014\pi\)
0.143968 + 0.989582i \(0.454014\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 40.0000 1.27710
\(982\) 0 0
\(983\) −33.0000 −1.05254 −0.526268 0.850319i \(-0.676409\pi\)
−0.526268 + 0.850319i \(0.676409\pi\)
\(984\) 0 0
\(985\) 54.0000 1.72058
\(986\) 0 0
\(987\) 0 0
\(988\) 16.0000 0.509028
\(989\) −24.0000 −0.763156
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 0 0
\(993\) 1.00000 0.0317340
\(994\) 0 0
\(995\) −48.0000 −1.52170
\(996\) 24.0000 0.760469
\(997\) −28.0000 −0.886769 −0.443384 0.896332i \(-0.646222\pi\)
−0.443384 + 0.896332i \(0.646222\pi\)
\(998\) 0 0
\(999\) 55.0000 1.74012
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 5929.2.a.d.1.1 1
7.6 odd 2 847.2.a.c.1.1 1
11.10 odd 2 539.2.a.b.1.1 1
21.20 even 2 7623.2.a.i.1.1 1
33.32 even 2 4851.2.a.k.1.1 1
44.43 even 2 8624.2.a.s.1.1 1
77.6 even 10 847.2.f.f.729.1 4
77.10 even 6 539.2.e.d.177.1 2
77.13 even 10 847.2.f.f.323.1 4
77.20 odd 10 847.2.f.g.323.1 4
77.27 odd 10 847.2.f.g.729.1 4
77.32 odd 6 539.2.e.e.177.1 2
77.41 even 10 847.2.f.f.372.1 4
77.48 odd 10 847.2.f.g.148.1 4
77.54 even 6 539.2.e.d.67.1 2
77.62 even 10 847.2.f.f.148.1 4
77.65 odd 6 539.2.e.e.67.1 2
77.69 odd 10 847.2.f.g.372.1 4
77.76 even 2 77.2.a.b.1.1 1
231.230 odd 2 693.2.a.b.1.1 1
308.307 odd 2 1232.2.a.d.1.1 1
385.153 odd 4 1925.2.b.g.1849.2 2
385.307 odd 4 1925.2.b.g.1849.1 2
385.384 even 2 1925.2.a.f.1.1 1
616.307 odd 2 4928.2.a.x.1.1 1
616.461 even 2 4928.2.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
77.2.a.b.1.1 1 77.76 even 2
539.2.a.b.1.1 1 11.10 odd 2
539.2.e.d.67.1 2 77.54 even 6
539.2.e.d.177.1 2 77.10 even 6
539.2.e.e.67.1 2 77.65 odd 6
539.2.e.e.177.1 2 77.32 odd 6
693.2.a.b.1.1 1 231.230 odd 2
847.2.a.c.1.1 1 7.6 odd 2
847.2.f.f.148.1 4 77.62 even 10
847.2.f.f.323.1 4 77.13 even 10
847.2.f.f.372.1 4 77.41 even 10
847.2.f.f.729.1 4 77.6 even 10
847.2.f.g.148.1 4 77.48 odd 10
847.2.f.g.323.1 4 77.20 odd 10
847.2.f.g.372.1 4 77.69 odd 10
847.2.f.g.729.1 4 77.27 odd 10
1232.2.a.d.1.1 1 308.307 odd 2
1925.2.a.f.1.1 1 385.384 even 2
1925.2.b.g.1849.1 2 385.307 odd 4
1925.2.b.g.1849.2 2 385.153 odd 4
4851.2.a.k.1.1 1 33.32 even 2
4928.2.a.i.1.1 1 616.461 even 2
4928.2.a.x.1.1 1 616.307 odd 2
5929.2.a.d.1.1 1 1.1 even 1 trivial
7623.2.a.i.1.1 1 21.20 even 2
8624.2.a.s.1.1 1 44.43 even 2