Properties

Label 77.2.a.b.1.1
Level $77$
Weight $2$
Character 77.1
Self dual yes
Analytic conductor $0.615$
Analytic rank $0$
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] = [77,2,Mod(1,77)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(77, 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("77.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 77 = 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 77.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: \(0.614848095564\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 77.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -2.00000 q^{4} +3.00000 q^{5} +1.00000 q^{7} -2.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -2.00000 q^{4} +3.00000 q^{5} +1.00000 q^{7} -2.00000 q^{9} -1.00000 q^{11} -2.00000 q^{12} -4.00000 q^{13} +3.00000 q^{15} +4.00000 q^{16} -6.00000 q^{17} +2.00000 q^{19} -6.00000 q^{20} +1.00000 q^{21} +3.00000 q^{23} +4.00000 q^{25} -5.00000 q^{27} -2.00000 q^{28} -6.00000 q^{29} +5.00000 q^{31} -1.00000 q^{33} +3.00000 q^{35} +4.00000 q^{36} +11.0000 q^{37} -4.00000 q^{39} +6.00000 q^{41} +8.00000 q^{43} +2.00000 q^{44} -6.00000 q^{45} +4.00000 q^{48} +1.00000 q^{49} -6.00000 q^{51} +8.00000 q^{52} -6.00000 q^{53} -3.00000 q^{55} +2.00000 q^{57} -9.00000 q^{59} -6.00000 q^{60} -10.0000 q^{61} -2.00000 q^{63} -8.00000 q^{64} -12.0000 q^{65} +5.00000 q^{67} +12.0000 q^{68} +3.00000 q^{69} +9.00000 q^{71} +2.00000 q^{73} +4.00000 q^{75} -4.00000 q^{76} -1.00000 q^{77} -10.0000 q^{79} +12.0000 q^{80} +1.00000 q^{81} +12.0000 q^{83} -2.00000 q^{84} -18.0000 q^{85} -6.00000 q^{87} -3.00000 q^{89} -4.00000 q^{91} -6.00000 q^{92} +5.00000 q^{93} +6.00000 q^{95} -1.00000 q^{97} +2.00000 q^{99} +O(q^{100})\)

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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 1.00000 0.577350 0.288675 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) −2.00000 −1.00000
\(5\) 3.00000 1.34164 0.670820 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) −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
\(21\) 1.00000 0.218218
\(22\) 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
\(28\) −2.00000 −0.377964
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 5.00000 0.898027 0.449013 0.893525i \(-0.351776\pi\)
0.449013 + 0.893525i \(0.351776\pi\)
\(32\) 0 0
\(33\) −1.00000 −0.174078
\(34\) 0 0
\(35\) 3.00000 0.507093
\(36\) 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 \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 2.00000 0.301511
\(45\) −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
\(49\) 1.00000 0.142857
\(50\) 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
\(55\) −3.00000 −0.404520
\(56\) 0 0
\(57\) 2.00000 0.264906
\(58\) 0 0
\(59\) −9.00000 −1.17170 −0.585850 0.810419i \(-0.699239\pi\)
−0.585850 + 0.810419i \(0.699239\pi\)
\(60\) −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
\(63\) −2.00000 −0.251976
\(64\) −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
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) −10.0000 −1.12509 −0.562544 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 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 \(0.271155\pi\)
\(84\) −2.00000 −0.218218
\(85\) −18.0000 −1.95237
\(86\) 0 0
\(87\) −6.00000 −0.643268
\(88\) 0 0
\(89\) −3.00000 −0.317999 −0.159000 0.987279i \(-0.550827\pi\)
−0.159000 + 0.987279i \(0.550827\pi\)
\(90\) 0 0
\(91\) −4.00000 −0.419314
\(92\) −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 \(-0.516167\pi\)
−0.0507673 + 0.998711i \(0.516167\pi\)
\(98\) 0 0
\(99\) 2.00000 0.201008
\(100\) −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 \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 0 0
\(105\) 3.00000 0.292770
\(106\) 0 0
\(107\) 6.00000 0.580042 0.290021 0.957020i \(-0.406338\pi\)
0.290021 + 0.957020i \(0.406338\pi\)
\(108\) 10.0000 0.962250
\(109\) 20.0000 1.91565 0.957826 0.287348i \(-0.0927736\pi\)
0.957826 + 0.287348i \(0.0927736\pi\)
\(110\) 0 0
\(111\) 11.0000 1.04407
\(112\) 4.00000 0.377964
\(113\) −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
\(119\) −6.00000 −0.550019
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 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 \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) 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 \(0.584419\pi\)
\(132\) 2.00000 0.174078
\(133\) 2.00000 0.173422
\(134\) 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 \(0.297654\pi\)
\(140\) −6.00000 −0.507093
\(141\) 0 0
\(142\) 0 0
\(143\) 4.00000 0.334497
\(144\) −8.00000 −0.666667
\(145\) −18.0000 −1.49482
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) −22.0000 −1.80839
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) −10.0000 −0.813788 −0.406894 0.913475i \(-0.633388\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(152\) 0 0
\(153\) 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 \(-0.673605\pi\)
−0.518756 + 0.854922i \(0.673605\pi\)
\(158\) 0 0
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) 3.00000 0.236433
\(162\) 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
\(165\) −3.00000 −0.233550
\(166\) 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
\(175\) 4.00000 0.302372
\(176\) −4.00000 −0.301511
\(177\) −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 \(-0.583773\pi\)
−0.260153 + 0.965567i \(0.583773\pi\)
\(182\) 0 0
\(183\) −10.0000 −0.739221
\(184\) 0 0
\(185\) 33.0000 2.42621
\(186\) 0 0
\(187\) 6.00000 0.438763
\(188\) 0 0
\(189\) −5.00000 −0.363696
\(190\) 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 \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) 0 0
\(195\) −12.0000 −0.859338
\(196\) −2.00000 −0.142857
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) 5.00000 0.352673
\(202\) 0 0
\(203\) −6.00000 −0.421117
\(204\) 12.0000 0.840168
\(205\) 18.0000 1.25717
\(206\) 0 0
\(207\) −6.00000 −0.417029
\(208\) −16.0000 −1.10940
\(209\) −2.00000 −0.138343
\(210\) 0 0
\(211\) 14.0000 0.963800 0.481900 0.876226i \(-0.339947\pi\)
0.481900 + 0.876226i \(0.339947\pi\)
\(212\) 12.0000 0.824163
\(213\) 9.00000 0.616670
\(214\) 0 0
\(215\) 24.0000 1.63679
\(216\) 0 0
\(217\) 5.00000 0.339422
\(218\) 0 0
\(219\) 2.00000 0.135147
\(220\) 6.00000 0.404520
\(221\) 24.0000 1.61441
\(222\) 0 0
\(223\) −19.0000 −1.27233 −0.636167 0.771551i \(-0.719481\pi\)
−0.636167 + 0.771551i \(0.719481\pi\)
\(224\) 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 \(-0.447172\pi\)
0.165205 + 0.986259i \(0.447172\pi\)
\(230\) 0 0
\(231\) −1.00000 −0.0657952
\(232\) 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
\(236\) 18.0000 1.17170
\(237\) −10.0000 −0.649570
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 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
\(245\) 3.00000 0.191663
\(246\) 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 \(-0.591674\pi\)
−0.284037 + 0.958813i \(0.591674\pi\)
\(252\) 4.00000 0.251976
\(253\) −3.00000 −0.188608
\(254\) 0 0
\(255\) −18.0000 −1.12720
\(256\) 16.0000 1.00000
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 0 0
\(259\) 11.0000 0.683507
\(260\) 24.0000 1.48842
\(261\) 12.0000 0.742781
\(262\) 0 0
\(263\) −30.0000 −1.84988 −0.924940 0.380114i \(-0.875885\pi\)
−0.924940 + 0.380114i \(0.875885\pi\)
\(264\) 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 \(-0.132532\pi\)
0.914566 + 0.404436i \(0.132532\pi\)
\(270\) 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
\(273\) −4.00000 −0.242091
\(274\) 0 0
\(275\) −4.00000 −0.241209
\(276\) −6.00000 −0.361158
\(277\) 8.00000 0.480673 0.240337 0.970690i \(-0.422742\pi\)
0.240337 + 0.970690i \(0.422742\pi\)
\(278\) 0 0
\(279\) −10.0000 −0.598684
\(280\) 0 0
\(281\) 12.0000 0.715860 0.357930 0.933748i \(-0.383483\pi\)
0.357930 + 0.933748i \(0.383483\pi\)
\(282\) 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
\(287\) 6.00000 0.354169
\(288\) 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
\(297\) 5.00000 0.290129
\(298\) 0 0
\(299\) −12.0000 −0.693978
\(300\) −8.00000 −0.461880
\(301\) 8.00000 0.461112
\(302\) 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 \(0.306660\pi\)
\(308\) 2.00000 0.113961
\(309\) −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 \(-0.680432\pi\)
−0.536972 + 0.843600i \(0.680432\pi\)
\(314\) 0 0
\(315\) −6.00000 −0.338062
\(316\) 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
\(319\) 6.00000 0.335936
\(320\) −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
\(336\) 4.00000 0.218218
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 0 0
\(339\) −3.00000 −0.162938
\(340\) 36.0000 1.95237
\(341\) −5.00000 −0.270765
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 9.00000 0.484544
\(346\) 0 0
\(347\) −18.0000 −0.966291 −0.483145 0.875540i \(-0.660506\pi\)
−0.483145 + 0.875540i \(0.660506\pi\)
\(348\) 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 \(-0.525440\pi\)
−0.0798369 + 0.996808i \(0.525440\pi\)
\(354\) 0 0
\(355\) 27.0000 1.43301
\(356\) 6.00000 0.317999
\(357\) −6.00000 −0.317554
\(358\) 0 0
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 1.00000 0.0524864
\(364\) 8.00000 0.419314
\(365\) 6.00000 0.314054
\(366\) 0 0
\(367\) 17.0000 0.887393 0.443696 0.896177i \(-0.353667\pi\)
0.443696 + 0.896177i \(0.353667\pi\)
\(368\) 12.0000 0.625543
\(369\) −12.0000 −0.624695
\(370\) 0 0
\(371\) −6.00000 −0.311504
\(372\) −10.0000 −0.518476
\(373\) −4.00000 −0.207112 −0.103556 0.994624i \(-0.533022\pi\)
−0.103556 + 0.994624i \(0.533022\pi\)
\(374\) 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 \(-0.319737\pi\)
0.536525 + 0.843884i \(0.319737\pi\)
\(384\) 0 0
\(385\) −3.00000 −0.152894
\(386\) 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
\(396\) −4.00000 −0.201008
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) 0 0
\(399\) 2.00000 0.100125
\(400\) 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
\(407\) −11.0000 −0.545250
\(408\) 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
\(413\) −9.00000 −0.442861
\(414\) 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 \(-0.300608\pi\)
0.586238 + 0.810139i \(0.300608\pi\)
\(420\) −6.00000 −0.292770
\(421\) −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
\(427\) −10.0000 −0.483934
\(428\) −12.0000 −0.580042
\(429\) 4.00000 0.193122
\(430\) 0 0
\(431\) 12.0000 0.578020 0.289010 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) −20.0000 −0.962250
\(433\) 11.0000 0.528626 0.264313 0.964437i \(-0.414855\pi\)
0.264313 + 0.964437i \(0.414855\pi\)
\(434\) 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
\(441\) −2.00000 −0.0952381
\(442\) 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
\(448\) −8.00000 −0.377964
\(449\) −9.00000 −0.424736 −0.212368 0.977190i \(-0.568118\pi\)
−0.212368 + 0.977190i \(0.568118\pi\)
\(450\) 0 0
\(451\) −6.00000 −0.282529
\(452\) 6.00000 0.282216
\(453\) −10.0000 −0.469841
\(454\) 0 0
\(455\) −12.0000 −0.562569
\(456\) 0 0
\(457\) 8.00000 0.374224 0.187112 0.982339i \(-0.440087\pi\)
0.187112 + 0.982339i \(0.440087\pi\)
\(458\) 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 \(-0.387180\pi\)
0.347059 + 0.937843i \(0.387180\pi\)
\(468\) −16.0000 −0.739600
\(469\) 5.00000 0.230879
\(470\) 0 0
\(471\) −13.0000 −0.599008
\(472\) 0 0
\(473\) −8.00000 −0.367840
\(474\) 0 0
\(475\) 8.00000 0.367065
\(476\) 12.0000 0.550019
\(477\) 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
\(483\) 3.00000 0.136505
\(484\) −2.00000 −0.0909091
\(485\) −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 \(-0.736695\pi\)
−0.676941 + 0.736038i \(0.736695\pi\)
\(492\) −12.0000 −0.541002
\(493\) 36.0000 1.62136
\(494\) 0 0
\(495\) 6.00000 0.269680
\(496\) 20.0000 0.898027
\(497\) 9.00000 0.403705
\(498\) 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 \(-0.345909\pi\)
0.465404 + 0.885098i \(0.345909\pi\)
\(510\) 0 0
\(511\) 2.00000 0.0884748
\(512\) 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 \(-0.479067\pi\)
0.0657162 + 0.997838i \(0.479067\pi\)
\(522\) 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
\(525\) 4.00000 0.174574
\(526\) 0 0
\(527\) −30.0000 −1.30682
\(528\) −4.00000 −0.174078
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) 18.0000 0.781133
\(532\) −4.00000 −0.173422
\(533\) −24.0000 −1.03956
\(534\) 0 0
\(535\) 18.0000 0.778208
\(536\) 0 0
\(537\) −15.0000 −0.647298
\(538\) 0 0
\(539\) −1.00000 −0.0430730
\(540\) 30.0000 1.29099
\(541\) −16.0000 −0.687894 −0.343947 0.938989i \(-0.611764\pi\)
−0.343947 + 0.938989i \(0.611764\pi\)
\(542\) 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 \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) 6.00000 0.256307
\(549\) 20.0000 0.853579
\(550\) 0 0
\(551\) −12.0000 −0.511217
\(552\) 0 0
\(553\) −10.0000 −0.425243
\(554\) 0 0
\(555\) 33.0000 1.40077
\(556\) −28.0000 −1.18746
\(557\) 30.0000 1.27114 0.635570 0.772043i \(-0.280765\pi\)
0.635570 + 0.772043i \(0.280765\pi\)
\(558\) 0 0
\(559\) −32.0000 −1.35346
\(560\) 12.0000 0.507093
\(561\) 6.00000 0.253320
\(562\) 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
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −18.0000 −0.754599 −0.377300 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) −8.00000 −0.334497
\(573\) −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 \(-0.426465\pi\)
0.228968 + 0.973434i \(0.426465\pi\)
\(578\) 0 0
\(579\) 14.0000 0.581820
\(580\) 36.0000 1.49482
\(581\) 12.0000 0.497844
\(582\) 0 0
\(583\) 6.00000 0.248495
\(584\) 0 0
\(585\) 24.0000 0.992278
\(586\) 0 0
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) −2.00000 −0.0824786
\(589\) 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
\(595\) −18.0000 −0.737928
\(596\) 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
\(605\) 3.00000 0.121967
\(606\) 0 0
\(607\) 14.0000 0.568242 0.284121 0.958788i \(-0.408298\pi\)
0.284121 + 0.958788i \(0.408298\pi\)
\(608\) 0 0
\(609\) −6.00000 −0.243132
\(610\) 0 0
\(611\) 0 0
\(612\) −24.0000 −0.970143
\(613\) −16.0000 −0.646234 −0.323117 0.946359i \(-0.604731\pi\)
−0.323117 + 0.946359i \(0.604731\pi\)
\(614\) 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 \(-0.624709\pi\)
−0.381837 + 0.924230i \(0.624709\pi\)
\(620\) −30.0000 −1.20483
\(621\) −15.0000 −0.601929
\(622\) 0 0
\(623\) −3.00000 −0.120192
\(624\) −16.0000 −0.640513
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) −2.00000 −0.0798723
\(628\) 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
\(637\) −4.00000 −0.158486
\(638\) 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 \(-0.916987\pi\)
−0.966186 + 0.257847i \(0.916987\pi\)
\(644\) −6.00000 −0.236433
\(645\) 24.0000 0.944999
\(646\) 0 0
\(647\) −33.0000 −1.29736 −0.648682 0.761060i \(-0.724679\pi\)
−0.648682 + 0.761060i \(0.724679\pi\)
\(648\) 0 0
\(649\) 9.00000 0.353281
\(650\) 0 0
\(651\) 5.00000 0.195965
\(652\) −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 \(-0.301362\pi\)
0.584317 + 0.811525i \(0.301362\pi\)
\(660\) 6.00000 0.233550
\(661\) −49.0000 −1.90588 −0.952940 0.303160i \(-0.901958\pi\)
−0.952940 + 0.303160i \(0.901958\pi\)
\(662\) 0 0
\(663\) 24.0000 0.932083
\(664\) 0 0
\(665\) 6.00000 0.232670
\(666\) 0 0
\(667\) −18.0000 −0.696963
\(668\) −12.0000 −0.464294
\(669\) −19.0000 −0.734582
\(670\) 0 0
\(671\) 10.0000 0.386046
\(672\) 0 0
\(673\) −28.0000 −1.07932 −0.539660 0.841883i \(-0.681447\pi\)
−0.539660 + 0.841883i \(0.681447\pi\)
\(674\) 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
\(679\) −1.00000 −0.0383765
\(680\) 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 \(-0.268120\pi\)
0.665731 + 0.746191i \(0.268120\pi\)
\(692\) 0 0
\(693\) 2.00000 0.0759737
\(694\) 0 0
\(695\) 42.0000 1.59315
\(696\) 0 0
\(697\) −36.0000 −1.36360
\(698\) 0 0
\(699\) 6.00000 0.226941
\(700\) −8.00000 −0.302372
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 22.0000 0.829746
\(704\) 8.00000 0.301511
\(705\) 0 0
\(706\) 0 0
\(707\) −12.0000 −0.451306
\(708\) 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
\(715\) 12.0000 0.448775
\(716\) 30.0000 1.12115
\(717\) −12.0000 −0.448148
\(718\) 0 0
\(719\) −39.0000 −1.45445 −0.727227 0.686397i \(-0.759191\pi\)
−0.727227 + 0.686397i \(0.759191\pi\)
\(720\) −24.0000 −0.894427
\(721\) −4.00000 −0.148968
\(722\) 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 \(-0.397912\pi\)
0.315248 + 0.949009i \(0.397912\pi\)
\(728\) 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
\(735\) 3.00000 0.110657
\(736\) 0 0
\(737\) −5.00000 −0.184177
\(738\) 0 0
\(739\) −34.0000 −1.25071 −0.625355 0.780340i \(-0.715046\pi\)
−0.625355 + 0.780340i \(0.715046\pi\)
\(740\) −66.0000 −2.42621
\(741\) −8.00000 −0.293887
\(742\) 0 0
\(743\) 24.0000 0.880475 0.440237 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) 0 0
\(745\) −18.0000 −0.659469
\(746\) 0 0
\(747\) −24.0000 −0.878114
\(748\) −12.0000 −0.438763
\(749\) 6.00000 0.219235
\(750\) 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
\(756\) 10.0000 0.363696
\(757\) 38.0000 1.38113 0.690567 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) 0 0
\(759\) −3.00000 −0.108893
\(760\) 0 0
\(761\) 48.0000 1.74000 0.869999 0.493053i \(-0.164119\pi\)
0.869999 + 0.493053i \(0.164119\pi\)
\(762\) 0 0
\(763\) 20.0000 0.724049
\(764\) 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 \(-0.534413\pi\)
−0.107903 + 0.994161i \(0.534413\pi\)
\(774\) 0 0
\(775\) 20.0000 0.718421
\(776\) 0 0
\(777\) 11.0000 0.394623
\(778\) 0 0
\(779\) 12.0000 0.429945
\(780\) 24.0000 0.859338
\(781\) −9.00000 −0.322045
\(782\) 0 0
\(783\) 30.0000 1.07211
\(784\) 4.00000 0.142857
\(785\) −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
\(791\) −3.00000 −0.106668
\(792\) 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 \(-0.621304\pi\)
−0.371929 + 0.928261i \(0.621304\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) 0 0
\(803\) −2.00000 −0.0705785
\(804\) −10.0000 −0.352673
\(805\) 9.00000 0.317208
\(806\) 0 0
\(807\) 30.0000 1.05605
\(808\) 0 0
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) 2.00000 0.0702295 0.0351147 0.999383i \(-0.488820\pi\)
0.0351147 + 0.999383i \(0.488820\pi\)
\(812\) 12.0000 0.421117
\(813\) −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
\(819\) 8.00000 0.279543
\(820\) −36.0000 −1.25717
\(821\) 18.0000 0.628204 0.314102 0.949389i \(-0.398297\pi\)
0.314102 + 0.949389i \(0.398297\pi\)
\(822\) 0 0
\(823\) 23.0000 0.801730 0.400865 0.916137i \(-0.368710\pi\)
0.400865 + 0.916137i \(0.368710\pi\)
\(824\) 0 0
\(825\) −4.00000 −0.139262
\(826\) 0 0
\(827\) 36.0000 1.25184 0.625921 0.779886i \(-0.284723\pi\)
0.625921 + 0.779886i \(0.284723\pi\)
\(828\) 12.0000 0.417029
\(829\) −25.0000 −0.868286 −0.434143 0.900844i \(-0.642949\pi\)
−0.434143 + 0.900844i \(0.642949\pi\)
\(830\) 0 0
\(831\) 8.00000 0.277517
\(832\) 32.0000 1.10940
\(833\) −6.00000 −0.207888
\(834\) 0 0
\(835\) 18.0000 0.622916
\(836\) 4.00000 0.138343
\(837\) −25.0000 −0.864126
\(838\) 0 0
\(839\) −15.0000 −0.517858 −0.258929 0.965896i \(-0.583369\pi\)
−0.258929 + 0.965896i \(0.583369\pi\)
\(840\) 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
\(847\) 1.00000 0.0343604
\(848\) −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 \(-0.571186\pi\)
−0.221777 + 0.975097i \(0.571186\pi\)
\(860\) −48.0000 −1.63679
\(861\) 6.00000 0.204479
\(862\) 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
\(868\) −10.0000 −0.339422
\(869\) 10.0000 0.339227
\(870\) 0 0
\(871\) −20.0000 −0.677674
\(872\) 0 0
\(873\) 2.00000 0.0676897
\(874\) 0 0
\(875\) −3.00000 −0.101419
\(876\) −4.00000 −0.135147
\(877\) −22.0000 −0.742887 −0.371444 0.928456i \(-0.621137\pi\)
−0.371444 + 0.928456i \(0.621137\pi\)
\(878\) 0 0
\(879\) −30.0000 −1.01187
\(880\) −12.0000 −0.404520
\(881\) −9.00000 −0.303218 −0.151609 0.988441i \(-0.548445\pi\)
−0.151609 + 0.988441i \(0.548445\pi\)
\(882\) 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
\(889\) 2.00000 0.0670778
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 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
\(903\) 8.00000 0.266223
\(904\) 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
\(913\) −12.0000 −0.397142
\(914\) 0 0
\(915\) −30.0000 −0.991769
\(916\) −10.0000 −0.330409
\(917\) −6.00000 −0.198137
\(918\) 0 0
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 0 0
\(921\) 20.0000 0.659022
\(922\) 0 0
\(923\) −36.0000 −1.18495
\(924\) 2.00000 0.0657952
\(925\) 44.0000 1.44671
\(926\) 0 0
\(927\) 8.00000 0.262754
\(928\) 0 0
\(929\) 18.0000 0.590561 0.295280 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) 0 0
\(931\) 2.00000 0.0655474
\(932\) −12.0000 −0.393073
\(933\) 0 0
\(934\) 0 0
\(935\) 18.0000 0.588663
\(936\) 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
\(945\) −15.0000 −0.487950
\(946\) 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 \(-0.698151\pi\)
−0.583077 + 0.812417i \(0.698151\pi\)
\(954\) 0 0
\(955\) −81.0000 −2.62110
\(956\) 24.0000 0.776215
\(957\) 6.00000 0.193952
\(958\) 0 0
\(959\) −3.00000 −0.0968751
\(960\) −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 \(-0.427728\pi\)
0.225105 + 0.974335i \(0.427728\pi\)
\(968\) 0 0
\(969\) −12.0000 −0.385496
\(970\) 0 0
\(971\) −39.0000 −1.25157 −0.625785 0.779996i \(-0.715221\pi\)
−0.625785 + 0.779996i \(0.715221\pi\)
\(972\) −32.0000 −1.02640
\(973\) 14.0000 0.448819
\(974\) 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
\(979\) 3.00000 0.0958804
\(980\) −6.00000 −0.191663
\(981\) −40.0000 −1.27710
\(982\) 0 0
\(983\) 33.0000 1.05254 0.526268 0.850319i \(-0.323591\pi\)
0.526268 + 0.850319i \(0.323591\pi\)
\(984\) 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 77.2.a.b.1.1 1
3.2 odd 2 693.2.a.b.1.1 1
4.3 odd 2 1232.2.a.d.1.1 1
5.2 odd 4 1925.2.b.g.1849.1 2
5.3 odd 4 1925.2.b.g.1849.2 2
5.4 even 2 1925.2.a.f.1.1 1
7.2 even 3 539.2.e.d.67.1 2
7.3 odd 6 539.2.e.e.177.1 2
7.4 even 3 539.2.e.d.177.1 2
7.5 odd 6 539.2.e.e.67.1 2
7.6 odd 2 539.2.a.b.1.1 1
8.3 odd 2 4928.2.a.x.1.1 1
8.5 even 2 4928.2.a.i.1.1 1
11.2 odd 10 847.2.f.g.323.1 4
11.3 even 5 847.2.f.f.372.1 4
11.4 even 5 847.2.f.f.148.1 4
11.5 even 5 847.2.f.f.729.1 4
11.6 odd 10 847.2.f.g.729.1 4
11.7 odd 10 847.2.f.g.148.1 4
11.8 odd 10 847.2.f.g.372.1 4
11.9 even 5 847.2.f.f.323.1 4
11.10 odd 2 847.2.a.c.1.1 1
21.20 even 2 4851.2.a.k.1.1 1
28.27 even 2 8624.2.a.s.1.1 1
33.32 even 2 7623.2.a.i.1.1 1
77.76 even 2 5929.2.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
77.2.a.b.1.1 1 1.1 even 1 trivial
539.2.a.b.1.1 1 7.6 odd 2
539.2.e.d.67.1 2 7.2 even 3
539.2.e.d.177.1 2 7.4 even 3
539.2.e.e.67.1 2 7.5 odd 6
539.2.e.e.177.1 2 7.3 odd 6
693.2.a.b.1.1 1 3.2 odd 2
847.2.a.c.1.1 1 11.10 odd 2
847.2.f.f.148.1 4 11.4 even 5
847.2.f.f.323.1 4 11.9 even 5
847.2.f.f.372.1 4 11.3 even 5
847.2.f.f.729.1 4 11.5 even 5
847.2.f.g.148.1 4 11.7 odd 10
847.2.f.g.323.1 4 11.2 odd 10
847.2.f.g.372.1 4 11.8 odd 10
847.2.f.g.729.1 4 11.6 odd 10
1232.2.a.d.1.1 1 4.3 odd 2
1925.2.a.f.1.1 1 5.4 even 2
1925.2.b.g.1849.1 2 5.2 odd 4
1925.2.b.g.1849.2 2 5.3 odd 4
4851.2.a.k.1.1 1 21.20 even 2
4928.2.a.i.1.1 1 8.5 even 2
4928.2.a.x.1.1 1 8.3 odd 2
5929.2.a.d.1.1 1 77.76 even 2
7623.2.a.i.1.1 1 33.32 even 2
8624.2.a.s.1.1 1 28.27 even 2