Properties

Label 1710.2.a.g.1.1
Level $1710$
Weight $2$
Character 1710.1
Self dual yes
Analytic conductor $13.654$
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] = [1710,2,Mod(1,1710)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1710, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1710.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1710.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: \(13.6544187456\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 190)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1710.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -5.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -5.00000 q^{7} -1.00000 q^{8} -1.00000 q^{10} +4.00000 q^{11} -1.00000 q^{13} +5.00000 q^{14} +1.00000 q^{16} +3.00000 q^{17} +1.00000 q^{19} +1.00000 q^{20} -4.00000 q^{22} -7.00000 q^{23} +1.00000 q^{25} +1.00000 q^{26} -5.00000 q^{28} +3.00000 q^{29} -2.00000 q^{31} -1.00000 q^{32} -3.00000 q^{34} -5.00000 q^{35} -2.00000 q^{37} -1.00000 q^{38} -1.00000 q^{40} +6.00000 q^{41} +6.00000 q^{43} +4.00000 q^{44} +7.00000 q^{46} +18.0000 q^{49} -1.00000 q^{50} -1.00000 q^{52} +13.0000 q^{53} +4.00000 q^{55} +5.00000 q^{56} -3.00000 q^{58} +9.00000 q^{59} -12.0000 q^{61} +2.00000 q^{62} +1.00000 q^{64} -1.00000 q^{65} -3.00000 q^{67} +3.00000 q^{68} +5.00000 q^{70} +11.0000 q^{73} +2.00000 q^{74} +1.00000 q^{76} -20.0000 q^{77} -2.00000 q^{79} +1.00000 q^{80} -6.00000 q^{82} +10.0000 q^{83} +3.00000 q^{85} -6.00000 q^{86} -4.00000 q^{88} -2.00000 q^{89} +5.00000 q^{91} -7.00000 q^{92} +1.00000 q^{95} -2.00000 q^{97} -18.0000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −5.00000 −1.88982 −0.944911 0.327327i \(-0.893852\pi\)
−0.944911 + 0.327327i \(0.893852\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350 −0.138675 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) 5.00000 1.33631
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) −4.00000 −0.852803
\(23\) −7.00000 −1.45960 −0.729800 0.683660i \(-0.760387\pi\)
−0.729800 + 0.683660i \(0.760387\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 1.00000 0.196116
\(27\) 0 0
\(28\) −5.00000 −0.944911
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −3.00000 −0.514496
\(35\) −5.00000 −0.845154
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) −1.00000 −0.162221
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 6.00000 0.914991 0.457496 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) 7.00000 1.03209
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 18.0000 2.57143
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −1.00000 −0.138675
\(53\) 13.0000 1.78569 0.892844 0.450367i \(-0.148707\pi\)
0.892844 + 0.450367i \(0.148707\pi\)
\(54\) 0 0
\(55\) 4.00000 0.539360
\(56\) 5.00000 0.668153
\(57\) 0 0
\(58\) −3.00000 −0.393919
\(59\) 9.00000 1.17170 0.585850 0.810419i \(-0.300761\pi\)
0.585850 + 0.810419i \(0.300761\pi\)
\(60\) 0 0
\(61\) −12.0000 −1.53644 −0.768221 0.640184i \(-0.778858\pi\)
−0.768221 + 0.640184i \(0.778858\pi\)
\(62\) 2.00000 0.254000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) −3.00000 −0.366508 −0.183254 0.983066i \(-0.558663\pi\)
−0.183254 + 0.983066i \(0.558663\pi\)
\(68\) 3.00000 0.363803
\(69\) 0 0
\(70\) 5.00000 0.597614
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 11.0000 1.28745 0.643726 0.765256i \(-0.277388\pi\)
0.643726 + 0.765256i \(0.277388\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) −20.0000 −2.27921
\(78\) 0 0
\(79\) −2.00000 −0.225018 −0.112509 0.993651i \(-0.535889\pi\)
−0.112509 + 0.993651i \(0.535889\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) −6.00000 −0.662589
\(83\) 10.0000 1.09764 0.548821 0.835940i \(-0.315077\pi\)
0.548821 + 0.835940i \(0.315077\pi\)
\(84\) 0 0
\(85\) 3.00000 0.325396
\(86\) −6.00000 −0.646997
\(87\) 0 0
\(88\) −4.00000 −0.426401
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) 5.00000 0.524142
\(92\) −7.00000 −0.729800
\(93\) 0 0
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) −18.0000 −1.81827
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 8.00000 0.796030 0.398015 0.917379i \(-0.369699\pi\)
0.398015 + 0.917379i \(0.369699\pi\)
\(102\) 0 0
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) −13.0000 −1.26267
\(107\) 13.0000 1.25676 0.628379 0.777908i \(-0.283719\pi\)
0.628379 + 0.777908i \(0.283719\pi\)
\(108\) 0 0
\(109\) 19.0000 1.81987 0.909935 0.414751i \(-0.136131\pi\)
0.909935 + 0.414751i \(0.136131\pi\)
\(110\) −4.00000 −0.381385
\(111\) 0 0
\(112\) −5.00000 −0.472456
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) −7.00000 −0.652753
\(116\) 3.00000 0.278543
\(117\) 0 0
\(118\) −9.00000 −0.828517
\(119\) −15.0000 −1.37505
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 12.0000 1.08643
\(123\) 0 0
\(124\) −2.00000 −0.179605
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −6.00000 −0.532414 −0.266207 0.963916i \(-0.585770\pi\)
−0.266207 + 0.963916i \(0.585770\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 1.00000 0.0877058
\(131\) −16.0000 −1.39793 −0.698963 0.715158i \(-0.746355\pi\)
−0.698963 + 0.715158i \(0.746355\pi\)
\(132\) 0 0
\(133\) −5.00000 −0.433555
\(134\) 3.00000 0.259161
\(135\) 0 0
\(136\) −3.00000 −0.257248
\(137\) −9.00000 −0.768922 −0.384461 0.923141i \(-0.625613\pi\)
−0.384461 + 0.923141i \(0.625613\pi\)
\(138\) 0 0
\(139\) 16.0000 1.35710 0.678551 0.734553i \(-0.262608\pi\)
0.678551 + 0.734553i \(0.262608\pi\)
\(140\) −5.00000 −0.422577
\(141\) 0 0
\(142\) 0 0
\(143\) −4.00000 −0.334497
\(144\) 0 0
\(145\) 3.00000 0.249136
\(146\) −11.0000 −0.910366
\(147\) 0 0
\(148\) −2.00000 −0.164399
\(149\) 4.00000 0.327693 0.163846 0.986486i \(-0.447610\pi\)
0.163846 + 0.986486i \(0.447610\pi\)
\(150\) 0 0
\(151\) −10.0000 −0.813788 −0.406894 0.913475i \(-0.633388\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 0 0
\(154\) 20.0000 1.61165
\(155\) −2.00000 −0.160644
\(156\) 0 0
\(157\) 6.00000 0.478852 0.239426 0.970915i \(-0.423041\pi\)
0.239426 + 0.970915i \(0.423041\pi\)
\(158\) 2.00000 0.159111
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) 35.0000 2.75839
\(162\) 0 0
\(163\) 22.0000 1.72317 0.861586 0.507611i \(-0.169471\pi\)
0.861586 + 0.507611i \(0.169471\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) −10.0000 −0.776151
\(167\) 2.00000 0.154765 0.0773823 0.997001i \(-0.475344\pi\)
0.0773823 + 0.997001i \(0.475344\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) −3.00000 −0.230089
\(171\) 0 0
\(172\) 6.00000 0.457496
\(173\) 14.0000 1.06440 0.532200 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) 0 0
\(175\) −5.00000 −0.377964
\(176\) 4.00000 0.301511
\(177\) 0 0
\(178\) 2.00000 0.149906
\(179\) 8.00000 0.597948 0.298974 0.954261i \(-0.403356\pi\)
0.298974 + 0.954261i \(0.403356\pi\)
\(180\) 0 0
\(181\) 26.0000 1.93256 0.966282 0.257485i \(-0.0828937\pi\)
0.966282 + 0.257485i \(0.0828937\pi\)
\(182\) −5.00000 −0.370625
\(183\) 0 0
\(184\) 7.00000 0.516047
\(185\) −2.00000 −0.147043
\(186\) 0 0
\(187\) 12.0000 0.877527
\(188\) 0 0
\(189\) 0 0
\(190\) −1.00000 −0.0725476
\(191\) −9.00000 −0.651217 −0.325609 0.945505i \(-0.605569\pi\)
−0.325609 + 0.945505i \(0.605569\pi\)
\(192\) 0 0
\(193\) 10.0000 0.719816 0.359908 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) 18.0000 1.28571
\(197\) 22.0000 1.56744 0.783718 0.621117i \(-0.213321\pi\)
0.783718 + 0.621117i \(0.213321\pi\)
\(198\) 0 0
\(199\) −15.0000 −1.06332 −0.531661 0.846957i \(-0.678432\pi\)
−0.531661 + 0.846957i \(0.678432\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) −8.00000 −0.562878
\(203\) −15.0000 −1.05279
\(204\) 0 0
\(205\) 6.00000 0.419058
\(206\) −4.00000 −0.278693
\(207\) 0 0
\(208\) −1.00000 −0.0693375
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) −5.00000 −0.344214 −0.172107 0.985078i \(-0.555058\pi\)
−0.172107 + 0.985078i \(0.555058\pi\)
\(212\) 13.0000 0.892844
\(213\) 0 0
\(214\) −13.0000 −0.888662
\(215\) 6.00000 0.409197
\(216\) 0 0
\(217\) 10.0000 0.678844
\(218\) −19.0000 −1.28684
\(219\) 0 0
\(220\) 4.00000 0.269680
\(221\) −3.00000 −0.201802
\(222\) 0 0
\(223\) −2.00000 −0.133930 −0.0669650 0.997755i \(-0.521332\pi\)
−0.0669650 + 0.997755i \(0.521332\pi\)
\(224\) 5.00000 0.334077
\(225\) 0 0
\(226\) 0 0
\(227\) −5.00000 −0.331862 −0.165931 0.986137i \(-0.553063\pi\)
−0.165931 + 0.986137i \(0.553063\pi\)
\(228\) 0 0
\(229\) −6.00000 −0.396491 −0.198246 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) 7.00000 0.461566
\(231\) 0 0
\(232\) −3.00000 −0.196960
\(233\) −10.0000 −0.655122 −0.327561 0.944830i \(-0.606227\pi\)
−0.327561 + 0.944830i \(0.606227\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 9.00000 0.585850
\(237\) 0 0
\(238\) 15.0000 0.972306
\(239\) 11.0000 0.711531 0.355765 0.934575i \(-0.384220\pi\)
0.355765 + 0.934575i \(0.384220\pi\)
\(240\) 0 0
\(241\) −12.0000 −0.772988 −0.386494 0.922292i \(-0.626314\pi\)
−0.386494 + 0.922292i \(0.626314\pi\)
\(242\) −5.00000 −0.321412
\(243\) 0 0
\(244\) −12.0000 −0.768221
\(245\) 18.0000 1.14998
\(246\) 0 0
\(247\) −1.00000 −0.0636285
\(248\) 2.00000 0.127000
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) −28.0000 −1.76034
\(254\) 6.00000 0.376473
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −22.0000 −1.37232 −0.686161 0.727450i \(-0.740706\pi\)
−0.686161 + 0.727450i \(0.740706\pi\)
\(258\) 0 0
\(259\) 10.0000 0.621370
\(260\) −1.00000 −0.0620174
\(261\) 0 0
\(262\) 16.0000 0.988483
\(263\) −8.00000 −0.493301 −0.246651 0.969104i \(-0.579330\pi\)
−0.246651 + 0.969104i \(0.579330\pi\)
\(264\) 0 0
\(265\) 13.0000 0.798584
\(266\) 5.00000 0.306570
\(267\) 0 0
\(268\) −3.00000 −0.183254
\(269\) −2.00000 −0.121942 −0.0609711 0.998140i \(-0.519420\pi\)
−0.0609711 + 0.998140i \(0.519420\pi\)
\(270\) 0 0
\(271\) −27.0000 −1.64013 −0.820067 0.572268i \(-0.806064\pi\)
−0.820067 + 0.572268i \(0.806064\pi\)
\(272\) 3.00000 0.181902
\(273\) 0 0
\(274\) 9.00000 0.543710
\(275\) 4.00000 0.241209
\(276\) 0 0
\(277\) −8.00000 −0.480673 −0.240337 0.970690i \(-0.577258\pi\)
−0.240337 + 0.970690i \(0.577258\pi\)
\(278\) −16.0000 −0.959616
\(279\) 0 0
\(280\) 5.00000 0.298807
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 0 0
\(283\) −2.00000 −0.118888 −0.0594438 0.998232i \(-0.518933\pi\)
−0.0594438 + 0.998232i \(0.518933\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 4.00000 0.236525
\(287\) −30.0000 −1.77084
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) −3.00000 −0.176166
\(291\) 0 0
\(292\) 11.0000 0.643726
\(293\) 27.0000 1.57736 0.788678 0.614806i \(-0.210766\pi\)
0.788678 + 0.614806i \(0.210766\pi\)
\(294\) 0 0
\(295\) 9.00000 0.524000
\(296\) 2.00000 0.116248
\(297\) 0 0
\(298\) −4.00000 −0.231714
\(299\) 7.00000 0.404820
\(300\) 0 0
\(301\) −30.0000 −1.72917
\(302\) 10.0000 0.575435
\(303\) 0 0
\(304\) 1.00000 0.0573539
\(305\) −12.0000 −0.687118
\(306\) 0 0
\(307\) 4.00000 0.228292 0.114146 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(308\) −20.0000 −1.13961
\(309\) 0 0
\(310\) 2.00000 0.113592
\(311\) −25.0000 −1.41762 −0.708810 0.705399i \(-0.750768\pi\)
−0.708810 + 0.705399i \(0.750768\pi\)
\(312\) 0 0
\(313\) −1.00000 −0.0565233 −0.0282617 0.999601i \(-0.508997\pi\)
−0.0282617 + 0.999601i \(0.508997\pi\)
\(314\) −6.00000 −0.338600
\(315\) 0 0
\(316\) −2.00000 −0.112509
\(317\) −9.00000 −0.505490 −0.252745 0.967533i \(-0.581333\pi\)
−0.252745 + 0.967533i \(0.581333\pi\)
\(318\) 0 0
\(319\) 12.0000 0.671871
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) −35.0000 −1.95047
\(323\) 3.00000 0.166924
\(324\) 0 0
\(325\) −1.00000 −0.0554700
\(326\) −22.0000 −1.21847
\(327\) 0 0
\(328\) −6.00000 −0.331295
\(329\) 0 0
\(330\) 0 0
\(331\) −7.00000 −0.384755 −0.192377 0.981321i \(-0.561620\pi\)
−0.192377 + 0.981321i \(0.561620\pi\)
\(332\) 10.0000 0.548821
\(333\) 0 0
\(334\) −2.00000 −0.109435
\(335\) −3.00000 −0.163908
\(336\) 0 0
\(337\) 6.00000 0.326841 0.163420 0.986557i \(-0.447747\pi\)
0.163420 + 0.986557i \(0.447747\pi\)
\(338\) 12.0000 0.652714
\(339\) 0 0
\(340\) 3.00000 0.162698
\(341\) −8.00000 −0.433224
\(342\) 0 0
\(343\) −55.0000 −2.96972
\(344\) −6.00000 −0.323498
\(345\) 0 0
\(346\) −14.0000 −0.752645
\(347\) 6.00000 0.322097 0.161048 0.986947i \(-0.448512\pi\)
0.161048 + 0.986947i \(0.448512\pi\)
\(348\) 0 0
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 5.00000 0.267261
\(351\) 0 0
\(352\) −4.00000 −0.213201
\(353\) −7.00000 −0.372572 −0.186286 0.982496i \(-0.559645\pi\)
−0.186286 + 0.982496i \(0.559645\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −2.00000 −0.106000
\(357\) 0 0
\(358\) −8.00000 −0.422813
\(359\) 5.00000 0.263890 0.131945 0.991257i \(-0.457878\pi\)
0.131945 + 0.991257i \(0.457878\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −26.0000 −1.36653
\(363\) 0 0
\(364\) 5.00000 0.262071
\(365\) 11.0000 0.575766
\(366\) 0 0
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) −7.00000 −0.364900
\(369\) 0 0
\(370\) 2.00000 0.103975
\(371\) −65.0000 −3.37463
\(372\) 0 0
\(373\) −23.0000 −1.19089 −0.595447 0.803394i \(-0.703025\pi\)
−0.595447 + 0.803394i \(0.703025\pi\)
\(374\) −12.0000 −0.620505
\(375\) 0 0
\(376\) 0 0
\(377\) −3.00000 −0.154508
\(378\) 0 0
\(379\) −33.0000 −1.69510 −0.847548 0.530719i \(-0.821922\pi\)
−0.847548 + 0.530719i \(0.821922\pi\)
\(380\) 1.00000 0.0512989
\(381\) 0 0
\(382\) 9.00000 0.460480
\(383\) 4.00000 0.204390 0.102195 0.994764i \(-0.467413\pi\)
0.102195 + 0.994764i \(0.467413\pi\)
\(384\) 0 0
\(385\) −20.0000 −1.01929
\(386\) −10.0000 −0.508987
\(387\) 0 0
\(388\) −2.00000 −0.101535
\(389\) 4.00000 0.202808 0.101404 0.994845i \(-0.467667\pi\)
0.101404 + 0.994845i \(0.467667\pi\)
\(390\) 0 0
\(391\) −21.0000 −1.06202
\(392\) −18.0000 −0.909137
\(393\) 0 0
\(394\) −22.0000 −1.10834
\(395\) −2.00000 −0.100631
\(396\) 0 0
\(397\) −16.0000 −0.803017 −0.401508 0.915855i \(-0.631514\pi\)
−0.401508 + 0.915855i \(0.631514\pi\)
\(398\) 15.0000 0.751882
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 6.00000 0.299626 0.149813 0.988714i \(-0.452133\pi\)
0.149813 + 0.988714i \(0.452133\pi\)
\(402\) 0 0
\(403\) 2.00000 0.0996271
\(404\) 8.00000 0.398015
\(405\) 0 0
\(406\) 15.0000 0.744438
\(407\) −8.00000 −0.396545
\(408\) 0 0
\(409\) 22.0000 1.08783 0.543915 0.839140i \(-0.316941\pi\)
0.543915 + 0.839140i \(0.316941\pi\)
\(410\) −6.00000 −0.296319
\(411\) 0 0
\(412\) 4.00000 0.197066
\(413\) −45.0000 −2.21431
\(414\) 0 0
\(415\) 10.0000 0.490881
\(416\) 1.00000 0.0490290
\(417\) 0 0
\(418\) −4.00000 −0.195646
\(419\) −14.0000 −0.683945 −0.341972 0.939710i \(-0.611095\pi\)
−0.341972 + 0.939710i \(0.611095\pi\)
\(420\) 0 0
\(421\) 1.00000 0.0487370 0.0243685 0.999703i \(-0.492242\pi\)
0.0243685 + 0.999703i \(0.492242\pi\)
\(422\) 5.00000 0.243396
\(423\) 0 0
\(424\) −13.0000 −0.631336
\(425\) 3.00000 0.145521
\(426\) 0 0
\(427\) 60.0000 2.90360
\(428\) 13.0000 0.628379
\(429\) 0 0
\(430\) −6.00000 −0.289346
\(431\) 36.0000 1.73406 0.867029 0.498257i \(-0.166026\pi\)
0.867029 + 0.498257i \(0.166026\pi\)
\(432\) 0 0
\(433\) −16.0000 −0.768911 −0.384455 0.923144i \(-0.625611\pi\)
−0.384455 + 0.923144i \(0.625611\pi\)
\(434\) −10.0000 −0.480015
\(435\) 0 0
\(436\) 19.0000 0.909935
\(437\) −7.00000 −0.334855
\(438\) 0 0
\(439\) 26.0000 1.24091 0.620456 0.784241i \(-0.286947\pi\)
0.620456 + 0.784241i \(0.286947\pi\)
\(440\) −4.00000 −0.190693
\(441\) 0 0
\(442\) 3.00000 0.142695
\(443\) −36.0000 −1.71041 −0.855206 0.518289i \(-0.826569\pi\)
−0.855206 + 0.518289i \(0.826569\pi\)
\(444\) 0 0
\(445\) −2.00000 −0.0948091
\(446\) 2.00000 0.0947027
\(447\) 0 0
\(448\) −5.00000 −0.236228
\(449\) 22.0000 1.03824 0.519122 0.854700i \(-0.326259\pi\)
0.519122 + 0.854700i \(0.326259\pi\)
\(450\) 0 0
\(451\) 24.0000 1.13012
\(452\) 0 0
\(453\) 0 0
\(454\) 5.00000 0.234662
\(455\) 5.00000 0.234404
\(456\) 0 0
\(457\) −29.0000 −1.35656 −0.678281 0.734802i \(-0.737275\pi\)
−0.678281 + 0.734802i \(0.737275\pi\)
\(458\) 6.00000 0.280362
\(459\) 0 0
\(460\) −7.00000 −0.326377
\(461\) −18.0000 −0.838344 −0.419172 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) 0 0
\(463\) −8.00000 −0.371792 −0.185896 0.982569i \(-0.559519\pi\)
−0.185896 + 0.982569i \(0.559519\pi\)
\(464\) 3.00000 0.139272
\(465\) 0 0
\(466\) 10.0000 0.463241
\(467\) 8.00000 0.370196 0.185098 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) 0 0
\(469\) 15.0000 0.692636
\(470\) 0 0
\(471\) 0 0
\(472\) −9.00000 −0.414259
\(473\) 24.0000 1.10352
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) −15.0000 −0.687524
\(477\) 0 0
\(478\) −11.0000 −0.503128
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) 12.0000 0.546585
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) −2.00000 −0.0908153
\(486\) 0 0
\(487\) −38.0000 −1.72194 −0.860972 0.508652i \(-0.830144\pi\)
−0.860972 + 0.508652i \(0.830144\pi\)
\(488\) 12.0000 0.543214
\(489\) 0 0
\(490\) −18.0000 −0.813157
\(491\) −18.0000 −0.812329 −0.406164 0.913800i \(-0.633134\pi\)
−0.406164 + 0.913800i \(0.633134\pi\)
\(492\) 0 0
\(493\) 9.00000 0.405340
\(494\) 1.00000 0.0449921
\(495\) 0 0
\(496\) −2.00000 −0.0898027
\(497\) 0 0
\(498\) 0 0
\(499\) 42.0000 1.88018 0.940089 0.340929i \(-0.110742\pi\)
0.940089 + 0.340929i \(0.110742\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) −12.0000 −0.535586
\(503\) 21.0000 0.936344 0.468172 0.883637i \(-0.344913\pi\)
0.468172 + 0.883637i \(0.344913\pi\)
\(504\) 0 0
\(505\) 8.00000 0.355995
\(506\) 28.0000 1.24475
\(507\) 0 0
\(508\) −6.00000 −0.266207
\(509\) −2.00000 −0.0886484 −0.0443242 0.999017i \(-0.514113\pi\)
−0.0443242 + 0.999017i \(0.514113\pi\)
\(510\) 0 0
\(511\) −55.0000 −2.43306
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 22.0000 0.970378
\(515\) 4.00000 0.176261
\(516\) 0 0
\(517\) 0 0
\(518\) −10.0000 −0.439375
\(519\) 0 0
\(520\) 1.00000 0.0438529
\(521\) −24.0000 −1.05146 −0.525730 0.850652i \(-0.676208\pi\)
−0.525730 + 0.850652i \(0.676208\pi\)
\(522\) 0 0
\(523\) −9.00000 −0.393543 −0.196771 0.980449i \(-0.563046\pi\)
−0.196771 + 0.980449i \(0.563046\pi\)
\(524\) −16.0000 −0.698963
\(525\) 0 0
\(526\) 8.00000 0.348817
\(527\) −6.00000 −0.261364
\(528\) 0 0
\(529\) 26.0000 1.13043
\(530\) −13.0000 −0.564684
\(531\) 0 0
\(532\) −5.00000 −0.216777
\(533\) −6.00000 −0.259889
\(534\) 0 0
\(535\) 13.0000 0.562039
\(536\) 3.00000 0.129580
\(537\) 0 0
\(538\) 2.00000 0.0862261
\(539\) 72.0000 3.10126
\(540\) 0 0
\(541\) 16.0000 0.687894 0.343947 0.938989i \(-0.388236\pi\)
0.343947 + 0.938989i \(0.388236\pi\)
\(542\) 27.0000 1.15975
\(543\) 0 0
\(544\) −3.00000 −0.128624
\(545\) 19.0000 0.813871
\(546\) 0 0
\(547\) 20.0000 0.855138 0.427569 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) −9.00000 −0.384461
\(549\) 0 0
\(550\) −4.00000 −0.170561
\(551\) 3.00000 0.127804
\(552\) 0 0
\(553\) 10.0000 0.425243
\(554\) 8.00000 0.339887
\(555\) 0 0
\(556\) 16.0000 0.678551
\(557\) −12.0000 −0.508456 −0.254228 0.967144i \(-0.581821\pi\)
−0.254228 + 0.967144i \(0.581821\pi\)
\(558\) 0 0
\(559\) −6.00000 −0.253773
\(560\) −5.00000 −0.211289
\(561\) 0 0
\(562\) −18.0000 −0.759284
\(563\) −20.0000 −0.842900 −0.421450 0.906852i \(-0.638479\pi\)
−0.421450 + 0.906852i \(0.638479\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 2.00000 0.0840663
\(567\) 0 0
\(568\) 0 0
\(569\) −24.0000 −1.00613 −0.503066 0.864248i \(-0.667795\pi\)
−0.503066 + 0.864248i \(0.667795\pi\)
\(570\) 0 0
\(571\) 6.00000 0.251092 0.125546 0.992088i \(-0.459932\pi\)
0.125546 + 0.992088i \(0.459932\pi\)
\(572\) −4.00000 −0.167248
\(573\) 0 0
\(574\) 30.0000 1.25218
\(575\) −7.00000 −0.291920
\(576\) 0 0
\(577\) −7.00000 −0.291414 −0.145707 0.989328i \(-0.546546\pi\)
−0.145707 + 0.989328i \(0.546546\pi\)
\(578\) 8.00000 0.332756
\(579\) 0 0
\(580\) 3.00000 0.124568
\(581\) −50.0000 −2.07435
\(582\) 0 0
\(583\) 52.0000 2.15362
\(584\) −11.0000 −0.455183
\(585\) 0 0
\(586\) −27.0000 −1.11536
\(587\) −18.0000 −0.742940 −0.371470 0.928445i \(-0.621146\pi\)
−0.371470 + 0.928445i \(0.621146\pi\)
\(588\) 0 0
\(589\) −2.00000 −0.0824086
\(590\) −9.00000 −0.370524
\(591\) 0 0
\(592\) −2.00000 −0.0821995
\(593\) −30.0000 −1.23195 −0.615976 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) 0 0
\(595\) −15.0000 −0.614940
\(596\) 4.00000 0.163846
\(597\) 0 0
\(598\) −7.00000 −0.286251
\(599\) 26.0000 1.06233 0.531166 0.847268i \(-0.321754\pi\)
0.531166 + 0.847268i \(0.321754\pi\)
\(600\) 0 0
\(601\) 42.0000 1.71322 0.856608 0.515968i \(-0.172568\pi\)
0.856608 + 0.515968i \(0.172568\pi\)
\(602\) 30.0000 1.22271
\(603\) 0 0
\(604\) −10.0000 −0.406894
\(605\) 5.00000 0.203279
\(606\) 0 0
\(607\) −26.0000 −1.05531 −0.527654 0.849460i \(-0.676928\pi\)
−0.527654 + 0.849460i \(0.676928\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 0 0
\(610\) 12.0000 0.485866
\(611\) 0 0
\(612\) 0 0
\(613\) −20.0000 −0.807792 −0.403896 0.914805i \(-0.632344\pi\)
−0.403896 + 0.914805i \(0.632344\pi\)
\(614\) −4.00000 −0.161427
\(615\) 0 0
\(616\) 20.0000 0.805823
\(617\) −14.0000 −0.563619 −0.281809 0.959470i \(-0.590935\pi\)
−0.281809 + 0.959470i \(0.590935\pi\)
\(618\) 0 0
\(619\) −24.0000 −0.964641 −0.482321 0.875995i \(-0.660206\pi\)
−0.482321 + 0.875995i \(0.660206\pi\)
\(620\) −2.00000 −0.0803219
\(621\) 0 0
\(622\) 25.0000 1.00241
\(623\) 10.0000 0.400642
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 1.00000 0.0399680
\(627\) 0 0
\(628\) 6.00000 0.239426
\(629\) −6.00000 −0.239236
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 2.00000 0.0795557
\(633\) 0 0
\(634\) 9.00000 0.357436
\(635\) −6.00000 −0.238103
\(636\) 0 0
\(637\) −18.0000 −0.713186
\(638\) −12.0000 −0.475085
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) −8.00000 −0.315981 −0.157991 0.987441i \(-0.550502\pi\)
−0.157991 + 0.987441i \(0.550502\pi\)
\(642\) 0 0
\(643\) 26.0000 1.02534 0.512670 0.858586i \(-0.328656\pi\)
0.512670 + 0.858586i \(0.328656\pi\)
\(644\) 35.0000 1.37919
\(645\) 0 0
\(646\) −3.00000 −0.118033
\(647\) 21.0000 0.825595 0.412798 0.910823i \(-0.364552\pi\)
0.412798 + 0.910823i \(0.364552\pi\)
\(648\) 0 0
\(649\) 36.0000 1.41312
\(650\) 1.00000 0.0392232
\(651\) 0 0
\(652\) 22.0000 0.861586
\(653\) 16.0000 0.626128 0.313064 0.949732i \(-0.398644\pi\)
0.313064 + 0.949732i \(0.398644\pi\)
\(654\) 0 0
\(655\) −16.0000 −0.625172
\(656\) 6.00000 0.234261
\(657\) 0 0
\(658\) 0 0
\(659\) −33.0000 −1.28550 −0.642749 0.766077i \(-0.722206\pi\)
−0.642749 + 0.766077i \(0.722206\pi\)
\(660\) 0 0
\(661\) 15.0000 0.583432 0.291716 0.956505i \(-0.405774\pi\)
0.291716 + 0.956505i \(0.405774\pi\)
\(662\) 7.00000 0.272063
\(663\) 0 0
\(664\) −10.0000 −0.388075
\(665\) −5.00000 −0.193892
\(666\) 0 0
\(667\) −21.0000 −0.813123
\(668\) 2.00000 0.0773823
\(669\) 0 0
\(670\) 3.00000 0.115900
\(671\) −48.0000 −1.85302
\(672\) 0 0
\(673\) −44.0000 −1.69608 −0.848038 0.529936i \(-0.822216\pi\)
−0.848038 + 0.529936i \(0.822216\pi\)
\(674\) −6.00000 −0.231111
\(675\) 0 0
\(676\) −12.0000 −0.461538
\(677\) 39.0000 1.49889 0.749446 0.662066i \(-0.230320\pi\)
0.749446 + 0.662066i \(0.230320\pi\)
\(678\) 0 0
\(679\) 10.0000 0.383765
\(680\) −3.00000 −0.115045
\(681\) 0 0
\(682\) 8.00000 0.306336
\(683\) 44.0000 1.68361 0.841807 0.539779i \(-0.181492\pi\)
0.841807 + 0.539779i \(0.181492\pi\)
\(684\) 0 0
\(685\) −9.00000 −0.343872
\(686\) 55.0000 2.09991
\(687\) 0 0
\(688\) 6.00000 0.228748
\(689\) −13.0000 −0.495261
\(690\) 0 0
\(691\) −42.0000 −1.59776 −0.798878 0.601494i \(-0.794573\pi\)
−0.798878 + 0.601494i \(0.794573\pi\)
\(692\) 14.0000 0.532200
\(693\) 0 0
\(694\) −6.00000 −0.227757
\(695\) 16.0000 0.606915
\(696\) 0 0
\(697\) 18.0000 0.681799
\(698\) −14.0000 −0.529908
\(699\) 0 0
\(700\) −5.00000 −0.188982
\(701\) −24.0000 −0.906467 −0.453234 0.891392i \(-0.649730\pi\)
−0.453234 + 0.891392i \(0.649730\pi\)
\(702\) 0 0
\(703\) −2.00000 −0.0754314
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) 7.00000 0.263448
\(707\) −40.0000 −1.50435
\(708\) 0 0
\(709\) 2.00000 0.0751116 0.0375558 0.999295i \(-0.488043\pi\)
0.0375558 + 0.999295i \(0.488043\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 2.00000 0.0749532
\(713\) 14.0000 0.524304
\(714\) 0 0
\(715\) −4.00000 −0.149592
\(716\) 8.00000 0.298974
\(717\) 0 0
\(718\) −5.00000 −0.186598
\(719\) 27.0000 1.00693 0.503465 0.864016i \(-0.332058\pi\)
0.503465 + 0.864016i \(0.332058\pi\)
\(720\) 0 0
\(721\) −20.0000 −0.744839
\(722\) −1.00000 −0.0372161
\(723\) 0 0
\(724\) 26.0000 0.966282
\(725\) 3.00000 0.111417
\(726\) 0 0
\(727\) 23.0000 0.853023 0.426511 0.904482i \(-0.359742\pi\)
0.426511 + 0.904482i \(0.359742\pi\)
\(728\) −5.00000 −0.185312
\(729\) 0 0
\(730\) −11.0000 −0.407128
\(731\) 18.0000 0.665754
\(732\) 0 0
\(733\) 36.0000 1.32969 0.664845 0.746981i \(-0.268498\pi\)
0.664845 + 0.746981i \(0.268498\pi\)
\(734\) −8.00000 −0.295285
\(735\) 0 0
\(736\) 7.00000 0.258023
\(737\) −12.0000 −0.442026
\(738\) 0 0
\(739\) −10.0000 −0.367856 −0.183928 0.982940i \(-0.558881\pi\)
−0.183928 + 0.982940i \(0.558881\pi\)
\(740\) −2.00000 −0.0735215
\(741\) 0 0
\(742\) 65.0000 2.38623
\(743\) 18.0000 0.660356 0.330178 0.943919i \(-0.392891\pi\)
0.330178 + 0.943919i \(0.392891\pi\)
\(744\) 0 0
\(745\) 4.00000 0.146549
\(746\) 23.0000 0.842090
\(747\) 0 0
\(748\) 12.0000 0.438763
\(749\) −65.0000 −2.37505
\(750\) 0 0
\(751\) −26.0000 −0.948753 −0.474377 0.880322i \(-0.657327\pi\)
−0.474377 + 0.880322i \(0.657327\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 3.00000 0.109254
\(755\) −10.0000 −0.363937
\(756\) 0 0
\(757\) −6.00000 −0.218074 −0.109037 0.994038i \(-0.534777\pi\)
−0.109037 + 0.994038i \(0.534777\pi\)
\(758\) 33.0000 1.19861
\(759\) 0 0
\(760\) −1.00000 −0.0362738
\(761\) −11.0000 −0.398750 −0.199375 0.979923i \(-0.563891\pi\)
−0.199375 + 0.979923i \(0.563891\pi\)
\(762\) 0 0
\(763\) −95.0000 −3.43923
\(764\) −9.00000 −0.325609
\(765\) 0 0
\(766\) −4.00000 −0.144526
\(767\) −9.00000 −0.324971
\(768\) 0 0
\(769\) −47.0000 −1.69486 −0.847432 0.530904i \(-0.821852\pi\)
−0.847432 + 0.530904i \(0.821852\pi\)
\(770\) 20.0000 0.720750
\(771\) 0 0
\(772\) 10.0000 0.359908
\(773\) 51.0000 1.83434 0.917171 0.398493i \(-0.130467\pi\)
0.917171 + 0.398493i \(0.130467\pi\)
\(774\) 0 0
\(775\) −2.00000 −0.0718421
\(776\) 2.00000 0.0717958
\(777\) 0 0
\(778\) −4.00000 −0.143407
\(779\) 6.00000 0.214972
\(780\) 0 0
\(781\) 0 0
\(782\) 21.0000 0.750958
\(783\) 0 0
\(784\) 18.0000 0.642857
\(785\) 6.00000 0.214149
\(786\) 0 0
\(787\) −39.0000 −1.39020 −0.695100 0.718913i \(-0.744640\pi\)
−0.695100 + 0.718913i \(0.744640\pi\)
\(788\) 22.0000 0.783718
\(789\) 0 0
\(790\) 2.00000 0.0711568
\(791\) 0 0
\(792\) 0 0
\(793\) 12.0000 0.426132
\(794\) 16.0000 0.567819
\(795\) 0 0
\(796\) −15.0000 −0.531661
\(797\) −31.0000 −1.09808 −0.549038 0.835797i \(-0.685006\pi\)
−0.549038 + 0.835797i \(0.685006\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) −6.00000 −0.211867
\(803\) 44.0000 1.55273
\(804\) 0 0
\(805\) 35.0000 1.23359
\(806\) −2.00000 −0.0704470
\(807\) 0 0
\(808\) −8.00000 −0.281439
\(809\) −25.0000 −0.878953 −0.439477 0.898254i \(-0.644836\pi\)
−0.439477 + 0.898254i \(0.644836\pi\)
\(810\) 0 0
\(811\) 37.0000 1.29925 0.649623 0.760257i \(-0.274927\pi\)
0.649623 + 0.760257i \(0.274927\pi\)
\(812\) −15.0000 −0.526397
\(813\) 0 0
\(814\) 8.00000 0.280400
\(815\) 22.0000 0.770626
\(816\) 0 0
\(817\) 6.00000 0.209913
\(818\) −22.0000 −0.769212
\(819\) 0 0
\(820\) 6.00000 0.209529
\(821\) 52.0000 1.81481 0.907406 0.420255i \(-0.138059\pi\)
0.907406 + 0.420255i \(0.138059\pi\)
\(822\) 0 0
\(823\) −43.0000 −1.49889 −0.749443 0.662069i \(-0.769679\pi\)
−0.749443 + 0.662069i \(0.769679\pi\)
\(824\) −4.00000 −0.139347
\(825\) 0 0
\(826\) 45.0000 1.56575
\(827\) 3.00000 0.104320 0.0521601 0.998639i \(-0.483389\pi\)
0.0521601 + 0.998639i \(0.483389\pi\)
\(828\) 0 0
\(829\) 35.0000 1.21560 0.607800 0.794090i \(-0.292052\pi\)
0.607800 + 0.794090i \(0.292052\pi\)
\(830\) −10.0000 −0.347105
\(831\) 0 0
\(832\) −1.00000 −0.0346688
\(833\) 54.0000 1.87099
\(834\) 0 0
\(835\) 2.00000 0.0692129
\(836\) 4.00000 0.138343
\(837\) 0 0
\(838\) 14.0000 0.483622
\(839\) 4.00000 0.138095 0.0690477 0.997613i \(-0.478004\pi\)
0.0690477 + 0.997613i \(0.478004\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) −1.00000 −0.0344623
\(843\) 0 0
\(844\) −5.00000 −0.172107
\(845\) −12.0000 −0.412813
\(846\) 0 0
\(847\) −25.0000 −0.859010
\(848\) 13.0000 0.446422
\(849\) 0 0
\(850\) −3.00000 −0.102899
\(851\) 14.0000 0.479914
\(852\) 0 0
\(853\) 42.0000 1.43805 0.719026 0.694983i \(-0.244588\pi\)
0.719026 + 0.694983i \(0.244588\pi\)
\(854\) −60.0000 −2.05316
\(855\) 0 0
\(856\) −13.0000 −0.444331
\(857\) 40.0000 1.36637 0.683187 0.730243i \(-0.260593\pi\)
0.683187 + 0.730243i \(0.260593\pi\)
\(858\) 0 0
\(859\) −28.0000 −0.955348 −0.477674 0.878537i \(-0.658520\pi\)
−0.477674 + 0.878537i \(0.658520\pi\)
\(860\) 6.00000 0.204598
\(861\) 0 0
\(862\) −36.0000 −1.22616
\(863\) −56.0000 −1.90626 −0.953131 0.302558i \(-0.902160\pi\)
−0.953131 + 0.302558i \(0.902160\pi\)
\(864\) 0 0
\(865\) 14.0000 0.476014
\(866\) 16.0000 0.543702
\(867\) 0 0
\(868\) 10.0000 0.339422
\(869\) −8.00000 −0.271381
\(870\) 0 0
\(871\) 3.00000 0.101651
\(872\) −19.0000 −0.643421
\(873\) 0 0
\(874\) 7.00000 0.236779
\(875\) −5.00000 −0.169031
\(876\) 0 0
\(877\) 33.0000 1.11433 0.557165 0.830402i \(-0.311889\pi\)
0.557165 + 0.830402i \(0.311889\pi\)
\(878\) −26.0000 −0.877457
\(879\) 0 0
\(880\) 4.00000 0.134840
\(881\) −10.0000 −0.336909 −0.168454 0.985709i \(-0.553878\pi\)
−0.168454 + 0.985709i \(0.553878\pi\)
\(882\) 0 0
\(883\) 30.0000 1.00958 0.504790 0.863242i \(-0.331570\pi\)
0.504790 + 0.863242i \(0.331570\pi\)
\(884\) −3.00000 −0.100901
\(885\) 0 0
\(886\) 36.0000 1.20944
\(887\) 28.0000 0.940148 0.470074 0.882627i \(-0.344227\pi\)
0.470074 + 0.882627i \(0.344227\pi\)
\(888\) 0 0
\(889\) 30.0000 1.00617
\(890\) 2.00000 0.0670402
\(891\) 0 0
\(892\) −2.00000 −0.0669650
\(893\) 0 0
\(894\) 0 0
\(895\) 8.00000 0.267411
\(896\) 5.00000 0.167038
\(897\) 0 0
\(898\) −22.0000 −0.734150
\(899\) −6.00000 −0.200111
\(900\) 0 0
\(901\) 39.0000 1.29928
\(902\) −24.0000 −0.799113
\(903\) 0 0
\(904\) 0 0
\(905\) 26.0000 0.864269
\(906\) 0 0
\(907\) −1.00000 −0.0332045 −0.0166022 0.999862i \(-0.505285\pi\)
−0.0166022 + 0.999862i \(0.505285\pi\)
\(908\) −5.00000 −0.165931
\(909\) 0 0
\(910\) −5.00000 −0.165748
\(911\) 12.0000 0.397578 0.198789 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) 0 0
\(913\) 40.0000 1.32381
\(914\) 29.0000 0.959235
\(915\) 0 0
\(916\) −6.00000 −0.198246
\(917\) 80.0000 2.64183
\(918\) 0 0
\(919\) −5.00000 −0.164935 −0.0824674 0.996594i \(-0.526280\pi\)
−0.0824674 + 0.996594i \(0.526280\pi\)
\(920\) 7.00000 0.230783
\(921\) 0 0
\(922\) 18.0000 0.592798
\(923\) 0 0
\(924\) 0 0
\(925\) −2.00000 −0.0657596
\(926\) 8.00000 0.262896
\(927\) 0 0
\(928\) −3.00000 −0.0984798
\(929\) 3.00000 0.0984268 0.0492134 0.998788i \(-0.484329\pi\)
0.0492134 + 0.998788i \(0.484329\pi\)
\(930\) 0 0
\(931\) 18.0000 0.589926
\(932\) −10.0000 −0.327561
\(933\) 0 0
\(934\) −8.00000 −0.261768
\(935\) 12.0000 0.392442
\(936\) 0 0
\(937\) 47.0000 1.53542 0.767712 0.640796i \(-0.221395\pi\)
0.767712 + 0.640796i \(0.221395\pi\)
\(938\) −15.0000 −0.489767
\(939\) 0 0
\(940\) 0 0
\(941\) 51.0000 1.66255 0.831276 0.555860i \(-0.187611\pi\)
0.831276 + 0.555860i \(0.187611\pi\)
\(942\) 0 0
\(943\) −42.0000 −1.36771
\(944\) 9.00000 0.292925
\(945\) 0 0
\(946\) −24.0000 −0.780307
\(947\) 24.0000 0.779895 0.389948 0.920837i \(-0.372493\pi\)
0.389948 + 0.920837i \(0.372493\pi\)
\(948\) 0 0
\(949\) −11.0000 −0.357075
\(950\) −1.00000 −0.0324443
\(951\) 0 0
\(952\) 15.0000 0.486153
\(953\) 24.0000 0.777436 0.388718 0.921357i \(-0.372918\pi\)
0.388718 + 0.921357i \(0.372918\pi\)
\(954\) 0 0
\(955\) −9.00000 −0.291233
\(956\) 11.0000 0.355765
\(957\) 0 0
\(958\) 0 0
\(959\) 45.0000 1.45313
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) −2.00000 −0.0644826
\(963\) 0 0
\(964\) −12.0000 −0.386494
\(965\) 10.0000 0.321911
\(966\) 0 0
\(967\) −44.0000 −1.41494 −0.707472 0.706741i \(-0.750165\pi\)
−0.707472 + 0.706741i \(0.750165\pi\)
\(968\) −5.00000 −0.160706
\(969\) 0 0
\(970\) 2.00000 0.0642161
\(971\) −28.0000 −0.898563 −0.449281 0.893390i \(-0.648320\pi\)
−0.449281 + 0.893390i \(0.648320\pi\)
\(972\) 0 0
\(973\) −80.0000 −2.56468
\(974\) 38.0000 1.21760
\(975\) 0 0
\(976\) −12.0000 −0.384111
\(977\) 62.0000 1.98356 0.991778 0.127971i \(-0.0408466\pi\)
0.991778 + 0.127971i \(0.0408466\pi\)
\(978\) 0 0
\(979\) −8.00000 −0.255681
\(980\) 18.0000 0.574989
\(981\) 0 0
\(982\) 18.0000 0.574403
\(983\) 42.0000 1.33959 0.669796 0.742545i \(-0.266382\pi\)
0.669796 + 0.742545i \(0.266382\pi\)
\(984\) 0 0
\(985\) 22.0000 0.700978
\(986\) −9.00000 −0.286618
\(987\) 0 0
\(988\) −1.00000 −0.0318142
\(989\) −42.0000 −1.33552
\(990\) 0 0
\(991\) 30.0000 0.952981 0.476491 0.879180i \(-0.341909\pi\)
0.476491 + 0.879180i \(0.341909\pi\)
\(992\) 2.00000 0.0635001
\(993\) 0 0
\(994\) 0 0
\(995\) −15.0000 −0.475532
\(996\) 0 0
\(997\) −50.0000 −1.58352 −0.791758 0.610835i \(-0.790834\pi\)
−0.791758 + 0.610835i \(0.790834\pi\)
\(998\) −42.0000 −1.32949
\(999\) 0 0
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 1710.2.a.g.1.1 1
3.2 odd 2 190.2.a.b.1.1 1
5.4 even 2 8550.2.a.bm.1.1 1
12.11 even 2 1520.2.a.j.1.1 1
15.2 even 4 950.2.b.a.799.2 2
15.8 even 4 950.2.b.a.799.1 2
15.14 odd 2 950.2.a.c.1.1 1
21.20 even 2 9310.2.a.u.1.1 1
24.5 odd 2 6080.2.a.x.1.1 1
24.11 even 2 6080.2.a.b.1.1 1
57.56 even 2 3610.2.a.e.1.1 1
60.59 even 2 7600.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
190.2.a.b.1.1 1 3.2 odd 2
950.2.a.c.1.1 1 15.14 odd 2
950.2.b.a.799.1 2 15.8 even 4
950.2.b.a.799.2 2 15.2 even 4
1520.2.a.j.1.1 1 12.11 even 2
1710.2.a.g.1.1 1 1.1 even 1 trivial
3610.2.a.e.1.1 1 57.56 even 2
6080.2.a.b.1.1 1 24.11 even 2
6080.2.a.x.1.1 1 24.5 odd 2
7600.2.a.a.1.1 1 60.59 even 2
8550.2.a.bm.1.1 1 5.4 even 2
9310.2.a.u.1.1 1 21.20 even 2