Properties

Label 1638.2.a.d.1.1
Level $1638$
Weight $2$
Character 1638.1
Self dual yes
Analytic conductor $13.079$
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] = [1638,2,Mod(1,1638)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1638, 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("1638.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1638.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: \(13.0794958511\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1638.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +2.00000 q^{10} +4.00000 q^{11} -1.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -6.00000 q^{17} -4.00000 q^{19} -2.00000 q^{20} -4.00000 q^{22} -1.00000 q^{25} +1.00000 q^{26} +1.00000 q^{28} +6.00000 q^{29} -8.00000 q^{31} -1.00000 q^{32} +6.00000 q^{34} -2.00000 q^{35} +10.0000 q^{37} +4.00000 q^{38} +2.00000 q^{40} +6.00000 q^{41} +4.00000 q^{43} +4.00000 q^{44} -4.00000 q^{47} +1.00000 q^{49} +1.00000 q^{50} -1.00000 q^{52} -10.0000 q^{53} -8.00000 q^{55} -1.00000 q^{56} -6.00000 q^{58} -4.00000 q^{59} -6.00000 q^{61} +8.00000 q^{62} +1.00000 q^{64} +2.00000 q^{65} -8.00000 q^{67} -6.00000 q^{68} +2.00000 q^{70} -10.0000 q^{73} -10.0000 q^{74} -4.00000 q^{76} +4.00000 q^{77} -8.00000 q^{79} -2.00000 q^{80} -6.00000 q^{82} -4.00000 q^{83} +12.0000 q^{85} -4.00000 q^{86} -4.00000 q^{88} +6.00000 q^{89} -1.00000 q^{91} +4.00000 q^{94} +8.00000 q^{95} -2.00000 q^{97} -1.00000 q^{98} +O(q^{100})\)

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
\(5\) −2.00000 −0.894427 −0.447214 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 2.00000 0.632456
\(11\) 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
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) −4.00000 −0.852803
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 1.00000 0.196116
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) −2.00000 −0.338062
\(36\) 0 0
\(37\) 10.0000 1.64399 0.821995 0.569495i \(-0.192861\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(38\) 4.00000 0.648886
\(39\) 0 0
\(40\) 2.00000 0.316228
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) 0 0
\(47\) −4.00000 −0.583460 −0.291730 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −1.00000 −0.138675
\(53\) −10.0000 −1.37361 −0.686803 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) 0 0
\(55\) −8.00000 −1.07872
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 8.00000 1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) −6.00000 −0.727607
\(69\) 0 0
\(70\) 2.00000 0.239046
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −10.0000 −1.17041 −0.585206 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) −10.0000 −1.16248
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) 4.00000 0.455842
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) −2.00000 −0.223607
\(81\) 0 0
\(82\) −6.00000 −0.662589
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 12.0000 1.30158
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) −4.00000 −0.426401
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 0 0
\(93\) 0 0
\(94\) 4.00000 0.412568
\(95\) 8.00000 0.820783
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 0 0
\(103\) −12.0000 −1.18240 −0.591198 0.806527i \(-0.701345\pi\)
−0.591198 + 0.806527i \(0.701345\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) 10.0000 0.971286
\(107\) −16.0000 −1.54678 −0.773389 0.633932i \(-0.781440\pi\)
−0.773389 + 0.633932i \(0.781440\pi\)
\(108\) 0 0
\(109\) −6.00000 −0.574696 −0.287348 0.957826i \(-0.592774\pi\)
−0.287348 + 0.957826i \(0.592774\pi\)
\(110\) 8.00000 0.762770
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) −2.00000 −0.188144 −0.0940721 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) 0 0
\(118\) 4.00000 0.368230
\(119\) −6.00000 −0.550019
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 6.00000 0.543214
\(123\) 0 0
\(124\) −8.00000 −0.718421
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −2.00000 −0.175412
\(131\) 20.0000 1.74741 0.873704 0.486458i \(-0.161711\pi\)
0.873704 + 0.486458i \(0.161711\pi\)
\(132\) 0 0
\(133\) −4.00000 −0.346844
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) −10.0000 −0.854358 −0.427179 0.904167i \(-0.640493\pi\)
−0.427179 + 0.904167i \(0.640493\pi\)
\(138\) 0 0
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) −2.00000 −0.169031
\(141\) 0 0
\(142\) 0 0
\(143\) −4.00000 −0.334497
\(144\) 0 0
\(145\) −12.0000 −0.996546
\(146\) 10.0000 0.827606
\(147\) 0 0
\(148\) 10.0000 0.821995
\(149\) −14.0000 −1.14692 −0.573462 0.819232i \(-0.694400\pi\)
−0.573462 + 0.819232i \(0.694400\pi\)
\(150\) 0 0
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 4.00000 0.324443
\(153\) 0 0
\(154\) −4.00000 −0.322329
\(155\) 16.0000 1.28515
\(156\) 0 0
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) 8.00000 0.636446
\(159\) 0 0
\(160\) 2.00000 0.158114
\(161\) 0 0
\(162\) 0 0
\(163\) 16.0000 1.25322 0.626608 0.779334i \(-0.284443\pi\)
0.626608 + 0.779334i \(0.284443\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) −12.0000 −0.920358
\(171\) 0 0
\(172\) 4.00000 0.304997
\(173\) −18.0000 −1.36851 −0.684257 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 4.00000 0.301511
\(177\) 0 0
\(178\) −6.00000 −0.449719
\(179\) −24.0000 −1.79384 −0.896922 0.442189i \(-0.854202\pi\)
−0.896922 + 0.442189i \(0.854202\pi\)
\(180\) 0 0
\(181\) −22.0000 −1.63525 −0.817624 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) 1.00000 0.0741249
\(183\) 0 0
\(184\) 0 0
\(185\) −20.0000 −1.47043
\(186\) 0 0
\(187\) −24.0000 −1.75505
\(188\) −4.00000 −0.291730
\(189\) 0 0
\(190\) −8.00000 −0.580381
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) −6.00000 −0.431889 −0.215945 0.976406i \(-0.569283\pi\)
−0.215945 + 0.976406i \(0.569283\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 0 0
\(199\) 20.0000 1.41776 0.708881 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) 10.0000 0.703598
\(203\) 6.00000 0.421117
\(204\) 0 0
\(205\) −12.0000 −0.838116
\(206\) 12.0000 0.836080
\(207\) 0 0
\(208\) −1.00000 −0.0693375
\(209\) −16.0000 −1.10674
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) −10.0000 −0.686803
\(213\) 0 0
\(214\) 16.0000 1.09374
\(215\) −8.00000 −0.545595
\(216\) 0 0
\(217\) −8.00000 −0.543075
\(218\) 6.00000 0.406371
\(219\) 0 0
\(220\) −8.00000 −0.539360
\(221\) 6.00000 0.403604
\(222\) 0 0
\(223\) −8.00000 −0.535720 −0.267860 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 2.00000 0.133038
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) −6.00000 −0.396491 −0.198246 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 0 0
\(235\) 8.00000 0.521862
\(236\) −4.00000 −0.260378
\(237\) 0 0
\(238\) 6.00000 0.388922
\(239\) 24.0000 1.55243 0.776215 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) 0 0
\(241\) 30.0000 1.93247 0.966235 0.257663i \(-0.0829523\pi\)
0.966235 + 0.257663i \(0.0829523\pi\)
\(242\) −5.00000 −0.321412
\(243\) 0 0
\(244\) −6.00000 −0.384111
\(245\) −2.00000 −0.127775
\(246\) 0 0
\(247\) 4.00000 0.254514
\(248\) 8.00000 0.508001
\(249\) 0 0
\(250\) −12.0000 −0.758947
\(251\) 4.00000 0.252478 0.126239 0.992000i \(-0.459709\pi\)
0.126239 + 0.992000i \(0.459709\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 16.0000 1.00393
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 0 0
\(259\) 10.0000 0.621370
\(260\) 2.00000 0.124035
\(261\) 0 0
\(262\) −20.0000 −1.23560
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 0 0
\(265\) 20.0000 1.22859
\(266\) 4.00000 0.245256
\(267\) 0 0
\(268\) −8.00000 −0.488678
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) −6.00000 −0.363803
\(273\) 0 0
\(274\) 10.0000 0.604122
\(275\) −4.00000 −0.241209
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 12.0000 0.719712
\(279\) 0 0
\(280\) 2.00000 0.119523
\(281\) −10.0000 −0.596550 −0.298275 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) 0 0
\(283\) 28.0000 1.66443 0.832214 0.554455i \(-0.187073\pi\)
0.832214 + 0.554455i \(0.187073\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 4.00000 0.236525
\(287\) 6.00000 0.354169
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 12.0000 0.704664
\(291\) 0 0
\(292\) −10.0000 −0.585206
\(293\) −2.00000 −0.116841 −0.0584206 0.998292i \(-0.518606\pi\)
−0.0584206 + 0.998292i \(0.518606\pi\)
\(294\) 0 0
\(295\) 8.00000 0.465778
\(296\) −10.0000 −0.581238
\(297\) 0 0
\(298\) 14.0000 0.810998
\(299\) 0 0
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) −16.0000 −0.920697
\(303\) 0 0
\(304\) −4.00000 −0.229416
\(305\) 12.0000 0.687118
\(306\) 0 0
\(307\) −12.0000 −0.684876 −0.342438 0.939540i \(-0.611253\pi\)
−0.342438 + 0.939540i \(0.611253\pi\)
\(308\) 4.00000 0.227921
\(309\) 0 0
\(310\) −16.0000 −0.908739
\(311\) 16.0000 0.907277 0.453638 0.891186i \(-0.350126\pi\)
0.453638 + 0.891186i \(0.350126\pi\)
\(312\) 0 0
\(313\) 10.0000 0.565233 0.282617 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) −18.0000 −1.01580
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 0 0
\(319\) 24.0000 1.34374
\(320\) −2.00000 −0.111803
\(321\) 0 0
\(322\) 0 0
\(323\) 24.0000 1.33540
\(324\) 0 0
\(325\) 1.00000 0.0554700
\(326\) −16.0000 −0.886158
\(327\) 0 0
\(328\) −6.00000 −0.331295
\(329\) −4.00000 −0.220527
\(330\) 0 0
\(331\) −8.00000 −0.439720 −0.219860 0.975531i \(-0.570560\pi\)
−0.219860 + 0.975531i \(0.570560\pi\)
\(332\) −4.00000 −0.219529
\(333\) 0 0
\(334\) −12.0000 −0.656611
\(335\) 16.0000 0.874173
\(336\) 0 0
\(337\) 34.0000 1.85210 0.926049 0.377403i \(-0.123183\pi\)
0.926049 + 0.377403i \(0.123183\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 0 0
\(340\) 12.0000 0.650791
\(341\) −32.0000 −1.73290
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) 18.0000 0.967686
\(347\) −8.00000 −0.429463 −0.214731 0.976673i \(-0.568888\pi\)
−0.214731 + 0.976673i \(0.568888\pi\)
\(348\) 0 0
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 1.00000 0.0534522
\(351\) 0 0
\(352\) −4.00000 −0.213201
\(353\) 6.00000 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) 24.0000 1.26844
\(359\) 16.0000 0.844448 0.422224 0.906492i \(-0.361250\pi\)
0.422224 + 0.906492i \(0.361250\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 22.0000 1.15629
\(363\) 0 0
\(364\) −1.00000 −0.0524142
\(365\) 20.0000 1.04685
\(366\) 0 0
\(367\) −28.0000 −1.46159 −0.730794 0.682598i \(-0.760850\pi\)
−0.730794 + 0.682598i \(0.760850\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 20.0000 1.03975
\(371\) −10.0000 −0.519174
\(372\) 0 0
\(373\) 14.0000 0.724893 0.362446 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) 24.0000 1.24101
\(375\) 0 0
\(376\) 4.00000 0.206284
\(377\) −6.00000 −0.309016
\(378\) 0 0
\(379\) 16.0000 0.821865 0.410932 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) 8.00000 0.410391
\(381\) 0 0
\(382\) 0 0
\(383\) −4.00000 −0.204390 −0.102195 0.994764i \(-0.532587\pi\)
−0.102195 + 0.994764i \(0.532587\pi\)
\(384\) 0 0
\(385\) −8.00000 −0.407718
\(386\) 6.00000 0.305392
\(387\) 0 0
\(388\) −2.00000 −0.101535
\(389\) −34.0000 −1.72387 −0.861934 0.507020i \(-0.830747\pi\)
−0.861934 + 0.507020i \(0.830747\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) 6.00000 0.302276
\(395\) 16.0000 0.805047
\(396\) 0 0
\(397\) 18.0000 0.903394 0.451697 0.892171i \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(398\) −20.0000 −1.00251
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) 0 0
\(403\) 8.00000 0.398508
\(404\) −10.0000 −0.497519
\(405\) 0 0
\(406\) −6.00000 −0.297775
\(407\) 40.0000 1.98273
\(408\) 0 0
\(409\) 22.0000 1.08783 0.543915 0.839140i \(-0.316941\pi\)
0.543915 + 0.839140i \(0.316941\pi\)
\(410\) 12.0000 0.592638
\(411\) 0 0
\(412\) −12.0000 −0.591198
\(413\) −4.00000 −0.196827
\(414\) 0 0
\(415\) 8.00000 0.392705
\(416\) 1.00000 0.0490290
\(417\) 0 0
\(418\) 16.0000 0.782586
\(419\) −28.0000 −1.36789 −0.683945 0.729534i \(-0.739737\pi\)
−0.683945 + 0.729534i \(0.739737\pi\)
\(420\) 0 0
\(421\) −30.0000 −1.46211 −0.731055 0.682318i \(-0.760972\pi\)
−0.731055 + 0.682318i \(0.760972\pi\)
\(422\) −12.0000 −0.584151
\(423\) 0 0
\(424\) 10.0000 0.485643
\(425\) 6.00000 0.291043
\(426\) 0 0
\(427\) −6.00000 −0.290360
\(428\) −16.0000 −0.773389
\(429\) 0 0
\(430\) 8.00000 0.385794
\(431\) 32.0000 1.54139 0.770693 0.637207i \(-0.219910\pi\)
0.770693 + 0.637207i \(0.219910\pi\)
\(432\) 0 0
\(433\) 34.0000 1.63394 0.816968 0.576683i \(-0.195653\pi\)
0.816968 + 0.576683i \(0.195653\pi\)
\(434\) 8.00000 0.384012
\(435\) 0 0
\(436\) −6.00000 −0.287348
\(437\) 0 0
\(438\) 0 0
\(439\) −20.0000 −0.954548 −0.477274 0.878755i \(-0.658375\pi\)
−0.477274 + 0.878755i \(0.658375\pi\)
\(440\) 8.00000 0.381385
\(441\) 0 0
\(442\) −6.00000 −0.285391
\(443\) −24.0000 −1.14027 −0.570137 0.821549i \(-0.693110\pi\)
−0.570137 + 0.821549i \(0.693110\pi\)
\(444\) 0 0
\(445\) −12.0000 −0.568855
\(446\) 8.00000 0.378811
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) −2.00000 −0.0943858 −0.0471929 0.998886i \(-0.515028\pi\)
−0.0471929 + 0.998886i \(0.515028\pi\)
\(450\) 0 0
\(451\) 24.0000 1.13012
\(452\) −2.00000 −0.0940721
\(453\) 0 0
\(454\) −12.0000 −0.563188
\(455\) 2.00000 0.0937614
\(456\) 0 0
\(457\) −14.0000 −0.654892 −0.327446 0.944870i \(-0.606188\pi\)
−0.327446 + 0.944870i \(0.606188\pi\)
\(458\) 6.00000 0.280362
\(459\) 0 0
\(460\) 0 0
\(461\) 14.0000 0.652045 0.326023 0.945362i \(-0.394291\pi\)
0.326023 + 0.945362i \(0.394291\pi\)
\(462\) 0 0
\(463\) −24.0000 −1.11537 −0.557687 0.830051i \(-0.688311\pi\)
−0.557687 + 0.830051i \(0.688311\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) 18.0000 0.833834
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 0 0
\(469\) −8.00000 −0.369406
\(470\) −8.00000 −0.369012
\(471\) 0 0
\(472\) 4.00000 0.184115
\(473\) 16.0000 0.735681
\(474\) 0 0
\(475\) 4.00000 0.183533
\(476\) −6.00000 −0.275010
\(477\) 0 0
\(478\) −24.0000 −1.09773
\(479\) 4.00000 0.182765 0.0913823 0.995816i \(-0.470871\pi\)
0.0913823 + 0.995816i \(0.470871\pi\)
\(480\) 0 0
\(481\) −10.0000 −0.455961
\(482\) −30.0000 −1.36646
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) 4.00000 0.181631
\(486\) 0 0
\(487\) 8.00000 0.362515 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) 6.00000 0.271607
\(489\) 0 0
\(490\) 2.00000 0.0903508
\(491\) 32.0000 1.44414 0.722070 0.691820i \(-0.243191\pi\)
0.722070 + 0.691820i \(0.243191\pi\)
\(492\) 0 0
\(493\) −36.0000 −1.62136
\(494\) −4.00000 −0.179969
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) 0 0
\(498\) 0 0
\(499\) 32.0000 1.43252 0.716258 0.697835i \(-0.245853\pi\)
0.716258 + 0.697835i \(0.245853\pi\)
\(500\) 12.0000 0.536656
\(501\) 0 0
\(502\) −4.00000 −0.178529
\(503\) 24.0000 1.07011 0.535054 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) 0 0
\(505\) 20.0000 0.889988
\(506\) 0 0
\(507\) 0 0
\(508\) −16.0000 −0.709885
\(509\) 14.0000 0.620539 0.310270 0.950649i \(-0.399581\pi\)
0.310270 + 0.950649i \(0.399581\pi\)
\(510\) 0 0
\(511\) −10.0000 −0.442374
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 6.00000 0.264649
\(515\) 24.0000 1.05757
\(516\) 0 0
\(517\) −16.0000 −0.703679
\(518\) −10.0000 −0.439375
\(519\) 0 0
\(520\) −2.00000 −0.0877058
\(521\) 26.0000 1.13908 0.569540 0.821963i \(-0.307121\pi\)
0.569540 + 0.821963i \(0.307121\pi\)
\(522\) 0 0
\(523\) 4.00000 0.174908 0.0874539 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) 20.0000 0.873704
\(525\) 0 0
\(526\) 24.0000 1.04645
\(527\) 48.0000 2.09091
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) −20.0000 −0.868744
\(531\) 0 0
\(532\) −4.00000 −0.173422
\(533\) −6.00000 −0.259889
\(534\) 0 0
\(535\) 32.0000 1.38348
\(536\) 8.00000 0.345547
\(537\) 0 0
\(538\) 18.0000 0.776035
\(539\) 4.00000 0.172292
\(540\) 0 0
\(541\) −46.0000 −1.97769 −0.988847 0.148933i \(-0.952416\pi\)
−0.988847 + 0.148933i \(0.952416\pi\)
\(542\) −8.00000 −0.343629
\(543\) 0 0
\(544\) 6.00000 0.257248
\(545\) 12.0000 0.514024
\(546\) 0 0
\(547\) 12.0000 0.513083 0.256541 0.966533i \(-0.417417\pi\)
0.256541 + 0.966533i \(0.417417\pi\)
\(548\) −10.0000 −0.427179
\(549\) 0 0
\(550\) 4.00000 0.170561
\(551\) −24.0000 −1.02243
\(552\) 0 0
\(553\) −8.00000 −0.340195
\(554\) 10.0000 0.424859
\(555\) 0 0
\(556\) −12.0000 −0.508913
\(557\) −22.0000 −0.932170 −0.466085 0.884740i \(-0.654336\pi\)
−0.466085 + 0.884740i \(0.654336\pi\)
\(558\) 0 0
\(559\) −4.00000 −0.169182
\(560\) −2.00000 −0.0845154
\(561\) 0 0
\(562\) 10.0000 0.421825
\(563\) 12.0000 0.505740 0.252870 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 0 0
\(565\) 4.00000 0.168281
\(566\) −28.0000 −1.17693
\(567\) 0 0
\(568\) 0 0
\(569\) 22.0000 0.922288 0.461144 0.887325i \(-0.347439\pi\)
0.461144 + 0.887325i \(0.347439\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) −4.00000 −0.167248
\(573\) 0 0
\(574\) −6.00000 −0.250435
\(575\) 0 0
\(576\) 0 0
\(577\) 22.0000 0.915872 0.457936 0.888985i \(-0.348589\pi\)
0.457936 + 0.888985i \(0.348589\pi\)
\(578\) −19.0000 −0.790296
\(579\) 0 0
\(580\) −12.0000 −0.498273
\(581\) −4.00000 −0.165948
\(582\) 0 0
\(583\) −40.0000 −1.65663
\(584\) 10.0000 0.413803
\(585\) 0 0
\(586\) 2.00000 0.0826192
\(587\) 28.0000 1.15568 0.577842 0.816149i \(-0.303895\pi\)
0.577842 + 0.816149i \(0.303895\pi\)
\(588\) 0 0
\(589\) 32.0000 1.31854
\(590\) −8.00000 −0.329355
\(591\) 0 0
\(592\) 10.0000 0.410997
\(593\) −18.0000 −0.739171 −0.369586 0.929197i \(-0.620500\pi\)
−0.369586 + 0.929197i \(0.620500\pi\)
\(594\) 0 0
\(595\) 12.0000 0.491952
\(596\) −14.0000 −0.573462
\(597\) 0 0
\(598\) 0 0
\(599\) −8.00000 −0.326871 −0.163436 0.986554i \(-0.552258\pi\)
−0.163436 + 0.986554i \(0.552258\pi\)
\(600\) 0 0
\(601\) 2.00000 0.0815817 0.0407909 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) −4.00000 −0.163028
\(603\) 0 0
\(604\) 16.0000 0.651031
\(605\) −10.0000 −0.406558
\(606\) 0 0
\(607\) −28.0000 −1.13648 −0.568242 0.822861i \(-0.692376\pi\)
−0.568242 + 0.822861i \(0.692376\pi\)
\(608\) 4.00000 0.162221
\(609\) 0 0
\(610\) −12.0000 −0.485866
\(611\) 4.00000 0.161823
\(612\) 0 0
\(613\) −30.0000 −1.21169 −0.605844 0.795583i \(-0.707165\pi\)
−0.605844 + 0.795583i \(0.707165\pi\)
\(614\) 12.0000 0.484281
\(615\) 0 0
\(616\) −4.00000 −0.161165
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 0 0
\(619\) 28.0000 1.12542 0.562708 0.826656i \(-0.309760\pi\)
0.562708 + 0.826656i \(0.309760\pi\)
\(620\) 16.0000 0.642575
\(621\) 0 0
\(622\) −16.0000 −0.641542
\(623\) 6.00000 0.240385
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) −10.0000 −0.399680
\(627\) 0 0
\(628\) 18.0000 0.718278
\(629\) −60.0000 −2.39236
\(630\) 0 0
\(631\) 16.0000 0.636950 0.318475 0.947931i \(-0.396829\pi\)
0.318475 + 0.947931i \(0.396829\pi\)
\(632\) 8.00000 0.318223
\(633\) 0 0
\(634\) −18.0000 −0.714871
\(635\) 32.0000 1.26988
\(636\) 0 0
\(637\) −1.00000 −0.0396214
\(638\) −24.0000 −0.950169
\(639\) 0 0
\(640\) 2.00000 0.0790569
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) 0 0
\(643\) −28.0000 −1.10421 −0.552106 0.833774i \(-0.686176\pi\)
−0.552106 + 0.833774i \(0.686176\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −24.0000 −0.944267
\(647\) −8.00000 −0.314512 −0.157256 0.987558i \(-0.550265\pi\)
−0.157256 + 0.987558i \(0.550265\pi\)
\(648\) 0 0
\(649\) −16.0000 −0.628055
\(650\) −1.00000 −0.0392232
\(651\) 0 0
\(652\) 16.0000 0.626608
\(653\) 14.0000 0.547862 0.273931 0.961749i \(-0.411676\pi\)
0.273931 + 0.961749i \(0.411676\pi\)
\(654\) 0 0
\(655\) −40.0000 −1.56293
\(656\) 6.00000 0.234261
\(657\) 0 0
\(658\) 4.00000 0.155936
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −38.0000 −1.47803 −0.739014 0.673690i \(-0.764708\pi\)
−0.739014 + 0.673690i \(0.764708\pi\)
\(662\) 8.00000 0.310929
\(663\) 0 0
\(664\) 4.00000 0.155230
\(665\) 8.00000 0.310227
\(666\) 0 0
\(667\) 0 0
\(668\) 12.0000 0.464294
\(669\) 0 0
\(670\) −16.0000 −0.618134
\(671\) −24.0000 −0.926510
\(672\) 0 0
\(673\) 18.0000 0.693849 0.346925 0.937893i \(-0.387226\pi\)
0.346925 + 0.937893i \(0.387226\pi\)
\(674\) −34.0000 −1.30963
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) 38.0000 1.46046 0.730229 0.683202i \(-0.239413\pi\)
0.730229 + 0.683202i \(0.239413\pi\)
\(678\) 0 0
\(679\) −2.00000 −0.0767530
\(680\) −12.0000 −0.460179
\(681\) 0 0
\(682\) 32.0000 1.22534
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) 0 0
\(685\) 20.0000 0.764161
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) 4.00000 0.152499
\(689\) 10.0000 0.380970
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) −18.0000 −0.684257
\(693\) 0 0
\(694\) 8.00000 0.303676
\(695\) 24.0000 0.910372
\(696\) 0 0
\(697\) −36.0000 −1.36360
\(698\) −2.00000 −0.0757011
\(699\) 0 0
\(700\) −1.00000 −0.0377964
\(701\) −50.0000 −1.88847 −0.944237 0.329267i \(-0.893198\pi\)
−0.944237 + 0.329267i \(0.893198\pi\)
\(702\) 0 0
\(703\) −40.0000 −1.50863
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) −10.0000 −0.376089
\(708\) 0 0
\(709\) −6.00000 −0.225335 −0.112667 0.993633i \(-0.535939\pi\)
−0.112667 + 0.993633i \(0.535939\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −6.00000 −0.224860
\(713\) 0 0
\(714\) 0 0
\(715\) 8.00000 0.299183
\(716\) −24.0000 −0.896922
\(717\) 0 0
\(718\) −16.0000 −0.597115
\(719\) −16.0000 −0.596699 −0.298350 0.954457i \(-0.596436\pi\)
−0.298350 + 0.954457i \(0.596436\pi\)
\(720\) 0 0
\(721\) −12.0000 −0.446903
\(722\) 3.00000 0.111648
\(723\) 0 0
\(724\) −22.0000 −0.817624
\(725\) −6.00000 −0.222834
\(726\) 0 0
\(727\) 28.0000 1.03846 0.519231 0.854634i \(-0.326218\pi\)
0.519231 + 0.854634i \(0.326218\pi\)
\(728\) 1.00000 0.0370625
\(729\) 0 0
\(730\) −20.0000 −0.740233
\(731\) −24.0000 −0.887672
\(732\) 0 0
\(733\) 2.00000 0.0738717 0.0369358 0.999318i \(-0.488240\pi\)
0.0369358 + 0.999318i \(0.488240\pi\)
\(734\) 28.0000 1.03350
\(735\) 0 0
\(736\) 0 0
\(737\) −32.0000 −1.17874
\(738\) 0 0
\(739\) 40.0000 1.47142 0.735712 0.677295i \(-0.236848\pi\)
0.735712 + 0.677295i \(0.236848\pi\)
\(740\) −20.0000 −0.735215
\(741\) 0 0
\(742\) 10.0000 0.367112
\(743\) 8.00000 0.293492 0.146746 0.989174i \(-0.453120\pi\)
0.146746 + 0.989174i \(0.453120\pi\)
\(744\) 0 0
\(745\) 28.0000 1.02584
\(746\) −14.0000 −0.512576
\(747\) 0 0
\(748\) −24.0000 −0.877527
\(749\) −16.0000 −0.584627
\(750\) 0 0
\(751\) 16.0000 0.583848 0.291924 0.956441i \(-0.405705\pi\)
0.291924 + 0.956441i \(0.405705\pi\)
\(752\) −4.00000 −0.145865
\(753\) 0 0
\(754\) 6.00000 0.218507
\(755\) −32.0000 −1.16460
\(756\) 0 0
\(757\) −26.0000 −0.944986 −0.472493 0.881334i \(-0.656646\pi\)
−0.472493 + 0.881334i \(0.656646\pi\)
\(758\) −16.0000 −0.581146
\(759\) 0 0
\(760\) −8.00000 −0.290191
\(761\) −26.0000 −0.942499 −0.471250 0.882000i \(-0.656197\pi\)
−0.471250 + 0.882000i \(0.656197\pi\)
\(762\) 0 0
\(763\) −6.00000 −0.217215
\(764\) 0 0
\(765\) 0 0
\(766\) 4.00000 0.144526
\(767\) 4.00000 0.144432
\(768\) 0 0
\(769\) −50.0000 −1.80305 −0.901523 0.432731i \(-0.857550\pi\)
−0.901523 + 0.432731i \(0.857550\pi\)
\(770\) 8.00000 0.288300
\(771\) 0 0
\(772\) −6.00000 −0.215945
\(773\) −18.0000 −0.647415 −0.323708 0.946157i \(-0.604929\pi\)
−0.323708 + 0.946157i \(0.604929\pi\)
\(774\) 0 0
\(775\) 8.00000 0.287368
\(776\) 2.00000 0.0717958
\(777\) 0 0
\(778\) 34.0000 1.21896
\(779\) −24.0000 −0.859889
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) −36.0000 −1.28490
\(786\) 0 0
\(787\) −12.0000 −0.427754 −0.213877 0.976861i \(-0.568609\pi\)
−0.213877 + 0.976861i \(0.568609\pi\)
\(788\) −6.00000 −0.213741
\(789\) 0 0
\(790\) −16.0000 −0.569254
\(791\) −2.00000 −0.0711118
\(792\) 0 0
\(793\) 6.00000 0.213066
\(794\) −18.0000 −0.638796
\(795\) 0 0
\(796\) 20.0000 0.708881
\(797\) 30.0000 1.06265 0.531327 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) 0 0
\(799\) 24.0000 0.849059
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) −30.0000 −1.05934
\(803\) −40.0000 −1.41157
\(804\) 0 0
\(805\) 0 0
\(806\) −8.00000 −0.281788
\(807\) 0 0
\(808\) 10.0000 0.351799
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) 6.00000 0.210559
\(813\) 0 0
\(814\) −40.0000 −1.40200
\(815\) −32.0000 −1.12091
\(816\) 0 0
\(817\) −16.0000 −0.559769
\(818\) −22.0000 −0.769212
\(819\) 0 0
\(820\) −12.0000 −0.419058
\(821\) 2.00000 0.0698005 0.0349002 0.999391i \(-0.488889\pi\)
0.0349002 + 0.999391i \(0.488889\pi\)
\(822\) 0 0
\(823\) 32.0000 1.11545 0.557725 0.830026i \(-0.311674\pi\)
0.557725 + 0.830026i \(0.311674\pi\)
\(824\) 12.0000 0.418040
\(825\) 0 0
\(826\) 4.00000 0.139178
\(827\) −20.0000 −0.695468 −0.347734 0.937593i \(-0.613049\pi\)
−0.347734 + 0.937593i \(0.613049\pi\)
\(828\) 0 0
\(829\) −38.0000 −1.31979 −0.659897 0.751356i \(-0.729400\pi\)
−0.659897 + 0.751356i \(0.729400\pi\)
\(830\) −8.00000 −0.277684
\(831\) 0 0
\(832\) −1.00000 −0.0346688
\(833\) −6.00000 −0.207888
\(834\) 0 0
\(835\) −24.0000 −0.830554
\(836\) −16.0000 −0.553372
\(837\) 0 0
\(838\) 28.0000 0.967244
\(839\) 4.00000 0.138095 0.0690477 0.997613i \(-0.478004\pi\)
0.0690477 + 0.997613i \(0.478004\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 30.0000 1.03387
\(843\) 0 0
\(844\) 12.0000 0.413057
\(845\) −2.00000 −0.0688021
\(846\) 0 0
\(847\) 5.00000 0.171802
\(848\) −10.0000 −0.343401
\(849\) 0 0
\(850\) −6.00000 −0.205798
\(851\) 0 0
\(852\) 0 0
\(853\) −22.0000 −0.753266 −0.376633 0.926363i \(-0.622918\pi\)
−0.376633 + 0.926363i \(0.622918\pi\)
\(854\) 6.00000 0.205316
\(855\) 0 0
\(856\) 16.0000 0.546869
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 0 0
\(859\) 28.0000 0.955348 0.477674 0.878537i \(-0.341480\pi\)
0.477674 + 0.878537i \(0.341480\pi\)
\(860\) −8.00000 −0.272798
\(861\) 0 0
\(862\) −32.0000 −1.08992
\(863\) −56.0000 −1.90626 −0.953131 0.302558i \(-0.902160\pi\)
−0.953131 + 0.302558i \(0.902160\pi\)
\(864\) 0 0
\(865\) 36.0000 1.22404
\(866\) −34.0000 −1.15537
\(867\) 0 0
\(868\) −8.00000 −0.271538
\(869\) −32.0000 −1.08553
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) 6.00000 0.203186
\(873\) 0 0
\(874\) 0 0
\(875\) 12.0000 0.405674
\(876\) 0 0
\(877\) −6.00000 −0.202606 −0.101303 0.994856i \(-0.532301\pi\)
−0.101303 + 0.994856i \(0.532301\pi\)
\(878\) 20.0000 0.674967
\(879\) 0 0
\(880\) −8.00000 −0.269680
\(881\) 18.0000 0.606435 0.303218 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) 0 0
\(883\) 12.0000 0.403832 0.201916 0.979403i \(-0.435283\pi\)
0.201916 + 0.979403i \(0.435283\pi\)
\(884\) 6.00000 0.201802
\(885\) 0 0
\(886\) 24.0000 0.806296
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) −16.0000 −0.536623
\(890\) 12.0000 0.402241
\(891\) 0 0
\(892\) −8.00000 −0.267860
\(893\) 16.0000 0.535420
\(894\) 0 0
\(895\) 48.0000 1.60446
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 2.00000 0.0667409
\(899\) −48.0000 −1.60089
\(900\) 0 0
\(901\) 60.0000 1.99889
\(902\) −24.0000 −0.799113
\(903\) 0 0
\(904\) 2.00000 0.0665190
\(905\) 44.0000 1.46261
\(906\) 0 0
\(907\) 12.0000 0.398453 0.199227 0.979953i \(-0.436157\pi\)
0.199227 + 0.979953i \(0.436157\pi\)
\(908\) 12.0000 0.398234
\(909\) 0 0
\(910\) −2.00000 −0.0662994
\(911\) 40.0000 1.32526 0.662630 0.748947i \(-0.269440\pi\)
0.662630 + 0.748947i \(0.269440\pi\)
\(912\) 0 0
\(913\) −16.0000 −0.529523
\(914\) 14.0000 0.463079
\(915\) 0 0
\(916\) −6.00000 −0.198246
\(917\) 20.0000 0.660458
\(918\) 0 0
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −14.0000 −0.461065
\(923\) 0 0
\(924\) 0 0
\(925\) −10.0000 −0.328798
\(926\) 24.0000 0.788689
\(927\) 0 0
\(928\) −6.00000 −0.196960
\(929\) −34.0000 −1.11550 −0.557752 0.830008i \(-0.688336\pi\)
−0.557752 + 0.830008i \(0.688336\pi\)
\(930\) 0 0
\(931\) −4.00000 −0.131095
\(932\) −18.0000 −0.589610
\(933\) 0 0
\(934\) 12.0000 0.392652
\(935\) 48.0000 1.56977
\(936\) 0 0
\(937\) −22.0000 −0.718709 −0.359354 0.933201i \(-0.617003\pi\)
−0.359354 + 0.933201i \(0.617003\pi\)
\(938\) 8.00000 0.261209
\(939\) 0 0
\(940\) 8.00000 0.260931
\(941\) 54.0000 1.76035 0.880175 0.474650i \(-0.157425\pi\)
0.880175 + 0.474650i \(0.157425\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) −16.0000 −0.520205
\(947\) −12.0000 −0.389948 −0.194974 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) 0 0
\(949\) 10.0000 0.324614
\(950\) −4.00000 −0.129777
\(951\) 0 0
\(952\) 6.00000 0.194461
\(953\) 14.0000 0.453504 0.226752 0.973952i \(-0.427189\pi\)
0.226752 + 0.973952i \(0.427189\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 24.0000 0.776215
\(957\) 0 0
\(958\) −4.00000 −0.129234
\(959\) −10.0000 −0.322917
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 10.0000 0.322413
\(963\) 0 0
\(964\) 30.0000 0.966235
\(965\) 12.0000 0.386294
\(966\) 0 0
\(967\) 16.0000 0.514525 0.257263 0.966342i \(-0.417179\pi\)
0.257263 + 0.966342i \(0.417179\pi\)
\(968\) −5.00000 −0.160706
\(969\) 0 0
\(970\) −4.00000 −0.128432
\(971\) −4.00000 −0.128366 −0.0641831 0.997938i \(-0.520444\pi\)
−0.0641831 + 0.997938i \(0.520444\pi\)
\(972\) 0 0
\(973\) −12.0000 −0.384702
\(974\) −8.00000 −0.256337
\(975\) 0 0
\(976\) −6.00000 −0.192055
\(977\) 30.0000 0.959785 0.479893 0.877327i \(-0.340676\pi\)
0.479893 + 0.877327i \(0.340676\pi\)
\(978\) 0 0
\(979\) 24.0000 0.767043
\(980\) −2.00000 −0.0638877
\(981\) 0 0
\(982\) −32.0000 −1.02116
\(983\) −4.00000 −0.127580 −0.0637901 0.997963i \(-0.520319\pi\)
−0.0637901 + 0.997963i \(0.520319\pi\)
\(984\) 0 0
\(985\) 12.0000 0.382352
\(986\) 36.0000 1.14647
\(987\) 0 0
\(988\) 4.00000 0.127257
\(989\) 0 0
\(990\) 0 0
\(991\) 40.0000 1.27064 0.635321 0.772248i \(-0.280868\pi\)
0.635321 + 0.772248i \(0.280868\pi\)
\(992\) 8.00000 0.254000
\(993\) 0 0
\(994\) 0 0
\(995\) −40.0000 −1.26809
\(996\) 0 0
\(997\) 2.00000 0.0633406 0.0316703 0.999498i \(-0.489917\pi\)
0.0316703 + 0.999498i \(0.489917\pi\)
\(998\) −32.0000 −1.01294
\(999\) 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 1638.2.a.d.1.1 1
3.2 odd 2 546.2.a.g.1.1 1
12.11 even 2 4368.2.a.k.1.1 1
21.20 even 2 3822.2.a.t.1.1 1
39.38 odd 2 7098.2.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.a.g.1.1 1 3.2 odd 2
1638.2.a.d.1.1 1 1.1 even 1 trivial
3822.2.a.t.1.1 1 21.20 even 2
4368.2.a.k.1.1 1 12.11 even 2
7098.2.a.j.1.1 1 39.38 odd 2