Properties

Label 5586.2.a.z.1.1
Level $5586$
Weight $2$
Character 5586.1
Self dual yes
Analytic conductor $44.604$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5586,2,Mod(1,5586)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5586, 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("5586.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5586 = 2 \cdot 3 \cdot 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5586.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: \(44.6044345691\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 5586.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} -2.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} -2.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} +1.00000 q^{19} -2.00000 q^{20} -2.00000 q^{22} -6.00000 q^{23} +1.00000 q^{24} -1.00000 q^{25} +2.00000 q^{26} +1.00000 q^{27} -6.00000 q^{29} -2.00000 q^{30} +1.00000 q^{32} -2.00000 q^{33} -2.00000 q^{34} +1.00000 q^{36} -10.0000 q^{37} +1.00000 q^{38} +2.00000 q^{39} -2.00000 q^{40} +8.00000 q^{41} -2.00000 q^{44} -2.00000 q^{45} -6.00000 q^{46} +1.00000 q^{48} -1.00000 q^{50} -2.00000 q^{51} +2.00000 q^{52} -2.00000 q^{53} +1.00000 q^{54} +4.00000 q^{55} +1.00000 q^{57} -6.00000 q^{58} -12.0000 q^{59} -2.00000 q^{60} +8.00000 q^{61} +1.00000 q^{64} -4.00000 q^{65} -2.00000 q^{66} -2.00000 q^{67} -2.00000 q^{68} -6.00000 q^{69} -4.00000 q^{71} +1.00000 q^{72} -8.00000 q^{73} -10.0000 q^{74} -1.00000 q^{75} +1.00000 q^{76} +2.00000 q^{78} +2.00000 q^{79} -2.00000 q^{80} +1.00000 q^{81} +8.00000 q^{82} -4.00000 q^{83} +4.00000 q^{85} -6.00000 q^{87} -2.00000 q^{88} -2.00000 q^{90} -6.00000 q^{92} -2.00000 q^{95} +1.00000 q^{96} +6.00000 q^{97} -2.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) −2.00000 −0.894427 −0.447214 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −2.00000 −0.632456
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 1.00000 0.288675
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) −2.00000 −0.516398
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 1.00000 0.235702
\(19\) 1.00000 0.229416
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) −2.00000 −0.426401
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 1.00000 0.204124
\(25\) −1.00000 −0.200000
\(26\) 2.00000 0.392232
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) −2.00000 −0.365148
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000 0.176777
\(33\) −2.00000 −0.348155
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −10.0000 −1.64399 −0.821995 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 1.00000 0.162221
\(39\) 2.00000 0.320256
\(40\) −2.00000 −0.316228
\(41\) 8.00000 1.24939 0.624695 0.780869i \(-0.285223\pi\)
0.624695 + 0.780869i \(0.285223\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) −2.00000 −0.301511
\(45\) −2.00000 −0.298142
\(46\) −6.00000 −0.884652
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) −1.00000 −0.141421
\(51\) −2.00000 −0.280056
\(52\) 2.00000 0.277350
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 1.00000 0.136083
\(55\) 4.00000 0.539360
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) −6.00000 −0.787839
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) −2.00000 −0.258199
\(61\) 8.00000 1.02430 0.512148 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −4.00000 −0.496139
\(66\) −2.00000 −0.246183
\(67\) −2.00000 −0.244339 −0.122169 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) −2.00000 −0.242536
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) −4.00000 −0.474713 −0.237356 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) 1.00000 0.117851
\(73\) −8.00000 −0.936329 −0.468165 0.883641i \(-0.655085\pi\)
−0.468165 + 0.883641i \(0.655085\pi\)
\(74\) −10.0000 −1.16248
\(75\) −1.00000 −0.115470
\(76\) 1.00000 0.114708
\(77\) 0 0
\(78\) 2.00000 0.226455
\(79\) 2.00000 0.225018 0.112509 0.993651i \(-0.464111\pi\)
0.112509 + 0.993651i \(0.464111\pi\)
\(80\) −2.00000 −0.223607
\(81\) 1.00000 0.111111
\(82\) 8.00000 0.883452
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 4.00000 0.433861
\(86\) 0 0
\(87\) −6.00000 −0.643268
\(88\) −2.00000 −0.213201
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) −2.00000 −0.210819
\(91\) 0 0
\(92\) −6.00000 −0.625543
\(93\) 0 0
\(94\) 0 0
\(95\) −2.00000 −0.205196
\(96\) 1.00000 0.102062
\(97\) 6.00000 0.609208 0.304604 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) 0 0
\(99\) −2.00000 −0.201008
\(100\) −1.00000 −0.100000
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) −2.00000 −0.198030
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 1.00000 0.0962250
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 4.00000 0.381385
\(111\) −10.0000 −0.949158
\(112\) 0 0
\(113\) −10.0000 −0.940721 −0.470360 0.882474i \(-0.655876\pi\)
−0.470360 + 0.882474i \(0.655876\pi\)
\(114\) 1.00000 0.0936586
\(115\) 12.0000 1.11901
\(116\) −6.00000 −0.557086
\(117\) 2.00000 0.184900
\(118\) −12.0000 −1.10469
\(119\) 0 0
\(120\) −2.00000 −0.182574
\(121\) −7.00000 −0.636364
\(122\) 8.00000 0.724286
\(123\) 8.00000 0.721336
\(124\) 0 0
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) 22.0000 1.95218 0.976092 0.217357i \(-0.0697436\pi\)
0.976092 + 0.217357i \(0.0697436\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −4.00000 −0.350823
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) −2.00000 −0.174078
\(133\) 0 0
\(134\) −2.00000 −0.172774
\(135\) −2.00000 −0.172133
\(136\) −2.00000 −0.171499
\(137\) −10.0000 −0.854358 −0.427179 0.904167i \(-0.640493\pi\)
−0.427179 + 0.904167i \(0.640493\pi\)
\(138\) −6.00000 −0.510754
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −4.00000 −0.335673
\(143\) −4.00000 −0.334497
\(144\) 1.00000 0.0833333
\(145\) 12.0000 0.996546
\(146\) −8.00000 −0.662085
\(147\) 0 0
\(148\) −10.0000 −0.821995
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 2.00000 0.162758 0.0813788 0.996683i \(-0.474068\pi\)
0.0813788 + 0.996683i \(0.474068\pi\)
\(152\) 1.00000 0.0811107
\(153\) −2.00000 −0.161690
\(154\) 0 0
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) −12.0000 −0.957704 −0.478852 0.877896i \(-0.658947\pi\)
−0.478852 + 0.877896i \(0.658947\pi\)
\(158\) 2.00000 0.159111
\(159\) −2.00000 −0.158610
\(160\) −2.00000 −0.158114
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −8.00000 −0.626608 −0.313304 0.949653i \(-0.601436\pi\)
−0.313304 + 0.949653i \(0.601436\pi\)
\(164\) 8.00000 0.624695
\(165\) 4.00000 0.311400
\(166\) −4.00000 −0.310460
\(167\) 4.00000 0.309529 0.154765 0.987951i \(-0.450538\pi\)
0.154765 + 0.987951i \(0.450538\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 4.00000 0.306786
\(171\) 1.00000 0.0764719
\(172\) 0 0
\(173\) −24.0000 −1.82469 −0.912343 0.409426i \(-0.865729\pi\)
−0.912343 + 0.409426i \(0.865729\pi\)
\(174\) −6.00000 −0.454859
\(175\) 0 0
\(176\) −2.00000 −0.150756
\(177\) −12.0000 −0.901975
\(178\) 0 0
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) −2.00000 −0.149071
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 0 0
\(183\) 8.00000 0.591377
\(184\) −6.00000 −0.442326
\(185\) 20.0000 1.47043
\(186\) 0 0
\(187\) 4.00000 0.292509
\(188\) 0 0
\(189\) 0 0
\(190\) −2.00000 −0.145095
\(191\) −6.00000 −0.434145 −0.217072 0.976156i \(-0.569651\pi\)
−0.217072 + 0.976156i \(0.569651\pi\)
\(192\) 1.00000 0.0721688
\(193\) 10.0000 0.719816 0.359908 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) 6.00000 0.430775
\(195\) −4.00000 −0.286446
\(196\) 0 0
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) −2.00000 −0.142134
\(199\) −20.0000 −1.41776 −0.708881 0.705328i \(-0.750800\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −2.00000 −0.141069
\(202\) −2.00000 −0.140720
\(203\) 0 0
\(204\) −2.00000 −0.140028
\(205\) −16.0000 −1.11749
\(206\) 8.00000 0.557386
\(207\) −6.00000 −0.417029
\(208\) 2.00000 0.138675
\(209\) −2.00000 −0.138343
\(210\) 0 0
\(211\) −10.0000 −0.688428 −0.344214 0.938891i \(-0.611855\pi\)
−0.344214 + 0.938891i \(0.611855\pi\)
\(212\) −2.00000 −0.137361
\(213\) −4.00000 −0.274075
\(214\) −4.00000 −0.273434
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) −2.00000 −0.135457
\(219\) −8.00000 −0.540590
\(220\) 4.00000 0.269680
\(221\) −4.00000 −0.269069
\(222\) −10.0000 −0.671156
\(223\) −8.00000 −0.535720 −0.267860 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) −10.0000 −0.665190
\(227\) 8.00000 0.530979 0.265489 0.964114i \(-0.414466\pi\)
0.265489 + 0.964114i \(0.414466\pi\)
\(228\) 1.00000 0.0662266
\(229\) 8.00000 0.528655 0.264327 0.964433i \(-0.414850\pi\)
0.264327 + 0.964433i \(0.414850\pi\)
\(230\) 12.0000 0.791257
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) −12.0000 −0.781133
\(237\) 2.00000 0.129914
\(238\) 0 0
\(239\) 18.0000 1.16432 0.582162 0.813073i \(-0.302207\pi\)
0.582162 + 0.813073i \(0.302207\pi\)
\(240\) −2.00000 −0.129099
\(241\) −6.00000 −0.386494 −0.193247 0.981150i \(-0.561902\pi\)
−0.193247 + 0.981150i \(0.561902\pi\)
\(242\) −7.00000 −0.449977
\(243\) 1.00000 0.0641500
\(244\) 8.00000 0.512148
\(245\) 0 0
\(246\) 8.00000 0.510061
\(247\) 2.00000 0.127257
\(248\) 0 0
\(249\) −4.00000 −0.253490
\(250\) 12.0000 0.758947
\(251\) −20.0000 −1.26239 −0.631194 0.775625i \(-0.717435\pi\)
−0.631194 + 0.775625i \(0.717435\pi\)
\(252\) 0 0
\(253\) 12.0000 0.754434
\(254\) 22.0000 1.38040
\(255\) 4.00000 0.250490
\(256\) 1.00000 0.0625000
\(257\) −4.00000 −0.249513 −0.124757 0.992187i \(-0.539815\pi\)
−0.124757 + 0.992187i \(0.539815\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −4.00000 −0.248069
\(261\) −6.00000 −0.371391
\(262\) −4.00000 −0.247121
\(263\) 30.0000 1.84988 0.924940 0.380114i \(-0.124115\pi\)
0.924940 + 0.380114i \(0.124115\pi\)
\(264\) −2.00000 −0.123091
\(265\) 4.00000 0.245718
\(266\) 0 0
\(267\) 0 0
\(268\) −2.00000 −0.122169
\(269\) 24.0000 1.46331 0.731653 0.681677i \(-0.238749\pi\)
0.731653 + 0.681677i \(0.238749\pi\)
\(270\) −2.00000 −0.121716
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) −2.00000 −0.121268
\(273\) 0 0
\(274\) −10.0000 −0.604122
\(275\) 2.00000 0.120605
\(276\) −6.00000 −0.361158
\(277\) −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
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) 20.0000 1.18888 0.594438 0.804141i \(-0.297374\pi\)
0.594438 + 0.804141i \(0.297374\pi\)
\(284\) −4.00000 −0.237356
\(285\) −2.00000 −0.118470
\(286\) −4.00000 −0.236525
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −13.0000 −0.764706
\(290\) 12.0000 0.704664
\(291\) 6.00000 0.351726
\(292\) −8.00000 −0.468165
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 24.0000 1.39733
\(296\) −10.0000 −0.581238
\(297\) −2.00000 −0.116052
\(298\) −6.00000 −0.347571
\(299\) −12.0000 −0.693978
\(300\) −1.00000 −0.0577350
\(301\) 0 0
\(302\) 2.00000 0.115087
\(303\) −2.00000 −0.114897
\(304\) 1.00000 0.0573539
\(305\) −16.0000 −0.916157
\(306\) −2.00000 −0.114332
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 2.00000 0.113228
\(313\) 8.00000 0.452187 0.226093 0.974106i \(-0.427405\pi\)
0.226093 + 0.974106i \(0.427405\pi\)
\(314\) −12.0000 −0.677199
\(315\) 0 0
\(316\) 2.00000 0.112509
\(317\) −30.0000 −1.68497 −0.842484 0.538721i \(-0.818908\pi\)
−0.842484 + 0.538721i \(0.818908\pi\)
\(318\) −2.00000 −0.112154
\(319\) 12.0000 0.671871
\(320\) −2.00000 −0.111803
\(321\) −4.00000 −0.223258
\(322\) 0 0
\(323\) −2.00000 −0.111283
\(324\) 1.00000 0.0555556
\(325\) −2.00000 −0.110940
\(326\) −8.00000 −0.443079
\(327\) −2.00000 −0.110600
\(328\) 8.00000 0.441726
\(329\) 0 0
\(330\) 4.00000 0.220193
\(331\) −14.0000 −0.769510 −0.384755 0.923019i \(-0.625714\pi\)
−0.384755 + 0.923019i \(0.625714\pi\)
\(332\) −4.00000 −0.219529
\(333\) −10.0000 −0.547997
\(334\) 4.00000 0.218870
\(335\) 4.00000 0.218543
\(336\) 0 0
\(337\) 22.0000 1.19842 0.599208 0.800593i \(-0.295482\pi\)
0.599208 + 0.800593i \(0.295482\pi\)
\(338\) −9.00000 −0.489535
\(339\) −10.0000 −0.543125
\(340\) 4.00000 0.216930
\(341\) 0 0
\(342\) 1.00000 0.0540738
\(343\) 0 0
\(344\) 0 0
\(345\) 12.0000 0.646058
\(346\) −24.0000 −1.29025
\(347\) 18.0000 0.966291 0.483145 0.875540i \(-0.339494\pi\)
0.483145 + 0.875540i \(0.339494\pi\)
\(348\) −6.00000 −0.321634
\(349\) −4.00000 −0.214115 −0.107058 0.994253i \(-0.534143\pi\)
−0.107058 + 0.994253i \(0.534143\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) −2.00000 −0.106600
\(353\) 6.00000 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) −12.0000 −0.637793
\(355\) 8.00000 0.424596
\(356\) 0 0
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) 6.00000 0.316668 0.158334 0.987386i \(-0.449388\pi\)
0.158334 + 0.987386i \(0.449388\pi\)
\(360\) −2.00000 −0.105409
\(361\) 1.00000 0.0526316
\(362\) 14.0000 0.735824
\(363\) −7.00000 −0.367405
\(364\) 0 0
\(365\) 16.0000 0.837478
\(366\) 8.00000 0.418167
\(367\) 24.0000 1.25279 0.626395 0.779506i \(-0.284530\pi\)
0.626395 + 0.779506i \(0.284530\pi\)
\(368\) −6.00000 −0.312772
\(369\) 8.00000 0.416463
\(370\) 20.0000 1.03975
\(371\) 0 0
\(372\) 0 0
\(373\) −14.0000 −0.724893 −0.362446 0.932005i \(-0.618058\pi\)
−0.362446 + 0.932005i \(0.618058\pi\)
\(374\) 4.00000 0.206835
\(375\) 12.0000 0.619677
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) −6.00000 −0.308199 −0.154100 0.988055i \(-0.549248\pi\)
−0.154100 + 0.988055i \(0.549248\pi\)
\(380\) −2.00000 −0.102598
\(381\) 22.0000 1.12709
\(382\) −6.00000 −0.306987
\(383\) −24.0000 −1.22634 −0.613171 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 10.0000 0.508987
\(387\) 0 0
\(388\) 6.00000 0.304604
\(389\) −10.0000 −0.507020 −0.253510 0.967333i \(-0.581585\pi\)
−0.253510 + 0.967333i \(0.581585\pi\)
\(390\) −4.00000 −0.202548
\(391\) 12.0000 0.606866
\(392\) 0 0
\(393\) −4.00000 −0.201773
\(394\) 18.0000 0.906827
\(395\) −4.00000 −0.201262
\(396\) −2.00000 −0.100504
\(397\) 20.0000 1.00377 0.501886 0.864934i \(-0.332640\pi\)
0.501886 + 0.864934i \(0.332640\pi\)
\(398\) −20.0000 −1.00251
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) −2.00000 −0.0997509
\(403\) 0 0
\(404\) −2.00000 −0.0995037
\(405\) −2.00000 −0.0993808
\(406\) 0 0
\(407\) 20.0000 0.991363
\(408\) −2.00000 −0.0990148
\(409\) −2.00000 −0.0988936 −0.0494468 0.998777i \(-0.515746\pi\)
−0.0494468 + 0.998777i \(0.515746\pi\)
\(410\) −16.0000 −0.790184
\(411\) −10.0000 −0.493264
\(412\) 8.00000 0.394132
\(413\) 0 0
\(414\) −6.00000 −0.294884
\(415\) 8.00000 0.392705
\(416\) 2.00000 0.0980581
\(417\) −12.0000 −0.587643
\(418\) −2.00000 −0.0978232
\(419\) 20.0000 0.977064 0.488532 0.872546i \(-0.337533\pi\)
0.488532 + 0.872546i \(0.337533\pi\)
\(420\) 0 0
\(421\) 34.0000 1.65706 0.828529 0.559946i \(-0.189178\pi\)
0.828529 + 0.559946i \(0.189178\pi\)
\(422\) −10.0000 −0.486792
\(423\) 0 0
\(424\) −2.00000 −0.0971286
\(425\) 2.00000 0.0970143
\(426\) −4.00000 −0.193801
\(427\) 0 0
\(428\) −4.00000 −0.193347
\(429\) −4.00000 −0.193122
\(430\) 0 0
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 1.00000 0.0481125
\(433\) 30.0000 1.44171 0.720854 0.693087i \(-0.243750\pi\)
0.720854 + 0.693087i \(0.243750\pi\)
\(434\) 0 0
\(435\) 12.0000 0.575356
\(436\) −2.00000 −0.0957826
\(437\) −6.00000 −0.287019
\(438\) −8.00000 −0.382255
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 4.00000 0.190693
\(441\) 0 0
\(442\) −4.00000 −0.190261
\(443\) 6.00000 0.285069 0.142534 0.989790i \(-0.454475\pi\)
0.142534 + 0.989790i \(0.454475\pi\)
\(444\) −10.0000 −0.474579
\(445\) 0 0
\(446\) −8.00000 −0.378811
\(447\) −6.00000 −0.283790
\(448\) 0 0
\(449\) −2.00000 −0.0943858 −0.0471929 0.998886i \(-0.515028\pi\)
−0.0471929 + 0.998886i \(0.515028\pi\)
\(450\) −1.00000 −0.0471405
\(451\) −16.0000 −0.753411
\(452\) −10.0000 −0.470360
\(453\) 2.00000 0.0939682
\(454\) 8.00000 0.375459
\(455\) 0 0
\(456\) 1.00000 0.0468293
\(457\) 38.0000 1.77757 0.888783 0.458329i \(-0.151552\pi\)
0.888783 + 0.458329i \(0.151552\pi\)
\(458\) 8.00000 0.373815
\(459\) −2.00000 −0.0933520
\(460\) 12.0000 0.559503
\(461\) 22.0000 1.02464 0.512321 0.858794i \(-0.328786\pi\)
0.512321 + 0.858794i \(0.328786\pi\)
\(462\) 0 0
\(463\) −8.00000 −0.371792 −0.185896 0.982569i \(-0.559519\pi\)
−0.185896 + 0.982569i \(0.559519\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) −18.0000 −0.833834
\(467\) 20.0000 0.925490 0.462745 0.886492i \(-0.346865\pi\)
0.462745 + 0.886492i \(0.346865\pi\)
\(468\) 2.00000 0.0924500
\(469\) 0 0
\(470\) 0 0
\(471\) −12.0000 −0.552931
\(472\) −12.0000 −0.552345
\(473\) 0 0
\(474\) 2.00000 0.0918630
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) −2.00000 −0.0915737
\(478\) 18.0000 0.823301
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) −2.00000 −0.0912871
\(481\) −20.0000 −0.911922
\(482\) −6.00000 −0.273293
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) −12.0000 −0.544892
\(486\) 1.00000 0.0453609
\(487\) −42.0000 −1.90320 −0.951601 0.307337i \(-0.900562\pi\)
−0.951601 + 0.307337i \(0.900562\pi\)
\(488\) 8.00000 0.362143
\(489\) −8.00000 −0.361773
\(490\) 0 0
\(491\) −14.0000 −0.631811 −0.315906 0.948791i \(-0.602308\pi\)
−0.315906 + 0.948791i \(0.602308\pi\)
\(492\) 8.00000 0.360668
\(493\) 12.0000 0.540453
\(494\) 2.00000 0.0899843
\(495\) 4.00000 0.179787
\(496\) 0 0
\(497\) 0 0
\(498\) −4.00000 −0.179244
\(499\) −16.0000 −0.716258 −0.358129 0.933672i \(-0.616585\pi\)
−0.358129 + 0.933672i \(0.616585\pi\)
\(500\) 12.0000 0.536656
\(501\) 4.00000 0.178707
\(502\) −20.0000 −0.892644
\(503\) −8.00000 −0.356702 −0.178351 0.983967i \(-0.557076\pi\)
−0.178351 + 0.983967i \(0.557076\pi\)
\(504\) 0 0
\(505\) 4.00000 0.177998
\(506\) 12.0000 0.533465
\(507\) −9.00000 −0.399704
\(508\) 22.0000 0.976092
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 4.00000 0.177123
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 1.00000 0.0441511
\(514\) −4.00000 −0.176432
\(515\) −16.0000 −0.705044
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −24.0000 −1.05348
\(520\) −4.00000 −0.175412
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) −6.00000 −0.262613
\(523\) 36.0000 1.57417 0.787085 0.616844i \(-0.211589\pi\)
0.787085 + 0.616844i \(0.211589\pi\)
\(524\) −4.00000 −0.174741
\(525\) 0 0
\(526\) 30.0000 1.30806
\(527\) 0 0
\(528\) −2.00000 −0.0870388
\(529\) 13.0000 0.565217
\(530\) 4.00000 0.173749
\(531\) −12.0000 −0.520756
\(532\) 0 0
\(533\) 16.0000 0.693037
\(534\) 0 0
\(535\) 8.00000 0.345870
\(536\) −2.00000 −0.0863868
\(537\) −12.0000 −0.517838
\(538\) 24.0000 1.03471
\(539\) 0 0
\(540\) −2.00000 −0.0860663
\(541\) 34.0000 1.46177 0.730887 0.682498i \(-0.239107\pi\)
0.730887 + 0.682498i \(0.239107\pi\)
\(542\) −8.00000 −0.343629
\(543\) 14.0000 0.600798
\(544\) −2.00000 −0.0857493
\(545\) 4.00000 0.171341
\(546\) 0 0
\(547\) −26.0000 −1.11168 −0.555840 0.831289i \(-0.687603\pi\)
−0.555840 + 0.831289i \(0.687603\pi\)
\(548\) −10.0000 −0.427179
\(549\) 8.00000 0.341432
\(550\) 2.00000 0.0852803
\(551\) −6.00000 −0.255609
\(552\) −6.00000 −0.255377
\(553\) 0 0
\(554\) −10.0000 −0.424859
\(555\) 20.0000 0.848953
\(556\) −12.0000 −0.508913
\(557\) 42.0000 1.77960 0.889799 0.456354i \(-0.150845\pi\)
0.889799 + 0.456354i \(0.150845\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 4.00000 0.168880
\(562\) −6.00000 −0.253095
\(563\) −28.0000 −1.18006 −0.590030 0.807382i \(-0.700884\pi\)
−0.590030 + 0.807382i \(0.700884\pi\)
\(564\) 0 0
\(565\) 20.0000 0.841406
\(566\) 20.0000 0.840663
\(567\) 0 0
\(568\) −4.00000 −0.167836
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) −2.00000 −0.0837708
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) −4.00000 −0.167248
\(573\) −6.00000 −0.250654
\(574\) 0 0
\(575\) 6.00000 0.250217
\(576\) 1.00000 0.0416667
\(577\) −32.0000 −1.33218 −0.666089 0.745873i \(-0.732033\pi\)
−0.666089 + 0.745873i \(0.732033\pi\)
\(578\) −13.0000 −0.540729
\(579\) 10.0000 0.415586
\(580\) 12.0000 0.498273
\(581\) 0 0
\(582\) 6.00000 0.248708
\(583\) 4.00000 0.165663
\(584\) −8.00000 −0.331042
\(585\) −4.00000 −0.165380
\(586\) 0 0
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 24.0000 0.988064
\(591\) 18.0000 0.740421
\(592\) −10.0000 −0.410997
\(593\) −14.0000 −0.574911 −0.287456 0.957794i \(-0.592809\pi\)
−0.287456 + 0.957794i \(0.592809\pi\)
\(594\) −2.00000 −0.0820610
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) −20.0000 −0.818546
\(598\) −12.0000 −0.490716
\(599\) 36.0000 1.47092 0.735460 0.677568i \(-0.236966\pi\)
0.735460 + 0.677568i \(0.236966\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 30.0000 1.22373 0.611863 0.790964i \(-0.290420\pi\)
0.611863 + 0.790964i \(0.290420\pi\)
\(602\) 0 0
\(603\) −2.00000 −0.0814463
\(604\) 2.00000 0.0813788
\(605\) 14.0000 0.569181
\(606\) −2.00000 −0.0812444
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 1.00000 0.0405554
\(609\) 0 0
\(610\) −16.0000 −0.647821
\(611\) 0 0
\(612\) −2.00000 −0.0808452
\(613\) −14.0000 −0.565455 −0.282727 0.959200i \(-0.591239\pi\)
−0.282727 + 0.959200i \(0.591239\pi\)
\(614\) 12.0000 0.484281
\(615\) −16.0000 −0.645182
\(616\) 0 0
\(617\) 14.0000 0.563619 0.281809 0.959470i \(-0.409065\pi\)
0.281809 + 0.959470i \(0.409065\pi\)
\(618\) 8.00000 0.321807
\(619\) −24.0000 −0.964641 −0.482321 0.875995i \(-0.660206\pi\)
−0.482321 + 0.875995i \(0.660206\pi\)
\(620\) 0 0
\(621\) −6.00000 −0.240772
\(622\) 0 0
\(623\) 0 0
\(624\) 2.00000 0.0800641
\(625\) −19.0000 −0.760000
\(626\) 8.00000 0.319744
\(627\) −2.00000 −0.0798723
\(628\) −12.0000 −0.478852
\(629\) 20.0000 0.797452
\(630\) 0 0
\(631\) 4.00000 0.159237 0.0796187 0.996825i \(-0.474630\pi\)
0.0796187 + 0.996825i \(0.474630\pi\)
\(632\) 2.00000 0.0795557
\(633\) −10.0000 −0.397464
\(634\) −30.0000 −1.19145
\(635\) −44.0000 −1.74609
\(636\) −2.00000 −0.0793052
\(637\) 0 0
\(638\) 12.0000 0.475085
\(639\) −4.00000 −0.158238
\(640\) −2.00000 −0.0790569
\(641\) 10.0000 0.394976 0.197488 0.980305i \(-0.436722\pi\)
0.197488 + 0.980305i \(0.436722\pi\)
\(642\) −4.00000 −0.157867
\(643\) −8.00000 −0.315489 −0.157745 0.987480i \(-0.550422\pi\)
−0.157745 + 0.987480i \(0.550422\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −2.00000 −0.0786889
\(647\) 24.0000 0.943537 0.471769 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) 1.00000 0.0392837
\(649\) 24.0000 0.942082
\(650\) −2.00000 −0.0784465
\(651\) 0 0
\(652\) −8.00000 −0.313304
\(653\) 46.0000 1.80012 0.900060 0.435767i \(-0.143523\pi\)
0.900060 + 0.435767i \(0.143523\pi\)
\(654\) −2.00000 −0.0782062
\(655\) 8.00000 0.312586
\(656\) 8.00000 0.312348
\(657\) −8.00000 −0.312110
\(658\) 0 0
\(659\) 20.0000 0.779089 0.389545 0.921008i \(-0.372632\pi\)
0.389545 + 0.921008i \(0.372632\pi\)
\(660\) 4.00000 0.155700
\(661\) −34.0000 −1.32245 −0.661223 0.750189i \(-0.729962\pi\)
−0.661223 + 0.750189i \(0.729962\pi\)
\(662\) −14.0000 −0.544125
\(663\) −4.00000 −0.155347
\(664\) −4.00000 −0.155230
\(665\) 0 0
\(666\) −10.0000 −0.387492
\(667\) 36.0000 1.39393
\(668\) 4.00000 0.154765
\(669\) −8.00000 −0.309298
\(670\) 4.00000 0.154533
\(671\) −16.0000 −0.617673
\(672\) 0 0
\(673\) 14.0000 0.539660 0.269830 0.962908i \(-0.413032\pi\)
0.269830 + 0.962908i \(0.413032\pi\)
\(674\) 22.0000 0.847408
\(675\) −1.00000 −0.0384900
\(676\) −9.00000 −0.346154
\(677\) −20.0000 −0.768662 −0.384331 0.923195i \(-0.625568\pi\)
−0.384331 + 0.923195i \(0.625568\pi\)
\(678\) −10.0000 −0.384048
\(679\) 0 0
\(680\) 4.00000 0.153393
\(681\) 8.00000 0.306561
\(682\) 0 0
\(683\) 16.0000 0.612223 0.306111 0.951996i \(-0.400972\pi\)
0.306111 + 0.951996i \(0.400972\pi\)
\(684\) 1.00000 0.0382360
\(685\) 20.0000 0.764161
\(686\) 0 0
\(687\) 8.00000 0.305219
\(688\) 0 0
\(689\) −4.00000 −0.152388
\(690\) 12.0000 0.456832
\(691\) 28.0000 1.06517 0.532585 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) −24.0000 −0.912343
\(693\) 0 0
\(694\) 18.0000 0.683271
\(695\) 24.0000 0.910372
\(696\) −6.00000 −0.227429
\(697\) −16.0000 −0.606043
\(698\) −4.00000 −0.151402
\(699\) −18.0000 −0.680823
\(700\) 0 0
\(701\) 6.00000 0.226617 0.113308 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) 2.00000 0.0754851
\(703\) −10.0000 −0.377157
\(704\) −2.00000 −0.0753778
\(705\) 0 0
\(706\) 6.00000 0.225813
\(707\) 0 0
\(708\) −12.0000 −0.450988
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 8.00000 0.300235
\(711\) 2.00000 0.0750059
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 8.00000 0.299183
\(716\) −12.0000 −0.448461
\(717\) 18.0000 0.672222
\(718\) 6.00000 0.223918
\(719\) −40.0000 −1.49175 −0.745874 0.666087i \(-0.767968\pi\)
−0.745874 + 0.666087i \(0.767968\pi\)
\(720\) −2.00000 −0.0745356
\(721\) 0 0
\(722\) 1.00000 0.0372161
\(723\) −6.00000 −0.223142
\(724\) 14.0000 0.520306
\(725\) 6.00000 0.222834
\(726\) −7.00000 −0.259794
\(727\) −8.00000 −0.296704 −0.148352 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 16.0000 0.592187
\(731\) 0 0
\(732\) 8.00000 0.295689
\(733\) 44.0000 1.62518 0.812589 0.582838i \(-0.198058\pi\)
0.812589 + 0.582838i \(0.198058\pi\)
\(734\) 24.0000 0.885856
\(735\) 0 0
\(736\) −6.00000 −0.221163
\(737\) 4.00000 0.147342
\(738\) 8.00000 0.294484
\(739\) −4.00000 −0.147142 −0.0735712 0.997290i \(-0.523440\pi\)
−0.0735712 + 0.997290i \(0.523440\pi\)
\(740\) 20.0000 0.735215
\(741\) 2.00000 0.0734718
\(742\) 0 0
\(743\) 24.0000 0.880475 0.440237 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) 0 0
\(745\) 12.0000 0.439646
\(746\) −14.0000 −0.512576
\(747\) −4.00000 −0.146352
\(748\) 4.00000 0.146254
\(749\) 0 0
\(750\) 12.0000 0.438178
\(751\) −10.0000 −0.364905 −0.182453 0.983215i \(-0.558404\pi\)
−0.182453 + 0.983215i \(0.558404\pi\)
\(752\) 0 0
\(753\) −20.0000 −0.728841
\(754\) −12.0000 −0.437014
\(755\) −4.00000 −0.145575
\(756\) 0 0
\(757\) 18.0000 0.654221 0.327111 0.944986i \(-0.393925\pi\)
0.327111 + 0.944986i \(0.393925\pi\)
\(758\) −6.00000 −0.217930
\(759\) 12.0000 0.435572
\(760\) −2.00000 −0.0725476
\(761\) −22.0000 −0.797499 −0.398750 0.917060i \(-0.630556\pi\)
−0.398750 + 0.917060i \(0.630556\pi\)
\(762\) 22.0000 0.796976
\(763\) 0 0
\(764\) −6.00000 −0.217072
\(765\) 4.00000 0.144620
\(766\) −24.0000 −0.867155
\(767\) −24.0000 −0.866590
\(768\) 1.00000 0.0360844
\(769\) 16.0000 0.576975 0.288487 0.957484i \(-0.406848\pi\)
0.288487 + 0.957484i \(0.406848\pi\)
\(770\) 0 0
\(771\) −4.00000 −0.144056
\(772\) 10.0000 0.359908
\(773\) −24.0000 −0.863220 −0.431610 0.902060i \(-0.642054\pi\)
−0.431610 + 0.902060i \(0.642054\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 6.00000 0.215387
\(777\) 0 0
\(778\) −10.0000 −0.358517
\(779\) 8.00000 0.286630
\(780\) −4.00000 −0.143223
\(781\) 8.00000 0.286263
\(782\) 12.0000 0.429119
\(783\) −6.00000 −0.214423
\(784\) 0 0
\(785\) 24.0000 0.856597
\(786\) −4.00000 −0.142675
\(787\) −36.0000 −1.28326 −0.641631 0.767014i \(-0.721742\pi\)
−0.641631 + 0.767014i \(0.721742\pi\)
\(788\) 18.0000 0.641223
\(789\) 30.0000 1.06803
\(790\) −4.00000 −0.142314
\(791\) 0 0
\(792\) −2.00000 −0.0710669
\(793\) 16.0000 0.568177
\(794\) 20.0000 0.709773
\(795\) 4.00000 0.141865
\(796\) −20.0000 −0.708881
\(797\) −48.0000 −1.70025 −0.850124 0.526583i \(-0.823473\pi\)
−0.850124 + 0.526583i \(0.823473\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) −18.0000 −0.635602
\(803\) 16.0000 0.564628
\(804\) −2.00000 −0.0705346
\(805\) 0 0
\(806\) 0 0
\(807\) 24.0000 0.844840
\(808\) −2.00000 −0.0703598
\(809\) 2.00000 0.0703163 0.0351581 0.999382i \(-0.488807\pi\)
0.0351581 + 0.999382i \(0.488807\pi\)
\(810\) −2.00000 −0.0702728
\(811\) −20.0000 −0.702295 −0.351147 0.936320i \(-0.614208\pi\)
−0.351147 + 0.936320i \(0.614208\pi\)
\(812\) 0 0
\(813\) −8.00000 −0.280572
\(814\) 20.0000 0.701000
\(815\) 16.0000 0.560456
\(816\) −2.00000 −0.0700140
\(817\) 0 0
\(818\) −2.00000 −0.0699284
\(819\) 0 0
\(820\) −16.0000 −0.558744
\(821\) −38.0000 −1.32621 −0.663105 0.748527i \(-0.730762\pi\)
−0.663105 + 0.748527i \(0.730762\pi\)
\(822\) −10.0000 −0.348790
\(823\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(824\) 8.00000 0.278693
\(825\) 2.00000 0.0696311
\(826\) 0 0
\(827\) −28.0000 −0.973655 −0.486828 0.873498i \(-0.661846\pi\)
−0.486828 + 0.873498i \(0.661846\pi\)
\(828\) −6.00000 −0.208514
\(829\) 34.0000 1.18087 0.590434 0.807086i \(-0.298956\pi\)
0.590434 + 0.807086i \(0.298956\pi\)
\(830\) 8.00000 0.277684
\(831\) −10.0000 −0.346896
\(832\) 2.00000 0.0693375
\(833\) 0 0
\(834\) −12.0000 −0.415526
\(835\) −8.00000 −0.276851
\(836\) −2.00000 −0.0691714
\(837\) 0 0
\(838\) 20.0000 0.690889
\(839\) 12.0000 0.414286 0.207143 0.978311i \(-0.433583\pi\)
0.207143 + 0.978311i \(0.433583\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 34.0000 1.17172
\(843\) −6.00000 −0.206651
\(844\) −10.0000 −0.344214
\(845\) 18.0000 0.619219
\(846\) 0 0
\(847\) 0 0
\(848\) −2.00000 −0.0686803
\(849\) 20.0000 0.686398
\(850\) 2.00000 0.0685994
\(851\) 60.0000 2.05677
\(852\) −4.00000 −0.137038
\(853\) −8.00000 −0.273915 −0.136957 0.990577i \(-0.543732\pi\)
−0.136957 + 0.990577i \(0.543732\pi\)
\(854\) 0 0
\(855\) −2.00000 −0.0683986
\(856\) −4.00000 −0.136717
\(857\) 32.0000 1.09310 0.546550 0.837427i \(-0.315941\pi\)
0.546550 + 0.837427i \(0.315941\pi\)
\(858\) −4.00000 −0.136558
\(859\) 56.0000 1.91070 0.955348 0.295484i \(-0.0954809\pi\)
0.955348 + 0.295484i \(0.0954809\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −12.0000 −0.408722
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 1.00000 0.0340207
\(865\) 48.0000 1.63205
\(866\) 30.0000 1.01944
\(867\) −13.0000 −0.441503
\(868\) 0 0
\(869\) −4.00000 −0.135691
\(870\) 12.0000 0.406838
\(871\) −4.00000 −0.135535
\(872\) −2.00000 −0.0677285
\(873\) 6.00000 0.203069
\(874\) −6.00000 −0.202953
\(875\) 0 0
\(876\) −8.00000 −0.270295
\(877\) −2.00000 −0.0675352 −0.0337676 0.999430i \(-0.510751\pi\)
−0.0337676 + 0.999430i \(0.510751\pi\)
\(878\) 8.00000 0.269987
\(879\) 0 0
\(880\) 4.00000 0.134840
\(881\) −34.0000 −1.14549 −0.572745 0.819734i \(-0.694121\pi\)
−0.572745 + 0.819734i \(0.694121\pi\)
\(882\) 0 0
\(883\) 8.00000 0.269221 0.134611 0.990899i \(-0.457022\pi\)
0.134611 + 0.990899i \(0.457022\pi\)
\(884\) −4.00000 −0.134535
\(885\) 24.0000 0.806751
\(886\) 6.00000 0.201574
\(887\) −8.00000 −0.268614 −0.134307 0.990940i \(-0.542881\pi\)
−0.134307 + 0.990940i \(0.542881\pi\)
\(888\) −10.0000 −0.335578
\(889\) 0 0
\(890\) 0 0
\(891\) −2.00000 −0.0670025
\(892\) −8.00000 −0.267860
\(893\) 0 0
\(894\) −6.00000 −0.200670
\(895\) 24.0000 0.802232
\(896\) 0 0
\(897\) −12.0000 −0.400668
\(898\) −2.00000 −0.0667409
\(899\) 0 0
\(900\) −1.00000 −0.0333333
\(901\) 4.00000 0.133259
\(902\) −16.0000 −0.532742
\(903\) 0 0
\(904\) −10.0000 −0.332595
\(905\) −28.0000 −0.930751
\(906\) 2.00000 0.0664455
\(907\) −6.00000 −0.199227 −0.0996134 0.995026i \(-0.531761\pi\)
−0.0996134 + 0.995026i \(0.531761\pi\)
\(908\) 8.00000 0.265489
\(909\) −2.00000 −0.0663358
\(910\) 0 0
\(911\) −20.0000 −0.662630 −0.331315 0.943520i \(-0.607492\pi\)
−0.331315 + 0.943520i \(0.607492\pi\)
\(912\) 1.00000 0.0331133
\(913\) 8.00000 0.264761
\(914\) 38.0000 1.25693
\(915\) −16.0000 −0.528944
\(916\) 8.00000 0.264327
\(917\) 0 0
\(918\) −2.00000 −0.0660098
\(919\) 40.0000 1.31948 0.659739 0.751495i \(-0.270667\pi\)
0.659739 + 0.751495i \(0.270667\pi\)
\(920\) 12.0000 0.395628
\(921\) 12.0000 0.395413
\(922\) 22.0000 0.724531
\(923\) −8.00000 −0.263323
\(924\) 0 0
\(925\) 10.0000 0.328798
\(926\) −8.00000 −0.262896
\(927\) 8.00000 0.262754
\(928\) −6.00000 −0.196960
\(929\) −42.0000 −1.37798 −0.688988 0.724773i \(-0.741945\pi\)
−0.688988 + 0.724773i \(0.741945\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −18.0000 −0.589610
\(933\) 0 0
\(934\) 20.0000 0.654420
\(935\) −8.00000 −0.261628
\(936\) 2.00000 0.0653720
\(937\) 28.0000 0.914720 0.457360 0.889282i \(-0.348795\pi\)
0.457360 + 0.889282i \(0.348795\pi\)
\(938\) 0 0
\(939\) 8.00000 0.261070
\(940\) 0 0
\(941\) 24.0000 0.782378 0.391189 0.920310i \(-0.372064\pi\)
0.391189 + 0.920310i \(0.372064\pi\)
\(942\) −12.0000 −0.390981
\(943\) −48.0000 −1.56310
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) 0 0
\(947\) 18.0000 0.584921 0.292461 0.956278i \(-0.405526\pi\)
0.292461 + 0.956278i \(0.405526\pi\)
\(948\) 2.00000 0.0649570
\(949\) −16.0000 −0.519382
\(950\) −1.00000 −0.0324443
\(951\) −30.0000 −0.972817
\(952\) 0 0
\(953\) −42.0000 −1.36051 −0.680257 0.732974i \(-0.738132\pi\)
−0.680257 + 0.732974i \(0.738132\pi\)
\(954\) −2.00000 −0.0647524
\(955\) 12.0000 0.388311
\(956\) 18.0000 0.582162
\(957\) 12.0000 0.387905
\(958\) 24.0000 0.775405
\(959\) 0 0
\(960\) −2.00000 −0.0645497
\(961\) −31.0000 −1.00000
\(962\) −20.0000 −0.644826
\(963\) −4.00000 −0.128898
\(964\) −6.00000 −0.193247
\(965\) −20.0000 −0.643823
\(966\) 0 0
\(967\) −44.0000 −1.41494 −0.707472 0.706741i \(-0.750165\pi\)
−0.707472 + 0.706741i \(0.750165\pi\)
\(968\) −7.00000 −0.224989
\(969\) −2.00000 −0.0642493
\(970\) −12.0000 −0.385297
\(971\) −16.0000 −0.513464 −0.256732 0.966483i \(-0.582646\pi\)
−0.256732 + 0.966483i \(0.582646\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −42.0000 −1.34577
\(975\) −2.00000 −0.0640513
\(976\) 8.00000 0.256074
\(977\) −18.0000 −0.575871 −0.287936 0.957650i \(-0.592969\pi\)
−0.287936 + 0.957650i \(0.592969\pi\)
\(978\) −8.00000 −0.255812
\(979\) 0 0
\(980\) 0 0
\(981\) −2.00000 −0.0638551
\(982\) −14.0000 −0.446758
\(983\) −60.0000 −1.91370 −0.956851 0.290578i \(-0.906153\pi\)
−0.956851 + 0.290578i \(0.906153\pi\)
\(984\) 8.00000 0.255031
\(985\) −36.0000 −1.14706
\(986\) 12.0000 0.382158
\(987\) 0 0
\(988\) 2.00000 0.0636285
\(989\) 0 0
\(990\) 4.00000 0.127128
\(991\) 18.0000 0.571789 0.285894 0.958261i \(-0.407709\pi\)
0.285894 + 0.958261i \(0.407709\pi\)
\(992\) 0 0
\(993\) −14.0000 −0.444277
\(994\) 0 0
\(995\) 40.0000 1.26809
\(996\) −4.00000 −0.126745
\(997\) −4.00000 −0.126681 −0.0633406 0.997992i \(-0.520175\pi\)
−0.0633406 + 0.997992i \(0.520175\pi\)
\(998\) −16.0000 −0.506471
\(999\) −10.0000 −0.316386
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 5586.2.a.z.1.1 yes 1
7.6 odd 2 5586.2.a.w.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5586.2.a.w.1.1 1 7.6 odd 2
5586.2.a.z.1.1 yes 1 1.1 even 1 trivial