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Properties

Label 5586.2.a.w.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

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

Display 100 coefficients 10 coefficients

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 \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) −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 \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) −2.00000 −0.516398
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 − 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 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 \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 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 \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) −2.00000 −0.258199
\(61\) −8.00000 −1.02430 −0.512148 − 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 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 \(-0.344915\pi\)
0.468165 + 0.883641i \(0.344915\pi\)
\(74\) −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 \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 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 \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 0 0
\(99\) −2.00000 −0.201008
\(100\) −1.00000 −0.100000
\(101\) 2.00000 0.199007 0.0995037 − 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) −2.00000 −0.198030
\(103\) −8.00000 −0.788263 −0.394132 − 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) −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 \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 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 \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 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 \(-0.341053\pi\)
0.478852 + 0.877896i \(0.341053\pi\)
\(158\) 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 \(-0.549462\pi\)
−0.154765 + 0.987951i \(0.549462\pi\)
\(168\) 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 \(-0.134271\pi\)
0.912343 + 0.409426i \(0.134271\pi\)
\(174\) 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 \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) 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 \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) −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 \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) −10.0000 −0.665190
\(227\) −8.00000 −0.530979 −0.265489 − 0.964114i \(-0.585534\pi\)
−0.265489 + 0.964114i \(0.585534\pi\)
\(228\) 1.00000 0.0662266
\(229\) −8.00000 −0.528655 −0.264327 − 0.964433i \(-0.585150\pi\)
−0.264327 + 0.964433i \(0.585150\pi\)
\(230\) −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 \(-0.438098\pi\)
0.193247 + 0.981150i \(0.438098\pi\)
\(242\) −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 \(-0.282565\pi\)
0.631194 + 0.775625i \(0.282565\pi\)
\(252\) 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 \(-0.460185\pi\)
0.124757 + 0.992187i \(0.460185\pi\)
\(258\) 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 \(-0.761251\pi\)
−0.731653 + 0.681677i \(0.761251\pi\)
\(270\) −2.00000 −0.121716
\(271\) 8.00000 0.485965 0.242983 − 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 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 \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) −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 \(-0.611253\pi\)
−0.342438 + 0.939540i \(0.611253\pi\)
\(308\) 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 \(-0.572595\pi\)
−0.226093 + 0.974106i \(0.572595\pi\)
\(314\) 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 \(-0.465857\pi\)
0.107058 + 0.994253i \(0.465857\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) −2.00000 −0.106600
\(353\) −6.00000 −0.319348 −0.159674 − 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) −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 \(-0.715470\pi\)
−0.626395 + 0.779506i \(0.715470\pi\)
\(368\) −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 \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) −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 \(-0.667360\pi\)
−0.501886 + 0.864934i \(0.667360\pi\)
\(398\) 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 \(-0.484254\pi\)
0.0494468 + 0.998777i \(0.484254\pi\)
\(410\) −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 \(-0.662467\pi\)
−0.488532 + 0.872546i \(0.662467\pi\)
\(420\) 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 \(-0.756250\pi\)
−0.720854 + 0.693087i \(0.756250\pi\)
\(434\) 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 \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) −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 \(-0.671214\pi\)
−0.512321 + 0.858794i \(0.671214\pi\)
\(462\) 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 \(-0.653135\pi\)
−0.462745 + 0.886492i \(0.653135\pi\)
\(468\) −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 \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) −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 \(-0.442924\pi\)
0.178351 + 0.983967i \(0.442924\pi\)
\(504\) 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 \(-0.788411\pi\)
−0.787085 + 0.616844i \(0.788411\pi\)
\(524\) 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 \(-0.299116\pi\)
0.590030 + 0.807382i \(0.299116\pi\)
\(564\) 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 \(-0.267967\pi\)
0.666089 + 0.745873i \(0.267967\pi\)
\(578\) −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 \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 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 \(-0.407191\pi\)
0.287456 + 0.957794i \(0.407191\pi\)
\(594\) 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 \(-0.709580\pi\)
−0.611863 + 0.790964i \(0.709580\pi\)
\(602\) 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 \(-0.339794\pi\)
0.482321 + 0.875995i \(0.339794\pi\)
\(620\) 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 \(-0.449578\pi\)
0.157745 + 0.987480i \(0.449578\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −2.00000 −0.0786889
\(647\) −24.0000 −0.943537 −0.471769 − 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) 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 \(-0.270038\pi\)
0.661223 + 0.750189i \(0.270038\pi\)
\(662\) −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 \(-0.374432\pi\)
0.384331 + 0.923195i \(0.374432\pi\)
\(678\) 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 \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) 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 \(-0.232032\pi\)
0.745874 + 0.666087i \(0.232032\pi\)
\(720\) 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 \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) 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 \(-0.801942\pi\)
−0.812589 + 0.582838i \(0.801942\pi\)
\(734\) −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 \(-0.369444\pi\)
0.398750 + 0.917060i \(0.369444\pi\)
\(762\) −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 \(-0.593152\pi\)
−0.288487 + 0.957484i \(0.593152\pi\)
\(770\) 0 0
\(771\) −4.00000 −0.144056
\(772\) 10.0000 0.359908
\(773\) 24.0000 0.863220 0.431610 − 0.902060i \(-0.357946\pi\)
0.431610 + 0.902060i \(0.357946\pi\)
\(774\) 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 \(-0.278258\pi\)
0.641631 + 0.767014i \(0.278258\pi\)
\(788\) 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 \(-0.176527\pi\)
0.850124 + 0.526583i \(0.176527\pi\)
\(798\) 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 \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) 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 \(-0.701044\pi\)
−0.590434 + 0.807086i \(0.701044\pi\)
\(830\) 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 \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) 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 \(-0.456268\pi\)
0.136957 + 0.990577i \(0.456268\pi\)
\(854\) 0 0
\(855\) −2.00000 −0.0683986
\(856\) −4.00000 −0.136717
\(857\) −32.0000 −1.09310 −0.546550 − 0.837427i \(-0.684059\pi\)
−0.546550 + 0.837427i \(0.684059\pi\)
\(858\) −4.00000 −0.136558
\(859\) −56.0000 −1.91070 −0.955348 − 0.295484i \(-0.904519\pi\)
−0.955348 + 0.295484i \(0.904519\pi\)
\(860\) 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 \(-0.305879\pi\)
0.572745 + 0.819734i \(0.305879\pi\)
\(882\) 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 \(-0.457119\pi\)
0.134307 + 0.990940i \(0.457119\pi\)
\(888\) 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 \(-0.258055\pi\)
0.688988 + 0.724773i \(0.258055\pi\)
\(930\) 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 \(-0.651205\pi\)
−0.457360 + 0.889282i \(0.651205\pi\)
\(938\) 0 0
\(939\) 8.00000 0.261070
\(940\) 0 0
\(941\) −24.0000 −0.782378 −0.391189 − 0.920310i \(-0.627936\pi\)
−0.391189 + 0.920310i \(0.627936\pi\)
\(942\) −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 \(-0.417354\pi\)
0.256732 + 0.966483i \(0.417354\pi\)
\(972\) −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 \(-0.0938475\pi\)
0.956851 + 0.290578i \(0.0938475\pi\)
\(984\) 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 \(-0.479825\pi\)
0.0633406 + 0.997992i \(0.479825\pi\)
\(998\) −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.w.1.1 ✓ 1
7.6 odd 2 5586.2.a.z.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5586.2.a.w.1.1 ✓ 1 1.1 even 1 trivial
5586.2.a.z.1.1 yes 1 7.6 odd 2