Properties

Label 117.2.a.a.1.1
Level $117$
Weight $2$
Character 117.1
Self dual yes
Analytic conductor $0.934$
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] = [117,2,Mod(1,117)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(117, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("117.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 117.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(0.934249703649\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 117.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{4} -2.00000 q^{5} -4.00000 q^{7} +3.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{4} -2.00000 q^{5} -4.00000 q^{7} +3.00000 q^{8} +2.00000 q^{10} -4.00000 q^{11} +1.00000 q^{13} +4.00000 q^{14} -1.00000 q^{16} -2.00000 q^{17} +2.00000 q^{20} +4.00000 q^{22} -1.00000 q^{25} -1.00000 q^{26} +4.00000 q^{28} +10.0000 q^{29} +4.00000 q^{31} -5.00000 q^{32} +2.00000 q^{34} +8.00000 q^{35} -2.00000 q^{37} -6.00000 q^{40} -6.00000 q^{41} -12.0000 q^{43} +4.00000 q^{44} +9.00000 q^{49} +1.00000 q^{50} -1.00000 q^{52} -6.00000 q^{53} +8.00000 q^{55} -12.0000 q^{56} -10.0000 q^{58} -12.0000 q^{59} -2.00000 q^{61} -4.00000 q^{62} +7.00000 q^{64} -2.00000 q^{65} -8.00000 q^{67} +2.00000 q^{68} -8.00000 q^{70} +2.00000 q^{73} +2.00000 q^{74} +16.0000 q^{77} +8.00000 q^{79} +2.00000 q^{80} +6.00000 q^{82} -4.00000 q^{83} +4.00000 q^{85} +12.0000 q^{86} -12.0000 q^{88} +2.00000 q^{89} -4.00000 q^{91} +10.0000 q^{97} -9.00000 q^{98} +O(q^{100})\)

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 −0.353553 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) 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
\(7\) −4.00000 −1.51186 −0.755929 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 3.00000 1.06066
\(9\) 0 0
\(10\) 2.00000 0.632456
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 4.00000 1.06904
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 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
\(28\) 4.00000 0.755929
\(29\) 10.0000 1.85695 0.928477 0.371391i \(-0.121119\pi\)
0.928477 + 0.371391i \(0.121119\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) −5.00000 −0.883883
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) 8.00000 1.35225
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −6.00000 −0.948683
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) −12.0000 −1.82998 −0.914991 0.403473i \(-0.867803\pi\)
−0.914991 + 0.403473i \(0.867803\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −1.00000 −0.138675
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 8.00000 1.07872
\(56\) −12.0000 −1.60357
\(57\) 0 0
\(58\) −10.0000 −1.31306
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) −4.00000 −0.508001
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 2.00000 0.242536
\(69\) 0 0
\(70\) −8.00000 −0.956183
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) 0 0
\(77\) 16.0000 1.82337
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 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
\(85\) 4.00000 0.433861
\(86\) 12.0000 1.29399
\(87\) 0 0
\(88\) −12.0000 −1.27920
\(89\) 2.00000 0.212000 0.106000 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 0 0
\(91\) −4.00000 −0.419314
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 10.0000 1.01535 0.507673 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) −9.00000 −0.909137
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 18.0000 1.79107 0.895533 0.444994i \(-0.146794\pi\)
0.895533 + 0.444994i \(0.146794\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 3.00000 0.294174
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) −8.00000 −0.762770
\(111\) 0 0
\(112\) 4.00000 0.377964
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −10.0000 −0.928477
\(117\) 0 0
\(118\) 12.0000 1.10469
\(119\) 8.00000 0.733359
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 2.00000 0.181071
\(123\) 0 0
\(124\) −4.00000 −0.359211
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 3.00000 0.265165
\(129\) 0 0
\(130\) 2.00000 0.175412
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) −8.00000 −0.676123
\(141\) 0 0
\(142\) 0 0
\(143\) −4.00000 −0.334497
\(144\) 0 0
\(145\) −20.0000 −1.66091
\(146\) −2.00000 −0.165521
\(147\) 0 0
\(148\) 2.00000 0.164399
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) 4.00000 0.325515 0.162758 0.986666i \(-0.447961\pi\)
0.162758 + 0.986666i \(0.447961\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −16.0000 −1.28932
\(155\) −8.00000 −0.642575
\(156\) 0 0
\(157\) −18.0000 −1.43656 −0.718278 0.695756i \(-0.755069\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) −8.00000 −0.636446
\(159\) 0 0
\(160\) 10.0000 0.790569
\(161\) 0 0
\(162\) 0 0
\(163\) 8.00000 0.626608 0.313304 0.949653i \(-0.398564\pi\)
0.313304 + 0.949653i \(0.398564\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) −4.00000 −0.306786
\(171\) 0 0
\(172\) 12.0000 0.914991
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) 4.00000 0.302372
\(176\) 4.00000 0.301511
\(177\) 0 0
\(178\) −2.00000 −0.149906
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 4.00000 0.296500
\(183\) 0 0
\(184\) 0 0
\(185\) 4.00000 0.294086
\(186\) 0 0
\(187\) 8.00000 0.585018
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 0 0
\(193\) 18.0000 1.29567 0.647834 0.761781i \(-0.275675\pi\)
0.647834 + 0.761781i \(0.275675\pi\)
\(194\) −10.0000 −0.717958
\(195\) 0 0
\(196\) −9.00000 −0.642857
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) −3.00000 −0.212132
\(201\) 0 0
\(202\) −18.0000 −1.26648
\(203\) −40.0000 −2.80745
\(204\) 0 0
\(205\) 12.0000 0.838116
\(206\) 0 0
\(207\) 0 0
\(208\) −1.00000 −0.0693375
\(209\) 0 0
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 6.00000 0.412082
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) 24.0000 1.63679
\(216\) 0 0
\(217\) −16.0000 −1.08615
\(218\) 2.00000 0.135457
\(219\) 0 0
\(220\) −8.00000 −0.539360
\(221\) −2.00000 −0.134535
\(222\) 0 0
\(223\) 4.00000 0.267860 0.133930 0.990991i \(-0.457240\pi\)
0.133930 + 0.990991i \(0.457240\pi\)
\(224\) 20.0000 1.33631
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) 20.0000 1.32745 0.663723 0.747978i \(-0.268975\pi\)
0.663723 + 0.747978i \(0.268975\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 30.0000 1.96960
\(233\) 14.0000 0.917170 0.458585 0.888650i \(-0.348356\pi\)
0.458585 + 0.888650i \(0.348356\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 12.0000 0.781133
\(237\) 0 0
\(238\) −8.00000 −0.518563
\(239\) 24.0000 1.55243 0.776215 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) −5.00000 −0.321412
\(243\) 0 0
\(244\) 2.00000 0.128037
\(245\) −18.0000 −1.14998
\(246\) 0 0
\(247\) 0 0
\(248\) 12.0000 0.762001
\(249\) 0 0
\(250\) −12.0000 −0.758947
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 16.0000 1.00393
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) −26.0000 −1.62184 −0.810918 0.585160i \(-0.801032\pi\)
−0.810918 + 0.585160i \(0.801032\pi\)
\(258\) 0 0
\(259\) 8.00000 0.497096
\(260\) 2.00000 0.124035
\(261\) 0 0
\(262\) 4.00000 0.247121
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 0 0
\(265\) 12.0000 0.737154
\(266\) 0 0
\(267\) 0 0
\(268\) 8.00000 0.488678
\(269\) −22.0000 −1.34136 −0.670682 0.741745i \(-0.733998\pi\)
−0.670682 + 0.741745i \(0.733998\pi\)
\(270\) 0 0
\(271\) −12.0000 −0.728948 −0.364474 0.931214i \(-0.618751\pi\)
−0.364474 + 0.931214i \(0.618751\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) 6.00000 0.362473
\(275\) 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
\(280\) 24.0000 1.43427
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) 12.0000 0.713326 0.356663 0.934233i \(-0.383914\pi\)
0.356663 + 0.934233i \(0.383914\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 4.00000 0.236525
\(287\) 24.0000 1.41668
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 20.0000 1.17444
\(291\) 0 0
\(292\) −2.00000 −0.117041
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) 24.0000 1.39733
\(296\) −6.00000 −0.348743
\(297\) 0 0
\(298\) −6.00000 −0.347571
\(299\) 0 0
\(300\) 0 0
\(301\) 48.0000 2.76667
\(302\) −4.00000 −0.230174
\(303\) 0 0
\(304\) 0 0
\(305\) 4.00000 0.229039
\(306\) 0 0
\(307\) −16.0000 −0.913168 −0.456584 0.889680i \(-0.650927\pi\)
−0.456584 + 0.889680i \(0.650927\pi\)
\(308\) −16.0000 −0.911685
\(309\) 0 0
\(310\) 8.00000 0.454369
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 18.0000 1.01580
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) −26.0000 −1.46031 −0.730153 0.683284i \(-0.760551\pi\)
−0.730153 + 0.683284i \(0.760551\pi\)
\(318\) 0 0
\(319\) −40.0000 −2.23957
\(320\) −14.0000 −0.782624
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −1.00000 −0.0554700
\(326\) −8.00000 −0.443079
\(327\) 0 0
\(328\) −18.0000 −0.993884
\(329\) 0 0
\(330\) 0 0
\(331\) −16.0000 −0.879440 −0.439720 0.898135i \(-0.644922\pi\)
−0.439720 + 0.898135i \(0.644922\pi\)
\(332\) 4.00000 0.219529
\(333\) 0 0
\(334\) −8.00000 −0.437741
\(335\) 16.0000 0.874173
\(336\) 0 0
\(337\) 18.0000 0.980522 0.490261 0.871576i \(-0.336901\pi\)
0.490261 + 0.871576i \(0.336901\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 0 0
\(340\) −4.00000 −0.216930
\(341\) −16.0000 −0.866449
\(342\) 0 0
\(343\) −8.00000 −0.431959
\(344\) −36.0000 −1.94099
\(345\) 0 0
\(346\) 6.00000 0.322562
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 0 0
\(349\) −26.0000 −1.39175 −0.695874 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) −4.00000 −0.213809
\(351\) 0 0
\(352\) 20.0000 1.06600
\(353\) 2.00000 0.106449 0.0532246 0.998583i \(-0.483050\pi\)
0.0532246 + 0.998583i \(0.483050\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −2.00000 −0.106000
\(357\) 0 0
\(358\) 4.00000 0.211407
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 10.0000 0.525588
\(363\) 0 0
\(364\) 4.00000 0.209657
\(365\) −4.00000 −0.209370
\(366\) 0 0
\(367\) 16.0000 0.835193 0.417597 0.908633i \(-0.362873\pi\)
0.417597 + 0.908633i \(0.362873\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) −4.00000 −0.207950
\(371\) 24.0000 1.24602
\(372\) 0 0
\(373\) −26.0000 −1.34623 −0.673114 0.739538i \(-0.735044\pi\)
−0.673114 + 0.739538i \(0.735044\pi\)
\(374\) −8.00000 −0.413670
\(375\) 0 0
\(376\) 0 0
\(377\) 10.0000 0.515026
\(378\) 0 0
\(379\) −24.0000 −1.23280 −0.616399 0.787434i \(-0.711409\pi\)
−0.616399 + 0.787434i \(0.711409\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 8.00000 0.409316
\(383\) 16.0000 0.817562 0.408781 0.912633i \(-0.365954\pi\)
0.408781 + 0.912633i \(0.365954\pi\)
\(384\) 0 0
\(385\) −32.0000 −1.63087
\(386\) −18.0000 −0.916176
\(387\) 0 0
\(388\) −10.0000 −0.507673
\(389\) −22.0000 −1.11544 −0.557722 0.830028i \(-0.688325\pi\)
−0.557722 + 0.830028i \(0.688325\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 27.0000 1.36371
\(393\) 0 0
\(394\) 18.0000 0.906827
\(395\) −16.0000 −0.805047
\(396\) 0 0
\(397\) 38.0000 1.90717 0.953583 0.301131i \(-0.0973643\pi\)
0.953583 + 0.301131i \(0.0973643\pi\)
\(398\) −8.00000 −0.401004
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −22.0000 −1.09863 −0.549314 0.835616i \(-0.685111\pi\)
−0.549314 + 0.835616i \(0.685111\pi\)
\(402\) 0 0
\(403\) 4.00000 0.199254
\(404\) −18.0000 −0.895533
\(405\) 0 0
\(406\) 40.0000 1.98517
\(407\) 8.00000 0.396545
\(408\) 0 0
\(409\) 34.0000 1.68119 0.840596 0.541663i \(-0.182205\pi\)
0.840596 + 0.541663i \(0.182205\pi\)
\(410\) −12.0000 −0.592638
\(411\) 0 0
\(412\) 0 0
\(413\) 48.0000 2.36193
\(414\) 0 0
\(415\) 8.00000 0.392705
\(416\) −5.00000 −0.245145
\(417\) 0 0
\(418\) 0 0
\(419\) −4.00000 −0.195413 −0.0977064 0.995215i \(-0.531151\pi\)
−0.0977064 + 0.995215i \(0.531151\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 20.0000 0.973585
\(423\) 0 0
\(424\) −18.0000 −0.874157
\(425\) 2.00000 0.0970143
\(426\) 0 0
\(427\) 8.00000 0.387147
\(428\) 12.0000 0.580042
\(429\) 0 0
\(430\) −24.0000 −1.15738
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 34.0000 1.63394 0.816968 0.576683i \(-0.195653\pi\)
0.816968 + 0.576683i \(0.195653\pi\)
\(434\) 16.0000 0.768025
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) 0 0
\(438\) 0 0
\(439\) 32.0000 1.52728 0.763638 0.645644i \(-0.223411\pi\)
0.763638 + 0.645644i \(0.223411\pi\)
\(440\) 24.0000 1.14416
\(441\) 0 0
\(442\) 2.00000 0.0951303
\(443\) 4.00000 0.190046 0.0950229 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) 0 0
\(445\) −4.00000 −0.189618
\(446\) −4.00000 −0.189405
\(447\) 0 0
\(448\) −28.0000 −1.32288
\(449\) −22.0000 −1.03824 −0.519122 0.854700i \(-0.673741\pi\)
−0.519122 + 0.854700i \(0.673741\pi\)
\(450\) 0 0
\(451\) 24.0000 1.13012
\(452\) −6.00000 −0.282216
\(453\) 0 0
\(454\) −20.0000 −0.938647
\(455\) 8.00000 0.375046
\(456\) 0 0
\(457\) 2.00000 0.0935561 0.0467780 0.998905i \(-0.485105\pi\)
0.0467780 + 0.998905i \(0.485105\pi\)
\(458\) 10.0000 0.467269
\(459\) 0 0
\(460\) 0 0
\(461\) 38.0000 1.76984 0.884918 0.465746i \(-0.154214\pi\)
0.884918 + 0.465746i \(0.154214\pi\)
\(462\) 0 0
\(463\) 4.00000 0.185896 0.0929479 0.995671i \(-0.470371\pi\)
0.0929479 + 0.995671i \(0.470371\pi\)
\(464\) −10.0000 −0.464238
\(465\) 0 0
\(466\) −14.0000 −0.648537
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 0 0
\(469\) 32.0000 1.47762
\(470\) 0 0
\(471\) 0 0
\(472\) −36.0000 −1.65703
\(473\) 48.0000 2.20704
\(474\) 0 0
\(475\) 0 0
\(476\) −8.00000 −0.366679
\(477\) 0 0
\(478\) −24.0000 −1.09773
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) −2.00000 −0.0911922
\(482\) −10.0000 −0.455488
\(483\) 0 0
\(484\) −5.00000 −0.227273
\(485\) −20.0000 −0.908153
\(486\) 0 0
\(487\) 12.0000 0.543772 0.271886 0.962329i \(-0.412353\pi\)
0.271886 + 0.962329i \(0.412353\pi\)
\(488\) −6.00000 −0.271607
\(489\) 0 0
\(490\) 18.0000 0.813157
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) −20.0000 −0.900755
\(494\) 0 0
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) 0 0
\(498\) 0 0
\(499\) −24.0000 −1.07439 −0.537194 0.843459i \(-0.680516\pi\)
−0.537194 + 0.843459i \(0.680516\pi\)
\(500\) −12.0000 −0.536656
\(501\) 0 0
\(502\) −12.0000 −0.535586
\(503\) 8.00000 0.356702 0.178351 0.983967i \(-0.442924\pi\)
0.178351 + 0.983967i \(0.442924\pi\)
\(504\) 0 0
\(505\) −36.0000 −1.60198
\(506\) 0 0
\(507\) 0 0
\(508\) 16.0000 0.709885
\(509\) −10.0000 −0.443242 −0.221621 0.975133i \(-0.571135\pi\)
−0.221621 + 0.975133i \(0.571135\pi\)
\(510\) 0 0
\(511\) −8.00000 −0.353899
\(512\) 11.0000 0.486136
\(513\) 0 0
\(514\) 26.0000 1.14681
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) −8.00000 −0.351500
\(519\) 0 0
\(520\) −6.00000 −0.263117
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 0 0
\(523\) 44.0000 1.92399 0.961993 0.273075i \(-0.0880406\pi\)
0.961993 + 0.273075i \(0.0880406\pi\)
\(524\) 4.00000 0.174741
\(525\) 0 0
\(526\) 24.0000 1.04645
\(527\) −8.00000 −0.348485
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) −12.0000 −0.521247
\(531\) 0 0
\(532\) 0 0
\(533\) −6.00000 −0.259889
\(534\) 0 0
\(535\) 24.0000 1.03761
\(536\) −24.0000 −1.03664
\(537\) 0 0
\(538\) 22.0000 0.948487
\(539\) −36.0000 −1.55063
\(540\) 0 0
\(541\) 30.0000 1.28980 0.644900 0.764267i \(-0.276899\pi\)
0.644900 + 0.764267i \(0.276899\pi\)
\(542\) 12.0000 0.515444
\(543\) 0 0
\(544\) 10.0000 0.428746
\(545\) 4.00000 0.171341
\(546\) 0 0
\(547\) 4.00000 0.171028 0.0855138 0.996337i \(-0.472747\pi\)
0.0855138 + 0.996337i \(0.472747\pi\)
\(548\) 6.00000 0.256307
\(549\) 0 0
\(550\) −4.00000 −0.170561
\(551\) 0 0
\(552\) 0 0
\(553\) −32.0000 −1.36078
\(554\) 10.0000 0.424859
\(555\) 0 0
\(556\) −12.0000 −0.508913
\(557\) −18.0000 −0.762684 −0.381342 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) 0 0
\(559\) −12.0000 −0.507546
\(560\) −8.00000 −0.338062
\(561\) 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
\(565\) −12.0000 −0.504844
\(566\) −12.0000 −0.504398
\(567\) 0 0
\(568\) 0 0
\(569\) −34.0000 −1.42535 −0.712677 0.701492i \(-0.752517\pi\)
−0.712677 + 0.701492i \(0.752517\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 4.00000 0.167248
\(573\) 0 0
\(574\) −24.0000 −1.00174
\(575\) 0 0
\(576\) 0 0
\(577\) −46.0000 −1.91501 −0.957503 0.288425i \(-0.906868\pi\)
−0.957503 + 0.288425i \(0.906868\pi\)
\(578\) 13.0000 0.540729
\(579\) 0 0
\(580\) 20.0000 0.830455
\(581\) 16.0000 0.663792
\(582\) 0 0
\(583\) 24.0000 0.993978
\(584\) 6.00000 0.248282
\(585\) 0 0
\(586\) −6.00000 −0.247858
\(587\) −28.0000 −1.15568 −0.577842 0.816149i \(-0.696105\pi\)
−0.577842 + 0.816149i \(0.696105\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) −24.0000 −0.988064
\(591\) 0 0
\(592\) 2.00000 0.0821995
\(593\) 26.0000 1.06769 0.533846 0.845582i \(-0.320746\pi\)
0.533846 + 0.845582i \(0.320746\pi\)
\(594\) 0 0
\(595\) −16.0000 −0.655936
\(596\) −6.00000 −0.245770
\(597\) 0 0
\(598\) 0 0
\(599\) 40.0000 1.63436 0.817178 0.576386i \(-0.195537\pi\)
0.817178 + 0.576386i \(0.195537\pi\)
\(600\) 0 0
\(601\) −38.0000 −1.55005 −0.775026 0.631929i \(-0.782263\pi\)
−0.775026 + 0.631929i \(0.782263\pi\)
\(602\) −48.0000 −1.95633
\(603\) 0 0
\(604\) −4.00000 −0.162758
\(605\) −10.0000 −0.406558
\(606\) 0 0
\(607\) −16.0000 −0.649420 −0.324710 0.945814i \(-0.605267\pi\)
−0.324710 + 0.945814i \(0.605267\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) −4.00000 −0.161955
\(611\) 0 0
\(612\) 0 0
\(613\) −2.00000 −0.0807792 −0.0403896 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) 16.0000 0.645707
\(615\) 0 0
\(616\) 48.0000 1.93398
\(617\) −22.0000 −0.885687 −0.442843 0.896599i \(-0.646030\pi\)
−0.442843 + 0.896599i \(0.646030\pi\)
\(618\) 0 0
\(619\) 24.0000 0.964641 0.482321 0.875995i \(-0.339794\pi\)
0.482321 + 0.875995i \(0.339794\pi\)
\(620\) 8.00000 0.321288
\(621\) 0 0
\(622\) 0 0
\(623\) −8.00000 −0.320513
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 6.00000 0.239808
\(627\) 0 0
\(628\) 18.0000 0.718278
\(629\) 4.00000 0.159490
\(630\) 0 0
\(631\) 20.0000 0.796187 0.398094 0.917345i \(-0.369672\pi\)
0.398094 + 0.917345i \(0.369672\pi\)
\(632\) 24.0000 0.954669
\(633\) 0 0
\(634\) 26.0000 1.03259
\(635\) 32.0000 1.26988
\(636\) 0 0
\(637\) 9.00000 0.356593
\(638\) 40.0000 1.58362
\(639\) 0 0
\(640\) −6.00000 −0.237171
\(641\) −2.00000 −0.0789953 −0.0394976 0.999220i \(-0.512576\pi\)
−0.0394976 + 0.999220i \(0.512576\pi\)
\(642\) 0 0
\(643\) −40.0000 −1.57745 −0.788723 0.614749i \(-0.789257\pi\)
−0.788723 + 0.614749i \(0.789257\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 8.00000 0.314512 0.157256 0.987558i \(-0.449735\pi\)
0.157256 + 0.987558i \(0.449735\pi\)
\(648\) 0 0
\(649\) 48.0000 1.88416
\(650\) 1.00000 0.0392232
\(651\) 0 0
\(652\) −8.00000 −0.313304
\(653\) −6.00000 −0.234798 −0.117399 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) 0 0
\(655\) 8.00000 0.312586
\(656\) 6.00000 0.234261
\(657\) 0 0
\(658\) 0 0
\(659\) −28.0000 −1.09073 −0.545363 0.838200i \(-0.683608\pi\)
−0.545363 + 0.838200i \(0.683608\pi\)
\(660\) 0 0
\(661\) 30.0000 1.16686 0.583432 0.812162i \(-0.301709\pi\)
0.583432 + 0.812162i \(0.301709\pi\)
\(662\) 16.0000 0.621858
\(663\) 0 0
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) −8.00000 −0.309529
\(669\) 0 0
\(670\) −16.0000 −0.618134
\(671\) 8.00000 0.308837
\(672\) 0 0
\(673\) −14.0000 −0.539660 −0.269830 0.962908i \(-0.586968\pi\)
−0.269830 + 0.962908i \(0.586968\pi\)
\(674\) −18.0000 −0.693334
\(675\) 0 0
\(676\) −1.00000 −0.0384615
\(677\) −22.0000 −0.845529 −0.422764 0.906240i \(-0.638940\pi\)
−0.422764 + 0.906240i \(0.638940\pi\)
\(678\) 0 0
\(679\) −40.0000 −1.53506
\(680\) 12.0000 0.460179
\(681\) 0 0
\(682\) 16.0000 0.612672
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 0 0
\(685\) 12.0000 0.458496
\(686\) 8.00000 0.305441
\(687\) 0 0
\(688\) 12.0000 0.457496
\(689\) −6.00000 −0.228582
\(690\) 0 0
\(691\) −24.0000 −0.913003 −0.456502 0.889723i \(-0.650898\pi\)
−0.456502 + 0.889723i \(0.650898\pi\)
\(692\) 6.00000 0.228086
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) −24.0000 −0.910372
\(696\) 0 0
\(697\) 12.0000 0.454532
\(698\) 26.0000 0.984115
\(699\) 0 0
\(700\) −4.00000 −0.151186
\(701\) 34.0000 1.28416 0.642081 0.766637i \(-0.278071\pi\)
0.642081 + 0.766637i \(0.278071\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −28.0000 −1.05529
\(705\) 0 0
\(706\) −2.00000 −0.0752710
\(707\) −72.0000 −2.70784
\(708\) 0 0
\(709\) −26.0000 −0.976450 −0.488225 0.872718i \(-0.662356\pi\)
−0.488225 + 0.872718i \(0.662356\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 6.00000 0.224860
\(713\) 0 0
\(714\) 0 0
\(715\) 8.00000 0.299183
\(716\) 4.00000 0.149487
\(717\) 0 0
\(718\) 24.0000 0.895672
\(719\) 24.0000 0.895049 0.447524 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 19.0000 0.707107
\(723\) 0 0
\(724\) 10.0000 0.371647
\(725\) −10.0000 −0.371391
\(726\) 0 0
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) −12.0000 −0.444750
\(729\) 0 0
\(730\) 4.00000 0.148047
\(731\) 24.0000 0.887672
\(732\) 0 0
\(733\) 30.0000 1.10808 0.554038 0.832492i \(-0.313086\pi\)
0.554038 + 0.832492i \(0.313086\pi\)
\(734\) −16.0000 −0.590571
\(735\) 0 0
\(736\) 0 0
\(737\) 32.0000 1.17874
\(738\) 0 0
\(739\) 32.0000 1.17714 0.588570 0.808447i \(-0.299691\pi\)
0.588570 + 0.808447i \(0.299691\pi\)
\(740\) −4.00000 −0.147043
\(741\) 0 0
\(742\) −24.0000 −0.881068
\(743\) −48.0000 −1.76095 −0.880475 0.474093i \(-0.842776\pi\)
−0.880475 + 0.474093i \(0.842776\pi\)
\(744\) 0 0
\(745\) −12.0000 −0.439646
\(746\) 26.0000 0.951928
\(747\) 0 0
\(748\) −8.00000 −0.292509
\(749\) 48.0000 1.75388
\(750\) 0 0
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) −10.0000 −0.364179
\(755\) −8.00000 −0.291150
\(756\) 0 0
\(757\) 22.0000 0.799604 0.399802 0.916602i \(-0.369079\pi\)
0.399802 + 0.916602i \(0.369079\pi\)
\(758\) 24.0000 0.871719
\(759\) 0 0
\(760\) 0 0
\(761\) 10.0000 0.362500 0.181250 0.983437i \(-0.441986\pi\)
0.181250 + 0.983437i \(0.441986\pi\)
\(762\) 0 0
\(763\) 8.00000 0.289619
\(764\) 8.00000 0.289430
\(765\) 0 0
\(766\) −16.0000 −0.578103
\(767\) −12.0000 −0.433295
\(768\) 0 0
\(769\) −30.0000 −1.08183 −0.540914 0.841078i \(-0.681921\pi\)
−0.540914 + 0.841078i \(0.681921\pi\)
\(770\) 32.0000 1.15320
\(771\) 0 0
\(772\) −18.0000 −0.647834
\(773\) −10.0000 −0.359675 −0.179838 0.983696i \(-0.557557\pi\)
−0.179838 + 0.983696i \(0.557557\pi\)
\(774\) 0 0
\(775\) −4.00000 −0.143684
\(776\) 30.0000 1.07694
\(777\) 0 0
\(778\) 22.0000 0.788738
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −9.00000 −0.321429
\(785\) 36.0000 1.28490
\(786\) 0 0
\(787\) −32.0000 −1.14068 −0.570338 0.821410i \(-0.693188\pi\)
−0.570338 + 0.821410i \(0.693188\pi\)
\(788\) 18.0000 0.641223
\(789\) 0 0
\(790\) 16.0000 0.569254
\(791\) −24.0000 −0.853342
\(792\) 0 0
\(793\) −2.00000 −0.0710221
\(794\) −38.0000 −1.34857
\(795\) 0 0
\(796\) −8.00000 −0.283552
\(797\) −46.0000 −1.62940 −0.814702 0.579880i \(-0.803099\pi\)
−0.814702 + 0.579880i \(0.803099\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 5.00000 0.176777
\(801\) 0 0
\(802\) 22.0000 0.776847
\(803\) −8.00000 −0.282314
\(804\) 0 0
\(805\) 0 0
\(806\) −4.00000 −0.140894
\(807\) 0 0
\(808\) 54.0000 1.89971
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) 8.00000 0.280918 0.140459 0.990086i \(-0.455142\pi\)
0.140459 + 0.990086i \(0.455142\pi\)
\(812\) 40.0000 1.40372
\(813\) 0 0
\(814\) −8.00000 −0.280400
\(815\) −16.0000 −0.560456
\(816\) 0 0
\(817\) 0 0
\(818\) −34.0000 −1.18878
\(819\) 0 0
\(820\) −12.0000 −0.419058
\(821\) 22.0000 0.767805 0.383903 0.923374i \(-0.374580\pi\)
0.383903 + 0.923374i \(0.374580\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) −48.0000 −1.67013
\(827\) −4.00000 −0.139094 −0.0695468 0.997579i \(-0.522155\pi\)
−0.0695468 + 0.997579i \(0.522155\pi\)
\(828\) 0 0
\(829\) −34.0000 −1.18087 −0.590434 0.807086i \(-0.701044\pi\)
−0.590434 + 0.807086i \(0.701044\pi\)
\(830\) −8.00000 −0.277684
\(831\) 0 0
\(832\) 7.00000 0.242681
\(833\) −18.0000 −0.623663
\(834\) 0 0
\(835\) −16.0000 −0.553703
\(836\) 0 0
\(837\) 0 0
\(838\) 4.00000 0.138178
\(839\) −48.0000 −1.65714 −0.828572 0.559883i \(-0.810846\pi\)
−0.828572 + 0.559883i \(0.810846\pi\)
\(840\) 0 0
\(841\) 71.0000 2.44828
\(842\) 10.0000 0.344623
\(843\) 0 0
\(844\) 20.0000 0.688428
\(845\) −2.00000 −0.0688021
\(846\) 0 0
\(847\) −20.0000 −0.687208
\(848\) 6.00000 0.206041
\(849\) 0 0
\(850\) −2.00000 −0.0685994
\(851\) 0 0
\(852\) 0 0
\(853\) 30.0000 1.02718 0.513590 0.858036i \(-0.328315\pi\)
0.513590 + 0.858036i \(0.328315\pi\)
\(854\) −8.00000 −0.273754
\(855\) 0 0
\(856\) −36.0000 −1.23045
\(857\) 46.0000 1.57133 0.785665 0.618652i \(-0.212321\pi\)
0.785665 + 0.618652i \(0.212321\pi\)
\(858\) 0 0
\(859\) −44.0000 −1.50126 −0.750630 0.660722i \(-0.770250\pi\)
−0.750630 + 0.660722i \(0.770250\pi\)
\(860\) −24.0000 −0.818393
\(861\) 0 0
\(862\) 0 0
\(863\) −16.0000 −0.544646 −0.272323 0.962206i \(-0.587792\pi\)
−0.272323 + 0.962206i \(0.587792\pi\)
\(864\) 0 0
\(865\) 12.0000 0.408012
\(866\) −34.0000 −1.15537
\(867\) 0 0
\(868\) 16.0000 0.543075
\(869\) −32.0000 −1.08553
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) −6.00000 −0.203186
\(873\) 0 0
\(874\) 0 0
\(875\) −48.0000 −1.62270
\(876\) 0 0
\(877\) −10.0000 −0.337676 −0.168838 0.985644i \(-0.554001\pi\)
−0.168838 + 0.985644i \(0.554001\pi\)
\(878\) −32.0000 −1.07995
\(879\) 0 0
\(880\) −8.00000 −0.269680
\(881\) −58.0000 −1.95407 −0.977035 0.213080i \(-0.931651\pi\)
−0.977035 + 0.213080i \(0.931651\pi\)
\(882\) 0 0
\(883\) −44.0000 −1.48072 −0.740359 0.672212i \(-0.765344\pi\)
−0.740359 + 0.672212i \(0.765344\pi\)
\(884\) 2.00000 0.0672673
\(885\) 0 0
\(886\) −4.00000 −0.134383
\(887\) 48.0000 1.61168 0.805841 0.592132i \(-0.201714\pi\)
0.805841 + 0.592132i \(0.201714\pi\)
\(888\) 0 0
\(889\) 64.0000 2.14649
\(890\) 4.00000 0.134080
\(891\) 0 0
\(892\) −4.00000 −0.133930
\(893\) 0 0
\(894\) 0 0
\(895\) 8.00000 0.267411
\(896\) −12.0000 −0.400892
\(897\) 0 0
\(898\) 22.0000 0.734150
\(899\) 40.0000 1.33407
\(900\) 0 0
\(901\) 12.0000 0.399778
\(902\) −24.0000 −0.799113
\(903\) 0 0
\(904\) 18.0000 0.598671
\(905\) 20.0000 0.664822
\(906\) 0 0
\(907\) 12.0000 0.398453 0.199227 0.979953i \(-0.436157\pi\)
0.199227 + 0.979953i \(0.436157\pi\)
\(908\) −20.0000 −0.663723
\(909\) 0 0
\(910\) −8.00000 −0.265197
\(911\) 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
\(914\) −2.00000 −0.0661541
\(915\) 0 0
\(916\) 10.0000 0.330409
\(917\) 16.0000 0.528367
\(918\) 0 0
\(919\) −48.0000 −1.58337 −0.791687 0.610927i \(-0.790797\pi\)
−0.791687 + 0.610927i \(0.790797\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −38.0000 −1.25146
\(923\) 0 0
\(924\) 0 0
\(925\) 2.00000 0.0657596
\(926\) −4.00000 −0.131448
\(927\) 0 0
\(928\) −50.0000 −1.64133
\(929\) −30.0000 −0.984268 −0.492134 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −14.0000 −0.458585
\(933\) 0 0
\(934\) 12.0000 0.392652
\(935\) −16.0000 −0.523256
\(936\) 0 0
\(937\) 26.0000 0.849383 0.424691 0.905338i \(-0.360383\pi\)
0.424691 + 0.905338i \(0.360383\pi\)
\(938\) −32.0000 −1.04484
\(939\) 0 0
\(940\) 0 0
\(941\) 14.0000 0.456387 0.228193 0.973616i \(-0.426718\pi\)
0.228193 + 0.973616i \(0.426718\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) −48.0000 −1.56061
\(947\) 60.0000 1.94974 0.974869 0.222779i \(-0.0715128\pi\)
0.974869 + 0.222779i \(0.0715128\pi\)
\(948\) 0 0
\(949\) 2.00000 0.0649227
\(950\) 0 0
\(951\) 0 0
\(952\) 24.0000 0.777844
\(953\) 30.0000 0.971795 0.485898 0.874016i \(-0.338493\pi\)
0.485898 + 0.874016i \(0.338493\pi\)
\(954\) 0 0
\(955\) 16.0000 0.517748
\(956\) −24.0000 −0.776215
\(957\) 0 0
\(958\) −24.0000 −0.775405
\(959\) 24.0000 0.775000
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 2.00000 0.0644826
\(963\) 0 0
\(964\) −10.0000 −0.322078
\(965\) −36.0000 −1.15888
\(966\) 0 0
\(967\) −52.0000 −1.67221 −0.836104 0.548572i \(-0.815172\pi\)
−0.836104 + 0.548572i \(0.815172\pi\)
\(968\) 15.0000 0.482118
\(969\) 0 0
\(970\) 20.0000 0.642161
\(971\) 60.0000 1.92549 0.962746 0.270408i \(-0.0871586\pi\)
0.962746 + 0.270408i \(0.0871586\pi\)
\(972\) 0 0
\(973\) −48.0000 −1.53881
\(974\) −12.0000 −0.384505
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) 42.0000 1.34370 0.671850 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) 0 0
\(979\) −8.00000 −0.255681
\(980\) 18.0000 0.574989
\(981\) 0 0
\(982\) −12.0000 −0.382935
\(983\) −16.0000 −0.510321 −0.255160 0.966899i \(-0.582128\pi\)
−0.255160 + 0.966899i \(0.582128\pi\)
\(984\) 0 0
\(985\) 36.0000 1.14706
\(986\) 20.0000 0.636930
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 16.0000 0.508257 0.254128 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) −20.0000 −0.635001
\(993\) 0 0
\(994\) 0 0
\(995\) −16.0000 −0.507234
\(996\) 0 0
\(997\) −26.0000 −0.823428 −0.411714 0.911313i \(-0.635070\pi\)
−0.411714 + 0.911313i \(0.635070\pi\)
\(998\) 24.0000 0.759707
\(999\) 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 117.2.a.a.1.1 1
3.2 odd 2 39.2.a.a.1.1 1
4.3 odd 2 1872.2.a.h.1.1 1
5.2 odd 4 2925.2.c.e.2224.1 2
5.3 odd 4 2925.2.c.e.2224.2 2
5.4 even 2 2925.2.a.p.1.1 1
7.6 odd 2 5733.2.a.e.1.1 1
8.3 odd 2 7488.2.a.by.1.1 1
8.5 even 2 7488.2.a.bl.1.1 1
9.2 odd 6 1053.2.e.b.352.1 2
9.4 even 3 1053.2.e.d.703.1 2
9.5 odd 6 1053.2.e.b.703.1 2
9.7 even 3 1053.2.e.d.352.1 2
12.11 even 2 624.2.a.i.1.1 1
13.5 odd 4 1521.2.b.b.1351.2 2
13.8 odd 4 1521.2.b.b.1351.1 2
13.12 even 2 1521.2.a.e.1.1 1
15.2 even 4 975.2.c.f.274.2 2
15.8 even 4 975.2.c.f.274.1 2
15.14 odd 2 975.2.a.f.1.1 1
21.20 even 2 1911.2.a.f.1.1 1
24.5 odd 2 2496.2.a.q.1.1 1
24.11 even 2 2496.2.a.e.1.1 1
33.32 even 2 4719.2.a.c.1.1 1
39.2 even 12 507.2.j.e.316.2 4
39.5 even 4 507.2.b.a.337.1 2
39.8 even 4 507.2.b.a.337.2 2
39.11 even 12 507.2.j.e.316.1 4
39.17 odd 6 507.2.e.b.484.1 2
39.20 even 12 507.2.j.e.361.2 4
39.23 odd 6 507.2.e.b.22.1 2
39.29 odd 6 507.2.e.a.22.1 2
39.32 even 12 507.2.j.e.361.1 4
39.35 odd 6 507.2.e.a.484.1 2
39.38 odd 2 507.2.a.a.1.1 1
156.155 even 2 8112.2.a.s.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.2.a.a.1.1 1 3.2 odd 2
117.2.a.a.1.1 1 1.1 even 1 trivial
507.2.a.a.1.1 1 39.38 odd 2
507.2.b.a.337.1 2 39.5 even 4
507.2.b.a.337.2 2 39.8 even 4
507.2.e.a.22.1 2 39.29 odd 6
507.2.e.a.484.1 2 39.35 odd 6
507.2.e.b.22.1 2 39.23 odd 6
507.2.e.b.484.1 2 39.17 odd 6
507.2.j.e.316.1 4 39.11 even 12
507.2.j.e.316.2 4 39.2 even 12
507.2.j.e.361.1 4 39.32 even 12
507.2.j.e.361.2 4 39.20 even 12
624.2.a.i.1.1 1 12.11 even 2
975.2.a.f.1.1 1 15.14 odd 2
975.2.c.f.274.1 2 15.8 even 4
975.2.c.f.274.2 2 15.2 even 4
1053.2.e.b.352.1 2 9.2 odd 6
1053.2.e.b.703.1 2 9.5 odd 6
1053.2.e.d.352.1 2 9.7 even 3
1053.2.e.d.703.1 2 9.4 even 3
1521.2.a.e.1.1 1 13.12 even 2
1521.2.b.b.1351.1 2 13.8 odd 4
1521.2.b.b.1351.2 2 13.5 odd 4
1872.2.a.h.1.1 1 4.3 odd 2
1911.2.a.f.1.1 1 21.20 even 2
2496.2.a.e.1.1 1 24.11 even 2
2496.2.a.q.1.1 1 24.5 odd 2
2925.2.a.p.1.1 1 5.4 even 2
2925.2.c.e.2224.1 2 5.2 odd 4
2925.2.c.e.2224.2 2 5.3 odd 4
4719.2.a.c.1.1 1 33.32 even 2
5733.2.a.e.1.1 1 7.6 odd 2
7488.2.a.bl.1.1 1 8.5 even 2
7488.2.a.by.1.1 1 8.3 odd 2
8112.2.a.s.1.1 1 156.155 even 2