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Properties

Label 189.2.a.a.1.1
Level $189$
Weight $2$
Character 189.1
Self dual yes
Analytic conductor $1.509$
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] = [189,2,Mod(1,189)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(189, 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("189.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 189 = 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 189.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: \(1.50917259820\)
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 \(\chi\) \(=\) 189.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} +2.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +O(q^{10})\) \(q-2.00000 q^{2} +2.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +2.00000 q^{10} -4.00000 q^{11} -2.00000 q^{13} +2.00000 q^{14} -4.00000 q^{16} +3.00000 q^{17} -8.00000 q^{19} -2.00000 q^{20} +8.00000 q^{22} -6.00000 q^{23} -4.00000 q^{25} +4.00000 q^{26} -2.00000 q^{28} -4.00000 q^{29} +6.00000 q^{31} +8.00000 q^{32} -6.00000 q^{34} +1.00000 q^{35} -3.00000 q^{37} +16.0000 q^{38} +1.00000 q^{41} +11.0000 q^{43} -8.00000 q^{44} +12.0000 q^{46} +9.00000 q^{47} +1.00000 q^{49} +8.00000 q^{50} -4.00000 q^{52} +6.00000 q^{53} +4.00000 q^{55} +8.00000 q^{58} -15.0000 q^{59} +4.00000 q^{61} -12.0000 q^{62} -8.00000 q^{64} +2.00000 q^{65} -8.00000 q^{67} +6.00000 q^{68} -2.00000 q^{70} -12.0000 q^{71} +6.00000 q^{73} +6.00000 q^{74} -16.0000 q^{76} +4.00000 q^{77} -1.00000 q^{79} +4.00000 q^{80} -2.00000 q^{82} -9.00000 q^{83} -3.00000 q^{85} -22.0000 q^{86} +2.00000 q^{89} +2.00000 q^{91} -12.0000 q^{92} -18.0000 q^{94} +8.00000 q^{95} +12.0000 q^{97} -2.00000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −1.41421 −0.707107 − 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) 0 0
\(4\) 2.00000 1.00000
\(5\) −1.00000 −0.447214 −0.223607 − 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 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
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) 3.00000 0.727607 0.363803 − 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) −8.00000 −1.83533 −0.917663 − 0.397360i \(-0.869927\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) 8.00000 1.70561
\(23\) −6.00000 −1.25109 −0.625543 − 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 4.00000 0.784465
\(27\) 0 0
\(28\) −2.00000 −0.377964
\(29\) −4.00000 −0.742781 −0.371391 − 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 0 0
\(31\) 6.00000 1.07763 0.538816 − 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) 8.00000 1.41421
\(33\) 0 0
\(34\) −6.00000 −1.02899
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) −3.00000 −0.493197 −0.246598 − 0.969118i \(-0.579313\pi\)
−0.246598 + 0.969118i \(0.579313\pi\)
\(38\) 16.0000 2.59554
\(39\) 0 0
\(40\) 0 0
\(41\) 1.00000 0.156174 0.0780869 − 0.996947i \(-0.475119\pi\)
0.0780869 + 0.996947i \(0.475119\pi\)
\(42\) 0 0
\(43\) 11.0000 1.67748 0.838742 − 0.544529i \(-0.183292\pi\)
0.838742 + 0.544529i \(0.183292\pi\)
\(44\) −8.00000 −1.20605
\(45\) 0 0
\(46\) 12.0000 1.76930
\(47\) 9.00000 1.31278 0.656392 − 0.754420i \(-0.272082\pi\)
0.656392 + 0.754420i \(0.272082\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 8.00000 1.13137
\(51\) 0 0
\(52\) −4.00000 −0.554700
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 4.00000 0.539360
\(56\) 0 0
\(57\) 0 0
\(58\) 8.00000 1.05045
\(59\) −15.0000 −1.95283 −0.976417 − 0.215894i \(-0.930733\pi\)
−0.976417 + 0.215894i \(0.930733\pi\)
\(60\) 0 0
\(61\) 4.00000 0.512148 0.256074 − 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) −12.0000 −1.52400
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 − 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 6.00000 0.727607
\(69\) 0 0
\(70\) −2.00000 −0.239046
\(71\) −12.0000 −1.42414 −0.712069 − 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 − 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 6.00000 0.697486
\(75\) 0 0
\(76\) −16.0000 −1.83533
\(77\) 4.00000 0.455842
\(78\) 0 0
\(79\) −1.00000 −0.112509 −0.0562544 − 0.998416i \(-0.517916\pi\)
−0.0562544 + 0.998416i \(0.517916\pi\)
\(80\) 4.00000 0.447214
\(81\) 0 0
\(82\) −2.00000 −0.220863
\(83\) −9.00000 −0.987878 −0.493939 − 0.869496i \(-0.664443\pi\)
−0.493939 + 0.869496i \(0.664443\pi\)
\(84\) 0 0
\(85\) −3.00000 −0.325396
\(86\) −22.0000 −2.37232
\(87\) 0 0
\(88\) 0 0
\(89\) 2.00000 0.212000 0.106000 − 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) −12.0000 −1.25109
\(93\) 0 0
\(94\) −18.0000 −1.85656
\(95\) 8.00000 0.820783
\(96\) 0 0
\(97\) 12.0000 1.21842 0.609208 − 0.793011i \(-0.291488\pi\)
0.609208 + 0.793011i \(0.291488\pi\)
\(98\) −2.00000 −0.202031
\(99\) 0 0
\(100\) −8.00000 −0.800000
\(101\) 10.0000 0.995037 0.497519 − 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 0 0
\(103\) 2.00000 0.197066 0.0985329 − 0.995134i \(-0.468585\pi\)
0.0985329 + 0.995134i \(0.468585\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −12.0000 −1.16554
\(107\) 2.00000 0.193347 0.0966736 − 0.995316i \(-0.469180\pi\)
0.0966736 + 0.995316i \(0.469180\pi\)
\(108\) 0 0
\(109\) −9.00000 −0.862044 −0.431022 − 0.902342i \(-0.641847\pi\)
−0.431022 + 0.902342i \(0.641847\pi\)
\(110\) −8.00000 −0.762770
\(111\) 0 0
\(112\) 4.00000 0.377964
\(113\) 2.00000 0.188144 0.0940721 − 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) 0 0
\(115\) 6.00000 0.559503
\(116\) −8.00000 −0.742781
\(117\) 0 0
\(118\) 30.0000 2.76172
\(119\) −3.00000 −0.275010
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) −8.00000 −0.724286
\(123\) 0 0
\(124\) 12.0000 1.07763
\(125\) 9.00000 0.804984
\(126\) 0 0
\(127\) −15.0000 −1.33103 −0.665517 − 0.746382i \(-0.731789\pi\)
−0.665517 + 0.746382i \(0.731789\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) −4.00000 −0.350823
\(131\) −4.00000 −0.349482 −0.174741 − 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 0 0
\(133\) 8.00000 0.693688
\(134\) 16.0000 1.38219
\(135\) 0 0
\(136\) 0 0
\(137\) −18.0000 −1.53784 −0.768922 − 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 0 0
\(139\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 2.00000 0.169031
\(141\) 0 0
\(142\) 24.0000 2.01404
\(143\) 8.00000 0.668994
\(144\) 0 0
\(145\) 4.00000 0.332182
\(146\) −12.0000 −0.993127
\(147\) 0 0
\(148\) −6.00000 −0.493197
\(149\) 12.0000 0.983078 0.491539 − 0.870855i \(-0.336434\pi\)
0.491539 + 0.870855i \(0.336434\pi\)
\(150\) 0 0
\(151\) 5.00000 0.406894 0.203447 − 0.979086i \(-0.434786\pi\)
0.203447 + 0.979086i \(0.434786\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −8.00000 −0.644658
\(155\) −6.00000 −0.481932
\(156\) 0 0
\(157\) −8.00000 −0.638470 −0.319235 − 0.947676i \(-0.603426\pi\)
−0.319235 + 0.947676i \(0.603426\pi\)
\(158\) 2.00000 0.159111
\(159\) 0 0
\(160\) −8.00000 −0.632456
\(161\) 6.00000 0.472866
\(162\) 0 0
\(163\) −11.0000 −0.861586 −0.430793 − 0.902451i \(-0.641766\pi\)
−0.430793 + 0.902451i \(0.641766\pi\)
\(164\) 2.00000 0.156174
\(165\) 0 0
\(166\) 18.0000 1.39707
\(167\) 17.0000 1.31550 0.657750 − 0.753237i \(-0.271508\pi\)
0.657750 + 0.753237i \(0.271508\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 6.00000 0.460179
\(171\) 0 0
\(172\) 22.0000 1.67748
\(173\) 22.0000 1.67263 0.836315 − 0.548250i \(-0.184706\pi\)
0.836315 + 0.548250i \(0.184706\pi\)
\(174\) 0 0
\(175\) 4.00000 0.302372
\(176\) 16.0000 1.20605
\(177\) 0 0
\(178\) −4.00000 −0.299813
\(179\) −2.00000 −0.149487 −0.0747435 − 0.997203i \(-0.523814\pi\)
−0.0747435 + 0.997203i \(0.523814\pi\)
\(180\) 0 0
\(181\) 16.0000 1.18927 0.594635 − 0.803996i \(-0.297296\pi\)
0.594635 + 0.803996i \(0.297296\pi\)
\(182\) −4.00000 −0.296500
\(183\) 0 0
\(184\) 0 0
\(185\) 3.00000 0.220564
\(186\) 0 0
\(187\) −12.0000 −0.877527
\(188\) 18.0000 1.31278
\(189\) 0 0
\(190\) −16.0000 −1.16076
\(191\) 2.00000 0.144715 0.0723575 − 0.997379i \(-0.476948\pi\)
0.0723575 + 0.997379i \(0.476948\pi\)
\(192\) 0 0
\(193\) −25.0000 −1.79954 −0.899770 − 0.436365i \(-0.856266\pi\)
−0.899770 + 0.436365i \(0.856266\pi\)
\(194\) −24.0000 −1.72310
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) −4.00000 −0.284988 −0.142494 − 0.989796i \(-0.545512\pi\)
−0.142494 + 0.989796i \(0.545512\pi\)
\(198\) 0 0
\(199\) −12.0000 −0.850657 −0.425329 − 0.905039i \(-0.639842\pi\)
−0.425329 + 0.905039i \(0.639842\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −20.0000 −1.40720
\(203\) 4.00000 0.280745
\(204\) 0 0
\(205\) −1.00000 −0.0698430
\(206\) −4.00000 −0.278693
\(207\) 0 0
\(208\) 8.00000 0.554700
\(209\) 32.0000 2.21349
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 − 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 12.0000 0.824163
\(213\) 0 0
\(214\) −4.00000 −0.273434
\(215\) −11.0000 −0.750194
\(216\) 0 0
\(217\) −6.00000 −0.407307
\(218\) 18.0000 1.21911
\(219\) 0 0
\(220\) 8.00000 0.539360
\(221\) −6.00000 −0.403604
\(222\) 0 0
\(223\) −2.00000 −0.133930 −0.0669650 − 0.997755i \(-0.521332\pi\)
−0.0669650 + 0.997755i \(0.521332\pi\)
\(224\) −8.00000 −0.534522
\(225\) 0 0
\(226\) −4.00000 −0.266076
\(227\) −24.0000 −1.59294 −0.796468 − 0.604681i \(-0.793301\pi\)
−0.796468 + 0.604681i \(0.793301\pi\)
\(228\) 0 0
\(229\) −4.00000 −0.264327 −0.132164 − 0.991228i \(-0.542192\pi\)
−0.132164 + 0.991228i \(0.542192\pi\)
\(230\) −12.0000 −0.791257
\(231\) 0 0
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 − 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) −9.00000 −0.587095
\(236\) −30.0000 −1.95283
\(237\) 0 0
\(238\) 6.00000 0.388922
\(239\) −18.0000 −1.16432 −0.582162 − 0.813073i \(-0.697793\pi\)
−0.582162 + 0.813073i \(0.697793\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 − 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) −10.0000 −0.642824
\(243\) 0 0
\(244\) 8.00000 0.512148
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) 16.0000 1.01806
\(248\) 0 0
\(249\) 0 0
\(250\) −18.0000 −1.13842
\(251\) −25.0000 −1.57799 −0.788993 − 0.614402i \(-0.789397\pi\)
−0.788993 + 0.614402i \(0.789397\pi\)
\(252\) 0 0
\(253\) 24.0000 1.50887
\(254\) 30.0000 1.88237
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 10.0000 0.623783 0.311891 − 0.950118i \(-0.399037\pi\)
0.311891 + 0.950118i \(0.399037\pi\)
\(258\) 0 0
\(259\) 3.00000 0.186411
\(260\) 4.00000 0.248069
\(261\) 0 0
\(262\) 8.00000 0.494242
\(263\) 14.0000 0.863277 0.431638 − 0.902047i \(-0.357936\pi\)
0.431638 + 0.902047i \(0.357936\pi\)
\(264\) 0 0
\(265\) −6.00000 −0.368577
\(266\) −16.0000 −0.981023
\(267\) 0 0
\(268\) −16.0000 −0.977356
\(269\) −21.0000 −1.28039 −0.640196 − 0.768211i \(-0.721147\pi\)
−0.640196 + 0.768211i \(0.721147\pi\)
\(270\) 0 0
\(271\) −2.00000 −0.121491 −0.0607457 − 0.998153i \(-0.519348\pi\)
−0.0607457 + 0.998153i \(0.519348\pi\)
\(272\) −12.0000 −0.727607
\(273\) 0 0
\(274\) 36.0000 2.17484
\(275\) 16.0000 0.964836
\(276\) 0 0
\(277\) 13.0000 0.781094 0.390547 − 0.920583i \(-0.372286\pi\)
0.390547 + 0.920583i \(0.372286\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 − 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) 28.0000 1.66443 0.832214 − 0.554455i \(-0.187073\pi\)
0.832214 + 0.554455i \(0.187073\pi\)
\(284\) −24.0000 −1.42414
\(285\) 0 0
\(286\) −16.0000 −0.946100
\(287\) −1.00000 −0.0590281
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) −8.00000 −0.469776
\(291\) 0 0
\(292\) 12.0000 0.702247
\(293\) 1.00000 0.0584206 0.0292103 − 0.999573i \(-0.490701\pi\)
0.0292103 + 0.999573i \(0.490701\pi\)
\(294\) 0 0
\(295\) 15.0000 0.873334
\(296\) 0 0
\(297\) 0 0
\(298\) −24.0000 −1.39028
\(299\) 12.0000 0.693978
\(300\) 0 0
\(301\) −11.0000 −0.634029
\(302\) −10.0000 −0.575435
\(303\) 0 0
\(304\) 32.0000 1.83533
\(305\) −4.00000 −0.229039
\(306\) 0 0
\(307\) −2.00000 −0.114146 −0.0570730 − 0.998370i \(-0.518177\pi\)
−0.0570730 + 0.998370i \(0.518177\pi\)
\(308\) 8.00000 0.455842
\(309\) 0 0
\(310\) 12.0000 0.681554
\(311\) −3.00000 −0.170114 −0.0850572 − 0.996376i \(-0.527107\pi\)
−0.0850572 + 0.996376i \(0.527107\pi\)
\(312\) 0 0
\(313\) −16.0000 −0.904373 −0.452187 − 0.891923i \(-0.649356\pi\)
−0.452187 + 0.891923i \(0.649356\pi\)
\(314\) 16.0000 0.902932
\(315\) 0 0
\(316\) −2.00000 −0.112509
\(317\) −18.0000 −1.01098 −0.505490 − 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 0 0
\(319\) 16.0000 0.895828
\(320\) 8.00000 0.447214
\(321\) 0 0
\(322\) −12.0000 −0.668734
\(323\) −24.0000 −1.33540
\(324\) 0 0
\(325\) 8.00000 0.443760
\(326\) 22.0000 1.21847
\(327\) 0 0
\(328\) 0 0
\(329\) −9.00000 −0.496186
\(330\) 0 0
\(331\) −1.00000 −0.0549650 −0.0274825 − 0.999622i \(-0.508749\pi\)
−0.0274825 + 0.999622i \(0.508749\pi\)
\(332\) −18.0000 −0.987878
\(333\) 0 0
\(334\) −34.0000 −1.86040
\(335\) 8.00000 0.437087
\(336\) 0 0
\(337\) 7.00000 0.381314 0.190657 − 0.981657i \(-0.438938\pi\)
0.190657 + 0.981657i \(0.438938\pi\)
\(338\) 18.0000 0.979071
\(339\) 0 0
\(340\) −6.00000 −0.325396
\(341\) −24.0000 −1.29967
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) −44.0000 −2.36545
\(347\) −2.00000 −0.107366 −0.0536828 − 0.998558i \(-0.517096\pi\)
−0.0536828 + 0.998558i \(0.517096\pi\)
\(348\) 0 0
\(349\) −20.0000 −1.07058 −0.535288 − 0.844670i \(-0.679797\pi\)
−0.535288 + 0.844670i \(0.679797\pi\)
\(350\) −8.00000 −0.427618
\(351\) 0 0
\(352\) −32.0000 −1.70561
\(353\) −25.0000 −1.33062 −0.665308 − 0.746569i \(-0.731700\pi\)
−0.665308 + 0.746569i \(0.731700\pi\)
\(354\) 0 0
\(355\) 12.0000 0.636894
\(356\) 4.00000 0.212000
\(357\) 0 0
\(358\) 4.00000 0.211407
\(359\) 34.0000 1.79445 0.897226 − 0.441572i \(-0.145579\pi\)
0.897226 + 0.441572i \(0.145579\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) −32.0000 −1.68188
\(363\) 0 0
\(364\) 4.00000 0.209657
\(365\) −6.00000 −0.314054
\(366\) 0 0
\(367\) −30.0000 −1.56599 −0.782994 − 0.622030i \(-0.786308\pi\)
−0.782994 + 0.622030i \(0.786308\pi\)
\(368\) 24.0000 1.25109
\(369\) 0 0
\(370\) −6.00000 −0.311925
\(371\) −6.00000 −0.311504
\(372\) 0 0
\(373\) −13.0000 −0.673114 −0.336557 − 0.941663i \(-0.609263\pi\)
−0.336557 + 0.941663i \(0.609263\pi\)
\(374\) 24.0000 1.24101
\(375\) 0 0
\(376\) 0 0
\(377\) 8.00000 0.412021
\(378\) 0 0
\(379\) −3.00000 −0.154100 −0.0770498 − 0.997027i \(-0.524550\pi\)
−0.0770498 + 0.997027i \(0.524550\pi\)
\(380\) 16.0000 0.820783
\(381\) 0 0
\(382\) −4.00000 −0.204658
\(383\) 21.0000 1.07305 0.536525 − 0.843884i \(-0.319737\pi\)
0.536525 + 0.843884i \(0.319737\pi\)
\(384\) 0 0
\(385\) −4.00000 −0.203859
\(386\) 50.0000 2.54493
\(387\) 0 0
\(388\) 24.0000 1.21842
\(389\) −24.0000 −1.21685 −0.608424 − 0.793612i \(-0.708198\pi\)
−0.608424 + 0.793612i \(0.708198\pi\)
\(390\) 0 0
\(391\) −18.0000 −0.910299
\(392\) 0 0
\(393\) 0 0
\(394\) 8.00000 0.403034
\(395\) 1.00000 0.0503155
\(396\) 0 0
\(397\) 6.00000 0.301131 0.150566 − 0.988600i \(-0.451890\pi\)
0.150566 + 0.988600i \(0.451890\pi\)
\(398\) 24.0000 1.20301
\(399\) 0 0
\(400\) 16.0000 0.800000
\(401\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) −12.0000 −0.597763
\(404\) 20.0000 0.995037
\(405\) 0 0
\(406\) −8.00000 −0.397033
\(407\) 12.0000 0.594818
\(408\) 0 0
\(409\) 32.0000 1.58230 0.791149 − 0.611623i \(-0.209483\pi\)
0.791149 + 0.611623i \(0.209483\pi\)
\(410\) 2.00000 0.0987730
\(411\) 0 0
\(412\) 4.00000 0.197066
\(413\) 15.0000 0.738102
\(414\) 0 0
\(415\) 9.00000 0.441793
\(416\) −16.0000 −0.784465
\(417\) 0 0
\(418\) −64.0000 −3.13034
\(419\) −33.0000 −1.61216 −0.806078 − 0.591810i \(-0.798414\pi\)
−0.806078 + 0.591810i \(0.798414\pi\)
\(420\) 0 0
\(421\) 14.0000 0.682318 0.341159 − 0.940006i \(-0.389181\pi\)
0.341159 + 0.940006i \(0.389181\pi\)
\(422\) −8.00000 −0.389434
\(423\) 0 0
\(424\) 0 0
\(425\) −12.0000 −0.582086
\(426\) 0 0
\(427\) −4.00000 −0.193574
\(428\) 4.00000 0.193347
\(429\) 0 0
\(430\) 22.0000 1.06093
\(431\) 24.0000 1.15604 0.578020 − 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) 0 0
\(433\) −2.00000 −0.0961139 −0.0480569 − 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 12.0000 0.576018
\(435\) 0 0
\(436\) −18.0000 −0.862044
\(437\) 48.0000 2.29615
\(438\) 0 0
\(439\) 24.0000 1.14546 0.572729 − 0.819745i \(-0.305885\pi\)
0.572729 + 0.819745i \(0.305885\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 12.0000 0.570782
\(443\) −24.0000 −1.14027 −0.570137 − 0.821549i \(-0.693110\pi\)
−0.570137 + 0.821549i \(0.693110\pi\)
\(444\) 0 0
\(445\) −2.00000 −0.0948091
\(446\) 4.00000 0.189405
\(447\) 0 0
\(448\) 8.00000 0.377964
\(449\) −6.00000 −0.283158 −0.141579 − 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 0 0
\(451\) −4.00000 −0.188353
\(452\) 4.00000 0.188144
\(453\) 0 0
\(454\) 48.0000 2.25275
\(455\) −2.00000 −0.0937614
\(456\) 0 0
\(457\) −14.0000 −0.654892 −0.327446 − 0.944870i \(-0.606188\pi\)
−0.327446 + 0.944870i \(0.606188\pi\)
\(458\) 8.00000 0.373815
\(459\) 0 0
\(460\) 12.0000 0.559503
\(461\) 1.00000 0.0465746 0.0232873 − 0.999729i \(-0.492587\pi\)
0.0232873 + 0.999729i \(0.492587\pi\)
\(462\) 0 0
\(463\) −23.0000 −1.06890 −0.534450 − 0.845200i \(-0.679481\pi\)
−0.534450 + 0.845200i \(0.679481\pi\)
\(464\) 16.0000 0.742781
\(465\) 0 0
\(466\) −12.0000 −0.555889
\(467\) 12.0000 0.555294 0.277647 − 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 0 0
\(469\) 8.00000 0.369406
\(470\) 18.0000 0.830278
\(471\) 0 0
\(472\) 0 0
\(473\) −44.0000 −2.02312
\(474\) 0 0
\(475\) 32.0000 1.46826
\(476\) −6.00000 −0.275010
\(477\) 0 0
\(478\) 36.0000 1.64660
\(479\) −17.0000 −0.776750 −0.388375 − 0.921501i \(-0.626963\pi\)
−0.388375 + 0.921501i \(0.626963\pi\)
\(480\) 0 0
\(481\) 6.00000 0.273576
\(482\) 20.0000 0.910975
\(483\) 0 0
\(484\) 10.0000 0.454545
\(485\) −12.0000 −0.544892
\(486\) 0 0
\(487\) 16.0000 0.725029 0.362515 − 0.931978i \(-0.381918\pi\)
0.362515 + 0.931978i \(0.381918\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 2.00000 0.0903508
\(491\) −26.0000 −1.17336 −0.586682 − 0.809818i \(-0.699566\pi\)
−0.586682 + 0.809818i \(0.699566\pi\)
\(492\) 0 0
\(493\) −12.0000 −0.540453
\(494\) −32.0000 −1.43975
\(495\) 0 0
\(496\) −24.0000 −1.07763
\(497\) 12.0000 0.538274
\(498\) 0 0
\(499\) 13.0000 0.581960 0.290980 − 0.956729i \(-0.406019\pi\)
0.290980 + 0.956729i \(0.406019\pi\)
\(500\) 18.0000 0.804984
\(501\) 0 0
\(502\) 50.0000 2.23161
\(503\) 21.0000 0.936344 0.468172 − 0.883637i \(-0.344913\pi\)
0.468172 + 0.883637i \(0.344913\pi\)
\(504\) 0 0
\(505\) −10.0000 −0.444994
\(506\) −48.0000 −2.13386
\(507\) 0 0
\(508\) −30.0000 −1.33103
\(509\) −41.0000 −1.81729 −0.908647 − 0.417566i \(-0.862883\pi\)
−0.908647 + 0.417566i \(0.862883\pi\)
\(510\) 0 0
\(511\) −6.00000 −0.265424
\(512\) −32.0000 −1.41421
\(513\) 0 0
\(514\) −20.0000 −0.882162
\(515\) −2.00000 −0.0881305
\(516\) 0 0
\(517\) −36.0000 −1.58328
\(518\) −6.00000 −0.263625
\(519\) 0 0
\(520\) 0 0
\(521\) 15.0000 0.657162 0.328581 − 0.944476i \(-0.393430\pi\)
0.328581 + 0.944476i \(0.393430\pi\)
\(522\) 0 0
\(523\) −26.0000 −1.13690 −0.568450 − 0.822718i \(-0.692457\pi\)
−0.568450 + 0.822718i \(0.692457\pi\)
\(524\) −8.00000 −0.349482
\(525\) 0 0
\(526\) −28.0000 −1.22086
\(527\) 18.0000 0.784092
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 12.0000 0.521247
\(531\) 0 0
\(532\) 16.0000 0.693688
\(533\) −2.00000 −0.0866296
\(534\) 0 0
\(535\) −2.00000 −0.0864675
\(536\) 0 0
\(537\) 0 0
\(538\) 42.0000 1.81075
\(539\) −4.00000 −0.172292
\(540\) 0 0
\(541\) 29.0000 1.24681 0.623404 − 0.781900i \(-0.285749\pi\)
0.623404 + 0.781900i \(0.285749\pi\)
\(542\) 4.00000 0.171815
\(543\) 0 0
\(544\) 24.0000 1.02899
\(545\) 9.00000 0.385518
\(546\) 0 0
\(547\) −29.0000 −1.23995 −0.619975 − 0.784621i \(-0.712857\pi\)
−0.619975 + 0.784621i \(0.712857\pi\)
\(548\) −36.0000 −1.53784
\(549\) 0 0
\(550\) −32.0000 −1.36448
\(551\) 32.0000 1.36325
\(552\) 0 0
\(553\) 1.00000 0.0425243
\(554\) −26.0000 −1.10463
\(555\) 0 0
\(556\) 0 0
\(557\) −4.00000 −0.169485 −0.0847427 − 0.996403i \(-0.527007\pi\)
−0.0847427 + 0.996403i \(0.527007\pi\)
\(558\) 0 0
\(559\) −22.0000 −0.930501
\(560\) −4.00000 −0.169031
\(561\) 0 0
\(562\) −20.0000 −0.843649
\(563\) −4.00000 −0.168580 −0.0842900 − 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) 0 0
\(565\) −2.00000 −0.0841406
\(566\) −56.0000 −2.35386
\(567\) 0 0
\(568\) 0 0
\(569\) 2.00000 0.0838444 0.0419222 − 0.999121i \(-0.486652\pi\)
0.0419222 + 0.999121i \(0.486652\pi\)
\(570\) 0 0
\(571\) 11.0000 0.460336 0.230168 − 0.973151i \(-0.426072\pi\)
0.230168 + 0.973151i \(0.426072\pi\)
\(572\) 16.0000 0.668994
\(573\) 0 0
\(574\) 2.00000 0.0834784
\(575\) 24.0000 1.00087
\(576\) 0 0
\(577\) −20.0000 −0.832611 −0.416305 − 0.909225i \(-0.636675\pi\)
−0.416305 + 0.909225i \(0.636675\pi\)
\(578\) 16.0000 0.665512
\(579\) 0 0
\(580\) 8.00000 0.332182
\(581\) 9.00000 0.373383
\(582\) 0 0
\(583\) −24.0000 −0.993978
\(584\) 0 0
\(585\) 0 0
\(586\) −2.00000 −0.0826192
\(587\) 20.0000 0.825488 0.412744 − 0.910847i \(-0.364570\pi\)
0.412744 + 0.910847i \(0.364570\pi\)
\(588\) 0 0
\(589\) −48.0000 −1.97781
\(590\) −30.0000 −1.23508
\(591\) 0 0
\(592\) 12.0000 0.493197
\(593\) −33.0000 −1.35515 −0.677574 − 0.735455i \(-0.736969\pi\)
−0.677574 + 0.735455i \(0.736969\pi\)
\(594\) 0 0
\(595\) 3.00000 0.122988
\(596\) 24.0000 0.983078
\(597\) 0 0
\(598\) −24.0000 −0.981433
\(599\) 36.0000 1.47092 0.735460 − 0.677568i \(-0.236966\pi\)
0.735460 + 0.677568i \(0.236966\pi\)
\(600\) 0 0
\(601\) −18.0000 −0.734235 −0.367118 − 0.930175i \(-0.619655\pi\)
−0.367118 + 0.930175i \(0.619655\pi\)
\(602\) 22.0000 0.896653
\(603\) 0 0
\(604\) 10.0000 0.406894
\(605\) −5.00000 −0.203279
\(606\) 0 0
\(607\) 14.0000 0.568242 0.284121 − 0.958788i \(-0.408298\pi\)
0.284121 + 0.958788i \(0.408298\pi\)
\(608\) −64.0000 −2.59554
\(609\) 0 0
\(610\) 8.00000 0.323911
\(611\) −18.0000 −0.728202
\(612\) 0 0
\(613\) −26.0000 −1.05013 −0.525065 − 0.851062i \(-0.675959\pi\)
−0.525065 + 0.851062i \(0.675959\pi\)
\(614\) 4.00000 0.161427
\(615\) 0 0
\(616\) 0 0
\(617\) −6.00000 −0.241551 −0.120775 − 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 0 0
\(619\) −44.0000 −1.76851 −0.884255 − 0.467005i \(-0.845333\pi\)
−0.884255 + 0.467005i \(0.845333\pi\)
\(620\) −12.0000 −0.481932
\(621\) 0 0
\(622\) 6.00000 0.240578
\(623\) −2.00000 −0.0801283
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 32.0000 1.27898
\(627\) 0 0
\(628\) −16.0000 −0.638470
\(629\) −9.00000 −0.358854
\(630\) 0 0
\(631\) −19.0000 −0.756378 −0.378189 − 0.925728i \(-0.623453\pi\)
−0.378189 + 0.925728i \(0.623453\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 36.0000 1.42974
\(635\) 15.0000 0.595257
\(636\) 0 0
\(637\) −2.00000 −0.0792429
\(638\) −32.0000 −1.26689
\(639\) 0 0
\(640\) 0 0
\(641\) 12.0000 0.473972 0.236986 − 0.971513i \(-0.423841\pi\)
0.236986 + 0.971513i \(0.423841\pi\)
\(642\) 0 0
\(643\) 26.0000 1.02534 0.512670 − 0.858586i \(-0.328656\pi\)
0.512670 + 0.858586i \(0.328656\pi\)
\(644\) 12.0000 0.472866
\(645\) 0 0
\(646\) 48.0000 1.88853
\(647\) −8.00000 −0.314512 −0.157256 − 0.987558i \(-0.550265\pi\)
−0.157256 + 0.987558i \(0.550265\pi\)
\(648\) 0 0
\(649\) 60.0000 2.35521
\(650\) −16.0000 −0.627572
\(651\) 0 0
\(652\) −22.0000 −0.861586
\(653\) −42.0000 −1.64359 −0.821794 − 0.569785i \(-0.807026\pi\)
−0.821794 + 0.569785i \(0.807026\pi\)
\(654\) 0 0
\(655\) 4.00000 0.156293
\(656\) −4.00000 −0.156174
\(657\) 0 0
\(658\) 18.0000 0.701713
\(659\) 12.0000 0.467454 0.233727 − 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 0 0
\(661\) −14.0000 −0.544537 −0.272268 − 0.962221i \(-0.587774\pi\)
−0.272268 + 0.962221i \(0.587774\pi\)
\(662\) 2.00000 0.0777322
\(663\) 0 0
\(664\) 0 0
\(665\) −8.00000 −0.310227
\(666\) 0 0
\(667\) 24.0000 0.929284
\(668\) 34.0000 1.31550
\(669\) 0 0
\(670\) −16.0000 −0.618134
\(671\) −16.0000 −0.617673
\(672\) 0 0
\(673\) −2.00000 −0.0770943 −0.0385472 − 0.999257i \(-0.512273\pi\)
−0.0385472 + 0.999257i \(0.512273\pi\)
\(674\) −14.0000 −0.539260
\(675\) 0 0
\(676\) −18.0000 −0.692308
\(677\) 30.0000 1.15299 0.576497 − 0.817099i \(-0.304419\pi\)
0.576497 + 0.817099i \(0.304419\pi\)
\(678\) 0 0
\(679\) −12.0000 −0.460518
\(680\) 0 0
\(681\) 0 0
\(682\) 48.0000 1.83801
\(683\) 42.0000 1.60709 0.803543 − 0.595247i \(-0.202946\pi\)
0.803543 + 0.595247i \(0.202946\pi\)
\(684\) 0 0
\(685\) 18.0000 0.687745
\(686\) 2.00000 0.0763604
\(687\) 0 0
\(688\) −44.0000 −1.67748
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) 50.0000 1.90209 0.951045 − 0.309053i \(-0.100012\pi\)
0.951045 + 0.309053i \(0.100012\pi\)
\(692\) 44.0000 1.67263
\(693\) 0 0
\(694\) 4.00000 0.151838
\(695\) 0 0
\(696\) 0 0
\(697\) 3.00000 0.113633
\(698\) 40.0000 1.51402
\(699\) 0 0
\(700\) 8.00000 0.302372
\(701\) 18.0000 0.679851 0.339925 − 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) 0 0
\(703\) 24.0000 0.905177
\(704\) 32.0000 1.20605
\(705\) 0 0
\(706\) 50.0000 1.88177
\(707\) −10.0000 −0.376089
\(708\) 0 0
\(709\) 15.0000 0.563337 0.281668 − 0.959512i \(-0.409112\pi\)
0.281668 + 0.959512i \(0.409112\pi\)
\(710\) −24.0000 −0.900704
\(711\) 0 0
\(712\) 0 0
\(713\) −36.0000 −1.34821
\(714\) 0 0
\(715\) −8.00000 −0.299183
\(716\) −4.00000 −0.149487
\(717\) 0 0
\(718\) −68.0000 −2.53774
\(719\) 33.0000 1.23069 0.615346 − 0.788257i \(-0.289016\pi\)
0.615346 + 0.788257i \(0.289016\pi\)
\(720\) 0 0
\(721\) −2.00000 −0.0744839
\(722\) −90.0000 −3.34945
\(723\) 0 0
\(724\) 32.0000 1.18927
\(725\) 16.0000 0.594225
\(726\) 0 0
\(727\) −4.00000 −0.148352 −0.0741759 − 0.997245i \(-0.523633\pi\)
−0.0741759 + 0.997245i \(0.523633\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 12.0000 0.444140
\(731\) 33.0000 1.22055
\(732\) 0 0
\(733\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) 60.0000 2.21464
\(735\) 0 0
\(736\) −48.0000 −1.76930
\(737\) 32.0000 1.17874
\(738\) 0 0
\(739\) 12.0000 0.441427 0.220714 − 0.975339i \(-0.429161\pi\)
0.220714 + 0.975339i \(0.429161\pi\)
\(740\) 6.00000 0.220564
\(741\) 0 0
\(742\) 12.0000 0.440534
\(743\) 36.0000 1.32071 0.660356 − 0.750953i \(-0.270405\pi\)
0.660356 + 0.750953i \(0.270405\pi\)
\(744\) 0 0
\(745\) −12.0000 −0.439646
\(746\) 26.0000 0.951928
\(747\) 0 0
\(748\) −24.0000 −0.877527
\(749\) −2.00000 −0.0730784
\(750\) 0 0
\(751\) 16.0000 0.583848 0.291924 − 0.956441i \(-0.405705\pi\)
0.291924 + 0.956441i \(0.405705\pi\)
\(752\) −36.0000 −1.31278
\(753\) 0 0
\(754\) −16.0000 −0.582686
\(755\) −5.00000 −0.181969
\(756\) 0 0
\(757\) 23.0000 0.835949 0.417975 − 0.908459i \(-0.362740\pi\)
0.417975 + 0.908459i \(0.362740\pi\)
\(758\) 6.00000 0.217930
\(759\) 0 0
\(760\) 0 0
\(761\) −3.00000 −0.108750 −0.0543750 − 0.998521i \(-0.517317\pi\)
−0.0543750 + 0.998521i \(0.517317\pi\)
\(762\) 0 0
\(763\) 9.00000 0.325822
\(764\) 4.00000 0.144715
\(765\) 0 0
\(766\) −42.0000 −1.51752
\(767\) 30.0000 1.08324
\(768\) 0 0
\(769\) 44.0000 1.58668 0.793340 − 0.608778i \(-0.208340\pi\)
0.793340 + 0.608778i \(0.208340\pi\)
\(770\) 8.00000 0.288300
\(771\) 0 0
\(772\) −50.0000 −1.79954
\(773\) 49.0000 1.76241 0.881204 − 0.472737i \(-0.156734\pi\)
0.881204 + 0.472737i \(0.156734\pi\)
\(774\) 0 0
\(775\) −24.0000 −0.862105
\(776\) 0 0
\(777\) 0 0
\(778\) 48.0000 1.72088
\(779\) −8.00000 −0.286630
\(780\) 0 0
\(781\) 48.0000 1.71758
\(782\) 36.0000 1.28736
\(783\) 0 0
\(784\) −4.00000 −0.142857
\(785\) 8.00000 0.285532
\(786\) 0 0
\(787\) −2.00000 −0.0712923 −0.0356462 − 0.999364i \(-0.511349\pi\)
−0.0356462 + 0.999364i \(0.511349\pi\)
\(788\) −8.00000 −0.284988
\(789\) 0 0
\(790\) −2.00000 −0.0711568
\(791\) −2.00000 −0.0711118
\(792\) 0 0
\(793\) −8.00000 −0.284088
\(794\) −12.0000 −0.425864
\(795\) 0 0
\(796\) −24.0000 −0.850657
\(797\) 2.00000 0.0708436 0.0354218 − 0.999372i \(-0.488723\pi\)
0.0354218 + 0.999372i \(0.488723\pi\)
\(798\) 0 0
\(799\) 27.0000 0.955191
\(800\) −32.0000 −1.13137
\(801\) 0 0
\(802\) 0 0
\(803\) −24.0000 −0.846942
\(804\) 0 0
\(805\) −6.00000 −0.211472
\(806\) 24.0000 0.845364
\(807\) 0 0
\(808\) 0 0
\(809\) −42.0000 −1.47664 −0.738321 − 0.674450i \(-0.764381\pi\)
−0.738321 + 0.674450i \(0.764381\pi\)
\(810\) 0 0
\(811\) −46.0000 −1.61528 −0.807639 − 0.589677i \(-0.799255\pi\)
−0.807639 + 0.589677i \(0.799255\pi\)
\(812\) 8.00000 0.280745
\(813\) 0 0
\(814\) −24.0000 −0.841200
\(815\) 11.0000 0.385313
\(816\) 0 0
\(817\) −88.0000 −3.07873
\(818\) −64.0000 −2.23771
\(819\) 0 0
\(820\) −2.00000 −0.0698430
\(821\) −8.00000 −0.279202 −0.139601 − 0.990208i \(-0.544582\pi\)
−0.139601 + 0.990208i \(0.544582\pi\)
\(822\) 0 0
\(823\) −3.00000 −0.104573 −0.0522867 − 0.998632i \(-0.516651\pi\)
−0.0522867 + 0.998632i \(0.516651\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) −30.0000 −1.04383
\(827\) −18.0000 −0.625921 −0.312961 − 0.949766i \(-0.601321\pi\)
−0.312961 + 0.949766i \(0.601321\pi\)
\(828\) 0 0
\(829\) −16.0000 −0.555703 −0.277851 − 0.960624i \(-0.589622\pi\)
−0.277851 + 0.960624i \(0.589622\pi\)
\(830\) −18.0000 −0.624789
\(831\) 0 0
\(832\) 16.0000 0.554700
\(833\) 3.00000 0.103944
\(834\) 0 0
\(835\) −17.0000 −0.588309
\(836\) 64.0000 2.21349
\(837\) 0 0
\(838\) 66.0000 2.27993
\(839\) −55.0000 −1.89881 −0.949405 − 0.314053i \(-0.898313\pi\)
−0.949405 + 0.314053i \(0.898313\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) −28.0000 −0.964944
\(843\) 0 0
\(844\) 8.00000 0.275371
\(845\) 9.00000 0.309609
\(846\) 0 0
\(847\) −5.00000 −0.171802
\(848\) −24.0000 −0.824163
\(849\) 0 0
\(850\) 24.0000 0.823193
\(851\) 18.0000 0.617032
\(852\) 0 0
\(853\) 8.00000 0.273915 0.136957 − 0.990577i \(-0.456268\pi\)
0.136957 + 0.990577i \(0.456268\pi\)
\(854\) 8.00000 0.273754
\(855\) 0 0
\(856\) 0 0
\(857\) 17.0000 0.580709 0.290354 − 0.956919i \(-0.406227\pi\)
0.290354 + 0.956919i \(0.406227\pi\)
\(858\) 0 0
\(859\) 2.00000 0.0682391 0.0341196 − 0.999418i \(-0.489137\pi\)
0.0341196 + 0.999418i \(0.489137\pi\)
\(860\) −22.0000 −0.750194
\(861\) 0 0
\(862\) −48.0000 −1.63489
\(863\) 6.00000 0.204242 0.102121 − 0.994772i \(-0.467437\pi\)
0.102121 + 0.994772i \(0.467437\pi\)
\(864\) 0 0
\(865\) −22.0000 −0.748022
\(866\) 4.00000 0.135926
\(867\) 0 0
\(868\) −12.0000 −0.407307
\(869\) 4.00000 0.135691
\(870\) 0 0
\(871\) 16.0000 0.542139
\(872\) 0 0
\(873\) 0 0
\(874\) −96.0000 −3.24725
\(875\) −9.00000 −0.304256
\(876\) 0 0
\(877\) 37.0000 1.24940 0.624701 − 0.780864i \(-0.285221\pi\)
0.624701 + 0.780864i \(0.285221\pi\)
\(878\) −48.0000 −1.61992
\(879\) 0 0
\(880\) −16.0000 −0.539360
\(881\) 30.0000 1.01073 0.505363 − 0.862907i \(-0.331359\pi\)
0.505363 + 0.862907i \(0.331359\pi\)
\(882\) 0 0
\(883\) 41.0000 1.37976 0.689880 − 0.723924i \(-0.257663\pi\)
0.689880 + 0.723924i \(0.257663\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) 48.0000 1.61259
\(887\) 13.0000 0.436497 0.218249 − 0.975893i \(-0.429966\pi\)
0.218249 + 0.975893i \(0.429966\pi\)
\(888\) 0 0
\(889\) 15.0000 0.503084
\(890\) 4.00000 0.134080
\(891\) 0 0
\(892\) −4.00000 −0.133930
\(893\) −72.0000 −2.40939
\(894\) 0 0
\(895\) 2.00000 0.0668526
\(896\) 0 0
\(897\) 0 0
\(898\) 12.0000 0.400445
\(899\) −24.0000 −0.800445
\(900\) 0 0
\(901\) 18.0000 0.599667
\(902\) 8.00000 0.266371
\(903\) 0 0
\(904\) 0 0
\(905\) −16.0000 −0.531858
\(906\) 0 0
\(907\) 17.0000 0.564476 0.282238 − 0.959344i \(-0.408923\pi\)
0.282238 + 0.959344i \(0.408923\pi\)
\(908\) −48.0000 −1.59294
\(909\) 0 0
\(910\) 4.00000 0.132599
\(911\) 12.0000 0.397578 0.198789 − 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) 0 0
\(913\) 36.0000 1.19143
\(914\) 28.0000 0.926158
\(915\) 0 0
\(916\) −8.00000 −0.264327
\(917\) 4.00000 0.132092
\(918\) 0 0
\(919\) −43.0000 −1.41844 −0.709220 − 0.704988i \(-0.750953\pi\)
−0.709220 + 0.704988i \(0.750953\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −2.00000 −0.0658665
\(923\) 24.0000 0.789970
\(924\) 0 0
\(925\) 12.0000 0.394558
\(926\) 46.0000 1.51165
\(927\) 0 0
\(928\) −32.0000 −1.05045
\(929\) −35.0000 −1.14831 −0.574156 − 0.818746i \(-0.694670\pi\)
−0.574156 + 0.818746i \(0.694670\pi\)
\(930\) 0 0
\(931\) −8.00000 −0.262189
\(932\) 12.0000 0.393073
\(933\) 0 0
\(934\) −24.0000 −0.785304
\(935\) 12.0000 0.392442
\(936\) 0 0
\(937\) −12.0000 −0.392023 −0.196011 − 0.980602i \(-0.562799\pi\)
−0.196011 + 0.980602i \(0.562799\pi\)
\(938\) −16.0000 −0.522419
\(939\) 0 0
\(940\) −18.0000 −0.587095
\(941\) −17.0000 −0.554184 −0.277092 − 0.960843i \(-0.589371\pi\)
−0.277092 + 0.960843i \(0.589371\pi\)
\(942\) 0 0
\(943\) −6.00000 −0.195387
\(944\) 60.0000 1.95283
\(945\) 0 0
\(946\) 88.0000 2.86113
\(947\) 28.0000 0.909878 0.454939 − 0.890523i \(-0.349661\pi\)
0.454939 + 0.890523i \(0.349661\pi\)
\(948\) 0 0
\(949\) −12.0000 −0.389536
\(950\) −64.0000 −2.07643
\(951\) 0 0
\(952\) 0 0
\(953\) −2.00000 −0.0647864 −0.0323932 − 0.999475i \(-0.510313\pi\)
−0.0323932 + 0.999475i \(0.510313\pi\)
\(954\) 0 0
\(955\) −2.00000 −0.0647185
\(956\) −36.0000 −1.16432
\(957\) 0 0
\(958\) 34.0000 1.09849
\(959\) 18.0000 0.581250
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) −12.0000 −0.386896
\(963\) 0 0
\(964\) −20.0000 −0.644157
\(965\) 25.0000 0.804778
\(966\) 0 0
\(967\) 40.0000 1.28631 0.643157 − 0.765735i \(-0.277624\pi\)
0.643157 + 0.765735i \(0.277624\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 24.0000 0.770594
\(971\) −45.0000 −1.44412 −0.722059 − 0.691831i \(-0.756804\pi\)
−0.722059 + 0.691831i \(0.756804\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −32.0000 −1.02535
\(975\) 0 0
\(976\) −16.0000 −0.512148
\(977\) −12.0000 −0.383914 −0.191957 − 0.981403i \(-0.561483\pi\)
−0.191957 + 0.981403i \(0.561483\pi\)
\(978\) 0 0
\(979\) −8.00000 −0.255681
\(980\) −2.00000 −0.0638877
\(981\) 0 0
\(982\) 52.0000 1.65939
\(983\) 21.0000 0.669796 0.334898 − 0.942254i \(-0.391298\pi\)
0.334898 + 0.942254i \(0.391298\pi\)
\(984\) 0 0
\(985\) 4.00000 0.127451
\(986\) 24.0000 0.764316
\(987\) 0 0
\(988\) 32.0000 1.01806
\(989\) −66.0000 −2.09868
\(990\) 0 0
\(991\) 11.0000 0.349427 0.174713 − 0.984619i \(-0.444100\pi\)
0.174713 + 0.984619i \(0.444100\pi\)
\(992\) 48.0000 1.52400
\(993\) 0 0
\(994\) −24.0000 −0.761234
\(995\) 12.0000 0.380426
\(996\) 0 0
\(997\) 40.0000 1.26681 0.633406 − 0.773819i \(-0.281656\pi\)
0.633406 + 0.773819i \(0.281656\pi\)
\(998\) −26.0000 −0.823016
\(999\) 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 189.2.a.a.1.1 ✓ 1
3.2 odd 2 189.2.a.d.1.1 yes 1
4.3 odd 2 3024.2.a.l.1.1 1
5.4 even 2 4725.2.a.s.1.1 1
7.6 odd 2 1323.2.a.b.1.1 1
9.2 odd 6 567.2.f.a.190.1 2
9.4 even 3 567.2.f.h.379.1 2
9.5 odd 6 567.2.f.a.379.1 2
9.7 even 3 567.2.f.h.190.1 2
12.11 even 2 3024.2.a.u.1.1 1
15.14 odd 2 4725.2.a.c.1.1 1
21.20 even 2 1323.2.a.r.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.a.a.1.1 ✓ 1 1.1 even 1 trivial
189.2.a.d.1.1 yes 1 3.2 odd 2
567.2.f.a.190.1 2 9.2 odd 6
567.2.f.a.379.1 2 9.5 odd 6
567.2.f.h.190.1 2 9.7 even 3
567.2.f.h.379.1 2 9.4 even 3
1323.2.a.b.1.1 1 7.6 odd 2
1323.2.a.r.1.1 1 21.20 even 2
3024.2.a.l.1.1 1 4.3 odd 2
3024.2.a.u.1.1 1 12.11 even 2
4725.2.a.c.1.1 1 15.14 odd 2
4725.2.a.s.1.1 1 5.4 even 2