Properties

Label 1890.2.a.p.1.1
Level $1890$
Weight $2$
Character 1890.1
Self dual yes
Analytic conductor $15.092$
Analytic rank $0$
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] = [1890,2,Mod(1,1890)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1890, 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("1890.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1890 = 2 \cdot 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1890.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: \(15.0917259820\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 1890.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} +1.00000 q^{11} +3.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +8.00000 q^{17} -3.00000 q^{19} -1.00000 q^{20} +1.00000 q^{22} -6.00000 q^{23} +1.00000 q^{25} +3.00000 q^{26} -1.00000 q^{28} +6.00000 q^{29} -4.00000 q^{31} +1.00000 q^{32} +8.00000 q^{34} +1.00000 q^{35} +2.00000 q^{37} -3.00000 q^{38} -1.00000 q^{40} +11.0000 q^{41} +1.00000 q^{43} +1.00000 q^{44} -6.00000 q^{46} -1.00000 q^{47} +1.00000 q^{49} +1.00000 q^{50} +3.00000 q^{52} +1.00000 q^{53} -1.00000 q^{55} -1.00000 q^{56} +6.00000 q^{58} +10.0000 q^{59} +4.00000 q^{61} -4.00000 q^{62} +1.00000 q^{64} -3.00000 q^{65} -3.00000 q^{67} +8.00000 q^{68} +1.00000 q^{70} +8.00000 q^{71} +11.0000 q^{73} +2.00000 q^{74} -3.00000 q^{76} -1.00000 q^{77} +4.00000 q^{79} -1.00000 q^{80} +11.0000 q^{82} +11.0000 q^{83} -8.00000 q^{85} +1.00000 q^{86} +1.00000 q^{88} +7.00000 q^{89} -3.00000 q^{91} -6.00000 q^{92} -1.00000 q^{94} +3.00000 q^{95} +2.00000 q^{97} +1.00000 q^{98} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) 1.00000 0.301511 0.150756 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) 0 0
\(13\) 3.00000 0.832050 0.416025 0.909353i \(-0.363423\pi\)
0.416025 + 0.909353i \(0.363423\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 8.00000 1.94029 0.970143 0.242536i \(-0.0779791\pi\)
0.970143 + 0.242536i \(0.0779791\pi\)
\(18\) 0 0
\(19\) −3.00000 −0.688247 −0.344124 0.938924i \(-0.611824\pi\)
−0.344124 + 0.938924i \(0.611824\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 3.00000 0.588348
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 8.00000 1.37199
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) −3.00000 −0.486664
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 11.0000 1.71791 0.858956 0.512050i \(-0.171114\pi\)
0.858956 + 0.512050i \(0.171114\pi\)
\(42\) 0 0
\(43\) 1.00000 0.152499 0.0762493 0.997089i \(-0.475706\pi\)
0.0762493 + 0.997089i \(0.475706\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) −6.00000 −0.884652
\(47\) −1.00000 −0.145865 −0.0729325 0.997337i \(-0.523236\pi\)
−0.0729325 + 0.997337i \(0.523236\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 3.00000 0.416025
\(53\) 1.00000 0.137361 0.0686803 0.997639i \(-0.478121\pi\)
0.0686803 + 0.997639i \(0.478121\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) 6.00000 0.787839
\(59\) 10.0000 1.30189 0.650945 0.759125i \(-0.274373\pi\)
0.650945 + 0.759125i \(0.274373\pi\)
\(60\) 0 0
\(61\) 4.00000 0.512148 0.256074 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) −4.00000 −0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −3.00000 −0.372104
\(66\) 0 0
\(67\) −3.00000 −0.366508 −0.183254 0.983066i \(-0.558663\pi\)
−0.183254 + 0.983066i \(0.558663\pi\)
\(68\) 8.00000 0.970143
\(69\) 0 0
\(70\) 1.00000 0.119523
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) 11.0000 1.28745 0.643726 0.765256i \(-0.277388\pi\)
0.643726 + 0.765256i \(0.277388\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) −3.00000 −0.344124
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) 11.0000 1.21475
\(83\) 11.0000 1.20741 0.603703 0.797209i \(-0.293691\pi\)
0.603703 + 0.797209i \(0.293691\pi\)
\(84\) 0 0
\(85\) −8.00000 −0.867722
\(86\) 1.00000 0.107833
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) 7.00000 0.741999 0.370999 0.928633i \(-0.379015\pi\)
0.370999 + 0.928633i \(0.379015\pi\)
\(90\) 0 0
\(91\) −3.00000 −0.314485
\(92\) −6.00000 −0.625543
\(93\) 0 0
\(94\) −1.00000 −0.103142
\(95\) 3.00000 0.307794
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −15.0000 −1.49256 −0.746278 0.665635i \(-0.768161\pi\)
−0.746278 + 0.665635i \(0.768161\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 3.00000 0.294174
\(105\) 0 0
\(106\) 1.00000 0.0971286
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) −9.00000 −0.862044 −0.431022 0.902342i \(-0.641847\pi\)
−0.431022 + 0.902342i \(0.641847\pi\)
\(110\) −1.00000 −0.0953463
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) −13.0000 −1.22294 −0.611469 0.791269i \(-0.709421\pi\)
−0.611469 + 0.791269i \(0.709421\pi\)
\(114\) 0 0
\(115\) 6.00000 0.559503
\(116\) 6.00000 0.557086
\(117\) 0 0
\(118\) 10.0000 0.920575
\(119\) −8.00000 −0.733359
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 4.00000 0.362143
\(123\) 0 0
\(124\) −4.00000 −0.359211
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −5.00000 −0.443678 −0.221839 0.975083i \(-0.571206\pi\)
−0.221839 + 0.975083i \(0.571206\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −3.00000 −0.263117
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 0 0
\(133\) 3.00000 0.260133
\(134\) −3.00000 −0.259161
\(135\) 0 0
\(136\) 8.00000 0.685994
\(137\) −3.00000 −0.256307 −0.128154 0.991754i \(-0.540905\pi\)
−0.128154 + 0.991754i \(0.540905\pi\)
\(138\) 0 0
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) 1.00000 0.0845154
\(141\) 0 0
\(142\) 8.00000 0.671345
\(143\) 3.00000 0.250873
\(144\) 0 0
\(145\) −6.00000 −0.498273
\(146\) 11.0000 0.910366
\(147\) 0 0
\(148\) 2.00000 0.164399
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 0 0
\(151\) −10.0000 −0.813788 −0.406894 0.913475i \(-0.633388\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(152\) −3.00000 −0.243332
\(153\) 0 0
\(154\) −1.00000 −0.0805823
\(155\) 4.00000 0.321288
\(156\) 0 0
\(157\) 22.0000 1.75579 0.877896 0.478852i \(-0.158947\pi\)
0.877896 + 0.478852i \(0.158947\pi\)
\(158\) 4.00000 0.318223
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) 6.00000 0.472866
\(162\) 0 0
\(163\) −16.0000 −1.25322 −0.626608 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) 11.0000 0.858956
\(165\) 0 0
\(166\) 11.0000 0.853766
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) −4.00000 −0.307692
\(170\) −8.00000 −0.613572
\(171\) 0 0
\(172\) 1.00000 0.0762493
\(173\) 2.00000 0.152057 0.0760286 0.997106i \(-0.475776\pi\)
0.0760286 + 0.997106i \(0.475776\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) 7.00000 0.524672
\(179\) 3.00000 0.224231 0.112115 0.993695i \(-0.464237\pi\)
0.112115 + 0.993695i \(0.464237\pi\)
\(180\) 0 0
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) −3.00000 −0.222375
\(183\) 0 0
\(184\) −6.00000 −0.442326
\(185\) −2.00000 −0.147043
\(186\) 0 0
\(187\) 8.00000 0.585018
\(188\) −1.00000 −0.0729325
\(189\) 0 0
\(190\) 3.00000 0.217643
\(191\) −3.00000 −0.217072 −0.108536 0.994092i \(-0.534616\pi\)
−0.108536 + 0.994092i \(0.534616\pi\)
\(192\) 0 0
\(193\) −10.0000 −0.719816 −0.359908 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 11.0000 0.783718 0.391859 0.920025i \(-0.371832\pi\)
0.391859 + 0.920025i \(0.371832\pi\)
\(198\) 0 0
\(199\) 23.0000 1.63043 0.815213 0.579161i \(-0.196620\pi\)
0.815213 + 0.579161i \(0.196620\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) −15.0000 −1.05540
\(203\) −6.00000 −0.421117
\(204\) 0 0
\(205\) −11.0000 −0.768273
\(206\) −8.00000 −0.557386
\(207\) 0 0
\(208\) 3.00000 0.208013
\(209\) −3.00000 −0.207514
\(210\) 0 0
\(211\) −26.0000 −1.78991 −0.894957 0.446153i \(-0.852794\pi\)
−0.894957 + 0.446153i \(0.852794\pi\)
\(212\) 1.00000 0.0686803
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) −1.00000 −0.0681994
\(216\) 0 0
\(217\) 4.00000 0.271538
\(218\) −9.00000 −0.609557
\(219\) 0 0
\(220\) −1.00000 −0.0674200
\(221\) 24.0000 1.61441
\(222\) 0 0
\(223\) −2.00000 −0.133930 −0.0669650 0.997755i \(-0.521332\pi\)
−0.0669650 + 0.997755i \(0.521332\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) −13.0000 −0.864747
\(227\) 11.0000 0.730096 0.365048 0.930989i \(-0.381053\pi\)
0.365048 + 0.930989i \(0.381053\pi\)
\(228\) 0 0
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 6.00000 0.395628
\(231\) 0 0
\(232\) 6.00000 0.393919
\(233\) −9.00000 −0.589610 −0.294805 0.955557i \(-0.595255\pi\)
−0.294805 + 0.955557i \(0.595255\pi\)
\(234\) 0 0
\(235\) 1.00000 0.0652328
\(236\) 10.0000 0.650945
\(237\) 0 0
\(238\) −8.00000 −0.518563
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 0 0
\(241\) −20.0000 −1.28831 −0.644157 0.764894i \(-0.722792\pi\)
−0.644157 + 0.764894i \(0.722792\pi\)
\(242\) −10.0000 −0.642824
\(243\) 0 0
\(244\) 4.00000 0.256074
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) −9.00000 −0.572656
\(248\) −4.00000 −0.254000
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) 30.0000 1.89358 0.946792 0.321847i \(-0.104304\pi\)
0.946792 + 0.321847i \(0.104304\pi\)
\(252\) 0 0
\(253\) −6.00000 −0.377217
\(254\) −5.00000 −0.313728
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 20.0000 1.24757 0.623783 0.781598i \(-0.285595\pi\)
0.623783 + 0.781598i \(0.285595\pi\)
\(258\) 0 0
\(259\) −2.00000 −0.124274
\(260\) −3.00000 −0.186052
\(261\) 0 0
\(262\) −4.00000 −0.247121
\(263\) −16.0000 −0.986602 −0.493301 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) 0 0
\(265\) −1.00000 −0.0614295
\(266\) 3.00000 0.183942
\(267\) 0 0
\(268\) −3.00000 −0.183254
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) −27.0000 −1.64013 −0.820067 0.572268i \(-0.806064\pi\)
−0.820067 + 0.572268i \(0.806064\pi\)
\(272\) 8.00000 0.485071
\(273\) 0 0
\(274\) −3.00000 −0.181237
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) 18.0000 1.08152 0.540758 0.841178i \(-0.318138\pi\)
0.540758 + 0.841178i \(0.318138\pi\)
\(278\) −20.0000 −1.19952
\(279\) 0 0
\(280\) 1.00000 0.0597614
\(281\) −10.0000 −0.596550 −0.298275 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) 0 0
\(283\) −22.0000 −1.30776 −0.653882 0.756596i \(-0.726861\pi\)
−0.653882 + 0.756596i \(0.726861\pi\)
\(284\) 8.00000 0.474713
\(285\) 0 0
\(286\) 3.00000 0.177394
\(287\) −11.0000 −0.649309
\(288\) 0 0
\(289\) 47.0000 2.76471
\(290\) −6.00000 −0.352332
\(291\) 0 0
\(292\) 11.0000 0.643726
\(293\) −14.0000 −0.817889 −0.408944 0.912559i \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(294\) 0 0
\(295\) −10.0000 −0.582223
\(296\) 2.00000 0.116248
\(297\) 0 0
\(298\) −18.0000 −1.04271
\(299\) −18.0000 −1.04097
\(300\) 0 0
\(301\) −1.00000 −0.0576390
\(302\) −10.0000 −0.575435
\(303\) 0 0
\(304\) −3.00000 −0.172062
\(305\) −4.00000 −0.229039
\(306\) 0 0
\(307\) 28.0000 1.59804 0.799022 0.601302i \(-0.205351\pi\)
0.799022 + 0.601302i \(0.205351\pi\)
\(308\) −1.00000 −0.0569803
\(309\) 0 0
\(310\) 4.00000 0.227185
\(311\) −18.0000 −1.02069 −0.510343 0.859971i \(-0.670482\pi\)
−0.510343 + 0.859971i \(0.670482\pi\)
\(312\) 0 0
\(313\) 29.0000 1.63918 0.819588 0.572953i \(-0.194202\pi\)
0.819588 + 0.572953i \(0.194202\pi\)
\(314\) 22.0000 1.24153
\(315\) 0 0
\(316\) 4.00000 0.225018
\(317\) −23.0000 −1.29181 −0.645904 0.763418i \(-0.723520\pi\)
−0.645904 + 0.763418i \(0.723520\pi\)
\(318\) 0 0
\(319\) 6.00000 0.335936
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) 6.00000 0.334367
\(323\) −24.0000 −1.33540
\(324\) 0 0
\(325\) 3.00000 0.166410
\(326\) −16.0000 −0.886158
\(327\) 0 0
\(328\) 11.0000 0.607373
\(329\) 1.00000 0.0551318
\(330\) 0 0
\(331\) 34.0000 1.86881 0.934405 0.356214i \(-0.115932\pi\)
0.934405 + 0.356214i \(0.115932\pi\)
\(332\) 11.0000 0.603703
\(333\) 0 0
\(334\) 12.0000 0.656611
\(335\) 3.00000 0.163908
\(336\) 0 0
\(337\) −8.00000 −0.435788 −0.217894 0.975972i \(-0.569919\pi\)
−0.217894 + 0.975972i \(0.569919\pi\)
\(338\) −4.00000 −0.217571
\(339\) 0 0
\(340\) −8.00000 −0.433861
\(341\) −4.00000 −0.216612
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 1.00000 0.0539164
\(345\) 0 0
\(346\) 2.00000 0.107521
\(347\) −2.00000 −0.107366 −0.0536828 0.998558i \(-0.517096\pi\)
−0.0536828 + 0.998558i \(0.517096\pi\)
\(348\) 0 0
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) −8.00000 −0.424596
\(356\) 7.00000 0.370999
\(357\) 0 0
\(358\) 3.00000 0.158555
\(359\) −1.00000 −0.0527780 −0.0263890 0.999652i \(-0.508401\pi\)
−0.0263890 + 0.999652i \(0.508401\pi\)
\(360\) 0 0
\(361\) −10.0000 −0.526316
\(362\) −14.0000 −0.735824
\(363\) 0 0
\(364\) −3.00000 −0.157243
\(365\) −11.0000 −0.575766
\(366\) 0 0
\(367\) 10.0000 0.521996 0.260998 0.965339i \(-0.415948\pi\)
0.260998 + 0.965339i \(0.415948\pi\)
\(368\) −6.00000 −0.312772
\(369\) 0 0
\(370\) −2.00000 −0.103975
\(371\) −1.00000 −0.0519174
\(372\) 0 0
\(373\) 32.0000 1.65690 0.828449 0.560065i \(-0.189224\pi\)
0.828449 + 0.560065i \(0.189224\pi\)
\(374\) 8.00000 0.413670
\(375\) 0 0
\(376\) −1.00000 −0.0515711
\(377\) 18.0000 0.927047
\(378\) 0 0
\(379\) 12.0000 0.616399 0.308199 0.951322i \(-0.400274\pi\)
0.308199 + 0.951322i \(0.400274\pi\)
\(380\) 3.00000 0.153897
\(381\) 0 0
\(382\) −3.00000 −0.153493
\(383\) −24.0000 −1.22634 −0.613171 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 0 0
\(385\) 1.00000 0.0509647
\(386\) −10.0000 −0.508987
\(387\) 0 0
\(388\) 2.00000 0.101535
\(389\) 26.0000 1.31825 0.659126 0.752032i \(-0.270926\pi\)
0.659126 + 0.752032i \(0.270926\pi\)
\(390\) 0 0
\(391\) −48.0000 −2.42746
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) 11.0000 0.554172
\(395\) −4.00000 −0.201262
\(396\) 0 0
\(397\) −14.0000 −0.702640 −0.351320 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) 23.0000 1.15289
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −20.0000 −0.998752 −0.499376 0.866385i \(-0.666437\pi\)
−0.499376 + 0.866385i \(0.666437\pi\)
\(402\) 0 0
\(403\) −12.0000 −0.597763
\(404\) −15.0000 −0.746278
\(405\) 0 0
\(406\) −6.00000 −0.297775
\(407\) 2.00000 0.0991363
\(408\) 0 0
\(409\) −18.0000 −0.890043 −0.445021 0.895520i \(-0.646804\pi\)
−0.445021 + 0.895520i \(0.646804\pi\)
\(410\) −11.0000 −0.543251
\(411\) 0 0
\(412\) −8.00000 −0.394132
\(413\) −10.0000 −0.492068
\(414\) 0 0
\(415\) −11.0000 −0.539969
\(416\) 3.00000 0.147087
\(417\) 0 0
\(418\) −3.00000 −0.146735
\(419\) −18.0000 −0.879358 −0.439679 0.898155i \(-0.644908\pi\)
−0.439679 + 0.898155i \(0.644908\pi\)
\(420\) 0 0
\(421\) −1.00000 −0.0487370 −0.0243685 0.999703i \(-0.507758\pi\)
−0.0243685 + 0.999703i \(0.507758\pi\)
\(422\) −26.0000 −1.26566
\(423\) 0 0
\(424\) 1.00000 0.0485643
\(425\) 8.00000 0.388057
\(426\) 0 0
\(427\) −4.00000 −0.193574
\(428\) 12.0000 0.580042
\(429\) 0 0
\(430\) −1.00000 −0.0482243
\(431\) 9.00000 0.433515 0.216757 0.976226i \(-0.430452\pi\)
0.216757 + 0.976226i \(0.430452\pi\)
\(432\) 0 0
\(433\) −7.00000 −0.336399 −0.168199 0.985753i \(-0.553795\pi\)
−0.168199 + 0.985753i \(0.553795\pi\)
\(434\) 4.00000 0.192006
\(435\) 0 0
\(436\) −9.00000 −0.431022
\(437\) 18.0000 0.861057
\(438\) 0 0
\(439\) −11.0000 −0.525001 −0.262501 0.964932i \(-0.584547\pi\)
−0.262501 + 0.964932i \(0.584547\pi\)
\(440\) −1.00000 −0.0476731
\(441\) 0 0
\(442\) 24.0000 1.14156
\(443\) 16.0000 0.760183 0.380091 0.924949i \(-0.375893\pi\)
0.380091 + 0.924949i \(0.375893\pi\)
\(444\) 0 0
\(445\) −7.00000 −0.331832
\(446\) −2.00000 −0.0947027
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) −36.0000 −1.69895 −0.849473 0.527633i \(-0.823080\pi\)
−0.849473 + 0.527633i \(0.823080\pi\)
\(450\) 0 0
\(451\) 11.0000 0.517970
\(452\) −13.0000 −0.611469
\(453\) 0 0
\(454\) 11.0000 0.516256
\(455\) 3.00000 0.140642
\(456\) 0 0
\(457\) 16.0000 0.748448 0.374224 0.927338i \(-0.377909\pi\)
0.374224 + 0.927338i \(0.377909\pi\)
\(458\) −14.0000 −0.654177
\(459\) 0 0
\(460\) 6.00000 0.279751
\(461\) 1.00000 0.0465746 0.0232873 0.999729i \(-0.492587\pi\)
0.0232873 + 0.999729i \(0.492587\pi\)
\(462\) 0 0
\(463\) 17.0000 0.790057 0.395029 0.918669i \(-0.370735\pi\)
0.395029 + 0.918669i \(0.370735\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) −9.00000 −0.416917
\(467\) −28.0000 −1.29569 −0.647843 0.761774i \(-0.724329\pi\)
−0.647843 + 0.761774i \(0.724329\pi\)
\(468\) 0 0
\(469\) 3.00000 0.138527
\(470\) 1.00000 0.0461266
\(471\) 0 0
\(472\) 10.0000 0.460287
\(473\) 1.00000 0.0459800
\(474\) 0 0
\(475\) −3.00000 −0.137649
\(476\) −8.00000 −0.366679
\(477\) 0 0
\(478\) −8.00000 −0.365911
\(479\) −12.0000 −0.548294 −0.274147 0.961688i \(-0.588395\pi\)
−0.274147 + 0.961688i \(0.588395\pi\)
\(480\) 0 0
\(481\) 6.00000 0.273576
\(482\) −20.0000 −0.910975
\(483\) 0 0
\(484\) −10.0000 −0.454545
\(485\) −2.00000 −0.0908153
\(486\) 0 0
\(487\) 11.0000 0.498458 0.249229 0.968445i \(-0.419823\pi\)
0.249229 + 0.968445i \(0.419823\pi\)
\(488\) 4.00000 0.181071
\(489\) 0 0
\(490\) −1.00000 −0.0451754
\(491\) −36.0000 −1.62466 −0.812329 0.583200i \(-0.801800\pi\)
−0.812329 + 0.583200i \(0.801800\pi\)
\(492\) 0 0
\(493\) 48.0000 2.16181
\(494\) −9.00000 −0.404929
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) −8.00000 −0.358849
\(498\) 0 0
\(499\) 18.0000 0.805791 0.402895 0.915246i \(-0.368004\pi\)
0.402895 + 0.915246i \(0.368004\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) 30.0000 1.33897
\(503\) 21.0000 0.936344 0.468172 0.883637i \(-0.344913\pi\)
0.468172 + 0.883637i \(0.344913\pi\)
\(504\) 0 0
\(505\) 15.0000 0.667491
\(506\) −6.00000 −0.266733
\(507\) 0 0
\(508\) −5.00000 −0.221839
\(509\) −26.0000 −1.15243 −0.576215 0.817298i \(-0.695471\pi\)
−0.576215 + 0.817298i \(0.695471\pi\)
\(510\) 0 0
\(511\) −11.0000 −0.486611
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 20.0000 0.882162
\(515\) 8.00000 0.352522
\(516\) 0 0
\(517\) −1.00000 −0.0439799
\(518\) −2.00000 −0.0878750
\(519\) 0 0
\(520\) −3.00000 −0.131559
\(521\) 35.0000 1.53338 0.766689 0.642019i \(-0.221903\pi\)
0.766689 + 0.642019i \(0.221903\pi\)
\(522\) 0 0
\(523\) −26.0000 −1.13690 −0.568450 0.822718i \(-0.692457\pi\)
−0.568450 + 0.822718i \(0.692457\pi\)
\(524\) −4.00000 −0.174741
\(525\) 0 0
\(526\) −16.0000 −0.697633
\(527\) −32.0000 −1.39394
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) −1.00000 −0.0434372
\(531\) 0 0
\(532\) 3.00000 0.130066
\(533\) 33.0000 1.42939
\(534\) 0 0
\(535\) −12.0000 −0.518805
\(536\) −3.00000 −0.129580
\(537\) 0 0
\(538\) −6.00000 −0.258678
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) 9.00000 0.386940 0.193470 0.981106i \(-0.438026\pi\)
0.193470 + 0.981106i \(0.438026\pi\)
\(542\) −27.0000 −1.15975
\(543\) 0 0
\(544\) 8.00000 0.342997
\(545\) 9.00000 0.385518
\(546\) 0 0
\(547\) −24.0000 −1.02617 −0.513083 0.858339i \(-0.671497\pi\)
−0.513083 + 0.858339i \(0.671497\pi\)
\(548\) −3.00000 −0.128154
\(549\) 0 0
\(550\) 1.00000 0.0426401
\(551\) −18.0000 −0.766826
\(552\) 0 0
\(553\) −4.00000 −0.170097
\(554\) 18.0000 0.764747
\(555\) 0 0
\(556\) −20.0000 −0.848189
\(557\) 46.0000 1.94908 0.974541 0.224208i \(-0.0719796\pi\)
0.974541 + 0.224208i \(0.0719796\pi\)
\(558\) 0 0
\(559\) 3.00000 0.126886
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) −10.0000 −0.421825
\(563\) 11.0000 0.463595 0.231797 0.972764i \(-0.425539\pi\)
0.231797 + 0.972764i \(0.425539\pi\)
\(564\) 0 0
\(565\) 13.0000 0.546914
\(566\) −22.0000 −0.924729
\(567\) 0 0
\(568\) 8.00000 0.335673
\(569\) −18.0000 −0.754599 −0.377300 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) 0 0
\(571\) 36.0000 1.50655 0.753277 0.657704i \(-0.228472\pi\)
0.753277 + 0.657704i \(0.228472\pi\)
\(572\) 3.00000 0.125436
\(573\) 0 0
\(574\) −11.0000 −0.459131
\(575\) −6.00000 −0.250217
\(576\) 0 0
\(577\) 25.0000 1.04076 0.520382 0.853934i \(-0.325790\pi\)
0.520382 + 0.853934i \(0.325790\pi\)
\(578\) 47.0000 1.95494
\(579\) 0 0
\(580\) −6.00000 −0.249136
\(581\) −11.0000 −0.456357
\(582\) 0 0
\(583\) 1.00000 0.0414158
\(584\) 11.0000 0.455183
\(585\) 0 0
\(586\) −14.0000 −0.578335
\(587\) 20.0000 0.825488 0.412744 0.910847i \(-0.364570\pi\)
0.412744 + 0.910847i \(0.364570\pi\)
\(588\) 0 0
\(589\) 12.0000 0.494451
\(590\) −10.0000 −0.411693
\(591\) 0 0
\(592\) 2.00000 0.0821995
\(593\) −48.0000 −1.97112 −0.985562 0.169316i \(-0.945844\pi\)
−0.985562 + 0.169316i \(0.945844\pi\)
\(594\) 0 0
\(595\) 8.00000 0.327968
\(596\) −18.0000 −0.737309
\(597\) 0 0
\(598\) −18.0000 −0.736075
\(599\) −9.00000 −0.367730 −0.183865 0.982952i \(-0.558861\pi\)
−0.183865 + 0.982952i \(0.558861\pi\)
\(600\) 0 0
\(601\) 12.0000 0.489490 0.244745 0.969587i \(-0.421296\pi\)
0.244745 + 0.969587i \(0.421296\pi\)
\(602\) −1.00000 −0.0407570
\(603\) 0 0
\(604\) −10.0000 −0.406894
\(605\) 10.0000 0.406558
\(606\) 0 0
\(607\) 34.0000 1.38002 0.690009 0.723801i \(-0.257607\pi\)
0.690009 + 0.723801i \(0.257607\pi\)
\(608\) −3.00000 −0.121666
\(609\) 0 0
\(610\) −4.00000 −0.161955
\(611\) −3.00000 −0.121367
\(612\) 0 0
\(613\) −36.0000 −1.45403 −0.727013 0.686624i \(-0.759092\pi\)
−0.727013 + 0.686624i \(0.759092\pi\)
\(614\) 28.0000 1.12999
\(615\) 0 0
\(616\) −1.00000 −0.0402911
\(617\) 14.0000 0.563619 0.281809 0.959470i \(-0.409065\pi\)
0.281809 + 0.959470i \(0.409065\pi\)
\(618\) 0 0
\(619\) 11.0000 0.442127 0.221064 0.975259i \(-0.429047\pi\)
0.221064 + 0.975259i \(0.429047\pi\)
\(620\) 4.00000 0.160644
\(621\) 0 0
\(622\) −18.0000 −0.721734
\(623\) −7.00000 −0.280449
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 29.0000 1.15907
\(627\) 0 0
\(628\) 22.0000 0.877896
\(629\) 16.0000 0.637962
\(630\) 0 0
\(631\) −34.0000 −1.35352 −0.676759 0.736204i \(-0.736616\pi\)
−0.676759 + 0.736204i \(0.736616\pi\)
\(632\) 4.00000 0.159111
\(633\) 0 0
\(634\) −23.0000 −0.913447
\(635\) 5.00000 0.198419
\(636\) 0 0
\(637\) 3.00000 0.118864
\(638\) 6.00000 0.237542
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) −28.0000 −1.10593 −0.552967 0.833203i \(-0.686504\pi\)
−0.552967 + 0.833203i \(0.686504\pi\)
\(642\) 0 0
\(643\) 16.0000 0.630978 0.315489 0.948929i \(-0.397831\pi\)
0.315489 + 0.948929i \(0.397831\pi\)
\(644\) 6.00000 0.236433
\(645\) 0 0
\(646\) −24.0000 −0.944267
\(647\) 7.00000 0.275198 0.137599 0.990488i \(-0.456061\pi\)
0.137599 + 0.990488i \(0.456061\pi\)
\(648\) 0 0
\(649\) 10.0000 0.392534
\(650\) 3.00000 0.117670
\(651\) 0 0
\(652\) −16.0000 −0.626608
\(653\) −22.0000 −0.860927 −0.430463 0.902608i \(-0.641650\pi\)
−0.430463 + 0.902608i \(0.641650\pi\)
\(654\) 0 0
\(655\) 4.00000 0.156293
\(656\) 11.0000 0.429478
\(657\) 0 0
\(658\) 1.00000 0.0389841
\(659\) −8.00000 −0.311636 −0.155818 0.987786i \(-0.549801\pi\)
−0.155818 + 0.987786i \(0.549801\pi\)
\(660\) 0 0
\(661\) −14.0000 −0.544537 −0.272268 0.962221i \(-0.587774\pi\)
−0.272268 + 0.962221i \(0.587774\pi\)
\(662\) 34.0000 1.32145
\(663\) 0 0
\(664\) 11.0000 0.426883
\(665\) −3.00000 −0.116335
\(666\) 0 0
\(667\) −36.0000 −1.39393
\(668\) 12.0000 0.464294
\(669\) 0 0
\(670\) 3.00000 0.115900
\(671\) 4.00000 0.154418
\(672\) 0 0
\(673\) −32.0000 −1.23351 −0.616755 0.787155i \(-0.711553\pi\)
−0.616755 + 0.787155i \(0.711553\pi\)
\(674\) −8.00000 −0.308148
\(675\) 0 0
\(676\) −4.00000 −0.153846
\(677\) −30.0000 −1.15299 −0.576497 0.817099i \(-0.695581\pi\)
−0.576497 + 0.817099i \(0.695581\pi\)
\(678\) 0 0
\(679\) −2.00000 −0.0767530
\(680\) −8.00000 −0.306786
\(681\) 0 0
\(682\) −4.00000 −0.153168
\(683\) 2.00000 0.0765279 0.0382639 0.999268i \(-0.487817\pi\)
0.0382639 + 0.999268i \(0.487817\pi\)
\(684\) 0 0
\(685\) 3.00000 0.114624
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) 1.00000 0.0381246
\(689\) 3.00000 0.114291
\(690\) 0 0
\(691\) −35.0000 −1.33146 −0.665731 0.746191i \(-0.731880\pi\)
−0.665731 + 0.746191i \(0.731880\pi\)
\(692\) 2.00000 0.0760286
\(693\) 0 0
\(694\) −2.00000 −0.0759190
\(695\) 20.0000 0.758643
\(696\) 0 0
\(697\) 88.0000 3.33324
\(698\) 10.0000 0.378506
\(699\) 0 0
\(700\) −1.00000 −0.0377964
\(701\) −32.0000 −1.20862 −0.604312 0.796748i \(-0.706552\pi\)
−0.604312 + 0.796748i \(0.706552\pi\)
\(702\) 0 0
\(703\) −6.00000 −0.226294
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) 0 0
\(707\) 15.0000 0.564133
\(708\) 0 0
\(709\) 25.0000 0.938895 0.469447 0.882960i \(-0.344453\pi\)
0.469447 + 0.882960i \(0.344453\pi\)
\(710\) −8.00000 −0.300235
\(711\) 0 0
\(712\) 7.00000 0.262336
\(713\) 24.0000 0.898807
\(714\) 0 0
\(715\) −3.00000 −0.112194
\(716\) 3.00000 0.112115
\(717\) 0 0
\(718\) −1.00000 −0.0373197
\(719\) 48.0000 1.79010 0.895049 0.445968i \(-0.147140\pi\)
0.895049 + 0.445968i \(0.147140\pi\)
\(720\) 0 0
\(721\) 8.00000 0.297936
\(722\) −10.0000 −0.372161
\(723\) 0 0
\(724\) −14.0000 −0.520306
\(725\) 6.00000 0.222834
\(726\) 0 0
\(727\) −24.0000 −0.890111 −0.445055 0.895503i \(-0.646816\pi\)
−0.445055 + 0.895503i \(0.646816\pi\)
\(728\) −3.00000 −0.111187
\(729\) 0 0
\(730\) −11.0000 −0.407128
\(731\) 8.00000 0.295891
\(732\) 0 0
\(733\) −5.00000 −0.184679 −0.0923396 0.995728i \(-0.529435\pi\)
−0.0923396 + 0.995728i \(0.529435\pi\)
\(734\) 10.0000 0.369107
\(735\) 0 0
\(736\) −6.00000 −0.221163
\(737\) −3.00000 −0.110506
\(738\) 0 0
\(739\) −8.00000 −0.294285 −0.147142 0.989115i \(-0.547008\pi\)
−0.147142 + 0.989115i \(0.547008\pi\)
\(740\) −2.00000 −0.0735215
\(741\) 0 0
\(742\) −1.00000 −0.0367112
\(743\) 6.00000 0.220119 0.110059 0.993925i \(-0.464896\pi\)
0.110059 + 0.993925i \(0.464896\pi\)
\(744\) 0 0
\(745\) 18.0000 0.659469
\(746\) 32.0000 1.17160
\(747\) 0 0
\(748\) 8.00000 0.292509
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) −34.0000 −1.24068 −0.620339 0.784334i \(-0.713005\pi\)
−0.620339 + 0.784334i \(0.713005\pi\)
\(752\) −1.00000 −0.0364662
\(753\) 0 0
\(754\) 18.0000 0.655521
\(755\) 10.0000 0.363937
\(756\) 0 0
\(757\) 28.0000 1.01768 0.508839 0.860862i \(-0.330075\pi\)
0.508839 + 0.860862i \(0.330075\pi\)
\(758\) 12.0000 0.435860
\(759\) 0 0
\(760\) 3.00000 0.108821
\(761\) 42.0000 1.52250 0.761249 0.648459i \(-0.224586\pi\)
0.761249 + 0.648459i \(0.224586\pi\)
\(762\) 0 0
\(763\) 9.00000 0.325822
\(764\) −3.00000 −0.108536
\(765\) 0 0
\(766\) −24.0000 −0.867155
\(767\) 30.0000 1.08324
\(768\) 0 0
\(769\) 4.00000 0.144244 0.0721218 0.997396i \(-0.477023\pi\)
0.0721218 + 0.997396i \(0.477023\pi\)
\(770\) 1.00000 0.0360375
\(771\) 0 0
\(772\) −10.0000 −0.359908
\(773\) −6.00000 −0.215805 −0.107903 0.994161i \(-0.534413\pi\)
−0.107903 + 0.994161i \(0.534413\pi\)
\(774\) 0 0
\(775\) −4.00000 −0.143684
\(776\) 2.00000 0.0717958
\(777\) 0 0
\(778\) 26.0000 0.932145
\(779\) −33.0000 −1.18235
\(780\) 0 0
\(781\) 8.00000 0.286263
\(782\) −48.0000 −1.71648
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) −22.0000 −0.785214
\(786\) 0 0
\(787\) 28.0000 0.998092 0.499046 0.866575i \(-0.333684\pi\)
0.499046 + 0.866575i \(0.333684\pi\)
\(788\) 11.0000 0.391859
\(789\) 0 0
\(790\) −4.00000 −0.142314
\(791\) 13.0000 0.462227
\(792\) 0 0
\(793\) 12.0000 0.426132
\(794\) −14.0000 −0.496841
\(795\) 0 0
\(796\) 23.0000 0.815213
\(797\) 12.0000 0.425062 0.212531 0.977154i \(-0.431829\pi\)
0.212531 + 0.977154i \(0.431829\pi\)
\(798\) 0 0
\(799\) −8.00000 −0.283020
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) −20.0000 −0.706225
\(803\) 11.0000 0.388182
\(804\) 0 0
\(805\) −6.00000 −0.211472
\(806\) −12.0000 −0.422682
\(807\) 0 0
\(808\) −15.0000 −0.527698
\(809\) 18.0000 0.632846 0.316423 0.948618i \(-0.397518\pi\)
0.316423 + 0.948618i \(0.397518\pi\)
\(810\) 0 0
\(811\) 29.0000 1.01833 0.509164 0.860670i \(-0.329955\pi\)
0.509164 + 0.860670i \(0.329955\pi\)
\(812\) −6.00000 −0.210559
\(813\) 0 0
\(814\) 2.00000 0.0701000
\(815\) 16.0000 0.560456
\(816\) 0 0
\(817\) −3.00000 −0.104957
\(818\) −18.0000 −0.629355
\(819\) 0 0
\(820\) −11.0000 −0.384137
\(821\) 42.0000 1.46581 0.732905 0.680331i \(-0.238164\pi\)
0.732905 + 0.680331i \(0.238164\pi\)
\(822\) 0 0
\(823\) 37.0000 1.28974 0.644869 0.764293i \(-0.276912\pi\)
0.644869 + 0.764293i \(0.276912\pi\)
\(824\) −8.00000 −0.278693
\(825\) 0 0
\(826\) −10.0000 −0.347945
\(827\) 32.0000 1.11275 0.556375 0.830932i \(-0.312192\pi\)
0.556375 + 0.830932i \(0.312192\pi\)
\(828\) 0 0
\(829\) −16.0000 −0.555703 −0.277851 0.960624i \(-0.589622\pi\)
−0.277851 + 0.960624i \(0.589622\pi\)
\(830\) −11.0000 −0.381816
\(831\) 0 0
\(832\) 3.00000 0.104006
\(833\) 8.00000 0.277184
\(834\) 0 0
\(835\) −12.0000 −0.415277
\(836\) −3.00000 −0.103757
\(837\) 0 0
\(838\) −18.0000 −0.621800
\(839\) 50.0000 1.72619 0.863096 0.505040i \(-0.168522\pi\)
0.863096 + 0.505040i \(0.168522\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −1.00000 −0.0344623
\(843\) 0 0
\(844\) −26.0000 −0.894957
\(845\) 4.00000 0.137604
\(846\) 0 0
\(847\) 10.0000 0.343604
\(848\) 1.00000 0.0343401
\(849\) 0 0
\(850\) 8.00000 0.274398
\(851\) −12.0000 −0.411355
\(852\) 0 0
\(853\) −22.0000 −0.753266 −0.376633 0.926363i \(-0.622918\pi\)
−0.376633 + 0.926363i \(0.622918\pi\)
\(854\) −4.00000 −0.136877
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) −18.0000 −0.614868 −0.307434 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 0 0
\(859\) −3.00000 −0.102359 −0.0511793 0.998689i \(-0.516298\pi\)
−0.0511793 + 0.998689i \(0.516298\pi\)
\(860\) −1.00000 −0.0340997
\(861\) 0 0
\(862\) 9.00000 0.306541
\(863\) −14.0000 −0.476566 −0.238283 0.971196i \(-0.576585\pi\)
−0.238283 + 0.971196i \(0.576585\pi\)
\(864\) 0 0
\(865\) −2.00000 −0.0680020
\(866\) −7.00000 −0.237870
\(867\) 0 0
\(868\) 4.00000 0.135769
\(869\) 4.00000 0.135691
\(870\) 0 0
\(871\) −9.00000 −0.304953
\(872\) −9.00000 −0.304778
\(873\) 0 0
\(874\) 18.0000 0.608859
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) −28.0000 −0.945493 −0.472746 0.881199i \(-0.656737\pi\)
−0.472746 + 0.881199i \(0.656737\pi\)
\(878\) −11.0000 −0.371232
\(879\) 0 0
\(880\) −1.00000 −0.0337100
\(881\) 50.0000 1.68454 0.842271 0.539054i \(-0.181218\pi\)
0.842271 + 0.539054i \(0.181218\pi\)
\(882\) 0 0
\(883\) 36.0000 1.21150 0.605748 0.795656i \(-0.292874\pi\)
0.605748 + 0.795656i \(0.292874\pi\)
\(884\) 24.0000 0.807207
\(885\) 0 0
\(886\) 16.0000 0.537531
\(887\) −37.0000 −1.24234 −0.621169 0.783676i \(-0.713342\pi\)
−0.621169 + 0.783676i \(0.713342\pi\)
\(888\) 0 0
\(889\) 5.00000 0.167695
\(890\) −7.00000 −0.234641
\(891\) 0 0
\(892\) −2.00000 −0.0669650
\(893\) 3.00000 0.100391
\(894\) 0 0
\(895\) −3.00000 −0.100279
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) −36.0000 −1.20134
\(899\) −24.0000 −0.800445
\(900\) 0 0
\(901\) 8.00000 0.266519
\(902\) 11.0000 0.366260
\(903\) 0 0
\(904\) −13.0000 −0.432374
\(905\) 14.0000 0.465376
\(906\) 0 0
\(907\) 17.0000 0.564476 0.282238 0.959344i \(-0.408923\pi\)
0.282238 + 0.959344i \(0.408923\pi\)
\(908\) 11.0000 0.365048
\(909\) 0 0
\(910\) 3.00000 0.0994490
\(911\) −33.0000 −1.09334 −0.546669 0.837349i \(-0.684105\pi\)
−0.546669 + 0.837349i \(0.684105\pi\)
\(912\) 0 0
\(913\) 11.0000 0.364047
\(914\) 16.0000 0.529233
\(915\) 0 0
\(916\) −14.0000 −0.462573
\(917\) 4.00000 0.132092
\(918\) 0 0
\(919\) −28.0000 −0.923635 −0.461817 0.886975i \(-0.652802\pi\)
−0.461817 + 0.886975i \(0.652802\pi\)
\(920\) 6.00000 0.197814
\(921\) 0 0
\(922\) 1.00000 0.0329332
\(923\) 24.0000 0.789970
\(924\) 0 0
\(925\) 2.00000 0.0657596
\(926\) 17.0000 0.558655
\(927\) 0 0
\(928\) 6.00000 0.196960
\(929\) −35.0000 −1.14831 −0.574156 0.818746i \(-0.694670\pi\)
−0.574156 + 0.818746i \(0.694670\pi\)
\(930\) 0 0
\(931\) −3.00000 −0.0983210
\(932\) −9.00000 −0.294805
\(933\) 0 0
\(934\) −28.0000 −0.916188
\(935\) −8.00000 −0.261628
\(936\) 0 0
\(937\) −37.0000 −1.20874 −0.604369 0.796705i \(-0.706575\pi\)
−0.604369 + 0.796705i \(0.706575\pi\)
\(938\) 3.00000 0.0979535
\(939\) 0 0
\(940\) 1.00000 0.0326164
\(941\) 23.0000 0.749779 0.374889 0.927070i \(-0.377681\pi\)
0.374889 + 0.927070i \(0.377681\pi\)
\(942\) 0 0
\(943\) −66.0000 −2.14926
\(944\) 10.0000 0.325472
\(945\) 0 0
\(946\) 1.00000 0.0325128
\(947\) 8.00000 0.259965 0.129983 0.991516i \(-0.458508\pi\)
0.129983 + 0.991516i \(0.458508\pi\)
\(948\) 0 0
\(949\) 33.0000 1.07123
\(950\) −3.00000 −0.0973329
\(951\) 0 0
\(952\) −8.00000 −0.259281
\(953\) −42.0000 −1.36051 −0.680257 0.732974i \(-0.738132\pi\)
−0.680257 + 0.732974i \(0.738132\pi\)
\(954\) 0 0
\(955\) 3.00000 0.0970777
\(956\) −8.00000 −0.258738
\(957\) 0 0
\(958\) −12.0000 −0.387702
\(959\) 3.00000 0.0968751
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 6.00000 0.193448
\(963\) 0 0
\(964\) −20.0000 −0.644157
\(965\) 10.0000 0.321911
\(966\) 0 0
\(967\) −35.0000 −1.12552 −0.562762 0.826619i \(-0.690261\pi\)
−0.562762 + 0.826619i \(0.690261\pi\)
\(968\) −10.0000 −0.321412
\(969\) 0 0
\(970\) −2.00000 −0.0642161
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 20.0000 0.641171
\(974\) 11.0000 0.352463
\(975\) 0 0
\(976\) 4.00000 0.128037
\(977\) 3.00000 0.0959785 0.0479893 0.998848i \(-0.484719\pi\)
0.0479893 + 0.998848i \(0.484719\pi\)
\(978\) 0 0
\(979\) 7.00000 0.223721
\(980\) −1.00000 −0.0319438
\(981\) 0 0
\(982\) −36.0000 −1.14881
\(983\) 56.0000 1.78612 0.893061 0.449935i \(-0.148553\pi\)
0.893061 + 0.449935i \(0.148553\pi\)
\(984\) 0 0
\(985\) −11.0000 −0.350489
\(986\) 48.0000 1.52863
\(987\) 0 0
\(988\) −9.00000 −0.286328
\(989\) −6.00000 −0.190789
\(990\) 0 0
\(991\) −44.0000 −1.39771 −0.698853 0.715265i \(-0.746306\pi\)
−0.698853 + 0.715265i \(0.746306\pi\)
\(992\) −4.00000 −0.127000
\(993\) 0 0
\(994\) −8.00000 −0.253745
\(995\) −23.0000 −0.729149
\(996\) 0 0
\(997\) −10.0000 −0.316703 −0.158352 0.987383i \(-0.550618\pi\)
−0.158352 + 0.987383i \(0.550618\pi\)
\(998\) 18.0000 0.569780
\(999\) 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 1890.2.a.p.1.1 yes 1
3.2 odd 2 1890.2.a.f.1.1 1
5.4 even 2 9450.2.a.bp.1.1 1
15.14 odd 2 9450.2.a.dj.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1890.2.a.f.1.1 1 3.2 odd 2
1890.2.a.p.1.1 yes 1 1.1 even 1 trivial
9450.2.a.bp.1.1 1 5.4 even 2
9450.2.a.dj.1.1 1 15.14 odd 2