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Properties

Label 2574.2.a.d.1.1
Level $2574$
Weight $2$
Character 2574.1
Self dual yes
Analytic conductor $20.553$
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] = [2574,2,Mod(1,2574)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2574, 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("2574.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2574 = 2 \cdot 3^{2} \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2574.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: \(20.5534934803\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 858)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2574.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} +4.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} +4.00000 q^{7} -1.00000 q^{8} +2.00000 q^{10} -1.00000 q^{11} -1.00000 q^{13} -4.00000 q^{14} +1.00000 q^{16} +8.00000 q^{17} -6.00000 q^{19} -2.00000 q^{20} +1.00000 q^{22} +6.00000 q^{23} -1.00000 q^{25} +1.00000 q^{26} +4.00000 q^{28} +2.00000 q^{29} -2.00000 q^{31} -1.00000 q^{32} -8.00000 q^{34} -8.00000 q^{35} +4.00000 q^{37} +6.00000 q^{38} +2.00000 q^{40} -2.00000 q^{41} +4.00000 q^{43} -1.00000 q^{44} -6.00000 q^{46} -4.00000 q^{47} +9.00000 q^{49} +1.00000 q^{50} -1.00000 q^{52} +8.00000 q^{53} +2.00000 q^{55} -4.00000 q^{56} -2.00000 q^{58} -12.0000 q^{59} -2.00000 q^{61} +2.00000 q^{62} +1.00000 q^{64} +2.00000 q^{65} +12.0000 q^{67} +8.00000 q^{68} +8.00000 q^{70} +16.0000 q^{71} -4.00000 q^{74} -6.00000 q^{76} -4.00000 q^{77} -8.00000 q^{79} -2.00000 q^{80} +2.00000 q^{82} -4.00000 q^{83} -16.0000 q^{85} -4.00000 q^{86} +1.00000 q^{88} -10.0000 q^{89} -4.00000 q^{91} +6.00000 q^{92} +4.00000 q^{94} +12.0000 q^{95} +10.0000 q^{97} -9.00000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 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 \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 2.00000 0.632456
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) −4.00000 −1.06904
\(15\) 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
\(19\) −6.00000 −1.37649 −0.688247 − 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 6.00000 1.25109 0.625543 − 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 1.00000 0.196116
\(27\) 0 0
\(28\) 4.00000 0.755929
\(29\) 2.00000 0.371391 0.185695 − 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 − 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −8.00000 −1.37199
\(35\) −8.00000 −1.35225
\(36\) 0 0
\(37\) 4.00000 0.657596 0.328798 − 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 6.00000 0.973329
\(39\) 0 0
\(40\) 2.00000 0.316228
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) −6.00000 −0.884652
\(47\) −4.00000 −0.583460 −0.291730 − 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −1.00000 −0.138675
\(53\) 8.00000 1.09888 0.549442 − 0.835532i \(-0.314840\pi\)
0.549442 + 0.835532i \(0.314840\pi\)
\(54\) 0 0
\(55\) 2.00000 0.269680
\(56\) −4.00000 −0.534522
\(57\) 0 0
\(58\) −2.00000 −0.262613
\(59\) −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 \(0.540868\pi\)
\(62\) 2.00000 0.254000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) 12.0000 1.46603 0.733017 − 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) 8.00000 0.970143
\(69\) 0 0
\(70\) 8.00000 0.956183
\(71\) 16.0000 1.89885 0.949425 − 0.313993i \(-0.101667\pi\)
0.949425 + 0.313993i \(0.101667\pi\)
\(72\) 0 0
\(73\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) −4.00000 −0.464991
\(75\) 0 0
\(76\) −6.00000 −0.688247
\(77\) −4.00000 −0.455842
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 − 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) −2.00000 −0.223607
\(81\) 0 0
\(82\) 2.00000 0.220863
\(83\) −4.00000 −0.439057 −0.219529 − 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) −16.0000 −1.73544
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) −10.0000 −1.06000 −0.529999 − 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) −4.00000 −0.419314
\(92\) 6.00000 0.625543
\(93\) 0 0
\(94\) 4.00000 0.412568
\(95\) 12.0000 1.23117
\(96\) 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
\(101\) −14.0000 −1.39305 −0.696526 − 0.717532i \(-0.745272\pi\)
−0.696526 + 0.717532i \(0.745272\pi\)
\(102\) 0 0
\(103\) 4.00000 0.394132 0.197066 − 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) −8.00000 −0.777029
\(107\) 18.0000 1.74013 0.870063 − 0.492941i \(-0.164078\pi\)
0.870063 + 0.492941i \(0.164078\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 − 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) −2.00000 −0.190693
\(111\) 0 0
\(112\) 4.00000 0.377964
\(113\) −6.00000 −0.564433 −0.282216 − 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) −12.0000 −1.11901
\(116\) 2.00000 0.185695
\(117\) 0 0
\(118\) 12.0000 1.10469
\(119\) 32.0000 2.93344
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 2.00000 0.181071
\(123\) 0 0
\(124\) −2.00000 −0.179605
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 − 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −2.00000 −0.175412
\(131\) 6.00000 0.524222 0.262111 − 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) 0 0
\(133\) −24.0000 −2.08106
\(134\) −12.0000 −1.03664
\(135\) 0 0
\(136\) −8.00000 −0.685994
\(137\) 2.00000 0.170872 0.0854358 − 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) 0 0
\(139\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) −8.00000 −0.676123
\(141\) 0 0
\(142\) −16.0000 −1.34269
\(143\) 1.00000 0.0836242
\(144\) 0 0
\(145\) −4.00000 −0.332182
\(146\) 0 0
\(147\) 0 0
\(148\) 4.00000 0.328798
\(149\) −2.00000 −0.163846 −0.0819232 − 0.996639i \(-0.526106\pi\)
−0.0819232 + 0.996639i \(0.526106\pi\)
\(150\) 0 0
\(151\) 20.0000 1.62758 0.813788 − 0.581161i \(-0.197401\pi\)
0.813788 + 0.581161i \(0.197401\pi\)
\(152\) 6.00000 0.486664
\(153\) 0 0
\(154\) 4.00000 0.322329
\(155\) 4.00000 0.321288
\(156\) 0 0
\(157\) 18.0000 1.43656 0.718278 − 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) 8.00000 0.636446
\(159\) 0 0
\(160\) 2.00000 0.158114
\(161\) 24.0000 1.89146
\(162\) 0 0
\(163\) −8.00000 −0.626608 −0.313304 − 0.949653i \(-0.601436\pi\)
−0.313304 + 0.949653i \(0.601436\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 16.0000 1.22714
\(171\) 0 0
\(172\) 4.00000 0.304997
\(173\) 6.00000 0.456172 0.228086 − 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) −4.00000 −0.302372
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) 10.0000 0.749532
\(179\) 12.0000 0.896922 0.448461 − 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) 26.0000 1.93256 0.966282 − 0.257485i \(-0.0828937\pi\)
0.966282 + 0.257485i \(0.0828937\pi\)
\(182\) 4.00000 0.296500
\(183\) 0 0
\(184\) −6.00000 −0.442326
\(185\) −8.00000 −0.588172
\(186\) 0 0
\(187\) −8.00000 −0.585018
\(188\) −4.00000 −0.291730
\(189\) 0 0
\(190\) −12.0000 −0.870572
\(191\) −10.0000 −0.723575 −0.361787 − 0.932261i \(-0.617833\pi\)
−0.361787 + 0.932261i \(0.617833\pi\)
\(192\) 0 0
\(193\) 20.0000 1.43963 0.719816 − 0.694165i \(-0.244226\pi\)
0.719816 + 0.694165i \(0.244226\pi\)
\(194\) −10.0000 −0.717958
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) 6.00000 0.427482 0.213741 − 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 − 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) 14.0000 0.985037
\(203\) 8.00000 0.561490
\(204\) 0 0
\(205\) 4.00000 0.279372
\(206\) −4.00000 −0.278693
\(207\) 0 0
\(208\) −1.00000 −0.0693375
\(209\) 6.00000 0.415029
\(210\) 0 0
\(211\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) 8.00000 0.549442
\(213\) 0 0
\(214\) −18.0000 −1.23045
\(215\) −8.00000 −0.545595
\(216\) 0 0
\(217\) −8.00000 −0.543075
\(218\) −2.00000 −0.135457
\(219\) 0 0
\(220\) 2.00000 0.134840
\(221\) −8.00000 −0.538138
\(222\) 0 0
\(223\) 14.0000 0.937509 0.468755 − 0.883328i \(-0.344703\pi\)
0.468755 + 0.883328i \(0.344703\pi\)
\(224\) −4.00000 −0.267261
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) −20.0000 −1.32164 −0.660819 − 0.750546i \(-0.729791\pi\)
−0.660819 + 0.750546i \(0.729791\pi\)
\(230\) 12.0000 0.791257
\(231\) 0 0
\(232\) −2.00000 −0.131306
\(233\) 12.0000 0.786146 0.393073 − 0.919507i \(-0.371412\pi\)
0.393073 + 0.919507i \(0.371412\pi\)
\(234\) 0 0
\(235\) 8.00000 0.521862
\(236\) −12.0000 −0.781133
\(237\) 0 0
\(238\) −32.0000 −2.07425
\(239\) 8.00000 0.517477 0.258738 − 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 0 0
\(241\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 0 0
\(244\) −2.00000 −0.128037
\(245\) −18.0000 −1.14998
\(246\) 0 0
\(247\) 6.00000 0.381771
\(248\) 2.00000 0.127000
\(249\) 0 0
\(250\) −12.0000 −0.758947
\(251\) 20.0000 1.26239 0.631194 − 0.775625i \(-0.282565\pi\)
0.631194 + 0.775625i \(0.282565\pi\)
\(252\) 0 0
\(253\) −6.00000 −0.377217
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −18.0000 −1.12281 −0.561405 − 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) 16.0000 0.994192
\(260\) 2.00000 0.124035
\(261\) 0 0
\(262\) −6.00000 −0.370681
\(263\) 16.0000 0.986602 0.493301 − 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) 0 0
\(265\) −16.0000 −0.982872
\(266\) 24.0000 1.47153
\(267\) 0 0
\(268\) 12.0000 0.733017
\(269\) −28.0000 −1.70719 −0.853595 − 0.520937i \(-0.825583\pi\)
−0.853595 + 0.520937i \(0.825583\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 − 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 8.00000 0.485071
\(273\) 0 0
\(274\) −2.00000 −0.120824
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) 2.00000 0.120168 0.0600842 − 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 8.00000 0.478091
\(281\) 14.0000 0.835170 0.417585 − 0.908638i \(-0.362877\pi\)
0.417585 + 0.908638i \(0.362877\pi\)
\(282\) 0 0
\(283\) −20.0000 −1.18888 −0.594438 − 0.804141i \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) 16.0000 0.949425
\(285\) 0 0
\(286\) −1.00000 −0.0591312
\(287\) −8.00000 −0.472225
\(288\) 0 0
\(289\) 47.0000 2.76471
\(290\) 4.00000 0.234888
\(291\) 0 0
\(292\) 0 0
\(293\) 22.0000 1.28525 0.642627 − 0.766179i \(-0.277845\pi\)
0.642627 + 0.766179i \(0.277845\pi\)
\(294\) 0 0
\(295\) 24.0000 1.39733
\(296\) −4.00000 −0.232495
\(297\) 0 0
\(298\) 2.00000 0.115857
\(299\) −6.00000 −0.346989
\(300\) 0 0
\(301\) 16.0000 0.922225
\(302\) −20.0000 −1.15087
\(303\) 0 0
\(304\) −6.00000 −0.344124
\(305\) 4.00000 0.229039
\(306\) 0 0
\(307\) −34.0000 −1.94048 −0.970241 − 0.242140i \(-0.922151\pi\)
−0.970241 + 0.242140i \(0.922151\pi\)
\(308\) −4.00000 −0.227921
\(309\) 0 0
\(310\) −4.00000 −0.227185
\(311\) 10.0000 0.567048 0.283524 − 0.958965i \(-0.408496\pi\)
0.283524 + 0.958965i \(0.408496\pi\)
\(312\) 0 0
\(313\) 6.00000 0.339140 0.169570 − 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) −18.0000 −1.01580
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) −10.0000 −0.561656 −0.280828 − 0.959758i \(-0.590609\pi\)
−0.280828 + 0.959758i \(0.590609\pi\)
\(318\) 0 0
\(319\) −2.00000 −0.111979
\(320\) −2.00000 −0.111803
\(321\) 0 0
\(322\) −24.0000 −1.33747
\(323\) −48.0000 −2.67079
\(324\) 0 0
\(325\) 1.00000 0.0554700
\(326\) 8.00000 0.443079
\(327\) 0 0
\(328\) 2.00000 0.110432
\(329\) −16.0000 −0.882109
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 − 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) −4.00000 −0.219529
\(333\) 0 0
\(334\) 0 0
\(335\) −24.0000 −1.31126
\(336\) 0 0
\(337\) −10.0000 −0.544735 −0.272367 − 0.962193i \(-0.587807\pi\)
−0.272367 + 0.962193i \(0.587807\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 0 0
\(340\) −16.0000 −0.867722
\(341\) 2.00000 0.108306
\(342\) 0 0
\(343\) 8.00000 0.431959
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) 18.0000 0.966291 0.483145 − 0.875540i \(-0.339494\pi\)
0.483145 + 0.875540i \(0.339494\pi\)
\(348\) 0 0
\(349\) 22.0000 1.17763 0.588817 − 0.808267i \(-0.299594\pi\)
0.588817 + 0.808267i \(0.299594\pi\)
\(350\) 4.00000 0.213809
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) 30.0000 1.59674 0.798369 − 0.602168i \(-0.205696\pi\)
0.798369 + 0.602168i \(0.205696\pi\)
\(354\) 0 0
\(355\) −32.0000 −1.69838
\(356\) −10.0000 −0.529999
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) −16.0000 −0.844448 −0.422224 − 0.906492i \(-0.638750\pi\)
−0.422224 + 0.906492i \(0.638750\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) −26.0000 −1.36653
\(363\) 0 0
\(364\) −4.00000 −0.209657
\(365\) 0 0
\(366\) 0 0
\(367\) 20.0000 1.04399 0.521996 − 0.852948i \(-0.325188\pi\)
0.521996 + 0.852948i \(0.325188\pi\)
\(368\) 6.00000 0.312772
\(369\) 0 0
\(370\) 8.00000 0.415900
\(371\) 32.0000 1.66136
\(372\) 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
\(376\) 4.00000 0.206284
\(377\) −2.00000 −0.103005
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 − 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 12.0000 0.615587
\(381\) 0 0
\(382\) 10.0000 0.511645
\(383\) 36.0000 1.83951 0.919757 − 0.392488i \(-0.128386\pi\)
0.919757 + 0.392488i \(0.128386\pi\)
\(384\) 0 0
\(385\) 8.00000 0.407718
\(386\) −20.0000 −1.01797
\(387\) 0 0
\(388\) 10.0000 0.507673
\(389\) −16.0000 −0.811232 −0.405616 − 0.914044i \(-0.632943\pi\)
−0.405616 + 0.914044i \(0.632943\pi\)
\(390\) 0 0
\(391\) 48.0000 2.42746
\(392\) −9.00000 −0.454569
\(393\) 0 0
\(394\) −6.00000 −0.302276
\(395\) 16.0000 0.805047
\(396\) 0 0
\(397\) 24.0000 1.20453 0.602263 − 0.798298i \(-0.294266\pi\)
0.602263 + 0.798298i \(0.294266\pi\)
\(398\) 16.0000 0.802008
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) −30.0000 −1.49813 −0.749064 − 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 0 0
\(403\) 2.00000 0.0996271
\(404\) −14.0000 −0.696526
\(405\) 0 0
\(406\) −8.00000 −0.397033
\(407\) −4.00000 −0.198273
\(408\) 0 0
\(409\) −4.00000 −0.197787 −0.0988936 − 0.995098i \(-0.531530\pi\)
−0.0988936 + 0.995098i \(0.531530\pi\)
\(410\) −4.00000 −0.197546
\(411\) 0 0
\(412\) 4.00000 0.197066
\(413\) −48.0000 −2.36193
\(414\) 0 0
\(415\) 8.00000 0.392705
\(416\) 1.00000 0.0490290
\(417\) 0 0
\(418\) −6.00000 −0.293470
\(419\) −36.0000 −1.75872 −0.879358 − 0.476162i \(-0.842028\pi\)
−0.879358 + 0.476162i \(0.842028\pi\)
\(420\) 0 0
\(421\) −36.0000 −1.75453 −0.877266 − 0.480004i \(-0.840635\pi\)
−0.877266 + 0.480004i \(0.840635\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) −8.00000 −0.388514
\(425\) −8.00000 −0.388057
\(426\) 0 0
\(427\) −8.00000 −0.387147
\(428\) 18.0000 0.870063
\(429\) 0 0
\(430\) 8.00000 0.385794
\(431\) −8.00000 −0.385346 −0.192673 − 0.981263i \(-0.561716\pi\)
−0.192673 + 0.981263i \(0.561716\pi\)
\(432\) 0 0
\(433\) 30.0000 1.44171 0.720854 − 0.693087i \(-0.243750\pi\)
0.720854 + 0.693087i \(0.243750\pi\)
\(434\) 8.00000 0.384012
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) −36.0000 −1.72211
\(438\) 0 0
\(439\) −8.00000 −0.381819 −0.190910 − 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) −2.00000 −0.0953463
\(441\) 0 0
\(442\) 8.00000 0.380521
\(443\) −24.0000 −1.14027 −0.570137 − 0.821549i \(-0.693110\pi\)
−0.570137 + 0.821549i \(0.693110\pi\)
\(444\) 0 0
\(445\) 20.0000 0.948091
\(446\) −14.0000 −0.662919
\(447\) 0 0
\(448\) 4.00000 0.188982
\(449\) 18.0000 0.849473 0.424736 − 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 0 0
\(451\) 2.00000 0.0941763
\(452\) −6.00000 −0.282216
\(453\) 0 0
\(454\) 0 0
\(455\) 8.00000 0.375046
\(456\) 0 0
\(457\) 28.0000 1.30978 0.654892 − 0.755722i \(-0.272714\pi\)
0.654892 + 0.755722i \(0.272714\pi\)
\(458\) 20.0000 0.934539
\(459\) 0 0
\(460\) −12.0000 −0.559503
\(461\) 18.0000 0.838344 0.419172 − 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) −38.0000 −1.76601 −0.883005 − 0.469364i \(-0.844483\pi\)
−0.883005 + 0.469364i \(0.844483\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) −12.0000 −0.555889
\(467\) 8.00000 0.370196 0.185098 − 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) 0 0
\(469\) 48.0000 2.21643
\(470\) −8.00000 −0.369012
\(471\) 0 0
\(472\) 12.0000 0.552345
\(473\) −4.00000 −0.183920
\(474\) 0 0
\(475\) 6.00000 0.275299
\(476\) 32.0000 1.46672
\(477\) 0 0
\(478\) −8.00000 −0.365911
\(479\) 8.00000 0.365529 0.182765 − 0.983157i \(-0.441495\pi\)
0.182765 + 0.983157i \(0.441495\pi\)
\(480\) 0 0
\(481\) −4.00000 −0.182384
\(482\) 0 0
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −20.0000 −0.908153
\(486\) 0 0
\(487\) −14.0000 −0.634401 −0.317200 − 0.948359i \(-0.602743\pi\)
−0.317200 + 0.948359i \(0.602743\pi\)
\(488\) 2.00000 0.0905357
\(489\) 0 0
\(490\) 18.0000 0.813157
\(491\) 30.0000 1.35388 0.676941 − 0.736038i \(-0.263305\pi\)
0.676941 + 0.736038i \(0.263305\pi\)
\(492\) 0 0
\(493\) 16.0000 0.720604
\(494\) −6.00000 −0.269953
\(495\) 0 0
\(496\) −2.00000 −0.0898027
\(497\) 64.0000 2.87079
\(498\) 0 0
\(499\) −20.0000 −0.895323 −0.447661 − 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) 12.0000 0.536656
\(501\) 0 0
\(502\) −20.0000 −0.892644
\(503\) −16.0000 −0.713405 −0.356702 − 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) 0 0
\(505\) 28.0000 1.24598
\(506\) 6.00000 0.266733
\(507\) 0 0
\(508\) 8.00000 0.354943
\(509\) −34.0000 −1.50702 −0.753512 − 0.657434i \(-0.771642\pi\)
−0.753512 + 0.657434i \(0.771642\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 18.0000 0.793946
\(515\) −8.00000 −0.352522
\(516\) 0 0
\(517\) 4.00000 0.175920
\(518\) −16.0000 −0.703000
\(519\) 0 0
\(520\) −2.00000 −0.0877058
\(521\) 30.0000 1.31432 0.657162 − 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) 0 0
\(523\) −4.00000 −0.174908 −0.0874539 − 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) 6.00000 0.262111
\(525\) 0 0
\(526\) −16.0000 −0.697633
\(527\) −16.0000 −0.696971
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 16.0000 0.694996
\(531\) 0 0
\(532\) −24.0000 −1.04053
\(533\) 2.00000 0.0866296
\(534\) 0 0
\(535\) −36.0000 −1.55642
\(536\) −12.0000 −0.518321
\(537\) 0 0
\(538\) 28.0000 1.20717
\(539\) −9.00000 −0.387657
\(540\) 0 0
\(541\) −30.0000 −1.28980 −0.644900 − 0.764267i \(-0.723101\pi\)
−0.644900 + 0.764267i \(0.723101\pi\)
\(542\) −16.0000 −0.687259
\(543\) 0 0
\(544\) −8.00000 −0.342997
\(545\) −4.00000 −0.171341
\(546\) 0 0
\(547\) 40.0000 1.71028 0.855138 − 0.518400i \(-0.173472\pi\)
0.855138 + 0.518400i \(0.173472\pi\)
\(548\) 2.00000 0.0854358
\(549\) 0 0
\(550\) −1.00000 −0.0426401
\(551\) −12.0000 −0.511217
\(552\) 0 0
\(553\) −32.0000 −1.36078
\(554\) −2.00000 −0.0849719
\(555\) 0 0
\(556\) 0 0
\(557\) −14.0000 −0.593199 −0.296600 − 0.955002i \(-0.595853\pi\)
−0.296600 + 0.955002i \(0.595853\pi\)
\(558\) 0 0
\(559\) −4.00000 −0.169182
\(560\) −8.00000 −0.338062
\(561\) 0 0
\(562\) −14.0000 −0.590554
\(563\) −42.0000 −1.77009 −0.885044 − 0.465506i \(-0.845872\pi\)
−0.885044 + 0.465506i \(0.845872\pi\)
\(564\) 0 0
\(565\) 12.0000 0.504844
\(566\) 20.0000 0.840663
\(567\) 0 0
\(568\) −16.0000 −0.671345
\(569\) 4.00000 0.167689 0.0838444 − 0.996479i \(-0.473280\pi\)
0.0838444 + 0.996479i \(0.473280\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 − 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 1.00000 0.0418121
\(573\) 0 0
\(574\) 8.00000 0.333914
\(575\) −6.00000 −0.250217
\(576\) 0 0
\(577\) 46.0000 1.91501 0.957503 − 0.288425i \(-0.0931316\pi\)
0.957503 + 0.288425i \(0.0931316\pi\)
\(578\) −47.0000 −1.95494
\(579\) 0 0
\(580\) −4.00000 −0.166091
\(581\) −16.0000 −0.663792
\(582\) 0 0
\(583\) −8.00000 −0.331326
\(584\) 0 0
\(585\) 0 0
\(586\) −22.0000 −0.908812
\(587\) −44.0000 −1.81607 −0.908037 − 0.418890i \(-0.862419\pi\)
−0.908037 + 0.418890i \(0.862419\pi\)
\(588\) 0 0
\(589\) 12.0000 0.494451
\(590\) −24.0000 −0.988064
\(591\) 0 0
\(592\) 4.00000 0.164399
\(593\) −14.0000 −0.574911 −0.287456 − 0.957794i \(-0.592809\pi\)
−0.287456 + 0.957794i \(0.592809\pi\)
\(594\) 0 0
\(595\) −64.0000 −2.62374
\(596\) −2.00000 −0.0819232
\(597\) 0 0
\(598\) 6.00000 0.245358
\(599\) −30.0000 −1.22577 −0.612883 − 0.790173i \(-0.709990\pi\)
−0.612883 + 0.790173i \(0.709990\pi\)
\(600\) 0 0
\(601\) −42.0000 −1.71322 −0.856608 − 0.515968i \(-0.827432\pi\)
−0.856608 + 0.515968i \(0.827432\pi\)
\(602\) −16.0000 −0.652111
\(603\) 0 0
\(604\) 20.0000 0.813788
\(605\) −2.00000 −0.0813116
\(606\) 0 0
\(607\) −40.0000 −1.62355 −0.811775 − 0.583970i \(-0.801498\pi\)
−0.811775 + 0.583970i \(0.801498\pi\)
\(608\) 6.00000 0.243332
\(609\) 0 0
\(610\) −4.00000 −0.161955
\(611\) 4.00000 0.161823
\(612\) 0 0
\(613\) 2.00000 0.0807792 0.0403896 − 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) 34.0000 1.37213
\(615\) 0 0
\(616\) 4.00000 0.161165
\(617\) −26.0000 −1.04672 −0.523360 − 0.852111i \(-0.675322\pi\)
−0.523360 + 0.852111i \(0.675322\pi\)
\(618\) 0 0
\(619\) −32.0000 −1.28619 −0.643094 − 0.765787i \(-0.722350\pi\)
−0.643094 + 0.765787i \(0.722350\pi\)
\(620\) 4.00000 0.160644
\(621\) 0 0
\(622\) −10.0000 −0.400963
\(623\) −40.0000 −1.60257
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) −6.00000 −0.239808
\(627\) 0 0
\(628\) 18.0000 0.718278
\(629\) 32.0000 1.27592
\(630\) 0 0
\(631\) −18.0000 −0.716569 −0.358284 − 0.933613i \(-0.616638\pi\)
−0.358284 + 0.933613i \(0.616638\pi\)
\(632\) 8.00000 0.318223
\(633\) 0 0
\(634\) 10.0000 0.397151
\(635\) −16.0000 −0.634941
\(636\) 0 0
\(637\) −9.00000 −0.356593
\(638\) 2.00000 0.0791808
\(639\) 0 0
\(640\) 2.00000 0.0790569
\(641\) 18.0000 0.710957 0.355479 − 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 0 0
\(643\) −28.0000 −1.10421 −0.552106 − 0.833774i \(-0.686176\pi\)
−0.552106 + 0.833774i \(0.686176\pi\)
\(644\) 24.0000 0.945732
\(645\) 0 0
\(646\) 48.0000 1.88853
\(647\) −50.0000 −1.96570 −0.982851 − 0.184399i \(-0.940966\pi\)
−0.982851 + 0.184399i \(0.940966\pi\)
\(648\) 0 0
\(649\) 12.0000 0.471041
\(650\) −1.00000 −0.0392232
\(651\) 0 0
\(652\) −8.00000 −0.313304
\(653\) −48.0000 −1.87839 −0.939193 − 0.343391i \(-0.888424\pi\)
−0.939193 + 0.343391i \(0.888424\pi\)
\(654\) 0 0
\(655\) −12.0000 −0.468879
\(656\) −2.00000 −0.0780869
\(657\) 0 0
\(658\) 16.0000 0.623745
\(659\) 38.0000 1.48027 0.740135 − 0.672458i \(-0.234762\pi\)
0.740135 + 0.672458i \(0.234762\pi\)
\(660\) 0 0
\(661\) 12.0000 0.466746 0.233373 − 0.972387i \(-0.425024\pi\)
0.233373 + 0.972387i \(0.425024\pi\)
\(662\) 4.00000 0.155464
\(663\) 0 0
\(664\) 4.00000 0.155230
\(665\) 48.0000 1.86136
\(666\) 0 0
\(667\) 12.0000 0.464642
\(668\) 0 0
\(669\) 0 0
\(670\) 24.0000 0.927201
\(671\) 2.00000 0.0772091
\(672\) 0 0
\(673\) −22.0000 −0.848038 −0.424019 − 0.905653i \(-0.639381\pi\)
−0.424019 + 0.905653i \(0.639381\pi\)
\(674\) 10.0000 0.385186
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) 22.0000 0.845529 0.422764 − 0.906240i \(-0.361060\pi\)
0.422764 + 0.906240i \(0.361060\pi\)
\(678\) 0 0
\(679\) 40.0000 1.53506
\(680\) 16.0000 0.613572
\(681\) 0 0
\(682\) −2.00000 −0.0765840
\(683\) 28.0000 1.07139 0.535695 − 0.844411i \(-0.320050\pi\)
0.535695 + 0.844411i \(0.320050\pi\)
\(684\) 0 0
\(685\) −4.00000 −0.152832
\(686\) −8.00000 −0.305441
\(687\) 0 0
\(688\) 4.00000 0.152499
\(689\) −8.00000 −0.304776
\(690\) 0 0
\(691\) −32.0000 −1.21734 −0.608669 − 0.793424i \(-0.708296\pi\)
−0.608669 + 0.793424i \(0.708296\pi\)
\(692\) 6.00000 0.228086
\(693\) 0 0
\(694\) −18.0000 −0.683271
\(695\) 0 0
\(696\) 0 0
\(697\) −16.0000 −0.606043
\(698\) −22.0000 −0.832712
\(699\) 0 0
\(700\) −4.00000 −0.151186
\(701\) 30.0000 1.13308 0.566542 − 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) 0 0
\(703\) −24.0000 −0.905177
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) −30.0000 −1.12906
\(707\) −56.0000 −2.10610
\(708\) 0 0
\(709\) 4.00000 0.150223 0.0751116 − 0.997175i \(-0.476069\pi\)
0.0751116 + 0.997175i \(0.476069\pi\)
\(710\) 32.0000 1.20094
\(711\) 0 0
\(712\) 10.0000 0.374766
\(713\) −12.0000 −0.449404
\(714\) 0 0
\(715\) −2.00000 −0.0747958
\(716\) 12.0000 0.448461
\(717\) 0 0
\(718\) 16.0000 0.597115
\(719\) 2.00000 0.0745874 0.0372937 − 0.999304i \(-0.488126\pi\)
0.0372937 + 0.999304i \(0.488126\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) −17.0000 −0.632674
\(723\) 0 0
\(724\) 26.0000 0.966282
\(725\) −2.00000 −0.0742781
\(726\) 0 0
\(727\) −44.0000 −1.63187 −0.815935 − 0.578144i \(-0.803777\pi\)
−0.815935 + 0.578144i \(0.803777\pi\)
\(728\) 4.00000 0.148250
\(729\) 0 0
\(730\) 0 0
\(731\) 32.0000 1.18356
\(732\) 0 0
\(733\) −14.0000 −0.517102 −0.258551 − 0.965998i \(-0.583245\pi\)
−0.258551 + 0.965998i \(0.583245\pi\)
\(734\) −20.0000 −0.738213
\(735\) 0 0
\(736\) −6.00000 −0.221163
\(737\) −12.0000 −0.442026
\(738\) 0 0
\(739\) −34.0000 −1.25071 −0.625355 − 0.780340i \(-0.715046\pi\)
−0.625355 + 0.780340i \(0.715046\pi\)
\(740\) −8.00000 −0.294086
\(741\) 0 0
\(742\) −32.0000 −1.17476
\(743\) −8.00000 −0.293492 −0.146746 − 0.989174i \(-0.546880\pi\)
−0.146746 + 0.989174i \(0.546880\pi\)
\(744\) 0 0
\(745\) 4.00000 0.146549
\(746\) 26.0000 0.951928
\(747\) 0 0
\(748\) −8.00000 −0.292509
\(749\) 72.0000 2.63082
\(750\) 0 0
\(751\) −8.00000 −0.291924 −0.145962 − 0.989290i \(-0.546628\pi\)
−0.145962 + 0.989290i \(0.546628\pi\)
\(752\) −4.00000 −0.145865
\(753\) 0 0
\(754\) 2.00000 0.0728357
\(755\) −40.0000 −1.45575
\(756\) 0 0
\(757\) −6.00000 −0.218074 −0.109037 − 0.994038i \(-0.534777\pi\)
−0.109037 + 0.994038i \(0.534777\pi\)
\(758\) 20.0000 0.726433
\(759\) 0 0
\(760\) −12.0000 −0.435286
\(761\) −10.0000 −0.362500 −0.181250 − 0.983437i \(-0.558014\pi\)
−0.181250 + 0.983437i \(0.558014\pi\)
\(762\) 0 0
\(763\) 8.00000 0.289619
\(764\) −10.0000 −0.361787
\(765\) 0 0
\(766\) −36.0000 −1.30073
\(767\) 12.0000 0.433295
\(768\) 0 0
\(769\) 52.0000 1.87517 0.937584 − 0.347759i \(-0.113057\pi\)
0.937584 + 0.347759i \(0.113057\pi\)
\(770\) −8.00000 −0.288300
\(771\) 0 0
\(772\) 20.0000 0.719816
\(773\) 2.00000 0.0719350 0.0359675 − 0.999353i \(-0.488549\pi\)
0.0359675 + 0.999353i \(0.488549\pi\)
\(774\) 0 0
\(775\) 2.00000 0.0718421
\(776\) −10.0000 −0.358979
\(777\) 0 0
\(778\) 16.0000 0.573628
\(779\) 12.0000 0.429945
\(780\) 0 0
\(781\) −16.0000 −0.572525
\(782\) −48.0000 −1.71648
\(783\) 0 0
\(784\) 9.00000 0.321429
\(785\) −36.0000 −1.28490
\(786\) 0 0
\(787\) 2.00000 0.0712923 0.0356462 − 0.999364i \(-0.488651\pi\)
0.0356462 + 0.999364i \(0.488651\pi\)
\(788\) 6.00000 0.213741
\(789\) 0 0
\(790\) −16.0000 −0.569254
\(791\) −24.0000 −0.853342
\(792\) 0 0
\(793\) 2.00000 0.0710221
\(794\) −24.0000 −0.851728
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) 8.00000 0.283375 0.141687 − 0.989911i \(-0.454747\pi\)
0.141687 + 0.989911i \(0.454747\pi\)
\(798\) 0 0
\(799\) −32.0000 −1.13208
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) 30.0000 1.05934
\(803\) 0 0
\(804\) 0 0
\(805\) −48.0000 −1.69178
\(806\) −2.00000 −0.0704470
\(807\) 0 0
\(808\) 14.0000 0.492518
\(809\) −4.00000 −0.140633 −0.0703163 − 0.997525i \(-0.522401\pi\)
−0.0703163 + 0.997525i \(0.522401\pi\)
\(810\) 0 0
\(811\) −30.0000 −1.05344 −0.526721 − 0.850038i \(-0.676579\pi\)
−0.526721 + 0.850038i \(0.676579\pi\)
\(812\) 8.00000 0.280745
\(813\) 0 0
\(814\) 4.00000 0.140200
\(815\) 16.0000 0.560456
\(816\) 0 0
\(817\) −24.0000 −0.839654
\(818\) 4.00000 0.139857
\(819\) 0 0
\(820\) 4.00000 0.139686
\(821\) −22.0000 −0.767805 −0.383903 − 0.923374i \(-0.625420\pi\)
−0.383903 + 0.923374i \(0.625420\pi\)
\(822\) 0 0
\(823\) −24.0000 −0.836587 −0.418294 − 0.908312i \(-0.637372\pi\)
−0.418294 + 0.908312i \(0.637372\pi\)
\(824\) −4.00000 −0.139347
\(825\) 0 0
\(826\) 48.0000 1.67013
\(827\) −24.0000 −0.834562 −0.417281 − 0.908778i \(-0.637017\pi\)
−0.417281 + 0.908778i \(0.637017\pi\)
\(828\) 0 0
\(829\) 6.00000 0.208389 0.104194 − 0.994557i \(-0.466774\pi\)
0.104194 + 0.994557i \(0.466774\pi\)
\(830\) −8.00000 −0.277684
\(831\) 0 0
\(832\) −1.00000 −0.0346688
\(833\) 72.0000 2.49465
\(834\) 0 0
\(835\) 0 0
\(836\) 6.00000 0.207514
\(837\) 0 0
\(838\) 36.0000 1.24360
\(839\) −12.0000 −0.414286 −0.207143 − 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 36.0000 1.24064
\(843\) 0 0
\(844\) 0 0
\(845\) −2.00000 −0.0688021
\(846\) 0 0
\(847\) 4.00000 0.137442
\(848\) 8.00000 0.274721
\(849\) 0 0
\(850\) 8.00000 0.274398
\(851\) 24.0000 0.822709
\(852\) 0 0
\(853\) −6.00000 −0.205436 −0.102718 − 0.994711i \(-0.532754\pi\)
−0.102718 + 0.994711i \(0.532754\pi\)
\(854\) 8.00000 0.273754
\(855\) 0 0
\(856\) −18.0000 −0.615227
\(857\) 28.0000 0.956462 0.478231 − 0.878234i \(-0.341278\pi\)
0.478231 + 0.878234i \(0.341278\pi\)
\(858\) 0 0
\(859\) 4.00000 0.136478 0.0682391 − 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) −8.00000 −0.272798
\(861\) 0 0
\(862\) 8.00000 0.272481
\(863\) 24.0000 0.816970 0.408485 − 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) 0 0
\(865\) −12.0000 −0.408012
\(866\) −30.0000 −1.01944
\(867\) 0 0
\(868\) −8.00000 −0.271538
\(869\) 8.00000 0.271381
\(870\) 0 0
\(871\) −12.0000 −0.406604
\(872\) −2.00000 −0.0677285
\(873\) 0 0
\(874\) 36.0000 1.21772
\(875\) 48.0000 1.62270
\(876\) 0 0
\(877\) −50.0000 −1.68838 −0.844190 − 0.536044i \(-0.819918\pi\)
−0.844190 + 0.536044i \(0.819918\pi\)
\(878\) 8.00000 0.269987
\(879\) 0 0
\(880\) 2.00000 0.0674200
\(881\) −22.0000 −0.741199 −0.370599 − 0.928793i \(-0.620848\pi\)
−0.370599 + 0.928793i \(0.620848\pi\)
\(882\) 0 0
\(883\) −4.00000 −0.134611 −0.0673054 − 0.997732i \(-0.521440\pi\)
−0.0673054 + 0.997732i \(0.521440\pi\)
\(884\) −8.00000 −0.269069
\(885\) 0 0
\(886\) 24.0000 0.806296
\(887\) 36.0000 1.20876 0.604381 − 0.796696i \(-0.293421\pi\)
0.604381 + 0.796696i \(0.293421\pi\)
\(888\) 0 0
\(889\) 32.0000 1.07325
\(890\) −20.0000 −0.670402
\(891\) 0 0
\(892\) 14.0000 0.468755
\(893\) 24.0000 0.803129
\(894\) 0 0
\(895\) −24.0000 −0.802232
\(896\) −4.00000 −0.133631
\(897\) 0 0
\(898\) −18.0000 −0.600668
\(899\) −4.00000 −0.133407
\(900\) 0 0
\(901\) 64.0000 2.13215
\(902\) −2.00000 −0.0665927
\(903\) 0 0
\(904\) 6.00000 0.199557
\(905\) −52.0000 −1.72854
\(906\) 0 0
\(907\) −36.0000 −1.19536 −0.597680 − 0.801735i \(-0.703911\pi\)
−0.597680 + 0.801735i \(0.703911\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) −8.00000 −0.265197
\(911\) −26.0000 −0.861418 −0.430709 − 0.902491i \(-0.641737\pi\)
−0.430709 + 0.902491i \(0.641737\pi\)
\(912\) 0 0
\(913\) 4.00000 0.132381
\(914\) −28.0000 −0.926158
\(915\) 0 0
\(916\) −20.0000 −0.660819
\(917\) 24.0000 0.792550
\(918\) 0 0
\(919\) 56.0000 1.84727 0.923635 − 0.383274i \(-0.125203\pi\)
0.923635 + 0.383274i \(0.125203\pi\)
\(920\) 12.0000 0.395628
\(921\) 0 0
\(922\) −18.0000 −0.592798
\(923\) −16.0000 −0.526646
\(924\) 0 0
\(925\) −4.00000 −0.131519
\(926\) 38.0000 1.24876
\(927\) 0 0
\(928\) −2.00000 −0.0656532
\(929\) −6.00000 −0.196854 −0.0984268 − 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) 0 0
\(931\) −54.0000 −1.76978
\(932\) 12.0000 0.393073
\(933\) 0 0
\(934\) −8.00000 −0.261768
\(935\) 16.0000 0.523256
\(936\) 0 0
\(937\) 34.0000 1.11073 0.555366 − 0.831606i \(-0.312578\pi\)
0.555366 + 0.831606i \(0.312578\pi\)
\(938\) −48.0000 −1.56726
\(939\) 0 0
\(940\) 8.00000 0.260931
\(941\) 18.0000 0.586783 0.293392 − 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) 0 0
\(943\) −12.0000 −0.390774
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) 4.00000 0.130051
\(947\) −4.00000 −0.129983 −0.0649913 − 0.997886i \(-0.520702\pi\)
−0.0649913 + 0.997886i \(0.520702\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −6.00000 −0.194666
\(951\) 0 0
\(952\) −32.0000 −1.03713
\(953\) 36.0000 1.16615 0.583077 − 0.812417i \(-0.301849\pi\)
0.583077 + 0.812417i \(0.301849\pi\)
\(954\) 0 0
\(955\) 20.0000 0.647185
\(956\) 8.00000 0.258738
\(957\) 0 0
\(958\) −8.00000 −0.258468
\(959\) 8.00000 0.258333
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 4.00000 0.128965
\(963\) 0 0
\(964\) 0 0
\(965\) −40.0000 −1.28765
\(966\) 0 0
\(967\) −4.00000 −0.128631 −0.0643157 − 0.997930i \(-0.520486\pi\)
−0.0643157 + 0.997930i \(0.520486\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 0 0
\(970\) 20.0000 0.642161
\(971\) 40.0000 1.28366 0.641831 − 0.766846i \(-0.278175\pi\)
0.641831 + 0.766846i \(0.278175\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 14.0000 0.448589
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) −2.00000 −0.0639857 −0.0319928 − 0.999488i \(-0.510185\pi\)
−0.0319928 + 0.999488i \(0.510185\pi\)
\(978\) 0 0
\(979\) 10.0000 0.319601
\(980\) −18.0000 −0.574989
\(981\) 0 0
\(982\) −30.0000 −0.957338
\(983\) −8.00000 −0.255160 −0.127580 − 0.991828i \(-0.540721\pi\)
−0.127580 + 0.991828i \(0.540721\pi\)
\(984\) 0 0
\(985\) −12.0000 −0.382352
\(986\) −16.0000 −0.509544
\(987\) 0 0
\(988\) 6.00000 0.190885
\(989\) 24.0000 0.763156
\(990\) 0 0
\(991\) 16.0000 0.508257 0.254128 − 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) 2.00000 0.0635001
\(993\) 0 0
\(994\) −64.0000 −2.02996
\(995\) 32.0000 1.01447
\(996\) 0 0
\(997\) 38.0000 1.20347 0.601736 − 0.798695i \(-0.294476\pi\)
0.601736 + 0.798695i \(0.294476\pi\)
\(998\) 20.0000 0.633089
\(999\) 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 2574.2.a.d.1.1 1
3.2 odd 2 858.2.a.l.1.1 ✓ 1
12.11 even 2 6864.2.a.k.1.1 1
33.32 even 2 9438.2.a.o.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
858.2.a.l.1.1 ✓ 1 3.2 odd 2
2574.2.a.d.1.1 1 1.1 even 1 trivial
6864.2.a.k.1.1 1 12.11 even 2
9438.2.a.o.1.1 1 33.32 even 2