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Properties

Label 5824.2.a.h.1.1
Level $5824$
Weight $2$
Character 5824.1
Self dual yes
Analytic conductor $46.505$
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] = [5824,2,Mod(1,5824)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5824, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("5824.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5824 = 2^{6} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5824.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: \(46.5048741372\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2912)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 5824.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -4.00000 q^{5} +1.00000 q^{7} -2.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -4.00000 q^{5} +1.00000 q^{7} -2.00000 q^{9} -5.00000 q^{11} -1.00000 q^{13} +4.00000 q^{15} +2.00000 q^{19} -1.00000 q^{21} +1.00000 q^{23} +11.0000 q^{25} +5.00000 q^{27} +9.00000 q^{31} +5.00000 q^{33} -4.00000 q^{35} +11.0000 q^{37} +1.00000 q^{39} -5.00000 q^{41} +8.00000 q^{43} +8.00000 q^{45} +9.00000 q^{47} +1.00000 q^{49} -8.00000 q^{53} +20.0000 q^{55} -2.00000 q^{57} -6.00000 q^{59} +5.00000 q^{61} -2.00000 q^{63} +4.00000 q^{65} -13.0000 q^{67} -1.00000 q^{69} -12.0000 q^{71} -13.0000 q^{73} -11.0000 q^{75} -5.00000 q^{77} +3.00000 q^{79} +1.00000 q^{81} -4.00000 q^{83} -2.00000 q^{89} -1.00000 q^{91} -9.00000 q^{93} -8.00000 q^{95} +1.00000 q^{97} +10.0000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350 −0.288675 − 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) 0 0
\(5\) −4.00000 −1.78885 −0.894427 − 0.447214i \(-0.852416\pi\)
−0.894427 + 0.447214i \(0.852416\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) −5.00000 −1.50756 −0.753778 − 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 4.00000 1.03280
\(16\) 0 0
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 − 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 1.00000 0.208514 0.104257 − 0.994550i \(-0.466753\pi\)
0.104257 + 0.994550i \(0.466753\pi\)
\(24\) 0 0
\(25\) 11.0000 2.20000
\(26\) 0 0
\(27\) 5.00000 0.962250
\(28\) 0 0
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 9.00000 1.61645 0.808224 − 0.588875i \(-0.200429\pi\)
0.808224 + 0.588875i \(0.200429\pi\)
\(32\) 0 0
\(33\) 5.00000 0.870388
\(34\) 0 0
\(35\) −4.00000 −0.676123
\(36\) 0 0
\(37\) 11.0000 1.80839 0.904194 − 0.427121i \(-0.140472\pi\)
0.904194 + 0.427121i \(0.140472\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) −5.00000 −0.780869 −0.390434 − 0.920631i \(-0.627675\pi\)
−0.390434 + 0.920631i \(0.627675\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 − 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) 8.00000 1.19257
\(46\) 0 0
\(47\) 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
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −8.00000 −1.09888 −0.549442 − 0.835532i \(-0.685160\pi\)
−0.549442 + 0.835532i \(0.685160\pi\)
\(54\) 0 0
\(55\) 20.0000 2.69680
\(56\) 0 0
\(57\) −2.00000 −0.264906
\(58\) 0 0
\(59\) −6.00000 −0.781133 −0.390567 − 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) 5.00000 0.640184 0.320092 − 0.947386i \(-0.396286\pi\)
0.320092 + 0.947386i \(0.396286\pi\)
\(62\) 0 0
\(63\) −2.00000 −0.251976
\(64\) 0 0
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) −13.0000 −1.58820 −0.794101 − 0.607785i \(-0.792058\pi\)
−0.794101 + 0.607785i \(0.792058\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 − 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) −13.0000 −1.52153 −0.760767 − 0.649025i \(-0.775177\pi\)
−0.760767 + 0.649025i \(0.775177\pi\)
\(74\) 0 0
\(75\) −11.0000 −1.27017
\(76\) 0 0
\(77\) −5.00000 −0.569803
\(78\) 0 0
\(79\) 3.00000 0.337526 0.168763 − 0.985657i \(-0.446023\pi\)
0.168763 + 0.985657i \(0.446023\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −4.00000 −0.439057 −0.219529 − 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −2.00000 −0.212000 −0.106000 − 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 0 0
\(93\) −9.00000 −0.933257
\(94\) 0 0
\(95\) −8.00000 −0.820783
\(96\) 0 0
\(97\) 1.00000 0.101535 0.0507673 − 0.998711i \(-0.483833\pi\)
0.0507673 + 0.998711i \(0.483833\pi\)
\(98\) 0 0
\(99\) 10.0000 1.00504
\(100\) 0 0
\(101\) 7.00000 0.696526 0.348263 − 0.937397i \(-0.386772\pi\)
0.348263 + 0.937397i \(0.386772\pi\)
\(102\) 0 0
\(103\) −14.0000 −1.37946 −0.689730 − 0.724066i \(-0.742271\pi\)
−0.689730 + 0.724066i \(0.742271\pi\)
\(104\) 0 0
\(105\) 4.00000 0.390360
\(106\) 0 0
\(107\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 14.0000 1.34096 0.670478 − 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 0 0
\(111\) −11.0000 −1.04407
\(112\) 0 0
\(113\) 9.00000 0.846649 0.423324 − 0.905978i \(-0.360863\pi\)
0.423324 + 0.905978i \(0.360863\pi\)
\(114\) 0 0
\(115\) −4.00000 −0.373002
\(116\) 0 0
\(117\) 2.00000 0.184900
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 0 0
\(123\) 5.00000 0.450835
\(124\) 0 0
\(125\) −24.0000 −2.14663
\(126\) 0 0
\(127\) 5.00000 0.443678 0.221839 − 0.975083i \(-0.428794\pi\)
0.221839 + 0.975083i \(0.428794\pi\)
\(128\) 0 0
\(129\) −8.00000 −0.704361
\(130\) 0 0
\(131\) −16.0000 −1.39793 −0.698963 − 0.715158i \(-0.746355\pi\)
−0.698963 + 0.715158i \(0.746355\pi\)
\(132\) 0 0
\(133\) 2.00000 0.173422
\(134\) 0 0
\(135\) −20.0000 −1.72133
\(136\) 0 0
\(137\) −2.00000 −0.170872 −0.0854358 − 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) 0 0
\(139\) 12.0000 1.01783 0.508913 − 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 0 0
\(141\) −9.00000 −0.757937
\(142\) 0 0
\(143\) 5.00000 0.418121
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) 19.0000 1.55654 0.778270 − 0.627929i \(-0.216097\pi\)
0.778270 + 0.627929i \(0.216097\pi\)
\(150\) 0 0
\(151\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −36.0000 −2.89159
\(156\) 0 0
\(157\) −7.00000 −0.558661 −0.279330 − 0.960195i \(-0.590112\pi\)
−0.279330 + 0.960195i \(0.590112\pi\)
\(158\) 0 0
\(159\) 8.00000 0.634441
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 − 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 0 0
\(165\) −20.0000 −1.55700
\(166\) 0 0
\(167\) −16.0000 −1.23812 −0.619059 − 0.785345i \(-0.712486\pi\)
−0.619059 + 0.785345i \(0.712486\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) 0 0
\(173\) 2.00000 0.152057 0.0760286 − 0.997106i \(-0.475776\pi\)
0.0760286 + 0.997106i \(0.475776\pi\)
\(174\) 0 0
\(175\) 11.0000 0.831522
\(176\) 0 0
\(177\) 6.00000 0.450988
\(178\) 0 0
\(179\) −10.0000 −0.747435 −0.373718 − 0.927543i \(-0.621917\pi\)
−0.373718 + 0.927543i \(0.621917\pi\)
\(180\) 0 0
\(181\) −5.00000 −0.371647 −0.185824 − 0.982583i \(-0.559495\pi\)
−0.185824 + 0.982583i \(0.559495\pi\)
\(182\) 0 0
\(183\) −5.00000 −0.369611
\(184\) 0 0
\(185\) −44.0000 −3.23494
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 5.00000 0.363696
\(190\) 0 0
\(191\) 24.0000 1.73658 0.868290 − 0.496058i \(-0.165220\pi\)
0.868290 + 0.496058i \(0.165220\pi\)
\(192\) 0 0
\(193\) −8.00000 −0.575853 −0.287926 − 0.957653i \(-0.592966\pi\)
−0.287926 + 0.957653i \(0.592966\pi\)
\(194\) 0 0
\(195\) −4.00000 −0.286446
\(196\) 0 0
\(197\) 15.0000 1.06871 0.534353 − 0.845262i \(-0.320555\pi\)
0.534353 + 0.845262i \(0.320555\pi\)
\(198\) 0 0
\(199\) −28.0000 −1.98487 −0.992434 − 0.122782i \(-0.960818\pi\)
−0.992434 + 0.122782i \(0.960818\pi\)
\(200\) 0 0
\(201\) 13.0000 0.916949
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 20.0000 1.39686
\(206\) 0 0
\(207\) −2.00000 −0.139010
\(208\) 0 0
\(209\) −10.0000 −0.691714
\(210\) 0 0
\(211\) 10.0000 0.688428 0.344214 − 0.938891i \(-0.388145\pi\)
0.344214 + 0.938891i \(0.388145\pi\)
\(212\) 0 0
\(213\) 12.0000 0.822226
\(214\) 0 0
\(215\) −32.0000 −2.18238
\(216\) 0 0
\(217\) 9.00000 0.610960
\(218\) 0 0
\(219\) 13.0000 0.878459
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 3.00000 0.200895 0.100447 − 0.994942i \(-0.467973\pi\)
0.100447 + 0.994942i \(0.467973\pi\)
\(224\) 0 0
\(225\) −22.0000 −1.46667
\(226\) 0 0
\(227\) −4.00000 −0.265489 −0.132745 − 0.991150i \(-0.542379\pi\)
−0.132745 + 0.991150i \(0.542379\pi\)
\(228\) 0 0
\(229\) 20.0000 1.32164 0.660819 − 0.750546i \(-0.270209\pi\)
0.660819 + 0.750546i \(0.270209\pi\)
\(230\) 0 0
\(231\) 5.00000 0.328976
\(232\) 0 0
\(233\) 27.0000 1.76883 0.884414 − 0.466702i \(-0.154558\pi\)
0.884414 + 0.466702i \(0.154558\pi\)
\(234\) 0 0
\(235\) −36.0000 −2.34838
\(236\) 0 0
\(237\) −3.00000 −0.194871
\(238\) 0 0
\(239\) 22.0000 1.42306 0.711531 − 0.702655i \(-0.248002\pi\)
0.711531 + 0.702655i \(0.248002\pi\)
\(240\) 0 0
\(241\) −6.00000 −0.386494 −0.193247 − 0.981150i \(-0.561902\pi\)
−0.193247 + 0.981150i \(0.561902\pi\)
\(242\) 0 0
\(243\) −16.0000 −1.02640
\(244\) 0 0
\(245\) −4.00000 −0.255551
\(246\) 0 0
\(247\) −2.00000 −0.127257
\(248\) 0 0
\(249\) 4.00000 0.253490
\(250\) 0 0
\(251\) −9.00000 −0.568075 −0.284037 − 0.958813i \(-0.591674\pi\)
−0.284037 + 0.958813i \(0.591674\pi\)
\(252\) 0 0
\(253\) −5.00000 −0.314347
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 20.0000 1.24757 0.623783 − 0.781598i \(-0.285595\pi\)
0.623783 + 0.781598i \(0.285595\pi\)
\(258\) 0 0
\(259\) 11.0000 0.683507
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 32.0000 1.96574
\(266\) 0 0
\(267\) 2.00000 0.122398
\(268\) 0 0
\(269\) 21.0000 1.28039 0.640196 − 0.768211i \(-0.278853\pi\)
0.640196 + 0.768211i \(0.278853\pi\)
\(270\) 0 0
\(271\) −25.0000 −1.51864 −0.759321 − 0.650716i \(-0.774469\pi\)
−0.759321 + 0.650716i \(0.774469\pi\)
\(272\) 0 0
\(273\) 1.00000 0.0605228
\(274\) 0 0
\(275\) −55.0000 −3.31662
\(276\) 0 0
\(277\) −26.0000 −1.56219 −0.781094 − 0.624413i \(-0.785338\pi\)
−0.781094 + 0.624413i \(0.785338\pi\)
\(278\) 0 0
\(279\) −18.0000 −1.07763
\(280\) 0 0
\(281\) 24.0000 1.43172 0.715860 − 0.698244i \(-0.246035\pi\)
0.715860 + 0.698244i \(0.246035\pi\)
\(282\) 0 0
\(283\) −25.0000 −1.48610 −0.743048 − 0.669238i \(-0.766621\pi\)
−0.743048 + 0.669238i \(0.766621\pi\)
\(284\) 0 0
\(285\) 8.00000 0.473879
\(286\) 0 0
\(287\) −5.00000 −0.295141
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) −1.00000 −0.0586210
\(292\) 0 0
\(293\) −6.00000 −0.350524 −0.175262 − 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 0 0
\(295\) 24.0000 1.39733
\(296\) 0 0
\(297\) −25.0000 −1.45065
\(298\) 0 0
\(299\) −1.00000 −0.0578315
\(300\) 0 0
\(301\) 8.00000 0.461112
\(302\) 0 0
\(303\) −7.00000 −0.402139
\(304\) 0 0
\(305\) −20.0000 −1.14520
\(306\) 0 0
\(307\) −12.0000 −0.684876 −0.342438 − 0.939540i \(-0.611253\pi\)
−0.342438 + 0.939540i \(0.611253\pi\)
\(308\) 0 0
\(309\) 14.0000 0.796432
\(310\) 0 0
\(311\) 22.0000 1.24751 0.623753 − 0.781622i \(-0.285607\pi\)
0.623753 + 0.781622i \(0.285607\pi\)
\(312\) 0 0
\(313\) −22.0000 −1.24351 −0.621757 − 0.783210i \(-0.713581\pi\)
−0.621757 + 0.783210i \(0.713581\pi\)
\(314\) 0 0
\(315\) 8.00000 0.450749
\(316\) 0 0
\(317\) 7.00000 0.393159 0.196580 − 0.980488i \(-0.437017\pi\)
0.196580 + 0.980488i \(0.437017\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −11.0000 −0.610170
\(326\) 0 0
\(327\) −14.0000 −0.774202
\(328\) 0 0
\(329\) 9.00000 0.496186
\(330\) 0 0
\(331\) −29.0000 −1.59398 −0.796992 − 0.603990i \(-0.793577\pi\)
−0.796992 + 0.603990i \(0.793577\pi\)
\(332\) 0 0
\(333\) −22.0000 −1.20559
\(334\) 0 0
\(335\) 52.0000 2.84106
\(336\) 0 0
\(337\) −7.00000 −0.381314 −0.190657 − 0.981657i \(-0.561062\pi\)
−0.190657 + 0.981657i \(0.561062\pi\)
\(338\) 0 0
\(339\) −9.00000 −0.488813
\(340\) 0 0
\(341\) −45.0000 −2.43689
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 4.00000 0.215353
\(346\) 0 0
\(347\) 12.0000 0.644194 0.322097 − 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 0 0
\(349\) −14.0000 −0.749403 −0.374701 − 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) −5.00000 −0.266880
\(352\) 0 0
\(353\) −17.0000 −0.904819 −0.452409 − 0.891810i \(-0.649435\pi\)
−0.452409 + 0.891810i \(0.649435\pi\)
\(354\) 0 0
\(355\) 48.0000 2.54758
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −36.0000 −1.90001 −0.950004 − 0.312239i \(-0.898921\pi\)
−0.950004 + 0.312239i \(0.898921\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) −14.0000 −0.734809
\(364\) 0 0
\(365\) 52.0000 2.72180
\(366\) 0 0
\(367\) −20.0000 −1.04399 −0.521996 − 0.852948i \(-0.674812\pi\)
−0.521996 + 0.852948i \(0.674812\pi\)
\(368\) 0 0
\(369\) 10.0000 0.520579
\(370\) 0 0
\(371\) −8.00000 −0.415339
\(372\) 0 0
\(373\) 24.0000 1.24267 0.621336 − 0.783544i \(-0.286590\pi\)
0.621336 + 0.783544i \(0.286590\pi\)
\(374\) 0 0
\(375\) 24.0000 1.23935
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) −5.00000 −0.256158
\(382\) 0 0
\(383\) −9.00000 −0.459879 −0.229939 − 0.973205i \(-0.573853\pi\)
−0.229939 + 0.973205i \(0.573853\pi\)
\(384\) 0 0
\(385\) 20.0000 1.01929
\(386\) 0 0
\(387\) −16.0000 −0.813326
\(388\) 0 0
\(389\) −32.0000 −1.62246 −0.811232 − 0.584724i \(-0.801203\pi\)
−0.811232 + 0.584724i \(0.801203\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 16.0000 0.807093
\(394\) 0 0
\(395\) −12.0000 −0.603786
\(396\) 0 0
\(397\) −4.00000 −0.200754 −0.100377 − 0.994949i \(-0.532005\pi\)
−0.100377 + 0.994949i \(0.532005\pi\)
\(398\) 0 0
\(399\) −2.00000 −0.100125
\(400\) 0 0
\(401\) 4.00000 0.199750 0.0998752 − 0.995000i \(-0.468156\pi\)
0.0998752 + 0.995000i \(0.468156\pi\)
\(402\) 0 0
\(403\) −9.00000 −0.448322
\(404\) 0 0
\(405\) −4.00000 −0.198762
\(406\) 0 0
\(407\) −55.0000 −2.72625
\(408\) 0 0
\(409\) 34.0000 1.68119 0.840596 − 0.541663i \(-0.182205\pi\)
0.840596 + 0.541663i \(0.182205\pi\)
\(410\) 0 0
\(411\) 2.00000 0.0986527
\(412\) 0 0
\(413\) −6.00000 −0.295241
\(414\) 0 0
\(415\) 16.0000 0.785409
\(416\) 0 0
\(417\) −12.0000 −0.587643
\(418\) 0 0
\(419\) 19.0000 0.928211 0.464105 − 0.885780i \(-0.346376\pi\)
0.464105 + 0.885780i \(0.346376\pi\)
\(420\) 0 0
\(421\) −15.0000 −0.731055 −0.365528 − 0.930800i \(-0.619111\pi\)
−0.365528 + 0.930800i \(0.619111\pi\)
\(422\) 0 0
\(423\) −18.0000 −0.875190
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 5.00000 0.241967
\(428\) 0 0
\(429\) −5.00000 −0.241402
\(430\) 0 0
\(431\) −10.0000 −0.481683 −0.240842 − 0.970564i \(-0.577423\pi\)
−0.240842 + 0.970564i \(0.577423\pi\)
\(432\) 0 0
\(433\) −6.00000 −0.288342 −0.144171 − 0.989553i \(-0.546051\pi\)
−0.144171 + 0.989553i \(0.546051\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 2.00000 0.0956730
\(438\) 0 0
\(439\) 34.0000 1.62273 0.811366 − 0.584539i \(-0.198725\pi\)
0.811366 + 0.584539i \(0.198725\pi\)
\(440\) 0 0
\(441\) −2.00000 −0.0952381
\(442\) 0 0
\(443\) 28.0000 1.33032 0.665160 − 0.746701i \(-0.268363\pi\)
0.665160 + 0.746701i \(0.268363\pi\)
\(444\) 0 0
\(445\) 8.00000 0.379236
\(446\) 0 0
\(447\) −19.0000 −0.898669
\(448\) 0 0
\(449\) −14.0000 −0.660701 −0.330350 − 0.943858i \(-0.607167\pi\)
−0.330350 + 0.943858i \(0.607167\pi\)
\(450\) 0 0
\(451\) 25.0000 1.17720
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 4.00000 0.187523
\(456\) 0 0
\(457\) 10.0000 0.467780 0.233890 − 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 4.00000 0.185896 0.0929479 − 0.995671i \(-0.470371\pi\)
0.0929479 + 0.995671i \(0.470371\pi\)
\(464\) 0 0
\(465\) 36.0000 1.66946
\(466\) 0 0
\(467\) 28.0000 1.29569 0.647843 − 0.761774i \(-0.275671\pi\)
0.647843 + 0.761774i \(0.275671\pi\)
\(468\) 0 0
\(469\) −13.0000 −0.600284
\(470\) 0 0
\(471\) 7.00000 0.322543
\(472\) 0 0
\(473\) −40.0000 −1.83920
\(474\) 0 0
\(475\) 22.0000 1.00943
\(476\) 0 0
\(477\) 16.0000 0.732590
\(478\) 0 0
\(479\) −24.0000 −1.09659 −0.548294 − 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) −11.0000 −0.501557
\(482\) 0 0
\(483\) −1.00000 −0.0455016
\(484\) 0 0
\(485\) −4.00000 −0.181631
\(486\) 0 0
\(487\) −14.0000 −0.634401 −0.317200 − 0.948359i \(-0.602743\pi\)
−0.317200 + 0.948359i \(0.602743\pi\)
\(488\) 0 0
\(489\) 4.00000 0.180886
\(490\) 0 0
\(491\) 24.0000 1.08310 0.541552 − 0.840667i \(-0.317837\pi\)
0.541552 + 0.840667i \(0.317837\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −40.0000 −1.79787
\(496\) 0 0
\(497\) −12.0000 −0.538274
\(498\) 0 0
\(499\) −7.00000 −0.313363 −0.156682 − 0.987649i \(-0.550080\pi\)
−0.156682 + 0.987649i \(0.550080\pi\)
\(500\) 0 0
\(501\) 16.0000 0.714827
\(502\) 0 0
\(503\) −8.00000 −0.356702 −0.178351 − 0.983967i \(-0.557076\pi\)
−0.178351 + 0.983967i \(0.557076\pi\)
\(504\) 0 0
\(505\) −28.0000 −1.24598
\(506\) 0 0
\(507\) −1.00000 −0.0444116
\(508\) 0 0
\(509\) −34.0000 −1.50702 −0.753512 − 0.657434i \(-0.771642\pi\)
−0.753512 + 0.657434i \(0.771642\pi\)
\(510\) 0 0
\(511\) −13.0000 −0.575086
\(512\) 0 0
\(513\) 10.0000 0.441511
\(514\) 0 0
\(515\) 56.0000 2.46765
\(516\) 0 0
\(517\) −45.0000 −1.97910
\(518\) 0 0
\(519\) −2.00000 −0.0877903
\(520\) 0 0
\(521\) −6.00000 −0.262865 −0.131432 − 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 0 0
\(523\) −1.00000 −0.0437269 −0.0218635 − 0.999761i \(-0.506960\pi\)
−0.0218635 + 0.999761i \(0.506960\pi\)
\(524\) 0 0
\(525\) −11.0000 −0.480079
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −22.0000 −0.956522
\(530\) 0 0
\(531\) 12.0000 0.520756
\(532\) 0 0
\(533\) 5.00000 0.216574
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 10.0000 0.431532
\(538\) 0 0
\(539\) −5.00000 −0.215365
\(540\) 0 0
\(541\) 10.0000 0.429934 0.214967 − 0.976621i \(-0.431036\pi\)
0.214967 + 0.976621i \(0.431036\pi\)
\(542\) 0 0
\(543\) 5.00000 0.214571
\(544\) 0 0
\(545\) −56.0000 −2.39878
\(546\) 0 0
\(547\) 4.00000 0.171028 0.0855138 − 0.996337i \(-0.472747\pi\)
0.0855138 + 0.996337i \(0.472747\pi\)
\(548\) 0 0
\(549\) −10.0000 −0.426790
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 3.00000 0.127573
\(554\) 0 0
\(555\) 44.0000 1.86770
\(556\) 0 0
\(557\) 43.0000 1.82197 0.910984 − 0.412441i \(-0.135324\pi\)
0.910984 + 0.412441i \(0.135324\pi\)
\(558\) 0 0
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −25.0000 −1.05362 −0.526812 − 0.849982i \(-0.676613\pi\)
−0.526812 + 0.849982i \(0.676613\pi\)
\(564\) 0 0
\(565\) −36.0000 −1.51453
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) 21.0000 0.880366 0.440183 − 0.897908i \(-0.354914\pi\)
0.440183 + 0.897908i \(0.354914\pi\)
\(570\) 0 0
\(571\) 38.0000 1.59025 0.795125 − 0.606445i \(-0.207405\pi\)
0.795125 + 0.606445i \(0.207405\pi\)
\(572\) 0 0
\(573\) −24.0000 −1.00261
\(574\) 0 0
\(575\) 11.0000 0.458732
\(576\) 0 0
\(577\) 2.00000 0.0832611 0.0416305 − 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 0 0
\(579\) 8.00000 0.332469
\(580\) 0 0
\(581\) −4.00000 −0.165948
\(582\) 0 0
\(583\) 40.0000 1.65663
\(584\) 0 0
\(585\) −8.00000 −0.330759
\(586\) 0 0
\(587\) −14.0000 −0.577842 −0.288921 − 0.957353i \(-0.593296\pi\)
−0.288921 + 0.957353i \(0.593296\pi\)
\(588\) 0 0
\(589\) 18.0000 0.741677
\(590\) 0 0
\(591\) −15.0000 −0.617018
\(592\) 0 0
\(593\) −30.0000 −1.23195 −0.615976 − 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 28.0000 1.14596
\(598\) 0 0
\(599\) −45.0000 −1.83865 −0.919325 − 0.393499i \(-0.871265\pi\)
−0.919325 + 0.393499i \(0.871265\pi\)
\(600\) 0 0
\(601\) 26.0000 1.06056 0.530281 − 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) 0 0
\(603\) 26.0000 1.05880
\(604\) 0 0
\(605\) −56.0000 −2.27672
\(606\) 0 0
\(607\) 34.0000 1.38002 0.690009 − 0.723801i \(-0.257607\pi\)
0.690009 + 0.723801i \(0.257607\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −9.00000 −0.364101
\(612\) 0 0
\(613\) −11.0000 −0.444286 −0.222143 − 0.975014i \(-0.571305\pi\)
−0.222143 + 0.975014i \(0.571305\pi\)
\(614\) 0 0
\(615\) −20.0000 −0.806478
\(616\) 0 0
\(617\) −6.00000 −0.241551 −0.120775 − 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 0 0
\(619\) 34.0000 1.36658 0.683288 − 0.730149i \(-0.260549\pi\)
0.683288 + 0.730149i \(0.260549\pi\)
\(620\) 0 0
\(621\) 5.00000 0.200643
\(622\) 0 0
\(623\) −2.00000 −0.0801283
\(624\) 0 0
\(625\) 41.0000 1.64000
\(626\) 0 0
\(627\) 10.0000 0.399362
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −10.0000 −0.398094 −0.199047 − 0.979990i \(-0.563785\pi\)
−0.199047 + 0.979990i \(0.563785\pi\)
\(632\) 0 0
\(633\) −10.0000 −0.397464
\(634\) 0 0
\(635\) −20.0000 −0.793676
\(636\) 0 0
\(637\) −1.00000 −0.0396214
\(638\) 0 0
\(639\) 24.0000 0.949425
\(640\) 0 0
\(641\) −1.00000 −0.0394976 −0.0197488 − 0.999805i \(-0.506287\pi\)
−0.0197488 + 0.999805i \(0.506287\pi\)
\(642\) 0 0
\(643\) 8.00000 0.315489 0.157745 − 0.987480i \(-0.449578\pi\)
0.157745 + 0.987480i \(0.449578\pi\)
\(644\) 0 0
\(645\) 32.0000 1.26000
\(646\) 0 0
\(647\) −12.0000 −0.471769 −0.235884 − 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) 0 0
\(649\) 30.0000 1.17760
\(650\) 0 0
\(651\) −9.00000 −0.352738
\(652\) 0 0
\(653\) −36.0000 −1.40879 −0.704394 − 0.709809i \(-0.748781\pi\)
−0.704394 + 0.709809i \(0.748781\pi\)
\(654\) 0 0
\(655\) 64.0000 2.50069
\(656\) 0 0
\(657\) 26.0000 1.01436
\(658\) 0 0
\(659\) −6.00000 −0.233727 −0.116863 − 0.993148i \(-0.537284\pi\)
−0.116863 + 0.993148i \(0.537284\pi\)
\(660\) 0 0
\(661\) −12.0000 −0.466746 −0.233373 − 0.972387i \(-0.574976\pi\)
−0.233373 + 0.972387i \(0.574976\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −8.00000 −0.310227
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −3.00000 −0.115987
\(670\) 0 0
\(671\) −25.0000 −0.965114
\(672\) 0 0
\(673\) −41.0000 −1.58043 −0.790217 − 0.612827i \(-0.790032\pi\)
−0.790217 + 0.612827i \(0.790032\pi\)
\(674\) 0 0
\(675\) 55.0000 2.11695
\(676\) 0 0
\(677\) 43.0000 1.65262 0.826312 − 0.563212i \(-0.190435\pi\)
0.826312 + 0.563212i \(0.190435\pi\)
\(678\) 0 0
\(679\) 1.00000 0.0383765
\(680\) 0 0
\(681\) 4.00000 0.153280
\(682\) 0 0
\(683\) 11.0000 0.420903 0.210452 − 0.977604i \(-0.432507\pi\)
0.210452 + 0.977604i \(0.432507\pi\)
\(684\) 0 0
\(685\) 8.00000 0.305664
\(686\) 0 0
\(687\) −20.0000 −0.763048
\(688\) 0 0
\(689\) 8.00000 0.304776
\(690\) 0 0
\(691\) −32.0000 −1.21734 −0.608669 − 0.793424i \(-0.708296\pi\)
−0.608669 + 0.793424i \(0.708296\pi\)
\(692\) 0 0
\(693\) 10.0000 0.379869
\(694\) 0 0
\(695\) −48.0000 −1.82074
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −27.0000 −1.02123
\(700\) 0 0
\(701\) 36.0000 1.35970 0.679851 − 0.733351i \(-0.262045\pi\)
0.679851 + 0.733351i \(0.262045\pi\)
\(702\) 0 0
\(703\) 22.0000 0.829746
\(704\) 0 0
\(705\) 36.0000 1.35584
\(706\) 0 0
\(707\) 7.00000 0.263262
\(708\) 0 0
\(709\) 1.00000 0.0375558 0.0187779 − 0.999824i \(-0.494022\pi\)
0.0187779 + 0.999824i \(0.494022\pi\)
\(710\) 0 0
\(711\) −6.00000 −0.225018
\(712\) 0 0
\(713\) 9.00000 0.337053
\(714\) 0 0
\(715\) −20.0000 −0.747958
\(716\) 0 0
\(717\) −22.0000 −0.821605
\(718\) 0 0
\(719\) −50.0000 −1.86469 −0.932343 − 0.361576i \(-0.882239\pi\)
−0.932343 + 0.361576i \(0.882239\pi\)
\(720\) 0 0
\(721\) −14.0000 −0.521387
\(722\) 0 0
\(723\) 6.00000 0.223142
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 6.00000 0.222528 0.111264 − 0.993791i \(-0.464510\pi\)
0.111264 + 0.993791i \(0.464510\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −40.0000 −1.47743 −0.738717 − 0.674016i \(-0.764568\pi\)
−0.738717 + 0.674016i \(0.764568\pi\)
\(734\) 0 0
\(735\) 4.00000 0.147542
\(736\) 0 0
\(737\) 65.0000 2.39431
\(738\) 0 0
\(739\) 32.0000 1.17714 0.588570 − 0.808447i \(-0.299691\pi\)
0.588570 + 0.808447i \(0.299691\pi\)
\(740\) 0 0
\(741\) 2.00000 0.0734718
\(742\) 0 0
\(743\) 4.00000 0.146746 0.0733729 − 0.997305i \(-0.476624\pi\)
0.0733729 + 0.997305i \(0.476624\pi\)
\(744\) 0 0
\(745\) −76.0000 −2.78442
\(746\) 0 0
\(747\) 8.00000 0.292705
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −19.0000 −0.693320 −0.346660 − 0.937991i \(-0.612684\pi\)
−0.346660 + 0.937991i \(0.612684\pi\)
\(752\) 0 0
\(753\) 9.00000 0.327978
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 30.0000 1.09037 0.545184 − 0.838316i \(-0.316460\pi\)
0.545184 + 0.838316i \(0.316460\pi\)
\(758\) 0 0
\(759\) 5.00000 0.181489
\(760\) 0 0
\(761\) 19.0000 0.688749 0.344375 − 0.938832i \(-0.388091\pi\)
0.344375 + 0.938832i \(0.388091\pi\)
\(762\) 0 0
\(763\) 14.0000 0.506834
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 6.00000 0.216647
\(768\) 0 0
\(769\) −13.0000 −0.468792 −0.234396 − 0.972141i \(-0.575311\pi\)
−0.234396 + 0.972141i \(0.575311\pi\)
\(770\) 0 0
\(771\) −20.0000 −0.720282
\(772\) 0 0
\(773\) 42.0000 1.51064 0.755318 − 0.655359i \(-0.227483\pi\)
0.755318 + 0.655359i \(0.227483\pi\)
\(774\) 0 0
\(775\) 99.0000 3.55618
\(776\) 0 0
\(777\) −11.0000 −0.394623
\(778\) 0 0
\(779\) −10.0000 −0.358287
\(780\) 0 0
\(781\) 60.0000 2.14697
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 28.0000 0.999363
\(786\) 0 0
\(787\) 8.00000 0.285169 0.142585 − 0.989783i \(-0.454459\pi\)
0.142585 + 0.989783i \(0.454459\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 9.00000 0.320003
\(792\) 0 0
\(793\) −5.00000 −0.177555
\(794\) 0 0
\(795\) −32.0000 −1.13492
\(796\) 0 0
\(797\) 43.0000 1.52314 0.761569 − 0.648084i \(-0.224429\pi\)
0.761569 + 0.648084i \(0.224429\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 4.00000 0.141333
\(802\) 0 0
\(803\) 65.0000 2.29380
\(804\) 0 0
\(805\) −4.00000 −0.140981
\(806\) 0 0
\(807\) −21.0000 −0.739235
\(808\) 0 0
\(809\) 6.00000 0.210949 0.105474 − 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) −40.0000 −1.40459 −0.702295 − 0.711886i \(-0.747841\pi\)
−0.702295 + 0.711886i \(0.747841\pi\)
\(812\) 0 0
\(813\) 25.0000 0.876788
\(814\) 0 0
\(815\) 16.0000 0.560456
\(816\) 0 0
\(817\) 16.0000 0.559769
\(818\) 0 0
\(819\) 2.00000 0.0698857
\(820\) 0 0
\(821\) −26.0000 −0.907406 −0.453703 − 0.891153i \(-0.649897\pi\)
−0.453703 + 0.891153i \(0.649897\pi\)
\(822\) 0 0
\(823\) 45.0000 1.56860 0.784301 − 0.620381i \(-0.213022\pi\)
0.784301 + 0.620381i \(0.213022\pi\)
\(824\) 0 0
\(825\) 55.0000 1.91485
\(826\) 0 0
\(827\) −12.0000 −0.417281 −0.208640 − 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 0 0
\(829\) 26.0000 0.903017 0.451509 − 0.892267i \(-0.350886\pi\)
0.451509 + 0.892267i \(0.350886\pi\)
\(830\) 0 0
\(831\) 26.0000 0.901930
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 64.0000 2.21481
\(836\) 0 0
\(837\) 45.0000 1.55543
\(838\) 0 0
\(839\) −33.0000 −1.13929 −0.569643 − 0.821892i \(-0.692919\pi\)
−0.569643 + 0.821892i \(0.692919\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 0 0
\(843\) −24.0000 −0.826604
\(844\) 0 0
\(845\) −4.00000 −0.137604
\(846\) 0 0
\(847\) 14.0000 0.481046
\(848\) 0 0
\(849\) 25.0000 0.857998
\(850\) 0 0
\(851\) 11.0000 0.377075
\(852\) 0 0
\(853\) −28.0000 −0.958702 −0.479351 − 0.877623i \(-0.659128\pi\)
−0.479351 + 0.877623i \(0.659128\pi\)
\(854\) 0 0
\(855\) 16.0000 0.547188
\(856\) 0 0
\(857\) −6.00000 −0.204956 −0.102478 − 0.994735i \(-0.532677\pi\)
−0.102478 + 0.994735i \(0.532677\pi\)
\(858\) 0 0
\(859\) 1.00000 0.0341196 0.0170598 − 0.999854i \(-0.494569\pi\)
0.0170598 + 0.999854i \(0.494569\pi\)
\(860\) 0 0
\(861\) 5.00000 0.170400
\(862\) 0 0
\(863\) −16.0000 −0.544646 −0.272323 − 0.962206i \(-0.587792\pi\)
−0.272323 + 0.962206i \(0.587792\pi\)
\(864\) 0 0
\(865\) −8.00000 −0.272008
\(866\) 0 0
\(867\) 17.0000 0.577350
\(868\) 0 0
\(869\) −15.0000 −0.508840
\(870\) 0 0
\(871\) 13.0000 0.440488
\(872\) 0 0
\(873\) −2.00000 −0.0676897
\(874\) 0 0
\(875\) −24.0000 −0.811348
\(876\) 0 0
\(877\) −15.0000 −0.506514 −0.253257 − 0.967399i \(-0.581502\pi\)
−0.253257 + 0.967399i \(0.581502\pi\)
\(878\) 0 0
\(879\) 6.00000 0.202375
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 − 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) 0 0
\(883\) 38.0000 1.27880 0.639401 − 0.768874i \(-0.279182\pi\)
0.639401 + 0.768874i \(0.279182\pi\)
\(884\) 0 0
\(885\) −24.0000 −0.806751
\(886\) 0 0
\(887\) 10.0000 0.335767 0.167884 − 0.985807i \(-0.446307\pi\)
0.167884 + 0.985807i \(0.446307\pi\)
\(888\) 0 0
\(889\) 5.00000 0.167695
\(890\) 0 0
\(891\) −5.00000 −0.167506
\(892\) 0 0
\(893\) 18.0000 0.602347
\(894\) 0 0
\(895\) 40.0000 1.33705
\(896\) 0 0
\(897\) 1.00000 0.0333890
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) −8.00000 −0.266223
\(904\) 0 0
\(905\) 20.0000 0.664822
\(906\) 0 0
\(907\) −58.0000 −1.92586 −0.962929 − 0.269754i \(-0.913058\pi\)
−0.962929 + 0.269754i \(0.913058\pi\)
\(908\) 0 0
\(909\) −14.0000 −0.464351
\(910\) 0 0
\(911\) −12.0000 −0.397578 −0.198789 − 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) 0 0
\(913\) 20.0000 0.661903
\(914\) 0 0
\(915\) 20.0000 0.661180
\(916\) 0 0
\(917\) −16.0000 −0.528367
\(918\) 0 0
\(919\) −29.0000 −0.956622 −0.478311 − 0.878191i \(-0.658751\pi\)
−0.478311 + 0.878191i \(0.658751\pi\)
\(920\) 0 0
\(921\) 12.0000 0.395413
\(922\) 0 0
\(923\) 12.0000 0.394985
\(924\) 0 0
\(925\) 121.000 3.97846
\(926\) 0 0
\(927\) 28.0000 0.919641
\(928\) 0 0
\(929\) 15.0000 0.492134 0.246067 − 0.969253i \(-0.420862\pi\)
0.246067 + 0.969253i \(0.420862\pi\)
\(930\) 0 0
\(931\) 2.00000 0.0655474
\(932\) 0 0
\(933\) −22.0000 −0.720248
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −28.0000 −0.914720 −0.457360 − 0.889282i \(-0.651205\pi\)
−0.457360 + 0.889282i \(0.651205\pi\)
\(938\) 0 0
\(939\) 22.0000 0.717943
\(940\) 0 0
\(941\) −54.0000 −1.76035 −0.880175 − 0.474650i \(-0.842575\pi\)
−0.880175 + 0.474650i \(0.842575\pi\)
\(942\) 0 0
\(943\) −5.00000 −0.162822
\(944\) 0 0
\(945\) −20.0000 −0.650600
\(946\) 0 0
\(947\) −48.0000 −1.55979 −0.779895 − 0.625910i \(-0.784728\pi\)
−0.779895 + 0.625910i \(0.784728\pi\)
\(948\) 0 0
\(949\) 13.0000 0.421998
\(950\) 0 0
\(951\) −7.00000 −0.226991
\(952\) 0 0
\(953\) −26.0000 −0.842223 −0.421111 − 0.907009i \(-0.638360\pi\)
−0.421111 + 0.907009i \(0.638360\pi\)
\(954\) 0 0
\(955\) −96.0000 −3.10649
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −2.00000 −0.0645834
\(960\) 0 0
\(961\) 50.0000 1.61290
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 32.0000 1.03012
\(966\) 0 0
\(967\) −26.0000 −0.836104 −0.418052 − 0.908423i \(-0.637287\pi\)
−0.418052 + 0.908423i \(0.637287\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −35.0000 −1.12320 −0.561602 − 0.827408i \(-0.689815\pi\)
−0.561602 + 0.827408i \(0.689815\pi\)
\(972\) 0 0
\(973\) 12.0000 0.384702
\(974\) 0 0
\(975\) 11.0000 0.352282
\(976\) 0 0
\(977\) 54.0000 1.72761 0.863807 − 0.503824i \(-0.168074\pi\)
0.863807 + 0.503824i \(0.168074\pi\)
\(978\) 0 0
\(979\) 10.0000 0.319601
\(980\) 0 0
\(981\) −28.0000 −0.893971
\(982\) 0 0
\(983\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) −60.0000 −1.91176
\(986\) 0 0
\(987\) −9.00000 −0.286473
\(988\) 0 0
\(989\) 8.00000 0.254385
\(990\) 0 0
\(991\) −1.00000 −0.0317660 −0.0158830 − 0.999874i \(-0.505056\pi\)
−0.0158830 + 0.999874i \(0.505056\pi\)
\(992\) 0 0
\(993\) 29.0000 0.920287
\(994\) 0 0
\(995\) 112.000 3.55064
\(996\) 0 0
\(997\) −7.00000 −0.221692 −0.110846 − 0.993838i \(-0.535356\pi\)
−0.110846 + 0.993838i \(0.535356\pi\)
\(998\) 0 0
\(999\) 55.0000 1.74012
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 5824.2.a.h.1.1 1
4.3 odd 2 5824.2.a.v.1.1 1
8.3 odd 2 2912.2.a.b.1.1 ✓ 1
8.5 even 2 2912.2.a.f.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2912.2.a.b.1.1 ✓ 1 8.3 odd 2
2912.2.a.f.1.1 yes 1 8.5 even 2
5824.2.a.h.1.1 1 1.1 even 1 trivial
5824.2.a.v.1.1 1 4.3 odd 2