Properties

Label 162.4.a.b.1.1
Level $162$
Weight $4$
Character 162.1
Self dual yes
Analytic conductor $9.558$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [162,4,Mod(1,162)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(162, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("162.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 162.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: \(9.55830942093\)
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 \(\chi\) \(=\) 162.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} +4.00000 q^{4} +21.0000 q^{5} +8.00000 q^{7} -8.00000 q^{8} +O(q^{10})\) \(q-2.00000 q^{2} +4.00000 q^{4} +21.0000 q^{5} +8.00000 q^{7} -8.00000 q^{8} -42.0000 q^{10} +36.0000 q^{11} -49.0000 q^{13} -16.0000 q^{14} +16.0000 q^{16} +21.0000 q^{17} -112.000 q^{19} +84.0000 q^{20} -72.0000 q^{22} +180.000 q^{23} +316.000 q^{25} +98.0000 q^{26} +32.0000 q^{28} -135.000 q^{29} +308.000 q^{31} -32.0000 q^{32} -42.0000 q^{34} +168.000 q^{35} -1.00000 q^{37} +224.000 q^{38} -168.000 q^{40} -42.0000 q^{41} +20.0000 q^{43} +144.000 q^{44} -360.000 q^{46} +84.0000 q^{47} -279.000 q^{49} -632.000 q^{50} -196.000 q^{52} -174.000 q^{53} +756.000 q^{55} -64.0000 q^{56} +270.000 q^{58} +504.000 q^{59} -385.000 q^{61} -616.000 q^{62} +64.0000 q^{64} -1029.00 q^{65} +272.000 q^{67} +84.0000 q^{68} -336.000 q^{70} -888.000 q^{71} +371.000 q^{73} +2.00000 q^{74} -448.000 q^{76} +288.000 q^{77} -652.000 q^{79} +336.000 q^{80} +84.0000 q^{82} +84.0000 q^{83} +441.000 q^{85} -40.0000 q^{86} -288.000 q^{88} +21.0000 q^{89} -392.000 q^{91} +720.000 q^{92} -168.000 q^{94} -2352.00 q^{95} -1246.00 q^{97} +558.000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) 0 0
\(4\) 4.00000 0.500000
\(5\) 21.0000 1.87830 0.939149 0.343511i \(-0.111616\pi\)
0.939149 + 0.343511i \(0.111616\pi\)
\(6\) 0 0
\(7\) 8.00000 0.431959 0.215980 0.976398i \(-0.430705\pi\)
0.215980 + 0.976398i \(0.430705\pi\)
\(8\) −8.00000 −0.353553
\(9\) 0 0
\(10\) −42.0000 −1.32816
\(11\) 36.0000 0.986764 0.493382 0.869813i \(-0.335760\pi\)
0.493382 + 0.869813i \(0.335760\pi\)
\(12\) 0 0
\(13\) −49.0000 −1.04540 −0.522698 0.852518i \(-0.675075\pi\)
−0.522698 + 0.852518i \(0.675075\pi\)
\(14\) −16.0000 −0.305441
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 21.0000 0.299603 0.149801 0.988716i \(-0.452137\pi\)
0.149801 + 0.988716i \(0.452137\pi\)
\(18\) 0 0
\(19\) −112.000 −1.35235 −0.676173 0.736743i \(-0.736363\pi\)
−0.676173 + 0.736743i \(0.736363\pi\)
\(20\) 84.0000 0.939149
\(21\) 0 0
\(22\) −72.0000 −0.697748
\(23\) 180.000 1.63185 0.815926 0.578156i \(-0.196228\pi\)
0.815926 + 0.578156i \(0.196228\pi\)
\(24\) 0 0
\(25\) 316.000 2.52800
\(26\) 98.0000 0.739207
\(27\) 0 0
\(28\) 32.0000 0.215980
\(29\) −135.000 −0.864444 −0.432222 0.901767i \(-0.642270\pi\)
−0.432222 + 0.901767i \(0.642270\pi\)
\(30\) 0 0
\(31\) 308.000 1.78447 0.892233 0.451576i \(-0.149138\pi\)
0.892233 + 0.451576i \(0.149138\pi\)
\(32\) −32.0000 −0.176777
\(33\) 0 0
\(34\) −42.0000 −0.211851
\(35\) 168.000 0.811348
\(36\) 0 0
\(37\) −1.00000 −0.00444322 −0.00222161 0.999998i \(-0.500707\pi\)
−0.00222161 + 0.999998i \(0.500707\pi\)
\(38\) 224.000 0.956253
\(39\) 0 0
\(40\) −168.000 −0.664078
\(41\) −42.0000 −0.159983 −0.0799914 0.996796i \(-0.525489\pi\)
−0.0799914 + 0.996796i \(0.525489\pi\)
\(42\) 0 0
\(43\) 20.0000 0.0709296 0.0354648 0.999371i \(-0.488709\pi\)
0.0354648 + 0.999371i \(0.488709\pi\)
\(44\) 144.000 0.493382
\(45\) 0 0
\(46\) −360.000 −1.15389
\(47\) 84.0000 0.260695 0.130347 0.991468i \(-0.458391\pi\)
0.130347 + 0.991468i \(0.458391\pi\)
\(48\) 0 0
\(49\) −279.000 −0.813411
\(50\) −632.000 −1.78757
\(51\) 0 0
\(52\) −196.000 −0.522698
\(53\) −174.000 −0.450957 −0.225479 0.974248i \(-0.572395\pi\)
−0.225479 + 0.974248i \(0.572395\pi\)
\(54\) 0 0
\(55\) 756.000 1.85344
\(56\) −64.0000 −0.152721
\(57\) 0 0
\(58\) 270.000 0.611254
\(59\) 504.000 1.11212 0.556061 0.831141i \(-0.312312\pi\)
0.556061 + 0.831141i \(0.312312\pi\)
\(60\) 0 0
\(61\) −385.000 −0.808102 −0.404051 0.914737i \(-0.632398\pi\)
−0.404051 + 0.914737i \(0.632398\pi\)
\(62\) −616.000 −1.26181
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) −1029.00 −1.96357
\(66\) 0 0
\(67\) 272.000 0.495971 0.247986 0.968764i \(-0.420231\pi\)
0.247986 + 0.968764i \(0.420231\pi\)
\(68\) 84.0000 0.149801
\(69\) 0 0
\(70\) −336.000 −0.573710
\(71\) −888.000 −1.48431 −0.742156 0.670227i \(-0.766197\pi\)
−0.742156 + 0.670227i \(0.766197\pi\)
\(72\) 0 0
\(73\) 371.000 0.594826 0.297413 0.954749i \(-0.403876\pi\)
0.297413 + 0.954749i \(0.403876\pi\)
\(74\) 2.00000 0.00314183
\(75\) 0 0
\(76\) −448.000 −0.676173
\(77\) 288.000 0.426242
\(78\) 0 0
\(79\) −652.000 −0.928554 −0.464277 0.885690i \(-0.653686\pi\)
−0.464277 + 0.885690i \(0.653686\pi\)
\(80\) 336.000 0.469574
\(81\) 0 0
\(82\) 84.0000 0.113125
\(83\) 84.0000 0.111087 0.0555434 0.998456i \(-0.482311\pi\)
0.0555434 + 0.998456i \(0.482311\pi\)
\(84\) 0 0
\(85\) 441.000 0.562743
\(86\) −40.0000 −0.0501548
\(87\) 0 0
\(88\) −288.000 −0.348874
\(89\) 21.0000 0.0250112 0.0125056 0.999922i \(-0.496019\pi\)
0.0125056 + 0.999922i \(0.496019\pi\)
\(90\) 0 0
\(91\) −392.000 −0.451569
\(92\) 720.000 0.815926
\(93\) 0 0
\(94\) −168.000 −0.184339
\(95\) −2352.00 −2.54011
\(96\) 0 0
\(97\) −1246.00 −1.30425 −0.652124 0.758112i \(-0.726122\pi\)
−0.652124 + 0.758112i \(0.726122\pi\)
\(98\) 558.000 0.575168
\(99\) 0 0
\(100\) 1264.00 1.26400
\(101\) 546.000 0.537911 0.268956 0.963153i \(-0.413322\pi\)
0.268956 + 0.963153i \(0.413322\pi\)
\(102\) 0 0
\(103\) −196.000 −0.187500 −0.0937498 0.995596i \(-0.529885\pi\)
−0.0937498 + 0.995596i \(0.529885\pi\)
\(104\) 392.000 0.369603
\(105\) 0 0
\(106\) 348.000 0.318875
\(107\) −300.000 −0.271048 −0.135524 0.990774i \(-0.543272\pi\)
−0.135524 + 0.990774i \(0.543272\pi\)
\(108\) 0 0
\(109\) −1069.00 −0.939373 −0.469686 0.882833i \(-0.655633\pi\)
−0.469686 + 0.882833i \(0.655633\pi\)
\(110\) −1512.00 −1.31058
\(111\) 0 0
\(112\) 128.000 0.107990
\(113\) 897.000 0.746749 0.373375 0.927681i \(-0.378201\pi\)
0.373375 + 0.927681i \(0.378201\pi\)
\(114\) 0 0
\(115\) 3780.00 3.06510
\(116\) −540.000 −0.432222
\(117\) 0 0
\(118\) −1008.00 −0.786389
\(119\) 168.000 0.129416
\(120\) 0 0
\(121\) −35.0000 −0.0262960
\(122\) 770.000 0.571414
\(123\) 0 0
\(124\) 1232.00 0.892233
\(125\) 4011.00 2.87004
\(126\) 0 0
\(127\) 1532.00 1.07042 0.535209 0.844720i \(-0.320233\pi\)
0.535209 + 0.844720i \(0.320233\pi\)
\(128\) −128.000 −0.0883883
\(129\) 0 0
\(130\) 2058.00 1.38845
\(131\) 840.000 0.560238 0.280119 0.959965i \(-0.409626\pi\)
0.280119 + 0.959965i \(0.409626\pi\)
\(132\) 0 0
\(133\) −896.000 −0.584158
\(134\) −544.000 −0.350705
\(135\) 0 0
\(136\) −168.000 −0.105926
\(137\) 729.000 0.454618 0.227309 0.973823i \(-0.427007\pi\)
0.227309 + 0.973823i \(0.427007\pi\)
\(138\) 0 0
\(139\) −2044.00 −1.24726 −0.623632 0.781718i \(-0.714344\pi\)
−0.623632 + 0.781718i \(0.714344\pi\)
\(140\) 672.000 0.405674
\(141\) 0 0
\(142\) 1776.00 1.04957
\(143\) −1764.00 −1.03156
\(144\) 0 0
\(145\) −2835.00 −1.62368
\(146\) −742.000 −0.420605
\(147\) 0 0
\(148\) −4.00000 −0.00222161
\(149\) −1287.00 −0.707618 −0.353809 0.935318i \(-0.615114\pi\)
−0.353809 + 0.935318i \(0.615114\pi\)
\(150\) 0 0
\(151\) −736.000 −0.396655 −0.198327 0.980136i \(-0.563551\pi\)
−0.198327 + 0.980136i \(0.563551\pi\)
\(152\) 896.000 0.478126
\(153\) 0 0
\(154\) −576.000 −0.301399
\(155\) 6468.00 3.35176
\(156\) 0 0
\(157\) −2149.00 −1.09241 −0.546207 0.837650i \(-0.683929\pi\)
−0.546207 + 0.837650i \(0.683929\pi\)
\(158\) 1304.00 0.656587
\(159\) 0 0
\(160\) −672.000 −0.332039
\(161\) 1440.00 0.704894
\(162\) 0 0
\(163\) −3088.00 −1.48387 −0.741935 0.670472i \(-0.766092\pi\)
−0.741935 + 0.670472i \(0.766092\pi\)
\(164\) −168.000 −0.0799914
\(165\) 0 0
\(166\) −168.000 −0.0785502
\(167\) 168.000 0.0778457 0.0389228 0.999242i \(-0.487607\pi\)
0.0389228 + 0.999242i \(0.487607\pi\)
\(168\) 0 0
\(169\) 204.000 0.0928539
\(170\) −882.000 −0.397919
\(171\) 0 0
\(172\) 80.0000 0.0354648
\(173\) −3003.00 −1.31973 −0.659867 0.751383i \(-0.729387\pi\)
−0.659867 + 0.751383i \(0.729387\pi\)
\(174\) 0 0
\(175\) 2528.00 1.09199
\(176\) 576.000 0.246691
\(177\) 0 0
\(178\) −42.0000 −0.0176856
\(179\) −1164.00 −0.486042 −0.243021 0.970021i \(-0.578138\pi\)
−0.243021 + 0.970021i \(0.578138\pi\)
\(180\) 0 0
\(181\) −1666.00 −0.684159 −0.342080 0.939671i \(-0.611131\pi\)
−0.342080 + 0.939671i \(0.611131\pi\)
\(182\) 784.000 0.319307
\(183\) 0 0
\(184\) −1440.00 −0.576947
\(185\) −21.0000 −0.00834568
\(186\) 0 0
\(187\) 756.000 0.295637
\(188\) 336.000 0.130347
\(189\) 0 0
\(190\) 4704.00 1.79613
\(191\) −2064.00 −0.781915 −0.390958 0.920409i \(-0.627856\pi\)
−0.390958 + 0.920409i \(0.627856\pi\)
\(192\) 0 0
\(193\) −565.000 −0.210723 −0.105362 0.994434i \(-0.533600\pi\)
−0.105362 + 0.994434i \(0.533600\pi\)
\(194\) 2492.00 0.922243
\(195\) 0 0
\(196\) −1116.00 −0.406706
\(197\) −4731.00 −1.71101 −0.855507 0.517791i \(-0.826754\pi\)
−0.855507 + 0.517791i \(0.826754\pi\)
\(198\) 0 0
\(199\) 4676.00 1.66569 0.832846 0.553504i \(-0.186710\pi\)
0.832846 + 0.553504i \(0.186710\pi\)
\(200\) −2528.00 −0.893783
\(201\) 0 0
\(202\) −1092.00 −0.380361
\(203\) −1080.00 −0.373405
\(204\) 0 0
\(205\) −882.000 −0.300495
\(206\) 392.000 0.132582
\(207\) 0 0
\(208\) −784.000 −0.261349
\(209\) −4032.00 −1.33445
\(210\) 0 0
\(211\) 3380.00 1.10279 0.551395 0.834244i \(-0.314096\pi\)
0.551395 + 0.834244i \(0.314096\pi\)
\(212\) −696.000 −0.225479
\(213\) 0 0
\(214\) 600.000 0.191660
\(215\) 420.000 0.133227
\(216\) 0 0
\(217\) 2464.00 0.770817
\(218\) 2138.00 0.664237
\(219\) 0 0
\(220\) 3024.00 0.926718
\(221\) −1029.00 −0.313204
\(222\) 0 0
\(223\) −5236.00 −1.57233 −0.786163 0.618020i \(-0.787935\pi\)
−0.786163 + 0.618020i \(0.787935\pi\)
\(224\) −256.000 −0.0763604
\(225\) 0 0
\(226\) −1794.00 −0.528031
\(227\) 3864.00 1.12979 0.564896 0.825162i \(-0.308916\pi\)
0.564896 + 0.825162i \(0.308916\pi\)
\(228\) 0 0
\(229\) −3913.00 −1.12916 −0.564581 0.825377i \(-0.690962\pi\)
−0.564581 + 0.825377i \(0.690962\pi\)
\(230\) −7560.00 −2.16735
\(231\) 0 0
\(232\) 1080.00 0.305627
\(233\) 6333.00 1.78064 0.890319 0.455337i \(-0.150481\pi\)
0.890319 + 0.455337i \(0.150481\pi\)
\(234\) 0 0
\(235\) 1764.00 0.489662
\(236\) 2016.00 0.556061
\(237\) 0 0
\(238\) −336.000 −0.0915111
\(239\) −3828.00 −1.03604 −0.518018 0.855370i \(-0.673330\pi\)
−0.518018 + 0.855370i \(0.673330\pi\)
\(240\) 0 0
\(241\) −1477.00 −0.394780 −0.197390 0.980325i \(-0.563246\pi\)
−0.197390 + 0.980325i \(0.563246\pi\)
\(242\) 70.0000 0.0185941
\(243\) 0 0
\(244\) −1540.00 −0.404051
\(245\) −5859.00 −1.52783
\(246\) 0 0
\(247\) 5488.00 1.41374
\(248\) −2464.00 −0.630904
\(249\) 0 0
\(250\) −8022.00 −2.02942
\(251\) −3612.00 −0.908316 −0.454158 0.890921i \(-0.650060\pi\)
−0.454158 + 0.890921i \(0.650060\pi\)
\(252\) 0 0
\(253\) 6480.00 1.61025
\(254\) −3064.00 −0.756899
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) −399.000 −0.0968441 −0.0484221 0.998827i \(-0.515419\pi\)
−0.0484221 + 0.998827i \(0.515419\pi\)
\(258\) 0 0
\(259\) −8.00000 −0.00191929
\(260\) −4116.00 −0.981783
\(261\) 0 0
\(262\) −1680.00 −0.396148
\(263\) 3228.00 0.756833 0.378416 0.925635i \(-0.376469\pi\)
0.378416 + 0.925635i \(0.376469\pi\)
\(264\) 0 0
\(265\) −3654.00 −0.847032
\(266\) 1792.00 0.413062
\(267\) 0 0
\(268\) 1088.00 0.247986
\(269\) −147.000 −0.0333188 −0.0166594 0.999861i \(-0.505303\pi\)
−0.0166594 + 0.999861i \(0.505303\pi\)
\(270\) 0 0
\(271\) 3332.00 0.746880 0.373440 0.927654i \(-0.378178\pi\)
0.373440 + 0.927654i \(0.378178\pi\)
\(272\) 336.000 0.0749007
\(273\) 0 0
\(274\) −1458.00 −0.321464
\(275\) 11376.0 2.49454
\(276\) 0 0
\(277\) 2414.00 0.523622 0.261811 0.965119i \(-0.415680\pi\)
0.261811 + 0.965119i \(0.415680\pi\)
\(278\) 4088.00 0.881949
\(279\) 0 0
\(280\) −1344.00 −0.286855
\(281\) −3555.00 −0.754710 −0.377355 0.926069i \(-0.623166\pi\)
−0.377355 + 0.926069i \(0.623166\pi\)
\(282\) 0 0
\(283\) 5348.00 1.12334 0.561671 0.827361i \(-0.310159\pi\)
0.561671 + 0.827361i \(0.310159\pi\)
\(284\) −3552.00 −0.742156
\(285\) 0 0
\(286\) 3528.00 0.729423
\(287\) −336.000 −0.0691061
\(288\) 0 0
\(289\) −4472.00 −0.910238
\(290\) 5670.00 1.14812
\(291\) 0 0
\(292\) 1484.00 0.297413
\(293\) 6489.00 1.29383 0.646914 0.762563i \(-0.276059\pi\)
0.646914 + 0.762563i \(0.276059\pi\)
\(294\) 0 0
\(295\) 10584.0 2.08890
\(296\) 8.00000 0.00157091
\(297\) 0 0
\(298\) 2574.00 0.500362
\(299\) −8820.00 −1.70593
\(300\) 0 0
\(301\) 160.000 0.0306387
\(302\) 1472.00 0.280477
\(303\) 0 0
\(304\) −1792.00 −0.338086
\(305\) −8085.00 −1.51785
\(306\) 0 0
\(307\) −1204.00 −0.223830 −0.111915 0.993718i \(-0.535698\pi\)
−0.111915 + 0.993718i \(0.535698\pi\)
\(308\) 1152.00 0.213121
\(309\) 0 0
\(310\) −12936.0 −2.37005
\(311\) −3192.00 −0.581999 −0.291000 0.956723i \(-0.593988\pi\)
−0.291000 + 0.956723i \(0.593988\pi\)
\(312\) 0 0
\(313\) −3241.00 −0.585278 −0.292639 0.956223i \(-0.594533\pi\)
−0.292639 + 0.956223i \(0.594533\pi\)
\(314\) 4298.00 0.772453
\(315\) 0 0
\(316\) −2608.00 −0.464277
\(317\) 5325.00 0.943476 0.471738 0.881739i \(-0.343627\pi\)
0.471738 + 0.881739i \(0.343627\pi\)
\(318\) 0 0
\(319\) −4860.00 −0.853002
\(320\) 1344.00 0.234787
\(321\) 0 0
\(322\) −2880.00 −0.498435
\(323\) −2352.00 −0.405167
\(324\) 0 0
\(325\) −15484.0 −2.64276
\(326\) 6176.00 1.04925
\(327\) 0 0
\(328\) 336.000 0.0565625
\(329\) 672.000 0.112610
\(330\) 0 0
\(331\) 968.000 0.160743 0.0803717 0.996765i \(-0.474389\pi\)
0.0803717 + 0.996765i \(0.474389\pi\)
\(332\) 336.000 0.0555434
\(333\) 0 0
\(334\) −336.000 −0.0550452
\(335\) 5712.00 0.931582
\(336\) 0 0
\(337\) 9890.00 1.59864 0.799321 0.600904i \(-0.205192\pi\)
0.799321 + 0.600904i \(0.205192\pi\)
\(338\) −408.000 −0.0656576
\(339\) 0 0
\(340\) 1764.00 0.281372
\(341\) 11088.0 1.76085
\(342\) 0 0
\(343\) −4976.00 −0.783320
\(344\) −160.000 −0.0250774
\(345\) 0 0
\(346\) 6006.00 0.933192
\(347\) −1560.00 −0.241341 −0.120670 0.992693i \(-0.538504\pi\)
−0.120670 + 0.992693i \(0.538504\pi\)
\(348\) 0 0
\(349\) 2870.00 0.440194 0.220097 0.975478i \(-0.429363\pi\)
0.220097 + 0.975478i \(0.429363\pi\)
\(350\) −5056.00 −0.772156
\(351\) 0 0
\(352\) −1152.00 −0.174437
\(353\) 7182.00 1.08289 0.541444 0.840737i \(-0.317878\pi\)
0.541444 + 0.840737i \(0.317878\pi\)
\(354\) 0 0
\(355\) −18648.0 −2.78798
\(356\) 84.0000 0.0125056
\(357\) 0 0
\(358\) 2328.00 0.343683
\(359\) 8100.00 1.19081 0.595406 0.803425i \(-0.296991\pi\)
0.595406 + 0.803425i \(0.296991\pi\)
\(360\) 0 0
\(361\) 5685.00 0.828838
\(362\) 3332.00 0.483774
\(363\) 0 0
\(364\) −1568.00 −0.225784
\(365\) 7791.00 1.11726
\(366\) 0 0
\(367\) 11144.0 1.58505 0.792523 0.609842i \(-0.208767\pi\)
0.792523 + 0.609842i \(0.208767\pi\)
\(368\) 2880.00 0.407963
\(369\) 0 0
\(370\) 42.0000 0.00590129
\(371\) −1392.00 −0.194795
\(372\) 0 0
\(373\) 13838.0 1.92092 0.960462 0.278412i \(-0.0898080\pi\)
0.960462 + 0.278412i \(0.0898080\pi\)
\(374\) −1512.00 −0.209047
\(375\) 0 0
\(376\) −672.000 −0.0921696
\(377\) 6615.00 0.903687
\(378\) 0 0
\(379\) 1196.00 0.162096 0.0810480 0.996710i \(-0.474173\pi\)
0.0810480 + 0.996710i \(0.474173\pi\)
\(380\) −9408.00 −1.27005
\(381\) 0 0
\(382\) 4128.00 0.552898
\(383\) 3864.00 0.515512 0.257756 0.966210i \(-0.417017\pi\)
0.257756 + 0.966210i \(0.417017\pi\)
\(384\) 0 0
\(385\) 6048.00 0.800609
\(386\) 1130.00 0.149004
\(387\) 0 0
\(388\) −4984.00 −0.652124
\(389\) −5070.00 −0.660821 −0.330410 0.943837i \(-0.607187\pi\)
−0.330410 + 0.943837i \(0.607187\pi\)
\(390\) 0 0
\(391\) 3780.00 0.488907
\(392\) 2232.00 0.287584
\(393\) 0 0
\(394\) 9462.00 1.20987
\(395\) −13692.0 −1.74410
\(396\) 0 0
\(397\) 15239.0 1.92651 0.963254 0.268593i \(-0.0865587\pi\)
0.963254 + 0.268593i \(0.0865587\pi\)
\(398\) −9352.00 −1.17782
\(399\) 0 0
\(400\) 5056.00 0.632000
\(401\) −1707.00 −0.212577 −0.106289 0.994335i \(-0.533897\pi\)
−0.106289 + 0.994335i \(0.533897\pi\)
\(402\) 0 0
\(403\) −15092.0 −1.86547
\(404\) 2184.00 0.268956
\(405\) 0 0
\(406\) 2160.00 0.264037
\(407\) −36.0000 −0.00438441
\(408\) 0 0
\(409\) −13321.0 −1.61047 −0.805234 0.592958i \(-0.797960\pi\)
−0.805234 + 0.592958i \(0.797960\pi\)
\(410\) 1764.00 0.212482
\(411\) 0 0
\(412\) −784.000 −0.0937498
\(413\) 4032.00 0.480392
\(414\) 0 0
\(415\) 1764.00 0.208654
\(416\) 1568.00 0.184802
\(417\) 0 0
\(418\) 8064.00 0.943596
\(419\) −13944.0 −1.62580 −0.812899 0.582405i \(-0.802112\pi\)
−0.812899 + 0.582405i \(0.802112\pi\)
\(420\) 0 0
\(421\) −10837.0 −1.25454 −0.627272 0.778800i \(-0.715829\pi\)
−0.627272 + 0.778800i \(0.715829\pi\)
\(422\) −6760.00 −0.779791
\(423\) 0 0
\(424\) 1392.00 0.159437
\(425\) 6636.00 0.757396
\(426\) 0 0
\(427\) −3080.00 −0.349067
\(428\) −1200.00 −0.135524
\(429\) 0 0
\(430\) −840.000 −0.0942056
\(431\) 12612.0 1.40951 0.704755 0.709451i \(-0.251057\pi\)
0.704755 + 0.709451i \(0.251057\pi\)
\(432\) 0 0
\(433\) −9709.00 −1.07756 −0.538781 0.842446i \(-0.681115\pi\)
−0.538781 + 0.842446i \(0.681115\pi\)
\(434\) −4928.00 −0.545050
\(435\) 0 0
\(436\) −4276.00 −0.469686
\(437\) −20160.0 −2.20683
\(438\) 0 0
\(439\) 10388.0 1.12937 0.564684 0.825307i \(-0.308998\pi\)
0.564684 + 0.825307i \(0.308998\pi\)
\(440\) −6048.00 −0.655289
\(441\) 0 0
\(442\) 2058.00 0.221469
\(443\) −2508.00 −0.268981 −0.134491 0.990915i \(-0.542940\pi\)
−0.134491 + 0.990915i \(0.542940\pi\)
\(444\) 0 0
\(445\) 441.000 0.0469784
\(446\) 10472.0 1.11180
\(447\) 0 0
\(448\) 512.000 0.0539949
\(449\) −13698.0 −1.43975 −0.719876 0.694103i \(-0.755801\pi\)
−0.719876 + 0.694103i \(0.755801\pi\)
\(450\) 0 0
\(451\) −1512.00 −0.157865
\(452\) 3588.00 0.373375
\(453\) 0 0
\(454\) −7728.00 −0.798883
\(455\) −8232.00 −0.848180
\(456\) 0 0
\(457\) −9745.00 −0.997488 −0.498744 0.866749i \(-0.666205\pi\)
−0.498744 + 0.866749i \(0.666205\pi\)
\(458\) 7826.00 0.798439
\(459\) 0 0
\(460\) 15120.0 1.53255
\(461\) 17514.0 1.76943 0.884716 0.466130i \(-0.154352\pi\)
0.884716 + 0.466130i \(0.154352\pi\)
\(462\) 0 0
\(463\) 4640.00 0.465743 0.232872 0.972507i \(-0.425188\pi\)
0.232872 + 0.972507i \(0.425188\pi\)
\(464\) −2160.00 −0.216111
\(465\) 0 0
\(466\) −12666.0 −1.25910
\(467\) −4368.00 −0.432820 −0.216410 0.976303i \(-0.569435\pi\)
−0.216410 + 0.976303i \(0.569435\pi\)
\(468\) 0 0
\(469\) 2176.00 0.214240
\(470\) −3528.00 −0.346244
\(471\) 0 0
\(472\) −4032.00 −0.393195
\(473\) 720.000 0.0699908
\(474\) 0 0
\(475\) −35392.0 −3.41873
\(476\) 672.000 0.0647081
\(477\) 0 0
\(478\) 7656.00 0.732588
\(479\) −18816.0 −1.79483 −0.897416 0.441184i \(-0.854558\pi\)
−0.897416 + 0.441184i \(0.854558\pi\)
\(480\) 0 0
\(481\) 49.0000 0.00464492
\(482\) 2954.00 0.279151
\(483\) 0 0
\(484\) −140.000 −0.0131480
\(485\) −26166.0 −2.44977
\(486\) 0 0
\(487\) −13756.0 −1.27997 −0.639983 0.768389i \(-0.721059\pi\)
−0.639983 + 0.768389i \(0.721059\pi\)
\(488\) 3080.00 0.285707
\(489\) 0 0
\(490\) 11718.0 1.08034
\(491\) 7740.00 0.711408 0.355704 0.934599i \(-0.384241\pi\)
0.355704 + 0.934599i \(0.384241\pi\)
\(492\) 0 0
\(493\) −2835.00 −0.258990
\(494\) −10976.0 −0.999663
\(495\) 0 0
\(496\) 4928.00 0.446116
\(497\) −7104.00 −0.641163
\(498\) 0 0
\(499\) 2396.00 0.214949 0.107475 0.994208i \(-0.465724\pi\)
0.107475 + 0.994208i \(0.465724\pi\)
\(500\) 16044.0 1.43502
\(501\) 0 0
\(502\) 7224.00 0.642277
\(503\) 12096.0 1.07223 0.536117 0.844144i \(-0.319890\pi\)
0.536117 + 0.844144i \(0.319890\pi\)
\(504\) 0 0
\(505\) 11466.0 1.01036
\(506\) −12960.0 −1.13862
\(507\) 0 0
\(508\) 6128.00 0.535209
\(509\) 1722.00 0.149953 0.0749767 0.997185i \(-0.476112\pi\)
0.0749767 + 0.997185i \(0.476112\pi\)
\(510\) 0 0
\(511\) 2968.00 0.256940
\(512\) −512.000 −0.0441942
\(513\) 0 0
\(514\) 798.000 0.0684791
\(515\) −4116.00 −0.352180
\(516\) 0 0
\(517\) 3024.00 0.257244
\(518\) 16.0000 0.00135714
\(519\) 0 0
\(520\) 8232.00 0.694225
\(521\) 2982.00 0.250756 0.125378 0.992109i \(-0.459986\pi\)
0.125378 + 0.992109i \(0.459986\pi\)
\(522\) 0 0
\(523\) 812.000 0.0678896 0.0339448 0.999424i \(-0.489193\pi\)
0.0339448 + 0.999424i \(0.489193\pi\)
\(524\) 3360.00 0.280119
\(525\) 0 0
\(526\) −6456.00 −0.535162
\(527\) 6468.00 0.534631
\(528\) 0 0
\(529\) 20233.0 1.66294
\(530\) 7308.00 0.598942
\(531\) 0 0
\(532\) −3584.00 −0.292079
\(533\) 2058.00 0.167246
\(534\) 0 0
\(535\) −6300.00 −0.509108
\(536\) −2176.00 −0.175352
\(537\) 0 0
\(538\) 294.000 0.0235599
\(539\) −10044.0 −0.802645
\(540\) 0 0
\(541\) 7055.00 0.560662 0.280331 0.959903i \(-0.409556\pi\)
0.280331 + 0.959903i \(0.409556\pi\)
\(542\) −6664.00 −0.528124
\(543\) 0 0
\(544\) −672.000 −0.0529628
\(545\) −22449.0 −1.76442
\(546\) 0 0
\(547\) −14596.0 −1.14091 −0.570457 0.821328i \(-0.693234\pi\)
−0.570457 + 0.821328i \(0.693234\pi\)
\(548\) 2916.00 0.227309
\(549\) 0 0
\(550\) −22752.0 −1.76391
\(551\) 15120.0 1.16903
\(552\) 0 0
\(553\) −5216.00 −0.401097
\(554\) −4828.00 −0.370256
\(555\) 0 0
\(556\) −8176.00 −0.623632
\(557\) −7755.00 −0.589928 −0.294964 0.955508i \(-0.595308\pi\)
−0.294964 + 0.955508i \(0.595308\pi\)
\(558\) 0 0
\(559\) −980.000 −0.0741495
\(560\) 2688.00 0.202837
\(561\) 0 0
\(562\) 7110.00 0.533661
\(563\) 16044.0 1.20102 0.600510 0.799617i \(-0.294964\pi\)
0.600510 + 0.799617i \(0.294964\pi\)
\(564\) 0 0
\(565\) 18837.0 1.40262
\(566\) −10696.0 −0.794322
\(567\) 0 0
\(568\) 7104.00 0.524784
\(569\) 17025.0 1.25435 0.627175 0.778878i \(-0.284211\pi\)
0.627175 + 0.778878i \(0.284211\pi\)
\(570\) 0 0
\(571\) 3320.00 0.243323 0.121662 0.992572i \(-0.461178\pi\)
0.121662 + 0.992572i \(0.461178\pi\)
\(572\) −7056.00 −0.515780
\(573\) 0 0
\(574\) 672.000 0.0488654
\(575\) 56880.0 4.12532
\(576\) 0 0
\(577\) 1127.00 0.0813130 0.0406565 0.999173i \(-0.487055\pi\)
0.0406565 + 0.999173i \(0.487055\pi\)
\(578\) 8944.00 0.643636
\(579\) 0 0
\(580\) −11340.0 −0.811841
\(581\) 672.000 0.0479850
\(582\) 0 0
\(583\) −6264.00 −0.444989
\(584\) −2968.00 −0.210303
\(585\) 0 0
\(586\) −12978.0 −0.914874
\(587\) −84.0000 −0.00590639 −0.00295320 0.999996i \(-0.500940\pi\)
−0.00295320 + 0.999996i \(0.500940\pi\)
\(588\) 0 0
\(589\) −34496.0 −2.41321
\(590\) −21168.0 −1.47707
\(591\) 0 0
\(592\) −16.0000 −0.00111080
\(593\) −1743.00 −0.120702 −0.0603511 0.998177i \(-0.519222\pi\)
−0.0603511 + 0.998177i \(0.519222\pi\)
\(594\) 0 0
\(595\) 3528.00 0.243082
\(596\) −5148.00 −0.353809
\(597\) 0 0
\(598\) 17640.0 1.20628
\(599\) −16092.0 −1.09766 −0.548832 0.835932i \(-0.684928\pi\)
−0.548832 + 0.835932i \(0.684928\pi\)
\(600\) 0 0
\(601\) 21035.0 1.42768 0.713840 0.700309i \(-0.246954\pi\)
0.713840 + 0.700309i \(0.246954\pi\)
\(602\) −320.000 −0.0216648
\(603\) 0 0
\(604\) −2944.00 −0.198327
\(605\) −735.000 −0.0493917
\(606\) 0 0
\(607\) 6776.00 0.453096 0.226548 0.974000i \(-0.427256\pi\)
0.226548 + 0.974000i \(0.427256\pi\)
\(608\) 3584.00 0.239063
\(609\) 0 0
\(610\) 16170.0 1.07329
\(611\) −4116.00 −0.272530
\(612\) 0 0
\(613\) −23794.0 −1.56775 −0.783875 0.620919i \(-0.786760\pi\)
−0.783875 + 0.620919i \(0.786760\pi\)
\(614\) 2408.00 0.158272
\(615\) 0 0
\(616\) −2304.00 −0.150699
\(617\) 21621.0 1.41074 0.705372 0.708838i \(-0.250780\pi\)
0.705372 + 0.708838i \(0.250780\pi\)
\(618\) 0 0
\(619\) 22232.0 1.44359 0.721793 0.692109i \(-0.243318\pi\)
0.721793 + 0.692109i \(0.243318\pi\)
\(620\) 25872.0 1.67588
\(621\) 0 0
\(622\) 6384.00 0.411535
\(623\) 168.000 0.0108038
\(624\) 0 0
\(625\) 44731.0 2.86278
\(626\) 6482.00 0.413854
\(627\) 0 0
\(628\) −8596.00 −0.546207
\(629\) −21.0000 −0.00133120
\(630\) 0 0
\(631\) −9280.00 −0.585469 −0.292735 0.956194i \(-0.594565\pi\)
−0.292735 + 0.956194i \(0.594565\pi\)
\(632\) 5216.00 0.328293
\(633\) 0 0
\(634\) −10650.0 −0.667138
\(635\) 32172.0 2.01056
\(636\) 0 0
\(637\) 13671.0 0.850337
\(638\) 9720.00 0.603164
\(639\) 0 0
\(640\) −2688.00 −0.166020
\(641\) −19179.0 −1.18179 −0.590893 0.806750i \(-0.701225\pi\)
−0.590893 + 0.806750i \(0.701225\pi\)
\(642\) 0 0
\(643\) −3220.00 −0.197487 −0.0987437 0.995113i \(-0.531482\pi\)
−0.0987437 + 0.995113i \(0.531482\pi\)
\(644\) 5760.00 0.352447
\(645\) 0 0
\(646\) 4704.00 0.286496
\(647\) 14112.0 0.857496 0.428748 0.903424i \(-0.358955\pi\)
0.428748 + 0.903424i \(0.358955\pi\)
\(648\) 0 0
\(649\) 18144.0 1.09740
\(650\) 30968.0 1.86872
\(651\) 0 0
\(652\) −12352.0 −0.741935
\(653\) 22842.0 1.36888 0.684438 0.729071i \(-0.260047\pi\)
0.684438 + 0.729071i \(0.260047\pi\)
\(654\) 0 0
\(655\) 17640.0 1.05229
\(656\) −672.000 −0.0399957
\(657\) 0 0
\(658\) −1344.00 −0.0796270
\(659\) −21720.0 −1.28390 −0.641951 0.766746i \(-0.721875\pi\)
−0.641951 + 0.766746i \(0.721875\pi\)
\(660\) 0 0
\(661\) 26327.0 1.54917 0.774585 0.632470i \(-0.217959\pi\)
0.774585 + 0.632470i \(0.217959\pi\)
\(662\) −1936.00 −0.113663
\(663\) 0 0
\(664\) −672.000 −0.0392751
\(665\) −18816.0 −1.09722
\(666\) 0 0
\(667\) −24300.0 −1.41064
\(668\) 672.000 0.0389228
\(669\) 0 0
\(670\) −11424.0 −0.658728
\(671\) −13860.0 −0.797406
\(672\) 0 0
\(673\) −19741.0 −1.13070 −0.565349 0.824852i \(-0.691258\pi\)
−0.565349 + 0.824852i \(0.691258\pi\)
\(674\) −19780.0 −1.13041
\(675\) 0 0
\(676\) 816.000 0.0464269
\(677\) 12642.0 0.717683 0.358842 0.933398i \(-0.383172\pi\)
0.358842 + 0.933398i \(0.383172\pi\)
\(678\) 0 0
\(679\) −9968.00 −0.563383
\(680\) −3528.00 −0.198960
\(681\) 0 0
\(682\) −22176.0 −1.24511
\(683\) −26172.0 −1.46624 −0.733121 0.680098i \(-0.761937\pi\)
−0.733121 + 0.680098i \(0.761937\pi\)
\(684\) 0 0
\(685\) 15309.0 0.853908
\(686\) 9952.00 0.553891
\(687\) 0 0
\(688\) 320.000 0.0177324
\(689\) 8526.00 0.471429
\(690\) 0 0
\(691\) −9520.00 −0.524107 −0.262053 0.965053i \(-0.584400\pi\)
−0.262053 + 0.965053i \(0.584400\pi\)
\(692\) −12012.0 −0.659867
\(693\) 0 0
\(694\) 3120.00 0.170654
\(695\) −42924.0 −2.34273
\(696\) 0 0
\(697\) −882.000 −0.0479313
\(698\) −5740.00 −0.311264
\(699\) 0 0
\(700\) 10112.0 0.545997
\(701\) 16773.0 0.903720 0.451860 0.892089i \(-0.350761\pi\)
0.451860 + 0.892089i \(0.350761\pi\)
\(702\) 0 0
\(703\) 112.000 0.00600876
\(704\) 2304.00 0.123346
\(705\) 0 0
\(706\) −14364.0 −0.765717
\(707\) 4368.00 0.232356
\(708\) 0 0
\(709\) 12767.0 0.676269 0.338135 0.941098i \(-0.390204\pi\)
0.338135 + 0.941098i \(0.390204\pi\)
\(710\) 37296.0 1.97140
\(711\) 0 0
\(712\) −168.000 −0.00884279
\(713\) 55440.0 2.91198
\(714\) 0 0
\(715\) −37044.0 −1.93758
\(716\) −4656.00 −0.243021
\(717\) 0 0
\(718\) −16200.0 −0.842032
\(719\) −24948.0 −1.29402 −0.647012 0.762480i \(-0.723982\pi\)
−0.647012 + 0.762480i \(0.723982\pi\)
\(720\) 0 0
\(721\) −1568.00 −0.0809922
\(722\) −11370.0 −0.586077
\(723\) 0 0
\(724\) −6664.00 −0.342080
\(725\) −42660.0 −2.18531
\(726\) 0 0
\(727\) 56.0000 0.00285684 0.00142842 0.999999i \(-0.499545\pi\)
0.00142842 + 0.999999i \(0.499545\pi\)
\(728\) 3136.00 0.159654
\(729\) 0 0
\(730\) −15582.0 −0.790021
\(731\) 420.000 0.0212507
\(732\) 0 0
\(733\) 1190.00 0.0599641 0.0299820 0.999550i \(-0.490455\pi\)
0.0299820 + 0.999550i \(0.490455\pi\)
\(734\) −22288.0 −1.12080
\(735\) 0 0
\(736\) −5760.00 −0.288473
\(737\) 9792.00 0.489407
\(738\) 0 0
\(739\) −26692.0 −1.32866 −0.664331 0.747439i \(-0.731283\pi\)
−0.664331 + 0.747439i \(0.731283\pi\)
\(740\) −84.0000 −0.00417284
\(741\) 0 0
\(742\) 2784.00 0.137741
\(743\) 18852.0 0.930838 0.465419 0.885090i \(-0.345904\pi\)
0.465419 + 0.885090i \(0.345904\pi\)
\(744\) 0 0
\(745\) −27027.0 −1.32912
\(746\) −27676.0 −1.35830
\(747\) 0 0
\(748\) 3024.00 0.147819
\(749\) −2400.00 −0.117082
\(750\) 0 0
\(751\) 22616.0 1.09889 0.549447 0.835528i \(-0.314838\pi\)
0.549447 + 0.835528i \(0.314838\pi\)
\(752\) 1344.00 0.0651737
\(753\) 0 0
\(754\) −13230.0 −0.639003
\(755\) −15456.0 −0.745035
\(756\) 0 0
\(757\) 9326.00 0.447766 0.223883 0.974616i \(-0.428127\pi\)
0.223883 + 0.974616i \(0.428127\pi\)
\(758\) −2392.00 −0.114619
\(759\) 0 0
\(760\) 18816.0 0.898063
\(761\) 20769.0 0.989324 0.494662 0.869085i \(-0.335292\pi\)
0.494662 + 0.869085i \(0.335292\pi\)
\(762\) 0 0
\(763\) −8552.00 −0.405771
\(764\) −8256.00 −0.390958
\(765\) 0 0
\(766\) −7728.00 −0.364522
\(767\) −24696.0 −1.16261
\(768\) 0 0
\(769\) −301.000 −0.0141149 −0.00705744 0.999975i \(-0.502246\pi\)
−0.00705744 + 0.999975i \(0.502246\pi\)
\(770\) −12096.0 −0.566116
\(771\) 0 0
\(772\) −2260.00 −0.105362
\(773\) −17955.0 −0.835442 −0.417721 0.908575i \(-0.637171\pi\)
−0.417721 + 0.908575i \(0.637171\pi\)
\(774\) 0 0
\(775\) 97328.0 4.51113
\(776\) 9968.00 0.461122
\(777\) 0 0
\(778\) 10140.0 0.467271
\(779\) 4704.00 0.216352
\(780\) 0 0
\(781\) −31968.0 −1.46467
\(782\) −7560.00 −0.345710
\(783\) 0 0
\(784\) −4464.00 −0.203353
\(785\) −45129.0 −2.05188
\(786\) 0 0
\(787\) −27412.0 −1.24159 −0.620796 0.783972i \(-0.713190\pi\)
−0.620796 + 0.783972i \(0.713190\pi\)
\(788\) −18924.0 −0.855507
\(789\) 0 0
\(790\) 27384.0 1.23326
\(791\) 7176.00 0.322565
\(792\) 0 0
\(793\) 18865.0 0.844787
\(794\) −30478.0 −1.36225
\(795\) 0 0
\(796\) 18704.0 0.832846
\(797\) 22533.0 1.00146 0.500728 0.865605i \(-0.333066\pi\)
0.500728 + 0.865605i \(0.333066\pi\)
\(798\) 0 0
\(799\) 1764.00 0.0781049
\(800\) −10112.0 −0.446891
\(801\) 0 0
\(802\) 3414.00 0.150315
\(803\) 13356.0 0.586953
\(804\) 0 0
\(805\) 30240.0 1.32400
\(806\) 30184.0 1.31909
\(807\) 0 0
\(808\) −4368.00 −0.190180
\(809\) 6105.00 0.265316 0.132658 0.991162i \(-0.457649\pi\)
0.132658 + 0.991162i \(0.457649\pi\)
\(810\) 0 0
\(811\) −3472.00 −0.150331 −0.0751655 0.997171i \(-0.523949\pi\)
−0.0751655 + 0.997171i \(0.523949\pi\)
\(812\) −4320.00 −0.186702
\(813\) 0 0
\(814\) 72.0000 0.00310024
\(815\) −64848.0 −2.78715
\(816\) 0 0
\(817\) −2240.00 −0.0959213
\(818\) 26642.0 1.13877
\(819\) 0 0
\(820\) −3528.00 −0.150248
\(821\) 4929.00 0.209529 0.104764 0.994497i \(-0.466591\pi\)
0.104764 + 0.994497i \(0.466591\pi\)
\(822\) 0 0
\(823\) 39524.0 1.67402 0.837011 0.547186i \(-0.184301\pi\)
0.837011 + 0.547186i \(0.184301\pi\)
\(824\) 1568.00 0.0662911
\(825\) 0 0
\(826\) −8064.00 −0.339688
\(827\) 38676.0 1.62623 0.813117 0.582100i \(-0.197769\pi\)
0.813117 + 0.582100i \(0.197769\pi\)
\(828\) 0 0
\(829\) 16646.0 0.697394 0.348697 0.937236i \(-0.386624\pi\)
0.348697 + 0.937236i \(0.386624\pi\)
\(830\) −3528.00 −0.147541
\(831\) 0 0
\(832\) −3136.00 −0.130675
\(833\) −5859.00 −0.243700
\(834\) 0 0
\(835\) 3528.00 0.146217
\(836\) −16128.0 −0.667223
\(837\) 0 0
\(838\) 27888.0 1.14961
\(839\) 6636.00 0.273063 0.136532 0.990636i \(-0.456404\pi\)
0.136532 + 0.990636i \(0.456404\pi\)
\(840\) 0 0
\(841\) −6164.00 −0.252737
\(842\) 21674.0 0.887097
\(843\) 0 0
\(844\) 13520.0 0.551395
\(845\) 4284.00 0.174407
\(846\) 0 0
\(847\) −280.000 −0.0113588
\(848\) −2784.00 −0.112739
\(849\) 0 0
\(850\) −13272.0 −0.535560
\(851\) −180.000 −0.00725067
\(852\) 0 0
\(853\) −9394.00 −0.377074 −0.188537 0.982066i \(-0.560375\pi\)
−0.188537 + 0.982066i \(0.560375\pi\)
\(854\) 6160.00 0.246828
\(855\) 0 0
\(856\) 2400.00 0.0958298
\(857\) −7287.00 −0.290454 −0.145227 0.989398i \(-0.546391\pi\)
−0.145227 + 0.989398i \(0.546391\pi\)
\(858\) 0 0
\(859\) −25732.0 −1.02208 −0.511039 0.859558i \(-0.670739\pi\)
−0.511039 + 0.859558i \(0.670739\pi\)
\(860\) 1680.00 0.0666134
\(861\) 0 0
\(862\) −25224.0 −0.996674
\(863\) −7188.00 −0.283525 −0.141763 0.989901i \(-0.545277\pi\)
−0.141763 + 0.989901i \(0.545277\pi\)
\(864\) 0 0
\(865\) −63063.0 −2.47885
\(866\) 19418.0 0.761952
\(867\) 0 0
\(868\) 9856.00 0.385408
\(869\) −23472.0 −0.916264
\(870\) 0 0
\(871\) −13328.0 −0.518487
\(872\) 8552.00 0.332118
\(873\) 0 0
\(874\) 40320.0 1.56046
\(875\) 32088.0 1.23974
\(876\) 0 0
\(877\) −12601.0 −0.485183 −0.242592 0.970129i \(-0.577997\pi\)
−0.242592 + 0.970129i \(0.577997\pi\)
\(878\) −20776.0 −0.798583
\(879\) 0 0
\(880\) 12096.0 0.463359
\(881\) −210.000 −0.00803074 −0.00401537 0.999992i \(-0.501278\pi\)
−0.00401537 + 0.999992i \(0.501278\pi\)
\(882\) 0 0
\(883\) −47524.0 −1.81122 −0.905612 0.424108i \(-0.860588\pi\)
−0.905612 + 0.424108i \(0.860588\pi\)
\(884\) −4116.00 −0.156602
\(885\) 0 0
\(886\) 5016.00 0.190198
\(887\) −41748.0 −1.58034 −0.790169 0.612888i \(-0.790008\pi\)
−0.790169 + 0.612888i \(0.790008\pi\)
\(888\) 0 0
\(889\) 12256.0 0.462377
\(890\) −882.000 −0.0332188
\(891\) 0 0
\(892\) −20944.0 −0.786163
\(893\) −9408.00 −0.352550
\(894\) 0 0
\(895\) −24444.0 −0.912931
\(896\) −1024.00 −0.0381802
\(897\) 0 0
\(898\) 27396.0 1.01806
\(899\) −41580.0 −1.54257
\(900\) 0 0
\(901\) −3654.00 −0.135108
\(902\) 3024.00 0.111628
\(903\) 0 0
\(904\) −7176.00 −0.264016
\(905\) −34986.0 −1.28505
\(906\) 0 0
\(907\) 14720.0 0.538886 0.269443 0.963016i \(-0.413160\pi\)
0.269443 + 0.963016i \(0.413160\pi\)
\(908\) 15456.0 0.564896
\(909\) 0 0
\(910\) 16464.0 0.599754
\(911\) 17652.0 0.641972 0.320986 0.947084i \(-0.395986\pi\)
0.320986 + 0.947084i \(0.395986\pi\)
\(912\) 0 0
\(913\) 3024.00 0.109616
\(914\) 19490.0 0.705330
\(915\) 0 0
\(916\) −15652.0 −0.564581
\(917\) 6720.00 0.242000
\(918\) 0 0
\(919\) 17156.0 0.615804 0.307902 0.951418i \(-0.400373\pi\)
0.307902 + 0.951418i \(0.400373\pi\)
\(920\) −30240.0 −1.08368
\(921\) 0 0
\(922\) −35028.0 −1.25118
\(923\) 43512.0 1.55170
\(924\) 0 0
\(925\) −316.000 −0.0112324
\(926\) −9280.00 −0.329330
\(927\) 0 0
\(928\) 4320.00 0.152814
\(929\) 24213.0 0.855116 0.427558 0.903988i \(-0.359374\pi\)
0.427558 + 0.903988i \(0.359374\pi\)
\(930\) 0 0
\(931\) 31248.0 1.10001
\(932\) 25332.0 0.890319
\(933\) 0 0
\(934\) 8736.00 0.306050
\(935\) 15876.0 0.555295
\(936\) 0 0
\(937\) −15085.0 −0.525940 −0.262970 0.964804i \(-0.584702\pi\)
−0.262970 + 0.964804i \(0.584702\pi\)
\(938\) −4352.00 −0.151490
\(939\) 0 0
\(940\) 7056.00 0.244831
\(941\) 5229.00 0.181148 0.0905741 0.995890i \(-0.471130\pi\)
0.0905741 + 0.995890i \(0.471130\pi\)
\(942\) 0 0
\(943\) −7560.00 −0.261068
\(944\) 8064.00 0.278031
\(945\) 0 0
\(946\) −1440.00 −0.0494909
\(947\) 28488.0 0.977546 0.488773 0.872411i \(-0.337445\pi\)
0.488773 + 0.872411i \(0.337445\pi\)
\(948\) 0 0
\(949\) −18179.0 −0.621829
\(950\) 70784.0 2.41741
\(951\) 0 0
\(952\) −1344.00 −0.0457556
\(953\) −1143.00 −0.0388514 −0.0194257 0.999811i \(-0.506184\pi\)
−0.0194257 + 0.999811i \(0.506184\pi\)
\(954\) 0 0
\(955\) −43344.0 −1.46867
\(956\) −15312.0 −0.518018
\(957\) 0 0
\(958\) 37632.0 1.26914
\(959\) 5832.00 0.196377
\(960\) 0 0
\(961\) 65073.0 2.18432
\(962\) −98.0000 −0.00328446
\(963\) 0 0
\(964\) −5908.00 −0.197390
\(965\) −11865.0 −0.395801
\(966\) 0 0
\(967\) 36416.0 1.21102 0.605512 0.795836i \(-0.292968\pi\)
0.605512 + 0.795836i \(0.292968\pi\)
\(968\) 280.000 0.00929705
\(969\) 0 0
\(970\) 52332.0 1.73225
\(971\) 25200.0 0.832859 0.416430 0.909168i \(-0.363281\pi\)
0.416430 + 0.909168i \(0.363281\pi\)
\(972\) 0 0
\(973\) −16352.0 −0.538768
\(974\) 27512.0 0.905073
\(975\) 0 0
\(976\) −6160.00 −0.202025
\(977\) 23742.0 0.777455 0.388728 0.921353i \(-0.372915\pi\)
0.388728 + 0.921353i \(0.372915\pi\)
\(978\) 0 0
\(979\) 756.000 0.0246801
\(980\) −23436.0 −0.763914
\(981\) 0 0
\(982\) −15480.0 −0.503041
\(983\) 4704.00 0.152629 0.0763145 0.997084i \(-0.475685\pi\)
0.0763145 + 0.997084i \(0.475685\pi\)
\(984\) 0 0
\(985\) −99351.0 −3.21379
\(986\) 5670.00 0.183133
\(987\) 0 0
\(988\) 21952.0 0.706869
\(989\) 3600.00 0.115747
\(990\) 0 0
\(991\) −6280.00 −0.201302 −0.100651 0.994922i \(-0.532093\pi\)
−0.100651 + 0.994922i \(0.532093\pi\)
\(992\) −9856.00 −0.315452
\(993\) 0 0
\(994\) 14208.0 0.453371
\(995\) 98196.0 3.12867
\(996\) 0 0
\(997\) −8701.00 −0.276393 −0.138196 0.990405i \(-0.544130\pi\)
−0.138196 + 0.990405i \(0.544130\pi\)
\(998\) −4792.00 −0.151992
\(999\) 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 162.4.a.b.1.1 1
3.2 odd 2 162.4.a.c.1.1 yes 1
4.3 odd 2 1296.4.a.h.1.1 1
9.2 odd 6 162.4.c.d.109.1 2
9.4 even 3 162.4.c.e.55.1 2
9.5 odd 6 162.4.c.d.55.1 2
9.7 even 3 162.4.c.e.109.1 2
12.11 even 2 1296.4.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
162.4.a.b.1.1 1 1.1 even 1 trivial
162.4.a.c.1.1 yes 1 3.2 odd 2
162.4.c.d.55.1 2 9.5 odd 6
162.4.c.d.109.1 2 9.2 odd 6
162.4.c.e.55.1 2 9.4 even 3
162.4.c.e.109.1 2 9.7 even 3
1296.4.a.a.1.1 1 12.11 even 2
1296.4.a.h.1.1 1 4.3 odd 2