Properties

Label 189.4.a.d.1.1
Level $189$
Weight $4$
Character 189.1
Self dual yes
Analytic conductor $11.151$
Analytic rank $1$
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] = [189,4,Mod(1,189)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(189, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("189.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 189 = 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 189.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(11.1513609911\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 189.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{2} +1.00000 q^{4} -12.0000 q^{5} +7.00000 q^{7} -21.0000 q^{8} +O(q^{10})\) \(q+3.00000 q^{2} +1.00000 q^{4} -12.0000 q^{5} +7.00000 q^{7} -21.0000 q^{8} -36.0000 q^{10} +12.0000 q^{11} -61.0000 q^{13} +21.0000 q^{14} -71.0000 q^{16} -117.000 q^{17} +2.00000 q^{19} -12.0000 q^{20} +36.0000 q^{22} -75.0000 q^{23} +19.0000 q^{25} -183.000 q^{26} +7.00000 q^{28} +3.00000 q^{29} +263.000 q^{31} -45.0000 q^{32} -351.000 q^{34} -84.0000 q^{35} +218.000 q^{37} +6.00000 q^{38} +252.000 q^{40} -246.000 q^{41} +515.000 q^{43} +12.0000 q^{44} -225.000 q^{46} +318.000 q^{47} +49.0000 q^{49} +57.0000 q^{50} -61.0000 q^{52} -459.000 q^{53} -144.000 q^{55} -147.000 q^{56} +9.00000 q^{58} -255.000 q^{59} -862.000 q^{61} +789.000 q^{62} +433.000 q^{64} +732.000 q^{65} +479.000 q^{67} -117.000 q^{68} -252.000 q^{70} -117.000 q^{71} -430.000 q^{73} +654.000 q^{74} +2.00000 q^{76} +84.0000 q^{77} -646.000 q^{79} +852.000 q^{80} -738.000 q^{82} -348.000 q^{83} +1404.00 q^{85} +1545.00 q^{86} -252.000 q^{88} -585.000 q^{89} -427.000 q^{91} -75.0000 q^{92} +954.000 q^{94} -24.0000 q^{95} -376.000 q^{97} +147.000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.00000 1.06066 0.530330 0.847791i \(-0.322068\pi\)
0.530330 + 0.847791i \(0.322068\pi\)
\(3\) 0 0
\(4\) 1.00000 0.125000
\(5\) −12.0000 −1.07331 −0.536656 0.843801i \(-0.680313\pi\)
−0.536656 + 0.843801i \(0.680313\pi\)
\(6\) 0 0
\(7\) 7.00000 0.377964
\(8\) −21.0000 −0.928078
\(9\) 0 0
\(10\) −36.0000 −1.13842
\(11\) 12.0000 0.328921 0.164461 0.986384i \(-0.447412\pi\)
0.164461 + 0.986384i \(0.447412\pi\)
\(12\) 0 0
\(13\) −61.0000 −1.30141 −0.650706 0.759330i \(-0.725527\pi\)
−0.650706 + 0.759330i \(0.725527\pi\)
\(14\) 21.0000 0.400892
\(15\) 0 0
\(16\) −71.0000 −1.10938
\(17\) −117.000 −1.66922 −0.834608 0.550845i \(-0.814306\pi\)
−0.834608 + 0.550845i \(0.814306\pi\)
\(18\) 0 0
\(19\) 2.00000 0.0241490 0.0120745 0.999927i \(-0.496156\pi\)
0.0120745 + 0.999927i \(0.496156\pi\)
\(20\) −12.0000 −0.134164
\(21\) 0 0
\(22\) 36.0000 0.348874
\(23\) −75.0000 −0.679938 −0.339969 0.940437i \(-0.610417\pi\)
−0.339969 + 0.940437i \(0.610417\pi\)
\(24\) 0 0
\(25\) 19.0000 0.152000
\(26\) −183.000 −1.38036
\(27\) 0 0
\(28\) 7.00000 0.0472456
\(29\) 3.00000 0.0192099 0.00960493 0.999954i \(-0.496943\pi\)
0.00960493 + 0.999954i \(0.496943\pi\)
\(30\) 0 0
\(31\) 263.000 1.52375 0.761874 0.647725i \(-0.224279\pi\)
0.761874 + 0.647725i \(0.224279\pi\)
\(32\) −45.0000 −0.248592
\(33\) 0 0
\(34\) −351.000 −1.77047
\(35\) −84.0000 −0.405674
\(36\) 0 0
\(37\) 218.000 0.968621 0.484311 0.874896i \(-0.339070\pi\)
0.484311 + 0.874896i \(0.339070\pi\)
\(38\) 6.00000 0.0256139
\(39\) 0 0
\(40\) 252.000 0.996117
\(41\) −246.000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 515.000 1.82644 0.913218 0.407471i \(-0.133589\pi\)
0.913218 + 0.407471i \(0.133589\pi\)
\(44\) 12.0000 0.0411152
\(45\) 0 0
\(46\) −225.000 −0.721183
\(47\) 318.000 0.986916 0.493458 0.869770i \(-0.335733\pi\)
0.493458 + 0.869770i \(0.335733\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 57.0000 0.161220
\(51\) 0 0
\(52\) −61.0000 −0.162676
\(53\) −459.000 −1.18959 −0.594797 0.803876i \(-0.702768\pi\)
−0.594797 + 0.803876i \(0.702768\pi\)
\(54\) 0 0
\(55\) −144.000 −0.353036
\(56\) −147.000 −0.350780
\(57\) 0 0
\(58\) 9.00000 0.0203751
\(59\) −255.000 −0.562681 −0.281340 0.959608i \(-0.590779\pi\)
−0.281340 + 0.959608i \(0.590779\pi\)
\(60\) 0 0
\(61\) −862.000 −1.80931 −0.904654 0.426147i \(-0.859871\pi\)
−0.904654 + 0.426147i \(0.859871\pi\)
\(62\) 789.000 1.61618
\(63\) 0 0
\(64\) 433.000 0.845703
\(65\) 732.000 1.39682
\(66\) 0 0
\(67\) 479.000 0.873420 0.436710 0.899602i \(-0.356144\pi\)
0.436710 + 0.899602i \(0.356144\pi\)
\(68\) −117.000 −0.208652
\(69\) 0 0
\(70\) −252.000 −0.430282
\(71\) −117.000 −0.195568 −0.0977841 0.995208i \(-0.531175\pi\)
−0.0977841 + 0.995208i \(0.531175\pi\)
\(72\) 0 0
\(73\) −430.000 −0.689420 −0.344710 0.938709i \(-0.612023\pi\)
−0.344710 + 0.938709i \(0.612023\pi\)
\(74\) 654.000 1.02738
\(75\) 0 0
\(76\) 2.00000 0.00301863
\(77\) 84.0000 0.124321
\(78\) 0 0
\(79\) −646.000 −0.920009 −0.460004 0.887917i \(-0.652152\pi\)
−0.460004 + 0.887917i \(0.652152\pi\)
\(80\) 852.000 1.19071
\(81\) 0 0
\(82\) −738.000 −0.993884
\(83\) −348.000 −0.460216 −0.230108 0.973165i \(-0.573908\pi\)
−0.230108 + 0.973165i \(0.573908\pi\)
\(84\) 0 0
\(85\) 1404.00 1.79159
\(86\) 1545.00 1.93723
\(87\) 0 0
\(88\) −252.000 −0.305265
\(89\) −585.000 −0.696740 −0.348370 0.937357i \(-0.613265\pi\)
−0.348370 + 0.937357i \(0.613265\pi\)
\(90\) 0 0
\(91\) −427.000 −0.491888
\(92\) −75.0000 −0.0849923
\(93\) 0 0
\(94\) 954.000 1.04678
\(95\) −24.0000 −0.0259195
\(96\) 0 0
\(97\) −376.000 −0.393577 −0.196789 0.980446i \(-0.563051\pi\)
−0.196789 + 0.980446i \(0.563051\pi\)
\(98\) 147.000 0.151523
\(99\) 0 0
\(100\) 19.0000 0.0190000
\(101\) −978.000 −0.963511 −0.481756 0.876306i \(-0.660001\pi\)
−0.481756 + 0.876306i \(0.660001\pi\)
\(102\) 0 0
\(103\) −655.000 −0.626593 −0.313296 0.949655i \(-0.601433\pi\)
−0.313296 + 0.949655i \(0.601433\pi\)
\(104\) 1281.00 1.20781
\(105\) 0 0
\(106\) −1377.00 −1.26176
\(107\) −108.000 −0.0975771 −0.0487886 0.998809i \(-0.515536\pi\)
−0.0487886 + 0.998809i \(0.515536\pi\)
\(108\) 0 0
\(109\) 272.000 0.239017 0.119509 0.992833i \(-0.461868\pi\)
0.119509 + 0.992833i \(0.461868\pi\)
\(110\) −432.000 −0.374451
\(111\) 0 0
\(112\) −497.000 −0.419304
\(113\) 2256.00 1.87811 0.939056 0.343765i \(-0.111702\pi\)
0.939056 + 0.343765i \(0.111702\pi\)
\(114\) 0 0
\(115\) 900.000 0.729786
\(116\) 3.00000 0.00240123
\(117\) 0 0
\(118\) −765.000 −0.596813
\(119\) −819.000 −0.630904
\(120\) 0 0
\(121\) −1187.00 −0.891811
\(122\) −2586.00 −1.91906
\(123\) 0 0
\(124\) 263.000 0.190469
\(125\) 1272.00 0.910169
\(126\) 0 0
\(127\) 1784.00 1.24649 0.623246 0.782026i \(-0.285814\pi\)
0.623246 + 0.782026i \(0.285814\pi\)
\(128\) 1659.00 1.14560
\(129\) 0 0
\(130\) 2196.00 1.48155
\(131\) −1749.00 −1.16649 −0.583247 0.812295i \(-0.698218\pi\)
−0.583247 + 0.812295i \(0.698218\pi\)
\(132\) 0 0
\(133\) 14.0000 0.00912747
\(134\) 1437.00 0.926402
\(135\) 0 0
\(136\) 2457.00 1.54916
\(137\) −1374.00 −0.856852 −0.428426 0.903577i \(-0.640932\pi\)
−0.428426 + 0.903577i \(0.640932\pi\)
\(138\) 0 0
\(139\) 2450.00 1.49501 0.747505 0.664257i \(-0.231252\pi\)
0.747505 + 0.664257i \(0.231252\pi\)
\(140\) −84.0000 −0.0507093
\(141\) 0 0
\(142\) −351.000 −0.207431
\(143\) −732.000 −0.428062
\(144\) 0 0
\(145\) −36.0000 −0.0206182
\(146\) −1290.00 −0.731241
\(147\) 0 0
\(148\) 218.000 0.121078
\(149\) −921.000 −0.506384 −0.253192 0.967416i \(-0.581480\pi\)
−0.253192 + 0.967416i \(0.581480\pi\)
\(150\) 0 0
\(151\) 650.000 0.350306 0.175153 0.984541i \(-0.443958\pi\)
0.175153 + 0.984541i \(0.443958\pi\)
\(152\) −42.0000 −0.0224122
\(153\) 0 0
\(154\) 252.000 0.131862
\(155\) −3156.00 −1.63546
\(156\) 0 0
\(157\) −871.000 −0.442760 −0.221380 0.975188i \(-0.571056\pi\)
−0.221380 + 0.975188i \(0.571056\pi\)
\(158\) −1938.00 −0.975816
\(159\) 0 0
\(160\) 540.000 0.266817
\(161\) −525.000 −0.256993
\(162\) 0 0
\(163\) −3697.00 −1.77651 −0.888256 0.459349i \(-0.848083\pi\)
−0.888256 + 0.459349i \(0.848083\pi\)
\(164\) −246.000 −0.117130
\(165\) 0 0
\(166\) −1044.00 −0.488133
\(167\) 1824.00 0.845182 0.422591 0.906321i \(-0.361121\pi\)
0.422591 + 0.906321i \(0.361121\pi\)
\(168\) 0 0
\(169\) 1524.00 0.693673
\(170\) 4212.00 1.90027
\(171\) 0 0
\(172\) 515.000 0.228305
\(173\) −744.000 −0.326967 −0.163483 0.986546i \(-0.552273\pi\)
−0.163483 + 0.986546i \(0.552273\pi\)
\(174\) 0 0
\(175\) 133.000 0.0574506
\(176\) −852.000 −0.364897
\(177\) 0 0
\(178\) −1755.00 −0.739005
\(179\) −4572.00 −1.90909 −0.954546 0.298065i \(-0.903659\pi\)
−0.954546 + 0.298065i \(0.903659\pi\)
\(180\) 0 0
\(181\) −3481.00 −1.42951 −0.714753 0.699377i \(-0.753461\pi\)
−0.714753 + 0.699377i \(0.753461\pi\)
\(182\) −1281.00 −0.521725
\(183\) 0 0
\(184\) 1575.00 0.631036
\(185\) −2616.00 −1.03963
\(186\) 0 0
\(187\) −1404.00 −0.549041
\(188\) 318.000 0.123365
\(189\) 0 0
\(190\) −72.0000 −0.0274917
\(191\) 3756.00 1.42290 0.711452 0.702735i \(-0.248038\pi\)
0.711452 + 0.702735i \(0.248038\pi\)
\(192\) 0 0
\(193\) −4867.00 −1.81520 −0.907602 0.419832i \(-0.862089\pi\)
−0.907602 + 0.419832i \(0.862089\pi\)
\(194\) −1128.00 −0.417452
\(195\) 0 0
\(196\) 49.0000 0.0178571
\(197\) −18.0000 −0.00650988 −0.00325494 0.999995i \(-0.501036\pi\)
−0.00325494 + 0.999995i \(0.501036\pi\)
\(198\) 0 0
\(199\) −1807.00 −0.643693 −0.321846 0.946792i \(-0.604303\pi\)
−0.321846 + 0.946792i \(0.604303\pi\)
\(200\) −399.000 −0.141068
\(201\) 0 0
\(202\) −2934.00 −1.02196
\(203\) 21.0000 0.00726065
\(204\) 0 0
\(205\) 2952.00 1.00574
\(206\) −1965.00 −0.664602
\(207\) 0 0
\(208\) 4331.00 1.44375
\(209\) 24.0000 0.00794313
\(210\) 0 0
\(211\) −4273.00 −1.39415 −0.697075 0.716999i \(-0.745515\pi\)
−0.697075 + 0.716999i \(0.745515\pi\)
\(212\) −459.000 −0.148699
\(213\) 0 0
\(214\) −324.000 −0.103496
\(215\) −6180.00 −1.96034
\(216\) 0 0
\(217\) 1841.00 0.575923
\(218\) 816.000 0.253516
\(219\) 0 0
\(220\) −144.000 −0.0441294
\(221\) 7137.00 2.17234
\(222\) 0 0
\(223\) 2648.00 0.795171 0.397586 0.917565i \(-0.369848\pi\)
0.397586 + 0.917565i \(0.369848\pi\)
\(224\) −315.000 −0.0939590
\(225\) 0 0
\(226\) 6768.00 1.99204
\(227\) 5493.00 1.60609 0.803047 0.595916i \(-0.203211\pi\)
0.803047 + 0.595916i \(0.203211\pi\)
\(228\) 0 0
\(229\) −790.000 −0.227968 −0.113984 0.993483i \(-0.536361\pi\)
−0.113984 + 0.993483i \(0.536361\pi\)
\(230\) 2700.00 0.774055
\(231\) 0 0
\(232\) −63.0000 −0.0178282
\(233\) 6228.00 1.75112 0.875558 0.483114i \(-0.160494\pi\)
0.875558 + 0.483114i \(0.160494\pi\)
\(234\) 0 0
\(235\) −3816.00 −1.05927
\(236\) −255.000 −0.0703351
\(237\) 0 0
\(238\) −2457.00 −0.669175
\(239\) 4812.00 1.30235 0.651177 0.758926i \(-0.274276\pi\)
0.651177 + 0.758926i \(0.274276\pi\)
\(240\) 0 0
\(241\) 1622.00 0.433536 0.216768 0.976223i \(-0.430448\pi\)
0.216768 + 0.976223i \(0.430448\pi\)
\(242\) −3561.00 −0.945908
\(243\) 0 0
\(244\) −862.000 −0.226164
\(245\) −588.000 −0.153330
\(246\) 0 0
\(247\) −122.000 −0.0314278
\(248\) −5523.00 −1.41416
\(249\) 0 0
\(250\) 3816.00 0.965380
\(251\) −2880.00 −0.724239 −0.362119 0.932132i \(-0.617947\pi\)
−0.362119 + 0.932132i \(0.617947\pi\)
\(252\) 0 0
\(253\) −900.000 −0.223646
\(254\) 5352.00 1.32210
\(255\) 0 0
\(256\) 1513.00 0.369385
\(257\) −2190.00 −0.531550 −0.265775 0.964035i \(-0.585628\pi\)
−0.265775 + 0.964035i \(0.585628\pi\)
\(258\) 0 0
\(259\) 1526.00 0.366104
\(260\) 732.000 0.174603
\(261\) 0 0
\(262\) −5247.00 −1.23725
\(263\) −1869.00 −0.438203 −0.219102 0.975702i \(-0.570313\pi\)
−0.219102 + 0.975702i \(0.570313\pi\)
\(264\) 0 0
\(265\) 5508.00 1.27681
\(266\) 42.0000 0.00968115
\(267\) 0 0
\(268\) 479.000 0.109178
\(269\) −2268.00 −0.514061 −0.257030 0.966403i \(-0.582744\pi\)
−0.257030 + 0.966403i \(0.582744\pi\)
\(270\) 0 0
\(271\) 2027.00 0.454360 0.227180 0.973853i \(-0.427049\pi\)
0.227180 + 0.973853i \(0.427049\pi\)
\(272\) 8307.00 1.85179
\(273\) 0 0
\(274\) −4122.00 −0.908829
\(275\) 228.000 0.0499961
\(276\) 0 0
\(277\) 3944.00 0.855495 0.427747 0.903898i \(-0.359307\pi\)
0.427747 + 0.903898i \(0.359307\pi\)
\(278\) 7350.00 1.58570
\(279\) 0 0
\(280\) 1764.00 0.376497
\(281\) 4602.00 0.976983 0.488492 0.872569i \(-0.337547\pi\)
0.488492 + 0.872569i \(0.337547\pi\)
\(282\) 0 0
\(283\) −3220.00 −0.676357 −0.338179 0.941082i \(-0.609811\pi\)
−0.338179 + 0.941082i \(0.609811\pi\)
\(284\) −117.000 −0.0244460
\(285\) 0 0
\(286\) −2196.00 −0.454029
\(287\) −1722.00 −0.354169
\(288\) 0 0
\(289\) 8776.00 1.78628
\(290\) −108.000 −0.0218689
\(291\) 0 0
\(292\) −430.000 −0.0861776
\(293\) 6810.00 1.35783 0.678915 0.734216i \(-0.262450\pi\)
0.678915 + 0.734216i \(0.262450\pi\)
\(294\) 0 0
\(295\) 3060.00 0.603933
\(296\) −4578.00 −0.898956
\(297\) 0 0
\(298\) −2763.00 −0.537102
\(299\) 4575.00 0.884880
\(300\) 0 0
\(301\) 3605.00 0.690328
\(302\) 1950.00 0.371556
\(303\) 0 0
\(304\) −142.000 −0.0267903
\(305\) 10344.0 1.94195
\(306\) 0 0
\(307\) 758.000 0.140916 0.0704582 0.997515i \(-0.477554\pi\)
0.0704582 + 0.997515i \(0.477554\pi\)
\(308\) 84.0000 0.0155401
\(309\) 0 0
\(310\) −9468.00 −1.73467
\(311\) −9480.00 −1.72849 −0.864247 0.503068i \(-0.832204\pi\)
−0.864247 + 0.503068i \(0.832204\pi\)
\(312\) 0 0
\(313\) 3350.00 0.604962 0.302481 0.953155i \(-0.402185\pi\)
0.302481 + 0.953155i \(0.402185\pi\)
\(314\) −2613.00 −0.469618
\(315\) 0 0
\(316\) −646.000 −0.115001
\(317\) 426.000 0.0754781 0.0377390 0.999288i \(-0.487984\pi\)
0.0377390 + 0.999288i \(0.487984\pi\)
\(318\) 0 0
\(319\) 36.0000 0.00631854
\(320\) −5196.00 −0.907704
\(321\) 0 0
\(322\) −1575.00 −0.272582
\(323\) −234.000 −0.0403099
\(324\) 0 0
\(325\) −1159.00 −0.197815
\(326\) −11091.0 −1.88428
\(327\) 0 0
\(328\) 5166.00 0.869648
\(329\) 2226.00 0.373019
\(330\) 0 0
\(331\) −3103.00 −0.515276 −0.257638 0.966242i \(-0.582944\pi\)
−0.257638 + 0.966242i \(0.582944\pi\)
\(332\) −348.000 −0.0575271
\(333\) 0 0
\(334\) 5472.00 0.896451
\(335\) −5748.00 −0.937453
\(336\) 0 0
\(337\) −11725.0 −1.89526 −0.947628 0.319375i \(-0.896527\pi\)
−0.947628 + 0.319375i \(0.896527\pi\)
\(338\) 4572.00 0.735752
\(339\) 0 0
\(340\) 1404.00 0.223949
\(341\) 3156.00 0.501193
\(342\) 0 0
\(343\) 343.000 0.0539949
\(344\) −10815.0 −1.69507
\(345\) 0 0
\(346\) −2232.00 −0.346801
\(347\) −1506.00 −0.232987 −0.116493 0.993191i \(-0.537165\pi\)
−0.116493 + 0.993191i \(0.537165\pi\)
\(348\) 0 0
\(349\) 10289.0 1.57810 0.789051 0.614328i \(-0.210573\pi\)
0.789051 + 0.614328i \(0.210573\pi\)
\(350\) 399.000 0.0609356
\(351\) 0 0
\(352\) −540.000 −0.0817673
\(353\) 12693.0 1.91382 0.956912 0.290376i \(-0.0937805\pi\)
0.956912 + 0.290376i \(0.0937805\pi\)
\(354\) 0 0
\(355\) 1404.00 0.209906
\(356\) −585.000 −0.0870925
\(357\) 0 0
\(358\) −13716.0 −2.02490
\(359\) 5031.00 0.739627 0.369813 0.929106i \(-0.379422\pi\)
0.369813 + 0.929106i \(0.379422\pi\)
\(360\) 0 0
\(361\) −6855.00 −0.999417
\(362\) −10443.0 −1.51622
\(363\) 0 0
\(364\) −427.000 −0.0614859
\(365\) 5160.00 0.739964
\(366\) 0 0
\(367\) 9479.00 1.34823 0.674114 0.738627i \(-0.264526\pi\)
0.674114 + 0.738627i \(0.264526\pi\)
\(368\) 5325.00 0.754307
\(369\) 0 0
\(370\) −7848.00 −1.10270
\(371\) −3213.00 −0.449624
\(372\) 0 0
\(373\) −8566.00 −1.18909 −0.594545 0.804062i \(-0.702668\pi\)
−0.594545 + 0.804062i \(0.702668\pi\)
\(374\) −4212.00 −0.582346
\(375\) 0 0
\(376\) −6678.00 −0.915935
\(377\) −183.000 −0.0249999
\(378\) 0 0
\(379\) −2320.00 −0.314434 −0.157217 0.987564i \(-0.550252\pi\)
−0.157217 + 0.987564i \(0.550252\pi\)
\(380\) −24.0000 −0.00323993
\(381\) 0 0
\(382\) 11268.0 1.50922
\(383\) −1092.00 −0.145688 −0.0728441 0.997343i \(-0.523208\pi\)
−0.0728441 + 0.997343i \(0.523208\pi\)
\(384\) 0 0
\(385\) −1008.00 −0.133435
\(386\) −14601.0 −1.92531
\(387\) 0 0
\(388\) −376.000 −0.0491972
\(389\) −12570.0 −1.63837 −0.819183 0.573532i \(-0.805573\pi\)
−0.819183 + 0.573532i \(0.805573\pi\)
\(390\) 0 0
\(391\) 8775.00 1.13496
\(392\) −1029.00 −0.132583
\(393\) 0 0
\(394\) −54.0000 −0.00690477
\(395\) 7752.00 0.987457
\(396\) 0 0
\(397\) −6802.00 −0.859906 −0.429953 0.902851i \(-0.641470\pi\)
−0.429953 + 0.902851i \(0.641470\pi\)
\(398\) −5421.00 −0.682739
\(399\) 0 0
\(400\) −1349.00 −0.168625
\(401\) −8526.00 −1.06177 −0.530883 0.847445i \(-0.678140\pi\)
−0.530883 + 0.847445i \(0.678140\pi\)
\(402\) 0 0
\(403\) −16043.0 −1.98302
\(404\) −978.000 −0.120439
\(405\) 0 0
\(406\) 63.0000 0.00770108
\(407\) 2616.00 0.318600
\(408\) 0 0
\(409\) 13466.0 1.62800 0.813999 0.580867i \(-0.197286\pi\)
0.813999 + 0.580867i \(0.197286\pi\)
\(410\) 8856.00 1.06675
\(411\) 0 0
\(412\) −655.000 −0.0783241
\(413\) −1785.00 −0.212673
\(414\) 0 0
\(415\) 4176.00 0.493956
\(416\) 2745.00 0.323521
\(417\) 0 0
\(418\) 72.0000 0.00842496
\(419\) −15339.0 −1.78845 −0.894223 0.447621i \(-0.852271\pi\)
−0.894223 + 0.447621i \(0.852271\pi\)
\(420\) 0 0
\(421\) −4102.00 −0.474868 −0.237434 0.971404i \(-0.576306\pi\)
−0.237434 + 0.971404i \(0.576306\pi\)
\(422\) −12819.0 −1.47872
\(423\) 0 0
\(424\) 9639.00 1.10404
\(425\) −2223.00 −0.253721
\(426\) 0 0
\(427\) −6034.00 −0.683854
\(428\) −108.000 −0.0121971
\(429\) 0 0
\(430\) −18540.0 −2.07925
\(431\) −2592.00 −0.289680 −0.144840 0.989455i \(-0.546267\pi\)
−0.144840 + 0.989455i \(0.546267\pi\)
\(432\) 0 0
\(433\) −9448.00 −1.04860 −0.524298 0.851535i \(-0.675672\pi\)
−0.524298 + 0.851535i \(0.675672\pi\)
\(434\) 5523.00 0.610858
\(435\) 0 0
\(436\) 272.000 0.0298772
\(437\) −150.000 −0.0164198
\(438\) 0 0
\(439\) 11369.0 1.23602 0.618010 0.786170i \(-0.287939\pi\)
0.618010 + 0.786170i \(0.287939\pi\)
\(440\) 3024.00 0.327644
\(441\) 0 0
\(442\) 21411.0 2.30411
\(443\) −10842.0 −1.16280 −0.581398 0.813619i \(-0.697494\pi\)
−0.581398 + 0.813619i \(0.697494\pi\)
\(444\) 0 0
\(445\) 7020.00 0.747820
\(446\) 7944.00 0.843407
\(447\) 0 0
\(448\) 3031.00 0.319646
\(449\) 13140.0 1.38110 0.690551 0.723284i \(-0.257368\pi\)
0.690551 + 0.723284i \(0.257368\pi\)
\(450\) 0 0
\(451\) −2952.00 −0.308213
\(452\) 2256.00 0.234764
\(453\) 0 0
\(454\) 16479.0 1.70352
\(455\) 5124.00 0.527949
\(456\) 0 0
\(457\) 1379.00 0.141153 0.0705765 0.997506i \(-0.477516\pi\)
0.0705765 + 0.997506i \(0.477516\pi\)
\(458\) −2370.00 −0.241797
\(459\) 0 0
\(460\) 900.000 0.0912233
\(461\) 678.000 0.0684981 0.0342490 0.999413i \(-0.489096\pi\)
0.0342490 + 0.999413i \(0.489096\pi\)
\(462\) 0 0
\(463\) 15734.0 1.57931 0.789655 0.613550i \(-0.210259\pi\)
0.789655 + 0.613550i \(0.210259\pi\)
\(464\) −213.000 −0.0213109
\(465\) 0 0
\(466\) 18684.0 1.85734
\(467\) −7416.00 −0.734843 −0.367421 0.930055i \(-0.619759\pi\)
−0.367421 + 0.930055i \(0.619759\pi\)
\(468\) 0 0
\(469\) 3353.00 0.330122
\(470\) −11448.0 −1.12353
\(471\) 0 0
\(472\) 5355.00 0.522212
\(473\) 6180.00 0.600754
\(474\) 0 0
\(475\) 38.0000 0.00367065
\(476\) −819.000 −0.0788630
\(477\) 0 0
\(478\) 14436.0 1.38135
\(479\) −1248.00 −0.119045 −0.0595225 0.998227i \(-0.518958\pi\)
−0.0595225 + 0.998227i \(0.518958\pi\)
\(480\) 0 0
\(481\) −13298.0 −1.26058
\(482\) 4866.00 0.459834
\(483\) 0 0
\(484\) −1187.00 −0.111476
\(485\) 4512.00 0.422432
\(486\) 0 0
\(487\) −20950.0 −1.94935 −0.974677 0.223619i \(-0.928213\pi\)
−0.974677 + 0.223619i \(0.928213\pi\)
\(488\) 18102.0 1.67918
\(489\) 0 0
\(490\) −1764.00 −0.162631
\(491\) −768.000 −0.0705893 −0.0352947 0.999377i \(-0.511237\pi\)
−0.0352947 + 0.999377i \(0.511237\pi\)
\(492\) 0 0
\(493\) −351.000 −0.0320654
\(494\) −366.000 −0.0333342
\(495\) 0 0
\(496\) −18673.0 −1.69041
\(497\) −819.000 −0.0739178
\(498\) 0 0
\(499\) 4772.00 0.428104 0.214052 0.976822i \(-0.431334\pi\)
0.214052 + 0.976822i \(0.431334\pi\)
\(500\) 1272.00 0.113771
\(501\) 0 0
\(502\) −8640.00 −0.768171
\(503\) 11088.0 0.982882 0.491441 0.870911i \(-0.336470\pi\)
0.491441 + 0.870911i \(0.336470\pi\)
\(504\) 0 0
\(505\) 11736.0 1.03415
\(506\) −2700.00 −0.237213
\(507\) 0 0
\(508\) 1784.00 0.155811
\(509\) 16854.0 1.46766 0.733831 0.679332i \(-0.237730\pi\)
0.733831 + 0.679332i \(0.237730\pi\)
\(510\) 0 0
\(511\) −3010.00 −0.260576
\(512\) −8733.00 −0.753804
\(513\) 0 0
\(514\) −6570.00 −0.563794
\(515\) 7860.00 0.672530
\(516\) 0 0
\(517\) 3816.00 0.324618
\(518\) 4578.00 0.388312
\(519\) 0 0
\(520\) −15372.0 −1.29636
\(521\) 3195.00 0.268667 0.134333 0.990936i \(-0.457111\pi\)
0.134333 + 0.990936i \(0.457111\pi\)
\(522\) 0 0
\(523\) 920.000 0.0769193 0.0384596 0.999260i \(-0.487755\pi\)
0.0384596 + 0.999260i \(0.487755\pi\)
\(524\) −1749.00 −0.145812
\(525\) 0 0
\(526\) −5607.00 −0.464785
\(527\) −30771.0 −2.54346
\(528\) 0 0
\(529\) −6542.00 −0.537684
\(530\) 16524.0 1.35426
\(531\) 0 0
\(532\) 14.0000 0.00114093
\(533\) 15006.0 1.21948
\(534\) 0 0
\(535\) 1296.00 0.104731
\(536\) −10059.0 −0.810602
\(537\) 0 0
\(538\) −6804.00 −0.545244
\(539\) 588.000 0.0469888
\(540\) 0 0
\(541\) −7180.00 −0.570596 −0.285298 0.958439i \(-0.592093\pi\)
−0.285298 + 0.958439i \(0.592093\pi\)
\(542\) 6081.00 0.481921
\(543\) 0 0
\(544\) 5265.00 0.414954
\(545\) −3264.00 −0.256540
\(546\) 0 0
\(547\) 11612.0 0.907666 0.453833 0.891087i \(-0.350056\pi\)
0.453833 + 0.891087i \(0.350056\pi\)
\(548\) −1374.00 −0.107107
\(549\) 0 0
\(550\) 684.000 0.0530288
\(551\) 6.00000 0.000463899 0
\(552\) 0 0
\(553\) −4522.00 −0.347731
\(554\) 11832.0 0.907389
\(555\) 0 0
\(556\) 2450.00 0.186876
\(557\) 9819.00 0.746938 0.373469 0.927643i \(-0.378168\pi\)
0.373469 + 0.927643i \(0.378168\pi\)
\(558\) 0 0
\(559\) −31415.0 −2.37695
\(560\) 5964.00 0.450045
\(561\) 0 0
\(562\) 13806.0 1.03625
\(563\) 15801.0 1.18283 0.591415 0.806368i \(-0.298570\pi\)
0.591415 + 0.806368i \(0.298570\pi\)
\(564\) 0 0
\(565\) −27072.0 −2.01580
\(566\) −9660.00 −0.717385
\(567\) 0 0
\(568\) 2457.00 0.181503
\(569\) −8340.00 −0.614466 −0.307233 0.951634i \(-0.599403\pi\)
−0.307233 + 0.951634i \(0.599403\pi\)
\(570\) 0 0
\(571\) 1073.00 0.0786404 0.0393202 0.999227i \(-0.487481\pi\)
0.0393202 + 0.999227i \(0.487481\pi\)
\(572\) −732.000 −0.0535078
\(573\) 0 0
\(574\) −5166.00 −0.375653
\(575\) −1425.00 −0.103351
\(576\) 0 0
\(577\) 22466.0 1.62092 0.810461 0.585793i \(-0.199217\pi\)
0.810461 + 0.585793i \(0.199217\pi\)
\(578\) 26328.0 1.89464
\(579\) 0 0
\(580\) −36.0000 −0.00257727
\(581\) −2436.00 −0.173945
\(582\) 0 0
\(583\) −5508.00 −0.391283
\(584\) 9030.00 0.639836
\(585\) 0 0
\(586\) 20430.0 1.44020
\(587\) −3219.00 −0.226341 −0.113171 0.993576i \(-0.536101\pi\)
−0.113171 + 0.993576i \(0.536101\pi\)
\(588\) 0 0
\(589\) 526.000 0.0367970
\(590\) 9180.00 0.640567
\(591\) 0 0
\(592\) −15478.0 −1.07456
\(593\) −9342.00 −0.646931 −0.323465 0.946240i \(-0.604848\pi\)
−0.323465 + 0.946240i \(0.604848\pi\)
\(594\) 0 0
\(595\) 9828.00 0.677158
\(596\) −921.000 −0.0632980
\(597\) 0 0
\(598\) 13725.0 0.938557
\(599\) 6927.00 0.472503 0.236252 0.971692i \(-0.424081\pi\)
0.236252 + 0.971692i \(0.424081\pi\)
\(600\) 0 0
\(601\) −15172.0 −1.02975 −0.514874 0.857266i \(-0.672161\pi\)
−0.514874 + 0.857266i \(0.672161\pi\)
\(602\) 10815.0 0.732203
\(603\) 0 0
\(604\) 650.000 0.0437883
\(605\) 14244.0 0.957192
\(606\) 0 0
\(607\) −4921.00 −0.329056 −0.164528 0.986372i \(-0.552610\pi\)
−0.164528 + 0.986372i \(0.552610\pi\)
\(608\) −90.0000 −0.00600326
\(609\) 0 0
\(610\) 31032.0 2.05975
\(611\) −19398.0 −1.28438
\(612\) 0 0
\(613\) 218.000 0.0143637 0.00718184 0.999974i \(-0.497714\pi\)
0.00718184 + 0.999974i \(0.497714\pi\)
\(614\) 2274.00 0.149464
\(615\) 0 0
\(616\) −1764.00 −0.115379
\(617\) −22350.0 −1.45831 −0.729155 0.684349i \(-0.760087\pi\)
−0.729155 + 0.684349i \(0.760087\pi\)
\(618\) 0 0
\(619\) 9290.00 0.603226 0.301613 0.953431i \(-0.402475\pi\)
0.301613 + 0.953431i \(0.402475\pi\)
\(620\) −3156.00 −0.204432
\(621\) 0 0
\(622\) −28440.0 −1.83334
\(623\) −4095.00 −0.263343
\(624\) 0 0
\(625\) −17639.0 −1.12890
\(626\) 10050.0 0.641659
\(627\) 0 0
\(628\) −871.000 −0.0553450
\(629\) −25506.0 −1.61684
\(630\) 0 0
\(631\) 13988.0 0.882494 0.441247 0.897386i \(-0.354536\pi\)
0.441247 + 0.897386i \(0.354536\pi\)
\(632\) 13566.0 0.853839
\(633\) 0 0
\(634\) 1278.00 0.0800566
\(635\) −21408.0 −1.33787
\(636\) 0 0
\(637\) −2989.00 −0.185916
\(638\) 108.000 0.00670182
\(639\) 0 0
\(640\) −19908.0 −1.22958
\(641\) 11388.0 0.701714 0.350857 0.936429i \(-0.385890\pi\)
0.350857 + 0.936429i \(0.385890\pi\)
\(642\) 0 0
\(643\) 9956.00 0.610616 0.305308 0.952254i \(-0.401241\pi\)
0.305308 + 0.952254i \(0.401241\pi\)
\(644\) −525.000 −0.0321241
\(645\) 0 0
\(646\) −702.000 −0.0427551
\(647\) 7830.00 0.475779 0.237890 0.971292i \(-0.423544\pi\)
0.237890 + 0.971292i \(0.423544\pi\)
\(648\) 0 0
\(649\) −3060.00 −0.185078
\(650\) −3477.00 −0.209814
\(651\) 0 0
\(652\) −3697.00 −0.222064
\(653\) 3633.00 0.217719 0.108859 0.994057i \(-0.465280\pi\)
0.108859 + 0.994057i \(0.465280\pi\)
\(654\) 0 0
\(655\) 20988.0 1.25201
\(656\) 17466.0 1.03953
\(657\) 0 0
\(658\) 6678.00 0.395647
\(659\) −6036.00 −0.356797 −0.178398 0.983958i \(-0.557092\pi\)
−0.178398 + 0.983958i \(0.557092\pi\)
\(660\) 0 0
\(661\) −5110.00 −0.300690 −0.150345 0.988634i \(-0.548038\pi\)
−0.150345 + 0.988634i \(0.548038\pi\)
\(662\) −9309.00 −0.546533
\(663\) 0 0
\(664\) 7308.00 0.427117
\(665\) −168.000 −0.00979663
\(666\) 0 0
\(667\) −225.000 −0.0130615
\(668\) 1824.00 0.105648
\(669\) 0 0
\(670\) −17244.0 −0.994319
\(671\) −10344.0 −0.595120
\(672\) 0 0
\(673\) −12553.0 −0.718993 −0.359497 0.933146i \(-0.617052\pi\)
−0.359497 + 0.933146i \(0.617052\pi\)
\(674\) −35175.0 −2.01022
\(675\) 0 0
\(676\) 1524.00 0.0867091
\(677\) 22872.0 1.29844 0.649219 0.760602i \(-0.275096\pi\)
0.649219 + 0.760602i \(0.275096\pi\)
\(678\) 0 0
\(679\) −2632.00 −0.148758
\(680\) −29484.0 −1.66273
\(681\) 0 0
\(682\) 9468.00 0.531596
\(683\) −21672.0 −1.21414 −0.607069 0.794649i \(-0.707655\pi\)
−0.607069 + 0.794649i \(0.707655\pi\)
\(684\) 0 0
\(685\) 16488.0 0.919670
\(686\) 1029.00 0.0572703
\(687\) 0 0
\(688\) −36565.0 −2.02620
\(689\) 27999.0 1.54815
\(690\) 0 0
\(691\) 17120.0 0.942512 0.471256 0.881997i \(-0.343801\pi\)
0.471256 + 0.881997i \(0.343801\pi\)
\(692\) −744.000 −0.0408709
\(693\) 0 0
\(694\) −4518.00 −0.247120
\(695\) −29400.0 −1.60461
\(696\) 0 0
\(697\) 28782.0 1.56413
\(698\) 30867.0 1.67383
\(699\) 0 0
\(700\) 133.000 0.00718132
\(701\) 5562.00 0.299677 0.149839 0.988710i \(-0.452125\pi\)
0.149839 + 0.988710i \(0.452125\pi\)
\(702\) 0 0
\(703\) 436.000 0.0233913
\(704\) 5196.00 0.278170
\(705\) 0 0
\(706\) 38079.0 2.02992
\(707\) −6846.00 −0.364173
\(708\) 0 0
\(709\) 16256.0 0.861082 0.430541 0.902571i \(-0.358323\pi\)
0.430541 + 0.902571i \(0.358323\pi\)
\(710\) 4212.00 0.222639
\(711\) 0 0
\(712\) 12285.0 0.646629
\(713\) −19725.0 −1.03605
\(714\) 0 0
\(715\) 8784.00 0.459445
\(716\) −4572.00 −0.238636
\(717\) 0 0
\(718\) 15093.0 0.784493
\(719\) −20016.0 −1.03821 −0.519104 0.854711i \(-0.673734\pi\)
−0.519104 + 0.854711i \(0.673734\pi\)
\(720\) 0 0
\(721\) −4585.00 −0.236830
\(722\) −20565.0 −1.06004
\(723\) 0 0
\(724\) −3481.00 −0.178688
\(725\) 57.0000 0.00291990
\(726\) 0 0
\(727\) 3539.00 0.180542 0.0902711 0.995917i \(-0.471227\pi\)
0.0902711 + 0.995917i \(0.471227\pi\)
\(728\) 8967.00 0.456510
\(729\) 0 0
\(730\) 15480.0 0.784850
\(731\) −60255.0 −3.04872
\(732\) 0 0
\(733\) −19663.0 −0.990818 −0.495409 0.868660i \(-0.664982\pi\)
−0.495409 + 0.868660i \(0.664982\pi\)
\(734\) 28437.0 1.43001
\(735\) 0 0
\(736\) 3375.00 0.169027
\(737\) 5748.00 0.287287
\(738\) 0 0
\(739\) −17116.0 −0.851992 −0.425996 0.904725i \(-0.640076\pi\)
−0.425996 + 0.904725i \(0.640076\pi\)
\(740\) −2616.00 −0.129954
\(741\) 0 0
\(742\) −9639.00 −0.476899
\(743\) 31641.0 1.56231 0.781155 0.624338i \(-0.214631\pi\)
0.781155 + 0.624338i \(0.214631\pi\)
\(744\) 0 0
\(745\) 11052.0 0.543509
\(746\) −25698.0 −1.26122
\(747\) 0 0
\(748\) −1404.00 −0.0686301
\(749\) −756.000 −0.0368807
\(750\) 0 0
\(751\) −17314.0 −0.841274 −0.420637 0.907229i \(-0.638193\pi\)
−0.420637 + 0.907229i \(0.638193\pi\)
\(752\) −22578.0 −1.09486
\(753\) 0 0
\(754\) −549.000 −0.0265164
\(755\) −7800.00 −0.375988
\(756\) 0 0
\(757\) 12476.0 0.599007 0.299503 0.954095i \(-0.403179\pi\)
0.299503 + 0.954095i \(0.403179\pi\)
\(758\) −6960.00 −0.333507
\(759\) 0 0
\(760\) 504.000 0.0240553
\(761\) −21855.0 −1.04106 −0.520528 0.853845i \(-0.674265\pi\)
−0.520528 + 0.853845i \(0.674265\pi\)
\(762\) 0 0
\(763\) 1904.00 0.0903400
\(764\) 3756.00 0.177863
\(765\) 0 0
\(766\) −3276.00 −0.154526
\(767\) 15555.0 0.732280
\(768\) 0 0
\(769\) −5812.00 −0.272544 −0.136272 0.990671i \(-0.543512\pi\)
−0.136272 + 0.990671i \(0.543512\pi\)
\(770\) −3024.00 −0.141529
\(771\) 0 0
\(772\) −4867.00 −0.226900
\(773\) −252.000 −0.0117255 −0.00586275 0.999983i \(-0.501866\pi\)
−0.00586275 + 0.999983i \(0.501866\pi\)
\(774\) 0 0
\(775\) 4997.00 0.231610
\(776\) 7896.00 0.365270
\(777\) 0 0
\(778\) −37710.0 −1.73775
\(779\) −492.000 −0.0226287
\(780\) 0 0
\(781\) −1404.00 −0.0643266
\(782\) 26325.0 1.20381
\(783\) 0 0
\(784\) −3479.00 −0.158482
\(785\) 10452.0 0.475220
\(786\) 0 0
\(787\) −12886.0 −0.583655 −0.291827 0.956471i \(-0.594263\pi\)
−0.291827 + 0.956471i \(0.594263\pi\)
\(788\) −18.0000 −0.000813735 0
\(789\) 0 0
\(790\) 23256.0 1.04736
\(791\) 15792.0 0.709860
\(792\) 0 0
\(793\) 52582.0 2.35466
\(794\) −20406.0 −0.912068
\(795\) 0 0
\(796\) −1807.00 −0.0804616
\(797\) −39342.0 −1.74851 −0.874257 0.485464i \(-0.838651\pi\)
−0.874257 + 0.485464i \(0.838651\pi\)
\(798\) 0 0
\(799\) −37206.0 −1.64738
\(800\) −855.000 −0.0377860
\(801\) 0 0
\(802\) −25578.0 −1.12617
\(803\) −5160.00 −0.226765
\(804\) 0 0
\(805\) 6300.00 0.275833
\(806\) −48129.0 −2.10331
\(807\) 0 0
\(808\) 20538.0 0.894213
\(809\) 33264.0 1.44561 0.722806 0.691051i \(-0.242852\pi\)
0.722806 + 0.691051i \(0.242852\pi\)
\(810\) 0 0
\(811\) −10258.0 −0.444152 −0.222076 0.975029i \(-0.571283\pi\)
−0.222076 + 0.975029i \(0.571283\pi\)
\(812\) 21.0000 0.000907581 0
\(813\) 0 0
\(814\) 7848.00 0.337927
\(815\) 44364.0 1.90675
\(816\) 0 0
\(817\) 1030.00 0.0441067
\(818\) 40398.0 1.72675
\(819\) 0 0
\(820\) 2952.00 0.125717
\(821\) −33063.0 −1.40549 −0.702745 0.711442i \(-0.748042\pi\)
−0.702745 + 0.711442i \(0.748042\pi\)
\(822\) 0 0
\(823\) −13102.0 −0.554930 −0.277465 0.960736i \(-0.589494\pi\)
−0.277465 + 0.960736i \(0.589494\pi\)
\(824\) 13755.0 0.581527
\(825\) 0 0
\(826\) −5355.00 −0.225574
\(827\) 25182.0 1.05884 0.529422 0.848359i \(-0.322409\pi\)
0.529422 + 0.848359i \(0.322409\pi\)
\(828\) 0 0
\(829\) −17170.0 −0.719347 −0.359674 0.933078i \(-0.617112\pi\)
−0.359674 + 0.933078i \(0.617112\pi\)
\(830\) 12528.0 0.523920
\(831\) 0 0
\(832\) −26413.0 −1.10061
\(833\) −5733.00 −0.238459
\(834\) 0 0
\(835\) −21888.0 −0.907144
\(836\) 24.0000 0.000992892 0
\(837\) 0 0
\(838\) −46017.0 −1.89693
\(839\) 4458.00 0.183441 0.0917207 0.995785i \(-0.470763\pi\)
0.0917207 + 0.995785i \(0.470763\pi\)
\(840\) 0 0
\(841\) −24380.0 −0.999631
\(842\) −12306.0 −0.503673
\(843\) 0 0
\(844\) −4273.00 −0.174269
\(845\) −18288.0 −0.744528
\(846\) 0 0
\(847\) −8309.00 −0.337073
\(848\) 32589.0 1.31971
\(849\) 0 0
\(850\) −6669.00 −0.269112
\(851\) −16350.0 −0.658603
\(852\) 0 0
\(853\) 39773.0 1.59648 0.798242 0.602336i \(-0.205763\pi\)
0.798242 + 0.602336i \(0.205763\pi\)
\(854\) −18102.0 −0.725337
\(855\) 0 0
\(856\) 2268.00 0.0905592
\(857\) −4191.00 −0.167050 −0.0835250 0.996506i \(-0.526618\pi\)
−0.0835250 + 0.996506i \(0.526618\pi\)
\(858\) 0 0
\(859\) 19946.0 0.792257 0.396128 0.918195i \(-0.370353\pi\)
0.396128 + 0.918195i \(0.370353\pi\)
\(860\) −6180.00 −0.245042
\(861\) 0 0
\(862\) −7776.00 −0.307252
\(863\) 41067.0 1.61986 0.809929 0.586528i \(-0.199506\pi\)
0.809929 + 0.586528i \(0.199506\pi\)
\(864\) 0 0
\(865\) 8928.00 0.350938
\(866\) −28344.0 −1.11220
\(867\) 0 0
\(868\) 1841.00 0.0719903
\(869\) −7752.00 −0.302611
\(870\) 0 0
\(871\) −29219.0 −1.13668
\(872\) −5712.00 −0.221827
\(873\) 0 0
\(874\) −450.000 −0.0174159
\(875\) 8904.00 0.344012
\(876\) 0 0
\(877\) −32056.0 −1.23427 −0.617135 0.786858i \(-0.711707\pi\)
−0.617135 + 0.786858i \(0.711707\pi\)
\(878\) 34107.0 1.31100
\(879\) 0 0
\(880\) 10224.0 0.391649
\(881\) −1899.00 −0.0726208 −0.0363104 0.999341i \(-0.511561\pi\)
−0.0363104 + 0.999341i \(0.511561\pi\)
\(882\) 0 0
\(883\) 8021.00 0.305694 0.152847 0.988250i \(-0.451156\pi\)
0.152847 + 0.988250i \(0.451156\pi\)
\(884\) 7137.00 0.271542
\(885\) 0 0
\(886\) −32526.0 −1.23333
\(887\) 6540.00 0.247567 0.123783 0.992309i \(-0.460497\pi\)
0.123783 + 0.992309i \(0.460497\pi\)
\(888\) 0 0
\(889\) 12488.0 0.471129
\(890\) 21060.0 0.793183
\(891\) 0 0
\(892\) 2648.00 0.0993964
\(893\) 636.000 0.0238331
\(894\) 0 0
\(895\) 54864.0 2.04905
\(896\) 11613.0 0.432995
\(897\) 0 0
\(898\) 39420.0 1.46488
\(899\) 789.000 0.0292710
\(900\) 0 0
\(901\) 53703.0 1.98569
\(902\) −8856.00 −0.326910
\(903\) 0 0
\(904\) −47376.0 −1.74303
\(905\) 41772.0 1.53431
\(906\) 0 0
\(907\) 5780.00 0.211601 0.105800 0.994387i \(-0.466260\pi\)
0.105800 + 0.994387i \(0.466260\pi\)
\(908\) 5493.00 0.200762
\(909\) 0 0
\(910\) 15372.0 0.559975
\(911\) 27480.0 0.999400 0.499700 0.866199i \(-0.333444\pi\)
0.499700 + 0.866199i \(0.333444\pi\)
\(912\) 0 0
\(913\) −4176.00 −0.151375
\(914\) 4137.00 0.149715
\(915\) 0 0
\(916\) −790.000 −0.0284960
\(917\) −12243.0 −0.440894
\(918\) 0 0
\(919\) −7450.00 −0.267413 −0.133707 0.991021i \(-0.542688\pi\)
−0.133707 + 0.991021i \(0.542688\pi\)
\(920\) −18900.0 −0.677298
\(921\) 0 0
\(922\) 2034.00 0.0726532
\(923\) 7137.00 0.254515
\(924\) 0 0
\(925\) 4142.00 0.147230
\(926\) 47202.0 1.67511
\(927\) 0 0
\(928\) −135.000 −0.00477542
\(929\) −7602.00 −0.268475 −0.134238 0.990949i \(-0.542859\pi\)
−0.134238 + 0.990949i \(0.542859\pi\)
\(930\) 0 0
\(931\) 98.0000 0.00344986
\(932\) 6228.00 0.218889
\(933\) 0 0
\(934\) −22248.0 −0.779418
\(935\) 16848.0 0.589293
\(936\) 0 0
\(937\) −6316.00 −0.220208 −0.110104 0.993920i \(-0.535118\pi\)
−0.110104 + 0.993920i \(0.535118\pi\)
\(938\) 10059.0 0.350147
\(939\) 0 0
\(940\) −3816.00 −0.132409
\(941\) 25170.0 0.871964 0.435982 0.899955i \(-0.356401\pi\)
0.435982 + 0.899955i \(0.356401\pi\)
\(942\) 0 0
\(943\) 18450.0 0.637131
\(944\) 18105.0 0.624224
\(945\) 0 0
\(946\) 18540.0 0.637196
\(947\) 20874.0 0.716277 0.358138 0.933669i \(-0.383412\pi\)
0.358138 + 0.933669i \(0.383412\pi\)
\(948\) 0 0
\(949\) 26230.0 0.897220
\(950\) 114.000 0.00389331
\(951\) 0 0
\(952\) 17199.0 0.585528
\(953\) −31824.0 −1.08172 −0.540861 0.841112i \(-0.681901\pi\)
−0.540861 + 0.841112i \(0.681901\pi\)
\(954\) 0 0
\(955\) −45072.0 −1.52722
\(956\) 4812.00 0.162794
\(957\) 0 0
\(958\) −3744.00 −0.126266
\(959\) −9618.00 −0.323860
\(960\) 0 0
\(961\) 39378.0 1.32181
\(962\) −39894.0 −1.33704
\(963\) 0 0
\(964\) 1622.00 0.0541920
\(965\) 58404.0 1.94828
\(966\) 0 0
\(967\) 7040.00 0.234117 0.117058 0.993125i \(-0.462653\pi\)
0.117058 + 0.993125i \(0.462653\pi\)
\(968\) 24927.0 0.827670
\(969\) 0 0
\(970\) 13536.0 0.448056
\(971\) −27189.0 −0.898596 −0.449298 0.893382i \(-0.648326\pi\)
−0.449298 + 0.893382i \(0.648326\pi\)
\(972\) 0 0
\(973\) 17150.0 0.565060
\(974\) −62850.0 −2.06760
\(975\) 0 0
\(976\) 61202.0 2.00720
\(977\) −33438.0 −1.09496 −0.547480 0.836819i \(-0.684413\pi\)
−0.547480 + 0.836819i \(0.684413\pi\)
\(978\) 0 0
\(979\) −7020.00 −0.229173
\(980\) −588.000 −0.0191663
\(981\) 0 0
\(982\) −2304.00 −0.0748713
\(983\) 16212.0 0.526025 0.263012 0.964792i \(-0.415284\pi\)
0.263012 + 0.964792i \(0.415284\pi\)
\(984\) 0 0
\(985\) 216.000 0.00698714
\(986\) −1053.00 −0.0340105
\(987\) 0 0
\(988\) −122.000 −0.00392848
\(989\) −38625.0 −1.24186
\(990\) 0 0
\(991\) 24194.0 0.775527 0.387764 0.921759i \(-0.373248\pi\)
0.387764 + 0.921759i \(0.373248\pi\)
\(992\) −11835.0 −0.378792
\(993\) 0 0
\(994\) −2457.00 −0.0784017
\(995\) 21684.0 0.690883
\(996\) 0 0
\(997\) −27565.0 −0.875619 −0.437810 0.899068i \(-0.644246\pi\)
−0.437810 + 0.899068i \(0.644246\pi\)
\(998\) 14316.0 0.454073
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 189.4.a.d.1.1 yes 1
3.2 odd 2 189.4.a.a.1.1 1
7.6 odd 2 1323.4.a.n.1.1 1
21.20 even 2 1323.4.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.4.a.a.1.1 1 3.2 odd 2
189.4.a.d.1.1 yes 1 1.1 even 1 trivial
1323.4.a.a.1.1 1 21.20 even 2
1323.4.a.n.1.1 1 7.6 odd 2