Properties

Label 1089.4.a.c.1.1
Level $1089$
Weight $4$
Character 1089.1
Self dual yes
Analytic conductor $64.253$
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] = [1089,4,Mod(1,1089)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1089, 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("1089.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1089.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: \(64.2530799963\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 363)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1089.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000 q^{2} +1.00000 q^{4} +12.0000 q^{5} +12.0000 q^{7} +21.0000 q^{8} +O(q^{10})\) \(q-3.00000 q^{2} +1.00000 q^{4} +12.0000 q^{5} +12.0000 q^{7} +21.0000 q^{8} -36.0000 q^{10} -66.0000 q^{13} -36.0000 q^{14} -71.0000 q^{16} +114.000 q^{17} +42.0000 q^{19} +12.0000 q^{20} -18.0000 q^{23} +19.0000 q^{25} +198.000 q^{26} +12.0000 q^{28} -186.000 q^{29} -308.000 q^{31} +45.0000 q^{32} -342.000 q^{34} +144.000 q^{35} -146.000 q^{37} -126.000 q^{38} +252.000 q^{40} -42.0000 q^{41} -366.000 q^{43} +54.0000 q^{46} -618.000 q^{47} -199.000 q^{49} -57.0000 q^{50} -66.0000 q^{52} +408.000 q^{53} +252.000 q^{56} +558.000 q^{58} +132.000 q^{59} +630.000 q^{61} +924.000 q^{62} +433.000 q^{64} -792.000 q^{65} -452.000 q^{67} +114.000 q^{68} -432.000 q^{70} +282.000 q^{71} +684.000 q^{73} +438.000 q^{74} +42.0000 q^{76} +1272.00 q^{79} -852.000 q^{80} +126.000 q^{82} +432.000 q^{83} +1368.00 q^{85} +1098.00 q^{86} -954.000 q^{89} -792.000 q^{91} -18.0000 q^{92} +1854.00 q^{94} +504.000 q^{95} +326.000 q^{97} +597.000 q^{98} +O(q^{100})\)

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 \(-0.677932\pi\)
−0.530330 + 0.847791i \(0.677932\pi\)
\(3\) 0 0
\(4\) 1.00000 0.125000
\(5\) 12.0000 1.07331 0.536656 0.843801i \(-0.319687\pi\)
0.536656 + 0.843801i \(0.319687\pi\)
\(6\) 0 0
\(7\) 12.0000 0.647939 0.323970 0.946068i \(-0.394982\pi\)
0.323970 + 0.946068i \(0.394982\pi\)
\(8\) 21.0000 0.928078
\(9\) 0 0
\(10\) −36.0000 −1.13842
\(11\) 0 0
\(12\) 0 0
\(13\) −66.0000 −1.40809 −0.704043 0.710158i \(-0.748624\pi\)
−0.704043 + 0.710158i \(0.748624\pi\)
\(14\) −36.0000 −0.687243
\(15\) 0 0
\(16\) −71.0000 −1.10938
\(17\) 114.000 1.62642 0.813208 0.581974i \(-0.197719\pi\)
0.813208 + 0.581974i \(0.197719\pi\)
\(18\) 0 0
\(19\) 42.0000 0.507130 0.253565 0.967318i \(-0.418397\pi\)
0.253565 + 0.967318i \(0.418397\pi\)
\(20\) 12.0000 0.134164
\(21\) 0 0
\(22\) 0 0
\(23\) −18.0000 −0.163185 −0.0815926 0.996666i \(-0.526001\pi\)
−0.0815926 + 0.996666i \(0.526001\pi\)
\(24\) 0 0
\(25\) 19.0000 0.152000
\(26\) 198.000 1.49350
\(27\) 0 0
\(28\) 12.0000 0.0809924
\(29\) −186.000 −1.19101 −0.595506 0.803351i \(-0.703048\pi\)
−0.595506 + 0.803351i \(0.703048\pi\)
\(30\) 0 0
\(31\) −308.000 −1.78447 −0.892233 0.451576i \(-0.850862\pi\)
−0.892233 + 0.451576i \(0.850862\pi\)
\(32\) 45.0000 0.248592
\(33\) 0 0
\(34\) −342.000 −1.72507
\(35\) 144.000 0.695441
\(36\) 0 0
\(37\) −146.000 −0.648710 −0.324355 0.945936i \(-0.605147\pi\)
−0.324355 + 0.945936i \(0.605147\pi\)
\(38\) −126.000 −0.537892
\(39\) 0 0
\(40\) 252.000 0.996117
\(41\) −42.0000 −0.159983 −0.0799914 0.996796i \(-0.525489\pi\)
−0.0799914 + 0.996796i \(0.525489\pi\)
\(42\) 0 0
\(43\) −366.000 −1.29801 −0.649006 0.760784i \(-0.724815\pi\)
−0.649006 + 0.760784i \(0.724815\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 54.0000 0.173084
\(47\) −618.000 −1.91797 −0.958985 0.283458i \(-0.908518\pi\)
−0.958985 + 0.283458i \(0.908518\pi\)
\(48\) 0 0
\(49\) −199.000 −0.580175
\(50\) −57.0000 −0.161220
\(51\) 0 0
\(52\) −66.0000 −0.176011
\(53\) 408.000 1.05742 0.528709 0.848803i \(-0.322676\pi\)
0.528709 + 0.848803i \(0.322676\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 252.000 0.601338
\(57\) 0 0
\(58\) 558.000 1.26326
\(59\) 132.000 0.291270 0.145635 0.989338i \(-0.453477\pi\)
0.145635 + 0.989338i \(0.453477\pi\)
\(60\) 0 0
\(61\) 630.000 1.32235 0.661174 0.750233i \(-0.270058\pi\)
0.661174 + 0.750233i \(0.270058\pi\)
\(62\) 924.000 1.89271
\(63\) 0 0
\(64\) 433.000 0.845703
\(65\) −792.000 −1.51132
\(66\) 0 0
\(67\) −452.000 −0.824188 −0.412094 0.911141i \(-0.635202\pi\)
−0.412094 + 0.911141i \(0.635202\pi\)
\(68\) 114.000 0.203302
\(69\) 0 0
\(70\) −432.000 −0.737627
\(71\) 282.000 0.471370 0.235685 0.971830i \(-0.424267\pi\)
0.235685 + 0.971830i \(0.424267\pi\)
\(72\) 0 0
\(73\) 684.000 1.09666 0.548330 0.836262i \(-0.315264\pi\)
0.548330 + 0.836262i \(0.315264\pi\)
\(74\) 438.000 0.688060
\(75\) 0 0
\(76\) 42.0000 0.0633912
\(77\) 0 0
\(78\) 0 0
\(79\) 1272.00 1.81153 0.905767 0.423776i \(-0.139296\pi\)
0.905767 + 0.423776i \(0.139296\pi\)
\(80\) −852.000 −1.19071
\(81\) 0 0
\(82\) 126.000 0.169687
\(83\) 432.000 0.571303 0.285652 0.958334i \(-0.407790\pi\)
0.285652 + 0.958334i \(0.407790\pi\)
\(84\) 0 0
\(85\) 1368.00 1.74565
\(86\) 1098.00 1.37675
\(87\) 0 0
\(88\) 0 0
\(89\) −954.000 −1.13622 −0.568111 0.822952i \(-0.692326\pi\)
−0.568111 + 0.822952i \(0.692326\pi\)
\(90\) 0 0
\(91\) −792.000 −0.912353
\(92\) −18.0000 −0.0203981
\(93\) 0 0
\(94\) 1854.00 2.03431
\(95\) 504.000 0.544309
\(96\) 0 0
\(97\) 326.000 0.341240 0.170620 0.985337i \(-0.445423\pi\)
0.170620 + 0.985337i \(0.445423\pi\)
\(98\) 597.000 0.615368
\(99\) 0 0
\(100\) 19.0000 0.0190000
\(101\) −1326.00 −1.30636 −0.653178 0.757205i \(-0.726565\pi\)
−0.653178 + 0.757205i \(0.726565\pi\)
\(102\) 0 0
\(103\) −1564.00 −1.49617 −0.748085 0.663603i \(-0.769026\pi\)
−0.748085 + 0.663603i \(0.769026\pi\)
\(104\) −1386.00 −1.30681
\(105\) 0 0
\(106\) −1224.00 −1.12156
\(107\) 84.0000 0.0758933 0.0379467 0.999280i \(-0.487918\pi\)
0.0379467 + 0.999280i \(0.487918\pi\)
\(108\) 0 0
\(109\) −1122.00 −0.985946 −0.492973 0.870045i \(-0.664090\pi\)
−0.492973 + 0.870045i \(0.664090\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −852.000 −0.718807
\(113\) 906.000 0.754242 0.377121 0.926164i \(-0.376914\pi\)
0.377121 + 0.926164i \(0.376914\pi\)
\(114\) 0 0
\(115\) −216.000 −0.175149
\(116\) −186.000 −0.148876
\(117\) 0 0
\(118\) −396.000 −0.308939
\(119\) 1368.00 1.05382
\(120\) 0 0
\(121\) 0 0
\(122\) −1890.00 −1.40256
\(123\) 0 0
\(124\) −308.000 −0.223058
\(125\) −1272.00 −0.910169
\(126\) 0 0
\(127\) −1992.00 −1.39182 −0.695911 0.718128i \(-0.744999\pi\)
−0.695911 + 0.718128i \(0.744999\pi\)
\(128\) −1659.00 −1.14560
\(129\) 0 0
\(130\) 2376.00 1.60299
\(131\) −1068.00 −0.712302 −0.356151 0.934428i \(-0.615911\pi\)
−0.356151 + 0.934428i \(0.615911\pi\)
\(132\) 0 0
\(133\) 504.000 0.328589
\(134\) 1356.00 0.874183
\(135\) 0 0
\(136\) 2394.00 1.50944
\(137\) 462.000 0.288112 0.144056 0.989570i \(-0.453985\pi\)
0.144056 + 0.989570i \(0.453985\pi\)
\(138\) 0 0
\(139\) 2274.00 1.38761 0.693806 0.720162i \(-0.255932\pi\)
0.693806 + 0.720162i \(0.255932\pi\)
\(140\) 144.000 0.0869302
\(141\) 0 0
\(142\) −846.000 −0.499963
\(143\) 0 0
\(144\) 0 0
\(145\) −2232.00 −1.27833
\(146\) −2052.00 −1.16318
\(147\) 0 0
\(148\) −146.000 −0.0810887
\(149\) 750.000 0.412365 0.206183 0.978514i \(-0.433896\pi\)
0.206183 + 0.978514i \(0.433896\pi\)
\(150\) 0 0
\(151\) −720.000 −0.388032 −0.194016 0.980998i \(-0.562151\pi\)
−0.194016 + 0.980998i \(0.562151\pi\)
\(152\) 882.000 0.470656
\(153\) 0 0
\(154\) 0 0
\(155\) −3696.00 −1.91529
\(156\) 0 0
\(157\) −3206.00 −1.62972 −0.814862 0.579655i \(-0.803187\pi\)
−0.814862 + 0.579655i \(0.803187\pi\)
\(158\) −3816.00 −1.92142
\(159\) 0 0
\(160\) 540.000 0.266817
\(161\) −216.000 −0.105734
\(162\) 0 0
\(163\) 740.000 0.355591 0.177795 0.984067i \(-0.443104\pi\)
0.177795 + 0.984067i \(0.443104\pi\)
\(164\) −42.0000 −0.0199979
\(165\) 0 0
\(166\) −1296.00 −0.605958
\(167\) 1968.00 0.911907 0.455953 0.890004i \(-0.349298\pi\)
0.455953 + 0.890004i \(0.349298\pi\)
\(168\) 0 0
\(169\) 2159.00 0.982704
\(170\) −4104.00 −1.85154
\(171\) 0 0
\(172\) −366.000 −0.162251
\(173\) −2898.00 −1.27359 −0.636794 0.771034i \(-0.719740\pi\)
−0.636794 + 0.771034i \(0.719740\pi\)
\(174\) 0 0
\(175\) 228.000 0.0984867
\(176\) 0 0
\(177\) 0 0
\(178\) 2862.00 1.20515
\(179\) −120.000 −0.0501074 −0.0250537 0.999686i \(-0.507976\pi\)
−0.0250537 + 0.999686i \(0.507976\pi\)
\(180\) 0 0
\(181\) −1150.00 −0.472259 −0.236129 0.971722i \(-0.575879\pi\)
−0.236129 + 0.971722i \(0.575879\pi\)
\(182\) 2376.00 0.967697
\(183\) 0 0
\(184\) −378.000 −0.151449
\(185\) −1752.00 −0.696268
\(186\) 0 0
\(187\) 0 0
\(188\) −618.000 −0.239746
\(189\) 0 0
\(190\) −1512.00 −0.577326
\(191\) 1518.00 0.575071 0.287536 0.957770i \(-0.407164\pi\)
0.287536 + 0.957770i \(0.407164\pi\)
\(192\) 0 0
\(193\) −2052.00 −0.765317 −0.382659 0.923890i \(-0.624991\pi\)
−0.382659 + 0.923890i \(0.624991\pi\)
\(194\) −978.000 −0.361940
\(195\) 0 0
\(196\) −199.000 −0.0725219
\(197\) 3282.00 1.18697 0.593484 0.804846i \(-0.297752\pi\)
0.593484 + 0.804846i \(0.297752\pi\)
\(198\) 0 0
\(199\) −704.000 −0.250780 −0.125390 0.992108i \(-0.540018\pi\)
−0.125390 + 0.992108i \(0.540018\pi\)
\(200\) 399.000 0.141068
\(201\) 0 0
\(202\) 3978.00 1.38560
\(203\) −2232.00 −0.771703
\(204\) 0 0
\(205\) −504.000 −0.171712
\(206\) 4692.00 1.58693
\(207\) 0 0
\(208\) 4686.00 1.56209
\(209\) 0 0
\(210\) 0 0
\(211\) −1986.00 −0.647971 −0.323985 0.946062i \(-0.605023\pi\)
−0.323985 + 0.946062i \(0.605023\pi\)
\(212\) 408.000 0.132177
\(213\) 0 0
\(214\) −252.000 −0.0804970
\(215\) −4392.00 −1.39317
\(216\) 0 0
\(217\) −3696.00 −1.15623
\(218\) 3366.00 1.04575
\(219\) 0 0
\(220\) 0 0
\(221\) −7524.00 −2.29013
\(222\) 0 0
\(223\) −2248.00 −0.675055 −0.337527 0.941316i \(-0.609591\pi\)
−0.337527 + 0.941316i \(0.609591\pi\)
\(224\) 540.000 0.161073
\(225\) 0 0
\(226\) −2718.00 −0.799994
\(227\) 2796.00 0.817520 0.408760 0.912642i \(-0.365961\pi\)
0.408760 + 0.912642i \(0.365961\pi\)
\(228\) 0 0
\(229\) −2198.00 −0.634270 −0.317135 0.948380i \(-0.602721\pi\)
−0.317135 + 0.948380i \(0.602721\pi\)
\(230\) 648.000 0.185773
\(231\) 0 0
\(232\) −3906.00 −1.10535
\(233\) −5802.00 −1.63134 −0.815669 0.578519i \(-0.803631\pi\)
−0.815669 + 0.578519i \(0.803631\pi\)
\(234\) 0 0
\(235\) −7416.00 −2.05858
\(236\) 132.000 0.0364088
\(237\) 0 0
\(238\) −4104.00 −1.11774
\(239\) −1104.00 −0.298794 −0.149397 0.988777i \(-0.547733\pi\)
−0.149397 + 0.988777i \(0.547733\pi\)
\(240\) 0 0
\(241\) 5208.00 1.39202 0.696010 0.718032i \(-0.254957\pi\)
0.696010 + 0.718032i \(0.254957\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 630.000 0.165294
\(245\) −2388.00 −0.622709
\(246\) 0 0
\(247\) −2772.00 −0.714082
\(248\) −6468.00 −1.65612
\(249\) 0 0
\(250\) 3816.00 0.965380
\(251\) 2880.00 0.724239 0.362119 0.932132i \(-0.382053\pi\)
0.362119 + 0.932132i \(0.382053\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 5976.00 1.47625
\(255\) 0 0
\(256\) 1513.00 0.369385
\(257\) −3762.00 −0.913102 −0.456551 0.889697i \(-0.650915\pi\)
−0.456551 + 0.889697i \(0.650915\pi\)
\(258\) 0 0
\(259\) −1752.00 −0.420324
\(260\) −792.000 −0.188914
\(261\) 0 0
\(262\) 3204.00 0.755511
\(263\) 1920.00 0.450161 0.225080 0.974340i \(-0.427736\pi\)
0.225080 + 0.974340i \(0.427736\pi\)
\(264\) 0 0
\(265\) 4896.00 1.13494
\(266\) −1512.00 −0.348521
\(267\) 0 0
\(268\) −452.000 −0.103023
\(269\) 6144.00 1.39259 0.696294 0.717756i \(-0.254831\pi\)
0.696294 + 0.717756i \(0.254831\pi\)
\(270\) 0 0
\(271\) −5904.00 −1.32340 −0.661702 0.749767i \(-0.730166\pi\)
−0.661702 + 0.749767i \(0.730166\pi\)
\(272\) −8094.00 −1.80430
\(273\) 0 0
\(274\) −1386.00 −0.305589
\(275\) 0 0
\(276\) 0 0
\(277\) −3630.00 −0.787385 −0.393692 0.919242i \(-0.628802\pi\)
−0.393692 + 0.919242i \(0.628802\pi\)
\(278\) −6822.00 −1.47179
\(279\) 0 0
\(280\) 3024.00 0.645423
\(281\) −1842.00 −0.391048 −0.195524 0.980699i \(-0.562641\pi\)
−0.195524 + 0.980699i \(0.562641\pi\)
\(282\) 0 0
\(283\) −2382.00 −0.500336 −0.250168 0.968202i \(-0.580486\pi\)
−0.250168 + 0.968202i \(0.580486\pi\)
\(284\) 282.000 0.0589212
\(285\) 0 0
\(286\) 0 0
\(287\) −504.000 −0.103659
\(288\) 0 0
\(289\) 8083.00 1.64523
\(290\) 6696.00 1.35587
\(291\) 0 0
\(292\) 684.000 0.137082
\(293\) −2238.00 −0.446230 −0.223115 0.974792i \(-0.571623\pi\)
−0.223115 + 0.974792i \(0.571623\pi\)
\(294\) 0 0
\(295\) 1584.00 0.312624
\(296\) −3066.00 −0.602053
\(297\) 0 0
\(298\) −2250.00 −0.437379
\(299\) 1188.00 0.229779
\(300\) 0 0
\(301\) −4392.00 −0.841032
\(302\) 2160.00 0.411570
\(303\) 0 0
\(304\) −2982.00 −0.562597
\(305\) 7560.00 1.41929
\(306\) 0 0
\(307\) −558.000 −0.103735 −0.0518677 0.998654i \(-0.516517\pi\)
−0.0518677 + 0.998654i \(0.516517\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 11088.0 2.03147
\(311\) 4062.00 0.740627 0.370313 0.928907i \(-0.379250\pi\)
0.370313 + 0.928907i \(0.379250\pi\)
\(312\) 0 0
\(313\) −3098.00 −0.559455 −0.279727 0.960079i \(-0.590244\pi\)
−0.279727 + 0.960079i \(0.590244\pi\)
\(314\) 9618.00 1.72858
\(315\) 0 0
\(316\) 1272.00 0.226442
\(317\) 3600.00 0.637843 0.318921 0.947781i \(-0.396679\pi\)
0.318921 + 0.947781i \(0.396679\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 5196.00 0.907704
\(321\) 0 0
\(322\) 648.000 0.112148
\(323\) 4788.00 0.824803
\(324\) 0 0
\(325\) −1254.00 −0.214029
\(326\) −2220.00 −0.377161
\(327\) 0 0
\(328\) −882.000 −0.148477
\(329\) −7416.00 −1.24273
\(330\) 0 0
\(331\) −4156.00 −0.690134 −0.345067 0.938578i \(-0.612144\pi\)
−0.345067 + 0.938578i \(0.612144\pi\)
\(332\) 432.000 0.0714129
\(333\) 0 0
\(334\) −5904.00 −0.967223
\(335\) −5424.00 −0.884611
\(336\) 0 0
\(337\) 2748.00 0.444193 0.222097 0.975025i \(-0.428710\pi\)
0.222097 + 0.975025i \(0.428710\pi\)
\(338\) −6477.00 −1.04231
\(339\) 0 0
\(340\) 1368.00 0.218207
\(341\) 0 0
\(342\) 0 0
\(343\) −6504.00 −1.02386
\(344\) −7686.00 −1.20466
\(345\) 0 0
\(346\) 8694.00 1.35084
\(347\) −696.000 −0.107675 −0.0538375 0.998550i \(-0.517145\pi\)
−0.0538375 + 0.998550i \(0.517145\pi\)
\(348\) 0 0
\(349\) −3246.00 −0.497864 −0.248932 0.968521i \(-0.580080\pi\)
−0.248932 + 0.968521i \(0.580080\pi\)
\(350\) −684.000 −0.104461
\(351\) 0 0
\(352\) 0 0
\(353\) 10926.0 1.64740 0.823700 0.567026i \(-0.191906\pi\)
0.823700 + 0.567026i \(0.191906\pi\)
\(354\) 0 0
\(355\) 3384.00 0.505927
\(356\) −954.000 −0.142028
\(357\) 0 0
\(358\) 360.000 0.0531469
\(359\) 1584.00 0.232870 0.116435 0.993198i \(-0.462853\pi\)
0.116435 + 0.993198i \(0.462853\pi\)
\(360\) 0 0
\(361\) −5095.00 −0.742820
\(362\) 3450.00 0.500906
\(363\) 0 0
\(364\) −792.000 −0.114044
\(365\) 8208.00 1.17706
\(366\) 0 0
\(367\) −232.000 −0.0329981 −0.0164990 0.999864i \(-0.505252\pi\)
−0.0164990 + 0.999864i \(0.505252\pi\)
\(368\) 1278.00 0.181034
\(369\) 0 0
\(370\) 5256.00 0.738504
\(371\) 4896.00 0.685142
\(372\) 0 0
\(373\) −10806.0 −1.50004 −0.750018 0.661417i \(-0.769955\pi\)
−0.750018 + 0.661417i \(0.769955\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −12978.0 −1.78002
\(377\) 12276.0 1.67705
\(378\) 0 0
\(379\) 164.000 0.0222272 0.0111136 0.999938i \(-0.496462\pi\)
0.0111136 + 0.999938i \(0.496462\pi\)
\(380\) 504.000 0.0680386
\(381\) 0 0
\(382\) −4554.00 −0.609955
\(383\) −8058.00 −1.07505 −0.537526 0.843247i \(-0.680641\pi\)
−0.537526 + 0.843247i \(0.680641\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 6156.00 0.811741
\(387\) 0 0
\(388\) 326.000 0.0426550
\(389\) 708.000 0.0922803 0.0461401 0.998935i \(-0.485308\pi\)
0.0461401 + 0.998935i \(0.485308\pi\)
\(390\) 0 0
\(391\) −2052.00 −0.265407
\(392\) −4179.00 −0.538447
\(393\) 0 0
\(394\) −9846.00 −1.25897
\(395\) 15264.0 1.94434
\(396\) 0 0
\(397\) −286.000 −0.0361560 −0.0180780 0.999837i \(-0.505755\pi\)
−0.0180780 + 0.999837i \(0.505755\pi\)
\(398\) 2112.00 0.265992
\(399\) 0 0
\(400\) −1349.00 −0.168625
\(401\) 5262.00 0.655291 0.327646 0.944801i \(-0.393745\pi\)
0.327646 + 0.944801i \(0.393745\pi\)
\(402\) 0 0
\(403\) 20328.0 2.51268
\(404\) −1326.00 −0.163294
\(405\) 0 0
\(406\) 6696.00 0.818515
\(407\) 0 0
\(408\) 0 0
\(409\) −2280.00 −0.275645 −0.137822 0.990457i \(-0.544010\pi\)
−0.137822 + 0.990457i \(0.544010\pi\)
\(410\) 1512.00 0.182128
\(411\) 0 0
\(412\) −1564.00 −0.187021
\(413\) 1584.00 0.188725
\(414\) 0 0
\(415\) 5184.00 0.613187
\(416\) −2970.00 −0.350039
\(417\) 0 0
\(418\) 0 0
\(419\) −4272.00 −0.498093 −0.249046 0.968492i \(-0.580117\pi\)
−0.249046 + 0.968492i \(0.580117\pi\)
\(420\) 0 0
\(421\) 10150.0 1.17501 0.587507 0.809219i \(-0.300110\pi\)
0.587507 + 0.809219i \(0.300110\pi\)
\(422\) 5958.00 0.687277
\(423\) 0 0
\(424\) 8568.00 0.981365
\(425\) 2166.00 0.247215
\(426\) 0 0
\(427\) 7560.00 0.856801
\(428\) 84.0000 0.00948667
\(429\) 0 0
\(430\) 13176.0 1.47768
\(431\) −13056.0 −1.45913 −0.729565 0.683911i \(-0.760278\pi\)
−0.729565 + 0.683911i \(0.760278\pi\)
\(432\) 0 0
\(433\) 13682.0 1.51851 0.759255 0.650793i \(-0.225563\pi\)
0.759255 + 0.650793i \(0.225563\pi\)
\(434\) 11088.0 1.22636
\(435\) 0 0
\(436\) −1122.00 −0.123243
\(437\) −756.000 −0.0827560
\(438\) 0 0
\(439\) −7020.00 −0.763203 −0.381602 0.924327i \(-0.624627\pi\)
−0.381602 + 0.924327i \(0.624627\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 22572.0 2.42905
\(443\) 8916.00 0.956235 0.478117 0.878296i \(-0.341319\pi\)
0.478117 + 0.878296i \(0.341319\pi\)
\(444\) 0 0
\(445\) −11448.0 −1.21952
\(446\) 6744.00 0.716004
\(447\) 0 0
\(448\) 5196.00 0.547964
\(449\) −10110.0 −1.06263 −0.531314 0.847175i \(-0.678302\pi\)
−0.531314 + 0.847175i \(0.678302\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 906.000 0.0942802
\(453\) 0 0
\(454\) −8388.00 −0.867111
\(455\) −9504.00 −0.979240
\(456\) 0 0
\(457\) 1824.00 0.186703 0.0933513 0.995633i \(-0.470242\pi\)
0.0933513 + 0.995633i \(0.470242\pi\)
\(458\) 6594.00 0.672745
\(459\) 0 0
\(460\) −216.000 −0.0218936
\(461\) −17118.0 −1.72942 −0.864712 0.502267i \(-0.832499\pi\)
−0.864712 + 0.502267i \(0.832499\pi\)
\(462\) 0 0
\(463\) 596.000 0.0598239 0.0299120 0.999553i \(-0.490477\pi\)
0.0299120 + 0.999553i \(0.490477\pi\)
\(464\) 13206.0 1.32128
\(465\) 0 0
\(466\) 17406.0 1.73029
\(467\) −5856.00 −0.580264 −0.290132 0.956987i \(-0.593699\pi\)
−0.290132 + 0.956987i \(0.593699\pi\)
\(468\) 0 0
\(469\) −5424.00 −0.534024
\(470\) 22248.0 2.18345
\(471\) 0 0
\(472\) 2772.00 0.270321
\(473\) 0 0
\(474\) 0 0
\(475\) 798.000 0.0770837
\(476\) 1368.00 0.131727
\(477\) 0 0
\(478\) 3312.00 0.316919
\(479\) −3264.00 −0.311349 −0.155674 0.987808i \(-0.549755\pi\)
−0.155674 + 0.987808i \(0.549755\pi\)
\(480\) 0 0
\(481\) 9636.00 0.913438
\(482\) −15624.0 −1.47646
\(483\) 0 0
\(484\) 0 0
\(485\) 3912.00 0.366257
\(486\) 0 0
\(487\) −9920.00 −0.923035 −0.461518 0.887131i \(-0.652695\pi\)
−0.461518 + 0.887131i \(0.652695\pi\)
\(488\) 13230.0 1.22724
\(489\) 0 0
\(490\) 7164.00 0.660483
\(491\) 10620.0 0.976118 0.488059 0.872811i \(-0.337705\pi\)
0.488059 + 0.872811i \(0.337705\pi\)
\(492\) 0 0
\(493\) −21204.0 −1.93708
\(494\) 8316.00 0.757398
\(495\) 0 0
\(496\) 21868.0 1.97964
\(497\) 3384.00 0.305419
\(498\) 0 0
\(499\) 52.0000 0.00466501 0.00233250 0.999997i \(-0.499258\pi\)
0.00233250 + 0.999997i \(0.499258\pi\)
\(500\) −1272.00 −0.113771
\(501\) 0 0
\(502\) −8640.00 −0.768171
\(503\) −8568.00 −0.759499 −0.379750 0.925089i \(-0.623990\pi\)
−0.379750 + 0.925089i \(0.623990\pi\)
\(504\) 0 0
\(505\) −15912.0 −1.40213
\(506\) 0 0
\(507\) 0 0
\(508\) −1992.00 −0.173978
\(509\) 21096.0 1.83706 0.918530 0.395351i \(-0.129377\pi\)
0.918530 + 0.395351i \(0.129377\pi\)
\(510\) 0 0
\(511\) 8208.00 0.710569
\(512\) 8733.00 0.753804
\(513\) 0 0
\(514\) 11286.0 0.968491
\(515\) −18768.0 −1.60586
\(516\) 0 0
\(517\) 0 0
\(518\) 5256.00 0.445821
\(519\) 0 0
\(520\) −16632.0 −1.40262
\(521\) −15762.0 −1.32542 −0.662712 0.748874i \(-0.730595\pi\)
−0.662712 + 0.748874i \(0.730595\pi\)
\(522\) 0 0
\(523\) 23778.0 1.98803 0.994015 0.109247i \(-0.0348438\pi\)
0.994015 + 0.109247i \(0.0348438\pi\)
\(524\) −1068.00 −0.0890378
\(525\) 0 0
\(526\) −5760.00 −0.477468
\(527\) −35112.0 −2.90228
\(528\) 0 0
\(529\) −11843.0 −0.973371
\(530\) −14688.0 −1.20378
\(531\) 0 0
\(532\) 504.000 0.0410736
\(533\) 2772.00 0.225270
\(534\) 0 0
\(535\) 1008.00 0.0814573
\(536\) −9492.00 −0.764910
\(537\) 0 0
\(538\) −18432.0 −1.47706
\(539\) 0 0
\(540\) 0 0
\(541\) 5478.00 0.435338 0.217669 0.976023i \(-0.430155\pi\)
0.217669 + 0.976023i \(0.430155\pi\)
\(542\) 17712.0 1.40368
\(543\) 0 0
\(544\) 5130.00 0.404314
\(545\) −13464.0 −1.05823
\(546\) 0 0
\(547\) −6714.00 −0.524808 −0.262404 0.964958i \(-0.584515\pi\)
−0.262404 + 0.964958i \(0.584515\pi\)
\(548\) 462.000 0.0360140
\(549\) 0 0
\(550\) 0 0
\(551\) −7812.00 −0.603997
\(552\) 0 0
\(553\) 15264.0 1.17376
\(554\) 10890.0 0.835148
\(555\) 0 0
\(556\) 2274.00 0.173452
\(557\) 7218.00 0.549078 0.274539 0.961576i \(-0.411475\pi\)
0.274539 + 0.961576i \(0.411475\pi\)
\(558\) 0 0
\(559\) 24156.0 1.82771
\(560\) −10224.0 −0.771505
\(561\) 0 0
\(562\) 5526.00 0.414769
\(563\) −552.000 −0.0413215 −0.0206608 0.999787i \(-0.506577\pi\)
−0.0206608 + 0.999787i \(0.506577\pi\)
\(564\) 0 0
\(565\) 10872.0 0.809537
\(566\) 7146.00 0.530687
\(567\) 0 0
\(568\) 5922.00 0.437468
\(569\) −5766.00 −0.424821 −0.212411 0.977180i \(-0.568131\pi\)
−0.212411 + 0.977180i \(0.568131\pi\)
\(570\) 0 0
\(571\) 17070.0 1.25106 0.625532 0.780199i \(-0.284882\pi\)
0.625532 + 0.780199i \(0.284882\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 1512.00 0.109947
\(575\) −342.000 −0.0248041
\(576\) 0 0
\(577\) 26170.0 1.88817 0.944083 0.329709i \(-0.106951\pi\)
0.944083 + 0.329709i \(0.106951\pi\)
\(578\) −24249.0 −1.74503
\(579\) 0 0
\(580\) −2232.00 −0.159791
\(581\) 5184.00 0.370170
\(582\) 0 0
\(583\) 0 0
\(584\) 14364.0 1.01779
\(585\) 0 0
\(586\) 6714.00 0.473298
\(587\) −27996.0 −1.96852 −0.984258 0.176739i \(-0.943445\pi\)
−0.984258 + 0.176739i \(0.943445\pi\)
\(588\) 0 0
\(589\) −12936.0 −0.904955
\(590\) −4752.00 −0.331588
\(591\) 0 0
\(592\) 10366.0 0.719662
\(593\) 28398.0 1.96655 0.983277 0.182118i \(-0.0582954\pi\)
0.983277 + 0.182118i \(0.0582954\pi\)
\(594\) 0 0
\(595\) 16416.0 1.13108
\(596\) 750.000 0.0515456
\(597\) 0 0
\(598\) −3564.00 −0.243717
\(599\) −13482.0 −0.919632 −0.459816 0.888014i \(-0.652085\pi\)
−0.459816 + 0.888014i \(0.652085\pi\)
\(600\) 0 0
\(601\) 4488.00 0.304608 0.152304 0.988334i \(-0.451331\pi\)
0.152304 + 0.988334i \(0.451331\pi\)
\(602\) 13176.0 0.892049
\(603\) 0 0
\(604\) −720.000 −0.0485039
\(605\) 0 0
\(606\) 0 0
\(607\) −600.000 −0.0401207 −0.0200603 0.999799i \(-0.506386\pi\)
−0.0200603 + 0.999799i \(0.506386\pi\)
\(608\) 1890.00 0.126068
\(609\) 0 0
\(610\) −22680.0 −1.50539
\(611\) 40788.0 2.70066
\(612\) 0 0
\(613\) −1674.00 −0.110297 −0.0551486 0.998478i \(-0.517563\pi\)
−0.0551486 + 0.998478i \(0.517563\pi\)
\(614\) 1674.00 0.110028
\(615\) 0 0
\(616\) 0 0
\(617\) 18834.0 1.22890 0.614448 0.788958i \(-0.289379\pi\)
0.614448 + 0.788958i \(0.289379\pi\)
\(618\) 0 0
\(619\) 7436.00 0.482840 0.241420 0.970421i \(-0.422387\pi\)
0.241420 + 0.970421i \(0.422387\pi\)
\(620\) −3696.00 −0.239411
\(621\) 0 0
\(622\) −12186.0 −0.785553
\(623\) −11448.0 −0.736203
\(624\) 0 0
\(625\) −17639.0 −1.12890
\(626\) 9294.00 0.593391
\(627\) 0 0
\(628\) −3206.00 −0.203715
\(629\) −16644.0 −1.05507
\(630\) 0 0
\(631\) −23852.0 −1.50481 −0.752403 0.658703i \(-0.771106\pi\)
−0.752403 + 0.658703i \(0.771106\pi\)
\(632\) 26712.0 1.68124
\(633\) 0 0
\(634\) −10800.0 −0.676534
\(635\) −23904.0 −1.49386
\(636\) 0 0
\(637\) 13134.0 0.816936
\(638\) 0 0
\(639\) 0 0
\(640\) −19908.0 −1.22958
\(641\) −10998.0 −0.677683 −0.338842 0.940843i \(-0.610035\pi\)
−0.338842 + 0.940843i \(0.610035\pi\)
\(642\) 0 0
\(643\) 16724.0 1.02571 0.512854 0.858476i \(-0.328588\pi\)
0.512854 + 0.858476i \(0.328588\pi\)
\(644\) −216.000 −0.0132168
\(645\) 0 0
\(646\) −14364.0 −0.874836
\(647\) −26226.0 −1.59359 −0.796793 0.604252i \(-0.793472\pi\)
−0.796793 + 0.604252i \(0.793472\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 3762.00 0.227012
\(651\) 0 0
\(652\) 740.000 0.0444488
\(653\) 1884.00 0.112904 0.0564522 0.998405i \(-0.482021\pi\)
0.0564522 + 0.998405i \(0.482021\pi\)
\(654\) 0 0
\(655\) −12816.0 −0.764523
\(656\) 2982.00 0.177481
\(657\) 0 0
\(658\) 22248.0 1.31811
\(659\) 22584.0 1.33497 0.667487 0.744622i \(-0.267370\pi\)
0.667487 + 0.744622i \(0.267370\pi\)
\(660\) 0 0
\(661\) 2266.00 0.133339 0.0666696 0.997775i \(-0.478763\pi\)
0.0666696 + 0.997775i \(0.478763\pi\)
\(662\) 12468.0 0.731998
\(663\) 0 0
\(664\) 9072.00 0.530214
\(665\) 6048.00 0.352679
\(666\) 0 0
\(667\) 3348.00 0.194355
\(668\) 1968.00 0.113988
\(669\) 0 0
\(670\) 16272.0 0.938272
\(671\) 0 0
\(672\) 0 0
\(673\) 23604.0 1.35196 0.675979 0.736921i \(-0.263721\pi\)
0.675979 + 0.736921i \(0.263721\pi\)
\(674\) −8244.00 −0.471138
\(675\) 0 0
\(676\) 2159.00 0.122838
\(677\) 6594.00 0.374340 0.187170 0.982328i \(-0.440069\pi\)
0.187170 + 0.982328i \(0.440069\pi\)
\(678\) 0 0
\(679\) 3912.00 0.221103
\(680\) 28728.0 1.62010
\(681\) 0 0
\(682\) 0 0
\(683\) −16368.0 −0.916990 −0.458495 0.888697i \(-0.651611\pi\)
−0.458495 + 0.888697i \(0.651611\pi\)
\(684\) 0 0
\(685\) 5544.00 0.309234
\(686\) 19512.0 1.08596
\(687\) 0 0
\(688\) 25986.0 1.43998
\(689\) −26928.0 −1.48893
\(690\) 0 0
\(691\) −11860.0 −0.652931 −0.326466 0.945209i \(-0.605858\pi\)
−0.326466 + 0.945209i \(0.605858\pi\)
\(692\) −2898.00 −0.159199
\(693\) 0 0
\(694\) 2088.00 0.114207
\(695\) 27288.0 1.48934
\(696\) 0 0
\(697\) −4788.00 −0.260199
\(698\) 9738.00 0.528064
\(699\) 0 0
\(700\) 228.000 0.0123108
\(701\) 28698.0 1.54623 0.773116 0.634265i \(-0.218697\pi\)
0.773116 + 0.634265i \(0.218697\pi\)
\(702\) 0 0
\(703\) −6132.00 −0.328980
\(704\) 0 0
\(705\) 0 0
\(706\) −32778.0 −1.74733
\(707\) −15912.0 −0.846439
\(708\) 0 0
\(709\) −10766.0 −0.570276 −0.285138 0.958486i \(-0.592039\pi\)
−0.285138 + 0.958486i \(0.592039\pi\)
\(710\) −10152.0 −0.536617
\(711\) 0 0
\(712\) −20034.0 −1.05450
\(713\) 5544.00 0.291198
\(714\) 0 0
\(715\) 0 0
\(716\) −120.000 −0.00626342
\(717\) 0 0
\(718\) −4752.00 −0.246996
\(719\) 20226.0 1.04910 0.524550 0.851380i \(-0.324234\pi\)
0.524550 + 0.851380i \(0.324234\pi\)
\(720\) 0 0
\(721\) −18768.0 −0.969427
\(722\) 15285.0 0.787879
\(723\) 0 0
\(724\) −1150.00 −0.0590323
\(725\) −3534.00 −0.181034
\(726\) 0 0
\(727\) −29576.0 −1.50882 −0.754411 0.656403i \(-0.772077\pi\)
−0.754411 + 0.656403i \(0.772077\pi\)
\(728\) −16632.0 −0.846735
\(729\) 0 0
\(730\) −24624.0 −1.24846
\(731\) −41724.0 −2.11111
\(732\) 0 0
\(733\) 16878.0 0.850482 0.425241 0.905080i \(-0.360189\pi\)
0.425241 + 0.905080i \(0.360189\pi\)
\(734\) 696.000 0.0349998
\(735\) 0 0
\(736\) −810.000 −0.0405666
\(737\) 0 0
\(738\) 0 0
\(739\) −6726.00 −0.334804 −0.167402 0.985889i \(-0.553538\pi\)
−0.167402 + 0.985889i \(0.553538\pi\)
\(740\) −1752.00 −0.0870335
\(741\) 0 0
\(742\) −14688.0 −0.726703
\(743\) −20592.0 −1.01675 −0.508376 0.861135i \(-0.669754\pi\)
−0.508376 + 0.861135i \(0.669754\pi\)
\(744\) 0 0
\(745\) 9000.00 0.442597
\(746\) 32418.0 1.59103
\(747\) 0 0
\(748\) 0 0
\(749\) 1008.00 0.0491743
\(750\) 0 0
\(751\) 8764.00 0.425836 0.212918 0.977070i \(-0.431703\pi\)
0.212918 + 0.977070i \(0.431703\pi\)
\(752\) 43878.0 2.12775
\(753\) 0 0
\(754\) −36828.0 −1.77878
\(755\) −8640.00 −0.416479
\(756\) 0 0
\(757\) −11810.0 −0.567030 −0.283515 0.958968i \(-0.591501\pi\)
−0.283515 + 0.958968i \(0.591501\pi\)
\(758\) −492.000 −0.0235755
\(759\) 0 0
\(760\) 10584.0 0.505161
\(761\) 29346.0 1.39789 0.698943 0.715177i \(-0.253654\pi\)
0.698943 + 0.715177i \(0.253654\pi\)
\(762\) 0 0
\(763\) −13464.0 −0.638833
\(764\) 1518.00 0.0718839
\(765\) 0 0
\(766\) 24174.0 1.14026
\(767\) −8712.00 −0.410133
\(768\) 0 0
\(769\) 15684.0 0.735474 0.367737 0.929930i \(-0.380133\pi\)
0.367737 + 0.929930i \(0.380133\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −2052.00 −0.0956646
\(773\) 25632.0 1.19265 0.596325 0.802743i \(-0.296627\pi\)
0.596325 + 0.802743i \(0.296627\pi\)
\(774\) 0 0
\(775\) −5852.00 −0.271239
\(776\) 6846.00 0.316697
\(777\) 0 0
\(778\) −2124.00 −0.0978780
\(779\) −1764.00 −0.0811320
\(780\) 0 0
\(781\) 0 0
\(782\) 6156.00 0.281507
\(783\) 0 0
\(784\) 14129.0 0.643632
\(785\) −38472.0 −1.74920
\(786\) 0 0
\(787\) −37854.0 −1.71455 −0.857274 0.514860i \(-0.827844\pi\)
−0.857274 + 0.514860i \(0.827844\pi\)
\(788\) 3282.00 0.148371
\(789\) 0 0
\(790\) −45792.0 −2.06229
\(791\) 10872.0 0.488703
\(792\) 0 0
\(793\) −41580.0 −1.86198
\(794\) 858.000 0.0383492
\(795\) 0 0
\(796\) −704.000 −0.0313475
\(797\) 23484.0 1.04372 0.521861 0.853031i \(-0.325238\pi\)
0.521861 + 0.853031i \(0.325238\pi\)
\(798\) 0 0
\(799\) −70452.0 −3.11942
\(800\) 855.000 0.0377860
\(801\) 0 0
\(802\) −15786.0 −0.695041
\(803\) 0 0
\(804\) 0 0
\(805\) −2592.00 −0.113486
\(806\) −60984.0 −2.66510
\(807\) 0 0
\(808\) −27846.0 −1.21240
\(809\) −9054.00 −0.393476 −0.196738 0.980456i \(-0.563035\pi\)
−0.196738 + 0.980456i \(0.563035\pi\)
\(810\) 0 0
\(811\) −9858.00 −0.426833 −0.213416 0.976961i \(-0.568459\pi\)
−0.213416 + 0.976961i \(0.568459\pi\)
\(812\) −2232.00 −0.0964629
\(813\) 0 0
\(814\) 0 0
\(815\) 8880.00 0.381660
\(816\) 0 0
\(817\) −15372.0 −0.658260
\(818\) 6840.00 0.292366
\(819\) 0 0
\(820\) −504.000 −0.0214640
\(821\) 18210.0 0.774097 0.387048 0.922059i \(-0.373495\pi\)
0.387048 + 0.922059i \(0.373495\pi\)
\(822\) 0 0
\(823\) 41240.0 1.74670 0.873351 0.487091i \(-0.161942\pi\)
0.873351 + 0.487091i \(0.161942\pi\)
\(824\) −32844.0 −1.38856
\(825\) 0 0
\(826\) −4752.00 −0.200173
\(827\) −37680.0 −1.58436 −0.792178 0.610290i \(-0.791053\pi\)
−0.792178 + 0.610290i \(0.791053\pi\)
\(828\) 0 0
\(829\) −11626.0 −0.487078 −0.243539 0.969891i \(-0.578308\pi\)
−0.243539 + 0.969891i \(0.578308\pi\)
\(830\) −15552.0 −0.650383
\(831\) 0 0
\(832\) −28578.0 −1.19082
\(833\) −22686.0 −0.943605
\(834\) 0 0
\(835\) 23616.0 0.978761
\(836\) 0 0
\(837\) 0 0
\(838\) 12816.0 0.528307
\(839\) −12150.0 −0.499958 −0.249979 0.968251i \(-0.580424\pi\)
−0.249979 + 0.968251i \(0.580424\pi\)
\(840\) 0 0
\(841\) 10207.0 0.418508
\(842\) −30450.0 −1.24629
\(843\) 0 0
\(844\) −1986.00 −0.0809964
\(845\) 25908.0 1.05475
\(846\) 0 0
\(847\) 0 0
\(848\) −28968.0 −1.17307
\(849\) 0 0
\(850\) −6498.00 −0.262211
\(851\) 2628.00 0.105860
\(852\) 0 0
\(853\) 41022.0 1.64662 0.823310 0.567592i \(-0.192125\pi\)
0.823310 + 0.567592i \(0.192125\pi\)
\(854\) −22680.0 −0.908775
\(855\) 0 0
\(856\) 1764.00 0.0704349
\(857\) 9234.00 0.368060 0.184030 0.982921i \(-0.441086\pi\)
0.184030 + 0.982921i \(0.441086\pi\)
\(858\) 0 0
\(859\) −30004.0 −1.19176 −0.595881 0.803073i \(-0.703197\pi\)
−0.595881 + 0.803073i \(0.703197\pi\)
\(860\) −4392.00 −0.174146
\(861\) 0 0
\(862\) 39168.0 1.54764
\(863\) −14958.0 −0.590007 −0.295004 0.955496i \(-0.595321\pi\)
−0.295004 + 0.955496i \(0.595321\pi\)
\(864\) 0 0
\(865\) −34776.0 −1.36696
\(866\) −41046.0 −1.61062
\(867\) 0 0
\(868\) −3696.00 −0.144528
\(869\) 0 0
\(870\) 0 0
\(871\) 29832.0 1.16053
\(872\) −23562.0 −0.915034
\(873\) 0 0
\(874\) 2268.00 0.0877760
\(875\) −15264.0 −0.589734
\(876\) 0 0
\(877\) 3714.00 0.143002 0.0715011 0.997441i \(-0.477221\pi\)
0.0715011 + 0.997441i \(0.477221\pi\)
\(878\) 21060.0 0.809500
\(879\) 0 0
\(880\) 0 0
\(881\) −45390.0 −1.73579 −0.867893 0.496751i \(-0.834526\pi\)
−0.867893 + 0.496751i \(0.834526\pi\)
\(882\) 0 0
\(883\) 23452.0 0.893797 0.446898 0.894585i \(-0.352529\pi\)
0.446898 + 0.894585i \(0.352529\pi\)
\(884\) −7524.00 −0.286266
\(885\) 0 0
\(886\) −26748.0 −1.01424
\(887\) −27072.0 −1.02479 −0.512395 0.858750i \(-0.671242\pi\)
−0.512395 + 0.858750i \(0.671242\pi\)
\(888\) 0 0
\(889\) −23904.0 −0.901816
\(890\) 34344.0 1.29350
\(891\) 0 0
\(892\) −2248.00 −0.0843818
\(893\) −25956.0 −0.972659
\(894\) 0 0
\(895\) −1440.00 −0.0537809
\(896\) −19908.0 −0.742276
\(897\) 0 0
\(898\) 30330.0 1.12709
\(899\) 57288.0 2.12532
\(900\) 0 0
\(901\) 46512.0 1.71980
\(902\) 0 0
\(903\) 0 0
\(904\) 19026.0 0.699995
\(905\) −13800.0 −0.506881
\(906\) 0 0
\(907\) −29716.0 −1.08788 −0.543938 0.839125i \(-0.683067\pi\)
−0.543938 + 0.839125i \(0.683067\pi\)
\(908\) 2796.00 0.102190
\(909\) 0 0
\(910\) 28512.0 1.03864
\(911\) −17094.0 −0.621679 −0.310839 0.950462i \(-0.600610\pi\)
−0.310839 + 0.950462i \(0.600610\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −5472.00 −0.198028
\(915\) 0 0
\(916\) −2198.00 −0.0792838
\(917\) −12816.0 −0.461528
\(918\) 0 0
\(919\) 24840.0 0.891617 0.445808 0.895128i \(-0.352916\pi\)
0.445808 + 0.895128i \(0.352916\pi\)
\(920\) −4536.00 −0.162552
\(921\) 0 0
\(922\) 51354.0 1.83433
\(923\) −18612.0 −0.663729
\(924\) 0 0
\(925\) −2774.00 −0.0986038
\(926\) −1788.00 −0.0634528
\(927\) 0 0
\(928\) −8370.00 −0.296076
\(929\) −10578.0 −0.373577 −0.186788 0.982400i \(-0.559808\pi\)
−0.186788 + 0.982400i \(0.559808\pi\)
\(930\) 0 0
\(931\) −8358.00 −0.294224
\(932\) −5802.00 −0.203917
\(933\) 0 0
\(934\) 17568.0 0.615463
\(935\) 0 0
\(936\) 0 0
\(937\) −38208.0 −1.33212 −0.666062 0.745896i \(-0.732022\pi\)
−0.666062 + 0.745896i \(0.732022\pi\)
\(938\) 16272.0 0.566418
\(939\) 0 0
\(940\) −7416.00 −0.257323
\(941\) 2154.00 0.0746210 0.0373105 0.999304i \(-0.488121\pi\)
0.0373105 + 0.999304i \(0.488121\pi\)
\(942\) 0 0
\(943\) 756.000 0.0261068
\(944\) −9372.00 −0.323128
\(945\) 0 0
\(946\) 0 0
\(947\) −6564.00 −0.225239 −0.112620 0.993638i \(-0.535924\pi\)
−0.112620 + 0.993638i \(0.535924\pi\)
\(948\) 0 0
\(949\) −45144.0 −1.54419
\(950\) −2394.00 −0.0817596
\(951\) 0 0
\(952\) 28728.0 0.978025
\(953\) 36162.0 1.22917 0.614587 0.788849i \(-0.289323\pi\)
0.614587 + 0.788849i \(0.289323\pi\)
\(954\) 0 0
\(955\) 18216.0 0.617231
\(956\) −1104.00 −0.0373493
\(957\) 0 0
\(958\) 9792.00 0.330235
\(959\) 5544.00 0.186679
\(960\) 0 0
\(961\) 65073.0 2.18432
\(962\) −28908.0 −0.968848
\(963\) 0 0
\(964\) 5208.00 0.174002
\(965\) −24624.0 −0.821424
\(966\) 0 0
\(967\) −37728.0 −1.25465 −0.627327 0.778756i \(-0.715851\pi\)
−0.627327 + 0.778756i \(0.715851\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −11736.0 −0.388474
\(971\) −24156.0 −0.798355 −0.399178 0.916874i \(-0.630704\pi\)
−0.399178 + 0.916874i \(0.630704\pi\)
\(972\) 0 0
\(973\) 27288.0 0.899089
\(974\) 29760.0 0.979027
\(975\) 0 0
\(976\) −44730.0 −1.46698
\(977\) 23742.0 0.777455 0.388728 0.921353i \(-0.372915\pi\)
0.388728 + 0.921353i \(0.372915\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −2388.00 −0.0778386
\(981\) 0 0
\(982\) −31860.0 −1.03533
\(983\) −11562.0 −0.375148 −0.187574 0.982250i \(-0.560062\pi\)
−0.187574 + 0.982250i \(0.560062\pi\)
\(984\) 0 0
\(985\) 39384.0 1.27399
\(986\) 63612.0 2.05458
\(987\) 0 0
\(988\) −2772.00 −0.0892602
\(989\) 6588.00 0.211816
\(990\) 0 0
\(991\) 52180.0 1.67261 0.836303 0.548268i \(-0.184713\pi\)
0.836303 + 0.548268i \(0.184713\pi\)
\(992\) −13860.0 −0.443604
\(993\) 0 0
\(994\) −10152.0 −0.323946
\(995\) −8448.00 −0.269165
\(996\) 0 0
\(997\) 15714.0 0.499165 0.249582 0.968354i \(-0.419707\pi\)
0.249582 + 0.968354i \(0.419707\pi\)
\(998\) −156.000 −0.00494799
\(999\) 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 1089.4.a.c.1.1 1
3.2 odd 2 363.4.a.f.1.1 yes 1
11.10 odd 2 1089.4.a.i.1.1 1
33.32 even 2 363.4.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
363.4.a.b.1.1 1 33.32 even 2
363.4.a.f.1.1 yes 1 3.2 odd 2
1089.4.a.c.1.1 1 1.1 even 1 trivial
1089.4.a.i.1.1 1 11.10 odd 2