Properties

Label 64.4.a.a.1.1
Level $64$
Weight $4$
Character 64.1
Self dual yes
Analytic conductor $3.776$
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] = [64,4,Mod(1,64)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(64, 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("64.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 64 = 2^{6} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 64.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: \(3.77612224037\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 32)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 64.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-8.00000 q^{3} +10.0000 q^{5} +16.0000 q^{7} +37.0000 q^{9} +O(q^{10})\) \(q-8.00000 q^{3} +10.0000 q^{5} +16.0000 q^{7} +37.0000 q^{9} +40.0000 q^{11} +50.0000 q^{13} -80.0000 q^{15} -30.0000 q^{17} -40.0000 q^{19} -128.000 q^{21} +48.0000 q^{23} -25.0000 q^{25} -80.0000 q^{27} +34.0000 q^{29} +320.000 q^{31} -320.000 q^{33} +160.000 q^{35} -310.000 q^{37} -400.000 q^{39} +410.000 q^{41} -152.000 q^{43} +370.000 q^{45} -416.000 q^{47} -87.0000 q^{49} +240.000 q^{51} +410.000 q^{53} +400.000 q^{55} +320.000 q^{57} +200.000 q^{59} -30.0000 q^{61} +592.000 q^{63} +500.000 q^{65} -776.000 q^{67} -384.000 q^{69} +400.000 q^{71} -630.000 q^{73} +200.000 q^{75} +640.000 q^{77} -1120.00 q^{79} -359.000 q^{81} -552.000 q^{83} -300.000 q^{85} -272.000 q^{87} -326.000 q^{89} +800.000 q^{91} -2560.00 q^{93} -400.000 q^{95} -110.000 q^{97} +1480.00 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −8.00000 −1.53960 −0.769800 0.638285i \(-0.779644\pi\)
−0.769800 + 0.638285i \(0.779644\pi\)
\(4\) 0 0
\(5\) 10.0000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) 16.0000 0.863919 0.431959 0.901893i \(-0.357822\pi\)
0.431959 + 0.901893i \(0.357822\pi\)
\(8\) 0 0
\(9\) 37.0000 1.37037
\(10\) 0 0
\(11\) 40.0000 1.09640 0.548202 0.836346i \(-0.315312\pi\)
0.548202 + 0.836346i \(0.315312\pi\)
\(12\) 0 0
\(13\) 50.0000 1.06673 0.533366 0.845885i \(-0.320927\pi\)
0.533366 + 0.845885i \(0.320927\pi\)
\(14\) 0 0
\(15\) −80.0000 −1.37706
\(16\) 0 0
\(17\) −30.0000 −0.428004 −0.214002 0.976833i \(-0.568650\pi\)
−0.214002 + 0.976833i \(0.568650\pi\)
\(18\) 0 0
\(19\) −40.0000 −0.482980 −0.241490 0.970403i \(-0.577636\pi\)
−0.241490 + 0.970403i \(0.577636\pi\)
\(20\) 0 0
\(21\) −128.000 −1.33009
\(22\) 0 0
\(23\) 48.0000 0.435161 0.217580 0.976042i \(-0.430184\pi\)
0.217580 + 0.976042i \(0.430184\pi\)
\(24\) 0 0
\(25\) −25.0000 −0.200000
\(26\) 0 0
\(27\) −80.0000 −0.570222
\(28\) 0 0
\(29\) 34.0000 0.217712 0.108856 0.994058i \(-0.465281\pi\)
0.108856 + 0.994058i \(0.465281\pi\)
\(30\) 0 0
\(31\) 320.000 1.85399 0.926995 0.375073i \(-0.122383\pi\)
0.926995 + 0.375073i \(0.122383\pi\)
\(32\) 0 0
\(33\) −320.000 −1.68803
\(34\) 0 0
\(35\) 160.000 0.772712
\(36\) 0 0
\(37\) −310.000 −1.37740 −0.688698 0.725048i \(-0.741818\pi\)
−0.688698 + 0.725048i \(0.741818\pi\)
\(38\) 0 0
\(39\) −400.000 −1.64234
\(40\) 0 0
\(41\) 410.000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) −152.000 −0.539065 −0.269532 0.962991i \(-0.586869\pi\)
−0.269532 + 0.962991i \(0.586869\pi\)
\(44\) 0 0
\(45\) 370.000 1.22570
\(46\) 0 0
\(47\) −416.000 −1.29106 −0.645530 0.763735i \(-0.723364\pi\)
−0.645530 + 0.763735i \(0.723364\pi\)
\(48\) 0 0
\(49\) −87.0000 −0.253644
\(50\) 0 0
\(51\) 240.000 0.658955
\(52\) 0 0
\(53\) 410.000 1.06260 0.531300 0.847184i \(-0.321704\pi\)
0.531300 + 0.847184i \(0.321704\pi\)
\(54\) 0 0
\(55\) 400.000 0.980654
\(56\) 0 0
\(57\) 320.000 0.743597
\(58\) 0 0
\(59\) 200.000 0.441318 0.220659 0.975351i \(-0.429179\pi\)
0.220659 + 0.975351i \(0.429179\pi\)
\(60\) 0 0
\(61\) −30.0000 −0.0629690 −0.0314845 0.999504i \(-0.510023\pi\)
−0.0314845 + 0.999504i \(0.510023\pi\)
\(62\) 0 0
\(63\) 592.000 1.18389
\(64\) 0 0
\(65\) 500.000 0.954113
\(66\) 0 0
\(67\) −776.000 −1.41498 −0.707489 0.706725i \(-0.750172\pi\)
−0.707489 + 0.706725i \(0.750172\pi\)
\(68\) 0 0
\(69\) −384.000 −0.669973
\(70\) 0 0
\(71\) 400.000 0.668609 0.334305 0.942465i \(-0.391499\pi\)
0.334305 + 0.942465i \(0.391499\pi\)
\(72\) 0 0
\(73\) −630.000 −1.01008 −0.505041 0.863096i \(-0.668522\pi\)
−0.505041 + 0.863096i \(0.668522\pi\)
\(74\) 0 0
\(75\) 200.000 0.307920
\(76\) 0 0
\(77\) 640.000 0.947205
\(78\) 0 0
\(79\) −1120.00 −1.59506 −0.797531 0.603278i \(-0.793861\pi\)
−0.797531 + 0.603278i \(0.793861\pi\)
\(80\) 0 0
\(81\) −359.000 −0.492455
\(82\) 0 0
\(83\) −552.000 −0.729998 −0.364999 0.931008i \(-0.618931\pi\)
−0.364999 + 0.931008i \(0.618931\pi\)
\(84\) 0 0
\(85\) −300.000 −0.382818
\(86\) 0 0
\(87\) −272.000 −0.335189
\(88\) 0 0
\(89\) −326.000 −0.388269 −0.194134 0.980975i \(-0.562190\pi\)
−0.194134 + 0.980975i \(0.562190\pi\)
\(90\) 0 0
\(91\) 800.000 0.921569
\(92\) 0 0
\(93\) −2560.00 −2.85440
\(94\) 0 0
\(95\) −400.000 −0.431991
\(96\) 0 0
\(97\) −110.000 −0.115142 −0.0575712 0.998341i \(-0.518336\pi\)
−0.0575712 + 0.998341i \(0.518336\pi\)
\(98\) 0 0
\(99\) 1480.00 1.50248
\(100\) 0 0
\(101\) 1098.00 1.08173 0.540867 0.841108i \(-0.318096\pi\)
0.540867 + 0.841108i \(0.318096\pi\)
\(102\) 0 0
\(103\) −48.0000 −0.0459183 −0.0229591 0.999736i \(-0.507309\pi\)
−0.0229591 + 0.999736i \(0.507309\pi\)
\(104\) 0 0
\(105\) −1280.00 −1.18967
\(106\) 0 0
\(107\) −664.000 −0.599919 −0.299959 0.953952i \(-0.596973\pi\)
−0.299959 + 0.953952i \(0.596973\pi\)
\(108\) 0 0
\(109\) 370.000 0.325134 0.162567 0.986698i \(-0.448023\pi\)
0.162567 + 0.986698i \(0.448023\pi\)
\(110\) 0 0
\(111\) 2480.00 2.12064
\(112\) 0 0
\(113\) 1490.00 1.24042 0.620210 0.784436i \(-0.287047\pi\)
0.620210 + 0.784436i \(0.287047\pi\)
\(114\) 0 0
\(115\) 480.000 0.389219
\(116\) 0 0
\(117\) 1850.00 1.46182
\(118\) 0 0
\(119\) −480.000 −0.369761
\(120\) 0 0
\(121\) 269.000 0.202104
\(122\) 0 0
\(123\) −3280.00 −2.40445
\(124\) 0 0
\(125\) −1500.00 −1.07331
\(126\) 0 0
\(127\) −1024.00 −0.715475 −0.357737 0.933822i \(-0.616452\pi\)
−0.357737 + 0.933822i \(0.616452\pi\)
\(128\) 0 0
\(129\) 1216.00 0.829944
\(130\) 0 0
\(131\) −1160.00 −0.773662 −0.386831 0.922151i \(-0.626430\pi\)
−0.386831 + 0.922151i \(0.626430\pi\)
\(132\) 0 0
\(133\) −640.000 −0.417256
\(134\) 0 0
\(135\) −800.000 −0.510022
\(136\) 0 0
\(137\) 570.000 0.355463 0.177731 0.984079i \(-0.443124\pi\)
0.177731 + 0.984079i \(0.443124\pi\)
\(138\) 0 0
\(139\) 1960.00 1.19601 0.598004 0.801493i \(-0.295961\pi\)
0.598004 + 0.801493i \(0.295961\pi\)
\(140\) 0 0
\(141\) 3328.00 1.98772
\(142\) 0 0
\(143\) 2000.00 1.16957
\(144\) 0 0
\(145\) 340.000 0.194727
\(146\) 0 0
\(147\) 696.000 0.390511
\(148\) 0 0
\(149\) 2010.00 1.10514 0.552569 0.833467i \(-0.313648\pi\)
0.552569 + 0.833467i \(0.313648\pi\)
\(150\) 0 0
\(151\) −720.000 −0.388032 −0.194016 0.980998i \(-0.562151\pi\)
−0.194016 + 0.980998i \(0.562151\pi\)
\(152\) 0 0
\(153\) −1110.00 −0.586524
\(154\) 0 0
\(155\) 3200.00 1.65826
\(156\) 0 0
\(157\) −1790.00 −0.909921 −0.454960 0.890512i \(-0.650347\pi\)
−0.454960 + 0.890512i \(0.650347\pi\)
\(158\) 0 0
\(159\) −3280.00 −1.63598
\(160\) 0 0
\(161\) 768.000 0.375943
\(162\) 0 0
\(163\) 1208.00 0.580478 0.290239 0.956954i \(-0.406265\pi\)
0.290239 + 0.956954i \(0.406265\pi\)
\(164\) 0 0
\(165\) −3200.00 −1.50982
\(166\) 0 0
\(167\) 2896.00 1.34191 0.670956 0.741497i \(-0.265884\pi\)
0.670956 + 0.741497i \(0.265884\pi\)
\(168\) 0 0
\(169\) 303.000 0.137915
\(170\) 0 0
\(171\) −1480.00 −0.661862
\(172\) 0 0
\(173\) −750.000 −0.329604 −0.164802 0.986327i \(-0.552698\pi\)
−0.164802 + 0.986327i \(0.552698\pi\)
\(174\) 0 0
\(175\) −400.000 −0.172784
\(176\) 0 0
\(177\) −1600.00 −0.679454
\(178\) 0 0
\(179\) −2280.00 −0.952040 −0.476020 0.879434i \(-0.657921\pi\)
−0.476020 + 0.879434i \(0.657921\pi\)
\(180\) 0 0
\(181\) 442.000 0.181512 0.0907558 0.995873i \(-0.471072\pi\)
0.0907558 + 0.995873i \(0.471072\pi\)
\(182\) 0 0
\(183\) 240.000 0.0969471
\(184\) 0 0
\(185\) −3100.00 −1.23198
\(186\) 0 0
\(187\) −1200.00 −0.469266
\(188\) 0 0
\(189\) −1280.00 −0.492626
\(190\) 0 0
\(191\) −1920.00 −0.727363 −0.363681 0.931523i \(-0.618480\pi\)
−0.363681 + 0.931523i \(0.618480\pi\)
\(192\) 0 0
\(193\) −5070.00 −1.89091 −0.945457 0.325746i \(-0.894385\pi\)
−0.945457 + 0.325746i \(0.894385\pi\)
\(194\) 0 0
\(195\) −4000.00 −1.46895
\(196\) 0 0
\(197\) −1910.00 −0.690771 −0.345385 0.938461i \(-0.612252\pi\)
−0.345385 + 0.938461i \(0.612252\pi\)
\(198\) 0 0
\(199\) 2960.00 1.05442 0.527208 0.849736i \(-0.323239\pi\)
0.527208 + 0.849736i \(0.323239\pi\)
\(200\) 0 0
\(201\) 6208.00 2.17850
\(202\) 0 0
\(203\) 544.000 0.188085
\(204\) 0 0
\(205\) 4100.00 1.39686
\(206\) 0 0
\(207\) 1776.00 0.596331
\(208\) 0 0
\(209\) −1600.00 −0.529542
\(210\) 0 0
\(211\) −40.0000 −0.0130508 −0.00652539 0.999979i \(-0.502077\pi\)
−0.00652539 + 0.999979i \(0.502077\pi\)
\(212\) 0 0
\(213\) −3200.00 −1.02939
\(214\) 0 0
\(215\) −1520.00 −0.482154
\(216\) 0 0
\(217\) 5120.00 1.60170
\(218\) 0 0
\(219\) 5040.00 1.55512
\(220\) 0 0
\(221\) −1500.00 −0.456565
\(222\) 0 0
\(223\) 4288.00 1.28765 0.643824 0.765173i \(-0.277347\pi\)
0.643824 + 0.765173i \(0.277347\pi\)
\(224\) 0 0
\(225\) −925.000 −0.274074
\(226\) 0 0
\(227\) 6456.00 1.88766 0.943832 0.330425i \(-0.107192\pi\)
0.943832 + 0.330425i \(0.107192\pi\)
\(228\) 0 0
\(229\) 1066.00 0.307613 0.153806 0.988101i \(-0.450847\pi\)
0.153806 + 0.988101i \(0.450847\pi\)
\(230\) 0 0
\(231\) −5120.00 −1.45832
\(232\) 0 0
\(233\) −5910.00 −1.66170 −0.830852 0.556494i \(-0.812146\pi\)
−0.830852 + 0.556494i \(0.812146\pi\)
\(234\) 0 0
\(235\) −4160.00 −1.15476
\(236\) 0 0
\(237\) 8960.00 2.45576
\(238\) 0 0
\(239\) −3360.00 −0.909374 −0.454687 0.890651i \(-0.650249\pi\)
−0.454687 + 0.890651i \(0.650249\pi\)
\(240\) 0 0
\(241\) 3970.00 1.06112 0.530561 0.847647i \(-0.321981\pi\)
0.530561 + 0.847647i \(0.321981\pi\)
\(242\) 0 0
\(243\) 5032.00 1.32841
\(244\) 0 0
\(245\) −870.000 −0.226866
\(246\) 0 0
\(247\) −2000.00 −0.515210
\(248\) 0 0
\(249\) 4416.00 1.12391
\(250\) 0 0
\(251\) −6840.00 −1.72007 −0.860034 0.510237i \(-0.829558\pi\)
−0.860034 + 0.510237i \(0.829558\pi\)
\(252\) 0 0
\(253\) 1920.00 0.477112
\(254\) 0 0
\(255\) 2400.00 0.589388
\(256\) 0 0
\(257\) 4610.00 1.11893 0.559463 0.828855i \(-0.311007\pi\)
0.559463 + 0.828855i \(0.311007\pi\)
\(258\) 0 0
\(259\) −4960.00 −1.18996
\(260\) 0 0
\(261\) 1258.00 0.298346
\(262\) 0 0
\(263\) −4848.00 −1.13666 −0.568328 0.822802i \(-0.692409\pi\)
−0.568328 + 0.822802i \(0.692409\pi\)
\(264\) 0 0
\(265\) 4100.00 0.950419
\(266\) 0 0
\(267\) 2608.00 0.597779
\(268\) 0 0
\(269\) −5550.00 −1.25795 −0.628977 0.777424i \(-0.716526\pi\)
−0.628977 + 0.777424i \(0.716526\pi\)
\(270\) 0 0
\(271\) −480.000 −0.107594 −0.0537969 0.998552i \(-0.517132\pi\)
−0.0537969 + 0.998552i \(0.517132\pi\)
\(272\) 0 0
\(273\) −6400.00 −1.41885
\(274\) 0 0
\(275\) −1000.00 −0.219281
\(276\) 0 0
\(277\) −1030.00 −0.223418 −0.111709 0.993741i \(-0.535632\pi\)
−0.111709 + 0.993741i \(0.535632\pi\)
\(278\) 0 0
\(279\) 11840.0 2.54065
\(280\) 0 0
\(281\) −3270.00 −0.694206 −0.347103 0.937827i \(-0.612835\pi\)
−0.347103 + 0.937827i \(0.612835\pi\)
\(282\) 0 0
\(283\) −2168.00 −0.455386 −0.227693 0.973733i \(-0.573118\pi\)
−0.227693 + 0.973733i \(0.573118\pi\)
\(284\) 0 0
\(285\) 3200.00 0.665093
\(286\) 0 0
\(287\) 6560.00 1.34921
\(288\) 0 0
\(289\) −4013.00 −0.816813
\(290\) 0 0
\(291\) 880.000 0.177273
\(292\) 0 0
\(293\) −2070.00 −0.412733 −0.206366 0.978475i \(-0.566164\pi\)
−0.206366 + 0.978475i \(0.566164\pi\)
\(294\) 0 0
\(295\) 2000.00 0.394727
\(296\) 0 0
\(297\) −3200.00 −0.625195
\(298\) 0 0
\(299\) 2400.00 0.464199
\(300\) 0 0
\(301\) −2432.00 −0.465708
\(302\) 0 0
\(303\) −8784.00 −1.66544
\(304\) 0 0
\(305\) −300.000 −0.0563211
\(306\) 0 0
\(307\) −1896.00 −0.352477 −0.176238 0.984347i \(-0.556393\pi\)
−0.176238 + 0.984347i \(0.556393\pi\)
\(308\) 0 0
\(309\) 384.000 0.0706958
\(310\) 0 0
\(311\) −1680.00 −0.306315 −0.153158 0.988202i \(-0.548944\pi\)
−0.153158 + 0.988202i \(0.548944\pi\)
\(312\) 0 0
\(313\) 970.000 0.175168 0.0875841 0.996157i \(-0.472085\pi\)
0.0875841 + 0.996157i \(0.472085\pi\)
\(314\) 0 0
\(315\) 5920.00 1.05890
\(316\) 0 0
\(317\) −7230.00 −1.28100 −0.640500 0.767958i \(-0.721273\pi\)
−0.640500 + 0.767958i \(0.721273\pi\)
\(318\) 0 0
\(319\) 1360.00 0.238700
\(320\) 0 0
\(321\) 5312.00 0.923635
\(322\) 0 0
\(323\) 1200.00 0.206718
\(324\) 0 0
\(325\) −1250.00 −0.213346
\(326\) 0 0
\(327\) −2960.00 −0.500576
\(328\) 0 0
\(329\) −6656.00 −1.11537
\(330\) 0 0
\(331\) 5800.00 0.963132 0.481566 0.876410i \(-0.340068\pi\)
0.481566 + 0.876410i \(0.340068\pi\)
\(332\) 0 0
\(333\) −11470.0 −1.88754
\(334\) 0 0
\(335\) −7760.00 −1.26559
\(336\) 0 0
\(337\) −1870.00 −0.302271 −0.151136 0.988513i \(-0.548293\pi\)
−0.151136 + 0.988513i \(0.548293\pi\)
\(338\) 0 0
\(339\) −11920.0 −1.90975
\(340\) 0 0
\(341\) 12800.0 2.03272
\(342\) 0 0
\(343\) −6880.00 −1.08305
\(344\) 0 0
\(345\) −3840.00 −0.599242
\(346\) 0 0
\(347\) −376.000 −0.0581693 −0.0290846 0.999577i \(-0.509259\pi\)
−0.0290846 + 0.999577i \(0.509259\pi\)
\(348\) 0 0
\(349\) 7586.00 1.16352 0.581761 0.813360i \(-0.302364\pi\)
0.581761 + 0.813360i \(0.302364\pi\)
\(350\) 0 0
\(351\) −4000.00 −0.608274
\(352\) 0 0
\(353\) 2530.00 0.381468 0.190734 0.981642i \(-0.438913\pi\)
0.190734 + 0.981642i \(0.438913\pi\)
\(354\) 0 0
\(355\) 4000.00 0.598022
\(356\) 0 0
\(357\) 3840.00 0.569284
\(358\) 0 0
\(359\) 9680.00 1.42309 0.711547 0.702638i \(-0.247995\pi\)
0.711547 + 0.702638i \(0.247995\pi\)
\(360\) 0 0
\(361\) −5259.00 −0.766730
\(362\) 0 0
\(363\) −2152.00 −0.311159
\(364\) 0 0
\(365\) −6300.00 −0.903444
\(366\) 0 0
\(367\) 2784.00 0.395977 0.197989 0.980204i \(-0.436559\pi\)
0.197989 + 0.980204i \(0.436559\pi\)
\(368\) 0 0
\(369\) 15170.0 2.14016
\(370\) 0 0
\(371\) 6560.00 0.918001
\(372\) 0 0
\(373\) −7910.00 −1.09803 −0.549014 0.835813i \(-0.684997\pi\)
−0.549014 + 0.835813i \(0.684997\pi\)
\(374\) 0 0
\(375\) 12000.0 1.65247
\(376\) 0 0
\(377\) 1700.00 0.232240
\(378\) 0 0
\(379\) −1720.00 −0.233115 −0.116557 0.993184i \(-0.537186\pi\)
−0.116557 + 0.993184i \(0.537186\pi\)
\(380\) 0 0
\(381\) 8192.00 1.10155
\(382\) 0 0
\(383\) 11008.0 1.46862 0.734311 0.678813i \(-0.237505\pi\)
0.734311 + 0.678813i \(0.237505\pi\)
\(384\) 0 0
\(385\) 6400.00 0.847206
\(386\) 0 0
\(387\) −5624.00 −0.738718
\(388\) 0 0
\(389\) 12330.0 1.60708 0.803542 0.595248i \(-0.202946\pi\)
0.803542 + 0.595248i \(0.202946\pi\)
\(390\) 0 0
\(391\) −1440.00 −0.186250
\(392\) 0 0
\(393\) 9280.00 1.19113
\(394\) 0 0
\(395\) −11200.0 −1.42667
\(396\) 0 0
\(397\) 4370.00 0.552453 0.276227 0.961093i \(-0.410916\pi\)
0.276227 + 0.961093i \(0.410916\pi\)
\(398\) 0 0
\(399\) 5120.00 0.642408
\(400\) 0 0
\(401\) 3298.00 0.410709 0.205354 0.978688i \(-0.434165\pi\)
0.205354 + 0.978688i \(0.434165\pi\)
\(402\) 0 0
\(403\) 16000.0 1.97771
\(404\) 0 0
\(405\) −3590.00 −0.440466
\(406\) 0 0
\(407\) −12400.0 −1.51018
\(408\) 0 0
\(409\) −9110.00 −1.10137 −0.550685 0.834713i \(-0.685634\pi\)
−0.550685 + 0.834713i \(0.685634\pi\)
\(410\) 0 0
\(411\) −4560.00 −0.547271
\(412\) 0 0
\(413\) 3200.00 0.381263
\(414\) 0 0
\(415\) −5520.00 −0.652930
\(416\) 0 0
\(417\) −15680.0 −1.84137
\(418\) 0 0
\(419\) −7880.00 −0.918767 −0.459383 0.888238i \(-0.651930\pi\)
−0.459383 + 0.888238i \(0.651930\pi\)
\(420\) 0 0
\(421\) 5290.00 0.612396 0.306198 0.951968i \(-0.400943\pi\)
0.306198 + 0.951968i \(0.400943\pi\)
\(422\) 0 0
\(423\) −15392.0 −1.76923
\(424\) 0 0
\(425\) 750.000 0.0856008
\(426\) 0 0
\(427\) −480.000 −0.0544001
\(428\) 0 0
\(429\) −16000.0 −1.80067
\(430\) 0 0
\(431\) 13920.0 1.55569 0.777845 0.628456i \(-0.216313\pi\)
0.777845 + 0.628456i \(0.216313\pi\)
\(432\) 0 0
\(433\) 4930.00 0.547161 0.273580 0.961849i \(-0.411792\pi\)
0.273580 + 0.961849i \(0.411792\pi\)
\(434\) 0 0
\(435\) −2720.00 −0.299802
\(436\) 0 0
\(437\) −1920.00 −0.210174
\(438\) 0 0
\(439\) −10640.0 −1.15676 −0.578382 0.815766i \(-0.696316\pi\)
−0.578382 + 0.815766i \(0.696316\pi\)
\(440\) 0 0
\(441\) −3219.00 −0.347587
\(442\) 0 0
\(443\) 9288.00 0.996131 0.498066 0.867139i \(-0.334044\pi\)
0.498066 + 0.867139i \(0.334044\pi\)
\(444\) 0 0
\(445\) −3260.00 −0.347278
\(446\) 0 0
\(447\) −16080.0 −1.70147
\(448\) 0 0
\(449\) 12850.0 1.35062 0.675311 0.737533i \(-0.264010\pi\)
0.675311 + 0.737533i \(0.264010\pi\)
\(450\) 0 0
\(451\) 16400.0 1.71230
\(452\) 0 0
\(453\) 5760.00 0.597414
\(454\) 0 0
\(455\) 8000.00 0.824276
\(456\) 0 0
\(457\) 10490.0 1.07375 0.536873 0.843663i \(-0.319606\pi\)
0.536873 + 0.843663i \(0.319606\pi\)
\(458\) 0 0
\(459\) 2400.00 0.244058
\(460\) 0 0
\(461\) −11118.0 −1.12325 −0.561624 0.827393i \(-0.689823\pi\)
−0.561624 + 0.827393i \(0.689823\pi\)
\(462\) 0 0
\(463\) 5792.00 0.581376 0.290688 0.956818i \(-0.406116\pi\)
0.290688 + 0.956818i \(0.406116\pi\)
\(464\) 0 0
\(465\) −25600.0 −2.55306
\(466\) 0 0
\(467\) −2216.00 −0.219581 −0.109790 0.993955i \(-0.535018\pi\)
−0.109790 + 0.993955i \(0.535018\pi\)
\(468\) 0 0
\(469\) −12416.0 −1.22243
\(470\) 0 0
\(471\) 14320.0 1.40091
\(472\) 0 0
\(473\) −6080.00 −0.591033
\(474\) 0 0
\(475\) 1000.00 0.0965961
\(476\) 0 0
\(477\) 15170.0 1.45616
\(478\) 0 0
\(479\) −10560.0 −1.00730 −0.503652 0.863907i \(-0.668011\pi\)
−0.503652 + 0.863907i \(0.668011\pi\)
\(480\) 0 0
\(481\) −15500.0 −1.46931
\(482\) 0 0
\(483\) −6144.00 −0.578803
\(484\) 0 0
\(485\) −1100.00 −0.102986
\(486\) 0 0
\(487\) 13264.0 1.23419 0.617094 0.786890i \(-0.288310\pi\)
0.617094 + 0.786890i \(0.288310\pi\)
\(488\) 0 0
\(489\) −9664.00 −0.893704
\(490\) 0 0
\(491\) 4840.00 0.444860 0.222430 0.974949i \(-0.428601\pi\)
0.222430 + 0.974949i \(0.428601\pi\)
\(492\) 0 0
\(493\) −1020.00 −0.0931815
\(494\) 0 0
\(495\) 14800.0 1.34386
\(496\) 0 0
\(497\) 6400.00 0.577624
\(498\) 0 0
\(499\) −19560.0 −1.75476 −0.877381 0.479795i \(-0.840711\pi\)
−0.877381 + 0.479795i \(0.840711\pi\)
\(500\) 0 0
\(501\) −23168.0 −2.06601
\(502\) 0 0
\(503\) −528.000 −0.0468039 −0.0234019 0.999726i \(-0.507450\pi\)
−0.0234019 + 0.999726i \(0.507450\pi\)
\(504\) 0 0
\(505\) 10980.0 0.967532
\(506\) 0 0
\(507\) −2424.00 −0.212335
\(508\) 0 0
\(509\) 19554.0 1.70278 0.851391 0.524532i \(-0.175760\pi\)
0.851391 + 0.524532i \(0.175760\pi\)
\(510\) 0 0
\(511\) −10080.0 −0.872628
\(512\) 0 0
\(513\) 3200.00 0.275406
\(514\) 0 0
\(515\) −480.000 −0.0410705
\(516\) 0 0
\(517\) −16640.0 −1.41552
\(518\) 0 0
\(519\) 6000.00 0.507458
\(520\) 0 0
\(521\) 15162.0 1.27497 0.637485 0.770463i \(-0.279975\pi\)
0.637485 + 0.770463i \(0.279975\pi\)
\(522\) 0 0
\(523\) −10968.0 −0.917012 −0.458506 0.888691i \(-0.651615\pi\)
−0.458506 + 0.888691i \(0.651615\pi\)
\(524\) 0 0
\(525\) 3200.00 0.266018
\(526\) 0 0
\(527\) −9600.00 −0.793515
\(528\) 0 0
\(529\) −9863.00 −0.810635
\(530\) 0 0
\(531\) 7400.00 0.604770
\(532\) 0 0
\(533\) 20500.0 1.66595
\(534\) 0 0
\(535\) −6640.00 −0.536584
\(536\) 0 0
\(537\) 18240.0 1.46576
\(538\) 0 0
\(539\) −3480.00 −0.278097
\(540\) 0 0
\(541\) 6722.00 0.534198 0.267099 0.963669i \(-0.413935\pi\)
0.267099 + 0.963669i \(0.413935\pi\)
\(542\) 0 0
\(543\) −3536.00 −0.279455
\(544\) 0 0
\(545\) 3700.00 0.290808
\(546\) 0 0
\(547\) −20424.0 −1.59647 −0.798233 0.602348i \(-0.794232\pi\)
−0.798233 + 0.602348i \(0.794232\pi\)
\(548\) 0 0
\(549\) −1110.00 −0.0862908
\(550\) 0 0
\(551\) −1360.00 −0.105151
\(552\) 0 0
\(553\) −17920.0 −1.37800
\(554\) 0 0
\(555\) 24800.0 1.89676
\(556\) 0 0
\(557\) 6610.00 0.502827 0.251414 0.967880i \(-0.419105\pi\)
0.251414 + 0.967880i \(0.419105\pi\)
\(558\) 0 0
\(559\) −7600.00 −0.575037
\(560\) 0 0
\(561\) 9600.00 0.722482
\(562\) 0 0
\(563\) 2712.00 0.203015 0.101507 0.994835i \(-0.467633\pi\)
0.101507 + 0.994835i \(0.467633\pi\)
\(564\) 0 0
\(565\) 14900.0 1.10946
\(566\) 0 0
\(567\) −5744.00 −0.425441
\(568\) 0 0
\(569\) 3530.00 0.260080 0.130040 0.991509i \(-0.458489\pi\)
0.130040 + 0.991509i \(0.458489\pi\)
\(570\) 0 0
\(571\) 13640.0 0.999678 0.499839 0.866118i \(-0.333392\pi\)
0.499839 + 0.866118i \(0.333392\pi\)
\(572\) 0 0
\(573\) 15360.0 1.11985
\(574\) 0 0
\(575\) −1200.00 −0.0870321
\(576\) 0 0
\(577\) −6270.00 −0.452380 −0.226190 0.974083i \(-0.572627\pi\)
−0.226190 + 0.974083i \(0.572627\pi\)
\(578\) 0 0
\(579\) 40560.0 2.91125
\(580\) 0 0
\(581\) −8832.00 −0.630659
\(582\) 0 0
\(583\) 16400.0 1.16504
\(584\) 0 0
\(585\) 18500.0 1.30749
\(586\) 0 0
\(587\) 8616.00 0.605827 0.302913 0.953018i \(-0.402041\pi\)
0.302913 + 0.953018i \(0.402041\pi\)
\(588\) 0 0
\(589\) −12800.0 −0.895441
\(590\) 0 0
\(591\) 15280.0 1.06351
\(592\) 0 0
\(593\) 5490.00 0.380181 0.190090 0.981767i \(-0.439122\pi\)
0.190090 + 0.981767i \(0.439122\pi\)
\(594\) 0 0
\(595\) −4800.00 −0.330724
\(596\) 0 0
\(597\) −23680.0 −1.62338
\(598\) 0 0
\(599\) −15440.0 −1.05319 −0.526595 0.850116i \(-0.676532\pi\)
−0.526595 + 0.850116i \(0.676532\pi\)
\(600\) 0 0
\(601\) 8890.00 0.603379 0.301689 0.953406i \(-0.402449\pi\)
0.301689 + 0.953406i \(0.402449\pi\)
\(602\) 0 0
\(603\) −28712.0 −1.93904
\(604\) 0 0
\(605\) 2690.00 0.180767
\(606\) 0 0
\(607\) 23744.0 1.58771 0.793854 0.608108i \(-0.208071\pi\)
0.793854 + 0.608108i \(0.208071\pi\)
\(608\) 0 0
\(609\) −4352.00 −0.289576
\(610\) 0 0
\(611\) −20800.0 −1.37721
\(612\) 0 0
\(613\) 15210.0 1.00216 0.501082 0.865400i \(-0.332936\pi\)
0.501082 + 0.865400i \(0.332936\pi\)
\(614\) 0 0
\(615\) −32800.0 −2.15061
\(616\) 0 0
\(617\) −12630.0 −0.824092 −0.412046 0.911163i \(-0.635186\pi\)
−0.412046 + 0.911163i \(0.635186\pi\)
\(618\) 0 0
\(619\) −11160.0 −0.724650 −0.362325 0.932052i \(-0.618017\pi\)
−0.362325 + 0.932052i \(0.618017\pi\)
\(620\) 0 0
\(621\) −3840.00 −0.248138
\(622\) 0 0
\(623\) −5216.00 −0.335433
\(624\) 0 0
\(625\) −11875.0 −0.760000
\(626\) 0 0
\(627\) 12800.0 0.815284
\(628\) 0 0
\(629\) 9300.00 0.589531
\(630\) 0 0
\(631\) 13040.0 0.822685 0.411342 0.911481i \(-0.365060\pi\)
0.411342 + 0.911481i \(0.365060\pi\)
\(632\) 0 0
\(633\) 320.000 0.0200930
\(634\) 0 0
\(635\) −10240.0 −0.639940
\(636\) 0 0
\(637\) −4350.00 −0.270570
\(638\) 0 0
\(639\) 14800.0 0.916242
\(640\) 0 0
\(641\) −16910.0 −1.04197 −0.520987 0.853565i \(-0.674436\pi\)
−0.520987 + 0.853565i \(0.674436\pi\)
\(642\) 0 0
\(643\) −4488.00 −0.275256 −0.137628 0.990484i \(-0.543948\pi\)
−0.137628 + 0.990484i \(0.543948\pi\)
\(644\) 0 0
\(645\) 12160.0 0.742325
\(646\) 0 0
\(647\) 2064.00 0.125416 0.0627080 0.998032i \(-0.480026\pi\)
0.0627080 + 0.998032i \(0.480026\pi\)
\(648\) 0 0
\(649\) 8000.00 0.483864
\(650\) 0 0
\(651\) −40960.0 −2.46597
\(652\) 0 0
\(653\) −4270.00 −0.255893 −0.127946 0.991781i \(-0.540839\pi\)
−0.127946 + 0.991781i \(0.540839\pi\)
\(654\) 0 0
\(655\) −11600.0 −0.691984
\(656\) 0 0
\(657\) −23310.0 −1.38419
\(658\) 0 0
\(659\) 19800.0 1.17041 0.585204 0.810886i \(-0.301015\pi\)
0.585204 + 0.810886i \(0.301015\pi\)
\(660\) 0 0
\(661\) −27110.0 −1.59524 −0.797622 0.603157i \(-0.793909\pi\)
−0.797622 + 0.603157i \(0.793909\pi\)
\(662\) 0 0
\(663\) 12000.0 0.702928
\(664\) 0 0
\(665\) −6400.00 −0.373205
\(666\) 0 0
\(667\) 1632.00 0.0947396
\(668\) 0 0
\(669\) −34304.0 −1.98247
\(670\) 0 0
\(671\) −1200.00 −0.0690395
\(672\) 0 0
\(673\) 32210.0 1.84488 0.922440 0.386140i \(-0.126192\pi\)
0.922440 + 0.386140i \(0.126192\pi\)
\(674\) 0 0
\(675\) 2000.00 0.114044
\(676\) 0 0
\(677\) −27190.0 −1.54357 −0.771785 0.635884i \(-0.780636\pi\)
−0.771785 + 0.635884i \(0.780636\pi\)
\(678\) 0 0
\(679\) −1760.00 −0.0994736
\(680\) 0 0
\(681\) −51648.0 −2.90625
\(682\) 0 0
\(683\) 20328.0 1.13884 0.569421 0.822046i \(-0.307167\pi\)
0.569421 + 0.822046i \(0.307167\pi\)
\(684\) 0 0
\(685\) 5700.00 0.317935
\(686\) 0 0
\(687\) −8528.00 −0.473600
\(688\) 0 0
\(689\) 20500.0 1.13351
\(690\) 0 0
\(691\) −12520.0 −0.689267 −0.344633 0.938737i \(-0.611997\pi\)
−0.344633 + 0.938737i \(0.611997\pi\)
\(692\) 0 0
\(693\) 23680.0 1.29802
\(694\) 0 0
\(695\) 19600.0 1.06974
\(696\) 0 0
\(697\) −12300.0 −0.668430
\(698\) 0 0
\(699\) 47280.0 2.55836
\(700\) 0 0
\(701\) −11550.0 −0.622307 −0.311154 0.950360i \(-0.600715\pi\)
−0.311154 + 0.950360i \(0.600715\pi\)
\(702\) 0 0
\(703\) 12400.0 0.665256
\(704\) 0 0
\(705\) 33280.0 1.77787
\(706\) 0 0
\(707\) 17568.0 0.934530
\(708\) 0 0
\(709\) 34154.0 1.80914 0.904570 0.426325i \(-0.140192\pi\)
0.904570 + 0.426325i \(0.140192\pi\)
\(710\) 0 0
\(711\) −41440.0 −2.18582
\(712\) 0 0
\(713\) 15360.0 0.806783
\(714\) 0 0
\(715\) 20000.0 1.04609
\(716\) 0 0
\(717\) 26880.0 1.40007
\(718\) 0 0
\(719\) −22880.0 −1.18676 −0.593380 0.804923i \(-0.702207\pi\)
−0.593380 + 0.804923i \(0.702207\pi\)
\(720\) 0 0
\(721\) −768.000 −0.0396696
\(722\) 0 0
\(723\) −31760.0 −1.63370
\(724\) 0 0
\(725\) −850.000 −0.0435424
\(726\) 0 0
\(727\) 10416.0 0.531373 0.265686 0.964060i \(-0.414401\pi\)
0.265686 + 0.964060i \(0.414401\pi\)
\(728\) 0 0
\(729\) −30563.0 −1.55276
\(730\) 0 0
\(731\) 4560.00 0.230722
\(732\) 0 0
\(733\) −14750.0 −0.743252 −0.371626 0.928383i \(-0.621200\pi\)
−0.371626 + 0.928383i \(0.621200\pi\)
\(734\) 0 0
\(735\) 6960.00 0.349284
\(736\) 0 0
\(737\) −31040.0 −1.55139
\(738\) 0 0
\(739\) 2360.00 0.117475 0.0587375 0.998273i \(-0.481293\pi\)
0.0587375 + 0.998273i \(0.481293\pi\)
\(740\) 0 0
\(741\) 16000.0 0.793218
\(742\) 0 0
\(743\) 32208.0 1.59031 0.795153 0.606409i \(-0.207391\pi\)
0.795153 + 0.606409i \(0.207391\pi\)
\(744\) 0 0
\(745\) 20100.0 0.988466
\(746\) 0 0
\(747\) −20424.0 −1.00037
\(748\) 0 0
\(749\) −10624.0 −0.518281
\(750\) 0 0
\(751\) −36640.0 −1.78031 −0.890155 0.455658i \(-0.849404\pi\)
−0.890155 + 0.455658i \(0.849404\pi\)
\(752\) 0 0
\(753\) 54720.0 2.64822
\(754\) 0 0
\(755\) −7200.00 −0.347066
\(756\) 0 0
\(757\) 12090.0 0.580474 0.290237 0.956955i \(-0.406266\pi\)
0.290237 + 0.956955i \(0.406266\pi\)
\(758\) 0 0
\(759\) −15360.0 −0.734562
\(760\) 0 0
\(761\) −3318.00 −0.158052 −0.0790259 0.996873i \(-0.525181\pi\)
−0.0790259 + 0.996873i \(0.525181\pi\)
\(762\) 0 0
\(763\) 5920.00 0.280889
\(764\) 0 0
\(765\) −11100.0 −0.524603
\(766\) 0 0
\(767\) 10000.0 0.470768
\(768\) 0 0
\(769\) 11506.0 0.539554 0.269777 0.962923i \(-0.413050\pi\)
0.269777 + 0.962923i \(0.413050\pi\)
\(770\) 0 0
\(771\) −36880.0 −1.72270
\(772\) 0 0
\(773\) −22230.0 −1.03436 −0.517178 0.855878i \(-0.673018\pi\)
−0.517178 + 0.855878i \(0.673018\pi\)
\(774\) 0 0
\(775\) −8000.00 −0.370798
\(776\) 0 0
\(777\) 39680.0 1.83206
\(778\) 0 0
\(779\) −16400.0 −0.754289
\(780\) 0 0
\(781\) 16000.0 0.733067
\(782\) 0 0
\(783\) −2720.00 −0.124144
\(784\) 0 0
\(785\) −17900.0 −0.813858
\(786\) 0 0
\(787\) 21336.0 0.966387 0.483193 0.875514i \(-0.339477\pi\)
0.483193 + 0.875514i \(0.339477\pi\)
\(788\) 0 0
\(789\) 38784.0 1.75000
\(790\) 0 0
\(791\) 23840.0 1.07162
\(792\) 0 0
\(793\) −1500.00 −0.0671709
\(794\) 0 0
\(795\) −32800.0 −1.46327
\(796\) 0 0
\(797\) 7170.00 0.318663 0.159332 0.987225i \(-0.449066\pi\)
0.159332 + 0.987225i \(0.449066\pi\)
\(798\) 0 0
\(799\) 12480.0 0.552579
\(800\) 0 0
\(801\) −12062.0 −0.532072
\(802\) 0 0
\(803\) −25200.0 −1.10746
\(804\) 0 0
\(805\) 7680.00 0.336254
\(806\) 0 0
\(807\) 44400.0 1.93675
\(808\) 0 0
\(809\) −23654.0 −1.02797 −0.513987 0.857798i \(-0.671832\pi\)
−0.513987 + 0.857798i \(0.671832\pi\)
\(810\) 0 0
\(811\) 30440.0 1.31799 0.658997 0.752146i \(-0.270981\pi\)
0.658997 + 0.752146i \(0.270981\pi\)
\(812\) 0 0
\(813\) 3840.00 0.165652
\(814\) 0 0
\(815\) 12080.0 0.519195
\(816\) 0 0
\(817\) 6080.00 0.260358
\(818\) 0 0
\(819\) 29600.0 1.26289
\(820\) 0 0
\(821\) 19930.0 0.847213 0.423606 0.905846i \(-0.360764\pi\)
0.423606 + 0.905846i \(0.360764\pi\)
\(822\) 0 0
\(823\) −9872.00 −0.418124 −0.209062 0.977902i \(-0.567041\pi\)
−0.209062 + 0.977902i \(0.567041\pi\)
\(824\) 0 0
\(825\) 8000.00 0.337605
\(826\) 0 0
\(827\) 5704.00 0.239840 0.119920 0.992784i \(-0.461736\pi\)
0.119920 + 0.992784i \(0.461736\pi\)
\(828\) 0 0
\(829\) −27230.0 −1.14082 −0.570408 0.821361i \(-0.693215\pi\)
−0.570408 + 0.821361i \(0.693215\pi\)
\(830\) 0 0
\(831\) 8240.00 0.343974
\(832\) 0 0
\(833\) 2610.00 0.108561
\(834\) 0 0
\(835\) 28960.0 1.20024
\(836\) 0 0
\(837\) −25600.0 −1.05719
\(838\) 0 0
\(839\) −18800.0 −0.773597 −0.386799 0.922164i \(-0.626419\pi\)
−0.386799 + 0.922164i \(0.626419\pi\)
\(840\) 0 0
\(841\) −23233.0 −0.952602
\(842\) 0 0
\(843\) 26160.0 1.06880
\(844\) 0 0
\(845\) 3030.00 0.123355
\(846\) 0 0
\(847\) 4304.00 0.174601
\(848\) 0 0
\(849\) 17344.0 0.701113
\(850\) 0 0
\(851\) −14880.0 −0.599389
\(852\) 0 0
\(853\) 12090.0 0.485292 0.242646 0.970115i \(-0.421985\pi\)
0.242646 + 0.970115i \(0.421985\pi\)
\(854\) 0 0
\(855\) −14800.0 −0.591988
\(856\) 0 0
\(857\) −470.000 −0.0187338 −0.00936692 0.999956i \(-0.502982\pi\)
−0.00936692 + 0.999956i \(0.502982\pi\)
\(858\) 0 0
\(859\) −24440.0 −0.970759 −0.485380 0.874304i \(-0.661319\pi\)
−0.485380 + 0.874304i \(0.661319\pi\)
\(860\) 0 0
\(861\) −52480.0 −2.07725
\(862\) 0 0
\(863\) −22592.0 −0.891125 −0.445562 0.895251i \(-0.646996\pi\)
−0.445562 + 0.895251i \(0.646996\pi\)
\(864\) 0 0
\(865\) −7500.00 −0.294807
\(866\) 0 0
\(867\) 32104.0 1.25757
\(868\) 0 0
\(869\) −44800.0 −1.74883
\(870\) 0 0
\(871\) −38800.0 −1.50940
\(872\) 0 0
\(873\) −4070.00 −0.157788
\(874\) 0 0
\(875\) −24000.0 −0.927255
\(876\) 0 0
\(877\) 17330.0 0.667266 0.333633 0.942703i \(-0.391725\pi\)
0.333633 + 0.942703i \(0.391725\pi\)
\(878\) 0 0
\(879\) 16560.0 0.635444
\(880\) 0 0
\(881\) −31470.0 −1.20346 −0.601732 0.798698i \(-0.705522\pi\)
−0.601732 + 0.798698i \(0.705522\pi\)
\(882\) 0 0
\(883\) 3352.00 0.127751 0.0638753 0.997958i \(-0.479654\pi\)
0.0638753 + 0.997958i \(0.479654\pi\)
\(884\) 0 0
\(885\) −16000.0 −0.607722
\(886\) 0 0
\(887\) −48144.0 −1.82245 −0.911227 0.411904i \(-0.864864\pi\)
−0.911227 + 0.411904i \(0.864864\pi\)
\(888\) 0 0
\(889\) −16384.0 −0.618112
\(890\) 0 0
\(891\) −14360.0 −0.539931
\(892\) 0 0
\(893\) 16640.0 0.623557
\(894\) 0 0
\(895\) −22800.0 −0.851531
\(896\) 0 0
\(897\) −19200.0 −0.714682
\(898\) 0 0
\(899\) 10880.0 0.403636
\(900\) 0 0
\(901\) −12300.0 −0.454797
\(902\) 0 0
\(903\) 19456.0 0.717005
\(904\) 0 0
\(905\) 4420.00 0.162349
\(906\) 0 0
\(907\) −16216.0 −0.593653 −0.296827 0.954931i \(-0.595928\pi\)
−0.296827 + 0.954931i \(0.595928\pi\)
\(908\) 0 0
\(909\) 40626.0 1.48238
\(910\) 0 0
\(911\) 49440.0 1.79805 0.899023 0.437901i \(-0.144278\pi\)
0.899023 + 0.437901i \(0.144278\pi\)
\(912\) 0 0
\(913\) −22080.0 −0.800374
\(914\) 0 0
\(915\) 2400.00 0.0867121
\(916\) 0 0
\(917\) −18560.0 −0.668381
\(918\) 0 0
\(919\) −16080.0 −0.577182 −0.288591 0.957452i \(-0.593187\pi\)
−0.288591 + 0.957452i \(0.593187\pi\)
\(920\) 0 0
\(921\) 15168.0 0.542674
\(922\) 0 0
\(923\) 20000.0 0.713226
\(924\) 0 0
\(925\) 7750.00 0.275479
\(926\) 0 0
\(927\) −1776.00 −0.0629250
\(928\) 0 0
\(929\) −11310.0 −0.399428 −0.199714 0.979854i \(-0.564001\pi\)
−0.199714 + 0.979854i \(0.564001\pi\)
\(930\) 0 0
\(931\) 3480.00 0.122505
\(932\) 0 0
\(933\) 13440.0 0.471603
\(934\) 0 0
\(935\) −12000.0 −0.419724
\(936\) 0 0
\(937\) 25130.0 0.876159 0.438080 0.898936i \(-0.355659\pi\)
0.438080 + 0.898936i \(0.355659\pi\)
\(938\) 0 0
\(939\) −7760.00 −0.269689
\(940\) 0 0
\(941\) 22322.0 0.773301 0.386651 0.922226i \(-0.373632\pi\)
0.386651 + 0.922226i \(0.373632\pi\)
\(942\) 0 0
\(943\) 19680.0 0.679607
\(944\) 0 0
\(945\) −12800.0 −0.440618
\(946\) 0 0
\(947\) −36456.0 −1.25096 −0.625481 0.780239i \(-0.715097\pi\)
−0.625481 + 0.780239i \(0.715097\pi\)
\(948\) 0 0
\(949\) −31500.0 −1.07749
\(950\) 0 0
\(951\) 57840.0 1.97223
\(952\) 0 0
\(953\) 40650.0 1.38172 0.690862 0.722987i \(-0.257231\pi\)
0.690862 + 0.722987i \(0.257231\pi\)
\(954\) 0 0
\(955\) −19200.0 −0.650573
\(956\) 0 0
\(957\) −10880.0 −0.367503
\(958\) 0 0
\(959\) 9120.00 0.307091
\(960\) 0 0
\(961\) 72609.0 2.43728
\(962\) 0 0
\(963\) −24568.0 −0.822111
\(964\) 0 0
\(965\) −50700.0 −1.69129
\(966\) 0 0
\(967\) 34704.0 1.15409 0.577045 0.816712i \(-0.304206\pi\)
0.577045 + 0.816712i \(0.304206\pi\)
\(968\) 0 0
\(969\) −9600.00 −0.318263
\(970\) 0 0
\(971\) 30760.0 1.01662 0.508309 0.861175i \(-0.330271\pi\)
0.508309 + 0.861175i \(0.330271\pi\)
\(972\) 0 0
\(973\) 31360.0 1.03325
\(974\) 0 0
\(975\) 10000.0 0.328468
\(976\) 0 0
\(977\) −38110.0 −1.24795 −0.623975 0.781444i \(-0.714483\pi\)
−0.623975 + 0.781444i \(0.714483\pi\)
\(978\) 0 0
\(979\) −13040.0 −0.425700
\(980\) 0 0
\(981\) 13690.0 0.445554
\(982\) 0 0
\(983\) 19632.0 0.636992 0.318496 0.947924i \(-0.396822\pi\)
0.318496 + 0.947924i \(0.396822\pi\)
\(984\) 0 0
\(985\) −19100.0 −0.617844
\(986\) 0 0
\(987\) 53248.0 1.71723
\(988\) 0 0
\(989\) −7296.00 −0.234580
\(990\) 0 0
\(991\) −47680.0 −1.52836 −0.764180 0.645003i \(-0.776856\pi\)
−0.764180 + 0.645003i \(0.776856\pi\)
\(992\) 0 0
\(993\) −46400.0 −1.48284
\(994\) 0 0
\(995\) 29600.0 0.943099
\(996\) 0 0
\(997\) 39690.0 1.26078 0.630389 0.776280i \(-0.282896\pi\)
0.630389 + 0.776280i \(0.282896\pi\)
\(998\) 0 0
\(999\) 24800.0 0.785423
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 64.4.a.a.1.1 1
3.2 odd 2 576.4.a.h.1.1 1
4.3 odd 2 64.4.a.e.1.1 1
5.4 even 2 1600.4.a.bw.1.1 1
8.3 odd 2 32.4.a.a.1.1 1
8.5 even 2 32.4.a.c.1.1 yes 1
12.11 even 2 576.4.a.g.1.1 1
16.3 odd 4 256.4.b.e.129.2 2
16.5 even 4 256.4.b.c.129.2 2
16.11 odd 4 256.4.b.e.129.1 2
16.13 even 4 256.4.b.c.129.1 2
20.19 odd 2 1600.4.a.e.1.1 1
24.5 odd 2 288.4.a.i.1.1 1
24.11 even 2 288.4.a.h.1.1 1
40.3 even 4 800.4.c.b.449.1 2
40.13 odd 4 800.4.c.a.449.2 2
40.19 odd 2 800.4.a.k.1.1 1
40.27 even 4 800.4.c.b.449.2 2
40.29 even 2 800.4.a.a.1.1 1
40.37 odd 4 800.4.c.a.449.1 2
56.13 odd 2 1568.4.a.c.1.1 1
56.27 even 2 1568.4.a.o.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
32.4.a.a.1.1 1 8.3 odd 2
32.4.a.c.1.1 yes 1 8.5 even 2
64.4.a.a.1.1 1 1.1 even 1 trivial
64.4.a.e.1.1 1 4.3 odd 2
256.4.b.c.129.1 2 16.13 even 4
256.4.b.c.129.2 2 16.5 even 4
256.4.b.e.129.1 2 16.11 odd 4
256.4.b.e.129.2 2 16.3 odd 4
288.4.a.h.1.1 1 24.11 even 2
288.4.a.i.1.1 1 24.5 odd 2
576.4.a.g.1.1 1 12.11 even 2
576.4.a.h.1.1 1 3.2 odd 2
800.4.a.a.1.1 1 40.29 even 2
800.4.a.k.1.1 1 40.19 odd 2
800.4.c.a.449.1 2 40.37 odd 4
800.4.c.a.449.2 2 40.13 odd 4
800.4.c.b.449.1 2 40.3 even 4
800.4.c.b.449.2 2 40.27 even 4
1568.4.a.c.1.1 1 56.13 odd 2
1568.4.a.o.1.1 1 56.27 even 2
1600.4.a.e.1.1 1 20.19 odd 2
1600.4.a.bw.1.1 1 5.4 even 2