Properties

Label 256.4.a.f.1.1
Level $256$
Weight $4$
Character 256.1
Self dual yes
Analytic conductor $15.104$
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] = [256,4,Mod(1,256)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(256, 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("256.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 256.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: \(15.1044889615\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 128)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 256.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+8.00000 q^{3} -12.0000 q^{5} -32.0000 q^{7} +37.0000 q^{9} +O(q^{10})\) \(q+8.00000 q^{3} -12.0000 q^{5} -32.0000 q^{7} +37.0000 q^{9} -8.00000 q^{11} +20.0000 q^{13} -96.0000 q^{15} -98.0000 q^{17} -88.0000 q^{19} -256.000 q^{21} +32.0000 q^{23} +19.0000 q^{25} +80.0000 q^{27} -172.000 q^{29} +256.000 q^{31} -64.0000 q^{33} +384.000 q^{35} -92.0000 q^{37} +160.000 q^{39} +102.000 q^{41} -296.000 q^{43} -444.000 q^{45} +320.000 q^{47} +681.000 q^{49} -784.000 q^{51} -76.0000 q^{53} +96.0000 q^{55} -704.000 q^{57} +408.000 q^{59} -636.000 q^{61} -1184.00 q^{63} -240.000 q^{65} +552.000 q^{67} +256.000 q^{69} -416.000 q^{71} +138.000 q^{73} +152.000 q^{75} +256.000 q^{77} +64.0000 q^{79} -359.000 q^{81} +392.000 q^{83} +1176.00 q^{85} -1376.00 q^{87} -582.000 q^{89} -640.000 q^{91} +2048.00 q^{93} +1056.00 q^{95} +238.000 q^{97} -296.000 q^{99} +O(q^{100})\)

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 \(-0.220356\pi\)
0.769800 + 0.638285i \(0.220356\pi\)
\(4\) 0 0
\(5\) −12.0000 −1.07331 −0.536656 0.843801i \(-0.680313\pi\)
−0.536656 + 0.843801i \(0.680313\pi\)
\(6\) 0 0
\(7\) −32.0000 −1.72784 −0.863919 0.503631i \(-0.831997\pi\)
−0.863919 + 0.503631i \(0.831997\pi\)
\(8\) 0 0
\(9\) 37.0000 1.37037
\(10\) 0 0
\(11\) −8.00000 −0.219281 −0.109640 0.993971i \(-0.534970\pi\)
−0.109640 + 0.993971i \(0.534970\pi\)
\(12\) 0 0
\(13\) 20.0000 0.426692 0.213346 0.976977i \(-0.431564\pi\)
0.213346 + 0.976977i \(0.431564\pi\)
\(14\) 0 0
\(15\) −96.0000 −1.65247
\(16\) 0 0
\(17\) −98.0000 −1.39815 −0.699073 0.715050i \(-0.746404\pi\)
−0.699073 + 0.715050i \(0.746404\pi\)
\(18\) 0 0
\(19\) −88.0000 −1.06256 −0.531279 0.847197i \(-0.678288\pi\)
−0.531279 + 0.847197i \(0.678288\pi\)
\(20\) 0 0
\(21\) −256.000 −2.66018
\(22\) 0 0
\(23\) 32.0000 0.290107 0.145054 0.989424i \(-0.453665\pi\)
0.145054 + 0.989424i \(0.453665\pi\)
\(24\) 0 0
\(25\) 19.0000 0.152000
\(26\) 0 0
\(27\) 80.0000 0.570222
\(28\) 0 0
\(29\) −172.000 −1.10137 −0.550683 0.834715i \(-0.685633\pi\)
−0.550683 + 0.834715i \(0.685633\pi\)
\(30\) 0 0
\(31\) 256.000 1.48319 0.741596 0.670847i \(-0.234069\pi\)
0.741596 + 0.670847i \(0.234069\pi\)
\(32\) 0 0
\(33\) −64.0000 −0.337605
\(34\) 0 0
\(35\) 384.000 1.85451
\(36\) 0 0
\(37\) −92.0000 −0.408776 −0.204388 0.978890i \(-0.565520\pi\)
−0.204388 + 0.978890i \(0.565520\pi\)
\(38\) 0 0
\(39\) 160.000 0.656936
\(40\) 0 0
\(41\) 102.000 0.388530 0.194265 0.980949i \(-0.437768\pi\)
0.194265 + 0.980949i \(0.437768\pi\)
\(42\) 0 0
\(43\) −296.000 −1.04976 −0.524879 0.851177i \(-0.675889\pi\)
−0.524879 + 0.851177i \(0.675889\pi\)
\(44\) 0 0
\(45\) −444.000 −1.47084
\(46\) 0 0
\(47\) 320.000 0.993123 0.496562 0.868001i \(-0.334596\pi\)
0.496562 + 0.868001i \(0.334596\pi\)
\(48\) 0 0
\(49\) 681.000 1.98542
\(50\) 0 0
\(51\) −784.000 −2.15259
\(52\) 0 0
\(53\) −76.0000 −0.196970 −0.0984849 0.995139i \(-0.531400\pi\)
−0.0984849 + 0.995139i \(0.531400\pi\)
\(54\) 0 0
\(55\) 96.0000 0.235357
\(56\) 0 0
\(57\) −704.000 −1.63591
\(58\) 0 0
\(59\) 408.000 0.900289 0.450145 0.892956i \(-0.351372\pi\)
0.450145 + 0.892956i \(0.351372\pi\)
\(60\) 0 0
\(61\) −636.000 −1.33494 −0.667471 0.744636i \(-0.732623\pi\)
−0.667471 + 0.744636i \(0.732623\pi\)
\(62\) 0 0
\(63\) −1184.00 −2.36778
\(64\) 0 0
\(65\) −240.000 −0.457974
\(66\) 0 0
\(67\) 552.000 1.00653 0.503265 0.864132i \(-0.332132\pi\)
0.503265 + 0.864132i \(0.332132\pi\)
\(68\) 0 0
\(69\) 256.000 0.446649
\(70\) 0 0
\(71\) −416.000 −0.695354 −0.347677 0.937614i \(-0.613029\pi\)
−0.347677 + 0.937614i \(0.613029\pi\)
\(72\) 0 0
\(73\) 138.000 0.221256 0.110628 0.993862i \(-0.464714\pi\)
0.110628 + 0.993862i \(0.464714\pi\)
\(74\) 0 0
\(75\) 152.000 0.234019
\(76\) 0 0
\(77\) 256.000 0.378882
\(78\) 0 0
\(79\) 64.0000 0.0911464 0.0455732 0.998961i \(-0.485489\pi\)
0.0455732 + 0.998961i \(0.485489\pi\)
\(80\) 0 0
\(81\) −359.000 −0.492455
\(82\) 0 0
\(83\) 392.000 0.518405 0.259202 0.965823i \(-0.416540\pi\)
0.259202 + 0.965823i \(0.416540\pi\)
\(84\) 0 0
\(85\) 1176.00 1.50065
\(86\) 0 0
\(87\) −1376.00 −1.69566
\(88\) 0 0
\(89\) −582.000 −0.693167 −0.346584 0.938019i \(-0.612658\pi\)
−0.346584 + 0.938019i \(0.612658\pi\)
\(90\) 0 0
\(91\) −640.000 −0.737255
\(92\) 0 0
\(93\) 2048.00 2.28352
\(94\) 0 0
\(95\) 1056.00 1.14046
\(96\) 0 0
\(97\) 238.000 0.249126 0.124563 0.992212i \(-0.460247\pi\)
0.124563 + 0.992212i \(0.460247\pi\)
\(98\) 0 0
\(99\) −296.000 −0.300496
\(100\) 0 0
\(101\) −1468.00 −1.44625 −0.723126 0.690716i \(-0.757295\pi\)
−0.723126 + 0.690716i \(0.757295\pi\)
\(102\) 0 0
\(103\) 992.000 0.948977 0.474489 0.880262i \(-0.342633\pi\)
0.474489 + 0.880262i \(0.342633\pi\)
\(104\) 0 0
\(105\) 3072.00 2.85520
\(106\) 0 0
\(107\) −584.000 −0.527639 −0.263820 0.964572i \(-0.584982\pi\)
−0.263820 + 0.964572i \(0.584982\pi\)
\(108\) 0 0
\(109\) 740.000 0.650267 0.325134 0.945668i \(-0.394591\pi\)
0.325134 + 0.945668i \(0.394591\pi\)
\(110\) 0 0
\(111\) −736.000 −0.629352
\(112\) 0 0
\(113\) −302.000 −0.251414 −0.125707 0.992067i \(-0.540120\pi\)
−0.125707 + 0.992067i \(0.540120\pi\)
\(114\) 0 0
\(115\) −384.000 −0.311376
\(116\) 0 0
\(117\) 740.000 0.584727
\(118\) 0 0
\(119\) 3136.00 2.41577
\(120\) 0 0
\(121\) −1267.00 −0.951916
\(122\) 0 0
\(123\) 816.000 0.598181
\(124\) 0 0
\(125\) 1272.00 0.910169
\(126\) 0 0
\(127\) 1664.00 1.16265 0.581323 0.813673i \(-0.302535\pi\)
0.581323 + 0.813673i \(0.302535\pi\)
\(128\) 0 0
\(129\) −2368.00 −1.61621
\(130\) 0 0
\(131\) −2328.00 −1.55266 −0.776329 0.630327i \(-0.782921\pi\)
−0.776329 + 0.630327i \(0.782921\pi\)
\(132\) 0 0
\(133\) 2816.00 1.83593
\(134\) 0 0
\(135\) −960.000 −0.612027
\(136\) 0 0
\(137\) 1734.00 1.08135 0.540677 0.841230i \(-0.318168\pi\)
0.540677 + 0.841230i \(0.318168\pi\)
\(138\) 0 0
\(139\) 3032.00 1.85015 0.925075 0.379784i \(-0.124002\pi\)
0.925075 + 0.379784i \(0.124002\pi\)
\(140\) 0 0
\(141\) 2560.00 1.52901
\(142\) 0 0
\(143\) −160.000 −0.0935655
\(144\) 0 0
\(145\) 2064.00 1.18211
\(146\) 0 0
\(147\) 5448.00 3.05676
\(148\) 0 0
\(149\) −1788.00 −0.983078 −0.491539 0.870855i \(-0.663566\pi\)
−0.491539 + 0.870855i \(0.663566\pi\)
\(150\) 0 0
\(151\) −480.000 −0.258688 −0.129344 0.991600i \(-0.541287\pi\)
−0.129344 + 0.991600i \(0.541287\pi\)
\(152\) 0 0
\(153\) −3626.00 −1.91598
\(154\) 0 0
\(155\) −3072.00 −1.59193
\(156\) 0 0
\(157\) −2300.00 −1.16917 −0.584586 0.811332i \(-0.698743\pi\)
−0.584586 + 0.811332i \(0.698743\pi\)
\(158\) 0 0
\(159\) −608.000 −0.303255
\(160\) 0 0
\(161\) −1024.00 −0.501258
\(162\) 0 0
\(163\) −1592.00 −0.765000 −0.382500 0.923955i \(-0.624937\pi\)
−0.382500 + 0.923955i \(0.624937\pi\)
\(164\) 0 0
\(165\) 768.000 0.362356
\(166\) 0 0
\(167\) −2208.00 −1.02311 −0.511557 0.859249i \(-0.670931\pi\)
−0.511557 + 0.859249i \(0.670931\pi\)
\(168\) 0 0
\(169\) −1797.00 −0.817934
\(170\) 0 0
\(171\) −3256.00 −1.45610
\(172\) 0 0
\(173\) −3948.00 −1.73503 −0.867517 0.497408i \(-0.834285\pi\)
−0.867517 + 0.497408i \(0.834285\pi\)
\(174\) 0 0
\(175\) −608.000 −0.262631
\(176\) 0 0
\(177\) 3264.00 1.38609
\(178\) 0 0
\(179\) −2104.00 −0.878549 −0.439275 0.898353i \(-0.644765\pi\)
−0.439275 + 0.898353i \(0.644765\pi\)
\(180\) 0 0
\(181\) 1412.00 0.579852 0.289926 0.957049i \(-0.406369\pi\)
0.289926 + 0.957049i \(0.406369\pi\)
\(182\) 0 0
\(183\) −5088.00 −2.05528
\(184\) 0 0
\(185\) 1104.00 0.438744
\(186\) 0 0
\(187\) 784.000 0.306587
\(188\) 0 0
\(189\) −2560.00 −0.985252
\(190\) 0 0
\(191\) −3712.00 −1.40624 −0.703118 0.711074i \(-0.748209\pi\)
−0.703118 + 0.711074i \(0.748209\pi\)
\(192\) 0 0
\(193\) 1614.00 0.601960 0.300980 0.953630i \(-0.402686\pi\)
0.300980 + 0.953630i \(0.402686\pi\)
\(194\) 0 0
\(195\) −1920.00 −0.705098
\(196\) 0 0
\(197\) −684.000 −0.247376 −0.123688 0.992321i \(-0.539472\pi\)
−0.123688 + 0.992321i \(0.539472\pi\)
\(198\) 0 0
\(199\) 4064.00 1.44769 0.723843 0.689965i \(-0.242374\pi\)
0.723843 + 0.689965i \(0.242374\pi\)
\(200\) 0 0
\(201\) 4416.00 1.54965
\(202\) 0 0
\(203\) 5504.00 1.90298
\(204\) 0 0
\(205\) −1224.00 −0.417014
\(206\) 0 0
\(207\) 1184.00 0.397554
\(208\) 0 0
\(209\) 704.000 0.232999
\(210\) 0 0
\(211\) 2120.00 0.691691 0.345846 0.938291i \(-0.387592\pi\)
0.345846 + 0.938291i \(0.387592\pi\)
\(212\) 0 0
\(213\) −3328.00 −1.07057
\(214\) 0 0
\(215\) 3552.00 1.12672
\(216\) 0 0
\(217\) −8192.00 −2.56272
\(218\) 0 0
\(219\) 1104.00 0.340646
\(220\) 0 0
\(221\) −1960.00 −0.596579
\(222\) 0 0
\(223\) −2816.00 −0.845620 −0.422810 0.906218i \(-0.638956\pi\)
−0.422810 + 0.906218i \(0.638956\pi\)
\(224\) 0 0
\(225\) 703.000 0.208296
\(226\) 0 0
\(227\) 3848.00 1.12511 0.562557 0.826759i \(-0.309818\pi\)
0.562557 + 0.826759i \(0.309818\pi\)
\(228\) 0 0
\(229\) −652.000 −0.188146 −0.0940729 0.995565i \(-0.529989\pi\)
−0.0940729 + 0.995565i \(0.529989\pi\)
\(230\) 0 0
\(231\) 2048.00 0.583327
\(232\) 0 0
\(233\) 3050.00 0.857563 0.428781 0.903408i \(-0.358943\pi\)
0.428781 + 0.903408i \(0.358943\pi\)
\(234\) 0 0
\(235\) −3840.00 −1.06593
\(236\) 0 0
\(237\) 512.000 0.140329
\(238\) 0 0
\(239\) 6336.00 1.71482 0.857410 0.514635i \(-0.172072\pi\)
0.857410 + 0.514635i \(0.172072\pi\)
\(240\) 0 0
\(241\) −4610.00 −1.23218 −0.616092 0.787674i \(-0.711285\pi\)
−0.616092 + 0.787674i \(0.711285\pi\)
\(242\) 0 0
\(243\) −5032.00 −1.32841
\(244\) 0 0
\(245\) −8172.00 −2.13098
\(246\) 0 0
\(247\) −1760.00 −0.453385
\(248\) 0 0
\(249\) 3136.00 0.798136
\(250\) 0 0
\(251\) 792.000 0.199166 0.0995829 0.995029i \(-0.468249\pi\)
0.0995829 + 0.995029i \(0.468249\pi\)
\(252\) 0 0
\(253\) −256.000 −0.0636149
\(254\) 0 0
\(255\) 9408.00 2.31040
\(256\) 0 0
\(257\) −5374.00 −1.30436 −0.652181 0.758063i \(-0.726146\pi\)
−0.652181 + 0.758063i \(0.726146\pi\)
\(258\) 0 0
\(259\) 2944.00 0.706298
\(260\) 0 0
\(261\) −6364.00 −1.50928
\(262\) 0 0
\(263\) −3488.00 −0.817792 −0.408896 0.912581i \(-0.634086\pi\)
−0.408896 + 0.912581i \(0.634086\pi\)
\(264\) 0 0
\(265\) 912.000 0.211410
\(266\) 0 0
\(267\) −4656.00 −1.06720
\(268\) 0 0
\(269\) −4764.00 −1.07980 −0.539900 0.841729i \(-0.681538\pi\)
−0.539900 + 0.841729i \(0.681538\pi\)
\(270\) 0 0
\(271\) −1344.00 −0.301263 −0.150631 0.988590i \(-0.548131\pi\)
−0.150631 + 0.988590i \(0.548131\pi\)
\(272\) 0 0
\(273\) −5120.00 −1.13508
\(274\) 0 0
\(275\) −152.000 −0.0333307
\(276\) 0 0
\(277\) 8596.00 1.86456 0.932281 0.361735i \(-0.117816\pi\)
0.932281 + 0.361735i \(0.117816\pi\)
\(278\) 0 0
\(279\) 9472.00 2.03252
\(280\) 0 0
\(281\) 2874.00 0.610137 0.305068 0.952330i \(-0.401321\pi\)
0.305068 + 0.952330i \(0.401321\pi\)
\(282\) 0 0
\(283\) −2888.00 −0.606621 −0.303311 0.952892i \(-0.598092\pi\)
−0.303311 + 0.952892i \(0.598092\pi\)
\(284\) 0 0
\(285\) 8448.00 1.75585
\(286\) 0 0
\(287\) −3264.00 −0.671316
\(288\) 0 0
\(289\) 4691.00 0.954814
\(290\) 0 0
\(291\) 1904.00 0.383555
\(292\) 0 0
\(293\) −6540.00 −1.30400 −0.651998 0.758221i \(-0.726069\pi\)
−0.651998 + 0.758221i \(0.726069\pi\)
\(294\) 0 0
\(295\) −4896.00 −0.966292
\(296\) 0 0
\(297\) −640.000 −0.125039
\(298\) 0 0
\(299\) 640.000 0.123786
\(300\) 0 0
\(301\) 9472.00 1.81381
\(302\) 0 0
\(303\) −11744.0 −2.22665
\(304\) 0 0
\(305\) 7632.00 1.43281
\(306\) 0 0
\(307\) −10584.0 −1.96762 −0.983812 0.179202i \(-0.942649\pi\)
−0.983812 + 0.179202i \(0.942649\pi\)
\(308\) 0 0
\(309\) 7936.00 1.46105
\(310\) 0 0
\(311\) −6368.00 −1.16108 −0.580540 0.814231i \(-0.697159\pi\)
−0.580540 + 0.814231i \(0.697159\pi\)
\(312\) 0 0
\(313\) 8758.00 1.58157 0.790785 0.612094i \(-0.209673\pi\)
0.790785 + 0.612094i \(0.209673\pi\)
\(314\) 0 0
\(315\) 14208.0 2.54137
\(316\) 0 0
\(317\) −716.000 −0.126860 −0.0634299 0.997986i \(-0.520204\pi\)
−0.0634299 + 0.997986i \(0.520204\pi\)
\(318\) 0 0
\(319\) 1376.00 0.241508
\(320\) 0 0
\(321\) −4672.00 −0.812354
\(322\) 0 0
\(323\) 8624.00 1.48561
\(324\) 0 0
\(325\) 380.000 0.0648573
\(326\) 0 0
\(327\) 5920.00 1.00115
\(328\) 0 0
\(329\) −10240.0 −1.71596
\(330\) 0 0
\(331\) 4408.00 0.731981 0.365990 0.930619i \(-0.380730\pi\)
0.365990 + 0.930619i \(0.380730\pi\)
\(332\) 0 0
\(333\) −3404.00 −0.560174
\(334\) 0 0
\(335\) −6624.00 −1.08032
\(336\) 0 0
\(337\) 1202.00 0.194294 0.0971471 0.995270i \(-0.469028\pi\)
0.0971471 + 0.995270i \(0.469028\pi\)
\(338\) 0 0
\(339\) −2416.00 −0.387077
\(340\) 0 0
\(341\) −2048.00 −0.325236
\(342\) 0 0
\(343\) −10816.0 −1.70265
\(344\) 0 0
\(345\) −3072.00 −0.479394
\(346\) 0 0
\(347\) −5160.00 −0.798280 −0.399140 0.916890i \(-0.630691\pi\)
−0.399140 + 0.916890i \(0.630691\pi\)
\(348\) 0 0
\(349\) −4876.00 −0.747869 −0.373935 0.927455i \(-0.621992\pi\)
−0.373935 + 0.927455i \(0.621992\pi\)
\(350\) 0 0
\(351\) 1600.00 0.243310
\(352\) 0 0
\(353\) 4834.00 0.728861 0.364430 0.931231i \(-0.381264\pi\)
0.364430 + 0.931231i \(0.381264\pi\)
\(354\) 0 0
\(355\) 4992.00 0.746332
\(356\) 0 0
\(357\) 25088.0 3.71932
\(358\) 0 0
\(359\) −4128.00 −0.606873 −0.303437 0.952852i \(-0.598134\pi\)
−0.303437 + 0.952852i \(0.598134\pi\)
\(360\) 0 0
\(361\) 885.000 0.129028
\(362\) 0 0
\(363\) −10136.0 −1.46557
\(364\) 0 0
\(365\) −1656.00 −0.237477
\(366\) 0 0
\(367\) 4416.00 0.628102 0.314051 0.949406i \(-0.398314\pi\)
0.314051 + 0.949406i \(0.398314\pi\)
\(368\) 0 0
\(369\) 3774.00 0.532430
\(370\) 0 0
\(371\) 2432.00 0.340332
\(372\) 0 0
\(373\) 4180.00 0.580247 0.290124 0.956989i \(-0.406304\pi\)
0.290124 + 0.956989i \(0.406304\pi\)
\(374\) 0 0
\(375\) 10176.0 1.40130
\(376\) 0 0
\(377\) −3440.00 −0.469944
\(378\) 0 0
\(379\) −13736.0 −1.86166 −0.930832 0.365446i \(-0.880916\pi\)
−0.930832 + 0.365446i \(0.880916\pi\)
\(380\) 0 0
\(381\) 13312.0 1.79001
\(382\) 0 0
\(383\) −512.000 −0.0683080 −0.0341540 0.999417i \(-0.510874\pi\)
−0.0341540 + 0.999417i \(0.510874\pi\)
\(384\) 0 0
\(385\) −3072.00 −0.406659
\(386\) 0 0
\(387\) −10952.0 −1.43856
\(388\) 0 0
\(389\) 1732.00 0.225748 0.112874 0.993609i \(-0.463994\pi\)
0.112874 + 0.993609i \(0.463994\pi\)
\(390\) 0 0
\(391\) −3136.00 −0.405612
\(392\) 0 0
\(393\) −18624.0 −2.39047
\(394\) 0 0
\(395\) −768.000 −0.0978285
\(396\) 0 0
\(397\) 10436.0 1.31931 0.659657 0.751567i \(-0.270701\pi\)
0.659657 + 0.751567i \(0.270701\pi\)
\(398\) 0 0
\(399\) 22528.0 2.82659
\(400\) 0 0
\(401\) −12130.0 −1.51058 −0.755291 0.655390i \(-0.772504\pi\)
−0.755291 + 0.655390i \(0.772504\pi\)
\(402\) 0 0
\(403\) 5120.00 0.632867
\(404\) 0 0
\(405\) 4308.00 0.528559
\(406\) 0 0
\(407\) 736.000 0.0896368
\(408\) 0 0
\(409\) 5014.00 0.606177 0.303088 0.952962i \(-0.401982\pi\)
0.303088 + 0.952962i \(0.401982\pi\)
\(410\) 0 0
\(411\) 13872.0 1.66485
\(412\) 0 0
\(413\) −13056.0 −1.55555
\(414\) 0 0
\(415\) −4704.00 −0.556410
\(416\) 0 0
\(417\) 24256.0 2.84849
\(418\) 0 0
\(419\) −8024.00 −0.935556 −0.467778 0.883846i \(-0.654945\pi\)
−0.467778 + 0.883846i \(0.654945\pi\)
\(420\) 0 0
\(421\) −2348.00 −0.271816 −0.135908 0.990721i \(-0.543395\pi\)
−0.135908 + 0.990721i \(0.543395\pi\)
\(422\) 0 0
\(423\) 11840.0 1.36095
\(424\) 0 0
\(425\) −1862.00 −0.212518
\(426\) 0 0
\(427\) 20352.0 2.30656
\(428\) 0 0
\(429\) −1280.00 −0.144054
\(430\) 0 0
\(431\) 1728.00 0.193120 0.0965601 0.995327i \(-0.469216\pi\)
0.0965601 + 0.995327i \(0.469216\pi\)
\(432\) 0 0
\(433\) 62.0000 0.00688113 0.00344057 0.999994i \(-0.498905\pi\)
0.00344057 + 0.999994i \(0.498905\pi\)
\(434\) 0 0
\(435\) 16512.0 1.81998
\(436\) 0 0
\(437\) −2816.00 −0.308255
\(438\) 0 0
\(439\) 14112.0 1.53423 0.767117 0.641507i \(-0.221690\pi\)
0.767117 + 0.641507i \(0.221690\pi\)
\(440\) 0 0
\(441\) 25197.0 2.72076
\(442\) 0 0
\(443\) −4488.00 −0.481335 −0.240667 0.970608i \(-0.577366\pi\)
−0.240667 + 0.970608i \(0.577366\pi\)
\(444\) 0 0
\(445\) 6984.00 0.743985
\(446\) 0 0
\(447\) −14304.0 −1.51355
\(448\) 0 0
\(449\) −2482.00 −0.260875 −0.130437 0.991457i \(-0.541638\pi\)
−0.130437 + 0.991457i \(0.541638\pi\)
\(450\) 0 0
\(451\) −816.000 −0.0851972
\(452\) 0 0
\(453\) −3840.00 −0.398276
\(454\) 0 0
\(455\) 7680.00 0.791305
\(456\) 0 0
\(457\) 5894.00 0.603303 0.301652 0.953418i \(-0.402462\pi\)
0.301652 + 0.953418i \(0.402462\pi\)
\(458\) 0 0
\(459\) −7840.00 −0.797255
\(460\) 0 0
\(461\) −7068.00 −0.714077 −0.357039 0.934090i \(-0.616214\pi\)
−0.357039 + 0.934090i \(0.616214\pi\)
\(462\) 0 0
\(463\) −7616.00 −0.764461 −0.382231 0.924067i \(-0.624844\pi\)
−0.382231 + 0.924067i \(0.624844\pi\)
\(464\) 0 0
\(465\) −24576.0 −2.45093
\(466\) 0 0
\(467\) −13080.0 −1.29608 −0.648041 0.761606i \(-0.724411\pi\)
−0.648041 + 0.761606i \(0.724411\pi\)
\(468\) 0 0
\(469\) −17664.0 −1.73912
\(470\) 0 0
\(471\) −18400.0 −1.80006
\(472\) 0 0
\(473\) 2368.00 0.230192
\(474\) 0 0
\(475\) −1672.00 −0.161509
\(476\) 0 0
\(477\) −2812.00 −0.269922
\(478\) 0 0
\(479\) 13568.0 1.29423 0.647117 0.762391i \(-0.275975\pi\)
0.647117 + 0.762391i \(0.275975\pi\)
\(480\) 0 0
\(481\) −1840.00 −0.174422
\(482\) 0 0
\(483\) −8192.00 −0.771737
\(484\) 0 0
\(485\) −2856.00 −0.267390
\(486\) 0 0
\(487\) −1696.00 −0.157809 −0.0789046 0.996882i \(-0.525142\pi\)
−0.0789046 + 0.996882i \(0.525142\pi\)
\(488\) 0 0
\(489\) −12736.0 −1.17780
\(490\) 0 0
\(491\) 3096.00 0.284563 0.142282 0.989826i \(-0.454556\pi\)
0.142282 + 0.989826i \(0.454556\pi\)
\(492\) 0 0
\(493\) 16856.0 1.53987
\(494\) 0 0
\(495\) 3552.00 0.322526
\(496\) 0 0
\(497\) 13312.0 1.20146
\(498\) 0 0
\(499\) 19208.0 1.72318 0.861591 0.507603i \(-0.169468\pi\)
0.861591 + 0.507603i \(0.169468\pi\)
\(500\) 0 0
\(501\) −17664.0 −1.57519
\(502\) 0 0
\(503\) −16224.0 −1.43816 −0.719078 0.694929i \(-0.755436\pi\)
−0.719078 + 0.694929i \(0.755436\pi\)
\(504\) 0 0
\(505\) 17616.0 1.55228
\(506\) 0 0
\(507\) −14376.0 −1.25929
\(508\) 0 0
\(509\) −11292.0 −0.983318 −0.491659 0.870788i \(-0.663609\pi\)
−0.491659 + 0.870788i \(0.663609\pi\)
\(510\) 0 0
\(511\) −4416.00 −0.382294
\(512\) 0 0
\(513\) −7040.00 −0.605894
\(514\) 0 0
\(515\) −11904.0 −1.01855
\(516\) 0 0
\(517\) −2560.00 −0.217773
\(518\) 0 0
\(519\) −31584.0 −2.67126
\(520\) 0 0
\(521\) −5178.00 −0.435417 −0.217709 0.976014i \(-0.569858\pi\)
−0.217709 + 0.976014i \(0.569858\pi\)
\(522\) 0 0
\(523\) −6856.00 −0.573216 −0.286608 0.958048i \(-0.592528\pi\)
−0.286608 + 0.958048i \(0.592528\pi\)
\(524\) 0 0
\(525\) −4864.00 −0.404347
\(526\) 0 0
\(527\) −25088.0 −2.07372
\(528\) 0 0
\(529\) −11143.0 −0.915838
\(530\) 0 0
\(531\) 15096.0 1.23373
\(532\) 0 0
\(533\) 2040.00 0.165783
\(534\) 0 0
\(535\) 7008.00 0.566322
\(536\) 0 0
\(537\) −16832.0 −1.35262
\(538\) 0 0
\(539\) −5448.00 −0.435365
\(540\) 0 0
\(541\) 13732.0 1.09128 0.545642 0.838018i \(-0.316286\pi\)
0.545642 + 0.838018i \(0.316286\pi\)
\(542\) 0 0
\(543\) 11296.0 0.892740
\(544\) 0 0
\(545\) −8880.00 −0.697940
\(546\) 0 0
\(547\) −10968.0 −0.857327 −0.428663 0.903464i \(-0.641015\pi\)
−0.428663 + 0.903464i \(0.641015\pi\)
\(548\) 0 0
\(549\) −23532.0 −1.82936
\(550\) 0 0
\(551\) 15136.0 1.17026
\(552\) 0 0
\(553\) −2048.00 −0.157486
\(554\) 0 0
\(555\) 8832.00 0.675491
\(556\) 0 0
\(557\) −25612.0 −1.94832 −0.974161 0.225855i \(-0.927482\pi\)
−0.974161 + 0.225855i \(0.927482\pi\)
\(558\) 0 0
\(559\) −5920.00 −0.447924
\(560\) 0 0
\(561\) 6272.00 0.472021
\(562\) 0 0
\(563\) 9768.00 0.731212 0.365606 0.930770i \(-0.380862\pi\)
0.365606 + 0.930770i \(0.380862\pi\)
\(564\) 0 0
\(565\) 3624.00 0.269846
\(566\) 0 0
\(567\) 11488.0 0.850883
\(568\) 0 0
\(569\) 22838.0 1.68263 0.841317 0.540542i \(-0.181781\pi\)
0.841317 + 0.540542i \(0.181781\pi\)
\(570\) 0 0
\(571\) 9208.00 0.674856 0.337428 0.941351i \(-0.390443\pi\)
0.337428 + 0.941351i \(0.390443\pi\)
\(572\) 0 0
\(573\) −29696.0 −2.16504
\(574\) 0 0
\(575\) 608.000 0.0440963
\(576\) 0 0
\(577\) −10878.0 −0.784848 −0.392424 0.919785i \(-0.628363\pi\)
−0.392424 + 0.919785i \(0.628363\pi\)
\(578\) 0 0
\(579\) 12912.0 0.926778
\(580\) 0 0
\(581\) −12544.0 −0.895719
\(582\) 0 0
\(583\) 608.000 0.0431917
\(584\) 0 0
\(585\) −8880.00 −0.627595
\(586\) 0 0
\(587\) 18040.0 1.26847 0.634234 0.773141i \(-0.281316\pi\)
0.634234 + 0.773141i \(0.281316\pi\)
\(588\) 0 0
\(589\) −22528.0 −1.57598
\(590\) 0 0
\(591\) −5472.00 −0.380860
\(592\) 0 0
\(593\) 26994.0 1.86933 0.934663 0.355534i \(-0.115701\pi\)
0.934663 + 0.355534i \(0.115701\pi\)
\(594\) 0 0
\(595\) −37632.0 −2.59288
\(596\) 0 0
\(597\) 32512.0 2.22886
\(598\) 0 0
\(599\) 18336.0 1.25073 0.625366 0.780331i \(-0.284950\pi\)
0.625366 + 0.780331i \(0.284950\pi\)
\(600\) 0 0
\(601\) −9286.00 −0.630256 −0.315128 0.949049i \(-0.602047\pi\)
−0.315128 + 0.949049i \(0.602047\pi\)
\(602\) 0 0
\(603\) 20424.0 1.37932
\(604\) 0 0
\(605\) 15204.0 1.02170
\(606\) 0 0
\(607\) 17536.0 1.17259 0.586297 0.810096i \(-0.300585\pi\)
0.586297 + 0.810096i \(0.300585\pi\)
\(608\) 0 0
\(609\) 44032.0 2.92983
\(610\) 0 0
\(611\) 6400.00 0.423758
\(612\) 0 0
\(613\) −5868.00 −0.386633 −0.193317 0.981136i \(-0.561924\pi\)
−0.193317 + 0.981136i \(0.561924\pi\)
\(614\) 0 0
\(615\) −9792.00 −0.642035
\(616\) 0 0
\(617\) −19286.0 −1.25839 −0.629194 0.777248i \(-0.716615\pi\)
−0.629194 + 0.777248i \(0.716615\pi\)
\(618\) 0 0
\(619\) 5240.00 0.340248 0.170124 0.985423i \(-0.445583\pi\)
0.170124 + 0.985423i \(0.445583\pi\)
\(620\) 0 0
\(621\) 2560.00 0.165426
\(622\) 0 0
\(623\) 18624.0 1.19768
\(624\) 0 0
\(625\) −17639.0 −1.12890
\(626\) 0 0
\(627\) 5632.00 0.358725
\(628\) 0 0
\(629\) 9016.00 0.571529
\(630\) 0 0
\(631\) 15520.0 0.979147 0.489573 0.871962i \(-0.337153\pi\)
0.489573 + 0.871962i \(0.337153\pi\)
\(632\) 0 0
\(633\) 16960.0 1.06493
\(634\) 0 0
\(635\) −19968.0 −1.24788
\(636\) 0 0
\(637\) 13620.0 0.847165
\(638\) 0 0
\(639\) −15392.0 −0.952892
\(640\) 0 0
\(641\) 654.000 0.0402987 0.0201493 0.999797i \(-0.493586\pi\)
0.0201493 + 0.999797i \(0.493586\pi\)
\(642\) 0 0
\(643\) 8232.00 0.504881 0.252440 0.967612i \(-0.418767\pi\)
0.252440 + 0.967612i \(0.418767\pi\)
\(644\) 0 0
\(645\) 28416.0 1.73470
\(646\) 0 0
\(647\) 24672.0 1.49916 0.749580 0.661914i \(-0.230256\pi\)
0.749580 + 0.661914i \(0.230256\pi\)
\(648\) 0 0
\(649\) −3264.00 −0.197416
\(650\) 0 0
\(651\) −65536.0 −3.94556
\(652\) 0 0
\(653\) 22052.0 1.32153 0.660767 0.750591i \(-0.270231\pi\)
0.660767 + 0.750591i \(0.270231\pi\)
\(654\) 0 0
\(655\) 27936.0 1.66649
\(656\) 0 0
\(657\) 5106.00 0.303202
\(658\) 0 0
\(659\) −12024.0 −0.710757 −0.355378 0.934723i \(-0.615648\pi\)
−0.355378 + 0.934723i \(0.615648\pi\)
\(660\) 0 0
\(661\) −19100.0 −1.12391 −0.561955 0.827168i \(-0.689950\pi\)
−0.561955 + 0.827168i \(0.689950\pi\)
\(662\) 0 0
\(663\) −15680.0 −0.918493
\(664\) 0 0
\(665\) −33792.0 −1.97052
\(666\) 0 0
\(667\) −5504.00 −0.319514
\(668\) 0 0
\(669\) −22528.0 −1.30192
\(670\) 0 0
\(671\) 5088.00 0.292727
\(672\) 0 0
\(673\) 9902.00 0.567153 0.283577 0.958950i \(-0.408479\pi\)
0.283577 + 0.958950i \(0.408479\pi\)
\(674\) 0 0
\(675\) 1520.00 0.0866738
\(676\) 0 0
\(677\) 19684.0 1.11746 0.558728 0.829351i \(-0.311290\pi\)
0.558728 + 0.829351i \(0.311290\pi\)
\(678\) 0 0
\(679\) −7616.00 −0.430450
\(680\) 0 0
\(681\) 30784.0 1.73223
\(682\) 0 0
\(683\) 19864.0 1.11285 0.556424 0.830899i \(-0.312173\pi\)
0.556424 + 0.830899i \(0.312173\pi\)
\(684\) 0 0
\(685\) −20808.0 −1.16063
\(686\) 0 0
\(687\) −5216.00 −0.289669
\(688\) 0 0
\(689\) −1520.00 −0.0840456
\(690\) 0 0
\(691\) −3256.00 −0.179253 −0.0896267 0.995975i \(-0.528567\pi\)
−0.0896267 + 0.995975i \(0.528567\pi\)
\(692\) 0 0
\(693\) 9472.00 0.519209
\(694\) 0 0
\(695\) −36384.0 −1.98579
\(696\) 0 0
\(697\) −9996.00 −0.543222
\(698\) 0 0
\(699\) 24400.0 1.32030
\(700\) 0 0
\(701\) −2876.00 −0.154957 −0.0774786 0.996994i \(-0.524687\pi\)
−0.0774786 + 0.996994i \(0.524687\pi\)
\(702\) 0 0
\(703\) 8096.00 0.434348
\(704\) 0 0
\(705\) −30720.0 −1.64111
\(706\) 0 0
\(707\) 46976.0 2.49889
\(708\) 0 0
\(709\) 7300.00 0.386682 0.193341 0.981132i \(-0.438068\pi\)
0.193341 + 0.981132i \(0.438068\pi\)
\(710\) 0 0
\(711\) 2368.00 0.124904
\(712\) 0 0
\(713\) 8192.00 0.430284
\(714\) 0 0
\(715\) 1920.00 0.100425
\(716\) 0 0
\(717\) 50688.0 2.64014
\(718\) 0 0
\(719\) 2880.00 0.149382 0.0746912 0.997207i \(-0.476203\pi\)
0.0746912 + 0.997207i \(0.476203\pi\)
\(720\) 0 0
\(721\) −31744.0 −1.63968
\(722\) 0 0
\(723\) −36880.0 −1.89707
\(724\) 0 0
\(725\) −3268.00 −0.167408
\(726\) 0 0
\(727\) −8800.00 −0.448933 −0.224466 0.974482i \(-0.572064\pi\)
−0.224466 + 0.974482i \(0.572064\pi\)
\(728\) 0 0
\(729\) −30563.0 −1.55276
\(730\) 0 0
\(731\) 29008.0 1.46771
\(732\) 0 0
\(733\) 21076.0 1.06202 0.531009 0.847366i \(-0.321813\pi\)
0.531009 + 0.847366i \(0.321813\pi\)
\(734\) 0 0
\(735\) −65376.0 −3.28086
\(736\) 0 0
\(737\) −4416.00 −0.220713
\(738\) 0 0
\(739\) 19336.0 0.962498 0.481249 0.876584i \(-0.340183\pi\)
0.481249 + 0.876584i \(0.340183\pi\)
\(740\) 0 0
\(741\) −14080.0 −0.698032
\(742\) 0 0
\(743\) 13664.0 0.674675 0.337338 0.941384i \(-0.390474\pi\)
0.337338 + 0.941384i \(0.390474\pi\)
\(744\) 0 0
\(745\) 21456.0 1.05515
\(746\) 0 0
\(747\) 14504.0 0.710406
\(748\) 0 0
\(749\) 18688.0 0.911675
\(750\) 0 0
\(751\) −19520.0 −0.948462 −0.474231 0.880400i \(-0.657274\pi\)
−0.474231 + 0.880400i \(0.657274\pi\)
\(752\) 0 0
\(753\) 6336.00 0.306636
\(754\) 0 0
\(755\) 5760.00 0.277653
\(756\) 0 0
\(757\) 20004.0 0.960446 0.480223 0.877146i \(-0.340556\pi\)
0.480223 + 0.877146i \(0.340556\pi\)
\(758\) 0 0
\(759\) −2048.00 −0.0979416
\(760\) 0 0
\(761\) 31478.0 1.49944 0.749722 0.661753i \(-0.230187\pi\)
0.749722 + 0.661753i \(0.230187\pi\)
\(762\) 0 0
\(763\) −23680.0 −1.12356
\(764\) 0 0
\(765\) 43512.0 2.05644
\(766\) 0 0
\(767\) 8160.00 0.384147
\(768\) 0 0
\(769\) 7054.00 0.330785 0.165393 0.986228i \(-0.447111\pi\)
0.165393 + 0.986228i \(0.447111\pi\)
\(770\) 0 0
\(771\) −42992.0 −2.00820
\(772\) 0 0
\(773\) 9604.00 0.446872 0.223436 0.974719i \(-0.428273\pi\)
0.223436 + 0.974719i \(0.428273\pi\)
\(774\) 0 0
\(775\) 4864.00 0.225445
\(776\) 0 0
\(777\) 23552.0 1.08742
\(778\) 0 0
\(779\) −8976.00 −0.412835
\(780\) 0 0
\(781\) 3328.00 0.152478
\(782\) 0 0
\(783\) −13760.0 −0.628023
\(784\) 0 0
\(785\) 27600.0 1.25489
\(786\) 0 0
\(787\) 3144.00 0.142403 0.0712017 0.997462i \(-0.477317\pi\)
0.0712017 + 0.997462i \(0.477317\pi\)
\(788\) 0 0
\(789\) −27904.0 −1.25907
\(790\) 0 0
\(791\) 9664.00 0.434402
\(792\) 0 0
\(793\) −12720.0 −0.569610
\(794\) 0 0
\(795\) 7296.00 0.325487
\(796\) 0 0
\(797\) 22084.0 0.981500 0.490750 0.871300i \(-0.336723\pi\)
0.490750 + 0.871300i \(0.336723\pi\)
\(798\) 0 0
\(799\) −31360.0 −1.38853
\(800\) 0 0
\(801\) −21534.0 −0.949896
\(802\) 0 0
\(803\) −1104.00 −0.0485172
\(804\) 0 0
\(805\) 12288.0 0.538006
\(806\) 0 0
\(807\) −38112.0 −1.66246
\(808\) 0 0
\(809\) 14950.0 0.649708 0.324854 0.945764i \(-0.394685\pi\)
0.324854 + 0.945764i \(0.394685\pi\)
\(810\) 0 0
\(811\) −23432.0 −1.01456 −0.507280 0.861781i \(-0.669349\pi\)
−0.507280 + 0.861781i \(0.669349\pi\)
\(812\) 0 0
\(813\) −10752.0 −0.463824
\(814\) 0 0
\(815\) 19104.0 0.821085
\(816\) 0 0
\(817\) 26048.0 1.11543
\(818\) 0 0
\(819\) −23680.0 −1.01031
\(820\) 0 0
\(821\) 1044.00 0.0443798 0.0221899 0.999754i \(-0.492936\pi\)
0.0221899 + 0.999754i \(0.492936\pi\)
\(822\) 0 0
\(823\) 18208.0 0.771192 0.385596 0.922668i \(-0.373996\pi\)
0.385596 + 0.922668i \(0.373996\pi\)
\(824\) 0 0
\(825\) −1216.00 −0.0513160
\(826\) 0 0
\(827\) −12488.0 −0.525091 −0.262546 0.964920i \(-0.584562\pi\)
−0.262546 + 0.964920i \(0.584562\pi\)
\(828\) 0 0
\(829\) −30172.0 −1.26407 −0.632037 0.774938i \(-0.717781\pi\)
−0.632037 + 0.774938i \(0.717781\pi\)
\(830\) 0 0
\(831\) 68768.0 2.87068
\(832\) 0 0
\(833\) −66738.0 −2.77591
\(834\) 0 0
\(835\) 26496.0 1.09812
\(836\) 0 0
\(837\) 20480.0 0.845750
\(838\) 0 0
\(839\) −32544.0 −1.33915 −0.669573 0.742746i \(-0.733523\pi\)
−0.669573 + 0.742746i \(0.733523\pi\)
\(840\) 0 0
\(841\) 5195.00 0.213006
\(842\) 0 0
\(843\) 22992.0 0.939367
\(844\) 0 0
\(845\) 21564.0 0.877898
\(846\) 0 0
\(847\) 40544.0 1.64476
\(848\) 0 0
\(849\) −23104.0 −0.933954
\(850\) 0 0
\(851\) −2944.00 −0.118589
\(852\) 0 0
\(853\) −26156.0 −1.04990 −0.524950 0.851133i \(-0.675916\pi\)
−0.524950 + 0.851133i \(0.675916\pi\)
\(854\) 0 0
\(855\) 39072.0 1.56285
\(856\) 0 0
\(857\) 18646.0 0.743215 0.371607 0.928390i \(-0.378807\pi\)
0.371607 + 0.928390i \(0.378807\pi\)
\(858\) 0 0
\(859\) −5800.00 −0.230377 −0.115188 0.993344i \(-0.536747\pi\)
−0.115188 + 0.993344i \(0.536747\pi\)
\(860\) 0 0
\(861\) −26112.0 −1.03356
\(862\) 0 0
\(863\) 25088.0 0.989578 0.494789 0.869013i \(-0.335245\pi\)
0.494789 + 0.869013i \(0.335245\pi\)
\(864\) 0 0
\(865\) 47376.0 1.86223
\(866\) 0 0
\(867\) 37528.0 1.47003
\(868\) 0 0
\(869\) −512.000 −0.0199867
\(870\) 0 0
\(871\) 11040.0 0.429479
\(872\) 0 0
\(873\) 8806.00 0.341395
\(874\) 0 0
\(875\) −40704.0 −1.57262
\(876\) 0 0
\(877\) −3004.00 −0.115665 −0.0578323 0.998326i \(-0.518419\pi\)
−0.0578323 + 0.998326i \(0.518419\pi\)
\(878\) 0 0
\(879\) −52320.0 −2.00763
\(880\) 0 0
\(881\) 43282.0 1.65517 0.827587 0.561338i \(-0.189713\pi\)
0.827587 + 0.561338i \(0.189713\pi\)
\(882\) 0 0
\(883\) 27880.0 1.06256 0.531278 0.847198i \(-0.321712\pi\)
0.531278 + 0.847198i \(0.321712\pi\)
\(884\) 0 0
\(885\) −39168.0 −1.48770
\(886\) 0 0
\(887\) −7392.00 −0.279819 −0.139909 0.990164i \(-0.544681\pi\)
−0.139909 + 0.990164i \(0.544681\pi\)
\(888\) 0 0
\(889\) −53248.0 −2.00886
\(890\) 0 0
\(891\) 2872.00 0.107986
\(892\) 0 0
\(893\) −28160.0 −1.05525
\(894\) 0 0
\(895\) 25248.0 0.942958
\(896\) 0 0
\(897\) 5120.00 0.190582
\(898\) 0 0
\(899\) −44032.0 −1.63354
\(900\) 0 0
\(901\) 7448.00 0.275393
\(902\) 0 0
\(903\) 75776.0 2.79254
\(904\) 0 0
\(905\) −16944.0 −0.622362
\(906\) 0 0
\(907\) 29080.0 1.06459 0.532296 0.846558i \(-0.321329\pi\)
0.532296 + 0.846558i \(0.321329\pi\)
\(908\) 0 0
\(909\) −54316.0 −1.98190
\(910\) 0 0
\(911\) −26688.0 −0.970596 −0.485298 0.874349i \(-0.661289\pi\)
−0.485298 + 0.874349i \(0.661289\pi\)
\(912\) 0 0
\(913\) −3136.00 −0.113676
\(914\) 0 0
\(915\) 61056.0 2.20596
\(916\) 0 0
\(917\) 74496.0 2.68274
\(918\) 0 0
\(919\) −19680.0 −0.706402 −0.353201 0.935547i \(-0.614907\pi\)
−0.353201 + 0.935547i \(0.614907\pi\)
\(920\) 0 0
\(921\) −84672.0 −3.02936
\(922\) 0 0
\(923\) −8320.00 −0.296702
\(924\) 0 0
\(925\) −1748.00 −0.0621339
\(926\) 0 0
\(927\) 36704.0 1.30045
\(928\) 0 0
\(929\) −48466.0 −1.71164 −0.855822 0.517270i \(-0.826948\pi\)
−0.855822 + 0.517270i \(0.826948\pi\)
\(930\) 0 0
\(931\) −59928.0 −2.10962
\(932\) 0 0
\(933\) −50944.0 −1.78760
\(934\) 0 0
\(935\) −9408.00 −0.329064
\(936\) 0 0
\(937\) 13610.0 0.474514 0.237257 0.971447i \(-0.423752\pi\)
0.237257 + 0.971447i \(0.423752\pi\)
\(938\) 0 0
\(939\) 70064.0 2.43499
\(940\) 0 0
\(941\) 30692.0 1.06326 0.531632 0.846976i \(-0.321579\pi\)
0.531632 + 0.846976i \(0.321579\pi\)
\(942\) 0 0
\(943\) 3264.00 0.112715
\(944\) 0 0
\(945\) 30720.0 1.05748
\(946\) 0 0
\(947\) −4824.00 −0.165532 −0.0827661 0.996569i \(-0.526375\pi\)
−0.0827661 + 0.996569i \(0.526375\pi\)
\(948\) 0 0
\(949\) 2760.00 0.0944082
\(950\) 0 0
\(951\) −5728.00 −0.195313
\(952\) 0 0
\(953\) −22986.0 −0.781311 −0.390656 0.920537i \(-0.627752\pi\)
−0.390656 + 0.920537i \(0.627752\pi\)
\(954\) 0 0
\(955\) 44544.0 1.50933
\(956\) 0 0
\(957\) 11008.0 0.371827
\(958\) 0 0
\(959\) −55488.0 −1.86841
\(960\) 0 0
\(961\) 35745.0 1.19986
\(962\) 0 0
\(963\) −21608.0 −0.723061
\(964\) 0 0
\(965\) −19368.0 −0.646091
\(966\) 0 0
\(967\) −17184.0 −0.571458 −0.285729 0.958310i \(-0.592236\pi\)
−0.285729 + 0.958310i \(0.592236\pi\)
\(968\) 0 0
\(969\) 68992.0 2.28725
\(970\) 0 0
\(971\) −2920.00 −0.0965059 −0.0482530 0.998835i \(-0.515365\pi\)
−0.0482530 + 0.998835i \(0.515365\pi\)
\(972\) 0 0
\(973\) −97024.0 −3.19676
\(974\) 0 0
\(975\) 3040.00 0.0998543
\(976\) 0 0
\(977\) −27042.0 −0.885517 −0.442759 0.896641i \(-0.646000\pi\)
−0.442759 + 0.896641i \(0.646000\pi\)
\(978\) 0 0
\(979\) 4656.00 0.151998
\(980\) 0 0
\(981\) 27380.0 0.891107
\(982\) 0 0
\(983\) 44192.0 1.43388 0.716941 0.697134i \(-0.245542\pi\)
0.716941 + 0.697134i \(0.245542\pi\)
\(984\) 0 0
\(985\) 8208.00 0.265511
\(986\) 0 0
\(987\) −81920.0 −2.64189
\(988\) 0 0
\(989\) −9472.00 −0.304542
\(990\) 0 0
\(991\) −29824.0 −0.955995 −0.477997 0.878361i \(-0.658637\pi\)
−0.477997 + 0.878361i \(0.658637\pi\)
\(992\) 0 0
\(993\) 35264.0 1.12696
\(994\) 0 0
\(995\) −48768.0 −1.55382
\(996\) 0 0
\(997\) −11612.0 −0.368862 −0.184431 0.982845i \(-0.559044\pi\)
−0.184431 + 0.982845i \(0.559044\pi\)
\(998\) 0 0
\(999\) −7360.00 −0.233093
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 256.4.a.f.1.1 1
3.2 odd 2 2304.4.a.m.1.1 1
4.3 odd 2 256.4.a.b.1.1 1
8.3 odd 2 256.4.a.g.1.1 1
8.5 even 2 256.4.a.c.1.1 1
12.11 even 2 2304.4.a.n.1.1 1
16.3 odd 4 128.4.b.a.65.1 2
16.5 even 4 128.4.b.d.65.1 yes 2
16.11 odd 4 128.4.b.a.65.2 yes 2
16.13 even 4 128.4.b.d.65.2 yes 2
24.5 odd 2 2304.4.a.c.1.1 1
24.11 even 2 2304.4.a.d.1.1 1
48.5 odd 4 1152.4.d.h.577.2 2
48.11 even 4 1152.4.d.a.577.2 2
48.29 odd 4 1152.4.d.h.577.1 2
48.35 even 4 1152.4.d.a.577.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.4.b.a.65.1 2 16.3 odd 4
128.4.b.a.65.2 yes 2 16.11 odd 4
128.4.b.d.65.1 yes 2 16.5 even 4
128.4.b.d.65.2 yes 2 16.13 even 4
256.4.a.b.1.1 1 4.3 odd 2
256.4.a.c.1.1 1 8.5 even 2
256.4.a.f.1.1 1 1.1 even 1 trivial
256.4.a.g.1.1 1 8.3 odd 2
1152.4.d.a.577.1 2 48.35 even 4
1152.4.d.a.577.2 2 48.11 even 4
1152.4.d.h.577.1 2 48.29 odd 4
1152.4.d.h.577.2 2 48.5 odd 4
2304.4.a.c.1.1 1 24.5 odd 2
2304.4.a.d.1.1 1 24.11 even 2
2304.4.a.m.1.1 1 3.2 odd 2
2304.4.a.n.1.1 1 12.11 even 2