Properties

Label 4032.2.b.q.3583.8
Level $4032$
Weight $2$
Character 4032.3583
Analytic conductor $32.196$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4032,2,Mod(3583,4032)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4032, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4032.3583");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4032.b (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(32.1956820950\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.836829184.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 14x^{6} + 61x^{4} + 84x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: no (minimal twist has level 672)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3583.8
Root \(2.06644i\) of defining polynomial
Character \(\chi\) \(=\) 4032.3583
Dual form 4032.2.b.q.3583.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.33660i q^{5} +(1.65222 + 2.06644i) q^{7} +O(q^{10})\) \(q+4.33660i q^{5} +(1.65222 + 2.06644i) q^{7} -3.79628i q^{11} +2.82843i q^{13} -4.33660i q^{17} -2.54032 q^{19} -5.64104i q^{23} -13.8061 q^{25} -9.50162 q^{29} +1.84476 q^{31} +(-8.96130 + 7.16502i) q^{35} -5.11654 q^{37} +1.32026i q^{41} +2.47602i q^{43} -12.2657 q^{47} +(-1.54032 + 6.82843i) q^{49} +1.23587 q^{53} +16.4629 q^{55} -1.65685 q^{59} +3.50162i q^{61} -12.2657 q^{65} +11.1492i q^{67} +3.03215i q^{71} -3.01634i q^{73} +(7.84476 - 6.27230i) q^{77} -10.1972i q^{79} +3.04796 q^{83} +18.8061 q^{85} -6.53804i q^{89} +(-5.84476 + 4.67319i) q^{91} -11.0163i q^{95} -6.33005i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{7} - 8 q^{19} - 16 q^{25} - 16 q^{31} - 8 q^{35} - 8 q^{37} - 16 q^{47} + 16 q^{53} - 8 q^{55} + 32 q^{59} - 16 q^{65} + 32 q^{77} + 16 q^{83} + 56 q^{85} - 16 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(1793\) \(3781\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 4.33660i 1.93938i 0.244329 + 0.969692i \(0.421432\pi\)
−0.244329 + 0.969692i \(0.578568\pi\)
\(6\) 0 0
\(7\) 1.65222 + 2.06644i 0.624482 + 0.781040i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.79628i 1.14462i −0.820037 0.572310i \(-0.806047\pi\)
0.820037 0.572310i \(-0.193953\pi\)
\(12\) 0 0
\(13\) 2.82843i 0.784465i 0.919866 + 0.392232i \(0.128297\pi\)
−0.919866 + 0.392232i \(0.871703\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.33660i 1.05178i −0.850553 0.525890i \(-0.823733\pi\)
0.850553 0.525890i \(-0.176267\pi\)
\(18\) 0 0
\(19\) −2.54032 −0.582789 −0.291395 0.956603i \(-0.594119\pi\)
−0.291395 + 0.956603i \(0.594119\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.64104i 1.17624i −0.808774 0.588119i \(-0.799869\pi\)
0.808774 0.588119i \(-0.200131\pi\)
\(24\) 0 0
\(25\) −13.8061 −2.76121
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −9.50162 −1.76441 −0.882203 0.470869i \(-0.843940\pi\)
−0.882203 + 0.470869i \(0.843940\pi\)
\(30\) 0 0
\(31\) 1.84476 0.331330 0.165665 0.986182i \(-0.447023\pi\)
0.165665 + 0.986182i \(0.447023\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −8.96130 + 7.16502i −1.51474 + 1.21111i
\(36\) 0 0
\(37\) −5.11654 −0.841153 −0.420577 0.907257i \(-0.638172\pi\)
−0.420577 + 0.907257i \(0.638172\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.32026i 0.206190i 0.994672 + 0.103095i \(0.0328745\pi\)
−0.994672 + 0.103095i \(0.967125\pi\)
\(42\) 0 0
\(43\) 2.47602i 0.377589i 0.982017 + 0.188795i \(0.0604581\pi\)
−0.982017 + 0.188795i \(0.939542\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −12.2657 −1.78914 −0.894571 0.446925i \(-0.852519\pi\)
−0.894571 + 0.446925i \(0.852519\pi\)
\(48\) 0 0
\(49\) −1.54032 + 6.82843i −0.220046 + 0.975490i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.23587 0.169760 0.0848801 0.996391i \(-0.472949\pi\)
0.0848801 + 0.996391i \(0.472949\pi\)
\(54\) 0 0
\(55\) 16.4629 2.21986
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −1.65685 −0.215704 −0.107852 0.994167i \(-0.534397\pi\)
−0.107852 + 0.994167i \(0.534397\pi\)
\(60\) 0 0
\(61\) 3.50162i 0.448336i 0.974550 + 0.224168i \(0.0719665\pi\)
−0.974550 + 0.224168i \(0.928034\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −12.2657 −1.52138
\(66\) 0 0
\(67\) 11.1492i 1.36209i 0.732240 + 0.681046i \(0.238475\pi\)
−0.732240 + 0.681046i \(0.761525\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 3.03215i 0.359850i 0.983680 + 0.179925i \(0.0575855\pi\)
−0.983680 + 0.179925i \(0.942414\pi\)
\(72\) 0 0
\(73\) 3.01634i 0.353036i −0.984297 0.176518i \(-0.943517\pi\)
0.984297 0.176518i \(-0.0564833\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 7.84476 6.27230i 0.893994 0.714794i
\(78\) 0 0
\(79\) 10.1972i 1.14727i −0.819110 0.573636i \(-0.805532\pi\)
0.819110 0.573636i \(-0.194468\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 3.04796 0.334557 0.167279 0.985910i \(-0.446502\pi\)
0.167279 + 0.985910i \(0.446502\pi\)
\(84\) 0 0
\(85\) 18.8061 2.03980
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.53804i 0.693031i −0.938044 0.346515i \(-0.887365\pi\)
0.938044 0.346515i \(-0.112635\pi\)
\(90\) 0 0
\(91\) −5.84476 + 4.67319i −0.612698 + 0.489884i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 11.0163i 1.13025i
\(96\) 0 0
\(97\) 6.33005i 0.642719i −0.946957 0.321359i \(-0.895860\pi\)
0.946957 0.321359i \(-0.104140\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 6.97711i 0.694249i 0.937819 + 0.347124i \(0.112842\pi\)
−0.937819 + 0.347124i \(0.887158\pi\)
\(102\) 0 0
\(103\) −9.84476 −0.970034 −0.485017 0.874505i \(-0.661187\pi\)
−0.485017 + 0.874505i \(0.661187\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.46947i 0.432080i −0.976385 0.216040i \(-0.930686\pi\)
0.976385 0.216040i \(-0.0693141\pi\)
\(108\) 0 0
\(109\) −4.60889 −0.441452 −0.220726 0.975336i \(-0.570843\pi\)
−0.220726 + 0.975336i \(0.570843\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −17.9553 −1.68909 −0.844545 0.535484i \(-0.820129\pi\)
−0.844545 + 0.535484i \(0.820129\pi\)
\(114\) 0 0
\(115\) 24.4629 2.28118
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 8.96130 7.16502i 0.821481 0.656817i
\(120\) 0 0
\(121\) −3.41172 −0.310156
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 38.1883i 3.41567i
\(126\) 0 0
\(127\) 9.11654i 0.808962i 0.914546 + 0.404481i \(0.132548\pi\)
−0.914546 + 0.404481i \(0.867452\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 4.57622 0.399826 0.199913 0.979814i \(-0.435934\pi\)
0.199913 + 0.979814i \(0.435934\pi\)
\(132\) 0 0
\(133\) −4.19717 5.24941i −0.363941 0.455181i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −11.9226 −1.01862 −0.509308 0.860584i \(-0.670099\pi\)
−0.509308 + 0.860584i \(0.670099\pi\)
\(138\) 0 0
\(139\) −7.31371 −0.620341 −0.310170 0.950681i \(-0.600386\pi\)
−0.310170 + 0.950681i \(0.600386\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 10.7375 0.897914
\(144\) 0 0
\(145\) 41.2047i 3.42186i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −16.6867 −1.36703 −0.683515 0.729937i \(-0.739550\pi\)
−0.683515 + 0.729937i \(0.739550\pi\)
\(150\) 0 0
\(151\) 6.10020i 0.496427i 0.968705 + 0.248214i \(0.0798435\pi\)
−0.968705 + 0.248214i \(0.920157\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 8.00000i 0.642575i
\(156\) 0 0
\(157\) 7.81209i 0.623473i −0.950169 0.311736i \(-0.899089\pi\)
0.950169 0.311736i \(-0.100911\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 11.6569 9.32026i 0.918689 0.734539i
\(162\) 0 0
\(163\) 14.4629i 1.13282i 0.824122 + 0.566412i \(0.191669\pi\)
−0.824122 + 0.566412i \(0.808331\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 18.2984 1.41597 0.707987 0.706225i \(-0.249603\pi\)
0.707987 + 0.706225i \(0.249603\pi\)
\(168\) 0 0
\(169\) 5.00000 0.384615
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 17.3203i 1.31683i 0.752653 + 0.658417i \(0.228774\pi\)
−0.752653 + 0.658417i \(0.771226\pi\)
\(174\) 0 0
\(175\) −22.8107 28.5294i −1.72433 2.15662i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 10.2680i 0.767468i −0.923444 0.383734i \(-0.874638\pi\)
0.923444 0.383734i \(-0.125362\pi\)
\(180\) 0 0
\(181\) 14.9570i 1.11175i 0.831267 + 0.555874i \(0.187616\pi\)
−0.831267 + 0.555874i \(0.812384\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 22.1883i 1.63132i
\(186\) 0 0
\(187\) −16.4629 −1.20389
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 24.9231i 1.80337i 0.432389 + 0.901687i \(0.357671\pi\)
−0.432389 + 0.901687i \(0.642329\pi\)
\(192\) 0 0
\(193\) 1.49236 0.107422 0.0537111 0.998557i \(-0.482895\pi\)
0.0537111 + 0.998557i \(0.482895\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 6.89273 0.491087 0.245543 0.969386i \(-0.421034\pi\)
0.245543 + 0.969386i \(0.421034\pi\)
\(198\) 0 0
\(199\) 22.9940 1.63000 0.815000 0.579461i \(-0.196737\pi\)
0.815000 + 0.579461i \(0.196737\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −15.6988 19.6345i −1.10184 1.37807i
\(204\) 0 0
\(205\) −5.72543 −0.399881
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 9.64375i 0.667072i
\(210\) 0 0
\(211\) 6.60462i 0.454681i 0.973815 + 0.227340i \(0.0730030\pi\)
−0.973815 + 0.227340i \(0.926997\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −10.7375 −0.732291
\(216\) 0 0
\(217\) 3.04796 + 3.81209i 0.206909 + 0.258781i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 12.2657 0.825083
\(222\) 0 0
\(223\) 16.0093 1.07206 0.536030 0.844199i \(-0.319923\pi\)
0.536030 + 0.844199i \(0.319923\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −0.310470 −0.0206066 −0.0103033 0.999947i \(-0.503280\pi\)
−0.0103033 + 0.999947i \(0.503280\pi\)
\(228\) 0 0
\(229\) 22.5180i 1.48803i −0.668164 0.744014i \(-0.732919\pi\)
0.668164 0.744014i \(-0.267081\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 11.6569 0.763666 0.381833 0.924231i \(-0.375293\pi\)
0.381833 + 0.924231i \(0.375293\pi\)
\(234\) 0 0
\(235\) 53.1916i 3.46984i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 3.88721i 0.251443i 0.992066 + 0.125721i \(0.0401245\pi\)
−0.992066 + 0.125721i \(0.959875\pi\)
\(240\) 0 0
\(241\) 4.54459i 0.292743i −0.989230 0.146371i \(-0.953241\pi\)
0.989230 0.146371i \(-0.0467595\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −29.6121 6.67974i −1.89185 0.426753i
\(246\) 0 0
\(247\) 7.18511i 0.457177i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 5.92260 0.373831 0.186916 0.982376i \(-0.440151\pi\)
0.186916 + 0.982376i \(0.440151\pi\)
\(252\) 0 0
\(253\) −21.4150 −1.34635
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0.0261258i 0.00162968i −1.00000 0.000814840i \(-0.999741\pi\)
1.00000 0.000814840i \(-0.000259372\pi\)
\(258\) 0 0
\(259\) −8.45366 10.5730i −0.525285 0.656974i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 12.3785i 0.763293i 0.924308 + 0.381647i \(0.124643\pi\)
−0.924308 + 0.381647i \(0.875357\pi\)
\(264\) 0 0
\(265\) 5.35948i 0.329230i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 2.13515i 0.130182i 0.997879 + 0.0650912i \(0.0207338\pi\)
−0.997879 + 0.0650912i \(0.979266\pi\)
\(270\) 0 0
\(271\) −26.3763 −1.60224 −0.801122 0.598501i \(-0.795763\pi\)
−0.801122 + 0.598501i \(0.795763\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 52.4116i 3.16054i
\(276\) 0 0
\(277\) 24.1198 1.44922 0.724608 0.689161i \(-0.242021\pi\)
0.724608 + 0.689161i \(0.242021\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 10.6416 0.634823 0.317411 0.948288i \(-0.397186\pi\)
0.317411 + 0.948288i \(0.397186\pi\)
\(282\) 0 0
\(283\) 17.5436 1.04286 0.521428 0.853295i \(-0.325399\pi\)
0.521428 + 0.853295i \(0.325399\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −2.72823 + 2.18136i −0.161042 + 0.128762i
\(288\) 0 0
\(289\) −1.80606 −0.106239
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 8.64707i 0.505167i −0.967575 0.252584i \(-0.918720\pi\)
0.967575 0.252584i \(-0.0812802\pi\)
\(294\) 0 0
\(295\) 7.18511i 0.418333i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 15.9553 0.922717
\(300\) 0 0
\(301\) −5.11654 + 4.09093i −0.294912 + 0.235798i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −15.1851 −0.869497
\(306\) 0 0
\(307\) −23.3823 −1.33450 −0.667249 0.744835i \(-0.732528\pi\)
−0.667249 + 0.744835i \(0.732528\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 22.1883 1.25819 0.629093 0.777330i \(-0.283427\pi\)
0.629093 + 0.777330i \(0.283427\pi\)
\(312\) 0 0
\(313\) 17.8779i 1.01052i 0.862968 + 0.505259i \(0.168603\pi\)
−0.862968 + 0.505259i \(0.831397\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −15.1585 −0.851385 −0.425692 0.904868i \(-0.639969\pi\)
−0.425692 + 0.904868i \(0.639969\pi\)
\(318\) 0 0
\(319\) 36.0708i 2.01958i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 11.0163i 0.612965i
\(324\) 0 0
\(325\) 39.0494i 2.16607i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −20.2657 25.3464i −1.11729 1.39739i
\(330\) 0 0
\(331\) 8.57194i 0.471157i 0.971855 + 0.235578i \(0.0756984\pi\)
−0.971855 + 0.235578i \(0.924302\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −48.3496 −2.64162
\(336\) 0 0
\(337\) −22.3943 −1.21990 −0.609949 0.792441i \(-0.708810\pi\)
−0.609949 + 0.792441i \(0.708810\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 7.00324i 0.379247i
\(342\) 0 0
\(343\) −16.6555 + 8.09911i −0.899310 + 0.437311i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 11.4205i 0.613082i −0.951857 0.306541i \(-0.900828\pi\)
0.951857 0.306541i \(-0.0991717\pi\)
\(348\) 0 0
\(349\) 5.65081i 0.302481i 0.988497 + 0.151241i \(0.0483268\pi\)
−0.988497 + 0.151241i \(0.951673\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 27.3398i 1.45515i 0.686027 + 0.727576i \(0.259353\pi\)
−0.686027 + 0.727576i \(0.740647\pi\)
\(354\) 0 0
\(355\) −13.1492 −0.697888
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 23.2848i 1.22892i −0.788946 0.614462i \(-0.789373\pi\)
0.788946 0.614462i \(-0.210627\pi\)
\(360\) 0 0
\(361\) −12.5468 −0.660357
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 13.0806 0.684672
\(366\) 0 0
\(367\) −16.1464 −0.842836 −0.421418 0.906867i \(-0.638467\pi\)
−0.421418 + 0.906867i \(0.638467\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 2.04194 + 2.55385i 0.106012 + 0.132589i
\(372\) 0 0
\(373\) −27.4750 −1.42260 −0.711300 0.702888i \(-0.751893\pi\)
−0.711300 + 0.702888i \(0.751893\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 26.8746i 1.38411i
\(378\) 0 0
\(379\) 4.90304i 0.251852i −0.992040 0.125926i \(-0.959810\pi\)
0.992040 0.125926i \(-0.0401902\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 2.73749 0.139879 0.0699397 0.997551i \(-0.477719\pi\)
0.0699397 + 0.997551i \(0.477719\pi\)
\(384\) 0 0
\(385\) 27.2004 + 34.0196i 1.38626 + 1.73380i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −31.4896 −1.59658 −0.798292 0.602270i \(-0.794263\pi\)
−0.798292 + 0.602270i \(0.794263\pi\)
\(390\) 0 0
\(391\) −24.4629 −1.23714
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 44.2210 2.22500
\(396\) 0 0
\(397\) 1.15847i 0.0581421i 0.999577 + 0.0290711i \(0.00925491\pi\)
−0.999577 + 0.0290711i \(0.990745\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 2.70482 0.135072 0.0675361 0.997717i \(-0.478486\pi\)
0.0675361 + 0.997717i \(0.478486\pi\)
\(402\) 0 0
\(403\) 5.21778i 0.259916i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 19.4238i 0.962801i
\(408\) 0 0
\(409\) 28.8942i 1.42873i 0.699775 + 0.714363i \(0.253284\pi\)
−0.699775 + 0.714363i \(0.746716\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −2.73749 3.42378i −0.134703 0.168473i
\(414\) 0 0
\(415\) 13.2178i 0.648835i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 29.3464 1.43366 0.716832 0.697246i \(-0.245591\pi\)
0.716832 + 0.697246i \(0.245591\pi\)
\(420\) 0 0
\(421\) −21.8867 −1.06669 −0.533346 0.845897i \(-0.679066\pi\)
−0.533346 + 0.845897i \(0.679066\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 59.8713i 2.90419i
\(426\) 0 0
\(427\) −7.23587 + 5.78546i −0.350168 + 0.279978i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 2.76640i 0.133253i −0.997778 0.0666265i \(-0.978776\pi\)
0.997778 0.0666265i \(-0.0212236\pi\)
\(432\) 0 0
\(433\) 23.8181i 1.14463i 0.820035 + 0.572313i \(0.193954\pi\)
−0.820035 + 0.572313i \(0.806046\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 14.3300i 0.685499i
\(438\) 0 0
\(439\) 12.8327 0.612471 0.306236 0.951956i \(-0.400930\pi\)
0.306236 + 0.951956i \(0.400930\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 22.1917i 1.05436i −0.849754 0.527179i \(-0.823250\pi\)
0.849754 0.527179i \(-0.176750\pi\)
\(444\) 0 0
\(445\) 28.3528 1.34405
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 9.11331 0.430084 0.215042 0.976605i \(-0.431011\pi\)
0.215042 + 0.976605i \(0.431011\pi\)
\(450\) 0 0
\(451\) 5.01207 0.236009
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −20.2657 25.3464i −0.950073 1.18826i
\(456\) 0 0
\(457\) 13.4509 0.629204 0.314602 0.949224i \(-0.398129\pi\)
0.314602 + 0.949224i \(0.398129\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 2.17532i 0.101315i 0.998716 + 0.0506574i \(0.0161316\pi\)
−0.998716 + 0.0506574i \(0.983868\pi\)
\(462\) 0 0
\(463\) 30.3343i 1.40976i 0.709329 + 0.704878i \(0.248998\pi\)
−0.709329 + 0.704878i \(0.751002\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −35.1404 −1.62610 −0.813051 0.582192i \(-0.802195\pi\)
−0.813051 + 0.582192i \(0.802195\pi\)
\(468\) 0 0
\(469\) −23.0391 + 18.4210i −1.06385 + 0.850602i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 9.39965 0.432196
\(474\) 0 0
\(475\) 35.0718 1.60921
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 17.9673 0.820948 0.410474 0.911872i \(-0.365363\pi\)
0.410474 + 0.911872i \(0.365363\pi\)
\(480\) 0 0
\(481\) 14.4717i 0.659855i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 27.4509 1.24648
\(486\) 0 0
\(487\) 34.4313i 1.56023i −0.625636 0.780115i \(-0.715160\pi\)
0.625636 0.780115i \(-0.284840\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 13.8737i 0.626110i 0.949735 + 0.313055i \(0.101352\pi\)
−0.949735 + 0.313055i \(0.898648\pi\)
\(492\) 0 0
\(493\) 41.2047i 1.85577i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −6.26575 + 5.00979i −0.281057 + 0.224720i
\(498\) 0 0
\(499\) 38.2494i 1.71228i −0.516744 0.856140i \(-0.672856\pi\)
0.516744 0.856140i \(-0.327144\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0.420546 0.0187512 0.00937560 0.999956i \(-0.497016\pi\)
0.00937560 + 0.999956i \(0.497016\pi\)
\(504\) 0 0
\(505\) −30.2569 −1.34642
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 13.8247i 0.612768i 0.951908 + 0.306384i \(0.0991192\pi\)
−0.951908 + 0.306384i \(0.900881\pi\)
\(510\) 0 0
\(511\) 6.23307 4.98366i 0.275735 0.220464i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 42.6928i 1.88127i
\(516\) 0 0
\(517\) 46.5642i 2.04789i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 9.55438i 0.418585i 0.977853 + 0.209292i \(0.0671161\pi\)
−0.977853 + 0.209292i \(0.932884\pi\)
\(522\) 0 0
\(523\) 32.2210 1.40893 0.704463 0.709740i \(-0.251188\pi\)
0.704463 + 0.709740i \(0.251188\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 8.00000i 0.348485i
\(528\) 0 0
\(529\) −8.82135 −0.383537
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −3.73425 −0.161749
\(534\) 0 0
\(535\) 19.3823 0.837969
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 25.9226 + 5.84748i 1.11657 + 0.251869i
\(540\) 0 0
\(541\) −9.20041 −0.395557 −0.197778 0.980247i \(-0.563373\pi\)
−0.197778 + 0.980247i \(0.563373\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 19.9869i 0.856145i
\(546\) 0 0
\(547\) 25.5109i 1.09077i −0.838187 0.545383i \(-0.816384\pi\)
0.838187 0.545383i \(-0.183616\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 24.1371 1.02828
\(552\) 0 0
\(553\) 21.0718 16.8480i 0.896065 0.716450i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −29.3730 −1.24457 −0.622287 0.782789i \(-0.713796\pi\)
−0.622287 + 0.782789i \(0.713796\pi\)
\(558\) 0 0
\(559\) −7.00324 −0.296205
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −44.8626 −1.89073 −0.945366 0.326010i \(-0.894296\pi\)
−0.945366 + 0.326010i \(0.894296\pi\)
\(564\) 0 0
\(565\) 77.8648i 3.27580i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 28.5642 1.19747 0.598736 0.800946i \(-0.295670\pi\)
0.598736 + 0.800946i \(0.295670\pi\)
\(570\) 0 0
\(571\) 24.1645i 1.01125i −0.862753 0.505626i \(-0.831261\pi\)
0.862753 0.505626i \(-0.168739\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 77.8806i 3.24785i
\(576\) 0 0
\(577\) 4.84196i 0.201574i 0.994908 + 0.100787i \(0.0321360\pi\)
−0.994908 + 0.100787i \(0.967864\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 5.03591 + 6.29842i 0.208925 + 0.261303i
\(582\) 0 0
\(583\) 4.69172i 0.194311i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −15.1766 −0.626404 −0.313202 0.949687i \(-0.601402\pi\)
−0.313202 + 0.949687i \(0.601402\pi\)
\(588\) 0 0
\(589\) −4.68629 −0.193095
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 22.9139i 0.940960i 0.882411 + 0.470480i \(0.155919\pi\)
−0.882411 + 0.470480i \(0.844081\pi\)
\(594\) 0 0
\(595\) 31.0718 + 38.8615i 1.27382 + 1.59317i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 8.31318i 0.339667i 0.985473 + 0.169834i \(0.0543231\pi\)
−0.985473 + 0.169834i \(0.945677\pi\)
\(600\) 0 0
\(601\) 21.8508i 0.891313i −0.895204 0.445656i \(-0.852970\pi\)
0.895204 0.445656i \(-0.147030\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 14.7952i 0.601512i
\(606\) 0 0
\(607\) 40.5408 1.64550 0.822749 0.568405i \(-0.192439\pi\)
0.822749 + 0.568405i \(0.192439\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 34.6928i 1.40352i
\(612\) 0 0
\(613\) 11.0153 0.444903 0.222452 0.974944i \(-0.428594\pi\)
0.222452 + 0.974944i \(0.428594\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 2.32900 0.0937618 0.0468809 0.998900i \(-0.485072\pi\)
0.0468809 + 0.998900i \(0.485072\pi\)
\(618\) 0 0
\(619\) 25.7493 1.03495 0.517475 0.855698i \(-0.326872\pi\)
0.517475 + 0.855698i \(0.326872\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 13.5104 10.8023i 0.541285 0.432785i
\(624\) 0 0
\(625\) 96.5771 3.86308
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 22.1883i 0.884707i
\(630\) 0 0
\(631\) 9.93143i 0.395364i 0.980266 + 0.197682i \(0.0633413\pi\)
−0.980266 + 0.197682i \(0.936659\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −39.5347 −1.56889
\(636\) 0 0
\(637\) −19.3137 4.35668i −0.765237 0.172618i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −19.7855 −0.781478 −0.390739 0.920501i \(-0.627781\pi\)
−0.390739 + 0.920501i \(0.627781\pi\)
\(642\) 0 0
\(643\) 3.07181 0.121140 0.0605702 0.998164i \(-0.480708\pi\)
0.0605702 + 0.998164i \(0.480708\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −15.6242 −0.614250 −0.307125 0.951669i \(-0.599367\pi\)
−0.307125 + 0.951669i \(0.599367\pi\)
\(648\) 0 0
\(649\) 6.28988i 0.246899i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 10.1432 0.396934 0.198467 0.980108i \(-0.436404\pi\)
0.198467 + 0.980108i \(0.436404\pi\)
\(654\) 0 0
\(655\) 19.8452i 0.775416i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 4.68296i 0.182422i 0.995832 + 0.0912112i \(0.0290738\pi\)
−0.995832 + 0.0912112i \(0.970926\pi\)
\(660\) 0 0
\(661\) 7.37302i 0.286777i 0.989666 + 0.143389i \(0.0457999\pi\)
−0.989666 + 0.143389i \(0.954200\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 22.7646 18.2014i 0.882772 0.705822i
\(666\) 0 0
\(667\) 53.5990i 2.07536i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 13.2931 0.513175
\(672\) 0 0
\(673\) 6.96734 0.268571 0.134286 0.990943i \(-0.457126\pi\)
0.134286 + 0.990943i \(0.457126\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 20.0130i 0.769163i −0.923091 0.384582i \(-0.874346\pi\)
0.923091 0.384582i \(-0.125654\pi\)
\(678\) 0 0
\(679\) 13.0806 10.4586i 0.501989 0.401366i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 19.2155i 0.735261i −0.929972 0.367630i \(-0.880169\pi\)
0.929972 0.367630i \(-0.119831\pi\)
\(684\) 0 0
\(685\) 51.7035i 1.97549i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 3.49558i 0.133171i
\(690\) 0 0
\(691\) −49.3735 −1.87825 −0.939127 0.343569i \(-0.888364\pi\)
−0.939127 + 0.343569i \(0.888364\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 31.7166i 1.20308i
\(696\) 0 0
\(697\) 5.72543 0.216866
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 16.0004 0.604328 0.302164 0.953256i \(-0.402291\pi\)
0.302164 + 0.953256i \(0.402291\pi\)
\(702\) 0 0
\(703\) 12.9976 0.490215
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −14.4178 + 11.5277i −0.542236 + 0.433545i
\(708\) 0 0
\(709\) 22.2210 0.834528 0.417264 0.908785i \(-0.362989\pi\)
0.417264 + 0.908785i \(0.362989\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 10.4064i 0.389723i
\(714\) 0 0
\(715\) 46.5642i 1.74140i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 1.96732 0.0733688 0.0366844 0.999327i \(-0.488320\pi\)
0.0366844 + 0.999327i \(0.488320\pi\)
\(720\) 0 0
\(721\) −16.2657 20.3436i −0.605768 0.757635i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 131.180 4.87190
\(726\) 0 0
\(727\) −33.4689 −1.24129 −0.620647 0.784090i \(-0.713130\pi\)
−0.620647 + 0.784090i \(0.713130\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 10.7375 0.397141
\(732\) 0 0
\(733\) 39.4253i 1.45621i 0.685468 + 0.728103i \(0.259598\pi\)
−0.685468 + 0.728103i \(0.740402\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 42.3255 1.55908
\(738\) 0 0
\(739\) 33.5109i 1.23272i −0.787465 0.616359i \(-0.788607\pi\)
0.787465 0.616359i \(-0.211393\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 30.7040i 1.12642i −0.826313 0.563211i \(-0.809566\pi\)
0.826313 0.563211i \(-0.190434\pi\)
\(744\) 0 0
\(745\) 72.3636i 2.65120i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 9.23587 7.38456i 0.337471 0.269826i
\(750\) 0 0
\(751\) 23.7123i 0.865275i −0.901568 0.432638i \(-0.857583\pi\)
0.901568 0.432638i \(-0.142417\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −26.4541 −0.962763
\(756\) 0 0
\(757\) 16.4717 0.598676 0.299338 0.954147i \(-0.403234\pi\)
0.299338 + 0.954147i \(0.403234\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 32.8812i 1.19194i 0.803006 + 0.595971i \(0.203233\pi\)
−0.803006 + 0.595971i \(0.796767\pi\)
\(762\) 0 0
\(763\) −7.61492 9.52398i −0.275678 0.344791i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 4.68629i 0.169212i
\(768\) 0 0
\(769\) 41.8779i 1.51015i 0.655636 + 0.755077i \(0.272401\pi\)
−0.655636 + 0.755077i \(0.727599\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 20.4521i 0.735611i −0.929903 0.367805i \(-0.880109\pi\)
0.929903 0.367805i \(-0.119891\pi\)
\(774\) 0 0
\(775\) −25.4689 −0.914871
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 3.35388i 0.120165i
\(780\) 0 0
\(781\) 11.5109 0.411892
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 33.8779 1.20915
\(786\) 0 0
\(787\) −37.6960 −1.34372 −0.671859 0.740679i \(-0.734504\pi\)
−0.671859 + 0.740679i \(0.734504\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −29.6661 37.1034i −1.05481 1.31925i
\(792\) 0 0
\(793\) −9.90407 −0.351704
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 27.8713i 0.987253i 0.869674 + 0.493627i \(0.164329\pi\)
−0.869674 + 0.493627i \(0.835671\pi\)
\(798\) 0 0
\(799\) 53.1916i 1.88178i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −11.4509 −0.404092
\(804\) 0 0
\(805\) 40.4182 + 50.5511i 1.42455 + 1.78169i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 12.2899 0.432089 0.216045 0.976384i \(-0.430684\pi\)
0.216045 + 0.976384i \(0.430684\pi\)
\(810\) 0 0
\(811\) 52.1557 1.83143 0.915717 0.401825i \(-0.131624\pi\)
0.915717 + 0.401825i \(0.131624\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −62.7198 −2.19698
\(816\) 0 0
\(817\) 6.28988i 0.220055i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −24.9254 −0.869903 −0.434951 0.900454i \(-0.643234\pi\)
−0.434951 + 0.900454i \(0.643234\pi\)
\(822\) 0 0
\(823\) 32.8331i 1.14449i 0.820082 + 0.572246i \(0.193928\pi\)
−0.820082 + 0.572246i \(0.806072\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 22.5469i 0.784031i 0.919959 + 0.392016i \(0.128222\pi\)
−0.919959 + 0.392016i \(0.871778\pi\)
\(828\) 0 0
\(829\) 54.6104i 1.89670i −0.317231 0.948348i \(-0.602753\pi\)
0.317231 0.948348i \(-0.397247\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 29.6121 + 6.67974i 1.02600 + 0.231439i
\(834\) 0 0
\(835\) 79.3529i 2.74612i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −16.1492 −0.557533 −0.278767 0.960359i \(-0.589926\pi\)
−0.278767 + 0.960359i \(0.589926\pi\)
\(840\) 0 0
\(841\) 61.2808 2.11313
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 21.6830i 0.745917i
\(846\) 0 0
\(847\) −5.63692 7.05010i −0.193687 0.242244i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 28.8626i 0.989397i
\(852\) 0 0
\(853\) 24.2803i 0.831343i 0.909515 + 0.415671i \(0.136453\pi\)
−0.909515 + 0.415671i \(0.863547\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 28.1949i 0.963119i 0.876413 + 0.481560i \(0.159930\pi\)
−0.876413 + 0.481560i \(0.840070\pi\)
\(858\) 0 0
\(859\) 28.0871 0.958319 0.479160 0.877728i \(-0.340941\pi\)
0.479160 + 0.877728i \(0.340941\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 30.2042i 1.02816i 0.857742 + 0.514081i \(0.171867\pi\)
−0.857742 + 0.514081i \(0.828133\pi\)
\(864\) 0 0
\(865\) −75.1110 −2.55385
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −38.7113 −1.31319
\(870\) 0 0
\(871\) −31.5347 −1.06851
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 78.9138 63.0957i 2.66777 2.13302i
\(876\) 0 0
\(877\) 38.7340 1.30795 0.653977 0.756515i \(-0.273099\pi\)
0.653977 + 0.756515i \(0.273099\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 14.2407i 0.479780i 0.970800 + 0.239890i \(0.0771114\pi\)
−0.970800 + 0.239890i \(0.922889\pi\)
\(882\) 0 0
\(883\) 33.8551i 1.13931i 0.821883 + 0.569657i \(0.192924\pi\)
−0.821883 + 0.569657i \(0.807076\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 48.3049 1.62192 0.810960 0.585102i \(-0.198945\pi\)
0.810960 + 0.585102i \(0.198945\pi\)
\(888\) 0 0
\(889\) −18.8387 + 15.0625i −0.631831 + 0.505182i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 31.1589 1.04269
\(894\) 0 0
\(895\) 44.5283 1.48842
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −17.5283 −0.584600
\(900\) 0 0
\(901\) 5.35948i 0.178550i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −64.8626 −2.15611
\(906\) 0 0
\(907\) 40.5543i 1.34658i −0.739377 0.673292i \(-0.764880\pi\)
0.739377 0.673292i \(-0.235120\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 32.2499i 1.06849i 0.845330 + 0.534244i \(0.179404\pi\)
−0.845330 + 0.534244i \(0.820596\pi\)
\(912\) 0 0
\(913\) 11.5709i 0.382941i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 7.56093 + 9.45646i 0.249684 + 0.312280i
\(918\) 0 0
\(919\) 30.3659i 1.00168i −0.865540 0.500840i \(-0.833024\pi\)
0.865540 0.500840i \(-0.166976\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −8.57622 −0.282290
\(924\) 0 0
\(925\) 70.6392 2.32260
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 31.2112i 1.02401i −0.858983 0.512004i \(-0.828903\pi\)
0.858983 0.512004i \(-0.171097\pi\)
\(930\) 0 0
\(931\) 3.91290 17.3464i 0.128240 0.568505i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 71.3930i 2.33480i
\(936\) 0 0
\(937\) 2.43557i 0.0795665i −0.999208 0.0397832i \(-0.987333\pi\)
0.999208 0.0397832i \(-0.0126667\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 22.5903i 0.736422i 0.929742 + 0.368211i \(0.120030\pi\)
−0.929742 + 0.368211i \(0.879970\pi\)
\(942\) 0 0
\(943\) 7.44763 0.242528
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 53.0848i 1.72503i 0.506035 + 0.862513i \(0.331111\pi\)
−0.506035 + 0.862513i \(0.668889\pi\)
\(948\) 0 0
\(949\) 8.53149 0.276944
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −60.7113 −1.96663 −0.983316 0.181906i \(-0.941773\pi\)
−0.983316 + 0.181906i \(0.941773\pi\)
\(954\) 0 0
\(955\) −108.082 −3.49744
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −19.6988 24.6373i −0.636107 0.795580i
\(960\) 0 0
\(961\) −27.5968 −0.890221
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 6.47175i 0.208333i
\(966\) 0 0
\(967\) 21.4476i 0.689709i −0.938656 0.344855i \(-0.887928\pi\)
0.938656 0.344855i \(-0.112072\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 35.2775 1.13211 0.566055 0.824368i \(-0.308469\pi\)
0.566055 + 0.824368i \(0.308469\pi\)
\(972\) 0 0
\(973\) −12.0839 15.1133i −0.387391 0.484511i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −27.2092 −0.870501 −0.435250 0.900309i \(-0.643340\pi\)
−0.435250 + 0.900309i \(0.643340\pi\)
\(978\) 0 0
\(979\) −24.8202 −0.793258
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −39.0218 −1.24460 −0.622300 0.782778i \(-0.713802\pi\)
−0.622300 + 0.782778i \(0.713802\pi\)
\(984\) 0 0
\(985\) 29.8910i 0.952406i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 13.9673 0.444135
\(990\) 0 0
\(991\) 31.1176i 0.988483i −0.869325 0.494241i \(-0.835446\pi\)
0.869325 0.494241i \(-0.164554\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 99.7156i 3.16120i
\(996\) 0 0
\(997\) 59.5678i 1.88653i 0.332037 + 0.943266i \(0.392264\pi\)
−0.332037 + 0.943266i \(0.607736\pi\)
\(998\) 0 0
\(999\) 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 4032.2.b.q.3583.8 8
3.2 odd 2 1344.2.b.g.895.1 8
4.3 odd 2 4032.2.b.o.3583.8 8
7.6 odd 2 4032.2.b.o.3583.1 8
8.3 odd 2 2016.2.b.a.1567.1 8
8.5 even 2 2016.2.b.c.1567.1 8
12.11 even 2 1344.2.b.h.895.1 8
21.20 even 2 1344.2.b.h.895.8 8
24.5 odd 2 672.2.b.b.223.8 yes 8
24.11 even 2 672.2.b.a.223.8 yes 8
28.27 even 2 inner 4032.2.b.q.3583.1 8
56.13 odd 2 2016.2.b.a.1567.8 8
56.27 even 2 2016.2.b.c.1567.8 8
84.83 odd 2 1344.2.b.g.895.8 8
168.83 odd 2 672.2.b.b.223.1 yes 8
168.125 even 2 672.2.b.a.223.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
672.2.b.a.223.1 8 168.125 even 2
672.2.b.a.223.8 yes 8 24.11 even 2
672.2.b.b.223.1 yes 8 168.83 odd 2
672.2.b.b.223.8 yes 8 24.5 odd 2
1344.2.b.g.895.1 8 3.2 odd 2
1344.2.b.g.895.8 8 84.83 odd 2
1344.2.b.h.895.1 8 12.11 even 2
1344.2.b.h.895.8 8 21.20 even 2
2016.2.b.a.1567.1 8 8.3 odd 2
2016.2.b.a.1567.8 8 56.13 odd 2
2016.2.b.c.1567.1 8 8.5 even 2
2016.2.b.c.1567.8 8 56.27 even 2
4032.2.b.o.3583.1 8 7.6 odd 2
4032.2.b.o.3583.8 8 4.3 odd 2
4032.2.b.q.3583.1 8 28.27 even 2 inner
4032.2.b.q.3583.8 8 1.1 even 1 trivial