Properties

Label 4032.2.p.i.1567.3
Level $4032$
Weight $2$
Character 4032.1567
Analytic conductor $32.196$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4032,2,Mod(1567,4032)]
 
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, 1, 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("4032.1567");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4032.p (of order \(2\), degree \(1\), 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: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1567.3
Root \(-0.258819 + 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 4032.1567
Dual form 4032.2.p.i.1567.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.82843 q^{5} +(1.73205 - 2.00000i) q^{7} +O(q^{10})\) \(q-2.82843 q^{5} +(1.73205 - 2.00000i) q^{7} +4.89898 q^{11} +6.00000 q^{13} -4.89898i q^{17} +8.48528i q^{23} +3.00000 q^{25} +3.46410 q^{31} +(-4.89898 + 5.65685i) q^{35} +6.92820i q^{37} -4.89898i q^{41} +3.46410 q^{43} -9.79796 q^{47} +(-1.00000 - 6.92820i) q^{49} +9.79796i q^{53} -13.8564 q^{55} -5.65685i q^{59} +6.00000 q^{61} -16.9706 q^{65} +10.3923 q^{67} -8.48528i q^{71} -13.8564i q^{73} +(8.48528 - 9.79796i) q^{77} -4.00000i q^{79} +5.65685i q^{83} +13.8564i q^{85} +14.6969i q^{89} +(10.3923 - 12.0000i) q^{91} +13.8564i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 48 q^{13} + 24 q^{25} - 8 q^{49} + 48 q^{61}+O(q^{100}) \) 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
\(5\) −2.82843 −1.26491 −0.632456 0.774597i \(-0.717953\pi\)
−0.632456 + 0.774597i \(0.717953\pi\)
\(6\) 0 0
\(7\) 1.73205 2.00000i 0.654654 0.755929i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.89898 1.47710 0.738549 0.674200i \(-0.235511\pi\)
0.738549 + 0.674200i \(0.235511\pi\)
\(12\) 0 0
\(13\) 6.00000 1.66410 0.832050 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.89898i 1.18818i −0.804400 0.594089i \(-0.797513\pi\)
0.804400 0.594089i \(-0.202487\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 8.48528i 1.76930i 0.466252 + 0.884652i \(0.345604\pi\)
−0.466252 + 0.884652i \(0.654396\pi\)
\(24\) 0 0
\(25\) 3.00000 0.600000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 3.46410 0.622171 0.311086 0.950382i \(-0.399307\pi\)
0.311086 + 0.950382i \(0.399307\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −4.89898 + 5.65685i −0.828079 + 0.956183i
\(36\) 0 0
\(37\) 6.92820i 1.13899i 0.821995 + 0.569495i \(0.192861\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.89898i 0.765092i −0.923936 0.382546i \(-0.875047\pi\)
0.923936 0.382546i \(-0.124953\pi\)
\(42\) 0 0
\(43\) 3.46410 0.528271 0.264135 0.964486i \(-0.414913\pi\)
0.264135 + 0.964486i \(0.414913\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −9.79796 −1.42918 −0.714590 0.699544i \(-0.753387\pi\)
−0.714590 + 0.699544i \(0.753387\pi\)
\(48\) 0 0
\(49\) −1.00000 6.92820i −0.142857 0.989743i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 9.79796i 1.34585i 0.739709 + 0.672927i \(0.234963\pi\)
−0.739709 + 0.672927i \(0.765037\pi\)
\(54\) 0 0
\(55\) −13.8564 −1.86840
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 5.65685i 0.736460i −0.929735 0.368230i \(-0.879964\pi\)
0.929735 0.368230i \(-0.120036\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −16.9706 −2.10494
\(66\) 0 0
\(67\) 10.3923 1.26962 0.634811 0.772667i \(-0.281078\pi\)
0.634811 + 0.772667i \(0.281078\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 8.48528i 1.00702i −0.863990 0.503509i \(-0.832042\pi\)
0.863990 0.503509i \(-0.167958\pi\)
\(72\) 0 0
\(73\) 13.8564i 1.62177i −0.585206 0.810885i \(-0.698986\pi\)
0.585206 0.810885i \(-0.301014\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 8.48528 9.79796i 0.966988 1.11658i
\(78\) 0 0
\(79\) 4.00000i 0.450035i −0.974355 0.225018i \(-0.927756\pi\)
0.974355 0.225018i \(-0.0722440\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 5.65685i 0.620920i 0.950586 + 0.310460i \(0.100483\pi\)
−0.950586 + 0.310460i \(0.899517\pi\)
\(84\) 0 0
\(85\) 13.8564i 1.50294i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 14.6969i 1.55787i 0.627103 + 0.778936i \(0.284240\pi\)
−0.627103 + 0.778936i \(0.715760\pi\)
\(90\) 0 0
\(91\) 10.3923 12.0000i 1.08941 1.25794i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 13.8564i 1.40690i 0.710742 + 0.703452i \(0.248359\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −14.1421 −1.40720 −0.703598 0.710599i \(-0.748424\pi\)
−0.703598 + 0.710599i \(0.748424\pi\)
\(102\) 0 0
\(103\) −10.3923 −1.02398 −0.511992 0.858990i \(-0.671092\pi\)
−0.511992 + 0.858990i \(0.671092\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.89898 0.473602 0.236801 0.971558i \(-0.423901\pi\)
0.236801 + 0.971558i \(0.423901\pi\)
\(108\) 0 0
\(109\) 6.92820i 0.663602i 0.943349 + 0.331801i \(0.107656\pi\)
−0.943349 + 0.331801i \(0.892344\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 16.9706 1.59646 0.798228 0.602355i \(-0.205771\pi\)
0.798228 + 0.602355i \(0.205771\pi\)
\(114\) 0 0
\(115\) 24.0000i 2.23801i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −9.79796 8.48528i −0.898177 0.777844i
\(120\) 0 0
\(121\) 13.0000 1.18182
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 5.65685 0.505964
\(126\) 0 0
\(127\) 4.00000i 0.354943i −0.984126 0.177471i \(-0.943208\pi\)
0.984126 0.177471i \(-0.0567917\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 5.65685i 0.494242i −0.968985 0.247121i \(-0.920516\pi\)
0.968985 0.247121i \(-0.0794845\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 29.3939 2.45804
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 9.79796i 0.802680i 0.915929 + 0.401340i \(0.131455\pi\)
−0.915929 + 0.401340i \(0.868545\pi\)
\(150\) 0 0
\(151\) 20.0000i 1.62758i −0.581161 0.813788i \(-0.697401\pi\)
0.581161 0.813788i \(-0.302599\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −9.79796 −0.786991
\(156\) 0 0
\(157\) 6.00000 0.478852 0.239426 0.970915i \(-0.423041\pi\)
0.239426 + 0.970915i \(0.423041\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 16.9706 + 14.6969i 1.33747 + 1.15828i
\(162\) 0 0
\(163\) 17.3205 1.35665 0.678323 0.734763i \(-0.262707\pi\)
0.678323 + 0.734763i \(0.262707\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 19.5959 1.51638 0.758189 0.652035i \(-0.226085\pi\)
0.758189 + 0.652035i \(0.226085\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 14.1421 1.07521 0.537603 0.843198i \(-0.319330\pi\)
0.537603 + 0.843198i \(0.319330\pi\)
\(174\) 0 0
\(175\) 5.19615 6.00000i 0.392792 0.453557i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −4.89898 −0.366167 −0.183083 0.983097i \(-0.558608\pi\)
−0.183083 + 0.983097i \(0.558608\pi\)
\(180\) 0 0
\(181\) 18.0000 1.33793 0.668965 0.743294i \(-0.266738\pi\)
0.668965 + 0.743294i \(0.266738\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 19.5959i 1.44072i
\(186\) 0 0
\(187\) 24.0000i 1.75505i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 8.48528i 0.613973i 0.951714 + 0.306987i \(0.0993207\pi\)
−0.951714 + 0.306987i \(0.900679\pi\)
\(192\) 0 0
\(193\) −10.0000 −0.719816 −0.359908 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 19.5959i 1.39615i −0.716024 0.698076i \(-0.754040\pi\)
0.716024 0.698076i \(-0.245960\pi\)
\(198\) 0 0
\(199\) −3.46410 −0.245564 −0.122782 0.992434i \(-0.539182\pi\)
−0.122782 + 0.992434i \(0.539182\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 13.8564i 0.967773i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −10.3923 −0.715436 −0.357718 0.933830i \(-0.616445\pi\)
−0.357718 + 0.933830i \(0.616445\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −9.79796 −0.668215
\(216\) 0 0
\(217\) 6.00000 6.92820i 0.407307 0.470317i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 29.3939i 1.97725i
\(222\) 0 0
\(223\) −10.3923 −0.695920 −0.347960 0.937509i \(-0.613126\pi\)
−0.347960 + 0.937509i \(0.613126\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 22.6274i 1.50183i −0.660396 0.750917i \(-0.729612\pi\)
0.660396 0.750917i \(-0.270388\pi\)
\(228\) 0 0
\(229\) 18.0000 1.18947 0.594737 0.803921i \(-0.297256\pi\)
0.594737 + 0.803921i \(0.297256\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 16.9706 1.11178 0.555889 0.831256i \(-0.312378\pi\)
0.555889 + 0.831256i \(0.312378\pi\)
\(234\) 0 0
\(235\) 27.7128 1.80778
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 8.48528i 0.548867i −0.961606 0.274434i \(-0.911510\pi\)
0.961606 0.274434i \(-0.0884904\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 2.82843 + 19.5959i 0.180702 + 1.25194i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 22.6274i 1.42823i −0.700028 0.714115i \(-0.746829\pi\)
0.700028 0.714115i \(-0.253171\pi\)
\(252\) 0 0
\(253\) 41.5692i 2.61343i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 14.6969i 0.916770i 0.888754 + 0.458385i \(0.151572\pi\)
−0.888754 + 0.458385i \(0.848428\pi\)
\(258\) 0 0
\(259\) 13.8564 + 12.0000i 0.860995 + 0.745644i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 8.48528i 0.523225i 0.965173 + 0.261612i \(0.0842542\pi\)
−0.965173 + 0.261612i \(0.915746\pi\)
\(264\) 0 0
\(265\) 27.7128i 1.70238i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −2.82843 −0.172452 −0.0862261 0.996276i \(-0.527481\pi\)
−0.0862261 + 0.996276i \(0.527481\pi\)
\(270\) 0 0
\(271\) −3.46410 −0.210429 −0.105215 0.994450i \(-0.533553\pi\)
−0.105215 + 0.994450i \(0.533553\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 14.6969 0.886259
\(276\) 0 0
\(277\) 6.92820i 0.416275i −0.978100 0.208138i \(-0.933260\pi\)
0.978100 0.208138i \(-0.0667402\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −16.9706 −1.01238 −0.506189 0.862422i \(-0.668946\pi\)
−0.506189 + 0.862422i \(0.668946\pi\)
\(282\) 0 0
\(283\) 24.0000i 1.42665i 0.700832 + 0.713326i \(0.252812\pi\)
−0.700832 + 0.713326i \(0.747188\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −9.79796 8.48528i −0.578355 0.500870i
\(288\) 0 0
\(289\) −7.00000 −0.411765
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −19.7990 −1.15667 −0.578335 0.815800i \(-0.696297\pi\)
−0.578335 + 0.815800i \(0.696297\pi\)
\(294\) 0 0
\(295\) 16.0000i 0.931556i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 50.9117i 2.94430i
\(300\) 0 0
\(301\) 6.00000 6.92820i 0.345834 0.399335i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −16.9706 −0.971732
\(306\) 0 0
\(307\) 24.0000i 1.36975i −0.728659 0.684876i \(-0.759856\pi\)
0.728659 0.684876i \(-0.240144\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 9.79796 0.555591 0.277796 0.960640i \(-0.410396\pi\)
0.277796 + 0.960640i \(0.410396\pi\)
\(312\) 0 0
\(313\) 27.7128i 1.56642i −0.621757 0.783210i \(-0.713581\pi\)
0.621757 0.783210i \(-0.286419\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 9.79796i 0.550308i 0.961400 + 0.275154i \(0.0887289\pi\)
−0.961400 + 0.275154i \(0.911271\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 18.0000 0.998460
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −16.9706 + 19.5959i −0.935617 + 1.08036i
\(330\) 0 0
\(331\) −3.46410 −0.190404 −0.0952021 0.995458i \(-0.530350\pi\)
−0.0952021 + 0.995458i \(0.530350\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −29.3939 −1.60596
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 16.9706 0.919007
\(342\) 0 0
\(343\) −15.5885 10.0000i −0.841698 0.539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −24.4949 −1.31495 −0.657477 0.753474i \(-0.728377\pi\)
−0.657477 + 0.753474i \(0.728377\pi\)
\(348\) 0 0
\(349\) −6.00000 −0.321173 −0.160586 0.987022i \(-0.551338\pi\)
−0.160586 + 0.987022i \(0.551338\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 34.2929i 1.82522i −0.408826 0.912612i \(-0.634062\pi\)
0.408826 0.912612i \(-0.365938\pi\)
\(354\) 0 0
\(355\) 24.0000i 1.27379i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 25.4558i 1.34351i −0.740774 0.671754i \(-0.765541\pi\)
0.740774 0.671754i \(-0.234459\pi\)
\(360\) 0 0
\(361\) 19.0000 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 39.1918i 2.05139i
\(366\) 0 0
\(367\) −17.3205 −0.904123 −0.452062 0.891987i \(-0.649311\pi\)
−0.452062 + 0.891987i \(0.649311\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 19.5959 + 16.9706i 1.01737 + 0.881068i
\(372\) 0 0
\(373\) 6.92820i 0.358729i 0.983783 + 0.179364i \(0.0574041\pi\)
−0.983783 + 0.179364i \(0.942596\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −24.2487 −1.24557 −0.622786 0.782392i \(-0.713999\pi\)
−0.622786 + 0.782392i \(0.713999\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −9.79796 −0.500652 −0.250326 0.968162i \(-0.580538\pi\)
−0.250326 + 0.968162i \(0.580538\pi\)
\(384\) 0 0
\(385\) −24.0000 + 27.7128i −1.22315 + 1.41238i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 19.5959i 0.993552i 0.867879 + 0.496776i \(0.165483\pi\)
−0.867879 + 0.496776i \(0.834517\pi\)
\(390\) 0 0
\(391\) 41.5692 2.10225
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 11.3137i 0.569254i
\(396\) 0 0
\(397\) −6.00000 −0.301131 −0.150566 0.988600i \(-0.548110\pi\)
−0.150566 + 0.988600i \(0.548110\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 20.7846 1.03536
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 33.9411i 1.68240i
\(408\) 0 0
\(409\) 13.8564i 0.685155i 0.939490 + 0.342578i \(0.111300\pi\)
−0.939490 + 0.342578i \(0.888700\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −11.3137 9.79796i −0.556711 0.482126i
\(414\) 0 0
\(415\) 16.0000i 0.785409i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 5.65685i 0.276355i 0.990407 + 0.138178i \(0.0441245\pi\)
−0.990407 + 0.138178i \(0.955875\pi\)
\(420\) 0 0
\(421\) 20.7846i 1.01298i 0.862246 + 0.506490i \(0.169057\pi\)
−0.862246 + 0.506490i \(0.830943\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 14.6969i 0.712906i
\(426\) 0 0
\(427\) 10.3923 12.0000i 0.502919 0.580721i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 8.48528i 0.408722i 0.978896 + 0.204361i \(0.0655116\pi\)
−0.978896 + 0.204361i \(0.934488\pi\)
\(432\) 0 0
\(433\) 13.8564i 0.665896i −0.942945 0.332948i \(-0.891957\pi\)
0.942945 0.332948i \(-0.108043\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 24.2487 1.15733 0.578664 0.815566i \(-0.303574\pi\)
0.578664 + 0.815566i \(0.303574\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −14.6969 −0.698273 −0.349136 0.937072i \(-0.613525\pi\)
−0.349136 + 0.937072i \(0.613525\pi\)
\(444\) 0 0
\(445\) 41.5692i 1.97057i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −16.9706 −0.800890 −0.400445 0.916321i \(-0.631145\pi\)
−0.400445 + 0.916321i \(0.631145\pi\)
\(450\) 0 0
\(451\) 24.0000i 1.13012i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −29.3939 + 33.9411i −1.37801 + 1.59118i
\(456\) 0 0
\(457\) 22.0000 1.02912 0.514558 0.857455i \(-0.327956\pi\)
0.514558 + 0.857455i \(0.327956\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −19.7990 −0.922131 −0.461065 0.887366i \(-0.652533\pi\)
−0.461065 + 0.887366i \(0.652533\pi\)
\(462\) 0 0
\(463\) 28.0000i 1.30127i 0.759390 + 0.650635i \(0.225497\pi\)
−0.759390 + 0.650635i \(0.774503\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 22.6274i 1.04707i −0.852004 0.523536i \(-0.824613\pi\)
0.852004 0.523536i \(-0.175387\pi\)
\(468\) 0 0
\(469\) 18.0000 20.7846i 0.831163 0.959744i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 16.9706 0.780307
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 9.79796 0.447680 0.223840 0.974626i \(-0.428141\pi\)
0.223840 + 0.974626i \(0.428141\pi\)
\(480\) 0 0
\(481\) 41.5692i 1.89539i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 39.1918i 1.77961i
\(486\) 0 0
\(487\) 20.0000i 0.906287i −0.891438 0.453143i \(-0.850303\pi\)
0.891438 0.453143i \(-0.149697\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 34.2929 1.54761 0.773807 0.633421i \(-0.218350\pi\)
0.773807 + 0.633421i \(0.218350\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −16.9706 14.6969i −0.761234 0.659248i
\(498\) 0 0
\(499\) −38.1051 −1.70582 −0.852910 0.522059i \(-0.825164\pi\)
−0.852910 + 0.522059i \(0.825164\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 9.79796 0.436869 0.218435 0.975852i \(-0.429905\pi\)
0.218435 + 0.975852i \(0.429905\pi\)
\(504\) 0 0
\(505\) 40.0000 1.77998
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 2.82843 0.125368 0.0626839 0.998033i \(-0.480034\pi\)
0.0626839 + 0.998033i \(0.480034\pi\)
\(510\) 0 0
\(511\) −27.7128 24.0000i −1.22594 1.06170i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 29.3939 1.29525
\(516\) 0 0
\(517\) −48.0000 −2.11104
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 4.89898i 0.214628i −0.994225 0.107314i \(-0.965775\pi\)
0.994225 0.107314i \(-0.0342250\pi\)
\(522\) 0 0
\(523\) 24.0000i 1.04945i −0.851273 0.524723i \(-0.824169\pi\)
0.851273 0.524723i \(-0.175831\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 16.9706i 0.739249i
\(528\) 0 0
\(529\) −49.0000 −2.13043
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 29.3939i 1.27319i
\(534\) 0 0
\(535\) −13.8564 −0.599065
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −4.89898 33.9411i −0.211014 1.46195i
\(540\) 0 0
\(541\) 6.92820i 0.297867i 0.988847 + 0.148933i \(0.0475840\pi\)
−0.988847 + 0.148933i \(0.952416\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 19.5959i 0.839397i
\(546\) 0 0
\(547\) −38.1051 −1.62926 −0.814629 0.579983i \(-0.803059\pi\)
−0.814629 + 0.579983i \(0.803059\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −8.00000 6.92820i −0.340195 0.294617i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 9.79796i 0.415153i 0.978219 + 0.207576i \(0.0665576\pi\)
−0.978219 + 0.207576i \(0.933442\pi\)
\(558\) 0 0
\(559\) 20.7846 0.879095
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 5.65685i 0.238408i 0.992870 + 0.119204i \(0.0380342\pi\)
−0.992870 + 0.119204i \(0.961966\pi\)
\(564\) 0 0
\(565\) −48.0000 −2.01938
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 16.9706 0.711443 0.355722 0.934592i \(-0.384235\pi\)
0.355722 + 0.934592i \(0.384235\pi\)
\(570\) 0 0
\(571\) 3.46410 0.144968 0.0724841 0.997370i \(-0.476907\pi\)
0.0724841 + 0.997370i \(0.476907\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 25.4558i 1.06158i
\(576\) 0 0
\(577\) 41.5692i 1.73055i 0.501298 + 0.865275i \(0.332856\pi\)
−0.501298 + 0.865275i \(0.667144\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 11.3137 + 9.79796i 0.469372 + 0.406488i
\(582\) 0 0
\(583\) 48.0000i 1.98796i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 11.3137i 0.466967i −0.972361 0.233483i \(-0.924988\pi\)
0.972361 0.233483i \(-0.0750124\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 4.89898i 0.201177i 0.994928 + 0.100588i \(0.0320726\pi\)
−0.994928 + 0.100588i \(0.967927\pi\)
\(594\) 0 0
\(595\) 27.7128 + 24.0000i 1.13611 + 0.983904i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 42.4264i 1.73350i −0.498746 0.866748i \(-0.666206\pi\)
0.498746 0.866748i \(-0.333794\pi\)
\(600\) 0 0
\(601\) 13.8564i 0.565215i −0.959236 0.282607i \(-0.908801\pi\)
0.959236 0.282607i \(-0.0911993\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −36.7696 −1.49489
\(606\) 0 0
\(607\) −10.3923 −0.421811 −0.210905 0.977506i \(-0.567641\pi\)
−0.210905 + 0.977506i \(0.567641\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −58.7878 −2.37830
\(612\) 0 0
\(613\) 20.7846i 0.839482i −0.907644 0.419741i \(-0.862121\pi\)
0.907644 0.419741i \(-0.137879\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) 48.0000i 1.92928i 0.263566 + 0.964641i \(0.415101\pi\)
−0.263566 + 0.964641i \(0.584899\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 29.3939 + 25.4558i 1.17764 + 1.01987i
\(624\) 0 0
\(625\) −31.0000 −1.24000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 33.9411 1.35332
\(630\) 0 0
\(631\) 4.00000i 0.159237i 0.996825 + 0.0796187i \(0.0253703\pi\)
−0.996825 + 0.0796187i \(0.974630\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 11.3137i 0.448971i
\(636\) 0 0
\(637\) −6.00000 41.5692i −0.237729 1.64703i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 33.9411 1.34059 0.670297 0.742093i \(-0.266167\pi\)
0.670297 + 0.742093i \(0.266167\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 27.7128i 1.08782i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 9.79796i 0.383424i −0.981451 0.191712i \(-0.938596\pi\)
0.981451 0.191712i \(-0.0614039\pi\)
\(654\) 0 0
\(655\) 16.0000i 0.625172i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −4.89898 −0.190837 −0.0954186 0.995437i \(-0.530419\pi\)
−0.0954186 + 0.995437i \(0.530419\pi\)
\(660\) 0 0
\(661\) −30.0000 −1.16686 −0.583432 0.812162i \(-0.698291\pi\)
−0.583432 + 0.812162i \(0.698291\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 29.3939 1.13474
\(672\) 0 0
\(673\) −22.0000 −0.848038 −0.424019 0.905653i \(-0.639381\pi\)
−0.424019 + 0.905653i \(0.639381\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 2.82843 0.108705 0.0543526 0.998522i \(-0.482690\pi\)
0.0543526 + 0.998522i \(0.482690\pi\)
\(678\) 0 0
\(679\) 27.7128 + 24.0000i 1.06352 + 0.921035i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 44.0908 1.68709 0.843544 0.537060i \(-0.180465\pi\)
0.843544 + 0.537060i \(0.180465\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 58.7878i 2.23964i
\(690\) 0 0
\(691\) 24.0000i 0.913003i 0.889723 + 0.456502i \(0.150898\pi\)
−0.889723 + 0.456502i \(0.849102\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −24.0000 −0.909065
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 48.9898i 1.85032i 0.379579 + 0.925160i \(0.376069\pi\)
−0.379579 + 0.925160i \(0.623931\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −24.4949 + 28.2843i −0.921225 + 1.06374i
\(708\) 0 0
\(709\) 6.92820i 0.260194i −0.991501 0.130097i \(-0.958471\pi\)
0.991501 0.130097i \(-0.0415289\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 29.3939i 1.10081i
\(714\) 0 0
\(715\) −83.1384 −3.10920
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 9.79796 0.365402 0.182701 0.983169i \(-0.441516\pi\)
0.182701 + 0.983169i \(0.441516\pi\)
\(720\) 0 0
\(721\) −18.0000 + 20.7846i −0.670355 + 0.774059i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −17.3205 −0.642382 −0.321191 0.947014i \(-0.604083\pi\)
−0.321191 + 0.947014i \(0.604083\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 16.9706i 0.627679i
\(732\) 0 0
\(733\) 6.00000 0.221615 0.110808 0.993842i \(-0.464656\pi\)
0.110808 + 0.993842i \(0.464656\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 50.9117 1.87536
\(738\) 0 0
\(739\) 10.3923 0.382287 0.191144 0.981562i \(-0.438780\pi\)
0.191144 + 0.981562i \(0.438780\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 42.4264i 1.55647i −0.627971 0.778237i \(-0.716114\pi\)
0.627971 0.778237i \(-0.283886\pi\)
\(744\) 0 0
\(745\) 27.7128i 1.01532i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 8.48528 9.79796i 0.310045 0.358010i
\(750\) 0 0
\(751\) 20.0000i 0.729810i 0.931045 + 0.364905i \(0.118899\pi\)
−0.931045 + 0.364905i \(0.881101\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 56.5685i 2.05874i
\(756\) 0 0
\(757\) 48.4974i 1.76267i 0.472493 + 0.881334i \(0.343354\pi\)
−0.472493 + 0.881334i \(0.656646\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 4.89898i 0.177588i 0.996050 + 0.0887939i \(0.0283013\pi\)
−0.996050 + 0.0887939i \(0.971699\pi\)
\(762\) 0 0
\(763\) 13.8564 + 12.0000i 0.501636 + 0.434429i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 33.9411i 1.22554i
\(768\) 0 0
\(769\) 27.7128i 0.999350i 0.866213 + 0.499675i \(0.166547\pi\)
−0.866213 + 0.499675i \(0.833453\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 48.0833 1.72943 0.864717 0.502259i \(-0.167498\pi\)
0.864717 + 0.502259i \(0.167498\pi\)
\(774\) 0 0
\(775\) 10.3923 0.373303
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 41.5692i 1.48746i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −16.9706 −0.605705
\(786\) 0 0
\(787\) 24.0000i 0.855508i 0.903895 + 0.427754i \(0.140695\pi\)
−0.903895 + 0.427754i \(0.859305\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 29.3939 33.9411i 1.04513 1.20681i
\(792\) 0 0
\(793\) 36.0000 1.27840
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 19.7990 0.701316 0.350658 0.936504i \(-0.385958\pi\)
0.350658 + 0.936504i \(0.385958\pi\)
\(798\) 0 0
\(799\) 48.0000i 1.69812i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 67.8823i 2.39551i
\(804\) 0 0
\(805\) −48.0000 41.5692i −1.69178 1.46512i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −16.9706 −0.596653 −0.298327 0.954464i \(-0.596428\pi\)
−0.298327 + 0.954464i \(0.596428\pi\)
\(810\) 0 0
\(811\) 24.0000i 0.842754i 0.906886 + 0.421377i \(0.138453\pi\)
−0.906886 + 0.421377i \(0.861547\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −48.9898 −1.71604
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 29.3939i 1.02585i −0.858432 0.512927i \(-0.828561\pi\)
0.858432 0.512927i \(-0.171439\pi\)
\(822\) 0 0
\(823\) 44.0000i 1.53374i −0.641800 0.766872i \(-0.721812\pi\)
0.641800 0.766872i \(-0.278188\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −14.6969 −0.511063 −0.255531 0.966801i \(-0.582250\pi\)
−0.255531 + 0.966801i \(0.582250\pi\)
\(828\) 0 0
\(829\) −6.00000 −0.208389 −0.104194 0.994557i \(-0.533226\pi\)
−0.104194 + 0.994557i \(0.533226\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −33.9411 + 4.89898i −1.17599 + 0.169740i
\(834\) 0 0
\(835\) −55.4256 −1.91808
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −19.5959 −0.676526 −0.338263 0.941052i \(-0.609839\pi\)
−0.338263 + 0.941052i \(0.609839\pi\)
\(840\) 0 0
\(841\) 29.0000 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −65.0538 −2.23792
\(846\) 0 0
\(847\) 22.5167 26.0000i 0.773682 0.893371i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −58.7878 −2.01522
\(852\) 0 0
\(853\) 30.0000 1.02718 0.513590 0.858036i \(-0.328315\pi\)
0.513590 + 0.858036i \(0.328315\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 24.4949i 0.836730i 0.908279 + 0.418365i \(0.137397\pi\)
−0.908279 + 0.418365i \(0.862603\pi\)
\(858\) 0 0
\(859\) 48.0000i 1.63774i 0.573980 + 0.818869i \(0.305399\pi\)
−0.573980 + 0.818869i \(0.694601\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 25.4558i 0.866527i −0.901267 0.433264i \(-0.857362\pi\)
0.901267 0.433264i \(-0.142638\pi\)
\(864\) 0 0
\(865\) −40.0000 −1.36004
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 19.5959i 0.664746i
\(870\) 0 0
\(871\) 62.3538 2.11278
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 9.79796 11.3137i 0.331231 0.382473i
\(876\) 0 0
\(877\) 34.6410i 1.16974i −0.811126 0.584872i \(-0.801145\pi\)
0.811126 0.584872i \(-0.198855\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 14.6969i 0.495152i 0.968868 + 0.247576i \(0.0796341\pi\)
−0.968868 + 0.247576i \(0.920366\pi\)
\(882\) 0 0
\(883\) −45.0333 −1.51549 −0.757746 0.652550i \(-0.773699\pi\)
−0.757746 + 0.652550i \(0.773699\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 19.5959 0.657967 0.328983 0.944336i \(-0.393294\pi\)
0.328983 + 0.944336i \(0.393294\pi\)
\(888\) 0 0
\(889\) −8.00000 6.92820i −0.268311 0.232364i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 13.8564 0.463169
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 48.0000 1.59911
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −50.9117 −1.69236
\(906\) 0 0
\(907\) 3.46410 0.115024 0.0575118 0.998345i \(-0.481683\pi\)
0.0575118 + 0.998345i \(0.481683\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 8.48528i 0.281130i −0.990071 0.140565i \(-0.955108\pi\)
0.990071 0.140565i \(-0.0448919\pi\)
\(912\) 0 0
\(913\) 27.7128i 0.917160i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −11.3137 9.79796i −0.373612 0.323557i
\(918\) 0 0
\(919\) 4.00000i 0.131948i −0.997821 0.0659739i \(-0.978985\pi\)
0.997821 0.0659739i \(-0.0210154\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 50.9117i 1.67578i
\(924\) 0 0
\(925\) 20.7846i 0.683394i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 4.89898i 0.160730i 0.996765 + 0.0803652i \(0.0256086\pi\)
−0.996765 + 0.0803652i \(0.974391\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 67.8823i 2.21999i
\(936\) 0 0
\(937\) 27.7128i 0.905338i 0.891679 + 0.452669i \(0.149528\pi\)
−0.891679 + 0.452669i \(0.850472\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 53.7401 1.75188 0.875939 0.482422i \(-0.160243\pi\)
0.875939 + 0.482422i \(0.160243\pi\)
\(942\) 0 0
\(943\) 41.5692 1.35368
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 53.8888 1.75115 0.875575 0.483082i \(-0.160483\pi\)
0.875575 + 0.483082i \(0.160483\pi\)
\(948\) 0 0
\(949\) 83.1384i 2.69879i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 33.9411 1.09946 0.549730 0.835342i \(-0.314730\pi\)
0.549730 + 0.835342i \(0.314730\pi\)
\(954\) 0 0
\(955\) 24.0000i 0.776622i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −19.0000 −0.612903
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 28.2843 0.910503
\(966\) 0 0
\(967\) 52.0000i 1.67221i 0.548572 + 0.836104i \(0.315172\pi\)
−0.548572 + 0.836104i \(0.684828\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 11.3137i 0.363074i 0.983384 + 0.181537i \(0.0581072\pi\)
−0.983384 + 0.181537i \(0.941893\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 16.9706 0.542936 0.271468 0.962447i \(-0.412491\pi\)
0.271468 + 0.962447i \(0.412491\pi\)
\(978\) 0 0
\(979\) 72.0000i 2.30113i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −19.5959 −0.625013 −0.312506 0.949916i \(-0.601169\pi\)
−0.312506 + 0.949916i \(0.601169\pi\)
\(984\) 0 0
\(985\) 55.4256i 1.76601i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 29.3939i 0.934671i
\(990\) 0 0
\(991\) 4.00000i 0.127064i 0.997980 + 0.0635321i \(0.0202365\pi\)
−0.997980 + 0.0635321i \(0.979763\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 9.79796 0.310616
\(996\) 0 0
\(997\) −18.0000 −0.570066 −0.285033 0.958518i \(-0.592005\pi\)
−0.285033 + 0.958518i \(0.592005\pi\)
\(998\) 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.p.i.1567.3 yes 8
3.2 odd 2 inner 4032.2.p.i.1567.7 yes 8
4.3 odd 2 inner 4032.2.p.i.1567.2 yes 8
7.6 odd 2 4032.2.p.e.1567.5 yes 8
8.3 odd 2 4032.2.p.e.1567.6 yes 8
8.5 even 2 4032.2.p.e.1567.7 yes 8
12.11 even 2 inner 4032.2.p.i.1567.6 yes 8
21.20 even 2 4032.2.p.e.1567.1 8
24.5 odd 2 4032.2.p.e.1567.3 yes 8
24.11 even 2 4032.2.p.e.1567.2 yes 8
28.27 even 2 4032.2.p.e.1567.8 yes 8
56.13 odd 2 inner 4032.2.p.i.1567.1 yes 8
56.27 even 2 inner 4032.2.p.i.1567.4 yes 8
84.83 odd 2 4032.2.p.e.1567.4 yes 8
168.83 odd 2 inner 4032.2.p.i.1567.8 yes 8
168.125 even 2 inner 4032.2.p.i.1567.5 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4032.2.p.e.1567.1 8 21.20 even 2
4032.2.p.e.1567.2 yes 8 24.11 even 2
4032.2.p.e.1567.3 yes 8 24.5 odd 2
4032.2.p.e.1567.4 yes 8 84.83 odd 2
4032.2.p.e.1567.5 yes 8 7.6 odd 2
4032.2.p.e.1567.6 yes 8 8.3 odd 2
4032.2.p.e.1567.7 yes 8 8.5 even 2
4032.2.p.e.1567.8 yes 8 28.27 even 2
4032.2.p.i.1567.1 yes 8 56.13 odd 2 inner
4032.2.p.i.1567.2 yes 8 4.3 odd 2 inner
4032.2.p.i.1567.3 yes 8 1.1 even 1 trivial
4032.2.p.i.1567.4 yes 8 56.27 even 2 inner
4032.2.p.i.1567.5 yes 8 168.125 even 2 inner
4032.2.p.i.1567.6 yes 8 12.11 even 2 inner
4032.2.p.i.1567.7 yes 8 3.2 odd 2 inner
4032.2.p.i.1567.8 yes 8 168.83 odd 2 inner