Properties

Label 4032.2.i.b.1889.4
Level $4032$
Weight $2$
Character 4032.1889
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(1889,4032)]
 
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([0, 1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4032.1889");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4032.i (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: 8.0.40960000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 7x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1889.4
Root \(-1.14412 - 1.14412i\) of defining polynomial
Character \(\chi\) \(=\) 4032.1889
Dual form 4032.2.i.b.1889.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000i q^{5} +(2.23607 + 1.41421i) q^{7} +O(q^{10})\) \(q-2.00000i q^{5} +(2.23607 + 1.41421i) q^{7} +1.41421 q^{11} +4.47214 q^{17} -6.32456 q^{19} +3.16228i q^{23} +1.00000 q^{25} +3.16228 q^{29} -5.65685i q^{31} +(2.82843 - 4.47214i) q^{35} -4.47214i q^{37} +4.47214 q^{41} +4.00000i q^{43} +8.00000 q^{47} +(3.00000 + 6.32456i) q^{49} +9.48683 q^{53} -2.82843i q^{55} +8.94427i q^{59} -8.48528 q^{61} -2.00000i q^{67} +9.48683i q^{71} -6.32456i q^{73} +(3.16228 + 2.00000i) q^{77} +8.94427 q^{79} -8.94427i q^{85} -13.4164 q^{89} +12.6491i q^{95} +6.32456i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{25} + 64 q^{47} + 24 q^{49}+O(q^{100}) \) 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
\(5\) 2.00000i 0.894427i −0.894427 0.447214i \(-0.852416\pi\)
0.894427 0.447214i \(-0.147584\pi\)
\(6\) 0 0
\(7\) 2.23607 + 1.41421i 0.845154 + 0.534522i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.41421 0.426401 0.213201 0.977008i \(-0.431611\pi\)
0.213201 + 0.977008i \(0.431611\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.47214 1.08465 0.542326 0.840168i \(-0.317544\pi\)
0.542326 + 0.840168i \(0.317544\pi\)
\(18\) 0 0
\(19\) −6.32456 −1.45095 −0.725476 0.688247i \(-0.758380\pi\)
−0.725476 + 0.688247i \(0.758380\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.16228i 0.659380i 0.944089 + 0.329690i \(0.106944\pi\)
−0.944089 + 0.329690i \(0.893056\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.16228 0.587220 0.293610 0.955925i \(-0.405143\pi\)
0.293610 + 0.955925i \(0.405143\pi\)
\(30\) 0 0
\(31\) 5.65685i 1.01600i −0.861357 0.508001i \(-0.830385\pi\)
0.861357 0.508001i \(-0.169615\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.82843 4.47214i 0.478091 0.755929i
\(36\) 0 0
\(37\) 4.47214i 0.735215i −0.929981 0.367607i \(-0.880177\pi\)
0.929981 0.367607i \(-0.119823\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.47214 0.698430 0.349215 0.937043i \(-0.386448\pi\)
0.349215 + 0.937043i \(0.386448\pi\)
\(42\) 0 0
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) 3.00000 + 6.32456i 0.428571 + 0.903508i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 9.48683 1.30312 0.651558 0.758599i \(-0.274116\pi\)
0.651558 + 0.758599i \(0.274116\pi\)
\(54\) 0 0
\(55\) 2.82843i 0.381385i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 8.94427i 1.16445i 0.813029 + 0.582223i \(0.197817\pi\)
−0.813029 + 0.582223i \(0.802183\pi\)
\(60\) 0 0
\(61\) −8.48528 −1.08643 −0.543214 0.839594i \(-0.682793\pi\)
−0.543214 + 0.839594i \(0.682793\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 2.00000i 0.244339i −0.992509 0.122169i \(-0.961015\pi\)
0.992509 0.122169i \(-0.0389851\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 9.48683i 1.12588i 0.826498 + 0.562940i \(0.190330\pi\)
−0.826498 + 0.562940i \(0.809670\pi\)
\(72\) 0 0
\(73\) 6.32456i 0.740233i −0.928985 0.370117i \(-0.879318\pi\)
0.928985 0.370117i \(-0.120682\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.16228 + 2.00000i 0.360375 + 0.227921i
\(78\) 0 0
\(79\) 8.94427 1.00631 0.503155 0.864196i \(-0.332173\pi\)
0.503155 + 0.864196i \(0.332173\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 8.94427i 0.970143i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −13.4164 −1.42214 −0.711068 0.703123i \(-0.751788\pi\)
−0.711068 + 0.703123i \(0.751788\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 12.6491i 1.29777i
\(96\) 0 0
\(97\) 6.32456i 0.642161i 0.947052 + 0.321081i \(0.104046\pi\)
−0.947052 + 0.321081i \(0.895954\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 18.0000i 1.79107i −0.444994 0.895533i \(-0.646794\pi\)
0.444994 0.895533i \(-0.353206\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −7.07107 −0.683586 −0.341793 0.939775i \(-0.611034\pi\)
−0.341793 + 0.939775i \(0.611034\pi\)
\(108\) 0 0
\(109\) 4.47214i 0.428353i −0.976795 0.214176i \(-0.931293\pi\)
0.976795 0.214176i \(-0.0687068\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 12.7279i 1.19734i −0.800995 0.598671i \(-0.795696\pi\)
0.800995 0.598671i \(-0.204304\pi\)
\(114\) 0 0
\(115\) 6.32456 0.589768
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 10.0000 + 6.32456i 0.916698 + 0.579771i
\(120\) 0 0
\(121\) −9.00000 −0.818182
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 12.0000i 1.07331i
\(126\) 0 0
\(127\) 8.94427 0.793676 0.396838 0.917889i \(-0.370108\pi\)
0.396838 + 0.917889i \(0.370108\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) −14.1421 8.94427i −1.22628 0.775567i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 9.89949i 0.845771i 0.906183 + 0.422885i \(0.138983\pi\)
−0.906183 + 0.422885i \(0.861017\pi\)
\(138\) 0 0
\(139\) 12.6491 1.07288 0.536442 0.843937i \(-0.319768\pi\)
0.536442 + 0.843937i \(0.319768\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 6.32456i 0.525226i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 22.1359 1.81345 0.906724 0.421725i \(-0.138575\pi\)
0.906724 + 0.421725i \(0.138575\pi\)
\(150\) 0 0
\(151\) 13.4164 1.09181 0.545906 0.837846i \(-0.316186\pi\)
0.545906 + 0.837846i \(0.316186\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −11.3137 −0.908739
\(156\) 0 0
\(157\) 14.1421 1.12867 0.564333 0.825547i \(-0.309134\pi\)
0.564333 + 0.825547i \(0.309134\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −4.47214 + 7.07107i −0.352454 + 0.557278i
\(162\) 0 0
\(163\) 6.00000i 0.469956i −0.972001 0.234978i \(-0.924498\pi\)
0.972001 0.234978i \(-0.0755019\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 6.00000i 0.456172i 0.973641 + 0.228086i \(0.0732467\pi\)
−0.973641 + 0.228086i \(0.926753\pi\)
\(174\) 0 0
\(175\) 2.23607 + 1.41421i 0.169031 + 0.106904i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 4.24264 0.317110 0.158555 0.987350i \(-0.449317\pi\)
0.158555 + 0.987350i \(0.449317\pi\)
\(180\) 0 0
\(181\) −5.65685 −0.420471 −0.210235 0.977651i \(-0.567423\pi\)
−0.210235 + 0.977651i \(0.567423\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −8.94427 −0.657596
\(186\) 0 0
\(187\) 6.32456 0.462497
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 15.8114i 1.14407i 0.820229 + 0.572036i \(0.193846\pi\)
−0.820229 + 0.572036i \(0.806154\pi\)
\(192\) 0 0
\(193\) 4.00000 0.287926 0.143963 0.989583i \(-0.454015\pi\)
0.143963 + 0.989583i \(0.454015\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 22.1359 1.57712 0.788560 0.614957i \(-0.210827\pi\)
0.788560 + 0.614957i \(0.210827\pi\)
\(198\) 0 0
\(199\) 25.4558i 1.80452i −0.431196 0.902258i \(-0.641908\pi\)
0.431196 0.902258i \(-0.358092\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 7.07107 + 4.47214i 0.496292 + 0.313882i
\(204\) 0 0
\(205\) 8.94427i 0.624695i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −8.94427 −0.618688
\(210\) 0 0
\(211\) 12.0000i 0.826114i 0.910705 + 0.413057i \(0.135539\pi\)
−0.910705 + 0.413057i \(0.864461\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 8.00000 0.545595
\(216\) 0 0
\(217\) 8.00000 12.6491i 0.543075 0.858678i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 19.7990i 1.32584i −0.748691 0.662919i \(-0.769317\pi\)
0.748691 0.662919i \(-0.230683\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 17.8885i 1.18730i −0.804722 0.593652i \(-0.797686\pi\)
0.804722 0.593652i \(-0.202314\pi\)
\(228\) 0 0
\(229\) −16.9706 −1.12145 −0.560723 0.828003i \(-0.689477\pi\)
−0.560723 + 0.828003i \(0.689477\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1.41421i 0.0926482i 0.998926 + 0.0463241i \(0.0147507\pi\)
−0.998926 + 0.0463241i \(0.985249\pi\)
\(234\) 0 0
\(235\) 16.0000i 1.04372i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 9.48683i 0.613652i 0.951766 + 0.306826i \(0.0992670\pi\)
−0.951766 + 0.306826i \(0.900733\pi\)
\(240\) 0 0
\(241\) 6.32456i 0.407400i 0.979033 + 0.203700i \(0.0652968\pi\)
−0.979033 + 0.203700i \(0.934703\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 12.6491 6.00000i 0.808122 0.383326i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 8.94427i 0.564557i −0.959332 0.282279i \(-0.908910\pi\)
0.959332 0.282279i \(-0.0910903\pi\)
\(252\) 0 0
\(253\) 4.47214i 0.281161i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 31.3050 1.95275 0.976375 0.216085i \(-0.0693287\pi\)
0.976375 + 0.216085i \(0.0693287\pi\)
\(258\) 0 0
\(259\) 6.32456 10.0000i 0.392989 0.621370i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 9.48683i 0.584983i 0.956268 + 0.292492i \(0.0944843\pi\)
−0.956268 + 0.292492i \(0.905516\pi\)
\(264\) 0 0
\(265\) 18.9737i 1.16554i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 6.00000i 0.365826i −0.983129 0.182913i \(-0.941447\pi\)
0.983129 0.182913i \(-0.0585527\pi\)
\(270\) 0 0
\(271\) 22.6274i 1.37452i 0.726413 + 0.687259i \(0.241186\pi\)
−0.726413 + 0.687259i \(0.758814\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.41421 0.0852803
\(276\) 0 0
\(277\) 26.8328i 1.61223i −0.591761 0.806114i \(-0.701567\pi\)
0.591761 0.806114i \(-0.298433\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 15.5563i 0.928014i −0.885832 0.464007i \(-0.846411\pi\)
0.885832 0.464007i \(-0.153589\pi\)
\(282\) 0 0
\(283\) −18.9737 −1.12787 −0.563934 0.825820i \(-0.690713\pi\)
−0.563934 + 0.825820i \(0.690713\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 10.0000 + 6.32456i 0.590281 + 0.373327i
\(288\) 0 0
\(289\) 3.00000 0.176471
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 14.0000i 0.817889i −0.912559 0.408944i \(-0.865897\pi\)
0.912559 0.408944i \(-0.134103\pi\)
\(294\) 0 0
\(295\) 17.8885 1.04151
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −5.65685 + 8.94427i −0.326056 + 0.515539i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 16.9706i 0.971732i
\(306\) 0 0
\(307\) −31.6228 −1.80481 −0.902404 0.430892i \(-0.858199\pi\)
−0.902404 + 0.430892i \(0.858199\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) 12.6491i 0.714970i 0.933919 + 0.357485i \(0.116366\pi\)
−0.933919 + 0.357485i \(0.883634\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −9.48683 −0.532834 −0.266417 0.963858i \(-0.585840\pi\)
−0.266417 + 0.963858i \(0.585840\pi\)
\(318\) 0 0
\(319\) 4.47214 0.250392
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −28.2843 −1.57378
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 17.8885 + 11.3137i 0.986227 + 0.623745i
\(330\) 0 0
\(331\) 12.0000i 0.659580i 0.944054 + 0.329790i \(0.106978\pi\)
−0.944054 + 0.329790i \(0.893022\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) −8.00000 −0.435788 −0.217894 0.975972i \(-0.569919\pi\)
−0.217894 + 0.975972i \(0.569919\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 8.00000i 0.433224i
\(342\) 0 0
\(343\) −2.23607 + 18.3848i −0.120736 + 0.992685i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 35.3553 1.89797 0.948987 0.315315i \(-0.102110\pi\)
0.948987 + 0.315315i \(0.102110\pi\)
\(348\) 0 0
\(349\) 2.82843 0.151402 0.0757011 0.997131i \(-0.475881\pi\)
0.0757011 + 0.997131i \(0.475881\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −4.47214 −0.238028 −0.119014 0.992893i \(-0.537973\pi\)
−0.119014 + 0.992893i \(0.537973\pi\)
\(354\) 0 0
\(355\) 18.9737 1.00702
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 9.48683i 0.500696i 0.968156 + 0.250348i \(0.0805450\pi\)
−0.968156 + 0.250348i \(0.919455\pi\)
\(360\) 0 0
\(361\) 21.0000 1.10526
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −12.6491 −0.662085
\(366\) 0 0
\(367\) 14.1421i 0.738213i −0.929387 0.369107i \(-0.879664\pi\)
0.929387 0.369107i \(-0.120336\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 21.2132 + 13.4164i 1.10133 + 0.696545i
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 20.0000i 1.02733i 0.857991 + 0.513665i \(0.171713\pi\)
−0.857991 + 0.513665i \(0.828287\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 0 0
\(385\) 4.00000 6.32456i 0.203859 0.322329i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 22.1359 1.12234 0.561168 0.827702i \(-0.310352\pi\)
0.561168 + 0.827702i \(0.310352\pi\)
\(390\) 0 0
\(391\) 14.1421i 0.715199i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 17.8885i 0.900070i
\(396\) 0 0
\(397\) 2.82843 0.141955 0.0709773 0.997478i \(-0.477388\pi\)
0.0709773 + 0.997478i \(0.477388\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 15.5563i 0.776847i −0.921481 0.388424i \(-0.873020\pi\)
0.921481 0.388424i \(-0.126980\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 6.32456i 0.313497i
\(408\) 0 0
\(409\) 31.6228i 1.56365i 0.623501 + 0.781823i \(0.285710\pi\)
−0.623501 + 0.781823i \(0.714290\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −12.6491 + 20.0000i −0.622422 + 0.984136i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 26.8328i 1.31087i 0.755252 + 0.655434i \(0.227514\pi\)
−0.755252 + 0.655434i \(0.772486\pi\)
\(420\) 0 0
\(421\) 26.8328i 1.30775i −0.756602 0.653876i \(-0.773142\pi\)
0.756602 0.653876i \(-0.226858\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 4.47214 0.216930
\(426\) 0 0
\(427\) −18.9737 12.0000i −0.918200 0.580721i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 15.8114i 0.761608i −0.924656 0.380804i \(-0.875647\pi\)
0.924656 0.380804i \(-0.124353\pi\)
\(432\) 0 0
\(433\) 12.6491i 0.607877i 0.952692 + 0.303939i \(0.0983018\pi\)
−0.952692 + 0.303939i \(0.901698\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 20.0000i 0.956730i
\(438\) 0 0
\(439\) 25.4558i 1.21494i −0.794342 0.607471i \(-0.792184\pi\)
0.794342 0.607471i \(-0.207816\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −7.07107 −0.335957 −0.167978 0.985791i \(-0.553724\pi\)
−0.167978 + 0.985791i \(0.553724\pi\)
\(444\) 0 0
\(445\) 26.8328i 1.27200i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 32.5269i 1.53504i −0.641025 0.767520i \(-0.721491\pi\)
0.641025 0.767520i \(-0.278509\pi\)
\(450\) 0 0
\(451\) 6.32456 0.297812
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 8.00000 0.374224 0.187112 0.982339i \(-0.440087\pi\)
0.187112 + 0.982339i \(0.440087\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 30.0000i 1.39724i 0.715493 + 0.698620i \(0.246202\pi\)
−0.715493 + 0.698620i \(0.753798\pi\)
\(462\) 0 0
\(463\) −26.8328 −1.24703 −0.623513 0.781813i \(-0.714295\pi\)
−0.623513 + 0.781813i \(0.714295\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 8.94427i 0.413892i 0.978352 + 0.206946i \(0.0663524\pi\)
−0.978352 + 0.206946i \(0.933648\pi\)
\(468\) 0 0
\(469\) 2.82843 4.47214i 0.130605 0.206504i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 5.65685i 0.260102i
\(474\) 0 0
\(475\) −6.32456 −0.290191
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −4.00000 −0.182765 −0.0913823 0.995816i \(-0.529129\pi\)
−0.0913823 + 0.995816i \(0.529129\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 12.6491 0.574367
\(486\) 0 0
\(487\) −13.4164 −0.607955 −0.303978 0.952679i \(-0.598315\pi\)
−0.303978 + 0.952679i \(0.598315\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −26.8701 −1.21263 −0.606314 0.795225i \(-0.707353\pi\)
−0.606314 + 0.795225i \(0.707353\pi\)
\(492\) 0 0
\(493\) 14.1421 0.636930
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −13.4164 + 21.2132i −0.601808 + 0.951542i
\(498\) 0 0
\(499\) 20.0000i 0.895323i 0.894203 + 0.447661i \(0.147743\pi\)
−0.894203 + 0.447661i \(0.852257\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −16.0000 −0.713405 −0.356702 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) 0 0
\(505\) −36.0000 −1.60198
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 14.0000i 0.620539i 0.950649 + 0.310270i \(0.100419\pi\)
−0.950649 + 0.310270i \(0.899581\pi\)
\(510\) 0 0
\(511\) 8.94427 14.1421i 0.395671 0.625611i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 11.3137 0.497576
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 4.47214 0.195928 0.0979639 0.995190i \(-0.468767\pi\)
0.0979639 + 0.995190i \(0.468767\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 25.2982i 1.10201i
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 14.1421i 0.611418i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 4.24264 + 8.94427i 0.182743 + 0.385257i
\(540\) 0 0
\(541\) 26.8328i 1.15363i 0.816874 + 0.576816i \(0.195705\pi\)
−0.816874 + 0.576816i \(0.804295\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −8.94427 −0.383131
\(546\) 0 0
\(547\) 22.0000i 0.940652i 0.882493 + 0.470326i \(0.155864\pi\)
−0.882493 + 0.470326i \(0.844136\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −20.0000 −0.852029
\(552\) 0 0
\(553\) 20.0000 + 12.6491i 0.850487 + 0.537895i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −34.7851 −1.47389 −0.736945 0.675953i \(-0.763732\pi\)
−0.736945 + 0.675953i \(0.763732\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 35.7771i 1.50782i 0.656975 + 0.753912i \(0.271836\pi\)
−0.656975 + 0.753912i \(0.728164\pi\)
\(564\) 0 0
\(565\) −25.4558 −1.07094
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 24.0416i 1.00788i −0.863739 0.503939i \(-0.831884\pi\)
0.863739 0.503939i \(-0.168116\pi\)
\(570\) 0 0
\(571\) 22.0000i 0.920671i −0.887745 0.460336i \(-0.847729\pi\)
0.887745 0.460336i \(-0.152271\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 3.16228i 0.131876i
\(576\) 0 0
\(577\) 25.2982i 1.05318i −0.850120 0.526589i \(-0.823471\pi\)
0.850120 0.526589i \(-0.176529\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 13.4164 0.555651
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 8.94427i 0.369170i −0.982817 0.184585i \(-0.940906\pi\)
0.982817 0.184585i \(-0.0590940\pi\)
\(588\) 0 0
\(589\) 35.7771i 1.47417i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −22.3607 −0.918243 −0.459122 0.888373i \(-0.651836\pi\)
−0.459122 + 0.888373i \(0.651836\pi\)
\(594\) 0 0
\(595\) 12.6491 20.0000i 0.518563 0.819920i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 9.48683i 0.387621i −0.981039 0.193811i \(-0.937915\pi\)
0.981039 0.193811i \(-0.0620848\pi\)
\(600\) 0 0
\(601\) 12.6491i 0.515968i 0.966149 + 0.257984i \(0.0830582\pi\)
−0.966149 + 0.257984i \(0.916942\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 18.0000i 0.731804i
\(606\) 0 0
\(607\) 25.4558i 1.03322i 0.856221 + 0.516610i \(0.172806\pi\)
−0.856221 + 0.516610i \(0.827194\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 17.8885i 0.722511i 0.932467 + 0.361256i \(0.117652\pi\)
−0.932467 + 0.361256i \(0.882348\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 35.3553i 1.42335i −0.702508 0.711676i \(-0.747936\pi\)
0.702508 0.711676i \(-0.252064\pi\)
\(618\) 0 0
\(619\) 12.6491 0.508411 0.254205 0.967150i \(-0.418186\pi\)
0.254205 + 0.967150i \(0.418186\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −30.0000 18.9737i −1.20192 0.760164i
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 20.0000i 0.797452i
\(630\) 0 0
\(631\) −26.8328 −1.06820 −0.534099 0.845422i \(-0.679349\pi\)
−0.534099 + 0.845422i \(0.679349\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 17.8885i 0.709885i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 43.8406i 1.73160i 0.500390 + 0.865800i \(0.333190\pi\)
−0.500390 + 0.865800i \(0.666810\pi\)
\(642\) 0 0
\(643\) 44.2719 1.74591 0.872956 0.487798i \(-0.162200\pi\)
0.872956 + 0.487798i \(0.162200\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 8.00000 0.314512 0.157256 0.987558i \(-0.449735\pi\)
0.157256 + 0.987558i \(0.449735\pi\)
\(648\) 0 0
\(649\) 12.6491i 0.496521i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −47.4342 −1.85624 −0.928121 0.372278i \(-0.878577\pi\)
−0.928121 + 0.372278i \(0.878577\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −46.6690 −1.81797 −0.908984 0.416831i \(-0.863141\pi\)
−0.908984 + 0.416831i \(0.863141\pi\)
\(660\) 0 0
\(661\) 8.48528 0.330039 0.165020 0.986290i \(-0.447231\pi\)
0.165020 + 0.986290i \(0.447231\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −17.8885 + 28.2843i −0.693688 + 1.09682i
\(666\) 0 0
\(667\) 10.0000i 0.387202i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −12.0000 −0.463255
\(672\) 0 0
\(673\) −36.0000 −1.38770 −0.693849 0.720121i \(-0.744086\pi\)
−0.693849 + 0.720121i \(0.744086\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 2.00000i 0.0768662i −0.999261 0.0384331i \(-0.987763\pi\)
0.999261 0.0384331i \(-0.0122367\pi\)
\(678\) 0 0
\(679\) −8.94427 + 14.1421i −0.343250 + 0.542725i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 41.0122 1.56929 0.784644 0.619947i \(-0.212846\pi\)
0.784644 + 0.619947i \(0.212846\pi\)
\(684\) 0 0
\(685\) 19.7990 0.756481
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −25.2982 −0.962390 −0.481195 0.876614i \(-0.659797\pi\)
−0.481195 + 0.876614i \(0.659797\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 25.2982i 0.959616i
\(696\) 0 0
\(697\) 20.0000 0.757554
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −41.1096 −1.55269 −0.776344 0.630309i \(-0.782928\pi\)
−0.776344 + 0.630309i \(0.782928\pi\)
\(702\) 0 0
\(703\) 28.2843i 1.06676i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 25.4558 40.2492i 0.957366 1.51373i
\(708\) 0 0
\(709\) 40.2492i 1.51159i −0.654808 0.755796i \(-0.727250\pi\)
0.654808 0.755796i \(-0.272750\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 17.8885 0.669931
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −20.0000 −0.745874 −0.372937 0.927857i \(-0.621649\pi\)
−0.372937 + 0.927857i \(0.621649\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 3.16228 0.117444
\(726\) 0 0
\(727\) 16.9706i 0.629403i 0.949191 + 0.314702i \(0.101904\pi\)
−0.949191 + 0.314702i \(0.898096\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 17.8885i 0.661632i
\(732\) 0 0
\(733\) 50.9117 1.88047 0.940233 0.340532i \(-0.110607\pi\)
0.940233 + 0.340532i \(0.110607\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 2.82843i 0.104186i
\(738\) 0 0
\(739\) 10.0000i 0.367856i −0.982940 0.183928i \(-0.941119\pi\)
0.982940 0.183928i \(-0.0588813\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 28.4605i 1.04411i −0.852910 0.522057i \(-0.825165\pi\)
0.852910 0.522057i \(-0.174835\pi\)
\(744\) 0 0
\(745\) 44.2719i 1.62200i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −15.8114 10.0000i −0.577736 0.365392i
\(750\) 0 0
\(751\) −49.1935 −1.79510 −0.897548 0.440917i \(-0.854653\pi\)
−0.897548 + 0.440917i \(0.854653\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 26.8328i 0.976546i
\(756\) 0 0
\(757\) 22.3607i 0.812713i 0.913715 + 0.406356i \(0.133201\pi\)
−0.913715 + 0.406356i \(0.866799\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −13.4164 −0.486344 −0.243172 0.969983i \(-0.578188\pi\)
−0.243172 + 0.969983i \(0.578188\pi\)
\(762\) 0 0
\(763\) 6.32456 10.0000i 0.228964 0.362024i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 37.9473i 1.36842i −0.729287 0.684208i \(-0.760148\pi\)
0.729287 0.684208i \(-0.239852\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 46.0000i 1.65451i 0.561830 + 0.827253i \(0.310097\pi\)
−0.561830 + 0.827253i \(0.689903\pi\)
\(774\) 0 0
\(775\) 5.65685i 0.203200i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −28.2843 −1.01339
\(780\) 0 0
\(781\) 13.4164i 0.480077i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 28.2843i 1.00951i
\(786\) 0 0
\(787\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 18.0000 28.4605i 0.640006 1.01194i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 42.0000i 1.48772i 0.668338 + 0.743858i \(0.267006\pi\)
−0.668338 + 0.743858i \(0.732994\pi\)
\(798\) 0 0
\(799\) 35.7771 1.26570
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 8.94427i 0.315637i
\(804\) 0 0
\(805\) 14.1421 + 8.94427i 0.498445 + 0.315244i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 24.0416i 0.845259i 0.906303 + 0.422629i \(0.138893\pi\)
−0.906303 + 0.422629i \(0.861107\pi\)
\(810\) 0 0
\(811\) 12.6491 0.444170 0.222085 0.975027i \(-0.428714\pi\)
0.222085 + 0.975027i \(0.428714\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −12.0000 −0.420342
\(816\) 0 0
\(817\) 25.2982i 0.885073i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −9.48683 −0.331093 −0.165546 0.986202i \(-0.552939\pi\)
−0.165546 + 0.986202i \(0.552939\pi\)
\(822\) 0 0
\(823\) −44.7214 −1.55889 −0.779444 0.626472i \(-0.784498\pi\)
−0.779444 + 0.626472i \(0.784498\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −32.5269 −1.13107 −0.565536 0.824724i \(-0.691331\pi\)
−0.565536 + 0.824724i \(0.691331\pi\)
\(828\) 0 0
\(829\) −11.3137 −0.392941 −0.196471 0.980510i \(-0.562948\pi\)
−0.196471 + 0.980510i \(0.562948\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 13.4164 + 28.2843i 0.464851 + 0.979992i
\(834\) 0 0
\(835\) 24.0000i 0.830554i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 36.0000 1.24286 0.621429 0.783470i \(-0.286552\pi\)
0.621429 + 0.783470i \(0.286552\pi\)
\(840\) 0 0
\(841\) −19.0000 −0.655172
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 26.0000i 0.894427i
\(846\) 0 0
\(847\) −20.1246 12.7279i −0.691490 0.437337i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 14.1421 0.484786
\(852\) 0 0
\(853\) −48.0833 −1.64634 −0.823170 0.567795i \(-0.807796\pi\)
−0.823170 + 0.567795i \(0.807796\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 40.2492 1.37489 0.687444 0.726238i \(-0.258733\pi\)
0.687444 + 0.726238i \(0.258733\pi\)
\(858\) 0 0
\(859\) −31.6228 −1.07896 −0.539478 0.842000i \(-0.681378\pi\)
−0.539478 + 0.842000i \(0.681378\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 15.8114i 0.538226i −0.963109 0.269113i \(-0.913270\pi\)
0.963109 0.269113i \(-0.0867305\pi\)
\(864\) 0 0
\(865\) 12.0000 0.408012
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 12.6491 0.429092
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 16.9706 26.8328i 0.573710 0.907115i
\(876\) 0 0
\(877\) 53.6656i 1.81216i 0.423107 + 0.906080i \(0.360939\pi\)
−0.423107 + 0.906080i \(0.639061\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 22.3607 0.753350 0.376675 0.926345i \(-0.377067\pi\)
0.376675 + 0.926345i \(0.377067\pi\)
\(882\) 0 0
\(883\) 44.0000i 1.48072i 0.672212 + 0.740359i \(0.265344\pi\)
−0.672212 + 0.740359i \(0.734656\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −52.0000 −1.74599 −0.872995 0.487730i \(-0.837825\pi\)
−0.872995 + 0.487730i \(0.837825\pi\)
\(888\) 0 0
\(889\) 20.0000 + 12.6491i 0.670778 + 0.424238i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −50.5964 −1.69315
\(894\) 0 0
\(895\) 8.48528i 0.283632i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 17.8885i 0.596616i
\(900\) 0 0
\(901\) 42.4264 1.41343
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 11.3137i 0.376080i
\(906\) 0 0
\(907\) 52.0000i 1.72663i −0.504664 0.863316i \(-0.668384\pi\)
0.504664 0.863316i \(-0.331616\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 47.4342i 1.57156i −0.618504 0.785782i \(-0.712261\pi\)
0.618504 0.785782i \(-0.287739\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −22.3607 −0.737611 −0.368805 0.929507i \(-0.620233\pi\)
−0.368805 + 0.929507i \(0.620233\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 4.47214i 0.147043i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −49.1935 −1.61399 −0.806993 0.590561i \(-0.798907\pi\)
−0.806993 + 0.590561i \(0.798907\pi\)
\(930\) 0 0
\(931\) −18.9737 40.0000i −0.621837 1.31095i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 12.6491i 0.413670i
\(936\) 0 0
\(937\) 37.9473i 1.23969i 0.784726 + 0.619843i \(0.212804\pi\)
−0.784726 + 0.619843i \(0.787196\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 30.0000i 0.977972i 0.872292 + 0.488986i \(0.162633\pi\)
−0.872292 + 0.488986i \(0.837367\pi\)
\(942\) 0 0
\(943\) 14.1421i 0.460531i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 18.3848 0.597425 0.298712 0.954343i \(-0.403443\pi\)
0.298712 + 0.954343i \(0.403443\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 35.3553i 1.14527i −0.819810 0.572636i \(-0.805921\pi\)
0.819810 0.572636i \(-0.194079\pi\)
\(954\) 0 0
\(955\) 31.6228 1.02329
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −14.0000 + 22.1359i −0.452084 + 0.714807i
\(960\) 0 0
\(961\) −1.00000 −0.0322581
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 8.00000i 0.257529i
\(966\) 0 0
\(967\) −26.8328 −0.862885 −0.431443 0.902140i \(-0.641995\pi\)
−0.431443 + 0.902140i \(0.641995\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 17.8885i 0.574071i 0.957920 + 0.287035i \(0.0926697\pi\)
−0.957920 + 0.287035i \(0.907330\pi\)
\(972\) 0 0
\(973\) 28.2843 + 17.8885i 0.906752 + 0.573480i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 24.0416i 0.769160i 0.923092 + 0.384580i \(0.125654\pi\)
−0.923092 + 0.384580i \(0.874346\pi\)
\(978\) 0 0
\(979\) −18.9737 −0.606401
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −36.0000 −1.14822 −0.574111 0.818778i \(-0.694652\pi\)
−0.574111 + 0.818778i \(0.694652\pi\)
\(984\) 0 0
\(985\) 44.2719i 1.41062i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −12.6491 −0.402218
\(990\) 0 0
\(991\) −13.4164 −0.426186 −0.213093 0.977032i \(-0.568354\pi\)
−0.213093 + 0.977032i \(0.568354\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −50.9117 −1.61401
\(996\) 0 0
\(997\) −53.7401 −1.70197 −0.850983 0.525193i \(-0.823993\pi\)
−0.850983 + 0.525193i \(0.823993\pi\)
\(998\) 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.i.b.1889.4 yes 8
3.2 odd 2 4032.2.i.a.1889.8 yes 8
4.3 odd 2 4032.2.i.a.1889.1 8
7.6 odd 2 4032.2.i.a.1889.7 yes 8
8.3 odd 2 4032.2.i.a.1889.5 yes 8
8.5 even 2 inner 4032.2.i.b.1889.8 yes 8
12.11 even 2 inner 4032.2.i.b.1889.5 yes 8
21.20 even 2 inner 4032.2.i.b.1889.3 yes 8
24.5 odd 2 4032.2.i.a.1889.4 yes 8
24.11 even 2 inner 4032.2.i.b.1889.1 yes 8
28.27 even 2 inner 4032.2.i.b.1889.6 yes 8
56.13 odd 2 4032.2.i.a.1889.3 yes 8
56.27 even 2 inner 4032.2.i.b.1889.2 yes 8
84.83 odd 2 4032.2.i.a.1889.2 yes 8
168.83 odd 2 4032.2.i.a.1889.6 yes 8
168.125 even 2 inner 4032.2.i.b.1889.7 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4032.2.i.a.1889.1 8 4.3 odd 2
4032.2.i.a.1889.2 yes 8 84.83 odd 2
4032.2.i.a.1889.3 yes 8 56.13 odd 2
4032.2.i.a.1889.4 yes 8 24.5 odd 2
4032.2.i.a.1889.5 yes 8 8.3 odd 2
4032.2.i.a.1889.6 yes 8 168.83 odd 2
4032.2.i.a.1889.7 yes 8 7.6 odd 2
4032.2.i.a.1889.8 yes 8 3.2 odd 2
4032.2.i.b.1889.1 yes 8 24.11 even 2 inner
4032.2.i.b.1889.2 yes 8 56.27 even 2 inner
4032.2.i.b.1889.3 yes 8 21.20 even 2 inner
4032.2.i.b.1889.4 yes 8 1.1 even 1 trivial
4032.2.i.b.1889.5 yes 8 12.11 even 2 inner
4032.2.i.b.1889.6 yes 8 28.27 even 2 inner
4032.2.i.b.1889.7 yes 8 168.125 even 2 inner
4032.2.i.b.1889.8 yes 8 8.5 even 2 inner