Properties

Label 3584.2.a.b.1.1
Level $3584$
Weight $2$
Character 3584.1
Self dual yes
Analytic conductor $28.618$
Analytic rank $1$
Dimension $2$
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] = [3584,2,Mod(1,3584)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3584, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3584.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3584 = 2^{9} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3584.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(28.6183840844\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 3584.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.41421 q^{3} +1.00000 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.41421 q^{3} +1.00000 q^{7} -1.00000 q^{9} +1.41421 q^{11} +1.41421 q^{19} -1.41421 q^{21} -4.00000 q^{23} -5.00000 q^{25} +5.65685 q^{27} +2.82843 q^{29} -4.00000 q^{31} -2.00000 q^{33} +2.82843 q^{37} -6.00000 q^{41} +4.24264 q^{43} +4.00000 q^{47} +1.00000 q^{49} -8.48528 q^{53} -2.00000 q^{57} +7.07107 q^{59} -5.65685 q^{61} -1.00000 q^{63} -1.41421 q^{67} +5.65685 q^{69} +8.00000 q^{71} -8.00000 q^{73} +7.07107 q^{75} +1.41421 q^{77} +4.00000 q^{79} -5.00000 q^{81} -7.07107 q^{83} -4.00000 q^{87} -8.00000 q^{89} +5.65685 q^{93} -8.00000 q^{97} -1.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{7} - 2 q^{9} - 8 q^{23} - 10 q^{25} - 8 q^{31} - 4 q^{33} - 12 q^{41} + 8 q^{47} + 2 q^{49} - 4 q^{57} - 2 q^{63} + 16 q^{71} - 16 q^{73} + 8 q^{79} - 10 q^{81} - 8 q^{87} - 16 q^{89} - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −1.41421 −0.816497 −0.408248 0.912871i \(-0.633860\pi\)
−0.408248 + 0.912871i \(0.633860\pi\)
\(4\) 0 0
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 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
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 1.41421 0.324443 0.162221 0.986754i \(-0.448134\pi\)
0.162221 + 0.986754i \(0.448134\pi\)
\(20\) 0 0
\(21\) −1.41421 −0.308607
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) 5.65685 1.08866
\(28\) 0 0
\(29\) 2.82843 0.525226 0.262613 0.964901i \(-0.415416\pi\)
0.262613 + 0.964901i \(0.415416\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) −2.00000 −0.348155
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.82843 0.464991 0.232495 0.972598i \(-0.425311\pi\)
0.232495 + 0.972598i \(0.425311\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 4.24264 0.646997 0.323498 0.946229i \(-0.395141\pi\)
0.323498 + 0.946229i \(0.395141\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.00000 0.583460 0.291730 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −8.48528 −1.16554 −0.582772 0.812636i \(-0.698032\pi\)
−0.582772 + 0.812636i \(0.698032\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −2.00000 −0.264906
\(58\) 0 0
\(59\) 7.07107 0.920575 0.460287 0.887770i \(-0.347746\pi\)
0.460287 + 0.887770i \(0.347746\pi\)
\(60\) 0 0
\(61\) −5.65685 −0.724286 −0.362143 0.932123i \(-0.617955\pi\)
−0.362143 + 0.932123i \(0.617955\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −1.41421 −0.172774 −0.0863868 0.996262i \(-0.527532\pi\)
−0.0863868 + 0.996262i \(0.527532\pi\)
\(68\) 0 0
\(69\) 5.65685 0.681005
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) −8.00000 −0.936329 −0.468165 0.883641i \(-0.655085\pi\)
−0.468165 + 0.883641i \(0.655085\pi\)
\(74\) 0 0
\(75\) 7.07107 0.816497
\(76\) 0 0
\(77\) 1.41421 0.161165
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) −5.00000 −0.555556
\(82\) 0 0
\(83\) −7.07107 −0.776151 −0.388075 0.921628i \(-0.626860\pi\)
−0.388075 + 0.921628i \(0.626860\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −4.00000 −0.428845
\(88\) 0 0
\(89\) −8.00000 −0.847998 −0.423999 0.905663i \(-0.639374\pi\)
−0.423999 + 0.905663i \(0.639374\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 5.65685 0.586588
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −8.00000 −0.812277 −0.406138 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) 0 0
\(99\) −1.41421 −0.142134
\(100\) 0 0
\(101\) 11.3137 1.12576 0.562878 0.826540i \(-0.309694\pi\)
0.562878 + 0.826540i \(0.309694\pi\)
\(102\) 0 0
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 7.07107 0.683586 0.341793 0.939775i \(-0.388966\pi\)
0.341793 + 0.939775i \(0.388966\pi\)
\(108\) 0 0
\(109\) −8.48528 −0.812743 −0.406371 0.913708i \(-0.633206\pi\)
−0.406371 + 0.913708i \(0.633206\pi\)
\(110\) 0 0
\(111\) −4.00000 −0.379663
\(112\) 0 0
\(113\) −10.0000 −0.940721 −0.470360 0.882474i \(-0.655876\pi\)
−0.470360 + 0.882474i \(0.655876\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −9.00000 −0.818182
\(122\) 0 0
\(123\) 8.48528 0.765092
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) −6.00000 −0.528271
\(130\) 0 0
\(131\) −9.89949 −0.864923 −0.432461 0.901652i \(-0.642355\pi\)
−0.432461 + 0.901652i \(0.642355\pi\)
\(132\) 0 0
\(133\) 1.41421 0.122628
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) 0 0
\(139\) 1.41421 0.119952 0.0599760 0.998200i \(-0.480898\pi\)
0.0599760 + 0.998200i \(0.480898\pi\)
\(140\) 0 0
\(141\) −5.65685 −0.476393
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −1.41421 −0.116642
\(148\) 0 0
\(149\) −19.7990 −1.62200 −0.810998 0.585049i \(-0.801075\pi\)
−0.810998 + 0.585049i \(0.801075\pi\)
\(150\) 0 0
\(151\) −20.0000 −1.62758 −0.813788 0.581161i \(-0.802599\pi\)
−0.813788 + 0.581161i \(0.802599\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 16.9706 1.35440 0.677199 0.735800i \(-0.263194\pi\)
0.677199 + 0.735800i \(0.263194\pi\)
\(158\) 0 0
\(159\) 12.0000 0.951662
\(160\) 0 0
\(161\) −4.00000 −0.315244
\(162\) 0 0
\(163\) −9.89949 −0.775388 −0.387694 0.921788i \(-0.626728\pi\)
−0.387694 + 0.921788i \(0.626728\pi\)
\(164\) 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
\(171\) −1.41421 −0.108148
\(172\) 0 0
\(173\) −11.3137 −0.860165 −0.430083 0.902790i \(-0.641516\pi\)
−0.430083 + 0.902790i \(0.641516\pi\)
\(174\) 0 0
\(175\) −5.00000 −0.377964
\(176\) 0 0
\(177\) −10.0000 −0.751646
\(178\) 0 0
\(179\) −15.5563 −1.16274 −0.581368 0.813641i \(-0.697482\pi\)
−0.581368 + 0.813641i \(0.697482\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 8.00000 0.591377
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 5.65685 0.411476
\(190\) 0 0
\(191\) −4.00000 −0.289430 −0.144715 0.989473i \(-0.546227\pi\)
−0.144715 + 0.989473i \(0.546227\pi\)
\(192\) 0 0
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 14.1421 1.00759 0.503793 0.863825i \(-0.331938\pi\)
0.503793 + 0.863825i \(0.331938\pi\)
\(198\) 0 0
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 0 0
\(201\) 2.00000 0.141069
\(202\) 0 0
\(203\) 2.82843 0.198517
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 4.00000 0.278019
\(208\) 0 0
\(209\) 2.00000 0.138343
\(210\) 0 0
\(211\) −15.5563 −1.07094 −0.535472 0.844553i \(-0.679866\pi\)
−0.535472 + 0.844553i \(0.679866\pi\)
\(212\) 0 0
\(213\) −11.3137 −0.775203
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −4.00000 −0.271538
\(218\) 0 0
\(219\) 11.3137 0.764510
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 0 0
\(225\) 5.00000 0.333333
\(226\) 0 0
\(227\) −18.3848 −1.22024 −0.610120 0.792309i \(-0.708879\pi\)
−0.610120 + 0.792309i \(0.708879\pi\)
\(228\) 0 0
\(229\) 16.9706 1.12145 0.560723 0.828003i \(-0.310523\pi\)
0.560723 + 0.828003i \(0.310523\pi\)
\(230\) 0 0
\(231\) −2.00000 −0.131590
\(232\) 0 0
\(233\) 4.00000 0.262049 0.131024 0.991379i \(-0.458173\pi\)
0.131024 + 0.991379i \(0.458173\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −5.65685 −0.367452
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −16.0000 −1.03065 −0.515325 0.856995i \(-0.672329\pi\)
−0.515325 + 0.856995i \(0.672329\pi\)
\(242\) 0 0
\(243\) −9.89949 −0.635053
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 10.0000 0.633724
\(250\) 0 0
\(251\) −9.89949 −0.624851 −0.312425 0.949942i \(-0.601141\pi\)
−0.312425 + 0.949942i \(0.601141\pi\)
\(252\) 0 0
\(253\) −5.65685 −0.355643
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) 2.82843 0.175750
\(260\) 0 0
\(261\) −2.82843 −0.175075
\(262\) 0 0
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 11.3137 0.692388
\(268\) 0 0
\(269\) 11.3137 0.689809 0.344904 0.938638i \(-0.387911\pi\)
0.344904 + 0.938638i \(0.387911\pi\)
\(270\) 0 0
\(271\) 24.0000 1.45790 0.728948 0.684569i \(-0.240010\pi\)
0.728948 + 0.684569i \(0.240010\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −7.07107 −0.426401
\(276\) 0 0
\(277\) −31.1127 −1.86938 −0.934690 0.355463i \(-0.884323\pi\)
−0.934690 + 0.355463i \(0.884323\pi\)
\(278\) 0 0
\(279\) 4.00000 0.239474
\(280\) 0 0
\(281\) 4.00000 0.238620 0.119310 0.992857i \(-0.461932\pi\)
0.119310 + 0.992857i \(0.461932\pi\)
\(282\) 0 0
\(283\) −1.41421 −0.0840663 −0.0420331 0.999116i \(-0.513384\pi\)
−0.0420331 + 0.999116i \(0.513384\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −6.00000 −0.354169
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 11.3137 0.663221
\(292\) 0 0
\(293\) 28.2843 1.65238 0.826192 0.563388i \(-0.190502\pi\)
0.826192 + 0.563388i \(0.190502\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 8.00000 0.464207
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 4.24264 0.244542
\(302\) 0 0
\(303\) −16.0000 −0.919176
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 1.41421 0.0807134 0.0403567 0.999185i \(-0.487151\pi\)
0.0403567 + 0.999185i \(0.487151\pi\)
\(308\) 0 0
\(309\) −5.65685 −0.321807
\(310\) 0 0
\(311\) 8.00000 0.453638 0.226819 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2.82843 0.158860 0.0794301 0.996840i \(-0.474690\pi\)
0.0794301 + 0.996840i \(0.474690\pi\)
\(318\) 0 0
\(319\) 4.00000 0.223957
\(320\) 0 0
\(321\) −10.0000 −0.558146
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 12.0000 0.663602
\(328\) 0 0
\(329\) 4.00000 0.220527
\(330\) 0 0
\(331\) −21.2132 −1.16598 −0.582992 0.812478i \(-0.698118\pi\)
−0.582992 + 0.812478i \(0.698118\pi\)
\(332\) 0 0
\(333\) −2.82843 −0.154997
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −26.0000 −1.41631 −0.708155 0.706057i \(-0.750472\pi\)
−0.708155 + 0.706057i \(0.750472\pi\)
\(338\) 0 0
\(339\) 14.1421 0.768095
\(340\) 0 0
\(341\) −5.65685 −0.306336
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −7.07107 −0.379595 −0.189797 0.981823i \(-0.560783\pi\)
−0.189797 + 0.981823i \(0.560783\pi\)
\(348\) 0 0
\(349\) 16.9706 0.908413 0.454207 0.890896i \(-0.349923\pi\)
0.454207 + 0.890896i \(0.349923\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 18.0000 0.958043 0.479022 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −20.0000 −1.05556 −0.527780 0.849381i \(-0.676975\pi\)
−0.527780 + 0.849381i \(0.676975\pi\)
\(360\) 0 0
\(361\) −17.0000 −0.894737
\(362\) 0 0
\(363\) 12.7279 0.668043
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −24.0000 −1.25279 −0.626395 0.779506i \(-0.715470\pi\)
−0.626395 + 0.779506i \(0.715470\pi\)
\(368\) 0 0
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) −8.48528 −0.440534
\(372\) 0 0
\(373\) −8.48528 −0.439351 −0.219676 0.975573i \(-0.570500\pi\)
−0.219676 + 0.975573i \(0.570500\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −7.07107 −0.363216 −0.181608 0.983371i \(-0.558130\pi\)
−0.181608 + 0.983371i \(0.558130\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 20.0000 1.02195 0.510976 0.859595i \(-0.329284\pi\)
0.510976 + 0.859595i \(0.329284\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −4.24264 −0.215666
\(388\) 0 0
\(389\) −19.7990 −1.00385 −0.501924 0.864912i \(-0.667374\pi\)
−0.501924 + 0.864912i \(0.667374\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 14.0000 0.706207
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −39.5980 −1.98737 −0.993683 0.112225i \(-0.964202\pi\)
−0.993683 + 0.112225i \(0.964202\pi\)
\(398\) 0 0
\(399\) −2.00000 −0.100125
\(400\) 0 0
\(401\) 20.0000 0.998752 0.499376 0.866385i \(-0.333563\pi\)
0.499376 + 0.866385i \(0.333563\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 4.00000 0.198273
\(408\) 0 0
\(409\) −6.00000 −0.296681 −0.148340 0.988936i \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) 0 0
\(411\) 2.82843 0.139516
\(412\) 0 0
\(413\) 7.07107 0.347945
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −2.00000 −0.0979404
\(418\) 0 0
\(419\) −4.24264 −0.207267 −0.103633 0.994616i \(-0.533047\pi\)
−0.103633 + 0.994616i \(0.533047\pi\)
\(420\) 0 0
\(421\) 36.7696 1.79204 0.896019 0.444015i \(-0.146446\pi\)
0.896019 + 0.444015i \(0.146446\pi\)
\(422\) 0 0
\(423\) −4.00000 −0.194487
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −5.65685 −0.273754
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 16.0000 0.768911 0.384455 0.923144i \(-0.374389\pi\)
0.384455 + 0.923144i \(0.374389\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −5.65685 −0.270604
\(438\) 0 0
\(439\) 32.0000 1.52728 0.763638 0.645644i \(-0.223411\pi\)
0.763638 + 0.645644i \(0.223411\pi\)
\(440\) 0 0
\(441\) −1.00000 −0.0476190
\(442\) 0 0
\(443\) 7.07107 0.335957 0.167978 0.985791i \(-0.446276\pi\)
0.167978 + 0.985791i \(0.446276\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 28.0000 1.32435
\(448\) 0 0
\(449\) 28.0000 1.32140 0.660701 0.750649i \(-0.270259\pi\)
0.660701 + 0.750649i \(0.270259\pi\)
\(450\) 0 0
\(451\) −8.48528 −0.399556
\(452\) 0 0
\(453\) 28.2843 1.32891
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −6.00000 −0.280668 −0.140334 0.990104i \(-0.544818\pi\)
−0.140334 + 0.990104i \(0.544818\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 5.65685 0.263466 0.131733 0.991285i \(-0.457946\pi\)
0.131733 + 0.991285i \(0.457946\pi\)
\(462\) 0 0
\(463\) 20.0000 0.929479 0.464739 0.885448i \(-0.346148\pi\)
0.464739 + 0.885448i \(0.346148\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 24.0416 1.11251 0.556257 0.831010i \(-0.312237\pi\)
0.556257 + 0.831010i \(0.312237\pi\)
\(468\) 0 0
\(469\) −1.41421 −0.0653023
\(470\) 0 0
\(471\) −24.0000 −1.10586
\(472\) 0 0
\(473\) 6.00000 0.275880
\(474\) 0 0
\(475\) −7.07107 −0.324443
\(476\) 0 0
\(477\) 8.48528 0.388514
\(478\) 0 0
\(479\) −28.0000 −1.27935 −0.639676 0.768644i \(-0.720932\pi\)
−0.639676 + 0.768644i \(0.720932\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 5.65685 0.257396
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −28.0000 −1.26880 −0.634401 0.773004i \(-0.718753\pi\)
−0.634401 + 0.773004i \(0.718753\pi\)
\(488\) 0 0
\(489\) 14.0000 0.633102
\(490\) 0 0
\(491\) −12.7279 −0.574403 −0.287202 0.957870i \(-0.592725\pi\)
−0.287202 + 0.957870i \(0.592725\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 8.00000 0.358849
\(498\) 0 0
\(499\) 29.6985 1.32949 0.664743 0.747072i \(-0.268541\pi\)
0.664743 + 0.747072i \(0.268541\pi\)
\(500\) 0 0
\(501\) −16.9706 −0.758189
\(502\) 0 0
\(503\) 40.0000 1.78351 0.891756 0.452517i \(-0.149474\pi\)
0.891756 + 0.452517i \(0.149474\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 18.3848 0.816497
\(508\) 0 0
\(509\) −39.5980 −1.75515 −0.877575 0.479440i \(-0.840840\pi\)
−0.877575 + 0.479440i \(0.840840\pi\)
\(510\) 0 0
\(511\) −8.00000 −0.353899
\(512\) 0 0
\(513\) 8.00000 0.353209
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 5.65685 0.248788
\(518\) 0 0
\(519\) 16.0000 0.702322
\(520\) 0 0
\(521\) −10.0000 −0.438108 −0.219054 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) 0 0
\(523\) −12.7279 −0.556553 −0.278277 0.960501i \(-0.589763\pi\)
−0.278277 + 0.960501i \(0.589763\pi\)
\(524\) 0 0
\(525\) 7.07107 0.308607
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −7.07107 −0.306858
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 22.0000 0.949370
\(538\) 0 0
\(539\) 1.41421 0.0609145
\(540\) 0 0
\(541\) 19.7990 0.851225 0.425613 0.904906i \(-0.360059\pi\)
0.425613 + 0.904906i \(0.360059\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 38.1838 1.63262 0.816310 0.577614i \(-0.196016\pi\)
0.816310 + 0.577614i \(0.196016\pi\)
\(548\) 0 0
\(549\) 5.65685 0.241429
\(550\) 0 0
\(551\) 4.00000 0.170406
\(552\) 0 0
\(553\) 4.00000 0.170097
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 14.1421 0.599222 0.299611 0.954062i \(-0.403143\pi\)
0.299611 + 0.954062i \(0.403143\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 29.6985 1.25164 0.625821 0.779967i \(-0.284764\pi\)
0.625821 + 0.779967i \(0.284764\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −5.00000 −0.209980
\(568\) 0 0
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) −21.2132 −0.887745 −0.443872 0.896090i \(-0.646396\pi\)
−0.443872 + 0.896090i \(0.646396\pi\)
\(572\) 0 0
\(573\) 5.65685 0.236318
\(574\) 0 0
\(575\) 20.0000 0.834058
\(576\) 0 0
\(577\) 14.0000 0.582828 0.291414 0.956597i \(-0.405874\pi\)
0.291414 + 0.956597i \(0.405874\pi\)
\(578\) 0 0
\(579\) 5.65685 0.235091
\(580\) 0 0
\(581\) −7.07107 −0.293357
\(582\) 0 0
\(583\) −12.0000 −0.496989
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 15.5563 0.642079 0.321040 0.947066i \(-0.395968\pi\)
0.321040 + 0.947066i \(0.395968\pi\)
\(588\) 0 0
\(589\) −5.65685 −0.233087
\(590\) 0 0
\(591\) −20.0000 −0.822690
\(592\) 0 0
\(593\) 34.0000 1.39621 0.698106 0.715994i \(-0.254026\pi\)
0.698106 + 0.715994i \(0.254026\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −5.65685 −0.231520
\(598\) 0 0
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) −24.0000 −0.978980 −0.489490 0.872009i \(-0.662817\pi\)
−0.489490 + 0.872009i \(0.662817\pi\)
\(602\) 0 0
\(603\) 1.41421 0.0575912
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 16.0000 0.649420 0.324710 0.945814i \(-0.394733\pi\)
0.324710 + 0.945814i \(0.394733\pi\)
\(608\) 0 0
\(609\) −4.00000 −0.162088
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −14.1421 −0.571195 −0.285598 0.958350i \(-0.592192\pi\)
−0.285598 + 0.958350i \(0.592192\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −12.0000 −0.483102 −0.241551 0.970388i \(-0.577656\pi\)
−0.241551 + 0.970388i \(0.577656\pi\)
\(618\) 0 0
\(619\) −35.3553 −1.42105 −0.710526 0.703671i \(-0.751543\pi\)
−0.710526 + 0.703671i \(0.751543\pi\)
\(620\) 0 0
\(621\) −22.6274 −0.908007
\(622\) 0 0
\(623\) −8.00000 −0.320513
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) −2.82843 −0.112956
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 32.0000 1.27390 0.636950 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) 0 0
\(633\) 22.0000 0.874421
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −8.00000 −0.316475
\(640\) 0 0
\(641\) 36.0000 1.42191 0.710957 0.703235i \(-0.248262\pi\)
0.710957 + 0.703235i \(0.248262\pi\)
\(642\) 0 0
\(643\) 29.6985 1.17119 0.585597 0.810602i \(-0.300860\pi\)
0.585597 + 0.810602i \(0.300860\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −12.0000 −0.471769 −0.235884 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) 0 0
\(649\) 10.0000 0.392534
\(650\) 0 0
\(651\) 5.65685 0.221710
\(652\) 0 0
\(653\) 8.48528 0.332055 0.166027 0.986121i \(-0.446906\pi\)
0.166027 + 0.986121i \(0.446906\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 8.00000 0.312110
\(658\) 0 0
\(659\) −9.89949 −0.385630 −0.192815 0.981235i \(-0.561762\pi\)
−0.192815 + 0.981235i \(0.561762\pi\)
\(660\) 0 0
\(661\) −11.3137 −0.440052 −0.220026 0.975494i \(-0.570614\pi\)
−0.220026 + 0.975494i \(0.570614\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −11.3137 −0.438069
\(668\) 0 0
\(669\) 22.6274 0.874826
\(670\) 0 0
\(671\) −8.00000 −0.308837
\(672\) 0 0
\(673\) 36.0000 1.38770 0.693849 0.720121i \(-0.255914\pi\)
0.693849 + 0.720121i \(0.255914\pi\)
\(674\) 0 0
\(675\) −28.2843 −1.08866
\(676\) 0 0
\(677\) 28.2843 1.08705 0.543526 0.839392i \(-0.317089\pi\)
0.543526 + 0.839392i \(0.317089\pi\)
\(678\) 0 0
\(679\) −8.00000 −0.307012
\(680\) 0 0
\(681\) 26.0000 0.996322
\(682\) 0 0
\(683\) 32.5269 1.24461 0.622304 0.782776i \(-0.286197\pi\)
0.622304 + 0.782776i \(0.286197\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −24.0000 −0.915657
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 15.5563 0.591791 0.295896 0.955220i \(-0.404382\pi\)
0.295896 + 0.955220i \(0.404382\pi\)
\(692\) 0 0
\(693\) −1.41421 −0.0537215
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −5.65685 −0.213962
\(700\) 0 0
\(701\) −8.48528 −0.320485 −0.160242 0.987078i \(-0.551228\pi\)
−0.160242 + 0.987078i \(0.551228\pi\)
\(702\) 0 0
\(703\) 4.00000 0.150863
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 11.3137 0.425496
\(708\) 0 0
\(709\) −25.4558 −0.956014 −0.478007 0.878356i \(-0.658641\pi\)
−0.478007 + 0.878356i \(0.658641\pi\)
\(710\) 0 0
\(711\) −4.00000 −0.150012
\(712\) 0 0
\(713\) 16.0000 0.599205
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 36.0000 1.34257 0.671287 0.741198i \(-0.265742\pi\)
0.671287 + 0.741198i \(0.265742\pi\)
\(720\) 0 0
\(721\) 4.00000 0.148968
\(722\) 0 0
\(723\) 22.6274 0.841523
\(724\) 0 0
\(725\) −14.1421 −0.525226
\(726\) 0 0
\(727\) −12.0000 −0.445055 −0.222528 0.974926i \(-0.571431\pi\)
−0.222528 + 0.974926i \(0.571431\pi\)
\(728\) 0 0
\(729\) 29.0000 1.07407
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −16.9706 −0.626822 −0.313411 0.949618i \(-0.601472\pi\)
−0.313411 + 0.949618i \(0.601472\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −2.00000 −0.0736709
\(738\) 0 0
\(739\) 46.6690 1.71675 0.858374 0.513024i \(-0.171475\pi\)
0.858374 + 0.513024i \(0.171475\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −4.00000 −0.146746 −0.0733729 0.997305i \(-0.523376\pi\)
−0.0733729 + 0.997305i \(0.523376\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 7.07107 0.258717
\(748\) 0 0
\(749\) 7.07107 0.258371
\(750\) 0 0
\(751\) −24.0000 −0.875772 −0.437886 0.899030i \(-0.644273\pi\)
−0.437886 + 0.899030i \(0.644273\pi\)
\(752\) 0 0
\(753\) 14.0000 0.510188
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 42.4264 1.54201 0.771007 0.636827i \(-0.219753\pi\)
0.771007 + 0.636827i \(0.219753\pi\)
\(758\) 0 0
\(759\) 8.00000 0.290382
\(760\) 0 0
\(761\) −22.0000 −0.797499 −0.398750 0.917060i \(-0.630556\pi\)
−0.398750 + 0.917060i \(0.630556\pi\)
\(762\) 0 0
\(763\) −8.48528 −0.307188
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −16.0000 −0.576975 −0.288487 0.957484i \(-0.593152\pi\)
−0.288487 + 0.957484i \(0.593152\pi\)
\(770\) 0 0
\(771\) 25.4558 0.916770
\(772\) 0 0
\(773\) −5.65685 −0.203463 −0.101731 0.994812i \(-0.532438\pi\)
−0.101731 + 0.994812i \(0.532438\pi\)
\(774\) 0 0
\(775\) 20.0000 0.718421
\(776\) 0 0
\(777\) −4.00000 −0.143499
\(778\) 0 0
\(779\) −8.48528 −0.304017
\(780\) 0 0
\(781\) 11.3137 0.404836
\(782\) 0 0
\(783\) 16.0000 0.571793
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 49.4975 1.76439 0.882197 0.470880i \(-0.156064\pi\)
0.882197 + 0.470880i \(0.156064\pi\)
\(788\) 0 0
\(789\) 33.9411 1.20834
\(790\) 0 0
\(791\) −10.0000 −0.355559
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −11.3137 −0.400752 −0.200376 0.979719i \(-0.564216\pi\)
−0.200376 + 0.979719i \(0.564216\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 8.00000 0.282666
\(802\) 0 0
\(803\) −11.3137 −0.399252
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −16.0000 −0.563227
\(808\) 0 0
\(809\) 42.0000 1.47664 0.738321 0.674450i \(-0.235619\pi\)
0.738321 + 0.674450i \(0.235619\pi\)
\(810\) 0 0
\(811\) −46.6690 −1.63877 −0.819386 0.573242i \(-0.805685\pi\)
−0.819386 + 0.573242i \(0.805685\pi\)
\(812\) 0 0
\(813\) −33.9411 −1.19037
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 6.00000 0.209913
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −31.1127 −1.08584 −0.542920 0.839784i \(-0.682681\pi\)
−0.542920 + 0.839784i \(0.682681\pi\)
\(822\) 0 0
\(823\) −16.0000 −0.557725 −0.278862 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) 0 0
\(825\) 10.0000 0.348155
\(826\) 0 0
\(827\) −15.5563 −0.540947 −0.270474 0.962727i \(-0.587180\pi\)
−0.270474 + 0.962727i \(0.587180\pi\)
\(828\) 0 0
\(829\) 45.2548 1.57177 0.785883 0.618376i \(-0.212209\pi\)
0.785883 + 0.618376i \(0.212209\pi\)
\(830\) 0 0
\(831\) 44.0000 1.52634
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −22.6274 −0.782118
\(838\) 0 0
\(839\) 20.0000 0.690477 0.345238 0.938515i \(-0.387798\pi\)
0.345238 + 0.938515i \(0.387798\pi\)
\(840\) 0 0
\(841\) −21.0000 −0.724138
\(842\) 0 0
\(843\) −5.65685 −0.194832
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −9.00000 −0.309244
\(848\) 0 0
\(849\) 2.00000 0.0686398
\(850\) 0 0
\(851\) −11.3137 −0.387829
\(852\) 0 0
\(853\) 5.65685 0.193687 0.0968435 0.995300i \(-0.469125\pi\)
0.0968435 + 0.995300i \(0.469125\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 22.0000 0.751506 0.375753 0.926720i \(-0.377384\pi\)
0.375753 + 0.926720i \(0.377384\pi\)
\(858\) 0 0
\(859\) −21.2132 −0.723785 −0.361893 0.932220i \(-0.617869\pi\)
−0.361893 + 0.932220i \(0.617869\pi\)
\(860\) 0 0
\(861\) 8.48528 0.289178
\(862\) 0 0
\(863\) −44.0000 −1.49778 −0.748889 0.662696i \(-0.769412\pi\)
−0.748889 + 0.662696i \(0.769412\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 24.0416 0.816497
\(868\) 0 0
\(869\) 5.65685 0.191896
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 8.00000 0.270759
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 8.48528 0.286528 0.143264 0.989685i \(-0.454240\pi\)
0.143264 + 0.989685i \(0.454240\pi\)
\(878\) 0 0
\(879\) −40.0000 −1.34917
\(880\) 0 0
\(881\) 30.0000 1.01073 0.505363 0.862907i \(-0.331359\pi\)
0.505363 + 0.862907i \(0.331359\pi\)
\(882\) 0 0
\(883\) −12.7279 −0.428329 −0.214164 0.976798i \(-0.568703\pi\)
−0.214164 + 0.976798i \(0.568703\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 44.0000 1.47738 0.738688 0.674048i \(-0.235446\pi\)
0.738688 + 0.674048i \(0.235446\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −7.07107 −0.236890
\(892\) 0 0
\(893\) 5.65685 0.189299
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −11.3137 −0.377333
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) −6.00000 −0.199667
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −35.3553 −1.17395 −0.586977 0.809603i \(-0.699682\pi\)
−0.586977 + 0.809603i \(0.699682\pi\)
\(908\) 0 0
\(909\) −11.3137 −0.375252
\(910\) 0 0
\(911\) 32.0000 1.06021 0.530104 0.847933i \(-0.322153\pi\)
0.530104 + 0.847933i \(0.322153\pi\)
\(912\) 0 0
\(913\) −10.0000 −0.330952
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −9.89949 −0.326910
\(918\) 0 0
\(919\) −8.00000 −0.263896 −0.131948 0.991257i \(-0.542123\pi\)
−0.131948 + 0.991257i \(0.542123\pi\)
\(920\) 0 0
\(921\) −2.00000 −0.0659022
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −14.1421 −0.464991
\(926\) 0 0
\(927\) −4.00000 −0.131377
\(928\) 0 0
\(929\) 8.00000 0.262471 0.131236 0.991351i \(-0.458106\pi\)
0.131236 + 0.991351i \(0.458106\pi\)
\(930\) 0 0
\(931\) 1.41421 0.0463490
\(932\) 0 0
\(933\) −11.3137 −0.370394
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(938\) 0 0
\(939\) 8.48528 0.276907
\(940\) 0 0
\(941\) 39.5980 1.29086 0.645429 0.763821i \(-0.276679\pi\)
0.645429 + 0.763821i \(0.276679\pi\)
\(942\) 0 0
\(943\) 24.0000 0.781548
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −7.07107 −0.229779 −0.114889 0.993378i \(-0.536651\pi\)
−0.114889 + 0.993378i \(0.536651\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −4.00000 −0.129709
\(952\) 0 0
\(953\) 30.0000 0.971795 0.485898 0.874016i \(-0.338493\pi\)
0.485898 + 0.874016i \(0.338493\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −5.65685 −0.182860
\(958\) 0 0
\(959\) −2.00000 −0.0645834
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) −7.07107 −0.227862
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 4.00000 0.128631 0.0643157 0.997930i \(-0.479514\pi\)
0.0643157 + 0.997930i \(0.479514\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −38.1838 −1.22538 −0.612688 0.790325i \(-0.709912\pi\)
−0.612688 + 0.790325i \(0.709912\pi\)
\(972\) 0 0
\(973\) 1.41421 0.0453376
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −12.0000 −0.383914 −0.191957 0.981403i \(-0.561483\pi\)
−0.191957 + 0.981403i \(0.561483\pi\)
\(978\) 0 0
\(979\) −11.3137 −0.361588
\(980\) 0 0
\(981\) 8.48528 0.270914
\(982\) 0 0
\(983\) −44.0000 −1.40338 −0.701691 0.712481i \(-0.747571\pi\)
−0.701691 + 0.712481i \(0.747571\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −5.65685 −0.180060
\(988\) 0 0
\(989\) −16.9706 −0.539633
\(990\) 0 0
\(991\) −52.0000 −1.65183 −0.825917 0.563791i \(-0.809342\pi\)
−0.825917 + 0.563791i \(0.809342\pi\)
\(992\) 0 0
\(993\) 30.0000 0.952021
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −45.2548 −1.43323 −0.716617 0.697466i \(-0.754311\pi\)
−0.716617 + 0.697466i \(0.754311\pi\)
\(998\) 0 0
\(999\) 16.0000 0.506218
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 3584.2.a.b.1.1 yes 2
4.3 odd 2 3584.2.a.a.1.2 yes 2
8.3 odd 2 3584.2.a.a.1.1 2
8.5 even 2 inner 3584.2.a.b.1.2 yes 2
16.3 odd 4 3584.2.b.b.1793.2 2
16.5 even 4 3584.2.b.a.1793.2 2
16.11 odd 4 3584.2.b.b.1793.1 2
16.13 even 4 3584.2.b.a.1793.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3584.2.a.a.1.1 2 8.3 odd 2
3584.2.a.a.1.2 yes 2 4.3 odd 2
3584.2.a.b.1.1 yes 2 1.1 even 1 trivial
3584.2.a.b.1.2 yes 2 8.5 even 2 inner
3584.2.b.a.1793.1 2 16.13 even 4
3584.2.b.a.1793.2 2 16.5 even 4
3584.2.b.b.1793.1 2 16.11 odd 4
3584.2.b.b.1793.2 2 16.3 odd 4