Properties

Label 4056.2.a.j.1.1
Level $4056$
Weight $2$
Character 4056.1
Self dual yes
Analytic conductor $32.387$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4056,2,Mod(1,4056)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4056, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4056.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4056 = 2^{3} \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4056.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: \(32.3873230598\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 312)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4056.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +3.00000 q^{5} -4.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +3.00000 q^{5} -4.00000 q^{7} +1.00000 q^{9} +4.00000 q^{11} -3.00000 q^{15} +3.00000 q^{17} -4.00000 q^{19} +4.00000 q^{21} -8.00000 q^{23} +4.00000 q^{25} -1.00000 q^{27} -5.00000 q^{29} -8.00000 q^{31} -4.00000 q^{33} -12.0000 q^{35} +7.00000 q^{37} -9.00000 q^{41} +8.00000 q^{43} +3.00000 q^{45} -4.00000 q^{47} +9.00000 q^{49} -3.00000 q^{51} -5.00000 q^{53} +12.0000 q^{55} +4.00000 q^{57} +4.00000 q^{59} -5.00000 q^{61} -4.00000 q^{63} +8.00000 q^{67} +8.00000 q^{69} -4.00000 q^{71} +11.0000 q^{73} -4.00000 q^{75} -16.0000 q^{77} -4.00000 q^{79} +1.00000 q^{81} +9.00000 q^{85} +5.00000 q^{87} -6.00000 q^{89} +8.00000 q^{93} -12.0000 q^{95} -14.0000 q^{97} +4.00000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 3.00000 1.34164 0.670820 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) 0 0
\(7\) −4.00000 −1.51186 −0.755929 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) −3.00000 −0.774597
\(16\) 0 0
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 4.00000 0.872872
\(22\) 0 0
\(23\) −8.00000 −1.66812 −0.834058 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −5.00000 −0.928477 −0.464238 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 0 0
\(33\) −4.00000 −0.696311
\(34\) 0 0
\(35\) −12.0000 −2.02837
\(36\) 0 0
\(37\) 7.00000 1.15079 0.575396 0.817875i \(-0.304848\pi\)
0.575396 + 0.817875i \(0.304848\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −9.00000 −1.40556 −0.702782 0.711405i \(-0.748059\pi\)
−0.702782 + 0.711405i \(0.748059\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) 3.00000 0.447214
\(46\) 0 0
\(47\) −4.00000 −0.583460 −0.291730 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) −3.00000 −0.420084
\(52\) 0 0
\(53\) −5.00000 −0.686803 −0.343401 0.939189i \(-0.611579\pi\)
−0.343401 + 0.939189i \(0.611579\pi\)
\(54\) 0 0
\(55\) 12.0000 1.61808
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) 0 0
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) −5.00000 −0.640184 −0.320092 0.947386i \(-0.603714\pi\)
−0.320092 + 0.947386i \(0.603714\pi\)
\(62\) 0 0
\(63\) −4.00000 −0.503953
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 0 0
\(69\) 8.00000 0.963087
\(70\) 0 0
\(71\) −4.00000 −0.474713 −0.237356 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) 0 0
\(73\) 11.0000 1.28745 0.643726 0.765256i \(-0.277388\pi\)
0.643726 + 0.765256i \(0.277388\pi\)
\(74\) 0 0
\(75\) −4.00000 −0.461880
\(76\) 0 0
\(77\) −16.0000 −1.82337
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 9.00000 0.976187
\(86\) 0 0
\(87\) 5.00000 0.536056
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 8.00000 0.829561
\(94\) 0 0
\(95\) −12.0000 −1.23117
\(96\) 0 0
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 0 0
\(99\) 4.00000 0.402015
\(100\) 0 0
\(101\) 11.0000 1.09454 0.547270 0.836956i \(-0.315667\pi\)
0.547270 + 0.836956i \(0.315667\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 0 0
\(105\) 12.0000 1.17108
\(106\) 0 0
\(107\) 8.00000 0.773389 0.386695 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) 0 0
\(109\) −18.0000 −1.72409 −0.862044 0.506834i \(-0.830816\pi\)
−0.862044 + 0.506834i \(0.830816\pi\)
\(110\) 0 0
\(111\) −7.00000 −0.664411
\(112\) 0 0
\(113\) −9.00000 −0.846649 −0.423324 0.905978i \(-0.639137\pi\)
−0.423324 + 0.905978i \(0.639137\pi\)
\(114\) 0 0
\(115\) −24.0000 −2.23801
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −12.0000 −1.10004
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) 9.00000 0.811503
\(124\) 0 0
\(125\) −3.00000 −0.268328
\(126\) 0 0
\(127\) −20.0000 −1.77471 −0.887357 0.461084i \(-0.847461\pi\)
−0.887357 + 0.461084i \(0.847461\pi\)
\(128\) 0 0
\(129\) −8.00000 −0.704361
\(130\) 0 0
\(131\) 8.00000 0.698963 0.349482 0.936943i \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(132\) 0 0
\(133\) 16.0000 1.38738
\(134\) 0 0
\(135\) −3.00000 −0.258199
\(136\) 0 0
\(137\) −9.00000 −0.768922 −0.384461 0.923141i \(-0.625613\pi\)
−0.384461 + 0.923141i \(0.625613\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 4.00000 0.336861
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −15.0000 −1.24568
\(146\) 0 0
\(147\) −9.00000 −0.742307
\(148\) 0 0
\(149\) 3.00000 0.245770 0.122885 0.992421i \(-0.460785\pi\)
0.122885 + 0.992421i \(0.460785\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) 3.00000 0.242536
\(154\) 0 0
\(155\) −24.0000 −1.92773
\(156\) 0 0
\(157\) 23.0000 1.83560 0.917800 0.397043i \(-0.129964\pi\)
0.917800 + 0.397043i \(0.129964\pi\)
\(158\) 0 0
\(159\) 5.00000 0.396526
\(160\) 0 0
\(161\) 32.0000 2.52195
\(162\) 0 0
\(163\) −8.00000 −0.626608 −0.313304 0.949653i \(-0.601436\pi\)
−0.313304 + 0.949653i \(0.601436\pi\)
\(164\) 0 0
\(165\) −12.0000 −0.934199
\(166\) 0 0
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) 0 0
\(173\) 14.0000 1.06440 0.532200 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) 0 0
\(175\) −16.0000 −1.20949
\(176\) 0 0
\(177\) −4.00000 −0.300658
\(178\) 0 0
\(179\) 16.0000 1.19590 0.597948 0.801535i \(-0.295983\pi\)
0.597948 + 0.801535i \(0.295983\pi\)
\(180\) 0 0
\(181\) −1.00000 −0.0743294 −0.0371647 0.999309i \(-0.511833\pi\)
−0.0371647 + 0.999309i \(0.511833\pi\)
\(182\) 0 0
\(183\) 5.00000 0.369611
\(184\) 0 0
\(185\) 21.0000 1.54395
\(186\) 0 0
\(187\) 12.0000 0.877527
\(188\) 0 0
\(189\) 4.00000 0.290957
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 0 0
\(193\) −13.0000 −0.935760 −0.467880 0.883792i \(-0.654982\pi\)
−0.467880 + 0.883792i \(0.654982\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −26.0000 −1.85242 −0.926212 0.377004i \(-0.876954\pi\)
−0.926212 + 0.377004i \(0.876954\pi\)
\(198\) 0 0
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 0 0
\(201\) −8.00000 −0.564276
\(202\) 0 0
\(203\) 20.0000 1.40372
\(204\) 0 0
\(205\) −27.0000 −1.88576
\(206\) 0 0
\(207\) −8.00000 −0.556038
\(208\) 0 0
\(209\) −16.0000 −1.10674
\(210\) 0 0
\(211\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) 0 0
\(213\) 4.00000 0.274075
\(214\) 0 0
\(215\) 24.0000 1.63679
\(216\) 0 0
\(217\) 32.0000 2.17230
\(218\) 0 0
\(219\) −11.0000 −0.743311
\(220\) 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
\(225\) 4.00000 0.266667
\(226\) 0 0
\(227\) −4.00000 −0.265489 −0.132745 0.991150i \(-0.542379\pi\)
−0.132745 + 0.991150i \(0.542379\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) 16.0000 1.05272
\(232\) 0 0
\(233\) 10.0000 0.655122 0.327561 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) 0 0
\(235\) −12.0000 −0.782794
\(236\) 0 0
\(237\) 4.00000 0.259828
\(238\) 0 0
\(239\) −28.0000 −1.81117 −0.905585 0.424165i \(-0.860568\pi\)
−0.905585 + 0.424165i \(0.860568\pi\)
\(240\) 0 0
\(241\) −5.00000 −0.322078 −0.161039 0.986948i \(-0.551485\pi\)
−0.161039 + 0.986948i \(0.551485\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 27.0000 1.72497
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) −32.0000 −2.01182
\(254\) 0 0
\(255\) −9.00000 −0.563602
\(256\) 0 0
\(257\) 7.00000 0.436648 0.218324 0.975876i \(-0.429941\pi\)
0.218324 + 0.975876i \(0.429941\pi\)
\(258\) 0 0
\(259\) −28.0000 −1.73984
\(260\) 0 0
\(261\) −5.00000 −0.309492
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) −15.0000 −0.921443
\(266\) 0 0
\(267\) 6.00000 0.367194
\(268\) 0 0
\(269\) −2.00000 −0.121942 −0.0609711 0.998140i \(-0.519420\pi\)
−0.0609711 + 0.998140i \(0.519420\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 16.0000 0.964836
\(276\) 0 0
\(277\) 19.0000 1.14160 0.570800 0.821089i \(-0.306633\pi\)
0.570800 + 0.821089i \(0.306633\pi\)
\(278\) 0 0
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) −13.0000 −0.775515 −0.387757 0.921761i \(-0.626750\pi\)
−0.387757 + 0.921761i \(0.626750\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 0 0
\(285\) 12.0000 0.710819
\(286\) 0 0
\(287\) 36.0000 2.12501
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 14.0000 0.820695
\(292\) 0 0
\(293\) −9.00000 −0.525786 −0.262893 0.964825i \(-0.584677\pi\)
−0.262893 + 0.964825i \(0.584677\pi\)
\(294\) 0 0
\(295\) 12.0000 0.698667
\(296\) 0 0
\(297\) −4.00000 −0.232104
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −32.0000 −1.84445
\(302\) 0 0
\(303\) −11.0000 −0.631933
\(304\) 0 0
\(305\) −15.0000 −0.858898
\(306\) 0 0
\(307\) 8.00000 0.456584 0.228292 0.973593i \(-0.426686\pi\)
0.228292 + 0.973593i \(0.426686\pi\)
\(308\) 0 0
\(309\) 4.00000 0.227552
\(310\) 0 0
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 0 0
\(315\) −12.0000 −0.676123
\(316\) 0 0
\(317\) −25.0000 −1.40414 −0.702070 0.712108i \(-0.747741\pi\)
−0.702070 + 0.712108i \(0.747741\pi\)
\(318\) 0 0
\(319\) −20.0000 −1.11979
\(320\) 0 0
\(321\) −8.00000 −0.446516
\(322\) 0 0
\(323\) −12.0000 −0.667698
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 18.0000 0.995402
\(328\) 0 0
\(329\) 16.0000 0.882109
\(330\) 0 0
\(331\) 8.00000 0.439720 0.219860 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) 0 0
\(333\) 7.00000 0.383598
\(334\) 0 0
\(335\) 24.0000 1.31126
\(336\) 0 0
\(337\) 23.0000 1.25289 0.626445 0.779466i \(-0.284509\pi\)
0.626445 + 0.779466i \(0.284509\pi\)
\(338\) 0 0
\(339\) 9.00000 0.488813
\(340\) 0 0
\(341\) −32.0000 −1.73290
\(342\) 0 0
\(343\) −8.00000 −0.431959
\(344\) 0 0
\(345\) 24.0000 1.29212
\(346\) 0 0
\(347\) −16.0000 −0.858925 −0.429463 0.903085i \(-0.641297\pi\)
−0.429463 + 0.903085i \(0.641297\pi\)
\(348\) 0 0
\(349\) 30.0000 1.60586 0.802932 0.596071i \(-0.203272\pi\)
0.802932 + 0.596071i \(0.203272\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −1.00000 −0.0532246 −0.0266123 0.999646i \(-0.508472\pi\)
−0.0266123 + 0.999646i \(0.508472\pi\)
\(354\) 0 0
\(355\) −12.0000 −0.636894
\(356\) 0 0
\(357\) 12.0000 0.635107
\(358\) 0 0
\(359\) 8.00000 0.422224 0.211112 0.977462i \(-0.432292\pi\)
0.211112 + 0.977462i \(0.432292\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) −5.00000 −0.262432
\(364\) 0 0
\(365\) 33.0000 1.72730
\(366\) 0 0
\(367\) 32.0000 1.67039 0.835193 0.549957i \(-0.185356\pi\)
0.835193 + 0.549957i \(0.185356\pi\)
\(368\) 0 0
\(369\) −9.00000 −0.468521
\(370\) 0 0
\(371\) 20.0000 1.03835
\(372\) 0 0
\(373\) −9.00000 −0.466002 −0.233001 0.972476i \(-0.574855\pi\)
−0.233001 + 0.972476i \(0.574855\pi\)
\(374\) 0 0
\(375\) 3.00000 0.154919
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −12.0000 −0.616399 −0.308199 0.951322i \(-0.599726\pi\)
−0.308199 + 0.951322i \(0.599726\pi\)
\(380\) 0 0
\(381\) 20.0000 1.02463
\(382\) 0 0
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 0 0
\(385\) −48.0000 −2.44631
\(386\) 0 0
\(387\) 8.00000 0.406663
\(388\) 0 0
\(389\) 27.0000 1.36895 0.684477 0.729034i \(-0.260031\pi\)
0.684477 + 0.729034i \(0.260031\pi\)
\(390\) 0 0
\(391\) −24.0000 −1.21373
\(392\) 0 0
\(393\) −8.00000 −0.403547
\(394\) 0 0
\(395\) −12.0000 −0.603786
\(396\) 0 0
\(397\) −18.0000 −0.903394 −0.451697 0.892171i \(-0.649181\pi\)
−0.451697 + 0.892171i \(0.649181\pi\)
\(398\) 0 0
\(399\) −16.0000 −0.801002
\(400\) 0 0
\(401\) −29.0000 −1.44819 −0.724095 0.689700i \(-0.757743\pi\)
−0.724095 + 0.689700i \(0.757743\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 3.00000 0.149071
\(406\) 0 0
\(407\) 28.0000 1.38791
\(408\) 0 0
\(409\) −25.0000 −1.23617 −0.618085 0.786111i \(-0.712091\pi\)
−0.618085 + 0.786111i \(0.712091\pi\)
\(410\) 0 0
\(411\) 9.00000 0.443937
\(412\) 0 0
\(413\) −16.0000 −0.787309
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −1.00000 −0.0487370 −0.0243685 0.999703i \(-0.507758\pi\)
−0.0243685 + 0.999703i \(0.507758\pi\)
\(422\) 0 0
\(423\) −4.00000 −0.194487
\(424\) 0 0
\(425\) 12.0000 0.582086
\(426\) 0 0
\(427\) 20.0000 0.967868
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 8.00000 0.385346 0.192673 0.981263i \(-0.438284\pi\)
0.192673 + 0.981263i \(0.438284\pi\)
\(432\) 0 0
\(433\) −29.0000 −1.39365 −0.696826 0.717241i \(-0.745405\pi\)
−0.696826 + 0.717241i \(0.745405\pi\)
\(434\) 0 0
\(435\) 15.0000 0.719195
\(436\) 0 0
\(437\) 32.0000 1.53077
\(438\) 0 0
\(439\) −32.0000 −1.52728 −0.763638 0.645644i \(-0.776589\pi\)
−0.763638 + 0.645644i \(0.776589\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) 0 0
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) −18.0000 −0.853282
\(446\) 0 0
\(447\) −3.00000 −0.141895
\(448\) 0 0
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 0 0
\(451\) −36.0000 −1.69517
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −9.00000 −0.421002 −0.210501 0.977594i \(-0.567510\pi\)
−0.210501 + 0.977594i \(0.567510\pi\)
\(458\) 0 0
\(459\) −3.00000 −0.140028
\(460\) 0 0
\(461\) −5.00000 −0.232873 −0.116437 0.993198i \(-0.537147\pi\)
−0.116437 + 0.993198i \(0.537147\pi\)
\(462\) 0 0
\(463\) 32.0000 1.48717 0.743583 0.668644i \(-0.233125\pi\)
0.743583 + 0.668644i \(0.233125\pi\)
\(464\) 0 0
\(465\) 24.0000 1.11297
\(466\) 0 0
\(467\) 28.0000 1.29569 0.647843 0.761774i \(-0.275671\pi\)
0.647843 + 0.761774i \(0.275671\pi\)
\(468\) 0 0
\(469\) −32.0000 −1.47762
\(470\) 0 0
\(471\) −23.0000 −1.05978
\(472\) 0 0
\(473\) 32.0000 1.47136
\(474\) 0 0
\(475\) −16.0000 −0.734130
\(476\) 0 0
\(477\) −5.00000 −0.228934
\(478\) 0 0
\(479\) −36.0000 −1.64488 −0.822441 0.568850i \(-0.807388\pi\)
−0.822441 + 0.568850i \(0.807388\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) −32.0000 −1.45605
\(484\) 0 0
\(485\) −42.0000 −1.90712
\(486\) 0 0
\(487\) 40.0000 1.81257 0.906287 0.422664i \(-0.138905\pi\)
0.906287 + 0.422664i \(0.138905\pi\)
\(488\) 0 0
\(489\) 8.00000 0.361773
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) −15.0000 −0.675566
\(494\) 0 0
\(495\) 12.0000 0.539360
\(496\) 0 0
\(497\) 16.0000 0.717698
\(498\) 0 0
\(499\) −36.0000 −1.61158 −0.805791 0.592200i \(-0.798259\pi\)
−0.805791 + 0.592200i \(0.798259\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 33.0000 1.46848
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 31.0000 1.37405 0.687025 0.726633i \(-0.258916\pi\)
0.687025 + 0.726633i \(0.258916\pi\)
\(510\) 0 0
\(511\) −44.0000 −1.94645
\(512\) 0 0
\(513\) 4.00000 0.176604
\(514\) 0 0
\(515\) −12.0000 −0.528783
\(516\) 0 0
\(517\) −16.0000 −0.703679
\(518\) 0 0
\(519\) −14.0000 −0.614532
\(520\) 0 0
\(521\) −9.00000 −0.394297 −0.197149 0.980374i \(-0.563168\pi\)
−0.197149 + 0.980374i \(0.563168\pi\)
\(522\) 0 0
\(523\) −20.0000 −0.874539 −0.437269 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) 0 0
\(525\) 16.0000 0.698297
\(526\) 0 0
\(527\) −24.0000 −1.04546
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) 0 0
\(531\) 4.00000 0.173585
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 24.0000 1.03761
\(536\) 0 0
\(537\) −16.0000 −0.690451
\(538\) 0 0
\(539\) 36.0000 1.55063
\(540\) 0 0
\(541\) 27.0000 1.16082 0.580410 0.814324i \(-0.302892\pi\)
0.580410 + 0.814324i \(0.302892\pi\)
\(542\) 0 0
\(543\) 1.00000 0.0429141
\(544\) 0 0
\(545\) −54.0000 −2.31311
\(546\) 0 0
\(547\) −4.00000 −0.171028 −0.0855138 0.996337i \(-0.527253\pi\)
−0.0855138 + 0.996337i \(0.527253\pi\)
\(548\) 0 0
\(549\) −5.00000 −0.213395
\(550\) 0 0
\(551\) 20.0000 0.852029
\(552\) 0 0
\(553\) 16.0000 0.680389
\(554\) 0 0
\(555\) −21.0000 −0.891400
\(556\) 0 0
\(557\) 35.0000 1.48300 0.741499 0.670954i \(-0.234115\pi\)
0.741499 + 0.670954i \(0.234115\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −12.0000 −0.506640
\(562\) 0 0
\(563\) −36.0000 −1.51722 −0.758610 0.651546i \(-0.774121\pi\)
−0.758610 + 0.651546i \(0.774121\pi\)
\(564\) 0 0
\(565\) −27.0000 −1.13590
\(566\) 0 0
\(567\) −4.00000 −0.167984
\(568\) 0 0
\(569\) −22.0000 −0.922288 −0.461144 0.887325i \(-0.652561\pi\)
−0.461144 + 0.887325i \(0.652561\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 0 0
\(573\) 8.00000 0.334205
\(574\) 0 0
\(575\) −32.0000 −1.33449
\(576\) 0 0
\(577\) −9.00000 −0.374675 −0.187337 0.982296i \(-0.559986\pi\)
−0.187337 + 0.982296i \(0.559986\pi\)
\(578\) 0 0
\(579\) 13.0000 0.540262
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −20.0000 −0.828315
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 48.0000 1.98117 0.990586 0.136892i \(-0.0437113\pi\)
0.990586 + 0.136892i \(0.0437113\pi\)
\(588\) 0 0
\(589\) 32.0000 1.31854
\(590\) 0 0
\(591\) 26.0000 1.06950
\(592\) 0 0
\(593\) 7.00000 0.287456 0.143728 0.989617i \(-0.454091\pi\)
0.143728 + 0.989617i \(0.454091\pi\)
\(594\) 0 0
\(595\) −36.0000 −1.47586
\(596\) 0 0
\(597\) 8.00000 0.327418
\(598\) 0 0
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) 35.0000 1.42768 0.713840 0.700309i \(-0.246954\pi\)
0.713840 + 0.700309i \(0.246954\pi\)
\(602\) 0 0
\(603\) 8.00000 0.325785
\(604\) 0 0
\(605\) 15.0000 0.609837
\(606\) 0 0
\(607\) 28.0000 1.13648 0.568242 0.822861i \(-0.307624\pi\)
0.568242 + 0.822861i \(0.307624\pi\)
\(608\) 0 0
\(609\) −20.0000 −0.810441
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 31.0000 1.25208 0.626039 0.779792i \(-0.284675\pi\)
0.626039 + 0.779792i \(0.284675\pi\)
\(614\) 0 0
\(615\) 27.0000 1.08875
\(616\) 0 0
\(617\) 27.0000 1.08698 0.543490 0.839416i \(-0.317103\pi\)
0.543490 + 0.839416i \(0.317103\pi\)
\(618\) 0 0
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 0 0
\(621\) 8.00000 0.321029
\(622\) 0 0
\(623\) 24.0000 0.961540
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) 16.0000 0.638978
\(628\) 0 0
\(629\) 21.0000 0.837325
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −60.0000 −2.38103
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −4.00000 −0.158238
\(640\) 0 0
\(641\) 35.0000 1.38242 0.691208 0.722655i \(-0.257079\pi\)
0.691208 + 0.722655i \(0.257079\pi\)
\(642\) 0 0
\(643\) −44.0000 −1.73519 −0.867595 0.497271i \(-0.834335\pi\)
−0.867595 + 0.497271i \(0.834335\pi\)
\(644\) 0 0
\(645\) −24.0000 −0.944999
\(646\) 0 0
\(647\) 8.00000 0.314512 0.157256 0.987558i \(-0.449735\pi\)
0.157256 + 0.987558i \(0.449735\pi\)
\(648\) 0 0
\(649\) 16.0000 0.628055
\(650\) 0 0
\(651\) −32.0000 −1.25418
\(652\) 0 0
\(653\) −18.0000 −0.704394 −0.352197 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) 0 0
\(655\) 24.0000 0.937758
\(656\) 0 0
\(657\) 11.0000 0.429151
\(658\) 0 0
\(659\) 40.0000 1.55818 0.779089 0.626913i \(-0.215682\pi\)
0.779089 + 0.626913i \(0.215682\pi\)
\(660\) 0 0
\(661\) −5.00000 −0.194477 −0.0972387 0.995261i \(-0.531001\pi\)
−0.0972387 + 0.995261i \(0.531001\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 48.0000 1.86136
\(666\) 0 0
\(667\) 40.0000 1.54881
\(668\) 0 0
\(669\) 16.0000 0.618596
\(670\) 0 0
\(671\) −20.0000 −0.772091
\(672\) 0 0
\(673\) 43.0000 1.65753 0.828764 0.559598i \(-0.189045\pi\)
0.828764 + 0.559598i \(0.189045\pi\)
\(674\) 0 0
\(675\) −4.00000 −0.153960
\(676\) 0 0
\(677\) 38.0000 1.46046 0.730229 0.683202i \(-0.239413\pi\)
0.730229 + 0.683202i \(0.239413\pi\)
\(678\) 0 0
\(679\) 56.0000 2.14908
\(680\) 0 0
\(681\) 4.00000 0.153280
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) −27.0000 −1.03162
\(686\) 0 0
\(687\) 10.0000 0.381524
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) 0 0
\(693\) −16.0000 −0.607790
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −27.0000 −1.02270
\(698\) 0 0
\(699\) −10.0000 −0.378235
\(700\) 0 0
\(701\) −2.00000 −0.0755390 −0.0377695 0.999286i \(-0.512025\pi\)
−0.0377695 + 0.999286i \(0.512025\pi\)
\(702\) 0 0
\(703\) −28.0000 −1.05604
\(704\) 0 0
\(705\) 12.0000 0.451946
\(706\) 0 0
\(707\) −44.0000 −1.65479
\(708\) 0 0
\(709\) −9.00000 −0.338002 −0.169001 0.985616i \(-0.554054\pi\)
−0.169001 + 0.985616i \(0.554054\pi\)
\(710\) 0 0
\(711\) −4.00000 −0.150012
\(712\) 0 0
\(713\) 64.0000 2.39682
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 28.0000 1.04568
\(718\) 0 0
\(719\) 4.00000 0.149175 0.0745874 0.997214i \(-0.476236\pi\)
0.0745874 + 0.997214i \(0.476236\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) 0 0
\(723\) 5.00000 0.185952
\(724\) 0 0
\(725\) −20.0000 −0.742781
\(726\) 0 0
\(727\) 28.0000 1.03846 0.519231 0.854634i \(-0.326218\pi\)
0.519231 + 0.854634i \(0.326218\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 24.0000 0.887672
\(732\) 0 0
\(733\) −13.0000 −0.480166 −0.240083 0.970752i \(-0.577175\pi\)
−0.240083 + 0.970752i \(0.577175\pi\)
\(734\) 0 0
\(735\) −27.0000 −0.995910
\(736\) 0 0
\(737\) 32.0000 1.17874
\(738\) 0 0
\(739\) −16.0000 −0.588570 −0.294285 0.955718i \(-0.595081\pi\)
−0.294285 + 0.955718i \(0.595081\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 8.00000 0.293492 0.146746 0.989174i \(-0.453120\pi\)
0.146746 + 0.989174i \(0.453120\pi\)
\(744\) 0 0
\(745\) 9.00000 0.329734
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −32.0000 −1.16925
\(750\) 0 0
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −42.0000 −1.52652 −0.763258 0.646094i \(-0.776401\pi\)
−0.763258 + 0.646094i \(0.776401\pi\)
\(758\) 0 0
\(759\) 32.0000 1.16153
\(760\) 0 0
\(761\) 42.0000 1.52250 0.761249 0.648459i \(-0.224586\pi\)
0.761249 + 0.648459i \(0.224586\pi\)
\(762\) 0 0
\(763\) 72.0000 2.60658
\(764\) 0 0
\(765\) 9.00000 0.325396
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 34.0000 1.22607 0.613036 0.790055i \(-0.289948\pi\)
0.613036 + 0.790055i \(0.289948\pi\)
\(770\) 0 0
\(771\) −7.00000 −0.252099
\(772\) 0 0
\(773\) 38.0000 1.36677 0.683383 0.730061i \(-0.260508\pi\)
0.683383 + 0.730061i \(0.260508\pi\)
\(774\) 0 0
\(775\) −32.0000 −1.14947
\(776\) 0 0
\(777\) 28.0000 1.00449
\(778\) 0 0
\(779\) 36.0000 1.28983
\(780\) 0 0
\(781\) −16.0000 −0.572525
\(782\) 0 0
\(783\) 5.00000 0.178685
\(784\) 0 0
\(785\) 69.0000 2.46272
\(786\) 0 0
\(787\) 16.0000 0.570338 0.285169 0.958477i \(-0.407950\pi\)
0.285169 + 0.958477i \(0.407950\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 36.0000 1.28001
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 15.0000 0.531995
\(796\) 0 0
\(797\) −18.0000 −0.637593 −0.318796 0.947823i \(-0.603279\pi\)
−0.318796 + 0.947823i \(0.603279\pi\)
\(798\) 0 0
\(799\) −12.0000 −0.424529
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 0 0
\(803\) 44.0000 1.55273
\(804\) 0 0
\(805\) 96.0000 3.38356
\(806\) 0 0
\(807\) 2.00000 0.0704033
\(808\) 0 0
\(809\) −9.00000 −0.316423 −0.158212 0.987405i \(-0.550573\pi\)
−0.158212 + 0.987405i \(0.550573\pi\)
\(810\) 0 0
\(811\) −24.0000 −0.842754 −0.421377 0.906886i \(-0.638453\pi\)
−0.421377 + 0.906886i \(0.638453\pi\)
\(812\) 0 0
\(813\) −20.0000 −0.701431
\(814\) 0 0
\(815\) −24.0000 −0.840683
\(816\) 0 0
\(817\) −32.0000 −1.11954
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −42.0000 −1.46581 −0.732905 0.680331i \(-0.761836\pi\)
−0.732905 + 0.680331i \(0.761836\pi\)
\(822\) 0 0
\(823\) 4.00000 0.139431 0.0697156 0.997567i \(-0.477791\pi\)
0.0697156 + 0.997567i \(0.477791\pi\)
\(824\) 0 0
\(825\) −16.0000 −0.557048
\(826\) 0 0
\(827\) 36.0000 1.25184 0.625921 0.779886i \(-0.284723\pi\)
0.625921 + 0.779886i \(0.284723\pi\)
\(828\) 0 0
\(829\) 11.0000 0.382046 0.191023 0.981586i \(-0.438820\pi\)
0.191023 + 0.981586i \(0.438820\pi\)
\(830\) 0 0
\(831\) −19.0000 −0.659103
\(832\) 0 0
\(833\) 27.0000 0.935495
\(834\) 0 0
\(835\) −36.0000 −1.24583
\(836\) 0 0
\(837\) 8.00000 0.276520
\(838\) 0 0
\(839\) 20.0000 0.690477 0.345238 0.938515i \(-0.387798\pi\)
0.345238 + 0.938515i \(0.387798\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 0 0
\(843\) 13.0000 0.447744
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −20.0000 −0.687208
\(848\) 0 0
\(849\) 4.00000 0.137280
\(850\) 0 0
\(851\) −56.0000 −1.91966
\(852\) 0 0
\(853\) 19.0000 0.650548 0.325274 0.945620i \(-0.394544\pi\)
0.325274 + 0.945620i \(0.394544\pi\)
\(854\) 0 0
\(855\) −12.0000 −0.410391
\(856\) 0 0
\(857\) −33.0000 −1.12726 −0.563629 0.826028i \(-0.690595\pi\)
−0.563629 + 0.826028i \(0.690595\pi\)
\(858\) 0 0
\(859\) −8.00000 −0.272956 −0.136478 0.990643i \(-0.543578\pi\)
−0.136478 + 0.990643i \(0.543578\pi\)
\(860\) 0 0
\(861\) −36.0000 −1.22688
\(862\) 0 0
\(863\) 44.0000 1.49778 0.748889 0.662696i \(-0.230588\pi\)
0.748889 + 0.662696i \(0.230588\pi\)
\(864\) 0 0
\(865\) 42.0000 1.42804
\(866\) 0 0
\(867\) 8.00000 0.271694
\(868\) 0 0
\(869\) −16.0000 −0.542763
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −14.0000 −0.473828
\(874\) 0 0
\(875\) 12.0000 0.405674
\(876\) 0 0
\(877\) 35.0000 1.18187 0.590933 0.806721i \(-0.298760\pi\)
0.590933 + 0.806721i \(0.298760\pi\)
\(878\) 0 0
\(879\) 9.00000 0.303562
\(880\) 0 0
\(881\) 15.0000 0.505363 0.252681 0.967550i \(-0.418688\pi\)
0.252681 + 0.967550i \(0.418688\pi\)
\(882\) 0 0
\(883\) 52.0000 1.74994 0.874970 0.484178i \(-0.160881\pi\)
0.874970 + 0.484178i \(0.160881\pi\)
\(884\) 0 0
\(885\) −12.0000 −0.403376
\(886\) 0 0
\(887\) 28.0000 0.940148 0.470074 0.882627i \(-0.344227\pi\)
0.470074 + 0.882627i \(0.344227\pi\)
\(888\) 0 0
\(889\) 80.0000 2.68311
\(890\) 0 0
\(891\) 4.00000 0.134005
\(892\) 0 0
\(893\) 16.0000 0.535420
\(894\) 0 0
\(895\) 48.0000 1.60446
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 40.0000 1.33407
\(900\) 0 0
\(901\) −15.0000 −0.499722
\(902\) 0 0
\(903\) 32.0000 1.06489
\(904\) 0 0
\(905\) −3.00000 −0.0997234
\(906\) 0 0
\(907\) −8.00000 −0.265636 −0.132818 0.991140i \(-0.542403\pi\)
−0.132818 + 0.991140i \(0.542403\pi\)
\(908\) 0 0
\(909\) 11.0000 0.364847
\(910\) 0 0
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 15.0000 0.495885
\(916\) 0 0
\(917\) −32.0000 −1.05673
\(918\) 0 0
\(919\) −12.0000 −0.395843 −0.197922 0.980218i \(-0.563419\pi\)
−0.197922 + 0.980218i \(0.563419\pi\)
\(920\) 0 0
\(921\) −8.00000 −0.263609
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 28.0000 0.920634
\(926\) 0 0
\(927\) −4.00000 −0.131377
\(928\) 0 0
\(929\) 11.0000 0.360898 0.180449 0.983584i \(-0.442245\pi\)
0.180449 + 0.983584i \(0.442245\pi\)
\(930\) 0 0
\(931\) −36.0000 −1.17985
\(932\) 0 0
\(933\) 12.0000 0.392862
\(934\) 0 0
\(935\) 36.0000 1.17733
\(936\) 0 0
\(937\) 59.0000 1.92745 0.963723 0.266904i \(-0.0860008\pi\)
0.963723 + 0.266904i \(0.0860008\pi\)
\(938\) 0 0
\(939\) 6.00000 0.195803
\(940\) 0 0
\(941\) −18.0000 −0.586783 −0.293392 0.955992i \(-0.594784\pi\)
−0.293392 + 0.955992i \(0.594784\pi\)
\(942\) 0 0
\(943\) 72.0000 2.34464
\(944\) 0 0
\(945\) 12.0000 0.390360
\(946\) 0 0
\(947\) 36.0000 1.16984 0.584921 0.811090i \(-0.301125\pi\)
0.584921 + 0.811090i \(0.301125\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 25.0000 0.810681
\(952\) 0 0
\(953\) 42.0000 1.36051 0.680257 0.732974i \(-0.261868\pi\)
0.680257 + 0.732974i \(0.261868\pi\)
\(954\) 0 0
\(955\) −24.0000 −0.776622
\(956\) 0 0
\(957\) 20.0000 0.646508
\(958\) 0 0
\(959\) 36.0000 1.16250
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) 8.00000 0.257796
\(964\) 0 0
\(965\) −39.0000 −1.25545
\(966\) 0 0
\(967\) −12.0000 −0.385894 −0.192947 0.981209i \(-0.561805\pi\)
−0.192947 + 0.981209i \(0.561805\pi\)
\(968\) 0 0
\(969\) 12.0000 0.385496
\(970\) 0 0
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −29.0000 −0.927792 −0.463896 0.885890i \(-0.653549\pi\)
−0.463896 + 0.885890i \(0.653549\pi\)
\(978\) 0 0
\(979\) −24.0000 −0.767043
\(980\) 0 0
\(981\) −18.0000 −0.574696
\(982\) 0 0
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) −78.0000 −2.48529
\(986\) 0 0
\(987\) −16.0000 −0.509286
\(988\) 0 0
\(989\) −64.0000 −2.03508
\(990\) 0 0
\(991\) −36.0000 −1.14358 −0.571789 0.820401i \(-0.693750\pi\)
−0.571789 + 0.820401i \(0.693750\pi\)
\(992\) 0 0
\(993\) −8.00000 −0.253872
\(994\) 0 0
\(995\) −24.0000 −0.760851
\(996\) 0 0
\(997\) 35.0000 1.10846 0.554231 0.832363i \(-0.313013\pi\)
0.554231 + 0.832363i \(0.313013\pi\)
\(998\) 0 0
\(999\) −7.00000 −0.221470
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 4056.2.a.j.1.1 1
4.3 odd 2 8112.2.a.bh.1.1 1
13.3 even 3 312.2.q.c.217.1 2
13.5 odd 4 4056.2.c.b.337.1 2
13.8 odd 4 4056.2.c.b.337.2 2
13.9 even 3 312.2.q.c.289.1 yes 2
13.12 even 2 4056.2.a.b.1.1 1
39.29 odd 6 936.2.t.b.217.1 2
39.35 odd 6 936.2.t.b.289.1 2
52.3 odd 6 624.2.q.e.529.1 2
52.35 odd 6 624.2.q.e.289.1 2
52.51 odd 2 8112.2.a.r.1.1 1
156.35 even 6 1872.2.t.a.289.1 2
156.107 even 6 1872.2.t.a.1153.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
312.2.q.c.217.1 2 13.3 even 3
312.2.q.c.289.1 yes 2 13.9 even 3
624.2.q.e.289.1 2 52.35 odd 6
624.2.q.e.529.1 2 52.3 odd 6
936.2.t.b.217.1 2 39.29 odd 6
936.2.t.b.289.1 2 39.35 odd 6
1872.2.t.a.289.1 2 156.35 even 6
1872.2.t.a.1153.1 2 156.107 even 6
4056.2.a.b.1.1 1 13.12 even 2
4056.2.a.j.1.1 1 1.1 even 1 trivial
4056.2.c.b.337.1 2 13.5 odd 4
4056.2.c.b.337.2 2 13.8 odd 4
8112.2.a.r.1.1 1 52.51 odd 2
8112.2.a.bh.1.1 1 4.3 odd 2