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Properties

Label 3952.2.a.c.1.1
Level $3952$
Weight $2$
Character 3952.1
Self dual yes
Analytic conductor $31.557$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3952,2,Mod(1,3952)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3952, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3952.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 3952 = 2^{4} \cdot 13 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3952.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,-2,0,4,0,-2,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(31.5568788788\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 988)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3952.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{3} +4.00000 q^{5} -2.00000 q^{7} +1.00000 q^{9} -1.00000 q^{13} -8.00000 q^{15} -3.00000 q^{17} +1.00000 q^{19} +4.00000 q^{21} -3.00000 q^{23} +11.0000 q^{25} +4.00000 q^{27} -2.00000 q^{29} +11.0000 q^{31} -8.00000 q^{35} -5.00000 q^{37} +2.00000 q^{39} -5.00000 q^{41} -11.0000 q^{43} +4.00000 q^{45} +6.00000 q^{47} -3.00000 q^{49} +6.00000 q^{51} -14.0000 q^{53} -2.00000 q^{57} -11.0000 q^{59} +7.00000 q^{61} -2.00000 q^{63} -4.00000 q^{65} +7.00000 q^{67} +6.00000 q^{69} -12.0000 q^{71} +4.00000 q^{73} -22.0000 q^{75} +2.00000 q^{79} -11.0000 q^{81} +2.00000 q^{83} -12.0000 q^{85} +4.00000 q^{87} +2.00000 q^{89} +2.00000 q^{91} -22.0000 q^{93} +4.00000 q^{95} +17.0000 q^{97} +O(q^{100})\)

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
\(3\) −2.00000 −1.15470 −0.577350 − 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 0 0
\(5\) 4.00000 1.78885 0.894427 − 0.447214i \(-0.147584\pi\)
0.894427 + 0.447214i \(0.147584\pi\)
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 − 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) −8.00000 −2.06559
\(16\) 0 0
\(17\) −3.00000 −0.727607 −0.363803 − 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 4.00000 0.872872
\(22\) 0 0
\(23\) −3.00000 −0.625543 −0.312772 − 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) 0 0
\(25\) 11.0000 2.20000
\(26\) 0 0
\(27\) 4.00000 0.769800
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 − 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 11.0000 1.97566 0.987829 − 0.155543i \(-0.0497126\pi\)
0.987829 + 0.155543i \(0.0497126\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −8.00000 −1.35225
\(36\) 0 0
\(37\) −5.00000 −0.821995 −0.410997 − 0.911636i \(-0.634819\pi\)
−0.410997 + 0.911636i \(0.634819\pi\)
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) −5.00000 −0.780869 −0.390434 − 0.920631i \(-0.627675\pi\)
−0.390434 + 0.920631i \(0.627675\pi\)
\(42\) 0 0
\(43\) −11.0000 −1.67748 −0.838742 − 0.544529i \(-0.816708\pi\)
−0.838742 + 0.544529i \(0.816708\pi\)
\(44\) 0 0
\(45\) 4.00000 0.596285
\(46\) 0 0
\(47\) 6.00000 0.875190 0.437595 − 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) 0 0
\(53\) −14.0000 −1.92305 −0.961524 − 0.274721i \(-0.911414\pi\)
−0.961524 + 0.274721i \(0.911414\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −2.00000 −0.264906
\(58\) 0 0
\(59\) −11.0000 −1.43208 −0.716039 − 0.698060i \(-0.754047\pi\)
−0.716039 + 0.698060i \(0.754047\pi\)
\(60\) 0 0
\(61\) 7.00000 0.896258 0.448129 − 0.893969i \(-0.352090\pi\)
0.448129 + 0.893969i \(0.352090\pi\)
\(62\) 0 0
\(63\) −2.00000 −0.251976
\(64\) 0 0
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) 7.00000 0.855186 0.427593 − 0.903971i \(-0.359362\pi\)
0.427593 + 0.903971i \(0.359362\pi\)
\(68\) 0 0
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 − 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) 4.00000 0.468165 0.234082 − 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) 0 0
\(75\) −22.0000 −2.54034
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 2.00000 0.225018 0.112509 − 0.993651i \(-0.464111\pi\)
0.112509 + 0.993651i \(0.464111\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) 2.00000 0.219529 0.109764 − 0.993958i \(-0.464990\pi\)
0.109764 + 0.993958i \(0.464990\pi\)
\(84\) 0 0
\(85\) −12.0000 −1.30158
\(86\) 0 0
\(87\) 4.00000 0.428845
\(88\) 0 0
\(89\) 2.00000 0.212000 0.106000 − 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 0 0
\(93\) −22.0000 −2.28129
\(94\) 0 0
\(95\) 4.00000 0.410391
\(96\) 0 0
\(97\) 17.0000 1.72609 0.863044 − 0.505128i \(-0.168555\pi\)
0.863044 + 0.505128i \(0.168555\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 15.0000 1.49256 0.746278 − 0.665635i \(-0.231839\pi\)
0.746278 + 0.665635i \(0.231839\pi\)
\(102\) 0 0
\(103\) −14.0000 −1.37946 −0.689730 − 0.724066i \(-0.742271\pi\)
−0.689730 + 0.724066i \(0.742271\pi\)
\(104\) 0 0
\(105\) 16.0000 1.56144
\(106\) 0 0
\(107\) −2.00000 −0.193347 −0.0966736 − 0.995316i \(-0.530820\pi\)
−0.0966736 + 0.995316i \(0.530820\pi\)
\(108\) 0 0
\(109\) −5.00000 −0.478913 −0.239457 − 0.970907i \(-0.576969\pi\)
−0.239457 + 0.970907i \(0.576969\pi\)
\(110\) 0 0
\(111\) 10.0000 0.949158
\(112\) 0 0
\(113\) −10.0000 −0.940721 −0.470360 − 0.882474i \(-0.655876\pi\)
−0.470360 + 0.882474i \(0.655876\pi\)
\(114\) 0 0
\(115\) −12.0000 −1.11901
\(116\) 0 0
\(117\) −1.00000 −0.0924500
\(118\) 0 0
\(119\) 6.00000 0.550019
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 10.0000 0.901670
\(124\) 0 0
\(125\) 24.0000 2.14663
\(126\) 0 0
\(127\) −16.0000 −1.41977 −0.709885 − 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 0 0
\(129\) 22.0000 1.93699
\(130\) 0 0
\(131\) −1.00000 −0.0873704 −0.0436852 − 0.999045i \(-0.513910\pi\)
−0.0436852 + 0.999045i \(0.513910\pi\)
\(132\) 0 0
\(133\) −2.00000 −0.173422
\(134\) 0 0
\(135\) 16.0000 1.37706
\(136\) 0 0
\(137\) 4.00000 0.341743 0.170872 − 0.985293i \(-0.445342\pi\)
0.170872 + 0.985293i \(0.445342\pi\)
\(138\) 0 0
\(139\) 13.0000 1.10265 0.551323 − 0.834292i \(-0.314123\pi\)
0.551323 + 0.834292i \(0.314123\pi\)
\(140\) 0 0
\(141\) −12.0000 −1.01058
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −8.00000 −0.664364
\(146\) 0 0
\(147\) 6.00000 0.494872
\(148\) 0 0
\(149\) 10.0000 0.819232 0.409616 − 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) −7.00000 −0.569652 −0.284826 − 0.958579i \(-0.591936\pi\)
−0.284826 + 0.958579i \(0.591936\pi\)
\(152\) 0 0
\(153\) −3.00000 −0.242536
\(154\) 0 0
\(155\) 44.0000 3.53417
\(156\) 0 0
\(157\) −17.0000 −1.35675 −0.678374 − 0.734717i \(-0.737315\pi\)
−0.678374 + 0.734717i \(0.737315\pi\)
\(158\) 0 0
\(159\) 28.0000 2.22054
\(160\) 0 0
\(161\) 6.00000 0.472866
\(162\) 0 0
\(163\) −6.00000 −0.469956 −0.234978 − 0.972001i \(-0.575502\pi\)
−0.234978 + 0.972001i \(0.575502\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −9.00000 −0.696441 −0.348220 − 0.937413i \(-0.613214\pi\)
−0.348220 + 0.937413i \(0.613214\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) 0 0
\(173\) −4.00000 −0.304114 −0.152057 − 0.988372i \(-0.548590\pi\)
−0.152057 + 0.988372i \(0.548590\pi\)
\(174\) 0 0
\(175\) −22.0000 −1.66304
\(176\) 0 0
\(177\) 22.0000 1.65362
\(178\) 0 0
\(179\) 2.00000 0.149487 0.0747435 − 0.997203i \(-0.476186\pi\)
0.0747435 + 0.997203i \(0.476186\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 − 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) −14.0000 −1.03491
\(184\) 0 0
\(185\) −20.0000 −1.47043
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −8.00000 −0.581914
\(190\) 0 0
\(191\) −5.00000 −0.361787 −0.180894 − 0.983503i \(-0.557899\pi\)
−0.180894 + 0.983503i \(0.557899\pi\)
\(192\) 0 0
\(193\) 2.00000 0.143963 0.0719816 − 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 0 0
\(195\) 8.00000 0.572892
\(196\) 0 0
\(197\) −8.00000 −0.569976 −0.284988 − 0.958531i \(-0.591990\pi\)
−0.284988 + 0.958531i \(0.591990\pi\)
\(198\) 0 0
\(199\) −13.0000 −0.921546 −0.460773 − 0.887518i \(-0.652428\pi\)
−0.460773 + 0.887518i \(0.652428\pi\)
\(200\) 0 0
\(201\) −14.0000 −0.987484
\(202\) 0 0
\(203\) 4.00000 0.280745
\(204\) 0 0
\(205\) −20.0000 −1.39686
\(206\) 0 0
\(207\) −3.00000 −0.208514
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 2.00000 0.137686 0.0688428 − 0.997628i \(-0.478069\pi\)
0.0688428 + 0.997628i \(0.478069\pi\)
\(212\) 0 0
\(213\) 24.0000 1.64445
\(214\) 0 0
\(215\) −44.0000 −3.00078
\(216\) 0 0
\(217\) −22.0000 −1.49346
\(218\) 0 0
\(219\) −8.00000 −0.540590
\(220\) 0 0
\(221\) 3.00000 0.201802
\(222\) 0 0
\(223\) 8.00000 0.535720 0.267860 − 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 0 0
\(225\) 11.0000 0.733333
\(226\) 0 0
\(227\) −4.00000 −0.265489 −0.132745 − 0.991150i \(-0.542379\pi\)
−0.132745 + 0.991150i \(0.542379\pi\)
\(228\) 0 0
\(229\) 2.00000 0.132164 0.0660819 − 0.997814i \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −27.0000 −1.76883 −0.884414 − 0.466702i \(-0.845442\pi\)
−0.884414 + 0.466702i \(0.845442\pi\)
\(234\) 0 0
\(235\) 24.0000 1.56559
\(236\) 0 0
\(237\) −4.00000 −0.259828
\(238\) 0 0
\(239\) −26.0000 −1.68180 −0.840900 − 0.541190i \(-0.817974\pi\)
−0.840900 + 0.541190i \(0.817974\pi\)
\(240\) 0 0
\(241\) −21.0000 −1.35273 −0.676364 − 0.736567i \(-0.736446\pi\)
−0.676364 + 0.736567i \(0.736446\pi\)
\(242\) 0 0
\(243\) 10.0000 0.641500
\(244\) 0 0
\(245\) −12.0000 −0.766652
\(246\) 0 0
\(247\) −1.00000 −0.0636285
\(248\) 0 0
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) 4.00000 0.252478 0.126239 − 0.992000i \(-0.459709\pi\)
0.126239 + 0.992000i \(0.459709\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 24.0000 1.50294
\(256\) 0 0
\(257\) −8.00000 −0.499026 −0.249513 − 0.968371i \(-0.580271\pi\)
−0.249513 + 0.968371i \(0.580271\pi\)
\(258\) 0 0
\(259\) 10.0000 0.621370
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) 0 0
\(263\) −24.0000 −1.47990 −0.739952 − 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 0 0
\(265\) −56.0000 −3.44005
\(266\) 0 0
\(267\) −4.00000 −0.244796
\(268\) 0 0
\(269\) 6.00000 0.365826 0.182913 − 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 − 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 0 0
\(273\) −4.00000 −0.242091
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −9.00000 −0.540758 −0.270379 − 0.962754i \(-0.587149\pi\)
−0.270379 + 0.962754i \(0.587149\pi\)
\(278\) 0 0
\(279\) 11.0000 0.658553
\(280\) 0 0
\(281\) 17.0000 1.01413 0.507067 − 0.861906i \(-0.330729\pi\)
0.507067 + 0.861906i \(0.330729\pi\)
\(282\) 0 0
\(283\) 28.0000 1.66443 0.832214 − 0.554455i \(-0.187073\pi\)
0.832214 + 0.554455i \(0.187073\pi\)
\(284\) 0 0
\(285\) −8.00000 −0.473879
\(286\) 0 0
\(287\) 10.0000 0.590281
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) −34.0000 −1.99312
\(292\) 0 0
\(293\) 23.0000 1.34367 0.671837 − 0.740699i \(-0.265505\pi\)
0.671837 + 0.740699i \(0.265505\pi\)
\(294\) 0 0
\(295\) −44.0000 −2.56178
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 3.00000 0.173494
\(300\) 0 0
\(301\) 22.0000 1.26806
\(302\) 0 0
\(303\) −30.0000 −1.72345
\(304\) 0 0
\(305\) 28.0000 1.60328
\(306\) 0 0
\(307\) −17.0000 −0.970241 −0.485121 − 0.874447i \(-0.661224\pi\)
−0.485121 + 0.874447i \(0.661224\pi\)
\(308\) 0 0
\(309\) 28.0000 1.59286
\(310\) 0 0
\(311\) 8.00000 0.453638 0.226819 − 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) 0 0
\(313\) −15.0000 −0.847850 −0.423925 − 0.905697i \(-0.639348\pi\)
−0.423925 + 0.905697i \(0.639348\pi\)
\(314\) 0 0
\(315\) −8.00000 −0.450749
\(316\) 0 0
\(317\) −30.0000 −1.68497 −0.842484 − 0.538721i \(-0.818908\pi\)
−0.842484 + 0.538721i \(0.818908\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 4.00000 0.223258
\(322\) 0 0
\(323\) −3.00000 −0.166924
\(324\) 0 0
\(325\) −11.0000 −0.610170
\(326\) 0 0
\(327\) 10.0000 0.553001
\(328\) 0 0
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) 19.0000 1.04433 0.522167 − 0.852843i \(-0.325124\pi\)
0.522167 + 0.852843i \(0.325124\pi\)
\(332\) 0 0
\(333\) −5.00000 −0.273998
\(334\) 0 0
\(335\) 28.0000 1.52980
\(336\) 0 0
\(337\) −2.00000 −0.108947 −0.0544735 − 0.998515i \(-0.517348\pi\)
−0.0544735 + 0.998515i \(0.517348\pi\)
\(338\) 0 0
\(339\) 20.0000 1.08625
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 0 0
\(345\) 24.0000 1.29212
\(346\) 0 0
\(347\) −4.00000 −0.214731 −0.107366 − 0.994220i \(-0.534242\pi\)
−0.107366 + 0.994220i \(0.534242\pi\)
\(348\) 0 0
\(349\) 4.00000 0.214115 0.107058 − 0.994253i \(-0.465857\pi\)
0.107058 + 0.994253i \(0.465857\pi\)
\(350\) 0 0
\(351\) −4.00000 −0.213504
\(352\) 0 0
\(353\) 4.00000 0.212899 0.106449 − 0.994318i \(-0.466052\pi\)
0.106449 + 0.994318i \(0.466052\pi\)
\(354\) 0 0
\(355\) −48.0000 −2.54758
\(356\) 0 0
\(357\) −12.0000 −0.635107
\(358\) 0 0
\(359\) 4.00000 0.211112 0.105556 − 0.994413i \(-0.466338\pi\)
0.105556 + 0.994413i \(0.466338\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 22.0000 1.15470
\(364\) 0 0
\(365\) 16.0000 0.837478
\(366\) 0 0
\(367\) 29.0000 1.51379 0.756894 − 0.653538i \(-0.226716\pi\)
0.756894 + 0.653538i \(0.226716\pi\)
\(368\) 0 0
\(369\) −5.00000 −0.260290
\(370\) 0 0
\(371\) 28.0000 1.45369
\(372\) 0 0
\(373\) −34.0000 −1.76045 −0.880227 − 0.474554i \(-0.842610\pi\)
−0.880227 + 0.474554i \(0.842610\pi\)
\(374\) 0 0
\(375\) −48.0000 −2.47871
\(376\) 0 0
\(377\) 2.00000 0.103005
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 − 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) 32.0000 1.63941
\(382\) 0 0
\(383\) 1.00000 0.0510976 0.0255488 − 0.999674i \(-0.491867\pi\)
0.0255488 + 0.999674i \(0.491867\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −11.0000 −0.559161
\(388\) 0 0
\(389\) 39.0000 1.97738 0.988689 − 0.149979i \(-0.0479205\pi\)
0.988689 + 0.149979i \(0.0479205\pi\)
\(390\) 0 0
\(391\) 9.00000 0.455150
\(392\) 0 0
\(393\) 2.00000 0.100887
\(394\) 0 0
\(395\) 8.00000 0.402524
\(396\) 0 0
\(397\) −30.0000 −1.50566 −0.752828 − 0.658217i \(-0.771311\pi\)
−0.752828 + 0.658217i \(0.771311\pi\)
\(398\) 0 0
\(399\) 4.00000 0.200250
\(400\) 0 0
\(401\) −22.0000 −1.09863 −0.549314 − 0.835616i \(-0.685111\pi\)
−0.549314 + 0.835616i \(0.685111\pi\)
\(402\) 0 0
\(403\) −11.0000 −0.547949
\(404\) 0 0
\(405\) −44.0000 −2.18638
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 25.0000 1.23617 0.618085 − 0.786111i \(-0.287909\pi\)
0.618085 + 0.786111i \(0.287909\pi\)
\(410\) 0 0
\(411\) −8.00000 −0.394611
\(412\) 0 0
\(413\) 22.0000 1.08255
\(414\) 0 0
\(415\) 8.00000 0.392705
\(416\) 0 0
\(417\) −26.0000 −1.27323
\(418\) 0 0
\(419\) −29.0000 −1.41674 −0.708371 − 0.705840i \(-0.750570\pi\)
−0.708371 + 0.705840i \(0.750570\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 − 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 0 0
\(423\) 6.00000 0.291730
\(424\) 0 0
\(425\) −33.0000 −1.60074
\(426\) 0 0
\(427\) −14.0000 −0.677507
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 1.00000 0.0481683 0.0240842 − 0.999710i \(-0.492333\pi\)
0.0240842 + 0.999710i \(0.492333\pi\)
\(432\) 0 0
\(433\) 2.00000 0.0961139 0.0480569 − 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) 16.0000 0.767141
\(436\) 0 0
\(437\) −3.00000 −0.143509
\(438\) 0 0
\(439\) −22.0000 −1.05000 −0.525001 − 0.851101i \(-0.675935\pi\)
−0.525001 + 0.851101i \(0.675935\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) −20.0000 −0.950229 −0.475114 − 0.879924i \(-0.657593\pi\)
−0.475114 + 0.879924i \(0.657593\pi\)
\(444\) 0 0
\(445\) 8.00000 0.379236
\(446\) 0 0
\(447\) −20.0000 −0.945968
\(448\) 0 0
\(449\) 11.0000 0.519122 0.259561 − 0.965727i \(-0.416422\pi\)
0.259561 + 0.965727i \(0.416422\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 14.0000 0.657777
\(454\) 0 0
\(455\) 8.00000 0.375046
\(456\) 0 0
\(457\) 32.0000 1.49690 0.748448 − 0.663193i \(-0.230799\pi\)
0.748448 + 0.663193i \(0.230799\pi\)
\(458\) 0 0
\(459\) −12.0000 −0.560112
\(460\) 0 0
\(461\) 36.0000 1.67669 0.838344 − 0.545142i \(-0.183524\pi\)
0.838344 + 0.545142i \(0.183524\pi\)
\(462\) 0 0
\(463\) −14.0000 −0.650635 −0.325318 − 0.945605i \(-0.605471\pi\)
−0.325318 + 0.945605i \(0.605471\pi\)
\(464\) 0 0
\(465\) −88.0000 −4.08090
\(466\) 0 0
\(467\) −25.0000 −1.15686 −0.578431 − 0.815731i \(-0.696335\pi\)
−0.578431 + 0.815731i \(0.696335\pi\)
\(468\) 0 0
\(469\) −14.0000 −0.646460
\(470\) 0 0
\(471\) 34.0000 1.56664
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 11.0000 0.504715
\(476\) 0 0
\(477\) −14.0000 −0.641016
\(478\) 0 0
\(479\) −24.0000 −1.09659 −0.548294 − 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) 5.00000 0.227980
\(482\) 0 0
\(483\) −12.0000 −0.546019
\(484\) 0 0
\(485\) 68.0000 3.08772
\(486\) 0 0
\(487\) 16.0000 0.725029 0.362515 − 0.931978i \(-0.381918\pi\)
0.362515 + 0.931978i \(0.381918\pi\)
\(488\) 0 0
\(489\) 12.0000 0.542659
\(490\) 0 0
\(491\) 33.0000 1.48927 0.744635 − 0.667472i \(-0.232624\pi\)
0.744635 + 0.667472i \(0.232624\pi\)
\(492\) 0 0
\(493\) 6.00000 0.270226
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 24.0000 1.07655
\(498\) 0 0
\(499\) 16.0000 0.716258 0.358129 − 0.933672i \(-0.383415\pi\)
0.358129 + 0.933672i \(0.383415\pi\)
\(500\) 0 0
\(501\) 18.0000 0.804181
\(502\) 0 0
\(503\) −24.0000 −1.07011 −0.535054 − 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) 60.0000 2.66996
\(506\) 0 0
\(507\) −2.00000 −0.0888231
\(508\) 0 0
\(509\) 25.0000 1.10811 0.554053 − 0.832482i \(-0.313081\pi\)
0.554053 + 0.832482i \(0.313081\pi\)
\(510\) 0 0
\(511\) −8.00000 −0.353899
\(512\) 0 0
\(513\) 4.00000 0.176604
\(514\) 0 0
\(515\) −56.0000 −2.46765
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 8.00000 0.351161
\(520\) 0 0
\(521\) −18.0000 −0.788594 −0.394297 − 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 0 0
\(523\) −26.0000 −1.13690 −0.568450 − 0.822718i \(-0.692457\pi\)
−0.568450 + 0.822718i \(0.692457\pi\)
\(524\) 0 0
\(525\) 44.0000 1.92032
\(526\) 0 0
\(527\) −33.0000 −1.43750
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) −11.0000 −0.477359
\(532\) 0 0
\(533\) 5.00000 0.216574
\(534\) 0 0
\(535\) −8.00000 −0.345870
\(536\) 0 0
\(537\) −4.00000 −0.172613
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −34.0000 −1.46177 −0.730887 − 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) 0 0
\(543\) 20.0000 0.858282
\(544\) 0 0
\(545\) −20.0000 −0.856706
\(546\) 0 0
\(547\) 22.0000 0.940652 0.470326 − 0.882493i \(-0.344136\pi\)
0.470326 + 0.882493i \(0.344136\pi\)
\(548\) 0 0
\(549\) 7.00000 0.298753
\(550\) 0 0
\(551\) −2.00000 −0.0852029
\(552\) 0 0
\(553\) −4.00000 −0.170097
\(554\) 0 0
\(555\) 40.0000 1.69791
\(556\) 0 0
\(557\) 36.0000 1.52537 0.762684 − 0.646771i \(-0.223881\pi\)
0.762684 + 0.646771i \(0.223881\pi\)
\(558\) 0 0
\(559\) 11.0000 0.465250
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −32.0000 −1.34864 −0.674320 − 0.738440i \(-0.735563\pi\)
−0.674320 + 0.738440i \(0.735563\pi\)
\(564\) 0 0
\(565\) −40.0000 −1.68281
\(566\) 0 0
\(567\) 22.0000 0.923913
\(568\) 0 0
\(569\) −8.00000 −0.335377 −0.167689 − 0.985840i \(-0.553630\pi\)
−0.167689 + 0.985840i \(0.553630\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 − 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 0 0
\(573\) 10.0000 0.417756
\(574\) 0 0
\(575\) −33.0000 −1.37620
\(576\) 0 0
\(577\) 22.0000 0.915872 0.457936 − 0.888985i \(-0.348589\pi\)
0.457936 + 0.888985i \(0.348589\pi\)
\(578\) 0 0
\(579\) −4.00000 −0.166234
\(580\) 0 0
\(581\) −4.00000 −0.165948
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −4.00000 −0.165380
\(586\) 0 0
\(587\) 42.0000 1.73353 0.866763 − 0.498721i \(-0.166197\pi\)
0.866763 + 0.498721i \(0.166197\pi\)
\(588\) 0 0
\(589\) 11.0000 0.453247
\(590\) 0 0
\(591\) 16.0000 0.658152
\(592\) 0 0
\(593\) 42.0000 1.72473 0.862367 − 0.506284i \(-0.168981\pi\)
0.862367 + 0.506284i \(0.168981\pi\)
\(594\) 0 0
\(595\) 24.0000 0.983904
\(596\) 0 0
\(597\) 26.0000 1.06411
\(598\) 0 0
\(599\) −18.0000 −0.735460 −0.367730 − 0.929933i \(-0.619865\pi\)
−0.367730 + 0.929933i \(0.619865\pi\)
\(600\) 0 0
\(601\) −40.0000 −1.63163 −0.815817 − 0.578310i \(-0.803712\pi\)
−0.815817 + 0.578310i \(0.803712\pi\)
\(602\) 0 0
\(603\) 7.00000 0.285062
\(604\) 0 0
\(605\) −44.0000 −1.78885
\(606\) 0 0
\(607\) −42.0000 −1.70473 −0.852364 − 0.522949i \(-0.824832\pi\)
−0.852364 + 0.522949i \(0.824832\pi\)
\(608\) 0 0
\(609\) −8.00000 −0.324176
\(610\) 0 0
\(611\) −6.00000 −0.242734
\(612\) 0 0
\(613\) 18.0000 0.727013 0.363507 − 0.931592i \(-0.381579\pi\)
0.363507 + 0.931592i \(0.381579\pi\)
\(614\) 0 0
\(615\) 40.0000 1.61296
\(616\) 0 0
\(617\) −6.00000 −0.241551 −0.120775 − 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 0 0
\(619\) 6.00000 0.241160 0.120580 − 0.992704i \(-0.461525\pi\)
0.120580 + 0.992704i \(0.461525\pi\)
\(620\) 0 0
\(621\) −12.0000 −0.481543
\(622\) 0 0
\(623\) −4.00000 −0.160257
\(624\) 0 0
\(625\) 41.0000 1.64000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 15.0000 0.598089
\(630\) 0 0
\(631\) 22.0000 0.875806 0.437903 − 0.899022i \(-0.355721\pi\)
0.437903 + 0.899022i \(0.355721\pi\)
\(632\) 0 0
\(633\) −4.00000 −0.158986
\(634\) 0 0
\(635\) −64.0000 −2.53976
\(636\) 0 0
\(637\) 3.00000 0.118864
\(638\) 0 0
\(639\) −12.0000 −0.474713
\(640\) 0 0
\(641\) 10.0000 0.394976 0.197488 − 0.980305i \(-0.436722\pi\)
0.197488 + 0.980305i \(0.436722\pi\)
\(642\) 0 0
\(643\) 36.0000 1.41970 0.709851 − 0.704352i \(-0.248762\pi\)
0.709851 + 0.704352i \(0.248762\pi\)
\(644\) 0 0
\(645\) 88.0000 3.46500
\(646\) 0 0
\(647\) −8.00000 −0.314512 −0.157256 − 0.987558i \(-0.550265\pi\)
−0.157256 + 0.987558i \(0.550265\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 44.0000 1.72450
\(652\) 0 0
\(653\) 3.00000 0.117399 0.0586995 − 0.998276i \(-0.481305\pi\)
0.0586995 + 0.998276i \(0.481305\pi\)
\(654\) 0 0
\(655\) −4.00000 −0.156293
\(656\) 0 0
\(657\) 4.00000 0.156055
\(658\) 0 0
\(659\) −16.0000 −0.623272 −0.311636 − 0.950202i \(-0.600877\pi\)
−0.311636 + 0.950202i \(0.600877\pi\)
\(660\) 0 0
\(661\) −19.0000 −0.739014 −0.369507 − 0.929228i \(-0.620473\pi\)
−0.369507 + 0.929228i \(0.620473\pi\)
\(662\) 0 0
\(663\) −6.00000 −0.233021
\(664\) 0 0
\(665\) −8.00000 −0.310227
\(666\) 0 0
\(667\) 6.00000 0.232321
\(668\) 0 0
\(669\) −16.0000 −0.618596
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −20.0000 −0.770943 −0.385472 − 0.922720i \(-0.625961\pi\)
−0.385472 + 0.922720i \(0.625961\pi\)
\(674\) 0 0
\(675\) 44.0000 1.69356
\(676\) 0 0
\(677\) −12.0000 −0.461197 −0.230599 − 0.973049i \(-0.574068\pi\)
−0.230599 + 0.973049i \(0.574068\pi\)
\(678\) 0 0
\(679\) −34.0000 −1.30480
\(680\) 0 0
\(681\) 8.00000 0.306561
\(682\) 0 0
\(683\) −20.0000 −0.765279 −0.382639 − 0.923898i \(-0.624985\pi\)
−0.382639 + 0.923898i \(0.624985\pi\)
\(684\) 0 0
\(685\) 16.0000 0.611329
\(686\) 0 0
\(687\) −4.00000 −0.152610
\(688\) 0 0
\(689\) 14.0000 0.533358
\(690\) 0 0
\(691\) 8.00000 0.304334 0.152167 − 0.988355i \(-0.451375\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 52.0000 1.97247
\(696\) 0 0
\(697\) 15.0000 0.568166
\(698\) 0 0
\(699\) 54.0000 2.04247
\(700\) 0 0
\(701\) −15.0000 −0.566542 −0.283271 − 0.959040i \(-0.591420\pi\)
−0.283271 + 0.959040i \(0.591420\pi\)
\(702\) 0 0
\(703\) −5.00000 −0.188579
\(704\) 0 0
\(705\) −48.0000 −1.80778
\(706\) 0 0
\(707\) −30.0000 −1.12827
\(708\) 0 0
\(709\) 44.0000 1.65245 0.826227 − 0.563337i \(-0.190483\pi\)
0.826227 + 0.563337i \(0.190483\pi\)
\(710\) 0 0
\(711\) 2.00000 0.0750059
\(712\) 0 0
\(713\) −33.0000 −1.23586
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 52.0000 1.94198
\(718\) 0 0
\(719\) 47.0000 1.75280 0.876402 − 0.481580i \(-0.159937\pi\)
0.876402 + 0.481580i \(0.159937\pi\)
\(720\) 0 0
\(721\) 28.0000 1.04277
\(722\) 0 0
\(723\) 42.0000 1.56200
\(724\) 0 0
\(725\) −22.0000 −0.817059
\(726\) 0 0
\(727\) 33.0000 1.22390 0.611951 − 0.790896i \(-0.290385\pi\)
0.611951 + 0.790896i \(0.290385\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 33.0000 1.22055
\(732\) 0 0
\(733\) 44.0000 1.62518 0.812589 − 0.582838i \(-0.198058\pi\)
0.812589 + 0.582838i \(0.198058\pi\)
\(734\) 0 0
\(735\) 24.0000 0.885253
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 8.00000 0.294285 0.147142 − 0.989115i \(-0.452992\pi\)
0.147142 + 0.989115i \(0.452992\pi\)
\(740\) 0 0
\(741\) 2.00000 0.0734718
\(742\) 0 0
\(743\) 35.0000 1.28403 0.642013 − 0.766694i \(-0.278100\pi\)
0.642013 + 0.766694i \(0.278100\pi\)
\(744\) 0 0
\(745\) 40.0000 1.46549
\(746\) 0 0
\(747\) 2.00000 0.0731762
\(748\) 0 0
\(749\) 4.00000 0.146157
\(750\) 0 0
\(751\) 16.0000 0.583848 0.291924 − 0.956441i \(-0.405705\pi\)
0.291924 + 0.956441i \(0.405705\pi\)
\(752\) 0 0
\(753\) −8.00000 −0.291536
\(754\) 0 0
\(755\) −28.0000 −1.01902
\(756\) 0 0
\(757\) 35.0000 1.27210 0.636048 − 0.771649i \(-0.280568\pi\)
0.636048 + 0.771649i \(0.280568\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −36.0000 −1.30500 −0.652499 − 0.757789i \(-0.726280\pi\)
−0.652499 + 0.757789i \(0.726280\pi\)
\(762\) 0 0
\(763\) 10.0000 0.362024
\(764\) 0 0
\(765\) −12.0000 −0.433861
\(766\) 0 0
\(767\) 11.0000 0.397187
\(768\) 0 0
\(769\) 26.0000 0.937584 0.468792 − 0.883309i \(-0.344689\pi\)
0.468792 + 0.883309i \(0.344689\pi\)
\(770\) 0 0
\(771\) 16.0000 0.576226
\(772\) 0 0
\(773\) 23.0000 0.827253 0.413626 − 0.910447i \(-0.364262\pi\)
0.413626 + 0.910447i \(0.364262\pi\)
\(774\) 0 0
\(775\) 121.000 4.34645
\(776\) 0 0
\(777\) −20.0000 −0.717496
\(778\) 0 0
\(779\) −5.00000 −0.179144
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −8.00000 −0.285897
\(784\) 0 0
\(785\) −68.0000 −2.42702
\(786\) 0 0
\(787\) 31.0000 1.10503 0.552515 − 0.833503i \(-0.313668\pi\)
0.552515 + 0.833503i \(0.313668\pi\)
\(788\) 0 0
\(789\) 48.0000 1.70885
\(790\) 0 0
\(791\) 20.0000 0.711118
\(792\) 0 0
\(793\) −7.00000 −0.248577
\(794\) 0 0
\(795\) 112.000 3.97223
\(796\) 0 0
\(797\) 8.00000 0.283375 0.141687 − 0.989911i \(-0.454747\pi\)
0.141687 + 0.989911i \(0.454747\pi\)
\(798\) 0 0
\(799\) −18.0000 −0.636794
\(800\) 0 0
\(801\) 2.00000 0.0706665
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 24.0000 0.845889
\(806\) 0 0
\(807\) −12.0000 −0.422420
\(808\) 0 0
\(809\) −51.0000 −1.79306 −0.896532 − 0.442978i \(-0.853922\pi\)
−0.896532 + 0.442978i \(0.853922\pi\)
\(810\) 0 0
\(811\) −5.00000 −0.175574 −0.0877869 − 0.996139i \(-0.527979\pi\)
−0.0877869 + 0.996139i \(0.527979\pi\)
\(812\) 0 0
\(813\) −16.0000 −0.561144
\(814\) 0 0
\(815\) −24.0000 −0.840683
\(816\) 0 0
\(817\) −11.0000 −0.384841
\(818\) 0 0
\(819\) 2.00000 0.0698857
\(820\) 0 0
\(821\) −20.0000 −0.698005 −0.349002 − 0.937122i \(-0.613479\pi\)
−0.349002 + 0.937122i \(0.613479\pi\)
\(822\) 0 0
\(823\) 32.0000 1.11545 0.557725 − 0.830026i \(-0.311674\pi\)
0.557725 + 0.830026i \(0.311674\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 3.00000 0.104320 0.0521601 − 0.998639i \(-0.483389\pi\)
0.0521601 + 0.998639i \(0.483389\pi\)
\(828\) 0 0
\(829\) −28.0000 −0.972480 −0.486240 − 0.873825i \(-0.661632\pi\)
−0.486240 + 0.873825i \(0.661632\pi\)
\(830\) 0 0
\(831\) 18.0000 0.624413
\(832\) 0 0
\(833\) 9.00000 0.311832
\(834\) 0 0
\(835\) −36.0000 −1.24583
\(836\) 0 0
\(837\) 44.0000 1.52086
\(838\) 0 0
\(839\) 5.00000 0.172619 0.0863096 − 0.996268i \(-0.472493\pi\)
0.0863096 + 0.996268i \(0.472493\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) −34.0000 −1.17102
\(844\) 0 0
\(845\) 4.00000 0.137604
\(846\) 0 0
\(847\) 22.0000 0.755929
\(848\) 0 0
\(849\) −56.0000 −1.92192
\(850\) 0 0
\(851\) 15.0000 0.514193
\(852\) 0 0
\(853\) 52.0000 1.78045 0.890223 − 0.455525i \(-0.150548\pi\)
0.890223 + 0.455525i \(0.150548\pi\)
\(854\) 0 0
\(855\) 4.00000 0.136797
\(856\) 0 0
\(857\) −42.0000 −1.43469 −0.717346 − 0.696717i \(-0.754643\pi\)
−0.717346 + 0.696717i \(0.754643\pi\)
\(858\) 0 0
\(859\) 39.0000 1.33066 0.665331 − 0.746548i \(-0.268290\pi\)
0.665331 + 0.746548i \(0.268290\pi\)
\(860\) 0 0
\(861\) −20.0000 −0.681598
\(862\) 0 0
\(863\) 19.0000 0.646768 0.323384 − 0.946268i \(-0.395180\pi\)
0.323384 + 0.946268i \(0.395180\pi\)
\(864\) 0 0
\(865\) −16.0000 −0.544016
\(866\) 0 0
\(867\) 16.0000 0.543388
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −7.00000 −0.237186
\(872\) 0 0
\(873\) 17.0000 0.575363
\(874\) 0 0
\(875\) −48.0000 −1.62270
\(876\) 0 0
\(877\) 49.0000 1.65461 0.827306 − 0.561751i \(-0.189872\pi\)
0.827306 + 0.561751i \(0.189872\pi\)
\(878\) 0 0
\(879\) −46.0000 −1.55154
\(880\) 0 0
\(881\) −1.00000 −0.0336909 −0.0168454 − 0.999858i \(-0.505362\pi\)
−0.0168454 + 0.999858i \(0.505362\pi\)
\(882\) 0 0
\(883\) −1.00000 −0.0336527 −0.0168263 − 0.999858i \(-0.505356\pi\)
−0.0168263 + 0.999858i \(0.505356\pi\)
\(884\) 0 0
\(885\) 88.0000 2.95809
\(886\) 0 0
\(887\) 28.0000 0.940148 0.470074 − 0.882627i \(-0.344227\pi\)
0.470074 + 0.882627i \(0.344227\pi\)
\(888\) 0 0
\(889\) 32.0000 1.07325
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 6.00000 0.200782
\(894\) 0 0
\(895\) 8.00000 0.267411
\(896\) 0 0
\(897\) −6.00000 −0.200334
\(898\) 0 0
\(899\) −22.0000 −0.733741
\(900\) 0 0
\(901\) 42.0000 1.39922
\(902\) 0 0
\(903\) −44.0000 −1.46423
\(904\) 0 0
\(905\) −40.0000 −1.32964
\(906\) 0 0
\(907\) −10.0000 −0.332045 −0.166022 − 0.986122i \(-0.553092\pi\)
−0.166022 + 0.986122i \(0.553092\pi\)
\(908\) 0 0
\(909\) 15.0000 0.497519
\(910\) 0 0
\(911\) 48.0000 1.59031 0.795155 − 0.606406i \(-0.207389\pi\)
0.795155 + 0.606406i \(0.207389\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −56.0000 −1.85130
\(916\) 0 0
\(917\) 2.00000 0.0660458
\(918\) 0 0
\(919\) −7.00000 −0.230909 −0.115454 − 0.993313i \(-0.536832\pi\)
−0.115454 + 0.993313i \(0.536832\pi\)
\(920\) 0 0
\(921\) 34.0000 1.12034
\(922\) 0 0
\(923\) 12.0000 0.394985
\(924\) 0 0
\(925\) −55.0000 −1.80839
\(926\) 0 0
\(927\) −14.0000 −0.459820
\(928\) 0 0
\(929\) 24.0000 0.787414 0.393707 − 0.919236i \(-0.371192\pi\)
0.393707 + 0.919236i \(0.371192\pi\)
\(930\) 0 0
\(931\) −3.00000 −0.0983210
\(932\) 0 0
\(933\) −16.0000 −0.523816
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −11.0000 −0.359354 −0.179677 − 0.983726i \(-0.557505\pi\)
−0.179677 + 0.983726i \(0.557505\pi\)
\(938\) 0 0
\(939\) 30.0000 0.979013
\(940\) 0 0
\(941\) 34.0000 1.10837 0.554184 − 0.832394i \(-0.313030\pi\)
0.554184 + 0.832394i \(0.313030\pi\)
\(942\) 0 0
\(943\) 15.0000 0.488467
\(944\) 0 0
\(945\) −32.0000 −1.04096
\(946\) 0 0
\(947\) −8.00000 −0.259965 −0.129983 − 0.991516i \(-0.541492\pi\)
−0.129983 + 0.991516i \(0.541492\pi\)
\(948\) 0 0
\(949\) −4.00000 −0.129845
\(950\) 0 0
\(951\) 60.0000 1.94563
\(952\) 0 0
\(953\) 24.0000 0.777436 0.388718 − 0.921357i \(-0.372918\pi\)
0.388718 + 0.921357i \(0.372918\pi\)
\(954\) 0 0
\(955\) −20.0000 −0.647185
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −8.00000 −0.258333
\(960\) 0 0
\(961\) 90.0000 2.90323
\(962\) 0 0
\(963\) −2.00000 −0.0644491
\(964\) 0 0
\(965\) 8.00000 0.257529
\(966\) 0 0
\(967\) −48.0000 −1.54358 −0.771788 − 0.635880i \(-0.780637\pi\)
−0.771788 + 0.635880i \(0.780637\pi\)
\(968\) 0 0
\(969\) 6.00000 0.192748
\(970\) 0 0
\(971\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) −26.0000 −0.833522
\(974\) 0 0
\(975\) 22.0000 0.704564
\(976\) 0 0
\(977\) −19.0000 −0.607864 −0.303932 − 0.952694i \(-0.598300\pi\)
−0.303932 + 0.952694i \(0.598300\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −5.00000 −0.159638
\(982\) 0 0
\(983\) −48.0000 −1.53096 −0.765481 − 0.643458i \(-0.777499\pi\)
−0.765481 + 0.643458i \(0.777499\pi\)
\(984\) 0 0
\(985\) −32.0000 −1.01960
\(986\) 0 0
\(987\) 24.0000 0.763928
\(988\) 0 0
\(989\) 33.0000 1.04934
\(990\) 0 0
\(991\) −2.00000 −0.0635321 −0.0317660 − 0.999495i \(-0.510113\pi\)
−0.0317660 + 0.999495i \(0.510113\pi\)
\(992\) 0 0
\(993\) −38.0000 −1.20589
\(994\) 0 0
\(995\) −52.0000 −1.64851
\(996\) 0 0
\(997\) 22.0000 0.696747 0.348373 − 0.937356i \(-0.386734\pi\)
0.348373 + 0.937356i \(0.386734\pi\)
\(998\) 0 0
\(999\) −20.0000 −0.632772
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 3952.2.a.c.1.1 1
4.3 odd 2 988.2.a.d.1.1 ✓ 1
12.11 even 2 8892.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
988.2.a.d.1.1 ✓ 1 4.3 odd 2
3952.2.a.c.1.1 1 1.1 even 1 trivial
8892.2.a.a.1.1 1 12.11 even 2