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Properties

Label 3344.2.a.c.1.1
Level $3344$
Weight $2$
Character 3344.1
Self dual yes
Analytic conductor $26.702$
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] = [3344,2,Mod(1,3344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3344, 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("3344.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3344.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: \(26.7019744359\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1672)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3344.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -3.00000 q^{5} -2.00000 q^{7} -2.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -3.00000 q^{5} -2.00000 q^{7} -2.00000 q^{9} +1.00000 q^{11} +4.00000 q^{13} +3.00000 q^{15} -2.00000 q^{17} +1.00000 q^{19} +2.00000 q^{21} +7.00000 q^{23} +4.00000 q^{25} +5.00000 q^{27} +4.00000 q^{29} +3.00000 q^{31} -1.00000 q^{33} +6.00000 q^{35} -5.00000 q^{37} -4.00000 q^{39} +4.00000 q^{41} +2.00000 q^{43} +6.00000 q^{45} -3.00000 q^{49} +2.00000 q^{51} -14.0000 q^{53} -3.00000 q^{55} -1.00000 q^{57} +5.00000 q^{59} -8.00000 q^{61} +4.00000 q^{63} -12.0000 q^{65} -7.00000 q^{67} -7.00000 q^{69} +9.00000 q^{71} +8.00000 q^{73} -4.00000 q^{75} -2.00000 q^{77} -6.00000 q^{79} +1.00000 q^{81} -2.00000 q^{83} +6.00000 q^{85} -4.00000 q^{87} -13.0000 q^{89} -8.00000 q^{91} -3.00000 q^{93} -3.00000 q^{95} -11.0000 q^{97} -2.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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 −0.288675 − 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) 0 0
\(5\) −3.00000 −1.34164 −0.670820 − 0.741620i \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 − 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 − 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 3.00000 0.774597
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 0 0
\(23\) 7.00000 1.45960 0.729800 − 0.683660i \(-0.239613\pi\)
0.729800 + 0.683660i \(0.239613\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) 5.00000 0.962250
\(28\) 0 0
\(29\) 4.00000 0.742781 0.371391 − 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 0 0
\(31\) 3.00000 0.538816 0.269408 − 0.963026i \(-0.413172\pi\)
0.269408 + 0.963026i \(0.413172\pi\)
\(32\) 0 0
\(33\) −1.00000 −0.174078
\(34\) 0 0
\(35\) 6.00000 1.01419
\(36\) 0 0
\(37\) −5.00000 −0.821995 −0.410997 − 0.911636i \(-0.634819\pi\)
−0.410997 + 0.911636i \(0.634819\pi\)
\(38\) 0 0
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) 4.00000 0.624695 0.312348 − 0.949968i \(-0.398885\pi\)
0.312348 + 0.949968i \(0.398885\pi\)
\(42\) 0 0
\(43\) 2.00000 0.304997 0.152499 − 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) 0 0
\(45\) 6.00000 0.894427
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) 0 0
\(53\) −14.0000 −1.92305 −0.961524 − 0.274721i \(-0.911414\pi\)
−0.961524 + 0.274721i \(0.911414\pi\)
\(54\) 0 0
\(55\) −3.00000 −0.404520
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 0 0
\(59\) 5.00000 0.650945 0.325472 − 0.945552i \(-0.394477\pi\)
0.325472 + 0.945552i \(0.394477\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 − 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 0 0
\(63\) 4.00000 0.503953
\(64\) 0 0
\(65\) −12.0000 −1.48842
\(66\) 0 0
\(67\) −7.00000 −0.855186 −0.427593 − 0.903971i \(-0.640638\pi\)
−0.427593 + 0.903971i \(0.640638\pi\)
\(68\) 0 0
\(69\) −7.00000 −0.842701
\(70\) 0 0
\(71\) 9.00000 1.06810 0.534052 − 0.845452i \(-0.320669\pi\)
0.534052 + 0.845452i \(0.320669\pi\)
\(72\) 0 0
\(73\) 8.00000 0.936329 0.468165 − 0.883641i \(-0.344915\pi\)
0.468165 + 0.883641i \(0.344915\pi\)
\(74\) 0 0
\(75\) −4.00000 −0.461880
\(76\) 0 0
\(77\) −2.00000 −0.227921
\(78\) 0 0
\(79\) −6.00000 −0.675053 −0.337526 − 0.941316i \(-0.609590\pi\)
−0.337526 + 0.941316i \(0.609590\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −2.00000 −0.219529 −0.109764 − 0.993958i \(-0.535010\pi\)
−0.109764 + 0.993958i \(0.535010\pi\)
\(84\) 0 0
\(85\) 6.00000 0.650791
\(86\) 0 0
\(87\) −4.00000 −0.428845
\(88\) 0 0
\(89\) −13.0000 −1.37800 −0.688999 − 0.724763i \(-0.741949\pi\)
−0.688999 + 0.724763i \(0.741949\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) 0 0
\(93\) −3.00000 −0.311086
\(94\) 0 0
\(95\) −3.00000 −0.307794
\(96\) 0 0
\(97\) −11.0000 −1.11688 −0.558440 − 0.829545i \(-0.688600\pi\)
−0.558440 + 0.829545i \(0.688600\pi\)
\(98\) 0 0
\(99\) −2.00000 −0.201008
\(100\) 0 0
\(101\) −14.0000 −1.39305 −0.696526 − 0.717532i \(-0.745272\pi\)
−0.696526 + 0.717532i \(0.745272\pi\)
\(102\) 0 0
\(103\) 16.0000 1.57653 0.788263 − 0.615338i \(-0.210980\pi\)
0.788263 + 0.615338i \(0.210980\pi\)
\(104\) 0 0
\(105\) −6.00000 −0.585540
\(106\) 0 0
\(107\) 6.00000 0.580042 0.290021 − 0.957020i \(-0.406338\pi\)
0.290021 + 0.957020i \(0.406338\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 − 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) 5.00000 0.474579
\(112\) 0 0
\(113\) 13.0000 1.22294 0.611469 − 0.791269i \(-0.290579\pi\)
0.611469 + 0.791269i \(0.290579\pi\)
\(114\) 0 0
\(115\) −21.0000 −1.95826
\(116\) 0 0
\(117\) −8.00000 −0.739600
\(118\) 0 0
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −4.00000 −0.360668
\(124\) 0 0
\(125\) 3.00000 0.268328
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 − 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 0 0
\(129\) −2.00000 −0.176090
\(130\) 0 0
\(131\) −6.00000 −0.524222 −0.262111 − 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 0 0
\(133\) −2.00000 −0.173422
\(134\) 0 0
\(135\) −15.0000 −1.29099
\(136\) 0 0
\(137\) −15.0000 −1.28154 −0.640768 − 0.767734i \(-0.721384\pi\)
−0.640768 + 0.767734i \(0.721384\pi\)
\(138\) 0 0
\(139\) −14.0000 −1.18746 −0.593732 − 0.804663i \(-0.702346\pi\)
−0.593732 + 0.804663i \(0.702346\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 4.00000 0.334497
\(144\) 0 0
\(145\) −12.0000 −0.996546
\(146\) 0 0
\(147\) 3.00000 0.247436
\(148\) 0 0
\(149\) 10.0000 0.819232 0.409616 − 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) −6.00000 −0.488273 −0.244137 − 0.969741i \(-0.578505\pi\)
−0.244137 + 0.969741i \(0.578505\pi\)
\(152\) 0 0
\(153\) 4.00000 0.323381
\(154\) 0 0
\(155\) −9.00000 −0.722897
\(156\) 0 0
\(157\) 5.00000 0.399043 0.199522 − 0.979893i \(-0.436061\pi\)
0.199522 + 0.979893i \(0.436061\pi\)
\(158\) 0 0
\(159\) 14.0000 1.11027
\(160\) 0 0
\(161\) −14.0000 −1.10335
\(162\) 0 0
\(163\) −20.0000 −1.56652 −0.783260 − 0.621694i \(-0.786445\pi\)
−0.783260 + 0.621694i \(0.786445\pi\)
\(164\) 0 0
\(165\) 3.00000 0.233550
\(166\) 0 0
\(167\) 12.0000 0.928588 0.464294 − 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) −2.00000 −0.152944
\(172\) 0 0
\(173\) 6.00000 0.456172 0.228086 − 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) −8.00000 −0.604743
\(176\) 0 0
\(177\) −5.00000 −0.375823
\(178\) 0 0
\(179\) −23.0000 −1.71910 −0.859550 − 0.511051i \(-0.829256\pi\)
−0.859550 + 0.511051i \(0.829256\pi\)
\(180\) 0 0
\(181\) −17.0000 −1.26360 −0.631800 − 0.775131i \(-0.717684\pi\)
−0.631800 + 0.775131i \(0.717684\pi\)
\(182\) 0 0
\(183\) 8.00000 0.591377
\(184\) 0 0
\(185\) 15.0000 1.10282
\(186\) 0 0
\(187\) −2.00000 −0.146254
\(188\) 0 0
\(189\) −10.0000 −0.727393
\(190\) 0 0
\(191\) 17.0000 1.23008 0.615038 − 0.788497i \(-0.289140\pi\)
0.615038 + 0.788497i \(0.289140\pi\)
\(192\) 0 0
\(193\) −12.0000 −0.863779 −0.431889 − 0.901927i \(-0.642153\pi\)
−0.431889 + 0.901927i \(0.642153\pi\)
\(194\) 0 0
\(195\) 12.0000 0.859338
\(196\) 0 0
\(197\) 18.0000 1.28245 0.641223 − 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 0 0
\(199\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 7.00000 0.493742
\(202\) 0 0
\(203\) −8.00000 −0.561490
\(204\) 0 0
\(205\) −12.0000 −0.838116
\(206\) 0 0
\(207\) −14.0000 −0.973067
\(208\) 0 0
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) 0 0
\(213\) −9.00000 −0.616670
\(214\) 0 0
\(215\) −6.00000 −0.409197
\(216\) 0 0
\(217\) −6.00000 −0.407307
\(218\) 0 0
\(219\) −8.00000 −0.540590
\(220\) 0 0
\(221\) −8.00000 −0.538138
\(222\) 0 0
\(223\) −9.00000 −0.602685 −0.301342 − 0.953516i \(-0.597435\pi\)
−0.301342 + 0.953516i \(0.597435\pi\)
\(224\) 0 0
\(225\) −8.00000 −0.533333
\(226\) 0 0
\(227\) −22.0000 −1.46019 −0.730096 − 0.683345i \(-0.760525\pi\)
−0.730096 + 0.683345i \(0.760525\pi\)
\(228\) 0 0
\(229\) 11.0000 0.726900 0.363450 − 0.931614i \(-0.381599\pi\)
0.363450 + 0.931614i \(0.381599\pi\)
\(230\) 0 0
\(231\) 2.00000 0.131590
\(232\) 0 0
\(233\) 20.0000 1.31024 0.655122 − 0.755523i \(-0.272617\pi\)
0.655122 + 0.755523i \(0.272617\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 6.00000 0.389742
\(238\) 0 0
\(239\) 10.0000 0.646846 0.323423 − 0.946254i \(-0.395166\pi\)
0.323423 + 0.946254i \(0.395166\pi\)
\(240\) 0 0
\(241\) −8.00000 −0.515325 −0.257663 − 0.966235i \(-0.582952\pi\)
−0.257663 + 0.966235i \(0.582952\pi\)
\(242\) 0 0
\(243\) −16.0000 −1.02640
\(244\) 0 0
\(245\) 9.00000 0.574989
\(246\) 0 0
\(247\) 4.00000 0.254514
\(248\) 0 0
\(249\) 2.00000 0.126745
\(250\) 0 0
\(251\) 5.00000 0.315597 0.157799 − 0.987471i \(-0.449560\pi\)
0.157799 + 0.987471i \(0.449560\pi\)
\(252\) 0 0
\(253\) 7.00000 0.440086
\(254\) 0 0
\(255\) −6.00000 −0.375735
\(256\) 0 0
\(257\) 14.0000 0.873296 0.436648 − 0.899632i \(-0.356166\pi\)
0.436648 + 0.899632i \(0.356166\pi\)
\(258\) 0 0
\(259\) 10.0000 0.621370
\(260\) 0 0
\(261\) −8.00000 −0.495188
\(262\) 0 0
\(263\) 26.0000 1.60323 0.801614 − 0.597841i \(-0.203975\pi\)
0.801614 + 0.597841i \(0.203975\pi\)
\(264\) 0 0
\(265\) 42.0000 2.58004
\(266\) 0 0
\(267\) 13.0000 0.795587
\(268\) 0 0
\(269\) −22.0000 −1.34136 −0.670682 − 0.741745i \(-0.733998\pi\)
−0.670682 + 0.741745i \(0.733998\pi\)
\(270\) 0 0
\(271\) 24.0000 1.45790 0.728948 − 0.684569i \(-0.240010\pi\)
0.728948 + 0.684569i \(0.240010\pi\)
\(272\) 0 0
\(273\) 8.00000 0.484182
\(274\) 0 0
\(275\) 4.00000 0.241209
\(276\) 0 0
\(277\) 22.0000 1.32185 0.660926 − 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) 0 0
\(279\) −6.00000 −0.359211
\(280\) 0 0
\(281\) −2.00000 −0.119310 −0.0596550 − 0.998219i \(-0.519000\pi\)
−0.0596550 + 0.998219i \(0.519000\pi\)
\(282\) 0 0
\(283\) −16.0000 −0.951101 −0.475551 − 0.879688i \(-0.657751\pi\)
−0.475551 + 0.879688i \(0.657751\pi\)
\(284\) 0 0
\(285\) 3.00000 0.177705
\(286\) 0 0
\(287\) −8.00000 −0.472225
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 11.0000 0.644831
\(292\) 0 0
\(293\) −8.00000 −0.467365 −0.233682 − 0.972313i \(-0.575078\pi\)
−0.233682 + 0.972313i \(0.575078\pi\)
\(294\) 0 0
\(295\) −15.0000 −0.873334
\(296\) 0 0
\(297\) 5.00000 0.290129
\(298\) 0 0
\(299\) 28.0000 1.61928
\(300\) 0 0
\(301\) −4.00000 −0.230556
\(302\) 0 0
\(303\) 14.0000 0.804279
\(304\) 0 0
\(305\) 24.0000 1.37424
\(306\) 0 0
\(307\) −16.0000 −0.913168 −0.456584 − 0.889680i \(-0.650927\pi\)
−0.456584 + 0.889680i \(0.650927\pi\)
\(308\) 0 0
\(309\) −16.0000 −0.910208
\(310\) 0 0
\(311\) −4.00000 −0.226819 −0.113410 − 0.993548i \(-0.536177\pi\)
−0.113410 + 0.993548i \(0.536177\pi\)
\(312\) 0 0
\(313\) −9.00000 −0.508710 −0.254355 − 0.967111i \(-0.581863\pi\)
−0.254355 + 0.967111i \(0.581863\pi\)
\(314\) 0 0
\(315\) −12.0000 −0.676123
\(316\) 0 0
\(317\) −27.0000 −1.51647 −0.758236 − 0.651981i \(-0.773938\pi\)
−0.758236 + 0.651981i \(0.773938\pi\)
\(318\) 0 0
\(319\) 4.00000 0.223957
\(320\) 0 0
\(321\) −6.00000 −0.334887
\(322\) 0 0
\(323\) −2.00000 −0.111283
\(324\) 0 0
\(325\) 16.0000 0.887520
\(326\) 0 0
\(327\) −2.00000 −0.110600
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −9.00000 −0.494685 −0.247342 − 0.968928i \(-0.579557\pi\)
−0.247342 + 0.968928i \(0.579557\pi\)
\(332\) 0 0
\(333\) 10.0000 0.547997
\(334\) 0 0
\(335\) 21.0000 1.14735
\(336\) 0 0
\(337\) −14.0000 −0.762629 −0.381314 − 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) 0 0
\(339\) −13.0000 −0.706063
\(340\) 0 0
\(341\) 3.00000 0.162459
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 0 0
\(345\) 21.0000 1.13060
\(346\) 0 0
\(347\) −28.0000 −1.50312 −0.751559 − 0.659665i \(-0.770698\pi\)
−0.751559 + 0.659665i \(0.770698\pi\)
\(348\) 0 0
\(349\) 14.0000 0.749403 0.374701 − 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 0 0
\(351\) 20.0000 1.06752
\(352\) 0 0
\(353\) 3.00000 0.159674 0.0798369 − 0.996808i \(-0.474560\pi\)
0.0798369 + 0.996808i \(0.474560\pi\)
\(354\) 0 0
\(355\) −27.0000 −1.43301
\(356\) 0 0
\(357\) −4.00000 −0.211702
\(358\) 0 0
\(359\) −8.00000 −0.422224 −0.211112 − 0.977462i \(-0.567708\pi\)
−0.211112 + 0.977462i \(0.567708\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −1.00000 −0.0524864
\(364\) 0 0
\(365\) −24.0000 −1.25622
\(366\) 0 0
\(367\) −1.00000 −0.0521996 −0.0260998 − 0.999659i \(-0.508309\pi\)
−0.0260998 + 0.999659i \(0.508309\pi\)
\(368\) 0 0
\(369\) −8.00000 −0.416463
\(370\) 0 0
\(371\) 28.0000 1.45369
\(372\) 0 0
\(373\) −22.0000 −1.13912 −0.569558 − 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) 0 0
\(375\) −3.00000 −0.154919
\(376\) 0 0
\(377\) 16.0000 0.824042
\(378\) 0 0
\(379\) −13.0000 −0.667765 −0.333883 − 0.942615i \(-0.608359\pi\)
−0.333883 + 0.942615i \(0.608359\pi\)
\(380\) 0 0
\(381\) 8.00000 0.409852
\(382\) 0 0
\(383\) −13.0000 −0.664269 −0.332134 − 0.943232i \(-0.607769\pi\)
−0.332134 + 0.943232i \(0.607769\pi\)
\(384\) 0 0
\(385\) 6.00000 0.305788
\(386\) 0 0
\(387\) −4.00000 −0.203331
\(388\) 0 0
\(389\) −19.0000 −0.963338 −0.481669 − 0.876353i \(-0.659969\pi\)
−0.481669 + 0.876353i \(0.659969\pi\)
\(390\) 0 0
\(391\) −14.0000 −0.708010
\(392\) 0 0
\(393\) 6.00000 0.302660
\(394\) 0 0
\(395\) 18.0000 0.905678
\(396\) 0 0
\(397\) −26.0000 −1.30490 −0.652451 − 0.757831i \(-0.726259\pi\)
−0.652451 + 0.757831i \(0.726259\pi\)
\(398\) 0 0
\(399\) 2.00000 0.100125
\(400\) 0 0
\(401\) −6.00000 −0.299626 −0.149813 − 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 0 0
\(403\) 12.0000 0.597763
\(404\) 0 0
\(405\) −3.00000 −0.149071
\(406\) 0 0
\(407\) −5.00000 −0.247841
\(408\) 0 0
\(409\) −10.0000 −0.494468 −0.247234 − 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 0 0
\(411\) 15.0000 0.739895
\(412\) 0 0
\(413\) −10.0000 −0.492068
\(414\) 0 0
\(415\) 6.00000 0.294528
\(416\) 0 0
\(417\) 14.0000 0.685583
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 − 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) 6.00000 0.292422 0.146211 − 0.989253i \(-0.453292\pi\)
0.146211 + 0.989253i \(0.453292\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −8.00000 −0.388057
\(426\) 0 0
\(427\) 16.0000 0.774294
\(428\) 0 0
\(429\) −4.00000 −0.193122
\(430\) 0 0
\(431\) −30.0000 −1.44505 −0.722525 − 0.691345i \(-0.757018\pi\)
−0.722525 + 0.691345i \(0.757018\pi\)
\(432\) 0 0
\(433\) 25.0000 1.20142 0.600712 − 0.799466i \(-0.294884\pi\)
0.600712 + 0.799466i \(0.294884\pi\)
\(434\) 0 0
\(435\) 12.0000 0.575356
\(436\) 0 0
\(437\) 7.00000 0.334855
\(438\) 0 0
\(439\) −12.0000 −0.572729 −0.286364 − 0.958121i \(-0.592447\pi\)
−0.286364 + 0.958121i \(0.592447\pi\)
\(440\) 0 0
\(441\) 6.00000 0.285714
\(442\) 0 0
\(443\) 1.00000 0.0475114 0.0237557 − 0.999718i \(-0.492438\pi\)
0.0237557 + 0.999718i \(0.492438\pi\)
\(444\) 0 0
\(445\) 39.0000 1.84878
\(446\) 0 0
\(447\) −10.0000 −0.472984
\(448\) 0 0
\(449\) −9.00000 −0.424736 −0.212368 − 0.977190i \(-0.568118\pi\)
−0.212368 + 0.977190i \(0.568118\pi\)
\(450\) 0 0
\(451\) 4.00000 0.188353
\(452\) 0 0
\(453\) 6.00000 0.281905
\(454\) 0 0
\(455\) 24.0000 1.12514
\(456\) 0 0
\(457\) −16.0000 −0.748448 −0.374224 − 0.927338i \(-0.622091\pi\)
−0.374224 + 0.927338i \(0.622091\pi\)
\(458\) 0 0
\(459\) −10.0000 −0.466760
\(460\) 0 0
\(461\) 36.0000 1.67669 0.838344 − 0.545142i \(-0.183524\pi\)
0.838344 + 0.545142i \(0.183524\pi\)
\(462\) 0 0
\(463\) −43.0000 −1.99838 −0.999190 − 0.0402476i \(-0.987185\pi\)
−0.999190 + 0.0402476i \(0.987185\pi\)
\(464\) 0 0
\(465\) 9.00000 0.417365
\(466\) 0 0
\(467\) −39.0000 −1.80470 −0.902352 − 0.430999i \(-0.858161\pi\)
−0.902352 + 0.430999i \(0.858161\pi\)
\(468\) 0 0
\(469\) 14.0000 0.646460
\(470\) 0 0
\(471\) −5.00000 −0.230388
\(472\) 0 0
\(473\) 2.00000 0.0919601
\(474\) 0 0
\(475\) 4.00000 0.183533
\(476\) 0 0
\(477\) 28.0000 1.28203
\(478\) 0 0
\(479\) −12.0000 −0.548294 −0.274147 − 0.961688i \(-0.588395\pi\)
−0.274147 + 0.961688i \(0.588395\pi\)
\(480\) 0 0
\(481\) −20.0000 −0.911922
\(482\) 0 0
\(483\) 14.0000 0.637022
\(484\) 0 0
\(485\) 33.0000 1.49845
\(486\) 0 0
\(487\) 11.0000 0.498458 0.249229 − 0.968445i \(-0.419823\pi\)
0.249229 + 0.968445i \(0.419823\pi\)
\(488\) 0 0
\(489\) 20.0000 0.904431
\(490\) 0 0
\(491\) 36.0000 1.62466 0.812329 − 0.583200i \(-0.198200\pi\)
0.812329 + 0.583200i \(0.198200\pi\)
\(492\) 0 0
\(493\) −8.00000 −0.360302
\(494\) 0 0
\(495\) 6.00000 0.269680
\(496\) 0 0
\(497\) −18.0000 −0.807410
\(498\) 0 0
\(499\) −12.0000 −0.537194 −0.268597 − 0.963253i \(-0.586560\pi\)
−0.268597 + 0.963253i \(0.586560\pi\)
\(500\) 0 0
\(501\) −12.0000 −0.536120
\(502\) 0 0
\(503\) 22.0000 0.980932 0.490466 − 0.871460i \(-0.336827\pi\)
0.490466 + 0.871460i \(0.336827\pi\)
\(504\) 0 0
\(505\) 42.0000 1.86898
\(506\) 0 0
\(507\) −3.00000 −0.133235
\(508\) 0 0
\(509\) 39.0000 1.72864 0.864322 − 0.502938i \(-0.167748\pi\)
0.864322 + 0.502938i \(0.167748\pi\)
\(510\) 0 0
\(511\) −16.0000 −0.707798
\(512\) 0 0
\(513\) 5.00000 0.220755
\(514\) 0 0
\(515\) −48.0000 −2.11513
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) 25.0000 1.09527 0.547635 − 0.836717i \(-0.315528\pi\)
0.547635 + 0.836717i \(0.315528\pi\)
\(522\) 0 0
\(523\) 20.0000 0.874539 0.437269 − 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 0 0
\(525\) 8.00000 0.349149
\(526\) 0 0
\(527\) −6.00000 −0.261364
\(528\) 0 0
\(529\) 26.0000 1.13043
\(530\) 0 0
\(531\) −10.0000 −0.433963
\(532\) 0 0
\(533\) 16.0000 0.693037
\(534\) 0 0
\(535\) −18.0000 −0.778208
\(536\) 0 0
\(537\) 23.0000 0.992523
\(538\) 0 0
\(539\) −3.00000 −0.129219
\(540\) 0 0
\(541\) −8.00000 −0.343947 −0.171973 − 0.985102i \(-0.555014\pi\)
−0.171973 + 0.985102i \(0.555014\pi\)
\(542\) 0 0
\(543\) 17.0000 0.729540
\(544\) 0 0
\(545\) −6.00000 −0.257012
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 − 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) 0 0
\(549\) 16.0000 0.682863
\(550\) 0 0
\(551\) 4.00000 0.170406
\(552\) 0 0
\(553\) 12.0000 0.510292
\(554\) 0 0
\(555\) −15.0000 −0.636715
\(556\) 0 0
\(557\) −14.0000 −0.593199 −0.296600 − 0.955002i \(-0.595853\pi\)
−0.296600 + 0.955002i \(0.595853\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) 2.00000 0.0844401
\(562\) 0 0
\(563\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) −39.0000 −1.64074
\(566\) 0 0
\(567\) −2.00000 −0.0839921
\(568\) 0 0
\(569\) −20.0000 −0.838444 −0.419222 − 0.907884i \(-0.637697\pi\)
−0.419222 + 0.907884i \(0.637697\pi\)
\(570\) 0 0
\(571\) −32.0000 −1.33916 −0.669579 − 0.742741i \(-0.733526\pi\)
−0.669579 + 0.742741i \(0.733526\pi\)
\(572\) 0 0
\(573\) −17.0000 −0.710185
\(574\) 0 0
\(575\) 28.0000 1.16768
\(576\) 0 0
\(577\) −23.0000 −0.957503 −0.478751 − 0.877951i \(-0.658910\pi\)
−0.478751 + 0.877951i \(0.658910\pi\)
\(578\) 0 0
\(579\) 12.0000 0.498703
\(580\) 0 0
\(581\) 4.00000 0.165948
\(582\) 0 0
\(583\) −14.0000 −0.579821
\(584\) 0 0
\(585\) 24.0000 0.992278
\(586\) 0 0
\(587\) 4.00000 0.165098 0.0825488 − 0.996587i \(-0.473694\pi\)
0.0825488 + 0.996587i \(0.473694\pi\)
\(588\) 0 0
\(589\) 3.00000 0.123613
\(590\) 0 0
\(591\) −18.0000 −0.740421
\(592\) 0 0
\(593\) 20.0000 0.821302 0.410651 − 0.911793i \(-0.365302\pi\)
0.410651 + 0.911793i \(0.365302\pi\)
\(594\) 0 0
\(595\) −12.0000 −0.491952
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 24.0000 0.980613 0.490307 − 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) 2.00000 0.0815817 0.0407909 − 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) 0 0
\(603\) 14.0000 0.570124
\(604\) 0 0
\(605\) −3.00000 −0.121967
\(606\) 0 0
\(607\) 22.0000 0.892952 0.446476 − 0.894795i \(-0.352679\pi\)
0.446476 + 0.894795i \(0.352679\pi\)
\(608\) 0 0
\(609\) 8.00000 0.324176
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 4.00000 0.161558 0.0807792 − 0.996732i \(-0.474259\pi\)
0.0807792 + 0.996732i \(0.474259\pi\)
\(614\) 0 0
\(615\) 12.0000 0.483887
\(616\) 0 0
\(617\) −6.00000 −0.241551 −0.120775 − 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 0 0
\(619\) 35.0000 1.40677 0.703384 − 0.710810i \(-0.251671\pi\)
0.703384 + 0.710810i \(0.251671\pi\)
\(620\) 0 0
\(621\) 35.0000 1.40450
\(622\) 0 0
\(623\) 26.0000 1.04167
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) −1.00000 −0.0399362
\(628\) 0 0
\(629\) 10.0000 0.398726
\(630\) 0 0
\(631\) −17.0000 −0.676759 −0.338380 − 0.941010i \(-0.609879\pi\)
−0.338380 + 0.941010i \(0.609879\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 24.0000 0.952411
\(636\) 0 0
\(637\) −12.0000 −0.475457
\(638\) 0 0
\(639\) −18.0000 −0.712069
\(640\) 0 0
\(641\) −21.0000 −0.829450 −0.414725 − 0.909947i \(-0.636122\pi\)
−0.414725 + 0.909947i \(0.636122\pi\)
\(642\) 0 0
\(643\) 33.0000 1.30139 0.650696 − 0.759338i \(-0.274477\pi\)
0.650696 + 0.759338i \(0.274477\pi\)
\(644\) 0 0
\(645\) 6.00000 0.236250
\(646\) 0 0
\(647\) 9.00000 0.353827 0.176913 − 0.984226i \(-0.443389\pi\)
0.176913 + 0.984226i \(0.443389\pi\)
\(648\) 0 0
\(649\) 5.00000 0.196267
\(650\) 0 0
\(651\) 6.00000 0.235159
\(652\) 0 0
\(653\) −29.0000 −1.13486 −0.567429 − 0.823422i \(-0.692062\pi\)
−0.567429 + 0.823422i \(0.692062\pi\)
\(654\) 0 0
\(655\) 18.0000 0.703318
\(656\) 0 0
\(657\) −16.0000 −0.624219
\(658\) 0 0
\(659\) −30.0000 −1.16863 −0.584317 − 0.811525i \(-0.698638\pi\)
−0.584317 + 0.811525i \(0.698638\pi\)
\(660\) 0 0
\(661\) −43.0000 −1.67251 −0.836253 − 0.548344i \(-0.815259\pi\)
−0.836253 + 0.548344i \(0.815259\pi\)
\(662\) 0 0
\(663\) 8.00000 0.310694
\(664\) 0 0
\(665\) 6.00000 0.232670
\(666\) 0 0
\(667\) 28.0000 1.08416
\(668\) 0 0
\(669\) 9.00000 0.347960
\(670\) 0 0
\(671\) −8.00000 −0.308837
\(672\) 0 0
\(673\) −34.0000 −1.31060 −0.655302 − 0.755367i \(-0.727459\pi\)
−0.655302 + 0.755367i \(0.727459\pi\)
\(674\) 0 0
\(675\) 20.0000 0.769800
\(676\) 0 0
\(677\) 18.0000 0.691796 0.345898 − 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) 0 0
\(679\) 22.0000 0.844283
\(680\) 0 0
\(681\) 22.0000 0.843042
\(682\) 0 0
\(683\) 32.0000 1.22445 0.612223 − 0.790685i \(-0.290275\pi\)
0.612223 + 0.790685i \(0.290275\pi\)
\(684\) 0 0
\(685\) 45.0000 1.71936
\(686\) 0 0
\(687\) −11.0000 −0.419676
\(688\) 0 0
\(689\) −56.0000 −2.13343
\(690\) 0 0
\(691\) 5.00000 0.190209 0.0951045 − 0.995467i \(-0.469681\pi\)
0.0951045 + 0.995467i \(0.469681\pi\)
\(692\) 0 0
\(693\) 4.00000 0.151947
\(694\) 0 0
\(695\) 42.0000 1.59315
\(696\) 0 0
\(697\) −8.00000 −0.303022
\(698\) 0 0
\(699\) −20.0000 −0.756469
\(700\) 0 0
\(701\) −50.0000 −1.88847 −0.944237 − 0.329267i \(-0.893198\pi\)
−0.944237 + 0.329267i \(0.893198\pi\)
\(702\) 0 0
\(703\) −5.00000 −0.188579
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 28.0000 1.05305
\(708\) 0 0
\(709\) 35.0000 1.31445 0.657226 − 0.753693i \(-0.271730\pi\)
0.657226 + 0.753693i \(0.271730\pi\)
\(710\) 0 0
\(711\) 12.0000 0.450035
\(712\) 0 0
\(713\) 21.0000 0.786456
\(714\) 0 0
\(715\) −12.0000 −0.448775
\(716\) 0 0
\(717\) −10.0000 −0.373457
\(718\) 0 0
\(719\) −1.00000 −0.0372937 −0.0186469 − 0.999826i \(-0.505936\pi\)
−0.0186469 + 0.999826i \(0.505936\pi\)
\(720\) 0 0
\(721\) −32.0000 −1.19174
\(722\) 0 0
\(723\) 8.00000 0.297523
\(724\) 0 0
\(725\) 16.0000 0.594225
\(726\) 0 0
\(727\) −13.0000 −0.482143 −0.241072 − 0.970507i \(-0.577499\pi\)
−0.241072 + 0.970507i \(0.577499\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −4.00000 −0.147945
\(732\) 0 0
\(733\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) 0 0
\(735\) −9.00000 −0.331970
\(736\) 0 0
\(737\) −7.00000 −0.257848
\(738\) 0 0
\(739\) −18.0000 −0.662141 −0.331070 − 0.943606i \(-0.607410\pi\)
−0.331070 + 0.943606i \(0.607410\pi\)
\(740\) 0 0
\(741\) −4.00000 −0.146944
\(742\) 0 0
\(743\) −24.0000 −0.880475 −0.440237 − 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) −30.0000 −1.09911
\(746\) 0 0
\(747\) 4.00000 0.146352
\(748\) 0 0
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) 21.0000 0.766301 0.383150 − 0.923686i \(-0.374839\pi\)
0.383150 + 0.923686i \(0.374839\pi\)
\(752\) 0 0
\(753\) −5.00000 −0.182210
\(754\) 0 0
\(755\) 18.0000 0.655087
\(756\) 0 0
\(757\) −38.0000 −1.38113 −0.690567 − 0.723269i \(-0.742639\pi\)
−0.690567 + 0.723269i \(0.742639\pi\)
\(758\) 0 0
\(759\) −7.00000 −0.254084
\(760\) 0 0
\(761\) −40.0000 −1.45000 −0.724999 − 0.688749i \(-0.758160\pi\)
−0.724999 + 0.688749i \(0.758160\pi\)
\(762\) 0 0
\(763\) −4.00000 −0.144810
\(764\) 0 0
\(765\) −12.0000 −0.433861
\(766\) 0 0
\(767\) 20.0000 0.722158
\(768\) 0 0
\(769\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(770\) 0 0
\(771\) −14.0000 −0.504198
\(772\) 0 0
\(773\) 26.0000 0.935155 0.467578 − 0.883952i \(-0.345127\pi\)
0.467578 + 0.883952i \(0.345127\pi\)
\(774\) 0 0
\(775\) 12.0000 0.431053
\(776\) 0 0
\(777\) −10.0000 −0.358748
\(778\) 0 0
\(779\) 4.00000 0.143315
\(780\) 0 0
\(781\) 9.00000 0.322045
\(782\) 0 0
\(783\) 20.0000 0.714742
\(784\) 0 0
\(785\) −15.0000 −0.535373
\(786\) 0 0
\(787\) 32.0000 1.14068 0.570338 − 0.821410i \(-0.306812\pi\)
0.570338 + 0.821410i \(0.306812\pi\)
\(788\) 0 0
\(789\) −26.0000 −0.925625
\(790\) 0 0
\(791\) −26.0000 −0.924454
\(792\) 0 0
\(793\) −32.0000 −1.13635
\(794\) 0 0
\(795\) −42.0000 −1.48959
\(796\) 0 0
\(797\) 53.0000 1.87736 0.938678 − 0.344795i \(-0.112051\pi\)
0.938678 + 0.344795i \(0.112051\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 26.0000 0.918665
\(802\) 0 0
\(803\) 8.00000 0.282314
\(804\) 0 0
\(805\) 42.0000 1.48031
\(806\) 0 0
\(807\) 22.0000 0.774437
\(808\) 0 0
\(809\) 12.0000 0.421898 0.210949 − 0.977497i \(-0.432345\pi\)
0.210949 + 0.977497i \(0.432345\pi\)
\(810\) 0 0
\(811\) 22.0000 0.772524 0.386262 − 0.922389i \(-0.373766\pi\)
0.386262 + 0.922389i \(0.373766\pi\)
\(812\) 0 0
\(813\) −24.0000 −0.841717
\(814\) 0 0
\(815\) 60.0000 2.10171
\(816\) 0 0
\(817\) 2.00000 0.0699711
\(818\) 0 0
\(819\) 16.0000 0.559085
\(820\) 0 0
\(821\) −54.0000 −1.88461 −0.942306 − 0.334751i \(-0.891348\pi\)
−0.942306 + 0.334751i \(0.891348\pi\)
\(822\) 0 0
\(823\) −25.0000 −0.871445 −0.435723 − 0.900081i \(-0.643507\pi\)
−0.435723 + 0.900081i \(0.643507\pi\)
\(824\) 0 0
\(825\) −4.00000 −0.139262
\(826\) 0 0
\(827\) 12.0000 0.417281 0.208640 − 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) 0 0
\(829\) 9.00000 0.312583 0.156291 − 0.987711i \(-0.450046\pi\)
0.156291 + 0.987711i \(0.450046\pi\)
\(830\) 0 0
\(831\) −22.0000 −0.763172
\(832\) 0 0
\(833\) 6.00000 0.207888
\(834\) 0 0
\(835\) −36.0000 −1.24583
\(836\) 0 0
\(837\) 15.0000 0.518476
\(838\) 0 0
\(839\) 23.0000 0.794048 0.397024 − 0.917808i \(-0.370043\pi\)
0.397024 + 0.917808i \(0.370043\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 0 0
\(843\) 2.00000 0.0688837
\(844\) 0 0
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) −2.00000 −0.0687208
\(848\) 0 0
\(849\) 16.0000 0.549119
\(850\) 0 0
\(851\) −35.0000 −1.19978
\(852\) 0 0
\(853\) −2.00000 −0.0684787 −0.0342393 − 0.999414i \(-0.510901\pi\)
−0.0342393 + 0.999414i \(0.510901\pi\)
\(854\) 0 0
\(855\) 6.00000 0.205196
\(856\) 0 0
\(857\) 12.0000 0.409912 0.204956 − 0.978771i \(-0.434295\pi\)
0.204956 + 0.978771i \(0.434295\pi\)
\(858\) 0 0
\(859\) −19.0000 −0.648272 −0.324136 − 0.946011i \(-0.605073\pi\)
−0.324136 + 0.946011i \(0.605073\pi\)
\(860\) 0 0
\(861\) 8.00000 0.272639
\(862\) 0 0
\(863\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) −18.0000 −0.612018
\(866\) 0 0
\(867\) 13.0000 0.441503
\(868\) 0 0
\(869\) −6.00000 −0.203536
\(870\) 0 0
\(871\) −28.0000 −0.948744
\(872\) 0 0
\(873\) 22.0000 0.744587
\(874\) 0 0
\(875\) −6.00000 −0.202837
\(876\) 0 0
\(877\) −40.0000 −1.35070 −0.675352 − 0.737496i \(-0.736008\pi\)
−0.675352 + 0.737496i \(0.736008\pi\)
\(878\) 0 0
\(879\) 8.00000 0.269833
\(880\) 0 0
\(881\) 21.0000 0.707508 0.353754 − 0.935339i \(-0.384905\pi\)
0.353754 + 0.935339i \(0.384905\pi\)
\(882\) 0 0
\(883\) 36.0000 1.21150 0.605748 − 0.795656i \(-0.292874\pi\)
0.605748 + 0.795656i \(0.292874\pi\)
\(884\) 0 0
\(885\) 15.0000 0.504219
\(886\) 0 0
\(887\) 42.0000 1.41022 0.705111 − 0.709097i \(-0.250897\pi\)
0.705111 + 0.709097i \(0.250897\pi\)
\(888\) 0 0
\(889\) 16.0000 0.536623
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 69.0000 2.30642
\(896\) 0 0
\(897\) −28.0000 −0.934893
\(898\) 0 0
\(899\) 12.0000 0.400222
\(900\) 0 0
\(901\) 28.0000 0.932815
\(902\) 0 0
\(903\) 4.00000 0.133112
\(904\) 0 0
\(905\) 51.0000 1.69530
\(906\) 0 0
\(907\) −20.0000 −0.664089 −0.332045 − 0.943264i \(-0.607738\pi\)
−0.332045 + 0.943264i \(0.607738\pi\)
\(908\) 0 0
\(909\) 28.0000 0.928701
\(910\) 0 0
\(911\) 36.0000 1.19273 0.596367 − 0.802712i \(-0.296610\pi\)
0.596367 + 0.802712i \(0.296610\pi\)
\(912\) 0 0
\(913\) −2.00000 −0.0661903
\(914\) 0 0
\(915\) −24.0000 −0.793416
\(916\) 0 0
\(917\) 12.0000 0.396275
\(918\) 0 0
\(919\) −10.0000 −0.329870 −0.164935 − 0.986304i \(-0.552741\pi\)
−0.164935 + 0.986304i \(0.552741\pi\)
\(920\) 0 0
\(921\) 16.0000 0.527218
\(922\) 0 0
\(923\) 36.0000 1.18495
\(924\) 0 0
\(925\) −20.0000 −0.657596
\(926\) 0 0
\(927\) −32.0000 −1.05102
\(928\) 0 0
\(929\) −38.0000 −1.24674 −0.623370 − 0.781927i \(-0.714237\pi\)
−0.623370 + 0.781927i \(0.714237\pi\)
\(930\) 0 0
\(931\) −3.00000 −0.0983210
\(932\) 0 0
\(933\) 4.00000 0.130954
\(934\) 0 0
\(935\) 6.00000 0.196221
\(936\) 0 0
\(937\) 24.0000 0.784046 0.392023 − 0.919955i \(-0.371775\pi\)
0.392023 + 0.919955i \(0.371775\pi\)
\(938\) 0 0
\(939\) 9.00000 0.293704
\(940\) 0 0
\(941\) −34.0000 −1.10837 −0.554184 − 0.832394i \(-0.686970\pi\)
−0.554184 + 0.832394i \(0.686970\pi\)
\(942\) 0 0
\(943\) 28.0000 0.911805
\(944\) 0 0
\(945\) 30.0000 0.975900
\(946\) 0 0
\(947\) 57.0000 1.85225 0.926126 − 0.377215i \(-0.123118\pi\)
0.926126 + 0.377215i \(0.123118\pi\)
\(948\) 0 0
\(949\) 32.0000 1.03876
\(950\) 0 0
\(951\) 27.0000 0.875535
\(952\) 0 0
\(953\) −38.0000 −1.23094 −0.615470 − 0.788160i \(-0.711034\pi\)
−0.615470 + 0.788160i \(0.711034\pi\)
\(954\) 0 0
\(955\) −51.0000 −1.65032
\(956\) 0 0
\(957\) −4.00000 −0.129302
\(958\) 0 0
\(959\) 30.0000 0.968751
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) 0 0
\(963\) −12.0000 −0.386695
\(964\) 0 0
\(965\) 36.0000 1.15888
\(966\) 0 0
\(967\) −4.00000 −0.128631 −0.0643157 − 0.997930i \(-0.520486\pi\)
−0.0643157 + 0.997930i \(0.520486\pi\)
\(968\) 0 0
\(969\) 2.00000 0.0642493
\(970\) 0 0
\(971\) 15.0000 0.481373 0.240686 − 0.970603i \(-0.422627\pi\)
0.240686 + 0.970603i \(0.422627\pi\)
\(972\) 0 0
\(973\) 28.0000 0.897639
\(974\) 0 0
\(975\) −16.0000 −0.512410
\(976\) 0 0
\(977\) 33.0000 1.05576 0.527882 − 0.849318i \(-0.322986\pi\)
0.527882 + 0.849318i \(0.322986\pi\)
\(978\) 0 0
\(979\) −13.0000 −0.415482
\(980\) 0 0
\(981\) −4.00000 −0.127710
\(982\) 0 0
\(983\) 51.0000 1.62665 0.813324 − 0.581811i \(-0.197656\pi\)
0.813324 + 0.581811i \(0.197656\pi\)
\(984\) 0 0
\(985\) −54.0000 −1.72058
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 14.0000 0.445174
\(990\) 0 0
\(991\) −32.0000 −1.01651 −0.508257 − 0.861206i \(-0.669710\pi\)
−0.508257 + 0.861206i \(0.669710\pi\)
\(992\) 0 0
\(993\) 9.00000 0.285606
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 6.00000 0.190022 0.0950110 − 0.995476i \(-0.469711\pi\)
0.0950110 + 0.995476i \(0.469711\pi\)
\(998\) 0 0
\(999\) −25.0000 −0.790965
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 3344.2.a.c.1.1 1
4.3 odd 2 1672.2.a.b.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1672.2.a.b.1.1 ✓ 1 4.3 odd 2
3344.2.a.c.1.1 1 1.1 even 1 trivial