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Properties

Label 2070.2.a.d.1.1
Level $2070$
Weight $2$
Character 2070.1
Self dual yes
Analytic conductor $16.529$
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] = [2070,2,Mod(1,2070)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2070, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2070.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2070.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: \(16.5290332184\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2070.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -4.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -4.00000 q^{7} -1.00000 q^{8} -1.00000 q^{10} +4.00000 q^{14} +1.00000 q^{16} -4.00000 q^{17} +8.00000 q^{19} +1.00000 q^{20} +1.00000 q^{23} +1.00000 q^{25} -4.00000 q^{28} +2.00000 q^{29} +4.00000 q^{31} -1.00000 q^{32} +4.00000 q^{34} -4.00000 q^{35} -2.00000 q^{37} -8.00000 q^{38} -1.00000 q^{40} -8.00000 q^{41} -4.00000 q^{43} -1.00000 q^{46} -8.00000 q^{47} +9.00000 q^{49} -1.00000 q^{50} -6.00000 q^{53} +4.00000 q^{56} -2.00000 q^{58} -10.0000 q^{59} -8.00000 q^{61} -4.00000 q^{62} +1.00000 q^{64} +12.0000 q^{67} -4.00000 q^{68} +4.00000 q^{70} -16.0000 q^{71} -2.00000 q^{73} +2.00000 q^{74} +8.00000 q^{76} -6.00000 q^{79} +1.00000 q^{80} +8.00000 q^{82} +2.00000 q^{83} -4.00000 q^{85} +4.00000 q^{86} -6.00000 q^{89} +1.00000 q^{92} +8.00000 q^{94} +8.00000 q^{95} -10.0000 q^{97} -9.00000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −4.00000 −1.51186 −0.755929 − 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 4.00000 1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −4.00000 −0.970143 −0.485071 − 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 0 0
\(19\) 8.00000 1.83533 0.917663 − 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) −4.00000 −0.755929
\(29\) 2.00000 0.371391 0.185695 − 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 − 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 4.00000 0.685994
\(35\) −4.00000 −0.676123
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 − 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) −8.00000 −1.29777
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) −8.00000 −1.24939 −0.624695 − 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) −8.00000 −1.16692 −0.583460 − 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 4.00000 0.534522
\(57\) 0 0
\(58\) −2.00000 −0.262613
\(59\) −10.0000 −1.30189 −0.650945 − 0.759125i \(-0.725627\pi\)
−0.650945 + 0.759125i \(0.725627\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 − 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) −4.00000 −0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 12.0000 1.46603 0.733017 − 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) −4.00000 −0.485071
\(69\) 0 0
\(70\) 4.00000 0.478091
\(71\) −16.0000 −1.89885 −0.949425 − 0.313993i \(-0.898333\pi\)
−0.949425 + 0.313993i \(0.898333\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 − 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) 8.00000 0.917663
\(77\) 0 0
\(78\) 0 0
\(79\) −6.00000 −0.675053 −0.337526 − 0.941316i \(-0.609590\pi\)
−0.337526 + 0.941316i \(0.609590\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) 8.00000 0.883452
\(83\) 2.00000 0.219529 0.109764 − 0.993958i \(-0.464990\pi\)
0.109764 + 0.993958i \(0.464990\pi\)
\(84\) 0 0
\(85\) −4.00000 −0.433861
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) 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
\(92\) 1.00000 0.104257
\(93\) 0 0
\(94\) 8.00000 0.825137
\(95\) 8.00000 0.820783
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 − 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) −9.00000 −0.909137
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 2.00000 0.199007 0.0995037 − 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 − 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) 2.00000 0.193347 0.0966736 − 0.995316i \(-0.469180\pi\)
0.0966736 + 0.995316i \(0.469180\pi\)
\(108\) 0 0
\(109\) −4.00000 −0.383131 −0.191565 − 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −4.00000 −0.377964
\(113\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 1.00000 0.0932505
\(116\) 2.00000 0.185695
\(117\) 0 0
\(118\) 10.0000 0.920575
\(119\) 16.0000 1.46672
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 8.00000 0.724286
\(123\) 0 0
\(124\) 4.00000 0.359211
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −10.0000 −0.887357 −0.443678 − 0.896186i \(-0.646327\pi\)
−0.443678 + 0.896186i \(0.646327\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) 6.00000 0.524222 0.262111 − 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) 0 0
\(133\) −32.0000 −2.77475
\(134\) −12.0000 −1.03664
\(135\) 0 0
\(136\) 4.00000 0.342997
\(137\) −12.0000 −1.02523 −0.512615 − 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 0 0
\(139\) 20.0000 1.69638 0.848189 − 0.529694i \(-0.177693\pi\)
0.848189 + 0.529694i \(0.177693\pi\)
\(140\) −4.00000 −0.338062
\(141\) 0 0
\(142\) 16.0000 1.34269
\(143\) 0 0
\(144\) 0 0
\(145\) 2.00000 0.166091
\(146\) 2.00000 0.165521
\(147\) 0 0
\(148\) −2.00000 −0.164399
\(149\) 6.00000 0.491539 0.245770 − 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) 20.0000 1.62758 0.813788 − 0.581161i \(-0.197401\pi\)
0.813788 + 0.581161i \(0.197401\pi\)
\(152\) −8.00000 −0.648886
\(153\) 0 0
\(154\) 0 0
\(155\) 4.00000 0.321288
\(156\) 0 0
\(157\) 10.0000 0.798087 0.399043 − 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) 6.00000 0.477334
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) −4.00000 −0.315244
\(162\) 0 0
\(163\) −8.00000 −0.626608 −0.313304 − 0.949653i \(-0.601436\pi\)
−0.313304 + 0.949653i \(0.601436\pi\)
\(164\) −8.00000 −0.624695
\(165\) 0 0
\(166\) −2.00000 −0.155230
\(167\) 16.0000 1.23812 0.619059 − 0.785345i \(-0.287514\pi\)
0.619059 + 0.785345i \(0.287514\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 4.00000 0.306786
\(171\) 0 0
\(172\) −4.00000 −0.304997
\(173\) −2.00000 −0.152057 −0.0760286 − 0.997106i \(-0.524224\pi\)
−0.0760286 + 0.997106i \(0.524224\pi\)
\(174\) 0 0
\(175\) −4.00000 −0.302372
\(176\) 0 0
\(177\) 0 0
\(178\) 6.00000 0.449719
\(179\) −6.00000 −0.448461 −0.224231 − 0.974536i \(-0.571987\pi\)
−0.224231 + 0.974536i \(0.571987\pi\)
\(180\) 0 0
\(181\) −4.00000 −0.297318 −0.148659 − 0.988889i \(-0.547496\pi\)
−0.148659 + 0.988889i \(0.547496\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −1.00000 −0.0737210
\(185\) −2.00000 −0.147043
\(186\) 0 0
\(187\) 0 0
\(188\) −8.00000 −0.583460
\(189\) 0 0
\(190\) −8.00000 −0.580381
\(191\) −24.0000 −1.73658 −0.868290 − 0.496058i \(-0.834780\pi\)
−0.868290 + 0.496058i \(0.834780\pi\)
\(192\) 0 0
\(193\) −18.0000 −1.29567 −0.647834 − 0.761781i \(-0.724325\pi\)
−0.647834 + 0.761781i \(0.724325\pi\)
\(194\) 10.0000 0.717958
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) −2.00000 −0.142494 −0.0712470 − 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 0 0
\(199\) −18.0000 −1.27599 −0.637993 − 0.770042i \(-0.720235\pi\)
−0.637993 + 0.770042i \(0.720235\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) −2.00000 −0.140720
\(203\) −8.00000 −0.561490
\(204\) 0 0
\(205\) −8.00000 −0.558744
\(206\) 4.00000 0.278693
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 − 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) −2.00000 −0.136717
\(215\) −4.00000 −0.272798
\(216\) 0 0
\(217\) −16.0000 −1.08615
\(218\) 4.00000 0.270914
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 6.00000 0.401790 0.200895 − 0.979613i \(-0.435615\pi\)
0.200895 + 0.979613i \(0.435615\pi\)
\(224\) 4.00000 0.267261
\(225\) 0 0
\(226\) 0 0
\(227\) 22.0000 1.46019 0.730096 − 0.683345i \(-0.239475\pi\)
0.730096 + 0.683345i \(0.239475\pi\)
\(228\) 0 0
\(229\) −24.0000 −1.58596 −0.792982 − 0.609245i \(-0.791473\pi\)
−0.792982 + 0.609245i \(0.791473\pi\)
\(230\) −1.00000 −0.0659380
\(231\) 0 0
\(232\) −2.00000 −0.131306
\(233\) −2.00000 −0.131024 −0.0655122 − 0.997852i \(-0.520868\pi\)
−0.0655122 + 0.997852i \(0.520868\pi\)
\(234\) 0 0
\(235\) −8.00000 −0.521862
\(236\) −10.0000 −0.650945
\(237\) 0 0
\(238\) −16.0000 −1.03713
\(239\) 12.0000 0.776215 0.388108 − 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 − 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 11.0000 0.707107
\(243\) 0 0
\(244\) −8.00000 −0.512148
\(245\) 9.00000 0.574989
\(246\) 0 0
\(247\) 0 0
\(248\) −4.00000 −0.254000
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) −20.0000 −1.26239 −0.631194 − 0.775625i \(-0.717435\pi\)
−0.631194 + 0.775625i \(0.717435\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 10.0000 0.627456
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −22.0000 −1.37232 −0.686161 − 0.727450i \(-0.740706\pi\)
−0.686161 + 0.727450i \(0.740706\pi\)
\(258\) 0 0
\(259\) 8.00000 0.497096
\(260\) 0 0
\(261\) 0 0
\(262\) −6.00000 −0.370681
\(263\) −4.00000 −0.246651 −0.123325 − 0.992366i \(-0.539356\pi\)
−0.123325 + 0.992366i \(0.539356\pi\)
\(264\) 0 0
\(265\) −6.00000 −0.368577
\(266\) 32.0000 1.96205
\(267\) 0 0
\(268\) 12.0000 0.733017
\(269\) 18.0000 1.09748 0.548740 − 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 0 0
\(271\) −20.0000 −1.21491 −0.607457 − 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) −4.00000 −0.242536
\(273\) 0 0
\(274\) 12.0000 0.724947
\(275\) 0 0
\(276\) 0 0
\(277\) −8.00000 −0.480673 −0.240337 − 0.970690i \(-0.577258\pi\)
−0.240337 + 0.970690i \(0.577258\pi\)
\(278\) −20.0000 −1.19952
\(279\) 0 0
\(280\) 4.00000 0.239046
\(281\) −6.00000 −0.357930 −0.178965 − 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) 12.0000 0.713326 0.356663 − 0.934233i \(-0.383914\pi\)
0.356663 + 0.934233i \(0.383914\pi\)
\(284\) −16.0000 −0.949425
\(285\) 0 0
\(286\) 0 0
\(287\) 32.0000 1.88890
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) −2.00000 −0.117444
\(291\) 0 0
\(292\) −2.00000 −0.117041
\(293\) −6.00000 −0.350524 −0.175262 − 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 0 0
\(295\) −10.0000 −0.582223
\(296\) 2.00000 0.116248
\(297\) 0 0
\(298\) −6.00000 −0.347571
\(299\) 0 0
\(300\) 0 0
\(301\) 16.0000 0.922225
\(302\) −20.0000 −1.15087
\(303\) 0 0
\(304\) 8.00000 0.458831
\(305\) −8.00000 −0.458079
\(306\) 0 0
\(307\) 12.0000 0.684876 0.342438 − 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −4.00000 −0.227185
\(311\) −32.0000 −1.81455 −0.907277 − 0.420534i \(-0.861843\pi\)
−0.907277 + 0.420534i \(0.861843\pi\)
\(312\) 0 0
\(313\) −14.0000 −0.791327 −0.395663 − 0.918396i \(-0.629485\pi\)
−0.395663 + 0.918396i \(0.629485\pi\)
\(314\) −10.0000 −0.564333
\(315\) 0 0
\(316\) −6.00000 −0.337526
\(317\) −30.0000 −1.68497 −0.842484 − 0.538721i \(-0.818908\pi\)
−0.842484 + 0.538721i \(0.818908\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) 4.00000 0.222911
\(323\) −32.0000 −1.78053
\(324\) 0 0
\(325\) 0 0
\(326\) 8.00000 0.443079
\(327\) 0 0
\(328\) 8.00000 0.441726
\(329\) 32.0000 1.76422
\(330\) 0 0
\(331\) 28.0000 1.53902 0.769510 − 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) 2.00000 0.109764
\(333\) 0 0
\(334\) −16.0000 −0.875481
\(335\) 12.0000 0.655630
\(336\) 0 0
\(337\) 18.0000 0.980522 0.490261 − 0.871576i \(-0.336901\pi\)
0.490261 + 0.871576i \(0.336901\pi\)
\(338\) 13.0000 0.707107
\(339\) 0 0
\(340\) −4.00000 −0.216930
\(341\) 0 0
\(342\) 0 0
\(343\) −8.00000 −0.431959
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) 2.00000 0.107521
\(347\) 28.0000 1.50312 0.751559 − 0.659665i \(-0.229302\pi\)
0.751559 + 0.659665i \(0.229302\pi\)
\(348\) 0 0
\(349\) 14.0000 0.749403 0.374701 − 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 4.00000 0.213809
\(351\) 0 0
\(352\) 0 0
\(353\) 18.0000 0.958043 0.479022 − 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) 0 0
\(355\) −16.0000 −0.849192
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) 6.00000 0.317110
\(359\) 32.0000 1.68890 0.844448 − 0.535638i \(-0.179929\pi\)
0.844448 + 0.535638i \(0.179929\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) 4.00000 0.210235
\(363\) 0 0
\(364\) 0 0
\(365\) −2.00000 −0.104685
\(366\) 0 0
\(367\) −32.0000 −1.67039 −0.835193 − 0.549957i \(-0.814644\pi\)
−0.835193 + 0.549957i \(0.814644\pi\)
\(368\) 1.00000 0.0521286
\(369\) 0 0
\(370\) 2.00000 0.103975
\(371\) 24.0000 1.24602
\(372\) 0 0
\(373\) −6.00000 −0.310668 −0.155334 − 0.987862i \(-0.549645\pi\)
−0.155334 + 0.987862i \(0.549645\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 8.00000 0.412568
\(377\) 0 0
\(378\) 0 0
\(379\) −4.00000 −0.205466 −0.102733 − 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 8.00000 0.410391
\(381\) 0 0
\(382\) 24.0000 1.22795
\(383\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 18.0000 0.916176
\(387\) 0 0
\(388\) −10.0000 −0.507673
\(389\) 18.0000 0.912636 0.456318 − 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 0 0
\(391\) −4.00000 −0.202289
\(392\) −9.00000 −0.454569
\(393\) 0 0
\(394\) 2.00000 0.100759
\(395\) −6.00000 −0.301893
\(396\) 0 0
\(397\) 36.0000 1.80679 0.903394 − 0.428811i \(-0.141067\pi\)
0.903394 + 0.428811i \(0.141067\pi\)
\(398\) 18.0000 0.902258
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 2.00000 0.0998752 0.0499376 − 0.998752i \(-0.484098\pi\)
0.0499376 + 0.998752i \(0.484098\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 2.00000 0.0995037
\(405\) 0 0
\(406\) 8.00000 0.397033
\(407\) 0 0
\(408\) 0 0
\(409\) −18.0000 −0.890043 −0.445021 − 0.895520i \(-0.646804\pi\)
−0.445021 + 0.895520i \(0.646804\pi\)
\(410\) 8.00000 0.395092
\(411\) 0 0
\(412\) −4.00000 −0.197066
\(413\) 40.0000 1.96827
\(414\) 0 0
\(415\) 2.00000 0.0981761
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 24.0000 1.17248 0.586238 − 0.810139i \(-0.300608\pi\)
0.586238 + 0.810139i \(0.300608\pi\)
\(420\) 0 0
\(421\) 4.00000 0.194948 0.0974740 − 0.995238i \(-0.468924\pi\)
0.0974740 + 0.995238i \(0.468924\pi\)
\(422\) −20.0000 −0.973585
\(423\) 0 0
\(424\) 6.00000 0.291386
\(425\) −4.00000 −0.194029
\(426\) 0 0
\(427\) 32.0000 1.54859
\(428\) 2.00000 0.0966736
\(429\) 0 0
\(430\) 4.00000 0.192897
\(431\) −8.00000 −0.385346 −0.192673 − 0.981263i \(-0.561716\pi\)
−0.192673 + 0.981263i \(0.561716\pi\)
\(432\) 0 0
\(433\) −2.00000 −0.0961139 −0.0480569 − 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 16.0000 0.768025
\(435\) 0 0
\(436\) −4.00000 −0.191565
\(437\) 8.00000 0.382692
\(438\) 0 0
\(439\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 24.0000 1.14027 0.570137 − 0.821549i \(-0.306890\pi\)
0.570137 + 0.821549i \(0.306890\pi\)
\(444\) 0 0
\(445\) −6.00000 −0.284427
\(446\) −6.00000 −0.284108
\(447\) 0 0
\(448\) −4.00000 −0.188982
\(449\) −20.0000 −0.943858 −0.471929 − 0.881636i \(-0.656442\pi\)
−0.471929 + 0.881636i \(0.656442\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) −22.0000 −1.03251
\(455\) 0 0
\(456\) 0 0
\(457\) 10.0000 0.467780 0.233890 − 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) 24.0000 1.12145
\(459\) 0 0
\(460\) 1.00000 0.0466252
\(461\) 30.0000 1.39724 0.698620 − 0.715493i \(-0.253798\pi\)
0.698620 + 0.715493i \(0.253798\pi\)
\(462\) 0 0
\(463\) −26.0000 −1.20832 −0.604161 − 0.796862i \(-0.706492\pi\)
−0.604161 + 0.796862i \(0.706492\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) 2.00000 0.0926482
\(467\) −2.00000 −0.0925490 −0.0462745 − 0.998929i \(-0.514735\pi\)
−0.0462745 + 0.998929i \(0.514735\pi\)
\(468\) 0 0
\(469\) −48.0000 −2.21643
\(470\) 8.00000 0.369012
\(471\) 0 0
\(472\) 10.0000 0.460287
\(473\) 0 0
\(474\) 0 0
\(475\) 8.00000 0.367065
\(476\) 16.0000 0.733359
\(477\) 0 0
\(478\) −12.0000 −0.548867
\(479\) −8.00000 −0.365529 −0.182765 − 0.983157i \(-0.558505\pi\)
−0.182765 + 0.983157i \(0.558505\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −2.00000 −0.0910975
\(483\) 0 0
\(484\) −11.0000 −0.500000
\(485\) −10.0000 −0.454077
\(486\) 0 0
\(487\) 30.0000 1.35943 0.679715 − 0.733476i \(-0.262104\pi\)
0.679715 + 0.733476i \(0.262104\pi\)
\(488\) 8.00000 0.362143
\(489\) 0 0
\(490\) −9.00000 −0.406579
\(491\) 6.00000 0.270776 0.135388 − 0.990793i \(-0.456772\pi\)
0.135388 + 0.990793i \(0.456772\pi\)
\(492\) 0 0
\(493\) −8.00000 −0.360302
\(494\) 0 0
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) 64.0000 2.87079
\(498\) 0 0
\(499\) 36.0000 1.61158 0.805791 − 0.592200i \(-0.201741\pi\)
0.805791 + 0.592200i \(0.201741\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) 20.0000 0.892644
\(503\) −24.0000 −1.07011 −0.535054 − 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) 2.00000 0.0889988
\(506\) 0 0
\(507\) 0 0
\(508\) −10.0000 −0.443678
\(509\) −6.00000 −0.265945 −0.132973 − 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) 0 0
\(511\) 8.00000 0.353899
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 22.0000 0.970378
\(515\) −4.00000 −0.176261
\(516\) 0 0
\(517\) 0 0
\(518\) −8.00000 −0.351500
\(519\) 0 0
\(520\) 0 0
\(521\) 38.0000 1.66481 0.832405 − 0.554168i \(-0.186963\pi\)
0.832405 + 0.554168i \(0.186963\pi\)
\(522\) 0 0
\(523\) −28.0000 −1.22435 −0.612177 − 0.790721i \(-0.709706\pi\)
−0.612177 + 0.790721i \(0.709706\pi\)
\(524\) 6.00000 0.262111
\(525\) 0 0
\(526\) 4.00000 0.174408
\(527\) −16.0000 −0.696971
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 6.00000 0.260623
\(531\) 0 0
\(532\) −32.0000 −1.38738
\(533\) 0 0
\(534\) 0 0
\(535\) 2.00000 0.0864675
\(536\) −12.0000 −0.518321
\(537\) 0 0
\(538\) −18.0000 −0.776035
\(539\) 0 0
\(540\) 0 0
\(541\) 22.0000 0.945854 0.472927 − 0.881102i \(-0.343197\pi\)
0.472927 + 0.881102i \(0.343197\pi\)
\(542\) 20.0000 0.859074
\(543\) 0 0
\(544\) 4.00000 0.171499
\(545\) −4.00000 −0.171341
\(546\) 0 0
\(547\) 8.00000 0.342055 0.171028 − 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) −12.0000 −0.512615
\(549\) 0 0
\(550\) 0 0
\(551\) 16.0000 0.681623
\(552\) 0 0
\(553\) 24.0000 1.02058
\(554\) 8.00000 0.339887
\(555\) 0 0
\(556\) 20.0000 0.848189
\(557\) −18.0000 −0.762684 −0.381342 − 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −4.00000 −0.169031
\(561\) 0 0
\(562\) 6.00000 0.253095
\(563\) −34.0000 −1.43293 −0.716465 − 0.697623i \(-0.754241\pi\)
−0.716465 + 0.697623i \(0.754241\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −12.0000 −0.504398
\(567\) 0 0
\(568\) 16.0000 0.671345
\(569\) −6.00000 −0.251533 −0.125767 − 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 − 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −32.0000 −1.33565
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) −14.0000 −0.582828 −0.291414 − 0.956597i \(-0.594126\pi\)
−0.291414 + 0.956597i \(0.594126\pi\)
\(578\) 1.00000 0.0415945
\(579\) 0 0
\(580\) 2.00000 0.0830455
\(581\) −8.00000 −0.331896
\(582\) 0 0
\(583\) 0 0
\(584\) 2.00000 0.0827606
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) 32.0000 1.32078 0.660391 − 0.750922i \(-0.270391\pi\)
0.660391 + 0.750922i \(0.270391\pi\)
\(588\) 0 0
\(589\) 32.0000 1.31854
\(590\) 10.0000 0.411693
\(591\) 0 0
\(592\) −2.00000 −0.0821995
\(593\) 10.0000 0.410651 0.205325 − 0.978694i \(-0.434175\pi\)
0.205325 + 0.978694i \(0.434175\pi\)
\(594\) 0 0
\(595\) 16.0000 0.655936
\(596\) 6.00000 0.245770
\(597\) 0 0
\(598\) 0 0
\(599\) −40.0000 −1.63436 −0.817178 − 0.576386i \(-0.804463\pi\)
−0.817178 + 0.576386i \(0.804463\pi\)
\(600\) 0 0
\(601\) −2.00000 −0.0815817 −0.0407909 − 0.999168i \(-0.512988\pi\)
−0.0407909 + 0.999168i \(0.512988\pi\)
\(602\) −16.0000 −0.652111
\(603\) 0 0
\(604\) 20.0000 0.813788
\(605\) −11.0000 −0.447214
\(606\) 0 0
\(607\) 26.0000 1.05531 0.527654 − 0.849460i \(-0.323072\pi\)
0.527654 + 0.849460i \(0.323072\pi\)
\(608\) −8.00000 −0.324443
\(609\) 0 0
\(610\) 8.00000 0.323911
\(611\) 0 0
\(612\) 0 0
\(613\) −46.0000 −1.85792 −0.928961 − 0.370177i \(-0.879297\pi\)
−0.928961 + 0.370177i \(0.879297\pi\)
\(614\) −12.0000 −0.484281
\(615\) 0 0
\(616\) 0 0
\(617\) 36.0000 1.44931 0.724653 − 0.689114i \(-0.242000\pi\)
0.724653 + 0.689114i \(0.242000\pi\)
\(618\) 0 0
\(619\) −20.0000 −0.803868 −0.401934 − 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 4.00000 0.160644
\(621\) 0 0
\(622\) 32.0000 1.28308
\(623\) 24.0000 0.961540
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 14.0000 0.559553
\(627\) 0 0
\(628\) 10.0000 0.399043
\(629\) 8.00000 0.318981
\(630\) 0 0
\(631\) −18.0000 −0.716569 −0.358284 − 0.933613i \(-0.616638\pi\)
−0.358284 + 0.933613i \(0.616638\pi\)
\(632\) 6.00000 0.238667
\(633\) 0 0
\(634\) 30.0000 1.19145
\(635\) −10.0000 −0.396838
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) −18.0000 −0.710957 −0.355479 − 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 0 0
\(643\) −20.0000 −0.788723 −0.394362 − 0.918955i \(-0.629034\pi\)
−0.394362 + 0.918955i \(0.629034\pi\)
\(644\) −4.00000 −0.157622
\(645\) 0 0
\(646\) 32.0000 1.25902
\(647\) −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
\(651\) 0 0
\(652\) −8.00000 −0.313304
\(653\) 14.0000 0.547862 0.273931 − 0.961749i \(-0.411676\pi\)
0.273931 + 0.961749i \(0.411676\pi\)
\(654\) 0 0
\(655\) 6.00000 0.234439
\(656\) −8.00000 −0.312348
\(657\) 0 0
\(658\) −32.0000 −1.24749
\(659\) −20.0000 −0.779089 −0.389545 − 0.921008i \(-0.627368\pi\)
−0.389545 + 0.921008i \(0.627368\pi\)
\(660\) 0 0
\(661\) −48.0000 −1.86698 −0.933492 − 0.358599i \(-0.883255\pi\)
−0.933492 + 0.358599i \(0.883255\pi\)
\(662\) −28.0000 −1.08825
\(663\) 0 0
\(664\) −2.00000 −0.0776151
\(665\) −32.0000 −1.24091
\(666\) 0 0
\(667\) 2.00000 0.0774403
\(668\) 16.0000 0.619059
\(669\) 0 0
\(670\) −12.0000 −0.463600
\(671\) 0 0
\(672\) 0 0
\(673\) 14.0000 0.539660 0.269830 − 0.962908i \(-0.413032\pi\)
0.269830 + 0.962908i \(0.413032\pi\)
\(674\) −18.0000 −0.693334
\(675\) 0 0
\(676\) −13.0000 −0.500000
\(677\) 6.00000 0.230599 0.115299 − 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 0 0
\(679\) 40.0000 1.53506
\(680\) 4.00000 0.153393
\(681\) 0 0
\(682\) 0 0
\(683\) 8.00000 0.306111 0.153056 − 0.988218i \(-0.451089\pi\)
0.153056 + 0.988218i \(0.451089\pi\)
\(684\) 0 0
\(685\) −12.0000 −0.458496
\(686\) 8.00000 0.305441
\(687\) 0 0
\(688\) −4.00000 −0.152499
\(689\) 0 0
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 − 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) −2.00000 −0.0760286
\(693\) 0 0
\(694\) −28.0000 −1.06287
\(695\) 20.0000 0.758643
\(696\) 0 0
\(697\) 32.0000 1.21209
\(698\) −14.0000 −0.529908
\(699\) 0 0
\(700\) −4.00000 −0.151186
\(701\) −46.0000 −1.73740 −0.868698 − 0.495342i \(-0.835043\pi\)
−0.868698 + 0.495342i \(0.835043\pi\)
\(702\) 0 0
\(703\) −16.0000 −0.603451
\(704\) 0 0
\(705\) 0 0
\(706\) −18.0000 −0.677439
\(707\) −8.00000 −0.300871
\(708\) 0 0
\(709\) −20.0000 −0.751116 −0.375558 − 0.926799i \(-0.622549\pi\)
−0.375558 + 0.926799i \(0.622549\pi\)
\(710\) 16.0000 0.600469
\(711\) 0 0
\(712\) 6.00000 0.224860
\(713\) 4.00000 0.149801
\(714\) 0 0
\(715\) 0 0
\(716\) −6.00000 −0.224231
\(717\) 0 0
\(718\) −32.0000 −1.19423
\(719\) 28.0000 1.04422 0.522112 − 0.852877i \(-0.325144\pi\)
0.522112 + 0.852877i \(0.325144\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) −45.0000 −1.67473
\(723\) 0 0
\(724\) −4.00000 −0.148659
\(725\) 2.00000 0.0742781
\(726\) 0 0
\(727\) −32.0000 −1.18681 −0.593407 − 0.804902i \(-0.702218\pi\)
−0.593407 + 0.804902i \(0.702218\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 2.00000 0.0740233
\(731\) 16.0000 0.591781
\(732\) 0 0
\(733\) −14.0000 −0.517102 −0.258551 − 0.965998i \(-0.583245\pi\)
−0.258551 + 0.965998i \(0.583245\pi\)
\(734\) 32.0000 1.18114
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 0 0
\(738\) 0 0
\(739\) −4.00000 −0.147142 −0.0735712 − 0.997290i \(-0.523440\pi\)
−0.0735712 + 0.997290i \(0.523440\pi\)
\(740\) −2.00000 −0.0735215
\(741\) 0 0
\(742\) −24.0000 −0.881068
\(743\) −16.0000 −0.586983 −0.293492 − 0.955962i \(-0.594817\pi\)
−0.293492 + 0.955962i \(0.594817\pi\)
\(744\) 0 0
\(745\) 6.00000 0.219823
\(746\) 6.00000 0.219676
\(747\) 0 0
\(748\) 0 0
\(749\) −8.00000 −0.292314
\(750\) 0 0
\(751\) 42.0000 1.53260 0.766301 − 0.642482i \(-0.222095\pi\)
0.766301 + 0.642482i \(0.222095\pi\)
\(752\) −8.00000 −0.291730
\(753\) 0 0
\(754\) 0 0
\(755\) 20.0000 0.727875
\(756\) 0 0
\(757\) −10.0000 −0.363456 −0.181728 − 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) 4.00000 0.145287
\(759\) 0 0
\(760\) −8.00000 −0.290191
\(761\) −48.0000 −1.74000 −0.869999 − 0.493053i \(-0.835881\pi\)
−0.869999 + 0.493053i \(0.835881\pi\)
\(762\) 0 0
\(763\) 16.0000 0.579239
\(764\) −24.0000 −0.868290
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 54.0000 1.94729 0.973645 − 0.228069i \(-0.0732413\pi\)
0.973645 + 0.228069i \(0.0732413\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −18.0000 −0.647834
\(773\) 34.0000 1.22290 0.611448 − 0.791285i \(-0.290588\pi\)
0.611448 + 0.791285i \(0.290588\pi\)
\(774\) 0 0
\(775\) 4.00000 0.143684
\(776\) 10.0000 0.358979
\(777\) 0 0
\(778\) −18.0000 −0.645331
\(779\) −64.0000 −2.29304
\(780\) 0 0
\(781\) 0 0
\(782\) 4.00000 0.143040
\(783\) 0 0
\(784\) 9.00000 0.321429
\(785\) 10.0000 0.356915
\(786\) 0 0
\(787\) −20.0000 −0.712923 −0.356462 − 0.934310i \(-0.616017\pi\)
−0.356462 + 0.934310i \(0.616017\pi\)
\(788\) −2.00000 −0.0712470
\(789\) 0 0
\(790\) 6.00000 0.213470
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) −36.0000 −1.27759
\(795\) 0 0
\(796\) −18.0000 −0.637993
\(797\) 30.0000 1.06265 0.531327 − 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) 0 0
\(799\) 32.0000 1.13208
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) −2.00000 −0.0706225
\(803\) 0 0
\(804\) 0 0
\(805\) −4.00000 −0.140981
\(806\) 0 0
\(807\) 0 0
\(808\) −2.00000 −0.0703598
\(809\) −28.0000 −0.984428 −0.492214 − 0.870474i \(-0.663812\pi\)
−0.492214 + 0.870474i \(0.663812\pi\)
\(810\) 0 0
\(811\) 4.00000 0.140459 0.0702295 − 0.997531i \(-0.477627\pi\)
0.0702295 + 0.997531i \(0.477627\pi\)
\(812\) −8.00000 −0.280745
\(813\) 0 0
\(814\) 0 0
\(815\) −8.00000 −0.280228
\(816\) 0 0
\(817\) −32.0000 −1.11954
\(818\) 18.0000 0.629355
\(819\) 0 0
\(820\) −8.00000 −0.279372
\(821\) 50.0000 1.74501 0.872506 − 0.488603i \(-0.162493\pi\)
0.872506 + 0.488603i \(0.162493\pi\)
\(822\) 0 0
\(823\) 38.0000 1.32460 0.662298 − 0.749240i \(-0.269581\pi\)
0.662298 + 0.749240i \(0.269581\pi\)
\(824\) 4.00000 0.139347
\(825\) 0 0
\(826\) −40.0000 −1.39178
\(827\) 34.0000 1.18230 0.591148 − 0.806563i \(-0.298675\pi\)
0.591148 + 0.806563i \(0.298675\pi\)
\(828\) 0 0
\(829\) 30.0000 1.04194 0.520972 − 0.853574i \(-0.325570\pi\)
0.520972 + 0.853574i \(0.325570\pi\)
\(830\) −2.00000 −0.0694210
\(831\) 0 0
\(832\) 0 0
\(833\) −36.0000 −1.24733
\(834\) 0 0
\(835\) 16.0000 0.553703
\(836\) 0 0
\(837\) 0 0
\(838\) −24.0000 −0.829066
\(839\) 16.0000 0.552381 0.276191 − 0.961103i \(-0.410928\pi\)
0.276191 + 0.961103i \(0.410928\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) −4.00000 −0.137849
\(843\) 0 0
\(844\) 20.0000 0.688428
\(845\) −13.0000 −0.447214
\(846\) 0 0
\(847\) 44.0000 1.51186
\(848\) −6.00000 −0.206041
\(849\) 0 0
\(850\) 4.00000 0.137199
\(851\) −2.00000 −0.0685591
\(852\) 0 0
\(853\) 12.0000 0.410872 0.205436 − 0.978671i \(-0.434139\pi\)
0.205436 + 0.978671i \(0.434139\pi\)
\(854\) −32.0000 −1.09502
\(855\) 0 0
\(856\) −2.00000 −0.0683586
\(857\) −42.0000 −1.43469 −0.717346 − 0.696717i \(-0.754643\pi\)
−0.717346 + 0.696717i \(0.754643\pi\)
\(858\) 0 0
\(859\) −20.0000 −0.682391 −0.341196 − 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) −4.00000 −0.136399
\(861\) 0 0
\(862\) 8.00000 0.272481
\(863\) −48.0000 −1.63394 −0.816970 − 0.576681i \(-0.804348\pi\)
−0.816970 + 0.576681i \(0.804348\pi\)
\(864\) 0 0
\(865\) −2.00000 −0.0680020
\(866\) 2.00000 0.0679628
\(867\) 0 0
\(868\) −16.0000 −0.543075
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 4.00000 0.135457
\(873\) 0 0
\(874\) −8.00000 −0.270604
\(875\) −4.00000 −0.135225
\(876\) 0 0
\(877\) −44.0000 −1.48577 −0.742887 − 0.669417i \(-0.766544\pi\)
−0.742887 + 0.669417i \(0.766544\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 18.0000 0.606435 0.303218 − 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) 0 0
\(883\) −20.0000 −0.673054 −0.336527 − 0.941674i \(-0.609252\pi\)
−0.336527 + 0.941674i \(0.609252\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −24.0000 −0.806296
\(887\) 48.0000 1.61168 0.805841 − 0.592132i \(-0.201714\pi\)
0.805841 + 0.592132i \(0.201714\pi\)
\(888\) 0 0
\(889\) 40.0000 1.34156
\(890\) 6.00000 0.201120
\(891\) 0 0
\(892\) 6.00000 0.200895
\(893\) −64.0000 −2.14168
\(894\) 0 0
\(895\) −6.00000 −0.200558
\(896\) 4.00000 0.133631
\(897\) 0 0
\(898\) 20.0000 0.667409
\(899\) 8.00000 0.266815
\(900\) 0 0
\(901\) 24.0000 0.799556
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −4.00000 −0.132964
\(906\) 0 0
\(907\) 28.0000 0.929725 0.464862 − 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) 22.0000 0.730096
\(909\) 0 0
\(910\) 0 0
\(911\) −8.00000 −0.265052 −0.132526 − 0.991180i \(-0.542309\pi\)
−0.132526 + 0.991180i \(0.542309\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −10.0000 −0.330771
\(915\) 0 0
\(916\) −24.0000 −0.792982
\(917\) −24.0000 −0.792550
\(918\) 0 0
\(919\) 34.0000 1.12156 0.560778 − 0.827966i \(-0.310502\pi\)
0.560778 + 0.827966i \(0.310502\pi\)
\(920\) −1.00000 −0.0329690
\(921\) 0 0
\(922\) −30.0000 −0.987997
\(923\) 0 0
\(924\) 0 0
\(925\) −2.00000 −0.0657596
\(926\) 26.0000 0.854413
\(927\) 0 0
\(928\) −2.00000 −0.0656532
\(929\) 52.0000 1.70606 0.853032 − 0.521858i \(-0.174761\pi\)
0.853032 + 0.521858i \(0.174761\pi\)
\(930\) 0 0
\(931\) 72.0000 2.35970
\(932\) −2.00000 −0.0655122
\(933\) 0 0
\(934\) 2.00000 0.0654420
\(935\) 0 0
\(936\) 0 0
\(937\) 42.0000 1.37208 0.686040 − 0.727564i \(-0.259347\pi\)
0.686040 + 0.727564i \(0.259347\pi\)
\(938\) 48.0000 1.56726
\(939\) 0 0
\(940\) −8.00000 −0.260931
\(941\) −46.0000 −1.49956 −0.749779 − 0.661689i \(-0.769840\pi\)
−0.749779 + 0.661689i \(0.769840\pi\)
\(942\) 0 0
\(943\) −8.00000 −0.260516
\(944\) −10.0000 −0.325472
\(945\) 0 0
\(946\) 0 0
\(947\) −36.0000 −1.16984 −0.584921 − 0.811090i \(-0.698875\pi\)
−0.584921 + 0.811090i \(0.698875\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −8.00000 −0.259554
\(951\) 0 0
\(952\) −16.0000 −0.518563
\(953\) 48.0000 1.55487 0.777436 − 0.628962i \(-0.216520\pi\)
0.777436 + 0.628962i \(0.216520\pi\)
\(954\) 0 0
\(955\) −24.0000 −0.776622
\(956\) 12.0000 0.388108
\(957\) 0 0
\(958\) 8.00000 0.258468
\(959\) 48.0000 1.55000
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 0 0
\(964\) 2.00000 0.0644157
\(965\) −18.0000 −0.579441
\(966\) 0 0
\(967\) 58.0000 1.86515 0.932577 − 0.360971i \(-0.117555\pi\)
0.932577 + 0.360971i \(0.117555\pi\)
\(968\) 11.0000 0.353553
\(969\) 0 0
\(970\) 10.0000 0.321081
\(971\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) −80.0000 −2.56468
\(974\) −30.0000 −0.961262
\(975\) 0 0
\(976\) −8.00000 −0.256074
\(977\) 32.0000 1.02377 0.511885 − 0.859054i \(-0.328947\pi\)
0.511885 + 0.859054i \(0.328947\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 9.00000 0.287494
\(981\) 0 0
\(982\) −6.00000 −0.191468
\(983\) −24.0000 −0.765481 −0.382741 − 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) 0 0
\(985\) −2.00000 −0.0637253
\(986\) 8.00000 0.254772
\(987\) 0 0
\(988\) 0 0
\(989\) −4.00000 −0.127193
\(990\) 0 0
\(991\) 32.0000 1.01651 0.508257 − 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) −4.00000 −0.127000
\(993\) 0 0
\(994\) −64.0000 −2.02996
\(995\) −18.0000 −0.570638
\(996\) 0 0
\(997\) −28.0000 −0.886769 −0.443384 − 0.896332i \(-0.646222\pi\)
−0.443384 + 0.896332i \(0.646222\pi\)
\(998\) −36.0000 −1.13956
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2070.2.a.d.1.1 ✓ 1
3.2 odd 2 2070.2.a.j.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2070.2.a.d.1.1 ✓ 1 1.1 even 1 trivial
2070.2.a.j.1.1 yes 1 3.2 odd 2