Properties

Label 819.2.a.a.1.1
Level $819$
Weight $2$
Character 819.1
Self dual yes
Analytic conductor $6.540$
Analytic rank $0$
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] = [819,2,Mod(1,819)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(819, 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("819.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.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: \(6.53974792554\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 273)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 819.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} +2.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +O(q^{10})\) \(q-2.00000 q^{2} +2.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +2.00000 q^{10} +2.00000 q^{11} -1.00000 q^{13} +2.00000 q^{14} -4.00000 q^{16} +1.00000 q^{19} -2.00000 q^{20} -4.00000 q^{22} -3.00000 q^{23} -4.00000 q^{25} +2.00000 q^{26} -2.00000 q^{28} +5.00000 q^{29} +9.00000 q^{31} +8.00000 q^{32} +1.00000 q^{35} -2.00000 q^{38} -2.00000 q^{41} -1.00000 q^{43} +4.00000 q^{44} +6.00000 q^{46} -3.00000 q^{47} +1.00000 q^{49} +8.00000 q^{50} -2.00000 q^{52} +9.00000 q^{53} -2.00000 q^{55} -10.0000 q^{58} -2.00000 q^{61} -18.0000 q^{62} -8.00000 q^{64} +1.00000 q^{65} +10.0000 q^{67} -2.00000 q^{70} +12.0000 q^{71} +15.0000 q^{73} +2.00000 q^{76} -2.00000 q^{77} +11.0000 q^{79} +4.00000 q^{80} +4.00000 q^{82} -3.00000 q^{83} +2.00000 q^{86} +17.0000 q^{89} +1.00000 q^{91} -6.00000 q^{92} +6.00000 q^{94} -1.00000 q^{95} +3.00000 q^{97} -2.00000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) 0 0
\(4\) 2.00000 1.00000
\(5\) −1.00000 −0.447214 −0.223607 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 2.00000 0.632456
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416 0.114708 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) −4.00000 −0.852803
\(23\) −3.00000 −0.625543 −0.312772 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 2.00000 0.392232
\(27\) 0 0
\(28\) −2.00000 −0.377964
\(29\) 5.00000 0.928477 0.464238 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 0 0
\(31\) 9.00000 1.61645 0.808224 0.588875i \(-0.200429\pi\)
0.808224 + 0.588875i \(0.200429\pi\)
\(32\) 8.00000 1.41421
\(33\) 0 0
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) −2.00000 −0.324443
\(39\) 0 0
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499 −0.0762493 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) 6.00000 0.884652
\(47\) −3.00000 −0.437595 −0.218797 0.975770i \(-0.570213\pi\)
−0.218797 + 0.975770i \(0.570213\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 8.00000 1.13137
\(51\) 0 0
\(52\) −2.00000 −0.277350
\(53\) 9.00000 1.23625 0.618123 0.786082i \(-0.287894\pi\)
0.618123 + 0.786082i \(0.287894\pi\)
\(54\) 0 0
\(55\) −2.00000 −0.269680
\(56\) 0 0
\(57\) 0 0
\(58\) −10.0000 −1.31306
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) −18.0000 −2.28600
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 1.00000 0.124035
\(66\) 0 0
\(67\) 10.0000 1.22169 0.610847 0.791748i \(-0.290829\pi\)
0.610847 + 0.791748i \(0.290829\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) −2.00000 −0.239046
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) 15.0000 1.75562 0.877809 0.479012i \(-0.159005\pi\)
0.877809 + 0.479012i \(0.159005\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) −2.00000 −0.227921
\(78\) 0 0
\(79\) 11.0000 1.23760 0.618798 0.785550i \(-0.287620\pi\)
0.618798 + 0.785550i \(0.287620\pi\)
\(80\) 4.00000 0.447214
\(81\) 0 0
\(82\) 4.00000 0.441726
\(83\) −3.00000 −0.329293 −0.164646 0.986353i \(-0.552648\pi\)
−0.164646 + 0.986353i \(0.552648\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 2.00000 0.215666
\(87\) 0 0
\(88\) 0 0
\(89\) 17.0000 1.80200 0.900998 0.433823i \(-0.142836\pi\)
0.900998 + 0.433823i \(0.142836\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) −6.00000 −0.625543
\(93\) 0 0
\(94\) 6.00000 0.618853
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) 3.00000 0.304604 0.152302 0.988334i \(-0.451331\pi\)
0.152302 + 0.988334i \(0.451331\pi\)
\(98\) −2.00000 −0.202031
\(99\) 0 0
\(100\) −8.00000 −0.800000
\(101\) −14.0000 −1.39305 −0.696526 0.717532i \(-0.745272\pi\)
−0.696526 + 0.717532i \(0.745272\pi\)
\(102\) 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
\(106\) −18.0000 −1.74831
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 0 0
\(109\) 18.0000 1.72409 0.862044 0.506834i \(-0.169184\pi\)
0.862044 + 0.506834i \(0.169184\pi\)
\(110\) 4.00000 0.381385
\(111\) 0 0
\(112\) 4.00000 0.377964
\(113\) 11.0000 1.03479 0.517396 0.855746i \(-0.326901\pi\)
0.517396 + 0.855746i \(0.326901\pi\)
\(114\) 0 0
\(115\) 3.00000 0.279751
\(116\) 10.0000 0.928477
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 4.00000 0.362143
\(123\) 0 0
\(124\) 18.0000 1.61645
\(125\) 9.00000 0.804984
\(126\) 0 0
\(127\) 12.0000 1.06483 0.532414 0.846484i \(-0.321285\pi\)
0.532414 + 0.846484i \(0.321285\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) −2.00000 −0.175412
\(131\) 8.00000 0.698963 0.349482 0.936943i \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(132\) 0 0
\(133\) −1.00000 −0.0867110
\(134\) −20.0000 −1.72774
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) −6.00000 −0.508913 −0.254457 0.967084i \(-0.581897\pi\)
−0.254457 + 0.967084i \(0.581897\pi\)
\(140\) 2.00000 0.169031
\(141\) 0 0
\(142\) −24.0000 −2.01404
\(143\) −2.00000 −0.167248
\(144\) 0 0
\(145\) −5.00000 −0.415227
\(146\) −30.0000 −2.48282
\(147\) 0 0
\(148\) 0 0
\(149\) 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 \(0.197401\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 4.00000 0.322329
\(155\) −9.00000 −0.722897
\(156\) 0 0
\(157\) 4.00000 0.319235 0.159617 0.987179i \(-0.448974\pi\)
0.159617 + 0.987179i \(0.448974\pi\)
\(158\) −22.0000 −1.75023
\(159\) 0 0
\(160\) −8.00000 −0.632456
\(161\) 3.00000 0.236433
\(162\) 0 0
\(163\) −20.0000 −1.56652 −0.783260 0.621694i \(-0.786445\pi\)
−0.783260 + 0.621694i \(0.786445\pi\)
\(164\) −4.00000 −0.312348
\(165\) 0 0
\(166\) 6.00000 0.465690
\(167\) −25.0000 −1.93456 −0.967279 0.253715i \(-0.918347\pi\)
−0.967279 + 0.253715i \(0.918347\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) −2.00000 −0.152499
\(173\) 4.00000 0.304114 0.152057 0.988372i \(-0.451410\pi\)
0.152057 + 0.988372i \(0.451410\pi\)
\(174\) 0 0
\(175\) 4.00000 0.302372
\(176\) −8.00000 −0.603023
\(177\) 0 0
\(178\) −34.0000 −2.54841
\(179\) 1.00000 0.0747435 0.0373718 0.999301i \(-0.488101\pi\)
0.0373718 + 0.999301i \(0.488101\pi\)
\(180\) 0 0
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) −2.00000 −0.148250
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) −6.00000 −0.437595
\(189\) 0 0
\(190\) 2.00000 0.145095
\(191\) −16.0000 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) 0 0
\(193\) −10.0000 −0.719816 −0.359908 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) −6.00000 −0.430775
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) −10.0000 −0.712470 −0.356235 0.934396i \(-0.615940\pi\)
−0.356235 + 0.934396i \(0.615940\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 28.0000 1.97007
\(203\) −5.00000 −0.350931
\(204\) 0 0
\(205\) 2.00000 0.139686
\(206\) 8.00000 0.557386
\(207\) 0 0
\(208\) 4.00000 0.277350
\(209\) 2.00000 0.138343
\(210\) 0 0
\(211\) −5.00000 −0.344214 −0.172107 0.985078i \(-0.555058\pi\)
−0.172107 + 0.985078i \(0.555058\pi\)
\(212\) 18.0000 1.23625
\(213\) 0 0
\(214\) 8.00000 0.546869
\(215\) 1.00000 0.0681994
\(216\) 0 0
\(217\) −9.00000 −0.610960
\(218\) −36.0000 −2.43823
\(219\) 0 0
\(220\) −4.00000 −0.269680
\(221\) 0 0
\(222\) 0 0
\(223\) −29.0000 −1.94198 −0.970992 0.239113i \(-0.923143\pi\)
−0.970992 + 0.239113i \(0.923143\pi\)
\(224\) −8.00000 −0.534522
\(225\) 0 0
\(226\) −22.0000 −1.46342
\(227\) −12.0000 −0.796468 −0.398234 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 0 0
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) −6.00000 −0.395628
\(231\) 0 0
\(232\) 0 0
\(233\) 9.00000 0.589610 0.294805 0.955557i \(-0.404745\pi\)
0.294805 + 0.955557i \(0.404745\pi\)
\(234\) 0 0
\(235\) 3.00000 0.195698
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) −13.0000 −0.837404 −0.418702 0.908124i \(-0.637515\pi\)
−0.418702 + 0.908124i \(0.637515\pi\)
\(242\) 14.0000 0.899954
\(243\) 0 0
\(244\) −4.00000 −0.256074
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) −1.00000 −0.0636285
\(248\) 0 0
\(249\) 0 0
\(250\) −18.0000 −1.13842
\(251\) 14.0000 0.883672 0.441836 0.897096i \(-0.354327\pi\)
0.441836 + 0.897096i \(0.354327\pi\)
\(252\) 0 0
\(253\) −6.00000 −0.377217
\(254\) −24.0000 −1.50589
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) −2.00000 −0.124757 −0.0623783 0.998053i \(-0.519869\pi\)
−0.0623783 + 0.998053i \(0.519869\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 2.00000 0.124035
\(261\) 0 0
\(262\) −16.0000 −0.988483
\(263\) 23.0000 1.41824 0.709120 0.705087i \(-0.249092\pi\)
0.709120 + 0.705087i \(0.249092\pi\)
\(264\) 0 0
\(265\) −9.00000 −0.552866
\(266\) 2.00000 0.122628
\(267\) 0 0
\(268\) 20.0000 1.22169
\(269\) 24.0000 1.46331 0.731653 0.681677i \(-0.238749\pi\)
0.731653 + 0.681677i \(0.238749\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −8.00000 −0.482418
\(276\) 0 0
\(277\) −23.0000 −1.38194 −0.690968 0.722885i \(-0.742815\pi\)
−0.690968 + 0.722885i \(0.742815\pi\)
\(278\) 12.0000 0.719712
\(279\) 0 0
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) −8.00000 −0.475551 −0.237775 0.971320i \(-0.576418\pi\)
−0.237775 + 0.971320i \(0.576418\pi\)
\(284\) 24.0000 1.42414
\(285\) 0 0
\(286\) 4.00000 0.236525
\(287\) 2.00000 0.118056
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 10.0000 0.587220
\(291\) 0 0
\(292\) 30.0000 1.75562
\(293\) 31.0000 1.81104 0.905520 0.424304i \(-0.139481\pi\)
0.905520 + 0.424304i \(0.139481\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) −12.0000 −0.695141
\(299\) 3.00000 0.173494
\(300\) 0 0
\(301\) 1.00000 0.0576390
\(302\) −40.0000 −2.30174
\(303\) 0 0
\(304\) −4.00000 −0.229416
\(305\) 2.00000 0.114520
\(306\) 0 0
\(307\) −5.00000 −0.285365 −0.142683 0.989769i \(-0.545573\pi\)
−0.142683 + 0.989769i \(0.545573\pi\)
\(308\) −4.00000 −0.227921
\(309\) 0 0
\(310\) 18.0000 1.02233
\(311\) −6.00000 −0.340229 −0.170114 0.985424i \(-0.554414\pi\)
−0.170114 + 0.985424i \(0.554414\pi\)
\(312\) 0 0
\(313\) −22.0000 −1.24351 −0.621757 0.783210i \(-0.713581\pi\)
−0.621757 + 0.783210i \(0.713581\pi\)
\(314\) −8.00000 −0.451466
\(315\) 0 0
\(316\) 22.0000 1.23760
\(317\) 24.0000 1.34797 0.673987 0.738743i \(-0.264580\pi\)
0.673987 + 0.738743i \(0.264580\pi\)
\(318\) 0 0
\(319\) 10.0000 0.559893
\(320\) 8.00000 0.447214
\(321\) 0 0
\(322\) −6.00000 −0.334367
\(323\) 0 0
\(324\) 0 0
\(325\) 4.00000 0.221880
\(326\) 40.0000 2.21540
\(327\) 0 0
\(328\) 0 0
\(329\) 3.00000 0.165395
\(330\) 0 0
\(331\) 14.0000 0.769510 0.384755 0.923019i \(-0.374286\pi\)
0.384755 + 0.923019i \(0.374286\pi\)
\(332\) −6.00000 −0.329293
\(333\) 0 0
\(334\) 50.0000 2.73588
\(335\) −10.0000 −0.546358
\(336\) 0 0
\(337\) 1.00000 0.0544735 0.0272367 0.999629i \(-0.491329\pi\)
0.0272367 + 0.999629i \(0.491329\pi\)
\(338\) −2.00000 −0.108786
\(339\) 0 0
\(340\) 0 0
\(341\) 18.0000 0.974755
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) −8.00000 −0.430083
\(347\) −32.0000 −1.71785 −0.858925 0.512101i \(-0.828867\pi\)
−0.858925 + 0.512101i \(0.828867\pi\)
\(348\) 0 0
\(349\) 31.0000 1.65939 0.829696 0.558216i \(-0.188514\pi\)
0.829696 + 0.558216i \(0.188514\pi\)
\(350\) −8.00000 −0.427618
\(351\) 0 0
\(352\) 16.0000 0.852803
\(353\) −22.0000 −1.17094 −0.585471 0.810693i \(-0.699090\pi\)
−0.585471 + 0.810693i \(0.699090\pi\)
\(354\) 0 0
\(355\) −12.0000 −0.636894
\(356\) 34.0000 1.80200
\(357\) 0 0
\(358\) −2.00000 −0.105703
\(359\) 16.0000 0.844448 0.422224 0.906492i \(-0.361250\pi\)
0.422224 + 0.906492i \(0.361250\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 28.0000 1.47165
\(363\) 0 0
\(364\) 2.00000 0.104828
\(365\) −15.0000 −0.785136
\(366\) 0 0
\(367\) 18.0000 0.939592 0.469796 0.882775i \(-0.344327\pi\)
0.469796 + 0.882775i \(0.344327\pi\)
\(368\) 12.0000 0.625543
\(369\) 0 0
\(370\) 0 0
\(371\) −9.00000 −0.467257
\(372\) 0 0
\(373\) 38.0000 1.96757 0.983783 0.179364i \(-0.0574041\pi\)
0.983783 + 0.179364i \(0.0574041\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −5.00000 −0.257513
\(378\) 0 0
\(379\) −18.0000 −0.924598 −0.462299 0.886724i \(-0.652975\pi\)
−0.462299 + 0.886724i \(0.652975\pi\)
\(380\) −2.00000 −0.102598
\(381\) 0 0
\(382\) 32.0000 1.63726
\(383\) −36.0000 −1.83951 −0.919757 0.392488i \(-0.871614\pi\)
−0.919757 + 0.392488i \(0.871614\pi\)
\(384\) 0 0
\(385\) 2.00000 0.101929
\(386\) 20.0000 1.01797
\(387\) 0 0
\(388\) 6.00000 0.304604
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 20.0000 1.00759
\(395\) −11.0000 −0.553470
\(396\) 0 0
\(397\) 15.0000 0.752828 0.376414 0.926451i \(-0.377157\pi\)
0.376414 + 0.926451i \(0.377157\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 16.0000 0.800000
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) −9.00000 −0.448322
\(404\) −28.0000 −1.39305
\(405\) 0 0
\(406\) 10.0000 0.496292
\(407\) 0 0
\(408\) 0 0
\(409\) 23.0000 1.13728 0.568638 0.822588i \(-0.307470\pi\)
0.568638 + 0.822588i \(0.307470\pi\)
\(410\) −4.00000 −0.197546
\(411\) 0 0
\(412\) −8.00000 −0.394132
\(413\) 0 0
\(414\) 0 0
\(415\) 3.00000 0.147264
\(416\) −8.00000 −0.392232
\(417\) 0 0
\(418\) −4.00000 −0.195646
\(419\) 30.0000 1.46560 0.732798 0.680446i \(-0.238214\pi\)
0.732798 + 0.680446i \(0.238214\pi\)
\(420\) 0 0
\(421\) 8.00000 0.389896 0.194948 0.980814i \(-0.437546\pi\)
0.194948 + 0.980814i \(0.437546\pi\)
\(422\) 10.0000 0.486792
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 2.00000 0.0967868
\(428\) −8.00000 −0.386695
\(429\) 0 0
\(430\) −2.00000 −0.0964486
\(431\) −30.0000 −1.44505 −0.722525 0.691345i \(-0.757018\pi\)
−0.722525 + 0.691345i \(0.757018\pi\)
\(432\) 0 0
\(433\) −20.0000 −0.961139 −0.480569 0.876957i \(-0.659570\pi\)
−0.480569 + 0.876957i \(0.659570\pi\)
\(434\) 18.0000 0.864028
\(435\) 0 0
\(436\) 36.0000 1.72409
\(437\) −3.00000 −0.143509
\(438\) 0 0
\(439\) −6.00000 −0.286364 −0.143182 0.989696i \(-0.545733\pi\)
−0.143182 + 0.989696i \(0.545733\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −27.0000 −1.28281 −0.641404 0.767203i \(-0.721648\pi\)
−0.641404 + 0.767203i \(0.721648\pi\)
\(444\) 0 0
\(445\) −17.0000 −0.805877
\(446\) 58.0000 2.74638
\(447\) 0 0
\(448\) 8.00000 0.377964
\(449\) 12.0000 0.566315 0.283158 0.959073i \(-0.408618\pi\)
0.283158 + 0.959073i \(0.408618\pi\)
\(450\) 0 0
\(451\) −4.00000 −0.188353
\(452\) 22.0000 1.03479
\(453\) 0 0
\(454\) 24.0000 1.12638
\(455\) −1.00000 −0.0468807
\(456\) 0 0
\(457\) 28.0000 1.30978 0.654892 0.755722i \(-0.272714\pi\)
0.654892 + 0.755722i \(0.272714\pi\)
\(458\) −28.0000 −1.30835
\(459\) 0 0
\(460\) 6.00000 0.279751
\(461\) −2.00000 −0.0931493 −0.0465746 0.998915i \(-0.514831\pi\)
−0.0465746 + 0.998915i \(0.514831\pi\)
\(462\) 0 0
\(463\) 22.0000 1.02243 0.511213 0.859454i \(-0.329196\pi\)
0.511213 + 0.859454i \(0.329196\pi\)
\(464\) −20.0000 −0.928477
\(465\) 0 0
\(466\) −18.0000 −0.833834
\(467\) −6.00000 −0.277647 −0.138823 0.990317i \(-0.544332\pi\)
−0.138823 + 0.990317i \(0.544332\pi\)
\(468\) 0 0
\(469\) −10.0000 −0.461757
\(470\) −6.00000 −0.276759
\(471\) 0 0
\(472\) 0 0
\(473\) −2.00000 −0.0919601
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) 0 0
\(477\) 0 0
\(478\) −24.0000 −1.09773
\(479\) −17.0000 −0.776750 −0.388375 0.921501i \(-0.626963\pi\)
−0.388375 + 0.921501i \(0.626963\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 26.0000 1.18427
\(483\) 0 0
\(484\) −14.0000 −0.636364
\(485\) −3.00000 −0.136223
\(486\) 0 0
\(487\) −2.00000 −0.0906287 −0.0453143 0.998973i \(-0.514429\pi\)
−0.0453143 + 0.998973i \(0.514429\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 2.00000 0.0903508
\(491\) 28.0000 1.26362 0.631811 0.775122i \(-0.282312\pi\)
0.631811 + 0.775122i \(0.282312\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 2.00000 0.0899843
\(495\) 0 0
\(496\) −36.0000 −1.61645
\(497\) −12.0000 −0.538274
\(498\) 0 0
\(499\) 28.0000 1.25345 0.626726 0.779240i \(-0.284395\pi\)
0.626726 + 0.779240i \(0.284395\pi\)
\(500\) 18.0000 0.804984
\(501\) 0 0
\(502\) −28.0000 −1.24970
\(503\) 6.00000 0.267527 0.133763 0.991013i \(-0.457294\pi\)
0.133763 + 0.991013i \(0.457294\pi\)
\(504\) 0 0
\(505\) 14.0000 0.622992
\(506\) 12.0000 0.533465
\(507\) 0 0
\(508\) 24.0000 1.06483
\(509\) −41.0000 −1.81729 −0.908647 0.417566i \(-0.862883\pi\)
−0.908647 + 0.417566i \(0.862883\pi\)
\(510\) 0 0
\(511\) −15.0000 −0.663561
\(512\) −32.0000 −1.41421
\(513\) 0 0
\(514\) 4.00000 0.176432
\(515\) 4.00000 0.176261
\(516\) 0 0
\(517\) −6.00000 −0.263880
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −24.0000 −1.05146 −0.525730 0.850652i \(-0.676208\pi\)
−0.525730 + 0.850652i \(0.676208\pi\)
\(522\) 0 0
\(523\) −14.0000 −0.612177 −0.306089 0.952003i \(-0.599020\pi\)
−0.306089 + 0.952003i \(0.599020\pi\)
\(524\) 16.0000 0.698963
\(525\) 0 0
\(526\) −46.0000 −2.00570
\(527\) 0 0
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) 18.0000 0.781870
\(531\) 0 0
\(532\) −2.00000 −0.0867110
\(533\) 2.00000 0.0866296
\(534\) 0 0
\(535\) 4.00000 0.172935
\(536\) 0 0
\(537\) 0 0
\(538\) −48.0000 −2.06943
\(539\) 2.00000 0.0861461
\(540\) 0 0
\(541\) −4.00000 −0.171973 −0.0859867 0.996296i \(-0.527404\pi\)
−0.0859867 + 0.996296i \(0.527404\pi\)
\(542\) 16.0000 0.687259
\(543\) 0 0
\(544\) 0 0
\(545\) −18.0000 −0.771035
\(546\) 0 0
\(547\) 1.00000 0.0427569 0.0213785 0.999771i \(-0.493195\pi\)
0.0213785 + 0.999771i \(0.493195\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 16.0000 0.682242
\(551\) 5.00000 0.213007
\(552\) 0 0
\(553\) −11.0000 −0.467768
\(554\) 46.0000 1.95435
\(555\) 0 0
\(556\) −12.0000 −0.508913
\(557\) 44.0000 1.86434 0.932170 0.362021i \(-0.117913\pi\)
0.932170 + 0.362021i \(0.117913\pi\)
\(558\) 0 0
\(559\) 1.00000 0.0422955
\(560\) −4.00000 −0.169031
\(561\) 0 0
\(562\) −20.0000 −0.843649
\(563\) −4.00000 −0.168580 −0.0842900 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) 0 0
\(565\) −11.0000 −0.462773
\(566\) 16.0000 0.672530
\(567\) 0 0
\(568\) 0 0
\(569\) 41.0000 1.71881 0.859405 0.511296i \(-0.170834\pi\)
0.859405 + 0.511296i \(0.170834\pi\)
\(570\) 0 0
\(571\) −25.0000 −1.04622 −0.523109 0.852266i \(-0.675228\pi\)
−0.523109 + 0.852266i \(0.675228\pi\)
\(572\) −4.00000 −0.167248
\(573\) 0 0
\(574\) −4.00000 −0.166957
\(575\) 12.0000 0.500435
\(576\) 0 0
\(577\) −14.0000 −0.582828 −0.291414 0.956597i \(-0.594126\pi\)
−0.291414 + 0.956597i \(0.594126\pi\)
\(578\) 34.0000 1.41421
\(579\) 0 0
\(580\) −10.0000 −0.415227
\(581\) 3.00000 0.124461
\(582\) 0 0
\(583\) 18.0000 0.745484
\(584\) 0 0
\(585\) 0 0
\(586\) −62.0000 −2.56120
\(587\) −19.0000 −0.784214 −0.392107 0.919920i \(-0.628254\pi\)
−0.392107 + 0.919920i \(0.628254\pi\)
\(588\) 0 0
\(589\) 9.00000 0.370839
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −33.0000 −1.35515 −0.677574 0.735455i \(-0.736969\pi\)
−0.677574 + 0.735455i \(0.736969\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 12.0000 0.491539
\(597\) 0 0
\(598\) −6.00000 −0.245358
\(599\) −27.0000 −1.10319 −0.551595 0.834112i \(-0.685981\pi\)
−0.551595 + 0.834112i \(0.685981\pi\)
\(600\) 0 0
\(601\) 30.0000 1.22373 0.611863 0.790964i \(-0.290420\pi\)
0.611863 + 0.790964i \(0.290420\pi\)
\(602\) −2.00000 −0.0815139
\(603\) 0 0
\(604\) 40.0000 1.62758
\(605\) 7.00000 0.284590
\(606\) 0 0
\(607\) 38.0000 1.54237 0.771186 0.636610i \(-0.219664\pi\)
0.771186 + 0.636610i \(0.219664\pi\)
\(608\) 8.00000 0.324443
\(609\) 0 0
\(610\) −4.00000 −0.161955
\(611\) 3.00000 0.121367
\(612\) 0 0
\(613\) 16.0000 0.646234 0.323117 0.946359i \(-0.395269\pi\)
0.323117 + 0.946359i \(0.395269\pi\)
\(614\) 10.0000 0.403567
\(615\) 0 0
\(616\) 0 0
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) 0 0
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) −18.0000 −0.722897
\(621\) 0 0
\(622\) 12.0000 0.481156
\(623\) −17.0000 −0.681091
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 44.0000 1.75859
\(627\) 0 0
\(628\) 8.00000 0.319235
\(629\) 0 0
\(630\) 0 0
\(631\) −34.0000 −1.35352 −0.676759 0.736204i \(-0.736616\pi\)
−0.676759 + 0.736204i \(0.736616\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −48.0000 −1.90632
\(635\) −12.0000 −0.476205
\(636\) 0 0
\(637\) −1.00000 −0.0396214
\(638\) −20.0000 −0.791808
\(639\) 0 0
\(640\) 0 0
\(641\) 15.0000 0.592464 0.296232 0.955116i \(-0.404270\pi\)
0.296232 + 0.955116i \(0.404270\pi\)
\(642\) 0 0
\(643\) −16.0000 −0.630978 −0.315489 0.948929i \(-0.602169\pi\)
−0.315489 + 0.948929i \(0.602169\pi\)
\(644\) 6.00000 0.236433
\(645\) 0 0
\(646\) 0 0
\(647\) −38.0000 −1.49393 −0.746967 0.664861i \(-0.768491\pi\)
−0.746967 + 0.664861i \(0.768491\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −8.00000 −0.313786
\(651\) 0 0
\(652\) −40.0000 −1.56652
\(653\) −18.0000 −0.704394 −0.352197 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) 0 0
\(655\) −8.00000 −0.312586
\(656\) 8.00000 0.312348
\(657\) 0 0
\(658\) −6.00000 −0.233904
\(659\) 39.0000 1.51922 0.759612 0.650376i \(-0.225389\pi\)
0.759612 + 0.650376i \(0.225389\pi\)
\(660\) 0 0
\(661\) −5.00000 −0.194477 −0.0972387 0.995261i \(-0.531001\pi\)
−0.0972387 + 0.995261i \(0.531001\pi\)
\(662\) −28.0000 −1.08825
\(663\) 0 0
\(664\) 0 0
\(665\) 1.00000 0.0387783
\(666\) 0 0
\(667\) −15.0000 −0.580802
\(668\) −50.0000 −1.93456
\(669\) 0 0
\(670\) 20.0000 0.772667
\(671\) −4.00000 −0.154418
\(672\) 0 0
\(673\) 49.0000 1.88881 0.944406 0.328783i \(-0.106638\pi\)
0.944406 + 0.328783i \(0.106638\pi\)
\(674\) −2.00000 −0.0770371
\(675\) 0 0
\(676\) 2.00000 0.0769231
\(677\) −42.0000 −1.61419 −0.807096 0.590421i \(-0.798962\pi\)
−0.807096 + 0.590421i \(0.798962\pi\)
\(678\) 0 0
\(679\) −3.00000 −0.115129
\(680\) 0 0
\(681\) 0 0
\(682\) −36.0000 −1.37851
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 2.00000 0.0763604
\(687\) 0 0
\(688\) 4.00000 0.152499
\(689\) −9.00000 −0.342873
\(690\) 0 0
\(691\) −25.0000 −0.951045 −0.475522 0.879704i \(-0.657741\pi\)
−0.475522 + 0.879704i \(0.657741\pi\)
\(692\) 8.00000 0.304114
\(693\) 0 0
\(694\) 64.0000 2.42941
\(695\) 6.00000 0.227593
\(696\) 0 0
\(697\) 0 0
\(698\) −62.0000 −2.34673
\(699\) 0 0
\(700\) 8.00000 0.302372
\(701\) −21.0000 −0.793159 −0.396580 0.918000i \(-0.629803\pi\)
−0.396580 + 0.918000i \(0.629803\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −16.0000 −0.603023
\(705\) 0 0
\(706\) 44.0000 1.65596
\(707\) 14.0000 0.526524
\(708\) 0 0
\(709\) −30.0000 −1.12667 −0.563337 0.826227i \(-0.690483\pi\)
−0.563337 + 0.826227i \(0.690483\pi\)
\(710\) 24.0000 0.900704
\(711\) 0 0
\(712\) 0 0
\(713\) −27.0000 −1.01116
\(714\) 0 0
\(715\) 2.00000 0.0747958
\(716\) 2.00000 0.0747435
\(717\) 0 0
\(718\) −32.0000 −1.19423
\(719\) −30.0000 −1.11881 −0.559406 0.828894i \(-0.688971\pi\)
−0.559406 + 0.828894i \(0.688971\pi\)
\(720\) 0 0
\(721\) 4.00000 0.148968
\(722\) 36.0000 1.33978
\(723\) 0 0
\(724\) −28.0000 −1.04061
\(725\) −20.0000 −0.742781
\(726\) 0 0
\(727\) 38.0000 1.40934 0.704671 0.709534i \(-0.251095\pi\)
0.704671 + 0.709534i \(0.251095\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 30.0000 1.11035
\(731\) 0 0
\(732\) 0 0
\(733\) 15.0000 0.554038 0.277019 0.960864i \(-0.410654\pi\)
0.277019 + 0.960864i \(0.410654\pi\)
\(734\) −36.0000 −1.32878
\(735\) 0 0
\(736\) −24.0000 −0.884652
\(737\) 20.0000 0.736709
\(738\) 0 0
\(739\) −6.00000 −0.220714 −0.110357 0.993892i \(-0.535199\pi\)
−0.110357 + 0.993892i \(0.535199\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 18.0000 0.660801
\(743\) 12.0000 0.440237 0.220119 0.975473i \(-0.429356\pi\)
0.220119 + 0.975473i \(0.429356\pi\)
\(744\) 0 0
\(745\) −6.00000 −0.219823
\(746\) −76.0000 −2.78256
\(747\) 0 0
\(748\) 0 0
\(749\) 4.00000 0.146157
\(750\) 0 0
\(751\) −41.0000 −1.49611 −0.748056 0.663636i \(-0.769012\pi\)
−0.748056 + 0.663636i \(0.769012\pi\)
\(752\) 12.0000 0.437595
\(753\) 0 0
\(754\) 10.0000 0.364179
\(755\) −20.0000 −0.727875
\(756\) 0 0
\(757\) 41.0000 1.49017 0.745085 0.666969i \(-0.232409\pi\)
0.745085 + 0.666969i \(0.232409\pi\)
\(758\) 36.0000 1.30758
\(759\) 0 0
\(760\) 0 0
\(761\) −21.0000 −0.761249 −0.380625 0.924730i \(-0.624291\pi\)
−0.380625 + 0.924730i \(0.624291\pi\)
\(762\) 0 0
\(763\) −18.0000 −0.651644
\(764\) −32.0000 −1.15772
\(765\) 0 0
\(766\) 72.0000 2.60147
\(767\) 0 0
\(768\) 0 0
\(769\) −7.00000 −0.252426 −0.126213 0.992003i \(-0.540282\pi\)
−0.126213 + 0.992003i \(0.540282\pi\)
\(770\) −4.00000 −0.144150
\(771\) 0 0
\(772\) −20.0000 −0.719816
\(773\) 34.0000 1.22290 0.611448 0.791285i \(-0.290588\pi\)
0.611448 + 0.791285i \(0.290588\pi\)
\(774\) 0 0
\(775\) −36.0000 −1.29316
\(776\) 0 0
\(777\) 0 0
\(778\) −36.0000 −1.29066
\(779\) −2.00000 −0.0716574
\(780\) 0 0
\(781\) 24.0000 0.858788
\(782\) 0 0
\(783\) 0 0
\(784\) −4.00000 −0.142857
\(785\) −4.00000 −0.142766
\(786\) 0 0
\(787\) −47.0000 −1.67537 −0.837685 0.546154i \(-0.816091\pi\)
−0.837685 + 0.546154i \(0.816091\pi\)
\(788\) −20.0000 −0.712470
\(789\) 0 0
\(790\) 22.0000 0.782725
\(791\) −11.0000 −0.391115
\(792\) 0 0
\(793\) 2.00000 0.0710221
\(794\) −30.0000 −1.06466
\(795\) 0 0
\(796\) 0 0
\(797\) 38.0000 1.34603 0.673015 0.739629i \(-0.264999\pi\)
0.673015 + 0.739629i \(0.264999\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −32.0000 −1.13137
\(801\) 0 0
\(802\) 0 0
\(803\) 30.0000 1.05868
\(804\) 0 0
\(805\) −3.00000 −0.105736
\(806\) 18.0000 0.634023
\(807\) 0 0
\(808\) 0 0
\(809\) 15.0000 0.527372 0.263686 0.964609i \(-0.415062\pi\)
0.263686 + 0.964609i \(0.415062\pi\)
\(810\) 0 0
\(811\) −4.00000 −0.140459 −0.0702295 0.997531i \(-0.522373\pi\)
−0.0702295 + 0.997531i \(0.522373\pi\)
\(812\) −10.0000 −0.350931
\(813\) 0 0
\(814\) 0 0
\(815\) 20.0000 0.700569
\(816\) 0 0
\(817\) −1.00000 −0.0349856
\(818\) −46.0000 −1.60835
\(819\) 0 0
\(820\) 4.00000 0.139686
\(821\) 46.0000 1.60541 0.802706 0.596376i \(-0.203393\pi\)
0.802706 + 0.596376i \(0.203393\pi\)
\(822\) 0 0
\(823\) 24.0000 0.836587 0.418294 0.908312i \(-0.362628\pi\)
0.418294 + 0.908312i \(0.362628\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 12.0000 0.417281 0.208640 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) 0 0
\(829\) 26.0000 0.903017 0.451509 0.892267i \(-0.350886\pi\)
0.451509 + 0.892267i \(0.350886\pi\)
\(830\) −6.00000 −0.208263
\(831\) 0 0
\(832\) 8.00000 0.277350
\(833\) 0 0
\(834\) 0 0
\(835\) 25.0000 0.865161
\(836\) 4.00000 0.138343
\(837\) 0 0
\(838\) −60.0000 −2.07267
\(839\) 32.0000 1.10476 0.552381 0.833592i \(-0.313719\pi\)
0.552381 + 0.833592i \(0.313719\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) −16.0000 −0.551396
\(843\) 0 0
\(844\) −10.0000 −0.344214
\(845\) −1.00000 −0.0344010
\(846\) 0 0
\(847\) 7.00000 0.240523
\(848\) −36.0000 −1.23625
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 17.0000 0.582069 0.291034 0.956713i \(-0.406001\pi\)
0.291034 + 0.956713i \(0.406001\pi\)
\(854\) −4.00000 −0.136877
\(855\) 0 0
\(856\) 0 0
\(857\) 38.0000 1.29806 0.649028 0.760765i \(-0.275176\pi\)
0.649028 + 0.760765i \(0.275176\pi\)
\(858\) 0 0
\(859\) −58.0000 −1.97893 −0.989467 0.144757i \(-0.953760\pi\)
−0.989467 + 0.144757i \(0.953760\pi\)
\(860\) 2.00000 0.0681994
\(861\) 0 0
\(862\) 60.0000 2.04361
\(863\) −48.0000 −1.63394 −0.816970 0.576681i \(-0.804348\pi\)
−0.816970 + 0.576681i \(0.804348\pi\)
\(864\) 0 0
\(865\) −4.00000 −0.136004
\(866\) 40.0000 1.35926
\(867\) 0 0
\(868\) −18.0000 −0.610960
\(869\) 22.0000 0.746299
\(870\) 0 0
\(871\) −10.0000 −0.338837
\(872\) 0 0
\(873\) 0 0
\(874\) 6.00000 0.202953
\(875\) −9.00000 −0.304256
\(876\) 0 0
\(877\) 4.00000 0.135070 0.0675352 0.997717i \(-0.478487\pi\)
0.0675352 + 0.997717i \(0.478487\pi\)
\(878\) 12.0000 0.404980
\(879\) 0 0
\(880\) 8.00000 0.269680
\(881\) −42.0000 −1.41502 −0.707508 0.706705i \(-0.750181\pi\)
−0.707508 + 0.706705i \(0.750181\pi\)
\(882\) 0 0
\(883\) 44.0000 1.48072 0.740359 0.672212i \(-0.234656\pi\)
0.740359 + 0.672212i \(0.234656\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 54.0000 1.81417
\(887\) −20.0000 −0.671534 −0.335767 0.941945i \(-0.608996\pi\)
−0.335767 + 0.941945i \(0.608996\pi\)
\(888\) 0 0
\(889\) −12.0000 −0.402467
\(890\) 34.0000 1.13968
\(891\) 0 0
\(892\) −58.0000 −1.94198
\(893\) −3.00000 −0.100391
\(894\) 0 0
\(895\) −1.00000 −0.0334263
\(896\) 0 0
\(897\) 0 0
\(898\) −24.0000 −0.800890
\(899\) 45.0000 1.50083
\(900\) 0 0
\(901\) 0 0
\(902\) 8.00000 0.266371
\(903\) 0 0
\(904\) 0 0
\(905\) 14.0000 0.465376
\(906\) 0 0
\(907\) −25.0000 −0.830111 −0.415056 0.909796i \(-0.636238\pi\)
−0.415056 + 0.909796i \(0.636238\pi\)
\(908\) −24.0000 −0.796468
\(909\) 0 0
\(910\) 2.00000 0.0662994
\(911\) −9.00000 −0.298183 −0.149092 0.988823i \(-0.547635\pi\)
−0.149092 + 0.988823i \(0.547635\pi\)
\(912\) 0 0
\(913\) −6.00000 −0.198571
\(914\) −56.0000 −1.85232
\(915\) 0 0
\(916\) 28.0000 0.925146
\(917\) −8.00000 −0.264183
\(918\) 0 0
\(919\) −40.0000 −1.31948 −0.659739 0.751495i \(-0.729333\pi\)
−0.659739 + 0.751495i \(0.729333\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 4.00000 0.131733
\(923\) −12.0000 −0.394985
\(924\) 0 0
\(925\) 0 0
\(926\) −44.0000 −1.44593
\(927\) 0 0
\(928\) 40.0000 1.31306
\(929\) −41.0000 −1.34517 −0.672583 0.740022i \(-0.734815\pi\)
−0.672583 + 0.740022i \(0.734815\pi\)
\(930\) 0 0
\(931\) 1.00000 0.0327737
\(932\) 18.0000 0.589610
\(933\) 0 0
\(934\) 12.0000 0.392652
\(935\) 0 0
\(936\) 0 0
\(937\) 36.0000 1.17607 0.588034 0.808836i \(-0.299902\pi\)
0.588034 + 0.808836i \(0.299902\pi\)
\(938\) 20.0000 0.653023
\(939\) 0 0
\(940\) 6.00000 0.195698
\(941\) 13.0000 0.423788 0.211894 0.977293i \(-0.432037\pi\)
0.211894 + 0.977293i \(0.432037\pi\)
\(942\) 0 0
\(943\) 6.00000 0.195387
\(944\) 0 0
\(945\) 0 0
\(946\) 4.00000 0.130051
\(947\) 58.0000 1.88475 0.942373 0.334563i \(-0.108589\pi\)
0.942373 + 0.334563i \(0.108589\pi\)
\(948\) 0 0
\(949\) −15.0000 −0.486921
\(950\) 8.00000 0.259554
\(951\) 0 0
\(952\) 0 0
\(953\) −41.0000 −1.32812 −0.664060 0.747679i \(-0.731168\pi\)
−0.664060 + 0.747679i \(0.731168\pi\)
\(954\) 0 0
\(955\) 16.0000 0.517748
\(956\) 24.0000 0.776215
\(957\) 0 0
\(958\) 34.0000 1.09849
\(959\) 0 0
\(960\) 0 0
\(961\) 50.0000 1.61290
\(962\) 0 0
\(963\) 0 0
\(964\) −26.0000 −0.837404
\(965\) 10.0000 0.321911
\(966\) 0 0
\(967\) 34.0000 1.09337 0.546683 0.837340i \(-0.315890\pi\)
0.546683 + 0.837340i \(0.315890\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 6.00000 0.192648
\(971\) 42.0000 1.34784 0.673922 0.738802i \(-0.264608\pi\)
0.673922 + 0.738802i \(0.264608\pi\)
\(972\) 0 0
\(973\) 6.00000 0.192351
\(974\) 4.00000 0.128168
\(975\) 0 0
\(976\) 8.00000 0.256074
\(977\) −30.0000 −0.959785 −0.479893 0.877327i \(-0.659324\pi\)
−0.479893 + 0.877327i \(0.659324\pi\)
\(978\) 0 0
\(979\) 34.0000 1.08664
\(980\) −2.00000 −0.0638877
\(981\) 0 0
\(982\) −56.0000 −1.78703
\(983\) −45.0000 −1.43528 −0.717639 0.696416i \(-0.754777\pi\)
−0.717639 + 0.696416i \(0.754777\pi\)
\(984\) 0 0
\(985\) 10.0000 0.318626
\(986\) 0 0
\(987\) 0 0
\(988\) −2.00000 −0.0636285
\(989\) 3.00000 0.0953945
\(990\) 0 0
\(991\) 20.0000 0.635321 0.317660 0.948205i \(-0.397103\pi\)
0.317660 + 0.948205i \(0.397103\pi\)
\(992\) 72.0000 2.28600
\(993\) 0 0
\(994\) 24.0000 0.761234
\(995\) 0 0
\(996\) 0 0
\(997\) −20.0000 −0.633406 −0.316703 0.948525i \(-0.602576\pi\)
−0.316703 + 0.948525i \(0.602576\pi\)
\(998\) −56.0000 −1.77265
\(999\) 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 819.2.a.a.1.1 1
3.2 odd 2 273.2.a.b.1.1 1
7.6 odd 2 5733.2.a.a.1.1 1
12.11 even 2 4368.2.a.i.1.1 1
15.14 odd 2 6825.2.a.a.1.1 1
21.20 even 2 1911.2.a.g.1.1 1
39.38 odd 2 3549.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
273.2.a.b.1.1 1 3.2 odd 2
819.2.a.a.1.1 1 1.1 even 1 trivial
1911.2.a.g.1.1 1 21.20 even 2
3549.2.a.b.1.1 1 39.38 odd 2
4368.2.a.i.1.1 1 12.11 even 2
5733.2.a.a.1.1 1 7.6 odd 2
6825.2.a.a.1.1 1 15.14 odd 2