Properties

Label 1089.2.a.b.1.1
Level $1089$
Weight $2$
Character 1089.1
Self dual yes
Analytic conductor $8.696$
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] = [1089,2,Mod(1,1089)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1089, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1089.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1089.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: \(8.69570878012\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 11)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1089.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} +2.00000 q^{4} -1.00000 q^{5} +2.00000 q^{7} +O(q^{10})\) \(q-2.00000 q^{2} +2.00000 q^{4} -1.00000 q^{5} +2.00000 q^{7} +2.00000 q^{10} -4.00000 q^{13} -4.00000 q^{14} -4.00000 q^{16} -2.00000 q^{17} -2.00000 q^{20} +1.00000 q^{23} -4.00000 q^{25} +8.00000 q^{26} +4.00000 q^{28} +7.00000 q^{31} +8.00000 q^{32} +4.00000 q^{34} -2.00000 q^{35} +3.00000 q^{37} -8.00000 q^{41} +6.00000 q^{43} -2.00000 q^{46} -8.00000 q^{47} -3.00000 q^{49} +8.00000 q^{50} -8.00000 q^{52} +6.00000 q^{53} -5.00000 q^{59} -12.0000 q^{61} -14.0000 q^{62} -8.00000 q^{64} +4.00000 q^{65} -7.00000 q^{67} -4.00000 q^{68} +4.00000 q^{70} +3.00000 q^{71} -4.00000 q^{73} -6.00000 q^{74} +10.0000 q^{79} +4.00000 q^{80} +16.0000 q^{82} -6.00000 q^{83} +2.00000 q^{85} -12.0000 q^{86} -15.0000 q^{89} -8.00000 q^{91} +2.00000 q^{92} +16.0000 q^{94} -7.00000 q^{97} +6.00000 q^{98} +O(q^{100})\)

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
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 2.00000 0.632456
\(11\) 0 0
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) −4.00000 −1.06904
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) 0 0
\(23\) 1.00000 0.208514 0.104257 0.994550i \(-0.466753\pi\)
0.104257 + 0.994550i \(0.466753\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 8.00000 1.56893
\(27\) 0 0
\(28\) 4.00000 0.755929
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 7.00000 1.25724 0.628619 0.777714i \(-0.283621\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) 8.00000 1.41421
\(33\) 0 0
\(34\) 4.00000 0.685994
\(35\) −2.00000 −0.338062
\(36\) 0 0
\(37\) 3.00000 0.493197 0.246598 0.969118i \(-0.420687\pi\)
0.246598 + 0.969118i \(0.420687\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −8.00000 −1.24939 −0.624695 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 0 0
\(43\) 6.00000 0.914991 0.457496 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −2.00000 −0.294884
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 8.00000 1.13137
\(51\) 0 0
\(52\) −8.00000 −1.10940
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −5.00000 −0.650945 −0.325472 0.945552i \(-0.605523\pi\)
−0.325472 + 0.945552i \(0.605523\pi\)
\(60\) 0 0
\(61\) −12.0000 −1.53644 −0.768221 0.640184i \(-0.778858\pi\)
−0.768221 + 0.640184i \(0.778858\pi\)
\(62\) −14.0000 −1.77800
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) −7.00000 −0.855186 −0.427593 0.903971i \(-0.640638\pi\)
−0.427593 + 0.903971i \(0.640638\pi\)
\(68\) −4.00000 −0.485071
\(69\) 0 0
\(70\) 4.00000 0.478091
\(71\) 3.00000 0.356034 0.178017 0.984027i \(-0.443032\pi\)
0.178017 + 0.984027i \(0.443032\pi\)
\(72\) 0 0
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) −6.00000 −0.697486
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 4.00000 0.447214
\(81\) 0 0
\(82\) 16.0000 1.76690
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) −12.0000 −1.29399
\(87\) 0 0
\(88\) 0 0
\(89\) −15.0000 −1.59000 −0.794998 0.606612i \(-0.792528\pi\)
−0.794998 + 0.606612i \(0.792528\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) 2.00000 0.208514
\(93\) 0 0
\(94\) 16.0000 1.65027
\(95\) 0 0
\(96\) 0 0
\(97\) −7.00000 −0.710742 −0.355371 0.934725i \(-0.615646\pi\)
−0.355371 + 0.934725i \(0.615646\pi\)
\(98\) 6.00000 0.606092
\(99\) 0 0
\(100\) −8.00000 −0.800000
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) 0 0
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −12.0000 −1.16554
\(107\) 18.0000 1.74013 0.870063 0.492941i \(-0.164078\pi\)
0.870063 + 0.492941i \(0.164078\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −8.00000 −0.755929
\(113\) −9.00000 −0.846649 −0.423324 0.905978i \(-0.639137\pi\)
−0.423324 + 0.905978i \(0.639137\pi\)
\(114\) 0 0
\(115\) −1.00000 −0.0932505
\(116\) 0 0
\(117\) 0 0
\(118\) 10.0000 0.920575
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) 0 0
\(122\) 24.0000 2.17286
\(123\) 0 0
\(124\) 14.0000 1.25724
\(125\) 9.00000 0.804984
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) −8.00000 −0.701646
\(131\) −18.0000 −1.57267 −0.786334 0.617802i \(-0.788023\pi\)
−0.786334 + 0.617802i \(0.788023\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 14.0000 1.20942
\(135\) 0 0
\(136\) 0 0
\(137\) 7.00000 0.598050 0.299025 0.954245i \(-0.403339\pi\)
0.299025 + 0.954245i \(0.403339\pi\)
\(138\) 0 0
\(139\) −10.0000 −0.848189 −0.424094 0.905618i \(-0.639408\pi\)
−0.424094 + 0.905618i \(0.639408\pi\)
\(140\) −4.00000 −0.338062
\(141\) 0 0
\(142\) −6.00000 −0.503509
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 8.00000 0.662085
\(147\) 0 0
\(148\) 6.00000 0.493197
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) −2.00000 −0.162758 −0.0813788 0.996683i \(-0.525932\pi\)
−0.0813788 + 0.996683i \(0.525932\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −7.00000 −0.562254
\(156\) 0 0
\(157\) −7.00000 −0.558661 −0.279330 0.960195i \(-0.590112\pi\)
−0.279330 + 0.960195i \(0.590112\pi\)
\(158\) −20.0000 −1.59111
\(159\) 0 0
\(160\) −8.00000 −0.632456
\(161\) 2.00000 0.157622
\(162\) 0 0
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) −16.0000 −1.24939
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) −4.00000 −0.306786
\(171\) 0 0
\(172\) 12.0000 0.914991
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) −8.00000 −0.604743
\(176\) 0 0
\(177\) 0 0
\(178\) 30.0000 2.24860
\(179\) 15.0000 1.12115 0.560576 0.828103i \(-0.310580\pi\)
0.560576 + 0.828103i \(0.310580\pi\)
\(180\) 0 0
\(181\) 7.00000 0.520306 0.260153 0.965567i \(-0.416227\pi\)
0.260153 + 0.965567i \(0.416227\pi\)
\(182\) 16.0000 1.18600
\(183\) 0 0
\(184\) 0 0
\(185\) −3.00000 −0.220564
\(186\) 0 0
\(187\) 0 0
\(188\) −16.0000 −1.16692
\(189\) 0 0
\(190\) 0 0
\(191\) −17.0000 −1.23008 −0.615038 0.788497i \(-0.710860\pi\)
−0.615038 + 0.788497i \(0.710860\pi\)
\(192\) 0 0
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) 14.0000 1.00514
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −4.00000 −0.281439
\(203\) 0 0
\(204\) 0 0
\(205\) 8.00000 0.558744
\(206\) 32.0000 2.22955
\(207\) 0 0
\(208\) 16.0000 1.10940
\(209\) 0 0
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 12.0000 0.824163
\(213\) 0 0
\(214\) −36.0000 −2.46091
\(215\) −6.00000 −0.409197
\(216\) 0 0
\(217\) 14.0000 0.950382
\(218\) 20.0000 1.35457
\(219\) 0 0
\(220\) 0 0
\(221\) 8.00000 0.538138
\(222\) 0 0
\(223\) 19.0000 1.27233 0.636167 0.771551i \(-0.280519\pi\)
0.636167 + 0.771551i \(0.280519\pi\)
\(224\) 16.0000 1.06904
\(225\) 0 0
\(226\) 18.0000 1.19734
\(227\) 18.0000 1.19470 0.597351 0.801980i \(-0.296220\pi\)
0.597351 + 0.801980i \(0.296220\pi\)
\(228\) 0 0
\(229\) 15.0000 0.991228 0.495614 0.868543i \(-0.334943\pi\)
0.495614 + 0.868543i \(0.334943\pi\)
\(230\) 2.00000 0.131876
\(231\) 0 0
\(232\) 0 0
\(233\) 24.0000 1.57229 0.786146 0.618041i \(-0.212073\pi\)
0.786146 + 0.618041i \(0.212073\pi\)
\(234\) 0 0
\(235\) 8.00000 0.521862
\(236\) −10.0000 −0.650945
\(237\) 0 0
\(238\) 8.00000 0.518563
\(239\) −30.0000 −1.94054 −0.970269 0.242028i \(-0.922188\pi\)
−0.970269 + 0.242028i \(0.922188\pi\)
\(240\) 0 0
\(241\) 8.00000 0.515325 0.257663 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −24.0000 −1.53644
\(245\) 3.00000 0.191663
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) −18.0000 −1.13842
\(251\) 23.0000 1.45175 0.725874 0.687828i \(-0.241436\pi\)
0.725874 + 0.687828i \(0.241436\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 16.0000 1.00393
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 2.00000 0.124757 0.0623783 0.998053i \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) 0 0
\(259\) 6.00000 0.372822
\(260\) 8.00000 0.496139
\(261\) 0 0
\(262\) 36.0000 2.22409
\(263\) 14.0000 0.863277 0.431638 0.902047i \(-0.357936\pi\)
0.431638 + 0.902047i \(0.357936\pi\)
\(264\) 0 0
\(265\) −6.00000 −0.368577
\(266\) 0 0
\(267\) 0 0
\(268\) −14.0000 −0.855186
\(269\) −10.0000 −0.609711 −0.304855 0.952399i \(-0.598608\pi\)
−0.304855 + 0.952399i \(0.598608\pi\)
\(270\) 0 0
\(271\) 28.0000 1.70088 0.850439 0.526073i \(-0.176336\pi\)
0.850439 + 0.526073i \(0.176336\pi\)
\(272\) 8.00000 0.485071
\(273\) 0 0
\(274\) −14.0000 −0.845771
\(275\) 0 0
\(276\) 0 0
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) 20.0000 1.19952
\(279\) 0 0
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 6.00000 0.356034
\(285\) 0 0
\(286\) 0 0
\(287\) −16.0000 −0.944450
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) −8.00000 −0.468165
\(293\) 24.0000 1.40209 0.701047 0.713115i \(-0.252716\pi\)
0.701047 + 0.713115i \(0.252716\pi\)
\(294\) 0 0
\(295\) 5.00000 0.291111
\(296\) 0 0
\(297\) 0 0
\(298\) 20.0000 1.15857
\(299\) −4.00000 −0.231326
\(300\) 0 0
\(301\) 12.0000 0.691669
\(302\) 4.00000 0.230174
\(303\) 0 0
\(304\) 0 0
\(305\) 12.0000 0.687118
\(306\) 0 0
\(307\) −8.00000 −0.456584 −0.228292 0.973593i \(-0.573314\pi\)
−0.228292 + 0.973593i \(0.573314\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 14.0000 0.795147
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) −1.00000 −0.0565233 −0.0282617 0.999601i \(-0.508997\pi\)
−0.0282617 + 0.999601i \(0.508997\pi\)
\(314\) 14.0000 0.790066
\(315\) 0 0
\(316\) 20.0000 1.12509
\(317\) −13.0000 −0.730153 −0.365076 0.930978i \(-0.618957\pi\)
−0.365076 + 0.930978i \(0.618957\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 8.00000 0.447214
\(321\) 0 0
\(322\) −4.00000 −0.222911
\(323\) 0 0
\(324\) 0 0
\(325\) 16.0000 0.887520
\(326\) −8.00000 −0.443079
\(327\) 0 0
\(328\) 0 0
\(329\) −16.0000 −0.882109
\(330\) 0 0
\(331\) 7.00000 0.384755 0.192377 0.981321i \(-0.438380\pi\)
0.192377 + 0.981321i \(0.438380\pi\)
\(332\) −12.0000 −0.658586
\(333\) 0 0
\(334\) 24.0000 1.31322
\(335\) 7.00000 0.382451
\(336\) 0 0
\(337\) 22.0000 1.19842 0.599208 0.800593i \(-0.295482\pi\)
0.599208 + 0.800593i \(0.295482\pi\)
\(338\) −6.00000 −0.326357
\(339\) 0 0
\(340\) 4.00000 0.216930
\(341\) 0 0
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 0 0
\(345\) 0 0
\(346\) 12.0000 0.645124
\(347\) 28.0000 1.50312 0.751559 0.659665i \(-0.229302\pi\)
0.751559 + 0.659665i \(0.229302\pi\)
\(348\) 0 0
\(349\) −30.0000 −1.60586 −0.802932 0.596071i \(-0.796728\pi\)
−0.802932 + 0.596071i \(0.796728\pi\)
\(350\) 16.0000 0.855236
\(351\) 0 0
\(352\) 0 0
\(353\) 21.0000 1.11772 0.558859 0.829263i \(-0.311239\pi\)
0.558859 + 0.829263i \(0.311239\pi\)
\(354\) 0 0
\(355\) −3.00000 −0.159223
\(356\) −30.0000 −1.59000
\(357\) 0 0
\(358\) −30.0000 −1.58555
\(359\) −20.0000 −1.05556 −0.527780 0.849381i \(-0.676975\pi\)
−0.527780 + 0.849381i \(0.676975\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) −14.0000 −0.735824
\(363\) 0 0
\(364\) −16.0000 −0.838628
\(365\) 4.00000 0.209370
\(366\) 0 0
\(367\) −17.0000 −0.887393 −0.443696 0.896177i \(-0.646333\pi\)
−0.443696 + 0.896177i \(0.646333\pi\)
\(368\) −4.00000 −0.208514
\(369\) 0 0
\(370\) 6.00000 0.311925
\(371\) 12.0000 0.623009
\(372\) 0 0
\(373\) 26.0000 1.34623 0.673114 0.739538i \(-0.264956\pi\)
0.673114 + 0.739538i \(0.264956\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −5.00000 −0.256833 −0.128416 0.991720i \(-0.540989\pi\)
−0.128416 + 0.991720i \(0.540989\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 34.0000 1.73959
\(383\) 1.00000 0.0510976 0.0255488 0.999674i \(-0.491867\pi\)
0.0255488 + 0.999674i \(0.491867\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 8.00000 0.407189
\(387\) 0 0
\(388\) −14.0000 −0.710742
\(389\) 15.0000 0.760530 0.380265 0.924878i \(-0.375833\pi\)
0.380265 + 0.924878i \(0.375833\pi\)
\(390\) 0 0
\(391\) −2.00000 −0.101144
\(392\) 0 0
\(393\) 0 0
\(394\) 4.00000 0.201517
\(395\) −10.0000 −0.503155
\(396\) 0 0
\(397\) −2.00000 −0.100377 −0.0501886 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 16.0000 0.800000
\(401\) −2.00000 −0.0998752 −0.0499376 0.998752i \(-0.515902\pi\)
−0.0499376 + 0.998752i \(0.515902\pi\)
\(402\) 0 0
\(403\) −28.0000 −1.39478
\(404\) 4.00000 0.199007
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 30.0000 1.48340 0.741702 0.670729i \(-0.234019\pi\)
0.741702 + 0.670729i \(0.234019\pi\)
\(410\) −16.0000 −0.790184
\(411\) 0 0
\(412\) −32.0000 −1.57653
\(413\) −10.0000 −0.492068
\(414\) 0 0
\(415\) 6.00000 0.294528
\(416\) −32.0000 −1.56893
\(417\) 0 0
\(418\) 0 0
\(419\) −20.0000 −0.977064 −0.488532 0.872546i \(-0.662467\pi\)
−0.488532 + 0.872546i \(0.662467\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) 24.0000 1.16830
\(423\) 0 0
\(424\) 0 0
\(425\) 8.00000 0.388057
\(426\) 0 0
\(427\) −24.0000 −1.16144
\(428\) 36.0000 1.74013
\(429\) 0 0
\(430\) 12.0000 0.578691
\(431\) −18.0000 −0.867029 −0.433515 0.901146i \(-0.642727\pi\)
−0.433515 + 0.901146i \(0.642727\pi\)
\(432\) 0 0
\(433\) −11.0000 −0.528626 −0.264313 0.964437i \(-0.585145\pi\)
−0.264313 + 0.964437i \(0.585145\pi\)
\(434\) −28.0000 −1.34404
\(435\) 0 0
\(436\) −20.0000 −0.957826
\(437\) 0 0
\(438\) 0 0
\(439\) −40.0000 −1.90910 −0.954548 0.298057i \(-0.903661\pi\)
−0.954548 + 0.298057i \(0.903661\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −16.0000 −0.761042
\(443\) 11.0000 0.522626 0.261313 0.965254i \(-0.415845\pi\)
0.261313 + 0.965254i \(0.415845\pi\)
\(444\) 0 0
\(445\) 15.0000 0.711068
\(446\) −38.0000 −1.79935
\(447\) 0 0
\(448\) −16.0000 −0.755929
\(449\) −35.0000 −1.65175 −0.825876 0.563852i \(-0.809319\pi\)
−0.825876 + 0.563852i \(0.809319\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −18.0000 −0.846649
\(453\) 0 0
\(454\) −36.0000 −1.68956
\(455\) 8.00000 0.375046
\(456\) 0 0
\(457\) 12.0000 0.561336 0.280668 0.959805i \(-0.409444\pi\)
0.280668 + 0.959805i \(0.409444\pi\)
\(458\) −30.0000 −1.40181
\(459\) 0 0
\(460\) −2.00000 −0.0932505
\(461\) 12.0000 0.558896 0.279448 0.960161i \(-0.409849\pi\)
0.279448 + 0.960161i \(0.409849\pi\)
\(462\) 0 0
\(463\) −11.0000 −0.511213 −0.255607 0.966781i \(-0.582275\pi\)
−0.255607 + 0.966781i \(0.582275\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −48.0000 −2.22356
\(467\) 27.0000 1.24941 0.624705 0.780860i \(-0.285219\pi\)
0.624705 + 0.780860i \(0.285219\pi\)
\(468\) 0 0
\(469\) −14.0000 −0.646460
\(470\) −16.0000 −0.738025
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) −8.00000 −0.366679
\(477\) 0 0
\(478\) 60.0000 2.74434
\(479\) 20.0000 0.913823 0.456912 0.889512i \(-0.348956\pi\)
0.456912 + 0.889512i \(0.348956\pi\)
\(480\) 0 0
\(481\) −12.0000 −0.547153
\(482\) −16.0000 −0.728780
\(483\) 0 0
\(484\) 0 0
\(485\) 7.00000 0.317854
\(486\) 0 0
\(487\) 23.0000 1.04223 0.521115 0.853487i \(-0.325516\pi\)
0.521115 + 0.853487i \(0.325516\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) −6.00000 −0.271052
\(491\) −8.00000 −0.361035 −0.180517 0.983572i \(-0.557777\pi\)
−0.180517 + 0.983572i \(0.557777\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −28.0000 −1.25724
\(497\) 6.00000 0.269137
\(498\) 0 0
\(499\) 20.0000 0.895323 0.447661 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) 18.0000 0.804984
\(501\) 0 0
\(502\) −46.0000 −2.05308
\(503\) −26.0000 −1.15928 −0.579641 0.814872i \(-0.696807\pi\)
−0.579641 + 0.814872i \(0.696807\pi\)
\(504\) 0 0
\(505\) −2.00000 −0.0889988
\(506\) 0 0
\(507\) 0 0
\(508\) −16.0000 −0.709885
\(509\) −15.0000 −0.664863 −0.332432 0.943127i \(-0.607869\pi\)
−0.332432 + 0.943127i \(0.607869\pi\)
\(510\) 0 0
\(511\) −8.00000 −0.353899
\(512\) −32.0000 −1.41421
\(513\) 0 0
\(514\) −4.00000 −0.176432
\(515\) 16.0000 0.705044
\(516\) 0 0
\(517\) 0 0
\(518\) −12.0000 −0.527250
\(519\) 0 0
\(520\) 0 0
\(521\) 3.00000 0.131432 0.0657162 0.997838i \(-0.479067\pi\)
0.0657162 + 0.997838i \(0.479067\pi\)
\(522\) 0 0
\(523\) 16.0000 0.699631 0.349816 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) −36.0000 −1.57267
\(525\) 0 0
\(526\) −28.0000 −1.22086
\(527\) −14.0000 −0.609850
\(528\) 0 0
\(529\) −22.0000 −0.956522
\(530\) 12.0000 0.521247
\(531\) 0 0
\(532\) 0 0
\(533\) 32.0000 1.38607
\(534\) 0 0
\(535\) −18.0000 −0.778208
\(536\) 0 0
\(537\) 0 0
\(538\) 20.0000 0.862261
\(539\) 0 0
\(540\) 0 0
\(541\) 8.00000 0.343947 0.171973 0.985102i \(-0.444986\pi\)
0.171973 + 0.985102i \(0.444986\pi\)
\(542\) −56.0000 −2.40541
\(543\) 0 0
\(544\) −16.0000 −0.685994
\(545\) 10.0000 0.428353
\(546\) 0 0
\(547\) −8.00000 −0.342055 −0.171028 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) 14.0000 0.598050
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 20.0000 0.850487
\(554\) −4.00000 −0.169944
\(555\) 0 0
\(556\) −20.0000 −0.848189
\(557\) −2.00000 −0.0847427 −0.0423714 0.999102i \(-0.513491\pi\)
−0.0423714 + 0.999102i \(0.513491\pi\)
\(558\) 0 0
\(559\) −24.0000 −1.01509
\(560\) 8.00000 0.338062
\(561\) 0 0
\(562\) 36.0000 1.51857
\(563\) 4.00000 0.168580 0.0842900 0.996441i \(-0.473138\pi\)
0.0842900 + 0.996441i \(0.473138\pi\)
\(564\) 0 0
\(565\) 9.00000 0.378633
\(566\) 8.00000 0.336265
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 28.0000 1.17176 0.585882 0.810397i \(-0.300748\pi\)
0.585882 + 0.810397i \(0.300748\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 32.0000 1.33565
\(575\) −4.00000 −0.166812
\(576\) 0 0
\(577\) 33.0000 1.37381 0.686904 0.726748i \(-0.258969\pi\)
0.686904 + 0.726748i \(0.258969\pi\)
\(578\) 26.0000 1.08146
\(579\) 0 0
\(580\) 0 0
\(581\) −12.0000 −0.497844
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) −48.0000 −1.98286
\(587\) −28.0000 −1.15568 −0.577842 0.816149i \(-0.696105\pi\)
−0.577842 + 0.816149i \(0.696105\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) −10.0000 −0.411693
\(591\) 0 0
\(592\) −12.0000 −0.493197
\(593\) 44.0000 1.80686 0.903432 0.428732i \(-0.141040\pi\)
0.903432 + 0.428732i \(0.141040\pi\)
\(594\) 0 0
\(595\) 4.00000 0.163984
\(596\) −20.0000 −0.819232
\(597\) 0 0
\(598\) 8.00000 0.327144
\(599\) −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 \(0.512988\pi\)
\(602\) −24.0000 −0.978167
\(603\) 0 0
\(604\) −4.00000 −0.162758
\(605\) 0 0
\(606\) 0 0
\(607\) 22.0000 0.892952 0.446476 0.894795i \(-0.352679\pi\)
0.446476 + 0.894795i \(0.352679\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) −24.0000 −0.971732
\(611\) 32.0000 1.29458
\(612\) 0 0
\(613\) 16.0000 0.646234 0.323117 0.946359i \(-0.395269\pi\)
0.323117 + 0.946359i \(0.395269\pi\)
\(614\) 16.0000 0.645707
\(615\) 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
\(619\) −25.0000 −1.00483 −0.502417 0.864625i \(-0.667556\pi\)
−0.502417 + 0.864625i \(0.667556\pi\)
\(620\) −14.0000 −0.562254
\(621\) 0 0
\(622\) 24.0000 0.962312
\(623\) −30.0000 −1.20192
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 2.00000 0.0799361
\(627\) 0 0
\(628\) −14.0000 −0.558661
\(629\) −6.00000 −0.239236
\(630\) 0 0
\(631\) 7.00000 0.278666 0.139333 0.990246i \(-0.455504\pi\)
0.139333 + 0.990246i \(0.455504\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 26.0000 1.03259
\(635\) 8.00000 0.317470
\(636\) 0 0
\(637\) 12.0000 0.475457
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 33.0000 1.30342 0.651711 0.758468i \(-0.274052\pi\)
0.651711 + 0.758468i \(0.274052\pi\)
\(642\) 0 0
\(643\) 29.0000 1.14365 0.571824 0.820376i \(-0.306236\pi\)
0.571824 + 0.820376i \(0.306236\pi\)
\(644\) 4.00000 0.157622
\(645\) 0 0
\(646\) 0 0
\(647\) 7.00000 0.275198 0.137599 0.990488i \(-0.456061\pi\)
0.137599 + 0.990488i \(0.456061\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −32.0000 −1.25514
\(651\) 0 0
\(652\) 8.00000 0.313304
\(653\) 41.0000 1.60445 0.802227 0.597019i \(-0.203648\pi\)
0.802227 + 0.597019i \(0.203648\pi\)
\(654\) 0 0
\(655\) 18.0000 0.703318
\(656\) 32.0000 1.24939
\(657\) 0 0
\(658\) 32.0000 1.24749
\(659\) 10.0000 0.389545 0.194772 0.980848i \(-0.437603\pi\)
0.194772 + 0.980848i \(0.437603\pi\)
\(660\) 0 0
\(661\) 37.0000 1.43913 0.719567 0.694423i \(-0.244340\pi\)
0.719567 + 0.694423i \(0.244340\pi\)
\(662\) −14.0000 −0.544125
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) −24.0000 −0.928588
\(669\) 0 0
\(670\) −14.0000 −0.540867
\(671\) 0 0
\(672\) 0 0
\(673\) −14.0000 −0.539660 −0.269830 0.962908i \(-0.586968\pi\)
−0.269830 + 0.962908i \(0.586968\pi\)
\(674\) −44.0000 −1.69482
\(675\) 0 0
\(676\) 6.00000 0.230769
\(677\) −42.0000 −1.61419 −0.807096 0.590421i \(-0.798962\pi\)
−0.807096 + 0.590421i \(0.798962\pi\)
\(678\) 0 0
\(679\) −14.0000 −0.537271
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 16.0000 0.612223 0.306111 0.951996i \(-0.400972\pi\)
0.306111 + 0.951996i \(0.400972\pi\)
\(684\) 0 0
\(685\) −7.00000 −0.267456
\(686\) 40.0000 1.52721
\(687\) 0 0
\(688\) −24.0000 −0.914991
\(689\) −24.0000 −0.914327
\(690\) 0 0
\(691\) 17.0000 0.646710 0.323355 0.946278i \(-0.395189\pi\)
0.323355 + 0.946278i \(0.395189\pi\)
\(692\) −12.0000 −0.456172
\(693\) 0 0
\(694\) −56.0000 −2.12573
\(695\) 10.0000 0.379322
\(696\) 0 0
\(697\) 16.0000 0.606043
\(698\) 60.0000 2.27103
\(699\) 0 0
\(700\) −16.0000 −0.604743
\(701\) 2.00000 0.0755390 0.0377695 0.999286i \(-0.487975\pi\)
0.0377695 + 0.999286i \(0.487975\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) −42.0000 −1.58069
\(707\) 4.00000 0.150435
\(708\) 0 0
\(709\) −25.0000 −0.938895 −0.469447 0.882960i \(-0.655547\pi\)
−0.469447 + 0.882960i \(0.655547\pi\)
\(710\) 6.00000 0.225176
\(711\) 0 0
\(712\) 0 0
\(713\) 7.00000 0.262152
\(714\) 0 0
\(715\) 0 0
\(716\) 30.0000 1.12115
\(717\) 0 0
\(718\) 40.0000 1.49279
\(719\) −15.0000 −0.559406 −0.279703 0.960087i \(-0.590236\pi\)
−0.279703 + 0.960087i \(0.590236\pi\)
\(720\) 0 0
\(721\) −32.0000 −1.19174
\(722\) 38.0000 1.41421
\(723\) 0 0
\(724\) 14.0000 0.520306
\(725\) 0 0
\(726\) 0 0
\(727\) 3.00000 0.111264 0.0556319 0.998451i \(-0.482283\pi\)
0.0556319 + 0.998451i \(0.482283\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −8.00000 −0.296093
\(731\) −12.0000 −0.443836
\(732\) 0 0
\(733\) 36.0000 1.32969 0.664845 0.746981i \(-0.268498\pi\)
0.664845 + 0.746981i \(0.268498\pi\)
\(734\) 34.0000 1.25496
\(735\) 0 0
\(736\) 8.00000 0.294884
\(737\) 0 0
\(738\) 0 0
\(739\) −50.0000 −1.83928 −0.919640 0.392763i \(-0.871519\pi\)
−0.919640 + 0.392763i \(0.871519\pi\)
\(740\) −6.00000 −0.220564
\(741\) 0 0
\(742\) −24.0000 −0.881068
\(743\) 4.00000 0.146746 0.0733729 0.997305i \(-0.476624\pi\)
0.0733729 + 0.997305i \(0.476624\pi\)
\(744\) 0 0
\(745\) 10.0000 0.366372
\(746\) −52.0000 −1.90386
\(747\) 0 0
\(748\) 0 0
\(749\) 36.0000 1.31541
\(750\) 0 0
\(751\) −23.0000 −0.839282 −0.419641 0.907690i \(-0.637844\pi\)
−0.419641 + 0.907690i \(0.637844\pi\)
\(752\) 32.0000 1.16692
\(753\) 0 0
\(754\) 0 0
\(755\) 2.00000 0.0727875
\(756\) 0 0
\(757\) −22.0000 −0.799604 −0.399802 0.916602i \(-0.630921\pi\)
−0.399802 + 0.916602i \(0.630921\pi\)
\(758\) 10.0000 0.363216
\(759\) 0 0
\(760\) 0 0
\(761\) 12.0000 0.435000 0.217500 0.976060i \(-0.430210\pi\)
0.217500 + 0.976060i \(0.430210\pi\)
\(762\) 0 0
\(763\) −20.0000 −0.724049
\(764\) −34.0000 −1.23008
\(765\) 0 0
\(766\) −2.00000 −0.0722629
\(767\) 20.0000 0.722158
\(768\) 0 0
\(769\) −20.0000 −0.721218 −0.360609 0.932717i \(-0.617431\pi\)
−0.360609 + 0.932717i \(0.617431\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −8.00000 −0.287926
\(773\) 6.00000 0.215805 0.107903 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) 0 0
\(775\) −28.0000 −1.00579
\(776\) 0 0
\(777\) 0 0
\(778\) −30.0000 −1.07555
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 4.00000 0.143040
\(783\) 0 0
\(784\) 12.0000 0.428571
\(785\) 7.00000 0.249841
\(786\) 0 0
\(787\) 32.0000 1.14068 0.570338 0.821410i \(-0.306812\pi\)
0.570338 + 0.821410i \(0.306812\pi\)
\(788\) −4.00000 −0.142494
\(789\) 0 0
\(790\) 20.0000 0.711568
\(791\) −18.0000 −0.640006
\(792\) 0 0
\(793\) 48.0000 1.70453
\(794\) 4.00000 0.141955
\(795\) 0 0
\(796\) 0 0
\(797\) −53.0000 −1.87736 −0.938678 0.344795i \(-0.887949\pi\)
−0.938678 + 0.344795i \(0.887949\pi\)
\(798\) 0 0
\(799\) 16.0000 0.566039
\(800\) −32.0000 −1.13137
\(801\) 0 0
\(802\) 4.00000 0.141245
\(803\) 0 0
\(804\) 0 0
\(805\) −2.00000 −0.0704907
\(806\) 56.0000 1.97252
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) 38.0000 1.33436 0.667180 0.744896i \(-0.267501\pi\)
0.667180 + 0.744896i \(0.267501\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −4.00000 −0.140114
\(816\) 0 0
\(817\) 0 0
\(818\) −60.0000 −2.09785
\(819\) 0 0
\(820\) 16.0000 0.558744
\(821\) 22.0000 0.767805 0.383903 0.923374i \(-0.374580\pi\)
0.383903 + 0.923374i \(0.374580\pi\)
\(822\) 0 0
\(823\) 39.0000 1.35945 0.679727 0.733465i \(-0.262098\pi\)
0.679727 + 0.733465i \(0.262098\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 20.0000 0.695889
\(827\) −52.0000 −1.80822 −0.904109 0.427303i \(-0.859464\pi\)
−0.904109 + 0.427303i \(0.859464\pi\)
\(828\) 0 0
\(829\) 25.0000 0.868286 0.434143 0.900844i \(-0.357051\pi\)
0.434143 + 0.900844i \(0.357051\pi\)
\(830\) −12.0000 −0.416526
\(831\) 0 0
\(832\) 32.0000 1.10940
\(833\) 6.00000 0.207888
\(834\) 0 0
\(835\) 12.0000 0.415277
\(836\) 0 0
\(837\) 0 0
\(838\) 40.0000 1.38178
\(839\) 5.00000 0.172619 0.0863096 0.996268i \(-0.472493\pi\)
0.0863096 + 0.996268i \(0.472493\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) −44.0000 −1.51634
\(843\) 0 0
\(844\) −24.0000 −0.826114
\(845\) −3.00000 −0.103203
\(846\) 0 0
\(847\) 0 0
\(848\) −24.0000 −0.824163
\(849\) 0 0
\(850\) −16.0000 −0.548795
\(851\) 3.00000 0.102839
\(852\) 0 0
\(853\) −14.0000 −0.479351 −0.239675 0.970853i \(-0.577041\pi\)
−0.239675 + 0.970853i \(0.577041\pi\)
\(854\) 48.0000 1.64253
\(855\) 0 0
\(856\) 0 0
\(857\) 8.00000 0.273275 0.136637 0.990621i \(-0.456370\pi\)
0.136637 + 0.990621i \(0.456370\pi\)
\(858\) 0 0
\(859\) −15.0000 −0.511793 −0.255897 0.966704i \(-0.582371\pi\)
−0.255897 + 0.966704i \(0.582371\pi\)
\(860\) −12.0000 −0.409197
\(861\) 0 0
\(862\) 36.0000 1.22616
\(863\) −24.0000 −0.816970 −0.408485 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) 0 0
\(865\) 6.00000 0.204006
\(866\) 22.0000 0.747590
\(867\) 0 0
\(868\) 28.0000 0.950382
\(869\) 0 0
\(870\) 0 0
\(871\) 28.0000 0.948744
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 18.0000 0.608511
\(876\) 0 0
\(877\) 12.0000 0.405211 0.202606 0.979260i \(-0.435059\pi\)
0.202606 + 0.979260i \(0.435059\pi\)
\(878\) 80.0000 2.69987
\(879\) 0 0
\(880\) 0 0
\(881\) 43.0000 1.44871 0.724353 0.689429i \(-0.242138\pi\)
0.724353 + 0.689429i \(0.242138\pi\)
\(882\) 0 0
\(883\) 4.00000 0.134611 0.0673054 0.997732i \(-0.478560\pi\)
0.0673054 + 0.997732i \(0.478560\pi\)
\(884\) 16.0000 0.538138
\(885\) 0 0
\(886\) −22.0000 −0.739104
\(887\) −22.0000 −0.738688 −0.369344 0.929293i \(-0.620418\pi\)
−0.369344 + 0.929293i \(0.620418\pi\)
\(888\) 0 0
\(889\) −16.0000 −0.536623
\(890\) −30.0000 −1.00560
\(891\) 0 0
\(892\) 38.0000 1.27233
\(893\) 0 0
\(894\) 0 0
\(895\) −15.0000 −0.501395
\(896\) 0 0
\(897\) 0 0
\(898\) 70.0000 2.33593
\(899\) 0 0
\(900\) 0 0
\(901\) −12.0000 −0.399778
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −7.00000 −0.232688
\(906\) 0 0
\(907\) −12.0000 −0.398453 −0.199227 0.979953i \(-0.563843\pi\)
−0.199227 + 0.979953i \(0.563843\pi\)
\(908\) 36.0000 1.19470
\(909\) 0 0
\(910\) −16.0000 −0.530395
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −24.0000 −0.793849
\(915\) 0 0
\(916\) 30.0000 0.991228
\(917\) −36.0000 −1.18882
\(918\) 0 0
\(919\) −10.0000 −0.329870 −0.164935 0.986304i \(-0.552741\pi\)
−0.164935 + 0.986304i \(0.552741\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −24.0000 −0.790398
\(923\) −12.0000 −0.394985
\(924\) 0 0
\(925\) −12.0000 −0.394558
\(926\) 22.0000 0.722965
\(927\) 0 0
\(928\) 0 0
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 48.0000 1.57229
\(933\) 0 0
\(934\) −54.0000 −1.76693
\(935\) 0 0
\(936\) 0 0
\(937\) −8.00000 −0.261349 −0.130674 0.991425i \(-0.541714\pi\)
−0.130674 + 0.991425i \(0.541714\pi\)
\(938\) 28.0000 0.914232
\(939\) 0 0
\(940\) 16.0000 0.521862
\(941\) 42.0000 1.36916 0.684580 0.728937i \(-0.259985\pi\)
0.684580 + 0.728937i \(0.259985\pi\)
\(942\) 0 0
\(943\) −8.00000 −0.260516
\(944\) 20.0000 0.650945
\(945\) 0 0
\(946\) 0 0
\(947\) 27.0000 0.877382 0.438691 0.898638i \(-0.355442\pi\)
0.438691 + 0.898638i \(0.355442\pi\)
\(948\) 0 0
\(949\) 16.0000 0.519382
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 34.0000 1.10137 0.550684 0.834714i \(-0.314367\pi\)
0.550684 + 0.834714i \(0.314367\pi\)
\(954\) 0 0
\(955\) 17.0000 0.550107
\(956\) −60.0000 −1.94054
\(957\) 0 0
\(958\) −40.0000 −1.29234
\(959\) 14.0000 0.452084
\(960\) 0 0
\(961\) 18.0000 0.580645
\(962\) 24.0000 0.773791
\(963\) 0 0
\(964\) 16.0000 0.515325
\(965\) 4.00000 0.128765
\(966\) 0 0
\(967\) 32.0000 1.02905 0.514525 0.857475i \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −14.0000 −0.449513
\(971\) −47.0000 −1.50830 −0.754151 0.656701i \(-0.771951\pi\)
−0.754151 + 0.656701i \(0.771951\pi\)
\(972\) 0 0
\(973\) −20.0000 −0.641171
\(974\) −46.0000 −1.47394
\(975\) 0 0
\(976\) 48.0000 1.53644
\(977\) 27.0000 0.863807 0.431903 0.901920i \(-0.357842\pi\)
0.431903 + 0.901920i \(0.357842\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 6.00000 0.191663
\(981\) 0 0
\(982\) 16.0000 0.510581
\(983\) −39.0000 −1.24391 −0.621953 0.783054i \(-0.713661\pi\)
−0.621953 + 0.783054i \(0.713661\pi\)
\(984\) 0 0
\(985\) 2.00000 0.0637253
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 6.00000 0.190789
\(990\) 0 0
\(991\) −8.00000 −0.254128 −0.127064 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) 56.0000 1.77800
\(993\) 0 0
\(994\) −12.0000 −0.380617
\(995\) 0 0
\(996\) 0 0
\(997\) −38.0000 −1.20347 −0.601736 0.798695i \(-0.705524\pi\)
−0.601736 + 0.798695i \(0.705524\pi\)
\(998\) −40.0000 −1.26618
\(999\) 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 1089.2.a.b.1.1 1
3.2 odd 2 121.2.a.d.1.1 1
11.10 odd 2 99.2.a.d.1.1 1
12.11 even 2 1936.2.a.i.1.1 1
15.14 odd 2 3025.2.a.a.1.1 1
21.20 even 2 5929.2.a.h.1.1 1
24.5 odd 2 7744.2.a.x.1.1 1
24.11 even 2 7744.2.a.k.1.1 1
33.2 even 10 121.2.c.e.81.1 4
33.5 odd 10 121.2.c.a.3.1 4
33.8 even 10 121.2.c.e.9.1 4
33.14 odd 10 121.2.c.a.9.1 4
33.17 even 10 121.2.c.e.3.1 4
33.20 odd 10 121.2.c.a.81.1 4
33.26 odd 10 121.2.c.a.27.1 4
33.29 even 10 121.2.c.e.27.1 4
33.32 even 2 11.2.a.a.1.1 1
44.43 even 2 1584.2.a.g.1.1 1
55.32 even 4 2475.2.c.a.199.2 2
55.43 even 4 2475.2.c.a.199.1 2
55.54 odd 2 2475.2.a.a.1.1 1
77.76 even 2 4851.2.a.t.1.1 1
88.21 odd 2 6336.2.a.br.1.1 1
88.43 even 2 6336.2.a.bu.1.1 1
99.32 even 6 891.2.e.k.298.1 2
99.43 odd 6 891.2.e.b.595.1 2
99.65 even 6 891.2.e.k.595.1 2
99.76 odd 6 891.2.e.b.298.1 2
132.131 odd 2 176.2.a.b.1.1 1
165.32 odd 4 275.2.b.a.199.1 2
165.98 odd 4 275.2.b.a.199.2 2
165.164 even 2 275.2.a.b.1.1 1
231.32 even 6 539.2.e.h.177.1 2
231.65 even 6 539.2.e.h.67.1 2
231.131 odd 6 539.2.e.g.67.1 2
231.164 odd 6 539.2.e.g.177.1 2
231.230 odd 2 539.2.a.a.1.1 1
264.131 odd 2 704.2.a.c.1.1 1
264.197 even 2 704.2.a.h.1.1 1
429.428 even 2 1859.2.a.b.1.1 1
528.131 odd 4 2816.2.c.f.1409.2 2
528.197 even 4 2816.2.c.j.1409.2 2
528.395 odd 4 2816.2.c.f.1409.1 2
528.461 even 4 2816.2.c.j.1409.1 2
561.560 even 2 3179.2.a.a.1.1 1
627.626 odd 2 3971.2.a.b.1.1 1
660.263 even 4 4400.2.b.h.4049.2 2
660.527 even 4 4400.2.b.h.4049.1 2
660.659 odd 2 4400.2.a.i.1.1 1
759.758 odd 2 5819.2.a.a.1.1 1
924.923 even 2 8624.2.a.j.1.1 1
957.956 even 2 9251.2.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.2.a.a.1.1 1 33.32 even 2
99.2.a.d.1.1 1 11.10 odd 2
121.2.a.d.1.1 1 3.2 odd 2
121.2.c.a.3.1 4 33.5 odd 10
121.2.c.a.9.1 4 33.14 odd 10
121.2.c.a.27.1 4 33.26 odd 10
121.2.c.a.81.1 4 33.20 odd 10
121.2.c.e.3.1 4 33.17 even 10
121.2.c.e.9.1 4 33.8 even 10
121.2.c.e.27.1 4 33.29 even 10
121.2.c.e.81.1 4 33.2 even 10
176.2.a.b.1.1 1 132.131 odd 2
275.2.a.b.1.1 1 165.164 even 2
275.2.b.a.199.1 2 165.32 odd 4
275.2.b.a.199.2 2 165.98 odd 4
539.2.a.a.1.1 1 231.230 odd 2
539.2.e.g.67.1 2 231.131 odd 6
539.2.e.g.177.1 2 231.164 odd 6
539.2.e.h.67.1 2 231.65 even 6
539.2.e.h.177.1 2 231.32 even 6
704.2.a.c.1.1 1 264.131 odd 2
704.2.a.h.1.1 1 264.197 even 2
891.2.e.b.298.1 2 99.76 odd 6
891.2.e.b.595.1 2 99.43 odd 6
891.2.e.k.298.1 2 99.32 even 6
891.2.e.k.595.1 2 99.65 even 6
1089.2.a.b.1.1 1 1.1 even 1 trivial
1584.2.a.g.1.1 1 44.43 even 2
1859.2.a.b.1.1 1 429.428 even 2
1936.2.a.i.1.1 1 12.11 even 2
2475.2.a.a.1.1 1 55.54 odd 2
2475.2.c.a.199.1 2 55.43 even 4
2475.2.c.a.199.2 2 55.32 even 4
2816.2.c.f.1409.1 2 528.395 odd 4
2816.2.c.f.1409.2 2 528.131 odd 4
2816.2.c.j.1409.1 2 528.461 even 4
2816.2.c.j.1409.2 2 528.197 even 4
3025.2.a.a.1.1 1 15.14 odd 2
3179.2.a.a.1.1 1 561.560 even 2
3971.2.a.b.1.1 1 627.626 odd 2
4400.2.a.i.1.1 1 660.659 odd 2
4400.2.b.h.4049.1 2 660.527 even 4
4400.2.b.h.4049.2 2 660.263 even 4
4851.2.a.t.1.1 1 77.76 even 2
5819.2.a.a.1.1 1 759.758 odd 2
5929.2.a.h.1.1 1 21.20 even 2
6336.2.a.br.1.1 1 88.21 odd 2
6336.2.a.bu.1.1 1 88.43 even 2
7744.2.a.k.1.1 1 24.11 even 2
7744.2.a.x.1.1 1 24.5 odd 2
8624.2.a.j.1.1 1 924.923 even 2
9251.2.a.d.1.1 1 957.956 even 2