Properties

Label 1638.2.a.g.1.1
Level $1638$
Weight $2$
Character 1638.1
Self dual yes
Analytic conductor $13.079$
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] = [1638,2,Mod(1,1638)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1638, 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("1638.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1638.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: \(13.0794958511\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 1638.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} -1.00000 q^{10} +5.00000 q^{11} +1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{19} +1.00000 q^{20} -5.00000 q^{22} +5.00000 q^{23} -4.00000 q^{25} -1.00000 q^{26} -1.00000 q^{28} +1.00000 q^{29} -6.00000 q^{31} -1.00000 q^{32} -1.00000 q^{34} -1.00000 q^{35} +7.00000 q^{37} +1.00000 q^{38} -1.00000 q^{40} +2.00000 q^{41} +1.00000 q^{43} +5.00000 q^{44} -5.00000 q^{46} +1.00000 q^{49} +4.00000 q^{50} +1.00000 q^{52} +6.00000 q^{53} +5.00000 q^{55} +1.00000 q^{56} -1.00000 q^{58} +6.00000 q^{59} -7.00000 q^{61} +6.00000 q^{62} +1.00000 q^{64} +1.00000 q^{65} -8.00000 q^{67} +1.00000 q^{68} +1.00000 q^{70} +10.0000 q^{71} +9.00000 q^{73} -7.00000 q^{74} -1.00000 q^{76} -5.00000 q^{77} +2.00000 q^{79} +1.00000 q^{80} -2.00000 q^{82} +2.00000 q^{83} +1.00000 q^{85} -1.00000 q^{86} -5.00000 q^{88} -6.00000 q^{89} -1.00000 q^{91} +5.00000 q^{92} -1.00000 q^{95} +14.0000 q^{97} -1.00000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214 0.223607 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) 5.00000 1.50756 0.753778 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 1.00000 0.242536 0.121268 0.992620i \(-0.461304\pi\)
0.121268 + 0.992620i \(0.461304\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) −5.00000 −1.06600
\(23\) 5.00000 1.04257 0.521286 0.853382i \(-0.325452\pi\)
0.521286 + 0.853382i \(0.325452\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) −1.00000 −0.196116
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) 1.00000 0.185695 0.0928477 0.995680i \(-0.470403\pi\)
0.0928477 + 0.995680i \(0.470403\pi\)
\(30\) 0 0
\(31\) −6.00000 −1.07763 −0.538816 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −1.00000 −0.171499
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) 7.00000 1.15079 0.575396 0.817875i \(-0.304848\pi\)
0.575396 + 0.817875i \(0.304848\pi\)
\(38\) 1.00000 0.162221
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 1.00000 0.152499 0.0762493 0.997089i \(-0.475706\pi\)
0.0762493 + 0.997089i \(0.475706\pi\)
\(44\) 5.00000 0.753778
\(45\) 0 0
\(46\) −5.00000 −0.737210
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 4.00000 0.565685
\(51\) 0 0
\(52\) 1.00000 0.138675
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 5.00000 0.674200
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) −1.00000 −0.131306
\(59\) 6.00000 0.781133 0.390567 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) −7.00000 −0.896258 −0.448129 0.893969i \(-0.647910\pi\)
−0.448129 + 0.893969i \(0.647910\pi\)
\(62\) 6.00000 0.762001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 1.00000 0.124035
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 1.00000 0.121268
\(69\) 0 0
\(70\) 1.00000 0.119523
\(71\) 10.0000 1.18678 0.593391 0.804914i \(-0.297789\pi\)
0.593391 + 0.804914i \(0.297789\pi\)
\(72\) 0 0
\(73\) 9.00000 1.05337 0.526685 0.850060i \(-0.323435\pi\)
0.526685 + 0.850060i \(0.323435\pi\)
\(74\) −7.00000 −0.813733
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) −5.00000 −0.569803
\(78\) 0 0
\(79\) 2.00000 0.225018 0.112509 0.993651i \(-0.464111\pi\)
0.112509 + 0.993651i \(0.464111\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) −2.00000 −0.220863
\(83\) 2.00000 0.219529 0.109764 0.993958i \(-0.464990\pi\)
0.109764 + 0.993958i \(0.464990\pi\)
\(84\) 0 0
\(85\) 1.00000 0.108465
\(86\) −1.00000 −0.107833
\(87\) 0 0
\(88\) −5.00000 −0.533002
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 5.00000 0.521286
\(93\) 0 0
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) 14.0000 1.42148 0.710742 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) −4.00000 −0.400000
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) 7.00000 0.689730 0.344865 0.938652i \(-0.387925\pi\)
0.344865 + 0.938652i \(0.387925\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) 0 0
\(109\) −7.00000 −0.670478 −0.335239 0.942133i \(-0.608817\pi\)
−0.335239 + 0.942133i \(0.608817\pi\)
\(110\) −5.00000 −0.476731
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) 16.0000 1.50515 0.752577 0.658505i \(-0.228811\pi\)
0.752577 + 0.658505i \(0.228811\pi\)
\(114\) 0 0
\(115\) 5.00000 0.466252
\(116\) 1.00000 0.0928477
\(117\) 0 0
\(118\) −6.00000 −0.552345
\(119\) −1.00000 −0.0916698
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 7.00000 0.633750
\(123\) 0 0
\(124\) −6.00000 −0.538816
\(125\) −9.00000 −0.804984
\(126\) 0 0
\(127\) 4.00000 0.354943 0.177471 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −1.00000 −0.0877058
\(131\) −9.00000 −0.786334 −0.393167 0.919467i \(-0.628621\pi\)
−0.393167 + 0.919467i \(0.628621\pi\)
\(132\) 0 0
\(133\) 1.00000 0.0867110
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) −1.00000 −0.0857493
\(137\) 13.0000 1.11066 0.555332 0.831628i \(-0.312591\pi\)
0.555332 + 0.831628i \(0.312591\pi\)
\(138\) 0 0
\(139\) −2.00000 −0.169638 −0.0848189 0.996396i \(-0.527031\pi\)
−0.0848189 + 0.996396i \(0.527031\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 0 0
\(142\) −10.0000 −0.839181
\(143\) 5.00000 0.418121
\(144\) 0 0
\(145\) 1.00000 0.0830455
\(146\) −9.00000 −0.744845
\(147\) 0 0
\(148\) 7.00000 0.575396
\(149\) 24.0000 1.96616 0.983078 0.183186i \(-0.0586410\pi\)
0.983078 + 0.183186i \(0.0586410\pi\)
\(150\) 0 0
\(151\) −7.00000 −0.569652 −0.284826 0.958579i \(-0.591936\pi\)
−0.284826 + 0.958579i \(0.591936\pi\)
\(152\) 1.00000 0.0811107
\(153\) 0 0
\(154\) 5.00000 0.402911
\(155\) −6.00000 −0.481932
\(156\) 0 0
\(157\) 13.0000 1.03751 0.518756 0.854922i \(-0.326395\pi\)
0.518756 + 0.854922i \(0.326395\pi\)
\(158\) −2.00000 −0.159111
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) −5.00000 −0.394055
\(162\) 0 0
\(163\) −6.00000 −0.469956 −0.234978 0.972001i \(-0.575502\pi\)
−0.234978 + 0.972001i \(0.575502\pi\)
\(164\) 2.00000 0.156174
\(165\) 0 0
\(166\) −2.00000 −0.155230
\(167\) 3.00000 0.232147 0.116073 0.993241i \(-0.462969\pi\)
0.116073 + 0.993241i \(0.462969\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) −1.00000 −0.0766965
\(171\) 0 0
\(172\) 1.00000 0.0762493
\(173\) 2.00000 0.152057 0.0760286 0.997106i \(-0.475776\pi\)
0.0760286 + 0.997106i \(0.475776\pi\)
\(174\) 0 0
\(175\) 4.00000 0.302372
\(176\) 5.00000 0.376889
\(177\) 0 0
\(178\) 6.00000 0.449719
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 10.0000 0.743294 0.371647 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 1.00000 0.0741249
\(183\) 0 0
\(184\) −5.00000 −0.368605
\(185\) 7.00000 0.514650
\(186\) 0 0
\(187\) 5.00000 0.365636
\(188\) 0 0
\(189\) 0 0
\(190\) 1.00000 0.0725476
\(191\) −15.0000 −1.08536 −0.542681 0.839939i \(-0.682591\pi\)
−0.542681 + 0.839939i \(0.682591\pi\)
\(192\) 0 0
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) −14.0000 −1.00514
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) 25.0000 1.77220 0.886102 0.463491i \(-0.153403\pi\)
0.886102 + 0.463491i \(0.153403\pi\)
\(200\) 4.00000 0.282843
\(201\) 0 0
\(202\) 6.00000 0.422159
\(203\) −1.00000 −0.0701862
\(204\) 0 0
\(205\) 2.00000 0.139686
\(206\) −7.00000 −0.487713
\(207\) 0 0
\(208\) 1.00000 0.0693375
\(209\) −5.00000 −0.345857
\(210\) 0 0
\(211\) −5.00000 −0.344214 −0.172107 0.985078i \(-0.555058\pi\)
−0.172107 + 0.985078i \(0.555058\pi\)
\(212\) 6.00000 0.412082
\(213\) 0 0
\(214\) 6.00000 0.410152
\(215\) 1.00000 0.0681994
\(216\) 0 0
\(217\) 6.00000 0.407307
\(218\) 7.00000 0.474100
\(219\) 0 0
\(220\) 5.00000 0.337100
\(221\) 1.00000 0.0672673
\(222\) 0 0
\(223\) −12.0000 −0.803579 −0.401790 0.915732i \(-0.631612\pi\)
−0.401790 + 0.915732i \(0.631612\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) −16.0000 −1.06430
\(227\) 10.0000 0.663723 0.331862 0.943328i \(-0.392323\pi\)
0.331862 + 0.943328i \(0.392323\pi\)
\(228\) 0 0
\(229\) 22.0000 1.45380 0.726900 0.686743i \(-0.240960\pi\)
0.726900 + 0.686743i \(0.240960\pi\)
\(230\) −5.00000 −0.329690
\(231\) 0 0
\(232\) −1.00000 −0.0656532
\(233\) −4.00000 −0.262049 −0.131024 0.991379i \(-0.541827\pi\)
−0.131024 + 0.991379i \(0.541827\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 6.00000 0.390567
\(237\) 0 0
\(238\) 1.00000 0.0648204
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) −14.0000 −0.899954
\(243\) 0 0
\(244\) −7.00000 −0.448129
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) −1.00000 −0.0636285
\(248\) 6.00000 0.381000
\(249\) 0 0
\(250\) 9.00000 0.569210
\(251\) −25.0000 −1.57799 −0.788993 0.614402i \(-0.789397\pi\)
−0.788993 + 0.614402i \(0.789397\pi\)
\(252\) 0 0
\(253\) 25.0000 1.57174
\(254\) −4.00000 −0.250982
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) −7.00000 −0.434959
\(260\) 1.00000 0.0620174
\(261\) 0 0
\(262\) 9.00000 0.556022
\(263\) −8.00000 −0.493301 −0.246651 0.969104i \(-0.579330\pi\)
−0.246651 + 0.969104i \(0.579330\pi\)
\(264\) 0 0
\(265\) 6.00000 0.368577
\(266\) −1.00000 −0.0613139
\(267\) 0 0
\(268\) −8.00000 −0.488678
\(269\) −30.0000 −1.82913 −0.914566 0.404436i \(-0.867468\pi\)
−0.914566 + 0.404436i \(0.867468\pi\)
\(270\) 0 0
\(271\) −30.0000 −1.82237 −0.911185 0.411997i \(-0.864831\pi\)
−0.911185 + 0.411997i \(0.864831\pi\)
\(272\) 1.00000 0.0606339
\(273\) 0 0
\(274\) −13.0000 −0.785359
\(275\) −20.0000 −1.20605
\(276\) 0 0
\(277\) −8.00000 −0.480673 −0.240337 0.970690i \(-0.577258\pi\)
−0.240337 + 0.970690i \(0.577258\pi\)
\(278\) 2.00000 0.119952
\(279\) 0 0
\(280\) 1.00000 0.0597614
\(281\) −2.00000 −0.119310 −0.0596550 0.998219i \(-0.519000\pi\)
−0.0596550 + 0.998219i \(0.519000\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 10.0000 0.593391
\(285\) 0 0
\(286\) −5.00000 −0.295656
\(287\) −2.00000 −0.118056
\(288\) 0 0
\(289\) −16.0000 −0.941176
\(290\) −1.00000 −0.0587220
\(291\) 0 0
\(292\) 9.00000 0.526685
\(293\) 26.0000 1.51894 0.759468 0.650545i \(-0.225459\pi\)
0.759468 + 0.650545i \(0.225459\pi\)
\(294\) 0 0
\(295\) 6.00000 0.349334
\(296\) −7.00000 −0.406867
\(297\) 0 0
\(298\) −24.0000 −1.39028
\(299\) 5.00000 0.289157
\(300\) 0 0
\(301\) −1.00000 −0.0576390
\(302\) 7.00000 0.402805
\(303\) 0 0
\(304\) −1.00000 −0.0573539
\(305\) −7.00000 −0.400819
\(306\) 0 0
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) −5.00000 −0.284901
\(309\) 0 0
\(310\) 6.00000 0.340777
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 0 0
\(313\) −30.0000 −1.69570 −0.847850 0.530236i \(-0.822103\pi\)
−0.847850 + 0.530236i \(0.822103\pi\)
\(314\) −13.0000 −0.733632
\(315\) 0 0
\(316\) 2.00000 0.112509
\(317\) −10.0000 −0.561656 −0.280828 0.959758i \(-0.590609\pi\)
−0.280828 + 0.959758i \(0.590609\pi\)
\(318\) 0 0
\(319\) 5.00000 0.279946
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) 5.00000 0.278639
\(323\) −1.00000 −0.0556415
\(324\) 0 0
\(325\) −4.00000 −0.221880
\(326\) 6.00000 0.332309
\(327\) 0 0
\(328\) −2.00000 −0.110432
\(329\) 0 0
\(330\) 0 0
\(331\) −18.0000 −0.989369 −0.494685 0.869072i \(-0.664716\pi\)
−0.494685 + 0.869072i \(0.664716\pi\)
\(332\) 2.00000 0.109764
\(333\) 0 0
\(334\) −3.00000 −0.164153
\(335\) −8.00000 −0.437087
\(336\) 0 0
\(337\) −9.00000 −0.490261 −0.245131 0.969490i \(-0.578831\pi\)
−0.245131 + 0.969490i \(0.578831\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 0 0
\(340\) 1.00000 0.0542326
\(341\) −30.0000 −1.62459
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −1.00000 −0.0539164
\(345\) 0 0
\(346\) −2.00000 −0.107521
\(347\) −20.0000 −1.07366 −0.536828 0.843692i \(-0.680378\pi\)
−0.536828 + 0.843692i \(0.680378\pi\)
\(348\) 0 0
\(349\) −4.00000 −0.214115 −0.107058 0.994253i \(-0.534143\pi\)
−0.107058 + 0.994253i \(0.534143\pi\)
\(350\) −4.00000 −0.213809
\(351\) 0 0
\(352\) −5.00000 −0.266501
\(353\) 10.0000 0.532246 0.266123 0.963939i \(-0.414257\pi\)
0.266123 + 0.963939i \(0.414257\pi\)
\(354\) 0 0
\(355\) 10.0000 0.530745
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) 0 0
\(359\) −30.0000 −1.58334 −0.791670 0.610949i \(-0.790788\pi\)
−0.791670 + 0.610949i \(0.790788\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) −10.0000 −0.525588
\(363\) 0 0
\(364\) −1.00000 −0.0524142
\(365\) 9.00000 0.471082
\(366\) 0 0
\(367\) −24.0000 −1.25279 −0.626395 0.779506i \(-0.715470\pi\)
−0.626395 + 0.779506i \(0.715470\pi\)
\(368\) 5.00000 0.260643
\(369\) 0 0
\(370\) −7.00000 −0.363913
\(371\) −6.00000 −0.311504
\(372\) 0 0
\(373\) 34.0000 1.76045 0.880227 0.474554i \(-0.157390\pi\)
0.880227 + 0.474554i \(0.157390\pi\)
\(374\) −5.00000 −0.258544
\(375\) 0 0
\(376\) 0 0
\(377\) 1.00000 0.0515026
\(378\) 0 0
\(379\) −2.00000 −0.102733 −0.0513665 0.998680i \(-0.516358\pi\)
−0.0513665 + 0.998680i \(0.516358\pi\)
\(380\) −1.00000 −0.0512989
\(381\) 0 0
\(382\) 15.0000 0.767467
\(383\) −9.00000 −0.459879 −0.229939 0.973205i \(-0.573853\pi\)
−0.229939 + 0.973205i \(0.573853\pi\)
\(384\) 0 0
\(385\) −5.00000 −0.254824
\(386\) −14.0000 −0.712581
\(387\) 0 0
\(388\) 14.0000 0.710742
\(389\) −10.0000 −0.507020 −0.253510 0.967333i \(-0.581585\pi\)
−0.253510 + 0.967333i \(0.581585\pi\)
\(390\) 0 0
\(391\) 5.00000 0.252861
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) −6.00000 −0.302276
\(395\) 2.00000 0.100631
\(396\) 0 0
\(397\) −4.00000 −0.200754 −0.100377 0.994949i \(-0.532005\pi\)
−0.100377 + 0.994949i \(0.532005\pi\)
\(398\) −25.0000 −1.25314
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) −10.0000 −0.499376 −0.249688 0.968326i \(-0.580328\pi\)
−0.249688 + 0.968326i \(0.580328\pi\)
\(402\) 0 0
\(403\) −6.00000 −0.298881
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) 1.00000 0.0496292
\(407\) 35.0000 1.73489
\(408\) 0 0
\(409\) −13.0000 −0.642809 −0.321404 0.946942i \(-0.604155\pi\)
−0.321404 + 0.946942i \(0.604155\pi\)
\(410\) −2.00000 −0.0987730
\(411\) 0 0
\(412\) 7.00000 0.344865
\(413\) −6.00000 −0.295241
\(414\) 0 0
\(415\) 2.00000 0.0981761
\(416\) −1.00000 −0.0490290
\(417\) 0 0
\(418\) 5.00000 0.244558
\(419\) −19.0000 −0.928211 −0.464105 0.885780i \(-0.653624\pi\)
−0.464105 + 0.885780i \(0.653624\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) 5.00000 0.243396
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) −4.00000 −0.194029
\(426\) 0 0
\(427\) 7.00000 0.338754
\(428\) −6.00000 −0.290021
\(429\) 0 0
\(430\) −1.00000 −0.0482243
\(431\) 26.0000 1.25238 0.626188 0.779672i \(-0.284614\pi\)
0.626188 + 0.779672i \(0.284614\pi\)
\(432\) 0 0
\(433\) 30.0000 1.44171 0.720854 0.693087i \(-0.243750\pi\)
0.720854 + 0.693087i \(0.243750\pi\)
\(434\) −6.00000 −0.288009
\(435\) 0 0
\(436\) −7.00000 −0.335239
\(437\) −5.00000 −0.239182
\(438\) 0 0
\(439\) −41.0000 −1.95682 −0.978412 0.206666i \(-0.933739\pi\)
−0.978412 + 0.206666i \(0.933739\pi\)
\(440\) −5.00000 −0.238366
\(441\) 0 0
\(442\) −1.00000 −0.0475651
\(443\) −6.00000 −0.285069 −0.142534 0.989790i \(-0.545525\pi\)
−0.142534 + 0.989790i \(0.545525\pi\)
\(444\) 0 0
\(445\) −6.00000 −0.284427
\(446\) 12.0000 0.568216
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) 33.0000 1.55737 0.778683 0.627417i \(-0.215888\pi\)
0.778683 + 0.627417i \(0.215888\pi\)
\(450\) 0 0
\(451\) 10.0000 0.470882
\(452\) 16.0000 0.752577
\(453\) 0 0
\(454\) −10.0000 −0.469323
\(455\) −1.00000 −0.0468807
\(456\) 0 0
\(457\) 12.0000 0.561336 0.280668 0.959805i \(-0.409444\pi\)
0.280668 + 0.959805i \(0.409444\pi\)
\(458\) −22.0000 −1.02799
\(459\) 0 0
\(460\) 5.00000 0.233126
\(461\) −29.0000 −1.35066 −0.675332 0.737514i \(-0.736000\pi\)
−0.675332 + 0.737514i \(0.736000\pi\)
\(462\) 0 0
\(463\) 21.0000 0.975953 0.487976 0.872857i \(-0.337735\pi\)
0.487976 + 0.872857i \(0.337735\pi\)
\(464\) 1.00000 0.0464238
\(465\) 0 0
\(466\) 4.00000 0.185296
\(467\) −29.0000 −1.34196 −0.670980 0.741475i \(-0.734126\pi\)
−0.670980 + 0.741475i \(0.734126\pi\)
\(468\) 0 0
\(469\) 8.00000 0.369406
\(470\) 0 0
\(471\) 0 0
\(472\) −6.00000 −0.276172
\(473\) 5.00000 0.229900
\(474\) 0 0
\(475\) 4.00000 0.183533
\(476\) −1.00000 −0.0458349
\(477\) 0 0
\(478\) −12.0000 −0.548867
\(479\) 33.0000 1.50781 0.753904 0.656984i \(-0.228168\pi\)
0.753904 + 0.656984i \(0.228168\pi\)
\(480\) 0 0
\(481\) 7.00000 0.319173
\(482\) 10.0000 0.455488
\(483\) 0 0
\(484\) 14.0000 0.636364
\(485\) 14.0000 0.635707
\(486\) 0 0
\(487\) −40.0000 −1.81257 −0.906287 0.422664i \(-0.861095\pi\)
−0.906287 + 0.422664i \(0.861095\pi\)
\(488\) 7.00000 0.316875
\(489\) 0 0
\(490\) −1.00000 −0.0451754
\(491\) 16.0000 0.722070 0.361035 0.932552i \(-0.382424\pi\)
0.361035 + 0.932552i \(0.382424\pi\)
\(492\) 0 0
\(493\) 1.00000 0.0450377
\(494\) 1.00000 0.0449921
\(495\) 0 0
\(496\) −6.00000 −0.269408
\(497\) −10.0000 −0.448561
\(498\) 0 0
\(499\) −8.00000 −0.358129 −0.179065 0.983837i \(-0.557307\pi\)
−0.179065 + 0.983837i \(0.557307\pi\)
\(500\) −9.00000 −0.402492
\(501\) 0 0
\(502\) 25.0000 1.11580
\(503\) −28.0000 −1.24846 −0.624229 0.781241i \(-0.714587\pi\)
−0.624229 + 0.781241i \(0.714587\pi\)
\(504\) 0 0
\(505\) −6.00000 −0.266996
\(506\) −25.0000 −1.11139
\(507\) 0 0
\(508\) 4.00000 0.177471
\(509\) −21.0000 −0.930809 −0.465404 0.885098i \(-0.654091\pi\)
−0.465404 + 0.885098i \(0.654091\pi\)
\(510\) 0 0
\(511\) −9.00000 −0.398137
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 18.0000 0.793946
\(515\) 7.00000 0.308457
\(516\) 0 0
\(517\) 0 0
\(518\) 7.00000 0.307562
\(519\) 0 0
\(520\) −1.00000 −0.0438529
\(521\) 15.0000 0.657162 0.328581 0.944476i \(-0.393430\pi\)
0.328581 + 0.944476i \(0.393430\pi\)
\(522\) 0 0
\(523\) −42.0000 −1.83653 −0.918266 0.395964i \(-0.870410\pi\)
−0.918266 + 0.395964i \(0.870410\pi\)
\(524\) −9.00000 −0.393167
\(525\) 0 0
\(526\) 8.00000 0.348817
\(527\) −6.00000 −0.261364
\(528\) 0 0
\(529\) 2.00000 0.0869565
\(530\) −6.00000 −0.260623
\(531\) 0 0
\(532\) 1.00000 0.0433555
\(533\) 2.00000 0.0866296
\(534\) 0 0
\(535\) −6.00000 −0.259403
\(536\) 8.00000 0.345547
\(537\) 0 0
\(538\) 30.0000 1.29339
\(539\) 5.00000 0.215365
\(540\) 0 0
\(541\) 25.0000 1.07483 0.537417 0.843317i \(-0.319400\pi\)
0.537417 + 0.843317i \(0.319400\pi\)
\(542\) 30.0000 1.28861
\(543\) 0 0
\(544\) −1.00000 −0.0428746
\(545\) −7.00000 −0.299847
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 13.0000 0.555332
\(549\) 0 0
\(550\) 20.0000 0.852803
\(551\) −1.00000 −0.0426014
\(552\) 0 0
\(553\) −2.00000 −0.0850487
\(554\) 8.00000 0.339887
\(555\) 0 0
\(556\) −2.00000 −0.0848189
\(557\) −14.0000 −0.593199 −0.296600 0.955002i \(-0.595853\pi\)
−0.296600 + 0.955002i \(0.595853\pi\)
\(558\) 0 0
\(559\) 1.00000 0.0422955
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) 2.00000 0.0843649
\(563\) 23.0000 0.969334 0.484667 0.874699i \(-0.338941\pi\)
0.484667 + 0.874699i \(0.338941\pi\)
\(564\) 0 0
\(565\) 16.0000 0.673125
\(566\) 4.00000 0.168133
\(567\) 0 0
\(568\) −10.0000 −0.419591
\(569\) −22.0000 −0.922288 −0.461144 0.887325i \(-0.652561\pi\)
−0.461144 + 0.887325i \(0.652561\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) 5.00000 0.209061
\(573\) 0 0
\(574\) 2.00000 0.0834784
\(575\) −20.0000 −0.834058
\(576\) 0 0
\(577\) −42.0000 −1.74848 −0.874241 0.485491i \(-0.838641\pi\)
−0.874241 + 0.485491i \(0.838641\pi\)
\(578\) 16.0000 0.665512
\(579\) 0 0
\(580\) 1.00000 0.0415227
\(581\) −2.00000 −0.0829740
\(582\) 0 0
\(583\) 30.0000 1.24247
\(584\) −9.00000 −0.372423
\(585\) 0 0
\(586\) −26.0000 −1.07405
\(587\) 42.0000 1.73353 0.866763 0.498721i \(-0.166197\pi\)
0.866763 + 0.498721i \(0.166197\pi\)
\(588\) 0 0
\(589\) 6.00000 0.247226
\(590\) −6.00000 −0.247016
\(591\) 0 0
\(592\) 7.00000 0.287698
\(593\) −40.0000 −1.64260 −0.821302 0.570494i \(-0.806752\pi\)
−0.821302 + 0.570494i \(0.806752\pi\)
\(594\) 0 0
\(595\) −1.00000 −0.0409960
\(596\) 24.0000 0.983078
\(597\) 0 0
\(598\) −5.00000 −0.204465
\(599\) 15.0000 0.612883 0.306442 0.951889i \(-0.400862\pi\)
0.306442 + 0.951889i \(0.400862\pi\)
\(600\) 0 0
\(601\) 2.00000 0.0815817 0.0407909 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) 1.00000 0.0407570
\(603\) 0 0
\(604\) −7.00000 −0.284826
\(605\) 14.0000 0.569181
\(606\) 0 0
\(607\) −37.0000 −1.50178 −0.750892 0.660425i \(-0.770376\pi\)
−0.750892 + 0.660425i \(0.770376\pi\)
\(608\) 1.00000 0.0405554
\(609\) 0 0
\(610\) 7.00000 0.283422
\(611\) 0 0
\(612\) 0 0
\(613\) −7.00000 −0.282727 −0.141364 0.989958i \(-0.545149\pi\)
−0.141364 + 0.989958i \(0.545149\pi\)
\(614\) −12.0000 −0.484281
\(615\) 0 0
\(616\) 5.00000 0.201456
\(617\) 43.0000 1.73111 0.865557 0.500810i \(-0.166964\pi\)
0.865557 + 0.500810i \(0.166964\pi\)
\(618\) 0 0
\(619\) −37.0000 −1.48716 −0.743578 0.668649i \(-0.766873\pi\)
−0.743578 + 0.668649i \(0.766873\pi\)
\(620\) −6.00000 −0.240966
\(621\) 0 0
\(622\) −12.0000 −0.481156
\(623\) 6.00000 0.240385
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 30.0000 1.19904
\(627\) 0 0
\(628\) 13.0000 0.518756
\(629\) 7.00000 0.279108
\(630\) 0 0
\(631\) 25.0000 0.995234 0.497617 0.867397i \(-0.334208\pi\)
0.497617 + 0.867397i \(0.334208\pi\)
\(632\) −2.00000 −0.0795557
\(633\) 0 0
\(634\) 10.0000 0.397151
\(635\) 4.00000 0.158735
\(636\) 0 0
\(637\) 1.00000 0.0396214
\(638\) −5.00000 −0.197952
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) 14.0000 0.552967 0.276483 0.961019i \(-0.410831\pi\)
0.276483 + 0.961019i \(0.410831\pi\)
\(642\) 0 0
\(643\) 5.00000 0.197181 0.0985904 0.995128i \(-0.468567\pi\)
0.0985904 + 0.995128i \(0.468567\pi\)
\(644\) −5.00000 −0.197028
\(645\) 0 0
\(646\) 1.00000 0.0393445
\(647\) 24.0000 0.943537 0.471769 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) 0 0
\(649\) 30.0000 1.17760
\(650\) 4.00000 0.156893
\(651\) 0 0
\(652\) −6.00000 −0.234978
\(653\) 3.00000 0.117399 0.0586995 0.998276i \(-0.481305\pi\)
0.0586995 + 0.998276i \(0.481305\pi\)
\(654\) 0 0
\(655\) −9.00000 −0.351659
\(656\) 2.00000 0.0780869
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −20.0000 −0.777910 −0.388955 0.921257i \(-0.627164\pi\)
−0.388955 + 0.921257i \(0.627164\pi\)
\(662\) 18.0000 0.699590
\(663\) 0 0
\(664\) −2.00000 −0.0776151
\(665\) 1.00000 0.0387783
\(666\) 0 0
\(667\) 5.00000 0.193601
\(668\) 3.00000 0.116073
\(669\) 0 0
\(670\) 8.00000 0.309067
\(671\) −35.0000 −1.35116
\(672\) 0 0
\(673\) −21.0000 −0.809491 −0.404745 0.914429i \(-0.632640\pi\)
−0.404745 + 0.914429i \(0.632640\pi\)
\(674\) 9.00000 0.346667
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) −2.00000 −0.0768662 −0.0384331 0.999261i \(-0.512237\pi\)
−0.0384331 + 0.999261i \(0.512237\pi\)
\(678\) 0 0
\(679\) −14.0000 −0.537271
\(680\) −1.00000 −0.0383482
\(681\) 0 0
\(682\) 30.0000 1.14876
\(683\) −5.00000 −0.191320 −0.0956598 0.995414i \(-0.530496\pi\)
−0.0956598 + 0.995414i \(0.530496\pi\)
\(684\) 0 0
\(685\) 13.0000 0.496704
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) 1.00000 0.0381246
\(689\) 6.00000 0.228582
\(690\) 0 0
\(691\) −8.00000 −0.304334 −0.152167 0.988355i \(-0.548625\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) 2.00000 0.0760286
\(693\) 0 0
\(694\) 20.0000 0.759190
\(695\) −2.00000 −0.0758643
\(696\) 0 0
\(697\) 2.00000 0.0757554
\(698\) 4.00000 0.151402
\(699\) 0 0
\(700\) 4.00000 0.151186
\(701\) −6.00000 −0.226617 −0.113308 0.993560i \(-0.536145\pi\)
−0.113308 + 0.993560i \(0.536145\pi\)
\(702\) 0 0
\(703\) −7.00000 −0.264010
\(704\) 5.00000 0.188445
\(705\) 0 0
\(706\) −10.0000 −0.376355
\(707\) 6.00000 0.225653
\(708\) 0 0
\(709\) 6.00000 0.225335 0.112667 0.993633i \(-0.464061\pi\)
0.112667 + 0.993633i \(0.464061\pi\)
\(710\) −10.0000 −0.375293
\(711\) 0 0
\(712\) 6.00000 0.224860
\(713\) −30.0000 −1.12351
\(714\) 0 0
\(715\) 5.00000 0.186989
\(716\) 0 0
\(717\) 0 0
\(718\) 30.0000 1.11959
\(719\) 44.0000 1.64092 0.820462 0.571702i \(-0.193717\pi\)
0.820462 + 0.571702i \(0.193717\pi\)
\(720\) 0 0
\(721\) −7.00000 −0.260694
\(722\) 18.0000 0.669891
\(723\) 0 0
\(724\) 10.0000 0.371647
\(725\) −4.00000 −0.148556
\(726\) 0 0
\(727\) 3.00000 0.111264 0.0556319 0.998451i \(-0.482283\pi\)
0.0556319 + 0.998451i \(0.482283\pi\)
\(728\) 1.00000 0.0370625
\(729\) 0 0
\(730\) −9.00000 −0.333105
\(731\) 1.00000 0.0369863
\(732\) 0 0
\(733\) 12.0000 0.443230 0.221615 0.975134i \(-0.428867\pi\)
0.221615 + 0.975134i \(0.428867\pi\)
\(734\) 24.0000 0.885856
\(735\) 0 0
\(736\) −5.00000 −0.184302
\(737\) −40.0000 −1.47342
\(738\) 0 0
\(739\) 46.0000 1.69214 0.846069 0.533074i \(-0.178963\pi\)
0.846069 + 0.533074i \(0.178963\pi\)
\(740\) 7.00000 0.257325
\(741\) 0 0
\(742\) 6.00000 0.220267
\(743\) 28.0000 1.02722 0.513610 0.858024i \(-0.328308\pi\)
0.513610 + 0.858024i \(0.328308\pi\)
\(744\) 0 0
\(745\) 24.0000 0.879292
\(746\) −34.0000 −1.24483
\(747\) 0 0
\(748\) 5.00000 0.182818
\(749\) 6.00000 0.219235
\(750\) 0 0
\(751\) 28.0000 1.02173 0.510867 0.859660i \(-0.329324\pi\)
0.510867 + 0.859660i \(0.329324\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) −1.00000 −0.0364179
\(755\) −7.00000 −0.254756
\(756\) 0 0
\(757\) −32.0000 −1.16306 −0.581530 0.813525i \(-0.697546\pi\)
−0.581530 + 0.813525i \(0.697546\pi\)
\(758\) 2.00000 0.0726433
\(759\) 0 0
\(760\) 1.00000 0.0362738
\(761\) 28.0000 1.01500 0.507500 0.861652i \(-0.330570\pi\)
0.507500 + 0.861652i \(0.330570\pi\)
\(762\) 0 0
\(763\) 7.00000 0.253417
\(764\) −15.0000 −0.542681
\(765\) 0 0
\(766\) 9.00000 0.325183
\(767\) 6.00000 0.216647
\(768\) 0 0
\(769\) −29.0000 −1.04577 −0.522883 0.852404i \(-0.675144\pi\)
−0.522883 + 0.852404i \(0.675144\pi\)
\(770\) 5.00000 0.180187
\(771\) 0 0
\(772\) 14.0000 0.503871
\(773\) 11.0000 0.395643 0.197821 0.980238i \(-0.436613\pi\)
0.197821 + 0.980238i \(0.436613\pi\)
\(774\) 0 0
\(775\) 24.0000 0.862105
\(776\) −14.0000 −0.502571
\(777\) 0 0
\(778\) 10.0000 0.358517
\(779\) −2.00000 −0.0716574
\(780\) 0 0
\(781\) 50.0000 1.78914
\(782\) −5.00000 −0.178800
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 13.0000 0.463990
\(786\) 0 0
\(787\) −37.0000 −1.31891 −0.659454 0.751745i \(-0.729212\pi\)
−0.659454 + 0.751745i \(0.729212\pi\)
\(788\) 6.00000 0.213741
\(789\) 0 0
\(790\) −2.00000 −0.0711568
\(791\) −16.0000 −0.568895
\(792\) 0 0
\(793\) −7.00000 −0.248577
\(794\) 4.00000 0.141955
\(795\) 0 0
\(796\) 25.0000 0.886102
\(797\) 12.0000 0.425062 0.212531 0.977154i \(-0.431829\pi\)
0.212531 + 0.977154i \(0.431829\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 4.00000 0.141421
\(801\) 0 0
\(802\) 10.0000 0.353112
\(803\) 45.0000 1.58802
\(804\) 0 0
\(805\) −5.00000 −0.176227
\(806\) 6.00000 0.211341
\(807\) 0 0
\(808\) 6.00000 0.211079
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 0 0
\(811\) −37.0000 −1.29925 −0.649623 0.760257i \(-0.725073\pi\)
−0.649623 + 0.760257i \(0.725073\pi\)
\(812\) −1.00000 −0.0350931
\(813\) 0 0
\(814\) −35.0000 −1.22675
\(815\) −6.00000 −0.210171
\(816\) 0 0
\(817\) −1.00000 −0.0349856
\(818\) 13.0000 0.454534
\(819\) 0 0
\(820\) 2.00000 0.0698430
\(821\) 4.00000 0.139601 0.0698005 0.997561i \(-0.477764\pi\)
0.0698005 + 0.997561i \(0.477764\pi\)
\(822\) 0 0
\(823\) −2.00000 −0.0697156 −0.0348578 0.999392i \(-0.511098\pi\)
−0.0348578 + 0.999392i \(0.511098\pi\)
\(824\) −7.00000 −0.243857
\(825\) 0 0
\(826\) 6.00000 0.208767
\(827\) 13.0000 0.452054 0.226027 0.974121i \(-0.427426\pi\)
0.226027 + 0.974121i \(0.427426\pi\)
\(828\) 0 0
\(829\) 33.0000 1.14614 0.573069 0.819507i \(-0.305753\pi\)
0.573069 + 0.819507i \(0.305753\pi\)
\(830\) −2.00000 −0.0694210
\(831\) 0 0
\(832\) 1.00000 0.0346688
\(833\) 1.00000 0.0346479
\(834\) 0 0
\(835\) 3.00000 0.103819
\(836\) −5.00000 −0.172929
\(837\) 0 0
\(838\) 19.0000 0.656344
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) −22.0000 −0.758170
\(843\) 0 0
\(844\) −5.00000 −0.172107
\(845\) 1.00000 0.0344010
\(846\) 0 0
\(847\) −14.0000 −0.481046
\(848\) 6.00000 0.206041
\(849\) 0 0
\(850\) 4.00000 0.137199
\(851\) 35.0000 1.19978
\(852\) 0 0
\(853\) 16.0000 0.547830 0.273915 0.961754i \(-0.411681\pi\)
0.273915 + 0.961754i \(0.411681\pi\)
\(854\) −7.00000 −0.239535
\(855\) 0 0
\(856\) 6.00000 0.205076
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 0 0
\(859\) −14.0000 −0.477674 −0.238837 0.971060i \(-0.576766\pi\)
−0.238837 + 0.971060i \(0.576766\pi\)
\(860\) 1.00000 0.0340997
\(861\) 0 0
\(862\) −26.0000 −0.885564
\(863\) 24.0000 0.816970 0.408485 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) 0 0
\(865\) 2.00000 0.0680020
\(866\) −30.0000 −1.01944
\(867\) 0 0
\(868\) 6.00000 0.203653
\(869\) 10.0000 0.339227
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) 7.00000 0.237050
\(873\) 0 0
\(874\) 5.00000 0.169128
\(875\) 9.00000 0.304256
\(876\) 0 0
\(877\) −10.0000 −0.337676 −0.168838 0.985644i \(-0.554001\pi\)
−0.168838 + 0.985644i \(0.554001\pi\)
\(878\) 41.0000 1.38368
\(879\) 0 0
\(880\) 5.00000 0.168550
\(881\) 1.00000 0.0336909 0.0168454 0.999858i \(-0.494638\pi\)
0.0168454 + 0.999858i \(0.494638\pi\)
\(882\) 0 0
\(883\) 29.0000 0.975928 0.487964 0.872864i \(-0.337740\pi\)
0.487964 + 0.872864i \(0.337740\pi\)
\(884\) 1.00000 0.0336336
\(885\) 0 0
\(886\) 6.00000 0.201574
\(887\) −10.0000 −0.335767 −0.167884 0.985807i \(-0.553693\pi\)
−0.167884 + 0.985807i \(0.553693\pi\)
\(888\) 0 0
\(889\) −4.00000 −0.134156
\(890\) 6.00000 0.201120
\(891\) 0 0
\(892\) −12.0000 −0.401790
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) −33.0000 −1.10122
\(899\) −6.00000 −0.200111
\(900\) 0 0
\(901\) 6.00000 0.199889
\(902\) −10.0000 −0.332964
\(903\) 0 0
\(904\) −16.0000 −0.532152
\(905\) 10.0000 0.332411
\(906\) 0 0
\(907\) −12.0000 −0.398453 −0.199227 0.979953i \(-0.563843\pi\)
−0.199227 + 0.979953i \(0.563843\pi\)
\(908\) 10.0000 0.331862
\(909\) 0 0
\(910\) 1.00000 0.0331497
\(911\) −45.0000 −1.49092 −0.745458 0.666552i \(-0.767769\pi\)
−0.745458 + 0.666552i \(0.767769\pi\)
\(912\) 0 0
\(913\) 10.0000 0.330952
\(914\) −12.0000 −0.396925
\(915\) 0 0
\(916\) 22.0000 0.726900
\(917\) 9.00000 0.297206
\(918\) 0 0
\(919\) −10.0000 −0.329870 −0.164935 0.986304i \(-0.552741\pi\)
−0.164935 + 0.986304i \(0.552741\pi\)
\(920\) −5.00000 −0.164845
\(921\) 0 0
\(922\) 29.0000 0.955064
\(923\) 10.0000 0.329154
\(924\) 0 0
\(925\) −28.0000 −0.920634
\(926\) −21.0000 −0.690103
\(927\) 0 0
\(928\) −1.00000 −0.0328266
\(929\) 6.00000 0.196854 0.0984268 0.995144i \(-0.468619\pi\)
0.0984268 + 0.995144i \(0.468619\pi\)
\(930\) 0 0
\(931\) −1.00000 −0.0327737
\(932\) −4.00000 −0.131024
\(933\) 0 0
\(934\) 29.0000 0.948909
\(935\) 5.00000 0.163517
\(936\) 0 0
\(937\) 6.00000 0.196011 0.0980057 0.995186i \(-0.468754\pi\)
0.0980057 + 0.995186i \(0.468754\pi\)
\(938\) −8.00000 −0.261209
\(939\) 0 0
\(940\) 0 0
\(941\) −46.0000 −1.49956 −0.749779 0.661689i \(-0.769840\pi\)
−0.749779 + 0.661689i \(0.769840\pi\)
\(942\) 0 0
\(943\) 10.0000 0.325645
\(944\) 6.00000 0.195283
\(945\) 0 0
\(946\) −5.00000 −0.162564
\(947\) 61.0000 1.98223 0.991117 0.132994i \(-0.0424591\pi\)
0.991117 + 0.132994i \(0.0424591\pi\)
\(948\) 0 0
\(949\) 9.00000 0.292152
\(950\) −4.00000 −0.129777
\(951\) 0 0
\(952\) 1.00000 0.0324102
\(953\) 56.0000 1.81402 0.907009 0.421111i \(-0.138360\pi\)
0.907009 + 0.421111i \(0.138360\pi\)
\(954\) 0 0
\(955\) −15.0000 −0.485389
\(956\) 12.0000 0.388108
\(957\) 0 0
\(958\) −33.0000 −1.06618
\(959\) −13.0000 −0.419792
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) −7.00000 −0.225689
\(963\) 0 0
\(964\) −10.0000 −0.322078
\(965\) 14.0000 0.450676
\(966\) 0 0
\(967\) −21.0000 −0.675314 −0.337657 0.941269i \(-0.609634\pi\)
−0.337657 + 0.941269i \(0.609634\pi\)
\(968\) −14.0000 −0.449977
\(969\) 0 0
\(970\) −14.0000 −0.449513
\(971\) −48.0000 −1.54039 −0.770197 0.637806i \(-0.779842\pi\)
−0.770197 + 0.637806i \(0.779842\pi\)
\(972\) 0 0
\(973\) 2.00000 0.0641171
\(974\) 40.0000 1.28168
\(975\) 0 0
\(976\) −7.00000 −0.224065
\(977\) 51.0000 1.63163 0.815817 0.578310i \(-0.196287\pi\)
0.815817 + 0.578310i \(0.196287\pi\)
\(978\) 0 0
\(979\) −30.0000 −0.958804
\(980\) 1.00000 0.0319438
\(981\) 0 0
\(982\) −16.0000 −0.510581
\(983\) −43.0000 −1.37149 −0.685744 0.727843i \(-0.740523\pi\)
−0.685744 + 0.727843i \(0.740523\pi\)
\(984\) 0 0
\(985\) 6.00000 0.191176
\(986\) −1.00000 −0.0318465
\(987\) 0 0
\(988\) −1.00000 −0.0318142
\(989\) 5.00000 0.158991
\(990\) 0 0
\(991\) −6.00000 −0.190596 −0.0952981 0.995449i \(-0.530380\pi\)
−0.0952981 + 0.995449i \(0.530380\pi\)
\(992\) 6.00000 0.190500
\(993\) 0 0
\(994\) 10.0000 0.317181
\(995\) 25.0000 0.792553
\(996\) 0 0
\(997\) −26.0000 −0.823428 −0.411714 0.911313i \(-0.635070\pi\)
−0.411714 + 0.911313i \(0.635070\pi\)
\(998\) 8.00000 0.253236
\(999\) 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 1638.2.a.g.1.1 1
3.2 odd 2 1638.2.a.n.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1638.2.a.g.1.1 1 1.1 even 1 trivial
1638.2.a.n.1.1 yes 1 3.2 odd 2