Properties

Label 201.2.a.a.1.1
Level $201$
Weight $2$
Character 201.1
Self dual yes
Analytic conductor $1.605$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [201,2,Mod(1,201)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(201, 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("201.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 201 = 3 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 201.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: \(1.60499308063\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 201.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} +2.00000 q^{6} +1.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} +2.00000 q^{6} +1.00000 q^{9} -6.00000 q^{11} -2.00000 q^{12} +4.00000 q^{13} -4.00000 q^{16} -7.00000 q^{17} -2.00000 q^{18} -5.00000 q^{19} +12.0000 q^{22} -1.00000 q^{23} -5.00000 q^{25} -8.00000 q^{26} -1.00000 q^{27} +1.00000 q^{29} -4.00000 q^{31} +8.00000 q^{32} +6.00000 q^{33} +14.0000 q^{34} +2.00000 q^{36} +3.00000 q^{37} +10.0000 q^{38} -4.00000 q^{39} -6.00000 q^{43} -12.0000 q^{44} +2.00000 q^{46} +9.00000 q^{47} +4.00000 q^{48} -7.00000 q^{49} +10.0000 q^{50} +7.00000 q^{51} +8.00000 q^{52} +10.0000 q^{53} +2.00000 q^{54} +5.00000 q^{57} -2.00000 q^{58} +3.00000 q^{59} +2.00000 q^{61} +8.00000 q^{62} -8.00000 q^{64} -12.0000 q^{66} -1.00000 q^{67} -14.0000 q^{68} +1.00000 q^{69} -16.0000 q^{71} -7.00000 q^{73} -6.00000 q^{74} +5.00000 q^{75} -10.0000 q^{76} +8.00000 q^{78} +8.00000 q^{79} +1.00000 q^{81} -4.00000 q^{83} +12.0000 q^{86} -1.00000 q^{87} -15.0000 q^{89} -2.00000 q^{92} +4.00000 q^{93} -18.0000 q^{94} -8.00000 q^{96} +4.00000 q^{97} +14.0000 q^{98} -6.00000 q^{99} +O(q^{100})\)

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 \(0.750000\pi\)
\(3\) −1.00000 −0.577350
\(4\) 2.00000 1.00000
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 2.00000 0.816497
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −6.00000 −1.80907 −0.904534 0.426401i \(-0.859781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) −2.00000 −0.577350
\(13\) 4.00000 1.10940 0.554700 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) −7.00000 −1.69775 −0.848875 0.528594i \(-0.822719\pi\)
−0.848875 + 0.528594i \(0.822719\pi\)
\(18\) −2.00000 −0.471405
\(19\) −5.00000 −1.14708 −0.573539 0.819178i \(-0.694430\pi\)
−0.573539 + 0.819178i \(0.694430\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 12.0000 2.55841
\(23\) −1.00000 −0.208514 −0.104257 0.994550i \(-0.533247\pi\)
−0.104257 + 0.994550i \(0.533247\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) −8.00000 −1.56893
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.00000 0.185695 0.0928477 0.995680i \(-0.470403\pi\)
0.0928477 + 0.995680i \(0.470403\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 8.00000 1.41421
\(33\) 6.00000 1.04447
\(34\) 14.0000 2.40098
\(35\) 0 0
\(36\) 2.00000 0.333333
\(37\) 3.00000 0.493197 0.246598 0.969118i \(-0.420687\pi\)
0.246598 + 0.969118i \(0.420687\pi\)
\(38\) 10.0000 1.62221
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −6.00000 −0.914991 −0.457496 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) −12.0000 −1.80907
\(45\) 0 0
\(46\) 2.00000 0.294884
\(47\) 9.00000 1.31278 0.656392 0.754420i \(-0.272082\pi\)
0.656392 + 0.754420i \(0.272082\pi\)
\(48\) 4.00000 0.577350
\(49\) −7.00000 −1.00000
\(50\) 10.0000 1.41421
\(51\) 7.00000 0.980196
\(52\) 8.00000 1.10940
\(53\) 10.0000 1.37361 0.686803 0.726844i \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) 2.00000 0.272166
\(55\) 0 0
\(56\) 0 0
\(57\) 5.00000 0.662266
\(58\) −2.00000 −0.262613
\(59\) 3.00000 0.390567 0.195283 0.980747i \(-0.437437\pi\)
0.195283 + 0.980747i \(0.437437\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 8.00000 1.01600
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) −12.0000 −1.47710
\(67\) −1.00000 −0.122169
\(68\) −14.0000 −1.69775
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −16.0000 −1.89885 −0.949425 0.313993i \(-0.898333\pi\)
−0.949425 + 0.313993i \(0.898333\pi\)
\(72\) 0 0
\(73\) −7.00000 −0.819288 −0.409644 0.912245i \(-0.634347\pi\)
−0.409644 + 0.912245i \(0.634347\pi\)
\(74\) −6.00000 −0.697486
\(75\) 5.00000 0.577350
\(76\) −10.0000 −1.14708
\(77\) 0 0
\(78\) 8.00000 0.905822
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 12.0000 1.29399
\(87\) −1.00000 −0.107211
\(88\) 0 0
\(89\) −15.0000 −1.59000 −0.794998 0.606612i \(-0.792528\pi\)
−0.794998 + 0.606612i \(0.792528\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −2.00000 −0.208514
\(93\) 4.00000 0.414781
\(94\) −18.0000 −1.85656
\(95\) 0 0
\(96\) −8.00000 −0.816497
\(97\) 4.00000 0.406138 0.203069 0.979164i \(-0.434908\pi\)
0.203069 + 0.979164i \(0.434908\pi\)
\(98\) 14.0000 1.41421
\(99\) −6.00000 −0.603023
\(100\) −10.0000 −1.00000
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) −14.0000 −1.38621
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −20.0000 −1.94257
\(107\) 7.00000 0.676716 0.338358 0.941018i \(-0.390129\pi\)
0.338358 + 0.941018i \(0.390129\pi\)
\(108\) −2.00000 −0.192450
\(109\) 4.00000 0.383131 0.191565 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) 0 0
\(111\) −3.00000 −0.284747
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) −10.0000 −0.936586
\(115\) 0 0
\(116\) 2.00000 0.185695
\(117\) 4.00000 0.369800
\(118\) −6.00000 −0.552345
\(119\) 0 0
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) −4.00000 −0.362143
\(123\) 0 0
\(124\) −8.00000 −0.718421
\(125\) 0 0
\(126\) 0 0
\(127\) −9.00000 −0.798621 −0.399310 0.916816i \(-0.630750\pi\)
−0.399310 + 0.916816i \(0.630750\pi\)
\(128\) 0 0
\(129\) 6.00000 0.528271
\(130\) 0 0
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 12.0000 1.04447
\(133\) 0 0
\(134\) 2.00000 0.172774
\(135\) 0 0
\(136\) 0 0
\(137\) 14.0000 1.19610 0.598050 0.801459i \(-0.295942\pi\)
0.598050 + 0.801459i \(0.295942\pi\)
\(138\) −2.00000 −0.170251
\(139\) 2.00000 0.169638 0.0848189 0.996396i \(-0.472969\pi\)
0.0848189 + 0.996396i \(0.472969\pi\)
\(140\) 0 0
\(141\) −9.00000 −0.757937
\(142\) 32.0000 2.68538
\(143\) −24.0000 −2.00698
\(144\) −4.00000 −0.333333
\(145\) 0 0
\(146\) 14.0000 1.15865
\(147\) 7.00000 0.577350
\(148\) 6.00000 0.493197
\(149\) 19.0000 1.55654 0.778270 0.627929i \(-0.216097\pi\)
0.778270 + 0.627929i \(0.216097\pi\)
\(150\) −10.0000 −0.816497
\(151\) −21.0000 −1.70896 −0.854478 0.519488i \(-0.826123\pi\)
−0.854478 + 0.519488i \(0.826123\pi\)
\(152\) 0 0
\(153\) −7.00000 −0.565916
\(154\) 0 0
\(155\) 0 0
\(156\) −8.00000 −0.640513
\(157\) 17.0000 1.35675 0.678374 0.734717i \(-0.262685\pi\)
0.678374 + 0.734717i \(0.262685\pi\)
\(158\) −16.0000 −1.27289
\(159\) −10.0000 −0.793052
\(160\) 0 0
\(161\) 0 0
\(162\) −2.00000 −0.157135
\(163\) −13.0000 −1.01824 −0.509119 0.860696i \(-0.670029\pi\)
−0.509119 + 0.860696i \(0.670029\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 8.00000 0.620920
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) −5.00000 −0.382360
\(172\) −12.0000 −0.914991
\(173\) −11.0000 −0.836315 −0.418157 0.908375i \(-0.637324\pi\)
−0.418157 + 0.908375i \(0.637324\pi\)
\(174\) 2.00000 0.151620
\(175\) 0 0
\(176\) 24.0000 1.80907
\(177\) −3.00000 −0.225494
\(178\) 30.0000 2.24860
\(179\) 26.0000 1.94333 0.971666 0.236360i \(-0.0759544\pi\)
0.971666 + 0.236360i \(0.0759544\pi\)
\(180\) 0 0
\(181\) 23.0000 1.70958 0.854788 0.518977i \(-0.173687\pi\)
0.854788 + 0.518977i \(0.173687\pi\)
\(182\) 0 0
\(183\) −2.00000 −0.147844
\(184\) 0 0
\(185\) 0 0
\(186\) −8.00000 −0.586588
\(187\) 42.0000 3.07134
\(188\) 18.0000 1.31278
\(189\) 0 0
\(190\) 0 0
\(191\) −24.0000 −1.73658 −0.868290 0.496058i \(-0.834780\pi\)
−0.868290 + 0.496058i \(0.834780\pi\)
\(192\) 8.00000 0.577350
\(193\) −15.0000 −1.07972 −0.539862 0.841754i \(-0.681524\pi\)
−0.539862 + 0.841754i \(0.681524\pi\)
\(194\) −8.00000 −0.574367
\(195\) 0 0
\(196\) −14.0000 −1.00000
\(197\) −22.0000 −1.56744 −0.783718 0.621117i \(-0.786679\pi\)
−0.783718 + 0.621117i \(0.786679\pi\)
\(198\) 12.0000 0.852803
\(199\) 7.00000 0.496217 0.248108 0.968732i \(-0.420191\pi\)
0.248108 + 0.968732i \(0.420191\pi\)
\(200\) 0 0
\(201\) 1.00000 0.0705346
\(202\) 0 0
\(203\) 0 0
\(204\) 14.0000 0.980196
\(205\) 0 0
\(206\) −8.00000 −0.557386
\(207\) −1.00000 −0.0695048
\(208\) −16.0000 −1.10940
\(209\) 30.0000 2.07514
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 20.0000 1.37361
\(213\) 16.0000 1.09630
\(214\) −14.0000 −0.957020
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −8.00000 −0.541828
\(219\) 7.00000 0.473016
\(220\) 0 0
\(221\) −28.0000 −1.88348
\(222\) 6.00000 0.402694
\(223\) 15.0000 1.00447 0.502237 0.864730i \(-0.332510\pi\)
0.502237 + 0.864730i \(0.332510\pi\)
\(224\) 0 0
\(225\) −5.00000 −0.333333
\(226\) −12.0000 −0.798228
\(227\) −19.0000 −1.26107 −0.630537 0.776159i \(-0.717165\pi\)
−0.630537 + 0.776159i \(0.717165\pi\)
\(228\) 10.0000 0.662266
\(229\) −18.0000 −1.18947 −0.594737 0.803921i \(-0.702744\pi\)
−0.594737 + 0.803921i \(0.702744\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −16.0000 −1.04819 −0.524097 0.851658i \(-0.675597\pi\)
−0.524097 + 0.851658i \(0.675597\pi\)
\(234\) −8.00000 −0.522976
\(235\) 0 0
\(236\) 6.00000 0.390567
\(237\) −8.00000 −0.519656
\(238\) 0 0
\(239\) 16.0000 1.03495 0.517477 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) 0 0
\(241\) 17.0000 1.09507 0.547533 0.836784i \(-0.315567\pi\)
0.547533 + 0.836784i \(0.315567\pi\)
\(242\) −50.0000 −3.21412
\(243\) −1.00000 −0.0641500
\(244\) 4.00000 0.256074
\(245\) 0 0
\(246\) 0 0
\(247\) −20.0000 −1.27257
\(248\) 0 0
\(249\) 4.00000 0.253490
\(250\) 0 0
\(251\) −2.00000 −0.126239 −0.0631194 0.998006i \(-0.520105\pi\)
−0.0631194 + 0.998006i \(0.520105\pi\)
\(252\) 0 0
\(253\) 6.00000 0.377217
\(254\) 18.0000 1.12942
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) −19.0000 −1.18519 −0.592594 0.805502i \(-0.701896\pi\)
−0.592594 + 0.805502i \(0.701896\pi\)
\(258\) −12.0000 −0.747087
\(259\) 0 0
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) −8.00000 −0.494242
\(263\) −8.00000 −0.493301 −0.246651 0.969104i \(-0.579330\pi\)
−0.246651 + 0.969104i \(0.579330\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 15.0000 0.917985
\(268\) −2.00000 −0.122169
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) −22.0000 −1.33640 −0.668202 0.743980i \(-0.732936\pi\)
−0.668202 + 0.743980i \(0.732936\pi\)
\(272\) 28.0000 1.69775
\(273\) 0 0
\(274\) −28.0000 −1.69154
\(275\) 30.0000 1.80907
\(276\) 2.00000 0.120386
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) −4.00000 −0.239904
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 18.0000 1.07188
\(283\) −19.0000 −1.12943 −0.564716 0.825285i \(-0.691014\pi\)
−0.564716 + 0.825285i \(0.691014\pi\)
\(284\) −32.0000 −1.89885
\(285\) 0 0
\(286\) 48.0000 2.83830
\(287\) 0 0
\(288\) 8.00000 0.471405
\(289\) 32.0000 1.88235
\(290\) 0 0
\(291\) −4.00000 −0.234484
\(292\) −14.0000 −0.819288
\(293\) −10.0000 −0.584206 −0.292103 0.956387i \(-0.594355\pi\)
−0.292103 + 0.956387i \(0.594355\pi\)
\(294\) −14.0000 −0.816497
\(295\) 0 0
\(296\) 0 0
\(297\) 6.00000 0.348155
\(298\) −38.0000 −2.20128
\(299\) −4.00000 −0.231326
\(300\) 10.0000 0.577350
\(301\) 0 0
\(302\) 42.0000 2.41683
\(303\) 0 0
\(304\) 20.0000 1.14708
\(305\) 0 0
\(306\) 14.0000 0.800327
\(307\) 23.0000 1.31268 0.656340 0.754466i \(-0.272104\pi\)
0.656340 + 0.754466i \(0.272104\pi\)
\(308\) 0 0
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) 26.0000 1.47432 0.737162 0.675716i \(-0.236165\pi\)
0.737162 + 0.675716i \(0.236165\pi\)
\(312\) 0 0
\(313\) −18.0000 −1.01742 −0.508710 0.860938i \(-0.669877\pi\)
−0.508710 + 0.860938i \(0.669877\pi\)
\(314\) −34.0000 −1.91873
\(315\) 0 0
\(316\) 16.0000 0.900070
\(317\) −14.0000 −0.786318 −0.393159 0.919470i \(-0.628618\pi\)
−0.393159 + 0.919470i \(0.628618\pi\)
\(318\) 20.0000 1.12154
\(319\) −6.00000 −0.335936
\(320\) 0 0
\(321\) −7.00000 −0.390702
\(322\) 0 0
\(323\) 35.0000 1.94745
\(324\) 2.00000 0.111111
\(325\) −20.0000 −1.10940
\(326\) 26.0000 1.44001
\(327\) −4.00000 −0.221201
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) −8.00000 −0.439057
\(333\) 3.00000 0.164399
\(334\) −24.0000 −1.31322
\(335\) 0 0
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) −6.00000 −0.326357
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) 24.0000 1.29967
\(342\) 10.0000 0.540738
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 22.0000 1.18273
\(347\) −6.00000 −0.322097 −0.161048 0.986947i \(-0.551488\pi\)
−0.161048 + 0.986947i \(0.551488\pi\)
\(348\) −2.00000 −0.107211
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 0 0
\(351\) −4.00000 −0.213504
\(352\) −48.0000 −2.55841
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 6.00000 0.318896
\(355\) 0 0
\(356\) −30.0000 −1.59000
\(357\) 0 0
\(358\) −52.0000 −2.74829
\(359\) 5.00000 0.263890 0.131945 0.991257i \(-0.457878\pi\)
0.131945 + 0.991257i \(0.457878\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) −46.0000 −2.41771
\(363\) −25.0000 −1.31216
\(364\) 0 0
\(365\) 0 0
\(366\) 4.00000 0.209083
\(367\) 28.0000 1.46159 0.730794 0.682598i \(-0.239150\pi\)
0.730794 + 0.682598i \(0.239150\pi\)
\(368\) 4.00000 0.208514
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 8.00000 0.414781
\(373\) 8.00000 0.414224 0.207112 0.978317i \(-0.433593\pi\)
0.207112 + 0.978317i \(0.433593\pi\)
\(374\) −84.0000 −4.34354
\(375\) 0 0
\(376\) 0 0
\(377\) 4.00000 0.206010
\(378\) 0 0
\(379\) −10.0000 −0.513665 −0.256833 0.966456i \(-0.582679\pi\)
−0.256833 + 0.966456i \(0.582679\pi\)
\(380\) 0 0
\(381\) 9.00000 0.461084
\(382\) 48.0000 2.45589
\(383\) 12.0000 0.613171 0.306586 0.951843i \(-0.400813\pi\)
0.306586 + 0.951843i \(0.400813\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 30.0000 1.52696
\(387\) −6.00000 −0.304997
\(388\) 8.00000 0.406138
\(389\) −10.0000 −0.507020 −0.253510 0.967333i \(-0.581585\pi\)
−0.253510 + 0.967333i \(0.581585\pi\)
\(390\) 0 0
\(391\) 7.00000 0.354005
\(392\) 0 0
\(393\) −4.00000 −0.201773
\(394\) 44.0000 2.21669
\(395\) 0 0
\(396\) −12.0000 −0.603023
\(397\) 29.0000 1.45547 0.727734 0.685859i \(-0.240573\pi\)
0.727734 + 0.685859i \(0.240573\pi\)
\(398\) −14.0000 −0.701757
\(399\) 0 0
\(400\) 20.0000 1.00000
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) −2.00000 −0.0997509
\(403\) −16.0000 −0.797017
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −18.0000 −0.892227
\(408\) 0 0
\(409\) −32.0000 −1.58230 −0.791149 0.611623i \(-0.790517\pi\)
−0.791149 + 0.611623i \(0.790517\pi\)
\(410\) 0 0
\(411\) −14.0000 −0.690569
\(412\) 8.00000 0.394132
\(413\) 0 0
\(414\) 2.00000 0.0982946
\(415\) 0 0
\(416\) 32.0000 1.56893
\(417\) −2.00000 −0.0979404
\(418\) −60.0000 −2.93470
\(419\) −7.00000 −0.341972 −0.170986 0.985273i \(-0.554695\pi\)
−0.170986 + 0.985273i \(0.554695\pi\)
\(420\) 0 0
\(421\) 19.0000 0.926003 0.463002 0.886357i \(-0.346772\pi\)
0.463002 + 0.886357i \(0.346772\pi\)
\(422\) 40.0000 1.94717
\(423\) 9.00000 0.437595
\(424\) 0 0
\(425\) 35.0000 1.69775
\(426\) −32.0000 −1.55041
\(427\) 0 0
\(428\) 14.0000 0.676716
\(429\) 24.0000 1.15873
\(430\) 0 0
\(431\) 5.00000 0.240842 0.120421 0.992723i \(-0.461576\pi\)
0.120421 + 0.992723i \(0.461576\pi\)
\(432\) 4.00000 0.192450
\(433\) 20.0000 0.961139 0.480569 0.876957i \(-0.340430\pi\)
0.480569 + 0.876957i \(0.340430\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 8.00000 0.383131
\(437\) 5.00000 0.239182
\(438\) −14.0000 −0.668946
\(439\) −29.0000 −1.38409 −0.692047 0.721852i \(-0.743291\pi\)
−0.692047 + 0.721852i \(0.743291\pi\)
\(440\) 0 0
\(441\) −7.00000 −0.333333
\(442\) 56.0000 2.66365
\(443\) 4.00000 0.190046 0.0950229 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) −6.00000 −0.284747
\(445\) 0 0
\(446\) −30.0000 −1.42054
\(447\) −19.0000 −0.898669
\(448\) 0 0
\(449\) −27.0000 −1.27421 −0.637104 0.770778i \(-0.719868\pi\)
−0.637104 + 0.770778i \(0.719868\pi\)
\(450\) 10.0000 0.471405
\(451\) 0 0
\(452\) 12.0000 0.564433
\(453\) 21.0000 0.986666
\(454\) 38.0000 1.78343
\(455\) 0 0
\(456\) 0 0
\(457\) −35.0000 −1.63723 −0.818615 0.574342i \(-0.805258\pi\)
−0.818615 + 0.574342i \(0.805258\pi\)
\(458\) 36.0000 1.68217
\(459\) 7.00000 0.326732
\(460\) 0 0
\(461\) −21.0000 −0.978068 −0.489034 0.872265i \(-0.662651\pi\)
−0.489034 + 0.872265i \(0.662651\pi\)
\(462\) 0 0
\(463\) 28.0000 1.30127 0.650635 0.759390i \(-0.274503\pi\)
0.650635 + 0.759390i \(0.274503\pi\)
\(464\) −4.00000 −0.185695
\(465\) 0 0
\(466\) 32.0000 1.48237
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 8.00000 0.369800
\(469\) 0 0
\(470\) 0 0
\(471\) −17.0000 −0.783319
\(472\) 0 0
\(473\) 36.0000 1.65528
\(474\) 16.0000 0.734904
\(475\) 25.0000 1.14708
\(476\) 0 0
\(477\) 10.0000 0.457869
\(478\) −32.0000 −1.46365
\(479\) −35.0000 −1.59919 −0.799595 0.600539i \(-0.794953\pi\)
−0.799595 + 0.600539i \(0.794953\pi\)
\(480\) 0 0
\(481\) 12.0000 0.547153
\(482\) −34.0000 −1.54866
\(483\) 0 0
\(484\) 50.0000 2.27273
\(485\) 0 0
\(486\) 2.00000 0.0907218
\(487\) 4.00000 0.181257 0.0906287 0.995885i \(-0.471112\pi\)
0.0906287 + 0.995885i \(0.471112\pi\)
\(488\) 0 0
\(489\) 13.0000 0.587880
\(490\) 0 0
\(491\) 5.00000 0.225647 0.112823 0.993615i \(-0.464011\pi\)
0.112823 + 0.993615i \(0.464011\pi\)
\(492\) 0 0
\(493\) −7.00000 −0.315264
\(494\) 40.0000 1.79969
\(495\) 0 0
\(496\) 16.0000 0.718421
\(497\) 0 0
\(498\) −8.00000 −0.358489
\(499\) −30.0000 −1.34298 −0.671492 0.741012i \(-0.734346\pi\)
−0.671492 + 0.741012i \(0.734346\pi\)
\(500\) 0 0
\(501\) −12.0000 −0.536120
\(502\) 4.00000 0.178529
\(503\) −8.00000 −0.356702 −0.178351 0.983967i \(-0.557076\pi\)
−0.178351 + 0.983967i \(0.557076\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −12.0000 −0.533465
\(507\) −3.00000 −0.133235
\(508\) −18.0000 −0.798621
\(509\) 15.0000 0.664863 0.332432 0.943127i \(-0.392131\pi\)
0.332432 + 0.943127i \(0.392131\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −32.0000 −1.41421
\(513\) 5.00000 0.220755
\(514\) 38.0000 1.67611
\(515\) 0 0
\(516\) 12.0000 0.528271
\(517\) −54.0000 −2.37492
\(518\) 0 0
\(519\) 11.0000 0.482846
\(520\) 0 0
\(521\) 30.0000 1.31432 0.657162 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) −2.00000 −0.0875376
\(523\) 41.0000 1.79280 0.896402 0.443241i \(-0.146171\pi\)
0.896402 + 0.443241i \(0.146171\pi\)
\(524\) 8.00000 0.349482
\(525\) 0 0
\(526\) 16.0000 0.697633
\(527\) 28.0000 1.21970
\(528\) −24.0000 −1.04447
\(529\) −22.0000 −0.956522
\(530\) 0 0
\(531\) 3.00000 0.130189
\(532\) 0 0
\(533\) 0 0
\(534\) −30.0000 −1.29823
\(535\) 0 0
\(536\) 0 0
\(537\) −26.0000 −1.12198
\(538\) 12.0000 0.517357
\(539\) 42.0000 1.80907
\(540\) 0 0
\(541\) −40.0000 −1.71973 −0.859867 0.510518i \(-0.829454\pi\)
−0.859867 + 0.510518i \(0.829454\pi\)
\(542\) 44.0000 1.88996
\(543\) −23.0000 −0.987024
\(544\) −56.0000 −2.40098
\(545\) 0 0
\(546\) 0 0
\(547\) −2.00000 −0.0855138 −0.0427569 0.999086i \(-0.513614\pi\)
−0.0427569 + 0.999086i \(0.513614\pi\)
\(548\) 28.0000 1.19610
\(549\) 2.00000 0.0853579
\(550\) −60.0000 −2.55841
\(551\) −5.00000 −0.213007
\(552\) 0 0
\(553\) 0 0
\(554\) 4.00000 0.169944
\(555\) 0 0
\(556\) 4.00000 0.169638
\(557\) −14.0000 −0.593199 −0.296600 0.955002i \(-0.595853\pi\)
−0.296600 + 0.955002i \(0.595853\pi\)
\(558\) 8.00000 0.338667
\(559\) −24.0000 −1.01509
\(560\) 0 0
\(561\) −42.0000 −1.77324
\(562\) 36.0000 1.51857
\(563\) 8.00000 0.337160 0.168580 0.985688i \(-0.446082\pi\)
0.168580 + 0.985688i \(0.446082\pi\)
\(564\) −18.0000 −0.757937
\(565\) 0 0
\(566\) 38.0000 1.59726
\(567\) 0 0
\(568\) 0 0
\(569\) 15.0000 0.628833 0.314416 0.949285i \(-0.398191\pi\)
0.314416 + 0.949285i \(0.398191\pi\)
\(570\) 0 0
\(571\) −7.00000 −0.292941 −0.146470 0.989215i \(-0.546791\pi\)
−0.146470 + 0.989215i \(0.546791\pi\)
\(572\) −48.0000 −2.00698
\(573\) 24.0000 1.00261
\(574\) 0 0
\(575\) 5.00000 0.208514
\(576\) −8.00000 −0.333333
\(577\) −12.0000 −0.499567 −0.249783 0.968302i \(-0.580359\pi\)
−0.249783 + 0.968302i \(0.580359\pi\)
\(578\) −64.0000 −2.66205
\(579\) 15.0000 0.623379
\(580\) 0 0
\(581\) 0 0
\(582\) 8.00000 0.331611
\(583\) −60.0000 −2.48495
\(584\) 0 0
\(585\) 0 0
\(586\) 20.0000 0.826192
\(587\) −24.0000 −0.990586 −0.495293 0.868726i \(-0.664939\pi\)
−0.495293 + 0.868726i \(0.664939\pi\)
\(588\) 14.0000 0.577350
\(589\) 20.0000 0.824086
\(590\) 0 0
\(591\) 22.0000 0.904959
\(592\) −12.0000 −0.493197
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) −12.0000 −0.492366
\(595\) 0 0
\(596\) 38.0000 1.55654
\(597\) −7.00000 −0.286491
\(598\) 8.00000 0.327144
\(599\) −6.00000 −0.245153 −0.122577 0.992459i \(-0.539116\pi\)
−0.122577 + 0.992459i \(0.539116\pi\)
\(600\) 0 0
\(601\) −45.0000 −1.83559 −0.917794 0.397057i \(-0.870032\pi\)
−0.917794 + 0.397057i \(0.870032\pi\)
\(602\) 0 0
\(603\) −1.00000 −0.0407231
\(604\) −42.0000 −1.70896
\(605\) 0 0
\(606\) 0 0
\(607\) 8.00000 0.324710 0.162355 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) −40.0000 −1.62221
\(609\) 0 0
\(610\) 0 0
\(611\) 36.0000 1.45640
\(612\) −14.0000 −0.565916
\(613\) −33.0000 −1.33286 −0.666429 0.745569i \(-0.732178\pi\)
−0.666429 + 0.745569i \(0.732178\pi\)
\(614\) −46.0000 −1.85641
\(615\) 0 0
\(616\) 0 0
\(617\) −39.0000 −1.57008 −0.785040 0.619445i \(-0.787358\pi\)
−0.785040 + 0.619445i \(0.787358\pi\)
\(618\) 8.00000 0.321807
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) −52.0000 −2.08501
\(623\) 0 0
\(624\) 16.0000 0.640513
\(625\) 25.0000 1.00000
\(626\) 36.0000 1.43885
\(627\) −30.0000 −1.19808
\(628\) 34.0000 1.35675
\(629\) −21.0000 −0.837325
\(630\) 0 0
\(631\) −6.00000 −0.238856 −0.119428 0.992843i \(-0.538106\pi\)
−0.119428 + 0.992843i \(0.538106\pi\)
\(632\) 0 0
\(633\) 20.0000 0.794929
\(634\) 28.0000 1.11202
\(635\) 0 0
\(636\) −20.0000 −0.793052
\(637\) −28.0000 −1.10940
\(638\) 12.0000 0.475085
\(639\) −16.0000 −0.632950
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 14.0000 0.552536
\(643\) 36.0000 1.41970 0.709851 0.704352i \(-0.248762\pi\)
0.709851 + 0.704352i \(0.248762\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −70.0000 −2.75411
\(647\) 18.0000 0.707653 0.353827 0.935311i \(-0.384880\pi\)
0.353827 + 0.935311i \(0.384880\pi\)
\(648\) 0 0
\(649\) −18.0000 −0.706562
\(650\) 40.0000 1.56893
\(651\) 0 0
\(652\) −26.0000 −1.01824
\(653\) 10.0000 0.391330 0.195665 0.980671i \(-0.437313\pi\)
0.195665 + 0.980671i \(0.437313\pi\)
\(654\) 8.00000 0.312825
\(655\) 0 0
\(656\) 0 0
\(657\) −7.00000 −0.273096
\(658\) 0 0
\(659\) 33.0000 1.28550 0.642749 0.766077i \(-0.277794\pi\)
0.642749 + 0.766077i \(0.277794\pi\)
\(660\) 0 0
\(661\) −12.0000 −0.466746 −0.233373 0.972387i \(-0.574976\pi\)
−0.233373 + 0.972387i \(0.574976\pi\)
\(662\) 8.00000 0.310929
\(663\) 28.0000 1.08743
\(664\) 0 0
\(665\) 0 0
\(666\) −6.00000 −0.232495
\(667\) −1.00000 −0.0387202
\(668\) 24.0000 0.928588
\(669\) −15.0000 −0.579934
\(670\) 0 0
\(671\) −12.0000 −0.463255
\(672\) 0 0
\(673\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(674\) −4.00000 −0.154074
\(675\) 5.00000 0.192450
\(676\) 6.00000 0.230769
\(677\) −30.0000 −1.15299 −0.576497 0.817099i \(-0.695581\pi\)
−0.576497 + 0.817099i \(0.695581\pi\)
\(678\) 12.0000 0.460857
\(679\) 0 0
\(680\) 0 0
\(681\) 19.0000 0.728082
\(682\) −48.0000 −1.83801
\(683\) −14.0000 −0.535695 −0.267848 0.963461i \(-0.586312\pi\)
−0.267848 + 0.963461i \(0.586312\pi\)
\(684\) −10.0000 −0.382360
\(685\) 0 0
\(686\) 0 0
\(687\) 18.0000 0.686743
\(688\) 24.0000 0.914991
\(689\) 40.0000 1.52388
\(690\) 0 0
\(691\) −5.00000 −0.190209 −0.0951045 0.995467i \(-0.530319\pi\)
−0.0951045 + 0.995467i \(0.530319\pi\)
\(692\) −22.0000 −0.836315
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) −28.0000 −1.05982
\(699\) 16.0000 0.605176
\(700\) 0 0
\(701\) 14.0000 0.528773 0.264386 0.964417i \(-0.414831\pi\)
0.264386 + 0.964417i \(0.414831\pi\)
\(702\) 8.00000 0.301941
\(703\) −15.0000 −0.565736
\(704\) 48.0000 1.80907
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) −6.00000 −0.225494
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 0 0
\(713\) 4.00000 0.149801
\(714\) 0 0
\(715\) 0 0
\(716\) 52.0000 1.94333
\(717\) −16.0000 −0.597531
\(718\) −10.0000 −0.373197
\(719\) −1.00000 −0.0372937 −0.0186469 0.999826i \(-0.505936\pi\)
−0.0186469 + 0.999826i \(0.505936\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −12.0000 −0.446594
\(723\) −17.0000 −0.632237
\(724\) 46.0000 1.70958
\(725\) −5.00000 −0.185695
\(726\) 50.0000 1.85567
\(727\) 2.00000 0.0741759 0.0370879 0.999312i \(-0.488192\pi\)
0.0370879 + 0.999312i \(0.488192\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 42.0000 1.55343
\(732\) −4.00000 −0.147844
\(733\) 34.0000 1.25582 0.627909 0.778287i \(-0.283911\pi\)
0.627909 + 0.778287i \(0.283911\pi\)
\(734\) −56.0000 −2.06700
\(735\) 0 0
\(736\) −8.00000 −0.294884
\(737\) 6.00000 0.221013
\(738\) 0 0
\(739\) 44.0000 1.61857 0.809283 0.587419i \(-0.199856\pi\)
0.809283 + 0.587419i \(0.199856\pi\)
\(740\) 0 0
\(741\) 20.0000 0.734718
\(742\) 0 0
\(743\) 40.0000 1.46746 0.733729 0.679442i \(-0.237778\pi\)
0.733729 + 0.679442i \(0.237778\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −16.0000 −0.585802
\(747\) −4.00000 −0.146352
\(748\) 84.0000 3.07134
\(749\) 0 0
\(750\) 0 0
\(751\) −49.0000 −1.78804 −0.894018 0.448032i \(-0.852125\pi\)
−0.894018 + 0.448032i \(0.852125\pi\)
\(752\) −36.0000 −1.31278
\(753\) 2.00000 0.0728841
\(754\) −8.00000 −0.291343
\(755\) 0 0
\(756\) 0 0
\(757\) 50.0000 1.81728 0.908640 0.417579i \(-0.137121\pi\)
0.908640 + 0.417579i \(0.137121\pi\)
\(758\) 20.0000 0.726433
\(759\) −6.00000 −0.217786
\(760\) 0 0
\(761\) −3.00000 −0.108750 −0.0543750 0.998521i \(-0.517317\pi\)
−0.0543750 + 0.998521i \(0.517317\pi\)
\(762\) −18.0000 −0.652071
\(763\) 0 0
\(764\) −48.0000 −1.73658
\(765\) 0 0
\(766\) −24.0000 −0.867155
\(767\) 12.0000 0.433295
\(768\) −16.0000 −0.577350
\(769\) 34.0000 1.22607 0.613036 0.790055i \(-0.289948\pi\)
0.613036 + 0.790055i \(0.289948\pi\)
\(770\) 0 0
\(771\) 19.0000 0.684268
\(772\) −30.0000 −1.07972
\(773\) −33.0000 −1.18693 −0.593464 0.804861i \(-0.702240\pi\)
−0.593464 + 0.804861i \(0.702240\pi\)
\(774\) 12.0000 0.431331
\(775\) 20.0000 0.718421
\(776\) 0 0
\(777\) 0 0
\(778\) 20.0000 0.717035
\(779\) 0 0
\(780\) 0 0
\(781\) 96.0000 3.43515
\(782\) −14.0000 −0.500639
\(783\) −1.00000 −0.0357371
\(784\) 28.0000 1.00000
\(785\) 0 0
\(786\) 8.00000 0.285351
\(787\) −4.00000 −0.142585 −0.0712923 0.997455i \(-0.522712\pi\)
−0.0712923 + 0.997455i \(0.522712\pi\)
\(788\) −44.0000 −1.56744
\(789\) 8.00000 0.284808
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 8.00000 0.284088
\(794\) −58.0000 −2.05834
\(795\) 0 0
\(796\) 14.0000 0.496217
\(797\) 34.0000 1.20434 0.602171 0.798367i \(-0.294303\pi\)
0.602171 + 0.798367i \(0.294303\pi\)
\(798\) 0 0
\(799\) −63.0000 −2.22878
\(800\) −40.0000 −1.41421
\(801\) −15.0000 −0.529999
\(802\) 60.0000 2.11867
\(803\) 42.0000 1.48215
\(804\) 2.00000 0.0705346
\(805\) 0 0
\(806\) 32.0000 1.12715
\(807\) 6.00000 0.211210
\(808\) 0 0
\(809\) −20.0000 −0.703163 −0.351581 0.936157i \(-0.614356\pi\)
−0.351581 + 0.936157i \(0.614356\pi\)
\(810\) 0 0
\(811\) −2.00000 −0.0702295 −0.0351147 0.999383i \(-0.511180\pi\)
−0.0351147 + 0.999383i \(0.511180\pi\)
\(812\) 0 0
\(813\) 22.0000 0.771574
\(814\) 36.0000 1.26180
\(815\) 0 0
\(816\) −28.0000 −0.980196
\(817\) 30.0000 1.04957
\(818\) 64.0000 2.23771
\(819\) 0 0
\(820\) 0 0
\(821\) −27.0000 −0.942306 −0.471153 0.882051i \(-0.656162\pi\)
−0.471153 + 0.882051i \(0.656162\pi\)
\(822\) 28.0000 0.976612
\(823\) −17.0000 −0.592583 −0.296291 0.955098i \(-0.595750\pi\)
−0.296291 + 0.955098i \(0.595750\pi\)
\(824\) 0 0
\(825\) −30.0000 −1.04447
\(826\) 0 0
\(827\) −21.0000 −0.730242 −0.365121 0.930960i \(-0.618972\pi\)
−0.365121 + 0.930960i \(0.618972\pi\)
\(828\) −2.00000 −0.0695048
\(829\) 45.0000 1.56291 0.781457 0.623959i \(-0.214477\pi\)
0.781457 + 0.623959i \(0.214477\pi\)
\(830\) 0 0
\(831\) 2.00000 0.0693792
\(832\) −32.0000 −1.10940
\(833\) 49.0000 1.69775
\(834\) 4.00000 0.138509
\(835\) 0 0
\(836\) 60.0000 2.07514
\(837\) 4.00000 0.138260
\(838\) 14.0000 0.483622
\(839\) 47.0000 1.62262 0.811310 0.584616i \(-0.198755\pi\)
0.811310 + 0.584616i \(0.198755\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) −38.0000 −1.30957
\(843\) 18.0000 0.619953
\(844\) −40.0000 −1.37686
\(845\) 0 0
\(846\) −18.0000 −0.618853
\(847\) 0 0
\(848\) −40.0000 −1.37361
\(849\) 19.0000 0.652078
\(850\) −70.0000 −2.40098
\(851\) −3.00000 −0.102839
\(852\) 32.0000 1.09630
\(853\) −11.0000 −0.376633 −0.188316 0.982108i \(-0.560303\pi\)
−0.188316 + 0.982108i \(0.560303\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −48.0000 −1.63965 −0.819824 0.572615i \(-0.805929\pi\)
−0.819824 + 0.572615i \(0.805929\pi\)
\(858\) −48.0000 −1.63869
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −10.0000 −0.340601
\(863\) 3.00000 0.102121 0.0510606 0.998696i \(-0.483740\pi\)
0.0510606 + 0.998696i \(0.483740\pi\)
\(864\) −8.00000 −0.272166
\(865\) 0 0
\(866\) −40.0000 −1.35926
\(867\) −32.0000 −1.08678
\(868\) 0 0
\(869\) −48.0000 −1.62829
\(870\) 0 0
\(871\) −4.00000 −0.135535
\(872\) 0 0
\(873\) 4.00000 0.135379
\(874\) −10.0000 −0.338255
\(875\) 0 0
\(876\) 14.0000 0.473016
\(877\) −19.0000 −0.641584 −0.320792 0.947150i \(-0.603949\pi\)
−0.320792 + 0.947150i \(0.603949\pi\)
\(878\) 58.0000 1.95741
\(879\) 10.0000 0.337292
\(880\) 0 0
\(881\) 21.0000 0.707508 0.353754 0.935339i \(-0.384905\pi\)
0.353754 + 0.935339i \(0.384905\pi\)
\(882\) 14.0000 0.471405
\(883\) −20.0000 −0.673054 −0.336527 0.941674i \(-0.609252\pi\)
−0.336527 + 0.941674i \(0.609252\pi\)
\(884\) −56.0000 −1.88348
\(885\) 0 0
\(886\) −8.00000 −0.268765
\(887\) −37.0000 −1.24234 −0.621169 0.783676i \(-0.713342\pi\)
−0.621169 + 0.783676i \(0.713342\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −6.00000 −0.201008
\(892\) 30.0000 1.00447
\(893\) −45.0000 −1.50587
\(894\) 38.0000 1.27091
\(895\) 0 0
\(896\) 0 0
\(897\) 4.00000 0.133556
\(898\) 54.0000 1.80200
\(899\) −4.00000 −0.133407
\(900\) −10.0000 −0.333333
\(901\) −70.0000 −2.33204
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) −42.0000 −1.39536
\(907\) 37.0000 1.22856 0.614282 0.789086i \(-0.289446\pi\)
0.614282 + 0.789086i \(0.289446\pi\)
\(908\) −38.0000 −1.26107
\(909\) 0 0
\(910\) 0 0
\(911\) −9.00000 −0.298183 −0.149092 0.988823i \(-0.547635\pi\)
−0.149092 + 0.988823i \(0.547635\pi\)
\(912\) −20.0000 −0.662266
\(913\) 24.0000 0.794284
\(914\) 70.0000 2.31539
\(915\) 0 0
\(916\) −36.0000 −1.18947
\(917\) 0 0
\(918\) −14.0000 −0.462069
\(919\) 42.0000 1.38545 0.692726 0.721201i \(-0.256409\pi\)
0.692726 + 0.721201i \(0.256409\pi\)
\(920\) 0 0
\(921\) −23.0000 −0.757876
\(922\) 42.0000 1.38320
\(923\) −64.0000 −2.10659
\(924\) 0 0
\(925\) −15.0000 −0.493197
\(926\) −56.0000 −1.84027
\(927\) 4.00000 0.131377
\(928\) 8.00000 0.262613
\(929\) −20.0000 −0.656179 −0.328089 0.944647i \(-0.606405\pi\)
−0.328089 + 0.944647i \(0.606405\pi\)
\(930\) 0 0
\(931\) 35.0000 1.14708
\(932\) −32.0000 −1.04819
\(933\) −26.0000 −0.851202
\(934\) −24.0000 −0.785304
\(935\) 0 0
\(936\) 0 0
\(937\) 8.00000 0.261349 0.130674 0.991425i \(-0.458286\pi\)
0.130674 + 0.991425i \(0.458286\pi\)
\(938\) 0 0
\(939\) 18.0000 0.587408
\(940\) 0 0
\(941\) −10.0000 −0.325991 −0.162995 0.986627i \(-0.552116\pi\)
−0.162995 + 0.986627i \(0.552116\pi\)
\(942\) 34.0000 1.10778
\(943\) 0 0
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) −72.0000 −2.34092
\(947\) 47.0000 1.52729 0.763647 0.645634i \(-0.223407\pi\)
0.763647 + 0.645634i \(0.223407\pi\)
\(948\) −16.0000 −0.519656
\(949\) −28.0000 −0.908918
\(950\) −50.0000 −1.62221
\(951\) 14.0000 0.453981
\(952\) 0 0
\(953\) 11.0000 0.356325 0.178162 0.984001i \(-0.442985\pi\)
0.178162 + 0.984001i \(0.442985\pi\)
\(954\) −20.0000 −0.647524
\(955\) 0 0
\(956\) 32.0000 1.03495
\(957\) 6.00000 0.193952
\(958\) 70.0000 2.26160
\(959\) 0 0
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) −24.0000 −0.773791
\(963\) 7.00000 0.225572
\(964\) 34.0000 1.09507
\(965\) 0 0
\(966\) 0 0
\(967\) −48.0000 −1.54358 −0.771788 0.635880i \(-0.780637\pi\)
−0.771788 + 0.635880i \(0.780637\pi\)
\(968\) 0 0
\(969\) −35.0000 −1.12436
\(970\) 0 0
\(971\) 55.0000 1.76503 0.882517 0.470281i \(-0.155847\pi\)
0.882517 + 0.470281i \(0.155847\pi\)
\(972\) −2.00000 −0.0641500
\(973\) 0 0
\(974\) −8.00000 −0.256337
\(975\) 20.0000 0.640513
\(976\) −8.00000 −0.256074
\(977\) 29.0000 0.927792 0.463896 0.885890i \(-0.346451\pi\)
0.463896 + 0.885890i \(0.346451\pi\)
\(978\) −26.0000 −0.831388
\(979\) 90.0000 2.87641
\(980\) 0 0
\(981\) 4.00000 0.127710
\(982\) −10.0000 −0.319113
\(983\) 56.0000 1.78612 0.893061 0.449935i \(-0.148553\pi\)
0.893061 + 0.449935i \(0.148553\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 14.0000 0.445851
\(987\) 0 0
\(988\) −40.0000 −1.27257
\(989\) 6.00000 0.190789
\(990\) 0 0
\(991\) 46.0000 1.46124 0.730619 0.682785i \(-0.239232\pi\)
0.730619 + 0.682785i \(0.239232\pi\)
\(992\) −32.0000 −1.01600
\(993\) 4.00000 0.126936
\(994\) 0 0
\(995\) 0 0
\(996\) 8.00000 0.253490
\(997\) 30.0000 0.950110 0.475055 0.879956i \(-0.342428\pi\)
0.475055 + 0.879956i \(0.342428\pi\)
\(998\) 60.0000 1.89927
\(999\) −3.00000 −0.0949158
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 201.2.a.a.1.1 1
3.2 odd 2 603.2.a.f.1.1 1
4.3 odd 2 3216.2.a.j.1.1 1
5.4 even 2 5025.2.a.i.1.1 1
7.6 odd 2 9849.2.a.b.1.1 1
12.11 even 2 9648.2.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
201.2.a.a.1.1 1 1.1 even 1 trivial
603.2.a.f.1.1 1 3.2 odd 2
3216.2.a.j.1.1 1 4.3 odd 2
5025.2.a.i.1.1 1 5.4 even 2
9648.2.a.j.1.1 1 12.11 even 2
9849.2.a.b.1.1 1 7.6 odd 2