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Properties

Label 6042.2.a.h.1.1
Level $6042$
Weight $2$
Character 6042.1
Self dual yes
Analytic conductor $48.246$
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] = [6042,2,Mod(1,6042)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6042, 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("6042.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6042 = 2 \cdot 3 \cdot 19 \cdot 53 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6042.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: \(48.2456129013\)
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 \(\chi\) \(=\) 6042.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{11} +1.00000 q^{12} +6.00000 q^{13} +1.00000 q^{16} +6.00000 q^{17} -1.00000 q^{18} -1.00000 q^{19} -2.00000 q^{22} -8.00000 q^{23} -1.00000 q^{24} -5.00000 q^{25} -6.00000 q^{26} +1.00000 q^{27} -2.00000 q^{29} +8.00000 q^{31} -1.00000 q^{32} +2.00000 q^{33} -6.00000 q^{34} +1.00000 q^{36} +6.00000 q^{37} +1.00000 q^{38} +6.00000 q^{39} +2.00000 q^{41} -4.00000 q^{43} +2.00000 q^{44} +8.00000 q^{46} +6.00000 q^{47} +1.00000 q^{48} -7.00000 q^{49} +5.00000 q^{50} +6.00000 q^{51} +6.00000 q^{52} -1.00000 q^{53} -1.00000 q^{54} -1.00000 q^{57} +2.00000 q^{58} +8.00000 q^{59} +14.0000 q^{61} -8.00000 q^{62} +1.00000 q^{64} -2.00000 q^{66} +12.0000 q^{67} +6.00000 q^{68} -8.00000 q^{69} -1.00000 q^{72} +2.00000 q^{73} -6.00000 q^{74} -5.00000 q^{75} -1.00000 q^{76} -6.00000 q^{78} +4.00000 q^{79} +1.00000 q^{81} -2.00000 q^{82} +4.00000 q^{86} -2.00000 q^{87} -2.00000 q^{88} -10.0000 q^{89} -8.00000 q^{92} +8.00000 q^{93} -6.00000 q^{94} -1.00000 q^{96} -10.0000 q^{97} +7.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 − 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 1.00000 0.288675
\(13\) 6.00000 1.66410 0.832050 − 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 − 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) −1.00000 −0.235702
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) −2.00000 −0.426401
\(23\) −8.00000 −1.66812 −0.834058 − 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) −1.00000 −0.204124
\(25\) −5.00000 −1.00000
\(26\) −6.00000 −1.17670
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 − 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 − 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) −1.00000 −0.176777
\(33\) 2.00000 0.348155
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 6.00000 0.986394 0.493197 − 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 1.00000 0.162221
\(39\) 6.00000 0.960769
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 − 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) 8.00000 1.17954
\(47\) 6.00000 0.875190 0.437595 − 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 1.00000 0.144338
\(49\) −7.00000 −1.00000
\(50\) 5.00000 0.707107
\(51\) 6.00000 0.840168
\(52\) 6.00000 0.832050
\(53\) −1.00000 −0.137361
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 2.00000 0.262613
\(59\) 8.00000 1.04151 0.520756 − 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 0 0
\(61\) 14.0000 1.79252 0.896258 − 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) −8.00000 −1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −2.00000 −0.246183
\(67\) 12.0000 1.46603 0.733017 − 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) 6.00000 0.727607
\(69\) −8.00000 −0.963087
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −1.00000 −0.117851
\(73\) 2.00000 0.234082 0.117041 − 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) −6.00000 −0.697486
\(75\) −5.00000 −0.577350
\(76\) −1.00000 −0.114708
\(77\) 0 0
\(78\) −6.00000 −0.679366
\(79\) 4.00000 0.450035 0.225018 − 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −2.00000 −0.220863
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) −2.00000 −0.214423
\(88\) −2.00000 −0.213201
\(89\) −10.0000 −1.06000 −0.529999 − 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −8.00000 −0.834058
\(93\) 8.00000 0.829561
\(94\) −6.00000 −0.618853
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −10.0000 −1.01535 −0.507673 − 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 7.00000 0.707107
\(99\) 2.00000 0.201008
\(100\) −5.00000 −0.500000
\(101\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) −6.00000 −0.594089
\(103\) −4.00000 −0.394132 −0.197066 − 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) 1.00000 0.0971286
\(107\) −8.00000 −0.773389 −0.386695 − 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) 1.00000 0.0962250
\(109\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) 6.00000 0.569495
\(112\) 0 0
\(113\) 2.00000 0.188144 0.0940721 − 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) 1.00000 0.0936586
\(115\) 0 0
\(116\) −2.00000 −0.185695
\(117\) 6.00000 0.554700
\(118\) −8.00000 −0.736460
\(119\) 0 0
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) −14.0000 −1.26750
\(123\) 2.00000 0.180334
\(124\) 8.00000 0.718421
\(125\) 0 0
\(126\) 0 0
\(127\) −4.00000 −0.354943 −0.177471 − 0.984126i \(-0.556792\pi\)
−0.177471 + 0.984126i \(0.556792\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) −10.0000 −0.873704 −0.436852 − 0.899533i \(-0.643907\pi\)
−0.436852 + 0.899533i \(0.643907\pi\)
\(132\) 2.00000 0.174078
\(133\) 0 0
\(134\) −12.0000 −1.03664
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 8.00000 0.681005
\(139\) −12.0000 −1.01783 −0.508913 − 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 0 0
\(141\) 6.00000 0.505291
\(142\) 0 0
\(143\) 12.0000 1.00349
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −2.00000 −0.165521
\(147\) −7.00000 −0.577350
\(148\) 6.00000 0.493197
\(149\) −22.0000 −1.80231 −0.901155 − 0.433497i \(-0.857280\pi\)
−0.901155 + 0.433497i \(0.857280\pi\)
\(150\) 5.00000 0.408248
\(151\) 20.0000 1.62758 0.813788 − 0.581161i \(-0.197401\pi\)
0.813788 + 0.581161i \(0.197401\pi\)
\(152\) 1.00000 0.0811107
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) 0 0
\(156\) 6.00000 0.480384
\(157\) −14.0000 −1.11732 −0.558661 − 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) −4.00000 −0.318223
\(159\) −1.00000 −0.0793052
\(160\) 0 0
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 20.0000 1.56652 0.783260 − 0.621694i \(-0.213555\pi\)
0.783260 + 0.621694i \(0.213555\pi\)
\(164\) 2.00000 0.156174
\(165\) 0 0
\(166\) 0 0
\(167\) 8.00000 0.619059 0.309529 − 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) −4.00000 −0.304997
\(173\) −10.0000 −0.760286 −0.380143 − 0.924928i \(-0.624125\pi\)
−0.380143 + 0.924928i \(0.624125\pi\)
\(174\) 2.00000 0.151620
\(175\) 0 0
\(176\) 2.00000 0.150756
\(177\) 8.00000 0.601317
\(178\) 10.0000 0.749532
\(179\) 12.0000 0.896922 0.448461 − 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) −16.0000 −1.18927 −0.594635 − 0.803996i \(-0.702704\pi\)
−0.594635 + 0.803996i \(0.702704\pi\)
\(182\) 0 0
\(183\) 14.0000 1.03491
\(184\) 8.00000 0.589768
\(185\) 0 0
\(186\) −8.00000 −0.586588
\(187\) 12.0000 0.877527
\(188\) 6.00000 0.437595
\(189\) 0 0
\(190\) 0 0
\(191\) −4.00000 −0.289430 −0.144715 − 0.989473i \(-0.546227\pi\)
−0.144715 + 0.989473i \(0.546227\pi\)
\(192\) 1.00000 0.0721688
\(193\) 16.0000 1.15171 0.575853 − 0.817554i \(-0.304670\pi\)
0.575853 + 0.817554i \(0.304670\pi\)
\(194\) 10.0000 0.717958
\(195\) 0 0
\(196\) −7.00000 −0.500000
\(197\) −18.0000 −1.28245 −0.641223 − 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) −2.00000 −0.142134
\(199\) 20.0000 1.41776 0.708881 − 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 5.00000 0.353553
\(201\) 12.0000 0.846415
\(202\) 0 0
\(203\) 0 0
\(204\) 6.00000 0.420084
\(205\) 0 0
\(206\) 4.00000 0.278693
\(207\) −8.00000 −0.556038
\(208\) 6.00000 0.416025
\(209\) −2.00000 −0.138343
\(210\) 0 0
\(211\) −18.0000 −1.23917 −0.619586 − 0.784929i \(-0.712699\pi\)
−0.619586 + 0.784929i \(0.712699\pi\)
\(212\) −1.00000 −0.0686803
\(213\) 0 0
\(214\) 8.00000 0.546869
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 0 0
\(219\) 2.00000 0.135147
\(220\) 0 0
\(221\) 36.0000 2.42162
\(222\) −6.00000 −0.402694
\(223\) 14.0000 0.937509 0.468755 − 0.883328i \(-0.344703\pi\)
0.468755 + 0.883328i \(0.344703\pi\)
\(224\) 0 0
\(225\) −5.00000 −0.333333
\(226\) −2.00000 −0.133038
\(227\) 8.00000 0.530979 0.265489 − 0.964114i \(-0.414466\pi\)
0.265489 + 0.964114i \(0.414466\pi\)
\(228\) −1.00000 −0.0662266
\(229\) −14.0000 −0.925146 −0.462573 − 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 2.00000 0.131306
\(233\) −20.0000 −1.31024 −0.655122 − 0.755523i \(-0.727383\pi\)
−0.655122 + 0.755523i \(0.727383\pi\)
\(234\) −6.00000 −0.392232
\(235\) 0 0
\(236\) 8.00000 0.520756
\(237\) 4.00000 0.259828
\(238\) 0 0
\(239\) 28.0000 1.81117 0.905585 − 0.424165i \(-0.139432\pi\)
0.905585 + 0.424165i \(0.139432\pi\)
\(240\) 0 0
\(241\) 30.0000 1.93247 0.966235 − 0.257663i \(-0.0829523\pi\)
0.966235 + 0.257663i \(0.0829523\pi\)
\(242\) 7.00000 0.449977
\(243\) 1.00000 0.0641500
\(244\) 14.0000 0.896258
\(245\) 0 0
\(246\) −2.00000 −0.127515
\(247\) −6.00000 −0.381771
\(248\) −8.00000 −0.508001
\(249\) 0 0
\(250\) 0 0
\(251\) 24.0000 1.51487 0.757433 − 0.652913i \(-0.226453\pi\)
0.757433 + 0.652913i \(0.226453\pi\)
\(252\) 0 0
\(253\) −16.0000 −1.00591
\(254\) 4.00000 0.250982
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 6.00000 0.374270 0.187135 − 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 4.00000 0.249029
\(259\) 0 0
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) 10.0000 0.617802
\(263\) 28.0000 1.72655 0.863277 − 0.504730i \(-0.168408\pi\)
0.863277 + 0.504730i \(0.168408\pi\)
\(264\) −2.00000 −0.123091
\(265\) 0 0
\(266\) 0 0
\(267\) −10.0000 −0.611990
\(268\) 12.0000 0.733017
\(269\) −6.00000 −0.365826 −0.182913 − 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 − 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 6.00000 0.363803
\(273\) 0 0
\(274\) 0 0
\(275\) −10.0000 −0.603023
\(276\) −8.00000 −0.481543
\(277\) −2.00000 −0.120168 −0.0600842 − 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 12.0000 0.719712
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) −26.0000 −1.55103 −0.775515 − 0.631329i \(-0.782510\pi\)
−0.775515 + 0.631329i \(0.782510\pi\)
\(282\) −6.00000 −0.357295
\(283\) 20.0000 1.18888 0.594438 − 0.804141i \(-0.297374\pi\)
0.594438 + 0.804141i \(0.297374\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) −12.0000 −0.709575
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) −10.0000 −0.586210
\(292\) 2.00000 0.117041
\(293\) 6.00000 0.350524 0.175262 − 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 7.00000 0.408248
\(295\) 0 0
\(296\) −6.00000 −0.348743
\(297\) 2.00000 0.116052
\(298\) 22.0000 1.27443
\(299\) −48.0000 −2.77591
\(300\) −5.00000 −0.288675
\(301\) 0 0
\(302\) −20.0000 −1.15087
\(303\) 0 0
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) −6.00000 −0.342997
\(307\) 2.00000 0.114146 0.0570730 − 0.998370i \(-0.481823\pi\)
0.0570730 + 0.998370i \(0.481823\pi\)
\(308\) 0 0
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) 6.00000 0.340229 0.170114 − 0.985424i \(-0.445586\pi\)
0.170114 + 0.985424i \(0.445586\pi\)
\(312\) −6.00000 −0.339683
\(313\) 18.0000 1.01742 0.508710 − 0.860938i \(-0.330123\pi\)
0.508710 + 0.860938i \(0.330123\pi\)
\(314\) 14.0000 0.790066
\(315\) 0 0
\(316\) 4.00000 0.225018
\(317\) 18.0000 1.01098 0.505490 − 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 1.00000 0.0560772
\(319\) −4.00000 −0.223957
\(320\) 0 0
\(321\) −8.00000 −0.446516
\(322\) 0 0
\(323\) −6.00000 −0.333849
\(324\) 1.00000 0.0555556
\(325\) −30.0000 −1.66410
\(326\) −20.0000 −1.10770
\(327\) 0 0
\(328\) −2.00000 −0.110432
\(329\) 0 0
\(330\) 0 0
\(331\) 2.00000 0.109930 0.0549650 − 0.998488i \(-0.482495\pi\)
0.0549650 + 0.998488i \(0.482495\pi\)
\(332\) 0 0
\(333\) 6.00000 0.328798
\(334\) −8.00000 −0.437741
\(335\) 0 0
\(336\) 0 0
\(337\) 16.0000 0.871576 0.435788 − 0.900049i \(-0.356470\pi\)
0.435788 + 0.900049i \(0.356470\pi\)
\(338\) −23.0000 −1.25104
\(339\) 2.00000 0.108625
\(340\) 0 0
\(341\) 16.0000 0.866449
\(342\) 1.00000 0.0540738
\(343\) 0 0
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) 10.0000 0.537603
\(347\) −14.0000 −0.751559 −0.375780 − 0.926709i \(-0.622625\pi\)
−0.375780 + 0.926709i \(0.622625\pi\)
\(348\) −2.00000 −0.107211
\(349\) 10.0000 0.535288 0.267644 − 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 0 0
\(351\) 6.00000 0.320256
\(352\) −2.00000 −0.106600
\(353\) −24.0000 −1.27739 −0.638696 − 0.769460i \(-0.720526\pi\)
−0.638696 + 0.769460i \(0.720526\pi\)
\(354\) −8.00000 −0.425195
\(355\) 0 0
\(356\) −10.0000 −0.529999
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) −24.0000 −1.26667 −0.633336 − 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 16.0000 0.840941
\(363\) −7.00000 −0.367405
\(364\) 0 0
\(365\) 0 0
\(366\) −14.0000 −0.731792
\(367\) −12.0000 −0.626395 −0.313197 − 0.949688i \(-0.601400\pi\)
−0.313197 + 0.949688i \(0.601400\pi\)
\(368\) −8.00000 −0.417029
\(369\) 2.00000 0.104116
\(370\) 0 0
\(371\) 0 0
\(372\) 8.00000 0.414781
\(373\) −20.0000 −1.03556 −0.517780 − 0.855514i \(-0.673242\pi\)
−0.517780 + 0.855514i \(0.673242\pi\)
\(374\) −12.0000 −0.620505
\(375\) 0 0
\(376\) −6.00000 −0.309426
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) 28.0000 1.43826 0.719132 − 0.694874i \(-0.244540\pi\)
0.719132 + 0.694874i \(0.244540\pi\)
\(380\) 0 0
\(381\) −4.00000 −0.204926
\(382\) 4.00000 0.204658
\(383\) −24.0000 −1.22634 −0.613171 − 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −16.0000 −0.814379
\(387\) −4.00000 −0.203331
\(388\) −10.0000 −0.507673
\(389\) 20.0000 1.01404 0.507020 − 0.861934i \(-0.330747\pi\)
0.507020 + 0.861934i \(0.330747\pi\)
\(390\) 0 0
\(391\) −48.0000 −2.42746
\(392\) 7.00000 0.353553
\(393\) −10.0000 −0.504433
\(394\) 18.0000 0.906827
\(395\) 0 0
\(396\) 2.00000 0.100504
\(397\) 18.0000 0.903394 0.451697 − 0.892171i \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(398\) −20.0000 −1.00251
\(399\) 0 0
\(400\) −5.00000 −0.250000
\(401\) 18.0000 0.898877 0.449439 − 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) −12.0000 −0.598506
\(403\) 48.0000 2.39105
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 12.0000 0.594818
\(408\) −6.00000 −0.297044
\(409\) −14.0000 −0.692255 −0.346128 − 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −4.00000 −0.197066
\(413\) 0 0
\(414\) 8.00000 0.393179
\(415\) 0 0
\(416\) −6.00000 −0.294174
\(417\) −12.0000 −0.587643
\(418\) 2.00000 0.0978232
\(419\) −8.00000 −0.390826 −0.195413 − 0.980721i \(-0.562605\pi\)
−0.195413 + 0.980721i \(0.562605\pi\)
\(420\) 0 0
\(421\) 20.0000 0.974740 0.487370 − 0.873195i \(-0.337956\pi\)
0.487370 + 0.873195i \(0.337956\pi\)
\(422\) 18.0000 0.876226
\(423\) 6.00000 0.291730
\(424\) 1.00000 0.0485643
\(425\) −30.0000 −1.45521
\(426\) 0 0
\(427\) 0 0
\(428\) −8.00000 −0.386695
\(429\) 12.0000 0.579365
\(430\) 0 0
\(431\) −32.0000 −1.54139 −0.770693 − 0.637207i \(-0.780090\pi\)
−0.770693 + 0.637207i \(0.780090\pi\)
\(432\) 1.00000 0.0481125
\(433\) −2.00000 −0.0961139 −0.0480569 − 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 8.00000 0.382692
\(438\) −2.00000 −0.0955637
\(439\) −2.00000 −0.0954548 −0.0477274 − 0.998860i \(-0.515198\pi\)
−0.0477274 + 0.998860i \(0.515198\pi\)
\(440\) 0 0
\(441\) −7.00000 −0.333333
\(442\) −36.0000 −1.71235
\(443\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 6.00000 0.284747
\(445\) 0 0
\(446\) −14.0000 −0.662919
\(447\) −22.0000 −1.04056
\(448\) 0 0
\(449\) −22.0000 −1.03824 −0.519122 − 0.854700i \(-0.673741\pi\)
−0.519122 + 0.854700i \(0.673741\pi\)
\(450\) 5.00000 0.235702
\(451\) 4.00000 0.188353
\(452\) 2.00000 0.0940721
\(453\) 20.0000 0.939682
\(454\) −8.00000 −0.375459
\(455\) 0 0
\(456\) 1.00000 0.0468293
\(457\) 22.0000 1.02912 0.514558 − 0.857455i \(-0.327956\pi\)
0.514558 + 0.857455i \(0.327956\pi\)
\(458\) 14.0000 0.654177
\(459\) 6.00000 0.280056
\(460\) 0 0
\(461\) 2.00000 0.0931493 0.0465746 − 0.998915i \(-0.485169\pi\)
0.0465746 + 0.998915i \(0.485169\pi\)
\(462\) 0 0
\(463\) 16.0000 0.743583 0.371792 − 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) 20.0000 0.926482
\(467\) 38.0000 1.75843 0.879215 − 0.476425i \(-0.158068\pi\)
0.879215 + 0.476425i \(0.158068\pi\)
\(468\) 6.00000 0.277350
\(469\) 0 0
\(470\) 0 0
\(471\) −14.0000 −0.645086
\(472\) −8.00000 −0.368230
\(473\) −8.00000 −0.367840
\(474\) −4.00000 −0.183726
\(475\) 5.00000 0.229416
\(476\) 0 0
\(477\) −1.00000 −0.0457869
\(478\) −28.0000 −1.28069
\(479\) −36.0000 −1.64488 −0.822441 − 0.568850i \(-0.807388\pi\)
−0.822441 + 0.568850i \(0.807388\pi\)
\(480\) 0 0
\(481\) 36.0000 1.64146
\(482\) −30.0000 −1.36646
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 2.00000 0.0906287 0.0453143 − 0.998973i \(-0.485571\pi\)
0.0453143 + 0.998973i \(0.485571\pi\)
\(488\) −14.0000 −0.633750
\(489\) 20.0000 0.904431
\(490\) 0 0
\(491\) 20.0000 0.902587 0.451294 − 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) 2.00000 0.0901670
\(493\) −12.0000 −0.540453
\(494\) 6.00000 0.269953
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) 0 0
\(498\) 0 0
\(499\) 36.0000 1.61158 0.805791 − 0.592200i \(-0.201741\pi\)
0.805791 + 0.592200i \(0.201741\pi\)
\(500\) 0 0
\(501\) 8.00000 0.357414
\(502\) −24.0000 −1.07117
\(503\) 12.0000 0.535054 0.267527 − 0.963550i \(-0.413794\pi\)
0.267527 + 0.963550i \(0.413794\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 16.0000 0.711287
\(507\) 23.0000 1.02147
\(508\) −4.00000 −0.177471
\(509\) −2.00000 −0.0886484 −0.0443242 − 0.999017i \(-0.514113\pi\)
−0.0443242 + 0.999017i \(0.514113\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) −1.00000 −0.0441511
\(514\) −6.00000 −0.264649
\(515\) 0 0
\(516\) −4.00000 −0.176090
\(517\) 12.0000 0.527759
\(518\) 0 0
\(519\) −10.0000 −0.438951
\(520\) 0 0
\(521\) 42.0000 1.84005 0.920027 − 0.391856i \(-0.128167\pi\)
0.920027 + 0.391856i \(0.128167\pi\)
\(522\) 2.00000 0.0875376
\(523\) −26.0000 −1.13690 −0.568450 − 0.822718i \(-0.692457\pi\)
−0.568450 + 0.822718i \(0.692457\pi\)
\(524\) −10.0000 −0.436852
\(525\) 0 0
\(526\) −28.0000 −1.22086
\(527\) 48.0000 2.09091
\(528\) 2.00000 0.0870388
\(529\) 41.0000 1.78261
\(530\) 0 0
\(531\) 8.00000 0.347170
\(532\) 0 0
\(533\) 12.0000 0.519778
\(534\) 10.0000 0.432742
\(535\) 0 0
\(536\) −12.0000 −0.518321
\(537\) 12.0000 0.517838
\(538\) 6.00000 0.258678
\(539\) −14.0000 −0.603023
\(540\) 0 0
\(541\) −2.00000 −0.0859867 −0.0429934 − 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) −20.0000 −0.859074
\(543\) −16.0000 −0.686626
\(544\) −6.00000 −0.257248
\(545\) 0 0
\(546\) 0 0
\(547\) −34.0000 −1.45374 −0.726868 − 0.686778i \(-0.759025\pi\)
−0.726868 + 0.686778i \(0.759025\pi\)
\(548\) 0 0
\(549\) 14.0000 0.597505
\(550\) 10.0000 0.426401
\(551\) 2.00000 0.0852029
\(552\) 8.00000 0.340503
\(553\) 0 0
\(554\) 2.00000 0.0849719
\(555\) 0 0
\(556\) −12.0000 −0.508913
\(557\) 40.0000 1.69485 0.847427 − 0.530912i \(-0.178150\pi\)
0.847427 + 0.530912i \(0.178150\pi\)
\(558\) −8.00000 −0.338667
\(559\) −24.0000 −1.01509
\(560\) 0 0
\(561\) 12.0000 0.506640
\(562\) 26.0000 1.09674
\(563\) 12.0000 0.505740 0.252870 − 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 6.00000 0.252646
\(565\) 0 0
\(566\) −20.0000 −0.840663
\(567\) 0 0
\(568\) 0 0
\(569\) 30.0000 1.25767 0.628833 − 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 − 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 12.0000 0.501745
\(573\) −4.00000 −0.167102
\(574\) 0 0
\(575\) 40.0000 1.66812
\(576\) 1.00000 0.0416667
\(577\) 18.0000 0.749350 0.374675 − 0.927156i \(-0.377754\pi\)
0.374675 + 0.927156i \(0.377754\pi\)
\(578\) −19.0000 −0.790296
\(579\) 16.0000 0.664937
\(580\) 0 0
\(581\) 0 0
\(582\) 10.0000 0.414513
\(583\) −2.00000 −0.0828315
\(584\) −2.00000 −0.0827606
\(585\) 0 0
\(586\) −6.00000 −0.247858
\(587\) −30.0000 −1.23823 −0.619116 − 0.785299i \(-0.712509\pi\)
−0.619116 + 0.785299i \(0.712509\pi\)
\(588\) −7.00000 −0.288675
\(589\) −8.00000 −0.329634
\(590\) 0 0
\(591\) −18.0000 −0.740421
\(592\) 6.00000 0.246598
\(593\) −10.0000 −0.410651 −0.205325 − 0.978694i \(-0.565825\pi\)
−0.205325 + 0.978694i \(0.565825\pi\)
\(594\) −2.00000 −0.0820610
\(595\) 0 0
\(596\) −22.0000 −0.901155
\(597\) 20.0000 0.818546
\(598\) 48.0000 1.96287
\(599\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 5.00000 0.204124
\(601\) −4.00000 −0.163163 −0.0815817 − 0.996667i \(-0.525997\pi\)
−0.0815817 + 0.996667i \(0.525997\pi\)
\(602\) 0 0
\(603\) 12.0000 0.488678
\(604\) 20.0000 0.813788
\(605\) 0 0
\(606\) 0 0
\(607\) −22.0000 −0.892952 −0.446476 − 0.894795i \(-0.647321\pi\)
−0.446476 + 0.894795i \(0.647321\pi\)
\(608\) 1.00000 0.0405554
\(609\) 0 0
\(610\) 0 0
\(611\) 36.0000 1.45640
\(612\) 6.00000 0.242536
\(613\) −38.0000 −1.53481 −0.767403 − 0.641165i \(-0.778451\pi\)
−0.767403 + 0.641165i \(0.778451\pi\)
\(614\) −2.00000 −0.0807134
\(615\) 0 0
\(616\) 0 0
\(617\) 12.0000 0.483102 0.241551 − 0.970388i \(-0.422344\pi\)
0.241551 + 0.970388i \(0.422344\pi\)
\(618\) 4.00000 0.160904
\(619\) −32.0000 −1.28619 −0.643094 − 0.765787i \(-0.722350\pi\)
−0.643094 + 0.765787i \(0.722350\pi\)
\(620\) 0 0
\(621\) −8.00000 −0.321029
\(622\) −6.00000 −0.240578
\(623\) 0 0
\(624\) 6.00000 0.240192
\(625\) 25.0000 1.00000
\(626\) −18.0000 −0.719425
\(627\) −2.00000 −0.0798723
\(628\) −14.0000 −0.558661
\(629\) 36.0000 1.43541
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 − 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) −4.00000 −0.159111
\(633\) −18.0000 −0.715436
\(634\) −18.0000 −0.714871
\(635\) 0 0
\(636\) −1.00000 −0.0396526
\(637\) −42.0000 −1.66410
\(638\) 4.00000 0.158362
\(639\) 0 0
\(640\) 0 0
\(641\) −30.0000 −1.18493 −0.592464 − 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 8.00000 0.315735
\(643\) 32.0000 1.26196 0.630978 − 0.775800i \(-0.282654\pi\)
0.630978 + 0.775800i \(0.282654\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 6.00000 0.236067
\(647\) −18.0000 −0.707653 −0.353827 − 0.935311i \(-0.615120\pi\)
−0.353827 + 0.935311i \(0.615120\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 16.0000 0.628055
\(650\) 30.0000 1.17670
\(651\) 0 0
\(652\) 20.0000 0.783260
\(653\) −30.0000 −1.17399 −0.586995 − 0.809590i \(-0.699689\pi\)
−0.586995 + 0.809590i \(0.699689\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 2.00000 0.0780869
\(657\) 2.00000 0.0780274
\(658\) 0 0
\(659\) 28.0000 1.09073 0.545363 − 0.838200i \(-0.316392\pi\)
0.545363 + 0.838200i \(0.316392\pi\)
\(660\) 0 0
\(661\) −38.0000 −1.47803 −0.739014 − 0.673690i \(-0.764708\pi\)
−0.739014 + 0.673690i \(0.764708\pi\)
\(662\) −2.00000 −0.0777322
\(663\) 36.0000 1.39812
\(664\) 0 0
\(665\) 0 0
\(666\) −6.00000 −0.232495
\(667\) 16.0000 0.619522
\(668\) 8.00000 0.309529
\(669\) 14.0000 0.541271
\(670\) 0 0
\(671\) 28.0000 1.08093
\(672\) 0 0
\(673\) −14.0000 −0.539660 −0.269830 − 0.962908i \(-0.586968\pi\)
−0.269830 + 0.962908i \(0.586968\pi\)
\(674\) −16.0000 −0.616297
\(675\) −5.00000 −0.192450
\(676\) 23.0000 0.884615
\(677\) −30.0000 −1.15299 −0.576497 − 0.817099i \(-0.695581\pi\)
−0.576497 + 0.817099i \(0.695581\pi\)
\(678\) −2.00000 −0.0768095
\(679\) 0 0
\(680\) 0 0
\(681\) 8.00000 0.306561
\(682\) −16.0000 −0.612672
\(683\) −24.0000 −0.918334 −0.459167 − 0.888350i \(-0.651852\pi\)
−0.459167 + 0.888350i \(0.651852\pi\)
\(684\) −1.00000 −0.0382360
\(685\) 0 0
\(686\) 0 0
\(687\) −14.0000 −0.534133
\(688\) −4.00000 −0.152499
\(689\) −6.00000 −0.228582
\(690\) 0 0
\(691\) 28.0000 1.06517 0.532585 − 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) −10.0000 −0.380143
\(693\) 0 0
\(694\) 14.0000 0.531433
\(695\) 0 0
\(696\) 2.00000 0.0758098
\(697\) 12.0000 0.454532
\(698\) −10.0000 −0.378506
\(699\) −20.0000 −0.756469
\(700\) 0 0
\(701\) 20.0000 0.755390 0.377695 − 0.925930i \(-0.376717\pi\)
0.377695 + 0.925930i \(0.376717\pi\)
\(702\) −6.00000 −0.226455
\(703\) −6.00000 −0.226294
\(704\) 2.00000 0.0753778
\(705\) 0 0
\(706\) 24.0000 0.903252
\(707\) 0 0
\(708\) 8.00000 0.300658
\(709\) 46.0000 1.72757 0.863783 − 0.503864i \(-0.168089\pi\)
0.863783 + 0.503864i \(0.168089\pi\)
\(710\) 0 0
\(711\) 4.00000 0.150012
\(712\) 10.0000 0.374766
\(713\) −64.0000 −2.39682
\(714\) 0 0
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) 28.0000 1.04568
\(718\) 24.0000 0.895672
\(719\) −24.0000 −0.895049 −0.447524 − 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −1.00000 −0.0372161
\(723\) 30.0000 1.11571
\(724\) −16.0000 −0.594635
\(725\) 10.0000 0.371391
\(726\) 7.00000 0.259794
\(727\) 28.0000 1.03846 0.519231 − 0.854634i \(-0.326218\pi\)
0.519231 + 0.854634i \(0.326218\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −24.0000 −0.887672
\(732\) 14.0000 0.517455
\(733\) −30.0000 −1.10808 −0.554038 − 0.832492i \(-0.686914\pi\)
−0.554038 + 0.832492i \(0.686914\pi\)
\(734\) 12.0000 0.442928
\(735\) 0 0
\(736\) 8.00000 0.294884
\(737\) 24.0000 0.884051
\(738\) −2.00000 −0.0736210
\(739\) −52.0000 −1.91285 −0.956425 − 0.291977i \(-0.905687\pi\)
−0.956425 + 0.291977i \(0.905687\pi\)
\(740\) 0 0
\(741\) −6.00000 −0.220416
\(742\) 0 0
\(743\) −48.0000 −1.76095 −0.880475 − 0.474093i \(-0.842776\pi\)
−0.880475 + 0.474093i \(0.842776\pi\)
\(744\) −8.00000 −0.293294
\(745\) 0 0
\(746\) 20.0000 0.732252
\(747\) 0 0
\(748\) 12.0000 0.438763
\(749\) 0 0
\(750\) 0 0
\(751\) −26.0000 −0.948753 −0.474377 − 0.880322i \(-0.657327\pi\)
−0.474377 + 0.880322i \(0.657327\pi\)
\(752\) 6.00000 0.218797
\(753\) 24.0000 0.874609
\(754\) 12.0000 0.437014
\(755\) 0 0
\(756\) 0 0
\(757\) −42.0000 −1.52652 −0.763258 − 0.646094i \(-0.776401\pi\)
−0.763258 + 0.646094i \(0.776401\pi\)
\(758\) −28.0000 −1.01701
\(759\) −16.0000 −0.580763
\(760\) 0 0
\(761\) 24.0000 0.869999 0.435000 − 0.900431i \(-0.356748\pi\)
0.435000 + 0.900431i \(0.356748\pi\)
\(762\) 4.00000 0.144905
\(763\) 0 0
\(764\) −4.00000 −0.144715
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) 48.0000 1.73318
\(768\) 1.00000 0.0360844
\(769\) −38.0000 −1.37032 −0.685158 − 0.728395i \(-0.740267\pi\)
−0.685158 + 0.728395i \(0.740267\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) 16.0000 0.575853
\(773\) −26.0000 −0.935155 −0.467578 − 0.883952i \(-0.654873\pi\)
−0.467578 + 0.883952i \(0.654873\pi\)
\(774\) 4.00000 0.143777
\(775\) −40.0000 −1.43684
\(776\) 10.0000 0.358979
\(777\) 0 0
\(778\) −20.0000 −0.717035
\(779\) −2.00000 −0.0716574
\(780\) 0 0
\(781\) 0 0
\(782\) 48.0000 1.71648
\(783\) −2.00000 −0.0714742
\(784\) −7.00000 −0.250000
\(785\) 0 0
\(786\) 10.0000 0.356688
\(787\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) −18.0000 −0.641223
\(789\) 28.0000 0.996826
\(790\) 0 0
\(791\) 0 0
\(792\) −2.00000 −0.0710669
\(793\) 84.0000 2.98293
\(794\) −18.0000 −0.638796
\(795\) 0 0
\(796\) 20.0000 0.708881
\(797\) −18.0000 −0.637593 −0.318796 − 0.947823i \(-0.603279\pi\)
−0.318796 + 0.947823i \(0.603279\pi\)
\(798\) 0 0
\(799\) 36.0000 1.27359
\(800\) 5.00000 0.176777
\(801\) −10.0000 −0.353333
\(802\) −18.0000 −0.635602
\(803\) 4.00000 0.141157
\(804\) 12.0000 0.423207
\(805\) 0 0
\(806\) −48.0000 −1.69073
\(807\) −6.00000 −0.211210
\(808\) 0 0
\(809\) 12.0000 0.421898 0.210949 − 0.977497i \(-0.432345\pi\)
0.210949 + 0.977497i \(0.432345\pi\)
\(810\) 0 0
\(811\) −2.00000 −0.0702295 −0.0351147 − 0.999383i \(-0.511180\pi\)
−0.0351147 + 0.999383i \(0.511180\pi\)
\(812\) 0 0
\(813\) 20.0000 0.701431
\(814\) −12.0000 −0.420600
\(815\) 0 0
\(816\) 6.00000 0.210042
\(817\) 4.00000 0.139942
\(818\) 14.0000 0.489499
\(819\) 0 0
\(820\) 0 0
\(821\) −24.0000 −0.837606 −0.418803 − 0.908077i \(-0.637550\pi\)
−0.418803 + 0.908077i \(0.637550\pi\)
\(822\) 0 0
\(823\) 12.0000 0.418294 0.209147 − 0.977884i \(-0.432931\pi\)
0.209147 + 0.977884i \(0.432931\pi\)
\(824\) 4.00000 0.139347
\(825\) −10.0000 −0.348155
\(826\) 0 0
\(827\) −44.0000 −1.53003 −0.765015 − 0.644013i \(-0.777268\pi\)
−0.765015 + 0.644013i \(0.777268\pi\)
\(828\) −8.00000 −0.278019
\(829\) −44.0000 −1.52818 −0.764092 − 0.645108i \(-0.776812\pi\)
−0.764092 + 0.645108i \(0.776812\pi\)
\(830\) 0 0
\(831\) −2.00000 −0.0693792
\(832\) 6.00000 0.208013
\(833\) −42.0000 −1.45521
\(834\) 12.0000 0.415526
\(835\) 0 0
\(836\) −2.00000 −0.0691714
\(837\) 8.00000 0.276520
\(838\) 8.00000 0.276355
\(839\) −56.0000 −1.93333 −0.966667 − 0.256036i \(-0.917584\pi\)
−0.966667 + 0.256036i \(0.917584\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) −20.0000 −0.689246
\(843\) −26.0000 −0.895488
\(844\) −18.0000 −0.619586
\(845\) 0 0
\(846\) −6.00000 −0.206284
\(847\) 0 0
\(848\) −1.00000 −0.0343401
\(849\) 20.0000 0.686398
\(850\) 30.0000 1.02899
\(851\) −48.0000 −1.64542
\(852\) 0 0
\(853\) 38.0000 1.30110 0.650548 − 0.759465i \(-0.274539\pi\)
0.650548 + 0.759465i \(0.274539\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 8.00000 0.273434
\(857\) 42.0000 1.43469 0.717346 − 0.696717i \(-0.245357\pi\)
0.717346 + 0.696717i \(0.245357\pi\)
\(858\) −12.0000 −0.409673
\(859\) −4.00000 −0.136478 −0.0682391 − 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 32.0000 1.08992
\(863\) −48.0000 −1.63394 −0.816970 − 0.576681i \(-0.804348\pi\)
−0.816970 + 0.576681i \(0.804348\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) 2.00000 0.0679628
\(867\) 19.0000 0.645274
\(868\) 0 0
\(869\) 8.00000 0.271381
\(870\) 0 0
\(871\) 72.0000 2.43963
\(872\) 0 0
\(873\) −10.0000 −0.338449
\(874\) −8.00000 −0.270604
\(875\) 0 0
\(876\) 2.00000 0.0675737
\(877\) −2.00000 −0.0675352 −0.0337676 − 0.999430i \(-0.510751\pi\)
−0.0337676 + 0.999430i \(0.510751\pi\)
\(878\) 2.00000 0.0674967
\(879\) 6.00000 0.202375
\(880\) 0 0
\(881\) −8.00000 −0.269527 −0.134763 − 0.990878i \(-0.543027\pi\)
−0.134763 + 0.990878i \(0.543027\pi\)
\(882\) 7.00000 0.235702
\(883\) 12.0000 0.403832 0.201916 − 0.979403i \(-0.435283\pi\)
0.201916 + 0.979403i \(0.435283\pi\)
\(884\) 36.0000 1.21081
\(885\) 0 0
\(886\) 0 0
\(887\) 56.0000 1.88030 0.940148 − 0.340766i \(-0.110687\pi\)
0.940148 + 0.340766i \(0.110687\pi\)
\(888\) −6.00000 −0.201347
\(889\) 0 0
\(890\) 0 0
\(891\) 2.00000 0.0670025
\(892\) 14.0000 0.468755
\(893\) −6.00000 −0.200782
\(894\) 22.0000 0.735790
\(895\) 0 0
\(896\) 0 0
\(897\) −48.0000 −1.60267
\(898\) 22.0000 0.734150
\(899\) −16.0000 −0.533630
\(900\) −5.00000 −0.166667
\(901\) −6.00000 −0.199889
\(902\) −4.00000 −0.133185
\(903\) 0 0
\(904\) −2.00000 −0.0665190
\(905\) 0 0
\(906\) −20.0000 −0.664455
\(907\) 34.0000 1.12895 0.564476 − 0.825450i \(-0.309078\pi\)
0.564476 + 0.825450i \(0.309078\pi\)
\(908\) 8.00000 0.265489
\(909\) 0 0
\(910\) 0 0
\(911\) 24.0000 0.795155 0.397578 − 0.917568i \(-0.369851\pi\)
0.397578 + 0.917568i \(0.369851\pi\)
\(912\) −1.00000 −0.0331133
\(913\) 0 0
\(914\) −22.0000 −0.727695
\(915\) 0 0
\(916\) −14.0000 −0.462573
\(917\) 0 0
\(918\) −6.00000 −0.198030
\(919\) −32.0000 −1.05558 −0.527791 − 0.849374i \(-0.676980\pi\)
−0.527791 + 0.849374i \(0.676980\pi\)
\(920\) 0 0
\(921\) 2.00000 0.0659022
\(922\) −2.00000 −0.0658665
\(923\) 0 0
\(924\) 0 0
\(925\) −30.0000 −0.986394
\(926\) −16.0000 −0.525793
\(927\) −4.00000 −0.131377
\(928\) 2.00000 0.0656532
\(929\) 18.0000 0.590561 0.295280 − 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) 0 0
\(931\) 7.00000 0.229416
\(932\) −20.0000 −0.655122
\(933\) 6.00000 0.196431
\(934\) −38.0000 −1.24340
\(935\) 0 0
\(936\) −6.00000 −0.196116
\(937\) 2.00000 0.0653372 0.0326686 − 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) 0 0
\(939\) 18.0000 0.587408
\(940\) 0 0
\(941\) 46.0000 1.49956 0.749779 − 0.661689i \(-0.230160\pi\)
0.749779 + 0.661689i \(0.230160\pi\)
\(942\) 14.0000 0.456145
\(943\) −16.0000 −0.521032
\(944\) 8.00000 0.260378
\(945\) 0 0
\(946\) 8.00000 0.260102
\(947\) −10.0000 −0.324956 −0.162478 − 0.986712i \(-0.551949\pi\)
−0.162478 + 0.986712i \(0.551949\pi\)
\(948\) 4.00000 0.129914
\(949\) 12.0000 0.389536
\(950\) −5.00000 −0.162221
\(951\) 18.0000 0.583690
\(952\) 0 0
\(953\) 54.0000 1.74923 0.874616 − 0.484817i \(-0.161114\pi\)
0.874616 + 0.484817i \(0.161114\pi\)
\(954\) 1.00000 0.0323762
\(955\) 0 0
\(956\) 28.0000 0.905585
\(957\) −4.00000 −0.129302
\(958\) 36.0000 1.16311
\(959\) 0 0
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) −36.0000 −1.16069
\(963\) −8.00000 −0.257796
\(964\) 30.0000 0.966235
\(965\) 0 0
\(966\) 0 0
\(967\) 52.0000 1.67221 0.836104 − 0.548572i \(-0.184828\pi\)
0.836104 + 0.548572i \(0.184828\pi\)
\(968\) 7.00000 0.224989
\(969\) −6.00000 −0.192748
\(970\) 0 0
\(971\) −52.0000 −1.66876 −0.834380 − 0.551190i \(-0.814174\pi\)
−0.834380 + 0.551190i \(0.814174\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −2.00000 −0.0640841
\(975\) −30.0000 −0.960769
\(976\) 14.0000 0.448129
\(977\) 2.00000 0.0639857 0.0319928 − 0.999488i \(-0.489815\pi\)
0.0319928 + 0.999488i \(0.489815\pi\)
\(978\) −20.0000 −0.639529
\(979\) −20.0000 −0.639203
\(980\) 0 0
\(981\) 0 0
\(982\) −20.0000 −0.638226
\(983\) −4.00000 −0.127580 −0.0637901 − 0.997963i \(-0.520319\pi\)
−0.0637901 + 0.997963i \(0.520319\pi\)
\(984\) −2.00000 −0.0637577
\(985\) 0 0
\(986\) 12.0000 0.382158
\(987\) 0 0
\(988\) −6.00000 −0.190885
\(989\) 32.0000 1.01754
\(990\) 0 0
\(991\) 2.00000 0.0635321 0.0317660 − 0.999495i \(-0.489887\pi\)
0.0317660 + 0.999495i \(0.489887\pi\)
\(992\) −8.00000 −0.254000
\(993\) 2.00000 0.0634681
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −6.00000 −0.190022 −0.0950110 − 0.995476i \(-0.530289\pi\)
−0.0950110 + 0.995476i \(0.530289\pi\)
\(998\) −36.0000 −1.13956
\(999\) 6.00000 0.189832
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 6042.2.a.h.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6042.2.a.h.1.1 ✓ 1 1.1 even 1 trivial