Properties

Label 1083.2.a.b.1.1
Level $1083$
Weight $2$
Character 1083.1
Self dual yes
Analytic conductor $8.648$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1083,2,Mod(1,1083)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1083, 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("1083.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1083 = 3 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1083.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(8.64779853890\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 57)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1083.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} -2.00000 q^{11} -1.00000 q^{12} -5.00000 q^{13} -1.00000 q^{14} -1.00000 q^{16} -4.00000 q^{17} -1.00000 q^{18} +1.00000 q^{21} +2.00000 q^{22} -4.00000 q^{23} +3.00000 q^{24} -5.00000 q^{25} +5.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} +8.00000 q^{29} +3.00000 q^{31} -5.00000 q^{32} -2.00000 q^{33} +4.00000 q^{34} -1.00000 q^{36} -3.00000 q^{37} -5.00000 q^{39} +12.0000 q^{41} -1.00000 q^{42} -1.00000 q^{43} +2.00000 q^{44} +4.00000 q^{46} -6.00000 q^{47} -1.00000 q^{48} -6.00000 q^{49} +5.00000 q^{50} -4.00000 q^{51} +5.00000 q^{52} -4.00000 q^{53} -1.00000 q^{54} +3.00000 q^{56} -8.00000 q^{58} -10.0000 q^{59} -13.0000 q^{61} -3.00000 q^{62} +1.00000 q^{63} +7.00000 q^{64} +2.00000 q^{66} -11.0000 q^{67} +4.00000 q^{68} -4.00000 q^{69} -6.00000 q^{71} +3.00000 q^{72} -11.0000 q^{73} +3.00000 q^{74} -5.00000 q^{75} -2.00000 q^{77} +5.00000 q^{78} -1.00000 q^{79} +1.00000 q^{81} -12.0000 q^{82} -1.00000 q^{84} +1.00000 q^{86} +8.00000 q^{87} -6.00000 q^{88} +6.00000 q^{89} -5.00000 q^{91} +4.00000 q^{92} +3.00000 q^{93} +6.00000 q^{94} -5.00000 q^{96} -2.00000 q^{97} +6.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107 −0.353553 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) 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
\(7\) 1.00000 0.377964 0.188982 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) 3.00000 1.06066
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) −1.00000 −0.288675
\(13\) −5.00000 −1.38675 −0.693375 0.720577i \(-0.743877\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) −1.00000 −0.235702
\(19\) 0 0
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 2.00000 0.426401
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 3.00000 0.612372
\(25\) −5.00000 −1.00000
\(26\) 5.00000 0.980581
\(27\) 1.00000 0.192450
\(28\) −1.00000 −0.188982
\(29\) 8.00000 1.48556 0.742781 0.669534i \(-0.233506\pi\)
0.742781 + 0.669534i \(0.233506\pi\)
\(30\) 0 0
\(31\) 3.00000 0.538816 0.269408 0.963026i \(-0.413172\pi\)
0.269408 + 0.963026i \(0.413172\pi\)
\(32\) −5.00000 −0.883883
\(33\) −2.00000 −0.348155
\(34\) 4.00000 0.685994
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) −3.00000 −0.493197 −0.246598 0.969118i \(-0.579313\pi\)
−0.246598 + 0.969118i \(0.579313\pi\)
\(38\) 0 0
\(39\) −5.00000 −0.800641
\(40\) 0 0
\(41\) 12.0000 1.87409 0.937043 0.349215i \(-0.113552\pi\)
0.937043 + 0.349215i \(0.113552\pi\)
\(42\) −1.00000 −0.154303
\(43\) −1.00000 −0.152499 −0.0762493 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) −1.00000 −0.144338
\(49\) −6.00000 −0.857143
\(50\) 5.00000 0.707107
\(51\) −4.00000 −0.560112
\(52\) 5.00000 0.693375
\(53\) −4.00000 −0.549442 −0.274721 0.961524i \(-0.588586\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 3.00000 0.400892
\(57\) 0 0
\(58\) −8.00000 −1.05045
\(59\) −10.0000 −1.30189 −0.650945 0.759125i \(-0.725627\pi\)
−0.650945 + 0.759125i \(0.725627\pi\)
\(60\) 0 0
\(61\) −13.0000 −1.66448 −0.832240 0.554416i \(-0.812942\pi\)
−0.832240 + 0.554416i \(0.812942\pi\)
\(62\) −3.00000 −0.381000
\(63\) 1.00000 0.125988
\(64\) 7.00000 0.875000
\(65\) 0 0
\(66\) 2.00000 0.246183
\(67\) −11.0000 −1.34386 −0.671932 0.740613i \(-0.734535\pi\)
−0.671932 + 0.740613i \(0.734535\pi\)
\(68\) 4.00000 0.485071
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 3.00000 0.353553
\(73\) −11.0000 −1.28745 −0.643726 0.765256i \(-0.722612\pi\)
−0.643726 + 0.765256i \(0.722612\pi\)
\(74\) 3.00000 0.348743
\(75\) −5.00000 −0.577350
\(76\) 0 0
\(77\) −2.00000 −0.227921
\(78\) 5.00000 0.566139
\(79\) −1.00000 −0.112509 −0.0562544 0.998416i \(-0.517916\pi\)
−0.0562544 + 0.998416i \(0.517916\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −12.0000 −1.32518
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) −1.00000 −0.109109
\(85\) 0 0
\(86\) 1.00000 0.107833
\(87\) 8.00000 0.857690
\(88\) −6.00000 −0.639602
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −5.00000 −0.524142
\(92\) 4.00000 0.417029
\(93\) 3.00000 0.311086
\(94\) 6.00000 0.618853
\(95\) 0 0
\(96\) −5.00000 −0.510310
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 6.00000 0.606092
\(99\) −2.00000 −0.201008
\(100\) 5.00000 0.500000
\(101\) 14.0000 1.39305 0.696526 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) 4.00000 0.396059
\(103\) −13.0000 −1.28093 −0.640464 0.767988i \(-0.721258\pi\)
−0.640464 + 0.767988i \(0.721258\pi\)
\(104\) −15.0000 −1.47087
\(105\) 0 0
\(106\) 4.00000 0.388514
\(107\) 18.0000 1.74013 0.870063 0.492941i \(-0.164078\pi\)
0.870063 + 0.492941i \(0.164078\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) −3.00000 −0.284747
\(112\) −1.00000 −0.0944911
\(113\) −2.00000 −0.188144 −0.0940721 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −8.00000 −0.742781
\(117\) −5.00000 −0.462250
\(118\) 10.0000 0.920575
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 13.0000 1.17696
\(123\) 12.0000 1.08200
\(124\) −3.00000 −0.269408
\(125\) 0 0
\(126\) −1.00000 −0.0890871
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 3.00000 0.265165
\(129\) −1.00000 −0.0880451
\(130\) 0 0
\(131\) −14.0000 −1.22319 −0.611593 0.791173i \(-0.709471\pi\)
−0.611593 + 0.791173i \(0.709471\pi\)
\(132\) 2.00000 0.174078
\(133\) 0 0
\(134\) 11.0000 0.950255
\(135\) 0 0
\(136\) −12.0000 −1.02899
\(137\) 18.0000 1.53784 0.768922 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) 4.00000 0.340503
\(139\) 5.00000 0.424094 0.212047 0.977259i \(-0.431987\pi\)
0.212047 + 0.977259i \(0.431987\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) 6.00000 0.503509
\(143\) 10.0000 0.836242
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 11.0000 0.910366
\(147\) −6.00000 −0.494872
\(148\) 3.00000 0.246598
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 5.00000 0.408248
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) 0 0
\(153\) −4.00000 −0.323381
\(154\) 2.00000 0.161165
\(155\) 0 0
\(156\) 5.00000 0.400320
\(157\) 3.00000 0.239426 0.119713 0.992809i \(-0.461803\pi\)
0.119713 + 0.992809i \(0.461803\pi\)
\(158\) 1.00000 0.0795557
\(159\) −4.00000 −0.317221
\(160\) 0 0
\(161\) −4.00000 −0.315244
\(162\) −1.00000 −0.0785674
\(163\) 3.00000 0.234978 0.117489 0.993074i \(-0.462515\pi\)
0.117489 + 0.993074i \(0.462515\pi\)
\(164\) −12.0000 −0.937043
\(165\) 0 0
\(166\) 0 0
\(167\) 2.00000 0.154765 0.0773823 0.997001i \(-0.475344\pi\)
0.0773823 + 0.997001i \(0.475344\pi\)
\(168\) 3.00000 0.231455
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) 0 0
\(172\) 1.00000 0.0762493
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) −8.00000 −0.606478
\(175\) −5.00000 −0.377964
\(176\) 2.00000 0.150756
\(177\) −10.0000 −0.751646
\(178\) −6.00000 −0.449719
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 5.00000 0.370625
\(183\) −13.0000 −0.960988
\(184\) −12.0000 −0.884652
\(185\) 0 0
\(186\) −3.00000 −0.219971
\(187\) 8.00000 0.585018
\(188\) 6.00000 0.437595
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 24.0000 1.73658 0.868290 0.496058i \(-0.165220\pi\)
0.868290 + 0.496058i \(0.165220\pi\)
\(192\) 7.00000 0.505181
\(193\) −19.0000 −1.36765 −0.683825 0.729646i \(-0.739685\pi\)
−0.683825 + 0.729646i \(0.739685\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) 6.00000 0.428571
\(197\) −26.0000 −1.85242 −0.926212 0.377004i \(-0.876954\pi\)
−0.926212 + 0.377004i \(0.876954\pi\)
\(198\) 2.00000 0.142134
\(199\) 21.0000 1.48865 0.744325 0.667817i \(-0.232771\pi\)
0.744325 + 0.667817i \(0.232771\pi\)
\(200\) −15.0000 −1.06066
\(201\) −11.0000 −0.775880
\(202\) −14.0000 −0.985037
\(203\) 8.00000 0.561490
\(204\) 4.00000 0.280056
\(205\) 0 0
\(206\) 13.0000 0.905753
\(207\) −4.00000 −0.278019
\(208\) 5.00000 0.346688
\(209\) 0 0
\(210\) 0 0
\(211\) 15.0000 1.03264 0.516321 0.856395i \(-0.327301\pi\)
0.516321 + 0.856395i \(0.327301\pi\)
\(212\) 4.00000 0.274721
\(213\) −6.00000 −0.411113
\(214\) −18.0000 −1.23045
\(215\) 0 0
\(216\) 3.00000 0.204124
\(217\) 3.00000 0.203653
\(218\) −2.00000 −0.135457
\(219\) −11.0000 −0.743311
\(220\) 0 0
\(221\) 20.0000 1.34535
\(222\) 3.00000 0.201347
\(223\) −9.00000 −0.602685 −0.301342 0.953516i \(-0.597435\pi\)
−0.301342 + 0.953516i \(0.597435\pi\)
\(224\) −5.00000 −0.334077
\(225\) −5.00000 −0.333333
\(226\) 2.00000 0.133038
\(227\) 24.0000 1.59294 0.796468 0.604681i \(-0.206699\pi\)
0.796468 + 0.604681i \(0.206699\pi\)
\(228\) 0 0
\(229\) 13.0000 0.859064 0.429532 0.903052i \(-0.358679\pi\)
0.429532 + 0.903052i \(0.358679\pi\)
\(230\) 0 0
\(231\) −2.00000 −0.131590
\(232\) 24.0000 1.57568
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 5.00000 0.326860
\(235\) 0 0
\(236\) 10.0000 0.650945
\(237\) −1.00000 −0.0649570
\(238\) 4.00000 0.259281
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) 7.00000 0.450910 0.225455 0.974254i \(-0.427613\pi\)
0.225455 + 0.974254i \(0.427613\pi\)
\(242\) 7.00000 0.449977
\(243\) 1.00000 0.0641500
\(244\) 13.0000 0.832240
\(245\) 0 0
\(246\) −12.0000 −0.765092
\(247\) 0 0
\(248\) 9.00000 0.571501
\(249\) 0 0
\(250\) 0 0
\(251\) −2.00000 −0.126239 −0.0631194 0.998006i \(-0.520105\pi\)
−0.0631194 + 0.998006i \(0.520105\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 8.00000 0.502956
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) 20.0000 1.24757 0.623783 0.781598i \(-0.285595\pi\)
0.623783 + 0.781598i \(0.285595\pi\)
\(258\) 1.00000 0.0622573
\(259\) −3.00000 −0.186411
\(260\) 0 0
\(261\) 8.00000 0.495188
\(262\) 14.0000 0.864923
\(263\) 26.0000 1.60323 0.801614 0.597841i \(-0.203975\pi\)
0.801614 + 0.597841i \(0.203975\pi\)
\(264\) −6.00000 −0.369274
\(265\) 0 0
\(266\) 0 0
\(267\) 6.00000 0.367194
\(268\) 11.0000 0.671932
\(269\) −10.0000 −0.609711 −0.304855 0.952399i \(-0.598608\pi\)
−0.304855 + 0.952399i \(0.598608\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 4.00000 0.242536
\(273\) −5.00000 −0.302614
\(274\) −18.0000 −1.08742
\(275\) 10.0000 0.603023
\(276\) 4.00000 0.240772
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) −5.00000 −0.299880
\(279\) 3.00000 0.179605
\(280\) 0 0
\(281\) 8.00000 0.477240 0.238620 0.971113i \(-0.423305\pi\)
0.238620 + 0.971113i \(0.423305\pi\)
\(282\) 6.00000 0.357295
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 6.00000 0.356034
\(285\) 0 0
\(286\) −10.0000 −0.591312
\(287\) 12.0000 0.708338
\(288\) −5.00000 −0.294628
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) −2.00000 −0.117242
\(292\) 11.0000 0.643726
\(293\) −14.0000 −0.817889 −0.408944 0.912559i \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(294\) 6.00000 0.349927
\(295\) 0 0
\(296\) −9.00000 −0.523114
\(297\) −2.00000 −0.116052
\(298\) 6.00000 0.347571
\(299\) 20.0000 1.15663
\(300\) 5.00000 0.288675
\(301\) −1.00000 −0.0576390
\(302\) 12.0000 0.690522
\(303\) 14.0000 0.804279
\(304\) 0 0
\(305\) 0 0
\(306\) 4.00000 0.228665
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) 2.00000 0.113961
\(309\) −13.0000 −0.739544
\(310\) 0 0
\(311\) 18.0000 1.02069 0.510343 0.859971i \(-0.329518\pi\)
0.510343 + 0.859971i \(0.329518\pi\)
\(312\) −15.0000 −0.849208
\(313\) 22.0000 1.24351 0.621757 0.783210i \(-0.286419\pi\)
0.621757 + 0.783210i \(0.286419\pi\)
\(314\) −3.00000 −0.169300
\(315\) 0 0
\(316\) 1.00000 0.0562544
\(317\) 4.00000 0.224662 0.112331 0.993671i \(-0.464168\pi\)
0.112331 + 0.993671i \(0.464168\pi\)
\(318\) 4.00000 0.224309
\(319\) −16.0000 −0.895828
\(320\) 0 0
\(321\) 18.0000 1.00466
\(322\) 4.00000 0.222911
\(323\) 0 0
\(324\) −1.00000 −0.0555556
\(325\) 25.0000 1.38675
\(326\) −3.00000 −0.166155
\(327\) 2.00000 0.110600
\(328\) 36.0000 1.98777
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) 25.0000 1.37412 0.687062 0.726599i \(-0.258900\pi\)
0.687062 + 0.726599i \(0.258900\pi\)
\(332\) 0 0
\(333\) −3.00000 −0.164399
\(334\) −2.00000 −0.109435
\(335\) 0 0
\(336\) −1.00000 −0.0545545
\(337\) −13.0000 −0.708155 −0.354078 0.935216i \(-0.615205\pi\)
−0.354078 + 0.935216i \(0.615205\pi\)
\(338\) −12.0000 −0.652714
\(339\) −2.00000 −0.108625
\(340\) 0 0
\(341\) −6.00000 −0.324918
\(342\) 0 0
\(343\) −13.0000 −0.701934
\(344\) −3.00000 −0.161749
\(345\) 0 0
\(346\) 0 0
\(347\) −16.0000 −0.858925 −0.429463 0.903085i \(-0.641297\pi\)
−0.429463 + 0.903085i \(0.641297\pi\)
\(348\) −8.00000 −0.428845
\(349\) −21.0000 −1.12410 −0.562052 0.827102i \(-0.689988\pi\)
−0.562052 + 0.827102i \(0.689988\pi\)
\(350\) 5.00000 0.267261
\(351\) −5.00000 −0.266880
\(352\) 10.0000 0.533002
\(353\) 4.00000 0.212899 0.106449 0.994318i \(-0.466052\pi\)
0.106449 + 0.994318i \(0.466052\pi\)
\(354\) 10.0000 0.531494
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) −4.00000 −0.211702
\(358\) 12.0000 0.634220
\(359\) −20.0000 −1.05556 −0.527780 0.849381i \(-0.676975\pi\)
−0.527780 + 0.849381i \(0.676975\pi\)
\(360\) 0 0
\(361\) 0 0
\(362\) −2.00000 −0.105118
\(363\) −7.00000 −0.367405
\(364\) 5.00000 0.262071
\(365\) 0 0
\(366\) 13.0000 0.679521
\(367\) 7.00000 0.365397 0.182699 0.983169i \(-0.441517\pi\)
0.182699 + 0.983169i \(0.441517\pi\)
\(368\) 4.00000 0.208514
\(369\) 12.0000 0.624695
\(370\) 0 0
\(371\) −4.00000 −0.207670
\(372\) −3.00000 −0.155543
\(373\) −10.0000 −0.517780 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) −8.00000 −0.413670
\(375\) 0 0
\(376\) −18.0000 −0.928279
\(377\) −40.0000 −2.06010
\(378\) −1.00000 −0.0514344
\(379\) 5.00000 0.256833 0.128416 0.991720i \(-0.459011\pi\)
0.128416 + 0.991720i \(0.459011\pi\)
\(380\) 0 0
\(381\) 8.00000 0.409852
\(382\) −24.0000 −1.22795
\(383\) −14.0000 −0.715367 −0.357683 0.933843i \(-0.616433\pi\)
−0.357683 + 0.933843i \(0.616433\pi\)
\(384\) 3.00000 0.153093
\(385\) 0 0
\(386\) 19.0000 0.967075
\(387\) −1.00000 −0.0508329
\(388\) 2.00000 0.101535
\(389\) −24.0000 −1.21685 −0.608424 0.793612i \(-0.708198\pi\)
−0.608424 + 0.793612i \(0.708198\pi\)
\(390\) 0 0
\(391\) 16.0000 0.809155
\(392\) −18.0000 −0.909137
\(393\) −14.0000 −0.706207
\(394\) 26.0000 1.30986
\(395\) 0 0
\(396\) 2.00000 0.100504
\(397\) 7.00000 0.351320 0.175660 0.984451i \(-0.443794\pi\)
0.175660 + 0.984451i \(0.443794\pi\)
\(398\) −21.0000 −1.05263
\(399\) 0 0
\(400\) 5.00000 0.250000
\(401\) 6.00000 0.299626 0.149813 0.988714i \(-0.452133\pi\)
0.149813 + 0.988714i \(0.452133\pi\)
\(402\) 11.0000 0.548630
\(403\) −15.0000 −0.747203
\(404\) −14.0000 −0.696526
\(405\) 0 0
\(406\) −8.00000 −0.397033
\(407\) 6.00000 0.297409
\(408\) −12.0000 −0.594089
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 0 0
\(411\) 18.0000 0.887875
\(412\) 13.0000 0.640464
\(413\) −10.0000 −0.492068
\(414\) 4.00000 0.196589
\(415\) 0 0
\(416\) 25.0000 1.22573
\(417\) 5.00000 0.244851
\(418\) 0 0
\(419\) −14.0000 −0.683945 −0.341972 0.939710i \(-0.611095\pi\)
−0.341972 + 0.939710i \(0.611095\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) −15.0000 −0.730189
\(423\) −6.00000 −0.291730
\(424\) −12.0000 −0.582772
\(425\) 20.0000 0.970143
\(426\) 6.00000 0.290701
\(427\) −13.0000 −0.629114
\(428\) −18.0000 −0.870063
\(429\) 10.0000 0.482805
\(430\) 0 0
\(431\) 10.0000 0.481683 0.240842 0.970564i \(-0.422577\pi\)
0.240842 + 0.970564i \(0.422577\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 9.00000 0.432512 0.216256 0.976337i \(-0.430615\pi\)
0.216256 + 0.976337i \(0.430615\pi\)
\(434\) −3.00000 −0.144005
\(435\) 0 0
\(436\) −2.00000 −0.0957826
\(437\) 0 0
\(438\) 11.0000 0.525600
\(439\) −35.0000 −1.67046 −0.835229 0.549902i \(-0.814665\pi\)
−0.835229 + 0.549902i \(0.814665\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) −20.0000 −0.951303
\(443\) −32.0000 −1.52037 −0.760183 0.649709i \(-0.774891\pi\)
−0.760183 + 0.649709i \(0.774891\pi\)
\(444\) 3.00000 0.142374
\(445\) 0 0
\(446\) 9.00000 0.426162
\(447\) −6.00000 −0.283790
\(448\) 7.00000 0.330719
\(449\) 12.0000 0.566315 0.283158 0.959073i \(-0.408618\pi\)
0.283158 + 0.959073i \(0.408618\pi\)
\(450\) 5.00000 0.235702
\(451\) −24.0000 −1.13012
\(452\) 2.00000 0.0940721
\(453\) −12.0000 −0.563809
\(454\) −24.0000 −1.12638
\(455\) 0 0
\(456\) 0 0
\(457\) −11.0000 −0.514558 −0.257279 0.966337i \(-0.582826\pi\)
−0.257279 + 0.966337i \(0.582826\pi\)
\(458\) −13.0000 −0.607450
\(459\) −4.00000 −0.186704
\(460\) 0 0
\(461\) 30.0000 1.39724 0.698620 0.715493i \(-0.253798\pi\)
0.698620 + 0.715493i \(0.253798\pi\)
\(462\) 2.00000 0.0930484
\(463\) 23.0000 1.06890 0.534450 0.845200i \(-0.320519\pi\)
0.534450 + 0.845200i \(0.320519\pi\)
\(464\) −8.00000 −0.371391
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) −8.00000 −0.370196 −0.185098 0.982720i \(-0.559260\pi\)
−0.185098 + 0.982720i \(0.559260\pi\)
\(468\) 5.00000 0.231125
\(469\) −11.0000 −0.507933
\(470\) 0 0
\(471\) 3.00000 0.138233
\(472\) −30.0000 −1.38086
\(473\) 2.00000 0.0919601
\(474\) 1.00000 0.0459315
\(475\) 0 0
\(476\) 4.00000 0.183340
\(477\) −4.00000 −0.183147
\(478\) 12.0000 0.548867
\(479\) 14.0000 0.639676 0.319838 0.947472i \(-0.396371\pi\)
0.319838 + 0.947472i \(0.396371\pi\)
\(480\) 0 0
\(481\) 15.0000 0.683941
\(482\) −7.00000 −0.318841
\(483\) −4.00000 −0.182006
\(484\) 7.00000 0.318182
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 16.0000 0.725029 0.362515 0.931978i \(-0.381918\pi\)
0.362515 + 0.931978i \(0.381918\pi\)
\(488\) −39.0000 −1.76545
\(489\) 3.00000 0.135665
\(490\) 0 0
\(491\) 6.00000 0.270776 0.135388 0.990793i \(-0.456772\pi\)
0.135388 + 0.990793i \(0.456772\pi\)
\(492\) −12.0000 −0.541002
\(493\) −32.0000 −1.44121
\(494\) 0 0
\(495\) 0 0
\(496\) −3.00000 −0.134704
\(497\) −6.00000 −0.269137
\(498\) 0 0
\(499\) 29.0000 1.29822 0.649109 0.760695i \(-0.275142\pi\)
0.649109 + 0.760695i \(0.275142\pi\)
\(500\) 0 0
\(501\) 2.00000 0.0893534
\(502\) 2.00000 0.0892644
\(503\) 40.0000 1.78351 0.891756 0.452517i \(-0.149474\pi\)
0.891756 + 0.452517i \(0.149474\pi\)
\(504\) 3.00000 0.133631
\(505\) 0 0
\(506\) −8.00000 −0.355643
\(507\) 12.0000 0.532939
\(508\) −8.00000 −0.354943
\(509\) 6.00000 0.265945 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 0 0
\(511\) −11.0000 −0.486611
\(512\) 11.0000 0.486136
\(513\) 0 0
\(514\) −20.0000 −0.882162
\(515\) 0 0
\(516\) 1.00000 0.0440225
\(517\) 12.0000 0.527759
\(518\) 3.00000 0.131812
\(519\) 0 0
\(520\) 0 0
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) −8.00000 −0.350150
\(523\) −29.0000 −1.26808 −0.634041 0.773300i \(-0.718605\pi\)
−0.634041 + 0.773300i \(0.718605\pi\)
\(524\) 14.0000 0.611593
\(525\) −5.00000 −0.218218
\(526\) −26.0000 −1.13365
\(527\) −12.0000 −0.522728
\(528\) 2.00000 0.0870388
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −10.0000 −0.433963
\(532\) 0 0
\(533\) −60.0000 −2.59889
\(534\) −6.00000 −0.259645
\(535\) 0 0
\(536\) −33.0000 −1.42538
\(537\) −12.0000 −0.517838
\(538\) 10.0000 0.431131
\(539\) 12.0000 0.516877
\(540\) 0 0
\(541\) −39.0000 −1.67674 −0.838370 0.545101i \(-0.816491\pi\)
−0.838370 + 0.545101i \(0.816491\pi\)
\(542\) 0 0
\(543\) 2.00000 0.0858282
\(544\) 20.0000 0.857493
\(545\) 0 0
\(546\) 5.00000 0.213980
\(547\) 29.0000 1.23995 0.619975 0.784621i \(-0.287143\pi\)
0.619975 + 0.784621i \(0.287143\pi\)
\(548\) −18.0000 −0.768922
\(549\) −13.0000 −0.554826
\(550\) −10.0000 −0.426401
\(551\) 0 0
\(552\) −12.0000 −0.510754
\(553\) −1.00000 −0.0425243
\(554\) 2.00000 0.0849719
\(555\) 0 0
\(556\) −5.00000 −0.212047
\(557\) −38.0000 −1.61011 −0.805056 0.593199i \(-0.797865\pi\)
−0.805056 + 0.593199i \(0.797865\pi\)
\(558\) −3.00000 −0.127000
\(559\) 5.00000 0.211477
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) −8.00000 −0.337460
\(563\) −18.0000 −0.758610 −0.379305 0.925272i \(-0.623837\pi\)
−0.379305 + 0.925272i \(0.623837\pi\)
\(564\) 6.00000 0.252646
\(565\) 0 0
\(566\) 4.00000 0.168133
\(567\) 1.00000 0.0419961
\(568\) −18.0000 −0.755263
\(569\) −24.0000 −1.00613 −0.503066 0.864248i \(-0.667795\pi\)
−0.503066 + 0.864248i \(0.667795\pi\)
\(570\) 0 0
\(571\) 33.0000 1.38101 0.690504 0.723329i \(-0.257389\pi\)
0.690504 + 0.723329i \(0.257389\pi\)
\(572\) −10.0000 −0.418121
\(573\) 24.0000 1.00261
\(574\) −12.0000 −0.500870
\(575\) 20.0000 0.834058
\(576\) 7.00000 0.291667
\(577\) 18.0000 0.749350 0.374675 0.927156i \(-0.377754\pi\)
0.374675 + 0.927156i \(0.377754\pi\)
\(578\) 1.00000 0.0415945
\(579\) −19.0000 −0.789613
\(580\) 0 0
\(581\) 0 0
\(582\) 2.00000 0.0829027
\(583\) 8.00000 0.331326
\(584\) −33.0000 −1.36555
\(585\) 0 0
\(586\) 14.0000 0.578335
\(587\) −26.0000 −1.07313 −0.536567 0.843857i \(-0.680279\pi\)
−0.536567 + 0.843857i \(0.680279\pi\)
\(588\) 6.00000 0.247436
\(589\) 0 0
\(590\) 0 0
\(591\) −26.0000 −1.06950
\(592\) 3.00000 0.123299
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 2.00000 0.0820610
\(595\) 0 0
\(596\) 6.00000 0.245770
\(597\) 21.0000 0.859473
\(598\) −20.0000 −0.817861
\(599\) −18.0000 −0.735460 −0.367730 0.929933i \(-0.619865\pi\)
−0.367730 + 0.929933i \(0.619865\pi\)
\(600\) −15.0000 −0.612372
\(601\) 21.0000 0.856608 0.428304 0.903635i \(-0.359111\pi\)
0.428304 + 0.903635i \(0.359111\pi\)
\(602\) 1.00000 0.0407570
\(603\) −11.0000 −0.447955
\(604\) 12.0000 0.488273
\(605\) 0 0
\(606\) −14.0000 −0.568711
\(607\) 43.0000 1.74532 0.872658 0.488332i \(-0.162394\pi\)
0.872658 + 0.488332i \(0.162394\pi\)
\(608\) 0 0
\(609\) 8.00000 0.324176
\(610\) 0 0
\(611\) 30.0000 1.21367
\(612\) 4.00000 0.161690
\(613\) −30.0000 −1.21169 −0.605844 0.795583i \(-0.707165\pi\)
−0.605844 + 0.795583i \(0.707165\pi\)
\(614\) −12.0000 −0.484281
\(615\) 0 0
\(616\) −6.00000 −0.241747
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 13.0000 0.522937
\(619\) 25.0000 1.00483 0.502417 0.864625i \(-0.332444\pi\)
0.502417 + 0.864625i \(0.332444\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) −18.0000 −0.721734
\(623\) 6.00000 0.240385
\(624\) 5.00000 0.200160
\(625\) 25.0000 1.00000
\(626\) −22.0000 −0.879297
\(627\) 0 0
\(628\) −3.00000 −0.119713
\(629\) 12.0000 0.478471
\(630\) 0 0
\(631\) −15.0000 −0.597141 −0.298570 0.954388i \(-0.596510\pi\)
−0.298570 + 0.954388i \(0.596510\pi\)
\(632\) −3.00000 −0.119334
\(633\) 15.0000 0.596196
\(634\) −4.00000 −0.158860
\(635\) 0 0
\(636\) 4.00000 0.158610
\(637\) 30.0000 1.18864
\(638\) 16.0000 0.633446
\(639\) −6.00000 −0.237356
\(640\) 0 0
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) −18.0000 −0.710403
\(643\) −19.0000 −0.749287 −0.374643 0.927169i \(-0.622235\pi\)
−0.374643 + 0.927169i \(0.622235\pi\)
\(644\) 4.00000 0.157622
\(645\) 0 0
\(646\) 0 0
\(647\) −24.0000 −0.943537 −0.471769 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) 3.00000 0.117851
\(649\) 20.0000 0.785069
\(650\) −25.0000 −0.980581
\(651\) 3.00000 0.117579
\(652\) −3.00000 −0.117489
\(653\) −18.0000 −0.704394 −0.352197 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) −2.00000 −0.0782062
\(655\) 0 0
\(656\) −12.0000 −0.468521
\(657\) −11.0000 −0.429151
\(658\) 6.00000 0.233904
\(659\) −34.0000 −1.32445 −0.662226 0.749304i \(-0.730388\pi\)
−0.662226 + 0.749304i \(0.730388\pi\)
\(660\) 0 0
\(661\) −42.0000 −1.63361 −0.816805 0.576913i \(-0.804257\pi\)
−0.816805 + 0.576913i \(0.804257\pi\)
\(662\) −25.0000 −0.971653
\(663\) 20.0000 0.776736
\(664\) 0 0
\(665\) 0 0
\(666\) 3.00000 0.116248
\(667\) −32.0000 −1.23904
\(668\) −2.00000 −0.0773823
\(669\) −9.00000 −0.347960
\(670\) 0 0
\(671\) 26.0000 1.00372
\(672\) −5.00000 −0.192879
\(673\) 33.0000 1.27206 0.636028 0.771666i \(-0.280576\pi\)
0.636028 + 0.771666i \(0.280576\pi\)
\(674\) 13.0000 0.500741
\(675\) −5.00000 −0.192450
\(676\) −12.0000 −0.461538
\(677\) −34.0000 −1.30673 −0.653363 0.757045i \(-0.726642\pi\)
−0.653363 + 0.757045i \(0.726642\pi\)
\(678\) 2.00000 0.0768095
\(679\) −2.00000 −0.0767530
\(680\) 0 0
\(681\) 24.0000 0.919682
\(682\) 6.00000 0.229752
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 13.0000 0.496342
\(687\) 13.0000 0.495981
\(688\) 1.00000 0.0381246
\(689\) 20.0000 0.761939
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) 0 0
\(693\) −2.00000 −0.0759737
\(694\) 16.0000 0.607352
\(695\) 0 0
\(696\) 24.0000 0.909718
\(697\) −48.0000 −1.81813
\(698\) 21.0000 0.794862
\(699\) −6.00000 −0.226941
\(700\) 5.00000 0.188982
\(701\) 20.0000 0.755390 0.377695 0.925930i \(-0.376717\pi\)
0.377695 + 0.925930i \(0.376717\pi\)
\(702\) 5.00000 0.188713
\(703\) 0 0
\(704\) −14.0000 −0.527645
\(705\) 0 0
\(706\) −4.00000 −0.150542
\(707\) 14.0000 0.526524
\(708\) 10.0000 0.375823
\(709\) −39.0000 −1.46468 −0.732338 0.680941i \(-0.761571\pi\)
−0.732338 + 0.680941i \(0.761571\pi\)
\(710\) 0 0
\(711\) −1.00000 −0.0375029
\(712\) 18.0000 0.674579
\(713\) −12.0000 −0.449404
\(714\) 4.00000 0.149696
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) −12.0000 −0.448148
\(718\) 20.0000 0.746393
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) −13.0000 −0.484145
\(722\) 0 0
\(723\) 7.00000 0.260333
\(724\) −2.00000 −0.0743294
\(725\) −40.0000 −1.48556
\(726\) 7.00000 0.259794
\(727\) −47.0000 −1.74313 −0.871567 0.490277i \(-0.836896\pi\)
−0.871567 + 0.490277i \(0.836896\pi\)
\(728\) −15.0000 −0.555937
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 4.00000 0.147945
\(732\) 13.0000 0.480494
\(733\) −50.0000 −1.84679 −0.923396 0.383849i \(-0.874598\pi\)
−0.923396 + 0.383849i \(0.874598\pi\)
\(734\) −7.00000 −0.258375
\(735\) 0 0
\(736\) 20.0000 0.737210
\(737\) 22.0000 0.810380
\(738\) −12.0000 −0.441726
\(739\) 19.0000 0.698926 0.349463 0.936950i \(-0.386364\pi\)
0.349463 + 0.936950i \(0.386364\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 4.00000 0.146845
\(743\) 50.0000 1.83432 0.917161 0.398517i \(-0.130475\pi\)
0.917161 + 0.398517i \(0.130475\pi\)
\(744\) 9.00000 0.329956
\(745\) 0 0
\(746\) 10.0000 0.366126
\(747\) 0 0
\(748\) −8.00000 −0.292509
\(749\) 18.0000 0.657706
\(750\) 0 0
\(751\) −45.0000 −1.64207 −0.821037 0.570875i \(-0.806604\pi\)
−0.821037 + 0.570875i \(0.806604\pi\)
\(752\) 6.00000 0.218797
\(753\) −2.00000 −0.0728841
\(754\) 40.0000 1.45671
\(755\) 0 0
\(756\) −1.00000 −0.0363696
\(757\) 13.0000 0.472493 0.236247 0.971693i \(-0.424083\pi\)
0.236247 + 0.971693i \(0.424083\pi\)
\(758\) −5.00000 −0.181608
\(759\) 8.00000 0.290382
\(760\) 0 0
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) −8.00000 −0.289809
\(763\) 2.00000 0.0724049
\(764\) −24.0000 −0.868290
\(765\) 0 0
\(766\) 14.0000 0.505841
\(767\) 50.0000 1.80540
\(768\) −17.0000 −0.613435
\(769\) −37.0000 −1.33425 −0.667127 0.744944i \(-0.732476\pi\)
−0.667127 + 0.744944i \(0.732476\pi\)
\(770\) 0 0
\(771\) 20.0000 0.720282
\(772\) 19.0000 0.683825
\(773\) −14.0000 −0.503545 −0.251773 0.967786i \(-0.581013\pi\)
−0.251773 + 0.967786i \(0.581013\pi\)
\(774\) 1.00000 0.0359443
\(775\) −15.0000 −0.538816
\(776\) −6.00000 −0.215387
\(777\) −3.00000 −0.107624
\(778\) 24.0000 0.860442
\(779\) 0 0
\(780\) 0 0
\(781\) 12.0000 0.429394
\(782\) −16.0000 −0.572159
\(783\) 8.00000 0.285897
\(784\) 6.00000 0.214286
\(785\) 0 0
\(786\) 14.0000 0.499363
\(787\) −25.0000 −0.891154 −0.445577 0.895244i \(-0.647001\pi\)
−0.445577 + 0.895244i \(0.647001\pi\)
\(788\) 26.0000 0.926212
\(789\) 26.0000 0.925625
\(790\) 0 0
\(791\) −2.00000 −0.0711118
\(792\) −6.00000 −0.213201
\(793\) 65.0000 2.30822
\(794\) −7.00000 −0.248421
\(795\) 0 0
\(796\) −21.0000 −0.744325
\(797\) 14.0000 0.495905 0.247953 0.968772i \(-0.420242\pi\)
0.247953 + 0.968772i \(0.420242\pi\)
\(798\) 0 0
\(799\) 24.0000 0.849059
\(800\) 25.0000 0.883883
\(801\) 6.00000 0.212000
\(802\) −6.00000 −0.211867
\(803\) 22.0000 0.776363
\(804\) 11.0000 0.387940
\(805\) 0 0
\(806\) 15.0000 0.528352
\(807\) −10.0000 −0.352017
\(808\) 42.0000 1.47755
\(809\) −16.0000 −0.562530 −0.281265 0.959630i \(-0.590754\pi\)
−0.281265 + 0.959630i \(0.590754\pi\)
\(810\) 0 0
\(811\) 44.0000 1.54505 0.772524 0.634985i \(-0.218994\pi\)
0.772524 + 0.634985i \(0.218994\pi\)
\(812\) −8.00000 −0.280745
\(813\) 0 0
\(814\) −6.00000 −0.210300
\(815\) 0 0
\(816\) 4.00000 0.140028
\(817\) 0 0
\(818\) −14.0000 −0.489499
\(819\) −5.00000 −0.174714
\(820\) 0 0
\(821\) 24.0000 0.837606 0.418803 0.908077i \(-0.362450\pi\)
0.418803 + 0.908077i \(0.362450\pi\)
\(822\) −18.0000 −0.627822
\(823\) 4.00000 0.139431 0.0697156 0.997567i \(-0.477791\pi\)
0.0697156 + 0.997567i \(0.477791\pi\)
\(824\) −39.0000 −1.35863
\(825\) 10.0000 0.348155
\(826\) 10.0000 0.347945
\(827\) 36.0000 1.25184 0.625921 0.779886i \(-0.284723\pi\)
0.625921 + 0.779886i \(0.284723\pi\)
\(828\) 4.00000 0.139010
\(829\) −25.0000 −0.868286 −0.434143 0.900844i \(-0.642949\pi\)
−0.434143 + 0.900844i \(0.642949\pi\)
\(830\) 0 0
\(831\) −2.00000 −0.0693792
\(832\) −35.0000 −1.21341
\(833\) 24.0000 0.831551
\(834\) −5.00000 −0.173136
\(835\) 0 0
\(836\) 0 0
\(837\) 3.00000 0.103695
\(838\) 14.0000 0.483622
\(839\) 12.0000 0.414286 0.207143 0.978311i \(-0.433583\pi\)
0.207143 + 0.978311i \(0.433583\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) −22.0000 −0.758170
\(843\) 8.00000 0.275535
\(844\) −15.0000 −0.516321
\(845\) 0 0
\(846\) 6.00000 0.206284
\(847\) −7.00000 −0.240523
\(848\) 4.00000 0.137361
\(849\) −4.00000 −0.137280
\(850\) −20.0000 −0.685994
\(851\) 12.0000 0.411355
\(852\) 6.00000 0.205557
\(853\) −5.00000 −0.171197 −0.0855984 0.996330i \(-0.527280\pi\)
−0.0855984 + 0.996330i \(0.527280\pi\)
\(854\) 13.0000 0.444851
\(855\) 0 0
\(856\) 54.0000 1.84568
\(857\) −10.0000 −0.341593 −0.170797 0.985306i \(-0.554634\pi\)
−0.170797 + 0.985306i \(0.554634\pi\)
\(858\) −10.0000 −0.341394
\(859\) 15.0000 0.511793 0.255897 0.966704i \(-0.417629\pi\)
0.255897 + 0.966704i \(0.417629\pi\)
\(860\) 0 0
\(861\) 12.0000 0.408959
\(862\) −10.0000 −0.340601
\(863\) −10.0000 −0.340404 −0.170202 0.985409i \(-0.554442\pi\)
−0.170202 + 0.985409i \(0.554442\pi\)
\(864\) −5.00000 −0.170103
\(865\) 0 0
\(866\) −9.00000 −0.305832
\(867\) −1.00000 −0.0339618
\(868\) −3.00000 −0.101827
\(869\) 2.00000 0.0678454
\(870\) 0 0
\(871\) 55.0000 1.86360
\(872\) 6.00000 0.203186
\(873\) −2.00000 −0.0676897
\(874\) 0 0
\(875\) 0 0
\(876\) 11.0000 0.371656
\(877\) −15.0000 −0.506514 −0.253257 0.967399i \(-0.581502\pi\)
−0.253257 + 0.967399i \(0.581502\pi\)
\(878\) 35.0000 1.18119
\(879\) −14.0000 −0.472208
\(880\) 0 0
\(881\) −28.0000 −0.943344 −0.471672 0.881774i \(-0.656349\pi\)
−0.471672 + 0.881774i \(0.656349\pi\)
\(882\) 6.00000 0.202031
\(883\) −1.00000 −0.0336527 −0.0168263 0.999858i \(-0.505356\pi\)
−0.0168263 + 0.999858i \(0.505356\pi\)
\(884\) −20.0000 −0.672673
\(885\) 0 0
\(886\) 32.0000 1.07506
\(887\) 6.00000 0.201460 0.100730 0.994914i \(-0.467882\pi\)
0.100730 + 0.994914i \(0.467882\pi\)
\(888\) −9.00000 −0.302020
\(889\) 8.00000 0.268311
\(890\) 0 0
\(891\) −2.00000 −0.0670025
\(892\) 9.00000 0.301342
\(893\) 0 0
\(894\) 6.00000 0.200670
\(895\) 0 0
\(896\) 3.00000 0.100223
\(897\) 20.0000 0.667781
\(898\) −12.0000 −0.400445
\(899\) 24.0000 0.800445
\(900\) 5.00000 0.166667
\(901\) 16.0000 0.533037
\(902\) 24.0000 0.799113
\(903\) −1.00000 −0.0332779
\(904\) −6.00000 −0.199557
\(905\) 0 0
\(906\) 12.0000 0.398673
\(907\) 44.0000 1.46100 0.730498 0.682915i \(-0.239288\pi\)
0.730498 + 0.682915i \(0.239288\pi\)
\(908\) −24.0000 −0.796468
\(909\) 14.0000 0.464351
\(910\) 0 0
\(911\) 12.0000 0.397578 0.198789 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 11.0000 0.363848
\(915\) 0 0
\(916\) −13.0000 −0.429532
\(917\) −14.0000 −0.462321
\(918\) 4.00000 0.132020
\(919\) −25.0000 −0.824674 −0.412337 0.911031i \(-0.635287\pi\)
−0.412337 + 0.911031i \(0.635287\pi\)
\(920\) 0 0
\(921\) 12.0000 0.395413
\(922\) −30.0000 −0.987997
\(923\) 30.0000 0.987462
\(924\) 2.00000 0.0657952
\(925\) 15.0000 0.493197
\(926\) −23.0000 −0.755827
\(927\) −13.0000 −0.426976
\(928\) −40.0000 −1.31306
\(929\) 28.0000 0.918650 0.459325 0.888268i \(-0.348091\pi\)
0.459325 + 0.888268i \(0.348091\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 6.00000 0.196537
\(933\) 18.0000 0.589294
\(934\) 8.00000 0.261768
\(935\) 0 0
\(936\) −15.0000 −0.490290
\(937\) −9.00000 −0.294017 −0.147009 0.989135i \(-0.546964\pi\)
−0.147009 + 0.989135i \(0.546964\pi\)
\(938\) 11.0000 0.359163
\(939\) 22.0000 0.717943
\(940\) 0 0
\(941\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) −3.00000 −0.0977453
\(943\) −48.0000 −1.56310
\(944\) 10.0000 0.325472
\(945\) 0 0
\(946\) −2.00000 −0.0650256
\(947\) 50.0000 1.62478 0.812391 0.583113i \(-0.198166\pi\)
0.812391 + 0.583113i \(0.198166\pi\)
\(948\) 1.00000 0.0324785
\(949\) 55.0000 1.78538
\(950\) 0 0
\(951\) 4.00000 0.129709
\(952\) −12.0000 −0.388922
\(953\) −34.0000 −1.10137 −0.550684 0.834714i \(-0.685633\pi\)
−0.550684 + 0.834714i \(0.685633\pi\)
\(954\) 4.00000 0.129505
\(955\) 0 0
\(956\) 12.0000 0.388108
\(957\) −16.0000 −0.517207
\(958\) −14.0000 −0.452319
\(959\) 18.0000 0.581250
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) −15.0000 −0.483619
\(963\) 18.0000 0.580042
\(964\) −7.00000 −0.225455
\(965\) 0 0
\(966\) 4.00000 0.128698
\(967\) −29.0000 −0.932577 −0.466289 0.884633i \(-0.654409\pi\)
−0.466289 + 0.884633i \(0.654409\pi\)
\(968\) −21.0000 −0.674966
\(969\) 0 0
\(970\) 0 0
\(971\) 42.0000 1.34784 0.673922 0.738802i \(-0.264608\pi\)
0.673922 + 0.738802i \(0.264608\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 5.00000 0.160293
\(974\) −16.0000 −0.512673
\(975\) 25.0000 0.800641
\(976\) 13.0000 0.416120
\(977\) −54.0000 −1.72761 −0.863807 0.503824i \(-0.831926\pi\)
−0.863807 + 0.503824i \(0.831926\pi\)
\(978\) −3.00000 −0.0959294
\(979\) −12.0000 −0.383522
\(980\) 0 0
\(981\) 2.00000 0.0638551
\(982\) −6.00000 −0.191468
\(983\) −10.0000 −0.318950 −0.159475 0.987202i \(-0.550980\pi\)
−0.159475 + 0.987202i \(0.550980\pi\)
\(984\) 36.0000 1.14764
\(985\) 0 0
\(986\) 32.0000 1.01909
\(987\) −6.00000 −0.190982
\(988\) 0 0
\(989\) 4.00000 0.127193
\(990\) 0 0
\(991\) −47.0000 −1.49300 −0.746502 0.665383i \(-0.768268\pi\)
−0.746502 + 0.665383i \(0.768268\pi\)
\(992\) −15.0000 −0.476250
\(993\) 25.0000 0.793351
\(994\) 6.00000 0.190308
\(995\) 0 0
\(996\) 0 0
\(997\) 7.00000 0.221692 0.110846 0.993838i \(-0.464644\pi\)
0.110846 + 0.993838i \(0.464644\pi\)
\(998\) −29.0000 −0.917979
\(999\) −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 1083.2.a.b.1.1 1
3.2 odd 2 3249.2.a.f.1.1 1
19.8 odd 6 57.2.e.a.7.1 2
19.12 odd 6 57.2.e.a.49.1 yes 2
19.18 odd 2 1083.2.a.c.1.1 1
57.8 even 6 171.2.f.a.64.1 2
57.50 even 6 171.2.f.a.163.1 2
57.56 even 2 3249.2.a.c.1.1 1
76.27 even 6 912.2.q.a.577.1 2
76.31 even 6 912.2.q.a.49.1 2
228.107 odd 6 2736.2.s.j.1873.1 2
228.179 odd 6 2736.2.s.j.577.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
57.2.e.a.7.1 2 19.8 odd 6
57.2.e.a.49.1 yes 2 19.12 odd 6
171.2.f.a.64.1 2 57.8 even 6
171.2.f.a.163.1 2 57.50 even 6
912.2.q.a.49.1 2 76.31 even 6
912.2.q.a.577.1 2 76.27 even 6
1083.2.a.b.1.1 1 1.1 even 1 trivial
1083.2.a.c.1.1 1 19.18 odd 2
2736.2.s.j.577.1 2 228.179 odd 6
2736.2.s.j.1873.1 2 228.107 odd 6
3249.2.a.c.1.1 1 57.56 even 2
3249.2.a.f.1.1 1 3.2 odd 2