Properties

Label 1001.2.a.a.1.1
Level $1001$
Weight $2$
Character 1001.1
Self dual yes
Analytic conductor $7.993$
Analytic rank $2$
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] = [1001,2,Mod(1,1001)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1001, 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("1001.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1001 = 7 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1001.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: \(7.99302524233\)
Analytic rank: \(2\)
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 \(\chi\) \(=\) 1001.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} -3.00000 q^{3} +2.00000 q^{4} -3.00000 q^{5} +6.00000 q^{6} -1.00000 q^{7} +6.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} -3.00000 q^{3} +2.00000 q^{4} -3.00000 q^{5} +6.00000 q^{6} -1.00000 q^{7} +6.00000 q^{9} +6.00000 q^{10} +1.00000 q^{11} -6.00000 q^{12} -1.00000 q^{13} +2.00000 q^{14} +9.00000 q^{15} -4.00000 q^{16} -8.00000 q^{17} -12.0000 q^{18} -4.00000 q^{19} -6.00000 q^{20} +3.00000 q^{21} -2.00000 q^{22} -9.00000 q^{23} +4.00000 q^{25} +2.00000 q^{26} -9.00000 q^{27} -2.00000 q^{28} -8.00000 q^{29} -18.0000 q^{30} +3.00000 q^{31} +8.00000 q^{32} -3.00000 q^{33} +16.0000 q^{34} +3.00000 q^{35} +12.0000 q^{36} -7.00000 q^{37} +8.00000 q^{38} +3.00000 q^{39} -6.00000 q^{42} +2.00000 q^{43} +2.00000 q^{44} -18.0000 q^{45} +18.0000 q^{46} -8.00000 q^{47} +12.0000 q^{48} +1.00000 q^{49} -8.00000 q^{50} +24.0000 q^{51} -2.00000 q^{52} +6.00000 q^{53} +18.0000 q^{54} -3.00000 q^{55} +12.0000 q^{57} +16.0000 q^{58} -7.00000 q^{59} +18.0000 q^{60} -4.00000 q^{61} -6.00000 q^{62} -6.00000 q^{63} -8.00000 q^{64} +3.00000 q^{65} +6.00000 q^{66} -9.00000 q^{67} -16.0000 q^{68} +27.0000 q^{69} -6.00000 q^{70} -5.00000 q^{71} -10.0000 q^{73} +14.0000 q^{74} -12.0000 q^{75} -8.00000 q^{76} -1.00000 q^{77} -6.00000 q^{78} +12.0000 q^{79} +12.0000 q^{80} +9.00000 q^{81} -6.00000 q^{83} +6.00000 q^{84} +24.0000 q^{85} -4.00000 q^{86} +24.0000 q^{87} -9.00000 q^{89} +36.0000 q^{90} +1.00000 q^{91} -18.0000 q^{92} -9.00000 q^{93} +16.0000 q^{94} +12.0000 q^{95} -24.0000 q^{96} +1.00000 q^{97} -2.00000 q^{98} +6.00000 q^{99} +O(q^{100})\)

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 \(0.750000\pi\)
\(3\) −3.00000 −1.73205 −0.866025 0.500000i \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(4\) 2.00000 1.00000
\(5\) −3.00000 −1.34164 −0.670820 0.741620i \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) 6.00000 2.44949
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 6.00000 2.00000
\(10\) 6.00000 1.89737
\(11\) 1.00000 0.301511
\(12\) −6.00000 −1.73205
\(13\) −1.00000 −0.277350
\(14\) 2.00000 0.534522
\(15\) 9.00000 2.32379
\(16\) −4.00000 −1.00000
\(17\) −8.00000 −1.94029 −0.970143 0.242536i \(-0.922021\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) −12.0000 −2.82843
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) −6.00000 −1.34164
\(21\) 3.00000 0.654654
\(22\) −2.00000 −0.426401
\(23\) −9.00000 −1.87663 −0.938315 0.345782i \(-0.887614\pi\)
−0.938315 + 0.345782i \(0.887614\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 2.00000 0.392232
\(27\) −9.00000 −1.73205
\(28\) −2.00000 −0.377964
\(29\) −8.00000 −1.48556 −0.742781 0.669534i \(-0.766494\pi\)
−0.742781 + 0.669534i \(0.766494\pi\)
\(30\) −18.0000 −3.28634
\(31\) 3.00000 0.538816 0.269408 0.963026i \(-0.413172\pi\)
0.269408 + 0.963026i \(0.413172\pi\)
\(32\) 8.00000 1.41421
\(33\) −3.00000 −0.522233
\(34\) 16.0000 2.74398
\(35\) 3.00000 0.507093
\(36\) 12.0000 2.00000
\(37\) −7.00000 −1.15079 −0.575396 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) 8.00000 1.29777
\(39\) 3.00000 0.480384
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) −6.00000 −0.925820
\(43\) 2.00000 0.304997 0.152499 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) 2.00000 0.301511
\(45\) −18.0000 −2.68328
\(46\) 18.0000 2.65396
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 12.0000 1.73205
\(49\) 1.00000 0.142857
\(50\) −8.00000 −1.13137
\(51\) 24.0000 3.36067
\(52\) −2.00000 −0.277350
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 18.0000 2.44949
\(55\) −3.00000 −0.404520
\(56\) 0 0
\(57\) 12.0000 1.58944
\(58\) 16.0000 2.10090
\(59\) −7.00000 −0.911322 −0.455661 0.890153i \(-0.650597\pi\)
−0.455661 + 0.890153i \(0.650597\pi\)
\(60\) 18.0000 2.32379
\(61\) −4.00000 −0.512148 −0.256074 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) −6.00000 −0.762001
\(63\) −6.00000 −0.755929
\(64\) −8.00000 −1.00000
\(65\) 3.00000 0.372104
\(66\) 6.00000 0.738549
\(67\) −9.00000 −1.09952 −0.549762 0.835321i \(-0.685282\pi\)
−0.549762 + 0.835321i \(0.685282\pi\)
\(68\) −16.0000 −1.94029
\(69\) 27.0000 3.25042
\(70\) −6.00000 −0.717137
\(71\) −5.00000 −0.593391 −0.296695 0.954972i \(-0.595885\pi\)
−0.296695 + 0.954972i \(0.595885\pi\)
\(72\) 0 0
\(73\) −10.0000 −1.17041 −0.585206 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 14.0000 1.62747
\(75\) −12.0000 −1.38564
\(76\) −8.00000 −0.917663
\(77\) −1.00000 −0.113961
\(78\) −6.00000 −0.679366
\(79\) 12.0000 1.35011 0.675053 0.737769i \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) 12.0000 1.34164
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 6.00000 0.654654
\(85\) 24.0000 2.60317
\(86\) −4.00000 −0.431331
\(87\) 24.0000 2.57307
\(88\) 0 0
\(89\) −9.00000 −0.953998 −0.476999 0.878904i \(-0.658275\pi\)
−0.476999 + 0.878904i \(0.658275\pi\)
\(90\) 36.0000 3.79473
\(91\) 1.00000 0.104828
\(92\) −18.0000 −1.87663
\(93\) −9.00000 −0.933257
\(94\) 16.0000 1.65027
\(95\) 12.0000 1.23117
\(96\) −24.0000 −2.44949
\(97\) 1.00000 0.101535 0.0507673 0.998711i \(-0.483833\pi\)
0.0507673 + 0.998711i \(0.483833\pi\)
\(98\) −2.00000 −0.202031
\(99\) 6.00000 0.603023
\(100\) 8.00000 0.800000
\(101\) −8.00000 −0.796030 −0.398015 0.917379i \(-0.630301\pi\)
−0.398015 + 0.917379i \(0.630301\pi\)
\(102\) −48.0000 −4.75271
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 0 0
\(105\) −9.00000 −0.878310
\(106\) −12.0000 −1.16554
\(107\) 8.00000 0.773389 0.386695 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) −18.0000 −1.73205
\(109\) −14.0000 −1.34096 −0.670478 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 6.00000 0.572078
\(111\) 21.0000 1.99323
\(112\) 4.00000 0.377964
\(113\) 9.00000 0.846649 0.423324 0.905978i \(-0.360863\pi\)
0.423324 + 0.905978i \(0.360863\pi\)
\(114\) −24.0000 −2.24781
\(115\) 27.0000 2.51776
\(116\) −16.0000 −1.48556
\(117\) −6.00000 −0.554700
\(118\) 14.0000 1.28880
\(119\) 8.00000 0.733359
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 8.00000 0.724286
\(123\) 0 0
\(124\) 6.00000 0.538816
\(125\) 3.00000 0.268328
\(126\) 12.0000 1.06904
\(127\) 14.0000 1.24230 0.621150 0.783692i \(-0.286666\pi\)
0.621150 + 0.783692i \(0.286666\pi\)
\(128\) 0 0
\(129\) −6.00000 −0.528271
\(130\) −6.00000 −0.526235
\(131\) −10.0000 −0.873704 −0.436852 0.899533i \(-0.643907\pi\)
−0.436852 + 0.899533i \(0.643907\pi\)
\(132\) −6.00000 −0.522233
\(133\) 4.00000 0.346844
\(134\) 18.0000 1.55496
\(135\) 27.0000 2.32379
\(136\) 0 0
\(137\) −17.0000 −1.45241 −0.726204 0.687479i \(-0.758717\pi\)
−0.726204 + 0.687479i \(0.758717\pi\)
\(138\) −54.0000 −4.59679
\(139\) 6.00000 0.508913 0.254457 0.967084i \(-0.418103\pi\)
0.254457 + 0.967084i \(0.418103\pi\)
\(140\) 6.00000 0.507093
\(141\) 24.0000 2.02116
\(142\) 10.0000 0.839181
\(143\) −1.00000 −0.0836242
\(144\) −24.0000 −2.00000
\(145\) 24.0000 1.99309
\(146\) 20.0000 1.65521
\(147\) −3.00000 −0.247436
\(148\) −14.0000 −1.15079
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 24.0000 1.95959
\(151\) 12.0000 0.976546 0.488273 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) 0 0
\(153\) −48.0000 −3.88057
\(154\) 2.00000 0.161165
\(155\) −9.00000 −0.722897
\(156\) 6.00000 0.480384
\(157\) 7.00000 0.558661 0.279330 0.960195i \(-0.409888\pi\)
0.279330 + 0.960195i \(0.409888\pi\)
\(158\) −24.0000 −1.90934
\(159\) −18.0000 −1.42749
\(160\) −24.0000 −1.89737
\(161\) 9.00000 0.709299
\(162\) −18.0000 −1.41421
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 0 0
\(165\) 9.00000 0.700649
\(166\) 12.0000 0.931381
\(167\) 14.0000 1.08335 0.541676 0.840587i \(-0.317790\pi\)
0.541676 + 0.840587i \(0.317790\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) −48.0000 −3.68143
\(171\) −24.0000 −1.83533
\(172\) 4.00000 0.304997
\(173\) −14.0000 −1.06440 −0.532200 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) −48.0000 −3.63887
\(175\) −4.00000 −0.302372
\(176\) −4.00000 −0.301511
\(177\) 21.0000 1.57846
\(178\) 18.0000 1.34916
\(179\) 5.00000 0.373718 0.186859 0.982387i \(-0.440169\pi\)
0.186859 + 0.982387i \(0.440169\pi\)
\(180\) −36.0000 −2.68328
\(181\) −19.0000 −1.41226 −0.706129 0.708083i \(-0.749560\pi\)
−0.706129 + 0.708083i \(0.749560\pi\)
\(182\) −2.00000 −0.148250
\(183\) 12.0000 0.887066
\(184\) 0 0
\(185\) 21.0000 1.54395
\(186\) 18.0000 1.31982
\(187\) −8.00000 −0.585018
\(188\) −16.0000 −1.16692
\(189\) 9.00000 0.654654
\(190\) −24.0000 −1.74114
\(191\) 25.0000 1.80894 0.904468 0.426541i \(-0.140268\pi\)
0.904468 + 0.426541i \(0.140268\pi\)
\(192\) 24.0000 1.73205
\(193\) 10.0000 0.719816 0.359908 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) −2.00000 −0.143592
\(195\) −9.00000 −0.644503
\(196\) 2.00000 0.142857
\(197\) −22.0000 −1.56744 −0.783718 0.621117i \(-0.786679\pi\)
−0.783718 + 0.621117i \(0.786679\pi\)
\(198\) −12.0000 −0.852803
\(199\) 28.0000 1.98487 0.992434 0.122782i \(-0.0391815\pi\)
0.992434 + 0.122782i \(0.0391815\pi\)
\(200\) 0 0
\(201\) 27.0000 1.90443
\(202\) 16.0000 1.12576
\(203\) 8.00000 0.561490
\(204\) 48.0000 3.36067
\(205\) 0 0
\(206\) −16.0000 −1.11477
\(207\) −54.0000 −3.75326
\(208\) 4.00000 0.277350
\(209\) −4.00000 −0.276686
\(210\) 18.0000 1.24212
\(211\) −26.0000 −1.78991 −0.894957 0.446153i \(-0.852794\pi\)
−0.894957 + 0.446153i \(0.852794\pi\)
\(212\) 12.0000 0.824163
\(213\) 15.0000 1.02778
\(214\) −16.0000 −1.09374
\(215\) −6.00000 −0.409197
\(216\) 0 0
\(217\) −3.00000 −0.203653
\(218\) 28.0000 1.89640
\(219\) 30.0000 2.02721
\(220\) −6.00000 −0.404520
\(221\) 8.00000 0.538138
\(222\) −42.0000 −2.81886
\(223\) −9.00000 −0.602685 −0.301342 0.953516i \(-0.597435\pi\)
−0.301342 + 0.953516i \(0.597435\pi\)
\(224\) −8.00000 −0.534522
\(225\) 24.0000 1.60000
\(226\) −18.0000 −1.19734
\(227\) −22.0000 −1.46019 −0.730096 0.683345i \(-0.760525\pi\)
−0.730096 + 0.683345i \(0.760525\pi\)
\(228\) 24.0000 1.58944
\(229\) −1.00000 −0.0660819 −0.0330409 0.999454i \(-0.510519\pi\)
−0.0330409 + 0.999454i \(0.510519\pi\)
\(230\) −54.0000 −3.56065
\(231\) 3.00000 0.197386
\(232\) 0 0
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 12.0000 0.784465
\(235\) 24.0000 1.56559
\(236\) −14.0000 −0.911322
\(237\) −36.0000 −2.33845
\(238\) −16.0000 −1.03713
\(239\) −4.00000 −0.258738 −0.129369 0.991596i \(-0.541295\pi\)
−0.129369 + 0.991596i \(0.541295\pi\)
\(240\) −36.0000 −2.32379
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) −2.00000 −0.128565
\(243\) 0 0
\(244\) −8.00000 −0.512148
\(245\) −3.00000 −0.191663
\(246\) 0 0
\(247\) 4.00000 0.254514
\(248\) 0 0
\(249\) 18.0000 1.14070
\(250\) −6.00000 −0.379473
\(251\) −5.00000 −0.315597 −0.157799 0.987471i \(-0.550440\pi\)
−0.157799 + 0.987471i \(0.550440\pi\)
\(252\) −12.0000 −0.755929
\(253\) −9.00000 −0.565825
\(254\) −28.0000 −1.75688
\(255\) −72.0000 −4.50881
\(256\) 16.0000 1.00000
\(257\) −2.00000 −0.124757 −0.0623783 0.998053i \(-0.519869\pi\)
−0.0623783 + 0.998053i \(0.519869\pi\)
\(258\) 12.0000 0.747087
\(259\) 7.00000 0.434959
\(260\) 6.00000 0.372104
\(261\) −48.0000 −2.97113
\(262\) 20.0000 1.23560
\(263\) −6.00000 −0.369976 −0.184988 0.982741i \(-0.559225\pi\)
−0.184988 + 0.982741i \(0.559225\pi\)
\(264\) 0 0
\(265\) −18.0000 −1.10573
\(266\) −8.00000 −0.490511
\(267\) 27.0000 1.65237
\(268\) −18.0000 −1.09952
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) −54.0000 −3.28634
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 32.0000 1.94029
\(273\) −3.00000 −0.181568
\(274\) 34.0000 2.05402
\(275\) 4.00000 0.241209
\(276\) 54.0000 3.25042
\(277\) −32.0000 −1.92269 −0.961347 0.275340i \(-0.911209\pi\)
−0.961347 + 0.275340i \(0.911209\pi\)
\(278\) −12.0000 −0.719712
\(279\) 18.0000 1.07763
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) −48.0000 −2.85836
\(283\) −2.00000 −0.118888 −0.0594438 0.998232i \(-0.518933\pi\)
−0.0594438 + 0.998232i \(0.518933\pi\)
\(284\) −10.0000 −0.593391
\(285\) −36.0000 −2.13246
\(286\) 2.00000 0.118262
\(287\) 0 0
\(288\) 48.0000 2.82843
\(289\) 47.0000 2.76471
\(290\) −48.0000 −2.81866
\(291\) −3.00000 −0.175863
\(292\) −20.0000 −1.17041
\(293\) 32.0000 1.86946 0.934730 0.355359i \(-0.115641\pi\)
0.934730 + 0.355359i \(0.115641\pi\)
\(294\) 6.00000 0.349927
\(295\) 21.0000 1.22267
\(296\) 0 0
\(297\) −9.00000 −0.522233
\(298\) 12.0000 0.695141
\(299\) 9.00000 0.520483
\(300\) −24.0000 −1.38564
\(301\) −2.00000 −0.115278
\(302\) −24.0000 −1.38104
\(303\) 24.0000 1.37876
\(304\) 16.0000 0.917663
\(305\) 12.0000 0.687118
\(306\) 96.0000 5.48795
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) −2.00000 −0.113961
\(309\) −24.0000 −1.36531
\(310\) 18.0000 1.02233
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 0 0
\(313\) −23.0000 −1.30004 −0.650018 0.759918i \(-0.725239\pi\)
−0.650018 + 0.759918i \(0.725239\pi\)
\(314\) −14.0000 −0.790066
\(315\) 18.0000 1.01419
\(316\) 24.0000 1.35011
\(317\) 3.00000 0.168497 0.0842484 0.996445i \(-0.473151\pi\)
0.0842484 + 0.996445i \(0.473151\pi\)
\(318\) 36.0000 2.01878
\(319\) −8.00000 −0.447914
\(320\) 24.0000 1.34164
\(321\) −24.0000 −1.33955
\(322\) −18.0000 −1.00310
\(323\) 32.0000 1.78053
\(324\) 18.0000 1.00000
\(325\) −4.00000 −0.221880
\(326\) 8.00000 0.443079
\(327\) 42.0000 2.32261
\(328\) 0 0
\(329\) 8.00000 0.441054
\(330\) −18.0000 −0.990867
\(331\) 1.00000 0.0549650 0.0274825 0.999622i \(-0.491251\pi\)
0.0274825 + 0.999622i \(0.491251\pi\)
\(332\) −12.0000 −0.658586
\(333\) −42.0000 −2.30159
\(334\) −28.0000 −1.53209
\(335\) 27.0000 1.47517
\(336\) −12.0000 −0.654654
\(337\) −4.00000 −0.217894 −0.108947 0.994048i \(-0.534748\pi\)
−0.108947 + 0.994048i \(0.534748\pi\)
\(338\) −2.00000 −0.108786
\(339\) −27.0000 −1.46644
\(340\) 48.0000 2.60317
\(341\) 3.00000 0.162459
\(342\) 48.0000 2.59554
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −81.0000 −4.36089
\(346\) 28.0000 1.50529
\(347\) 4.00000 0.214731 0.107366 0.994220i \(-0.465758\pi\)
0.107366 + 0.994220i \(0.465758\pi\)
\(348\) 48.0000 2.57307
\(349\) −16.0000 −0.856460 −0.428230 0.903670i \(-0.640863\pi\)
−0.428230 + 0.903670i \(0.640863\pi\)
\(350\) 8.00000 0.427618
\(351\) 9.00000 0.480384
\(352\) 8.00000 0.426401
\(353\) 11.0000 0.585471 0.292735 0.956193i \(-0.405434\pi\)
0.292735 + 0.956193i \(0.405434\pi\)
\(354\) −42.0000 −2.23227
\(355\) 15.0000 0.796117
\(356\) −18.0000 −0.953998
\(357\) −24.0000 −1.27021
\(358\) −10.0000 −0.528516
\(359\) −10.0000 −0.527780 −0.263890 0.964553i \(-0.585006\pi\)
−0.263890 + 0.964553i \(0.585006\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 38.0000 1.99724
\(363\) −3.00000 −0.157459
\(364\) 2.00000 0.104828
\(365\) 30.0000 1.57027
\(366\) −24.0000 −1.25450
\(367\) −7.00000 −0.365397 −0.182699 0.983169i \(-0.558483\pi\)
−0.182699 + 0.983169i \(0.558483\pi\)
\(368\) 36.0000 1.87663
\(369\) 0 0
\(370\) −42.0000 −2.18348
\(371\) −6.00000 −0.311504
\(372\) −18.0000 −0.933257
\(373\) 12.0000 0.621336 0.310668 0.950518i \(-0.399447\pi\)
0.310668 + 0.950518i \(0.399447\pi\)
\(374\) 16.0000 0.827340
\(375\) −9.00000 −0.464758
\(376\) 0 0
\(377\) 8.00000 0.412021
\(378\) −18.0000 −0.925820
\(379\) −27.0000 −1.38690 −0.693448 0.720506i \(-0.743909\pi\)
−0.693448 + 0.720506i \(0.743909\pi\)
\(380\) 24.0000 1.23117
\(381\) −42.0000 −2.15173
\(382\) −50.0000 −2.55822
\(383\) 27.0000 1.37964 0.689818 0.723983i \(-0.257691\pi\)
0.689818 + 0.723983i \(0.257691\pi\)
\(384\) 0 0
\(385\) 3.00000 0.152894
\(386\) −20.0000 −1.01797
\(387\) 12.0000 0.609994
\(388\) 2.00000 0.101535
\(389\) 21.0000 1.06474 0.532371 0.846511i \(-0.321301\pi\)
0.532371 + 0.846511i \(0.321301\pi\)
\(390\) 18.0000 0.911465
\(391\) 72.0000 3.64120
\(392\) 0 0
\(393\) 30.0000 1.51330
\(394\) 44.0000 2.21669
\(395\) −36.0000 −1.81136
\(396\) 12.0000 0.603023
\(397\) −10.0000 −0.501886 −0.250943 0.968002i \(-0.580741\pi\)
−0.250943 + 0.968002i \(0.580741\pi\)
\(398\) −56.0000 −2.80703
\(399\) −12.0000 −0.600751
\(400\) −16.0000 −0.800000
\(401\) 34.0000 1.69788 0.848939 0.528490i \(-0.177242\pi\)
0.848939 + 0.528490i \(0.177242\pi\)
\(402\) −54.0000 −2.69328
\(403\) −3.00000 −0.149441
\(404\) −16.0000 −0.796030
\(405\) −27.0000 −1.34164
\(406\) −16.0000 −0.794067
\(407\) −7.00000 −0.346977
\(408\) 0 0
\(409\) −22.0000 −1.08783 −0.543915 0.839140i \(-0.683059\pi\)
−0.543915 + 0.839140i \(0.683059\pi\)
\(410\) 0 0
\(411\) 51.0000 2.51564
\(412\) 16.0000 0.788263
\(413\) 7.00000 0.344447
\(414\) 108.000 5.30791
\(415\) 18.0000 0.883585
\(416\) −8.00000 −0.392232
\(417\) −18.0000 −0.881464
\(418\) 8.00000 0.391293
\(419\) −4.00000 −0.195413 −0.0977064 0.995215i \(-0.531151\pi\)
−0.0977064 + 0.995215i \(0.531151\pi\)
\(420\) −18.0000 −0.878310
\(421\) −30.0000 −1.46211 −0.731055 0.682318i \(-0.760972\pi\)
−0.731055 + 0.682318i \(0.760972\pi\)
\(422\) 52.0000 2.53132
\(423\) −48.0000 −2.33384
\(424\) 0 0
\(425\) −32.0000 −1.55223
\(426\) −30.0000 −1.45350
\(427\) 4.00000 0.193574
\(428\) 16.0000 0.773389
\(429\) 3.00000 0.144841
\(430\) 12.0000 0.578691
\(431\) −30.0000 −1.44505 −0.722525 0.691345i \(-0.757018\pi\)
−0.722525 + 0.691345i \(0.757018\pi\)
\(432\) 36.0000 1.73205
\(433\) −9.00000 −0.432512 −0.216256 0.976337i \(-0.569385\pi\)
−0.216256 + 0.976337i \(0.569385\pi\)
\(434\) 6.00000 0.288009
\(435\) −72.0000 −3.45214
\(436\) −28.0000 −1.34096
\(437\) 36.0000 1.72211
\(438\) −60.0000 −2.86691
\(439\) −28.0000 −1.33637 −0.668184 0.743996i \(-0.732928\pi\)
−0.668184 + 0.743996i \(0.732928\pi\)
\(440\) 0 0
\(441\) 6.00000 0.285714
\(442\) −16.0000 −0.761042
\(443\) −11.0000 −0.522626 −0.261313 0.965254i \(-0.584155\pi\)
−0.261313 + 0.965254i \(0.584155\pi\)
\(444\) 42.0000 1.99323
\(445\) 27.0000 1.27992
\(446\) 18.0000 0.852325
\(447\) 18.0000 0.851371
\(448\) 8.00000 0.377964
\(449\) −3.00000 −0.141579 −0.0707894 0.997491i \(-0.522552\pi\)
−0.0707894 + 0.997491i \(0.522552\pi\)
\(450\) −48.0000 −2.26274
\(451\) 0 0
\(452\) 18.0000 0.846649
\(453\) −36.0000 −1.69143
\(454\) 44.0000 2.06502
\(455\) −3.00000 −0.140642
\(456\) 0 0
\(457\) −36.0000 −1.68401 −0.842004 0.539471i \(-0.818624\pi\)
−0.842004 + 0.539471i \(0.818624\pi\)
\(458\) 2.00000 0.0934539
\(459\) 72.0000 3.36067
\(460\) 54.0000 2.51776
\(461\) 8.00000 0.372597 0.186299 0.982493i \(-0.440351\pi\)
0.186299 + 0.982493i \(0.440351\pi\)
\(462\) −6.00000 −0.279145
\(463\) 7.00000 0.325318 0.162659 0.986682i \(-0.447993\pi\)
0.162659 + 0.986682i \(0.447993\pi\)
\(464\) 32.0000 1.48556
\(465\) 27.0000 1.25210
\(466\) 12.0000 0.555889
\(467\) 11.0000 0.509019 0.254510 0.967070i \(-0.418086\pi\)
0.254510 + 0.967070i \(0.418086\pi\)
\(468\) −12.0000 −0.554700
\(469\) 9.00000 0.415581
\(470\) −48.0000 −2.21407
\(471\) −21.0000 −0.967629
\(472\) 0 0
\(473\) 2.00000 0.0919601
\(474\) 72.0000 3.30707
\(475\) −16.0000 −0.734130
\(476\) 16.0000 0.733359
\(477\) 36.0000 1.64833
\(478\) 8.00000 0.365911
\(479\) 4.00000 0.182765 0.0913823 0.995816i \(-0.470871\pi\)
0.0913823 + 0.995816i \(0.470871\pi\)
\(480\) 72.0000 3.28634
\(481\) 7.00000 0.319173
\(482\) −20.0000 −0.910975
\(483\) −27.0000 −1.22854
\(484\) 2.00000 0.0909091
\(485\) −3.00000 −0.136223
\(486\) 0 0
\(487\) 1.00000 0.0453143 0.0226572 0.999743i \(-0.492787\pi\)
0.0226572 + 0.999743i \(0.492787\pi\)
\(488\) 0 0
\(489\) 12.0000 0.542659
\(490\) 6.00000 0.271052
\(491\) −30.0000 −1.35388 −0.676941 0.736038i \(-0.736695\pi\)
−0.676941 + 0.736038i \(0.736695\pi\)
\(492\) 0 0
\(493\) 64.0000 2.88242
\(494\) −8.00000 −0.359937
\(495\) −18.0000 −0.809040
\(496\) −12.0000 −0.538816
\(497\) 5.00000 0.224281
\(498\) −36.0000 −1.61320
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 6.00000 0.268328
\(501\) −42.0000 −1.87642
\(502\) 10.0000 0.446322
\(503\) −4.00000 −0.178351 −0.0891756 0.996016i \(-0.528423\pi\)
−0.0891756 + 0.996016i \(0.528423\pi\)
\(504\) 0 0
\(505\) 24.0000 1.06799
\(506\) 18.0000 0.800198
\(507\) −3.00000 −0.133235
\(508\) 28.0000 1.24230
\(509\) 35.0000 1.55135 0.775674 0.631134i \(-0.217410\pi\)
0.775674 + 0.631134i \(0.217410\pi\)
\(510\) 144.000 6.37643
\(511\) 10.0000 0.442374
\(512\) −32.0000 −1.41421
\(513\) 36.0000 1.58944
\(514\) 4.00000 0.176432
\(515\) −24.0000 −1.05757
\(516\) −12.0000 −0.528271
\(517\) −8.00000 −0.351840
\(518\) −14.0000 −0.615125
\(519\) 42.0000 1.84360
\(520\) 0 0
\(521\) −29.0000 −1.27051 −0.635257 0.772301i \(-0.719106\pi\)
−0.635257 + 0.772301i \(0.719106\pi\)
\(522\) 96.0000 4.20181
\(523\) 34.0000 1.48672 0.743358 0.668894i \(-0.233232\pi\)
0.743358 + 0.668894i \(0.233232\pi\)
\(524\) −20.0000 −0.873704
\(525\) 12.0000 0.523723
\(526\) 12.0000 0.523225
\(527\) −24.0000 −1.04546
\(528\) 12.0000 0.522233
\(529\) 58.0000 2.52174
\(530\) 36.0000 1.56374
\(531\) −42.0000 −1.82264
\(532\) 8.00000 0.346844
\(533\) 0 0
\(534\) −54.0000 −2.33681
\(535\) −24.0000 −1.03761
\(536\) 0 0
\(537\) −15.0000 −0.647298
\(538\) −12.0000 −0.517357
\(539\) 1.00000 0.0430730
\(540\) 54.0000 2.32379
\(541\) 14.0000 0.601907 0.300954 0.953639i \(-0.402695\pi\)
0.300954 + 0.953639i \(0.402695\pi\)
\(542\) −16.0000 −0.687259
\(543\) 57.0000 2.44610
\(544\) −64.0000 −2.74398
\(545\) 42.0000 1.79908
\(546\) 6.00000 0.256776
\(547\) 8.00000 0.342055 0.171028 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) −34.0000 −1.45241
\(549\) −24.0000 −1.02430
\(550\) −8.00000 −0.341121
\(551\) 32.0000 1.36325
\(552\) 0 0
\(553\) −12.0000 −0.510292
\(554\) 64.0000 2.71910
\(555\) −63.0000 −2.67420
\(556\) 12.0000 0.508913
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) −36.0000 −1.52400
\(559\) −2.00000 −0.0845910
\(560\) −12.0000 −0.507093
\(561\) 24.0000 1.01328
\(562\) 0 0
\(563\) 10.0000 0.421450 0.210725 0.977545i \(-0.432418\pi\)
0.210725 + 0.977545i \(0.432418\pi\)
\(564\) 48.0000 2.02116
\(565\) −27.0000 −1.13590
\(566\) 4.00000 0.168133
\(567\) −9.00000 −0.377964
\(568\) 0 0
\(569\) −14.0000 −0.586911 −0.293455 0.955973i \(-0.594805\pi\)
−0.293455 + 0.955973i \(0.594805\pi\)
\(570\) 72.0000 3.01575
\(571\) −8.00000 −0.334790 −0.167395 0.985890i \(-0.553535\pi\)
−0.167395 + 0.985890i \(0.553535\pi\)
\(572\) −2.00000 −0.0836242
\(573\) −75.0000 −3.13317
\(574\) 0 0
\(575\) −36.0000 −1.50130
\(576\) −48.0000 −2.00000
\(577\) −19.0000 −0.790980 −0.395490 0.918470i \(-0.629425\pi\)
−0.395490 + 0.918470i \(0.629425\pi\)
\(578\) −94.0000 −3.90988
\(579\) −30.0000 −1.24676
\(580\) 48.0000 1.99309
\(581\) 6.00000 0.248922
\(582\) 6.00000 0.248708
\(583\) 6.00000 0.248495
\(584\) 0 0
\(585\) 18.0000 0.744208
\(586\) −64.0000 −2.64382
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) −6.00000 −0.247436
\(589\) −12.0000 −0.494451
\(590\) −42.0000 −1.72911
\(591\) 66.0000 2.71488
\(592\) 28.0000 1.15079
\(593\) 18.0000 0.739171 0.369586 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) 18.0000 0.738549
\(595\) −24.0000 −0.983904
\(596\) −12.0000 −0.491539
\(597\) −84.0000 −3.43789
\(598\) −18.0000 −0.736075
\(599\) −40.0000 −1.63436 −0.817178 0.576386i \(-0.804463\pi\)
−0.817178 + 0.576386i \(0.804463\pi\)
\(600\) 0 0
\(601\) −28.0000 −1.14214 −0.571072 0.820900i \(-0.693472\pi\)
−0.571072 + 0.820900i \(0.693472\pi\)
\(602\) 4.00000 0.163028
\(603\) −54.0000 −2.19905
\(604\) 24.0000 0.976546
\(605\) −3.00000 −0.121967
\(606\) −48.0000 −1.94987
\(607\) −34.0000 −1.38002 −0.690009 0.723801i \(-0.742393\pi\)
−0.690009 + 0.723801i \(0.742393\pi\)
\(608\) −32.0000 −1.29777
\(609\) −24.0000 −0.972529
\(610\) −24.0000 −0.971732
\(611\) 8.00000 0.323645
\(612\) −96.0000 −3.88057
\(613\) −22.0000 −0.888572 −0.444286 0.895885i \(-0.646543\pi\)
−0.444286 + 0.895885i \(0.646543\pi\)
\(614\) −24.0000 −0.968561
\(615\) 0 0
\(616\) 0 0
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) 48.0000 1.93084
\(619\) −13.0000 −0.522514 −0.261257 0.965269i \(-0.584137\pi\)
−0.261257 + 0.965269i \(0.584137\pi\)
\(620\) −18.0000 −0.722897
\(621\) 81.0000 3.25042
\(622\) −48.0000 −1.92462
\(623\) 9.00000 0.360577
\(624\) −12.0000 −0.480384
\(625\) −29.0000 −1.16000
\(626\) 46.0000 1.83853
\(627\) 12.0000 0.479234
\(628\) 14.0000 0.558661
\(629\) 56.0000 2.23287
\(630\) −36.0000 −1.43427
\(631\) −47.0000 −1.87104 −0.935520 0.353273i \(-0.885069\pi\)
−0.935520 + 0.353273i \(0.885069\pi\)
\(632\) 0 0
\(633\) 78.0000 3.10022
\(634\) −6.00000 −0.238290
\(635\) −42.0000 −1.66672
\(636\) −36.0000 −1.42749
\(637\) −1.00000 −0.0396214
\(638\) 16.0000 0.633446
\(639\) −30.0000 −1.18678
\(640\) 0 0
\(641\) −45.0000 −1.77739 −0.888697 0.458496i \(-0.848388\pi\)
−0.888697 + 0.458496i \(0.848388\pi\)
\(642\) 48.0000 1.89441
\(643\) 29.0000 1.14365 0.571824 0.820376i \(-0.306236\pi\)
0.571824 + 0.820376i \(0.306236\pi\)
\(644\) 18.0000 0.709299
\(645\) 18.0000 0.708749
\(646\) −64.0000 −2.51805
\(647\) 39.0000 1.53325 0.766624 0.642096i \(-0.221935\pi\)
0.766624 + 0.642096i \(0.221935\pi\)
\(648\) 0 0
\(649\) −7.00000 −0.274774
\(650\) 8.00000 0.313786
\(651\) 9.00000 0.352738
\(652\) −8.00000 −0.313304
\(653\) 39.0000 1.52619 0.763094 0.646288i \(-0.223679\pi\)
0.763094 + 0.646288i \(0.223679\pi\)
\(654\) −84.0000 −3.28466
\(655\) 30.0000 1.17220
\(656\) 0 0
\(657\) −60.0000 −2.34082
\(658\) −16.0000 −0.623745
\(659\) −28.0000 −1.09073 −0.545363 0.838200i \(-0.683608\pi\)
−0.545363 + 0.838200i \(0.683608\pi\)
\(660\) 18.0000 0.700649
\(661\) 9.00000 0.350059 0.175030 0.984563i \(-0.443998\pi\)
0.175030 + 0.984563i \(0.443998\pi\)
\(662\) −2.00000 −0.0777322
\(663\) −24.0000 −0.932083
\(664\) 0 0
\(665\) −12.0000 −0.465340
\(666\) 84.0000 3.25493
\(667\) 72.0000 2.78785
\(668\) 28.0000 1.08335
\(669\) 27.0000 1.04388
\(670\) −54.0000 −2.08620
\(671\) −4.00000 −0.154418
\(672\) 24.0000 0.925820
\(673\) 10.0000 0.385472 0.192736 0.981251i \(-0.438264\pi\)
0.192736 + 0.981251i \(0.438264\pi\)
\(674\) 8.00000 0.308148
\(675\) −36.0000 −1.38564
\(676\) 2.00000 0.0769231
\(677\) −2.00000 −0.0768662 −0.0384331 0.999261i \(-0.512237\pi\)
−0.0384331 + 0.999261i \(0.512237\pi\)
\(678\) 54.0000 2.07386
\(679\) −1.00000 −0.0383765
\(680\) 0 0
\(681\) 66.0000 2.52913
\(682\) −6.00000 −0.229752
\(683\) 24.0000 0.918334 0.459167 0.888350i \(-0.348148\pi\)
0.459167 + 0.888350i \(0.348148\pi\)
\(684\) −48.0000 −1.83533
\(685\) 51.0000 1.94861
\(686\) 2.00000 0.0763604
\(687\) 3.00000 0.114457
\(688\) −8.00000 −0.304997
\(689\) −6.00000 −0.228582
\(690\) 162.000 6.16723
\(691\) 17.0000 0.646710 0.323355 0.946278i \(-0.395189\pi\)
0.323355 + 0.946278i \(0.395189\pi\)
\(692\) −28.0000 −1.06440
\(693\) −6.00000 −0.227921
\(694\) −8.00000 −0.303676
\(695\) −18.0000 −0.682779
\(696\) 0 0
\(697\) 0 0
\(698\) 32.0000 1.21122
\(699\) 18.0000 0.680823
\(700\) −8.00000 −0.302372
\(701\) −24.0000 −0.906467 −0.453234 0.891392i \(-0.649730\pi\)
−0.453234 + 0.891392i \(0.649730\pi\)
\(702\) −18.0000 −0.679366
\(703\) 28.0000 1.05604
\(704\) −8.00000 −0.301511
\(705\) −72.0000 −2.71168
\(706\) −22.0000 −0.827981
\(707\) 8.00000 0.300871
\(708\) 42.0000 1.57846
\(709\) 5.00000 0.187779 0.0938895 0.995583i \(-0.470070\pi\)
0.0938895 + 0.995583i \(0.470070\pi\)
\(710\) −30.0000 −1.12588
\(711\) 72.0000 2.70021
\(712\) 0 0
\(713\) −27.0000 −1.01116
\(714\) 48.0000 1.79635
\(715\) 3.00000 0.112194
\(716\) 10.0000 0.373718
\(717\) 12.0000 0.448148
\(718\) 20.0000 0.746393
\(719\) −3.00000 −0.111881 −0.0559406 0.998434i \(-0.517816\pi\)
−0.0559406 + 0.998434i \(0.517816\pi\)
\(720\) 72.0000 2.68328
\(721\) −8.00000 −0.297936
\(722\) 6.00000 0.223297
\(723\) −30.0000 −1.11571
\(724\) −38.0000 −1.41226
\(725\) −32.0000 −1.18845
\(726\) 6.00000 0.222681
\(727\) 37.0000 1.37225 0.686127 0.727482i \(-0.259309\pi\)
0.686127 + 0.727482i \(0.259309\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) −60.0000 −2.22070
\(731\) −16.0000 −0.591781
\(732\) 24.0000 0.887066
\(733\) 18.0000 0.664845 0.332423 0.943131i \(-0.392134\pi\)
0.332423 + 0.943131i \(0.392134\pi\)
\(734\) 14.0000 0.516749
\(735\) 9.00000 0.331970
\(736\) −72.0000 −2.65396
\(737\) −9.00000 −0.331519
\(738\) 0 0
\(739\) −14.0000 −0.514998 −0.257499 0.966279i \(-0.582898\pi\)
−0.257499 + 0.966279i \(0.582898\pi\)
\(740\) 42.0000 1.54395
\(741\) −12.0000 −0.440831
\(742\) 12.0000 0.440534
\(743\) −6.00000 −0.220119 −0.110059 0.993925i \(-0.535104\pi\)
−0.110059 + 0.993925i \(0.535104\pi\)
\(744\) 0 0
\(745\) 18.0000 0.659469
\(746\) −24.0000 −0.878702
\(747\) −36.0000 −1.31717
\(748\) −16.0000 −0.585018
\(749\) −8.00000 −0.292314
\(750\) 18.0000 0.657267
\(751\) 25.0000 0.912263 0.456131 0.889912i \(-0.349235\pi\)
0.456131 + 0.889912i \(0.349235\pi\)
\(752\) 32.0000 1.16692
\(753\) 15.0000 0.546630
\(754\) −16.0000 −0.582686
\(755\) −36.0000 −1.31017
\(756\) 18.0000 0.654654
\(757\) 6.00000 0.218074 0.109037 0.994038i \(-0.465223\pi\)
0.109037 + 0.994038i \(0.465223\pi\)
\(758\) 54.0000 1.96137
\(759\) 27.0000 0.980038
\(760\) 0 0
\(761\) −6.00000 −0.217500 −0.108750 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) 84.0000 3.04300
\(763\) 14.0000 0.506834
\(764\) 50.0000 1.80894
\(765\) 144.000 5.20633
\(766\) −54.0000 −1.95110
\(767\) 7.00000 0.252755
\(768\) −48.0000 −1.73205
\(769\) −44.0000 −1.58668 −0.793340 0.608778i \(-0.791660\pi\)
−0.793340 + 0.608778i \(0.791660\pi\)
\(770\) −6.00000 −0.216225
\(771\) 6.00000 0.216085
\(772\) 20.0000 0.719816
\(773\) −6.00000 −0.215805 −0.107903 0.994161i \(-0.534413\pi\)
−0.107903 + 0.994161i \(0.534413\pi\)
\(774\) −24.0000 −0.862662
\(775\) 12.0000 0.431053
\(776\) 0 0
\(777\) −21.0000 −0.753371
\(778\) −42.0000 −1.50577
\(779\) 0 0
\(780\) −18.0000 −0.644503
\(781\) −5.00000 −0.178914
\(782\) −144.000 −5.14943
\(783\) 72.0000 2.57307
\(784\) −4.00000 −0.142857
\(785\) −21.0000 −0.749522
\(786\) −60.0000 −2.14013
\(787\) −50.0000 −1.78231 −0.891154 0.453701i \(-0.850103\pi\)
−0.891154 + 0.453701i \(0.850103\pi\)
\(788\) −44.0000 −1.56744
\(789\) 18.0000 0.640817
\(790\) 72.0000 2.56165
\(791\) −9.00000 −0.320003
\(792\) 0 0
\(793\) 4.00000 0.142044
\(794\) 20.0000 0.709773
\(795\) 54.0000 1.91518
\(796\) 56.0000 1.98487
\(797\) −21.0000 −0.743858 −0.371929 0.928261i \(-0.621304\pi\)
−0.371929 + 0.928261i \(0.621304\pi\)
\(798\) 24.0000 0.849591
\(799\) 64.0000 2.26416
\(800\) 32.0000 1.13137
\(801\) −54.0000 −1.90800
\(802\) −68.0000 −2.40116
\(803\) −10.0000 −0.352892
\(804\) 54.0000 1.90443
\(805\) −27.0000 −0.951625
\(806\) 6.00000 0.211341
\(807\) −18.0000 −0.633630
\(808\) 0 0
\(809\) 2.00000 0.0703163 0.0351581 0.999382i \(-0.488807\pi\)
0.0351581 + 0.999382i \(0.488807\pi\)
\(810\) 54.0000 1.89737
\(811\) −28.0000 −0.983213 −0.491606 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) 16.0000 0.561490
\(813\) −24.0000 −0.841717
\(814\) 14.0000 0.490700
\(815\) 12.0000 0.420342
\(816\) −96.0000 −3.36067
\(817\) −8.00000 −0.279885
\(818\) 44.0000 1.53842
\(819\) 6.00000 0.209657
\(820\) 0 0
\(821\) 48.0000 1.67521 0.837606 0.546275i \(-0.183955\pi\)
0.837606 + 0.546275i \(0.183955\pi\)
\(822\) −102.000 −3.55766
\(823\) 31.0000 1.08059 0.540296 0.841475i \(-0.318312\pi\)
0.540296 + 0.841475i \(0.318312\pi\)
\(824\) 0 0
\(825\) −12.0000 −0.417786
\(826\) −14.0000 −0.487122
\(827\) 22.0000 0.765015 0.382507 0.923952i \(-0.375061\pi\)
0.382507 + 0.923952i \(0.375061\pi\)
\(828\) −108.000 −3.75326
\(829\) −37.0000 −1.28506 −0.642532 0.766259i \(-0.722116\pi\)
−0.642532 + 0.766259i \(0.722116\pi\)
\(830\) −36.0000 −1.24958
\(831\) 96.0000 3.33020
\(832\) 8.00000 0.277350
\(833\) −8.00000 −0.277184
\(834\) 36.0000 1.24658
\(835\) −42.0000 −1.45347
\(836\) −8.00000 −0.276686
\(837\) −27.0000 −0.933257
\(838\) 8.00000 0.276355
\(839\) −29.0000 −1.00119 −0.500596 0.865681i \(-0.666886\pi\)
−0.500596 + 0.865681i \(0.666886\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) 60.0000 2.06774
\(843\) 0 0
\(844\) −52.0000 −1.78991
\(845\) −3.00000 −0.103203
\(846\) 96.0000 3.30055
\(847\) −1.00000 −0.0343604
\(848\) −24.0000 −0.824163
\(849\) 6.00000 0.205919
\(850\) 64.0000 2.19518
\(851\) 63.0000 2.15961
\(852\) 30.0000 1.02778
\(853\) −42.0000 −1.43805 −0.719026 0.694983i \(-0.755412\pi\)
−0.719026 + 0.694983i \(0.755412\pi\)
\(854\) −8.00000 −0.273754
\(855\) 72.0000 2.46235
\(856\) 0 0
\(857\) −12.0000 −0.409912 −0.204956 0.978771i \(-0.565705\pi\)
−0.204956 + 0.978771i \(0.565705\pi\)
\(858\) −6.00000 −0.204837
\(859\) 7.00000 0.238837 0.119418 0.992844i \(-0.461897\pi\)
0.119418 + 0.992844i \(0.461897\pi\)
\(860\) −12.0000 −0.409197
\(861\) 0 0
\(862\) 60.0000 2.04361
\(863\) −56.0000 −1.90626 −0.953131 0.302558i \(-0.902160\pi\)
−0.953131 + 0.302558i \(0.902160\pi\)
\(864\) −72.0000 −2.44949
\(865\) 42.0000 1.42804
\(866\) 18.0000 0.611665
\(867\) −141.000 −4.78861
\(868\) −6.00000 −0.203653
\(869\) 12.0000 0.407072
\(870\) 144.000 4.88206
\(871\) 9.00000 0.304953
\(872\) 0 0
\(873\) 6.00000 0.203069
\(874\) −72.0000 −2.43544
\(875\) −3.00000 −0.101419
\(876\) 60.0000 2.02721
\(877\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(878\) 56.0000 1.88991
\(879\) −96.0000 −3.23800
\(880\) 12.0000 0.404520
\(881\) −21.0000 −0.707508 −0.353754 0.935339i \(-0.615095\pi\)
−0.353754 + 0.935339i \(0.615095\pi\)
\(882\) −12.0000 −0.404061
\(883\) −4.00000 −0.134611 −0.0673054 0.997732i \(-0.521440\pi\)
−0.0673054 + 0.997732i \(0.521440\pi\)
\(884\) 16.0000 0.538138
\(885\) −63.0000 −2.11772
\(886\) 22.0000 0.739104
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) −14.0000 −0.469545
\(890\) −54.0000 −1.81008
\(891\) 9.00000 0.301511
\(892\) −18.0000 −0.602685
\(893\) 32.0000 1.07084
\(894\) −36.0000 −1.20402
\(895\) −15.0000 −0.501395
\(896\) 0 0
\(897\) −27.0000 −0.901504
\(898\) 6.00000 0.200223
\(899\) −24.0000 −0.800445
\(900\) 48.0000 1.60000
\(901\) −48.0000 −1.59911
\(902\) 0 0
\(903\) 6.00000 0.199667
\(904\) 0 0
\(905\) 57.0000 1.89474
\(906\) 72.0000 2.39204
\(907\) −32.0000 −1.06254 −0.531271 0.847202i \(-0.678286\pi\)
−0.531271 + 0.847202i \(0.678286\pi\)
\(908\) −44.0000 −1.46019
\(909\) −48.0000 −1.59206
\(910\) 6.00000 0.198898
\(911\) 24.0000 0.795155 0.397578 0.917568i \(-0.369851\pi\)
0.397578 + 0.917568i \(0.369851\pi\)
\(912\) −48.0000 −1.58944
\(913\) −6.00000 −0.198571
\(914\) 72.0000 2.38155
\(915\) −36.0000 −1.19012
\(916\) −2.00000 −0.0660819
\(917\) 10.0000 0.330229
\(918\) −144.000 −4.75271
\(919\) 38.0000 1.25350 0.626752 0.779219i \(-0.284384\pi\)
0.626752 + 0.779219i \(0.284384\pi\)
\(920\) 0 0
\(921\) −36.0000 −1.18624
\(922\) −16.0000 −0.526932
\(923\) 5.00000 0.164577
\(924\) 6.00000 0.197386
\(925\) −28.0000 −0.920634
\(926\) −14.0000 −0.460069
\(927\) 48.0000 1.57653
\(928\) −64.0000 −2.10090
\(929\) 14.0000 0.459325 0.229663 0.973270i \(-0.426238\pi\)
0.229663 + 0.973270i \(0.426238\pi\)
\(930\) −54.0000 −1.77073
\(931\) −4.00000 −0.131095
\(932\) −12.0000 −0.393073
\(933\) −72.0000 −2.35717
\(934\) −22.0000 −0.719862
\(935\) 24.0000 0.784884
\(936\) 0 0
\(937\) −8.00000 −0.261349 −0.130674 0.991425i \(-0.541714\pi\)
−0.130674 + 0.991425i \(0.541714\pi\)
\(938\) −18.0000 −0.587721
\(939\) 69.0000 2.25173
\(940\) 48.0000 1.56559
\(941\) 4.00000 0.130396 0.0651981 0.997872i \(-0.479232\pi\)
0.0651981 + 0.997872i \(0.479232\pi\)
\(942\) 42.0000 1.36843
\(943\) 0 0
\(944\) 28.0000 0.911322
\(945\) −27.0000 −0.878310
\(946\) −4.00000 −0.130051
\(947\) 3.00000 0.0974869 0.0487435 0.998811i \(-0.484478\pi\)
0.0487435 + 0.998811i \(0.484478\pi\)
\(948\) −72.0000 −2.33845
\(949\) 10.0000 0.324614
\(950\) 32.0000 1.03822
\(951\) −9.00000 −0.291845
\(952\) 0 0
\(953\) 24.0000 0.777436 0.388718 0.921357i \(-0.372918\pi\)
0.388718 + 0.921357i \(0.372918\pi\)
\(954\) −72.0000 −2.33109
\(955\) −75.0000 −2.42694
\(956\) −8.00000 −0.258738
\(957\) 24.0000 0.775810
\(958\) −8.00000 −0.258468
\(959\) 17.0000 0.548959
\(960\) −72.0000 −2.32379
\(961\) −22.0000 −0.709677
\(962\) −14.0000 −0.451378
\(963\) 48.0000 1.54678
\(964\) 20.0000 0.644157
\(965\) −30.0000 −0.965734
\(966\) 54.0000 1.73742
\(967\) 10.0000 0.321578 0.160789 0.986989i \(-0.448596\pi\)
0.160789 + 0.986989i \(0.448596\pi\)
\(968\) 0 0
\(969\) −96.0000 −3.08396
\(970\) 6.00000 0.192648
\(971\) 17.0000 0.545556 0.272778 0.962077i \(-0.412058\pi\)
0.272778 + 0.962077i \(0.412058\pi\)
\(972\) 0 0
\(973\) −6.00000 −0.192351
\(974\) −2.00000 −0.0640841
\(975\) 12.0000 0.384308
\(976\) 16.0000 0.512148
\(977\) −41.0000 −1.31171 −0.655853 0.754889i \(-0.727691\pi\)
−0.655853 + 0.754889i \(0.727691\pi\)
\(978\) −24.0000 −0.767435
\(979\) −9.00000 −0.287641
\(980\) −6.00000 −0.191663
\(981\) −84.0000 −2.68191
\(982\) 60.0000 1.91468
\(983\) −1.00000 −0.0318950 −0.0159475 0.999873i \(-0.505076\pi\)
−0.0159475 + 0.999873i \(0.505076\pi\)
\(984\) 0 0
\(985\) 66.0000 2.10293
\(986\) −128.000 −4.07635
\(987\) −24.0000 −0.763928
\(988\) 8.00000 0.254514
\(989\) −18.0000 −0.572367
\(990\) 36.0000 1.14416
\(991\) −8.00000 −0.254128 −0.127064 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) 24.0000 0.762001
\(993\) −3.00000 −0.0952021
\(994\) −10.0000 −0.317181
\(995\) −84.0000 −2.66298
\(996\) 36.0000 1.14070
\(997\) −40.0000 −1.26681 −0.633406 0.773819i \(-0.718344\pi\)
−0.633406 + 0.773819i \(0.718344\pi\)
\(998\) 8.00000 0.253236
\(999\) 63.0000 1.99323
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 1001.2.a.a.1.1 1
3.2 odd 2 9009.2.a.n.1.1 1
7.6 odd 2 7007.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1001.2.a.a.1.1 1 1.1 even 1 trivial
7007.2.a.a.1.1 1 7.6 odd 2
9009.2.a.n.1.1 1 3.2 odd 2