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Properties

Label 4002.2.a.q.1.1
Level $4002$
Weight $2$
Character 4002.1
Self dual yes
Analytic conductor $31.956$
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] = [4002,2,Mod(1,4002)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4002, 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("4002.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4002 = 2 \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4002.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: \(31.9561308889\)
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 \(\chi\) \(=\) 4002.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{12} -2.00000 q^{13} +4.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} +1.00000 q^{18} +8.00000 q^{19} +4.00000 q^{21} -1.00000 q^{23} +1.00000 q^{24} -5.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} +4.00000 q^{28} +1.00000 q^{29} +4.00000 q^{31} +1.00000 q^{32} +4.00000 q^{34} +1.00000 q^{36} -10.0000 q^{37} +8.00000 q^{38} -2.00000 q^{39} -2.00000 q^{41} +4.00000 q^{42} -8.00000 q^{43} -1.00000 q^{46} +8.00000 q^{47} +1.00000 q^{48} +9.00000 q^{49} -5.00000 q^{50} +4.00000 q^{51} -2.00000 q^{52} -8.00000 q^{53} +1.00000 q^{54} +4.00000 q^{56} +8.00000 q^{57} +1.00000 q^{58} +12.0000 q^{59} -14.0000 q^{61} +4.00000 q^{62} +4.00000 q^{63} +1.00000 q^{64} +2.00000 q^{67} +4.00000 q^{68} -1.00000 q^{69} -8.00000 q^{71} +1.00000 q^{72} -2.00000 q^{73} -10.0000 q^{74} -5.00000 q^{75} +8.00000 q^{76} -2.00000 q^{78} -2.00000 q^{79} +1.00000 q^{81} -2.00000 q^{82} +14.0000 q^{83} +4.00000 q^{84} -8.00000 q^{86} +1.00000 q^{87} -8.00000 q^{91} -1.00000 q^{92} +4.00000 q^{93} +8.00000 q^{94} +1.00000 q^{96} +12.0000 q^{97} +9.00000 q^{98} +O(q^{100})\)

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
\(7\) 4.00000 1.51186 0.755929 − 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 1.00000 0.288675
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 4.00000 1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 − 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 1.00000 0.235702
\(19\) 8.00000 1.83533 0.917663 − 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 0 0
\(21\) 4.00000 0.872872
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 1.00000 0.204124
\(25\) −5.00000 −1.00000
\(26\) −2.00000 −0.392232
\(27\) 1.00000 0.192450
\(28\) 4.00000 0.755929
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 − 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 4.00000 0.685994
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −10.0000 −1.64399 −0.821995 − 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 8.00000 1.29777
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 4.00000 0.617213
\(43\) −8.00000 −1.21999 −0.609994 − 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) 8.00000 1.16692 0.583460 − 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 1.00000 0.144338
\(49\) 9.00000 1.28571
\(50\) −5.00000 −0.707107
\(51\) 4.00000 0.560112
\(52\) −2.00000 −0.277350
\(53\) −8.00000 −1.09888 −0.549442 − 0.835532i \(-0.685160\pi\)
−0.549442 + 0.835532i \(0.685160\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 4.00000 0.534522
\(57\) 8.00000 1.05963
\(58\) 1.00000 0.131306
\(59\) 12.0000 1.56227 0.781133 − 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) −14.0000 −1.79252 −0.896258 − 0.443533i \(-0.853725\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 4.00000 0.508001
\(63\) 4.00000 0.503953
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 2.00000 0.244339 0.122169 − 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) 4.00000 0.485071
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 1.00000 0.117851
\(73\) −2.00000 −0.234082 −0.117041 − 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) −10.0000 −1.16248
\(75\) −5.00000 −0.577350
\(76\) 8.00000 0.917663
\(77\) 0 0
\(78\) −2.00000 −0.226455
\(79\) −2.00000 −0.225018 −0.112509 − 0.993651i \(-0.535889\pi\)
−0.112509 + 0.993651i \(0.535889\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −2.00000 −0.220863
\(83\) 14.0000 1.53670 0.768350 − 0.640030i \(-0.221078\pi\)
0.768350 + 0.640030i \(0.221078\pi\)
\(84\) 4.00000 0.436436
\(85\) 0 0
\(86\) −8.00000 −0.862662
\(87\) 1.00000 0.107211
\(88\) 0 0
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) −1.00000 −0.104257
\(93\) 4.00000 0.414781
\(94\) 8.00000 0.825137
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 12.0000 1.21842 0.609208 − 0.793011i \(-0.291488\pi\)
0.609208 + 0.793011i \(0.291488\pi\)
\(98\) 9.00000 0.909137
\(99\) 0 0
\(100\) −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
\(103\) 4.00000 0.394132 0.197066 − 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) −8.00000 −0.777029
\(107\) 10.0000 0.966736 0.483368 − 0.875417i \(-0.339413\pi\)
0.483368 + 0.875417i \(0.339413\pi\)
\(108\) 1.00000 0.0962250
\(109\) 8.00000 0.766261 0.383131 − 0.923694i \(-0.374846\pi\)
0.383131 + 0.923694i \(0.374846\pi\)
\(110\) 0 0
\(111\) −10.0000 −0.949158
\(112\) 4.00000 0.377964
\(113\) −12.0000 −1.12887 −0.564433 − 0.825479i \(-0.690905\pi\)
−0.564433 + 0.825479i \(0.690905\pi\)
\(114\) 8.00000 0.749269
\(115\) 0 0
\(116\) 1.00000 0.0928477
\(117\) −2.00000 −0.184900
\(118\) 12.0000 1.10469
\(119\) 16.0000 1.46672
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) −14.0000 −1.26750
\(123\) −2.00000 −0.180334
\(124\) 4.00000 0.359211
\(125\) 0 0
\(126\) 4.00000 0.356348
\(127\) −8.00000 −0.709885 −0.354943 − 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 1.00000 0.0883883
\(129\) −8.00000 −0.704361
\(130\) 0 0
\(131\) 4.00000 0.349482 0.174741 − 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 0 0
\(133\) 32.0000 2.77475
\(134\) 2.00000 0.172774
\(135\) 0 0
\(136\) 4.00000 0.342997
\(137\) −12.0000 −1.02523 −0.512615 − 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) −1.00000 −0.0851257
\(139\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 8.00000 0.673722
\(142\) −8.00000 −0.671345
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −2.00000 −0.165521
\(147\) 9.00000 0.742307
\(148\) −10.0000 −0.821995
\(149\) 4.00000 0.327693 0.163846 − 0.986486i \(-0.447610\pi\)
0.163846 + 0.986486i \(0.447610\pi\)
\(150\) −5.00000 −0.408248
\(151\) −16.0000 −1.30206 −0.651031 − 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 8.00000 0.648886
\(153\) 4.00000 0.323381
\(154\) 0 0
\(155\) 0 0
\(156\) −2.00000 −0.160128
\(157\) −10.0000 −0.798087 −0.399043 − 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) −2.00000 −0.159111
\(159\) −8.00000 −0.634441
\(160\) 0 0
\(161\) −4.00000 −0.315244
\(162\) 1.00000 0.0785674
\(163\) −4.00000 −0.313304 −0.156652 − 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) 14.0000 1.08661
\(167\) 8.00000 0.619059 0.309529 − 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 4.00000 0.308607
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 8.00000 0.611775
\(172\) −8.00000 −0.609994
\(173\) −10.0000 −0.760286 −0.380143 − 0.924928i \(-0.624125\pi\)
−0.380143 + 0.924928i \(0.624125\pi\)
\(174\) 1.00000 0.0758098
\(175\) −20.0000 −1.51186
\(176\) 0 0
\(177\) 12.0000 0.901975
\(178\) 0 0
\(179\) 12.0000 0.896922 0.448461 − 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) −12.0000 −0.891953 −0.445976 − 0.895045i \(-0.647144\pi\)
−0.445976 + 0.895045i \(0.647144\pi\)
\(182\) −8.00000 −0.592999
\(183\) −14.0000 −1.03491
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) 4.00000 0.293294
\(187\) 0 0
\(188\) 8.00000 0.583460
\(189\) 4.00000 0.290957
\(190\) 0 0
\(191\) −2.00000 −0.144715 −0.0723575 − 0.997379i \(-0.523052\pi\)
−0.0723575 + 0.997379i \(0.523052\pi\)
\(192\) 1.00000 0.0721688
\(193\) 6.00000 0.431889 0.215945 − 0.976406i \(-0.430717\pi\)
0.215945 + 0.976406i \(0.430717\pi\)
\(194\) 12.0000 0.861550
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) −2.00000 −0.142494 −0.0712470 − 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 0 0
\(199\) −8.00000 −0.567105 −0.283552 − 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) −5.00000 −0.353553
\(201\) 2.00000 0.141069
\(202\) 14.0000 0.985037
\(203\) 4.00000 0.280745
\(204\) 4.00000 0.280056
\(205\) 0 0
\(206\) 4.00000 0.278693
\(207\) −1.00000 −0.0695048
\(208\) −2.00000 −0.138675
\(209\) 0 0
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 − 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) −8.00000 −0.549442
\(213\) −8.00000 −0.548151
\(214\) 10.0000 0.683586
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 16.0000 1.08615
\(218\) 8.00000 0.541828
\(219\) −2.00000 −0.135147
\(220\) 0 0
\(221\) −8.00000 −0.538138
\(222\) −10.0000 −0.671156
\(223\) 16.0000 1.07144 0.535720 − 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) 4.00000 0.267261
\(225\) −5.00000 −0.333333
\(226\) −12.0000 −0.798228
\(227\) 2.00000 0.132745 0.0663723 − 0.997795i \(-0.478857\pi\)
0.0663723 + 0.997795i \(0.478857\pi\)
\(228\) 8.00000 0.529813
\(229\) −2.00000 −0.132164 −0.0660819 − 0.997814i \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 1.00000 0.0656532
\(233\) −10.0000 −0.655122 −0.327561 − 0.944830i \(-0.606227\pi\)
−0.327561 + 0.944830i \(0.606227\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) 12.0000 0.781133
\(237\) −2.00000 −0.129914
\(238\) 16.0000 1.03713
\(239\) −16.0000 −1.03495 −0.517477 − 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 0 0
\(241\) −30.0000 −1.93247 −0.966235 − 0.257663i \(-0.917048\pi\)
−0.966235 + 0.257663i \(0.917048\pi\)
\(242\) −11.0000 −0.707107
\(243\) 1.00000 0.0641500
\(244\) −14.0000 −0.896258
\(245\) 0 0
\(246\) −2.00000 −0.127515
\(247\) −16.0000 −1.01806
\(248\) 4.00000 0.254000
\(249\) 14.0000 0.887214
\(250\) 0 0
\(251\) 28.0000 1.76734 0.883672 − 0.468106i \(-0.155064\pi\)
0.883672 + 0.468106i \(0.155064\pi\)
\(252\) 4.00000 0.251976
\(253\) 0 0
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −2.00000 −0.124757 −0.0623783 − 0.998053i \(-0.519869\pi\)
−0.0623783 + 0.998053i \(0.519869\pi\)
\(258\) −8.00000 −0.498058
\(259\) −40.0000 −2.48548
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) 4.00000 0.247121
\(263\) 18.0000 1.10993 0.554964 − 0.831875i \(-0.312732\pi\)
0.554964 + 0.831875i \(0.312732\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 32.0000 1.96205
\(267\) 0 0
\(268\) 2.00000 0.122169
\(269\) 2.00000 0.121942 0.0609711 − 0.998140i \(-0.480580\pi\)
0.0609711 + 0.998140i \(0.480580\pi\)
\(270\) 0 0
\(271\) −24.0000 −1.45790 −0.728948 − 0.684569i \(-0.759990\pi\)
−0.728948 + 0.684569i \(0.759990\pi\)
\(272\) 4.00000 0.242536
\(273\) −8.00000 −0.484182
\(274\) −12.0000 −0.724947
\(275\) 0 0
\(276\) −1.00000 −0.0601929
\(277\) 26.0000 1.56219 0.781094 − 0.624413i \(-0.214662\pi\)
0.781094 + 0.624413i \(0.214662\pi\)
\(278\) 0 0
\(279\) 4.00000 0.239474
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 − 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 8.00000 0.476393
\(283\) 18.0000 1.06999 0.534994 − 0.844856i \(-0.320314\pi\)
0.534994 + 0.844856i \(0.320314\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) 0 0
\(287\) −8.00000 −0.472225
\(288\) 1.00000 0.0589256
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 12.0000 0.703452
\(292\) −2.00000 −0.117041
\(293\) 6.00000 0.350524 0.175262 − 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 9.00000 0.524891
\(295\) 0 0
\(296\) −10.0000 −0.581238
\(297\) 0 0
\(298\) 4.00000 0.231714
\(299\) 2.00000 0.115663
\(300\) −5.00000 −0.288675
\(301\) −32.0000 −1.84445
\(302\) −16.0000 −0.920697
\(303\) 14.0000 0.804279
\(304\) 8.00000 0.458831
\(305\) 0 0
\(306\) 4.00000 0.228665
\(307\) 20.0000 1.14146 0.570730 − 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 0 0
\(309\) 4.00000 0.227552
\(310\) 0 0
\(311\) 12.0000 0.680458 0.340229 − 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) −2.00000 −0.113228
\(313\) 26.0000 1.46961 0.734803 − 0.678280i \(-0.237274\pi\)
0.734803 + 0.678280i \(0.237274\pi\)
\(314\) −10.0000 −0.564333
\(315\) 0 0
\(316\) −2.00000 −0.112509
\(317\) −6.00000 −0.336994 −0.168497 − 0.985702i \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) −8.00000 −0.448618
\(319\) 0 0
\(320\) 0 0
\(321\) 10.0000 0.558146
\(322\) −4.00000 −0.222911
\(323\) 32.0000 1.78053
\(324\) 1.00000 0.0555556
\(325\) 10.0000 0.554700
\(326\) −4.00000 −0.221540
\(327\) 8.00000 0.442401
\(328\) −2.00000 −0.110432
\(329\) 32.0000 1.76422
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 − 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) 14.0000 0.768350
\(333\) −10.0000 −0.547997
\(334\) 8.00000 0.437741
\(335\) 0 0
\(336\) 4.00000 0.218218
\(337\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(338\) −9.00000 −0.489535
\(339\) −12.0000 −0.651751
\(340\) 0 0
\(341\) 0 0
\(342\) 8.00000 0.432590
\(343\) 8.00000 0.431959
\(344\) −8.00000 −0.431331
\(345\) 0 0
\(346\) −10.0000 −0.537603
\(347\) 8.00000 0.429463 0.214731 − 0.976673i \(-0.431112\pi\)
0.214731 + 0.976673i \(0.431112\pi\)
\(348\) 1.00000 0.0536056
\(349\) −6.00000 −0.321173 −0.160586 − 0.987022i \(-0.551338\pi\)
−0.160586 + 0.987022i \(0.551338\pi\)
\(350\) −20.0000 −1.06904
\(351\) −2.00000 −0.106752
\(352\) 0 0
\(353\) −26.0000 −1.38384 −0.691920 − 0.721974i \(-0.743235\pi\)
−0.691920 + 0.721974i \(0.743235\pi\)
\(354\) 12.0000 0.637793
\(355\) 0 0
\(356\) 0 0
\(357\) 16.0000 0.846810
\(358\) 12.0000 0.634220
\(359\) 26.0000 1.37223 0.686114 − 0.727494i \(-0.259315\pi\)
0.686114 + 0.727494i \(0.259315\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) −12.0000 −0.630706
\(363\) −11.0000 −0.577350
\(364\) −8.00000 −0.419314
\(365\) 0 0
\(366\) −14.0000 −0.731792
\(367\) 22.0000 1.14839 0.574195 − 0.818718i \(-0.305315\pi\)
0.574195 + 0.818718i \(0.305315\pi\)
\(368\) −1.00000 −0.0521286
\(369\) −2.00000 −0.104116
\(370\) 0 0
\(371\) −32.0000 −1.66136
\(372\) 4.00000 0.207390
\(373\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 8.00000 0.412568
\(377\) −2.00000 −0.103005
\(378\) 4.00000 0.205738
\(379\) 16.0000 0.821865 0.410932 − 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) −2.00000 −0.102329
\(383\) −36.0000 −1.83951 −0.919757 − 0.392488i \(-0.871614\pi\)
−0.919757 + 0.392488i \(0.871614\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 6.00000 0.305392
\(387\) −8.00000 −0.406663
\(388\) 12.0000 0.609208
\(389\) −22.0000 −1.11544 −0.557722 − 0.830028i \(-0.688325\pi\)
−0.557722 + 0.830028i \(0.688325\pi\)
\(390\) 0 0
\(391\) −4.00000 −0.202289
\(392\) 9.00000 0.454569
\(393\) 4.00000 0.201773
\(394\) −2.00000 −0.100759
\(395\) 0 0
\(396\) 0 0
\(397\) −2.00000 −0.100377 −0.0501886 − 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) −8.00000 −0.401004
\(399\) 32.0000 1.60200
\(400\) −5.00000 −0.250000
\(401\) −10.0000 −0.499376 −0.249688 − 0.968326i \(-0.580328\pi\)
−0.249688 + 0.968326i \(0.580328\pi\)
\(402\) 2.00000 0.0997509
\(403\) −8.00000 −0.398508
\(404\) 14.0000 0.696526
\(405\) 0 0
\(406\) 4.00000 0.198517
\(407\) 0 0
\(408\) 4.00000 0.198030
\(409\) −18.0000 −0.890043 −0.445021 − 0.895520i \(-0.646804\pi\)
−0.445021 + 0.895520i \(0.646804\pi\)
\(410\) 0 0
\(411\) −12.0000 −0.591916
\(412\) 4.00000 0.197066
\(413\) 48.0000 2.36193
\(414\) −1.00000 −0.0491473
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) 0 0
\(418\) 0 0
\(419\) 30.0000 1.46560 0.732798 − 0.680446i \(-0.238214\pi\)
0.732798 + 0.680446i \(0.238214\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 − 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 4.00000 0.194717
\(423\) 8.00000 0.388973
\(424\) −8.00000 −0.388514
\(425\) −20.0000 −0.970143
\(426\) −8.00000 −0.387601
\(427\) −56.0000 −2.71003
\(428\) 10.0000 0.483368
\(429\) 0 0
\(430\) 0 0
\(431\) −28.0000 −1.34871 −0.674356 − 0.738406i \(-0.735579\pi\)
−0.674356 + 0.738406i \(0.735579\pi\)
\(432\) 1.00000 0.0481125
\(433\) −24.0000 −1.15337 −0.576683 − 0.816968i \(-0.695653\pi\)
−0.576683 + 0.816968i \(0.695653\pi\)
\(434\) 16.0000 0.768025
\(435\) 0 0
\(436\) 8.00000 0.383131
\(437\) −8.00000 −0.382692
\(438\) −2.00000 −0.0955637
\(439\) 8.00000 0.381819 0.190910 − 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) −8.00000 −0.380521
\(443\) −36.0000 −1.71041 −0.855206 − 0.518289i \(-0.826569\pi\)
−0.855206 + 0.518289i \(0.826569\pi\)
\(444\) −10.0000 −0.474579
\(445\) 0 0
\(446\) 16.0000 0.757622
\(447\) 4.00000 0.189194
\(448\) 4.00000 0.188982
\(449\) −18.0000 −0.849473 −0.424736 − 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) −5.00000 −0.235702
\(451\) 0 0
\(452\) −12.0000 −0.564433
\(453\) −16.0000 −0.751746
\(454\) 2.00000 0.0938647
\(455\) 0 0
\(456\) 8.00000 0.374634
\(457\) −10.0000 −0.467780 −0.233890 − 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) −2.00000 −0.0934539
\(459\) 4.00000 0.186704
\(460\) 0 0
\(461\) −14.0000 −0.652045 −0.326023 − 0.945362i \(-0.605709\pi\)
−0.326023 + 0.945362i \(0.605709\pi\)
\(462\) 0 0
\(463\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) 1.00000 0.0464238
\(465\) 0 0
\(466\) −10.0000 −0.463241
\(467\) 8.00000 0.370196 0.185098 − 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 8.00000 0.369406
\(470\) 0 0
\(471\) −10.0000 −0.460776
\(472\) 12.0000 0.552345
\(473\) 0 0
\(474\) −2.00000 −0.0918630
\(475\) −40.0000 −1.83533
\(476\) 16.0000 0.733359
\(477\) −8.00000 −0.366295
\(478\) −16.0000 −0.731823
\(479\) −42.0000 −1.91903 −0.959514 − 0.281659i \(-0.909115\pi\)
−0.959514 + 0.281659i \(0.909115\pi\)
\(480\) 0 0
\(481\) 20.0000 0.911922
\(482\) −30.0000 −1.36646
\(483\) −4.00000 −0.182006
\(484\) −11.0000 −0.500000
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 16.0000 0.725029 0.362515 − 0.931978i \(-0.381918\pi\)
0.362515 + 0.931978i \(0.381918\pi\)
\(488\) −14.0000 −0.633750
\(489\) −4.00000 −0.180886
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 − 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) −2.00000 −0.0901670
\(493\) 4.00000 0.180151
\(494\) −16.0000 −0.719874
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) −32.0000 −1.43540
\(498\) 14.0000 0.627355
\(499\) −12.0000 −0.537194 −0.268597 − 0.963253i \(-0.586560\pi\)
−0.268597 + 0.963253i \(0.586560\pi\)
\(500\) 0 0
\(501\) 8.00000 0.357414
\(502\) 28.0000 1.24970
\(503\) 14.0000 0.624229 0.312115 − 0.950044i \(-0.398963\pi\)
0.312115 + 0.950044i \(0.398963\pi\)
\(504\) 4.00000 0.178174
\(505\) 0 0
\(506\) 0 0
\(507\) −9.00000 −0.399704
\(508\) −8.00000 −0.354943
\(509\) −2.00000 −0.0886484 −0.0443242 − 0.999017i \(-0.514113\pi\)
−0.0443242 + 0.999017i \(0.514113\pi\)
\(510\) 0 0
\(511\) −8.00000 −0.353899
\(512\) 1.00000 0.0441942
\(513\) 8.00000 0.353209
\(514\) −2.00000 −0.0882162
\(515\) 0 0
\(516\) −8.00000 −0.352180
\(517\) 0 0
\(518\) −40.0000 −1.75750
\(519\) −10.0000 −0.438951
\(520\) 0 0
\(521\) −10.0000 −0.438108 −0.219054 − 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) 1.00000 0.0437688
\(523\) −6.00000 −0.262362 −0.131181 − 0.991358i \(-0.541877\pi\)
−0.131181 + 0.991358i \(0.541877\pi\)
\(524\) 4.00000 0.174741
\(525\) −20.0000 −0.872872
\(526\) 18.0000 0.784837
\(527\) 16.0000 0.696971
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 12.0000 0.520756
\(532\) 32.0000 1.38738
\(533\) 4.00000 0.173259
\(534\) 0 0
\(535\) 0 0
\(536\) 2.00000 0.0863868
\(537\) 12.0000 0.517838
\(538\) 2.00000 0.0862261
\(539\) 0 0
\(540\) 0 0
\(541\) −30.0000 −1.28980 −0.644900 − 0.764267i \(-0.723101\pi\)
−0.644900 + 0.764267i \(0.723101\pi\)
\(542\) −24.0000 −1.03089
\(543\) −12.0000 −0.514969
\(544\) 4.00000 0.171499
\(545\) 0 0
\(546\) −8.00000 −0.342368
\(547\) −20.0000 −0.855138 −0.427569 − 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) −12.0000 −0.512615
\(549\) −14.0000 −0.597505
\(550\) 0 0
\(551\) 8.00000 0.340811
\(552\) −1.00000 −0.0425628
\(553\) −8.00000 −0.340195
\(554\) 26.0000 1.10463
\(555\) 0 0
\(556\) 0 0
\(557\) −12.0000 −0.508456 −0.254228 − 0.967144i \(-0.581821\pi\)
−0.254228 + 0.967144i \(0.581821\pi\)
\(558\) 4.00000 0.169334
\(559\) 16.0000 0.676728
\(560\) 0 0
\(561\) 0 0
\(562\) −6.00000 −0.253095
\(563\) −36.0000 −1.51722 −0.758610 − 0.651546i \(-0.774121\pi\)
−0.758610 + 0.651546i \(0.774121\pi\)
\(564\) 8.00000 0.336861
\(565\) 0 0
\(566\) 18.0000 0.756596
\(567\) 4.00000 0.167984
\(568\) −8.00000 −0.335673
\(569\) 40.0000 1.67689 0.838444 − 0.544988i \(-0.183466\pi\)
0.838444 + 0.544988i \(0.183466\pi\)
\(570\) 0 0
\(571\) −18.0000 −0.753277 −0.376638 − 0.926360i \(-0.622920\pi\)
−0.376638 + 0.926360i \(0.622920\pi\)
\(572\) 0 0
\(573\) −2.00000 −0.0835512
\(574\) −8.00000 −0.333914
\(575\) 5.00000 0.208514
\(576\) 1.00000 0.0416667
\(577\) −26.0000 −1.08239 −0.541197 − 0.840896i \(-0.682029\pi\)
−0.541197 + 0.840896i \(0.682029\pi\)
\(578\) −1.00000 −0.0415945
\(579\) 6.00000 0.249351
\(580\) 0 0
\(581\) 56.0000 2.32327
\(582\) 12.0000 0.497416
\(583\) 0 0
\(584\) −2.00000 −0.0827606
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) −32.0000 −1.32078 −0.660391 − 0.750922i \(-0.729609\pi\)
−0.660391 + 0.750922i \(0.729609\pi\)
\(588\) 9.00000 0.371154
\(589\) 32.0000 1.31854
\(590\) 0 0
\(591\) −2.00000 −0.0822690
\(592\) −10.0000 −0.410997
\(593\) 18.0000 0.739171 0.369586 − 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 4.00000 0.163846
\(597\) −8.00000 −0.327418
\(598\) 2.00000 0.0817861
\(599\) −12.0000 −0.490307 −0.245153 − 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) −5.00000 −0.204124
\(601\) −2.00000 −0.0815817 −0.0407909 − 0.999168i \(-0.512988\pi\)
−0.0407909 + 0.999168i \(0.512988\pi\)
\(602\) −32.0000 −1.30422
\(603\) 2.00000 0.0814463
\(604\) −16.0000 −0.651031
\(605\) 0 0
\(606\) 14.0000 0.568711
\(607\) −32.0000 −1.29884 −0.649420 − 0.760430i \(-0.724988\pi\)
−0.649420 + 0.760430i \(0.724988\pi\)
\(608\) 8.00000 0.324443
\(609\) 4.00000 0.162088
\(610\) 0 0
\(611\) −16.0000 −0.647291
\(612\) 4.00000 0.161690
\(613\) 16.0000 0.646234 0.323117 − 0.946359i \(-0.395269\pi\)
0.323117 + 0.946359i \(0.395269\pi\)
\(614\) 20.0000 0.807134
\(615\) 0 0
\(616\) 0 0
\(617\) −36.0000 −1.44931 −0.724653 − 0.689114i \(-0.758000\pi\)
−0.724653 + 0.689114i \(0.758000\pi\)
\(618\) 4.00000 0.160904
\(619\) −20.0000 −0.803868 −0.401934 − 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 12.0000 0.481156
\(623\) 0 0
\(624\) −2.00000 −0.0800641
\(625\) 25.0000 1.00000
\(626\) 26.0000 1.03917
\(627\) 0 0
\(628\) −10.0000 −0.399043
\(629\) −40.0000 −1.59490
\(630\) 0 0
\(631\) 12.0000 0.477712 0.238856 − 0.971055i \(-0.423228\pi\)
0.238856 + 0.971055i \(0.423228\pi\)
\(632\) −2.00000 −0.0795557
\(633\) 4.00000 0.158986
\(634\) −6.00000 −0.238290
\(635\) 0 0
\(636\) −8.00000 −0.317221
\(637\) −18.0000 −0.713186
\(638\) 0 0
\(639\) −8.00000 −0.316475
\(640\) 0 0
\(641\) 16.0000 0.631962 0.315981 − 0.948766i \(-0.397666\pi\)
0.315981 + 0.948766i \(0.397666\pi\)
\(642\) 10.0000 0.394669
\(643\) 46.0000 1.81406 0.907031 − 0.421063i \(-0.138343\pi\)
0.907031 + 0.421063i \(0.138343\pi\)
\(644\) −4.00000 −0.157622
\(645\) 0 0
\(646\) 32.0000 1.25902
\(647\) 32.0000 1.25805 0.629025 − 0.777385i \(-0.283454\pi\)
0.629025 + 0.777385i \(0.283454\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) 10.0000 0.392232
\(651\) 16.0000 0.627089
\(652\) −4.00000 −0.156652
\(653\) 18.0000 0.704394 0.352197 − 0.935926i \(-0.385435\pi\)
0.352197 + 0.935926i \(0.385435\pi\)
\(654\) 8.00000 0.312825
\(655\) 0 0
\(656\) −2.00000 −0.0780869
\(657\) −2.00000 −0.0780274
\(658\) 32.0000 1.24749
\(659\) 8.00000 0.311636 0.155818 − 0.987786i \(-0.450199\pi\)
0.155818 + 0.987786i \(0.450199\pi\)
\(660\) 0 0
\(661\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(662\) −4.00000 −0.155464
\(663\) −8.00000 −0.310694
\(664\) 14.0000 0.543305
\(665\) 0 0
\(666\) −10.0000 −0.387492
\(667\) −1.00000 −0.0387202
\(668\) 8.00000 0.309529
\(669\) 16.0000 0.618596
\(670\) 0 0
\(671\) 0 0
\(672\) 4.00000 0.154303
\(673\) −42.0000 −1.61898 −0.809491 − 0.587133i \(-0.800257\pi\)
−0.809491 + 0.587133i \(0.800257\pi\)
\(674\) 0 0
\(675\) −5.00000 −0.192450
\(676\) −9.00000 −0.346154
\(677\) 14.0000 0.538064 0.269032 − 0.963131i \(-0.413296\pi\)
0.269032 + 0.963131i \(0.413296\pi\)
\(678\) −12.0000 −0.460857
\(679\) 48.0000 1.84207
\(680\) 0 0
\(681\) 2.00000 0.0766402
\(682\) 0 0
\(683\) 24.0000 0.918334 0.459167 − 0.888350i \(-0.348148\pi\)
0.459167 + 0.888350i \(0.348148\pi\)
\(684\) 8.00000 0.305888
\(685\) 0 0
\(686\) 8.00000 0.305441
\(687\) −2.00000 −0.0763048
\(688\) −8.00000 −0.304997
\(689\) 16.0000 0.609551
\(690\) 0 0
\(691\) −8.00000 −0.304334 −0.152167 − 0.988355i \(-0.548625\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) −10.0000 −0.380143
\(693\) 0 0
\(694\) 8.00000 0.303676
\(695\) 0 0
\(696\) 1.00000 0.0379049
\(697\) −8.00000 −0.303022
\(698\) −6.00000 −0.227103
\(699\) −10.0000 −0.378235
\(700\) −20.0000 −0.755929
\(701\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) −2.00000 −0.0754851
\(703\) −80.0000 −3.01726
\(704\) 0 0
\(705\) 0 0
\(706\) −26.0000 −0.978523
\(707\) 56.0000 2.10610
\(708\) 12.0000 0.450988
\(709\) −16.0000 −0.600893 −0.300446 − 0.953799i \(-0.597136\pi\)
−0.300446 + 0.953799i \(0.597136\pi\)
\(710\) 0 0
\(711\) −2.00000 −0.0750059
\(712\) 0 0
\(713\) −4.00000 −0.149801
\(714\) 16.0000 0.598785
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) −16.0000 −0.597531
\(718\) 26.0000 0.970311
\(719\) 24.0000 0.895049 0.447524 − 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) 45.0000 1.67473
\(723\) −30.0000 −1.11571
\(724\) −12.0000 −0.445976
\(725\) −5.00000 −0.185695
\(726\) −11.0000 −0.408248
\(727\) 2.00000 0.0741759 0.0370879 − 0.999312i \(-0.488192\pi\)
0.0370879 + 0.999312i \(0.488192\pi\)
\(728\) −8.00000 −0.296500
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −32.0000 −1.18356
\(732\) −14.0000 −0.517455
\(733\) −6.00000 −0.221615 −0.110808 − 0.993842i \(-0.535344\pi\)
−0.110808 + 0.993842i \(0.535344\pi\)
\(734\) 22.0000 0.812035
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 0 0
\(738\) −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
\(741\) −16.0000 −0.587775
\(742\) −32.0000 −1.17476
\(743\) 26.0000 0.953847 0.476924 − 0.878945i \(-0.341752\pi\)
0.476924 + 0.878945i \(0.341752\pi\)
\(744\) 4.00000 0.146647
\(745\) 0 0
\(746\) 0 0
\(747\) 14.0000 0.512233
\(748\) 0 0
\(749\) 40.0000 1.46157
\(750\) 0 0
\(751\) −14.0000 −0.510867 −0.255434 − 0.966827i \(-0.582218\pi\)
−0.255434 + 0.966827i \(0.582218\pi\)
\(752\) 8.00000 0.291730
\(753\) 28.0000 1.02038
\(754\) −2.00000 −0.0728357
\(755\) 0 0
\(756\) 4.00000 0.145479
\(757\) −10.0000 −0.363456 −0.181728 − 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) 16.0000 0.581146
\(759\) 0 0
\(760\) 0 0
\(761\) −18.0000 −0.652499 −0.326250 − 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) −8.00000 −0.289809
\(763\) 32.0000 1.15848
\(764\) −2.00000 −0.0723575
\(765\) 0 0
\(766\) −36.0000 −1.30073
\(767\) −24.0000 −0.866590
\(768\) 1.00000 0.0360844
\(769\) −32.0000 −1.15395 −0.576975 − 0.816762i \(-0.695767\pi\)
−0.576975 + 0.816762i \(0.695767\pi\)
\(770\) 0 0
\(771\) −2.00000 −0.0720282
\(772\) 6.00000 0.215945
\(773\) 14.0000 0.503545 0.251773 − 0.967786i \(-0.418987\pi\)
0.251773 + 0.967786i \(0.418987\pi\)
\(774\) −8.00000 −0.287554
\(775\) −20.0000 −0.718421
\(776\) 12.0000 0.430775
\(777\) −40.0000 −1.43499
\(778\) −22.0000 −0.788738
\(779\) −16.0000 −0.573259
\(780\) 0 0
\(781\) 0 0
\(782\) −4.00000 −0.143040
\(783\) 1.00000 0.0357371
\(784\) 9.00000 0.321429
\(785\) 0 0
\(786\) 4.00000 0.142675
\(787\) −18.0000 −0.641631 −0.320815 − 0.947142i \(-0.603957\pi\)
−0.320815 + 0.947142i \(0.603957\pi\)
\(788\) −2.00000 −0.0712470
\(789\) 18.0000 0.640817
\(790\) 0 0
\(791\) −48.0000 −1.70668
\(792\) 0 0
\(793\) 28.0000 0.994309
\(794\) −2.00000 −0.0709773
\(795\) 0 0
\(796\) −8.00000 −0.283552
\(797\) −18.0000 −0.637593 −0.318796 − 0.947823i \(-0.603279\pi\)
−0.318796 + 0.947823i \(0.603279\pi\)
\(798\) 32.0000 1.13279
\(799\) 32.0000 1.13208
\(800\) −5.00000 −0.176777
\(801\) 0 0
\(802\) −10.0000 −0.353112
\(803\) 0 0
\(804\) 2.00000 0.0705346
\(805\) 0 0
\(806\) −8.00000 −0.281788
\(807\) 2.00000 0.0704033
\(808\) 14.0000 0.492518
\(809\) 2.00000 0.0703163 0.0351581 − 0.999382i \(-0.488807\pi\)
0.0351581 + 0.999382i \(0.488807\pi\)
\(810\) 0 0
\(811\) 28.0000 0.983213 0.491606 − 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) 4.00000 0.140372
\(813\) −24.0000 −0.841717
\(814\) 0 0
\(815\) 0 0
\(816\) 4.00000 0.140028
\(817\) −64.0000 −2.23908
\(818\) −18.0000 −0.629355
\(819\) −8.00000 −0.279543
\(820\) 0 0
\(821\) −14.0000 −0.488603 −0.244302 − 0.969699i \(-0.578559\pi\)
−0.244302 + 0.969699i \(0.578559\pi\)
\(822\) −12.0000 −0.418548
\(823\) −16.0000 −0.557725 −0.278862 − 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) 4.00000 0.139347
\(825\) 0 0
\(826\) 48.0000 1.67013
\(827\) 24.0000 0.834562 0.417281 − 0.908778i \(-0.362983\pi\)
0.417281 + 0.908778i \(0.362983\pi\)
\(828\) −1.00000 −0.0347524
\(829\) −10.0000 −0.347314 −0.173657 − 0.984806i \(-0.555558\pi\)
−0.173657 + 0.984806i \(0.555558\pi\)
\(830\) 0 0
\(831\) 26.0000 0.901930
\(832\) −2.00000 −0.0693375
\(833\) 36.0000 1.24733
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 4.00000 0.138260
\(838\) 30.0000 1.03633
\(839\) 22.0000 0.759524 0.379762 − 0.925084i \(-0.376006\pi\)
0.379762 + 0.925084i \(0.376006\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) −10.0000 −0.344623
\(843\) −6.00000 −0.206651
\(844\) 4.00000 0.137686
\(845\) 0 0
\(846\) 8.00000 0.275046
\(847\) −44.0000 −1.51186
\(848\) −8.00000 −0.274721
\(849\) 18.0000 0.617758
\(850\) −20.0000 −0.685994
\(851\) 10.0000 0.342796
\(852\) −8.00000 −0.274075
\(853\) −6.00000 −0.205436 −0.102718 − 0.994711i \(-0.532754\pi\)
−0.102718 + 0.994711i \(0.532754\pi\)
\(854\) −56.0000 −1.91628
\(855\) 0 0
\(856\) 10.0000 0.341793
\(857\) 6.00000 0.204956 0.102478 − 0.994735i \(-0.467323\pi\)
0.102478 + 0.994735i \(0.467323\pi\)
\(858\) 0 0
\(859\) 28.0000 0.955348 0.477674 − 0.878537i \(-0.341480\pi\)
0.477674 + 0.878537i \(0.341480\pi\)
\(860\) 0 0
\(861\) −8.00000 −0.272639
\(862\) −28.0000 −0.953684
\(863\) 8.00000 0.272323 0.136162 − 0.990687i \(-0.456523\pi\)
0.136162 + 0.990687i \(0.456523\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) −24.0000 −0.815553
\(867\) −1.00000 −0.0339618
\(868\) 16.0000 0.543075
\(869\) 0 0
\(870\) 0 0
\(871\) −4.00000 −0.135535
\(872\) 8.00000 0.270914
\(873\) 12.0000 0.406138
\(874\) −8.00000 −0.270604
\(875\) 0 0
\(876\) −2.00000 −0.0675737
\(877\) 42.0000 1.41824 0.709120 − 0.705088i \(-0.249093\pi\)
0.709120 + 0.705088i \(0.249093\pi\)
\(878\) 8.00000 0.269987
\(879\) 6.00000 0.202375
\(880\) 0 0
\(881\) 32.0000 1.07811 0.539054 − 0.842271i \(-0.318782\pi\)
0.539054 + 0.842271i \(0.318782\pi\)
\(882\) 9.00000 0.303046
\(883\) −24.0000 −0.807664 −0.403832 − 0.914833i \(-0.632322\pi\)
−0.403832 + 0.914833i \(0.632322\pi\)
\(884\) −8.00000 −0.269069
\(885\) 0 0
\(886\) −36.0000 −1.20944
\(887\) −28.0000 −0.940148 −0.470074 − 0.882627i \(-0.655773\pi\)
−0.470074 + 0.882627i \(0.655773\pi\)
\(888\) −10.0000 −0.335578
\(889\) −32.0000 −1.07325
\(890\) 0 0
\(891\) 0 0
\(892\) 16.0000 0.535720
\(893\) 64.0000 2.14168
\(894\) 4.00000 0.133780
\(895\) 0 0
\(896\) 4.00000 0.133631
\(897\) 2.00000 0.0667781
\(898\) −18.0000 −0.600668
\(899\) 4.00000 0.133407
\(900\) −5.00000 −0.166667
\(901\) −32.0000 −1.06607
\(902\) 0 0
\(903\) −32.0000 −1.06489
\(904\) −12.0000 −0.399114
\(905\) 0 0
\(906\) −16.0000 −0.531564
\(907\) 32.0000 1.06254 0.531271 − 0.847202i \(-0.321714\pi\)
0.531271 + 0.847202i \(0.321714\pi\)
\(908\) 2.00000 0.0663723
\(909\) 14.0000 0.464351
\(910\) 0 0
\(911\) 30.0000 0.993944 0.496972 − 0.867766i \(-0.334445\pi\)
0.496972 + 0.867766i \(0.334445\pi\)
\(912\) 8.00000 0.264906
\(913\) 0 0
\(914\) −10.0000 −0.330771
\(915\) 0 0
\(916\) −2.00000 −0.0660819
\(917\) 16.0000 0.528367
\(918\) 4.00000 0.132020
\(919\) −20.0000 −0.659739 −0.329870 − 0.944027i \(-0.607005\pi\)
−0.329870 + 0.944027i \(0.607005\pi\)
\(920\) 0 0
\(921\) 20.0000 0.659022
\(922\) −14.0000 −0.461065
\(923\) 16.0000 0.526646
\(924\) 0 0
\(925\) 50.0000 1.64399
\(926\) 0 0
\(927\) 4.00000 0.131377
\(928\) 1.00000 0.0328266
\(929\) 18.0000 0.590561 0.295280 − 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) 0 0
\(931\) 72.0000 2.35970
\(932\) −10.0000 −0.327561
\(933\) 12.0000 0.392862
\(934\) 8.00000 0.261768
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) 50.0000 1.63343 0.816714 − 0.577042i \(-0.195793\pi\)
0.816714 + 0.577042i \(0.195793\pi\)
\(938\) 8.00000 0.261209
\(939\) 26.0000 0.848478
\(940\) 0 0
\(941\) 16.0000 0.521585 0.260793 − 0.965395i \(-0.416016\pi\)
0.260793 + 0.965395i \(0.416016\pi\)
\(942\) −10.0000 −0.325818
\(943\) 2.00000 0.0651290
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) 0 0
\(947\) −28.0000 −0.909878 −0.454939 − 0.890523i \(-0.650339\pi\)
−0.454939 + 0.890523i \(0.650339\pi\)
\(948\) −2.00000 −0.0649570
\(949\) 4.00000 0.129845
\(950\) −40.0000 −1.29777
\(951\) −6.00000 −0.194563
\(952\) 16.0000 0.518563
\(953\) 38.0000 1.23094 0.615470 − 0.788160i \(-0.288966\pi\)
0.615470 + 0.788160i \(0.288966\pi\)
\(954\) −8.00000 −0.259010
\(955\) 0 0
\(956\) −16.0000 −0.517477
\(957\) 0 0
\(958\) −42.0000 −1.35696
\(959\) −48.0000 −1.55000
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 20.0000 0.644826
\(963\) 10.0000 0.322245
\(964\) −30.0000 −0.966235
\(965\) 0 0
\(966\) −4.00000 −0.128698
\(967\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) −11.0000 −0.353553
\(969\) 32.0000 1.02799
\(970\) 0 0
\(971\) −36.0000 −1.15529 −0.577647 − 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) 16.0000 0.512673
\(975\) 10.0000 0.320256
\(976\) −14.0000 −0.448129
\(977\) 18.0000 0.575871 0.287936 − 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) −4.00000 −0.127906
\(979\) 0 0
\(980\) 0 0
\(981\) 8.00000 0.255420
\(982\) −12.0000 −0.382935
\(983\) −46.0000 −1.46717 −0.733586 − 0.679597i \(-0.762155\pi\)
−0.733586 + 0.679597i \(0.762155\pi\)
\(984\) −2.00000 −0.0637577
\(985\) 0 0
\(986\) 4.00000 0.127386
\(987\) 32.0000 1.01857
\(988\) −16.0000 −0.509028
\(989\) 8.00000 0.254385
\(990\) 0 0
\(991\) −56.0000 −1.77890 −0.889449 − 0.457034i \(-0.848912\pi\)
−0.889449 + 0.457034i \(0.848912\pi\)
\(992\) 4.00000 0.127000
\(993\) −4.00000 −0.126936
\(994\) −32.0000 −1.01498
\(995\) 0 0
\(996\) 14.0000 0.443607
\(997\) −18.0000 −0.570066 −0.285033 − 0.958518i \(-0.592005\pi\)
−0.285033 + 0.958518i \(0.592005\pi\)
\(998\) −12.0000 −0.379853
\(999\) −10.0000 −0.316386
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 4002.2.a.q.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4002.2.a.q.1.1 ✓ 1 1.1 even 1 trivial