↑

Properties

Label 8470.2.a.be.1.1
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8470,2,Mod(1,8470)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8470, 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("8470.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8470.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: \(67.6332905120\)
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 \(\chi\) \(=\) 8470.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} +1.00000 q^{10} +1.00000 q^{12} -5.00000 q^{13} -1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -2.00000 q^{18} -5.00000 q^{19} +1.00000 q^{20} -1.00000 q^{21} +9.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} -5.00000 q^{26} -5.00000 q^{27} -1.00000 q^{28} +6.00000 q^{29} +1.00000 q^{30} +8.00000 q^{31} +1.00000 q^{32} -1.00000 q^{35} -2.00000 q^{36} +8.00000 q^{37} -5.00000 q^{38} -5.00000 q^{39} +1.00000 q^{40} -6.00000 q^{41} -1.00000 q^{42} +10.0000 q^{43} -2.00000 q^{45} +9.00000 q^{46} +1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} -5.00000 q^{52} +6.00000 q^{53} -5.00000 q^{54} -1.00000 q^{56} -5.00000 q^{57} +6.00000 q^{58} -3.00000 q^{59} +1.00000 q^{60} -2.00000 q^{61} +8.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} -5.00000 q^{65} -10.0000 q^{67} +9.00000 q^{69} -1.00000 q^{70} +12.0000 q^{71} -2.00000 q^{72} +10.0000 q^{73} +8.00000 q^{74} +1.00000 q^{75} -5.00000 q^{76} -5.00000 q^{78} +13.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} +3.00000 q^{83} -1.00000 q^{84} +10.0000 q^{86} +6.00000 q^{87} +12.0000 q^{89} -2.00000 q^{90} +5.00000 q^{91} +9.00000 q^{92} +8.00000 q^{93} -5.00000 q^{95} +1.00000 q^{96} -10.0000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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 0.288675 − 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 1.00000 0.408248
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) −2.00000 −0.666667
\(10\) 1.00000 0.316228
\(11\) 0 0
\(12\) 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
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −2.00000 −0.471405
\(19\) −5.00000 −1.14708 −0.573539 − 0.819178i \(-0.694430\pi\)
−0.573539 + 0.819178i \(0.694430\pi\)
\(20\) 1.00000 0.223607
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 9.00000 1.87663 0.938315 − 0.345782i \(-0.112386\pi\)
0.938315 + 0.345782i \(0.112386\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) −5.00000 −0.980581
\(27\) −5.00000 −0.962250
\(28\) −1.00000 −0.188982
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 1.00000 0.182574
\(31\) 8.00000 1.43684 0.718421 − 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 0 0
\(35\) −1.00000 −0.169031
\(36\) −2.00000 −0.333333
\(37\) 8.00000 1.31519 0.657596 − 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) −5.00000 −0.811107
\(39\) −5.00000 −0.800641
\(40\) 1.00000 0.158114
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) −1.00000 −0.154303
\(43\) 10.0000 1.52499 0.762493 − 0.646997i \(-0.223975\pi\)
0.762493 + 0.646997i \(0.223975\pi\)
\(44\) 0 0
\(45\) −2.00000 −0.298142
\(46\) 9.00000 1.32698
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −5.00000 −0.693375
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) −5.00000 −0.680414
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) −5.00000 −0.662266
\(58\) 6.00000 0.787839
\(59\) −3.00000 −0.390567 −0.195283 − 0.980747i \(-0.562563\pi\)
−0.195283 + 0.980747i \(0.562563\pi\)
\(60\) 1.00000 0.129099
\(61\) −2.00000 −0.256074 −0.128037 − 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 8.00000 1.01600
\(63\) 2.00000 0.251976
\(64\) 1.00000 0.125000
\(65\) −5.00000 −0.620174
\(66\) 0 0
\(67\) −10.0000 −1.22169 −0.610847 − 0.791748i \(-0.709171\pi\)
−0.610847 + 0.791748i \(0.709171\pi\)
\(68\) 0 0
\(69\) 9.00000 1.08347
\(70\) −1.00000 −0.119523
\(71\) 12.0000 1.42414 0.712069 − 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) −2.00000 −0.235702
\(73\) 10.0000 1.17041 0.585206 − 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 8.00000 0.929981
\(75\) 1.00000 0.115470
\(76\) −5.00000 −0.573539
\(77\) 0 0
\(78\) −5.00000 −0.566139
\(79\) 13.0000 1.46261 0.731307 − 0.682048i \(-0.238911\pi\)
0.731307 + 0.682048i \(0.238911\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) 3.00000 0.329293 0.164646 − 0.986353i \(-0.447352\pi\)
0.164646 + 0.986353i \(0.447352\pi\)
\(84\) −1.00000 −0.109109
\(85\) 0 0
\(86\) 10.0000 1.07833
\(87\) 6.00000 0.643268
\(88\) 0 0
\(89\) 12.0000 1.27200 0.635999 − 0.771690i \(-0.280588\pi\)
0.635999 + 0.771690i \(0.280588\pi\)
\(90\) −2.00000 −0.210819
\(91\) 5.00000 0.524142
\(92\) 9.00000 0.938315
\(93\) 8.00000 0.829561
\(94\) 0 0
\(95\) −5.00000 −0.512989
\(96\) 1.00000 0.102062
\(97\) −10.0000 −1.01535 −0.507673 − 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 3.00000 0.298511 0.149256 − 0.988799i \(-0.452312\pi\)
0.149256 + 0.988799i \(0.452312\pi\)
\(102\) 0 0
\(103\) −16.0000 −1.57653 −0.788263 − 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) −5.00000 −0.490290
\(105\) −1.00000 −0.0975900
\(106\) 6.00000 0.582772
\(107\) −12.0000 −1.16008 −0.580042 − 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) −5.00000 −0.481125
\(109\) 16.0000 1.53252 0.766261 − 0.642529i \(-0.222115\pi\)
0.766261 + 0.642529i \(0.222115\pi\)
\(110\) 0 0
\(111\) 8.00000 0.759326
\(112\) −1.00000 −0.0944911
\(113\) −3.00000 −0.282216 −0.141108 − 0.989994i \(-0.545067\pi\)
−0.141108 + 0.989994i \(0.545067\pi\)
\(114\) −5.00000 −0.468293
\(115\) 9.00000 0.839254
\(116\) 6.00000 0.557086
\(117\) 10.0000 0.924500
\(118\) −3.00000 −0.276172
\(119\) 0 0
\(120\) 1.00000 0.0912871
\(121\) 0 0
\(122\) −2.00000 −0.181071
\(123\) −6.00000 −0.541002
\(124\) 8.00000 0.718421
\(125\) 1.00000 0.0894427
\(126\) 2.00000 0.178174
\(127\) 19.0000 1.68598 0.842989 − 0.537931i \(-0.180794\pi\)
0.842989 + 0.537931i \(0.180794\pi\)
\(128\) 1.00000 0.0883883
\(129\) 10.0000 0.880451
\(130\) −5.00000 −0.438529
\(131\) −9.00000 −0.786334 −0.393167 − 0.919467i \(-0.628621\pi\)
−0.393167 + 0.919467i \(0.628621\pi\)
\(132\) 0 0
\(133\) 5.00000 0.433555
\(134\) −10.0000 −0.863868
\(135\) −5.00000 −0.430331
\(136\) 0 0
\(137\) 9.00000 0.768922 0.384461 − 0.923141i \(-0.374387\pi\)
0.384461 + 0.923141i \(0.374387\pi\)
\(138\) 9.00000 0.766131
\(139\) −5.00000 −0.424094 −0.212047 − 0.977259i \(-0.568013\pi\)
−0.212047 + 0.977259i \(0.568013\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 0 0
\(142\) 12.0000 1.00702
\(143\) 0 0
\(144\) −2.00000 −0.166667
\(145\) 6.00000 0.498273
\(146\) 10.0000 0.827606
\(147\) 1.00000 0.0824786
\(148\) 8.00000 0.657596
\(149\) 24.0000 1.96616 0.983078 − 0.183186i \(-0.0586410\pi\)
0.983078 + 0.183186i \(0.0586410\pi\)
\(150\) 1.00000 0.0816497
\(151\) −17.0000 −1.38344 −0.691720 − 0.722166i \(-0.743147\pi\)
−0.691720 + 0.722166i \(0.743147\pi\)
\(152\) −5.00000 −0.405554
\(153\) 0 0
\(154\) 0 0
\(155\) 8.00000 0.642575
\(156\) −5.00000 −0.400320
\(157\) 17.0000 1.35675 0.678374 − 0.734717i \(-0.262685\pi\)
0.678374 + 0.734717i \(0.262685\pi\)
\(158\) 13.0000 1.03422
\(159\) 6.00000 0.475831
\(160\) 1.00000 0.0790569
\(161\) −9.00000 −0.709299
\(162\) 1.00000 0.0785674
\(163\) 14.0000 1.09656 0.548282 − 0.836293i \(-0.315282\pi\)
0.548282 + 0.836293i \(0.315282\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 3.00000 0.232845
\(167\) 6.00000 0.464294 0.232147 − 0.972681i \(-0.425425\pi\)
0.232147 + 0.972681i \(0.425425\pi\)
\(168\) −1.00000 −0.0771517
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) 10.0000 0.764719
\(172\) 10.0000 0.762493
\(173\) −6.00000 −0.456172 −0.228086 − 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 6.00000 0.454859
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) −3.00000 −0.225494
\(178\) 12.0000 0.899438
\(179\) 18.0000 1.34538 0.672692 − 0.739923i \(-0.265138\pi\)
0.672692 + 0.739923i \(0.265138\pi\)
\(180\) −2.00000 −0.149071
\(181\) −13.0000 −0.966282 −0.483141 − 0.875542i \(-0.660504\pi\)
−0.483141 + 0.875542i \(0.660504\pi\)
\(182\) 5.00000 0.370625
\(183\) −2.00000 −0.147844
\(184\) 9.00000 0.663489
\(185\) 8.00000 0.588172
\(186\) 8.00000 0.586588
\(187\) 0 0
\(188\) 0 0
\(189\) 5.00000 0.363696
\(190\) −5.00000 −0.362738
\(191\) 3.00000 0.217072 0.108536 − 0.994092i \(-0.465384\pi\)
0.108536 + 0.994092i \(0.465384\pi\)
\(192\) 1.00000 0.0721688
\(193\) 1.00000 0.0719816 0.0359908 − 0.999352i \(-0.488541\pi\)
0.0359908 + 0.999352i \(0.488541\pi\)
\(194\) −10.0000 −0.717958
\(195\) −5.00000 −0.358057
\(196\) 1.00000 0.0714286
\(197\) −18.0000 −1.28245 −0.641223 − 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) 2.00000 0.141776 0.0708881 − 0.997484i \(-0.477417\pi\)
0.0708881 + 0.997484i \(0.477417\pi\)
\(200\) 1.00000 0.0707107
\(201\) −10.0000 −0.705346
\(202\) 3.00000 0.211079
\(203\) −6.00000 −0.421117
\(204\) 0 0
\(205\) −6.00000 −0.419058
\(206\) −16.0000 −1.11477
\(207\) −18.0000 −1.25109
\(208\) −5.00000 −0.346688
\(209\) 0 0
\(210\) −1.00000 −0.0690066
\(211\) −14.0000 −0.963800 −0.481900 − 0.876226i \(-0.660053\pi\)
−0.481900 + 0.876226i \(0.660053\pi\)
\(212\) 6.00000 0.412082
\(213\) 12.0000 0.822226
\(214\) −12.0000 −0.820303
\(215\) 10.0000 0.681994
\(216\) −5.00000 −0.340207
\(217\) −8.00000 −0.543075
\(218\) 16.0000 1.08366
\(219\) 10.0000 0.675737
\(220\) 0 0
\(221\) 0 0
\(222\) 8.00000 0.536925
\(223\) 2.00000 0.133930 0.0669650 − 0.997755i \(-0.478668\pi\)
0.0669650 + 0.997755i \(0.478668\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −2.00000 −0.133333
\(226\) −3.00000 −0.199557
\(227\) −24.0000 −1.59294 −0.796468 − 0.604681i \(-0.793301\pi\)
−0.796468 + 0.604681i \(0.793301\pi\)
\(228\) −5.00000 −0.331133
\(229\) −22.0000 −1.45380 −0.726900 − 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 9.00000 0.593442
\(231\) 0 0
\(232\) 6.00000 0.393919
\(233\) −3.00000 −0.196537 −0.0982683 − 0.995160i \(-0.531330\pi\)
−0.0982683 + 0.995160i \(0.531330\pi\)
\(234\) 10.0000 0.653720
\(235\) 0 0
\(236\) −3.00000 −0.195283
\(237\) 13.0000 0.844441
\(238\) 0 0
\(239\) −21.0000 −1.35838 −0.679189 − 0.733964i \(-0.737668\pi\)
−0.679189 + 0.733964i \(0.737668\pi\)
\(240\) 1.00000 0.0645497
\(241\) 10.0000 0.644157 0.322078 − 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 0 0
\(243\) 16.0000 1.02640
\(244\) −2.00000 −0.128037
\(245\) 1.00000 0.0638877
\(246\) −6.00000 −0.382546
\(247\) 25.0000 1.59071
\(248\) 8.00000 0.508001
\(249\) 3.00000 0.190117
\(250\) 1.00000 0.0632456
\(251\) 12.0000 0.757433 0.378717 − 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 2.00000 0.125988
\(253\) 0 0
\(254\) 19.0000 1.19217
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −6.00000 −0.374270 −0.187135 − 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 10.0000 0.622573
\(259\) −8.00000 −0.497096
\(260\) −5.00000 −0.310087
\(261\) −12.0000 −0.742781
\(262\) −9.00000 −0.556022
\(263\) −3.00000 −0.184988 −0.0924940 − 0.995713i \(-0.529484\pi\)
−0.0924940 + 0.995713i \(0.529484\pi\)
\(264\) 0 0
\(265\) 6.00000 0.368577
\(266\) 5.00000 0.306570
\(267\) 12.0000 0.734388
\(268\) −10.0000 −0.610847
\(269\) 3.00000 0.182913 0.0914566 − 0.995809i \(-0.470848\pi\)
0.0914566 + 0.995809i \(0.470848\pi\)
\(270\) −5.00000 −0.304290
\(271\) 16.0000 0.971931 0.485965 − 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 0 0
\(273\) 5.00000 0.302614
\(274\) 9.00000 0.543710
\(275\) 0 0
\(276\) 9.00000 0.541736
\(277\) −2.00000 −0.120168 −0.0600842 − 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) −5.00000 −0.299880
\(279\) −16.0000 −0.957895
\(280\) −1.00000 −0.0597614
\(281\) −33.0000 −1.96861 −0.984307 − 0.176462i \(-0.943535\pi\)
−0.984307 + 0.176462i \(0.943535\pi\)
\(282\) 0 0
\(283\) −17.0000 −1.01055 −0.505273 − 0.862960i \(-0.668608\pi\)
−0.505273 + 0.862960i \(0.668608\pi\)
\(284\) 12.0000 0.712069
\(285\) −5.00000 −0.296174
\(286\) 0 0
\(287\) 6.00000 0.354169
\(288\) −2.00000 −0.117851
\(289\) −17.0000 −1.00000
\(290\) 6.00000 0.352332
\(291\) −10.0000 −0.586210
\(292\) 10.0000 0.585206
\(293\) 9.00000 0.525786 0.262893 − 0.964825i \(-0.415323\pi\)
0.262893 + 0.964825i \(0.415323\pi\)
\(294\) 1.00000 0.0583212
\(295\) −3.00000 −0.174667
\(296\) 8.00000 0.464991
\(297\) 0 0
\(298\) 24.0000 1.39028
\(299\) −45.0000 −2.60242
\(300\) 1.00000 0.0577350
\(301\) −10.0000 −0.576390
\(302\) −17.0000 −0.978240
\(303\) 3.00000 0.172345
\(304\) −5.00000 −0.286770
\(305\) −2.00000 −0.114520
\(306\) 0 0
\(307\) 4.00000 0.228292 0.114146 − 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(308\) 0 0
\(309\) −16.0000 −0.910208
\(310\) 8.00000 0.454369
\(311\) 24.0000 1.36092 0.680458 − 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) −5.00000 −0.283069
\(313\) 26.0000 1.46961 0.734803 − 0.678280i \(-0.237274\pi\)
0.734803 + 0.678280i \(0.237274\pi\)
\(314\) 17.0000 0.959366
\(315\) 2.00000 0.112687
\(316\) 13.0000 0.731307
\(317\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 6.00000 0.336463
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) −12.0000 −0.669775
\(322\) −9.00000 −0.501550
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) −5.00000 −0.277350
\(326\) 14.0000 0.775388
\(327\) 16.0000 0.884802
\(328\) −6.00000 −0.331295
\(329\) 0 0
\(330\) 0 0
\(331\) −22.0000 −1.20923 −0.604615 − 0.796518i \(-0.706673\pi\)
−0.604615 + 0.796518i \(0.706673\pi\)
\(332\) 3.00000 0.164646
\(333\) −16.0000 −0.876795
\(334\) 6.00000 0.328305
\(335\) −10.0000 −0.546358
\(336\) −1.00000 −0.0545545
\(337\) −23.0000 −1.25289 −0.626445 − 0.779466i \(-0.715491\pi\)
−0.626445 + 0.779466i \(0.715491\pi\)
\(338\) 12.0000 0.652714
\(339\) −3.00000 −0.162938
\(340\) 0 0
\(341\) 0 0
\(342\) 10.0000 0.540738
\(343\) −1.00000 −0.0539949
\(344\) 10.0000 0.539164
\(345\) 9.00000 0.484544
\(346\) −6.00000 −0.322562
\(347\) −18.0000 −0.966291 −0.483145 − 0.875540i \(-0.660506\pi\)
−0.483145 + 0.875540i \(0.660506\pi\)
\(348\) 6.00000 0.321634
\(349\) 13.0000 0.695874 0.347937 − 0.937518i \(-0.386882\pi\)
0.347937 + 0.937518i \(0.386882\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 25.0000 1.33440
\(352\) 0 0
\(353\) −30.0000 −1.59674 −0.798369 − 0.602168i \(-0.794304\pi\)
−0.798369 + 0.602168i \(0.794304\pi\)
\(354\) −3.00000 −0.159448
\(355\) 12.0000 0.636894
\(356\) 12.0000 0.635999
\(357\) 0 0
\(358\) 18.0000 0.951330
\(359\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) −2.00000 −0.105409
\(361\) 6.00000 0.315789
\(362\) −13.0000 −0.683265
\(363\) 0 0
\(364\) 5.00000 0.262071
\(365\) 10.0000 0.523424
\(366\) −2.00000 −0.104542
\(367\) 26.0000 1.35719 0.678594 − 0.734513i \(-0.262589\pi\)
0.678594 + 0.734513i \(0.262589\pi\)
\(368\) 9.00000 0.469157
\(369\) 12.0000 0.624695
\(370\) 8.00000 0.415900
\(371\) −6.00000 −0.311504
\(372\) 8.00000 0.414781
\(373\) 4.00000 0.207112 0.103556 − 0.994624i \(-0.466978\pi\)
0.103556 + 0.994624i \(0.466978\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −30.0000 −1.54508
\(378\) 5.00000 0.257172
\(379\) −16.0000 −0.821865 −0.410932 − 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) −5.00000 −0.256495
\(381\) 19.0000 0.973399
\(382\) 3.00000 0.153493
\(383\) −24.0000 −1.22634 −0.613171 − 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 1.00000 0.0508987
\(387\) −20.0000 −1.01666
\(388\) −10.0000 −0.507673
\(389\) 6.00000 0.304212 0.152106 − 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) −5.00000 −0.253185
\(391\) 0 0
\(392\) 1.00000 0.0505076
\(393\) −9.00000 −0.453990
\(394\) −18.0000 −0.906827
\(395\) 13.0000 0.654101
\(396\) 0 0
\(397\) 38.0000 1.90717 0.953583 − 0.301131i \(-0.0973643\pi\)
0.953583 + 0.301131i \(0.0973643\pi\)
\(398\) 2.00000 0.100251
\(399\) 5.00000 0.250313
\(400\) 1.00000 0.0500000
\(401\) −18.0000 −0.898877 −0.449439 − 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) −10.0000 −0.498755
\(403\) −40.0000 −1.99254
\(404\) 3.00000 0.149256
\(405\) 1.00000 0.0496904
\(406\) −6.00000 −0.297775
\(407\) 0 0
\(408\) 0 0
\(409\) −14.0000 −0.692255 −0.346128 − 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) −6.00000 −0.296319
\(411\) 9.00000 0.443937
\(412\) −16.0000 −0.788263
\(413\) 3.00000 0.147620
\(414\) −18.0000 −0.884652
\(415\) 3.00000 0.147264
\(416\) −5.00000 −0.245145
\(417\) −5.00000 −0.244851
\(418\) 0 0
\(419\) −27.0000 −1.31904 −0.659518 − 0.751689i \(-0.729240\pi\)
−0.659518 + 0.751689i \(0.729240\pi\)
\(420\) −1.00000 −0.0487950
\(421\) 8.00000 0.389896 0.194948 − 0.980814i \(-0.437546\pi\)
0.194948 + 0.980814i \(0.437546\pi\)
\(422\) −14.0000 −0.681509
\(423\) 0 0
\(424\) 6.00000 0.291386
\(425\) 0 0
\(426\) 12.0000 0.581402
\(427\) 2.00000 0.0967868
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) 10.0000 0.482243
\(431\) 3.00000 0.144505 0.0722525 − 0.997386i \(-0.476981\pi\)
0.0722525 + 0.997386i \(0.476981\pi\)
\(432\) −5.00000 −0.240563
\(433\) 32.0000 1.53782 0.768911 − 0.639356i \(-0.220799\pi\)
0.768911 + 0.639356i \(0.220799\pi\)
\(434\) −8.00000 −0.384012
\(435\) 6.00000 0.287678
\(436\) 16.0000 0.766261
\(437\) −45.0000 −2.15264
\(438\) 10.0000 0.477818
\(439\) 28.0000 1.33637 0.668184 − 0.743996i \(-0.267072\pi\)
0.668184 + 0.743996i \(0.267072\pi\)
\(440\) 0 0
\(441\) −2.00000 −0.0952381
\(442\) 0 0
\(443\) −12.0000 −0.570137 −0.285069 − 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 8.00000 0.379663
\(445\) 12.0000 0.568855
\(446\) 2.00000 0.0947027
\(447\) 24.0000 1.13516
\(448\) −1.00000 −0.0472456
\(449\) 9.00000 0.424736 0.212368 − 0.977190i \(-0.431882\pi\)
0.212368 + 0.977190i \(0.431882\pi\)
\(450\) −2.00000 −0.0942809
\(451\) 0 0
\(452\) −3.00000 −0.141108
\(453\) −17.0000 −0.798730
\(454\) −24.0000 −1.12638
\(455\) 5.00000 0.234404
\(456\) −5.00000 −0.234146
\(457\) 37.0000 1.73079 0.865393 − 0.501093i \(-0.167069\pi\)
0.865393 + 0.501093i \(0.167069\pi\)
\(458\) −22.0000 −1.02799
\(459\) 0 0
\(460\) 9.00000 0.419627
\(461\) 6.00000 0.279448 0.139724 − 0.990190i \(-0.455378\pi\)
0.139724 + 0.990190i \(0.455378\pi\)
\(462\) 0 0
\(463\) 5.00000 0.232370 0.116185 − 0.993228i \(-0.462933\pi\)
0.116185 + 0.993228i \(0.462933\pi\)
\(464\) 6.00000 0.278543
\(465\) 8.00000 0.370991
\(466\) −3.00000 −0.138972
\(467\) −27.0000 −1.24941 −0.624705 − 0.780860i \(-0.714781\pi\)
−0.624705 + 0.780860i \(0.714781\pi\)
\(468\) 10.0000 0.462250
\(469\) 10.0000 0.461757
\(470\) 0 0
\(471\) 17.0000 0.783319
\(472\) −3.00000 −0.138086
\(473\) 0 0
\(474\) 13.0000 0.597110
\(475\) −5.00000 −0.229416
\(476\) 0 0
\(477\) −12.0000 −0.549442
\(478\) −21.0000 −0.960518
\(479\) 30.0000 1.37073 0.685367 − 0.728197i \(-0.259642\pi\)
0.685367 + 0.728197i \(0.259642\pi\)
\(480\) 1.00000 0.0456435
\(481\) −40.0000 −1.82384
\(482\) 10.0000 0.455488
\(483\) −9.00000 −0.409514
\(484\) 0 0
\(485\) −10.0000 −0.454077
\(486\) 16.0000 0.725775
\(487\) −25.0000 −1.13286 −0.566429 − 0.824110i \(-0.691675\pi\)
−0.566429 + 0.824110i \(0.691675\pi\)
\(488\) −2.00000 −0.0905357
\(489\) 14.0000 0.633102
\(490\) 1.00000 0.0451754
\(491\) −18.0000 −0.812329 −0.406164 − 0.913800i \(-0.633134\pi\)
−0.406164 + 0.913800i \(0.633134\pi\)
\(492\) −6.00000 −0.270501
\(493\) 0 0
\(494\) 25.0000 1.12480
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) −12.0000 −0.538274
\(498\) 3.00000 0.134433
\(499\) 32.0000 1.43252 0.716258 − 0.697835i \(-0.245853\pi\)
0.716258 + 0.697835i \(0.245853\pi\)
\(500\) 1.00000 0.0447214
\(501\) 6.00000 0.268060
\(502\) 12.0000 0.535586
\(503\) 36.0000 1.60516 0.802580 − 0.596544i \(-0.203460\pi\)
0.802580 + 0.596544i \(0.203460\pi\)
\(504\) 2.00000 0.0890871
\(505\) 3.00000 0.133498
\(506\) 0 0
\(507\) 12.0000 0.532939
\(508\) 19.0000 0.842989
\(509\) 15.0000 0.664863 0.332432 − 0.943127i \(-0.392131\pi\)
0.332432 + 0.943127i \(0.392131\pi\)
\(510\) 0 0
\(511\) −10.0000 −0.442374
\(512\) 1.00000 0.0441942
\(513\) 25.0000 1.10378
\(514\) −6.00000 −0.264649
\(515\) −16.0000 −0.705044
\(516\) 10.0000 0.440225
\(517\) 0 0
\(518\) −8.00000 −0.351500
\(519\) −6.00000 −0.263371
\(520\) −5.00000 −0.219265
\(521\) −42.0000 −1.84005 −0.920027 − 0.391856i \(-0.871833\pi\)
−0.920027 + 0.391856i \(0.871833\pi\)
\(522\) −12.0000 −0.525226
\(523\) 31.0000 1.35554 0.677768 − 0.735276i \(-0.262948\pi\)
0.677768 + 0.735276i \(0.262948\pi\)
\(524\) −9.00000 −0.393167
\(525\) −1.00000 −0.0436436
\(526\) −3.00000 −0.130806
\(527\) 0 0
\(528\) 0 0
\(529\) 58.0000 2.52174
\(530\) 6.00000 0.260623
\(531\) 6.00000 0.260378
\(532\) 5.00000 0.216777
\(533\) 30.0000 1.29944
\(534\) 12.0000 0.519291
\(535\) −12.0000 −0.518805
\(536\) −10.0000 −0.431934
\(537\) 18.0000 0.776757
\(538\) 3.00000 0.129339
\(539\) 0 0
\(540\) −5.00000 −0.215166
\(541\) −2.00000 −0.0859867 −0.0429934 − 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) 16.0000 0.687259
\(543\) −13.0000 −0.557883
\(544\) 0 0
\(545\) 16.0000 0.685365
\(546\) 5.00000 0.213980
\(547\) −26.0000 −1.11168 −0.555840 − 0.831289i \(-0.687603\pi\)
−0.555840 + 0.831289i \(0.687603\pi\)
\(548\) 9.00000 0.384461
\(549\) 4.00000 0.170716
\(550\) 0 0
\(551\) −30.0000 −1.27804
\(552\) 9.00000 0.383065
\(553\) −13.0000 −0.552816
\(554\) −2.00000 −0.0849719
\(555\) 8.00000 0.339581
\(556\) −5.00000 −0.212047
\(557\) −12.0000 −0.508456 −0.254228 − 0.967144i \(-0.581821\pi\)
−0.254228 + 0.967144i \(0.581821\pi\)
\(558\) −16.0000 −0.677334
\(559\) −50.0000 −2.11477
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) −33.0000 −1.39202
\(563\) −21.0000 −0.885044 −0.442522 − 0.896758i \(-0.645916\pi\)
−0.442522 + 0.896758i \(0.645916\pi\)
\(564\) 0 0
\(565\) −3.00000 −0.126211
\(566\) −17.0000 −0.714563
\(567\) −1.00000 −0.0419961
\(568\) 12.0000 0.503509
\(569\) −27.0000 −1.13190 −0.565949 − 0.824440i \(-0.691490\pi\)
−0.565949 + 0.824440i \(0.691490\pi\)
\(570\) −5.00000 −0.209427
\(571\) 40.0000 1.67395 0.836974 − 0.547243i \(-0.184323\pi\)
0.836974 + 0.547243i \(0.184323\pi\)
\(572\) 0 0
\(573\) 3.00000 0.125327
\(574\) 6.00000 0.250435
\(575\) 9.00000 0.375326
\(576\) −2.00000 −0.0833333
\(577\) 14.0000 0.582828 0.291414 − 0.956597i \(-0.405874\pi\)
0.291414 + 0.956597i \(0.405874\pi\)
\(578\) −17.0000 −0.707107
\(579\) 1.00000 0.0415586
\(580\) 6.00000 0.249136
\(581\) −3.00000 −0.124461
\(582\) −10.0000 −0.414513
\(583\) 0 0
\(584\) 10.0000 0.413803
\(585\) 10.0000 0.413449
\(586\) 9.00000 0.371787
\(587\) 3.00000 0.123823 0.0619116 − 0.998082i \(-0.480280\pi\)
0.0619116 + 0.998082i \(0.480280\pi\)
\(588\) 1.00000 0.0412393
\(589\) −40.0000 −1.64817
\(590\) −3.00000 −0.123508
\(591\) −18.0000 −0.740421
\(592\) 8.00000 0.328798
\(593\) 48.0000 1.97112 0.985562 − 0.169316i \(-0.0541557\pi\)
0.985562 + 0.169316i \(0.0541557\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 24.0000 0.983078
\(597\) 2.00000 0.0818546
\(598\) −45.0000 −1.84019
\(599\) −21.0000 −0.858037 −0.429018 − 0.903296i \(-0.641140\pi\)
−0.429018 + 0.903296i \(0.641140\pi\)
\(600\) 1.00000 0.0408248
\(601\) −2.00000 −0.0815817 −0.0407909 − 0.999168i \(-0.512988\pi\)
−0.0407909 + 0.999168i \(0.512988\pi\)
\(602\) −10.0000 −0.407570
\(603\) 20.0000 0.814463
\(604\) −17.0000 −0.691720
\(605\) 0 0
\(606\) 3.00000 0.121867
\(607\) 22.0000 0.892952 0.446476 − 0.894795i \(-0.352679\pi\)
0.446476 + 0.894795i \(0.352679\pi\)
\(608\) −5.00000 −0.202777
\(609\) −6.00000 −0.243132
\(610\) −2.00000 −0.0809776
\(611\) 0 0
\(612\) 0 0
\(613\) −8.00000 −0.323117 −0.161558 − 0.986863i \(-0.551652\pi\)
−0.161558 + 0.986863i \(0.551652\pi\)
\(614\) 4.00000 0.161427
\(615\) −6.00000 −0.241943
\(616\) 0 0
\(617\) 30.0000 1.20775 0.603877 − 0.797077i \(-0.293622\pi\)
0.603877 + 0.797077i \(0.293622\pi\)
\(618\) −16.0000 −0.643614
\(619\) −1.00000 −0.0401934 −0.0200967 − 0.999798i \(-0.506397\pi\)
−0.0200967 + 0.999798i \(0.506397\pi\)
\(620\) 8.00000 0.321288
\(621\) −45.0000 −1.80579
\(622\) 24.0000 0.962312
\(623\) −12.0000 −0.480770
\(624\) −5.00000 −0.200160
\(625\) 1.00000 0.0400000
\(626\) 26.0000 1.03917
\(627\) 0 0
\(628\) 17.0000 0.678374
\(629\) 0 0
\(630\) 2.00000 0.0796819
\(631\) 20.0000 0.796187 0.398094 − 0.917345i \(-0.369672\pi\)
0.398094 + 0.917345i \(0.369672\pi\)
\(632\) 13.0000 0.517112
\(633\) −14.0000 −0.556450
\(634\) 0 0
\(635\) 19.0000 0.753992
\(636\) 6.00000 0.237915
\(637\) −5.00000 −0.198107
\(638\) 0 0
\(639\) −24.0000 −0.949425
\(640\) 1.00000 0.0395285
\(641\) −9.00000 −0.355479 −0.177739 − 0.984078i \(-0.556878\pi\)
−0.177739 + 0.984078i \(0.556878\pi\)
\(642\) −12.0000 −0.473602
\(643\) 20.0000 0.788723 0.394362 − 0.918955i \(-0.370966\pi\)
0.394362 + 0.918955i \(0.370966\pi\)
\(644\) −9.00000 −0.354650
\(645\) 10.0000 0.393750
\(646\) 0 0
\(647\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) −5.00000 −0.196116
\(651\) −8.00000 −0.313545
\(652\) 14.0000 0.548282
\(653\) −6.00000 −0.234798 −0.117399 − 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) 16.0000 0.625650
\(655\) −9.00000 −0.351659
\(656\) −6.00000 −0.234261
\(657\) −20.0000 −0.780274
\(658\) 0 0
\(659\) −6.00000 −0.233727 −0.116863 − 0.993148i \(-0.537284\pi\)
−0.116863 + 0.993148i \(0.537284\pi\)
\(660\) 0 0
\(661\) −7.00000 −0.272268 −0.136134 − 0.990690i \(-0.543468\pi\)
−0.136134 + 0.990690i \(0.543468\pi\)
\(662\) −22.0000 −0.855054
\(663\) 0 0
\(664\) 3.00000 0.116423
\(665\) 5.00000 0.193892
\(666\) −16.0000 −0.619987
\(667\) 54.0000 2.09089
\(668\) 6.00000 0.232147
\(669\) 2.00000 0.0773245
\(670\) −10.0000 −0.386334
\(671\) 0 0
\(672\) −1.00000 −0.0385758
\(673\) −41.0000 −1.58043 −0.790217 − 0.612827i \(-0.790032\pi\)
−0.790217 + 0.612827i \(0.790032\pi\)
\(674\) −23.0000 −0.885927
\(675\) −5.00000 −0.192450
\(676\) 12.0000 0.461538
\(677\) −45.0000 −1.72949 −0.864745 − 0.502211i \(-0.832520\pi\)
−0.864745 + 0.502211i \(0.832520\pi\)
\(678\) −3.00000 −0.115214
\(679\) 10.0000 0.383765
\(680\) 0 0
\(681\) −24.0000 −0.919682
\(682\) 0 0
\(683\) 30.0000 1.14792 0.573959 − 0.818884i \(-0.305407\pi\)
0.573959 + 0.818884i \(0.305407\pi\)
\(684\) 10.0000 0.382360
\(685\) 9.00000 0.343872
\(686\) −1.00000 −0.0381802
\(687\) −22.0000 −0.839352
\(688\) 10.0000 0.381246
\(689\) −30.0000 −1.14291
\(690\) 9.00000 0.342624
\(691\) −4.00000 −0.152167 −0.0760836 − 0.997101i \(-0.524242\pi\)
−0.0760836 + 0.997101i \(0.524242\pi\)
\(692\) −6.00000 −0.228086
\(693\) 0 0
\(694\) −18.0000 −0.683271
\(695\) −5.00000 −0.189661
\(696\) 6.00000 0.227429
\(697\) 0 0
\(698\) 13.0000 0.492057
\(699\) −3.00000 −0.113470
\(700\) −1.00000 −0.0377964
\(701\) −12.0000 −0.453234 −0.226617 − 0.973984i \(-0.572767\pi\)
−0.226617 + 0.973984i \(0.572767\pi\)
\(702\) 25.0000 0.943564
\(703\) −40.0000 −1.50863
\(704\) 0 0
\(705\) 0 0
\(706\) −30.0000 −1.12906
\(707\) −3.00000 −0.112827
\(708\) −3.00000 −0.112747
\(709\) −28.0000 −1.05156 −0.525781 − 0.850620i \(-0.676227\pi\)
−0.525781 + 0.850620i \(0.676227\pi\)
\(710\) 12.0000 0.450352
\(711\) −26.0000 −0.975076
\(712\) 12.0000 0.449719
\(713\) 72.0000 2.69642
\(714\) 0 0
\(715\) 0 0
\(716\) 18.0000 0.672692
\(717\) −21.0000 −0.784259
\(718\) 0 0
\(719\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) −2.00000 −0.0745356
\(721\) 16.0000 0.595871
\(722\) 6.00000 0.223297
\(723\) 10.0000 0.371904
\(724\) −13.0000 −0.483141
\(725\) 6.00000 0.222834
\(726\) 0 0
\(727\) 26.0000 0.964287 0.482143 − 0.876092i \(-0.339858\pi\)
0.482143 + 0.876092i \(0.339858\pi\)
\(728\) 5.00000 0.185312
\(729\) 13.0000 0.481481
\(730\) 10.0000 0.370117
\(731\) 0 0
\(732\) −2.00000 −0.0739221
\(733\) −23.0000 −0.849524 −0.424762 − 0.905305i \(-0.639642\pi\)
−0.424762 + 0.905305i \(0.639642\pi\)
\(734\) 26.0000 0.959678
\(735\) 1.00000 0.0368856
\(736\) 9.00000 0.331744
\(737\) 0 0
\(738\) 12.0000 0.441726
\(739\) 10.0000 0.367856 0.183928 − 0.982940i \(-0.441119\pi\)
0.183928 + 0.982940i \(0.441119\pi\)
\(740\) 8.00000 0.294086
\(741\) 25.0000 0.918398
\(742\) −6.00000 −0.220267
\(743\) −24.0000 −0.880475 −0.440237 − 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 8.00000 0.293294
\(745\) 24.0000 0.879292
\(746\) 4.00000 0.146450
\(747\) −6.00000 −0.219529
\(748\) 0 0
\(749\) 12.0000 0.438470
\(750\) 1.00000 0.0365148
\(751\) 11.0000 0.401396 0.200698 − 0.979653i \(-0.435679\pi\)
0.200698 + 0.979653i \(0.435679\pi\)
\(752\) 0 0
\(753\) 12.0000 0.437304
\(754\) −30.0000 −1.09254
\(755\) −17.0000 −0.618693
\(756\) 5.00000 0.181848
\(757\) 2.00000 0.0726912 0.0363456 − 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) −16.0000 −0.581146
\(759\) 0 0
\(760\) −5.00000 −0.181369
\(761\) 12.0000 0.435000 0.217500 − 0.976060i \(-0.430210\pi\)
0.217500 + 0.976060i \(0.430210\pi\)
\(762\) 19.0000 0.688297
\(763\) −16.0000 −0.579239
\(764\) 3.00000 0.108536
\(765\) 0 0
\(766\) −24.0000 −0.867155
\(767\) 15.0000 0.541619
\(768\) 1.00000 0.0360844
\(769\) −8.00000 −0.288487 −0.144244 − 0.989542i \(-0.546075\pi\)
−0.144244 + 0.989542i \(0.546075\pi\)
\(770\) 0 0
\(771\) −6.00000 −0.216085
\(772\) 1.00000 0.0359908
\(773\) −21.0000 −0.755318 −0.377659 − 0.925945i \(-0.623271\pi\)
−0.377659 + 0.925945i \(0.623271\pi\)
\(774\) −20.0000 −0.718885
\(775\) 8.00000 0.287368
\(776\) −10.0000 −0.358979
\(777\) −8.00000 −0.286998
\(778\) 6.00000 0.215110
\(779\) 30.0000 1.07486
\(780\) −5.00000 −0.179029
\(781\) 0 0
\(782\) 0 0
\(783\) −30.0000 −1.07211
\(784\) 1.00000 0.0357143
\(785\) 17.0000 0.606756
\(786\) −9.00000 −0.321019
\(787\) 4.00000 0.142585 0.0712923 − 0.997455i \(-0.477288\pi\)
0.0712923 + 0.997455i \(0.477288\pi\)
\(788\) −18.0000 −0.641223
\(789\) −3.00000 −0.106803
\(790\) 13.0000 0.462519
\(791\) 3.00000 0.106668
\(792\) 0 0
\(793\) 10.0000 0.355110
\(794\) 38.0000 1.34857
\(795\) 6.00000 0.212798
\(796\) 2.00000 0.0708881
\(797\) 9.00000 0.318796 0.159398 − 0.987214i \(-0.449045\pi\)
0.159398 + 0.987214i \(0.449045\pi\)
\(798\) 5.00000 0.176998
\(799\) 0 0
\(800\) 1.00000 0.0353553
\(801\) −24.0000 −0.847998
\(802\) −18.0000 −0.635602
\(803\) 0 0
\(804\) −10.0000 −0.352673
\(805\) −9.00000 −0.317208
\(806\) −40.0000 −1.40894
\(807\) 3.00000 0.105605
\(808\) 3.00000 0.105540
\(809\) 30.0000 1.05474 0.527372 − 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 1.00000 0.0351364
\(811\) 16.0000 0.561836 0.280918 − 0.959732i \(-0.409361\pi\)
0.280918 + 0.959732i \(0.409361\pi\)
\(812\) −6.00000 −0.210559
\(813\) 16.0000 0.561144
\(814\) 0 0
\(815\) 14.0000 0.490399
\(816\) 0 0
\(817\) −50.0000 −1.74928
\(818\) −14.0000 −0.489499
\(819\) −10.0000 −0.349428
\(820\) −6.00000 −0.209529
\(821\) 18.0000 0.628204 0.314102 − 0.949389i \(-0.398297\pi\)
0.314102 + 0.949389i \(0.398297\pi\)
\(822\) 9.00000 0.313911
\(823\) −4.00000 −0.139431 −0.0697156 − 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) −16.0000 −0.557386
\(825\) 0 0
\(826\) 3.00000 0.104383
\(827\) 42.0000 1.46048 0.730242 − 0.683189i \(-0.239408\pi\)
0.730242 + 0.683189i \(0.239408\pi\)
\(828\) −18.0000 −0.625543
\(829\) −55.0000 −1.91023 −0.955114 − 0.296237i \(-0.904268\pi\)
−0.955114 + 0.296237i \(0.904268\pi\)
\(830\) 3.00000 0.104132
\(831\) −2.00000 −0.0693792
\(832\) −5.00000 −0.173344
\(833\) 0 0
\(834\) −5.00000 −0.173136
\(835\) 6.00000 0.207639
\(836\) 0 0
\(837\) −40.0000 −1.38260
\(838\) −27.0000 −0.932700
\(839\) 24.0000 0.828572 0.414286 − 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) −1.00000 −0.0345033
\(841\) 7.00000 0.241379
\(842\) 8.00000 0.275698
\(843\) −33.0000 −1.13658
\(844\) −14.0000 −0.481900
\(845\) 12.0000 0.412813
\(846\) 0 0
\(847\) 0 0
\(848\) 6.00000 0.206041
\(849\) −17.0000 −0.583438
\(850\) 0 0
\(851\) 72.0000 2.46813
\(852\) 12.0000 0.411113
\(853\) 7.00000 0.239675 0.119838 − 0.992793i \(-0.461763\pi\)
0.119838 + 0.992793i \(0.461763\pi\)
\(854\) 2.00000 0.0684386
\(855\) 10.0000 0.341993
\(856\) −12.0000 −0.410152
\(857\) 36.0000 1.22974 0.614868 − 0.788630i \(-0.289209\pi\)
0.614868 + 0.788630i \(0.289209\pi\)
\(858\) 0 0
\(859\) −52.0000 −1.77422 −0.887109 − 0.461561i \(-0.847290\pi\)
−0.887109 + 0.461561i \(0.847290\pi\)
\(860\) 10.0000 0.340997
\(861\) 6.00000 0.204479
\(862\) 3.00000 0.102180
\(863\) 12.0000 0.408485 0.204242 − 0.978920i \(-0.434527\pi\)
0.204242 + 0.978920i \(0.434527\pi\)
\(864\) −5.00000 −0.170103
\(865\) −6.00000 −0.204006
\(866\) 32.0000 1.08740
\(867\) −17.0000 −0.577350
\(868\) −8.00000 −0.271538
\(869\) 0 0
\(870\) 6.00000 0.203419
\(871\) 50.0000 1.69419
\(872\) 16.0000 0.541828
\(873\) 20.0000 0.676897
\(874\) −45.0000 −1.52215
\(875\) −1.00000 −0.0338062
\(876\) 10.0000 0.337869
\(877\) 10.0000 0.337676 0.168838 − 0.985644i \(-0.445999\pi\)
0.168838 + 0.985644i \(0.445999\pi\)
\(878\) 28.0000 0.944954
\(879\) 9.00000 0.303562
\(880\) 0 0
\(881\) −36.0000 −1.21287 −0.606435 − 0.795133i \(-0.707401\pi\)
−0.606435 + 0.795133i \(0.707401\pi\)
\(882\) −2.00000 −0.0673435
\(883\) −16.0000 −0.538443 −0.269221 − 0.963078i \(-0.586766\pi\)
−0.269221 + 0.963078i \(0.586766\pi\)
\(884\) 0 0
\(885\) −3.00000 −0.100844
\(886\) −12.0000 −0.403148
\(887\) 12.0000 0.402921 0.201460 − 0.979497i \(-0.435431\pi\)
0.201460 + 0.979497i \(0.435431\pi\)
\(888\) 8.00000 0.268462
\(889\) −19.0000 −0.637240
\(890\) 12.0000 0.402241
\(891\) 0 0
\(892\) 2.00000 0.0669650
\(893\) 0 0
\(894\) 24.0000 0.802680
\(895\) 18.0000 0.601674
\(896\) −1.00000 −0.0334077
\(897\) −45.0000 −1.50251
\(898\) 9.00000 0.300334
\(899\) 48.0000 1.60089
\(900\) −2.00000 −0.0666667
\(901\) 0 0
\(902\) 0 0
\(903\) −10.0000 −0.332779
\(904\) −3.00000 −0.0997785
\(905\) −13.0000 −0.432135
\(906\) −17.0000 −0.564787
\(907\) −4.00000 −0.132818 −0.0664089 − 0.997792i \(-0.521154\pi\)
−0.0664089 + 0.997792i \(0.521154\pi\)
\(908\) −24.0000 −0.796468
\(909\) −6.00000 −0.199007
\(910\) 5.00000 0.165748
\(911\) 45.0000 1.49092 0.745458 − 0.666552i \(-0.232231\pi\)
0.745458 + 0.666552i \(0.232231\pi\)
\(912\) −5.00000 −0.165567
\(913\) 0 0
\(914\) 37.0000 1.22385
\(915\) −2.00000 −0.0661180
\(916\) −22.0000 −0.726900
\(917\) 9.00000 0.297206
\(918\) 0 0
\(919\) 16.0000 0.527791 0.263896 − 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 9.00000 0.296721
\(921\) 4.00000 0.131804
\(922\) 6.00000 0.197599
\(923\) −60.0000 −1.97492
\(924\) 0 0
\(925\) 8.00000 0.263038
\(926\) 5.00000 0.164310
\(927\) 32.0000 1.05102
\(928\) 6.00000 0.196960
\(929\) −12.0000 −0.393707 −0.196854 − 0.980433i \(-0.563072\pi\)
−0.196854 + 0.980433i \(0.563072\pi\)
\(930\) 8.00000 0.262330
\(931\) −5.00000 −0.163868
\(932\) −3.00000 −0.0982683
\(933\) 24.0000 0.785725
\(934\) −27.0000 −0.883467
\(935\) 0 0
\(936\) 10.0000 0.326860
\(937\) −38.0000 −1.24141 −0.620703 − 0.784046i \(-0.713153\pi\)
−0.620703 + 0.784046i \(0.713153\pi\)
\(938\) 10.0000 0.326512
\(939\) 26.0000 0.848478
\(940\) 0 0
\(941\) −30.0000 −0.977972 −0.488986 − 0.872292i \(-0.662633\pi\)
−0.488986 + 0.872292i \(0.662633\pi\)
\(942\) 17.0000 0.553890
\(943\) −54.0000 −1.75848
\(944\) −3.00000 −0.0976417
\(945\) 5.00000 0.162650
\(946\) 0 0
\(947\) −18.0000 −0.584921 −0.292461 − 0.956278i \(-0.594474\pi\)
−0.292461 + 0.956278i \(0.594474\pi\)
\(948\) 13.0000 0.422220
\(949\) −50.0000 −1.62307
\(950\) −5.00000 −0.162221
\(951\) 0 0
\(952\) 0 0
\(953\) −39.0000 −1.26333 −0.631667 − 0.775240i \(-0.717629\pi\)
−0.631667 + 0.775240i \(0.717629\pi\)
\(954\) −12.0000 −0.388514
\(955\) 3.00000 0.0970777
\(956\) −21.0000 −0.679189
\(957\) 0 0
\(958\) 30.0000 0.969256
\(959\) −9.00000 −0.290625
\(960\) 1.00000 0.0322749
\(961\) 33.0000 1.06452
\(962\) −40.0000 −1.28965
\(963\) 24.0000 0.773389
\(964\) 10.0000 0.322078
\(965\) 1.00000 0.0321911
\(966\) −9.00000 −0.289570
\(967\) 52.0000 1.67221 0.836104 − 0.548572i \(-0.184828\pi\)
0.836104 + 0.548572i \(0.184828\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −10.0000 −0.321081
\(971\) 27.0000 0.866471 0.433236 − 0.901281i \(-0.357372\pi\)
0.433236 + 0.901281i \(0.357372\pi\)
\(972\) 16.0000 0.513200
\(973\) 5.00000 0.160293
\(974\) −25.0000 −0.801052
\(975\) −5.00000 −0.160128
\(976\) −2.00000 −0.0640184
\(977\) −51.0000 −1.63163 −0.815817 − 0.578310i \(-0.803713\pi\)
−0.815817 + 0.578310i \(0.803713\pi\)
\(978\) 14.0000 0.447671
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) −32.0000 −1.02168
\(982\) −18.0000 −0.574403
\(983\) 36.0000 1.14822 0.574111 − 0.818778i \(-0.305348\pi\)
0.574111 + 0.818778i \(0.305348\pi\)
\(984\) −6.00000 −0.191273
\(985\) −18.0000 −0.573528
\(986\) 0 0
\(987\) 0 0
\(988\) 25.0000 0.795356
\(989\) 90.0000 2.86183
\(990\) 0 0
\(991\) 11.0000 0.349427 0.174713 − 0.984619i \(-0.444100\pi\)
0.174713 + 0.984619i \(0.444100\pi\)
\(992\) 8.00000 0.254000
\(993\) −22.0000 −0.698149
\(994\) −12.0000 −0.380617
\(995\) 2.00000 0.0634043
\(996\) 3.00000 0.0950586
\(997\) 49.0000 1.55185 0.775923 − 0.630828i \(-0.217285\pi\)
0.775923 + 0.630828i \(0.217285\pi\)
\(998\) 32.0000 1.01294
\(999\) −40.0000 −1.26554
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 8470.2.a.be.1.1 yes 1
11.10 odd 2 8470.2.a.o.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.o.1.1 ✓ 1 11.10 odd 2
8470.2.a.be.1.1 yes 1 1.1 even 1 trivial