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Properties

Label 8470.2.a.p.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

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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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -3.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +6.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -3.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +6.00000 q^{9} -1.00000 q^{10} +3.00000 q^{12} -3.00000 q^{13} -1.00000 q^{14} +3.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} -6.00000 q^{18} -1.00000 q^{19} +1.00000 q^{20} +3.00000 q^{21} -1.00000 q^{23} -3.00000 q^{24} +1.00000 q^{25} +3.00000 q^{26} +9.00000 q^{27} +1.00000 q^{28} +6.00000 q^{29} -3.00000 q^{30} +6.00000 q^{31} -1.00000 q^{32} +2.00000 q^{34} +1.00000 q^{35} +6.00000 q^{36} -8.00000 q^{37} +1.00000 q^{38} -9.00000 q^{39} -1.00000 q^{40} -3.00000 q^{42} +8.00000 q^{43} +6.00000 q^{45} +1.00000 q^{46} +10.0000 q^{47} +3.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} -6.00000 q^{51} -3.00000 q^{52} +4.00000 q^{53} -9.00000 q^{54} -1.00000 q^{56} -3.00000 q^{57} -6.00000 q^{58} +3.00000 q^{59} +3.00000 q^{60} -2.00000 q^{61} -6.00000 q^{62} +6.00000 q^{63} +1.00000 q^{64} -3.00000 q^{65} +16.0000 q^{67} -2.00000 q^{68} -3.00000 q^{69} -1.00000 q^{70} -8.00000 q^{71} -6.00000 q^{72} -4.00000 q^{73} +8.00000 q^{74} +3.00000 q^{75} -1.00000 q^{76} +9.00000 q^{78} +1.00000 q^{79} +1.00000 q^{80} +9.00000 q^{81} +3.00000 q^{83} +3.00000 q^{84} -2.00000 q^{85} -8.00000 q^{86} +18.0000 q^{87} +10.0000 q^{89} -6.00000 q^{90} -3.00000 q^{91} -1.00000 q^{92} +18.0000 q^{93} -10.0000 q^{94} -1.00000 q^{95} -3.00000 q^{96} -6.00000 q^{97} -1.00000 q^{98} +O(q^{100})\)

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
\(3\) 3.00000 1.73205 0.866025 − 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −3.00000 −1.22474
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 6.00000 2.00000
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) 3.00000 0.866025
\(13\) −3.00000 −0.832050 −0.416025 − 0.909353i \(-0.636577\pi\)
−0.416025 + 0.909353i \(0.636577\pi\)
\(14\) −1.00000 −0.267261
\(15\) 3.00000 0.774597
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) −6.00000 −1.41421
\(19\) −1.00000 −0.229416 −0.114708 − 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 1.00000 0.223607
\(21\) 3.00000 0.654654
\(22\) 0 0
\(23\) −1.00000 −0.208514 −0.104257 − 0.994550i \(-0.533247\pi\)
−0.104257 + 0.994550i \(0.533247\pi\)
\(24\) −3.00000 −0.612372
\(25\) 1.00000 0.200000
\(26\) 3.00000 0.588348
\(27\) 9.00000 1.73205
\(28\) 1.00000 0.188982
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) −3.00000 −0.547723
\(31\) 6.00000 1.07763 0.538816 − 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) 1.00000 0.169031
\(36\) 6.00000 1.00000
\(37\) −8.00000 −1.31519 −0.657596 − 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 1.00000 0.162221
\(39\) −9.00000 −1.44115
\(40\) −1.00000 −0.158114
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) −3.00000 −0.462910
\(43\) 8.00000 1.21999 0.609994 − 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) 6.00000 0.894427
\(46\) 1.00000 0.147442
\(47\) 10.0000 1.45865 0.729325 − 0.684167i \(-0.239834\pi\)
0.729325 + 0.684167i \(0.239834\pi\)
\(48\) 3.00000 0.433013
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) −6.00000 −0.840168
\(52\) −3.00000 −0.416025
\(53\) 4.00000 0.549442 0.274721 − 0.961524i \(-0.411414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) −9.00000 −1.22474
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) −3.00000 −0.397360
\(58\) −6.00000 −0.787839
\(59\) 3.00000 0.390567 0.195283 − 0.980747i \(-0.437437\pi\)
0.195283 + 0.980747i \(0.437437\pi\)
\(60\) 3.00000 0.387298
\(61\) −2.00000 −0.256074 −0.128037 − 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) −6.00000 −0.762001
\(63\) 6.00000 0.755929
\(64\) 1.00000 0.125000
\(65\) −3.00000 −0.372104
\(66\) 0 0
\(67\) 16.0000 1.95471 0.977356 − 0.211604i \(-0.0678686\pi\)
0.977356 + 0.211604i \(0.0678686\pi\)
\(68\) −2.00000 −0.242536
\(69\) −3.00000 −0.361158
\(70\) −1.00000 −0.119523
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) −6.00000 −0.707107
\(73\) −4.00000 −0.468165 −0.234082 − 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 8.00000 0.929981
\(75\) 3.00000 0.346410
\(76\) −1.00000 −0.114708
\(77\) 0 0
\(78\) 9.00000 1.01905
\(79\) 1.00000 0.112509 0.0562544 − 0.998416i \(-0.482084\pi\)
0.0562544 + 0.998416i \(0.482084\pi\)
\(80\) 1.00000 0.111803
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 3.00000 0.329293 0.164646 − 0.986353i \(-0.447352\pi\)
0.164646 + 0.986353i \(0.447352\pi\)
\(84\) 3.00000 0.327327
\(85\) −2.00000 −0.216930
\(86\) −8.00000 −0.862662
\(87\) 18.0000 1.92980
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 − 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) −6.00000 −0.632456
\(91\) −3.00000 −0.314485
\(92\) −1.00000 −0.104257
\(93\) 18.0000 1.86651
\(94\) −10.0000 −1.03142
\(95\) −1.00000 −0.102598
\(96\) −3.00000 −0.306186
\(97\) −6.00000 −0.609208 −0.304604 − 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −19.0000 −1.89057 −0.945285 − 0.326245i \(-0.894217\pi\)
−0.945285 + 0.326245i \(0.894217\pi\)
\(102\) 6.00000 0.594089
\(103\) 14.0000 1.37946 0.689730 − 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) 3.00000 0.294174
\(105\) 3.00000 0.292770
\(106\) −4.00000 −0.388514
\(107\) 16.0000 1.54678 0.773389 − 0.633932i \(-0.218560\pi\)
0.773389 + 0.633932i \(0.218560\pi\)
\(108\) 9.00000 0.866025
\(109\) 14.0000 1.34096 0.670478 − 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 0 0
\(111\) −24.0000 −2.27798
\(112\) 1.00000 0.0944911
\(113\) −3.00000 −0.282216 −0.141108 − 0.989994i \(-0.545067\pi\)
−0.141108 + 0.989994i \(0.545067\pi\)
\(114\) 3.00000 0.280976
\(115\) −1.00000 −0.0932505
\(116\) 6.00000 0.557086
\(117\) −18.0000 −1.66410
\(118\) −3.00000 −0.276172
\(119\) −2.00000 −0.183340
\(120\) −3.00000 −0.273861
\(121\) 0 0
\(122\) 2.00000 0.181071
\(123\) 0 0
\(124\) 6.00000 0.538816
\(125\) 1.00000 0.0894427
\(126\) −6.00000 −0.534522
\(127\) 7.00000 0.621150 0.310575 − 0.950549i \(-0.399478\pi\)
0.310575 + 0.950549i \(0.399478\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 24.0000 2.11308
\(130\) 3.00000 0.263117
\(131\) 3.00000 0.262111 0.131056 − 0.991375i \(-0.458163\pi\)
0.131056 + 0.991375i \(0.458163\pi\)
\(132\) 0 0
\(133\) −1.00000 −0.0867110
\(134\) −16.0000 −1.38219
\(135\) 9.00000 0.774597
\(136\) 2.00000 0.171499
\(137\) −3.00000 −0.256307 −0.128154 − 0.991754i \(-0.540905\pi\)
−0.128154 + 0.991754i \(0.540905\pi\)
\(138\) 3.00000 0.255377
\(139\) −5.00000 −0.424094 −0.212047 − 0.977259i \(-0.568013\pi\)
−0.212047 + 0.977259i \(0.568013\pi\)
\(140\) 1.00000 0.0845154
\(141\) 30.0000 2.52646
\(142\) 8.00000 0.671345
\(143\) 0 0
\(144\) 6.00000 0.500000
\(145\) 6.00000 0.498273
\(146\) 4.00000 0.331042
\(147\) 3.00000 0.247436
\(148\) −8.00000 −0.657596
\(149\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) −3.00000 −0.244949
\(151\) 23.0000 1.87171 0.935857 − 0.352381i \(-0.114628\pi\)
0.935857 + 0.352381i \(0.114628\pi\)
\(152\) 1.00000 0.0811107
\(153\) −12.0000 −0.970143
\(154\) 0 0
\(155\) 6.00000 0.481932
\(156\) −9.00000 −0.720577
\(157\) 13.0000 1.03751 0.518756 − 0.854922i \(-0.326395\pi\)
0.518756 + 0.854922i \(0.326395\pi\)
\(158\) −1.00000 −0.0795557
\(159\) 12.0000 0.951662
\(160\) −1.00000 −0.0790569
\(161\) −1.00000 −0.0788110
\(162\) −9.00000 −0.707107
\(163\) −10.0000 −0.783260 −0.391630 − 0.920123i \(-0.628089\pi\)
−0.391630 + 0.920123i \(0.628089\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) −3.00000 −0.232845
\(167\) 24.0000 1.85718 0.928588 − 0.371113i \(-0.121024\pi\)
0.928588 + 0.371113i \(0.121024\pi\)
\(168\) −3.00000 −0.231455
\(169\) −4.00000 −0.307692
\(170\) 2.00000 0.153393
\(171\) −6.00000 −0.458831
\(172\) 8.00000 0.609994
\(173\) 22.0000 1.67263 0.836315 − 0.548250i \(-0.184706\pi\)
0.836315 + 0.548250i \(0.184706\pi\)
\(174\) −18.0000 −1.36458
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 9.00000 0.676481
\(178\) −10.0000 −0.749532
\(179\) −16.0000 −1.19590 −0.597948 − 0.801535i \(-0.704017\pi\)
−0.597948 + 0.801535i \(0.704017\pi\)
\(180\) 6.00000 0.447214
\(181\) 3.00000 0.222988 0.111494 − 0.993765i \(-0.464436\pi\)
0.111494 + 0.993765i \(0.464436\pi\)
\(182\) 3.00000 0.222375
\(183\) −6.00000 −0.443533
\(184\) 1.00000 0.0737210
\(185\) −8.00000 −0.588172
\(186\) −18.0000 −1.31982
\(187\) 0 0
\(188\) 10.0000 0.729325
\(189\) 9.00000 0.654654
\(190\) 1.00000 0.0725476
\(191\) 5.00000 0.361787 0.180894 − 0.983503i \(-0.442101\pi\)
0.180894 + 0.983503i \(0.442101\pi\)
\(192\) 3.00000 0.216506
\(193\) −9.00000 −0.647834 −0.323917 − 0.946085i \(-0.605000\pi\)
−0.323917 + 0.946085i \(0.605000\pi\)
\(194\) 6.00000 0.430775
\(195\) −9.00000 −0.644503
\(196\) 1.00000 0.0714286
\(197\) 6.00000 0.427482 0.213741 − 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) −18.0000 −1.27599 −0.637993 − 0.770042i \(-0.720235\pi\)
−0.637993 + 0.770042i \(0.720235\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 48.0000 3.38566
\(202\) 19.0000 1.33684
\(203\) 6.00000 0.421117
\(204\) −6.00000 −0.420084
\(205\) 0 0
\(206\) −14.0000 −0.975426
\(207\) −6.00000 −0.417029
\(208\) −3.00000 −0.208013
\(209\) 0 0
\(210\) −3.00000 −0.207020
\(211\) −14.0000 −0.963800 −0.481900 − 0.876226i \(-0.660053\pi\)
−0.481900 + 0.876226i \(0.660053\pi\)
\(212\) 4.00000 0.274721
\(213\) −24.0000 −1.64445
\(214\) −16.0000 −1.09374
\(215\) 8.00000 0.545595
\(216\) −9.00000 −0.612372
\(217\) 6.00000 0.407307
\(218\) −14.0000 −0.948200
\(219\) −12.0000 −0.810885
\(220\) 0 0
\(221\) 6.00000 0.403604
\(222\) 24.0000 1.61077
\(223\) −4.00000 −0.267860 −0.133930 − 0.990991i \(-0.542760\pi\)
−0.133930 + 0.990991i \(0.542760\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 6.00000 0.400000
\(226\) 3.00000 0.199557
\(227\) −20.0000 −1.32745 −0.663723 − 0.747978i \(-0.731025\pi\)
−0.663723 + 0.747978i \(0.731025\pi\)
\(228\) −3.00000 −0.198680
\(229\) −30.0000 −1.98246 −0.991228 − 0.132164i \(-0.957808\pi\)
−0.991228 + 0.132164i \(0.957808\pi\)
\(230\) 1.00000 0.0659380
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) −1.00000 −0.0655122 −0.0327561 − 0.999463i \(-0.510428\pi\)
−0.0327561 + 0.999463i \(0.510428\pi\)
\(234\) 18.0000 1.17670
\(235\) 10.0000 0.652328
\(236\) 3.00000 0.195283
\(237\) 3.00000 0.194871
\(238\) 2.00000 0.129641
\(239\) −9.00000 −0.582162 −0.291081 − 0.956698i \(-0.594015\pi\)
−0.291081 + 0.956698i \(0.594015\pi\)
\(240\) 3.00000 0.193649
\(241\) −22.0000 −1.41714 −0.708572 − 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −2.00000 −0.128037
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) 3.00000 0.190885
\(248\) −6.00000 −0.381000
\(249\) 9.00000 0.570352
\(250\) −1.00000 −0.0632456
\(251\) 20.0000 1.26239 0.631194 − 0.775625i \(-0.282565\pi\)
0.631194 + 0.775625i \(0.282565\pi\)
\(252\) 6.00000 0.377964
\(253\) 0 0
\(254\) −7.00000 −0.439219
\(255\) −6.00000 −0.375735
\(256\) 1.00000 0.0625000
\(257\) −24.0000 −1.49708 −0.748539 − 0.663090i \(-0.769245\pi\)
−0.748539 + 0.663090i \(0.769245\pi\)
\(258\) −24.0000 −1.49417
\(259\) −8.00000 −0.497096
\(260\) −3.00000 −0.186052
\(261\) 36.0000 2.22834
\(262\) −3.00000 −0.185341
\(263\) −23.0000 −1.41824 −0.709120 − 0.705087i \(-0.750908\pi\)
−0.709120 + 0.705087i \(0.750908\pi\)
\(264\) 0 0
\(265\) 4.00000 0.245718
\(266\) 1.00000 0.0613139
\(267\) 30.0000 1.83597
\(268\) 16.0000 0.977356
\(269\) −5.00000 −0.304855 −0.152428 − 0.988315i \(-0.548709\pi\)
−0.152428 + 0.988315i \(0.548709\pi\)
\(270\) −9.00000 −0.547723
\(271\) 12.0000 0.728948 0.364474 − 0.931214i \(-0.381249\pi\)
0.364474 + 0.931214i \(0.381249\pi\)
\(272\) −2.00000 −0.121268
\(273\) −9.00000 −0.544705
\(274\) 3.00000 0.181237
\(275\) 0 0
\(276\) −3.00000 −0.180579
\(277\) 8.00000 0.480673 0.240337 − 0.970690i \(-0.422742\pi\)
0.240337 + 0.970690i \(0.422742\pi\)
\(278\) 5.00000 0.299880
\(279\) 36.0000 2.15526
\(280\) −1.00000 −0.0597614
\(281\) 9.00000 0.536895 0.268447 − 0.963294i \(-0.413489\pi\)
0.268447 + 0.963294i \(0.413489\pi\)
\(282\) −30.0000 −1.78647
\(283\) 19.0000 1.12943 0.564716 − 0.825285i \(-0.308986\pi\)
0.564716 + 0.825285i \(0.308986\pi\)
\(284\) −8.00000 −0.474713
\(285\) −3.00000 −0.177705
\(286\) 0 0
\(287\) 0 0
\(288\) −6.00000 −0.353553
\(289\) −13.0000 −0.764706
\(290\) −6.00000 −0.352332
\(291\) −18.0000 −1.05518
\(292\) −4.00000 −0.234082
\(293\) 7.00000 0.408944 0.204472 − 0.978872i \(-0.434452\pi\)
0.204472 + 0.978872i \(0.434452\pi\)
\(294\) −3.00000 −0.174964
\(295\) 3.00000 0.174667
\(296\) 8.00000 0.464991
\(297\) 0 0
\(298\) 0 0
\(299\) 3.00000 0.173494
\(300\) 3.00000 0.173205
\(301\) 8.00000 0.461112
\(302\) −23.0000 −1.32350
\(303\) −57.0000 −3.27456
\(304\) −1.00000 −0.0573539
\(305\) −2.00000 −0.114520
\(306\) 12.0000 0.685994
\(307\) 16.0000 0.913168 0.456584 − 0.889680i \(-0.349073\pi\)
0.456584 + 0.889680i \(0.349073\pi\)
\(308\) 0 0
\(309\) 42.0000 2.38930
\(310\) −6.00000 −0.340777
\(311\) 30.0000 1.70114 0.850572 − 0.525859i \(-0.176256\pi\)
0.850572 + 0.525859i \(0.176256\pi\)
\(312\) 9.00000 0.509525
\(313\) 22.0000 1.24351 0.621757 − 0.783210i \(-0.286419\pi\)
0.621757 + 0.783210i \(0.286419\pi\)
\(314\) −13.0000 −0.733632
\(315\) 6.00000 0.338062
\(316\) 1.00000 0.0562544
\(317\) −22.0000 −1.23564 −0.617822 − 0.786318i \(-0.711985\pi\)
−0.617822 + 0.786318i \(0.711985\pi\)
\(318\) −12.0000 −0.672927
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) 48.0000 2.67910
\(322\) 1.00000 0.0557278
\(323\) 2.00000 0.111283
\(324\) 9.00000 0.500000
\(325\) −3.00000 −0.166410
\(326\) 10.0000 0.553849
\(327\) 42.0000 2.32261
\(328\) 0 0
\(329\) 10.0000 0.551318
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 − 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) 3.00000 0.164646
\(333\) −48.0000 −2.63038
\(334\) −24.0000 −1.31322
\(335\) 16.0000 0.874173
\(336\) 3.00000 0.163663
\(337\) −17.0000 −0.926049 −0.463025 − 0.886345i \(-0.653236\pi\)
−0.463025 + 0.886345i \(0.653236\pi\)
\(338\) 4.00000 0.217571
\(339\) −9.00000 −0.488813
\(340\) −2.00000 −0.108465
\(341\) 0 0
\(342\) 6.00000 0.324443
\(343\) 1.00000 0.0539949
\(344\) −8.00000 −0.431331
\(345\) −3.00000 −0.161515
\(346\) −22.0000 −1.18273
\(347\) −22.0000 −1.18102 −0.590511 − 0.807030i \(-0.701074\pi\)
−0.590511 + 0.807030i \(0.701074\pi\)
\(348\) 18.0000 0.964901
\(349\) −1.00000 −0.0535288 −0.0267644 − 0.999642i \(-0.508520\pi\)
−0.0267644 + 0.999642i \(0.508520\pi\)
\(350\) −1.00000 −0.0534522
\(351\) −27.0000 −1.44115
\(352\) 0 0
\(353\) 16.0000 0.851594 0.425797 − 0.904819i \(-0.359994\pi\)
0.425797 + 0.904819i \(0.359994\pi\)
\(354\) −9.00000 −0.478345
\(355\) −8.00000 −0.424596
\(356\) 10.0000 0.529999
\(357\) −6.00000 −0.317554
\(358\) 16.0000 0.845626
\(359\) 32.0000 1.68890 0.844448 − 0.535638i \(-0.179929\pi\)
0.844448 + 0.535638i \(0.179929\pi\)
\(360\) −6.00000 −0.316228
\(361\) −18.0000 −0.947368
\(362\) −3.00000 −0.157676
\(363\) 0 0
\(364\) −3.00000 −0.157243
\(365\) −4.00000 −0.209370
\(366\) 6.00000 0.313625
\(367\) −16.0000 −0.835193 −0.417597 − 0.908633i \(-0.637127\pi\)
−0.417597 + 0.908633i \(0.637127\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 0 0
\(370\) 8.00000 0.415900
\(371\) 4.00000 0.207670
\(372\) 18.0000 0.933257
\(373\) −32.0000 −1.65690 −0.828449 − 0.560065i \(-0.810776\pi\)
−0.828449 + 0.560065i \(0.810776\pi\)
\(374\) 0 0
\(375\) 3.00000 0.154919
\(376\) −10.0000 −0.515711
\(377\) −18.0000 −0.927047
\(378\) −9.00000 −0.462910
\(379\) −12.0000 −0.616399 −0.308199 − 0.951322i \(-0.599726\pi\)
−0.308199 + 0.951322i \(0.599726\pi\)
\(380\) −1.00000 −0.0512989
\(381\) 21.0000 1.07586
\(382\) −5.00000 −0.255822
\(383\) 26.0000 1.32854 0.664269 − 0.747494i \(-0.268743\pi\)
0.664269 + 0.747494i \(0.268743\pi\)
\(384\) −3.00000 −0.153093
\(385\) 0 0
\(386\) 9.00000 0.458088
\(387\) 48.0000 2.43998
\(388\) −6.00000 −0.304604
\(389\) 12.0000 0.608424 0.304212 − 0.952604i \(-0.401607\pi\)
0.304212 + 0.952604i \(0.401607\pi\)
\(390\) 9.00000 0.455733
\(391\) 2.00000 0.101144
\(392\) −1.00000 −0.0505076
\(393\) 9.00000 0.453990
\(394\) −6.00000 −0.302276
\(395\) 1.00000 0.0503155
\(396\) 0 0
\(397\) 14.0000 0.702640 0.351320 − 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) 18.0000 0.902258
\(399\) −3.00000 −0.150188
\(400\) 1.00000 0.0500000
\(401\) 22.0000 1.09863 0.549314 − 0.835616i \(-0.314889\pi\)
0.549314 + 0.835616i \(0.314889\pi\)
\(402\) −48.0000 −2.39402
\(403\) −18.0000 −0.896644
\(404\) −19.0000 −0.945285
\(405\) 9.00000 0.447214
\(406\) −6.00000 −0.297775
\(407\) 0 0
\(408\) 6.00000 0.297044
\(409\) −14.0000 −0.692255 −0.346128 − 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 0 0
\(411\) −9.00000 −0.443937
\(412\) 14.0000 0.689730
\(413\) 3.00000 0.147620
\(414\) 6.00000 0.294884
\(415\) 3.00000 0.147264
\(416\) 3.00000 0.147087
\(417\) −15.0000 −0.734553
\(418\) 0 0
\(419\) −21.0000 −1.02592 −0.512959 − 0.858413i \(-0.671451\pi\)
−0.512959 + 0.858413i \(0.671451\pi\)
\(420\) 3.00000 0.146385
\(421\) −32.0000 −1.55958 −0.779792 − 0.626038i \(-0.784675\pi\)
−0.779792 + 0.626038i \(0.784675\pi\)
\(422\) 14.0000 0.681509
\(423\) 60.0000 2.91730
\(424\) −4.00000 −0.194257
\(425\) −2.00000 −0.0970143
\(426\) 24.0000 1.16280
\(427\) −2.00000 −0.0967868
\(428\) 16.0000 0.773389
\(429\) 0 0
\(430\) −8.00000 −0.385794
\(431\) −37.0000 −1.78223 −0.891114 − 0.453780i \(-0.850075\pi\)
−0.891114 + 0.453780i \(0.850075\pi\)
\(432\) 9.00000 0.433013
\(433\) −18.0000 −0.865025 −0.432512 − 0.901628i \(-0.642373\pi\)
−0.432512 + 0.901628i \(0.642373\pi\)
\(434\) −6.00000 −0.288009
\(435\) 18.0000 0.863034
\(436\) 14.0000 0.670478
\(437\) 1.00000 0.0478365
\(438\) 12.0000 0.573382
\(439\) −30.0000 −1.43182 −0.715911 − 0.698192i \(-0.753988\pi\)
−0.715911 + 0.698192i \(0.753988\pi\)
\(440\) 0 0
\(441\) 6.00000 0.285714
\(442\) −6.00000 −0.285391
\(443\) −32.0000 −1.52037 −0.760183 − 0.649709i \(-0.774891\pi\)
−0.760183 + 0.649709i \(0.774891\pi\)
\(444\) −24.0000 −1.13899
\(445\) 10.0000 0.474045
\(446\) 4.00000 0.189405
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) 41.0000 1.93491 0.967455 − 0.253044i \(-0.0814317\pi\)
0.967455 + 0.253044i \(0.0814317\pi\)
\(450\) −6.00000 −0.282843
\(451\) 0 0
\(452\) −3.00000 −0.141108
\(453\) 69.0000 3.24190
\(454\) 20.0000 0.938647
\(455\) −3.00000 −0.140642
\(456\) 3.00000 0.140488
\(457\) −17.0000 −0.795226 −0.397613 − 0.917553i \(-0.630161\pi\)
−0.397613 + 0.917553i \(0.630161\pi\)
\(458\) 30.0000 1.40181
\(459\) −18.0000 −0.840168
\(460\) −1.00000 −0.0466252
\(461\) −6.00000 −0.279448 −0.139724 − 0.990190i \(-0.544622\pi\)
−0.139724 + 0.990190i \(0.544622\pi\)
\(462\) 0 0
\(463\) 23.0000 1.06890 0.534450 − 0.845200i \(-0.320519\pi\)
0.534450 + 0.845200i \(0.320519\pi\)
\(464\) 6.00000 0.278543
\(465\) 18.0000 0.834730
\(466\) 1.00000 0.0463241
\(467\) 27.0000 1.24941 0.624705 − 0.780860i \(-0.285219\pi\)
0.624705 + 0.780860i \(0.285219\pi\)
\(468\) −18.0000 −0.832050
\(469\) 16.0000 0.738811
\(470\) −10.0000 −0.461266
\(471\) 39.0000 1.79703
\(472\) −3.00000 −0.138086
\(473\) 0 0
\(474\) −3.00000 −0.137795
\(475\) −1.00000 −0.0458831
\(476\) −2.00000 −0.0916698
\(477\) 24.0000 1.09888
\(478\) 9.00000 0.411650
\(479\) −24.0000 −1.09659 −0.548294 − 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) −3.00000 −0.136931
\(481\) 24.0000 1.09431
\(482\) 22.0000 1.00207
\(483\) −3.00000 −0.136505
\(484\) 0 0
\(485\) −6.00000 −0.272446
\(486\) 0 0
\(487\) −19.0000 −0.860972 −0.430486 − 0.902597i \(-0.641658\pi\)
−0.430486 + 0.902597i \(0.641658\pi\)
\(488\) 2.00000 0.0905357
\(489\) −30.0000 −1.35665
\(490\) −1.00000 −0.0451754
\(491\) 8.00000 0.361035 0.180517 − 0.983572i \(-0.442223\pi\)
0.180517 + 0.983572i \(0.442223\pi\)
\(492\) 0 0
\(493\) −12.0000 −0.540453
\(494\) −3.00000 −0.134976
\(495\) 0 0
\(496\) 6.00000 0.269408
\(497\) −8.00000 −0.358849
\(498\) −9.00000 −0.403300
\(499\) −8.00000 −0.358129 −0.179065 − 0.983837i \(-0.557307\pi\)
−0.179065 + 0.983837i \(0.557307\pi\)
\(500\) 1.00000 0.0447214
\(501\) 72.0000 3.21672
\(502\) −20.0000 −0.892644
\(503\) −14.0000 −0.624229 −0.312115 − 0.950044i \(-0.601037\pi\)
−0.312115 + 0.950044i \(0.601037\pi\)
\(504\) −6.00000 −0.267261
\(505\) −19.0000 −0.845489
\(506\) 0 0
\(507\) −12.0000 −0.532939
\(508\) 7.00000 0.310575
\(509\) −41.0000 −1.81729 −0.908647 − 0.417566i \(-0.862883\pi\)
−0.908647 + 0.417566i \(0.862883\pi\)
\(510\) 6.00000 0.265684
\(511\) −4.00000 −0.176950
\(512\) −1.00000 −0.0441942
\(513\) −9.00000 −0.397360
\(514\) 24.0000 1.05859
\(515\) 14.0000 0.616914
\(516\) 24.0000 1.05654
\(517\) 0 0
\(518\) 8.00000 0.351500
\(519\) 66.0000 2.89708
\(520\) 3.00000 0.131559
\(521\) −10.0000 −0.438108 −0.219054 − 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) −36.0000 −1.57568
\(523\) −21.0000 −0.918266 −0.459133 − 0.888368i \(-0.651840\pi\)
−0.459133 + 0.888368i \(0.651840\pi\)
\(524\) 3.00000 0.131056
\(525\) 3.00000 0.130931
\(526\) 23.0000 1.00285
\(527\) −12.0000 −0.522728
\(528\) 0 0
\(529\) −22.0000 −0.956522
\(530\) −4.00000 −0.173749
\(531\) 18.0000 0.781133
\(532\) −1.00000 −0.0433555
\(533\) 0 0
\(534\) −30.0000 −1.29823
\(535\) 16.0000 0.691740
\(536\) −16.0000 −0.691095
\(537\) −48.0000 −2.07135
\(538\) 5.00000 0.215565
\(539\) 0 0
\(540\) 9.00000 0.387298
\(541\) 6.00000 0.257960 0.128980 − 0.991647i \(-0.458830\pi\)
0.128980 + 0.991647i \(0.458830\pi\)
\(542\) −12.0000 −0.515444
\(543\) 9.00000 0.386227
\(544\) 2.00000 0.0857493
\(545\) 14.0000 0.599694
\(546\) 9.00000 0.385164
\(547\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(548\) −3.00000 −0.128154
\(549\) −12.0000 −0.512148
\(550\) 0 0
\(551\) −6.00000 −0.255609
\(552\) 3.00000 0.127688
\(553\) 1.00000 0.0425243
\(554\) −8.00000 −0.339887
\(555\) −24.0000 −1.01874
\(556\) −5.00000 −0.212047
\(557\) 42.0000 1.77960 0.889799 − 0.456354i \(-0.150845\pi\)
0.889799 + 0.456354i \(0.150845\pi\)
\(558\) −36.0000 −1.52400
\(559\) −24.0000 −1.01509
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) −9.00000 −0.379642
\(563\) −25.0000 −1.05362 −0.526812 − 0.849982i \(-0.676613\pi\)
−0.526812 + 0.849982i \(0.676613\pi\)
\(564\) 30.0000 1.26323
\(565\) −3.00000 −0.126211
\(566\) −19.0000 −0.798630
\(567\) 9.00000 0.377964
\(568\) 8.00000 0.335673
\(569\) −9.00000 −0.377300 −0.188650 − 0.982044i \(-0.560411\pi\)
−0.188650 + 0.982044i \(0.560411\pi\)
\(570\) 3.00000 0.125656
\(571\) 28.0000 1.17176 0.585882 − 0.810397i \(-0.300748\pi\)
0.585882 + 0.810397i \(0.300748\pi\)
\(572\) 0 0
\(573\) 15.0000 0.626634
\(574\) 0 0
\(575\) −1.00000 −0.0417029
\(576\) 6.00000 0.250000
\(577\) 16.0000 0.666089 0.333044 − 0.942911i \(-0.391924\pi\)
0.333044 + 0.942911i \(0.391924\pi\)
\(578\) 13.0000 0.540729
\(579\) −27.0000 −1.12208
\(580\) 6.00000 0.249136
\(581\) 3.00000 0.124461
\(582\) 18.0000 0.746124
\(583\) 0 0
\(584\) 4.00000 0.165521
\(585\) −18.0000 −0.744208
\(586\) −7.00000 −0.289167
\(587\) 5.00000 0.206372 0.103186 − 0.994662i \(-0.467096\pi\)
0.103186 + 0.994662i \(0.467096\pi\)
\(588\) 3.00000 0.123718
\(589\) −6.00000 −0.247226
\(590\) −3.00000 −0.123508
\(591\) 18.0000 0.740421
\(592\) −8.00000 −0.328798
\(593\) 24.0000 0.985562 0.492781 − 0.870153i \(-0.335980\pi\)
0.492781 + 0.870153i \(0.335980\pi\)
\(594\) 0 0
\(595\) −2.00000 −0.0819920
\(596\) 0 0
\(597\) −54.0000 −2.21007
\(598\) −3.00000 −0.122679
\(599\) −43.0000 −1.75693 −0.878466 − 0.477805i \(-0.841433\pi\)
−0.878466 + 0.477805i \(0.841433\pi\)
\(600\) −3.00000 −0.122474
\(601\) 20.0000 0.815817 0.407909 − 0.913023i \(-0.366258\pi\)
0.407909 + 0.913023i \(0.366258\pi\)
\(602\) −8.00000 −0.326056
\(603\) 96.0000 3.90942
\(604\) 23.0000 0.935857
\(605\) 0 0
\(606\) 57.0000 2.31547
\(607\) 10.0000 0.405887 0.202944 − 0.979190i \(-0.434949\pi\)
0.202944 + 0.979190i \(0.434949\pi\)
\(608\) 1.00000 0.0405554
\(609\) 18.0000 0.729397
\(610\) 2.00000 0.0809776
\(611\) −30.0000 −1.21367
\(612\) −12.0000 −0.485071
\(613\) −26.0000 −1.05013 −0.525065 − 0.851062i \(-0.675959\pi\)
−0.525065 + 0.851062i \(0.675959\pi\)
\(614\) −16.0000 −0.645707
\(615\) 0 0
\(616\) 0 0
\(617\) 22.0000 0.885687 0.442843 − 0.896599i \(-0.353970\pi\)
0.442843 + 0.896599i \(0.353970\pi\)
\(618\) −42.0000 −1.68949
\(619\) 25.0000 1.00483 0.502417 − 0.864625i \(-0.332444\pi\)
0.502417 + 0.864625i \(0.332444\pi\)
\(620\) 6.00000 0.240966
\(621\) −9.00000 −0.361158
\(622\) −30.0000 −1.20289
\(623\) 10.0000 0.400642
\(624\) −9.00000 −0.360288
\(625\) 1.00000 0.0400000
\(626\) −22.0000 −0.879297
\(627\) 0 0
\(628\) 13.0000 0.518756
\(629\) 16.0000 0.637962
\(630\) −6.00000 −0.239046
\(631\) −16.0000 −0.636950 −0.318475 − 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) −1.00000 −0.0397779
\(633\) −42.0000 −1.66935
\(634\) 22.0000 0.873732
\(635\) 7.00000 0.277787
\(636\) 12.0000 0.475831
\(637\) −3.00000 −0.118864
\(638\) 0 0
\(639\) −48.0000 −1.89885
\(640\) −1.00000 −0.0395285
\(641\) −9.00000 −0.355479 −0.177739 − 0.984078i \(-0.556878\pi\)
−0.177739 + 0.984078i \(0.556878\pi\)
\(642\) −48.0000 −1.89441
\(643\) −28.0000 −1.10421 −0.552106 − 0.833774i \(-0.686176\pi\)
−0.552106 + 0.833774i \(0.686176\pi\)
\(644\) −1.00000 −0.0394055
\(645\) 24.0000 0.944999
\(646\) −2.00000 −0.0786889
\(647\) −30.0000 −1.17942 −0.589711 − 0.807614i \(-0.700758\pi\)
−0.589711 + 0.807614i \(0.700758\pi\)
\(648\) −9.00000 −0.353553
\(649\) 0 0
\(650\) 3.00000 0.117670
\(651\) 18.0000 0.705476
\(652\) −10.0000 −0.391630
\(653\) −24.0000 −0.939193 −0.469596 − 0.882881i \(-0.655601\pi\)
−0.469596 + 0.882881i \(0.655601\pi\)
\(654\) −42.0000 −1.64233
\(655\) 3.00000 0.117220
\(656\) 0 0
\(657\) −24.0000 −0.936329
\(658\) −10.0000 −0.389841
\(659\) −36.0000 −1.40236 −0.701180 − 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 0 0
\(661\) −11.0000 −0.427850 −0.213925 − 0.976850i \(-0.568625\pi\)
−0.213925 + 0.976850i \(0.568625\pi\)
\(662\) 4.00000 0.155464
\(663\) 18.0000 0.699062
\(664\) −3.00000 −0.116423
\(665\) −1.00000 −0.0387783
\(666\) 48.0000 1.85996
\(667\) −6.00000 −0.232321
\(668\) 24.0000 0.928588
\(669\) −12.0000 −0.463947
\(670\) −16.0000 −0.618134
\(671\) 0 0
\(672\) −3.00000 −0.115728
\(673\) −7.00000 −0.269830 −0.134915 − 0.990857i \(-0.543076\pi\)
−0.134915 + 0.990857i \(0.543076\pi\)
\(674\) 17.0000 0.654816
\(675\) 9.00000 0.346410
\(676\) −4.00000 −0.153846
\(677\) 13.0000 0.499631 0.249815 − 0.968294i \(-0.419630\pi\)
0.249815 + 0.968294i \(0.419630\pi\)
\(678\) 9.00000 0.345643
\(679\) −6.00000 −0.230259
\(680\) 2.00000 0.0766965
\(681\) −60.0000 −2.29920
\(682\) 0 0
\(683\) −30.0000 −1.14792 −0.573959 − 0.818884i \(-0.694593\pi\)
−0.573959 + 0.818884i \(0.694593\pi\)
\(684\) −6.00000 −0.229416
\(685\) −3.00000 −0.114624
\(686\) −1.00000 −0.0381802
\(687\) −90.0000 −3.43371
\(688\) 8.00000 0.304997
\(689\) −12.0000 −0.457164
\(690\) 3.00000 0.114208
\(691\) 12.0000 0.456502 0.228251 − 0.973602i \(-0.426699\pi\)
0.228251 + 0.973602i \(0.426699\pi\)
\(692\) 22.0000 0.836315
\(693\) 0 0
\(694\) 22.0000 0.835109
\(695\) −5.00000 −0.189661
\(696\) −18.0000 −0.682288
\(697\) 0 0
\(698\) 1.00000 0.0378506
\(699\) −3.00000 −0.113470
\(700\) 1.00000 0.0377964
\(701\) 18.0000 0.679851 0.339925 − 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) 27.0000 1.01905
\(703\) 8.00000 0.301726
\(704\) 0 0
\(705\) 30.0000 1.12987
\(706\) −16.0000 −0.602168
\(707\) −19.0000 −0.714569
\(708\) 9.00000 0.338241
\(709\) −44.0000 −1.65245 −0.826227 − 0.563337i \(-0.809517\pi\)
−0.826227 + 0.563337i \(0.809517\pi\)
\(710\) 8.00000 0.300235
\(711\) 6.00000 0.225018
\(712\) −10.0000 −0.374766
\(713\) −6.00000 −0.224702
\(714\) 6.00000 0.224544
\(715\) 0 0
\(716\) −16.0000 −0.597948
\(717\) −27.0000 −1.00833
\(718\) −32.0000 −1.19423
\(719\) 8.00000 0.298350 0.149175 − 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(720\) 6.00000 0.223607
\(721\) 14.0000 0.521387
\(722\) 18.0000 0.669891
\(723\) −66.0000 −2.45457
\(724\) 3.00000 0.111494
\(725\) 6.00000 0.222834
\(726\) 0 0
\(727\) 6.00000 0.222528 0.111264 − 0.993791i \(-0.464510\pi\)
0.111264 + 0.993791i \(0.464510\pi\)
\(728\) 3.00000 0.111187
\(729\) −27.0000 −1.00000
\(730\) 4.00000 0.148047
\(731\) −16.0000 −0.591781
\(732\) −6.00000 −0.221766
\(733\) −13.0000 −0.480166 −0.240083 − 0.970752i \(-0.577175\pi\)
−0.240083 + 0.970752i \(0.577175\pi\)
\(734\) 16.0000 0.590571
\(735\) 3.00000 0.110657
\(736\) 1.00000 0.0368605
\(737\) 0 0
\(738\) 0 0
\(739\) 36.0000 1.32428 0.662141 − 0.749380i \(-0.269648\pi\)
0.662141 + 0.749380i \(0.269648\pi\)
\(740\) −8.00000 −0.294086
\(741\) 9.00000 0.330623
\(742\) −4.00000 −0.146845
\(743\) 24.0000 0.880475 0.440237 − 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) −18.0000 −0.659912
\(745\) 0 0
\(746\) 32.0000 1.17160
\(747\) 18.0000 0.658586
\(748\) 0 0
\(749\) 16.0000 0.584627
\(750\) −3.00000 −0.109545
\(751\) 25.0000 0.912263 0.456131 − 0.889912i \(-0.349235\pi\)
0.456131 + 0.889912i \(0.349235\pi\)
\(752\) 10.0000 0.364662
\(753\) 60.0000 2.18652
\(754\) 18.0000 0.655521
\(755\) 23.0000 0.837056
\(756\) 9.00000 0.327327
\(757\) −48.0000 −1.74459 −0.872295 − 0.488980i \(-0.837369\pi\)
−0.872295 + 0.488980i \(0.837369\pi\)
\(758\) 12.0000 0.435860
\(759\) 0 0
\(760\) 1.00000 0.0362738
\(761\) −14.0000 −0.507500 −0.253750 − 0.967270i \(-0.581664\pi\)
−0.253750 + 0.967270i \(0.581664\pi\)
\(762\) −21.0000 −0.760750
\(763\) 14.0000 0.506834
\(764\) 5.00000 0.180894
\(765\) −12.0000 −0.433861
\(766\) −26.0000 −0.939418
\(767\) −9.00000 −0.324971
\(768\) 3.00000 0.108253
\(769\) −32.0000 −1.15395 −0.576975 − 0.816762i \(-0.695767\pi\)
−0.576975 + 0.816762i \(0.695767\pi\)
\(770\) 0 0
\(771\) −72.0000 −2.59302
\(772\) −9.00000 −0.323917
\(773\) 3.00000 0.107903 0.0539513 − 0.998544i \(-0.482818\pi\)
0.0539513 + 0.998544i \(0.482818\pi\)
\(774\) −48.0000 −1.72532
\(775\) 6.00000 0.215526
\(776\) 6.00000 0.215387
\(777\) −24.0000 −0.860995
\(778\) −12.0000 −0.430221
\(779\) 0 0
\(780\) −9.00000 −0.322252
\(781\) 0 0
\(782\) −2.00000 −0.0715199
\(783\) 54.0000 1.92980
\(784\) 1.00000 0.0357143
\(785\) 13.0000 0.463990
\(786\) −9.00000 −0.321019
\(787\) −20.0000 −0.712923 −0.356462 − 0.934310i \(-0.616017\pi\)
−0.356462 + 0.934310i \(0.616017\pi\)
\(788\) 6.00000 0.213741
\(789\) −69.0000 −2.45647
\(790\) −1.00000 −0.0355784
\(791\) −3.00000 −0.106668
\(792\) 0 0
\(793\) 6.00000 0.213066
\(794\) −14.0000 −0.496841
\(795\) 12.0000 0.425596
\(796\) −18.0000 −0.637993
\(797\) −7.00000 −0.247953 −0.123976 − 0.992285i \(-0.539565\pi\)
−0.123976 + 0.992285i \(0.539565\pi\)
\(798\) 3.00000 0.106199
\(799\) −20.0000 −0.707549
\(800\) −1.00000 −0.0353553
\(801\) 60.0000 2.12000
\(802\) −22.0000 −0.776847
\(803\) 0 0
\(804\) 48.0000 1.69283
\(805\) −1.00000 −0.0352454
\(806\) 18.0000 0.634023
\(807\) −15.0000 −0.528025
\(808\) 19.0000 0.668418
\(809\) 38.0000 1.33601 0.668004 − 0.744157i \(-0.267149\pi\)
0.668004 + 0.744157i \(0.267149\pi\)
\(810\) −9.00000 −0.316228
\(811\) −20.0000 −0.702295 −0.351147 − 0.936320i \(-0.614208\pi\)
−0.351147 + 0.936320i \(0.614208\pi\)
\(812\) 6.00000 0.210559
\(813\) 36.0000 1.26258
\(814\) 0 0
\(815\) −10.0000 −0.350285
\(816\) −6.00000 −0.210042
\(817\) −8.00000 −0.279885
\(818\) 14.0000 0.489499
\(819\) −18.0000 −0.628971
\(820\) 0 0
\(821\) 6.00000 0.209401 0.104701 − 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) 9.00000 0.313911
\(823\) −32.0000 −1.11545 −0.557725 − 0.830026i \(-0.688326\pi\)
−0.557725 + 0.830026i \(0.688326\pi\)
\(824\) −14.0000 −0.487713
\(825\) 0 0
\(826\) −3.00000 −0.104383
\(827\) 24.0000 0.834562 0.417281 − 0.908778i \(-0.362983\pi\)
0.417281 + 0.908778i \(0.362983\pi\)
\(828\) −6.00000 −0.208514
\(829\) −15.0000 −0.520972 −0.260486 − 0.965478i \(-0.583883\pi\)
−0.260486 + 0.965478i \(0.583883\pi\)
\(830\) −3.00000 −0.104132
\(831\) 24.0000 0.832551
\(832\) −3.00000 −0.104006
\(833\) −2.00000 −0.0692959
\(834\) 15.0000 0.519408
\(835\) 24.0000 0.830554
\(836\) 0 0
\(837\) 54.0000 1.86651
\(838\) 21.0000 0.725433
\(839\) 26.0000 0.897620 0.448810 − 0.893627i \(-0.351848\pi\)
0.448810 + 0.893627i \(0.351848\pi\)
\(840\) −3.00000 −0.103510
\(841\) 7.00000 0.241379
\(842\) 32.0000 1.10279
\(843\) 27.0000 0.929929
\(844\) −14.0000 −0.481900
\(845\) −4.00000 −0.137604
\(846\) −60.0000 −2.06284
\(847\) 0 0
\(848\) 4.00000 0.137361
\(849\) 57.0000 1.95623
\(850\) 2.00000 0.0685994
\(851\) 8.00000 0.274236
\(852\) −24.0000 −0.822226
\(853\) 21.0000 0.719026 0.359513 − 0.933140i \(-0.382943\pi\)
0.359513 + 0.933140i \(0.382943\pi\)
\(854\) 2.00000 0.0684386
\(855\) −6.00000 −0.205196
\(856\) −16.0000 −0.546869
\(857\) −12.0000 −0.409912 −0.204956 − 0.978771i \(-0.565705\pi\)
−0.204956 + 0.978771i \(0.565705\pi\)
\(858\) 0 0
\(859\) 52.0000 1.77422 0.887109 − 0.461561i \(-0.152710\pi\)
0.887109 + 0.461561i \(0.152710\pi\)
\(860\) 8.00000 0.272798
\(861\) 0 0
\(862\) 37.0000 1.26023
\(863\) 20.0000 0.680808 0.340404 − 0.940279i \(-0.389436\pi\)
0.340404 + 0.940279i \(0.389436\pi\)
\(864\) −9.00000 −0.306186
\(865\) 22.0000 0.748022
\(866\) 18.0000 0.611665
\(867\) −39.0000 −1.32451
\(868\) 6.00000 0.203653
\(869\) 0 0
\(870\) −18.0000 −0.610257
\(871\) −48.0000 −1.62642
\(872\) −14.0000 −0.474100
\(873\) −36.0000 −1.21842
\(874\) −1.00000 −0.0338255
\(875\) 1.00000 0.0338062
\(876\) −12.0000 −0.405442
\(877\) 52.0000 1.75592 0.877958 − 0.478738i \(-0.158906\pi\)
0.877958 + 0.478738i \(0.158906\pi\)
\(878\) 30.0000 1.01245
\(879\) 21.0000 0.708312
\(880\) 0 0
\(881\) −52.0000 −1.75192 −0.875962 − 0.482380i \(-0.839773\pi\)
−0.875962 + 0.482380i \(0.839773\pi\)
\(882\) −6.00000 −0.202031
\(883\) 44.0000 1.48072 0.740359 − 0.672212i \(-0.234656\pi\)
0.740359 + 0.672212i \(0.234656\pi\)
\(884\) 6.00000 0.201802
\(885\) 9.00000 0.302532
\(886\) 32.0000 1.07506
\(887\) 54.0000 1.81314 0.906571 − 0.422053i \(-0.138690\pi\)
0.906571 + 0.422053i \(0.138690\pi\)
\(888\) 24.0000 0.805387
\(889\) 7.00000 0.234772
\(890\) −10.0000 −0.335201
\(891\) 0 0
\(892\) −4.00000 −0.133930
\(893\) −10.0000 −0.334637
\(894\) 0 0
\(895\) −16.0000 −0.534821
\(896\) −1.00000 −0.0334077
\(897\) 9.00000 0.300501
\(898\) −41.0000 −1.36819
\(899\) 36.0000 1.20067
\(900\) 6.00000 0.200000
\(901\) −8.00000 −0.266519
\(902\) 0 0
\(903\) 24.0000 0.798670
\(904\) 3.00000 0.0997785
\(905\) 3.00000 0.0997234
\(906\) −69.0000 −2.29237
\(907\) −16.0000 −0.531271 −0.265636 − 0.964073i \(-0.585582\pi\)
−0.265636 + 0.964073i \(0.585582\pi\)
\(908\) −20.0000 −0.663723
\(909\) −114.000 −3.78114
\(910\) 3.00000 0.0994490
\(911\) −1.00000 −0.0331315 −0.0165657 − 0.999863i \(-0.505273\pi\)
−0.0165657 + 0.999863i \(0.505273\pi\)
\(912\) −3.00000 −0.0993399
\(913\) 0 0
\(914\) 17.0000 0.562310
\(915\) −6.00000 −0.198354
\(916\) −30.0000 −0.991228
\(917\) 3.00000 0.0990687
\(918\) 18.0000 0.594089
\(919\) −40.0000 −1.31948 −0.659739 − 0.751495i \(-0.729333\pi\)
−0.659739 + 0.751495i \(0.729333\pi\)
\(920\) 1.00000 0.0329690
\(921\) 48.0000 1.58165
\(922\) 6.00000 0.197599
\(923\) 24.0000 0.789970
\(924\) 0 0
\(925\) −8.00000 −0.263038
\(926\) −23.0000 −0.755827
\(927\) 84.0000 2.75892
\(928\) −6.00000 −0.196960
\(929\) 12.0000 0.393707 0.196854 − 0.980433i \(-0.436928\pi\)
0.196854 + 0.980433i \(0.436928\pi\)
\(930\) −18.0000 −0.590243
\(931\) −1.00000 −0.0327737
\(932\) −1.00000 −0.0327561
\(933\) 90.0000 2.94647
\(934\) −27.0000 −0.883467
\(935\) 0 0
\(936\) 18.0000 0.588348
\(937\) −38.0000 −1.24141 −0.620703 − 0.784046i \(-0.713153\pi\)
−0.620703 + 0.784046i \(0.713153\pi\)
\(938\) −16.0000 −0.522419
\(939\) 66.0000 2.15383
\(940\) 10.0000 0.326164
\(941\) 14.0000 0.456387 0.228193 − 0.973616i \(-0.426718\pi\)
0.228193 + 0.973616i \(0.426718\pi\)
\(942\) −39.0000 −1.27069
\(943\) 0 0
\(944\) 3.00000 0.0976417
\(945\) 9.00000 0.292770
\(946\) 0 0
\(947\) −36.0000 −1.16984 −0.584921 − 0.811090i \(-0.698875\pi\)
−0.584921 + 0.811090i \(0.698875\pi\)
\(948\) 3.00000 0.0974355
\(949\) 12.0000 0.389536
\(950\) 1.00000 0.0324443
\(951\) −66.0000 −2.14020
\(952\) 2.00000 0.0648204
\(953\) 31.0000 1.00419 0.502094 − 0.864813i \(-0.332563\pi\)
0.502094 + 0.864813i \(0.332563\pi\)
\(954\) −24.0000 −0.777029
\(955\) 5.00000 0.161796
\(956\) −9.00000 −0.291081
\(957\) 0 0
\(958\) 24.0000 0.775405
\(959\) −3.00000 −0.0968751
\(960\) 3.00000 0.0968246
\(961\) 5.00000 0.161290
\(962\) −24.0000 −0.773791
\(963\) 96.0000 3.09356
\(964\) −22.0000 −0.708572
\(965\) −9.00000 −0.289720
\(966\) 3.00000 0.0965234
\(967\) −32.0000 −1.02905 −0.514525 − 0.857475i \(-0.672032\pi\)
−0.514525 + 0.857475i \(0.672032\pi\)
\(968\) 0 0
\(969\) 6.00000 0.192748
\(970\) 6.00000 0.192648
\(971\) 41.0000 1.31575 0.657876 − 0.753126i \(-0.271455\pi\)
0.657876 + 0.753126i \(0.271455\pi\)
\(972\) 0 0
\(973\) −5.00000 −0.160293
\(974\) 19.0000 0.608799
\(975\) −9.00000 −0.288231
\(976\) −2.00000 −0.0640184
\(977\) −3.00000 −0.0959785 −0.0479893 − 0.998848i \(-0.515281\pi\)
−0.0479893 + 0.998848i \(0.515281\pi\)
\(978\) 30.0000 0.959294
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) 84.0000 2.68191
\(982\) −8.00000 −0.255290
\(983\) −16.0000 −0.510321 −0.255160 − 0.966899i \(-0.582128\pi\)
−0.255160 + 0.966899i \(0.582128\pi\)
\(984\) 0 0
\(985\) 6.00000 0.191176
\(986\) 12.0000 0.382158
\(987\) 30.0000 0.954911
\(988\) 3.00000 0.0954427
\(989\) −8.00000 −0.254385
\(990\) 0 0
\(991\) 5.00000 0.158830 0.0794151 − 0.996842i \(-0.474695\pi\)
0.0794151 + 0.996842i \(0.474695\pi\)
\(992\) −6.00000 −0.190500
\(993\) −12.0000 −0.380808
\(994\) 8.00000 0.253745
\(995\) −18.0000 −0.570638
\(996\) 9.00000 0.285176
\(997\) 55.0000 1.74187 0.870934 − 0.491400i \(-0.163515\pi\)
0.870934 + 0.491400i \(0.163515\pi\)
\(998\) 8.00000 0.253236
\(999\) −72.0000 −2.27798
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.p.1.1 ✓ 1
11.10 odd 2 8470.2.a.bh.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.p.1.1 ✓ 1 1.1 even 1 trivial
8470.2.a.bh.1.1 yes 1 11.10 odd 2