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Properties

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

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [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: \(1\)
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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +1.00000 q^{10} +1.00000 q^{12} -1.00000 q^{13} -1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +6.00000 q^{17} +2.00000 q^{18} -7.00000 q^{19} -1.00000 q^{20} +1.00000 q^{21} +3.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{26} -5.00000 q^{27} +1.00000 q^{28} +6.00000 q^{29} +1.00000 q^{30} +2.00000 q^{31} -1.00000 q^{32} -6.00000 q^{34} -1.00000 q^{35} -2.00000 q^{36} -4.00000 q^{37} +7.00000 q^{38} -1.00000 q^{39} +1.00000 q^{40} -1.00000 q^{42} -4.00000 q^{43} +2.00000 q^{45} -3.00000 q^{46} -6.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} +6.00000 q^{51} -1.00000 q^{52} +5.00000 q^{54} -1.00000 q^{56} -7.00000 q^{57} -6.00000 q^{58} -3.00000 q^{59} -1.00000 q^{60} +2.00000 q^{61} -2.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} +1.00000 q^{65} -4.00000 q^{67} +6.00000 q^{68} +3.00000 q^{69} +1.00000 q^{70} +2.00000 q^{72} -4.00000 q^{73} +4.00000 q^{74} +1.00000 q^{75} -7.00000 q^{76} +1.00000 q^{78} +17.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +9.00000 q^{83} +1.00000 q^{84} -6.00000 q^{85} +4.00000 q^{86} +6.00000 q^{87} -18.0000 q^{89} -2.00000 q^{90} -1.00000 q^{91} +3.00000 q^{92} +2.00000 q^{93} +6.00000 q^{94} +7.00000 q^{95} -1.00000 q^{96} -10.0000 q^{97} -1.00000 q^{98} +O(q^{100})\)

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
\(13\) −1.00000 −0.277350 −0.138675 − 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) −1.00000 −0.267261
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 − 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 2.00000 0.471405
\(19\) −7.00000 −1.60591 −0.802955 − 0.596040i \(-0.796740\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) −1.00000 −0.223607
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 3.00000 0.625543 0.312772 − 0.949828i \(-0.398743\pi\)
0.312772 + 0.949828i \(0.398743\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 1.00000 0.196116
\(27\) −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
\(31\) 2.00000 0.359211 0.179605 − 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −6.00000 −1.02899
\(35\) −1.00000 −0.169031
\(36\) −2.00000 −0.333333
\(37\) −4.00000 −0.657596 −0.328798 − 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 7.00000 1.13555
\(39\) −1.00000 −0.160128
\(40\) 1.00000 0.158114
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) −1.00000 −0.154303
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 2.00000 0.298142
\(46\) −3.00000 −0.442326
\(47\) −6.00000 −0.875190 −0.437595 − 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 6.00000 0.840168
\(52\) −1.00000 −0.138675
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 5.00000 0.680414
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) −7.00000 −0.927173
\(58\) −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 \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −2.00000 −0.254000
\(63\) −2.00000 −0.251976
\(64\) 1.00000 0.125000
\(65\) 1.00000 0.124035
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 6.00000 0.727607
\(69\) 3.00000 0.361158
\(70\) 1.00000 0.119523
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 2.00000 0.235702
\(73\) −4.00000 −0.468165 −0.234082 − 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 4.00000 0.464991
\(75\) 1.00000 0.115470
\(76\) −7.00000 −0.802955
\(77\) 0 0
\(78\) 1.00000 0.113228
\(79\) 17.0000 1.91265 0.956325 − 0.292306i \(-0.0944227\pi\)
0.956325 + 0.292306i \(0.0944227\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 9.00000 0.987878 0.493939 − 0.869496i \(-0.335557\pi\)
0.493939 + 0.869496i \(0.335557\pi\)
\(84\) 1.00000 0.109109
\(85\) −6.00000 −0.650791
\(86\) 4.00000 0.431331
\(87\) 6.00000 0.643268
\(88\) 0 0
\(89\) −18.0000 −1.90800 −0.953998 − 0.299813i \(-0.903076\pi\)
−0.953998 + 0.299813i \(0.903076\pi\)
\(90\) −2.00000 −0.210819
\(91\) −1.00000 −0.104828
\(92\) 3.00000 0.312772
\(93\) 2.00000 0.207390
\(94\) 6.00000 0.618853
\(95\) 7.00000 0.718185
\(96\) −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 \(0.452312\pi\)
\(102\) −6.00000 −0.594089
\(103\) 14.0000 1.37946 0.689730 − 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) 1.00000 0.0980581
\(105\) −1.00000 −0.0975900
\(106\) 0 0
\(107\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) −5.00000 −0.481125
\(109\) 2.00000 0.191565 0.0957826 − 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) −4.00000 −0.379663
\(112\) 1.00000 0.0944911
\(113\) −15.0000 −1.41108 −0.705541 − 0.708669i \(-0.749296\pi\)
−0.705541 + 0.708669i \(0.749296\pi\)
\(114\) 7.00000 0.655610
\(115\) −3.00000 −0.279751
\(116\) 6.00000 0.557086
\(117\) 2.00000 0.184900
\(118\) 3.00000 0.276172
\(119\) 6.00000 0.550019
\(120\) 1.00000 0.0912871
\(121\) 0 0
\(122\) −2.00000 −0.181071
\(123\) 0 0
\(124\) 2.00000 0.179605
\(125\) −1.00000 −0.0894427
\(126\) 2.00000 0.178174
\(127\) −13.0000 −1.15356 −0.576782 − 0.816898i \(-0.695692\pi\)
−0.576782 + 0.816898i \(0.695692\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −4.00000 −0.352180
\(130\) −1.00000 −0.0877058
\(131\) −3.00000 −0.262111 −0.131056 − 0.991375i \(-0.541837\pi\)
−0.131056 + 0.991375i \(0.541837\pi\)
\(132\) 0 0
\(133\) −7.00000 −0.606977
\(134\) 4.00000 0.345547
\(135\) 5.00000 0.430331
\(136\) −6.00000 −0.514496
\(137\) −15.0000 −1.28154 −0.640768 − 0.767734i \(-0.721384\pi\)
−0.640768 + 0.767734i \(0.721384\pi\)
\(138\) −3.00000 −0.255377
\(139\) 5.00000 0.424094 0.212047 − 0.977259i \(-0.431987\pi\)
0.212047 + 0.977259i \(0.431987\pi\)
\(140\) −1.00000 −0.0845154
\(141\) −6.00000 −0.505291
\(142\) 0 0
\(143\) 0 0
\(144\) −2.00000 −0.166667
\(145\) −6.00000 −0.498273
\(146\) 4.00000 0.331042
\(147\) 1.00000 0.0824786
\(148\) −4.00000 −0.328798
\(149\) 12.0000 0.983078 0.491539 − 0.870855i \(-0.336434\pi\)
0.491539 + 0.870855i \(0.336434\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −1.00000 −0.0813788 −0.0406894 − 0.999172i \(-0.512955\pi\)
−0.0406894 + 0.999172i \(0.512955\pi\)
\(152\) 7.00000 0.567775
\(153\) −12.0000 −0.970143
\(154\) 0 0
\(155\) −2.00000 −0.160644
\(156\) −1.00000 −0.0800641
\(157\) −1.00000 −0.0798087 −0.0399043 − 0.999204i \(-0.512705\pi\)
−0.0399043 + 0.999204i \(0.512705\pi\)
\(158\) −17.0000 −1.35245
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) 3.00000 0.236433
\(162\) −1.00000 −0.0785674
\(163\) 14.0000 1.09656 0.548282 − 0.836293i \(-0.315282\pi\)
0.548282 + 0.836293i \(0.315282\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) −9.00000 −0.698535
\(167\) −12.0000 −0.928588 −0.464294 − 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) −1.00000 −0.0771517
\(169\) −12.0000 −0.923077
\(170\) 6.00000 0.460179
\(171\) 14.0000 1.07061
\(172\) −4.00000 −0.304997
\(173\) −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
\(178\) 18.0000 1.34916
\(179\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 2.00000 0.149071
\(181\) −19.0000 −1.41226 −0.706129 − 0.708083i \(-0.749560\pi\)
−0.706129 + 0.708083i \(0.749560\pi\)
\(182\) 1.00000 0.0741249
\(183\) 2.00000 0.147844
\(184\) −3.00000 −0.221163
\(185\) 4.00000 0.294086
\(186\) −2.00000 −0.146647
\(187\) 0 0
\(188\) −6.00000 −0.437595
\(189\) −5.00000 −0.363696
\(190\) −7.00000 −0.507833
\(191\) −3.00000 −0.217072 −0.108536 − 0.994092i \(-0.534616\pi\)
−0.108536 + 0.994092i \(0.534616\pi\)
\(192\) 1.00000 0.0721688
\(193\) 11.0000 0.791797 0.395899 − 0.918294i \(-0.370433\pi\)
0.395899 + 0.918294i \(0.370433\pi\)
\(194\) 10.0000 0.717958
\(195\) 1.00000 0.0716115
\(196\) 1.00000 0.0714286
\(197\) 18.0000 1.28245 0.641223 − 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 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
\(201\) −4.00000 −0.282138
\(202\) −3.00000 −0.211079
\(203\) 6.00000 0.421117
\(204\) 6.00000 0.420084
\(205\) 0 0
\(206\) −14.0000 −0.975426
\(207\) −6.00000 −0.417029
\(208\) −1.00000 −0.0693375
\(209\) 0 0
\(210\) 1.00000 0.0690066
\(211\) −22.0000 −1.51454 −0.757271 − 0.653101i \(-0.773468\pi\)
−0.757271 + 0.653101i \(0.773468\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 4.00000 0.272798
\(216\) 5.00000 0.340207
\(217\) 2.00000 0.135769
\(218\) −2.00000 −0.135457
\(219\) −4.00000 −0.270295
\(220\) 0 0
\(221\) −6.00000 −0.403604
\(222\) 4.00000 0.268462
\(223\) −16.0000 −1.07144 −0.535720 − 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −2.00000 −0.133333
\(226\) 15.0000 0.997785
\(227\) −12.0000 −0.796468 −0.398234 − 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) −7.00000 −0.463586
\(229\) −10.0000 −0.660819 −0.330409 − 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 3.00000 0.197814
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) −21.0000 −1.37576 −0.687878 − 0.725826i \(-0.741458\pi\)
−0.687878 + 0.725826i \(0.741458\pi\)
\(234\) −2.00000 −0.130744
\(235\) 6.00000 0.391397
\(236\) −3.00000 −0.195283
\(237\) 17.0000 1.10427
\(238\) −6.00000 −0.388922
\(239\) −9.00000 −0.582162 −0.291081 − 0.956698i \(-0.594015\pi\)
−0.291081 + 0.956698i \(0.594015\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 26.0000 1.67481 0.837404 − 0.546585i \(-0.184072\pi\)
0.837404 + 0.546585i \(0.184072\pi\)
\(242\) 0 0
\(243\) 16.0000 1.02640
\(244\) 2.00000 0.128037
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) 7.00000 0.445399
\(248\) −2.00000 −0.127000
\(249\) 9.00000 0.570352
\(250\) 1.00000 0.0632456
\(251\) −12.0000 −0.757433 −0.378717 − 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) −2.00000 −0.125988
\(253\) 0 0
\(254\) 13.0000 0.815693
\(255\) −6.00000 −0.375735
\(256\) 1.00000 0.0625000
\(257\) −12.0000 −0.748539 −0.374270 − 0.927320i \(-0.622107\pi\)
−0.374270 + 0.927320i \(0.622107\pi\)
\(258\) 4.00000 0.249029
\(259\) −4.00000 −0.248548
\(260\) 1.00000 0.0620174
\(261\) −12.0000 −0.742781
\(262\) 3.00000 0.185341
\(263\) 21.0000 1.29492 0.647458 − 0.762101i \(-0.275832\pi\)
0.647458 + 0.762101i \(0.275832\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 7.00000 0.429198
\(267\) −18.0000 −1.10158
\(268\) −4.00000 −0.244339
\(269\) −27.0000 −1.64622 −0.823110 − 0.567883i \(-0.807763\pi\)
−0.823110 + 0.567883i \(0.807763\pi\)
\(270\) −5.00000 −0.304290
\(271\) −16.0000 −0.971931 −0.485965 − 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 6.00000 0.363803
\(273\) −1.00000 −0.0605228
\(274\) 15.0000 0.906183
\(275\) 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
\(279\) −4.00000 −0.239474
\(280\) 1.00000 0.0597614
\(281\) 9.00000 0.536895 0.268447 − 0.963294i \(-0.413489\pi\)
0.268447 + 0.963294i \(0.413489\pi\)
\(282\) 6.00000 0.357295
\(283\) −31.0000 −1.84276 −0.921379 − 0.388664i \(-0.872937\pi\)
−0.921379 + 0.388664i \(0.872937\pi\)
\(284\) 0 0
\(285\) 7.00000 0.414644
\(286\) 0 0
\(287\) 0 0
\(288\) 2.00000 0.117851
\(289\) 19.0000 1.11765
\(290\) 6.00000 0.352332
\(291\) −10.0000 −0.586210
\(292\) −4.00000 −0.234082
\(293\) −3.00000 −0.175262 −0.0876309 − 0.996153i \(-0.527930\pi\)
−0.0876309 + 0.996153i \(0.527930\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 3.00000 0.174667
\(296\) 4.00000 0.232495
\(297\) 0 0
\(298\) −12.0000 −0.695141
\(299\) −3.00000 −0.173494
\(300\) 1.00000 0.0577350
\(301\) −4.00000 −0.230556
\(302\) 1.00000 0.0575435
\(303\) 3.00000 0.172345
\(304\) −7.00000 −0.401478
\(305\) −2.00000 −0.114520
\(306\) 12.0000 0.685994
\(307\) 8.00000 0.456584 0.228292 − 0.973593i \(-0.426686\pi\)
0.228292 + 0.973593i \(0.426686\pi\)
\(308\) 0 0
\(309\) 14.0000 0.796432
\(310\) 2.00000 0.113592
\(311\) −6.00000 −0.340229 −0.170114 − 0.985424i \(-0.554414\pi\)
−0.170114 + 0.985424i \(0.554414\pi\)
\(312\) 1.00000 0.0566139
\(313\) 14.0000 0.791327 0.395663 − 0.918396i \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) 1.00000 0.0564333
\(315\) 2.00000 0.112687
\(316\) 17.0000 0.956325
\(317\) −18.0000 −1.01098 −0.505490 − 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) −3.00000 −0.167183
\(323\) −42.0000 −2.33694
\(324\) 1.00000 0.0555556
\(325\) −1.00000 −0.0554700
\(326\) −14.0000 −0.775388
\(327\) 2.00000 0.110600
\(328\) 0 0
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) −16.0000 −0.879440 −0.439720 − 0.898135i \(-0.644922\pi\)
−0.439720 + 0.898135i \(0.644922\pi\)
\(332\) 9.00000 0.493939
\(333\) 8.00000 0.438397
\(334\) 12.0000 0.656611
\(335\) 4.00000 0.218543
\(336\) 1.00000 0.0545545
\(337\) −13.0000 −0.708155 −0.354078 − 0.935216i \(-0.615205\pi\)
−0.354078 + 0.935216i \(0.615205\pi\)
\(338\) 12.0000 0.652714
\(339\) −15.0000 −0.814688
\(340\) −6.00000 −0.325396
\(341\) 0 0
\(342\) −14.0000 −0.757033
\(343\) 1.00000 0.0539949
\(344\) 4.00000 0.215666
\(345\) −3.00000 −0.161515
\(346\) 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
\(349\) −31.0000 −1.65939 −0.829696 − 0.558216i \(-0.811486\pi\)
−0.829696 + 0.558216i \(0.811486\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 5.00000 0.266880
\(352\) 0 0
\(353\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 3.00000 0.159448
\(355\) 0 0
\(356\) −18.0000 −0.953998
\(357\) 6.00000 0.317554
\(358\) 0 0
\(359\) −24.0000 −1.26667 −0.633336 − 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) −2.00000 −0.105409
\(361\) 30.0000 1.57895
\(362\) 19.0000 0.998618
\(363\) 0 0
\(364\) −1.00000 −0.0524142
\(365\) 4.00000 0.209370
\(366\) −2.00000 −0.104542
\(367\) 32.0000 1.67039 0.835193 − 0.549957i \(-0.185356\pi\)
0.835193 + 0.549957i \(0.185356\pi\)
\(368\) 3.00000 0.156386
\(369\) 0 0
\(370\) −4.00000 −0.207950
\(371\) 0 0
\(372\) 2.00000 0.103695
\(373\) 32.0000 1.65690 0.828449 − 0.560065i \(-0.189224\pi\)
0.828449 + 0.560065i \(0.189224\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 6.00000 0.309426
\(377\) −6.00000 −0.309016
\(378\) 5.00000 0.257172
\(379\) −4.00000 −0.205466 −0.102733 − 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 7.00000 0.359092
\(381\) −13.0000 −0.666010
\(382\) 3.00000 0.153493
\(383\) 6.00000 0.306586 0.153293 − 0.988181i \(-0.451012\pi\)
0.153293 + 0.988181i \(0.451012\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −11.0000 −0.559885
\(387\) 8.00000 0.406663
\(388\) −10.0000 −0.507673
\(389\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) −1.00000 −0.0506370
\(391\) 18.0000 0.910299
\(392\) −1.00000 −0.0505076
\(393\) −3.00000 −0.151330
\(394\) −18.0000 −0.906827
\(395\) −17.0000 −0.855363
\(396\) 0 0
\(397\) 2.00000 0.100377 0.0501886 − 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) −2.00000 −0.100251
\(399\) −7.00000 −0.350438
\(400\) 1.00000 0.0500000
\(401\) −18.0000 −0.898877 −0.449439 − 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 4.00000 0.199502
\(403\) −2.00000 −0.0996271
\(404\) 3.00000 0.149256
\(405\) −1.00000 −0.0496904
\(406\) −6.00000 −0.297775
\(407\) 0 0
\(408\) −6.00000 −0.297044
\(409\) −22.0000 −1.08783 −0.543915 − 0.839140i \(-0.683059\pi\)
−0.543915 + 0.839140i \(0.683059\pi\)
\(410\) 0 0
\(411\) −15.0000 −0.739895
\(412\) 14.0000 0.689730
\(413\) −3.00000 −0.147620
\(414\) 6.00000 0.294884
\(415\) −9.00000 −0.441793
\(416\) 1.00000 0.0490290
\(417\) 5.00000 0.244851
\(418\) 0 0
\(419\) 21.0000 1.02592 0.512959 − 0.858413i \(-0.328549\pi\)
0.512959 + 0.858413i \(0.328549\pi\)
\(420\) −1.00000 −0.0487950
\(421\) 8.00000 0.389896 0.194948 − 0.980814i \(-0.437546\pi\)
0.194948 + 0.980814i \(0.437546\pi\)
\(422\) 22.0000 1.07094
\(423\) 12.0000 0.583460
\(424\) 0 0
\(425\) 6.00000 0.291043
\(426\) 0 0
\(427\) 2.00000 0.0967868
\(428\) 0 0
\(429\) 0 0
\(430\) −4.00000 −0.192897
\(431\) 3.00000 0.144505 0.0722525 − 0.997386i \(-0.476981\pi\)
0.0722525 + 0.997386i \(0.476981\pi\)
\(432\) −5.00000 −0.240563
\(433\) 26.0000 1.24948 0.624740 − 0.780833i \(-0.285205\pi\)
0.624740 + 0.780833i \(0.285205\pi\)
\(434\) −2.00000 −0.0960031
\(435\) −6.00000 −0.287678
\(436\) 2.00000 0.0957826
\(437\) −21.0000 −1.00457
\(438\) 4.00000 0.191127
\(439\) −10.0000 −0.477274 −0.238637 − 0.971109i \(-0.576701\pi\)
−0.238637 + 0.971109i \(0.576701\pi\)
\(440\) 0 0
\(441\) −2.00000 −0.0952381
\(442\) 6.00000 0.285391
\(443\) −12.0000 −0.570137 −0.285069 − 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) −4.00000 −0.189832
\(445\) 18.0000 0.853282
\(446\) 16.0000 0.757622
\(447\) 12.0000 0.567581
\(448\) 1.00000 0.0472456
\(449\) −15.0000 −0.707894 −0.353947 − 0.935266i \(-0.615161\pi\)
−0.353947 + 0.935266i \(0.615161\pi\)
\(450\) 2.00000 0.0942809
\(451\) 0 0
\(452\) −15.0000 −0.705541
\(453\) −1.00000 −0.0469841
\(454\) 12.0000 0.563188
\(455\) 1.00000 0.0468807
\(456\) 7.00000 0.327805
\(457\) 35.0000 1.63723 0.818615 − 0.574342i \(-0.194742\pi\)
0.818615 + 0.574342i \(0.194742\pi\)
\(458\) 10.0000 0.467269
\(459\) −30.0000 −1.40028
\(460\) −3.00000 −0.139876
\(461\) −18.0000 −0.838344 −0.419172 − 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) 0 0
\(463\) −13.0000 −0.604161 −0.302081 − 0.953282i \(-0.597681\pi\)
−0.302081 + 0.953282i \(0.597681\pi\)
\(464\) 6.00000 0.278543
\(465\) −2.00000 −0.0927478
\(466\) 21.0000 0.972806
\(467\) 9.00000 0.416470 0.208235 − 0.978079i \(-0.433228\pi\)
0.208235 + 0.978079i \(0.433228\pi\)
\(468\) 2.00000 0.0924500
\(469\) −4.00000 −0.184703
\(470\) −6.00000 −0.276759
\(471\) −1.00000 −0.0460776
\(472\) 3.00000 0.138086
\(473\) 0 0
\(474\) −17.0000 −0.780836
\(475\) −7.00000 −0.321182
\(476\) 6.00000 0.275010
\(477\) 0 0
\(478\) 9.00000 0.411650
\(479\) −12.0000 −0.548294 −0.274147 − 0.961688i \(-0.588395\pi\)
−0.274147 + 0.961688i \(0.588395\pi\)
\(480\) 1.00000 0.0456435
\(481\) 4.00000 0.182384
\(482\) −26.0000 −1.18427
\(483\) 3.00000 0.136505
\(484\) 0 0
\(485\) 10.0000 0.454077
\(486\) −16.0000 −0.725775
\(487\) 17.0000 0.770344 0.385172 − 0.922845i \(-0.374142\pi\)
0.385172 + 0.922845i \(0.374142\pi\)
\(488\) −2.00000 −0.0905357
\(489\) 14.0000 0.633102
\(490\) 1.00000 0.0451754
\(491\) −36.0000 −1.62466 −0.812329 − 0.583200i \(-0.801800\pi\)
−0.812329 + 0.583200i \(0.801800\pi\)
\(492\) 0 0
\(493\) 36.0000 1.62136
\(494\) −7.00000 −0.314945
\(495\) 0 0
\(496\) 2.00000 0.0898027
\(497\) 0 0
\(498\) −9.00000 −0.403300
\(499\) −40.0000 −1.79065 −0.895323 − 0.445418i \(-0.853055\pi\)
−0.895323 + 0.445418i \(0.853055\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −12.0000 −0.536120
\(502\) 12.0000 0.535586
\(503\) −6.00000 −0.267527 −0.133763 − 0.991013i \(-0.542706\pi\)
−0.133763 + 0.991013i \(0.542706\pi\)
\(504\) 2.00000 0.0890871
\(505\) −3.00000 −0.133498
\(506\) 0 0
\(507\) −12.0000 −0.532939
\(508\) −13.0000 −0.576782
\(509\) −15.0000 −0.664863 −0.332432 − 0.943127i \(-0.607869\pi\)
−0.332432 + 0.943127i \(0.607869\pi\)
\(510\) 6.00000 0.265684
\(511\) −4.00000 −0.176950
\(512\) −1.00000 −0.0441942
\(513\) 35.0000 1.54529
\(514\) 12.0000 0.529297
\(515\) −14.0000 −0.616914
\(516\) −4.00000 −0.176090
\(517\) 0 0
\(518\) 4.00000 0.175750
\(519\) −6.00000 −0.263371
\(520\) −1.00000 −0.0438529
\(521\) 18.0000 0.788594 0.394297 − 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) 12.0000 0.525226
\(523\) 17.0000 0.743358 0.371679 − 0.928361i \(-0.378782\pi\)
0.371679 + 0.928361i \(0.378782\pi\)
\(524\) −3.00000 −0.131056
\(525\) 1.00000 0.0436436
\(526\) −21.0000 −0.915644
\(527\) 12.0000 0.522728
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) 6.00000 0.260378
\(532\) −7.00000 −0.303488
\(533\) 0 0
\(534\) 18.0000 0.778936
\(535\) 0 0
\(536\) 4.00000 0.172774
\(537\) 0 0
\(538\) 27.0000 1.16405
\(539\) 0 0
\(540\) 5.00000 0.215166
\(541\) −22.0000 −0.945854 −0.472927 − 0.881102i \(-0.656803\pi\)
−0.472927 + 0.881102i \(0.656803\pi\)
\(542\) 16.0000 0.687259
\(543\) −19.0000 −0.815368
\(544\) −6.00000 −0.257248
\(545\) −2.00000 −0.0856706
\(546\) 1.00000 0.0427960
\(547\) 8.00000 0.342055 0.171028 − 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) −15.0000 −0.640768
\(549\) −4.00000 −0.170716
\(550\) 0 0
\(551\) −42.0000 −1.78926
\(552\) −3.00000 −0.127688
\(553\) 17.0000 0.722914
\(554\) −8.00000 −0.339887
\(555\) 4.00000 0.169791
\(556\) 5.00000 0.212047
\(557\) −30.0000 −1.27114 −0.635570 − 0.772043i \(-0.719235\pi\)
−0.635570 + 0.772043i \(0.719235\pi\)
\(558\) 4.00000 0.169334
\(559\) 4.00000 0.169182
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) −9.00000 −0.379642
\(563\) 21.0000 0.885044 0.442522 − 0.896758i \(-0.354084\pi\)
0.442522 + 0.896758i \(0.354084\pi\)
\(564\) −6.00000 −0.252646
\(565\) 15.0000 0.631055
\(566\) 31.0000 1.30303
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −33.0000 −1.38343 −0.691716 − 0.722170i \(-0.743145\pi\)
−0.691716 + 0.722170i \(0.743145\pi\)
\(570\) −7.00000 −0.293198
\(571\) 32.0000 1.33916 0.669579 − 0.742741i \(-0.266474\pi\)
0.669579 + 0.742741i \(0.266474\pi\)
\(572\) 0 0
\(573\) −3.00000 −0.125327
\(574\) 0 0
\(575\) 3.00000 0.125109
\(576\) −2.00000 −0.0833333
\(577\) 8.00000 0.333044 0.166522 − 0.986038i \(-0.446746\pi\)
0.166522 + 0.986038i \(0.446746\pi\)
\(578\) −19.0000 −0.790296
\(579\) 11.0000 0.457144
\(580\) −6.00000 −0.249136
\(581\) 9.00000 0.373383
\(582\) 10.0000 0.414513
\(583\) 0 0
\(584\) 4.00000 0.165521
\(585\) −2.00000 −0.0826898
\(586\) 3.00000 0.123929
\(587\) −33.0000 −1.36206 −0.681028 − 0.732257i \(-0.738467\pi\)
−0.681028 + 0.732257i \(0.738467\pi\)
\(588\) 1.00000 0.0412393
\(589\) −14.0000 −0.576860
\(590\) −3.00000 −0.123508
\(591\) 18.0000 0.740421
\(592\) −4.00000 −0.164399
\(593\) 24.0000 0.985562 0.492781 − 0.870153i \(-0.335980\pi\)
0.492781 + 0.870153i \(0.335980\pi\)
\(594\) 0 0
\(595\) −6.00000 −0.245976
\(596\) 12.0000 0.491539
\(597\) 2.00000 0.0818546
\(598\) 3.00000 0.122679
\(599\) 21.0000 0.858037 0.429018 − 0.903296i \(-0.358860\pi\)
0.429018 + 0.903296i \(0.358860\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 44.0000 1.79480 0.897399 − 0.441221i \(-0.145454\pi\)
0.897399 + 0.441221i \(0.145454\pi\)
\(602\) 4.00000 0.163028
\(603\) 8.00000 0.325785
\(604\) −1.00000 −0.0406894
\(605\) 0 0
\(606\) −3.00000 −0.121867
\(607\) −22.0000 −0.892952 −0.446476 − 0.894795i \(-0.647321\pi\)
−0.446476 + 0.894795i \(0.647321\pi\)
\(608\) 7.00000 0.283887
\(609\) 6.00000 0.243132
\(610\) 2.00000 0.0809776
\(611\) 6.00000 0.242734
\(612\) −12.0000 −0.485071
\(613\) −46.0000 −1.85792 −0.928961 − 0.370177i \(-0.879297\pi\)
−0.928961 + 0.370177i \(0.879297\pi\)
\(614\) −8.00000 −0.322854
\(615\) 0 0
\(616\) 0 0
\(617\) 6.00000 0.241551 0.120775 − 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) −14.0000 −0.563163
\(619\) −49.0000 −1.96948 −0.984738 − 0.174042i \(-0.944317\pi\)
−0.984738 + 0.174042i \(0.944317\pi\)
\(620\) −2.00000 −0.0803219
\(621\) −15.0000 −0.601929
\(622\) 6.00000 0.240578
\(623\) −18.0000 −0.721155
\(624\) −1.00000 −0.0400320
\(625\) 1.00000 0.0400000
\(626\) −14.0000 −0.559553
\(627\) 0 0
\(628\) −1.00000 −0.0399043
\(629\) −24.0000 −0.956943
\(630\) −2.00000 −0.0796819
\(631\) −16.0000 −0.636950 −0.318475 − 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) −17.0000 −0.676224
\(633\) −22.0000 −0.874421
\(634\) 18.0000 0.714871
\(635\) 13.0000 0.515889
\(636\) 0 0
\(637\) −1.00000 −0.0396214
\(638\) 0 0
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) 39.0000 1.54041 0.770204 − 0.637798i \(-0.220155\pi\)
0.770204 + 0.637798i \(0.220155\pi\)
\(642\) 0 0
\(643\) 44.0000 1.73519 0.867595 − 0.497271i \(-0.165665\pi\)
0.867595 + 0.497271i \(0.165665\pi\)
\(644\) 3.00000 0.118217
\(645\) 4.00000 0.157500
\(646\) 42.0000 1.65247
\(647\) −42.0000 −1.65119 −0.825595 − 0.564263i \(-0.809160\pi\)
−0.825595 + 0.564263i \(0.809160\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) 1.00000 0.0392232
\(651\) 2.00000 0.0783862
\(652\) 14.0000 0.548282
\(653\) −12.0000 −0.469596 −0.234798 − 0.972044i \(-0.575443\pi\)
−0.234798 + 0.972044i \(0.575443\pi\)
\(654\) −2.00000 −0.0782062
\(655\) 3.00000 0.117220
\(656\) 0 0
\(657\) 8.00000 0.312110
\(658\) 6.00000 0.233904
\(659\) 12.0000 0.467454 0.233727 − 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 0 0
\(661\) 35.0000 1.36134 0.680671 − 0.732589i \(-0.261688\pi\)
0.680671 + 0.732589i \(0.261688\pi\)
\(662\) 16.0000 0.621858
\(663\) −6.00000 −0.233021
\(664\) −9.00000 −0.349268
\(665\) 7.00000 0.271448
\(666\) −8.00000 −0.309994
\(667\) 18.0000 0.696963
\(668\) −12.0000 −0.464294
\(669\) −16.0000 −0.618596
\(670\) −4.00000 −0.154533
\(671\) 0 0
\(672\) −1.00000 −0.0385758
\(673\) 29.0000 1.11787 0.558934 − 0.829212i \(-0.311211\pi\)
0.558934 + 0.829212i \(0.311211\pi\)
\(674\) 13.0000 0.500741
\(675\) −5.00000 −0.192450
\(676\) −12.0000 −0.461538
\(677\) 15.0000 0.576497 0.288248 − 0.957556i \(-0.406927\pi\)
0.288248 + 0.957556i \(0.406927\pi\)
\(678\) 15.0000 0.576072
\(679\) −10.0000 −0.383765
\(680\) 6.00000 0.230089
\(681\) −12.0000 −0.459841
\(682\) 0 0
\(683\) 30.0000 1.14792 0.573959 − 0.818884i \(-0.305407\pi\)
0.573959 + 0.818884i \(0.305407\pi\)
\(684\) 14.0000 0.535303
\(685\) 15.0000 0.573121
\(686\) −1.00000 −0.0381802
\(687\) −10.0000 −0.381524
\(688\) −4.00000 −0.152499
\(689\) 0 0
\(690\) 3.00000 0.114208
\(691\) 44.0000 1.67384 0.836919 − 0.547326i \(-0.184354\pi\)
0.836919 + 0.547326i \(0.184354\pi\)
\(692\) −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
\(698\) 31.0000 1.17337
\(699\) −21.0000 −0.794293
\(700\) 1.00000 0.0377964
\(701\) 6.00000 0.226617 0.113308 − 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) −5.00000 −0.188713
\(703\) 28.0000 1.05604
\(704\) 0 0
\(705\) 6.00000 0.225973
\(706\) 0 0
\(707\) 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 \(0.676227\pi\)
\(710\) 0 0
\(711\) −34.0000 −1.27510
\(712\) 18.0000 0.674579
\(713\) 6.00000 0.224702
\(714\) −6.00000 −0.224544
\(715\) 0 0
\(716\) 0 0
\(717\) −9.00000 −0.336111
\(718\) 24.0000 0.895672
\(719\) 24.0000 0.895049 0.447524 − 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 2.00000 0.0745356
\(721\) 14.0000 0.521387
\(722\) −30.0000 −1.11648
\(723\) 26.0000 0.966950
\(724\) −19.0000 −0.706129
\(725\) 6.00000 0.222834
\(726\) 0 0
\(727\) 38.0000 1.40934 0.704671 − 0.709534i \(-0.251095\pi\)
0.704671 + 0.709534i \(0.251095\pi\)
\(728\) 1.00000 0.0370625
\(729\) 13.0000 0.481481
\(730\) −4.00000 −0.148047
\(731\) −24.0000 −0.887672
\(732\) 2.00000 0.0739221
\(733\) −31.0000 −1.14501 −0.572506 − 0.819901i \(-0.694029\pi\)
−0.572506 + 0.819901i \(0.694029\pi\)
\(734\) −32.0000 −1.18114
\(735\) −1.00000 −0.0368856
\(736\) −3.00000 −0.110581
\(737\) 0 0
\(738\) 0 0
\(739\) −16.0000 −0.588570 −0.294285 − 0.955718i \(-0.595081\pi\)
−0.294285 + 0.955718i \(0.595081\pi\)
\(740\) 4.00000 0.147043
\(741\) 7.00000 0.257151
\(742\) 0 0
\(743\) −24.0000 −0.880475 −0.440237 − 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) −2.00000 −0.0733236
\(745\) −12.0000 −0.439646
\(746\) −32.0000 −1.17160
\(747\) −18.0000 −0.658586
\(748\) 0 0
\(749\) 0 0
\(750\) 1.00000 0.0365148
\(751\) 17.0000 0.620339 0.310169 − 0.950681i \(-0.399614\pi\)
0.310169 + 0.950681i \(0.399614\pi\)
\(752\) −6.00000 −0.218797
\(753\) −12.0000 −0.437304
\(754\) 6.00000 0.218507
\(755\) 1.00000 0.0363937
\(756\) −5.00000 −0.181848
\(757\) −52.0000 −1.88997 −0.944986 − 0.327111i \(-0.893925\pi\)
−0.944986 + 0.327111i \(0.893925\pi\)
\(758\) 4.00000 0.145287
\(759\) 0 0
\(760\) −7.00000 −0.253917
\(761\) −30.0000 −1.08750 −0.543750 − 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) 13.0000 0.470940
\(763\) 2.00000 0.0724049
\(764\) −3.00000 −0.108536
\(765\) 12.0000 0.433861
\(766\) −6.00000 −0.216789
\(767\) 3.00000 0.108324
\(768\) 1.00000 0.0360844
\(769\) −16.0000 −0.576975 −0.288487 − 0.957484i \(-0.593152\pi\)
−0.288487 + 0.957484i \(0.593152\pi\)
\(770\) 0 0
\(771\) −12.0000 −0.432169
\(772\) 11.0000 0.395899
\(773\) −39.0000 −1.40273 −0.701366 − 0.712801i \(-0.747426\pi\)
−0.701366 + 0.712801i \(0.747426\pi\)
\(774\) −8.00000 −0.287554
\(775\) 2.00000 0.0718421
\(776\) 10.0000 0.358979
\(777\) −4.00000 −0.143499
\(778\) 0 0
\(779\) 0 0
\(780\) 1.00000 0.0358057
\(781\) 0 0
\(782\) −18.0000 −0.643679
\(783\) −30.0000 −1.07211
\(784\) 1.00000 0.0357143
\(785\) 1.00000 0.0356915
\(786\) 3.00000 0.107006
\(787\) −4.00000 −0.142585 −0.0712923 − 0.997455i \(-0.522712\pi\)
−0.0712923 + 0.997455i \(0.522712\pi\)
\(788\) 18.0000 0.641223
\(789\) 21.0000 0.747620
\(790\) 17.0000 0.604833
\(791\) −15.0000 −0.533339
\(792\) 0 0
\(793\) −2.00000 −0.0710221
\(794\) −2.00000 −0.0709773
\(795\) 0 0
\(796\) 2.00000 0.0708881
\(797\) −21.0000 −0.743858 −0.371929 − 0.928261i \(-0.621304\pi\)
−0.371929 + 0.928261i \(0.621304\pi\)
\(798\) 7.00000 0.247797
\(799\) −36.0000 −1.27359
\(800\) −1.00000 −0.0353553
\(801\) 36.0000 1.27200
\(802\) 18.0000 0.635602
\(803\) 0 0
\(804\) −4.00000 −0.141069
\(805\) −3.00000 −0.105736
\(806\) 2.00000 0.0704470
\(807\) −27.0000 −0.950445
\(808\) −3.00000 −0.105540
\(809\) −42.0000 −1.47664 −0.738321 − 0.674450i \(-0.764381\pi\)
−0.738321 + 0.674450i \(0.764381\pi\)
\(810\) 1.00000 0.0351364
\(811\) 20.0000 0.702295 0.351147 − 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) 6.00000 0.210559
\(813\) −16.0000 −0.561144
\(814\) 0 0
\(815\) −14.0000 −0.490399
\(816\) 6.00000 0.210042
\(817\) 28.0000 0.979596
\(818\) 22.0000 0.769212
\(819\) 2.00000 0.0698857
\(820\) 0 0
\(821\) 42.0000 1.46581 0.732905 − 0.680331i \(-0.238164\pi\)
0.732905 + 0.680331i \(0.238164\pi\)
\(822\) 15.0000 0.523185
\(823\) 32.0000 1.11545 0.557725 − 0.830026i \(-0.311674\pi\)
0.557725 + 0.830026i \(0.311674\pi\)
\(824\) −14.0000 −0.487713
\(825\) 0 0
\(826\) 3.00000 0.104383
\(827\) −48.0000 −1.66912 −0.834562 − 0.550914i \(-0.814279\pi\)
−0.834562 + 0.550914i \(0.814279\pi\)
\(828\) −6.00000 −0.208514
\(829\) 47.0000 1.63238 0.816189 − 0.577785i \(-0.196083\pi\)
0.816189 + 0.577785i \(0.196083\pi\)
\(830\) 9.00000 0.312395
\(831\) 8.00000 0.277517
\(832\) −1.00000 −0.0346688
\(833\) 6.00000 0.207888
\(834\) −5.00000 −0.173136
\(835\) 12.0000 0.415277
\(836\) 0 0
\(837\) −10.0000 −0.345651
\(838\) −21.0000 −0.725433
\(839\) 54.0000 1.86429 0.932144 − 0.362089i \(-0.117936\pi\)
0.932144 + 0.362089i \(0.117936\pi\)
\(840\) 1.00000 0.0345033
\(841\) 7.00000 0.241379
\(842\) −8.00000 −0.275698
\(843\) 9.00000 0.309976
\(844\) −22.0000 −0.757271
\(845\) 12.0000 0.412813
\(846\) −12.0000 −0.412568
\(847\) 0 0
\(848\) 0 0
\(849\) −31.0000 −1.06392
\(850\) −6.00000 −0.205798
\(851\) −12.0000 −0.411355
\(852\) 0 0
\(853\) −49.0000 −1.67773 −0.838864 − 0.544341i \(-0.816780\pi\)
−0.838864 + 0.544341i \(0.816780\pi\)
\(854\) −2.00000 −0.0684386
\(855\) −14.0000 −0.478790
\(856\) 0 0
\(857\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) −52.0000 −1.77422 −0.887109 − 0.461561i \(-0.847290\pi\)
−0.887109 + 0.461561i \(0.847290\pi\)
\(860\) 4.00000 0.136399
\(861\) 0 0
\(862\) −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
\(866\) −26.0000 −0.883516
\(867\) 19.0000 0.645274
\(868\) 2.00000 0.0678844
\(869\) 0 0
\(870\) 6.00000 0.203419
\(871\) 4.00000 0.135535
\(872\) −2.00000 −0.0677285
\(873\) 20.0000 0.676897
\(874\) 21.0000 0.710336
\(875\) −1.00000 −0.0338062
\(876\) −4.00000 −0.135147
\(877\) 44.0000 1.48577 0.742887 − 0.669417i \(-0.233456\pi\)
0.742887 + 0.669417i \(0.233456\pi\)
\(878\) 10.0000 0.337484
\(879\) −3.00000 −0.101187
\(880\) 0 0
\(881\) −48.0000 −1.61716 −0.808581 − 0.588386i \(-0.799764\pi\)
−0.808581 + 0.588386i \(0.799764\pi\)
\(882\) 2.00000 0.0673435
\(883\) −16.0000 −0.538443 −0.269221 − 0.963078i \(-0.586766\pi\)
−0.269221 + 0.963078i \(0.586766\pi\)
\(884\) −6.00000 −0.201802
\(885\) 3.00000 0.100844
\(886\) 12.0000 0.403148
\(887\) −42.0000 −1.41022 −0.705111 − 0.709097i \(-0.749103\pi\)
−0.705111 + 0.709097i \(0.749103\pi\)
\(888\) 4.00000 0.134231
\(889\) −13.0000 −0.436006
\(890\) −18.0000 −0.603361
\(891\) 0 0
\(892\) −16.0000 −0.535720
\(893\) 42.0000 1.40548
\(894\) −12.0000 −0.401340
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) −3.00000 −0.100167
\(898\) 15.0000 0.500556
\(899\) 12.0000 0.400222
\(900\) −2.00000 −0.0666667
\(901\) 0 0
\(902\) 0 0
\(903\) −4.00000 −0.133112
\(904\) 15.0000 0.498893
\(905\) 19.0000 0.631581
\(906\) 1.00000 0.0332228
\(907\) −16.0000 −0.531271 −0.265636 − 0.964073i \(-0.585582\pi\)
−0.265636 + 0.964073i \(0.585582\pi\)
\(908\) −12.0000 −0.398234
\(909\) −6.00000 −0.199007
\(910\) −1.00000 −0.0331497
\(911\) −33.0000 −1.09334 −0.546669 − 0.837349i \(-0.684105\pi\)
−0.546669 + 0.837349i \(0.684105\pi\)
\(912\) −7.00000 −0.231793
\(913\) 0 0
\(914\) −35.0000 −1.15770
\(915\) −2.00000 −0.0661180
\(916\) −10.0000 −0.330409
\(917\) −3.00000 −0.0990687
\(918\) 30.0000 0.990148
\(919\) 56.0000 1.84727 0.923635 − 0.383274i \(-0.125203\pi\)
0.923635 + 0.383274i \(0.125203\pi\)
\(920\) 3.00000 0.0989071
\(921\) 8.00000 0.263609
\(922\) 18.0000 0.592798
\(923\) 0 0
\(924\) 0 0
\(925\) −4.00000 −0.131519
\(926\) 13.0000 0.427207
\(927\) −28.0000 −0.919641
\(928\) −6.00000 −0.196960
\(929\) −60.0000 −1.96854 −0.984268 − 0.176682i \(-0.943464\pi\)
−0.984268 + 0.176682i \(0.943464\pi\)
\(930\) 2.00000 0.0655826
\(931\) −7.00000 −0.229416
\(932\) −21.0000 −0.687878
\(933\) −6.00000 −0.196431
\(934\) −9.00000 −0.294489
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) 2.00000 0.0653372 0.0326686 − 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) 4.00000 0.130605
\(939\) 14.0000 0.456873
\(940\) 6.00000 0.195698
\(941\) −6.00000 −0.195594 −0.0977972 − 0.995206i \(-0.531180\pi\)
−0.0977972 + 0.995206i \(0.531180\pi\)
\(942\) 1.00000 0.0325818
\(943\) 0 0
\(944\) −3.00000 −0.0976417
\(945\) 5.00000 0.162650
\(946\) 0 0
\(947\) 48.0000 1.55979 0.779895 − 0.625910i \(-0.215272\pi\)
0.779895 + 0.625910i \(0.215272\pi\)
\(948\) 17.0000 0.552134
\(949\) 4.00000 0.129845
\(950\) 7.00000 0.227110
\(951\) −18.0000 −0.583690
\(952\) −6.00000 −0.194461
\(953\) 27.0000 0.874616 0.437308 − 0.899312i \(-0.355932\pi\)
0.437308 + 0.899312i \(0.355932\pi\)
\(954\) 0 0
\(955\) 3.00000 0.0970777
\(956\) −9.00000 −0.291081
\(957\) 0 0
\(958\) 12.0000 0.387702
\(959\) −15.0000 −0.484375
\(960\) −1.00000 −0.0322749
\(961\) −27.0000 −0.870968
\(962\) −4.00000 −0.128965
\(963\) 0 0
\(964\) 26.0000 0.837404
\(965\) −11.0000 −0.354103
\(966\) −3.00000 −0.0965234
\(967\) 8.00000 0.257263 0.128631 − 0.991692i \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) 0 0
\(969\) −42.0000 −1.34923
\(970\) −10.0000 −0.321081
\(971\) −9.00000 −0.288824 −0.144412 − 0.989518i \(-0.546129\pi\)
−0.144412 + 0.989518i \(0.546129\pi\)
\(972\) 16.0000 0.513200
\(973\) 5.00000 0.160293
\(974\) −17.0000 −0.544715
\(975\) −1.00000 −0.0320256
\(976\) 2.00000 0.0640184
\(977\) 33.0000 1.05576 0.527882 − 0.849318i \(-0.322986\pi\)
0.527882 + 0.849318i \(0.322986\pi\)
\(978\) −14.0000 −0.447671
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) −4.00000 −0.127710
\(982\) 36.0000 1.14881
\(983\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) −18.0000 −0.573528
\(986\) −36.0000 −1.14647
\(987\) −6.00000 −0.190982
\(988\) 7.00000 0.222700
\(989\) −12.0000 −0.381578
\(990\) 0 0
\(991\) −19.0000 −0.603555 −0.301777 − 0.953378i \(-0.597580\pi\)
−0.301777 + 0.953378i \(0.597580\pi\)
\(992\) −2.00000 −0.0635001
\(993\) −16.0000 −0.507745
\(994\) 0 0
\(995\) −2.00000 −0.0634043
\(996\) 9.00000 0.285176
\(997\) −19.0000 −0.601736 −0.300868 − 0.953666i \(-0.597276\pi\)
−0.300868 + 0.953666i \(0.597276\pi\)
\(998\) 40.0000 1.26618
\(999\) 20.0000 0.632772
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.m.1.1 ✓ 1
11.10 odd 2 8470.2.a.bb.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.m.1.1 ✓ 1 1.1 even 1 trivial
8470.2.a.bb.1.1 yes 1 11.10 odd 2