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Properties

Label 8670.2.a.r.1.1
Level $8670$
Weight $2$
Character 8670.1
Self dual yes
Analytic conductor $69.230$
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] = [8670,2,Mod(1,8670)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8670, 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("8670.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8670 = 2 \cdot 3 \cdot 5 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8670.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: \(69.2302985525\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 510)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 8670.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -4.00000 q^{11} -1.00000 q^{12} +2.00000 q^{13} -1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{18} -4.00000 q^{19} +1.00000 q^{20} -4.00000 q^{22} -4.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} -2.00000 q^{29} -1.00000 q^{30} +4.00000 q^{31} +1.00000 q^{32} +4.00000 q^{33} +1.00000 q^{36} +6.00000 q^{37} -4.00000 q^{38} -2.00000 q^{39} +1.00000 q^{40} +10.0000 q^{41} -8.00000 q^{43} -4.00000 q^{44} +1.00000 q^{45} -4.00000 q^{46} -1.00000 q^{48} -7.00000 q^{49} +1.00000 q^{50} +2.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} -4.00000 q^{55} +4.00000 q^{57} -2.00000 q^{58} -8.00000 q^{59} -1.00000 q^{60} -10.0000 q^{61} +4.00000 q^{62} +1.00000 q^{64} +2.00000 q^{65} +4.00000 q^{66} -8.00000 q^{67} +4.00000 q^{69} -8.00000 q^{71} +1.00000 q^{72} +2.00000 q^{73} +6.00000 q^{74} -1.00000 q^{75} -4.00000 q^{76} -2.00000 q^{78} -4.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} +4.00000 q^{83} -8.00000 q^{86} +2.00000 q^{87} -4.00000 q^{88} -14.0000 q^{89} +1.00000 q^{90} -4.00000 q^{92} -4.00000 q^{93} -4.00000 q^{95} -1.00000 q^{96} +10.0000 q^{97} -7.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −1.00000 −0.408248
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) −4.00000 −1.20605 −0.603023 − 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) −1.00000 −0.288675
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 0 0
\(18\) 1.00000 0.235702
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) −4.00000 −0.852803
\(23\) −4.00000 −0.834058 −0.417029 − 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 2.00000 0.392232
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 − 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) −1.00000 −0.182574
\(31\) 4.00000 0.718421 0.359211 − 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 1.00000 0.176777
\(33\) 4.00000 0.696311
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 6.00000 0.986394 0.493197 − 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) −4.00000 −0.648886
\(39\) −2.00000 −0.320256
\(40\) 1.00000 0.158114
\(41\) 10.0000 1.56174 0.780869 − 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 − 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) −4.00000 −0.603023
\(45\) 1.00000 0.149071
\(46\) −4.00000 −0.589768
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −1.00000 −0.144338
\(49\) −7.00000 −1.00000
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) −1.00000 −0.136083
\(55\) −4.00000 −0.539360
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) −2.00000 −0.262613
\(59\) −8.00000 −1.04151 −0.520756 − 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) −1.00000 −0.129099
\(61\) −10.0000 −1.28037 −0.640184 − 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 4.00000 0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) 4.00000 0.492366
\(67\) −8.00000 −0.977356 −0.488678 − 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 0 0
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 1.00000 0.117851
\(73\) 2.00000 0.234082 0.117041 − 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 6.00000 0.697486
\(75\) −1.00000 −0.115470
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) −2.00000 −0.226455
\(79\) −4.00000 −0.450035 −0.225018 − 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 10.0000 1.10432
\(83\) 4.00000 0.439057 0.219529 − 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −8.00000 −0.862662
\(87\) 2.00000 0.214423
\(88\) −4.00000 −0.426401
\(89\) −14.0000 −1.48400 −0.741999 − 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) −4.00000 −0.417029
\(93\) −4.00000 −0.414781
\(94\) 0 0
\(95\) −4.00000 −0.410391
\(96\) −1.00000 −0.102062
\(97\) 10.0000 1.01535 0.507673 − 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) −7.00000 −0.707107
\(99\) −4.00000 −0.402015
\(100\) 1.00000 0.100000
\(101\) −6.00000 −0.597022 −0.298511 − 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) 16.0000 1.57653 0.788263 − 0.615338i \(-0.210980\pi\)
0.788263 + 0.615338i \(0.210980\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) −12.0000 −1.16008 −0.580042 − 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 14.0000 1.34096 0.670478 − 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) −4.00000 −0.381385
\(111\) −6.00000 −0.569495
\(112\) 0 0
\(113\) −18.0000 −1.69330 −0.846649 − 0.532152i \(-0.821383\pi\)
−0.846649 + 0.532152i \(0.821383\pi\)
\(114\) 4.00000 0.374634
\(115\) −4.00000 −0.373002
\(116\) −2.00000 −0.185695
\(117\) 2.00000 0.184900
\(118\) −8.00000 −0.736460
\(119\) 0 0
\(120\) −1.00000 −0.0912871
\(121\) 5.00000 0.454545
\(122\) −10.0000 −0.905357
\(123\) −10.0000 −0.901670
\(124\) 4.00000 0.359211
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −16.0000 −1.41977 −0.709885 − 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 1.00000 0.0883883
\(129\) 8.00000 0.704361
\(130\) 2.00000 0.175412
\(131\) −4.00000 −0.349482 −0.174741 − 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 4.00000 0.348155
\(133\) 0 0
\(134\) −8.00000 −0.691095
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) −6.00000 −0.512615 −0.256307 − 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 4.00000 0.340503
\(139\) 4.00000 0.339276 0.169638 − 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −8.00000 −0.671345
\(143\) −8.00000 −0.668994
\(144\) 1.00000 0.0833333
\(145\) −2.00000 −0.166091
\(146\) 2.00000 0.165521
\(147\) 7.00000 0.577350
\(148\) 6.00000 0.493197
\(149\) 2.00000 0.163846 0.0819232 − 0.996639i \(-0.473894\pi\)
0.0819232 + 0.996639i \(0.473894\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) −4.00000 −0.324443
\(153\) 0 0
\(154\) 0 0
\(155\) 4.00000 0.321288
\(156\) −2.00000 −0.160128
\(157\) −6.00000 −0.478852 −0.239426 − 0.970915i \(-0.576959\pi\)
−0.239426 + 0.970915i \(0.576959\pi\)
\(158\) −4.00000 −0.318223
\(159\) −6.00000 −0.475831
\(160\) 1.00000 0.0790569
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −12.0000 −0.939913 −0.469956 − 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) 10.0000 0.780869
\(165\) 4.00000 0.311400
\(166\) 4.00000 0.310460
\(167\) −12.0000 −0.928588 −0.464294 − 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) −8.00000 −0.609994
\(173\) −2.00000 −0.152057 −0.0760286 − 0.997106i \(-0.524224\pi\)
−0.0760286 + 0.997106i \(0.524224\pi\)
\(174\) 2.00000 0.151620
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) 8.00000 0.601317
\(178\) −14.0000 −1.04934
\(179\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 1.00000 0.0745356
\(181\) 14.0000 1.04061 0.520306 − 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 0 0
\(183\) 10.0000 0.739221
\(184\) −4.00000 −0.294884
\(185\) 6.00000 0.441129
\(186\) −4.00000 −0.293294
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) −4.00000 −0.290191
\(191\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −6.00000 −0.431889 −0.215945 − 0.976406i \(-0.569283\pi\)
−0.215945 + 0.976406i \(0.569283\pi\)
\(194\) 10.0000 0.717958
\(195\) −2.00000 −0.143223
\(196\) −7.00000 −0.500000
\(197\) −18.0000 −1.28245 −0.641223 − 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) −4.00000 −0.284268
\(199\) 4.00000 0.283552 0.141776 − 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 1.00000 0.0707107
\(201\) 8.00000 0.564276
\(202\) −6.00000 −0.422159
\(203\) 0 0
\(204\) 0 0
\(205\) 10.0000 0.698430
\(206\) 16.0000 1.11477
\(207\) −4.00000 −0.278019
\(208\) 2.00000 0.138675
\(209\) 16.0000 1.10674
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 − 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 6.00000 0.412082
\(213\) 8.00000 0.548151
\(214\) −12.0000 −0.820303
\(215\) −8.00000 −0.545595
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 14.0000 0.948200
\(219\) −2.00000 −0.135147
\(220\) −4.00000 −0.269680
\(221\) 0 0
\(222\) −6.00000 −0.402694
\(223\) 24.0000 1.60716 0.803579 − 0.595198i \(-0.202926\pi\)
0.803579 + 0.595198i \(0.202926\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) −18.0000 −1.19734
\(227\) −12.0000 −0.796468 −0.398234 − 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 4.00000 0.264906
\(229\) 6.00000 0.396491 0.198246 − 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) −4.00000 −0.263752
\(231\) 0 0
\(232\) −2.00000 −0.131306
\(233\) 30.0000 1.96537 0.982683 − 0.185296i \(-0.0593245\pi\)
0.982683 + 0.185296i \(0.0593245\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) −8.00000 −0.520756
\(237\) 4.00000 0.259828
\(238\) 0 0
\(239\) 8.00000 0.517477 0.258738 − 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −10.0000 −0.644157 −0.322078 − 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 5.00000 0.321412
\(243\) −1.00000 −0.0641500
\(244\) −10.0000 −0.640184
\(245\) −7.00000 −0.447214
\(246\) −10.0000 −0.637577
\(247\) −8.00000 −0.509028
\(248\) 4.00000 0.254000
\(249\) −4.00000 −0.253490
\(250\) 1.00000 0.0632456
\(251\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 16.0000 1.00591
\(254\) −16.0000 −1.00393
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 18.0000 1.12281 0.561405 − 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 8.00000 0.498058
\(259\) 0 0
\(260\) 2.00000 0.124035
\(261\) −2.00000 −0.123797
\(262\) −4.00000 −0.247121
\(263\) 16.0000 0.986602 0.493301 − 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) 4.00000 0.246183
\(265\) 6.00000 0.368577
\(266\) 0 0
\(267\) 14.0000 0.856786
\(268\) −8.00000 −0.488678
\(269\) 14.0000 0.853595 0.426798 − 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(270\) −1.00000 −0.0608581
\(271\) −8.00000 −0.485965 −0.242983 − 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) −6.00000 −0.362473
\(275\) −4.00000 −0.241209
\(276\) 4.00000 0.240772
\(277\) −10.0000 −0.600842 −0.300421 − 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 4.00000 0.239904
\(279\) 4.00000 0.239474
\(280\) 0 0
\(281\) −30.0000 −1.78965 −0.894825 − 0.446417i \(-0.852700\pi\)
−0.894825 + 0.446417i \(0.852700\pi\)
\(282\) 0 0
\(283\) −20.0000 −1.18888 −0.594438 − 0.804141i \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) −8.00000 −0.474713
\(285\) 4.00000 0.236940
\(286\) −8.00000 −0.473050
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) 0 0
\(290\) −2.00000 −0.117444
\(291\) −10.0000 −0.586210
\(292\) 2.00000 0.117041
\(293\) 6.00000 0.350524 0.175262 − 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 7.00000 0.408248
\(295\) −8.00000 −0.465778
\(296\) 6.00000 0.348743
\(297\) 4.00000 0.232104
\(298\) 2.00000 0.115857
\(299\) −8.00000 −0.462652
\(300\) −1.00000 −0.0577350
\(301\) 0 0
\(302\) 0 0
\(303\) 6.00000 0.344691
\(304\) −4.00000 −0.229416
\(305\) −10.0000 −0.572598
\(306\) 0 0
\(307\) 16.0000 0.913168 0.456584 − 0.889680i \(-0.349073\pi\)
0.456584 + 0.889680i \(0.349073\pi\)
\(308\) 0 0
\(309\) −16.0000 −0.910208
\(310\) 4.00000 0.227185
\(311\) 8.00000 0.453638 0.226819 − 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) −2.00000 −0.113228
\(313\) 2.00000 0.113047 0.0565233 − 0.998401i \(-0.481998\pi\)
0.0565233 + 0.998401i \(0.481998\pi\)
\(314\) −6.00000 −0.338600
\(315\) 0 0
\(316\) −4.00000 −0.225018
\(317\) −26.0000 −1.46031 −0.730153 − 0.683284i \(-0.760551\pi\)
−0.730153 + 0.683284i \(0.760551\pi\)
\(318\) −6.00000 −0.336463
\(319\) 8.00000 0.447914
\(320\) 1.00000 0.0559017
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 2.00000 0.110940
\(326\) −12.0000 −0.664619
\(327\) −14.0000 −0.774202
\(328\) 10.0000 0.552158
\(329\) 0 0
\(330\) 4.00000 0.220193
\(331\) 20.0000 1.09930 0.549650 − 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 4.00000 0.219529
\(333\) 6.00000 0.328798
\(334\) −12.0000 −0.656611
\(335\) −8.00000 −0.437087
\(336\) 0 0
\(337\) −6.00000 −0.326841 −0.163420 − 0.986557i \(-0.552253\pi\)
−0.163420 + 0.986557i \(0.552253\pi\)
\(338\) −9.00000 −0.489535
\(339\) 18.0000 0.977626
\(340\) 0 0
\(341\) −16.0000 −0.866449
\(342\) −4.00000 −0.216295
\(343\) 0 0
\(344\) −8.00000 −0.431331
\(345\) 4.00000 0.215353
\(346\) −2.00000 −0.107521
\(347\) −4.00000 −0.214731 −0.107366 − 0.994220i \(-0.534242\pi\)
−0.107366 + 0.994220i \(0.534242\pi\)
\(348\) 2.00000 0.107211
\(349\) −34.0000 −1.81998 −0.909989 − 0.414632i \(-0.863910\pi\)
−0.909989 + 0.414632i \(0.863910\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) −4.00000 −0.213201
\(353\) −30.0000 −1.59674 −0.798369 − 0.602168i \(-0.794304\pi\)
−0.798369 + 0.602168i \(0.794304\pi\)
\(354\) 8.00000 0.425195
\(355\) −8.00000 −0.424596
\(356\) −14.0000 −0.741999
\(357\) 0 0
\(358\) 0 0
\(359\) 32.0000 1.68890 0.844448 − 0.535638i \(-0.179929\pi\)
0.844448 + 0.535638i \(0.179929\pi\)
\(360\) 1.00000 0.0527046
\(361\) −3.00000 −0.157895
\(362\) 14.0000 0.735824
\(363\) −5.00000 −0.262432
\(364\) 0 0
\(365\) 2.00000 0.104685
\(366\) 10.0000 0.522708
\(367\) −8.00000 −0.417597 −0.208798 − 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) −4.00000 −0.208514
\(369\) 10.0000 0.520579
\(370\) 6.00000 0.311925
\(371\) 0 0
\(372\) −4.00000 −0.207390
\(373\) 18.0000 0.932005 0.466002 − 0.884783i \(-0.345694\pi\)
0.466002 + 0.884783i \(0.345694\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −4.00000 −0.206010
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 − 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) −4.00000 −0.205196
\(381\) 16.0000 0.819705
\(382\) 0 0
\(383\) −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
\(386\) −6.00000 −0.305392
\(387\) −8.00000 −0.406663
\(388\) 10.0000 0.507673
\(389\) −30.0000 −1.52106 −0.760530 − 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) −2.00000 −0.101274
\(391\) 0 0
\(392\) −7.00000 −0.353553
\(393\) 4.00000 0.201773
\(394\) −18.0000 −0.906827
\(395\) −4.00000 −0.201262
\(396\) −4.00000 −0.201008
\(397\) −18.0000 −0.903394 −0.451697 − 0.892171i \(-0.649181\pi\)
−0.451697 + 0.892171i \(0.649181\pi\)
\(398\) 4.00000 0.200502
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −6.00000 −0.299626 −0.149813 − 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 8.00000 0.399004
\(403\) 8.00000 0.398508
\(404\) −6.00000 −0.298511
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) −24.0000 −1.18964
\(408\) 0 0
\(409\) 10.0000 0.494468 0.247234 − 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 10.0000 0.493865
\(411\) 6.00000 0.295958
\(412\) 16.0000 0.788263
\(413\) 0 0
\(414\) −4.00000 −0.196589
\(415\) 4.00000 0.196352
\(416\) 2.00000 0.0980581
\(417\) −4.00000 −0.195881
\(418\) 16.0000 0.782586
\(419\) −28.0000 −1.36789 −0.683945 − 0.729534i \(-0.739737\pi\)
−0.683945 + 0.729534i \(0.739737\pi\)
\(420\) 0 0
\(421\) 30.0000 1.46211 0.731055 − 0.682318i \(-0.239028\pi\)
0.731055 + 0.682318i \(0.239028\pi\)
\(422\) −12.0000 −0.584151
\(423\) 0 0
\(424\) 6.00000 0.291386
\(425\) 0 0
\(426\) 8.00000 0.387601
\(427\) 0 0
\(428\) −12.0000 −0.580042
\(429\) 8.00000 0.386244
\(430\) −8.00000 −0.385794
\(431\) 16.0000 0.770693 0.385346 − 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −6.00000 −0.288342 −0.144171 − 0.989553i \(-0.546051\pi\)
−0.144171 + 0.989553i \(0.546051\pi\)
\(434\) 0 0
\(435\) 2.00000 0.0958927
\(436\) 14.0000 0.670478
\(437\) 16.0000 0.765384
\(438\) −2.00000 −0.0955637
\(439\) −20.0000 −0.954548 −0.477274 − 0.878755i \(-0.658375\pi\)
−0.477274 + 0.878755i \(0.658375\pi\)
\(440\) −4.00000 −0.190693
\(441\) −7.00000 −0.333333
\(442\) 0 0
\(443\) −28.0000 −1.33032 −0.665160 − 0.746701i \(-0.731637\pi\)
−0.665160 + 0.746701i \(0.731637\pi\)
\(444\) −6.00000 −0.284747
\(445\) −14.0000 −0.663664
\(446\) 24.0000 1.13643
\(447\) −2.00000 −0.0945968
\(448\) 0 0
\(449\) 26.0000 1.22702 0.613508 − 0.789689i \(-0.289758\pi\)
0.613508 + 0.789689i \(0.289758\pi\)
\(450\) 1.00000 0.0471405
\(451\) −40.0000 −1.88353
\(452\) −18.0000 −0.846649
\(453\) 0 0
\(454\) −12.0000 −0.563188
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) 2.00000 0.0935561 0.0467780 − 0.998905i \(-0.485105\pi\)
0.0467780 + 0.998905i \(0.485105\pi\)
\(458\) 6.00000 0.280362
\(459\) 0 0
\(460\) −4.00000 −0.186501
\(461\) −38.0000 −1.76984 −0.884918 − 0.465746i \(-0.845786\pi\)
−0.884918 + 0.465746i \(0.845786\pi\)
\(462\) 0 0
\(463\) −24.0000 −1.11537 −0.557687 − 0.830051i \(-0.688311\pi\)
−0.557687 + 0.830051i \(0.688311\pi\)
\(464\) −2.00000 −0.0928477
\(465\) −4.00000 −0.185496
\(466\) 30.0000 1.38972
\(467\) −12.0000 −0.555294 −0.277647 − 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 2.00000 0.0924500
\(469\) 0 0
\(470\) 0 0
\(471\) 6.00000 0.276465
\(472\) −8.00000 −0.368230
\(473\) 32.0000 1.47136
\(474\) 4.00000 0.183726
\(475\) −4.00000 −0.183533
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) 8.00000 0.365911
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 12.0000 0.547153
\(482\) −10.0000 −0.455488
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) 10.0000 0.454077
\(486\) −1.00000 −0.0453609
\(487\) 40.0000 1.81257 0.906287 − 0.422664i \(-0.138905\pi\)
0.906287 + 0.422664i \(0.138905\pi\)
\(488\) −10.0000 −0.452679
\(489\) 12.0000 0.542659
\(490\) −7.00000 −0.316228
\(491\) 24.0000 1.08310 0.541552 − 0.840667i \(-0.317837\pi\)
0.541552 + 0.840667i \(0.317837\pi\)
\(492\) −10.0000 −0.450835
\(493\) 0 0
\(494\) −8.00000 −0.359937
\(495\) −4.00000 −0.179787
\(496\) 4.00000 0.179605
\(497\) 0 0
\(498\) −4.00000 −0.179244
\(499\) −28.0000 −1.25345 −0.626726 − 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) 1.00000 0.0447214
\(501\) 12.0000 0.536120
\(502\) 0 0
\(503\) 28.0000 1.24846 0.624229 − 0.781241i \(-0.285413\pi\)
0.624229 + 0.781241i \(0.285413\pi\)
\(504\) 0 0
\(505\) −6.00000 −0.266996
\(506\) 16.0000 0.711287
\(507\) 9.00000 0.399704
\(508\) −16.0000 −0.709885
\(509\) −6.00000 −0.265945 −0.132973 − 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 4.00000 0.176604
\(514\) 18.0000 0.793946
\(515\) 16.0000 0.705044
\(516\) 8.00000 0.352180
\(517\) 0 0
\(518\) 0 0
\(519\) 2.00000 0.0877903
\(520\) 2.00000 0.0877058
\(521\) −30.0000 −1.31432 −0.657162 − 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) −2.00000 −0.0875376
\(523\) −40.0000 −1.74908 −0.874539 − 0.484955i \(-0.838836\pi\)
−0.874539 + 0.484955i \(0.838836\pi\)
\(524\) −4.00000 −0.174741
\(525\) 0 0
\(526\) 16.0000 0.697633
\(527\) 0 0
\(528\) 4.00000 0.174078
\(529\) −7.00000 −0.304348
\(530\) 6.00000 0.260623
\(531\) −8.00000 −0.347170
\(532\) 0 0
\(533\) 20.0000 0.866296
\(534\) 14.0000 0.605839
\(535\) −12.0000 −0.518805
\(536\) −8.00000 −0.345547
\(537\) 0 0
\(538\) 14.0000 0.603583
\(539\) 28.0000 1.20605
\(540\) −1.00000 −0.0430331
\(541\) 22.0000 0.945854 0.472927 − 0.881102i \(-0.343197\pi\)
0.472927 + 0.881102i \(0.343197\pi\)
\(542\) −8.00000 −0.343629
\(543\) −14.0000 −0.600798
\(544\) 0 0
\(545\) 14.0000 0.599694
\(546\) 0 0
\(547\) 20.0000 0.855138 0.427569 − 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) −6.00000 −0.256307
\(549\) −10.0000 −0.426790
\(550\) −4.00000 −0.170561
\(551\) 8.00000 0.340811
\(552\) 4.00000 0.170251
\(553\) 0 0
\(554\) −10.0000 −0.424859
\(555\) −6.00000 −0.254686
\(556\) 4.00000 0.169638
\(557\) 30.0000 1.27114 0.635570 − 0.772043i \(-0.280765\pi\)
0.635570 + 0.772043i \(0.280765\pi\)
\(558\) 4.00000 0.169334
\(559\) −16.0000 −0.676728
\(560\) 0 0
\(561\) 0 0
\(562\) −30.0000 −1.26547
\(563\) 20.0000 0.842900 0.421450 − 0.906852i \(-0.361521\pi\)
0.421450 + 0.906852i \(0.361521\pi\)
\(564\) 0 0
\(565\) −18.0000 −0.757266
\(566\) −20.0000 −0.840663
\(567\) 0 0
\(568\) −8.00000 −0.335673
\(569\) 26.0000 1.08998 0.544988 − 0.838444i \(-0.316534\pi\)
0.544988 + 0.838444i \(0.316534\pi\)
\(570\) 4.00000 0.167542
\(571\) 44.0000 1.84134 0.920671 − 0.390339i \(-0.127642\pi\)
0.920671 + 0.390339i \(0.127642\pi\)
\(572\) −8.00000 −0.334497
\(573\) 0 0
\(574\) 0 0
\(575\) −4.00000 −0.166812
\(576\) 1.00000 0.0416667
\(577\) −6.00000 −0.249783 −0.124892 − 0.992170i \(-0.539858\pi\)
−0.124892 + 0.992170i \(0.539858\pi\)
\(578\) 0 0
\(579\) 6.00000 0.249351
\(580\) −2.00000 −0.0830455
\(581\) 0 0
\(582\) −10.0000 −0.414513
\(583\) −24.0000 −0.993978
\(584\) 2.00000 0.0827606
\(585\) 2.00000 0.0826898
\(586\) 6.00000 0.247858
\(587\) −12.0000 −0.495293 −0.247647 − 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 7.00000 0.288675
\(589\) −16.0000 −0.659269
\(590\) −8.00000 −0.329355
\(591\) 18.0000 0.740421
\(592\) 6.00000 0.246598
\(593\) −30.0000 −1.23195 −0.615976 − 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) 4.00000 0.164122
\(595\) 0 0
\(596\) 2.00000 0.0819232
\(597\) −4.00000 −0.163709
\(598\) −8.00000 −0.327144
\(599\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −34.0000 −1.38689 −0.693444 − 0.720510i \(-0.743908\pi\)
−0.693444 + 0.720510i \(0.743908\pi\)
\(602\) 0 0
\(603\) −8.00000 −0.325785
\(604\) 0 0
\(605\) 5.00000 0.203279
\(606\) 6.00000 0.243733
\(607\) 32.0000 1.29884 0.649420 − 0.760430i \(-0.275012\pi\)
0.649420 + 0.760430i \(0.275012\pi\)
\(608\) −4.00000 −0.162221
\(609\) 0 0
\(610\) −10.0000 −0.404888
\(611\) 0 0
\(612\) 0 0
\(613\) 10.0000 0.403896 0.201948 − 0.979396i \(-0.435273\pi\)
0.201948 + 0.979396i \(0.435273\pi\)
\(614\) 16.0000 0.645707
\(615\) −10.0000 −0.403239
\(616\) 0 0
\(617\) −2.00000 −0.0805170 −0.0402585 − 0.999189i \(-0.512818\pi\)
−0.0402585 + 0.999189i \(0.512818\pi\)
\(618\) −16.0000 −0.643614
\(619\) −44.0000 −1.76851 −0.884255 − 0.467005i \(-0.845333\pi\)
−0.884255 + 0.467005i \(0.845333\pi\)
\(620\) 4.00000 0.160644
\(621\) 4.00000 0.160514
\(622\) 8.00000 0.320771
\(623\) 0 0
\(624\) −2.00000 −0.0800641
\(625\) 1.00000 0.0400000
\(626\) 2.00000 0.0799361
\(627\) −16.0000 −0.638978
\(628\) −6.00000 −0.239426
\(629\) 0 0
\(630\) 0 0
\(631\) 32.0000 1.27390 0.636950 − 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) −4.00000 −0.159111
\(633\) 12.0000 0.476957
\(634\) −26.0000 −1.03259
\(635\) −16.0000 −0.634941
\(636\) −6.00000 −0.237915
\(637\) −14.0000 −0.554700
\(638\) 8.00000 0.316723
\(639\) −8.00000 −0.316475
\(640\) 1.00000 0.0395285
\(641\) 42.0000 1.65890 0.829450 − 0.558581i \(-0.188654\pi\)
0.829450 + 0.558581i \(0.188654\pi\)
\(642\) 12.0000 0.473602
\(643\) 20.0000 0.788723 0.394362 − 0.918955i \(-0.370966\pi\)
0.394362 + 0.918955i \(0.370966\pi\)
\(644\) 0 0
\(645\) 8.00000 0.315000
\(646\) 0 0
\(647\) −32.0000 −1.25805 −0.629025 − 0.777385i \(-0.716546\pi\)
−0.629025 + 0.777385i \(0.716546\pi\)
\(648\) 1.00000 0.0392837
\(649\) 32.0000 1.25611
\(650\) 2.00000 0.0784465
\(651\) 0 0
\(652\) −12.0000 −0.469956
\(653\) −34.0000 −1.33052 −0.665261 − 0.746611i \(-0.731680\pi\)
−0.665261 + 0.746611i \(0.731680\pi\)
\(654\) −14.0000 −0.547443
\(655\) −4.00000 −0.156293
\(656\) 10.0000 0.390434
\(657\) 2.00000 0.0780274
\(658\) 0 0
\(659\) 24.0000 0.934907 0.467454 − 0.884018i \(-0.345171\pi\)
0.467454 + 0.884018i \(0.345171\pi\)
\(660\) 4.00000 0.155700
\(661\) 22.0000 0.855701 0.427850 − 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) 20.0000 0.777322
\(663\) 0 0
\(664\) 4.00000 0.155230
\(665\) 0 0
\(666\) 6.00000 0.232495
\(667\) 8.00000 0.309761
\(668\) −12.0000 −0.464294
\(669\) −24.0000 −0.927894
\(670\) −8.00000 −0.309067
\(671\) 40.0000 1.54418
\(672\) 0 0
\(673\) 10.0000 0.385472 0.192736 − 0.981251i \(-0.438264\pi\)
0.192736 + 0.981251i \(0.438264\pi\)
\(674\) −6.00000 −0.231111
\(675\) −1.00000 −0.0384900
\(676\) −9.00000 −0.346154
\(677\) −18.0000 −0.691796 −0.345898 − 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) 18.0000 0.691286
\(679\) 0 0
\(680\) 0 0
\(681\) 12.0000 0.459841
\(682\) −16.0000 −0.612672
\(683\) −44.0000 −1.68361 −0.841807 − 0.539779i \(-0.818508\pi\)
−0.841807 + 0.539779i \(0.818508\pi\)
\(684\) −4.00000 −0.152944
\(685\) −6.00000 −0.229248
\(686\) 0 0
\(687\) −6.00000 −0.228914
\(688\) −8.00000 −0.304997
\(689\) 12.0000 0.457164
\(690\) 4.00000 0.152277
\(691\) 36.0000 1.36950 0.684752 − 0.728776i \(-0.259910\pi\)
0.684752 + 0.728776i \(0.259910\pi\)
\(692\) −2.00000 −0.0760286
\(693\) 0 0
\(694\) −4.00000 −0.151838
\(695\) 4.00000 0.151729
\(696\) 2.00000 0.0758098
\(697\) 0 0
\(698\) −34.0000 −1.28692
\(699\) −30.0000 −1.13470
\(700\) 0 0
\(701\) 10.0000 0.377695 0.188847 − 0.982006i \(-0.439525\pi\)
0.188847 + 0.982006i \(0.439525\pi\)
\(702\) −2.00000 −0.0754851
\(703\) −24.0000 −0.905177
\(704\) −4.00000 −0.150756
\(705\) 0 0
\(706\) −30.0000 −1.12906
\(707\) 0 0
\(708\) 8.00000 0.300658
\(709\) −10.0000 −0.375558 −0.187779 − 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) −8.00000 −0.300235
\(711\) −4.00000 −0.150012
\(712\) −14.0000 −0.524672
\(713\) −16.0000 −0.599205
\(714\) 0 0
\(715\) −8.00000 −0.299183
\(716\) 0 0
\(717\) −8.00000 −0.298765
\(718\) 32.0000 1.19423
\(719\) 48.0000 1.79010 0.895049 − 0.445968i \(-0.147140\pi\)
0.895049 + 0.445968i \(0.147140\pi\)
\(720\) 1.00000 0.0372678
\(721\) 0 0
\(722\) −3.00000 −0.111648
\(723\) 10.0000 0.371904
\(724\) 14.0000 0.520306
\(725\) −2.00000 −0.0742781
\(726\) −5.00000 −0.185567
\(727\) 40.0000 1.48352 0.741759 − 0.670667i \(-0.233992\pi\)
0.741759 + 0.670667i \(0.233992\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 2.00000 0.0740233
\(731\) 0 0
\(732\) 10.0000 0.369611
\(733\) 26.0000 0.960332 0.480166 − 0.877178i \(-0.340576\pi\)
0.480166 + 0.877178i \(0.340576\pi\)
\(734\) −8.00000 −0.295285
\(735\) 7.00000 0.258199
\(736\) −4.00000 −0.147442
\(737\) 32.0000 1.17874
\(738\) 10.0000 0.368105
\(739\) −44.0000 −1.61857 −0.809283 − 0.587419i \(-0.800144\pi\)
−0.809283 + 0.587419i \(0.800144\pi\)
\(740\) 6.00000 0.220564
\(741\) 8.00000 0.293887
\(742\) 0 0
\(743\) −28.0000 −1.02722 −0.513610 − 0.858024i \(-0.671692\pi\)
−0.513610 + 0.858024i \(0.671692\pi\)
\(744\) −4.00000 −0.146647
\(745\) 2.00000 0.0732743
\(746\) 18.0000 0.659027
\(747\) 4.00000 0.146352
\(748\) 0 0
\(749\) 0 0
\(750\) −1.00000 −0.0365148
\(751\) 4.00000 0.145962 0.0729810 − 0.997333i \(-0.476749\pi\)
0.0729810 + 0.997333i \(0.476749\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) −4.00000 −0.145671
\(755\) 0 0
\(756\) 0 0
\(757\) −14.0000 −0.508839 −0.254419 − 0.967094i \(-0.581884\pi\)
−0.254419 + 0.967094i \(0.581884\pi\)
\(758\) −20.0000 −0.726433
\(759\) −16.0000 −0.580763
\(760\) −4.00000 −0.145095
\(761\) 42.0000 1.52250 0.761249 − 0.648459i \(-0.224586\pi\)
0.761249 + 0.648459i \(0.224586\pi\)
\(762\) 16.0000 0.579619
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) −24.0000 −0.867155
\(767\) −16.0000 −0.577727
\(768\) −1.00000 −0.0360844
\(769\) 2.00000 0.0721218 0.0360609 − 0.999350i \(-0.488519\pi\)
0.0360609 + 0.999350i \(0.488519\pi\)
\(770\) 0 0
\(771\) −18.0000 −0.648254
\(772\) −6.00000 −0.215945
\(773\) 14.0000 0.503545 0.251773 − 0.967786i \(-0.418987\pi\)
0.251773 + 0.967786i \(0.418987\pi\)
\(774\) −8.00000 −0.287554
\(775\) 4.00000 0.143684
\(776\) 10.0000 0.358979
\(777\) 0 0
\(778\) −30.0000 −1.07555
\(779\) −40.0000 −1.43315
\(780\) −2.00000 −0.0716115
\(781\) 32.0000 1.14505
\(782\) 0 0
\(783\) 2.00000 0.0714742
\(784\) −7.00000 −0.250000
\(785\) −6.00000 −0.214149
\(786\) 4.00000 0.142675
\(787\) −4.00000 −0.142585 −0.0712923 − 0.997455i \(-0.522712\pi\)
−0.0712923 + 0.997455i \(0.522712\pi\)
\(788\) −18.0000 −0.641223
\(789\) −16.0000 −0.569615
\(790\) −4.00000 −0.142314
\(791\) 0 0
\(792\) −4.00000 −0.142134
\(793\) −20.0000 −0.710221
\(794\) −18.0000 −0.638796
\(795\) −6.00000 −0.212798
\(796\) 4.00000 0.141776
\(797\) 6.00000 0.212531 0.106265 − 0.994338i \(-0.466111\pi\)
0.106265 + 0.994338i \(0.466111\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 1.00000 0.0353553
\(801\) −14.0000 −0.494666
\(802\) −6.00000 −0.211867
\(803\) −8.00000 −0.282314
\(804\) 8.00000 0.282138
\(805\) 0 0
\(806\) 8.00000 0.281788
\(807\) −14.0000 −0.492823
\(808\) −6.00000 −0.211079
\(809\) −54.0000 −1.89854 −0.949269 − 0.314464i \(-0.898175\pi\)
−0.949269 + 0.314464i \(0.898175\pi\)
\(810\) 1.00000 0.0351364
\(811\) 28.0000 0.983213 0.491606 − 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) 0 0
\(813\) 8.00000 0.280572
\(814\) −24.0000 −0.841200
\(815\) −12.0000 −0.420342
\(816\) 0 0
\(817\) 32.0000 1.11954
\(818\) 10.0000 0.349642
\(819\) 0 0
\(820\) 10.0000 0.349215
\(821\) 6.00000 0.209401 0.104701 − 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) 6.00000 0.209274
\(823\) −16.0000 −0.557725 −0.278862 − 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) 16.0000 0.557386
\(825\) 4.00000 0.139262
\(826\) 0 0
\(827\) −36.0000 −1.25184 −0.625921 − 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) −4.00000 −0.139010
\(829\) 54.0000 1.87550 0.937749 − 0.347314i \(-0.112906\pi\)
0.937749 + 0.347314i \(0.112906\pi\)
\(830\) 4.00000 0.138842
\(831\) 10.0000 0.346896
\(832\) 2.00000 0.0693375
\(833\) 0 0
\(834\) −4.00000 −0.138509
\(835\) −12.0000 −0.415277
\(836\) 16.0000 0.553372
\(837\) −4.00000 −0.138260
\(838\) −28.0000 −0.967244
\(839\) −16.0000 −0.552381 −0.276191 − 0.961103i \(-0.589072\pi\)
−0.276191 + 0.961103i \(0.589072\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 30.0000 1.03387
\(843\) 30.0000 1.03325
\(844\) −12.0000 −0.413057
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) 0 0
\(848\) 6.00000 0.206041
\(849\) 20.0000 0.686398
\(850\) 0 0
\(851\) −24.0000 −0.822709
\(852\) 8.00000 0.274075
\(853\) 54.0000 1.84892 0.924462 − 0.381273i \(-0.124514\pi\)
0.924462 + 0.381273i \(0.124514\pi\)
\(854\) 0 0
\(855\) −4.00000 −0.136797
\(856\) −12.0000 −0.410152
\(857\) −26.0000 −0.888143 −0.444072 − 0.895991i \(-0.646466\pi\)
−0.444072 + 0.895991i \(0.646466\pi\)
\(858\) 8.00000 0.273115
\(859\) 20.0000 0.682391 0.341196 − 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) −8.00000 −0.272798
\(861\) 0 0
\(862\) 16.0000 0.544962
\(863\) −8.00000 −0.272323 −0.136162 − 0.990687i \(-0.543477\pi\)
−0.136162 + 0.990687i \(0.543477\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −2.00000 −0.0680020
\(866\) −6.00000 −0.203888
\(867\) 0 0
\(868\) 0 0
\(869\) 16.0000 0.542763
\(870\) 2.00000 0.0678064
\(871\) −16.0000 −0.542139
\(872\) 14.0000 0.474100
\(873\) 10.0000 0.338449
\(874\) 16.0000 0.541208
\(875\) 0 0
\(876\) −2.00000 −0.0675737
\(877\) −50.0000 −1.68838 −0.844190 − 0.536044i \(-0.819918\pi\)
−0.844190 + 0.536044i \(0.819918\pi\)
\(878\) −20.0000 −0.674967
\(879\) −6.00000 −0.202375
\(880\) −4.00000 −0.134840
\(881\) 2.00000 0.0673817 0.0336909 − 0.999432i \(-0.489274\pi\)
0.0336909 + 0.999432i \(0.489274\pi\)
\(882\) −7.00000 −0.235702
\(883\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(884\) 0 0
\(885\) 8.00000 0.268917
\(886\) −28.0000 −0.940678
\(887\) −12.0000 −0.402921 −0.201460 − 0.979497i \(-0.564569\pi\)
−0.201460 + 0.979497i \(0.564569\pi\)
\(888\) −6.00000 −0.201347
\(889\) 0 0
\(890\) −14.0000 −0.469281
\(891\) −4.00000 −0.134005
\(892\) 24.0000 0.803579
\(893\) 0 0
\(894\) −2.00000 −0.0668900
\(895\) 0 0
\(896\) 0 0
\(897\) 8.00000 0.267112
\(898\) 26.0000 0.867631
\(899\) −8.00000 −0.266815
\(900\) 1.00000 0.0333333
\(901\) 0 0
\(902\) −40.0000 −1.33185
\(903\) 0 0
\(904\) −18.0000 −0.598671
\(905\) 14.0000 0.465376
\(906\) 0 0
\(907\) 36.0000 1.19536 0.597680 − 0.801735i \(-0.296089\pi\)
0.597680 + 0.801735i \(0.296089\pi\)
\(908\) −12.0000 −0.398234
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) −48.0000 −1.59031 −0.795155 − 0.606406i \(-0.792611\pi\)
−0.795155 + 0.606406i \(0.792611\pi\)
\(912\) 4.00000 0.132453
\(913\) −16.0000 −0.529523
\(914\) 2.00000 0.0661541
\(915\) 10.0000 0.330590
\(916\) 6.00000 0.198246
\(917\) 0 0
\(918\) 0 0
\(919\) 16.0000 0.527791 0.263896 − 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) −4.00000 −0.131876
\(921\) −16.0000 −0.527218
\(922\) −38.0000 −1.25146
\(923\) −16.0000 −0.526646
\(924\) 0 0
\(925\) 6.00000 0.197279
\(926\) −24.0000 −0.788689
\(927\) 16.0000 0.525509
\(928\) −2.00000 −0.0656532
\(929\) 50.0000 1.64045 0.820223 − 0.572043i \(-0.193849\pi\)
0.820223 + 0.572043i \(0.193849\pi\)
\(930\) −4.00000 −0.131165
\(931\) 28.0000 0.917663
\(932\) 30.0000 0.982683
\(933\) −8.00000 −0.261908
\(934\) −12.0000 −0.392652
\(935\) 0 0
\(936\) 2.00000 0.0653720
\(937\) 2.00000 0.0653372 0.0326686 − 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) 0 0
\(939\) −2.00000 −0.0652675
\(940\) 0 0
\(941\) 30.0000 0.977972 0.488986 − 0.872292i \(-0.337367\pi\)
0.488986 + 0.872292i \(0.337367\pi\)
\(942\) 6.00000 0.195491
\(943\) −40.0000 −1.30258
\(944\) −8.00000 −0.260378
\(945\) 0 0
\(946\) 32.0000 1.04041
\(947\) 28.0000 0.909878 0.454939 − 0.890523i \(-0.349661\pi\)
0.454939 + 0.890523i \(0.349661\pi\)
\(948\) 4.00000 0.129914
\(949\) 4.00000 0.129845
\(950\) −4.00000 −0.129777
\(951\) 26.0000 0.843108
\(952\) 0 0
\(953\) −6.00000 −0.194359 −0.0971795 − 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) 6.00000 0.194257
\(955\) 0 0
\(956\) 8.00000 0.258738
\(957\) −8.00000 −0.258603
\(958\) 0 0
\(959\) 0 0
\(960\) −1.00000 −0.0322749
\(961\) −15.0000 −0.483871
\(962\) 12.0000 0.386896
\(963\) −12.0000 −0.386695
\(964\) −10.0000 −0.322078
\(965\) −6.00000 −0.193147
\(966\) 0 0
\(967\) 24.0000 0.771788 0.385894 − 0.922543i \(-0.373893\pi\)
0.385894 + 0.922543i \(0.373893\pi\)
\(968\) 5.00000 0.160706
\(969\) 0 0
\(970\) 10.0000 0.321081
\(971\) 32.0000 1.02693 0.513464 − 0.858111i \(-0.328362\pi\)
0.513464 + 0.858111i \(0.328362\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 40.0000 1.28168
\(975\) −2.00000 −0.0640513
\(976\) −10.0000 −0.320092
\(977\) 18.0000 0.575871 0.287936 − 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) 12.0000 0.383718
\(979\) 56.0000 1.78977
\(980\) −7.00000 −0.223607
\(981\) 14.0000 0.446986
\(982\) 24.0000 0.765871
\(983\) 44.0000 1.40338 0.701691 − 0.712481i \(-0.252429\pi\)
0.701691 + 0.712481i \(0.252429\pi\)
\(984\) −10.0000 −0.318788
\(985\) −18.0000 −0.573528
\(986\) 0 0
\(987\) 0 0
\(988\) −8.00000 −0.254514
\(989\) 32.0000 1.01754
\(990\) −4.00000 −0.127128
\(991\) 44.0000 1.39771 0.698853 − 0.715265i \(-0.253694\pi\)
0.698853 + 0.715265i \(0.253694\pi\)
\(992\) 4.00000 0.127000
\(993\) −20.0000 −0.634681
\(994\) 0 0
\(995\) 4.00000 0.126809
\(996\) −4.00000 −0.126745
\(997\) −10.0000 −0.316703 −0.158352 − 0.987383i \(-0.550618\pi\)
−0.158352 + 0.987383i \(0.550618\pi\)
\(998\) −28.0000 −0.886325
\(999\) −6.00000 −0.189832
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 8670.2.a.r.1.1 1
17.16 even 2 510.2.a.f.1.1 ✓ 1
51.50 odd 2 1530.2.a.f.1.1 1
68.67 odd 2 4080.2.a.c.1.1 1
85.33 odd 4 2550.2.d.s.2449.1 2
85.67 odd 4 2550.2.d.s.2449.2 2
85.84 even 2 2550.2.a.d.1.1 1
255.254 odd 2 7650.2.a.bw.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
510.2.a.f.1.1 ✓ 1 17.16 even 2
1530.2.a.f.1.1 1 51.50 odd 2
2550.2.a.d.1.1 1 85.84 even 2
2550.2.d.s.2449.1 2 85.33 odd 4
2550.2.d.s.2449.2 2 85.67 odd 4
4080.2.a.c.1.1 1 68.67 odd 2
7650.2.a.bw.1.1 1 255.254 odd 2
8670.2.a.r.1.1 1 1.1 even 1 trivial