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Properties

Label 8050.2.a.n.1.1
Level $8050$
Weight $2$
Character 8050.1
Self dual yes
Analytic conductor $64.280$
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] = [8050,2,Mod(1,8050)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8050, 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("8050.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8050 = 2 \cdot 5^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8050.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: \(64.2795736271\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 322)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 8050.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -2.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -2.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +4.00000 q^{11} -2.00000 q^{12} -1.00000 q^{14} +1.00000 q^{16} -6.00000 q^{17} +1.00000 q^{18} -6.00000 q^{19} +2.00000 q^{21} +4.00000 q^{22} +1.00000 q^{23} -2.00000 q^{24} +4.00000 q^{27} -1.00000 q^{28} +10.0000 q^{29} +4.00000 q^{31} +1.00000 q^{32} -8.00000 q^{33} -6.00000 q^{34} +1.00000 q^{36} +2.00000 q^{37} -6.00000 q^{38} -10.0000 q^{41} +2.00000 q^{42} +4.00000 q^{43} +4.00000 q^{44} +1.00000 q^{46} -12.0000 q^{47} -2.00000 q^{48} +1.00000 q^{49} +12.0000 q^{51} +6.00000 q^{53} +4.00000 q^{54} -1.00000 q^{56} +12.0000 q^{57} +10.0000 q^{58} -2.00000 q^{59} +4.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -8.00000 q^{66} -6.00000 q^{68} -2.00000 q^{69} -8.00000 q^{71} +1.00000 q^{72} +6.00000 q^{73} +2.00000 q^{74} -6.00000 q^{76} -4.00000 q^{77} -8.00000 q^{79} -11.0000 q^{81} -10.0000 q^{82} +14.0000 q^{83} +2.00000 q^{84} +4.00000 q^{86} -20.0000 q^{87} +4.00000 q^{88} -14.0000 q^{89} +1.00000 q^{92} -8.00000 q^{93} -12.0000 q^{94} -2.00000 q^{96} +2.00000 q^{97} +1.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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
\(3\) −2.00000 −1.15470 −0.577350 − 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −2.00000 −0.816497
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 − 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) −2.00000 −0.577350
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −6.00000 −1.45521 −0.727607 − 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 1.00000 0.235702
\(19\) −6.00000 −1.37649 −0.688247 − 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 4.00000 0.852803
\(23\) 1.00000 0.208514
\(24\) −2.00000 −0.408248
\(25\) 0 0
\(26\) 0 0
\(27\) 4.00000 0.769800
\(28\) −1.00000 −0.188982
\(29\) 10.0000 1.85695 0.928477 − 0.371391i \(-0.121119\pi\)
0.928477 + 0.371391i \(0.121119\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 − 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 1.00000 0.176777
\(33\) −8.00000 −1.39262
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) −6.00000 −0.973329
\(39\) 0 0
\(40\) 0 0
\(41\) −10.0000 −1.56174 −0.780869 − 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 2.00000 0.308607
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) −12.0000 −1.75038 −0.875190 − 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) −2.00000 −0.288675
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 12.0000 1.68034
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 4.00000 0.544331
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 12.0000 1.58944
\(58\) 10.0000 1.31306
\(59\) −2.00000 −0.260378 −0.130189 − 0.991489i \(-0.541558\pi\)
−0.130189 + 0.991489i \(0.541558\pi\)
\(60\) 0 0
\(61\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 4.00000 0.508001
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −8.00000 −0.984732
\(67\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) −6.00000 −0.727607
\(69\) −2.00000 −0.240772
\(70\) 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
\(73\) 6.00000 0.702247 0.351123 − 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) −6.00000 −0.688247
\(77\) −4.00000 −0.455842
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 − 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) −10.0000 −1.10432
\(83\) 14.0000 1.53670 0.768350 − 0.640030i \(-0.221078\pi\)
0.768350 + 0.640030i \(0.221078\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) −20.0000 −2.14423
\(88\) 4.00000 0.426401
\(89\) −14.0000 −1.48400 −0.741999 − 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1.00000 0.104257
\(93\) −8.00000 −0.829561
\(94\) −12.0000 −1.23771
\(95\) 0 0
\(96\) −2.00000 −0.204124
\(97\) 2.00000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 1.00000 0.101015
\(99\) 4.00000 0.402015
\(100\) 0 0
\(101\) −4.00000 −0.398015 −0.199007 − 0.979998i \(-0.563772\pi\)
−0.199007 + 0.979998i \(0.563772\pi\)
\(102\) 12.0000 1.18818
\(103\) 4.00000 0.394132 0.197066 − 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) −16.0000 −1.54678 −0.773389 − 0.633932i \(-0.781440\pi\)
−0.773389 + 0.633932i \(0.781440\pi\)
\(108\) 4.00000 0.384900
\(109\) −10.0000 −0.957826 −0.478913 − 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) −4.00000 −0.379663
\(112\) −1.00000 −0.0944911
\(113\) 2.00000 0.188144 0.0940721 − 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) 12.0000 1.12390
\(115\) 0 0
\(116\) 10.0000 0.928477
\(117\) 0 0
\(118\) −2.00000 −0.184115
\(119\) 6.00000 0.550019
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) 20.0000 1.80334
\(124\) 4.00000 0.359211
\(125\) 0 0
\(126\) −1.00000 −0.0890871
\(127\) −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
\(130\) 0 0
\(131\) 6.00000 0.524222 0.262111 − 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) −8.00000 −0.696311
\(133\) 6.00000 0.520266
\(134\) 0 0
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) −10.0000 −0.854358 −0.427179 − 0.904167i \(-0.640493\pi\)
−0.427179 + 0.904167i \(0.640493\pi\)
\(138\) −2.00000 −0.170251
\(139\) 10.0000 0.848189 0.424094 − 0.905618i \(-0.360592\pi\)
0.424094 + 0.905618i \(0.360592\pi\)
\(140\) 0 0
\(141\) 24.0000 2.02116
\(142\) −8.00000 −0.671345
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 6.00000 0.496564
\(147\) −2.00000 −0.164957
\(148\) 2.00000 0.164399
\(149\) 18.0000 1.47462 0.737309 − 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) 0 0
\(151\) 16.0000 1.30206 0.651031 − 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) −6.00000 −0.486664
\(153\) −6.00000 −0.485071
\(154\) −4.00000 −0.322329
\(155\) 0 0
\(156\) 0 0
\(157\) 12.0000 0.957704 0.478852 − 0.877896i \(-0.341053\pi\)
0.478852 + 0.877896i \(0.341053\pi\)
\(158\) −8.00000 −0.636446
\(159\) −12.0000 −0.951662
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) −11.0000 −0.864242
\(163\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(164\) −10.0000 −0.780869
\(165\) 0 0
\(166\) 14.0000 1.08661
\(167\) −12.0000 −0.928588 −0.464294 − 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 2.00000 0.154303
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) −6.00000 −0.458831
\(172\) 4.00000 0.304997
\(173\) 16.0000 1.21646 0.608229 − 0.793762i \(-0.291880\pi\)
0.608229 + 0.793762i \(0.291880\pi\)
\(174\) −20.0000 −1.51620
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) 4.00000 0.300658
\(178\) −14.0000 −1.04934
\(179\) 12.0000 0.896922 0.448461 − 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) −12.0000 −0.891953 −0.445976 − 0.895045i \(-0.647144\pi\)
−0.445976 + 0.895045i \(0.647144\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 1.00000 0.0737210
\(185\) 0 0
\(186\) −8.00000 −0.586588
\(187\) −24.0000 −1.75505
\(188\) −12.0000 −0.875190
\(189\) −4.00000 −0.290957
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 − 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) −2.00000 −0.144338
\(193\) −14.0000 −1.00774 −0.503871 − 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −6.00000 −0.427482 −0.213741 − 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 4.00000 0.284268
\(199\) 4.00000 0.283552 0.141776 − 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −4.00000 −0.281439
\(203\) −10.0000 −0.701862
\(204\) 12.0000 0.840168
\(205\) 0 0
\(206\) 4.00000 0.278693
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) −24.0000 −1.66011
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 − 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 6.00000 0.412082
\(213\) 16.0000 1.09630
\(214\) −16.0000 −1.09374
\(215\) 0 0
\(216\) 4.00000 0.272166
\(217\) −4.00000 −0.271538
\(218\) −10.0000 −0.677285
\(219\) −12.0000 −0.810885
\(220\) 0 0
\(221\) 0 0
\(222\) −4.00000 −0.268462
\(223\) −8.00000 −0.535720 −0.267860 − 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 2.00000 0.133038
\(227\) 14.0000 0.929213 0.464606 − 0.885517i \(-0.346196\pi\)
0.464606 + 0.885517i \(0.346196\pi\)
\(228\) 12.0000 0.794719
\(229\) −20.0000 −1.32164 −0.660819 − 0.750546i \(-0.729791\pi\)
−0.660819 + 0.750546i \(0.729791\pi\)
\(230\) 0 0
\(231\) 8.00000 0.526361
\(232\) 10.0000 0.656532
\(233\) −26.0000 −1.70332 −0.851658 − 0.524097i \(-0.824403\pi\)
−0.851658 + 0.524097i \(0.824403\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −2.00000 −0.130189
\(237\) 16.0000 1.03931
\(238\) 6.00000 0.388922
\(239\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −18.0000 −1.15948 −0.579741 − 0.814801i \(-0.696846\pi\)
−0.579741 + 0.814801i \(0.696846\pi\)
\(242\) 5.00000 0.321412
\(243\) 10.0000 0.641500
\(244\) 0 0
\(245\) 0 0
\(246\) 20.0000 1.27515
\(247\) 0 0
\(248\) 4.00000 0.254000
\(249\) −28.0000 −1.77443
\(250\) 0 0
\(251\) −2.00000 −0.126239 −0.0631194 − 0.998006i \(-0.520105\pi\)
−0.0631194 + 0.998006i \(0.520105\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 4.00000 0.251478
\(254\) −16.0000 −1.00393
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 22.0000 1.37232 0.686161 − 0.727450i \(-0.259294\pi\)
0.686161 + 0.727450i \(0.259294\pi\)
\(258\) −8.00000 −0.498058
\(259\) −2.00000 −0.124274
\(260\) 0 0
\(261\) 10.0000 0.618984
\(262\) 6.00000 0.370681
\(263\) −24.0000 −1.47990 −0.739952 − 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) −8.00000 −0.492366
\(265\) 0 0
\(266\) 6.00000 0.367884
\(267\) 28.0000 1.71357
\(268\) 0 0
\(269\) 16.0000 0.975537 0.487769 − 0.872973i \(-0.337811\pi\)
0.487769 + 0.872973i \(0.337811\pi\)
\(270\) 0 0
\(271\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) −6.00000 −0.363803
\(273\) 0 0
\(274\) −10.0000 −0.604122
\(275\) 0 0
\(276\) −2.00000 −0.120386
\(277\) −30.0000 −1.80253 −0.901263 − 0.433273i \(-0.857359\pi\)
−0.901263 + 0.433273i \(0.857359\pi\)
\(278\) 10.0000 0.599760
\(279\) 4.00000 0.239474
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 − 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 24.0000 1.42918
\(283\) −26.0000 −1.54554 −0.772770 − 0.634686i \(-0.781129\pi\)
−0.772770 + 0.634686i \(0.781129\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) 0 0
\(287\) 10.0000 0.590281
\(288\) 1.00000 0.0589256
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) −4.00000 −0.234484
\(292\) 6.00000 0.351123
\(293\) −8.00000 −0.467365 −0.233682 − 0.972313i \(-0.575078\pi\)
−0.233682 + 0.972313i \(0.575078\pi\)
\(294\) −2.00000 −0.116642
\(295\) 0 0
\(296\) 2.00000 0.116248
\(297\) 16.0000 0.928414
\(298\) 18.0000 1.04271
\(299\) 0 0
\(300\) 0 0
\(301\) −4.00000 −0.230556
\(302\) 16.0000 0.920697
\(303\) 8.00000 0.459588
\(304\) −6.00000 −0.344124
\(305\) 0 0
\(306\) −6.00000 −0.342997
\(307\) 2.00000 0.114146 0.0570730 − 0.998370i \(-0.481823\pi\)
0.0570730 + 0.998370i \(0.481823\pi\)
\(308\) −4.00000 −0.227921
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) 8.00000 0.453638 0.226819 − 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) 0 0
\(313\) −14.0000 −0.791327 −0.395663 − 0.918396i \(-0.629485\pi\)
−0.395663 + 0.918396i \(0.629485\pi\)
\(314\) 12.0000 0.677199
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) −30.0000 −1.68497 −0.842484 − 0.538721i \(-0.818908\pi\)
−0.842484 + 0.538721i \(0.818908\pi\)
\(318\) −12.0000 −0.672927
\(319\) 40.0000 2.23957
\(320\) 0 0
\(321\) 32.0000 1.78607
\(322\) −1.00000 −0.0557278
\(323\) 36.0000 2.00309
\(324\) −11.0000 −0.611111
\(325\) 0 0
\(326\) 0 0
\(327\) 20.0000 1.10600
\(328\) −10.0000 −0.552158
\(329\) 12.0000 0.661581
\(330\) 0 0
\(331\) −32.0000 −1.75888 −0.879440 − 0.476011i \(-0.842082\pi\)
−0.879440 + 0.476011i \(0.842082\pi\)
\(332\) 14.0000 0.768350
\(333\) 2.00000 0.109599
\(334\) −12.0000 −0.656611
\(335\) 0 0
\(336\) 2.00000 0.109109
\(337\) −22.0000 −1.19842 −0.599208 − 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) −13.0000 −0.707107
\(339\) −4.00000 −0.217250
\(340\) 0 0
\(341\) 16.0000 0.866449
\(342\) −6.00000 −0.324443
\(343\) −1.00000 −0.0539949
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) 16.0000 0.860165
\(347\) −24.0000 −1.28839 −0.644194 − 0.764862i \(-0.722807\pi\)
−0.644194 + 0.764862i \(0.722807\pi\)
\(348\) −20.0000 −1.07211
\(349\) −16.0000 −0.856460 −0.428230 − 0.903670i \(-0.640863\pi\)
−0.428230 + 0.903670i \(0.640863\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 4.00000 0.213201
\(353\) 14.0000 0.745145 0.372572 − 0.928003i \(-0.378476\pi\)
0.372572 + 0.928003i \(0.378476\pi\)
\(354\) 4.00000 0.212598
\(355\) 0 0
\(356\) −14.0000 −0.741999
\(357\) −12.0000 −0.635107
\(358\) 12.0000 0.634220
\(359\) −24.0000 −1.26667 −0.633336 − 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) −12.0000 −0.630706
\(363\) −10.0000 −0.524864
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −24.0000 −1.25279 −0.626395 − 0.779506i \(-0.715470\pi\)
−0.626395 + 0.779506i \(0.715470\pi\)
\(368\) 1.00000 0.0521286
\(369\) −10.0000 −0.520579
\(370\) 0 0
\(371\) −6.00000 −0.311504
\(372\) −8.00000 −0.414781
\(373\) 14.0000 0.724893 0.362446 − 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) −24.0000 −1.24101
\(375\) 0 0
\(376\) −12.0000 −0.618853
\(377\) 0 0
\(378\) −4.00000 −0.205738
\(379\) −20.0000 −1.02733 −0.513665 − 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) 32.0000 1.63941
\(382\) −8.00000 −0.409316
\(383\) −20.0000 −1.02195 −0.510976 − 0.859595i \(-0.670716\pi\)
−0.510976 + 0.859595i \(0.670716\pi\)
\(384\) −2.00000 −0.102062
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) 4.00000 0.203331
\(388\) 2.00000 0.101535
\(389\) −26.0000 −1.31825 −0.659126 − 0.752032i \(-0.729074\pi\)
−0.659126 + 0.752032i \(0.729074\pi\)
\(390\) 0 0
\(391\) −6.00000 −0.303433
\(392\) 1.00000 0.0505076
\(393\) −12.0000 −0.605320
\(394\) −6.00000 −0.302276
\(395\) 0 0
\(396\) 4.00000 0.201008
\(397\) −24.0000 −1.20453 −0.602263 − 0.798298i \(-0.705734\pi\)
−0.602263 + 0.798298i \(0.705734\pi\)
\(398\) 4.00000 0.200502
\(399\) −12.0000 −0.600751
\(400\) 0 0
\(401\) −18.0000 −0.898877 −0.449439 − 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −4.00000 −0.199007
\(405\) 0 0
\(406\) −10.0000 −0.496292
\(407\) 8.00000 0.396545
\(408\) 12.0000 0.594089
\(409\) 6.00000 0.296681 0.148340 − 0.988936i \(-0.452607\pi\)
0.148340 + 0.988936i \(0.452607\pi\)
\(410\) 0 0
\(411\) 20.0000 0.986527
\(412\) 4.00000 0.197066
\(413\) 2.00000 0.0984136
\(414\) 1.00000 0.0491473
\(415\) 0 0
\(416\) 0 0
\(417\) −20.0000 −0.979404
\(418\) −24.0000 −1.17388
\(419\) −26.0000 −1.27018 −0.635092 − 0.772437i \(-0.719038\pi\)
−0.635092 + 0.772437i \(0.719038\pi\)
\(420\) 0 0
\(421\) −22.0000 −1.07221 −0.536107 − 0.844150i \(-0.680106\pi\)
−0.536107 + 0.844150i \(0.680106\pi\)
\(422\) 12.0000 0.584151
\(423\) −12.0000 −0.583460
\(424\) 6.00000 0.291386
\(425\) 0 0
\(426\) 16.0000 0.775203
\(427\) 0 0
\(428\) −16.0000 −0.773389
\(429\) 0 0
\(430\) 0 0
\(431\) 24.0000 1.15604 0.578020 − 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) 4.00000 0.192450
\(433\) −30.0000 −1.44171 −0.720854 − 0.693087i \(-0.756250\pi\)
−0.720854 + 0.693087i \(0.756250\pi\)
\(434\) −4.00000 −0.192006
\(435\) 0 0
\(436\) −10.0000 −0.478913
\(437\) −6.00000 −0.287019
\(438\) −12.0000 −0.573382
\(439\) 8.00000 0.381819 0.190910 − 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 20.0000 0.950229 0.475114 − 0.879924i \(-0.342407\pi\)
0.475114 + 0.879924i \(0.342407\pi\)
\(444\) −4.00000 −0.189832
\(445\) 0 0
\(446\) −8.00000 −0.378811
\(447\) −36.0000 −1.70274
\(448\) −1.00000 −0.0472456
\(449\) −30.0000 −1.41579 −0.707894 − 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 0 0
\(451\) −40.0000 −1.88353
\(452\) 2.00000 0.0940721
\(453\) −32.0000 −1.50349
\(454\) 14.0000 0.657053
\(455\) 0 0
\(456\) 12.0000 0.561951
\(457\) 10.0000 0.467780 0.233890 − 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) −20.0000 −0.934539
\(459\) −24.0000 −1.12022
\(460\) 0 0
\(461\) −24.0000 −1.11779 −0.558896 − 0.829238i \(-0.688775\pi\)
−0.558896 + 0.829238i \(0.688775\pi\)
\(462\) 8.00000 0.372194
\(463\) −8.00000 −0.371792 −0.185896 − 0.982569i \(-0.559519\pi\)
−0.185896 + 0.982569i \(0.559519\pi\)
\(464\) 10.0000 0.464238
\(465\) 0 0
\(466\) −26.0000 −1.20443
\(467\) 30.0000 1.38823 0.694117 − 0.719862i \(-0.255795\pi\)
0.694117 + 0.719862i \(0.255795\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −24.0000 −1.10586
\(472\) −2.00000 −0.0920575
\(473\) 16.0000 0.735681
\(474\) 16.0000 0.734904
\(475\) 0 0
\(476\) 6.00000 0.275010
\(477\) 6.00000 0.274721
\(478\) 0 0
\(479\) 12.0000 0.548294 0.274147 − 0.961688i \(-0.411605\pi\)
0.274147 + 0.961688i \(0.411605\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −18.0000 −0.819878
\(483\) 2.00000 0.0910032
\(484\) 5.00000 0.227273
\(485\) 0 0
\(486\) 10.0000 0.453609
\(487\) 32.0000 1.45006 0.725029 − 0.688718i \(-0.241826\pi\)
0.725029 + 0.688718i \(0.241826\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 28.0000 1.26362 0.631811 − 0.775122i \(-0.282312\pi\)
0.631811 + 0.775122i \(0.282312\pi\)
\(492\) 20.0000 0.901670
\(493\) −60.0000 −2.70226
\(494\) 0 0
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) 8.00000 0.358849
\(498\) −28.0000 −1.25471
\(499\) 20.0000 0.895323 0.447661 − 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) 0 0
\(501\) 24.0000 1.07224
\(502\) −2.00000 −0.0892644
\(503\) −16.0000 −0.713405 −0.356702 − 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 0 0
\(506\) 4.00000 0.177822
\(507\) 26.0000 1.15470
\(508\) −16.0000 −0.709885
\(509\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) −6.00000 −0.265424
\(512\) 1.00000 0.0441942
\(513\) −24.0000 −1.05963
\(514\) 22.0000 0.970378
\(515\) 0 0
\(516\) −8.00000 −0.352180
\(517\) −48.0000 −2.11104
\(518\) −2.00000 −0.0878750
\(519\) −32.0000 −1.40464
\(520\) 0 0
\(521\) 22.0000 0.963837 0.481919 − 0.876216i \(-0.339940\pi\)
0.481919 + 0.876216i \(0.339940\pi\)
\(522\) 10.0000 0.437688
\(523\) 22.0000 0.961993 0.480996 − 0.876723i \(-0.340275\pi\)
0.480996 + 0.876723i \(0.340275\pi\)
\(524\) 6.00000 0.262111
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) −24.0000 −1.04546
\(528\) −8.00000 −0.348155
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −2.00000 −0.0867926
\(532\) 6.00000 0.260133
\(533\) 0 0
\(534\) 28.0000 1.21168
\(535\) 0 0
\(536\) 0 0
\(537\) −24.0000 −1.03568
\(538\) 16.0000 0.689809
\(539\) 4.00000 0.172292
\(540\) 0 0
\(541\) −18.0000 −0.773880 −0.386940 − 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) 0 0
\(543\) 24.0000 1.02994
\(544\) −6.00000 −0.257248
\(545\) 0 0
\(546\) 0 0
\(547\) 32.0000 1.36822 0.684111 − 0.729378i \(-0.260191\pi\)
0.684111 + 0.729378i \(0.260191\pi\)
\(548\) −10.0000 −0.427179
\(549\) 0 0
\(550\) 0 0
\(551\) −60.0000 −2.55609
\(552\) −2.00000 −0.0851257
\(553\) 8.00000 0.340195
\(554\) −30.0000 −1.27458
\(555\) 0 0
\(556\) 10.0000 0.424094
\(557\) −18.0000 −0.762684 −0.381342 − 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) 4.00000 0.169334
\(559\) 0 0
\(560\) 0 0
\(561\) 48.0000 2.02656
\(562\) 10.0000 0.421825
\(563\) −6.00000 −0.252870 −0.126435 − 0.991975i \(-0.540353\pi\)
−0.126435 + 0.991975i \(0.540353\pi\)
\(564\) 24.0000 1.01058
\(565\) 0 0
\(566\) −26.0000 −1.09286
\(567\) 11.0000 0.461957
\(568\) −8.00000 −0.335673
\(569\) −10.0000 −0.419222 −0.209611 − 0.977785i \(-0.567220\pi\)
−0.209611 + 0.977785i \(0.567220\pi\)
\(570\) 0 0
\(571\) 28.0000 1.17176 0.585882 − 0.810397i \(-0.300748\pi\)
0.585882 + 0.810397i \(0.300748\pi\)
\(572\) 0 0
\(573\) 16.0000 0.668410
\(574\) 10.0000 0.417392
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 38.0000 1.58196 0.790980 − 0.611842i \(-0.209571\pi\)
0.790980 + 0.611842i \(0.209571\pi\)
\(578\) 19.0000 0.790296
\(579\) 28.0000 1.16364
\(580\) 0 0
\(581\) −14.0000 −0.580818
\(582\) −4.00000 −0.165805
\(583\) 24.0000 0.993978
\(584\) 6.00000 0.248282
\(585\) 0 0
\(586\) −8.00000 −0.330477
\(587\) 14.0000 0.577842 0.288921 − 0.957353i \(-0.406704\pi\)
0.288921 + 0.957353i \(0.406704\pi\)
\(588\) −2.00000 −0.0824786
\(589\) −24.0000 −0.988903
\(590\) 0 0
\(591\) 12.0000 0.493614
\(592\) 2.00000 0.0821995
\(593\) −18.0000 −0.739171 −0.369586 − 0.929197i \(-0.620500\pi\)
−0.369586 + 0.929197i \(0.620500\pi\)
\(594\) 16.0000 0.656488
\(595\) 0 0
\(596\) 18.0000 0.737309
\(597\) −8.00000 −0.327418
\(598\) 0 0
\(599\) 24.0000 0.980613 0.490307 − 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) −30.0000 −1.22373 −0.611863 − 0.790964i \(-0.709580\pi\)
−0.611863 + 0.790964i \(0.709580\pi\)
\(602\) −4.00000 −0.163028
\(603\) 0 0
\(604\) 16.0000 0.651031
\(605\) 0 0
\(606\) 8.00000 0.324978
\(607\) −16.0000 −0.649420 −0.324710 − 0.945814i \(-0.605267\pi\)
−0.324710 + 0.945814i \(0.605267\pi\)
\(608\) −6.00000 −0.243332
\(609\) 20.0000 0.810441
\(610\) 0 0
\(611\) 0 0
\(612\) −6.00000 −0.242536
\(613\) 42.0000 1.69636 0.848182 − 0.529705i \(-0.177697\pi\)
0.848182 + 0.529705i \(0.177697\pi\)
\(614\) 2.00000 0.0807134
\(615\) 0 0
\(616\) −4.00000 −0.161165
\(617\) −30.0000 −1.20775 −0.603877 − 0.797077i \(-0.706378\pi\)
−0.603877 + 0.797077i \(0.706378\pi\)
\(618\) −8.00000 −0.321807
\(619\) 10.0000 0.401934 0.200967 − 0.979598i \(-0.435592\pi\)
0.200967 + 0.979598i \(0.435592\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) 8.00000 0.320771
\(623\) 14.0000 0.560898
\(624\) 0 0
\(625\) 0 0
\(626\) −14.0000 −0.559553
\(627\) 48.0000 1.91694
\(628\) 12.0000 0.478852
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) 40.0000 1.59237 0.796187 − 0.605050i \(-0.206847\pi\)
0.796187 + 0.605050i \(0.206847\pi\)
\(632\) −8.00000 −0.318223
\(633\) −24.0000 −0.953914
\(634\) −30.0000 −1.19145
\(635\) 0 0
\(636\) −12.0000 −0.475831
\(637\) 0 0
\(638\) 40.0000 1.58362
\(639\) −8.00000 −0.316475
\(640\) 0 0
\(641\) 46.0000 1.81689 0.908445 − 0.418004i \(-0.137270\pi\)
0.908445 + 0.418004i \(0.137270\pi\)
\(642\) 32.0000 1.26294
\(643\) −22.0000 −0.867595 −0.433798 − 0.901010i \(-0.642827\pi\)
−0.433798 + 0.901010i \(0.642827\pi\)
\(644\) −1.00000 −0.0394055
\(645\) 0 0
\(646\) 36.0000 1.41640
\(647\) 36.0000 1.41531 0.707653 − 0.706560i \(-0.249754\pi\)
0.707653 + 0.706560i \(0.249754\pi\)
\(648\) −11.0000 −0.432121
\(649\) −8.00000 −0.314027
\(650\) 0 0
\(651\) 8.00000 0.313545
\(652\) 0 0
\(653\) −10.0000 −0.391330 −0.195665 − 0.980671i \(-0.562687\pi\)
−0.195665 + 0.980671i \(0.562687\pi\)
\(654\) 20.0000 0.782062
\(655\) 0 0
\(656\) −10.0000 −0.390434
\(657\) 6.00000 0.234082
\(658\) 12.0000 0.467809
\(659\) −44.0000 −1.71400 −0.856998 − 0.515319i \(-0.827673\pi\)
−0.856998 + 0.515319i \(0.827673\pi\)
\(660\) 0 0
\(661\) −32.0000 −1.24466 −0.622328 − 0.782757i \(-0.713813\pi\)
−0.622328 + 0.782757i \(0.713813\pi\)
\(662\) −32.0000 −1.24372
\(663\) 0 0
\(664\) 14.0000 0.543305
\(665\) 0 0
\(666\) 2.00000 0.0774984
\(667\) 10.0000 0.387202
\(668\) −12.0000 −0.464294
\(669\) 16.0000 0.618596
\(670\) 0 0
\(671\) 0 0
\(672\) 2.00000 0.0771517
\(673\) −10.0000 −0.385472 −0.192736 − 0.981251i \(-0.561736\pi\)
−0.192736 + 0.981251i \(0.561736\pi\)
\(674\) −22.0000 −0.847408
\(675\) 0 0
\(676\) −13.0000 −0.500000
\(677\) 36.0000 1.38359 0.691796 − 0.722093i \(-0.256820\pi\)
0.691796 + 0.722093i \(0.256820\pi\)
\(678\) −4.00000 −0.153619
\(679\) −2.00000 −0.0767530
\(680\) 0 0
\(681\) −28.0000 −1.07296
\(682\) 16.0000 0.612672
\(683\) 28.0000 1.07139 0.535695 − 0.844411i \(-0.320050\pi\)
0.535695 + 0.844411i \(0.320050\pi\)
\(684\) −6.00000 −0.229416
\(685\) 0 0
\(686\) −1.00000 −0.0381802
\(687\) 40.0000 1.52610
\(688\) 4.00000 0.152499
\(689\) 0 0
\(690\) 0 0
\(691\) 6.00000 0.228251 0.114125 − 0.993466i \(-0.463593\pi\)
0.114125 + 0.993466i \(0.463593\pi\)
\(692\) 16.0000 0.608229
\(693\) −4.00000 −0.151947
\(694\) −24.0000 −0.911028
\(695\) 0 0
\(696\) −20.0000 −0.758098
\(697\) 60.0000 2.27266
\(698\) −16.0000 −0.605609
\(699\) 52.0000 1.96682
\(700\) 0 0
\(701\) 6.00000 0.226617 0.113308 − 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) 0 0
\(703\) −12.0000 −0.452589
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) 14.0000 0.526897
\(707\) 4.00000 0.150435
\(708\) 4.00000 0.150329
\(709\) −2.00000 −0.0751116 −0.0375558 − 0.999295i \(-0.511957\pi\)
−0.0375558 + 0.999295i \(0.511957\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) −14.0000 −0.524672
\(713\) 4.00000 0.149801
\(714\) −12.0000 −0.449089
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) 0 0
\(718\) −24.0000 −0.895672
\(719\) −28.0000 −1.04422 −0.522112 − 0.852877i \(-0.674856\pi\)
−0.522112 + 0.852877i \(0.674856\pi\)
\(720\) 0 0
\(721\) −4.00000 −0.148968
\(722\) 17.0000 0.632674
\(723\) 36.0000 1.33885
\(724\) −12.0000 −0.445976
\(725\) 0 0
\(726\) −10.0000 −0.371135
\(727\) −44.0000 −1.63187 −0.815935 − 0.578144i \(-0.803777\pi\)
−0.815935 + 0.578144i \(0.803777\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −24.0000 −0.887672
\(732\) 0 0
\(733\) 24.0000 0.886460 0.443230 − 0.896408i \(-0.353832\pi\)
0.443230 + 0.896408i \(0.353832\pi\)
\(734\) −24.0000 −0.885856
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) 0 0
\(738\) −10.0000 −0.368105
\(739\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −6.00000 −0.220267
\(743\) −32.0000 −1.17397 −0.586983 − 0.809599i \(-0.699684\pi\)
−0.586983 + 0.809599i \(0.699684\pi\)
\(744\) −8.00000 −0.293294
\(745\) 0 0
\(746\) 14.0000 0.512576
\(747\) 14.0000 0.512233
\(748\) −24.0000 −0.877527
\(749\) 16.0000 0.584627
\(750\) 0 0
\(751\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) −12.0000 −0.437595
\(753\) 4.00000 0.145768
\(754\) 0 0
\(755\) 0 0
\(756\) −4.00000 −0.145479
\(757\) 10.0000 0.363456 0.181728 − 0.983349i \(-0.441831\pi\)
0.181728 + 0.983349i \(0.441831\pi\)
\(758\) −20.0000 −0.726433
\(759\) −8.00000 −0.290382
\(760\) 0 0
\(761\) −10.0000 −0.362500 −0.181250 − 0.983437i \(-0.558014\pi\)
−0.181250 + 0.983437i \(0.558014\pi\)
\(762\) 32.0000 1.15924
\(763\) 10.0000 0.362024
\(764\) −8.00000 −0.289430
\(765\) 0 0
\(766\) −20.0000 −0.722629
\(767\) 0 0
\(768\) −2.00000 −0.0721688
\(769\) 14.0000 0.504853 0.252426 − 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) −44.0000 −1.58462
\(772\) −14.0000 −0.503871
\(773\) 16.0000 0.575480 0.287740 − 0.957709i \(-0.407096\pi\)
0.287740 + 0.957709i \(0.407096\pi\)
\(774\) 4.00000 0.143777
\(775\) 0 0
\(776\) 2.00000 0.0717958
\(777\) 4.00000 0.143499
\(778\) −26.0000 −0.932145
\(779\) 60.0000 2.14972
\(780\) 0 0
\(781\) −32.0000 −1.14505
\(782\) −6.00000 −0.214560
\(783\) 40.0000 1.42948
\(784\) 1.00000 0.0357143
\(785\) 0 0
\(786\) −12.0000 −0.428026
\(787\) 22.0000 0.784215 0.392108 − 0.919919i \(-0.371746\pi\)
0.392108 + 0.919919i \(0.371746\pi\)
\(788\) −6.00000 −0.213741
\(789\) 48.0000 1.70885
\(790\) 0 0
\(791\) −2.00000 −0.0711118
\(792\) 4.00000 0.142134
\(793\) 0 0
\(794\) −24.0000 −0.851728
\(795\) 0 0
\(796\) 4.00000 0.141776
\(797\) −20.0000 −0.708436 −0.354218 − 0.935163i \(-0.615253\pi\)
−0.354218 + 0.935163i \(0.615253\pi\)
\(798\) −12.0000 −0.424795
\(799\) 72.0000 2.54718
\(800\) 0 0
\(801\) −14.0000 −0.494666
\(802\) −18.0000 −0.635602
\(803\) 24.0000 0.846942
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −32.0000 −1.12645
\(808\) −4.00000 −0.140720
\(809\) 6.00000 0.210949 0.105474 − 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) 38.0000 1.33436 0.667180 − 0.744896i \(-0.267501\pi\)
0.667180 + 0.744896i \(0.267501\pi\)
\(812\) −10.0000 −0.350931
\(813\) 0 0
\(814\) 8.00000 0.280400
\(815\) 0 0
\(816\) 12.0000 0.420084
\(817\) −24.0000 −0.839654
\(818\) 6.00000 0.209785
\(819\) 0 0
\(820\) 0 0
\(821\) 6.00000 0.209401 0.104701 − 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) 20.0000 0.697580
\(823\) −40.0000 −1.39431 −0.697156 − 0.716919i \(-0.745552\pi\)
−0.697156 + 0.716919i \(0.745552\pi\)
\(824\) 4.00000 0.139347
\(825\) 0 0
\(826\) 2.00000 0.0695889
\(827\) 48.0000 1.66912 0.834562 − 0.550914i \(-0.185721\pi\)
0.834562 + 0.550914i \(0.185721\pi\)
\(828\) 1.00000 0.0347524
\(829\) −20.0000 −0.694629 −0.347314 − 0.937749i \(-0.612906\pi\)
−0.347314 + 0.937749i \(0.612906\pi\)
\(830\) 0 0
\(831\) 60.0000 2.08138
\(832\) 0 0
\(833\) −6.00000 −0.207888
\(834\) −20.0000 −0.692543
\(835\) 0 0
\(836\) −24.0000 −0.830057
\(837\) 16.0000 0.553041
\(838\) −26.0000 −0.898155
\(839\) −36.0000 −1.24286 −0.621429 − 0.783470i \(-0.713448\pi\)
−0.621429 + 0.783470i \(0.713448\pi\)
\(840\) 0 0
\(841\) 71.0000 2.44828
\(842\) −22.0000 −0.758170
\(843\) −20.0000 −0.688837
\(844\) 12.0000 0.413057
\(845\) 0 0
\(846\) −12.0000 −0.412568
\(847\) −5.00000 −0.171802
\(848\) 6.00000 0.206041
\(849\) 52.0000 1.78464
\(850\) 0 0
\(851\) 2.00000 0.0685591
\(852\) 16.0000 0.548151
\(853\) 16.0000 0.547830 0.273915 − 0.961754i \(-0.411681\pi\)
0.273915 + 0.961754i \(0.411681\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −16.0000 −0.546869
\(857\) −54.0000 −1.84460 −0.922302 − 0.386469i \(-0.873695\pi\)
−0.922302 + 0.386469i \(0.873695\pi\)
\(858\) 0 0
\(859\) 34.0000 1.16007 0.580033 − 0.814593i \(-0.303040\pi\)
0.580033 + 0.814593i \(0.303040\pi\)
\(860\) 0 0
\(861\) −20.0000 −0.681598
\(862\) 24.0000 0.817443
\(863\) −24.0000 −0.816970 −0.408485 − 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) 4.00000 0.136083
\(865\) 0 0
\(866\) −30.0000 −1.01944
\(867\) −38.0000 −1.29055
\(868\) −4.00000 −0.135769
\(869\) −32.0000 −1.08553
\(870\) 0 0
\(871\) 0 0
\(872\) −10.0000 −0.338643
\(873\) 2.00000 0.0676897
\(874\) −6.00000 −0.202953
\(875\) 0 0
\(876\) −12.0000 −0.405442
\(877\) 38.0000 1.28317 0.641584 − 0.767052i \(-0.278277\pi\)
0.641584 + 0.767052i \(0.278277\pi\)
\(878\) 8.00000 0.269987
\(879\) 16.0000 0.539667
\(880\) 0 0
\(881\) 26.0000 0.875962 0.437981 − 0.898984i \(-0.355694\pi\)
0.437981 + 0.898984i \(0.355694\pi\)
\(882\) 1.00000 0.0336718
\(883\) 12.0000 0.403832 0.201916 − 0.979403i \(-0.435283\pi\)
0.201916 + 0.979403i \(0.435283\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 20.0000 0.671913
\(887\) 28.0000 0.940148 0.470074 − 0.882627i \(-0.344227\pi\)
0.470074 + 0.882627i \(0.344227\pi\)
\(888\) −4.00000 −0.134231
\(889\) 16.0000 0.536623
\(890\) 0 0
\(891\) −44.0000 −1.47406
\(892\) −8.00000 −0.267860
\(893\) 72.0000 2.40939
\(894\) −36.0000 −1.20402
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) −30.0000 −1.00111
\(899\) 40.0000 1.33407
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) −40.0000 −1.33185
\(903\) 8.00000 0.266223
\(904\) 2.00000 0.0665190
\(905\) 0 0
\(906\) −32.0000 −1.06313
\(907\) 40.0000 1.32818 0.664089 − 0.747653i \(-0.268820\pi\)
0.664089 + 0.747653i \(0.268820\pi\)
\(908\) 14.0000 0.464606
\(909\) −4.00000 −0.132672
\(910\) 0 0
\(911\) 32.0000 1.06021 0.530104 − 0.847933i \(-0.322153\pi\)
0.530104 + 0.847933i \(0.322153\pi\)
\(912\) 12.0000 0.397360
\(913\) 56.0000 1.85333
\(914\) 10.0000 0.330771
\(915\) 0 0
\(916\) −20.0000 −0.660819
\(917\) −6.00000 −0.198137
\(918\) −24.0000 −0.792118
\(919\) −24.0000 −0.791687 −0.395843 − 0.918318i \(-0.629548\pi\)
−0.395843 + 0.918318i \(0.629548\pi\)
\(920\) 0 0
\(921\) −4.00000 −0.131804
\(922\) −24.0000 −0.790398
\(923\) 0 0
\(924\) 8.00000 0.263181
\(925\) 0 0
\(926\) −8.00000 −0.262896
\(927\) 4.00000 0.131377
\(928\) 10.0000 0.328266
\(929\) 54.0000 1.77168 0.885841 − 0.463988i \(-0.153582\pi\)
0.885841 + 0.463988i \(0.153582\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) −26.0000 −0.851658
\(933\) −16.0000 −0.523816
\(934\) 30.0000 0.981630
\(935\) 0 0
\(936\) 0 0
\(937\) 38.0000 1.24141 0.620703 − 0.784046i \(-0.286847\pi\)
0.620703 + 0.784046i \(0.286847\pi\)
\(938\) 0 0
\(939\) 28.0000 0.913745
\(940\) 0 0
\(941\) 24.0000 0.782378 0.391189 − 0.920310i \(-0.372064\pi\)
0.391189 + 0.920310i \(0.372064\pi\)
\(942\) −24.0000 −0.781962
\(943\) −10.0000 −0.325645
\(944\) −2.00000 −0.0650945
\(945\) 0 0
\(946\) 16.0000 0.520205
\(947\) −24.0000 −0.779895 −0.389948 − 0.920837i \(-0.627507\pi\)
−0.389948 + 0.920837i \(0.627507\pi\)
\(948\) 16.0000 0.519656
\(949\) 0 0
\(950\) 0 0
\(951\) 60.0000 1.94563
\(952\) 6.00000 0.194461
\(953\) −2.00000 −0.0647864 −0.0323932 − 0.999475i \(-0.510313\pi\)
−0.0323932 + 0.999475i \(0.510313\pi\)
\(954\) 6.00000 0.194257
\(955\) 0 0
\(956\) 0 0
\(957\) −80.0000 −2.58603
\(958\) 12.0000 0.387702
\(959\) 10.0000 0.322917
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) −16.0000 −0.515593
\(964\) −18.0000 −0.579741
\(965\) 0 0
\(966\) 2.00000 0.0643489
\(967\) 8.00000 0.257263 0.128631 − 0.991692i \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) 5.00000 0.160706
\(969\) −72.0000 −2.31297
\(970\) 0 0
\(971\) −22.0000 −0.706014 −0.353007 − 0.935621i \(-0.614841\pi\)
−0.353007 + 0.935621i \(0.614841\pi\)
\(972\) 10.0000 0.320750
\(973\) −10.0000 −0.320585
\(974\) 32.0000 1.02535
\(975\) 0 0
\(976\) 0 0
\(977\) −50.0000 −1.59964 −0.799821 − 0.600239i \(-0.795072\pi\)
−0.799821 + 0.600239i \(0.795072\pi\)
\(978\) 0 0
\(979\) −56.0000 −1.78977
\(980\) 0 0
\(981\) −10.0000 −0.319275
\(982\) 28.0000 0.893516
\(983\) −44.0000 −1.40338 −0.701691 − 0.712481i \(-0.747571\pi\)
−0.701691 + 0.712481i \(0.747571\pi\)
\(984\) 20.0000 0.637577
\(985\) 0 0
\(986\) −60.0000 −1.91079
\(987\) −24.0000 −0.763928
\(988\) 0 0
\(989\) 4.00000 0.127193
\(990\) 0 0
\(991\) 32.0000 1.01651 0.508257 − 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) 4.00000 0.127000
\(993\) 64.0000 2.03098
\(994\) 8.00000 0.253745
\(995\) 0 0
\(996\) −28.0000 −0.887214
\(997\) 20.0000 0.633406 0.316703 − 0.948525i \(-0.397424\pi\)
0.316703 + 0.948525i \(0.397424\pi\)
\(998\) 20.0000 0.633089
\(999\) 8.00000 0.253109
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 8050.2.a.n.1.1 1
5.4 even 2 322.2.a.b.1.1 ✓ 1
15.14 odd 2 2898.2.a.p.1.1 1
20.19 odd 2 2576.2.a.c.1.1 1
35.34 odd 2 2254.2.a.a.1.1 1
115.114 odd 2 7406.2.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
322.2.a.b.1.1 ✓ 1 5.4 even 2
2254.2.a.a.1.1 1 35.34 odd 2
2576.2.a.c.1.1 1 20.19 odd 2
2898.2.a.p.1.1 1 15.14 odd 2
7406.2.a.e.1.1 1 115.114 odd 2
8050.2.a.n.1.1 1 1.1 even 1 trivial