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Properties

Label 8850.2.a.be.1.1
Level $8850$
Weight $2$
Character 8850.1
Self dual yes
Analytic conductor $70.668$
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] = [8850,2,Mod(1,8850)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8850, 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("8850.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8850 = 2 \cdot 3 \cdot 5^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8850.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: \(70.6676057888\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 354)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 8850.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -5.00000 q^{11} +1.00000 q^{12} -1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -1.00000 q^{17} +1.00000 q^{18} +1.00000 q^{21} -5.00000 q^{22} +6.00000 q^{23} +1.00000 q^{24} -1.00000 q^{26} +1.00000 q^{27} +1.00000 q^{28} -10.0000 q^{29} -8.00000 q^{31} +1.00000 q^{32} -5.00000 q^{33} -1.00000 q^{34} +1.00000 q^{36} -9.00000 q^{37} -1.00000 q^{39} -5.00000 q^{41} +1.00000 q^{42} -3.00000 q^{43} -5.00000 q^{44} +6.00000 q^{46} +1.00000 q^{48} -6.00000 q^{49} -1.00000 q^{51} -1.00000 q^{52} -4.00000 q^{53} +1.00000 q^{54} +1.00000 q^{56} -10.0000 q^{58} -1.00000 q^{59} +6.00000 q^{61} -8.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -5.00000 q^{66} -4.00000 q^{67} -1.00000 q^{68} +6.00000 q^{69} +1.00000 q^{71} +1.00000 q^{72} -9.00000 q^{74} -5.00000 q^{77} -1.00000 q^{78} -3.00000 q^{79} +1.00000 q^{81} -5.00000 q^{82} -7.00000 q^{83} +1.00000 q^{84} -3.00000 q^{86} -10.0000 q^{87} -5.00000 q^{88} +16.0000 q^{89} -1.00000 q^{91} +6.00000 q^{92} -8.00000 q^{93} +1.00000 q^{96} +12.0000 q^{97} -6.00000 q^{98} -5.00000 q^{99} +O(q^{100})\)

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
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 1.00000 0.377964 0.188982 − 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −5.00000 −1.50756 −0.753778 − 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) 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
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −1.00000 −0.242536 −0.121268 − 0.992620i \(-0.538696\pi\)
−0.121268 + 0.992620i \(0.538696\pi\)
\(18\) 1.00000 0.235702
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) −5.00000 −1.06600
\(23\) 6.00000 1.25109 0.625543 − 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) −1.00000 −0.196116
\(27\) 1.00000 0.192450
\(28\) 1.00000 0.188982
\(29\) −10.0000 −1.85695 −0.928477 − 0.371391i \(-0.878881\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 − 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 1.00000 0.176777
\(33\) −5.00000 −0.870388
\(34\) −1.00000 −0.171499
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −9.00000 −1.47959 −0.739795 − 0.672832i \(-0.765078\pi\)
−0.739795 + 0.672832i \(0.765078\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) −5.00000 −0.780869 −0.390434 − 0.920631i \(-0.627675\pi\)
−0.390434 + 0.920631i \(0.627675\pi\)
\(42\) 1.00000 0.154303
\(43\) −3.00000 −0.457496 −0.228748 − 0.973486i \(-0.573463\pi\)
−0.228748 + 0.973486i \(0.573463\pi\)
\(44\) −5.00000 −0.753778
\(45\) 0 0
\(46\) 6.00000 0.884652
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.00000 0.144338
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) −1.00000 −0.140028
\(52\) −1.00000 −0.138675
\(53\) −4.00000 −0.549442 −0.274721 − 0.961524i \(-0.588586\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) −10.0000 −1.31306
\(59\) −1.00000 −0.130189
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 − 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) −8.00000 −1.01600
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −5.00000 −0.615457
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −1.00000 −0.121268
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) 1.00000 0.118678 0.0593391 − 0.998238i \(-0.481101\pi\)
0.0593391 + 0.998238i \(0.481101\pi\)
\(72\) 1.00000 0.117851
\(73\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) −9.00000 −1.04623
\(75\) 0 0
\(76\) 0 0
\(77\) −5.00000 −0.569803
\(78\) −1.00000 −0.113228
\(79\) −3.00000 −0.337526 −0.168763 − 0.985657i \(-0.553977\pi\)
−0.168763 + 0.985657i \(0.553977\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −5.00000 −0.552158
\(83\) −7.00000 −0.768350 −0.384175 − 0.923260i \(-0.625514\pi\)
−0.384175 + 0.923260i \(0.625514\pi\)
\(84\) 1.00000 0.109109
\(85\) 0 0
\(86\) −3.00000 −0.323498
\(87\) −10.0000 −1.07211
\(88\) −5.00000 −0.533002
\(89\) 16.0000 1.69600 0.847998 − 0.529999i \(-0.177808\pi\)
0.847998 + 0.529999i \(0.177808\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 6.00000 0.625543
\(93\) −8.00000 −0.829561
\(94\) 0 0
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 12.0000 1.21842 0.609208 − 0.793011i \(-0.291488\pi\)
0.609208 + 0.793011i \(0.291488\pi\)
\(98\) −6.00000 −0.606092
\(99\) −5.00000 −0.502519
\(100\) 0 0
\(101\) 17.0000 1.69156 0.845782 − 0.533529i \(-0.179135\pi\)
0.845782 + 0.533529i \(0.179135\pi\)
\(102\) −1.00000 −0.0990148
\(103\) −4.00000 −0.394132 −0.197066 − 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) −4.00000 −0.388514
\(107\) 6.00000 0.580042 0.290021 − 0.957020i \(-0.406338\pi\)
0.290021 + 0.957020i \(0.406338\pi\)
\(108\) 1.00000 0.0962250
\(109\) −10.0000 −0.957826 −0.478913 − 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) −9.00000 −0.854242
\(112\) 1.00000 0.0944911
\(113\) −6.00000 −0.564433 −0.282216 − 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −10.0000 −0.928477
\(117\) −1.00000 −0.0924500
\(118\) −1.00000 −0.0920575
\(119\) −1.00000 −0.0916698
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 6.00000 0.543214
\(123\) −5.00000 −0.450835
\(124\) −8.00000 −0.718421
\(125\) 0 0
\(126\) 1.00000 0.0890871
\(127\) 8.00000 0.709885 0.354943 − 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 1.00000 0.0883883
\(129\) −3.00000 −0.264135
\(130\) 0 0
\(131\) −20.0000 −1.74741 −0.873704 − 0.486458i \(-0.838289\pi\)
−0.873704 + 0.486458i \(0.838289\pi\)
\(132\) −5.00000 −0.435194
\(133\) 0 0
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) −1.00000 −0.0857493
\(137\) 5.00000 0.427179 0.213589 − 0.976924i \(-0.431485\pi\)
0.213589 + 0.976924i \(0.431485\pi\)
\(138\) 6.00000 0.510754
\(139\) 4.00000 0.339276 0.169638 − 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 1.00000 0.0839181
\(143\) 5.00000 0.418121
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 0 0
\(147\) −6.00000 −0.494872
\(148\) −9.00000 −0.739795
\(149\) 21.0000 1.72039 0.860194 − 0.509968i \(-0.170343\pi\)
0.860194 + 0.509968i \(0.170343\pi\)
\(150\) 0 0
\(151\) −10.0000 −0.813788 −0.406894 − 0.913475i \(-0.633388\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(152\) 0 0
\(153\) −1.00000 −0.0808452
\(154\) −5.00000 −0.402911
\(155\) 0 0
\(156\) −1.00000 −0.0800641
\(157\) 6.00000 0.478852 0.239426 − 0.970915i \(-0.423041\pi\)
0.239426 + 0.970915i \(0.423041\pi\)
\(158\) −3.00000 −0.238667
\(159\) −4.00000 −0.317221
\(160\) 0 0
\(161\) 6.00000 0.472866
\(162\) 1.00000 0.0785674
\(163\) −14.0000 −1.09656 −0.548282 − 0.836293i \(-0.684718\pi\)
−0.548282 + 0.836293i \(0.684718\pi\)
\(164\) −5.00000 −0.390434
\(165\) 0 0
\(166\) −7.00000 −0.543305
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 1.00000 0.0771517
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) 0 0
\(172\) −3.00000 −0.228748
\(173\) 9.00000 0.684257 0.342129 − 0.939653i \(-0.388852\pi\)
0.342129 + 0.939653i \(0.388852\pi\)
\(174\) −10.0000 −0.758098
\(175\) 0 0
\(176\) −5.00000 −0.376889
\(177\) −1.00000 −0.0751646
\(178\) 16.0000 1.19925
\(179\) 5.00000 0.373718 0.186859 − 0.982387i \(-0.440169\pi\)
0.186859 + 0.982387i \(0.440169\pi\)
\(180\) 0 0
\(181\) −22.0000 −1.63525 −0.817624 − 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) −1.00000 −0.0741249
\(183\) 6.00000 0.443533
\(184\) 6.00000 0.442326
\(185\) 0 0
\(186\) −8.00000 −0.586588
\(187\) 5.00000 0.365636
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −12.0000 −0.868290 −0.434145 − 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 1.00000 0.0721688
\(193\) −13.0000 −0.935760 −0.467880 − 0.883792i \(-0.654982\pi\)
−0.467880 + 0.883792i \(0.654982\pi\)
\(194\) 12.0000 0.861550
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) −5.00000 −0.355335
\(199\) −4.00000 −0.283552 −0.141776 − 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) 17.0000 1.19612
\(203\) −10.0000 −0.701862
\(204\) −1.00000 −0.0700140
\(205\) 0 0
\(206\) −4.00000 −0.278693
\(207\) 6.00000 0.417029
\(208\) −1.00000 −0.0693375
\(209\) 0 0
\(210\) 0 0
\(211\) −3.00000 −0.206529 −0.103264 − 0.994654i \(-0.532929\pi\)
−0.103264 + 0.994654i \(0.532929\pi\)
\(212\) −4.00000 −0.274721
\(213\) 1.00000 0.0685189
\(214\) 6.00000 0.410152
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) −8.00000 −0.543075
\(218\) −10.0000 −0.677285
\(219\) 0 0
\(220\) 0 0
\(221\) 1.00000 0.0672673
\(222\) −9.00000 −0.604040
\(223\) 9.00000 0.602685 0.301342 − 0.953516i \(-0.402565\pi\)
0.301342 + 0.953516i \(0.402565\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) −7.00000 −0.464606 −0.232303 − 0.972643i \(-0.574626\pi\)
−0.232303 + 0.972643i \(0.574626\pi\)
\(228\) 0 0
\(229\) −25.0000 −1.65205 −0.826023 − 0.563636i \(-0.809402\pi\)
−0.826023 + 0.563636i \(0.809402\pi\)
\(230\) 0 0
\(231\) −5.00000 −0.328976
\(232\) −10.0000 −0.656532
\(233\) −20.0000 −1.31024 −0.655122 − 0.755523i \(-0.727383\pi\)
−0.655122 + 0.755523i \(0.727383\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 0 0
\(236\) −1.00000 −0.0650945
\(237\) −3.00000 −0.194871
\(238\) −1.00000 −0.0648204
\(239\) 8.00000 0.517477 0.258738 − 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 0 0
\(241\) 15.0000 0.966235 0.483117 − 0.875556i \(-0.339504\pi\)
0.483117 + 0.875556i \(0.339504\pi\)
\(242\) 14.0000 0.899954
\(243\) 1.00000 0.0641500
\(244\) 6.00000 0.384111
\(245\) 0 0
\(246\) −5.00000 −0.318788
\(247\) 0 0
\(248\) −8.00000 −0.508001
\(249\) −7.00000 −0.443607
\(250\) 0 0
\(251\) 4.00000 0.252478 0.126239 − 0.992000i \(-0.459709\pi\)
0.126239 + 0.992000i \(0.459709\pi\)
\(252\) 1.00000 0.0629941
\(253\) −30.0000 −1.88608
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 21.0000 1.30994 0.654972 − 0.755653i \(-0.272680\pi\)
0.654972 + 0.755653i \(0.272680\pi\)
\(258\) −3.00000 −0.186772
\(259\) −9.00000 −0.559233
\(260\) 0 0
\(261\) −10.0000 −0.618984
\(262\) −20.0000 −1.23560
\(263\) −3.00000 −0.184988 −0.0924940 − 0.995713i \(-0.529484\pi\)
−0.0924940 + 0.995713i \(0.529484\pi\)
\(264\) −5.00000 −0.307729
\(265\) 0 0
\(266\) 0 0
\(267\) 16.0000 0.979184
\(268\) −4.00000 −0.244339
\(269\) 3.00000 0.182913 0.0914566 − 0.995809i \(-0.470848\pi\)
0.0914566 + 0.995809i \(0.470848\pi\)
\(270\) 0 0
\(271\) −15.0000 −0.911185 −0.455593 − 0.890188i \(-0.650573\pi\)
−0.455593 + 0.890188i \(0.650573\pi\)
\(272\) −1.00000 −0.0606339
\(273\) −1.00000 −0.0605228
\(274\) 5.00000 0.302061
\(275\) 0 0
\(276\) 6.00000 0.361158
\(277\) 26.0000 1.56219 0.781094 − 0.624413i \(-0.214662\pi\)
0.781094 + 0.624413i \(0.214662\pi\)
\(278\) 4.00000 0.239904
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) −7.00000 −0.417585 −0.208792 − 0.977960i \(-0.566953\pi\)
−0.208792 + 0.977960i \(0.566953\pi\)
\(282\) 0 0
\(283\) −31.0000 −1.84276 −0.921379 − 0.388664i \(-0.872937\pi\)
−0.921379 + 0.388664i \(0.872937\pi\)
\(284\) 1.00000 0.0593391
\(285\) 0 0
\(286\) 5.00000 0.295656
\(287\) −5.00000 −0.295141
\(288\) 1.00000 0.0589256
\(289\) −16.0000 −0.941176
\(290\) 0 0
\(291\) 12.0000 0.703452
\(292\) 0 0
\(293\) 12.0000 0.701047 0.350524 − 0.936554i \(-0.386004\pi\)
0.350524 + 0.936554i \(0.386004\pi\)
\(294\) −6.00000 −0.349927
\(295\) 0 0
\(296\) −9.00000 −0.523114
\(297\) −5.00000 −0.290129
\(298\) 21.0000 1.21650
\(299\) −6.00000 −0.346989
\(300\) 0 0
\(301\) −3.00000 −0.172917
\(302\) −10.0000 −0.575435
\(303\) 17.0000 0.976624
\(304\) 0 0
\(305\) 0 0
\(306\) −1.00000 −0.0571662
\(307\) −2.00000 −0.114146 −0.0570730 − 0.998370i \(-0.518177\pi\)
−0.0570730 + 0.998370i \(0.518177\pi\)
\(308\) −5.00000 −0.284901
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) 7.00000 0.396934 0.198467 − 0.980108i \(-0.436404\pi\)
0.198467 + 0.980108i \(0.436404\pi\)
\(312\) −1.00000 −0.0566139
\(313\) −14.0000 −0.791327 −0.395663 − 0.918396i \(-0.629485\pi\)
−0.395663 + 0.918396i \(0.629485\pi\)
\(314\) 6.00000 0.338600
\(315\) 0 0
\(316\) −3.00000 −0.168763
\(317\) −32.0000 −1.79730 −0.898650 − 0.438667i \(-0.855451\pi\)
−0.898650 + 0.438667i \(0.855451\pi\)
\(318\) −4.00000 −0.224309
\(319\) 50.0000 2.79946
\(320\) 0 0
\(321\) 6.00000 0.334887
\(322\) 6.00000 0.334367
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −14.0000 −0.775388
\(327\) −10.0000 −0.553001
\(328\) −5.00000 −0.276079
\(329\) 0 0
\(330\) 0 0
\(331\) 34.0000 1.86881 0.934405 − 0.356214i \(-0.115932\pi\)
0.934405 + 0.356214i \(0.115932\pi\)
\(332\) −7.00000 −0.384175
\(333\) −9.00000 −0.493197
\(334\) 0 0
\(335\) 0 0
\(336\) 1.00000 0.0545545
\(337\) −2.00000 −0.108947 −0.0544735 − 0.998515i \(-0.517348\pi\)
−0.0544735 + 0.998515i \(0.517348\pi\)
\(338\) −12.0000 −0.652714
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) 40.0000 2.16612
\(342\) 0 0
\(343\) −13.0000 −0.701934
\(344\) −3.00000 −0.161749
\(345\) 0 0
\(346\) 9.00000 0.483843
\(347\) 36.0000 1.93258 0.966291 − 0.257454i \(-0.0828835\pi\)
0.966291 + 0.257454i \(0.0828835\pi\)
\(348\) −10.0000 −0.536056
\(349\) −7.00000 −0.374701 −0.187351 − 0.982293i \(-0.559990\pi\)
−0.187351 + 0.982293i \(0.559990\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) −5.00000 −0.266501
\(353\) 24.0000 1.27739 0.638696 − 0.769460i \(-0.279474\pi\)
0.638696 + 0.769460i \(0.279474\pi\)
\(354\) −1.00000 −0.0531494
\(355\) 0 0
\(356\) 16.0000 0.847998
\(357\) −1.00000 −0.0529256
\(358\) 5.00000 0.264258
\(359\) 11.0000 0.580558 0.290279 − 0.956942i \(-0.406252\pi\)
0.290279 + 0.956942i \(0.406252\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) −22.0000 −1.15629
\(363\) 14.0000 0.734809
\(364\) −1.00000 −0.0524142
\(365\) 0 0
\(366\) 6.00000 0.313625
\(367\) 32.0000 1.67039 0.835193 − 0.549957i \(-0.185356\pi\)
0.835193 + 0.549957i \(0.185356\pi\)
\(368\) 6.00000 0.312772
\(369\) −5.00000 −0.260290
\(370\) 0 0
\(371\) −4.00000 −0.207670
\(372\) −8.00000 −0.414781
\(373\) −24.0000 −1.24267 −0.621336 − 0.783544i \(-0.713410\pi\)
−0.621336 + 0.783544i \(0.713410\pi\)
\(374\) 5.00000 0.258544
\(375\) 0 0
\(376\) 0 0
\(377\) 10.0000 0.515026
\(378\) 1.00000 0.0514344
\(379\) 14.0000 0.719132 0.359566 − 0.933120i \(-0.382925\pi\)
0.359566 + 0.933120i \(0.382925\pi\)
\(380\) 0 0
\(381\) 8.00000 0.409852
\(382\) −12.0000 −0.613973
\(383\) 21.0000 1.07305 0.536525 − 0.843884i \(-0.319737\pi\)
0.536525 + 0.843884i \(0.319737\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −13.0000 −0.661683
\(387\) −3.00000 −0.152499
\(388\) 12.0000 0.609208
\(389\) −22.0000 −1.11544 −0.557722 − 0.830028i \(-0.688325\pi\)
−0.557722 + 0.830028i \(0.688325\pi\)
\(390\) 0 0
\(391\) −6.00000 −0.303433
\(392\) −6.00000 −0.303046
\(393\) −20.0000 −1.00887
\(394\) 0 0
\(395\) 0 0
\(396\) −5.00000 −0.251259
\(397\) 22.0000 1.10415 0.552074 − 0.833795i \(-0.313837\pi\)
0.552074 + 0.833795i \(0.313837\pi\)
\(398\) −4.00000 −0.200502
\(399\) 0 0
\(400\) 0 0
\(401\) 16.0000 0.799002 0.399501 − 0.916733i \(-0.369183\pi\)
0.399501 + 0.916733i \(0.369183\pi\)
\(402\) −4.00000 −0.199502
\(403\) 8.00000 0.398508
\(404\) 17.0000 0.845782
\(405\) 0 0
\(406\) −10.0000 −0.496292
\(407\) 45.0000 2.23057
\(408\) −1.00000 −0.0495074
\(409\) 26.0000 1.28562 0.642809 − 0.766027i \(-0.277769\pi\)
0.642809 + 0.766027i \(0.277769\pi\)
\(410\) 0 0
\(411\) 5.00000 0.246632
\(412\) −4.00000 −0.197066
\(413\) −1.00000 −0.0492068
\(414\) 6.00000 0.294884
\(415\) 0 0
\(416\) −1.00000 −0.0490290
\(417\) 4.00000 0.195881
\(418\) 0 0
\(419\) −35.0000 −1.70986 −0.854931 − 0.518742i \(-0.826401\pi\)
−0.854931 + 0.518742i \(0.826401\pi\)
\(420\) 0 0
\(421\) 1.00000 0.0487370 0.0243685 − 0.999703i \(-0.492242\pi\)
0.0243685 + 0.999703i \(0.492242\pi\)
\(422\) −3.00000 −0.146038
\(423\) 0 0
\(424\) −4.00000 −0.194257
\(425\) 0 0
\(426\) 1.00000 0.0484502
\(427\) 6.00000 0.290360
\(428\) 6.00000 0.290021
\(429\) 5.00000 0.241402
\(430\) 0 0
\(431\) 26.0000 1.25238 0.626188 − 0.779672i \(-0.284614\pi\)
0.626188 + 0.779672i \(0.284614\pi\)
\(432\) 1.00000 0.0481125
\(433\) 31.0000 1.48976 0.744882 − 0.667196i \(-0.232506\pi\)
0.744882 + 0.667196i \(0.232506\pi\)
\(434\) −8.00000 −0.384012
\(435\) 0 0
\(436\) −10.0000 −0.478913
\(437\) 0 0
\(438\) 0 0
\(439\) −11.0000 −0.525001 −0.262501 − 0.964932i \(-0.584547\pi\)
−0.262501 + 0.964932i \(0.584547\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) 1.00000 0.0475651
\(443\) −17.0000 −0.807694 −0.403847 − 0.914826i \(-0.632327\pi\)
−0.403847 + 0.914826i \(0.632327\pi\)
\(444\) −9.00000 −0.427121
\(445\) 0 0
\(446\) 9.00000 0.426162
\(447\) 21.0000 0.993266
\(448\) 1.00000 0.0472456
\(449\) −31.0000 −1.46298 −0.731490 − 0.681852i \(-0.761175\pi\)
−0.731490 + 0.681852i \(0.761175\pi\)
\(450\) 0 0
\(451\) 25.0000 1.17720
\(452\) −6.00000 −0.282216
\(453\) −10.0000 −0.469841
\(454\) −7.00000 −0.328526
\(455\) 0 0
\(456\) 0 0
\(457\) 4.00000 0.187112 0.0935561 − 0.995614i \(-0.470177\pi\)
0.0935561 + 0.995614i \(0.470177\pi\)
\(458\) −25.0000 −1.16817
\(459\) −1.00000 −0.0466760
\(460\) 0 0
\(461\) −20.0000 −0.931493 −0.465746 − 0.884918i \(-0.654214\pi\)
−0.465746 + 0.884918i \(0.654214\pi\)
\(462\) −5.00000 −0.232621
\(463\) −20.0000 −0.929479 −0.464739 − 0.885448i \(-0.653852\pi\)
−0.464739 + 0.885448i \(0.653852\pi\)
\(464\) −10.0000 −0.464238
\(465\) 0 0
\(466\) −20.0000 −0.926482
\(467\) −13.0000 −0.601568 −0.300784 − 0.953692i \(-0.597248\pi\)
−0.300784 + 0.953692i \(0.597248\pi\)
\(468\) −1.00000 −0.0462250
\(469\) −4.00000 −0.184703
\(470\) 0 0
\(471\) 6.00000 0.276465
\(472\) −1.00000 −0.0460287
\(473\) 15.0000 0.689701
\(474\) −3.00000 −0.137795
\(475\) 0 0
\(476\) −1.00000 −0.0458349
\(477\) −4.00000 −0.183147
\(478\) 8.00000 0.365911
\(479\) −28.0000 −1.27935 −0.639676 − 0.768644i \(-0.720932\pi\)
−0.639676 + 0.768644i \(0.720932\pi\)
\(480\) 0 0
\(481\) 9.00000 0.410365
\(482\) 15.0000 0.683231
\(483\) 6.00000 0.273009
\(484\) 14.0000 0.636364
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 7.00000 0.317200 0.158600 − 0.987343i \(-0.449302\pi\)
0.158600 + 0.987343i \(0.449302\pi\)
\(488\) 6.00000 0.271607
\(489\) −14.0000 −0.633102
\(490\) 0 0
\(491\) 30.0000 1.35388 0.676941 − 0.736038i \(-0.263305\pi\)
0.676941 + 0.736038i \(0.263305\pi\)
\(492\) −5.00000 −0.225417
\(493\) 10.0000 0.450377
\(494\) 0 0
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) 1.00000 0.0448561
\(498\) −7.00000 −0.313678
\(499\) 24.0000 1.07439 0.537194 − 0.843459i \(-0.319484\pi\)
0.537194 + 0.843459i \(0.319484\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 4.00000 0.178529
\(503\) 26.0000 1.15928 0.579641 − 0.814872i \(-0.303193\pi\)
0.579641 + 0.814872i \(0.303193\pi\)
\(504\) 1.00000 0.0445435
\(505\) 0 0
\(506\) −30.0000 −1.33366
\(507\) −12.0000 −0.532939
\(508\) 8.00000 0.354943
\(509\) −10.0000 −0.443242 −0.221621 − 0.975133i \(-0.571135\pi\)
−0.221621 + 0.975133i \(0.571135\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 21.0000 0.926270
\(515\) 0 0
\(516\) −3.00000 −0.132068
\(517\) 0 0
\(518\) −9.00000 −0.395437
\(519\) 9.00000 0.395056
\(520\) 0 0
\(521\) −18.0000 −0.788594 −0.394297 − 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) −10.0000 −0.437688
\(523\) −28.0000 −1.22435 −0.612177 − 0.790721i \(-0.709706\pi\)
−0.612177 + 0.790721i \(0.709706\pi\)
\(524\) −20.0000 −0.873704
\(525\) 0 0
\(526\) −3.00000 −0.130806
\(527\) 8.00000 0.348485
\(528\) −5.00000 −0.217597
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) −1.00000 −0.0433963
\(532\) 0 0
\(533\) 5.00000 0.216574
\(534\) 16.0000 0.692388
\(535\) 0 0
\(536\) −4.00000 −0.172774
\(537\) 5.00000 0.215766
\(538\) 3.00000 0.129339
\(539\) 30.0000 1.29219
\(540\) 0 0
\(541\) −11.0000 −0.472927 −0.236463 − 0.971640i \(-0.575988\pi\)
−0.236463 + 0.971640i \(0.575988\pi\)
\(542\) −15.0000 −0.644305
\(543\) −22.0000 −0.944110
\(544\) −1.00000 −0.0428746
\(545\) 0 0
\(546\) −1.00000 −0.0427960
\(547\) 30.0000 1.28271 0.641354 − 0.767245i \(-0.278373\pi\)
0.641354 + 0.767245i \(0.278373\pi\)
\(548\) 5.00000 0.213589
\(549\) 6.00000 0.256074
\(550\) 0 0
\(551\) 0 0
\(552\) 6.00000 0.255377
\(553\) −3.00000 −0.127573
\(554\) 26.0000 1.10463
\(555\) 0 0
\(556\) 4.00000 0.169638
\(557\) 12.0000 0.508456 0.254228 − 0.967144i \(-0.418179\pi\)
0.254228 + 0.967144i \(0.418179\pi\)
\(558\) −8.00000 −0.338667
\(559\) 3.00000 0.126886
\(560\) 0 0
\(561\) 5.00000 0.211100
\(562\) −7.00000 −0.295277
\(563\) −24.0000 −1.01148 −0.505740 − 0.862686i \(-0.668780\pi\)
−0.505740 + 0.862686i \(0.668780\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −31.0000 −1.30303
\(567\) 1.00000 0.0419961
\(568\) 1.00000 0.0419591
\(569\) 30.0000 1.25767 0.628833 − 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 − 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 5.00000 0.209061
\(573\) −12.0000 −0.501307
\(574\) −5.00000 −0.208696
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 23.0000 0.957503 0.478751 − 0.877951i \(-0.341090\pi\)
0.478751 + 0.877951i \(0.341090\pi\)
\(578\) −16.0000 −0.665512
\(579\) −13.0000 −0.540262
\(580\) 0 0
\(581\) −7.00000 −0.290409
\(582\) 12.0000 0.497416
\(583\) 20.0000 0.828315
\(584\) 0 0
\(585\) 0 0
\(586\) 12.0000 0.495715
\(587\) 3.00000 0.123823 0.0619116 − 0.998082i \(-0.480280\pi\)
0.0619116 + 0.998082i \(0.480280\pi\)
\(588\) −6.00000 −0.247436
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −9.00000 −0.369898
\(593\) 2.00000 0.0821302 0.0410651 − 0.999156i \(-0.486925\pi\)
0.0410651 + 0.999156i \(0.486925\pi\)
\(594\) −5.00000 −0.205152
\(595\) 0 0
\(596\) 21.0000 0.860194
\(597\) −4.00000 −0.163709
\(598\) −6.00000 −0.245358
\(599\) −33.0000 −1.34834 −0.674172 − 0.738575i \(-0.735499\pi\)
−0.674172 + 0.738575i \(0.735499\pi\)
\(600\) 0 0
\(601\) −48.0000 −1.95796 −0.978980 − 0.203954i \(-0.934621\pi\)
−0.978980 + 0.203954i \(0.934621\pi\)
\(602\) −3.00000 −0.122271
\(603\) −4.00000 −0.162893
\(604\) −10.0000 −0.406894
\(605\) 0 0
\(606\) 17.0000 0.690578
\(607\) −37.0000 −1.50178 −0.750892 − 0.660425i \(-0.770376\pi\)
−0.750892 + 0.660425i \(0.770376\pi\)
\(608\) 0 0
\(609\) −10.0000 −0.405220
\(610\) 0 0
\(611\) 0 0
\(612\) −1.00000 −0.0404226
\(613\) −37.0000 −1.49442 −0.747208 − 0.664590i \(-0.768606\pi\)
−0.747208 + 0.664590i \(0.768606\pi\)
\(614\) −2.00000 −0.0807134
\(615\) 0 0
\(616\) −5.00000 −0.201456
\(617\) −3.00000 −0.120775 −0.0603877 − 0.998175i \(-0.519234\pi\)
−0.0603877 + 0.998175i \(0.519234\pi\)
\(618\) −4.00000 −0.160904
\(619\) 12.0000 0.482321 0.241160 − 0.970485i \(-0.422472\pi\)
0.241160 + 0.970485i \(0.422472\pi\)
\(620\) 0 0
\(621\) 6.00000 0.240772
\(622\) 7.00000 0.280674
\(623\) 16.0000 0.641026
\(624\) −1.00000 −0.0400320
\(625\) 0 0
\(626\) −14.0000 −0.559553
\(627\) 0 0
\(628\) 6.00000 0.239426
\(629\) 9.00000 0.358854
\(630\) 0 0
\(631\) −27.0000 −1.07485 −0.537427 − 0.843311i \(-0.680603\pi\)
−0.537427 + 0.843311i \(0.680603\pi\)
\(632\) −3.00000 −0.119334
\(633\) −3.00000 −0.119239
\(634\) −32.0000 −1.27088
\(635\) 0 0
\(636\) −4.00000 −0.158610
\(637\) 6.00000 0.237729
\(638\) 50.0000 1.97952
\(639\) 1.00000 0.0395594
\(640\) 0 0
\(641\) −14.0000 −0.552967 −0.276483 − 0.961019i \(-0.589169\pi\)
−0.276483 + 0.961019i \(0.589169\pi\)
\(642\) 6.00000 0.236801
\(643\) −16.0000 −0.630978 −0.315489 − 0.948929i \(-0.602169\pi\)
−0.315489 + 0.948929i \(0.602169\pi\)
\(644\) 6.00000 0.236433
\(645\) 0 0
\(646\) 0 0
\(647\) 24.0000 0.943537 0.471769 − 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) 1.00000 0.0392837
\(649\) 5.00000 0.196267
\(650\) 0 0
\(651\) −8.00000 −0.313545
\(652\) −14.0000 −0.548282
\(653\) 48.0000 1.87839 0.939193 − 0.343391i \(-0.111576\pi\)
0.939193 + 0.343391i \(0.111576\pi\)
\(654\) −10.0000 −0.391031
\(655\) 0 0
\(656\) −5.00000 −0.195217
\(657\) 0 0
\(658\) 0 0
\(659\) 36.0000 1.40236 0.701180 − 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) 0 0
\(661\) 2.00000 0.0777910 0.0388955 − 0.999243i \(-0.487616\pi\)
0.0388955 + 0.999243i \(0.487616\pi\)
\(662\) 34.0000 1.32145
\(663\) 1.00000 0.0388368
\(664\) −7.00000 −0.271653
\(665\) 0 0
\(666\) −9.00000 −0.348743
\(667\) −60.0000 −2.32321
\(668\) 0 0
\(669\) 9.00000 0.347960
\(670\) 0 0
\(671\) −30.0000 −1.15814
\(672\) 1.00000 0.0385758
\(673\) −46.0000 −1.77317 −0.886585 − 0.462566i \(-0.846929\pi\)
−0.886585 + 0.462566i \(0.846929\pi\)
\(674\) −2.00000 −0.0770371
\(675\) 0 0
\(676\) −12.0000 −0.461538
\(677\) −36.0000 −1.38359 −0.691796 − 0.722093i \(-0.743180\pi\)
−0.691796 + 0.722093i \(0.743180\pi\)
\(678\) −6.00000 −0.230429
\(679\) 12.0000 0.460518
\(680\) 0 0
\(681\) −7.00000 −0.268241
\(682\) 40.0000 1.53168
\(683\) −9.00000 −0.344375 −0.172188 − 0.985064i \(-0.555084\pi\)
−0.172188 + 0.985064i \(0.555084\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −13.0000 −0.496342
\(687\) −25.0000 −0.953809
\(688\) −3.00000 −0.114374
\(689\) 4.00000 0.152388
\(690\) 0 0
\(691\) 25.0000 0.951045 0.475522 − 0.879704i \(-0.342259\pi\)
0.475522 + 0.879704i \(0.342259\pi\)
\(692\) 9.00000 0.342129
\(693\) −5.00000 −0.189934
\(694\) 36.0000 1.36654
\(695\) 0 0
\(696\) −10.0000 −0.379049
\(697\) 5.00000 0.189389
\(698\) −7.00000 −0.264954
\(699\) −20.0000 −0.756469
\(700\) 0 0
\(701\) −47.0000 −1.77517 −0.887583 − 0.460648i \(-0.847617\pi\)
−0.887583 + 0.460648i \(0.847617\pi\)
\(702\) −1.00000 −0.0377426
\(703\) 0 0
\(704\) −5.00000 −0.188445
\(705\) 0 0
\(706\) 24.0000 0.903252
\(707\) 17.0000 0.639351
\(708\) −1.00000 −0.0375823
\(709\) −26.0000 −0.976450 −0.488225 − 0.872718i \(-0.662356\pi\)
−0.488225 + 0.872718i \(0.662356\pi\)
\(710\) 0 0
\(711\) −3.00000 −0.112509
\(712\) 16.0000 0.599625
\(713\) −48.0000 −1.79761
\(714\) −1.00000 −0.0374241
\(715\) 0 0
\(716\) 5.00000 0.186859
\(717\) 8.00000 0.298765
\(718\) 11.0000 0.410516
\(719\) −4.00000 −0.149175 −0.0745874 − 0.997214i \(-0.523764\pi\)
−0.0745874 + 0.997214i \(0.523764\pi\)
\(720\) 0 0
\(721\) −4.00000 −0.148968
\(722\) −19.0000 −0.707107
\(723\) 15.0000 0.557856
\(724\) −22.0000 −0.817624
\(725\) 0 0
\(726\) 14.0000 0.519589
\(727\) −11.0000 −0.407967 −0.203984 − 0.978974i \(-0.565389\pi\)
−0.203984 + 0.978974i \(0.565389\pi\)
\(728\) −1.00000 −0.0370625
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 3.00000 0.110959
\(732\) 6.00000 0.221766
\(733\) 50.0000 1.84679 0.923396 − 0.383849i \(-0.125402\pi\)
0.923396 + 0.383849i \(0.125402\pi\)
\(734\) 32.0000 1.18114
\(735\) 0 0
\(736\) 6.00000 0.221163
\(737\) 20.0000 0.736709
\(738\) −5.00000 −0.184053
\(739\) 37.0000 1.36107 0.680534 − 0.732717i \(-0.261748\pi\)
0.680534 + 0.732717i \(0.261748\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −4.00000 −0.146845
\(743\) −24.0000 −0.880475 −0.440237 − 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) −8.00000 −0.293294
\(745\) 0 0
\(746\) −24.0000 −0.878702
\(747\) −7.00000 −0.256117
\(748\) 5.00000 0.182818
\(749\) 6.00000 0.219235
\(750\) 0 0
\(751\) 50.0000 1.82453 0.912263 − 0.409605i \(-0.134333\pi\)
0.912263 + 0.409605i \(0.134333\pi\)
\(752\) 0 0
\(753\) 4.00000 0.145768
\(754\) 10.0000 0.364179
\(755\) 0 0
\(756\) 1.00000 0.0363696
\(757\) 40.0000 1.45382 0.726912 − 0.686730i \(-0.240955\pi\)
0.726912 + 0.686730i \(0.240955\pi\)
\(758\) 14.0000 0.508503
\(759\) −30.0000 −1.08893
\(760\) 0 0
\(761\) 34.0000 1.23250 0.616250 − 0.787551i \(-0.288651\pi\)
0.616250 + 0.787551i \(0.288651\pi\)
\(762\) 8.00000 0.289809
\(763\) −10.0000 −0.362024
\(764\) −12.0000 −0.434145
\(765\) 0 0
\(766\) 21.0000 0.758761
\(767\) 1.00000 0.0361079
\(768\) 1.00000 0.0360844
\(769\) 10.0000 0.360609 0.180305 − 0.983611i \(-0.442292\pi\)
0.180305 + 0.983611i \(0.442292\pi\)
\(770\) 0 0
\(771\) 21.0000 0.756297
\(772\) −13.0000 −0.467880
\(773\) 2.00000 0.0719350 0.0359675 − 0.999353i \(-0.488549\pi\)
0.0359675 + 0.999353i \(0.488549\pi\)
\(774\) −3.00000 −0.107833
\(775\) 0 0
\(776\) 12.0000 0.430775
\(777\) −9.00000 −0.322873
\(778\) −22.0000 −0.788738
\(779\) 0 0
\(780\) 0 0
\(781\) −5.00000 −0.178914
\(782\) −6.00000 −0.214560
\(783\) −10.0000 −0.357371
\(784\) −6.00000 −0.214286
\(785\) 0 0
\(786\) −20.0000 −0.713376
\(787\) 46.0000 1.63972 0.819861 − 0.572562i \(-0.194050\pi\)
0.819861 + 0.572562i \(0.194050\pi\)
\(788\) 0 0
\(789\) −3.00000 −0.106803
\(790\) 0 0
\(791\) −6.00000 −0.213335
\(792\) −5.00000 −0.177667
\(793\) −6.00000 −0.213066
\(794\) 22.0000 0.780751
\(795\) 0 0
\(796\) −4.00000 −0.141776
\(797\) −21.0000 −0.743858 −0.371929 − 0.928261i \(-0.621304\pi\)
−0.371929 + 0.928261i \(0.621304\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 16.0000 0.565332
\(802\) 16.0000 0.564980
\(803\) 0 0
\(804\) −4.00000 −0.141069
\(805\) 0 0
\(806\) 8.00000 0.281788
\(807\) 3.00000 0.105605
\(808\) 17.0000 0.598058
\(809\) −12.0000 −0.421898 −0.210949 − 0.977497i \(-0.567655\pi\)
−0.210949 + 0.977497i \(0.567655\pi\)
\(810\) 0 0
\(811\) −21.0000 −0.737410 −0.368705 − 0.929547i \(-0.620199\pi\)
−0.368705 + 0.929547i \(0.620199\pi\)
\(812\) −10.0000 −0.350931
\(813\) −15.0000 −0.526073
\(814\) 45.0000 1.57725
\(815\) 0 0
\(816\) −1.00000 −0.0350070
\(817\) 0 0
\(818\) 26.0000 0.909069
\(819\) −1.00000 −0.0349428
\(820\) 0 0
\(821\) −21.0000 −0.732905 −0.366453 − 0.930437i \(-0.619428\pi\)
−0.366453 + 0.930437i \(0.619428\pi\)
\(822\) 5.00000 0.174395
\(823\) −28.0000 −0.976019 −0.488009 − 0.872838i \(-0.662277\pi\)
−0.488009 + 0.872838i \(0.662277\pi\)
\(824\) −4.00000 −0.139347
\(825\) 0 0
\(826\) −1.00000 −0.0347945
\(827\) −26.0000 −0.904109 −0.452054 − 0.891990i \(-0.649309\pi\)
−0.452054 + 0.891990i \(0.649309\pi\)
\(828\) 6.00000 0.208514
\(829\) −28.0000 −0.972480 −0.486240 − 0.873825i \(-0.661632\pi\)
−0.486240 + 0.873825i \(0.661632\pi\)
\(830\) 0 0
\(831\) 26.0000 0.901930
\(832\) −1.00000 −0.0346688
\(833\) 6.00000 0.207888
\(834\) 4.00000 0.138509
\(835\) 0 0
\(836\) 0 0
\(837\) −8.00000 −0.276520
\(838\) −35.0000 −1.20905
\(839\) 54.0000 1.86429 0.932144 − 0.362089i \(-0.117936\pi\)
0.932144 + 0.362089i \(0.117936\pi\)
\(840\) 0 0
\(841\) 71.0000 2.44828
\(842\) 1.00000 0.0344623
\(843\) −7.00000 −0.241093
\(844\) −3.00000 −0.103264
\(845\) 0 0
\(846\) 0 0
\(847\) 14.0000 0.481046
\(848\) −4.00000 −0.137361
\(849\) −31.0000 −1.06392
\(850\) 0 0
\(851\) −54.0000 −1.85110
\(852\) 1.00000 0.0342594
\(853\) −44.0000 −1.50653 −0.753266 − 0.657716i \(-0.771523\pi\)
−0.753266 + 0.657716i \(0.771523\pi\)
\(854\) 6.00000 0.205316
\(855\) 0 0
\(856\) 6.00000 0.205076
\(857\) −2.00000 −0.0683187 −0.0341593 − 0.999416i \(-0.510875\pi\)
−0.0341593 + 0.999416i \(0.510875\pi\)
\(858\) 5.00000 0.170697
\(859\) −20.0000 −0.682391 −0.341196 − 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) 0 0
\(861\) −5.00000 −0.170400
\(862\) 26.0000 0.885564
\(863\) 16.0000 0.544646 0.272323 − 0.962206i \(-0.412208\pi\)
0.272323 + 0.962206i \(0.412208\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 31.0000 1.05342
\(867\) −16.0000 −0.543388
\(868\) −8.00000 −0.271538
\(869\) 15.0000 0.508840
\(870\) 0 0
\(871\) 4.00000 0.135535
\(872\) −10.0000 −0.338643
\(873\) 12.0000 0.406138
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 44.0000 1.48577 0.742887 − 0.669417i \(-0.233456\pi\)
0.742887 + 0.669417i \(0.233456\pi\)
\(878\) −11.0000 −0.371232
\(879\) 12.0000 0.404750
\(880\) 0 0
\(881\) 46.0000 1.54978 0.774890 − 0.632096i \(-0.217805\pi\)
0.774890 + 0.632096i \(0.217805\pi\)
\(882\) −6.00000 −0.202031
\(883\) 40.0000 1.34611 0.673054 − 0.739594i \(-0.264982\pi\)
0.673054 + 0.739594i \(0.264982\pi\)
\(884\) 1.00000 0.0336336
\(885\) 0 0
\(886\) −17.0000 −0.571126
\(887\) 42.0000 1.41022 0.705111 − 0.709097i \(-0.250897\pi\)
0.705111 + 0.709097i \(0.250897\pi\)
\(888\) −9.00000 −0.302020
\(889\) 8.00000 0.268311
\(890\) 0 0
\(891\) −5.00000 −0.167506
\(892\) 9.00000 0.301342
\(893\) 0 0
\(894\) 21.0000 0.702345
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) −6.00000 −0.200334
\(898\) −31.0000 −1.03448
\(899\) 80.0000 2.66815
\(900\) 0 0
\(901\) 4.00000 0.133259
\(902\) 25.0000 0.832409
\(903\) −3.00000 −0.0998337
\(904\) −6.00000 −0.199557
\(905\) 0 0
\(906\) −10.0000 −0.332228
\(907\) −44.0000 −1.46100 −0.730498 − 0.682915i \(-0.760712\pi\)
−0.730498 + 0.682915i \(0.760712\pi\)
\(908\) −7.00000 −0.232303
\(909\) 17.0000 0.563854
\(910\) 0 0
\(911\) 1.00000 0.0331315 0.0165657 − 0.999863i \(-0.494727\pi\)
0.0165657 + 0.999863i \(0.494727\pi\)
\(912\) 0 0
\(913\) 35.0000 1.15833
\(914\) 4.00000 0.132308
\(915\) 0 0
\(916\) −25.0000 −0.826023
\(917\) −20.0000 −0.660458
\(918\) −1.00000 −0.0330049
\(919\) −34.0000 −1.12156 −0.560778 − 0.827966i \(-0.689498\pi\)
−0.560778 + 0.827966i \(0.689498\pi\)
\(920\) 0 0
\(921\) −2.00000 −0.0659022
\(922\) −20.0000 −0.658665
\(923\) −1.00000 −0.0329154
\(924\) −5.00000 −0.164488
\(925\) 0 0
\(926\) −20.0000 −0.657241
\(927\) −4.00000 −0.131377
\(928\) −10.0000 −0.328266
\(929\) −32.0000 −1.04989 −0.524943 − 0.851137i \(-0.675913\pi\)
−0.524943 + 0.851137i \(0.675913\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −20.0000 −0.655122
\(933\) 7.00000 0.229170
\(934\) −13.0000 −0.425373
\(935\) 0 0
\(936\) −1.00000 −0.0326860
\(937\) −2.00000 −0.0653372 −0.0326686 − 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) −4.00000 −0.130605
\(939\) −14.0000 −0.456873
\(940\) 0 0
\(941\) −6.00000 −0.195594 −0.0977972 − 0.995206i \(-0.531180\pi\)
−0.0977972 + 0.995206i \(0.531180\pi\)
\(942\) 6.00000 0.195491
\(943\) −30.0000 −0.976934
\(944\) −1.00000 −0.0325472
\(945\) 0 0
\(946\) 15.0000 0.487692
\(947\) −48.0000 −1.55979 −0.779895 − 0.625910i \(-0.784728\pi\)
−0.779895 + 0.625910i \(0.784728\pi\)
\(948\) −3.00000 −0.0974355
\(949\) 0 0
\(950\) 0 0
\(951\) −32.0000 −1.03767
\(952\) −1.00000 −0.0324102
\(953\) −21.0000 −0.680257 −0.340128 − 0.940379i \(-0.610471\pi\)
−0.340128 + 0.940379i \(0.610471\pi\)
\(954\) −4.00000 −0.129505
\(955\) 0 0
\(956\) 8.00000 0.258738
\(957\) 50.0000 1.61627
\(958\) −28.0000 −0.904639
\(959\) 5.00000 0.161458
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 9.00000 0.290172
\(963\) 6.00000 0.193347
\(964\) 15.0000 0.483117
\(965\) 0 0
\(966\) 6.00000 0.193047
\(967\) 44.0000 1.41494 0.707472 − 0.706741i \(-0.249835\pi\)
0.707472 + 0.706741i \(0.249835\pi\)
\(968\) 14.0000 0.449977
\(969\) 0 0
\(970\) 0 0
\(971\) −18.0000 −0.577647 −0.288824 − 0.957382i \(-0.593264\pi\)
−0.288824 + 0.957382i \(0.593264\pi\)
\(972\) 1.00000 0.0320750
\(973\) 4.00000 0.128234
\(974\) 7.00000 0.224294
\(975\) 0 0
\(976\) 6.00000 0.192055
\(977\) 48.0000 1.53566 0.767828 − 0.640656i \(-0.221338\pi\)
0.767828 + 0.640656i \(0.221338\pi\)
\(978\) −14.0000 −0.447671
\(979\) −80.0000 −2.55681
\(980\) 0 0
\(981\) −10.0000 −0.319275
\(982\) 30.0000 0.957338
\(983\) 14.0000 0.446531 0.223265 − 0.974758i \(-0.428328\pi\)
0.223265 + 0.974758i \(0.428328\pi\)
\(984\) −5.00000 −0.159394
\(985\) 0 0
\(986\) 10.0000 0.318465
\(987\) 0 0
\(988\) 0 0
\(989\) −18.0000 −0.572367
\(990\) 0 0
\(991\) −20.0000 −0.635321 −0.317660 − 0.948205i \(-0.602897\pi\)
−0.317660 + 0.948205i \(0.602897\pi\)
\(992\) −8.00000 −0.254000
\(993\) 34.0000 1.07896
\(994\) 1.00000 0.0317181
\(995\) 0 0
\(996\) −7.00000 −0.221803
\(997\) 20.0000 0.633406 0.316703 − 0.948525i \(-0.397424\pi\)
0.316703 + 0.948525i \(0.397424\pi\)
\(998\) 24.0000 0.759707
\(999\) −9.00000 −0.284747
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 8850.2.a.be.1.1 1
5.4 even 2 354.2.a.a.1.1 ✓ 1
15.14 odd 2 1062.2.a.k.1.1 1
20.19 odd 2 2832.2.a.e.1.1 1
60.59 even 2 8496.2.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
354.2.a.a.1.1 ✓ 1 5.4 even 2
1062.2.a.k.1.1 1 15.14 odd 2
2832.2.a.e.1.1 1 20.19 odd 2
8496.2.a.j.1.1 1 60.59 even 2
8850.2.a.be.1.1 1 1.1 even 1 trivial