Properties

Label 4550.2.a.g.1.1
Level $4550$
Weight $2$
Character 4550.1
Self dual yes
Analytic conductor $36.332$
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] = [4550,2,Mod(1,4550)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4550, 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("4550.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4550 = 2 \cdot 5^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4550.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: \(36.3319329197\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 182)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4550.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} -1.00000 q^{8} -3.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} -1.00000 q^{8} -3.00000 q^{9} +4.00000 q^{11} +1.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +6.00000 q^{17} +3.00000 q^{18} -4.00000 q^{22} -8.00000 q^{23} -1.00000 q^{26} +1.00000 q^{28} -10.0000 q^{29} -8.00000 q^{31} -1.00000 q^{32} -6.00000 q^{34} -3.00000 q^{36} -6.00000 q^{37} -6.00000 q^{41} -4.00000 q^{43} +4.00000 q^{44} +8.00000 q^{46} +8.00000 q^{47} +1.00000 q^{49} +1.00000 q^{52} -6.00000 q^{53} -1.00000 q^{56} +10.0000 q^{58} +8.00000 q^{59} +10.0000 q^{61} +8.00000 q^{62} -3.00000 q^{63} +1.00000 q^{64} -4.00000 q^{67} +6.00000 q^{68} -8.00000 q^{71} +3.00000 q^{72} -2.00000 q^{73} +6.00000 q^{74} +4.00000 q^{77} +8.00000 q^{79} +9.00000 q^{81} +6.00000 q^{82} +4.00000 q^{86} -4.00000 q^{88} +18.0000 q^{89} +1.00000 q^{91} -8.00000 q^{92} -8.00000 q^{94} -2.00000 q^{97} -1.00000 q^{98} -12.0000 q^{99} +O(q^{100})\)

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
\(3\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 3.00000 0.707107
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −4.00000 −0.852803
\(23\) −8.00000 −1.66812 −0.834058 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −1.00000 −0.196116
\(27\) 0 0
\(28\) 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
\(33\) 0 0
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) −3.00000 −0.500000
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) 8.00000 1.17954
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 1.00000 0.138675
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) 10.0000 1.31306
\(59\) 8.00000 1.04151 0.520756 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 8.00000 1.01600
\(63\) −3.00000 −0.377964
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 6.00000 0.727607
\(69\) 0 0
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 3.00000 0.353553
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 6.00000 0.697486
\(75\) 0 0
\(76\) 0 0
\(77\) 4.00000 0.455842
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 6.00000 0.662589
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) −4.00000 −0.426401
\(89\) 18.0000 1.90800 0.953998 0.299813i \(-0.0969242\pi\)
0.953998 + 0.299813i \(0.0969242\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) −8.00000 −0.834058
\(93\) 0 0
\(94\) −8.00000 −0.825137
\(95\) 0 0
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) −1.00000 −0.101015
\(99\) −12.0000 −1.20605
\(100\) 0 0
\(101\) −14.0000 −1.39305 −0.696526 0.717532i \(-0.745272\pi\)
−0.696526 + 0.717532i \(0.745272\pi\)
\(102\) 0 0
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) −2.00000 −0.188144 −0.0940721 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −10.0000 −0.928477
\(117\) −3.00000 −0.277350
\(118\) −8.00000 −0.736460
\(119\) 6.00000 0.550019
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) −10.0000 −0.905357
\(123\) 0 0
\(124\) −8.00000 −0.718421
\(125\) 0 0
\(126\) 3.00000 0.267261
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) 8.00000 0.698963 0.349482 0.936943i \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) −8.00000 −0.678551 −0.339276 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 8.00000 0.671345
\(143\) 4.00000 0.334497
\(144\) −3.00000 −0.250000
\(145\) 0 0
\(146\) 2.00000 0.165521
\(147\) 0 0
\(148\) −6.00000 −0.493197
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) −18.0000 −1.45521
\(154\) −4.00000 −0.322329
\(155\) 0 0
\(156\) 0 0
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) −8.00000 −0.636446
\(159\) 0 0
\(160\) 0 0
\(161\) −8.00000 −0.630488
\(162\) −9.00000 −0.707107
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) −4.00000 −0.304997
\(173\) −2.00000 −0.152057 −0.0760286 0.997106i \(-0.524224\pi\)
−0.0760286 + 0.997106i \(0.524224\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) 0 0
\(178\) −18.0000 −1.34916
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) −6.00000 −0.445976 −0.222988 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) −1.00000 −0.0741249
\(183\) 0 0
\(184\) 8.00000 0.589768
\(185\) 0 0
\(186\) 0 0
\(187\) 24.0000 1.75505
\(188\) 8.00000 0.583460
\(189\) 0 0
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 0 0
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 12.0000 0.852803
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 14.0000 0.985037
\(203\) −10.0000 −0.701862
\(204\) 0 0
\(205\) 0 0
\(206\) 16.0000 1.11477
\(207\) 24.0000 1.66812
\(208\) 1.00000 0.0693375
\(209\) 0 0
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) −4.00000 −0.273434
\(215\) 0 0
\(216\) 0 0
\(217\) −8.00000 −0.543075
\(218\) 2.00000 0.135457
\(219\) 0 0
\(220\) 0 0
\(221\) 6.00000 0.403604
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 2.00000 0.133038
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) −6.00000 −0.396491 −0.198246 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 10.0000 0.656532
\(233\) −10.0000 −0.655122 −0.327561 0.944830i \(-0.606227\pi\)
−0.327561 + 0.944830i \(0.606227\pi\)
\(234\) 3.00000 0.196116
\(235\) 0 0
\(236\) 8.00000 0.520756
\(237\) 0 0
\(238\) −6.00000 −0.388922
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 0 0
\(241\) 18.0000 1.15948 0.579741 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) −5.00000 −0.321412
\(243\) 0 0
\(244\) 10.0000 0.640184
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 8.00000 0.508001
\(249\) 0 0
\(250\) 0 0
\(251\) 24.0000 1.51487 0.757433 0.652913i \(-0.226453\pi\)
0.757433 + 0.652913i \(0.226453\pi\)
\(252\) −3.00000 −0.188982
\(253\) −32.0000 −2.01182
\(254\) 16.0000 1.00393
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) −6.00000 −0.372822
\(260\) 0 0
\(261\) 30.0000 1.85695
\(262\) −8.00000 −0.494242
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −4.00000 −0.244339
\(269\) −30.0000 −1.82913 −0.914566 0.404436i \(-0.867468\pi\)
−0.914566 + 0.404436i \(0.867468\pi\)
\(270\) 0 0
\(271\) −32.0000 −1.94386 −0.971931 0.235267i \(-0.924404\pi\)
−0.971931 + 0.235267i \(0.924404\pi\)
\(272\) 6.00000 0.363803
\(273\) 0 0
\(274\) −6.00000 −0.362473
\(275\) 0 0
\(276\) 0 0
\(277\) −6.00000 −0.360505 −0.180253 0.983620i \(-0.557691\pi\)
−0.180253 + 0.983620i \(0.557691\pi\)
\(278\) 8.00000 0.479808
\(279\) 24.0000 1.43684
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) −16.0000 −0.951101 −0.475551 0.879688i \(-0.657751\pi\)
−0.475551 + 0.879688i \(0.657751\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) −4.00000 −0.236525
\(287\) −6.00000 −0.354169
\(288\) 3.00000 0.176777
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 0 0
\(292\) −2.00000 −0.117041
\(293\) 14.0000 0.817889 0.408944 0.912559i \(-0.365897\pi\)
0.408944 + 0.912559i \(0.365897\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 6.00000 0.348743
\(297\) 0 0
\(298\) 18.0000 1.04271
\(299\) −8.00000 −0.462652
\(300\) 0 0
\(301\) −4.00000 −0.230556
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 18.0000 1.02899
\(307\) −32.0000 −1.82634 −0.913168 0.407583i \(-0.866372\pi\)
−0.913168 + 0.407583i \(0.866372\pi\)
\(308\) 4.00000 0.227921
\(309\) 0 0
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 0 0
\(313\) −2.00000 −0.113047 −0.0565233 0.998401i \(-0.518002\pi\)
−0.0565233 + 0.998401i \(0.518002\pi\)
\(314\) 2.00000 0.112867
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) −6.00000 −0.336994 −0.168497 0.985702i \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) 0 0
\(319\) −40.0000 −2.23957
\(320\) 0 0
\(321\) 0 0
\(322\) 8.00000 0.445823
\(323\) 0 0
\(324\) 9.00000 0.500000
\(325\) 0 0
\(326\) −4.00000 −0.221540
\(327\) 0 0
\(328\) 6.00000 0.331295
\(329\) 8.00000 0.441054
\(330\) 0 0
\(331\) −28.0000 −1.53902 −0.769510 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) 0 0
\(333\) 18.0000 0.986394
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −2.00000 −0.108947 −0.0544735 0.998515i \(-0.517348\pi\)
−0.0544735 + 0.998515i \(0.517348\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 0 0
\(340\) 0 0
\(341\) −32.0000 −1.73290
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) 2.00000 0.107521
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 0 0
\(349\) −14.0000 −0.749403 −0.374701 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −4.00000 −0.213201
\(353\) −10.0000 −0.532246 −0.266123 0.963939i \(-0.585743\pi\)
−0.266123 + 0.963939i \(0.585743\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 18.0000 0.953998
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 6.00000 0.315353
\(363\) 0 0
\(364\) 1.00000 0.0524142
\(365\) 0 0
\(366\) 0 0
\(367\) 16.0000 0.835193 0.417597 0.908633i \(-0.362873\pi\)
0.417597 + 0.908633i \(0.362873\pi\)
\(368\) −8.00000 −0.417029
\(369\) 18.0000 0.937043
\(370\) 0 0
\(371\) −6.00000 −0.311504
\(372\) 0 0
\(373\) 10.0000 0.517780 0.258890 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) −24.0000 −1.24101
\(375\) 0 0
\(376\) −8.00000 −0.412568
\(377\) −10.0000 −0.515026
\(378\) 0 0
\(379\) 4.00000 0.205466 0.102733 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 8.00000 0.409316
\(383\) −24.0000 −1.22634 −0.613171 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 2.00000 0.101797
\(387\) 12.0000 0.609994
\(388\) −2.00000 −0.101535
\(389\) 30.0000 1.52106 0.760530 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) 0 0
\(391\) −48.0000 −2.42746
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) −18.0000 −0.906827
\(395\) 0 0
\(396\) −12.0000 −0.603023
\(397\) −2.00000 −0.100377 −0.0501886 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) 16.0000 0.802008
\(399\) 0 0
\(400\) 0 0
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 0 0
\(403\) −8.00000 −0.398508
\(404\) −14.0000 −0.696526
\(405\) 0 0
\(406\) 10.0000 0.496292
\(407\) −24.0000 −1.18964
\(408\) 0 0
\(409\) −38.0000 −1.87898 −0.939490 0.342578i \(-0.888700\pi\)
−0.939490 + 0.342578i \(0.888700\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −16.0000 −0.788263
\(413\) 8.00000 0.393654
\(414\) −24.0000 −1.17954
\(415\) 0 0
\(416\) −1.00000 −0.0490290
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −2.00000 −0.0974740 −0.0487370 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) −20.0000 −0.973585
\(423\) −24.0000 −1.16692
\(424\) 6.00000 0.291386
\(425\) 0 0
\(426\) 0 0
\(427\) 10.0000 0.483934
\(428\) 4.00000 0.193347
\(429\) 0 0
\(430\) 0 0
\(431\) −24.0000 −1.15604 −0.578020 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 0 0
\(433\) 38.0000 1.82616 0.913082 0.407777i \(-0.133696\pi\)
0.913082 + 0.407777i \(0.133696\pi\)
\(434\) 8.00000 0.384012
\(435\) 0 0
\(436\) −2.00000 −0.0957826
\(437\) 0 0
\(438\) 0 0
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) −6.00000 −0.285391
\(443\) 36.0000 1.71041 0.855206 0.518289i \(-0.173431\pi\)
0.855206 + 0.518289i \(0.173431\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) −14.0000 −0.660701 −0.330350 0.943858i \(-0.607167\pi\)
−0.330350 + 0.943858i \(0.607167\pi\)
\(450\) 0 0
\(451\) −24.0000 −1.13012
\(452\) −2.00000 −0.0940721
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) 6.00000 0.280362
\(459\) 0 0
\(460\) 0 0
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) −16.0000 −0.743583 −0.371792 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) −10.0000 −0.464238
\(465\) 0 0
\(466\) 10.0000 0.463241
\(467\) −8.00000 −0.370196 −0.185098 0.982720i \(-0.559260\pi\)
−0.185098 + 0.982720i \(0.559260\pi\)
\(468\) −3.00000 −0.138675
\(469\) −4.00000 −0.184703
\(470\) 0 0
\(471\) 0 0
\(472\) −8.00000 −0.368230
\(473\) −16.0000 −0.735681
\(474\) 0 0
\(475\) 0 0
\(476\) 6.00000 0.275010
\(477\) 18.0000 0.824163
\(478\) 24.0000 1.09773
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) −6.00000 −0.273576
\(482\) −18.0000 −0.819878
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) 0 0
\(486\) 0 0
\(487\) 16.0000 0.725029 0.362515 0.931978i \(-0.381918\pi\)
0.362515 + 0.931978i \(0.381918\pi\)
\(488\) −10.0000 −0.452679
\(489\) 0 0
\(490\) 0 0
\(491\) 28.0000 1.26362 0.631811 0.775122i \(-0.282312\pi\)
0.631811 + 0.775122i \(0.282312\pi\)
\(492\) 0 0
\(493\) −60.0000 −2.70226
\(494\) 0 0
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) −8.00000 −0.358849
\(498\) 0 0
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −24.0000 −1.07117
\(503\) 8.00000 0.356702 0.178351 0.983967i \(-0.442924\pi\)
0.178351 + 0.983967i \(0.442924\pi\)
\(504\) 3.00000 0.133631
\(505\) 0 0
\(506\) 32.0000 1.42257
\(507\) 0 0
\(508\) −16.0000 −0.709885
\(509\) −14.0000 −0.620539 −0.310270 0.950649i \(-0.600419\pi\)
−0.310270 + 0.950649i \(0.600419\pi\)
\(510\) 0 0
\(511\) −2.00000 −0.0884748
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 18.0000 0.793946
\(515\) 0 0
\(516\) 0 0
\(517\) 32.0000 1.40736
\(518\) 6.00000 0.263625
\(519\) 0 0
\(520\) 0 0
\(521\) −30.0000 −1.31432 −0.657162 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) −30.0000 −1.31306
\(523\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) 8.00000 0.349482
\(525\) 0 0
\(526\) 0 0
\(527\) −48.0000 −2.09091
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) 0 0
\(531\) −24.0000 −1.04151
\(532\) 0 0
\(533\) −6.00000 −0.259889
\(534\) 0 0
\(535\) 0 0
\(536\) 4.00000 0.172774
\(537\) 0 0
\(538\) 30.0000 1.29339
\(539\) 4.00000 0.172292
\(540\) 0 0
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) 32.0000 1.37452
\(543\) 0 0
\(544\) −6.00000 −0.257248
\(545\) 0 0
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 6.00000 0.256307
\(549\) −30.0000 −1.28037
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 8.00000 0.340195
\(554\) 6.00000 0.254916
\(555\) 0 0
\(556\) −8.00000 −0.339276
\(557\) 42.0000 1.77960 0.889799 0.456354i \(-0.150845\pi\)
0.889799 + 0.456354i \(0.150845\pi\)
\(558\) −24.0000 −1.01600
\(559\) −4.00000 −0.169182
\(560\) 0 0
\(561\) 0 0
\(562\) −10.0000 −0.421825
\(563\) 16.0000 0.674320 0.337160 0.941447i \(-0.390534\pi\)
0.337160 + 0.941447i \(0.390534\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 16.0000 0.672530
\(567\) 9.00000 0.377964
\(568\) 8.00000 0.335673
\(569\) −38.0000 −1.59304 −0.796521 0.604610i \(-0.793329\pi\)
−0.796521 + 0.604610i \(0.793329\pi\)
\(570\) 0 0
\(571\) −12.0000 −0.502184 −0.251092 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) 4.00000 0.167248
\(573\) 0 0
\(574\) 6.00000 0.250435
\(575\) 0 0
\(576\) −3.00000 −0.125000
\(577\) −42.0000 −1.74848 −0.874241 0.485491i \(-0.838641\pi\)
−0.874241 + 0.485491i \(0.838641\pi\)
\(578\) −19.0000 −0.790296
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −24.0000 −0.993978
\(584\) 2.00000 0.0827606
\(585\) 0 0
\(586\) −14.0000 −0.578335
\(587\) 16.0000 0.660391 0.330195 0.943913i \(-0.392885\pi\)
0.330195 + 0.943913i \(0.392885\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −6.00000 −0.246598
\(593\) 22.0000 0.903432 0.451716 0.892162i \(-0.350812\pi\)
0.451716 + 0.892162i \(0.350812\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −18.0000 −0.737309
\(597\) 0 0
\(598\) 8.00000 0.327144
\(599\) 16.0000 0.653742 0.326871 0.945069i \(-0.394006\pi\)
0.326871 + 0.945069i \(0.394006\pi\)
\(600\) 0 0
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) 4.00000 0.163028
\(603\) 12.0000 0.488678
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 48.0000 1.94826 0.974130 0.225989i \(-0.0725612\pi\)
0.974130 + 0.225989i \(0.0725612\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 8.00000 0.323645
\(612\) −18.0000 −0.727607
\(613\) −38.0000 −1.53481 −0.767403 0.641165i \(-0.778451\pi\)
−0.767403 + 0.641165i \(0.778451\pi\)
\(614\) 32.0000 1.29141
\(615\) 0 0
\(616\) −4.00000 −0.161165
\(617\) −10.0000 −0.402585 −0.201292 0.979531i \(-0.564514\pi\)
−0.201292 + 0.979531i \(0.564514\pi\)
\(618\) 0 0
\(619\) 40.0000 1.60774 0.803868 0.594808i \(-0.202772\pi\)
0.803868 + 0.594808i \(0.202772\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −24.0000 −0.962312
\(623\) 18.0000 0.721155
\(624\) 0 0
\(625\) 0 0
\(626\) 2.00000 0.0799361
\(627\) 0 0
\(628\) −2.00000 −0.0798087
\(629\) −36.0000 −1.43541
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) −8.00000 −0.318223
\(633\) 0 0
\(634\) 6.00000 0.238290
\(635\) 0 0
\(636\) 0 0
\(637\) 1.00000 0.0396214
\(638\) 40.0000 1.58362
\(639\) 24.0000 0.949425
\(640\) 0 0
\(641\) 34.0000 1.34292 0.671460 0.741041i \(-0.265668\pi\)
0.671460 + 0.741041i \(0.265668\pi\)
\(642\) 0 0
\(643\) −8.00000 −0.315489 −0.157745 0.987480i \(-0.550422\pi\)
−0.157745 + 0.987480i \(0.550422\pi\)
\(644\) −8.00000 −0.315244
\(645\) 0 0
\(646\) 0 0
\(647\) 32.0000 1.25805 0.629025 0.777385i \(-0.283454\pi\)
0.629025 + 0.777385i \(0.283454\pi\)
\(648\) −9.00000 −0.353553
\(649\) 32.0000 1.25611
\(650\) 0 0
\(651\) 0 0
\(652\) 4.00000 0.156652
\(653\) 42.0000 1.64359 0.821794 0.569785i \(-0.192974\pi\)
0.821794 + 0.569785i \(0.192974\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −6.00000 −0.234261
\(657\) 6.00000 0.234082
\(658\) −8.00000 −0.311872
\(659\) 12.0000 0.467454 0.233727 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 0 0
\(661\) 2.00000 0.0777910 0.0388955 0.999243i \(-0.487616\pi\)
0.0388955 + 0.999243i \(0.487616\pi\)
\(662\) 28.0000 1.08825
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) −18.0000 −0.697486
\(667\) 80.0000 3.09761
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 40.0000 1.54418
\(672\) 0 0
\(673\) 14.0000 0.539660 0.269830 0.962908i \(-0.413032\pi\)
0.269830 + 0.962908i \(0.413032\pi\)
\(674\) 2.00000 0.0770371
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) −42.0000 −1.61419 −0.807096 0.590421i \(-0.798962\pi\)
−0.807096 + 0.590421i \(0.798962\pi\)
\(678\) 0 0
\(679\) −2.00000 −0.0767530
\(680\) 0 0
\(681\) 0 0
\(682\) 32.0000 1.22534
\(683\) −28.0000 −1.07139 −0.535695 0.844411i \(-0.679950\pi\)
−0.535695 + 0.844411i \(0.679950\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) −4.00000 −0.152499
\(689\) −6.00000 −0.228582
\(690\) 0 0
\(691\) 16.0000 0.608669 0.304334 0.952565i \(-0.401566\pi\)
0.304334 + 0.952565i \(0.401566\pi\)
\(692\) −2.00000 −0.0760286
\(693\) −12.0000 −0.455842
\(694\) −12.0000 −0.455514
\(695\) 0 0
\(696\) 0 0
\(697\) −36.0000 −1.36360
\(698\) 14.0000 0.529908
\(699\) 0 0
\(700\) 0 0
\(701\) −42.0000 −1.58632 −0.793159 0.609015i \(-0.791565\pi\)
−0.793159 + 0.609015i \(0.791565\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) 10.0000 0.376355
\(707\) −14.0000 −0.526524
\(708\) 0 0
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 0 0
\(711\) −24.0000 −0.900070
\(712\) −18.0000 −0.674579
\(713\) 64.0000 2.39682
\(714\) 0 0
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) 0 0
\(718\) 0 0
\(719\) 8.00000 0.298350 0.149175 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(720\) 0 0
\(721\) −16.0000 −0.595871
\(722\) 19.0000 0.707107
\(723\) 0 0
\(724\) −6.00000 −0.222988
\(725\) 0 0
\(726\) 0 0
\(727\) −16.0000 −0.593407 −0.296704 0.954970i \(-0.595887\pi\)
−0.296704 + 0.954970i \(0.595887\pi\)
\(728\) −1.00000 −0.0370625
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) −24.0000 −0.887672
\(732\) 0 0
\(733\) −26.0000 −0.960332 −0.480166 0.877178i \(-0.659424\pi\)
−0.480166 + 0.877178i \(0.659424\pi\)
\(734\) −16.0000 −0.590571
\(735\) 0 0
\(736\) 8.00000 0.294884
\(737\) −16.0000 −0.589368
\(738\) −18.0000 −0.662589
\(739\) −36.0000 −1.32428 −0.662141 0.749380i \(-0.730352\pi\)
−0.662141 + 0.749380i \(0.730352\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 6.00000 0.220267
\(743\) −16.0000 −0.586983 −0.293492 0.955962i \(-0.594817\pi\)
−0.293492 + 0.955962i \(0.594817\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −10.0000 −0.366126
\(747\) 0 0
\(748\) 24.0000 0.877527
\(749\) 4.00000 0.146157
\(750\) 0 0
\(751\) 48.0000 1.75154 0.875772 0.482724i \(-0.160353\pi\)
0.875772 + 0.482724i \(0.160353\pi\)
\(752\) 8.00000 0.291730
\(753\) 0 0
\(754\) 10.0000 0.364179
\(755\) 0 0
\(756\) 0 0
\(757\) −30.0000 −1.09037 −0.545184 0.838316i \(-0.683540\pi\)
−0.545184 + 0.838316i \(0.683540\pi\)
\(758\) −4.00000 −0.145287
\(759\) 0 0
\(760\) 0 0
\(761\) −6.00000 −0.217500 −0.108750 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) 0 0
\(763\) −2.00000 −0.0724049
\(764\) −8.00000 −0.289430
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) 8.00000 0.288863
\(768\) 0 0
\(769\) −30.0000 −1.08183 −0.540914 0.841078i \(-0.681921\pi\)
−0.540914 + 0.841078i \(0.681921\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −2.00000 −0.0719816
\(773\) 46.0000 1.65451 0.827253 0.561830i \(-0.189903\pi\)
0.827253 + 0.561830i \(0.189903\pi\)
\(774\) −12.0000 −0.431331
\(775\) 0 0
\(776\) 2.00000 0.0717958
\(777\) 0 0
\(778\) −30.0000 −1.07555
\(779\) 0 0
\(780\) 0 0
\(781\) −32.0000 −1.14505
\(782\) 48.0000 1.71648
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 0 0
\(786\) 0 0
\(787\) −32.0000 −1.14068 −0.570338 0.821410i \(-0.693188\pi\)
−0.570338 + 0.821410i \(0.693188\pi\)
\(788\) 18.0000 0.641223
\(789\) 0 0
\(790\) 0 0
\(791\) −2.00000 −0.0711118
\(792\) 12.0000 0.426401
\(793\) 10.0000 0.355110
\(794\) 2.00000 0.0709773
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) −18.0000 −0.637593 −0.318796 0.947823i \(-0.603279\pi\)
−0.318796 + 0.947823i \(0.603279\pi\)
\(798\) 0 0
\(799\) 48.0000 1.69812
\(800\) 0 0
\(801\) −54.0000 −1.90800
\(802\) −18.0000 −0.635602
\(803\) −8.00000 −0.282314
\(804\) 0 0
\(805\) 0 0
\(806\) 8.00000 0.281788
\(807\) 0 0
\(808\) 14.0000 0.492518
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 0 0
\(811\) 8.00000 0.280918 0.140459 0.990086i \(-0.455142\pi\)
0.140459 + 0.990086i \(0.455142\pi\)
\(812\) −10.0000 −0.350931
\(813\) 0 0
\(814\) 24.0000 0.841200
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 38.0000 1.32864
\(819\) −3.00000 −0.104828
\(820\) 0 0
\(821\) 14.0000 0.488603 0.244302 0.969699i \(-0.421441\pi\)
0.244302 + 0.969699i \(0.421441\pi\)
\(822\) 0 0
\(823\) −48.0000 −1.67317 −0.836587 0.547833i \(-0.815453\pi\)
−0.836587 + 0.547833i \(0.815453\pi\)
\(824\) 16.0000 0.557386
\(825\) 0 0
\(826\) −8.00000 −0.278356
\(827\) 36.0000 1.25184 0.625921 0.779886i \(-0.284723\pi\)
0.625921 + 0.779886i \(0.284723\pi\)
\(828\) 24.0000 0.834058
\(829\) 10.0000 0.347314 0.173657 0.984806i \(-0.444442\pi\)
0.173657 + 0.984806i \(0.444442\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 1.00000 0.0346688
\(833\) 6.00000 0.207888
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 32.0000 1.10476 0.552381 0.833592i \(-0.313719\pi\)
0.552381 + 0.833592i \(0.313719\pi\)
\(840\) 0 0
\(841\) 71.0000 2.44828
\(842\) 2.00000 0.0689246
\(843\) 0 0
\(844\) 20.0000 0.688428
\(845\) 0 0
\(846\) 24.0000 0.825137
\(847\) 5.00000 0.171802
\(848\) −6.00000 −0.206041
\(849\) 0 0
\(850\) 0 0
\(851\) 48.0000 1.64542
\(852\) 0 0
\(853\) −10.0000 −0.342393 −0.171197 0.985237i \(-0.554763\pi\)
−0.171197 + 0.985237i \(0.554763\pi\)
\(854\) −10.0000 −0.342193
\(855\) 0 0
\(856\) −4.00000 −0.136717
\(857\) −34.0000 −1.16142 −0.580709 0.814111i \(-0.697225\pi\)
−0.580709 + 0.814111i \(0.697225\pi\)
\(858\) 0 0
\(859\) 8.00000 0.272956 0.136478 0.990643i \(-0.456422\pi\)
0.136478 + 0.990643i \(0.456422\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 24.0000 0.817443
\(863\) 32.0000 1.08929 0.544646 0.838666i \(-0.316664\pi\)
0.544646 + 0.838666i \(0.316664\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −38.0000 −1.29129
\(867\) 0 0
\(868\) −8.00000 −0.271538
\(869\) 32.0000 1.08553
\(870\) 0 0
\(871\) −4.00000 −0.135535
\(872\) 2.00000 0.0677285
\(873\) 6.00000 0.203069
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −30.0000 −1.01303 −0.506514 0.862232i \(-0.669066\pi\)
−0.506514 + 0.862232i \(0.669066\pi\)
\(878\) −8.00000 −0.269987
\(879\) 0 0
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) 3.00000 0.101015
\(883\) 44.0000 1.48072 0.740359 0.672212i \(-0.234656\pi\)
0.740359 + 0.672212i \(0.234656\pi\)
\(884\) 6.00000 0.201802
\(885\) 0 0
\(886\) −36.0000 −1.20944
\(887\) 32.0000 1.07445 0.537227 0.843437i \(-0.319472\pi\)
0.537227 + 0.843437i \(0.319472\pi\)
\(888\) 0 0
\(889\) −16.0000 −0.536623
\(890\) 0 0
\(891\) 36.0000 1.20605
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 14.0000 0.467186
\(899\) 80.0000 2.66815
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) 24.0000 0.799113
\(903\) 0 0
\(904\) 2.00000 0.0665190
\(905\) 0 0
\(906\) 0 0
\(907\) −12.0000 −0.398453 −0.199227 0.979953i \(-0.563843\pi\)
−0.199227 + 0.979953i \(0.563843\pi\)
\(908\) 0 0
\(909\) 42.0000 1.39305
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 10.0000 0.330771
\(915\) 0 0
\(916\) −6.00000 −0.198246
\(917\) 8.00000 0.264183
\(918\) 0 0
\(919\) 48.0000 1.58337 0.791687 0.610927i \(-0.209203\pi\)
0.791687 + 0.610927i \(0.209203\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −18.0000 −0.592798
\(923\) −8.00000 −0.263323
\(924\) 0 0
\(925\) 0 0
\(926\) 16.0000 0.525793
\(927\) 48.0000 1.57653
\(928\) 10.0000 0.328266
\(929\) −30.0000 −0.984268 −0.492134 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −10.0000 −0.327561
\(933\) 0 0
\(934\) 8.00000 0.261768
\(935\) 0 0
\(936\) 3.00000 0.0980581
\(937\) −42.0000 −1.37208 −0.686040 0.727564i \(-0.740653\pi\)
−0.686040 + 0.727564i \(0.740653\pi\)
\(938\) 4.00000 0.130605
\(939\) 0 0
\(940\) 0 0
\(941\) 10.0000 0.325991 0.162995 0.986627i \(-0.447884\pi\)
0.162995 + 0.986627i \(0.447884\pi\)
\(942\) 0 0
\(943\) 48.0000 1.56310
\(944\) 8.00000 0.260378
\(945\) 0 0
\(946\) 16.0000 0.520205
\(947\) −12.0000 −0.389948 −0.194974 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) 0 0
\(949\) −2.00000 −0.0649227
\(950\) 0 0
\(951\) 0 0
\(952\) −6.00000 −0.194461
\(953\) −26.0000 −0.842223 −0.421111 0.907009i \(-0.638360\pi\)
−0.421111 + 0.907009i \(0.638360\pi\)
\(954\) −18.0000 −0.582772
\(955\) 0 0
\(956\) −24.0000 −0.776215
\(957\) 0 0
\(958\) −24.0000 −0.775405
\(959\) 6.00000 0.193750
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 6.00000 0.193448
\(963\) −12.0000 −0.386695
\(964\) 18.0000 0.579741
\(965\) 0 0
\(966\) 0 0
\(967\) −32.0000 −1.02905 −0.514525 0.857475i \(-0.672032\pi\)
−0.514525 + 0.857475i \(0.672032\pi\)
\(968\) −5.00000 −0.160706
\(969\) 0 0
\(970\) 0 0
\(971\) 48.0000 1.54039 0.770197 0.637806i \(-0.220158\pi\)
0.770197 + 0.637806i \(0.220158\pi\)
\(972\) 0 0
\(973\) −8.00000 −0.256468
\(974\) −16.0000 −0.512673
\(975\) 0 0
\(976\) 10.0000 0.320092
\(977\) 30.0000 0.959785 0.479893 0.877327i \(-0.340676\pi\)
0.479893 + 0.877327i \(0.340676\pi\)
\(978\) 0 0
\(979\) 72.0000 2.30113
\(980\) 0 0
\(981\) 6.00000 0.191565
\(982\) −28.0000 −0.893516
\(983\) −32.0000 −1.02064 −0.510321 0.859984i \(-0.670473\pi\)
−0.510321 + 0.859984i \(0.670473\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 60.0000 1.91079
\(987\) 0 0
\(988\) 0 0
\(989\) 32.0000 1.01754
\(990\) 0 0
\(991\) 24.0000 0.762385 0.381193 0.924496i \(-0.375513\pi\)
0.381193 + 0.924496i \(0.375513\pi\)
\(992\) 8.00000 0.254000
\(993\) 0 0
\(994\) 8.00000 0.253745
\(995\) 0 0
\(996\) 0 0
\(997\) −2.00000 −0.0633406 −0.0316703 0.999498i \(-0.510083\pi\)
−0.0316703 + 0.999498i \(0.510083\pi\)
\(998\) −4.00000 −0.126618
\(999\) 0 0
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 4550.2.a.g.1.1 1
5.4 even 2 182.2.a.c.1.1 1
15.14 odd 2 1638.2.a.c.1.1 1
20.19 odd 2 1456.2.a.i.1.1 1
35.4 even 6 1274.2.f.f.79.1 2
35.9 even 6 1274.2.f.f.1145.1 2
35.19 odd 6 1274.2.f.g.1145.1 2
35.24 odd 6 1274.2.f.g.79.1 2
35.34 odd 2 1274.2.a.l.1.1 1
40.19 odd 2 5824.2.a.m.1.1 1
40.29 even 2 5824.2.a.l.1.1 1
65.34 odd 4 2366.2.d.d.337.2 2
65.44 odd 4 2366.2.d.d.337.1 2
65.64 even 2 2366.2.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
182.2.a.c.1.1 1 5.4 even 2
1274.2.a.l.1.1 1 35.34 odd 2
1274.2.f.f.79.1 2 35.4 even 6
1274.2.f.f.1145.1 2 35.9 even 6
1274.2.f.g.79.1 2 35.24 odd 6
1274.2.f.g.1145.1 2 35.19 odd 6
1456.2.a.i.1.1 1 20.19 odd 2
1638.2.a.c.1.1 1 15.14 odd 2
2366.2.a.d.1.1 1 65.64 even 2
2366.2.d.d.337.1 2 65.44 odd 4
2366.2.d.d.337.2 2 65.34 odd 4
4550.2.a.g.1.1 1 1.1 even 1 trivial
5824.2.a.l.1.1 1 40.29 even 2
5824.2.a.m.1.1 1 40.19 odd 2