Properties

Label 4840.2.a.bf.1.5
Level $4840$
Weight $2$
Character 4840.1
Self dual yes
Analytic conductor $38.648$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4840,2,Mod(1,4840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4840, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4840.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4840 = 2^{3} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4840.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: \(38.6475945783\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.45753625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 13x^{4} + 11x^{3} + 41x^{2} - 30x - 20 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 440)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(3.05921\) of defining polynomial
Character \(\chi\) \(=\) 4840.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.89070 q^{3} -1.00000 q^{5} -2.74075 q^{7} +0.574737 q^{9} +O(q^{10})\) \(q+1.89070 q^{3} -1.00000 q^{5} -2.74075 q^{7} +0.574737 q^{9} +0.196821 q^{13} -1.89070 q^{15} -1.46809 q^{17} -2.68154 q^{19} -5.18193 q^{21} +1.81736 q^{23} +1.00000 q^{25} -4.58544 q^{27} +1.37946 q^{29} +5.90411 q^{31} +2.74075 q^{35} +4.57623 q^{37} +0.372129 q^{39} +1.23428 q^{41} +2.96514 q^{43} -0.574737 q^{45} +9.30107 q^{47} +0.511708 q^{49} -2.77571 q^{51} +2.69388 q^{53} -5.06997 q^{57} +14.2092 q^{59} +7.54027 q^{61} -1.57521 q^{63} -0.196821 q^{65} +10.6500 q^{67} +3.43607 q^{69} -6.87672 q^{71} +10.4314 q^{73} +1.89070 q^{75} +1.87328 q^{79} -10.3939 q^{81} -5.79397 q^{83} +1.46809 q^{85} +2.60815 q^{87} +17.3060 q^{89} -0.539438 q^{91} +11.1629 q^{93} +2.68154 q^{95} -18.3188 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{3} - 6 q^{5} + 6 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 2 q^{3} - 6 q^{5} + 6 q^{7} + 10 q^{9} - 6 q^{13} - 2 q^{15} + 11 q^{17} - 11 q^{19} + 2 q^{21} + 18 q^{23} + 6 q^{25} - q^{27} - 6 q^{29} + q^{31} - 6 q^{35} + 4 q^{37} + 27 q^{39} - 4 q^{41} + 3 q^{43} - 10 q^{45} + 14 q^{47} + 8 q^{49} + 31 q^{51} + 14 q^{53} - 5 q^{57} + 2 q^{59} - 4 q^{61} - 16 q^{63} + 6 q^{65} + 11 q^{67} + 8 q^{69} + 7 q^{71} - 9 q^{73} + 2 q^{75} + 36 q^{79} + 30 q^{81} - 45 q^{83} - 11 q^{85} + 25 q^{87} + q^{89} - 8 q^{91} + 55 q^{93} + 11 q^{95} + 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.89070 1.09159 0.545797 0.837917i \(-0.316227\pi\)
0.545797 + 0.837917i \(0.316227\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −2.74075 −1.03591 −0.517953 0.855409i \(-0.673306\pi\)
−0.517953 + 0.855409i \(0.673306\pi\)
\(8\) 0 0
\(9\) 0.574737 0.191579
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) 0.196821 0.0545884 0.0272942 0.999627i \(-0.491311\pi\)
0.0272942 + 0.999627i \(0.491311\pi\)
\(14\) 0 0
\(15\) −1.89070 −0.488176
\(16\) 0 0
\(17\) −1.46809 −0.356063 −0.178032 0.984025i \(-0.556973\pi\)
−0.178032 + 0.984025i \(0.556973\pi\)
\(18\) 0 0
\(19\) −2.68154 −0.615187 −0.307593 0.951518i \(-0.599524\pi\)
−0.307593 + 0.951518i \(0.599524\pi\)
\(20\) 0 0
\(21\) −5.18193 −1.13079
\(22\) 0 0
\(23\) 1.81736 0.378945 0.189473 0.981886i \(-0.439322\pi\)
0.189473 + 0.981886i \(0.439322\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −4.58544 −0.882468
\(28\) 0 0
\(29\) 1.37946 0.256160 0.128080 0.991764i \(-0.459119\pi\)
0.128080 + 0.991764i \(0.459119\pi\)
\(30\) 0 0
\(31\) 5.90411 1.06041 0.530205 0.847870i \(-0.322115\pi\)
0.530205 + 0.847870i \(0.322115\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.74075 0.463271
\(36\) 0 0
\(37\) 4.57623 0.752328 0.376164 0.926553i \(-0.377243\pi\)
0.376164 + 0.926553i \(0.377243\pi\)
\(38\) 0 0
\(39\) 0.372129 0.0595884
\(40\) 0 0
\(41\) 1.23428 0.192762 0.0963812 0.995344i \(-0.469273\pi\)
0.0963812 + 0.995344i \(0.469273\pi\)
\(42\) 0 0
\(43\) 2.96514 0.452180 0.226090 0.974106i \(-0.427406\pi\)
0.226090 + 0.974106i \(0.427406\pi\)
\(44\) 0 0
\(45\) −0.574737 −0.0856768
\(46\) 0 0
\(47\) 9.30107 1.35670 0.678350 0.734739i \(-0.262695\pi\)
0.678350 + 0.734739i \(0.262695\pi\)
\(48\) 0 0
\(49\) 0.511708 0.0731012
\(50\) 0 0
\(51\) −2.77571 −0.388677
\(52\) 0 0
\(53\) 2.69388 0.370032 0.185016 0.982735i \(-0.440766\pi\)
0.185016 + 0.982735i \(0.440766\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −5.06997 −0.671535
\(58\) 0 0
\(59\) 14.2092 1.84988 0.924942 0.380108i \(-0.124113\pi\)
0.924942 + 0.380108i \(0.124113\pi\)
\(60\) 0 0
\(61\) 7.54027 0.965433 0.482716 0.875777i \(-0.339650\pi\)
0.482716 + 0.875777i \(0.339650\pi\)
\(62\) 0 0
\(63\) −1.57521 −0.198458
\(64\) 0 0
\(65\) −0.196821 −0.0244127
\(66\) 0 0
\(67\) 10.6500 1.30111 0.650553 0.759461i \(-0.274537\pi\)
0.650553 + 0.759461i \(0.274537\pi\)
\(68\) 0 0
\(69\) 3.43607 0.413655
\(70\) 0 0
\(71\) −6.87672 −0.816116 −0.408058 0.912956i \(-0.633794\pi\)
−0.408058 + 0.912956i \(0.633794\pi\)
\(72\) 0 0
\(73\) 10.4314 1.22090 0.610452 0.792053i \(-0.290988\pi\)
0.610452 + 0.792053i \(0.290988\pi\)
\(74\) 0 0
\(75\) 1.89070 0.218319
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 1.87328 0.210761 0.105380 0.994432i \(-0.466394\pi\)
0.105380 + 0.994432i \(0.466394\pi\)
\(80\) 0 0
\(81\) −10.3939 −1.15488
\(82\) 0 0
\(83\) −5.79397 −0.635971 −0.317986 0.948096i \(-0.603006\pi\)
−0.317986 + 0.948096i \(0.603006\pi\)
\(84\) 0 0
\(85\) 1.46809 0.159236
\(86\) 0 0
\(87\) 2.60815 0.279623
\(88\) 0 0
\(89\) 17.3060 1.83443 0.917214 0.398396i \(-0.130433\pi\)
0.917214 + 0.398396i \(0.130433\pi\)
\(90\) 0 0
\(91\) −0.539438 −0.0565484
\(92\) 0 0
\(93\) 11.1629 1.15754
\(94\) 0 0
\(95\) 2.68154 0.275120
\(96\) 0 0
\(97\) −18.3188 −1.86000 −0.929998 0.367564i \(-0.880192\pi\)
−0.929998 + 0.367564i \(0.880192\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 5.00039 0.497558 0.248779 0.968560i \(-0.419971\pi\)
0.248779 + 0.968560i \(0.419971\pi\)
\(102\) 0 0
\(103\) 9.17856 0.904391 0.452195 0.891919i \(-0.350641\pi\)
0.452195 + 0.891919i \(0.350641\pi\)
\(104\) 0 0
\(105\) 5.18193 0.505704
\(106\) 0 0
\(107\) −16.8116 −1.62524 −0.812621 0.582792i \(-0.801960\pi\)
−0.812621 + 0.582792i \(0.801960\pi\)
\(108\) 0 0
\(109\) −9.91076 −0.949279 −0.474639 0.880180i \(-0.657422\pi\)
−0.474639 + 0.880180i \(0.657422\pi\)
\(110\) 0 0
\(111\) 8.65227 0.821238
\(112\) 0 0
\(113\) 12.6713 1.19202 0.596009 0.802978i \(-0.296752\pi\)
0.596009 + 0.802978i \(0.296752\pi\)
\(114\) 0 0
\(115\) −1.81736 −0.169470
\(116\) 0 0
\(117\) 0.113120 0.0104580
\(118\) 0 0
\(119\) 4.02366 0.368848
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 2.33365 0.210418
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 10.4404 0.926437 0.463218 0.886244i \(-0.346695\pi\)
0.463218 + 0.886244i \(0.346695\pi\)
\(128\) 0 0
\(129\) 5.60619 0.493597
\(130\) 0 0
\(131\) −20.9537 −1.83073 −0.915367 0.402621i \(-0.868099\pi\)
−0.915367 + 0.402621i \(0.868099\pi\)
\(132\) 0 0
\(133\) 7.34942 0.637276
\(134\) 0 0
\(135\) 4.58544 0.394652
\(136\) 0 0
\(137\) 2.96083 0.252961 0.126481 0.991969i \(-0.459632\pi\)
0.126481 + 0.991969i \(0.459632\pi\)
\(138\) 0 0
\(139\) 16.4253 1.39318 0.696589 0.717470i \(-0.254700\pi\)
0.696589 + 0.717470i \(0.254700\pi\)
\(140\) 0 0
\(141\) 17.5855 1.48097
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −1.37946 −0.114558
\(146\) 0 0
\(147\) 0.967485 0.0797968
\(148\) 0 0
\(149\) −19.1836 −1.57158 −0.785790 0.618494i \(-0.787743\pi\)
−0.785790 + 0.618494i \(0.787743\pi\)
\(150\) 0 0
\(151\) −9.04992 −0.736472 −0.368236 0.929732i \(-0.620038\pi\)
−0.368236 + 0.929732i \(0.620038\pi\)
\(152\) 0 0
\(153\) −0.843764 −0.0682142
\(154\) 0 0
\(155\) −5.90411 −0.474230
\(156\) 0 0
\(157\) −18.1894 −1.45167 −0.725837 0.687867i \(-0.758547\pi\)
−0.725837 + 0.687867i \(0.758547\pi\)
\(158\) 0 0
\(159\) 5.09331 0.403925
\(160\) 0 0
\(161\) −4.98092 −0.392552
\(162\) 0 0
\(163\) 6.34677 0.497117 0.248559 0.968617i \(-0.420043\pi\)
0.248559 + 0.968617i \(0.420043\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 13.9261 1.07763 0.538817 0.842423i \(-0.318871\pi\)
0.538817 + 0.842423i \(0.318871\pi\)
\(168\) 0 0
\(169\) −12.9613 −0.997020
\(170\) 0 0
\(171\) −1.54118 −0.117857
\(172\) 0 0
\(173\) 24.2172 1.84120 0.920598 0.390511i \(-0.127701\pi\)
0.920598 + 0.390511i \(0.127701\pi\)
\(174\) 0 0
\(175\) −2.74075 −0.207181
\(176\) 0 0
\(177\) 26.8654 2.01932
\(178\) 0 0
\(179\) 1.41972 0.106115 0.0530573 0.998591i \(-0.483103\pi\)
0.0530573 + 0.998591i \(0.483103\pi\)
\(180\) 0 0
\(181\) 5.62889 0.418392 0.209196 0.977874i \(-0.432915\pi\)
0.209196 + 0.977874i \(0.432915\pi\)
\(182\) 0 0
\(183\) 14.2564 1.05386
\(184\) 0 0
\(185\) −4.57623 −0.336451
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 12.5675 0.914154
\(190\) 0 0
\(191\) −2.24259 −0.162268 −0.0811341 0.996703i \(-0.525854\pi\)
−0.0811341 + 0.996703i \(0.525854\pi\)
\(192\) 0 0
\(193\) −9.30699 −0.669932 −0.334966 0.942230i \(-0.608725\pi\)
−0.334966 + 0.942230i \(0.608725\pi\)
\(194\) 0 0
\(195\) −0.372129 −0.0266487
\(196\) 0 0
\(197\) 14.3935 1.02549 0.512747 0.858540i \(-0.328628\pi\)
0.512747 + 0.858540i \(0.328628\pi\)
\(198\) 0 0
\(199\) 3.00915 0.213313 0.106657 0.994296i \(-0.465985\pi\)
0.106657 + 0.994296i \(0.465985\pi\)
\(200\) 0 0
\(201\) 20.1360 1.42028
\(202\) 0 0
\(203\) −3.78076 −0.265358
\(204\) 0 0
\(205\) −1.23428 −0.0862059
\(206\) 0 0
\(207\) 1.04450 0.0725980
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 15.6428 1.07689 0.538447 0.842659i \(-0.319011\pi\)
0.538447 + 0.842659i \(0.319011\pi\)
\(212\) 0 0
\(213\) −13.0018 −0.890868
\(214\) 0 0
\(215\) −2.96514 −0.202221
\(216\) 0 0
\(217\) −16.1817 −1.09848
\(218\) 0 0
\(219\) 19.7226 1.33273
\(220\) 0 0
\(221\) −0.288950 −0.0194369
\(222\) 0 0
\(223\) 5.21134 0.348977 0.174488 0.984659i \(-0.444173\pi\)
0.174488 + 0.984659i \(0.444173\pi\)
\(224\) 0 0
\(225\) 0.574737 0.0383158
\(226\) 0 0
\(227\) −13.7838 −0.914861 −0.457430 0.889245i \(-0.651230\pi\)
−0.457430 + 0.889245i \(0.651230\pi\)
\(228\) 0 0
\(229\) 24.5800 1.62429 0.812147 0.583453i \(-0.198299\pi\)
0.812147 + 0.583453i \(0.198299\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 8.02886 0.525988 0.262994 0.964797i \(-0.415290\pi\)
0.262994 + 0.964797i \(0.415290\pi\)
\(234\) 0 0
\(235\) −9.30107 −0.606735
\(236\) 0 0
\(237\) 3.54181 0.230066
\(238\) 0 0
\(239\) −1.36215 −0.0881102 −0.0440551 0.999029i \(-0.514028\pi\)
−0.0440551 + 0.999029i \(0.514028\pi\)
\(240\) 0 0
\(241\) −2.94349 −0.189607 −0.0948035 0.995496i \(-0.530222\pi\)
−0.0948035 + 0.995496i \(0.530222\pi\)
\(242\) 0 0
\(243\) −5.89538 −0.378189
\(244\) 0 0
\(245\) −0.511708 −0.0326918
\(246\) 0 0
\(247\) −0.527783 −0.0335820
\(248\) 0 0
\(249\) −10.9547 −0.694223
\(250\) 0 0
\(251\) 15.7107 0.991651 0.495826 0.868422i \(-0.334866\pi\)
0.495826 + 0.868422i \(0.334866\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 2.77571 0.173821
\(256\) 0 0
\(257\) −8.90470 −0.555460 −0.277730 0.960659i \(-0.589582\pi\)
−0.277730 + 0.960659i \(0.589582\pi\)
\(258\) 0 0
\(259\) −12.5423 −0.779341
\(260\) 0 0
\(261\) 0.792829 0.0490749
\(262\) 0 0
\(263\) 0.671733 0.0414208 0.0207104 0.999786i \(-0.493407\pi\)
0.0207104 + 0.999786i \(0.493407\pi\)
\(264\) 0 0
\(265\) −2.69388 −0.165484
\(266\) 0 0
\(267\) 32.7203 2.00245
\(268\) 0 0
\(269\) 26.7105 1.62857 0.814283 0.580468i \(-0.197130\pi\)
0.814283 + 0.580468i \(0.197130\pi\)
\(270\) 0 0
\(271\) 22.3089 1.35517 0.677584 0.735446i \(-0.263027\pi\)
0.677584 + 0.735446i \(0.263027\pi\)
\(272\) 0 0
\(273\) −1.01991 −0.0617280
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 22.9001 1.37593 0.687967 0.725742i \(-0.258503\pi\)
0.687967 + 0.725742i \(0.258503\pi\)
\(278\) 0 0
\(279\) 3.39331 0.203152
\(280\) 0 0
\(281\) −28.3433 −1.69082 −0.845411 0.534117i \(-0.820644\pi\)
−0.845411 + 0.534117i \(0.820644\pi\)
\(282\) 0 0
\(283\) −31.0561 −1.84609 −0.923046 0.384690i \(-0.874308\pi\)
−0.923046 + 0.384690i \(0.874308\pi\)
\(284\) 0 0
\(285\) 5.06997 0.300319
\(286\) 0 0
\(287\) −3.38286 −0.199684
\(288\) 0 0
\(289\) −14.8447 −0.873219
\(290\) 0 0
\(291\) −34.6354 −2.03036
\(292\) 0 0
\(293\) 7.84981 0.458591 0.229295 0.973357i \(-0.426358\pi\)
0.229295 + 0.973357i \(0.426358\pi\)
\(294\) 0 0
\(295\) −14.2092 −0.827293
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0.357695 0.0206860
\(300\) 0 0
\(301\) −8.12671 −0.468416
\(302\) 0 0
\(303\) 9.45423 0.543131
\(304\) 0 0
\(305\) −7.54027 −0.431755
\(306\) 0 0
\(307\) 10.0601 0.574160 0.287080 0.957907i \(-0.407315\pi\)
0.287080 + 0.957907i \(0.407315\pi\)
\(308\) 0 0
\(309\) 17.3539 0.987228
\(310\) 0 0
\(311\) −22.3650 −1.26820 −0.634101 0.773250i \(-0.718630\pi\)
−0.634101 + 0.773250i \(0.718630\pi\)
\(312\) 0 0
\(313\) 18.1486 1.02582 0.512910 0.858442i \(-0.328567\pi\)
0.512910 + 0.858442i \(0.328567\pi\)
\(314\) 0 0
\(315\) 1.57521 0.0887531
\(316\) 0 0
\(317\) −2.51660 −0.141346 −0.0706732 0.997500i \(-0.522515\pi\)
−0.0706732 + 0.997500i \(0.522515\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −31.7857 −1.77411
\(322\) 0 0
\(323\) 3.93673 0.219045
\(324\) 0 0
\(325\) 0.196821 0.0109177
\(326\) 0 0
\(327\) −18.7383 −1.03623
\(328\) 0 0
\(329\) −25.4919 −1.40541
\(330\) 0 0
\(331\) −7.54546 −0.414736 −0.207368 0.978263i \(-0.566490\pi\)
−0.207368 + 0.978263i \(0.566490\pi\)
\(332\) 0 0
\(333\) 2.63013 0.144130
\(334\) 0 0
\(335\) −10.6500 −0.581872
\(336\) 0 0
\(337\) −23.0017 −1.25298 −0.626491 0.779429i \(-0.715510\pi\)
−0.626491 + 0.779429i \(0.715510\pi\)
\(338\) 0 0
\(339\) 23.9576 1.30120
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 17.7828 0.960180
\(344\) 0 0
\(345\) −3.43607 −0.184992
\(346\) 0 0
\(347\) 4.07466 0.218739 0.109370 0.994001i \(-0.465117\pi\)
0.109370 + 0.994001i \(0.465117\pi\)
\(348\) 0 0
\(349\) −23.8075 −1.27439 −0.637193 0.770704i \(-0.719905\pi\)
−0.637193 + 0.770704i \(0.719905\pi\)
\(350\) 0 0
\(351\) −0.902511 −0.0481725
\(352\) 0 0
\(353\) −7.00060 −0.372605 −0.186302 0.982492i \(-0.559650\pi\)
−0.186302 + 0.982492i \(0.559650\pi\)
\(354\) 0 0
\(355\) 6.87672 0.364978
\(356\) 0 0
\(357\) 7.60752 0.402632
\(358\) 0 0
\(359\) 36.4700 1.92482 0.962408 0.271609i \(-0.0875557\pi\)
0.962408 + 0.271609i \(0.0875557\pi\)
\(360\) 0 0
\(361\) −11.8094 −0.621545
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −10.4314 −0.546005
\(366\) 0 0
\(367\) −29.1854 −1.52346 −0.761732 0.647892i \(-0.775651\pi\)
−0.761732 + 0.647892i \(0.775651\pi\)
\(368\) 0 0
\(369\) 0.709387 0.0369292
\(370\) 0 0
\(371\) −7.38324 −0.383319
\(372\) 0 0
\(373\) 20.0215 1.03667 0.518337 0.855176i \(-0.326551\pi\)
0.518337 + 0.855176i \(0.326551\pi\)
\(374\) 0 0
\(375\) −1.89070 −0.0976352
\(376\) 0 0
\(377\) 0.271508 0.0139833
\(378\) 0 0
\(379\) 8.18971 0.420677 0.210338 0.977629i \(-0.432543\pi\)
0.210338 + 0.977629i \(0.432543\pi\)
\(380\) 0 0
\(381\) 19.7397 1.01129
\(382\) 0 0
\(383\) 20.2852 1.03653 0.518263 0.855221i \(-0.326579\pi\)
0.518263 + 0.855221i \(0.326579\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 1.70418 0.0866282
\(388\) 0 0
\(389\) 13.0260 0.660443 0.330221 0.943904i \(-0.392877\pi\)
0.330221 + 0.943904i \(0.392877\pi\)
\(390\) 0 0
\(391\) −2.66804 −0.134928
\(392\) 0 0
\(393\) −39.6171 −1.99842
\(394\) 0 0
\(395\) −1.87328 −0.0942552
\(396\) 0 0
\(397\) −0.593277 −0.0297757 −0.0148878 0.999889i \(-0.504739\pi\)
−0.0148878 + 0.999889i \(0.504739\pi\)
\(398\) 0 0
\(399\) 13.8955 0.695647
\(400\) 0 0
\(401\) −9.04664 −0.451768 −0.225884 0.974154i \(-0.572527\pi\)
−0.225884 + 0.974154i \(0.572527\pi\)
\(402\) 0 0
\(403\) 1.16205 0.0578860
\(404\) 0 0
\(405\) 10.3939 0.516476
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −17.7566 −0.878008 −0.439004 0.898485i \(-0.644669\pi\)
−0.439004 + 0.898485i \(0.644669\pi\)
\(410\) 0 0
\(411\) 5.59804 0.276131
\(412\) 0 0
\(413\) −38.9439 −1.91631
\(414\) 0 0
\(415\) 5.79397 0.284415
\(416\) 0 0
\(417\) 31.0553 1.52079
\(418\) 0 0
\(419\) −9.43701 −0.461028 −0.230514 0.973069i \(-0.574041\pi\)
−0.230514 + 0.973069i \(0.574041\pi\)
\(420\) 0 0
\(421\) −24.2461 −1.18168 −0.590842 0.806787i \(-0.701204\pi\)
−0.590842 + 0.806787i \(0.701204\pi\)
\(422\) 0 0
\(423\) 5.34567 0.259915
\(424\) 0 0
\(425\) −1.46809 −0.0712126
\(426\) 0 0
\(427\) −20.6660 −1.00010
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 9.61485 0.463131 0.231565 0.972819i \(-0.425615\pi\)
0.231565 + 0.972819i \(0.425615\pi\)
\(432\) 0 0
\(433\) −9.44277 −0.453791 −0.226895 0.973919i \(-0.572858\pi\)
−0.226895 + 0.973919i \(0.572858\pi\)
\(434\) 0 0
\(435\) −2.60815 −0.125051
\(436\) 0 0
\(437\) −4.87331 −0.233122
\(438\) 0 0
\(439\) 17.8211 0.850555 0.425278 0.905063i \(-0.360176\pi\)
0.425278 + 0.905063i \(0.360176\pi\)
\(440\) 0 0
\(441\) 0.294098 0.0140047
\(442\) 0 0
\(443\) −17.6167 −0.836995 −0.418497 0.908218i \(-0.637443\pi\)
−0.418497 + 0.908218i \(0.637443\pi\)
\(444\) 0 0
\(445\) −17.3060 −0.820381
\(446\) 0 0
\(447\) −36.2703 −1.71553
\(448\) 0 0
\(449\) −8.01122 −0.378073 −0.189036 0.981970i \(-0.560536\pi\)
−0.189036 + 0.981970i \(0.560536\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −17.1107 −0.803929
\(454\) 0 0
\(455\) 0.539438 0.0252892
\(456\) 0 0
\(457\) 8.21578 0.384318 0.192159 0.981364i \(-0.438451\pi\)
0.192159 + 0.981364i \(0.438451\pi\)
\(458\) 0 0
\(459\) 6.73182 0.314214
\(460\) 0 0
\(461\) 5.47559 0.255024 0.127512 0.991837i \(-0.459301\pi\)
0.127512 + 0.991837i \(0.459301\pi\)
\(462\) 0 0
\(463\) 27.1843 1.26336 0.631681 0.775228i \(-0.282365\pi\)
0.631681 + 0.775228i \(0.282365\pi\)
\(464\) 0 0
\(465\) −11.1629 −0.517667
\(466\) 0 0
\(467\) 7.05376 0.326409 0.163205 0.986592i \(-0.447817\pi\)
0.163205 + 0.986592i \(0.447817\pi\)
\(468\) 0 0
\(469\) −29.1890 −1.34782
\(470\) 0 0
\(471\) −34.3907 −1.58464
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −2.68154 −0.123037
\(476\) 0 0
\(477\) 1.54827 0.0708905
\(478\) 0 0
\(479\) −5.31048 −0.242642 −0.121321 0.992613i \(-0.538713\pi\)
−0.121321 + 0.992613i \(0.538713\pi\)
\(480\) 0 0
\(481\) 0.900700 0.0410684
\(482\) 0 0
\(483\) −9.41742 −0.428507
\(484\) 0 0
\(485\) 18.3188 0.831816
\(486\) 0 0
\(487\) 11.8480 0.536884 0.268442 0.963296i \(-0.413491\pi\)
0.268442 + 0.963296i \(0.413491\pi\)
\(488\) 0 0
\(489\) 11.9998 0.542651
\(490\) 0 0
\(491\) 38.3787 1.73201 0.866003 0.500038i \(-0.166681\pi\)
0.866003 + 0.500038i \(0.166681\pi\)
\(492\) 0 0
\(493\) −2.02517 −0.0912091
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 18.8474 0.845420
\(498\) 0 0
\(499\) 10.6869 0.478411 0.239205 0.970969i \(-0.423113\pi\)
0.239205 + 0.970969i \(0.423113\pi\)
\(500\) 0 0
\(501\) 26.3300 1.17634
\(502\) 0 0
\(503\) 35.5077 1.58321 0.791606 0.611032i \(-0.209245\pi\)
0.791606 + 0.611032i \(0.209245\pi\)
\(504\) 0 0
\(505\) −5.00039 −0.222515
\(506\) 0 0
\(507\) −24.5058 −1.08834
\(508\) 0 0
\(509\) −21.3976 −0.948433 −0.474216 0.880408i \(-0.657269\pi\)
−0.474216 + 0.880408i \(0.657269\pi\)
\(510\) 0 0
\(511\) −28.5899 −1.26474
\(512\) 0 0
\(513\) 12.2960 0.542883
\(514\) 0 0
\(515\) −9.17856 −0.404456
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 45.7873 2.00984
\(520\) 0 0
\(521\) −19.3712 −0.848669 −0.424334 0.905506i \(-0.639492\pi\)
−0.424334 + 0.905506i \(0.639492\pi\)
\(522\) 0 0
\(523\) 0.894305 0.0391052 0.0195526 0.999809i \(-0.493776\pi\)
0.0195526 + 0.999809i \(0.493776\pi\)
\(524\) 0 0
\(525\) −5.18193 −0.226158
\(526\) 0 0
\(527\) −8.66774 −0.377573
\(528\) 0 0
\(529\) −19.6972 −0.856400
\(530\) 0 0
\(531\) 8.16657 0.354399
\(532\) 0 0
\(533\) 0.242933 0.0105226
\(534\) 0 0
\(535\) 16.8116 0.726831
\(536\) 0 0
\(537\) 2.68425 0.115834
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −40.1527 −1.72630 −0.863150 0.504948i \(-0.831512\pi\)
−0.863150 + 0.504948i \(0.831512\pi\)
\(542\) 0 0
\(543\) 10.6425 0.456715
\(544\) 0 0
\(545\) 9.91076 0.424530
\(546\) 0 0
\(547\) −4.49729 −0.192290 −0.0961451 0.995367i \(-0.530651\pi\)
−0.0961451 + 0.995367i \(0.530651\pi\)
\(548\) 0 0
\(549\) 4.33367 0.184957
\(550\) 0 0
\(551\) −3.69908 −0.157586
\(552\) 0 0
\(553\) −5.13420 −0.218329
\(554\) 0 0
\(555\) −8.65227 −0.367269
\(556\) 0 0
\(557\) −23.9736 −1.01580 −0.507898 0.861417i \(-0.669577\pi\)
−0.507898 + 0.861417i \(0.669577\pi\)
\(558\) 0 0
\(559\) 0.583603 0.0246838
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 18.3442 0.773116 0.386558 0.922265i \(-0.373664\pi\)
0.386558 + 0.922265i \(0.373664\pi\)
\(564\) 0 0
\(565\) −12.6713 −0.533087
\(566\) 0 0
\(567\) 28.4870 1.19634
\(568\) 0 0
\(569\) −2.70511 −0.113404 −0.0567020 0.998391i \(-0.518058\pi\)
−0.0567020 + 0.998391i \(0.518058\pi\)
\(570\) 0 0
\(571\) 13.6593 0.571626 0.285813 0.958285i \(-0.407736\pi\)
0.285813 + 0.958285i \(0.407736\pi\)
\(572\) 0 0
\(573\) −4.24006 −0.177131
\(574\) 0 0
\(575\) 1.81736 0.0757891
\(576\) 0 0
\(577\) 8.23640 0.342886 0.171443 0.985194i \(-0.445157\pi\)
0.171443 + 0.985194i \(0.445157\pi\)
\(578\) 0 0
\(579\) −17.5967 −0.731294
\(580\) 0 0
\(581\) 15.8798 0.658806
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −0.113120 −0.00467696
\(586\) 0 0
\(587\) 22.3766 0.923581 0.461791 0.886989i \(-0.347207\pi\)
0.461791 + 0.886989i \(0.347207\pi\)
\(588\) 0 0
\(589\) −15.8321 −0.652350
\(590\) 0 0
\(591\) 27.2138 1.11942
\(592\) 0 0
\(593\) −44.5253 −1.82844 −0.914218 0.405223i \(-0.867194\pi\)
−0.914218 + 0.405223i \(0.867194\pi\)
\(594\) 0 0
\(595\) −4.02366 −0.164954
\(596\) 0 0
\(597\) 5.68940 0.232851
\(598\) 0 0
\(599\) 39.9292 1.63146 0.815732 0.578430i \(-0.196334\pi\)
0.815732 + 0.578430i \(0.196334\pi\)
\(600\) 0 0
\(601\) 30.9828 1.26381 0.631907 0.775044i \(-0.282273\pi\)
0.631907 + 0.775044i \(0.282273\pi\)
\(602\) 0 0
\(603\) 6.12096 0.249265
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 26.5634 1.07817 0.539087 0.842250i \(-0.318769\pi\)
0.539087 + 0.842250i \(0.318769\pi\)
\(608\) 0 0
\(609\) −7.14828 −0.289663
\(610\) 0 0
\(611\) 1.83065 0.0740601
\(612\) 0 0
\(613\) 28.2647 1.14160 0.570801 0.821089i \(-0.306633\pi\)
0.570801 + 0.821089i \(0.306633\pi\)
\(614\) 0 0
\(615\) −2.33365 −0.0941020
\(616\) 0 0
\(617\) −3.98722 −0.160520 −0.0802598 0.996774i \(-0.525575\pi\)
−0.0802598 + 0.996774i \(0.525575\pi\)
\(618\) 0 0
\(619\) 29.6186 1.19047 0.595236 0.803551i \(-0.297059\pi\)
0.595236 + 0.803551i \(0.297059\pi\)
\(620\) 0 0
\(621\) −8.33338 −0.334407
\(622\) 0 0
\(623\) −47.4313 −1.90029
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −6.71831 −0.267876
\(630\) 0 0
\(631\) −34.7749 −1.38437 −0.692183 0.721722i \(-0.743351\pi\)
−0.692183 + 0.721722i \(0.743351\pi\)
\(632\) 0 0
\(633\) 29.5758 1.17553
\(634\) 0 0
\(635\) −10.4404 −0.414315
\(636\) 0 0
\(637\) 0.100715 0.00399047
\(638\) 0 0
\(639\) −3.95231 −0.156351
\(640\) 0 0
\(641\) 27.0319 1.06770 0.533848 0.845580i \(-0.320745\pi\)
0.533848 + 0.845580i \(0.320745\pi\)
\(642\) 0 0
\(643\) −31.3855 −1.23772 −0.618861 0.785500i \(-0.712406\pi\)
−0.618861 + 0.785500i \(0.712406\pi\)
\(644\) 0 0
\(645\) −5.60619 −0.220743
\(646\) 0 0
\(647\) −43.1624 −1.69689 −0.848444 0.529284i \(-0.822460\pi\)
−0.848444 + 0.529284i \(0.822460\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −30.5947 −1.19910
\(652\) 0 0
\(653\) 9.89228 0.387115 0.193557 0.981089i \(-0.437997\pi\)
0.193557 + 0.981089i \(0.437997\pi\)
\(654\) 0 0
\(655\) 20.9537 0.818729
\(656\) 0 0
\(657\) 5.99532 0.233900
\(658\) 0 0
\(659\) 12.3139 0.479683 0.239842 0.970812i \(-0.422904\pi\)
0.239842 + 0.970812i \(0.422904\pi\)
\(660\) 0 0
\(661\) −26.4390 −1.02836 −0.514179 0.857683i \(-0.671903\pi\)
−0.514179 + 0.857683i \(0.671903\pi\)
\(662\) 0 0
\(663\) −0.546318 −0.0212172
\(664\) 0 0
\(665\) −7.34942 −0.284998
\(666\) 0 0
\(667\) 2.50698 0.0970706
\(668\) 0 0
\(669\) 9.85306 0.380941
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −14.6219 −0.563632 −0.281816 0.959469i \(-0.590937\pi\)
−0.281816 + 0.959469i \(0.590937\pi\)
\(674\) 0 0
\(675\) −4.58544 −0.176494
\(676\) 0 0
\(677\) −29.1645 −1.12088 −0.560441 0.828195i \(-0.689368\pi\)
−0.560441 + 0.828195i \(0.689368\pi\)
\(678\) 0 0
\(679\) 50.2074 1.92678
\(680\) 0 0
\(681\) −26.0609 −0.998657
\(682\) 0 0
\(683\) 25.7011 0.983427 0.491713 0.870757i \(-0.336371\pi\)
0.491713 + 0.870757i \(0.336371\pi\)
\(684\) 0 0
\(685\) −2.96083 −0.113128
\(686\) 0 0
\(687\) 46.4734 1.77307
\(688\) 0 0
\(689\) 0.530212 0.0201995
\(690\) 0 0
\(691\) 1.87725 0.0714139 0.0357070 0.999362i \(-0.488632\pi\)
0.0357070 + 0.999362i \(0.488632\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −16.4253 −0.623048
\(696\) 0 0
\(697\) −1.81203 −0.0686356
\(698\) 0 0
\(699\) 15.1801 0.574166
\(700\) 0 0
\(701\) 39.4735 1.49089 0.745447 0.666565i \(-0.232236\pi\)
0.745447 + 0.666565i \(0.232236\pi\)
\(702\) 0 0
\(703\) −12.2713 −0.462822
\(704\) 0 0
\(705\) −17.5855 −0.662308
\(706\) 0 0
\(707\) −13.7048 −0.515423
\(708\) 0 0
\(709\) −34.7390 −1.30465 −0.652325 0.757939i \(-0.726206\pi\)
−0.652325 + 0.757939i \(0.726206\pi\)
\(710\) 0 0
\(711\) 1.07665 0.0403774
\(712\) 0 0
\(713\) 10.7299 0.401837
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −2.57542 −0.0961807
\(718\) 0 0
\(719\) 21.7972 0.812897 0.406449 0.913674i \(-0.366767\pi\)
0.406449 + 0.913674i \(0.366767\pi\)
\(720\) 0 0
\(721\) −25.1561 −0.936864
\(722\) 0 0
\(723\) −5.56525 −0.206974
\(724\) 0 0
\(725\) 1.37946 0.0512320
\(726\) 0 0
\(727\) 33.0916 1.22730 0.613650 0.789578i \(-0.289701\pi\)
0.613650 + 0.789578i \(0.289701\pi\)
\(728\) 0 0
\(729\) 20.0353 0.742047
\(730\) 0 0
\(731\) −4.35308 −0.161005
\(732\) 0 0
\(733\) −37.8016 −1.39624 −0.698118 0.715983i \(-0.745979\pi\)
−0.698118 + 0.715983i \(0.745979\pi\)
\(734\) 0 0
\(735\) −0.967485 −0.0356862
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 27.8114 1.02306 0.511529 0.859266i \(-0.329079\pi\)
0.511529 + 0.859266i \(0.329079\pi\)
\(740\) 0 0
\(741\) −0.997878 −0.0366580
\(742\) 0 0
\(743\) 22.6766 0.831925 0.415963 0.909382i \(-0.363445\pi\)
0.415963 + 0.909382i \(0.363445\pi\)
\(744\) 0 0
\(745\) 19.1836 0.702832
\(746\) 0 0
\(747\) −3.33001 −0.121839
\(748\) 0 0
\(749\) 46.0765 1.68360
\(750\) 0 0
\(751\) 1.70546 0.0622330 0.0311165 0.999516i \(-0.490094\pi\)
0.0311165 + 0.999516i \(0.490094\pi\)
\(752\) 0 0
\(753\) 29.7042 1.08248
\(754\) 0 0
\(755\) 9.04992 0.329360
\(756\) 0 0
\(757\) −8.27807 −0.300872 −0.150436 0.988620i \(-0.548068\pi\)
−0.150436 + 0.988620i \(0.548068\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −14.2064 −0.514982 −0.257491 0.966281i \(-0.582896\pi\)
−0.257491 + 0.966281i \(0.582896\pi\)
\(762\) 0 0
\(763\) 27.1629 0.983364
\(764\) 0 0
\(765\) 0.843764 0.0305063
\(766\) 0 0
\(767\) 2.79668 0.100982
\(768\) 0 0
\(769\) −31.0174 −1.11852 −0.559258 0.828993i \(-0.688914\pi\)
−0.559258 + 0.828993i \(0.688914\pi\)
\(770\) 0 0
\(771\) −16.8361 −0.606337
\(772\) 0 0
\(773\) −12.2223 −0.439607 −0.219803 0.975544i \(-0.570542\pi\)
−0.219803 + 0.975544i \(0.570542\pi\)
\(774\) 0 0
\(775\) 5.90411 0.212082
\(776\) 0 0
\(777\) −23.7137 −0.850725
\(778\) 0 0
\(779\) −3.30977 −0.118585
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −6.32544 −0.226053
\(784\) 0 0
\(785\) 18.1894 0.649208
\(786\) 0 0
\(787\) 7.68049 0.273780 0.136890 0.990586i \(-0.456289\pi\)
0.136890 + 0.990586i \(0.456289\pi\)
\(788\) 0 0
\(789\) 1.27004 0.0452147
\(790\) 0 0
\(791\) −34.7289 −1.23482
\(792\) 0 0
\(793\) 1.48408 0.0527014
\(794\) 0 0
\(795\) −5.09331 −0.180641
\(796\) 0 0
\(797\) −1.14757 −0.0406491 −0.0203245 0.999793i \(-0.506470\pi\)
−0.0203245 + 0.999793i \(0.506470\pi\)
\(798\) 0 0
\(799\) −13.6548 −0.483071
\(800\) 0 0
\(801\) 9.94637 0.351438
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 4.98092 0.175554
\(806\) 0 0
\(807\) 50.5014 1.77773
\(808\) 0 0
\(809\) 0.494854 0.0173982 0.00869908 0.999962i \(-0.497231\pi\)
0.00869908 + 0.999962i \(0.497231\pi\)
\(810\) 0 0
\(811\) 23.7818 0.835091 0.417546 0.908656i \(-0.362890\pi\)
0.417546 + 0.908656i \(0.362890\pi\)
\(812\) 0 0
\(813\) 42.1793 1.47929
\(814\) 0 0
\(815\) −6.34677 −0.222318
\(816\) 0 0
\(817\) −7.95113 −0.278175
\(818\) 0 0
\(819\) −0.310035 −0.0108335
\(820\) 0 0
\(821\) 41.7817 1.45819 0.729096 0.684412i \(-0.239941\pi\)
0.729096 + 0.684412i \(0.239941\pi\)
\(822\) 0 0
\(823\) −23.3527 −0.814023 −0.407011 0.913423i \(-0.633429\pi\)
−0.407011 + 0.913423i \(0.633429\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −19.2469 −0.669280 −0.334640 0.942346i \(-0.608615\pi\)
−0.334640 + 0.942346i \(0.608615\pi\)
\(828\) 0 0
\(829\) 17.6334 0.612434 0.306217 0.951962i \(-0.400937\pi\)
0.306217 + 0.951962i \(0.400937\pi\)
\(830\) 0 0
\(831\) 43.2972 1.50196
\(832\) 0 0
\(833\) −0.751232 −0.0260286
\(834\) 0 0
\(835\) −13.9261 −0.481932
\(836\) 0 0
\(837\) −27.0729 −0.935778
\(838\) 0 0
\(839\) −29.8854 −1.03176 −0.515879 0.856662i \(-0.672535\pi\)
−0.515879 + 0.856662i \(0.672535\pi\)
\(840\) 0 0
\(841\) −27.0971 −0.934382
\(842\) 0 0
\(843\) −53.5887 −1.84569
\(844\) 0 0
\(845\) 12.9613 0.445881
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −58.7176 −2.01518
\(850\) 0 0
\(851\) 8.31666 0.285091
\(852\) 0 0
\(853\) −8.78136 −0.300668 −0.150334 0.988635i \(-0.548035\pi\)
−0.150334 + 0.988635i \(0.548035\pi\)
\(854\) 0 0
\(855\) 1.54118 0.0527072
\(856\) 0 0
\(857\) 25.6711 0.876909 0.438454 0.898753i \(-0.355526\pi\)
0.438454 + 0.898753i \(0.355526\pi\)
\(858\) 0 0
\(859\) −41.5338 −1.41711 −0.708557 0.705654i \(-0.750654\pi\)
−0.708557 + 0.705654i \(0.750654\pi\)
\(860\) 0 0
\(861\) −6.39596 −0.217974
\(862\) 0 0
\(863\) −2.83297 −0.0964353 −0.0482177 0.998837i \(-0.515354\pi\)
−0.0482177 + 0.998837i \(0.515354\pi\)
\(864\) 0 0
\(865\) −24.2172 −0.823408
\(866\) 0 0
\(867\) −28.0669 −0.953201
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 2.09615 0.0710253
\(872\) 0 0
\(873\) −10.5285 −0.356336
\(874\) 0 0
\(875\) 2.74075 0.0926542
\(876\) 0 0
\(877\) 26.0319 0.879035 0.439518 0.898234i \(-0.355149\pi\)
0.439518 + 0.898234i \(0.355149\pi\)
\(878\) 0 0
\(879\) 14.8416 0.500595
\(880\) 0 0
\(881\) −37.6862 −1.26968 −0.634841 0.772643i \(-0.718934\pi\)
−0.634841 + 0.772643i \(0.718934\pi\)
\(882\) 0 0
\(883\) −47.5341 −1.59965 −0.799825 0.600233i \(-0.795074\pi\)
−0.799825 + 0.600233i \(0.795074\pi\)
\(884\) 0 0
\(885\) −26.8654 −0.903069
\(886\) 0 0
\(887\) 47.7310 1.60265 0.801326 0.598228i \(-0.204128\pi\)
0.801326 + 0.598228i \(0.204128\pi\)
\(888\) 0 0
\(889\) −28.6146 −0.959701
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −24.9412 −0.834624
\(894\) 0 0
\(895\) −1.41972 −0.0474559
\(896\) 0 0
\(897\) 0.676292 0.0225807
\(898\) 0 0
\(899\) 8.14450 0.271634
\(900\) 0 0
\(901\) −3.95484 −0.131755
\(902\) 0 0
\(903\) −15.3651 −0.511320
\(904\) 0 0
\(905\) −5.62889 −0.187111
\(906\) 0 0
\(907\) 33.7132 1.11943 0.559714 0.828686i \(-0.310911\pi\)
0.559714 + 0.828686i \(0.310911\pi\)
\(908\) 0 0
\(909\) 2.87391 0.0953216
\(910\) 0 0
\(911\) 38.9640 1.29094 0.645468 0.763788i \(-0.276662\pi\)
0.645468 + 0.763788i \(0.276662\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −14.2564 −0.471301
\(916\) 0 0
\(917\) 57.4289 1.89647
\(918\) 0 0
\(919\) 12.5411 0.413692 0.206846 0.978374i \(-0.433680\pi\)
0.206846 + 0.978374i \(0.433680\pi\)
\(920\) 0 0
\(921\) 19.0206 0.626750
\(922\) 0 0
\(923\) −1.35348 −0.0445505
\(924\) 0 0
\(925\) 4.57623 0.150466
\(926\) 0 0
\(927\) 5.27526 0.173262
\(928\) 0 0
\(929\) 26.5557 0.871263 0.435631 0.900125i \(-0.356525\pi\)
0.435631 + 0.900125i \(0.356525\pi\)
\(930\) 0 0
\(931\) −1.37216 −0.0449709
\(932\) 0 0
\(933\) −42.2854 −1.38436
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0.566670 0.0185123 0.00925615 0.999957i \(-0.497054\pi\)
0.00925615 + 0.999957i \(0.497054\pi\)
\(938\) 0 0
\(939\) 34.3136 1.11978
\(940\) 0 0
\(941\) 11.4578 0.373514 0.186757 0.982406i \(-0.440202\pi\)
0.186757 + 0.982406i \(0.440202\pi\)
\(942\) 0 0
\(943\) 2.24313 0.0730464
\(944\) 0 0
\(945\) −12.5675 −0.408822
\(946\) 0 0
\(947\) 30.2412 0.982707 0.491354 0.870960i \(-0.336502\pi\)
0.491354 + 0.870960i \(0.336502\pi\)
\(948\) 0 0
\(949\) 2.05312 0.0666472
\(950\) 0 0
\(951\) −4.75813 −0.154293
\(952\) 0 0
\(953\) 41.7753 1.35324 0.676618 0.736334i \(-0.263445\pi\)
0.676618 + 0.736334i \(0.263445\pi\)
\(954\) 0 0
\(955\) 2.24259 0.0725685
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −8.11490 −0.262044
\(960\) 0 0
\(961\) 3.85852 0.124468
\(962\) 0 0
\(963\) −9.66227 −0.311362
\(964\) 0 0
\(965\) 9.30699 0.299603
\(966\) 0 0
\(967\) 13.9026 0.447078 0.223539 0.974695i \(-0.428239\pi\)
0.223539 + 0.974695i \(0.428239\pi\)
\(968\) 0 0
\(969\) 7.44316 0.239109
\(970\) 0 0
\(971\) −5.24997 −0.168480 −0.0842398 0.996446i \(-0.526846\pi\)
−0.0842398 + 0.996446i \(0.526846\pi\)
\(972\) 0 0
\(973\) −45.0177 −1.44320
\(974\) 0 0
\(975\) 0.372129 0.0119177
\(976\) 0 0
\(977\) 53.0523 1.69729 0.848646 0.528961i \(-0.177418\pi\)
0.848646 + 0.528961i \(0.177418\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −5.69608 −0.181862
\(982\) 0 0
\(983\) 1.43729 0.0458425 0.0229212 0.999737i \(-0.492703\pi\)
0.0229212 + 0.999737i \(0.492703\pi\)
\(984\) 0 0
\(985\) −14.3935 −0.458615
\(986\) 0 0
\(987\) −48.1975 −1.53414
\(988\) 0 0
\(989\) 5.38872 0.171351
\(990\) 0 0
\(991\) −22.6101 −0.718234 −0.359117 0.933292i \(-0.616922\pi\)
−0.359117 + 0.933292i \(0.616922\pi\)
\(992\) 0 0
\(993\) −14.2662 −0.452724
\(994\) 0 0
\(995\) −3.00915 −0.0953965
\(996\) 0 0
\(997\) −59.0567 −1.87034 −0.935172 0.354194i \(-0.884755\pi\)
−0.935172 + 0.354194i \(0.884755\pi\)
\(998\) 0 0
\(999\) −20.9840 −0.663906
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 4840.2.a.bf.1.5 6
4.3 odd 2 9680.2.a.cx.1.2 6
11.5 even 5 440.2.y.b.201.3 yes 12
11.9 even 5 440.2.y.b.81.3 12
11.10 odd 2 4840.2.a.be.1.5 6
44.27 odd 10 880.2.bo.j.641.1 12
44.31 odd 10 880.2.bo.j.81.1 12
44.43 even 2 9680.2.a.cy.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.b.81.3 12 11.9 even 5
440.2.y.b.201.3 yes 12 11.5 even 5
880.2.bo.j.81.1 12 44.31 odd 10
880.2.bo.j.641.1 12 44.27 odd 10
4840.2.a.be.1.5 6 11.10 odd 2
4840.2.a.bf.1.5 6 1.1 even 1 trivial
9680.2.a.cx.1.2 6 4.3 odd 2
9680.2.a.cy.1.2 6 44.43 even 2