Properties

Label 4840.2.a.bd.1.1
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: 6.6.22733568.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 8x^{4} - 2x^{3} + 16x^{2} + 8x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0.184585\) 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)
 
\(f(q)\) \(=\) \(q-2.96593 q^{3} +1.00000 q^{5} +1.02876 q^{7} +5.79673 q^{9} +O(q^{10})\) \(q-2.96593 q^{3} +1.00000 q^{5} +1.02876 q^{7} +5.79673 q^{9} -0.504296 q^{13} -2.96593 q^{15} -3.93186 q^{17} +2.50430 q^{19} -3.05124 q^{21} -5.73921 q^{23} +1.00000 q^{25} -8.29490 q^{27} +6.92327 q^{29} +4.47755 q^{31} +1.02876 q^{35} +4.78329 q^{37} +1.49570 q^{39} +11.8404 q^{41} -7.10060 q^{43} +5.79673 q^{45} -0.182642 q^{47} -5.94165 q^{49} +11.6616 q^{51} +12.4669 q^{53} -7.42756 q^{57} -10.5494 q^{59} -10.9209 q^{61} +5.96346 q^{63} -0.504296 q^{65} +9.76386 q^{67} +17.0221 q^{69} -13.9992 q^{71} -7.32321 q^{73} -2.96593 q^{75} +10.2743 q^{79} +7.21190 q^{81} +4.33368 q^{83} -3.93186 q^{85} -20.5339 q^{87} +17.2789 q^{89} -0.518800 q^{91} -13.2801 q^{93} +2.50430 q^{95} -7.80167 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{3} + 6 q^{5} + 4 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{3} + 6 q^{5} + 4 q^{7} + 4 q^{9} - 2 q^{15} + 8 q^{17} + 12 q^{19} - 8 q^{21} - 8 q^{23} + 6 q^{25} - 14 q^{27} + 16 q^{29} - 4 q^{31} + 4 q^{35} + 8 q^{37} + 12 q^{39} + 32 q^{41} - 4 q^{43} + 4 q^{45} - 6 q^{47} + 16 q^{49} + 40 q^{51} + 8 q^{53} - 16 q^{57} + 4 q^{59} + 16 q^{61} + 28 q^{63} - 2 q^{67} + 8 q^{69} - 28 q^{71} + 16 q^{73} - 2 q^{75} - 10 q^{81} + 12 q^{83} + 8 q^{85} - 24 q^{87} + 18 q^{89} - 24 q^{91} + 20 q^{93} + 12 q^{95}+O(q^{100}) \) 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
\(3\) −2.96593 −1.71238 −0.856190 0.516661i \(-0.827175\pi\)
−0.856190 + 0.516661i \(0.827175\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 1.02876 0.388836 0.194418 0.980919i \(-0.437718\pi\)
0.194418 + 0.980919i \(0.437718\pi\)
\(8\) 0 0
\(9\) 5.79673 1.93224
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) −0.504296 −0.139866 −0.0699332 0.997552i \(-0.522279\pi\)
−0.0699332 + 0.997552i \(0.522279\pi\)
\(14\) 0 0
\(15\) −2.96593 −0.765799
\(16\) 0 0
\(17\) −3.93186 −0.953615 −0.476808 0.879008i \(-0.658206\pi\)
−0.476808 + 0.879008i \(0.658206\pi\)
\(18\) 0 0
\(19\) 2.50430 0.574525 0.287262 0.957852i \(-0.407255\pi\)
0.287262 + 0.957852i \(0.407255\pi\)
\(20\) 0 0
\(21\) −3.05124 −0.665834
\(22\) 0 0
\(23\) −5.73921 −1.19671 −0.598354 0.801232i \(-0.704178\pi\)
−0.598354 + 0.801232i \(0.704178\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −8.29490 −1.59636
\(28\) 0 0
\(29\) 6.92327 1.28562 0.642809 0.766026i \(-0.277769\pi\)
0.642809 + 0.766026i \(0.277769\pi\)
\(30\) 0 0
\(31\) 4.47755 0.804191 0.402096 0.915598i \(-0.368282\pi\)
0.402096 + 0.915598i \(0.368282\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.02876 0.173893
\(36\) 0 0
\(37\) 4.78329 0.786367 0.393184 0.919460i \(-0.371374\pi\)
0.393184 + 0.919460i \(0.371374\pi\)
\(38\) 0 0
\(39\) 1.49570 0.239504
\(40\) 0 0
\(41\) 11.8404 1.84916 0.924582 0.380983i \(-0.124414\pi\)
0.924582 + 0.380983i \(0.124414\pi\)
\(42\) 0 0
\(43\) −7.10060 −1.08283 −0.541416 0.840755i \(-0.682111\pi\)
−0.541416 + 0.840755i \(0.682111\pi\)
\(44\) 0 0
\(45\) 5.79673 0.864126
\(46\) 0 0
\(47\) −0.182642 −0.0266410 −0.0133205 0.999911i \(-0.504240\pi\)
−0.0133205 + 0.999911i \(0.504240\pi\)
\(48\) 0 0
\(49\) −5.94165 −0.848807
\(50\) 0 0
\(51\) 11.6616 1.63295
\(52\) 0 0
\(53\) 12.4669 1.71246 0.856232 0.516591i \(-0.172799\pi\)
0.856232 + 0.516591i \(0.172799\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −7.42756 −0.983805
\(58\) 0 0
\(59\) −10.5494 −1.37341 −0.686706 0.726935i \(-0.740944\pi\)
−0.686706 + 0.726935i \(0.740944\pi\)
\(60\) 0 0
\(61\) −10.9209 −1.39827 −0.699136 0.714989i \(-0.746432\pi\)
−0.699136 + 0.714989i \(0.746432\pi\)
\(62\) 0 0
\(63\) 5.96346 0.751325
\(64\) 0 0
\(65\) −0.504296 −0.0625502
\(66\) 0 0
\(67\) 9.76386 1.19285 0.596423 0.802671i \(-0.296588\pi\)
0.596423 + 0.802671i \(0.296588\pi\)
\(68\) 0 0
\(69\) 17.0221 2.04922
\(70\) 0 0
\(71\) −13.9992 −1.66140 −0.830698 0.556723i \(-0.812058\pi\)
−0.830698 + 0.556723i \(0.812058\pi\)
\(72\) 0 0
\(73\) −7.32321 −0.857117 −0.428559 0.903514i \(-0.640978\pi\)
−0.428559 + 0.903514i \(0.640978\pi\)
\(74\) 0 0
\(75\) −2.96593 −0.342476
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 10.2743 1.15595 0.577973 0.816056i \(-0.303844\pi\)
0.577973 + 0.816056i \(0.303844\pi\)
\(80\) 0 0
\(81\) 7.21190 0.801322
\(82\) 0 0
\(83\) 4.33368 0.475683 0.237842 0.971304i \(-0.423560\pi\)
0.237842 + 0.971304i \(0.423560\pi\)
\(84\) 0 0
\(85\) −3.93186 −0.426470
\(86\) 0 0
\(87\) −20.5339 −2.20147
\(88\) 0 0
\(89\) 17.2789 1.83156 0.915779 0.401683i \(-0.131575\pi\)
0.915779 + 0.401683i \(0.131575\pi\)
\(90\) 0 0
\(91\) −0.518800 −0.0543851
\(92\) 0 0
\(93\) −13.2801 −1.37708
\(94\) 0 0
\(95\) 2.50430 0.256935
\(96\) 0 0
\(97\) −7.80167 −0.792139 −0.396070 0.918220i \(-0.629626\pi\)
−0.396070 + 0.918220i \(0.629626\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −6.91035 −0.687605 −0.343803 0.939042i \(-0.611715\pi\)
−0.343803 + 0.939042i \(0.611715\pi\)
\(102\) 0 0
\(103\) 1.07395 0.105819 0.0529096 0.998599i \(-0.483150\pi\)
0.0529096 + 0.998599i \(0.483150\pi\)
\(104\) 0 0
\(105\) −3.05124 −0.297770
\(106\) 0 0
\(107\) −15.6135 −1.50942 −0.754708 0.656061i \(-0.772222\pi\)
−0.754708 + 0.656061i \(0.772222\pi\)
\(108\) 0 0
\(109\) 10.5632 1.01177 0.505886 0.862600i \(-0.331166\pi\)
0.505886 + 0.862600i \(0.331166\pi\)
\(110\) 0 0
\(111\) −14.1869 −1.34656
\(112\) 0 0
\(113\) 1.72975 0.162721 0.0813605 0.996685i \(-0.474073\pi\)
0.0813605 + 0.996685i \(0.474073\pi\)
\(114\) 0 0
\(115\) −5.73921 −0.535184
\(116\) 0 0
\(117\) −2.92327 −0.270256
\(118\) 0 0
\(119\) −4.04495 −0.370800
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) −35.1179 −3.16647
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −9.80700 −0.870231 −0.435115 0.900375i \(-0.643292\pi\)
−0.435115 + 0.900375i \(0.643292\pi\)
\(128\) 0 0
\(129\) 21.0599 1.85422
\(130\) 0 0
\(131\) −3.79284 −0.331382 −0.165691 0.986178i \(-0.552985\pi\)
−0.165691 + 0.986178i \(0.552985\pi\)
\(132\) 0 0
\(133\) 2.57633 0.223396
\(134\) 0 0
\(135\) −8.29490 −0.713912
\(136\) 0 0
\(137\) −4.58496 −0.391719 −0.195860 0.980632i \(-0.562750\pi\)
−0.195860 + 0.980632i \(0.562750\pi\)
\(138\) 0 0
\(139\) 9.11697 0.773291 0.386646 0.922228i \(-0.373634\pi\)
0.386646 + 0.922228i \(0.373634\pi\)
\(140\) 0 0
\(141\) 0.541702 0.0456196
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 6.92327 0.574946
\(146\) 0 0
\(147\) 17.6225 1.45348
\(148\) 0 0
\(149\) 18.1001 1.48281 0.741407 0.671055i \(-0.234159\pi\)
0.741407 + 0.671055i \(0.234159\pi\)
\(150\) 0 0
\(151\) −10.6982 −0.870606 −0.435303 0.900284i \(-0.643359\pi\)
−0.435303 + 0.900284i \(0.643359\pi\)
\(152\) 0 0
\(153\) −22.7919 −1.84262
\(154\) 0 0
\(155\) 4.47755 0.359645
\(156\) 0 0
\(157\) −3.32321 −0.265221 −0.132611 0.991168i \(-0.542336\pi\)
−0.132611 + 0.991168i \(0.542336\pi\)
\(158\) 0 0
\(159\) −36.9760 −2.93239
\(160\) 0 0
\(161\) −5.90428 −0.465322
\(162\) 0 0
\(163\) −10.0868 −0.790058 −0.395029 0.918669i \(-0.629265\pi\)
−0.395029 + 0.918669i \(0.629265\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 7.39010 0.571863 0.285931 0.958250i \(-0.407697\pi\)
0.285931 + 0.958250i \(0.407697\pi\)
\(168\) 0 0
\(169\) −12.7457 −0.980437
\(170\) 0 0
\(171\) 14.5167 1.11012
\(172\) 0 0
\(173\) 18.5390 1.40949 0.704745 0.709460i \(-0.251061\pi\)
0.704745 + 0.709460i \(0.251061\pi\)
\(174\) 0 0
\(175\) 1.02876 0.0777671
\(176\) 0 0
\(177\) 31.2887 2.35180
\(178\) 0 0
\(179\) 26.2551 1.96240 0.981199 0.192997i \(-0.0618206\pi\)
0.981199 + 0.192997i \(0.0618206\pi\)
\(180\) 0 0
\(181\) 11.5551 0.858887 0.429444 0.903094i \(-0.358710\pi\)
0.429444 + 0.903094i \(0.358710\pi\)
\(182\) 0 0
\(183\) 32.3905 2.39437
\(184\) 0 0
\(185\) 4.78329 0.351674
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −8.53349 −0.620720
\(190\) 0 0
\(191\) 10.4804 0.758334 0.379167 0.925328i \(-0.376211\pi\)
0.379167 + 0.925328i \(0.376211\pi\)
\(192\) 0 0
\(193\) 19.1446 1.37806 0.689030 0.724733i \(-0.258037\pi\)
0.689030 + 0.724733i \(0.258037\pi\)
\(194\) 0 0
\(195\) 1.49570 0.107110
\(196\) 0 0
\(197\) 0.935844 0.0666761 0.0333381 0.999444i \(-0.489386\pi\)
0.0333381 + 0.999444i \(0.489386\pi\)
\(198\) 0 0
\(199\) −26.4159 −1.87257 −0.936286 0.351238i \(-0.885761\pi\)
−0.936286 + 0.351238i \(0.885761\pi\)
\(200\) 0 0
\(201\) −28.9589 −2.04260
\(202\) 0 0
\(203\) 7.12240 0.499894
\(204\) 0 0
\(205\) 11.8404 0.826971
\(206\) 0 0
\(207\) −33.2686 −2.31233
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 10.1888 0.701423 0.350712 0.936483i \(-0.385940\pi\)
0.350712 + 0.936483i \(0.385940\pi\)
\(212\) 0 0
\(213\) 41.5205 2.84494
\(214\) 0 0
\(215\) −7.10060 −0.484257
\(216\) 0 0
\(217\) 4.60633 0.312698
\(218\) 0 0
\(219\) 21.7201 1.46771
\(220\) 0 0
\(221\) 1.98282 0.133379
\(222\) 0 0
\(223\) −3.62471 −0.242728 −0.121364 0.992608i \(-0.538727\pi\)
−0.121364 + 0.992608i \(0.538727\pi\)
\(224\) 0 0
\(225\) 5.79673 0.386449
\(226\) 0 0
\(227\) 10.6472 0.706676 0.353338 0.935496i \(-0.385046\pi\)
0.353338 + 0.935496i \(0.385046\pi\)
\(228\) 0 0
\(229\) −2.43755 −0.161078 −0.0805388 0.996751i \(-0.525664\pi\)
−0.0805388 + 0.996751i \(0.525664\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 23.6962 1.55239 0.776193 0.630495i \(-0.217148\pi\)
0.776193 + 0.630495i \(0.217148\pi\)
\(234\) 0 0
\(235\) −0.182642 −0.0119142
\(236\) 0 0
\(237\) −30.4728 −1.97942
\(238\) 0 0
\(239\) 3.50301 0.226591 0.113295 0.993561i \(-0.463859\pi\)
0.113295 + 0.993561i \(0.463859\pi\)
\(240\) 0 0
\(241\) −7.01997 −0.452196 −0.226098 0.974105i \(-0.572597\pi\)
−0.226098 + 0.974105i \(0.572597\pi\)
\(242\) 0 0
\(243\) 3.49474 0.224188
\(244\) 0 0
\(245\) −5.94165 −0.379598
\(246\) 0 0
\(247\) −1.26291 −0.0803567
\(248\) 0 0
\(249\) −12.8534 −0.814550
\(250\) 0 0
\(251\) 21.8188 1.37719 0.688596 0.725145i \(-0.258228\pi\)
0.688596 + 0.725145i \(0.258228\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 11.6616 0.730278
\(256\) 0 0
\(257\) 1.66770 0.104029 0.0520143 0.998646i \(-0.483436\pi\)
0.0520143 + 0.998646i \(0.483436\pi\)
\(258\) 0 0
\(259\) 4.92087 0.305768
\(260\) 0 0
\(261\) 40.1323 2.48413
\(262\) 0 0
\(263\) 23.9489 1.47675 0.738376 0.674389i \(-0.235593\pi\)
0.738376 + 0.674389i \(0.235593\pi\)
\(264\) 0 0
\(265\) 12.4669 0.765837
\(266\) 0 0
\(267\) −51.2479 −3.13632
\(268\) 0 0
\(269\) 12.1425 0.740340 0.370170 0.928964i \(-0.379300\pi\)
0.370170 + 0.928964i \(0.379300\pi\)
\(270\) 0 0
\(271\) 18.3731 1.11608 0.558042 0.829813i \(-0.311553\pi\)
0.558042 + 0.829813i \(0.311553\pi\)
\(272\) 0 0
\(273\) 1.53872 0.0931279
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 19.3747 1.16411 0.582057 0.813148i \(-0.302248\pi\)
0.582057 + 0.813148i \(0.302248\pi\)
\(278\) 0 0
\(279\) 25.9551 1.55389
\(280\) 0 0
\(281\) 5.28855 0.315488 0.157744 0.987480i \(-0.449578\pi\)
0.157744 + 0.987480i \(0.449578\pi\)
\(282\) 0 0
\(283\) 13.3392 0.792936 0.396468 0.918049i \(-0.370236\pi\)
0.396468 + 0.918049i \(0.370236\pi\)
\(284\) 0 0
\(285\) −7.42756 −0.439971
\(286\) 0 0
\(287\) 12.1810 0.719021
\(288\) 0 0
\(289\) −1.54050 −0.0906178
\(290\) 0 0
\(291\) 23.1392 1.35644
\(292\) 0 0
\(293\) −9.70482 −0.566961 −0.283481 0.958978i \(-0.591489\pi\)
−0.283481 + 0.958978i \(0.591489\pi\)
\(294\) 0 0
\(295\) −10.5494 −0.614209
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 2.89426 0.167379
\(300\) 0 0
\(301\) −7.30483 −0.421043
\(302\) 0 0
\(303\) 20.4956 1.17744
\(304\) 0 0
\(305\) −10.9209 −0.625326
\(306\) 0 0
\(307\) 13.7220 0.783156 0.391578 0.920145i \(-0.371929\pi\)
0.391578 + 0.920145i \(0.371929\pi\)
\(308\) 0 0
\(309\) −3.18525 −0.181203
\(310\) 0 0
\(311\) −0.293496 −0.0166426 −0.00832131 0.999965i \(-0.502649\pi\)
−0.00832131 + 0.999965i \(0.502649\pi\)
\(312\) 0 0
\(313\) 1.45181 0.0820611 0.0410305 0.999158i \(-0.486936\pi\)
0.0410305 + 0.999158i \(0.486936\pi\)
\(314\) 0 0
\(315\) 5.96346 0.336003
\(316\) 0 0
\(317\) −27.4596 −1.54228 −0.771142 0.636664i \(-0.780314\pi\)
−0.771142 + 0.636664i \(0.780314\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 46.3086 2.58469
\(322\) 0 0
\(323\) −9.84653 −0.547876
\(324\) 0 0
\(325\) −0.504296 −0.0279733
\(326\) 0 0
\(327\) −31.3297 −1.73254
\(328\) 0 0
\(329\) −0.187895 −0.0103590
\(330\) 0 0
\(331\) 16.6968 0.917741 0.458871 0.888503i \(-0.348254\pi\)
0.458871 + 0.888503i \(0.348254\pi\)
\(332\) 0 0
\(333\) 27.7274 1.51945
\(334\) 0 0
\(335\) 9.76386 0.533457
\(336\) 0 0
\(337\) −9.17166 −0.499612 −0.249806 0.968296i \(-0.580367\pi\)
−0.249806 + 0.968296i \(0.580367\pi\)
\(338\) 0 0
\(339\) −5.13031 −0.278640
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −13.3139 −0.718882
\(344\) 0 0
\(345\) 17.0221 0.916438
\(346\) 0 0
\(347\) 23.7060 1.27260 0.636301 0.771441i \(-0.280463\pi\)
0.636301 + 0.771441i \(0.280463\pi\)
\(348\) 0 0
\(349\) 13.3834 0.716396 0.358198 0.933646i \(-0.383391\pi\)
0.358198 + 0.933646i \(0.383391\pi\)
\(350\) 0 0
\(351\) 4.18308 0.223276
\(352\) 0 0
\(353\) −22.7286 −1.20972 −0.604861 0.796331i \(-0.706771\pi\)
−0.604861 + 0.796331i \(0.706771\pi\)
\(354\) 0 0
\(355\) −13.9992 −0.742999
\(356\) 0 0
\(357\) 11.9970 0.634950
\(358\) 0 0
\(359\) 5.39337 0.284651 0.142325 0.989820i \(-0.454542\pi\)
0.142325 + 0.989820i \(0.454542\pi\)
\(360\) 0 0
\(361\) −12.7285 −0.669921
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −7.32321 −0.383314
\(366\) 0 0
\(367\) 8.58151 0.447951 0.223976 0.974595i \(-0.428096\pi\)
0.223976 + 0.974595i \(0.428096\pi\)
\(368\) 0 0
\(369\) 68.6358 3.57304
\(370\) 0 0
\(371\) 12.8255 0.665867
\(372\) 0 0
\(373\) 33.0945 1.71357 0.856784 0.515676i \(-0.172459\pi\)
0.856784 + 0.515676i \(0.172459\pi\)
\(374\) 0 0
\(375\) −2.96593 −0.153160
\(376\) 0 0
\(377\) −3.49137 −0.179815
\(378\) 0 0
\(379\) −7.28378 −0.374143 −0.187071 0.982346i \(-0.559900\pi\)
−0.187071 + 0.982346i \(0.559900\pi\)
\(380\) 0 0
\(381\) 29.0869 1.49017
\(382\) 0 0
\(383\) −8.90242 −0.454892 −0.227446 0.973791i \(-0.573038\pi\)
−0.227446 + 0.973791i \(0.573038\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −41.1603 −2.09229
\(388\) 0 0
\(389\) −3.12005 −0.158193 −0.0790963 0.996867i \(-0.525203\pi\)
−0.0790963 + 0.996867i \(0.525203\pi\)
\(390\) 0 0
\(391\) 22.5657 1.14120
\(392\) 0 0
\(393\) 11.2493 0.567452
\(394\) 0 0
\(395\) 10.2743 0.516955
\(396\) 0 0
\(397\) 28.9534 1.45313 0.726566 0.687097i \(-0.241115\pi\)
0.726566 + 0.687097i \(0.241115\pi\)
\(398\) 0 0
\(399\) −7.64120 −0.382538
\(400\) 0 0
\(401\) −22.7102 −1.13410 −0.567048 0.823685i \(-0.691915\pi\)
−0.567048 + 0.823685i \(0.691915\pi\)
\(402\) 0 0
\(403\) −2.25801 −0.112479
\(404\) 0 0
\(405\) 7.21190 0.358362
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −27.8432 −1.37676 −0.688379 0.725351i \(-0.741677\pi\)
−0.688379 + 0.725351i \(0.741677\pi\)
\(410\) 0 0
\(411\) 13.5986 0.670772
\(412\) 0 0
\(413\) −10.8528 −0.534032
\(414\) 0 0
\(415\) 4.33368 0.212732
\(416\) 0 0
\(417\) −27.0403 −1.32417
\(418\) 0 0
\(419\) −34.6969 −1.69506 −0.847528 0.530751i \(-0.821910\pi\)
−0.847528 + 0.530751i \(0.821910\pi\)
\(420\) 0 0
\(421\) 6.16073 0.300256 0.150128 0.988667i \(-0.452031\pi\)
0.150128 + 0.988667i \(0.452031\pi\)
\(422\) 0 0
\(423\) −1.05872 −0.0514770
\(424\) 0 0
\(425\) −3.93186 −0.190723
\(426\) 0 0
\(427\) −11.2350 −0.543698
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 25.9212 1.24858 0.624291 0.781192i \(-0.285388\pi\)
0.624291 + 0.781192i \(0.285388\pi\)
\(432\) 0 0
\(433\) −16.6393 −0.799634 −0.399817 0.916595i \(-0.630926\pi\)
−0.399817 + 0.916595i \(0.630926\pi\)
\(434\) 0 0
\(435\) −20.5339 −0.984526
\(436\) 0 0
\(437\) −14.3727 −0.687538
\(438\) 0 0
\(439\) 22.3385 1.06616 0.533079 0.846066i \(-0.321035\pi\)
0.533079 + 0.846066i \(0.321035\pi\)
\(440\) 0 0
\(441\) −34.4421 −1.64010
\(442\) 0 0
\(443\) 18.3803 0.873275 0.436637 0.899638i \(-0.356169\pi\)
0.436637 + 0.899638i \(0.356169\pi\)
\(444\) 0 0
\(445\) 17.2789 0.819098
\(446\) 0 0
\(447\) −53.6835 −2.53914
\(448\) 0 0
\(449\) 13.8228 0.652338 0.326169 0.945311i \(-0.394242\pi\)
0.326169 + 0.945311i \(0.394242\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 31.7301 1.49081
\(454\) 0 0
\(455\) −0.518800 −0.0243217
\(456\) 0 0
\(457\) −15.7689 −0.737638 −0.368819 0.929501i \(-0.620238\pi\)
−0.368819 + 0.929501i \(0.620238\pi\)
\(458\) 0 0
\(459\) 32.6144 1.52231
\(460\) 0 0
\(461\) 30.8758 1.43803 0.719014 0.694996i \(-0.244594\pi\)
0.719014 + 0.694996i \(0.244594\pi\)
\(462\) 0 0
\(463\) 19.1958 0.892106 0.446053 0.895007i \(-0.352829\pi\)
0.446053 + 0.895007i \(0.352829\pi\)
\(464\) 0 0
\(465\) −13.2801 −0.615849
\(466\) 0 0
\(467\) −38.8289 −1.79679 −0.898393 0.439193i \(-0.855264\pi\)
−0.898393 + 0.439193i \(0.855264\pi\)
\(468\) 0 0
\(469\) 10.0447 0.463821
\(470\) 0 0
\(471\) 9.85641 0.454159
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 2.50430 0.114905
\(476\) 0 0
\(477\) 72.2674 3.30890
\(478\) 0 0
\(479\) 9.12135 0.416765 0.208383 0.978047i \(-0.433180\pi\)
0.208383 + 0.978047i \(0.433180\pi\)
\(480\) 0 0
\(481\) −2.41219 −0.109986
\(482\) 0 0
\(483\) 17.5117 0.796809
\(484\) 0 0
\(485\) −7.80167 −0.354256
\(486\) 0 0
\(487\) 15.8103 0.716431 0.358216 0.933639i \(-0.383385\pi\)
0.358216 + 0.933639i \(0.383385\pi\)
\(488\) 0 0
\(489\) 29.9167 1.35288
\(490\) 0 0
\(491\) 8.09594 0.365365 0.182682 0.983172i \(-0.441522\pi\)
0.182682 + 0.983172i \(0.441522\pi\)
\(492\) 0 0
\(493\) −27.2213 −1.22599
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −14.4018 −0.646010
\(498\) 0 0
\(499\) 38.2856 1.71390 0.856948 0.515402i \(-0.172358\pi\)
0.856948 + 0.515402i \(0.172358\pi\)
\(500\) 0 0
\(501\) −21.9185 −0.979246
\(502\) 0 0
\(503\) −26.1242 −1.16482 −0.582410 0.812895i \(-0.697890\pi\)
−0.582410 + 0.812895i \(0.697890\pi\)
\(504\) 0 0
\(505\) −6.91035 −0.307506
\(506\) 0 0
\(507\) 37.8028 1.67888
\(508\) 0 0
\(509\) 38.3357 1.69920 0.849600 0.527427i \(-0.176843\pi\)
0.849600 + 0.527427i \(0.176843\pi\)
\(510\) 0 0
\(511\) −7.53385 −0.333278
\(512\) 0 0
\(513\) −20.7729 −0.917146
\(514\) 0 0
\(515\) 1.07395 0.0473238
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −54.9852 −2.41358
\(520\) 0 0
\(521\) −22.0519 −0.966111 −0.483055 0.875590i \(-0.660473\pi\)
−0.483055 + 0.875590i \(0.660473\pi\)
\(522\) 0 0
\(523\) 11.3086 0.494492 0.247246 0.968953i \(-0.420474\pi\)
0.247246 + 0.968953i \(0.420474\pi\)
\(524\) 0 0
\(525\) −3.05124 −0.133167
\(526\) 0 0
\(527\) −17.6051 −0.766889
\(528\) 0 0
\(529\) 9.93849 0.432108
\(530\) 0 0
\(531\) −61.1519 −2.65377
\(532\) 0 0
\(533\) −5.97107 −0.258636
\(534\) 0 0
\(535\) −15.6135 −0.675032
\(536\) 0 0
\(537\) −77.8708 −3.36037
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −13.9019 −0.597688 −0.298844 0.954302i \(-0.596601\pi\)
−0.298844 + 0.954302i \(0.596601\pi\)
\(542\) 0 0
\(543\) −34.2717 −1.47074
\(544\) 0 0
\(545\) 10.5632 0.452478
\(546\) 0 0
\(547\) −24.7035 −1.05625 −0.528123 0.849168i \(-0.677104\pi\)
−0.528123 + 0.849168i \(0.677104\pi\)
\(548\) 0 0
\(549\) −63.3053 −2.70180
\(550\) 0 0
\(551\) 17.3379 0.738620
\(552\) 0 0
\(553\) 10.5698 0.449473
\(554\) 0 0
\(555\) −14.1869 −0.602200
\(556\) 0 0
\(557\) 17.4258 0.738355 0.369177 0.929359i \(-0.379639\pi\)
0.369177 + 0.929359i \(0.379639\pi\)
\(558\) 0 0
\(559\) 3.58080 0.151452
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 6.29128 0.265146 0.132573 0.991173i \(-0.457676\pi\)
0.132573 + 0.991173i \(0.457676\pi\)
\(564\) 0 0
\(565\) 1.72975 0.0727711
\(566\) 0 0
\(567\) 7.41933 0.311583
\(568\) 0 0
\(569\) 23.8241 0.998759 0.499380 0.866383i \(-0.333561\pi\)
0.499380 + 0.866383i \(0.333561\pi\)
\(570\) 0 0
\(571\) 32.9400 1.37850 0.689248 0.724525i \(-0.257941\pi\)
0.689248 + 0.724525i \(0.257941\pi\)
\(572\) 0 0
\(573\) −31.0841 −1.29856
\(574\) 0 0
\(575\) −5.73921 −0.239341
\(576\) 0 0
\(577\) 6.98162 0.290649 0.145324 0.989384i \(-0.453577\pi\)
0.145324 + 0.989384i \(0.453577\pi\)
\(578\) 0 0
\(579\) −56.7816 −2.35976
\(580\) 0 0
\(581\) 4.45833 0.184963
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −2.92327 −0.120862
\(586\) 0 0
\(587\) −19.4035 −0.800867 −0.400433 0.916326i \(-0.631140\pi\)
−0.400433 + 0.916326i \(0.631140\pi\)
\(588\) 0 0
\(589\) 11.2131 0.462028
\(590\) 0 0
\(591\) −2.77565 −0.114175
\(592\) 0 0
\(593\) 1.22305 0.0502248 0.0251124 0.999685i \(-0.492006\pi\)
0.0251124 + 0.999685i \(0.492006\pi\)
\(594\) 0 0
\(595\) −4.04495 −0.165827
\(596\) 0 0
\(597\) 78.3476 3.20655
\(598\) 0 0
\(599\) −18.5791 −0.759122 −0.379561 0.925167i \(-0.623925\pi\)
−0.379561 + 0.925167i \(0.623925\pi\)
\(600\) 0 0
\(601\) 10.7275 0.437585 0.218793 0.975771i \(-0.429788\pi\)
0.218793 + 0.975771i \(0.429788\pi\)
\(602\) 0 0
\(603\) 56.5985 2.30487
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 20.8219 0.845133 0.422567 0.906332i \(-0.361129\pi\)
0.422567 + 0.906332i \(0.361129\pi\)
\(608\) 0 0
\(609\) −21.1245 −0.856009
\(610\) 0 0
\(611\) 0.0921054 0.00372619
\(612\) 0 0
\(613\) −15.4592 −0.624390 −0.312195 0.950018i \(-0.601064\pi\)
−0.312195 + 0.950018i \(0.601064\pi\)
\(614\) 0 0
\(615\) −35.1179 −1.41609
\(616\) 0 0
\(617\) 7.32523 0.294903 0.147451 0.989069i \(-0.452893\pi\)
0.147451 + 0.989069i \(0.452893\pi\)
\(618\) 0 0
\(619\) 18.8171 0.756322 0.378161 0.925740i \(-0.376557\pi\)
0.378161 + 0.925740i \(0.376557\pi\)
\(620\) 0 0
\(621\) 47.6062 1.91037
\(622\) 0 0
\(623\) 17.7759 0.712175
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −18.8072 −0.749892
\(630\) 0 0
\(631\) −40.5028 −1.61239 −0.806196 0.591648i \(-0.798477\pi\)
−0.806196 + 0.591648i \(0.798477\pi\)
\(632\) 0 0
\(633\) −30.2191 −1.20110
\(634\) 0 0
\(635\) −9.80700 −0.389179
\(636\) 0 0
\(637\) 2.99635 0.118720
\(638\) 0 0
\(639\) −81.1494 −3.21022
\(640\) 0 0
\(641\) 21.5110 0.849632 0.424816 0.905280i \(-0.360339\pi\)
0.424816 + 0.905280i \(0.360339\pi\)
\(642\) 0 0
\(643\) −7.91518 −0.312144 −0.156072 0.987746i \(-0.549883\pi\)
−0.156072 + 0.987746i \(0.549883\pi\)
\(644\) 0 0
\(645\) 21.0599 0.829231
\(646\) 0 0
\(647\) 19.1958 0.754666 0.377333 0.926078i \(-0.376841\pi\)
0.377333 + 0.926078i \(0.376841\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −13.6620 −0.535458
\(652\) 0 0
\(653\) −38.0894 −1.49055 −0.745277 0.666755i \(-0.767683\pi\)
−0.745277 + 0.666755i \(0.767683\pi\)
\(654\) 0 0
\(655\) −3.79284 −0.148199
\(656\) 0 0
\(657\) −42.4507 −1.65616
\(658\) 0 0
\(659\) 2.64662 0.103098 0.0515488 0.998670i \(-0.483584\pi\)
0.0515488 + 0.998670i \(0.483584\pi\)
\(660\) 0 0
\(661\) −7.55682 −0.293926 −0.146963 0.989142i \(-0.546950\pi\)
−0.146963 + 0.989142i \(0.546950\pi\)
\(662\) 0 0
\(663\) −5.88090 −0.228395
\(664\) 0 0
\(665\) 2.57633 0.0999056
\(666\) 0 0
\(667\) −39.7340 −1.53851
\(668\) 0 0
\(669\) 10.7506 0.415643
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 37.5542 1.44761 0.723803 0.690006i \(-0.242392\pi\)
0.723803 + 0.690006i \(0.242392\pi\)
\(674\) 0 0
\(675\) −8.29490 −0.319271
\(676\) 0 0
\(677\) −27.3287 −1.05033 −0.525163 0.851002i \(-0.675996\pi\)
−0.525163 + 0.851002i \(0.675996\pi\)
\(678\) 0 0
\(679\) −8.02606 −0.308012
\(680\) 0 0
\(681\) −31.5787 −1.21010
\(682\) 0 0
\(683\) 29.6633 1.13503 0.567516 0.823362i \(-0.307904\pi\)
0.567516 + 0.823362i \(0.307904\pi\)
\(684\) 0 0
\(685\) −4.58496 −0.175182
\(686\) 0 0
\(687\) 7.22959 0.275826
\(688\) 0 0
\(689\) −6.28702 −0.239516
\(690\) 0 0
\(691\) 38.5532 1.46663 0.733317 0.679887i \(-0.237971\pi\)
0.733317 + 0.679887i \(0.237971\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 9.11697 0.345826
\(696\) 0 0
\(697\) −46.5549 −1.76339
\(698\) 0 0
\(699\) −70.2811 −2.65828
\(700\) 0 0
\(701\) 27.6081 1.04274 0.521372 0.853330i \(-0.325420\pi\)
0.521372 + 0.853330i \(0.325420\pi\)
\(702\) 0 0
\(703\) 11.9788 0.451788
\(704\) 0 0
\(705\) 0.541702 0.0204017
\(706\) 0 0
\(707\) −7.10910 −0.267365
\(708\) 0 0
\(709\) 7.98510 0.299887 0.149943 0.988695i \(-0.452091\pi\)
0.149943 + 0.988695i \(0.452091\pi\)
\(710\) 0 0
\(711\) 59.5572 2.23357
\(712\) 0 0
\(713\) −25.6976 −0.962381
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −10.3897 −0.388010
\(718\) 0 0
\(719\) 7.46162 0.278271 0.139136 0.990273i \(-0.455568\pi\)
0.139136 + 0.990273i \(0.455568\pi\)
\(720\) 0 0
\(721\) 1.10484 0.0411463
\(722\) 0 0
\(723\) 20.8207 0.774331
\(724\) 0 0
\(725\) 6.92327 0.257124
\(726\) 0 0
\(727\) 30.2439 1.12168 0.560842 0.827923i \(-0.310478\pi\)
0.560842 + 0.827923i \(0.310478\pi\)
\(728\) 0 0
\(729\) −32.0008 −1.18522
\(730\) 0 0
\(731\) 27.9185 1.03260
\(732\) 0 0
\(733\) 16.0298 0.592074 0.296037 0.955176i \(-0.404335\pi\)
0.296037 + 0.955176i \(0.404335\pi\)
\(734\) 0 0
\(735\) 17.6225 0.650016
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 52.7498 1.94043 0.970217 0.242238i \(-0.0778814\pi\)
0.970217 + 0.242238i \(0.0778814\pi\)
\(740\) 0 0
\(741\) 3.74569 0.137601
\(742\) 0 0
\(743\) 52.3073 1.91897 0.959484 0.281762i \(-0.0909190\pi\)
0.959484 + 0.281762i \(0.0909190\pi\)
\(744\) 0 0
\(745\) 18.1001 0.663135
\(746\) 0 0
\(747\) 25.1212 0.919136
\(748\) 0 0
\(749\) −16.0626 −0.586915
\(750\) 0 0
\(751\) −11.3042 −0.412497 −0.206249 0.978500i \(-0.566126\pi\)
−0.206249 + 0.978500i \(0.566126\pi\)
\(752\) 0 0
\(753\) −64.7130 −2.35827
\(754\) 0 0
\(755\) −10.6982 −0.389347
\(756\) 0 0
\(757\) 26.3105 0.956271 0.478135 0.878286i \(-0.341313\pi\)
0.478135 + 0.878286i \(0.341313\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 45.5746 1.65208 0.826039 0.563612i \(-0.190589\pi\)
0.826039 + 0.563612i \(0.190589\pi\)
\(762\) 0 0
\(763\) 10.8670 0.393413
\(764\) 0 0
\(765\) −22.7919 −0.824043
\(766\) 0 0
\(767\) 5.32001 0.192094
\(768\) 0 0
\(769\) −52.2177 −1.88302 −0.941509 0.336987i \(-0.890592\pi\)
−0.941509 + 0.336987i \(0.890592\pi\)
\(770\) 0 0
\(771\) −4.94629 −0.178136
\(772\) 0 0
\(773\) 19.5400 0.702807 0.351403 0.936224i \(-0.385705\pi\)
0.351403 + 0.936224i \(0.385705\pi\)
\(774\) 0 0
\(775\) 4.47755 0.160838
\(776\) 0 0
\(777\) −14.5949 −0.523590
\(778\) 0 0
\(779\) 29.6519 1.06239
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −57.4278 −2.05230
\(784\) 0 0
\(785\) −3.32321 −0.118610
\(786\) 0 0
\(787\) −30.8849 −1.10093 −0.550464 0.834859i \(-0.685549\pi\)
−0.550464 + 0.834859i \(0.685549\pi\)
\(788\) 0 0
\(789\) −71.0307 −2.52876
\(790\) 0 0
\(791\) 1.77950 0.0632718
\(792\) 0 0
\(793\) 5.50734 0.195571
\(794\) 0 0
\(795\) −36.9760 −1.31140
\(796\) 0 0
\(797\) −23.6837 −0.838920 −0.419460 0.907774i \(-0.637781\pi\)
−0.419460 + 0.907774i \(0.637781\pi\)
\(798\) 0 0
\(799\) 0.718121 0.0254053
\(800\) 0 0
\(801\) 100.161 3.53902
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −5.90428 −0.208099
\(806\) 0 0
\(807\) −36.0137 −1.26774
\(808\) 0 0
\(809\) 3.57949 0.125848 0.0629240 0.998018i \(-0.479957\pi\)
0.0629240 + 0.998018i \(0.479957\pi\)
\(810\) 0 0
\(811\) −22.1256 −0.776933 −0.388467 0.921463i \(-0.626995\pi\)
−0.388467 + 0.921463i \(0.626995\pi\)
\(812\) 0 0
\(813\) −54.4932 −1.91116
\(814\) 0 0
\(815\) −10.0868 −0.353325
\(816\) 0 0
\(817\) −17.7820 −0.622113
\(818\) 0 0
\(819\) −3.00735 −0.105085
\(820\) 0 0
\(821\) 40.1154 1.40004 0.700019 0.714124i \(-0.253175\pi\)
0.700019 + 0.714124i \(0.253175\pi\)
\(822\) 0 0
\(823\) 25.9856 0.905801 0.452900 0.891561i \(-0.350389\pi\)
0.452900 + 0.891561i \(0.350389\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −32.5217 −1.13089 −0.565445 0.824786i \(-0.691296\pi\)
−0.565445 + 0.824786i \(0.691296\pi\)
\(828\) 0 0
\(829\) −4.05382 −0.140795 −0.0703975 0.997519i \(-0.522427\pi\)
−0.0703975 + 0.997519i \(0.522427\pi\)
\(830\) 0 0
\(831\) −57.4640 −1.99341
\(832\) 0 0
\(833\) 23.3617 0.809435
\(834\) 0 0
\(835\) 7.39010 0.255745
\(836\) 0 0
\(837\) −37.1408 −1.28377
\(838\) 0 0
\(839\) 14.1939 0.490029 0.245015 0.969519i \(-0.421207\pi\)
0.245015 + 0.969519i \(0.421207\pi\)
\(840\) 0 0
\(841\) 18.9316 0.652814
\(842\) 0 0
\(843\) −15.6855 −0.540236
\(844\) 0 0
\(845\) −12.7457 −0.438465
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −39.5633 −1.35781
\(850\) 0 0
\(851\) −27.4523 −0.941052
\(852\) 0 0
\(853\) −12.0363 −0.412115 −0.206057 0.978540i \(-0.566063\pi\)
−0.206057 + 0.978540i \(0.566063\pi\)
\(854\) 0 0
\(855\) 14.5167 0.496462
\(856\) 0 0
\(857\) 12.4433 0.425054 0.212527 0.977155i \(-0.431831\pi\)
0.212527 + 0.977155i \(0.431831\pi\)
\(858\) 0 0
\(859\) 46.4462 1.58473 0.792363 0.610050i \(-0.208851\pi\)
0.792363 + 0.610050i \(0.208851\pi\)
\(860\) 0 0
\(861\) −36.1279 −1.23124
\(862\) 0 0
\(863\) 11.1358 0.379066 0.189533 0.981874i \(-0.439303\pi\)
0.189533 + 0.981874i \(0.439303\pi\)
\(864\) 0 0
\(865\) 18.5390 0.630343
\(866\) 0 0
\(867\) 4.56902 0.155172
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −4.92387 −0.166839
\(872\) 0 0
\(873\) −45.2242 −1.53061
\(874\) 0 0
\(875\) 1.02876 0.0347785
\(876\) 0 0
\(877\) −57.3312 −1.93594 −0.967969 0.251071i \(-0.919217\pi\)
−0.967969 + 0.251071i \(0.919217\pi\)
\(878\) 0 0
\(879\) 28.7838 0.970853
\(880\) 0 0
\(881\) −22.0068 −0.741428 −0.370714 0.928747i \(-0.620887\pi\)
−0.370714 + 0.928747i \(0.620887\pi\)
\(882\) 0 0
\(883\) −4.31647 −0.145261 −0.0726304 0.997359i \(-0.523139\pi\)
−0.0726304 + 0.997359i \(0.523139\pi\)
\(884\) 0 0
\(885\) 31.2887 1.05176
\(886\) 0 0
\(887\) 19.2472 0.646258 0.323129 0.946355i \(-0.395265\pi\)
0.323129 + 0.946355i \(0.395265\pi\)
\(888\) 0 0
\(889\) −10.0891 −0.338377
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −0.457389 −0.0153059
\(894\) 0 0
\(895\) 26.2551 0.877611
\(896\) 0 0
\(897\) −8.58416 −0.286617
\(898\) 0 0
\(899\) 30.9992 1.03388
\(900\) 0 0
\(901\) −49.0182 −1.63303
\(902\) 0 0
\(903\) 21.6656 0.720986
\(904\) 0 0
\(905\) 11.5551 0.384106
\(906\) 0 0
\(907\) −20.5740 −0.683148 −0.341574 0.939855i \(-0.610960\pi\)
−0.341574 + 0.939855i \(0.610960\pi\)
\(908\) 0 0
\(909\) −40.0574 −1.32862
\(910\) 0 0
\(911\) 26.3198 0.872014 0.436007 0.899943i \(-0.356392\pi\)
0.436007 + 0.899943i \(0.356392\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 32.3905 1.07080
\(916\) 0 0
\(917\) −3.90194 −0.128853
\(918\) 0 0
\(919\) −38.9505 −1.28486 −0.642429 0.766345i \(-0.722073\pi\)
−0.642429 + 0.766345i \(0.722073\pi\)
\(920\) 0 0
\(921\) −40.6985 −1.34106
\(922\) 0 0
\(923\) 7.05972 0.232374
\(924\) 0 0
\(925\) 4.78329 0.157273
\(926\) 0 0
\(927\) 6.22538 0.204468
\(928\) 0 0
\(929\) −3.42694 −0.112434 −0.0562171 0.998419i \(-0.517904\pi\)
−0.0562171 + 0.998419i \(0.517904\pi\)
\(930\) 0 0
\(931\) −14.8796 −0.487661
\(932\) 0 0
\(933\) 0.870488 0.0284985
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −48.2956 −1.57775 −0.788875 0.614554i \(-0.789336\pi\)
−0.788875 + 0.614554i \(0.789336\pi\)
\(938\) 0 0
\(939\) −4.30596 −0.140520
\(940\) 0 0
\(941\) −33.1280 −1.07994 −0.539971 0.841684i \(-0.681565\pi\)
−0.539971 + 0.841684i \(0.681565\pi\)
\(942\) 0 0
\(943\) −67.9547 −2.21291
\(944\) 0 0
\(945\) −8.53349 −0.277594
\(946\) 0 0
\(947\) −38.8503 −1.26247 −0.631233 0.775593i \(-0.717451\pi\)
−0.631233 + 0.775593i \(0.717451\pi\)
\(948\) 0 0
\(949\) 3.69306 0.119882
\(950\) 0 0
\(951\) 81.4431 2.64097
\(952\) 0 0
\(953\) −38.7874 −1.25645 −0.628224 0.778033i \(-0.716218\pi\)
−0.628224 + 0.778033i \(0.716218\pi\)
\(954\) 0 0
\(955\) 10.4804 0.339137
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −4.71683 −0.152314
\(960\) 0 0
\(961\) −10.9516 −0.353277
\(962\) 0 0
\(963\) −90.5074 −2.91656
\(964\) 0 0
\(965\) 19.1446 0.616287
\(966\) 0 0
\(967\) −60.9782 −1.96093 −0.980464 0.196698i \(-0.936978\pi\)
−0.980464 + 0.196698i \(0.936978\pi\)
\(968\) 0 0
\(969\) 29.2041 0.938171
\(970\) 0 0
\(971\) 29.6498 0.951509 0.475754 0.879578i \(-0.342175\pi\)
0.475754 + 0.879578i \(0.342175\pi\)
\(972\) 0 0
\(973\) 9.37919 0.300683
\(974\) 0 0
\(975\) 1.49570 0.0479009
\(976\) 0 0
\(977\) 21.0925 0.674810 0.337405 0.941360i \(-0.390451\pi\)
0.337405 + 0.941360i \(0.390451\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 61.2321 1.95499
\(982\) 0 0
\(983\) 33.0051 1.05270 0.526350 0.850268i \(-0.323560\pi\)
0.526350 + 0.850268i \(0.323560\pi\)
\(984\) 0 0
\(985\) 0.935844 0.0298185
\(986\) 0 0
\(987\) 0.557283 0.0177385
\(988\) 0 0
\(989\) 40.7518 1.29583
\(990\) 0 0
\(991\) −19.6935 −0.625586 −0.312793 0.949821i \(-0.601265\pi\)
−0.312793 + 0.949821i \(0.601265\pi\)
\(992\) 0 0
\(993\) −49.5216 −1.57152
\(994\) 0 0
\(995\) −26.4159 −0.837440
\(996\) 0 0
\(997\) 9.30145 0.294580 0.147290 0.989093i \(-0.452945\pi\)
0.147290 + 0.989093i \(0.452945\pi\)
\(998\) 0 0
\(999\) −39.6769 −1.25532
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.bd.1.1 yes 6
4.3 odd 2 9680.2.a.da.1.6 6
11.10 odd 2 4840.2.a.bc.1.1 6
44.43 even 2 9680.2.a.db.1.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4840.2.a.bc.1.1 6 11.10 odd 2
4840.2.a.bd.1.1 yes 6 1.1 even 1 trivial
9680.2.a.da.1.6 6 4.3 odd 2
9680.2.a.db.1.6 6 44.43 even 2