Properties

Label 48.24.a.k.1.2
Level $48$
Weight $24$
Character 48.1
Self dual yes
Analytic conductor $160.898$
Analytic rank $0$
Dimension $3$
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] = [48,24,Mod(1,48)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(48, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 24, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("48.1");
 
S:= CuspForms(chi, 24);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 24 \)
Character orbit: \([\chi]\) \(=\) 48.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: \(160.897937926\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 510971349x - 545907620601 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{21}\cdot 3^{4}\cdot 5 \)
Twist minimal: no (minimal twist has level 24)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-22049.7\) of defining polynomial
Character \(\chi\) \(=\) 48.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-177147. q^{3} -2.66963e7 q^{5} -2.63708e9 q^{7} +3.13811e10 q^{9} +O(q^{10})\) \(q-177147. q^{3} -2.66963e7 q^{5} -2.63708e9 q^{7} +3.13811e10 q^{9} +6.90658e11 q^{11} +8.43948e12 q^{13} +4.72917e12 q^{15} +1.75506e13 q^{17} +5.46727e14 q^{19} +4.67151e14 q^{21} -4.16817e15 q^{23} -1.12082e16 q^{25} -5.55906e15 q^{27} +7.34605e16 q^{29} -1.44267e17 q^{31} -1.22348e17 q^{33} +7.04003e16 q^{35} +4.39423e16 q^{37} -1.49503e18 q^{39} +1.13798e18 q^{41} +3.65242e18 q^{43} -8.37758e17 q^{45} -1.31349e18 q^{47} -2.04146e19 q^{49} -3.10904e18 q^{51} +1.56940e18 q^{53} -1.84380e19 q^{55} -9.68511e19 q^{57} -1.30965e20 q^{59} +2.40655e20 q^{61} -8.27543e19 q^{63} -2.25303e20 q^{65} -2.77336e20 q^{67} +7.38379e20 q^{69} -2.66742e21 q^{71} +2.44977e20 q^{73} +1.98551e21 q^{75} -1.82132e21 q^{77} -4.73406e21 q^{79} +9.84771e20 q^{81} -3.82639e21 q^{83} -4.68537e20 q^{85} -1.30133e22 q^{87} +3.93473e22 q^{89} -2.22556e22 q^{91} +2.55564e22 q^{93} -1.45956e22 q^{95} +1.29544e23 q^{97} +2.16736e22 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 531441 q^{3} - 36400950 q^{5} + 418786392 q^{7} + 94143178827 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 531441 q^{3} - 36400950 q^{5} + 418786392 q^{7} + 94143178827 q^{9} - 1481178196140 q^{11} - 7482693510030 q^{13} + 6448319089650 q^{15} - 67650915008682 q^{17} - 408489373374228 q^{19} - 74186752983624 q^{21} + 39\!\cdots\!36 q^{23}+ \cdots - 46\!\cdots\!60 q^{99}+O(q^{100}) \) 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
\(3\) −177147. −0.577350
\(4\) 0 0
\(5\) −2.66963e7 −0.244510 −0.122255 0.992499i \(-0.539013\pi\)
−0.122255 + 0.992499i \(0.539013\pi\)
\(6\) 0 0
\(7\) −2.63708e9 −0.504075 −0.252038 0.967717i \(-0.581101\pi\)
−0.252038 + 0.967717i \(0.581101\pi\)
\(8\) 0 0
\(9\) 3.13811e10 0.333333
\(10\) 0 0
\(11\) 6.90658e11 0.729873 0.364936 0.931032i \(-0.381091\pi\)
0.364936 + 0.931032i \(0.381091\pi\)
\(12\) 0 0
\(13\) 8.43948e12 1.30607 0.653036 0.757327i \(-0.273495\pi\)
0.653036 + 0.757327i \(0.273495\pi\)
\(14\) 0 0
\(15\) 4.72917e12 0.141168
\(16\) 0 0
\(17\) 1.75506e13 0.124202 0.0621012 0.998070i \(-0.480220\pi\)
0.0621012 + 0.998070i \(0.480220\pi\)
\(18\) 0 0
\(19\) 5.46727e14 1.07672 0.538362 0.842714i \(-0.319043\pi\)
0.538362 + 0.842714i \(0.319043\pi\)
\(20\) 0 0
\(21\) 4.67151e14 0.291028
\(22\) 0 0
\(23\) −4.16817e15 −0.912169 −0.456084 0.889936i \(-0.650749\pi\)
−0.456084 + 0.889936i \(0.650749\pi\)
\(24\) 0 0
\(25\) −1.12082e16 −0.940215
\(26\) 0 0
\(27\) −5.55906e15 −0.192450
\(28\) 0 0
\(29\) 7.34605e16 1.11809 0.559045 0.829137i \(-0.311168\pi\)
0.559045 + 0.829137i \(0.311168\pi\)
\(30\) 0 0
\(31\) −1.44267e17 −1.01978 −0.509890 0.860239i \(-0.670314\pi\)
−0.509890 + 0.860239i \(0.670314\pi\)
\(32\) 0 0
\(33\) −1.22348e17 −0.421392
\(34\) 0 0
\(35\) 7.04003e16 0.123251
\(36\) 0 0
\(37\) 4.39423e16 0.0406034 0.0203017 0.999794i \(-0.493537\pi\)
0.0203017 + 0.999794i \(0.493537\pi\)
\(38\) 0 0
\(39\) −1.49503e18 −0.754061
\(40\) 0 0
\(41\) 1.13798e18 0.322938 0.161469 0.986878i \(-0.448377\pi\)
0.161469 + 0.986878i \(0.448377\pi\)
\(42\) 0 0
\(43\) 3.65242e18 0.599368 0.299684 0.954039i \(-0.403119\pi\)
0.299684 + 0.954039i \(0.403119\pi\)
\(44\) 0 0
\(45\) −8.37758e17 −0.0815032
\(46\) 0 0
\(47\) −1.31349e18 −0.0775001 −0.0387501 0.999249i \(-0.512338\pi\)
−0.0387501 + 0.999249i \(0.512338\pi\)
\(48\) 0 0
\(49\) −2.04146e19 −0.745908
\(50\) 0 0
\(51\) −3.10904e18 −0.0717083
\(52\) 0 0
\(53\) 1.56940e18 0.0232575 0.0116287 0.999932i \(-0.496298\pi\)
0.0116287 + 0.999932i \(0.496298\pi\)
\(54\) 0 0
\(55\) −1.84380e19 −0.178461
\(56\) 0 0
\(57\) −9.68511e19 −0.621647
\(58\) 0 0
\(59\) −1.30965e20 −0.565403 −0.282701 0.959208i \(-0.591231\pi\)
−0.282701 + 0.959208i \(0.591231\pi\)
\(60\) 0 0
\(61\) 2.40655e20 0.708111 0.354056 0.935224i \(-0.384802\pi\)
0.354056 + 0.935224i \(0.384802\pi\)
\(62\) 0 0
\(63\) −8.27543e19 −0.168025
\(64\) 0 0
\(65\) −2.25303e20 −0.319347
\(66\) 0 0
\(67\) −2.77336e20 −0.277425 −0.138713 0.990333i \(-0.544296\pi\)
−0.138713 + 0.990333i \(0.544296\pi\)
\(68\) 0 0
\(69\) 7.38379e20 0.526641
\(70\) 0 0
\(71\) −2.66742e21 −1.36968 −0.684842 0.728692i \(-0.740129\pi\)
−0.684842 + 0.728692i \(0.740129\pi\)
\(72\) 0 0
\(73\) 2.44977e20 0.0913929 0.0456965 0.998955i \(-0.485449\pi\)
0.0456965 + 0.998955i \(0.485449\pi\)
\(74\) 0 0
\(75\) 1.98551e21 0.542833
\(76\) 0 0
\(77\) −1.82132e21 −0.367911
\(78\) 0 0
\(79\) −4.73406e21 −0.712070 −0.356035 0.934473i \(-0.615872\pi\)
−0.356035 + 0.934473i \(0.615872\pi\)
\(80\) 0 0
\(81\) 9.84771e20 0.111111
\(82\) 0 0
\(83\) −3.82639e21 −0.326130 −0.163065 0.986615i \(-0.552138\pi\)
−0.163065 + 0.986615i \(0.552138\pi\)
\(84\) 0 0
\(85\) −4.68537e20 −0.0303687
\(86\) 0 0
\(87\) −1.30133e22 −0.645530
\(88\) 0 0
\(89\) 3.93473e22 1.50290 0.751450 0.659790i \(-0.229355\pi\)
0.751450 + 0.659790i \(0.229355\pi\)
\(90\) 0 0
\(91\) −2.22556e22 −0.658359
\(92\) 0 0
\(93\) 2.55564e22 0.588771
\(94\) 0 0
\(95\) −1.45956e22 −0.263269
\(96\) 0 0
\(97\) 1.29544e23 1.83883 0.919415 0.393290i \(-0.128663\pi\)
0.919415 + 0.393290i \(0.128663\pi\)
\(98\) 0 0
\(99\) 2.16736e22 0.243291
\(100\) 0 0
\(101\) 5.23956e22 0.467304 0.233652 0.972320i \(-0.424932\pi\)
0.233652 + 0.972320i \(0.424932\pi\)
\(102\) 0 0
\(103\) −9.71005e22 −0.691184 −0.345592 0.938385i \(-0.612322\pi\)
−0.345592 + 0.938385i \(0.612322\pi\)
\(104\) 0 0
\(105\) −1.24712e22 −0.0711592
\(106\) 0 0
\(107\) −2.60654e23 −1.19716 −0.598579 0.801064i \(-0.704268\pi\)
−0.598579 + 0.801064i \(0.704268\pi\)
\(108\) 0 0
\(109\) 1.69368e23 0.628675 0.314338 0.949311i \(-0.398218\pi\)
0.314338 + 0.949311i \(0.398218\pi\)
\(110\) 0 0
\(111\) −7.78424e21 −0.0234424
\(112\) 0 0
\(113\) 1.71661e23 0.420988 0.210494 0.977595i \(-0.432493\pi\)
0.210494 + 0.977595i \(0.432493\pi\)
\(114\) 0 0
\(115\) 1.11275e23 0.223034
\(116\) 0 0
\(117\) 2.64840e23 0.435357
\(118\) 0 0
\(119\) −4.62824e22 −0.0626074
\(120\) 0 0
\(121\) −4.18421e23 −0.467285
\(122\) 0 0
\(123\) −2.01589e23 −0.186448
\(124\) 0 0
\(125\) 6.17463e23 0.474401
\(126\) 0 0
\(127\) −1.61449e24 −1.03345 −0.516727 0.856150i \(-0.672850\pi\)
−0.516727 + 0.856150i \(0.672850\pi\)
\(128\) 0 0
\(129\) −6.47015e23 −0.346045
\(130\) 0 0
\(131\) 1.87162e24 0.838682 0.419341 0.907829i \(-0.362261\pi\)
0.419341 + 0.907829i \(0.362261\pi\)
\(132\) 0 0
\(133\) −1.44176e24 −0.542750
\(134\) 0 0
\(135\) 1.48406e23 0.0470559
\(136\) 0 0
\(137\) 1.89209e24 0.506589 0.253294 0.967389i \(-0.418486\pi\)
0.253294 + 0.967389i \(0.418486\pi\)
\(138\) 0 0
\(139\) 2.89527e24 0.656173 0.328086 0.944648i \(-0.393596\pi\)
0.328086 + 0.944648i \(0.393596\pi\)
\(140\) 0 0
\(141\) 2.32681e23 0.0447447
\(142\) 0 0
\(143\) 5.82880e24 0.953267
\(144\) 0 0
\(145\) −1.96112e24 −0.273384
\(146\) 0 0
\(147\) 3.61638e24 0.430650
\(148\) 0 0
\(149\) −1.58742e25 −1.61827 −0.809134 0.587625i \(-0.800063\pi\)
−0.809134 + 0.587625i \(0.800063\pi\)
\(150\) 0 0
\(151\) 1.60479e25 1.40340 0.701702 0.712471i \(-0.252424\pi\)
0.701702 + 0.712471i \(0.252424\pi\)
\(152\) 0 0
\(153\) 5.50757e23 0.0414008
\(154\) 0 0
\(155\) 3.85139e24 0.249346
\(156\) 0 0
\(157\) 7.83816e24 0.437893 0.218947 0.975737i \(-0.429738\pi\)
0.218947 + 0.975737i \(0.429738\pi\)
\(158\) 0 0
\(159\) −2.78015e23 −0.0134277
\(160\) 0 0
\(161\) 1.09918e25 0.459802
\(162\) 0 0
\(163\) 2.53086e25 0.918565 0.459283 0.888290i \(-0.348107\pi\)
0.459283 + 0.888290i \(0.348107\pi\)
\(164\) 0 0
\(165\) 3.26624e24 0.103035
\(166\) 0 0
\(167\) 3.85481e25 1.05868 0.529339 0.848411i \(-0.322440\pi\)
0.529339 + 0.848411i \(0.322440\pi\)
\(168\) 0 0
\(169\) 2.94710e25 0.705825
\(170\) 0 0
\(171\) 1.71569e25 0.358908
\(172\) 0 0
\(173\) 7.85513e25 1.43755 0.718775 0.695242i \(-0.244703\pi\)
0.718775 + 0.695242i \(0.244703\pi\)
\(174\) 0 0
\(175\) 2.95570e25 0.473939
\(176\) 0 0
\(177\) 2.32001e25 0.326435
\(178\) 0 0
\(179\) −1.00073e26 −1.23739 −0.618694 0.785632i \(-0.712338\pi\)
−0.618694 + 0.785632i \(0.712338\pi\)
\(180\) 0 0
\(181\) −8.03087e24 −0.0873894 −0.0436947 0.999045i \(-0.513913\pi\)
−0.0436947 + 0.999045i \(0.513913\pi\)
\(182\) 0 0
\(183\) −4.26313e25 −0.408828
\(184\) 0 0
\(185\) −1.17310e24 −0.00992793
\(186\) 0 0
\(187\) 1.21215e25 0.0906520
\(188\) 0 0
\(189\) 1.46597e25 0.0970094
\(190\) 0 0
\(191\) 2.60881e26 1.52953 0.764767 0.644307i \(-0.222854\pi\)
0.764767 + 0.644307i \(0.222854\pi\)
\(192\) 0 0
\(193\) 3.24177e26 1.68606 0.843029 0.537868i \(-0.180770\pi\)
0.843029 + 0.537868i \(0.180770\pi\)
\(194\) 0 0
\(195\) 3.99118e25 0.184375
\(196\) 0 0
\(197\) 3.86517e26 1.58784 0.793921 0.608021i \(-0.208036\pi\)
0.793921 + 0.608021i \(0.208036\pi\)
\(198\) 0 0
\(199\) −1.51969e26 −0.555834 −0.277917 0.960605i \(-0.589644\pi\)
−0.277917 + 0.960605i \(0.589644\pi\)
\(200\) 0 0
\(201\) 4.91292e25 0.160171
\(202\) 0 0
\(203\) −1.93721e26 −0.563602
\(204\) 0 0
\(205\) −3.03798e25 −0.0789614
\(206\) 0 0
\(207\) −1.30802e26 −0.304056
\(208\) 0 0
\(209\) 3.77602e26 0.785872
\(210\) 0 0
\(211\) 7.61152e26 1.41979 0.709894 0.704309i \(-0.248743\pi\)
0.709894 + 0.704309i \(0.248743\pi\)
\(212\) 0 0
\(213\) 4.72525e26 0.790787
\(214\) 0 0
\(215\) −9.75061e25 −0.146551
\(216\) 0 0
\(217\) 3.80443e26 0.514046
\(218\) 0 0
\(219\) −4.33970e25 −0.0527657
\(220\) 0 0
\(221\) 1.48118e26 0.162217
\(222\) 0 0
\(223\) 1.64289e27 1.62220 0.811098 0.584910i \(-0.198870\pi\)
0.811098 + 0.584910i \(0.198870\pi\)
\(224\) 0 0
\(225\) −3.51726e26 −0.313405
\(226\) 0 0
\(227\) 1.60467e27 1.29148 0.645742 0.763556i \(-0.276548\pi\)
0.645742 + 0.763556i \(0.276548\pi\)
\(228\) 0 0
\(229\) −2.58626e26 −0.188176 −0.0940880 0.995564i \(-0.529994\pi\)
−0.0940880 + 0.995564i \(0.529994\pi\)
\(230\) 0 0
\(231\) 3.22641e26 0.212414
\(232\) 0 0
\(233\) 8.31388e26 0.495691 0.247845 0.968800i \(-0.420278\pi\)
0.247845 + 0.968800i \(0.420278\pi\)
\(234\) 0 0
\(235\) 3.50654e25 0.0189495
\(236\) 0 0
\(237\) 8.38625e26 0.411114
\(238\) 0 0
\(239\) 2.57103e27 1.14428 0.572139 0.820157i \(-0.306114\pi\)
0.572139 + 0.820157i \(0.306114\pi\)
\(240\) 0 0
\(241\) −4.23649e27 −1.71321 −0.856605 0.515973i \(-0.827430\pi\)
−0.856605 + 0.515973i \(0.827430\pi\)
\(242\) 0 0
\(243\) −1.74449e26 −0.0641500
\(244\) 0 0
\(245\) 5.44993e26 0.182382
\(246\) 0 0
\(247\) 4.61409e27 1.40628
\(248\) 0 0
\(249\) 6.77833e26 0.188291
\(250\) 0 0
\(251\) −3.84249e27 −0.973566 −0.486783 0.873523i \(-0.661830\pi\)
−0.486783 + 0.873523i \(0.661830\pi\)
\(252\) 0 0
\(253\) −2.87878e27 −0.665768
\(254\) 0 0
\(255\) 8.29999e25 0.0175334
\(256\) 0 0
\(257\) −2.26731e27 −0.437804 −0.218902 0.975747i \(-0.570247\pi\)
−0.218902 + 0.975747i \(0.570247\pi\)
\(258\) 0 0
\(259\) −1.15879e26 −0.0204672
\(260\) 0 0
\(261\) 2.30527e27 0.372697
\(262\) 0 0
\(263\) −8.92637e27 −1.32186 −0.660928 0.750450i \(-0.729837\pi\)
−0.660928 + 0.750450i \(0.729837\pi\)
\(264\) 0 0
\(265\) −4.18973e25 −0.00568668
\(266\) 0 0
\(267\) −6.97026e27 −0.867700
\(268\) 0 0
\(269\) −6.64561e27 −0.759248 −0.379624 0.925141i \(-0.623947\pi\)
−0.379624 + 0.925141i \(0.623947\pi\)
\(270\) 0 0
\(271\) 4.12761e27 0.433064 0.216532 0.976276i \(-0.430525\pi\)
0.216532 + 0.976276i \(0.430525\pi\)
\(272\) 0 0
\(273\) 3.94251e27 0.380104
\(274\) 0 0
\(275\) −7.74106e27 −0.686238
\(276\) 0 0
\(277\) −1.21768e28 −0.993152 −0.496576 0.867993i \(-0.665410\pi\)
−0.496576 + 0.867993i \(0.665410\pi\)
\(278\) 0 0
\(279\) −4.52724e27 −0.339927
\(280\) 0 0
\(281\) 1.78160e28 1.23221 0.616107 0.787662i \(-0.288709\pi\)
0.616107 + 0.787662i \(0.288709\pi\)
\(282\) 0 0
\(283\) −6.48624e26 −0.0413475 −0.0206737 0.999786i \(-0.506581\pi\)
−0.0206737 + 0.999786i \(0.506581\pi\)
\(284\) 0 0
\(285\) 2.58557e27 0.151999
\(286\) 0 0
\(287\) −3.00093e27 −0.162785
\(288\) 0 0
\(289\) −1.96595e28 −0.984574
\(290\) 0 0
\(291\) −2.29483e28 −1.06165
\(292\) 0 0
\(293\) 2.82593e28 1.20833 0.604163 0.796861i \(-0.293508\pi\)
0.604163 + 0.796861i \(0.293508\pi\)
\(294\) 0 0
\(295\) 3.49629e27 0.138246
\(296\) 0 0
\(297\) −3.83941e27 −0.140464
\(298\) 0 0
\(299\) −3.51772e28 −1.19136
\(300\) 0 0
\(301\) −9.63172e27 −0.302127
\(302\) 0 0
\(303\) −9.28173e27 −0.269798
\(304\) 0 0
\(305\) −6.42459e27 −0.173140
\(306\) 0 0
\(307\) −1.80973e28 −0.452401 −0.226200 0.974081i \(-0.572630\pi\)
−0.226200 + 0.974081i \(0.572630\pi\)
\(308\) 0 0
\(309\) 1.72011e28 0.399055
\(310\) 0 0
\(311\) −5.26369e28 −1.13382 −0.566912 0.823778i \(-0.691862\pi\)
−0.566912 + 0.823778i \(0.691862\pi\)
\(312\) 0 0
\(313\) −5.71094e28 −1.14274 −0.571370 0.820693i \(-0.693588\pi\)
−0.571370 + 0.820693i \(0.693588\pi\)
\(314\) 0 0
\(315\) 2.20923e27 0.0410838
\(316\) 0 0
\(317\) −4.58206e28 −0.792281 −0.396140 0.918190i \(-0.629651\pi\)
−0.396140 + 0.918190i \(0.629651\pi\)
\(318\) 0 0
\(319\) 5.07361e28 0.816064
\(320\) 0 0
\(321\) 4.61741e28 0.691180
\(322\) 0 0
\(323\) 9.59541e27 0.133732
\(324\) 0 0
\(325\) −9.45917e28 −1.22799
\(326\) 0 0
\(327\) −3.00030e28 −0.362966
\(328\) 0 0
\(329\) 3.46378e27 0.0390659
\(330\) 0 0
\(331\) 1.14307e29 1.20240 0.601200 0.799098i \(-0.294689\pi\)
0.601200 + 0.799098i \(0.294689\pi\)
\(332\) 0 0
\(333\) 1.37896e27 0.0135345
\(334\) 0 0
\(335\) 7.40384e27 0.0678331
\(336\) 0 0
\(337\) −1.79475e29 −1.53553 −0.767767 0.640729i \(-0.778632\pi\)
−0.767767 + 0.640729i \(0.778632\pi\)
\(338\) 0 0
\(339\) −3.04093e28 −0.243058
\(340\) 0 0
\(341\) −9.96390e28 −0.744310
\(342\) 0 0
\(343\) 1.26008e29 0.880069
\(344\) 0 0
\(345\) −1.97120e28 −0.128769
\(346\) 0 0
\(347\) −3.77055e28 −0.230470 −0.115235 0.993338i \(-0.536762\pi\)
−0.115235 + 0.993338i \(0.536762\pi\)
\(348\) 0 0
\(349\) 2.42646e28 0.138829 0.0694146 0.997588i \(-0.477887\pi\)
0.0694146 + 0.997588i \(0.477887\pi\)
\(350\) 0 0
\(351\) −4.69156e28 −0.251354
\(352\) 0 0
\(353\) 2.01961e29 1.01358 0.506791 0.862069i \(-0.330832\pi\)
0.506791 + 0.862069i \(0.330832\pi\)
\(354\) 0 0
\(355\) 7.12102e28 0.334901
\(356\) 0 0
\(357\) 8.19879e27 0.0361464
\(358\) 0 0
\(359\) 2.65591e29 1.09806 0.549032 0.835801i \(-0.314996\pi\)
0.549032 + 0.835801i \(0.314996\pi\)
\(360\) 0 0
\(361\) 4.10810e28 0.159334
\(362\) 0 0
\(363\) 7.41221e28 0.269787
\(364\) 0 0
\(365\) −6.53998e27 −0.0223465
\(366\) 0 0
\(367\) −4.02820e29 −1.29256 −0.646281 0.763100i \(-0.723676\pi\)
−0.646281 + 0.763100i \(0.723676\pi\)
\(368\) 0 0
\(369\) 3.57109e28 0.107646
\(370\) 0 0
\(371\) −4.13864e27 −0.0117235
\(372\) 0 0
\(373\) 2.34018e29 0.623156 0.311578 0.950221i \(-0.399142\pi\)
0.311578 + 0.950221i \(0.399142\pi\)
\(374\) 0 0
\(375\) −1.09382e29 −0.273896
\(376\) 0 0
\(377\) 6.19969e29 1.46031
\(378\) 0 0
\(379\) 7.63581e29 1.69241 0.846203 0.532861i \(-0.178883\pi\)
0.846203 + 0.532861i \(0.178883\pi\)
\(380\) 0 0
\(381\) 2.86001e29 0.596665
\(382\) 0 0
\(383\) 1.50832e29 0.296283 0.148141 0.988966i \(-0.452671\pi\)
0.148141 + 0.988966i \(0.452671\pi\)
\(384\) 0 0
\(385\) 4.86225e28 0.0899578
\(386\) 0 0
\(387\) 1.14617e29 0.199789
\(388\) 0 0
\(389\) 7.88719e29 1.29569 0.647846 0.761771i \(-0.275670\pi\)
0.647846 + 0.761771i \(0.275670\pi\)
\(390\) 0 0
\(391\) −7.31540e28 −0.113294
\(392\) 0 0
\(393\) −3.31552e29 −0.484214
\(394\) 0 0
\(395\) 1.26382e29 0.174108
\(396\) 0 0
\(397\) 1.07099e30 1.39218 0.696090 0.717955i \(-0.254922\pi\)
0.696090 + 0.717955i \(0.254922\pi\)
\(398\) 0 0
\(399\) 2.55404e29 0.313357
\(400\) 0 0
\(401\) −3.53206e29 −0.409136 −0.204568 0.978852i \(-0.565579\pi\)
−0.204568 + 0.978852i \(0.565579\pi\)
\(402\) 0 0
\(403\) −1.21754e30 −1.33191
\(404\) 0 0
\(405\) −2.62897e28 −0.0271677
\(406\) 0 0
\(407\) 3.03491e28 0.0296353
\(408\) 0 0
\(409\) 1.43029e30 1.32010 0.660048 0.751223i \(-0.270536\pi\)
0.660048 + 0.751223i \(0.270536\pi\)
\(410\) 0 0
\(411\) −3.35178e29 −0.292479
\(412\) 0 0
\(413\) 3.45365e29 0.285006
\(414\) 0 0
\(415\) 1.02150e29 0.0797420
\(416\) 0 0
\(417\) −5.12888e29 −0.378842
\(418\) 0 0
\(419\) 1.74971e30 1.22322 0.611610 0.791160i \(-0.290522\pi\)
0.611610 + 0.791160i \(0.290522\pi\)
\(420\) 0 0
\(421\) −1.48338e30 −0.981767 −0.490883 0.871225i \(-0.663326\pi\)
−0.490883 + 0.871225i \(0.663326\pi\)
\(422\) 0 0
\(423\) −4.12188e28 −0.0258334
\(424\) 0 0
\(425\) −1.96712e29 −0.116777
\(426\) 0 0
\(427\) −6.34626e29 −0.356941
\(428\) 0 0
\(429\) −1.03255e30 −0.550369
\(430\) 0 0
\(431\) −2.96194e29 −0.149654 −0.0748269 0.997197i \(-0.523840\pi\)
−0.0748269 + 0.997197i \(0.523840\pi\)
\(432\) 0 0
\(433\) 2.35179e30 1.12665 0.563324 0.826236i \(-0.309522\pi\)
0.563324 + 0.826236i \(0.309522\pi\)
\(434\) 0 0
\(435\) 3.47407e29 0.157838
\(436\) 0 0
\(437\) −2.27885e30 −0.982154
\(438\) 0 0
\(439\) −2.38873e30 −0.976845 −0.488422 0.872607i \(-0.662427\pi\)
−0.488422 + 0.872607i \(0.662427\pi\)
\(440\) 0 0
\(441\) −6.40631e29 −0.248636
\(442\) 0 0
\(443\) 3.06602e30 1.12962 0.564810 0.825221i \(-0.308950\pi\)
0.564810 + 0.825221i \(0.308950\pi\)
\(444\) 0 0
\(445\) −1.05043e30 −0.367474
\(446\) 0 0
\(447\) 2.81207e30 0.934307
\(448\) 0 0
\(449\) −3.24613e30 −1.02455 −0.512274 0.858822i \(-0.671197\pi\)
−0.512274 + 0.858822i \(0.671197\pi\)
\(450\) 0 0
\(451\) 7.85953e29 0.235704
\(452\) 0 0
\(453\) −2.84283e30 −0.810255
\(454\) 0 0
\(455\) 5.94142e29 0.160975
\(456\) 0 0
\(457\) −3.51307e30 −0.905005 −0.452502 0.891763i \(-0.649469\pi\)
−0.452502 + 0.891763i \(0.649469\pi\)
\(458\) 0 0
\(459\) −9.75650e28 −0.0239028
\(460\) 0 0
\(461\) 1.41936e30 0.330774 0.165387 0.986229i \(-0.447113\pi\)
0.165387 + 0.986229i \(0.447113\pi\)
\(462\) 0 0
\(463\) −3.42125e30 −0.758584 −0.379292 0.925277i \(-0.623832\pi\)
−0.379292 + 0.925277i \(0.623832\pi\)
\(464\) 0 0
\(465\) −6.82262e29 −0.143960
\(466\) 0 0
\(467\) 3.68193e30 0.739489 0.369744 0.929134i \(-0.379445\pi\)
0.369744 + 0.929134i \(0.379445\pi\)
\(468\) 0 0
\(469\) 7.31356e29 0.139843
\(470\) 0 0
\(471\) −1.38851e30 −0.252818
\(472\) 0 0
\(473\) 2.52257e30 0.437462
\(474\) 0 0
\(475\) −6.12785e30 −1.01235
\(476\) 0 0
\(477\) 4.92496e28 0.00775249
\(478\) 0 0
\(479\) 2.88655e30 0.433033 0.216516 0.976279i \(-0.430531\pi\)
0.216516 + 0.976279i \(0.430531\pi\)
\(480\) 0 0
\(481\) 3.70850e29 0.0530310
\(482\) 0 0
\(483\) −1.94716e30 −0.265467
\(484\) 0 0
\(485\) −3.45834e30 −0.449612
\(486\) 0 0
\(487\) −9.49733e30 −1.17766 −0.588828 0.808259i \(-0.700410\pi\)
−0.588828 + 0.808259i \(0.700410\pi\)
\(488\) 0 0
\(489\) −4.48334e30 −0.530334
\(490\) 0 0
\(491\) −1.39737e31 −1.57716 −0.788580 0.614932i \(-0.789183\pi\)
−0.788580 + 0.614932i \(0.789183\pi\)
\(492\) 0 0
\(493\) 1.28928e30 0.138870
\(494\) 0 0
\(495\) −5.78605e29 −0.0594870
\(496\) 0 0
\(497\) 7.03419e30 0.690424
\(498\) 0 0
\(499\) −1.44217e31 −1.35164 −0.675821 0.737066i \(-0.736211\pi\)
−0.675821 + 0.737066i \(0.736211\pi\)
\(500\) 0 0
\(501\) −6.82868e30 −0.611228
\(502\) 0 0
\(503\) 1.49630e31 1.27934 0.639671 0.768649i \(-0.279071\pi\)
0.639671 + 0.768649i \(0.279071\pi\)
\(504\) 0 0
\(505\) −1.39877e30 −0.114260
\(506\) 0 0
\(507\) −5.22069e30 −0.407508
\(508\) 0 0
\(509\) 5.58922e30 0.416962 0.208481 0.978026i \(-0.433148\pi\)
0.208481 + 0.978026i \(0.433148\pi\)
\(510\) 0 0
\(511\) −6.46024e29 −0.0460689
\(512\) 0 0
\(513\) −3.03929e30 −0.207216
\(514\) 0 0
\(515\) 2.59223e30 0.169001
\(516\) 0 0
\(517\) −9.07174e29 −0.0565652
\(518\) 0 0
\(519\) −1.39151e31 −0.829970
\(520\) 0 0
\(521\) 1.30155e31 0.742726 0.371363 0.928488i \(-0.378891\pi\)
0.371363 + 0.928488i \(0.378891\pi\)
\(522\) 0 0
\(523\) −3.70459e30 −0.202288 −0.101144 0.994872i \(-0.532250\pi\)
−0.101144 + 0.994872i \(0.532250\pi\)
\(524\) 0 0
\(525\) −5.23593e30 −0.273629
\(526\) 0 0
\(527\) −2.53197e30 −0.126659
\(528\) 0 0
\(529\) −3.50683e30 −0.167948
\(530\) 0 0
\(531\) −4.10983e30 −0.188468
\(532\) 0 0
\(533\) 9.60393e30 0.421780
\(534\) 0 0
\(535\) 6.95851e30 0.292717
\(536\) 0 0
\(537\) 1.77276e31 0.714406
\(538\) 0 0
\(539\) −1.40995e31 −0.544418
\(540\) 0 0
\(541\) −3.85046e31 −1.42477 −0.712385 0.701789i \(-0.752385\pi\)
−0.712385 + 0.701789i \(0.752385\pi\)
\(542\) 0 0
\(543\) 1.42264e30 0.0504543
\(544\) 0 0
\(545\) −4.52150e30 −0.153717
\(546\) 0 0
\(547\) 2.89456e31 0.943471 0.471736 0.881740i \(-0.343628\pi\)
0.471736 + 0.881740i \(0.343628\pi\)
\(548\) 0 0
\(549\) 7.55200e30 0.236037
\(550\) 0 0
\(551\) 4.01629e31 1.20387
\(552\) 0 0
\(553\) 1.24841e31 0.358937
\(554\) 0 0
\(555\) 2.07811e29 0.00573189
\(556\) 0 0
\(557\) 6.55955e31 1.73596 0.867979 0.496601i \(-0.165419\pi\)
0.867979 + 0.496601i \(0.165419\pi\)
\(558\) 0 0
\(559\) 3.08245e31 0.782818
\(560\) 0 0
\(561\) −2.14729e30 −0.0523380
\(562\) 0 0
\(563\) 6.24113e31 1.46021 0.730107 0.683333i \(-0.239470\pi\)
0.730107 + 0.683333i \(0.239470\pi\)
\(564\) 0 0
\(565\) −4.58272e30 −0.102936
\(566\) 0 0
\(567\) −2.59692e30 −0.0560084
\(568\) 0 0
\(569\) 4.79693e31 0.993510 0.496755 0.867891i \(-0.334525\pi\)
0.496755 + 0.867891i \(0.334525\pi\)
\(570\) 0 0
\(571\) 7.43230e31 1.47845 0.739227 0.673456i \(-0.235191\pi\)
0.739227 + 0.673456i \(0.235191\pi\)
\(572\) 0 0
\(573\) −4.62143e31 −0.883076
\(574\) 0 0
\(575\) 4.67178e31 0.857635
\(576\) 0 0
\(577\) −7.78263e31 −1.37279 −0.686396 0.727228i \(-0.740808\pi\)
−0.686396 + 0.727228i \(0.740808\pi\)
\(578\) 0 0
\(579\) −5.74269e31 −0.973446
\(580\) 0 0
\(581\) 1.00905e31 0.164394
\(582\) 0 0
\(583\) 1.08392e30 0.0169750
\(584\) 0 0
\(585\) −7.07025e30 −0.106449
\(586\) 0 0
\(587\) 8.13383e31 1.17749 0.588744 0.808319i \(-0.299623\pi\)
0.588744 + 0.808319i \(0.299623\pi\)
\(588\) 0 0
\(589\) −7.88745e31 −1.09802
\(590\) 0 0
\(591\) −6.84704e31 −0.916741
\(592\) 0 0
\(593\) 1.15019e32 1.48129 0.740644 0.671898i \(-0.234521\pi\)
0.740644 + 0.671898i \(0.234521\pi\)
\(594\) 0 0
\(595\) 1.23557e30 0.0153081
\(596\) 0 0
\(597\) 2.69209e31 0.320911
\(598\) 0 0
\(599\) 2.95748e31 0.339244 0.169622 0.985509i \(-0.445745\pi\)
0.169622 + 0.985509i \(0.445745\pi\)
\(600\) 0 0
\(601\) 1.33359e31 0.147220 0.0736099 0.997287i \(-0.476548\pi\)
0.0736099 + 0.997287i \(0.476548\pi\)
\(602\) 0 0
\(603\) −8.70309e30 −0.0924750
\(604\) 0 0
\(605\) 1.11703e31 0.114256
\(606\) 0 0
\(607\) 8.41625e30 0.0828798 0.0414399 0.999141i \(-0.486805\pi\)
0.0414399 + 0.999141i \(0.486805\pi\)
\(608\) 0 0
\(609\) 3.43171e31 0.325396
\(610\) 0 0
\(611\) −1.10852e31 −0.101221
\(612\) 0 0
\(613\) −6.41255e31 −0.563943 −0.281972 0.959423i \(-0.590988\pi\)
−0.281972 + 0.959423i \(0.590988\pi\)
\(614\) 0 0
\(615\) 5.38169e30 0.0455884
\(616\) 0 0
\(617\) 2.07994e32 1.69735 0.848675 0.528915i \(-0.177401\pi\)
0.848675 + 0.528915i \(0.177401\pi\)
\(618\) 0 0
\(619\) −7.06394e31 −0.555398 −0.277699 0.960668i \(-0.589572\pi\)
−0.277699 + 0.960668i \(0.589572\pi\)
\(620\) 0 0
\(621\) 2.31711e31 0.175547
\(622\) 0 0
\(623\) −1.03762e32 −0.757575
\(624\) 0 0
\(625\) 1.17129e32 0.824219
\(626\) 0 0
\(627\) −6.68910e31 −0.453723
\(628\) 0 0
\(629\) 7.71215e29 0.00504305
\(630\) 0 0
\(631\) −7.38793e31 −0.465785 −0.232892 0.972503i \(-0.574819\pi\)
−0.232892 + 0.972503i \(0.574819\pi\)
\(632\) 0 0
\(633\) −1.34836e32 −0.819715
\(634\) 0 0
\(635\) 4.31008e31 0.252690
\(636\) 0 0
\(637\) −1.72288e32 −0.974210
\(638\) 0 0
\(639\) −8.37064e31 −0.456561
\(640\) 0 0
\(641\) 1.35614e32 0.713569 0.356785 0.934187i \(-0.383873\pi\)
0.356785 + 0.934187i \(0.383873\pi\)
\(642\) 0 0
\(643\) −1.52991e32 −0.776677 −0.388339 0.921517i \(-0.626951\pi\)
−0.388339 + 0.921517i \(0.626951\pi\)
\(644\) 0 0
\(645\) 1.72729e31 0.0846114
\(646\) 0 0
\(647\) −8.82951e31 −0.417386 −0.208693 0.977981i \(-0.566921\pi\)
−0.208693 + 0.977981i \(0.566921\pi\)
\(648\) 0 0
\(649\) −9.04522e31 −0.412672
\(650\) 0 0
\(651\) −6.73943e31 −0.296785
\(652\) 0 0
\(653\) 8.08654e31 0.343764 0.171882 0.985118i \(-0.445015\pi\)
0.171882 + 0.985118i \(0.445015\pi\)
\(654\) 0 0
\(655\) −4.99653e31 −0.205066
\(656\) 0 0
\(657\) 7.68764e30 0.0304643
\(658\) 0 0
\(659\) 3.02900e32 1.15909 0.579546 0.814940i \(-0.303230\pi\)
0.579546 + 0.814940i \(0.303230\pi\)
\(660\) 0 0
\(661\) −1.69718e32 −0.627207 −0.313604 0.949554i \(-0.601536\pi\)
−0.313604 + 0.949554i \(0.601536\pi\)
\(662\) 0 0
\(663\) −2.62387e31 −0.0936563
\(664\) 0 0
\(665\) 3.84897e31 0.132708
\(666\) 0 0
\(667\) −3.06196e32 −1.01989
\(668\) 0 0
\(669\) −2.91034e32 −0.936575
\(670\) 0 0
\(671\) 1.66210e32 0.516831
\(672\) 0 0
\(673\) 2.59259e32 0.779041 0.389520 0.921018i \(-0.372641\pi\)
0.389520 + 0.921018i \(0.372641\pi\)
\(674\) 0 0
\(675\) 6.23073e31 0.180944
\(676\) 0 0
\(677\) −3.80491e32 −1.06801 −0.534004 0.845482i \(-0.679313\pi\)
−0.534004 + 0.845482i \(0.679313\pi\)
\(678\) 0 0
\(679\) −3.41617e32 −0.926909
\(680\) 0 0
\(681\) −2.84263e32 −0.745638
\(682\) 0 0
\(683\) 1.34319e32 0.340643 0.170322 0.985389i \(-0.445519\pi\)
0.170322 + 0.985389i \(0.445519\pi\)
\(684\) 0 0
\(685\) −5.05118e31 −0.123866
\(686\) 0 0
\(687\) 4.58148e31 0.108643
\(688\) 0 0
\(689\) 1.32450e31 0.0303759
\(690\) 0 0
\(691\) 2.79228e32 0.619387 0.309694 0.950836i \(-0.399774\pi\)
0.309694 + 0.950836i \(0.399774\pi\)
\(692\) 0 0
\(693\) −5.71549e31 −0.122637
\(694\) 0 0
\(695\) −7.72929e31 −0.160441
\(696\) 0 0
\(697\) 1.99722e31 0.0401097
\(698\) 0 0
\(699\) −1.47278e32 −0.286187
\(700\) 0 0
\(701\) −7.05564e32 −1.32672 −0.663360 0.748301i \(-0.730870\pi\)
−0.663360 + 0.748301i \(0.730870\pi\)
\(702\) 0 0
\(703\) 2.40244e31 0.0437187
\(704\) 0 0
\(705\) −6.21173e30 −0.0109405
\(706\) 0 0
\(707\) −1.38171e32 −0.235556
\(708\) 0 0
\(709\) −4.54508e32 −0.750085 −0.375042 0.927008i \(-0.622372\pi\)
−0.375042 + 0.927008i \(0.622372\pi\)
\(710\) 0 0
\(711\) −1.48560e32 −0.237357
\(712\) 0 0
\(713\) 6.01328e32 0.930212
\(714\) 0 0
\(715\) −1.55607e32 −0.233083
\(716\) 0 0
\(717\) −4.55450e32 −0.660649
\(718\) 0 0
\(719\) 1.25674e33 1.76548 0.882738 0.469866i \(-0.155698\pi\)
0.882738 + 0.469866i \(0.155698\pi\)
\(720\) 0 0
\(721\) 2.56062e32 0.348409
\(722\) 0 0
\(723\) 7.50482e32 0.989122
\(724\) 0 0
\(725\) −8.23363e32 −1.05125
\(726\) 0 0
\(727\) −1.23753e33 −1.53077 −0.765384 0.643574i \(-0.777451\pi\)
−0.765384 + 0.643574i \(0.777451\pi\)
\(728\) 0 0
\(729\) 3.09032e31 0.0370370
\(730\) 0 0
\(731\) 6.41023e31 0.0744429
\(732\) 0 0
\(733\) −3.07938e32 −0.346551 −0.173275 0.984873i \(-0.555435\pi\)
−0.173275 + 0.984873i \(0.555435\pi\)
\(734\) 0 0
\(735\) −9.65440e31 −0.105298
\(736\) 0 0
\(737\) −1.91544e32 −0.202485
\(738\) 0 0
\(739\) −1.25293e33 −1.28386 −0.641928 0.766765i \(-0.721865\pi\)
−0.641928 + 0.766765i \(0.721865\pi\)
\(740\) 0 0
\(741\) −8.17373e32 −0.811916
\(742\) 0 0
\(743\) 8.96246e32 0.863089 0.431545 0.902092i \(-0.357969\pi\)
0.431545 + 0.902092i \(0.357969\pi\)
\(744\) 0 0
\(745\) 4.23783e32 0.395682
\(746\) 0 0
\(747\) −1.20076e32 −0.108710
\(748\) 0 0
\(749\) 6.87366e32 0.603458
\(750\) 0 0
\(751\) 6.16616e32 0.524996 0.262498 0.964933i \(-0.415454\pi\)
0.262498 + 0.964933i \(0.415454\pi\)
\(752\) 0 0
\(753\) 6.80686e32 0.562088
\(754\) 0 0
\(755\) −4.28419e32 −0.343146
\(756\) 0 0
\(757\) 1.99545e33 1.55038 0.775192 0.631726i \(-0.217653\pi\)
0.775192 + 0.631726i \(0.217653\pi\)
\(758\) 0 0
\(759\) 5.09967e32 0.384381
\(760\) 0 0
\(761\) 2.60859e33 1.90757 0.953787 0.300482i \(-0.0971475\pi\)
0.953787 + 0.300482i \(0.0971475\pi\)
\(762\) 0 0
\(763\) −4.46636e32 −0.316900
\(764\) 0 0
\(765\) −1.47032e31 −0.0101229
\(766\) 0 0
\(767\) −1.10528e33 −0.738457
\(768\) 0 0
\(769\) 2.98234e33 1.93377 0.966884 0.255216i \(-0.0821465\pi\)
0.966884 + 0.255216i \(0.0821465\pi\)
\(770\) 0 0
\(771\) 4.01647e32 0.252766
\(772\) 0 0
\(773\) 4.86325e32 0.297072 0.148536 0.988907i \(-0.452544\pi\)
0.148536 + 0.988907i \(0.452544\pi\)
\(774\) 0 0
\(775\) 1.61698e33 0.958813
\(776\) 0 0
\(777\) 2.05277e31 0.0118167
\(778\) 0 0
\(779\) 6.22163e32 0.347715
\(780\) 0 0
\(781\) −1.84227e33 −0.999695
\(782\) 0 0
\(783\) −4.08371e32 −0.215177
\(784\) 0 0
\(785\) −2.09250e32 −0.107069
\(786\) 0 0
\(787\) 5.77342e32 0.286896 0.143448 0.989658i \(-0.454181\pi\)
0.143448 + 0.989658i \(0.454181\pi\)
\(788\) 0 0
\(789\) 1.58128e33 0.763173
\(790\) 0 0
\(791\) −4.52684e32 −0.212210
\(792\) 0 0
\(793\) 2.03100e33 0.924844
\(794\) 0 0
\(795\) 7.42198e30 0.00328320
\(796\) 0 0
\(797\) −1.99971e33 −0.859402 −0.429701 0.902971i \(-0.641381\pi\)
−0.429701 + 0.902971i \(0.641381\pi\)
\(798\) 0 0
\(799\) −2.30526e31 −0.00962570
\(800\) 0 0
\(801\) 1.23476e33 0.500967
\(802\) 0 0
\(803\) 1.69195e32 0.0667052
\(804\) 0 0
\(805\) −2.93440e32 −0.112426
\(806\) 0 0
\(807\) 1.17725e33 0.438352
\(808\) 0 0
\(809\) 2.44035e33 0.883169 0.441584 0.897220i \(-0.354417\pi\)
0.441584 + 0.897220i \(0.354417\pi\)
\(810\) 0 0
\(811\) 2.28105e33 0.802406 0.401203 0.915989i \(-0.368592\pi\)
0.401203 + 0.915989i \(0.368592\pi\)
\(812\) 0 0
\(813\) −7.31194e32 −0.250029
\(814\) 0 0
\(815\) −6.75645e32 −0.224598
\(816\) 0 0
\(817\) 1.99688e33 0.645353
\(818\) 0 0
\(819\) −6.98404e32 −0.219453
\(820\) 0 0
\(821\) −5.89474e33 −1.80102 −0.900509 0.434838i \(-0.856806\pi\)
−0.900509 + 0.434838i \(0.856806\pi\)
\(822\) 0 0
\(823\) 5.31080e33 1.57784 0.788918 0.614498i \(-0.210642\pi\)
0.788918 + 0.614498i \(0.210642\pi\)
\(824\) 0 0
\(825\) 1.37131e33 0.396199
\(826\) 0 0
\(827\) −1.64278e33 −0.461601 −0.230801 0.973001i \(-0.574134\pi\)
−0.230801 + 0.973001i \(0.574134\pi\)
\(828\) 0 0
\(829\) −6.55887e33 −1.79247 −0.896235 0.443579i \(-0.853708\pi\)
−0.896235 + 0.443579i \(0.853708\pi\)
\(830\) 0 0
\(831\) 2.15708e33 0.573397
\(832\) 0 0
\(833\) −3.58289e32 −0.0926436
\(834\) 0 0
\(835\) −1.02909e33 −0.258857
\(836\) 0 0
\(837\) 8.01987e32 0.196257
\(838\) 0 0
\(839\) −1.63564e33 −0.389426 −0.194713 0.980860i \(-0.562378\pi\)
−0.194713 + 0.980860i \(0.562378\pi\)
\(840\) 0 0
\(841\) 1.07973e33 0.250126
\(842\) 0 0
\(843\) −3.15604e33 −0.711419
\(844\) 0 0
\(845\) −7.86766e32 −0.172581
\(846\) 0 0
\(847\) 1.10341e33 0.235547
\(848\) 0 0
\(849\) 1.14902e32 0.0238720
\(850\) 0 0
\(851\) −1.83159e32 −0.0370372
\(852\) 0 0
\(853\) −9.05910e33 −1.78308 −0.891541 0.452940i \(-0.850375\pi\)
−0.891541 + 0.452940i \(0.850375\pi\)
\(854\) 0 0
\(855\) −4.58025e32 −0.0877565
\(856\) 0 0
\(857\) 2.55960e33 0.477411 0.238706 0.971092i \(-0.423277\pi\)
0.238706 + 0.971092i \(0.423277\pi\)
\(858\) 0 0
\(859\) 6.48360e33 1.17732 0.588660 0.808381i \(-0.299656\pi\)
0.588660 + 0.808381i \(0.299656\pi\)
\(860\) 0 0
\(861\) 5.31606e32 0.0939840
\(862\) 0 0
\(863\) −5.16834e33 −0.889666 −0.444833 0.895614i \(-0.646737\pi\)
−0.444833 + 0.895614i \(0.646737\pi\)
\(864\) 0 0
\(865\) −2.09703e33 −0.351495
\(866\) 0 0
\(867\) 3.48263e33 0.568444
\(868\) 0 0
\(869\) −3.26962e33 −0.519720
\(870\) 0 0
\(871\) −2.34057e33 −0.362337
\(872\) 0 0
\(873\) 4.06522e33 0.612943
\(874\) 0 0
\(875\) −1.62830e33 −0.239134
\(876\) 0 0
\(877\) −1.42031e33 −0.203184 −0.101592 0.994826i \(-0.532394\pi\)
−0.101592 + 0.994826i \(0.532394\pi\)
\(878\) 0 0
\(879\) −5.00606e33 −0.697628
\(880\) 0 0
\(881\) −1.08884e34 −1.47822 −0.739111 0.673584i \(-0.764754\pi\)
−0.739111 + 0.673584i \(0.764754\pi\)
\(882\) 0 0
\(883\) 3.35808e33 0.444163 0.222082 0.975028i \(-0.428715\pi\)
0.222082 + 0.975028i \(0.428715\pi\)
\(884\) 0 0
\(885\) −6.19357e32 −0.0798166
\(886\) 0 0
\(887\) −7.80507e33 −0.980066 −0.490033 0.871704i \(-0.663015\pi\)
−0.490033 + 0.871704i \(0.663015\pi\)
\(888\) 0 0
\(889\) 4.25752e33 0.520939
\(890\) 0 0
\(891\) 6.80140e32 0.0810970
\(892\) 0 0
\(893\) −7.18122e32 −0.0834462
\(894\) 0 0
\(895\) 2.67157e33 0.302553
\(896\) 0 0
\(897\) 6.23153e33 0.687831
\(898\) 0 0
\(899\) −1.05979e34 −1.14021
\(900\) 0 0
\(901\) 2.75440e31 0.00288863
\(902\) 0 0
\(903\) 1.70623e33 0.174433
\(904\) 0 0
\(905\) 2.14394e32 0.0213676
\(906\) 0 0
\(907\) 1.04038e34 1.01090 0.505449 0.862856i \(-0.331327\pi\)
0.505449 + 0.862856i \(0.331327\pi\)
\(908\) 0 0
\(909\) 1.64423e33 0.155768
\(910\) 0 0
\(911\) −9.34626e33 −0.863330 −0.431665 0.902034i \(-0.642074\pi\)
−0.431665 + 0.902034i \(0.642074\pi\)
\(912\) 0 0
\(913\) −2.64273e33 −0.238034
\(914\) 0 0
\(915\) 1.13810e33 0.0999625
\(916\) 0 0
\(917\) −4.93561e33 −0.422759
\(918\) 0 0
\(919\) −1.07362e34 −0.896854 −0.448427 0.893819i \(-0.648016\pi\)
−0.448427 + 0.893819i \(0.648016\pi\)
\(920\) 0 0
\(921\) 3.20589e33 0.261194
\(922\) 0 0
\(923\) −2.25116e34 −1.78891
\(924\) 0 0
\(925\) −4.92515e32 −0.0381760
\(926\) 0 0
\(927\) −3.04712e33 −0.230395
\(928\) 0 0
\(929\) 9.90198e33 0.730367 0.365184 0.930935i \(-0.381006\pi\)
0.365184 + 0.930935i \(0.381006\pi\)
\(930\) 0 0
\(931\) −1.11612e34 −0.803137
\(932\) 0 0
\(933\) 9.32447e33 0.654614
\(934\) 0 0
\(935\) −3.23599e32 −0.0221653
\(936\) 0 0
\(937\) 2.21853e34 1.48272 0.741362 0.671106i \(-0.234180\pi\)
0.741362 + 0.671106i \(0.234180\pi\)
\(938\) 0 0
\(939\) 1.01168e34 0.659761
\(940\) 0 0
\(941\) −2.69613e34 −1.71578 −0.857889 0.513836i \(-0.828224\pi\)
−0.857889 + 0.513836i \(0.828224\pi\)
\(942\) 0 0
\(943\) −4.74328e33 −0.294574
\(944\) 0 0
\(945\) −3.91359e32 −0.0237197
\(946\) 0 0
\(947\) 1.08449e34 0.641503 0.320752 0.947163i \(-0.396064\pi\)
0.320752 + 0.947163i \(0.396064\pi\)
\(948\) 0 0
\(949\) 2.06748e33 0.119366
\(950\) 0 0
\(951\) 8.11698e33 0.457423
\(952\) 0 0
\(953\) 2.61511e33 0.143854 0.0719270 0.997410i \(-0.477085\pi\)
0.0719270 + 0.997410i \(0.477085\pi\)
\(954\) 0 0
\(955\) −6.96456e33 −0.373986
\(956\) 0 0
\(957\) −8.98775e33 −0.471155
\(958\) 0 0
\(959\) −4.98959e33 −0.255359
\(960\) 0 0
\(961\) 7.99577e32 0.0399522
\(962\) 0 0
\(963\) −8.17961e33 −0.399053
\(964\) 0 0
\(965\) −8.65432e33 −0.412258
\(966\) 0 0
\(967\) −1.67844e34 −0.780732 −0.390366 0.920660i \(-0.627652\pi\)
−0.390366 + 0.920660i \(0.627652\pi\)
\(968\) 0 0
\(969\) −1.69980e33 −0.0772100
\(970\) 0 0
\(971\) 2.05763e33 0.0912739 0.0456369 0.998958i \(-0.485468\pi\)
0.0456369 + 0.998958i \(0.485468\pi\)
\(972\) 0 0
\(973\) −7.63504e33 −0.330761
\(974\) 0 0
\(975\) 1.67566e34 0.708980
\(976\) 0 0
\(977\) 1.70711e34 0.705465 0.352732 0.935724i \(-0.385253\pi\)
0.352732 + 0.935724i \(0.385253\pi\)
\(978\) 0 0
\(979\) 2.71755e34 1.09693
\(980\) 0 0
\(981\) 5.31494e33 0.209558
\(982\) 0 0
\(983\) 4.02184e34 1.54903 0.774513 0.632558i \(-0.217995\pi\)
0.774513 + 0.632558i \(0.217995\pi\)
\(984\) 0 0
\(985\) −1.03186e34 −0.388243
\(986\) 0 0
\(987\) −6.13599e32 −0.0225547
\(988\) 0 0
\(989\) −1.52239e34 −0.546725
\(990\) 0 0
\(991\) 4.85503e34 1.70351 0.851756 0.523939i \(-0.175538\pi\)
0.851756 + 0.523939i \(0.175538\pi\)
\(992\) 0 0
\(993\) −2.02491e34 −0.694206
\(994\) 0 0
\(995\) 4.05702e33 0.135907
\(996\) 0 0
\(997\) 3.83432e34 1.25515 0.627573 0.778557i \(-0.284048\pi\)
0.627573 + 0.778557i \(0.284048\pi\)
\(998\) 0 0
\(999\) −2.44278e32 −0.00781413
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 48.24.a.k.1.2 3
4.3 odd 2 24.24.a.d.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
24.24.a.d.1.2 3 4.3 odd 2
48.24.a.k.1.2 3 1.1 even 1 trivial