Properties

Label 48.22.a.d.1.1
Level $48$
Weight $22$
Character 48.1
Self dual yes
Analytic conductor $134.149$
Analytic rank $1$
Dimension $1$
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] = [48,22,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, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("48.1");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 22 \)
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: \(134.149125258\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
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)
 
\(f(q)\) \(=\) \(q+59049.0 q^{3} -4.15128e7 q^{5} -5.38430e8 q^{7} +3.48678e9 q^{9} +O(q^{10})\) \(q+59049.0 q^{3} -4.15128e7 q^{5} -5.38430e8 q^{7} +3.48678e9 q^{9} +6.41130e10 q^{11} -1.30980e11 q^{13} -2.45129e12 q^{15} +8.24203e12 q^{17} -1.34921e13 q^{19} -3.17937e13 q^{21} +2.33185e14 q^{23} +1.24647e15 q^{25} +2.05891e14 q^{27} -2.02456e15 q^{29} +6.86919e15 q^{31} +3.78581e15 q^{33} +2.23517e16 q^{35} +3.44400e15 q^{37} -7.73424e15 q^{39} -2.18424e16 q^{41} +7.17928e16 q^{43} -1.44746e17 q^{45} -2.83545e17 q^{47} -2.68639e17 q^{49} +4.86684e17 q^{51} -2.17229e18 q^{53} -2.66151e18 q^{55} -7.96695e17 q^{57} -1.53483e18 q^{59} +4.31159e18 q^{61} -1.87739e18 q^{63} +5.43735e18 q^{65} -9.24391e18 q^{67} +1.37693e19 q^{69} +2.03874e19 q^{71} +1.66178e19 q^{73} +7.36030e19 q^{75} -3.45204e19 q^{77} -6.79403e19 q^{79} +1.21577e19 q^{81} -3.95037e19 q^{83} -3.42149e20 q^{85} -1.19548e20 q^{87} +4.16117e19 q^{89} +7.05236e19 q^{91} +4.05619e20 q^{93} +5.60095e20 q^{95} +5.71815e19 q^{97} +2.23548e20 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 59049.0 0.577350
\(4\) 0 0
\(5\) −4.15128e7 −1.90106 −0.950532 0.310627i \(-0.899461\pi\)
−0.950532 + 0.310627i \(0.899461\pi\)
\(6\) 0 0
\(7\) −5.38430e8 −0.720443 −0.360222 0.932867i \(-0.617299\pi\)
−0.360222 + 0.932867i \(0.617299\pi\)
\(8\) 0 0
\(9\) 3.48678e9 0.333333
\(10\) 0 0
\(11\) 6.41130e10 0.745286 0.372643 0.927975i \(-0.378452\pi\)
0.372643 + 0.927975i \(0.378452\pi\)
\(12\) 0 0
\(13\) −1.30980e11 −0.263512 −0.131756 0.991282i \(-0.542062\pi\)
−0.131756 + 0.991282i \(0.542062\pi\)
\(14\) 0 0
\(15\) −2.45129e12 −1.09758
\(16\) 0 0
\(17\) 8.24203e12 0.991563 0.495782 0.868447i \(-0.334882\pi\)
0.495782 + 0.868447i \(0.334882\pi\)
\(18\) 0 0
\(19\) −1.34921e13 −0.504855 −0.252428 0.967616i \(-0.581229\pi\)
−0.252428 + 0.967616i \(0.581229\pi\)
\(20\) 0 0
\(21\) −3.17937e13 −0.415948
\(22\) 0 0
\(23\) 2.33185e14 1.17370 0.586851 0.809695i \(-0.300367\pi\)
0.586851 + 0.809695i \(0.300367\pi\)
\(24\) 0 0
\(25\) 1.24647e15 2.61404
\(26\) 0 0
\(27\) 2.05891e14 0.192450
\(28\) 0 0
\(29\) −2.02456e15 −0.893618 −0.446809 0.894629i \(-0.647440\pi\)
−0.446809 + 0.894629i \(0.647440\pi\)
\(30\) 0 0
\(31\) 6.86919e15 1.50525 0.752624 0.658451i \(-0.228788\pi\)
0.752624 + 0.658451i \(0.228788\pi\)
\(32\) 0 0
\(33\) 3.78581e15 0.430291
\(34\) 0 0
\(35\) 2.23517e16 1.36961
\(36\) 0 0
\(37\) 3.44400e15 0.117746 0.0588728 0.998265i \(-0.481249\pi\)
0.0588728 + 0.998265i \(0.481249\pi\)
\(38\) 0 0
\(39\) −7.73424e15 −0.152139
\(40\) 0 0
\(41\) −2.18424e16 −0.254138 −0.127069 0.991894i \(-0.540557\pi\)
−0.127069 + 0.991894i \(0.540557\pi\)
\(42\) 0 0
\(43\) 7.17928e16 0.506597 0.253298 0.967388i \(-0.418485\pi\)
0.253298 + 0.967388i \(0.418485\pi\)
\(44\) 0 0
\(45\) −1.44746e17 −0.633688
\(46\) 0 0
\(47\) −2.83545e17 −0.786310 −0.393155 0.919472i \(-0.628617\pi\)
−0.393155 + 0.919472i \(0.628617\pi\)
\(48\) 0 0
\(49\) −2.68639e17 −0.480962
\(50\) 0 0
\(51\) 4.86684e17 0.572479
\(52\) 0 0
\(53\) −2.17229e18 −1.70616 −0.853081 0.521779i \(-0.825269\pi\)
−0.853081 + 0.521779i \(0.825269\pi\)
\(54\) 0 0
\(55\) −2.66151e18 −1.41684
\(56\) 0 0
\(57\) −7.96695e17 −0.291478
\(58\) 0 0
\(59\) −1.53483e18 −0.390944 −0.195472 0.980709i \(-0.562624\pi\)
−0.195472 + 0.980709i \(0.562624\pi\)
\(60\) 0 0
\(61\) 4.31159e18 0.773881 0.386940 0.922105i \(-0.373532\pi\)
0.386940 + 0.922105i \(0.373532\pi\)
\(62\) 0 0
\(63\) −1.87739e18 −0.240148
\(64\) 0 0
\(65\) 5.43735e18 0.500953
\(66\) 0 0
\(67\) −9.24391e18 −0.619541 −0.309771 0.950811i \(-0.600252\pi\)
−0.309771 + 0.950811i \(0.600252\pi\)
\(68\) 0 0
\(69\) 1.37693e19 0.677637
\(70\) 0 0
\(71\) 2.03874e19 0.743273 0.371636 0.928378i \(-0.378797\pi\)
0.371636 + 0.928378i \(0.378797\pi\)
\(72\) 0 0
\(73\) 1.66178e19 0.452566 0.226283 0.974062i \(-0.427343\pi\)
0.226283 + 0.974062i \(0.427343\pi\)
\(74\) 0 0
\(75\) 7.36030e19 1.50922
\(76\) 0 0
\(77\) −3.45204e19 −0.536936
\(78\) 0 0
\(79\) −6.79403e19 −0.807315 −0.403658 0.914910i \(-0.632261\pi\)
−0.403658 + 0.914910i \(0.632261\pi\)
\(80\) 0 0
\(81\) 1.21577e19 0.111111
\(82\) 0 0
\(83\) −3.95037e19 −0.279459 −0.139730 0.990190i \(-0.544623\pi\)
−0.139730 + 0.990190i \(0.544623\pi\)
\(84\) 0 0
\(85\) −3.42149e20 −1.88503
\(86\) 0 0
\(87\) −1.19548e20 −0.515931
\(88\) 0 0
\(89\) 4.16117e19 0.141456 0.0707278 0.997496i \(-0.477468\pi\)
0.0707278 + 0.997496i \(0.477468\pi\)
\(90\) 0 0
\(91\) 7.05236e19 0.189845
\(92\) 0 0
\(93\) 4.05619e20 0.869055
\(94\) 0 0
\(95\) 5.60095e20 0.959762
\(96\) 0 0
\(97\) 5.71815e19 0.0787322 0.0393661 0.999225i \(-0.487466\pi\)
0.0393661 + 0.999225i \(0.487466\pi\)
\(98\) 0 0
\(99\) 2.23548e20 0.248429
\(100\) 0 0
\(101\) 4.32417e20 0.389518 0.194759 0.980851i \(-0.437607\pi\)
0.194759 + 0.980851i \(0.437607\pi\)
\(102\) 0 0
\(103\) −1.84123e21 −1.34995 −0.674974 0.737841i \(-0.735845\pi\)
−0.674974 + 0.737841i \(0.735845\pi\)
\(104\) 0 0
\(105\) 1.31985e21 0.790744
\(106\) 0 0
\(107\) 2.43805e21 1.19815 0.599077 0.800691i \(-0.295534\pi\)
0.599077 + 0.800691i \(0.295534\pi\)
\(108\) 0 0
\(109\) −4.13676e21 −1.67372 −0.836859 0.547418i \(-0.815611\pi\)
−0.836859 + 0.547418i \(0.815611\pi\)
\(110\) 0 0
\(111\) 2.03365e20 0.0679805
\(112\) 0 0
\(113\) 3.47910e21 0.964146 0.482073 0.876131i \(-0.339884\pi\)
0.482073 + 0.876131i \(0.339884\pi\)
\(114\) 0 0
\(115\) −9.68015e21 −2.23128
\(116\) 0 0
\(117\) −4.56699e20 −0.0878373
\(118\) 0 0
\(119\) −4.43775e21 −0.714365
\(120\) 0 0
\(121\) −3.28977e21 −0.444548
\(122\) 0 0
\(123\) −1.28977e21 −0.146727
\(124\) 0 0
\(125\) −3.19497e22 −3.06840
\(126\) 0 0
\(127\) −1.37141e21 −0.111488 −0.0557438 0.998445i \(-0.517753\pi\)
−0.0557438 + 0.998445i \(0.517753\pi\)
\(128\) 0 0
\(129\) 4.23929e21 0.292484
\(130\) 0 0
\(131\) 2.45276e22 1.43981 0.719907 0.694071i \(-0.244185\pi\)
0.719907 + 0.694071i \(0.244185\pi\)
\(132\) 0 0
\(133\) 7.26455e21 0.363719
\(134\) 0 0
\(135\) −8.54711e21 −0.365860
\(136\) 0 0
\(137\) 1.02835e22 0.377204 0.188602 0.982054i \(-0.439604\pi\)
0.188602 + 0.982054i \(0.439604\pi\)
\(138\) 0 0
\(139\) −8.70692e21 −0.274289 −0.137145 0.990551i \(-0.543793\pi\)
−0.137145 + 0.990551i \(0.543793\pi\)
\(140\) 0 0
\(141\) −1.67430e22 −0.453977
\(142\) 0 0
\(143\) −8.39753e21 −0.196392
\(144\) 0 0
\(145\) 8.40452e22 1.69883
\(146\) 0 0
\(147\) −1.58629e22 −0.277683
\(148\) 0 0
\(149\) −9.03997e22 −1.37313 −0.686564 0.727069i \(-0.740882\pi\)
−0.686564 + 0.727069i \(0.740882\pi\)
\(150\) 0 0
\(151\) 4.75206e22 0.627514 0.313757 0.949503i \(-0.398412\pi\)
0.313757 + 0.949503i \(0.398412\pi\)
\(152\) 0 0
\(153\) 2.87382e22 0.330521
\(154\) 0 0
\(155\) −2.85159e23 −2.86157
\(156\) 0 0
\(157\) −1.50901e23 −1.32356 −0.661781 0.749697i \(-0.730199\pi\)
−0.661781 + 0.749697i \(0.730199\pi\)
\(158\) 0 0
\(159\) −1.28271e23 −0.985053
\(160\) 0 0
\(161\) −1.25554e23 −0.845586
\(162\) 0 0
\(163\) 4.83503e22 0.286042 0.143021 0.989720i \(-0.454318\pi\)
0.143021 + 0.989720i \(0.454318\pi\)
\(164\) 0 0
\(165\) −1.57159e23 −0.818011
\(166\) 0 0
\(167\) −4.78731e20 −0.00219568 −0.00109784 0.999999i \(-0.500349\pi\)
−0.00109784 + 0.999999i \(0.500349\pi\)
\(168\) 0 0
\(169\) −2.29909e23 −0.930562
\(170\) 0 0
\(171\) −4.70440e22 −0.168285
\(172\) 0 0
\(173\) −1.61804e23 −0.512277 −0.256139 0.966640i \(-0.582450\pi\)
−0.256139 + 0.966640i \(0.582450\pi\)
\(174\) 0 0
\(175\) −6.71138e23 −1.88327
\(176\) 0 0
\(177\) −9.06303e22 −0.225712
\(178\) 0 0
\(179\) 8.76377e22 0.193970 0.0969849 0.995286i \(-0.469080\pi\)
0.0969849 + 0.995286i \(0.469080\pi\)
\(180\) 0 0
\(181\) 9.36624e22 0.184476 0.0922381 0.995737i \(-0.470598\pi\)
0.0922381 + 0.995737i \(0.470598\pi\)
\(182\) 0 0
\(183\) 2.54595e23 0.446800
\(184\) 0 0
\(185\) −1.42970e23 −0.223842
\(186\) 0 0
\(187\) 5.28422e23 0.738999
\(188\) 0 0
\(189\) −1.10858e23 −0.138649
\(190\) 0 0
\(191\) −1.20858e24 −1.35340 −0.676699 0.736260i \(-0.736590\pi\)
−0.676699 + 0.736260i \(0.736590\pi\)
\(192\) 0 0
\(193\) −1.78822e24 −1.79502 −0.897509 0.440997i \(-0.854625\pi\)
−0.897509 + 0.440997i \(0.854625\pi\)
\(194\) 0 0
\(195\) 3.21070e23 0.289225
\(196\) 0 0
\(197\) 1.90963e24 1.54545 0.772723 0.634743i \(-0.218894\pi\)
0.772723 + 0.634743i \(0.218894\pi\)
\(198\) 0 0
\(199\) −1.44254e24 −1.04995 −0.524977 0.851116i \(-0.675926\pi\)
−0.524977 + 0.851116i \(0.675926\pi\)
\(200\) 0 0
\(201\) −5.45844e23 −0.357692
\(202\) 0 0
\(203\) 1.09008e24 0.643801
\(204\) 0 0
\(205\) 9.06739e23 0.483133
\(206\) 0 0
\(207\) 8.13065e23 0.391234
\(208\) 0 0
\(209\) −8.65020e23 −0.376262
\(210\) 0 0
\(211\) −3.98848e24 −1.56979 −0.784895 0.619629i \(-0.787283\pi\)
−0.784895 + 0.619629i \(0.787283\pi\)
\(212\) 0 0
\(213\) 1.20385e24 0.429129
\(214\) 0 0
\(215\) −2.98032e24 −0.963072
\(216\) 0 0
\(217\) −3.69858e24 −1.08445
\(218\) 0 0
\(219\) 9.81262e23 0.261289
\(220\) 0 0
\(221\) −1.07954e24 −0.261289
\(222\) 0 0
\(223\) 4.62963e24 1.01940 0.509700 0.860352i \(-0.329756\pi\)
0.509700 + 0.860352i \(0.329756\pi\)
\(224\) 0 0
\(225\) 4.34618e24 0.871348
\(226\) 0 0
\(227\) 3.43010e24 0.626664 0.313332 0.949644i \(-0.398555\pi\)
0.313332 + 0.949644i \(0.398555\pi\)
\(228\) 0 0
\(229\) 8.11792e23 0.135261 0.0676304 0.997710i \(-0.478456\pi\)
0.0676304 + 0.997710i \(0.478456\pi\)
\(230\) 0 0
\(231\) −2.03839e24 −0.310000
\(232\) 0 0
\(233\) 8.22188e23 0.114218 0.0571089 0.998368i \(-0.481812\pi\)
0.0571089 + 0.998368i \(0.481812\pi\)
\(234\) 0 0
\(235\) 1.17707e25 1.49483
\(236\) 0 0
\(237\) −4.01181e24 −0.466104
\(238\) 0 0
\(239\) 8.85525e24 0.941940 0.470970 0.882149i \(-0.343904\pi\)
0.470970 + 0.882149i \(0.343904\pi\)
\(240\) 0 0
\(241\) 7.46934e24 0.727953 0.363977 0.931408i \(-0.381419\pi\)
0.363977 + 0.931408i \(0.381419\pi\)
\(242\) 0 0
\(243\) 7.17898e23 0.0641500
\(244\) 0 0
\(245\) 1.11520e25 0.914339
\(246\) 0 0
\(247\) 1.76720e24 0.133035
\(248\) 0 0
\(249\) −2.33266e24 −0.161346
\(250\) 0 0
\(251\) −9.46474e23 −0.0601914 −0.0300957 0.999547i \(-0.509581\pi\)
−0.0300957 + 0.999547i \(0.509581\pi\)
\(252\) 0 0
\(253\) 1.49502e25 0.874744
\(254\) 0 0
\(255\) −2.02036e25 −1.08832
\(256\) 0 0
\(257\) −1.91825e25 −0.951936 −0.475968 0.879463i \(-0.657902\pi\)
−0.475968 + 0.879463i \(0.657902\pi\)
\(258\) 0 0
\(259\) −1.85435e24 −0.0848290
\(260\) 0 0
\(261\) −7.05921e24 −0.297873
\(262\) 0 0
\(263\) −8.88429e23 −0.0346009 −0.0173004 0.999850i \(-0.505507\pi\)
−0.0173004 + 0.999850i \(0.505507\pi\)
\(264\) 0 0
\(265\) 9.01776e25 3.24352
\(266\) 0 0
\(267\) 2.45713e24 0.0816694
\(268\) 0 0
\(269\) −2.13847e25 −0.657211 −0.328605 0.944467i \(-0.606579\pi\)
−0.328605 + 0.944467i \(0.606579\pi\)
\(270\) 0 0
\(271\) 1.56435e25 0.444791 0.222395 0.974957i \(-0.428612\pi\)
0.222395 + 0.974957i \(0.428612\pi\)
\(272\) 0 0
\(273\) 4.16435e24 0.109607
\(274\) 0 0
\(275\) 7.99152e25 1.94821
\(276\) 0 0
\(277\) −8.04973e25 −1.81863 −0.909313 0.416112i \(-0.863392\pi\)
−0.909313 + 0.416112i \(0.863392\pi\)
\(278\) 0 0
\(279\) 2.39514e25 0.501749
\(280\) 0 0
\(281\) 8.33171e25 1.61926 0.809632 0.586938i \(-0.199667\pi\)
0.809632 + 0.586938i \(0.199667\pi\)
\(282\) 0 0
\(283\) 4.46130e24 0.0804829 0.0402415 0.999190i \(-0.487187\pi\)
0.0402415 + 0.999190i \(0.487187\pi\)
\(284\) 0 0
\(285\) 3.30730e25 0.554119
\(286\) 0 0
\(287\) 1.17606e25 0.183092
\(288\) 0 0
\(289\) −1.16088e24 −0.0168020
\(290\) 0 0
\(291\) 3.37651e24 0.0454560
\(292\) 0 0
\(293\) 9.67128e25 1.21164 0.605820 0.795602i \(-0.292845\pi\)
0.605820 + 0.795602i \(0.292845\pi\)
\(294\) 0 0
\(295\) 6.37151e25 0.743209
\(296\) 0 0
\(297\) 1.32003e25 0.143430
\(298\) 0 0
\(299\) −3.05426e25 −0.309284
\(300\) 0 0
\(301\) −3.86554e25 −0.364974
\(302\) 0 0
\(303\) 2.55338e25 0.224889
\(304\) 0 0
\(305\) −1.78986e26 −1.47120
\(306\) 0 0
\(307\) −1.68163e26 −1.29056 −0.645278 0.763948i \(-0.723258\pi\)
−0.645278 + 0.763948i \(0.723258\pi\)
\(308\) 0 0
\(309\) −1.08723e26 −0.779393
\(310\) 0 0
\(311\) −2.30370e26 −1.54327 −0.771636 0.636065i \(-0.780561\pi\)
−0.771636 + 0.636065i \(0.780561\pi\)
\(312\) 0 0
\(313\) −2.79658e26 −1.75151 −0.875753 0.482759i \(-0.839635\pi\)
−0.875753 + 0.482759i \(0.839635\pi\)
\(314\) 0 0
\(315\) 7.79356e25 0.456536
\(316\) 0 0
\(317\) 2.98501e25 0.163615 0.0818075 0.996648i \(-0.473931\pi\)
0.0818075 + 0.996648i \(0.473931\pi\)
\(318\) 0 0
\(319\) −1.29801e26 −0.666002
\(320\) 0 0
\(321\) 1.43964e26 0.691755
\(322\) 0 0
\(323\) −1.11202e26 −0.500596
\(324\) 0 0
\(325\) −1.63263e26 −0.688831
\(326\) 0 0
\(327\) −2.44272e26 −0.966322
\(328\) 0 0
\(329\) 1.52669e26 0.566492
\(330\) 0 0
\(331\) −2.55594e26 −0.889933 −0.444967 0.895547i \(-0.646784\pi\)
−0.444967 + 0.895547i \(0.646784\pi\)
\(332\) 0 0
\(333\) 1.20085e25 0.0392485
\(334\) 0 0
\(335\) 3.83740e26 1.17779
\(336\) 0 0
\(337\) −4.91931e25 −0.141837 −0.0709187 0.997482i \(-0.522593\pi\)
−0.0709187 + 0.997482i \(0.522593\pi\)
\(338\) 0 0
\(339\) 2.05437e26 0.556650
\(340\) 0 0
\(341\) 4.40405e26 1.12184
\(342\) 0 0
\(343\) 4.45381e26 1.06695
\(344\) 0 0
\(345\) −5.71603e26 −1.28823
\(346\) 0 0
\(347\) −2.98136e26 −0.632345 −0.316173 0.948702i \(-0.602398\pi\)
−0.316173 + 0.948702i \(0.602398\pi\)
\(348\) 0 0
\(349\) 7.72834e26 1.54319 0.771595 0.636115i \(-0.219459\pi\)
0.771595 + 0.636115i \(0.219459\pi\)
\(350\) 0 0
\(351\) −2.69676e25 −0.0507129
\(352\) 0 0
\(353\) −7.30755e26 −1.29461 −0.647303 0.762233i \(-0.724103\pi\)
−0.647303 + 0.762233i \(0.724103\pi\)
\(354\) 0 0
\(355\) −8.46336e26 −1.41301
\(356\) 0 0
\(357\) −2.62045e26 −0.412439
\(358\) 0 0
\(359\) 1.58936e25 0.0235901 0.0117951 0.999930i \(-0.496245\pi\)
0.0117951 + 0.999930i \(0.496245\pi\)
\(360\) 0 0
\(361\) −5.32173e26 −0.745121
\(362\) 0 0
\(363\) −1.94258e26 −0.256660
\(364\) 0 0
\(365\) −6.89849e26 −0.860358
\(366\) 0 0
\(367\) 1.40734e27 1.65732 0.828660 0.559752i \(-0.189104\pi\)
0.828660 + 0.559752i \(0.189104\pi\)
\(368\) 0 0
\(369\) −7.61598e25 −0.0847127
\(370\) 0 0
\(371\) 1.16962e27 1.22919
\(372\) 0 0
\(373\) −9.30077e26 −0.923797 −0.461898 0.886933i \(-0.652831\pi\)
−0.461898 + 0.886933i \(0.652831\pi\)
\(374\) 0 0
\(375\) −1.88660e27 −1.77154
\(376\) 0 0
\(377\) 2.65177e26 0.235479
\(378\) 0 0
\(379\) −2.18541e27 −1.83578 −0.917892 0.396830i \(-0.870110\pi\)
−0.917892 + 0.396830i \(0.870110\pi\)
\(380\) 0 0
\(381\) −8.09801e25 −0.0643674
\(382\) 0 0
\(383\) −2.10347e27 −1.58252 −0.791258 0.611482i \(-0.790574\pi\)
−0.791258 + 0.611482i \(0.790574\pi\)
\(384\) 0 0
\(385\) 1.43304e27 1.02075
\(386\) 0 0
\(387\) 2.50326e26 0.168866
\(388\) 0 0
\(389\) −2.97815e26 −0.190316 −0.0951582 0.995462i \(-0.530336\pi\)
−0.0951582 + 0.995462i \(0.530336\pi\)
\(390\) 0 0
\(391\) 1.92192e27 1.16380
\(392\) 0 0
\(393\) 1.44833e27 0.831277
\(394\) 0 0
\(395\) 2.82039e27 1.53476
\(396\) 0 0
\(397\) 6.36504e26 0.328474 0.164237 0.986421i \(-0.447484\pi\)
0.164237 + 0.986421i \(0.447484\pi\)
\(398\) 0 0
\(399\) 4.28964e26 0.209993
\(400\) 0 0
\(401\) 2.43888e27 1.13286 0.566428 0.824111i \(-0.308325\pi\)
0.566428 + 0.824111i \(0.308325\pi\)
\(402\) 0 0
\(403\) −8.99728e26 −0.396651
\(404\) 0 0
\(405\) −5.04698e26 −0.211229
\(406\) 0 0
\(407\) 2.20805e26 0.0877542
\(408\) 0 0
\(409\) 5.48032e26 0.206876 0.103438 0.994636i \(-0.467016\pi\)
0.103438 + 0.994636i \(0.467016\pi\)
\(410\) 0 0
\(411\) 6.07232e26 0.217779
\(412\) 0 0
\(413\) 8.26399e26 0.281653
\(414\) 0 0
\(415\) 1.63991e27 0.531269
\(416\) 0 0
\(417\) −5.14135e26 −0.158361
\(418\) 0 0
\(419\) 6.08246e27 1.78169 0.890844 0.454309i \(-0.150114\pi\)
0.890844 + 0.454309i \(0.150114\pi\)
\(420\) 0 0
\(421\) −4.05990e27 −1.13124 −0.565618 0.824667i \(-0.691362\pi\)
−0.565618 + 0.824667i \(0.691362\pi\)
\(422\) 0 0
\(423\) −9.88659e26 −0.262103
\(424\) 0 0
\(425\) 1.02735e28 2.59199
\(426\) 0 0
\(427\) −2.32149e27 −0.557537
\(428\) 0 0
\(429\) −4.95866e26 −0.113387
\(430\) 0 0
\(431\) 7.87214e27 1.71428 0.857140 0.515084i \(-0.172239\pi\)
0.857140 + 0.515084i \(0.172239\pi\)
\(432\) 0 0
\(433\) −1.73785e27 −0.360486 −0.180243 0.983622i \(-0.557688\pi\)
−0.180243 + 0.983622i \(0.557688\pi\)
\(434\) 0 0
\(435\) 4.96278e27 0.980817
\(436\) 0 0
\(437\) −3.14615e27 −0.592550
\(438\) 0 0
\(439\) −8.37416e27 −1.50336 −0.751681 0.659526i \(-0.770757\pi\)
−0.751681 + 0.659526i \(0.770757\pi\)
\(440\) 0 0
\(441\) −9.36687e26 −0.160321
\(442\) 0 0
\(443\) 3.30286e25 0.00539077 0.00269539 0.999996i \(-0.499142\pi\)
0.00269539 + 0.999996i \(0.499142\pi\)
\(444\) 0 0
\(445\) −1.72742e27 −0.268916
\(446\) 0 0
\(447\) −5.33801e27 −0.792776
\(448\) 0 0
\(449\) 5.21713e27 0.739341 0.369670 0.929163i \(-0.379471\pi\)
0.369670 + 0.929163i \(0.379471\pi\)
\(450\) 0 0
\(451\) −1.40038e27 −0.189406
\(452\) 0 0
\(453\) 2.80604e27 0.362296
\(454\) 0 0
\(455\) −2.92763e27 −0.360908
\(456\) 0 0
\(457\) 2.15211e26 0.0253363 0.0126682 0.999920i \(-0.495967\pi\)
0.0126682 + 0.999920i \(0.495967\pi\)
\(458\) 0 0
\(459\) 1.69696e27 0.190826
\(460\) 0 0
\(461\) 1.68699e28 1.81239 0.906197 0.422855i \(-0.138972\pi\)
0.906197 + 0.422855i \(0.138972\pi\)
\(462\) 0 0
\(463\) 1.90352e28 1.95415 0.977074 0.212898i \(-0.0682901\pi\)
0.977074 + 0.212898i \(0.0682901\pi\)
\(464\) 0 0
\(465\) −1.68384e28 −1.65213
\(466\) 0 0
\(467\) 1.21027e28 1.13515 0.567576 0.823321i \(-0.307881\pi\)
0.567576 + 0.823321i \(0.307881\pi\)
\(468\) 0 0
\(469\) 4.97720e27 0.446344
\(470\) 0 0
\(471\) −8.91053e27 −0.764159
\(472\) 0 0
\(473\) 4.60286e27 0.377559
\(474\) 0 0
\(475\) −1.68175e28 −1.31971
\(476\) 0 0
\(477\) −7.57429e27 −0.568721
\(478\) 0 0
\(479\) −6.95253e27 −0.499597 −0.249798 0.968298i \(-0.580364\pi\)
−0.249798 + 0.968298i \(0.580364\pi\)
\(480\) 0 0
\(481\) −4.51095e26 −0.0310274
\(482\) 0 0
\(483\) −7.41382e27 −0.488199
\(484\) 0 0
\(485\) −2.37376e27 −0.149675
\(486\) 0 0
\(487\) 1.06412e28 0.642596 0.321298 0.946978i \(-0.395881\pi\)
0.321298 + 0.946978i \(0.395881\pi\)
\(488\) 0 0
\(489\) 2.85504e27 0.165146
\(490\) 0 0
\(491\) −1.68064e28 −0.931361 −0.465681 0.884953i \(-0.654190\pi\)
−0.465681 + 0.884953i \(0.654190\pi\)
\(492\) 0 0
\(493\) −1.66865e28 −0.886079
\(494\) 0 0
\(495\) −9.28011e27 −0.472279
\(496\) 0 0
\(497\) −1.09772e28 −0.535486
\(498\) 0 0
\(499\) 5.12285e27 0.239583 0.119792 0.992799i \(-0.461777\pi\)
0.119792 + 0.992799i \(0.461777\pi\)
\(500\) 0 0
\(501\) −2.82686e25 −0.00126768
\(502\) 0 0
\(503\) −1.99606e28 −0.858442 −0.429221 0.903200i \(-0.641212\pi\)
−0.429221 + 0.903200i \(0.641212\pi\)
\(504\) 0 0
\(505\) −1.79508e28 −0.740499
\(506\) 0 0
\(507\) −1.35759e28 −0.537260
\(508\) 0 0
\(509\) −2.57966e27 −0.0979550 −0.0489775 0.998800i \(-0.515596\pi\)
−0.0489775 + 0.998800i \(0.515596\pi\)
\(510\) 0 0
\(511\) −8.94749e27 −0.326048
\(512\) 0 0
\(513\) −2.77790e27 −0.0971594
\(514\) 0 0
\(515\) 7.64346e28 2.56634
\(516\) 0 0
\(517\) −1.81789e28 −0.586026
\(518\) 0 0
\(519\) −9.55436e27 −0.295763
\(520\) 0 0
\(521\) −2.61230e28 −0.776652 −0.388326 0.921522i \(-0.626947\pi\)
−0.388326 + 0.921522i \(0.626947\pi\)
\(522\) 0 0
\(523\) −6.70750e28 −1.91555 −0.957774 0.287523i \(-0.907168\pi\)
−0.957774 + 0.287523i \(0.907168\pi\)
\(524\) 0 0
\(525\) −3.96300e28 −1.08731
\(526\) 0 0
\(527\) 5.66161e28 1.49255
\(528\) 0 0
\(529\) 1.49036e28 0.377577
\(530\) 0 0
\(531\) −5.35163e27 −0.130315
\(532\) 0 0
\(533\) 2.86092e27 0.0669684
\(534\) 0 0
\(535\) −1.01210e29 −2.27777
\(536\) 0 0
\(537\) 5.17492e27 0.111988
\(538\) 0 0
\(539\) −1.72233e28 −0.358454
\(540\) 0 0
\(541\) −2.15196e28 −0.430787 −0.215394 0.976527i \(-0.569103\pi\)
−0.215394 + 0.976527i \(0.569103\pi\)
\(542\) 0 0
\(543\) 5.53067e27 0.106507
\(544\) 0 0
\(545\) 1.71728e29 3.18185
\(546\) 0 0
\(547\) 7.46789e28 1.33147 0.665734 0.746189i \(-0.268118\pi\)
0.665734 + 0.746189i \(0.268118\pi\)
\(548\) 0 0
\(549\) 1.50336e28 0.257960
\(550\) 0 0
\(551\) 2.73156e28 0.451148
\(552\) 0 0
\(553\) 3.65811e28 0.581625
\(554\) 0 0
\(555\) −8.44223e27 −0.129235
\(556\) 0 0
\(557\) −7.95166e28 −1.17214 −0.586068 0.810262i \(-0.699325\pi\)
−0.586068 + 0.810262i \(0.699325\pi\)
\(558\) 0 0
\(559\) −9.40343e27 −0.133494
\(560\) 0 0
\(561\) 3.12028e28 0.426661
\(562\) 0 0
\(563\) −5.46305e28 −0.719609 −0.359805 0.933028i \(-0.617157\pi\)
−0.359805 + 0.933028i \(0.617157\pi\)
\(564\) 0 0
\(565\) −1.44427e29 −1.83290
\(566\) 0 0
\(567\) −6.54605e27 −0.0800492
\(568\) 0 0
\(569\) 9.43478e28 1.11187 0.555933 0.831227i \(-0.312361\pi\)
0.555933 + 0.831227i \(0.312361\pi\)
\(570\) 0 0
\(571\) −8.05027e28 −0.914390 −0.457195 0.889367i \(-0.651146\pi\)
−0.457195 + 0.889367i \(0.651146\pi\)
\(572\) 0 0
\(573\) −7.13655e28 −0.781384
\(574\) 0 0
\(575\) 2.90659e29 3.06811
\(576\) 0 0
\(577\) −1.67132e28 −0.170104 −0.0850519 0.996377i \(-0.527106\pi\)
−0.0850519 + 0.996377i \(0.527106\pi\)
\(578\) 0 0
\(579\) −1.05592e29 −1.03635
\(580\) 0 0
\(581\) 2.12700e28 0.201334
\(582\) 0 0
\(583\) −1.39272e29 −1.27158
\(584\) 0 0
\(585\) 1.89589e28 0.166984
\(586\) 0 0
\(587\) −3.15730e28 −0.268297 −0.134149 0.990961i \(-0.542830\pi\)
−0.134149 + 0.990961i \(0.542830\pi\)
\(588\) 0 0
\(589\) −9.26799e28 −0.759932
\(590\) 0 0
\(591\) 1.12762e29 0.892264
\(592\) 0 0
\(593\) −5.48493e27 −0.0418887 −0.0209443 0.999781i \(-0.506667\pi\)
−0.0209443 + 0.999781i \(0.506667\pi\)
\(594\) 0 0
\(595\) 1.84223e29 1.35805
\(596\) 0 0
\(597\) −8.51805e28 −0.606191
\(598\) 0 0
\(599\) −1.25621e29 −0.863136 −0.431568 0.902080i \(-0.642040\pi\)
−0.431568 + 0.902080i \(0.642040\pi\)
\(600\) 0 0
\(601\) 3.99325e28 0.264938 0.132469 0.991187i \(-0.457709\pi\)
0.132469 + 0.991187i \(0.457709\pi\)
\(602\) 0 0
\(603\) −3.22315e28 −0.206514
\(604\) 0 0
\(605\) 1.36567e29 0.845115
\(606\) 0 0
\(607\) 2.46990e29 1.47638 0.738189 0.674594i \(-0.235681\pi\)
0.738189 + 0.674594i \(0.235681\pi\)
\(608\) 0 0
\(609\) 6.43684e28 0.371699
\(610\) 0 0
\(611\) 3.71387e28 0.207202
\(612\) 0 0
\(613\) 2.63911e28 0.142273 0.0711364 0.997467i \(-0.477337\pi\)
0.0711364 + 0.997467i \(0.477337\pi\)
\(614\) 0 0
\(615\) 5.35420e28 0.278937
\(616\) 0 0
\(617\) 3.09820e29 1.55997 0.779984 0.625800i \(-0.215227\pi\)
0.779984 + 0.625800i \(0.215227\pi\)
\(618\) 0 0
\(619\) 2.50758e29 1.22040 0.610202 0.792246i \(-0.291088\pi\)
0.610202 + 0.792246i \(0.291088\pi\)
\(620\) 0 0
\(621\) 4.80107e28 0.225879
\(622\) 0 0
\(623\) −2.24050e28 −0.101911
\(624\) 0 0
\(625\) 7.31956e29 3.21918
\(626\) 0 0
\(627\) −5.10785e28 −0.217235
\(628\) 0 0
\(629\) 2.83855e28 0.116752
\(630\) 0 0
\(631\) 4.32770e28 0.172167 0.0860833 0.996288i \(-0.472565\pi\)
0.0860833 + 0.996288i \(0.472565\pi\)
\(632\) 0 0
\(633\) −2.35516e29 −0.906318
\(634\) 0 0
\(635\) 5.69309e28 0.211945
\(636\) 0 0
\(637\) 3.51864e28 0.126739
\(638\) 0 0
\(639\) 7.10863e28 0.247758
\(640\) 0 0
\(641\) −8.73381e28 −0.294574 −0.147287 0.989094i \(-0.547054\pi\)
−0.147287 + 0.989094i \(0.547054\pi\)
\(642\) 0 0
\(643\) 4.72013e29 1.54077 0.770386 0.637578i \(-0.220063\pi\)
0.770386 + 0.637578i \(0.220063\pi\)
\(644\) 0 0
\(645\) −1.75985e29 −0.556030
\(646\) 0 0
\(647\) −1.26799e28 −0.0387812 −0.0193906 0.999812i \(-0.506173\pi\)
−0.0193906 + 0.999812i \(0.506173\pi\)
\(648\) 0 0
\(649\) −9.84027e28 −0.291365
\(650\) 0 0
\(651\) −2.18397e29 −0.626105
\(652\) 0 0
\(653\) −2.76226e29 −0.766790 −0.383395 0.923584i \(-0.625245\pi\)
−0.383395 + 0.923584i \(0.625245\pi\)
\(654\) 0 0
\(655\) −1.01821e30 −2.73718
\(656\) 0 0
\(657\) 5.79425e28 0.150855
\(658\) 0 0
\(659\) −6.46511e28 −0.163034 −0.0815172 0.996672i \(-0.525977\pi\)
−0.0815172 + 0.996672i \(0.525977\pi\)
\(660\) 0 0
\(661\) −2.27730e29 −0.556295 −0.278147 0.960538i \(-0.589720\pi\)
−0.278147 + 0.960538i \(0.589720\pi\)
\(662\) 0 0
\(663\) −6.37459e28 −0.150855
\(664\) 0 0
\(665\) −3.01572e29 −0.691454
\(666\) 0 0
\(667\) −4.72097e29 −1.04884
\(668\) 0 0
\(669\) 2.73375e29 0.588551
\(670\) 0 0
\(671\) 2.76429e29 0.576763
\(672\) 0 0
\(673\) −3.79243e29 −0.766936 −0.383468 0.923554i \(-0.625270\pi\)
−0.383468 + 0.923554i \(0.625270\pi\)
\(674\) 0 0
\(675\) 2.56638e29 0.503073
\(676\) 0 0
\(677\) −3.39717e29 −0.645559 −0.322780 0.946474i \(-0.604617\pi\)
−0.322780 + 0.946474i \(0.604617\pi\)
\(678\) 0 0
\(679\) −3.07882e28 −0.0567220
\(680\) 0 0
\(681\) 2.02544e29 0.361805
\(682\) 0 0
\(683\) −4.54742e29 −0.787677 −0.393838 0.919180i \(-0.628853\pi\)
−0.393838 + 0.919180i \(0.628853\pi\)
\(684\) 0 0
\(685\) −4.26897e29 −0.717088
\(686\) 0 0
\(687\) 4.79355e28 0.0780929
\(688\) 0 0
\(689\) 2.84526e29 0.449594
\(690\) 0 0
\(691\) 3.71838e29 0.569947 0.284974 0.958535i \(-0.408015\pi\)
0.284974 + 0.958535i \(0.408015\pi\)
\(692\) 0 0
\(693\) −1.20365e29 −0.178979
\(694\) 0 0
\(695\) 3.61448e29 0.521442
\(696\) 0 0
\(697\) −1.80026e29 −0.251994
\(698\) 0 0
\(699\) 4.85494e28 0.0659437
\(700\) 0 0
\(701\) −9.37969e29 −1.23637 −0.618186 0.786032i \(-0.712132\pi\)
−0.618186 + 0.786032i \(0.712132\pi\)
\(702\) 0 0
\(703\) −4.64668e28 −0.0594445
\(704\) 0 0
\(705\) 6.95050e29 0.863038
\(706\) 0 0
\(707\) −2.32826e29 −0.280626
\(708\) 0 0
\(709\) −7.54578e29 −0.882914 −0.441457 0.897282i \(-0.645538\pi\)
−0.441457 + 0.897282i \(0.645538\pi\)
\(710\) 0 0
\(711\) −2.36893e29 −0.269105
\(712\) 0 0
\(713\) 1.60179e30 1.76671
\(714\) 0 0
\(715\) 3.48605e29 0.373353
\(716\) 0 0
\(717\) 5.22894e29 0.543829
\(718\) 0 0
\(719\) −1.22754e30 −1.23989 −0.619944 0.784646i \(-0.712845\pi\)
−0.619944 + 0.784646i \(0.712845\pi\)
\(720\) 0 0
\(721\) 9.91374e29 0.972561
\(722\) 0 0
\(723\) 4.41057e29 0.420284
\(724\) 0 0
\(725\) −2.52356e30 −2.33596
\(726\) 0 0
\(727\) −9.04407e29 −0.813303 −0.406652 0.913583i \(-0.633304\pi\)
−0.406652 + 0.913583i \(0.633304\pi\)
\(728\) 0 0
\(729\) 4.23912e28 0.0370370
\(730\) 0 0
\(731\) 5.91719e29 0.502323
\(732\) 0 0
\(733\) −1.38874e30 −1.14559 −0.572795 0.819699i \(-0.694141\pi\)
−0.572795 + 0.819699i \(0.694141\pi\)
\(734\) 0 0
\(735\) 6.58512e29 0.527894
\(736\) 0 0
\(737\) −5.92655e29 −0.461736
\(738\) 0 0
\(739\) −2.11506e30 −1.60161 −0.800804 0.598927i \(-0.795594\pi\)
−0.800804 + 0.598927i \(0.795594\pi\)
\(740\) 0 0
\(741\) 1.04351e29 0.0768080
\(742\) 0 0
\(743\) −2.26805e30 −1.62282 −0.811411 0.584476i \(-0.801300\pi\)
−0.811411 + 0.584476i \(0.801300\pi\)
\(744\) 0 0
\(745\) 3.75274e30 2.61040
\(746\) 0 0
\(747\) −1.37741e29 −0.0931530
\(748\) 0 0
\(749\) −1.31272e30 −0.863202
\(750\) 0 0
\(751\) 6.98349e29 0.446533 0.223266 0.974757i \(-0.428328\pi\)
0.223266 + 0.974757i \(0.428328\pi\)
\(752\) 0 0
\(753\) −5.58883e28 −0.0347515
\(754\) 0 0
\(755\) −1.97271e30 −1.19294
\(756\) 0 0
\(757\) −3.79778e29 −0.223369 −0.111685 0.993744i \(-0.535625\pi\)
−0.111685 + 0.993744i \(0.535625\pi\)
\(758\) 0 0
\(759\) 8.82794e29 0.505034
\(760\) 0 0
\(761\) −7.50371e29 −0.417577 −0.208789 0.977961i \(-0.566952\pi\)
−0.208789 + 0.977961i \(0.566952\pi\)
\(762\) 0 0
\(763\) 2.22736e30 1.20582
\(764\) 0 0
\(765\) −1.19300e30 −0.628342
\(766\) 0 0
\(767\) 2.01032e29 0.103018
\(768\) 0 0
\(769\) 2.16884e29 0.108144 0.0540719 0.998537i \(-0.482780\pi\)
0.0540719 + 0.998537i \(0.482780\pi\)
\(770\) 0 0
\(771\) −1.13271e30 −0.549600
\(772\) 0 0
\(773\) −3.08128e30 −1.45495 −0.727473 0.686136i \(-0.759305\pi\)
−0.727473 + 0.686136i \(0.759305\pi\)
\(774\) 0 0
\(775\) 8.56227e30 3.93478
\(776\) 0 0
\(777\) −1.09498e29 −0.0489761
\(778\) 0 0
\(779\) 2.94700e29 0.128303
\(780\) 0 0
\(781\) 1.30710e30 0.553951
\(782\) 0 0
\(783\) −4.16839e29 −0.171977
\(784\) 0 0
\(785\) 6.26430e30 2.51618
\(786\) 0 0
\(787\) −1.46460e30 −0.572775 −0.286387 0.958114i \(-0.592454\pi\)
−0.286387 + 0.958114i \(0.592454\pi\)
\(788\) 0 0
\(789\) −5.24608e28 −0.0199768
\(790\) 0 0
\(791\) −1.87325e30 −0.694612
\(792\) 0 0
\(793\) −5.64732e29 −0.203927
\(794\) 0 0
\(795\) 5.32490e30 1.87265
\(796\) 0 0
\(797\) 1.27731e30 0.437505 0.218752 0.975780i \(-0.429801\pi\)
0.218752 + 0.975780i \(0.429801\pi\)
\(798\) 0 0
\(799\) −2.33698e30 −0.779677
\(800\) 0 0
\(801\) 1.45091e29 0.0471519
\(802\) 0 0
\(803\) 1.06541e30 0.337292
\(804\) 0 0
\(805\) 5.21208e30 1.60751
\(806\) 0 0
\(807\) −1.26275e30 −0.379441
\(808\) 0 0
\(809\) −4.24975e30 −1.24424 −0.622120 0.782922i \(-0.713728\pi\)
−0.622120 + 0.782922i \(0.713728\pi\)
\(810\) 0 0
\(811\) −2.05863e30 −0.587298 −0.293649 0.955913i \(-0.594870\pi\)
−0.293649 + 0.955913i \(0.594870\pi\)
\(812\) 0 0
\(813\) 9.23732e29 0.256800
\(814\) 0 0
\(815\) −2.00716e30 −0.543784
\(816\) 0 0
\(817\) −9.68636e29 −0.255758
\(818\) 0 0
\(819\) 2.45901e29 0.0632818
\(820\) 0 0
\(821\) −1.80717e29 −0.0453309 −0.0226655 0.999743i \(-0.507215\pi\)
−0.0226655 + 0.999743i \(0.507215\pi\)
\(822\) 0 0
\(823\) 6.23532e30 1.52462 0.762308 0.647214i \(-0.224066\pi\)
0.762308 + 0.647214i \(0.224066\pi\)
\(824\) 0 0
\(825\) 4.71891e30 1.12480
\(826\) 0 0
\(827\) −3.37179e30 −0.783524 −0.391762 0.920067i \(-0.628134\pi\)
−0.391762 + 0.920067i \(0.628134\pi\)
\(828\) 0 0
\(829\) −2.70711e29 −0.0613315 −0.0306658 0.999530i \(-0.509763\pi\)
−0.0306658 + 0.999530i \(0.509763\pi\)
\(830\) 0 0
\(831\) −4.75328e30 −1.04998
\(832\) 0 0
\(833\) −2.21413e30 −0.476904
\(834\) 0 0
\(835\) 1.98735e28 0.00417413
\(836\) 0 0
\(837\) 1.41431e30 0.289685
\(838\) 0 0
\(839\) 7.45457e30 1.48909 0.744547 0.667570i \(-0.232666\pi\)
0.744547 + 0.667570i \(0.232666\pi\)
\(840\) 0 0
\(841\) −1.03399e30 −0.201446
\(842\) 0 0
\(843\) 4.91979e30 0.934882
\(844\) 0 0
\(845\) 9.54415e30 1.76906
\(846\) 0 0
\(847\) 1.77131e30 0.320272
\(848\) 0 0
\(849\) 2.63435e29 0.0464668
\(850\) 0 0
\(851\) 8.03088e29 0.138198
\(852\) 0 0
\(853\) −6.10653e30 −1.02525 −0.512625 0.858613i \(-0.671327\pi\)
−0.512625 + 0.858613i \(0.671327\pi\)
\(854\) 0 0
\(855\) 1.95293e30 0.319921
\(856\) 0 0
\(857\) −4.08307e30 −0.652662 −0.326331 0.945256i \(-0.605812\pi\)
−0.326331 + 0.945256i \(0.605812\pi\)
\(858\) 0 0
\(859\) −5.27189e30 −0.822316 −0.411158 0.911564i \(-0.634875\pi\)
−0.411158 + 0.911564i \(0.634875\pi\)
\(860\) 0 0
\(861\) 6.94452e29 0.105708
\(862\) 0 0
\(863\) −1.05537e31 −1.56780 −0.783902 0.620884i \(-0.786774\pi\)
−0.783902 + 0.620884i \(0.786774\pi\)
\(864\) 0 0
\(865\) 6.71693e30 0.973872
\(866\) 0 0
\(867\) −6.85488e28 −0.00970062
\(868\) 0 0
\(869\) −4.35586e30 −0.601681
\(870\) 0 0
\(871\) 1.21077e30 0.163256
\(872\) 0 0
\(873\) 1.99379e29 0.0262441
\(874\) 0 0
\(875\) 1.72027e31 2.21061
\(876\) 0 0
\(877\) 7.16069e30 0.898378 0.449189 0.893437i \(-0.351713\pi\)
0.449189 + 0.893437i \(0.351713\pi\)
\(878\) 0 0
\(879\) 5.71079e30 0.699541
\(880\) 0 0
\(881\) −1.05943e31 −1.26714 −0.633568 0.773687i \(-0.718410\pi\)
−0.633568 + 0.773687i \(0.718410\pi\)
\(882\) 0 0
\(883\) −6.60744e30 −0.771695 −0.385848 0.922562i \(-0.626091\pi\)
−0.385848 + 0.922562i \(0.626091\pi\)
\(884\) 0 0
\(885\) 3.76231e30 0.429092
\(886\) 0 0
\(887\) 2.74467e30 0.305698 0.152849 0.988250i \(-0.451155\pi\)
0.152849 + 0.988250i \(0.451155\pi\)
\(888\) 0 0
\(889\) 7.38406e29 0.0803205
\(890\) 0 0
\(891\) 7.79465e29 0.0828096
\(892\) 0 0
\(893\) 3.82561e30 0.396973
\(894\) 0 0
\(895\) −3.63808e30 −0.368749
\(896\) 0 0
\(897\) −1.80351e30 −0.178565
\(898\) 0 0
\(899\) −1.39071e31 −1.34512
\(900\) 0 0
\(901\) −1.79040e31 −1.69177
\(902\) 0 0
\(903\) −2.28256e30 −0.210718
\(904\) 0 0
\(905\) −3.88819e30 −0.350701
\(906\) 0 0
\(907\) −5.45568e30 −0.480809 −0.240404 0.970673i \(-0.577280\pi\)
−0.240404 + 0.970673i \(0.577280\pi\)
\(908\) 0 0
\(909\) 1.50774e30 0.129839
\(910\) 0 0
\(911\) 8.75739e30 0.736940 0.368470 0.929640i \(-0.379882\pi\)
0.368470 + 0.929640i \(0.379882\pi\)
\(912\) 0 0
\(913\) −2.53270e30 −0.208277
\(914\) 0 0
\(915\) −1.05689e31 −0.849396
\(916\) 0 0
\(917\) −1.32064e31 −1.03730
\(918\) 0 0
\(919\) −1.24295e31 −0.954206 −0.477103 0.878847i \(-0.658313\pi\)
−0.477103 + 0.878847i \(0.658313\pi\)
\(920\) 0 0
\(921\) −9.92984e30 −0.745102
\(922\) 0 0
\(923\) −2.67034e30 −0.195861
\(924\) 0 0
\(925\) 4.29285e30 0.307792
\(926\) 0 0
\(927\) −6.41997e30 −0.449983
\(928\) 0 0
\(929\) 2.24310e31 1.53703 0.768517 0.639829i \(-0.220995\pi\)
0.768517 + 0.639829i \(0.220995\pi\)
\(930\) 0 0
\(931\) 3.62451e30 0.242816
\(932\) 0 0
\(933\) −1.36031e31 −0.891008
\(934\) 0 0
\(935\) −2.19362e31 −1.40488
\(936\) 0 0
\(937\) −1.10052e31 −0.689176 −0.344588 0.938754i \(-0.611981\pi\)
−0.344588 + 0.938754i \(0.611981\pi\)
\(938\) 0 0
\(939\) −1.65135e31 −1.01123
\(940\) 0 0
\(941\) −2.38036e31 −1.42545 −0.712725 0.701444i \(-0.752539\pi\)
−0.712725 + 0.701444i \(0.752539\pi\)
\(942\) 0 0
\(943\) −5.09332e30 −0.298283
\(944\) 0 0
\(945\) 4.60202e30 0.263581
\(946\) 0 0
\(947\) −8.09762e30 −0.453610 −0.226805 0.973940i \(-0.572828\pi\)
−0.226805 + 0.973940i \(0.572828\pi\)
\(948\) 0 0
\(949\) −2.17660e30 −0.119257
\(950\) 0 0
\(951\) 1.76262e30 0.0944631
\(952\) 0 0
\(953\) −3.42232e30 −0.179410 −0.0897048 0.995968i \(-0.528592\pi\)
−0.0897048 + 0.995968i \(0.528592\pi\)
\(954\) 0 0
\(955\) 5.01715e31 2.57290
\(956\) 0 0
\(957\) −7.66461e30 −0.384516
\(958\) 0 0
\(959\) −5.53695e30 −0.271754
\(960\) 0 0
\(961\) 2.63603e31 1.26577
\(962\) 0 0
\(963\) 8.50095e30 0.399385
\(964\) 0 0
\(965\) 7.42339e31 3.41244
\(966\) 0 0
\(967\) −1.33121e31 −0.598780 −0.299390 0.954131i \(-0.596783\pi\)
−0.299390 + 0.954131i \(0.596783\pi\)
\(968\) 0 0
\(969\) −6.56638e30 −0.289019
\(970\) 0 0
\(971\) −9.71774e30 −0.418565 −0.209283 0.977855i \(-0.567113\pi\)
−0.209283 + 0.977855i \(0.567113\pi\)
\(972\) 0 0
\(973\) 4.68806e30 0.197610
\(974\) 0 0
\(975\) −9.64053e30 −0.397697
\(976\) 0 0
\(977\) −1.40637e31 −0.567816 −0.283908 0.958851i \(-0.591631\pi\)
−0.283908 + 0.958851i \(0.591631\pi\)
\(978\) 0 0
\(979\) 2.66785e30 0.105425
\(980\) 0 0
\(981\) −1.44240e31 −0.557906
\(982\) 0 0
\(983\) 4.87291e31 1.84492 0.922458 0.386098i \(-0.126177\pi\)
0.922458 + 0.386098i \(0.126177\pi\)
\(984\) 0 0
\(985\) −7.92741e31 −2.93799
\(986\) 0 0
\(987\) 9.01495e30 0.327064
\(988\) 0 0
\(989\) 1.67410e31 0.594594
\(990\) 0 0
\(991\) −1.17847e30 −0.0409773 −0.0204887 0.999790i \(-0.506522\pi\)
−0.0204887 + 0.999790i \(0.506522\pi\)
\(992\) 0 0
\(993\) −1.50926e31 −0.513803
\(994\) 0 0
\(995\) 5.98838e31 1.99603
\(996\) 0 0
\(997\) 5.05438e31 1.64956 0.824781 0.565453i \(-0.191299\pi\)
0.824781 + 0.565453i \(0.191299\pi\)
\(998\) 0 0
\(999\) 7.09089e29 0.0226602
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 48.22.a.d.1.1 1
4.3 odd 2 3.22.a.b.1.1 1
12.11 even 2 9.22.a.a.1.1 1
20.3 even 4 75.22.b.b.49.1 2
20.7 even 4 75.22.b.b.49.2 2
20.19 odd 2 75.22.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.22.a.b.1.1 1 4.3 odd 2
9.22.a.a.1.1 1 12.11 even 2
48.22.a.d.1.1 1 1.1 even 1 trivial
75.22.a.a.1.1 1 20.19 odd 2
75.22.b.b.49.1 2 20.3 even 4
75.22.b.b.49.2 2 20.7 even 4