Properties

Label 197.12.a.b.1.18
Level $197$
Weight $12$
Character 197.1
Self dual yes
Analytic conductor $151.364$
Analytic rank $0$
Dimension $92$
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] = [197,12,Mod(1,197)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(197, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("197.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 197 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 197.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: \(151.363606570\)
Analytic rank: \(0\)
Dimension: \(92\)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.18
Character \(\chi\) \(=\) 197.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-59.6321 q^{2} +54.3091 q^{3} +1507.99 q^{4} -3681.72 q^{5} -3238.57 q^{6} +14583.9 q^{7} +32202.1 q^{8} -174198. q^{9} +O(q^{10})\) \(q-59.6321 q^{2} +54.3091 q^{3} +1507.99 q^{4} -3681.72 q^{5} -3238.57 q^{6} +14583.9 q^{7} +32202.1 q^{8} -174198. q^{9} +219549. q^{10} -156448. q^{11} +81897.5 q^{12} -1.74648e6 q^{13} -869667. q^{14} -199951. q^{15} -5.00864e6 q^{16} +3.88025e6 q^{17} +1.03878e7 q^{18} -1.62315e7 q^{19} -5.55199e6 q^{20} +792037. q^{21} +9.32934e6 q^{22} -1.86347e6 q^{23} +1.74887e6 q^{24} -3.52731e7 q^{25} +1.04146e8 q^{26} -1.90812e7 q^{27} +2.19923e7 q^{28} -8.33015e7 q^{29} +1.19235e7 q^{30} +3.26662e7 q^{31} +2.32726e8 q^{32} -8.49657e6 q^{33} -2.31388e8 q^{34} -5.36938e7 q^{35} -2.62688e8 q^{36} +1.22917e8 q^{37} +9.67919e8 q^{38} -9.48497e7 q^{39} -1.18559e8 q^{40} -8.31993e8 q^{41} -4.72309e7 q^{42} -3.95017e8 q^{43} -2.35922e8 q^{44} +6.41347e8 q^{45} +1.11123e8 q^{46} -5.91028e7 q^{47} -2.72015e8 q^{48} -1.76464e9 q^{49} +2.10341e9 q^{50} +2.10733e8 q^{51} -2.63367e9 q^{52} -3.30628e8 q^{53} +1.13785e9 q^{54} +5.75999e8 q^{55} +4.69631e8 q^{56} -8.81519e8 q^{57} +4.96744e9 q^{58} +3.02707e9 q^{59} -3.01524e8 q^{60} +4.16365e9 q^{61} -1.94795e9 q^{62} -2.54047e9 q^{63} -3.62023e9 q^{64} +6.43005e9 q^{65} +5.06668e8 q^{66} -1.12005e10 q^{67} +5.85137e9 q^{68} -1.01203e8 q^{69} +3.20187e9 q^{70} +1.50655e10 q^{71} -5.60952e9 q^{72} -1.97083e10 q^{73} -7.32979e9 q^{74} -1.91565e9 q^{75} -2.44769e10 q^{76} -2.28162e9 q^{77} +5.65609e9 q^{78} -3.50481e10 q^{79} +1.84404e10 q^{80} +2.98223e10 q^{81} +4.96135e10 q^{82} -4.16870e10 q^{83} +1.19438e9 q^{84} -1.42860e10 q^{85} +2.35557e10 q^{86} -4.52403e9 q^{87} -5.03796e9 q^{88} -5.19236e10 q^{89} -3.82449e10 q^{90} -2.54704e10 q^{91} -2.81009e9 q^{92} +1.77407e9 q^{93} +3.52442e9 q^{94} +5.97599e10 q^{95} +1.26391e10 q^{96} -1.13549e10 q^{97} +1.05229e11 q^{98} +2.72529e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 92 q + 64 q^{2} + 2420 q^{3} + 98304 q^{4} + 16604 q^{5} + 46656 q^{6} + 305891 q^{7} + 234027 q^{8} + 5900444 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 92 q + 64 q^{2} + 2420 q^{3} + 98304 q^{4} + 16604 q^{5} + 46656 q^{6} + 305891 q^{7} + 234027 q^{8} + 5900444 q^{9} + 1074277 q^{10} + 595928 q^{11} + 4956160 q^{12} + 7463810 q^{13} + 4915769 q^{14} + 6749159 q^{15} + 109051904 q^{16} + 9869683 q^{17} + 15324721 q^{18} + 71500013 q^{19} + 79804779 q^{20} + 55741034 q^{21} + 120367289 q^{22} + 38597564 q^{23} + 104015637 q^{24} + 1039976880 q^{25} + 51802726 q^{26} + 665312351 q^{27} + 713160630 q^{28} - 2827541 q^{29} - 43013765 q^{30} + 939775728 q^{31} + 723479980 q^{32} + 978717002 q^{33} + 1063528738 q^{34} + 843197112 q^{35} + 6613742458 q^{36} + 2009031770 q^{37} + 918086794 q^{38} + 2279970607 q^{39} + 3093842646 q^{40} - 30014736 q^{41} + 3159549307 q^{42} + 6320365127 q^{43} + 3019176802 q^{44} + 5538230697 q^{45} + 4344764621 q^{46} + 2853606373 q^{47} + 15616060178 q^{48} + 31617715853 q^{49} - 28784136763 q^{50} - 7216660253 q^{51} + 19141188860 q^{52} + 3389732468 q^{53} + 14059623295 q^{54} + 23123173075 q^{55} + 75592019872 q^{56} + 27508365203 q^{57} + 42079603023 q^{58} + 40032802875 q^{59} + 99482549469 q^{60} + 31816702886 q^{61} + 35401555571 q^{62} + 79170604993 q^{63} + 168463042533 q^{64} + 50313809737 q^{65} + 93218912814 q^{66} + 129966589578 q^{67} + 73640879491 q^{68} - 9850635033 q^{69} + 62387700391 q^{70} + 32189826123 q^{71} + 47439469795 q^{72} + 60612500511 q^{73} - 110473527245 q^{74} + 48823210110 q^{75} + 48170982110 q^{76} - 9381499362 q^{77} - 251751253299 q^{78} + 40812909551 q^{79} - 36932560002 q^{80} + 294126355776 q^{81} + 29914326881 q^{82} + 112077199076 q^{83} - 242466547988 q^{84} - 65194167314 q^{85} - 384596985360 q^{86} + 30701235703 q^{87} + 102229734783 q^{88} - 47595308310 q^{89} - 957657692329 q^{90} + 217575098757 q^{91} - 762245602306 q^{92} - 47762776839 q^{93} - 251435906373 q^{94} - 84935429021 q^{95} - 1089429573223 q^{96} + 432665748880 q^{97} - 404245603446 q^{98} + 9542377031 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −59.6321 −1.31770 −0.658848 0.752276i \(-0.728956\pi\)
−0.658848 + 0.752276i \(0.728956\pi\)
\(3\) 54.3091 0.129035 0.0645173 0.997917i \(-0.479449\pi\)
0.0645173 + 0.997917i \(0.479449\pi\)
\(4\) 1507.99 0.736322
\(5\) −3681.72 −0.526885 −0.263443 0.964675i \(-0.584858\pi\)
−0.263443 + 0.964675i \(0.584858\pi\)
\(6\) −3238.57 −0.170028
\(7\) 14583.9 0.327970 0.163985 0.986463i \(-0.447565\pi\)
0.163985 + 0.986463i \(0.447565\pi\)
\(8\) 32202.1 0.347447
\(9\) −174198. −0.983350
\(10\) 219549. 0.694274
\(11\) −156448. −0.292894 −0.146447 0.989218i \(-0.546784\pi\)
−0.146447 + 0.989218i \(0.546784\pi\)
\(12\) 81897.5 0.0950109
\(13\) −1.74648e6 −1.30459 −0.652296 0.757964i \(-0.726194\pi\)
−0.652296 + 0.757964i \(0.726194\pi\)
\(14\) −869667. −0.432164
\(15\) −199951. −0.0679864
\(16\) −5.00864e6 −1.19415
\(17\) 3.88025e6 0.662813 0.331406 0.943488i \(-0.392477\pi\)
0.331406 + 0.943488i \(0.392477\pi\)
\(18\) 1.03878e7 1.29576
\(19\) −1.62315e7 −1.50388 −0.751941 0.659230i \(-0.770882\pi\)
−0.751941 + 0.659230i \(0.770882\pi\)
\(20\) −5.55199e6 −0.387957
\(21\) 792037. 0.0423194
\(22\) 9.32934e6 0.385946
\(23\) −1.86347e6 −0.0603697 −0.0301849 0.999544i \(-0.509610\pi\)
−0.0301849 + 0.999544i \(0.509610\pi\)
\(24\) 1.74887e6 0.0448327
\(25\) −3.52731e7 −0.722392
\(26\) 1.04146e8 1.71906
\(27\) −1.90812e7 −0.255921
\(28\) 2.19923e7 0.241491
\(29\) −8.33015e7 −0.754160 −0.377080 0.926181i \(-0.623072\pi\)
−0.377080 + 0.926181i \(0.623072\pi\)
\(30\) 1.19235e7 0.0895853
\(31\) 3.26662e7 0.204932 0.102466 0.994737i \(-0.467327\pi\)
0.102466 + 0.994737i \(0.467327\pi\)
\(32\) 2.32726e8 1.22608
\(33\) −8.49657e6 −0.0377935
\(34\) −2.31388e8 −0.873386
\(35\) −5.36938e7 −0.172802
\(36\) −2.62688e8 −0.724062
\(37\) 1.22917e8 0.291408 0.145704 0.989328i \(-0.453455\pi\)
0.145704 + 0.989328i \(0.453455\pi\)
\(38\) 9.67919e8 1.98166
\(39\) −9.48497e7 −0.168337
\(40\) −1.18559e8 −0.183065
\(41\) −8.31993e8 −1.12152 −0.560762 0.827977i \(-0.689492\pi\)
−0.560762 + 0.827977i \(0.689492\pi\)
\(42\) −4.72309e7 −0.0557641
\(43\) −3.95017e8 −0.409769 −0.204885 0.978786i \(-0.565682\pi\)
−0.204885 + 0.978786i \(0.565682\pi\)
\(44\) −2.35922e8 −0.215665
\(45\) 6.41347e8 0.518113
\(46\) 1.11123e8 0.0795489
\(47\) −5.91028e7 −0.0375898 −0.0187949 0.999823i \(-0.505983\pi\)
−0.0187949 + 0.999823i \(0.505983\pi\)
\(48\) −2.72015e8 −0.154087
\(49\) −1.76464e9 −0.892436
\(50\) 2.10341e9 0.951893
\(51\) 2.10733e8 0.0855257
\(52\) −2.63367e9 −0.960600
\(53\) −3.30628e8 −0.108598 −0.0542991 0.998525i \(-0.517292\pi\)
−0.0542991 + 0.998525i \(0.517292\pi\)
\(54\) 1.13785e9 0.337225
\(55\) 5.75999e8 0.154322
\(56\) 4.69631e8 0.113952
\(57\) −8.81519e8 −0.194053
\(58\) 4.96744e9 0.993754
\(59\) 3.02707e9 0.551234 0.275617 0.961268i \(-0.411118\pi\)
0.275617 + 0.961268i \(0.411118\pi\)
\(60\) −3.01524e8 −0.0500599
\(61\) 4.16365e9 0.631190 0.315595 0.948894i \(-0.397796\pi\)
0.315595 + 0.948894i \(0.397796\pi\)
\(62\) −1.94795e9 −0.270037
\(63\) −2.54047e9 −0.322509
\(64\) −3.62023e9 −0.421450
\(65\) 6.43005e9 0.687370
\(66\) 5.06668e8 0.0498003
\(67\) −1.12005e10 −1.01351 −0.506753 0.862091i \(-0.669154\pi\)
−0.506753 + 0.862091i \(0.669154\pi\)
\(68\) 5.85137e9 0.488044
\(69\) −1.01203e8 −0.00778978
\(70\) 3.20187e9 0.227701
\(71\) 1.50655e10 0.990974 0.495487 0.868615i \(-0.334990\pi\)
0.495487 + 0.868615i \(0.334990\pi\)
\(72\) −5.60952e9 −0.341663
\(73\) −1.97083e10 −1.11269 −0.556344 0.830952i \(-0.687796\pi\)
−0.556344 + 0.830952i \(0.687796\pi\)
\(74\) −7.32979e9 −0.383988
\(75\) −1.91565e9 −0.0932135
\(76\) −2.44769e10 −1.10734
\(77\) −2.28162e9 −0.0960605
\(78\) 5.65609e9 0.221818
\(79\) −3.50481e10 −1.28149 −0.640745 0.767753i \(-0.721375\pi\)
−0.640745 + 0.767753i \(0.721375\pi\)
\(80\) 1.84404e10 0.629181
\(81\) 2.98223e10 0.950328
\(82\) 4.96135e10 1.47783
\(83\) −4.16870e10 −1.16164 −0.580820 0.814032i \(-0.697268\pi\)
−0.580820 + 0.814032i \(0.697268\pi\)
\(84\) 1.19438e9 0.0311607
\(85\) −1.42860e10 −0.349226
\(86\) 2.35557e10 0.539951
\(87\) −4.52403e9 −0.0973127
\(88\) −5.03796e9 −0.101765
\(89\) −5.19236e10 −0.985643 −0.492821 0.870130i \(-0.664034\pi\)
−0.492821 + 0.870130i \(0.664034\pi\)
\(90\) −3.82449e10 −0.682715
\(91\) −2.54704e10 −0.427867
\(92\) −2.81009e9 −0.0444515
\(93\) 1.77407e9 0.0264432
\(94\) 3.52442e9 0.0495319
\(95\) 5.97599e10 0.792373
\(96\) 1.26391e10 0.158207
\(97\) −1.13549e10 −0.134257 −0.0671286 0.997744i \(-0.521384\pi\)
−0.0671286 + 0.997744i \(0.521384\pi\)
\(98\) 1.05229e11 1.17596
\(99\) 2.72529e10 0.288018
\(100\) −5.31913e10 −0.531913
\(101\) −2.49096e10 −0.235830 −0.117915 0.993024i \(-0.537621\pi\)
−0.117915 + 0.993024i \(0.537621\pi\)
\(102\) −1.25665e10 −0.112697
\(103\) −1.98283e11 −1.68531 −0.842656 0.538452i \(-0.819009\pi\)
−0.842656 + 0.538452i \(0.819009\pi\)
\(104\) −5.62403e10 −0.453277
\(105\) −2.91606e9 −0.0222975
\(106\) 1.97161e10 0.143099
\(107\) −2.27153e11 −1.56570 −0.782848 0.622213i \(-0.786234\pi\)
−0.782848 + 0.622213i \(0.786234\pi\)
\(108\) −2.87742e10 −0.188440
\(109\) −2.79407e11 −1.73937 −0.869684 0.493609i \(-0.835677\pi\)
−0.869684 + 0.493609i \(0.835677\pi\)
\(110\) −3.43480e10 −0.203349
\(111\) 6.67551e9 0.0376017
\(112\) −7.30453e10 −0.391646
\(113\) −2.19301e11 −1.11972 −0.559859 0.828588i \(-0.689145\pi\)
−0.559859 + 0.828588i \(0.689145\pi\)
\(114\) 5.25668e10 0.255703
\(115\) 6.86078e9 0.0318079
\(116\) −1.25618e11 −0.555305
\(117\) 3.04232e11 1.28287
\(118\) −1.80510e11 −0.726359
\(119\) 5.65891e10 0.217383
\(120\) −6.43884e9 −0.0236217
\(121\) −2.60836e11 −0.914213
\(122\) −2.48287e11 −0.831716
\(123\) −4.51848e10 −0.144715
\(124\) 4.92602e10 0.150896
\(125\) 3.09637e11 0.907503
\(126\) 1.51494e11 0.424969
\(127\) 2.07166e11 0.556415 0.278207 0.960521i \(-0.410260\pi\)
0.278207 + 0.960521i \(0.410260\pi\)
\(128\) −2.60740e11 −0.670738
\(129\) −2.14530e10 −0.0528744
\(130\) −3.83437e11 −0.905745
\(131\) 1.28725e10 0.0291522 0.0145761 0.999894i \(-0.495360\pi\)
0.0145761 + 0.999894i \(0.495360\pi\)
\(132\) −1.28127e10 −0.0278282
\(133\) −2.36718e11 −0.493228
\(134\) 6.67910e11 1.33549
\(135\) 7.02517e10 0.134841
\(136\) 1.24952e11 0.230293
\(137\) 2.14830e10 0.0380304 0.0190152 0.999819i \(-0.493947\pi\)
0.0190152 + 0.999819i \(0.493947\pi\)
\(138\) 6.03497e9 0.0102646
\(139\) −1.52919e11 −0.249965 −0.124983 0.992159i \(-0.539887\pi\)
−0.124983 + 0.992159i \(0.539887\pi\)
\(140\) −8.09695e10 −0.127238
\(141\) −3.20982e9 −0.00485038
\(142\) −8.98387e11 −1.30580
\(143\) 2.73234e11 0.382108
\(144\) 8.72492e11 1.17427
\(145\) 3.06693e11 0.397356
\(146\) 1.17525e12 1.46618
\(147\) −9.58359e10 −0.115155
\(148\) 1.85357e11 0.214570
\(149\) 2.26761e11 0.252956 0.126478 0.991969i \(-0.459633\pi\)
0.126478 + 0.991969i \(0.459633\pi\)
\(150\) 1.14234e11 0.122827
\(151\) 1.72647e12 1.78972 0.894862 0.446344i \(-0.147274\pi\)
0.894862 + 0.446344i \(0.147274\pi\)
\(152\) −5.22688e11 −0.522520
\(153\) −6.75930e11 −0.651777
\(154\) 1.36058e11 0.126579
\(155\) −1.20268e11 −0.107975
\(156\) −1.43032e11 −0.123951
\(157\) 1.64135e12 1.37326 0.686631 0.727006i \(-0.259089\pi\)
0.686631 + 0.727006i \(0.259089\pi\)
\(158\) 2.08999e12 1.68861
\(159\) −1.79561e10 −0.0140129
\(160\) −8.56831e11 −0.646004
\(161\) −2.71766e10 −0.0197994
\(162\) −1.77837e12 −1.25224
\(163\) −4.11704e11 −0.280255 −0.140127 0.990133i \(-0.544751\pi\)
−0.140127 + 0.990133i \(0.544751\pi\)
\(164\) −1.25464e12 −0.825803
\(165\) 3.12820e10 0.0199128
\(166\) 2.48589e12 1.53069
\(167\) −1.19365e12 −0.711109 −0.355555 0.934656i \(-0.615708\pi\)
−0.355555 + 0.934656i \(0.615708\pi\)
\(168\) 2.55053e10 0.0147038
\(169\) 1.25803e12 0.701961
\(170\) 8.51905e11 0.460174
\(171\) 2.82749e12 1.47884
\(172\) −5.95680e11 −0.301722
\(173\) −1.17438e12 −0.576175 −0.288087 0.957604i \(-0.593019\pi\)
−0.288087 + 0.957604i \(0.593019\pi\)
\(174\) 2.69777e11 0.128229
\(175\) −5.14418e11 −0.236923
\(176\) 7.83593e11 0.349761
\(177\) 1.64397e11 0.0711282
\(178\) 3.09631e12 1.29878
\(179\) 2.76587e12 1.12497 0.562484 0.826808i \(-0.309846\pi\)
0.562484 + 0.826808i \(0.309846\pi\)
\(180\) 9.67143e11 0.381498
\(181\) 1.68018e12 0.642869 0.321435 0.946932i \(-0.395835\pi\)
0.321435 + 0.946932i \(0.395835\pi\)
\(182\) 1.51885e12 0.563798
\(183\) 2.26124e11 0.0814453
\(184\) −6.00076e10 −0.0209753
\(185\) −4.52546e11 −0.153539
\(186\) −1.05792e11 −0.0348441
\(187\) −6.07059e11 −0.194134
\(188\) −8.91262e10 −0.0276782
\(189\) −2.78278e11 −0.0839342
\(190\) −3.56361e12 −1.04411
\(191\) 7.94093e11 0.226041 0.113021 0.993593i \(-0.463947\pi\)
0.113021 + 0.993593i \(0.463947\pi\)
\(192\) −1.96611e11 −0.0543816
\(193\) 1.06857e12 0.287234 0.143617 0.989633i \(-0.454127\pi\)
0.143617 + 0.989633i \(0.454127\pi\)
\(194\) 6.77115e11 0.176910
\(195\) 3.49210e11 0.0886945
\(196\) −2.66105e12 −0.657120
\(197\) −2.96709e11 −0.0712470
\(198\) −1.62515e12 −0.379520
\(199\) 1.15668e12 0.262736 0.131368 0.991334i \(-0.458063\pi\)
0.131368 + 0.991334i \(0.458063\pi\)
\(200\) −1.13587e12 −0.250993
\(201\) −6.08290e11 −0.130777
\(202\) 1.48541e12 0.310753
\(203\) −1.21486e12 −0.247342
\(204\) 3.17783e11 0.0629745
\(205\) 3.06317e12 0.590914
\(206\) 1.18240e13 2.22073
\(207\) 3.24612e11 0.0593646
\(208\) 8.74748e12 1.55788
\(209\) 2.53939e12 0.440479
\(210\) 1.73891e11 0.0293813
\(211\) 6.19328e12 1.01945 0.509727 0.860336i \(-0.329746\pi\)
0.509727 + 0.860336i \(0.329746\pi\)
\(212\) −4.98583e11 −0.0799633
\(213\) 8.18193e11 0.127870
\(214\) 1.35456e13 2.06311
\(215\) 1.45434e12 0.215901
\(216\) −6.14455e11 −0.0889190
\(217\) 4.76399e11 0.0672113
\(218\) 1.66616e13 2.29196
\(219\) −1.07034e12 −0.143575
\(220\) 8.68600e11 0.113630
\(221\) −6.77678e12 −0.864701
\(222\) −3.98075e11 −0.0495477
\(223\) 1.17986e13 1.43269 0.716346 0.697745i \(-0.245813\pi\)
0.716346 + 0.697745i \(0.245813\pi\)
\(224\) 3.39404e12 0.402118
\(225\) 6.14448e12 0.710364
\(226\) 1.30774e13 1.47545
\(227\) −1.25970e13 −1.38715 −0.693577 0.720382i \(-0.743966\pi\)
−0.693577 + 0.720382i \(0.743966\pi\)
\(228\) −1.32932e12 −0.142885
\(229\) −4.02361e12 −0.422203 −0.211101 0.977464i \(-0.567705\pi\)
−0.211101 + 0.977464i \(0.567705\pi\)
\(230\) −4.09122e11 −0.0419131
\(231\) −1.23913e11 −0.0123951
\(232\) −2.68248e12 −0.262031
\(233\) −1.93281e13 −1.84388 −0.921939 0.387334i \(-0.873396\pi\)
−0.921939 + 0.387334i \(0.873396\pi\)
\(234\) −1.81420e13 −1.69043
\(235\) 2.17600e11 0.0198055
\(236\) 4.56478e12 0.405886
\(237\) −1.90343e12 −0.165357
\(238\) −3.37453e12 −0.286444
\(239\) −1.17143e13 −0.971693 −0.485846 0.874044i \(-0.661489\pi\)
−0.485846 + 0.874044i \(0.661489\pi\)
\(240\) 1.00148e12 0.0811861
\(241\) 1.19550e13 0.947230 0.473615 0.880732i \(-0.342949\pi\)
0.473615 + 0.880732i \(0.342949\pi\)
\(242\) 1.55542e13 1.20465
\(243\) 4.99980e12 0.378546
\(244\) 6.27873e12 0.464759
\(245\) 6.49690e12 0.470211
\(246\) 2.69447e12 0.190691
\(247\) 2.83480e13 1.96195
\(248\) 1.05192e12 0.0712029
\(249\) −2.26399e12 −0.149892
\(250\) −1.84643e13 −1.19581
\(251\) −1.50205e13 −0.951653 −0.475826 0.879539i \(-0.657851\pi\)
−0.475826 + 0.879539i \(0.657851\pi\)
\(252\) −3.83100e12 −0.237470
\(253\) 2.91537e11 0.0176820
\(254\) −1.23538e13 −0.733186
\(255\) −7.75861e11 −0.0450622
\(256\) 2.29627e13 1.30528
\(257\) 2.14129e13 1.19136 0.595680 0.803222i \(-0.296883\pi\)
0.595680 + 0.803222i \(0.296883\pi\)
\(258\) 1.27929e12 0.0696723
\(259\) 1.79260e12 0.0955731
\(260\) 9.69643e12 0.506126
\(261\) 1.45109e13 0.741604
\(262\) −7.67614e11 −0.0384137
\(263\) 2.38120e13 1.16691 0.583457 0.812144i \(-0.301700\pi\)
0.583457 + 0.812144i \(0.301700\pi\)
\(264\) −2.73607e11 −0.0131313
\(265\) 1.21728e12 0.0572188
\(266\) 1.41160e13 0.649924
\(267\) −2.81992e12 −0.127182
\(268\) −1.68902e13 −0.746267
\(269\) 8.47485e12 0.366855 0.183428 0.983033i \(-0.441281\pi\)
0.183428 + 0.983033i \(0.441281\pi\)
\(270\) −4.18926e12 −0.177679
\(271\) −1.46872e13 −0.610389 −0.305194 0.952290i \(-0.598721\pi\)
−0.305194 + 0.952290i \(0.598721\pi\)
\(272\) −1.94348e13 −0.791499
\(273\) −1.38328e12 −0.0552096
\(274\) −1.28107e12 −0.0501125
\(275\) 5.51841e12 0.211585
\(276\) −1.52613e11 −0.00573578
\(277\) 2.52911e13 0.931814 0.465907 0.884834i \(-0.345728\pi\)
0.465907 + 0.884834i \(0.345728\pi\)
\(278\) 9.11887e12 0.329378
\(279\) −5.69036e12 −0.201519
\(280\) −1.72905e12 −0.0600397
\(281\) −1.30811e13 −0.445410 −0.222705 0.974886i \(-0.571489\pi\)
−0.222705 + 0.974886i \(0.571489\pi\)
\(282\) 1.91408e11 0.00639132
\(283\) 2.72947e12 0.0893825 0.0446912 0.999001i \(-0.485770\pi\)
0.0446912 + 0.999001i \(0.485770\pi\)
\(284\) 2.27186e13 0.729676
\(285\) 3.24551e12 0.102244
\(286\) −1.62935e13 −0.503502
\(287\) −1.21337e13 −0.367826
\(288\) −4.05402e13 −1.20567
\(289\) −1.92155e13 −0.560679
\(290\) −1.82887e13 −0.523594
\(291\) −6.16673e11 −0.0173238
\(292\) −2.97198e13 −0.819296
\(293\) 2.53825e13 0.686693 0.343347 0.939209i \(-0.388439\pi\)
0.343347 + 0.939209i \(0.388439\pi\)
\(294\) 5.71490e12 0.151739
\(295\) −1.11448e13 −0.290437
\(296\) 3.95818e12 0.101249
\(297\) 2.98522e12 0.0749577
\(298\) −1.35223e13 −0.333319
\(299\) 3.25451e12 0.0787579
\(300\) −2.88877e12 −0.0686351
\(301\) −5.76088e12 −0.134392
\(302\) −1.02953e14 −2.35831
\(303\) −1.35282e12 −0.0304303
\(304\) 8.12977e13 1.79586
\(305\) −1.53294e13 −0.332564
\(306\) 4.03072e13 0.858844
\(307\) −7.61067e13 −1.59280 −0.796401 0.604768i \(-0.793266\pi\)
−0.796401 + 0.604768i \(0.793266\pi\)
\(308\) −3.44066e12 −0.0707315
\(309\) −1.07686e13 −0.217464
\(310\) 7.17182e12 0.142279
\(311\) 1.59752e12 0.0311361 0.0155680 0.999879i \(-0.495044\pi\)
0.0155680 + 0.999879i \(0.495044\pi\)
\(312\) −3.05436e12 −0.0584884
\(313\) 3.92075e13 0.737693 0.368846 0.929490i \(-0.379753\pi\)
0.368846 + 0.929490i \(0.379753\pi\)
\(314\) −9.78772e13 −1.80954
\(315\) 9.35332e12 0.169925
\(316\) −5.28521e13 −0.943590
\(317\) 6.08389e13 1.06747 0.533735 0.845652i \(-0.320788\pi\)
0.533735 + 0.845652i \(0.320788\pi\)
\(318\) 1.07076e12 0.0184648
\(319\) 1.30324e13 0.220889
\(320\) 1.33287e13 0.222056
\(321\) −1.23365e13 −0.202029
\(322\) 1.62060e12 0.0260896
\(323\) −6.29823e13 −0.996793
\(324\) 4.49716e13 0.699747
\(325\) 6.16036e13 0.942427
\(326\) 2.45508e13 0.369291
\(327\) −1.51743e13 −0.224439
\(328\) −2.67919e13 −0.389671
\(329\) −8.61947e11 −0.0123283
\(330\) −1.86541e12 −0.0262391
\(331\) −1.10951e14 −1.53489 −0.767447 0.641112i \(-0.778473\pi\)
−0.767447 + 0.641112i \(0.778473\pi\)
\(332\) −6.28635e13 −0.855341
\(333\) −2.14118e13 −0.286556
\(334\) 7.11798e13 0.937026
\(335\) 4.12372e13 0.534001
\(336\) −3.96703e12 −0.0505358
\(337\) 6.79224e13 0.851233 0.425616 0.904904i \(-0.360057\pi\)
0.425616 + 0.904904i \(0.360057\pi\)
\(338\) −7.50188e13 −0.924971
\(339\) −1.19100e13 −0.144482
\(340\) −2.15431e13 −0.257143
\(341\) −5.11057e12 −0.0600233
\(342\) −1.68609e14 −1.94867
\(343\) −5.45723e13 −0.620662
\(344\) −1.27204e13 −0.142373
\(345\) 3.72603e11 0.00410432
\(346\) 7.00306e13 0.759223
\(347\) −6.94224e13 −0.740777 −0.370389 0.928877i \(-0.620776\pi\)
−0.370389 + 0.928877i \(0.620776\pi\)
\(348\) −6.82218e12 −0.0716535
\(349\) −5.29484e13 −0.547411 −0.273705 0.961814i \(-0.588249\pi\)
−0.273705 + 0.961814i \(0.588249\pi\)
\(350\) 3.06758e13 0.312192
\(351\) 3.33249e13 0.333872
\(352\) −3.64095e13 −0.359112
\(353\) 9.49390e13 0.921900 0.460950 0.887426i \(-0.347509\pi\)
0.460950 + 0.887426i \(0.347509\pi\)
\(354\) −9.80336e12 −0.0937254
\(355\) −5.54669e13 −0.522129
\(356\) −7.83001e13 −0.725750
\(357\) 3.07331e12 0.0280499
\(358\) −1.64935e14 −1.48236
\(359\) 3.28215e13 0.290496 0.145248 0.989395i \(-0.453602\pi\)
0.145248 + 0.989395i \(0.453602\pi\)
\(360\) 2.06527e13 0.180017
\(361\) 1.46972e14 1.26166
\(362\) −1.00192e14 −0.847106
\(363\) −1.41658e13 −0.117965
\(364\) −3.84091e13 −0.315048
\(365\) 7.25604e13 0.586258
\(366\) −1.34843e13 −0.107320
\(367\) −9.38110e12 −0.0735513 −0.0367757 0.999324i \(-0.511709\pi\)
−0.0367757 + 0.999324i \(0.511709\pi\)
\(368\) 9.33344e12 0.0720906
\(369\) 1.44931e14 1.10285
\(370\) 2.69863e13 0.202317
\(371\) −4.82184e12 −0.0356169
\(372\) 2.67528e12 0.0194707
\(373\) −7.34751e13 −0.526917 −0.263458 0.964671i \(-0.584863\pi\)
−0.263458 + 0.964671i \(0.584863\pi\)
\(374\) 3.62002e13 0.255810
\(375\) 1.68161e13 0.117099
\(376\) −1.90323e12 −0.0130605
\(377\) 1.45484e14 0.983872
\(378\) 1.65943e13 0.110600
\(379\) 2.22745e14 1.46316 0.731580 0.681756i \(-0.238783\pi\)
0.731580 + 0.681756i \(0.238783\pi\)
\(380\) 9.01171e13 0.583442
\(381\) 1.12510e13 0.0717967
\(382\) −4.73534e13 −0.297854
\(383\) −1.03751e13 −0.0643280 −0.0321640 0.999483i \(-0.510240\pi\)
−0.0321640 + 0.999483i \(0.510240\pi\)
\(384\) −1.41606e13 −0.0865484
\(385\) 8.40030e12 0.0506129
\(386\) −6.37208e13 −0.378487
\(387\) 6.88110e13 0.402946
\(388\) −1.71230e13 −0.0988566
\(389\) 2.32767e14 1.32495 0.662474 0.749085i \(-0.269506\pi\)
0.662474 + 0.749085i \(0.269506\pi\)
\(390\) −2.08241e13 −0.116872
\(391\) −7.23073e12 −0.0400138
\(392\) −5.68250e13 −0.310075
\(393\) 6.99094e11 0.00376163
\(394\) 1.76934e13 0.0938819
\(395\) 1.29037e14 0.675198
\(396\) 4.10970e13 0.212074
\(397\) 2.47962e14 1.26193 0.630967 0.775810i \(-0.282658\pi\)
0.630967 + 0.775810i \(0.282658\pi\)
\(398\) −6.89751e13 −0.346207
\(399\) −1.28560e13 −0.0636434
\(400\) 1.76670e14 0.862646
\(401\) 1.70142e14 0.819438 0.409719 0.912212i \(-0.365627\pi\)
0.409719 + 0.912212i \(0.365627\pi\)
\(402\) 3.62736e13 0.172325
\(403\) −5.70507e13 −0.267352
\(404\) −3.75634e13 −0.173647
\(405\) −1.09797e14 −0.500713
\(406\) 7.24446e13 0.325921
\(407\) −1.92301e13 −0.0853519
\(408\) 6.78605e12 0.0297157
\(409\) −1.08911e14 −0.470535 −0.235267 0.971931i \(-0.575597\pi\)
−0.235267 + 0.971931i \(0.575597\pi\)
\(410\) −1.82663e14 −0.778645
\(411\) 1.16672e12 0.00490724
\(412\) −2.99008e14 −1.24093
\(413\) 4.41464e13 0.180788
\(414\) −1.93573e13 −0.0782244
\(415\) 1.53480e14 0.612051
\(416\) −4.06450e14 −1.59954
\(417\) −8.30488e12 −0.0322541
\(418\) −1.51429e14 −0.580417
\(419\) 1.61974e14 0.612728 0.306364 0.951914i \(-0.400888\pi\)
0.306364 + 0.951914i \(0.400888\pi\)
\(420\) −4.39738e12 −0.0164181
\(421\) 3.20398e14 1.18070 0.590349 0.807148i \(-0.298990\pi\)
0.590349 + 0.807148i \(0.298990\pi\)
\(422\) −3.69319e14 −1.34333
\(423\) 1.02956e13 0.0369639
\(424\) −1.06469e13 −0.0377322
\(425\) −1.36868e14 −0.478811
\(426\) −4.87906e13 −0.168494
\(427\) 6.07221e13 0.207011
\(428\) −3.42544e14 −1.15286
\(429\) 1.48391e13 0.0493051
\(430\) −8.67255e13 −0.284492
\(431\) −1.16939e14 −0.378735 −0.189368 0.981906i \(-0.560644\pi\)
−0.189368 + 0.981906i \(0.560644\pi\)
\(432\) 9.55709e13 0.305608
\(433\) −5.72084e14 −1.80624 −0.903122 0.429385i \(-0.858730\pi\)
−0.903122 + 0.429385i \(0.858730\pi\)
\(434\) −2.84087e13 −0.0885641
\(435\) 1.66562e13 0.0512726
\(436\) −4.21342e14 −1.28073
\(437\) 3.02469e13 0.0907890
\(438\) 6.38266e13 0.189188
\(439\) −3.90439e13 −0.114287 −0.0571437 0.998366i \(-0.518199\pi\)
−0.0571437 + 0.998366i \(0.518199\pi\)
\(440\) 1.85484e13 0.0536187
\(441\) 3.07395e14 0.877577
\(442\) 4.04114e14 1.13941
\(443\) −1.17408e14 −0.326946 −0.163473 0.986548i \(-0.552270\pi\)
−0.163473 + 0.986548i \(0.552270\pi\)
\(444\) 1.00666e13 0.0276870
\(445\) 1.91168e14 0.519321
\(446\) −7.03574e14 −1.88785
\(447\) 1.23152e13 0.0326400
\(448\) −5.27970e13 −0.138223
\(449\) −1.70371e14 −0.440597 −0.220298 0.975433i \(-0.570703\pi\)
−0.220298 + 0.975433i \(0.570703\pi\)
\(450\) −3.66408e14 −0.936044
\(451\) 1.30164e14 0.328488
\(452\) −3.30703e14 −0.824473
\(453\) 9.37631e13 0.230936
\(454\) 7.51185e14 1.82785
\(455\) 9.37750e13 0.225437
\(456\) −2.83868e13 −0.0674232
\(457\) −7.02155e14 −1.64776 −0.823880 0.566764i \(-0.808195\pi\)
−0.823880 + 0.566764i \(0.808195\pi\)
\(458\) 2.39936e14 0.556334
\(459\) −7.40399e13 −0.169627
\(460\) 1.03460e13 0.0234209
\(461\) −2.37670e14 −0.531642 −0.265821 0.964022i \(-0.585643\pi\)
−0.265821 + 0.964022i \(0.585643\pi\)
\(462\) 7.38919e12 0.0163330
\(463\) 2.98063e14 0.651048 0.325524 0.945534i \(-0.394459\pi\)
0.325524 + 0.945534i \(0.394459\pi\)
\(464\) 4.17227e14 0.900582
\(465\) −6.53163e12 −0.0139326
\(466\) 1.15258e15 2.42967
\(467\) −7.66413e14 −1.59669 −0.798344 0.602202i \(-0.794290\pi\)
−0.798344 + 0.602202i \(0.794290\pi\)
\(468\) 4.58778e14 0.944606
\(469\) −1.63347e14 −0.332399
\(470\) −1.29759e13 −0.0260976
\(471\) 8.91403e13 0.177198
\(472\) 9.74779e13 0.191525
\(473\) 6.17997e13 0.120019
\(474\) 1.13506e14 0.217890
\(475\) 5.72535e14 1.08639
\(476\) 8.53357e13 0.160064
\(477\) 5.75946e13 0.106790
\(478\) 6.98550e14 1.28040
\(479\) −1.94492e14 −0.352417 −0.176209 0.984353i \(-0.556383\pi\)
−0.176209 + 0.984353i \(0.556383\pi\)
\(480\) −4.65337e13 −0.0833568
\(481\) −2.14672e14 −0.380169
\(482\) −7.12901e14 −1.24816
\(483\) −1.47594e12 −0.00255481
\(484\) −3.93337e14 −0.673155
\(485\) 4.18055e13 0.0707382
\(486\) −2.98149e14 −0.498808
\(487\) −8.15070e14 −1.34830 −0.674148 0.738596i \(-0.735489\pi\)
−0.674148 + 0.738596i \(0.735489\pi\)
\(488\) 1.34078e14 0.219305
\(489\) −2.23593e13 −0.0361626
\(490\) −3.87424e14 −0.619595
\(491\) −5.34800e14 −0.845752 −0.422876 0.906188i \(-0.638979\pi\)
−0.422876 + 0.906188i \(0.638979\pi\)
\(492\) −6.81381e13 −0.106557
\(493\) −3.23231e14 −0.499867
\(494\) −1.69045e15 −2.58526
\(495\) −1.00338e14 −0.151752
\(496\) −1.63613e14 −0.244719
\(497\) 2.19713e14 0.325009
\(498\) 1.35006e14 0.197512
\(499\) 2.42900e14 0.351459 0.175729 0.984439i \(-0.443772\pi\)
0.175729 + 0.984439i \(0.443772\pi\)
\(500\) 4.66929e14 0.668214
\(501\) −6.48261e13 −0.0917576
\(502\) 8.95703e14 1.25399
\(503\) −4.30988e14 −0.596816 −0.298408 0.954438i \(-0.596456\pi\)
−0.298408 + 0.954438i \(0.596456\pi\)
\(504\) −8.18086e13 −0.112055
\(505\) 9.17104e13 0.124256
\(506\) −1.73849e13 −0.0232994
\(507\) 6.83223e13 0.0905772
\(508\) 3.12404e14 0.409701
\(509\) 4.42162e14 0.573633 0.286816 0.957986i \(-0.407403\pi\)
0.286816 + 0.957986i \(0.407403\pi\)
\(510\) 4.62662e13 0.0593783
\(511\) −2.87423e14 −0.364928
\(512\) −8.35319e14 −1.04922
\(513\) 3.09717e14 0.384875
\(514\) −1.27690e15 −1.56985
\(515\) 7.30022e14 0.887966
\(516\) −3.23509e13 −0.0389325
\(517\) 9.24653e12 0.0110098
\(518\) −1.06897e14 −0.125936
\(519\) −6.37794e13 −0.0743464
\(520\) 2.07061e14 0.238825
\(521\) 1.50039e14 0.171237 0.0856186 0.996328i \(-0.472713\pi\)
0.0856186 + 0.996328i \(0.472713\pi\)
\(522\) −8.65316e14 −0.977208
\(523\) 5.71183e14 0.638287 0.319143 0.947706i \(-0.396605\pi\)
0.319143 + 0.947706i \(0.396605\pi\)
\(524\) 1.94116e13 0.0214654
\(525\) −2.79376e13 −0.0305712
\(526\) −1.41996e15 −1.53764
\(527\) 1.26753e14 0.135831
\(528\) 4.25562e13 0.0451312
\(529\) −9.49337e14 −0.996355
\(530\) −7.25891e13 −0.0753970
\(531\) −5.27308e14 −0.542056
\(532\) −3.56968e14 −0.363175
\(533\) 1.45306e15 1.46313
\(534\) 1.68158e14 0.167587
\(535\) 8.36314e14 0.824942
\(536\) −3.60680e14 −0.352140
\(537\) 1.50212e14 0.145160
\(538\) −5.05373e14 −0.483404
\(539\) 2.76075e14 0.261390
\(540\) 1.05939e14 0.0992862
\(541\) 7.30107e14 0.677332 0.338666 0.940907i \(-0.390024\pi\)
0.338666 + 0.940907i \(0.390024\pi\)
\(542\) 8.75826e14 0.804307
\(543\) 9.12489e13 0.0829523
\(544\) 9.03034e14 0.812663
\(545\) 1.02870e15 0.916447
\(546\) 8.24877e13 0.0727494
\(547\) 1.13981e15 0.995181 0.497590 0.867412i \(-0.334218\pi\)
0.497590 + 0.867412i \(0.334218\pi\)
\(548\) 3.23960e13 0.0280026
\(549\) −7.25297e14 −0.620680
\(550\) −3.29074e14 −0.278804
\(551\) 1.35211e15 1.13417
\(552\) −3.25896e12 −0.00270654
\(553\) −5.11137e14 −0.420290
\(554\) −1.50816e15 −1.22785
\(555\) −2.45774e13 −0.0198118
\(556\) −2.30600e14 −0.184055
\(557\) 4.30559e14 0.340274 0.170137 0.985420i \(-0.445579\pi\)
0.170137 + 0.985420i \(0.445579\pi\)
\(558\) 3.39328e14 0.265541
\(559\) 6.89889e14 0.534582
\(560\) 2.68933e14 0.206352
\(561\) −3.29688e13 −0.0250500
\(562\) 7.80055e14 0.586915
\(563\) 1.38321e15 1.03060 0.515301 0.857009i \(-0.327680\pi\)
0.515301 + 0.857009i \(0.327680\pi\)
\(564\) −4.84037e12 −0.00357144
\(565\) 8.07404e14 0.589963
\(566\) −1.62764e14 −0.117779
\(567\) 4.34924e14 0.311679
\(568\) 4.85140e14 0.344311
\(569\) −2.76214e15 −1.94146 −0.970730 0.240172i \(-0.922796\pi\)
−0.970730 + 0.240172i \(0.922796\pi\)
\(570\) −1.93536e14 −0.134726
\(571\) 7.65857e14 0.528018 0.264009 0.964520i \(-0.414955\pi\)
0.264009 + 0.964520i \(0.414955\pi\)
\(572\) 4.12033e14 0.281354
\(573\) 4.31265e13 0.0291671
\(574\) 7.23557e14 0.484683
\(575\) 6.57302e13 0.0436106
\(576\) 6.30635e14 0.414433
\(577\) 1.78338e15 1.16085 0.580425 0.814313i \(-0.302886\pi\)
0.580425 + 0.814313i \(0.302886\pi\)
\(578\) 1.14586e15 0.738804
\(579\) 5.80329e13 0.0370631
\(580\) 4.62489e14 0.292582
\(581\) −6.07959e14 −0.380983
\(582\) 3.67735e13 0.0228275
\(583\) 5.17263e13 0.0318078
\(584\) −6.34648e14 −0.386600
\(585\) −1.12010e15 −0.675926
\(586\) −1.51361e15 −0.904853
\(587\) −1.76590e14 −0.104582 −0.0522909 0.998632i \(-0.516652\pi\)
−0.0522909 + 0.998632i \(0.516652\pi\)
\(588\) −1.44519e14 −0.0847912
\(589\) −5.30221e14 −0.308193
\(590\) 6.64589e14 0.382708
\(591\) −1.61140e13 −0.00919333
\(592\) −6.15646e14 −0.347986
\(593\) 2.06251e15 1.15503 0.577517 0.816379i \(-0.304022\pi\)
0.577517 + 0.816379i \(0.304022\pi\)
\(594\) −1.78015e14 −0.0987715
\(595\) −2.08345e14 −0.114536
\(596\) 3.41953e14 0.186257
\(597\) 6.28181e13 0.0339021
\(598\) −1.94073e14 −0.103779
\(599\) 2.68938e14 0.142497 0.0712483 0.997459i \(-0.477302\pi\)
0.0712483 + 0.997459i \(0.477302\pi\)
\(600\) −6.16879e13 −0.0323868
\(601\) 3.19450e15 1.66186 0.830928 0.556380i \(-0.187810\pi\)
0.830928 + 0.556380i \(0.187810\pi\)
\(602\) 3.43533e14 0.177088
\(603\) 1.95110e15 0.996631
\(604\) 2.60350e15 1.31781
\(605\) 9.60324e14 0.481685
\(606\) 8.06715e13 0.0400978
\(607\) −9.76362e14 −0.480920 −0.240460 0.970659i \(-0.577298\pi\)
−0.240460 + 0.970659i \(0.577298\pi\)
\(608\) −3.77749e15 −1.84388
\(609\) −6.59779e13 −0.0319156
\(610\) 9.14124e14 0.438219
\(611\) 1.03222e14 0.0490393
\(612\) −1.01929e15 −0.479918
\(613\) 1.06345e15 0.496233 0.248116 0.968730i \(-0.420188\pi\)
0.248116 + 0.968730i \(0.420188\pi\)
\(614\) 4.53840e15 2.09883
\(615\) 1.66358e14 0.0762483
\(616\) −7.34730e13 −0.0333760
\(617\) −8.68888e14 −0.391197 −0.195598 0.980684i \(-0.562665\pi\)
−0.195598 + 0.980684i \(0.562665\pi\)
\(618\) 6.42152e14 0.286551
\(619\) 5.77597e14 0.255462 0.127731 0.991809i \(-0.459231\pi\)
0.127731 + 0.991809i \(0.459231\pi\)
\(620\) −1.81362e14 −0.0795046
\(621\) 3.55573e13 0.0154499
\(622\) −9.52635e13 −0.0410279
\(623\) −7.57247e14 −0.323261
\(624\) 4.75068e14 0.201020
\(625\) 5.82319e14 0.244242
\(626\) −2.33803e15 −0.972054
\(627\) 1.37912e14 0.0568370
\(628\) 2.47514e15 1.01116
\(629\) 4.76949e14 0.193149
\(630\) −5.57758e14 −0.223910
\(631\) −1.25642e15 −0.500004 −0.250002 0.968245i \(-0.580431\pi\)
−0.250002 + 0.968245i \(0.580431\pi\)
\(632\) −1.12862e15 −0.445251
\(633\) 3.36352e14 0.131545
\(634\) −3.62795e15 −1.40660
\(635\) −7.62729e14 −0.293167
\(636\) −2.70776e13 −0.0103180
\(637\) 3.08190e15 1.16426
\(638\) −7.77148e14 −0.291065
\(639\) −2.62437e15 −0.974474
\(640\) 9.59973e14 0.353402
\(641\) −1.84677e15 −0.674051 −0.337026 0.941496i \(-0.609421\pi\)
−0.337026 + 0.941496i \(0.609421\pi\)
\(642\) 7.35650e14 0.266213
\(643\) −4.17247e15 −1.49704 −0.748518 0.663114i \(-0.769234\pi\)
−0.748518 + 0.663114i \(0.769234\pi\)
\(644\) −4.09820e13 −0.0145788
\(645\) 7.89841e13 0.0278587
\(646\) 3.75577e15 1.31347
\(647\) −2.50532e15 −0.868740 −0.434370 0.900735i \(-0.643029\pi\)
−0.434370 + 0.900735i \(0.643029\pi\)
\(648\) 9.60340e14 0.330189
\(649\) −4.73580e14 −0.161453
\(650\) −3.67355e15 −1.24183
\(651\) 2.58728e13 0.00867258
\(652\) −6.20844e14 −0.206358
\(653\) −1.09329e15 −0.360341 −0.180170 0.983635i \(-0.557665\pi\)
−0.180170 + 0.983635i \(0.557665\pi\)
\(654\) 9.04878e14 0.295742
\(655\) −4.73930e13 −0.0153598
\(656\) 4.16715e15 1.33927
\(657\) 3.43313e15 1.09416
\(658\) 5.13997e13 0.0162449
\(659\) 2.81792e15 0.883201 0.441600 0.897212i \(-0.354411\pi\)
0.441600 + 0.897212i \(0.354411\pi\)
\(660\) 4.71729e13 0.0146623
\(661\) 3.94426e15 1.21579 0.607894 0.794018i \(-0.292014\pi\)
0.607894 + 0.794018i \(0.292014\pi\)
\(662\) 6.61626e15 2.02252
\(663\) −3.68041e14 −0.111576
\(664\) −1.34241e15 −0.403609
\(665\) 8.71531e14 0.259874
\(666\) 1.27683e15 0.377594
\(667\) 1.55230e14 0.0455285
\(668\) −1.80001e15 −0.523605
\(669\) 6.40770e14 0.184867
\(670\) −2.45906e15 −0.703651
\(671\) −6.51396e14 −0.184872
\(672\) 1.84327e14 0.0518870
\(673\) 4.49130e15 1.25398 0.626988 0.779029i \(-0.284288\pi\)
0.626988 + 0.779029i \(0.284288\pi\)
\(674\) −4.05035e15 −1.12167
\(675\) 6.73053e14 0.184875
\(676\) 1.89709e15 0.516869
\(677\) 6.65605e15 1.79878 0.899392 0.437142i \(-0.144009\pi\)
0.899392 + 0.437142i \(0.144009\pi\)
\(678\) 7.10220e14 0.190384
\(679\) −1.65598e14 −0.0440323
\(680\) −4.60039e14 −0.121338
\(681\) −6.84132e14 −0.178991
\(682\) 3.04754e14 0.0790925
\(683\) −3.87670e15 −0.998041 −0.499020 0.866590i \(-0.666307\pi\)
−0.499020 + 0.866590i \(0.666307\pi\)
\(684\) 4.26382e15 1.08890
\(685\) −7.90943e13 −0.0200377
\(686\) 3.25426e15 0.817843
\(687\) −2.18519e14 −0.0544787
\(688\) 1.97850e15 0.489327
\(689\) 5.77435e14 0.141676
\(690\) −2.22191e13 −0.00540824
\(691\) 7.31622e15 1.76668 0.883339 0.468735i \(-0.155290\pi\)
0.883339 + 0.468735i \(0.155290\pi\)
\(692\) −1.77095e15 −0.424250
\(693\) 3.97453e14 0.0944611
\(694\) 4.13981e15 0.976119
\(695\) 5.63004e14 0.131703
\(696\) −1.45683e14 −0.0338111
\(697\) −3.22834e15 −0.743361
\(698\) 3.15743e15 0.721321
\(699\) −1.04969e15 −0.237924
\(700\) −7.75735e14 −0.174451
\(701\) −7.67409e15 −1.71229 −0.856146 0.516734i \(-0.827147\pi\)
−0.856146 + 0.516734i \(0.827147\pi\)
\(702\) −1.98724e15 −0.439942
\(703\) −1.99513e15 −0.438244
\(704\) 5.66379e14 0.123440
\(705\) 1.18177e13 0.00255559
\(706\) −5.66141e15 −1.21478
\(707\) −3.63279e14 −0.0773452
\(708\) 2.47909e14 0.0523733
\(709\) 7.60848e15 1.59494 0.797469 0.603360i \(-0.206172\pi\)
0.797469 + 0.603360i \(0.206172\pi\)
\(710\) 3.30761e15 0.688008
\(711\) 6.10529e15 1.26015
\(712\) −1.67205e15 −0.342459
\(713\) −6.08724e13 −0.0123717
\(714\) −1.83268e14 −0.0369612
\(715\) −1.00597e15 −0.201327
\(716\) 4.17089e15 0.828338
\(717\) −6.36195e14 −0.125382
\(718\) −1.95722e15 −0.382785
\(719\) 1.29989e15 0.252289 0.126144 0.992012i \(-0.459740\pi\)
0.126144 + 0.992012i \(0.459740\pi\)
\(720\) −3.21227e15 −0.618705
\(721\) −2.89173e15 −0.552732
\(722\) −8.76422e15 −1.66249
\(723\) 6.49265e14 0.122225
\(724\) 2.53368e15 0.473359
\(725\) 2.93830e15 0.544800
\(726\) 8.44734e14 0.155442
\(727\) −5.30782e15 −0.969342 −0.484671 0.874696i \(-0.661061\pi\)
−0.484671 + 0.874696i \(0.661061\pi\)
\(728\) −8.20201e14 −0.148661
\(729\) −5.01139e15 −0.901482
\(730\) −4.32693e15 −0.772510
\(731\) −1.53277e15 −0.271600
\(732\) 3.40992e14 0.0599699
\(733\) −9.22269e14 −0.160985 −0.0804926 0.996755i \(-0.525649\pi\)
−0.0804926 + 0.996755i \(0.525649\pi\)
\(734\) 5.59415e14 0.0969183
\(735\) 3.52841e14 0.0606735
\(736\) −4.33677e14 −0.0740182
\(737\) 1.75230e15 0.296850
\(738\) −8.64255e15 −1.45322
\(739\) 2.80660e15 0.468420 0.234210 0.972186i \(-0.424750\pi\)
0.234210 + 0.972186i \(0.424750\pi\)
\(740\) −6.82433e14 −0.113054
\(741\) 1.53955e15 0.253160
\(742\) 2.87537e14 0.0469323
\(743\) −6.63578e15 −1.07511 −0.537556 0.843228i \(-0.680652\pi\)
−0.537556 + 0.843228i \(0.680652\pi\)
\(744\) 5.71288e13 0.00918764
\(745\) −8.34873e14 −0.133279
\(746\) 4.38148e15 0.694316
\(747\) 7.26178e15 1.14230
\(748\) −9.15437e14 −0.142945
\(749\) −3.31277e15 −0.513501
\(750\) −1.00278e15 −0.154301
\(751\) 5.83015e15 0.890554 0.445277 0.895393i \(-0.353105\pi\)
0.445277 + 0.895393i \(0.353105\pi\)
\(752\) 2.96024e14 0.0448879
\(753\) −8.15749e14 −0.122796
\(754\) −8.67553e15 −1.29644
\(755\) −6.35638e15 −0.942979
\(756\) −4.19640e14 −0.0618026
\(757\) −7.70624e15 −1.12672 −0.563359 0.826212i \(-0.690491\pi\)
−0.563359 + 0.826212i \(0.690491\pi\)
\(758\) −1.32827e16 −1.92800
\(759\) 1.58331e13 0.00228158
\(760\) 1.92439e15 0.275308
\(761\) −1.82857e15 −0.259714 −0.129857 0.991533i \(-0.541452\pi\)
−0.129857 + 0.991533i \(0.541452\pi\)
\(762\) −6.70922e14 −0.0946063
\(763\) −4.07484e15 −0.570460
\(764\) 1.19748e15 0.166439
\(765\) 2.48859e15 0.343412
\(766\) 6.18690e14 0.0847647
\(767\) −5.28671e15 −0.719136
\(768\) 1.24708e15 0.168426
\(769\) −6.76987e15 −0.907790 −0.453895 0.891055i \(-0.649966\pi\)
−0.453895 + 0.891055i \(0.649966\pi\)
\(770\) −5.00928e14 −0.0666923
\(771\) 1.16292e15 0.153727
\(772\) 1.61138e15 0.211497
\(773\) −5.41600e15 −0.705816 −0.352908 0.935658i \(-0.614807\pi\)
−0.352908 + 0.935658i \(0.614807\pi\)
\(774\) −4.10334e15 −0.530961
\(775\) −1.15224e15 −0.148041
\(776\) −3.65651e14 −0.0466473
\(777\) 9.73548e13 0.0123322
\(778\) −1.38804e16 −1.74588
\(779\) 1.35045e16 1.68664
\(780\) 5.26605e14 0.0653077
\(781\) −2.35697e15 −0.290251
\(782\) 4.31184e14 0.0527261
\(783\) 1.58949e15 0.193005
\(784\) 8.83843e15 1.06570
\(785\) −6.04299e15 −0.723551
\(786\) −4.16885e13 −0.00495669
\(787\) −1.32704e16 −1.56683 −0.783416 0.621498i \(-0.786524\pi\)
−0.783416 + 0.621498i \(0.786524\pi\)
\(788\) −4.47434e14 −0.0524608
\(789\) 1.29321e15 0.150572
\(790\) −7.69477e15 −0.889706
\(791\) −3.19825e15 −0.367233
\(792\) 8.77601e14 0.100071
\(793\) −7.27172e15 −0.823445
\(794\) −1.47865e16 −1.66285
\(795\) 6.61095e13 0.00738320
\(796\) 1.74425e15 0.193459
\(797\) 1.10962e16 1.22223 0.611117 0.791540i \(-0.290721\pi\)
0.611117 + 0.791540i \(0.290721\pi\)
\(798\) 7.66628e14 0.0838627
\(799\) −2.29334e14 −0.0249150
\(800\) −8.20894e15 −0.885711
\(801\) 9.04496e15 0.969232
\(802\) −1.01459e16 −1.07977
\(803\) 3.08333e15 0.325900
\(804\) −9.17293e14 −0.0962942
\(805\) 1.00057e14 0.0104320
\(806\) 3.40206e15 0.352289
\(807\) 4.60262e14 0.0473370
\(808\) −8.02143e14 −0.0819387
\(809\) 1.30343e16 1.32242 0.661211 0.750200i \(-0.270043\pi\)
0.661211 + 0.750200i \(0.270043\pi\)
\(810\) 6.54745e15 0.659788
\(811\) −1.50810e16 −1.50943 −0.754717 0.656051i \(-0.772226\pi\)
−0.754717 + 0.656051i \(0.772226\pi\)
\(812\) −1.83199e15 −0.182123
\(813\) −7.97646e14 −0.0787612
\(814\) 1.14673e15 0.112468
\(815\) 1.51578e15 0.147662
\(816\) −1.05549e15 −0.102131
\(817\) 6.41172e15 0.616245
\(818\) 6.49456e15 0.620022
\(819\) 4.43688e15 0.420743
\(820\) 4.61922e15 0.435103
\(821\) 9.41101e14 0.0880540 0.0440270 0.999030i \(-0.485981\pi\)
0.0440270 + 0.999030i \(0.485981\pi\)
\(822\) −6.95740e13 −0.00646624
\(823\) 3.69420e15 0.341053 0.170526 0.985353i \(-0.445453\pi\)
0.170526 + 0.985353i \(0.445453\pi\)
\(824\) −6.38512e15 −0.585558
\(825\) 2.99700e14 0.0273017
\(826\) −2.63254e15 −0.238224
\(827\) −1.75606e16 −1.57855 −0.789276 0.614038i \(-0.789544\pi\)
−0.789276 + 0.614038i \(0.789544\pi\)
\(828\) 4.89510e14 0.0437114
\(829\) −7.62266e15 −0.676171 −0.338085 0.941115i \(-0.609779\pi\)
−0.338085 + 0.941115i \(0.609779\pi\)
\(830\) −9.15234e15 −0.806497
\(831\) 1.37354e15 0.120236
\(832\) 6.32265e15 0.549821
\(833\) −6.84724e15 −0.591518
\(834\) 4.95238e14 0.0425011
\(835\) 4.39469e15 0.374673
\(836\) 3.82937e15 0.324334
\(837\) −6.23310e14 −0.0524462
\(838\) −9.65884e15 −0.807389
\(839\) −1.55469e16 −1.29108 −0.645539 0.763728i \(-0.723367\pi\)
−0.645539 + 0.763728i \(0.723367\pi\)
\(840\) −9.39033e13 −0.00774720
\(841\) −5.26137e15 −0.431242
\(842\) −1.91060e16 −1.55580
\(843\) −7.10424e14 −0.0574733
\(844\) 9.33939e15 0.750646
\(845\) −4.63170e15 −0.369853
\(846\) −6.13945e14 −0.0487072
\(847\) −3.80399e15 −0.299834
\(848\) 1.65600e15 0.129683
\(849\) 1.48235e14 0.0115334
\(850\) 8.16175e15 0.630927
\(851\) −2.29052e14 −0.0175922
\(852\) 1.23383e15 0.0941534
\(853\) 4.18469e15 0.317280 0.158640 0.987336i \(-0.449289\pi\)
0.158640 + 0.987336i \(0.449289\pi\)
\(854\) −3.62099e15 −0.272778
\(855\) −1.04100e16 −0.779181
\(856\) −7.31480e15 −0.543997
\(857\) −3.28964e15 −0.243083 −0.121541 0.992586i \(-0.538784\pi\)
−0.121541 + 0.992586i \(0.538784\pi\)
\(858\) −8.84886e14 −0.0649691
\(859\) −1.11698e16 −0.814859 −0.407430 0.913237i \(-0.633575\pi\)
−0.407430 + 0.913237i \(0.633575\pi\)
\(860\) 2.19313e15 0.158973
\(861\) −6.58970e14 −0.0474622
\(862\) 6.97334e15 0.499058
\(863\) −2.90899e15 −0.206863 −0.103432 0.994637i \(-0.532982\pi\)
−0.103432 + 0.994637i \(0.532982\pi\)
\(864\) −4.44069e15 −0.313780
\(865\) 4.32373e15 0.303578
\(866\) 3.41146e16 2.38008
\(867\) −1.04358e15 −0.0723469
\(868\) 7.18404e14 0.0494892
\(869\) 5.48322e15 0.375342
\(870\) −9.93246e14 −0.0675617
\(871\) 1.95614e16 1.32221
\(872\) −8.99749e15 −0.604339
\(873\) 1.97799e15 0.132022
\(874\) −1.80369e15 −0.119632
\(875\) 4.51571e15 0.297633
\(876\) −1.61406e15 −0.105717
\(877\) 4.54147e14 0.0295596 0.0147798 0.999891i \(-0.495295\pi\)
0.0147798 + 0.999891i \(0.495295\pi\)
\(878\) 2.32827e15 0.150596
\(879\) 1.37850e15 0.0886071
\(880\) −2.88497e15 −0.184284
\(881\) −2.13080e16 −1.35262 −0.676308 0.736619i \(-0.736421\pi\)
−0.676308 + 0.736619i \(0.736421\pi\)
\(882\) −1.83306e16 −1.15638
\(883\) 2.43871e16 1.52889 0.764446 0.644688i \(-0.223013\pi\)
0.764446 + 0.644688i \(0.223013\pi\)
\(884\) −1.02193e16 −0.636698
\(885\) −6.05266e14 −0.0374764
\(886\) 7.00128e15 0.430816
\(887\) −2.95854e15 −0.180925 −0.0904624 0.995900i \(-0.528834\pi\)
−0.0904624 + 0.995900i \(0.528834\pi\)
\(888\) 2.14965e14 0.0130646
\(889\) 3.02129e15 0.182487
\(890\) −1.13998e16 −0.684307
\(891\) −4.66565e15 −0.278346
\(892\) 1.77921e16 1.05492
\(893\) 9.59327e14 0.0565306
\(894\) −7.34382e14 −0.0430096
\(895\) −1.01832e16 −0.592729
\(896\) −3.80260e15 −0.219982
\(897\) 1.76750e14 0.0101625
\(898\) 1.01596e16 0.580572
\(899\) −2.72114e15 −0.154551
\(900\) 9.26579e15 0.523057
\(901\) −1.28292e15 −0.0719803
\(902\) −7.76195e15 −0.432847
\(903\) −3.12868e14 −0.0173412
\(904\) −7.06194e15 −0.389043
\(905\) −6.18594e15 −0.338718
\(906\) −5.59129e15 −0.304303
\(907\) 2.55989e16 1.38478 0.692392 0.721522i \(-0.256557\pi\)
0.692392 + 0.721522i \(0.256557\pi\)
\(908\) −1.89961e16 −1.02139
\(909\) 4.33920e15 0.231904
\(910\) −5.59200e15 −0.297057
\(911\) 1.66008e16 0.876554 0.438277 0.898840i \(-0.355589\pi\)
0.438277 + 0.898840i \(0.355589\pi\)
\(912\) 4.41521e15 0.231729
\(913\) 6.52187e15 0.340238
\(914\) 4.18710e16 2.17125
\(915\) −8.32526e14 −0.0429123
\(916\) −6.06755e15 −0.310877
\(917\) 1.87731e14 0.00956102
\(918\) 4.41516e15 0.223517
\(919\) 3.18784e16 1.60421 0.802105 0.597184i \(-0.203714\pi\)
0.802105 + 0.597184i \(0.203714\pi\)
\(920\) 2.20931e14 0.0110516
\(921\) −4.13329e15 −0.205527
\(922\) 1.41727e16 0.700542
\(923\) −2.63115e16 −1.29282
\(924\) −1.86859e14 −0.00912680
\(925\) −4.33565e15 −0.210511
\(926\) −1.77741e16 −0.857883
\(927\) 3.45404e16 1.65725
\(928\) −1.93864e16 −0.924662
\(929\) −2.37259e16 −1.12496 −0.562479 0.826811i \(-0.690152\pi\)
−0.562479 + 0.826811i \(0.690152\pi\)
\(930\) 3.89495e14 0.0183589
\(931\) 2.86427e16 1.34212
\(932\) −2.91466e16 −1.35769
\(933\) 8.67599e13 0.00401763
\(934\) 4.57028e16 2.10395
\(935\) 2.23502e15 0.102286
\(936\) 9.79691e15 0.445730
\(937\) 2.63273e15 0.119080 0.0595400 0.998226i \(-0.481037\pi\)
0.0595400 + 0.998226i \(0.481037\pi\)
\(938\) 9.74071e15 0.438001
\(939\) 2.12933e15 0.0951878
\(940\) 3.28138e14 0.0145832
\(941\) 1.32794e16 0.586726 0.293363 0.956001i \(-0.405226\pi\)
0.293363 + 0.956001i \(0.405226\pi\)
\(942\) −5.31562e15 −0.233493
\(943\) 1.55039e15 0.0677061
\(944\) −1.51615e16 −0.658257
\(945\) 1.02454e15 0.0442237
\(946\) −3.68525e15 −0.158149
\(947\) 3.47962e16 1.48459 0.742296 0.670072i \(-0.233737\pi\)
0.742296 + 0.670072i \(0.233737\pi\)
\(948\) −2.87035e15 −0.121756
\(949\) 3.44201e16 1.45160
\(950\) −3.41414e16 −1.43154
\(951\) 3.30411e15 0.137740
\(952\) 1.82229e15 0.0755290
\(953\) 1.20181e16 0.495249 0.247625 0.968856i \(-0.420350\pi\)
0.247625 + 0.968856i \(0.420350\pi\)
\(954\) −3.43449e15 −0.140717
\(955\) −2.92363e15 −0.119098
\(956\) −1.76651e16 −0.715479
\(957\) 7.07777e14 0.0285024
\(958\) 1.15980e16 0.464379
\(959\) 3.13305e14 0.0124728
\(960\) 7.23869e14 0.0286529
\(961\) −2.43414e16 −0.958003
\(962\) 1.28013e16 0.500947
\(963\) 3.95695e16 1.53963
\(964\) 1.80280e16 0.697466
\(965\) −3.93416e15 −0.151339
\(966\) 8.80133e13 0.00336646
\(967\) −3.16167e15 −0.120246 −0.0601231 0.998191i \(-0.519149\pi\)
−0.0601231 + 0.998191i \(0.519149\pi\)
\(968\) −8.39945e15 −0.317641
\(969\) −3.42052e15 −0.128621
\(970\) −2.49295e15 −0.0932114
\(971\) −1.09754e16 −0.408051 −0.204025 0.978966i \(-0.565403\pi\)
−0.204025 + 0.978966i \(0.565403\pi\)
\(972\) 7.53964e15 0.278731
\(973\) −2.23015e15 −0.0819810
\(974\) 4.86043e16 1.77664
\(975\) 3.34564e15 0.121606
\(976\) −2.08542e16 −0.753736
\(977\) −4.62054e15 −0.166063 −0.0830315 0.996547i \(-0.526460\pi\)
−0.0830315 + 0.996547i \(0.526460\pi\)
\(978\) 1.33333e15 0.0476512
\(979\) 8.12336e15 0.288689
\(980\) 9.79725e15 0.346227
\(981\) 4.86720e16 1.71041
\(982\) 3.18912e16 1.11444
\(983\) −6.60643e15 −0.229574 −0.114787 0.993390i \(-0.536619\pi\)
−0.114787 + 0.993390i \(0.536619\pi\)
\(984\) −1.45505e15 −0.0502810
\(985\) 1.09240e15 0.0375390
\(986\) 1.92749e16 0.658673
\(987\) −4.68116e13 −0.00159078
\(988\) 4.27484e16 1.44463
\(989\) 7.36102e14 0.0247376
\(990\) 5.98334e15 0.199963
\(991\) −4.54195e16 −1.50951 −0.754757 0.656004i \(-0.772245\pi\)
−0.754757 + 0.656004i \(0.772245\pi\)
\(992\) 7.60225e15 0.251263
\(993\) −6.02567e15 −0.198054
\(994\) −1.31020e16 −0.428264
\(995\) −4.25856e15 −0.138432
\(996\) −3.41406e15 −0.110368
\(997\) 3.73274e16 1.20006 0.600032 0.799976i \(-0.295154\pi\)
0.600032 + 0.799976i \(0.295154\pi\)
\(998\) −1.44846e16 −0.463116
\(999\) −2.34540e15 −0.0745774
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 197.12.a.b.1.18 92
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
197.12.a.b.1.18 92 1.1 even 1 trivial