Properties

Label 3.16.a.b.1.1
Level $3$
Weight $16$
Character 3.1
Self dual yes
Analytic conductor $4.281$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3,16,Mod(1,3)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 16, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3.1");
 
S:= CuspForms(chi, 16);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 3.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: \(4.28080515300\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-72.0000 q^{2} +2187.00 q^{3} -27584.0 q^{4} -221490. q^{5} -157464. q^{6} -2.14900e6 q^{7} +4.34534e6 q^{8} +4.78297e6 q^{9} +O(q^{10})\) \(q-72.0000 q^{2} +2187.00 q^{3} -27584.0 q^{4} -221490. q^{5} -157464. q^{6} -2.14900e6 q^{7} +4.34534e6 q^{8} +4.78297e6 q^{9} +1.59473e7 q^{10} +3.71693e7 q^{11} -6.03262e7 q^{12} -2.79974e8 q^{13} +1.54728e8 q^{14} -4.84399e8 q^{15} +5.91008e8 q^{16} +2.49291e9 q^{17} -3.44374e8 q^{18} -4.66978e9 q^{19} +6.10958e9 q^{20} -4.69986e9 q^{21} -2.67619e9 q^{22} -1.84679e10 q^{23} +9.50327e9 q^{24} +1.85402e10 q^{25} +2.01581e10 q^{26} +1.04604e10 q^{27} +5.92780e10 q^{28} -1.15953e11 q^{29} +3.48767e10 q^{30} -5.61870e10 q^{31} -1.84941e11 q^{32} +8.12893e10 q^{33} -1.79490e11 q^{34} +4.75982e11 q^{35} -1.31933e11 q^{36} +6.14765e11 q^{37} +3.36224e11 q^{38} -6.12304e11 q^{39} -9.62450e11 q^{40} +5.49860e11 q^{41} +3.38390e11 q^{42} -9.82884e11 q^{43} -1.02528e12 q^{44} -1.05938e12 q^{45} +1.32969e12 q^{46} +2.07614e12 q^{47} +1.29253e12 q^{48} -1.29361e11 q^{49} -1.33490e12 q^{50} +5.45200e12 q^{51} +7.72281e12 q^{52} -1.20484e13 q^{53} -7.53145e11 q^{54} -8.23263e12 q^{55} -9.33814e12 q^{56} -1.02128e13 q^{57} +8.34865e12 q^{58} +2.30879e13 q^{59} +1.33617e13 q^{60} -8.50581e12 q^{61} +4.04547e12 q^{62} -1.02786e13 q^{63} -6.05040e12 q^{64} +6.20115e13 q^{65} -5.85283e12 q^{66} -1.23310e13 q^{67} -6.87645e13 q^{68} -4.03894e13 q^{69} -3.42707e13 q^{70} +5.89892e13 q^{71} +2.07836e13 q^{72} -5.60983e12 q^{73} -4.42631e13 q^{74} +4.05475e13 q^{75} +1.28811e14 q^{76} -7.98769e13 q^{77} +4.40859e13 q^{78} +1.59919e14 q^{79} -1.30902e14 q^{80} +2.28768e13 q^{81} -3.95899e13 q^{82} +5.76759e13 q^{83} +1.29641e14 q^{84} -5.52155e14 q^{85} +7.07677e13 q^{86} -2.53590e14 q^{87} +1.61513e14 q^{88} -3.62288e14 q^{89} +7.62753e13 q^{90} +6.01665e14 q^{91} +5.09419e14 q^{92} -1.22881e14 q^{93} -1.49482e14 q^{94} +1.03431e15 q^{95} -4.04466e14 q^{96} -5.39787e14 q^{97} +9.31396e12 q^{98} +1.77780e14 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −72.0000 −0.397748 −0.198874 0.980025i \(-0.563728\pi\)
−0.198874 + 0.980025i \(0.563728\pi\)
\(3\) 2187.00 0.577350
\(4\) −27584.0 −0.841797
\(5\) −221490. −1.26788 −0.633941 0.773381i \(-0.718564\pi\)
−0.633941 + 0.773381i \(0.718564\pi\)
\(6\) −157464. −0.229640
\(7\) −2.14900e6 −0.986282 −0.493141 0.869949i \(-0.664151\pi\)
−0.493141 + 0.869949i \(0.664151\pi\)
\(8\) 4.34534e6 0.732570
\(9\) 4.78297e6 0.333333
\(10\) 1.59473e7 0.504297
\(11\) 3.71693e7 0.575095 0.287547 0.957766i \(-0.407160\pi\)
0.287547 + 0.957766i \(0.407160\pi\)
\(12\) −6.03262e7 −0.486012
\(13\) −2.79974e8 −1.23749 −0.618747 0.785590i \(-0.712359\pi\)
−0.618747 + 0.785590i \(0.712359\pi\)
\(14\) 1.54728e8 0.392291
\(15\) −4.84399e8 −0.732012
\(16\) 5.91008e8 0.550419
\(17\) 2.49291e9 1.47347 0.736733 0.676184i \(-0.236367\pi\)
0.736733 + 0.676184i \(0.236367\pi\)
\(18\) −3.44374e8 −0.132583
\(19\) −4.66978e9 −1.19852 −0.599259 0.800555i \(-0.704538\pi\)
−0.599259 + 0.800555i \(0.704538\pi\)
\(20\) 6.10958e9 1.06730
\(21\) −4.69986e9 −0.569430
\(22\) −2.67619e9 −0.228743
\(23\) −1.84679e10 −1.13099 −0.565496 0.824751i \(-0.691315\pi\)
−0.565496 + 0.824751i \(0.691315\pi\)
\(24\) 9.50327e9 0.422950
\(25\) 1.85402e10 0.607527
\(26\) 2.01581e10 0.492210
\(27\) 1.04604e10 0.192450
\(28\) 5.92780e10 0.830249
\(29\) −1.15953e11 −1.24824 −0.624121 0.781328i \(-0.714543\pi\)
−0.624121 + 0.781328i \(0.714543\pi\)
\(30\) 3.48767e10 0.291156
\(31\) −5.61870e10 −0.366795 −0.183397 0.983039i \(-0.558710\pi\)
−0.183397 + 0.983039i \(0.558710\pi\)
\(32\) −1.84941e11 −0.951498
\(33\) 8.12893e10 0.332031
\(34\) −1.79490e11 −0.586068
\(35\) 4.75982e11 1.25049
\(36\) −1.31933e11 −0.280599
\(37\) 6.14765e11 1.06462 0.532312 0.846548i \(-0.321323\pi\)
0.532312 + 0.846548i \(0.321323\pi\)
\(38\) 3.36224e11 0.476708
\(39\) −6.12304e11 −0.714467
\(40\) −9.62450e11 −0.928813
\(41\) 5.49860e11 0.440933 0.220467 0.975394i \(-0.429242\pi\)
0.220467 + 0.975394i \(0.429242\pi\)
\(42\) 3.38390e11 0.226489
\(43\) −9.82884e11 −0.551428 −0.275714 0.961240i \(-0.588914\pi\)
−0.275714 + 0.961240i \(0.588914\pi\)
\(44\) −1.02528e12 −0.484113
\(45\) −1.05938e12 −0.422628
\(46\) 1.32969e12 0.449849
\(47\) 2.07614e12 0.597756 0.298878 0.954291i \(-0.403388\pi\)
0.298878 + 0.954291i \(0.403388\pi\)
\(48\) 1.29253e12 0.317784
\(49\) −1.29361e11 −0.0272478
\(50\) −1.33490e12 −0.241642
\(51\) 5.45200e12 0.850706
\(52\) 7.72281e12 1.04172
\(53\) −1.20484e13 −1.40883 −0.704417 0.709787i \(-0.748791\pi\)
−0.704417 + 0.709787i \(0.748791\pi\)
\(54\) −7.53145e11 −0.0765466
\(55\) −8.23263e12 −0.729153
\(56\) −9.33814e12 −0.722521
\(57\) −1.02128e13 −0.691965
\(58\) 8.34865e12 0.496485
\(59\) 2.30879e13 1.20780 0.603899 0.797061i \(-0.293613\pi\)
0.603899 + 0.797061i \(0.293613\pi\)
\(60\) 1.33617e13 0.616206
\(61\) −8.50581e12 −0.346531 −0.173265 0.984875i \(-0.555432\pi\)
−0.173265 + 0.984875i \(0.555432\pi\)
\(62\) 4.04547e12 0.145892
\(63\) −1.02786e13 −0.328761
\(64\) −6.05040e12 −0.171963
\(65\) 6.20115e13 1.56900
\(66\) −5.85283e12 −0.132065
\(67\) −1.23310e13 −0.248564 −0.124282 0.992247i \(-0.539663\pi\)
−0.124282 + 0.992247i \(0.539663\pi\)
\(68\) −6.87645e13 −1.24036
\(69\) −4.03894e13 −0.652979
\(70\) −3.42707e13 −0.497379
\(71\) 5.89892e13 0.769724 0.384862 0.922974i \(-0.374249\pi\)
0.384862 + 0.922974i \(0.374249\pi\)
\(72\) 2.07836e13 0.244190
\(73\) −5.60983e12 −0.0594331 −0.0297165 0.999558i \(-0.509460\pi\)
−0.0297165 + 0.999558i \(0.509460\pi\)
\(74\) −4.42631e13 −0.423452
\(75\) 4.05475e13 0.350756
\(76\) 1.28811e14 1.00891
\(77\) −7.98769e13 −0.567206
\(78\) 4.40859e13 0.284178
\(79\) 1.59919e14 0.936906 0.468453 0.883488i \(-0.344812\pi\)
0.468453 + 0.883488i \(0.344812\pi\)
\(80\) −1.30902e14 −0.697867
\(81\) 2.28768e13 0.111111
\(82\) −3.95899e13 −0.175380
\(83\) 5.76759e13 0.233297 0.116648 0.993173i \(-0.462785\pi\)
0.116648 + 0.993173i \(0.462785\pi\)
\(84\) 1.29641e14 0.479345
\(85\) −5.52155e14 −1.86818
\(86\) 7.07677e13 0.219329
\(87\) −2.53590e14 −0.720673
\(88\) 1.61513e14 0.421297
\(89\) −3.62288e14 −0.868217 −0.434109 0.900861i \(-0.642937\pi\)
−0.434109 + 0.900861i \(0.642937\pi\)
\(90\) 7.62753e13 0.168099
\(91\) 6.01665e14 1.22052
\(92\) 5.09419e14 0.952066
\(93\) −1.22881e14 −0.211769
\(94\) −1.49482e14 −0.237756
\(95\) 1.03431e15 1.51958
\(96\) −4.04466e14 −0.549348
\(97\) −5.39787e14 −0.678319 −0.339160 0.940729i \(-0.610143\pi\)
−0.339160 + 0.940729i \(0.610143\pi\)
\(98\) 9.31396e12 0.0108377
\(99\) 1.77780e14 0.191698
\(100\) −5.11414e14 −0.511414
\(101\) −4.22221e14 −0.391859 −0.195929 0.980618i \(-0.562772\pi\)
−0.195929 + 0.980618i \(0.562772\pi\)
\(102\) −3.92544e14 −0.338366
\(103\) −1.45689e14 −0.116720 −0.0583602 0.998296i \(-0.518587\pi\)
−0.0583602 + 0.998296i \(0.518587\pi\)
\(104\) −1.21658e15 −0.906551
\(105\) 1.04097e15 0.721971
\(106\) 8.67483e14 0.560360
\(107\) −4.08696e14 −0.246049 −0.123024 0.992404i \(-0.539259\pi\)
−0.123024 + 0.992404i \(0.539259\pi\)
\(108\) −2.88538e14 −0.162004
\(109\) 1.10134e15 0.577060 0.288530 0.957471i \(-0.406833\pi\)
0.288530 + 0.957471i \(0.406833\pi\)
\(110\) 5.92749e14 0.290019
\(111\) 1.34449e15 0.614661
\(112\) −1.27008e15 −0.542868
\(113\) −2.13023e15 −0.851802 −0.425901 0.904770i \(-0.640043\pi\)
−0.425901 + 0.904770i \(0.640043\pi\)
\(114\) 7.35323e14 0.275227
\(115\) 4.09046e15 1.43397
\(116\) 3.19846e15 1.05077
\(117\) −1.33911e15 −0.412498
\(118\) −1.66233e15 −0.480399
\(119\) −5.35727e15 −1.45325
\(120\) −2.10488e15 −0.536251
\(121\) −2.79569e15 −0.669266
\(122\) 6.12418e14 0.137832
\(123\) 1.20254e15 0.254573
\(124\) 1.54986e15 0.308767
\(125\) 2.65286e15 0.497610
\(126\) 7.40059e14 0.130764
\(127\) −6.25962e15 −1.04236 −0.521182 0.853445i \(-0.674509\pi\)
−0.521182 + 0.853445i \(0.674509\pi\)
\(128\) 6.49577e15 1.01990
\(129\) −2.14957e15 −0.318367
\(130\) −4.46483e15 −0.624065
\(131\) 1.01059e16 1.33365 0.666825 0.745214i \(-0.267653\pi\)
0.666825 + 0.745214i \(0.267653\pi\)
\(132\) −2.24228e15 −0.279503
\(133\) 1.00354e16 1.18208
\(134\) 8.87833e14 0.0988656
\(135\) −2.31686e15 −0.244004
\(136\) 1.08326e16 1.07942
\(137\) −9.93739e15 −0.937277 −0.468638 0.883390i \(-0.655255\pi\)
−0.468638 + 0.883390i \(0.655255\pi\)
\(138\) 2.90803e15 0.259721
\(139\) −1.95078e16 −1.65043 −0.825216 0.564817i \(-0.808947\pi\)
−0.825216 + 0.564817i \(0.808947\pi\)
\(140\) −1.31295e16 −1.05266
\(141\) 4.54053e15 0.345114
\(142\) −4.24722e15 −0.306156
\(143\) −1.04065e16 −0.711676
\(144\) 2.82677e15 0.183473
\(145\) 2.56825e16 1.58262
\(146\) 4.03908e14 0.0236394
\(147\) −2.82911e14 −0.0157315
\(148\) −1.69577e16 −0.896197
\(149\) 1.58935e16 0.798589 0.399295 0.916823i \(-0.369255\pi\)
0.399295 + 0.916823i \(0.369255\pi\)
\(150\) −2.91942e15 −0.139512
\(151\) 3.10597e16 1.41211 0.706057 0.708155i \(-0.250472\pi\)
0.706057 + 0.708155i \(0.250472\pi\)
\(152\) −2.02918e16 −0.877999
\(153\) 1.19235e16 0.491155
\(154\) 5.75113e15 0.225605
\(155\) 1.24449e16 0.465053
\(156\) 1.68898e16 0.601436
\(157\) −5.41372e16 −1.83759 −0.918794 0.394738i \(-0.870835\pi\)
−0.918794 + 0.394738i \(0.870835\pi\)
\(158\) −1.15141e16 −0.372652
\(159\) −2.63498e16 −0.813390
\(160\) 4.09625e16 1.20639
\(161\) 3.96876e16 1.11548
\(162\) −1.64713e15 −0.0441942
\(163\) −5.54612e16 −1.42096 −0.710481 0.703717i \(-0.751522\pi\)
−0.710481 + 0.703717i \(0.751522\pi\)
\(164\) −1.51673e16 −0.371176
\(165\) −1.80048e16 −0.420977
\(166\) −4.15266e15 −0.0927932
\(167\) −5.83552e15 −0.124654 −0.0623270 0.998056i \(-0.519852\pi\)
−0.0623270 + 0.998056i \(0.519852\pi\)
\(168\) −2.04225e16 −0.417148
\(169\) 2.71997e16 0.531390
\(170\) 3.97552e16 0.743065
\(171\) −2.23354e16 −0.399506
\(172\) 2.71119e16 0.464191
\(173\) 4.40958e16 0.722855 0.361428 0.932400i \(-0.382289\pi\)
0.361428 + 0.932400i \(0.382289\pi\)
\(174\) 1.82585e16 0.286646
\(175\) −3.98430e16 −0.599193
\(176\) 2.19674e16 0.316543
\(177\) 5.04932e16 0.697323
\(178\) 2.60847e16 0.345331
\(179\) −5.10594e16 −0.648153 −0.324077 0.946031i \(-0.605054\pi\)
−0.324077 + 0.946031i \(0.605054\pi\)
\(180\) 2.92219e16 0.355767
\(181\) 3.16143e16 0.369228 0.184614 0.982811i \(-0.440897\pi\)
0.184614 + 0.982811i \(0.440897\pi\)
\(182\) −4.33199e16 −0.485458
\(183\) −1.86022e16 −0.200070
\(184\) −8.02495e16 −0.828531
\(185\) −1.36164e17 −1.34982
\(186\) 8.84743e15 0.0842307
\(187\) 9.26599e16 0.847383
\(188\) −5.72684e16 −0.503189
\(189\) −2.24793e16 −0.189810
\(190\) −7.44703e16 −0.604409
\(191\) 9.44291e16 0.736810 0.368405 0.929665i \(-0.379904\pi\)
0.368405 + 0.929665i \(0.379904\pi\)
\(192\) −1.32322e16 −0.0992828
\(193\) 4.17744e16 0.301460 0.150730 0.988575i \(-0.451838\pi\)
0.150730 + 0.988575i \(0.451838\pi\)
\(194\) 3.88646e16 0.269800
\(195\) 1.35619e17 0.905861
\(196\) 3.56828e15 0.0229371
\(197\) −1.48383e17 −0.918094 −0.459047 0.888412i \(-0.651809\pi\)
−0.459047 + 0.888412i \(0.651809\pi\)
\(198\) −1.28001e16 −0.0762475
\(199\) 1.97555e17 1.13316 0.566579 0.824007i \(-0.308267\pi\)
0.566579 + 0.824007i \(0.308267\pi\)
\(200\) 8.05637e16 0.445056
\(201\) −2.69679e16 −0.143508
\(202\) 3.03999e16 0.155861
\(203\) 2.49184e17 1.23112
\(204\) −1.50388e17 −0.716122
\(205\) −1.21788e17 −0.559052
\(206\) 1.04896e16 0.0464253
\(207\) −8.83316e16 −0.376997
\(208\) −1.65467e17 −0.681140
\(209\) −1.73573e17 −0.689262
\(210\) −7.49500e16 −0.287162
\(211\) −4.48570e17 −1.65849 −0.829244 0.558887i \(-0.811228\pi\)
−0.829244 + 0.558887i \(0.811228\pi\)
\(212\) 3.32342e17 1.18595
\(213\) 1.29009e17 0.444400
\(214\) 2.94261e16 0.0978654
\(215\) 2.17699e17 0.699146
\(216\) 4.54538e16 0.140983
\(217\) 1.20746e17 0.361763
\(218\) −7.92962e16 −0.229524
\(219\) −1.22687e16 −0.0343137
\(220\) 2.27089e17 0.613799
\(221\) −6.97951e17 −1.82341
\(222\) −9.68033e16 −0.244480
\(223\) 1.59977e17 0.390634 0.195317 0.980740i \(-0.437426\pi\)
0.195317 + 0.980740i \(0.437426\pi\)
\(224\) 3.97438e17 0.938445
\(225\) 8.86774e16 0.202509
\(226\) 1.53377e17 0.338802
\(227\) 1.45803e17 0.311582 0.155791 0.987790i \(-0.450207\pi\)
0.155791 + 0.987790i \(0.450207\pi\)
\(228\) 2.81710e17 0.582494
\(229\) 6.81194e17 1.36303 0.681514 0.731805i \(-0.261322\pi\)
0.681514 + 0.731805i \(0.261322\pi\)
\(230\) −2.94513e17 −0.570356
\(231\) −1.74691e17 −0.327476
\(232\) −5.03858e17 −0.914425
\(233\) −3.89035e17 −0.683628 −0.341814 0.939768i \(-0.611041\pi\)
−0.341814 + 0.939768i \(0.611041\pi\)
\(234\) 9.64158e16 0.164070
\(235\) −4.59845e17 −0.757884
\(236\) −6.36857e17 −1.01672
\(237\) 3.49742e17 0.540923
\(238\) 3.85723e17 0.578028
\(239\) 6.76227e16 0.0981993 0.0490996 0.998794i \(-0.484365\pi\)
0.0490996 + 0.998794i \(0.484365\pi\)
\(240\) −2.86283e17 −0.402913
\(241\) −9.00528e17 −1.22848 −0.614242 0.789118i \(-0.710538\pi\)
−0.614242 + 0.789118i \(0.710538\pi\)
\(242\) 2.01290e17 0.266199
\(243\) 5.00315e16 0.0641500
\(244\) 2.34624e17 0.291709
\(245\) 2.86521e16 0.0345470
\(246\) −8.65831e16 −0.101256
\(247\) 1.30742e18 1.48316
\(248\) −2.44152e17 −0.268703
\(249\) 1.26137e17 0.134694
\(250\) −1.91006e17 −0.197923
\(251\) 1.58929e18 1.59827 0.799134 0.601153i \(-0.205292\pi\)
0.799134 + 0.601153i \(0.205292\pi\)
\(252\) 2.83525e17 0.276750
\(253\) −6.86440e17 −0.650428
\(254\) 4.50692e17 0.414598
\(255\) −1.20756e18 −1.07860
\(256\) −2.69436e17 −0.233698
\(257\) 2.29738e18 1.93524 0.967618 0.252418i \(-0.0812258\pi\)
0.967618 + 0.252418i \(0.0812258\pi\)
\(258\) 1.54769e17 0.126630
\(259\) −1.32113e18 −1.05002
\(260\) −1.71053e18 −1.32078
\(261\) −5.54602e17 −0.416081
\(262\) −7.27628e17 −0.530456
\(263\) −5.05657e17 −0.358252 −0.179126 0.983826i \(-0.557327\pi\)
−0.179126 + 0.983826i \(0.557327\pi\)
\(264\) 3.53230e17 0.243236
\(265\) 2.66860e18 1.78624
\(266\) −7.22546e17 −0.470168
\(267\) −7.92323e17 −0.501265
\(268\) 3.40139e17 0.209240
\(269\) −2.44731e18 −1.46402 −0.732009 0.681295i \(-0.761417\pi\)
−0.732009 + 0.681295i \(0.761417\pi\)
\(270\) 1.66814e17 0.0970521
\(271\) −1.53807e18 −0.870373 −0.435187 0.900340i \(-0.643318\pi\)
−0.435187 + 0.900340i \(0.643318\pi\)
\(272\) 1.47333e18 0.811024
\(273\) 1.31584e18 0.704666
\(274\) 7.15492e17 0.372800
\(275\) 6.89128e17 0.349385
\(276\) 1.11410e18 0.549675
\(277\) −7.89959e17 −0.379321 −0.189660 0.981850i \(-0.560739\pi\)
−0.189660 + 0.981850i \(0.560739\pi\)
\(278\) 1.40456e18 0.656456
\(279\) −2.68741e17 −0.122265
\(280\) 2.06831e18 0.916072
\(281\) −6.15303e17 −0.265333 −0.132667 0.991161i \(-0.542354\pi\)
−0.132667 + 0.991161i \(0.542354\pi\)
\(282\) −3.26918e17 −0.137268
\(283\) −1.26279e18 −0.516337 −0.258169 0.966100i \(-0.583119\pi\)
−0.258169 + 0.966100i \(0.583119\pi\)
\(284\) −1.62716e18 −0.647951
\(285\) 2.26204e18 0.877330
\(286\) 7.49265e17 0.283067
\(287\) −1.18165e18 −0.434885
\(288\) −8.84566e17 −0.317166
\(289\) 3.35219e18 1.17110
\(290\) −1.84914e18 −0.629485
\(291\) −1.18051e18 −0.391628
\(292\) 1.54742e17 0.0500306
\(293\) −3.49864e18 −1.10253 −0.551267 0.834329i \(-0.685855\pi\)
−0.551267 + 0.834329i \(0.685855\pi\)
\(294\) 2.03696e16 0.00625717
\(295\) −5.11374e18 −1.53135
\(296\) 2.67137e18 0.779912
\(297\) 3.88804e17 0.110677
\(298\) −1.14433e18 −0.317637
\(299\) 5.17055e18 1.39960
\(300\) −1.11846e18 −0.295265
\(301\) 2.11222e18 0.543864
\(302\) −2.23630e18 −0.561665
\(303\) −9.23398e17 −0.226240
\(304\) −2.75988e18 −0.659687
\(305\) 1.88395e18 0.439360
\(306\) −8.58494e17 −0.195356
\(307\) 1.28861e18 0.286142 0.143071 0.989712i \(-0.454302\pi\)
0.143071 + 0.989712i \(0.454302\pi\)
\(308\) 2.20332e18 0.477472
\(309\) −3.18621e17 −0.0673886
\(310\) −8.96030e17 −0.184974
\(311\) 8.43030e18 1.69879 0.849396 0.527756i \(-0.176967\pi\)
0.849396 + 0.527756i \(0.176967\pi\)
\(312\) −2.66067e18 −0.523397
\(313\) 7.88359e18 1.51405 0.757027 0.653383i \(-0.226651\pi\)
0.757027 + 0.653383i \(0.226651\pi\)
\(314\) 3.89788e18 0.730896
\(315\) 2.27661e18 0.416830
\(316\) −4.41120e18 −0.788685
\(317\) −8.71539e18 −1.52175 −0.760873 0.648900i \(-0.775229\pi\)
−0.760873 + 0.648900i \(0.775229\pi\)
\(318\) 1.89719e18 0.323524
\(319\) −4.30991e18 −0.717857
\(320\) 1.34010e18 0.218029
\(321\) −8.93817e17 −0.142056
\(322\) −2.85751e18 −0.443678
\(323\) −1.16414e19 −1.76598
\(324\) −6.31033e17 −0.0935330
\(325\) −5.19079e18 −0.751810
\(326\) 3.99321e18 0.565184
\(327\) 2.40862e18 0.333166
\(328\) 2.38933e18 0.323015
\(329\) −4.46163e18 −0.589556
\(330\) 1.29634e18 0.167442
\(331\) 1.26995e19 1.60353 0.801763 0.597643i \(-0.203896\pi\)
0.801763 + 0.597643i \(0.203896\pi\)
\(332\) −1.59093e18 −0.196388
\(333\) 2.94040e18 0.354875
\(334\) 4.20157e17 0.0495808
\(335\) 2.73120e18 0.315150
\(336\) −2.77766e18 −0.313425
\(337\) −6.64250e18 −0.733005 −0.366503 0.930417i \(-0.619445\pi\)
−0.366503 + 0.930417i \(0.619445\pi\)
\(338\) −1.95838e18 −0.211359
\(339\) −4.65882e18 −0.491788
\(340\) 1.52307e19 1.57263
\(341\) −2.08843e18 −0.210942
\(342\) 1.60815e18 0.158903
\(343\) 1.04805e19 1.01316
\(344\) −4.27097e18 −0.403960
\(345\) 8.94584e18 0.827900
\(346\) −3.17490e18 −0.287514
\(347\) 1.87835e18 0.166458 0.0832289 0.996530i \(-0.473477\pi\)
0.0832289 + 0.996530i \(0.473477\pi\)
\(348\) 6.99503e18 0.606660
\(349\) −1.48545e19 −1.26086 −0.630432 0.776245i \(-0.717122\pi\)
−0.630432 + 0.776245i \(0.717122\pi\)
\(350\) 2.86869e18 0.238327
\(351\) −2.92863e18 −0.238156
\(352\) −6.87412e18 −0.547202
\(353\) −8.23380e18 −0.641638 −0.320819 0.947140i \(-0.603958\pi\)
−0.320819 + 0.947140i \(0.603958\pi\)
\(354\) −3.63551e18 −0.277358
\(355\) −1.30655e19 −0.975919
\(356\) 9.99334e18 0.730863
\(357\) −1.17163e19 −0.839036
\(358\) 3.67627e18 0.257801
\(359\) 1.87417e19 1.28706 0.643532 0.765419i \(-0.277468\pi\)
0.643532 + 0.765419i \(0.277468\pi\)
\(360\) −4.60337e18 −0.309604
\(361\) 6.62574e18 0.436446
\(362\) −2.27623e18 −0.146859
\(363\) −6.11417e18 −0.386401
\(364\) −1.65963e19 −1.02743
\(365\) 1.24252e18 0.0753542
\(366\) 1.33936e18 0.0795772
\(367\) 8.63117e18 0.502428 0.251214 0.967932i \(-0.419170\pi\)
0.251214 + 0.967932i \(0.419170\pi\)
\(368\) −1.09147e19 −0.622519
\(369\) 2.62996e18 0.146978
\(370\) 9.80383e18 0.536887
\(371\) 2.58920e19 1.38951
\(372\) 3.38955e18 0.178267
\(373\) −9.95975e18 −0.513372 −0.256686 0.966495i \(-0.582631\pi\)
−0.256686 + 0.966495i \(0.582631\pi\)
\(374\) −6.67151e18 −0.337044
\(375\) 5.80181e18 0.287295
\(376\) 9.02156e18 0.437898
\(377\) 3.24640e19 1.54469
\(378\) 1.61851e18 0.0754965
\(379\) 1.00949e19 0.461645 0.230823 0.972996i \(-0.425858\pi\)
0.230823 + 0.972996i \(0.425858\pi\)
\(380\) −2.85304e19 −1.27918
\(381\) −1.36898e19 −0.601810
\(382\) −6.79890e18 −0.293065
\(383\) −2.71764e19 −1.14869 −0.574343 0.818614i \(-0.694743\pi\)
−0.574343 + 0.818614i \(0.694743\pi\)
\(384\) 1.42062e19 0.588837
\(385\) 1.76919e19 0.719150
\(386\) −3.00775e18 −0.119905
\(387\) −4.70111e18 −0.183809
\(388\) 1.48895e19 0.571007
\(389\) −9.05285e18 −0.340536 −0.170268 0.985398i \(-0.554463\pi\)
−0.170268 + 0.985398i \(0.554463\pi\)
\(390\) −9.76458e18 −0.360304
\(391\) −4.60389e19 −1.66648
\(392\) −5.62116e17 −0.0199609
\(393\) 2.21017e19 0.769984
\(394\) 1.06836e19 0.365170
\(395\) −3.54204e19 −1.18789
\(396\) −4.90387e18 −0.161371
\(397\) −1.76495e19 −0.569908 −0.284954 0.958541i \(-0.591978\pi\)
−0.284954 + 0.958541i \(0.591978\pi\)
\(398\) −1.42240e19 −0.450711
\(399\) 2.19473e19 0.682472
\(400\) 1.09574e19 0.334394
\(401\) 2.81135e19 0.842040 0.421020 0.907051i \(-0.361672\pi\)
0.421020 + 0.907051i \(0.361672\pi\)
\(402\) 1.94169e18 0.0570801
\(403\) 1.57309e19 0.453906
\(404\) 1.16465e19 0.329866
\(405\) −5.06698e18 −0.140876
\(406\) −1.79412e19 −0.489674
\(407\) 2.28504e19 0.612260
\(408\) 2.36908e19 0.623202
\(409\) 6.42955e19 1.66056 0.830282 0.557343i \(-0.188179\pi\)
0.830282 + 0.557343i \(0.188179\pi\)
\(410\) 8.76877e18 0.222362
\(411\) −2.17331e19 −0.541137
\(412\) 4.01868e18 0.0982549
\(413\) −4.96159e19 −1.19123
\(414\) 6.35987e18 0.149950
\(415\) −1.27746e19 −0.295793
\(416\) 5.17787e19 1.17747
\(417\) −4.26636e19 −0.952878
\(418\) 1.24972e19 0.274152
\(419\) −2.34474e19 −0.505230 −0.252615 0.967567i \(-0.581291\pi\)
−0.252615 + 0.967567i \(0.581291\pi\)
\(420\) −2.87142e19 −0.607753
\(421\) −7.28227e19 −1.51409 −0.757044 0.653364i \(-0.773357\pi\)
−0.757044 + 0.653364i \(0.773357\pi\)
\(422\) 3.22971e19 0.659659
\(423\) 9.93013e18 0.199252
\(424\) −5.23543e19 −1.03207
\(425\) 4.62192e19 0.895170
\(426\) −9.28867e18 −0.176759
\(427\) 1.82790e19 0.341777
\(428\) 1.12735e19 0.207123
\(429\) −2.27589e19 −0.410886
\(430\) −1.56743e19 −0.278084
\(431\) −4.71831e19 −0.822635 −0.411317 0.911492i \(-0.634931\pi\)
−0.411317 + 0.911492i \(0.634931\pi\)
\(432\) 6.18215e18 0.105928
\(433\) 3.79099e19 0.638400 0.319200 0.947687i \(-0.396586\pi\)
0.319200 + 0.947687i \(0.396586\pi\)
\(434\) −8.69371e18 −0.143890
\(435\) 5.61677e19 0.913728
\(436\) −3.03793e19 −0.485768
\(437\) 8.62412e19 1.35551
\(438\) 8.83346e17 0.0136482
\(439\) −2.26832e19 −0.344525 −0.172262 0.985051i \(-0.555108\pi\)
−0.172262 + 0.985051i \(0.555108\pi\)
\(440\) −3.57736e19 −0.534156
\(441\) −6.18727e17 −0.00908259
\(442\) 5.02525e19 0.725255
\(443\) −9.94888e19 −1.41171 −0.705856 0.708355i \(-0.749438\pi\)
−0.705856 + 0.708355i \(0.749438\pi\)
\(444\) −3.70864e19 −0.517420
\(445\) 8.02431e19 1.10080
\(446\) −1.15183e19 −0.155374
\(447\) 3.47592e19 0.461066
\(448\) 1.30023e19 0.169604
\(449\) 1.30159e20 1.66965 0.834825 0.550515i \(-0.185568\pi\)
0.834825 + 0.550515i \(0.185568\pi\)
\(450\) −6.38477e18 −0.0805474
\(451\) 2.04379e19 0.253579
\(452\) 5.87604e19 0.717045
\(453\) 6.79275e19 0.815285
\(454\) −1.04978e19 −0.123931
\(455\) −1.33263e20 −1.54747
\(456\) −4.43782e19 −0.506913
\(457\) −1.01910e20 −1.14510 −0.572550 0.819870i \(-0.694046\pi\)
−0.572550 + 0.819870i \(0.694046\pi\)
\(458\) −4.90460e19 −0.542141
\(459\) 2.60767e19 0.283569
\(460\) −1.12831e20 −1.20711
\(461\) −1.08142e20 −1.13825 −0.569123 0.822253i \(-0.692717\pi\)
−0.569123 + 0.822253i \(0.692717\pi\)
\(462\) 1.25777e19 0.130253
\(463\) 1.69113e20 1.72314 0.861569 0.507641i \(-0.169482\pi\)
0.861569 + 0.507641i \(0.169482\pi\)
\(464\) −6.85294e19 −0.687056
\(465\) 2.72169e19 0.268498
\(466\) 2.80105e19 0.271911
\(467\) −4.58418e19 −0.437910 −0.218955 0.975735i \(-0.570265\pi\)
−0.218955 + 0.975735i \(0.570265\pi\)
\(468\) 3.69380e19 0.347239
\(469\) 2.64993e19 0.245154
\(470\) 3.31089e19 0.301446
\(471\) −1.18398e20 −1.06093
\(472\) 1.00325e20 0.884797
\(473\) −3.65331e19 −0.317124
\(474\) −2.51814e19 −0.215151
\(475\) −8.65789e19 −0.728132
\(476\) 1.47775e20 1.22334
\(477\) −5.76270e19 −0.469611
\(478\) −4.86884e18 −0.0390585
\(479\) 1.31513e20 1.03861 0.519303 0.854590i \(-0.326192\pi\)
0.519303 + 0.854590i \(0.326192\pi\)
\(480\) 8.95851e19 0.696508
\(481\) −1.72118e20 −1.31747
\(482\) 6.48380e19 0.488626
\(483\) 8.67968e19 0.644021
\(484\) 7.71163e19 0.563386
\(485\) 1.19557e20 0.860029
\(486\) −3.60227e18 −0.0255155
\(487\) 6.13671e19 0.428024 0.214012 0.976831i \(-0.431347\pi\)
0.214012 + 0.976831i \(0.431347\pi\)
\(488\) −3.69607e19 −0.253858
\(489\) −1.21294e20 −0.820392
\(490\) −2.06295e18 −0.0137410
\(491\) 2.42057e20 1.58784 0.793918 0.608024i \(-0.208038\pi\)
0.793918 + 0.608024i \(0.208038\pi\)
\(492\) −3.31710e19 −0.214299
\(493\) −2.89062e20 −1.83924
\(494\) −9.41342e19 −0.589923
\(495\) −3.93764e19 −0.243051
\(496\) −3.32070e19 −0.201891
\(497\) −1.26768e20 −0.759165
\(498\) −9.08188e18 −0.0535742
\(499\) 1.46049e20 0.848680 0.424340 0.905503i \(-0.360506\pi\)
0.424340 + 0.905503i \(0.360506\pi\)
\(500\) −7.31765e19 −0.418887
\(501\) −1.27623e19 −0.0719690
\(502\) −1.14429e20 −0.635707
\(503\) −1.68041e20 −0.919720 −0.459860 0.887991i \(-0.652100\pi\)
−0.459860 + 0.887991i \(0.652100\pi\)
\(504\) −4.46641e19 −0.240840
\(505\) 9.35178e19 0.496831
\(506\) 4.94237e19 0.258706
\(507\) 5.94857e19 0.306798
\(508\) 1.72665e20 0.877459
\(509\) 7.37709e19 0.369404 0.184702 0.982795i \(-0.440868\pi\)
0.184702 + 0.982795i \(0.440868\pi\)
\(510\) 8.69446e19 0.429009
\(511\) 1.20555e19 0.0586178
\(512\) −1.93454e20 −0.926943
\(513\) −4.88476e19 −0.230655
\(514\) −1.65411e20 −0.769736
\(515\) 3.22686e19 0.147988
\(516\) 5.92937e19 0.268001
\(517\) 7.71689e19 0.343766
\(518\) 9.51213e19 0.417643
\(519\) 9.64375e19 0.417341
\(520\) 2.69461e20 1.14940
\(521\) 3.22023e20 1.35396 0.676978 0.736003i \(-0.263289\pi\)
0.676978 + 0.736003i \(0.263289\pi\)
\(522\) 3.99313e19 0.165495
\(523\) −1.19064e19 −0.0486429 −0.0243214 0.999704i \(-0.507743\pi\)
−0.0243214 + 0.999704i \(0.507743\pi\)
\(524\) −2.78762e20 −1.12266
\(525\) −8.71366e19 −0.345944
\(526\) 3.64073e19 0.142494
\(527\) −1.40069e20 −0.540460
\(528\) 4.80426e19 0.182756
\(529\) 7.44293e19 0.279143
\(530\) −1.92139e20 −0.710471
\(531\) 1.10429e20 0.402599
\(532\) −2.76815e20 −0.995069
\(533\) −1.53947e20 −0.545652
\(534\) 5.70473e19 0.199377
\(535\) 9.05220e19 0.311961
\(536\) −5.35825e19 −0.182090
\(537\) −1.11667e20 −0.374211
\(538\) 1.76206e20 0.582309
\(539\) −4.80824e18 −0.0156701
\(540\) 6.39084e19 0.205402
\(541\) 1.59033e20 0.504089 0.252045 0.967716i \(-0.418897\pi\)
0.252045 + 0.967716i \(0.418897\pi\)
\(542\) 1.10741e20 0.346189
\(543\) 6.91405e19 0.213174
\(544\) −4.61041e20 −1.40200
\(545\) −2.43935e20 −0.731645
\(546\) −9.47405e19 −0.280279
\(547\) −4.25405e20 −1.24136 −0.620680 0.784064i \(-0.713143\pi\)
−0.620680 + 0.784064i \(0.713143\pi\)
\(548\) 2.74113e20 0.788997
\(549\) −4.06830e19 −0.115510
\(550\) −4.96172e19 −0.138967
\(551\) 5.41477e20 1.49604
\(552\) −1.75506e20 −0.478353
\(553\) −3.43665e20 −0.924054
\(554\) 5.68771e19 0.150874
\(555\) −2.97791e20 −0.779318
\(556\) 5.38104e20 1.38933
\(557\) 2.71579e20 0.691803 0.345901 0.938271i \(-0.387573\pi\)
0.345901 + 0.938271i \(0.387573\pi\)
\(558\) 1.93493e19 0.0486306
\(559\) 2.75182e20 0.682389
\(560\) 2.81309e20 0.688293
\(561\) 2.02647e20 0.489237
\(562\) 4.43018e19 0.105536
\(563\) −2.68438e20 −0.631002 −0.315501 0.948925i \(-0.602173\pi\)
−0.315501 + 0.948925i \(0.602173\pi\)
\(564\) −1.25246e20 −0.290516
\(565\) 4.71825e20 1.07999
\(566\) 9.09210e19 0.205372
\(567\) −4.91622e19 −0.109587
\(568\) 2.56328e20 0.563877
\(569\) −1.34893e20 −0.292853 −0.146426 0.989222i \(-0.546777\pi\)
−0.146426 + 0.989222i \(0.546777\pi\)
\(570\) −1.62867e20 −0.348956
\(571\) −6.81153e20 −1.44037 −0.720185 0.693782i \(-0.755943\pi\)
−0.720185 + 0.693782i \(0.755943\pi\)
\(572\) 2.87052e20 0.599087
\(573\) 2.06517e20 0.425398
\(574\) 8.50787e19 0.172974
\(575\) −3.42400e20 −0.687108
\(576\) −2.89389e19 −0.0573209
\(577\) −2.42927e20 −0.474961 −0.237480 0.971392i \(-0.576322\pi\)
−0.237480 + 0.971392i \(0.576322\pi\)
\(578\) −2.41358e20 −0.465803
\(579\) 9.13605e19 0.174048
\(580\) −7.08427e20 −1.33225
\(581\) −1.23945e20 −0.230096
\(582\) 8.49970e19 0.155769
\(583\) −4.47830e20 −0.810213
\(584\) −2.43766e19 −0.0435389
\(585\) 2.96599e20 0.522999
\(586\) 2.51902e20 0.438530
\(587\) −1.06241e20 −0.182602 −0.0913012 0.995823i \(-0.529103\pi\)
−0.0913012 + 0.995823i \(0.529103\pi\)
\(588\) 7.80383e18 0.0132427
\(589\) 2.62381e20 0.439610
\(590\) 3.68189e20 0.609089
\(591\) −3.24513e20 −0.530062
\(592\) 3.63331e20 0.585989
\(593\) −7.70415e20 −1.22692 −0.613458 0.789727i \(-0.710222\pi\)
−0.613458 + 0.789727i \(0.710222\pi\)
\(594\) −2.79939e19 −0.0440215
\(595\) 1.18658e21 1.84255
\(596\) −4.38407e20 −0.672250
\(597\) 4.32053e20 0.654229
\(598\) −3.72279e20 −0.556686
\(599\) −7.59796e19 −0.112201 −0.0561004 0.998425i \(-0.517867\pi\)
−0.0561004 + 0.998425i \(0.517867\pi\)
\(600\) 1.76193e20 0.256953
\(601\) 7.78003e20 1.12053 0.560264 0.828314i \(-0.310700\pi\)
0.560264 + 0.828314i \(0.310700\pi\)
\(602\) −1.52080e20 −0.216321
\(603\) −5.89788e19 −0.0828546
\(604\) −8.56750e20 −1.18871
\(605\) 6.19217e20 0.848551
\(606\) 6.64846e19 0.0899863
\(607\) 3.21867e20 0.430290 0.215145 0.976582i \(-0.430978\pi\)
0.215145 + 0.976582i \(0.430978\pi\)
\(608\) 8.63633e20 1.14039
\(609\) 5.44965e20 0.710787
\(610\) −1.35645e20 −0.174755
\(611\) −5.81267e20 −0.739719
\(612\) −3.28898e20 −0.413453
\(613\) 6.68867e20 0.830589 0.415294 0.909687i \(-0.363679\pi\)
0.415294 + 0.909687i \(0.363679\pi\)
\(614\) −9.27796e19 −0.113812
\(615\) −2.66351e20 −0.322769
\(616\) −3.47092e20 −0.415518
\(617\) −8.47904e20 −1.00279 −0.501393 0.865220i \(-0.667179\pi\)
−0.501393 + 0.865220i \(0.667179\pi\)
\(618\) 2.29407e19 0.0268036
\(619\) −3.51027e20 −0.405192 −0.202596 0.979262i \(-0.564938\pi\)
−0.202596 + 0.979262i \(0.564938\pi\)
\(620\) −3.43279e20 −0.391480
\(621\) −1.93181e20 −0.217660
\(622\) −6.06982e20 −0.675690
\(623\) 7.78556e20 0.856307
\(624\) −3.61876e20 −0.393256
\(625\) −1.15339e21 −1.23844
\(626\) −5.67619e20 −0.602212
\(627\) −3.79603e20 −0.397945
\(628\) 1.49332e21 1.54688
\(629\) 1.53256e21 1.56869
\(630\) −1.63916e20 −0.165793
\(631\) −1.30718e21 −1.30652 −0.653260 0.757134i \(-0.726599\pi\)
−0.653260 + 0.757134i \(0.726599\pi\)
\(632\) 6.94902e20 0.686349
\(633\) −9.81023e20 −0.957528
\(634\) 6.27508e20 0.605271
\(635\) 1.38644e21 1.32160
\(636\) 7.26833e20 0.684709
\(637\) 3.62176e19 0.0337190
\(638\) 3.10314e20 0.285526
\(639\) 2.82143e20 0.256575
\(640\) −1.43875e21 −1.29311
\(641\) −1.77975e21 −1.58097 −0.790486 0.612480i \(-0.790172\pi\)
−0.790486 + 0.612480i \(0.790172\pi\)
\(642\) 6.43548e19 0.0565026
\(643\) −5.18269e19 −0.0449752 −0.0224876 0.999747i \(-0.507159\pi\)
−0.0224876 + 0.999747i \(0.507159\pi\)
\(644\) −1.09474e21 −0.939005
\(645\) 4.76108e20 0.403652
\(646\) 8.38178e20 0.702413
\(647\) 1.80268e21 1.49327 0.746634 0.665235i \(-0.231669\pi\)
0.746634 + 0.665235i \(0.231669\pi\)
\(648\) 9.94075e19 0.0813967
\(649\) 8.58162e20 0.694599
\(650\) 3.73737e20 0.299031
\(651\) 2.64071e20 0.208864
\(652\) 1.52984e21 1.19616
\(653\) 6.01117e20 0.464633 0.232317 0.972640i \(-0.425369\pi\)
0.232317 + 0.972640i \(0.425369\pi\)
\(654\) −1.73421e20 −0.132516
\(655\) −2.23837e21 −1.69091
\(656\) 3.24971e20 0.242698
\(657\) −2.68316e19 −0.0198110
\(658\) 3.21238e20 0.234494
\(659\) −1.34492e21 −0.970635 −0.485318 0.874338i \(-0.661296\pi\)
−0.485318 + 0.874338i \(0.661296\pi\)
\(660\) 4.96643e20 0.354377
\(661\) 3.00301e20 0.211858 0.105929 0.994374i \(-0.466218\pi\)
0.105929 + 0.994374i \(0.466218\pi\)
\(662\) −9.14362e20 −0.637798
\(663\) −1.52642e21 −1.05274
\(664\) 2.50622e20 0.170906
\(665\) −2.22273e21 −1.49873
\(666\) −2.11709e20 −0.141151
\(667\) 2.14142e21 1.41175
\(668\) 1.60967e20 0.104933
\(669\) 3.49869e20 0.225533
\(670\) −1.96646e20 −0.125350
\(671\) −3.16155e20 −0.199288
\(672\) 8.69196e20 0.541812
\(673\) −2.48252e21 −1.53031 −0.765156 0.643845i \(-0.777338\pi\)
−0.765156 + 0.643845i \(0.777338\pi\)
\(674\) 4.78260e20 0.291551
\(675\) 1.93937e20 0.116919
\(676\) −7.50276e20 −0.447323
\(677\) −1.84000e21 −1.08493 −0.542467 0.840077i \(-0.682509\pi\)
−0.542467 + 0.840077i \(0.682509\pi\)
\(678\) 3.35435e20 0.195608
\(679\) 1.16000e21 0.669014
\(680\) −2.39930e21 −1.36857
\(681\) 3.18871e20 0.179892
\(682\) 1.50367e20 0.0839016
\(683\) 2.19144e21 1.20941 0.604705 0.796450i \(-0.293291\pi\)
0.604705 + 0.796450i \(0.293291\pi\)
\(684\) 6.16100e20 0.336303
\(685\) 2.20103e21 1.18836
\(686\) −7.54596e20 −0.402980
\(687\) 1.48977e21 0.786945
\(688\) −5.80892e20 −0.303517
\(689\) 3.37324e21 1.74342
\(690\) −6.44101e20 −0.329295
\(691\) −5.25387e20 −0.265701 −0.132851 0.991136i \(-0.542413\pi\)
−0.132851 + 0.991136i \(0.542413\pi\)
\(692\) −1.21634e21 −0.608497
\(693\) −3.82049e20 −0.189069
\(694\) −1.35241e20 −0.0662082
\(695\) 4.32079e21 2.09256
\(696\) −1.10194e21 −0.527943
\(697\) 1.37075e21 0.649701
\(698\) 1.06953e21 0.501505
\(699\) −8.50820e20 −0.394693
\(700\) 1.09903e21 0.504398
\(701\) −3.78227e21 −1.71739 −0.858693 0.512491i \(-0.828723\pi\)
−0.858693 + 0.512491i \(0.828723\pi\)
\(702\) 2.10861e20 0.0947259
\(703\) −2.87082e21 −1.27597
\(704\) −2.24889e20 −0.0988949
\(705\) −1.00568e21 −0.437564
\(706\) 5.92834e20 0.255210
\(707\) 9.07353e20 0.386483
\(708\) −1.39281e21 −0.587004
\(709\) −4.24647e21 −1.77085 −0.885424 0.464784i \(-0.846132\pi\)
−0.885424 + 0.464784i \(0.846132\pi\)
\(710\) 9.40717e20 0.388170
\(711\) 7.64886e20 0.312302
\(712\) −1.57426e21 −0.636030
\(713\) 1.03766e21 0.414842
\(714\) 8.43577e20 0.333725
\(715\) 2.30493e21 0.902322
\(716\) 1.40842e21 0.545613
\(717\) 1.47891e20 0.0566954
\(718\) −1.34940e21 −0.511927
\(719\) −3.17360e20 −0.119148 −0.0595738 0.998224i \(-0.518974\pi\)
−0.0595738 + 0.998224i \(0.518974\pi\)
\(720\) −6.26102e20 −0.232622
\(721\) 3.13085e20 0.115119
\(722\) −4.77053e20 −0.173595
\(723\) −1.96946e21 −0.709266
\(724\) −8.72049e20 −0.310815
\(725\) −2.14981e21 −0.758340
\(726\) 4.40221e20 0.153690
\(727\) 1.25112e21 0.432307 0.216154 0.976359i \(-0.430649\pi\)
0.216154 + 0.976359i \(0.430649\pi\)
\(728\) 2.61444e21 0.894115
\(729\) 1.09419e20 0.0370370
\(730\) −8.94615e19 −0.0299719
\(731\) −2.45025e21 −0.812511
\(732\) 5.13123e20 0.168418
\(733\) 4.26993e21 1.38721 0.693603 0.720357i \(-0.256022\pi\)
0.693603 + 0.720357i \(0.256022\pi\)
\(734\) −6.21444e20 −0.199840
\(735\) 6.26621e19 0.0199457
\(736\) 3.41547e21 1.07614
\(737\) −4.58335e20 −0.142948
\(738\) −1.89357e20 −0.0584601
\(739\) 1.73302e21 0.529627 0.264813 0.964300i \(-0.414690\pi\)
0.264813 + 0.964300i \(0.414690\pi\)
\(740\) 3.75596e21 1.13627
\(741\) 2.85933e21 0.856302
\(742\) −1.86422e21 −0.552673
\(743\) −2.63739e21 −0.774032 −0.387016 0.922073i \(-0.626494\pi\)
−0.387016 + 0.922073i \(0.626494\pi\)
\(744\) −5.33960e20 −0.155136
\(745\) −3.52026e21 −1.01252
\(746\) 7.17102e20 0.204192
\(747\) 2.75862e20 0.0777656
\(748\) −2.55593e21 −0.713324
\(749\) 8.78287e20 0.242674
\(750\) −4.17730e20 −0.114271
\(751\) 2.27078e21 0.615001 0.307500 0.951548i \(-0.400507\pi\)
0.307500 + 0.951548i \(0.400507\pi\)
\(752\) 1.22702e21 0.329016
\(753\) 3.47577e21 0.922761
\(754\) −2.33741e21 −0.614397
\(755\) −6.87941e21 −1.79040
\(756\) 6.20069e20 0.159782
\(757\) −1.72959e21 −0.441291 −0.220645 0.975354i \(-0.570816\pi\)
−0.220645 + 0.975354i \(0.570816\pi\)
\(758\) −7.26834e20 −0.183618
\(759\) −1.50125e21 −0.375525
\(760\) 4.49443e21 1.11320
\(761\) −2.06739e20 −0.0507035 −0.0253518 0.999679i \(-0.508071\pi\)
−0.0253518 + 0.999679i \(0.508071\pi\)
\(762\) 9.85664e20 0.239368
\(763\) −2.36677e21 −0.569144
\(764\) −2.60473e21 −0.620245
\(765\) −2.64094e21 −0.622727
\(766\) 1.95670e21 0.456887
\(767\) −6.46402e21 −1.49464
\(768\) −5.89256e20 −0.134926
\(769\) 3.01490e21 0.683636 0.341818 0.939766i \(-0.388957\pi\)
0.341818 + 0.939766i \(0.388957\pi\)
\(770\) −1.27382e21 −0.286040
\(771\) 5.02437e21 1.11731
\(772\) −1.15230e21 −0.253768
\(773\) 8.99357e21 1.96149 0.980745 0.195293i \(-0.0625658\pi\)
0.980745 + 0.195293i \(0.0625658\pi\)
\(774\) 3.38480e20 0.0731098
\(775\) −1.04172e21 −0.222838
\(776\) −2.34556e21 −0.496916
\(777\) −2.88931e21 −0.606229
\(778\) 6.51805e20 0.135447
\(779\) −2.56773e21 −0.528467
\(780\) −3.74092e21 −0.762551
\(781\) 2.19259e21 0.442664
\(782\) 3.31480e21 0.662838
\(783\) −1.21291e21 −0.240224
\(784\) −7.64531e19 −0.0149977
\(785\) 1.19908e22 2.32985
\(786\) −1.59132e21 −0.306259
\(787\) −5.92807e21 −1.13006 −0.565032 0.825069i \(-0.691136\pi\)
−0.565032 + 0.825069i \(0.691136\pi\)
\(788\) 4.09299e21 0.772849
\(789\) −1.10587e21 −0.206837
\(790\) 2.55027e21 0.472479
\(791\) 4.57787e21 0.840117
\(792\) 7.72514e20 0.140432
\(793\) 2.38141e21 0.428830
\(794\) 1.27077e21 0.226680
\(795\) 5.83622e21 1.03128
\(796\) −5.44936e21 −0.953889
\(797\) −3.83837e21 −0.665593 −0.332797 0.942999i \(-0.607992\pi\)
−0.332797 + 0.942999i \(0.607992\pi\)
\(798\) −1.58021e21 −0.271452
\(799\) 5.17565e21 0.880773
\(800\) −3.42885e21 −0.578060
\(801\) −1.73281e21 −0.289406
\(802\) −2.02417e21 −0.334919
\(803\) −2.08513e20 −0.0341796
\(804\) 7.43883e20 0.120805
\(805\) −8.79040e21 −1.41429
\(806\) −1.13263e21 −0.180540
\(807\) −5.35226e21 −0.845251
\(808\) −1.83470e21 −0.287064
\(809\) −6.17948e21 −0.957940 −0.478970 0.877831i \(-0.658990\pi\)
−0.478970 + 0.877831i \(0.658990\pi\)
\(810\) 3.64823e20 0.0560330
\(811\) 1.87862e21 0.285879 0.142939 0.989731i \(-0.454345\pi\)
0.142939 + 0.989731i \(0.454345\pi\)
\(812\) −6.87349e21 −1.03635
\(813\) −3.36375e21 −0.502510
\(814\) −1.64523e21 −0.243525
\(815\) 1.22841e22 1.80161
\(816\) 3.22217e21 0.468245
\(817\) 4.58986e21 0.660897
\(818\) −4.62927e21 −0.660485
\(819\) 2.87774e21 0.406839
\(820\) 3.35941e21 0.470608
\(821\) 3.93227e21 0.545845 0.272923 0.962036i \(-0.412010\pi\)
0.272923 + 0.962036i \(0.412010\pi\)
\(822\) 1.56478e21 0.215236
\(823\) −7.92927e21 −1.08077 −0.540386 0.841417i \(-0.681722\pi\)
−0.540386 + 0.841417i \(0.681722\pi\)
\(824\) −6.33068e20 −0.0855059
\(825\) 1.50712e21 0.201718
\(826\) 3.57235e21 0.473809
\(827\) −1.97038e21 −0.258976 −0.129488 0.991581i \(-0.541333\pi\)
−0.129488 + 0.991581i \(0.541333\pi\)
\(828\) 2.43654e21 0.317355
\(829\) 2.43930e21 0.314851 0.157426 0.987531i \(-0.449681\pi\)
0.157426 + 0.987531i \(0.449681\pi\)
\(830\) 9.19774e20 0.117651
\(831\) −1.72764e21 −0.219001
\(832\) 1.69396e21 0.212803
\(833\) −3.22484e20 −0.0401487
\(834\) 3.07178e21 0.379005
\(835\) 1.29251e21 0.158047
\(836\) 4.78783e21 0.580218
\(837\) −5.87736e20 −0.0705897
\(838\) 1.68821e21 0.200954
\(839\) −1.35318e22 −1.59639 −0.798196 0.602398i \(-0.794212\pi\)
−0.798196 + 0.602398i \(0.794212\pi\)
\(840\) 4.52338e21 0.528894
\(841\) 4.81601e21 0.558107
\(842\) 5.24323e21 0.602225
\(843\) −1.34567e21 −0.153190
\(844\) 1.23734e22 1.39611
\(845\) −6.02446e21 −0.673741
\(846\) −7.14970e20 −0.0792519
\(847\) 6.00794e21 0.660085
\(848\) −7.12068e21 −0.775448
\(849\) −2.76173e21 −0.298107
\(850\) −3.32778e21 −0.356052
\(851\) −1.13534e22 −1.20408
\(852\) −3.55859e21 −0.374095
\(853\) −7.30038e21 −0.760725 −0.380363 0.924837i \(-0.624201\pi\)
−0.380363 + 0.924837i \(0.624201\pi\)
\(854\) −1.31609e21 −0.135941
\(855\) 4.94707e21 0.506527
\(856\) −1.77592e21 −0.180248
\(857\) 1.16475e22 1.17187 0.585934 0.810359i \(-0.300728\pi\)
0.585934 + 0.810359i \(0.300728\pi\)
\(858\) 1.63864e21 0.163429
\(859\) −1.05473e22 −1.04278 −0.521388 0.853320i \(-0.674586\pi\)
−0.521388 + 0.853320i \(0.674586\pi\)
\(860\) −6.00501e21 −0.588539
\(861\) −2.58427e21 −0.251081
\(862\) 3.39718e21 0.327201
\(863\) 1.25514e22 1.19843 0.599215 0.800588i \(-0.295480\pi\)
0.599215 + 0.800588i \(0.295480\pi\)
\(864\) −1.93455e21 −0.183116
\(865\) −9.76678e21 −0.916496
\(866\) −2.72951e21 −0.253922
\(867\) 7.33124e21 0.676136
\(868\) −3.33066e21 −0.304531
\(869\) 5.94407e21 0.538810
\(870\) −4.04407e21 −0.363433
\(871\) 3.45237e21 0.307596
\(872\) 4.78569e21 0.422737
\(873\) −2.58178e21 −0.226106
\(874\) −6.20937e21 −0.539153
\(875\) −5.70100e21 −0.490784
\(876\) 3.38420e20 0.0288852
\(877\) −1.57294e22 −1.33112 −0.665558 0.746346i \(-0.731807\pi\)
−0.665558 + 0.746346i \(0.731807\pi\)
\(878\) 1.63319e21 0.137034
\(879\) −7.65152e21 −0.636548
\(880\) −4.86555e21 −0.401339
\(881\) 1.80412e22 1.47553 0.737764 0.675059i \(-0.235882\pi\)
0.737764 + 0.675059i \(0.235882\pi\)
\(882\) 4.45484e19 0.00361258
\(883\) 2.41866e22 1.94478 0.972388 0.233371i \(-0.0749757\pi\)
0.972388 + 0.233371i \(0.0749757\pi\)
\(884\) 1.92523e22 1.53494
\(885\) −1.11837e22 −0.884123
\(886\) 7.16320e21 0.561505
\(887\) 7.05392e21 0.548282 0.274141 0.961690i \(-0.411607\pi\)
0.274141 + 0.961690i \(0.411607\pi\)
\(888\) 5.84228e21 0.450282
\(889\) 1.34519e22 1.02807
\(890\) −5.77750e21 −0.437840
\(891\) 8.50315e20 0.0638994
\(892\) −4.41279e21 −0.328834
\(893\) −9.69514e21 −0.716421
\(894\) −2.50266e21 −0.183388
\(895\) 1.13091e22 0.821782
\(896\) −1.39594e22 −1.00590
\(897\) 1.13080e22 0.808057
\(898\) −9.37143e21 −0.664100
\(899\) 6.51508e21 0.457849
\(900\) −2.44608e21 −0.170471
\(901\) −3.00356e22 −2.07587
\(902\) −1.47153e21 −0.100860
\(903\) 4.61942e21 0.314000
\(904\) −9.25660e21 −0.624005
\(905\) −7.00226e21 −0.468138
\(906\) −4.89078e21 −0.324277
\(907\) 2.48498e22 1.63406 0.817030 0.576596i \(-0.195619\pi\)
0.817030 + 0.576596i \(0.195619\pi\)
\(908\) −4.02183e21 −0.262289
\(909\) −2.01947e21 −0.130620
\(910\) 9.59492e21 0.615504
\(911\) −2.83848e21 −0.180592 −0.0902958 0.995915i \(-0.528781\pi\)
−0.0902958 + 0.995915i \(0.528781\pi\)
\(912\) −6.03585e21 −0.380870
\(913\) 2.14377e21 0.134168
\(914\) 7.33749e21 0.455461
\(915\) 4.12020e21 0.253665
\(916\) −1.87901e22 −1.14739
\(917\) −2.17177e22 −1.31536
\(918\) −1.87753e21 −0.112789
\(919\) −1.87762e22 −1.11877 −0.559387 0.828907i \(-0.688963\pi\)
−0.559387 + 0.828907i \(0.688963\pi\)
\(920\) 1.77745e22 1.05048
\(921\) 2.81818e21 0.165204
\(922\) 7.78620e21 0.452734
\(923\) −1.65155e22 −0.952528
\(924\) 4.81867e21 0.275669
\(925\) 1.13979e22 0.646787
\(926\) −1.21761e22 −0.685374
\(927\) −6.96825e20 −0.0389068
\(928\) 2.14445e22 1.18770
\(929\) −2.61729e21 −0.143792 −0.0718958 0.997412i \(-0.522905\pi\)
−0.0718958 + 0.997412i \(0.522905\pi\)
\(930\) −1.95962e21 −0.106795
\(931\) 6.04085e20 0.0326570
\(932\) 1.07311e22 0.575476
\(933\) 1.84371e22 0.980798
\(934\) 3.30061e21 0.174178
\(935\) −2.05232e22 −1.07438
\(936\) −5.81889e21 −0.302184
\(937\) 1.24366e22 0.640701 0.320350 0.947299i \(-0.396199\pi\)
0.320350 + 0.947299i \(0.396199\pi\)
\(938\) −1.90795e21 −0.0975094
\(939\) 1.72414e22 0.874140
\(940\) 1.26844e22 0.637984
\(941\) 1.71266e22 0.854574 0.427287 0.904116i \(-0.359469\pi\)
0.427287 + 0.904116i \(0.359469\pi\)
\(942\) 8.52465e21 0.421983
\(943\) −1.01548e22 −0.498692
\(944\) 1.36451e22 0.664795
\(945\) 4.97894e21 0.240657
\(946\) 2.63039e21 0.126135
\(947\) 1.26035e21 0.0599607 0.0299804 0.999550i \(-0.490456\pi\)
0.0299804 + 0.999550i \(0.490456\pi\)
\(948\) −9.64729e21 −0.455347
\(949\) 1.57061e21 0.0735480
\(950\) 6.23368e21 0.289613
\(951\) −1.90606e22 −0.878581
\(952\) −2.32792e22 −1.06461
\(953\) −2.19887e22 −0.997706 −0.498853 0.866687i \(-0.666245\pi\)
−0.498853 + 0.866687i \(0.666245\pi\)
\(954\) 4.14915e21 0.186787
\(955\) −2.09151e22 −0.934189
\(956\) −1.86531e21 −0.0826639
\(957\) −9.42577e21 −0.414455
\(958\) −9.46890e21 −0.413103
\(959\) 2.13554e22 0.924419
\(960\) 2.93081e21 0.125879
\(961\) −2.03083e22 −0.865461
\(962\) 1.23925e22 0.524019
\(963\) −1.95478e21 −0.0820163
\(964\) 2.48402e22 1.03413
\(965\) −9.25260e21 −0.382216
\(966\) −6.24937e21 −0.256158
\(967\) 1.21130e21 0.0492666 0.0246333 0.999697i \(-0.492158\pi\)
0.0246333 + 0.999697i \(0.492158\pi\)
\(968\) −1.21482e22 −0.490284
\(969\) −2.54597e22 −1.01959
\(970\) −8.60813e21 −0.342075
\(971\) 1.26825e22 0.500103 0.250051 0.968233i \(-0.419552\pi\)
0.250051 + 0.968233i \(0.419552\pi\)
\(972\) −1.38007e21 −0.0540013
\(973\) 4.19223e22 1.62779
\(974\) −4.41843e21 −0.170245
\(975\) −1.13523e22 −0.434058
\(976\) −5.02700e21 −0.190737
\(977\) −3.29367e21 −0.124014 −0.0620070 0.998076i \(-0.519750\pi\)
−0.0620070 + 0.998076i \(0.519750\pi\)
\(978\) 8.73314e21 0.326309
\(979\) −1.34660e22 −0.499307
\(980\) −7.90338e20 −0.0290815
\(981\) 5.26766e21 0.192353
\(982\) −1.74281e22 −0.631558
\(983\) 3.74166e22 1.34559 0.672796 0.739828i \(-0.265093\pi\)
0.672796 + 0.739828i \(0.265093\pi\)
\(984\) 5.22546e21 0.186493
\(985\) 3.28653e22 1.16404
\(986\) 2.08125e22 0.731554
\(987\) −9.75759e21 −0.340380
\(988\) −3.60638e22 −1.24852
\(989\) 1.81518e22 0.623661
\(990\) 2.83510e21 0.0966729
\(991\) −4.40305e22 −1.49005 −0.745026 0.667035i \(-0.767563\pi\)
−0.745026 + 0.667035i \(0.767563\pi\)
\(992\) 1.03913e22 0.349005
\(993\) 2.77738e22 0.925796
\(994\) 9.12728e21 0.301956
\(995\) −4.37565e22 −1.43671
\(996\) −3.47937e21 −0.113385
\(997\) 2.62067e22 0.847614 0.423807 0.905753i \(-0.360694\pi\)
0.423807 + 0.905753i \(0.360694\pi\)
\(998\) −1.05155e22 −0.337560
\(999\) 6.43066e21 0.204887
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 3.16.a.b.1.1 1
3.2 odd 2 9.16.a.c.1.1 1
4.3 odd 2 48.16.a.a.1.1 1
5.2 odd 4 75.16.b.b.49.1 2
5.3 odd 4 75.16.b.b.49.2 2
5.4 even 2 75.16.a.a.1.1 1
7.6 odd 2 147.16.a.b.1.1 1
12.11 even 2 144.16.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.16.a.b.1.1 1 1.1 even 1 trivial
9.16.a.c.1.1 1 3.2 odd 2
48.16.a.a.1.1 1 4.3 odd 2
75.16.a.a.1.1 1 5.4 even 2
75.16.b.b.49.1 2 5.2 odd 4
75.16.b.b.49.2 2 5.3 odd 4
144.16.a.l.1.1 1 12.11 even 2
147.16.a.b.1.1 1 7.6 odd 2