Properties

Label 10.16.a.d.1.2
Level $10$
Weight $16$
Character 10.1
Self dual yes
Analytic conductor $14.269$
Analytic rank $0$
Dimension $2$
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] = [10,16,Mod(1,10)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(10, 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("10.1");
 
S:= CuspForms(chi, 16);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 10.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: \(14.2693505100\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{239569}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 59892 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2}\cdot 5 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-244.229\) of defining polynomial
Character \(\chi\) \(=\) 10.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+128.000 q^{2} +3972.58 q^{3} +16384.0 q^{4} +78125.0 q^{5} +508490. q^{6} +1.57794e6 q^{7} +2.09715e6 q^{8} +1.43247e6 q^{9} +1.00000e7 q^{10} -2.98824e6 q^{11} +6.50867e7 q^{12} +4.92637e6 q^{13} +2.01976e8 q^{14} +3.10358e8 q^{15} +2.68435e8 q^{16} +6.45145e8 q^{17} +1.83357e8 q^{18} +2.41911e9 q^{19} +1.28000e9 q^{20} +6.26849e9 q^{21} -3.82495e8 q^{22} +2.33075e10 q^{23} +8.33110e9 q^{24} +6.10352e9 q^{25} +6.30575e8 q^{26} -5.13115e10 q^{27} +2.58530e10 q^{28} -1.80269e11 q^{29} +3.97258e10 q^{30} -2.85505e11 q^{31} +3.43597e10 q^{32} -1.18710e10 q^{33} +8.25786e10 q^{34} +1.23277e11 q^{35} +2.34697e10 q^{36} +1.04185e11 q^{37} +3.09646e11 q^{38} +1.95704e10 q^{39} +1.63840e11 q^{40} -1.28393e12 q^{41} +8.02367e11 q^{42} -5.11969e11 q^{43} -4.89593e10 q^{44} +1.11912e11 q^{45} +2.98336e12 q^{46} +5.22662e12 q^{47} +1.06638e12 q^{48} -2.25766e12 q^{49} +7.81250e11 q^{50} +2.56289e12 q^{51} +8.07136e10 q^{52} -1.39001e13 q^{53} -6.56788e12 q^{54} -2.33456e11 q^{55} +3.30918e12 q^{56} +9.61011e12 q^{57} -2.30744e13 q^{58} +9.11699e12 q^{59} +5.08490e12 q^{60} -1.50270e13 q^{61} -3.65446e13 q^{62} +2.26036e12 q^{63} +4.39805e12 q^{64} +3.84872e11 q^{65} -1.51949e12 q^{66} +2.18097e13 q^{67} +1.05701e13 q^{68} +9.25908e13 q^{69} +1.57794e13 q^{70} -6.45213e13 q^{71} +3.00412e12 q^{72} +9.76233e13 q^{73} +1.33356e13 q^{74} +2.42467e13 q^{75} +3.96347e13 q^{76} -4.71526e12 q^{77} +2.50501e12 q^{78} +2.35831e14 q^{79} +2.09715e13 q^{80} -2.24394e14 q^{81} -1.64343e14 q^{82} -4.13505e14 q^{83} +1.02703e14 q^{84} +5.04020e13 q^{85} -6.55320e13 q^{86} -7.16133e14 q^{87} -6.26679e12 q^{88} +5.45429e14 q^{89} +1.43247e13 q^{90} +7.77352e12 q^{91} +3.81870e14 q^{92} -1.13419e15 q^{93} +6.69007e14 q^{94} +1.88993e14 q^{95} +1.36497e14 q^{96} +4.71267e14 q^{97} -2.88981e14 q^{98} -4.28057e12 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 256 q^{2} - 1844 q^{3} + 32768 q^{4} + 156250 q^{5} - 236032 q^{6} - 984932 q^{7} + 4194304 q^{8} + 20916154 q^{9} + 20000000 q^{10} + 111552144 q^{11} - 30212096 q^{12} + 289313596 q^{13} - 126071296 q^{14}+ \cdots + 22\!\cdots\!88 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 128.000 0.707107
\(3\) 3972.58 1.04873 0.524364 0.851494i \(-0.324303\pi\)
0.524364 + 0.851494i \(0.324303\pi\)
\(4\) 16384.0 0.500000
\(5\) 78125.0 0.447214
\(6\) 508490. 0.741563
\(7\) 1.57794e6 0.724195 0.362097 0.932140i \(-0.382061\pi\)
0.362097 + 0.932140i \(0.382061\pi\)
\(8\) 2.09715e6 0.353553
\(9\) 1.43247e6 0.0998316
\(10\) 1.00000e7 0.316228
\(11\) −2.98824e6 −0.0462349 −0.0231175 0.999733i \(-0.507359\pi\)
−0.0231175 + 0.999733i \(0.507359\pi\)
\(12\) 6.50867e7 0.524364
\(13\) 4.92637e6 0.0217747 0.0108873 0.999941i \(-0.496534\pi\)
0.0108873 + 0.999941i \(0.496534\pi\)
\(14\) 2.01976e8 0.512083
\(15\) 3.10358e8 0.469006
\(16\) 2.68435e8 0.250000
\(17\) 6.45145e8 0.381321 0.190660 0.981656i \(-0.438937\pi\)
0.190660 + 0.981656i \(0.438937\pi\)
\(18\) 1.83357e8 0.0705916
\(19\) 2.41911e9 0.620874 0.310437 0.950594i \(-0.399525\pi\)
0.310437 + 0.950594i \(0.399525\pi\)
\(20\) 1.28000e9 0.223607
\(21\) 6.26849e9 0.759484
\(22\) −3.82495e8 −0.0326930
\(23\) 2.33075e10 1.42737 0.713685 0.700467i \(-0.247025\pi\)
0.713685 + 0.700467i \(0.247025\pi\)
\(24\) 8.33110e9 0.370782
\(25\) 6.10352e9 0.200000
\(26\) 6.30575e8 0.0153970
\(27\) −5.13115e10 −0.944032
\(28\) 2.58530e10 0.362097
\(29\) −1.80269e11 −1.94060 −0.970300 0.241904i \(-0.922228\pi\)
−0.970300 + 0.241904i \(0.922228\pi\)
\(30\) 3.97258e10 0.331637
\(31\) −2.85505e11 −1.86381 −0.931904 0.362706i \(-0.881853\pi\)
−0.931904 + 0.362706i \(0.881853\pi\)
\(32\) 3.43597e10 0.176777
\(33\) −1.18710e10 −0.0484879
\(34\) 8.25786e10 0.269635
\(35\) 1.23277e11 0.323870
\(36\) 2.34697e10 0.0499158
\(37\) 1.04185e11 0.180422 0.0902112 0.995923i \(-0.471246\pi\)
0.0902112 + 0.995923i \(0.471246\pi\)
\(38\) 3.09646e11 0.439025
\(39\) 1.95704e10 0.0228357
\(40\) 1.63840e11 0.158114
\(41\) −1.28393e12 −1.02958 −0.514792 0.857315i \(-0.672131\pi\)
−0.514792 + 0.857315i \(0.672131\pi\)
\(42\) 8.02367e11 0.537036
\(43\) −5.11969e11 −0.287230 −0.143615 0.989634i \(-0.545873\pi\)
−0.143615 + 0.989634i \(0.545873\pi\)
\(44\) −4.89593e10 −0.0231175
\(45\) 1.11912e11 0.0446460
\(46\) 2.98336e12 1.00930
\(47\) 5.22662e12 1.50483 0.752414 0.658690i \(-0.228889\pi\)
0.752414 + 0.658690i \(0.228889\pi\)
\(48\) 1.06638e12 0.262182
\(49\) −2.25766e12 −0.475542
\(50\) 7.81250e11 0.141421
\(51\) 2.56289e12 0.399902
\(52\) 8.07136e10 0.0108873
\(53\) −1.39001e13 −1.62536 −0.812680 0.582710i \(-0.801992\pi\)
−0.812680 + 0.582710i \(0.801992\pi\)
\(54\) −6.56788e12 −0.667532
\(55\) −2.33456e11 −0.0206769
\(56\) 3.30918e12 0.256042
\(57\) 9.61011e12 0.651129
\(58\) −2.30744e13 −1.37221
\(59\) 9.11699e12 0.476938 0.238469 0.971150i \(-0.423355\pi\)
0.238469 + 0.971150i \(0.423355\pi\)
\(60\) 5.08490e12 0.234503
\(61\) −1.50270e13 −0.612206 −0.306103 0.951998i \(-0.599025\pi\)
−0.306103 + 0.951998i \(0.599025\pi\)
\(62\) −3.65446e13 −1.31791
\(63\) 2.26036e12 0.0722975
\(64\) 4.39805e12 0.125000
\(65\) 3.84872e11 0.00973793
\(66\) −1.51949e12 −0.0342861
\(67\) 2.18097e13 0.439631 0.219816 0.975541i \(-0.429454\pi\)
0.219816 + 0.975541i \(0.429454\pi\)
\(68\) 1.05701e13 0.190660
\(69\) 9.25908e13 1.49692
\(70\) 1.57794e13 0.229011
\(71\) −6.45213e13 −0.841909 −0.420955 0.907082i \(-0.638305\pi\)
−0.420955 + 0.907082i \(0.638305\pi\)
\(72\) 3.00412e12 0.0352958
\(73\) 9.76233e13 1.03427 0.517133 0.855905i \(-0.326999\pi\)
0.517133 + 0.855905i \(0.326999\pi\)
\(74\) 1.33356e13 0.127578
\(75\) 2.42467e13 0.209746
\(76\) 3.96347e13 0.310437
\(77\) −4.71526e12 −0.0334831
\(78\) 2.50501e12 0.0161473
\(79\) 2.35831e14 1.38165 0.690823 0.723024i \(-0.257248\pi\)
0.690823 + 0.723024i \(0.257248\pi\)
\(80\) 2.09715e13 0.111803
\(81\) −2.24394e14 −1.08987
\(82\) −1.64343e14 −0.728026
\(83\) −4.13505e14 −1.67261 −0.836305 0.548264i \(-0.815289\pi\)
−0.836305 + 0.548264i \(0.815289\pi\)
\(84\) 1.02703e14 0.379742
\(85\) 5.04020e13 0.170532
\(86\) −6.55320e13 −0.203102
\(87\) −7.16133e14 −2.03516
\(88\) −6.26679e12 −0.0163465
\(89\) 5.45429e14 1.30711 0.653556 0.756878i \(-0.273276\pi\)
0.653556 + 0.756878i \(0.273276\pi\)
\(90\) 1.43247e13 0.0315695
\(91\) 7.77352e12 0.0157691
\(92\) 3.81870e14 0.713685
\(93\) −1.13419e15 −1.95463
\(94\) 6.69007e14 1.06407
\(95\) 1.88993e14 0.277664
\(96\) 1.36497e14 0.185391
\(97\) 4.71267e14 0.592214 0.296107 0.955155i \(-0.404311\pi\)
0.296107 + 0.955155i \(0.404311\pi\)
\(98\) −2.88981e14 −0.336259
\(99\) −4.28057e12 −0.00461571
\(100\) 1.00000e14 0.100000
\(101\) 8.62387e14 0.800372 0.400186 0.916434i \(-0.368945\pi\)
0.400186 + 0.916434i \(0.368945\pi\)
\(102\) 3.28050e14 0.282773
\(103\) −1.08890e15 −0.872385 −0.436192 0.899853i \(-0.643673\pi\)
−0.436192 + 0.899853i \(0.643673\pi\)
\(104\) 1.03313e13 0.00769851
\(105\) 4.89726e14 0.339651
\(106\) −1.77922e15 −1.14930
\(107\) 1.24168e15 0.747536 0.373768 0.927522i \(-0.378066\pi\)
0.373768 + 0.927522i \(0.378066\pi\)
\(108\) −8.40688e14 −0.472016
\(109\) 2.12985e15 1.11596 0.557982 0.829853i \(-0.311576\pi\)
0.557982 + 0.829853i \(0.311576\pi\)
\(110\) −2.98824e13 −0.0146208
\(111\) 4.13881e14 0.189214
\(112\) 4.23575e14 0.181049
\(113\) 6.89269e14 0.275613 0.137807 0.990459i \(-0.455995\pi\)
0.137807 + 0.990459i \(0.455995\pi\)
\(114\) 1.23009e15 0.460418
\(115\) 1.82090e15 0.638339
\(116\) −2.95353e15 −0.970300
\(117\) 7.05689e12 0.00217380
\(118\) 1.16698e15 0.337246
\(119\) 1.01800e15 0.276151
\(120\) 6.50867e14 0.165819
\(121\) −4.16832e15 −0.997862
\(122\) −1.92345e15 −0.432895
\(123\) −5.10050e15 −1.07975
\(124\) −4.67771e15 −0.931904
\(125\) 4.76837e14 0.0894427
\(126\) 2.89326e14 0.0511221
\(127\) −6.43459e15 −1.07150 −0.535751 0.844376i \(-0.679971\pi\)
−0.535751 + 0.844376i \(0.679971\pi\)
\(128\) 5.62950e14 0.0883883
\(129\) −2.03384e15 −0.301226
\(130\) 4.92637e13 0.00688576
\(131\) 2.77378e15 0.366047 0.183023 0.983109i \(-0.441412\pi\)
0.183023 + 0.983109i \(0.441412\pi\)
\(132\) −1.94495e14 −0.0242439
\(133\) 3.81721e15 0.449634
\(134\) 2.79164e15 0.310866
\(135\) −4.00871e15 −0.422184
\(136\) 1.35297e15 0.134817
\(137\) 1.42960e16 1.34837 0.674185 0.738563i \(-0.264495\pi\)
0.674185 + 0.738563i \(0.264495\pi\)
\(138\) 1.18516e16 1.05849
\(139\) −4.57651e15 −0.387189 −0.193595 0.981082i \(-0.562015\pi\)
−0.193595 + 0.981082i \(0.562015\pi\)
\(140\) 2.01976e15 0.161935
\(141\) 2.07632e16 1.57816
\(142\) −8.25872e15 −0.595320
\(143\) −1.47212e13 −0.00100675
\(144\) 3.84527e14 0.0249579
\(145\) −1.40835e16 −0.867863
\(146\) 1.24958e16 0.731336
\(147\) −8.96875e15 −0.498714
\(148\) 1.70696e15 0.0902112
\(149\) 1.32935e16 0.667947 0.333974 0.942582i \(-0.391610\pi\)
0.333974 + 0.942582i \(0.391610\pi\)
\(150\) 3.10358e15 0.148313
\(151\) −1.93100e16 −0.877922 −0.438961 0.898506i \(-0.644653\pi\)
−0.438961 + 0.898506i \(0.644653\pi\)
\(152\) 5.07324e15 0.219512
\(153\) 9.24154e14 0.0380679
\(154\) −6.03554e14 −0.0236761
\(155\) −2.23051e16 −0.833520
\(156\) 3.20641e14 0.0114179
\(157\) 1.06880e16 0.362784 0.181392 0.983411i \(-0.441940\pi\)
0.181392 + 0.983411i \(0.441940\pi\)
\(158\) 3.01863e16 0.976971
\(159\) −5.52193e16 −1.70456
\(160\) 2.68435e15 0.0790569
\(161\) 3.67778e16 1.03369
\(162\) −2.87224e16 −0.770651
\(163\) −3.29026e16 −0.842991 −0.421496 0.906830i \(-0.638495\pi\)
−0.421496 + 0.906830i \(0.638495\pi\)
\(164\) −2.10359e16 −0.514792
\(165\) −9.27423e14 −0.0216844
\(166\) −5.29286e16 −1.18271
\(167\) 3.14439e16 0.671681 0.335841 0.941919i \(-0.390980\pi\)
0.335841 + 0.941919i \(0.390980\pi\)
\(168\) 1.31460e16 0.268518
\(169\) −5.11616e16 −0.999526
\(170\) 6.45145e15 0.120584
\(171\) 3.46531e15 0.0619829
\(172\) −8.38809e15 −0.143615
\(173\) 1.19867e17 1.96496 0.982482 0.186359i \(-0.0596686\pi\)
0.982482 + 0.186359i \(0.0596686\pi\)
\(174\) −9.16650e16 −1.43908
\(175\) 9.63099e15 0.144839
\(176\) −8.02149e14 −0.0115587
\(177\) 3.62180e16 0.500178
\(178\) 6.98149e16 0.924268
\(179\) 4.18778e16 0.531602 0.265801 0.964028i \(-0.414364\pi\)
0.265801 + 0.964028i \(0.414364\pi\)
\(180\) 1.83357e15 0.0223230
\(181\) 1.25193e17 1.46214 0.731071 0.682301i \(-0.239021\pi\)
0.731071 + 0.682301i \(0.239021\pi\)
\(182\) 9.95010e14 0.0111504
\(183\) −5.96959e16 −0.642038
\(184\) 4.88793e16 0.504652
\(185\) 8.13942e15 0.0806874
\(186\) −1.45176e17 −1.38213
\(187\) −1.92785e15 −0.0176303
\(188\) 8.56330e16 0.752414
\(189\) −8.09666e16 −0.683663
\(190\) 2.41911e16 0.196338
\(191\) 7.75948e16 0.605456 0.302728 0.953077i \(-0.402103\pi\)
0.302728 + 0.953077i \(0.402103\pi\)
\(192\) 1.74716e16 0.131091
\(193\) −2.25828e17 −1.62967 −0.814834 0.579695i \(-0.803172\pi\)
−0.814834 + 0.579695i \(0.803172\pi\)
\(194\) 6.03221e16 0.418759
\(195\) 1.52894e15 0.0102124
\(196\) −3.69896e16 −0.237771
\(197\) 5.36977e16 0.332246 0.166123 0.986105i \(-0.446875\pi\)
0.166123 + 0.986105i \(0.446875\pi\)
\(198\) −5.47914e14 −0.00326380
\(199\) 1.41350e17 0.810771 0.405386 0.914146i \(-0.367137\pi\)
0.405386 + 0.914146i \(0.367137\pi\)
\(200\) 1.28000e16 0.0707107
\(201\) 8.66407e16 0.461054
\(202\) 1.10386e17 0.565949
\(203\) −2.84454e17 −1.40537
\(204\) 4.19904e16 0.199951
\(205\) −1.00307e17 −0.460444
\(206\) −1.39379e17 −0.616869
\(207\) 3.33874e16 0.142497
\(208\) 1.32241e15 0.00544367
\(209\) −7.22888e15 −0.0287061
\(210\) 6.26849e16 0.240170
\(211\) −4.41718e16 −0.163315 −0.0816577 0.996660i \(-0.526021\pi\)
−0.0816577 + 0.996660i \(0.526021\pi\)
\(212\) −2.27740e17 −0.812680
\(213\) −2.56316e17 −0.882934
\(214\) 1.58935e17 0.528587
\(215\) −3.99975e16 −0.128453
\(216\) −1.07608e17 −0.333766
\(217\) −4.50510e17 −1.34976
\(218\) 2.72621e17 0.789105
\(219\) 3.87816e17 1.08466
\(220\) −3.82495e15 −0.0103384
\(221\) 3.17822e15 0.00830314
\(222\) 5.29768e16 0.133795
\(223\) −5.00066e17 −1.22107 −0.610535 0.791989i \(-0.709046\pi\)
−0.610535 + 0.791989i \(0.709046\pi\)
\(224\) 5.42176e16 0.128021
\(225\) 8.74313e15 0.0199663
\(226\) 8.82264e16 0.194888
\(227\) −1.16741e17 −0.249477 −0.124738 0.992190i \(-0.539809\pi\)
−0.124738 + 0.992190i \(0.539809\pi\)
\(228\) 1.57452e17 0.325564
\(229\) 6.72239e17 1.34511 0.672555 0.740047i \(-0.265197\pi\)
0.672555 + 0.740047i \(0.265197\pi\)
\(230\) 2.33075e17 0.451374
\(231\) −1.87318e16 −0.0351147
\(232\) −3.78052e17 −0.686106
\(233\) 4.08177e17 0.717265 0.358632 0.933479i \(-0.383243\pi\)
0.358632 + 0.933479i \(0.383243\pi\)
\(234\) 9.03282e14 0.00153711
\(235\) 4.08330e17 0.672980
\(236\) 1.49373e17 0.238469
\(237\) 9.36855e17 1.44897
\(238\) 1.30304e17 0.195268
\(239\) 1.27317e18 1.84885 0.924424 0.381366i \(-0.124546\pi\)
0.924424 + 0.381366i \(0.124546\pi\)
\(240\) 8.33110e16 0.117251
\(241\) −4.00654e17 −0.546565 −0.273282 0.961934i \(-0.588109\pi\)
−0.273282 + 0.961934i \(0.588109\pi\)
\(242\) −5.33545e17 −0.705595
\(243\) −1.55157e17 −0.198940
\(244\) −2.46202e17 −0.306103
\(245\) −1.76380e17 −0.212669
\(246\) −6.52864e17 −0.763501
\(247\) 1.19174e16 0.0135193
\(248\) −5.98747e17 −0.658955
\(249\) −1.64268e18 −1.75411
\(250\) 6.10352e16 0.0632456
\(251\) −3.69701e17 −0.371790 −0.185895 0.982570i \(-0.559518\pi\)
−0.185895 + 0.982570i \(0.559518\pi\)
\(252\) 3.70337e16 0.0361488
\(253\) −6.96483e16 −0.0659944
\(254\) −8.23628e17 −0.757666
\(255\) 2.00226e17 0.178842
\(256\) 7.20576e16 0.0625000
\(257\) −5.68963e16 −0.0479276 −0.0239638 0.999713i \(-0.507629\pi\)
−0.0239638 + 0.999713i \(0.507629\pi\)
\(258\) −2.60331e17 −0.212999
\(259\) 1.64397e17 0.130661
\(260\) 6.30575e15 0.00486896
\(261\) −2.58231e17 −0.193733
\(262\) 3.55043e17 0.258834
\(263\) 4.31202e16 0.0305501 0.0152751 0.999883i \(-0.495138\pi\)
0.0152751 + 0.999883i \(0.495138\pi\)
\(264\) −2.48953e16 −0.0171431
\(265\) −1.08595e18 −0.726883
\(266\) 4.88603e17 0.317939
\(267\) 2.16676e18 1.37081
\(268\) 3.57330e17 0.219816
\(269\) −2.57678e18 −1.54147 −0.770737 0.637154i \(-0.780112\pi\)
−0.770737 + 0.637154i \(0.780112\pi\)
\(270\) −5.13115e17 −0.298529
\(271\) 1.81142e18 1.02506 0.512530 0.858669i \(-0.328708\pi\)
0.512530 + 0.858669i \(0.328708\pi\)
\(272\) 1.73180e17 0.0953302
\(273\) 3.08809e16 0.0165375
\(274\) 1.82988e18 0.953442
\(275\) −1.82388e16 −0.00924699
\(276\) 1.51701e18 0.748462
\(277\) −1.36071e18 −0.653381 −0.326690 0.945131i \(-0.605933\pi\)
−0.326690 + 0.945131i \(0.605933\pi\)
\(278\) −5.85793e17 −0.273784
\(279\) −4.08979e17 −0.186067
\(280\) 2.58530e17 0.114505
\(281\) 2.12909e17 0.0918115 0.0459057 0.998946i \(-0.485383\pi\)
0.0459057 + 0.998946i \(0.485383\pi\)
\(282\) 2.65768e18 1.11593
\(283\) 1.23798e18 0.506193 0.253096 0.967441i \(-0.418551\pi\)
0.253096 + 0.967441i \(0.418551\pi\)
\(284\) −1.05712e18 −0.420955
\(285\) 7.50790e17 0.291194
\(286\) −1.88431e15 −0.000711880 0
\(287\) −2.02596e18 −0.745619
\(288\) 4.92194e16 0.0176479
\(289\) −2.44621e18 −0.854594
\(290\) −1.80269e18 −0.613672
\(291\) 1.87214e18 0.621072
\(292\) 1.59946e18 0.517133
\(293\) 3.88932e18 1.22565 0.612825 0.790219i \(-0.290033\pi\)
0.612825 + 0.790219i \(0.290033\pi\)
\(294\) −1.14800e18 −0.352644
\(295\) 7.12265e17 0.213293
\(296\) 2.18491e17 0.0637890
\(297\) 1.53331e17 0.0436473
\(298\) 1.70157e18 0.472310
\(299\) 1.14821e17 0.0310805
\(300\) 3.97258e17 0.104873
\(301\) −8.07856e17 −0.208011
\(302\) −2.47169e18 −0.620785
\(303\) 3.42590e18 0.839373
\(304\) 6.49375e17 0.155219
\(305\) −1.17398e18 −0.273787
\(306\) 1.18292e17 0.0269180
\(307\) −7.12129e18 −1.58132 −0.790662 0.612253i \(-0.790263\pi\)
−0.790662 + 0.612253i \(0.790263\pi\)
\(308\) −7.72549e16 −0.0167415
\(309\) −4.32573e18 −0.914895
\(310\) −2.85505e18 −0.589388
\(311\) −8.62738e18 −1.73851 −0.869253 0.494367i \(-0.835400\pi\)
−0.869253 + 0.494367i \(0.835400\pi\)
\(312\) 4.10421e16 0.00807365
\(313\) 8.40829e18 1.61482 0.807411 0.589989i \(-0.200868\pi\)
0.807411 + 0.589989i \(0.200868\pi\)
\(314\) 1.36806e18 0.256527
\(315\) 1.76591e17 0.0323324
\(316\) 3.86385e18 0.690823
\(317\) −5.66897e18 −0.989828 −0.494914 0.868942i \(-0.664801\pi\)
−0.494914 + 0.868942i \(0.664801\pi\)
\(318\) −7.06807e18 −1.20531
\(319\) 5.38687e17 0.0897235
\(320\) 3.43597e17 0.0559017
\(321\) 4.93268e18 0.783962
\(322\) 4.70756e18 0.730932
\(323\) 1.56068e18 0.236752
\(324\) −3.67646e18 −0.544933
\(325\) 3.00682e16 0.00435493
\(326\) −4.21153e18 −0.596085
\(327\) 8.46099e18 1.17034
\(328\) −2.69259e18 −0.364013
\(329\) 8.24730e18 1.08979
\(330\) −1.18710e17 −0.0153332
\(331\) −6.94615e18 −0.877070 −0.438535 0.898714i \(-0.644502\pi\)
−0.438535 + 0.898714i \(0.644502\pi\)
\(332\) −6.77486e18 −0.836305
\(333\) 1.49242e17 0.0180119
\(334\) 4.02482e18 0.474950
\(335\) 1.70388e18 0.196609
\(336\) 1.68269e18 0.189871
\(337\) 1.06418e19 1.17433 0.587166 0.809466i \(-0.300244\pi\)
0.587166 + 0.809466i \(0.300244\pi\)
\(338\) −6.54869e18 −0.706772
\(339\) 2.73817e18 0.289044
\(340\) 8.25786e17 0.0852659
\(341\) 8.53157e17 0.0861730
\(342\) 4.43560e17 0.0438285
\(343\) −1.10538e19 −1.06858
\(344\) −1.07368e18 −0.101551
\(345\) 7.23366e18 0.669445
\(346\) 1.53430e19 1.38944
\(347\) 1.32412e18 0.117342 0.0586712 0.998277i \(-0.481314\pi\)
0.0586712 + 0.998277i \(0.481314\pi\)
\(348\) −1.17331e19 −1.01758
\(349\) 7.66248e18 0.650397 0.325198 0.945646i \(-0.394569\pi\)
0.325198 + 0.945646i \(0.394569\pi\)
\(350\) 1.23277e18 0.102417
\(351\) −2.52780e17 −0.0205560
\(352\) −1.02675e17 −0.00817326
\(353\) 2.74207e18 0.213682 0.106841 0.994276i \(-0.465926\pi\)
0.106841 + 0.994276i \(0.465926\pi\)
\(354\) 4.63590e18 0.353679
\(355\) −5.04072e18 −0.376513
\(356\) 8.93630e18 0.653556
\(357\) 4.04409e18 0.289607
\(358\) 5.36036e18 0.375899
\(359\) 2.87954e18 0.197749 0.0988746 0.995100i \(-0.468476\pi\)
0.0988746 + 0.995100i \(0.468476\pi\)
\(360\) 2.34697e17 0.0157848
\(361\) −9.32903e18 −0.614515
\(362\) 1.60247e19 1.03389
\(363\) −1.65590e19 −1.04649
\(364\) 1.27361e17 0.00788455
\(365\) 7.62682e18 0.462538
\(366\) −7.64107e18 −0.453990
\(367\) −2.01936e19 −1.17549 −0.587744 0.809047i \(-0.699984\pi\)
−0.587744 + 0.809047i \(0.699984\pi\)
\(368\) 6.25656e18 0.356843
\(369\) −1.83919e18 −0.102785
\(370\) 1.04185e18 0.0570546
\(371\) −2.19336e19 −1.17708
\(372\) −1.85826e19 −0.977314
\(373\) −3.62151e19 −1.86670 −0.933348 0.358972i \(-0.883127\pi\)
−0.933348 + 0.358972i \(0.883127\pi\)
\(374\) −2.46765e17 −0.0124665
\(375\) 1.89427e18 0.0938011
\(376\) 1.09610e19 0.532037
\(377\) −8.88071e17 −0.0422559
\(378\) −1.03637e19 −0.483423
\(379\) −1.59928e19 −0.731359 −0.365679 0.930741i \(-0.619163\pi\)
−0.365679 + 0.930741i \(0.619163\pi\)
\(380\) 3.09646e18 0.138832
\(381\) −2.55619e19 −1.12371
\(382\) 9.93214e18 0.428122
\(383\) 3.56761e19 1.50795 0.753975 0.656903i \(-0.228134\pi\)
0.753975 + 0.656903i \(0.228134\pi\)
\(384\) 2.23636e18 0.0926954
\(385\) −3.68380e17 −0.0149741
\(386\) −2.89060e19 −1.15235
\(387\) −7.33382e17 −0.0286746
\(388\) 7.72123e18 0.296107
\(389\) 4.67525e19 1.75866 0.879331 0.476211i \(-0.157990\pi\)
0.879331 + 0.476211i \(0.157990\pi\)
\(390\) 1.95704e17 0.00722129
\(391\) 1.50367e19 0.544286
\(392\) −4.73467e18 −0.168129
\(393\) 1.10190e19 0.383884
\(394\) 6.87331e18 0.234933
\(395\) 1.84243e19 0.617891
\(396\) −7.01329e16 −0.00230785
\(397\) 1.61556e18 0.0521668 0.0260834 0.999660i \(-0.491696\pi\)
0.0260834 + 0.999660i \(0.491696\pi\)
\(398\) 1.80928e19 0.573302
\(399\) 1.51642e19 0.471544
\(400\) 1.63840e18 0.0500000
\(401\) 3.22657e19 0.966402 0.483201 0.875509i \(-0.339474\pi\)
0.483201 + 0.875509i \(0.339474\pi\)
\(402\) 1.10900e19 0.326014
\(403\) −1.40650e18 −0.0405838
\(404\) 1.41294e19 0.400186
\(405\) −1.75307e19 −0.487403
\(406\) −3.64101e19 −0.993749
\(407\) −3.11328e17 −0.00834182
\(408\) 5.37477e18 0.141387
\(409\) −2.09116e19 −0.540084 −0.270042 0.962849i \(-0.587038\pi\)
−0.270042 + 0.962849i \(0.587038\pi\)
\(410\) −1.28393e19 −0.325583
\(411\) 5.67918e19 1.41407
\(412\) −1.78405e19 −0.436192
\(413\) 1.43861e19 0.345396
\(414\) 4.27358e18 0.100760
\(415\) −3.23051e19 −0.748014
\(416\) 1.69269e17 0.00384925
\(417\) −1.81805e19 −0.406057
\(418\) −9.25297e17 −0.0202983
\(419\) −4.31574e19 −0.929930 −0.464965 0.885329i \(-0.653933\pi\)
−0.464965 + 0.885329i \(0.653933\pi\)
\(420\) 8.02367e18 0.169826
\(421\) 3.44152e19 0.715541 0.357770 0.933810i \(-0.383537\pi\)
0.357770 + 0.933810i \(0.383537\pi\)
\(422\) −5.65399e18 −0.115481
\(423\) 7.48700e18 0.150229
\(424\) −2.91507e19 −0.574652
\(425\) 3.93765e18 0.0762642
\(426\) −3.28084e19 −0.624329
\(427\) −2.37117e19 −0.443357
\(428\) 2.03437e19 0.373768
\(429\) −5.84810e16 −0.00105581
\(430\) −5.11969e18 −0.0908301
\(431\) 7.95134e19 1.38631 0.693156 0.720788i \(-0.256220\pi\)
0.693156 + 0.720788i \(0.256220\pi\)
\(432\) −1.37738e19 −0.236008
\(433\) −6.99185e19 −1.17742 −0.588712 0.808343i \(-0.700365\pi\)
−0.588712 + 0.808343i \(0.700365\pi\)
\(434\) −5.76653e19 −0.954424
\(435\) −5.59479e19 −0.910153
\(436\) 3.48954e19 0.557982
\(437\) 5.63834e19 0.886218
\(438\) 4.96405e19 0.766973
\(439\) −5.37736e19 −0.816743 −0.408372 0.912816i \(-0.633903\pi\)
−0.408372 + 0.912816i \(0.633903\pi\)
\(440\) −4.89593e17 −0.00731038
\(441\) −3.23405e18 −0.0474741
\(442\) 4.06812e17 0.00587121
\(443\) 1.46107e19 0.207321 0.103660 0.994613i \(-0.466945\pi\)
0.103660 + 0.994613i \(0.466945\pi\)
\(444\) 6.78103e18 0.0946071
\(445\) 4.26116e19 0.584558
\(446\) −6.40084e19 −0.863427
\(447\) 5.28094e19 0.700495
\(448\) 6.93986e18 0.0905243
\(449\) −7.03318e19 −0.902203 −0.451101 0.892473i \(-0.648969\pi\)
−0.451101 + 0.892473i \(0.648969\pi\)
\(450\) 1.11912e18 0.0141183
\(451\) 3.83668e18 0.0476027
\(452\) 1.12930e19 0.137807
\(453\) −7.67107e19 −0.920702
\(454\) −1.49429e19 −0.176407
\(455\) 6.07306e17 0.00705216
\(456\) 2.01539e19 0.230209
\(457\) 4.35839e19 0.489728 0.244864 0.969557i \(-0.421257\pi\)
0.244864 + 0.969557i \(0.421257\pi\)
\(458\) 8.60466e19 0.951136
\(459\) −3.31034e19 −0.359979
\(460\) 2.98336e19 0.319170
\(461\) −1.24140e20 −1.30663 −0.653317 0.757084i \(-0.726623\pi\)
−0.653317 + 0.757084i \(0.726623\pi\)
\(462\) −2.39767e18 −0.0248298
\(463\) 6.58546e19 0.671009 0.335505 0.942039i \(-0.391093\pi\)
0.335505 + 0.942039i \(0.391093\pi\)
\(464\) −4.83906e19 −0.485150
\(465\) −8.86087e19 −0.874136
\(466\) 5.22467e19 0.507183
\(467\) 1.28108e19 0.122377 0.0611886 0.998126i \(-0.480511\pi\)
0.0611886 + 0.998126i \(0.480511\pi\)
\(468\) 1.15620e17 0.00108690
\(469\) 3.44144e19 0.318379
\(470\) 5.22662e19 0.475869
\(471\) 4.24589e19 0.380462
\(472\) 1.91197e19 0.168623
\(473\) 1.52988e18 0.0132801
\(474\) 1.19917e20 1.02458
\(475\) 1.47651e19 0.124175
\(476\) 1.66789e19 0.138075
\(477\) −1.99116e19 −0.162262
\(478\) 1.62965e20 1.30733
\(479\) 2.89091e19 0.228307 0.114153 0.993463i \(-0.463584\pi\)
0.114153 + 0.993463i \(0.463584\pi\)
\(480\) 1.06638e19 0.0829093
\(481\) 5.13251e17 0.00392864
\(482\) −5.12837e19 −0.386480
\(483\) 1.46103e20 1.08406
\(484\) −6.82937e19 −0.498931
\(485\) 3.68177e19 0.264846
\(486\) −1.98600e19 −0.140672
\(487\) 1.96397e20 1.36983 0.684916 0.728622i \(-0.259839\pi\)
0.684916 + 0.728622i \(0.259839\pi\)
\(488\) −3.15139e19 −0.216448
\(489\) −1.30708e20 −0.884069
\(490\) −2.25766e19 −0.150380
\(491\) −1.38375e20 −0.907707 −0.453854 0.891076i \(-0.649951\pi\)
−0.453854 + 0.891076i \(0.649951\pi\)
\(492\) −8.35666e19 −0.539877
\(493\) −1.16300e20 −0.739992
\(494\) 1.52543e18 0.00955962
\(495\) −3.34420e17 −0.00206421
\(496\) −7.66397e19 −0.465952
\(497\) −1.01811e20 −0.609706
\(498\) −2.10263e20 −1.24035
\(499\) 1.95546e20 1.13630 0.568152 0.822924i \(-0.307659\pi\)
0.568152 + 0.822924i \(0.307659\pi\)
\(500\) 7.81250e18 0.0447214
\(501\) 1.24913e20 0.704411
\(502\) −4.73218e19 −0.262895
\(503\) −1.06861e20 −0.584869 −0.292434 0.956286i \(-0.594465\pi\)
−0.292434 + 0.956286i \(0.594465\pi\)
\(504\) 4.74032e18 0.0255610
\(505\) 6.73740e19 0.357937
\(506\) −8.91499e18 −0.0466651
\(507\) −2.03244e20 −1.04823
\(508\) −1.05424e20 −0.535751
\(509\) 1.52058e19 0.0761424 0.0380712 0.999275i \(-0.487879\pi\)
0.0380712 + 0.999275i \(0.487879\pi\)
\(510\) 2.56289e19 0.126460
\(511\) 1.54044e20 0.749010
\(512\) 9.22337e18 0.0441942
\(513\) −1.24128e20 −0.586126
\(514\) −7.28272e18 −0.0338899
\(515\) −8.50702e19 −0.390142
\(516\) −3.33224e19 −0.150613
\(517\) −1.56184e19 −0.0695756
\(518\) 2.10428e19 0.0923913
\(519\) 4.76182e20 2.06071
\(520\) 8.07136e17 0.00344288
\(521\) −1.96345e20 −0.825536 −0.412768 0.910836i \(-0.635438\pi\)
−0.412768 + 0.910836i \(0.635438\pi\)
\(522\) −3.30535e19 −0.136990
\(523\) −4.20870e20 −1.71943 −0.859716 0.510773i \(-0.829359\pi\)
−0.859716 + 0.510773i \(0.829359\pi\)
\(524\) 4.54455e19 0.183023
\(525\) 3.82598e19 0.151897
\(526\) 5.51939e18 0.0216022
\(527\) −1.84192e20 −0.710709
\(528\) −3.18660e18 −0.0121220
\(529\) 2.76604e20 1.03739
\(530\) −1.39001e20 −0.513984
\(531\) 1.30599e19 0.0476134
\(532\) 6.25412e19 0.224817
\(533\) −6.32510e18 −0.0224188
\(534\) 2.77345e20 0.969306
\(535\) 9.70064e19 0.334308
\(536\) 4.57382e19 0.155433
\(537\) 1.66363e20 0.557506
\(538\) −3.29828e20 −1.08999
\(539\) 6.74644e18 0.0219866
\(540\) −6.56788e19 −0.211092
\(541\) 3.30654e20 1.04808 0.524040 0.851694i \(-0.324424\pi\)
0.524040 + 0.851694i \(0.324424\pi\)
\(542\) 2.31862e20 0.724827
\(543\) 4.97338e20 1.53339
\(544\) 2.21670e19 0.0674086
\(545\) 1.66394e20 0.499074
\(546\) 3.95276e18 0.0116938
\(547\) −3.44218e20 −1.00445 −0.502225 0.864737i \(-0.667485\pi\)
−0.502225 + 0.864737i \(0.667485\pi\)
\(548\) 2.34225e20 0.674185
\(549\) −2.15258e19 −0.0611175
\(550\) −2.33456e18 −0.00653861
\(551\) −4.36091e20 −1.20487
\(552\) 1.94177e20 0.529243
\(553\) 3.72127e20 1.00058
\(554\) −1.74170e20 −0.462010
\(555\) 3.23345e19 0.0846191
\(556\) −7.49816e19 −0.193595
\(557\) 4.03852e20 1.02875 0.514374 0.857566i \(-0.328024\pi\)
0.514374 + 0.857566i \(0.328024\pi\)
\(558\) −5.23493e19 −0.131569
\(559\) −2.52215e18 −0.00625434
\(560\) 3.30918e19 0.0809674
\(561\) −7.65853e18 −0.0184894
\(562\) 2.72524e19 0.0649205
\(563\) −3.78722e20 −0.890240 −0.445120 0.895471i \(-0.646839\pi\)
−0.445120 + 0.895471i \(0.646839\pi\)
\(564\) 3.40184e20 0.789078
\(565\) 5.38491e19 0.123258
\(566\) 1.58462e20 0.357932
\(567\) −3.54080e20 −0.789275
\(568\) −1.35311e20 −0.297660
\(569\) −5.32798e20 −1.15670 −0.578350 0.815789i \(-0.696303\pi\)
−0.578350 + 0.815789i \(0.696303\pi\)
\(570\) 9.61011e19 0.205905
\(571\) −6.45129e20 −1.36419 −0.682097 0.731262i \(-0.738932\pi\)
−0.682097 + 0.731262i \(0.738932\pi\)
\(572\) −2.41192e17 −0.000503375 0
\(573\) 3.08251e20 0.634959
\(574\) −2.59323e20 −0.527232
\(575\) 1.42258e20 0.285474
\(576\) 6.30009e18 0.0124789
\(577\) 5.38124e18 0.0105212 0.00526058 0.999986i \(-0.498325\pi\)
0.00526058 + 0.999986i \(0.498325\pi\)
\(578\) −3.13115e20 −0.604289
\(579\) −8.97121e20 −1.70908
\(580\) −2.30744e20 −0.433931
\(581\) −6.52486e20 −1.21130
\(582\) 2.39634e20 0.439164
\(583\) 4.15369e19 0.0751484
\(584\) 2.04731e20 0.365668
\(585\) 5.51320e17 0.000972153 0
\(586\) 4.97833e20 0.866665
\(587\) −6.26245e20 −1.07636 −0.538181 0.842829i \(-0.680888\pi\)
−0.538181 + 0.842829i \(0.680888\pi\)
\(588\) −1.46944e20 −0.249357
\(589\) −6.90668e20 −1.15719
\(590\) 9.11699e19 0.150821
\(591\) 2.13318e20 0.348435
\(592\) 2.79668e19 0.0451056
\(593\) 6.78491e20 1.08052 0.540262 0.841497i \(-0.318325\pi\)
0.540262 + 0.841497i \(0.318325\pi\)
\(594\) 1.96264e19 0.0308633
\(595\) 7.95313e19 0.123498
\(596\) 2.17801e20 0.333974
\(597\) 5.61525e20 0.850279
\(598\) 1.46971e19 0.0219772
\(599\) 1.26446e21 1.86725 0.933625 0.358251i \(-0.116627\pi\)
0.933625 + 0.358251i \(0.116627\pi\)
\(600\) 5.08490e19 0.0741563
\(601\) 9.57115e20 1.37850 0.689248 0.724526i \(-0.257941\pi\)
0.689248 + 0.724526i \(0.257941\pi\)
\(602\) −1.03406e20 −0.147086
\(603\) 3.12418e19 0.0438891
\(604\) −3.16376e20 −0.438961
\(605\) −3.25650e20 −0.446258
\(606\) 4.38515e20 0.593526
\(607\) 1.00033e20 0.133730 0.0668650 0.997762i \(-0.478700\pi\)
0.0668650 + 0.997762i \(0.478700\pi\)
\(608\) 8.31200e19 0.109756
\(609\) −1.13002e21 −1.47385
\(610\) −1.50270e20 −0.193597
\(611\) 2.57483e19 0.0327672
\(612\) 1.51413e19 0.0190339
\(613\) 7.36761e20 0.914898 0.457449 0.889236i \(-0.348763\pi\)
0.457449 + 0.889236i \(0.348763\pi\)
\(614\) −9.11525e20 −1.11816
\(615\) −3.98477e20 −0.482881
\(616\) −9.88863e18 −0.0118381
\(617\) 1.36626e21 1.61582 0.807910 0.589305i \(-0.200599\pi\)
0.807910 + 0.589305i \(0.200599\pi\)
\(618\) −5.53694e20 −0.646928
\(619\) 4.69623e20 0.542087 0.271044 0.962567i \(-0.412631\pi\)
0.271044 + 0.962567i \(0.412631\pi\)
\(620\) −3.65446e20 −0.416760
\(621\) −1.19594e21 −1.34748
\(622\) −1.10431e21 −1.22931
\(623\) 8.60654e20 0.946604
\(624\) 5.25338e18 0.00570893
\(625\) 3.72529e19 0.0400000
\(626\) 1.07626e21 1.14185
\(627\) −2.87173e19 −0.0301049
\(628\) 1.75112e20 0.181392
\(629\) 6.72142e19 0.0687988
\(630\) 2.26036e19 0.0228625
\(631\) 1.56542e21 1.56463 0.782315 0.622882i \(-0.214039\pi\)
0.782315 + 0.622882i \(0.214039\pi\)
\(632\) 4.94572e20 0.488486
\(633\) −1.75476e20 −0.171273
\(634\) −7.25629e20 −0.699914
\(635\) −5.02702e20 −0.479190
\(636\) −9.04713e20 −0.852281
\(637\) −1.11221e19 −0.0103548
\(638\) 6.89519e19 0.0634441
\(639\) −9.24250e19 −0.0840491
\(640\) 4.39805e19 0.0395285
\(641\) 1.07910e21 0.958578 0.479289 0.877657i \(-0.340895\pi\)
0.479289 + 0.877657i \(0.340895\pi\)
\(642\) 6.31383e20 0.554345
\(643\) 7.31630e20 0.634906 0.317453 0.948274i \(-0.397172\pi\)
0.317453 + 0.948274i \(0.397172\pi\)
\(644\) 6.02568e20 0.516847
\(645\) −1.58893e20 −0.134713
\(646\) 1.99767e20 0.167409
\(647\) −3.86121e20 −0.319846 −0.159923 0.987129i \(-0.551125\pi\)
−0.159923 + 0.987129i \(0.551125\pi\)
\(648\) −4.70587e20 −0.385326
\(649\) −2.72438e19 −0.0220512
\(650\) 3.84872e18 0.00307940
\(651\) −1.78969e21 −1.41553
\(652\) −5.39076e20 −0.421496
\(653\) −8.04784e20 −0.622057 −0.311029 0.950401i \(-0.600674\pi\)
−0.311029 + 0.950401i \(0.600674\pi\)
\(654\) 1.08301e21 0.827557
\(655\) 2.16701e20 0.163701
\(656\) −3.44652e20 −0.257396
\(657\) 1.39843e20 0.103252
\(658\) 1.05565e21 0.770597
\(659\) 1.07785e21 0.777889 0.388945 0.921261i \(-0.372840\pi\)
0.388945 + 0.921261i \(0.372840\pi\)
\(660\) −1.51949e19 −0.0108422
\(661\) −8.11863e20 −0.572759 −0.286379 0.958116i \(-0.592452\pi\)
−0.286379 + 0.958116i \(0.592452\pi\)
\(662\) −8.89107e20 −0.620182
\(663\) 1.26257e19 0.00870774
\(664\) −8.67183e20 −0.591357
\(665\) 2.98220e20 0.201082
\(666\) 1.91029e19 0.0127363
\(667\) −4.20162e21 −2.76996
\(668\) 5.15177e20 0.335841
\(669\) −1.98655e21 −1.28057
\(670\) 2.18097e20 0.139024
\(671\) 4.49042e19 0.0283053
\(672\) 2.15384e20 0.134259
\(673\) −5.57984e20 −0.343961 −0.171980 0.985100i \(-0.555017\pi\)
−0.171980 + 0.985100i \(0.555017\pi\)
\(674\) 1.36215e21 0.830378
\(675\) −3.13181e20 −0.188806
\(676\) −8.38232e20 −0.499763
\(677\) −1.86877e20 −0.110190 −0.0550948 0.998481i \(-0.517546\pi\)
−0.0550948 + 0.998481i \(0.517546\pi\)
\(678\) 3.50486e20 0.204385
\(679\) 7.43631e20 0.428878
\(680\) 1.05701e20 0.0602921
\(681\) −4.63763e20 −0.261633
\(682\) 1.09204e20 0.0609335
\(683\) −1.32462e21 −0.731034 −0.365517 0.930805i \(-0.619108\pi\)
−0.365517 + 0.930805i \(0.619108\pi\)
\(684\) 5.67757e19 0.0309914
\(685\) 1.11687e21 0.603009
\(686\) −1.41489e21 −0.755600
\(687\) 2.67052e21 1.41065
\(688\) −1.37431e20 −0.0718075
\(689\) −6.84771e19 −0.0353917
\(690\) 9.25908e20 0.473369
\(691\) 1.97901e21 1.00084 0.500418 0.865784i \(-0.333180\pi\)
0.500418 + 0.865784i \(0.333180\pi\)
\(692\) 1.96390e21 0.982482
\(693\) −6.75449e18 −0.00334267
\(694\) 1.69487e20 0.0829736
\(695\) −3.57540e20 −0.173156
\(696\) −1.50184e21 −0.719539
\(697\) −8.28320e20 −0.392602
\(698\) 9.80797e20 0.459900
\(699\) 1.62152e21 0.752216
\(700\) 1.57794e20 0.0724195
\(701\) 2.58448e21 1.17351 0.586757 0.809763i \(-0.300404\pi\)
0.586757 + 0.809763i \(0.300404\pi\)
\(702\) −3.23558e19 −0.0145353
\(703\) 2.52034e20 0.112020
\(704\) −1.31424e19 −0.00577937
\(705\) 1.62212e21 0.705773
\(706\) 3.50985e20 0.151096
\(707\) 1.36080e21 0.579625
\(708\) 5.93395e20 0.250089
\(709\) −2.35138e21 −0.980562 −0.490281 0.871564i \(-0.663106\pi\)
−0.490281 + 0.871564i \(0.663106\pi\)
\(710\) −6.45213e20 −0.266235
\(711\) 3.37821e20 0.137932
\(712\) 1.14385e21 0.462134
\(713\) −6.65440e21 −2.66034
\(714\) 5.17643e20 0.204783
\(715\) −1.15009e18 −0.000450232 0
\(716\) 6.86126e20 0.265801
\(717\) 5.05776e21 1.93894
\(718\) 3.68581e20 0.139830
\(719\) 5.01526e21 1.88290 0.941450 0.337153i \(-0.109464\pi\)
0.941450 + 0.337153i \(0.109464\pi\)
\(720\) 3.00412e19 0.0111615
\(721\) −1.71822e21 −0.631776
\(722\) −1.19412e21 −0.434528
\(723\) −1.59163e21 −0.573198
\(724\) 2.05116e21 0.731071
\(725\) −1.10027e21 −0.388120
\(726\) −2.11955e21 −0.739978
\(727\) −2.68041e21 −0.926175 −0.463087 0.886313i \(-0.653258\pi\)
−0.463087 + 0.886313i \(0.653258\pi\)
\(728\) 1.63022e19 0.00557522
\(729\) 2.60343e21 0.881231
\(730\) 9.76233e20 0.327063
\(731\) −3.30294e20 −0.109527
\(732\) −9.78057e20 −0.321019
\(733\) −5.84752e21 −1.89973 −0.949865 0.312661i \(-0.898780\pi\)
−0.949865 + 0.312661i \(0.898780\pi\)
\(734\) −2.58478e21 −0.831195
\(735\) −7.00684e20 −0.223032
\(736\) 8.00839e20 0.252326
\(737\) −6.51726e19 −0.0203263
\(738\) −2.35417e20 −0.0726799
\(739\) −2.19735e20 −0.0671530 −0.0335765 0.999436i \(-0.510690\pi\)
−0.0335765 + 0.999436i \(0.510690\pi\)
\(740\) 1.33356e20 0.0403437
\(741\) 4.73429e19 0.0141781
\(742\) −2.80750e21 −0.832319
\(743\) 1.85608e21 0.544730 0.272365 0.962194i \(-0.412194\pi\)
0.272365 + 0.962194i \(0.412194\pi\)
\(744\) −2.37857e21 −0.691065
\(745\) 1.03855e21 0.298715
\(746\) −4.63554e21 −1.31995
\(747\) −5.92335e20 −0.166979
\(748\) −3.15859e19 −0.00881517
\(749\) 1.95930e21 0.541361
\(750\) 2.42467e20 0.0663274
\(751\) 4.78115e21 1.29489 0.647445 0.762112i \(-0.275837\pi\)
0.647445 + 0.762112i \(0.275837\pi\)
\(752\) 1.40301e21 0.376207
\(753\) −1.46867e21 −0.389907
\(754\) −1.13673e20 −0.0298795
\(755\) −1.50860e21 −0.392619
\(756\) −1.32656e21 −0.341832
\(757\) 2.41955e21 0.617326 0.308663 0.951171i \(-0.400118\pi\)
0.308663 + 0.951171i \(0.400118\pi\)
\(758\) −2.04708e21 −0.517149
\(759\) −2.76683e20 −0.0692102
\(760\) 3.96347e20 0.0981689
\(761\) −3.48193e21 −0.853955 −0.426978 0.904262i \(-0.640422\pi\)
−0.426978 + 0.904262i \(0.640422\pi\)
\(762\) −3.27193e21 −0.794586
\(763\) 3.36077e21 0.808175
\(764\) 1.27131e21 0.302728
\(765\) 7.21995e19 0.0170245
\(766\) 4.56654e21 1.06628
\(767\) 4.49137e19 0.0103852
\(768\) 2.86254e20 0.0655455
\(769\) −2.58771e21 −0.586770 −0.293385 0.955994i \(-0.594782\pi\)
−0.293385 + 0.955994i \(0.594782\pi\)
\(770\) −4.71526e19 −0.0105883
\(771\) −2.26025e20 −0.0502630
\(772\) −3.69997e21 −0.814834
\(773\) 3.07396e21 0.670429 0.335214 0.942142i \(-0.391191\pi\)
0.335214 + 0.942142i \(0.391191\pi\)
\(774\) −9.38729e19 −0.0202760
\(775\) −1.74258e21 −0.372762
\(776\) 9.88318e20 0.209379
\(777\) 6.53080e20 0.137028
\(778\) 5.98431e21 1.24356
\(779\) −3.10596e21 −0.639242
\(780\) 2.50501e19 0.00510622
\(781\) 1.92805e20 0.0389256
\(782\) 1.92470e21 0.384868
\(783\) 9.24988e21 1.83199
\(784\) −6.06037e20 −0.118885
\(785\) 8.34999e20 0.162242
\(786\) 1.41044e21 0.271447
\(787\) −5.67154e21 −1.08116 −0.540581 0.841292i \(-0.681795\pi\)
−0.540581 + 0.841292i \(0.681795\pi\)
\(788\) 8.79784e20 0.166123
\(789\) 1.71298e20 0.0320388
\(790\) 2.35831e21 0.436915
\(791\) 1.08763e21 0.199598
\(792\) −8.97702e18 −0.00163190
\(793\) −7.40284e19 −0.0133306
\(794\) 2.06792e20 0.0368875
\(795\) −4.31401e21 −0.762303
\(796\) 2.31588e21 0.405386
\(797\) 6.99506e21 1.21298 0.606491 0.795091i \(-0.292577\pi\)
0.606491 + 0.795091i \(0.292577\pi\)
\(798\) 1.94102e21 0.333432
\(799\) 3.37193e21 0.573823
\(800\) 2.09715e20 0.0353553
\(801\) 7.81313e20 0.130491
\(802\) 4.13001e21 0.683350
\(803\) −2.91722e20 −0.0478192
\(804\) 1.41952e21 0.230527
\(805\) 2.87327e21 0.462282
\(806\) −1.80032e20 −0.0286971
\(807\) −1.02365e22 −1.61659
\(808\) 1.80856e21 0.282974
\(809\) −1.02572e22 −1.59007 −0.795036 0.606562i \(-0.792548\pi\)
−0.795036 + 0.606562i \(0.792548\pi\)
\(810\) −2.24394e21 −0.344646
\(811\) 5.19989e20 0.0791294 0.0395647 0.999217i \(-0.487403\pi\)
0.0395647 + 0.999217i \(0.487403\pi\)
\(812\) −4.66049e21 −0.702686
\(813\) 7.19601e21 1.07501
\(814\) −3.98500e19 −0.00589856
\(815\) −2.57051e21 −0.376997
\(816\) 6.87971e20 0.0999755
\(817\) −1.23851e21 −0.178334
\(818\) −2.67668e21 −0.381897
\(819\) 1.11354e19 0.00157425
\(820\) −1.64343e21 −0.230222
\(821\) −1.18946e22 −1.65111 −0.825557 0.564319i \(-0.809139\pi\)
−0.825557 + 0.564319i \(0.809139\pi\)
\(822\) 7.26935e21 0.999901
\(823\) −4.67215e21 −0.636822 −0.318411 0.947953i \(-0.603149\pi\)
−0.318411 + 0.947953i \(0.603149\pi\)
\(824\) −2.28358e21 −0.308435
\(825\) −7.24549e19 −0.00969758
\(826\) 1.84142e21 0.244232
\(827\) −6.43261e19 −0.00845465 −0.00422733 0.999991i \(-0.501346\pi\)
−0.00422733 + 0.999991i \(0.501346\pi\)
\(828\) 5.47019e20 0.0712483
\(829\) −1.59621e21 −0.206030 −0.103015 0.994680i \(-0.532849\pi\)
−0.103015 + 0.994680i \(0.532849\pi\)
\(830\) −4.13505e21 −0.528926
\(831\) −5.40551e21 −0.685219
\(832\) 2.16664e19 0.00272183
\(833\) −1.45652e21 −0.181334
\(834\) −2.32711e21 −0.287125
\(835\) 2.45656e21 0.300385
\(836\) −1.18438e20 −0.0143530
\(837\) 1.46497e22 1.75949
\(838\) −5.52415e21 −0.657560
\(839\) −1.03172e22 −1.21715 −0.608577 0.793495i \(-0.708259\pi\)
−0.608577 + 0.793495i \(0.708259\pi\)
\(840\) 1.02703e21 0.120085
\(841\) 2.38677e22 2.76593
\(842\) 4.40514e21 0.505964
\(843\) 8.45799e20 0.0962853
\(844\) −7.23711e20 −0.0816577
\(845\) −3.99700e21 −0.447002
\(846\) 9.58336e20 0.106228
\(847\) −6.57736e21 −0.722647
\(848\) −3.73129e21 −0.406340
\(849\) 4.91798e21 0.530859
\(850\) 5.04020e20 0.0539269
\(851\) 2.42828e21 0.257530
\(852\) −4.19948e21 −0.441467
\(853\) −2.65492e21 −0.276652 −0.138326 0.990387i \(-0.544172\pi\)
−0.138326 + 0.990387i \(0.544172\pi\)
\(854\) −3.03510e21 −0.313501
\(855\) 2.70728e20 0.0277196
\(856\) 2.60399e21 0.264294
\(857\) 9.17781e21 0.923385 0.461693 0.887040i \(-0.347242\pi\)
0.461693 + 0.887040i \(0.347242\pi\)
\(858\) −7.48557e18 −0.000746569 0
\(859\) −1.85419e22 −1.83318 −0.916592 0.399824i \(-0.869071\pi\)
−0.916592 + 0.399824i \(0.869071\pi\)
\(860\) −6.55320e20 −0.0642266
\(861\) −8.04829e21 −0.781952
\(862\) 1.01777e22 0.980270
\(863\) 3.65598e21 0.349078 0.174539 0.984650i \(-0.444156\pi\)
0.174539 + 0.984650i \(0.444156\pi\)
\(864\) −1.76305e21 −0.166883
\(865\) 9.36462e21 0.878758
\(866\) −8.94957e21 −0.832565
\(867\) −9.71776e21 −0.896238
\(868\) −7.38116e21 −0.674880
\(869\) −7.04718e20 −0.0638803
\(870\) −7.16133e21 −0.643575
\(871\) 1.07443e20 0.00957282
\(872\) 4.46662e21 0.394553
\(873\) 6.75077e20 0.0591217
\(874\) 7.21707e21 0.626651
\(875\) 7.52421e20 0.0647740
\(876\) 6.35398e21 0.542332
\(877\) −1.66541e21 −0.140937 −0.0704683 0.997514i \(-0.522449\pi\)
−0.0704683 + 0.997514i \(0.522449\pi\)
\(878\) −6.88302e21 −0.577525
\(879\) 1.54506e22 1.28537
\(880\) −6.26679e19 −0.00516922
\(881\) 4.64406e21 0.379821 0.189910 0.981801i \(-0.439180\pi\)
0.189910 + 0.981801i \(0.439180\pi\)
\(882\) −4.13958e20 −0.0335693
\(883\) 1.60318e22 1.28907 0.644536 0.764574i \(-0.277051\pi\)
0.644536 + 0.764574i \(0.277051\pi\)
\(884\) 5.20720e19 0.00415157
\(885\) 2.82953e21 0.223686
\(886\) 1.87017e21 0.146598
\(887\) −1.91774e21 −0.149061 −0.0745304 0.997219i \(-0.523746\pi\)
−0.0745304 + 0.997219i \(0.523746\pi\)
\(888\) 8.67972e20 0.0668973
\(889\) −1.01534e22 −0.775976
\(890\) 5.45429e21 0.413345
\(891\) 6.70542e20 0.0503898
\(892\) −8.19308e21 −0.610535
\(893\) 1.26438e22 0.934310
\(894\) 6.75961e21 0.495325
\(895\) 3.27170e21 0.237739
\(896\) 8.88302e20 0.0640104
\(897\) 4.56136e20 0.0325950
\(898\) −9.00247e21 −0.637954
\(899\) 5.14677e22 3.61691
\(900\) 1.43247e20 0.00998316
\(901\) −8.96760e21 −0.619784
\(902\) 4.91095e20 0.0336602
\(903\) −3.20927e21 −0.218147
\(904\) 1.44550e21 0.0974440
\(905\) 9.78068e21 0.653890
\(906\) −9.81896e21 −0.651035
\(907\) −1.43639e22 −0.944532 −0.472266 0.881456i \(-0.656564\pi\)
−0.472266 + 0.881456i \(0.656564\pi\)
\(908\) −1.91269e21 −0.124738
\(909\) 1.23535e21 0.0799024
\(910\) 7.77352e19 0.00498663
\(911\) −1.53974e22 −0.979626 −0.489813 0.871828i \(-0.662935\pi\)
−0.489813 + 0.871828i \(0.662935\pi\)
\(912\) 2.57969e21 0.162782
\(913\) 1.23565e21 0.0773330
\(914\) 5.57874e21 0.346290
\(915\) −4.66374e21 −0.287128
\(916\) 1.10140e22 0.672555
\(917\) 4.37685e21 0.265089
\(918\) −4.23724e21 −0.254544
\(919\) −9.59237e21 −0.571557 −0.285779 0.958296i \(-0.592252\pi\)
−0.285779 + 0.958296i \(0.592252\pi\)
\(920\) 3.81870e21 0.225687
\(921\) −2.82899e22 −1.65838
\(922\) −1.58899e22 −0.923930
\(923\) −3.17855e20 −0.0183323
\(924\) −3.06901e20 −0.0175573
\(925\) 6.35892e20 0.0360845
\(926\) 8.42938e21 0.474475
\(927\) −1.55982e21 −0.0870915
\(928\) −6.19400e21 −0.343053
\(929\) −2.12844e22 −1.16935 −0.584673 0.811269i \(-0.698777\pi\)
−0.584673 + 0.811269i \(0.698777\pi\)
\(930\) −1.13419e22 −0.618108
\(931\) −5.46154e21 −0.295252
\(932\) 6.68758e21 0.358632
\(933\) −3.42730e22 −1.82322
\(934\) 1.63979e21 0.0865337
\(935\) −1.50613e20 −0.00788453
\(936\) 1.47994e19 0.000768554 0
\(937\) −3.01568e22 −1.55360 −0.776799 0.629748i \(-0.783158\pi\)
−0.776799 + 0.629748i \(0.783158\pi\)
\(938\) 4.40504e21 0.225128
\(939\) 3.34026e22 1.69351
\(940\) 6.69007e21 0.336490
\(941\) −1.85558e22 −0.925885 −0.462943 0.886388i \(-0.653206\pi\)
−0.462943 + 0.886388i \(0.653206\pi\)
\(942\) 5.43473e21 0.269027
\(943\) −2.99251e22 −1.46960
\(944\) 2.44732e21 0.119234
\(945\) −6.32551e21 −0.305744
\(946\) 1.95825e20 0.00939042
\(947\) 1.73295e22 0.824444 0.412222 0.911083i \(-0.364753\pi\)
0.412222 + 0.911083i \(0.364753\pi\)
\(948\) 1.53494e22 0.724486
\(949\) 4.80928e20 0.0225208
\(950\) 1.88993e21 0.0878049
\(951\) −2.25204e22 −1.03806
\(952\) 2.13490e21 0.0976340
\(953\) 2.51658e22 1.14186 0.570931 0.820998i \(-0.306583\pi\)
0.570931 + 0.820998i \(0.306583\pi\)
\(954\) −2.54868e21 −0.114737
\(955\) 6.06209e21 0.270768
\(956\) 2.08596e22 0.924424
\(957\) 2.13998e21 0.0940956
\(958\) 3.70037e21 0.161437
\(959\) 2.25582e22 0.976482
\(960\) 1.36497e21 0.0586257
\(961\) 5.80479e22 2.47378
\(962\) 6.56962e19 0.00277797
\(963\) 1.77868e21 0.0746277
\(964\) −6.56432e21 −0.273282
\(965\) −1.76429e22 −0.728809
\(966\) 1.87012e22 0.766549
\(967\) −9.74068e21 −0.396178 −0.198089 0.980184i \(-0.563474\pi\)
−0.198089 + 0.980184i \(0.563474\pi\)
\(968\) −8.74160e21 −0.352798
\(969\) 6.19992e21 0.248289
\(970\) 4.71267e21 0.187275
\(971\) −1.21314e22 −0.478373 −0.239187 0.970974i \(-0.576881\pi\)
−0.239187 + 0.970974i \(0.576881\pi\)
\(972\) −2.54209e21 −0.0994702
\(973\) −7.22146e21 −0.280401
\(974\) 2.51388e22 0.968618
\(975\) 1.19448e20 0.00456714
\(976\) −4.03377e21 −0.153052
\(977\) 1.82562e22 0.687388 0.343694 0.939082i \(-0.388322\pi\)
0.343694 + 0.939082i \(0.388322\pi\)
\(978\) −1.67306e22 −0.625131
\(979\) −1.62987e21 −0.0604343
\(980\) −2.88981e21 −0.106334
\(981\) 3.05095e21 0.111408
\(982\) −1.77120e22 −0.641846
\(983\) 4.33715e21 0.155974 0.0779872 0.996954i \(-0.475151\pi\)
0.0779872 + 0.996954i \(0.475151\pi\)
\(984\) −1.06965e22 −0.381751
\(985\) 4.19513e21 0.148585
\(986\) −1.48864e22 −0.523253
\(987\) 3.27630e22 1.14289
\(988\) 1.95255e20 0.00675967
\(989\) −1.19327e22 −0.409984
\(990\) −4.28057e19 −0.00145961
\(991\) 5.10575e22 1.72786 0.863928 0.503615i \(-0.167997\pi\)
0.863928 + 0.503615i \(0.167997\pi\)
\(992\) −9.80988e21 −0.329478
\(993\) −2.75941e22 −0.919808
\(994\) −1.30318e22 −0.431127
\(995\) 1.10430e22 0.362588
\(996\) −2.69137e22 −0.877057
\(997\) 3.95869e22 1.28038 0.640188 0.768219i \(-0.278857\pi\)
0.640188 + 0.768219i \(0.278857\pi\)
\(998\) 2.50299e22 0.803488
\(999\) −5.34587e21 −0.170325
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 10.16.a.d.1.2 2
3.2 odd 2 90.16.a.j.1.2 2
4.3 odd 2 80.16.a.f.1.1 2
5.2 odd 4 50.16.b.e.49.3 4
5.3 odd 4 50.16.b.e.49.2 4
5.4 even 2 50.16.a.f.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.16.a.d.1.2 2 1.1 even 1 trivial
50.16.a.f.1.1 2 5.4 even 2
50.16.b.e.49.2 4 5.3 odd 4
50.16.b.e.49.3 4 5.2 odd 4
80.16.a.f.1.1 2 4.3 odd 2
90.16.a.j.1.2 2 3.2 odd 2