Properties

Label 2.38.a.a.1.1
Level $2$
Weight $38$
Character 2.1
Self dual yes
Analytic conductor $17.343$
Analytic rank $1$
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] = [2,38,Mod(1,2)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2, base_ring=CyclotomicField(1))
 
chi = DirichletCharacter(H, H._module([]))
 
N = Newforms(chi, 38, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2.1");
 
S:= CuspForms(chi, 38);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 38 \)
Character orbit: \([\chi]\) \(=\) 2.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: \(17.3428076249\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 756643680 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{8}\cdot 3^{3}\cdot 5 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(27507.7\) of defining polynomial
Character \(\chi\) \(=\) 2.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-262144. q^{2} -7.39112e8 q^{3} +6.87195e10 q^{4} -3.47974e12 q^{5} +1.93754e14 q^{6} +6.42679e15 q^{7} -1.80144e16 q^{8} +9.60023e16 q^{9} +O(q^{10})\) \(q-262144. q^{2} -7.39112e8 q^{3} +6.87195e10 q^{4} -3.47974e12 q^{5} +1.93754e14 q^{6} +6.42679e15 q^{7} -1.80144e16 q^{8} +9.60023e16 q^{9} +9.12192e17 q^{10} +2.60208e19 q^{11} -5.07914e19 q^{12} -1.70663e20 q^{13} -1.68474e21 q^{14} +2.57191e21 q^{15} +4.72237e21 q^{16} -6.89990e22 q^{17} -2.51664e22 q^{18} -6.71981e23 q^{19} -2.39126e23 q^{20} -4.75011e24 q^{21} -6.82120e24 q^{22} +1.55339e25 q^{23} +1.33147e25 q^{24} -6.06510e25 q^{25} +4.47383e25 q^{26} +2.61854e26 q^{27} +4.41646e26 q^{28} +1.41680e27 q^{29} -6.74212e26 q^{30} -3.59446e27 q^{31} -1.23794e27 q^{32} -1.92323e28 q^{33} +1.80877e28 q^{34} -2.23635e28 q^{35} +6.59722e27 q^{36} -3.04432e28 q^{37} +1.76156e29 q^{38} +1.26139e29 q^{39} +6.26853e28 q^{40} -7.95085e28 q^{41} +1.24521e30 q^{42} -2.87613e30 q^{43} +1.78814e30 q^{44} -3.34062e29 q^{45} -4.07212e30 q^{46} -1.27820e31 q^{47} -3.49036e30 q^{48} +2.27415e31 q^{49} +1.58993e31 q^{50} +5.09980e31 q^{51} -1.17279e31 q^{52} +1.70666e31 q^{53} -6.86434e31 q^{54} -9.05455e31 q^{55} -1.15775e32 q^{56} +4.96669e32 q^{57} -3.71406e32 q^{58} -7.46370e32 q^{59} +1.76741e32 q^{60} -2.64957e32 q^{61} +9.42266e32 q^{62} +6.16986e32 q^{63} +3.24519e32 q^{64} +5.93862e32 q^{65} +5.04163e33 q^{66} +1.82198e33 q^{67} -4.74157e33 q^{68} -1.14813e34 q^{69} +5.86246e33 q^{70} -7.50959e33 q^{71} -1.72942e33 q^{72} -4.19120e34 q^{73} +7.98050e33 q^{74} +4.48279e34 q^{75} -4.61782e34 q^{76} +1.67230e35 q^{77} -3.30666e34 q^{78} +1.22253e35 q^{79} -1.64326e34 q^{80} -2.36767e35 q^{81} +2.08427e34 q^{82} -7.58400e34 q^{83} -3.26425e35 q^{84} +2.40098e35 q^{85} +7.53961e35 q^{86} -1.04717e36 q^{87} -4.68749e35 q^{88} +2.32524e35 q^{89} +8.75725e34 q^{90} -1.09682e36 q^{91} +1.06748e36 q^{92} +2.65671e36 q^{93} +3.35071e36 q^{94} +2.33832e36 q^{95} +9.14976e35 q^{96} -8.69459e36 q^{97} -5.96154e36 q^{98} +2.49806e36 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 524288 q^{2} + 423071208 q^{3} + 137438953472 q^{4} - 13507530555540 q^{5} - 110905578749952 q^{6} + 31\!\cdots\!56 q^{7}+ \cdots + 99\!\cdots\!06 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 524288 q^{2} + 423071208 q^{3} + 137438953472 q^{4} - 13507530555540 q^{5} - 110905578749952 q^{6} + 31\!\cdots\!56 q^{7}+ \cdots - 17\!\cdots\!88 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −262144. −0.707107
\(3\) −7.39112e8 −1.10146 −0.550728 0.834685i \(-0.685650\pi\)
−0.550728 + 0.834685i \(0.685650\pi\)
\(4\) 6.87195e10 0.500000
\(5\) −3.47974e12 −0.407945 −0.203972 0.978977i \(-0.565385\pi\)
−0.203972 + 0.978977i \(0.565385\pi\)
\(6\) 1.93754e14 0.778847
\(7\) 6.42679e15 1.49170 0.745848 0.666116i \(-0.232045\pi\)
0.745848 + 0.666116i \(0.232045\pi\)
\(8\) −1.80144e16 −0.353553
\(9\) 9.60023e16 0.213204
\(10\) 9.12192e17 0.288460
\(11\) 2.60208e19 1.41109 0.705547 0.708663i \(-0.250701\pi\)
0.705547 + 0.708663i \(0.250701\pi\)
\(12\) −5.07914e19 −0.550728
\(13\) −1.70663e20 −0.420908 −0.210454 0.977604i \(-0.567494\pi\)
−0.210454 + 0.977604i \(0.567494\pi\)
\(14\) −1.68474e21 −1.05479
\(15\) 2.57191e21 0.449333
\(16\) 4.72237e21 0.250000
\(17\) −6.89990e22 −1.18997 −0.594987 0.803735i \(-0.702843\pi\)
−0.594987 + 0.803735i \(0.702843\pi\)
\(18\) −2.51664e22 −0.150758
\(19\) −6.71981e23 −1.48052 −0.740261 0.672319i \(-0.765298\pi\)
−0.740261 + 0.672319i \(0.765298\pi\)
\(20\) −2.39126e23 −0.203972
\(21\) −4.75011e24 −1.64304
\(22\) −6.82120e24 −0.997794
\(23\) 1.55339e25 0.998424 0.499212 0.866480i \(-0.333623\pi\)
0.499212 + 0.866480i \(0.333623\pi\)
\(24\) 1.33147e25 0.389423
\(25\) −6.06510e25 −0.833581
\(26\) 4.47383e25 0.297627
\(27\) 2.61854e26 0.866621
\(28\) 4.41646e26 0.745848
\(29\) 1.41680e27 1.25010 0.625052 0.780583i \(-0.285078\pi\)
0.625052 + 0.780583i \(0.285078\pi\)
\(30\) −6.74212e26 −0.317726
\(31\) −3.59446e27 −0.923511 −0.461756 0.887007i \(-0.652780\pi\)
−0.461756 + 0.887007i \(0.652780\pi\)
\(32\) −1.23794e27 −0.176777
\(33\) −1.92323e28 −1.55426
\(34\) 1.80877e28 0.841439
\(35\) −2.23635e28 −0.608529
\(36\) 6.59722e27 0.106602
\(37\) −3.04432e28 −0.296318 −0.148159 0.988964i \(-0.547335\pi\)
−0.148159 + 0.988964i \(0.547335\pi\)
\(38\) 1.76156e29 1.04689
\(39\) 1.26139e29 0.463612
\(40\) 6.26853e28 0.144230
\(41\) −7.95085e28 −0.115854 −0.0579271 0.998321i \(-0.518449\pi\)
−0.0579271 + 0.998321i \(0.518449\pi\)
\(42\) 1.24521e30 1.16180
\(43\) −2.87613e30 −1.73637 −0.868186 0.496240i \(-0.834714\pi\)
−0.868186 + 0.496240i \(0.834714\pi\)
\(44\) 1.78814e30 0.705547
\(45\) −3.34062e29 −0.0869754
\(46\) −4.07212e30 −0.705993
\(47\) −1.27820e31 −1.48863 −0.744315 0.667829i \(-0.767224\pi\)
−0.744315 + 0.667829i \(0.767224\pi\)
\(48\) −3.49036e30 −0.275364
\(49\) 2.27415e31 1.22516
\(50\) 1.58993e31 0.589431
\(51\) 5.09980e31 1.31070
\(52\) −1.17279e31 −0.210454
\(53\) 1.70666e31 0.215300 0.107650 0.994189i \(-0.465667\pi\)
0.107650 + 0.994189i \(0.465667\pi\)
\(54\) −6.86434e31 −0.612793
\(55\) −9.05455e31 −0.575648
\(56\) −1.15775e32 −0.527394
\(57\) 4.96669e32 1.63073
\(58\) −3.71406e32 −0.883957
\(59\) −7.46370e32 −1.29477 −0.647384 0.762164i \(-0.724137\pi\)
−0.647384 + 0.762164i \(0.724137\pi\)
\(60\) 1.76741e32 0.224666
\(61\) −2.64957e32 −0.248070 −0.124035 0.992278i \(-0.539584\pi\)
−0.124035 + 0.992278i \(0.539584\pi\)
\(62\) 9.42266e32 0.653021
\(63\) 6.16986e32 0.318035
\(64\) 3.24519e32 0.125000
\(65\) 5.93862e32 0.171707
\(66\) 5.04163e33 1.09903
\(67\) 1.82198e33 0.300718 0.150359 0.988631i \(-0.451957\pi\)
0.150359 + 0.988631i \(0.451957\pi\)
\(68\) −4.74157e33 −0.594987
\(69\) −1.14813e34 −1.09972
\(70\) 5.86246e33 0.430295
\(71\) −7.50959e33 −0.423972 −0.211986 0.977273i \(-0.567993\pi\)
−0.211986 + 0.977273i \(0.567993\pi\)
\(72\) −1.72942e33 −0.0753789
\(73\) −4.19120e34 −1.41536 −0.707678 0.706535i \(-0.750257\pi\)
−0.707678 + 0.706535i \(0.750257\pi\)
\(74\) 7.98050e33 0.209529
\(75\) 4.48279e34 0.918152
\(76\) −4.61782e34 −0.740261
\(77\) 1.67230e35 2.10492
\(78\) −3.30666e34 −0.327823
\(79\) 1.22253e35 0.957543 0.478772 0.877939i \(-0.341082\pi\)
0.478772 + 0.877939i \(0.341082\pi\)
\(80\) −1.64326e34 −0.101986
\(81\) −2.36767e35 −1.16775
\(82\) 2.08427e34 0.0819214
\(83\) −7.58400e34 −0.238206 −0.119103 0.992882i \(-0.538002\pi\)
−0.119103 + 0.992882i \(0.538002\pi\)
\(84\) −3.26425e35 −0.821518
\(85\) 2.40098e35 0.485444
\(86\) 7.53961e35 1.22780
\(87\) −1.04717e36 −1.37693
\(88\) −4.68749e35 −0.498897
\(89\) 2.32524e35 0.200795 0.100398 0.994947i \(-0.467989\pi\)
0.100398 + 0.994947i \(0.467989\pi\)
\(90\) 8.75725e34 0.0615009
\(91\) −1.09682e36 −0.627867
\(92\) 1.06748e36 0.499212
\(93\) 2.65671e36 1.01721
\(94\) 3.35071e36 1.05262
\(95\) 2.33832e36 0.603971
\(96\) 9.14976e35 0.194712
\(97\) −8.69459e36 −1.52747 −0.763734 0.645531i \(-0.776636\pi\)
−0.763734 + 0.645531i \(0.776636\pi\)
\(98\) −5.96154e36 −0.866316
\(99\) 2.49806e36 0.300851
\(100\) −4.16791e36 −0.416791
\(101\) 3.03773e36 0.252699 0.126349 0.991986i \(-0.459674\pi\)
0.126349 + 0.991986i \(0.459674\pi\)
\(102\) −1.33688e37 −0.926807
\(103\) −1.95543e36 −0.113176 −0.0565878 0.998398i \(-0.518022\pi\)
−0.0565878 + 0.998398i \(0.518022\pi\)
\(104\) 3.07439e36 0.148814
\(105\) 1.65291e37 0.670268
\(106\) −4.47392e36 −0.152240
\(107\) 2.09964e37 0.600543 0.300271 0.953854i \(-0.402923\pi\)
0.300271 + 0.953854i \(0.402923\pi\)
\(108\) 1.79945e37 0.433310
\(109\) 9.65773e35 0.0196103 0.00980516 0.999952i \(-0.496879\pi\)
0.00980516 + 0.999952i \(0.496879\pi\)
\(110\) 2.37360e37 0.407045
\(111\) 2.25009e37 0.326381
\(112\) 3.03496e37 0.372924
\(113\) −8.80397e37 −0.917756 −0.458878 0.888499i \(-0.651749\pi\)
−0.458878 + 0.888499i \(0.651749\pi\)
\(114\) −1.30199e38 −1.15310
\(115\) −5.40538e37 −0.407302
\(116\) 9.73619e37 0.625052
\(117\) −1.63840e37 −0.0897392
\(118\) 1.95657e38 0.915540
\(119\) −4.43442e38 −1.77508
\(120\) −4.63315e37 −0.158863
\(121\) 3.37043e38 0.991187
\(122\) 6.94570e37 0.175412
\(123\) 5.87657e37 0.127608
\(124\) −2.47009e38 −0.461756
\(125\) 4.64234e38 0.748000
\(126\) −1.61739e38 −0.224885
\(127\) 1.95704e38 0.235088 0.117544 0.993068i \(-0.462498\pi\)
0.117544 + 0.993068i \(0.462498\pi\)
\(128\) −8.50706e37 −0.0883883
\(129\) 2.12578e39 1.91254
\(130\) −1.55677e38 −0.121415
\(131\) −2.24388e38 −0.151873 −0.0759367 0.997113i \(-0.524195\pi\)
−0.0759367 + 0.997113i \(0.524195\pi\)
\(132\) −1.32163e39 −0.777129
\(133\) −4.31868e39 −2.20849
\(134\) −4.77621e38 −0.212640
\(135\) −9.11182e38 −0.353533
\(136\) 1.24298e39 0.420720
\(137\) −4.45772e38 −0.131760 −0.0658798 0.997828i \(-0.520985\pi\)
−0.0658798 + 0.997828i \(0.520985\pi\)
\(138\) 3.00975e39 0.777619
\(139\) 1.52545e39 0.344846 0.172423 0.985023i \(-0.444840\pi\)
0.172423 + 0.985023i \(0.444840\pi\)
\(140\) −1.53681e39 −0.304265
\(141\) 9.44729e39 1.63966
\(142\) 1.96859e39 0.299793
\(143\) −4.44079e39 −0.593941
\(144\) 4.53358e38 0.0533010
\(145\) −4.93010e39 −0.509973
\(146\) 1.09870e40 1.00081
\(147\) −1.68085e40 −1.34945
\(148\) −2.09204e39 −0.148159
\(149\) 2.63069e40 1.64484 0.822421 0.568879i \(-0.192623\pi\)
0.822421 + 0.568879i \(0.192623\pi\)
\(150\) −1.17514e40 −0.649232
\(151\) −2.66162e40 −1.30039 −0.650193 0.759769i \(-0.725312\pi\)
−0.650193 + 0.759769i \(0.725312\pi\)
\(152\) 1.21053e40 0.523444
\(153\) −6.62406e39 −0.253707
\(154\) −4.38384e40 −1.48841
\(155\) 1.25078e40 0.376741
\(156\) 8.66821e39 0.231806
\(157\) 6.40937e40 1.52290 0.761451 0.648222i \(-0.224487\pi\)
0.761451 + 0.648222i \(0.224487\pi\)
\(158\) −3.20479e40 −0.677085
\(159\) −1.26141e40 −0.237143
\(160\) 4.30770e39 0.0721151
\(161\) 9.98330e40 1.48935
\(162\) 6.20672e40 0.825723
\(163\) −1.54833e41 −1.83820 −0.919099 0.394027i \(-0.871082\pi\)
−0.919099 + 0.394027i \(0.871082\pi\)
\(164\) −5.46378e39 −0.0579271
\(165\) 6.69233e40 0.634051
\(166\) 1.98810e40 0.168437
\(167\) 2.28769e40 0.173437 0.0867185 0.996233i \(-0.472362\pi\)
0.0867185 + 0.996233i \(0.472362\pi\)
\(168\) 8.55705e40 0.580901
\(169\) −1.35275e41 −0.822836
\(170\) −6.29403e40 −0.343261
\(171\) −6.45117e40 −0.315653
\(172\) −1.97646e41 −0.868186
\(173\) −2.46815e41 −0.973909 −0.486955 0.873427i \(-0.661892\pi\)
−0.486955 + 0.873427i \(0.661892\pi\)
\(174\) 2.74511e41 0.973639
\(175\) −3.89791e41 −1.24345
\(176\) 1.22880e41 0.352774
\(177\) 5.51651e41 1.42613
\(178\) −6.09549e40 −0.141984
\(179\) 2.79681e41 0.587330 0.293665 0.955908i \(-0.405125\pi\)
0.293665 + 0.955908i \(0.405125\pi\)
\(180\) −2.29566e40 −0.0434877
\(181\) 6.41750e41 1.09727 0.548633 0.836063i \(-0.315148\pi\)
0.548633 + 0.836063i \(0.315148\pi\)
\(182\) 2.87523e41 0.443969
\(183\) 1.95833e41 0.273238
\(184\) −2.79834e41 −0.352996
\(185\) 1.05934e41 0.120881
\(186\) −6.96440e41 −0.719274
\(187\) −1.79541e42 −1.67917
\(188\) −8.78369e41 −0.744315
\(189\) 1.68288e42 1.29273
\(190\) −6.12975e41 −0.427072
\(191\) 1.09565e42 0.692712 0.346356 0.938103i \(-0.387419\pi\)
0.346356 + 0.938103i \(0.387419\pi\)
\(192\) −2.39855e41 −0.137682
\(193\) −2.90412e42 −1.51427 −0.757136 0.653257i \(-0.773402\pi\)
−0.757136 + 0.653257i \(0.773402\pi\)
\(194\) 2.27923e42 1.08008
\(195\) −4.38931e41 −0.189128
\(196\) 1.56278e42 0.612578
\(197\) 1.05947e42 0.377977 0.188988 0.981979i \(-0.439479\pi\)
0.188988 + 0.981979i \(0.439479\pi\)
\(198\) −6.54850e41 −0.212734
\(199\) −2.86684e42 −0.848441 −0.424220 0.905559i \(-0.639452\pi\)
−0.424220 + 0.905559i \(0.639452\pi\)
\(200\) 1.09259e42 0.294715
\(201\) −1.34665e42 −0.331228
\(202\) −7.96321e41 −0.178685
\(203\) 9.10548e42 1.86477
\(204\) 3.50455e42 0.655352
\(205\) 2.76669e41 0.0472621
\(206\) 5.12603e41 0.0800272
\(207\) 1.49129e42 0.212868
\(208\) −8.05933e41 −0.105227
\(209\) −1.74855e43 −2.08916
\(210\) −4.33302e42 −0.473951
\(211\) −1.88800e42 −0.189137 −0.0945686 0.995518i \(-0.530147\pi\)
−0.0945686 + 0.995518i \(0.530147\pi\)
\(212\) 1.17281e42 0.107650
\(213\) 5.55043e42 0.466986
\(214\) −5.50407e42 −0.424648
\(215\) 1.00082e43 0.708343
\(216\) −4.71714e42 −0.306397
\(217\) −2.31008e43 −1.37760
\(218\) −2.53172e41 −0.0138666
\(219\) 3.09777e43 1.55895
\(220\) −6.22224e42 −0.287824
\(221\) 1.17756e43 0.500870
\(222\) −5.89848e42 −0.230786
\(223\) 4.29674e43 1.54703 0.773515 0.633777i \(-0.218496\pi\)
0.773515 + 0.633777i \(0.218496\pi\)
\(224\) −7.95598e42 −0.263697
\(225\) −5.82263e42 −0.177723
\(226\) 2.30791e43 0.648952
\(227\) −4.38424e43 −1.13610 −0.568049 0.822995i \(-0.692301\pi\)
−0.568049 + 0.822995i \(0.692301\pi\)
\(228\) 3.41308e43 0.815365
\(229\) 4.31986e43 0.951728 0.475864 0.879519i \(-0.342135\pi\)
0.475864 + 0.879519i \(0.342135\pi\)
\(230\) 1.41699e43 0.288006
\(231\) −1.23602e44 −2.31848
\(232\) −2.55228e43 −0.441978
\(233\) −7.61369e43 −1.21762 −0.608808 0.793317i \(-0.708352\pi\)
−0.608808 + 0.793317i \(0.708352\pi\)
\(234\) 4.29498e42 0.0634552
\(235\) 4.44778e43 0.607279
\(236\) −5.12902e43 −0.647384
\(237\) −9.03587e43 −1.05469
\(238\) 1.16246e44 1.25517
\(239\) 1.50442e44 1.50317 0.751586 0.659635i \(-0.229289\pi\)
0.751586 + 0.659635i \(0.229289\pi\)
\(240\) 1.21455e43 0.112333
\(241\) 6.61898e43 0.566860 0.283430 0.958993i \(-0.408528\pi\)
0.283430 + 0.958993i \(0.408528\pi\)
\(242\) −8.83537e43 −0.700875
\(243\) 5.70891e43 0.419601
\(244\) −1.82077e43 −0.124035
\(245\) −7.91344e43 −0.499796
\(246\) −1.54051e43 −0.0902327
\(247\) 1.14682e44 0.623164
\(248\) 6.47520e43 0.326511
\(249\) 5.60542e43 0.262374
\(250\) −1.21696e44 −0.528916
\(251\) −1.83205e44 −0.739559 −0.369780 0.929119i \(-0.620567\pi\)
−0.369780 + 0.929119i \(0.620567\pi\)
\(252\) 4.23990e43 0.159018
\(253\) 4.04204e44 1.40887
\(254\) −5.13025e43 −0.166232
\(255\) −1.77459e44 −0.534695
\(256\) 2.23007e43 0.0625000
\(257\) −8.05029e43 −0.209918 −0.104959 0.994477i \(-0.533471\pi\)
−0.104959 + 0.994477i \(0.533471\pi\)
\(258\) −5.57261e44 −1.35237
\(259\) −1.95652e44 −0.442016
\(260\) 4.08099e43 0.0858536
\(261\) 1.36016e44 0.266527
\(262\) 5.88221e43 0.107391
\(263\) −7.13601e44 −1.21416 −0.607078 0.794642i \(-0.707658\pi\)
−0.607078 + 0.794642i \(0.707658\pi\)
\(264\) 3.46458e44 0.549513
\(265\) −5.93874e43 −0.0878305
\(266\) 1.13212e45 1.56164
\(267\) −1.71861e44 −0.221167
\(268\) 1.25206e44 0.150359
\(269\) 1.26185e45 1.41445 0.707227 0.706986i \(-0.249946\pi\)
0.707227 + 0.706986i \(0.249946\pi\)
\(270\) 2.38861e44 0.249986
\(271\) 4.59894e43 0.0449496 0.0224748 0.999747i \(-0.492845\pi\)
0.0224748 + 0.999747i \(0.492845\pi\)
\(272\) −3.25838e44 −0.297494
\(273\) 8.10669e44 0.691567
\(274\) 1.16857e44 0.0931682
\(275\) −1.57819e45 −1.17626
\(276\) −7.88987e44 −0.549860
\(277\) −1.16921e45 −0.762106 −0.381053 0.924553i \(-0.624438\pi\)
−0.381053 + 0.924553i \(0.624438\pi\)
\(278\) −3.99889e44 −0.243843
\(279\) −3.45076e44 −0.196896
\(280\) 4.02865e44 0.215148
\(281\) 2.11366e45 1.05674 0.528370 0.849014i \(-0.322803\pi\)
0.528370 + 0.849014i \(0.322803\pi\)
\(282\) −2.47655e45 −1.15941
\(283\) 2.78101e44 0.121942 0.0609710 0.998140i \(-0.480580\pi\)
0.0609710 + 0.998140i \(0.480580\pi\)
\(284\) −5.16055e44 −0.211986
\(285\) −1.72828e45 −0.665247
\(286\) 1.16413e45 0.419980
\(287\) −5.10984e44 −0.172819
\(288\) −1.18845e44 −0.0376895
\(289\) 1.39876e45 0.416039
\(290\) 1.29239e45 0.360605
\(291\) 6.42627e45 1.68244
\(292\) −2.88017e45 −0.707678
\(293\) −3.24815e45 −0.749177 −0.374589 0.927191i \(-0.622216\pi\)
−0.374589 + 0.927191i \(0.622216\pi\)
\(294\) 4.40625e45 0.954208
\(295\) 2.59717e45 0.528194
\(296\) 5.48416e44 0.104764
\(297\) 6.81364e45 1.22288
\(298\) −6.89620e45 −1.16308
\(299\) −2.65106e45 −0.420245
\(300\) 3.08055e45 0.459076
\(301\) −1.84843e46 −2.59014
\(302\) 6.97728e45 0.919512
\(303\) −2.24522e45 −0.278336
\(304\) −3.17334e45 −0.370131
\(305\) 9.21982e44 0.101199
\(306\) 1.73646e45 0.179398
\(307\) −9.85048e44 −0.0958070 −0.0479035 0.998852i \(-0.515254\pi\)
−0.0479035 + 0.998852i \(0.515254\pi\)
\(308\) 1.14920e46 1.05246
\(309\) 1.44528e45 0.124658
\(310\) −3.27884e45 −0.266396
\(311\) −5.05506e45 −0.386954 −0.193477 0.981105i \(-0.561976\pi\)
−0.193477 + 0.981105i \(0.561976\pi\)
\(312\) −2.27232e45 −0.163911
\(313\) −6.66761e45 −0.453315 −0.226657 0.973975i \(-0.572780\pi\)
−0.226657 + 0.973975i \(0.572780\pi\)
\(314\) −1.68018e46 −1.07685
\(315\) −2.14695e45 −0.129741
\(316\) 8.40116e45 0.478772
\(317\) 2.02795e46 1.09009 0.545044 0.838408i \(-0.316513\pi\)
0.545044 + 0.838408i \(0.316513\pi\)
\(318\) 3.30672e45 0.167686
\(319\) 3.68663e46 1.76401
\(320\) −1.12924e45 −0.0509931
\(321\) −1.55187e46 −0.661471
\(322\) −2.61706e46 −1.05313
\(323\) 4.63660e46 1.76178
\(324\) −1.62705e46 −0.583874
\(325\) 1.03509e46 0.350861
\(326\) 4.05885e46 1.29980
\(327\) −7.13814e44 −0.0215999
\(328\) 1.43230e45 0.0409607
\(329\) −8.21469e46 −2.22058
\(330\) −1.75435e46 −0.448342
\(331\) −6.38864e46 −1.54380 −0.771899 0.635745i \(-0.780693\pi\)
−0.771899 + 0.635745i \(0.780693\pi\)
\(332\) −5.21168e45 −0.119103
\(333\) −2.92261e45 −0.0631761
\(334\) −5.99703e45 −0.122638
\(335\) −6.34001e45 −0.122676
\(336\) −2.24318e46 −0.410759
\(337\) 7.62763e46 1.32202 0.661008 0.750379i \(-0.270129\pi\)
0.661008 + 0.750379i \(0.270129\pi\)
\(338\) 3.54615e46 0.581833
\(339\) 6.50712e46 1.01087
\(340\) 1.64994e46 0.242722
\(341\) −9.35307e46 −1.30316
\(342\) 1.69113e46 0.223200
\(343\) 2.68599e46 0.335864
\(344\) 5.18118e46 0.613900
\(345\) 3.99518e46 0.448625
\(346\) 6.47011e46 0.688658
\(347\) −9.67434e46 −0.976171 −0.488086 0.872796i \(-0.662305\pi\)
−0.488086 + 0.872796i \(0.662305\pi\)
\(348\) −7.19613e46 −0.688467
\(349\) 1.11323e47 1.00998 0.504992 0.863124i \(-0.331495\pi\)
0.504992 + 0.863124i \(0.331495\pi\)
\(350\) 1.02181e47 0.879252
\(351\) −4.46887e46 −0.364768
\(352\) −3.22122e46 −0.249449
\(353\) −1.61697e47 −1.18815 −0.594074 0.804410i \(-0.702481\pi\)
−0.594074 + 0.804410i \(0.702481\pi\)
\(354\) −1.44612e47 −1.00843
\(355\) 2.61314e46 0.172957
\(356\) 1.59790e46 0.100398
\(357\) 3.27753e47 1.95517
\(358\) −7.33168e46 −0.415305
\(359\) −3.46444e47 −1.86374 −0.931870 0.362792i \(-0.881823\pi\)
−0.931870 + 0.362792i \(0.881823\pi\)
\(360\) 6.01793e45 0.0307504
\(361\) 2.45550e47 1.19195
\(362\) −1.68231e47 −0.775884
\(363\) −2.49112e47 −1.09175
\(364\) −7.53726e46 −0.313933
\(365\) 1.45843e47 0.577387
\(366\) −5.13365e46 −0.193208
\(367\) 3.11549e47 1.11482 0.557410 0.830237i \(-0.311795\pi\)
0.557410 + 0.830237i \(0.311795\pi\)
\(368\) 7.33567e46 0.249606
\(369\) −7.63299e45 −0.0247006
\(370\) −2.77700e46 −0.0854760
\(371\) 1.09684e47 0.321162
\(372\) 1.82568e47 0.508603
\(373\) −4.04595e46 −0.107253 −0.0536263 0.998561i \(-0.517078\pi\)
−0.0536263 + 0.998561i \(0.517078\pi\)
\(374\) 4.70656e47 1.18735
\(375\) −3.43121e47 −0.823888
\(376\) 2.30259e47 0.526310
\(377\) −2.41796e47 −0.526179
\(378\) −4.41156e47 −0.914101
\(379\) 4.45515e47 0.879097 0.439548 0.898219i \(-0.355139\pi\)
0.439548 + 0.898219i \(0.355139\pi\)
\(380\) 1.60688e47 0.301986
\(381\) −1.44647e47 −0.258939
\(382\) −2.87217e47 −0.489822
\(383\) 1.15561e47 0.187774 0.0938868 0.995583i \(-0.470071\pi\)
0.0938868 + 0.995583i \(0.470071\pi\)
\(384\) 6.28767e46 0.0973558
\(385\) −5.81917e47 −0.858692
\(386\) 7.61297e47 1.07075
\(387\) −2.76115e47 −0.370201
\(388\) −5.97488e47 −0.763734
\(389\) −5.77044e46 −0.0703301 −0.0351650 0.999382i \(-0.511196\pi\)
−0.0351650 + 0.999382i \(0.511196\pi\)
\(390\) 1.15063e47 0.133734
\(391\) −1.07182e48 −1.18810
\(392\) −4.09674e47 −0.433158
\(393\) 1.65848e47 0.167282
\(394\) −2.77735e47 −0.267270
\(395\) −4.25408e47 −0.390625
\(396\) 1.71665e47 0.150425
\(397\) 2.91353e47 0.243666 0.121833 0.992551i \(-0.461123\pi\)
0.121833 + 0.992551i \(0.461123\pi\)
\(398\) 7.51525e47 0.599938
\(399\) 3.19198e48 2.43255
\(400\) −2.86416e47 −0.208395
\(401\) 1.08285e48 0.752314 0.376157 0.926556i \(-0.377245\pi\)
0.376157 + 0.926556i \(0.377245\pi\)
\(402\) 3.53015e47 0.234213
\(403\) 6.13441e47 0.388714
\(404\) 2.08751e47 0.126349
\(405\) 8.23888e47 0.476376
\(406\) −2.38695e48 −1.31859
\(407\) −7.92156e47 −0.418133
\(408\) −9.18698e47 −0.463404
\(409\) −5.33536e47 −0.257207 −0.128603 0.991696i \(-0.541049\pi\)
−0.128603 + 0.991696i \(0.541049\pi\)
\(410\) −7.25270e46 −0.0334194
\(411\) 3.29476e47 0.145127
\(412\) −1.34376e47 −0.0565878
\(413\) −4.79676e48 −1.93140
\(414\) −3.90932e47 −0.150520
\(415\) 2.63903e47 0.0971750
\(416\) 2.11271e47 0.0744068
\(417\) −1.12748e48 −0.379833
\(418\) 4.58371e48 1.47726
\(419\) 3.03589e48 0.936109 0.468055 0.883700i \(-0.344955\pi\)
0.468055 + 0.883700i \(0.344955\pi\)
\(420\) 1.13587e48 0.335134
\(421\) −5.78685e48 −1.63389 −0.816946 0.576714i \(-0.804335\pi\)
−0.816946 + 0.576714i \(0.804335\pi\)
\(422\) 4.94928e47 0.133740
\(423\) −1.22710e48 −0.317382
\(424\) −3.07445e47 −0.0761201
\(425\) 4.18486e48 0.991940
\(426\) −1.45501e48 −0.330209
\(427\) −1.70283e48 −0.370045
\(428\) 1.44286e48 0.300271
\(429\) 3.28224e48 0.654200
\(430\) −2.62359e48 −0.500874
\(431\) 7.93042e47 0.145033 0.0725165 0.997367i \(-0.476897\pi\)
0.0725165 + 0.997367i \(0.476897\pi\)
\(432\) 1.23657e48 0.216655
\(433\) 2.27386e48 0.381714 0.190857 0.981618i \(-0.438873\pi\)
0.190857 + 0.981618i \(0.438873\pi\)
\(434\) 6.05574e48 0.974109
\(435\) 3.64389e48 0.561713
\(436\) 6.63674e46 0.00980516
\(437\) −1.04385e49 −1.47819
\(438\) −8.12061e48 −1.10235
\(439\) 1.05173e49 1.36871 0.684354 0.729150i \(-0.260084\pi\)
0.684354 + 0.729150i \(0.260084\pi\)
\(440\) 1.63112e48 0.203522
\(441\) 2.18323e48 0.261208
\(442\) −3.08690e48 −0.354169
\(443\) −1.96518e48 −0.216239 −0.108119 0.994138i \(-0.534483\pi\)
−0.108119 + 0.994138i \(0.534483\pi\)
\(444\) 1.54625e48 0.163191
\(445\) −8.09123e47 −0.0819133
\(446\) −1.12636e49 −1.09392
\(447\) −1.94438e49 −1.81172
\(448\) 2.08561e48 0.186462
\(449\) 2.75656e48 0.236488 0.118244 0.992985i \(-0.462273\pi\)
0.118244 + 0.992985i \(0.462273\pi\)
\(450\) 1.52637e48 0.125669
\(451\) −2.06887e48 −0.163481
\(452\) −6.05004e48 −0.458878
\(453\) 1.96723e49 1.43232
\(454\) 1.14930e49 0.803342
\(455\) 3.81663e48 0.256135
\(456\) −8.94719e48 −0.576550
\(457\) 8.54404e48 0.528705 0.264353 0.964426i \(-0.414842\pi\)
0.264353 + 0.964426i \(0.414842\pi\)
\(458\) −1.13242e49 −0.672973
\(459\) −1.80676e49 −1.03126
\(460\) −3.71455e48 −0.203651
\(461\) 3.55334e49 1.87141 0.935707 0.352778i \(-0.114763\pi\)
0.935707 + 0.352778i \(0.114763\pi\)
\(462\) 3.24015e49 1.63941
\(463\) −2.26251e49 −1.09987 −0.549937 0.835206i \(-0.685348\pi\)
−0.549937 + 0.835206i \(0.685348\pi\)
\(464\) 6.69066e48 0.312526
\(465\) −9.24464e48 −0.414964
\(466\) 1.99588e49 0.860985
\(467\) 1.49204e49 0.618612 0.309306 0.950963i \(-0.399903\pi\)
0.309306 + 0.950963i \(0.399903\pi\)
\(468\) −1.12590e48 −0.0448696
\(469\) 1.17095e49 0.448580
\(470\) −1.16596e49 −0.429411
\(471\) −4.73724e49 −1.67741
\(472\) 1.34454e49 0.457770
\(473\) −7.48393e49 −2.45018
\(474\) 2.36870e49 0.745779
\(475\) 4.07563e49 1.23414
\(476\) −3.04731e49 −0.887540
\(477\) 1.63844e48 0.0459028
\(478\) −3.94375e49 −1.06290
\(479\) 1.98243e49 0.514033 0.257016 0.966407i \(-0.417261\pi\)
0.257016 + 0.966407i \(0.417261\pi\)
\(480\) −3.18387e48 −0.0794316
\(481\) 5.19553e48 0.124723
\(482\) −1.73513e49 −0.400831
\(483\) −7.37877e49 −1.64045
\(484\) 2.31614e49 0.495594
\(485\) 3.02549e49 0.623122
\(486\) −1.49656e49 −0.296703
\(487\) 7.79986e49 1.48868 0.744339 0.667802i \(-0.232765\pi\)
0.744339 + 0.667802i \(0.232765\pi\)
\(488\) 4.77305e48 0.0877060
\(489\) 1.14439e50 2.02469
\(490\) 2.07446e49 0.353409
\(491\) 6.21402e49 1.01945 0.509724 0.860338i \(-0.329747\pi\)
0.509724 + 0.860338i \(0.329747\pi\)
\(492\) 4.03834e48 0.0638042
\(493\) −9.77579e49 −1.48759
\(494\) −3.00633e49 −0.440644
\(495\) −8.69257e48 −0.122730
\(496\) −1.69744e49 −0.230878
\(497\) −4.82626e49 −0.632437
\(498\) −1.46943e49 −0.185526
\(499\) −1.01715e48 −0.0123744 −0.00618718 0.999981i \(-0.501969\pi\)
−0.00618718 + 0.999981i \(0.501969\pi\)
\(500\) 3.19019e49 0.374000
\(501\) −1.69086e49 −0.191033
\(502\) 4.80260e49 0.522947
\(503\) −8.23991e48 −0.0864798 −0.0432399 0.999065i \(-0.513768\pi\)
−0.0432399 + 0.999065i \(0.513768\pi\)
\(504\) −1.11146e49 −0.112442
\(505\) −1.05705e49 −0.103087
\(506\) −1.05960e50 −0.996222
\(507\) 9.99833e49 0.906317
\(508\) 1.34486e49 0.117544
\(509\) 1.50535e50 1.26870 0.634350 0.773046i \(-0.281268\pi\)
0.634350 + 0.773046i \(0.281268\pi\)
\(510\) 4.65199e49 0.378086
\(511\) −2.69360e50 −2.11128
\(512\) −5.84601e48 −0.0441942
\(513\) −1.75961e50 −1.28305
\(514\) 2.11034e49 0.148434
\(515\) 6.80437e48 0.0461694
\(516\) 1.46083e50 0.956268
\(517\) −3.32597e50 −2.10060
\(518\) 5.12890e49 0.312553
\(519\) 1.82424e50 1.07272
\(520\) −1.06981e49 −0.0607077
\(521\) 1.38659e49 0.0759363 0.0379682 0.999279i \(-0.487911\pi\)
0.0379682 + 0.999279i \(0.487911\pi\)
\(522\) −3.56558e49 −0.188463
\(523\) 1.22782e50 0.626402 0.313201 0.949687i \(-0.398599\pi\)
0.313201 + 0.949687i \(0.398599\pi\)
\(524\) −1.54199e49 −0.0759367
\(525\) 2.88099e50 1.36960
\(526\) 1.87066e50 0.858538
\(527\) 2.48014e50 1.09895
\(528\) −9.08219e49 −0.388564
\(529\) −7.62218e47 −0.00314883
\(530\) 1.55680e49 0.0621056
\(531\) −7.16532e49 −0.276050
\(532\) −2.96777e50 −1.10424
\(533\) 1.35692e49 0.0487640
\(534\) 4.50525e49 0.156389
\(535\) −7.30618e49 −0.244988
\(536\) −3.28219e49 −0.106320
\(537\) −2.06716e50 −0.646918
\(538\) −3.30785e50 −1.00017
\(539\) 5.91752e50 1.72881
\(540\) −6.26159e49 −0.176767
\(541\) −4.12270e50 −1.12469 −0.562344 0.826904i \(-0.690100\pi\)
−0.562344 + 0.826904i \(0.690100\pi\)
\(542\) −1.20558e49 −0.0317842
\(543\) −4.74325e50 −1.20859
\(544\) 8.54166e49 0.210360
\(545\) −3.36063e48 −0.00799992
\(546\) −2.12512e50 −0.489012
\(547\) 4.73750e50 1.05386 0.526931 0.849908i \(-0.323343\pi\)
0.526931 + 0.849908i \(0.323343\pi\)
\(548\) −3.06332e49 −0.0658798
\(549\) −2.54365e49 −0.0528895
\(550\) 4.13713e50 0.831743
\(551\) −9.52063e50 −1.85081
\(552\) 2.06828e50 0.388810
\(553\) 7.85694e50 1.42836
\(554\) 3.06500e50 0.538890
\(555\) −7.82972e49 −0.133145
\(556\) 1.04828e50 0.172423
\(557\) −1.89090e50 −0.300848 −0.150424 0.988622i \(-0.548064\pi\)
−0.150424 + 0.988622i \(0.548064\pi\)
\(558\) 9.04597e49 0.139227
\(559\) 4.90850e50 0.730853
\(560\) −1.05609e50 −0.152132
\(561\) 1.32701e51 1.84953
\(562\) −5.54083e50 −0.747228
\(563\) −6.92011e50 −0.903041 −0.451521 0.892261i \(-0.649118\pi\)
−0.451521 + 0.892261i \(0.649118\pi\)
\(564\) 6.49213e50 0.819830
\(565\) 3.06355e50 0.374394
\(566\) −7.29024e49 −0.0862260
\(567\) −1.52165e51 −1.74192
\(568\) 1.35281e50 0.149897
\(569\) 2.37823e50 0.255080 0.127540 0.991833i \(-0.459292\pi\)
0.127540 + 0.991833i \(0.459292\pi\)
\(570\) 4.53057e50 0.470401
\(571\) −9.61787e50 −0.966742 −0.483371 0.875416i \(-0.660588\pi\)
−0.483371 + 0.875416i \(0.660588\pi\)
\(572\) −3.05169e50 −0.296971
\(573\) −8.09805e50 −0.762992
\(574\) 1.33951e50 0.122202
\(575\) −9.42146e50 −0.832268
\(576\) 3.11545e49 0.0266505
\(577\) 3.68489e50 0.305262 0.152631 0.988283i \(-0.451225\pi\)
0.152631 + 0.988283i \(0.451225\pi\)
\(578\) −3.66678e50 −0.294184
\(579\) 2.14647e51 1.66790
\(580\) −3.38794e50 −0.254987
\(581\) −4.87407e50 −0.355331
\(582\) −1.68461e51 −1.18966
\(583\) 4.44088e50 0.303809
\(584\) 7.55020e50 0.500404
\(585\) 5.70121e49 0.0366086
\(586\) 8.51482e50 0.529748
\(587\) 1.48680e51 0.896287 0.448144 0.893962i \(-0.352085\pi\)
0.448144 + 0.893962i \(0.352085\pi\)
\(588\) −1.15507e51 −0.674727
\(589\) 2.41541e51 1.36728
\(590\) −6.80833e50 −0.373490
\(591\) −7.83069e50 −0.416324
\(592\) −1.43764e50 −0.0740795
\(593\) −2.32937e51 −1.16339 −0.581696 0.813406i \(-0.697611\pi\)
−0.581696 + 0.813406i \(0.697611\pi\)
\(594\) −1.78616e51 −0.864709
\(595\) 1.54306e51 0.724134
\(596\) 1.80780e51 0.822421
\(597\) 2.11892e51 0.934520
\(598\) 6.94959e50 0.297158
\(599\) −1.32479e51 −0.549226 −0.274613 0.961555i \(-0.588550\pi\)
−0.274613 + 0.961555i \(0.588550\pi\)
\(600\) −8.07547e50 −0.324616
\(601\) 1.22549e51 0.477675 0.238837 0.971060i \(-0.423234\pi\)
0.238837 + 0.971060i \(0.423234\pi\)
\(602\) 4.84555e51 1.83150
\(603\) 1.74914e50 0.0641143
\(604\) −1.82905e51 −0.650193
\(605\) −1.17282e51 −0.404349
\(606\) 5.88571e50 0.196814
\(607\) −2.85022e51 −0.924460 −0.462230 0.886760i \(-0.652951\pi\)
−0.462230 + 0.886760i \(0.652951\pi\)
\(608\) 8.31872e50 0.261722
\(609\) −6.72997e51 −2.05397
\(610\) −2.41692e50 −0.0715584
\(611\) 2.18141e51 0.626577
\(612\) −4.55202e50 −0.126854
\(613\) −1.66565e51 −0.450365 −0.225182 0.974317i \(-0.572298\pi\)
−0.225182 + 0.974317i \(0.572298\pi\)
\(614\) 2.58224e50 0.0677458
\(615\) −2.04489e50 −0.0520571
\(616\) −3.01255e51 −0.744203
\(617\) −4.81221e51 −1.15364 −0.576818 0.816873i \(-0.695706\pi\)
−0.576818 + 0.816873i \(0.695706\pi\)
\(618\) −3.78871e50 −0.0881464
\(619\) 9.79976e50 0.221278 0.110639 0.993861i \(-0.464710\pi\)
0.110639 + 0.993861i \(0.464710\pi\)
\(620\) 8.59527e50 0.188371
\(621\) 4.06761e51 0.865255
\(622\) 1.32515e51 0.273618
\(623\) 1.49438e51 0.299525
\(624\) 5.95675e50 0.115903
\(625\) 2.79753e51 0.528439
\(626\) 1.74787e51 0.320542
\(627\) 1.29237e52 2.30111
\(628\) 4.40449e51 0.761451
\(629\) 2.10055e51 0.352611
\(630\) 5.62810e50 0.0917406
\(631\) −9.71711e51 −1.53813 −0.769066 0.639169i \(-0.779278\pi\)
−0.769066 + 0.639169i \(0.779278\pi\)
\(632\) −2.20231e51 −0.338543
\(633\) 1.39544e51 0.208326
\(634\) −5.31615e51 −0.770808
\(635\) −6.80997e50 −0.0959028
\(636\) −8.66838e50 −0.118572
\(637\) −3.88113e51 −0.515678
\(638\) −9.66428e51 −1.24735
\(639\) −7.20938e50 −0.0903924
\(640\) 2.96023e50 0.0360575
\(641\) 6.64686e51 0.786580 0.393290 0.919415i \(-0.371337\pi\)
0.393290 + 0.919415i \(0.371337\pi\)
\(642\) 4.06812e51 0.467731
\(643\) 1.19542e52 1.33542 0.667708 0.744423i \(-0.267275\pi\)
0.667708 + 0.744423i \(0.267275\pi\)
\(644\) 6.86047e51 0.744673
\(645\) −7.39717e51 −0.780208
\(646\) −1.21546e52 −1.24577
\(647\) −1.25086e52 −1.24589 −0.622947 0.782264i \(-0.714065\pi\)
−0.622947 + 0.782264i \(0.714065\pi\)
\(648\) 4.26522e51 0.412861
\(649\) −1.94212e52 −1.82704
\(650\) −2.71342e51 −0.248096
\(651\) 1.70741e52 1.51736
\(652\) −1.06400e52 −0.919099
\(653\) 1.24143e52 1.04239 0.521194 0.853438i \(-0.325487\pi\)
0.521194 + 0.853438i \(0.325487\pi\)
\(654\) 1.87122e50 0.0152734
\(655\) 7.80813e50 0.0619559
\(656\) −3.75468e50 −0.0289636
\(657\) −4.02365e51 −0.301759
\(658\) 2.15343e52 1.57019
\(659\) −1.09815e52 −0.778538 −0.389269 0.921124i \(-0.627272\pi\)
−0.389269 + 0.921124i \(0.627272\pi\)
\(660\) 4.59893e51 0.317025
\(661\) −6.97744e51 −0.467702 −0.233851 0.972272i \(-0.575133\pi\)
−0.233851 + 0.972272i \(0.575133\pi\)
\(662\) 1.67474e52 1.09163
\(663\) −8.70347e51 −0.551686
\(664\) 1.36621e51 0.0842187
\(665\) 1.50279e52 0.900941
\(666\) 7.66146e50 0.0446723
\(667\) 2.20084e52 1.24813
\(668\) 1.57209e51 0.0867185
\(669\) −3.17577e52 −1.70399
\(670\) 1.66200e51 0.0867453
\(671\) −6.89441e51 −0.350050
\(672\) 5.88036e51 0.290450
\(673\) 3.84641e52 1.84832 0.924160 0.382006i \(-0.124767\pi\)
0.924160 + 0.382006i \(0.124767\pi\)
\(674\) −1.99954e52 −0.934806
\(675\) −1.58817e52 −0.722399
\(676\) −9.29603e51 −0.411418
\(677\) 1.92251e52 0.827900 0.413950 0.910300i \(-0.364149\pi\)
0.413950 + 0.910300i \(0.364149\pi\)
\(678\) −1.70580e52 −0.714791
\(679\) −5.58783e52 −2.27852
\(680\) −4.32523e51 −0.171630
\(681\) 3.24044e52 1.25136
\(682\) 2.45185e52 0.921474
\(683\) 1.99791e52 0.730790 0.365395 0.930853i \(-0.380934\pi\)
0.365395 + 0.930853i \(0.380934\pi\)
\(684\) −4.43321e51 −0.157827
\(685\) 1.55117e51 0.0537507
\(686\) −7.04117e51 −0.237492
\(687\) −3.19286e52 −1.04829
\(688\) −1.35822e52 −0.434093
\(689\) −2.91264e51 −0.0906216
\(690\) −1.04731e52 −0.317226
\(691\) −1.88822e52 −0.556811 −0.278406 0.960464i \(-0.589806\pi\)
−0.278406 + 0.960464i \(0.589806\pi\)
\(692\) −1.69610e52 −0.486955
\(693\) 1.60545e52 0.448778
\(694\) 2.53607e52 0.690257
\(695\) −5.30818e51 −0.140678
\(696\) 1.88642e52 0.486820
\(697\) 5.48600e51 0.137864
\(698\) −2.91827e52 −0.714166
\(699\) 5.62737e52 1.34115
\(700\) −2.67862e52 −0.621725
\(701\) 2.68099e52 0.606054 0.303027 0.952982i \(-0.402003\pi\)
0.303027 + 0.952982i \(0.402003\pi\)
\(702\) 1.17149e52 0.257930
\(703\) 2.04572e52 0.438706
\(704\) 8.44423e51 0.176387
\(705\) −3.28741e52 −0.668890
\(706\) 4.23880e52 0.840148
\(707\) 1.95228e52 0.376950
\(708\) 3.79092e52 0.713065
\(709\) 5.13576e52 0.941129 0.470565 0.882366i \(-0.344050\pi\)
0.470565 + 0.882366i \(0.344050\pi\)
\(710\) −6.85019e51 −0.122299
\(711\) 1.17366e52 0.204152
\(712\) −4.18879e51 −0.0709918
\(713\) −5.58359e52 −0.922056
\(714\) −8.59185e52 −1.38251
\(715\) 1.54528e52 0.242295
\(716\) 1.92195e52 0.293665
\(717\) −1.11194e53 −1.65568
\(718\) 9.08181e52 1.31786
\(719\) −1.27331e53 −1.80074 −0.900368 0.435129i \(-0.856703\pi\)
−0.900368 + 0.435129i \(0.856703\pi\)
\(720\) −1.57757e51 −0.0217438
\(721\) −1.25671e52 −0.168824
\(722\) −6.43695e52 −0.842835
\(723\) −4.89217e52 −0.624371
\(724\) 4.41007e52 0.548633
\(725\) −8.59305e52 −1.04206
\(726\) 6.53033e52 0.771983
\(727\) 6.59926e51 0.0760517 0.0380258 0.999277i \(-0.487893\pi\)
0.0380258 + 0.999277i \(0.487893\pi\)
\(728\) 1.97585e52 0.221984
\(729\) 6.44174e52 0.705576
\(730\) −3.82318e52 −0.408274
\(731\) 1.98450e53 2.06624
\(732\) 1.34576e52 0.136619
\(733\) 1.54131e52 0.152569 0.0762845 0.997086i \(-0.475694\pi\)
0.0762845 + 0.997086i \(0.475694\pi\)
\(734\) −8.16708e52 −0.788297
\(735\) 5.84891e52 0.550503
\(736\) −1.92300e52 −0.176498
\(737\) 4.74094e52 0.424342
\(738\) 2.00094e51 0.0174659
\(739\) 5.46870e52 0.465545 0.232773 0.972531i \(-0.425220\pi\)
0.232773 + 0.972531i \(0.425220\pi\)
\(740\) 7.27974e51 0.0604407
\(741\) −8.47630e52 −0.686388
\(742\) −2.87529e52 −0.227096
\(743\) 1.56320e53 1.20426 0.602132 0.798397i \(-0.294318\pi\)
0.602132 + 0.798397i \(0.294318\pi\)
\(744\) −4.78590e52 −0.359637
\(745\) −9.15412e52 −0.671005
\(746\) 1.06062e52 0.0758390
\(747\) −7.28081e51 −0.0507865
\(748\) −1.23380e53 −0.839583
\(749\) 1.34939e53 0.895827
\(750\) 8.99470e52 0.582577
\(751\) −1.90633e53 −1.20465 −0.602324 0.798252i \(-0.705758\pi\)
−0.602324 + 0.798252i \(0.705758\pi\)
\(752\) −6.03611e52 −0.372158
\(753\) 1.35409e53 0.814592
\(754\) 6.33853e52 0.372065
\(755\) 9.26173e52 0.530486
\(756\) 1.15647e53 0.646367
\(757\) 7.39737e52 0.403463 0.201731 0.979441i \(-0.435343\pi\)
0.201731 + 0.979441i \(0.435343\pi\)
\(758\) −1.16789e53 −0.621615
\(759\) −2.98752e53 −1.55181
\(760\) −4.21233e52 −0.213536
\(761\) −3.31955e53 −1.64234 −0.821169 0.570685i \(-0.806678\pi\)
−0.821169 + 0.570685i \(0.806678\pi\)
\(762\) 3.79183e52 0.183097
\(763\) 6.20682e51 0.0292526
\(764\) 7.52922e52 0.346356
\(765\) 2.30500e52 0.103498
\(766\) −3.02937e52 −0.132776
\(767\) 1.27378e53 0.544979
\(768\) −1.64827e52 −0.0688410
\(769\) −3.78094e53 −1.54157 −0.770783 0.637098i \(-0.780135\pi\)
−0.770783 + 0.637098i \(0.780135\pi\)
\(770\) 1.52546e53 0.607187
\(771\) 5.95006e52 0.231215
\(772\) −1.99569e53 −0.757136
\(773\) 1.75568e53 0.650315 0.325158 0.945660i \(-0.394583\pi\)
0.325158 + 0.945660i \(0.394583\pi\)
\(774\) 7.23820e52 0.261772
\(775\) 2.18008e53 0.769822
\(776\) 1.56628e53 0.540041
\(777\) 1.44609e53 0.486861
\(778\) 1.51269e52 0.0497309
\(779\) 5.34282e52 0.171525
\(780\) −3.01631e52 −0.0945639
\(781\) −1.95406e53 −0.598264
\(782\) 2.80972e53 0.840113
\(783\) 3.70995e53 1.08337
\(784\) 1.07394e53 0.306289
\(785\) −2.23029e53 −0.621260
\(786\) −4.34761e52 −0.118286
\(787\) −5.64566e53 −1.50031 −0.750157 0.661260i \(-0.770022\pi\)
−0.750157 + 0.661260i \(0.770022\pi\)
\(788\) 7.28064e52 0.188988
\(789\) 5.27431e53 1.33734
\(790\) 1.11518e53 0.276213
\(791\) −5.65812e53 −1.36901
\(792\) −4.50010e52 −0.106367
\(793\) 4.52184e52 0.104415
\(794\) −7.63765e52 −0.172298
\(795\) 4.38939e52 0.0967414
\(796\) −1.97008e53 −0.424220
\(797\) 5.95809e53 1.25351 0.626755 0.779216i \(-0.284383\pi\)
0.626755 + 0.779216i \(0.284383\pi\)
\(798\) −8.36760e53 −1.72007
\(799\) 8.81942e53 1.77143
\(800\) 7.50823e52 0.147358
\(801\) 2.23229e52 0.0428103
\(802\) −2.83864e53 −0.531967
\(803\) −1.09058e54 −1.99720
\(804\) −9.25409e52 −0.165614
\(805\) −3.47392e53 −0.607570
\(806\) −1.60810e53 −0.274862
\(807\) −9.32645e53 −1.55796
\(808\) −5.47228e52 −0.0893425
\(809\) 5.07007e53 0.809033 0.404517 0.914531i \(-0.367440\pi\)
0.404517 + 0.914531i \(0.367440\pi\)
\(810\) −2.15977e53 −0.336849
\(811\) 4.73551e53 0.721906 0.360953 0.932584i \(-0.382451\pi\)
0.360953 + 0.932584i \(0.382451\pi\)
\(812\) 6.25724e53 0.932387
\(813\) −3.39913e52 −0.0495100
\(814\) 2.07659e53 0.295664
\(815\) 5.38777e53 0.749883
\(816\) 2.40831e53 0.327676
\(817\) 1.93271e54 2.57074
\(818\) 1.39863e53 0.181873
\(819\) −1.05297e53 −0.133864
\(820\) 1.90125e52 0.0236311
\(821\) −1.06816e54 −1.29804 −0.649020 0.760771i \(-0.724821\pi\)
−0.649020 + 0.760771i \(0.724821\pi\)
\(822\) −8.63700e52 −0.102621
\(823\) −6.19938e53 −0.720197 −0.360098 0.932914i \(-0.617257\pi\)
−0.360098 + 0.932914i \(0.617257\pi\)
\(824\) 3.52258e52 0.0400136
\(825\) 1.16646e54 1.29560
\(826\) 1.25744e54 1.36571
\(827\) −9.55114e53 −1.01439 −0.507193 0.861832i \(-0.669317\pi\)
−0.507193 + 0.861832i \(0.669317\pi\)
\(828\) 1.02481e53 0.106434
\(829\) −9.44288e53 −0.959060 −0.479530 0.877525i \(-0.659193\pi\)
−0.479530 + 0.877525i \(0.659193\pi\)
\(830\) −6.91806e52 −0.0687131
\(831\) 8.64174e53 0.839426
\(832\) −5.53833e52 −0.0526135
\(833\) −1.56914e54 −1.45790
\(834\) 2.95562e53 0.268582
\(835\) −7.96054e52 −0.0707527
\(836\) −1.20159e54 −1.04458
\(837\) −9.41223e53 −0.800334
\(838\) −7.95840e53 −0.661929
\(839\) 3.51965e53 0.286354 0.143177 0.989697i \(-0.454268\pi\)
0.143177 + 0.989697i \(0.454268\pi\)
\(840\) −2.97763e53 −0.236975
\(841\) 7.22851e53 0.562760
\(842\) 1.51699e54 1.15534
\(843\) −1.56223e54 −1.16395
\(844\) −1.29742e53 −0.0945686
\(845\) 4.70721e53 0.335672
\(846\) 3.21676e53 0.224423
\(847\) 2.16610e54 1.47855
\(848\) 8.05949e52 0.0538250
\(849\) −2.05548e53 −0.134314
\(850\) −1.09704e54 −0.701408
\(851\) −4.72901e53 −0.295851
\(852\) 3.81422e53 0.233493
\(853\) −2.00062e53 −0.119841 −0.0599207 0.998203i \(-0.519085\pi\)
−0.0599207 + 0.998203i \(0.519085\pi\)
\(854\) 4.46385e53 0.261661
\(855\) 2.24484e53 0.128769
\(856\) −3.78237e53 −0.212324
\(857\) −3.05443e54 −1.67797 −0.838987 0.544152i \(-0.816851\pi\)
−0.838987 + 0.544152i \(0.816851\pi\)
\(858\) −8.60419e53 −0.462589
\(859\) 2.43101e54 1.27912 0.639562 0.768740i \(-0.279116\pi\)
0.639562 + 0.768740i \(0.279116\pi\)
\(860\) 6.87757e53 0.354172
\(861\) 3.77674e53 0.190353
\(862\) −2.07891e53 −0.102554
\(863\) 3.92924e54 1.89718 0.948591 0.316506i \(-0.102510\pi\)
0.948591 + 0.316506i \(0.102510\pi\)
\(864\) −3.24159e53 −0.153198
\(865\) 8.58851e53 0.397301
\(866\) −5.96080e53 −0.269913
\(867\) −1.03384e54 −0.458249
\(868\) −1.58748e54 −0.688799
\(869\) 3.18112e54 1.35118
\(870\) −9.55224e53 −0.397191
\(871\) −3.10945e53 −0.126575
\(872\) −1.73978e52 −0.00693330
\(873\) −8.34700e53 −0.325662
\(874\) 2.73638e54 1.04524
\(875\) 2.98353e54 1.11579
\(876\) 2.12877e54 0.779476
\(877\) −5.07395e54 −1.81909 −0.909544 0.415607i \(-0.863569\pi\)
−0.909544 + 0.415607i \(0.863569\pi\)
\(878\) −2.75705e54 −0.967823
\(879\) 2.40074e54 0.825185
\(880\) −4.27589e53 −0.143912
\(881\) −2.84352e54 −0.937133 −0.468567 0.883428i \(-0.655229\pi\)
−0.468567 + 0.883428i \(0.655229\pi\)
\(882\) −5.72322e53 −0.184702
\(883\) −5.21769e54 −1.64894 −0.824471 0.565904i \(-0.808527\pi\)
−0.824471 + 0.565904i \(0.808527\pi\)
\(884\) 8.09211e53 0.250435
\(885\) −1.91960e54 −0.581782
\(886\) 5.15160e53 0.152904
\(887\) −3.57280e54 −1.03854 −0.519269 0.854611i \(-0.673796\pi\)
−0.519269 + 0.854611i \(0.673796\pi\)
\(888\) −4.05340e53 −0.115393
\(889\) 1.25775e54 0.350679
\(890\) 2.12107e53 0.0579215
\(891\) −6.16088e54 −1.64780
\(892\) 2.95270e54 0.773515
\(893\) 8.58922e54 2.20395
\(894\) 5.09707e54 1.28108
\(895\) −9.73217e53 −0.239598
\(896\) −5.46731e53 −0.131849
\(897\) 1.95943e54 0.462881
\(898\) −7.22614e53 −0.167222
\(899\) −5.09264e54 −1.15449
\(900\) −4.00128e53 −0.0888614
\(901\) −1.17758e54 −0.256202
\(902\) 5.42343e53 0.115599
\(903\) 1.36620e55 2.85292
\(904\) 1.58598e54 0.324476
\(905\) −2.23312e54 −0.447624
\(906\) −5.15699e54 −1.01280
\(907\) 3.79043e54 0.729380 0.364690 0.931129i \(-0.381175\pi\)
0.364690 + 0.931129i \(0.381175\pi\)
\(908\) −3.01282e54 −0.568049
\(909\) 2.91628e53 0.0538763
\(910\) −1.00051e54 −0.181115
\(911\) 5.53419e54 0.981667 0.490833 0.871253i \(-0.336692\pi\)
0.490833 + 0.871253i \(0.336692\pi\)
\(912\) 2.34545e54 0.407682
\(913\) −1.97342e54 −0.336132
\(914\) −2.23977e54 −0.373851
\(915\) −6.81448e53 −0.111466
\(916\) 2.96858e54 0.475864
\(917\) −1.44210e54 −0.226549
\(918\) 4.73632e54 0.729209
\(919\) −9.55457e54 −1.44170 −0.720849 0.693092i \(-0.756248\pi\)
−0.720849 + 0.693092i \(0.756248\pi\)
\(920\) 9.73747e53 0.144003
\(921\) 7.28060e53 0.105527
\(922\) −9.31487e54 −1.32329
\(923\) 1.28161e54 0.178453
\(924\) −8.49385e54 −1.15924
\(925\) 1.84641e54 0.247005
\(926\) 5.93104e54 0.777728
\(927\) −1.87725e53 −0.0241295
\(928\) −1.75392e54 −0.220989
\(929\) −2.71937e54 −0.335875 −0.167937 0.985798i \(-0.553711\pi\)
−0.167937 + 0.985798i \(0.553711\pi\)
\(930\) 2.42343e54 0.293424
\(931\) −1.52818e55 −1.81387
\(932\) −5.23209e54 −0.608808
\(933\) 3.73626e54 0.426213
\(934\) −3.91130e54 −0.437425
\(935\) 6.24755e54 0.685007
\(936\) 2.95149e53 0.0317276
\(937\) 9.91981e54 1.04549 0.522746 0.852488i \(-0.324907\pi\)
0.522746 + 0.852488i \(0.324907\pi\)
\(938\) −3.06957e54 −0.317194
\(939\) 4.92811e54 0.499306
\(940\) 3.05649e54 0.303639
\(941\) 1.26474e55 1.23195 0.615976 0.787765i \(-0.288762\pi\)
0.615976 + 0.787765i \(0.288762\pi\)
\(942\) 1.24184e55 1.18611
\(943\) −1.23508e54 −0.115672
\(944\) −3.52463e54 −0.323692
\(945\) −5.85597e54 −0.527364
\(946\) 1.96187e55 1.73254
\(947\) −1.03728e55 −0.898297 −0.449148 0.893457i \(-0.648273\pi\)
−0.449148 + 0.893457i \(0.648273\pi\)
\(948\) −6.20940e54 −0.527346
\(949\) 7.15283e54 0.595735
\(950\) −1.06840e55 −0.872666
\(951\) −1.49888e55 −1.20068
\(952\) 7.98834e54 0.627585
\(953\) 1.87492e55 1.44466 0.722329 0.691550i \(-0.243072\pi\)
0.722329 + 0.691550i \(0.243072\pi\)
\(954\) −4.29506e53 −0.0324582
\(955\) −3.81256e54 −0.282588
\(956\) 1.03383e55 0.751586
\(957\) −2.72483e55 −1.94298
\(958\) −5.19682e54 −0.363476
\(959\) −2.86488e54 −0.196545
\(960\) 8.34634e53 0.0561666
\(961\) −2.22882e54 −0.147127
\(962\) −1.36198e54 −0.0881923
\(963\) 2.01570e54 0.128038
\(964\) 4.54853e54 0.283430
\(965\) 1.01056e55 0.617739
\(966\) 1.93430e55 1.15997
\(967\) 2.38613e55 1.40380 0.701900 0.712276i \(-0.252335\pi\)
0.701900 + 0.712276i \(0.252335\pi\)
\(968\) −6.07162e54 −0.350438
\(969\) −3.42696e55 −1.94053
\(970\) −7.93113e54 −0.440614
\(971\) −1.05062e55 −0.572650 −0.286325 0.958133i \(-0.592434\pi\)
−0.286325 + 0.958133i \(0.592434\pi\)
\(972\) 3.92313e54 0.209801
\(973\) 9.80377e54 0.514406
\(974\) −2.04469e55 −1.05265
\(975\) −7.65046e54 −0.386458
\(976\) −1.25123e54 −0.0620175
\(977\) 2.06489e55 1.00426 0.502131 0.864792i \(-0.332549\pi\)
0.502131 + 0.864792i \(0.332549\pi\)
\(978\) −2.99994e55 −1.43167
\(979\) 6.05047e54 0.283341
\(980\) −5.43807e54 −0.249898
\(981\) 9.27164e52 0.00418100
\(982\) −1.62897e55 −0.720859
\(983\) −1.98554e55 −0.862259 −0.431130 0.902290i \(-0.641885\pi\)
−0.431130 + 0.902290i \(0.641885\pi\)
\(984\) −1.05863e54 −0.0451164
\(985\) −3.68669e54 −0.154193
\(986\) 2.56266e55 1.05189
\(987\) 6.07157e55 2.44587
\(988\) 7.88090e54 0.311582
\(989\) −4.46775e55 −1.73364
\(990\) 2.27871e54 0.0867835
\(991\) 3.30495e55 1.23539 0.617693 0.786419i \(-0.288067\pi\)
0.617693 + 0.786419i \(0.288067\pi\)
\(992\) 4.44973e54 0.163255
\(993\) 4.72192e55 1.70042
\(994\) 1.26517e55 0.447200
\(995\) 9.97585e54 0.346117
\(996\) 3.85202e54 0.131187
\(997\) −2.73187e55 −0.913269 −0.456635 0.889654i \(-0.650945\pi\)
−0.456635 + 0.889654i \(0.650945\pi\)
\(998\) 2.66639e53 0.00875000
\(999\) −7.97166e54 −0.256795
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 2.38.a.a.1.1 2
3.2 odd 2 18.38.a.f.1.1 2
4.3 odd 2 16.38.a.a.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.38.a.a.1.1 2 1.1 even 1 trivial
16.38.a.a.1.2 2 4.3 odd 2
18.38.a.f.1.1 2 3.2 odd 2