Properties

Label 50.26.a.l.1.6
Level $50$
Weight $26$
Character 50.1
Self dual yes
Analytic conductor $197.998$
Analytic rank $1$
Dimension $6$
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] = [50,26,Mod(1,50)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(50, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 26, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("50.1");
 
S:= CuspForms(chi, 26);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 50.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: \(197.998389976\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2 x^{5} - 27590779188 x^{4} + 26487255863952 x^{3} + \cdots - 30\!\cdots\!08 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{20}\cdot 3^{4}\cdot 5^{18} \)
Twist minimal: no (minimal twist has level 10)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-122054.\) of defining polynomial
Character \(\chi\) \(=\) 50.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4096.00 q^{2} +1.08697e6 q^{3} +1.67772e7 q^{4} +4.45223e9 q^{6} -4.89841e10 q^{7} +6.87195e10 q^{8} +3.34213e11 q^{9} +1.14584e13 q^{11} +1.82363e13 q^{12} +1.28077e14 q^{13} -2.00639e14 q^{14} +2.81475e14 q^{16} -3.78822e15 q^{17} +1.36894e15 q^{18} -1.02229e16 q^{19} -5.32442e16 q^{21} +4.69335e16 q^{22} -1.81999e17 q^{23} +7.46959e16 q^{24} +5.24604e17 q^{26} -5.57697e17 q^{27} -8.21816e17 q^{28} +2.19149e18 q^{29} +2.89689e18 q^{31} +1.15292e18 q^{32} +1.24549e19 q^{33} -1.55166e19 q^{34} +5.60717e18 q^{36} -4.56681e19 q^{37} -4.18730e19 q^{38} +1.39216e20 q^{39} -4.22120e18 q^{41} -2.18088e20 q^{42} +1.22773e19 q^{43} +1.92240e20 q^{44} -7.45466e20 q^{46} -4.60132e20 q^{47} +3.05955e20 q^{48} +1.05837e21 q^{49} -4.11768e21 q^{51} +2.14878e21 q^{52} -2.54574e21 q^{53} -2.28433e21 q^{54} -3.36616e21 q^{56} -1.11120e22 q^{57} +8.97634e21 q^{58} -2.20173e21 q^{59} +7.41676e21 q^{61} +1.18656e22 q^{62} -1.63711e22 q^{63} +4.72237e21 q^{64} +5.10153e22 q^{66} +6.01134e22 q^{67} -6.35558e22 q^{68} -1.97827e23 q^{69} +3.59036e22 q^{71} +2.29670e22 q^{72} -1.98356e23 q^{73} -1.87057e23 q^{74} -1.71512e23 q^{76} -5.61278e23 q^{77} +5.70229e23 q^{78} -7.84538e23 q^{79} -8.89375e23 q^{81} -1.72901e22 q^{82} -4.60438e23 q^{83} -8.93289e23 q^{84} +5.02877e22 q^{86} +2.38208e24 q^{87} +7.87413e23 q^{88} +5.94026e23 q^{89} -6.27374e24 q^{91} -3.05343e24 q^{92} +3.14882e24 q^{93} -1.88470e24 q^{94} +1.25319e24 q^{96} +8.71039e24 q^{97} +4.33508e24 q^{98} +3.82954e24 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 24576 q^{2} - 801416 q^{3} + 100663296 q^{4} - 3282599936 q^{6} - 34007705352 q^{7} + 412316860416 q^{8} + 541468782118 q^{9} + 9861544614312 q^{11} - 13445529337856 q^{12} - 30787386783696 q^{13}+ \cdots - 20\!\cdots\!64 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4096.00 0.707107
\(3\) 1.08697e6 1.18087 0.590434 0.807086i \(-0.298957\pi\)
0.590434 + 0.807086i \(0.298957\pi\)
\(4\) 1.67772e7 0.500000
\(5\) 0 0
\(6\) 4.45223e9 0.835000
\(7\) −4.89841e10 −1.33761 −0.668805 0.743438i \(-0.733194\pi\)
−0.668805 + 0.743438i \(0.733194\pi\)
\(8\) 6.87195e10 0.353553
\(9\) 3.34213e11 0.394450
\(10\) 0 0
\(11\) 1.14584e13 1.10082 0.550408 0.834896i \(-0.314472\pi\)
0.550408 + 0.834896i \(0.314472\pi\)
\(12\) 1.82363e13 0.590434
\(13\) 1.28077e14 1.52468 0.762342 0.647174i \(-0.224049\pi\)
0.762342 + 0.647174i \(0.224049\pi\)
\(14\) −2.00639e14 −0.945833
\(15\) 0 0
\(16\) 2.81475e14 0.250000
\(17\) −3.78822e15 −1.57697 −0.788486 0.615053i \(-0.789134\pi\)
−0.788486 + 0.615053i \(0.789134\pi\)
\(18\) 1.36894e15 0.278918
\(19\) −1.02229e16 −1.05963 −0.529815 0.848113i \(-0.677739\pi\)
−0.529815 + 0.848113i \(0.677739\pi\)
\(20\) 0 0
\(21\) −5.32442e16 −1.57954
\(22\) 4.69335e16 0.778394
\(23\) −1.81999e17 −1.73169 −0.865845 0.500312i \(-0.833218\pi\)
−0.865845 + 0.500312i \(0.833218\pi\)
\(24\) 7.46959e16 0.417500
\(25\) 0 0
\(26\) 5.24604e17 1.07811
\(27\) −5.57697e17 −0.715075
\(28\) −8.21816e17 −0.668805
\(29\) 2.19149e18 1.15018 0.575088 0.818092i \(-0.304968\pi\)
0.575088 + 0.818092i \(0.304968\pi\)
\(30\) 0 0
\(31\) 2.89689e18 0.660557 0.330278 0.943884i \(-0.392857\pi\)
0.330278 + 0.943884i \(0.392857\pi\)
\(32\) 1.15292e18 0.176777
\(33\) 1.24549e19 1.29992
\(34\) −1.55166e19 −1.11509
\(35\) 0 0
\(36\) 5.60717e18 0.197225
\(37\) −4.56681e19 −1.14049 −0.570245 0.821475i \(-0.693152\pi\)
−0.570245 + 0.821475i \(0.693152\pi\)
\(38\) −4.18730e19 −0.749271
\(39\) 1.39216e20 1.80045
\(40\) 0 0
\(41\) −4.22120e18 −0.0292172 −0.0146086 0.999893i \(-0.504650\pi\)
−0.0146086 + 0.999893i \(0.504650\pi\)
\(42\) −2.18088e20 −1.11690
\(43\) 1.22773e19 0.0468540 0.0234270 0.999726i \(-0.492542\pi\)
0.0234270 + 0.999726i \(0.492542\pi\)
\(44\) 1.92240e20 0.550408
\(45\) 0 0
\(46\) −7.45466e20 −1.22449
\(47\) −4.60132e20 −0.577642 −0.288821 0.957383i \(-0.593263\pi\)
−0.288821 + 0.957383i \(0.593263\pi\)
\(48\) 3.05955e20 0.295217
\(49\) 1.05837e21 0.789199
\(50\) 0 0
\(51\) −4.11768e21 −1.86220
\(52\) 2.14878e21 0.762342
\(53\) −2.54574e21 −0.711814 −0.355907 0.934521i \(-0.615828\pi\)
−0.355907 + 0.934521i \(0.615828\pi\)
\(54\) −2.28433e21 −0.505634
\(55\) 0 0
\(56\) −3.36616e21 −0.472916
\(57\) −1.11120e22 −1.25128
\(58\) 8.97634e21 0.813297
\(59\) −2.20173e21 −0.161107 −0.0805536 0.996750i \(-0.525669\pi\)
−0.0805536 + 0.996750i \(0.525669\pi\)
\(60\) 0 0
\(61\) 7.41676e21 0.357760 0.178880 0.983871i \(-0.442753\pi\)
0.178880 + 0.983871i \(0.442753\pi\)
\(62\) 1.18656e22 0.467084
\(63\) −1.63711e22 −0.527620
\(64\) 4.72237e21 0.125000
\(65\) 0 0
\(66\) 5.10153e22 0.919181
\(67\) 6.01134e22 0.897503 0.448752 0.893656i \(-0.351869\pi\)
0.448752 + 0.893656i \(0.351869\pi\)
\(68\) −6.35558e22 −0.788486
\(69\) −1.97827e23 −2.04490
\(70\) 0 0
\(71\) 3.59036e22 0.259662 0.129831 0.991536i \(-0.458557\pi\)
0.129831 + 0.991536i \(0.458557\pi\)
\(72\) 2.29670e22 0.139459
\(73\) −1.98356e23 −1.01370 −0.506851 0.862034i \(-0.669190\pi\)
−0.506851 + 0.862034i \(0.669190\pi\)
\(74\) −1.87057e23 −0.806448
\(75\) 0 0
\(76\) −1.71512e23 −0.529815
\(77\) −5.61278e23 −1.47246
\(78\) 5.70229e23 1.27311
\(79\) −7.84538e23 −1.49374 −0.746870 0.664970i \(-0.768444\pi\)
−0.746870 + 0.664970i \(0.768444\pi\)
\(80\) 0 0
\(81\) −8.89375e23 −1.23886
\(82\) −1.72901e22 −0.0206597
\(83\) −4.60438e23 −0.472819 −0.236409 0.971654i \(-0.575971\pi\)
−0.236409 + 0.971654i \(0.575971\pi\)
\(84\) −8.93289e23 −0.789770
\(85\) 0 0
\(86\) 5.02877e22 0.0331308
\(87\) 2.38208e24 1.35821
\(88\) 7.87413e23 0.389197
\(89\) 5.94026e23 0.254936 0.127468 0.991843i \(-0.459315\pi\)
0.127468 + 0.991843i \(0.459315\pi\)
\(90\) 0 0
\(91\) −6.27374e24 −2.03943
\(92\) −3.05343e24 −0.865845
\(93\) 3.14882e24 0.780031
\(94\) −1.88470e24 −0.408455
\(95\) 0 0
\(96\) 1.25319e24 0.208750
\(97\) 8.71039e24 1.27465 0.637325 0.770595i \(-0.280041\pi\)
0.637325 + 0.770595i \(0.280041\pi\)
\(98\) 4.33508e24 0.558048
\(99\) 3.82954e24 0.434217
\(100\) 0 0
\(101\) 7.81456e24 0.690061 0.345031 0.938591i \(-0.387869\pi\)
0.345031 + 0.938591i \(0.387869\pi\)
\(102\) −1.68660e25 −1.31677
\(103\) −1.05677e25 −0.730321 −0.365161 0.930945i \(-0.618986\pi\)
−0.365161 + 0.930945i \(0.618986\pi\)
\(104\) 8.80140e24 0.539057
\(105\) 0 0
\(106\) −1.04274e25 −0.503328
\(107\) 4.46639e24 0.191716 0.0958582 0.995395i \(-0.469440\pi\)
0.0958582 + 0.995395i \(0.469440\pi\)
\(108\) −9.35660e24 −0.357537
\(109\) −4.49825e25 −1.53184 −0.765918 0.642939i \(-0.777715\pi\)
−0.765918 + 0.642939i \(0.777715\pi\)
\(110\) 0 0
\(111\) −4.96398e25 −1.34677
\(112\) −1.37878e25 −0.334402
\(113\) −2.79039e25 −0.605598 −0.302799 0.953054i \(-0.597921\pi\)
−0.302799 + 0.953054i \(0.597921\pi\)
\(114\) −4.55146e25 −0.884791
\(115\) 0 0
\(116\) 3.67671e25 0.575088
\(117\) 4.28051e25 0.601412
\(118\) −9.01830e24 −0.113920
\(119\) 1.85562e26 2.10937
\(120\) 0 0
\(121\) 2.29472e25 0.211794
\(122\) 3.03790e25 0.252974
\(123\) −4.58832e24 −0.0345016
\(124\) 4.86017e25 0.330278
\(125\) 0 0
\(126\) −6.70561e25 −0.373084
\(127\) −1.91643e26 −0.965930 −0.482965 0.875640i \(-0.660440\pi\)
−0.482965 + 0.875640i \(0.660440\pi\)
\(128\) 1.93428e25 0.0883883
\(129\) 1.33450e25 0.0553284
\(130\) 0 0
\(131\) −2.22743e26 −0.761925 −0.380963 0.924590i \(-0.624407\pi\)
−0.380963 + 0.924590i \(0.624407\pi\)
\(132\) 2.08958e26 0.649959
\(133\) 5.00759e26 1.41737
\(134\) 2.46224e26 0.634631
\(135\) 0 0
\(136\) −2.60325e26 −0.557543
\(137\) −3.59939e26 −0.703432 −0.351716 0.936107i \(-0.614402\pi\)
−0.351716 + 0.936107i \(0.614402\pi\)
\(138\) −8.10299e26 −1.44596
\(139\) −5.18148e26 −0.844829 −0.422414 0.906403i \(-0.638817\pi\)
−0.422414 + 0.906403i \(0.638817\pi\)
\(140\) 0 0
\(141\) −5.00149e26 −0.682119
\(142\) 1.47061e26 0.183609
\(143\) 1.46756e27 1.67840
\(144\) 9.40727e25 0.0986126
\(145\) 0 0
\(146\) −8.12467e26 −0.716795
\(147\) 1.15042e27 0.931940
\(148\) −7.66184e26 −0.570245
\(149\) −1.91990e27 −1.31356 −0.656780 0.754082i \(-0.728082\pi\)
−0.656780 + 0.754082i \(0.728082\pi\)
\(150\) 0 0
\(151\) −1.20411e27 −0.697356 −0.348678 0.937243i \(-0.613369\pi\)
−0.348678 + 0.937243i \(0.613369\pi\)
\(152\) −7.02512e26 −0.374636
\(153\) −1.26607e27 −0.622037
\(154\) −2.29899e27 −1.04119
\(155\) 0 0
\(156\) 2.33566e27 0.900226
\(157\) −4.32983e27 −1.54073 −0.770363 0.637606i \(-0.779925\pi\)
−0.770363 + 0.637606i \(0.779925\pi\)
\(158\) −3.21347e27 −1.05623
\(159\) −2.76715e27 −0.840559
\(160\) 0 0
\(161\) 8.91503e27 2.31632
\(162\) −3.64288e27 −0.876006
\(163\) 5.02220e27 1.11828 0.559138 0.829075i \(-0.311132\pi\)
0.559138 + 0.829075i \(0.311132\pi\)
\(164\) −7.08201e25 −0.0146086
\(165\) 0 0
\(166\) −1.88595e27 −0.334333
\(167\) 1.18728e28 1.95253 0.976265 0.216580i \(-0.0694903\pi\)
0.976265 + 0.216580i \(0.0694903\pi\)
\(168\) −3.65891e27 −0.558452
\(169\) 9.34736e27 1.32466
\(170\) 0 0
\(171\) −3.41663e27 −0.417971
\(172\) 2.05979e26 0.0234270
\(173\) −4.16222e27 −0.440300 −0.220150 0.975466i \(-0.570655\pi\)
−0.220150 + 0.975466i \(0.570655\pi\)
\(174\) 9.75700e27 0.960397
\(175\) 0 0
\(176\) 3.22524e27 0.275204
\(177\) −2.39322e27 −0.190246
\(178\) 2.43313e27 0.180267
\(179\) 1.09350e28 0.755363 0.377681 0.925936i \(-0.376721\pi\)
0.377681 + 0.925936i \(0.376721\pi\)
\(180\) 0 0
\(181\) −8.64913e27 −0.519984 −0.259992 0.965611i \(-0.583720\pi\)
−0.259992 + 0.965611i \(0.583720\pi\)
\(182\) −2.56972e28 −1.44210
\(183\) 8.06179e27 0.422467
\(184\) −1.25068e28 −0.612245
\(185\) 0 0
\(186\) 1.28976e28 0.551565
\(187\) −4.34068e28 −1.73595
\(188\) −7.71973e27 −0.288821
\(189\) 2.73183e28 0.956490
\(190\) 0 0
\(191\) 1.76410e28 0.541511 0.270755 0.962648i \(-0.412727\pi\)
0.270755 + 0.962648i \(0.412727\pi\)
\(192\) 5.13307e27 0.147609
\(193\) −5.06807e27 −0.136577 −0.0682883 0.997666i \(-0.521754\pi\)
−0.0682883 + 0.997666i \(0.521754\pi\)
\(194\) 3.56778e28 0.901314
\(195\) 0 0
\(196\) 1.77565e28 0.394599
\(197\) −5.65172e27 −0.117856 −0.0589282 0.998262i \(-0.518768\pi\)
−0.0589282 + 0.998262i \(0.518768\pi\)
\(198\) 1.56858e28 0.307038
\(199\) 2.19231e28 0.402938 0.201469 0.979495i \(-0.435428\pi\)
0.201469 + 0.979495i \(0.435428\pi\)
\(200\) 0 0
\(201\) 6.53414e28 1.05983
\(202\) 3.20085e28 0.487947
\(203\) −1.07348e29 −1.53849
\(204\) −6.90832e28 −0.931098
\(205\) 0 0
\(206\) −4.32852e28 −0.516415
\(207\) −6.08263e28 −0.683066
\(208\) 3.60505e28 0.381171
\(209\) −1.17138e29 −1.16646
\(210\) 0 0
\(211\) 9.10994e28 0.805350 0.402675 0.915343i \(-0.368080\pi\)
0.402675 + 0.915343i \(0.368080\pi\)
\(212\) −4.27105e28 −0.355907
\(213\) 3.90261e28 0.306627
\(214\) 1.82943e28 0.135564
\(215\) 0 0
\(216\) −3.83247e28 −0.252817
\(217\) −1.41901e29 −0.883567
\(218\) −1.84248e29 −1.08317
\(219\) −2.15607e29 −1.19705
\(220\) 0 0
\(221\) −4.85185e29 −2.40438
\(222\) −2.03325e29 −0.952309
\(223\) 1.06006e29 0.469374 0.234687 0.972071i \(-0.424594\pi\)
0.234687 + 0.972071i \(0.424594\pi\)
\(224\) −5.64748e28 −0.236458
\(225\) 0 0
\(226\) −1.14294e29 −0.428223
\(227\) −5.01045e29 −1.77645 −0.888226 0.459407i \(-0.848062\pi\)
−0.888226 + 0.459407i \(0.848062\pi\)
\(228\) −1.86428e29 −0.625641
\(229\) −3.20108e28 −0.101708 −0.0508538 0.998706i \(-0.516194\pi\)
−0.0508538 + 0.998706i \(0.516194\pi\)
\(230\) 0 0
\(231\) −6.10091e29 −1.73878
\(232\) 1.50598e29 0.406648
\(233\) −6.19627e29 −1.58556 −0.792778 0.609510i \(-0.791366\pi\)
−0.792778 + 0.609510i \(0.791366\pi\)
\(234\) 1.75330e29 0.425263
\(235\) 0 0
\(236\) −3.69390e28 −0.0805536
\(237\) −8.52768e29 −1.76391
\(238\) 7.60064e29 1.49155
\(239\) 2.16736e29 0.403605 0.201803 0.979426i \(-0.435320\pi\)
0.201803 + 0.979426i \(0.435320\pi\)
\(240\) 0 0
\(241\) 7.46346e29 1.25236 0.626178 0.779681i \(-0.284619\pi\)
0.626178 + 0.779681i \(0.284619\pi\)
\(242\) 9.39918e28 0.149761
\(243\) −4.94192e29 −0.747855
\(244\) 1.24433e29 0.178880
\(245\) 0 0
\(246\) −1.87938e28 −0.0243963
\(247\) −1.30932e30 −1.61560
\(248\) 1.99072e29 0.233542
\(249\) −5.00481e29 −0.558337
\(250\) 0 0
\(251\) 7.46681e29 0.753726 0.376863 0.926269i \(-0.377003\pi\)
0.376863 + 0.926269i \(0.377003\pi\)
\(252\) −2.74662e29 −0.263810
\(253\) −2.08541e30 −1.90627
\(254\) −7.84968e29 −0.683016
\(255\) 0 0
\(256\) 7.92282e28 0.0625000
\(257\) −2.03309e29 −0.152754 −0.0763769 0.997079i \(-0.524335\pi\)
−0.0763769 + 0.997079i \(0.524335\pi\)
\(258\) 5.46612e28 0.0391231
\(259\) 2.23701e30 1.52553
\(260\) 0 0
\(261\) 7.32424e29 0.453687
\(262\) −9.12354e29 −0.538763
\(263\) 3.18718e30 1.79457 0.897284 0.441454i \(-0.145537\pi\)
0.897284 + 0.441454i \(0.145537\pi\)
\(264\) 8.55894e29 0.459590
\(265\) 0 0
\(266\) 2.05111e30 1.00223
\(267\) 6.45688e29 0.301046
\(268\) 1.00854e30 0.448752
\(269\) 1.08398e29 0.0460382 0.0230191 0.999735i \(-0.492672\pi\)
0.0230191 + 0.999735i \(0.492672\pi\)
\(270\) 0 0
\(271\) 8.40525e29 0.325413 0.162706 0.986675i \(-0.447978\pi\)
0.162706 + 0.986675i \(0.447978\pi\)
\(272\) −1.06629e30 −0.394243
\(273\) −6.81936e30 −2.40830
\(274\) −1.47431e30 −0.497402
\(275\) 0 0
\(276\) −3.31898e30 −1.02245
\(277\) −5.28289e30 −1.55551 −0.777757 0.628564i \(-0.783643\pi\)
−0.777757 + 0.628564i \(0.783643\pi\)
\(278\) −2.12233e30 −0.597384
\(279\) 9.68177e29 0.260557
\(280\) 0 0
\(281\) −5.89709e30 −1.45147 −0.725736 0.687973i \(-0.758501\pi\)
−0.725736 + 0.687973i \(0.758501\pi\)
\(282\) −2.04861e30 −0.482331
\(283\) 1.58325e30 0.356631 0.178315 0.983973i \(-0.442935\pi\)
0.178315 + 0.983973i \(0.442935\pi\)
\(284\) 6.02362e29 0.129831
\(285\) 0 0
\(286\) 6.01111e30 1.18680
\(287\) 2.06772e29 0.0390812
\(288\) 3.85322e29 0.0697296
\(289\) 8.57999e30 1.48684
\(290\) 0 0
\(291\) 9.46793e30 1.50519
\(292\) −3.32787e30 −0.506851
\(293\) 4.96361e30 0.724357 0.362179 0.932109i \(-0.382033\pi\)
0.362179 + 0.932109i \(0.382033\pi\)
\(294\) 4.71210e30 0.658981
\(295\) 0 0
\(296\) −3.13829e30 −0.403224
\(297\) −6.39030e30 −0.787165
\(298\) −7.86391e30 −0.928828
\(299\) −2.33099e31 −2.64028
\(300\) 0 0
\(301\) −6.01391e29 −0.0626723
\(302\) −4.93203e30 −0.493105
\(303\) 8.49419e30 0.814871
\(304\) −2.87749e30 −0.264907
\(305\) 0 0
\(306\) −5.18584e30 −0.439846
\(307\) 1.29347e31 1.05323 0.526617 0.850102i \(-0.323460\pi\)
0.526617 + 0.850102i \(0.323460\pi\)
\(308\) −9.41668e30 −0.736230
\(309\) −1.14867e31 −0.862413
\(310\) 0 0
\(311\) −1.68637e30 −0.116801 −0.0584007 0.998293i \(-0.518600\pi\)
−0.0584007 + 0.998293i \(0.518600\pi\)
\(312\) 9.56685e30 0.636556
\(313\) 2.70375e30 0.172847 0.0864236 0.996258i \(-0.472456\pi\)
0.0864236 + 0.996258i \(0.472456\pi\)
\(314\) −1.77350e31 −1.08946
\(315\) 0 0
\(316\) −1.31624e31 −0.746870
\(317\) 3.29106e31 1.79513 0.897563 0.440887i \(-0.145336\pi\)
0.897563 + 0.440887i \(0.145336\pi\)
\(318\) −1.13342e31 −0.594365
\(319\) 2.51109e31 1.26613
\(320\) 0 0
\(321\) 4.85483e30 0.226392
\(322\) 3.65160e31 1.63789
\(323\) 3.87266e31 1.67100
\(324\) −1.49212e31 −0.619430
\(325\) 0 0
\(326\) 2.05709e31 0.790740
\(327\) −4.88946e31 −1.80890
\(328\) −2.90079e29 −0.0103298
\(329\) 2.25391e31 0.772659
\(330\) 0 0
\(331\) −3.10367e31 −0.986337 −0.493168 0.869934i \(-0.664161\pi\)
−0.493168 + 0.869934i \(0.664161\pi\)
\(332\) −7.72486e30 −0.236409
\(333\) −1.52629e31 −0.449867
\(334\) 4.86310e31 1.38065
\(335\) 0 0
\(336\) −1.49869e31 −0.394885
\(337\) 3.19922e31 0.812214 0.406107 0.913825i \(-0.366886\pi\)
0.406107 + 0.913825i \(0.366886\pi\)
\(338\) 3.82868e31 0.936678
\(339\) −3.03307e31 −0.715132
\(340\) 0 0
\(341\) 3.31936e31 0.727151
\(342\) −1.39945e31 −0.295550
\(343\) 1.38477e31 0.281970
\(344\) 8.43688e29 0.0165654
\(345\) 0 0
\(346\) −1.70484e31 −0.311339
\(347\) 4.34215e31 0.764867 0.382434 0.923983i \(-0.375086\pi\)
0.382434 + 0.923983i \(0.375086\pi\)
\(348\) 3.99647e31 0.679103
\(349\) 9.72186e31 1.59379 0.796896 0.604117i \(-0.206474\pi\)
0.796896 + 0.604117i \(0.206474\pi\)
\(350\) 0 0
\(351\) −7.14283e31 −1.09026
\(352\) 1.32106e31 0.194598
\(353\) −1.03434e32 −1.47055 −0.735277 0.677767i \(-0.762948\pi\)
−0.735277 + 0.677767i \(0.762948\pi\)
\(354\) −9.80262e30 −0.134524
\(355\) 0 0
\(356\) 9.96611e30 0.127468
\(357\) 2.01701e32 2.49089
\(358\) 4.47897e31 0.534122
\(359\) 4.33845e31 0.499636 0.249818 0.968293i \(-0.419629\pi\)
0.249818 + 0.968293i \(0.419629\pi\)
\(360\) 0 0
\(361\) 1.14311e31 0.122814
\(362\) −3.54268e31 −0.367684
\(363\) 2.49429e31 0.250100
\(364\) −1.05256e32 −1.01972
\(365\) 0 0
\(366\) 3.30211e31 0.298729
\(367\) −4.71466e31 −0.412216 −0.206108 0.978529i \(-0.566080\pi\)
−0.206108 + 0.978529i \(0.566080\pi\)
\(368\) −5.12281e31 −0.432922
\(369\) −1.41078e30 −0.0115247
\(370\) 0 0
\(371\) 1.24701e32 0.952129
\(372\) 5.28285e31 0.390015
\(373\) 9.92589e31 0.708613 0.354306 0.935129i \(-0.384717\pi\)
0.354306 + 0.935129i \(0.384717\pi\)
\(374\) −1.77794e32 −1.22750
\(375\) 0 0
\(376\) −3.16200e31 −0.204227
\(377\) 2.80680e32 1.75365
\(378\) 1.11896e32 0.676341
\(379\) −6.28833e31 −0.367743 −0.183872 0.982950i \(-0.558863\pi\)
−0.183872 + 0.982950i \(0.558863\pi\)
\(380\) 0 0
\(381\) −2.08310e32 −1.14064
\(382\) 7.22577e31 0.382906
\(383\) −4.95371e31 −0.254065 −0.127033 0.991899i \(-0.540545\pi\)
−0.127033 + 0.991899i \(0.540545\pi\)
\(384\) 2.10250e31 0.104375
\(385\) 0 0
\(386\) −2.07588e31 −0.0965742
\(387\) 4.10323e30 0.0184816
\(388\) 1.46136e32 0.637325
\(389\) −4.10071e32 −1.73177 −0.865883 0.500246i \(-0.833243\pi\)
−0.865883 + 0.500246i \(0.833243\pi\)
\(390\) 0 0
\(391\) 6.89451e32 2.73082
\(392\) 7.27306e31 0.279024
\(393\) −2.42114e32 −0.899734
\(394\) −2.31495e31 −0.0833370
\(395\) 0 0
\(396\) 6.42490e31 0.217108
\(397\) −1.81115e32 −0.593026 −0.296513 0.955029i \(-0.595824\pi\)
−0.296513 + 0.955029i \(0.595824\pi\)
\(398\) 8.97970e31 0.284920
\(399\) 5.44310e32 1.67373
\(400\) 0 0
\(401\) 1.30258e32 0.376271 0.188136 0.982143i \(-0.439756\pi\)
0.188136 + 0.982143i \(0.439756\pi\)
\(402\) 2.67638e32 0.749415
\(403\) 3.71025e32 1.00714
\(404\) 1.31107e32 0.345031
\(405\) 0 0
\(406\) −4.39697e32 −1.08787
\(407\) −5.23282e32 −1.25547
\(408\) −2.82965e32 −0.658385
\(409\) 8.59252e32 1.93901 0.969503 0.245081i \(-0.0788144\pi\)
0.969503 + 0.245081i \(0.0788144\pi\)
\(410\) 0 0
\(411\) −3.91243e32 −0.830661
\(412\) −1.77296e32 −0.365161
\(413\) 1.07850e32 0.215498
\(414\) −2.49145e32 −0.483000
\(415\) 0 0
\(416\) 1.47663e32 0.269529
\(417\) −5.63211e32 −0.997631
\(418\) −4.79796e32 −0.824809
\(419\) −4.67029e32 −0.779235 −0.389618 0.920977i \(-0.627393\pi\)
−0.389618 + 0.920977i \(0.627393\pi\)
\(420\) 0 0
\(421\) −7.42072e32 −1.16659 −0.583297 0.812259i \(-0.698238\pi\)
−0.583297 + 0.812259i \(0.698238\pi\)
\(422\) 3.73143e32 0.569468
\(423\) −1.53782e32 −0.227851
\(424\) −1.74942e32 −0.251664
\(425\) 0 0
\(426\) 1.59851e32 0.216818
\(427\) −3.63303e32 −0.478543
\(428\) 7.49336e31 0.0958582
\(429\) 1.59519e33 1.98196
\(430\) 0 0
\(431\) 1.06420e33 1.24755 0.623774 0.781605i \(-0.285599\pi\)
0.623774 + 0.781605i \(0.285599\pi\)
\(432\) −1.56978e32 −0.178769
\(433\) 9.20340e32 1.01824 0.509119 0.860696i \(-0.329971\pi\)
0.509119 + 0.860696i \(0.329971\pi\)
\(434\) −5.81227e32 −0.624776
\(435\) 0 0
\(436\) −7.54681e32 −0.765918
\(437\) 1.86055e33 1.83495
\(438\) −8.83127e32 −0.846441
\(439\) −3.84917e32 −0.358559 −0.179279 0.983798i \(-0.557377\pi\)
−0.179279 + 0.983798i \(0.557377\pi\)
\(440\) 0 0
\(441\) 3.53721e32 0.311300
\(442\) −1.98732e33 −1.70016
\(443\) −8.53492e32 −0.709827 −0.354914 0.934899i \(-0.615490\pi\)
−0.354914 + 0.934899i \(0.615490\pi\)
\(444\) −8.32818e32 −0.673384
\(445\) 0 0
\(446\) 4.34200e32 0.331897
\(447\) −2.08687e33 −1.55114
\(448\) −2.31321e32 −0.167201
\(449\) 3.42652e32 0.240865 0.120432 0.992722i \(-0.461572\pi\)
0.120432 + 0.992722i \(0.461572\pi\)
\(450\) 0 0
\(451\) −4.83681e31 −0.0321627
\(452\) −4.68150e32 −0.302799
\(453\) −1.30883e33 −0.823485
\(454\) −2.05228e33 −1.25614
\(455\) 0 0
\(456\) −7.63609e32 −0.442395
\(457\) 4.52087e32 0.254841 0.127421 0.991849i \(-0.459330\pi\)
0.127421 + 0.991849i \(0.459330\pi\)
\(458\) −1.31116e32 −0.0719181
\(459\) 2.11268e33 1.12765
\(460\) 0 0
\(461\) 2.47159e33 1.24944 0.624718 0.780850i \(-0.285214\pi\)
0.624718 + 0.780850i \(0.285214\pi\)
\(462\) −2.49893e33 −1.22950
\(463\) −1.72628e33 −0.826703 −0.413351 0.910572i \(-0.635642\pi\)
−0.413351 + 0.910572i \(0.635642\pi\)
\(464\) 6.16849e32 0.287544
\(465\) 0 0
\(466\) −2.53799e33 −1.12116
\(467\) −1.88292e33 −0.809787 −0.404893 0.914364i \(-0.632691\pi\)
−0.404893 + 0.914364i \(0.632691\pi\)
\(468\) 7.18150e32 0.300706
\(469\) −2.94460e33 −1.20051
\(470\) 0 0
\(471\) −4.70639e33 −1.81939
\(472\) −1.51302e32 −0.0569600
\(473\) 1.40678e32 0.0515775
\(474\) −3.49294e33 −1.24727
\(475\) 0 0
\(476\) 3.11322e33 1.05469
\(477\) −8.50822e32 −0.280775
\(478\) 8.87749e32 0.285392
\(479\) 1.07303e33 0.336061 0.168030 0.985782i \(-0.446259\pi\)
0.168030 + 0.985782i \(0.446259\pi\)
\(480\) 0 0
\(481\) −5.84904e33 −1.73889
\(482\) 3.05704e33 0.885549
\(483\) 9.69036e33 2.73527
\(484\) 3.84991e32 0.105897
\(485\) 0 0
\(486\) −2.02421e33 −0.528814
\(487\) −6.26292e33 −1.59465 −0.797324 0.603552i \(-0.793752\pi\)
−0.797324 + 0.603552i \(0.793752\pi\)
\(488\) 5.09676e32 0.126487
\(489\) 5.45897e33 1.32054
\(490\) 0 0
\(491\) 3.10431e33 0.713586 0.356793 0.934183i \(-0.383870\pi\)
0.356793 + 0.934183i \(0.383870\pi\)
\(492\) −7.69792e31 −0.0172508
\(493\) −8.30184e33 −1.81379
\(494\) −5.36298e33 −1.14240
\(495\) 0 0
\(496\) 8.15401e32 0.165139
\(497\) −1.75870e33 −0.347326
\(498\) −2.04997e33 −0.394804
\(499\) 7.12443e33 1.33811 0.669057 0.743211i \(-0.266698\pi\)
0.669057 + 0.743211i \(0.266698\pi\)
\(500\) 0 0
\(501\) 1.29054e34 2.30568
\(502\) 3.05840e33 0.532965
\(503\) −1.03191e34 −1.75406 −0.877028 0.480439i \(-0.840477\pi\)
−0.877028 + 0.480439i \(0.840477\pi\)
\(504\) −1.12501e33 −0.186542
\(505\) 0 0
\(506\) −8.54183e33 −1.34794
\(507\) 1.01603e34 1.56425
\(508\) −3.21523e33 −0.482965
\(509\) −4.04421e32 −0.0592736 −0.0296368 0.999561i \(-0.509435\pi\)
−0.0296368 + 0.999561i \(0.509435\pi\)
\(510\) 0 0
\(511\) 9.71629e33 1.35594
\(512\) 3.24519e32 0.0441942
\(513\) 5.70128e33 0.757714
\(514\) −8.32753e32 −0.108013
\(515\) 0 0
\(516\) 2.23892e32 0.0276642
\(517\) −5.27236e33 −0.635877
\(518\) 9.16279e33 1.07871
\(519\) −4.52420e33 −0.519936
\(520\) 0 0
\(521\) 9.14369e33 1.00150 0.500749 0.865593i \(-0.333058\pi\)
0.500749 + 0.865593i \(0.333058\pi\)
\(522\) 3.00001e33 0.320805
\(523\) 1.50595e33 0.157231 0.0786156 0.996905i \(-0.474950\pi\)
0.0786156 + 0.996905i \(0.474950\pi\)
\(524\) −3.73700e33 −0.380963
\(525\) 0 0
\(526\) 1.30547e34 1.26895
\(527\) −1.09740e34 −1.04168
\(528\) 3.50574e33 0.324979
\(529\) 2.20777e34 1.99875
\(530\) 0 0
\(531\) −7.35849e32 −0.0635487
\(532\) 8.40134e33 0.708685
\(533\) −5.40640e32 −0.0445470
\(534\) 2.64474e33 0.212871
\(535\) 0 0
\(536\) 4.13096e33 0.317315
\(537\) 1.18860e34 0.891984
\(538\) 4.43999e32 0.0325539
\(539\) 1.21272e34 0.868762
\(540\) 0 0
\(541\) 7.92779e32 0.0542234 0.0271117 0.999632i \(-0.491369\pi\)
0.0271117 + 0.999632i \(0.491369\pi\)
\(542\) 3.44279e33 0.230102
\(543\) −9.40134e33 −0.614033
\(544\) −4.36752e33 −0.278772
\(545\) 0 0
\(546\) −2.79321e34 −1.70293
\(547\) 2.56023e34 1.52559 0.762794 0.646641i \(-0.223827\pi\)
0.762794 + 0.646641i \(0.223827\pi\)
\(548\) −6.03878e33 −0.351716
\(549\) 2.47878e33 0.141118
\(550\) 0 0
\(551\) −2.24034e34 −1.21876
\(552\) −1.35946e34 −0.722981
\(553\) 3.84298e34 1.99804
\(554\) −2.16387e34 −1.09992
\(555\) 0 0
\(556\) −8.69308e33 −0.422414
\(557\) 2.77017e34 1.31618 0.658091 0.752938i \(-0.271364\pi\)
0.658091 + 0.752938i \(0.271364\pi\)
\(558\) 3.96565e33 0.184241
\(559\) 1.57244e33 0.0714375
\(560\) 0 0
\(561\) −4.71819e34 −2.04993
\(562\) −2.41545e34 −1.02635
\(563\) −1.80222e33 −0.0748951 −0.0374476 0.999299i \(-0.511923\pi\)
−0.0374476 + 0.999299i \(0.511923\pi\)
\(564\) −8.39110e33 −0.341060
\(565\) 0 0
\(566\) 6.48499e33 0.252176
\(567\) 4.35652e34 1.65711
\(568\) 2.46728e33 0.0918044
\(569\) −2.00804e34 −0.730920 −0.365460 0.930827i \(-0.619088\pi\)
−0.365460 + 0.930827i \(0.619088\pi\)
\(570\) 0 0
\(571\) 3.27237e34 1.14002 0.570008 0.821639i \(-0.306940\pi\)
0.570008 + 0.821639i \(0.306940\pi\)
\(572\) 2.46215e34 0.839198
\(573\) 1.91753e34 0.639453
\(574\) 8.46937e32 0.0276346
\(575\) 0 0
\(576\) 1.57828e33 0.0493063
\(577\) −1.86866e34 −0.571257 −0.285629 0.958340i \(-0.592202\pi\)
−0.285629 + 0.958340i \(0.592202\pi\)
\(578\) 3.51436e34 1.05135
\(579\) −5.50883e33 −0.161279
\(580\) 0 0
\(581\) 2.25541e34 0.632447
\(582\) 3.87806e34 1.06433
\(583\) −2.91701e34 −0.783575
\(584\) −1.36309e34 −0.358398
\(585\) 0 0
\(586\) 2.03310e34 0.512198
\(587\) 2.44688e33 0.0603443 0.0301721 0.999545i \(-0.490394\pi\)
0.0301721 + 0.999545i \(0.490394\pi\)
\(588\) 1.93008e34 0.465970
\(589\) −2.96146e34 −0.699945
\(590\) 0 0
\(591\) −6.14325e33 −0.139173
\(592\) −1.28544e34 −0.285122
\(593\) −3.48844e34 −0.757613 −0.378806 0.925476i \(-0.623665\pi\)
−0.378806 + 0.925476i \(0.623665\pi\)
\(594\) −2.61747e34 −0.556610
\(595\) 0 0
\(596\) −3.22106e34 −0.656780
\(597\) 2.38297e34 0.475817
\(598\) −9.54772e34 −1.86696
\(599\) 6.48903e34 1.24264 0.621319 0.783558i \(-0.286597\pi\)
0.621319 + 0.783558i \(0.286597\pi\)
\(600\) 0 0
\(601\) −2.18034e34 −0.400492 −0.200246 0.979746i \(-0.564174\pi\)
−0.200246 + 0.979746i \(0.564174\pi\)
\(602\) −2.46330e33 −0.0443160
\(603\) 2.00907e34 0.354020
\(604\) −2.02016e34 −0.348678
\(605\) 0 0
\(606\) 3.47922e34 0.576201
\(607\) −7.37348e34 −1.19623 −0.598114 0.801411i \(-0.704083\pi\)
−0.598114 + 0.801411i \(0.704083\pi\)
\(608\) −1.17862e34 −0.187318
\(609\) −1.16684e35 −1.81675
\(610\) 0 0
\(611\) −5.89324e34 −0.880722
\(612\) −2.12412e34 −0.311018
\(613\) 1.05162e35 1.50870 0.754349 0.656473i \(-0.227953\pi\)
0.754349 + 0.656473i \(0.227953\pi\)
\(614\) 5.29804e34 0.744750
\(615\) 0 0
\(616\) −3.85707e34 −0.520593
\(617\) 8.12769e34 1.07499 0.537493 0.843268i \(-0.319372\pi\)
0.537493 + 0.843268i \(0.319372\pi\)
\(618\) −4.70496e34 −0.609818
\(619\) 4.81305e34 0.611346 0.305673 0.952137i \(-0.401119\pi\)
0.305673 + 0.952137i \(0.401119\pi\)
\(620\) 0 0
\(621\) 1.01500e35 1.23829
\(622\) −6.90737e33 −0.0825910
\(623\) −2.90978e34 −0.341004
\(624\) 3.91858e34 0.450113
\(625\) 0 0
\(626\) 1.10746e34 0.122221
\(627\) −1.27325e35 −1.37743
\(628\) −7.26425e34 −0.770363
\(629\) 1.73001e35 1.79852
\(630\) 0 0
\(631\) 6.11675e34 0.611158 0.305579 0.952167i \(-0.401150\pi\)
0.305579 + 0.952167i \(0.401150\pi\)
\(632\) −5.39130e34 −0.528117
\(633\) 9.90222e34 0.951012
\(634\) 1.34802e35 1.26935
\(635\) 0 0
\(636\) −4.64250e34 −0.420279
\(637\) 1.35553e35 1.20328
\(638\) 1.02854e35 0.895289
\(639\) 1.19995e34 0.102424
\(640\) 0 0
\(641\) 9.15866e34 0.751807 0.375904 0.926659i \(-0.377332\pi\)
0.375904 + 0.926659i \(0.377332\pi\)
\(642\) 1.98854e34 0.160083
\(643\) −5.09730e34 −0.402442 −0.201221 0.979546i \(-0.564491\pi\)
−0.201221 + 0.979546i \(0.564491\pi\)
\(644\) 1.49569e35 1.15816
\(645\) 0 0
\(646\) 1.58624e35 1.18158
\(647\) −2.28658e35 −1.67064 −0.835320 0.549765i \(-0.814717\pi\)
−0.835320 + 0.549765i \(0.814717\pi\)
\(648\) −6.11174e34 −0.438003
\(649\) −2.52283e34 −0.177349
\(650\) 0 0
\(651\) −1.54242e35 −1.04338
\(652\) 8.42585e34 0.559138
\(653\) 1.93736e35 1.26124 0.630618 0.776093i \(-0.282801\pi\)
0.630618 + 0.776093i \(0.282801\pi\)
\(654\) −2.00272e35 −1.27908
\(655\) 0 0
\(656\) −1.18816e33 −0.00730429
\(657\) −6.62933e34 −0.399855
\(658\) 9.23202e34 0.546353
\(659\) −2.14078e35 −1.24310 −0.621548 0.783376i \(-0.713496\pi\)
−0.621548 + 0.783376i \(0.713496\pi\)
\(660\) 0 0
\(661\) 1.52863e35 0.854641 0.427320 0.904100i \(-0.359458\pi\)
0.427320 + 0.904100i \(0.359458\pi\)
\(662\) −1.27126e35 −0.697445
\(663\) −5.27381e35 −2.83926
\(664\) −3.16410e34 −0.167167
\(665\) 0 0
\(666\) −6.25168e34 −0.318104
\(667\) −3.98848e35 −1.99175
\(668\) 1.99193e35 0.976265
\(669\) 1.15225e35 0.554268
\(670\) 0 0
\(671\) 8.49840e34 0.393827
\(672\) −6.13863e34 −0.279226
\(673\) −2.04386e35 −0.912564 −0.456282 0.889835i \(-0.650819\pi\)
−0.456282 + 0.889835i \(0.650819\pi\)
\(674\) 1.31040e35 0.574322
\(675\) 0 0
\(676\) 1.56823e35 0.662331
\(677\) 4.21908e34 0.174928 0.0874640 0.996168i \(-0.472124\pi\)
0.0874640 + 0.996168i \(0.472124\pi\)
\(678\) −1.24235e35 −0.505675
\(679\) −4.26671e35 −1.70498
\(680\) 0 0
\(681\) −5.44620e35 −2.09776
\(682\) 1.35961e35 0.514173
\(683\) −2.99380e35 −1.11164 −0.555820 0.831303i \(-0.687595\pi\)
−0.555820 + 0.831303i \(0.687595\pi\)
\(684\) −5.73215e34 −0.208986
\(685\) 0 0
\(686\) 5.67203e34 0.199383
\(687\) −3.47948e34 −0.120103
\(688\) 3.45575e33 0.0117135
\(689\) −3.26052e35 −1.08529
\(690\) 0 0
\(691\) 8.03613e34 0.257971 0.128986 0.991646i \(-0.458828\pi\)
0.128986 + 0.991646i \(0.458828\pi\)
\(692\) −6.98304e34 −0.220150
\(693\) −1.87586e35 −0.580812
\(694\) 1.77854e35 0.540843
\(695\) 0 0
\(696\) 1.63695e35 0.480198
\(697\) 1.59909e34 0.0460746
\(698\) 3.98207e35 1.12698
\(699\) −6.73516e35 −1.87233
\(700\) 0 0
\(701\) −4.53710e35 −1.21704 −0.608518 0.793540i \(-0.708235\pi\)
−0.608518 + 0.793540i \(0.708235\pi\)
\(702\) −2.92570e35 −0.770932
\(703\) 4.66860e35 1.20850
\(704\) 5.41106e34 0.137602
\(705\) 0 0
\(706\) −4.23667e35 −1.03984
\(707\) −3.82789e35 −0.923032
\(708\) −4.01515e34 −0.0951232
\(709\) −1.72814e35 −0.402255 −0.201128 0.979565i \(-0.564461\pi\)
−0.201128 + 0.979565i \(0.564461\pi\)
\(710\) 0 0
\(711\) −2.62203e35 −0.589206
\(712\) 4.08212e34 0.0901334
\(713\) −5.27229e35 −1.14388
\(714\) 8.26166e35 1.76133
\(715\) 0 0
\(716\) 1.83459e35 0.377681
\(717\) 2.35585e35 0.476604
\(718\) 1.77703e35 0.353296
\(719\) 9.50282e35 1.85670 0.928349 0.371709i \(-0.121228\pi\)
0.928349 + 0.371709i \(0.121228\pi\)
\(720\) 0 0
\(721\) 5.17647e35 0.976884
\(722\) 4.68219e34 0.0868429
\(723\) 8.11256e35 1.47887
\(724\) −1.45108e35 −0.259992
\(725\) 0 0
\(726\) 1.02166e35 0.176848
\(727\) 1.71833e35 0.292367 0.146183 0.989258i \(-0.453301\pi\)
0.146183 + 0.989258i \(0.453301\pi\)
\(728\) −4.31128e35 −0.721048
\(729\) 2.16385e35 0.355741
\(730\) 0 0
\(731\) −4.65090e34 −0.0738873
\(732\) 1.35254e35 0.211233
\(733\) −5.57098e35 −0.855327 −0.427664 0.903938i \(-0.640663\pi\)
−0.427664 + 0.903938i \(0.640663\pi\)
\(734\) −1.93113e35 −0.291481
\(735\) 0 0
\(736\) −2.09830e35 −0.306122
\(737\) 6.88801e35 0.987985
\(738\) −5.77856e33 −0.00814921
\(739\) 6.81418e35 0.944839 0.472420 0.881374i \(-0.343381\pi\)
0.472420 + 0.881374i \(0.343381\pi\)
\(740\) 0 0
\(741\) −1.42319e36 −1.90781
\(742\) 5.10775e35 0.673257
\(743\) 7.94423e35 1.02965 0.514827 0.857294i \(-0.327856\pi\)
0.514827 + 0.857294i \(0.327856\pi\)
\(744\) 2.16386e35 0.275782
\(745\) 0 0
\(746\) 4.06564e35 0.501065
\(747\) −1.53884e35 −0.186503
\(748\) −7.28246e35 −0.867977
\(749\) −2.18782e35 −0.256442
\(750\) 0 0
\(751\) 9.09984e35 1.03166 0.515828 0.856692i \(-0.327484\pi\)
0.515828 + 0.856692i \(0.327484\pi\)
\(752\) −1.29516e35 −0.144411
\(753\) 8.11619e35 0.890051
\(754\) 1.14966e36 1.24002
\(755\) 0 0
\(756\) 4.58325e35 0.478245
\(757\) 9.30502e35 0.955034 0.477517 0.878623i \(-0.341537\pi\)
0.477517 + 0.878623i \(0.341537\pi\)
\(758\) −2.57570e35 −0.260034
\(759\) −2.26677e36 −2.25105
\(760\) 0 0
\(761\) 1.13441e36 1.09009 0.545045 0.838407i \(-0.316513\pi\)
0.545045 + 0.838407i \(0.316513\pi\)
\(762\) −8.53236e35 −0.806552
\(763\) 2.20342e36 2.04900
\(764\) 2.95967e35 0.270755
\(765\) 0 0
\(766\) −2.02904e35 −0.179651
\(767\) −2.81992e35 −0.245637
\(768\) 8.61186e34 0.0738043
\(769\) −2.29001e36 −1.93090 −0.965448 0.260597i \(-0.916081\pi\)
−0.965448 + 0.260597i \(0.916081\pi\)
\(770\) 0 0
\(771\) −2.20990e35 −0.180382
\(772\) −8.50281e34 −0.0682883
\(773\) −3.21550e35 −0.254100 −0.127050 0.991896i \(-0.540551\pi\)
−0.127050 + 0.991896i \(0.540551\pi\)
\(774\) 1.68068e34 0.0130684
\(775\) 0 0
\(776\) 5.98574e35 0.450657
\(777\) 2.43156e36 1.80145
\(778\) −1.67965e36 −1.22454
\(779\) 4.31529e34 0.0309594
\(780\) 0 0
\(781\) 4.11397e35 0.285840
\(782\) 2.82399e36 1.93098
\(783\) −1.22219e36 −0.822461
\(784\) 2.97905e35 0.197300
\(785\) 0 0
\(786\) −9.91701e35 −0.636208
\(787\) −3.25977e35 −0.205827 −0.102914 0.994690i \(-0.532817\pi\)
−0.102914 + 0.994690i \(0.532817\pi\)
\(788\) −9.48202e34 −0.0589282
\(789\) 3.46437e36 2.11915
\(790\) 0 0
\(791\) 1.36685e36 0.810054
\(792\) 2.63164e35 0.153519
\(793\) 9.49918e35 0.545470
\(794\) −7.41849e35 −0.419333
\(795\) 0 0
\(796\) 3.67809e35 0.201469
\(797\) 8.54851e35 0.460958 0.230479 0.973077i \(-0.425971\pi\)
0.230479 + 0.973077i \(0.425971\pi\)
\(798\) 2.22949e36 1.18350
\(799\) 1.74308e36 0.910925
\(800\) 0 0
\(801\) 1.98532e35 0.100559
\(802\) 5.33538e35 0.266064
\(803\) −2.27284e36 −1.11590
\(804\) 1.09625e36 0.529917
\(805\) 0 0
\(806\) 1.51972e36 0.712156
\(807\) 1.17825e35 0.0543651
\(808\) 5.37013e35 0.243973
\(809\) −3.81958e36 −1.70867 −0.854336 0.519721i \(-0.826036\pi\)
−0.854336 + 0.519721i \(0.826036\pi\)
\(810\) 0 0
\(811\) −2.53614e36 −1.10005 −0.550025 0.835148i \(-0.685382\pi\)
−0.550025 + 0.835148i \(0.685382\pi\)
\(812\) −1.80100e36 −0.769243
\(813\) 9.13625e35 0.384270
\(814\) −2.14336e36 −0.887750
\(815\) 0 0
\(816\) −1.15902e36 −0.465549
\(817\) −1.25509e35 −0.0496478
\(818\) 3.51949e36 1.37108
\(819\) −2.09677e36 −0.804455
\(820\) 0 0
\(821\) −6.03536e35 −0.224602 −0.112301 0.993674i \(-0.535822\pi\)
−0.112301 + 0.993674i \(0.535822\pi\)
\(822\) −1.60253e36 −0.587366
\(823\) 3.78844e36 1.36761 0.683804 0.729665i \(-0.260324\pi\)
0.683804 + 0.729665i \(0.260324\pi\)
\(824\) −7.26205e35 −0.258208
\(825\) 0 0
\(826\) 4.41753e35 0.152380
\(827\) 3.11766e36 1.05928 0.529639 0.848223i \(-0.322328\pi\)
0.529639 + 0.848223i \(0.322328\pi\)
\(828\) −1.02050e36 −0.341533
\(829\) 5.42365e36 1.78797 0.893984 0.448099i \(-0.147899\pi\)
0.893984 + 0.448099i \(0.147899\pi\)
\(830\) 0 0
\(831\) −5.74233e36 −1.83686
\(832\) 6.04827e35 0.190586
\(833\) −4.00934e36 −1.24454
\(834\) −2.30691e36 −0.705432
\(835\) 0 0
\(836\) −1.96525e36 −0.583228
\(837\) −1.61558e36 −0.472347
\(838\) −1.91295e36 −0.551003
\(839\) −3.97800e35 −0.112886 −0.0564431 0.998406i \(-0.517976\pi\)
−0.0564431 + 0.998406i \(0.517976\pi\)
\(840\) 0 0
\(841\) 1.17226e36 0.322904
\(842\) −3.03953e36 −0.824907
\(843\) −6.40996e36 −1.71400
\(844\) 1.52839e36 0.402675
\(845\) 0 0
\(846\) −6.29891e35 −0.161115
\(847\) −1.12405e36 −0.283297
\(848\) −7.16563e35 −0.177953
\(849\) 1.72094e36 0.421134
\(850\) 0 0
\(851\) 8.31153e36 1.97497
\(852\) 6.54749e35 0.153313
\(853\) 2.49788e36 0.576379 0.288189 0.957573i \(-0.406947\pi\)
0.288189 + 0.957573i \(0.406947\pi\)
\(854\) −1.48809e36 −0.338381
\(855\) 0 0
\(856\) 3.06928e35 0.0677820
\(857\) −5.58799e36 −1.21617 −0.608086 0.793871i \(-0.708062\pi\)
−0.608086 + 0.793871i \(0.708062\pi\)
\(858\) 6.53389e36 1.40146
\(859\) 4.00214e36 0.846015 0.423007 0.906126i \(-0.360974\pi\)
0.423007 + 0.906126i \(0.360974\pi\)
\(860\) 0 0
\(861\) 2.24755e35 0.0461497
\(862\) 4.35896e36 0.882149
\(863\) 1.01320e36 0.202097 0.101048 0.994882i \(-0.467780\pi\)
0.101048 + 0.994882i \(0.467780\pi\)
\(864\) −6.42981e35 −0.126409
\(865\) 0 0
\(866\) 3.76971e36 0.720003
\(867\) 9.32618e36 1.75576
\(868\) −2.38071e36 −0.441783
\(869\) −8.98952e36 −1.64433
\(870\) 0 0
\(871\) 7.69915e36 1.36841
\(872\) −3.09117e36 −0.541586
\(873\) 2.91113e36 0.502786
\(874\) 7.62083e36 1.29751
\(875\) 0 0
\(876\) −3.61729e36 −0.598524
\(877\) −1.04808e37 −1.70962 −0.854808 0.518944i \(-0.826325\pi\)
−0.854808 + 0.518944i \(0.826325\pi\)
\(878\) −1.57662e36 −0.253539
\(879\) 5.39529e36 0.855370
\(880\) 0 0
\(881\) −7.82632e36 −1.20603 −0.603016 0.797729i \(-0.706035\pi\)
−0.603016 + 0.797729i \(0.706035\pi\)
\(882\) 1.44884e36 0.220122
\(883\) −1.14075e37 −1.70876 −0.854379 0.519651i \(-0.826062\pi\)
−0.854379 + 0.519651i \(0.826062\pi\)
\(884\) −8.14005e36 −1.20219
\(885\) 0 0
\(886\) −3.49590e36 −0.501924
\(887\) −2.08825e35 −0.0295622 −0.0147811 0.999891i \(-0.504705\pi\)
−0.0147811 + 0.999891i \(0.504705\pi\)
\(888\) −3.41122e36 −0.476155
\(889\) 9.38744e36 1.29204
\(890\) 0 0
\(891\) −1.01908e37 −1.36375
\(892\) 1.77848e36 0.234687
\(893\) 4.70388e36 0.612086
\(894\) −8.54783e36 −1.09682
\(895\) 0 0
\(896\) −9.47490e35 −0.118229
\(897\) −2.53371e37 −3.11782
\(898\) 1.40350e36 0.170317
\(899\) 6.34849e36 0.759756
\(900\) 0 0
\(901\) 9.64384e36 1.12251
\(902\) −1.98116e35 −0.0227425
\(903\) −6.53693e35 −0.0740077
\(904\) −1.91754e36 −0.214111
\(905\) 0 0
\(906\) −5.36097e36 −0.582292
\(907\) −1.26800e37 −1.35841 −0.679203 0.733950i \(-0.737674\pi\)
−0.679203 + 0.733950i \(0.737674\pi\)
\(908\) −8.40614e36 −0.888226
\(909\) 2.61173e36 0.272195
\(910\) 0 0
\(911\) −9.78070e36 −0.991722 −0.495861 0.868402i \(-0.665148\pi\)
−0.495861 + 0.868402i \(0.665148\pi\)
\(912\) −3.12774e36 −0.312821
\(913\) −5.27586e36 −0.520486
\(914\) 1.85175e36 0.180200
\(915\) 0 0
\(916\) −5.37053e35 −0.0508538
\(917\) 1.09108e37 1.01916
\(918\) 8.65354e36 0.797370
\(919\) −7.03823e36 −0.639764 −0.319882 0.947457i \(-0.603643\pi\)
−0.319882 + 0.947457i \(0.603643\pi\)
\(920\) 0 0
\(921\) 1.40596e37 1.24373
\(922\) 1.01236e37 0.883485
\(923\) 4.59843e36 0.395903
\(924\) −1.02356e37 −0.869391
\(925\) 0 0
\(926\) −7.07085e36 −0.584567
\(927\) −3.53185e36 −0.288075
\(928\) 2.52661e36 0.203324
\(929\) 2.91517e36 0.231455 0.115728 0.993281i \(-0.463080\pi\)
0.115728 + 0.993281i \(0.463080\pi\)
\(930\) 0 0
\(931\) −1.08196e37 −0.836258
\(932\) −1.03956e37 −0.792778
\(933\) −1.83303e36 −0.137927
\(934\) −7.71244e36 −0.572606
\(935\) 0 0
\(936\) 2.94154e36 0.212631
\(937\) −1.15106e37 −0.821016 −0.410508 0.911857i \(-0.634649\pi\)
−0.410508 + 0.911857i \(0.634649\pi\)
\(938\) −1.20611e37 −0.848888
\(939\) 2.93889e36 0.204110
\(940\) 0 0
\(941\) 1.44936e37 0.980179 0.490090 0.871672i \(-0.336964\pi\)
0.490090 + 0.871672i \(0.336964\pi\)
\(942\) −1.92774e37 −1.28651
\(943\) 7.68253e35 0.0505951
\(944\) −6.19733e35 −0.0402768
\(945\) 0 0
\(946\) 5.76215e35 0.0364708
\(947\) 2.53109e37 1.58100 0.790502 0.612459i \(-0.209820\pi\)
0.790502 + 0.612459i \(0.209820\pi\)
\(948\) −1.43071e37 −0.881956
\(949\) −2.54049e37 −1.54557
\(950\) 0 0
\(951\) 3.57728e37 2.11981
\(952\) 1.27518e37 0.745775
\(953\) 8.20660e36 0.473698 0.236849 0.971546i \(-0.423885\pi\)
0.236849 + 0.971546i \(0.423885\pi\)
\(954\) −3.48496e36 −0.198538
\(955\) 0 0
\(956\) 3.63622e36 0.201803
\(957\) 2.72948e37 1.49513
\(958\) 4.39513e36 0.237631
\(959\) 1.76313e37 0.940917
\(960\) 0 0
\(961\) −1.08408e37 −0.563665
\(962\) −2.39577e37 −1.22958
\(963\) 1.49273e36 0.0756226
\(964\) 1.25216e37 0.626178
\(965\) 0 0
\(966\) 3.96917e37 1.93413
\(967\) 2.92106e37 1.40511 0.702555 0.711630i \(-0.252043\pi\)
0.702555 + 0.711630i \(0.252043\pi\)
\(968\) 1.57692e36 0.0748804
\(969\) 4.20946e37 1.97324
\(970\) 0 0
\(971\) 9.97325e36 0.455613 0.227807 0.973706i \(-0.426845\pi\)
0.227807 + 0.973706i \(0.426845\pi\)
\(972\) −8.29117e36 −0.373928
\(973\) 2.53810e37 1.13005
\(974\) −2.56529e37 −1.12759
\(975\) 0 0
\(976\) 2.08763e36 0.0894399
\(977\) −3.07257e37 −1.29963 −0.649816 0.760092i \(-0.725154\pi\)
−0.649816 + 0.760092i \(0.725154\pi\)
\(978\) 2.23600e37 0.933760
\(979\) 6.80658e36 0.280637
\(980\) 0 0
\(981\) −1.50337e37 −0.604233
\(982\) 1.27153e37 0.504582
\(983\) 5.34152e36 0.209289 0.104644 0.994510i \(-0.466630\pi\)
0.104644 + 0.994510i \(0.466630\pi\)
\(984\) −3.15307e35 −0.0121982
\(985\) 0 0
\(986\) −3.40043e37 −1.28255
\(987\) 2.44993e37 0.912409
\(988\) −2.19667e37 −0.807800
\(989\) −2.23445e36 −0.0811365
\(990\) 0 0
\(991\) −4.37784e36 −0.155003 −0.0775013 0.996992i \(-0.524694\pi\)
−0.0775013 + 0.996992i \(0.524694\pi\)
\(992\) 3.33988e36 0.116771
\(993\) −3.37359e37 −1.16473
\(994\) −7.20365e36 −0.245597
\(995\) 0 0
\(996\) −8.39668e36 −0.279168
\(997\) 2.23560e37 0.734015 0.367008 0.930218i \(-0.380382\pi\)
0.367008 + 0.930218i \(0.380382\pi\)
\(998\) 2.91817e37 0.946190
\(999\) 2.54690e37 0.815535
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 50.26.a.l.1.6 6
5.2 odd 4 10.26.b.a.9.7 yes 12
5.3 odd 4 10.26.b.a.9.6 12
5.4 even 2 50.26.a.k.1.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.26.b.a.9.6 12 5.3 odd 4
10.26.b.a.9.7 yes 12 5.2 odd 4
50.26.a.k.1.1 6 5.4 even 2
50.26.a.l.1.6 6 1.1 even 1 trivial