Properties

Label 50.26.a.l.1.4
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 1.4
Root \(-1691.05\) 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)
 
\(f(q)\) \(=\) \(q+4096.00 q^{2} -116655. q^{3} +1.67772e7 q^{4} -4.77821e8 q^{6} +1.34370e10 q^{7} +6.87195e10 q^{8} -8.33680e11 q^{9} +4.67882e11 q^{11} -1.95715e12 q^{12} +9.38695e13 q^{13} +5.50381e13 q^{14} +2.81475e14 q^{16} -6.14184e14 q^{17} -3.41475e15 q^{18} -1.66661e16 q^{19} -1.56750e15 q^{21} +1.91644e15 q^{22} +1.60865e17 q^{23} -8.01650e15 q^{24} +3.84489e17 q^{26} +1.96094e17 q^{27} +2.25436e17 q^{28} -1.05009e18 q^{29} -3.00725e18 q^{31} +1.15292e18 q^{32} -5.45810e16 q^{33} -2.51570e18 q^{34} -1.39868e19 q^{36} +3.36937e19 q^{37} -6.82643e19 q^{38} -1.09504e19 q^{39} +5.09874e19 q^{41} -6.42050e18 q^{42} -2.92822e20 q^{43} +7.84976e18 q^{44} +6.58901e20 q^{46} +9.77356e20 q^{47} -3.28356e19 q^{48} -1.16051e21 q^{49} +7.16479e19 q^{51} +1.57487e21 q^{52} -5.26216e21 q^{53} +8.03202e20 q^{54} +9.23387e20 q^{56} +1.94419e21 q^{57} -4.30117e21 q^{58} +4.10935e20 q^{59} -5.44766e21 q^{61} -1.23177e22 q^{62} -1.12022e22 q^{63} +4.72237e21 q^{64} -2.23564e20 q^{66} -6.32396e22 q^{67} -1.03043e22 q^{68} -1.87657e22 q^{69} -1.81308e22 q^{71} -5.72901e22 q^{72} +3.48574e23 q^{73} +1.38010e23 q^{74} -2.79610e23 q^{76} +6.28695e21 q^{77} -4.48528e22 q^{78} +3.43193e23 q^{79} +6.83492e23 q^{81} +2.08844e23 q^{82} -6.85739e23 q^{83} -2.62984e22 q^{84} -1.19940e24 q^{86} +1.22499e23 q^{87} +3.21526e22 q^{88} -4.29595e24 q^{89} +1.26133e24 q^{91} +2.69886e24 q^{92} +3.50812e23 q^{93} +4.00325e24 q^{94} -1.34495e23 q^{96} +2.90151e24 q^{97} -4.75347e24 q^{98} -3.90064e23 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 24576 q^{2} - 801416 q^{3} + 100663296 q^{4} - 3282599936 q^{6} - 34007705352 q^{7} + 412316860416 q^{8} + 541468782118 q^{9} + 9861544614312 q^{11} - 13445529337856 q^{12} - 30787386783696 q^{13}+ \cdots - 20\!\cdots\!64 q^{99}+O(q^{100}) \) 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
\(3\) −116655. −0.126733 −0.0633664 0.997990i \(-0.520184\pi\)
−0.0633664 + 0.997990i \(0.520184\pi\)
\(4\) 1.67772e7 0.500000
\(5\) 0 0
\(6\) −4.77821e8 −0.0896137
\(7\) 1.34370e10 0.366926 0.183463 0.983027i \(-0.441269\pi\)
0.183463 + 0.983027i \(0.441269\pi\)
\(8\) 6.87195e10 0.353553
\(9\) −8.33680e11 −0.983939
\(10\) 0 0
\(11\) 4.67882e11 0.0449498 0.0224749 0.999747i \(-0.492845\pi\)
0.0224749 + 0.999747i \(0.492845\pi\)
\(12\) −1.95715e12 −0.0633664
\(13\) 9.38695e13 1.11746 0.558731 0.829349i \(-0.311289\pi\)
0.558731 + 0.829349i \(0.311289\pi\)
\(14\) 5.50381e13 0.259456
\(15\) 0 0
\(16\) 2.81475e14 0.250000
\(17\) −6.14184e14 −0.255674 −0.127837 0.991795i \(-0.540803\pi\)
−0.127837 + 0.991795i \(0.540803\pi\)
\(18\) −3.41475e15 −0.695750
\(19\) −1.66661e16 −1.72748 −0.863741 0.503936i \(-0.831885\pi\)
−0.863741 + 0.503936i \(0.831885\pi\)
\(20\) 0 0
\(21\) −1.56750e15 −0.0465016
\(22\) 1.91644e15 0.0317843
\(23\) 1.60865e17 1.53060 0.765301 0.643673i \(-0.222590\pi\)
0.765301 + 0.643673i \(0.222590\pi\)
\(24\) −8.01650e15 −0.0448068
\(25\) 0 0
\(26\) 3.84489e17 0.790165
\(27\) 1.96094e17 0.251430
\(28\) 2.25436e17 0.183463
\(29\) −1.05009e18 −0.551126 −0.275563 0.961283i \(-0.588864\pi\)
−0.275563 + 0.961283i \(0.588864\pi\)
\(30\) 0 0
\(31\) −3.00725e18 −0.685722 −0.342861 0.939386i \(-0.611396\pi\)
−0.342861 + 0.939386i \(0.611396\pi\)
\(32\) 1.15292e18 0.176777
\(33\) −5.45810e16 −0.00569662
\(34\) −2.51570e18 −0.180789
\(35\) 0 0
\(36\) −1.39868e19 −0.491969
\(37\) 3.36937e19 0.841449 0.420724 0.907189i \(-0.361776\pi\)
0.420724 + 0.907189i \(0.361776\pi\)
\(38\) −6.82643e19 −1.22151
\(39\) −1.09504e19 −0.141619
\(40\) 0 0
\(41\) 5.09874e19 0.352911 0.176455 0.984309i \(-0.443537\pi\)
0.176455 + 0.984309i \(0.443537\pi\)
\(42\) −6.42050e18 −0.0328816
\(43\) −2.92822e20 −1.11750 −0.558751 0.829335i \(-0.688719\pi\)
−0.558751 + 0.829335i \(0.688719\pi\)
\(44\) 7.84976e18 0.0224749
\(45\) 0 0
\(46\) 6.58901e20 1.08230
\(47\) 9.77356e20 1.22696 0.613479 0.789711i \(-0.289770\pi\)
0.613479 + 0.789711i \(0.289770\pi\)
\(48\) −3.28356e19 −0.0316832
\(49\) −1.16051e21 −0.865365
\(50\) 0 0
\(51\) 7.16479e19 0.0324023
\(52\) 1.57487e21 0.558731
\(53\) −5.26216e21 −1.47135 −0.735675 0.677335i \(-0.763135\pi\)
−0.735675 + 0.677335i \(0.763135\pi\)
\(54\) 8.03202e20 0.177788
\(55\) 0 0
\(56\) 9.23387e20 0.129728
\(57\) 1.94419e21 0.218929
\(58\) −4.30117e21 −0.389705
\(59\) 4.10935e20 0.0300693 0.0150346 0.999887i \(-0.495214\pi\)
0.0150346 + 0.999887i \(0.495214\pi\)
\(60\) 0 0
\(61\) −5.44766e21 −0.262777 −0.131388 0.991331i \(-0.541943\pi\)
−0.131388 + 0.991331i \(0.541943\pi\)
\(62\) −1.23177e22 −0.484879
\(63\) −1.12022e22 −0.361032
\(64\) 4.72237e21 0.125000
\(65\) 0 0
\(66\) −2.23564e20 −0.00402812
\(67\) −6.32396e22 −0.944178 −0.472089 0.881551i \(-0.656500\pi\)
−0.472089 + 0.881551i \(0.656500\pi\)
\(68\) −1.03043e22 −0.127837
\(69\) −1.87657e22 −0.193978
\(70\) 0 0
\(71\) −1.81308e22 −0.131126 −0.0655628 0.997848i \(-0.520884\pi\)
−0.0655628 + 0.997848i \(0.520884\pi\)
\(72\) −5.72901e22 −0.347875
\(73\) 3.48574e23 1.78139 0.890694 0.454603i \(-0.150219\pi\)
0.890694 + 0.454603i \(0.150219\pi\)
\(74\) 1.38010e23 0.594994
\(75\) 0 0
\(76\) −2.79610e23 −0.863741
\(77\) 6.28695e21 0.0164932
\(78\) −4.48528e22 −0.100140
\(79\) 3.43193e23 0.653432 0.326716 0.945123i \(-0.394058\pi\)
0.326716 + 0.945123i \(0.394058\pi\)
\(80\) 0 0
\(81\) 6.83492e23 0.952074
\(82\) 2.08844e23 0.249545
\(83\) −6.85739e23 −0.704178 −0.352089 0.935967i \(-0.614529\pi\)
−0.352089 + 0.935967i \(0.614529\pi\)
\(84\) −2.62984e22 −0.0232508
\(85\) 0 0
\(86\) −1.19940e24 −0.790194
\(87\) 1.22499e23 0.0698458
\(88\) 3.21526e22 0.0158922
\(89\) −4.29595e24 −1.84367 −0.921837 0.387579i \(-0.873312\pi\)
−0.921837 + 0.387579i \(0.873312\pi\)
\(90\) 0 0
\(91\) 1.26133e24 0.410025
\(92\) 2.69886e24 0.765301
\(93\) 3.50812e23 0.0869035
\(94\) 4.00325e24 0.867590
\(95\) 0 0
\(96\) −1.34495e23 −0.0224034
\(97\) 2.90151e24 0.424598 0.212299 0.977205i \(-0.431905\pi\)
0.212299 + 0.977205i \(0.431905\pi\)
\(98\) −4.75347e24 −0.611906
\(99\) −3.90064e23 −0.0442279
\(100\) 0 0
\(101\) −8.46099e24 −0.747144 −0.373572 0.927601i \(-0.621867\pi\)
−0.373572 + 0.927601i \(0.621867\pi\)
\(102\) 2.93470e23 0.0229119
\(103\) 1.06290e25 0.734562 0.367281 0.930110i \(-0.380289\pi\)
0.367281 + 0.930110i \(0.380289\pi\)
\(104\) 6.45066e24 0.395082
\(105\) 0 0
\(106\) −2.15538e25 −1.04040
\(107\) −2.28121e25 −0.979193 −0.489597 0.871949i \(-0.662856\pi\)
−0.489597 + 0.871949i \(0.662856\pi\)
\(108\) 3.28991e24 0.125715
\(109\) −3.17871e25 −1.08248 −0.541240 0.840868i \(-0.682045\pi\)
−0.541240 + 0.840868i \(0.682045\pi\)
\(110\) 0 0
\(111\) −3.93056e24 −0.106639
\(112\) 3.78219e24 0.0917314
\(113\) −9.52597e24 −0.206742 −0.103371 0.994643i \(-0.532963\pi\)
−0.103371 + 0.994643i \(0.532963\pi\)
\(114\) 7.96340e24 0.154806
\(115\) 0 0
\(116\) −1.76176e25 −0.275563
\(117\) −7.82571e25 −1.09951
\(118\) 1.68319e24 0.0212622
\(119\) −8.25281e24 −0.0938134
\(120\) 0 0
\(121\) −1.08128e26 −0.997980
\(122\) −2.23136e25 −0.185811
\(123\) −5.94796e24 −0.0447254
\(124\) −5.04532e25 −0.342861
\(125\) 0 0
\(126\) −4.58842e25 −0.255289
\(127\) 2.95859e26 1.49121 0.745604 0.666390i \(-0.232161\pi\)
0.745604 + 0.666390i \(0.232161\pi\)
\(128\) 1.93428e25 0.0883883
\(129\) 3.41593e25 0.141624
\(130\) 0 0
\(131\) −2.19768e26 −0.751749 −0.375874 0.926671i \(-0.622658\pi\)
−0.375874 + 0.926671i \(0.622658\pi\)
\(132\) −9.15717e23 −0.00284831
\(133\) −2.23943e26 −0.633857
\(134\) −2.59029e26 −0.667635
\(135\) 0 0
\(136\) −4.22064e25 −0.0903945
\(137\) 3.42423e25 0.0669200 0.0334600 0.999440i \(-0.489347\pi\)
0.0334600 + 0.999440i \(0.489347\pi\)
\(138\) −7.68644e25 −0.137163
\(139\) −2.79093e26 −0.455055 −0.227528 0.973772i \(-0.573064\pi\)
−0.227528 + 0.973772i \(0.573064\pi\)
\(140\) 0 0
\(141\) −1.14014e26 −0.155496
\(142\) −7.42638e25 −0.0927199
\(143\) 4.39198e25 0.0502297
\(144\) −2.34660e26 −0.245985
\(145\) 0 0
\(146\) 1.42776e27 1.25963
\(147\) 1.35380e26 0.109670
\(148\) 5.65287e26 0.420724
\(149\) 1.42906e27 0.977739 0.488870 0.872357i \(-0.337409\pi\)
0.488870 + 0.872357i \(0.337409\pi\)
\(150\) 0 0
\(151\) −2.51226e27 −1.45496 −0.727482 0.686127i \(-0.759310\pi\)
−0.727482 + 0.686127i \(0.759310\pi\)
\(152\) −1.14528e27 −0.610757
\(153\) 5.12033e26 0.251568
\(154\) 2.57513e25 0.0116625
\(155\) 0 0
\(156\) −1.83717e26 −0.0708096
\(157\) −3.34741e27 −1.19114 −0.595571 0.803302i \(-0.703074\pi\)
−0.595571 + 0.803302i \(0.703074\pi\)
\(158\) 1.40572e27 0.462046
\(159\) 6.13860e26 0.186468
\(160\) 0 0
\(161\) 2.16154e27 0.561617
\(162\) 2.79958e27 0.673218
\(163\) −8.42668e26 −0.187634 −0.0938170 0.995589i \(-0.529907\pi\)
−0.0938170 + 0.995589i \(0.529907\pi\)
\(164\) 8.55426e26 0.176455
\(165\) 0 0
\(166\) −2.80879e27 −0.497929
\(167\) −3.08789e27 −0.507815 −0.253907 0.967229i \(-0.581716\pi\)
−0.253907 + 0.967229i \(0.581716\pi\)
\(168\) −1.07718e26 −0.0164408
\(169\) 1.75507e27 0.248720
\(170\) 0 0
\(171\) 1.38942e28 1.69974
\(172\) −4.91274e27 −0.558751
\(173\) −9.76292e27 −1.03277 −0.516385 0.856357i \(-0.672723\pi\)
−0.516385 + 0.856357i \(0.672723\pi\)
\(174\) 5.01754e26 0.0493885
\(175\) 0 0
\(176\) 1.31697e26 0.0112375
\(177\) −4.79378e25 −0.00381077
\(178\) −1.75962e28 −1.30367
\(179\) −1.48035e28 −1.02259 −0.511294 0.859406i \(-0.670834\pi\)
−0.511294 + 0.859406i \(0.670834\pi\)
\(180\) 0 0
\(181\) 1.20192e28 0.722591 0.361295 0.932451i \(-0.382335\pi\)
0.361295 + 0.932451i \(0.382335\pi\)
\(182\) 5.16640e27 0.289932
\(183\) 6.35499e26 0.0333025
\(184\) 1.10545e28 0.541150
\(185\) 0 0
\(186\) 1.43693e27 0.0614501
\(187\) −2.87366e26 −0.0114925
\(188\) 1.63973e28 0.613479
\(189\) 2.63493e27 0.0922563
\(190\) 0 0
\(191\) 8.66884e27 0.266099 0.133050 0.991109i \(-0.457523\pi\)
0.133050 + 0.991109i \(0.457523\pi\)
\(192\) −5.50890e26 −0.0158416
\(193\) 1.60225e28 0.431782 0.215891 0.976417i \(-0.430734\pi\)
0.215891 + 0.976417i \(0.430734\pi\)
\(194\) 1.18846e28 0.300236
\(195\) 0 0
\(196\) −1.94702e28 −0.432683
\(197\) −6.69730e28 −1.39660 −0.698300 0.715805i \(-0.746060\pi\)
−0.698300 + 0.715805i \(0.746060\pi\)
\(198\) −1.59770e27 −0.0312738
\(199\) 1.81836e28 0.334208 0.167104 0.985939i \(-0.446558\pi\)
0.167104 + 0.985939i \(0.446558\pi\)
\(200\) 0 0
\(201\) 7.37724e27 0.119658
\(202\) −3.46562e28 −0.528310
\(203\) −1.41101e28 −0.202222
\(204\) 1.20205e27 0.0162012
\(205\) 0 0
\(206\) 4.35365e28 0.519414
\(207\) −1.34110e29 −1.50602
\(208\) 2.64219e28 0.279365
\(209\) −7.79776e27 −0.0776500
\(210\) 0 0
\(211\) −1.59042e29 −1.40598 −0.702992 0.711197i \(-0.748153\pi\)
−0.702992 + 0.711197i \(0.748153\pi\)
\(212\) −8.82844e28 −0.735675
\(213\) 2.11506e27 0.0166179
\(214\) −9.34384e28 −0.692394
\(215\) 0 0
\(216\) 1.34755e28 0.0888940
\(217\) −4.04085e28 −0.251609
\(218\) −1.30200e29 −0.765428
\(219\) −4.06630e28 −0.225761
\(220\) 0 0
\(221\) −5.76531e28 −0.285706
\(222\) −1.60996e28 −0.0754053
\(223\) −3.74438e29 −1.65794 −0.828970 0.559293i \(-0.811073\pi\)
−0.828970 + 0.559293i \(0.811073\pi\)
\(224\) 1.54919e28 0.0648639
\(225\) 0 0
\(226\) −3.90184e28 −0.146189
\(227\) −3.09492e29 −1.09730 −0.548651 0.836052i \(-0.684858\pi\)
−0.548651 + 0.836052i \(0.684858\pi\)
\(228\) 3.26181e28 0.109464
\(229\) 4.36225e29 1.38601 0.693006 0.720932i \(-0.256286\pi\)
0.693006 + 0.720932i \(0.256286\pi\)
\(230\) 0 0
\(231\) −7.33407e26 −0.00209024
\(232\) −7.21616e28 −0.194853
\(233\) −2.03884e29 −0.521715 −0.260858 0.965377i \(-0.584005\pi\)
−0.260858 + 0.965377i \(0.584005\pi\)
\(234\) −3.20541e29 −0.777474
\(235\) 0 0
\(236\) 6.89435e27 0.0150346
\(237\) −4.00354e28 −0.0828113
\(238\) −3.38035e28 −0.0663361
\(239\) −7.19381e29 −1.33963 −0.669816 0.742527i \(-0.733627\pi\)
−0.669816 + 0.742527i \(0.733627\pi\)
\(240\) 0 0
\(241\) −5.94932e28 −0.0998285 −0.0499142 0.998754i \(-0.515895\pi\)
−0.0499142 + 0.998754i \(0.515895\pi\)
\(242\) −4.42893e29 −0.705678
\(243\) −2.45881e29 −0.372089
\(244\) −9.13966e28 −0.131388
\(245\) 0 0
\(246\) −2.43628e28 −0.0316256
\(247\) −1.56444e30 −1.93039
\(248\) −2.06656e29 −0.242439
\(249\) 7.99951e28 0.0892425
\(250\) 0 0
\(251\) −1.11692e30 −1.12745 −0.563727 0.825961i \(-0.690633\pi\)
−0.563727 + 0.825961i \(0.690633\pi\)
\(252\) −1.87942e29 −0.180516
\(253\) 7.52656e28 0.0688003
\(254\) 1.21184e30 1.05444
\(255\) 0 0
\(256\) 7.92282e28 0.0625000
\(257\) −2.23332e30 −1.67798 −0.838991 0.544146i \(-0.816854\pi\)
−0.838991 + 0.544146i \(0.816854\pi\)
\(258\) 1.39917e29 0.100144
\(259\) 4.52744e29 0.308749
\(260\) 0 0
\(261\) 8.75438e29 0.542275
\(262\) −9.00168e29 −0.531566
\(263\) 1.67388e30 0.942489 0.471244 0.882003i \(-0.343805\pi\)
0.471244 + 0.882003i \(0.343805\pi\)
\(264\) −3.75078e27 −0.00201406
\(265\) 0 0
\(266\) −9.17270e29 −0.448205
\(267\) 5.01146e29 0.233654
\(268\) −1.06098e30 −0.472089
\(269\) −1.51842e30 −0.644895 −0.322448 0.946587i \(-0.604506\pi\)
−0.322448 + 0.946587i \(0.604506\pi\)
\(270\) 0 0
\(271\) 3.73589e30 1.44637 0.723183 0.690657i \(-0.242678\pi\)
0.723183 + 0.690657i \(0.242678\pi\)
\(272\) −1.72877e29 −0.0639185
\(273\) −1.47141e29 −0.0519637
\(274\) 1.40256e29 0.0473196
\(275\) 0 0
\(276\) −3.14837e29 −0.0969888
\(277\) −4.88475e30 −1.43829 −0.719143 0.694863i \(-0.755465\pi\)
−0.719143 + 0.694863i \(0.755465\pi\)
\(278\) −1.14316e30 −0.321772
\(279\) 2.50708e30 0.674708
\(280\) 0 0
\(281\) 4.07868e30 1.00390 0.501950 0.864896i \(-0.332616\pi\)
0.501950 + 0.864896i \(0.332616\pi\)
\(282\) −4.67001e29 −0.109952
\(283\) −8.77909e30 −1.97751 −0.988756 0.149537i \(-0.952222\pi\)
−0.988756 + 0.149537i \(0.952222\pi\)
\(284\) −3.04184e29 −0.0655628
\(285\) 0 0
\(286\) 1.79896e29 0.0355177
\(287\) 6.85120e29 0.129492
\(288\) −9.61168e29 −0.173937
\(289\) −5.39341e30 −0.934631
\(290\) 0 0
\(291\) −3.38477e29 −0.0538105
\(292\) 5.84809e30 0.890694
\(293\) −4.02063e30 −0.586744 −0.293372 0.955998i \(-0.594778\pi\)
−0.293372 + 0.955998i \(0.594778\pi\)
\(294\) 5.54518e29 0.0775486
\(295\) 0 0
\(296\) 2.31542e30 0.297497
\(297\) 9.17489e28 0.0113017
\(298\) 5.85344e30 0.691366
\(299\) 1.51003e31 1.71039
\(300\) 0 0
\(301\) −3.93467e30 −0.410040
\(302\) −1.02902e31 −1.02882
\(303\) 9.87021e29 0.0946877
\(304\) −4.69108e30 −0.431870
\(305\) 0 0
\(306\) 2.09729e30 0.177885
\(307\) −3.20834e29 −0.0261247 −0.0130623 0.999915i \(-0.504158\pi\)
−0.0130623 + 0.999915i \(0.504158\pi\)
\(308\) 1.05478e29 0.00824662
\(309\) −1.23993e30 −0.0930932
\(310\) 0 0
\(311\) −1.26721e31 −0.877692 −0.438846 0.898562i \(-0.644613\pi\)
−0.438846 + 0.898562i \(0.644613\pi\)
\(312\) −7.52505e29 −0.0500699
\(313\) 2.76579e31 1.76813 0.884066 0.467361i \(-0.154795\pi\)
0.884066 + 0.467361i \(0.154795\pi\)
\(314\) −1.37110e31 −0.842265
\(315\) 0 0
\(316\) 5.75783e30 0.326716
\(317\) 8.12742e30 0.443315 0.221657 0.975125i \(-0.428853\pi\)
0.221657 + 0.975125i \(0.428853\pi\)
\(318\) 2.51437e30 0.131853
\(319\) −4.91318e29 −0.0247730
\(320\) 0 0
\(321\) 2.66116e30 0.124096
\(322\) 8.85368e30 0.397123
\(323\) 1.02360e31 0.441672
\(324\) 1.14671e31 0.476037
\(325\) 0 0
\(326\) −3.45157e30 −0.132677
\(327\) 3.70814e30 0.137186
\(328\) 3.50383e30 0.124773
\(329\) 1.31328e31 0.450203
\(330\) 0 0
\(331\) −3.24680e31 −1.03182 −0.515912 0.856642i \(-0.672547\pi\)
−0.515912 + 0.856642i \(0.672547\pi\)
\(332\) −1.15048e31 −0.352089
\(333\) −2.80898e31 −0.827934
\(334\) −1.26480e31 −0.359079
\(335\) 0 0
\(336\) −4.41213e29 −0.0116254
\(337\) 6.16105e31 1.56416 0.782080 0.623178i \(-0.214159\pi\)
0.782080 + 0.623178i \(0.214159\pi\)
\(338\) 7.18877e30 0.175872
\(339\) 1.11126e30 0.0262010
\(340\) 0 0
\(341\) −1.40704e30 −0.0308231
\(342\) 5.69106e31 1.20189
\(343\) −3.36139e31 −0.684451
\(344\) −2.01226e31 −0.395097
\(345\) 0 0
\(346\) −3.99889e31 −0.730279
\(347\) 8.34408e31 1.46981 0.734903 0.678172i \(-0.237228\pi\)
0.734903 + 0.678172i \(0.237228\pi\)
\(348\) 2.05519e30 0.0349229
\(349\) −1.07311e32 −1.75924 −0.879621 0.475676i \(-0.842204\pi\)
−0.879621 + 0.475676i \(0.842204\pi\)
\(350\) 0 0
\(351\) 1.84073e31 0.280964
\(352\) 5.39431e29 0.00794608
\(353\) 1.05139e32 1.49479 0.747394 0.664381i \(-0.231305\pi\)
0.747394 + 0.664381i \(0.231305\pi\)
\(354\) −1.96353e29 −0.00269462
\(355\) 0 0
\(356\) −7.20741e31 −0.921837
\(357\) 9.62736e29 0.0118892
\(358\) −6.06350e31 −0.723079
\(359\) 1.15543e32 1.33064 0.665321 0.746557i \(-0.268295\pi\)
0.665321 + 0.746557i \(0.268295\pi\)
\(360\) 0 0
\(361\) 1.84682e32 1.98419
\(362\) 4.92305e31 0.510949
\(363\) 1.26137e31 0.126477
\(364\) 2.11616e31 0.205013
\(365\) 0 0
\(366\) 2.60300e30 0.0235484
\(367\) 5.65157e31 0.494133 0.247066 0.968999i \(-0.420533\pi\)
0.247066 + 0.968999i \(0.420533\pi\)
\(368\) 4.52793e31 0.382651
\(369\) −4.25072e31 −0.347242
\(370\) 0 0
\(371\) −7.07079e31 −0.539876
\(372\) 5.88565e30 0.0434518
\(373\) 1.41123e31 0.100748 0.0503741 0.998730i \(-0.483959\pi\)
0.0503741 + 0.998730i \(0.483959\pi\)
\(374\) −1.17705e30 −0.00812643
\(375\) 0 0
\(376\) 6.71634e31 0.433795
\(377\) −9.85713e31 −0.615862
\(378\) 1.07927e31 0.0652350
\(379\) 2.60316e32 1.52234 0.761169 0.648554i \(-0.224626\pi\)
0.761169 + 0.648554i \(0.224626\pi\)
\(380\) 0 0
\(381\) −3.45135e31 −0.188985
\(382\) 3.55076e31 0.188161
\(383\) 2.22460e32 1.14095 0.570475 0.821315i \(-0.306759\pi\)
0.570475 + 0.821315i \(0.306759\pi\)
\(384\) −2.25644e30 −0.0112017
\(385\) 0 0
\(386\) 6.56283e31 0.305316
\(387\) 2.44120e32 1.09955
\(388\) 4.86793e31 0.212299
\(389\) 2.83579e31 0.119758 0.0598789 0.998206i \(-0.480929\pi\)
0.0598789 + 0.998206i \(0.480929\pi\)
\(390\) 0 0
\(391\) −9.88004e31 −0.391335
\(392\) −7.97499e31 −0.305953
\(393\) 2.56371e31 0.0952713
\(394\) −2.74322e32 −0.987545
\(395\) 0 0
\(396\) −6.54419e30 −0.0221139
\(397\) 1.97865e32 0.647870 0.323935 0.946079i \(-0.394994\pi\)
0.323935 + 0.946079i \(0.394994\pi\)
\(398\) 7.44802e31 0.236321
\(399\) 2.61242e31 0.0803306
\(400\) 0 0
\(401\) 5.91837e32 1.70961 0.854806 0.518947i \(-0.173676\pi\)
0.854806 + 0.518947i \(0.173676\pi\)
\(402\) 3.02172e31 0.0846113
\(403\) −2.82289e32 −0.766268
\(404\) −1.41952e32 −0.373572
\(405\) 0 0
\(406\) −5.77949e31 −0.142993
\(407\) 1.57647e31 0.0378229
\(408\) 4.92361e30 0.0114560
\(409\) 5.10098e32 1.15110 0.575549 0.817767i \(-0.304788\pi\)
0.575549 + 0.817767i \(0.304788\pi\)
\(410\) 0 0
\(411\) −3.99455e30 −0.00848097
\(412\) 1.78326e32 0.367281
\(413\) 5.52175e30 0.0110332
\(414\) −5.49313e32 −1.06492
\(415\) 0 0
\(416\) 1.08224e32 0.197541
\(417\) 3.25577e31 0.0576704
\(418\) −3.19396e31 −0.0549068
\(419\) 3.45646e32 0.576709 0.288354 0.957524i \(-0.406892\pi\)
0.288354 + 0.957524i \(0.406892\pi\)
\(420\) 0 0
\(421\) 1.15256e31 0.0181191 0.00905955 0.999959i \(-0.497116\pi\)
0.00905955 + 0.999959i \(0.497116\pi\)
\(422\) −6.51435e32 −0.994181
\(423\) −8.14803e32 −1.20725
\(424\) −3.61613e32 −0.520201
\(425\) 0 0
\(426\) 8.66328e30 0.0117507
\(427\) −7.32004e31 −0.0964196
\(428\) −3.82724e32 −0.489597
\(429\) −5.12349e30 −0.00636575
\(430\) 0 0
\(431\) −1.12581e33 −1.31978 −0.659888 0.751364i \(-0.729396\pi\)
−0.659888 + 0.751364i \(0.729396\pi\)
\(432\) 5.51956e31 0.0628576
\(433\) −5.31109e32 −0.587604 −0.293802 0.955866i \(-0.594921\pi\)
−0.293802 + 0.955866i \(0.594921\pi\)
\(434\) −1.65513e32 −0.177914
\(435\) 0 0
\(436\) −5.33299e32 −0.541240
\(437\) −2.68098e33 −2.64409
\(438\) −1.66556e32 −0.159637
\(439\) 7.89847e32 0.735760 0.367880 0.929873i \(-0.380084\pi\)
0.367880 + 0.929873i \(0.380084\pi\)
\(440\) 0 0
\(441\) 9.67498e32 0.851467
\(442\) −2.36147e32 −0.202025
\(443\) 1.82758e33 1.51995 0.759974 0.649953i \(-0.225212\pi\)
0.759974 + 0.649953i \(0.225212\pi\)
\(444\) −6.59438e31 −0.0533196
\(445\) 0 0
\(446\) −1.53370e33 −1.17234
\(447\) −1.66708e32 −0.123912
\(448\) 6.34546e31 0.0458657
\(449\) −8.32692e31 −0.0585336 −0.0292668 0.999572i \(-0.509317\pi\)
−0.0292668 + 0.999572i \(0.509317\pi\)
\(450\) 0 0
\(451\) 2.38561e31 0.0158633
\(452\) −1.59819e32 −0.103371
\(453\) 2.93069e32 0.184392
\(454\) −1.26768e33 −0.775909
\(455\) 0 0
\(456\) 1.33604e32 0.0774030
\(457\) −2.02366e33 −1.14074 −0.570368 0.821389i \(-0.693199\pi\)
−0.570368 + 0.821389i \(0.693199\pi\)
\(458\) 1.78678e33 0.980058
\(459\) −1.20438e32 −0.0642842
\(460\) 0 0
\(461\) −8.37646e32 −0.423447 −0.211724 0.977330i \(-0.567908\pi\)
−0.211724 + 0.977330i \(0.567908\pi\)
\(462\) −3.00404e30 −0.00147802
\(463\) −3.37029e33 −1.61401 −0.807003 0.590547i \(-0.798912\pi\)
−0.807003 + 0.590547i \(0.798912\pi\)
\(464\) −2.95574e32 −0.137782
\(465\) 0 0
\(466\) −8.35108e32 −0.368909
\(467\) −2.79564e33 −1.20232 −0.601161 0.799128i \(-0.705295\pi\)
−0.601161 + 0.799128i \(0.705295\pi\)
\(468\) −1.31294e33 −0.549757
\(469\) −8.49753e32 −0.346443
\(470\) 0 0
\(471\) 3.90494e32 0.150957
\(472\) 2.82392e31 0.0106311
\(473\) −1.37006e32 −0.0502315
\(474\) −1.63985e32 −0.0585564
\(475\) 0 0
\(476\) −1.38459e32 −0.0469067
\(477\) 4.38696e33 1.44772
\(478\) −2.94659e33 −0.947263
\(479\) 6.03192e33 1.88913 0.944565 0.328324i \(-0.106484\pi\)
0.944565 + 0.328324i \(0.106484\pi\)
\(480\) 0 0
\(481\) 3.16281e33 0.940286
\(482\) −2.43684e32 −0.0705894
\(483\) −2.52156e32 −0.0711754
\(484\) −1.81409e33 −0.498990
\(485\) 0 0
\(486\) −1.00713e33 −0.263107
\(487\) −2.63729e33 −0.671500 −0.335750 0.941951i \(-0.608990\pi\)
−0.335750 + 0.941951i \(0.608990\pi\)
\(488\) −3.74360e32 −0.0929056
\(489\) 9.83018e31 0.0237794
\(490\) 0 0
\(491\) 4.59365e33 1.05594 0.527970 0.849263i \(-0.322953\pi\)
0.527970 + 0.849263i \(0.322953\pi\)
\(492\) −9.97902e31 −0.0223627
\(493\) 6.44948e32 0.140909
\(494\) −6.40793e33 −1.36499
\(495\) 0 0
\(496\) −8.46465e32 −0.171430
\(497\) −2.43624e32 −0.0481134
\(498\) 3.27660e32 0.0631040
\(499\) 4.28804e33 0.805382 0.402691 0.915336i \(-0.368075\pi\)
0.402691 + 0.915336i \(0.368075\pi\)
\(500\) 0 0
\(501\) 3.60219e32 0.0643569
\(502\) −4.57488e33 −0.797230
\(503\) 1.27163e33 0.216152 0.108076 0.994143i \(-0.465531\pi\)
0.108076 + 0.994143i \(0.465531\pi\)
\(504\) −7.69809e32 −0.127644
\(505\) 0 0
\(506\) 3.08288e32 0.0486491
\(507\) −2.04739e32 −0.0315210
\(508\) 4.96369e33 0.745604
\(509\) 8.49642e33 1.24527 0.622635 0.782512i \(-0.286062\pi\)
0.622635 + 0.782512i \(0.286062\pi\)
\(510\) 0 0
\(511\) 4.68380e33 0.653637
\(512\) 3.24519e32 0.0441942
\(513\) −3.26812e33 −0.434341
\(514\) −9.14768e33 −1.18651
\(515\) 0 0
\(516\) 5.73098e32 0.0708122
\(517\) 4.57287e32 0.0551515
\(518\) 1.85444e33 0.218319
\(519\) 1.13890e33 0.130886
\(520\) 0 0
\(521\) −1.16997e34 −1.28146 −0.640730 0.767767i \(-0.721368\pi\)
−0.640730 + 0.767767i \(0.721368\pi\)
\(522\) 3.58580e33 0.383446
\(523\) 2.58251e33 0.269631 0.134816 0.990871i \(-0.456956\pi\)
0.134816 + 0.990871i \(0.456956\pi\)
\(524\) −3.68709e33 −0.375874
\(525\) 0 0
\(526\) 6.85619e33 0.666440
\(527\) 1.84700e33 0.175321
\(528\) −1.53632e31 −0.00142415
\(529\) 1.48316e34 1.34274
\(530\) 0 0
\(531\) −3.42588e32 −0.0295863
\(532\) −3.75714e33 −0.316929
\(533\) 4.78616e33 0.394364
\(534\) 2.05269e33 0.165218
\(535\) 0 0
\(536\) −4.34579e33 −0.333817
\(537\) 1.72691e33 0.129596
\(538\) −6.21946e33 −0.456010
\(539\) −5.42984e32 −0.0388980
\(540\) 0 0
\(541\) 8.08434e33 0.552941 0.276470 0.961022i \(-0.410835\pi\)
0.276470 + 0.961022i \(0.410835\pi\)
\(542\) 1.53022e34 1.02274
\(543\) −1.40210e33 −0.0915760
\(544\) −7.08106e32 −0.0451972
\(545\) 0 0
\(546\) −6.02689e32 −0.0367439
\(547\) 3.07411e34 1.83180 0.915900 0.401408i \(-0.131479\pi\)
0.915900 + 0.401408i \(0.131479\pi\)
\(548\) 5.74491e32 0.0334600
\(549\) 4.54160e33 0.258556
\(550\) 0 0
\(551\) 1.75009e34 0.952061
\(552\) −1.28957e33 −0.0685814
\(553\) 4.61150e33 0.239761
\(554\) −2.00079e34 −1.01702
\(555\) 0 0
\(556\) −4.68240e33 −0.227528
\(557\) 2.42852e34 1.15385 0.576927 0.816796i \(-0.304252\pi\)
0.576927 + 0.816796i \(0.304252\pi\)
\(558\) 1.02690e34 0.477091
\(559\) −2.74871e34 −1.24877
\(560\) 0 0
\(561\) 3.35228e31 0.00145648
\(562\) 1.67063e34 0.709865
\(563\) 2.54352e34 1.05701 0.528507 0.848929i \(-0.322752\pi\)
0.528507 + 0.848929i \(0.322752\pi\)
\(564\) −1.91284e33 −0.0777480
\(565\) 0 0
\(566\) −3.59592e34 −1.39831
\(567\) 9.18411e33 0.349341
\(568\) −1.24594e33 −0.0463599
\(569\) 9.11823e33 0.331900 0.165950 0.986134i \(-0.446931\pi\)
0.165950 + 0.986134i \(0.446931\pi\)
\(570\) 0 0
\(571\) −7.94351e33 −0.276733 −0.138366 0.990381i \(-0.544185\pi\)
−0.138366 + 0.990381i \(0.544185\pi\)
\(572\) 7.36853e32 0.0251148
\(573\) −1.01127e33 −0.0337236
\(574\) 2.80625e33 0.0915646
\(575\) 0 0
\(576\) −3.93694e33 −0.122992
\(577\) −3.99407e33 −0.122101 −0.0610503 0.998135i \(-0.519445\pi\)
−0.0610503 + 0.998135i \(0.519445\pi\)
\(578\) −2.20914e34 −0.660884
\(579\) −1.86912e33 −0.0547210
\(580\) 0 0
\(581\) −9.21430e33 −0.258381
\(582\) −1.38640e33 −0.0380498
\(583\) −2.46207e33 −0.0661369
\(584\) 2.39538e34 0.629816
\(585\) 0 0
\(586\) −1.64685e34 −0.414891
\(587\) 1.28546e34 0.317017 0.158508 0.987358i \(-0.449332\pi\)
0.158508 + 0.987358i \(0.449332\pi\)
\(588\) 2.27131e33 0.0548351
\(589\) 5.01190e34 1.18457
\(590\) 0 0
\(591\) 7.81277e33 0.176995
\(592\) 9.48394e33 0.210362
\(593\) −1.35249e34 −0.293732 −0.146866 0.989156i \(-0.546919\pi\)
−0.146866 + 0.989156i \(0.546919\pi\)
\(594\) 3.75804e32 0.00799154
\(595\) 0 0
\(596\) 2.39757e34 0.488870
\(597\) −2.12122e33 −0.0423552
\(598\) 6.18507e34 1.20943
\(599\) −4.11263e34 −0.787561 −0.393781 0.919204i \(-0.628833\pi\)
−0.393781 + 0.919204i \(0.628833\pi\)
\(600\) 0 0
\(601\) −8.66879e34 −1.59231 −0.796155 0.605092i \(-0.793136\pi\)
−0.796155 + 0.605092i \(0.793136\pi\)
\(602\) −1.61164e34 −0.289942
\(603\) 5.27216e34 0.929013
\(604\) −4.21487e34 −0.727482
\(605\) 0 0
\(606\) 4.04284e33 0.0669543
\(607\) −1.06263e35 −1.72394 −0.861972 0.506956i \(-0.830771\pi\)
−0.861972 + 0.506956i \(0.830771\pi\)
\(608\) −1.92147e34 −0.305378
\(609\) 1.64602e33 0.0256282
\(610\) 0 0
\(611\) 9.17439e34 1.37108
\(612\) 8.59049e33 0.125784
\(613\) −3.14879e34 −0.451740 −0.225870 0.974157i \(-0.572522\pi\)
−0.225870 + 0.974157i \(0.572522\pi\)
\(614\) −1.31414e33 −0.0184729
\(615\) 0 0
\(616\) 4.32036e32 0.00583124
\(617\) 1.34862e35 1.78371 0.891856 0.452320i \(-0.149403\pi\)
0.891856 + 0.452320i \(0.149403\pi\)
\(618\) −5.07877e33 −0.0658268
\(619\) 2.48732e34 0.315936 0.157968 0.987444i \(-0.449506\pi\)
0.157968 + 0.987444i \(0.449506\pi\)
\(620\) 0 0
\(621\) 3.15446e34 0.384840
\(622\) −5.19048e34 −0.620622
\(623\) −5.77248e34 −0.676491
\(624\) −3.08226e33 −0.0354048
\(625\) 0 0
\(626\) 1.13287e35 1.25026
\(627\) 9.09651e32 0.00984080
\(628\) −5.61603e34 −0.595571
\(629\) −2.06941e34 −0.215137
\(630\) 0 0
\(631\) −9.95340e34 −0.994499 −0.497250 0.867607i \(-0.665657\pi\)
−0.497250 + 0.867607i \(0.665657\pi\)
\(632\) 2.35841e34 0.231023
\(633\) 1.85531e34 0.178184
\(634\) 3.32899e34 0.313471
\(635\) 0 0
\(636\) 1.02989e34 0.0932342
\(637\) −1.08937e35 −0.967013
\(638\) −2.01244e33 −0.0175172
\(639\) 1.51153e34 0.129020
\(640\) 0 0
\(641\) 2.11836e33 0.0173890 0.00869452 0.999962i \(-0.497232\pi\)
0.00869452 + 0.999962i \(0.497232\pi\)
\(642\) 1.09001e34 0.0877491
\(643\) −5.48861e34 −0.433336 −0.216668 0.976245i \(-0.569519\pi\)
−0.216668 + 0.976245i \(0.569519\pi\)
\(644\) 3.62647e34 0.280809
\(645\) 0 0
\(646\) 4.19268e34 0.312310
\(647\) 8.23657e34 0.601787 0.300894 0.953658i \(-0.402715\pi\)
0.300894 + 0.953658i \(0.402715\pi\)
\(648\) 4.69692e34 0.336609
\(649\) 1.92269e32 0.00135161
\(650\) 0 0
\(651\) 4.71387e33 0.0318871
\(652\) −1.41376e34 −0.0938170
\(653\) −2.06557e35 −1.34470 −0.672349 0.740235i \(-0.734714\pi\)
−0.672349 + 0.740235i \(0.734714\pi\)
\(654\) 1.51885e34 0.0970050
\(655\) 0 0
\(656\) 1.43517e34 0.0882276
\(657\) −2.90599e35 −1.75278
\(658\) 5.37919e34 0.318341
\(659\) −1.48751e35 −0.863756 −0.431878 0.901932i \(-0.642149\pi\)
−0.431878 + 0.901932i \(0.642149\pi\)
\(660\) 0 0
\(661\) 1.93614e35 1.08248 0.541238 0.840869i \(-0.317956\pi\)
0.541238 + 0.840869i \(0.317956\pi\)
\(662\) −1.32989e35 −0.729610
\(663\) 6.72555e33 0.0362084
\(664\) −4.71236e34 −0.248965
\(665\) 0 0
\(666\) −1.15056e35 −0.585438
\(667\) −1.68922e35 −0.843555
\(668\) −5.18061e34 −0.253907
\(669\) 4.36802e34 0.210116
\(670\) 0 0
\(671\) −2.54886e33 −0.0118118
\(672\) −1.80721e33 −0.00822039
\(673\) 2.68004e35 1.19661 0.598305 0.801268i \(-0.295841\pi\)
0.598305 + 0.801268i \(0.295841\pi\)
\(674\) 2.52357e35 1.10603
\(675\) 0 0
\(676\) 2.94452e34 0.124360
\(677\) −2.37470e35 −0.984578 −0.492289 0.870432i \(-0.663840\pi\)
−0.492289 + 0.870432i \(0.663840\pi\)
\(678\) 4.55170e33 0.0185269
\(679\) 3.89878e34 0.155796
\(680\) 0 0
\(681\) 3.61039e34 0.139064
\(682\) −5.76322e33 −0.0217952
\(683\) 2.30743e35 0.856779 0.428390 0.903594i \(-0.359081\pi\)
0.428390 + 0.903594i \(0.359081\pi\)
\(684\) 2.33106e35 0.849868
\(685\) 0 0
\(686\) −1.37682e35 −0.483980
\(687\) −5.08880e34 −0.175653
\(688\) −8.24222e34 −0.279376
\(689\) −4.93957e35 −1.64418
\(690\) 0 0
\(691\) −1.65717e35 −0.531974 −0.265987 0.963977i \(-0.585698\pi\)
−0.265987 + 0.963977i \(0.585698\pi\)
\(692\) −1.63795e35 −0.516385
\(693\) −5.24131e33 −0.0162283
\(694\) 3.41774e35 1.03931
\(695\) 0 0
\(696\) 8.41804e33 0.0246942
\(697\) −3.13156e34 −0.0902301
\(698\) −4.39545e35 −1.24397
\(699\) 2.37842e34 0.0661185
\(700\) 0 0
\(701\) −4.91864e35 −1.31938 −0.659690 0.751538i \(-0.729312\pi\)
−0.659690 + 0.751538i \(0.729312\pi\)
\(702\) 7.53961e34 0.198671
\(703\) −5.61542e35 −1.45359
\(704\) 2.20951e33 0.00561873
\(705\) 0 0
\(706\) 4.30649e35 1.05697
\(707\) −1.13691e35 −0.274146
\(708\) −8.04263e32 −0.00190538
\(709\) −7.85100e35 −1.82746 −0.913728 0.406326i \(-0.866810\pi\)
−0.913728 + 0.406326i \(0.866810\pi\)
\(710\) 0 0
\(711\) −2.86114e35 −0.642937
\(712\) −2.95215e35 −0.651837
\(713\) −4.83759e35 −1.04957
\(714\) 3.94337e33 0.00840697
\(715\) 0 0
\(716\) −2.48361e35 −0.511294
\(717\) 8.39197e34 0.169775
\(718\) 4.73262e35 0.940906
\(719\) 2.69785e33 0.00527118 0.00263559 0.999997i \(-0.499161\pi\)
0.00263559 + 0.999997i \(0.499161\pi\)
\(720\) 0 0
\(721\) 1.42823e35 0.269530
\(722\) 7.56456e35 1.40304
\(723\) 6.94021e33 0.0126516
\(724\) 2.01648e35 0.361295
\(725\) 0 0
\(726\) 5.16659e34 0.0894326
\(727\) 5.83647e35 0.993048 0.496524 0.868023i \(-0.334609\pi\)
0.496524 + 0.868023i \(0.334609\pi\)
\(728\) 8.66778e34 0.144966
\(729\) −5.50432e35 −0.904918
\(730\) 0 0
\(731\) 1.79847e35 0.285717
\(732\) 1.06619e34 0.0166512
\(733\) 6.31572e35 0.969668 0.484834 0.874606i \(-0.338880\pi\)
0.484834 + 0.874606i \(0.338880\pi\)
\(734\) 2.31488e35 0.349405
\(735\) 0 0
\(736\) 1.85464e35 0.270575
\(737\) −2.95887e34 −0.0424406
\(738\) −1.74109e35 −0.245537
\(739\) 6.44973e35 0.894305 0.447153 0.894458i \(-0.352438\pi\)
0.447153 + 0.894458i \(0.352438\pi\)
\(740\) 0 0
\(741\) 1.82500e35 0.244644
\(742\) −2.89620e35 −0.381750
\(743\) 5.26299e35 0.682138 0.341069 0.940038i \(-0.389211\pi\)
0.341069 + 0.940038i \(0.389211\pi\)
\(744\) 2.41076e34 0.0307250
\(745\) 0 0
\(746\) 5.78040e34 0.0712398
\(747\) 5.71687e35 0.692868
\(748\) −4.82119e33 −0.00574625
\(749\) −3.06527e35 −0.359291
\(750\) 0 0
\(751\) −1.61425e36 −1.83009 −0.915044 0.403355i \(-0.867844\pi\)
−0.915044 + 0.403355i \(0.867844\pi\)
\(752\) 2.75101e35 0.306740
\(753\) 1.30294e35 0.142885
\(754\) −4.03748e35 −0.435481
\(755\) 0 0
\(756\) 4.42067e34 0.0461281
\(757\) 6.23656e35 0.640098 0.320049 0.947401i \(-0.396301\pi\)
0.320049 + 0.947401i \(0.396301\pi\)
\(758\) 1.06625e36 1.07645
\(759\) −8.78014e33 −0.00871926
\(760\) 0 0
\(761\) 4.69451e35 0.451110 0.225555 0.974230i \(-0.427580\pi\)
0.225555 + 0.974230i \(0.427580\pi\)
\(762\) −1.41367e35 −0.133633
\(763\) −4.27125e35 −0.397190
\(764\) 1.45439e35 0.133050
\(765\) 0 0
\(766\) 9.11195e35 0.806774
\(767\) 3.85743e34 0.0336013
\(768\) −9.24240e33 −0.00792081
\(769\) 1.07024e36 0.902402 0.451201 0.892422i \(-0.350996\pi\)
0.451201 + 0.892422i \(0.350996\pi\)
\(770\) 0 0
\(771\) 2.60529e35 0.212655
\(772\) 2.68814e35 0.215891
\(773\) 1.40143e36 1.10746 0.553732 0.832695i \(-0.313203\pi\)
0.553732 + 0.832695i \(0.313203\pi\)
\(774\) 9.99916e35 0.777502
\(775\) 0 0
\(776\) 1.99390e35 0.150118
\(777\) −5.28151e34 −0.0391287
\(778\) 1.16154e35 0.0846816
\(779\) −8.49760e35 −0.609646
\(780\) 0 0
\(781\) −8.48308e33 −0.00589407
\(782\) −4.04686e35 −0.276716
\(783\) −2.05916e35 −0.138570
\(784\) −3.26656e35 −0.216341
\(785\) 0 0
\(786\) 1.05010e35 0.0673670
\(787\) 2.04262e36 1.28974 0.644870 0.764292i \(-0.276911\pi\)
0.644870 + 0.764292i \(0.276911\pi\)
\(788\) −1.12362e36 −0.698300
\(789\) −1.95267e35 −0.119444
\(790\) 0 0
\(791\) −1.28001e35 −0.0758589
\(792\) −2.68050e34 −0.0156369
\(793\) −5.11369e35 −0.293643
\(794\) 8.10456e35 0.458114
\(795\) 0 0
\(796\) 3.05071e35 0.167104
\(797\) 4.60347e35 0.248231 0.124115 0.992268i \(-0.460391\pi\)
0.124115 + 0.992268i \(0.460391\pi\)
\(798\) 1.07005e35 0.0568023
\(799\) −6.00276e35 −0.313701
\(800\) 0 0
\(801\) 3.58145e36 1.81406
\(802\) 2.42417e36 1.20888
\(803\) 1.63091e35 0.0800731
\(804\) 1.23770e35 0.0598292
\(805\) 0 0
\(806\) −1.15625e36 −0.541833
\(807\) 1.77132e35 0.0817294
\(808\) −5.81435e35 −0.264155
\(809\) 2.81919e35 0.126115 0.0630576 0.998010i \(-0.479915\pi\)
0.0630576 + 0.998010i \(0.479915\pi\)
\(810\) 0 0
\(811\) 8.66756e35 0.375955 0.187978 0.982173i \(-0.439807\pi\)
0.187978 + 0.982173i \(0.439807\pi\)
\(812\) −2.36728e35 −0.101111
\(813\) −4.35812e35 −0.183302
\(814\) 6.45722e34 0.0267449
\(815\) 0 0
\(816\) 2.01671e34 0.00810058
\(817\) 4.88020e36 1.93047
\(818\) 2.08936e36 0.813950
\(819\) −1.05154e36 −0.403440
\(820\) 0 0
\(821\) −2.61490e36 −0.973120 −0.486560 0.873647i \(-0.661748\pi\)
−0.486560 + 0.873647i \(0.661748\pi\)
\(822\) −1.63617e34 −0.00599695
\(823\) −3.63703e35 −0.131295 −0.0656477 0.997843i \(-0.520911\pi\)
−0.0656477 + 0.997843i \(0.520911\pi\)
\(824\) 7.30422e35 0.259707
\(825\) 0 0
\(826\) 2.26171e34 0.00780165
\(827\) −3.87599e36 −1.31693 −0.658467 0.752610i \(-0.728795\pi\)
−0.658467 + 0.752610i \(0.728795\pi\)
\(828\) −2.24998e36 −0.753009
\(829\) 1.44009e36 0.474743 0.237371 0.971419i \(-0.423714\pi\)
0.237371 + 0.971419i \(0.423714\pi\)
\(830\) 0 0
\(831\) 5.69832e35 0.182278
\(832\) 4.43286e35 0.139683
\(833\) 7.12769e35 0.221252
\(834\) 1.33356e35 0.0407792
\(835\) 0 0
\(836\) −1.30825e35 −0.0388250
\(837\) −5.89704e35 −0.172411
\(838\) 1.41577e36 0.407795
\(839\) −2.94352e36 −0.835300 −0.417650 0.908608i \(-0.637146\pi\)
−0.417650 + 0.908608i \(0.637146\pi\)
\(840\) 0 0
\(841\) −2.52767e36 −0.696260
\(842\) 4.72088e34 0.0128121
\(843\) −4.75800e35 −0.127227
\(844\) −2.66828e36 −0.702992
\(845\) 0 0
\(846\) −3.33743e36 −0.853656
\(847\) −1.45292e36 −0.366184
\(848\) −1.48117e36 −0.367837
\(849\) 1.02413e36 0.250616
\(850\) 0 0
\(851\) 5.42013e36 1.28792
\(852\) 3.54848e34 0.00830897
\(853\) 1.60861e36 0.371183 0.185591 0.982627i \(-0.440580\pi\)
0.185591 + 0.982627i \(0.440580\pi\)
\(854\) −2.99829e35 −0.0681789
\(855\) 0 0
\(856\) −1.56764e36 −0.346197
\(857\) −7.04998e36 −1.53436 −0.767180 0.641432i \(-0.778341\pi\)
−0.767180 + 0.641432i \(0.778341\pi\)
\(858\) −2.09858e34 −0.00450127
\(859\) 6.59563e36 1.39425 0.697126 0.716948i \(-0.254462\pi\)
0.697126 + 0.716948i \(0.254462\pi\)
\(860\) 0 0
\(861\) −7.99230e34 −0.0164109
\(862\) −4.61133e36 −0.933222
\(863\) 9.41312e36 1.87758 0.938791 0.344487i \(-0.111947\pi\)
0.938791 + 0.344487i \(0.111947\pi\)
\(864\) 2.26081e35 0.0444470
\(865\) 0 0
\(866\) −2.17542e36 −0.415499
\(867\) 6.29170e35 0.118448
\(868\) −6.77942e35 −0.125805
\(869\) 1.60574e35 0.0293716
\(870\) 0 0
\(871\) −5.93627e36 −1.05508
\(872\) −2.18439e36 −0.382714
\(873\) −2.41893e36 −0.417778
\(874\) −1.09813e37 −1.86965
\(875\) 0 0
\(876\) −6.82212e35 −0.112880
\(877\) −4.23262e36 −0.690422 −0.345211 0.938525i \(-0.612193\pi\)
−0.345211 + 0.938525i \(0.612193\pi\)
\(878\) 3.23521e36 0.520261
\(879\) 4.69029e35 0.0743598
\(880\) 0 0
\(881\) −9.23339e36 −1.42286 −0.711431 0.702756i \(-0.751952\pi\)
−0.711431 + 0.702756i \(0.751952\pi\)
\(882\) 3.96287e36 0.602078
\(883\) −9.76831e36 −1.46322 −0.731612 0.681721i \(-0.761232\pi\)
−0.731612 + 0.681721i \(0.761232\pi\)
\(884\) −9.67259e35 −0.142853
\(885\) 0 0
\(886\) 7.48575e36 1.07477
\(887\) 1.62985e36 0.230729 0.115365 0.993323i \(-0.463196\pi\)
0.115365 + 0.993323i \(0.463196\pi\)
\(888\) −2.70106e35 −0.0377027
\(889\) 3.97547e36 0.547162
\(890\) 0 0
\(891\) 3.19794e35 0.0427955
\(892\) −6.28202e36 −0.828970
\(893\) −1.62887e37 −2.11955
\(894\) −6.82836e35 −0.0876188
\(895\) 0 0
\(896\) 2.59910e35 0.0324320
\(897\) −1.76153e36 −0.216763
\(898\) −3.41071e35 −0.0413895
\(899\) 3.15788e36 0.377919
\(900\) 0 0
\(901\) 3.23194e36 0.376186
\(902\) 9.77145e34 0.0112170
\(903\) 4.59000e35 0.0519656
\(904\) −6.54619e35 −0.0730943
\(905\) 0 0
\(906\) 1.20041e36 0.130385
\(907\) −4.47773e36 −0.479697 −0.239848 0.970810i \(-0.577098\pi\)
−0.239848 + 0.970810i \(0.577098\pi\)
\(908\) −5.19241e36 −0.548651
\(909\) 7.05376e36 0.735144
\(910\) 0 0
\(911\) 9.15716e36 0.928498 0.464249 0.885705i \(-0.346324\pi\)
0.464249 + 0.885705i \(0.346324\pi\)
\(912\) 5.47241e35 0.0547322
\(913\) −3.20845e35 −0.0316527
\(914\) −8.28891e36 −0.806622
\(915\) 0 0
\(916\) 7.31864e36 0.693006
\(917\) −2.95303e36 −0.275836
\(918\) −4.93313e35 −0.0454558
\(919\) 1.60416e36 0.145816 0.0729078 0.997339i \(-0.476772\pi\)
0.0729078 + 0.997339i \(0.476772\pi\)
\(920\) 0 0
\(921\) 3.74270e34 0.00331085
\(922\) −3.43100e36 −0.299422
\(923\) −1.70193e36 −0.146528
\(924\) −1.23045e34 −0.00104512
\(925\) 0 0
\(926\) −1.38047e37 −1.14127
\(927\) −8.86121e36 −0.722764
\(928\) −1.21067e36 −0.0974263
\(929\) −8.64983e35 −0.0686770 −0.0343385 0.999410i \(-0.510932\pi\)
−0.0343385 + 0.999410i \(0.510932\pi\)
\(930\) 0 0
\(931\) 1.93412e37 1.49490
\(932\) −3.42060e36 −0.260858
\(933\) 1.47826e36 0.111232
\(934\) −1.14510e37 −0.850170
\(935\) 0 0
\(936\) −5.37779e36 −0.388737
\(937\) 7.42933e36 0.529913 0.264956 0.964260i \(-0.414642\pi\)
0.264956 + 0.964260i \(0.414642\pi\)
\(938\) −3.48059e36 −0.244972
\(939\) −3.22644e36 −0.224081
\(940\) 0 0
\(941\) 6.99775e36 0.473247 0.236623 0.971601i \(-0.423959\pi\)
0.236623 + 0.971601i \(0.423959\pi\)
\(942\) 1.59946e36 0.106743
\(943\) 8.20206e36 0.540166
\(944\) 1.15668e35 0.00751732
\(945\) 0 0
\(946\) −5.61178e35 −0.0355190
\(947\) −9.94818e36 −0.621397 −0.310698 0.950509i \(-0.600563\pi\)
−0.310698 + 0.950509i \(0.600563\pi\)
\(948\) −6.71682e35 −0.0414057
\(949\) 3.27204e37 1.99063
\(950\) 0 0
\(951\) −9.48108e35 −0.0561825
\(952\) −5.67129e35 −0.0331681
\(953\) 1.89836e37 1.09577 0.547883 0.836555i \(-0.315434\pi\)
0.547883 + 0.836555i \(0.315434\pi\)
\(954\) 1.79690e37 1.02369
\(955\) 0 0
\(956\) −1.20692e37 −0.669816
\(957\) 5.73149e34 0.00313956
\(958\) 2.47068e37 1.33582
\(959\) 4.60115e35 0.0245547
\(960\) 0 0
\(961\) −1.01893e37 −0.529785
\(962\) 1.29549e37 0.664883
\(963\) 1.90180e37 0.963466
\(964\) −9.98131e35 −0.0499142
\(965\) 0 0
\(966\) −1.03283e36 −0.0503286
\(967\) −1.99094e37 −0.957693 −0.478846 0.877899i \(-0.658945\pi\)
−0.478846 + 0.877899i \(0.658945\pi\)
\(968\) −7.43051e36 −0.352839
\(969\) −1.19409e36 −0.0559744
\(970\) 0 0
\(971\) 1.95878e37 0.894841 0.447421 0.894324i \(-0.352343\pi\)
0.447421 + 0.894324i \(0.352343\pi\)
\(972\) −4.12521e36 −0.186045
\(973\) −3.75018e36 −0.166971
\(974\) −1.08023e37 −0.474822
\(975\) 0 0
\(976\) −1.53338e36 −0.0656942
\(977\) 2.97587e37 1.25873 0.629365 0.777110i \(-0.283315\pi\)
0.629365 + 0.777110i \(0.283315\pi\)
\(978\) 4.02644e35 0.0168146
\(979\) −2.01000e36 −0.0828727
\(980\) 0 0
\(981\) 2.65003e37 1.06509
\(982\) 1.88156e37 0.746663
\(983\) −4.05260e37 −1.58787 −0.793934 0.608004i \(-0.791970\pi\)
−0.793934 + 0.608004i \(0.791970\pi\)
\(984\) −4.08741e35 −0.0158128
\(985\) 0 0
\(986\) 2.64171e36 0.0996376
\(987\) −1.53201e36 −0.0570555
\(988\) −2.62469e37 −0.965197
\(989\) −4.71047e37 −1.71045
\(990\) 0 0
\(991\) −3.58276e37 −1.26852 −0.634260 0.773120i \(-0.718695\pi\)
−0.634260 + 0.773120i \(0.718695\pi\)
\(992\) −3.46712e36 −0.121220
\(993\) 3.78757e36 0.130766
\(994\) −9.97886e35 −0.0340213
\(995\) 0 0
\(996\) 1.34210e36 0.0446213
\(997\) 9.64459e36 0.316661 0.158330 0.987386i \(-0.449389\pi\)
0.158330 + 0.987386i \(0.449389\pi\)
\(998\) 1.75638e37 0.569491
\(999\) 6.60714e36 0.211566
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.1.4 6
5.2 odd 4 10.26.b.a.9.9 yes 12
5.3 odd 4 10.26.b.a.9.4 12
5.4 even 2 50.26.a.k.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.26.b.a.9.4 12 5.3 odd 4
10.26.b.a.9.9 yes 12 5.2 odd 4
50.26.a.k.1.3 6 5.4 even 2
50.26.a.l.1.4 6 1.1 even 1 trivial