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Properties

Label 1.122.a.a.1.6
Level $1$
Weight $122$
Character 1.1
Self dual yes
Analytic conductor $92.717$
Analytic rank $1$
Dimension $9$
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] = [1,122,Mod(1,1)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1, base_ring=CyclotomicField(1))
 
chi = DirichletCharacter(H, H._module([]))
 
N = Newforms(chi, 122, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1.1");
 
S:= CuspForms(chi, 122);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 122 \)
Character orbit: \([\chi]\) \(=\) 1.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: \(92.7173263878\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 2 x^{8} + \cdots + 32\!\cdots\!74 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: multiple of \( 2^{145}\cdot 3^{53}\cdot 5^{20}\cdot 7^{8}\cdot 11^{6}\cdot 13^{2}\cdot 17^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(1.54621e16\) of defining polynomial
Character \(\chi\) \(=\) 1.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.81626e17 q^{2} -2.96062e28 q^{3} -2.42649e36 q^{4} +2.39673e42 q^{5} -1.42591e46 q^{6} +2.39069e50 q^{7} -2.44905e54 q^{8} -4.51451e57 q^{9} +O(q^{10})\) \(q+4.81626e17 q^{2} -2.96062e28 q^{3} -2.42649e36 q^{4} +2.39673e42 q^{5} -1.42591e46 q^{6} +2.39069e50 q^{7} -2.44905e54 q^{8} -4.51451e57 q^{9} +1.15433e60 q^{10} +9.14826e62 q^{11} +7.18391e64 q^{12} -4.47361e66 q^{13} +1.15142e68 q^{14} -7.09578e70 q^{15} +5.27120e72 q^{16} -1.65789e74 q^{17} -2.17431e75 q^{18} -2.27364e77 q^{19} -5.81563e78 q^{20} -7.07793e78 q^{21} +4.40605e80 q^{22} +4.15067e82 q^{23} +7.25068e82 q^{24} +1.98271e84 q^{25} -2.15461e84 q^{26} +2.93265e86 q^{27} -5.80100e86 q^{28} -3.28021e88 q^{29} -3.41752e88 q^{30} +2.45303e90 q^{31} +9.04943e90 q^{32} -2.70845e91 q^{33} -7.98482e91 q^{34} +5.72984e92 q^{35} +1.09544e94 q^{36} -4.62213e94 q^{37} -1.09505e95 q^{38} +1.32446e95 q^{39} -5.86969e96 q^{40} +2.48576e97 q^{41} -3.40892e96 q^{42} +5.78279e98 q^{43} -2.21982e99 q^{44} -1.08200e100 q^{45} +1.99907e100 q^{46} +3.23763e100 q^{47} -1.56060e101 q^{48} -1.74945e102 q^{49} +9.54925e101 q^{50} +4.90837e102 q^{51} +1.08552e103 q^{52} -3.25936e104 q^{53} +1.41244e104 q^{54} +2.19259e105 q^{55} -5.85492e104 q^{56} +6.73139e105 q^{57} -1.57984e106 q^{58} -2.10859e106 q^{59} +1.72179e107 q^{60} -4.56941e107 q^{61} +1.18144e108 q^{62} -1.07928e108 q^{63} -9.65480e108 q^{64} -1.07220e109 q^{65} -1.30446e109 q^{66} +2.01613e110 q^{67} +4.02285e110 q^{68} -1.22885e111 q^{69} +2.75964e110 q^{70} +2.53468e111 q^{71} +1.10562e112 q^{72} -5.38692e112 q^{73} -2.22614e112 q^{74} -5.87004e112 q^{75} +5.51698e113 q^{76} +2.18707e113 q^{77} +6.37896e112 q^{78} -9.50620e114 q^{79} +1.26336e115 q^{80} +1.56554e115 q^{81} +1.19721e115 q^{82} -1.55348e116 q^{83} +1.71745e115 q^{84} -3.97350e116 q^{85} +2.78515e116 q^{86} +9.71145e116 q^{87} -2.24045e117 q^{88} -3.29777e117 q^{89} -5.21121e117 q^{90} -1.06950e117 q^{91} -1.00716e119 q^{92} -7.26248e118 q^{93} +1.55933e118 q^{94} -5.44930e119 q^{95} -2.67919e119 q^{96} -2.32897e120 q^{97} -8.42581e119 q^{98} -4.12999e120 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 23\!\cdots\!28 q^{2}+ \cdots + 79\!\cdots\!17 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 23\!\cdots\!28 q^{2}+ \cdots - 44\!\cdots\!96 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.81626e17 0.295390 0.147695 − 0.989033i \(-0.452815\pi\)
0.147695 + 0.989033i \(0.452815\pi\)
\(3\) −2.96062e28 −0.403224 −0.201612 − 0.979465i \(-0.564618\pi\)
−0.201612 + 0.979465i \(0.564618\pi\)
\(4\) −2.42649e36 −0.912745
\(5\) 2.39673e42 1.23576 0.617878 − 0.786274i \(-0.287992\pi\)
0.617878 + 0.786274i \(0.287992\pi\)
\(6\) −1.42591e46 −0.119108
\(7\) 2.39069e50 0.177866 0.0889329 − 0.996038i \(-0.471654\pi\)
0.0889329 + 0.996038i \(0.471654\pi\)
\(8\) −2.44905e54 −0.565005
\(9\) −4.51451e57 −0.837411
\(10\) 1.15433e60 0.365030
\(11\) 9.14826e62 0.905902 0.452951 − 0.891535i \(-0.350371\pi\)
0.452951 + 0.891535i \(0.350371\pi\)
\(12\) 7.18391e64 0.368041
\(13\) −4.47361e66 −0.180753 −0.0903764 − 0.995908i \(-0.528807\pi\)
−0.0903764 + 0.995908i \(0.528807\pi\)
\(14\) 1.15142e68 0.0525398
\(15\) −7.09578e70 −0.498287
\(16\) 5.27120e72 0.745848
\(17\) −1.65789e74 −0.598957 −0.299478 − 0.954103i \(-0.596813\pi\)
−0.299478 + 0.954103i \(0.596813\pi\)
\(18\) −2.17431e75 −0.247363
\(19\) −2.27364e77 −0.982041 −0.491020 − 0.871148i \(-0.663376\pi\)
−0.491020 + 0.871148i \(0.663376\pi\)
\(20\) −5.81563e78 −1.12793
\(21\) −7.07793e78 −0.0717198
\(22\) 4.40605e80 0.267594
\(23\) 4.15067e82 1.71232 0.856158 − 0.516714i \(-0.172845\pi\)
0.856158 + 0.516714i \(0.172845\pi\)
\(24\) 7.25068e82 0.227824
\(25\) 1.98271e84 0.527095
\(26\) −2.15461e84 −0.0533925
\(27\) 2.93265e86 0.740888
\(28\) −5.80100e86 −0.162346
\(29\) −3.28021e88 −1.09856 −0.549279 − 0.835639i \(-0.685098\pi\)
−0.549279 + 0.835639i \(0.685098\pi\)
\(30\) −3.41752e88 −0.147189
\(31\) 2.45303e90 1.45318 0.726591 − 0.687070i \(-0.241103\pi\)
0.726591 + 0.687070i \(0.241103\pi\)
\(32\) 9.04943e90 0.785321
\(33\) −2.70845e91 −0.365281
\(34\) −7.98482e91 −0.176926
\(35\) 5.72984e92 0.219799
\(36\) 1.09544e94 0.764342
\(37\) −4.62213e94 −0.614664 −0.307332 − 0.951602i \(-0.599436\pi\)
−0.307332 + 0.951602i \(0.599436\pi\)
\(38\) −1.09505e95 −0.290085
\(39\) 1.32446e95 0.0728838
\(40\) −5.86969e96 −0.698209
\(41\) 2.48576e97 0.663799 0.331899 − 0.943315i \(-0.392311\pi\)
0.331899 + 0.943315i \(0.392311\pi\)
\(42\) −3.40892e96 −0.0211853
\(43\) 5.78279e98 0.865558 0.432779 − 0.901500i \(-0.357533\pi\)
0.432779 + 0.901500i \(0.357533\pi\)
\(44\) −2.21982e99 −0.826857
\(45\) −1.08200e100 −1.03484
\(46\) 1.99907e100 0.505801
\(47\) 3.23763e100 0.223001 0.111501 − 0.993764i \(-0.464434\pi\)
0.111501 + 0.993764i \(0.464434\pi\)
\(48\) −1.56060e101 −0.300744
\(49\) −1.74945e102 −0.968364
\(50\) 9.54925e101 0.155698
\(51\) 4.90837e102 0.241514
\(52\) 1.08552e103 0.164981
\(53\) −3.25936e104 −1.56474 −0.782370 − 0.622814i \(-0.785989\pi\)
−0.782370 + 0.622814i \(0.785989\pi\)
\(54\) 1.41244e104 0.218851
\(55\) 2.19259e105 1.11947
\(56\) −5.85492e104 −0.100495
\(57\) 6.73139e105 0.395982
\(58\) −1.57984e106 −0.324503
\(59\) −2.10859e106 −0.153973 −0.0769866 − 0.997032i \(-0.524530\pi\)
−0.0769866 + 0.997032i \(0.524530\pi\)
\(60\) 1.72179e107 0.454809
\(61\) −4.56941e107 −0.444022 −0.222011 − 0.975044i \(-0.571262\pi\)
−0.222011 + 0.975044i \(0.571262\pi\)
\(62\) 1.18144e108 0.429255
\(63\) −1.07928e108 −0.148947
\(64\) −9.65480e108 −0.513872
\(65\) −1.07220e109 −0.223366
\(66\) −1.30446e109 −0.107900
\(67\) 2.01613e110 0.671419 0.335710 − 0.941966i \(-0.391024\pi\)
0.335710 + 0.941966i \(0.391024\pi\)
\(68\) 4.02285e110 0.546695
\(69\) −1.22885e111 −0.690447
\(70\) 2.75964e110 0.0649264
\(71\) 2.53468e111 0.252809 0.126404 − 0.991979i \(-0.459656\pi\)
0.126404 + 0.991979i \(0.459656\pi\)
\(72\) 1.10562e112 0.473141
\(73\) −5.38692e112 −1.00071 −0.500354 − 0.865821i \(-0.666797\pi\)
−0.500354 + 0.865821i \(0.666797\pi\)
\(74\) −2.22614e112 −0.181565
\(75\) −5.87004e112 −0.212537
\(76\) 5.51698e113 0.896353
\(77\) 2.18707e113 0.161129
\(78\) 6.37896e112 0.0215291
\(79\) −9.50620e114 −1.48446 −0.742229 − 0.670147i \(-0.766231\pi\)
−0.742229 + 0.670147i \(0.766231\pi\)
\(80\) 1.26336e115 0.921687
\(81\) 1.56554e115 0.538667
\(82\) 1.19721e115 0.196079
\(83\) −1.55348e116 −1.22203 −0.611013 − 0.791621i \(-0.709238\pi\)
−0.611013 + 0.791621i \(0.709238\pi\)
\(84\) 1.71745e115 0.0654618
\(85\) −3.97350e116 −0.740165
\(86\) 2.78515e116 0.255677
\(87\) 9.71145e116 0.442965
\(88\) −2.24045e117 −0.511839
\(89\) −3.29777e117 −0.380298 −0.190149 − 0.981755i \(-0.560897\pi\)
−0.190149 + 0.981755i \(0.560897\pi\)
\(90\) −5.21121e117 −0.305680
\(91\) −1.06950e117 −0.0321497
\(92\) −1.00716e119 −1.56291
\(93\) −7.26248e118 −0.585958
\(94\) 1.55933e118 0.0658723
\(95\) −5.44930e119 −1.21356
\(96\) −2.67919e119 −0.316660
\(97\) −2.32897e120 −1.47053 −0.735266 − 0.677778i \(-0.762943\pi\)
−0.735266 + 0.677778i \(0.762943\pi\)
\(98\) −8.42581e119 −0.286045
\(99\) −4.12999e120 −0.758612
\(100\) −4.81103e120 −0.481103
\(101\) −3.31517e121 −1.81578 −0.907889 − 0.419211i \(-0.862307\pi\)
−0.907889 + 0.419211i \(0.862307\pi\)
\(102\) 2.36400e120 0.0713407
\(103\) −1.01157e122 −1.69178 −0.845888 − 0.533361i \(-0.820929\pi\)
−0.845888 + 0.533361i \(0.820929\pi\)
\(104\) 1.09561e121 0.102126
\(105\) −1.69638e121 −0.0886282
\(106\) −1.56979e122 −0.462208
\(107\) −2.97148e122 −0.495740 −0.247870 − 0.968793i \(-0.579731\pi\)
−0.247870 + 0.968793i \(0.579731\pi\)
\(108\) −7.11605e122 −0.676241
\(109\) 3.57150e123 1.94334 0.971668 − 0.236348i \(-0.0759506\pi\)
0.971668 + 0.236348i \(0.0759506\pi\)
\(110\) 1.05601e123 0.330681
\(111\) 1.36843e123 0.247847
\(112\) 1.26018e123 0.132661
\(113\) −1.06541e123 −0.0655042 −0.0327521 − 0.999464i \(-0.510427\pi\)
−0.0327521 + 0.999464i \(0.510427\pi\)
\(114\) 3.24201e123 0.116969
\(115\) 9.94801e124 2.11601
\(116\) 7.95941e124 1.00270
\(117\) 2.01961e124 0.151364
\(118\) −1.01555e124 −0.0454821
\(119\) −3.96350e124 −0.106534
\(120\) 1.73779e125 0.281535
\(121\) −1.82893e125 −0.179342
\(122\) −2.20075e125 −0.131160
\(123\) −7.35939e125 −0.267659
\(124\) −5.95225e126 −1.32638
\(125\) −4.26347e126 −0.584396
\(126\) −5.19810e125 −0.0439974
\(127\) −2.78614e126 −0.146177 −0.0730884 − 0.997325i \(-0.523286\pi\)
−0.0730884 + 0.997325i \(0.523286\pi\)
\(128\) −2.87075e127 −0.937114
\(129\) −1.71206e127 −0.349014
\(130\) −5.16400e126 −0.0659802
\(131\) 8.65837e126 0.0695861 0.0347931 − 0.999395i \(-0.488923\pi\)
0.0347931 + 0.999395i \(0.488923\pi\)
\(132\) 6.57203e127 0.333409
\(133\) −5.43559e127 −0.174672
\(134\) 9.71023e127 0.198330
\(135\) 7.02875e128 0.915557
\(136\) 4.06024e128 0.338414
\(137\) 2.18306e129 1.16808 0.584040 − 0.811725i \(-0.301471\pi\)
0.584040 + 0.811725i \(0.301471\pi\)
\(138\) −5.91848e128 −0.203951
\(139\) 5.08860e129 1.13293 0.566464 − 0.824086i \(-0.308311\pi\)
0.566464 + 0.824086i \(0.308311\pi\)
\(140\) −1.39034e129 −0.200620
\(141\) −9.58538e128 −0.0899194
\(142\) 1.22077e129 0.0746772
\(143\) −4.09257e129 −0.163744
\(144\) −2.37969e130 −0.624581
\(145\) −7.86177e130 −1.35755
\(146\) −2.59448e130 −0.295599
\(147\) 5.17945e130 0.390467
\(148\) 1.12156e131 0.561031
\(149\) −4.00517e131 −1.33307 −0.666533 − 0.745475i \(-0.732223\pi\)
−0.666533 + 0.745475i \(0.732223\pi\)
\(150\) −2.82717e130 −0.0627813
\(151\) −8.46130e131 −1.25699 −0.628495 − 0.777814i \(-0.716329\pi\)
−0.628495 + 0.777814i \(0.716329\pi\)
\(152\) 5.56826e131 0.554858
\(153\) 7.48454e131 0.501573
\(154\) 1.05335e131 0.0475959
\(155\) 5.87924e132 1.79578
\(156\) −3.21380e131 −0.0665243
\(157\) 1.48703e132 0.209118 0.104559 − 0.994519i \(-0.466657\pi\)
0.104559 + 0.994519i \(0.466657\pi\)
\(158\) −4.57843e132 −0.438494
\(159\) 9.64971e132 0.630940
\(160\) 2.16890e133 0.970466
\(161\) 9.92298e132 0.304563
\(162\) 7.54005e132 0.159117
\(163\) −9.68673e133 −1.40873 −0.704364 − 0.709839i \(-0.748768\pi\)
−0.704364 + 0.709839i \(0.748768\pi\)
\(164\) −6.03168e133 −0.605879
\(165\) −6.49141e133 −0.451399
\(166\) −7.48196e133 −0.360974
\(167\) 9.17548e132 0.0307810 0.0153905 − 0.999882i \(-0.495101\pi\)
0.0153905 + 0.999882i \(0.495101\pi\)
\(168\) 1.73342e133 0.0405220
\(169\) −5.92543e134 −0.967328
\(170\) −1.91374e134 −0.218637
\(171\) 1.02644e135 0.822371
\(172\) −1.40319e135 −0.790033
\(173\) 2.80319e135 1.11138 0.555691 − 0.831389i \(-0.312454\pi\)
0.555691 + 0.831389i \(0.312454\pi\)
\(174\) 4.67729e134 0.130847
\(175\) 4.74005e134 0.0937521
\(176\) 4.82223e135 0.675665
\(177\) 6.24272e134 0.0620857
\(178\) −1.58829e135 −0.112336
\(179\) 2.54719e136 1.28367 0.641834 − 0.766844i \(-0.278174\pi\)
0.641834 + 0.766844i \(0.278174\pi\)
\(180\) 2.62547e136 0.944541
\(181\) −5.58765e136 −1.43772 −0.718862 − 0.695153i \(-0.755337\pi\)
−0.718862 + 0.695153i \(0.755337\pi\)
\(182\) −5.15101e134 −0.00949671
\(183\) 1.35283e136 0.179040
\(184\) −1.01652e137 −0.967468
\(185\) −1.10780e137 −0.759575
\(186\) −3.49780e136 −0.173086
\(187\) −1.51668e137 −0.542596
\(188\) −7.85609e136 −0.203543
\(189\) 7.01107e136 0.131779
\(190\) −2.62453e137 −0.358474
\(191\) 8.90666e137 0.885516 0.442758 − 0.896641i \(-0.354000\pi\)
0.442758 + 0.896641i \(0.354000\pi\)
\(192\) 2.85842e137 0.207205
\(193\) 5.31475e137 0.281362 0.140681 − 0.990055i \(-0.455071\pi\)
0.140681 + 0.990055i \(0.455071\pi\)
\(194\) −1.12170e138 −0.434380
\(195\) 3.17437e137 0.0900667
\(196\) 4.24502e138 0.883869
\(197\) 1.00309e139 1.53509 0.767547 − 0.640992i \(-0.221477\pi\)
0.767547 + 0.640992i \(0.221477\pi\)
\(198\) −1.98911e138 −0.224086
\(199\) −2.30953e138 −0.191828 −0.0959140 − 0.995390i \(-0.530577\pi\)
−0.0959140 + 0.995390i \(0.530577\pi\)
\(200\) −4.85575e138 −0.297811
\(201\) −5.96900e138 −0.270732
\(202\) −1.59667e139 −0.536362
\(203\) −7.84198e138 −0.195396
\(204\) −1.19101e139 −0.220440
\(205\) 5.95769e139 0.820294
\(206\) −4.87198e139 −0.499733
\(207\) −1.87382e140 −1.43391
\(208\) −2.35813e139 −0.134814
\(209\) −2.07999e140 −0.889633
\(210\) −8.17024e138 −0.0261799
\(211\) 4.18462e140 1.00593 0.502965 − 0.864307i \(-0.332243\pi\)
0.502965 + 0.864307i \(0.332243\pi\)
\(212\) 7.90881e140 1.42821
\(213\) −7.50420e139 −0.101939
\(214\) −1.43114e140 −0.146436
\(215\) 1.38598e141 1.06962
\(216\) −7.18219e140 −0.418606
\(217\) 5.86444e140 0.258472
\(218\) 1.72013e141 0.574042
\(219\) 1.59486e141 0.403509
\(220\) −5.32030e141 −1.02179
\(221\) 7.41674e140 0.108263
\(222\) 6.59074e140 0.0732115
\(223\) −1.83108e142 −1.54976 −0.774879 − 0.632110i \(-0.782189\pi\)
−0.774879 + 0.632110i \(0.782189\pi\)
\(224\) 2.16344e141 0.139682
\(225\) −8.95095e141 −0.441395
\(226\) −5.13128e140 −0.0193493
\(227\) 3.32910e142 0.961087 0.480544 − 0.876971i \(-0.340439\pi\)
0.480544 + 0.876971i \(0.340439\pi\)
\(228\) −1.63337e142 −0.361431
\(229\) −1.02387e143 −1.73860 −0.869301 − 0.494283i \(-0.835431\pi\)
−0.869301 + 0.494283i \(0.835431\pi\)
\(230\) 4.79122e142 0.625047
\(231\) −6.47508e141 −0.0649711
\(232\) 8.03339e142 0.620691
\(233\) 1.97016e143 1.17346 0.586729 − 0.809783i \(-0.300415\pi\)
0.586729 + 0.809783i \(0.300415\pi\)
\(234\) 9.72699e141 0.0447115
\(235\) 7.75971e142 0.275575
\(236\) 5.11647e142 0.140538
\(237\) 2.81442e143 0.598569
\(238\) −1.90893e142 −0.0314690
\(239\) −1.98705e142 −0.0254176 −0.0127088 − 0.999919i \(-0.504045\pi\)
−0.0127088 + 0.999919i \(0.504045\pi\)
\(240\) −3.74033e143 −0.371646
\(241\) −1.96735e143 −0.152002 −0.0760011 − 0.997108i \(-0.524215\pi\)
−0.0760011 + 0.997108i \(0.524215\pi\)
\(242\) −8.80859e142 −0.0529757
\(243\) −2.04450e144 −0.958091
\(244\) 1.10876e144 0.405279
\(245\) −4.19295e144 −1.19666
\(246\) −3.54448e143 −0.0790639
\(247\) 1.01714e144 0.177507
\(248\) −6.00758e144 −0.821056
\(249\) 4.59925e144 0.492750
\(250\) −2.05340e144 −0.172625
\(251\) 8.51453e144 0.562213 0.281106 − 0.959677i \(-0.409299\pi\)
0.281106 + 0.959677i \(0.409299\pi\)
\(252\) 2.61886e144 0.135950
\(253\) 3.79714e145 1.55119
\(254\) −1.34188e144 −0.0431791
\(255\) 1.17640e145 0.298452
\(256\) 1.18406e145 0.237058
\(257\) −5.10254e145 −0.806923 −0.403462 − 0.914997i \(-0.632193\pi\)
−0.403462 + 0.914997i \(0.632193\pi\)
\(258\) −8.24575e144 −0.103095
\(259\) −1.10501e145 −0.109328
\(260\) 2.60169e145 0.203877
\(261\) 1.48085e146 0.919944
\(262\) 4.17010e144 0.0205550
\(263\) −2.72719e146 −1.06756 −0.533779 − 0.845624i \(-0.679229\pi\)
−0.533779 + 0.845624i \(0.679229\pi\)
\(264\) 6.63312e145 0.206386
\(265\) −7.81179e146 −1.93364
\(266\) −2.61792e145 −0.0515962
\(267\) 9.76342e145 0.153345
\(268\) −4.89213e146 −0.612834
\(269\) 7.34919e146 0.734898 0.367449 − 0.930044i \(-0.380231\pi\)
0.367449 + 0.930044i \(0.380231\pi\)
\(270\) 3.38523e146 0.270446
\(271\) 1.09818e147 0.701507 0.350753 − 0.936468i \(-0.385926\pi\)
0.350753 + 0.936468i \(0.385926\pi\)
\(272\) −8.73905e146 −0.446731
\(273\) 3.16639e145 0.0129635
\(274\) 1.05142e147 0.345039
\(275\) 1.81384e147 0.477496
\(276\) 2.98180e147 0.630202
\(277\) 6.14463e147 1.04344 0.521722 − 0.853115i \(-0.325290\pi\)
0.521722 + 0.853115i \(0.325290\pi\)
\(278\) 2.45080e147 0.334656
\(279\) −1.10742e148 −1.21691
\(280\) −1.40326e147 −0.124188
\(281\) −3.44821e147 −0.245958 −0.122979 − 0.992409i \(-0.539245\pi\)
−0.122979 + 0.992409i \(0.539245\pi\)
\(282\) −4.61657e146 −0.0265613
\(283\) −2.64734e148 −1.22951 −0.614753 − 0.788719i \(-0.710744\pi\)
−0.614753 + 0.788719i \(0.710744\pi\)
\(284\) −6.15037e147 −0.230750
\(285\) 1.61333e148 0.489338
\(286\) −1.97109e147 −0.0483684
\(287\) 5.94270e147 0.118067
\(288\) −4.08537e148 −0.657636
\(289\) −4.91301e148 −0.641251
\(290\) −3.78643e148 −0.401007
\(291\) 6.89520e148 0.592954
\(292\) 1.30713e149 0.913391
\(293\) 1.41343e149 0.803124 0.401562 − 0.915832i \(-0.368467\pi\)
0.401562 + 0.915832i \(0.368467\pi\)
\(294\) 2.49456e148 0.115340
\(295\) −5.05371e148 −0.190273
\(296\) 1.13198e149 0.347288
\(297\) 2.68286e149 0.671172
\(298\) −1.92900e149 −0.393774
\(299\) −1.85685e149 −0.309506
\(300\) 1.42436e149 0.193992
\(301\) 1.38249e149 0.153953
\(302\) −4.07519e149 −0.371302
\(303\) 9.81495e149 0.732165
\(304\) −1.19848e150 −0.732453
\(305\) −1.09516e150 −0.548703
\(306\) 3.60475e149 0.148159
\(307\) −5.37885e150 −1.81475 −0.907376 − 0.420319i \(-0.861918\pi\)
−0.907376 + 0.420319i \(0.861918\pi\)
\(308\) −5.30691e149 −0.147070
\(309\) 2.99486e150 0.682164
\(310\) 2.83160e150 0.530455
\(311\) −8.80263e150 −1.35709 −0.678546 − 0.734558i \(-0.737389\pi\)
−0.678546 + 0.734558i \(0.737389\pi\)
\(312\) −3.24367e149 −0.0411798
\(313\) 1.75005e151 1.83071 0.915354 − 0.402650i \(-0.131911\pi\)
0.915354 + 0.402650i \(0.131911\pi\)
\(314\) 7.16195e149 0.0617714
\(315\) −2.58674e150 −0.184062
\(316\) 2.30667e151 1.35493
\(317\) 1.26354e151 0.613061 0.306531 − 0.951861i \(-0.400832\pi\)
0.306531 + 0.951861i \(0.400832\pi\)
\(318\) 4.64756e150 0.186373
\(319\) −3.00082e151 −0.995186
\(320\) −2.31399e151 −0.635021
\(321\) 8.79740e150 0.199894
\(322\) 4.77917e150 0.0899647
\(323\) 3.76945e151 0.588200
\(324\) −3.79877e151 −0.491665
\(325\) −8.86986e150 −0.0952738
\(326\) −4.66539e151 −0.416124
\(327\) −1.05739e152 −0.783600
\(328\) −6.08775e151 −0.375050
\(329\) 7.74019e150 0.0396643
\(330\) −3.12643e151 −0.133339
\(331\) −2.85034e152 −1.01229 −0.506143 − 0.862450i \(-0.668929\pi\)
−0.506143 + 0.862450i \(0.668929\pi\)
\(332\) 3.76950e152 1.11540
\(333\) 2.08666e152 0.514726
\(334\) 4.41915e150 0.00909239
\(335\) 4.83212e152 0.829711
\(336\) −3.73092e151 −0.0534920
\(337\) −1.67229e152 −0.200310 −0.100155 − 0.994972i \(-0.531934\pi\)
−0.100155 + 0.994972i \(0.531934\pi\)
\(338\) −2.85384e152 −0.285739
\(339\) 3.15426e151 0.0264129
\(340\) 9.64167e152 0.675581
\(341\) 2.24410e153 1.31644
\(342\) 4.94360e152 0.242920
\(343\) −8.50143e152 −0.350105
\(344\) −1.41623e153 −0.489045
\(345\) −2.94522e153 −0.853224
\(346\) 1.35009e153 0.328291
\(347\) −6.33399e153 −1.29343 −0.646716 − 0.762731i \(-0.723858\pi\)
−0.646716 + 0.762731i \(0.723858\pi\)
\(348\) −2.35647e153 −0.404314
\(349\) 1.82728e153 0.263553 0.131776 − 0.991279i \(-0.457932\pi\)
0.131776 + 0.991279i \(0.457932\pi\)
\(350\) 2.28293e152 0.0276934
\(351\) −1.31195e153 −0.133918
\(352\) 8.27865e153 0.711424
\(353\) −2.01982e154 −1.46198 −0.730992 − 0.682386i \(-0.760942\pi\)
−0.730992 + 0.682386i \(0.760942\pi\)
\(354\) 3.00666e152 0.0183395
\(355\) 6.07492e153 0.312410
\(356\) 8.00200e153 0.347115
\(357\) 1.17344e153 0.0429570
\(358\) 1.22679e154 0.379182
\(359\) 6.28753e154 1.64159 0.820797 − 0.571219i \(-0.193529\pi\)
0.820797 + 0.571219i \(0.193529\pi\)
\(360\) 2.64987e154 0.584688
\(361\) −1.90803e153 −0.0355958
\(362\) −2.69116e154 −0.424689
\(363\) 5.41475e153 0.0723148
\(364\) 2.59514e153 0.0293445
\(365\) −1.29110e155 −1.23663
\(366\) 6.51557e153 0.0528867
\(367\) −1.09367e155 −0.752646 −0.376323 − 0.926489i \(-0.622812\pi\)
−0.376323 + 0.926489i \(0.622812\pi\)
\(368\) 2.18790e155 1.27713
\(369\) −1.12220e155 −0.555872
\(370\) −5.33544e154 −0.224371
\(371\) −7.79213e154 −0.278314
\(372\) 1.76223e155 0.534830
\(373\) −5.40298e155 −1.39396 −0.696978 − 0.717092i \(-0.745473\pi\)
−0.696978 + 0.717092i \(0.745473\pi\)
\(374\) −7.30473e154 −0.160277
\(375\) 1.26225e155 0.235642
\(376\) −7.92911e154 −0.125997
\(377\) 1.46744e155 0.198567
\(378\) 3.37672e154 0.0389261
\(379\) 1.53602e155 0.150912 0.0754561 − 0.997149i \(-0.475959\pi\)
0.0754561 + 0.997149i \(0.475959\pi\)
\(380\) 1.32227e156 1.10767
\(381\) 8.24870e154 0.0589420
\(382\) 4.28968e155 0.261572
\(383\) 4.80126e155 0.249937 0.124968 − 0.992161i \(-0.460117\pi\)
0.124968 + 0.992161i \(0.460117\pi\)
\(384\) 8.49919e155 0.377867
\(385\) 5.24181e155 0.199116
\(386\) 2.55972e155 0.0831115
\(387\) −2.61065e156 −0.724827
\(388\) 5.65124e156 1.34222
\(389\) −8.82510e156 −1.79378 −0.896889 − 0.442256i \(-0.854178\pi\)
−0.896889 + 0.442256i \(0.854178\pi\)
\(390\) 1.52886e155 0.0266048
\(391\) −6.88134e156 −1.02560
\(392\) 4.28448e156 0.547131
\(393\) −2.56341e155 −0.0280588
\(394\) 4.83117e156 0.453451
\(395\) −2.27837e157 −1.83443
\(396\) 1.00214e157 0.692419
\(397\) −1.10795e157 −0.657194 −0.328597 − 0.944470i \(-0.606576\pi\)
−0.328597 + 0.944470i \(0.606576\pi\)
\(398\) −1.11233e156 −0.0566640
\(399\) 1.60927e156 0.0704317
\(400\) 1.04513e157 0.393132
\(401\) 2.86879e157 0.927821 0.463910 − 0.885882i \(-0.346446\pi\)
0.463910 + 0.885882i \(0.346446\pi\)
\(402\) −2.87483e156 −0.0799715
\(403\) −1.09739e157 −0.262667
\(404\) 8.04423e157 1.65734
\(405\) 3.75217e157 0.665661
\(406\) −3.77691e156 −0.0577180
\(407\) −4.22844e157 −0.556825
\(408\) −1.20208e157 −0.136456
\(409\) −9.77901e157 −0.957270 −0.478635 − 0.878014i \(-0.658868\pi\)
−0.478635 + 0.878014i \(0.658868\pi\)
\(410\) 2.86938e157 0.242306
\(411\) −6.46322e157 −0.470997
\(412\) 2.45456e158 1.54416
\(413\) −5.04099e156 −0.0273866
\(414\) −9.02482e157 −0.423563
\(415\) −3.72326e158 −1.51013
\(416\) −4.04836e157 −0.141949
\(417\) −1.50654e158 −0.456824
\(418\) −1.00178e158 −0.262788
\(419\) 1.30636e158 0.296562 0.148281 − 0.988945i \(-0.452626\pi\)
0.148281 + 0.988945i \(0.452626\pi\)
\(420\) 4.11626e157 0.0808949
\(421\) 7.55579e157 0.128592 0.0642959 − 0.997931i \(-0.479520\pi\)
0.0642959 + 0.997931i \(0.479520\pi\)
\(422\) 2.01542e158 0.297141
\(423\) −1.46163e158 −0.186743
\(424\) 7.98232e158 0.884086
\(425\) −3.28711e158 −0.315707
\(426\) −3.61422e157 −0.0301116
\(427\) −1.09240e158 −0.0789764
\(428\) 7.21026e158 0.452484
\(429\) 1.21165e158 0.0660256
\(430\) 6.67523e158 0.315955
\(431\) 2.34221e159 0.963276 0.481638 − 0.876370i \(-0.340042\pi\)
0.481638 + 0.876370i \(0.340042\pi\)
\(432\) 1.54586e159 0.552590
\(433\) −6.08282e158 −0.189055 −0.0945273 − 0.995522i \(-0.530134\pi\)
−0.0945273 + 0.995522i \(0.530134\pi\)
\(434\) 2.82447e158 0.0763499
\(435\) 2.32757e159 0.547397
\(436\) −8.66623e159 −1.77377
\(437\) −9.43714e159 −1.68156
\(438\) 7.68127e158 0.119193
\(439\) −2.07161e159 −0.280030 −0.140015 − 0.990149i \(-0.544715\pi\)
−0.140015 + 0.990149i \(0.544715\pi\)
\(440\) −5.36975e159 −0.632509
\(441\) 7.89790e159 0.810918
\(442\) 3.57210e158 0.0319798
\(443\) 1.00029e160 0.781088 0.390544 − 0.920584i \(-0.372287\pi\)
0.390544 + 0.920584i \(0.372287\pi\)
\(444\) −3.32050e159 −0.226221
\(445\) −7.90384e159 −0.469956
\(446\) −8.81898e159 −0.457783
\(447\) 1.18578e160 0.537524
\(448\) −2.30817e159 −0.0914003
\(449\) 2.84866e160 0.985683 0.492841 − 0.870119i \(-0.335958\pi\)
0.492841 + 0.870119i \(0.335958\pi\)
\(450\) −4.31102e159 −0.130383
\(451\) 2.27404e160 0.601336
\(452\) 2.58520e159 0.0597886
\(453\) 2.50507e160 0.506848
\(454\) 1.60338e160 0.283895
\(455\) −2.56330e159 −0.0397293
\(456\) −1.64855e160 −0.223732
\(457\) 4.02628e160 0.478601 0.239300 − 0.970946i \(-0.423082\pi\)
0.239300 + 0.970946i \(0.423082\pi\)
\(458\) −4.93125e160 −0.513565
\(459\) −4.86200e160 −0.443760
\(460\) −2.41388e161 −1.93137
\(461\) 2.03800e161 1.42987 0.714936 − 0.699189i \(-0.246456\pi\)
0.714936 + 0.699189i \(0.246456\pi\)
\(462\) −3.11857e159 −0.0191918
\(463\) −6.61773e159 −0.0357322 −0.0178661 − 0.999840i \(-0.505687\pi\)
−0.0178661 + 0.999840i \(0.505687\pi\)
\(464\) −1.72906e161 −0.819357
\(465\) −1.74062e161 −0.724101
\(466\) 9.48879e160 0.346627
\(467\) 8.39556e160 0.269388 0.134694 − 0.990887i \(-0.456995\pi\)
0.134694 + 0.990887i \(0.456995\pi\)
\(468\) −4.90057e160 −0.138157
\(469\) 4.81996e160 0.119423
\(470\) 3.73728e160 0.0814021
\(471\) −4.40254e160 −0.0843215
\(472\) 5.16403e160 0.0869957
\(473\) 5.29025e161 0.784110
\(474\) 1.35550e161 0.176811
\(475\) −4.50798e161 −0.517628
\(476\) 9.61741e160 0.0972383
\(477\) 1.47144e162 1.31033
\(478\) −9.57014e159 −0.00750811
\(479\) 9.49769e161 0.656628 0.328314 − 0.944569i \(-0.393520\pi\)
0.328314 + 0.944569i \(0.393520\pi\)
\(480\) −6.42128e161 −0.391315
\(481\) 2.06776e161 0.111102
\(482\) −9.47526e160 −0.0448999
\(483\) −2.93781e161 −0.122807
\(484\) 4.43787e161 0.163693
\(485\) −5.58191e162 −1.81722
\(486\) −9.84683e161 −0.283010
\(487\) −2.50586e162 −0.635996 −0.317998 − 0.948091i \(-0.603010\pi\)
−0.317998 + 0.948091i \(0.603010\pi\)
\(488\) 1.11907e162 0.250875
\(489\) 2.86787e162 0.568033
\(490\) −2.01944e162 −0.353482
\(491\) −5.34167e162 −0.826507 −0.413253 − 0.910616i \(-0.635608\pi\)
−0.413253 + 0.910616i \(0.635608\pi\)
\(492\) 1.78575e162 0.244305
\(493\) 5.43822e162 0.657989
\(494\) 4.89881e161 0.0524336
\(495\) −9.89845e162 −0.937460
\(496\) 1.29304e163 1.08385
\(497\) 6.05963e161 0.0449661
\(498\) 2.21512e162 0.145553
\(499\) 1.90646e163 1.10954 0.554771 − 0.832003i \(-0.312806\pi\)
0.554771 + 0.832003i \(0.312806\pi\)
\(500\) 1.03453e163 0.533404
\(501\) −2.71651e161 −0.0124116
\(502\) 4.10082e162 0.166072
\(503\) 7.84083e162 0.281513 0.140757 − 0.990044i \(-0.455046\pi\)
0.140757 + 0.990044i \(0.455046\pi\)
\(504\) 2.64321e162 0.0841557
\(505\) −7.94555e163 −2.24386
\(506\) 1.82880e163 0.458206
\(507\) 1.75429e163 0.390050
\(508\) 6.76056e162 0.133422
\(509\) −2.55914e163 −0.448402 −0.224201 − 0.974543i \(-0.571977\pi\)
−0.224201 + 0.974543i \(0.571977\pi\)
\(510\) 5.66586e162 0.0881597
\(511\) −1.28785e163 −0.177992
\(512\) 8.20204e163 1.00714
\(513\) −6.66780e163 −0.727582
\(514\) −2.45752e163 −0.238357
\(515\) −2.42445e164 −2.09062
\(516\) 4.15431e163 0.318560
\(517\) 2.96187e163 0.202017
\(518\) −5.32202e162 −0.0322943
\(519\) −8.29917e163 −0.448136
\(520\) 2.62587e163 0.126203
\(521\) 4.22014e164 1.80570 0.902849 − 0.429959i \(-0.141472\pi\)
0.902849 + 0.429959i \(0.141472\pi\)
\(522\) 7.13218e163 0.271742
\(523\) 2.86419e164 0.971963 0.485981 − 0.873969i \(-0.338462\pi\)
0.485981 + 0.873969i \(0.338462\pi\)
\(524\) −2.10095e163 −0.0635144
\(525\) −1.40335e163 −0.0378031
\(526\) −1.31348e164 −0.315346
\(527\) −4.06685e164 −0.870393
\(528\) −1.42768e164 −0.272444
\(529\) 1.13522e165 1.93203
\(530\) −3.76236e164 −0.571177
\(531\) 9.51924e163 0.128939
\(532\) 1.31894e164 0.159431
\(533\) −1.11203e164 −0.119983
\(534\) 4.70232e163 0.0452967
\(535\) −7.12181e164 −0.612614
\(536\) −4.93760e164 −0.379355
\(537\) −7.54124e164 −0.517605
\(538\) 3.53957e164 0.217081
\(539\) −1.60044e165 −0.877243
\(540\) −1.70552e165 −0.835670
\(541\) −1.18166e165 −0.517674 −0.258837 − 0.965921i \(-0.583339\pi\)
−0.258837 + 0.965921i \(0.583339\pi\)
\(542\) 5.28913e164 0.207218
\(543\) 1.65429e165 0.579725
\(544\) −1.50029e165 −0.470373
\(545\) 8.55992e165 2.40149
\(546\) 1.52502e163 0.00382930
\(547\) 5.43964e164 0.122275 0.0611374 − 0.998129i \(-0.480527\pi\)
0.0611374 + 0.998129i \(0.480527\pi\)
\(548\) −5.29719e165 −1.06616
\(549\) 2.06286e165 0.371829
\(550\) 8.73591e164 0.141047
\(551\) 7.45803e165 1.07883
\(552\) 3.00952e165 0.390106
\(553\) −2.27264e165 −0.264034
\(554\) 2.95941e165 0.308223
\(555\) 3.27976e165 0.306279
\(556\) −1.23474e166 −1.03407
\(557\) −2.50940e165 −0.188508 −0.0942541 − 0.995548i \(-0.530047\pi\)
−0.0942541 + 0.995548i \(0.530047\pi\)
\(558\) −5.33363e165 −0.359463
\(559\) −2.58699e165 −0.156452
\(560\) 3.02031e165 0.163937
\(561\) 4.49031e165 0.218788
\(562\) −1.66075e165 −0.0726536
\(563\) −2.41225e166 −0.947687 −0.473843 − 0.880609i \(-0.657134\pi\)
−0.473843 + 0.880609i \(0.657134\pi\)
\(564\) 2.32589e165 0.0820734
\(565\) −2.55349e165 −0.0809473
\(566\) −1.27503e166 −0.363184
\(567\) 3.74273e165 0.0958104
\(568\) −6.20754e165 −0.142838
\(569\) −3.21596e166 −0.665301 −0.332651 − 0.943050i \(-0.607943\pi\)
−0.332651 + 0.943050i \(0.607943\pi\)
\(570\) 7.77022e165 0.144545
\(571\) 5.87485e166 0.982907 0.491453 − 0.870904i \(-0.336466\pi\)
0.491453 + 0.870904i \(0.336466\pi\)
\(572\) 9.93060e165 0.149457
\(573\) −2.63692e166 −0.357061
\(574\) 2.86216e165 0.0348758
\(575\) 8.22957e166 0.902553
\(576\) 4.35867e166 0.430322
\(577\) 2.24498e166 0.199561 0.0997805 − 0.995009i \(-0.468186\pi\)
0.0997805 + 0.995009i \(0.468186\pi\)
\(578\) −2.36623e166 −0.189419
\(579\) −1.57349e166 −0.113452
\(580\) 1.90765e167 1.23910
\(581\) −3.71389e166 −0.217357
\(582\) 3.32091e166 0.175153
\(583\) −2.98175e167 −1.41750
\(584\) 1.31928e167 0.565406
\(585\) 4.84046e166 0.187049
\(586\) 6.80744e166 0.237235
\(587\) −1.58460e167 −0.498097 −0.249049 − 0.968491i \(-0.580118\pi\)
−0.249049 + 0.968491i \(0.580118\pi\)
\(588\) −1.25679e167 −0.356397
\(589\) −5.57731e167 −1.42708
\(590\) −2.43400e166 −0.0562049
\(591\) −2.96978e167 −0.618987
\(592\) −2.43641e167 −0.458446
\(593\) 3.77197e167 0.640854 0.320427 − 0.947273i \(-0.396174\pi\)
0.320427 + 0.947273i \(0.396174\pi\)
\(594\) 1.29214e167 0.198257
\(595\) −9.49943e166 −0.131650
\(596\) 9.71852e167 1.21675
\(597\) 6.83762e166 0.0773496
\(598\) −8.94306e166 −0.0914249
\(599\) −3.97379e167 −0.367183 −0.183591 − 0.983003i \(-0.558772\pi\)
−0.183591 + 0.983003i \(0.558772\pi\)
\(600\) 1.43760e167 0.120085
\(601\) 8.01744e167 0.605522 0.302761 − 0.953067i \(-0.402092\pi\)
0.302761 + 0.953067i \(0.402092\pi\)
\(602\) 6.65843e166 0.0454762
\(603\) −9.10185e167 −0.562253
\(604\) 2.05313e168 1.14731
\(605\) −4.38343e167 −0.221623
\(606\) 4.72714e167 0.216274
\(607\) 4.08681e168 1.69227 0.846133 − 0.532973i \(-0.178925\pi\)
0.846133 + 0.532973i \(0.178925\pi\)
\(608\) −2.05752e168 −0.771218
\(609\) 2.32171e167 0.0787883
\(610\) −5.27458e167 −0.162081
\(611\) −1.44839e167 −0.0403081
\(612\) −1.81612e168 −0.457808
\(613\) −7.73383e168 −1.76618 −0.883092 − 0.469200i \(-0.844542\pi\)
−0.883092 + 0.469200i \(0.844542\pi\)
\(614\) −2.59059e168 −0.536059
\(615\) −1.76384e168 −0.330762
\(616\) −5.35623e167 −0.0910388
\(617\) 3.39431e168 0.522997 0.261499 − 0.965204i \(-0.415783\pi\)
0.261499 + 0.965204i \(0.415783\pi\)
\(618\) 1.44240e168 0.201504
\(619\) 1.05699e169 1.33902 0.669512 − 0.742801i \(-0.266503\pi\)
0.669512 + 0.742801i \(0.266503\pi\)
\(620\) −1.42659e169 −1.63909
\(621\) 1.21725e169 1.26863
\(622\) −4.23958e168 −0.400871
\(623\) −7.88395e167 −0.0676421
\(624\) 6.98151e167 0.0543603
\(625\) −1.76765e169 −1.24927
\(626\) 8.42872e168 0.540773
\(627\) 6.15805e168 0.358721
\(628\) −3.60828e168 −0.190872
\(629\) 7.66297e168 0.368157
\(630\) −1.24584e168 −0.0543700
\(631\) 4.48469e169 1.77810 0.889050 − 0.457810i \(-0.151366\pi\)
0.889050 + 0.457810i \(0.151366\pi\)
\(632\) 2.32811e169 0.838726
\(633\) −1.23891e169 −0.405615
\(634\) 6.08554e168 0.181092
\(635\) −6.67762e168 −0.180639
\(636\) −2.34149e169 −0.575888
\(637\) 7.82635e168 0.175034
\(638\) −1.44528e169 −0.293968
\(639\) −1.14428e169 −0.211705
\(640\) −6.88040e169 −1.15804
\(641\) 4.22451e169 0.646942 0.323471 − 0.946238i \(-0.395150\pi\)
0.323471 + 0.946238i \(0.395150\pi\)
\(642\) 4.23706e168 0.0590467
\(643\) −5.41963e169 −0.687394 −0.343697 − 0.939081i \(-0.611679\pi\)
−0.343697 + 0.939081i \(0.611679\pi\)
\(644\) −2.40780e169 −0.277988
\(645\) −4.10334e169 −0.431296
\(646\) 1.81546e169 0.173748
\(647\) −2.95000e169 −0.257107 −0.128553 − 0.991703i \(-0.541033\pi\)
−0.128553 + 0.991703i \(0.541033\pi\)
\(648\) −3.83408e169 −0.304350
\(649\) −1.92899e169 −0.139485
\(650\) −4.27196e168 −0.0281429
\(651\) −1.73624e169 −0.104222
\(652\) 2.35048e170 1.28581
\(653\) 1.01219e170 0.504676 0.252338 − 0.967639i \(-0.418801\pi\)
0.252338 + 0.967639i \(0.418801\pi\)
\(654\) −5.09265e169 −0.231467
\(655\) 2.07517e169 0.0859915
\(656\) 1.31029e170 0.495093
\(657\) 2.43193e170 0.838004
\(658\) 3.72788e168 0.0117164
\(659\) −1.17124e170 −0.335799 −0.167899 − 0.985804i \(-0.553698\pi\)
−0.167899 + 0.985804i \(0.553698\pi\)
\(660\) 1.57514e170 0.412012
\(661\) −8.13381e169 −0.194136 −0.0970680 − 0.995278i \(-0.530946\pi\)
−0.0970680 + 0.995278i \(0.530946\pi\)
\(662\) −1.37280e170 −0.299019
\(663\) −2.19581e169 −0.0436542
\(664\) 3.80454e170 0.690451
\(665\) −1.30276e170 −0.215852
\(666\) 1.00499e170 0.152045
\(667\) −1.36151e171 −1.88108
\(668\) −2.22642e169 −0.0280952
\(669\) 5.42114e170 0.624899
\(670\) 2.32728e170 0.245088
\(671\) −4.18021e170 −0.402241
\(672\) −6.40512e169 −0.0563231
\(673\) −2.24386e171 −1.80337 −0.901685 − 0.432394i \(-0.857669\pi\)
−0.901685 + 0.432394i \(0.857669\pi\)
\(674\) −8.05417e169 −0.0591694
\(675\) 5.81459e170 0.390518
\(676\) 1.43780e171 0.882924
\(677\) 1.77502e170 0.0996754 0.0498377 − 0.998757i \(-0.484130\pi\)
0.0498377 + 0.998757i \(0.484130\pi\)
\(678\) 1.51917e169 0.00780209
\(679\) −5.56786e170 −0.261558
\(680\) 9.73128e170 0.418197
\(681\) −9.85620e170 −0.387533
\(682\) 1.08082e171 0.388863
\(683\) 9.74351e170 0.320821 0.160411 − 0.987050i \(-0.448718\pi\)
0.160411 + 0.987050i \(0.448718\pi\)
\(684\) −2.49064e171 −0.750615
\(685\) 5.23221e171 1.44346
\(686\) −4.09452e170 −0.103417
\(687\) 3.03130e171 0.701046
\(688\) 3.04822e171 0.645575
\(689\) 1.45811e171 0.282831
\(690\) −1.41850e171 −0.252034
\(691\) −4.14222e171 −0.674235 −0.337117 − 0.941463i \(-0.609452\pi\)
−0.337117 + 0.941463i \(0.609452\pi\)
\(692\) −6.80192e171 −1.01441
\(693\) −9.87354e170 −0.134931
\(694\) −3.05061e171 −0.382067
\(695\) 1.21960e172 1.40002
\(696\) −2.37838e171 −0.250277
\(697\) −4.12112e171 −0.397587
\(698\) 8.80067e170 0.0778507
\(699\) −5.83288e171 −0.473166
\(700\) −1.15017e171 −0.0855718
\(701\) −1.71159e172 −1.16805 −0.584024 − 0.811736i \(-0.698523\pi\)
−0.584024 + 0.811736i \(0.698523\pi\)
\(702\) −6.31871e170 −0.0395579
\(703\) 1.05091e172 0.603625
\(704\) −8.83247e171 −0.465518
\(705\) −2.29735e171 −0.111118
\(706\) −9.72798e171 −0.431855
\(707\) −7.92556e171 −0.322965
\(708\) −1.51479e171 −0.0566684
\(709\) 1.74143e172 0.598148 0.299074 − 0.954230i \(-0.403322\pi\)
0.299074 + 0.954230i \(0.403322\pi\)
\(710\) 2.92584e171 0.0922828
\(711\) 4.29158e172 1.24310
\(712\) 8.07638e171 0.214871
\(713\) 1.01817e173 2.48831
\(714\) 5.65160e170 0.0126891
\(715\) −9.80877e171 −0.202348
\(716\) −6.18073e172 −1.17166
\(717\) 5.88288e170 0.0102490
\(718\) 3.02824e172 0.484910
\(719\) −5.83912e172 −0.859507 −0.429754 − 0.902946i \(-0.641400\pi\)
−0.429754 + 0.902946i \(0.641400\pi\)
\(720\) −5.70345e172 −0.771830
\(721\) −2.41835e172 −0.300909
\(722\) −9.18955e170 −0.0105146
\(723\) 5.82456e171 0.0612909
\(724\) 1.35584e173 1.31228
\(725\) −6.50371e172 −0.579044
\(726\) 2.60788e171 0.0213611
\(727\) −9.12339e172 −0.687582 −0.343791 − 0.939046i \(-0.611711\pi\)
−0.343791 + 0.939046i \(0.611711\pi\)
\(728\) 2.61926e171 0.0181648
\(729\) −2.38690e172 −0.152342
\(730\) −6.21826e172 −0.365289
\(731\) −9.58722e172 −0.518432
\(732\) −3.28262e172 −0.163418
\(733\) −1.79942e173 −0.824789 −0.412395 − 0.911005i \(-0.635308\pi\)
−0.412395 + 0.911005i \(0.635308\pi\)
\(734\) −5.26742e172 −0.222324
\(735\) 1.24137e173 0.482523
\(736\) 3.75612e173 1.34472
\(737\) 1.84441e173 0.608240
\(738\) −5.40481e172 −0.164199
\(739\) −1.10849e173 −0.310273 −0.155137 − 0.987893i \(-0.549582\pi\)
−0.155137 + 0.987893i \(0.549582\pi\)
\(740\) 2.68806e173 0.693298
\(741\) −3.01136e172 −0.0715749
\(742\) −3.75290e172 −0.0822111
\(743\) 6.33225e173 1.27860 0.639300 − 0.768957i \(-0.279224\pi\)
0.639300 + 0.768957i \(0.279224\pi\)
\(744\) 1.77861e173 0.331069
\(745\) −9.59930e173 −1.64735
\(746\) −2.60222e173 −0.411761
\(747\) 7.01318e173 1.02334
\(748\) 3.68021e173 0.495252
\(749\) −7.10389e172 −0.0881752
\(750\) 6.07933e172 0.0696064
\(751\) 5.81349e172 0.0614074 0.0307037 − 0.999529i \(-0.490225\pi\)
0.0307037 + 0.999529i \(0.490225\pi\)
\(752\) 1.70662e173 0.166325
\(753\) −2.52083e173 −0.226698
\(754\) 7.06757e172 0.0586548
\(755\) −2.02794e174 −1.55333
\(756\) −1.70123e173 −0.120280
\(757\) −1.10200e174 −0.719253 −0.359626 − 0.933096i \(-0.617096\pi\)
−0.359626 + 0.933096i \(0.617096\pi\)
\(758\) 7.39787e172 0.0445779
\(759\) −1.12419e174 −0.625477
\(760\) 1.33456e174 0.685670
\(761\) −2.30609e173 −0.109422 −0.0547111 − 0.998502i \(-0.517424\pi\)
−0.0547111 + 0.998502i \(0.517424\pi\)
\(762\) 3.97279e172 0.0174109
\(763\) 8.53837e173 0.345653
\(764\) −2.16119e174 −0.808250
\(765\) 1.79384e174 0.619822
\(766\) 2.31242e173 0.0738287
\(767\) 9.43300e172 0.0278311
\(768\) −3.50554e173 −0.0955875
\(769\) 4.18167e174 1.05392 0.526958 − 0.849891i \(-0.323332\pi\)
0.526958 + 0.849891i \(0.323332\pi\)
\(770\) 2.52459e173 0.0588169
\(771\) 1.51067e174 0.325371
\(772\) −1.28962e174 −0.256812
\(773\) 2.86308e174 0.527198 0.263599 − 0.964632i \(-0.415091\pi\)
0.263599 + 0.964632i \(0.415091\pi\)
\(774\) −1.25736e174 −0.214107
\(775\) 4.86364e174 0.765965
\(776\) 5.70376e174 0.830859
\(777\) 3.27151e173 0.0440835
\(778\) −4.25040e174 −0.529864
\(779\) −5.65174e174 −0.651877
\(780\) −7.70259e173 −0.0822079
\(781\) 2.31879e174 0.229020
\(782\) −3.31424e174 −0.302953
\(783\) −9.61971e174 −0.813908
\(784\) −9.22169e174 −0.722252
\(785\) 3.56401e174 0.258419
\(786\) −1.23461e173 −0.00828828
\(787\) 4.94968e174 0.307685 0.153842 − 0.988095i \(-0.450835\pi\)
0.153842 + 0.988095i \(0.450835\pi\)
\(788\) −2.43400e175 −1.40115
\(789\) 8.07415e174 0.430465
\(790\) −1.09733e175 −0.541871
\(791\) −2.54706e173 −0.0116510
\(792\) 1.01145e175 0.428620
\(793\) 2.04417e174 0.0802582
\(794\) −5.33617e174 −0.194128
\(795\) 2.31277e175 0.779689
\(796\) 5.60405e174 0.175090
\(797\) 1.80868e175 0.523761 0.261881 − 0.965100i \(-0.415657\pi\)
0.261881 + 0.965100i \(0.415657\pi\)
\(798\) 7.75066e173 0.0208048
\(799\) −5.36763e174 −0.133568
\(800\) 1.79424e175 0.413939
\(801\) 1.48878e175 0.318466
\(802\) 1.38168e175 0.274069
\(803\) −4.92810e175 −0.906544
\(804\) 1.44837e175 0.247109
\(805\) 2.37827e175 0.376365
\(806\) −5.28531e174 −0.0775891
\(807\) −2.17581e175 −0.296328
\(808\) 8.11900e175 1.02592
\(809\) −1.14018e176 −1.33686 −0.668432 − 0.743773i \(-0.733034\pi\)
−0.668432 + 0.743773i \(0.733034\pi\)
\(810\) 1.80714e175 0.196630
\(811\) 1.55514e176 1.57039 0.785194 − 0.619249i \(-0.212563\pi\)
0.785194 + 0.619249i \(0.212563\pi\)
\(812\) 1.90285e175 0.178347
\(813\) −3.25129e175 −0.282864
\(814\) −2.03653e175 −0.164480
\(815\) −2.32164e176 −1.74085
\(816\) 2.58730e175 0.180132
\(817\) −1.31480e176 −0.850013
\(818\) −4.70983e175 −0.282768
\(819\) 4.82827e174 0.0269225
\(820\) −1.44563e176 −0.748719
\(821\) −1.87691e176 −0.902988 −0.451494 − 0.892274i \(-0.649109\pi\)
−0.451494 + 0.892274i \(0.649109\pi\)
\(822\) −3.11286e175 −0.139128
\(823\) 3.51655e176 1.46025 0.730125 − 0.683314i \(-0.239462\pi\)
0.730125 + 0.683314i \(0.239462\pi\)
\(824\) 2.47737e176 0.955862
\(825\) −5.37007e175 −0.192538
\(826\) −2.42787e174 −0.00808972
\(827\) 1.37907e176 0.427075 0.213538 − 0.976935i \(-0.431501\pi\)
0.213538 + 0.976935i \(0.431501\pi\)
\(828\) 4.54681e176 1.30880
\(829\) −1.82240e176 −0.487635 −0.243818 − 0.969821i \(-0.578400\pi\)
−0.243818 + 0.969821i \(0.578400\pi\)
\(830\) −1.79322e176 −0.446076
\(831\) −1.81919e176 −0.420742
\(832\) 4.31918e175 0.0928838
\(833\) 2.90039e176 0.580008
\(834\) −7.25589e175 −0.134941
\(835\) 2.19911e175 0.0380378
\(836\) 5.04708e176 0.812008
\(837\) 7.19387e176 1.07665
\(838\) 6.29178e175 0.0876013
\(839\) −8.58257e176 −1.11178 −0.555890 − 0.831256i \(-0.687622\pi\)
−0.555890 + 0.831256i \(0.687622\pi\)
\(840\) 4.15452e175 0.0500754
\(841\) 1.84404e176 0.206829
\(842\) 3.63907e175 0.0379847
\(843\) 1.02088e176 0.0991763
\(844\) −1.01539e177 −0.918157
\(845\) −1.42016e177 −1.19538
\(846\) −7.03960e175 −0.0551621
\(847\) −4.37240e175 −0.0318987
\(848\) −1.71807e177 −1.16706
\(849\) 7.83777e176 0.495767
\(850\) −1.58316e176 −0.0932566
\(851\) −1.91849e177 −1.05250
\(852\) 1.82089e176 0.0930439
\(853\) 3.71056e177 1.76613 0.883066 − 0.469250i \(-0.155476\pi\)
0.883066 + 0.469250i \(0.155476\pi\)
\(854\) −5.26131e175 −0.0233288
\(855\) 2.46009e177 1.01625
\(856\) 7.27728e176 0.280096
\(857\) −3.84852e177 −1.38024 −0.690120 − 0.723695i \(-0.742442\pi\)
−0.690120 + 0.723695i \(0.742442\pi\)
\(858\) 5.83564e175 0.0195033
\(859\) 2.55767e177 0.796633 0.398317 − 0.917248i \(-0.369595\pi\)
0.398317 + 0.917248i \(0.369595\pi\)
\(860\) −3.36306e177 −0.976289
\(861\) −1.75941e176 −0.0476075
\(862\) 1.12807e177 0.284542
\(863\) 3.25487e177 0.765386 0.382693 − 0.923876i \(-0.374997\pi\)
0.382693 + 0.923876i \(0.374997\pi\)
\(864\) 2.65388e177 0.581835
\(865\) 6.71848e177 1.37340
\(866\) −2.92965e176 −0.0558448
\(867\) 1.45455e177 0.258568
\(868\) −1.42300e177 −0.235919
\(869\) −8.69652e177 −1.34477
\(870\) 1.12102e177 0.161695
\(871\) −9.01939e176 −0.121361
\(872\) −8.74678e177 −1.09800
\(873\) 1.05142e178 1.23144
\(874\) −4.54518e177 −0.496717
\(875\) −1.01926e177 −0.103944
\(876\) −3.86992e177 −0.368301
\(877\) −2.61321e177 −0.232113 −0.116056 − 0.993243i \(-0.537025\pi\)
−0.116056 + 0.993243i \(0.537025\pi\)
\(878\) −9.97742e176 −0.0827180
\(879\) −4.18462e177 −0.323839
\(880\) 1.15576e178 0.834958
\(881\) −1.32074e178 −0.890790 −0.445395 − 0.895334i \(-0.646937\pi\)
−0.445395 + 0.895334i \(0.646937\pi\)
\(882\) 3.80384e177 0.239537
\(883\) −6.59348e177 −0.387697 −0.193848 − 0.981031i \(-0.562097\pi\)
−0.193848 + 0.981031i \(0.562097\pi\)
\(884\) −1.79967e177 −0.0988165
\(885\) 1.49621e177 0.0767228
\(886\) 4.81765e177 0.230725
\(887\) −1.39753e178 −0.625148 −0.312574 − 0.949893i \(-0.601191\pi\)
−0.312574 + 0.949893i \(0.601191\pi\)
\(888\) −3.35136e177 −0.140035
\(889\) −6.66082e176 −0.0259999
\(890\) −3.80670e177 −0.138820
\(891\) 1.43220e178 0.487979
\(892\) 4.44311e178 1.41453
\(893\) −7.36122e177 −0.218996
\(894\) 5.71102e177 0.158779
\(895\) 6.10491e178 1.58630
\(896\) −6.86309e177 −0.166681
\(897\) 5.49741e177 0.124800
\(898\) 1.37199e178 0.291161
\(899\) −8.04645e178 −1.59641
\(900\) 2.17194e178 0.402881
\(901\) 5.40365e178 0.937211
\(902\) 1.09524e178 0.177629
\(903\) −4.09302e177 −0.0620776
\(904\) 2.60923e177 0.0370102
\(905\) −1.33921e179 −1.77668
\(906\) 1.20651e178 0.149718
\(907\) −6.92455e178 −0.803805 −0.401903 − 0.915682i \(-0.631651\pi\)
−0.401903 + 0.915682i \(0.631651\pi\)
\(908\) −8.07804e178 −0.877228
\(909\) 1.49664e179 1.52055
\(910\) −1.23455e177 −0.0117356
\(911\) −1.73278e179 −1.54128 −0.770638 − 0.637273i \(-0.780062\pi\)
−0.770638 + 0.637273i \(0.780062\pi\)
\(912\) 3.54825e178 0.295343
\(913\) −1.42116e179 −1.10704
\(914\) 1.93916e178 0.141374
\(915\) 3.24235e178 0.221250
\(916\) 2.48442e179 1.58690
\(917\) 2.06995e177 0.0123770
\(918\) −2.34167e178 −0.131082
\(919\) 6.96141e178 0.364845 0.182423 − 0.983220i \(-0.441606\pi\)
0.182423 + 0.983220i \(0.441606\pi\)
\(920\) −2.43631e179 −1.19555
\(921\) 1.59247e179 0.731751
\(922\) 9.81553e178 0.422370
\(923\) −1.13391e178 −0.0456959
\(924\) 1.57117e178 0.0593020
\(925\) −9.16434e178 −0.323986
\(926\) −3.18727e177 −0.0105549
\(927\) 4.56673e179 1.41671
\(928\) −2.96840e179 −0.862721
\(929\) 5.18727e179 1.41250 0.706250 − 0.707963i \(-0.250385\pi\)
0.706250 + 0.707963i \(0.250385\pi\)
\(930\) −8.38327e178 −0.213892
\(931\) 3.97762e179 0.950973
\(932\) −4.78057e179 −1.07107
\(933\) 2.60612e179 0.547212
\(934\) 4.04352e178 0.0795745
\(935\) −3.63506e179 −0.670517
\(936\) −4.94612e178 −0.0855216
\(937\) −3.25947e179 −0.528325 −0.264163 − 0.964478i \(-0.585096\pi\)
−0.264163 + 0.964478i \(0.585096\pi\)
\(938\) 2.32142e178 0.0352762
\(939\) −5.18124e179 −0.738185
\(940\) −1.88289e179 −0.251530
\(941\) −1.27019e180 −1.59109 −0.795546 − 0.605894i \(-0.792816\pi\)
−0.795546 + 0.605894i \(0.792816\pi\)
\(942\) −2.12038e178 −0.0249077
\(943\) 1.03176e180 1.13663
\(944\) −1.11148e179 −0.114841
\(945\) 1.68036e179 0.162846
\(946\) 2.54792e179 0.231618
\(947\) 8.00110e179 0.682301 0.341150 − 0.940009i \(-0.389183\pi\)
0.341150 + 0.940009i \(0.389183\pi\)
\(948\) −6.82917e179 −0.546340
\(949\) 2.40990e179 0.180881
\(950\) −2.17116e179 −0.152902
\(951\) −3.74086e179 −0.247201
\(952\) 9.70680e178 0.0601922
\(953\) −2.56024e180 −1.48991 −0.744955 − 0.667114i \(-0.767529\pi\)
−0.744955 + 0.667114i \(0.767529\pi\)
\(954\) 7.08684e179 0.387058
\(955\) 2.13468e180 1.09428
\(956\) 4.82155e178 0.0231998
\(957\) 8.88429e179 0.401283
\(958\) 4.57434e179 0.193961
\(959\) 5.21904e179 0.207761
\(960\) 6.85084e179 0.256056
\(961\) 3.16787e180 1.11174
\(962\) 9.95887e178 0.0328185
\(963\) 1.34147e180 0.415138
\(964\) 4.77375e179 0.138739
\(965\) 1.27380e180 0.347695
\(966\) −1.41493e179 −0.0362759
\(967\) −7.71155e180 −1.85712 −0.928559 − 0.371184i \(-0.878952\pi\)
−0.928559 + 0.371184i \(0.878952\pi\)
\(968\) 4.47912e179 0.101329
\(969\) −1.11599e180 −0.237176
\(970\) −2.68840e180 −0.536789
\(971\) −4.67788e180 −0.877578 −0.438789 − 0.898590i \(-0.644593\pi\)
−0.438789 + 0.898590i \(0.644593\pi\)
\(972\) 4.96095e180 0.874493
\(973\) 1.21653e180 0.201509
\(974\) −1.20689e180 −0.187867
\(975\) 2.62603e179 0.0384167
\(976\) −2.40862e180 −0.331173
\(977\) −6.75817e180 −0.873390 −0.436695 − 0.899610i \(-0.643851\pi\)
−0.436695 + 0.899610i \(0.643851\pi\)
\(978\) 1.38124e180 0.167791
\(979\) −3.01688e180 −0.344513
\(980\) 1.01742e181 1.09225
\(981\) −1.61236e181 −1.62737
\(982\) −2.57269e180 −0.244142
\(983\) −4.15955e180 −0.371158 −0.185579 − 0.982629i \(-0.559416\pi\)
−0.185579 + 0.982629i \(0.559416\pi\)
\(984\) 1.80235e180 0.151229
\(985\) 2.40414e181 1.89700
\(986\) 2.61919e180 0.194363
\(987\) −2.29157e179 −0.0159936
\(988\) −2.46808e180 −0.162018
\(989\) 2.40025e181 1.48211
\(990\) −4.76735e180 −0.276916
\(991\) −2.82765e180 −0.154515 −0.0772574 − 0.997011i \(-0.524616\pi\)
−0.0772574 + 0.997011i \(0.524616\pi\)
\(992\) 2.21985e181 1.14122
\(993\) 8.43877e180 0.408178
\(994\) 2.91848e179 0.0132825
\(995\) −5.53530e180 −0.237053
\(996\) −1.11600e181 −0.449755
\(997\) −3.23979e181 −1.22874 −0.614371 − 0.789018i \(-0.710590\pi\)
−0.614371 + 0.789018i \(0.710590\pi\)
\(998\) 9.18200e180 0.327747
\(999\) −1.35551e181 −0.455397
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 1.122.a.a.1.6 ✓ 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.122.a.a.1.6 ✓ 9 1.1 even 1 trivial