Properties

Label 1.118.a.a.1.6
Level $1$
Weight $118$
Character 1.1
Self dual yes
Analytic conductor $86.689$
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] = [1,118,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, 118, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1.1");
 
S:= CuspForms(chi, 118);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 118 \)
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: \(86.6887159558\)
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: \( x^{9} - 4 x^{8} + \cdots - 93\!\cdots\!48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{151}\cdot 3^{56}\cdot 5^{18}\cdot 7^{7}\cdot 11^{4}\cdot 13^{4}\cdot 17^{2}\cdot 19\cdot 23\cdot 29^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-7.00565e14\) 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)
 
\(f(q)\) \(=\) \(q+2.06254e17 q^{2} -1.02253e28 q^{3} -1.23613e35 q^{4} -6.31988e40 q^{5} -2.10900e45 q^{6} +1.11715e49 q^{7} -5.97654e52 q^{8} +3.80005e55 q^{9} +O(q^{10})\) \(q+2.06254e17 q^{2} -1.02253e28 q^{3} -1.23613e35 q^{4} -6.31988e40 q^{5} -2.10900e45 q^{6} +1.11715e49 q^{7} -5.97654e52 q^{8} +3.80005e55 q^{9} -1.30350e58 q^{10} +2.60121e60 q^{11} +1.26398e63 q^{12} -1.57235e65 q^{13} +2.30416e66 q^{14} +6.46226e68 q^{15} +8.21190e69 q^{16} +3.00946e71 q^{17} +7.83774e72 q^{18} +5.05945e74 q^{19} +7.81220e75 q^{20} -1.14232e77 q^{21} +5.36510e77 q^{22} +4.56056e79 q^{23} +6.11118e80 q^{24} -2.02444e81 q^{25} -3.24303e82 q^{26} +2.91988e83 q^{27} -1.38094e84 q^{28} +6.67232e85 q^{29} +1.33286e86 q^{30} +2.24272e87 q^{31} +1.16240e88 q^{32} -2.65981e88 q^{33} +6.20713e88 q^{34} -7.06026e89 q^{35} -4.69735e90 q^{36} -8.85763e91 q^{37} +1.04353e92 q^{38} +1.60777e93 q^{39} +3.77710e93 q^{40} -2.18111e94 q^{41} -2.35607e94 q^{42} -3.81160e95 q^{43} -3.21544e95 q^{44} -2.40159e96 q^{45} +9.40631e96 q^{46} -2.67847e97 q^{47} -8.39690e97 q^{48} -6.27636e98 q^{49} -4.17547e98 q^{50} -3.07726e99 q^{51} +1.94363e100 q^{52} +8.42877e100 q^{53} +6.02235e100 q^{54} -1.64394e101 q^{55} -6.67669e101 q^{56} -5.17343e102 q^{57} +1.37619e103 q^{58} +3.41944e103 q^{59} -7.98819e103 q^{60} -4.30059e104 q^{61} +4.62569e104 q^{62} +4.24522e104 q^{63} +1.03305e105 q^{64} +9.93708e105 q^{65} -5.48596e105 q^{66} +7.34075e105 q^{67} -3.72009e106 q^{68} -4.66330e107 q^{69} -1.45620e107 q^{70} +2.49365e108 q^{71} -2.27111e108 q^{72} +7.67304e108 q^{73} -1.82692e109 q^{74} +2.07004e109 q^{75} -6.25414e109 q^{76} +2.90595e109 q^{77} +3.31609e110 q^{78} -6.74284e110 q^{79} -5.18982e110 q^{80} -5.51481e111 q^{81} -4.49862e111 q^{82} +1.43081e112 q^{83} +1.41205e112 q^{84} -1.90195e112 q^{85} -7.86157e112 q^{86} -6.82264e113 q^{87} -1.55463e113 q^{88} -1.04979e114 q^{89} -4.95336e113 q^{90} -1.75655e114 q^{91} -5.63744e114 q^{92} -2.29324e115 q^{93} -5.52444e114 q^{94} -3.19752e115 q^{95} -1.18858e116 q^{96} -1.36089e116 q^{97} -1.29452e116 q^{98} +9.88474e115 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 40\!\cdots\!52 q^{2}+ \cdots + 22\!\cdots\!97 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 40\!\cdots\!52 q^{2}+ \cdots - 15\!\cdots\!16 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.06254e17 0.505996 0.252998 0.967467i \(-0.418583\pi\)
0.252998 + 0.967467i \(0.418583\pi\)
\(3\) −1.02253e28 −1.25338 −0.626689 0.779270i \(-0.715590\pi\)
−0.626689 + 0.779270i \(0.715590\pi\)
\(4\) −1.23613e35 −0.743968
\(5\) −6.31988e40 −0.814637 −0.407318 0.913286i \(-0.633536\pi\)
−0.407318 + 0.913286i \(0.633536\pi\)
\(6\) −2.10900e45 −0.634204
\(7\) 1.11715e49 0.407264 0.203632 0.979047i \(-0.434725\pi\)
0.203632 + 0.979047i \(0.434725\pi\)
\(8\) −5.97654e52 −0.882441
\(9\) 3.80005e55 0.570956
\(10\) −1.30350e58 −0.412203
\(11\) 2.60121e60 0.311676 0.155838 0.987783i \(-0.450192\pi\)
0.155838 + 0.987783i \(0.450192\pi\)
\(12\) 1.26398e63 0.932473
\(13\) −1.57235e65 −1.07365 −0.536826 0.843693i \(-0.680377\pi\)
−0.536826 + 0.843693i \(0.680377\pi\)
\(14\) 2.30416e66 0.206074
\(15\) 6.46226e68 1.02105
\(16\) 8.21190e69 0.297457
\(17\) 3.00946e71 0.314215 0.157108 0.987581i \(-0.449783\pi\)
0.157108 + 0.987581i \(0.449783\pi\)
\(18\) 7.83774e72 0.288901
\(19\) 5.05945e74 0.788893 0.394446 0.918919i \(-0.370936\pi\)
0.394446 + 0.918919i \(0.370936\pi\)
\(20\) 7.81220e75 0.606064
\(21\) −1.14232e77 −0.510456
\(22\) 5.36510e77 0.157707
\(23\) 4.56056e79 0.995266 0.497633 0.867388i \(-0.334203\pi\)
0.497633 + 0.867388i \(0.334203\pi\)
\(24\) 6.11118e80 1.10603
\(25\) −2.02444e81 −0.336367
\(26\) −3.24303e82 −0.543263
\(27\) 2.91988e83 0.537755
\(28\) −1.38094e84 −0.302992
\(29\) 6.67232e85 1.87929 0.939646 0.342149i \(-0.111155\pi\)
0.939646 + 0.342149i \(0.111155\pi\)
\(30\) 1.33286e86 0.516646
\(31\) 2.24272e87 1.27678 0.638389 0.769714i \(-0.279601\pi\)
0.638389 + 0.769714i \(0.279601\pi\)
\(32\) 1.16240e88 1.03295
\(33\) −2.65981e88 −0.390648
\(34\) 6.20713e88 0.158992
\(35\) −7.06026e89 −0.331772
\(36\) −4.69735e90 −0.424773
\(37\) −8.85763e91 −1.61256 −0.806281 0.591532i \(-0.798523\pi\)
−0.806281 + 0.591532i \(0.798523\pi\)
\(38\) 1.04353e92 0.399176
\(39\) 1.60777e93 1.34569
\(40\) 3.77710e93 0.718868
\(41\) −2.18111e94 −0.979088 −0.489544 0.871979i \(-0.662837\pi\)
−0.489544 + 0.871979i \(0.662837\pi\)
\(42\) −2.35607e94 −0.258288
\(43\) −3.81160e95 −1.05488 −0.527440 0.849592i \(-0.676848\pi\)
−0.527440 + 0.849592i \(0.676848\pi\)
\(44\) −3.21544e95 −0.231877
\(45\) −2.40159e96 −0.465121
\(46\) 9.40631e96 0.503601
\(47\) −2.67847e97 −0.407532 −0.203766 0.979020i \(-0.565318\pi\)
−0.203766 + 0.979020i \(0.565318\pi\)
\(48\) −8.39690e97 −0.372826
\(49\) −6.27636e98 −0.834136
\(50\) −4.17547e98 −0.170200
\(51\) −3.07726e99 −0.393830
\(52\) 1.94363e100 0.798763
\(53\) 8.42877e100 1.13665 0.568324 0.822805i \(-0.307592\pi\)
0.568324 + 0.822805i \(0.307592\pi\)
\(54\) 6.02235e100 0.272102
\(55\) −1.64394e101 −0.253903
\(56\) −6.67669e101 −0.359386
\(57\) −5.17343e102 −0.988780
\(58\) 1.37619e103 0.950913
\(59\) 3.41944e103 0.869184 0.434592 0.900627i \(-0.356893\pi\)
0.434592 + 0.900627i \(0.356893\pi\)
\(60\) −7.98819e103 −0.759627
\(61\) −4.30059e104 −1.55501 −0.777505 0.628877i \(-0.783515\pi\)
−0.777505 + 0.628877i \(0.783515\pi\)
\(62\) 4.62569e104 0.646045
\(63\) 4.24522e104 0.232530
\(64\) 1.03305e105 0.225212
\(65\) 9.93708e105 0.874636
\(66\) −5.48596e105 −0.197666
\(67\) 7.34075e105 0.109740 0.0548700 0.998494i \(-0.482526\pi\)
0.0548700 + 0.998494i \(0.482526\pi\)
\(68\) −3.72009e106 −0.233766
\(69\) −4.66330e107 −1.24744
\(70\) −1.45620e107 −0.167875
\(71\) 2.49365e108 1.25378 0.626890 0.779108i \(-0.284327\pi\)
0.626890 + 0.779108i \(0.284327\pi\)
\(72\) −2.27111e108 −0.503834
\(73\) 7.67304e108 0.759591 0.379796 0.925070i \(-0.375994\pi\)
0.379796 + 0.925070i \(0.375994\pi\)
\(74\) −1.82692e109 −0.815950
\(75\) 2.07004e109 0.421595
\(76\) −6.25414e109 −0.586911
\(77\) 2.90595e109 0.126935
\(78\) 3.31609e110 0.680914
\(79\) −6.74284e110 −0.657140 −0.328570 0.944480i \(-0.606567\pi\)
−0.328570 + 0.944480i \(0.606567\pi\)
\(80\) −5.18982e110 −0.242320
\(81\) −5.51481e111 −1.24497
\(82\) −4.49862e111 −0.495414
\(83\) 1.43081e112 0.775377 0.387689 0.921790i \(-0.373274\pi\)
0.387689 + 0.921790i \(0.373274\pi\)
\(84\) 1.41205e112 0.379763
\(85\) −1.90195e112 −0.255971
\(86\) −7.86157e112 −0.533764
\(87\) −6.82264e113 −2.35546
\(88\) −1.55463e113 −0.275036
\(89\) −1.04979e114 −0.958928 −0.479464 0.877562i \(-0.659169\pi\)
−0.479464 + 0.877562i \(0.659169\pi\)
\(90\) −4.95336e113 −0.235349
\(91\) −1.75655e114 −0.437260
\(92\) −5.63744e114 −0.740447
\(93\) −2.29324e115 −1.60029
\(94\) −5.52444e114 −0.206209
\(95\) −3.19752e115 −0.642661
\(96\) −1.18858e116 −1.29468
\(97\) −1.36089e116 −0.808492 −0.404246 0.914650i \(-0.632466\pi\)
−0.404246 + 0.914650i \(0.632466\pi\)
\(98\) −1.29452e116 −0.422069
\(99\) 9.88474e115 0.177953
\(100\) 2.50247e116 0.250247
\(101\) −2.33869e117 −1.30669 −0.653346 0.757060i \(-0.726635\pi\)
−0.653346 + 0.757060i \(0.726635\pi\)
\(102\) −6.34696e116 −0.199276
\(103\) 8.81025e117 1.56319 0.781593 0.623789i \(-0.214408\pi\)
0.781593 + 0.623789i \(0.214408\pi\)
\(104\) 9.39722e117 0.947434
\(105\) 7.21931e117 0.415836
\(106\) 1.73846e118 0.575139
\(107\) 1.43168e118 0.273461 0.136731 0.990608i \(-0.456341\pi\)
0.136731 + 0.990608i \(0.456341\pi\)
\(108\) −3.60934e118 −0.400072
\(109\) 1.62351e119 1.04955 0.524777 0.851240i \(-0.324149\pi\)
0.524777 + 0.851240i \(0.324149\pi\)
\(110\) −3.39068e118 −0.128474
\(111\) 9.05718e119 2.02115
\(112\) 9.17392e118 0.121144
\(113\) 1.94627e119 0.152797 0.0763983 0.997077i \(-0.475658\pi\)
0.0763983 + 0.997077i \(0.475658\pi\)
\(114\) −1.06704e120 −0.500319
\(115\) −2.88222e120 −0.810780
\(116\) −8.24785e120 −1.39813
\(117\) −5.97501e120 −0.613008
\(118\) 7.05271e120 0.439803
\(119\) 3.36202e120 0.127969
\(120\) −3.86219e121 −0.901013
\(121\) −6.28874e121 −0.902858
\(122\) −8.87013e121 −0.786828
\(123\) 2.23025e122 1.22717
\(124\) −2.77229e122 −0.949883
\(125\) 5.08306e122 1.08865
\(126\) 8.75593e121 0.117659
\(127\) 1.11458e123 0.943180 0.471590 0.881818i \(-0.343680\pi\)
0.471590 + 0.881818i \(0.343680\pi\)
\(128\) −1.71829e123 −0.918996
\(129\) 3.89747e123 1.32216
\(130\) 2.04956e123 0.442562
\(131\) 3.81645e123 0.526368 0.263184 0.964746i \(-0.415227\pi\)
0.263184 + 0.964746i \(0.415227\pi\)
\(132\) 3.28788e123 0.290630
\(133\) 5.65217e123 0.321288
\(134\) 1.51406e123 0.0555279
\(135\) −1.84533e124 −0.438075
\(136\) −1.79862e124 −0.277276
\(137\) −1.23314e125 −1.23839 −0.619196 0.785236i \(-0.712541\pi\)
−0.619196 + 0.785236i \(0.712541\pi\)
\(138\) −9.61822e124 −0.631202
\(139\) −2.80964e125 −1.20860 −0.604302 0.796755i \(-0.706548\pi\)
−0.604302 + 0.796755i \(0.706548\pi\)
\(140\) 8.72739e124 0.246828
\(141\) 2.73881e125 0.510791
\(142\) 5.14324e125 0.634408
\(143\) −4.09002e125 −0.334632
\(144\) 3.12056e125 0.169835
\(145\) −4.21683e126 −1.53094
\(146\) 1.58259e126 0.384350
\(147\) 6.41775e126 1.04549
\(148\) 1.09492e127 1.19970
\(149\) 8.13026e126 0.600769 0.300385 0.953818i \(-0.402885\pi\)
0.300385 + 0.953818i \(0.402885\pi\)
\(150\) 4.26954e126 0.213325
\(151\) 1.66189e127 0.562924 0.281462 0.959572i \(-0.409181\pi\)
0.281462 + 0.959572i \(0.409181\pi\)
\(152\) −3.02380e127 −0.696151
\(153\) 1.14361e127 0.179403
\(154\) 5.99362e126 0.0642284
\(155\) −1.41737e128 −1.04011
\(156\) −1.98742e128 −1.00115
\(157\) 3.29730e128 1.14296 0.571478 0.820618i \(-0.306370\pi\)
0.571478 + 0.820618i \(0.306370\pi\)
\(158\) −1.39073e128 −0.332510
\(159\) −8.61866e128 −1.42465
\(160\) −7.34621e128 −0.841481
\(161\) 5.09482e128 0.405336
\(162\) −1.13745e129 −0.629947
\(163\) 3.57156e129 1.38001 0.690006 0.723804i \(-0.257608\pi\)
0.690006 + 0.723804i \(0.257608\pi\)
\(164\) 2.69613e129 0.728410
\(165\) 1.68097e129 0.318236
\(166\) 2.95109e129 0.392338
\(167\) −5.12749e129 −0.479724 −0.239862 0.970807i \(-0.577102\pi\)
−0.239862 + 0.970807i \(0.577102\pi\)
\(168\) 6.82710e129 0.450447
\(169\) 3.27560e129 0.152728
\(170\) −3.92283e129 −0.129520
\(171\) 1.92262e130 0.450423
\(172\) 4.71163e130 0.784797
\(173\) −9.87059e129 −0.117124 −0.0585620 0.998284i \(-0.518652\pi\)
−0.0585620 + 0.998284i \(0.518652\pi\)
\(174\) −1.40719e131 −1.19185
\(175\) −2.26160e130 −0.136990
\(176\) 2.13609e130 0.0927104
\(177\) −3.49647e131 −1.08942
\(178\) −2.16523e131 −0.485214
\(179\) −9.05305e131 −1.46182 −0.730908 0.682476i \(-0.760903\pi\)
−0.730908 + 0.682476i \(0.760903\pi\)
\(180\) 2.96867e131 0.346036
\(181\) 1.41858e132 1.19580 0.597899 0.801571i \(-0.296002\pi\)
0.597899 + 0.801571i \(0.296002\pi\)
\(182\) −3.62295e131 −0.221252
\(183\) 4.39748e132 1.94901
\(184\) −2.72563e132 −0.878263
\(185\) 5.59792e132 1.31365
\(186\) −4.72990e132 −0.809738
\(187\) 7.82826e131 0.0979335
\(188\) 3.31093e132 0.303191
\(189\) 3.26194e132 0.219008
\(190\) −6.59499e132 −0.325184
\(191\) 4.37164e133 1.58560 0.792798 0.609485i \(-0.208624\pi\)
0.792798 + 0.609485i \(0.208624\pi\)
\(192\) −1.05632e133 −0.282276
\(193\) −9.31937e133 −1.83774 −0.918870 0.394562i \(-0.870896\pi\)
−0.918870 + 0.394562i \(0.870896\pi\)
\(194\) −2.80688e133 −0.409094
\(195\) −1.01609e134 −1.09625
\(196\) 7.75839e133 0.620571
\(197\) −8.84922e132 −0.0525570 −0.0262785 0.999655i \(-0.508366\pi\)
−0.0262785 + 0.999655i \(0.508366\pi\)
\(198\) 2.03876e133 0.0900437
\(199\) −3.89716e133 −0.128187 −0.0640934 0.997944i \(-0.520416\pi\)
−0.0640934 + 0.997944i \(0.520416\pi\)
\(200\) 1.20991e134 0.296824
\(201\) −7.50613e133 −0.137546
\(202\) −4.82364e134 −0.661180
\(203\) 7.45398e134 0.765368
\(204\) 3.80389e134 0.292997
\(205\) 1.37844e135 0.797601
\(206\) 1.81715e135 0.790965
\(207\) 1.73303e135 0.568253
\(208\) −1.29120e135 −0.319365
\(209\) 1.31607e135 0.245879
\(210\) 1.48901e135 0.210411
\(211\) −1.06499e136 −1.13978 −0.569892 0.821720i \(-0.693015\pi\)
−0.569892 + 0.821720i \(0.693015\pi\)
\(212\) −1.04191e136 −0.845630
\(213\) −2.54982e136 −1.57146
\(214\) 2.95290e135 0.138370
\(215\) 2.40889e136 0.859343
\(216\) −1.74507e136 −0.474536
\(217\) 2.50545e136 0.519986
\(218\) 3.34855e136 0.531070
\(219\) −7.84590e136 −0.952055
\(220\) 2.03212e136 0.188896
\(221\) −4.73193e136 −0.337358
\(222\) 1.86808e137 1.02269
\(223\) 8.11659e136 0.341617 0.170808 0.985304i \(-0.445362\pi\)
0.170808 + 0.985304i \(0.445362\pi\)
\(224\) 1.29857e137 0.420684
\(225\) −7.69296e136 −0.192051
\(226\) 4.01425e136 0.0773144
\(227\) 4.44026e137 0.660536 0.330268 0.943887i \(-0.392861\pi\)
0.330268 + 0.943887i \(0.392861\pi\)
\(228\) 6.39503e137 0.735621
\(229\) −4.74332e137 −0.422383 −0.211192 0.977445i \(-0.567734\pi\)
−0.211192 + 0.977445i \(0.567734\pi\)
\(230\) −5.94468e137 −0.410251
\(231\) −2.97141e137 −0.159097
\(232\) −3.98774e138 −1.65836
\(233\) −5.00182e137 −0.161736 −0.0808680 0.996725i \(-0.525769\pi\)
−0.0808680 + 0.996725i \(0.525769\pi\)
\(234\) −1.23237e138 −0.310179
\(235\) 1.69276e138 0.331990
\(236\) −4.22687e138 −0.646645
\(237\) 6.89474e138 0.823644
\(238\) 6.93429e137 0.0647515
\(239\) −5.26866e138 −0.384966 −0.192483 0.981300i \(-0.561654\pi\)
−0.192483 + 0.981300i \(0.561654\pi\)
\(240\) 5.30674e138 0.303718
\(241\) 6.24703e138 0.280334 0.140167 0.990128i \(-0.455236\pi\)
0.140167 + 0.990128i \(0.455236\pi\)
\(242\) −1.29708e139 −0.456842
\(243\) 3.69570e139 1.02266
\(244\) 5.31609e139 1.15688
\(245\) 3.96658e139 0.679518
\(246\) 4.59996e139 0.620941
\(247\) −7.95524e139 −0.846996
\(248\) −1.34037e140 −1.12668
\(249\) −1.46304e140 −0.971841
\(250\) 1.04840e140 0.550854
\(251\) −2.35375e140 −0.979145 −0.489572 0.871963i \(-0.662847\pi\)
−0.489572 + 0.871963i \(0.662847\pi\)
\(252\) −5.24765e139 −0.172995
\(253\) 1.18630e140 0.310201
\(254\) 2.29887e140 0.477245
\(255\) 1.94479e140 0.320829
\(256\) −5.26049e140 −0.690220
\(257\) −6.06950e140 −0.633965 −0.316982 0.948431i \(-0.602670\pi\)
−0.316982 + 0.948431i \(0.602670\pi\)
\(258\) 8.03867e140 0.669009
\(259\) −9.89530e140 −0.656739
\(260\) −1.22835e141 −0.650701
\(261\) 2.53551e141 1.07299
\(262\) 7.87157e140 0.266340
\(263\) 6.71043e141 1.81693 0.908466 0.417958i \(-0.137254\pi\)
0.908466 + 0.417958i \(0.137254\pi\)
\(264\) 1.58965e141 0.344724
\(265\) −5.32689e141 −0.925955
\(266\) 1.16578e141 0.162570
\(267\) 1.07344e142 1.20190
\(268\) −9.07412e140 −0.0816430
\(269\) −1.79907e142 −1.30178 −0.650892 0.759170i \(-0.725605\pi\)
−0.650892 + 0.759170i \(0.725605\pi\)
\(270\) −3.80606e141 −0.221664
\(271\) −3.65599e142 −1.71515 −0.857575 0.514360i \(-0.828030\pi\)
−0.857575 + 0.514360i \(0.828030\pi\)
\(272\) 2.47134e141 0.0934656
\(273\) 1.79612e142 0.548052
\(274\) −2.54339e142 −0.626621
\(275\) −5.26599e141 −0.104838
\(276\) 5.76444e142 0.928059
\(277\) −3.51238e142 −0.457652 −0.228826 0.973467i \(-0.573489\pi\)
−0.228826 + 0.973467i \(0.573489\pi\)
\(278\) −5.79498e142 −0.611549
\(279\) 8.52244e142 0.728984
\(280\) 4.21959e142 0.292769
\(281\) 2.99440e142 0.168651 0.0843257 0.996438i \(-0.473126\pi\)
0.0843257 + 0.996438i \(0.473126\pi\)
\(282\) 5.64889e142 0.258458
\(283\) 3.94255e143 1.46646 0.733231 0.679979i \(-0.238011\pi\)
0.733231 + 0.679979i \(0.238011\pi\)
\(284\) −3.08247e143 −0.932773
\(285\) 3.26955e143 0.805497
\(286\) −8.43582e142 −0.169322
\(287\) −2.43662e143 −0.398747
\(288\) 4.41716e143 0.589770
\(289\) −8.26758e143 −0.901269
\(290\) −8.69737e143 −0.774649
\(291\) 1.39155e144 1.01335
\(292\) −9.48487e143 −0.565112
\(293\) −3.46158e144 −1.68857 −0.844285 0.535895i \(-0.819974\pi\)
−0.844285 + 0.535895i \(0.819974\pi\)
\(294\) 1.32368e144 0.529012
\(295\) −2.16104e144 −0.708069
\(296\) 5.29380e144 1.42299
\(297\) 7.59522e143 0.167605
\(298\) 1.67690e144 0.303987
\(299\) −7.17080e144 −1.06857
\(300\) −2.55884e144 −0.313654
\(301\) −4.25813e144 −0.429614
\(302\) 3.42770e144 0.284837
\(303\) 2.39138e145 1.63778
\(304\) 4.15477e144 0.234662
\(305\) 2.71793e145 1.26677
\(306\) 2.35874e144 0.0907771
\(307\) 2.43352e145 0.773821 0.386910 0.922117i \(-0.373542\pi\)
0.386910 + 0.922117i \(0.373542\pi\)
\(308\) −3.59212e144 −0.0944353
\(309\) −9.00873e145 −1.95926
\(310\) −2.92338e145 −0.526292
\(311\) −5.29296e144 −0.0789252 −0.0394626 0.999221i \(-0.512565\pi\)
−0.0394626 + 0.999221i \(0.512565\pi\)
\(312\) −9.60892e145 −1.18749
\(313\) −4.02666e145 −0.412669 −0.206334 0.978482i \(-0.566153\pi\)
−0.206334 + 0.978482i \(0.566153\pi\)
\(314\) 6.80081e145 0.578331
\(315\) −2.68293e145 −0.189427
\(316\) 8.33502e145 0.488891
\(317\) 1.18600e146 0.578253 0.289126 0.957291i \(-0.406635\pi\)
0.289126 + 0.957291i \(0.406635\pi\)
\(318\) −1.77763e146 −0.720866
\(319\) 1.73561e146 0.585731
\(320\) −6.52875e145 −0.183466
\(321\) −1.46394e146 −0.342750
\(322\) 1.05083e146 0.205098
\(323\) 1.52262e146 0.247882
\(324\) 6.81702e146 0.926215
\(325\) 3.18313e146 0.361141
\(326\) 7.36647e146 0.698280
\(327\) −1.66008e147 −1.31549
\(328\) 1.30355e147 0.863987
\(329\) −2.99225e146 −0.165973
\(330\) 3.46707e146 0.161026
\(331\) −2.45081e146 −0.0953614 −0.0476807 0.998863i \(-0.515183\pi\)
−0.0476807 + 0.998863i \(0.515183\pi\)
\(332\) −1.76866e147 −0.576856
\(333\) −3.36594e147 −0.920702
\(334\) −1.05756e147 −0.242738
\(335\) −4.63927e146 −0.0893982
\(336\) −9.38059e146 −0.151839
\(337\) −1.19796e147 −0.162964 −0.0814822 0.996675i \(-0.525965\pi\)
−0.0814822 + 0.996675i \(0.525965\pi\)
\(338\) 6.75605e146 0.0772798
\(339\) −1.99011e147 −0.191512
\(340\) 2.35105e147 0.190434
\(341\) 5.83379e147 0.397942
\(342\) 3.96547e147 0.227912
\(343\) −1.54175e148 −0.746978
\(344\) 2.27802e148 0.930868
\(345\) 2.94715e148 1.01621
\(346\) −2.03584e147 −0.0592642
\(347\) −4.41566e148 −1.08573 −0.542864 0.839820i \(-0.682660\pi\)
−0.542864 + 0.839820i \(0.682660\pi\)
\(348\) 8.43366e148 1.75239
\(349\) 3.55967e148 0.625349 0.312675 0.949860i \(-0.398775\pi\)
0.312675 + 0.949860i \(0.398775\pi\)
\(350\) −4.66463e147 −0.0693165
\(351\) −4.59107e148 −0.577361
\(352\) 3.02364e148 0.321947
\(353\) −1.87564e149 −1.69172 −0.845860 0.533406i \(-0.820912\pi\)
−0.845860 + 0.533406i \(0.820912\pi\)
\(354\) −7.21160e148 −0.551240
\(355\) −1.57596e149 −1.02138
\(356\) 1.29767e149 0.713412
\(357\) −3.43776e148 −0.160393
\(358\) −1.86723e149 −0.739673
\(359\) 1.17918e149 0.396784 0.198392 0.980123i \(-0.436428\pi\)
0.198392 + 0.980123i \(0.436428\pi\)
\(360\) 1.43532e149 0.410442
\(361\) −1.55331e149 −0.377648
\(362\) 2.92588e149 0.605069
\(363\) 6.43041e149 1.13162
\(364\) 2.17133e149 0.325307
\(365\) −4.84927e149 −0.618791
\(366\) 9.06996e149 0.986193
\(367\) −1.64912e150 −1.52858 −0.764289 0.644874i \(-0.776910\pi\)
−0.764289 + 0.644874i \(0.776910\pi\)
\(368\) 3.74508e149 0.296049
\(369\) −8.28832e149 −0.559016
\(370\) 1.15459e150 0.664703
\(371\) 9.41620e149 0.462916
\(372\) 2.83475e150 1.19056
\(373\) 1.25728e150 0.451302 0.225651 0.974208i \(-0.427549\pi\)
0.225651 + 0.974208i \(0.427549\pi\)
\(374\) 1.61461e149 0.0495539
\(375\) −5.19758e150 −1.36449
\(376\) 1.60080e150 0.359623
\(377\) −1.04912e151 −2.01770
\(378\) 6.72787e149 0.110817
\(379\) −7.82834e150 −1.10478 −0.552390 0.833586i \(-0.686284\pi\)
−0.552390 + 0.833586i \(0.686284\pi\)
\(380\) 3.95254e150 0.478119
\(381\) −1.13969e151 −1.18216
\(382\) 9.01667e150 0.802305
\(383\) 1.36028e151 1.03872 0.519360 0.854555i \(-0.326170\pi\)
0.519360 + 0.854555i \(0.326170\pi\)
\(384\) 1.75700e151 1.15185
\(385\) −1.83652e150 −0.103406
\(386\) −1.92215e151 −0.929888
\(387\) −1.44843e151 −0.602289
\(388\) 1.68223e151 0.601493
\(389\) 5.70014e151 1.75321 0.876605 0.481211i \(-0.159803\pi\)
0.876605 + 0.481211i \(0.159803\pi\)
\(390\) −2.09573e151 −0.554697
\(391\) 1.37248e151 0.312728
\(392\) 3.75109e151 0.736075
\(393\) −3.90243e151 −0.659737
\(394\) −1.82518e150 −0.0265936
\(395\) 4.26140e151 0.535330
\(396\) −1.22188e151 −0.132392
\(397\) −1.28954e152 −1.20557 −0.602783 0.797905i \(-0.705941\pi\)
−0.602783 + 0.797905i \(0.705941\pi\)
\(398\) −8.03803e150 −0.0648619
\(399\) −5.77950e151 −0.402695
\(400\) −1.66245e151 −0.100055
\(401\) −1.35206e152 −0.703151 −0.351576 0.936159i \(-0.614354\pi\)
−0.351576 + 0.936159i \(0.614354\pi\)
\(402\) −1.54817e151 −0.0695975
\(403\) −3.52634e152 −1.37082
\(404\) 2.89093e152 0.972137
\(405\) 3.48530e152 1.01419
\(406\) 1.53741e152 0.387273
\(407\) −2.30406e152 −0.502598
\(408\) 1.83914e152 0.347532
\(409\) −3.83225e151 −0.0627539 −0.0313769 0.999508i \(-0.509989\pi\)
−0.0313769 + 0.999508i \(0.509989\pi\)
\(410\) 2.84307e152 0.403583
\(411\) 1.26092e153 1.55217
\(412\) −1.08906e153 −1.16296
\(413\) 3.82002e152 0.353987
\(414\) 3.57444e152 0.287534
\(415\) −9.04255e152 −0.631651
\(416\) −1.82769e153 −1.10903
\(417\) 2.87293e153 1.51484
\(418\) 2.71445e152 0.124414
\(419\) −1.20496e152 −0.0480232 −0.0240116 0.999712i \(-0.507644\pi\)
−0.0240116 + 0.999712i \(0.507644\pi\)
\(420\) −8.92401e152 −0.309369
\(421\) 2.08475e153 0.628855 0.314428 0.949281i \(-0.398187\pi\)
0.314428 + 0.949281i \(0.398187\pi\)
\(422\) −2.19658e153 −0.576726
\(423\) −1.01783e153 −0.232683
\(424\) −5.03749e153 −1.00302
\(425\) −6.09247e152 −0.105692
\(426\) −5.25911e153 −0.795152
\(427\) −4.80441e153 −0.633299
\(428\) −1.76975e153 −0.203447
\(429\) 4.18216e153 0.419420
\(430\) 4.96842e153 0.434824
\(431\) 1.44213e154 1.10175 0.550877 0.834586i \(-0.314293\pi\)
0.550877 + 0.834586i \(0.314293\pi\)
\(432\) 2.39777e153 0.159959
\(433\) 3.21884e153 0.187568 0.0937838 0.995593i \(-0.470104\pi\)
0.0937838 + 0.995593i \(0.470104\pi\)
\(434\) 5.16759e153 0.263111
\(435\) 4.31183e154 1.91885
\(436\) −2.00687e154 −0.780835
\(437\) 2.30739e154 0.785158
\(438\) −1.61824e154 −0.481736
\(439\) −4.91798e154 −1.28119 −0.640593 0.767881i \(-0.721311\pi\)
−0.640593 + 0.767881i \(0.721311\pi\)
\(440\) 9.82505e153 0.224054
\(441\) −2.38505e154 −0.476255
\(442\) −9.75978e153 −0.170702
\(443\) 5.06151e154 0.775644 0.387822 0.921734i \(-0.373228\pi\)
0.387822 + 0.921734i \(0.373228\pi\)
\(444\) −1.11958e155 −1.50367
\(445\) 6.63454e154 0.781178
\(446\) 1.67408e154 0.172857
\(447\) −8.31342e154 −0.752991
\(448\) 1.15407e154 0.0917209
\(449\) −3.80471e154 −0.265406 −0.132703 0.991156i \(-0.542366\pi\)
−0.132703 + 0.991156i \(0.542366\pi\)
\(450\) −1.58670e154 −0.0971769
\(451\) −5.67353e154 −0.305159
\(452\) −2.40584e154 −0.113676
\(453\) −1.69933e155 −0.705557
\(454\) 9.15821e154 0.334228
\(455\) 1.11012e155 0.356208
\(456\) 3.09192e155 0.872540
\(457\) 7.92549e155 1.96756 0.983781 0.179373i \(-0.0574067\pi\)
0.983781 + 0.179373i \(0.0574067\pi\)
\(458\) −9.78326e154 −0.213724
\(459\) 8.78726e154 0.168971
\(460\) 3.56280e155 0.603195
\(461\) −8.52437e155 −1.27104 −0.635518 0.772086i \(-0.719214\pi\)
−0.635518 + 0.772086i \(0.719214\pi\)
\(462\) −6.12864e154 −0.0805024
\(463\) −1.49464e156 −1.73001 −0.865007 0.501760i \(-0.832686\pi\)
−0.865007 + 0.501760i \(0.832686\pi\)
\(464\) 5.47924e155 0.559009
\(465\) 1.44930e156 1.30365
\(466\) −1.03164e155 −0.0818378
\(467\) −8.02419e155 −0.561518 −0.280759 0.959778i \(-0.590586\pi\)
−0.280759 + 0.959778i \(0.590586\pi\)
\(468\) 7.38589e155 0.456058
\(469\) 8.20072e154 0.0446931
\(470\) 3.49138e155 0.167986
\(471\) −3.37159e156 −1.43255
\(472\) −2.04364e156 −0.767003
\(473\) −9.91479e155 −0.328781
\(474\) 1.42207e156 0.416761
\(475\) −1.02425e156 −0.265358
\(476\) −4.15589e155 −0.0952045
\(477\) 3.20297e156 0.648975
\(478\) −1.08668e156 −0.194791
\(479\) −5.47633e156 −0.868684 −0.434342 0.900748i \(-0.643019\pi\)
−0.434342 + 0.900748i \(0.643019\pi\)
\(480\) 7.51171e156 1.05469
\(481\) 1.39273e157 1.73133
\(482\) 1.28847e156 0.141848
\(483\) −5.20960e156 −0.508039
\(484\) 7.77369e156 0.671698
\(485\) 8.60065e156 0.658627
\(486\) 7.62252e156 0.517460
\(487\) −2.05161e157 −1.23495 −0.617477 0.786589i \(-0.711845\pi\)
−0.617477 + 0.786589i \(0.711845\pi\)
\(488\) 2.57027e157 1.37220
\(489\) −3.65202e157 −1.72968
\(490\) 8.18122e156 0.343833
\(491\) −4.10785e157 −1.53231 −0.766156 0.642654i \(-0.777833\pi\)
−0.766156 + 0.642654i \(0.777833\pi\)
\(492\) −2.75687e157 −0.912973
\(493\) 2.00801e157 0.590502
\(494\) −1.64080e157 −0.428576
\(495\) −6.24704e156 −0.144967
\(496\) 1.84170e157 0.379787
\(497\) 2.78578e157 0.510620
\(498\) −3.01758e157 −0.491747
\(499\) 7.09122e157 1.02764 0.513818 0.857899i \(-0.328231\pi\)
0.513818 + 0.857899i \(0.328231\pi\)
\(500\) −6.28332e157 −0.809924
\(501\) 5.24301e157 0.601275
\(502\) −4.85469e157 −0.495443
\(503\) −1.50629e157 −0.136830 −0.0684152 0.997657i \(-0.521794\pi\)
−0.0684152 + 0.997657i \(0.521794\pi\)
\(504\) −2.53717e157 −0.205194
\(505\) 1.47803e158 1.06448
\(506\) 2.44678e157 0.156960
\(507\) −3.34940e157 −0.191426
\(508\) −1.37777e158 −0.701696
\(509\) 3.20519e158 1.45500 0.727502 0.686106i \(-0.240681\pi\)
0.727502 + 0.686106i \(0.240681\pi\)
\(510\) 4.01121e157 0.162338
\(511\) 8.57193e157 0.309354
\(512\) 1.77001e158 0.569747
\(513\) 1.47730e158 0.424231
\(514\) −1.25186e158 −0.320783
\(515\) −5.56797e158 −1.27343
\(516\) −4.81778e158 −0.983647
\(517\) −6.96727e157 −0.127018
\(518\) −2.04094e158 −0.332307
\(519\) 1.00930e158 0.146801
\(520\) −5.93893e158 −0.771814
\(521\) 5.48692e158 0.637267 0.318634 0.947878i \(-0.396776\pi\)
0.318634 + 0.947878i \(0.396776\pi\)
\(522\) 5.22959e158 0.542929
\(523\) −1.11172e159 −1.03193 −0.515963 0.856611i \(-0.672566\pi\)
−0.515963 + 0.856611i \(0.672566\pi\)
\(524\) −4.71763e158 −0.391601
\(525\) 2.31255e158 0.171701
\(526\) 1.38405e159 0.919360
\(527\) 6.74938e158 0.401183
\(528\) −2.18421e158 −0.116201
\(529\) −1.98311e157 −0.00944473
\(530\) −1.09869e159 −0.468529
\(531\) 1.29940e159 0.496266
\(532\) −6.98681e158 −0.239028
\(533\) 3.42947e159 1.05120
\(534\) 2.21400e159 0.608156
\(535\) −9.04807e158 −0.222772
\(536\) −4.38723e158 −0.0968390
\(537\) 9.25700e159 1.83221
\(538\) −3.71064e159 −0.658697
\(539\) −1.63261e159 −0.259981
\(540\) 2.28106e159 0.325914
\(541\) −1.13734e160 −1.45832 −0.729159 0.684345i \(-0.760088\pi\)
−0.729159 + 0.684345i \(0.760088\pi\)
\(542\) −7.54062e159 −0.867858
\(543\) −1.45054e160 −1.49879
\(544\) 3.49819e159 0.324569
\(545\) −1.02604e160 −0.855005
\(546\) 3.70457e159 0.277312
\(547\) 7.37062e159 0.495730 0.247865 0.968795i \(-0.420271\pi\)
0.247865 + 0.968795i \(0.420271\pi\)
\(548\) 1.52432e160 0.921324
\(549\) −1.63425e160 −0.887842
\(550\) −1.08613e159 −0.0530475
\(551\) 3.37583e160 1.48256
\(552\) 2.78704e160 1.10080
\(553\) −7.53276e159 −0.267629
\(554\) −7.24441e159 −0.231570
\(555\) −5.72403e160 −1.64650
\(556\) 3.47307e160 0.899164
\(557\) −1.61276e160 −0.375872 −0.187936 0.982181i \(-0.560180\pi\)
−0.187936 + 0.982181i \(0.560180\pi\)
\(558\) 1.75778e160 0.368863
\(559\) 5.99318e160 1.13257
\(560\) −5.79781e159 −0.0986880
\(561\) −8.00461e159 −0.122748
\(562\) 6.17606e159 0.0853369
\(563\) 3.35018e160 0.417184 0.208592 0.978003i \(-0.433112\pi\)
0.208592 + 0.978003i \(0.433112\pi\)
\(564\) −3.38552e160 −0.380013
\(565\) −1.23002e160 −0.124474
\(566\) 8.13166e160 0.742024
\(567\) −6.16087e160 −0.507030
\(568\) −1.49034e161 −1.10639
\(569\) −1.22970e161 −0.823626 −0.411813 0.911268i \(-0.635104\pi\)
−0.411813 + 0.911268i \(0.635104\pi\)
\(570\) 6.74357e160 0.407578
\(571\) 1.76853e161 0.964718 0.482359 0.875974i \(-0.339780\pi\)
0.482359 + 0.875974i \(0.339780\pi\)
\(572\) 5.05580e160 0.248956
\(573\) −4.47013e161 −1.98735
\(574\) −5.02563e160 −0.201764
\(575\) −9.23256e160 −0.334775
\(576\) 3.92564e160 0.128586
\(577\) −2.79681e161 −0.827708 −0.413854 0.910343i \(-0.635818\pi\)
−0.413854 + 0.910343i \(0.635818\pi\)
\(578\) −1.70522e161 −0.456038
\(579\) 9.52932e161 2.30338
\(580\) 5.21255e161 1.13897
\(581\) 1.59843e161 0.315783
\(582\) 2.87011e161 0.512749
\(583\) 2.19250e161 0.354266
\(584\) −4.58582e161 −0.670294
\(585\) 3.77614e161 0.499378
\(586\) −7.13964e161 −0.854409
\(587\) 3.81473e161 0.413176 0.206588 0.978428i \(-0.433764\pi\)
0.206588 + 0.978428i \(0.433764\pi\)
\(588\) −7.93317e161 −0.777810
\(589\) 1.13469e162 1.00724
\(590\) −4.45723e161 −0.358280
\(591\) 9.04857e160 0.0658738
\(592\) −7.27379e161 −0.479668
\(593\) −4.84288e161 −0.289337 −0.144669 0.989480i \(-0.546212\pi\)
−0.144669 + 0.989480i \(0.546212\pi\)
\(594\) 1.56654e161 0.0848076
\(595\) −2.12476e161 −0.104248
\(596\) −1.00501e162 −0.446953
\(597\) 3.98495e161 0.160666
\(598\) −1.47900e162 −0.540692
\(599\) −2.82842e162 −0.937725 −0.468862 0.883271i \(-0.655336\pi\)
−0.468862 + 0.883271i \(0.655336\pi\)
\(600\) −1.23717e162 −0.372033
\(601\) 2.49792e162 0.681432 0.340716 0.940166i \(-0.389331\pi\)
0.340716 + 0.940166i \(0.389331\pi\)
\(602\) −8.78255e161 −0.217383
\(603\) 2.78952e161 0.0626566
\(604\) −2.05431e162 −0.418798
\(605\) 3.97441e162 0.735501
\(606\) 4.93231e162 0.828708
\(607\) 6.61731e162 1.00958 0.504792 0.863241i \(-0.331569\pi\)
0.504792 + 0.863241i \(0.331569\pi\)
\(608\) 5.88109e162 0.814889
\(609\) −7.62191e162 −0.959295
\(610\) 5.60582e162 0.640979
\(611\) 4.21149e162 0.437547
\(612\) −1.41365e162 −0.133470
\(613\) −8.03381e162 −0.689420 −0.344710 0.938709i \(-0.612023\pi\)
−0.344710 + 0.938709i \(0.612023\pi\)
\(614\) 5.01923e162 0.391550
\(615\) −1.40949e163 −0.999695
\(616\) −1.73675e162 −0.112012
\(617\) 8.99712e162 0.527742 0.263871 0.964558i \(-0.415001\pi\)
0.263871 + 0.964558i \(0.415001\pi\)
\(618\) −1.85808e163 −0.991378
\(619\) −7.73941e162 −0.375669 −0.187834 0.982201i \(-0.560147\pi\)
−0.187834 + 0.982201i \(0.560147\pi\)
\(620\) 1.75206e163 0.773809
\(621\) 1.33163e163 0.535209
\(622\) −1.09169e162 −0.0399358
\(623\) −1.17277e163 −0.390537
\(624\) 1.32029e163 0.400286
\(625\) −1.99402e163 −0.550490
\(626\) −8.30514e162 −0.208809
\(627\) −1.34572e163 −0.308180
\(628\) −4.07589e163 −0.850323
\(629\) −2.66567e163 −0.506692
\(630\) −5.53365e162 −0.0958494
\(631\) −2.94001e163 −0.464121 −0.232061 0.972701i \(-0.574547\pi\)
−0.232061 + 0.972701i \(0.574547\pi\)
\(632\) 4.02988e163 0.579887
\(633\) 1.08898e164 1.42858
\(634\) 2.44617e163 0.292593
\(635\) −7.04403e163 −0.768349
\(636\) 1.06538e164 1.05989
\(637\) 9.86864e163 0.895572
\(638\) 3.57977e163 0.296377
\(639\) 9.47598e163 0.715853
\(640\) 1.08594e164 0.748648
\(641\) −8.78740e163 −0.552924 −0.276462 0.961025i \(-0.589162\pi\)
−0.276462 + 0.961025i \(0.589162\pi\)
\(642\) −3.01942e163 −0.173430
\(643\) −1.70500e164 −0.894094 −0.447047 0.894510i \(-0.647524\pi\)
−0.447047 + 0.894510i \(0.647524\pi\)
\(644\) −6.29786e163 −0.301557
\(645\) −2.46316e164 −1.07708
\(646\) 3.14047e163 0.125427
\(647\) −3.89635e164 −1.42154 −0.710768 0.703426i \(-0.751653\pi\)
−0.710768 + 0.703426i \(0.751653\pi\)
\(648\) 3.29595e164 1.09861
\(649\) 8.89468e163 0.270904
\(650\) 6.56531e163 0.182736
\(651\) −2.56190e164 −0.651739
\(652\) −4.41491e164 −1.02668
\(653\) 6.09081e164 1.29495 0.647477 0.762085i \(-0.275824\pi\)
0.647477 + 0.762085i \(0.275824\pi\)
\(654\) −3.42398e164 −0.665631
\(655\) −2.41195e164 −0.428798
\(656\) −1.79110e164 −0.291237
\(657\) 2.91579e164 0.433693
\(658\) −6.17162e163 −0.0839817
\(659\) 9.84938e164 1.22634 0.613172 0.789950i \(-0.289893\pi\)
0.613172 + 0.789950i \(0.289893\pi\)
\(660\) −2.07790e164 −0.236758
\(661\) 9.16804e164 0.956073 0.478036 0.878340i \(-0.341349\pi\)
0.478036 + 0.878340i \(0.341349\pi\)
\(662\) −5.05489e163 −0.0482525
\(663\) 4.83854e164 0.422837
\(664\) −8.55128e164 −0.684224
\(665\) −3.57210e164 −0.261733
\(666\) −6.94238e164 −0.465871
\(667\) 3.04295e165 1.87040
\(668\) 6.33824e164 0.356899
\(669\) −8.29944e164 −0.428175
\(670\) −9.56867e163 −0.0452351
\(671\) −1.11868e165 −0.484660
\(672\) −1.32783e165 −0.527276
\(673\) 1.96629e165 0.715758 0.357879 0.933768i \(-0.383500\pi\)
0.357879 + 0.933768i \(0.383500\pi\)
\(674\) −2.47083e164 −0.0824593
\(675\) −5.91110e164 −0.180883
\(676\) −4.04907e164 −0.113625
\(677\) −6.54449e165 −1.68437 −0.842187 0.539186i \(-0.818732\pi\)
−0.842187 + 0.539186i \(0.818732\pi\)
\(678\) −4.10468e164 −0.0969041
\(679\) −1.52032e165 −0.329270
\(680\) 1.13671e165 0.225879
\(681\) −4.54030e165 −0.827900
\(682\) 1.20324e165 0.201357
\(683\) 9.66733e165 1.48489 0.742447 0.669905i \(-0.233665\pi\)
0.742447 + 0.669905i \(0.233665\pi\)
\(684\) −2.37660e165 −0.335100
\(685\) 7.79328e165 1.00884
\(686\) −3.17991e165 −0.377967
\(687\) 4.85018e165 0.529406
\(688\) −3.13005e165 −0.313781
\(689\) −1.32530e166 −1.22036
\(690\) 6.07860e165 0.514200
\(691\) 2.25132e165 0.174973 0.0874866 0.996166i \(-0.472117\pi\)
0.0874866 + 0.996166i \(0.472117\pi\)
\(692\) 1.22013e165 0.0871365
\(693\) 1.10427e165 0.0724740
\(694\) −9.10746e165 −0.549374
\(695\) 1.77566e166 0.984573
\(696\) 4.07758e166 2.07855
\(697\) −6.56396e165 −0.307644
\(698\) 7.34195e165 0.316424
\(699\) 5.11450e165 0.202716
\(700\) 2.79563e165 0.101916
\(701\) −3.61920e166 −1.21369 −0.606846 0.794819i \(-0.707566\pi\)
−0.606846 + 0.794819i \(0.707566\pi\)
\(702\) −9.46925e165 −0.292142
\(703\) −4.48148e166 −1.27214
\(704\) 2.68718e165 0.0701934
\(705\) −1.73090e166 −0.416109
\(706\) −3.86857e166 −0.856003
\(707\) −2.61267e166 −0.532168
\(708\) 4.32209e166 0.810491
\(709\) −3.87666e166 −0.669349 −0.334674 0.942334i \(-0.608626\pi\)
−0.334674 + 0.942334i \(0.608626\pi\)
\(710\) −3.25047e166 −0.516812
\(711\) −2.56231e166 −0.375198
\(712\) 6.27410e166 0.846197
\(713\) 1.02280e167 1.27074
\(714\) −7.09051e165 −0.0811581
\(715\) 2.58485e166 0.272603
\(716\) 1.11907e167 1.08754
\(717\) 5.38735e166 0.482508
\(718\) 2.43210e166 0.200771
\(719\) −8.96599e166 −0.682273 −0.341137 0.940014i \(-0.610812\pi\)
−0.341137 + 0.940014i \(0.610812\pi\)
\(720\) −1.97216e166 −0.138354
\(721\) 9.84236e166 0.636629
\(722\) −3.20376e166 −0.191088
\(723\) −6.38776e166 −0.351365
\(724\) −1.75355e167 −0.889636
\(725\) −1.35077e167 −0.632132
\(726\) 1.32630e167 0.572596
\(727\) 3.81141e167 1.51818 0.759089 0.650987i \(-0.225645\pi\)
0.759089 + 0.650987i \(0.225645\pi\)
\(728\) 1.04981e167 0.385856
\(729\) −1.08525e166 −0.0368103
\(730\) −1.00018e167 −0.313106
\(731\) −1.14709e167 −0.331459
\(732\) −5.43585e167 −1.45000
\(733\) 3.88613e167 0.957051 0.478525 0.878074i \(-0.341171\pi\)
0.478525 + 0.878074i \(0.341171\pi\)
\(734\) −3.40137e167 −0.773454
\(735\) −4.05594e167 −0.851692
\(736\) 5.30117e167 1.02806
\(737\) 1.90949e166 0.0342033
\(738\) −1.70950e167 −0.282860
\(739\) −1.02588e168 −1.56819 −0.784095 0.620640i \(-0.786873\pi\)
−0.784095 + 0.620640i \(0.786873\pi\)
\(740\) −6.91975e167 −0.977316
\(741\) 8.13445e167 1.06161
\(742\) 1.94213e167 0.234233
\(743\) 4.99109e167 0.556352 0.278176 0.960530i \(-0.410270\pi\)
0.278176 + 0.960530i \(0.410270\pi\)
\(744\) 1.37057e168 1.41216
\(745\) −5.13823e167 −0.489408
\(746\) 2.59320e167 0.228357
\(747\) 5.43714e167 0.442706
\(748\) −9.67674e166 −0.0728594
\(749\) 1.59940e167 0.111371
\(750\) −1.07202e168 −0.690428
\(751\) 1.86758e168 1.11261 0.556305 0.830978i \(-0.312218\pi\)
0.556305 + 0.830978i \(0.312218\pi\)
\(752\) −2.19953e167 −0.121223
\(753\) 2.40677e168 1.22724
\(754\) −2.16385e168 −1.02095
\(755\) −1.05029e168 −0.458579
\(756\) −4.03218e167 −0.162935
\(757\) 3.18804e168 1.19238 0.596190 0.802843i \(-0.296680\pi\)
0.596190 + 0.802843i \(0.296680\pi\)
\(758\) −1.61462e168 −0.559014
\(759\) −1.21302e168 −0.388799
\(760\) 1.91101e168 0.567110
\(761\) −3.17720e168 −0.873057 −0.436528 0.899690i \(-0.643792\pi\)
−0.436528 + 0.899690i \(0.643792\pi\)
\(762\) −2.35066e168 −0.598168
\(763\) 1.81370e168 0.427446
\(764\) −5.40391e168 −1.17963
\(765\) −7.22749e167 −0.146148
\(766\) 2.80562e168 0.525588
\(767\) −5.37655e168 −0.933201
\(768\) 5.37900e168 0.865107
\(769\) −1.13101e169 −1.68568 −0.842841 0.538163i \(-0.819119\pi\)
−0.842841 + 0.538163i \(0.819119\pi\)
\(770\) −3.78790e167 −0.0523228
\(771\) 6.20623e168 0.794597
\(772\) 1.15199e169 1.36722
\(773\) −1.88326e168 −0.207209 −0.103605 0.994619i \(-0.533038\pi\)
−0.103605 + 0.994619i \(0.533038\pi\)
\(774\) −2.98743e168 −0.304756
\(775\) −4.54024e168 −0.429467
\(776\) 8.13340e168 0.713446
\(777\) 1.01182e169 0.823142
\(778\) 1.17567e169 0.887116
\(779\) −1.10352e169 −0.772395
\(780\) 1.25602e169 0.815575
\(781\) 6.48651e168 0.390774
\(782\) 2.83079e168 0.158239
\(783\) 1.94823e169 1.01060
\(784\) −5.15408e168 −0.248120
\(785\) −2.08386e169 −0.931093
\(786\) −8.04891e168 −0.333824
\(787\) −2.71141e169 −1.04393 −0.521967 0.852966i \(-0.674802\pi\)
−0.521967 + 0.852966i \(0.674802\pi\)
\(788\) 1.09388e168 0.0391007
\(789\) −6.86160e169 −2.27730
\(790\) 8.78928e168 0.270875
\(791\) 2.17427e168 0.0622285
\(792\) −5.90765e168 −0.157033
\(793\) 6.76204e169 1.66954
\(794\) −2.65973e169 −0.610011
\(795\) 5.44689e169 1.16057
\(796\) 4.81739e168 0.0953669
\(797\) −1.79020e169 −0.329298 −0.164649 0.986352i \(-0.552649\pi\)
−0.164649 + 0.986352i \(0.552649\pi\)
\(798\) −1.19204e169 −0.203762
\(799\) −8.06075e168 −0.128053
\(800\) −2.35320e169 −0.347451
\(801\) −3.98924e169 −0.547505
\(802\) −2.78866e169 −0.355792
\(803\) 1.99592e169 0.236747
\(804\) 9.27855e168 0.102330
\(805\) −3.21987e169 −0.330202
\(806\) −7.27321e169 −0.693627
\(807\) 1.83960e170 1.63163
\(808\) 1.39773e170 1.15308
\(809\) −1.04540e170 −0.802225 −0.401112 0.916029i \(-0.631376\pi\)
−0.401112 + 0.916029i \(0.631376\pi\)
\(810\) 7.18855e169 0.513178
\(811\) 1.67747e170 1.11413 0.557064 0.830470i \(-0.311928\pi\)
0.557064 + 0.830470i \(0.311928\pi\)
\(812\) −9.21409e169 −0.569409
\(813\) 3.73836e170 2.14973
\(814\) −4.75221e169 −0.254312
\(815\) −2.25718e170 −1.12421
\(816\) −2.52702e169 −0.117148
\(817\) −1.92846e170 −0.832187
\(818\) −7.90416e168 −0.0317532
\(819\) −6.67498e169 −0.249656
\(820\) −1.70392e170 −0.593390
\(821\) −3.20975e170 −1.04087 −0.520434 0.853902i \(-0.674230\pi\)
−0.520434 + 0.853902i \(0.674230\pi\)
\(822\) 2.60069e170 0.785393
\(823\) 1.90558e170 0.535965 0.267982 0.963424i \(-0.413643\pi\)
0.267982 + 0.963424i \(0.413643\pi\)
\(824\) −5.26548e170 −1.37942
\(825\) 5.38463e169 0.131401
\(826\) 7.87893e169 0.179116
\(827\) −6.00446e170 −1.27175 −0.635875 0.771792i \(-0.719360\pi\)
−0.635875 + 0.771792i \(0.719360\pi\)
\(828\) −2.14225e170 −0.422762
\(829\) 9.94678e170 1.82912 0.914560 0.404450i \(-0.132537\pi\)
0.914560 + 0.404450i \(0.132537\pi\)
\(830\) −1.86506e170 −0.319613
\(831\) 3.59151e170 0.573610
\(832\) −1.62432e170 −0.241800
\(833\) −1.88885e170 −0.262098
\(834\) 5.92553e170 0.766502
\(835\) 3.24052e170 0.390801
\(836\) −1.62684e170 −0.182926
\(837\) 6.54846e170 0.686594
\(838\) −2.48527e169 −0.0242995
\(839\) −1.66290e171 −1.51632 −0.758159 0.652070i \(-0.773901\pi\)
−0.758159 + 0.652070i \(0.773901\pi\)
\(840\) −4.31465e170 −0.366950
\(841\) 3.19142e171 2.53174
\(842\) 4.29987e170 0.318198
\(843\) −3.06186e170 −0.211384
\(844\) 1.31647e171 0.847963
\(845\) −2.07014e170 −0.124418
\(846\) −2.09931e170 −0.117736
\(847\) −7.02546e170 −0.367702
\(848\) 6.92162e170 0.338104
\(849\) −4.03137e171 −1.83803
\(850\) −1.25659e170 −0.0534795
\(851\) −4.03957e171 −1.60493
\(852\) 3.15191e171 1.16912
\(853\) −1.09954e171 −0.380797 −0.190399 0.981707i \(-0.560978\pi\)
−0.190399 + 0.981707i \(0.560978\pi\)
\(854\) −9.90926e170 −0.320447
\(855\) −1.21507e171 −0.366931
\(856\) −8.55651e170 −0.241313
\(857\) −5.62118e170 −0.148064 −0.0740320 0.997256i \(-0.523587\pi\)
−0.0740320 + 0.997256i \(0.523587\pi\)
\(858\) 8.62586e170 0.212225
\(859\) 5.39313e171 1.23948 0.619742 0.784805i \(-0.287237\pi\)
0.619742 + 0.784805i \(0.287237\pi\)
\(860\) −2.97770e171 −0.639324
\(861\) 2.49152e171 0.499781
\(862\) 2.97445e171 0.557483
\(863\) −6.37689e171 −1.11681 −0.558403 0.829570i \(-0.688586\pi\)
−0.558403 + 0.829570i \(0.688586\pi\)
\(864\) 3.39405e171 0.555475
\(865\) 6.23810e170 0.0954135
\(866\) 6.63898e170 0.0949084
\(867\) 8.45383e171 1.12963
\(868\) −3.09706e171 −0.386853
\(869\) −1.75396e171 −0.204815
\(870\) 8.89330e171 0.970927
\(871\) −1.15422e171 −0.117822
\(872\) −9.70297e171 −0.926169
\(873\) −5.17144e171 −0.461613
\(874\) 4.75908e171 0.397287
\(875\) 5.67854e171 0.443369
\(876\) 9.69855e171 0.708299
\(877\) −6.87419e171 −0.469619 −0.234809 0.972041i \(-0.575447\pi\)
−0.234809 + 0.972041i \(0.575447\pi\)
\(878\) −1.01435e172 −0.648274
\(879\) 3.53957e172 2.11642
\(880\) −1.34998e171 −0.0755253
\(881\) −3.04162e172 −1.59226 −0.796131 0.605125i \(-0.793123\pi\)
−0.796131 + 0.605125i \(0.793123\pi\)
\(882\) −4.91924e171 −0.240983
\(883\) −1.53410e172 −0.703322 −0.351661 0.936128i \(-0.614383\pi\)
−0.351661 + 0.936128i \(0.614383\pi\)
\(884\) 5.84928e171 0.250983
\(885\) 2.20973e172 0.887478
\(886\) 1.04396e172 0.392472
\(887\) −4.39366e172 −1.54630 −0.773152 0.634220i \(-0.781321\pi\)
−0.773152 + 0.634220i \(0.781321\pi\)
\(888\) −5.41306e172 −1.78355
\(889\) 1.24516e172 0.384123
\(890\) 1.36840e172 0.395273
\(891\) −1.43452e172 −0.388026
\(892\) −1.00332e172 −0.254152
\(893\) −1.35516e172 −0.321499
\(894\) −1.71467e172 −0.381010
\(895\) 5.72143e172 1.19085
\(896\) −1.91959e172 −0.374274
\(897\) 7.33234e172 1.33932
\(898\) −7.84734e171 −0.134294
\(899\) 1.49641e173 2.39944
\(900\) 9.50949e171 0.142880
\(901\) 2.53661e172 0.357152
\(902\) −1.17019e172 −0.154409
\(903\) 4.35406e172 0.538469
\(904\) −1.16319e172 −0.134834
\(905\) −8.96528e172 −0.974141
\(906\) −3.50492e172 −0.357009
\(907\) −3.61788e172 −0.345484 −0.172742 0.984967i \(-0.555263\pi\)
−0.172742 + 0.984967i \(0.555263\pi\)
\(908\) −5.48874e172 −0.491418
\(909\) −8.88715e172 −0.746063
\(910\) 2.28966e172 0.180240
\(911\) 4.18676e172 0.309067 0.154534 0.987988i \(-0.450613\pi\)
0.154534 + 0.987988i \(0.450613\pi\)
\(912\) −4.24837e172 −0.294120
\(913\) 3.72184e172 0.241667
\(914\) 1.63466e173 0.995578
\(915\) −2.77916e173 −1.58774
\(916\) 5.86335e172 0.314240
\(917\) 4.26355e172 0.214371
\(918\) 1.81240e172 0.0854984
\(919\) −7.07915e172 −0.313345 −0.156673 0.987651i \(-0.550077\pi\)
−0.156673 + 0.987651i \(0.550077\pi\)
\(920\) 1.72257e173 0.715465
\(921\) −2.48835e173 −0.969890
\(922\) −1.75818e173 −0.643139
\(923\) −3.92089e173 −1.34612
\(924\) 3.67305e172 0.118363
\(925\) 1.79317e173 0.542413
\(926\) −3.08275e173 −0.875379
\(927\) 3.34794e173 0.892509
\(928\) 7.75588e173 1.94122
\(929\) 3.48959e173 0.820077 0.410039 0.912068i \(-0.365515\pi\)
0.410039 + 0.912068i \(0.365515\pi\)
\(930\) 2.98924e173 0.659642
\(931\) −3.17549e173 −0.658044
\(932\) 6.18290e172 0.120327
\(933\) 5.41220e172 0.0989231
\(934\) −1.65502e173 −0.284126
\(935\) −4.94737e172 −0.0797802
\(936\) 3.57099e173 0.540943
\(937\) −2.76839e173 −0.393969 −0.196984 0.980407i \(-0.563115\pi\)
−0.196984 + 0.980407i \(0.563115\pi\)
\(938\) 1.69143e172 0.0226145
\(939\) 4.11738e173 0.517230
\(940\) −2.09247e173 −0.246990
\(941\) 1.65023e173 0.183043 0.0915213 0.995803i \(-0.470827\pi\)
0.0915213 + 0.995803i \(0.470827\pi\)
\(942\) −6.95402e173 −0.724867
\(943\) −9.94707e173 −0.974453
\(944\) 2.80801e173 0.258545
\(945\) −2.06151e173 −0.178412
\(946\) −2.04496e173 −0.166362
\(947\) 1.71987e174 1.31529 0.657647 0.753326i \(-0.271552\pi\)
0.657647 + 0.753326i \(0.271552\pi\)
\(948\) −8.52279e173 −0.612765
\(949\) −1.20647e174 −0.815537
\(950\) −2.11256e173 −0.134270
\(951\) −1.21272e174 −0.724769
\(952\) −2.00932e173 −0.112925
\(953\) 3.14844e174 1.66403 0.832013 0.554756i \(-0.187188\pi\)
0.832013 + 0.554756i \(0.187188\pi\)
\(954\) 6.60625e173 0.328379
\(955\) −2.76283e174 −1.29168
\(956\) 6.51274e173 0.286403
\(957\) −1.77471e174 −0.734142
\(958\) −1.12951e174 −0.439550
\(959\) −1.37760e174 −0.504353
\(960\) 6.67583e173 0.229952
\(961\) 1.94434e174 0.630164
\(962\) 2.87256e174 0.876046
\(963\) 5.44047e173 0.156134
\(964\) −7.72214e173 −0.208560
\(965\) 5.88973e174 1.49709
\(966\) −1.07450e174 −0.257066
\(967\) 6.89859e173 0.155350 0.0776750 0.996979i \(-0.475250\pi\)
0.0776750 + 0.996979i \(0.475250\pi\)
\(968\) 3.75849e174 0.796718
\(969\) −1.55693e174 −0.310690
\(970\) 1.77392e174 0.333263
\(971\) 1.00197e175 1.77227 0.886135 0.463426i \(-0.153380\pi\)
0.886135 + 0.463426i \(0.153380\pi\)
\(972\) −4.56837e174 −0.760825
\(973\) −3.13878e174 −0.492221
\(974\) −4.23152e174 −0.624881
\(975\) −3.25484e174 −0.452647
\(976\) −3.53160e174 −0.462549
\(977\) −5.22761e174 −0.644869 −0.322435 0.946592i \(-0.604501\pi\)
−0.322435 + 0.946592i \(0.604501\pi\)
\(978\) −7.53242e174 −0.875208
\(979\) −2.73072e174 −0.298875
\(980\) −4.90321e174 −0.505540
\(981\) 6.16942e174 0.599249
\(982\) −8.47259e174 −0.775344
\(983\) 1.49888e175 1.29237 0.646184 0.763182i \(-0.276364\pi\)
0.646184 + 0.763182i \(0.276364\pi\)
\(984\) −1.33291e175 −1.08290
\(985\) 5.59260e173 0.0428148
\(986\) 4.14159e174 0.298791
\(987\) 3.05966e174 0.208027
\(988\) 9.83370e174 0.630138
\(989\) −1.73830e175 −1.04989
\(990\) −1.28847e174 −0.0733529
\(991\) 8.39333e174 0.450428 0.225214 0.974309i \(-0.427692\pi\)
0.225214 + 0.974309i \(0.427692\pi\)
\(992\) 2.60693e175 1.31885
\(993\) 2.50603e174 0.119524
\(994\) 5.74577e174 0.258371
\(995\) 2.46296e174 0.104426
\(996\) 1.80851e175 0.723019
\(997\) −1.23005e175 −0.463718 −0.231859 0.972749i \(-0.574481\pi\)
−0.231859 + 0.972749i \(0.574481\pi\)
\(998\) 1.46259e175 0.519979
\(999\) −2.58632e175 −0.867163
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 1.118.a.a.1.6 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.118.a.a.1.6 9 1.1 even 1 trivial