Properties

Label 1.92.a.a.1.2
Level $1$
Weight $92$
Character 1.1
Self dual yes
Analytic conductor $52.442$
Analytic rank $0$
Dimension $7$
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] = [1,92,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, 92, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1.1");
 
S:= CuspForms(chi, 92);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 92 \)
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: \(52.4421558310\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 2 x^{6} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: multiple of \( 2^{83}\cdot 3^{31}\cdot 5^{8}\cdot 7^{6}\cdot 11\cdot 13^{3}\cdot 23 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.36516e12\) 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)
 
\(f(q)\) \(=\) \(q-5.62150e13 q^{2} -7.69269e21 q^{3} +6.84242e26 q^{4} -3.78165e31 q^{5} +4.32444e35 q^{6} +1.92974e38 q^{7} +1.00717e41 q^{8} +3.29936e43 q^{9} +O(q^{10})\) \(q-5.62150e13 q^{2} -7.69269e21 q^{3} +6.84242e26 q^{4} -3.78165e31 q^{5} +4.32444e35 q^{6} +1.92974e38 q^{7} +1.00717e41 q^{8} +3.29936e43 q^{9} +2.12585e45 q^{10} -3.23035e47 q^{11} -5.26366e48 q^{12} +6.38412e50 q^{13} -1.08480e52 q^{14} +2.90911e53 q^{15} -7.35590e54 q^{16} -2.02934e55 q^{17} -1.85474e57 q^{18} -2.40506e58 q^{19} -2.58756e58 q^{20} -1.48449e60 q^{21} +1.81594e61 q^{22} +9.50903e61 q^{23} -7.74784e62 q^{24} -2.60888e63 q^{25} -3.58883e64 q^{26} -5.23853e64 q^{27} +1.32041e65 q^{28} -5.71159e66 q^{29} -1.63535e67 q^{30} -5.62988e67 q^{31} +1.64148e68 q^{32} +2.48501e69 q^{33} +1.14079e69 q^{34} -7.29761e69 q^{35} +2.25756e70 q^{36} -3.40807e71 q^{37} +1.35200e72 q^{38} -4.91111e72 q^{39} -3.80876e72 q^{40} +2.61368e73 q^{41} +8.34506e73 q^{42} +6.31280e73 q^{43} -2.21034e74 q^{44} -1.24770e75 q^{45} -5.34550e75 q^{46} +1.01518e76 q^{47} +5.65867e76 q^{48} -4.29143e76 q^{49} +1.46658e77 q^{50} +1.56111e77 q^{51} +4.36828e77 q^{52} +1.79595e78 q^{53} +2.94484e78 q^{54} +1.22161e79 q^{55} +1.94358e79 q^{56} +1.85014e80 q^{57} +3.21077e80 q^{58} -2.69860e80 q^{59} +1.99053e80 q^{60} -9.80050e80 q^{61} +3.16483e81 q^{62} +6.36692e81 q^{63} +8.98472e81 q^{64} -2.41425e82 q^{65} -1.39695e83 q^{66} -4.61723e82 q^{67} -1.38856e82 q^{68} -7.31500e83 q^{69} +4.10235e83 q^{70} -2.60022e83 q^{71} +3.32302e84 q^{72} -8.39038e84 q^{73} +1.91584e85 q^{74} +2.00693e85 q^{75} -1.64564e85 q^{76} -6.23374e85 q^{77} +2.76078e86 q^{78} +4.28257e85 q^{79} +2.78174e86 q^{80} -4.60918e86 q^{81} -1.46928e87 q^{82} -3.77027e87 q^{83} -1.01575e87 q^{84} +7.67424e86 q^{85} -3.54874e87 q^{86} +4.39375e88 q^{87} -3.25351e88 q^{88} -5.46272e88 q^{89} +7.01397e88 q^{90} +1.23197e89 q^{91} +6.50647e88 q^{92} +4.33089e89 q^{93} -5.70686e89 q^{94} +9.09510e89 q^{95} -1.26274e90 q^{96} -2.52770e90 q^{97} +2.41243e90 q^{98} -1.06581e91 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 3841716838056 q^{2} + 62\!\cdots\!32 q^{3}+ \cdots + 38\!\cdots\!59 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 3841716838056 q^{2} + 62\!\cdots\!32 q^{3}+ \cdots - 23\!\cdots\!92 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.62150e13 −1.12976 −0.564881 0.825172i \(-0.691078\pi\)
−0.564881 + 0.825172i \(0.691078\pi\)
\(3\) −7.69269e21 −1.50335 −0.751677 0.659531i \(-0.770755\pi\)
−0.751677 + 0.659531i \(0.770755\pi\)
\(4\) 6.84242e26 0.276363
\(5\) −3.78165e31 −0.595040 −0.297520 0.954716i \(-0.596160\pi\)
−0.297520 + 0.954716i \(0.596160\pi\)
\(6\) 4.32444e35 1.69843
\(7\) 1.92974e38 0.681614 0.340807 0.940133i \(-0.389300\pi\)
0.340807 + 0.940133i \(0.389300\pi\)
\(8\) 1.00717e41 0.817538
\(9\) 3.29936e43 1.26007
\(10\) 2.12585e45 0.672254
\(11\) −3.23035e47 −1.33623 −0.668117 0.744056i \(-0.732899\pi\)
−0.668117 + 0.744056i \(0.732899\pi\)
\(12\) −5.26366e48 −0.415472
\(13\) 6.38412e50 1.32032 0.660159 0.751126i \(-0.270489\pi\)
0.660159 + 0.751126i \(0.270489\pi\)
\(14\) −1.08480e52 −0.770061
\(15\) 2.90911e53 0.894556
\(16\) −7.35590e54 −1.19999
\(17\) −2.02934e55 −0.209859 −0.104930 0.994480i \(-0.533462\pi\)
−0.104930 + 0.994480i \(0.533462\pi\)
\(18\) −1.85474e57 −1.42358
\(19\) −2.40506e58 −1.57702 −0.788510 0.615022i \(-0.789147\pi\)
−0.788510 + 0.615022i \(0.789147\pi\)
\(20\) −2.58756e58 −0.164447
\(21\) −1.48449e60 −1.02471
\(22\) 1.81594e61 1.50963
\(23\) 9.50903e61 1.04597 0.522986 0.852342i \(-0.324818\pi\)
0.522986 + 0.852342i \(0.324818\pi\)
\(24\) −7.74784e62 −1.22905
\(25\) −2.60888e63 −0.645927
\(26\) −3.58883e64 −1.49165
\(27\) −5.23853e64 −0.390983
\(28\) 1.32041e65 0.188373
\(29\) −5.71159e66 −1.65062 −0.825312 0.564677i \(-0.809001\pi\)
−0.825312 + 0.564677i \(0.809001\pi\)
\(30\) −1.63535e67 −1.01064
\(31\) −5.62988e67 −0.782604 −0.391302 0.920262i \(-0.627975\pi\)
−0.391302 + 0.920262i \(0.627975\pi\)
\(32\) 1.64148e68 0.538162
\(33\) 2.48501e69 2.00883
\(34\) 1.14079e69 0.237091
\(35\) −7.29761e69 −0.405587
\(36\) 2.25756e70 0.348238
\(37\) −3.40807e71 −1.51123 −0.755613 0.655018i \(-0.772661\pi\)
−0.755613 + 0.655018i \(0.772661\pi\)
\(38\) 1.35200e72 1.78166
\(39\) −4.91111e72 −1.98490
\(40\) −3.80876e72 −0.486468
\(41\) 2.61368e73 1.08539 0.542696 0.839929i \(-0.317404\pi\)
0.542696 + 0.839929i \(0.317404\pi\)
\(42\) 8.34506e73 1.15768
\(43\) 6.31280e73 0.300198 0.150099 0.988671i \(-0.452041\pi\)
0.150099 + 0.988671i \(0.452041\pi\)
\(44\) −2.21034e74 −0.369286
\(45\) −1.24770e75 −0.749795
\(46\) −5.34550e75 −1.18170
\(47\) 1.01518e76 0.843515 0.421757 0.906709i \(-0.361413\pi\)
0.421757 + 0.906709i \(0.361413\pi\)
\(48\) 5.65867e76 1.80400
\(49\) −4.29143e76 −0.535403
\(50\) 1.46658e77 0.729744
\(51\) 1.56111e77 0.315493
\(52\) 4.36828e77 0.364887
\(53\) 1.79595e78 0.630583 0.315291 0.948995i \(-0.397898\pi\)
0.315291 + 0.948995i \(0.397898\pi\)
\(54\) 2.94484e78 0.441718
\(55\) 1.22161e79 0.795113
\(56\) 1.94358e79 0.557245
\(57\) 1.85014e80 2.37082
\(58\) 3.21077e80 1.86481
\(59\) −2.69860e80 −0.720066 −0.360033 0.932940i \(-0.617234\pi\)
−0.360033 + 0.932940i \(0.617234\pi\)
\(60\) 1.99053e80 0.247222
\(61\) −9.80050e80 −0.573774 −0.286887 0.957964i \(-0.592620\pi\)
−0.286887 + 0.957964i \(0.592620\pi\)
\(62\) 3.16483e81 0.884156
\(63\) 6.36692e81 0.858883
\(64\) 8.98472e81 0.591992
\(65\) −2.41425e82 −0.785642
\(66\) −1.39695e83 −2.26950
\(67\) −4.61723e82 −0.378423 −0.189212 0.981936i \(-0.560593\pi\)
−0.189212 + 0.981936i \(0.560593\pi\)
\(68\) −1.38856e82 −0.0579973
\(69\) −7.31500e83 −1.57246
\(70\) 4.10235e83 0.458217
\(71\) −2.60022e83 −0.152320 −0.0761598 0.997096i \(-0.524266\pi\)
−0.0761598 + 0.997096i \(0.524266\pi\)
\(72\) 3.32302e84 1.03016
\(73\) −8.39038e84 −1.38865 −0.694323 0.719663i \(-0.744296\pi\)
−0.694323 + 0.719663i \(0.744296\pi\)
\(74\) 1.91584e85 1.70733
\(75\) 2.00693e85 0.971057
\(76\) −1.64564e85 −0.435830
\(77\) −6.23374e85 −0.910795
\(78\) 2.76078e86 2.24247
\(79\) 4.28257e85 0.194837 0.0974185 0.995244i \(-0.468941\pi\)
0.0974185 + 0.995244i \(0.468941\pi\)
\(80\) 2.78174e86 0.714040
\(81\) −4.60918e86 −0.672288
\(82\) −1.46928e87 −1.22623
\(83\) −3.77027e87 −1.81267 −0.906336 0.422559i \(-0.861132\pi\)
−0.906336 + 0.422559i \(0.861132\pi\)
\(84\) −1.01575e87 −0.283191
\(85\) 7.67424e86 0.124875
\(86\) −3.54874e87 −0.339152
\(87\) 4.39375e88 2.48147
\(88\) −3.25351e88 −1.09242
\(89\) −5.46272e88 −1.09689 −0.548445 0.836186i \(-0.684780\pi\)
−0.548445 + 0.836186i \(0.684780\pi\)
\(90\) 7.01397e88 0.847090
\(91\) 1.23197e89 0.899946
\(92\) 6.50647e88 0.289068
\(93\) 4.33089e89 1.17653
\(94\) −5.70686e89 −0.952971
\(95\) 9.09510e89 0.938390
\(96\) −1.26274e90 −0.809048
\(97\) −2.52770e90 −1.01067 −0.505337 0.862922i \(-0.668632\pi\)
−0.505337 + 0.862922i \(0.668632\pi\)
\(98\) 2.41243e90 0.604878
\(99\) −1.06581e91 −1.68375
\(100\) −1.78510e90 −0.178510
\(101\) −5.55608e90 −0.353302 −0.176651 0.984274i \(-0.556526\pi\)
−0.176651 + 0.984274i \(0.556526\pi\)
\(102\) −8.77575e90 −0.356432
\(103\) 6.03016e91 1.57121 0.785607 0.618726i \(-0.212351\pi\)
0.785607 + 0.618726i \(0.212351\pi\)
\(104\) 6.42988e91 1.07941
\(105\) 5.61383e91 0.609742
\(106\) −1.00959e92 −0.712408
\(107\) 7.40024e91 0.340631 0.170315 0.985390i \(-0.445521\pi\)
0.170315 + 0.985390i \(0.445521\pi\)
\(108\) −3.58442e91 −0.108053
\(109\) −2.17456e92 −0.430988 −0.215494 0.976505i \(-0.569136\pi\)
−0.215494 + 0.976505i \(0.569136\pi\)
\(110\) −6.86726e92 −0.898289
\(111\) 2.62172e93 2.27191
\(112\) −1.41950e93 −0.817927
\(113\) 1.03543e93 0.398153 0.199076 0.979984i \(-0.436206\pi\)
0.199076 + 0.979984i \(0.436206\pi\)
\(114\) −1.04005e94 −2.67846
\(115\) −3.59598e93 −0.622395
\(116\) −3.90811e93 −0.456171
\(117\) 2.10635e94 1.66370
\(118\) 1.51702e94 0.813503
\(119\) −3.91609e93 −0.143043
\(120\) 2.92996e94 0.731334
\(121\) 4.59085e94 0.785522
\(122\) 5.50935e94 0.648228
\(123\) −2.01062e95 −1.63173
\(124\) −3.85220e94 −0.216283
\(125\) 2.51398e95 0.979393
\(126\) −3.57916e95 −0.970334
\(127\) −6.88314e94 −0.130232 −0.0651160 0.997878i \(-0.520742\pi\)
−0.0651160 + 0.997878i \(0.520742\pi\)
\(128\) −9.11487e95 −1.20697
\(129\) −4.85624e95 −0.451303
\(130\) 1.35717e96 0.887589
\(131\) 1.10101e96 0.508098 0.254049 0.967191i \(-0.418238\pi\)
0.254049 + 0.967191i \(0.418238\pi\)
\(132\) 1.70035e96 0.555167
\(133\) −4.64114e96 −1.07492
\(134\) 2.59557e96 0.427529
\(135\) 1.98103e96 0.232651
\(136\) −2.04388e96 −0.171568
\(137\) 1.30170e97 0.782933 0.391466 0.920192i \(-0.371968\pi\)
0.391466 + 0.920192i \(0.371968\pi\)
\(138\) 4.11213e97 1.77651
\(139\) −2.37924e97 −0.740058 −0.370029 0.929020i \(-0.620652\pi\)
−0.370029 + 0.929020i \(0.620652\pi\)
\(140\) −4.99333e96 −0.112089
\(141\) −7.80951e97 −1.26810
\(142\) 1.46172e97 0.172085
\(143\) −2.06229e98 −1.76425
\(144\) −2.42698e98 −1.51207
\(145\) 2.15992e98 0.982187
\(146\) 4.71665e98 1.56884
\(147\) 3.30127e98 0.804900
\(148\) −2.33194e98 −0.417647
\(149\) 7.78787e98 1.02670 0.513350 0.858179i \(-0.328404\pi\)
0.513350 + 0.858179i \(0.328404\pi\)
\(150\) −1.12820e99 −1.09706
\(151\) 1.97778e99 1.42144 0.710718 0.703477i \(-0.248370\pi\)
0.710718 + 0.703477i \(0.248370\pi\)
\(152\) −2.42230e99 −1.28927
\(153\) −6.69551e98 −0.264438
\(154\) 3.50430e99 1.02898
\(155\) 2.12902e99 0.465681
\(156\) −3.36038e99 −0.548554
\(157\) −6.55275e99 −0.799815 −0.399907 0.916556i \(-0.630958\pi\)
−0.399907 + 0.916556i \(0.630958\pi\)
\(158\) −2.40744e99 −0.220119
\(159\) −1.38157e100 −0.947989
\(160\) −6.20752e99 −0.320228
\(161\) 1.83500e100 0.712948
\(162\) 2.59105e100 0.759525
\(163\) −2.29210e100 −0.507805 −0.253903 0.967230i \(-0.581714\pi\)
−0.253903 + 0.967230i \(0.581714\pi\)
\(164\) 1.78839e100 0.299962
\(165\) −9.39744e100 −1.19534
\(166\) 2.11945e101 2.04789
\(167\) 1.88522e101 1.38600 0.692999 0.720939i \(-0.256289\pi\)
0.692999 + 0.720939i \(0.256289\pi\)
\(168\) −1.49513e101 −0.837736
\(169\) 1.73769e101 0.743238
\(170\) −4.31407e100 −0.141079
\(171\) −7.93517e101 −1.98716
\(172\) 4.31948e100 0.0829635
\(173\) 4.26354e101 0.629032 0.314516 0.949252i \(-0.398158\pi\)
0.314516 + 0.949252i \(0.398158\pi\)
\(174\) −2.46995e102 −2.80347
\(175\) −5.03446e101 −0.440273
\(176\) 2.37621e102 1.60346
\(177\) 2.07595e102 1.08251
\(178\) 3.07086e102 1.23923
\(179\) −1.69826e102 −0.531117 −0.265558 0.964095i \(-0.585556\pi\)
−0.265558 + 0.964095i \(0.585556\pi\)
\(180\) −8.53731e101 −0.207216
\(181\) −1.51134e102 −0.285094 −0.142547 0.989788i \(-0.545529\pi\)
−0.142547 + 0.989788i \(0.545529\pi\)
\(182\) −6.92551e102 −1.01673
\(183\) 7.53923e102 0.862586
\(184\) 9.57720e102 0.855121
\(185\) 1.28881e103 0.899241
\(186\) −2.43461e103 −1.32920
\(187\) 6.55547e102 0.280421
\(188\) 6.94632e102 0.233116
\(189\) −1.01090e103 −0.266499
\(190\) −5.11281e103 −1.06016
\(191\) −5.22518e103 −0.853263 −0.426632 0.904425i \(-0.640300\pi\)
−0.426632 + 0.904425i \(0.640300\pi\)
\(192\) −6.91167e103 −0.889973
\(193\) 1.28561e103 0.130693 0.0653464 0.997863i \(-0.479185\pi\)
0.0653464 + 0.997863i \(0.479185\pi\)
\(194\) 1.42094e104 1.14182
\(195\) 1.85721e104 1.18110
\(196\) −2.93638e103 −0.147966
\(197\) −6.74933e103 −0.269803 −0.134902 0.990859i \(-0.543072\pi\)
−0.134902 + 0.990859i \(0.543072\pi\)
\(198\) 5.99145e104 1.90224
\(199\) −2.90888e104 −0.734362 −0.367181 0.930150i \(-0.619677\pi\)
−0.367181 + 0.930150i \(0.619677\pi\)
\(200\) −2.62758e104 −0.528070
\(201\) 3.55189e104 0.568904
\(202\) 3.12335e104 0.399148
\(203\) −1.10219e105 −1.12509
\(204\) 1.06817e104 0.0871905
\(205\) −9.88401e104 −0.645852
\(206\) −3.38985e105 −1.77510
\(207\) 3.13737e105 1.31800
\(208\) −4.69609e105 −1.58436
\(209\) 7.76919e105 2.10727
\(210\) −3.15581e105 −0.688863
\(211\) 6.80099e105 1.19597 0.597985 0.801507i \(-0.295968\pi\)
0.597985 + 0.801507i \(0.295968\pi\)
\(212\) 1.22886e105 0.174270
\(213\) 2.00027e105 0.228990
\(214\) −4.16004e105 −0.384832
\(215\) −2.38728e105 −0.178630
\(216\) −5.27608e105 −0.319644
\(217\) −1.08642e106 −0.533433
\(218\) 1.22243e106 0.486914
\(219\) 6.45447e106 2.08763
\(220\) 8.35874e105 0.219740
\(221\) −1.29555e106 −0.277081
\(222\) −1.47380e107 −2.56672
\(223\) −1.29916e106 −0.184413 −0.0922065 0.995740i \(-0.529392\pi\)
−0.0922065 + 0.995740i \(0.529392\pi\)
\(224\) 3.16764e106 0.366818
\(225\) −8.60764e106 −0.813916
\(226\) −5.82064e106 −0.449818
\(227\) −8.73941e106 −0.552466 −0.276233 0.961091i \(-0.589086\pi\)
−0.276233 + 0.961091i \(0.589086\pi\)
\(228\) 1.26594e107 0.655207
\(229\) 4.00186e107 1.69726 0.848629 0.528988i \(-0.177429\pi\)
0.848629 + 0.528988i \(0.177429\pi\)
\(230\) 2.02148e107 0.703158
\(231\) 4.79543e107 1.36925
\(232\) −5.75253e107 −1.34945
\(233\) 9.51391e107 1.83512 0.917560 0.397597i \(-0.130156\pi\)
0.917560 + 0.397597i \(0.130156\pi\)
\(234\) −1.18409e108 −1.87958
\(235\) −3.83908e107 −0.501925
\(236\) −1.84649e107 −0.198999
\(237\) −3.29445e107 −0.292909
\(238\) 2.20143e107 0.161604
\(239\) 1.76907e108 1.07310 0.536551 0.843868i \(-0.319727\pi\)
0.536551 + 0.843868i \(0.319727\pi\)
\(240\) −2.13991e108 −1.07346
\(241\) −2.39870e108 −0.995867 −0.497933 0.867215i \(-0.665908\pi\)
−0.497933 + 0.867215i \(0.665908\pi\)
\(242\) −2.58074e108 −0.887453
\(243\) 4.91735e108 1.40167
\(244\) −6.70591e107 −0.158570
\(245\) 1.62287e108 0.318586
\(246\) 1.13027e109 1.84346
\(247\) −1.53542e109 −2.08217
\(248\) −5.67024e108 −0.639808
\(249\) 2.90035e109 2.72509
\(250\) −1.41324e109 −1.10648
\(251\) 1.43365e109 0.936029 0.468014 0.883721i \(-0.344970\pi\)
0.468014 + 0.883721i \(0.344970\pi\)
\(252\) 4.35651e108 0.237364
\(253\) −3.07175e109 −1.39766
\(254\) 3.86936e108 0.147131
\(255\) −5.90356e108 −0.187731
\(256\) 2.89941e109 0.771600
\(257\) −1.79396e109 −0.399812 −0.199906 0.979815i \(-0.564064\pi\)
−0.199906 + 0.979815i \(0.564064\pi\)
\(258\) 2.72993e109 0.509866
\(259\) −6.57669e109 −1.03007
\(260\) −1.65193e109 −0.217122
\(261\) −1.88446e110 −2.07991
\(262\) −6.18933e109 −0.574030
\(263\) 1.38443e110 1.07965 0.539826 0.841777i \(-0.318490\pi\)
0.539826 + 0.841777i \(0.318490\pi\)
\(264\) 2.50282e110 1.64230
\(265\) −6.79165e109 −0.375222
\(266\) 2.60902e110 1.21440
\(267\) 4.20230e110 1.64902
\(268\) −3.15930e109 −0.104582
\(269\) −6.03377e109 −0.168601 −0.0843005 0.996440i \(-0.526866\pi\)
−0.0843005 + 0.996440i \(0.526866\pi\)
\(270\) −1.11363e110 −0.262840
\(271\) −6.44449e110 −1.28554 −0.642770 0.766059i \(-0.722215\pi\)
−0.642770 + 0.766059i \(0.722215\pi\)
\(272\) 1.49276e110 0.251828
\(273\) −9.47716e110 −1.35294
\(274\) −7.31749e110 −0.884528
\(275\) 8.42760e110 0.863110
\(276\) −5.00523e110 −0.434571
\(277\) 1.49744e111 1.10286 0.551429 0.834222i \(-0.314083\pi\)
0.551429 + 0.834222i \(0.314083\pi\)
\(278\) 1.33749e111 0.836089
\(279\) −1.85750e111 −0.986139
\(280\) −7.34992e110 −0.331583
\(281\) −2.59225e111 −0.994348 −0.497174 0.867651i \(-0.665629\pi\)
−0.497174 + 0.867651i \(0.665629\pi\)
\(282\) 4.39011e111 1.43265
\(283\) 3.11823e110 0.0866213 0.0433107 0.999062i \(-0.486209\pi\)
0.0433107 + 0.999062i \(0.486209\pi\)
\(284\) −1.77918e110 −0.0420955
\(285\) −6.99658e111 −1.41073
\(286\) 1.15932e112 1.99319
\(287\) 5.04372e111 0.739818
\(288\) 5.41586e111 0.678124
\(289\) −8.93905e111 −0.955959
\(290\) −1.21420e112 −1.10964
\(291\) 1.94448e112 1.51940
\(292\) −5.74105e111 −0.383771
\(293\) −1.10509e112 −0.632298 −0.316149 0.948710i \(-0.602390\pi\)
−0.316149 + 0.948710i \(0.602390\pi\)
\(294\) −1.85581e112 −0.909346
\(295\) 1.02052e112 0.428468
\(296\) −3.43250e112 −1.23548
\(297\) 1.69223e112 0.522445
\(298\) −4.37795e112 −1.15993
\(299\) 6.07068e112 1.38101
\(300\) 1.37323e112 0.268364
\(301\) 1.21821e112 0.204619
\(302\) −1.11181e113 −1.60588
\(303\) 4.27413e112 0.531138
\(304\) 1.76914e113 1.89240
\(305\) 3.70621e112 0.341419
\(306\) 3.76388e112 0.298752
\(307\) 2.66068e113 1.82053 0.910263 0.414031i \(-0.135879\pi\)
0.910263 + 0.414031i \(0.135879\pi\)
\(308\) −4.26539e112 −0.251710
\(309\) −4.63882e113 −2.36209
\(310\) −1.19683e113 −0.526109
\(311\) 2.35637e112 0.0894634 0.0447317 0.998999i \(-0.485757\pi\)
0.0447317 + 0.998999i \(0.485757\pi\)
\(312\) −4.94631e113 −1.62273
\(313\) 6.11078e112 0.173312 0.0866562 0.996238i \(-0.472382\pi\)
0.0866562 + 0.996238i \(0.472382\pi\)
\(314\) 3.68363e113 0.903601
\(315\) −2.40775e113 −0.511070
\(316\) 2.93031e112 0.0538457
\(317\) −6.37657e113 −1.01482 −0.507412 0.861704i \(-0.669398\pi\)
−0.507412 + 0.861704i \(0.669398\pi\)
\(318\) 7.76648e113 1.07100
\(319\) 1.84504e114 2.20562
\(320\) −3.39771e113 −0.352259
\(321\) −5.69278e113 −0.512089
\(322\) −1.03154e114 −0.805462
\(323\) 4.88067e113 0.330952
\(324\) −3.15379e113 −0.185795
\(325\) −1.66554e114 −0.852829
\(326\) 1.28850e114 0.573699
\(327\) 1.67282e114 0.647928
\(328\) 2.63241e114 0.887349
\(329\) 1.95904e114 0.574951
\(330\) 5.28277e114 1.35045
\(331\) −6.21240e114 −1.38384 −0.691919 0.721975i \(-0.743235\pi\)
−0.691919 + 0.721975i \(0.743235\pi\)
\(332\) −2.57977e114 −0.500955
\(333\) −1.12445e115 −1.90426
\(334\) −1.05978e115 −1.56585
\(335\) 1.74608e114 0.225177
\(336\) 1.09198e115 1.22963
\(337\) 6.30404e114 0.620095 0.310048 0.950721i \(-0.399655\pi\)
0.310048 + 0.950721i \(0.399655\pi\)
\(338\) −9.76844e114 −0.839683
\(339\) −7.96521e114 −0.598564
\(340\) 5.25103e113 0.0345107
\(341\) 1.81865e115 1.04574
\(342\) 4.46075e115 2.24502
\(343\) −2.37489e115 −1.04655
\(344\) 6.35805e114 0.245423
\(345\) 2.76628e115 0.935680
\(346\) −2.39675e115 −0.710657
\(347\) −2.09181e115 −0.543916 −0.271958 0.962309i \(-0.587671\pi\)
−0.271958 + 0.962309i \(0.587671\pi\)
\(348\) 3.00639e115 0.685787
\(349\) −2.95833e115 −0.592229 −0.296115 0.955152i \(-0.595691\pi\)
−0.296115 + 0.955152i \(0.595691\pi\)
\(350\) 2.83012e115 0.497404
\(351\) −3.34434e115 −0.516222
\(352\) −5.30257e115 −0.719110
\(353\) −1.07870e115 −0.128573 −0.0642865 0.997931i \(-0.520477\pi\)
−0.0642865 + 0.997931i \(0.520477\pi\)
\(354\) −1.16699e116 −1.22298
\(355\) 9.83314e114 0.0906363
\(356\) −3.73782e115 −0.303140
\(357\) 3.01253e115 0.215044
\(358\) 9.54674e115 0.600036
\(359\) 1.64225e116 0.909166 0.454583 0.890704i \(-0.349788\pi\)
0.454583 + 0.890704i \(0.349788\pi\)
\(360\) −1.25665e116 −0.612985
\(361\) 3.45849e116 1.48699
\(362\) 8.49600e115 0.322088
\(363\) −3.53160e116 −1.18092
\(364\) 8.42965e115 0.248712
\(365\) 3.17295e116 0.826301
\(366\) −4.23817e116 −0.974517
\(367\) 2.76048e116 0.560633 0.280317 0.959908i \(-0.409561\pi\)
0.280317 + 0.959908i \(0.409561\pi\)
\(368\) −6.99474e116 −1.25515
\(369\) 8.62347e116 1.36767
\(370\) −7.24505e116 −1.01593
\(371\) 3.46572e116 0.429814
\(372\) 2.96338e116 0.325150
\(373\) 1.11360e117 1.08138 0.540691 0.841221i \(-0.318163\pi\)
0.540691 + 0.841221i \(0.318163\pi\)
\(374\) −3.68515e116 −0.316809
\(375\) −1.93393e117 −1.47237
\(376\) 1.02246e117 0.689605
\(377\) −3.64635e117 −2.17935
\(378\) 5.68277e116 0.301081
\(379\) −2.45909e117 −1.15529 −0.577644 0.816289i \(-0.696028\pi\)
−0.577644 + 0.816289i \(0.696028\pi\)
\(380\) 6.22325e116 0.259336
\(381\) 5.29499e116 0.195785
\(382\) 2.93733e117 0.963985
\(383\) 1.93839e117 0.564802 0.282401 0.959296i \(-0.408869\pi\)
0.282401 + 0.959296i \(0.408869\pi\)
\(384\) 7.01179e117 1.81451
\(385\) 2.35738e117 0.541960
\(386\) −7.22705e116 −0.147652
\(387\) 2.08282e117 0.378271
\(388\) −1.72956e117 −0.279313
\(389\) 9.57143e116 0.137490 0.0687448 0.997634i \(-0.478101\pi\)
0.0687448 + 0.997634i \(0.478101\pi\)
\(390\) −1.04403e118 −1.33436
\(391\) −1.92970e117 −0.219507
\(392\) −4.32220e117 −0.437712
\(393\) −8.46974e117 −0.763851
\(394\) 3.79413e117 0.304814
\(395\) −1.61952e117 −0.115936
\(396\) −7.29272e117 −0.465327
\(397\) −1.97371e118 −1.12284 −0.561418 0.827533i \(-0.689744\pi\)
−0.561418 + 0.827533i \(0.689744\pi\)
\(398\) 1.63522e118 0.829654
\(399\) 3.57029e118 1.61598
\(400\) 1.91906e118 0.775104
\(401\) −1.80007e118 −0.648965 −0.324483 0.945892i \(-0.605190\pi\)
−0.324483 + 0.945892i \(0.605190\pi\)
\(402\) −1.99670e118 −0.642727
\(403\) −3.59418e118 −1.03329
\(404\) −3.80170e117 −0.0976397
\(405\) 1.74303e118 0.400038
\(406\) 6.19595e118 1.27108
\(407\) 1.10093e119 2.01935
\(408\) 1.57230e118 0.257927
\(409\) −1.29259e119 −1.89693 −0.948466 0.316879i \(-0.897365\pi\)
−0.948466 + 0.316879i \(0.897365\pi\)
\(410\) 5.55629e118 0.729659
\(411\) −1.00136e119 −1.17703
\(412\) 4.12609e118 0.434225
\(413\) −5.20760e118 −0.490806
\(414\) −1.76367e119 −1.48903
\(415\) 1.42578e119 1.07861
\(416\) 1.04794e119 0.710544
\(417\) 1.83028e119 1.11257
\(418\) −4.36745e119 −2.38071
\(419\) 1.38548e119 0.677426 0.338713 0.940890i \(-0.390008\pi\)
0.338713 + 0.940890i \(0.390008\pi\)
\(420\) 3.84121e118 0.168510
\(421\) −2.14976e119 −0.846359 −0.423179 0.906046i \(-0.639086\pi\)
−0.423179 + 0.906046i \(0.639086\pi\)
\(422\) −3.82317e119 −1.35116
\(423\) 3.34946e119 1.06289
\(424\) 1.80882e119 0.515525
\(425\) 5.29429e118 0.135554
\(426\) −1.12445e119 −0.258705
\(427\) −1.89124e119 −0.391092
\(428\) 5.06355e118 0.0941377
\(429\) 1.58646e120 2.65230
\(430\) 1.34201e119 0.201809
\(431\) −6.16830e119 −0.834545 −0.417272 0.908781i \(-0.637014\pi\)
−0.417272 + 0.908781i \(0.637014\pi\)
\(432\) 3.85341e119 0.469175
\(433\) 8.81345e119 0.965932 0.482966 0.875639i \(-0.339559\pi\)
0.482966 + 0.875639i \(0.339559\pi\)
\(434\) 6.10731e119 0.602653
\(435\) −1.66156e120 −1.47658
\(436\) −1.48792e119 −0.119109
\(437\) −2.28698e120 −1.64952
\(438\) −3.62838e120 −2.35852
\(439\) 5.21639e119 0.305657 0.152828 0.988253i \(-0.451162\pi\)
0.152828 + 0.988253i \(0.451162\pi\)
\(440\) 1.23036e120 0.650035
\(441\) −1.41590e120 −0.674647
\(442\) 7.28294e119 0.313035
\(443\) 3.27234e120 1.26908 0.634540 0.772890i \(-0.281190\pi\)
0.634540 + 0.772890i \(0.281190\pi\)
\(444\) 1.79389e120 0.627872
\(445\) 2.06581e120 0.652694
\(446\) 7.30322e119 0.208343
\(447\) −5.99097e120 −1.54349
\(448\) 1.73382e120 0.403509
\(449\) 8.16505e120 1.71692 0.858460 0.512880i \(-0.171421\pi\)
0.858460 + 0.512880i \(0.171421\pi\)
\(450\) 4.83878e120 0.919532
\(451\) −8.44309e120 −1.45034
\(452\) 7.08481e119 0.110035
\(453\) −1.52145e121 −2.13692
\(454\) 4.91285e120 0.624155
\(455\) −4.65888e120 −0.535504
\(456\) 1.86340e121 1.93823
\(457\) 1.92588e121 1.81318 0.906591 0.422011i \(-0.138676\pi\)
0.906591 + 0.422011i \(0.138676\pi\)
\(458\) −2.24965e121 −1.91750
\(459\) 1.06307e120 0.0820514
\(460\) −2.46052e120 −0.172007
\(461\) 4.05866e120 0.257034 0.128517 0.991707i \(-0.458978\pi\)
0.128517 + 0.991707i \(0.458978\pi\)
\(462\) −2.69575e121 −1.54692
\(463\) −2.28579e121 −1.18878 −0.594389 0.804178i \(-0.702606\pi\)
−0.594389 + 0.804178i \(0.702606\pi\)
\(464\) 4.20139e121 1.98073
\(465\) −1.63779e121 −0.700083
\(466\) −5.34824e121 −2.07325
\(467\) 4.07747e121 1.43375 0.716873 0.697203i \(-0.245573\pi\)
0.716873 + 0.697203i \(0.245573\pi\)
\(468\) 1.44125e121 0.459785
\(469\) −8.91006e120 −0.257939
\(470\) 2.15813e121 0.567056
\(471\) 5.04083e121 1.20240
\(472\) −2.71795e121 −0.588681
\(473\) −2.03926e121 −0.401134
\(474\) 1.85197e121 0.330918
\(475\) 6.27451e121 1.01864
\(476\) −2.67955e120 −0.0395317
\(477\) 5.92549e121 0.794581
\(478\) −9.94483e121 −1.21235
\(479\) 6.44914e121 0.714885 0.357443 0.933935i \(-0.383649\pi\)
0.357443 + 0.933935i \(0.383649\pi\)
\(480\) 4.77526e121 0.481416
\(481\) −2.17575e122 −1.99530
\(482\) 1.34843e122 1.12509
\(483\) −1.41161e122 −1.07181
\(484\) 3.14125e121 0.217089
\(485\) 9.55887e121 0.601392
\(486\) −2.76429e122 −1.58355
\(487\) −2.38886e122 −1.24630 −0.623152 0.782101i \(-0.714148\pi\)
−0.623152 + 0.782101i \(0.714148\pi\)
\(488\) −9.87076e121 −0.469082
\(489\) 1.76324e122 0.763411
\(490\) −9.12296e121 −0.359927
\(491\) 2.74613e122 0.987446 0.493723 0.869619i \(-0.335636\pi\)
0.493723 + 0.869619i \(0.335636\pi\)
\(492\) −1.37575e122 −0.450949
\(493\) 1.15907e122 0.346398
\(494\) 8.63135e122 2.35235
\(495\) 4.03052e122 1.00190
\(496\) 4.14128e122 0.939114
\(497\) −5.01776e121 −0.103823
\(498\) −1.63043e123 −3.07870
\(499\) 1.02145e123 1.76053 0.880266 0.474481i \(-0.157364\pi\)
0.880266 + 0.474481i \(0.157364\pi\)
\(500\) 1.72017e122 0.270668
\(501\) −1.45024e123 −2.08365
\(502\) −8.05927e122 −1.05749
\(503\) 1.17607e122 0.140958 0.0704789 0.997513i \(-0.477547\pi\)
0.0704789 + 0.997513i \(0.477547\pi\)
\(504\) 6.41256e122 0.702170
\(505\) 2.10112e122 0.210229
\(506\) 1.72678e123 1.57903
\(507\) −1.33675e123 −1.11735
\(508\) −4.70973e121 −0.0359913
\(509\) −7.10848e122 −0.496727 −0.248363 0.968667i \(-0.579893\pi\)
−0.248363 + 0.968667i \(0.579893\pi\)
\(510\) 3.31868e122 0.212091
\(511\) −1.61913e123 −0.946520
\(512\) 6.26829e122 0.335247
\(513\) 1.25990e123 0.616588
\(514\) 1.00847e123 0.451692
\(515\) −2.28040e123 −0.934936
\(516\) −3.32284e122 −0.124724
\(517\) −3.27940e123 −1.12713
\(518\) 3.69708e123 1.16374
\(519\) −3.27981e123 −0.945659
\(520\) −2.43156e123 −0.642292
\(521\) 7.73667e123 1.87257 0.936285 0.351242i \(-0.114241\pi\)
0.936285 + 0.351242i \(0.114241\pi\)
\(522\) 1.05935e124 2.34980
\(523\) 1.56809e123 0.318819 0.159410 0.987213i \(-0.449041\pi\)
0.159410 + 0.987213i \(0.449041\pi\)
\(524\) 7.53357e122 0.140420
\(525\) 3.87286e123 0.661886
\(526\) −7.78256e123 −1.21975
\(527\) 1.14249e123 0.164237
\(528\) −1.82795e124 −2.41057
\(529\) 7.77350e122 0.0940554
\(530\) 3.81793e123 0.423912
\(531\) −8.90366e123 −0.907336
\(532\) −3.17566e123 −0.297068
\(533\) 1.66860e124 1.43306
\(534\) −2.36232e124 −1.86300
\(535\) −2.79851e123 −0.202689
\(536\) −4.65033e123 −0.309375
\(537\) 1.30642e124 0.798457
\(538\) 3.39188e123 0.190479
\(539\) 1.38628e124 0.715424
\(540\) 1.35550e123 0.0642961
\(541\) −2.69933e124 −1.17701 −0.588506 0.808493i \(-0.700284\pi\)
−0.588506 + 0.808493i \(0.700284\pi\)
\(542\) 3.62277e124 1.45236
\(543\) 1.16263e124 0.428597
\(544\) −3.33112e123 −0.112938
\(545\) 8.22342e123 0.256455
\(546\) 5.32758e124 1.52850
\(547\) −3.22227e124 −0.850627 −0.425313 0.905046i \(-0.639836\pi\)
−0.425313 + 0.905046i \(0.639836\pi\)
\(548\) 8.90676e123 0.216374
\(549\) −3.23354e124 −0.722998
\(550\) −4.73757e124 −0.975109
\(551\) 1.37367e125 2.60307
\(552\) −7.36744e124 −1.28555
\(553\) 8.26424e123 0.132804
\(554\) −8.41785e124 −1.24597
\(555\) −9.91443e124 −1.35188
\(556\) −1.62798e124 −0.204525
\(557\) −8.11888e124 −0.939906 −0.469953 0.882691i \(-0.655729\pi\)
−0.469953 + 0.882691i \(0.655729\pi\)
\(558\) 1.04419e125 1.11410
\(559\) 4.03017e124 0.396356
\(560\) 5.36804e124 0.486699
\(561\) −5.04292e124 −0.421572
\(562\) 1.45723e125 1.12338
\(563\) 8.74092e124 0.621477 0.310738 0.950495i \(-0.399424\pi\)
0.310738 + 0.950495i \(0.399424\pi\)
\(564\) −5.34359e124 −0.350456
\(565\) −3.91562e124 −0.236917
\(566\) −1.75291e124 −0.0978615
\(567\) −8.89452e124 −0.458240
\(568\) −2.61887e124 −0.124527
\(569\) −3.58712e125 −1.57448 −0.787242 0.616644i \(-0.788492\pi\)
−0.787242 + 0.616644i \(0.788492\pi\)
\(570\) 3.93313e125 1.59379
\(571\) 2.83528e125 1.06085 0.530424 0.847732i \(-0.322033\pi\)
0.530424 + 0.847732i \(0.322033\pi\)
\(572\) −1.41111e125 −0.487574
\(573\) 4.01957e125 1.28276
\(574\) −2.83532e125 −0.835818
\(575\) −2.48079e125 −0.675621
\(576\) 2.96438e125 0.745953
\(577\) 2.80174e125 0.651521 0.325760 0.945452i \(-0.394380\pi\)
0.325760 + 0.945452i \(0.394380\pi\)
\(578\) 5.02508e125 1.08001
\(579\) −9.88980e124 −0.196478
\(580\) 1.47791e125 0.271440
\(581\) −7.27564e125 −1.23554
\(582\) −1.09309e126 −1.71656
\(583\) −5.80155e125 −0.842606
\(584\) −8.45053e125 −1.13527
\(585\) −7.96549e125 −0.989967
\(586\) 6.21227e125 0.714347
\(587\) 5.66199e125 0.602470 0.301235 0.953550i \(-0.402601\pi\)
0.301235 + 0.953550i \(0.402601\pi\)
\(588\) 2.25887e125 0.222445
\(589\) 1.35402e126 1.23418
\(590\) −5.73683e125 −0.484067
\(591\) 5.19205e125 0.405610
\(592\) 2.50694e126 1.81345
\(593\) 8.91060e125 0.596923 0.298461 0.954422i \(-0.403527\pi\)
0.298461 + 0.954422i \(0.403527\pi\)
\(594\) −9.51286e125 −0.590239
\(595\) 1.48093e125 0.0851162
\(596\) 5.32879e125 0.283742
\(597\) 2.23771e126 1.10401
\(598\) −3.41263e126 −1.56022
\(599\) −2.42971e126 −1.02952 −0.514760 0.857334i \(-0.672119\pi\)
−0.514760 + 0.857334i \(0.672119\pi\)
\(600\) 2.02132e126 0.793876
\(601\) 3.04722e125 0.110947 0.0554734 0.998460i \(-0.482333\pi\)
0.0554734 + 0.998460i \(0.482333\pi\)
\(602\) −6.84814e125 −0.231171
\(603\) −1.52339e126 −0.476841
\(604\) 1.35328e126 0.392832
\(605\) −1.73610e126 −0.467417
\(606\) −2.40270e126 −0.600060
\(607\) −3.83401e126 −0.888316 −0.444158 0.895948i \(-0.646497\pi\)
−0.444158 + 0.895948i \(0.646497\pi\)
\(608\) −3.94787e126 −0.848692
\(609\) 8.47880e126 1.69141
\(610\) −2.08344e126 −0.385722
\(611\) 6.48106e126 1.11371
\(612\) −4.58135e125 −0.0730809
\(613\) 2.58389e126 0.382668 0.191334 0.981525i \(-0.438719\pi\)
0.191334 + 0.981525i \(0.438719\pi\)
\(614\) −1.49570e127 −2.05676
\(615\) 7.60347e126 0.970944
\(616\) −6.27843e126 −0.744610
\(617\) −8.49357e126 −0.935654 −0.467827 0.883820i \(-0.654963\pi\)
−0.467827 + 0.883820i \(0.654963\pi\)
\(618\) 2.60771e127 2.66860
\(619\) 6.34151e126 0.602932 0.301466 0.953477i \(-0.402524\pi\)
0.301466 + 0.953477i \(0.402524\pi\)
\(620\) 1.45677e126 0.128697
\(621\) −4.98133e126 −0.408957
\(622\) −1.32463e126 −0.101072
\(623\) −1.05416e127 −0.747655
\(624\) 3.61256e127 2.38186
\(625\) 1.03017e126 0.0631490
\(626\) −3.43518e126 −0.195802
\(627\) −5.97660e127 −3.16797
\(628\) −4.48366e126 −0.221039
\(629\) 6.91611e126 0.317145
\(630\) 1.35351e127 0.577388
\(631\) 4.33929e127 1.72220 0.861098 0.508439i \(-0.169777\pi\)
0.861098 + 0.508439i \(0.169777\pi\)
\(632\) 4.31327e126 0.159287
\(633\) −5.23179e127 −1.79797
\(634\) 3.58459e127 1.14651
\(635\) 2.60296e126 0.0774932
\(636\) −9.45327e126 −0.261989
\(637\) −2.73970e127 −0.706902
\(638\) −1.03719e128 −2.49183
\(639\) −8.57909e126 −0.191934
\(640\) 3.44693e127 0.718197
\(641\) −2.44604e126 −0.0474705 −0.0237353 0.999718i \(-0.507556\pi\)
−0.0237353 + 0.999718i \(0.507556\pi\)
\(642\) 3.20019e127 0.578539
\(643\) 4.22084e127 0.710886 0.355443 0.934698i \(-0.384330\pi\)
0.355443 + 0.934698i \(0.384330\pi\)
\(644\) 1.25558e127 0.197032
\(645\) 1.83646e127 0.268544
\(646\) −2.74367e127 −0.373897
\(647\) 1.31481e128 1.67000 0.835001 0.550249i \(-0.185467\pi\)
0.835001 + 0.550249i \(0.185467\pi\)
\(648\) −4.64222e127 −0.549621
\(649\) 8.71743e127 0.962176
\(650\) 9.36282e127 0.963494
\(651\) 8.35750e127 0.801939
\(652\) −1.56835e127 −0.140339
\(653\) −4.67831e127 −0.390426 −0.195213 0.980761i \(-0.562540\pi\)
−0.195213 + 0.980761i \(0.562540\pi\)
\(654\) −9.40376e127 −0.732005
\(655\) −4.16364e127 −0.302339
\(656\) −1.92259e128 −1.30246
\(657\) −2.76829e128 −1.74980
\(658\) −1.10128e128 −0.649558
\(659\) −9.77042e127 −0.537808 −0.268904 0.963167i \(-0.586661\pi\)
−0.268904 + 0.963167i \(0.586661\pi\)
\(660\) −6.43012e127 −0.330347
\(661\) 2.06865e128 0.992018 0.496009 0.868317i \(-0.334798\pi\)
0.496009 + 0.868317i \(0.334798\pi\)
\(662\) 3.49230e128 1.56341
\(663\) 9.96628e127 0.416550
\(664\) −3.79730e128 −1.48193
\(665\) 1.75512e128 0.639619
\(666\) 6.32106e128 2.15136
\(667\) −5.43117e128 −1.72650
\(668\) 1.28995e128 0.383039
\(669\) 9.99404e127 0.277238
\(670\) −9.81556e127 −0.254397
\(671\) 3.16591e128 0.766697
\(672\) −2.43677e128 −0.551458
\(673\) 2.75253e127 0.0582167 0.0291083 0.999576i \(-0.490733\pi\)
0.0291083 + 0.999576i \(0.490733\pi\)
\(674\) −3.54381e128 −0.700560
\(675\) 1.36667e128 0.252547
\(676\) 1.18900e128 0.205404
\(677\) 3.28446e128 0.530493 0.265246 0.964181i \(-0.414547\pi\)
0.265246 + 0.964181i \(0.414547\pi\)
\(678\) 4.47764e128 0.676236
\(679\) −4.87780e128 −0.688889
\(680\) 7.72925e127 0.102090
\(681\) 6.72296e128 0.830552
\(682\) −1.02235e129 −1.18144
\(683\) 1.04661e129 1.13147 0.565736 0.824587i \(-0.308592\pi\)
0.565736 + 0.824587i \(0.308592\pi\)
\(684\) −5.42957e128 −0.549178
\(685\) −4.92257e128 −0.465877
\(686\) 1.33504e129 1.18235
\(687\) −3.07851e129 −2.55158
\(688\) −4.64363e128 −0.360233
\(689\) 1.14656e129 0.832569
\(690\) −1.55506e129 −1.05710
\(691\) 4.76039e128 0.302964 0.151482 0.988460i \(-0.451595\pi\)
0.151482 + 0.988460i \(0.451595\pi\)
\(692\) 2.91729e128 0.173841
\(693\) −2.05674e129 −1.14767
\(694\) 1.17591e129 0.614496
\(695\) 8.99747e128 0.440364
\(696\) 4.42525e129 2.02870
\(697\) −5.30403e128 −0.227779
\(698\) 1.66302e129 0.669078
\(699\) −7.31876e129 −2.75884
\(700\) −3.44479e128 −0.121675
\(701\) 1.83996e129 0.609029 0.304515 0.952508i \(-0.401506\pi\)
0.304515 + 0.952508i \(0.401506\pi\)
\(702\) 1.88002e129 0.583208
\(703\) 8.19660e129 2.38323
\(704\) −2.90238e129 −0.791039
\(705\) 2.95328e129 0.754571
\(706\) 6.06388e128 0.145257
\(707\) −1.07218e129 −0.240816
\(708\) 1.42045e129 0.299167
\(709\) −6.54379e129 −1.29248 −0.646242 0.763132i \(-0.723660\pi\)
−0.646242 + 0.763132i \(0.723660\pi\)
\(710\) −5.52770e128 −0.102398
\(711\) 1.41297e129 0.245509
\(712\) −5.50188e129 −0.896750
\(713\) −5.35347e129 −0.818581
\(714\) −1.69349e129 −0.242949
\(715\) 7.79888e129 1.04980
\(716\) −1.16202e129 −0.146781
\(717\) −1.36089e130 −1.61325
\(718\) −9.23193e129 −1.02714
\(719\) −2.08574e129 −0.217819 −0.108909 0.994052i \(-0.534736\pi\)
−0.108909 + 0.994052i \(0.534736\pi\)
\(720\) 9.17798e129 0.899743
\(721\) 1.16367e130 1.07096
\(722\) −1.94419e130 −1.67995
\(723\) 1.84525e130 1.49714
\(724\) −1.03412e129 −0.0787893
\(725\) 1.49008e130 1.06618
\(726\) 1.98529e130 1.33416
\(727\) 2.57362e130 1.62453 0.812267 0.583286i \(-0.198233\pi\)
0.812267 + 0.583286i \(0.198233\pi\)
\(728\) 1.24080e130 0.735740
\(729\) −2.57590e130 −1.43492
\(730\) −1.78367e130 −0.933523
\(731\) −1.28108e129 −0.0629992
\(732\) 5.15865e129 0.238387
\(733\) −2.63415e130 −1.14396 −0.571978 0.820269i \(-0.693824\pi\)
−0.571978 + 0.820269i \(0.693824\pi\)
\(734\) −1.55180e130 −0.633382
\(735\) −1.24842e130 −0.478948
\(736\) 1.56089e130 0.562902
\(737\) 1.49153e130 0.505662
\(738\) −4.84768e130 −1.54515
\(739\) −3.11762e129 −0.0934331 −0.0467166 0.998908i \(-0.514876\pi\)
−0.0467166 + 0.998908i \(0.514876\pi\)
\(740\) 8.81859e129 0.248517
\(741\) 1.18115e131 3.13023
\(742\) −1.94825e130 −0.485587
\(743\) −3.03372e130 −0.711189 −0.355595 0.934640i \(-0.615722\pi\)
−0.355595 + 0.934640i \(0.615722\pi\)
\(744\) 4.36194e130 0.961858
\(745\) −2.94510e130 −0.610928
\(746\) −6.26012e130 −1.22170
\(747\) −1.24395e131 −2.28410
\(748\) 4.48552e129 0.0774980
\(749\) 1.42805e130 0.232179
\(750\) 1.08716e131 1.66343
\(751\) 3.69393e130 0.531951 0.265976 0.963980i \(-0.414306\pi\)
0.265976 + 0.963980i \(0.414306\pi\)
\(752\) −7.46760e130 −1.01221
\(753\) −1.10286e131 −1.40718
\(754\) 2.04979e131 2.46214
\(755\) −7.47928e130 −0.845811
\(756\) −6.91700e129 −0.0736506
\(757\) −8.74343e130 −0.876636 −0.438318 0.898820i \(-0.644426\pi\)
−0.438318 + 0.898820i \(0.644426\pi\)
\(758\) 1.38238e131 1.30520
\(759\) 2.36300e131 2.10118
\(760\) 9.16030e130 0.767169
\(761\) 1.23497e130 0.0974213 0.0487106 0.998813i \(-0.484489\pi\)
0.0487106 + 0.998813i \(0.484489\pi\)
\(762\) −2.97658e130 −0.221190
\(763\) −4.19633e130 −0.293767
\(764\) −3.57529e130 −0.235810
\(765\) 2.53201e130 0.157351
\(766\) −1.08966e131 −0.638092
\(767\) −1.72282e131 −0.950715
\(768\) −2.23043e131 −1.15999
\(769\) −4.35333e130 −0.213390 −0.106695 0.994292i \(-0.534027\pi\)
−0.106695 + 0.994292i \(0.534027\pi\)
\(770\) −1.32520e131 −0.612286
\(771\) 1.38004e131 0.601059
\(772\) 8.79668e129 0.0361187
\(773\) 1.74484e130 0.0675441 0.0337721 0.999430i \(-0.489248\pi\)
0.0337721 + 0.999430i \(0.489248\pi\)
\(774\) −1.17086e131 −0.427357
\(775\) 1.46877e131 0.505505
\(776\) −2.54582e131 −0.826264
\(777\) 5.05924e131 1.54856
\(778\) −5.38058e130 −0.155331
\(779\) −6.28605e131 −1.71168
\(780\) 1.27078e131 0.326412
\(781\) 8.39964e130 0.203535
\(782\) 1.08478e131 0.247990
\(783\) 2.99203e131 0.645366
\(784\) 3.15673e131 0.642476
\(785\) 2.47802e131 0.475922
\(786\) 4.76126e131 0.862971
\(787\) 8.46104e131 1.44735 0.723675 0.690141i \(-0.242452\pi\)
0.723675 + 0.690141i \(0.242452\pi\)
\(788\) −4.61817e130 −0.0745637
\(789\) −1.06500e132 −1.62310
\(790\) 9.10411e130 0.130980
\(791\) 1.99810e131 0.271386
\(792\) −1.07345e132 −1.37653
\(793\) −6.25676e131 −0.757564
\(794\) 1.10952e132 1.26854
\(795\) 5.22461e131 0.564092
\(796\) −1.99037e131 −0.202950
\(797\) −4.56377e131 −0.439511 −0.219756 0.975555i \(-0.570526\pi\)
−0.219756 + 0.975555i \(0.570526\pi\)
\(798\) −2.00704e132 −1.82568
\(799\) −2.06015e131 −0.177019
\(800\) −4.28244e131 −0.347613
\(801\) −1.80235e132 −1.38216
\(802\) 1.01191e132 0.733176
\(803\) 2.71039e132 1.85556
\(804\) 2.43035e131 0.157224
\(805\) −6.93932e131 −0.424233
\(806\) 2.02047e132 1.16737
\(807\) 4.64159e131 0.253467
\(808\) −5.59591e131 −0.288838
\(809\) 1.78159e131 0.0869260 0.0434630 0.999055i \(-0.486161\pi\)
0.0434630 + 0.999055i \(0.486161\pi\)
\(810\) −9.79844e131 −0.451948
\(811\) −1.01353e132 −0.441966 −0.220983 0.975278i \(-0.570926\pi\)
−0.220983 + 0.975278i \(0.570926\pi\)
\(812\) −7.54164e131 −0.310933
\(813\) 4.95755e132 1.93262
\(814\) −6.18885e132 −2.28139
\(815\) 8.66791e131 0.302164
\(816\) −1.14833e132 −0.378587
\(817\) −1.51827e132 −0.473418
\(818\) 7.26631e132 2.14308
\(819\) 4.06472e132 1.13400
\(820\) −6.76306e131 −0.178489
\(821\) −5.58073e132 −1.39341 −0.696703 0.717360i \(-0.745350\pi\)
−0.696703 + 0.717360i \(0.745350\pi\)
\(822\) 5.62912e132 1.32976
\(823\) −1.66879e132 −0.373000 −0.186500 0.982455i \(-0.559714\pi\)
−0.186500 + 0.982455i \(0.559714\pi\)
\(824\) 6.07339e132 1.28453
\(825\) −6.48309e132 −1.29756
\(826\) 2.92745e132 0.554495
\(827\) 7.08680e131 0.127043 0.0635213 0.997980i \(-0.479767\pi\)
0.0635213 + 0.997980i \(0.479767\pi\)
\(828\) 2.14672e132 0.364247
\(829\) −7.66881e132 −1.23168 −0.615838 0.787873i \(-0.711183\pi\)
−0.615838 + 0.787873i \(0.711183\pi\)
\(830\) −8.01504e132 −1.21858
\(831\) −1.15193e133 −1.65799
\(832\) 5.73595e132 0.781617
\(833\) 8.70876e131 0.112359
\(834\) −1.02889e133 −1.25694
\(835\) −7.12925e132 −0.824725
\(836\) 5.31600e132 0.582371
\(837\) 2.94923e132 0.305985
\(838\) −7.78846e132 −0.765330
\(839\) 1.18559e133 1.10349 0.551743 0.834014i \(-0.313963\pi\)
0.551743 + 0.834014i \(0.313963\pi\)
\(840\) 5.65407e132 0.498487
\(841\) 2.06488e133 1.72456
\(842\) 1.20849e133 0.956184
\(843\) 1.99413e133 1.49486
\(844\) 4.65352e132 0.330522
\(845\) −6.57135e132 −0.442257
\(846\) −1.88290e133 −1.20081
\(847\) 8.85914e132 0.535422
\(848\) −1.32108e133 −0.756691
\(849\) −2.39876e132 −0.130223
\(850\) −2.97618e132 −0.153143
\(851\) −3.24074e133 −1.58070
\(852\) 1.36867e132 0.0632845
\(853\) 1.11576e133 0.489090 0.244545 0.969638i \(-0.421361\pi\)
0.244545 + 0.969638i \(0.421361\pi\)
\(854\) 1.06316e133 0.441841
\(855\) 3.00080e133 1.18244
\(856\) 7.45329e132 0.278479
\(857\) −1.41529e132 −0.0501438 −0.0250719 0.999686i \(-0.507981\pi\)
−0.0250719 + 0.999686i \(0.507981\pi\)
\(858\) −8.91828e133 −2.99647
\(859\) 4.54822e132 0.144928 0.0724641 0.997371i \(-0.476914\pi\)
0.0724641 + 0.997371i \(0.476914\pi\)
\(860\) −1.63348e132 −0.0493666
\(861\) −3.87998e133 −1.11221
\(862\) 3.46751e133 0.942837
\(863\) 4.19992e133 1.08330 0.541652 0.840603i \(-0.317799\pi\)
0.541652 + 0.840603i \(0.317799\pi\)
\(864\) −8.59897e132 −0.210412
\(865\) −1.61232e133 −0.374300
\(866\) −4.95448e133 −1.09127
\(867\) 6.87653e133 1.43715
\(868\) −7.43374e132 −0.147421
\(869\) −1.38342e133 −0.260348
\(870\) 9.34047e133 1.66818
\(871\) −2.94769e133 −0.499639
\(872\) −2.19015e133 −0.352349
\(873\) −8.33979e133 −1.27352
\(874\) 1.28562e134 1.86356
\(875\) 4.85134e133 0.667567
\(876\) 4.41641e133 0.576943
\(877\) −5.09327e133 −0.631706 −0.315853 0.948808i \(-0.602291\pi\)
−0.315853 + 0.948808i \(0.602291\pi\)
\(878\) −2.93239e133 −0.345320
\(879\) 8.50114e133 0.950568
\(880\) −8.98601e133 −0.954125
\(881\) −1.28135e134 −1.29201 −0.646003 0.763335i \(-0.723561\pi\)
−0.646003 + 0.763335i \(0.723561\pi\)
\(882\) 7.95948e133 0.762191
\(883\) 1.34221e134 1.22070 0.610351 0.792131i \(-0.291028\pi\)
0.610351 + 0.792131i \(0.291028\pi\)
\(884\) −8.86470e132 −0.0765748
\(885\) −7.85052e133 −0.644139
\(886\) −1.83955e134 −1.43376
\(887\) −9.27028e133 −0.686386 −0.343193 0.939265i \(-0.611508\pi\)
−0.343193 + 0.939265i \(0.611508\pi\)
\(888\) 2.64052e134 1.85737
\(889\) −1.32827e133 −0.0887679
\(890\) −1.16129e134 −0.737389
\(891\) 1.48893e134 0.898334
\(892\) −8.88940e132 −0.0509649
\(893\) −2.44158e134 −1.33024
\(894\) 3.36782e134 1.74378
\(895\) 6.42221e133 0.316036
\(896\) −1.75893e134 −0.822688
\(897\) −4.66999e134 −2.07615
\(898\) −4.58998e134 −1.93971
\(899\) 3.21556e134 1.29178
\(900\) −5.88971e133 −0.224936
\(901\) −3.64458e133 −0.132334
\(902\) 4.74628e134 1.63854
\(903\) −9.37129e133 −0.307614
\(904\) 1.04285e134 0.325505
\(905\) 5.71537e133 0.169642
\(906\) 8.55281e134 2.41421
\(907\) 4.24923e134 1.14072 0.570360 0.821395i \(-0.306804\pi\)
0.570360 + 0.821395i \(0.306804\pi\)
\(908\) −5.97987e133 −0.152681
\(909\) −1.83315e134 −0.445187
\(910\) 2.61899e134 0.604993
\(911\) 3.15914e134 0.694198 0.347099 0.937828i \(-0.387167\pi\)
0.347099 + 0.937828i \(0.387167\pi\)
\(912\) −1.36094e135 −2.84495
\(913\) 1.21793e135 2.42215
\(914\) −1.08263e135 −2.04846
\(915\) −2.85107e134 −0.513273
\(916\) 2.73824e134 0.469059
\(917\) 2.12467e134 0.346327
\(918\) −5.97606e133 −0.0926986
\(919\) −7.04232e134 −1.03958 −0.519792 0.854293i \(-0.673991\pi\)
−0.519792 + 0.854293i \(0.673991\pi\)
\(920\) −3.62176e134 −0.508831
\(921\) −2.04678e135 −2.73690
\(922\) −2.28157e134 −0.290387
\(923\) −1.66001e134 −0.201110
\(924\) 3.28123e134 0.378410
\(925\) 8.89123e134 0.976142
\(926\) 1.28495e135 1.34304
\(927\) 1.98957e135 1.97985
\(928\) −9.37549e134 −0.888303
\(929\) −8.78763e134 −0.792788 −0.396394 0.918081i \(-0.629739\pi\)
−0.396394 + 0.918081i \(0.629739\pi\)
\(930\) 9.20685e134 0.790928
\(931\) 1.03212e135 0.844341
\(932\) 6.50981e134 0.507159
\(933\) −1.81268e134 −0.134495
\(934\) −2.29215e135 −1.61979
\(935\) −2.47905e134 −0.166862
\(936\) 2.12145e135 1.36014
\(937\) −1.33059e135 −0.812633 −0.406316 0.913733i \(-0.633187\pi\)
−0.406316 + 0.913733i \(0.633187\pi\)
\(938\) 5.00879e134 0.291409
\(939\) −4.70084e134 −0.260550
\(940\) −2.62686e134 −0.138714
\(941\) 1.15538e135 0.581295 0.290648 0.956830i \(-0.406129\pi\)
0.290648 + 0.956830i \(0.406129\pi\)
\(942\) −2.83370e135 −1.35843
\(943\) 2.48535e135 1.13529
\(944\) 1.98506e135 0.864069
\(945\) 3.82287e134 0.158578
\(946\) 1.14637e135 0.453186
\(947\) −6.75941e134 −0.254674 −0.127337 0.991859i \(-0.540643\pi\)
−0.127337 + 0.991859i \(0.540643\pi\)
\(948\) −2.25420e134 −0.0809492
\(949\) −5.35652e135 −1.83345
\(950\) −3.52721e135 −1.15082
\(951\) 4.90530e135 1.52564
\(952\) −3.94416e134 −0.116943
\(953\) 3.89093e135 1.09983 0.549915 0.835220i \(-0.314660\pi\)
0.549915 + 0.835220i \(0.314660\pi\)
\(954\) −3.33101e135 −0.897687
\(955\) 1.97598e135 0.507726
\(956\) 1.21047e135 0.296566
\(957\) −1.41934e136 −3.31583
\(958\) −3.62538e135 −0.807650
\(959\) 2.51194e135 0.533658
\(960\) 2.61375e135 0.529570
\(961\) −2.00549e135 −0.387531
\(962\) 1.22310e136 2.25421
\(963\) 2.44161e135 0.429220
\(964\) −1.64129e135 −0.275221
\(965\) −4.86173e134 −0.0777675
\(966\) 7.93534e135 1.21089
\(967\) −2.80342e135 −0.408116 −0.204058 0.978959i \(-0.565413\pi\)
−0.204058 + 0.978959i \(0.565413\pi\)
\(968\) 4.62376e135 0.642194
\(969\) −3.75455e135 −0.497538
\(970\) −5.37351e135 −0.679430
\(971\) −6.04055e135 −0.728789 −0.364394 0.931245i \(-0.618724\pi\)
−0.364394 + 0.931245i \(0.618724\pi\)
\(972\) 3.36466e135 0.387370
\(973\) −4.59133e135 −0.504433
\(974\) 1.34290e136 1.40803
\(975\) 1.28125e136 1.28210
\(976\) 7.20915e135 0.688521
\(977\) −9.72367e135 −0.886396 −0.443198 0.896424i \(-0.646156\pi\)
−0.443198 + 0.896424i \(0.646156\pi\)
\(978\) −9.91205e135 −0.862473
\(979\) 1.76465e136 1.46570
\(980\) 1.11044e135 0.0880455
\(981\) −7.17466e135 −0.543077
\(982\) −1.54374e136 −1.11558
\(983\) −3.38448e135 −0.233511 −0.116755 0.993161i \(-0.537249\pi\)
−0.116755 + 0.993161i \(0.537249\pi\)
\(984\) −2.02504e136 −1.33400
\(985\) 2.55236e135 0.160544
\(986\) −6.51572e135 −0.391348
\(987\) −1.50703e136 −0.864355
\(988\) −1.05060e136 −0.575434
\(989\) 6.00286e135 0.313998
\(990\) −2.26576e136 −1.13191
\(991\) 1.21030e136 0.577486 0.288743 0.957407i \(-0.406763\pi\)
0.288743 + 0.957407i \(0.406763\pi\)
\(992\) −9.24136e135 −0.421167
\(993\) 4.77901e136 2.08040
\(994\) 2.82073e135 0.117295
\(995\) 1.10004e136 0.436975
\(996\) 1.98454e136 0.753113
\(997\) 4.44306e136 1.61084 0.805419 0.592705i \(-0.201940\pi\)
0.805419 + 0.592705i \(0.201940\pi\)
\(998\) −5.74210e136 −1.98898
\(999\) 1.78533e136 0.590864
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 1.92.a.a.1.2 7
3.2 odd 2 9.92.a.b.1.6 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.92.a.a.1.2 7 1.1 even 1 trivial
9.92.a.b.1.6 7 3.2 odd 2