Properties

Label 1.110.a.a.1.4
Level $1$
Weight $110$
Character 1.1
Self dual yes
Analytic conductor $75.239$
Analytic rank $1$
Dimension $8$
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] = [1,110,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, 110, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1.1");
 
S:= CuspForms(chi, 110);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 110 \)
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: \(75.2394221917\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2 x^{7} + \cdots + 46\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: multiple of \( 2^{118}\cdot 3^{40}\cdot 5^{14}\cdot 7^{6}\cdot 11^{3}\cdot 13 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(1.20500e13\) 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)
 
\(f(q)\) \(=\) \(q-2.02754e15 q^{2} -9.96254e25 q^{3} -6.44926e32 q^{4} -2.14431e38 q^{5} +2.01995e41 q^{6} -1.47165e46 q^{7} +2.62357e48 q^{8} -2.18949e50 q^{9} +O(q^{10})\) \(q-2.02754e15 q^{2} -9.96254e25 q^{3} -6.44926e32 q^{4} -2.14431e38 q^{5} +2.01995e41 q^{6} -1.47165e46 q^{7} +2.62357e48 q^{8} -2.18949e50 q^{9} +4.34769e53 q^{10} +5.24987e56 q^{11} +6.42510e58 q^{12} -3.46172e60 q^{13} +2.98383e61 q^{14} +2.13628e64 q^{15} +4.13262e65 q^{16} +1.50023e67 q^{17} +4.43928e65 q^{18} -4.60234e69 q^{19} +1.38292e71 q^{20} +1.46614e72 q^{21} -1.06444e72 q^{22} -1.72378e74 q^{23} -2.61374e74 q^{24} +3.05734e76 q^{25} +7.01879e75 q^{26} +1.03243e78 q^{27} +9.49105e78 q^{28} -4.10244e79 q^{29} -4.33141e79 q^{30} -8.97441e79 q^{31} -2.54070e81 q^{32} -5.23021e82 q^{33} -3.04178e82 q^{34} +3.15568e84 q^{35} +1.41206e83 q^{36} -4.96618e85 q^{37} +9.33144e84 q^{38} +3.44875e86 q^{39} -5.62576e86 q^{40} +5.38189e87 q^{41} -2.97266e87 q^{42} +3.74176e88 q^{43} -3.38578e89 q^{44} +4.69495e88 q^{45} +3.49505e89 q^{46} +1.16615e91 q^{47} -4.11714e91 q^{48} +8.60522e91 q^{49} -6.19890e91 q^{50} -1.49461e93 q^{51} +2.23255e93 q^{52} +1.39456e94 q^{53} -2.09330e93 q^{54} -1.12574e95 q^{55} -3.86097e94 q^{56} +4.58510e95 q^{57} +8.31788e94 q^{58} +2.10352e96 q^{59} -1.37774e97 q^{60} +1.91674e97 q^{61} +1.81960e95 q^{62} +3.22215e96 q^{63} -2.63071e98 q^{64} +7.42302e98 q^{65} +1.06045e98 q^{66} -2.19176e99 q^{67} -9.67536e99 q^{68} +1.71733e100 q^{69} -6.39828e99 q^{70} -1.48237e100 q^{71} -5.74426e98 q^{72} -4.73812e101 q^{73} +1.00692e101 q^{74} -3.04589e102 q^{75} +2.96817e102 q^{76} -7.72597e102 q^{77} -6.99250e101 q^{78} -3.66779e103 q^{79} -8.86163e103 q^{80} -1.00635e104 q^{81} -1.09120e103 q^{82} +1.15763e104 q^{83} -9.45550e104 q^{84} -3.21696e105 q^{85} -7.58658e103 q^{86} +4.08707e105 q^{87} +1.37734e105 q^{88} -7.77888e105 q^{89} -9.51921e103 q^{90} +5.09444e106 q^{91} +1.11171e107 q^{92} +8.94080e105 q^{93} -2.36441e106 q^{94} +9.86886e107 q^{95} +2.53118e107 q^{96} +2.38667e108 q^{97} -1.74475e107 q^{98} -1.14945e107 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 22\!\cdots\!00 q^{2}+ \cdots + 24\!\cdots\!84 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 22\!\cdots\!00 q^{2}+ \cdots + 71\!\cdots\!28 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.02754e15 −0.0795858 −0.0397929 0.999208i \(-0.512670\pi\)
−0.0397929 + 0.999208i \(0.512670\pi\)
\(3\) −9.96254e25 −0.989149 −0.494575 0.869135i \(-0.664676\pi\)
−0.494575 + 0.869135i \(0.664676\pi\)
\(4\) −6.44926e32 −0.993666
\(5\) −2.14431e38 −1.72752 −0.863761 0.503903i \(-0.831897\pi\)
−0.863761 + 0.503903i \(0.831897\pi\)
\(6\) 2.01995e41 0.0787222
\(7\) −1.47165e46 −1.28813 −0.644067 0.764969i \(-0.722754\pi\)
−0.644067 + 0.764969i \(0.722754\pi\)
\(8\) 2.62357e48 0.158668
\(9\) −2.18949e50 −0.0215837
\(10\) 4.34769e53 0.137486
\(11\) 5.24987e56 0.920974 0.460487 0.887666i \(-0.347675\pi\)
0.460487 + 0.887666i \(0.347675\pi\)
\(12\) 6.42510e58 0.982884
\(13\) −3.46172e60 −0.675117 −0.337559 0.941304i \(-0.609601\pi\)
−0.337559 + 0.941304i \(0.609601\pi\)
\(14\) 2.98383e61 0.102517
\(15\) 2.13628e64 1.70878
\(16\) 4.13262e65 0.981038
\(17\) 1.50023e67 1.30825 0.654125 0.756386i \(-0.273037\pi\)
0.654125 + 0.756386i \(0.273037\pi\)
\(18\) 4.43928e65 0.00171775
\(19\) −4.60234e69 −0.935206 −0.467603 0.883939i \(-0.654882\pi\)
−0.467603 + 0.883939i \(0.654882\pi\)
\(20\) 1.38292e71 1.71658
\(21\) 1.46614e72 1.27416
\(22\) −1.06444e72 −0.0732965
\(23\) −1.72378e74 −1.05273 −0.526364 0.850260i \(-0.676445\pi\)
−0.526364 + 0.850260i \(0.676445\pi\)
\(24\) −2.61374e74 −0.156946
\(25\) 3.05734e76 1.98433
\(26\) 7.01879e75 0.0537298
\(27\) 1.03243e78 1.01050
\(28\) 9.49105e78 1.27997
\(29\) −4.10244e79 −0.817243 −0.408622 0.912704i \(-0.633990\pi\)
−0.408622 + 0.912704i \(0.633990\pi\)
\(30\) −4.33141e79 −0.135994
\(31\) −8.97441e79 −0.0471839 −0.0235919 0.999722i \(-0.507510\pi\)
−0.0235919 + 0.999722i \(0.507510\pi\)
\(32\) −2.54070e81 −0.236744
\(33\) −5.23021e82 −0.910981
\(34\) −3.04178e82 −0.104118
\(35\) 3.15568e84 2.22528
\(36\) 1.41206e83 0.0214470
\(37\) −4.96618e85 −1.69445 −0.847225 0.531235i \(-0.821728\pi\)
−0.847225 + 0.531235i \(0.821728\pi\)
\(38\) 9.33144e84 0.0744291
\(39\) 3.44875e86 0.667792
\(40\) −5.62576e86 −0.274101
\(41\) 5.38189e87 0.682676 0.341338 0.939941i \(-0.389120\pi\)
0.341338 + 0.939941i \(0.389120\pi\)
\(42\) −2.97266e87 −0.101405
\(43\) 3.74176e88 0.354034 0.177017 0.984208i \(-0.443355\pi\)
0.177017 + 0.984208i \(0.443355\pi\)
\(44\) −3.38578e89 −0.915141
\(45\) 4.69495e88 0.0372863
\(46\) 3.49505e89 0.0837821
\(47\) 1.16615e91 0.865804 0.432902 0.901441i \(-0.357490\pi\)
0.432902 + 0.901441i \(0.357490\pi\)
\(48\) −4.11714e91 −0.970393
\(49\) 8.60522e91 0.659289
\(50\) −6.19890e91 −0.157924
\(51\) −1.49461e93 −1.29405
\(52\) 2.23255e93 0.670841
\(53\) 1.39456e94 1.48389 0.741946 0.670459i \(-0.233903\pi\)
0.741946 + 0.670459i \(0.233903\pi\)
\(54\) −2.09330e93 −0.0804213
\(55\) −1.12574e95 −1.59100
\(56\) −3.86097e94 −0.204385
\(57\) 4.58510e95 0.925058
\(58\) 8.31788e94 0.0650410
\(59\) 2.10352e96 0.647905 0.323952 0.946073i \(-0.394988\pi\)
0.323952 + 0.946073i \(0.394988\pi\)
\(60\) −1.37774e97 −1.69795
\(61\) 1.91674e97 0.959594 0.479797 0.877380i \(-0.340710\pi\)
0.479797 + 0.877380i \(0.340710\pi\)
\(62\) 1.81960e95 0.00375517
\(63\) 3.22215e96 0.0278027
\(64\) −2.63071e98 −0.962197
\(65\) 7.42302e98 1.16628
\(66\) 1.06045e98 0.0725012
\(67\) −2.19176e99 −0.660262 −0.330131 0.943935i \(-0.607093\pi\)
−0.330131 + 0.943935i \(0.607093\pi\)
\(68\) −9.67536e99 −1.29996
\(69\) 1.71733e100 1.04130
\(70\) −6.39828e99 −0.177101
\(71\) −1.48237e100 −0.189399 −0.0946993 0.995506i \(-0.530189\pi\)
−0.0946993 + 0.995506i \(0.530189\pi\)
\(72\) −5.74426e98 −0.00342463
\(73\) −4.73812e101 −1.33202 −0.666009 0.745944i \(-0.731999\pi\)
−0.666009 + 0.745944i \(0.731999\pi\)
\(74\) 1.00692e101 0.134854
\(75\) −3.04589e102 −1.96280
\(76\) 2.96817e102 0.929282
\(77\) −7.72597e102 −1.18634
\(78\) −6.99250e101 −0.0531468
\(79\) −3.66779e103 −1.39229 −0.696143 0.717904i \(-0.745102\pi\)
−0.696143 + 0.717904i \(0.745102\pi\)
\(80\) −8.86163e103 −1.69476
\(81\) −1.00635e104 −0.977950
\(82\) −1.09120e103 −0.0543313
\(83\) 1.15763e104 0.297724 0.148862 0.988858i \(-0.452439\pi\)
0.148862 + 0.988858i \(0.452439\pi\)
\(84\) −9.45550e104 −1.26609
\(85\) −3.21696e105 −2.26003
\(86\) −7.58658e103 −0.0281761
\(87\) 4.08707e105 0.808376
\(88\) 1.37734e105 0.146129
\(89\) −7.77888e105 −0.445822 −0.222911 0.974839i \(-0.571556\pi\)
−0.222911 + 0.974839i \(0.571556\pi\)
\(90\) −9.51921e103 −0.00296746
\(91\) 5.09444e106 0.869642
\(92\) 1.11171e107 1.04606
\(93\) 8.94080e105 0.0466719
\(94\) −2.36441e106 −0.0689057
\(95\) 9.86886e107 1.61559
\(96\) 2.53118e107 0.234175
\(97\) 2.38667e108 1.25526 0.627629 0.778513i \(-0.284026\pi\)
0.627629 + 0.778513i \(0.284026\pi\)
\(98\) −1.74475e107 −0.0524700
\(99\) −1.14945e107 −0.0198780
\(100\) −1.97176e109 −1.97176
\(101\) 1.79300e109 1.04248 0.521239 0.853411i \(-0.325470\pi\)
0.521239 + 0.853411i \(0.325470\pi\)
\(102\) 3.03038e108 0.102988
\(103\) −3.76956e109 −0.752769 −0.376385 0.926464i \(-0.622833\pi\)
−0.376385 + 0.926464i \(0.622833\pi\)
\(104\) −9.08206e108 −0.107119
\(105\) −3.14386e110 −2.20113
\(106\) −2.82753e109 −0.118097
\(107\) 2.26052e110 0.565968 0.282984 0.959125i \(-0.408676\pi\)
0.282984 + 0.959125i \(0.408676\pi\)
\(108\) −6.65842e110 −1.00410
\(109\) 4.65522e110 0.424811 0.212406 0.977182i \(-0.431870\pi\)
0.212406 + 0.977182i \(0.431870\pi\)
\(110\) 2.28248e110 0.126621
\(111\) 4.94758e111 1.67606
\(112\) −6.08176e111 −1.26371
\(113\) 5.07866e111 0.650091 0.325046 0.945698i \(-0.394620\pi\)
0.325046 + 0.945698i \(0.394620\pi\)
\(114\) −9.29649e110 −0.0736215
\(115\) 3.69634e112 1.81861
\(116\) 2.64577e112 0.812067
\(117\) 7.57939e110 0.0145715
\(118\) −4.26497e111 −0.0515640
\(119\) −2.20781e113 −1.68520
\(120\) 5.60468e112 0.271127
\(121\) −4.93278e112 −0.151806
\(122\) −3.88628e112 −0.0763700
\(123\) −5.36173e113 −0.675269
\(124\) 5.78783e112 0.0468850
\(125\) −3.25207e114 −1.70045
\(126\) −6.53306e111 −0.00221270
\(127\) 3.91250e114 0.861293 0.430647 0.902521i \(-0.358285\pi\)
0.430647 + 0.902521i \(0.358285\pi\)
\(128\) 2.18240e114 0.313321
\(129\) −3.72774e114 −0.350193
\(130\) −1.50505e114 −0.0928193
\(131\) 2.54602e115 1.03413 0.517066 0.855945i \(-0.327024\pi\)
0.517066 + 0.855945i \(0.327024\pi\)
\(132\) 3.37310e115 0.905211
\(133\) 6.77302e115 1.20467
\(134\) 4.44389e114 0.0525475
\(135\) −2.21386e116 −1.74566
\(136\) 3.93595e115 0.207577
\(137\) 1.85339e116 0.655686 0.327843 0.944732i \(-0.393678\pi\)
0.327843 + 0.944732i \(0.393678\pi\)
\(138\) −3.48196e115 −0.0828730
\(139\) 5.08781e116 0.817004 0.408502 0.912757i \(-0.366051\pi\)
0.408502 + 0.912757i \(0.366051\pi\)
\(140\) −2.03518e117 −2.21118
\(141\) −1.16178e117 −0.856410
\(142\) 3.00557e115 0.0150734
\(143\) −1.81736e117 −0.621766
\(144\) −9.04831e115 −0.0211744
\(145\) 8.79692e117 1.41181
\(146\) 9.60674e116 0.106010
\(147\) −8.57299e117 −0.652135
\(148\) 3.20282e118 1.68372
\(149\) −2.95585e118 −1.07654 −0.538270 0.842772i \(-0.680922\pi\)
−0.538270 + 0.842772i \(0.680922\pi\)
\(150\) 6.17568e117 0.156211
\(151\) 9.53184e118 1.67855 0.839273 0.543711i \(-0.182981\pi\)
0.839273 + 0.543711i \(0.182981\pi\)
\(152\) −1.20745e118 −0.148387
\(153\) −3.28473e117 −0.0282369
\(154\) 1.56647e118 0.0944157
\(155\) 1.92440e118 0.0815112
\(156\) −2.22419e119 −0.663562
\(157\) −6.55912e119 −1.38139 −0.690693 0.723148i \(-0.742694\pi\)
−0.690693 + 0.723148i \(0.742694\pi\)
\(158\) 7.43661e118 0.110806
\(159\) −1.38934e120 −1.46779
\(160\) 5.44806e119 0.408981
\(161\) 2.53680e120 1.35605
\(162\) 2.04043e119 0.0778310
\(163\) −3.98860e120 −1.08792 −0.543959 0.839112i \(-0.683075\pi\)
−0.543959 + 0.839112i \(0.683075\pi\)
\(164\) −3.47092e120 −0.678352
\(165\) 1.12152e121 1.57374
\(166\) −2.34715e119 −0.0236946
\(167\) 2.12485e121 1.54625 0.773126 0.634253i \(-0.218692\pi\)
0.773126 + 0.634253i \(0.218692\pi\)
\(168\) 3.84651e120 0.202167
\(169\) −1.43086e121 −0.544216
\(170\) 6.52253e120 0.179866
\(171\) 1.00768e120 0.0201852
\(172\) −2.41316e121 −0.351792
\(173\) 3.38854e121 0.360164 0.180082 0.983652i \(-0.442364\pi\)
0.180082 + 0.983652i \(0.442364\pi\)
\(174\) −8.28672e120 −0.0643352
\(175\) −4.49933e122 −2.55608
\(176\) 2.16957e122 0.903511
\(177\) −2.09564e122 −0.640875
\(178\) 1.57720e121 0.0354811
\(179\) 2.93078e122 0.485840 0.242920 0.970046i \(-0.421895\pi\)
0.242920 + 0.970046i \(0.421895\pi\)
\(180\) −3.02789e121 −0.0370501
\(181\) 5.78889e122 0.523737 0.261868 0.965104i \(-0.415661\pi\)
0.261868 + 0.965104i \(0.415661\pi\)
\(182\) −1.03292e122 −0.0692111
\(183\) −1.90956e123 −0.949181
\(184\) −4.52246e122 −0.167034
\(185\) 1.06491e124 2.92720
\(186\) −1.81279e121 −0.00371442
\(187\) 7.87600e123 1.20487
\(188\) −7.52078e123 −0.860320
\(189\) −1.51938e124 −1.30166
\(190\) −2.00095e123 −0.128578
\(191\) −3.20680e124 −1.54794 −0.773969 0.633224i \(-0.781731\pi\)
−0.773969 + 0.633224i \(0.781731\pi\)
\(192\) 2.62085e124 0.951756
\(193\) 1.82650e124 0.499742 0.249871 0.968279i \(-0.419612\pi\)
0.249871 + 0.968279i \(0.419612\pi\)
\(194\) −4.83908e123 −0.0999007
\(195\) −7.39522e124 −1.15362
\(196\) −5.54973e124 −0.655113
\(197\) 6.94971e124 0.621666 0.310833 0.950464i \(-0.399392\pi\)
0.310833 + 0.950464i \(0.399392\pi\)
\(198\) 2.33057e122 0.00158201
\(199\) −3.75935e123 −0.0193919 −0.00969595 0.999953i \(-0.503086\pi\)
−0.00969595 + 0.999953i \(0.503086\pi\)
\(200\) 8.02115e124 0.314848
\(201\) 2.18355e125 0.653097
\(202\) −3.63540e124 −0.0829664
\(203\) 6.03735e125 1.05272
\(204\) 9.63912e125 1.28586
\(205\) −1.15405e126 −1.17934
\(206\) 7.64295e124 0.0599097
\(207\) 3.77420e124 0.0227217
\(208\) −1.43060e126 −0.662316
\(209\) −2.41617e126 −0.861300
\(210\) 6.37431e125 0.175179
\(211\) 3.63035e126 0.770112 0.385056 0.922893i \(-0.374182\pi\)
0.385056 + 0.922893i \(0.374182\pi\)
\(212\) −8.99388e126 −1.47449
\(213\) 1.47682e126 0.187344
\(214\) −4.58330e125 −0.0450430
\(215\) −8.02351e126 −0.611601
\(216\) 2.70865e126 0.160333
\(217\) 1.32072e126 0.0607792
\(218\) −9.43866e125 −0.0338089
\(219\) 4.72037e127 1.31756
\(220\) 7.26018e127 1.58093
\(221\) −5.19337e127 −0.883223
\(222\) −1.00314e127 −0.133391
\(223\) 1.69776e127 0.176710 0.0883552 0.996089i \(-0.471839\pi\)
0.0883552 + 0.996089i \(0.471839\pi\)
\(224\) 3.73902e127 0.304958
\(225\) −6.69401e126 −0.0428291
\(226\) −1.02972e127 −0.0517380
\(227\) −2.63698e127 −0.104159 −0.0520795 0.998643i \(-0.516585\pi\)
−0.0520795 + 0.998643i \(0.516585\pi\)
\(228\) −2.95705e128 −0.919199
\(229\) −5.95327e128 −1.45788 −0.728940 0.684577i \(-0.759987\pi\)
−0.728940 + 0.684577i \(0.759987\pi\)
\(230\) −7.49448e127 −0.144735
\(231\) 7.69703e128 1.17347
\(232\) −1.07630e128 −0.129670
\(233\) 5.27747e128 0.502954 0.251477 0.967863i \(-0.419084\pi\)
0.251477 + 0.967863i \(0.419084\pi\)
\(234\) −1.53675e126 −0.00115969
\(235\) −2.50058e129 −1.49569
\(236\) −1.35661e129 −0.643801
\(237\) 3.65405e129 1.37718
\(238\) 4.47643e128 0.134118
\(239\) 6.00392e129 1.43136 0.715681 0.698427i \(-0.246116\pi\)
0.715681 + 0.698427i \(0.246116\pi\)
\(240\) 8.82844e129 1.67638
\(241\) −6.09889e129 −0.923255 −0.461627 0.887074i \(-0.652734\pi\)
−0.461627 + 0.887074i \(0.652734\pi\)
\(242\) 1.00014e128 0.0120816
\(243\) −4.47323e128 −0.0431598
\(244\) −1.23616e130 −0.953516
\(245\) −1.84523e130 −1.13894
\(246\) 1.08712e129 0.0537418
\(247\) 1.59320e130 0.631374
\(248\) −2.35450e128 −0.00748655
\(249\) −1.15329e130 −0.294494
\(250\) 6.59371e129 0.135332
\(251\) −6.08328e130 −1.00443 −0.502215 0.864743i \(-0.667482\pi\)
−0.502215 + 0.864743i \(0.667482\pi\)
\(252\) −2.07805e129 −0.0276266
\(253\) −9.04965e130 −0.969535
\(254\) −7.93276e129 −0.0685467
\(255\) 3.20491e131 2.23551
\(256\) 1.66318e131 0.937261
\(257\) −2.64025e131 −1.20307 −0.601533 0.798848i \(-0.705443\pi\)
−0.601533 + 0.798848i \(0.705443\pi\)
\(258\) 7.55817e129 0.0278704
\(259\) 7.30847e131 2.18268
\(260\) −4.78730e131 −1.15889
\(261\) 8.98224e129 0.0176391
\(262\) −5.16217e130 −0.0823023
\(263\) 6.99024e131 0.905532 0.452766 0.891629i \(-0.350437\pi\)
0.452766 + 0.891629i \(0.350437\pi\)
\(264\) −1.37218e131 −0.144543
\(265\) −2.99038e132 −2.56346
\(266\) −1.37326e131 −0.0958746
\(267\) 7.74974e131 0.440985
\(268\) 1.41352e132 0.656080
\(269\) −6.91264e131 −0.261905 −0.130953 0.991389i \(-0.541804\pi\)
−0.130953 + 0.991389i \(0.541804\pi\)
\(270\) 4.48869e131 0.138930
\(271\) −7.05558e132 −1.78528 −0.892640 0.450770i \(-0.851149\pi\)
−0.892640 + 0.450770i \(0.851149\pi\)
\(272\) 6.19986e132 1.28344
\(273\) −5.07536e132 −0.860205
\(274\) −3.75783e131 −0.0521833
\(275\) 1.60507e133 1.82752
\(276\) −1.10755e133 −1.03471
\(277\) 2.41655e133 1.85374 0.926870 0.375383i \(-0.122489\pi\)
0.926870 + 0.375383i \(0.122489\pi\)
\(278\) −1.03158e132 −0.0650219
\(279\) 1.96494e130 0.00101840
\(280\) 8.27914e132 0.353079
\(281\) −1.29415e133 −0.454453 −0.227226 0.973842i \(-0.572966\pi\)
−0.227226 + 0.973842i \(0.572966\pi\)
\(282\) 2.35556e132 0.0681580
\(283\) −1.51288e132 −0.0360948 −0.0180474 0.999837i \(-0.505745\pi\)
−0.0180474 + 0.999837i \(0.505745\pi\)
\(284\) 9.56020e132 0.188199
\(285\) −9.83189e133 −1.59806
\(286\) 3.68478e132 0.0494837
\(287\) −7.92026e133 −0.879379
\(288\) 5.56282e131 0.00510981
\(289\) 9.35662e133 0.711519
\(290\) −1.78362e133 −0.112360
\(291\) −2.37773e134 −1.24164
\(292\) 3.05573e134 1.32358
\(293\) 2.46161e134 0.884980 0.442490 0.896773i \(-0.354095\pi\)
0.442490 + 0.896773i \(0.354095\pi\)
\(294\) 1.73821e133 0.0519007
\(295\) −4.51060e134 −1.11927
\(296\) −1.30291e134 −0.268854
\(297\) 5.42013e134 0.930644
\(298\) 5.99311e133 0.0856773
\(299\) 5.96726e134 0.710714
\(300\) 1.96437e135 1.95037
\(301\) −5.50656e134 −0.456043
\(302\) −1.93262e134 −0.133588
\(303\) −1.78629e135 −1.03117
\(304\) −1.90197e135 −0.917473
\(305\) −4.11010e135 −1.65772
\(306\) 6.65993e132 0.00224725
\(307\) 2.70496e134 0.0764047 0.0382023 0.999270i \(-0.487837\pi\)
0.0382023 + 0.999270i \(0.487837\pi\)
\(308\) 4.98268e135 1.17882
\(309\) 3.75544e135 0.744601
\(310\) −3.90180e133 −0.00648713
\(311\) −1.26793e136 −1.76870 −0.884349 0.466826i \(-0.845397\pi\)
−0.884349 + 0.466826i \(0.845397\pi\)
\(312\) 9.04804e134 0.105957
\(313\) −3.22055e134 −0.0316784 −0.0158392 0.999875i \(-0.505042\pi\)
−0.0158392 + 0.999875i \(0.505042\pi\)
\(314\) 1.32989e135 0.109939
\(315\) −6.90931e134 −0.0480297
\(316\) 2.36546e136 1.38347
\(317\) 1.60712e136 0.791258 0.395629 0.918410i \(-0.370527\pi\)
0.395629 + 0.918410i \(0.370527\pi\)
\(318\) 2.81694e135 0.116815
\(319\) −2.15373e136 −0.752660
\(320\) 5.64107e136 1.66222
\(321\) −2.25205e136 −0.559827
\(322\) −5.14348e135 −0.107923
\(323\) −6.90455e136 −1.22348
\(324\) 6.49023e136 0.971756
\(325\) −1.05837e137 −1.33965
\(326\) 8.08706e135 0.0865828
\(327\) −4.63778e136 −0.420202
\(328\) 1.41198e136 0.108319
\(329\) −1.71616e137 −1.11527
\(330\) −2.27393e136 −0.125247
\(331\) 2.41606e137 1.12845 0.564226 0.825621i \(-0.309175\pi\)
0.564226 + 0.825621i \(0.309175\pi\)
\(332\) −7.46586e136 −0.295838
\(333\) 1.08734e136 0.0365724
\(334\) −4.30822e136 −0.123060
\(335\) 4.69982e137 1.14062
\(336\) 6.05898e137 1.25000
\(337\) −9.68101e137 −1.69860 −0.849300 0.527911i \(-0.822975\pi\)
−0.849300 + 0.527911i \(0.822975\pi\)
\(338\) 2.90113e136 0.0433119
\(339\) −5.05964e137 −0.643037
\(340\) 2.07470e138 2.24572
\(341\) −4.71145e136 −0.0434552
\(342\) −2.04311e135 −0.00160645
\(343\) 6.54450e137 0.438881
\(344\) 9.81676e136 0.0561737
\(345\) −3.68249e138 −1.79887
\(346\) −6.87042e136 −0.0286639
\(347\) −9.86471e137 −0.351664 −0.175832 0.984420i \(-0.556262\pi\)
−0.175832 + 0.984420i \(0.556262\pi\)
\(348\) −2.63586e138 −0.803255
\(349\) 1.74389e138 0.454498 0.227249 0.973837i \(-0.427027\pi\)
0.227249 + 0.973837i \(0.427027\pi\)
\(350\) 9.12260e137 0.203428
\(351\) −3.57399e138 −0.682205
\(352\) −1.33384e138 −0.218035
\(353\) 5.89853e137 0.0826081 0.0413040 0.999147i \(-0.486849\pi\)
0.0413040 + 0.999147i \(0.486849\pi\)
\(354\) 4.24900e137 0.0510045
\(355\) 3.17867e138 0.327190
\(356\) 5.01680e138 0.442998
\(357\) 2.19954e139 1.66692
\(358\) −5.94229e137 −0.0386660
\(359\) −1.00536e139 −0.561918 −0.280959 0.959720i \(-0.590653\pi\)
−0.280959 + 0.959720i \(0.590653\pi\)
\(360\) 1.23175e137 0.00591612
\(361\) −3.03674e138 −0.125391
\(362\) −1.17372e138 −0.0416820
\(363\) 4.91431e138 0.150159
\(364\) −3.28554e139 −0.864133
\(365\) 1.01600e140 2.30109
\(366\) 3.87172e138 0.0755413
\(367\) −6.47645e139 −1.08902 −0.544511 0.838754i \(-0.683285\pi\)
−0.544511 + 0.838754i \(0.683285\pi\)
\(368\) −7.12374e139 −1.03277
\(369\) −1.17836e138 −0.0147347
\(370\) −2.15914e139 −0.232963
\(371\) −2.05230e140 −1.91145
\(372\) −5.76616e138 −0.0463763
\(373\) 4.82923e139 0.335542 0.167771 0.985826i \(-0.446343\pi\)
0.167771 + 0.985826i \(0.446343\pi\)
\(374\) −1.59689e139 −0.0958901
\(375\) 3.23988e140 1.68200
\(376\) 3.05946e139 0.137375
\(377\) 1.42015e140 0.551735
\(378\) 3.08060e139 0.103593
\(379\) 4.91466e140 1.43106 0.715528 0.698584i \(-0.246186\pi\)
0.715528 + 0.698584i \(0.246186\pi\)
\(380\) −6.36469e140 −1.60535
\(381\) −3.89784e140 −0.851948
\(382\) 6.50193e139 0.123194
\(383\) 4.10831e140 0.675040 0.337520 0.941318i \(-0.390412\pi\)
0.337520 + 0.941318i \(0.390412\pi\)
\(384\) −2.17422e140 −0.309922
\(385\) 1.65669e141 2.04942
\(386\) −3.70331e139 −0.0397724
\(387\) −8.19253e138 −0.00764136
\(388\) −1.53923e141 −1.24731
\(389\) −6.99680e140 −0.492772 −0.246386 0.969172i \(-0.579243\pi\)
−0.246386 + 0.969172i \(0.579243\pi\)
\(390\) 1.49941e140 0.0918121
\(391\) −2.58607e141 −1.37723
\(392\) 2.25764e140 0.104608
\(393\) −2.53648e141 −1.02291
\(394\) −1.40908e140 −0.0494758
\(395\) 7.86490e141 2.40520
\(396\) 7.41312e139 0.0197521
\(397\) −7.26167e141 −1.68637 −0.843186 0.537621i \(-0.819323\pi\)
−0.843186 + 0.537621i \(0.819323\pi\)
\(398\) 7.62225e138 0.00154332
\(399\) −6.74765e141 −1.19160
\(400\) 1.26348e142 1.94670
\(401\) 1.37737e142 1.85218 0.926090 0.377302i \(-0.123148\pi\)
0.926090 + 0.377302i \(0.123148\pi\)
\(402\) −4.42724e140 −0.0519773
\(403\) 3.10669e140 0.0318547
\(404\) −1.15636e142 −1.03587
\(405\) 2.15794e142 1.68943
\(406\) −1.22410e141 −0.0837815
\(407\) −2.60718e142 −1.56054
\(408\) −3.92121e141 −0.205324
\(409\) 8.08649e141 0.370544 0.185272 0.982687i \(-0.440683\pi\)
0.185272 + 0.982687i \(0.440683\pi\)
\(410\) 2.33988e141 0.0938585
\(411\) −1.84645e142 −0.648572
\(412\) 2.43109e142 0.748001
\(413\) −3.09564e142 −0.834588
\(414\) −7.65236e139 −0.00180833
\(415\) −2.48232e142 −0.514325
\(416\) 8.79519e141 0.159830
\(417\) −5.06875e142 −0.808139
\(418\) 4.89889e141 0.0685473
\(419\) −1.03306e143 −1.26900 −0.634501 0.772922i \(-0.718795\pi\)
−0.634501 + 0.772922i \(0.718795\pi\)
\(420\) 2.02756e143 2.18719
\(421\) −1.72030e142 −0.163016 −0.0815081 0.996673i \(-0.525974\pi\)
−0.0815081 + 0.996673i \(0.525974\pi\)
\(422\) −7.36069e141 −0.0612900
\(423\) −2.55326e141 −0.0186872
\(424\) 3.65872e142 0.235445
\(425\) 4.58671e143 2.59600
\(426\) −2.99431e141 −0.0149099
\(427\) −2.82077e143 −1.23608
\(428\) −1.45787e143 −0.562383
\(429\) 1.81055e143 0.615019
\(430\) 1.62680e142 0.0486748
\(431\) −3.29879e143 −0.869647 −0.434824 0.900516i \(-0.643189\pi\)
−0.434824 + 0.900516i \(0.643189\pi\)
\(432\) 4.26664e143 0.991338
\(433\) −3.54235e143 −0.725606 −0.362803 0.931866i \(-0.618180\pi\)
−0.362803 + 0.931866i \(0.618180\pi\)
\(434\) −2.67782e141 −0.00483716
\(435\) −8.76397e143 −1.39649
\(436\) −3.00227e143 −0.422121
\(437\) 7.93343e143 0.984516
\(438\) −9.57076e142 −0.104859
\(439\) 1.32418e143 0.128124 0.0640619 0.997946i \(-0.479594\pi\)
0.0640619 + 0.997946i \(0.479594\pi\)
\(440\) −2.95345e143 −0.252440
\(441\) −1.88410e142 −0.0142299
\(442\) 1.05298e143 0.0702920
\(443\) −2.49424e142 −0.0147209 −0.00736045 0.999973i \(-0.502343\pi\)
−0.00736045 + 0.999973i \(0.502343\pi\)
\(444\) −3.19082e144 −1.66545
\(445\) 1.66804e144 0.770167
\(446\) −3.44229e142 −0.0140636
\(447\) 2.94478e144 1.06486
\(448\) 3.87148e144 1.23944
\(449\) 5.02265e144 1.42399 0.711996 0.702184i \(-0.247792\pi\)
0.711996 + 0.702184i \(0.247792\pi\)
\(450\) 1.35724e142 0.00340859
\(451\) 2.82543e144 0.628727
\(452\) −3.27536e144 −0.645974
\(453\) −9.49614e144 −1.66033
\(454\) 5.34658e142 0.00828958
\(455\) −1.09241e145 −1.50232
\(456\) 1.20293e144 0.146777
\(457\) 2.47069e144 0.267538 0.133769 0.991013i \(-0.457292\pi\)
0.133769 + 0.991013i \(0.457292\pi\)
\(458\) 1.20705e144 0.116027
\(459\) 1.54888e145 1.32199
\(460\) −2.38386e145 −1.80709
\(461\) 9.07133e144 0.610901 0.305450 0.952208i \(-0.401193\pi\)
0.305450 + 0.952208i \(0.401193\pi\)
\(462\) −1.56061e144 −0.0933912
\(463\) −2.19315e144 −0.116655 −0.0583277 0.998297i \(-0.518577\pi\)
−0.0583277 + 0.998297i \(0.518577\pi\)
\(464\) −1.69538e145 −0.801747
\(465\) −1.91719e144 −0.0806267
\(466\) −1.07003e144 −0.0400280
\(467\) 7.75309e144 0.258051 0.129025 0.991641i \(-0.458815\pi\)
0.129025 + 0.991641i \(0.458815\pi\)
\(468\) −4.88815e143 −0.0144792
\(469\) 3.22550e145 0.850505
\(470\) 5.07004e144 0.119036
\(471\) 6.53456e145 1.36640
\(472\) 5.51872e144 0.102801
\(473\) 1.96438e145 0.326056
\(474\) −7.40876e144 −0.109604
\(475\) −1.40709e146 −1.85576
\(476\) 1.42387e146 1.67453
\(477\) −3.05337e144 −0.0320279
\(478\) −1.21732e145 −0.113916
\(479\) −1.24921e146 −1.04316 −0.521578 0.853203i \(-0.674657\pi\)
−0.521578 + 0.853203i \(0.674657\pi\)
\(480\) −5.42765e145 −0.404543
\(481\) 1.71915e146 1.14395
\(482\) 1.23658e145 0.0734780
\(483\) −2.52730e146 −1.34134
\(484\) 3.18128e145 0.150845
\(485\) −5.11777e146 −2.16848
\(486\) 9.06966e143 0.00343490
\(487\) 2.45739e146 0.832044 0.416022 0.909354i \(-0.363424\pi\)
0.416022 + 0.909354i \(0.363424\pi\)
\(488\) 5.02870e145 0.152256
\(489\) 3.97366e146 1.07611
\(490\) 3.74129e145 0.0906431
\(491\) 2.18529e146 0.473771 0.236886 0.971538i \(-0.423873\pi\)
0.236886 + 0.971538i \(0.423873\pi\)
\(492\) 3.45792e146 0.670992
\(493\) −6.15459e146 −1.06916
\(494\) −3.23028e145 −0.0502484
\(495\) 2.46479e145 0.0343397
\(496\) −3.70878e145 −0.0462892
\(497\) 2.18153e146 0.243971
\(498\) 2.33836e145 0.0234375
\(499\) 1.46230e147 1.31388 0.656942 0.753941i \(-0.271850\pi\)
0.656942 + 0.753941i \(0.271850\pi\)
\(500\) 2.09734e147 1.68968
\(501\) −2.11689e147 −1.52947
\(502\) 1.23341e146 0.0799384
\(503\) −8.66865e146 −0.504076 −0.252038 0.967717i \(-0.581101\pi\)
−0.252038 + 0.967717i \(0.581101\pi\)
\(504\) 8.45354e144 0.00441138
\(505\) −3.84477e147 −1.80090
\(506\) 1.83486e146 0.0771612
\(507\) 1.42550e147 0.538311
\(508\) −2.52327e147 −0.855838
\(509\) −3.08181e147 −0.939046 −0.469523 0.882920i \(-0.655574\pi\)
−0.469523 + 0.882920i \(0.655574\pi\)
\(510\) −6.49810e146 −0.177915
\(511\) 6.97284e147 1.71582
\(512\) −1.75367e147 −0.387914
\(513\) −4.75159e147 −0.945024
\(514\) 5.35322e146 0.0957470
\(515\) 8.08312e147 1.30042
\(516\) 2.40412e147 0.347974
\(517\) 6.12212e147 0.797383
\(518\) −1.48183e147 −0.173710
\(519\) −3.37585e147 −0.356256
\(520\) 1.94748e147 0.185051
\(521\) 5.80957e147 0.497151 0.248575 0.968613i \(-0.420038\pi\)
0.248575 + 0.968613i \(0.420038\pi\)
\(522\) −1.82119e145 −0.00140382
\(523\) −1.36517e148 −0.948078 −0.474039 0.880504i \(-0.657204\pi\)
−0.474039 + 0.880504i \(0.657204\pi\)
\(524\) −1.64199e148 −1.02758
\(525\) 4.48248e148 2.52835
\(526\) −1.41730e147 −0.0720675
\(527\) −1.34637e147 −0.0617283
\(528\) −2.16145e148 −0.893708
\(529\) 2.90201e147 0.108234
\(530\) 6.06312e147 0.204015
\(531\) −4.60562e146 −0.0139842
\(532\) −4.36810e148 −1.19704
\(533\) −1.86306e148 −0.460887
\(534\) −1.57129e147 −0.0350961
\(535\) −4.84726e148 −0.977722
\(536\) −5.75023e147 −0.104762
\(537\) −2.91981e148 −0.480569
\(538\) 1.40157e147 0.0208440
\(539\) 4.51763e148 0.607188
\(540\) 1.42777e149 1.73460
\(541\) 1.34934e149 1.48208 0.741039 0.671462i \(-0.234333\pi\)
0.741039 + 0.671462i \(0.234333\pi\)
\(542\) 1.43055e148 0.142083
\(543\) −5.76721e148 −0.518054
\(544\) −3.81163e148 −0.309721
\(545\) −9.98226e148 −0.733870
\(546\) 1.02905e148 0.0684601
\(547\) 1.50039e149 0.903433 0.451717 0.892162i \(-0.350812\pi\)
0.451717 + 0.892162i \(0.350812\pi\)
\(548\) −1.19530e149 −0.651533
\(549\) −4.19668e147 −0.0207116
\(550\) −3.25434e148 −0.145444
\(551\) 1.88808e149 0.764291
\(552\) 4.50552e148 0.165221
\(553\) 5.39770e149 1.79345
\(554\) −4.89967e148 −0.147531
\(555\) −1.06092e150 −2.89543
\(556\) −3.28126e149 −0.811829
\(557\) −3.51708e149 −0.788994 −0.394497 0.918897i \(-0.629081\pi\)
−0.394497 + 0.918897i \(0.629081\pi\)
\(558\) −3.98399e145 −8.10503e−5 0
\(559\) −1.29529e149 −0.239015
\(560\) 1.30412e150 2.18308
\(561\) −7.84650e149 −1.19179
\(562\) 2.62394e148 0.0361680
\(563\) −4.35628e148 −0.0545014 −0.0272507 0.999629i \(-0.508675\pi\)
−0.0272507 + 0.999629i \(0.508675\pi\)
\(564\) 7.49261e149 0.850985
\(565\) −1.08903e150 −1.12305
\(566\) 3.06744e147 0.00287264
\(567\) 1.48100e150 1.25973
\(568\) −3.88910e148 −0.0300514
\(569\) 3.28942e149 0.230941 0.115470 0.993311i \(-0.463162\pi\)
0.115470 + 0.993311i \(0.463162\pi\)
\(570\) 1.99346e149 0.127183
\(571\) 3.01006e149 0.174545 0.0872725 0.996184i \(-0.472185\pi\)
0.0872725 + 0.996184i \(0.472185\pi\)
\(572\) 1.17206e150 0.617828
\(573\) 3.19479e150 1.53114
\(574\) 1.60587e149 0.0699860
\(575\) −5.27020e150 −2.08896
\(576\) 5.75990e148 0.0207677
\(577\) 1.14168e150 0.374511 0.187255 0.982311i \(-0.440041\pi\)
0.187255 + 0.982311i \(0.440041\pi\)
\(578\) −1.89710e149 −0.0566268
\(579\) −1.81966e150 −0.494320
\(580\) −5.67337e150 −1.40286
\(581\) −1.70363e150 −0.383508
\(582\) 4.82095e149 0.0988167
\(583\) 7.32127e150 1.36663
\(584\) −1.24308e150 −0.211348
\(585\) −1.62526e149 −0.0251726
\(586\) −4.99102e149 −0.0704318
\(587\) −7.65183e150 −0.983986 −0.491993 0.870599i \(-0.663731\pi\)
−0.491993 + 0.870599i \(0.663731\pi\)
\(588\) 5.52895e150 0.648005
\(589\) 4.13033e149 0.0441266
\(590\) 9.14545e149 0.0890779
\(591\) −6.92368e150 −0.614921
\(592\) −2.05233e151 −1.66232
\(593\) 9.96267e150 0.736029 0.368014 0.929820i \(-0.380038\pi\)
0.368014 + 0.929820i \(0.380038\pi\)
\(594\) −1.09896e150 −0.0740660
\(595\) 4.73424e151 2.91122
\(596\) 1.90630e151 1.06972
\(597\) 3.74527e149 0.0191815
\(598\) −1.20989e150 −0.0565628
\(599\) −4.15148e150 −0.177191 −0.0885955 0.996068i \(-0.528238\pi\)
−0.0885955 + 0.996068i \(0.528238\pi\)
\(600\) −7.99110e150 −0.311432
\(601\) −5.13527e150 −0.182770 −0.0913848 0.995816i \(-0.529129\pi\)
−0.0913848 + 0.995816i \(0.529129\pi\)
\(602\) 1.11648e150 0.0362946
\(603\) 4.79883e149 0.0142509
\(604\) −6.14733e151 −1.66791
\(605\) 1.05774e151 0.262248
\(606\) 3.62178e150 0.0820661
\(607\) −4.21841e150 −0.0873705 −0.0436853 0.999045i \(-0.513910\pi\)
−0.0436853 + 0.999045i \(0.513910\pi\)
\(608\) 1.16932e151 0.221405
\(609\) −6.01474e151 −1.04130
\(610\) 8.33340e150 0.131931
\(611\) −4.03687e151 −0.584519
\(612\) 2.11841e150 0.0280580
\(613\) 7.73237e150 0.0936951 0.0468475 0.998902i \(-0.485083\pi\)
0.0468475 + 0.998902i \(0.485083\pi\)
\(614\) −5.48443e149 −0.00608073
\(615\) 1.14972e152 1.16654
\(616\) −2.02696e151 −0.188233
\(617\) −2.11130e151 −0.179477 −0.0897385 0.995965i \(-0.528603\pi\)
−0.0897385 + 0.995965i \(0.528603\pi\)
\(618\) −7.61432e150 −0.0592597
\(619\) 9.83927e151 0.701169 0.350584 0.936531i \(-0.385983\pi\)
0.350584 + 0.936531i \(0.385983\pi\)
\(620\) −1.24109e151 −0.0809949
\(621\) −1.77969e152 −1.06378
\(622\) 2.57078e151 0.140763
\(623\) 1.14478e152 0.574279
\(624\) 1.42524e152 0.655130
\(625\) 2.26287e152 0.953232
\(626\) 6.52981e149 0.00252115
\(627\) 2.40712e152 0.851955
\(628\) 4.23015e152 1.37264
\(629\) −7.45040e152 −2.21676
\(630\) 1.40089e150 0.00382248
\(631\) −3.94557e152 −0.987439 −0.493719 0.869621i \(-0.664363\pi\)
−0.493719 + 0.869621i \(0.664363\pi\)
\(632\) −9.62270e151 −0.220910
\(633\) −3.61675e152 −0.761756
\(634\) −3.25850e151 −0.0629729
\(635\) −8.38962e152 −1.48790
\(636\) 8.96020e152 1.45849
\(637\) −2.97889e152 −0.445097
\(638\) 4.36678e151 0.0599011
\(639\) 3.24563e150 0.00408792
\(640\) −4.67974e152 −0.541269
\(641\) 1.84989e153 1.96510 0.982549 0.186004i \(-0.0595536\pi\)
0.982549 + 0.186004i \(0.0595536\pi\)
\(642\) 4.56613e151 0.0445543
\(643\) −5.38862e152 −0.483036 −0.241518 0.970396i \(-0.577645\pi\)
−0.241518 + 0.970396i \(0.577645\pi\)
\(644\) −1.63605e153 −1.34746
\(645\) 7.99346e152 0.604965
\(646\) 1.39993e152 0.0973719
\(647\) 7.98584e152 0.510549 0.255275 0.966869i \(-0.417834\pi\)
0.255275 + 0.966869i \(0.417834\pi\)
\(648\) −2.64024e152 −0.155169
\(649\) 1.10432e153 0.596704
\(650\) 2.14589e152 0.106618
\(651\) −1.31577e152 −0.0601197
\(652\) 2.57235e153 1.08103
\(653\) 2.62517e153 1.01482 0.507409 0.861705i \(-0.330603\pi\)
0.507409 + 0.861705i \(0.330603\pi\)
\(654\) 9.40331e151 0.0334421
\(655\) −5.45947e153 −1.78649
\(656\) 2.22413e153 0.669732
\(657\) 1.03740e152 0.0287498
\(658\) 3.47958e152 0.0887598
\(659\) −7.70190e153 −1.80860 −0.904300 0.426897i \(-0.859607\pi\)
−0.904300 + 0.426897i \(0.859607\pi\)
\(660\) −7.23299e153 −1.56377
\(661\) 9.26631e153 1.84471 0.922353 0.386348i \(-0.126264\pi\)
0.922353 + 0.386348i \(0.126264\pi\)
\(662\) −4.89868e152 −0.0898087
\(663\) 5.17392e153 0.873639
\(664\) 3.03712e152 0.0472391
\(665\) −1.45235e154 −2.08109
\(666\) −2.20463e151 −0.00291065
\(667\) 7.07172e153 0.860334
\(668\) −1.37037e154 −1.53646
\(669\) −1.69140e153 −0.174793
\(670\) −9.52910e152 −0.0907768
\(671\) 1.00626e154 0.883761
\(672\) −3.72501e153 −0.301649
\(673\) 2.43592e154 1.81904 0.909520 0.415660i \(-0.136449\pi\)
0.909520 + 0.415660i \(0.136449\pi\)
\(674\) 1.96287e153 0.135184
\(675\) 3.15649e154 2.00516
\(676\) 9.22799e153 0.540769
\(677\) −1.82490e154 −0.986634 −0.493317 0.869850i \(-0.664216\pi\)
−0.493317 + 0.869850i \(0.664216\pi\)
\(678\) 1.02586e153 0.0511766
\(679\) −3.51234e154 −1.61694
\(680\) −8.43991e153 −0.358593
\(681\) 2.62710e153 0.103029
\(682\) 9.55268e151 0.00345841
\(683\) 3.13386e154 1.04749 0.523747 0.851874i \(-0.324534\pi\)
0.523747 + 0.851874i \(0.324534\pi\)
\(684\) −6.49876e152 −0.0200573
\(685\) −3.97425e154 −1.13271
\(686\) −1.32693e153 −0.0349287
\(687\) 5.93097e154 1.44206
\(688\) 1.54633e154 0.347321
\(689\) −4.82758e154 −1.00180
\(690\) 7.46641e153 0.143165
\(691\) −8.30513e154 −1.47161 −0.735805 0.677193i \(-0.763196\pi\)
−0.735805 + 0.677193i \(0.763196\pi\)
\(692\) −2.18536e154 −0.357883
\(693\) 1.69159e153 0.0256055
\(694\) 2.00011e153 0.0279875
\(695\) −1.09099e155 −1.41139
\(696\) 1.07227e154 0.128263
\(697\) 8.07406e154 0.893112
\(698\) −3.53581e153 −0.0361716
\(699\) −5.25770e154 −0.497497
\(700\) 2.90174e155 2.53989
\(701\) −9.99671e154 −0.809516 −0.404758 0.914424i \(-0.632644\pi\)
−0.404758 + 0.914424i \(0.632644\pi\)
\(702\) 7.24642e153 0.0542939
\(703\) 2.28560e155 1.58466
\(704\) −1.38109e155 −0.886159
\(705\) 2.49122e155 1.47947
\(706\) −1.19595e153 −0.00657443
\(707\) −2.63867e155 −1.34285
\(708\) 1.35153e155 0.636816
\(709\) −1.82497e155 −0.796227 −0.398113 0.917336i \(-0.630335\pi\)
−0.398113 + 0.917336i \(0.630335\pi\)
\(710\) −6.44489e153 −0.0260397
\(711\) 8.03058e153 0.0300506
\(712\) −2.04084e154 −0.0707375
\(713\) 1.54700e154 0.0496718
\(714\) −4.45966e154 −0.132663
\(715\) 3.89699e155 1.07411
\(716\) −1.89014e155 −0.482763
\(717\) −5.98143e155 −1.41583
\(718\) 2.03841e154 0.0447207
\(719\) −5.62215e155 −1.14335 −0.571673 0.820481i \(-0.693706\pi\)
−0.571673 + 0.820481i \(0.693706\pi\)
\(720\) 1.94024e154 0.0365792
\(721\) 5.54747e155 0.969668
\(722\) 6.15712e153 0.00997930
\(723\) 6.07604e155 0.913237
\(724\) −3.73341e155 −0.520420
\(725\) −1.25426e156 −1.62168
\(726\) −9.96397e153 −0.0119505
\(727\) −1.84945e154 −0.0205787 −0.0102893 0.999947i \(-0.503275\pi\)
−0.0102893 + 0.999947i \(0.503275\pi\)
\(728\) 1.33656e155 0.137984
\(729\) 1.06543e156 1.02064
\(730\) −2.05999e155 −0.183134
\(731\) 5.61349e155 0.463165
\(732\) 1.23153e156 0.943169
\(733\) 8.12603e155 0.577713 0.288856 0.957372i \(-0.406725\pi\)
0.288856 + 0.957372i \(0.406725\pi\)
\(734\) 1.31313e155 0.0866707
\(735\) 1.83832e156 1.12658
\(736\) 4.37962e155 0.249227
\(737\) −1.15065e156 −0.608084
\(738\) 2.38917e153 0.00117267
\(739\) 1.31041e156 0.597426 0.298713 0.954343i \(-0.403443\pi\)
0.298713 + 0.954343i \(0.403443\pi\)
\(740\) −6.86785e156 −2.90866
\(741\) −1.58723e156 −0.624523
\(742\) 4.16113e155 0.152124
\(743\) −6.49975e155 −0.220804 −0.110402 0.993887i \(-0.535214\pi\)
−0.110402 + 0.993887i \(0.535214\pi\)
\(744\) 2.34568e154 0.00740532
\(745\) 6.33827e156 1.85975
\(746\) −9.79149e154 −0.0267044
\(747\) −2.53462e154 −0.00642598
\(748\) −5.07944e156 −1.19723
\(749\) −3.32669e156 −0.729043
\(750\) −6.56901e155 −0.133863
\(751\) −4.15514e156 −0.787425 −0.393713 0.919234i \(-0.628809\pi\)
−0.393713 + 0.919234i \(0.628809\pi\)
\(752\) 4.81923e156 0.849387
\(753\) 6.06049e156 0.993532
\(754\) −2.87942e155 −0.0439103
\(755\) −2.04393e157 −2.89972
\(756\) 9.79885e156 1.29341
\(757\) 1.53128e157 1.88074 0.940370 0.340154i \(-0.110479\pi\)
0.940370 + 0.340154i \(0.110479\pi\)
\(758\) −9.96469e155 −0.113892
\(759\) 9.01575e156 0.959015
\(760\) 2.58916e156 0.256341
\(761\) −9.88537e156 −0.911022 −0.455511 0.890230i \(-0.650543\pi\)
−0.455511 + 0.890230i \(0.650543\pi\)
\(762\) 7.90305e155 0.0678029
\(763\) −6.85085e156 −0.547214
\(764\) 2.06815e157 1.53813
\(765\) 7.04349e155 0.0487798
\(766\) −8.32977e155 −0.0537236
\(767\) −7.28179e156 −0.437412
\(768\) −1.65695e157 −0.927091
\(769\) 2.54160e157 1.32471 0.662356 0.749190i \(-0.269557\pi\)
0.662356 + 0.749190i \(0.269557\pi\)
\(770\) −3.35902e156 −0.163105
\(771\) 2.63036e157 1.19001
\(772\) −1.17796e157 −0.496577
\(773\) −4.34579e157 −1.70721 −0.853604 0.520922i \(-0.825588\pi\)
−0.853604 + 0.520922i \(0.825588\pi\)
\(774\) 1.66107e154 0.000608143 0
\(775\) −2.74379e156 −0.0936284
\(776\) 6.26159e156 0.199169
\(777\) −7.28110e157 −2.15899
\(778\) 1.41863e156 0.0392177
\(779\) −2.47693e157 −0.638443
\(780\) 4.76937e157 1.14632
\(781\) −7.78226e156 −0.174431
\(782\) 5.24337e156 0.109608
\(783\) −4.23549e157 −0.825823
\(784\) 3.55621e157 0.646788
\(785\) 1.40648e158 2.38637
\(786\) 5.14283e156 0.0814092
\(787\) 8.49250e157 1.25433 0.627166 0.778886i \(-0.284215\pi\)
0.627166 + 0.778886i \(0.284215\pi\)
\(788\) −4.48205e157 −0.617729
\(789\) −6.96406e157 −0.895707
\(790\) −1.59464e157 −0.191420
\(791\) −7.47401e157 −0.837405
\(792\) −3.01567e155 −0.00315399
\(793\) −6.63522e157 −0.647838
\(794\) 1.47233e157 0.134211
\(795\) 2.97918e158 2.53564
\(796\) 2.42450e156 0.0192691
\(797\) −4.32541e157 −0.321032 −0.160516 0.987033i \(-0.551316\pi\)
−0.160516 + 0.987033i \(0.551316\pi\)
\(798\) 1.36812e157 0.0948343
\(799\) 1.74948e158 1.13269
\(800\) −7.76779e157 −0.469778
\(801\) 1.70317e156 0.00962248
\(802\) −2.79269e157 −0.147407
\(803\) −2.48745e158 −1.22675
\(804\) −1.40823e158 −0.648961
\(805\) −5.43971e158 −2.34261
\(806\) −6.29895e155 −0.00253518
\(807\) 6.88675e157 0.259064
\(808\) 4.70407e157 0.165407
\(809\) 4.42811e158 1.45554 0.727769 0.685822i \(-0.240557\pi\)
0.727769 + 0.685822i \(0.240557\pi\)
\(810\) −4.37531e157 −0.134455
\(811\) −3.12561e158 −0.898046 −0.449023 0.893520i \(-0.648228\pi\)
−0.449023 + 0.893520i \(0.648228\pi\)
\(812\) −3.89365e158 −1.04605
\(813\) 7.02915e158 1.76591
\(814\) 5.28618e157 0.124197
\(815\) 8.55281e158 1.87940
\(816\) −6.17664e158 −1.26952
\(817\) −1.72208e158 −0.331095
\(818\) −1.63957e157 −0.0294900
\(819\) −1.11542e157 −0.0187701
\(820\) 7.44275e158 1.17187
\(821\) 5.69745e158 0.839418 0.419709 0.907659i \(-0.362132\pi\)
0.419709 + 0.907659i \(0.362132\pi\)
\(822\) 3.74376e157 0.0516171
\(823\) −4.03382e157 −0.0520505 −0.0260253 0.999661i \(-0.508285\pi\)
−0.0260253 + 0.999661i \(0.508285\pi\)
\(824\) −9.88969e157 −0.119440
\(825\) −1.59905e159 −1.80769
\(826\) 6.27654e157 0.0664214
\(827\) 9.76894e158 0.967825 0.483913 0.875116i \(-0.339215\pi\)
0.483913 + 0.875116i \(0.339215\pi\)
\(828\) −2.43408e157 −0.0225778
\(829\) 1.04176e159 0.904786 0.452393 0.891819i \(-0.350570\pi\)
0.452393 + 0.891819i \(0.350570\pi\)
\(830\) 5.03302e157 0.0409329
\(831\) −2.40750e159 −1.83363
\(832\) 9.10677e158 0.649596
\(833\) 1.29098e159 0.862515
\(834\) 1.02771e158 0.0643164
\(835\) −4.55634e159 −2.67118
\(836\) 1.55825e159 0.855845
\(837\) −9.26546e157 −0.0476793
\(838\) 2.09458e158 0.100995
\(839\) −1.51200e159 −0.683164 −0.341582 0.939852i \(-0.610963\pi\)
−0.341582 + 0.939852i \(0.610963\pi\)
\(840\) −8.24813e158 −0.349248
\(841\) −8.36890e158 −0.332113
\(842\) 3.48799e157 0.0129738
\(843\) 1.28930e159 0.449522
\(844\) −2.34130e159 −0.765235
\(845\) 3.06821e159 0.940145
\(846\) 5.17685e156 0.00148724
\(847\) 7.25932e158 0.195547
\(848\) 5.76318e159 1.45576
\(849\) 1.50722e158 0.0357032
\(850\) −9.29975e158 −0.206605
\(851\) 8.56062e159 1.78379
\(852\) −9.52439e158 −0.186157
\(853\) −3.07144e159 −0.563144 −0.281572 0.959540i \(-0.590856\pi\)
−0.281572 + 0.959540i \(0.590856\pi\)
\(854\) 5.71923e158 0.0983748
\(855\) −2.16077e158 −0.0348703
\(856\) 5.93062e158 0.0898007
\(857\) −8.89973e159 −1.26451 −0.632255 0.774761i \(-0.717870\pi\)
−0.632255 + 0.774761i \(0.717870\pi\)
\(858\) −3.67098e158 −0.0489468
\(859\) −4.57533e159 −0.572525 −0.286263 0.958151i \(-0.592413\pi\)
−0.286263 + 0.958151i \(0.592413\pi\)
\(860\) 5.17457e159 0.607728
\(861\) 7.89059e159 0.869837
\(862\) 6.68844e158 0.0692116
\(863\) −4.25641e159 −0.413481 −0.206740 0.978396i \(-0.566286\pi\)
−0.206740 + 0.978396i \(0.566286\pi\)
\(864\) −2.62310e159 −0.239230
\(865\) −7.26610e159 −0.622191
\(866\) 7.18227e158 0.0577480
\(867\) −9.32157e159 −0.703798
\(868\) −8.51766e158 −0.0603942
\(869\) −1.92554e160 −1.28226
\(870\) 1.77693e159 0.111140
\(871\) 7.58726e159 0.445754
\(872\) 1.22133e159 0.0674037
\(873\) −5.22558e158 −0.0270931
\(874\) −1.60854e159 −0.0783535
\(875\) 4.78590e160 2.19041
\(876\) −3.04429e160 −1.30922
\(877\) 4.15823e160 1.68047 0.840235 0.542222i \(-0.182417\pi\)
0.840235 + 0.542222i \(0.182417\pi\)
\(878\) −2.68483e158 −0.0101968
\(879\) −2.45239e160 −0.875377
\(880\) −4.65224e160 −1.56083
\(881\) 2.49774e160 0.787698 0.393849 0.919175i \(-0.371143\pi\)
0.393849 + 0.919175i \(0.371143\pi\)
\(882\) 3.82010e157 0.00113250
\(883\) 6.64721e159 0.185260 0.0926299 0.995701i \(-0.470473\pi\)
0.0926299 + 0.995701i \(0.470473\pi\)
\(884\) 3.34934e160 0.877628
\(885\) 4.49371e160 1.10712
\(886\) 5.05717e157 0.00117157
\(887\) 6.17454e160 1.34514 0.672570 0.740033i \(-0.265190\pi\)
0.672570 + 0.740033i \(0.265190\pi\)
\(888\) 1.29803e160 0.265937
\(889\) −5.75782e160 −1.10946
\(890\) −3.38202e159 −0.0612944
\(891\) −5.28323e160 −0.900667
\(892\) −1.09493e160 −0.175591
\(893\) −5.36700e160 −0.809705
\(894\) −5.97066e159 −0.0847477
\(895\) −6.28452e160 −0.839300
\(896\) −3.21172e160 −0.403600
\(897\) −5.94491e160 −0.703003
\(898\) −1.01836e160 −0.113329
\(899\) 3.68170e159 0.0385607
\(900\) 4.31714e159 0.0425578
\(901\) 2.09216e161 1.94130
\(902\) −5.72868e159 −0.0500378
\(903\) 5.48593e160 0.451095
\(904\) 1.33242e160 0.103148
\(905\) −1.24132e161 −0.904766
\(906\) 1.92538e160 0.132139
\(907\) −3.57082e160 −0.230765 −0.115383 0.993321i \(-0.536809\pi\)
−0.115383 + 0.993321i \(0.536809\pi\)
\(908\) 1.70065e160 0.103499
\(909\) −3.92576e159 −0.0225005
\(910\) 2.21490e160 0.119564
\(911\) −2.92874e161 −1.48912 −0.744559 0.667556i \(-0.767340\pi\)
−0.744559 + 0.667556i \(0.767340\pi\)
\(912\) 1.89484e161 0.907517
\(913\) 6.07741e160 0.274196
\(914\) −5.00944e159 −0.0212923
\(915\) 4.09470e161 1.63973
\(916\) 3.83942e161 1.44865
\(917\) −3.74685e161 −1.33210
\(918\) −3.14042e160 −0.105211
\(919\) 4.40405e161 1.39046 0.695228 0.718789i \(-0.255303\pi\)
0.695228 + 0.718789i \(0.255303\pi\)
\(920\) 9.69759e160 0.288554
\(921\) −2.69483e160 −0.0755756
\(922\) −1.83925e160 −0.0486190
\(923\) 5.13155e160 0.127866
\(924\) −4.96402e161 −1.16603
\(925\) −1.51833e162 −3.36234
\(926\) 4.44672e159 0.00928412
\(927\) 8.25340e159 0.0162475
\(928\) 1.04231e161 0.193478
\(929\) 6.33862e161 1.10953 0.554764 0.832008i \(-0.312809\pi\)
0.554764 + 0.832008i \(0.312809\pi\)
\(930\) 3.88719e159 0.00641674
\(931\) −3.96041e161 −0.616571
\(932\) −3.40358e161 −0.499768
\(933\) 1.26318e162 1.74951
\(934\) −1.57197e160 −0.0205372
\(935\) −1.68886e162 −2.08143
\(936\) 1.98850e159 0.00231203
\(937\) 1.10322e162 1.21019 0.605095 0.796153i \(-0.293135\pi\)
0.605095 + 0.796153i \(0.293135\pi\)
\(938\) −6.53984e160 −0.0676882
\(939\) 3.20849e160 0.0313347
\(940\) 1.61269e162 1.48622
\(941\) −8.53090e161 −0.741925 −0.370963 0.928648i \(-0.620972\pi\)
−0.370963 + 0.928648i \(0.620972\pi\)
\(942\) −1.32491e161 −0.108746
\(943\) −9.27722e161 −0.718672
\(944\) 8.69303e161 0.635620
\(945\) 3.25802e162 2.24864
\(946\) −3.98286e160 −0.0259495
\(947\) −5.35159e161 −0.329162 −0.164581 0.986364i \(-0.552627\pi\)
−0.164581 + 0.986364i \(0.552627\pi\)
\(948\) −2.35659e162 −1.36846
\(949\) 1.64020e162 0.899268
\(950\) 2.85294e161 0.147692
\(951\) −1.60110e162 −0.782672
\(952\) −5.79233e161 −0.267387
\(953\) −1.03103e162 −0.449477 −0.224739 0.974419i \(-0.572153\pi\)
−0.224739 + 0.974419i \(0.572153\pi\)
\(954\) 6.19084e159 0.00254896
\(955\) 6.87639e162 2.67409
\(956\) −3.87209e162 −1.42230
\(957\) 2.14566e162 0.744493
\(958\) 2.53283e161 0.0830205
\(959\) −2.72754e162 −0.844612
\(960\) −5.61994e162 −1.64418
\(961\) −3.60958e162 −0.997774
\(962\) −3.48566e161 −0.0910423
\(963\) −4.94937e160 −0.0122157
\(964\) 3.93333e162 0.917407
\(965\) −3.91659e162 −0.863315
\(966\) 5.12422e161 0.106752
\(967\) −3.03077e162 −0.596774 −0.298387 0.954445i \(-0.596449\pi\)
−0.298387 + 0.954445i \(0.596449\pi\)
\(968\) −1.29415e161 −0.0240867
\(969\) 6.87869e162 1.21021
\(970\) 1.03765e162 0.172580
\(971\) 6.89509e162 1.08416 0.542078 0.840328i \(-0.317638\pi\)
0.542078 + 0.840328i \(0.317638\pi\)
\(972\) 2.88490e161 0.0428864
\(973\) −7.48747e162 −1.05241
\(974\) −4.98248e161 −0.0662189
\(975\) 1.05440e163 1.32512
\(976\) 7.92115e162 0.941398
\(977\) −6.84252e162 −0.769064 −0.384532 0.923112i \(-0.625637\pi\)
−0.384532 + 0.923112i \(0.625637\pi\)
\(978\) −8.05677e161 −0.0856433
\(979\) −4.08381e162 −0.410591
\(980\) 1.19004e163 1.13172
\(981\) −1.01925e161 −0.00916899
\(982\) −4.43078e161 −0.0377055
\(983\) 2.23686e162 0.180083 0.0900415 0.995938i \(-0.471300\pi\)
0.0900415 + 0.995938i \(0.471300\pi\)
\(984\) −1.40669e162 −0.107143
\(985\) −1.49024e163 −1.07394
\(986\) 1.24787e162 0.0850899
\(987\) 1.70973e163 1.10317
\(988\) −1.02750e163 −0.627375
\(989\) −6.44999e162 −0.372701
\(990\) −4.99747e160 −0.00273295
\(991\) −9.24985e162 −0.478763 −0.239382 0.970926i \(-0.576945\pi\)
−0.239382 + 0.970926i \(0.576945\pi\)
\(992\) 2.28013e161 0.0111705
\(993\) −2.40702e163 −1.11621
\(994\) −4.42315e161 −0.0194166
\(995\) 8.06124e161 0.0334999
\(996\) 7.43790e162 0.292628
\(997\) 1.90575e163 0.709873 0.354937 0.934890i \(-0.384502\pi\)
0.354937 + 0.934890i \(0.384502\pi\)
\(998\) −2.96488e162 −0.104566
\(999\) −5.12724e163 −1.71224
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 1.110.a.a.1.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.110.a.a.1.4 8 1.1 even 1 trivial