Properties

Label 1.88.a.a.1.7
Level $1$
Weight $88$
Character 1.1
Self dual yes
Analytic conductor $47.933$
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] = [1,88,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, 88, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1.1");
 
S:= CuspForms(chi, 88);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 88 \)
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: \(47.9333631461\)
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: \( x^{7} - x^{6} + \cdots - 79\!\cdots\!56 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: multiple of \( 2^{76}\cdot 3^{35}\cdot 5^{8}\cdot 7^{4}\cdot 11^{2}\cdot 13\cdot 17\cdot 29^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(2.68228e11\) 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)
 
\(f(q)\) \(=\) \(q+2.19120e13 q^{2} +9.03577e20 q^{3} +3.25393e26 q^{4} -2.93989e30 q^{5} +1.97992e34 q^{6} +7.30934e36 q^{7} +3.73929e39 q^{8} +4.93193e41 q^{9} +O(q^{10})\) \(q+2.19120e13 q^{2} +9.03577e20 q^{3} +3.25393e26 q^{4} -2.93989e30 q^{5} +1.97992e34 q^{6} +7.30934e36 q^{7} +3.73929e39 q^{8} +4.93193e41 q^{9} -6.44188e43 q^{10} -1.23524e45 q^{11} +2.94018e47 q^{12} +1.56397e48 q^{13} +1.60162e50 q^{14} -2.65641e51 q^{15} +3.15832e52 q^{16} +4.26647e53 q^{17} +1.08069e55 q^{18} -1.72994e55 q^{19} -9.56618e56 q^{20} +6.60456e57 q^{21} -2.70667e58 q^{22} +1.74415e59 q^{23} +3.37874e60 q^{24} +2.18058e60 q^{25} +3.42697e61 q^{26} +1.53550e62 q^{27} +2.37841e63 q^{28} +1.40898e63 q^{29} -5.82073e64 q^{30} -6.33280e64 q^{31} +1.13424e65 q^{32} -1.11614e66 q^{33} +9.34869e66 q^{34} -2.14886e67 q^{35} +1.60482e68 q^{36} -2.97743e68 q^{37} -3.79065e68 q^{38} +1.41317e69 q^{39} -1.09931e70 q^{40} -4.83288e69 q^{41} +1.44719e71 q^{42} +1.18830e71 q^{43} -4.01940e71 q^{44} -1.44993e72 q^{45} +3.82179e72 q^{46} -2.96099e72 q^{47} +2.85379e73 q^{48} +2.00432e73 q^{49} +4.77809e73 q^{50} +3.85509e74 q^{51} +5.08905e74 q^{52} -1.89528e75 q^{53} +3.36458e75 q^{54} +3.63148e75 q^{55} +2.73318e76 q^{56} -1.56313e76 q^{57} +3.08735e76 q^{58} -2.41334e76 q^{59} -8.64378e77 q^{60} +4.87105e77 q^{61} -1.38764e78 q^{62} +3.60492e78 q^{63} -2.40192e78 q^{64} -4.59790e78 q^{65} -2.44568e79 q^{66} -1.66474e79 q^{67} +1.38828e80 q^{68} +1.57598e80 q^{69} -4.70859e80 q^{70} +5.91466e80 q^{71} +1.84419e81 q^{72} -3.16168e80 q^{73} -6.52415e81 q^{74} +1.97032e81 q^{75} -5.62911e81 q^{76} -9.02883e81 q^{77} +3.09653e82 q^{78} +9.23615e81 q^{79} -9.28511e82 q^{80} -2.06846e82 q^{81} -1.05898e83 q^{82} -2.59548e83 q^{83} +2.14908e84 q^{84} -1.25429e84 q^{85} +2.60381e84 q^{86} +1.27312e84 q^{87} -4.61894e84 q^{88} -1.03166e84 q^{89} -3.17709e85 q^{90} +1.14316e85 q^{91} +5.67536e85 q^{92} -5.72217e85 q^{93} -6.48811e85 q^{94} +5.08583e85 q^{95} +1.02487e86 q^{96} +1.01460e85 q^{97} +4.39187e86 q^{98} -6.09214e86 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 18197022042936 q^{2} - 75\!\cdots\!48 q^{3}+ \cdots + 67\!\cdots\!39 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 18197022042936 q^{2} - 75\!\cdots\!48 q^{3}+ \cdots + 15\!\cdots\!08 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.19120e13 1.76148 0.880739 0.473602i \(-0.157047\pi\)
0.880739 + 0.473602i \(0.157047\pi\)
\(3\) 9.03577e20 1.58924 0.794622 0.607105i \(-0.207669\pi\)
0.794622 + 0.607105i \(0.207669\pi\)
\(4\) 3.25393e26 2.10280
\(5\) −2.93989e30 −1.15647 −0.578236 0.815869i \(-0.696259\pi\)
−0.578236 + 0.815869i \(0.696259\pi\)
\(6\) 1.97992e34 2.79942
\(7\) 7.30934e36 1.26507 0.632534 0.774533i \(-0.282015\pi\)
0.632534 + 0.774533i \(0.282015\pi\)
\(8\) 3.73929e39 1.94256
\(9\) 4.93193e41 1.52570
\(10\) −6.44188e43 −2.03710
\(11\) −1.23524e45 −0.618260 −0.309130 0.951020i \(-0.600038\pi\)
−0.309130 + 0.951020i \(0.600038\pi\)
\(12\) 2.94018e47 3.34187
\(13\) 1.56397e48 0.546629 0.273315 0.961925i \(-0.411880\pi\)
0.273315 + 0.961925i \(0.411880\pi\)
\(14\) 1.60162e50 2.22839
\(15\) −2.65641e51 −1.83792
\(16\) 3.15832e52 1.31898
\(17\) 4.26647e53 1.27509 0.637545 0.770413i \(-0.279950\pi\)
0.637545 + 0.770413i \(0.279950\pi\)
\(18\) 1.08069e55 2.68748
\(19\) −1.72994e55 −0.409496 −0.204748 0.978815i \(-0.565637\pi\)
−0.204748 + 0.978815i \(0.565637\pi\)
\(20\) −9.56618e56 −2.43183
\(21\) 6.60456e57 2.01050
\(22\) −2.70667e58 −1.08905
\(23\) 1.74415e59 1.01490 0.507451 0.861681i \(-0.330588\pi\)
0.507451 + 0.861681i \(0.330588\pi\)
\(24\) 3.37874e60 3.08721
\(25\) 2.18058e60 0.337428
\(26\) 3.42697e61 0.962875
\(27\) 1.53550e62 0.835460
\(28\) 2.37841e63 2.66019
\(29\) 1.40898e63 0.342445 0.171223 0.985232i \(-0.445228\pi\)
0.171223 + 0.985232i \(0.445228\pi\)
\(30\) −5.82073e64 −3.23745
\(31\) −6.33280e64 −0.845984 −0.422992 0.906134i \(-0.639020\pi\)
−0.422992 + 0.906134i \(0.639020\pi\)
\(32\) 1.13424e65 0.380786
\(33\) −1.11614e66 −0.982566
\(34\) 9.34869e66 2.24604
\(35\) −2.14886e67 −1.46302
\(36\) 1.60482e68 3.20824
\(37\) −2.97743e68 −1.80745 −0.903727 0.428110i \(-0.859180\pi\)
−0.903727 + 0.428110i \(0.859180\pi\)
\(38\) −3.79065e68 −0.721318
\(39\) 1.41317e69 0.868727
\(40\) −1.09931e70 −2.24652
\(41\) −4.83288e69 −0.337371 −0.168686 0.985670i \(-0.553952\pi\)
−0.168686 + 0.985670i \(0.553952\pi\)
\(42\) 1.44719e71 3.54145
\(43\) 1.18830e71 1.04484 0.522419 0.852689i \(-0.325030\pi\)
0.522419 + 0.852689i \(0.325030\pi\)
\(44\) −4.01940e71 −1.30008
\(45\) −1.44993e72 −1.76443
\(46\) 3.82179e72 1.78773
\(47\) −2.96099e72 −0.543475 −0.271737 0.962371i \(-0.587598\pi\)
−0.271737 + 0.962371i \(0.587598\pi\)
\(48\) 2.85379e73 2.09618
\(49\) 2.00432e73 0.600396
\(50\) 4.77809e73 0.594373
\(51\) 3.85509e74 2.02643
\(52\) 5.08905e74 1.14945
\(53\) −1.89528e75 −1.86927 −0.934635 0.355609i \(-0.884273\pi\)
−0.934635 + 0.355609i \(0.884273\pi\)
\(54\) 3.36458e75 1.47164
\(55\) 3.63148e75 0.715000
\(56\) 2.73318e76 2.45747
\(57\) −1.56313e76 −0.650789
\(58\) 3.08735e76 0.603210
\(59\) −2.41334e76 −0.224159 −0.112080 0.993699i \(-0.535751\pi\)
−0.112080 + 0.993699i \(0.535751\pi\)
\(60\) −8.64378e77 −3.86478
\(61\) 4.87105e77 1.06114 0.530572 0.847640i \(-0.321977\pi\)
0.530572 + 0.847640i \(0.321977\pi\)
\(62\) −1.38764e78 −1.49018
\(63\) 3.60492e78 1.93011
\(64\) −2.40192e78 −0.648231
\(65\) −4.59790e78 −0.632162
\(66\) −2.44568e79 −1.73077
\(67\) −1.66474e79 −0.612482 −0.306241 0.951954i \(-0.599071\pi\)
−0.306241 + 0.951954i \(0.599071\pi\)
\(68\) 1.38828e80 2.68126
\(69\) 1.57598e80 1.61293
\(70\) −4.70859e80 −2.57707
\(71\) 5.91466e80 1.74659 0.873297 0.487188i \(-0.161977\pi\)
0.873297 + 0.487188i \(0.161977\pi\)
\(72\) 1.84419e81 2.96376
\(73\) −3.16168e80 −0.278852 −0.139426 0.990233i \(-0.544526\pi\)
−0.139426 + 0.990233i \(0.544526\pi\)
\(74\) −6.52415e81 −3.18379
\(75\) 1.97032e81 0.536256
\(76\) −5.62911e81 −0.861089
\(77\) −9.02883e81 −0.782140
\(78\) 3.09653e82 1.53024
\(79\) 9.23615e81 0.262248 0.131124 0.991366i \(-0.458141\pi\)
0.131124 + 0.991366i \(0.458141\pi\)
\(80\) −9.28511e82 −1.52536
\(81\) −2.06846e82 −0.197947
\(82\) −1.05898e83 −0.594272
\(83\) −2.59548e83 −0.859649 −0.429824 0.902912i \(-0.641425\pi\)
−0.429824 + 0.902912i \(0.641425\pi\)
\(84\) 2.14908e84 4.22769
\(85\) −1.25429e84 −1.47461
\(86\) 2.60381e84 1.84046
\(87\) 1.27312e84 0.544229
\(88\) −4.61894e84 −1.20101
\(89\) −1.03166e84 −0.164086 −0.0820432 0.996629i \(-0.526145\pi\)
−0.0820432 + 0.996629i \(0.526145\pi\)
\(90\) −3.17709e85 −3.10800
\(91\) 1.14316e85 0.691523
\(92\) 5.67536e85 2.13414
\(93\) −5.72217e85 −1.34447
\(94\) −6.48811e85 −0.957319
\(95\) 5.08583e85 0.473571
\(96\) 1.02487e86 0.605161
\(97\) 1.01460e85 0.0381701 0.0190851 0.999818i \(-0.493925\pi\)
0.0190851 + 0.999818i \(0.493925\pi\)
\(98\) 4.39187e86 1.05758
\(99\) −6.09214e86 −0.943277
\(100\) 7.09545e86 0.709545
\(101\) −1.74397e87 −1.13125 −0.565627 0.824661i \(-0.691366\pi\)
−0.565627 + 0.824661i \(0.691366\pi\)
\(102\) 8.44726e87 3.56951
\(103\) −5.83129e87 −1.61193 −0.805964 0.591965i \(-0.798352\pi\)
−0.805964 + 0.591965i \(0.798352\pi\)
\(104\) 5.84815e87 1.06186
\(105\) −1.94166e88 −2.32509
\(106\) −4.15293e88 −3.29268
\(107\) 2.16890e87 0.114300 0.0571498 0.998366i \(-0.481799\pi\)
0.0571498 + 0.998366i \(0.481799\pi\)
\(108\) 4.99640e88 1.75681
\(109\) 7.27972e88 1.71420 0.857101 0.515148i \(-0.172263\pi\)
0.857101 + 0.515148i \(0.172263\pi\)
\(110\) 7.95729e88 1.25946
\(111\) −2.69034e89 −2.87248
\(112\) 2.30853e89 1.66859
\(113\) 2.71121e89 1.33122 0.665611 0.746299i \(-0.268171\pi\)
0.665611 + 0.746299i \(0.268171\pi\)
\(114\) −3.42514e89 −1.14635
\(115\) −5.12762e89 −1.17371
\(116\) 4.58471e89 0.720095
\(117\) 7.71340e89 0.833991
\(118\) −5.28812e89 −0.394852
\(119\) 3.11851e90 1.61308
\(120\) −9.93310e90 −3.57027
\(121\) −2.46592e90 −0.617755
\(122\) 1.06734e91 1.86918
\(123\) −4.36688e90 −0.536165
\(124\) −2.06065e91 −1.77894
\(125\) 1.25879e91 0.766246
\(126\) 7.89910e91 3.39984
\(127\) −4.43227e89 −0.0135258 −0.00676292 0.999977i \(-0.502153\pi\)
−0.00676292 + 0.999977i \(0.502153\pi\)
\(128\) −7.01824e91 −1.52263
\(129\) 1.07372e92 1.66050
\(130\) −1.00749e92 −1.11354
\(131\) 1.52151e92 1.20496 0.602482 0.798133i \(-0.294179\pi\)
0.602482 + 0.798133i \(0.294179\pi\)
\(132\) −3.63184e92 −2.06614
\(133\) −1.26447e92 −0.518040
\(134\) −3.64778e92 −1.07887
\(135\) −4.51419e92 −0.966186
\(136\) 1.59536e93 2.47694
\(137\) −2.51594e92 −0.284025 −0.142012 0.989865i \(-0.545357\pi\)
−0.142012 + 0.989865i \(0.545357\pi\)
\(138\) 3.45328e93 2.84113
\(139\) −1.65051e93 −0.991912 −0.495956 0.868348i \(-0.665182\pi\)
−0.495956 + 0.868348i \(0.665182\pi\)
\(140\) −6.99225e93 −3.07643
\(141\) −2.67548e93 −0.863714
\(142\) 1.29602e94 3.07659
\(143\) −1.93189e93 −0.337959
\(144\) 1.55766e94 2.01236
\(145\) −4.14223e93 −0.396029
\(146\) −6.92787e93 −0.491191
\(147\) 1.81106e94 0.954176
\(148\) −9.68836e94 −3.80072
\(149\) −1.58122e94 −0.462796 −0.231398 0.972859i \(-0.574330\pi\)
−0.231398 + 0.972859i \(0.574330\pi\)
\(150\) 4.31737e94 0.944603
\(151\) 8.62448e93 0.141330 0.0706652 0.997500i \(-0.477488\pi\)
0.0706652 + 0.997500i \(0.477488\pi\)
\(152\) −6.46875e94 −0.795471
\(153\) 2.10420e95 1.94540
\(154\) −1.97840e95 −1.37772
\(155\) 1.86177e95 0.978357
\(156\) 4.59835e95 1.82676
\(157\) 1.43573e95 0.431955 0.215977 0.976398i \(-0.430706\pi\)
0.215977 + 0.976398i \(0.430706\pi\)
\(158\) 2.02382e95 0.461944
\(159\) −1.71253e96 −2.97073
\(160\) −3.33453e95 −0.440368
\(161\) 1.27486e96 1.28392
\(162\) −4.53240e95 −0.348679
\(163\) −3.09568e95 −0.182220 −0.0911099 0.995841i \(-0.529041\pi\)
−0.0911099 + 0.995841i \(0.529041\pi\)
\(164\) −1.57259e96 −0.709425
\(165\) 3.28132e96 1.13631
\(166\) −5.68722e96 −1.51425
\(167\) −4.38882e96 −0.899873 −0.449936 0.893061i \(-0.648553\pi\)
−0.449936 + 0.893061i \(0.648553\pi\)
\(168\) 2.46964e97 3.90552
\(169\) −5.73999e96 −0.701196
\(170\) −2.74841e97 −2.59749
\(171\) −8.53195e96 −0.624767
\(172\) 3.86665e97 2.19709
\(173\) 1.65137e97 0.729187 0.364594 0.931167i \(-0.381208\pi\)
0.364594 + 0.931167i \(0.381208\pi\)
\(174\) 2.78966e97 0.958647
\(175\) 1.59386e97 0.426870
\(176\) −3.90130e97 −0.815470
\(177\) −2.18064e97 −0.356244
\(178\) −2.26058e97 −0.289034
\(179\) −1.28269e97 −0.128533 −0.0642666 0.997933i \(-0.520471\pi\)
−0.0642666 + 0.997933i \(0.520471\pi\)
\(180\) −4.71798e98 −3.71024
\(181\) −9.38672e97 −0.580091 −0.290045 0.957013i \(-0.593670\pi\)
−0.290045 + 0.957013i \(0.593670\pi\)
\(182\) 2.50489e98 1.21810
\(183\) 4.40137e98 1.68642
\(184\) 6.52190e98 1.97151
\(185\) 8.75331e98 2.09027
\(186\) −1.25384e99 −2.36826
\(187\) −5.27014e98 −0.788337
\(188\) −9.63484e98 −1.14282
\(189\) 1.12235e99 1.05691
\(190\) 1.11441e99 0.834184
\(191\) 1.32496e99 0.789316 0.394658 0.918828i \(-0.370863\pi\)
0.394658 + 0.918828i \(0.370863\pi\)
\(192\) −2.17032e99 −1.03020
\(193\) −2.26498e99 −0.857672 −0.428836 0.903382i \(-0.641076\pi\)
−0.428836 + 0.903382i \(0.641076\pi\)
\(194\) 2.22319e98 0.0672358
\(195\) −4.15455e99 −1.00466
\(196\) 6.52192e99 1.26251
\(197\) −5.25147e98 −0.0814705 −0.0407353 0.999170i \(-0.512970\pi\)
−0.0407353 + 0.999170i \(0.512970\pi\)
\(198\) −1.33491e100 −1.66156
\(199\) 1.59307e100 1.59267 0.796335 0.604856i \(-0.206769\pi\)
0.796335 + 0.604856i \(0.206769\pi\)
\(200\) 8.15383e99 0.655476
\(201\) −1.50422e100 −0.973383
\(202\) −3.82139e100 −1.99268
\(203\) 1.02987e100 0.433216
\(204\) 1.25442e101 4.26118
\(205\) 1.42081e100 0.390161
\(206\) −1.27775e101 −2.83937
\(207\) 8.60206e100 1.54843
\(208\) 4.93953e100 0.720991
\(209\) 2.13690e100 0.253175
\(210\) −4.25457e101 −4.09559
\(211\) 2.48888e101 1.94857 0.974287 0.225311i \(-0.0723398\pi\)
0.974287 + 0.225311i \(0.0723398\pi\)
\(212\) −6.16709e101 −3.93071
\(213\) 5.34436e101 2.77576
\(214\) 4.75249e100 0.201336
\(215\) −3.49347e101 −1.20833
\(216\) 5.74168e101 1.62293
\(217\) −4.62886e101 −1.07023
\(218\) 1.59513e102 3.01953
\(219\) −2.85682e101 −0.443163
\(220\) 1.18166e102 1.50350
\(221\) 6.67264e101 0.697002
\(222\) −5.89507e102 −5.05982
\(223\) −3.78800e100 −0.0267392 −0.0133696 0.999911i \(-0.504256\pi\)
−0.0133696 + 0.999911i \(0.504256\pi\)
\(224\) 8.29055e101 0.481720
\(225\) 1.07545e102 0.514813
\(226\) 5.94079e102 2.34492
\(227\) −8.50779e101 −0.277135 −0.138568 0.990353i \(-0.544250\pi\)
−0.138568 + 0.990353i \(0.544250\pi\)
\(228\) −5.08633e102 −1.36848
\(229\) 6.43959e102 1.43224 0.716119 0.697979i \(-0.245917\pi\)
0.716119 + 0.697979i \(0.245917\pi\)
\(230\) −1.12356e103 −2.06746
\(231\) −8.15824e102 −1.24301
\(232\) 5.26858e102 0.665221
\(233\) −9.54865e102 −0.999907 −0.499954 0.866052i \(-0.666650\pi\)
−0.499954 + 0.866052i \(0.666650\pi\)
\(234\) 1.69016e103 1.46906
\(235\) 8.70496e102 0.628514
\(236\) −7.85285e102 −0.471363
\(237\) 8.34557e102 0.416776
\(238\) 6.83328e103 2.84140
\(239\) 4.88897e102 0.169398 0.0846990 0.996407i \(-0.473007\pi\)
0.0846990 + 0.996407i \(0.473007\pi\)
\(240\) −8.38981e103 −2.42417
\(241\) 3.07268e103 0.740927 0.370463 0.928847i \(-0.379199\pi\)
0.370463 + 0.928847i \(0.379199\pi\)
\(242\) −5.40333e103 −1.08816
\(243\) −6.83263e103 −1.15005
\(244\) 1.58500e104 2.23138
\(245\) −5.89247e103 −0.694341
\(246\) −9.56870e103 −0.944443
\(247\) −2.70558e103 −0.223843
\(248\) −2.36802e104 −1.64338
\(249\) −2.34522e104 −1.36619
\(250\) 2.75826e104 1.34972
\(251\) 3.90400e104 1.60584 0.802921 0.596085i \(-0.203278\pi\)
0.802921 + 0.596085i \(0.203278\pi\)
\(252\) 1.17302e105 4.05864
\(253\) −2.15446e104 −0.627473
\(254\) −9.71198e102 −0.0238255
\(255\) −1.13335e105 −2.34351
\(256\) −1.16616e105 −2.03385
\(257\) 7.62729e104 1.12274 0.561371 0.827565i \(-0.310274\pi\)
0.561371 + 0.827565i \(0.310274\pi\)
\(258\) 2.35274e105 2.92494
\(259\) −2.17631e105 −2.28655
\(260\) −1.49612e105 −1.32931
\(261\) 6.94899e104 0.522468
\(262\) 3.33394e105 2.12252
\(263\) −1.42711e105 −0.769808 −0.384904 0.922957i \(-0.625765\pi\)
−0.384904 + 0.922957i \(0.625765\pi\)
\(264\) −4.17357e105 −1.90869
\(265\) 5.57189e105 2.16176
\(266\) −2.77071e105 −0.912516
\(267\) −9.32187e104 −0.260773
\(268\) −5.41696e105 −1.28793
\(269\) 5.61895e105 1.13614 0.568069 0.822981i \(-0.307691\pi\)
0.568069 + 0.822981i \(0.307691\pi\)
\(270\) −9.89149e105 −1.70192
\(271\) 4.63708e105 0.679330 0.339665 0.940547i \(-0.389686\pi\)
0.339665 + 0.940547i \(0.389686\pi\)
\(272\) 1.34749e106 1.68181
\(273\) 1.03293e106 1.09900
\(274\) −5.51293e105 −0.500303
\(275\) −2.69355e105 −0.208618
\(276\) 5.12812e106 3.39167
\(277\) −1.39351e106 −0.787481 −0.393741 0.919222i \(-0.628819\pi\)
−0.393741 + 0.919222i \(0.628819\pi\)
\(278\) −3.61659e106 −1.74723
\(279\) −3.12329e106 −1.29071
\(280\) −8.03523e106 −2.84200
\(281\) 6.47868e106 1.96228 0.981142 0.193289i \(-0.0619155\pi\)
0.981142 + 0.193289i \(0.0619155\pi\)
\(282\) −5.86251e106 −1.52141
\(283\) −6.71456e106 −1.49385 −0.746927 0.664907i \(-0.768471\pi\)
−0.746927 + 0.664907i \(0.768471\pi\)
\(284\) 1.92459e107 3.67274
\(285\) 4.59544e106 0.752619
\(286\) −4.23315e106 −0.595307
\(287\) −3.53252e106 −0.426798
\(288\) 5.59399e106 0.580963
\(289\) 7.00697e106 0.625855
\(290\) −9.07646e106 −0.697595
\(291\) 9.16767e105 0.0606616
\(292\) −1.02879e107 −0.586370
\(293\) 2.71547e107 1.33384 0.666918 0.745131i \(-0.267613\pi\)
0.666918 + 0.745131i \(0.267613\pi\)
\(294\) 3.96839e107 1.68076
\(295\) 7.09496e106 0.259234
\(296\) −1.11335e108 −3.51109
\(297\) −1.89672e107 −0.516531
\(298\) −3.46477e107 −0.815205
\(299\) 2.72781e107 0.554775
\(300\) 6.41129e107 1.12764
\(301\) 8.68571e107 1.32179
\(302\) 1.88979e107 0.248950
\(303\) −1.57581e108 −1.79784
\(304\) −5.46371e107 −0.540116
\(305\) −1.43203e108 −1.22718
\(306\) 4.61071e108 3.42678
\(307\) −3.12996e107 −0.201846 −0.100923 0.994894i \(-0.532180\pi\)
−0.100923 + 0.994894i \(0.532180\pi\)
\(308\) −2.93792e108 −1.64469
\(309\) −5.26902e108 −2.56175
\(310\) 4.07951e108 1.72335
\(311\) 4.07185e108 1.49525 0.747626 0.664120i \(-0.231194\pi\)
0.747626 + 0.664120i \(0.231194\pi\)
\(312\) 5.28425e108 1.68756
\(313\) 4.81634e108 1.33825 0.669126 0.743149i \(-0.266668\pi\)
0.669126 + 0.743149i \(0.266668\pi\)
\(314\) 3.14597e108 0.760879
\(315\) −1.05981e109 −2.23212
\(316\) 3.00538e108 0.551456
\(317\) 1.26342e108 0.202055 0.101028 0.994884i \(-0.467787\pi\)
0.101028 + 0.994884i \(0.467787\pi\)
\(318\) −3.75249e109 −5.23287
\(319\) −1.74043e108 −0.211720
\(320\) 7.06138e108 0.749662
\(321\) 1.95977e108 0.181650
\(322\) 2.79348e109 2.26160
\(323\) −7.38075e108 −0.522144
\(324\) −6.73062e108 −0.416243
\(325\) 3.41036e108 0.184448
\(326\) −6.78326e108 −0.320976
\(327\) 6.57779e109 2.72429
\(328\) −1.80715e109 −0.655365
\(329\) −2.16429e109 −0.687532
\(330\) 7.19002e109 2.00158
\(331\) 4.12697e109 1.00720 0.503598 0.863938i \(-0.332009\pi\)
0.503598 + 0.863938i \(0.332009\pi\)
\(332\) −8.44552e109 −1.80767
\(333\) −1.46845e110 −2.75763
\(334\) −9.61678e109 −1.58511
\(335\) 4.89416e109 0.708319
\(336\) 2.08593e110 2.65180
\(337\) 7.52106e109 0.840191 0.420096 0.907480i \(-0.361997\pi\)
0.420096 + 0.907480i \(0.361997\pi\)
\(338\) −1.25775e110 −1.23514
\(339\) 2.44978e110 2.11564
\(340\) −4.08139e110 −3.10081
\(341\) 7.82255e109 0.523038
\(342\) −1.86952e110 −1.10051
\(343\) −9.75075e109 −0.505526
\(344\) 4.44341e110 2.02966
\(345\) −4.63320e110 −1.86531
\(346\) 3.61848e110 1.28445
\(347\) 1.33452e109 0.0417823 0.0208912 0.999782i \(-0.493350\pi\)
0.0208912 + 0.999782i \(0.493350\pi\)
\(348\) 4.14264e110 1.14441
\(349\) −3.38813e110 −0.826141 −0.413071 0.910699i \(-0.635544\pi\)
−0.413071 + 0.910699i \(0.635544\pi\)
\(350\) 3.49247e110 0.751921
\(351\) 2.40148e110 0.456687
\(352\) −1.40106e110 −0.235424
\(353\) 7.71568e110 1.14597 0.572987 0.819565i \(-0.305785\pi\)
0.572987 + 0.819565i \(0.305785\pi\)
\(354\) −4.77822e110 −0.627515
\(355\) −1.73884e111 −2.01989
\(356\) −3.35696e110 −0.345041
\(357\) 2.81782e111 2.56357
\(358\) −2.81063e110 −0.226408
\(359\) 1.36407e111 0.973261 0.486631 0.873608i \(-0.338226\pi\)
0.486631 + 0.873608i \(0.338226\pi\)
\(360\) −5.42172e111 −3.42751
\(361\) −1.48542e111 −0.832313
\(362\) −2.05682e111 −1.02182
\(363\) −2.22815e111 −0.981763
\(364\) 3.71976e111 1.45414
\(365\) 9.29497e110 0.322484
\(366\) 9.64427e111 2.97059
\(367\) −8.73210e110 −0.238861 −0.119430 0.992843i \(-0.538107\pi\)
−0.119430 + 0.992843i \(0.538107\pi\)
\(368\) 5.50860e111 1.33863
\(369\) −2.38354e111 −0.514726
\(370\) 1.91803e112 3.68196
\(371\) −1.38532e112 −2.36475
\(372\) −1.86195e112 −2.82716
\(373\) 2.10926e110 0.0284968 0.0142484 0.999898i \(-0.495464\pi\)
0.0142484 + 0.999898i \(0.495464\pi\)
\(374\) −1.15479e112 −1.38864
\(375\) 1.13741e112 1.21775
\(376\) −1.10720e112 −1.05573
\(377\) 2.20360e111 0.187191
\(378\) 2.45929e112 1.86173
\(379\) −2.78355e112 −1.87842 −0.939212 0.343337i \(-0.888443\pi\)
−0.939212 + 0.343337i \(0.888443\pi\)
\(380\) 1.65489e112 0.995826
\(381\) −4.00490e110 −0.0214959
\(382\) 2.90325e112 1.39036
\(383\) −3.62318e111 −0.154862 −0.0774308 0.996998i \(-0.524672\pi\)
−0.0774308 + 0.996998i \(0.524672\pi\)
\(384\) −6.34152e112 −2.41983
\(385\) 2.65437e112 0.904524
\(386\) −4.96302e112 −1.51077
\(387\) 5.86063e112 1.59411
\(388\) 3.30143e111 0.0802642
\(389\) 1.31636e112 0.286132 0.143066 0.989713i \(-0.454304\pi\)
0.143066 + 0.989713i \(0.454304\pi\)
\(390\) −9.10346e112 −1.76968
\(391\) 7.44139e112 1.29409
\(392\) 7.49474e112 1.16631
\(393\) 1.37480e113 1.91498
\(394\) −1.15070e112 −0.143508
\(395\) −2.71532e112 −0.303283
\(396\) −1.98234e113 −1.98353
\(397\) −1.77777e113 −1.59400 −0.797002 0.603977i \(-0.793582\pi\)
−0.797002 + 0.603977i \(0.793582\pi\)
\(398\) 3.49073e113 2.80545
\(399\) −1.14255e113 −0.823292
\(400\) 6.88697e112 0.445060
\(401\) −4.38593e112 −0.254262 −0.127131 0.991886i \(-0.540577\pi\)
−0.127131 + 0.991886i \(0.540577\pi\)
\(402\) −3.29605e113 −1.71459
\(403\) −9.90431e112 −0.462439
\(404\) −5.67477e113 −2.37880
\(405\) 6.08103e112 0.228920
\(406\) 2.25665e113 0.763101
\(407\) 3.67786e113 1.11748
\(408\) 1.44153e114 3.93647
\(409\) −1.55796e113 −0.382466 −0.191233 0.981545i \(-0.561249\pi\)
−0.191233 + 0.981545i \(0.561249\pi\)
\(410\) 3.11328e113 0.687259
\(411\) −2.27335e113 −0.451385
\(412\) −1.89746e114 −3.38957
\(413\) −1.76400e113 −0.283577
\(414\) 1.88488e114 2.72753
\(415\) 7.63042e113 0.994160
\(416\) 1.77392e113 0.208149
\(417\) −1.49136e114 −1.57639
\(418\) 4.68237e113 0.445962
\(419\) 1.49846e114 1.28628 0.643140 0.765748i \(-0.277631\pi\)
0.643140 + 0.765748i \(0.277631\pi\)
\(420\) −6.31804e114 −4.88920
\(421\) −1.30632e114 −0.911544 −0.455772 0.890096i \(-0.650637\pi\)
−0.455772 + 0.890096i \(0.650637\pi\)
\(422\) 5.45363e114 3.43237
\(423\) −1.46034e114 −0.829178
\(424\) −7.08699e114 −3.63117
\(425\) 9.30339e113 0.430252
\(426\) 1.17105e115 4.88945
\(427\) 3.56042e114 1.34242
\(428\) 7.05744e113 0.240349
\(429\) −1.74561e114 −0.537099
\(430\) −7.65490e114 −2.12844
\(431\) −5.31720e114 −1.33635 −0.668177 0.744002i \(-0.732925\pi\)
−0.668177 + 0.744002i \(0.732925\pi\)
\(432\) 4.84960e114 1.10195
\(433\) 2.52039e114 0.517899 0.258949 0.965891i \(-0.416624\pi\)
0.258949 + 0.965891i \(0.416624\pi\)
\(434\) −1.01428e115 −1.88518
\(435\) −3.74283e114 −0.629386
\(436\) 2.36877e115 3.60463
\(437\) −3.01728e114 −0.415598
\(438\) −6.25986e114 −0.780622
\(439\) 9.43647e113 0.106562 0.0532811 0.998580i \(-0.483032\pi\)
0.0532811 + 0.998580i \(0.483032\pi\)
\(440\) 1.35792e115 1.38893
\(441\) 9.88518e114 0.916022
\(442\) 1.46211e115 1.22775
\(443\) 2.77028e114 0.210844 0.105422 0.994428i \(-0.466381\pi\)
0.105422 + 0.994428i \(0.466381\pi\)
\(444\) −8.75418e115 −6.04027
\(445\) 3.03297e114 0.189761
\(446\) −8.30027e113 −0.0471005
\(447\) −1.42875e115 −0.735496
\(448\) −1.75565e115 −0.820057
\(449\) 2.39383e115 1.01479 0.507397 0.861712i \(-0.330608\pi\)
0.507397 + 0.861712i \(0.330608\pi\)
\(450\) 2.35652e115 0.906832
\(451\) 5.96979e114 0.208583
\(452\) 8.82207e115 2.79930
\(453\) 7.79288e114 0.224608
\(454\) −1.86423e115 −0.488168
\(455\) −3.36076e115 −0.799727
\(456\) −5.84502e115 −1.26420
\(457\) 5.47308e115 1.07616 0.538081 0.842893i \(-0.319150\pi\)
0.538081 + 0.842893i \(0.319150\pi\)
\(458\) 1.41104e116 2.52285
\(459\) 6.55116e115 1.06529
\(460\) −1.66849e116 −2.46807
\(461\) 3.15068e115 0.424047 0.212023 0.977265i \(-0.431995\pi\)
0.212023 + 0.977265i \(0.431995\pi\)
\(462\) −1.78763e116 −2.18954
\(463\) −6.34150e115 −0.706999 −0.353500 0.935435i \(-0.615008\pi\)
−0.353500 + 0.935435i \(0.615008\pi\)
\(464\) 4.45001e115 0.451677
\(465\) 1.68225e116 1.55485
\(466\) −2.09230e116 −1.76131
\(467\) −2.26650e116 −1.73808 −0.869042 0.494738i \(-0.835264\pi\)
−0.869042 + 0.494738i \(0.835264\pi\)
\(468\) 2.50989e116 1.75372
\(469\) −1.21682e116 −0.774831
\(470\) 1.90743e116 1.10711
\(471\) 1.29729e116 0.686482
\(472\) −9.02420e115 −0.435443
\(473\) −1.46784e116 −0.645982
\(474\) 1.82868e116 0.734142
\(475\) −3.77227e115 −0.138176
\(476\) 1.01474e117 3.39198
\(477\) −9.34737e116 −2.85194
\(478\) 1.07127e116 0.298391
\(479\) −2.73182e113 −0.000694795 0 −0.000347398 1.00000i \(-0.500111\pi\)
−0.000347398 1.00000i \(0.500111\pi\)
\(480\) −3.01301e116 −0.699852
\(481\) −4.65662e116 −0.988007
\(482\) 6.73285e116 1.30513
\(483\) 1.15194e117 2.04046
\(484\) −8.02394e116 −1.29902
\(485\) −2.98280e115 −0.0441427
\(486\) −1.49717e117 −2.02578
\(487\) 8.24893e116 1.02068 0.510338 0.859974i \(-0.329520\pi\)
0.510338 + 0.859974i \(0.329520\pi\)
\(488\) 1.82143e117 2.06134
\(489\) −2.79719e116 −0.289592
\(490\) −1.29116e117 −1.22307
\(491\) −7.22914e116 −0.626674 −0.313337 0.949642i \(-0.601447\pi\)
−0.313337 + 0.949642i \(0.601447\pi\)
\(492\) −1.42095e117 −1.12745
\(493\) 6.01137e116 0.436649
\(494\) −5.92846e116 −0.394294
\(495\) 1.79102e117 1.09087
\(496\) −2.00010e117 −1.11583
\(497\) 4.32323e117 2.20956
\(498\) −5.13884e117 −2.40652
\(499\) −3.07202e117 −1.31841 −0.659204 0.751964i \(-0.729107\pi\)
−0.659204 + 0.751964i \(0.729107\pi\)
\(500\) 4.09602e117 1.61126
\(501\) −3.96564e117 −1.43012
\(502\) 8.55445e117 2.82866
\(503\) 3.23349e116 0.0980538 0.0490269 0.998797i \(-0.484388\pi\)
0.0490269 + 0.998797i \(0.484388\pi\)
\(504\) 1.34799e118 3.74936
\(505\) 5.12708e117 1.30826
\(506\) −4.72084e117 −1.10528
\(507\) −5.18652e117 −1.11437
\(508\) −1.44223e116 −0.0284422
\(509\) −5.28702e117 −0.957166 −0.478583 0.878042i \(-0.658849\pi\)
−0.478583 + 0.878042i \(0.658849\pi\)
\(510\) −2.48340e118 −4.12804
\(511\) −2.31098e117 −0.352766
\(512\) −1.46926e118 −2.05995
\(513\) −2.65632e117 −0.342117
\(514\) 1.67129e118 1.97768
\(515\) 1.71433e118 1.86415
\(516\) 3.49382e118 3.49171
\(517\) 3.65754e117 0.336009
\(518\) −4.76873e118 −4.02771
\(519\) 1.49214e118 1.15886
\(520\) −1.71929e118 −1.22801
\(521\) 9.85155e117 0.647237 0.323619 0.946188i \(-0.395101\pi\)
0.323619 + 0.946188i \(0.395101\pi\)
\(522\) 1.52266e118 0.920315
\(523\) −2.26194e118 −1.25793 −0.628966 0.777433i \(-0.716521\pi\)
−0.628966 + 0.777433i \(0.716521\pi\)
\(524\) 4.95089e118 2.53380
\(525\) 1.44018e118 0.678400
\(526\) −3.12708e118 −1.35600
\(527\) −2.70187e118 −1.07871
\(528\) −3.52512e118 −1.29598
\(529\) 8.86796e116 0.0300263
\(530\) 1.22091e119 3.80789
\(531\) −1.19025e118 −0.341999
\(532\) −4.11451e118 −1.08934
\(533\) −7.55849e117 −0.184417
\(534\) −2.04261e118 −0.459346
\(535\) −6.37631e117 −0.132184
\(536\) −6.22496e118 −1.18978
\(537\) −1.15901e118 −0.204271
\(538\) 1.23122e119 2.00128
\(539\) −2.47583e118 −0.371201
\(540\) −1.46889e119 −2.03170
\(541\) 6.77058e118 0.864062 0.432031 0.901859i \(-0.357797\pi\)
0.432031 + 0.901859i \(0.357797\pi\)
\(542\) 1.01608e119 1.19662
\(543\) −8.48162e118 −0.921906
\(544\) 4.83920e118 0.485536
\(545\) −2.14015e119 −1.98243
\(546\) 2.26336e119 1.93586
\(547\) −1.64167e119 −1.29669 −0.648346 0.761346i \(-0.724539\pi\)
−0.648346 + 0.761346i \(0.724539\pi\)
\(548\) −8.18670e118 −0.597248
\(549\) 2.40237e119 1.61898
\(550\) −5.90210e118 −0.367477
\(551\) −2.43745e118 −0.140230
\(552\) 5.89304e119 3.13321
\(553\) 6.75102e118 0.331762
\(554\) −3.05346e119 −1.38713
\(555\) 7.90929e119 3.32195
\(556\) −5.37063e119 −2.08580
\(557\) −1.22711e119 −0.440739 −0.220369 0.975417i \(-0.570726\pi\)
−0.220369 + 0.975417i \(0.570726\pi\)
\(558\) −6.84376e119 −2.27356
\(559\) 1.85847e119 0.571139
\(560\) −6.78680e119 −1.92968
\(561\) −4.76197e119 −1.25286
\(562\) 1.41961e120 3.45652
\(563\) 3.15592e119 0.711232 0.355616 0.934632i \(-0.384271\pi\)
0.355616 + 0.934632i \(0.384271\pi\)
\(564\) −8.70582e119 −1.81622
\(565\) −7.97064e119 −1.53952
\(566\) −1.47129e120 −2.63139
\(567\) −1.51191e119 −0.250416
\(568\) 2.21167e120 3.39287
\(569\) 7.97402e119 1.13317 0.566583 0.824005i \(-0.308265\pi\)
0.566583 + 0.824005i \(0.308265\pi\)
\(570\) 1.00695e120 1.32572
\(571\) 7.69582e119 0.938825 0.469412 0.882979i \(-0.344466\pi\)
0.469412 + 0.882979i \(0.344466\pi\)
\(572\) −6.28622e119 −0.710661
\(573\) 1.19720e120 1.25442
\(574\) −7.74045e119 −0.751794
\(575\) 3.80327e119 0.342457
\(576\) −1.18461e120 −0.989004
\(577\) −3.60188e119 −0.278857 −0.139429 0.990232i \(-0.544527\pi\)
−0.139429 + 0.990232i \(0.544527\pi\)
\(578\) 1.53537e120 1.10243
\(579\) −2.04658e120 −1.36305
\(580\) −1.34785e120 −0.832770
\(581\) −1.89713e120 −1.08751
\(582\) 2.00882e119 0.106854
\(583\) 2.34113e120 1.15569
\(584\) −1.18224e120 −0.541687
\(585\) −2.26765e120 −0.964487
\(586\) 5.95013e120 2.34952
\(587\) −4.10740e120 −1.50595 −0.752973 0.658052i \(-0.771381\pi\)
−0.752973 + 0.658052i \(0.771381\pi\)
\(588\) 5.89306e120 2.00644
\(589\) 1.09554e120 0.346427
\(590\) 1.55465e120 0.456635
\(591\) −4.74511e119 −0.129477
\(592\) −9.40369e120 −2.38399
\(593\) −5.50645e119 −0.129716 −0.0648579 0.997895i \(-0.520659\pi\)
−0.0648579 + 0.997895i \(0.520659\pi\)
\(594\) −4.15608e120 −0.909858
\(595\) −9.16807e120 −1.86548
\(596\) −5.14518e120 −0.973169
\(597\) 1.43946e121 2.53114
\(598\) 5.97717e120 0.977224
\(599\) 7.02687e120 1.06831 0.534153 0.845388i \(-0.320631\pi\)
0.534153 + 0.845388i \(0.320631\pi\)
\(600\) 7.36761e120 1.04171
\(601\) −1.79955e120 −0.236659 −0.118330 0.992974i \(-0.537754\pi\)
−0.118330 + 0.992974i \(0.537754\pi\)
\(602\) 1.90321e121 2.32831
\(603\) −8.21041e120 −0.934462
\(604\) 2.80634e120 0.297190
\(605\) 7.24954e120 0.714416
\(606\) −3.45292e121 −3.16685
\(607\) −3.31580e120 −0.283062 −0.141531 0.989934i \(-0.545202\pi\)
−0.141531 + 0.989934i \(0.545202\pi\)
\(608\) −1.96217e120 −0.155930
\(609\) 9.30567e120 0.688487
\(610\) −3.13787e121 −2.16166
\(611\) −4.63090e120 −0.297079
\(612\) 6.84691e121 4.09079
\(613\) −2.68663e121 −1.49512 −0.747562 0.664192i \(-0.768776\pi\)
−0.747562 + 0.664192i \(0.768776\pi\)
\(614\) −6.85838e120 −0.355547
\(615\) 1.28381e121 0.620060
\(616\) −3.37614e121 −1.51936
\(617\) −9.07733e119 −0.0380674 −0.0190337 0.999819i \(-0.506059\pi\)
−0.0190337 + 0.999819i \(0.506059\pi\)
\(618\) −1.15455e122 −4.51246
\(619\) 1.74643e121 0.636222 0.318111 0.948053i \(-0.396952\pi\)
0.318111 + 0.948053i \(0.396952\pi\)
\(620\) 6.05807e121 2.05729
\(621\) 2.67815e121 0.847910
\(622\) 8.92223e121 2.63385
\(623\) −7.54078e120 −0.207580
\(624\) 4.46324e121 1.14583
\(625\) −5.10987e121 −1.22357
\(626\) 1.05536e122 2.35730
\(627\) 1.93085e121 0.402357
\(628\) 4.67177e121 0.908316
\(629\) −1.27031e122 −2.30467
\(630\) −2.32225e122 −3.93183
\(631\) 2.91383e121 0.460456 0.230228 0.973137i \(-0.426053\pi\)
0.230228 + 0.973137i \(0.426053\pi\)
\(632\) 3.45367e121 0.509433
\(633\) 2.24889e122 3.09676
\(634\) 2.76841e121 0.355916
\(635\) 1.30304e120 0.0156423
\(636\) −5.57244e122 −6.24685
\(637\) 3.13470e121 0.328194
\(638\) −3.81363e121 −0.372940
\(639\) 2.91707e122 2.66477
\(640\) 2.06328e122 1.76088
\(641\) −1.26041e122 −1.00505 −0.502524 0.864563i \(-0.667595\pi\)
−0.502524 + 0.864563i \(0.667595\pi\)
\(642\) 4.29424e121 0.319972
\(643\) −1.27618e121 −0.0888655 −0.0444327 0.999012i \(-0.514148\pi\)
−0.0444327 + 0.999012i \(0.514148\pi\)
\(644\) 4.14831e122 2.69983
\(645\) −3.15662e122 −1.92033
\(646\) −1.61727e122 −0.919745
\(647\) 8.91644e120 0.0474084 0.0237042 0.999719i \(-0.492454\pi\)
0.0237042 + 0.999719i \(0.492454\pi\)
\(648\) −7.73457e121 −0.384524
\(649\) 2.98107e121 0.138589
\(650\) 7.47279e121 0.324901
\(651\) −4.18253e122 −1.70085
\(652\) −1.00731e122 −0.383172
\(653\) 1.51404e122 0.538782 0.269391 0.963031i \(-0.413178\pi\)
0.269391 + 0.963031i \(0.413178\pi\)
\(654\) 1.44132e123 4.79877
\(655\) −4.47307e122 −1.39351
\(656\) −1.52638e122 −0.444985
\(657\) −1.55932e122 −0.425443
\(658\) −4.74238e122 −1.21107
\(659\) −2.23773e122 −0.534924 −0.267462 0.963568i \(-0.586185\pi\)
−0.267462 + 0.963568i \(0.586185\pi\)
\(660\) 1.06772e123 2.38944
\(661\) −3.07824e122 −0.644970 −0.322485 0.946575i \(-0.604518\pi\)
−0.322485 + 0.946575i \(0.604518\pi\)
\(662\) 9.04302e122 1.77415
\(663\) 6.02925e122 1.10771
\(664\) −9.70527e122 −1.66992
\(665\) 3.71741e122 0.599099
\(666\) −3.21767e123 −4.85749
\(667\) 2.45748e122 0.347548
\(668\) −1.42809e123 −1.89226
\(669\) −3.42275e121 −0.0424952
\(670\) 1.07241e123 1.24769
\(671\) −6.01693e122 −0.656063
\(672\) 7.49115e122 0.765570
\(673\) 9.53095e122 0.913022 0.456511 0.889718i \(-0.349099\pi\)
0.456511 + 0.889718i \(0.349099\pi\)
\(674\) 1.64801e123 1.47998
\(675\) 3.34828e122 0.281908
\(676\) −1.86775e123 −1.47448
\(677\) 2.56578e123 1.89938 0.949691 0.313188i \(-0.101397\pi\)
0.949691 + 0.313188i \(0.101397\pi\)
\(678\) 5.36796e123 3.72664
\(679\) 7.41604e121 0.0482878
\(680\) −4.69017e123 −2.86452
\(681\) −7.68745e122 −0.440436
\(682\) 1.71408e123 0.921319
\(683\) −2.88998e123 −1.45745 −0.728727 0.684805i \(-0.759888\pi\)
−0.728727 + 0.684805i \(0.759888\pi\)
\(684\) −2.77624e123 −1.31376
\(685\) 7.39659e122 0.328467
\(686\) −2.13658e123 −0.890473
\(687\) 5.81867e123 2.27617
\(688\) 3.75304e123 1.37812
\(689\) −2.96416e123 −1.02180
\(690\) −1.01523e124 −3.28569
\(691\) −1.42365e122 −0.0432621 −0.0216311 0.999766i \(-0.506886\pi\)
−0.0216311 + 0.999766i \(0.506886\pi\)
\(692\) 5.37344e123 1.53334
\(693\) −4.45296e123 −1.19331
\(694\) 2.92420e122 0.0735986
\(695\) 4.85230e123 1.14712
\(696\) 4.76057e123 1.05720
\(697\) −2.06194e123 −0.430179
\(698\) −7.42407e123 −1.45523
\(699\) −8.62794e123 −1.58910
\(700\) 5.18631e123 0.897623
\(701\) 6.07300e123 0.987801 0.493900 0.869518i \(-0.335571\pi\)
0.493900 + 0.869518i \(0.335571\pi\)
\(702\) 5.26211e123 0.804444
\(703\) 5.15078e123 0.740145
\(704\) 2.96696e123 0.400775
\(705\) 7.86560e123 0.998862
\(706\) 1.69066e124 2.01861
\(707\) −1.27473e124 −1.43111
\(708\) −7.09566e123 −0.749111
\(709\) −9.12953e123 −0.906436 −0.453218 0.891400i \(-0.649724\pi\)
−0.453218 + 0.891400i \(0.649724\pi\)
\(710\) −3.81015e124 −3.55799
\(711\) 4.55521e123 0.400111
\(712\) −3.85769e123 −0.318748
\(713\) −1.10454e124 −0.858591
\(714\) 6.17440e124 4.51567
\(715\) 5.67953e123 0.390840
\(716\) −4.17379e123 −0.270280
\(717\) 4.41756e123 0.269215
\(718\) 2.98896e124 1.71438
\(719\) 1.34804e124 0.727771 0.363885 0.931444i \(-0.381450\pi\)
0.363885 + 0.931444i \(0.381450\pi\)
\(720\) −4.57935e124 −2.32724
\(721\) −4.26229e124 −2.03920
\(722\) −3.25486e124 −1.46610
\(723\) 2.77640e124 1.17751
\(724\) −3.05437e124 −1.21982
\(725\) 3.07239e123 0.115551
\(726\) −4.88233e124 −1.72935
\(727\) 2.45578e124 0.819299 0.409650 0.912243i \(-0.365651\pi\)
0.409650 + 0.912243i \(0.365651\pi\)
\(728\) 4.27461e124 1.34333
\(729\) −5.50516e124 −1.62976
\(730\) 2.03671e124 0.568049
\(731\) 5.06986e124 1.33226
\(732\) 1.43217e125 3.54620
\(733\) −6.62584e123 −0.154603 −0.0773015 0.997008i \(-0.524630\pi\)
−0.0773015 + 0.997008i \(0.524630\pi\)
\(734\) −1.91338e124 −0.420748
\(735\) −5.32430e124 −1.10348
\(736\) 1.97829e124 0.386460
\(737\) 2.05636e124 0.378673
\(738\) −5.22282e124 −0.906679
\(739\) 5.98416e123 0.0979422 0.0489711 0.998800i \(-0.484406\pi\)
0.0489711 + 0.998800i \(0.484406\pi\)
\(740\) 2.84827e125 4.39543
\(741\) −2.44470e124 −0.355740
\(742\) −3.03552e125 −4.16546
\(743\) 4.85932e124 0.628872 0.314436 0.949279i \(-0.398185\pi\)
0.314436 + 0.949279i \(0.398185\pi\)
\(744\) −2.13969e125 −2.61173
\(745\) 4.64861e124 0.535211
\(746\) 4.62181e123 0.0501965
\(747\) −1.28007e125 −1.31156
\(748\) −1.71487e125 −1.65772
\(749\) 1.58532e124 0.144597
\(750\) 2.49230e125 2.14504
\(751\) −3.85338e124 −0.312971 −0.156485 0.987680i \(-0.550016\pi\)
−0.156485 + 0.987680i \(0.550016\pi\)
\(752\) −9.35175e124 −0.716831
\(753\) 3.52757e125 2.55208
\(754\) 4.82853e124 0.329732
\(755\) −2.53550e124 −0.163445
\(756\) 3.65204e125 2.22248
\(757\) 1.30143e125 0.747739 0.373870 0.927481i \(-0.378031\pi\)
0.373870 + 0.927481i \(0.378031\pi\)
\(758\) −6.09932e125 −3.30880
\(759\) −1.94672e125 −0.997208
\(760\) 1.90174e125 0.919941
\(761\) −8.62091e124 −0.393841 −0.196920 0.980419i \(-0.563094\pi\)
−0.196920 + 0.980419i \(0.563094\pi\)
\(762\) −8.77552e123 −0.0378645
\(763\) 5.32100e125 2.16858
\(764\) 4.31132e125 1.65978
\(765\) −6.18610e125 −2.24980
\(766\) −7.93912e124 −0.272785
\(767\) −3.77440e124 −0.122532
\(768\) −1.05371e126 −3.23228
\(769\) 4.22053e125 1.22341 0.611704 0.791087i \(-0.290484\pi\)
0.611704 + 0.791087i \(0.290484\pi\)
\(770\) 5.81626e125 1.59330
\(771\) 6.89185e125 1.78431
\(772\) −7.37008e125 −1.80351
\(773\) −1.70364e125 −0.394067 −0.197034 0.980397i \(-0.563131\pi\)
−0.197034 + 0.980397i \(0.563131\pi\)
\(774\) 1.28418e126 2.80798
\(775\) −1.38092e125 −0.285459
\(776\) 3.79388e124 0.0741478
\(777\) −1.96646e126 −3.63389
\(778\) 2.88440e125 0.504015
\(779\) 8.36060e124 0.138152
\(780\) −1.35186e126 −2.11260
\(781\) −7.30606e125 −1.07985
\(782\) 1.63056e126 2.27951
\(783\) 2.16348e125 0.286099
\(784\) 6.33029e125 0.791908
\(785\) −4.22089e125 −0.499544
\(786\) 3.01247e126 3.37320
\(787\) −1.36872e126 −1.45015 −0.725075 0.688670i \(-0.758195\pi\)
−0.725075 + 0.688670i \(0.758195\pi\)
\(788\) −1.70879e125 −0.171316
\(789\) −1.28950e126 −1.22341
\(790\) −5.94981e125 −0.534226
\(791\) 1.98171e126 1.68408
\(792\) −2.27803e126 −1.83237
\(793\) 7.61818e125 0.580053
\(794\) −3.89546e126 −2.80780
\(795\) 5.03463e126 3.43556
\(796\) 5.18374e126 3.34907
\(797\) 1.50722e126 0.922018 0.461009 0.887396i \(-0.347488\pi\)
0.461009 + 0.887396i \(0.347488\pi\)
\(798\) −2.50355e126 −1.45021
\(799\) −1.26330e126 −0.692980
\(800\) 2.47330e125 0.128488
\(801\) −5.08809e125 −0.250346
\(802\) −9.61044e125 −0.447877
\(803\) 3.90544e125 0.172403
\(804\) −4.89464e126 −2.04683
\(805\) −3.74795e126 −1.48482
\(806\) −2.17023e126 −0.814577
\(807\) 5.07716e126 1.80560
\(808\) −6.52122e126 −2.19753
\(809\) −3.29569e126 −1.05241 −0.526206 0.850357i \(-0.676386\pi\)
−0.526206 + 0.850357i \(0.676386\pi\)
\(810\) 1.33247e126 0.403237
\(811\) −1.12128e126 −0.321595 −0.160797 0.986987i \(-0.551407\pi\)
−0.160797 + 0.986987i \(0.551407\pi\)
\(812\) 3.35113e126 0.910969
\(813\) 4.18996e126 1.07962
\(814\) 8.05892e126 1.96841
\(815\) 9.10095e125 0.210732
\(816\) 1.21756e127 2.67281
\(817\) −2.05569e126 −0.427857
\(818\) −3.41380e126 −0.673705
\(819\) 5.63799e126 1.05505
\(820\) 4.62322e126 0.820431
\(821\) −5.03284e126 −0.847004 −0.423502 0.905895i \(-0.639199\pi\)
−0.423502 + 0.905895i \(0.639199\pi\)
\(822\) −4.98136e126 −0.795104
\(823\) −2.27045e126 −0.343732 −0.171866 0.985120i \(-0.554980\pi\)
−0.171866 + 0.985120i \(0.554980\pi\)
\(824\) −2.18049e127 −3.13127
\(825\) −2.43383e126 −0.331546
\(826\) −3.86527e126 −0.499514
\(827\) −5.03797e126 −0.617683 −0.308842 0.951114i \(-0.599941\pi\)
−0.308842 + 0.951114i \(0.599941\pi\)
\(828\) 2.79905e127 3.25605
\(829\) −1.43049e126 −0.157893 −0.0789466 0.996879i \(-0.525156\pi\)
−0.0789466 + 0.996879i \(0.525156\pi\)
\(830\) 1.67198e127 1.75119
\(831\) −1.25914e127 −1.25150
\(832\) −3.75654e126 −0.354342
\(833\) 8.55138e126 0.765559
\(834\) −3.26786e127 −2.77678
\(835\) 1.29026e127 1.04068
\(836\) 6.95332e126 0.532377
\(837\) −9.72400e126 −0.706785
\(838\) 3.28343e127 2.26575
\(839\) 1.26937e127 0.831654 0.415827 0.909444i \(-0.363492\pi\)
0.415827 + 0.909444i \(0.363492\pi\)
\(840\) −7.26045e127 −4.51663
\(841\) −1.49436e127 −0.882731
\(842\) −2.86240e127 −1.60566
\(843\) 5.85399e127 3.11855
\(844\) 8.09864e127 4.09747
\(845\) 1.68749e127 0.810914
\(846\) −3.19989e127 −1.46058
\(847\) −1.80243e127 −0.781502
\(848\) −5.98589e127 −2.46552
\(849\) −6.06713e127 −2.37410
\(850\) 2.03856e127 0.757879
\(851\) −5.19311e127 −1.83439
\(852\) 1.73902e128 5.83688
\(853\) −1.54493e127 −0.492748 −0.246374 0.969175i \(-0.579239\pi\)
−0.246374 + 0.969175i \(0.579239\pi\)
\(854\) 7.80158e127 2.36464
\(855\) 2.50830e127 0.722525
\(856\) 8.11014e126 0.222034
\(857\) −2.01359e127 −0.523967 −0.261984 0.965072i \(-0.584377\pi\)
−0.261984 + 0.965072i \(0.584377\pi\)
\(858\) −3.82498e127 −0.946088
\(859\) 4.07203e127 0.957432 0.478716 0.877970i \(-0.341102\pi\)
0.478716 + 0.877970i \(0.341102\pi\)
\(860\) −1.13675e128 −2.54087
\(861\) −3.19190e127 −0.678285
\(862\) −1.16510e128 −2.35396
\(863\) 6.84641e127 1.31521 0.657603 0.753365i \(-0.271571\pi\)
0.657603 + 0.753365i \(0.271571\pi\)
\(864\) 1.74162e127 0.318131
\(865\) −4.85484e127 −0.843285
\(866\) 5.52268e127 0.912267
\(867\) 6.33133e127 0.994637
\(868\) −1.50620e128 −2.25048
\(869\) −1.14089e127 −0.162138
\(870\) −8.20128e127 −1.10865
\(871\) −2.60361e127 −0.334801
\(872\) 2.72210e128 3.32994
\(873\) 5.00393e126 0.0582360
\(874\) −6.61147e127 −0.732067
\(875\) 9.20094e127 0.969353
\(876\) −9.29589e127 −0.931885
\(877\) −6.89675e127 −0.657904 −0.328952 0.944347i \(-0.606695\pi\)
−0.328952 + 0.944347i \(0.606695\pi\)
\(878\) 2.06772e127 0.187707
\(879\) 2.45363e128 2.11979
\(880\) 1.14694e128 0.943069
\(881\) −1.42995e128 −1.11910 −0.559551 0.828796i \(-0.689026\pi\)
−0.559551 + 0.828796i \(0.689026\pi\)
\(882\) 2.16604e128 1.61355
\(883\) −1.83260e128 −1.29950 −0.649750 0.760148i \(-0.725126\pi\)
−0.649750 + 0.760148i \(0.725126\pi\)
\(884\) 2.17123e128 1.46566
\(885\) 6.41084e127 0.411986
\(886\) 6.07023e127 0.371397
\(887\) 2.22818e128 1.29799 0.648997 0.760791i \(-0.275189\pi\)
0.648997 + 0.760791i \(0.275189\pi\)
\(888\) −1.00600e129 −5.57998
\(889\) −3.23970e126 −0.0171111
\(890\) 6.64584e127 0.334260
\(891\) 2.55505e127 0.122382
\(892\) −1.23259e127 −0.0562273
\(893\) 5.12233e127 0.222551
\(894\) −3.13069e128 −1.29556
\(895\) 3.77097e127 0.148645
\(896\) −5.12987e128 −1.92623
\(897\) 2.46478e128 0.881673
\(898\) 5.24536e128 1.78754
\(899\) −8.92277e127 −0.289703
\(900\) 3.49943e128 1.08255
\(901\) −8.08614e128 −2.38349
\(902\) 1.30810e128 0.367415
\(903\) 7.84821e128 2.10065
\(904\) 1.01380e129 2.58598
\(905\) 2.75959e128 0.670859
\(906\) 1.70757e128 0.395643
\(907\) −6.58794e128 −1.45490 −0.727449 0.686161i \(-0.759294\pi\)
−0.727449 + 0.686161i \(0.759294\pi\)
\(908\) −2.76838e128 −0.582761
\(909\) −8.60116e128 −1.72595
\(910\) −7.36410e128 −1.40870
\(911\) −6.80939e128 −1.24182 −0.620910 0.783882i \(-0.713237\pi\)
−0.620910 + 0.783882i \(0.713237\pi\)
\(912\) −4.93688e128 −0.858376
\(913\) 3.20605e128 0.531486
\(914\) 1.19926e129 1.89563
\(915\) −1.29395e129 −1.95030
\(916\) 2.09540e129 3.01171
\(917\) 1.11213e129 1.52436
\(918\) 1.43549e129 1.87648
\(919\) 1.02101e129 1.27293 0.636464 0.771306i \(-0.280396\pi\)
0.636464 + 0.771306i \(0.280396\pi\)
\(920\) −1.91737e129 −2.28000
\(921\) −2.82816e128 −0.320783
\(922\) 6.90376e128 0.746949
\(923\) 9.25037e128 0.954740
\(924\) −2.65463e129 −2.61381
\(925\) −6.49253e128 −0.609886
\(926\) −1.38955e129 −1.24536
\(927\) −2.87596e129 −2.45931
\(928\) 1.59812e128 0.130398
\(929\) 1.05096e128 0.0818281 0.0409141 0.999163i \(-0.486973\pi\)
0.0409141 + 0.999163i \(0.486973\pi\)
\(930\) 3.68615e129 2.73883
\(931\) −3.46736e128 −0.245860
\(932\) −3.10707e129 −2.10261
\(933\) 3.67923e129 2.37632
\(934\) −4.96635e129 −3.06160
\(935\) 1.54936e129 0.911690
\(936\) 2.88427e129 1.62008
\(937\) −1.30807e129 −0.701388 −0.350694 0.936490i \(-0.614054\pi\)
−0.350694 + 0.936490i \(0.614054\pi\)
\(938\) −2.66629e129 −1.36485
\(939\) 4.35193e129 2.12681
\(940\) 2.83253e129 1.32164
\(941\) 3.19454e129 1.42318 0.711590 0.702595i \(-0.247975\pi\)
0.711590 + 0.702595i \(0.247975\pi\)
\(942\) 2.84263e129 1.20922
\(943\) −8.42929e128 −0.342399
\(944\) −7.62212e128 −0.295661
\(945\) −3.29958e129 −1.22229
\(946\) −3.21634e129 −1.13788
\(947\) −1.49588e128 −0.0505442 −0.0252721 0.999681i \(-0.508045\pi\)
−0.0252721 + 0.999681i \(0.508045\pi\)
\(948\) 2.71559e129 0.876398
\(949\) −4.94477e128 −0.152428
\(950\) −8.26581e128 −0.243393
\(951\) 1.14160e129 0.321115
\(952\) 1.16610e130 3.13350
\(953\) −1.25754e129 −0.322835 −0.161417 0.986886i \(-0.551607\pi\)
−0.161417 + 0.986886i \(0.551607\pi\)
\(954\) −2.04820e130 −5.02362
\(955\) −3.89523e129 −0.912822
\(956\) 1.59084e129 0.356211
\(957\) −1.57261e129 −0.336475
\(958\) −5.98596e126 −0.00122387
\(959\) −1.83899e129 −0.359311
\(960\) 6.38050e129 1.19140
\(961\) −1.59317e129 −0.284312
\(962\) −1.02036e130 −1.74035
\(963\) 1.06969e129 0.174386
\(964\) 9.99827e129 1.55802
\(965\) 6.65878e129 0.991874
\(966\) 2.52412e130 3.59423
\(967\) −3.69822e129 −0.503433 −0.251717 0.967801i \(-0.580995\pi\)
−0.251717 + 0.967801i \(0.580995\pi\)
\(968\) −9.22081e129 −1.20003
\(969\) −6.66907e129 −0.829815
\(970\) −6.53591e128 −0.0777563
\(971\) 8.38667e129 0.954010 0.477005 0.878901i \(-0.341722\pi\)
0.477005 + 0.878901i \(0.341722\pi\)
\(972\) −2.22329e130 −2.41832
\(973\) −1.20641e130 −1.25484
\(974\) 1.80750e130 1.79790
\(975\) 3.08153e129 0.293133
\(976\) 1.53843e130 1.39963
\(977\) 8.47225e129 0.737200 0.368600 0.929588i \(-0.379837\pi\)
0.368600 + 0.929588i \(0.379837\pi\)
\(978\) −6.12920e129 −0.510109
\(979\) 1.27436e129 0.101448
\(980\) −1.91737e130 −1.46006
\(981\) 3.59031e130 2.61535
\(982\) −1.58405e130 −1.10387
\(983\) −2.47112e130 −1.64746 −0.823732 0.566979i \(-0.808112\pi\)
−0.823732 + 0.566979i \(0.808112\pi\)
\(984\) −1.63290e130 −1.04153
\(985\) 1.54387e129 0.0942184
\(986\) 1.31721e130 0.769147
\(987\) −1.95560e130 −1.09266
\(988\) −8.80376e129 −0.470697
\(989\) 2.07258e130 1.06041
\(990\) 3.92448e130 1.92155
\(991\) −4.15849e130 −1.94864 −0.974319 0.225171i \(-0.927706\pi\)
−0.974319 + 0.225171i \(0.927706\pi\)
\(992\) −7.18291e129 −0.322138
\(993\) 3.72904e130 1.60068
\(994\) 9.47306e130 3.89209
\(995\) −4.68344e130 −1.84188
\(996\) −7.63118e130 −2.87283
\(997\) 2.55616e130 0.921188 0.460594 0.887611i \(-0.347636\pi\)
0.460594 + 0.887611i \(0.347636\pi\)
\(998\) −6.73141e130 −2.32235
\(999\) −4.57184e130 −1.51005
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 1.88.a.a.1.7 7
3.2 odd 2 9.88.a.b.1.1 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.88.a.a.1.7 7 1.1 even 1 trivial
9.88.a.b.1.1 7 3.2 odd 2