Properties

Label 10.38.a.a.1.1
Level $10$
Weight $38$
Character 10.1
Self dual yes
Analytic conductor $86.714$
Analytic rank $1$
Dimension $2$
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] = [10,38,Mod(1,10)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(10, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 38, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("10.1");
 
S:= CuspForms(chi, 38);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 38 \)
Character orbit: \([\chi]\) \(=\) 10.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(86.7140381246\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1222518952080 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3\cdot 5 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.10568e6\) of defining polynomial
Character \(\chi\) \(=\) 10.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+262144. q^{2} -6.89815e8 q^{3} +6.87195e10 q^{4} +3.81470e12 q^{5} -1.80831e14 q^{6} +3.29617e15 q^{7} +1.80144e16 q^{8} +2.55614e16 q^{9} +1.00000e18 q^{10} -1.36311e19 q^{11} -4.74038e19 q^{12} -1.75195e20 q^{13} +8.64071e20 q^{14} -2.63144e21 q^{15} +4.72237e21 q^{16} -8.98224e20 q^{17} +6.70076e21 q^{18} +1.81344e23 q^{19} +2.62144e23 q^{20} -2.27375e24 q^{21} -3.57330e24 q^{22} -8.96395e24 q^{23} -1.24266e25 q^{24} +1.45519e25 q^{25} -4.59263e25 q^{26} +2.92980e26 q^{27} +2.26511e26 q^{28} +8.10751e26 q^{29} -6.89815e26 q^{30} +1.17491e27 q^{31} +1.23794e27 q^{32} +9.40292e27 q^{33} -2.35464e26 q^{34} +1.25739e28 q^{35} +1.75656e27 q^{36} +1.05137e29 q^{37} +4.75384e28 q^{38} +1.20852e29 q^{39} +6.87195e28 q^{40} -7.89765e29 q^{41} -5.96049e29 q^{42} -2.46365e30 q^{43} -9.36720e29 q^{44} +9.75088e28 q^{45} -2.34985e30 q^{46} -4.05145e30 q^{47} -3.25756e30 q^{48} -7.69740e30 q^{49} +3.81470e30 q^{50} +6.19609e29 q^{51} -1.20393e31 q^{52} -7.63383e31 q^{53} +7.68030e31 q^{54} -5.19984e31 q^{55} +5.93785e31 q^{56} -1.25094e32 q^{57} +2.12534e32 q^{58} +5.16782e30 q^{59} -1.80831e32 q^{60} -3.70977e32 q^{61} +3.07996e32 q^{62} +8.42545e31 q^{63} +3.24519e32 q^{64} -6.68315e32 q^{65} +2.46492e33 q^{66} -4.33067e33 q^{67} -6.17255e31 q^{68} +6.18347e33 q^{69} +3.29617e33 q^{70} -5.77653e31 q^{71} +4.60472e32 q^{72} -2.08780e34 q^{73} +2.75610e34 q^{74} -1.00381e34 q^{75} +1.24619e34 q^{76} -4.49303e34 q^{77} +3.16807e34 q^{78} +1.43759e35 q^{79} +1.80144e34 q^{80} -2.13612e35 q^{81} -2.07032e35 q^{82} -4.18848e35 q^{83} -1.56251e35 q^{84} -3.42645e33 q^{85} -6.45832e35 q^{86} -5.59269e35 q^{87} -2.45555e35 q^{88} -8.24770e35 q^{89} +2.55614e34 q^{90} -5.77472e35 q^{91} -6.15998e35 q^{92} -8.10471e35 q^{93} -1.06206e36 q^{94} +6.91774e35 q^{95} -8.53950e35 q^{96} +6.35165e36 q^{97} -2.01783e36 q^{98} -3.48428e35 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 524288 q^{2} - 185500908 q^{3} + 137438953472 q^{4} + 7629394531250 q^{5} - 48627950026752 q^{6} - 651580251842156 q^{7} + 36\!\cdots\!68 q^{8} - 17\!\cdots\!94 q^{9} + 20\!\cdots\!00 q^{10} - 15\!\cdots\!96 q^{11}+ \cdots + 78\!\cdots\!12 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 262144. 0.707107
\(3\) −6.89815e8 −1.02799 −0.513996 0.857793i \(-0.671835\pi\)
−0.513996 + 0.857793i \(0.671835\pi\)
\(4\) 6.87195e10 0.500000
\(5\) 3.81470e12 0.447214
\(6\) −1.80831e14 −0.726900
\(7\) 3.29617e15 0.765060 0.382530 0.923943i \(-0.375053\pi\)
0.382530 + 0.923943i \(0.375053\pi\)
\(8\) 1.80144e16 0.353553
\(9\) 2.55614e16 0.0567672
\(10\) 1.00000e18 0.316228
\(11\) −1.36311e19 −0.739205 −0.369603 0.929190i \(-0.620506\pi\)
−0.369603 + 0.929190i \(0.620506\pi\)
\(12\) −4.74038e19 −0.513996
\(13\) −1.75195e20 −0.432085 −0.216043 0.976384i \(-0.569315\pi\)
−0.216043 + 0.976384i \(0.569315\pi\)
\(14\) 8.64071e20 0.540979
\(15\) −2.63144e21 −0.459732
\(16\) 4.72237e21 0.250000
\(17\) −8.98224e20 −0.0154910 −0.00774550 0.999970i \(-0.502465\pi\)
−0.00774550 + 0.999970i \(0.502465\pi\)
\(18\) 6.70076e21 0.0401405
\(19\) 1.81344e23 0.399542 0.199771 0.979843i \(-0.435980\pi\)
0.199771 + 0.979843i \(0.435980\pi\)
\(20\) 2.62144e23 0.223607
\(21\) −2.27375e24 −0.786475
\(22\) −3.57330e24 −0.522697
\(23\) −8.96395e24 −0.576149 −0.288074 0.957608i \(-0.593015\pi\)
−0.288074 + 0.957608i \(0.593015\pi\)
\(24\) −1.24266e25 −0.363450
\(25\) 1.45519e25 0.200000
\(26\) −4.59263e25 −0.305530
\(27\) 2.92980e26 0.969636
\(28\) 2.26511e26 0.382530
\(29\) 8.10751e26 0.715360 0.357680 0.933844i \(-0.383568\pi\)
0.357680 + 0.933844i \(0.383568\pi\)
\(30\) −6.89815e26 −0.325080
\(31\) 1.17491e27 0.301865 0.150933 0.988544i \(-0.451772\pi\)
0.150933 + 0.988544i \(0.451772\pi\)
\(32\) 1.23794e27 0.176777
\(33\) 9.40292e27 0.759897
\(34\) −2.35464e26 −0.0109538
\(35\) 1.25739e28 0.342145
\(36\) 1.75656e27 0.0283836
\(37\) 1.05137e29 1.02335 0.511673 0.859180i \(-0.329026\pi\)
0.511673 + 0.859180i \(0.329026\pi\)
\(38\) 4.75384e28 0.282519
\(39\) 1.20852e29 0.444180
\(40\) 6.87195e28 0.158114
\(41\) −7.89765e29 −1.15079 −0.575396 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(42\) −5.96049e29 −0.556122
\(43\) −2.46365e30 −1.48735 −0.743675 0.668541i \(-0.766919\pi\)
−0.743675 + 0.668541i \(0.766919\pi\)
\(44\) −9.36720e29 −0.369603
\(45\) 9.75088e28 0.0253871
\(46\) −2.34985e30 −0.407399
\(47\) −4.05145e30 −0.471846 −0.235923 0.971772i \(-0.575811\pi\)
−0.235923 + 0.971772i \(0.575811\pi\)
\(48\) −3.25756e30 −0.256998
\(49\) −7.69740e30 −0.414683
\(50\) 3.81470e30 0.141421
\(51\) 6.19609e29 0.0159246
\(52\) −1.20393e31 −0.216043
\(53\) −7.63383e31 −0.963027 −0.481514 0.876439i \(-0.659913\pi\)
−0.481514 + 0.876439i \(0.659913\pi\)
\(54\) 7.68030e31 0.685636
\(55\) −5.19984e31 −0.330583
\(56\) 5.93785e31 0.270490
\(57\) −1.25094e32 −0.410726
\(58\) 2.12534e32 0.505836
\(59\) 5.16782e30 0.00896489 0.00448245 0.999990i \(-0.498573\pi\)
0.00448245 + 0.999990i \(0.498573\pi\)
\(60\) −1.80831e32 −0.229866
\(61\) −3.70977e32 −0.347332 −0.173666 0.984805i \(-0.555561\pi\)
−0.173666 + 0.984805i \(0.555561\pi\)
\(62\) 3.07996e32 0.213451
\(63\) 8.42545e31 0.0434303
\(64\) 3.24519e32 0.125000
\(65\) −6.68315e32 −0.193234
\(66\) 2.46492e33 0.537328
\(67\) −4.33067e33 −0.714778 −0.357389 0.933956i \(-0.616333\pi\)
−0.357389 + 0.933956i \(0.616333\pi\)
\(68\) −6.17255e31 −0.00774550
\(69\) 6.18347e33 0.592276
\(70\) 3.29617e33 0.241933
\(71\) −5.77653e31 −0.00326128 −0.00163064 0.999999i \(-0.500519\pi\)
−0.00163064 + 0.999999i \(0.500519\pi\)
\(72\) 4.60472e32 0.0200702
\(73\) −2.08780e34 −0.705043 −0.352522 0.935804i \(-0.614676\pi\)
−0.352522 + 0.935804i \(0.614676\pi\)
\(74\) 2.75610e34 0.723615
\(75\) −1.00381e34 −0.205598
\(76\) 1.24619e34 0.199771
\(77\) −4.49303e34 −0.565536
\(78\) 3.16807e34 0.314083
\(79\) 1.43759e35 1.12599 0.562994 0.826461i \(-0.309650\pi\)
0.562994 + 0.826461i \(0.309650\pi\)
\(80\) 1.80144e34 0.111803
\(81\) −2.13612e35 −1.05354
\(82\) −2.07032e35 −0.813732
\(83\) −4.18848e35 −1.31556 −0.657782 0.753208i \(-0.728505\pi\)
−0.657782 + 0.753208i \(0.728505\pi\)
\(84\) −1.56251e35 −0.393238
\(85\) −3.42645e33 −0.00692779
\(86\) −6.45832e35 −1.05172
\(87\) −5.59269e35 −0.735384
\(88\) −2.45555e35 −0.261349
\(89\) −8.24770e35 −0.712226 −0.356113 0.934443i \(-0.615898\pi\)
−0.356113 + 0.934443i \(0.615898\pi\)
\(90\) 2.55614e34 0.0179514
\(91\) −5.77472e35 −0.330571
\(92\) −6.15998e35 −0.288074
\(93\) −8.10471e35 −0.310315
\(94\) −1.06206e36 −0.333645
\(95\) 6.91774e35 0.178681
\(96\) −8.53950e35 −0.181725
\(97\) 6.35165e36 1.11586 0.557930 0.829888i \(-0.311596\pi\)
0.557930 + 0.829888i \(0.311596\pi\)
\(98\) −2.01783e36 −0.293225
\(99\) −3.48428e35 −0.0419626
\(100\) 1.00000e36 0.100000
\(101\) −6.22694e36 −0.518000 −0.259000 0.965877i \(-0.583393\pi\)
−0.259000 + 0.965877i \(0.583393\pi\)
\(102\) 1.62427e35 0.0112604
\(103\) −2.97899e37 −1.72417 −0.862085 0.506764i \(-0.830842\pi\)
−0.862085 + 0.506764i \(0.830842\pi\)
\(104\) −3.15603e36 −0.152765
\(105\) −8.67366e36 −0.351723
\(106\) −2.00116e37 −0.680963
\(107\) −5.21404e37 −1.49133 −0.745666 0.666320i \(-0.767869\pi\)
−0.745666 + 0.666320i \(0.767869\pi\)
\(108\) 2.01334e37 0.484818
\(109\) −6.52849e37 −1.32563 −0.662815 0.748783i \(-0.730639\pi\)
−0.662815 + 0.748783i \(0.730639\pi\)
\(110\) −1.36311e37 −0.233757
\(111\) −7.25250e37 −1.05199
\(112\) 1.55657e37 0.191265
\(113\) 2.53181e37 0.263925 0.131963 0.991255i \(-0.457872\pi\)
0.131963 + 0.991255i \(0.457872\pi\)
\(114\) −3.27927e37 −0.290427
\(115\) −3.41948e37 −0.257661
\(116\) 5.57144e37 0.357680
\(117\) −4.47822e36 −0.0245283
\(118\) 1.35471e36 0.00633914
\(119\) −2.96070e36 −0.0118515
\(120\) −4.74038e37 −0.162540
\(121\) −1.54234e38 −0.453576
\(122\) −9.72493e37 −0.245601
\(123\) 5.44792e38 1.18300
\(124\) 8.07392e37 0.150933
\(125\) 5.55112e37 0.0894427
\(126\) 2.20868e37 0.0307099
\(127\) 6.80429e38 0.817361 0.408681 0.912678i \(-0.365989\pi\)
0.408681 + 0.912678i \(0.365989\pi\)
\(128\) 8.50706e37 0.0883883
\(129\) 1.69947e39 1.52898
\(130\) −1.75195e38 −0.136637
\(131\) −1.33635e39 −0.904487 −0.452243 0.891895i \(-0.649376\pi\)
−0.452243 + 0.891895i \(0.649376\pi\)
\(132\) 6.46164e38 0.379948
\(133\) 5.97742e38 0.305674
\(134\) −1.13526e39 −0.505424
\(135\) 1.11763e39 0.433634
\(136\) −1.61810e37 −0.00547690
\(137\) 2.14881e39 0.635138 0.317569 0.948235i \(-0.397134\pi\)
0.317569 + 0.948235i \(0.397134\pi\)
\(138\) 1.62096e39 0.418802
\(139\) 6.21195e39 1.40428 0.702141 0.712038i \(-0.252228\pi\)
0.702141 + 0.712038i \(0.252228\pi\)
\(140\) 8.64071e38 0.171073
\(141\) 2.79475e39 0.485054
\(142\) −1.51428e37 −0.00230607
\(143\) 2.38809e39 0.319400
\(144\) 1.20710e38 0.0141918
\(145\) 3.09277e39 0.319919
\(146\) −5.47304e39 −0.498541
\(147\) 5.30978e39 0.426291
\(148\) 7.22494e39 0.511673
\(149\) −1.54514e40 −0.966100 −0.483050 0.875593i \(-0.660471\pi\)
−0.483050 + 0.875593i \(0.660471\pi\)
\(150\) −2.63144e39 −0.145380
\(151\) −2.44879e40 −1.19640 −0.598202 0.801345i \(-0.704118\pi\)
−0.598202 + 0.801345i \(0.704118\pi\)
\(152\) 3.26681e39 0.141260
\(153\) −2.29598e37 −0.000879381 0
\(154\) −1.17782e40 −0.399895
\(155\) 4.48193e39 0.134998
\(156\) 8.30489e39 0.222090
\(157\) −5.93872e40 −1.41107 −0.705536 0.708674i \(-0.749294\pi\)
−0.705536 + 0.708674i \(0.749294\pi\)
\(158\) 3.76855e40 0.796193
\(159\) 5.26593e40 0.989984
\(160\) 4.72237e39 0.0790569
\(161\) −2.95467e40 −0.440788
\(162\) −5.59971e40 −0.744969
\(163\) 1.27298e41 1.51131 0.755653 0.654972i \(-0.227320\pi\)
0.755653 + 0.654972i \(0.227320\pi\)
\(164\) −5.42722e40 −0.575396
\(165\) 3.58693e40 0.339836
\(166\) −1.09799e41 −0.930245
\(167\) −1.13007e41 −0.856742 −0.428371 0.903603i \(-0.640912\pi\)
−0.428371 + 0.903603i \(0.640912\pi\)
\(168\) −4.09602e40 −0.278061
\(169\) −1.33708e41 −0.813302
\(170\) −8.98224e38 −0.00489868
\(171\) 4.63541e39 0.0226809
\(172\) −1.69301e41 −0.743675
\(173\) 2.46449e41 0.972464 0.486232 0.873830i \(-0.338371\pi\)
0.486232 + 0.873830i \(0.338371\pi\)
\(174\) −1.46609e41 −0.519995
\(175\) 4.79655e40 0.153012
\(176\) −6.43709e40 −0.184801
\(177\) −3.56484e39 −0.00921584
\(178\) −2.16208e41 −0.503620
\(179\) −1.68898e41 −0.354685 −0.177343 0.984149i \(-0.556750\pi\)
−0.177343 + 0.984149i \(0.556750\pi\)
\(180\) 6.70076e39 0.0126935
\(181\) −6.06854e41 −1.03760 −0.518801 0.854895i \(-0.673621\pi\)
−0.518801 + 0.854895i \(0.673621\pi\)
\(182\) −1.51381e41 −0.233749
\(183\) 2.55905e41 0.357054
\(184\) −1.61480e41 −0.203699
\(185\) 4.01065e41 0.457655
\(186\) −2.12460e41 −0.219426
\(187\) 1.22437e40 0.0114510
\(188\) −2.78414e41 −0.235923
\(189\) 9.65712e41 0.741829
\(190\) 1.81344e41 0.126346
\(191\) 1.09994e42 0.695429 0.347714 0.937600i \(-0.386958\pi\)
0.347714 + 0.937600i \(0.386958\pi\)
\(192\) −2.23858e41 −0.128499
\(193\) 3.18819e42 1.66240 0.831198 0.555977i \(-0.187656\pi\)
0.831198 + 0.555977i \(0.187656\pi\)
\(194\) 1.66505e42 0.789032
\(195\) 4.61014e41 0.198643
\(196\) −5.28961e41 −0.207342
\(197\) 2.35999e42 0.841948 0.420974 0.907073i \(-0.361688\pi\)
0.420974 + 0.907073i \(0.361688\pi\)
\(198\) −9.13384e40 −0.0296720
\(199\) 5.36739e42 1.58848 0.794238 0.607606i \(-0.207870\pi\)
0.794238 + 0.607606i \(0.207870\pi\)
\(200\) 2.62144e41 0.0707107
\(201\) 2.98736e42 0.734786
\(202\) −1.63236e42 −0.366281
\(203\) 2.67237e42 0.547293
\(204\) 4.25792e40 0.00796231
\(205\) −3.01271e42 −0.514649
\(206\) −7.80924e42 −1.21917
\(207\) −2.29131e41 −0.0327063
\(208\) −8.27334e41 −0.108021
\(209\) −2.47192e42 −0.295344
\(210\) −2.27375e42 −0.248705
\(211\) 6.79782e42 0.680996 0.340498 0.940245i \(-0.389404\pi\)
0.340498 + 0.940245i \(0.389404\pi\)
\(212\) −5.24592e42 −0.481514
\(213\) 3.98474e40 0.00335257
\(214\) −1.36683e43 −1.05453
\(215\) −9.39810e42 −0.665163
\(216\) 5.27786e42 0.342818
\(217\) 3.87270e42 0.230945
\(218\) −1.71140e43 −0.937363
\(219\) 1.44020e43 0.724778
\(220\) −3.57330e42 −0.165291
\(221\) 1.57364e41 0.00669343
\(222\) −1.90120e43 −0.743871
\(223\) −3.83398e42 −0.138042 −0.0690208 0.997615i \(-0.521987\pi\)
−0.0690208 + 0.997615i \(0.521987\pi\)
\(224\) 4.08046e42 0.135245
\(225\) 3.71967e41 0.0113534
\(226\) 6.63700e42 0.186623
\(227\) 1.43677e42 0.0372313 0.0186157 0.999827i \(-0.494074\pi\)
0.0186157 + 0.999827i \(0.494074\pi\)
\(228\) −8.59641e42 −0.205363
\(229\) 2.80830e43 0.618710 0.309355 0.950947i \(-0.399887\pi\)
0.309355 + 0.950947i \(0.399887\pi\)
\(230\) −8.96395e42 −0.182194
\(231\) 3.09936e43 0.581367
\(232\) 1.46052e43 0.252918
\(233\) −1.02012e44 −1.63143 −0.815716 0.578453i \(-0.803657\pi\)
−0.815716 + 0.578453i \(0.803657\pi\)
\(234\) −1.17394e42 −0.0173441
\(235\) −1.54551e43 −0.211016
\(236\) 3.55130e41 0.00448245
\(237\) −9.91671e43 −1.15751
\(238\) −7.76129e41 −0.00838031
\(239\) 1.19632e43 0.119532 0.0597661 0.998212i \(-0.480965\pi\)
0.0597661 + 0.998212i \(0.480965\pi\)
\(240\) −1.24266e43 −0.114933
\(241\) −2.33793e43 −0.200224 −0.100112 0.994976i \(-0.531920\pi\)
−0.100112 + 0.994976i \(0.531920\pi\)
\(242\) −4.04314e43 −0.320726
\(243\) 1.54287e43 0.113400
\(244\) −2.54933e43 −0.173666
\(245\) −2.93632e43 −0.185452
\(246\) 1.42814e44 0.836510
\(247\) −3.17706e43 −0.172636
\(248\) 2.11653e43 0.106726
\(249\) 2.88928e44 1.35239
\(250\) 1.45519e43 0.0632456
\(251\) 1.57065e44 0.634040 0.317020 0.948419i \(-0.397318\pi\)
0.317020 + 0.948419i \(0.397318\pi\)
\(252\) 5.78993e42 0.0217152
\(253\) 1.22188e44 0.425892
\(254\) 1.78370e44 0.577962
\(255\) 2.36362e42 0.00712171
\(256\) 2.23007e43 0.0625000
\(257\) −6.40421e44 −1.66995 −0.834973 0.550291i \(-0.814517\pi\)
−0.834973 + 0.550291i \(0.814517\pi\)
\(258\) 4.45505e44 1.08116
\(259\) 3.46548e44 0.782922
\(260\) −4.59263e43 −0.0966172
\(261\) 2.07239e43 0.0406090
\(262\) −3.50317e44 −0.639569
\(263\) 6.42820e44 1.09372 0.546862 0.837223i \(-0.315822\pi\)
0.546862 + 0.837223i \(0.315822\pi\)
\(264\) 1.69388e44 0.268664
\(265\) −2.91207e44 −0.430679
\(266\) 1.56694e44 0.216144
\(267\) 5.68939e44 0.732162
\(268\) −2.97601e44 −0.357389
\(269\) −4.52122e44 −0.506803 −0.253401 0.967361i \(-0.581549\pi\)
−0.253401 + 0.967361i \(0.581549\pi\)
\(270\) 2.92980e44 0.306626
\(271\) −5.63666e44 −0.550922 −0.275461 0.961312i \(-0.588830\pi\)
−0.275461 + 0.961312i \(0.588830\pi\)
\(272\) −4.24174e42 −0.00387275
\(273\) 3.98349e44 0.339824
\(274\) 5.63298e44 0.449110
\(275\) −1.98358e44 −0.147841
\(276\) 4.24925e44 0.296138
\(277\) −8.72981e44 −0.569022 −0.284511 0.958673i \(-0.591831\pi\)
−0.284511 + 0.958673i \(0.591831\pi\)
\(278\) 1.62843e45 0.992977
\(279\) 3.00323e43 0.0171361
\(280\) 2.26511e44 0.120967
\(281\) −3.74401e45 −1.87184 −0.935922 0.352206i \(-0.885432\pi\)
−0.935922 + 0.352206i \(0.885432\pi\)
\(282\) 7.32628e44 0.342985
\(283\) 1.09059e45 0.478204 0.239102 0.970994i \(-0.423147\pi\)
0.239102 + 0.970994i \(0.423147\pi\)
\(284\) −3.96960e42 −0.00163064
\(285\) −4.77197e44 −0.183682
\(286\) 6.26024e44 0.225850
\(287\) −2.60320e45 −0.880424
\(288\) 3.16434e43 0.0100351
\(289\) −3.36129e45 −0.999760
\(290\) 8.10751e44 0.226217
\(291\) −4.38147e45 −1.14709
\(292\) −1.43472e45 −0.352522
\(293\) −1.50027e45 −0.346033 −0.173017 0.984919i \(-0.555351\pi\)
−0.173017 + 0.984919i \(0.555351\pi\)
\(294\) 1.39193e45 0.301433
\(295\) 1.97137e43 0.00400922
\(296\) 1.89398e45 0.361808
\(297\) −3.99363e45 −0.716760
\(298\) −4.05049e45 −0.683136
\(299\) 1.57044e45 0.248945
\(300\) −6.89815e44 −0.102799
\(301\) −8.12062e45 −1.13791
\(302\) −6.41936e45 −0.845986
\(303\) 4.29544e45 0.532500
\(304\) 8.56375e44 0.0998856
\(305\) −1.41516e45 −0.155332
\(306\) −6.01878e42 −0.000621816 0
\(307\) 8.88941e45 0.864595 0.432298 0.901731i \(-0.357703\pi\)
0.432298 + 0.901731i \(0.357703\pi\)
\(308\) −3.08758e45 −0.282768
\(309\) 2.05495e46 1.77243
\(310\) 1.17491e45 0.0954582
\(311\) 1.86301e46 1.42609 0.713045 0.701118i \(-0.247316\pi\)
0.713045 + 0.701118i \(0.247316\pi\)
\(312\) 2.17708e45 0.157041
\(313\) −1.03471e46 −0.703476 −0.351738 0.936098i \(-0.614409\pi\)
−0.351738 + 0.936098i \(0.614409\pi\)
\(314\) −1.55680e46 −0.997779
\(315\) 3.21405e44 0.0194226
\(316\) 9.87903e45 0.562994
\(317\) 3.08721e46 1.65947 0.829736 0.558157i \(-0.188491\pi\)
0.829736 + 0.558157i \(0.188491\pi\)
\(318\) 1.38043e46 0.700025
\(319\) −1.10514e46 −0.528798
\(320\) 1.23794e45 0.0559017
\(321\) 3.59672e46 1.53308
\(322\) −7.74549e45 −0.311684
\(323\) −1.62888e44 −0.00618931
\(324\) −1.46793e46 −0.526772
\(325\) −2.54942e45 −0.0864170
\(326\) 3.33705e46 1.06865
\(327\) 4.50345e46 1.36274
\(328\) −1.42271e46 −0.406866
\(329\) −1.33543e46 −0.360990
\(330\) 9.40292e45 0.240301
\(331\) −3.83499e46 −0.926716 −0.463358 0.886171i \(-0.653356\pi\)
−0.463358 + 0.886171i \(0.653356\pi\)
\(332\) −2.87830e46 −0.657782
\(333\) 2.68744e45 0.0580925
\(334\) −2.96241e46 −0.605808
\(335\) −1.65202e46 −0.319658
\(336\) −1.07375e46 −0.196619
\(337\) 3.97446e46 0.688850 0.344425 0.938814i \(-0.388074\pi\)
0.344425 + 0.938814i \(0.388074\pi\)
\(338\) −3.50506e46 −0.575092
\(339\) −1.74648e46 −0.271313
\(340\) −2.35464e44 −0.00346389
\(341\) −1.60153e46 −0.223141
\(342\) 1.21514e45 0.0160378
\(343\) −8.65557e46 −1.08232
\(344\) −4.43813e46 −0.525858
\(345\) 2.35881e46 0.264874
\(346\) 6.46050e46 0.687636
\(347\) 1.01071e47 1.01983 0.509917 0.860223i \(-0.329676\pi\)
0.509917 + 0.860223i \(0.329676\pi\)
\(348\) −3.84326e46 −0.367692
\(349\) 6.03446e46 0.547480 0.273740 0.961804i \(-0.411739\pi\)
0.273740 + 0.961804i \(0.411739\pi\)
\(350\) 1.25739e46 0.108196
\(351\) −5.13286e46 −0.418965
\(352\) −1.68744e46 −0.130674
\(353\) 6.49991e46 0.477612 0.238806 0.971067i \(-0.423244\pi\)
0.238806 + 0.971067i \(0.423244\pi\)
\(354\) −9.34502e44 −0.00651658
\(355\) −2.20357e44 −0.00145849
\(356\) −5.66777e46 −0.356113
\(357\) 2.04233e45 0.0121833
\(358\) −4.42756e46 −0.250801
\(359\) −1.34716e47 −0.724724 −0.362362 0.932038i \(-0.618030\pi\)
−0.362362 + 0.932038i \(0.618030\pi\)
\(360\) 1.75656e45 0.00897568
\(361\) −1.73122e47 −0.840366
\(362\) −1.59083e47 −0.733695
\(363\) 1.06393e47 0.466272
\(364\) −3.96835e46 −0.165286
\(365\) −7.96432e46 −0.315305
\(366\) 6.70841e46 0.252476
\(367\) 1.76042e47 0.629934 0.314967 0.949103i \(-0.398007\pi\)
0.314967 + 0.949103i \(0.398007\pi\)
\(368\) −4.23311e46 −0.144037
\(369\) −2.01875e46 −0.0653272
\(370\) 1.05137e47 0.323611
\(371\) −2.51624e47 −0.736774
\(372\) −5.56952e46 −0.155158
\(373\) 3.37762e47 0.895359 0.447679 0.894194i \(-0.352251\pi\)
0.447679 + 0.894194i \(0.352251\pi\)
\(374\) 3.20962e45 0.00809710
\(375\) −3.82924e46 −0.0919464
\(376\) −7.29844e46 −0.166823
\(377\) −1.42039e47 −0.309096
\(378\) 2.53156e47 0.524553
\(379\) −6.92554e47 −1.36656 −0.683279 0.730157i \(-0.739447\pi\)
−0.683279 + 0.730157i \(0.739447\pi\)
\(380\) 4.75384e46 0.0893404
\(381\) −4.69370e47 −0.840240
\(382\) 2.88343e47 0.491743
\(383\) 2.21003e47 0.359105 0.179553 0.983748i \(-0.442535\pi\)
0.179553 + 0.983748i \(0.442535\pi\)
\(384\) −5.86830e46 −0.0908625
\(385\) −1.71395e47 −0.252916
\(386\) 8.35765e47 1.17549
\(387\) −6.29743e46 −0.0844327
\(388\) 4.36482e47 0.557930
\(389\) 3.20256e47 0.390328 0.195164 0.980771i \(-0.437476\pi\)
0.195164 + 0.980771i \(0.437476\pi\)
\(390\) 1.20852e47 0.140462
\(391\) 8.05163e45 0.00892512
\(392\) −1.38664e47 −0.146613
\(393\) 9.21837e47 0.929805
\(394\) 6.18657e47 0.595347
\(395\) 5.48396e47 0.503557
\(396\) −2.39438e46 −0.0209813
\(397\) 2.55758e47 0.213897 0.106948 0.994265i \(-0.465892\pi\)
0.106948 + 0.994265i \(0.465892\pi\)
\(398\) 1.40703e48 1.12322
\(399\) −4.12331e47 −0.314230
\(400\) 6.87195e46 0.0500000
\(401\) 2.29575e48 1.59498 0.797488 0.603334i \(-0.206161\pi\)
0.797488 + 0.603334i \(0.206161\pi\)
\(402\) 7.83119e47 0.519572
\(403\) −2.05838e47 −0.130432
\(404\) −4.27912e47 −0.259000
\(405\) −8.14865e47 −0.471160
\(406\) 7.00546e47 0.386995
\(407\) −1.43313e48 −0.756463
\(408\) 1.11619e46 0.00563020
\(409\) −1.23961e48 −0.597592 −0.298796 0.954317i \(-0.596585\pi\)
−0.298796 + 0.954317i \(0.596585\pi\)
\(410\) −7.89765e47 −0.363912
\(411\) −1.48228e48 −0.652916
\(412\) −2.04714e48 −0.862085
\(413\) 1.70340e46 0.00685868
\(414\) −6.00652e46 −0.0231269
\(415\) −1.59778e48 −0.588338
\(416\) −2.16881e47 −0.0763826
\(417\) −4.28510e48 −1.44359
\(418\) −6.47998e47 −0.208840
\(419\) 3.37300e48 1.04006 0.520029 0.854149i \(-0.325921\pi\)
0.520029 + 0.854149i \(0.325921\pi\)
\(420\) −5.96049e47 −0.175861
\(421\) 2.03580e48 0.574798 0.287399 0.957811i \(-0.407209\pi\)
0.287399 + 0.957811i \(0.407209\pi\)
\(422\) 1.78201e48 0.481537
\(423\) −1.03561e47 −0.0267854
\(424\) −1.37519e48 −0.340482
\(425\) −1.30709e46 −0.00309820
\(426\) 1.04458e46 0.00237062
\(427\) −1.22280e48 −0.265730
\(428\) −3.58306e48 −0.745666
\(429\) −1.64734e48 −0.328340
\(430\) −2.46365e48 −0.470342
\(431\) 6.24281e48 1.14170 0.570848 0.821056i \(-0.306615\pi\)
0.570848 + 0.821056i \(0.306615\pi\)
\(432\) 1.38356e48 0.242409
\(433\) −4.71991e48 −0.792333 −0.396166 0.918179i \(-0.629660\pi\)
−0.396166 + 0.918179i \(0.629660\pi\)
\(434\) 1.01521e48 0.163303
\(435\) −2.13344e48 −0.328874
\(436\) −4.48635e48 −0.662815
\(437\) −1.62556e48 −0.230196
\(438\) 3.77539e48 0.512496
\(439\) −7.57651e48 −0.985997 −0.492999 0.870030i \(-0.664099\pi\)
−0.492999 + 0.870030i \(0.664099\pi\)
\(440\) −9.36720e47 −0.116879
\(441\) −1.96756e47 −0.0235404
\(442\) 4.12521e46 0.00473297
\(443\) 8.83812e48 0.972503 0.486252 0.873819i \(-0.338364\pi\)
0.486252 + 0.873819i \(0.338364\pi\)
\(444\) −4.98388e48 −0.525996
\(445\) −3.14625e48 −0.318517
\(446\) −1.00505e48 −0.0976101
\(447\) 1.06586e49 0.993143
\(448\) 1.06967e48 0.0956325
\(449\) −2.01986e49 −1.73286 −0.866429 0.499300i \(-0.833590\pi\)
−0.866429 + 0.499300i \(0.833590\pi\)
\(450\) 9.75088e46 0.00802809
\(451\) 1.07653e49 0.850671
\(452\) 1.73985e48 0.131963
\(453\) 1.68921e49 1.22989
\(454\) 3.76640e47 0.0263265
\(455\) −2.20288e48 −0.147836
\(456\) −2.25350e48 −0.145214
\(457\) 6.61916e48 0.409594 0.204797 0.978804i \(-0.434347\pi\)
0.204797 + 0.978804i \(0.434347\pi\)
\(458\) 7.36180e48 0.437494
\(459\) −2.63162e47 −0.0150206
\(460\) −2.34985e48 −0.128831
\(461\) 2.46377e49 1.29758 0.648788 0.760969i \(-0.275276\pi\)
0.648788 + 0.760969i \(0.275276\pi\)
\(462\) 8.12478e48 0.411088
\(463\) 3.33469e49 1.62109 0.810546 0.585675i \(-0.199171\pi\)
0.810546 + 0.585675i \(0.199171\pi\)
\(464\) 3.82866e48 0.178840
\(465\) −3.09170e48 −0.138777
\(466\) −2.67420e49 −1.15360
\(467\) 9.90311e48 0.410591 0.205295 0.978700i \(-0.434184\pi\)
0.205295 + 0.978700i \(0.434184\pi\)
\(468\) −3.07741e47 −0.0122641
\(469\) −1.42746e49 −0.546848
\(470\) −4.05145e48 −0.149211
\(471\) 4.09662e49 1.45057
\(472\) 9.30952e46 0.00316957
\(473\) 3.35822e49 1.09946
\(474\) −2.59961e49 −0.818480
\(475\) 2.63891e48 0.0799084
\(476\) −2.03457e47 −0.00592577
\(477\) −1.95131e48 −0.0546684
\(478\) 3.13607e48 0.0845221
\(479\) −3.54457e49 −0.919086 −0.459543 0.888156i \(-0.651987\pi\)
−0.459543 + 0.888156i \(0.651987\pi\)
\(480\) −3.25756e48 −0.0812699
\(481\) −1.84194e49 −0.442173
\(482\) −6.12874e48 −0.141580
\(483\) 2.03818e49 0.453127
\(484\) −1.05989e49 −0.226788
\(485\) 2.42296e49 0.499028
\(486\) 4.04453e48 0.0801857
\(487\) −5.61975e49 −1.07258 −0.536292 0.844032i \(-0.680175\pi\)
−0.536292 + 0.844032i \(0.680175\pi\)
\(488\) −6.68292e48 −0.122800
\(489\) −8.78124e49 −1.55361
\(490\) −7.69740e48 −0.131134
\(491\) −6.73087e49 −1.10424 −0.552120 0.833764i \(-0.686181\pi\)
−0.552120 + 0.833764i \(0.686181\pi\)
\(492\) 3.74378e49 0.591502
\(493\) −7.28236e47 −0.0110816
\(494\) −8.32848e48 −0.122072
\(495\) −1.32915e48 −0.0187662
\(496\) 5.54836e48 0.0754664
\(497\) −1.90404e47 −0.00249507
\(498\) 7.57408e49 0.956284
\(499\) −1.57170e50 −1.91209 −0.956045 0.293221i \(-0.905273\pi\)
−0.956045 + 0.293221i \(0.905273\pi\)
\(500\) 3.81470e48 0.0447214
\(501\) 7.79539e49 0.880724
\(502\) 4.11738e49 0.448334
\(503\) 8.33167e49 0.874429 0.437214 0.899357i \(-0.355965\pi\)
0.437214 + 0.899357i \(0.355965\pi\)
\(504\) 1.51779e48 0.0153549
\(505\) −2.37539e49 −0.231657
\(506\) 3.20309e49 0.301151
\(507\) 9.22336e49 0.836068
\(508\) 4.67587e49 0.408681
\(509\) −1.50351e50 −1.26715 −0.633575 0.773682i \(-0.718413\pi\)
−0.633575 + 0.773682i \(0.718413\pi\)
\(510\) 6.19609e47 0.00503581
\(511\) −6.88173e49 −0.539400
\(512\) 5.84601e48 0.0441942
\(513\) 5.31303e49 0.387410
\(514\) −1.67882e50 −1.18083
\(515\) −1.13639e50 −0.771072
\(516\) 1.16786e50 0.764492
\(517\) 5.52256e49 0.348791
\(518\) 9.08456e49 0.553609
\(519\) −1.70004e50 −0.999685
\(520\) −1.20393e49 −0.0683187
\(521\) 5.18958e49 0.284207 0.142103 0.989852i \(-0.454613\pi\)
0.142103 + 0.989852i \(0.454613\pi\)
\(522\) 5.43264e48 0.0287149
\(523\) −1.48076e50 −0.755445 −0.377723 0.925919i \(-0.623293\pi\)
−0.377723 + 0.925919i \(0.623293\pi\)
\(524\) −9.18335e49 −0.452243
\(525\) −3.30874e49 −0.157295
\(526\) 1.68511e50 0.773380
\(527\) −1.05533e48 −0.00467620
\(528\) 4.44040e49 0.189974
\(529\) −1.61711e50 −0.668053
\(530\) −7.63383e49 −0.304536
\(531\) 1.32096e47 0.000508912 0
\(532\) 4.10765e49 0.152837
\(533\) 1.38363e50 0.497240
\(534\) 1.49144e50 0.517717
\(535\) −1.98900e50 −0.666944
\(536\) −7.80144e49 −0.252712
\(537\) 1.16508e50 0.364614
\(538\) −1.18521e50 −0.358364
\(539\) 1.04924e50 0.306536
\(540\) 7.68030e49 0.216817
\(541\) 5.01469e50 1.36803 0.684013 0.729470i \(-0.260233\pi\)
0.684013 + 0.729470i \(0.260233\pi\)
\(542\) −1.47762e50 −0.389561
\(543\) 4.18617e50 1.06665
\(544\) −1.11195e48 −0.00273845
\(545\) −2.49042e50 −0.592840
\(546\) 1.04425e50 0.240292
\(547\) 4.97650e50 1.10703 0.553513 0.832840i \(-0.313287\pi\)
0.553513 + 0.832840i \(0.313287\pi\)
\(548\) 1.47665e50 0.317569
\(549\) −9.48266e48 −0.0197171
\(550\) −5.19984e49 −0.104539
\(551\) 1.47025e50 0.285816
\(552\) 1.11392e50 0.209401
\(553\) 4.73853e50 0.861448
\(554\) −2.28847e50 −0.402359
\(555\) −2.76661e50 −0.470465
\(556\) 4.26882e50 0.702141
\(557\) 6.67759e49 0.106243 0.0531213 0.998588i \(-0.483083\pi\)
0.0531213 + 0.998588i \(0.483083\pi\)
\(558\) 7.87279e48 0.0121170
\(559\) 4.31620e50 0.642662
\(560\) 5.93785e49 0.0855363
\(561\) −8.44592e48 −0.0117716
\(562\) −9.81469e50 −1.32359
\(563\) 1.01553e51 1.32522 0.662610 0.748964i \(-0.269449\pi\)
0.662610 + 0.748964i \(0.269449\pi\)
\(564\) 1.92054e50 0.242527
\(565\) 9.65810e49 0.118031
\(566\) 2.85892e50 0.338142
\(567\) −7.04101e50 −0.806025
\(568\) −1.04061e48 −0.00115304
\(569\) −4.88120e50 −0.523540 −0.261770 0.965130i \(-0.584306\pi\)
−0.261770 + 0.965130i \(0.584306\pi\)
\(570\) −1.25094e50 −0.129883
\(571\) −6.10676e49 −0.0613822 −0.0306911 0.999529i \(-0.509771\pi\)
−0.0306911 + 0.999529i \(0.509771\pi\)
\(572\) 1.64108e50 0.159700
\(573\) −7.58757e50 −0.714895
\(574\) −6.82413e50 −0.622554
\(575\) −1.30443e50 −0.115230
\(576\) 8.29513e48 0.00709590
\(577\) −5.54304e50 −0.459193 −0.229597 0.973286i \(-0.573741\pi\)
−0.229597 + 0.973286i \(0.573741\pi\)
\(578\) −8.81142e50 −0.706937
\(579\) −2.19926e51 −1.70893
\(580\) 2.12534e50 0.159959
\(581\) −1.38059e51 −1.00649
\(582\) −1.14858e51 −0.811118
\(583\) 1.04057e51 0.711875
\(584\) −3.76104e50 −0.249270
\(585\) −1.70830e49 −0.0109694
\(586\) −3.93286e50 −0.244682
\(587\) 7.00286e50 0.422154 0.211077 0.977469i \(-0.432303\pi\)
0.211077 + 0.977469i \(0.432303\pi\)
\(588\) 3.64885e50 0.213145
\(589\) 2.13064e50 0.120608
\(590\) 5.16782e48 0.00283495
\(591\) −1.62796e51 −0.865515
\(592\) 4.96494e50 0.255837
\(593\) 3.03179e51 1.51421 0.757106 0.653292i \(-0.226613\pi\)
0.757106 + 0.653292i \(0.226613\pi\)
\(594\) −1.04691e51 −0.506826
\(595\) −1.12942e49 −0.00530017
\(596\) −1.06181e51 −0.483050
\(597\) −3.70251e51 −1.63294
\(598\) 4.11681e50 0.176031
\(599\) 3.37693e51 1.39999 0.699997 0.714145i \(-0.253184\pi\)
0.699997 + 0.714145i \(0.253184\pi\)
\(600\) −1.80831e50 −0.0726900
\(601\) 3.37420e51 1.31521 0.657603 0.753365i \(-0.271571\pi\)
0.657603 + 0.753365i \(0.271571\pi\)
\(602\) −2.12877e51 −0.804626
\(603\) −1.10698e50 −0.0405759
\(604\) −1.68280e51 −0.598202
\(605\) −5.88355e50 −0.202845
\(606\) 1.12602e51 0.376534
\(607\) 5.86428e51 1.90206 0.951029 0.309101i \(-0.100028\pi\)
0.951029 + 0.309101i \(0.100028\pi\)
\(608\) 2.24494e50 0.0706298
\(609\) −1.84344e51 −0.562613
\(610\) −3.70977e50 −0.109836
\(611\) 7.09793e50 0.203878
\(612\) −1.57779e48 −0.000439690 0
\(613\) −6.86486e50 −0.185615 −0.0928075 0.995684i \(-0.529584\pi\)
−0.0928075 + 0.995684i \(0.529584\pi\)
\(614\) 2.33031e51 0.611361
\(615\) 2.07822e51 0.529055
\(616\) −8.09392e50 −0.199947
\(617\) −2.90878e51 −0.697326 −0.348663 0.937248i \(-0.613364\pi\)
−0.348663 + 0.937248i \(0.613364\pi\)
\(618\) 5.38693e51 1.25330
\(619\) 4.53284e51 1.02351 0.511756 0.859131i \(-0.328995\pi\)
0.511756 + 0.859131i \(0.328995\pi\)
\(620\) 3.07996e50 0.0674992
\(621\) −2.62626e51 −0.558654
\(622\) 4.88376e51 1.00840
\(623\) −2.71858e51 −0.544895
\(624\) 5.70708e50 0.111045
\(625\) 2.11758e50 0.0400000
\(626\) −2.71244e51 −0.497433
\(627\) 1.70517e51 0.303611
\(628\) −4.08106e51 −0.705536
\(629\) −9.44363e49 −0.0158527
\(630\) 8.42545e49 0.0137339
\(631\) −4.30708e50 −0.0681772 −0.0340886 0.999419i \(-0.510853\pi\)
−0.0340886 + 0.999419i \(0.510853\pi\)
\(632\) 2.58973e51 0.398097
\(633\) −4.68924e51 −0.700058
\(634\) 8.09293e51 1.17342
\(635\) 2.59563e51 0.365535
\(636\) 3.61872e51 0.494992
\(637\) 1.34854e51 0.179178
\(638\) −2.89706e51 −0.373916
\(639\) −1.47656e48 −0.000185133 0
\(640\) 3.24519e50 0.0395285
\(641\) −3.95135e51 −0.467597 −0.233799 0.972285i \(-0.575116\pi\)
−0.233799 + 0.972285i \(0.575116\pi\)
\(642\) 9.42860e51 1.08405
\(643\) −1.17106e52 −1.30821 −0.654103 0.756406i \(-0.726954\pi\)
−0.654103 + 0.756406i \(0.726954\pi\)
\(644\) −2.03043e51 −0.220394
\(645\) 6.48295e51 0.683783
\(646\) −4.27001e49 −0.00437650
\(647\) 2.47032e51 0.246051 0.123025 0.992404i \(-0.460740\pi\)
0.123025 + 0.992404i \(0.460740\pi\)
\(648\) −3.84809e51 −0.372484
\(649\) −7.04429e49 −0.00662690
\(650\) −6.68315e50 −0.0611061
\(651\) −2.67145e51 −0.237410
\(652\) 8.74788e51 0.755653
\(653\) −1.97344e52 −1.65703 −0.828513 0.559970i \(-0.810812\pi\)
−0.828513 + 0.559970i \(0.810812\pi\)
\(654\) 1.18055e52 0.963601
\(655\) −5.09778e51 −0.404499
\(656\) −3.72956e51 −0.287698
\(657\) −5.33669e50 −0.0400233
\(658\) −3.50074e51 −0.255259
\(659\) −1.00529e52 −0.712711 −0.356355 0.934351i \(-0.615981\pi\)
−0.356355 + 0.934351i \(0.615981\pi\)
\(660\) 2.46492e51 0.169918
\(661\) 1.30458e52 0.874468 0.437234 0.899348i \(-0.355958\pi\)
0.437234 + 0.899348i \(0.355958\pi\)
\(662\) −1.00532e52 −0.655287
\(663\) −1.08552e50 −0.00688079
\(664\) −7.54530e51 −0.465122
\(665\) 2.28020e51 0.136701
\(666\) 7.04496e50 0.0410776
\(667\) −7.26753e51 −0.412154
\(668\) −7.76577e51 −0.428371
\(669\) 2.64474e51 0.141906
\(670\) −4.33067e51 −0.226033
\(671\) 5.05681e51 0.256750
\(672\) −2.81476e51 −0.139031
\(673\) −2.06557e52 −0.992570 −0.496285 0.868160i \(-0.665303\pi\)
−0.496285 + 0.868160i \(0.665303\pi\)
\(674\) 1.04188e52 0.487091
\(675\) 4.26342e51 0.193927
\(676\) −9.18832e51 −0.406651
\(677\) −1.48357e52 −0.638876 −0.319438 0.947607i \(-0.603494\pi\)
−0.319438 + 0.947607i \(0.603494\pi\)
\(678\) −4.57830e51 −0.191847
\(679\) 2.09361e52 0.853700
\(680\) −6.17255e49 −0.00244934
\(681\) −9.91105e50 −0.0382735
\(682\) −4.19831e51 −0.157784
\(683\) −1.63537e51 −0.0598183 −0.0299092 0.999553i \(-0.509522\pi\)
−0.0299092 + 0.999553i \(0.509522\pi\)
\(684\) 3.18543e50 0.0113404
\(685\) 8.19706e51 0.284042
\(686\) −2.26901e52 −0.765314
\(687\) −1.93721e52 −0.636029
\(688\) −1.16343e52 −0.371838
\(689\) 1.33741e52 0.416110
\(690\) 6.18347e51 0.187294
\(691\) −2.87913e51 −0.0849018 −0.0424509 0.999099i \(-0.513517\pi\)
−0.0424509 + 0.999099i \(0.513517\pi\)
\(692\) 1.69358e52 0.486232
\(693\) −1.14848e51 −0.0321039
\(694\) 2.64951e52 0.721132
\(695\) 2.36967e52 0.628014
\(696\) −1.00749e52 −0.259998
\(697\) 7.09386e50 0.0178269
\(698\) 1.58190e52 0.387127
\(699\) 7.03698e52 1.67710
\(700\) 3.29617e51 0.0765060
\(701\) −4.36552e52 −0.986852 −0.493426 0.869788i \(-0.664256\pi\)
−0.493426 + 0.869788i \(0.664256\pi\)
\(702\) −1.34555e52 −0.296253
\(703\) 1.90660e52 0.408870
\(704\) −4.42353e51 −0.0924007
\(705\) 1.06611e52 0.216923
\(706\) 1.70391e52 0.337723
\(707\) −2.05251e52 −0.396301
\(708\) −2.44974e50 −0.00460792
\(709\) 6.12553e52 1.12251 0.561253 0.827645i \(-0.310320\pi\)
0.561253 + 0.827645i \(0.310320\pi\)
\(710\) −5.77653e49 −0.00103131
\(711\) 3.67467e51 0.0639191
\(712\) −1.48577e52 −0.251810
\(713\) −1.05318e52 −0.173919
\(714\) 5.35385e50 0.00861489
\(715\) 9.10985e51 0.142840
\(716\) −1.16066e52 −0.177343
\(717\) −8.25238e51 −0.122878
\(718\) −3.53150e52 −0.512457
\(719\) 6.31848e52 0.893569 0.446784 0.894642i \(-0.352569\pi\)
0.446784 + 0.894642i \(0.352569\pi\)
\(720\) 4.60472e50 0.00634677
\(721\) −9.81924e52 −1.31909
\(722\) −4.53828e52 −0.594228
\(723\) 1.61274e52 0.205829
\(724\) −4.17027e52 −0.518801
\(725\) 1.17980e52 0.143072
\(726\) 2.78902e52 0.329704
\(727\) 5.46946e52 0.630316 0.315158 0.949039i \(-0.397943\pi\)
0.315158 + 0.949039i \(0.397943\pi\)
\(728\) −1.04028e52 −0.116875
\(729\) 8.55432e52 0.936971
\(730\) −2.08780e52 −0.222954
\(731\) 2.21291e51 0.0230405
\(732\) 1.75857e52 0.178527
\(733\) 3.01068e52 0.298017 0.149008 0.988836i \(-0.452392\pi\)
0.149008 + 0.988836i \(0.452392\pi\)
\(734\) 4.61485e52 0.445431
\(735\) 2.02552e52 0.190643
\(736\) −1.10968e52 −0.101850
\(737\) 5.90316e52 0.528367
\(738\) −5.29202e51 −0.0461933
\(739\) 1.55405e53 1.32295 0.661476 0.749967i \(-0.269930\pi\)
0.661476 + 0.749967i \(0.269930\pi\)
\(740\) 2.75610e52 0.228827
\(741\) 2.19159e52 0.177469
\(742\) −6.59616e52 −0.520978
\(743\) 1.58003e53 1.21723 0.608613 0.793467i \(-0.291726\pi\)
0.608613 + 0.793467i \(0.291726\pi\)
\(744\) −1.46002e52 −0.109713
\(745\) −5.89424e52 −0.432053
\(746\) 8.85422e52 0.633114
\(747\) −1.07063e52 −0.0746809
\(748\) 8.41384e50 0.00572551
\(749\) −1.71863e53 −1.14096
\(750\) −1.00381e52 −0.0650159
\(751\) −7.41700e52 −0.468694 −0.234347 0.972153i \(-0.575295\pi\)
−0.234347 + 0.972153i \(0.575295\pi\)
\(752\) −1.91324e52 −0.117961
\(753\) −1.08346e53 −0.651788
\(754\) −3.72348e52 −0.218564
\(755\) −9.34139e52 −0.535048
\(756\) 6.63632e52 0.370915
\(757\) −2.14270e53 −1.16866 −0.584329 0.811517i \(-0.698642\pi\)
−0.584329 + 0.811517i \(0.698642\pi\)
\(758\) −1.81549e53 −0.966303
\(759\) −8.42873e52 −0.437814
\(760\) 1.24619e52 0.0631732
\(761\) −4.02061e53 −1.98919 −0.994594 0.103844i \(-0.966886\pi\)
−0.994594 + 0.103844i \(0.966886\pi\)
\(762\) −1.23043e53 −0.594140
\(763\) −2.15190e53 −1.01419
\(764\) 7.55875e52 0.347714
\(765\) −8.75847e49 −0.000393271 0
\(766\) 5.79347e52 0.253926
\(767\) −9.05375e50 −0.00387360
\(768\) −1.53834e52 −0.0642495
\(769\) −1.75938e53 −0.717335 −0.358667 0.933465i \(-0.616769\pi\)
−0.358667 + 0.933465i \(0.616769\pi\)
\(770\) −4.49303e52 −0.178838
\(771\) 4.41772e53 1.71669
\(772\) 2.19091e53 0.831198
\(773\) −2.25368e53 −0.834781 −0.417390 0.908727i \(-0.637055\pi\)
−0.417390 + 0.908727i \(0.637055\pi\)
\(774\) −1.65083e52 −0.0597029
\(775\) 1.70972e52 0.0603731
\(776\) 1.14421e53 0.394516
\(777\) −2.39054e53 −0.804837
\(778\) 8.39531e52 0.276003
\(779\) −1.43219e53 −0.459790
\(780\) 3.16807e52 0.0993217
\(781\) 7.87402e50 0.00241075
\(782\) 2.11069e51 0.00631101
\(783\) 2.37534e53 0.693638
\(784\) −3.63499e52 −0.103671
\(785\) −2.26544e53 −0.631051
\(786\) 2.41654e53 0.657471
\(787\) 5.34066e53 1.41926 0.709631 0.704574i \(-0.248862\pi\)
0.709631 + 0.704574i \(0.248862\pi\)
\(788\) 1.62177e53 0.420974
\(789\) −4.43427e53 −1.12434
\(790\) 1.43759e53 0.356068
\(791\) 8.34528e52 0.201918
\(792\) −6.27673e51 −0.0148360
\(793\) 6.49932e52 0.150077
\(794\) 6.70453e52 0.151248
\(795\) 2.00879e53 0.442734
\(796\) 3.68844e53 0.794238
\(797\) 6.52917e53 1.37366 0.686830 0.726818i \(-0.259002\pi\)
0.686830 + 0.726818i \(0.259002\pi\)
\(798\) −1.08090e53 −0.222194
\(799\) 3.63911e51 0.00730937
\(800\) 1.80144e52 0.0353553
\(801\) −2.10822e52 −0.0404311
\(802\) 6.01818e53 1.12782
\(803\) 2.84589e53 0.521171
\(804\) 2.05290e53 0.367393
\(805\) −1.12712e53 −0.197127
\(806\) −5.39593e52 −0.0922291
\(807\) 3.11881e53 0.520989
\(808\) −1.12175e53 −0.183141
\(809\) −5.66679e53 −0.904252 −0.452126 0.891954i \(-0.649334\pi\)
−0.452126 + 0.891954i \(0.649334\pi\)
\(810\) −2.13612e53 −0.333160
\(811\) 4.52502e53 0.689818 0.344909 0.938636i \(-0.387910\pi\)
0.344909 + 0.938636i \(0.387910\pi\)
\(812\) 1.83644e53 0.273647
\(813\) 3.88826e53 0.566343
\(814\) −3.75685e53 −0.534900
\(815\) 4.85605e53 0.675877
\(816\) 2.92602e51 0.00398116
\(817\) −4.46770e53 −0.594259
\(818\) −3.24957e53 −0.422561
\(819\) −1.47610e52 −0.0187656
\(820\) −2.07032e53 −0.257325
\(821\) −1.11490e54 −1.35484 −0.677419 0.735597i \(-0.736902\pi\)
−0.677419 + 0.735597i \(0.736902\pi\)
\(822\) −3.88572e53 −0.461681
\(823\) 1.69605e54 1.97034 0.985171 0.171575i \(-0.0548855\pi\)
0.985171 + 0.171575i \(0.0548855\pi\)
\(824\) −5.36647e53 −0.609586
\(825\) 1.36830e53 0.151979
\(826\) 4.46536e51 0.00484982
\(827\) −4.68182e53 −0.497236 −0.248618 0.968602i \(-0.579976\pi\)
−0.248618 + 0.968602i \(0.579976\pi\)
\(828\) −1.57457e52 −0.0163532
\(829\) −6.57785e53 −0.668075 −0.334037 0.942560i \(-0.608411\pi\)
−0.334037 + 0.942560i \(0.608411\pi\)
\(830\) −4.18848e53 −0.416018
\(831\) 6.02196e53 0.584950
\(832\) −5.68540e52 −0.0540106
\(833\) 6.91398e51 0.00642386
\(834\) −1.12331e54 −1.02077
\(835\) −4.31087e53 −0.383147
\(836\) −1.69869e53 −0.147672
\(837\) 3.44225e53 0.292700
\(838\) 8.84213e53 0.735432
\(839\) −1.37838e54 −1.12143 −0.560714 0.828009i \(-0.689473\pi\)
−0.560714 + 0.828009i \(0.689473\pi\)
\(840\) −1.56251e53 −0.124353
\(841\) −6.27159e53 −0.488260
\(842\) 5.33672e53 0.406444
\(843\) 2.58267e54 1.92424
\(844\) 4.67143e53 0.340498
\(845\) −5.10054e53 −0.363720
\(846\) −2.71478e52 −0.0189401
\(847\) −5.08380e53 −0.347013
\(848\) −3.60497e53 −0.240757
\(849\) −7.52308e53 −0.491590
\(850\) −3.42645e51 −0.00219076
\(851\) −9.42441e53 −0.589600
\(852\) 2.73829e51 0.00167628
\(853\) 3.21106e54 1.92350 0.961748 0.273937i \(-0.0883260\pi\)
0.961748 + 0.273937i \(0.0883260\pi\)
\(854\) −3.20550e53 −0.187899
\(855\) 1.76827e52 0.0101432
\(856\) −9.39278e53 −0.527265
\(857\) −3.02093e54 −1.65957 −0.829784 0.558085i \(-0.811536\pi\)
−0.829784 + 0.558085i \(0.811536\pi\)
\(858\) −4.31841e53 −0.232172
\(859\) −1.03473e54 −0.544444 −0.272222 0.962235i \(-0.587758\pi\)
−0.272222 + 0.962235i \(0.587758\pi\)
\(860\) −6.45832e53 −0.332582
\(861\) 1.79573e54 0.905069
\(862\) 1.63651e54 0.807301
\(863\) −2.58733e54 −1.24926 −0.624629 0.780921i \(-0.714750\pi\)
−0.624629 + 0.780921i \(0.714750\pi\)
\(864\) 3.62692e53 0.171409
\(865\) 9.40127e53 0.434899
\(866\) −1.23730e54 −0.560264
\(867\) 2.31867e54 1.02775
\(868\) 2.66130e53 0.115473
\(869\) −1.95959e54 −0.832336
\(870\) −5.59269e53 −0.232549
\(871\) 7.58711e53 0.308845
\(872\) −1.17607e54 −0.468681
\(873\) 1.62357e53 0.0633442
\(874\) −4.26132e53 −0.162773
\(875\) 1.82974e53 0.0684291
\(876\) 9.89695e53 0.362389
\(877\) −1.33643e54 −0.479132 −0.239566 0.970880i \(-0.577005\pi\)
−0.239566 + 0.970880i \(0.577005\pi\)
\(878\) −1.98614e54 −0.697205
\(879\) 1.03491e54 0.355719
\(880\) −2.45555e53 −0.0826457
\(881\) 1.19179e54 0.392776 0.196388 0.980526i \(-0.437079\pi\)
0.196388 + 0.980526i \(0.437079\pi\)
\(882\) −5.15784e52 −0.0166456
\(883\) 2.32196e54 0.733805 0.366903 0.930259i \(-0.380418\pi\)
0.366903 + 0.930259i \(0.380418\pi\)
\(884\) 1.08140e52 0.00334672
\(885\) −1.35988e52 −0.00412145
\(886\) 2.31686e54 0.687664
\(887\) −2.27100e54 −0.660131 −0.330065 0.943958i \(-0.607071\pi\)
−0.330065 + 0.943958i \(0.607071\pi\)
\(888\) −1.30649e54 −0.371935
\(889\) 2.24281e54 0.625330
\(890\) −8.24770e53 −0.225226
\(891\) 2.91176e54 0.778786
\(892\) −2.63469e53 −0.0690208
\(893\) −7.34708e53 −0.188522
\(894\) 2.79409e54 0.702258
\(895\) −6.44295e53 −0.158620
\(896\) 2.80407e53 0.0676224
\(897\) −1.08331e54 −0.255914
\(898\) −5.29493e54 −1.22532
\(899\) 9.52560e53 0.215942
\(900\) 2.55614e52 0.00567672
\(901\) 6.85688e52 0.0149183
\(902\) 2.82207e54 0.601515
\(903\) 5.60173e54 1.16976
\(904\) 4.56091e53 0.0933116
\(905\) −2.31497e54 −0.464030
\(906\) 4.42817e54 0.869666
\(907\) 9.99936e53 0.192414 0.0962072 0.995361i \(-0.469329\pi\)
0.0962072 + 0.995361i \(0.469329\pi\)
\(908\) 9.87340e52 0.0186157
\(909\) −1.59169e53 −0.0294054
\(910\) −5.77472e53 −0.104536
\(911\) 8.26172e52 0.0146548 0.00732741 0.999973i \(-0.497668\pi\)
0.00732741 + 0.999973i \(0.497668\pi\)
\(912\) −5.90741e53 −0.102682
\(913\) 5.70935e54 0.972472
\(914\) 1.73517e54 0.289627
\(915\) 9.76202e53 0.159680
\(916\) 1.92985e54 0.309355
\(917\) −4.40484e54 −0.691987
\(918\) −6.89863e52 −0.0106212
\(919\) 5.97032e54 0.900867 0.450433 0.892810i \(-0.351269\pi\)
0.450433 + 0.892810i \(0.351269\pi\)
\(920\) −6.15998e53 −0.0910971
\(921\) −6.13205e54 −0.888797
\(922\) 6.45862e54 0.917524
\(923\) 1.01202e52 0.00140915
\(924\) 2.12986e54 0.290683
\(925\) 1.52994e54 0.204669
\(926\) 8.74170e54 1.14628
\(927\) −7.61470e53 −0.0978763
\(928\) 1.00366e54 0.126459
\(929\) −1.17859e55 −1.45571 −0.727853 0.685734i \(-0.759482\pi\)
−0.727853 + 0.685734i \(0.759482\pi\)
\(930\) −8.10471e53 −0.0981303
\(931\) −1.39588e54 −0.165683
\(932\) −7.01025e54 −0.815716
\(933\) −1.28513e55 −1.46601
\(934\) 2.59604e54 0.290332
\(935\) 4.67062e52 0.00512106
\(936\) −8.06724e52 −0.00867205
\(937\) −1.35568e54 −0.142881 −0.0714405 0.997445i \(-0.522760\pi\)
−0.0714405 + 0.997445i \(0.522760\pi\)
\(938\) −3.74200e54 −0.386680
\(939\) 7.13761e54 0.723168
\(940\) −1.06206e54 −0.105508
\(941\) −1.66103e55 −1.61796 −0.808981 0.587835i \(-0.799980\pi\)
−0.808981 + 0.587835i \(0.799980\pi\)
\(942\) 1.07390e55 1.02571
\(943\) 7.07941e54 0.663027
\(944\) 2.44043e52 0.00224122
\(945\) 3.68390e54 0.331756
\(946\) 8.80338e54 0.777434
\(947\) −5.07081e54 −0.439139 −0.219570 0.975597i \(-0.570465\pi\)
−0.219570 + 0.975597i \(0.570465\pi\)
\(948\) −6.81471e54 −0.578753
\(949\) 3.65772e54 0.304639
\(950\) 6.91774e53 0.0565038
\(951\) −2.12960e55 −1.70592
\(952\) −5.33352e52 −0.00419015
\(953\) −1.58180e55 −1.21880 −0.609401 0.792862i \(-0.708590\pi\)
−0.609401 + 0.792862i \(0.708590\pi\)
\(954\) −5.11524e53 −0.0386564
\(955\) 4.19595e54 0.311005
\(956\) 8.22103e53 0.0597661
\(957\) 7.62342e54 0.543600
\(958\) −9.29187e54 −0.649892
\(959\) 7.08284e54 0.485918
\(960\) −8.53950e53 −0.0574665
\(961\) −1.37685e55 −0.908877
\(962\) −4.82854e54 −0.312663
\(963\) −1.33278e54 −0.0846587
\(964\) −1.60661e54 −0.100112
\(965\) 1.21620e55 0.743446
\(966\) 5.34296e54 0.320409
\(967\) 1.47153e55 0.865724 0.432862 0.901460i \(-0.357504\pi\)
0.432862 + 0.901460i \(0.357504\pi\)
\(968\) −2.77843e54 −0.160363
\(969\) 1.12363e53 0.00636256
\(970\) 6.35165e54 0.352866
\(971\) −3.18652e55 −1.73684 −0.868422 0.495825i \(-0.834866\pi\)
−0.868422 + 0.495825i \(0.834866\pi\)
\(972\) 1.06025e54 0.0566999
\(973\) 2.04756e55 1.07436
\(974\) −1.47318e55 −0.758432
\(975\) 1.75863e54 0.0888360
\(976\) −1.75189e54 −0.0868330
\(977\) −1.74536e55 −0.848857 −0.424428 0.905462i \(-0.639525\pi\)
−0.424428 + 0.905462i \(0.639525\pi\)
\(978\) −2.30195e55 −1.09857
\(979\) 1.12425e55 0.526481
\(980\) −2.01783e54 −0.0927260
\(981\) −1.66877e54 −0.0752523
\(982\) −1.76446e55 −0.780816
\(983\) −7.48657e54 −0.325119 −0.162560 0.986699i \(-0.551975\pi\)
−0.162560 + 0.986699i \(0.551975\pi\)
\(984\) 9.81410e54 0.418255
\(985\) 9.00265e54 0.376530
\(986\) −1.90903e53 −0.00783590
\(987\) 9.21197e54 0.371095
\(988\) −2.18326e54 −0.0863181
\(989\) 2.20841e55 0.856935
\(990\) −3.48428e53 −0.0132697
\(991\) −1.50174e55 −0.561349 −0.280674 0.959803i \(-0.590558\pi\)
−0.280674 + 0.959803i \(0.590558\pi\)
\(992\) 1.45447e54 0.0533628
\(993\) 2.64544e55 0.952656
\(994\) −4.99133e52 −0.00176428
\(995\) 2.04750e55 0.710388
\(996\) 1.98550e55 0.676195
\(997\) 2.10810e55 0.704743 0.352371 0.935860i \(-0.385375\pi\)
0.352371 + 0.935860i \(0.385375\pi\)
\(998\) −4.12010e55 −1.35205
\(999\) 3.08030e55 0.992273
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 10.38.a.a.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.38.a.a.1.1 2 1.1 even 1 trivial