Properties

Label 10.32.a.a.1.1
Level $10$
Weight $32$
Character 10.1
Self dual yes
Analytic conductor $60.877$
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] = [10,32,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, 32, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("10.1");
 
S:= CuspForms(chi, 32);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 32 \)
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: \(60.8771328190\)
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: \( x^{2} - x - 17573188320 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2}\cdot 5 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(132564.\) 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)
 
\(f(q)\) \(=\) \(q-32768.0 q^{2} -2.68139e7 q^{3} +1.07374e9 q^{4} +3.05176e10 q^{5} +8.78637e11 q^{6} +7.28747e11 q^{7} -3.51844e13 q^{8} +1.01311e14 q^{9} -1.00000e15 q^{10} -3.37576e15 q^{11} -2.87912e16 q^{12} -1.37589e17 q^{13} -2.38796e16 q^{14} -8.18295e17 q^{15} +1.15292e18 q^{16} +2.39641e18 q^{17} -3.31975e18 q^{18} +9.48888e18 q^{19} +3.27680e19 q^{20} -1.95405e19 q^{21} +1.10617e20 q^{22} +1.89426e21 q^{23} +9.43430e20 q^{24} +9.31323e20 q^{25} +4.50853e21 q^{26} +1.38457e22 q^{27} +7.82486e20 q^{28} +5.03519e21 q^{29} +2.68139e22 q^{30} -8.54182e22 q^{31} -3.77789e22 q^{32} +9.05173e22 q^{33} -7.85254e22 q^{34} +2.22396e22 q^{35} +1.08782e23 q^{36} +1.26772e24 q^{37} -3.10932e23 q^{38} +3.68931e24 q^{39} -1.07374e24 q^{40} +8.53929e24 q^{41} +6.40304e23 q^{42} +2.40945e25 q^{43} -3.62470e24 q^{44} +3.09176e24 q^{45} -6.20710e25 q^{46} -3.96314e25 q^{47} -3.09143e25 q^{48} -1.57244e26 q^{49} -3.05176e25 q^{50} -6.42569e25 q^{51} -1.47736e26 q^{52} +7.23147e25 q^{53} -4.53695e26 q^{54} -1.03020e26 q^{55} -2.56405e25 q^{56} -2.54434e26 q^{57} -1.64993e26 q^{58} -1.71371e27 q^{59} -8.78637e26 q^{60} +2.02558e27 q^{61} +2.79898e27 q^{62} +7.38299e25 q^{63} +1.23794e27 q^{64} -4.19890e27 q^{65} -2.96607e27 q^{66} -2.22923e28 q^{67} +2.57312e27 q^{68} -5.07923e28 q^{69} -7.28747e26 q^{70} -8.76737e28 q^{71} -3.56456e27 q^{72} -6.23103e28 q^{73} -4.15405e28 q^{74} -2.49724e28 q^{75} +1.01886e28 q^{76} -2.46008e27 q^{77} -1.20891e29 q^{78} +1.07042e29 q^{79} +3.51844e28 q^{80} -4.33834e29 q^{81} -2.79815e29 q^{82} -5.93551e28 q^{83} -2.09815e28 q^{84} +7.31325e28 q^{85} -7.89527e29 q^{86} -1.35013e29 q^{87} +1.18774e29 q^{88} +1.10405e30 q^{89} -1.01311e29 q^{90} -1.00268e29 q^{91} +2.03394e30 q^{92} +2.29039e30 q^{93} +1.29864e30 q^{94} +2.89578e29 q^{95} +1.01300e30 q^{96} +5.31660e30 q^{97} +5.15258e30 q^{98} -3.42001e29 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 65536 q^{2} - 5904756 q^{3} + 2147483648 q^{4} + 61035156250 q^{5} + 193487044608 q^{6} + 18016565093212 q^{7} - 70368744177664 q^{8} - 79171117705926 q^{9} - 20\!\cdots\!00 q^{10} - 17\!\cdots\!76 q^{11}+ \cdots + 21\!\cdots\!88 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −32768.0 −0.707107
\(3\) −2.68139e7 −1.07890 −0.539449 0.842018i \(-0.681367\pi\)
−0.539449 + 0.842018i \(0.681367\pi\)
\(4\) 1.07374e9 0.500000
\(5\) 3.05176e10 0.447214
\(6\) 8.78637e11 0.762896
\(7\) 7.28747e11 0.0580172 0.0290086 0.999579i \(-0.490765\pi\)
0.0290086 + 0.999579i \(0.490765\pi\)
\(8\) −3.51844e13 −0.353553
\(9\) 1.01311e14 0.164020
\(10\) −1.00000e15 −0.316228
\(11\) −3.37576e15 −0.243661 −0.121830 0.992551i \(-0.538876\pi\)
−0.121830 + 0.992551i \(0.538876\pi\)
\(12\) −2.87912e16 −0.539449
\(13\) −1.37589e17 −0.745527 −0.372763 0.927926i \(-0.621590\pi\)
−0.372763 + 0.927926i \(0.621590\pi\)
\(14\) −2.38796e16 −0.0410244
\(15\) −8.18295e17 −0.482498
\(16\) 1.15292e18 0.250000
\(17\) 2.39641e18 0.203050 0.101525 0.994833i \(-0.467628\pi\)
0.101525 + 0.994833i \(0.467628\pi\)
\(18\) −3.31975e18 −0.115980
\(19\) 9.48888e18 0.143395 0.0716975 0.997426i \(-0.477158\pi\)
0.0716975 + 0.997426i \(0.477158\pi\)
\(20\) 3.27680e19 0.223607
\(21\) −1.95405e19 −0.0625947
\(22\) 1.10617e20 0.172294
\(23\) 1.89426e21 1.48135 0.740674 0.671865i \(-0.234506\pi\)
0.740674 + 0.671865i \(0.234506\pi\)
\(24\) 9.43430e20 0.381448
\(25\) 9.31323e20 0.200000
\(26\) 4.50853e21 0.527167
\(27\) 1.38457e22 0.901937
\(28\) 7.82486e20 0.0290086
\(29\) 5.03519e21 0.108354 0.0541772 0.998531i \(-0.482746\pi\)
0.0541772 + 0.998531i \(0.482746\pi\)
\(30\) 2.68139e22 0.341177
\(31\) −8.54182e22 −0.653799 −0.326899 0.945059i \(-0.606004\pi\)
−0.326899 + 0.945059i \(0.606004\pi\)
\(32\) −3.77789e22 −0.176777
\(33\) 9.05173e22 0.262885
\(34\) −7.85254e22 −0.143578
\(35\) 2.22396e22 0.0259461
\(36\) 1.08782e23 0.0820100
\(37\) 1.26772e24 0.625022 0.312511 0.949914i \(-0.398830\pi\)
0.312511 + 0.949914i \(0.398830\pi\)
\(38\) −3.10932e23 −0.101396
\(39\) 3.68931e24 0.804347
\(40\) −1.07374e24 −0.158114
\(41\) 8.53929e24 0.857574 0.428787 0.903406i \(-0.358941\pi\)
0.428787 + 0.903406i \(0.358941\pi\)
\(42\) 6.40304e23 0.0442611
\(43\) 2.40945e25 1.15653 0.578264 0.815850i \(-0.303730\pi\)
0.578264 + 0.815850i \(0.303730\pi\)
\(44\) −3.62470e24 −0.121830
\(45\) 3.09176e24 0.0733520
\(46\) −6.20710e25 −1.04747
\(47\) −3.96314e25 −0.479207 −0.239603 0.970871i \(-0.577017\pi\)
−0.239603 + 0.970871i \(0.577017\pi\)
\(48\) −3.09143e25 −0.269724
\(49\) −1.57244e26 −0.996634
\(50\) −3.05176e25 −0.141421
\(51\) −6.42569e25 −0.219070
\(52\) −1.47736e26 −0.372763
\(53\) 7.23147e25 0.135816 0.0679080 0.997692i \(-0.478368\pi\)
0.0679080 + 0.997692i \(0.478368\pi\)
\(54\) −4.53695e26 −0.637766
\(55\) −1.03020e26 −0.108968
\(56\) −2.56405e25 −0.0205122
\(57\) −2.54434e26 −0.154709
\(58\) −1.64993e26 −0.0766182
\(59\) −1.71371e27 −0.610563 −0.305281 0.952262i \(-0.598751\pi\)
−0.305281 + 0.952262i \(0.598751\pi\)
\(60\) −8.78637e26 −0.241249
\(61\) 2.02558e27 0.430464 0.215232 0.976563i \(-0.430949\pi\)
0.215232 + 0.976563i \(0.430949\pi\)
\(62\) 2.79898e27 0.462305
\(63\) 7.38299e25 0.00951599
\(64\) 1.23794e27 0.125000
\(65\) −4.19890e27 −0.333410
\(66\) −2.96607e27 −0.185888
\(67\) −2.22923e28 −1.10661 −0.553307 0.832978i \(-0.686634\pi\)
−0.553307 + 0.832978i \(0.686634\pi\)
\(68\) 2.57312e27 0.101525
\(69\) −5.07923e28 −1.59822
\(70\) −7.28747e26 −0.0183467
\(71\) −8.76737e28 −1.77160 −0.885799 0.464069i \(-0.846389\pi\)
−0.885799 + 0.464069i \(0.846389\pi\)
\(72\) −3.56456e27 −0.0579898
\(73\) −6.23103e28 −0.818569 −0.409285 0.912407i \(-0.634222\pi\)
−0.409285 + 0.912407i \(0.634222\pi\)
\(74\) −4.15405e28 −0.441958
\(75\) −2.49724e28 −0.215780
\(76\) 1.01886e28 0.0716975
\(77\) −2.46008e27 −0.0141365
\(78\) −1.20891e29 −0.568759
\(79\) 1.07042e29 0.413367 0.206683 0.978408i \(-0.433733\pi\)
0.206683 + 0.978408i \(0.433733\pi\)
\(80\) 3.51844e28 0.111803
\(81\) −4.33834e29 −1.13712
\(82\) −2.79815e29 −0.606397
\(83\) −5.93551e28 −0.106598 −0.0532988 0.998579i \(-0.516974\pi\)
−0.0532988 + 0.998579i \(0.516974\pi\)
\(84\) −2.09815e28 −0.0312973
\(85\) 7.31325e28 0.0908065
\(86\) −7.89527e29 −0.817789
\(87\) −1.35013e29 −0.116903
\(88\) 1.18774e29 0.0861471
\(89\) 1.10405e30 0.672118 0.336059 0.941841i \(-0.390906\pi\)
0.336059 + 0.941841i \(0.390906\pi\)
\(90\) −1.01311e29 −0.0518677
\(91\) −1.00268e29 −0.0432534
\(92\) 2.03394e30 0.740674
\(93\) 2.29039e30 0.705382
\(94\) 1.29864e30 0.338850
\(95\) 2.89578e29 0.0641282
\(96\) 1.01300e30 0.190724
\(97\) 5.31660e30 0.852456 0.426228 0.904616i \(-0.359842\pi\)
0.426228 + 0.904616i \(0.359842\pi\)
\(98\) 5.15258e30 0.704727
\(99\) −3.42001e29 −0.0399652
\(100\) 1.00000e30 0.100000
\(101\) −6.37914e30 −0.546740 −0.273370 0.961909i \(-0.588138\pi\)
−0.273370 + 0.961909i \(0.588138\pi\)
\(102\) 2.10557e30 0.154906
\(103\) 1.27411e31 0.805803 0.402901 0.915243i \(-0.368002\pi\)
0.402901 + 0.915243i \(0.368002\pi\)
\(104\) 4.84100e30 0.263583
\(105\) −5.96330e29 −0.0279932
\(106\) −2.36961e30 −0.0960364
\(107\) −1.47464e31 −0.516698 −0.258349 0.966052i \(-0.583178\pi\)
−0.258349 + 0.966052i \(0.583178\pi\)
\(108\) 1.48667e31 0.450968
\(109\) −2.43352e31 −0.639917 −0.319958 0.947432i \(-0.603669\pi\)
−0.319958 + 0.947432i \(0.603669\pi\)
\(110\) 3.37576e30 0.0770523
\(111\) −3.39924e31 −0.674335
\(112\) 8.40188e29 0.0145043
\(113\) 1.29196e32 1.94327 0.971633 0.236495i \(-0.0759987\pi\)
0.971633 + 0.236495i \(0.0759987\pi\)
\(114\) 8.33729e30 0.109396
\(115\) 5.78081e31 0.662479
\(116\) 5.40649e30 0.0541772
\(117\) −1.39393e31 −0.122281
\(118\) 5.61548e31 0.431733
\(119\) 1.74637e30 0.0117804
\(120\) 2.87912e31 0.170589
\(121\) −1.80548e32 −0.940629
\(122\) −6.63742e31 −0.304384
\(123\) −2.28971e32 −0.925235
\(124\) −9.17171e31 −0.326899
\(125\) 2.84217e31 0.0894427
\(126\) −2.41926e30 −0.00672882
\(127\) −6.29352e32 −1.54859 −0.774294 0.632826i \(-0.781895\pi\)
−0.774294 + 0.632826i \(0.781895\pi\)
\(128\) −4.05648e31 −0.0883883
\(129\) −6.46066e32 −1.24778
\(130\) 1.37589e32 0.235756
\(131\) −1.13960e33 −1.73400 −0.866999 0.498310i \(-0.833954\pi\)
−0.866999 + 0.498310i \(0.833954\pi\)
\(132\) 9.71922e31 0.131442
\(133\) 6.91499e30 0.00831939
\(134\) 7.30475e32 0.782494
\(135\) 4.22537e32 0.403358
\(136\) −8.43160e31 −0.0717889
\(137\) −1.23195e33 −0.936318 −0.468159 0.883644i \(-0.655083\pi\)
−0.468159 + 0.883644i \(0.655083\pi\)
\(138\) 1.66436e33 1.13011
\(139\) −1.10200e33 −0.669042 −0.334521 0.942388i \(-0.608575\pi\)
−0.334521 + 0.942388i \(0.608575\pi\)
\(140\) 2.38796e31 0.0129730
\(141\) 1.06267e33 0.517015
\(142\) 2.87289e33 1.25271
\(143\) 4.64470e32 0.181656
\(144\) 1.16803e32 0.0410050
\(145\) 1.53662e32 0.0484576
\(146\) 2.04178e33 0.578816
\(147\) 4.21633e33 1.07527
\(148\) 1.36120e33 0.312511
\(149\) −1.57024e33 −0.324772 −0.162386 0.986727i \(-0.551919\pi\)
−0.162386 + 0.986727i \(0.551919\pi\)
\(150\) 8.18295e32 0.152579
\(151\) −1.72612e33 −0.290354 −0.145177 0.989406i \(-0.546375\pi\)
−0.145177 + 0.989406i \(0.546375\pi\)
\(152\) −3.33860e32 −0.0506978
\(153\) 2.42782e32 0.0333042
\(154\) 8.06118e31 0.00999603
\(155\) −2.60676e33 −0.292388
\(156\) 3.96136e33 0.402173
\(157\) 1.22917e34 1.13023 0.565114 0.825013i \(-0.308832\pi\)
0.565114 + 0.825013i \(0.308832\pi\)
\(158\) −3.50756e33 −0.292294
\(159\) −1.93904e33 −0.146531
\(160\) −1.15292e33 −0.0790569
\(161\) 1.38043e33 0.0859437
\(162\) 1.42159e34 0.804063
\(163\) −2.97139e34 −1.52775 −0.763874 0.645366i \(-0.776705\pi\)
−0.763874 + 0.645366i \(0.776705\pi\)
\(164\) 9.16899e33 0.428787
\(165\) 2.76237e33 0.117566
\(166\) 1.94495e33 0.0753759
\(167\) −1.23629e32 −0.00436531 −0.00218265 0.999998i \(-0.500695\pi\)
−0.00218265 + 0.999998i \(0.500695\pi\)
\(168\) 6.87521e32 0.0221306
\(169\) −1.51291e34 −0.444190
\(170\) −2.39641e33 −0.0642099
\(171\) 9.61326e32 0.0235197
\(172\) 2.58712e34 0.578264
\(173\) −8.47230e34 −1.73096 −0.865479 0.500945i \(-0.832986\pi\)
−0.865479 + 0.500945i \(0.832986\pi\)
\(174\) 4.42411e33 0.0826632
\(175\) 6.78698e32 0.0116034
\(176\) −3.89199e33 −0.0609152
\(177\) 4.59511e34 0.658735
\(178\) −3.61776e34 −0.475259
\(179\) −1.31638e35 −1.58547 −0.792736 0.609565i \(-0.791344\pi\)
−0.792736 + 0.609565i \(0.791344\pi\)
\(180\) 3.31975e33 0.0366760
\(181\) 7.25006e34 0.735062 0.367531 0.930011i \(-0.380203\pi\)
0.367531 + 0.930011i \(0.380203\pi\)
\(182\) 3.28558e33 0.0305848
\(183\) −5.43136e34 −0.464426
\(184\) −6.66482e34 −0.523736
\(185\) 3.86877e34 0.279519
\(186\) −7.50516e34 −0.498780
\(187\) −8.08970e33 −0.0494752
\(188\) −4.25539e34 −0.239603
\(189\) 1.00900e34 0.0523279
\(190\) −9.48888e33 −0.0453455
\(191\) 6.07467e34 0.267612 0.133806 0.991008i \(-0.457280\pi\)
0.133806 + 0.991008i \(0.457280\pi\)
\(192\) −3.31940e34 −0.134862
\(193\) 3.30662e35 1.23950 0.619750 0.784799i \(-0.287234\pi\)
0.619750 + 0.784799i \(0.287234\pi\)
\(194\) −1.74214e35 −0.602777
\(195\) 1.12589e35 0.359715
\(196\) −1.68840e35 −0.498317
\(197\) 3.74089e35 1.02035 0.510174 0.860071i \(-0.329581\pi\)
0.510174 + 0.860071i \(0.329581\pi\)
\(198\) 1.12067e34 0.0282597
\(199\) −1.54190e35 −0.359611 −0.179805 0.983702i \(-0.557547\pi\)
−0.179805 + 0.983702i \(0.557547\pi\)
\(200\) −3.27680e34 −0.0707107
\(201\) 5.97744e35 1.19392
\(202\) 2.09032e35 0.386604
\(203\) 3.66938e33 0.00628643
\(204\) −6.89954e34 −0.109535
\(205\) 2.60598e35 0.383519
\(206\) −4.17499e35 −0.569789
\(207\) 1.91909e35 0.242971
\(208\) −1.58630e35 −0.186382
\(209\) −3.20322e34 −0.0349397
\(210\) 1.95405e34 0.0197942
\(211\) −6.65945e35 −0.626701 −0.313350 0.949638i \(-0.601451\pi\)
−0.313350 + 0.949638i \(0.601451\pi\)
\(212\) 7.76473e34 0.0679080
\(213\) 2.35087e36 1.91137
\(214\) 4.83209e35 0.365361
\(215\) 7.35305e35 0.517215
\(216\) −4.87152e35 −0.318883
\(217\) −6.22482e34 −0.0379316
\(218\) 7.97415e35 0.452490
\(219\) 1.67078e36 0.883153
\(220\) −1.10617e35 −0.0544842
\(221\) −3.29720e35 −0.151379
\(222\) 1.11386e36 0.476827
\(223\) −1.97136e36 −0.787117 −0.393559 0.919299i \(-0.628756\pi\)
−0.393559 + 0.919299i \(0.628756\pi\)
\(224\) −2.75313e34 −0.0102561
\(225\) 9.43530e34 0.0328040
\(226\) −4.23348e36 −1.37410
\(227\) −5.24999e36 −1.59132 −0.795661 0.605743i \(-0.792876\pi\)
−0.795661 + 0.605743i \(0.792876\pi\)
\(228\) −2.73196e35 −0.0773543
\(229\) −5.83672e36 −1.54425 −0.772127 0.635468i \(-0.780807\pi\)
−0.772127 + 0.635468i \(0.780807\pi\)
\(230\) −1.89426e36 −0.468443
\(231\) 6.59642e34 0.0152519
\(232\) −1.77160e35 −0.0383091
\(233\) −2.74902e36 −0.556110 −0.278055 0.960565i \(-0.589690\pi\)
−0.278055 + 0.960565i \(0.589690\pi\)
\(234\) 4.56763e35 0.0864659
\(235\) −1.20946e36 −0.214308
\(236\) −1.84008e36 −0.305281
\(237\) −2.87022e36 −0.445980
\(238\) −5.72252e34 −0.00832998
\(239\) −6.06727e35 −0.0827610 −0.0413805 0.999143i \(-0.513176\pi\)
−0.0413805 + 0.999143i \(0.513176\pi\)
\(240\) −9.43430e35 −0.120624
\(241\) 9.96011e36 1.19399 0.596994 0.802246i \(-0.296362\pi\)
0.596994 + 0.802246i \(0.296362\pi\)
\(242\) 5.91619e36 0.665125
\(243\) 3.08065e36 0.324896
\(244\) 2.17495e36 0.215232
\(245\) −4.79872e36 −0.445708
\(246\) 7.50293e36 0.654240
\(247\) −1.30557e36 −0.106905
\(248\) 3.00538e36 0.231153
\(249\) 1.59154e36 0.115008
\(250\) −9.31323e35 −0.0632456
\(251\) 2.33773e37 1.49228 0.746141 0.665788i \(-0.231905\pi\)
0.746141 + 0.665788i \(0.231905\pi\)
\(252\) 7.92743e34 0.00475799
\(253\) −6.39456e36 −0.360946
\(254\) 2.06226e37 1.09502
\(255\) −1.96097e36 −0.0979709
\(256\) 1.32923e36 0.0625000
\(257\) −3.78920e37 −1.67720 −0.838599 0.544749i \(-0.816625\pi\)
−0.838599 + 0.544749i \(0.816625\pi\)
\(258\) 2.11703e37 0.882310
\(259\) 9.23845e35 0.0362621
\(260\) −4.50853e36 −0.166705
\(261\) 5.10119e35 0.0177723
\(262\) 3.73425e37 1.22612
\(263\) −1.23104e37 −0.381029 −0.190515 0.981684i \(-0.561016\pi\)
−0.190515 + 0.981684i \(0.561016\pi\)
\(264\) −3.18480e36 −0.0929439
\(265\) 2.20687e36 0.0607387
\(266\) −2.26590e35 −0.00588269
\(267\) −2.96040e37 −0.725146
\(268\) −2.39362e37 −0.553307
\(269\) −8.37019e36 −0.182631 −0.0913156 0.995822i \(-0.529107\pi\)
−0.0913156 + 0.995822i \(0.529107\pi\)
\(270\) −1.38457e37 −0.285217
\(271\) 6.44376e37 1.25347 0.626737 0.779231i \(-0.284390\pi\)
0.626737 + 0.779231i \(0.284390\pi\)
\(272\) 2.76287e36 0.0507624
\(273\) 2.68857e36 0.0466660
\(274\) 4.03685e37 0.662077
\(275\) −3.14393e36 −0.0487321
\(276\) −5.45379e37 −0.799111
\(277\) 3.75754e37 0.520556 0.260278 0.965534i \(-0.416186\pi\)
0.260278 + 0.965534i \(0.416186\pi\)
\(278\) 3.61104e37 0.473084
\(279\) −8.65378e36 −0.107236
\(280\) −7.82486e35 −0.00917333
\(281\) −3.31550e37 −0.367791 −0.183896 0.982946i \(-0.558871\pi\)
−0.183896 + 0.982946i \(0.558871\pi\)
\(282\) −3.48216e37 −0.365585
\(283\) 1.66415e38 1.65388 0.826938 0.562292i \(-0.190080\pi\)
0.826938 + 0.562292i \(0.190080\pi\)
\(284\) −9.41390e37 −0.885799
\(285\) −7.76470e36 −0.0691878
\(286\) −1.52197e37 −0.128450
\(287\) 6.22298e36 0.0497541
\(288\) −3.82741e36 −0.0289949
\(289\) −1.33546e38 −0.958771
\(290\) −5.03519e36 −0.0342647
\(291\) −1.42559e38 −0.919713
\(292\) −6.69052e37 −0.409285
\(293\) 5.69894e37 0.330633 0.165317 0.986241i \(-0.447135\pi\)
0.165317 + 0.986241i \(0.447135\pi\)
\(294\) −1.38161e38 −0.760328
\(295\) −5.22982e37 −0.273052
\(296\) −4.46038e37 −0.220979
\(297\) −4.67398e37 −0.219767
\(298\) 5.14536e37 0.229649
\(299\) −2.60630e38 −1.10438
\(300\) −2.68139e37 −0.107890
\(301\) 1.75588e37 0.0670986
\(302\) 5.65615e37 0.205311
\(303\) 1.71050e38 0.589877
\(304\) 1.09399e37 0.0358488
\(305\) 6.18158e37 0.192509
\(306\) −7.95547e36 −0.0235496
\(307\) 3.60072e38 1.01331 0.506657 0.862147i \(-0.330881\pi\)
0.506657 + 0.862147i \(0.330881\pi\)
\(308\) −2.64149e36 −0.00706826
\(309\) −3.41637e38 −0.869379
\(310\) 8.54182e37 0.206749
\(311\) 2.32592e37 0.0535560 0.0267780 0.999641i \(-0.491475\pi\)
0.0267780 + 0.999641i \(0.491475\pi\)
\(312\) −1.29806e38 −0.284380
\(313\) 7.58763e38 1.58186 0.790932 0.611904i \(-0.209596\pi\)
0.790932 + 0.611904i \(0.209596\pi\)
\(314\) −4.02773e38 −0.799191
\(315\) 2.25311e36 0.00425568
\(316\) 1.14936e38 0.206683
\(317\) 6.69462e38 1.14632 0.573162 0.819442i \(-0.305717\pi\)
0.573162 + 0.819442i \(0.305717\pi\)
\(318\) 6.35384e37 0.103613
\(319\) −1.69976e37 −0.0264017
\(320\) 3.77789e37 0.0559017
\(321\) 3.95408e38 0.557464
\(322\) −4.52340e37 −0.0607714
\(323\) 2.27392e37 0.0291163
\(324\) −4.65825e38 −0.568559
\(325\) −1.28140e38 −0.149105
\(326\) 9.73664e38 1.08028
\(327\) 6.52520e38 0.690405
\(328\) −3.00449e38 −0.303198
\(329\) −2.88813e37 −0.0278023
\(330\) −9.05173e37 −0.0831315
\(331\) 1.26121e39 1.10523 0.552615 0.833437i \(-0.313630\pi\)
0.552615 + 0.833437i \(0.313630\pi\)
\(332\) −6.37321e37 −0.0532988
\(333\) 1.28433e38 0.102516
\(334\) 4.05107e36 0.00308674
\(335\) −6.80308e38 −0.494893
\(336\) −2.25287e37 −0.0156487
\(337\) 1.05833e39 0.702036 0.351018 0.936369i \(-0.385836\pi\)
0.351018 + 0.936369i \(0.385836\pi\)
\(338\) 4.95750e38 0.314090
\(339\) −3.46424e39 −2.09658
\(340\) 7.85254e37 0.0454033
\(341\) 2.88352e38 0.159305
\(342\) −3.15007e37 −0.0166309
\(343\) −2.29570e38 −0.115839
\(344\) −8.47749e38 −0.408894
\(345\) −1.55006e39 −0.714747
\(346\) 2.77620e39 1.22397
\(347\) −5.76947e38 −0.243237 −0.121618 0.992577i \(-0.538808\pi\)
−0.121618 + 0.992577i \(0.538808\pi\)
\(348\) −1.44969e38 −0.0584517
\(349\) −2.40704e39 −0.928304 −0.464152 0.885756i \(-0.653641\pi\)
−0.464152 + 0.885756i \(0.653641\pi\)
\(350\) −2.22396e37 −0.00820488
\(351\) −1.90502e39 −0.672418
\(352\) 1.27533e38 0.0430735
\(353\) 3.14178e38 0.101547 0.0507736 0.998710i \(-0.483831\pi\)
0.0507736 + 0.998710i \(0.483831\pi\)
\(354\) −1.50573e39 −0.465796
\(355\) −2.67559e39 −0.792283
\(356\) 1.18547e39 0.336059
\(357\) −4.68270e37 −0.0127098
\(358\) 4.31351e39 1.12110
\(359\) −6.98872e39 −1.73954 −0.869768 0.493460i \(-0.835732\pi\)
−0.869768 + 0.493460i \(0.835732\pi\)
\(360\) −1.08782e38 −0.0259338
\(361\) −4.28883e39 −0.979438
\(362\) −2.37570e39 −0.519767
\(363\) 4.84118e39 1.01484
\(364\) −1.07662e38 −0.0216267
\(365\) −1.90156e39 −0.366075
\(366\) 1.77975e39 0.328399
\(367\) −2.43434e39 −0.430583 −0.215292 0.976550i \(-0.569070\pi\)
−0.215292 + 0.976550i \(0.569070\pi\)
\(368\) 2.18393e39 0.370337
\(369\) 8.65122e38 0.140659
\(370\) −1.26772e39 −0.197649
\(371\) 5.26991e37 0.00787966
\(372\) 2.45929e39 0.352691
\(373\) 1.04824e40 1.44203 0.721017 0.692917i \(-0.243675\pi\)
0.721017 + 0.692917i \(0.243675\pi\)
\(374\) 2.65083e38 0.0349842
\(375\) −7.62096e38 −0.0964995
\(376\) 1.39441e39 0.169425
\(377\) −6.92789e38 −0.0807812
\(378\) −3.30629e38 −0.0370014
\(379\) 6.74056e39 0.724083 0.362041 0.932162i \(-0.382080\pi\)
0.362041 + 0.932162i \(0.382080\pi\)
\(380\) 3.10932e38 0.0320641
\(381\) 1.68754e40 1.67077
\(382\) −1.99055e39 −0.189230
\(383\) 1.08157e40 0.987358 0.493679 0.869644i \(-0.335652\pi\)
0.493679 + 0.869644i \(0.335652\pi\)
\(384\) 1.08770e39 0.0953620
\(385\) −7.50756e37 −0.00632204
\(386\) −1.08351e40 −0.876459
\(387\) 2.44103e39 0.189694
\(388\) 5.70865e39 0.426228
\(389\) −2.00224e40 −1.43647 −0.718236 0.695800i \(-0.755050\pi\)
−0.718236 + 0.695800i \(0.755050\pi\)
\(390\) −3.68931e39 −0.254357
\(391\) 4.53940e39 0.300787
\(392\) 5.53254e39 0.352363
\(393\) 3.05572e40 1.87081
\(394\) −1.22581e40 −0.721495
\(395\) 3.26667e39 0.184863
\(396\) −3.67221e38 −0.0199826
\(397\) −2.20892e40 −1.15592 −0.577958 0.816066i \(-0.696150\pi\)
−0.577958 + 0.816066i \(0.696150\pi\)
\(398\) 5.05249e39 0.254283
\(399\) −1.85418e38 −0.00897577
\(400\) 1.07374e39 0.0500000
\(401\) −6.25993e39 −0.280435 −0.140217 0.990121i \(-0.544780\pi\)
−0.140217 + 0.990121i \(0.544780\pi\)
\(402\) −1.95869e40 −0.844231
\(403\) 1.17526e40 0.487424
\(404\) −6.84955e39 −0.273370
\(405\) −1.32395e40 −0.508534
\(406\) −1.20238e38 −0.00444518
\(407\) −4.27951e39 −0.152293
\(408\) 2.26084e39 0.0774528
\(409\) −3.54495e40 −1.16923 −0.584614 0.811311i \(-0.698754\pi\)
−0.584614 + 0.811311i \(0.698754\pi\)
\(410\) −8.53929e39 −0.271189
\(411\) 3.30333e40 1.01019
\(412\) 1.36806e40 0.402901
\(413\) −1.24886e39 −0.0354232
\(414\) −6.28846e39 −0.171806
\(415\) −1.81137e39 −0.0476719
\(416\) 5.19798e39 0.131792
\(417\) 2.95490e40 0.721828
\(418\) 1.04963e39 0.0247061
\(419\) −2.98080e40 −0.676106 −0.338053 0.941127i \(-0.609768\pi\)
−0.338053 + 0.941127i \(0.609768\pi\)
\(420\) −6.40304e38 −0.0139966
\(421\) 5.17658e40 1.09061 0.545307 0.838237i \(-0.316413\pi\)
0.545307 + 0.838237i \(0.316413\pi\)
\(422\) 2.18217e40 0.443144
\(423\) −4.01509e39 −0.0785995
\(424\) −2.54435e39 −0.0480182
\(425\) 2.23183e39 0.0406099
\(426\) −7.70334e40 −1.35154
\(427\) 1.47613e39 0.0249743
\(428\) −1.58338e40 −0.258349
\(429\) −1.24542e40 −0.195988
\(430\) −2.40945e40 −0.365726
\(431\) 3.26439e40 0.477973 0.238987 0.971023i \(-0.423185\pi\)
0.238987 + 0.971023i \(0.423185\pi\)
\(432\) 1.59630e40 0.225484
\(433\) −1.01421e41 −1.38218 −0.691092 0.722767i \(-0.742870\pi\)
−0.691092 + 0.722767i \(0.742870\pi\)
\(434\) 2.03975e39 0.0268217
\(435\) −4.12027e39 −0.0522808
\(436\) −2.61297e40 −0.319958
\(437\) 1.79744e40 0.212418
\(438\) −5.47482e40 −0.624483
\(439\) −9.77448e40 −1.07620 −0.538101 0.842881i \(-0.680858\pi\)
−0.538101 + 0.842881i \(0.680858\pi\)
\(440\) 3.62470e39 0.0385261
\(441\) −1.59305e40 −0.163468
\(442\) 1.08043e40 0.107041
\(443\) −8.28639e40 −0.792698 −0.396349 0.918100i \(-0.629723\pi\)
−0.396349 + 0.918100i \(0.629723\pi\)
\(444\) −3.64991e40 −0.337168
\(445\) 3.36931e40 0.300580
\(446\) 6.45974e40 0.556576
\(447\) 4.21042e40 0.350396
\(448\) 9.02145e38 0.00725215
\(449\) 1.71749e41 1.33375 0.666875 0.745169i \(-0.267631\pi\)
0.666875 + 0.745169i \(0.267631\pi\)
\(450\) −3.09176e39 −0.0231959
\(451\) −2.88266e40 −0.208957
\(452\) 1.38723e41 0.971633
\(453\) 4.62839e40 0.313262
\(454\) 1.72032e41 1.12523
\(455\) −3.05993e39 −0.0193435
\(456\) 8.95209e39 0.0546978
\(457\) 1.56497e41 0.924281 0.462140 0.886807i \(-0.347082\pi\)
0.462140 + 0.886807i \(0.347082\pi\)
\(458\) 1.91258e41 1.09195
\(459\) 3.31799e40 0.183138
\(460\) 6.20710e40 0.331239
\(461\) 8.09856e40 0.417872 0.208936 0.977929i \(-0.433000\pi\)
0.208936 + 0.977929i \(0.433000\pi\)
\(462\) −2.16152e39 −0.0107847
\(463\) −5.84222e40 −0.281885 −0.140943 0.990018i \(-0.545013\pi\)
−0.140943 + 0.990018i \(0.545013\pi\)
\(464\) 5.80518e39 0.0270886
\(465\) 6.98972e40 0.315456
\(466\) 9.00799e40 0.393229
\(467\) −1.90743e41 −0.805446 −0.402723 0.915322i \(-0.631936\pi\)
−0.402723 + 0.915322i \(0.631936\pi\)
\(468\) −1.49672e40 −0.0611406
\(469\) −1.62455e40 −0.0642027
\(470\) 3.96314e40 0.151538
\(471\) −3.29587e41 −1.21940
\(472\) 6.02957e40 0.215867
\(473\) −8.13372e40 −0.281800
\(474\) 9.40514e40 0.315356
\(475\) 8.83721e39 0.0286790
\(476\) 1.87515e39 0.00589019
\(477\) 7.32626e39 0.0222765
\(478\) 1.98812e40 0.0585209
\(479\) −1.74507e41 −0.497294 −0.248647 0.968594i \(-0.579986\pi\)
−0.248647 + 0.968594i \(0.579986\pi\)
\(480\) 3.09143e40 0.0852943
\(481\) −1.74424e41 −0.465971
\(482\) −3.26373e41 −0.844277
\(483\) −3.70148e40 −0.0927245
\(484\) −1.93862e41 −0.470315
\(485\) 1.62250e41 0.381230
\(486\) −1.00947e41 −0.229736
\(487\) −5.84025e41 −1.28746 −0.643728 0.765254i \(-0.722613\pi\)
−0.643728 + 0.765254i \(0.722613\pi\)
\(488\) −7.12687e40 −0.152192
\(489\) 7.96744e41 1.64828
\(490\) 1.57244e41 0.315163
\(491\) −5.87015e41 −1.13995 −0.569975 0.821662i \(-0.693047\pi\)
−0.569975 + 0.821662i \(0.693047\pi\)
\(492\) −2.45856e41 −0.462617
\(493\) 1.20664e40 0.0220013
\(494\) 4.27809e40 0.0755932
\(495\) −1.04371e40 −0.0178730
\(496\) −9.84805e40 −0.163450
\(497\) −6.38920e40 −0.102783
\(498\) −5.21516e40 −0.0813229
\(499\) 5.42200e41 0.819598 0.409799 0.912176i \(-0.365599\pi\)
0.409799 + 0.912176i \(0.365599\pi\)
\(500\) 3.05176e40 0.0447214
\(501\) 3.31497e39 0.00470972
\(502\) −7.66026e41 −1.05520
\(503\) −1.24021e42 −1.65651 −0.828253 0.560354i \(-0.810665\pi\)
−0.828253 + 0.560354i \(0.810665\pi\)
\(504\) −2.59766e39 −0.00336441
\(505\) −1.94676e41 −0.244510
\(506\) 2.09537e41 0.255228
\(507\) 4.05669e41 0.479235
\(508\) −6.75761e41 −0.774294
\(509\) 9.96253e41 1.10725 0.553623 0.832768i \(-0.313245\pi\)
0.553623 + 0.832768i \(0.313245\pi\)
\(510\) 6.42569e40 0.0692759
\(511\) −4.54084e40 −0.0474911
\(512\) −4.35561e40 −0.0441942
\(513\) 1.31380e41 0.129333
\(514\) 1.24165e42 1.18596
\(515\) 3.88826e41 0.360366
\(516\) −6.93708e41 −0.623888
\(517\) 1.33786e41 0.116764
\(518\) −3.02725e40 −0.0256412
\(519\) 2.27175e42 1.86753
\(520\) 1.47736e41 0.117878
\(521\) −4.81065e41 −0.372579 −0.186290 0.982495i \(-0.559646\pi\)
−0.186290 + 0.982495i \(0.559646\pi\)
\(522\) −1.67156e40 −0.0125669
\(523\) −4.21516e41 −0.307636 −0.153818 0.988099i \(-0.549157\pi\)
−0.153818 + 0.988099i \(0.549157\pi\)
\(524\) −1.22364e42 −0.866999
\(525\) −1.81985e40 −0.0125189
\(526\) 4.03388e41 0.269428
\(527\) −2.04697e41 −0.132754
\(528\) 1.04359e41 0.0657212
\(529\) 1.95303e42 1.19439
\(530\) −7.23147e40 −0.0429488
\(531\) −1.73617e41 −0.100145
\(532\) 7.42492e39 0.00415969
\(533\) −1.17492e42 −0.639345
\(534\) 9.70063e41 0.512756
\(535\) −4.50024e41 −0.231074
\(536\) 7.84342e41 0.391247
\(537\) 3.52972e42 1.71056
\(538\) 2.74274e41 0.129140
\(539\) 5.30820e41 0.242841
\(540\) 4.53695e41 0.201679
\(541\) 3.12292e42 1.34897 0.674486 0.738288i \(-0.264365\pi\)
0.674486 + 0.738288i \(0.264365\pi\)
\(542\) −2.11149e42 −0.886340
\(543\) −1.94402e42 −0.793056
\(544\) −9.05336e40 −0.0358944
\(545\) −7.42650e41 −0.286180
\(546\) −8.80991e40 −0.0329978
\(547\) 8.44341e41 0.307407 0.153704 0.988117i \(-0.450880\pi\)
0.153704 + 0.988117i \(0.450880\pi\)
\(548\) −1.32279e42 −0.468159
\(549\) 2.05213e41 0.0706047
\(550\) 1.03020e41 0.0344588
\(551\) 4.77783e40 0.0155375
\(552\) 1.78710e42 0.565057
\(553\) 7.80068e40 0.0239824
\(554\) −1.23127e42 −0.368089
\(555\) −1.03737e42 −0.301572
\(556\) −1.18327e42 −0.334521
\(557\) −4.72577e42 −1.29932 −0.649662 0.760223i \(-0.725089\pi\)
−0.649662 + 0.760223i \(0.725089\pi\)
\(558\) 2.83567e41 0.0758273
\(559\) −3.31514e42 −0.862223
\(560\) 2.56405e40 0.00648652
\(561\) 2.16916e41 0.0533787
\(562\) 1.08642e42 0.260068
\(563\) −5.45592e42 −1.27054 −0.635269 0.772291i \(-0.719111\pi\)
−0.635269 + 0.772291i \(0.719111\pi\)
\(564\) 1.14104e42 0.258508
\(565\) 3.94274e42 0.869055
\(566\) −5.45309e42 −1.16947
\(567\) −3.16155e41 −0.0659724
\(568\) 3.08475e42 0.626354
\(569\) −1.38927e42 −0.274503 −0.137251 0.990536i \(-0.543827\pi\)
−0.137251 + 0.990536i \(0.543827\pi\)
\(570\) 2.54434e41 0.0489232
\(571\) 7.44435e41 0.139305 0.0696526 0.997571i \(-0.477811\pi\)
0.0696526 + 0.997571i \(0.477811\pi\)
\(572\) 4.98720e41 0.0908278
\(573\) −1.62885e42 −0.288726
\(574\) −2.03915e41 −0.0351815
\(575\) 1.76416e42 0.296270
\(576\) 1.25417e41 0.0205025
\(577\) 3.47495e42 0.552998 0.276499 0.961014i \(-0.410826\pi\)
0.276499 + 0.961014i \(0.410826\pi\)
\(578\) 4.37604e42 0.677953
\(579\) −8.86634e42 −1.33729
\(580\) 1.64993e41 0.0242288
\(581\) −4.32549e40 −0.00618450
\(582\) 4.67136e42 0.650335
\(583\) −2.44117e41 −0.0330930
\(584\) 2.19235e42 0.289408
\(585\) −4.25394e41 −0.0546859
\(586\) −1.86743e42 −0.233793
\(587\) −1.21331e43 −1.47939 −0.739694 0.672944i \(-0.765030\pi\)
−0.739694 + 0.672944i \(0.765030\pi\)
\(588\) 4.52725e42 0.537633
\(589\) −8.10523e41 −0.0937515
\(590\) 1.71371e42 0.193077
\(591\) −1.00308e43 −1.10085
\(592\) 1.46158e42 0.156256
\(593\) 6.01048e42 0.625981 0.312991 0.949756i \(-0.398669\pi\)
0.312991 + 0.949756i \(0.398669\pi\)
\(594\) 1.53157e42 0.155398
\(595\) 5.32951e40 0.00526834
\(596\) −1.68603e42 −0.162386
\(597\) 4.13442e42 0.387983
\(598\) 8.54031e42 0.780918
\(599\) −1.66324e43 −1.48197 −0.740987 0.671520i \(-0.765642\pi\)
−0.740987 + 0.671520i \(0.765642\pi\)
\(600\) 8.78637e41 0.0762896
\(601\) 1.70666e43 1.44409 0.722043 0.691848i \(-0.243203\pi\)
0.722043 + 0.691848i \(0.243203\pi\)
\(602\) −5.75366e41 −0.0474459
\(603\) −2.25845e42 −0.181507
\(604\) −1.85341e42 −0.145177
\(605\) −5.50988e42 −0.420662
\(606\) −5.60495e42 −0.417106
\(607\) 7.20634e42 0.522745 0.261373 0.965238i \(-0.415825\pi\)
0.261373 + 0.965238i \(0.415825\pi\)
\(608\) −3.58480e41 −0.0253489
\(609\) −9.83903e40 −0.00678241
\(610\) −2.02558e42 −0.136125
\(611\) 5.45287e42 0.357261
\(612\) 2.60685e41 0.0166521
\(613\) 2.06526e43 1.28629 0.643144 0.765745i \(-0.277630\pi\)
0.643144 + 0.765745i \(0.277630\pi\)
\(614\) −1.17988e43 −0.716522
\(615\) −6.98765e42 −0.413778
\(616\) 8.65563e40 0.00499801
\(617\) 2.04236e43 1.15004 0.575018 0.818140i \(-0.304995\pi\)
0.575018 + 0.818140i \(0.304995\pi\)
\(618\) 1.11948e43 0.614744
\(619\) −2.85830e42 −0.153074 −0.0765372 0.997067i \(-0.524386\pi\)
−0.0765372 + 0.997067i \(0.524386\pi\)
\(620\) −2.79898e42 −0.146194
\(621\) 2.62273e43 1.33608
\(622\) −7.62157e41 −0.0378698
\(623\) 8.04576e41 0.0389944
\(624\) 4.25348e42 0.201087
\(625\) 8.67362e41 0.0400000
\(626\) −2.48632e43 −1.11855
\(627\) 8.58908e41 0.0376964
\(628\) 1.31981e43 0.565114
\(629\) 3.03796e42 0.126911
\(630\) −7.38299e40 −0.00300922
\(631\) 3.58197e43 1.42451 0.712257 0.701919i \(-0.247673\pi\)
0.712257 + 0.701919i \(0.247673\pi\)
\(632\) −3.76622e42 −0.146147
\(633\) 1.78566e43 0.676146
\(634\) −2.19369e43 −0.810573
\(635\) −1.92063e43 −0.692549
\(636\) −2.08203e42 −0.0732657
\(637\) 2.16352e43 0.743017
\(638\) 5.56978e41 0.0186688
\(639\) −8.88230e42 −0.290577
\(640\) −1.23794e42 −0.0395285
\(641\) 2.02115e43 0.629941 0.314971 0.949101i \(-0.398005\pi\)
0.314971 + 0.949101i \(0.398005\pi\)
\(642\) −1.29567e43 −0.394187
\(643\) −1.67149e43 −0.496401 −0.248200 0.968709i \(-0.579839\pi\)
−0.248200 + 0.968709i \(0.579839\pi\)
\(644\) 1.48223e42 0.0429719
\(645\) −1.97164e43 −0.558022
\(646\) −7.45118e41 −0.0205883
\(647\) −4.63260e43 −1.24971 −0.624853 0.780742i \(-0.714841\pi\)
−0.624853 + 0.780742i \(0.714841\pi\)
\(648\) 1.52642e43 0.402032
\(649\) 5.78507e42 0.148770
\(650\) 4.19890e42 0.105433
\(651\) 1.66912e42 0.0409243
\(652\) −3.19050e43 −0.763874
\(653\) −8.05067e43 −1.88225 −0.941126 0.338056i \(-0.890231\pi\)
−0.941126 + 0.338056i \(0.890231\pi\)
\(654\) −2.13818e43 −0.488190
\(655\) −3.47779e43 −0.775468
\(656\) 9.84513e42 0.214394
\(657\) −6.31271e42 −0.134262
\(658\) 9.46382e41 0.0196592
\(659\) −6.45090e43 −1.30887 −0.654435 0.756118i \(-0.727093\pi\)
−0.654435 + 0.756118i \(0.727093\pi\)
\(660\) 2.96607e42 0.0587829
\(661\) −7.74577e42 −0.149948 −0.0749742 0.997185i \(-0.523887\pi\)
−0.0749742 + 0.997185i \(0.523887\pi\)
\(662\) −4.13272e43 −0.781516
\(663\) 8.84108e42 0.163322
\(664\) 2.08837e42 0.0376880
\(665\) 2.11029e41 0.00372054
\(666\) −4.20851e42 −0.0724899
\(667\) 9.53794e42 0.160511
\(668\) −1.32746e41 −0.00218265
\(669\) 5.28597e43 0.849219
\(670\) 2.22923e43 0.349942
\(671\) −6.83787e42 −0.104887
\(672\) 7.38220e41 0.0110653
\(673\) 4.06603e43 0.595576 0.297788 0.954632i \(-0.403751\pi\)
0.297788 + 0.954632i \(0.403751\pi\)
\(674\) −3.46795e43 −0.496414
\(675\) 1.28948e43 0.180387
\(676\) −1.62447e43 −0.222095
\(677\) 9.74695e42 0.130240 0.0651200 0.997877i \(-0.479257\pi\)
0.0651200 + 0.997877i \(0.479257\pi\)
\(678\) 1.13516e44 1.48251
\(679\) 3.87445e42 0.0494571
\(680\) −2.57312e42 −0.0321050
\(681\) 1.40773e44 1.71687
\(682\) −9.44871e42 −0.112646
\(683\) 1.36490e43 0.159067 0.0795337 0.996832i \(-0.474657\pi\)
0.0795337 + 0.996832i \(0.474657\pi\)
\(684\) 1.03222e42 0.0117598
\(685\) −3.75961e43 −0.418734
\(686\) 7.52254e42 0.0819107
\(687\) 1.56505e44 1.66609
\(688\) 2.77790e43 0.289132
\(689\) −9.94974e42 −0.101254
\(690\) 5.07923e43 0.505402
\(691\) −1.10022e42 −0.0107046 −0.00535229 0.999986i \(-0.501704\pi\)
−0.00535229 + 0.999986i \(0.501704\pi\)
\(692\) −9.09706e43 −0.865479
\(693\) −2.49232e41 −0.00231867
\(694\) 1.89054e43 0.171994
\(695\) −3.36305e43 −0.299205
\(696\) 4.75035e42 0.0413316
\(697\) 2.04636e43 0.174130
\(698\) 7.88740e43 0.656410
\(699\) 7.37119e43 0.599986
\(700\) 7.28747e41 0.00580172
\(701\) −2.42614e44 −1.88924 −0.944619 0.328169i \(-0.893568\pi\)
−0.944619 + 0.328169i \(0.893568\pi\)
\(702\) 6.24237e43 0.475471
\(703\) 1.20292e43 0.0896251
\(704\) −4.17899e42 −0.0304576
\(705\) 3.24302e43 0.231216
\(706\) −1.02950e43 −0.0718048
\(707\) −4.64878e42 −0.0317204
\(708\) 4.93397e43 0.329367
\(709\) −1.50345e44 −0.981909 −0.490954 0.871185i \(-0.663352\pi\)
−0.490954 + 0.871185i \(0.663352\pi\)
\(710\) 8.76737e43 0.560228
\(711\) 1.08445e43 0.0678004
\(712\) −3.88454e43 −0.237629
\(713\) −1.61804e44 −0.968503
\(714\) 1.53443e42 0.00898720
\(715\) 1.41745e43 0.0812388
\(716\) −1.41345e44 −0.792736
\(717\) 1.62687e43 0.0892906
\(718\) 2.29006e44 1.23004
\(719\) 2.19794e43 0.115536 0.0577680 0.998330i \(-0.481602\pi\)
0.0577680 + 0.998330i \(0.481602\pi\)
\(720\) 3.56456e42 0.0183380
\(721\) 9.28501e42 0.0467505
\(722\) 1.40536e44 0.692567
\(723\) −2.67069e44 −1.28819
\(724\) 7.78469e43 0.367531
\(725\) 4.68939e42 0.0216709
\(726\) −1.58636e44 −0.717602
\(727\) 1.29172e43 0.0571987 0.0285993 0.999591i \(-0.490895\pi\)
0.0285993 + 0.999591i \(0.490895\pi\)
\(728\) 3.52786e42 0.0152924
\(729\) 1.85363e44 0.786587
\(730\) 6.23103e43 0.258854
\(731\) 5.77401e43 0.234833
\(732\) −5.83188e43 −0.232213
\(733\) −3.65349e44 −1.42428 −0.712141 0.702037i \(-0.752274\pi\)
−0.712141 + 0.702037i \(0.752274\pi\)
\(734\) 7.97684e43 0.304468
\(735\) 1.28672e44 0.480874
\(736\) −7.15629e43 −0.261868
\(737\) 7.52536e43 0.269638
\(738\) −2.83483e43 −0.0994612
\(739\) 1.65100e44 0.567230 0.283615 0.958938i \(-0.408466\pi\)
0.283615 + 0.958938i \(0.408466\pi\)
\(740\) 4.15405e43 0.139759
\(741\) 3.50074e43 0.115339
\(742\) −1.72684e42 −0.00557176
\(743\) −5.25353e44 −1.66006 −0.830031 0.557717i \(-0.811677\pi\)
−0.830031 + 0.557717i \(0.811677\pi\)
\(744\) −8.05860e43 −0.249390
\(745\) −4.79199e43 −0.145243
\(746\) −3.43489e44 −1.01967
\(747\) −6.01331e42 −0.0174841
\(748\) −8.68625e42 −0.0247376
\(749\) −1.07464e43 −0.0299774
\(750\) 2.49724e43 0.0682355
\(751\) 2.28749e44 0.612265 0.306132 0.951989i \(-0.400965\pi\)
0.306132 + 0.951989i \(0.400965\pi\)
\(752\) −4.56919e43 −0.119802
\(753\) −6.26835e44 −1.61002
\(754\) 2.27013e43 0.0571209
\(755\) −5.26770e43 −0.129850
\(756\) 1.08341e43 0.0261639
\(757\) 3.96084e43 0.0937134 0.0468567 0.998902i \(-0.485080\pi\)
0.0468567 + 0.998902i \(0.485080\pi\)
\(758\) −2.20875e44 −0.512004
\(759\) 1.71463e44 0.389424
\(760\) −1.01886e43 −0.0226728
\(761\) 4.33856e44 0.945984 0.472992 0.881067i \(-0.343174\pi\)
0.472992 + 0.881067i \(0.343174\pi\)
\(762\) −5.52972e44 −1.18141
\(763\) −1.77342e43 −0.0371262
\(764\) 6.52262e43 0.133806
\(765\) 7.40911e42 0.0148941
\(766\) −3.54410e44 −0.698168
\(767\) 2.35788e44 0.455191
\(768\) −3.56418e43 −0.0674311
\(769\) 9.78759e44 1.81475 0.907377 0.420318i \(-0.138082\pi\)
0.907377 + 0.420318i \(0.138082\pi\)
\(770\) 2.46008e42 0.00447036
\(771\) 1.01603e45 1.80953
\(772\) 3.55046e44 0.619750
\(773\) −4.51209e44 −0.771961 −0.385981 0.922507i \(-0.626137\pi\)
−0.385981 + 0.922507i \(0.626137\pi\)
\(774\) −7.99876e43 −0.134134
\(775\) −7.95519e43 −0.130760
\(776\) −1.87061e44 −0.301389
\(777\) −2.47719e43 −0.0391231
\(778\) 6.56094e44 1.01574
\(779\) 8.10283e43 0.122972
\(780\) 1.20891e44 0.179857
\(781\) 2.95966e44 0.431669
\(782\) −1.48747e44 −0.212689
\(783\) 6.97157e43 0.0977289
\(784\) −1.81290e44 −0.249159
\(785\) 3.75112e44 0.505453
\(786\) −1.00130e45 −1.32286
\(787\) 2.01056e44 0.260441 0.130220 0.991485i \(-0.458431\pi\)
0.130220 + 0.991485i \(0.458431\pi\)
\(788\) 4.01675e44 0.510174
\(789\) 3.30090e44 0.411092
\(790\) −1.07042e44 −0.130718
\(791\) 9.41509e43 0.112743
\(792\) 1.20331e43 0.0141298
\(793\) −2.78698e44 −0.320922
\(794\) 7.23818e44 0.817357
\(795\) −5.91747e43 −0.0655309
\(796\) −1.65560e44 −0.179805
\(797\) −1.47308e45 −1.56900 −0.784500 0.620129i \(-0.787080\pi\)
−0.784500 + 0.620129i \(0.787080\pi\)
\(798\) 6.07577e42 0.00634682
\(799\) −9.49730e43 −0.0973027
\(800\) −3.51844e43 −0.0353553
\(801\) 1.11853e44 0.110241
\(802\) 2.05125e44 0.198297
\(803\) 2.10345e44 0.199453
\(804\) 6.41823e44 0.596961
\(805\) 4.21275e43 0.0384352
\(806\) −3.85111e44 −0.344661
\(807\) 2.24437e44 0.197040
\(808\) 2.24446e44 0.193302
\(809\) 1.82842e45 1.54481 0.772404 0.635131i \(-0.219054\pi\)
0.772404 + 0.635131i \(0.219054\pi\)
\(810\) 4.33834e44 0.359588
\(811\) 8.48526e44 0.689988 0.344994 0.938605i \(-0.387881\pi\)
0.344994 + 0.938605i \(0.387881\pi\)
\(812\) 3.93997e42 0.00314321
\(813\) −1.72782e45 −1.35237
\(814\) 1.40231e44 0.107688
\(815\) −9.06796e44 −0.683229
\(816\) −7.40832e43 −0.0547674
\(817\) 2.28630e44 0.165840
\(818\) 1.16161e45 0.826769
\(819\) −1.01582e43 −0.00709442
\(820\) 2.79815e44 0.191759
\(821\) −2.50451e45 −1.68424 −0.842119 0.539291i \(-0.818692\pi\)
−0.842119 + 0.539291i \(0.818692\pi\)
\(822\) −1.08244e45 −0.714313
\(823\) 1.83581e45 1.18886 0.594430 0.804148i \(-0.297378\pi\)
0.594430 + 0.804148i \(0.297378\pi\)
\(824\) −4.48286e44 −0.284894
\(825\) 8.43008e43 0.0525770
\(826\) 4.09226e43 0.0250480
\(827\) 1.18525e43 0.00711988 0.00355994 0.999994i \(-0.498867\pi\)
0.00355994 + 0.999994i \(0.498867\pi\)
\(828\) 2.06060e44 0.121485
\(829\) −1.49150e45 −0.863034 −0.431517 0.902105i \(-0.642022\pi\)
−0.431517 + 0.902105i \(0.642022\pi\)
\(830\) 5.93551e43 0.0337091
\(831\) −1.00754e45 −0.561626
\(832\) −1.70328e44 −0.0931908
\(833\) −3.76821e44 −0.202366
\(834\) −9.68261e44 −0.510409
\(835\) −3.77285e42 −0.00195222
\(836\) −3.43943e43 −0.0174699
\(837\) −1.18267e45 −0.589685
\(838\) 9.76747e44 0.478079
\(839\) 7.17066e44 0.344547 0.172274 0.985049i \(-0.444889\pi\)
0.172274 + 0.985049i \(0.444889\pi\)
\(840\) 2.09815e43 0.00989708
\(841\) −2.13407e45 −0.988259
\(842\) −1.69626e45 −0.771180
\(843\) 8.89015e44 0.396809
\(844\) −7.15053e44 −0.313350
\(845\) −4.61703e44 −0.198648
\(846\) 1.31567e44 0.0555782
\(847\) −1.31574e44 −0.0545727
\(848\) 8.33732e43 0.0339540
\(849\) −4.46224e45 −1.78436
\(850\) −7.31325e43 −0.0287155
\(851\) 2.40138e45 0.925876
\(852\) 2.52423e45 0.955686
\(853\) −2.17378e45 −0.808176 −0.404088 0.914720i \(-0.632411\pi\)
−0.404088 + 0.914720i \(0.632411\pi\)
\(854\) −4.83700e43 −0.0176595
\(855\) 2.93373e43 0.0105183
\(856\) 5.18842e44 0.182680
\(857\) 1.46655e45 0.507102 0.253551 0.967322i \(-0.418401\pi\)
0.253551 + 0.967322i \(0.418401\pi\)
\(858\) 4.08100e44 0.138584
\(859\) −3.71283e45 −1.23826 −0.619129 0.785290i \(-0.712514\pi\)
−0.619129 + 0.785290i \(0.712514\pi\)
\(860\) 7.89527e44 0.258608
\(861\) −1.66862e44 −0.0536796
\(862\) −1.06967e45 −0.337978
\(863\) 2.28506e45 0.709138 0.354569 0.935030i \(-0.384628\pi\)
0.354569 + 0.935030i \(0.384628\pi\)
\(864\) −5.23075e44 −0.159441
\(865\) −2.58554e45 −0.774108
\(866\) 3.32337e45 0.977352
\(867\) 3.58089e45 1.03442
\(868\) −6.68385e43 −0.0189658
\(869\) −3.61350e44 −0.100721
\(870\) 1.35013e44 0.0369681
\(871\) 3.06719e45 0.825010
\(872\) 8.56218e44 0.226245
\(873\) 5.38629e44 0.139820
\(874\) −5.88984e44 −0.150202
\(875\) 2.07122e43 0.00518922
\(876\) 1.79399e45 0.441576
\(877\) −1.08696e45 −0.262858 −0.131429 0.991326i \(-0.541957\pi\)
−0.131429 + 0.991326i \(0.541957\pi\)
\(878\) 3.20290e45 0.760989
\(879\) −1.52811e45 −0.356719
\(880\) −1.18774e44 −0.0272421
\(881\) 5.54107e45 1.24873 0.624363 0.781134i \(-0.285359\pi\)
0.624363 + 0.781134i \(0.285359\pi\)
\(882\) 5.22012e44 0.115589
\(883\) 2.97545e45 0.647384 0.323692 0.946162i \(-0.395076\pi\)
0.323692 + 0.946162i \(0.395076\pi\)
\(884\) −3.54034e44 −0.0756894
\(885\) 1.40232e45 0.294595
\(886\) 2.71528e45 0.560522
\(887\) 9.12159e45 1.85035 0.925177 0.379537i \(-0.123917\pi\)
0.925177 + 0.379537i \(0.123917\pi\)
\(888\) 1.19600e45 0.238413
\(889\) −4.58638e44 −0.0898448
\(890\) −1.10405e45 −0.212542
\(891\) 1.46452e45 0.277071
\(892\) −2.11673e45 −0.393559
\(893\) −3.76058e44 −0.0687159
\(894\) −1.37967e45 −0.247767
\(895\) −4.01727e45 −0.709044
\(896\) −2.95615e43 −0.00512805
\(897\) 6.98849e45 1.19152
\(898\) −5.62786e45 −0.943104
\(899\) −4.30097e44 −0.0708420
\(900\) 1.01311e44 0.0164020
\(901\) 1.73295e44 0.0275774
\(902\) 9.44590e44 0.147755
\(903\) −4.70819e44 −0.0723925
\(904\) −4.54567e45 −0.687048
\(905\) 2.21254e45 0.328730
\(906\) −1.51663e45 −0.221510
\(907\) −5.68239e45 −0.815865 −0.407932 0.913012i \(-0.633750\pi\)
−0.407932 + 0.913012i \(0.633750\pi\)
\(908\) −5.63714e45 −0.795661
\(909\) −6.46276e44 −0.0896763
\(910\) 1.00268e44 0.0136779
\(911\) −2.79936e43 −0.00375426 −0.00187713 0.999998i \(-0.500598\pi\)
−0.00187713 + 0.999998i \(0.500598\pi\)
\(912\) −2.93342e44 −0.0386772
\(913\) 2.00369e44 0.0259737
\(914\) −5.12808e45 −0.653565
\(915\) −1.65752e45 −0.207698
\(916\) −6.26713e45 −0.772127
\(917\) −8.30482e44 −0.100602
\(918\) −1.08724e45 −0.129498
\(919\) −1.43630e46 −1.68211 −0.841055 0.540950i \(-0.818065\pi\)
−0.841055 + 0.540950i \(0.818065\pi\)
\(920\) −2.03394e45 −0.234222
\(921\) −9.65493e45 −1.09326
\(922\) −2.65374e45 −0.295480
\(923\) 1.20630e46 1.32077
\(924\) 7.08285e43 0.00762593
\(925\) 1.18065e45 0.125004
\(926\) 1.91438e45 0.199323
\(927\) 1.29081e45 0.132168
\(928\) −1.90224e44 −0.0191545
\(929\) 1.85223e46 1.83422 0.917108 0.398638i \(-0.130517\pi\)
0.917108 + 0.398638i \(0.130517\pi\)
\(930\) −2.29039e45 −0.223061
\(931\) −1.49207e45 −0.142912
\(932\) −2.95174e45 −0.278055
\(933\) −6.23669e44 −0.0577815
\(934\) 6.25026e45 0.569536
\(935\) −2.46878e44 −0.0221260
\(936\) 4.90445e44 0.0432330
\(937\) −7.97850e45 −0.691764 −0.345882 0.938278i \(-0.612420\pi\)
−0.345882 + 0.938278i \(0.612420\pi\)
\(938\) 5.32331e44 0.0453981
\(939\) −2.03454e46 −1.70667
\(940\) −1.29864e45 −0.107154
\(941\) 7.58451e45 0.615585 0.307793 0.951453i \(-0.400410\pi\)
0.307793 + 0.951453i \(0.400410\pi\)
\(942\) 1.07999e46 0.862246
\(943\) 1.61756e46 1.27037
\(944\) −1.97577e45 −0.152641
\(945\) 3.07922e44 0.0234017
\(946\) 2.66526e45 0.199263
\(947\) −1.64569e46 −1.21038 −0.605192 0.796079i \(-0.706904\pi\)
−0.605192 + 0.796079i \(0.706904\pi\)
\(948\) −3.08188e45 −0.222990
\(949\) 8.57324e45 0.610265
\(950\) −2.89578e44 −0.0202791
\(951\) −1.79509e46 −1.23677
\(952\) −6.14450e43 −0.00416499
\(953\) 1.88162e46 1.25485 0.627426 0.778676i \(-0.284109\pi\)
0.627426 + 0.778676i \(0.284109\pi\)
\(954\) −2.40067e44 −0.0157519
\(955\) 1.85384e45 0.119680
\(956\) −6.51468e44 −0.0413805
\(957\) 4.55772e44 0.0284848
\(958\) 5.71825e45 0.351640
\(959\) −8.97778e44 −0.0543226
\(960\) −1.01300e45 −0.0603122
\(961\) −9.77291e45 −0.572547
\(962\) 5.71554e45 0.329491
\(963\) −1.49397e45 −0.0847488
\(964\) 1.06946e46 0.596994
\(965\) 1.00910e46 0.554321
\(966\) 1.21290e45 0.0655661
\(967\) −2.62276e46 −1.39524 −0.697619 0.716469i \(-0.745757\pi\)
−0.697619 + 0.716469i \(0.745757\pi\)
\(968\) 6.35246e45 0.332563
\(969\) −6.09727e44 −0.0314135
\(970\) −5.31660e45 −0.269570
\(971\) 1.34449e46 0.670904 0.335452 0.942057i \(-0.391111\pi\)
0.335452 + 0.942057i \(0.391111\pi\)
\(972\) 3.30782e45 0.162448
\(973\) −8.03081e44 −0.0388160
\(974\) 1.91373e46 0.910369
\(975\) 3.43594e45 0.160869
\(976\) 2.33533e45 0.107616
\(977\) −1.98144e46 −0.898699 −0.449350 0.893356i \(-0.648344\pi\)
−0.449350 + 0.893356i \(0.648344\pi\)
\(978\) −2.61077e46 −1.16551
\(979\) −3.72703e45 −0.163769
\(980\) −5.15258e45 −0.222854
\(981\) −2.46541e45 −0.104959
\(982\) 1.92353e46 0.806067
\(983\) −5.61388e44 −0.0231571 −0.0115785 0.999933i \(-0.503686\pi\)
−0.0115785 + 0.999933i \(0.503686\pi\)
\(984\) 8.05621e45 0.327120
\(985\) 1.14163e46 0.456313
\(986\) −3.95390e44 −0.0155573
\(987\) 7.74419e44 0.0299958
\(988\) −1.40185e45 −0.0534524
\(989\) 4.56411e46 1.71322
\(990\) 3.42001e44 0.0126381
\(991\) 9.54885e45 0.347384 0.173692 0.984800i \(-0.444430\pi\)
0.173692 + 0.984800i \(0.444430\pi\)
\(992\) 3.22701e45 0.115576
\(993\) −3.38179e46 −1.19243
\(994\) 2.09361e45 0.0726787
\(995\) −4.70550e45 −0.160823
\(996\) 1.70890e45 0.0575040
\(997\) −4.00436e45 −0.132666 −0.0663328 0.997798i \(-0.521130\pi\)
−0.0663328 + 0.997798i \(0.521130\pi\)
\(998\) −1.77668e46 −0.579543
\(999\) 1.75524e46 0.563731
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 10.32.a.a.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.32.a.a.1.1 2 1.1 even 1 trivial