Properties

Label 10.22.a.b.1.2
Level $10$
Weight $22$
Character 10.1
Self dual yes
Analytic conductor $27.948$
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] = [10,22,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, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("10.1");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 22 \)
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: \(27.9477344287\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{157921}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 39480 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4}\cdot 5\cdot 7 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-198.196\) 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)
 
\(f(q)\) \(=\) \(q-1024.00 q^{2} +47924.0 q^{3} +1.04858e6 q^{4} -9.76562e6 q^{5} -4.90741e7 q^{6} +6.67863e8 q^{7} -1.07374e9 q^{8} -8.16365e9 q^{9} +1.00000e10 q^{10} -1.26793e11 q^{11} +5.02519e10 q^{12} +9.12750e11 q^{13} -6.83892e11 q^{14} -4.68008e11 q^{15} +1.09951e12 q^{16} +8.60406e12 q^{17} +8.35957e12 q^{18} -6.93702e12 q^{19} -1.02400e13 q^{20} +3.20066e13 q^{21} +1.29836e14 q^{22} -3.30974e14 q^{23} -5.14580e13 q^{24} +9.53674e13 q^{25} -9.34656e14 q^{26} -8.92536e14 q^{27} +7.00305e14 q^{28} -3.90534e15 q^{29} +4.79240e14 q^{30} +3.32185e15 q^{31} -1.12590e15 q^{32} -6.07642e15 q^{33} -8.81056e15 q^{34} -6.52210e15 q^{35} -8.56020e15 q^{36} -4.33804e16 q^{37} +7.10351e15 q^{38} +4.37426e16 q^{39} +1.04858e16 q^{40} -9.56620e16 q^{41} -3.27748e16 q^{42} -7.81706e16 q^{43} -1.32952e17 q^{44} +7.97231e16 q^{45} +3.38917e17 q^{46} -2.03634e17 q^{47} +5.26930e16 q^{48} -1.12505e17 q^{49} -9.76562e16 q^{50} +4.12341e17 q^{51} +9.57088e17 q^{52} +1.42121e18 q^{53} +9.13957e17 q^{54} +1.23821e18 q^{55} -7.17112e17 q^{56} -3.32449e17 q^{57} +3.99907e18 q^{58} +2.28919e18 q^{59} -4.90741e17 q^{60} -5.32798e18 q^{61} -3.40158e18 q^{62} -5.45220e18 q^{63} +1.15292e18 q^{64} -8.91357e18 q^{65} +6.22226e18 q^{66} -1.31401e19 q^{67} +9.02201e18 q^{68} -1.58616e19 q^{69} +6.67863e18 q^{70} -2.98159e19 q^{71} +8.76565e18 q^{72} +1.37094e18 q^{73} +4.44215e19 q^{74} +4.57039e18 q^{75} -7.27399e18 q^{76} -8.46803e19 q^{77} -4.47924e19 q^{78} +1.74281e19 q^{79} -1.07374e19 q^{80} +4.26208e19 q^{81} +9.79579e19 q^{82} +3.75717e19 q^{83} +3.35614e19 q^{84} -8.40240e19 q^{85} +8.00467e19 q^{86} -1.87159e20 q^{87} +1.36143e20 q^{88} +2.43231e20 q^{89} -8.16365e19 q^{90} +6.09592e20 q^{91} -3.47051e20 q^{92} +1.59196e20 q^{93} +2.08521e20 q^{94} +6.77443e19 q^{95} -5.39576e19 q^{96} -1.80740e20 q^{97} +1.15205e20 q^{98} +1.03509e21 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2048 q^{2} - 126692 q^{3} + 2097152 q^{4} - 19531250 q^{5} + 129732608 q^{6} - 292598684 q^{7} - 2147483648 q^{8} + 11866737826 q^{9} + 20000000000 q^{10} - 41326831776 q^{11} - 132846190592 q^{12}+ \cdots + 27\!\cdots\!12 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1024.00 −0.707107
\(3\) 47924.0 0.468576 0.234288 0.972167i \(-0.424724\pi\)
0.234288 + 0.972167i \(0.424724\pi\)
\(4\) 1.04858e6 0.500000
\(5\) −9.76562e6 −0.447214
\(6\) −4.90741e7 −0.331333
\(7\) 6.67863e8 0.893631 0.446815 0.894626i \(-0.352558\pi\)
0.446815 + 0.894626i \(0.352558\pi\)
\(8\) −1.07374e9 −0.353553
\(9\) −8.16365e9 −0.780437
\(10\) 1.00000e10 0.316228
\(11\) −1.26793e11 −1.47391 −0.736957 0.675940i \(-0.763738\pi\)
−0.736957 + 0.675940i \(0.763738\pi\)
\(12\) 5.02519e10 0.234288
\(13\) 9.12750e11 1.83631 0.918156 0.396219i \(-0.129678\pi\)
0.918156 + 0.396219i \(0.129678\pi\)
\(14\) −6.83892e11 −0.631892
\(15\) −4.68008e11 −0.209553
\(16\) 1.09951e12 0.250000
\(17\) 8.60406e12 1.03512 0.517559 0.855648i \(-0.326841\pi\)
0.517559 + 0.855648i \(0.326841\pi\)
\(18\) 8.35957e12 0.551852
\(19\) −6.93702e12 −0.259573 −0.129787 0.991542i \(-0.541429\pi\)
−0.129787 + 0.991542i \(0.541429\pi\)
\(20\) −1.02400e13 −0.223607
\(21\) 3.20066e13 0.418733
\(22\) 1.29836e14 1.04221
\(23\) −3.30974e14 −1.66591 −0.832955 0.553341i \(-0.813352\pi\)
−0.832955 + 0.553341i \(0.813352\pi\)
\(24\) −5.14580e13 −0.165666
\(25\) 9.53674e13 0.200000
\(26\) −9.34656e14 −1.29847
\(27\) −8.92536e14 −0.834269
\(28\) 7.00305e14 0.446815
\(29\) −3.90534e15 −1.72377 −0.861886 0.507103i \(-0.830716\pi\)
−0.861886 + 0.507103i \(0.830716\pi\)
\(30\) 4.79240e14 0.148177
\(31\) 3.32185e15 0.727918 0.363959 0.931415i \(-0.381425\pi\)
0.363959 + 0.931415i \(0.381425\pi\)
\(32\) −1.12590e15 −0.176777
\(33\) −6.07642e15 −0.690640
\(34\) −8.81056e15 −0.731939
\(35\) −6.52210e15 −0.399644
\(36\) −8.56020e15 −0.390218
\(37\) −4.33804e16 −1.48312 −0.741558 0.670889i \(-0.765913\pi\)
−0.741558 + 0.670889i \(0.765913\pi\)
\(38\) 7.10351e15 0.183546
\(39\) 4.37426e16 0.860451
\(40\) 1.04858e16 0.158114
\(41\) −9.56620e16 −1.11304 −0.556518 0.830836i \(-0.687863\pi\)
−0.556518 + 0.830836i \(0.687863\pi\)
\(42\) −3.27748e16 −0.296089
\(43\) −7.81706e16 −0.551601 −0.275800 0.961215i \(-0.588943\pi\)
−0.275800 + 0.961215i \(0.588943\pi\)
\(44\) −1.32952e17 −0.736957
\(45\) 7.97231e16 0.349022
\(46\) 3.38917e17 1.17798
\(47\) −2.03634e17 −0.564707 −0.282354 0.959310i \(-0.591115\pi\)
−0.282354 + 0.959310i \(0.591115\pi\)
\(48\) 5.26930e16 0.117144
\(49\) −1.12505e17 −0.201425
\(50\) −9.76562e16 −0.141421
\(51\) 4.12341e17 0.485031
\(52\) 9.57088e17 0.918156
\(53\) 1.42121e18 1.11625 0.558127 0.829756i \(-0.311520\pi\)
0.558127 + 0.829756i \(0.311520\pi\)
\(54\) 9.13957e17 0.589917
\(55\) 1.23821e18 0.659154
\(56\) −7.17112e17 −0.315946
\(57\) −3.32449e17 −0.121630
\(58\) 3.99907e18 1.21889
\(59\) 2.28919e18 0.583089 0.291545 0.956557i \(-0.405831\pi\)
0.291545 + 0.956557i \(0.405831\pi\)
\(60\) −4.90741e17 −0.104777
\(61\) −5.32798e18 −0.956311 −0.478155 0.878275i \(-0.658694\pi\)
−0.478155 + 0.878275i \(0.658694\pi\)
\(62\) −3.40158e18 −0.514716
\(63\) −5.45220e18 −0.697422
\(64\) 1.15292e18 0.125000
\(65\) −8.91357e18 −0.821224
\(66\) 6.22226e18 0.488356
\(67\) −1.31401e19 −0.880668 −0.440334 0.897834i \(-0.645140\pi\)
−0.440334 + 0.897834i \(0.645140\pi\)
\(68\) 9.02201e18 0.517559
\(69\) −1.58616e19 −0.780604
\(70\) 6.67863e18 0.282591
\(71\) −2.98159e19 −1.08702 −0.543508 0.839404i \(-0.682904\pi\)
−0.543508 + 0.839404i \(0.682904\pi\)
\(72\) 8.76565e18 0.275926
\(73\) 1.37094e18 0.0373361 0.0186681 0.999826i \(-0.494057\pi\)
0.0186681 + 0.999826i \(0.494057\pi\)
\(74\) 4.44215e19 1.04872
\(75\) 4.57039e18 0.0937151
\(76\) −7.27399e18 −0.129787
\(77\) −8.46803e19 −1.31713
\(78\) −4.47924e19 −0.608431
\(79\) 1.74281e19 0.207093 0.103546 0.994625i \(-0.466981\pi\)
0.103546 + 0.994625i \(0.466981\pi\)
\(80\) −1.07374e19 −0.111803
\(81\) 4.26208e19 0.389519
\(82\) 9.79579e19 0.787035
\(83\) 3.75717e19 0.265792 0.132896 0.991130i \(-0.457572\pi\)
0.132896 + 0.991130i \(0.457572\pi\)
\(84\) 3.35614e19 0.209367
\(85\) −8.40240e19 −0.462919
\(86\) 8.00467e19 0.390040
\(87\) −1.87159e20 −0.807717
\(88\) 1.36143e20 0.521107
\(89\) 2.43231e20 0.826845 0.413423 0.910539i \(-0.364333\pi\)
0.413423 + 0.910539i \(0.364333\pi\)
\(90\) −8.16365e19 −0.246796
\(91\) 6.09592e20 1.64098
\(92\) −3.47051e20 −0.832955
\(93\) 1.59196e20 0.341084
\(94\) 2.08521e20 0.399308
\(95\) 6.77443e19 0.116085
\(96\) −5.39576e19 −0.0828332
\(97\) −1.80740e20 −0.248858 −0.124429 0.992229i \(-0.539710\pi\)
−0.124429 + 0.992229i \(0.539710\pi\)
\(98\) 1.15205e20 0.142429
\(99\) 1.03509e21 1.15030
\(100\) 1.00000e20 0.100000
\(101\) −4.40260e20 −0.396583 −0.198292 0.980143i \(-0.563539\pi\)
−0.198292 + 0.980143i \(0.563539\pi\)
\(102\) −4.22237e20 −0.342969
\(103\) −1.91819e21 −1.40637 −0.703186 0.711005i \(-0.748240\pi\)
−0.703186 + 0.711005i \(0.748240\pi\)
\(104\) −9.80058e20 −0.649234
\(105\) −3.12565e20 −0.187263
\(106\) −1.45532e21 −0.789310
\(107\) 7.55037e20 0.371055 0.185528 0.982639i \(-0.440601\pi\)
0.185528 + 0.982639i \(0.440601\pi\)
\(108\) −9.35892e20 −0.417135
\(109\) −9.55072e20 −0.386418 −0.193209 0.981158i \(-0.561890\pi\)
−0.193209 + 0.981158i \(0.561890\pi\)
\(110\) −1.26793e21 −0.466092
\(111\) −2.07896e21 −0.694952
\(112\) 7.34323e20 0.223408
\(113\) 5.50602e21 1.52586 0.762929 0.646482i \(-0.223760\pi\)
0.762929 + 0.646482i \(0.223760\pi\)
\(114\) 3.40428e20 0.0860052
\(115\) 3.23217e21 0.745017
\(116\) −4.09504e21 −0.861886
\(117\) −7.45137e21 −1.43313
\(118\) −2.34413e21 −0.412306
\(119\) 5.74634e21 0.925013
\(120\) 5.02519e20 0.0740883
\(121\) 8.67620e21 1.17242
\(122\) 5.45585e21 0.676214
\(123\) −4.58450e21 −0.521541
\(124\) 3.48321e21 0.363959
\(125\) −9.31323e20 −0.0894427
\(126\) 5.58305e21 0.493152
\(127\) 2.04590e22 1.66320 0.831602 0.555371i \(-0.187424\pi\)
0.831602 + 0.555371i \(0.187424\pi\)
\(128\) −1.18059e21 −0.0883883
\(129\) −3.74625e21 −0.258467
\(130\) 9.12750e21 0.580693
\(131\) −1.03791e22 −0.609271 −0.304635 0.952469i \(-0.598535\pi\)
−0.304635 + 0.952469i \(0.598535\pi\)
\(132\) −6.37159e21 −0.345320
\(133\) −4.63298e21 −0.231963
\(134\) 1.34554e22 0.622727
\(135\) 8.71617e21 0.373097
\(136\) −9.23854e21 −0.365969
\(137\) −9.13014e21 −0.334897 −0.167449 0.985881i \(-0.553553\pi\)
−0.167449 + 0.985881i \(0.553553\pi\)
\(138\) 1.62423e22 0.551971
\(139\) −3.18076e22 −1.00202 −0.501009 0.865442i \(-0.667038\pi\)
−0.501009 + 0.865442i \(0.667038\pi\)
\(140\) −6.83892e21 −0.199822
\(141\) −9.75896e21 −0.264608
\(142\) 3.05315e22 0.768636
\(143\) −1.15730e23 −2.70656
\(144\) −8.97602e21 −0.195109
\(145\) 3.81381e22 0.770894
\(146\) −1.40384e21 −0.0264006
\(147\) −5.39168e21 −0.0943826
\(148\) −4.54876e22 −0.741558
\(149\) 1.17039e23 1.77777 0.888884 0.458133i \(-0.151482\pi\)
0.888884 + 0.458133i \(0.151482\pi\)
\(150\) −4.68008e21 −0.0662666
\(151\) −1.57736e22 −0.208293 −0.104146 0.994562i \(-0.533211\pi\)
−0.104146 + 0.994562i \(0.533211\pi\)
\(152\) 7.44857e21 0.0917730
\(153\) −7.02405e22 −0.807844
\(154\) 8.67126e22 0.931354
\(155\) −3.24399e22 −0.325535
\(156\) 4.58674e22 0.430226
\(157\) −9.37324e22 −0.822135 −0.411068 0.911605i \(-0.634844\pi\)
−0.411068 + 0.911605i \(0.634844\pi\)
\(158\) −1.78464e22 −0.146437
\(159\) 6.81102e22 0.523049
\(160\) 1.09951e22 0.0790569
\(161\) −2.21045e23 −1.48871
\(162\) −4.36437e22 −0.275431
\(163\) 1.15735e23 0.684691 0.342345 0.939574i \(-0.388779\pi\)
0.342345 + 0.939574i \(0.388779\pi\)
\(164\) −1.00309e23 −0.556518
\(165\) 5.93400e22 0.308863
\(166\) −3.84734e22 −0.187943
\(167\) 1.46022e23 0.669723 0.334862 0.942267i \(-0.391310\pi\)
0.334862 + 0.942267i \(0.391310\pi\)
\(168\) −3.43669e22 −0.148045
\(169\) 5.86048e23 2.37204
\(170\) 8.60406e22 0.327333
\(171\) 5.66314e22 0.202581
\(172\) −8.19678e22 −0.275800
\(173\) −2.62784e23 −0.831984 −0.415992 0.909368i \(-0.636566\pi\)
−0.415992 + 0.909368i \(0.636566\pi\)
\(174\) 1.91651e23 0.571142
\(175\) 6.36924e22 0.178726
\(176\) −1.39410e23 −0.368478
\(177\) 1.09707e23 0.273221
\(178\) −2.49069e23 −0.584668
\(179\) 8.92249e23 1.97483 0.987413 0.158160i \(-0.0505562\pi\)
0.987413 + 0.158160i \(0.0505562\pi\)
\(180\) 8.35957e22 0.174511
\(181\) 2.71393e23 0.534531 0.267266 0.963623i \(-0.413880\pi\)
0.267266 + 0.963623i \(0.413880\pi\)
\(182\) −6.24222e23 −1.16035
\(183\) −2.55338e23 −0.448104
\(184\) 3.55380e23 0.588988
\(185\) 4.23636e23 0.663269
\(186\) −1.63017e23 −0.241183
\(187\) −1.09093e24 −1.52567
\(188\) −2.13526e23 −0.282354
\(189\) −5.96092e23 −0.745528
\(190\) −6.93702e22 −0.0820843
\(191\) −1.16742e24 −1.30731 −0.653654 0.756794i \(-0.726765\pi\)
−0.653654 + 0.756794i \(0.726765\pi\)
\(192\) 5.52526e22 0.0585719
\(193\) 8.18854e22 0.0821968 0.0410984 0.999155i \(-0.486914\pi\)
0.0410984 + 0.999155i \(0.486914\pi\)
\(194\) 1.85078e23 0.175969
\(195\) −4.27174e23 −0.384805
\(196\) −1.17970e23 −0.100712
\(197\) −6.59417e23 −0.533660 −0.266830 0.963744i \(-0.585976\pi\)
−0.266830 + 0.963744i \(0.585976\pi\)
\(198\) −1.05993e24 −0.813382
\(199\) 1.82645e24 1.32938 0.664692 0.747117i \(-0.268563\pi\)
0.664692 + 0.747117i \(0.268563\pi\)
\(200\) −1.02400e23 −0.0707107
\(201\) −6.29725e23 −0.412660
\(202\) 4.50826e23 0.280427
\(203\) −2.60823e24 −1.54041
\(204\) 4.32371e23 0.242515
\(205\) 9.34199e23 0.497764
\(206\) 1.96423e24 0.994456
\(207\) 2.70195e24 1.30014
\(208\) 1.00358e24 0.459078
\(209\) 8.79565e23 0.382588
\(210\) 3.20066e23 0.132415
\(211\) −3.36470e24 −1.32428 −0.662140 0.749380i \(-0.730352\pi\)
−0.662140 + 0.749380i \(0.730352\pi\)
\(212\) 1.49025e24 0.558127
\(213\) −1.42890e24 −0.509349
\(214\) −7.73158e23 −0.262376
\(215\) 7.63385e23 0.246683
\(216\) 9.58353e23 0.294959
\(217\) 2.21854e24 0.650490
\(218\) 9.77993e23 0.273239
\(219\) 6.57010e22 0.0174948
\(220\) 1.29836e24 0.329577
\(221\) 7.85336e24 1.90080
\(222\) 2.12885e24 0.491405
\(223\) 1.44704e23 0.0318626 0.0159313 0.999873i \(-0.494929\pi\)
0.0159313 + 0.999873i \(0.494929\pi\)
\(224\) −7.51947e23 −0.157973
\(225\) −7.78546e23 −0.156087
\(226\) −5.63817e24 −1.07894
\(227\) −4.83544e24 −0.883415 −0.441707 0.897159i \(-0.645627\pi\)
−0.441707 + 0.897159i \(0.645627\pi\)
\(228\) −3.48598e23 −0.0608148
\(229\) −4.35342e24 −0.725367 −0.362684 0.931912i \(-0.618139\pi\)
−0.362684 + 0.931912i \(0.618139\pi\)
\(230\) −3.30974e24 −0.526807
\(231\) −4.05822e24 −0.617177
\(232\) 4.19332e24 0.609445
\(233\) 1.03556e23 0.0143860 0.00719300 0.999974i \(-0.497710\pi\)
0.00719300 + 0.999974i \(0.497710\pi\)
\(234\) 7.63020e24 1.01337
\(235\) 1.98862e24 0.252545
\(236\) 2.40039e24 0.291545
\(237\) 8.35223e23 0.0970387
\(238\) −5.88425e24 −0.654083
\(239\) −1.45089e25 −1.54332 −0.771660 0.636035i \(-0.780573\pi\)
−0.771660 + 0.636035i \(0.780573\pi\)
\(240\) −5.14580e23 −0.0523883
\(241\) −2.10504e23 −0.0205155 −0.0102578 0.999947i \(-0.503265\pi\)
−0.0102578 + 0.999947i \(0.503265\pi\)
\(242\) −8.88443e24 −0.829026
\(243\) 1.13788e25 1.01679
\(244\) −5.58679e24 −0.478155
\(245\) 1.09868e24 0.0900798
\(246\) 4.69453e24 0.368785
\(247\) −6.33176e24 −0.476658
\(248\) −3.56681e24 −0.257358
\(249\) 1.80059e24 0.124543
\(250\) 9.53674e23 0.0632456
\(251\) −8.79823e24 −0.559527 −0.279764 0.960069i \(-0.590256\pi\)
−0.279764 + 0.960069i \(0.590256\pi\)
\(252\) −5.71704e24 −0.348711
\(253\) 4.19651e25 2.45541
\(254\) −2.09500e25 −1.17606
\(255\) −4.02677e24 −0.216912
\(256\) 1.20893e24 0.0625000
\(257\) 2.93954e24 0.145875 0.0729376 0.997337i \(-0.476763\pi\)
0.0729376 + 0.997337i \(0.476763\pi\)
\(258\) 3.83616e24 0.182763
\(259\) −2.89721e25 −1.32536
\(260\) −9.34656e24 −0.410612
\(261\) 3.18818e25 1.34529
\(262\) 1.06282e25 0.430820
\(263\) −7.06799e23 −0.0275271 −0.0137636 0.999905i \(-0.504381\pi\)
−0.0137636 + 0.999905i \(0.504381\pi\)
\(264\) 6.52451e24 0.244178
\(265\) −1.38790e25 −0.499204
\(266\) 4.74417e24 0.164022
\(267\) 1.16566e25 0.387439
\(268\) −1.37784e25 −0.440334
\(269\) 4.29393e25 1.31964 0.659821 0.751423i \(-0.270632\pi\)
0.659821 + 0.751423i \(0.270632\pi\)
\(270\) −8.92536e24 −0.263819
\(271\) 1.66203e25 0.472564 0.236282 0.971685i \(-0.424071\pi\)
0.236282 + 0.971685i \(0.424071\pi\)
\(272\) 9.46027e24 0.258780
\(273\) 2.92141e25 0.768925
\(274\) 9.34927e24 0.236808
\(275\) −1.20919e25 −0.294783
\(276\) −1.66321e25 −0.390302
\(277\) −2.85979e25 −0.646095 −0.323048 0.946383i \(-0.604707\pi\)
−0.323048 + 0.946383i \(0.604707\pi\)
\(278\) 3.25710e25 0.708534
\(279\) −2.71184e25 −0.568094
\(280\) 7.00305e24 0.141295
\(281\) −5.58235e25 −1.08493 −0.542464 0.840079i \(-0.682509\pi\)
−0.542464 + 0.840079i \(0.682509\pi\)
\(282\) 9.99318e24 0.187106
\(283\) 6.54228e25 1.18024 0.590121 0.807315i \(-0.299080\pi\)
0.590121 + 0.807315i \(0.299080\pi\)
\(284\) −3.12643e25 −0.543508
\(285\) 3.24658e24 0.0543944
\(286\) 1.18508e26 1.91383
\(287\) −6.38891e25 −0.994642
\(288\) 9.19145e24 0.137963
\(289\) 4.93795e24 0.0714693
\(290\) −3.90534e25 −0.545104
\(291\) −8.66179e24 −0.116609
\(292\) 1.43754e24 0.0186681
\(293\) −1.10836e25 −0.138858 −0.0694289 0.997587i \(-0.522118\pi\)
−0.0694289 + 0.997587i \(0.522118\pi\)
\(294\) 5.52108e24 0.0667386
\(295\) −2.23553e25 −0.260765
\(296\) 4.65793e25 0.524361
\(297\) 1.13167e26 1.22964
\(298\) −1.19848e26 −1.25707
\(299\) −3.02096e26 −3.05913
\(300\) 4.79240e24 0.0468576
\(301\) −5.22073e25 −0.492927
\(302\) 1.61522e25 0.147285
\(303\) −2.10990e25 −0.185829
\(304\) −7.62733e24 −0.0648933
\(305\) 5.20310e25 0.427675
\(306\) 7.19263e25 0.571232
\(307\) −1.02459e26 −0.786315 −0.393157 0.919471i \(-0.628617\pi\)
−0.393157 + 0.919471i \(0.628617\pi\)
\(308\) −8.87937e25 −0.658567
\(309\) −9.19272e25 −0.658992
\(310\) 3.32185e25 0.230188
\(311\) −2.17194e26 −1.45500 −0.727502 0.686105i \(-0.759319\pi\)
−0.727502 + 0.686105i \(0.759319\pi\)
\(312\) −4.69683e25 −0.304215
\(313\) 2.19829e26 1.37679 0.688397 0.725334i \(-0.258315\pi\)
0.688397 + 0.725334i \(0.258315\pi\)
\(314\) 9.59820e25 0.581337
\(315\) 5.32441e25 0.311897
\(316\) 1.82747e25 0.103546
\(317\) 1.26596e26 0.693901 0.346950 0.937883i \(-0.387217\pi\)
0.346950 + 0.937883i \(0.387217\pi\)
\(318\) −6.97449e25 −0.369852
\(319\) 4.95169e26 2.54069
\(320\) −1.12590e25 −0.0559017
\(321\) 3.61844e25 0.173867
\(322\) 2.26350e26 1.05267
\(323\) −5.96865e25 −0.268689
\(324\) 4.46911e25 0.194759
\(325\) 8.70466e25 0.367262
\(326\) −1.18513e26 −0.484150
\(327\) −4.57708e25 −0.181066
\(328\) 1.02716e26 0.393517
\(329\) −1.36000e26 −0.504640
\(330\) −6.07642e25 −0.218399
\(331\) 1.93702e26 0.674433 0.337217 0.941427i \(-0.390514\pi\)
0.337217 + 0.941427i \(0.390514\pi\)
\(332\) 3.93968e25 0.132896
\(333\) 3.54142e26 1.15748
\(334\) −1.49527e26 −0.473566
\(335\) 1.28321e26 0.393847
\(336\) 3.51917e25 0.104683
\(337\) 1.34839e26 0.388779 0.194389 0.980924i \(-0.437727\pi\)
0.194389 + 0.980924i \(0.437727\pi\)
\(338\) −6.00113e26 −1.67729
\(339\) 2.63870e26 0.714980
\(340\) −8.81056e25 −0.231459
\(341\) −4.21187e26 −1.07289
\(342\) −5.79905e25 −0.143246
\(343\) −4.48170e26 −1.07363
\(344\) 8.39350e25 0.195020
\(345\) 1.54898e26 0.349097
\(346\) 2.69091e26 0.588302
\(347\) −6.73325e26 −1.42812 −0.714061 0.700084i \(-0.753146\pi\)
−0.714061 + 0.700084i \(0.753146\pi\)
\(348\) −1.96251e26 −0.403859
\(349\) 9.07724e26 1.81254 0.906268 0.422703i \(-0.138919\pi\)
0.906268 + 0.422703i \(0.138919\pi\)
\(350\) −6.52210e25 −0.126378
\(351\) −8.14662e26 −1.53198
\(352\) 1.42756e26 0.260553
\(353\) 4.40145e25 0.0779761 0.0389880 0.999240i \(-0.487587\pi\)
0.0389880 + 0.999240i \(0.487587\pi\)
\(354\) −1.12340e26 −0.193197
\(355\) 2.91171e26 0.486128
\(356\) 2.55046e26 0.413423
\(357\) 2.75387e26 0.433438
\(358\) −9.13663e26 −1.39641
\(359\) −3.93523e26 −0.584088 −0.292044 0.956405i \(-0.594335\pi\)
−0.292044 + 0.956405i \(0.594335\pi\)
\(360\) −8.56020e25 −0.123398
\(361\) −6.66087e26 −0.932622
\(362\) −2.77906e26 −0.377971
\(363\) 4.15798e26 0.549367
\(364\) 6.39203e26 0.820492
\(365\) −1.33881e25 −0.0166972
\(366\) 2.61466e26 0.316857
\(367\) 1.48020e25 0.0174312 0.00871559 0.999962i \(-0.497226\pi\)
0.00871559 + 0.999962i \(0.497226\pi\)
\(368\) −3.63910e26 −0.416477
\(369\) 7.80951e26 0.868654
\(370\) −4.33804e26 −0.469002
\(371\) 9.49176e26 0.997518
\(372\) 1.66929e26 0.170542
\(373\) −1.57080e27 −1.56019 −0.780094 0.625662i \(-0.784829\pi\)
−0.780094 + 0.625662i \(0.784829\pi\)
\(374\) 1.11712e27 1.07881
\(375\) −4.46327e25 −0.0419107
\(376\) 2.18651e26 0.199654
\(377\) −3.56460e27 −3.16538
\(378\) 6.10398e26 0.527168
\(379\) −2.27703e26 −0.191275 −0.0956373 0.995416i \(-0.530489\pi\)
−0.0956373 + 0.995416i \(0.530489\pi\)
\(380\) 7.10351e25 0.0580423
\(381\) 9.80478e26 0.779337
\(382\) 1.19544e27 0.924406
\(383\) 2.29376e27 1.72569 0.862843 0.505472i \(-0.168682\pi\)
0.862843 + 0.505472i \(0.168682\pi\)
\(384\) −5.65786e25 −0.0414166
\(385\) 8.26956e26 0.589040
\(386\) −8.38506e25 −0.0581219
\(387\) 6.38157e26 0.430489
\(388\) −1.89520e26 −0.124429
\(389\) 1.10508e27 0.706190 0.353095 0.935587i \(-0.385129\pi\)
0.353095 + 0.935587i \(0.385129\pi\)
\(390\) 4.37426e26 0.272099
\(391\) −2.84772e27 −1.72441
\(392\) 1.20801e26 0.0712143
\(393\) −4.97407e26 −0.285489
\(394\) 6.75243e26 0.377355
\(395\) −1.70196e26 −0.0926148
\(396\) 1.08537e27 0.575148
\(397\) 1.88707e26 0.0973839 0.0486919 0.998814i \(-0.484495\pi\)
0.0486919 + 0.998814i \(0.484495\pi\)
\(398\) −1.87029e27 −0.940017
\(399\) −2.22031e26 −0.108692
\(400\) 1.04858e26 0.0500000
\(401\) −1.74678e27 −0.811376 −0.405688 0.914012i \(-0.632968\pi\)
−0.405688 + 0.914012i \(0.632968\pi\)
\(402\) 6.44838e26 0.291794
\(403\) 3.03202e27 1.33668
\(404\) −4.61646e26 −0.198292
\(405\) −4.16218e26 −0.174198
\(406\) 2.67083e27 1.08924
\(407\) 5.50032e27 2.18598
\(408\) −4.42748e26 −0.171484
\(409\) −2.92958e27 −1.10588 −0.552942 0.833219i \(-0.686495\pi\)
−0.552942 + 0.833219i \(0.686495\pi\)
\(410\) −9.56620e26 −0.351973
\(411\) −4.37553e26 −0.156925
\(412\) −2.01137e27 −0.703186
\(413\) 1.52886e27 0.521066
\(414\) −2.76680e27 −0.919336
\(415\) −3.66911e26 −0.118866
\(416\) −1.02766e27 −0.324617
\(417\) −1.52435e27 −0.469521
\(418\) −9.00674e26 −0.270531
\(419\) −2.59534e27 −0.760234 −0.380117 0.924938i \(-0.624116\pi\)
−0.380117 + 0.924938i \(0.624116\pi\)
\(420\) −3.27748e26 −0.0936316
\(421\) −2.08416e27 −0.580723 −0.290362 0.956917i \(-0.593776\pi\)
−0.290362 + 0.956917i \(0.593776\pi\)
\(422\) 3.44545e27 0.936408
\(423\) 1.66240e27 0.440718
\(424\) −1.52602e27 −0.394655
\(425\) 8.20547e26 0.207024
\(426\) 1.46319e27 0.360164
\(427\) −3.55836e27 −0.854588
\(428\) 7.91713e26 0.185528
\(429\) −5.54625e27 −1.26823
\(430\) −7.81706e26 −0.174431
\(431\) 1.85265e27 0.403444 0.201722 0.979443i \(-0.435346\pi\)
0.201722 + 0.979443i \(0.435346\pi\)
\(432\) −9.81354e26 −0.208567
\(433\) 6.31486e26 0.130991 0.0654954 0.997853i \(-0.479137\pi\)
0.0654954 + 0.997853i \(0.479137\pi\)
\(434\) −2.27179e27 −0.459966
\(435\) 1.82773e27 0.361222
\(436\) −1.00147e27 −0.193209
\(437\) 2.29597e27 0.432425
\(438\) −6.72778e25 −0.0123707
\(439\) 3.19822e27 0.574158 0.287079 0.957907i \(-0.407316\pi\)
0.287079 + 0.957907i \(0.407316\pi\)
\(440\) −1.32952e27 −0.233046
\(441\) 9.18450e26 0.157199
\(442\) −8.04184e27 −1.34407
\(443\) 7.29283e26 0.119030 0.0595151 0.998227i \(-0.481045\pi\)
0.0595151 + 0.998227i \(0.481045\pi\)
\(444\) −2.17995e27 −0.347476
\(445\) −2.37530e27 −0.369776
\(446\) −1.48177e26 −0.0225302
\(447\) 5.60898e27 0.833018
\(448\) 7.69994e26 0.111704
\(449\) −1.59099e27 −0.225466 −0.112733 0.993625i \(-0.535961\pi\)
−0.112733 + 0.993625i \(0.535961\pi\)
\(450\) 7.97231e26 0.110370
\(451\) 1.21293e28 1.64052
\(452\) 5.77348e27 0.762929
\(453\) −7.55936e26 −0.0976009
\(454\) 4.95149e27 0.624669
\(455\) −5.95305e27 −0.733871
\(456\) 3.56965e26 0.0430026
\(457\) 8.94924e27 1.05358 0.526788 0.849997i \(-0.323396\pi\)
0.526788 + 0.849997i \(0.323396\pi\)
\(458\) 4.45790e27 0.512912
\(459\) −7.67944e27 −0.863567
\(460\) 3.38917e27 0.372509
\(461\) 1.32946e28 1.42829 0.714144 0.699999i \(-0.246816\pi\)
0.714144 + 0.699999i \(0.246816\pi\)
\(462\) 4.15561e27 0.436410
\(463\) 1.29662e28 1.33110 0.665551 0.746352i \(-0.268197\pi\)
0.665551 + 0.746352i \(0.268197\pi\)
\(464\) −4.29396e27 −0.430943
\(465\) −1.55465e27 −0.152538
\(466\) −1.06042e26 −0.0101724
\(467\) −5.02371e27 −0.471191 −0.235595 0.971851i \(-0.575704\pi\)
−0.235595 + 0.971851i \(0.575704\pi\)
\(468\) −7.81332e27 −0.716563
\(469\) −8.77577e27 −0.786992
\(470\) −2.03634e27 −0.178576
\(471\) −4.49203e27 −0.385232
\(472\) −2.45800e27 −0.206153
\(473\) 9.91148e27 0.813011
\(474\) −8.55269e26 −0.0686167
\(475\) −6.61565e26 −0.0519147
\(476\) 6.02547e27 0.462507
\(477\) −1.16023e28 −0.871165
\(478\) 1.48571e28 1.09129
\(479\) 1.23595e28 0.888132 0.444066 0.895994i \(-0.353536\pi\)
0.444066 + 0.895994i \(0.353536\pi\)
\(480\) 5.26930e26 0.0370441
\(481\) −3.95954e28 −2.72346
\(482\) 2.15556e26 0.0145067
\(483\) −1.05934e28 −0.697572
\(484\) 9.09765e27 0.586210
\(485\) 1.76504e27 0.111293
\(486\) −1.16519e28 −0.718978
\(487\) −2.33221e28 −1.40836 −0.704179 0.710023i \(-0.748685\pi\)
−0.704179 + 0.710023i \(0.748685\pi\)
\(488\) 5.72087e27 0.338107
\(489\) 5.54648e27 0.320829
\(490\) −1.12505e27 −0.0636960
\(491\) 2.65852e28 1.47328 0.736640 0.676286i \(-0.236411\pi\)
0.736640 + 0.676286i \(0.236411\pi\)
\(492\) −4.80720e27 −0.260771
\(493\) −3.36018e28 −1.78431
\(494\) 6.48372e27 0.337048
\(495\) −1.01083e28 −0.514428
\(496\) 3.65241e27 0.181979
\(497\) −1.99130e28 −0.971390
\(498\) −1.84380e27 −0.0880655
\(499\) −1.38803e28 −0.649148 −0.324574 0.945860i \(-0.605221\pi\)
−0.324574 + 0.945860i \(0.605221\pi\)
\(500\) −9.76563e26 −0.0447214
\(501\) 6.99795e27 0.313816
\(502\) 9.00939e27 0.395646
\(503\) −8.55444e27 −0.367898 −0.183949 0.982936i \(-0.558888\pi\)
−0.183949 + 0.982936i \(0.558888\pi\)
\(504\) 5.85425e27 0.246576
\(505\) 4.29941e27 0.177357
\(506\) −4.29723e28 −1.73623
\(507\) 2.80857e28 1.11148
\(508\) 2.14528e28 0.831602
\(509\) 3.82946e28 1.45412 0.727060 0.686574i \(-0.240886\pi\)
0.727060 + 0.686574i \(0.240886\pi\)
\(510\) 4.12341e27 0.153380
\(511\) 9.15601e26 0.0333647
\(512\) −1.23794e27 −0.0441942
\(513\) 6.19154e27 0.216554
\(514\) −3.01009e27 −0.103149
\(515\) 1.87323e28 0.628949
\(516\) −3.92822e27 −0.129233
\(517\) 2.58194e28 0.832329
\(518\) 2.96675e28 0.937169
\(519\) −1.25937e28 −0.389847
\(520\) 9.57088e27 0.290346
\(521\) 2.56460e28 0.762471 0.381236 0.924478i \(-0.375499\pi\)
0.381236 + 0.924478i \(0.375499\pi\)
\(522\) −3.26470e28 −0.951267
\(523\) −4.56980e28 −1.30506 −0.652529 0.757764i \(-0.726292\pi\)
−0.652529 + 0.757764i \(0.726292\pi\)
\(524\) −1.08833e28 −0.304635
\(525\) 3.05239e27 0.0837467
\(526\) 7.23762e26 0.0194646
\(527\) 2.85814e28 0.753481
\(528\) −6.68110e27 −0.172660
\(529\) 7.00720e28 1.77525
\(530\) 1.42121e28 0.352990
\(531\) −1.86881e28 −0.455064
\(532\) −4.85803e27 −0.115981
\(533\) −8.73155e28 −2.04388
\(534\) −1.19364e28 −0.273961
\(535\) −7.37341e27 −0.165941
\(536\) 1.41090e28 0.311363
\(537\) 4.27601e28 0.925356
\(538\) −4.39699e28 −0.933128
\(539\) 1.42648e28 0.296882
\(540\) 9.13957e27 0.186548
\(541\) 5.19261e28 1.03948 0.519738 0.854326i \(-0.326029\pi\)
0.519738 + 0.854326i \(0.326029\pi\)
\(542\) −1.70192e28 −0.334153
\(543\) 1.30062e28 0.250468
\(544\) −9.68731e27 −0.182985
\(545\) 9.32687e27 0.172812
\(546\) −2.99152e28 −0.543712
\(547\) −4.55529e28 −0.812173 −0.406087 0.913835i \(-0.633107\pi\)
−0.406087 + 0.913835i \(0.633107\pi\)
\(548\) −9.57365e27 −0.167449
\(549\) 4.34957e28 0.746340
\(550\) 1.23821e28 0.208443
\(551\) 2.70914e28 0.447445
\(552\) 1.70312e28 0.275985
\(553\) 1.16396e28 0.185065
\(554\) 2.92842e28 0.456858
\(555\) 2.03023e28 0.310792
\(556\) −3.33527e28 −0.501009
\(557\) 2.98975e28 0.440713 0.220356 0.975419i \(-0.429278\pi\)
0.220356 + 0.975419i \(0.429278\pi\)
\(558\) 2.77693e28 0.401703
\(559\) −7.13502e28 −1.01291
\(560\) −7.17112e27 −0.0999109
\(561\) −5.22819e28 −0.714894
\(562\) 5.71633e28 0.767160
\(563\) −4.42316e28 −0.582632 −0.291316 0.956627i \(-0.594093\pi\)
−0.291316 + 0.956627i \(0.594093\pi\)
\(564\) −1.02330e28 −0.132304
\(565\) −5.37697e28 −0.682385
\(566\) −6.69929e28 −0.834558
\(567\) 2.84648e28 0.348086
\(568\) 3.20146e28 0.384318
\(569\) 2.66278e28 0.313803 0.156901 0.987614i \(-0.449849\pi\)
0.156901 + 0.987614i \(0.449849\pi\)
\(570\) −3.32449e27 −0.0384627
\(571\) −5.39217e28 −0.612469 −0.306235 0.951956i \(-0.599069\pi\)
−0.306235 + 0.951956i \(0.599069\pi\)
\(572\) −1.21352e29 −1.35328
\(573\) −5.59475e28 −0.612572
\(574\) 6.54224e28 0.703318
\(575\) −3.15641e28 −0.333182
\(576\) −9.41204e27 −0.0975546
\(577\) −4.00855e28 −0.407983 −0.203991 0.978973i \(-0.565391\pi\)
−0.203991 + 0.978973i \(0.565391\pi\)
\(578\) −5.05646e27 −0.0505364
\(579\) 3.92427e27 0.0385154
\(580\) 3.99907e28 0.385447
\(581\) 2.50928e28 0.237519
\(582\) 8.86967e27 0.0824548
\(583\) −1.80200e29 −1.64526
\(584\) −1.47204e27 −0.0132003
\(585\) 7.27673e28 0.640913
\(586\) 1.13496e28 0.0981873
\(587\) 6.52023e28 0.554068 0.277034 0.960860i \(-0.410649\pi\)
0.277034 + 0.960860i \(0.410649\pi\)
\(588\) −5.65358e27 −0.0471913
\(589\) −2.30437e28 −0.188948
\(590\) 2.28919e28 0.184389
\(591\) −3.16019e28 −0.250060
\(592\) −4.76972e28 −0.370779
\(593\) 7.72360e28 0.589855 0.294928 0.955520i \(-0.404704\pi\)
0.294928 + 0.955520i \(0.404704\pi\)
\(594\) −1.15883e29 −0.869487
\(595\) −5.61166e28 −0.413678
\(596\) 1.22724e29 0.888884
\(597\) 8.75308e28 0.622917
\(598\) 3.09347e29 2.16313
\(599\) 8.22197e28 0.564930 0.282465 0.959278i \(-0.408848\pi\)
0.282465 + 0.959278i \(0.408848\pi\)
\(600\) −4.90741e27 −0.0331333
\(601\) 1.57712e29 1.04636 0.523181 0.852221i \(-0.324745\pi\)
0.523181 + 0.852221i \(0.324745\pi\)
\(602\) 5.34602e28 0.348552
\(603\) 1.07271e29 0.687306
\(604\) −1.65399e28 −0.104146
\(605\) −8.47285e28 −0.524322
\(606\) 2.16054e28 0.131401
\(607\) −3.31059e29 −1.97890 −0.989450 0.144877i \(-0.953721\pi\)
−0.989450 + 0.144877i \(0.953721\pi\)
\(608\) 7.81039e27 0.0458865
\(609\) −1.24997e29 −0.721801
\(610\) −5.32798e28 −0.302412
\(611\) −1.85867e29 −1.03698
\(612\) −7.36525e28 −0.403922
\(613\) 3.50723e29 1.89073 0.945363 0.326018i \(-0.105707\pi\)
0.945363 + 0.326018i \(0.105707\pi\)
\(614\) 1.04918e29 0.556008
\(615\) 4.47705e28 0.233240
\(616\) 9.09248e28 0.465677
\(617\) −2.45228e28 −0.123474 −0.0617371 0.998092i \(-0.519664\pi\)
−0.0617371 + 0.998092i \(0.519664\pi\)
\(618\) 9.41335e28 0.465978
\(619\) 1.47013e28 0.0715492 0.0357746 0.999360i \(-0.488610\pi\)
0.0357746 + 0.999360i \(0.488610\pi\)
\(620\) −3.40158e28 −0.162767
\(621\) 2.95406e29 1.38982
\(622\) 2.22407e29 1.02884
\(623\) 1.62445e29 0.738894
\(624\) 4.80955e28 0.215113
\(625\) 9.09495e27 0.0400000
\(626\) −2.25105e29 −0.973541
\(627\) 4.21522e28 0.179272
\(628\) −9.82855e28 −0.411068
\(629\) −3.73247e29 −1.53520
\(630\) −5.45220e28 −0.220544
\(631\) 1.42534e29 0.567034 0.283517 0.958967i \(-0.408499\pi\)
0.283517 + 0.958967i \(0.408499\pi\)
\(632\) −1.87133e28 −0.0732184
\(633\) −1.61250e29 −0.620526
\(634\) −1.29634e29 −0.490662
\(635\) −1.99795e29 −0.743808
\(636\) 7.14187e28 0.261525
\(637\) −1.02689e29 −0.369878
\(638\) −5.07053e29 −1.79654
\(639\) 2.43407e29 0.848347
\(640\) 1.15292e28 0.0395285
\(641\) 2.82277e28 0.0952064 0.0476032 0.998866i \(-0.484842\pi\)
0.0476032 + 0.998866i \(0.484842\pi\)
\(642\) −3.70528e28 −0.122943
\(643\) 1.52966e29 0.499321 0.249661 0.968333i \(-0.419681\pi\)
0.249661 + 0.968333i \(0.419681\pi\)
\(644\) −2.31783e29 −0.744354
\(645\) 3.65844e28 0.115590
\(646\) 6.11190e28 0.189992
\(647\) −9.19042e28 −0.281087 −0.140543 0.990075i \(-0.544885\pi\)
−0.140543 + 0.990075i \(0.544885\pi\)
\(648\) −4.57637e28 −0.137716
\(649\) −2.90253e29 −0.859423
\(650\) −8.91357e28 −0.259694
\(651\) 1.06321e29 0.304803
\(652\) 1.21357e29 0.342345
\(653\) −8.26379e28 −0.229399 −0.114699 0.993400i \(-0.536590\pi\)
−0.114699 + 0.993400i \(0.536590\pi\)
\(654\) 4.68693e28 0.128033
\(655\) 1.01358e29 0.272474
\(656\) −1.05181e29 −0.278259
\(657\) −1.11919e28 −0.0291385
\(658\) 1.39264e29 0.356834
\(659\) −7.76286e29 −1.95760 −0.978802 0.204807i \(-0.934343\pi\)
−0.978802 + 0.204807i \(0.934343\pi\)
\(660\) 6.22226e28 0.154432
\(661\) −6.63151e29 −1.61993 −0.809967 0.586476i \(-0.800515\pi\)
−0.809967 + 0.586476i \(0.800515\pi\)
\(662\) −1.98350e29 −0.476896
\(663\) 3.76364e29 0.890668
\(664\) −4.03423e28 −0.0939715
\(665\) 4.52439e28 0.103737
\(666\) −3.62641e29 −0.818461
\(667\) 1.29256e30 2.87165
\(668\) 1.53115e29 0.334862
\(669\) 6.93481e27 0.0149300
\(670\) −1.31401e29 −0.278492
\(671\) 6.75550e29 1.40952
\(672\) −3.60363e28 −0.0740223
\(673\) 3.60156e29 0.728337 0.364169 0.931333i \(-0.381353\pi\)
0.364169 + 0.931333i \(0.381353\pi\)
\(674\) −1.38075e29 −0.274908
\(675\) −8.51189e28 −0.166854
\(676\) 6.14516e29 1.18602
\(677\) −1.28335e28 −0.0243874 −0.0121937 0.999926i \(-0.503881\pi\)
−0.0121937 + 0.999926i \(0.503881\pi\)
\(678\) −2.70203e29 −0.505567
\(679\) −1.20710e29 −0.222387
\(680\) 9.02201e28 0.163667
\(681\) −2.31734e29 −0.413947
\(682\) 4.31296e29 0.758646
\(683\) −2.83065e29 −0.490308 −0.245154 0.969484i \(-0.578838\pi\)
−0.245154 + 0.969484i \(0.578838\pi\)
\(684\) 5.93823e28 0.101290
\(685\) 8.91616e28 0.149771
\(686\) 4.58926e29 0.759171
\(687\) −2.08633e29 −0.339889
\(688\) −8.59495e28 −0.137900
\(689\) 1.29721e30 2.04979
\(690\) −1.58616e29 −0.246849
\(691\) −6.08933e29 −0.933361 −0.466681 0.884426i \(-0.654550\pi\)
−0.466681 + 0.884426i \(0.654550\pi\)
\(692\) −2.75549e29 −0.415992
\(693\) 6.91300e29 1.02794
\(694\) 6.89485e29 1.00983
\(695\) 3.10621e29 0.448116
\(696\) 2.00961e29 0.285571
\(697\) −8.23082e29 −1.15212
\(698\) −9.29509e29 −1.28166
\(699\) 4.96284e27 0.00674093
\(700\) 6.67863e28 0.0893631
\(701\) −4.09891e29 −0.540292 −0.270146 0.962819i \(-0.587072\pi\)
−0.270146 + 0.962819i \(0.587072\pi\)
\(702\) 8.34214e29 1.08327
\(703\) 3.00930e29 0.384977
\(704\) −1.46182e29 −0.184239
\(705\) 9.53024e28 0.118336
\(706\) −4.50709e28 −0.0551374
\(707\) −2.94033e29 −0.354399
\(708\) 1.15036e29 0.136611
\(709\) −1.14539e30 −1.34020 −0.670098 0.742273i \(-0.733748\pi\)
−0.670098 + 0.742273i \(0.733748\pi\)
\(710\) −2.98159e29 −0.343745
\(711\) −1.42277e29 −0.161623
\(712\) −2.61168e29 −0.292334
\(713\) −1.09945e30 −1.21264
\(714\) −2.81996e29 −0.306487
\(715\) 1.13018e30 1.21041
\(716\) 9.35590e29 0.987413
\(717\) −6.95323e29 −0.723162
\(718\) 4.02968e29 0.413013
\(719\) 1.29783e30 1.31088 0.655440 0.755247i \(-0.272483\pi\)
0.655440 + 0.755247i \(0.272483\pi\)
\(720\) 8.76565e28 0.0872555
\(721\) −1.28109e30 −1.25678
\(722\) 6.82073e29 0.659463
\(723\) −1.00882e28 −0.00961306
\(724\) 2.84576e29 0.267266
\(725\) −3.72442e29 −0.344754
\(726\) −4.25777e29 −0.388461
\(727\) −1.67396e30 −1.50534 −0.752669 0.658399i \(-0.771234\pi\)
−0.752669 + 0.658399i \(0.771234\pi\)
\(728\) −6.54544e29 −0.580176
\(729\) 9.94890e28 0.0869233
\(730\) 1.37094e28 0.0118067
\(731\) −6.72585e29 −0.570972
\(732\) −2.67741e29 −0.224052
\(733\) 4.12599e29 0.340359 0.170179 0.985413i \(-0.445565\pi\)
0.170179 + 0.985413i \(0.445565\pi\)
\(734\) −1.51573e28 −0.0123257
\(735\) 5.26531e28 0.0422092
\(736\) 3.72643e29 0.294494
\(737\) 1.66607e30 1.29803
\(738\) −7.99693e29 −0.614231
\(739\) 1.73492e30 1.31375 0.656877 0.753998i \(-0.271877\pi\)
0.656877 + 0.753998i \(0.271877\pi\)
\(740\) 4.44215e29 0.331635
\(741\) −3.03443e29 −0.223350
\(742\) −9.71956e29 −0.705352
\(743\) −1.36497e30 −0.976656 −0.488328 0.872660i \(-0.662393\pi\)
−0.488328 + 0.872660i \(0.662393\pi\)
\(744\) −1.70936e29 −0.120592
\(745\) −1.14296e30 −0.795042
\(746\) 1.60849e30 1.10322
\(747\) −3.06722e29 −0.207434
\(748\) −1.14393e30 −0.762837
\(749\) 5.04261e29 0.331586
\(750\) 4.57039e28 0.0296353
\(751\) −1.35372e30 −0.865587 −0.432793 0.901493i \(-0.642472\pi\)
−0.432793 + 0.901493i \(0.642472\pi\)
\(752\) −2.23898e29 −0.141177
\(753\) −4.21646e29 −0.262181
\(754\) 3.65015e30 2.23826
\(755\) 1.54040e29 0.0931514
\(756\) −6.25048e29 −0.372764
\(757\) −8.02724e29 −0.472128 −0.236064 0.971738i \(-0.575857\pi\)
−0.236064 + 0.971738i \(0.575857\pi\)
\(758\) 2.33168e29 0.135252
\(759\) 2.01114e30 1.15054
\(760\) −7.27399e28 −0.0410421
\(761\) 2.59098e29 0.144186 0.0720932 0.997398i \(-0.477032\pi\)
0.0720932 + 0.997398i \(0.477032\pi\)
\(762\) −1.00401e30 −0.551075
\(763\) −6.37857e29 −0.345315
\(764\) −1.22413e30 −0.653654
\(765\) 6.85943e29 0.361279
\(766\) −2.34882e30 −1.22024
\(767\) 2.08945e30 1.07073
\(768\) 5.79365e28 0.0292860
\(769\) −9.50454e29 −0.473919 −0.236960 0.971519i \(-0.576151\pi\)
−0.236960 + 0.971519i \(0.576151\pi\)
\(770\) −8.46803e29 −0.416514
\(771\) 1.40874e29 0.0683535
\(772\) 8.58631e28 0.0410984
\(773\) 3.15209e30 1.48838 0.744191 0.667967i \(-0.232835\pi\)
0.744191 + 0.667967i \(0.232835\pi\)
\(774\) −6.53473e29 −0.304402
\(775\) 3.16796e29 0.145584
\(776\) 1.94068e29 0.0879846
\(777\) −1.38846e30 −0.621030
\(778\) −1.13160e30 −0.499352
\(779\) 6.63609e29 0.288914
\(780\) −4.47924e29 −0.192403
\(781\) 3.78045e30 1.60217
\(782\) 2.91606e30 1.21934
\(783\) 3.48565e30 1.43809
\(784\) −1.23700e29 −0.0503561
\(785\) 9.15355e29 0.367670
\(786\) 5.09345e29 0.201872
\(787\) −1.06937e30 −0.418211 −0.209105 0.977893i \(-0.567055\pi\)
−0.209105 + 0.977893i \(0.567055\pi\)
\(788\) −6.91449e29 −0.266830
\(789\) −3.38726e28 −0.0128985
\(790\) 1.74281e29 0.0654885
\(791\) 3.67727e30 1.36355
\(792\) −1.11142e30 −0.406691
\(793\) −4.86311e30 −1.75609
\(794\) −1.93236e29 −0.0688608
\(795\) −6.65139e29 −0.233915
\(796\) 1.91517e30 0.664692
\(797\) 2.66578e30 0.913086 0.456543 0.889701i \(-0.349087\pi\)
0.456543 + 0.889701i \(0.349087\pi\)
\(798\) 2.27359e29 0.0768568
\(799\) −1.75208e30 −0.584539
\(800\) −1.07374e29 −0.0353553
\(801\) −1.98565e30 −0.645301
\(802\) 1.78870e30 0.573730
\(803\) −1.73826e29 −0.0550302
\(804\) −6.60314e29 −0.206330
\(805\) 2.15864e30 0.665770
\(806\) −3.10479e30 −0.945179
\(807\) 2.05782e30 0.618352
\(808\) 4.72725e29 0.140213
\(809\) 1.67414e30 0.490153 0.245077 0.969504i \(-0.421187\pi\)
0.245077 + 0.969504i \(0.421187\pi\)
\(810\) 4.26208e29 0.123177
\(811\) −4.62583e30 −1.31969 −0.659843 0.751404i \(-0.729377\pi\)
−0.659843 + 0.751404i \(0.729377\pi\)
\(812\) −2.73493e30 −0.770207
\(813\) 7.96509e29 0.221432
\(814\) −5.63233e30 −1.54572
\(815\) −1.13022e30 −0.306203
\(816\) 4.53374e29 0.121258
\(817\) 5.42271e29 0.143181
\(818\) 2.99988e30 0.781979
\(819\) −4.97649e30 −1.28069
\(820\) 9.79579e29 0.248882
\(821\) −2.81912e30 −0.707147 −0.353573 0.935407i \(-0.615034\pi\)
−0.353573 + 0.935407i \(0.615034\pi\)
\(822\) 4.48054e29 0.110963
\(823\) −8.15621e29 −0.199430 −0.0997149 0.995016i \(-0.531793\pi\)
−0.0997149 + 0.995016i \(0.531793\pi\)
\(824\) 2.05964e30 0.497228
\(825\) −5.79493e29 −0.138128
\(826\) −1.56556e30 −0.368450
\(827\) −5.27710e30 −1.22627 −0.613137 0.789977i \(-0.710093\pi\)
−0.613137 + 0.789977i \(0.710093\pi\)
\(828\) 2.83320e30 0.650068
\(829\) −5.17029e30 −1.17137 −0.585683 0.810540i \(-0.699174\pi\)
−0.585683 + 0.810540i \(0.699174\pi\)
\(830\) 3.75717e29 0.0840507
\(831\) −1.37052e30 −0.302744
\(832\) 1.05233e30 0.229539
\(833\) −9.67999e29 −0.208498
\(834\) 1.56093e30 0.332002
\(835\) −1.42600e30 −0.299509
\(836\) 9.22291e29 0.191294
\(837\) −2.96487e30 −0.607279
\(838\) 2.65763e30 0.537567
\(839\) −9.14574e30 −1.82692 −0.913458 0.406934i \(-0.866598\pi\)
−0.913458 + 0.406934i \(0.866598\pi\)
\(840\) 3.35614e29 0.0662076
\(841\) 1.01188e31 1.97139
\(842\) 2.13418e30 0.410633
\(843\) −2.67529e30 −0.508371
\(844\) −3.52814e30 −0.662140
\(845\) −5.72312e30 −1.06081
\(846\) −1.70230e30 −0.311635
\(847\) 5.79451e30 1.04771
\(848\) 1.56264e30 0.279063
\(849\) 3.13532e30 0.553033
\(850\) −8.40240e29 −0.146388
\(851\) 1.43578e31 2.47074
\(852\) −1.49831e30 −0.254674
\(853\) 9.89405e30 1.66115 0.830576 0.556906i \(-0.188011\pi\)
0.830576 + 0.556906i \(0.188011\pi\)
\(854\) 3.64376e30 0.604285
\(855\) −5.53041e29 −0.0905968
\(856\) −8.10714e29 −0.131188
\(857\) 5.52749e30 0.883546 0.441773 0.897127i \(-0.354350\pi\)
0.441773 + 0.897127i \(0.354350\pi\)
\(858\) 5.67936e30 0.896774
\(859\) −3.34705e29 −0.0522076 −0.0261038 0.999659i \(-0.508310\pi\)
−0.0261038 + 0.999659i \(0.508310\pi\)
\(860\) 8.00467e29 0.123342
\(861\) −3.06182e30 −0.466065
\(862\) −1.89712e30 −0.285278
\(863\) −4.68654e30 −0.696208 −0.348104 0.937456i \(-0.613174\pi\)
−0.348104 + 0.937456i \(0.613174\pi\)
\(864\) 1.00491e30 0.147479
\(865\) 2.56625e30 0.372075
\(866\) −6.46641e29 −0.0926244
\(867\) 2.36646e29 0.0334888
\(868\) 2.32631e30 0.325245
\(869\) −2.20976e30 −0.305237
\(870\) −1.87159e30 −0.255423
\(871\) −1.19936e31 −1.61718
\(872\) 1.02550e30 0.136620
\(873\) 1.47550e30 0.194218
\(874\) −2.35107e30 −0.305771
\(875\) −6.21996e29 −0.0799287
\(876\) 6.88925e28 0.00874739
\(877\) 2.46570e30 0.309346 0.154673 0.987966i \(-0.450568\pi\)
0.154673 + 0.987966i \(0.450568\pi\)
\(878\) −3.27498e30 −0.405991
\(879\) −5.31170e29 −0.0650654
\(880\) 1.36143e30 0.164788
\(881\) 9.27878e30 1.10980 0.554899 0.831918i \(-0.312757\pi\)
0.554899 + 0.831918i \(0.312757\pi\)
\(882\) −9.40492e29 −0.111157
\(883\) 1.09864e29 0.0128313 0.00641564 0.999979i \(-0.497958\pi\)
0.00641564 + 0.999979i \(0.497958\pi\)
\(884\) 8.23484e30 0.950400
\(885\) −1.07136e30 −0.122188
\(886\) −7.46786e29 −0.0841670
\(887\) 9.09262e30 1.01272 0.506362 0.862321i \(-0.330990\pi\)
0.506362 + 0.862321i \(0.330990\pi\)
\(888\) 2.23227e30 0.245703
\(889\) 1.36638e31 1.48629
\(890\) 2.43231e30 0.261471
\(891\) −5.40401e30 −0.574117
\(892\) 1.51734e29 0.0159313
\(893\) 1.41261e30 0.146583
\(894\) −5.74359e30 −0.589033
\(895\) −8.71337e30 −0.883169
\(896\) −7.88474e29 −0.0789865
\(897\) −1.44777e31 −1.43343
\(898\) 1.62918e30 0.159429
\(899\) −1.29729e31 −1.25476
\(900\) −8.16365e29 −0.0780437
\(901\) 1.22282e31 1.15545
\(902\) −1.24204e31 −1.16002
\(903\) −2.50198e30 −0.230974
\(904\) −5.91205e30 −0.539472
\(905\) −2.65032e30 −0.239050
\(906\) 7.74078e29 0.0690143
\(907\) −4.70059e30 −0.414263 −0.207132 0.978313i \(-0.566413\pi\)
−0.207132 + 0.978313i \(0.566413\pi\)
\(908\) −5.07033e30 −0.441707
\(909\) 3.59412e30 0.309508
\(910\) 6.09592e30 0.518925
\(911\) 1.05639e31 0.888960 0.444480 0.895789i \(-0.353388\pi\)
0.444480 + 0.895789i \(0.353388\pi\)
\(912\) −3.65532e29 −0.0304074
\(913\) −4.76383e30 −0.391754
\(914\) −9.16402e30 −0.744991
\(915\) 2.49353e30 0.200398
\(916\) −4.56489e30 −0.362684
\(917\) −6.93181e30 −0.544463
\(918\) 7.86374e30 0.610634
\(919\) 2.15821e31 1.65684 0.828421 0.560105i \(-0.189239\pi\)
0.828421 + 0.560105i \(0.189239\pi\)
\(920\) −3.47051e30 −0.263403
\(921\) −4.91023e30 −0.368448
\(922\) −1.36137e31 −1.00995
\(923\) −2.72145e31 −1.99610
\(924\) −4.25535e30 −0.308588
\(925\) −4.13707e30 −0.296623
\(926\) −1.32774e31 −0.941231
\(927\) 1.56594e31 1.09759
\(928\) 4.39702e30 0.304723
\(929\) 8.91027e30 0.610557 0.305278 0.952263i \(-0.401250\pi\)
0.305278 + 0.952263i \(0.401250\pi\)
\(930\) 1.59196e30 0.107860
\(931\) 7.80448e29 0.0522844
\(932\) 1.08587e29 0.00719300
\(933\) −1.04088e31 −0.681780
\(934\) 5.14427e30 0.333182
\(935\) 1.06537e31 0.682302
\(936\) 8.00084e30 0.506687
\(937\) 8.67478e29 0.0543241 0.0271621 0.999631i \(-0.491353\pi\)
0.0271621 + 0.999631i \(0.491353\pi\)
\(938\) 8.98639e30 0.556487
\(939\) 1.05351e31 0.645132
\(940\) 2.08521e30 0.126272
\(941\) −1.58550e31 −0.949459 −0.474730 0.880132i \(-0.657454\pi\)
−0.474730 + 0.880132i \(0.657454\pi\)
\(942\) 4.59984e30 0.272401
\(943\) 3.16616e31 1.85422
\(944\) 2.51699e30 0.145772
\(945\) 5.82121e30 0.333410
\(946\) −1.01494e31 −0.574886
\(947\) −2.27081e31 −1.27206 −0.636028 0.771666i \(-0.719424\pi\)
−0.636028 + 0.771666i \(0.719424\pi\)
\(948\) 8.75795e29 0.0485193
\(949\) 1.25133e30 0.0685607
\(950\) 6.77443e29 0.0367092
\(951\) 6.06698e30 0.325145
\(952\) −6.17008e30 −0.327041
\(953\) −1.88217e31 −0.986696 −0.493348 0.869832i \(-0.664227\pi\)
−0.493348 + 0.869832i \(0.664227\pi\)
\(954\) 1.18807e31 0.616007
\(955\) 1.14006e31 0.584646
\(956\) −1.52137e31 −0.771660
\(957\) 2.37305e31 1.19050
\(958\) −1.26561e31 −0.628004
\(959\) −6.09769e30 −0.299274
\(960\) −5.39576e29 −0.0261942
\(961\) −9.79081e30 −0.470136
\(962\) 4.05457e31 1.92578
\(963\) −6.16385e30 −0.289585
\(964\) −2.20730e29 −0.0102578
\(965\) −7.99662e29 −0.0367595
\(966\) 1.08476e31 0.493258
\(967\) 1.59166e31 0.715934 0.357967 0.933734i \(-0.383470\pi\)
0.357967 + 0.933734i \(0.383470\pi\)
\(968\) −9.31600e30 −0.414513
\(969\) −2.86042e30 −0.125901
\(970\) −1.80740e30 −0.0786958
\(971\) −1.49572e31 −0.644239 −0.322119 0.946699i \(-0.604395\pi\)
−0.322119 + 0.946699i \(0.604395\pi\)
\(972\) 1.19315e31 0.508394
\(973\) −2.12431e31 −0.895434
\(974\) 2.38818e31 0.995859
\(975\) 4.17162e30 0.172090
\(976\) −5.85817e30 −0.239078
\(977\) 2.26547e31 0.914672 0.457336 0.889294i \(-0.348804\pi\)
0.457336 + 0.889294i \(0.348804\pi\)
\(978\) −5.67959e30 −0.226861
\(979\) −3.08400e31 −1.21870
\(980\) 1.15205e30 0.0450399
\(981\) 7.79687e30 0.301575
\(982\) −2.72233e31 −1.04177
\(983\) −7.38759e30 −0.279699 −0.139849 0.990173i \(-0.544662\pi\)
−0.139849 + 0.990173i \(0.544662\pi\)
\(984\) 4.92257e30 0.184393
\(985\) 6.43962e30 0.238660
\(986\) 3.44082e31 1.26170
\(987\) −6.51765e30 −0.236462
\(988\) −6.63933e30 −0.238329
\(989\) 2.58724e31 0.918916
\(990\) 1.03509e31 0.363756
\(991\) 1.82576e31 0.634849 0.317425 0.948284i \(-0.397182\pi\)
0.317425 + 0.948284i \(0.397182\pi\)
\(992\) −3.74007e30 −0.128679
\(993\) 9.28295e30 0.316023
\(994\) 2.03909e31 0.686877
\(995\) −1.78364e31 −0.594519
\(996\) 1.88805e30 0.0622717
\(997\) −3.18289e31 −1.03878 −0.519389 0.854538i \(-0.673840\pi\)
−0.519389 + 0.854538i \(0.673840\pi\)
\(998\) 1.42134e31 0.459017
\(999\) 3.87185e31 1.23732
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 10.22.a.b.1.2 2
4.3 odd 2 80.22.a.d.1.1 2
5.2 odd 4 50.22.b.f.49.1 4
5.3 odd 4 50.22.b.f.49.4 4
5.4 even 2 50.22.a.f.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.22.a.b.1.2 2 1.1 even 1 trivial
50.22.a.f.1.1 2 5.4 even 2
50.22.b.f.49.1 4 5.2 odd 4
50.22.b.f.49.4 4 5.3 odd 4
80.22.a.d.1.1 2 4.3 odd 2