Properties

Label 8042.2.a.d.1.11
Level $8042$
Weight $2$
Character 8042.1
Self dual yes
Analytic conductor $64.216$
Analytic rank $0$
Dimension $101$
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] = [8042,2,Mod(1,8042)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8042, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8042.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8042 = 2 \cdot 4021 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8042.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: \(64.2156933055\)
Analytic rank: \(0\)
Dimension: \(101\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.11
Character \(\chi\) \(=\) 8042.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.78808 q^{3} +1.00000 q^{4} +1.68662 q^{5} -2.78808 q^{6} -0.478055 q^{7} +1.00000 q^{8} +4.77338 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -2.78808 q^{3} +1.00000 q^{4} +1.68662 q^{5} -2.78808 q^{6} -0.478055 q^{7} +1.00000 q^{8} +4.77338 q^{9} +1.68662 q^{10} +2.20706 q^{11} -2.78808 q^{12} +5.56059 q^{13} -0.478055 q^{14} -4.70241 q^{15} +1.00000 q^{16} +1.64824 q^{17} +4.77338 q^{18} +4.26835 q^{19} +1.68662 q^{20} +1.33286 q^{21} +2.20706 q^{22} -6.71615 q^{23} -2.78808 q^{24} -2.15533 q^{25} +5.56059 q^{26} -4.94431 q^{27} -0.478055 q^{28} +3.62683 q^{29} -4.70241 q^{30} +2.87716 q^{31} +1.00000 q^{32} -6.15347 q^{33} +1.64824 q^{34} -0.806296 q^{35} +4.77338 q^{36} +1.83077 q^{37} +4.26835 q^{38} -15.5034 q^{39} +1.68662 q^{40} +6.78226 q^{41} +1.33286 q^{42} +1.02674 q^{43} +2.20706 q^{44} +8.05085 q^{45} -6.71615 q^{46} +10.3458 q^{47} -2.78808 q^{48} -6.77146 q^{49} -2.15533 q^{50} -4.59543 q^{51} +5.56059 q^{52} +12.9134 q^{53} -4.94431 q^{54} +3.72247 q^{55} -0.478055 q^{56} -11.9005 q^{57} +3.62683 q^{58} +6.42878 q^{59} -4.70241 q^{60} +4.33547 q^{61} +2.87716 q^{62} -2.28194 q^{63} +1.00000 q^{64} +9.37857 q^{65} -6.15347 q^{66} -9.68788 q^{67} +1.64824 q^{68} +18.7251 q^{69} -0.806296 q^{70} +4.36042 q^{71} +4.77338 q^{72} -0.189903 q^{73} +1.83077 q^{74} +6.00922 q^{75} +4.26835 q^{76} -1.05510 q^{77} -15.5034 q^{78} -12.0283 q^{79} +1.68662 q^{80} -0.535012 q^{81} +6.78226 q^{82} +5.45551 q^{83} +1.33286 q^{84} +2.77995 q^{85} +1.02674 q^{86} -10.1119 q^{87} +2.20706 q^{88} -12.7133 q^{89} +8.05085 q^{90} -2.65827 q^{91} -6.71615 q^{92} -8.02173 q^{93} +10.3458 q^{94} +7.19907 q^{95} -2.78808 q^{96} -5.00870 q^{97} -6.77146 q^{98} +10.5351 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 101 q + 101 q^{2} + 10 q^{3} + 101 q^{4} + 19 q^{5} + 10 q^{6} + 42 q^{7} + 101 q^{8} + 147 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 101 q + 101 q^{2} + 10 q^{3} + 101 q^{4} + 19 q^{5} + 10 q^{6} + 42 q^{7} + 101 q^{8} + 147 q^{9} + 19 q^{10} + 4 q^{11} + 10 q^{12} + 58 q^{13} + 42 q^{14} + 27 q^{15} + 101 q^{16} + 34 q^{17} + 147 q^{18} + 36 q^{19} + 19 q^{20} + 45 q^{21} + 4 q^{22} + 47 q^{23} + 10 q^{24} + 174 q^{25} + 58 q^{26} + 31 q^{27} + 42 q^{28} + 62 q^{29} + 27 q^{30} + 47 q^{31} + 101 q^{32} + 55 q^{33} + 34 q^{34} + 16 q^{35} + 147 q^{36} + 90 q^{37} + 36 q^{38} + 50 q^{39} + 19 q^{40} + 54 q^{41} + 45 q^{42} + 65 q^{43} + 4 q^{44} + 47 q^{45} + 47 q^{46} + 54 q^{47} + 10 q^{48} + 189 q^{49} + 174 q^{50} + 36 q^{51} + 58 q^{52} + 94 q^{53} + 31 q^{54} + 68 q^{55} + 42 q^{56} + 79 q^{57} + 62 q^{58} - 6 q^{59} + 27 q^{60} + 58 q^{61} + 47 q^{62} + 117 q^{63} + 101 q^{64} + 89 q^{65} + 55 q^{66} + 127 q^{67} + 34 q^{68} + 45 q^{69} + 16 q^{70} + 87 q^{71} + 147 q^{72} + 83 q^{73} + 90 q^{74} - 4 q^{75} + 36 q^{76} + 53 q^{77} + 50 q^{78} + 74 q^{79} + 19 q^{80} + 241 q^{81} + 54 q^{82} + 11 q^{83} + 45 q^{84} + 120 q^{85} + 65 q^{86} + 37 q^{87} + 4 q^{88} + 89 q^{89} + 47 q^{90} + 31 q^{91} + 47 q^{92} + 123 q^{93} + 54 q^{94} + 61 q^{95} + 10 q^{96} + 85 q^{97} + 189 q^{98} - 55 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −2.78808 −1.60970 −0.804849 0.593480i \(-0.797754\pi\)
−0.804849 + 0.593480i \(0.797754\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.68662 0.754277 0.377139 0.926157i \(-0.376908\pi\)
0.377139 + 0.926157i \(0.376908\pi\)
\(6\) −2.78808 −1.13823
\(7\) −0.478055 −0.180688 −0.0903440 0.995911i \(-0.528797\pi\)
−0.0903440 + 0.995911i \(0.528797\pi\)
\(8\) 1.00000 0.353553
\(9\) 4.77338 1.59113
\(10\) 1.68662 0.533355
\(11\) 2.20706 0.665455 0.332727 0.943023i \(-0.392031\pi\)
0.332727 + 0.943023i \(0.392031\pi\)
\(12\) −2.78808 −0.804849
\(13\) 5.56059 1.54223 0.771115 0.636696i \(-0.219699\pi\)
0.771115 + 0.636696i \(0.219699\pi\)
\(14\) −0.478055 −0.127766
\(15\) −4.70241 −1.21416
\(16\) 1.00000 0.250000
\(17\) 1.64824 0.399758 0.199879 0.979821i \(-0.435945\pi\)
0.199879 + 0.979821i \(0.435945\pi\)
\(18\) 4.77338 1.12510
\(19\) 4.26835 0.979227 0.489614 0.871939i \(-0.337138\pi\)
0.489614 + 0.871939i \(0.337138\pi\)
\(20\) 1.68662 0.377139
\(21\) 1.33286 0.290853
\(22\) 2.20706 0.470548
\(23\) −6.71615 −1.40041 −0.700207 0.713940i \(-0.746909\pi\)
−0.700207 + 0.713940i \(0.746909\pi\)
\(24\) −2.78808 −0.569114
\(25\) −2.15533 −0.431066
\(26\) 5.56059 1.09052
\(27\) −4.94431 −0.951533
\(28\) −0.478055 −0.0903440
\(29\) 3.62683 0.673485 0.336743 0.941597i \(-0.390675\pi\)
0.336743 + 0.941597i \(0.390675\pi\)
\(30\) −4.70241 −0.858539
\(31\) 2.87716 0.516752 0.258376 0.966044i \(-0.416813\pi\)
0.258376 + 0.966044i \(0.416813\pi\)
\(32\) 1.00000 0.176777
\(33\) −6.15347 −1.07118
\(34\) 1.64824 0.282671
\(35\) −0.806296 −0.136289
\(36\) 4.77338 0.795563
\(37\) 1.83077 0.300977 0.150489 0.988612i \(-0.451915\pi\)
0.150489 + 0.988612i \(0.451915\pi\)
\(38\) 4.26835 0.692418
\(39\) −15.5034 −2.48252
\(40\) 1.68662 0.266677
\(41\) 6.78226 1.05921 0.529606 0.848244i \(-0.322340\pi\)
0.529606 + 0.848244i \(0.322340\pi\)
\(42\) 1.33286 0.205664
\(43\) 1.02674 0.156576 0.0782882 0.996931i \(-0.475055\pi\)
0.0782882 + 0.996931i \(0.475055\pi\)
\(44\) 2.20706 0.332727
\(45\) 8.05085 1.20015
\(46\) −6.71615 −0.990242
\(47\) 10.3458 1.50909 0.754543 0.656251i \(-0.227859\pi\)
0.754543 + 0.656251i \(0.227859\pi\)
\(48\) −2.78808 −0.402424
\(49\) −6.77146 −0.967352
\(50\) −2.15533 −0.304809
\(51\) −4.59543 −0.643489
\(52\) 5.56059 0.771115
\(53\) 12.9134 1.77379 0.886894 0.461974i \(-0.152859\pi\)
0.886894 + 0.461974i \(0.152859\pi\)
\(54\) −4.94431 −0.672835
\(55\) 3.72247 0.501938
\(56\) −0.478055 −0.0638828
\(57\) −11.9005 −1.57626
\(58\) 3.62683 0.476226
\(59\) 6.42878 0.836955 0.418478 0.908227i \(-0.362564\pi\)
0.418478 + 0.908227i \(0.362564\pi\)
\(60\) −4.70241 −0.607079
\(61\) 4.33547 0.555100 0.277550 0.960711i \(-0.410478\pi\)
0.277550 + 0.960711i \(0.410478\pi\)
\(62\) 2.87716 0.365399
\(63\) −2.28194 −0.287497
\(64\) 1.00000 0.125000
\(65\) 9.37857 1.16327
\(66\) −6.15347 −0.757439
\(67\) −9.68788 −1.18356 −0.591781 0.806098i \(-0.701575\pi\)
−0.591781 + 0.806098i \(0.701575\pi\)
\(68\) 1.64824 0.199879
\(69\) 18.7251 2.25424
\(70\) −0.806296 −0.0963708
\(71\) 4.36042 0.517486 0.258743 0.965946i \(-0.416692\pi\)
0.258743 + 0.965946i \(0.416692\pi\)
\(72\) 4.77338 0.562548
\(73\) −0.189903 −0.0222265 −0.0111132 0.999938i \(-0.503538\pi\)
−0.0111132 + 0.999938i \(0.503538\pi\)
\(74\) 1.83077 0.212823
\(75\) 6.00922 0.693885
\(76\) 4.26835 0.489614
\(77\) −1.05510 −0.120240
\(78\) −15.5034 −1.75541
\(79\) −12.0283 −1.35329 −0.676647 0.736308i \(-0.736567\pi\)
−0.676647 + 0.736308i \(0.736567\pi\)
\(80\) 1.68662 0.188569
\(81\) −0.535012 −0.0594458
\(82\) 6.78226 0.748975
\(83\) 5.45551 0.598820 0.299410 0.954125i \(-0.403210\pi\)
0.299410 + 0.954125i \(0.403210\pi\)
\(84\) 1.33286 0.145426
\(85\) 2.77995 0.301528
\(86\) 1.02674 0.110716
\(87\) −10.1119 −1.08411
\(88\) 2.20706 0.235274
\(89\) −12.7133 −1.34761 −0.673803 0.738911i \(-0.735341\pi\)
−0.673803 + 0.738911i \(0.735341\pi\)
\(90\) 8.05085 0.848634
\(91\) −2.65827 −0.278662
\(92\) −6.71615 −0.700207
\(93\) −8.02173 −0.831815
\(94\) 10.3458 1.06708
\(95\) 7.19907 0.738609
\(96\) −2.78808 −0.284557
\(97\) −5.00870 −0.508556 −0.254278 0.967131i \(-0.581838\pi\)
−0.254278 + 0.967131i \(0.581838\pi\)
\(98\) −6.77146 −0.684021
\(99\) 10.5351 1.05882
\(100\) −2.15533 −0.215533
\(101\) −6.47702 −0.644488 −0.322244 0.946657i \(-0.604437\pi\)
−0.322244 + 0.946657i \(0.604437\pi\)
\(102\) −4.59543 −0.455015
\(103\) −3.67441 −0.362051 −0.181025 0.983478i \(-0.557942\pi\)
−0.181025 + 0.983478i \(0.557942\pi\)
\(104\) 5.56059 0.545261
\(105\) 2.24801 0.219384
\(106\) 12.9134 1.25426
\(107\) −16.3658 −1.58214 −0.791071 0.611724i \(-0.790476\pi\)
−0.791071 + 0.611724i \(0.790476\pi\)
\(108\) −4.94431 −0.475766
\(109\) −0.447873 −0.0428984 −0.0214492 0.999770i \(-0.506828\pi\)
−0.0214492 + 0.999770i \(0.506828\pi\)
\(110\) 3.72247 0.354923
\(111\) −5.10433 −0.484482
\(112\) −0.478055 −0.0451720
\(113\) 2.96760 0.279168 0.139584 0.990210i \(-0.455423\pi\)
0.139584 + 0.990210i \(0.455423\pi\)
\(114\) −11.9005 −1.11458
\(115\) −11.3276 −1.05630
\(116\) 3.62683 0.336743
\(117\) 26.5428 2.45388
\(118\) 6.42878 0.591817
\(119\) −0.787952 −0.0722314
\(120\) −4.70241 −0.429270
\(121\) −6.12887 −0.557170
\(122\) 4.33547 0.392515
\(123\) −18.9095 −1.70501
\(124\) 2.87716 0.258376
\(125\) −12.0683 −1.07942
\(126\) −2.28194 −0.203291
\(127\) −5.87319 −0.521161 −0.260581 0.965452i \(-0.583914\pi\)
−0.260581 + 0.965452i \(0.583914\pi\)
\(128\) 1.00000 0.0883883
\(129\) −2.86263 −0.252041
\(130\) 9.37857 0.822555
\(131\) −16.5788 −1.44850 −0.724248 0.689540i \(-0.757813\pi\)
−0.724248 + 0.689540i \(0.757813\pi\)
\(132\) −6.15347 −0.535590
\(133\) −2.04051 −0.176935
\(134\) −9.68788 −0.836905
\(135\) −8.33915 −0.717719
\(136\) 1.64824 0.141336
\(137\) 6.99325 0.597473 0.298737 0.954336i \(-0.403435\pi\)
0.298737 + 0.954336i \(0.403435\pi\)
\(138\) 18.7251 1.59399
\(139\) 0.329142 0.0279175 0.0139587 0.999903i \(-0.495557\pi\)
0.0139587 + 0.999903i \(0.495557\pi\)
\(140\) −0.806296 −0.0681444
\(141\) −28.8448 −2.42917
\(142\) 4.36042 0.365918
\(143\) 12.2726 1.02628
\(144\) 4.77338 0.397781
\(145\) 6.11706 0.507995
\(146\) −0.189903 −0.0157165
\(147\) 18.8794 1.55714
\(148\) 1.83077 0.150489
\(149\) 14.3445 1.17515 0.587574 0.809170i \(-0.300083\pi\)
0.587574 + 0.809170i \(0.300083\pi\)
\(150\) 6.00922 0.490651
\(151\) −19.9450 −1.62310 −0.811551 0.584281i \(-0.801376\pi\)
−0.811551 + 0.584281i \(0.801376\pi\)
\(152\) 4.26835 0.346209
\(153\) 7.86768 0.636065
\(154\) −1.05510 −0.0850223
\(155\) 4.85265 0.389775
\(156\) −15.5034 −1.24126
\(157\) 6.65329 0.530990 0.265495 0.964112i \(-0.414465\pi\)
0.265495 + 0.964112i \(0.414465\pi\)
\(158\) −12.0283 −0.956923
\(159\) −36.0035 −2.85526
\(160\) 1.68662 0.133339
\(161\) 3.21069 0.253038
\(162\) −0.535012 −0.0420345
\(163\) 17.4424 1.36619 0.683097 0.730328i \(-0.260633\pi\)
0.683097 + 0.730328i \(0.260633\pi\)
\(164\) 6.78226 0.529606
\(165\) −10.3785 −0.807967
\(166\) 5.45551 0.423430
\(167\) −4.36510 −0.337781 −0.168891 0.985635i \(-0.554018\pi\)
−0.168891 + 0.985635i \(0.554018\pi\)
\(168\) 1.33286 0.102832
\(169\) 17.9201 1.37847
\(170\) 2.77995 0.213213
\(171\) 20.3745 1.55807
\(172\) 1.02674 0.0782882
\(173\) −2.68710 −0.204297 −0.102148 0.994769i \(-0.532572\pi\)
−0.102148 + 0.994769i \(0.532572\pi\)
\(174\) −10.1119 −0.766579
\(175\) 1.03037 0.0778884
\(176\) 2.20706 0.166364
\(177\) −17.9239 −1.34724
\(178\) −12.7133 −0.952902
\(179\) 6.98478 0.522067 0.261033 0.965330i \(-0.415937\pi\)
0.261033 + 0.965330i \(0.415937\pi\)
\(180\) 8.05085 0.600075
\(181\) 8.12867 0.604199 0.302100 0.953276i \(-0.402313\pi\)
0.302100 + 0.953276i \(0.402313\pi\)
\(182\) −2.65827 −0.197044
\(183\) −12.0876 −0.893542
\(184\) −6.71615 −0.495121
\(185\) 3.08781 0.227020
\(186\) −8.02173 −0.588182
\(187\) 3.63778 0.266021
\(188\) 10.3458 0.754543
\(189\) 2.36365 0.171930
\(190\) 7.19907 0.522275
\(191\) −6.84039 −0.494954 −0.247477 0.968894i \(-0.579601\pi\)
−0.247477 + 0.968894i \(0.579601\pi\)
\(192\) −2.78808 −0.201212
\(193\) 20.5250 1.47742 0.738710 0.674023i \(-0.235435\pi\)
0.738710 + 0.674023i \(0.235435\pi\)
\(194\) −5.00870 −0.359603
\(195\) −26.1482 −1.87251
\(196\) −6.77146 −0.483676
\(197\) 23.6099 1.68214 0.841068 0.540929i \(-0.181927\pi\)
0.841068 + 0.540929i \(0.181927\pi\)
\(198\) 10.5351 0.748700
\(199\) −17.2752 −1.22461 −0.612304 0.790623i \(-0.709757\pi\)
−0.612304 + 0.790623i \(0.709757\pi\)
\(200\) −2.15533 −0.152405
\(201\) 27.0106 1.90518
\(202\) −6.47702 −0.455722
\(203\) −1.73382 −0.121691
\(204\) −4.59543 −0.321744
\(205\) 11.4391 0.798939
\(206\) −3.67441 −0.256009
\(207\) −32.0587 −2.22823
\(208\) 5.56059 0.385557
\(209\) 9.42053 0.651632
\(210\) 2.24801 0.155128
\(211\) 4.04319 0.278344 0.139172 0.990268i \(-0.455556\pi\)
0.139172 + 0.990268i \(0.455556\pi\)
\(212\) 12.9134 0.886894
\(213\) −12.1572 −0.832996
\(214\) −16.3658 −1.11874
\(215\) 1.73172 0.118102
\(216\) −4.94431 −0.336418
\(217\) −1.37544 −0.0933709
\(218\) −0.447873 −0.0303338
\(219\) 0.529464 0.0357779
\(220\) 3.72247 0.250969
\(221\) 9.16520 0.616518
\(222\) −5.10433 −0.342581
\(223\) 4.68255 0.313567 0.156783 0.987633i \(-0.449888\pi\)
0.156783 + 0.987633i \(0.449888\pi\)
\(224\) −0.478055 −0.0319414
\(225\) −10.2882 −0.685880
\(226\) 2.96760 0.197402
\(227\) 21.4011 1.42044 0.710221 0.703978i \(-0.248595\pi\)
0.710221 + 0.703978i \(0.248595\pi\)
\(228\) −11.9005 −0.788130
\(229\) 12.6702 0.837267 0.418634 0.908155i \(-0.362509\pi\)
0.418634 + 0.908155i \(0.362509\pi\)
\(230\) −11.3276 −0.746917
\(231\) 2.94170 0.193550
\(232\) 3.62683 0.238113
\(233\) 2.85283 0.186895 0.0934476 0.995624i \(-0.470211\pi\)
0.0934476 + 0.995624i \(0.470211\pi\)
\(234\) 26.5428 1.73516
\(235\) 17.4493 1.13827
\(236\) 6.42878 0.418478
\(237\) 33.5359 2.17839
\(238\) −0.787952 −0.0510753
\(239\) −13.6057 −0.880079 −0.440040 0.897978i \(-0.645036\pi\)
−0.440040 + 0.897978i \(0.645036\pi\)
\(240\) −4.70241 −0.303540
\(241\) 16.2213 1.04491 0.522454 0.852668i \(-0.325017\pi\)
0.522454 + 0.852668i \(0.325017\pi\)
\(242\) −6.12887 −0.393979
\(243\) 16.3246 1.04722
\(244\) 4.33547 0.277550
\(245\) −11.4209 −0.729652
\(246\) −18.9095 −1.20562
\(247\) 23.7346 1.51019
\(248\) 2.87716 0.182700
\(249\) −15.2104 −0.963919
\(250\) −12.0683 −0.763265
\(251\) 4.08474 0.257826 0.128913 0.991656i \(-0.458851\pi\)
0.128913 + 0.991656i \(0.458851\pi\)
\(252\) −2.28194 −0.143749
\(253\) −14.8230 −0.931912
\(254\) −5.87319 −0.368517
\(255\) −7.75072 −0.485369
\(256\) 1.00000 0.0625000
\(257\) −10.7789 −0.672372 −0.336186 0.941796i \(-0.609137\pi\)
−0.336186 + 0.941796i \(0.609137\pi\)
\(258\) −2.86263 −0.178220
\(259\) −0.875211 −0.0543829
\(260\) 9.37857 0.581635
\(261\) 17.3122 1.07160
\(262\) −16.5788 −1.02424
\(263\) −23.3509 −1.43988 −0.719940 0.694036i \(-0.755831\pi\)
−0.719940 + 0.694036i \(0.755831\pi\)
\(264\) −6.15347 −0.378720
\(265\) 21.7799 1.33793
\(266\) −2.04051 −0.125112
\(267\) 35.4457 2.16924
\(268\) −9.68788 −0.591781
\(269\) 4.24828 0.259022 0.129511 0.991578i \(-0.458659\pi\)
0.129511 + 0.991578i \(0.458659\pi\)
\(270\) −8.33915 −0.507504
\(271\) 8.46851 0.514426 0.257213 0.966355i \(-0.417196\pi\)
0.257213 + 0.966355i \(0.417196\pi\)
\(272\) 1.64824 0.0999394
\(273\) 7.41146 0.448562
\(274\) 6.99325 0.422478
\(275\) −4.75695 −0.286855
\(276\) 18.7251 1.12712
\(277\) 22.8520 1.37304 0.686521 0.727110i \(-0.259137\pi\)
0.686521 + 0.727110i \(0.259137\pi\)
\(278\) 0.329142 0.0197406
\(279\) 13.7337 0.822218
\(280\) −0.806296 −0.0481854
\(281\) 2.97082 0.177224 0.0886122 0.996066i \(-0.471757\pi\)
0.0886122 + 0.996066i \(0.471757\pi\)
\(282\) −28.8448 −1.71768
\(283\) 24.7400 1.47064 0.735319 0.677721i \(-0.237032\pi\)
0.735319 + 0.677721i \(0.237032\pi\)
\(284\) 4.36042 0.258743
\(285\) −20.0716 −1.18894
\(286\) 12.2726 0.725693
\(287\) −3.24230 −0.191387
\(288\) 4.77338 0.281274
\(289\) −14.2833 −0.840194
\(290\) 6.11706 0.359206
\(291\) 13.9646 0.818621
\(292\) −0.189903 −0.0111132
\(293\) 5.69462 0.332683 0.166342 0.986068i \(-0.446805\pi\)
0.166342 + 0.986068i \(0.446805\pi\)
\(294\) 18.8794 1.10107
\(295\) 10.8429 0.631296
\(296\) 1.83077 0.106411
\(297\) −10.9124 −0.633202
\(298\) 14.3445 0.830956
\(299\) −37.3457 −2.15976
\(300\) 6.00922 0.346943
\(301\) −0.490839 −0.0282915
\(302\) −19.9450 −1.14771
\(303\) 18.0584 1.03743
\(304\) 4.26835 0.244807
\(305\) 7.31226 0.418699
\(306\) 7.86768 0.449766
\(307\) −1.44560 −0.0825050 −0.0412525 0.999149i \(-0.513135\pi\)
−0.0412525 + 0.999149i \(0.513135\pi\)
\(308\) −1.05510 −0.0601199
\(309\) 10.2446 0.582792
\(310\) 4.85265 0.275612
\(311\) −25.2656 −1.43268 −0.716342 0.697750i \(-0.754185\pi\)
−0.716342 + 0.697750i \(0.754185\pi\)
\(312\) −15.5034 −0.877704
\(313\) 25.7334 1.45454 0.727268 0.686353i \(-0.240790\pi\)
0.727268 + 0.686353i \(0.240790\pi\)
\(314\) 6.65329 0.375467
\(315\) −3.84875 −0.216853
\(316\) −12.0283 −0.676647
\(317\) −2.01159 −0.112982 −0.0564911 0.998403i \(-0.517991\pi\)
−0.0564911 + 0.998403i \(0.517991\pi\)
\(318\) −36.0035 −2.01897
\(319\) 8.00464 0.448174
\(320\) 1.68662 0.0942847
\(321\) 45.6292 2.54677
\(322\) 3.21069 0.178925
\(323\) 7.03528 0.391454
\(324\) −0.535012 −0.0297229
\(325\) −11.9849 −0.664802
\(326\) 17.4424 0.966044
\(327\) 1.24870 0.0690535
\(328\) 6.78226 0.374488
\(329\) −4.94585 −0.272674
\(330\) −10.3785 −0.571319
\(331\) −17.5611 −0.965248 −0.482624 0.875828i \(-0.660316\pi\)
−0.482624 + 0.875828i \(0.660316\pi\)
\(332\) 5.45551 0.299410
\(333\) 8.73896 0.478892
\(334\) −4.36510 −0.238848
\(335\) −16.3397 −0.892735
\(336\) 1.33286 0.0727132
\(337\) −6.66556 −0.363096 −0.181548 0.983382i \(-0.558111\pi\)
−0.181548 + 0.983382i \(0.558111\pi\)
\(338\) 17.9201 0.974728
\(339\) −8.27389 −0.449376
\(340\) 2.77995 0.150764
\(341\) 6.35007 0.343875
\(342\) 20.3745 1.10172
\(343\) 6.58352 0.355477
\(344\) 1.02674 0.0553581
\(345\) 31.5821 1.70032
\(346\) −2.68710 −0.144460
\(347\) 17.9000 0.960922 0.480461 0.877016i \(-0.340469\pi\)
0.480461 + 0.877016i \(0.340469\pi\)
\(348\) −10.1119 −0.542054
\(349\) −10.0449 −0.537694 −0.268847 0.963183i \(-0.586643\pi\)
−0.268847 + 0.963183i \(0.586643\pi\)
\(350\) 1.03037 0.0550754
\(351\) −27.4933 −1.46748
\(352\) 2.20706 0.117637
\(353\) −4.49994 −0.239508 −0.119754 0.992804i \(-0.538211\pi\)
−0.119754 + 0.992804i \(0.538211\pi\)
\(354\) −17.9239 −0.952646
\(355\) 7.35434 0.390328
\(356\) −12.7133 −0.673803
\(357\) 2.19687 0.116271
\(358\) 6.98478 0.369157
\(359\) 7.05175 0.372177 0.186089 0.982533i \(-0.440419\pi\)
0.186089 + 0.982533i \(0.440419\pi\)
\(360\) 8.05085 0.424317
\(361\) −0.781163 −0.0411138
\(362\) 8.12867 0.427233
\(363\) 17.0878 0.896875
\(364\) −2.65827 −0.139331
\(365\) −0.320293 −0.0167649
\(366\) −12.0876 −0.631830
\(367\) 23.1937 1.21070 0.605352 0.795958i \(-0.293033\pi\)
0.605352 + 0.795958i \(0.293033\pi\)
\(368\) −6.71615 −0.350103
\(369\) 32.3743 1.68534
\(370\) 3.08781 0.160528
\(371\) −6.17330 −0.320502
\(372\) −8.02173 −0.415907
\(373\) −30.7680 −1.59311 −0.796553 0.604568i \(-0.793346\pi\)
−0.796553 + 0.604568i \(0.793346\pi\)
\(374\) 3.63778 0.188105
\(375\) 33.6473 1.73754
\(376\) 10.3458 0.533542
\(377\) 20.1673 1.03867
\(378\) 2.36365 0.121573
\(379\) 1.67234 0.0859022 0.0429511 0.999077i \(-0.486324\pi\)
0.0429511 + 0.999077i \(0.486324\pi\)
\(380\) 7.19907 0.369304
\(381\) 16.3749 0.838912
\(382\) −6.84039 −0.349985
\(383\) −3.82222 −0.195306 −0.0976531 0.995221i \(-0.531134\pi\)
−0.0976531 + 0.995221i \(0.531134\pi\)
\(384\) −2.78808 −0.142278
\(385\) −1.77955 −0.0906941
\(386\) 20.5250 1.04469
\(387\) 4.90102 0.249133
\(388\) −5.00870 −0.254278
\(389\) 3.29055 0.166838 0.0834188 0.996515i \(-0.473416\pi\)
0.0834188 + 0.996515i \(0.473416\pi\)
\(390\) −26.1482 −1.32407
\(391\) −11.0698 −0.559826
\(392\) −6.77146 −0.342011
\(393\) 46.2230 2.33164
\(394\) 23.6099 1.18945
\(395\) −20.2872 −1.02076
\(396\) 10.5351 0.529411
\(397\) −22.9518 −1.15192 −0.575958 0.817479i \(-0.695371\pi\)
−0.575958 + 0.817479i \(0.695371\pi\)
\(398\) −17.2752 −0.865928
\(399\) 5.68910 0.284811
\(400\) −2.15533 −0.107766
\(401\) −30.7638 −1.53627 −0.768136 0.640287i \(-0.778815\pi\)
−0.768136 + 0.640287i \(0.778815\pi\)
\(402\) 27.0106 1.34716
\(403\) 15.9987 0.796951
\(404\) −6.47702 −0.322244
\(405\) −0.902360 −0.0448386
\(406\) −1.73382 −0.0860483
\(407\) 4.04063 0.200287
\(408\) −4.59543 −0.227508
\(409\) 28.3717 1.40289 0.701446 0.712723i \(-0.252538\pi\)
0.701446 + 0.712723i \(0.252538\pi\)
\(410\) 11.4391 0.564935
\(411\) −19.4977 −0.961751
\(412\) −3.67441 −0.181025
\(413\) −3.07331 −0.151228
\(414\) −32.0587 −1.57560
\(415\) 9.20134 0.451676
\(416\) 5.56059 0.272630
\(417\) −0.917674 −0.0449387
\(418\) 9.42053 0.460773
\(419\) −14.8052 −0.723279 −0.361640 0.932318i \(-0.617783\pi\)
−0.361640 + 0.932318i \(0.617783\pi\)
\(420\) 2.24801 0.109692
\(421\) 34.4115 1.67711 0.838556 0.544816i \(-0.183400\pi\)
0.838556 + 0.544816i \(0.183400\pi\)
\(422\) 4.04319 0.196819
\(423\) 49.3842 2.40114
\(424\) 12.9134 0.627128
\(425\) −3.55251 −0.172322
\(426\) −12.1572 −0.589017
\(427\) −2.07259 −0.100300
\(428\) −16.3658 −0.791071
\(429\) −34.2169 −1.65201
\(430\) 1.73172 0.0835107
\(431\) −2.03361 −0.0979555 −0.0489778 0.998800i \(-0.515596\pi\)
−0.0489778 + 0.998800i \(0.515596\pi\)
\(432\) −4.94431 −0.237883
\(433\) −19.3295 −0.928914 −0.464457 0.885596i \(-0.653751\pi\)
−0.464457 + 0.885596i \(0.653751\pi\)
\(434\) −1.37544 −0.0660232
\(435\) −17.0548 −0.817717
\(436\) −0.447873 −0.0214492
\(437\) −28.6669 −1.37132
\(438\) 0.529464 0.0252988
\(439\) −15.0399 −0.717817 −0.358909 0.933373i \(-0.616851\pi\)
−0.358909 + 0.933373i \(0.616851\pi\)
\(440\) 3.72247 0.177462
\(441\) −32.3227 −1.53918
\(442\) 9.16520 0.435944
\(443\) 8.06470 0.383166 0.191583 0.981476i \(-0.438638\pi\)
0.191583 + 0.981476i \(0.438638\pi\)
\(444\) −5.10433 −0.242241
\(445\) −21.4424 −1.01647
\(446\) 4.68255 0.221725
\(447\) −39.9936 −1.89163
\(448\) −0.478055 −0.0225860
\(449\) −30.5246 −1.44055 −0.720274 0.693690i \(-0.755984\pi\)
−0.720274 + 0.693690i \(0.755984\pi\)
\(450\) −10.2882 −0.484990
\(451\) 14.9689 0.704857
\(452\) 2.96760 0.139584
\(453\) 55.6082 2.61270
\(454\) 21.4011 1.00440
\(455\) −4.48348 −0.210189
\(456\) −11.9005 −0.557292
\(457\) 35.7756 1.67351 0.836755 0.547577i \(-0.184450\pi\)
0.836755 + 0.547577i \(0.184450\pi\)
\(458\) 12.6702 0.592037
\(459\) −8.14942 −0.380382
\(460\) −11.3276 −0.528150
\(461\) 33.8197 1.57514 0.787571 0.616225i \(-0.211339\pi\)
0.787571 + 0.616225i \(0.211339\pi\)
\(462\) 2.94170 0.136860
\(463\) −26.5118 −1.23211 −0.616054 0.787704i \(-0.711270\pi\)
−0.616054 + 0.787704i \(0.711270\pi\)
\(464\) 3.62683 0.168371
\(465\) −13.5296 −0.627419
\(466\) 2.85283 0.132155
\(467\) 23.1866 1.07295 0.536473 0.843918i \(-0.319756\pi\)
0.536473 + 0.843918i \(0.319756\pi\)
\(468\) 26.5428 1.22694
\(469\) 4.63134 0.213856
\(470\) 17.4493 0.804878
\(471\) −18.5499 −0.854734
\(472\) 6.42878 0.295908
\(473\) 2.26608 0.104195
\(474\) 33.5359 1.54036
\(475\) −9.19970 −0.422111
\(476\) −0.787952 −0.0361157
\(477\) 61.6403 2.82232
\(478\) −13.6057 −0.622310
\(479\) −24.2960 −1.11011 −0.555056 0.831813i \(-0.687303\pi\)
−0.555056 + 0.831813i \(0.687303\pi\)
\(480\) −4.70241 −0.214635
\(481\) 10.1802 0.464176
\(482\) 16.2213 0.738861
\(483\) −8.95165 −0.407314
\(484\) −6.12887 −0.278585
\(485\) −8.44774 −0.383592
\(486\) 16.3246 0.740498
\(487\) 16.6398 0.754021 0.377011 0.926209i \(-0.376952\pi\)
0.377011 + 0.926209i \(0.376952\pi\)
\(488\) 4.33547 0.196257
\(489\) −48.6307 −2.19916
\(490\) −11.4209 −0.515942
\(491\) 6.51985 0.294237 0.147118 0.989119i \(-0.453000\pi\)
0.147118 + 0.989119i \(0.453000\pi\)
\(492\) −18.9095 −0.852505
\(493\) 5.97789 0.269231
\(494\) 23.7346 1.06787
\(495\) 17.7687 0.798645
\(496\) 2.87716 0.129188
\(497\) −2.08452 −0.0935035
\(498\) −15.2104 −0.681593
\(499\) 9.47873 0.424326 0.212163 0.977234i \(-0.431949\pi\)
0.212163 + 0.977234i \(0.431949\pi\)
\(500\) −12.0683 −0.539710
\(501\) 12.1702 0.543726
\(502\) 4.08474 0.182311
\(503\) 8.74160 0.389769 0.194884 0.980826i \(-0.437567\pi\)
0.194884 + 0.980826i \(0.437567\pi\)
\(504\) −2.28194 −0.101646
\(505\) −10.9242 −0.486122
\(506\) −14.8230 −0.658961
\(507\) −49.9628 −2.21892
\(508\) −5.87319 −0.260581
\(509\) 28.0557 1.24355 0.621774 0.783197i \(-0.286412\pi\)
0.621774 + 0.783197i \(0.286412\pi\)
\(510\) −7.75072 −0.343208
\(511\) 0.0907842 0.00401606
\(512\) 1.00000 0.0441942
\(513\) −21.1041 −0.931767
\(514\) −10.7789 −0.475439
\(515\) −6.19732 −0.273087
\(516\) −2.86263 −0.126020
\(517\) 22.8338 1.00423
\(518\) −0.875211 −0.0384545
\(519\) 7.49186 0.328856
\(520\) 9.37857 0.411278
\(521\) −0.755597 −0.0331033 −0.0165516 0.999863i \(-0.505269\pi\)
−0.0165516 + 0.999863i \(0.505269\pi\)
\(522\) 17.3122 0.757735
\(523\) 14.9165 0.652252 0.326126 0.945326i \(-0.394257\pi\)
0.326126 + 0.945326i \(0.394257\pi\)
\(524\) −16.5788 −0.724248
\(525\) −2.87274 −0.125377
\(526\) −23.3509 −1.01815
\(527\) 4.74225 0.206576
\(528\) −6.15347 −0.267795
\(529\) 22.1066 0.961157
\(530\) 21.7799 0.946058
\(531\) 30.6870 1.33170
\(532\) −2.04051 −0.0884673
\(533\) 37.7134 1.63355
\(534\) 35.4457 1.53388
\(535\) −27.6028 −1.19337
\(536\) −9.68788 −0.418453
\(537\) −19.4741 −0.840369
\(538\) 4.24828 0.183156
\(539\) −14.9451 −0.643729
\(540\) −8.33915 −0.358860
\(541\) 17.2044 0.739676 0.369838 0.929096i \(-0.379413\pi\)
0.369838 + 0.929096i \(0.379413\pi\)
\(542\) 8.46851 0.363754
\(543\) −22.6634 −0.972578
\(544\) 1.64824 0.0706678
\(545\) −0.755389 −0.0323573
\(546\) 7.41146 0.317181
\(547\) −11.2801 −0.482302 −0.241151 0.970488i \(-0.577525\pi\)
−0.241151 + 0.970488i \(0.577525\pi\)
\(548\) 6.99325 0.298737
\(549\) 20.6948 0.883233
\(550\) −4.75695 −0.202837
\(551\) 15.4806 0.659495
\(552\) 18.7251 0.796995
\(553\) 5.75021 0.244524
\(554\) 22.8520 0.970887
\(555\) −8.60905 −0.365434
\(556\) 0.329142 0.0139587
\(557\) −12.1698 −0.515650 −0.257825 0.966192i \(-0.583006\pi\)
−0.257825 + 0.966192i \(0.583006\pi\)
\(558\) 13.7337 0.581396
\(559\) 5.70928 0.241477
\(560\) −0.806296 −0.0340722
\(561\) −10.1424 −0.428213
\(562\) 2.97082 0.125317
\(563\) 27.9875 1.17953 0.589766 0.807574i \(-0.299220\pi\)
0.589766 + 0.807574i \(0.299220\pi\)
\(564\) −28.8448 −1.21459
\(565\) 5.00519 0.210570
\(566\) 24.7400 1.03990
\(567\) 0.255766 0.0107411
\(568\) 4.36042 0.182959
\(569\) 27.9075 1.16994 0.584971 0.811054i \(-0.301106\pi\)
0.584971 + 0.811054i \(0.301106\pi\)
\(570\) −20.0716 −0.840705
\(571\) 20.8409 0.872165 0.436083 0.899907i \(-0.356366\pi\)
0.436083 + 0.899907i \(0.356366\pi\)
\(572\) 12.2726 0.513142
\(573\) 19.0715 0.796725
\(574\) −3.24230 −0.135331
\(575\) 14.4755 0.603670
\(576\) 4.77338 0.198891
\(577\) −26.5218 −1.10412 −0.552059 0.833805i \(-0.686158\pi\)
−0.552059 + 0.833805i \(0.686158\pi\)
\(578\) −14.2833 −0.594107
\(579\) −57.2252 −2.37820
\(580\) 6.11706 0.253997
\(581\) −2.60804 −0.108200
\(582\) 13.9646 0.578853
\(583\) 28.5006 1.18038
\(584\) −0.189903 −0.00785824
\(585\) 44.7675 1.85091
\(586\) 5.69462 0.235243
\(587\) −12.7019 −0.524262 −0.262131 0.965032i \(-0.584425\pi\)
−0.262131 + 0.965032i \(0.584425\pi\)
\(588\) 18.8794 0.778572
\(589\) 12.2807 0.506018
\(590\) 10.8429 0.446394
\(591\) −65.8262 −2.70773
\(592\) 1.83077 0.0752443
\(593\) 0.655127 0.0269028 0.0134514 0.999910i \(-0.495718\pi\)
0.0134514 + 0.999910i \(0.495718\pi\)
\(594\) −10.9124 −0.447741
\(595\) −1.32897 −0.0544825
\(596\) 14.3445 0.587574
\(597\) 48.1646 1.97125
\(598\) −37.3457 −1.52718
\(599\) 42.1444 1.72197 0.860987 0.508627i \(-0.169847\pi\)
0.860987 + 0.508627i \(0.169847\pi\)
\(600\) 6.00922 0.245325
\(601\) −41.2275 −1.68170 −0.840852 0.541265i \(-0.817946\pi\)
−0.840852 + 0.541265i \(0.817946\pi\)
\(602\) −0.490839 −0.0200051
\(603\) −46.2439 −1.88320
\(604\) −19.9450 −0.811551
\(605\) −10.3370 −0.420261
\(606\) 18.0584 0.733574
\(607\) −10.1623 −0.412475 −0.206238 0.978502i \(-0.566122\pi\)
−0.206238 + 0.978502i \(0.566122\pi\)
\(608\) 4.26835 0.173105
\(609\) 4.83404 0.195885
\(610\) 7.31226 0.296065
\(611\) 57.5286 2.32736
\(612\) 7.86768 0.318032
\(613\) −19.9943 −0.807560 −0.403780 0.914856i \(-0.632304\pi\)
−0.403780 + 0.914856i \(0.632304\pi\)
\(614\) −1.44560 −0.0583398
\(615\) −31.8930 −1.28605
\(616\) −1.05510 −0.0425112
\(617\) 30.5781 1.23103 0.615515 0.788125i \(-0.288948\pi\)
0.615515 + 0.788125i \(0.288948\pi\)
\(618\) 10.2446 0.412096
\(619\) 11.2306 0.451394 0.225697 0.974198i \(-0.427534\pi\)
0.225697 + 0.974198i \(0.427534\pi\)
\(620\) 4.85265 0.194887
\(621\) 33.2067 1.33254
\(622\) −25.2656 −1.01306
\(623\) 6.07766 0.243496
\(624\) −15.5034 −0.620631
\(625\) −9.57792 −0.383117
\(626\) 25.7334 1.02851
\(627\) −26.2652 −1.04893
\(628\) 6.65329 0.265495
\(629\) 3.01756 0.120318
\(630\) −3.84875 −0.153338
\(631\) 24.6022 0.979399 0.489700 0.871891i \(-0.337106\pi\)
0.489700 + 0.871891i \(0.337106\pi\)
\(632\) −12.0283 −0.478461
\(633\) −11.2727 −0.448050
\(634\) −2.01159 −0.0798905
\(635\) −9.90581 −0.393100
\(636\) −36.0035 −1.42763
\(637\) −37.6533 −1.49188
\(638\) 8.00464 0.316907
\(639\) 20.8139 0.823385
\(640\) 1.68662 0.0666693
\(641\) 21.8401 0.862632 0.431316 0.902201i \(-0.358049\pi\)
0.431316 + 0.902201i \(0.358049\pi\)
\(642\) 45.6292 1.80084
\(643\) 13.4891 0.531960 0.265980 0.963979i \(-0.414305\pi\)
0.265980 + 0.963979i \(0.414305\pi\)
\(644\) 3.21069 0.126519
\(645\) −4.82816 −0.190108
\(646\) 7.03528 0.276800
\(647\) 22.7573 0.894683 0.447342 0.894363i \(-0.352371\pi\)
0.447342 + 0.894363i \(0.352371\pi\)
\(648\) −0.535012 −0.0210173
\(649\) 14.1887 0.556956
\(650\) −11.9849 −0.470086
\(651\) 3.83483 0.150299
\(652\) 17.4424 0.683097
\(653\) −12.0355 −0.470985 −0.235493 0.971876i \(-0.575670\pi\)
−0.235493 + 0.971876i \(0.575670\pi\)
\(654\) 1.24870 0.0488282
\(655\) −27.9620 −1.09257
\(656\) 6.78226 0.264803
\(657\) −0.906478 −0.0353651
\(658\) −4.94585 −0.192809
\(659\) 36.4807 1.42109 0.710543 0.703654i \(-0.248449\pi\)
0.710543 + 0.703654i \(0.248449\pi\)
\(660\) −10.3785 −0.403984
\(661\) 26.0746 1.01419 0.507093 0.861891i \(-0.330720\pi\)
0.507093 + 0.861891i \(0.330720\pi\)
\(662\) −17.5611 −0.682533
\(663\) −25.5533 −0.992408
\(664\) 5.45551 0.211715
\(665\) −3.44155 −0.133458
\(666\) 8.73896 0.338628
\(667\) −24.3583 −0.943157
\(668\) −4.36510 −0.168891
\(669\) −13.0553 −0.504747
\(670\) −16.3397 −0.631259
\(671\) 9.56865 0.369394
\(672\) 1.33286 0.0514160
\(673\) −25.9618 −1.00076 −0.500378 0.865807i \(-0.666806\pi\)
−0.500378 + 0.865807i \(0.666806\pi\)
\(674\) −6.66556 −0.256748
\(675\) 10.6566 0.410173
\(676\) 17.9201 0.689237
\(677\) −43.6384 −1.67716 −0.838580 0.544779i \(-0.816614\pi\)
−0.838580 + 0.544779i \(0.816614\pi\)
\(678\) −8.27389 −0.317757
\(679\) 2.39443 0.0918900
\(680\) 2.77995 0.106606
\(681\) −59.6680 −2.28648
\(682\) 6.35007 0.243157
\(683\) −2.25353 −0.0862290 −0.0431145 0.999070i \(-0.513728\pi\)
−0.0431145 + 0.999070i \(0.513728\pi\)
\(684\) 20.3745 0.779037
\(685\) 11.7949 0.450661
\(686\) 6.58352 0.251360
\(687\) −35.3254 −1.34775
\(688\) 1.02674 0.0391441
\(689\) 71.8059 2.73559
\(690\) 31.5821 1.20231
\(691\) −45.8134 −1.74282 −0.871412 0.490552i \(-0.836795\pi\)
−0.871412 + 0.490552i \(0.836795\pi\)
\(692\) −2.68710 −0.102148
\(693\) −5.03638 −0.191316
\(694\) 17.9000 0.679475
\(695\) 0.555136 0.0210575
\(696\) −10.1119 −0.383290
\(697\) 11.1788 0.423428
\(698\) −10.0449 −0.380207
\(699\) −7.95391 −0.300845
\(700\) 1.03037 0.0389442
\(701\) 9.67099 0.365268 0.182634 0.983181i \(-0.441538\pi\)
0.182634 + 0.983181i \(0.441538\pi\)
\(702\) −27.4933 −1.03767
\(703\) 7.81438 0.294725
\(704\) 2.20706 0.0831819
\(705\) −48.6501 −1.83227
\(706\) −4.49994 −0.169358
\(707\) 3.09637 0.116451
\(708\) −17.9239 −0.673622
\(709\) −29.5138 −1.10841 −0.554207 0.832379i \(-0.686978\pi\)
−0.554207 + 0.832379i \(0.686978\pi\)
\(710\) 7.35434 0.276004
\(711\) −57.4157 −2.15326
\(712\) −12.7133 −0.476451
\(713\) −19.3234 −0.723667
\(714\) 2.19687 0.0822158
\(715\) 20.6991 0.774103
\(716\) 6.98478 0.261033
\(717\) 37.9337 1.41666
\(718\) 7.05175 0.263169
\(719\) −7.43147 −0.277147 −0.138574 0.990352i \(-0.544252\pi\)
−0.138574 + 0.990352i \(0.544252\pi\)
\(720\) 8.05085 0.300037
\(721\) 1.75657 0.0654182
\(722\) −0.781163 −0.0290719
\(723\) −45.2263 −1.68198
\(724\) 8.12867 0.302100
\(725\) −7.81701 −0.290316
\(726\) 17.0878 0.634186
\(727\) −3.38859 −0.125676 −0.0628379 0.998024i \(-0.520015\pi\)
−0.0628379 + 0.998024i \(0.520015\pi\)
\(728\) −2.65827 −0.0985220
\(729\) −43.9092 −1.62627
\(730\) −0.320293 −0.0118546
\(731\) 1.69232 0.0625926
\(732\) −12.0876 −0.446771
\(733\) 40.9806 1.51365 0.756827 0.653615i \(-0.226748\pi\)
0.756827 + 0.653615i \(0.226748\pi\)
\(734\) 23.1937 0.856096
\(735\) 31.8422 1.17452
\(736\) −6.71615 −0.247560
\(737\) −21.3818 −0.787608
\(738\) 32.3743 1.19171
\(739\) −1.27282 −0.0468215 −0.0234107 0.999726i \(-0.507453\pi\)
−0.0234107 + 0.999726i \(0.507453\pi\)
\(740\) 3.08781 0.113510
\(741\) −66.1738 −2.43095
\(742\) −6.17330 −0.226629
\(743\) −6.06834 −0.222626 −0.111313 0.993785i \(-0.535506\pi\)
−0.111313 + 0.993785i \(0.535506\pi\)
\(744\) −8.02173 −0.294091
\(745\) 24.1937 0.886388
\(746\) −30.7680 −1.12650
\(747\) 26.0412 0.952797
\(748\) 3.63778 0.133010
\(749\) 7.82377 0.285874
\(750\) 33.6473 1.22863
\(751\) −37.1670 −1.35624 −0.678121 0.734950i \(-0.737206\pi\)
−0.678121 + 0.734950i \(0.737206\pi\)
\(752\) 10.3458 0.377271
\(753\) −11.3886 −0.415022
\(754\) 20.1673 0.734450
\(755\) −33.6396 −1.22427
\(756\) 2.36365 0.0859652
\(757\) 23.1285 0.840619 0.420310 0.907381i \(-0.361921\pi\)
0.420310 + 0.907381i \(0.361921\pi\)
\(758\) 1.67234 0.0607421
\(759\) 41.3276 1.50010
\(760\) 7.19907 0.261138
\(761\) −33.8114 −1.22566 −0.612830 0.790214i \(-0.709969\pi\)
−0.612830 + 0.790214i \(0.709969\pi\)
\(762\) 16.3749 0.593200
\(763\) 0.214108 0.00775123
\(764\) −6.84039 −0.247477
\(765\) 13.2698 0.479769
\(766\) −3.82222 −0.138102
\(767\) 35.7478 1.29078
\(768\) −2.78808 −0.100606
\(769\) 44.6314 1.60945 0.804724 0.593648i \(-0.202313\pi\)
0.804724 + 0.593648i \(0.202313\pi\)
\(770\) −1.77955 −0.0641304
\(771\) 30.0525 1.08232
\(772\) 20.5250 0.738710
\(773\) −37.9486 −1.36492 −0.682458 0.730925i \(-0.739089\pi\)
−0.682458 + 0.730925i \(0.739089\pi\)
\(774\) 4.90102 0.176163
\(775\) −6.20121 −0.222754
\(776\) −5.00870 −0.179802
\(777\) 2.44015 0.0875401
\(778\) 3.29055 0.117972
\(779\) 28.9491 1.03721
\(780\) −26.1482 −0.936255
\(781\) 9.62372 0.344364
\(782\) −11.0698 −0.395857
\(783\) −17.9322 −0.640843
\(784\) −6.77146 −0.241838
\(785\) 11.2215 0.400514
\(786\) 46.2230 1.64872
\(787\) 3.75804 0.133960 0.0669799 0.997754i \(-0.478664\pi\)
0.0669799 + 0.997754i \(0.478664\pi\)
\(788\) 23.6099 0.841068
\(789\) 65.1042 2.31777
\(790\) −20.2872 −0.721785
\(791\) −1.41868 −0.0504423
\(792\) 10.5351 0.374350
\(793\) 24.1077 0.856091
\(794\) −22.9518 −0.814527
\(795\) −60.7240 −2.15366
\(796\) −17.2752 −0.612304
\(797\) 10.0792 0.357023 0.178512 0.983938i \(-0.442872\pi\)
0.178512 + 0.983938i \(0.442872\pi\)
\(798\) 5.68910 0.201392
\(799\) 17.0523 0.603268
\(800\) −2.15533 −0.0762024
\(801\) −60.6854 −2.14421
\(802\) −30.7638 −1.08631
\(803\) −0.419128 −0.0147907
\(804\) 27.0106 0.952589
\(805\) 5.41520 0.190861
\(806\) 15.9987 0.563529
\(807\) −11.8445 −0.416947
\(808\) −6.47702 −0.227861
\(809\) −45.9563 −1.61574 −0.807869 0.589362i \(-0.799379\pi\)
−0.807869 + 0.589362i \(0.799379\pi\)
\(810\) −0.902360 −0.0317057
\(811\) −23.7415 −0.833677 −0.416838 0.908981i \(-0.636862\pi\)
−0.416838 + 0.908981i \(0.636862\pi\)
\(812\) −1.73382 −0.0608453
\(813\) −23.6109 −0.828070
\(814\) 4.04063 0.141624
\(815\) 29.4186 1.03049
\(816\) −4.59543 −0.160872
\(817\) 4.38249 0.153324
\(818\) 28.3717 0.991994
\(819\) −12.6889 −0.443387
\(820\) 11.4391 0.399469
\(821\) 4.39466 0.153375 0.0766873 0.997055i \(-0.475566\pi\)
0.0766873 + 0.997055i \(0.475566\pi\)
\(822\) −19.4977 −0.680061
\(823\) 4.72146 0.164580 0.0822898 0.996608i \(-0.473777\pi\)
0.0822898 + 0.996608i \(0.473777\pi\)
\(824\) −3.67441 −0.128004
\(825\) 13.2627 0.461749
\(826\) −3.07331 −0.106934
\(827\) −45.0152 −1.56533 −0.782666 0.622442i \(-0.786141\pi\)
−0.782666 + 0.622442i \(0.786141\pi\)
\(828\) −32.0587 −1.11412
\(829\) −10.8984 −0.378516 −0.189258 0.981927i \(-0.560608\pi\)
−0.189258 + 0.981927i \(0.560608\pi\)
\(830\) 9.20134 0.319383
\(831\) −63.7130 −2.21018
\(832\) 5.56059 0.192779
\(833\) −11.1610 −0.386706
\(834\) −0.917674 −0.0317765
\(835\) −7.36224 −0.254781
\(836\) 9.42053 0.325816
\(837\) −14.2255 −0.491707
\(838\) −14.8052 −0.511435
\(839\) −4.15441 −0.143426 −0.0717130 0.997425i \(-0.522847\pi\)
−0.0717130 + 0.997425i \(0.522847\pi\)
\(840\) 2.24801 0.0775639
\(841\) −15.8461 −0.546418
\(842\) 34.4115 1.18590
\(843\) −8.28288 −0.285278
\(844\) 4.04319 0.139172
\(845\) 30.2244 1.03975
\(846\) 49.3842 1.69787
\(847\) 2.92994 0.100674
\(848\) 12.9134 0.443447
\(849\) −68.9770 −2.36728
\(850\) −3.55251 −0.121850
\(851\) −12.2957 −0.421492
\(852\) −12.1572 −0.416498
\(853\) −38.8860 −1.33143 −0.665716 0.746205i \(-0.731874\pi\)
−0.665716 + 0.746205i \(0.731874\pi\)
\(854\) −2.07259 −0.0709227
\(855\) 34.3639 1.17522
\(856\) −16.3658 −0.559372
\(857\) 24.8510 0.848894 0.424447 0.905453i \(-0.360469\pi\)
0.424447 + 0.905453i \(0.360469\pi\)
\(858\) −34.2169 −1.16815
\(859\) 27.4246 0.935714 0.467857 0.883804i \(-0.345026\pi\)
0.467857 + 0.883804i \(0.345026\pi\)
\(860\) 1.73172 0.0590510
\(861\) 9.03977 0.308075
\(862\) −2.03361 −0.0692650
\(863\) 21.3701 0.727448 0.363724 0.931507i \(-0.381505\pi\)
0.363724 + 0.931507i \(0.381505\pi\)
\(864\) −4.94431 −0.168209
\(865\) −4.53211 −0.154096
\(866\) −19.3295 −0.656842
\(867\) 39.8229 1.35246
\(868\) −1.37544 −0.0466855
\(869\) −26.5473 −0.900555
\(870\) −17.0548 −0.578214
\(871\) −53.8703 −1.82533
\(872\) −0.447873 −0.0151669
\(873\) −23.9084 −0.809176
\(874\) −28.6669 −0.969672
\(875\) 5.76931 0.195038
\(876\) 0.529464 0.0178889
\(877\) 28.4297 0.960002 0.480001 0.877268i \(-0.340636\pi\)
0.480001 + 0.877268i \(0.340636\pi\)
\(878\) −15.0399 −0.507574
\(879\) −15.8770 −0.535519
\(880\) 3.72247 0.125484
\(881\) −13.5102 −0.455172 −0.227586 0.973758i \(-0.573083\pi\)
−0.227586 + 0.973758i \(0.573083\pi\)
\(882\) −32.3227 −1.08836
\(883\) −23.4769 −0.790061 −0.395030 0.918668i \(-0.629266\pi\)
−0.395030 + 0.918668i \(0.629266\pi\)
\(884\) 9.16520 0.308259
\(885\) −30.2308 −1.01620
\(886\) 8.06470 0.270939
\(887\) −15.2173 −0.510947 −0.255474 0.966816i \(-0.582231\pi\)
−0.255474 + 0.966816i \(0.582231\pi\)
\(888\) −5.10433 −0.171290
\(889\) 2.80771 0.0941676
\(890\) −21.4424 −0.718752
\(891\) −1.18081 −0.0395585
\(892\) 4.68255 0.156783
\(893\) 44.1594 1.47774
\(894\) −39.9936 −1.33759
\(895\) 11.7806 0.393783
\(896\) −0.478055 −0.0159707
\(897\) 104.123 3.47656
\(898\) −30.5246 −1.01862
\(899\) 10.4349 0.348025
\(900\) −10.2882 −0.342940
\(901\) 21.2844 0.709085
\(902\) 14.9689 0.498409
\(903\) 1.36850 0.0455407
\(904\) 2.96760 0.0987008
\(905\) 13.7099 0.455734
\(906\) 55.6082 1.84746
\(907\) −50.0011 −1.66026 −0.830130 0.557569i \(-0.811734\pi\)
−0.830130 + 0.557569i \(0.811734\pi\)
\(908\) 21.4011 0.710221
\(909\) −30.9173 −1.02546
\(910\) −4.48348 −0.148626
\(911\) −44.2728 −1.46682 −0.733412 0.679784i \(-0.762074\pi\)
−0.733412 + 0.679784i \(0.762074\pi\)
\(912\) −11.9005 −0.394065
\(913\) 12.0407 0.398488
\(914\) 35.7756 1.18335
\(915\) −20.3872 −0.673979
\(916\) 12.6702 0.418634
\(917\) 7.92558 0.261726
\(918\) −8.14942 −0.268971
\(919\) 29.2871 0.966092 0.483046 0.875595i \(-0.339530\pi\)
0.483046 + 0.875595i \(0.339530\pi\)
\(920\) −11.3276 −0.373458
\(921\) 4.03046 0.132808
\(922\) 33.8197 1.11379
\(923\) 24.2465 0.798082
\(924\) 2.94170 0.0967748
\(925\) −3.94592 −0.129741
\(926\) −26.5118 −0.871233
\(927\) −17.5394 −0.576068
\(928\) 3.62683 0.119056
\(929\) −24.3930 −0.800310 −0.400155 0.916448i \(-0.631044\pi\)
−0.400155 + 0.916448i \(0.631044\pi\)
\(930\) −13.5296 −0.443652
\(931\) −28.9030 −0.947257
\(932\) 2.85283 0.0934476
\(933\) 70.4426 2.30619
\(934\) 23.1866 0.758687
\(935\) 6.13553 0.200653
\(936\) 26.5428 0.867578
\(937\) −7.48744 −0.244604 −0.122302 0.992493i \(-0.539028\pi\)
−0.122302 + 0.992493i \(0.539028\pi\)
\(938\) 4.63134 0.151219
\(939\) −71.7466 −2.34136
\(940\) 17.4493 0.569134
\(941\) −46.2791 −1.50866 −0.754328 0.656498i \(-0.772037\pi\)
−0.754328 + 0.656498i \(0.772037\pi\)
\(942\) −18.5499 −0.604388
\(943\) −45.5507 −1.48333
\(944\) 6.42878 0.209239
\(945\) 3.98657 0.129683
\(946\) 2.26608 0.0736766
\(947\) 19.5323 0.634713 0.317357 0.948306i \(-0.397205\pi\)
0.317357 + 0.948306i \(0.397205\pi\)
\(948\) 33.5359 1.08920
\(949\) −1.05597 −0.0342783
\(950\) −9.19970 −0.298478
\(951\) 5.60847 0.181867
\(952\) −0.787952 −0.0255377
\(953\) 0.246261 0.00797717 0.00398859 0.999992i \(-0.498730\pi\)
0.00398859 + 0.999992i \(0.498730\pi\)
\(954\) 61.6403 1.99568
\(955\) −11.5371 −0.373332
\(956\) −13.6057 −0.440040
\(957\) −22.3176 −0.721424
\(958\) −24.2960 −0.784968
\(959\) −3.34316 −0.107956
\(960\) −4.70241 −0.151770
\(961\) −22.7220 −0.732967
\(962\) 10.1802 0.328222
\(963\) −78.1202 −2.51739
\(964\) 16.2213 0.522454
\(965\) 34.6178 1.11438
\(966\) −8.95165 −0.288015
\(967\) 9.27650 0.298312 0.149156 0.988814i \(-0.452344\pi\)
0.149156 + 0.988814i \(0.452344\pi\)
\(968\) −6.12887 −0.196989
\(969\) −19.6149 −0.630122
\(970\) −8.44774 −0.271241
\(971\) −33.1115 −1.06260 −0.531299 0.847185i \(-0.678296\pi\)
−0.531299 + 0.847185i \(0.678296\pi\)
\(972\) 16.3246 0.523611
\(973\) −0.157348 −0.00504435
\(974\) 16.6398 0.533173
\(975\) 33.4148 1.07013
\(976\) 4.33547 0.138775
\(977\) 35.0471 1.12126 0.560629 0.828067i \(-0.310560\pi\)
0.560629 + 0.828067i \(0.310560\pi\)
\(978\) −48.6307 −1.55504
\(979\) −28.0591 −0.896772
\(980\) −11.4209 −0.364826
\(981\) −2.13786 −0.0682567
\(982\) 6.51985 0.208057
\(983\) −26.8722 −0.857091 −0.428546 0.903520i \(-0.640974\pi\)
−0.428546 + 0.903520i \(0.640974\pi\)
\(984\) −18.9095 −0.602812
\(985\) 39.8208 1.26880
\(986\) 5.97789 0.190375
\(987\) 13.7894 0.438922
\(988\) 23.7346 0.755097
\(989\) −6.89573 −0.219272
\(990\) 17.7687 0.564728
\(991\) 49.6279 1.57648 0.788241 0.615367i \(-0.210992\pi\)
0.788241 + 0.615367i \(0.210992\pi\)
\(992\) 2.87716 0.0913498
\(993\) 48.9618 1.55376
\(994\) −2.08452 −0.0661170
\(995\) −29.1366 −0.923694
\(996\) −15.2104 −0.481959
\(997\) −5.68832 −0.180151 −0.0900755 0.995935i \(-0.528711\pi\)
−0.0900755 + 0.995935i \(0.528711\pi\)
\(998\) 9.47873 0.300044
\(999\) −9.05190 −0.286390
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 8042.2.a.d.1.11 101
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8042.2.a.d.1.11 101 1.1 even 1 trivial