Properties

Label 8038.2.a.a.1.14
Level $8038$
Weight $2$
Character 8038.1
Self dual yes
Analytic conductor $64.184$
Analytic rank $1$
Dimension $75$
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] = [8038,2,Mod(1,8038)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8038, 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("8038.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8038 = 2 \cdot 4019 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8038.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: \(64.1837531447\)
Analytic rank: \(1\)
Dimension: \(75\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.14
Character \(\chi\) \(=\) 8038.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.46931 q^{3} +1.00000 q^{4} +0.696459 q^{5} -2.46931 q^{6} -2.94189 q^{7} +1.00000 q^{8} +3.09749 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -2.46931 q^{3} +1.00000 q^{4} +0.696459 q^{5} -2.46931 q^{6} -2.94189 q^{7} +1.00000 q^{8} +3.09749 q^{9} +0.696459 q^{10} +6.47128 q^{11} -2.46931 q^{12} +0.247100 q^{13} -2.94189 q^{14} -1.71977 q^{15} +1.00000 q^{16} -4.32305 q^{17} +3.09749 q^{18} +5.40761 q^{19} +0.696459 q^{20} +7.26444 q^{21} +6.47128 q^{22} -7.44793 q^{23} -2.46931 q^{24} -4.51494 q^{25} +0.247100 q^{26} -0.240730 q^{27} -2.94189 q^{28} -9.62171 q^{29} -1.71977 q^{30} +1.40003 q^{31} +1.00000 q^{32} -15.9796 q^{33} -4.32305 q^{34} -2.04891 q^{35} +3.09749 q^{36} +10.6398 q^{37} +5.40761 q^{38} -0.610166 q^{39} +0.696459 q^{40} -0.621738 q^{41} +7.26444 q^{42} -1.89591 q^{43} +6.47128 q^{44} +2.15727 q^{45} -7.44793 q^{46} +2.21454 q^{47} -2.46931 q^{48} +1.65472 q^{49} -4.51494 q^{50} +10.6749 q^{51} +0.247100 q^{52} -5.99737 q^{53} -0.240730 q^{54} +4.50698 q^{55} -2.94189 q^{56} -13.3531 q^{57} -9.62171 q^{58} -3.90307 q^{59} -1.71977 q^{60} +9.72924 q^{61} +1.40003 q^{62} -9.11247 q^{63} +1.00000 q^{64} +0.172095 q^{65} -15.9796 q^{66} -7.84645 q^{67} -4.32305 q^{68} +18.3913 q^{69} -2.04891 q^{70} -8.12981 q^{71} +3.09749 q^{72} +9.56163 q^{73} +10.6398 q^{74} +11.1488 q^{75} +5.40761 q^{76} -19.0378 q^{77} -0.610166 q^{78} +13.6598 q^{79} +0.696459 q^{80} -8.69803 q^{81} -0.621738 q^{82} -13.2203 q^{83} +7.26444 q^{84} -3.01083 q^{85} -1.89591 q^{86} +23.7590 q^{87} +6.47128 q^{88} +5.90688 q^{89} +2.15727 q^{90} -0.726941 q^{91} -7.44793 q^{92} -3.45710 q^{93} +2.21454 q^{94} +3.76618 q^{95} -2.46931 q^{96} +4.65460 q^{97} +1.65472 q^{98} +20.0447 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 75 q + 75 q^{2} - 30 q^{3} + 75 q^{4} - 29 q^{5} - 30 q^{6} - 31 q^{7} + 75 q^{8} + 55 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 75 q + 75 q^{2} - 30 q^{3} + 75 q^{4} - 29 q^{5} - 30 q^{6} - 31 q^{7} + 75 q^{8} + 55 q^{9} - 29 q^{10} - 30 q^{11} - 30 q^{12} - 23 q^{13} - 31 q^{14} - 22 q^{15} + 75 q^{16} - 48 q^{17} + 55 q^{18} - 58 q^{19} - 29 q^{20} - 5 q^{21} - 30 q^{22} - 79 q^{23} - 30 q^{24} + 42 q^{25} - 23 q^{26} - 108 q^{27} - 31 q^{28} - 39 q^{29} - 22 q^{30} - 95 q^{31} + 75 q^{32} - 44 q^{33} - 48 q^{34} - 60 q^{35} + 55 q^{36} - 16 q^{37} - 58 q^{38} - 57 q^{39} - 29 q^{40} - 85 q^{41} - 5 q^{42} - 55 q^{43} - 30 q^{44} - 48 q^{45} - 79 q^{46} - 88 q^{47} - 30 q^{48} + 26 q^{49} + 42 q^{50} - 39 q^{51} - 23 q^{52} - 79 q^{53} - 108 q^{54} - 78 q^{55} - 31 q^{56} - 40 q^{57} - 39 q^{58} - 74 q^{59} - 22 q^{60} - 21 q^{61} - 95 q^{62} - 97 q^{63} + 75 q^{64} - 63 q^{65} - 44 q^{66} - 59 q^{67} - 48 q^{68} + 3 q^{69} - 60 q^{70} - 72 q^{71} + 55 q^{72} - 91 q^{73} - 16 q^{74} - 95 q^{75} - 58 q^{76} - 60 q^{77} - 57 q^{78} - 64 q^{79} - 29 q^{80} + 47 q^{81} - 85 q^{82} - 105 q^{83} - 5 q^{84} + 14 q^{85} - 55 q^{86} - 75 q^{87} - 30 q^{88} - 78 q^{89} - 48 q^{90} - 89 q^{91} - 79 q^{92} + 27 q^{93} - 88 q^{94} - 53 q^{95} - 30 q^{96} - 79 q^{97} + 26 q^{98} - 57 q^{99}+O(q^{100}) \) 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
\(3\) −2.46931 −1.42566 −0.712828 0.701339i \(-0.752586\pi\)
−0.712828 + 0.701339i \(0.752586\pi\)
\(4\) 1.00000 0.500000
\(5\) 0.696459 0.311466 0.155733 0.987799i \(-0.450226\pi\)
0.155733 + 0.987799i \(0.450226\pi\)
\(6\) −2.46931 −1.00809
\(7\) −2.94189 −1.11193 −0.555965 0.831206i \(-0.687651\pi\)
−0.555965 + 0.831206i \(0.687651\pi\)
\(8\) 1.00000 0.353553
\(9\) 3.09749 1.03250
\(10\) 0.696459 0.220240
\(11\) 6.47128 1.95116 0.975582 0.219637i \(-0.0704872\pi\)
0.975582 + 0.219637i \(0.0704872\pi\)
\(12\) −2.46931 −0.712828
\(13\) 0.247100 0.0685332 0.0342666 0.999413i \(-0.489090\pi\)
0.0342666 + 0.999413i \(0.489090\pi\)
\(14\) −2.94189 −0.786253
\(15\) −1.71977 −0.444044
\(16\) 1.00000 0.250000
\(17\) −4.32305 −1.04849 −0.524247 0.851567i \(-0.675653\pi\)
−0.524247 + 0.851567i \(0.675653\pi\)
\(18\) 3.09749 0.730085
\(19\) 5.40761 1.24059 0.620296 0.784368i \(-0.287013\pi\)
0.620296 + 0.784368i \(0.287013\pi\)
\(20\) 0.696459 0.155733
\(21\) 7.26444 1.58523
\(22\) 6.47128 1.37968
\(23\) −7.44793 −1.55300 −0.776501 0.630116i \(-0.783007\pi\)
−0.776501 + 0.630116i \(0.783007\pi\)
\(24\) −2.46931 −0.504046
\(25\) −4.51494 −0.902989
\(26\) 0.247100 0.0484603
\(27\) −0.240730 −0.0463284
\(28\) −2.94189 −0.555965
\(29\) −9.62171 −1.78671 −0.893353 0.449355i \(-0.851654\pi\)
−0.893353 + 0.449355i \(0.851654\pi\)
\(30\) −1.71977 −0.313986
\(31\) 1.40003 0.251452 0.125726 0.992065i \(-0.459874\pi\)
0.125726 + 0.992065i \(0.459874\pi\)
\(32\) 1.00000 0.176777
\(33\) −15.9796 −2.78169
\(34\) −4.32305 −0.741397
\(35\) −2.04891 −0.346328
\(36\) 3.09749 0.516248
\(37\) 10.6398 1.74917 0.874584 0.484874i \(-0.161135\pi\)
0.874584 + 0.484874i \(0.161135\pi\)
\(38\) 5.40761 0.877230
\(39\) −0.610166 −0.0977048
\(40\) 0.696459 0.110120
\(41\) −0.621738 −0.0970991 −0.0485495 0.998821i \(-0.515460\pi\)
−0.0485495 + 0.998821i \(0.515460\pi\)
\(42\) 7.26444 1.12093
\(43\) −1.89591 −0.289123 −0.144562 0.989496i \(-0.546177\pi\)
−0.144562 + 0.989496i \(0.546177\pi\)
\(44\) 6.47128 0.975582
\(45\) 2.15727 0.321588
\(46\) −7.44793 −1.09814
\(47\) 2.21454 0.323024 0.161512 0.986871i \(-0.448363\pi\)
0.161512 + 0.986871i \(0.448363\pi\)
\(48\) −2.46931 −0.356414
\(49\) 1.65472 0.236389
\(50\) −4.51494 −0.638510
\(51\) 10.6749 1.49479
\(52\) 0.247100 0.0342666
\(53\) −5.99737 −0.823802 −0.411901 0.911229i \(-0.635135\pi\)
−0.411901 + 0.911229i \(0.635135\pi\)
\(54\) −0.240730 −0.0327591
\(55\) 4.50698 0.607721
\(56\) −2.94189 −0.393127
\(57\) −13.3531 −1.76866
\(58\) −9.62171 −1.26339
\(59\) −3.90307 −0.508137 −0.254068 0.967186i \(-0.581769\pi\)
−0.254068 + 0.967186i \(0.581769\pi\)
\(60\) −1.71977 −0.222022
\(61\) 9.72924 1.24570 0.622851 0.782341i \(-0.285974\pi\)
0.622851 + 0.782341i \(0.285974\pi\)
\(62\) 1.40003 0.177803
\(63\) −9.11247 −1.14806
\(64\) 1.00000 0.125000
\(65\) 0.172095 0.0213458
\(66\) −15.9796 −1.96695
\(67\) −7.84645 −0.958597 −0.479298 0.877652i \(-0.659109\pi\)
−0.479298 + 0.877652i \(0.659109\pi\)
\(68\) −4.32305 −0.524247
\(69\) 18.3913 2.21405
\(70\) −2.04891 −0.244891
\(71\) −8.12981 −0.964831 −0.482415 0.875943i \(-0.660240\pi\)
−0.482415 + 0.875943i \(0.660240\pi\)
\(72\) 3.09749 0.365043
\(73\) 9.56163 1.11910 0.559552 0.828795i \(-0.310973\pi\)
0.559552 + 0.828795i \(0.310973\pi\)
\(74\) 10.6398 1.23685
\(75\) 11.1488 1.28735
\(76\) 5.40761 0.620296
\(77\) −19.0378 −2.16956
\(78\) −0.610166 −0.0690878
\(79\) 13.6598 1.53685 0.768424 0.639941i \(-0.221041\pi\)
0.768424 + 0.639941i \(0.221041\pi\)
\(80\) 0.696459 0.0778665
\(81\) −8.69803 −0.966448
\(82\) −0.621738 −0.0686594
\(83\) −13.2203 −1.45112 −0.725559 0.688160i \(-0.758419\pi\)
−0.725559 + 0.688160i \(0.758419\pi\)
\(84\) 7.26444 0.792615
\(85\) −3.01083 −0.326570
\(86\) −1.89591 −0.204441
\(87\) 23.7590 2.54723
\(88\) 6.47128 0.689840
\(89\) 5.90688 0.626128 0.313064 0.949732i \(-0.398645\pi\)
0.313064 + 0.949732i \(0.398645\pi\)
\(90\) 2.15727 0.227397
\(91\) −0.726941 −0.0762042
\(92\) −7.44793 −0.776501
\(93\) −3.45710 −0.358484
\(94\) 2.21454 0.228412
\(95\) 3.76618 0.386402
\(96\) −2.46931 −0.252023
\(97\) 4.65460 0.472603 0.236302 0.971680i \(-0.424065\pi\)
0.236302 + 0.971680i \(0.424065\pi\)
\(98\) 1.65472 0.167152
\(99\) 20.0447 2.01457
\(100\) −4.51494 −0.451494
\(101\) −13.8748 −1.38060 −0.690299 0.723524i \(-0.742521\pi\)
−0.690299 + 0.723524i \(0.742521\pi\)
\(102\) 10.6749 1.05698
\(103\) −3.48612 −0.343498 −0.171749 0.985141i \(-0.554942\pi\)
−0.171749 + 0.985141i \(0.554942\pi\)
\(104\) 0.247100 0.0242302
\(105\) 5.05939 0.493745
\(106\) −5.99737 −0.582516
\(107\) −10.7045 −1.03484 −0.517419 0.855732i \(-0.673107\pi\)
−0.517419 + 0.855732i \(0.673107\pi\)
\(108\) −0.240730 −0.0231642
\(109\) 17.3569 1.66249 0.831247 0.555904i \(-0.187628\pi\)
0.831247 + 0.555904i \(0.187628\pi\)
\(110\) 4.50698 0.429724
\(111\) −26.2729 −2.49371
\(112\) −2.94189 −0.277983
\(113\) 12.1686 1.14473 0.572364 0.820000i \(-0.306026\pi\)
0.572364 + 0.820000i \(0.306026\pi\)
\(114\) −13.3531 −1.25063
\(115\) −5.18718 −0.483707
\(116\) −9.62171 −0.893353
\(117\) 0.765390 0.0707603
\(118\) −3.90307 −0.359307
\(119\) 12.7179 1.16585
\(120\) −1.71977 −0.156993
\(121\) 30.8774 2.80704
\(122\) 9.72924 0.880844
\(123\) 1.53526 0.138430
\(124\) 1.40003 0.125726
\(125\) −6.62677 −0.592716
\(126\) −9.11247 −0.811804
\(127\) 2.44248 0.216735 0.108368 0.994111i \(-0.465438\pi\)
0.108368 + 0.994111i \(0.465438\pi\)
\(128\) 1.00000 0.0883883
\(129\) 4.68158 0.412190
\(130\) 0.172095 0.0150937
\(131\) −12.9309 −1.12978 −0.564890 0.825166i \(-0.691081\pi\)
−0.564890 + 0.825166i \(0.691081\pi\)
\(132\) −15.9796 −1.39084
\(133\) −15.9086 −1.37945
\(134\) −7.84645 −0.677830
\(135\) −0.167658 −0.0144297
\(136\) −4.32305 −0.370698
\(137\) −13.9638 −1.19301 −0.596506 0.802609i \(-0.703445\pi\)
−0.596506 + 0.802609i \(0.703445\pi\)
\(138\) 18.3913 1.56557
\(139\) −11.7947 −1.00041 −0.500207 0.865906i \(-0.666743\pi\)
−0.500207 + 0.865906i \(0.666743\pi\)
\(140\) −2.04891 −0.173164
\(141\) −5.46839 −0.460521
\(142\) −8.12981 −0.682238
\(143\) 1.59905 0.133720
\(144\) 3.09749 0.258124
\(145\) −6.70113 −0.556498
\(146\) 9.56163 0.791326
\(147\) −4.08602 −0.337009
\(148\) 10.6398 0.874584
\(149\) 15.3095 1.25420 0.627102 0.778937i \(-0.284241\pi\)
0.627102 + 0.778937i \(0.284241\pi\)
\(150\) 11.1488 0.910295
\(151\) 19.2314 1.56503 0.782514 0.622633i \(-0.213937\pi\)
0.782514 + 0.622633i \(0.213937\pi\)
\(152\) 5.40761 0.438615
\(153\) −13.3906 −1.08257
\(154\) −19.0378 −1.53411
\(155\) 0.975061 0.0783188
\(156\) −0.610166 −0.0488524
\(157\) −4.25207 −0.339352 −0.169676 0.985500i \(-0.554272\pi\)
−0.169676 + 0.985500i \(0.554272\pi\)
\(158\) 13.6598 1.08672
\(159\) 14.8094 1.17446
\(160\) 0.696459 0.0550599
\(161\) 21.9110 1.72683
\(162\) −8.69803 −0.683382
\(163\) 20.9568 1.64146 0.820732 0.571314i \(-0.193566\pi\)
0.820732 + 0.571314i \(0.193566\pi\)
\(164\) −0.621738 −0.0485495
\(165\) −11.1291 −0.866402
\(166\) −13.2203 −1.02609
\(167\) −10.0101 −0.774604 −0.387302 0.921953i \(-0.626593\pi\)
−0.387302 + 0.921953i \(0.626593\pi\)
\(168\) 7.26444 0.560464
\(169\) −12.9389 −0.995303
\(170\) −3.01083 −0.230920
\(171\) 16.7500 1.28091
\(172\) −1.89591 −0.144562
\(173\) 6.49479 0.493789 0.246895 0.969042i \(-0.420590\pi\)
0.246895 + 0.969042i \(0.420590\pi\)
\(174\) 23.7590 1.80116
\(175\) 13.2825 1.00406
\(176\) 6.47128 0.487791
\(177\) 9.63789 0.724428
\(178\) 5.90688 0.442740
\(179\) −3.91627 −0.292716 −0.146358 0.989232i \(-0.546755\pi\)
−0.146358 + 0.989232i \(0.546755\pi\)
\(180\) 2.15727 0.160794
\(181\) −19.7376 −1.46708 −0.733542 0.679644i \(-0.762134\pi\)
−0.733542 + 0.679644i \(0.762134\pi\)
\(182\) −0.726941 −0.0538845
\(183\) −24.0245 −1.77594
\(184\) −7.44793 −0.549069
\(185\) 7.41017 0.544806
\(186\) −3.45710 −0.253487
\(187\) −27.9756 −2.04578
\(188\) 2.21454 0.161512
\(189\) 0.708200 0.0515140
\(190\) 3.76618 0.273227
\(191\) −6.28499 −0.454766 −0.227383 0.973805i \(-0.573017\pi\)
−0.227383 + 0.973805i \(0.573017\pi\)
\(192\) −2.46931 −0.178207
\(193\) 4.98373 0.358737 0.179369 0.983782i \(-0.442595\pi\)
0.179369 + 0.983782i \(0.442595\pi\)
\(194\) 4.65460 0.334181
\(195\) −0.424956 −0.0304317
\(196\) 1.65472 0.118194
\(197\) −8.89301 −0.633601 −0.316800 0.948492i \(-0.602609\pi\)
−0.316800 + 0.948492i \(0.602609\pi\)
\(198\) 20.0447 1.42452
\(199\) −5.48201 −0.388609 −0.194305 0.980941i \(-0.562245\pi\)
−0.194305 + 0.980941i \(0.562245\pi\)
\(200\) −4.51494 −0.319255
\(201\) 19.3753 1.36663
\(202\) −13.8748 −0.976230
\(203\) 28.3060 1.98669
\(204\) 10.6749 0.747396
\(205\) −0.433015 −0.0302431
\(206\) −3.48612 −0.242890
\(207\) −23.0699 −1.60347
\(208\) 0.247100 0.0171333
\(209\) 34.9941 2.42060
\(210\) 5.05939 0.349131
\(211\) −23.9300 −1.64741 −0.823705 0.567018i \(-0.808097\pi\)
−0.823705 + 0.567018i \(0.808097\pi\)
\(212\) −5.99737 −0.411901
\(213\) 20.0750 1.37552
\(214\) −10.7045 −0.731742
\(215\) −1.32042 −0.0900521
\(216\) −0.240730 −0.0163796
\(217\) −4.11872 −0.279597
\(218\) 17.3569 1.17556
\(219\) −23.6106 −1.59546
\(220\) 4.50698 0.303861
\(221\) −1.06823 −0.0718566
\(222\) −26.2729 −1.76332
\(223\) −18.6303 −1.24758 −0.623789 0.781593i \(-0.714407\pi\)
−0.623789 + 0.781593i \(0.714407\pi\)
\(224\) −2.94189 −0.196563
\(225\) −13.9850 −0.932333
\(226\) 12.1686 0.809445
\(227\) −11.4522 −0.760108 −0.380054 0.924964i \(-0.624095\pi\)
−0.380054 + 0.924964i \(0.624095\pi\)
\(228\) −13.3531 −0.884328
\(229\) 8.16991 0.539883 0.269942 0.962877i \(-0.412996\pi\)
0.269942 + 0.962877i \(0.412996\pi\)
\(230\) −5.18718 −0.342033
\(231\) 47.0102 3.09304
\(232\) −9.62171 −0.631696
\(233\) −29.0524 −1.90328 −0.951642 0.307209i \(-0.900605\pi\)
−0.951642 + 0.307209i \(0.900605\pi\)
\(234\) 0.765390 0.0500351
\(235\) 1.54234 0.100611
\(236\) −3.90307 −0.254068
\(237\) −33.7303 −2.19102
\(238\) 12.7179 0.824381
\(239\) 6.29282 0.407049 0.203524 0.979070i \(-0.434760\pi\)
0.203524 + 0.979070i \(0.434760\pi\)
\(240\) −1.71977 −0.111011
\(241\) 1.21320 0.0781493 0.0390746 0.999236i \(-0.487559\pi\)
0.0390746 + 0.999236i \(0.487559\pi\)
\(242\) 30.8774 1.98488
\(243\) 22.2003 1.42415
\(244\) 9.72924 0.622851
\(245\) 1.15245 0.0736271
\(246\) 1.53526 0.0978847
\(247\) 1.33622 0.0850217
\(248\) 1.40003 0.0889017
\(249\) 32.6450 2.06879
\(250\) −6.62677 −0.419114
\(251\) −30.7423 −1.94044 −0.970218 0.242234i \(-0.922120\pi\)
−0.970218 + 0.242234i \(0.922120\pi\)
\(252\) −9.11247 −0.574032
\(253\) −48.1976 −3.03016
\(254\) 2.44248 0.153255
\(255\) 7.43466 0.465577
\(256\) 1.00000 0.0625000
\(257\) −22.7955 −1.42194 −0.710971 0.703222i \(-0.751744\pi\)
−0.710971 + 0.703222i \(0.751744\pi\)
\(258\) 4.68158 0.291463
\(259\) −31.3010 −1.94495
\(260\) 0.172095 0.0106729
\(261\) −29.8031 −1.84477
\(262\) −12.9309 −0.798875
\(263\) −18.6455 −1.14973 −0.574867 0.818247i \(-0.694946\pi\)
−0.574867 + 0.818247i \(0.694946\pi\)
\(264\) −15.9796 −0.983475
\(265\) −4.17692 −0.256586
\(266\) −15.9086 −0.975419
\(267\) −14.5859 −0.892644
\(268\) −7.84645 −0.479298
\(269\) −6.31721 −0.385167 −0.192584 0.981281i \(-0.561687\pi\)
−0.192584 + 0.981281i \(0.561687\pi\)
\(270\) −0.167658 −0.0102034
\(271\) −21.6233 −1.31352 −0.656759 0.754100i \(-0.728073\pi\)
−0.656759 + 0.754100i \(0.728073\pi\)
\(272\) −4.32305 −0.262123
\(273\) 1.79504 0.108641
\(274\) −13.9638 −0.843587
\(275\) −29.2175 −1.76188
\(276\) 18.3913 1.10702
\(277\) −18.4094 −1.10611 −0.553057 0.833144i \(-0.686539\pi\)
−0.553057 + 0.833144i \(0.686539\pi\)
\(278\) −11.7947 −0.707400
\(279\) 4.33656 0.259623
\(280\) −2.04891 −0.122446
\(281\) −16.6219 −0.991577 −0.495788 0.868443i \(-0.665121\pi\)
−0.495788 + 0.868443i \(0.665121\pi\)
\(282\) −5.46839 −0.325638
\(283\) 3.84401 0.228503 0.114251 0.993452i \(-0.463553\pi\)
0.114251 + 0.993452i \(0.463553\pi\)
\(284\) −8.12981 −0.482415
\(285\) −9.29987 −0.550876
\(286\) 1.59905 0.0945540
\(287\) 1.82908 0.107967
\(288\) 3.09749 0.182521
\(289\) 1.68875 0.0993380
\(290\) −6.70113 −0.393504
\(291\) −11.4937 −0.673770
\(292\) 9.56163 0.559552
\(293\) 31.2539 1.82587 0.912936 0.408103i \(-0.133809\pi\)
0.912936 + 0.408103i \(0.133809\pi\)
\(294\) −4.08602 −0.238301
\(295\) −2.71833 −0.158267
\(296\) 10.6398 0.618424
\(297\) −1.55783 −0.0903943
\(298\) 15.3095 0.886856
\(299\) −1.84038 −0.106432
\(300\) 11.1488 0.643676
\(301\) 5.57755 0.321485
\(302\) 19.2314 1.10664
\(303\) 34.2613 1.96826
\(304\) 5.40761 0.310148
\(305\) 6.77602 0.387994
\(306\) −13.3906 −0.765489
\(307\) 7.83720 0.447293 0.223646 0.974670i \(-0.428204\pi\)
0.223646 + 0.974670i \(0.428204\pi\)
\(308\) −19.0378 −1.08478
\(309\) 8.60832 0.489710
\(310\) 0.975061 0.0553798
\(311\) −29.8325 −1.69165 −0.845824 0.533462i \(-0.820891\pi\)
−0.845824 + 0.533462i \(0.820891\pi\)
\(312\) −0.610166 −0.0345439
\(313\) 16.7816 0.948551 0.474276 0.880376i \(-0.342710\pi\)
0.474276 + 0.880376i \(0.342710\pi\)
\(314\) −4.25207 −0.239958
\(315\) −6.34647 −0.357583
\(316\) 13.6598 0.768424
\(317\) −10.5987 −0.595284 −0.297642 0.954678i \(-0.596200\pi\)
−0.297642 + 0.954678i \(0.596200\pi\)
\(318\) 14.8094 0.830467
\(319\) −62.2647 −3.48616
\(320\) 0.696459 0.0389333
\(321\) 26.4326 1.47532
\(322\) 21.9110 1.22105
\(323\) −23.3774 −1.30075
\(324\) −8.69803 −0.483224
\(325\) −1.11564 −0.0618847
\(326\) 20.9568 1.16069
\(327\) −42.8596 −2.37014
\(328\) −0.621738 −0.0343297
\(329\) −6.51494 −0.359180
\(330\) −11.1291 −0.612638
\(331\) 35.8218 1.96894 0.984472 0.175541i \(-0.0561676\pi\)
0.984472 + 0.175541i \(0.0561676\pi\)
\(332\) −13.2203 −0.725559
\(333\) 32.9566 1.80601
\(334\) −10.0101 −0.547728
\(335\) −5.46473 −0.298570
\(336\) 7.26444 0.396308
\(337\) 5.65042 0.307798 0.153899 0.988087i \(-0.450817\pi\)
0.153899 + 0.988087i \(0.450817\pi\)
\(338\) −12.9389 −0.703786
\(339\) −30.0481 −1.63199
\(340\) −3.01083 −0.163285
\(341\) 9.05996 0.490624
\(342\) 16.7500 0.905737
\(343\) 15.7252 0.849082
\(344\) −1.89591 −0.102221
\(345\) 12.8088 0.689600
\(346\) 6.49479 0.349162
\(347\) −6.06311 −0.325485 −0.162742 0.986669i \(-0.552034\pi\)
−0.162742 + 0.986669i \(0.552034\pi\)
\(348\) 23.7590 1.27361
\(349\) 11.9537 0.639864 0.319932 0.947440i \(-0.396340\pi\)
0.319932 + 0.947440i \(0.396340\pi\)
\(350\) 13.2825 0.709978
\(351\) −0.0594843 −0.00317504
\(352\) 6.47128 0.344920
\(353\) −26.7512 −1.42382 −0.711911 0.702270i \(-0.752170\pi\)
−0.711911 + 0.702270i \(0.752170\pi\)
\(354\) 9.63789 0.512248
\(355\) −5.66208 −0.300512
\(356\) 5.90688 0.313064
\(357\) −31.4045 −1.66210
\(358\) −3.91627 −0.206981
\(359\) 3.02557 0.159683 0.0798417 0.996808i \(-0.474559\pi\)
0.0798417 + 0.996808i \(0.474559\pi\)
\(360\) 2.15727 0.113698
\(361\) 10.2423 0.539066
\(362\) −19.7376 −1.03738
\(363\) −76.2459 −4.00187
\(364\) −0.726941 −0.0381021
\(365\) 6.65929 0.348563
\(366\) −24.0245 −1.25578
\(367\) −19.2019 −1.00233 −0.501165 0.865351i \(-0.667095\pi\)
−0.501165 + 0.865351i \(0.667095\pi\)
\(368\) −7.44793 −0.388250
\(369\) −1.92582 −0.100254
\(370\) 7.41017 0.385236
\(371\) 17.6436 0.916010
\(372\) −3.45710 −0.179242
\(373\) −9.96496 −0.515966 −0.257983 0.966149i \(-0.583058\pi\)
−0.257983 + 0.966149i \(0.583058\pi\)
\(374\) −27.9756 −1.44659
\(375\) 16.3635 0.845010
\(376\) 2.21454 0.114206
\(377\) −2.37752 −0.122449
\(378\) 0.708200 0.0364259
\(379\) −1.37433 −0.0705944 −0.0352972 0.999377i \(-0.511238\pi\)
−0.0352972 + 0.999377i \(0.511238\pi\)
\(380\) 3.76618 0.193201
\(381\) −6.03124 −0.308990
\(382\) −6.28499 −0.321568
\(383\) 2.50631 0.128066 0.0640331 0.997948i \(-0.479604\pi\)
0.0640331 + 0.997948i \(0.479604\pi\)
\(384\) −2.46931 −0.126011
\(385\) −13.2590 −0.675744
\(386\) 4.98373 0.253665
\(387\) −5.87255 −0.298519
\(388\) 4.65460 0.236302
\(389\) −1.84245 −0.0934159 −0.0467080 0.998909i \(-0.514873\pi\)
−0.0467080 + 0.998909i \(0.514873\pi\)
\(390\) −0.424956 −0.0215185
\(391\) 32.1978 1.62831
\(392\) 1.65472 0.0835761
\(393\) 31.9305 1.61068
\(394\) −8.89301 −0.448023
\(395\) 9.51350 0.478676
\(396\) 20.0447 1.00728
\(397\) −10.8483 −0.544463 −0.272231 0.962232i \(-0.587762\pi\)
−0.272231 + 0.962232i \(0.587762\pi\)
\(398\) −5.48201 −0.274788
\(399\) 39.2833 1.96662
\(400\) −4.51494 −0.225747
\(401\) 26.6113 1.32890 0.664451 0.747331i \(-0.268665\pi\)
0.664451 + 0.747331i \(0.268665\pi\)
\(402\) 19.3753 0.966353
\(403\) 0.345947 0.0172328
\(404\) −13.8748 −0.690299
\(405\) −6.05782 −0.301016
\(406\) 28.3060 1.40480
\(407\) 68.8529 3.41291
\(408\) 10.6749 0.528488
\(409\) 26.8595 1.32811 0.664057 0.747682i \(-0.268833\pi\)
0.664057 + 0.747682i \(0.268833\pi\)
\(410\) −0.433015 −0.0213851
\(411\) 34.4810 1.70082
\(412\) −3.48612 −0.171749
\(413\) 11.4824 0.565013
\(414\) −23.0699 −1.13382
\(415\) −9.20741 −0.451974
\(416\) 0.247100 0.0121151
\(417\) 29.1248 1.42625
\(418\) 34.9941 1.71162
\(419\) −13.3434 −0.651866 −0.325933 0.945393i \(-0.605678\pi\)
−0.325933 + 0.945393i \(0.605678\pi\)
\(420\) 5.05939 0.246873
\(421\) −8.43153 −0.410928 −0.205464 0.978665i \(-0.565870\pi\)
−0.205464 + 0.978665i \(0.565870\pi\)
\(422\) −23.9300 −1.16490
\(423\) 6.85951 0.333521
\(424\) −5.99737 −0.291258
\(425\) 19.5183 0.946778
\(426\) 20.0750 0.972637
\(427\) −28.6224 −1.38513
\(428\) −10.7045 −0.517419
\(429\) −3.94856 −0.190638
\(430\) −1.32042 −0.0636764
\(431\) −0.278983 −0.0134381 −0.00671907 0.999977i \(-0.502139\pi\)
−0.00671907 + 0.999977i \(0.502139\pi\)
\(432\) −0.240730 −0.0115821
\(433\) −0.219535 −0.0105502 −0.00527509 0.999986i \(-0.501679\pi\)
−0.00527509 + 0.999986i \(0.501679\pi\)
\(434\) −4.11872 −0.197705
\(435\) 16.5472 0.793375
\(436\) 17.3569 0.831247
\(437\) −40.2755 −1.92664
\(438\) −23.6106 −1.12816
\(439\) −22.7700 −1.08675 −0.543377 0.839489i \(-0.682854\pi\)
−0.543377 + 0.839489i \(0.682854\pi\)
\(440\) 4.50698 0.214862
\(441\) 5.12548 0.244071
\(442\) −1.06823 −0.0508103
\(443\) 16.1828 0.768866 0.384433 0.923153i \(-0.374397\pi\)
0.384433 + 0.923153i \(0.374397\pi\)
\(444\) −26.2729 −1.24686
\(445\) 4.11390 0.195018
\(446\) −18.6303 −0.882171
\(447\) −37.8039 −1.78806
\(448\) −2.94189 −0.138991
\(449\) 30.1522 1.42297 0.711486 0.702700i \(-0.248022\pi\)
0.711486 + 0.702700i \(0.248022\pi\)
\(450\) −13.9850 −0.659259
\(451\) −4.02344 −0.189456
\(452\) 12.1686 0.572364
\(453\) −47.4882 −2.23119
\(454\) −11.4522 −0.537478
\(455\) −0.506285 −0.0237350
\(456\) −13.3531 −0.625315
\(457\) 12.3632 0.578324 0.289162 0.957280i \(-0.406623\pi\)
0.289162 + 0.957280i \(0.406623\pi\)
\(458\) 8.16991 0.381755
\(459\) 1.04069 0.0485750
\(460\) −5.18718 −0.241854
\(461\) −28.6537 −1.33453 −0.667267 0.744819i \(-0.732536\pi\)
−0.667267 + 0.744819i \(0.732536\pi\)
\(462\) 47.0102 2.18711
\(463\) 12.4083 0.576664 0.288332 0.957531i \(-0.406899\pi\)
0.288332 + 0.957531i \(0.406899\pi\)
\(464\) −9.62171 −0.446677
\(465\) −2.40773 −0.111656
\(466\) −29.0524 −1.34583
\(467\) 0.477760 0.0221081 0.0110541 0.999939i \(-0.496481\pi\)
0.0110541 + 0.999939i \(0.496481\pi\)
\(468\) 0.765390 0.0353801
\(469\) 23.0834 1.06589
\(470\) 1.54234 0.0711427
\(471\) 10.4997 0.483799
\(472\) −3.90307 −0.179653
\(473\) −12.2689 −0.564127
\(474\) −33.7303 −1.54928
\(475\) −24.4151 −1.12024
\(476\) 12.7179 0.582926
\(477\) −18.5768 −0.850572
\(478\) 6.29282 0.287827
\(479\) 32.8108 1.49916 0.749581 0.661913i \(-0.230255\pi\)
0.749581 + 0.661913i \(0.230255\pi\)
\(480\) −1.71977 −0.0784966
\(481\) 2.62909 0.119876
\(482\) 1.21320 0.0552599
\(483\) −54.1050 −2.46186
\(484\) 30.8774 1.40352
\(485\) 3.24174 0.147200
\(486\) 22.2003 1.00703
\(487\) −37.9687 −1.72053 −0.860264 0.509849i \(-0.829701\pi\)
−0.860264 + 0.509849i \(0.829701\pi\)
\(488\) 9.72924 0.440422
\(489\) −51.7488 −2.34016
\(490\) 1.15245 0.0520622
\(491\) 17.0848 0.771026 0.385513 0.922702i \(-0.374024\pi\)
0.385513 + 0.922702i \(0.374024\pi\)
\(492\) 1.53526 0.0692150
\(493\) 41.5951 1.87335
\(494\) 1.33622 0.0601194
\(495\) 13.9603 0.627470
\(496\) 1.40003 0.0628630
\(497\) 23.9170 1.07282
\(498\) 32.6450 1.46286
\(499\) −7.42496 −0.332387 −0.166193 0.986093i \(-0.553148\pi\)
−0.166193 + 0.986093i \(0.553148\pi\)
\(500\) −6.62677 −0.296358
\(501\) 24.7180 1.10432
\(502\) −30.7423 −1.37210
\(503\) −4.26539 −0.190184 −0.0950922 0.995468i \(-0.530315\pi\)
−0.0950922 + 0.995468i \(0.530315\pi\)
\(504\) −9.11247 −0.405902
\(505\) −9.66326 −0.430009
\(506\) −48.1976 −2.14265
\(507\) 31.9502 1.41896
\(508\) 2.44248 0.108368
\(509\) 24.5941 1.09012 0.545058 0.838398i \(-0.316508\pi\)
0.545058 + 0.838398i \(0.316508\pi\)
\(510\) 7.43466 0.329212
\(511\) −28.1293 −1.24437
\(512\) 1.00000 0.0441942
\(513\) −1.30177 −0.0574746
\(514\) −22.7955 −1.00546
\(515\) −2.42794 −0.106988
\(516\) 4.68158 0.206095
\(517\) 14.3309 0.630273
\(518\) −31.3010 −1.37529
\(519\) −16.0376 −0.703974
\(520\) 0.172095 0.00754687
\(521\) −27.1394 −1.18900 −0.594498 0.804097i \(-0.702649\pi\)
−0.594498 + 0.804097i \(0.702649\pi\)
\(522\) −29.8031 −1.30445
\(523\) −40.6113 −1.77581 −0.887903 0.460030i \(-0.847839\pi\)
−0.887903 + 0.460030i \(0.847839\pi\)
\(524\) −12.9309 −0.564890
\(525\) −32.7985 −1.43145
\(526\) −18.6455 −0.812984
\(527\) −6.05238 −0.263646
\(528\) −15.9796 −0.695422
\(529\) 32.4717 1.41181
\(530\) −4.17692 −0.181434
\(531\) −12.0897 −0.524649
\(532\) −15.9086 −0.689725
\(533\) −0.153631 −0.00665451
\(534\) −14.5859 −0.631195
\(535\) −7.45522 −0.322317
\(536\) −7.84645 −0.338915
\(537\) 9.67049 0.417312
\(538\) −6.31721 −0.272354
\(539\) 10.7082 0.461233
\(540\) −0.167658 −0.00721487
\(541\) 12.2652 0.527324 0.263662 0.964615i \(-0.415070\pi\)
0.263662 + 0.964615i \(0.415070\pi\)
\(542\) −21.6233 −0.928798
\(543\) 48.7382 2.09156
\(544\) −4.32305 −0.185349
\(545\) 12.0884 0.517810
\(546\) 1.79504 0.0768208
\(547\) −40.8139 −1.74508 −0.872538 0.488546i \(-0.837528\pi\)
−0.872538 + 0.488546i \(0.837528\pi\)
\(548\) −13.9638 −0.596506
\(549\) 30.1362 1.28618
\(550\) −29.2175 −1.24584
\(551\) −52.0305 −2.21657
\(552\) 18.3913 0.782784
\(553\) −40.1857 −1.70887
\(554\) −18.4094 −0.782140
\(555\) −18.2980 −0.776707
\(556\) −11.7947 −0.500207
\(557\) −29.3130 −1.24203 −0.621017 0.783797i \(-0.713280\pi\)
−0.621017 + 0.783797i \(0.713280\pi\)
\(558\) 4.33656 0.183581
\(559\) −0.468479 −0.0198146
\(560\) −2.04891 −0.0865821
\(561\) 69.0805 2.91658
\(562\) −16.6219 −0.701151
\(563\) −19.5949 −0.825828 −0.412914 0.910770i \(-0.635489\pi\)
−0.412914 + 0.910770i \(0.635489\pi\)
\(564\) −5.46839 −0.230261
\(565\) 8.47496 0.356544
\(566\) 3.84401 0.161576
\(567\) 25.5887 1.07462
\(568\) −8.12981 −0.341119
\(569\) 2.12297 0.0889996 0.0444998 0.999009i \(-0.485831\pi\)
0.0444998 + 0.999009i \(0.485831\pi\)
\(570\) −9.29987 −0.389528
\(571\) −34.2072 −1.43153 −0.715763 0.698343i \(-0.753921\pi\)
−0.715763 + 0.698343i \(0.753921\pi\)
\(572\) 1.59905 0.0668598
\(573\) 15.5196 0.648339
\(574\) 1.82908 0.0763445
\(575\) 33.6270 1.40234
\(576\) 3.09749 0.129062
\(577\) −27.7876 −1.15681 −0.578406 0.815749i \(-0.696325\pi\)
−0.578406 + 0.815749i \(0.696325\pi\)
\(578\) 1.68875 0.0702426
\(579\) −12.3064 −0.511436
\(580\) −6.70113 −0.278249
\(581\) 38.8927 1.61354
\(582\) −11.4937 −0.476427
\(583\) −38.8106 −1.60737
\(584\) 9.56163 0.395663
\(585\) 0.533063 0.0220394
\(586\) 31.2539 1.29109
\(587\) 32.8144 1.35440 0.677199 0.735800i \(-0.263194\pi\)
0.677199 + 0.735800i \(0.263194\pi\)
\(588\) −4.08602 −0.168505
\(589\) 7.57080 0.311949
\(590\) −2.71833 −0.111912
\(591\) 21.9596 0.903297
\(592\) 10.6398 0.437292
\(593\) −26.4635 −1.08673 −0.543364 0.839497i \(-0.682850\pi\)
−0.543364 + 0.839497i \(0.682850\pi\)
\(594\) −1.55783 −0.0639184
\(595\) 8.85752 0.363123
\(596\) 15.3095 0.627102
\(597\) 13.5368 0.554023
\(598\) −1.84038 −0.0752589
\(599\) −16.3840 −0.669434 −0.334717 0.942319i \(-0.608641\pi\)
−0.334717 + 0.942319i \(0.608641\pi\)
\(600\) 11.1488 0.455148
\(601\) 5.75079 0.234580 0.117290 0.993098i \(-0.462579\pi\)
0.117290 + 0.993098i \(0.462579\pi\)
\(602\) 5.57755 0.227324
\(603\) −24.3043 −0.989747
\(604\) 19.2314 0.782514
\(605\) 21.5049 0.874297
\(606\) 34.2613 1.39177
\(607\) −6.89203 −0.279739 −0.139869 0.990170i \(-0.544668\pi\)
−0.139869 + 0.990170i \(0.544668\pi\)
\(608\) 5.40761 0.219308
\(609\) −69.8963 −2.83234
\(610\) 6.77602 0.274353
\(611\) 0.547213 0.0221379
\(612\) −13.3906 −0.541283
\(613\) −38.3962 −1.55081 −0.775405 0.631465i \(-0.782454\pi\)
−0.775405 + 0.631465i \(0.782454\pi\)
\(614\) 7.83720 0.316284
\(615\) 1.06925 0.0431162
\(616\) −19.0378 −0.767054
\(617\) 30.5710 1.23074 0.615372 0.788237i \(-0.289006\pi\)
0.615372 + 0.788237i \(0.289006\pi\)
\(618\) 8.60832 0.346277
\(619\) −7.12931 −0.286551 −0.143276 0.989683i \(-0.545764\pi\)
−0.143276 + 0.989683i \(0.545764\pi\)
\(620\) 0.975061 0.0391594
\(621\) 1.79294 0.0719481
\(622\) −29.8325 −1.19618
\(623\) −17.3774 −0.696211
\(624\) −0.610166 −0.0244262
\(625\) 17.9594 0.718378
\(626\) 16.7816 0.670727
\(627\) −86.4114 −3.45094
\(628\) −4.25207 −0.169676
\(629\) −45.9962 −1.83399
\(630\) −6.34647 −0.252849
\(631\) 5.51858 0.219691 0.109846 0.993949i \(-0.464964\pi\)
0.109846 + 0.993949i \(0.464964\pi\)
\(632\) 13.6598 0.543358
\(633\) 59.0906 2.34864
\(634\) −10.5987 −0.420929
\(635\) 1.70109 0.0675057
\(636\) 14.8094 0.587229
\(637\) 0.408882 0.0162005
\(638\) −62.2647 −2.46508
\(639\) −25.1820 −0.996184
\(640\) 0.696459 0.0275300
\(641\) 24.0020 0.948020 0.474010 0.880519i \(-0.342806\pi\)
0.474010 + 0.880519i \(0.342806\pi\)
\(642\) 26.4326 1.04321
\(643\) −1.14708 −0.0452363 −0.0226182 0.999744i \(-0.507200\pi\)
−0.0226182 + 0.999744i \(0.507200\pi\)
\(644\) 21.9110 0.863415
\(645\) 3.26053 0.128383
\(646\) −23.3774 −0.919770
\(647\) 17.1933 0.675937 0.337969 0.941157i \(-0.390260\pi\)
0.337969 + 0.941157i \(0.390260\pi\)
\(648\) −8.69803 −0.341691
\(649\) −25.2579 −0.991458
\(650\) −1.11564 −0.0437591
\(651\) 10.1704 0.398610
\(652\) 20.9568 0.820732
\(653\) 29.6101 1.15873 0.579367 0.815067i \(-0.303300\pi\)
0.579367 + 0.815067i \(0.303300\pi\)
\(654\) −42.8596 −1.67594
\(655\) −9.00586 −0.351888
\(656\) −0.621738 −0.0242748
\(657\) 29.6170 1.15547
\(658\) −6.51494 −0.253979
\(659\) −25.6107 −0.997650 −0.498825 0.866703i \(-0.666235\pi\)
−0.498825 + 0.866703i \(0.666235\pi\)
\(660\) −11.1291 −0.433201
\(661\) 13.1870 0.512913 0.256457 0.966556i \(-0.417445\pi\)
0.256457 + 0.966556i \(0.417445\pi\)
\(662\) 35.8218 1.39225
\(663\) 2.63778 0.102443
\(664\) −13.2203 −0.513047
\(665\) −11.0797 −0.429652
\(666\) 32.9566 1.27704
\(667\) 71.6618 2.77476
\(668\) −10.0101 −0.387302
\(669\) 46.0040 1.77862
\(670\) −5.46473 −0.211121
\(671\) 62.9606 2.43057
\(672\) 7.26444 0.280232
\(673\) 40.5333 1.56245 0.781223 0.624252i \(-0.214596\pi\)
0.781223 + 0.624252i \(0.214596\pi\)
\(674\) 5.65042 0.217646
\(675\) 1.08688 0.0418341
\(676\) −12.9389 −0.497652
\(677\) −39.0396 −1.50041 −0.750207 0.661203i \(-0.770046\pi\)
−0.750207 + 0.661203i \(0.770046\pi\)
\(678\) −30.0481 −1.15399
\(679\) −13.6933 −0.525502
\(680\) −3.01083 −0.115460
\(681\) 28.2790 1.08365
\(682\) 9.05996 0.346924
\(683\) 25.0002 0.956605 0.478303 0.878195i \(-0.341252\pi\)
0.478303 + 0.878195i \(0.341252\pi\)
\(684\) 16.7500 0.640453
\(685\) −9.72525 −0.371583
\(686\) 15.7252 0.600392
\(687\) −20.1740 −0.769688
\(688\) −1.89591 −0.0722808
\(689\) −1.48195 −0.0564578
\(690\) 12.8088 0.487621
\(691\) −51.0081 −1.94044 −0.970219 0.242230i \(-0.922121\pi\)
−0.970219 + 0.242230i \(0.922121\pi\)
\(692\) 6.49479 0.246895
\(693\) −58.9693 −2.24006
\(694\) −6.06311 −0.230152
\(695\) −8.21454 −0.311595
\(696\) 23.7590 0.900582
\(697\) 2.68780 0.101808
\(698\) 11.9537 0.452452
\(699\) 71.7393 2.71343
\(700\) 13.2825 0.502030
\(701\) −37.1792 −1.40424 −0.702120 0.712059i \(-0.747763\pi\)
−0.702120 + 0.712059i \(0.747763\pi\)
\(702\) −0.0594843 −0.00224509
\(703\) 57.5357 2.17000
\(704\) 6.47128 0.243895
\(705\) −3.80851 −0.143437
\(706\) −26.7512 −1.00679
\(707\) 40.8183 1.53513
\(708\) 9.63789 0.362214
\(709\) 41.5877 1.56186 0.780930 0.624618i \(-0.214745\pi\)
0.780930 + 0.624618i \(0.214745\pi\)
\(710\) −5.66208 −0.212494
\(711\) 42.3111 1.58679
\(712\) 5.90688 0.221370
\(713\) −10.4273 −0.390505
\(714\) −31.4045 −1.17528
\(715\) 1.11368 0.0416491
\(716\) −3.91627 −0.146358
\(717\) −15.5389 −0.580312
\(718\) 3.02557 0.112913
\(719\) −22.0324 −0.821670 −0.410835 0.911710i \(-0.634763\pi\)
−0.410835 + 0.911710i \(0.634763\pi\)
\(720\) 2.15727 0.0803969
\(721\) 10.2558 0.381946
\(722\) 10.2423 0.381177
\(723\) −2.99577 −0.111414
\(724\) −19.7376 −0.733542
\(725\) 43.4415 1.61338
\(726\) −76.2459 −2.82975
\(727\) −8.48485 −0.314686 −0.157343 0.987544i \(-0.550293\pi\)
−0.157343 + 0.987544i \(0.550293\pi\)
\(728\) −0.726941 −0.0269422
\(729\) −28.7254 −1.06390
\(730\) 6.65929 0.246471
\(731\) 8.19610 0.303144
\(732\) −24.0245 −0.887971
\(733\) 28.9346 1.06872 0.534362 0.845256i \(-0.320552\pi\)
0.534362 + 0.845256i \(0.320552\pi\)
\(734\) −19.2019 −0.708755
\(735\) −2.84575 −0.104967
\(736\) −7.44793 −0.274534
\(737\) −50.7766 −1.87038
\(738\) −1.92582 −0.0708906
\(739\) −31.0629 −1.14267 −0.571333 0.820718i \(-0.693574\pi\)
−0.571333 + 0.820718i \(0.693574\pi\)
\(740\) 7.41017 0.272403
\(741\) −3.29954 −0.121212
\(742\) 17.6436 0.647717
\(743\) 12.7090 0.466247 0.233124 0.972447i \(-0.425105\pi\)
0.233124 + 0.972447i \(0.425105\pi\)
\(744\) −3.45710 −0.126743
\(745\) 10.6625 0.390642
\(746\) −9.96496 −0.364843
\(747\) −40.9498 −1.49827
\(748\) −27.9756 −1.02289
\(749\) 31.4913 1.15067
\(750\) 16.3635 0.597512
\(751\) 12.9330 0.471933 0.235967 0.971761i \(-0.424174\pi\)
0.235967 + 0.971761i \(0.424174\pi\)
\(752\) 2.21454 0.0807560
\(753\) 75.9122 2.76639
\(754\) −2.37752 −0.0865843
\(755\) 13.3939 0.487453
\(756\) 0.708200 0.0257570
\(757\) 12.1514 0.441651 0.220825 0.975313i \(-0.429125\pi\)
0.220825 + 0.975313i \(0.429125\pi\)
\(758\) −1.37433 −0.0499177
\(759\) 119.015 4.31997
\(760\) 3.76618 0.136614
\(761\) 22.4820 0.814971 0.407486 0.913212i \(-0.366406\pi\)
0.407486 + 0.913212i \(0.366406\pi\)
\(762\) −6.03124 −0.218489
\(763\) −51.0622 −1.84858
\(764\) −6.28499 −0.227383
\(765\) −9.32600 −0.337182
\(766\) 2.50631 0.0905565
\(767\) −0.964449 −0.0348243
\(768\) −2.46931 −0.0891035
\(769\) −39.1107 −1.41037 −0.705183 0.709025i \(-0.749135\pi\)
−0.705183 + 0.709025i \(0.749135\pi\)
\(770\) −13.2590 −0.477823
\(771\) 56.2890 2.02720
\(772\) 4.98373 0.179369
\(773\) −37.5363 −1.35009 −0.675043 0.737778i \(-0.735875\pi\)
−0.675043 + 0.737778i \(0.735875\pi\)
\(774\) −5.87255 −0.211085
\(775\) −6.32104 −0.227058
\(776\) 4.65460 0.167090
\(777\) 77.2920 2.77283
\(778\) −1.84245 −0.0660550
\(779\) −3.36211 −0.120460
\(780\) −0.424956 −0.0152159
\(781\) −52.6102 −1.88254
\(782\) 32.1978 1.15139
\(783\) 2.31623 0.0827753
\(784\) 1.65472 0.0590972
\(785\) −2.96139 −0.105697
\(786\) 31.9305 1.13892
\(787\) −40.7619 −1.45300 −0.726502 0.687164i \(-0.758855\pi\)
−0.726502 + 0.687164i \(0.758855\pi\)
\(788\) −8.89301 −0.316800
\(789\) 46.0416 1.63912
\(790\) 9.51350 0.338475
\(791\) −35.7988 −1.27286
\(792\) 20.0447 0.712258
\(793\) 2.40410 0.0853720
\(794\) −10.8483 −0.384993
\(795\) 10.3141 0.365804
\(796\) −5.48201 −0.194305
\(797\) 26.4335 0.936322 0.468161 0.883643i \(-0.344917\pi\)
0.468161 + 0.883643i \(0.344917\pi\)
\(798\) 39.2833 1.39061
\(799\) −9.57357 −0.338688
\(800\) −4.51494 −0.159627
\(801\) 18.2965 0.646475
\(802\) 26.6113 0.939676
\(803\) 61.8760 2.18356
\(804\) 19.3753 0.683315
\(805\) 15.2601 0.537849
\(806\) 0.345947 0.0121854
\(807\) 15.5992 0.549116
\(808\) −13.8748 −0.488115
\(809\) −14.4091 −0.506596 −0.253298 0.967388i \(-0.581515\pi\)
−0.253298 + 0.967388i \(0.581515\pi\)
\(810\) −6.05782 −0.212850
\(811\) 12.3035 0.432036 0.216018 0.976389i \(-0.430693\pi\)
0.216018 + 0.976389i \(0.430693\pi\)
\(812\) 28.3060 0.993346
\(813\) 53.3945 1.87263
\(814\) 68.8529 2.41329
\(815\) 14.5956 0.511260
\(816\) 10.6749 0.373698
\(817\) −10.2523 −0.358684
\(818\) 26.8595 0.939119
\(819\) −2.25169 −0.0786805
\(820\) −0.433015 −0.0151215
\(821\) 54.7857 1.91203 0.956017 0.293312i \(-0.0947575\pi\)
0.956017 + 0.293312i \(0.0947575\pi\)
\(822\) 34.4810 1.20266
\(823\) 9.26606 0.322994 0.161497 0.986873i \(-0.448368\pi\)
0.161497 + 0.986873i \(0.448368\pi\)
\(824\) −3.48612 −0.121445
\(825\) 72.1469 2.51183
\(826\) 11.4824 0.399524
\(827\) −40.6307 −1.41287 −0.706433 0.707780i \(-0.749697\pi\)
−0.706433 + 0.707780i \(0.749697\pi\)
\(828\) −23.0699 −0.801734
\(829\) −10.2587 −0.356300 −0.178150 0.984003i \(-0.557011\pi\)
−0.178150 + 0.984003i \(0.557011\pi\)
\(830\) −9.20741 −0.319594
\(831\) 45.4585 1.57694
\(832\) 0.247100 0.00856665
\(833\) −7.15344 −0.247852
\(834\) 29.1248 1.00851
\(835\) −6.97162 −0.241263
\(836\) 34.9941 1.21030
\(837\) −0.337028 −0.0116494
\(838\) −13.3434 −0.460939
\(839\) 25.3075 0.873711 0.436855 0.899532i \(-0.356092\pi\)
0.436855 + 0.899532i \(0.356092\pi\)
\(840\) 5.05939 0.174565
\(841\) 63.5773 2.19232
\(842\) −8.43153 −0.290570
\(843\) 41.0445 1.41365
\(844\) −23.9300 −0.823705
\(845\) −9.01145 −0.310003
\(846\) 6.85951 0.235835
\(847\) −90.8380 −3.12123
\(848\) −5.99737 −0.205950
\(849\) −9.49205 −0.325766
\(850\) 19.5183 0.669473
\(851\) −79.2443 −2.71646
\(852\) 20.0750 0.687759
\(853\) −35.4844 −1.21496 −0.607482 0.794334i \(-0.707820\pi\)
−0.607482 + 0.794334i \(0.707820\pi\)
\(854\) −28.6224 −0.979437
\(855\) 11.6657 0.398959
\(856\) −10.7045 −0.365871
\(857\) −34.9964 −1.19545 −0.597727 0.801699i \(-0.703929\pi\)
−0.597727 + 0.801699i \(0.703929\pi\)
\(858\) −3.94856 −0.134801
\(859\) 45.4198 1.54970 0.774852 0.632143i \(-0.217825\pi\)
0.774852 + 0.632143i \(0.217825\pi\)
\(860\) −1.32042 −0.0450260
\(861\) −4.51657 −0.153924
\(862\) −0.278983 −0.00950220
\(863\) 16.9731 0.577772 0.288886 0.957363i \(-0.406715\pi\)
0.288886 + 0.957363i \(0.406715\pi\)
\(864\) −0.240730 −0.00818979
\(865\) 4.52335 0.153799
\(866\) −0.219535 −0.00746010
\(867\) −4.17004 −0.141622
\(868\) −4.11872 −0.139799
\(869\) 88.3964 2.99864
\(870\) 16.5472 0.561001
\(871\) −1.93886 −0.0656957
\(872\) 17.3569 0.587780
\(873\) 14.4176 0.487961
\(874\) −40.2755 −1.36234
\(875\) 19.4952 0.659059
\(876\) −23.6106 −0.797729
\(877\) −12.9442 −0.437094 −0.218547 0.975826i \(-0.570132\pi\)
−0.218547 + 0.975826i \(0.570132\pi\)
\(878\) −22.7700 −0.768451
\(879\) −77.1755 −2.60307
\(880\) 4.50698 0.151930
\(881\) 1.04540 0.0352205 0.0176103 0.999845i \(-0.494394\pi\)
0.0176103 + 0.999845i \(0.494394\pi\)
\(882\) 5.12548 0.172584
\(883\) −16.3441 −0.550022 −0.275011 0.961441i \(-0.588682\pi\)
−0.275011 + 0.961441i \(0.588682\pi\)
\(884\) −1.06823 −0.0359283
\(885\) 6.71240 0.225635
\(886\) 16.1828 0.543670
\(887\) 27.6199 0.927385 0.463692 0.885996i \(-0.346524\pi\)
0.463692 + 0.885996i \(0.346524\pi\)
\(888\) −26.2729 −0.881660
\(889\) −7.18551 −0.240994
\(890\) 4.11390 0.137898
\(891\) −56.2874 −1.88570
\(892\) −18.6303 −0.623789
\(893\) 11.9754 0.400741
\(894\) −37.8039 −1.26435
\(895\) −2.72752 −0.0911711
\(896\) −2.94189 −0.0982817
\(897\) 4.54448 0.151736
\(898\) 30.1522 1.00619
\(899\) −13.4706 −0.449271
\(900\) −13.9850 −0.466166
\(901\) 25.9269 0.863750
\(902\) −4.02344 −0.133966
\(903\) −13.7727 −0.458327
\(904\) 12.1686 0.404723
\(905\) −13.7464 −0.456947
\(906\) −47.4882 −1.57769
\(907\) −9.40241 −0.312202 −0.156101 0.987741i \(-0.549893\pi\)
−0.156101 + 0.987741i \(0.549893\pi\)
\(908\) −11.4522 −0.380054
\(909\) −42.9772 −1.42546
\(910\) −0.506285 −0.0167832
\(911\) −23.0408 −0.763376 −0.381688 0.924291i \(-0.624657\pi\)
−0.381688 + 0.924291i \(0.624657\pi\)
\(912\) −13.3531 −0.442164
\(913\) −85.5523 −2.83137
\(914\) 12.3632 0.408937
\(915\) −16.7321 −0.553146
\(916\) 8.16991 0.269942
\(917\) 38.0414 1.25624
\(918\) 1.04069 0.0343477
\(919\) 3.25112 0.107245 0.0536223 0.998561i \(-0.482923\pi\)
0.0536223 + 0.998561i \(0.482923\pi\)
\(920\) −5.18718 −0.171016
\(921\) −19.3525 −0.637686
\(922\) −28.6537 −0.943658
\(923\) −2.00888 −0.0661230
\(924\) 47.0102 1.54652
\(925\) −48.0380 −1.57948
\(926\) 12.4083 0.407763
\(927\) −10.7982 −0.354660
\(928\) −9.62171 −0.315848
\(929\) −20.5269 −0.673466 −0.336733 0.941600i \(-0.609322\pi\)
−0.336733 + 0.941600i \(0.609322\pi\)
\(930\) −2.40773 −0.0789525
\(931\) 8.94809 0.293262
\(932\) −29.0524 −0.951642
\(933\) 73.6657 2.41171
\(934\) 0.477760 0.0156328
\(935\) −19.4839 −0.637192
\(936\) 0.765390 0.0250175
\(937\) 16.8798 0.551440 0.275720 0.961238i \(-0.411084\pi\)
0.275720 + 0.961238i \(0.411084\pi\)
\(938\) 23.0834 0.753700
\(939\) −41.4389 −1.35231
\(940\) 1.54234 0.0503055
\(941\) 31.8413 1.03800 0.518999 0.854775i \(-0.326305\pi\)
0.518999 + 0.854775i \(0.326305\pi\)
\(942\) 10.4997 0.342098
\(943\) 4.63066 0.150795
\(944\) −3.90307 −0.127034
\(945\) 0.493232 0.0160449
\(946\) −12.2689 −0.398898
\(947\) 15.7039 0.510307 0.255154 0.966901i \(-0.417874\pi\)
0.255154 + 0.966901i \(0.417874\pi\)
\(948\) −33.7303 −1.09551
\(949\) 2.36268 0.0766958
\(950\) −24.4151 −0.792129
\(951\) 26.1715 0.848670
\(952\) 12.7179 0.412191
\(953\) 20.5730 0.666425 0.333212 0.942852i \(-0.391867\pi\)
0.333212 + 0.942852i \(0.391867\pi\)
\(954\) −18.5768 −0.601445
\(955\) −4.37724 −0.141644
\(956\) 6.29282 0.203524
\(957\) 153.751 4.97006
\(958\) 32.8108 1.06007
\(959\) 41.0801 1.32655
\(960\) −1.71977 −0.0555054
\(961\) −29.0399 −0.936772
\(962\) 2.62909 0.0847652
\(963\) −33.1569 −1.06847
\(964\) 1.21320 0.0390746
\(965\) 3.47097 0.111734
\(966\) −54.1050 −1.74080
\(967\) 22.2325 0.714950 0.357475 0.933923i \(-0.383638\pi\)
0.357475 + 0.933923i \(0.383638\pi\)
\(968\) 30.8774 0.992438
\(969\) 57.7259 1.85442
\(970\) 3.24174 0.104086
\(971\) −29.5759 −0.949134 −0.474567 0.880219i \(-0.657395\pi\)
−0.474567 + 0.880219i \(0.657395\pi\)
\(972\) 22.2003 0.712075
\(973\) 34.6988 1.11239
\(974\) −37.9687 −1.21660
\(975\) 2.75487 0.0882264
\(976\) 9.72924 0.311425
\(977\) 46.6229 1.49160 0.745800 0.666170i \(-0.232067\pi\)
0.745800 + 0.666170i \(0.232067\pi\)
\(978\) −51.7488 −1.65474
\(979\) 38.2251 1.22168
\(980\) 1.15245 0.0368135
\(981\) 53.7629 1.71652
\(982\) 17.0848 0.545198
\(983\) 34.5085 1.10065 0.550326 0.834950i \(-0.314504\pi\)
0.550326 + 0.834950i \(0.314504\pi\)
\(984\) 1.53526 0.0489424
\(985\) −6.19362 −0.197345
\(986\) 41.5951 1.32466
\(987\) 16.0874 0.512067
\(988\) 1.33622 0.0425109
\(989\) 14.1206 0.449009
\(990\) 13.9603 0.443688
\(991\) 28.4311 0.903145 0.451573 0.892234i \(-0.350863\pi\)
0.451573 + 0.892234i \(0.350863\pi\)
\(992\) 1.40003 0.0444509
\(993\) −88.4551 −2.80704
\(994\) 23.9170 0.758601
\(995\) −3.81799 −0.121039
\(996\) 32.6450 1.03440
\(997\) −34.1080 −1.08021 −0.540105 0.841598i \(-0.681616\pi\)
−0.540105 + 0.841598i \(0.681616\pi\)
\(998\) −7.42496 −0.235033
\(999\) −2.56131 −0.0810362
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 8038.2.a.a.1.14 75
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8038.2.a.a.1.14 75 1.1 even 1 trivial