Properties

Label 2001.2.a.n.1.5
Level $2001$
Weight $2$
Character 2001.1
Self dual yes
Analytic conductor $15.978$
Analytic rank $0$
Dimension $16$
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] = [2001,2,Mod(1,2001)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2001, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2001.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2001 = 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2001.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: \(15.9780654445\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{15} - 28 x^{14} + 27 x^{13} + 316 x^{12} - 295 x^{11} - 1835 x^{10} + 1665 x^{9} + 5778 x^{8} - 5124 x^{7} - 9405 x^{6} + 8288 x^{5} + 6405 x^{4} - 6032 x^{3} + \cdots - 192 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(1.78406\) of defining polynomial
Character \(\chi\) \(=\) 2001.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.78406 q^{2} +1.00000 q^{3} +1.18288 q^{4} +3.42296 q^{5} -1.78406 q^{6} +3.39829 q^{7} +1.45779 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.78406 q^{2} +1.00000 q^{3} +1.18288 q^{4} +3.42296 q^{5} -1.78406 q^{6} +3.39829 q^{7} +1.45779 q^{8} +1.00000 q^{9} -6.10677 q^{10} +1.48855 q^{11} +1.18288 q^{12} +1.54078 q^{13} -6.06276 q^{14} +3.42296 q^{15} -4.96655 q^{16} +1.71252 q^{17} -1.78406 q^{18} +5.94186 q^{19} +4.04896 q^{20} +3.39829 q^{21} -2.65566 q^{22} -1.00000 q^{23} +1.45779 q^{24} +6.71663 q^{25} -2.74885 q^{26} +1.00000 q^{27} +4.01978 q^{28} -1.00000 q^{29} -6.10677 q^{30} -2.03180 q^{31} +5.94507 q^{32} +1.48855 q^{33} -3.05524 q^{34} +11.6322 q^{35} +1.18288 q^{36} -2.86662 q^{37} -10.6006 q^{38} +1.54078 q^{39} +4.98994 q^{40} +2.74906 q^{41} -6.06276 q^{42} +2.40752 q^{43} +1.76078 q^{44} +3.42296 q^{45} +1.78406 q^{46} +0.626003 q^{47} -4.96655 q^{48} +4.54837 q^{49} -11.9829 q^{50} +1.71252 q^{51} +1.82256 q^{52} -1.96409 q^{53} -1.78406 q^{54} +5.09523 q^{55} +4.95398 q^{56} +5.94186 q^{57} +1.78406 q^{58} -10.3160 q^{59} +4.04896 q^{60} -6.34964 q^{61} +3.62487 q^{62} +3.39829 q^{63} -0.673284 q^{64} +5.27402 q^{65} -2.65566 q^{66} -3.34937 q^{67} +2.02571 q^{68} -1.00000 q^{69} -20.7526 q^{70} -15.8477 q^{71} +1.45779 q^{72} -8.02067 q^{73} +5.11424 q^{74} +6.71663 q^{75} +7.02852 q^{76} +5.05851 q^{77} -2.74885 q^{78} +2.84758 q^{79} -17.0003 q^{80} +1.00000 q^{81} -4.90450 q^{82} -16.8010 q^{83} +4.01978 q^{84} +5.86188 q^{85} -4.29516 q^{86} -1.00000 q^{87} +2.16999 q^{88} -14.5368 q^{89} -6.10677 q^{90} +5.23601 q^{91} -1.18288 q^{92} -2.03180 q^{93} -1.11683 q^{94} +20.3387 q^{95} +5.94507 q^{96} +6.93709 q^{97} -8.11457 q^{98} +1.48855 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - q^{2} + 16 q^{3} + 25 q^{4} + 3 q^{5} - q^{6} + 13 q^{7} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - q^{2} + 16 q^{3} + 25 q^{4} + 3 q^{5} - q^{6} + 13 q^{7} + 16 q^{9} + 11 q^{10} + 8 q^{11} + 25 q^{12} + 19 q^{13} + 16 q^{14} + 3 q^{15} + 31 q^{16} - 4 q^{17} - q^{18} + 19 q^{19} + 16 q^{20} + 13 q^{21} + 6 q^{22} - 16 q^{23} + 23 q^{25} - 15 q^{26} + 16 q^{27} + 18 q^{28} - 16 q^{29} + 11 q^{30} + 24 q^{31} - 21 q^{32} + 8 q^{33} - 9 q^{34} - 13 q^{35} + 25 q^{36} + 26 q^{37} + 19 q^{39} - 22 q^{40} - 15 q^{41} + 16 q^{42} + 33 q^{43} + 6 q^{44} + 3 q^{45} + q^{46} + 13 q^{47} + 31 q^{48} + 41 q^{49} + 13 q^{50} - 4 q^{51} - 26 q^{52} + 5 q^{53} - q^{54} + 9 q^{55} + 40 q^{56} + 19 q^{57} + q^{58} + 2 q^{59} + 16 q^{60} + 29 q^{61} - 32 q^{62} + 13 q^{63} + 28 q^{64} + 18 q^{65} + 6 q^{66} + 32 q^{67} - 26 q^{68} - 16 q^{69} + 18 q^{70} + 29 q^{71} + 19 q^{73} - 16 q^{74} + 23 q^{75} + 64 q^{76} - 21 q^{77} - 15 q^{78} + 56 q^{79} + 16 q^{81} + 14 q^{82} + 5 q^{83} + 18 q^{84} + 16 q^{85} - 20 q^{86} - 16 q^{87} + q^{88} + 7 q^{89} + 11 q^{90} - 6 q^{91} - 25 q^{92} + 24 q^{93} - 11 q^{94} + 39 q^{95} - 21 q^{96} + 35 q^{97} - 109 q^{98} + 8 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.78406 −1.26152 −0.630762 0.775976i \(-0.717258\pi\)
−0.630762 + 0.775976i \(0.717258\pi\)
\(3\) 1.00000 0.577350
\(4\) 1.18288 0.591442
\(5\) 3.42296 1.53079 0.765396 0.643559i \(-0.222543\pi\)
0.765396 + 0.643559i \(0.222543\pi\)
\(6\) −1.78406 −0.728341
\(7\) 3.39829 1.28443 0.642216 0.766524i \(-0.278015\pi\)
0.642216 + 0.766524i \(0.278015\pi\)
\(8\) 1.45779 0.515406
\(9\) 1.00000 0.333333
\(10\) −6.10677 −1.93113
\(11\) 1.48855 0.448814 0.224407 0.974496i \(-0.427956\pi\)
0.224407 + 0.974496i \(0.427956\pi\)
\(12\) 1.18288 0.341469
\(13\) 1.54078 0.427335 0.213668 0.976906i \(-0.431459\pi\)
0.213668 + 0.976906i \(0.431459\pi\)
\(14\) −6.06276 −1.62034
\(15\) 3.42296 0.883804
\(16\) −4.96655 −1.24164
\(17\) 1.71252 0.415347 0.207673 0.978198i \(-0.433411\pi\)
0.207673 + 0.978198i \(0.433411\pi\)
\(18\) −1.78406 −0.420508
\(19\) 5.94186 1.36316 0.681578 0.731746i \(-0.261294\pi\)
0.681578 + 0.731746i \(0.261294\pi\)
\(20\) 4.04896 0.905375
\(21\) 3.39829 0.741567
\(22\) −2.65566 −0.566189
\(23\) −1.00000 −0.208514
\(24\) 1.45779 0.297570
\(25\) 6.71663 1.34333
\(26\) −2.74885 −0.539094
\(27\) 1.00000 0.192450
\(28\) 4.01978 0.759667
\(29\) −1.00000 −0.185695
\(30\) −6.10677 −1.11494
\(31\) −2.03180 −0.364923 −0.182461 0.983213i \(-0.558406\pi\)
−0.182461 + 0.983213i \(0.558406\pi\)
\(32\) 5.94507 1.05095
\(33\) 1.48855 0.259123
\(34\) −3.05524 −0.523970
\(35\) 11.6322 1.96620
\(36\) 1.18288 0.197147
\(37\) −2.86662 −0.471270 −0.235635 0.971842i \(-0.575717\pi\)
−0.235635 + 0.971842i \(0.575717\pi\)
\(38\) −10.6006 −1.71965
\(39\) 1.54078 0.246722
\(40\) 4.98994 0.788979
\(41\) 2.74906 0.429331 0.214665 0.976688i \(-0.431134\pi\)
0.214665 + 0.976688i \(0.431134\pi\)
\(42\) −6.06276 −0.935505
\(43\) 2.40752 0.367143 0.183571 0.983006i \(-0.441234\pi\)
0.183571 + 0.983006i \(0.441234\pi\)
\(44\) 1.76078 0.265447
\(45\) 3.42296 0.510264
\(46\) 1.78406 0.263046
\(47\) 0.626003 0.0913120 0.0456560 0.998957i \(-0.485462\pi\)
0.0456560 + 0.998957i \(0.485462\pi\)
\(48\) −4.96655 −0.716860
\(49\) 4.54837 0.649766
\(50\) −11.9829 −1.69464
\(51\) 1.71252 0.239801
\(52\) 1.82256 0.252744
\(53\) −1.96409 −0.269789 −0.134894 0.990860i \(-0.543070\pi\)
−0.134894 + 0.990860i \(0.543070\pi\)
\(54\) −1.78406 −0.242780
\(55\) 5.09523 0.687041
\(56\) 4.95398 0.662004
\(57\) 5.94186 0.787018
\(58\) 1.78406 0.234259
\(59\) −10.3160 −1.34303 −0.671514 0.740992i \(-0.734355\pi\)
−0.671514 + 0.740992i \(0.734355\pi\)
\(60\) 4.04896 0.522719
\(61\) −6.34964 −0.812988 −0.406494 0.913653i \(-0.633249\pi\)
−0.406494 + 0.913653i \(0.633249\pi\)
\(62\) 3.62487 0.460359
\(63\) 3.39829 0.428144
\(64\) −0.673284 −0.0841605
\(65\) 5.27402 0.654162
\(66\) −2.65566 −0.326890
\(67\) −3.34937 −0.409190 −0.204595 0.978847i \(-0.565588\pi\)
−0.204595 + 0.978847i \(0.565588\pi\)
\(68\) 2.02571 0.245654
\(69\) −1.00000 −0.120386
\(70\) −20.7526 −2.48041
\(71\) −15.8477 −1.88078 −0.940388 0.340104i \(-0.889538\pi\)
−0.940388 + 0.340104i \(0.889538\pi\)
\(72\) 1.45779 0.171802
\(73\) −8.02067 −0.938748 −0.469374 0.882999i \(-0.655520\pi\)
−0.469374 + 0.882999i \(0.655520\pi\)
\(74\) 5.11424 0.594518
\(75\) 6.71663 0.775570
\(76\) 7.02852 0.806227
\(77\) 5.05851 0.576471
\(78\) −2.74885 −0.311246
\(79\) 2.84758 0.320378 0.160189 0.987086i \(-0.448790\pi\)
0.160189 + 0.987086i \(0.448790\pi\)
\(80\) −17.0003 −1.90069
\(81\) 1.00000 0.111111
\(82\) −4.90450 −0.541611
\(83\) −16.8010 −1.84415 −0.922077 0.387006i \(-0.873509\pi\)
−0.922077 + 0.387006i \(0.873509\pi\)
\(84\) 4.01978 0.438594
\(85\) 5.86188 0.635810
\(86\) −4.29516 −0.463159
\(87\) −1.00000 −0.107211
\(88\) 2.16999 0.231321
\(89\) −14.5368 −1.54090 −0.770449 0.637502i \(-0.779968\pi\)
−0.770449 + 0.637502i \(0.779968\pi\)
\(90\) −6.10677 −0.643710
\(91\) 5.23601 0.548883
\(92\) −1.18288 −0.123324
\(93\) −2.03180 −0.210688
\(94\) −1.11683 −0.115192
\(95\) 20.3387 2.08671
\(96\) 5.94507 0.606767
\(97\) 6.93709 0.704355 0.352177 0.935933i \(-0.385441\pi\)
0.352177 + 0.935933i \(0.385441\pi\)
\(98\) −8.11457 −0.819696
\(99\) 1.48855 0.149605
\(100\) 7.94500 0.794500
\(101\) −17.0941 −1.70093 −0.850465 0.526031i \(-0.823680\pi\)
−0.850465 + 0.526031i \(0.823680\pi\)
\(102\) −3.05524 −0.302514
\(103\) 0.593783 0.0585071 0.0292536 0.999572i \(-0.490687\pi\)
0.0292536 + 0.999572i \(0.490687\pi\)
\(104\) 2.24613 0.220251
\(105\) 11.6322 1.13519
\(106\) 3.50407 0.340345
\(107\) −6.00249 −0.580283 −0.290141 0.956984i \(-0.593702\pi\)
−0.290141 + 0.956984i \(0.593702\pi\)
\(108\) 1.18288 0.113823
\(109\) 16.3993 1.57077 0.785385 0.619008i \(-0.212465\pi\)
0.785385 + 0.619008i \(0.212465\pi\)
\(110\) −9.09022 −0.866718
\(111\) −2.86662 −0.272088
\(112\) −16.8778 −1.59480
\(113\) 6.64814 0.625404 0.312702 0.949851i \(-0.398766\pi\)
0.312702 + 0.949851i \(0.398766\pi\)
\(114\) −10.6006 −0.992842
\(115\) −3.42296 −0.319192
\(116\) −1.18288 −0.109828
\(117\) 1.54078 0.142445
\(118\) 18.4044 1.69426
\(119\) 5.81963 0.533485
\(120\) 4.98994 0.455517
\(121\) −8.78423 −0.798566
\(122\) 11.3282 1.02560
\(123\) 2.74906 0.247874
\(124\) −2.40339 −0.215831
\(125\) 5.87596 0.525562
\(126\) −6.06276 −0.540114
\(127\) 2.85194 0.253069 0.126535 0.991962i \(-0.459615\pi\)
0.126535 + 0.991962i \(0.459615\pi\)
\(128\) −10.6890 −0.944780
\(129\) 2.40752 0.211970
\(130\) −9.40919 −0.825241
\(131\) −1.25044 −0.109251 −0.0546256 0.998507i \(-0.517397\pi\)
−0.0546256 + 0.998507i \(0.517397\pi\)
\(132\) 1.76078 0.153256
\(133\) 20.1921 1.75088
\(134\) 5.97549 0.516203
\(135\) 3.42296 0.294601
\(136\) 2.49649 0.214072
\(137\) −17.0060 −1.45292 −0.726459 0.687210i \(-0.758835\pi\)
−0.726459 + 0.687210i \(0.758835\pi\)
\(138\) 1.78406 0.151870
\(139\) −14.2040 −1.20477 −0.602386 0.798205i \(-0.705783\pi\)
−0.602386 + 0.798205i \(0.705783\pi\)
\(140\) 13.7595 1.16289
\(141\) 0.626003 0.0527190
\(142\) 28.2733 2.37264
\(143\) 2.29352 0.191794
\(144\) −4.96655 −0.413879
\(145\) −3.42296 −0.284261
\(146\) 14.3094 1.18425
\(147\) 4.54837 0.375143
\(148\) −3.39088 −0.278729
\(149\) 4.72788 0.387323 0.193661 0.981068i \(-0.437964\pi\)
0.193661 + 0.981068i \(0.437964\pi\)
\(150\) −11.9829 −0.978400
\(151\) 13.9866 1.13822 0.569108 0.822263i \(-0.307289\pi\)
0.569108 + 0.822263i \(0.307289\pi\)
\(152\) 8.66196 0.702578
\(153\) 1.71252 0.138449
\(154\) −9.02471 −0.727232
\(155\) −6.95478 −0.558621
\(156\) 1.82256 0.145922
\(157\) −3.91044 −0.312087 −0.156044 0.987750i \(-0.549874\pi\)
−0.156044 + 0.987750i \(0.549874\pi\)
\(158\) −5.08026 −0.404164
\(159\) −1.96409 −0.155763
\(160\) 20.3497 1.60879
\(161\) −3.39829 −0.267823
\(162\) −1.78406 −0.140169
\(163\) 24.1757 1.89359 0.946794 0.321840i \(-0.104301\pi\)
0.946794 + 0.321840i \(0.104301\pi\)
\(164\) 3.25182 0.253924
\(165\) 5.09523 0.396663
\(166\) 29.9741 2.32644
\(167\) −8.39892 −0.649928 −0.324964 0.945726i \(-0.605352\pi\)
−0.324964 + 0.945726i \(0.605352\pi\)
\(168\) 4.95398 0.382208
\(169\) −10.6260 −0.817385
\(170\) −10.4580 −0.802089
\(171\) 5.94186 0.454385
\(172\) 2.84781 0.217144
\(173\) 5.63120 0.428132 0.214066 0.976819i \(-0.431329\pi\)
0.214066 + 0.976819i \(0.431329\pi\)
\(174\) 1.78406 0.135250
\(175\) 22.8251 1.72541
\(176\) −7.39295 −0.557264
\(177\) −10.3160 −0.775397
\(178\) 25.9346 1.94388
\(179\) −14.2510 −1.06517 −0.532584 0.846377i \(-0.678779\pi\)
−0.532584 + 0.846377i \(0.678779\pi\)
\(180\) 4.04896 0.301792
\(181\) −3.49672 −0.259909 −0.129955 0.991520i \(-0.541483\pi\)
−0.129955 + 0.991520i \(0.541483\pi\)
\(182\) −9.34138 −0.692429
\(183\) −6.34964 −0.469379
\(184\) −1.45779 −0.107470
\(185\) −9.81232 −0.721416
\(186\) 3.62487 0.265788
\(187\) 2.54917 0.186413
\(188\) 0.740489 0.0540057
\(189\) 3.39829 0.247189
\(190\) −36.2856 −2.63243
\(191\) 8.41875 0.609159 0.304580 0.952487i \(-0.401484\pi\)
0.304580 + 0.952487i \(0.401484\pi\)
\(192\) −0.673284 −0.0485901
\(193\) 22.9526 1.65217 0.826083 0.563548i \(-0.190564\pi\)
0.826083 + 0.563548i \(0.190564\pi\)
\(194\) −12.3762 −0.888560
\(195\) 5.27402 0.377680
\(196\) 5.38019 0.384299
\(197\) 6.02811 0.429485 0.214743 0.976671i \(-0.431109\pi\)
0.214743 + 0.976671i \(0.431109\pi\)
\(198\) −2.65566 −0.188730
\(199\) −2.91164 −0.206401 −0.103200 0.994661i \(-0.532908\pi\)
−0.103200 + 0.994661i \(0.532908\pi\)
\(200\) 9.79143 0.692358
\(201\) −3.34937 −0.236246
\(202\) 30.4970 2.14576
\(203\) −3.39829 −0.238513
\(204\) 2.02571 0.141828
\(205\) 9.40991 0.657217
\(206\) −1.05935 −0.0738081
\(207\) −1.00000 −0.0695048
\(208\) −7.65236 −0.530596
\(209\) 8.84473 0.611803
\(210\) −20.7526 −1.43206
\(211\) −8.02009 −0.552126 −0.276063 0.961140i \(-0.589030\pi\)
−0.276063 + 0.961140i \(0.589030\pi\)
\(212\) −2.32329 −0.159564
\(213\) −15.8477 −1.08587
\(214\) 10.7088 0.732040
\(215\) 8.24083 0.562020
\(216\) 1.45779 0.0991899
\(217\) −6.90466 −0.468719
\(218\) −29.2574 −1.98156
\(219\) −8.02067 −0.541987
\(220\) 6.02707 0.406345
\(221\) 2.63861 0.177492
\(222\) 5.11424 0.343245
\(223\) 18.6751 1.25058 0.625288 0.780394i \(-0.284981\pi\)
0.625288 + 0.780394i \(0.284981\pi\)
\(224\) 20.2031 1.34987
\(225\) 6.71663 0.447776
\(226\) −11.8607 −0.788962
\(227\) 7.54846 0.501009 0.250504 0.968115i \(-0.419404\pi\)
0.250504 + 0.968115i \(0.419404\pi\)
\(228\) 7.02852 0.465475
\(229\) −6.17163 −0.407833 −0.203916 0.978988i \(-0.565367\pi\)
−0.203916 + 0.978988i \(0.565367\pi\)
\(230\) 6.10677 0.402669
\(231\) 5.05851 0.332826
\(232\) −1.45779 −0.0957084
\(233\) 1.49889 0.0981957 0.0490979 0.998794i \(-0.484365\pi\)
0.0490979 + 0.998794i \(0.484365\pi\)
\(234\) −2.74885 −0.179698
\(235\) 2.14278 0.139780
\(236\) −12.2026 −0.794322
\(237\) 2.84758 0.184970
\(238\) −10.3826 −0.673004
\(239\) 2.45089 0.158535 0.0792675 0.996853i \(-0.474742\pi\)
0.0792675 + 0.996853i \(0.474742\pi\)
\(240\) −17.0003 −1.09736
\(241\) −3.23042 −0.208089 −0.104045 0.994573i \(-0.533179\pi\)
−0.104045 + 0.994573i \(0.533179\pi\)
\(242\) 15.6716 1.00741
\(243\) 1.00000 0.0641500
\(244\) −7.51088 −0.480835
\(245\) 15.5689 0.994658
\(246\) −4.90450 −0.312699
\(247\) 9.15509 0.582524
\(248\) −2.96194 −0.188083
\(249\) −16.8010 −1.06472
\(250\) −10.4831 −0.663009
\(251\) 27.8680 1.75902 0.879508 0.475885i \(-0.157872\pi\)
0.879508 + 0.475885i \(0.157872\pi\)
\(252\) 4.01978 0.253222
\(253\) −1.48855 −0.0935842
\(254\) −5.08805 −0.319253
\(255\) 5.86188 0.367085
\(256\) 20.4164 1.27602
\(257\) 11.8216 0.737409 0.368704 0.929547i \(-0.379801\pi\)
0.368704 + 0.929547i \(0.379801\pi\)
\(258\) −4.29516 −0.267405
\(259\) −9.74161 −0.605314
\(260\) 6.23855 0.386899
\(261\) −1.00000 −0.0618984
\(262\) 2.23086 0.137823
\(263\) −11.6260 −0.716890 −0.358445 0.933551i \(-0.616693\pi\)
−0.358445 + 0.933551i \(0.616693\pi\)
\(264\) 2.16999 0.133553
\(265\) −6.72300 −0.412991
\(266\) −36.0241 −2.20878
\(267\) −14.5368 −0.889638
\(268\) −3.96191 −0.242012
\(269\) 23.7915 1.45060 0.725298 0.688435i \(-0.241702\pi\)
0.725298 + 0.688435i \(0.241702\pi\)
\(270\) −6.10677 −0.371646
\(271\) 19.3046 1.17267 0.586335 0.810069i \(-0.300570\pi\)
0.586335 + 0.810069i \(0.300570\pi\)
\(272\) −8.50532 −0.515711
\(273\) 5.23601 0.316898
\(274\) 30.3397 1.83289
\(275\) 9.99803 0.602904
\(276\) −1.18288 −0.0712012
\(277\) −5.11059 −0.307066 −0.153533 0.988144i \(-0.549065\pi\)
−0.153533 + 0.988144i \(0.549065\pi\)
\(278\) 25.3409 1.51985
\(279\) −2.03180 −0.121641
\(280\) 16.9573 1.01339
\(281\) 28.6341 1.70817 0.854084 0.520135i \(-0.174118\pi\)
0.854084 + 0.520135i \(0.174118\pi\)
\(282\) −1.11683 −0.0665062
\(283\) −16.4299 −0.976659 −0.488330 0.872659i \(-0.662394\pi\)
−0.488330 + 0.872659i \(0.662394\pi\)
\(284\) −18.7460 −1.11237
\(285\) 20.3387 1.20476
\(286\) −4.09179 −0.241953
\(287\) 9.34210 0.551447
\(288\) 5.94507 0.350317
\(289\) −14.0673 −0.827487
\(290\) 6.10677 0.358602
\(291\) 6.93709 0.406659
\(292\) −9.48752 −0.555215
\(293\) −5.69142 −0.332496 −0.166248 0.986084i \(-0.553165\pi\)
−0.166248 + 0.986084i \(0.553165\pi\)
\(294\) −8.11457 −0.473252
\(295\) −35.3112 −2.05590
\(296\) −4.17893 −0.242895
\(297\) 1.48855 0.0863743
\(298\) −8.43484 −0.488617
\(299\) −1.54078 −0.0891056
\(300\) 7.94500 0.458705
\(301\) 8.18144 0.471570
\(302\) −24.9530 −1.43589
\(303\) −17.0941 −0.982033
\(304\) −29.5105 −1.69255
\(305\) −21.7345 −1.24452
\(306\) −3.05524 −0.174657
\(307\) 34.0596 1.94389 0.971943 0.235218i \(-0.0755803\pi\)
0.971943 + 0.235218i \(0.0755803\pi\)
\(308\) 5.98363 0.340949
\(309\) 0.593783 0.0337791
\(310\) 12.4078 0.704714
\(311\) −3.80385 −0.215696 −0.107848 0.994167i \(-0.534396\pi\)
−0.107848 + 0.994167i \(0.534396\pi\)
\(312\) 2.24613 0.127162
\(313\) −6.17586 −0.349080 −0.174540 0.984650i \(-0.555844\pi\)
−0.174540 + 0.984650i \(0.555844\pi\)
\(314\) 6.97648 0.393706
\(315\) 11.6322 0.655400
\(316\) 3.36836 0.189485
\(317\) −0.190758 −0.0107140 −0.00535701 0.999986i \(-0.501705\pi\)
−0.00535701 + 0.999986i \(0.501705\pi\)
\(318\) 3.50407 0.196498
\(319\) −1.48855 −0.0833426
\(320\) −2.30462 −0.128832
\(321\) −6.00249 −0.335026
\(322\) 6.06276 0.337865
\(323\) 10.1755 0.566182
\(324\) 1.18288 0.0657158
\(325\) 10.3489 0.574051
\(326\) −43.1310 −2.38881
\(327\) 16.3993 0.906884
\(328\) 4.00754 0.221280
\(329\) 2.12734 0.117284
\(330\) −9.09022 −0.500400
\(331\) 27.8480 1.53066 0.765331 0.643637i \(-0.222575\pi\)
0.765331 + 0.643637i \(0.222575\pi\)
\(332\) −19.8737 −1.09071
\(333\) −2.86662 −0.157090
\(334\) 14.9842 0.819899
\(335\) −11.4647 −0.626386
\(336\) −16.8778 −0.920758
\(337\) −26.8796 −1.46423 −0.732113 0.681183i \(-0.761466\pi\)
−0.732113 + 0.681183i \(0.761466\pi\)
\(338\) 18.9575 1.03115
\(339\) 6.64814 0.361077
\(340\) 6.93392 0.376045
\(341\) −3.02444 −0.163782
\(342\) −10.6006 −0.573217
\(343\) −8.33136 −0.449851
\(344\) 3.50965 0.189228
\(345\) −3.42296 −0.184286
\(346\) −10.0464 −0.540099
\(347\) 19.4468 1.04396 0.521979 0.852958i \(-0.325194\pi\)
0.521979 + 0.852958i \(0.325194\pi\)
\(348\) −1.18288 −0.0634092
\(349\) 7.49870 0.401396 0.200698 0.979653i \(-0.435679\pi\)
0.200698 + 0.979653i \(0.435679\pi\)
\(350\) −40.7214 −2.17665
\(351\) 1.54078 0.0822407
\(352\) 8.84952 0.471681
\(353\) −13.1539 −0.700110 −0.350055 0.936729i \(-0.613837\pi\)
−0.350055 + 0.936729i \(0.613837\pi\)
\(354\) 18.4044 0.978182
\(355\) −54.2460 −2.87908
\(356\) −17.1953 −0.911351
\(357\) 5.81963 0.308008
\(358\) 25.4246 1.34373
\(359\) 4.78965 0.252788 0.126394 0.991980i \(-0.459660\pi\)
0.126394 + 0.991980i \(0.459660\pi\)
\(360\) 4.98994 0.262993
\(361\) 16.3056 0.858192
\(362\) 6.23837 0.327881
\(363\) −8.78423 −0.461052
\(364\) 6.19359 0.324633
\(365\) −27.4544 −1.43703
\(366\) 11.3282 0.592132
\(367\) −5.18576 −0.270695 −0.135347 0.990798i \(-0.543215\pi\)
−0.135347 + 0.990798i \(0.543215\pi\)
\(368\) 4.96655 0.258899
\(369\) 2.74906 0.143110
\(370\) 17.5058 0.910084
\(371\) −6.67455 −0.346526
\(372\) −2.40339 −0.124610
\(373\) −25.4860 −1.31962 −0.659808 0.751434i \(-0.729362\pi\)
−0.659808 + 0.751434i \(0.729362\pi\)
\(374\) −4.54787 −0.235165
\(375\) 5.87596 0.303433
\(376\) 0.912580 0.0470627
\(377\) −1.54078 −0.0793542
\(378\) −6.06276 −0.311835
\(379\) 19.7633 1.01517 0.507586 0.861601i \(-0.330538\pi\)
0.507586 + 0.861601i \(0.330538\pi\)
\(380\) 24.0583 1.23417
\(381\) 2.85194 0.146109
\(382\) −15.0196 −0.768469
\(383\) −0.518090 −0.0264732 −0.0132366 0.999912i \(-0.504213\pi\)
−0.0132366 + 0.999912i \(0.504213\pi\)
\(384\) −10.6890 −0.545469
\(385\) 17.3151 0.882458
\(386\) −40.9490 −2.08425
\(387\) 2.40752 0.122381
\(388\) 8.20577 0.416585
\(389\) −16.2237 −0.822574 −0.411287 0.911506i \(-0.634921\pi\)
−0.411287 + 0.911506i \(0.634921\pi\)
\(390\) −9.40919 −0.476453
\(391\) −1.71252 −0.0866058
\(392\) 6.63055 0.334893
\(393\) −1.25044 −0.0630762
\(394\) −10.7545 −0.541806
\(395\) 9.74714 0.490432
\(396\) 1.76078 0.0884824
\(397\) −11.7502 −0.589727 −0.294863 0.955539i \(-0.595274\pi\)
−0.294863 + 0.955539i \(0.595274\pi\)
\(398\) 5.19455 0.260379
\(399\) 20.1921 1.01087
\(400\) −33.3585 −1.66793
\(401\) −1.11434 −0.0556473 −0.0278236 0.999613i \(-0.508858\pi\)
−0.0278236 + 0.999613i \(0.508858\pi\)
\(402\) 5.97549 0.298030
\(403\) −3.13056 −0.155944
\(404\) −20.2204 −1.00600
\(405\) 3.42296 0.170088
\(406\) 6.06276 0.300890
\(407\) −4.26710 −0.211512
\(408\) 2.49649 0.123595
\(409\) −11.1891 −0.553267 −0.276634 0.960975i \(-0.589219\pi\)
−0.276634 + 0.960975i \(0.589219\pi\)
\(410\) −16.7879 −0.829095
\(411\) −17.0060 −0.838843
\(412\) 0.702376 0.0346036
\(413\) −35.0567 −1.72503
\(414\) 1.78406 0.0876820
\(415\) −57.5093 −2.82302
\(416\) 9.16005 0.449108
\(417\) −14.2040 −0.695575
\(418\) −15.7796 −0.771804
\(419\) −28.0240 −1.36906 −0.684531 0.728983i \(-0.739993\pi\)
−0.684531 + 0.728983i \(0.739993\pi\)
\(420\) 13.7595 0.671397
\(421\) 5.58458 0.272176 0.136088 0.990697i \(-0.456547\pi\)
0.136088 + 0.990697i \(0.456547\pi\)
\(422\) 14.3083 0.696519
\(423\) 0.626003 0.0304373
\(424\) −2.86323 −0.139051
\(425\) 11.5024 0.557947
\(426\) 28.2733 1.36985
\(427\) −21.5779 −1.04423
\(428\) −7.10025 −0.343203
\(429\) 2.29352 0.110732
\(430\) −14.7022 −0.709001
\(431\) 10.7167 0.516205 0.258102 0.966118i \(-0.416903\pi\)
0.258102 + 0.966118i \(0.416903\pi\)
\(432\) −4.96655 −0.238953
\(433\) 17.9099 0.860697 0.430349 0.902663i \(-0.358391\pi\)
0.430349 + 0.902663i \(0.358391\pi\)
\(434\) 12.3183 0.591300
\(435\) −3.42296 −0.164118
\(436\) 19.3985 0.929019
\(437\) −5.94186 −0.284237
\(438\) 14.3094 0.683729
\(439\) −15.0813 −0.719792 −0.359896 0.932992i \(-0.617188\pi\)
−0.359896 + 0.932992i \(0.617188\pi\)
\(440\) 7.42777 0.354105
\(441\) 4.54837 0.216589
\(442\) −4.70746 −0.223911
\(443\) 21.6438 1.02833 0.514164 0.857692i \(-0.328102\pi\)
0.514164 + 0.857692i \(0.328102\pi\)
\(444\) −3.39088 −0.160924
\(445\) −49.7588 −2.35880
\(446\) −33.3176 −1.57763
\(447\) 4.72788 0.223621
\(448\) −2.28801 −0.108098
\(449\) 24.8955 1.17489 0.587445 0.809264i \(-0.300134\pi\)
0.587445 + 0.809264i \(0.300134\pi\)
\(450\) −11.9829 −0.564880
\(451\) 4.09210 0.192690
\(452\) 7.86398 0.369890
\(453\) 13.9866 0.657149
\(454\) −13.4669 −0.632035
\(455\) 17.9226 0.840227
\(456\) 8.66196 0.405634
\(457\) 13.6695 0.639433 0.319717 0.947513i \(-0.396412\pi\)
0.319717 + 0.947513i \(0.396412\pi\)
\(458\) 11.0106 0.514491
\(459\) 1.71252 0.0799336
\(460\) −4.04896 −0.188784
\(461\) 30.3073 1.41155 0.705776 0.708435i \(-0.250598\pi\)
0.705776 + 0.708435i \(0.250598\pi\)
\(462\) −9.02471 −0.419867
\(463\) 21.9221 1.01881 0.509403 0.860528i \(-0.329866\pi\)
0.509403 + 0.860528i \(0.329866\pi\)
\(464\) 4.96655 0.230566
\(465\) −6.95478 −0.322520
\(466\) −2.67412 −0.123876
\(467\) 7.29438 0.337544 0.168772 0.985655i \(-0.446020\pi\)
0.168772 + 0.985655i \(0.446020\pi\)
\(468\) 1.82256 0.0842480
\(469\) −11.3821 −0.525577
\(470\) −3.82286 −0.176335
\(471\) −3.91044 −0.180184
\(472\) −15.0385 −0.692204
\(473\) 3.58370 0.164779
\(474\) −5.08026 −0.233344
\(475\) 39.9093 1.83116
\(476\) 6.88395 0.315525
\(477\) −1.96409 −0.0899296
\(478\) −4.37255 −0.199996
\(479\) −26.8328 −1.22602 −0.613012 0.790074i \(-0.710042\pi\)
−0.613012 + 0.790074i \(0.710042\pi\)
\(480\) 20.3497 0.928834
\(481\) −4.41683 −0.201390
\(482\) 5.76327 0.262510
\(483\) −3.39829 −0.154627
\(484\) −10.3907 −0.472305
\(485\) 23.7454 1.07822
\(486\) −1.78406 −0.0809268
\(487\) 11.3571 0.514638 0.257319 0.966326i \(-0.417161\pi\)
0.257319 + 0.966326i \(0.417161\pi\)
\(488\) −9.25642 −0.419019
\(489\) 24.1757 1.09326
\(490\) −27.7758 −1.25478
\(491\) −21.7557 −0.981821 −0.490911 0.871210i \(-0.663336\pi\)
−0.490911 + 0.871210i \(0.663336\pi\)
\(492\) 3.25182 0.146603
\(493\) −1.71252 −0.0771280
\(494\) −16.3333 −0.734868
\(495\) 5.09523 0.229014
\(496\) 10.0911 0.453102
\(497\) −53.8551 −2.41573
\(498\) 29.9741 1.34317
\(499\) 5.25062 0.235050 0.117525 0.993070i \(-0.462504\pi\)
0.117525 + 0.993070i \(0.462504\pi\)
\(500\) 6.95058 0.310840
\(501\) −8.39892 −0.375236
\(502\) −49.7184 −2.21904
\(503\) −29.2770 −1.30540 −0.652698 0.757618i \(-0.726363\pi\)
−0.652698 + 0.757618i \(0.726363\pi\)
\(504\) 4.95398 0.220668
\(505\) −58.5125 −2.60377
\(506\) 2.65566 0.118059
\(507\) −10.6260 −0.471917
\(508\) 3.37352 0.149676
\(509\) −39.0644 −1.73150 −0.865750 0.500477i \(-0.833158\pi\)
−0.865750 + 0.500477i \(0.833158\pi\)
\(510\) −10.4580 −0.463087
\(511\) −27.2565 −1.20576
\(512\) −15.0462 −0.664953
\(513\) 5.94186 0.262339
\(514\) −21.0904 −0.930259
\(515\) 2.03249 0.0895623
\(516\) 2.84781 0.125368
\(517\) 0.931835 0.0409821
\(518\) 17.3797 0.763618
\(519\) 5.63120 0.247182
\(520\) 7.68840 0.337159
\(521\) −7.75144 −0.339597 −0.169798 0.985479i \(-0.554312\pi\)
−0.169798 + 0.985479i \(0.554312\pi\)
\(522\) 1.78406 0.0780864
\(523\) 28.2941 1.23721 0.618607 0.785701i \(-0.287697\pi\)
0.618607 + 0.785701i \(0.287697\pi\)
\(524\) −1.47912 −0.0646157
\(525\) 22.8251 0.996167
\(526\) 20.7415 0.904373
\(527\) −3.47950 −0.151570
\(528\) −7.39295 −0.321737
\(529\) 1.00000 0.0434783
\(530\) 11.9943 0.520998
\(531\) −10.3160 −0.447676
\(532\) 23.8850 1.03554
\(533\) 4.23569 0.183468
\(534\) 25.9346 1.12230
\(535\) −20.5463 −0.888293
\(536\) −4.88267 −0.210899
\(537\) −14.2510 −0.614975
\(538\) −42.4456 −1.82996
\(539\) 6.77046 0.291624
\(540\) 4.04896 0.174240
\(541\) 14.5626 0.626093 0.313047 0.949738i \(-0.398650\pi\)
0.313047 + 0.949738i \(0.398650\pi\)
\(542\) −34.4406 −1.47935
\(543\) −3.49672 −0.150059
\(544\) 10.1811 0.436509
\(545\) 56.1342 2.40452
\(546\) −9.34138 −0.399774
\(547\) −19.0407 −0.814120 −0.407060 0.913401i \(-0.633446\pi\)
−0.407060 + 0.913401i \(0.633446\pi\)
\(548\) −20.1161 −0.859317
\(549\) −6.34964 −0.270996
\(550\) −17.8371 −0.760577
\(551\) −5.94186 −0.253132
\(552\) −1.45779 −0.0620476
\(553\) 9.67690 0.411504
\(554\) 9.11763 0.387371
\(555\) −9.81232 −0.416510
\(556\) −16.8017 −0.712552
\(557\) 23.4385 0.993121 0.496561 0.868002i \(-0.334596\pi\)
0.496561 + 0.868002i \(0.334596\pi\)
\(558\) 3.62487 0.153453
\(559\) 3.70945 0.156893
\(560\) −57.7719 −2.44131
\(561\) 2.54917 0.107626
\(562\) −51.0851 −2.15489
\(563\) 24.2755 1.02309 0.511546 0.859256i \(-0.329073\pi\)
0.511546 + 0.859256i \(0.329073\pi\)
\(564\) 0.740489 0.0311802
\(565\) 22.7563 0.957364
\(566\) 29.3121 1.23208
\(567\) 3.39829 0.142715
\(568\) −23.1026 −0.969363
\(569\) 3.68449 0.154462 0.0772309 0.997013i \(-0.475392\pi\)
0.0772309 + 0.997013i \(0.475392\pi\)
\(570\) −36.2856 −1.51984
\(571\) 8.23061 0.344440 0.172220 0.985058i \(-0.444906\pi\)
0.172220 + 0.985058i \(0.444906\pi\)
\(572\) 2.71297 0.113435
\(573\) 8.41875 0.351698
\(574\) −16.6669 −0.695663
\(575\) −6.71663 −0.280103
\(576\) −0.673284 −0.0280535
\(577\) −14.9312 −0.621595 −0.310798 0.950476i \(-0.600596\pi\)
−0.310798 + 0.950476i \(0.600596\pi\)
\(578\) 25.0969 1.04389
\(579\) 22.9526 0.953879
\(580\) −4.04896 −0.168124
\(581\) −57.0948 −2.36869
\(582\) −12.3762 −0.513010
\(583\) −2.92364 −0.121085
\(584\) −11.6924 −0.483836
\(585\) 5.27402 0.218054
\(586\) 10.1539 0.419452
\(587\) −32.3414 −1.33487 −0.667436 0.744667i \(-0.732608\pi\)
−0.667436 + 0.744667i \(0.732608\pi\)
\(588\) 5.38019 0.221875
\(589\) −12.0727 −0.497446
\(590\) 62.9974 2.59356
\(591\) 6.02811 0.247963
\(592\) 14.2372 0.585147
\(593\) −40.7038 −1.67150 −0.835752 0.549107i \(-0.814968\pi\)
−0.835752 + 0.549107i \(0.814968\pi\)
\(594\) −2.65566 −0.108963
\(595\) 19.9204 0.816655
\(596\) 5.59253 0.229079
\(597\) −2.91164 −0.119165
\(598\) 2.74885 0.112409
\(599\) 8.42483 0.344229 0.172115 0.985077i \(-0.444940\pi\)
0.172115 + 0.985077i \(0.444940\pi\)
\(600\) 9.79143 0.399733
\(601\) −5.26509 −0.214768 −0.107384 0.994218i \(-0.534247\pi\)
−0.107384 + 0.994218i \(0.534247\pi\)
\(602\) −14.5962 −0.594897
\(603\) −3.34937 −0.136397
\(604\) 16.5445 0.673188
\(605\) −30.0680 −1.22244
\(606\) 30.4970 1.23886
\(607\) 24.9444 1.01246 0.506231 0.862398i \(-0.331038\pi\)
0.506231 + 0.862398i \(0.331038\pi\)
\(608\) 35.3248 1.43261
\(609\) −3.39829 −0.137706
\(610\) 38.7758 1.56999
\(611\) 0.964533 0.0390208
\(612\) 2.02571 0.0818845
\(613\) −47.3822 −1.91375 −0.956874 0.290504i \(-0.906177\pi\)
−0.956874 + 0.290504i \(0.906177\pi\)
\(614\) −60.7645 −2.45226
\(615\) 9.40991 0.379444
\(616\) 7.37424 0.297116
\(617\) 18.4036 0.740900 0.370450 0.928852i \(-0.379203\pi\)
0.370450 + 0.928852i \(0.379203\pi\)
\(618\) −1.05935 −0.0426131
\(619\) 26.9757 1.08424 0.542122 0.840299i \(-0.317621\pi\)
0.542122 + 0.840299i \(0.317621\pi\)
\(620\) −8.22669 −0.330392
\(621\) −1.00000 −0.0401286
\(622\) 6.78631 0.272106
\(623\) −49.4002 −1.97918
\(624\) −7.65236 −0.306340
\(625\) −13.4700 −0.538800
\(626\) 11.0181 0.440373
\(627\) 8.84473 0.353225
\(628\) −4.62560 −0.184582
\(629\) −4.90915 −0.195740
\(630\) −20.7526 −0.826803
\(631\) −20.0249 −0.797179 −0.398589 0.917130i \(-0.630500\pi\)
−0.398589 + 0.917130i \(0.630500\pi\)
\(632\) 4.15117 0.165125
\(633\) −8.02009 −0.318770
\(634\) 0.340324 0.0135160
\(635\) 9.76208 0.387396
\(636\) −2.32329 −0.0921246
\(637\) 7.00803 0.277668
\(638\) 2.65566 0.105139
\(639\) −15.8477 −0.626925
\(640\) −36.5879 −1.44626
\(641\) −0.303011 −0.0119682 −0.00598410 0.999982i \(-0.501905\pi\)
−0.00598410 + 0.999982i \(0.501905\pi\)
\(642\) 10.7088 0.422644
\(643\) −13.8559 −0.546425 −0.273212 0.961954i \(-0.588086\pi\)
−0.273212 + 0.961954i \(0.588086\pi\)
\(644\) −4.01978 −0.158402
\(645\) 8.24083 0.324482
\(646\) −18.1538 −0.714252
\(647\) 2.92725 0.115082 0.0575411 0.998343i \(-0.481674\pi\)
0.0575411 + 0.998343i \(0.481674\pi\)
\(648\) 1.45779 0.0572673
\(649\) −15.3558 −0.602769
\(650\) −18.4630 −0.724179
\(651\) −6.90466 −0.270615
\(652\) 28.5971 1.11995
\(653\) −11.4806 −0.449270 −0.224635 0.974443i \(-0.572119\pi\)
−0.224635 + 0.974443i \(0.572119\pi\)
\(654\) −29.2574 −1.14406
\(655\) −4.28019 −0.167241
\(656\) −13.6534 −0.533074
\(657\) −8.02067 −0.312916
\(658\) −3.79531 −0.147957
\(659\) 16.8223 0.655304 0.327652 0.944798i \(-0.393743\pi\)
0.327652 + 0.944798i \(0.393743\pi\)
\(660\) 6.02707 0.234603
\(661\) 42.3254 1.64627 0.823134 0.567847i \(-0.192223\pi\)
0.823134 + 0.567847i \(0.192223\pi\)
\(662\) −49.6825 −1.93097
\(663\) 2.63861 0.102475
\(664\) −24.4924 −0.950488
\(665\) 69.1168 2.68024
\(666\) 5.11424 0.198173
\(667\) 1.00000 0.0387202
\(668\) −9.93494 −0.384394
\(669\) 18.6751 0.722021
\(670\) 20.4538 0.790200
\(671\) −9.45174 −0.364880
\(672\) 20.2031 0.779351
\(673\) −42.8345 −1.65115 −0.825575 0.564292i \(-0.809149\pi\)
−0.825575 + 0.564292i \(0.809149\pi\)
\(674\) 47.9550 1.84716
\(675\) 6.71663 0.258523
\(676\) −12.5693 −0.483435
\(677\) 0.731629 0.0281188 0.0140594 0.999901i \(-0.495525\pi\)
0.0140594 + 0.999901i \(0.495525\pi\)
\(678\) −11.8607 −0.455508
\(679\) 23.5742 0.904696
\(680\) 8.54537 0.327700
\(681\) 7.54846 0.289258
\(682\) 5.39579 0.206615
\(683\) 32.6387 1.24888 0.624442 0.781071i \(-0.285326\pi\)
0.624442 + 0.781071i \(0.285326\pi\)
\(684\) 7.02852 0.268742
\(685\) −58.2107 −2.22412
\(686\) 14.8637 0.567498
\(687\) −6.17163 −0.235462
\(688\) −11.9571 −0.455859
\(689\) −3.02623 −0.115290
\(690\) 6.10677 0.232481
\(691\) 34.7545 1.32212 0.661062 0.750331i \(-0.270106\pi\)
0.661062 + 0.750331i \(0.270106\pi\)
\(692\) 6.66105 0.253215
\(693\) 5.05851 0.192157
\(694\) −34.6943 −1.31698
\(695\) −48.6198 −1.84426
\(696\) −1.45779 −0.0552573
\(697\) 4.70782 0.178321
\(698\) −13.3782 −0.506371
\(699\) 1.49889 0.0566933
\(700\) 26.9994 1.02048
\(701\) 36.0044 1.35987 0.679934 0.733273i \(-0.262008\pi\)
0.679934 + 0.733273i \(0.262008\pi\)
\(702\) −2.74885 −0.103749
\(703\) −17.0331 −0.642414
\(704\) −1.00221 −0.0377724
\(705\) 2.14278 0.0807018
\(706\) 23.4674 0.883206
\(707\) −58.0908 −2.18473
\(708\) −12.2026 −0.458602
\(709\) −36.0363 −1.35337 −0.676687 0.736271i \(-0.736585\pi\)
−0.676687 + 0.736271i \(0.736585\pi\)
\(710\) 96.7783 3.63203
\(711\) 2.84758 0.106793
\(712\) −21.1916 −0.794187
\(713\) 2.03180 0.0760916
\(714\) −10.3826 −0.388559
\(715\) 7.85063 0.293597
\(716\) −16.8572 −0.629985
\(717\) 2.45089 0.0915302
\(718\) −8.54504 −0.318898
\(719\) 39.4319 1.47056 0.735281 0.677763i \(-0.237050\pi\)
0.735281 + 0.677763i \(0.237050\pi\)
\(720\) −17.0003 −0.633564
\(721\) 2.01784 0.0751485
\(722\) −29.0903 −1.08263
\(723\) −3.23042 −0.120140
\(724\) −4.13621 −0.153721
\(725\) −6.71663 −0.249450
\(726\) 15.6716 0.581628
\(727\) 27.5541 1.02193 0.510963 0.859603i \(-0.329289\pi\)
0.510963 + 0.859603i \(0.329289\pi\)
\(728\) 7.63299 0.282898
\(729\) 1.00000 0.0370370
\(730\) 48.9804 1.81285
\(731\) 4.12292 0.152492
\(732\) −7.51088 −0.277610
\(733\) −9.01914 −0.333129 −0.166565 0.986031i \(-0.553267\pi\)
−0.166565 + 0.986031i \(0.553267\pi\)
\(734\) 9.25174 0.341488
\(735\) 15.5689 0.574266
\(736\) −5.94507 −0.219138
\(737\) −4.98569 −0.183650
\(738\) −4.90450 −0.180537
\(739\) 31.5605 1.16097 0.580486 0.814271i \(-0.302863\pi\)
0.580486 + 0.814271i \(0.302863\pi\)
\(740\) −11.6068 −0.426676
\(741\) 9.15509 0.336321
\(742\) 11.9078 0.437150
\(743\) 8.31854 0.305177 0.152589 0.988290i \(-0.451239\pi\)
0.152589 + 0.988290i \(0.451239\pi\)
\(744\) −2.96194 −0.108590
\(745\) 16.1833 0.592911
\(746\) 45.4687 1.66473
\(747\) −16.8010 −0.614718
\(748\) 3.01537 0.110253
\(749\) −20.3982 −0.745334
\(750\) −10.4831 −0.382789
\(751\) 23.4686 0.856380 0.428190 0.903689i \(-0.359151\pi\)
0.428190 + 0.903689i \(0.359151\pi\)
\(752\) −3.10908 −0.113376
\(753\) 27.8680 1.01557
\(754\) 2.74885 0.100107
\(755\) 47.8756 1.74237
\(756\) 4.01978 0.146198
\(757\) 19.5805 0.711665 0.355832 0.934550i \(-0.384197\pi\)
0.355832 + 0.934550i \(0.384197\pi\)
\(758\) −35.2590 −1.28066
\(759\) −1.48855 −0.0540308
\(760\) 29.6495 1.07550
\(761\) 37.1441 1.34647 0.673236 0.739427i \(-0.264904\pi\)
0.673236 + 0.739427i \(0.264904\pi\)
\(762\) −5.08805 −0.184321
\(763\) 55.7296 2.01755
\(764\) 9.95840 0.360282
\(765\) 5.86188 0.211937
\(766\) 0.924306 0.0333965
\(767\) −15.8947 −0.573923
\(768\) 20.4164 0.736712
\(769\) −32.0627 −1.15621 −0.578106 0.815962i \(-0.696208\pi\)
−0.578106 + 0.815962i \(0.696208\pi\)
\(770\) −30.8912 −1.11324
\(771\) 11.8216 0.425743
\(772\) 27.1503 0.977161
\(773\) −15.9858 −0.574971 −0.287486 0.957785i \(-0.592819\pi\)
−0.287486 + 0.957785i \(0.592819\pi\)
\(774\) −4.29516 −0.154386
\(775\) −13.6469 −0.490210
\(776\) 10.1128 0.363028
\(777\) −9.74161 −0.349478
\(778\) 28.9441 1.03770
\(779\) 16.3345 0.585245
\(780\) 6.23855 0.223376
\(781\) −23.5900 −0.844118
\(782\) 3.05524 0.109255
\(783\) −1.00000 −0.0357371
\(784\) −22.5897 −0.806775
\(785\) −13.3853 −0.477741
\(786\) 2.23086 0.0795721
\(787\) 16.2988 0.580990 0.290495 0.956876i \(-0.406180\pi\)
0.290495 + 0.956876i \(0.406180\pi\)
\(788\) 7.13056 0.254016
\(789\) −11.6260 −0.413896
\(790\) −17.3895 −0.618691
\(791\) 22.5923 0.803290
\(792\) 2.16999 0.0771071
\(793\) −9.78339 −0.347418
\(794\) 20.9632 0.743954
\(795\) −6.72300 −0.238440
\(796\) −3.44413 −0.122074
\(797\) 2.65493 0.0940424 0.0470212 0.998894i \(-0.485027\pi\)
0.0470212 + 0.998894i \(0.485027\pi\)
\(798\) −36.0241 −1.27524
\(799\) 1.07204 0.0379261
\(800\) 39.9309 1.41177
\(801\) −14.5368 −0.513633
\(802\) 1.98805 0.0702003
\(803\) −11.9391 −0.421323
\(804\) −3.96191 −0.139726
\(805\) −11.6322 −0.409981
\(806\) 5.58512 0.196727
\(807\) 23.7915 0.837502
\(808\) −24.9196 −0.876669
\(809\) 40.5385 1.42526 0.712628 0.701542i \(-0.247505\pi\)
0.712628 + 0.701542i \(0.247505\pi\)
\(810\) −6.10677 −0.214570
\(811\) 39.5091 1.38735 0.693677 0.720287i \(-0.255990\pi\)
0.693677 + 0.720287i \(0.255990\pi\)
\(812\) −4.01978 −0.141067
\(813\) 19.3046 0.677042
\(814\) 7.61278 0.266828
\(815\) 82.7524 2.89869
\(816\) −8.50532 −0.297746
\(817\) 14.3051 0.500473
\(818\) 19.9621 0.697960
\(819\) 5.23601 0.182961
\(820\) 11.1308 0.388706
\(821\) 0.156877 0.00547506 0.00273753 0.999996i \(-0.499129\pi\)
0.00273753 + 0.999996i \(0.499129\pi\)
\(822\) 30.3397 1.05822
\(823\) 27.8701 0.971490 0.485745 0.874101i \(-0.338548\pi\)
0.485745 + 0.874101i \(0.338548\pi\)
\(824\) 0.865609 0.0301549
\(825\) 9.99803 0.348087
\(826\) 62.5434 2.17616
\(827\) −36.9020 −1.28321 −0.641605 0.767035i \(-0.721731\pi\)
−0.641605 + 0.767035i \(0.721731\pi\)
\(828\) −1.18288 −0.0411081
\(829\) 2.26800 0.0787709 0.0393855 0.999224i \(-0.487460\pi\)
0.0393855 + 0.999224i \(0.487460\pi\)
\(830\) 102.600 3.56131
\(831\) −5.11059 −0.177285
\(832\) −1.03738 −0.0359647
\(833\) 7.78916 0.269879
\(834\) 25.3409 0.877485
\(835\) −28.7491 −0.994905
\(836\) 10.4623 0.361846
\(837\) −2.03180 −0.0702294
\(838\) 49.9966 1.72711
\(839\) 20.0828 0.693334 0.346667 0.937988i \(-0.387313\pi\)
0.346667 + 0.937988i \(0.387313\pi\)
\(840\) 16.9573 0.585081
\(841\) 1.00000 0.0344828
\(842\) −9.96324 −0.343356
\(843\) 28.6341 0.986211
\(844\) −9.48683 −0.326550
\(845\) −36.3723 −1.25125
\(846\) −1.11683 −0.0383974
\(847\) −29.8513 −1.02570
\(848\) 9.75477 0.334980
\(849\) −16.4299 −0.563874
\(850\) −20.5210 −0.703863
\(851\) 2.86662 0.0982665
\(852\) −18.7460 −0.642227
\(853\) −24.1153 −0.825694 −0.412847 0.910800i \(-0.635466\pi\)
−0.412847 + 0.910800i \(0.635466\pi\)
\(854\) 38.4964 1.31732
\(855\) 20.3387 0.695569
\(856\) −8.75036 −0.299081
\(857\) −10.3653 −0.354071 −0.177036 0.984204i \(-0.556651\pi\)
−0.177036 + 0.984204i \(0.556651\pi\)
\(858\) −4.09179 −0.139691
\(859\) 17.5952 0.600339 0.300170 0.953886i \(-0.402957\pi\)
0.300170 + 0.953886i \(0.402957\pi\)
\(860\) 9.74794 0.332402
\(861\) 9.34210 0.318378
\(862\) −19.1193 −0.651205
\(863\) 47.9193 1.63119 0.815596 0.578622i \(-0.196409\pi\)
0.815596 + 0.578622i \(0.196409\pi\)
\(864\) 5.94507 0.202256
\(865\) 19.2753 0.655382
\(866\) −31.9525 −1.08579
\(867\) −14.0673 −0.477750
\(868\) −8.16740 −0.277220
\(869\) 4.23876 0.143790
\(870\) 6.10677 0.207039
\(871\) −5.16064 −0.174861
\(872\) 23.9067 0.809584
\(873\) 6.93709 0.234785
\(874\) 10.6006 0.358572
\(875\) 19.9682 0.675049
\(876\) −9.48752 −0.320554
\(877\) −42.4419 −1.43316 −0.716580 0.697505i \(-0.754293\pi\)
−0.716580 + 0.697505i \(0.754293\pi\)
\(878\) 26.9060 0.908035
\(879\) −5.69142 −0.191967
\(880\) −25.3057 −0.853057
\(881\) 38.4672 1.29599 0.647996 0.761644i \(-0.275607\pi\)
0.647996 + 0.761644i \(0.275607\pi\)
\(882\) −8.11457 −0.273232
\(883\) −20.5029 −0.689977 −0.344989 0.938607i \(-0.612117\pi\)
−0.344989 + 0.938607i \(0.612117\pi\)
\(884\) 3.12117 0.104976
\(885\) −35.3112 −1.18697
\(886\) −38.6139 −1.29726
\(887\) −16.1194 −0.541236 −0.270618 0.962687i \(-0.587228\pi\)
−0.270618 + 0.962687i \(0.587228\pi\)
\(888\) −4.17893 −0.140236
\(889\) 9.69173 0.325050
\(890\) 88.7729 2.97568
\(891\) 1.48855 0.0498682
\(892\) 22.0905 0.739643
\(893\) 3.71962 0.124472
\(894\) −8.43484 −0.282103
\(895\) −48.7804 −1.63055
\(896\) −36.3242 −1.21351
\(897\) −1.54078 −0.0514451
\(898\) −44.4151 −1.48215
\(899\) 2.03180 0.0677644
\(900\) 7.94500 0.264833
\(901\) −3.36355 −0.112056
\(902\) −7.30058 −0.243083
\(903\) 8.18144 0.272261
\(904\) 9.69157 0.322337
\(905\) −11.9691 −0.397867
\(906\) −24.9530 −0.829009
\(907\) −31.6215 −1.04997 −0.524987 0.851110i \(-0.675930\pi\)
−0.524987 + 0.851110i \(0.675930\pi\)
\(908\) 8.92895 0.296318
\(909\) −17.0941 −0.566977
\(910\) −31.9751 −1.05997
\(911\) 38.6264 1.27975 0.639874 0.768480i \(-0.278986\pi\)
0.639874 + 0.768480i \(0.278986\pi\)
\(912\) −29.5105 −0.977192
\(913\) −25.0092 −0.827682
\(914\) −24.3873 −0.806660
\(915\) −21.7345 −0.718522
\(916\) −7.30032 −0.241209
\(917\) −4.24934 −0.140326
\(918\) −3.05524 −0.100838
\(919\) −12.5300 −0.413326 −0.206663 0.978412i \(-0.566260\pi\)
−0.206663 + 0.978412i \(0.566260\pi\)
\(920\) −4.98994 −0.164514
\(921\) 34.0596 1.12230
\(922\) −54.0702 −1.78071
\(923\) −24.4178 −0.803722
\(924\) 5.98363 0.196847
\(925\) −19.2541 −0.633069
\(926\) −39.1104 −1.28525
\(927\) 0.593783 0.0195024
\(928\) −5.94507 −0.195157
\(929\) 29.9489 0.982592 0.491296 0.870993i \(-0.336523\pi\)
0.491296 + 0.870993i \(0.336523\pi\)
\(930\) 12.4078 0.406867
\(931\) 27.0257 0.885732
\(932\) 1.77302 0.0580771
\(933\) −3.80385 −0.124532
\(934\) −13.0136 −0.425819
\(935\) 8.72568 0.285360
\(936\) 2.24613 0.0734170
\(937\) −38.4575 −1.25635 −0.628177 0.778071i \(-0.716198\pi\)
−0.628177 + 0.778071i \(0.716198\pi\)
\(938\) 20.3064 0.663028
\(939\) −6.17586 −0.201542
\(940\) 2.53466 0.0826716
\(941\) 50.0232 1.63071 0.815354 0.578962i \(-0.196542\pi\)
0.815354 + 0.578962i \(0.196542\pi\)
\(942\) 6.97648 0.227306
\(943\) −2.74906 −0.0895217
\(944\) 51.2349 1.66755
\(945\) 11.6322 0.378395
\(946\) −6.39355 −0.207872
\(947\) −46.4362 −1.50897 −0.754486 0.656316i \(-0.772114\pi\)
−0.754486 + 0.656316i \(0.772114\pi\)
\(948\) 3.36836 0.109399
\(949\) −12.3581 −0.401160
\(950\) −71.2007 −2.31006
\(951\) −0.190758 −0.00618574
\(952\) 8.48379 0.274961
\(953\) 19.8670 0.643556 0.321778 0.946815i \(-0.395720\pi\)
0.321778 + 0.946815i \(0.395720\pi\)
\(954\) 3.50407 0.113448
\(955\) 28.8170 0.932497
\(956\) 2.89912 0.0937642
\(957\) −1.48855 −0.0481179
\(958\) 47.8715 1.54666
\(959\) −57.7912 −1.86618
\(960\) −2.30462 −0.0743813
\(961\) −26.8718 −0.866831
\(962\) 7.87991 0.254058
\(963\) −6.00249 −0.193428
\(964\) −3.82121 −0.123073
\(965\) 78.5659 2.52913
\(966\) 6.06276 0.195066
\(967\) −46.8673 −1.50715 −0.753575 0.657362i \(-0.771672\pi\)
−0.753575 + 0.657362i \(0.771672\pi\)
\(968\) −12.8055 −0.411586
\(969\) 10.1755 0.326885
\(970\) −42.3632 −1.36020
\(971\) −32.0173 −1.02748 −0.513742 0.857945i \(-0.671741\pi\)
−0.513742 + 0.857945i \(0.671741\pi\)
\(972\) 1.18288 0.0379410
\(973\) −48.2695 −1.54745
\(974\) −20.2618 −0.649228
\(975\) 10.3489 0.331428
\(976\) 31.5358 1.00944
\(977\) −6.57514 −0.210357 −0.105179 0.994453i \(-0.533541\pi\)
−0.105179 + 0.994453i \(0.533541\pi\)
\(978\) −43.1310 −1.37918
\(979\) −21.6387 −0.691576
\(980\) 18.4161 0.588282
\(981\) 16.3993 0.523590
\(982\) 38.8136 1.23859
\(983\) −39.0158 −1.24441 −0.622205 0.782854i \(-0.713763\pi\)
−0.622205 + 0.782854i \(0.713763\pi\)
\(984\) 4.00754 0.127756
\(985\) 20.6340 0.657453
\(986\) 3.05524 0.0972988
\(987\) 2.12734 0.0677140
\(988\) 10.8294 0.344529
\(989\) −2.40752 −0.0765546
\(990\) −9.09022 −0.288906
\(991\) 49.9715 1.58740 0.793699 0.608311i \(-0.208153\pi\)
0.793699 + 0.608311i \(0.208153\pi\)
\(992\) −12.0792 −0.383516
\(993\) 27.8480 0.883728
\(994\) 96.0809 3.04750
\(995\) −9.96641 −0.315957
\(996\) −19.8737 −0.629722
\(997\) 30.3821 0.962212 0.481106 0.876662i \(-0.340235\pi\)
0.481106 + 0.876662i \(0.340235\pi\)
\(998\) −9.36744 −0.296521
\(999\) −2.86662 −0.0906959
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 2001.2.a.n.1.5 16
3.2 odd 2 6003.2.a.r.1.12 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2001.2.a.n.1.5 16 1.1 even 1 trivial
6003.2.a.r.1.12 16 3.2 odd 2