Properties

Label 2010.2.a.r.1.4
Level $2010$
Weight $2$
Character 2010.1
Self dual yes
Analytic conductor $16.050$
Analytic rank $1$
Dimension $4$
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] = [2010,2,Mod(1,2010)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2010, 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("2010.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2010 = 2 \cdot 3 \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2010.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: \(16.0499308063\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.70292.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 8x^{2} + 10x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-2.88727\) of defining polynomial
Character \(\chi\) \(=\) 2010.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +4.33630 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +4.33630 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -4.74553 q^{11} -1.00000 q^{12} +0.409227 q^{13} -4.33630 q^{14} +1.00000 q^{15} +1.00000 q^{16} +5.77453 q^{17} -1.00000 q^{18} -5.77453 q^{19} -1.00000 q^{20} -4.33630 q^{21} +4.74553 q^{22} -9.36530 q^{23} +1.00000 q^{24} +1.00000 q^{25} -0.409227 q^{26} -1.00000 q^{27} +4.33630 q^{28} -1.30730 q^{29} -1.00000 q^{30} +2.40923 q^{31} -1.00000 q^{32} +4.74553 q^{33} -5.77453 q^{34} -4.33630 q^{35} +1.00000 q^{36} +3.02900 q^{37} +5.77453 q^{38} -0.409227 q^{39} +1.00000 q^{40} +1.71653 q^{41} +4.33630 q^{42} -9.49106 q^{43} -4.74553 q^{44} -1.00000 q^{45} +9.36530 q^{46} -1.51115 q^{47} -1.00000 q^{48} +11.8035 q^{49} -1.00000 q^{50} -5.77453 q^{51} +0.409227 q^{52} -10.6726 q^{53} +1.00000 q^{54} +4.74553 q^{55} -4.33630 q^{56} +5.77453 q^{57} +1.30730 q^{58} +1.71653 q^{59} +1.00000 q^{60} +6.33630 q^{61} -2.40923 q^{62} +4.33630 q^{63} +1.00000 q^{64} -0.409227 q^{65} -4.74553 q^{66} -1.00000 q^{67} +5.77453 q^{68} +9.36530 q^{69} +4.33630 q^{70} +8.11084 q^{71} -1.00000 q^{72} -8.05800 q^{73} -3.02900 q^{74} -1.00000 q^{75} -5.77453 q^{76} -20.5781 q^{77} +0.409227 q^{78} +7.64878 q^{79} -1.00000 q^{80} +1.00000 q^{81} -1.71653 q^{82} +8.57806 q^{83} -4.33630 q^{84} -5.77453 q^{85} +9.49106 q^{86} +1.30730 q^{87} +4.74553 q^{88} -9.08700 q^{89} +1.00000 q^{90} +1.77453 q^{91} -9.36530 q^{92} -2.40923 q^{93} +1.51115 q^{94} +5.77453 q^{95} +1.00000 q^{96} -4.74553 q^{97} -11.8035 q^{98} -4.74553 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} - 4 q^{5} + 4 q^{6} + q^{7} - 4 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} - 4 q^{5} + 4 q^{6} + q^{7} - 4 q^{8} + 4 q^{9} + 4 q^{10} - 3 q^{11} - 4 q^{12} + 2 q^{13} - q^{14} + 4 q^{15} + 4 q^{16} - 2 q^{17} - 4 q^{18} + 2 q^{19} - 4 q^{20} - q^{21} + 3 q^{22} - 12 q^{23} + 4 q^{24} + 4 q^{25} - 2 q^{26} - 4 q^{27} + q^{28} + 2 q^{29} - 4 q^{30} + 10 q^{31} - 4 q^{32} + 3 q^{33} + 2 q^{34} - q^{35} + 4 q^{36} + 3 q^{37} - 2 q^{38} - 2 q^{39} + 4 q^{40} + q^{42} - 6 q^{43} - 3 q^{44} - 4 q^{45} + 12 q^{46} - 14 q^{47} - 4 q^{48} + 13 q^{49} - 4 q^{50} + 2 q^{51} + 2 q^{52} - 10 q^{53} + 4 q^{54} + 3 q^{55} - q^{56} - 2 q^{57} - 2 q^{58} + 4 q^{60} + 9 q^{61} - 10 q^{62} + q^{63} + 4 q^{64} - 2 q^{65} - 3 q^{66} - 4 q^{67} - 2 q^{68} + 12 q^{69} + q^{70} - 9 q^{71} - 4 q^{72} - 14 q^{73} - 3 q^{74} - 4 q^{75} + 2 q^{76} - 23 q^{77} + 2 q^{78} + 12 q^{79} - 4 q^{80} + 4 q^{81} - 25 q^{83} - q^{84} + 2 q^{85} + 6 q^{86} - 2 q^{87} + 3 q^{88} - 9 q^{89} + 4 q^{90} - 18 q^{91} - 12 q^{92} - 10 q^{93} + 14 q^{94} - 2 q^{95} + 4 q^{96} - 3 q^{97} - 13 q^{98} - 3 q^{99}+O(q^{100}) \) 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
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 1.00000 0.408248
\(7\) 4.33630 1.63897 0.819484 0.573101i \(-0.194260\pi\)
0.819484 + 0.573101i \(0.194260\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) −4.74553 −1.43083 −0.715416 0.698699i \(-0.753763\pi\)
−0.715416 + 0.698699i \(0.753763\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0.409227 0.113499 0.0567495 0.998388i \(-0.481926\pi\)
0.0567495 + 0.998388i \(0.481926\pi\)
\(14\) −4.33630 −1.15893
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) 5.77453 1.40053 0.700265 0.713883i \(-0.253065\pi\)
0.700265 + 0.713883i \(0.253065\pi\)
\(18\) −1.00000 −0.235702
\(19\) −5.77453 −1.32477 −0.662384 0.749164i \(-0.730455\pi\)
−0.662384 + 0.749164i \(0.730455\pi\)
\(20\) −1.00000 −0.223607
\(21\) −4.33630 −0.946259
\(22\) 4.74553 1.01175
\(23\) −9.36530 −1.95280 −0.976401 0.215968i \(-0.930709\pi\)
−0.976401 + 0.215968i \(0.930709\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) −0.409227 −0.0802560
\(27\) −1.00000 −0.192450
\(28\) 4.33630 0.819484
\(29\) −1.30730 −0.242760 −0.121380 0.992606i \(-0.538732\pi\)
−0.121380 + 0.992606i \(0.538732\pi\)
\(30\) −1.00000 −0.182574
\(31\) 2.40923 0.432710 0.216355 0.976315i \(-0.430583\pi\)
0.216355 + 0.976315i \(0.430583\pi\)
\(32\) −1.00000 −0.176777
\(33\) 4.74553 0.826091
\(34\) −5.77453 −0.990324
\(35\) −4.33630 −0.732969
\(36\) 1.00000 0.166667
\(37\) 3.02900 0.497965 0.248982 0.968508i \(-0.419904\pi\)
0.248982 + 0.968508i \(0.419904\pi\)
\(38\) 5.77453 0.936753
\(39\) −0.409227 −0.0655287
\(40\) 1.00000 0.158114
\(41\) 1.71653 0.268077 0.134038 0.990976i \(-0.457205\pi\)
0.134038 + 0.990976i \(0.457205\pi\)
\(42\) 4.33630 0.669106
\(43\) −9.49106 −1.44737 −0.723687 0.690129i \(-0.757554\pi\)
−0.723687 + 0.690129i \(0.757554\pi\)
\(44\) −4.74553 −0.715416
\(45\) −1.00000 −0.149071
\(46\) 9.36530 1.38084
\(47\) −1.51115 −0.220424 −0.110212 0.993908i \(-0.535153\pi\)
−0.110212 + 0.993908i \(0.535153\pi\)
\(48\) −1.00000 −0.144338
\(49\) 11.8035 1.68622
\(50\) −1.00000 −0.141421
\(51\) −5.77453 −0.808596
\(52\) 0.409227 0.0567495
\(53\) −10.6726 −1.46600 −0.732998 0.680231i \(-0.761879\pi\)
−0.732998 + 0.680231i \(0.761879\pi\)
\(54\) 1.00000 0.136083
\(55\) 4.74553 0.639887
\(56\) −4.33630 −0.579463
\(57\) 5.77453 0.764855
\(58\) 1.30730 0.171657
\(59\) 1.71653 0.223473 0.111737 0.993738i \(-0.464359\pi\)
0.111737 + 0.993738i \(0.464359\pi\)
\(60\) 1.00000 0.129099
\(61\) 6.33630 0.811281 0.405640 0.914033i \(-0.367049\pi\)
0.405640 + 0.914033i \(0.367049\pi\)
\(62\) −2.40923 −0.305972
\(63\) 4.33630 0.546323
\(64\) 1.00000 0.125000
\(65\) −0.409227 −0.0507583
\(66\) −4.74553 −0.584134
\(67\) −1.00000 −0.122169
\(68\) 5.77453 0.700265
\(69\) 9.36530 1.12745
\(70\) 4.33630 0.518287
\(71\) 8.11084 0.962579 0.481290 0.876562i \(-0.340169\pi\)
0.481290 + 0.876562i \(0.340169\pi\)
\(72\) −1.00000 −0.117851
\(73\) −8.05800 −0.943118 −0.471559 0.881835i \(-0.656309\pi\)
−0.471559 + 0.881835i \(0.656309\pi\)
\(74\) −3.02900 −0.352114
\(75\) −1.00000 −0.115470
\(76\) −5.77453 −0.662384
\(77\) −20.5781 −2.34509
\(78\) 0.409227 0.0463358
\(79\) 7.64878 0.860554 0.430277 0.902697i \(-0.358416\pi\)
0.430277 + 0.902697i \(0.358416\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −1.71653 −0.189559
\(83\) 8.57806 0.941565 0.470782 0.882249i \(-0.343972\pi\)
0.470782 + 0.882249i \(0.343972\pi\)
\(84\) −4.33630 −0.473130
\(85\) −5.77453 −0.626336
\(86\) 9.49106 1.02345
\(87\) 1.30730 0.140158
\(88\) 4.74553 0.505875
\(89\) −9.08700 −0.963220 −0.481610 0.876386i \(-0.659948\pi\)
−0.481610 + 0.876386i \(0.659948\pi\)
\(90\) 1.00000 0.105409
\(91\) 1.77453 0.186021
\(92\) −9.36530 −0.976401
\(93\) −2.40923 −0.249825
\(94\) 1.51115 0.155863
\(95\) 5.77453 0.592454
\(96\) 1.00000 0.102062
\(97\) −4.74553 −0.481836 −0.240918 0.970546i \(-0.577448\pi\)
−0.240918 + 0.970546i \(0.577448\pi\)
\(98\) −11.8035 −1.19234
\(99\) −4.74553 −0.476944
\(100\) 1.00000 0.100000
\(101\) 8.52006 0.847778 0.423889 0.905714i \(-0.360665\pi\)
0.423889 + 0.905714i \(0.360665\pi\)
\(102\) 5.77453 0.571764
\(103\) −13.4911 −1.32931 −0.664657 0.747149i \(-0.731422\pi\)
−0.664657 + 0.747149i \(0.731422\pi\)
\(104\) −0.409227 −0.0401280
\(105\) 4.33630 0.423180
\(106\) 10.6726 1.03662
\(107\) −8.81845 −0.852512 −0.426256 0.904603i \(-0.640168\pi\)
−0.426256 + 0.904603i \(0.640168\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −17.8854 −1.71311 −0.856554 0.516058i \(-0.827399\pi\)
−0.856554 + 0.516058i \(0.827399\pi\)
\(110\) −4.74553 −0.452469
\(111\) −3.02900 −0.287500
\(112\) 4.33630 0.409742
\(113\) −3.86908 −0.363972 −0.181986 0.983301i \(-0.558253\pi\)
−0.181986 + 0.983301i \(0.558253\pi\)
\(114\) −5.77453 −0.540834
\(115\) 9.36530 0.873319
\(116\) −1.30730 −0.121380
\(117\) 0.409227 0.0378330
\(118\) −1.71653 −0.158019
\(119\) 25.0401 2.29542
\(120\) −1.00000 −0.0912871
\(121\) 11.5201 1.04728
\(122\) −6.33630 −0.573662
\(123\) −1.71653 −0.154774
\(124\) 2.40923 0.216355
\(125\) −1.00000 −0.0894427
\(126\) −4.33630 −0.386309
\(127\) −9.23955 −0.819877 −0.409939 0.912113i \(-0.634450\pi\)
−0.409939 + 0.912113i \(0.634450\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 9.49106 0.835641
\(130\) 0.409227 0.0358916
\(131\) −18.9561 −1.65620 −0.828100 0.560580i \(-0.810578\pi\)
−0.828100 + 0.560580i \(0.810578\pi\)
\(132\) 4.74553 0.413045
\(133\) −25.0401 −2.17125
\(134\) 1.00000 0.0863868
\(135\) 1.00000 0.0860663
\(136\) −5.77453 −0.495162
\(137\) −0.970999 −0.0829581 −0.0414790 0.999139i \(-0.513207\pi\)
−0.0414790 + 0.999139i \(0.513207\pi\)
\(138\) −9.36530 −0.797228
\(139\) 9.41144 0.798268 0.399134 0.916893i \(-0.369311\pi\)
0.399134 + 0.916893i \(0.369311\pi\)
\(140\) −4.33630 −0.366485
\(141\) 1.51115 0.127262
\(142\) −8.11084 −0.680646
\(143\) −1.94200 −0.162398
\(144\) 1.00000 0.0833333
\(145\) 1.30730 0.108566
\(146\) 8.05800 0.666885
\(147\) −11.8035 −0.973539
\(148\) 3.02900 0.248982
\(149\) −1.30730 −0.107098 −0.0535492 0.998565i \(-0.517053\pi\)
−0.0535492 + 0.998565i \(0.517053\pi\)
\(150\) 1.00000 0.0816497
\(151\) −15.3385 −1.24823 −0.624115 0.781332i \(-0.714540\pi\)
−0.624115 + 0.781332i \(0.714540\pi\)
\(152\) 5.77453 0.468376
\(153\) 5.77453 0.466843
\(154\) 20.5781 1.65823
\(155\) −2.40923 −0.193514
\(156\) −0.409227 −0.0327644
\(157\) −7.79615 −0.622201 −0.311100 0.950377i \(-0.600698\pi\)
−0.311100 + 0.950377i \(0.600698\pi\)
\(158\) −7.64878 −0.608504
\(159\) 10.6726 0.846393
\(160\) 1.00000 0.0790569
\(161\) −40.6108 −3.20058
\(162\) −1.00000 −0.0785674
\(163\) −6.97100 −0.546011 −0.273005 0.962013i \(-0.588018\pi\)
−0.273005 + 0.962013i \(0.588018\pi\)
\(164\) 1.71653 0.134038
\(165\) −4.74553 −0.369439
\(166\) −8.57806 −0.665787
\(167\) 0.546851 0.0423166 0.0211583 0.999776i \(-0.493265\pi\)
0.0211583 + 0.999776i \(0.493265\pi\)
\(168\) 4.33630 0.334553
\(169\) −12.8325 −0.987118
\(170\) 5.77453 0.442886
\(171\) −5.77453 −0.441589
\(172\) −9.49106 −0.723687
\(173\) −16.3147 −1.24038 −0.620191 0.784451i \(-0.712945\pi\)
−0.620191 + 0.784451i \(0.712945\pi\)
\(174\) −1.30730 −0.0991064
\(175\) 4.33630 0.327794
\(176\) −4.74553 −0.357708
\(177\) −1.71653 −0.129022
\(178\) 9.08700 0.681100
\(179\) −1.92038 −0.143536 −0.0717679 0.997421i \(-0.522864\pi\)
−0.0717679 + 0.997421i \(0.522864\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 17.4032 1.29357 0.646785 0.762672i \(-0.276113\pi\)
0.646785 + 0.762672i \(0.276113\pi\)
\(182\) −1.77453 −0.131537
\(183\) −6.33630 −0.468393
\(184\) 9.36530 0.690419
\(185\) −3.02900 −0.222697
\(186\) 2.40923 0.176653
\(187\) −27.4032 −2.00392
\(188\) −1.51115 −0.110212
\(189\) −4.33630 −0.315420
\(190\) −5.77453 −0.418929
\(191\) −2.50894 −0.181540 −0.0907702 0.995872i \(-0.528933\pi\)
−0.0907702 + 0.995872i \(0.528933\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 25.9382 1.86707 0.933536 0.358483i \(-0.116706\pi\)
0.933536 + 0.358483i \(0.116706\pi\)
\(194\) 4.74553 0.340709
\(195\) 0.409227 0.0293053
\(196\) 11.8035 0.843109
\(197\) −1.18155 −0.0841817 −0.0420909 0.999114i \(-0.513402\pi\)
−0.0420909 + 0.999114i \(0.513402\pi\)
\(198\) 4.74553 0.337250
\(199\) −16.5781 −1.17519 −0.587594 0.809156i \(-0.699925\pi\)
−0.587594 + 0.809156i \(0.699925\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 1.00000 0.0705346
\(202\) −8.52006 −0.599470
\(203\) −5.66886 −0.397876
\(204\) −5.77453 −0.404298
\(205\) −1.71653 −0.119888
\(206\) 13.4911 0.939967
\(207\) −9.36530 −0.650934
\(208\) 0.409227 0.0283748
\(209\) 27.4032 1.89552
\(210\) −4.33630 −0.299233
\(211\) −25.0617 −1.72532 −0.862661 0.505783i \(-0.831204\pi\)
−0.862661 + 0.505783i \(0.831204\pi\)
\(212\) −10.6726 −0.732998
\(213\) −8.11084 −0.555745
\(214\) 8.81845 0.602817
\(215\) 9.49106 0.647285
\(216\) 1.00000 0.0680414
\(217\) 10.4471 0.709198
\(218\) 17.8854 1.21135
\(219\) 8.05800 0.544509
\(220\) 4.74553 0.319944
\(221\) 2.36309 0.158959
\(222\) 3.02900 0.203293
\(223\) −25.8586 −1.73162 −0.865809 0.500374i \(-0.833196\pi\)
−0.865809 + 0.500374i \(0.833196\pi\)
\(224\) −4.33630 −0.289731
\(225\) 1.00000 0.0666667
\(226\) 3.86908 0.257367
\(227\) 17.7127 1.17564 0.587818 0.808993i \(-0.299987\pi\)
0.587818 + 0.808993i \(0.299987\pi\)
\(228\) 5.77453 0.382428
\(229\) −15.8274 −1.04590 −0.522951 0.852363i \(-0.675169\pi\)
−0.522951 + 0.852363i \(0.675169\pi\)
\(230\) −9.36530 −0.617530
\(231\) 20.5781 1.35394
\(232\) 1.30730 0.0858287
\(233\) −22.4322 −1.46958 −0.734792 0.678293i \(-0.762720\pi\)
−0.734792 + 0.678293i \(0.762720\pi\)
\(234\) −0.409227 −0.0267520
\(235\) 1.51115 0.0985766
\(236\) 1.71653 0.111737
\(237\) −7.64878 −0.496841
\(238\) −25.0401 −1.62311
\(239\) 9.23955 0.597657 0.298828 0.954307i \(-0.403404\pi\)
0.298828 + 0.954307i \(0.403404\pi\)
\(240\) 1.00000 0.0645497
\(241\) 6.17859 0.397998 0.198999 0.980000i \(-0.436231\pi\)
0.198999 + 0.980000i \(0.436231\pi\)
\(242\) −11.5201 −0.740538
\(243\) −1.00000 −0.0641500
\(244\) 6.33630 0.405640
\(245\) −11.8035 −0.754100
\(246\) 1.71653 0.109442
\(247\) −2.36309 −0.150360
\(248\) −2.40923 −0.152986
\(249\) −8.57806 −0.543613
\(250\) 1.00000 0.0632456
\(251\) −6.97100 −0.440006 −0.220003 0.975499i \(-0.570607\pi\)
−0.220003 + 0.975499i \(0.570607\pi\)
\(252\) 4.33630 0.273161
\(253\) 44.4433 2.79413
\(254\) 9.23955 0.579741
\(255\) 5.77453 0.361615
\(256\) 1.00000 0.0625000
\(257\) 14.7090 0.917522 0.458761 0.888560i \(-0.348293\pi\)
0.458761 + 0.888560i \(0.348293\pi\)
\(258\) −9.49106 −0.590888
\(259\) 13.1347 0.816149
\(260\) −0.409227 −0.0253792
\(261\) −1.30730 −0.0809200
\(262\) 18.9561 1.17111
\(263\) 8.79836 0.542530 0.271265 0.962505i \(-0.412558\pi\)
0.271265 + 0.962505i \(0.412558\pi\)
\(264\) −4.74553 −0.292067
\(265\) 10.6726 0.655613
\(266\) 25.0401 1.53531
\(267\) 9.08700 0.556116
\(268\) −1.00000 −0.0610847
\(269\) 28.8564 1.75940 0.879702 0.475526i \(-0.157742\pi\)
0.879702 + 0.475526i \(0.157742\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 13.8124 0.839046 0.419523 0.907745i \(-0.362197\pi\)
0.419523 + 0.907745i \(0.362197\pi\)
\(272\) 5.77453 0.350132
\(273\) −1.77453 −0.107400
\(274\) 0.970999 0.0586602
\(275\) −4.74553 −0.286166
\(276\) 9.36530 0.563725
\(277\) −3.28051 −0.197107 −0.0985535 0.995132i \(-0.531422\pi\)
−0.0985535 + 0.995132i \(0.531422\pi\)
\(278\) −9.41144 −0.564461
\(279\) 2.40923 0.144237
\(280\) 4.33630 0.259144
\(281\) 0.363093 0.0216603 0.0108302 0.999941i \(-0.496553\pi\)
0.0108302 + 0.999941i \(0.496553\pi\)
\(282\) −1.51115 −0.0899877
\(283\) 12.1570 0.722657 0.361328 0.932439i \(-0.382323\pi\)
0.361328 + 0.932439i \(0.382323\pi\)
\(284\) 8.11084 0.481290
\(285\) −5.77453 −0.342054
\(286\) 1.94200 0.114833
\(287\) 7.44340 0.439370
\(288\) −1.00000 −0.0589256
\(289\) 16.3452 0.961483
\(290\) −1.30730 −0.0767675
\(291\) 4.74553 0.278188
\(292\) −8.05800 −0.471559
\(293\) −0.871287 −0.0509012 −0.0254506 0.999676i \(-0.508102\pi\)
−0.0254506 + 0.999676i \(0.508102\pi\)
\(294\) 11.8035 0.688396
\(295\) −1.71653 −0.0999402
\(296\) −3.02900 −0.176057
\(297\) 4.74553 0.275364
\(298\) 1.30730 0.0757300
\(299\) −3.83253 −0.221641
\(300\) −1.00000 −0.0577350
\(301\) −41.1561 −2.37220
\(302\) 15.3385 0.882632
\(303\) −8.52006 −0.489465
\(304\) −5.77453 −0.331192
\(305\) −6.33630 −0.362816
\(306\) −5.77453 −0.330108
\(307\) 26.4583 1.51005 0.755026 0.655694i \(-0.227624\pi\)
0.755026 + 0.655694i \(0.227624\pi\)
\(308\) −20.5781 −1.17254
\(309\) 13.4911 0.767480
\(310\) 2.40923 0.136835
\(311\) −26.1637 −1.48361 −0.741803 0.670618i \(-0.766029\pi\)
−0.741803 + 0.670618i \(0.766029\pi\)
\(312\) 0.409227 0.0231679
\(313\) 12.4404 0.703175 0.351588 0.936155i \(-0.385642\pi\)
0.351588 + 0.936155i \(0.385642\pi\)
\(314\) 7.79615 0.439962
\(315\) −4.33630 −0.244323
\(316\) 7.64878 0.430277
\(317\) −31.2558 −1.75550 −0.877751 0.479116i \(-0.840957\pi\)
−0.877751 + 0.479116i \(0.840957\pi\)
\(318\) −10.6726 −0.598490
\(319\) 6.20385 0.347349
\(320\) −1.00000 −0.0559017
\(321\) 8.81845 0.492198
\(322\) 40.6108 2.26315
\(323\) −33.3452 −1.85538
\(324\) 1.00000 0.0555556
\(325\) 0.409227 0.0226998
\(326\) 6.97100 0.386088
\(327\) 17.8854 0.989063
\(328\) −1.71653 −0.0947795
\(329\) −6.55281 −0.361268
\(330\) 4.74553 0.261233
\(331\) 28.6041 1.57222 0.786112 0.618084i \(-0.212091\pi\)
0.786112 + 0.618084i \(0.212091\pi\)
\(332\) 8.57806 0.470782
\(333\) 3.02900 0.165988
\(334\) −0.546851 −0.0299224
\(335\) 1.00000 0.0546358
\(336\) −4.33630 −0.236565
\(337\) −17.5751 −0.957377 −0.478689 0.877985i \(-0.658888\pi\)
−0.478689 + 0.877985i \(0.658888\pi\)
\(338\) 12.8325 0.697998
\(339\) 3.86908 0.210139
\(340\) −5.77453 −0.313168
\(341\) −11.4331 −0.619135
\(342\) 5.77453 0.312251
\(343\) 20.8296 1.12469
\(344\) 9.49106 0.511724
\(345\) −9.36530 −0.504211
\(346\) 16.3147 0.877083
\(347\) −34.1122 −1.83124 −0.915620 0.402046i \(-0.868299\pi\)
−0.915620 + 0.402046i \(0.868299\pi\)
\(348\) 1.30730 0.0700788
\(349\) −12.9821 −0.694917 −0.347459 0.937695i \(-0.612955\pi\)
−0.347459 + 0.937695i \(0.612955\pi\)
\(350\) −4.33630 −0.231785
\(351\) −0.409227 −0.0218429
\(352\) 4.74553 0.252938
\(353\) 3.55286 0.189100 0.0945498 0.995520i \(-0.469859\pi\)
0.0945498 + 0.995520i \(0.469859\pi\)
\(354\) 1.71653 0.0912325
\(355\) −8.11084 −0.430478
\(356\) −9.08700 −0.481610
\(357\) −25.0401 −1.32526
\(358\) 1.92038 0.101495
\(359\) −22.0230 −1.16233 −0.581165 0.813786i \(-0.697403\pi\)
−0.581165 + 0.813786i \(0.697403\pi\)
\(360\) 1.00000 0.0527046
\(361\) 14.3452 0.755011
\(362\) −17.4032 −0.914693
\(363\) −11.5201 −0.604646
\(364\) 1.77453 0.0930107
\(365\) 8.05800 0.421775
\(366\) 6.33630 0.331204
\(367\) −2.34300 −0.122304 −0.0611519 0.998128i \(-0.519477\pi\)
−0.0611519 + 0.998128i \(0.519477\pi\)
\(368\) −9.36530 −0.488200
\(369\) 1.71653 0.0893590
\(370\) 3.02900 0.157470
\(371\) −46.2797 −2.40272
\(372\) −2.40923 −0.124913
\(373\) 16.9360 0.876912 0.438456 0.898753i \(-0.355525\pi\)
0.438456 + 0.898753i \(0.355525\pi\)
\(374\) 27.4032 1.41699
\(375\) 1.00000 0.0516398
\(376\) 1.51115 0.0779316
\(377\) −0.534983 −0.0275530
\(378\) 4.33630 0.223035
\(379\) 7.93378 0.407531 0.203765 0.979020i \(-0.434682\pi\)
0.203765 + 0.979020i \(0.434682\pi\)
\(380\) 5.77453 0.296227
\(381\) 9.23955 0.473356
\(382\) 2.50894 0.128368
\(383\) 26.3742 1.34766 0.673830 0.738887i \(-0.264648\pi\)
0.673830 + 0.738887i \(0.264648\pi\)
\(384\) 1.00000 0.0510310
\(385\) 20.5781 1.04876
\(386\) −25.9382 −1.32022
\(387\) −9.49106 −0.482458
\(388\) −4.74553 −0.240918
\(389\) 39.4277 1.99907 0.999533 0.0305693i \(-0.00973203\pi\)
0.999533 + 0.0305693i \(0.00973203\pi\)
\(390\) −0.409227 −0.0207220
\(391\) −54.0802 −2.73496
\(392\) −11.8035 −0.596168
\(393\) 18.9561 0.956208
\(394\) 1.18155 0.0595255
\(395\) −7.64878 −0.384852
\(396\) −4.74553 −0.238472
\(397\) −23.8073 −1.19485 −0.597426 0.801924i \(-0.703810\pi\)
−0.597426 + 0.801924i \(0.703810\pi\)
\(398\) 16.5781 0.830983
\(399\) 25.0401 1.25357
\(400\) 1.00000 0.0500000
\(401\) −17.8109 −0.889435 −0.444717 0.895671i \(-0.646696\pi\)
−0.444717 + 0.895671i \(0.646696\pi\)
\(402\) −1.00000 −0.0498755
\(403\) 0.985920 0.0491122
\(404\) 8.52006 0.423889
\(405\) −1.00000 −0.0496904
\(406\) 5.66886 0.281341
\(407\) −14.3742 −0.712503
\(408\) 5.77453 0.285882
\(409\) 9.18155 0.453998 0.226999 0.973895i \(-0.427109\pi\)
0.226999 + 0.973895i \(0.427109\pi\)
\(410\) 1.71653 0.0847734
\(411\) 0.970999 0.0478959
\(412\) −13.4911 −0.664657
\(413\) 7.44340 0.366266
\(414\) 9.36530 0.460280
\(415\) −8.57806 −0.421081
\(416\) −0.409227 −0.0200640
\(417\) −9.41144 −0.460880
\(418\) −27.4032 −1.34034
\(419\) −12.3013 −0.600958 −0.300479 0.953789i \(-0.597146\pi\)
−0.300479 + 0.953789i \(0.597146\pi\)
\(420\) 4.33630 0.211590
\(421\) 32.8363 1.60034 0.800171 0.599772i \(-0.204742\pi\)
0.800171 + 0.599772i \(0.204742\pi\)
\(422\) 25.0617 1.21999
\(423\) −1.51115 −0.0734746
\(424\) 10.6726 0.518308
\(425\) 5.77453 0.280106
\(426\) 8.11084 0.392971
\(427\) 27.4761 1.32966
\(428\) −8.81845 −0.426256
\(429\) 1.94200 0.0937606
\(430\) −9.49106 −0.457700
\(431\) −41.0051 −1.97515 −0.987573 0.157158i \(-0.949767\pi\)
−0.987573 + 0.157158i \(0.949767\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 23.5274 1.13066 0.565328 0.824866i \(-0.308749\pi\)
0.565328 + 0.824866i \(0.308749\pi\)
\(434\) −10.4471 −0.501479
\(435\) −1.30730 −0.0626804
\(436\) −17.8854 −0.856554
\(437\) 54.0802 2.58701
\(438\) −8.05800 −0.385026
\(439\) −22.0691 −1.05330 −0.526651 0.850082i \(-0.676553\pi\)
−0.526651 + 0.850082i \(0.676553\pi\)
\(440\) −4.74553 −0.226234
\(441\) 11.8035 0.562073
\(442\) −2.36309 −0.112401
\(443\) 7.54906 0.358667 0.179333 0.983788i \(-0.442606\pi\)
0.179333 + 0.983788i \(0.442606\pi\)
\(444\) −3.02900 −0.143750
\(445\) 9.08700 0.430765
\(446\) 25.8586 1.22444
\(447\) 1.30730 0.0618333
\(448\) 4.33630 0.204871
\(449\) 27.4398 1.29496 0.647481 0.762081i \(-0.275822\pi\)
0.647481 + 0.762081i \(0.275822\pi\)
\(450\) −1.00000 −0.0471405
\(451\) −8.14585 −0.383573
\(452\) −3.86908 −0.181986
\(453\) 15.3385 0.720666
\(454\) −17.7127 −0.831300
\(455\) −1.77453 −0.0831913
\(456\) −5.77453 −0.270417
\(457\) −14.3891 −0.673095 −0.336548 0.941666i \(-0.609259\pi\)
−0.336548 + 0.941666i \(0.609259\pi\)
\(458\) 15.8274 0.739564
\(459\) −5.77453 −0.269532
\(460\) 9.36530 0.436660
\(461\) −14.5365 −0.677033 −0.338517 0.940960i \(-0.609925\pi\)
−0.338517 + 0.940960i \(0.609925\pi\)
\(462\) −20.5781 −0.957378
\(463\) 11.6124 0.539674 0.269837 0.962906i \(-0.413030\pi\)
0.269837 + 0.962906i \(0.413030\pi\)
\(464\) −1.30730 −0.0606900
\(465\) 2.40923 0.111725
\(466\) 22.4322 1.03915
\(467\) 8.88316 0.411063 0.205532 0.978650i \(-0.434108\pi\)
0.205532 + 0.978650i \(0.434108\pi\)
\(468\) 0.409227 0.0189165
\(469\) −4.33630 −0.200232
\(470\) −1.51115 −0.0697042
\(471\) 7.79615 0.359228
\(472\) −1.71653 −0.0790097
\(473\) 45.0401 2.07095
\(474\) 7.64878 0.351320
\(475\) −5.77453 −0.264954
\(476\) 25.0401 1.14771
\(477\) −10.6726 −0.488665
\(478\) −9.23955 −0.422607
\(479\) 24.5319 1.12089 0.560446 0.828191i \(-0.310630\pi\)
0.560446 + 0.828191i \(0.310630\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 1.23955 0.0565185
\(482\) −6.17859 −0.281427
\(483\) 40.6108 1.84786
\(484\) 11.5201 0.523639
\(485\) 4.74553 0.215483
\(486\) 1.00000 0.0453609
\(487\) −12.9553 −0.587062 −0.293531 0.955950i \(-0.594830\pi\)
−0.293531 + 0.955950i \(0.594830\pi\)
\(488\) −6.33630 −0.286831
\(489\) 6.97100 0.315239
\(490\) 11.8035 0.533229
\(491\) 27.6911 1.24968 0.624841 0.780752i \(-0.285164\pi\)
0.624841 + 0.780752i \(0.285164\pi\)
\(492\) −1.71653 −0.0773871
\(493\) −7.54906 −0.339993
\(494\) 2.36309 0.106321
\(495\) 4.74553 0.213296
\(496\) 2.40923 0.108177
\(497\) 35.1710 1.57764
\(498\) 8.57806 0.384392
\(499\) 32.7737 1.46715 0.733576 0.679607i \(-0.237850\pi\)
0.733576 + 0.679607i \(0.237850\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −0.546851 −0.0244315
\(502\) 6.97100 0.311131
\(503\) 20.5677 0.917070 0.458535 0.888676i \(-0.348374\pi\)
0.458535 + 0.888676i \(0.348374\pi\)
\(504\) −4.33630 −0.193154
\(505\) −8.52006 −0.379138
\(506\) −44.4433 −1.97575
\(507\) 12.8325 0.569913
\(508\) −9.23955 −0.409939
\(509\) −1.97991 −0.0877580 −0.0438790 0.999037i \(-0.513972\pi\)
−0.0438790 + 0.999037i \(0.513972\pi\)
\(510\) −5.77453 −0.255701
\(511\) −34.9419 −1.54574
\(512\) −1.00000 −0.0441942
\(513\) 5.77453 0.254952
\(514\) −14.7090 −0.648786
\(515\) 13.4911 0.594487
\(516\) 9.49106 0.417821
\(517\) 7.17121 0.315389
\(518\) −13.1347 −0.577104
\(519\) 16.3147 0.716135
\(520\) 0.409227 0.0179458
\(521\) 22.6407 0.991905 0.495952 0.868350i \(-0.334819\pi\)
0.495952 + 0.868350i \(0.334819\pi\)
\(522\) 1.30730 0.0572191
\(523\) −23.8519 −1.04297 −0.521485 0.853261i \(-0.674622\pi\)
−0.521485 + 0.853261i \(0.674622\pi\)
\(524\) −18.9561 −0.828100
\(525\) −4.33630 −0.189252
\(526\) −8.79836 −0.383627
\(527\) 13.9122 0.606023
\(528\) 4.74553 0.206523
\(529\) 64.7089 2.81343
\(530\) −10.6726 −0.463588
\(531\) 1.71653 0.0744911
\(532\) −25.0401 −1.08563
\(533\) 0.702450 0.0304265
\(534\) −9.08700 −0.393233
\(535\) 8.81845 0.381255
\(536\) 1.00000 0.0431934
\(537\) 1.92038 0.0828704
\(538\) −28.8564 −1.24409
\(539\) −56.0140 −2.41269
\(540\) 1.00000 0.0430331
\(541\) −4.49919 −0.193435 −0.0967175 0.995312i \(-0.530834\pi\)
−0.0967175 + 0.995312i \(0.530834\pi\)
\(542\) −13.8124 −0.593295
\(543\) −17.4032 −0.746844
\(544\) −5.77453 −0.247581
\(545\) 17.8854 0.766125
\(546\) 1.77453 0.0759429
\(547\) 5.68015 0.242866 0.121433 0.992600i \(-0.461251\pi\)
0.121433 + 0.992600i \(0.461251\pi\)
\(548\) −0.970999 −0.0414790
\(549\) 6.33630 0.270427
\(550\) 4.74553 0.202350
\(551\) 7.54906 0.321601
\(552\) −9.36530 −0.398614
\(553\) 33.1674 1.41042
\(554\) 3.28051 0.139376
\(555\) 3.02900 0.128574
\(556\) 9.41144 0.399134
\(557\) −20.6412 −0.874597 −0.437299 0.899316i \(-0.644065\pi\)
−0.437299 + 0.899316i \(0.644065\pi\)
\(558\) −2.40923 −0.101991
\(559\) −3.88400 −0.164276
\(560\) −4.33630 −0.183242
\(561\) 27.4032 1.15696
\(562\) −0.363093 −0.0153162
\(563\) −2.50894 −0.105739 −0.0528696 0.998601i \(-0.516837\pi\)
−0.0528696 + 0.998601i \(0.516837\pi\)
\(564\) 1.51115 0.0636309
\(565\) 3.86908 0.162773
\(566\) −12.1570 −0.510996
\(567\) 4.33630 0.182108
\(568\) −8.11084 −0.340323
\(569\) 41.6138 1.74454 0.872270 0.489025i \(-0.162647\pi\)
0.872270 + 0.489025i \(0.162647\pi\)
\(570\) 5.77453 0.241869
\(571\) 11.5491 0.483313 0.241657 0.970362i \(-0.422309\pi\)
0.241657 + 0.970362i \(0.422309\pi\)
\(572\) −1.94200 −0.0811990
\(573\) 2.50894 0.104812
\(574\) −7.44340 −0.310681
\(575\) −9.36530 −0.390560
\(576\) 1.00000 0.0416667
\(577\) −40.6211 −1.69108 −0.845540 0.533912i \(-0.820721\pi\)
−0.845540 + 0.533912i \(0.820721\pi\)
\(578\) −16.3452 −0.679871
\(579\) −25.9382 −1.07795
\(580\) 1.30730 0.0542828
\(581\) 37.1971 1.54320
\(582\) −4.74553 −0.196709
\(583\) 50.6472 2.09759
\(584\) 8.05800 0.333442
\(585\) −0.409227 −0.0169194
\(586\) 0.871287 0.0359926
\(587\) 7.82220 0.322857 0.161428 0.986884i \(-0.448390\pi\)
0.161428 + 0.986884i \(0.448390\pi\)
\(588\) −11.8035 −0.486769
\(589\) −13.9122 −0.573240
\(590\) 1.71653 0.0706684
\(591\) 1.18155 0.0486023
\(592\) 3.02900 0.124491
\(593\) −12.4041 −0.509374 −0.254687 0.967024i \(-0.581972\pi\)
−0.254687 + 0.967024i \(0.581972\pi\)
\(594\) −4.74553 −0.194711
\(595\) −25.0401 −1.02655
\(596\) −1.30730 −0.0535492
\(597\) 16.5781 0.678495
\(598\) 3.83253 0.156724
\(599\) 20.5134 0.838153 0.419077 0.907951i \(-0.362354\pi\)
0.419077 + 0.907951i \(0.362354\pi\)
\(600\) 1.00000 0.0408248
\(601\) 20.4984 0.836149 0.418074 0.908413i \(-0.362705\pi\)
0.418074 + 0.908413i \(0.362705\pi\)
\(602\) 41.1561 1.67740
\(603\) −1.00000 −0.0407231
\(604\) −15.3385 −0.624115
\(605\) −11.5201 −0.468357
\(606\) 8.52006 0.346104
\(607\) 8.96430 0.363850 0.181925 0.983312i \(-0.441767\pi\)
0.181925 + 0.983312i \(0.441767\pi\)
\(608\) 5.77453 0.234188
\(609\) 5.66886 0.229714
\(610\) 6.33630 0.256549
\(611\) −0.618403 −0.0250179
\(612\) 5.77453 0.233422
\(613\) −28.6904 −1.15880 −0.579398 0.815045i \(-0.696712\pi\)
−0.579398 + 0.815045i \(0.696712\pi\)
\(614\) −26.4583 −1.06777
\(615\) 1.71653 0.0692172
\(616\) 20.5781 0.829114
\(617\) −2.85483 −0.114931 −0.0574656 0.998347i \(-0.518302\pi\)
−0.0574656 + 0.998347i \(0.518302\pi\)
\(618\) −13.4911 −0.542690
\(619\) −25.7745 −1.03597 −0.517983 0.855391i \(-0.673317\pi\)
−0.517983 + 0.855391i \(0.673317\pi\)
\(620\) −2.40923 −0.0967569
\(621\) 9.36530 0.375817
\(622\) 26.1637 1.04907
\(623\) −39.4040 −1.57869
\(624\) −0.409227 −0.0163822
\(625\) 1.00000 0.0400000
\(626\) −12.4404 −0.497220
\(627\) −27.4032 −1.09438
\(628\) −7.79615 −0.311100
\(629\) 17.4911 0.697414
\(630\) 4.33630 0.172762
\(631\) −29.1398 −1.16004 −0.580019 0.814603i \(-0.696955\pi\)
−0.580019 + 0.814603i \(0.696955\pi\)
\(632\) −7.64878 −0.304252
\(633\) 25.0617 0.996115
\(634\) 31.2558 1.24133
\(635\) 9.23955 0.366660
\(636\) 10.6726 0.423196
\(637\) 4.83032 0.191384
\(638\) −6.20385 −0.245613
\(639\) 8.11084 0.320860
\(640\) 1.00000 0.0395285
\(641\) 31.0535 1.22654 0.613270 0.789873i \(-0.289854\pi\)
0.613270 + 0.789873i \(0.289854\pi\)
\(642\) −8.81845 −0.348037
\(643\) −28.1273 −1.10923 −0.554616 0.832106i \(-0.687135\pi\)
−0.554616 + 0.832106i \(0.687135\pi\)
\(644\) −40.6108 −1.60029
\(645\) −9.49106 −0.373710
\(646\) 33.3452 1.31195
\(647\) 37.2641 1.46500 0.732501 0.680766i \(-0.238353\pi\)
0.732501 + 0.680766i \(0.238353\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −8.14585 −0.319752
\(650\) −0.409227 −0.0160512
\(651\) −10.4471 −0.409456
\(652\) −6.97100 −0.273005
\(653\) −11.1963 −0.438145 −0.219073 0.975709i \(-0.570303\pi\)
−0.219073 + 0.975709i \(0.570303\pi\)
\(654\) −17.8854 −0.699373
\(655\) 18.9561 0.740675
\(656\) 1.71653 0.0670192
\(657\) −8.05800 −0.314373
\(658\) 6.55281 0.255455
\(659\) −18.8727 −0.735174 −0.367587 0.929989i \(-0.619816\pi\)
−0.367587 + 0.929989i \(0.619816\pi\)
\(660\) −4.74553 −0.184720
\(661\) 13.5290 0.526216 0.263108 0.964766i \(-0.415252\pi\)
0.263108 + 0.964766i \(0.415252\pi\)
\(662\) −28.6041 −1.11173
\(663\) −2.36309 −0.0917749
\(664\) −8.57806 −0.332893
\(665\) 25.0401 0.971014
\(666\) −3.02900 −0.117371
\(667\) 12.2433 0.474062
\(668\) 0.546851 0.0211583
\(669\) 25.8586 0.999750
\(670\) −1.00000 −0.0386334
\(671\) −30.0691 −1.16081
\(672\) 4.33630 0.167277
\(673\) −43.2358 −1.66662 −0.833308 0.552809i \(-0.813556\pi\)
−0.833308 + 0.552809i \(0.813556\pi\)
\(674\) 17.5751 0.676968
\(675\) −1.00000 −0.0384900
\(676\) −12.8325 −0.493559
\(677\) 3.63691 0.139778 0.0698888 0.997555i \(-0.477736\pi\)
0.0698888 + 0.997555i \(0.477736\pi\)
\(678\) −3.86908 −0.148591
\(679\) −20.5781 −0.789714
\(680\) 5.77453 0.221443
\(681\) −17.7127 −0.678753
\(682\) 11.4331 0.437794
\(683\) 42.9523 1.64352 0.821762 0.569831i \(-0.192991\pi\)
0.821762 + 0.569831i \(0.192991\pi\)
\(684\) −5.77453 −0.220795
\(685\) 0.970999 0.0371000
\(686\) −20.8296 −0.795277
\(687\) 15.8274 0.603852
\(688\) −9.49106 −0.361843
\(689\) −4.36752 −0.166389
\(690\) 9.36530 0.356531
\(691\) 43.0683 1.63839 0.819197 0.573512i \(-0.194419\pi\)
0.819197 + 0.573512i \(0.194419\pi\)
\(692\) −16.3147 −0.620191
\(693\) −20.5781 −0.781696
\(694\) 34.1122 1.29488
\(695\) −9.41144 −0.356996
\(696\) −1.30730 −0.0495532
\(697\) 9.91216 0.375450
\(698\) 12.9821 0.491381
\(699\) 22.4322 0.848464
\(700\) 4.33630 0.163897
\(701\) 42.5379 1.60663 0.803317 0.595552i \(-0.203067\pi\)
0.803317 + 0.595552i \(0.203067\pi\)
\(702\) 0.409227 0.0154453
\(703\) −17.4911 −0.659688
\(704\) −4.74553 −0.178854
\(705\) −1.51115 −0.0569132
\(706\) −3.55286 −0.133714
\(707\) 36.9456 1.38948
\(708\) −1.71653 −0.0645111
\(709\) −22.8319 −0.857468 −0.428734 0.903431i \(-0.641040\pi\)
−0.428734 + 0.903431i \(0.641040\pi\)
\(710\) 8.11084 0.304394
\(711\) 7.64878 0.286851
\(712\) 9.08700 0.340550
\(713\) −22.5631 −0.844996
\(714\) 25.0401 0.937103
\(715\) 1.94200 0.0726266
\(716\) −1.92038 −0.0717679
\(717\) −9.23955 −0.345057
\(718\) 22.0230 0.821891
\(719\) −11.3869 −0.424661 −0.212330 0.977198i \(-0.568105\pi\)
−0.212330 + 0.977198i \(0.568105\pi\)
\(720\) −1.00000 −0.0372678
\(721\) −58.5013 −2.17870
\(722\) −14.3452 −0.533874
\(723\) −6.17859 −0.229784
\(724\) 17.4032 0.646785
\(725\) −1.30730 −0.0485520
\(726\) 11.5201 0.427550
\(727\) 8.38761 0.311079 0.155540 0.987830i \(-0.450288\pi\)
0.155540 + 0.987830i \(0.450288\pi\)
\(728\) −1.77453 −0.0657685
\(729\) 1.00000 0.0370370
\(730\) −8.05800 −0.298240
\(731\) −54.8064 −2.02709
\(732\) −6.33630 −0.234197
\(733\) −18.8244 −0.695295 −0.347648 0.937625i \(-0.613019\pi\)
−0.347648 + 0.937625i \(0.613019\pi\)
\(734\) 2.34300 0.0864819
\(735\) 11.8035 0.435380
\(736\) 9.36530 0.345210
\(737\) 4.74553 0.174804
\(738\) −1.71653 −0.0631863
\(739\) −1.86238 −0.0685086 −0.0342543 0.999413i \(-0.510906\pi\)
−0.0342543 + 0.999413i \(0.510906\pi\)
\(740\) −3.02900 −0.111348
\(741\) 2.36309 0.0868104
\(742\) 46.2797 1.69898
\(743\) 3.21504 0.117948 0.0589741 0.998260i \(-0.481217\pi\)
0.0589741 + 0.998260i \(0.481217\pi\)
\(744\) 2.40923 0.0883265
\(745\) 1.30730 0.0478959
\(746\) −16.9360 −0.620071
\(747\) 8.57806 0.313855
\(748\) −27.4032 −1.00196
\(749\) −38.2395 −1.39724
\(750\) −1.00000 −0.0365148
\(751\) 14.9821 0.546705 0.273353 0.961914i \(-0.411867\pi\)
0.273353 + 0.961914i \(0.411867\pi\)
\(752\) −1.51115 −0.0551060
\(753\) 6.97100 0.254037
\(754\) 0.534983 0.0194829
\(755\) 15.3385 0.558226
\(756\) −4.33630 −0.157710
\(757\) 47.8749 1.74004 0.870021 0.493015i \(-0.164105\pi\)
0.870021 + 0.493015i \(0.164105\pi\)
\(758\) −7.93378 −0.288168
\(759\) −44.4433 −1.61319
\(760\) −5.77453 −0.209464
\(761\) 10.3266 0.374337 0.187169 0.982328i \(-0.440069\pi\)
0.187169 + 0.982328i \(0.440069\pi\)
\(762\) −9.23955 −0.334714
\(763\) −77.5564 −2.80773
\(764\) −2.50894 −0.0907702
\(765\) −5.77453 −0.208779
\(766\) −26.3742 −0.952939
\(767\) 0.702450 0.0253640
\(768\) −1.00000 −0.0360844
\(769\) −9.60264 −0.346280 −0.173140 0.984897i \(-0.555391\pi\)
−0.173140 + 0.984897i \(0.555391\pi\)
\(770\) −20.5781 −0.741582
\(771\) −14.7090 −0.529731
\(772\) 25.9382 0.933536
\(773\) −32.7151 −1.17668 −0.588340 0.808613i \(-0.700218\pi\)
−0.588340 + 0.808613i \(0.700218\pi\)
\(774\) 9.49106 0.341149
\(775\) 2.40923 0.0865420
\(776\) 4.74553 0.170355
\(777\) −13.1347 −0.471204
\(778\) −39.4277 −1.41355
\(779\) −9.91216 −0.355140
\(780\) 0.409227 0.0146527
\(781\) −38.4902 −1.37729
\(782\) 54.0802 1.93391
\(783\) 1.30730 0.0467192
\(784\) 11.8035 0.421555
\(785\) 7.79615 0.278257
\(786\) −18.9561 −0.676141
\(787\) −15.3795 −0.548219 −0.274110 0.961698i \(-0.588383\pi\)
−0.274110 + 0.961698i \(0.588383\pi\)
\(788\) −1.18155 −0.0420909
\(789\) −8.79836 −0.313230
\(790\) 7.64878 0.272131
\(791\) −16.7775 −0.596539
\(792\) 4.74553 0.168625
\(793\) 2.59299 0.0920796
\(794\) 23.8073 0.844889
\(795\) −10.6726 −0.378518
\(796\) −16.5781 −0.587594
\(797\) −24.5260 −0.868756 −0.434378 0.900731i \(-0.643032\pi\)
−0.434378 + 0.900731i \(0.643032\pi\)
\(798\) −25.0401 −0.886411
\(799\) −8.72619 −0.308710
\(800\) −1.00000 −0.0353553
\(801\) −9.08700 −0.321073
\(802\) 17.8109 0.628925
\(803\) 38.2395 1.34944
\(804\) 1.00000 0.0352673
\(805\) 40.6108 1.43134
\(806\) −0.985920 −0.0347276
\(807\) −28.8564 −1.01579
\(808\) −8.52006 −0.299735
\(809\) 26.0000 0.914111 0.457056 0.889438i \(-0.348904\pi\)
0.457056 + 0.889438i \(0.348904\pi\)
\(810\) 1.00000 0.0351364
\(811\) −36.9329 −1.29689 −0.648445 0.761261i \(-0.724581\pi\)
−0.648445 + 0.761261i \(0.724581\pi\)
\(812\) −5.66886 −0.198938
\(813\) −13.8124 −0.484424
\(814\) 14.3742 0.503816
\(815\) 6.97100 0.244183
\(816\) −5.77453 −0.202149
\(817\) 54.8064 1.91743
\(818\) −9.18155 −0.321025
\(819\) 1.77453 0.0620072
\(820\) −1.71653 −0.0599438
\(821\) 45.4991 1.58793 0.793965 0.607963i \(-0.208013\pi\)
0.793965 + 0.607963i \(0.208013\pi\)
\(822\) −0.970999 −0.0338675
\(823\) 28.9241 1.00823 0.504116 0.863636i \(-0.331818\pi\)
0.504116 + 0.863636i \(0.331818\pi\)
\(824\) 13.4911 0.469983
\(825\) 4.74553 0.165218
\(826\) −7.44340 −0.258989
\(827\) −31.0767 −1.08064 −0.540321 0.841459i \(-0.681697\pi\)
−0.540321 + 0.841459i \(0.681697\pi\)
\(828\) −9.36530 −0.325467
\(829\) 10.2217 0.355013 0.177507 0.984120i \(-0.443197\pi\)
0.177507 + 0.984120i \(0.443197\pi\)
\(830\) 8.57806 0.297749
\(831\) 3.28051 0.113800
\(832\) 0.409227 0.0141874
\(833\) 68.1599 2.36160
\(834\) 9.41144 0.325891
\(835\) −0.546851 −0.0189246
\(836\) 27.4032 0.947760
\(837\) −2.40923 −0.0832751
\(838\) 12.3013 0.424941
\(839\) −55.7870 −1.92598 −0.962991 0.269533i \(-0.913131\pi\)
−0.962991 + 0.269533i \(0.913131\pi\)
\(840\) −4.33630 −0.149617
\(841\) −27.2910 −0.941068
\(842\) −32.8363 −1.13161
\(843\) −0.363093 −0.0125056
\(844\) −25.0617 −0.862661
\(845\) 12.8325 0.441453
\(846\) 1.51115 0.0519544
\(847\) 49.9545 1.71646
\(848\) −10.6726 −0.366499
\(849\) −12.1570 −0.417226
\(850\) −5.77453 −0.198065
\(851\) −28.3675 −0.972426
\(852\) −8.11084 −0.277873
\(853\) 31.9880 1.09525 0.547624 0.836725i \(-0.315533\pi\)
0.547624 + 0.836725i \(0.315533\pi\)
\(854\) −27.4761 −0.940214
\(855\) 5.77453 0.197485
\(856\) 8.81845 0.301409
\(857\) 29.4329 1.00541 0.502704 0.864458i \(-0.332338\pi\)
0.502704 + 0.864458i \(0.332338\pi\)
\(858\) −1.94200 −0.0662987
\(859\) −14.5306 −0.495776 −0.247888 0.968789i \(-0.579737\pi\)
−0.247888 + 0.968789i \(0.579737\pi\)
\(860\) 9.49106 0.323643
\(861\) −7.44340 −0.253670
\(862\) 41.0051 1.39664
\(863\) −20.9442 −0.712949 −0.356475 0.934305i \(-0.616021\pi\)
−0.356475 + 0.934305i \(0.616021\pi\)
\(864\) 1.00000 0.0340207
\(865\) 16.3147 0.554716
\(866\) −23.5274 −0.799495
\(867\) −16.3452 −0.555113
\(868\) 10.4471 0.354599
\(869\) −36.2975 −1.23131
\(870\) 1.30730 0.0443217
\(871\) −0.409227 −0.0138661
\(872\) 17.8854 0.605675
\(873\) −4.74553 −0.160612
\(874\) −54.0802 −1.82929
\(875\) −4.33630 −0.146594
\(876\) 8.05800 0.272255
\(877\) −42.5789 −1.43779 −0.718893 0.695121i \(-0.755351\pi\)
−0.718893 + 0.695121i \(0.755351\pi\)
\(878\) 22.0691 0.744797
\(879\) 0.871287 0.0293878
\(880\) 4.74553 0.159972
\(881\) −40.1851 −1.35387 −0.676936 0.736042i \(-0.736692\pi\)
−0.676936 + 0.736042i \(0.736692\pi\)
\(882\) −11.8035 −0.397446
\(883\) −38.1637 −1.28431 −0.642155 0.766575i \(-0.721959\pi\)
−0.642155 + 0.766575i \(0.721959\pi\)
\(884\) 2.36309 0.0794794
\(885\) 1.71653 0.0577005
\(886\) −7.54906 −0.253616
\(887\) 22.4353 0.753303 0.376651 0.926355i \(-0.377075\pi\)
0.376651 + 0.926355i \(0.377075\pi\)
\(888\) 3.02900 0.101647
\(889\) −40.0655 −1.34375
\(890\) −9.08700 −0.304597
\(891\) −4.74553 −0.158981
\(892\) −25.8586 −0.865809
\(893\) 8.72619 0.292011
\(894\) −1.30730 −0.0437228
\(895\) 1.92038 0.0641911
\(896\) −4.33630 −0.144866
\(897\) 3.83253 0.127965
\(898\) −27.4398 −0.915677
\(899\) −3.14959 −0.105045
\(900\) 1.00000 0.0333333
\(901\) −61.6293 −2.05317
\(902\) 8.14585 0.271227
\(903\) 41.1561 1.36959
\(904\) 3.86908 0.128684
\(905\) −17.4032 −0.578503
\(906\) −15.3385 −0.509588
\(907\) 45.2945 1.50398 0.751990 0.659174i \(-0.229094\pi\)
0.751990 + 0.659174i \(0.229094\pi\)
\(908\) 17.7127 0.587818
\(909\) 8.52006 0.282593
\(910\) 1.77453 0.0588252
\(911\) −6.75528 −0.223813 −0.111906 0.993719i \(-0.535696\pi\)
−0.111906 + 0.993719i \(0.535696\pi\)
\(912\) 5.77453 0.191214
\(913\) −40.7075 −1.34722
\(914\) 14.3891 0.475950
\(915\) 6.33630 0.209472
\(916\) −15.8274 −0.522951
\(917\) −82.1993 −2.71446
\(918\) 5.77453 0.190588
\(919\) −1.48511 −0.0489891 −0.0244946 0.999700i \(-0.507798\pi\)
−0.0244946 + 0.999700i \(0.507798\pi\)
\(920\) −9.36530 −0.308765
\(921\) −26.4583 −0.871829
\(922\) 14.5365 0.478735
\(923\) 3.31917 0.109252
\(924\) 20.5781 0.676969
\(925\) 3.02900 0.0995929
\(926\) −11.6124 −0.381607
\(927\) −13.4911 −0.443105
\(928\) 1.30730 0.0429143
\(929\) 30.3490 0.995717 0.497859 0.867258i \(-0.334120\pi\)
0.497859 + 0.867258i \(0.334120\pi\)
\(930\) −2.40923 −0.0790017
\(931\) −68.1599 −2.23385
\(932\) −22.4322 −0.734792
\(933\) 26.1637 0.856560
\(934\) −8.88316 −0.290666
\(935\) 27.4032 0.896181
\(936\) −0.409227 −0.0133760
\(937\) −53.1992 −1.73794 −0.868971 0.494863i \(-0.835218\pi\)
−0.868971 + 0.494863i \(0.835218\pi\)
\(938\) 4.33630 0.141585
\(939\) −12.4404 −0.405978
\(940\) 1.51115 0.0492883
\(941\) 32.2395 1.05098 0.525489 0.850801i \(-0.323882\pi\)
0.525489 + 0.850801i \(0.323882\pi\)
\(942\) −7.79615 −0.254012
\(943\) −16.0758 −0.523501
\(944\) 1.71653 0.0558683
\(945\) 4.33630 0.141060
\(946\) −45.0401 −1.46438
\(947\) −27.7060 −0.900325 −0.450163 0.892947i \(-0.648634\pi\)
−0.450163 + 0.892947i \(0.648634\pi\)
\(948\) −7.64878 −0.248421
\(949\) −3.29755 −0.107043
\(950\) 5.77453 0.187351
\(951\) 31.2558 1.01354
\(952\) −25.0401 −0.811555
\(953\) −50.3771 −1.63187 −0.815937 0.578140i \(-0.803779\pi\)
−0.815937 + 0.578140i \(0.803779\pi\)
\(954\) 10.6726 0.345538
\(955\) 2.50894 0.0811873
\(956\) 9.23955 0.298828
\(957\) −6.20385 −0.200542
\(958\) −24.5319 −0.792591
\(959\) −4.21055 −0.135966
\(960\) 1.00000 0.0322749
\(961\) −25.1956 −0.812762
\(962\) −1.23955 −0.0399646
\(963\) −8.81845 −0.284171
\(964\) 6.17859 0.198999
\(965\) −25.9382 −0.834980
\(966\) −40.6108 −1.30663
\(967\) 2.23675 0.0719291 0.0359646 0.999353i \(-0.488550\pi\)
0.0359646 + 0.999353i \(0.488550\pi\)
\(968\) −11.5201 −0.370269
\(969\) 33.3452 1.07120
\(970\) −4.74553 −0.152370
\(971\) 37.2878 1.19662 0.598312 0.801263i \(-0.295838\pi\)
0.598312 + 0.801263i \(0.295838\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 40.8109 1.30834
\(974\) 12.9553 0.415116
\(975\) −0.409227 −0.0131057
\(976\) 6.33630 0.202820
\(977\) 13.5261 0.432738 0.216369 0.976312i \(-0.430579\pi\)
0.216369 + 0.976312i \(0.430579\pi\)
\(978\) −6.97100 −0.222908
\(979\) 43.1227 1.37821
\(980\) −11.8035 −0.377050
\(981\) −17.8854 −0.571036
\(982\) −27.6911 −0.883659
\(983\) −59.9924 −1.91346 −0.956730 0.290976i \(-0.906020\pi\)
−0.956730 + 0.290976i \(0.906020\pi\)
\(984\) 1.71653 0.0547210
\(985\) 1.18155 0.0376472
\(986\) 7.54906 0.240411
\(987\) 6.55281 0.208578
\(988\) −2.36309 −0.0751800
\(989\) 88.8867 2.82643
\(990\) −4.74553 −0.150823
\(991\) 3.37353 0.107164 0.0535818 0.998563i \(-0.482936\pi\)
0.0535818 + 0.998563i \(0.482936\pi\)
\(992\) −2.40923 −0.0764930
\(993\) −28.6041 −0.907724
\(994\) −35.1710 −1.11556
\(995\) 16.5781 0.525560
\(996\) −8.57806 −0.271806
\(997\) −10.0647 −0.318752 −0.159376 0.987218i \(-0.550948\pi\)
−0.159376 + 0.987218i \(0.550948\pi\)
\(998\) −32.7737 −1.03743
\(999\) −3.02900 −0.0958333
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 2010.2.a.r.1.4 4
3.2 odd 2 6030.2.a.bu.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2010.2.a.r.1.4 4 1.1 even 1 trivial
6030.2.a.bu.1.4 4 3.2 odd 2