Properties

Label 538.2.a.a.1.1
Level $538$
Weight $2$
Character 538.1
Self dual yes
Analytic conductor $4.296$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [538,2,Mod(1,538)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(538, 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("538.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 538 = 2 \cdot 269 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 538.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: \(4.29595162874\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 538.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.61803 q^{3} +1.00000 q^{4} -1.38197 q^{5} -1.61803 q^{6} +0.236068 q^{7} +1.00000 q^{8} -0.381966 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.61803 q^{3} +1.00000 q^{4} -1.38197 q^{5} -1.61803 q^{6} +0.236068 q^{7} +1.00000 q^{8} -0.381966 q^{9} -1.38197 q^{10} -4.23607 q^{11} -1.61803 q^{12} -1.00000 q^{13} +0.236068 q^{14} +2.23607 q^{15} +1.00000 q^{16} -4.23607 q^{17} -0.381966 q^{18} +2.00000 q^{19} -1.38197 q^{20} -0.381966 q^{21} -4.23607 q^{22} -3.76393 q^{23} -1.61803 q^{24} -3.09017 q^{25} -1.00000 q^{26} +5.47214 q^{27} +0.236068 q^{28} -5.85410 q^{29} +2.23607 q^{30} -1.47214 q^{31} +1.00000 q^{32} +6.85410 q^{33} -4.23607 q^{34} -0.326238 q^{35} -0.381966 q^{36} +4.70820 q^{37} +2.00000 q^{38} +1.61803 q^{39} -1.38197 q^{40} -5.70820 q^{41} -0.381966 q^{42} -2.23607 q^{43} -4.23607 q^{44} +0.527864 q^{45} -3.76393 q^{46} -0.145898 q^{47} -1.61803 q^{48} -6.94427 q^{49} -3.09017 q^{50} +6.85410 q^{51} -1.00000 q^{52} +5.47214 q^{54} +5.85410 q^{55} +0.236068 q^{56} -3.23607 q^{57} -5.85410 q^{58} +13.5623 q^{59} +2.23607 q^{60} +3.47214 q^{61} -1.47214 q^{62} -0.0901699 q^{63} +1.00000 q^{64} +1.38197 q^{65} +6.85410 q^{66} +6.23607 q^{67} -4.23607 q^{68} +6.09017 q^{69} -0.326238 q^{70} -3.70820 q^{71} -0.381966 q^{72} +2.70820 q^{73} +4.70820 q^{74} +5.00000 q^{75} +2.00000 q^{76} -1.00000 q^{77} +1.61803 q^{78} +10.7984 q^{79} -1.38197 q^{80} -7.70820 q^{81} -5.70820 q^{82} -0.763932 q^{83} -0.381966 q^{84} +5.85410 q^{85} -2.23607 q^{86} +9.47214 q^{87} -4.23607 q^{88} +14.7984 q^{89} +0.527864 q^{90} -0.236068 q^{91} -3.76393 q^{92} +2.38197 q^{93} -0.145898 q^{94} -2.76393 q^{95} -1.61803 q^{96} -18.6525 q^{97} -6.94427 q^{98} +1.61803 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - q^{3} + 2 q^{4} - 5 q^{5} - q^{6} - 4 q^{7} + 2 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - q^{3} + 2 q^{4} - 5 q^{5} - q^{6} - 4 q^{7} + 2 q^{8} - 3 q^{9} - 5 q^{10} - 4 q^{11} - q^{12} - 2 q^{13} - 4 q^{14} + 2 q^{16} - 4 q^{17} - 3 q^{18} + 4 q^{19} - 5 q^{20} - 3 q^{21} - 4 q^{22} - 12 q^{23} - q^{24} + 5 q^{25} - 2 q^{26} + 2 q^{27} - 4 q^{28} - 5 q^{29} + 6 q^{31} + 2 q^{32} + 7 q^{33} - 4 q^{34} + 15 q^{35} - 3 q^{36} - 4 q^{37} + 4 q^{38} + q^{39} - 5 q^{40} + 2 q^{41} - 3 q^{42} - 4 q^{44} + 10 q^{45} - 12 q^{46} - 7 q^{47} - q^{48} + 4 q^{49} + 5 q^{50} + 7 q^{51} - 2 q^{52} + 2 q^{54} + 5 q^{55} - 4 q^{56} - 2 q^{57} - 5 q^{58} + 7 q^{59} - 2 q^{61} + 6 q^{62} + 11 q^{63} + 2 q^{64} + 5 q^{65} + 7 q^{66} + 8 q^{67} - 4 q^{68} + q^{69} + 15 q^{70} + 6 q^{71} - 3 q^{72} - 8 q^{73} - 4 q^{74} + 10 q^{75} + 4 q^{76} - 2 q^{77} + q^{78} - 3 q^{79} - 5 q^{80} - 2 q^{81} + 2 q^{82} - 6 q^{83} - 3 q^{84} + 5 q^{85} + 10 q^{87} - 4 q^{88} + 5 q^{89} + 10 q^{90} + 4 q^{91} - 12 q^{92} + 7 q^{93} - 7 q^{94} - 10 q^{95} - q^{96} - 6 q^{97} + 4 q^{98} + q^{99}+O(q^{100}) \) 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
\(3\) −1.61803 −0.934172 −0.467086 0.884212i \(-0.654696\pi\)
−0.467086 + 0.884212i \(0.654696\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.38197 −0.618034 −0.309017 0.951057i \(-0.600000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(6\) −1.61803 −0.660560
\(7\) 0.236068 0.0892253 0.0446127 0.999004i \(-0.485795\pi\)
0.0446127 + 0.999004i \(0.485795\pi\)
\(8\) 1.00000 0.353553
\(9\) −0.381966 −0.127322
\(10\) −1.38197 −0.437016
\(11\) −4.23607 −1.27722 −0.638611 0.769529i \(-0.720491\pi\)
−0.638611 + 0.769529i \(0.720491\pi\)
\(12\) −1.61803 −0.467086
\(13\) −1.00000 −0.277350 −0.138675 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) 0.236068 0.0630918
\(15\) 2.23607 0.577350
\(16\) 1.00000 0.250000
\(17\) −4.23607 −1.02740 −0.513699 0.857971i \(-0.671725\pi\)
−0.513699 + 0.857971i \(0.671725\pi\)
\(18\) −0.381966 −0.0900303
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) −1.38197 −0.309017
\(21\) −0.381966 −0.0833518
\(22\) −4.23607 −0.903133
\(23\) −3.76393 −0.784834 −0.392417 0.919787i \(-0.628361\pi\)
−0.392417 + 0.919787i \(0.628361\pi\)
\(24\) −1.61803 −0.330280
\(25\) −3.09017 −0.618034
\(26\) −1.00000 −0.196116
\(27\) 5.47214 1.05311
\(28\) 0.236068 0.0446127
\(29\) −5.85410 −1.08708 −0.543540 0.839383i \(-0.682916\pi\)
−0.543540 + 0.839383i \(0.682916\pi\)
\(30\) 2.23607 0.408248
\(31\) −1.47214 −0.264403 −0.132202 0.991223i \(-0.542205\pi\)
−0.132202 + 0.991223i \(0.542205\pi\)
\(32\) 1.00000 0.176777
\(33\) 6.85410 1.19315
\(34\) −4.23607 −0.726480
\(35\) −0.326238 −0.0551443
\(36\) −0.381966 −0.0636610
\(37\) 4.70820 0.774024 0.387012 0.922075i \(-0.373507\pi\)
0.387012 + 0.922075i \(0.373507\pi\)
\(38\) 2.00000 0.324443
\(39\) 1.61803 0.259093
\(40\) −1.38197 −0.218508
\(41\) −5.70820 −0.891472 −0.445736 0.895165i \(-0.647058\pi\)
−0.445736 + 0.895165i \(0.647058\pi\)
\(42\) −0.381966 −0.0589386
\(43\) −2.23607 −0.340997 −0.170499 0.985358i \(-0.554538\pi\)
−0.170499 + 0.985358i \(0.554538\pi\)
\(44\) −4.23607 −0.638611
\(45\) 0.527864 0.0786893
\(46\) −3.76393 −0.554962
\(47\) −0.145898 −0.0212814 −0.0106407 0.999943i \(-0.503387\pi\)
−0.0106407 + 0.999943i \(0.503387\pi\)
\(48\) −1.61803 −0.233543
\(49\) −6.94427 −0.992039
\(50\) −3.09017 −0.437016
\(51\) 6.85410 0.959766
\(52\) −1.00000 −0.138675
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 5.47214 0.744663
\(55\) 5.85410 0.789367
\(56\) 0.236068 0.0315459
\(57\) −3.23607 −0.428628
\(58\) −5.85410 −0.768681
\(59\) 13.5623 1.76566 0.882831 0.469691i \(-0.155635\pi\)
0.882831 + 0.469691i \(0.155635\pi\)
\(60\) 2.23607 0.288675
\(61\) 3.47214 0.444561 0.222281 0.974983i \(-0.428650\pi\)
0.222281 + 0.974983i \(0.428650\pi\)
\(62\) −1.47214 −0.186961
\(63\) −0.0901699 −0.0113603
\(64\) 1.00000 0.125000
\(65\) 1.38197 0.171412
\(66\) 6.85410 0.843682
\(67\) 6.23607 0.761857 0.380928 0.924605i \(-0.375604\pi\)
0.380928 + 0.924605i \(0.375604\pi\)
\(68\) −4.23607 −0.513699
\(69\) 6.09017 0.733170
\(70\) −0.326238 −0.0389929
\(71\) −3.70820 −0.440083 −0.220041 0.975491i \(-0.570619\pi\)
−0.220041 + 0.975491i \(0.570619\pi\)
\(72\) −0.381966 −0.0450151
\(73\) 2.70820 0.316971 0.158486 0.987361i \(-0.449339\pi\)
0.158486 + 0.987361i \(0.449339\pi\)
\(74\) 4.70820 0.547318
\(75\) 5.00000 0.577350
\(76\) 2.00000 0.229416
\(77\) −1.00000 −0.113961
\(78\) 1.61803 0.183206
\(79\) 10.7984 1.21491 0.607456 0.794353i \(-0.292190\pi\)
0.607456 + 0.794353i \(0.292190\pi\)
\(80\) −1.38197 −0.154508
\(81\) −7.70820 −0.856467
\(82\) −5.70820 −0.630366
\(83\) −0.763932 −0.0838524 −0.0419262 0.999121i \(-0.513349\pi\)
−0.0419262 + 0.999121i \(0.513349\pi\)
\(84\) −0.381966 −0.0416759
\(85\) 5.85410 0.634967
\(86\) −2.23607 −0.241121
\(87\) 9.47214 1.01552
\(88\) −4.23607 −0.451566
\(89\) 14.7984 1.56862 0.784312 0.620366i \(-0.213016\pi\)
0.784312 + 0.620366i \(0.213016\pi\)
\(90\) 0.527864 0.0556418
\(91\) −0.236068 −0.0247466
\(92\) −3.76393 −0.392417
\(93\) 2.38197 0.246998
\(94\) −0.145898 −0.0150482
\(95\) −2.76393 −0.283573
\(96\) −1.61803 −0.165140
\(97\) −18.6525 −1.89387 −0.946936 0.321422i \(-0.895839\pi\)
−0.946936 + 0.321422i \(0.895839\pi\)
\(98\) −6.94427 −0.701477
\(99\) 1.61803 0.162619
\(100\) −3.09017 −0.309017
\(101\) 6.09017 0.605995 0.302997 0.952991i \(-0.402013\pi\)
0.302997 + 0.952991i \(0.402013\pi\)
\(102\) 6.85410 0.678657
\(103\) 4.14590 0.408507 0.204254 0.978918i \(-0.434523\pi\)
0.204254 + 0.978918i \(0.434523\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0.527864 0.0515143
\(106\) 0 0
\(107\) −7.76393 −0.750568 −0.375284 0.926910i \(-0.622455\pi\)
−0.375284 + 0.926910i \(0.622455\pi\)
\(108\) 5.47214 0.526557
\(109\) 8.70820 0.834095 0.417047 0.908885i \(-0.363065\pi\)
0.417047 + 0.908885i \(0.363065\pi\)
\(110\) 5.85410 0.558167
\(111\) −7.61803 −0.723072
\(112\) 0.236068 0.0223063
\(113\) −17.6180 −1.65737 −0.828683 0.559719i \(-0.810909\pi\)
−0.828683 + 0.559719i \(0.810909\pi\)
\(114\) −3.23607 −0.303086
\(115\) 5.20163 0.485054
\(116\) −5.85410 −0.543540
\(117\) 0.381966 0.0353128
\(118\) 13.5623 1.24851
\(119\) −1.00000 −0.0916698
\(120\) 2.23607 0.204124
\(121\) 6.94427 0.631297
\(122\) 3.47214 0.314352
\(123\) 9.23607 0.832788
\(124\) −1.47214 −0.132202
\(125\) 11.1803 1.00000
\(126\) −0.0901699 −0.00803298
\(127\) −3.79837 −0.337051 −0.168526 0.985697i \(-0.553901\pi\)
−0.168526 + 0.985697i \(0.553901\pi\)
\(128\) 1.00000 0.0883883
\(129\) 3.61803 0.318550
\(130\) 1.38197 0.121206
\(131\) 7.23607 0.632218 0.316109 0.948723i \(-0.397623\pi\)
0.316109 + 0.948723i \(0.397623\pi\)
\(132\) 6.85410 0.596573
\(133\) 0.472136 0.0409394
\(134\) 6.23607 0.538714
\(135\) −7.56231 −0.650860
\(136\) −4.23607 −0.363240
\(137\) −13.3820 −1.14330 −0.571649 0.820498i \(-0.693696\pi\)
−0.571649 + 0.820498i \(0.693696\pi\)
\(138\) 6.09017 0.518430
\(139\) −9.56231 −0.811064 −0.405532 0.914081i \(-0.632914\pi\)
−0.405532 + 0.914081i \(0.632914\pi\)
\(140\) −0.326238 −0.0275721
\(141\) 0.236068 0.0198805
\(142\) −3.70820 −0.311186
\(143\) 4.23607 0.354238
\(144\) −0.381966 −0.0318305
\(145\) 8.09017 0.671852
\(146\) 2.70820 0.224133
\(147\) 11.2361 0.926735
\(148\) 4.70820 0.387012
\(149\) −15.3262 −1.25557 −0.627787 0.778385i \(-0.716039\pi\)
−0.627787 + 0.778385i \(0.716039\pi\)
\(150\) 5.00000 0.408248
\(151\) 7.94427 0.646496 0.323248 0.946314i \(-0.395225\pi\)
0.323248 + 0.946314i \(0.395225\pi\)
\(152\) 2.00000 0.162221
\(153\) 1.61803 0.130810
\(154\) −1.00000 −0.0805823
\(155\) 2.03444 0.163410
\(156\) 1.61803 0.129546
\(157\) −8.65248 −0.690543 −0.345271 0.938503i \(-0.612213\pi\)
−0.345271 + 0.938503i \(0.612213\pi\)
\(158\) 10.7984 0.859072
\(159\) 0 0
\(160\) −1.38197 −0.109254
\(161\) −0.888544 −0.0700271
\(162\) −7.70820 −0.605614
\(163\) 1.67376 0.131099 0.0655496 0.997849i \(-0.479120\pi\)
0.0655496 + 0.997849i \(0.479120\pi\)
\(164\) −5.70820 −0.445736
\(165\) −9.47214 −0.737405
\(166\) −0.763932 −0.0592926
\(167\) 1.38197 0.106940 0.0534699 0.998569i \(-0.482972\pi\)
0.0534699 + 0.998569i \(0.482972\pi\)
\(168\) −0.381966 −0.0294693
\(169\) −12.0000 −0.923077
\(170\) 5.85410 0.448989
\(171\) −0.763932 −0.0584193
\(172\) −2.23607 −0.170499
\(173\) 1.47214 0.111924 0.0559622 0.998433i \(-0.482177\pi\)
0.0559622 + 0.998433i \(0.482177\pi\)
\(174\) 9.47214 0.718081
\(175\) −0.729490 −0.0551443
\(176\) −4.23607 −0.319306
\(177\) −21.9443 −1.64943
\(178\) 14.7984 1.10919
\(179\) 5.29180 0.395527 0.197764 0.980250i \(-0.436632\pi\)
0.197764 + 0.980250i \(0.436632\pi\)
\(180\) 0.527864 0.0393447
\(181\) −7.41641 −0.551257 −0.275629 0.961264i \(-0.588886\pi\)
−0.275629 + 0.961264i \(0.588886\pi\)
\(182\) −0.236068 −0.0174985
\(183\) −5.61803 −0.415297
\(184\) −3.76393 −0.277481
\(185\) −6.50658 −0.478373
\(186\) 2.38197 0.174654
\(187\) 17.9443 1.31222
\(188\) −0.145898 −0.0106407
\(189\) 1.29180 0.0939643
\(190\) −2.76393 −0.200517
\(191\) −22.9443 −1.66019 −0.830095 0.557623i \(-0.811714\pi\)
−0.830095 + 0.557623i \(0.811714\pi\)
\(192\) −1.61803 −0.116772
\(193\) −3.76393 −0.270934 −0.135467 0.990782i \(-0.543253\pi\)
−0.135467 + 0.990782i \(0.543253\pi\)
\(194\) −18.6525 −1.33917
\(195\) −2.23607 −0.160128
\(196\) −6.94427 −0.496019
\(197\) −24.6180 −1.75396 −0.876981 0.480525i \(-0.840446\pi\)
−0.876981 + 0.480525i \(0.840446\pi\)
\(198\) 1.61803 0.114989
\(199\) −7.61803 −0.540028 −0.270014 0.962856i \(-0.587028\pi\)
−0.270014 + 0.962856i \(0.587028\pi\)
\(200\) −3.09017 −0.218508
\(201\) −10.0902 −0.711706
\(202\) 6.09017 0.428503
\(203\) −1.38197 −0.0969950
\(204\) 6.85410 0.479883
\(205\) 7.88854 0.550960
\(206\) 4.14590 0.288858
\(207\) 1.43769 0.0999266
\(208\) −1.00000 −0.0693375
\(209\) −8.47214 −0.586030
\(210\) 0.527864 0.0364261
\(211\) −5.85410 −0.403013 −0.201506 0.979487i \(-0.564584\pi\)
−0.201506 + 0.979487i \(0.564584\pi\)
\(212\) 0 0
\(213\) 6.00000 0.411113
\(214\) −7.76393 −0.530731
\(215\) 3.09017 0.210748
\(216\) 5.47214 0.372332
\(217\) −0.347524 −0.0235915
\(218\) 8.70820 0.589794
\(219\) −4.38197 −0.296106
\(220\) 5.85410 0.394683
\(221\) 4.23607 0.284949
\(222\) −7.61803 −0.511289
\(223\) 10.3262 0.691496 0.345748 0.938327i \(-0.387625\pi\)
0.345748 + 0.938327i \(0.387625\pi\)
\(224\) 0.236068 0.0157730
\(225\) 1.18034 0.0786893
\(226\) −17.6180 −1.17193
\(227\) 5.03444 0.334148 0.167074 0.985944i \(-0.446568\pi\)
0.167074 + 0.985944i \(0.446568\pi\)
\(228\) −3.23607 −0.214314
\(229\) 2.70820 0.178963 0.0894816 0.995988i \(-0.471479\pi\)
0.0894816 + 0.995988i \(0.471479\pi\)
\(230\) 5.20163 0.342985
\(231\) 1.61803 0.106459
\(232\) −5.85410 −0.384341
\(233\) −0.236068 −0.0154653 −0.00773266 0.999970i \(-0.502461\pi\)
−0.00773266 + 0.999970i \(0.502461\pi\)
\(234\) 0.381966 0.0249699
\(235\) 0.201626 0.0131526
\(236\) 13.5623 0.882831
\(237\) −17.4721 −1.13494
\(238\) −1.00000 −0.0648204
\(239\) 14.2705 0.923083 0.461541 0.887119i \(-0.347297\pi\)
0.461541 + 0.887119i \(0.347297\pi\)
\(240\) 2.23607 0.144338
\(241\) 8.70820 0.560945 0.280472 0.959862i \(-0.409509\pi\)
0.280472 + 0.959862i \(0.409509\pi\)
\(242\) 6.94427 0.446395
\(243\) −3.94427 −0.253025
\(244\) 3.47214 0.222281
\(245\) 9.59675 0.613114
\(246\) 9.23607 0.588870
\(247\) −2.00000 −0.127257
\(248\) −1.47214 −0.0934807
\(249\) 1.23607 0.0783326
\(250\) 11.1803 0.707107
\(251\) 12.9098 0.814861 0.407431 0.913236i \(-0.366425\pi\)
0.407431 + 0.913236i \(0.366425\pi\)
\(252\) −0.0901699 −0.00568017
\(253\) 15.9443 1.00241
\(254\) −3.79837 −0.238331
\(255\) −9.47214 −0.593168
\(256\) 1.00000 0.0625000
\(257\) 3.47214 0.216586 0.108293 0.994119i \(-0.465462\pi\)
0.108293 + 0.994119i \(0.465462\pi\)
\(258\) 3.61803 0.225249
\(259\) 1.11146 0.0690625
\(260\) 1.38197 0.0857059
\(261\) 2.23607 0.138409
\(262\) 7.23607 0.447046
\(263\) −10.2705 −0.633307 −0.316653 0.948541i \(-0.602559\pi\)
−0.316653 + 0.948541i \(0.602559\pi\)
\(264\) 6.85410 0.421841
\(265\) 0 0
\(266\) 0.472136 0.0289485
\(267\) −23.9443 −1.46537
\(268\) 6.23607 0.380928
\(269\) 1.00000 0.0609711
\(270\) −7.56231 −0.460227
\(271\) −16.1246 −0.979500 −0.489750 0.871863i \(-0.662912\pi\)
−0.489750 + 0.871863i \(0.662912\pi\)
\(272\) −4.23607 −0.256849
\(273\) 0.381966 0.0231176
\(274\) −13.3820 −0.808434
\(275\) 13.0902 0.789367
\(276\) 6.09017 0.366585
\(277\) −23.6180 −1.41907 −0.709535 0.704670i \(-0.751095\pi\)
−0.709535 + 0.704670i \(0.751095\pi\)
\(278\) −9.56231 −0.573509
\(279\) 0.562306 0.0336644
\(280\) −0.326238 −0.0194964
\(281\) −6.81966 −0.406827 −0.203413 0.979093i \(-0.565204\pi\)
−0.203413 + 0.979093i \(0.565204\pi\)
\(282\) 0.236068 0.0140576
\(283\) 10.2361 0.608471 0.304236 0.952597i \(-0.401599\pi\)
0.304236 + 0.952597i \(0.401599\pi\)
\(284\) −3.70820 −0.220041
\(285\) 4.47214 0.264906
\(286\) 4.23607 0.250484
\(287\) −1.34752 −0.0795418
\(288\) −0.381966 −0.0225076
\(289\) 0.944272 0.0555454
\(290\) 8.09017 0.475071
\(291\) 30.1803 1.76920
\(292\) 2.70820 0.158486
\(293\) 16.4721 0.962312 0.481156 0.876635i \(-0.340217\pi\)
0.481156 + 0.876635i \(0.340217\pi\)
\(294\) 11.2361 0.655301
\(295\) −18.7426 −1.09124
\(296\) 4.70820 0.273659
\(297\) −23.1803 −1.34506
\(298\) −15.3262 −0.887825
\(299\) 3.76393 0.217674
\(300\) 5.00000 0.288675
\(301\) −0.527864 −0.0304256
\(302\) 7.94427 0.457141
\(303\) −9.85410 −0.566103
\(304\) 2.00000 0.114708
\(305\) −4.79837 −0.274754
\(306\) 1.61803 0.0924968
\(307\) −8.12461 −0.463696 −0.231848 0.972752i \(-0.574477\pi\)
−0.231848 + 0.972752i \(0.574477\pi\)
\(308\) −1.00000 −0.0569803
\(309\) −6.70820 −0.381616
\(310\) 2.03444 0.115549
\(311\) −12.7639 −0.723776 −0.361888 0.932222i \(-0.617868\pi\)
−0.361888 + 0.932222i \(0.617868\pi\)
\(312\) 1.61803 0.0916031
\(313\) 19.8541 1.12222 0.561110 0.827741i \(-0.310374\pi\)
0.561110 + 0.827741i \(0.310374\pi\)
\(314\) −8.65248 −0.488287
\(315\) 0.124612 0.00702108
\(316\) 10.7984 0.607456
\(317\) −25.7639 −1.44705 −0.723523 0.690300i \(-0.757479\pi\)
−0.723523 + 0.690300i \(0.757479\pi\)
\(318\) 0 0
\(319\) 24.7984 1.38844
\(320\) −1.38197 −0.0772542
\(321\) 12.5623 0.701160
\(322\) −0.888544 −0.0495166
\(323\) −8.47214 −0.471402
\(324\) −7.70820 −0.428234
\(325\) 3.09017 0.171412
\(326\) 1.67376 0.0927011
\(327\) −14.0902 −0.779188
\(328\) −5.70820 −0.315183
\(329\) −0.0344419 −0.00189884
\(330\) −9.47214 −0.521424
\(331\) 23.1803 1.27411 0.637053 0.770820i \(-0.280153\pi\)
0.637053 + 0.770820i \(0.280153\pi\)
\(332\) −0.763932 −0.0419262
\(333\) −1.79837 −0.0985503
\(334\) 1.38197 0.0756178
\(335\) −8.61803 −0.470853
\(336\) −0.381966 −0.0208380
\(337\) 18.9787 1.03384 0.516918 0.856035i \(-0.327079\pi\)
0.516918 + 0.856035i \(0.327079\pi\)
\(338\) −12.0000 −0.652714
\(339\) 28.5066 1.54826
\(340\) 5.85410 0.317483
\(341\) 6.23607 0.337702
\(342\) −0.763932 −0.0413087
\(343\) −3.29180 −0.177740
\(344\) −2.23607 −0.120561
\(345\) −8.41641 −0.453124
\(346\) 1.47214 0.0791425
\(347\) −31.3607 −1.68353 −0.841765 0.539845i \(-0.818483\pi\)
−0.841765 + 0.539845i \(0.818483\pi\)
\(348\) 9.47214 0.507760
\(349\) −12.2705 −0.656825 −0.328413 0.944534i \(-0.606514\pi\)
−0.328413 + 0.944534i \(0.606514\pi\)
\(350\) −0.729490 −0.0389929
\(351\) −5.47214 −0.292081
\(352\) −4.23607 −0.225783
\(353\) −3.14590 −0.167439 −0.0837196 0.996489i \(-0.526680\pi\)
−0.0837196 + 0.996489i \(0.526680\pi\)
\(354\) −21.9443 −1.16632
\(355\) 5.12461 0.271986
\(356\) 14.7984 0.784312
\(357\) 1.61803 0.0856354
\(358\) 5.29180 0.279680
\(359\) 0.944272 0.0498368 0.0249184 0.999689i \(-0.492067\pi\)
0.0249184 + 0.999689i \(0.492067\pi\)
\(360\) 0.527864 0.0278209
\(361\) −15.0000 −0.789474
\(362\) −7.41641 −0.389798
\(363\) −11.2361 −0.589741
\(364\) −0.236068 −0.0123733
\(365\) −3.74265 −0.195899
\(366\) −5.61803 −0.293659
\(367\) −33.7984 −1.76426 −0.882130 0.471005i \(-0.843891\pi\)
−0.882130 + 0.471005i \(0.843891\pi\)
\(368\) −3.76393 −0.196209
\(369\) 2.18034 0.113504
\(370\) −6.50658 −0.338261
\(371\) 0 0
\(372\) 2.38197 0.123499
\(373\) −10.9443 −0.566673 −0.283336 0.959021i \(-0.591441\pi\)
−0.283336 + 0.959021i \(0.591441\pi\)
\(374\) 17.9443 0.927876
\(375\) −18.0902 −0.934172
\(376\) −0.145898 −0.00752412
\(377\) 5.85410 0.301502
\(378\) 1.29180 0.0664428
\(379\) 12.8541 0.660271 0.330135 0.943934i \(-0.392906\pi\)
0.330135 + 0.943934i \(0.392906\pi\)
\(380\) −2.76393 −0.141787
\(381\) 6.14590 0.314864
\(382\) −22.9443 −1.17393
\(383\) −19.6180 −1.00243 −0.501217 0.865321i \(-0.667114\pi\)
−0.501217 + 0.865321i \(0.667114\pi\)
\(384\) −1.61803 −0.0825700
\(385\) 1.38197 0.0704315
\(386\) −3.76393 −0.191579
\(387\) 0.854102 0.0434164
\(388\) −18.6525 −0.946936
\(389\) 11.0557 0.560548 0.280274 0.959920i \(-0.409575\pi\)
0.280274 + 0.959920i \(0.409575\pi\)
\(390\) −2.23607 −0.113228
\(391\) 15.9443 0.806336
\(392\) −6.94427 −0.350739
\(393\) −11.7082 −0.590601
\(394\) −24.6180 −1.24024
\(395\) −14.9230 −0.750857
\(396\) 1.61803 0.0813093
\(397\) −0.729490 −0.0366121 −0.0183060 0.999832i \(-0.505827\pi\)
−0.0183060 + 0.999832i \(0.505827\pi\)
\(398\) −7.61803 −0.381858
\(399\) −0.763932 −0.0382444
\(400\) −3.09017 −0.154508
\(401\) 29.2705 1.46170 0.730850 0.682538i \(-0.239124\pi\)
0.730850 + 0.682538i \(0.239124\pi\)
\(402\) −10.0902 −0.503252
\(403\) 1.47214 0.0733323
\(404\) 6.09017 0.302997
\(405\) 10.6525 0.529326
\(406\) −1.38197 −0.0685858
\(407\) −19.9443 −0.988601
\(408\) 6.85410 0.339329
\(409\) 19.6525 0.971752 0.485876 0.874028i \(-0.338501\pi\)
0.485876 + 0.874028i \(0.338501\pi\)
\(410\) 7.88854 0.389587
\(411\) 21.6525 1.06804
\(412\) 4.14590 0.204254
\(413\) 3.20163 0.157542
\(414\) 1.43769 0.0706588
\(415\) 1.05573 0.0518237
\(416\) −1.00000 −0.0490290
\(417\) 15.4721 0.757674
\(418\) −8.47214 −0.414386
\(419\) −37.5623 −1.83504 −0.917519 0.397691i \(-0.869812\pi\)
−0.917519 + 0.397691i \(0.869812\pi\)
\(420\) 0.527864 0.0257571
\(421\) 2.52786 0.123201 0.0616003 0.998101i \(-0.480380\pi\)
0.0616003 + 0.998101i \(0.480380\pi\)
\(422\) −5.85410 −0.284973
\(423\) 0.0557281 0.00270959
\(424\) 0 0
\(425\) 13.0902 0.634967
\(426\) 6.00000 0.290701
\(427\) 0.819660 0.0396661
\(428\) −7.76393 −0.375284
\(429\) −6.85410 −0.330919
\(430\) 3.09017 0.149021
\(431\) −6.52786 −0.314436 −0.157218 0.987564i \(-0.550253\pi\)
−0.157218 + 0.987564i \(0.550253\pi\)
\(432\) 5.47214 0.263278
\(433\) 36.7984 1.76842 0.884208 0.467092i \(-0.154698\pi\)
0.884208 + 0.467092i \(0.154698\pi\)
\(434\) −0.347524 −0.0166817
\(435\) −13.0902 −0.627626
\(436\) 8.70820 0.417047
\(437\) −7.52786 −0.360107
\(438\) −4.38197 −0.209378
\(439\) 25.8541 1.23395 0.616974 0.786983i \(-0.288358\pi\)
0.616974 + 0.786983i \(0.288358\pi\)
\(440\) 5.85410 0.279083
\(441\) 2.65248 0.126308
\(442\) 4.23607 0.201489
\(443\) 6.58359 0.312796 0.156398 0.987694i \(-0.450012\pi\)
0.156398 + 0.987694i \(0.450012\pi\)
\(444\) −7.61803 −0.361536
\(445\) −20.4508 −0.969463
\(446\) 10.3262 0.488962
\(447\) 24.7984 1.17292
\(448\) 0.236068 0.0111532
\(449\) 14.0902 0.664956 0.332478 0.943111i \(-0.392115\pi\)
0.332478 + 0.943111i \(0.392115\pi\)
\(450\) 1.18034 0.0556418
\(451\) 24.1803 1.13861
\(452\) −17.6180 −0.828683
\(453\) −12.8541 −0.603938
\(454\) 5.03444 0.236278
\(455\) 0.326238 0.0152943
\(456\) −3.23607 −0.151543
\(457\) 2.20163 0.102988 0.0514939 0.998673i \(-0.483602\pi\)
0.0514939 + 0.998673i \(0.483602\pi\)
\(458\) 2.70820 0.126546
\(459\) −23.1803 −1.08197
\(460\) 5.20163 0.242527
\(461\) −19.1246 −0.890722 −0.445361 0.895351i \(-0.646925\pi\)
−0.445361 + 0.895351i \(0.646925\pi\)
\(462\) 1.61803 0.0752778
\(463\) −8.90983 −0.414075 −0.207037 0.978333i \(-0.566382\pi\)
−0.207037 + 0.978333i \(0.566382\pi\)
\(464\) −5.85410 −0.271770
\(465\) −3.29180 −0.152653
\(466\) −0.236068 −0.0109356
\(467\) −33.4164 −1.54633 −0.773163 0.634207i \(-0.781327\pi\)
−0.773163 + 0.634207i \(0.781327\pi\)
\(468\) 0.381966 0.0176564
\(469\) 1.47214 0.0679769
\(470\) 0.201626 0.00930032
\(471\) 14.0000 0.645086
\(472\) 13.5623 0.624256
\(473\) 9.47214 0.435529
\(474\) −17.4721 −0.802522
\(475\) −6.18034 −0.283573
\(476\) −1.00000 −0.0458349
\(477\) 0 0
\(478\) 14.2705 0.652718
\(479\) 12.5967 0.575560 0.287780 0.957697i \(-0.407083\pi\)
0.287780 + 0.957697i \(0.407083\pi\)
\(480\) 2.23607 0.102062
\(481\) −4.70820 −0.214676
\(482\) 8.70820 0.396648
\(483\) 1.43769 0.0654173
\(484\) 6.94427 0.315649
\(485\) 25.7771 1.17048
\(486\) −3.94427 −0.178916
\(487\) 38.8328 1.75968 0.879841 0.475267i \(-0.157649\pi\)
0.879841 + 0.475267i \(0.157649\pi\)
\(488\) 3.47214 0.157176
\(489\) −2.70820 −0.122469
\(490\) 9.59675 0.433537
\(491\) 11.2361 0.507077 0.253538 0.967325i \(-0.418406\pi\)
0.253538 + 0.967325i \(0.418406\pi\)
\(492\) 9.23607 0.416394
\(493\) 24.7984 1.11686
\(494\) −2.00000 −0.0899843
\(495\) −2.23607 −0.100504
\(496\) −1.47214 −0.0661009
\(497\) −0.875388 −0.0392665
\(498\) 1.23607 0.0553895
\(499\) −5.94427 −0.266102 −0.133051 0.991109i \(-0.542477\pi\)
−0.133051 + 0.991109i \(0.542477\pi\)
\(500\) 11.1803 0.500000
\(501\) −2.23607 −0.0999001
\(502\) 12.9098 0.576194
\(503\) −10.3607 −0.461960 −0.230980 0.972959i \(-0.574193\pi\)
−0.230980 + 0.972959i \(0.574193\pi\)
\(504\) −0.0901699 −0.00401649
\(505\) −8.41641 −0.374525
\(506\) 15.9443 0.708809
\(507\) 19.4164 0.862313
\(508\) −3.79837 −0.168526
\(509\) 42.3262 1.87608 0.938039 0.346530i \(-0.112640\pi\)
0.938039 + 0.346530i \(0.112640\pi\)
\(510\) −9.47214 −0.419433
\(511\) 0.639320 0.0282819
\(512\) 1.00000 0.0441942
\(513\) 10.9443 0.483201
\(514\) 3.47214 0.153149
\(515\) −5.72949 −0.252472
\(516\) 3.61803 0.159275
\(517\) 0.618034 0.0271811
\(518\) 1.11146 0.0488346
\(519\) −2.38197 −0.104557
\(520\) 1.38197 0.0606032
\(521\) 3.58359 0.157000 0.0785000 0.996914i \(-0.474987\pi\)
0.0785000 + 0.996914i \(0.474987\pi\)
\(522\) 2.23607 0.0978700
\(523\) 25.3820 1.10988 0.554938 0.831892i \(-0.312742\pi\)
0.554938 + 0.831892i \(0.312742\pi\)
\(524\) 7.23607 0.316109
\(525\) 1.18034 0.0515143
\(526\) −10.2705 −0.447816
\(527\) 6.23607 0.271647
\(528\) 6.85410 0.298287
\(529\) −8.83282 −0.384035
\(530\) 0 0
\(531\) −5.18034 −0.224808
\(532\) 0.472136 0.0204697
\(533\) 5.70820 0.247250
\(534\) −23.9443 −1.03617
\(535\) 10.7295 0.463876
\(536\) 6.23607 0.269357
\(537\) −8.56231 −0.369491
\(538\) 1.00000 0.0431131
\(539\) 29.4164 1.26705
\(540\) −7.56231 −0.325430
\(541\) 11.5279 0.495622 0.247811 0.968808i \(-0.420289\pi\)
0.247811 + 0.968808i \(0.420289\pi\)
\(542\) −16.1246 −0.692611
\(543\) 12.0000 0.514969
\(544\) −4.23607 −0.181620
\(545\) −12.0344 −0.515499
\(546\) 0.381966 0.0163466
\(547\) 27.7984 1.18857 0.594286 0.804254i \(-0.297435\pi\)
0.594286 + 0.804254i \(0.297435\pi\)
\(548\) −13.3820 −0.571649
\(549\) −1.32624 −0.0566025
\(550\) 13.0902 0.558167
\(551\) −11.7082 −0.498786
\(552\) 6.09017 0.259215
\(553\) 2.54915 0.108401
\(554\) −23.6180 −1.00343
\(555\) 10.5279 0.446883
\(556\) −9.56231 −0.405532
\(557\) 41.7082 1.76723 0.883617 0.468211i \(-0.155101\pi\)
0.883617 + 0.468211i \(0.155101\pi\)
\(558\) 0.562306 0.0238043
\(559\) 2.23607 0.0945756
\(560\) −0.326238 −0.0137861
\(561\) −29.0344 −1.22584
\(562\) −6.81966 −0.287670
\(563\) 20.7082 0.872747 0.436373 0.899766i \(-0.356263\pi\)
0.436373 + 0.899766i \(0.356263\pi\)
\(564\) 0.236068 0.00994026
\(565\) 24.3475 1.02431
\(566\) 10.2361 0.430254
\(567\) −1.81966 −0.0764185
\(568\) −3.70820 −0.155593
\(569\) −17.0902 −0.716457 −0.358229 0.933634i \(-0.616619\pi\)
−0.358229 + 0.933634i \(0.616619\pi\)
\(570\) 4.47214 0.187317
\(571\) 10.0000 0.418487 0.209243 0.977864i \(-0.432900\pi\)
0.209243 + 0.977864i \(0.432900\pi\)
\(572\) 4.23607 0.177119
\(573\) 37.1246 1.55090
\(574\) −1.34752 −0.0562446
\(575\) 11.6312 0.485054
\(576\) −0.381966 −0.0159153
\(577\) 3.00000 0.124892 0.0624458 0.998048i \(-0.480110\pi\)
0.0624458 + 0.998048i \(0.480110\pi\)
\(578\) 0.944272 0.0392765
\(579\) 6.09017 0.253099
\(580\) 8.09017 0.335926
\(581\) −0.180340 −0.00748176
\(582\) 30.1803 1.25102
\(583\) 0 0
\(584\) 2.70820 0.112066
\(585\) −0.527864 −0.0218245
\(586\) 16.4721 0.680458
\(587\) −26.3262 −1.08660 −0.543300 0.839539i \(-0.682825\pi\)
−0.543300 + 0.839539i \(0.682825\pi\)
\(588\) 11.2361 0.463368
\(589\) −2.94427 −0.121317
\(590\) −18.7426 −0.771623
\(591\) 39.8328 1.63850
\(592\) 4.70820 0.193506
\(593\) −27.4164 −1.12586 −0.562928 0.826506i \(-0.690325\pi\)
−0.562928 + 0.826506i \(0.690325\pi\)
\(594\) −23.1803 −0.951101
\(595\) 1.38197 0.0566551
\(596\) −15.3262 −0.627787
\(597\) 12.3262 0.504479
\(598\) 3.76393 0.153919
\(599\) 33.5967 1.37273 0.686363 0.727259i \(-0.259206\pi\)
0.686363 + 0.727259i \(0.259206\pi\)
\(600\) 5.00000 0.204124
\(601\) 34.6869 1.41491 0.707454 0.706759i \(-0.249843\pi\)
0.707454 + 0.706759i \(0.249843\pi\)
\(602\) −0.527864 −0.0215141
\(603\) −2.38197 −0.0970012
\(604\) 7.94427 0.323248
\(605\) −9.59675 −0.390163
\(606\) −9.85410 −0.400296
\(607\) −48.5755 −1.97162 −0.985809 0.167873i \(-0.946310\pi\)
−0.985809 + 0.167873i \(0.946310\pi\)
\(608\) 2.00000 0.0811107
\(609\) 2.23607 0.0906100
\(610\) −4.79837 −0.194280
\(611\) 0.145898 0.00590240
\(612\) 1.61803 0.0654051
\(613\) −35.4721 −1.43271 −0.716353 0.697738i \(-0.754190\pi\)
−0.716353 + 0.697738i \(0.754190\pi\)
\(614\) −8.12461 −0.327883
\(615\) −12.7639 −0.514691
\(616\) −1.00000 −0.0402911
\(617\) −33.1591 −1.33493 −0.667467 0.744640i \(-0.732621\pi\)
−0.667467 + 0.744640i \(0.732621\pi\)
\(618\) −6.70820 −0.269844
\(619\) 12.8328 0.515794 0.257897 0.966172i \(-0.416970\pi\)
0.257897 + 0.966172i \(0.416970\pi\)
\(620\) 2.03444 0.0817052
\(621\) −20.5967 −0.826519
\(622\) −12.7639 −0.511787
\(623\) 3.49342 0.139961
\(624\) 1.61803 0.0647732
\(625\) 0 0
\(626\) 19.8541 0.793530
\(627\) 13.7082 0.547453
\(628\) −8.65248 −0.345271
\(629\) −19.9443 −0.795230
\(630\) 0.124612 0.00496465
\(631\) 29.7984 1.18625 0.593127 0.805109i \(-0.297893\pi\)
0.593127 + 0.805109i \(0.297893\pi\)
\(632\) 10.7984 0.429536
\(633\) 9.47214 0.376484
\(634\) −25.7639 −1.02322
\(635\) 5.24922 0.208309
\(636\) 0 0
\(637\) 6.94427 0.275142
\(638\) 24.7984 0.981777
\(639\) 1.41641 0.0560322
\(640\) −1.38197 −0.0546270
\(641\) −19.1803 −0.757578 −0.378789 0.925483i \(-0.623659\pi\)
−0.378789 + 0.925483i \(0.623659\pi\)
\(642\) 12.5623 0.495795
\(643\) −10.8541 −0.428044 −0.214022 0.976829i \(-0.568656\pi\)
−0.214022 + 0.976829i \(0.568656\pi\)
\(644\) −0.888544 −0.0350135
\(645\) −5.00000 −0.196875
\(646\) −8.47214 −0.333332
\(647\) −25.1803 −0.989941 −0.494971 0.868910i \(-0.664821\pi\)
−0.494971 + 0.868910i \(0.664821\pi\)
\(648\) −7.70820 −0.302807
\(649\) −57.4508 −2.25514
\(650\) 3.09017 0.121206
\(651\) 0.562306 0.0220385
\(652\) 1.67376 0.0655496
\(653\) 39.5410 1.54736 0.773680 0.633577i \(-0.218414\pi\)
0.773680 + 0.633577i \(0.218414\pi\)
\(654\) −14.0902 −0.550969
\(655\) −10.0000 −0.390732
\(656\) −5.70820 −0.222868
\(657\) −1.03444 −0.0403574
\(658\) −0.0344419 −0.00134268
\(659\) −15.6738 −0.610563 −0.305282 0.952262i \(-0.598751\pi\)
−0.305282 + 0.952262i \(0.598751\pi\)
\(660\) −9.47214 −0.368702
\(661\) −14.1459 −0.550212 −0.275106 0.961414i \(-0.588713\pi\)
−0.275106 + 0.961414i \(0.588713\pi\)
\(662\) 23.1803 0.900929
\(663\) −6.85410 −0.266191
\(664\) −0.763932 −0.0296463
\(665\) −0.652476 −0.0253019
\(666\) −1.79837 −0.0696856
\(667\) 22.0344 0.853177
\(668\) 1.38197 0.0534699
\(669\) −16.7082 −0.645976
\(670\) −8.61803 −0.332944
\(671\) −14.7082 −0.567804
\(672\) −0.381966 −0.0147347
\(673\) −2.70820 −0.104394 −0.0521968 0.998637i \(-0.516622\pi\)
−0.0521968 + 0.998637i \(0.516622\pi\)
\(674\) 18.9787 0.731033
\(675\) −16.9098 −0.650860
\(676\) −12.0000 −0.461538
\(677\) −37.5623 −1.44364 −0.721818 0.692083i \(-0.756693\pi\)
−0.721818 + 0.692083i \(0.756693\pi\)
\(678\) 28.5066 1.09479
\(679\) −4.40325 −0.168981
\(680\) 5.85410 0.224495
\(681\) −8.14590 −0.312151
\(682\) 6.23607 0.238791
\(683\) −16.7639 −0.641454 −0.320727 0.947172i \(-0.603927\pi\)
−0.320727 + 0.947172i \(0.603927\pi\)
\(684\) −0.763932 −0.0292097
\(685\) 18.4934 0.706597
\(686\) −3.29180 −0.125681
\(687\) −4.38197 −0.167182
\(688\) −2.23607 −0.0852493
\(689\) 0 0
\(690\) −8.41641 −0.320407
\(691\) 25.1246 0.955785 0.477893 0.878418i \(-0.341401\pi\)
0.477893 + 0.878418i \(0.341401\pi\)
\(692\) 1.47214 0.0559622
\(693\) 0.381966 0.0145097
\(694\) −31.3607 −1.19044
\(695\) 13.2148 0.501265
\(696\) 9.47214 0.359040
\(697\) 24.1803 0.915896
\(698\) −12.2705 −0.464446
\(699\) 0.381966 0.0144473
\(700\) −0.729490 −0.0275721
\(701\) 35.9787 1.35890 0.679449 0.733723i \(-0.262219\pi\)
0.679449 + 0.733723i \(0.262219\pi\)
\(702\) −5.47214 −0.206532
\(703\) 9.41641 0.355147
\(704\) −4.23607 −0.159653
\(705\) −0.326238 −0.0122868
\(706\) −3.14590 −0.118397
\(707\) 1.43769 0.0540701
\(708\) −21.9443 −0.824716
\(709\) −20.0557 −0.753209 −0.376604 0.926374i \(-0.622908\pi\)
−0.376604 + 0.926374i \(0.622908\pi\)
\(710\) 5.12461 0.192323
\(711\) −4.12461 −0.154685
\(712\) 14.7984 0.554593
\(713\) 5.54102 0.207513
\(714\) 1.61803 0.0605534
\(715\) −5.85410 −0.218931
\(716\) 5.29180 0.197764
\(717\) −23.0902 −0.862318
\(718\) 0.944272 0.0352399
\(719\) 19.8197 0.739149 0.369574 0.929201i \(-0.379504\pi\)
0.369574 + 0.929201i \(0.379504\pi\)
\(720\) 0.527864 0.0196723
\(721\) 0.978714 0.0364492
\(722\) −15.0000 −0.558242
\(723\) −14.0902 −0.524019
\(724\) −7.41641 −0.275629
\(725\) 18.0902 0.671852
\(726\) −11.2361 −0.417010
\(727\) −31.2918 −1.16055 −0.580274 0.814421i \(-0.697055\pi\)
−0.580274 + 0.814421i \(0.697055\pi\)
\(728\) −0.236068 −0.00874926
\(729\) 29.5066 1.09284
\(730\) −3.74265 −0.138522
\(731\) 9.47214 0.350340
\(732\) −5.61803 −0.207649
\(733\) 11.3607 0.419616 0.209808 0.977743i \(-0.432716\pi\)
0.209808 + 0.977743i \(0.432716\pi\)
\(734\) −33.7984 −1.24752
\(735\) −15.5279 −0.572754
\(736\) −3.76393 −0.138740
\(737\) −26.4164 −0.973061
\(738\) 2.18034 0.0802594
\(739\) 5.96556 0.219447 0.109723 0.993962i \(-0.465004\pi\)
0.109723 + 0.993962i \(0.465004\pi\)
\(740\) −6.50658 −0.239187
\(741\) 3.23607 0.118880
\(742\) 0 0
\(743\) −33.7984 −1.23994 −0.619971 0.784625i \(-0.712856\pi\)
−0.619971 + 0.784625i \(0.712856\pi\)
\(744\) 2.38197 0.0873271
\(745\) 21.1803 0.775988
\(746\) −10.9443 −0.400698
\(747\) 0.291796 0.0106763
\(748\) 17.9443 0.656108
\(749\) −1.83282 −0.0669696
\(750\) −18.0902 −0.660560
\(751\) −42.8673 −1.56425 −0.782124 0.623123i \(-0.785864\pi\)
−0.782124 + 0.623123i \(0.785864\pi\)
\(752\) −0.145898 −0.00532035
\(753\) −20.8885 −0.761221
\(754\) 5.85410 0.213194
\(755\) −10.9787 −0.399556
\(756\) 1.29180 0.0469822
\(757\) −21.0557 −0.765283 −0.382642 0.923897i \(-0.624986\pi\)
−0.382642 + 0.923897i \(0.624986\pi\)
\(758\) 12.8541 0.466882
\(759\) −25.7984 −0.936422
\(760\) −2.76393 −0.100258
\(761\) 25.4164 0.921344 0.460672 0.887570i \(-0.347608\pi\)
0.460672 + 0.887570i \(0.347608\pi\)
\(762\) 6.14590 0.222642
\(763\) 2.05573 0.0744224
\(764\) −22.9443 −0.830095
\(765\) −2.23607 −0.0808452
\(766\) −19.6180 −0.708828
\(767\) −13.5623 −0.489706
\(768\) −1.61803 −0.0583858
\(769\) −17.6738 −0.637332 −0.318666 0.947867i \(-0.603235\pi\)
−0.318666 + 0.947867i \(0.603235\pi\)
\(770\) 1.38197 0.0498026
\(771\) −5.61803 −0.202329
\(772\) −3.76393 −0.135467
\(773\) −1.59675 −0.0574310 −0.0287155 0.999588i \(-0.509142\pi\)
−0.0287155 + 0.999588i \(0.509142\pi\)
\(774\) 0.854102 0.0307001
\(775\) 4.54915 0.163410
\(776\) −18.6525 −0.669585
\(777\) −1.79837 −0.0645163
\(778\) 11.0557 0.396367
\(779\) −11.4164 −0.409035
\(780\) −2.23607 −0.0800641
\(781\) 15.7082 0.562084
\(782\) 15.9443 0.570166
\(783\) −32.0344 −1.14482
\(784\) −6.94427 −0.248010
\(785\) 11.9574 0.426779
\(786\) −11.7082 −0.417618
\(787\) −33.6525 −1.19958 −0.599791 0.800157i \(-0.704749\pi\)
−0.599791 + 0.800157i \(0.704749\pi\)
\(788\) −24.6180 −0.876981
\(789\) 16.6180 0.591618
\(790\) −14.9230 −0.530936
\(791\) −4.15905 −0.147879
\(792\) 1.61803 0.0574943
\(793\) −3.47214 −0.123299
\(794\) −0.729490 −0.0258886
\(795\) 0 0
\(796\) −7.61803 −0.270014
\(797\) 37.7984 1.33889 0.669444 0.742863i \(-0.266533\pi\)
0.669444 + 0.742863i \(0.266533\pi\)
\(798\) −0.763932 −0.0270429
\(799\) 0.618034 0.0218645
\(800\) −3.09017 −0.109254
\(801\) −5.65248 −0.199720
\(802\) 29.2705 1.03358
\(803\) −11.4721 −0.404843
\(804\) −10.0902 −0.355853
\(805\) 1.22794 0.0432791
\(806\) 1.47214 0.0518538
\(807\) −1.61803 −0.0569575
\(808\) 6.09017 0.214251
\(809\) 24.4721 0.860394 0.430197 0.902735i \(-0.358444\pi\)
0.430197 + 0.902735i \(0.358444\pi\)
\(810\) 10.6525 0.374290
\(811\) 1.59675 0.0560694 0.0280347 0.999607i \(-0.491075\pi\)
0.0280347 + 0.999607i \(0.491075\pi\)
\(812\) −1.38197 −0.0484975
\(813\) 26.0902 0.915022
\(814\) −19.9443 −0.699046
\(815\) −2.31308 −0.0810237
\(816\) 6.85410 0.239942
\(817\) −4.47214 −0.156460
\(818\) 19.6525 0.687133
\(819\) 0.0901699 0.00315079
\(820\) 7.88854 0.275480
\(821\) −6.36068 −0.221989 −0.110995 0.993821i \(-0.535404\pi\)
−0.110995 + 0.993821i \(0.535404\pi\)
\(822\) 21.6525 0.755217
\(823\) −15.1115 −0.526752 −0.263376 0.964693i \(-0.584836\pi\)
−0.263376 + 0.964693i \(0.584836\pi\)
\(824\) 4.14590 0.144429
\(825\) −21.1803 −0.737405
\(826\) 3.20163 0.111399
\(827\) 4.74265 0.164918 0.0824590 0.996594i \(-0.473723\pi\)
0.0824590 + 0.996594i \(0.473723\pi\)
\(828\) 1.43769 0.0499633
\(829\) 19.2016 0.666900 0.333450 0.942768i \(-0.391787\pi\)
0.333450 + 0.942768i \(0.391787\pi\)
\(830\) 1.05573 0.0366449
\(831\) 38.2148 1.32566
\(832\) −1.00000 −0.0346688
\(833\) 29.4164 1.01922
\(834\) 15.4721 0.535756
\(835\) −1.90983 −0.0660924
\(836\) −8.47214 −0.293015
\(837\) −8.05573 −0.278447
\(838\) −37.5623 −1.29757
\(839\) −41.4508 −1.43104 −0.715521 0.698591i \(-0.753811\pi\)
−0.715521 + 0.698591i \(0.753811\pi\)
\(840\) 0.527864 0.0182130
\(841\) 5.27051 0.181742
\(842\) 2.52786 0.0871159
\(843\) 11.0344 0.380046
\(844\) −5.85410 −0.201506
\(845\) 16.5836 0.570493
\(846\) 0.0557281 0.00191597
\(847\) 1.63932 0.0563277
\(848\) 0 0
\(849\) −16.5623 −0.568417
\(850\) 13.0902 0.448989
\(851\) −17.7214 −0.607480
\(852\) 6.00000 0.205557
\(853\) −48.8115 −1.67127 −0.835637 0.549281i \(-0.814902\pi\)
−0.835637 + 0.549281i \(0.814902\pi\)
\(854\) 0.819660 0.0280482
\(855\) 1.05573 0.0361051
\(856\) −7.76393 −0.265366
\(857\) 16.0689 0.548903 0.274451 0.961601i \(-0.411504\pi\)
0.274451 + 0.961601i \(0.411504\pi\)
\(858\) −6.85410 −0.233995
\(859\) 29.7426 1.01481 0.507403 0.861709i \(-0.330606\pi\)
0.507403 + 0.861709i \(0.330606\pi\)
\(860\) 3.09017 0.105374
\(861\) 2.18034 0.0743058
\(862\) −6.52786 −0.222340
\(863\) 1.67376 0.0569755 0.0284878 0.999594i \(-0.490931\pi\)
0.0284878 + 0.999594i \(0.490931\pi\)
\(864\) 5.47214 0.186166
\(865\) −2.03444 −0.0691731
\(866\) 36.7984 1.25046
\(867\) −1.52786 −0.0518890
\(868\) −0.347524 −0.0117957
\(869\) −45.7426 −1.55171
\(870\) −13.0902 −0.443798
\(871\) −6.23607 −0.211301
\(872\) 8.70820 0.294897
\(873\) 7.12461 0.241132
\(874\) −7.52786 −0.254634
\(875\) 2.63932 0.0892253
\(876\) −4.38197 −0.148053
\(877\) −53.5755 −1.80911 −0.904557 0.426352i \(-0.859799\pi\)
−0.904557 + 0.426352i \(0.859799\pi\)
\(878\) 25.8541 0.872534
\(879\) −26.6525 −0.898966
\(880\) 5.85410 0.197342
\(881\) −6.49342 −0.218769 −0.109384 0.994000i \(-0.534888\pi\)
−0.109384 + 0.994000i \(0.534888\pi\)
\(882\) 2.65248 0.0893135
\(883\) 48.1935 1.62184 0.810920 0.585157i \(-0.198967\pi\)
0.810920 + 0.585157i \(0.198967\pi\)
\(884\) 4.23607 0.142474
\(885\) 30.3262 1.01941
\(886\) 6.58359 0.221180
\(887\) 39.8328 1.33746 0.668728 0.743508i \(-0.266839\pi\)
0.668728 + 0.743508i \(0.266839\pi\)
\(888\) −7.61803 −0.255644
\(889\) −0.896674 −0.0300735
\(890\) −20.4508 −0.685514
\(891\) 32.6525 1.09390
\(892\) 10.3262 0.345748
\(893\) −0.291796 −0.00976458
\(894\) 24.7984 0.829382
\(895\) −7.31308 −0.244449
\(896\) 0.236068 0.00788648
\(897\) −6.09017 −0.203345
\(898\) 14.0902 0.470195
\(899\) 8.61803 0.287428
\(900\) 1.18034 0.0393447
\(901\) 0 0
\(902\) 24.1803 0.805117
\(903\) 0.854102 0.0284227
\(904\) −17.6180 −0.585967
\(905\) 10.2492 0.340696
\(906\) −12.8541 −0.427049
\(907\) 26.1246 0.867453 0.433727 0.901044i \(-0.357198\pi\)
0.433727 + 0.901044i \(0.357198\pi\)
\(908\) 5.03444 0.167074
\(909\) −2.32624 −0.0771564
\(910\) 0.326238 0.0108147
\(911\) −44.3394 −1.46903 −0.734515 0.678593i \(-0.762590\pi\)
−0.734515 + 0.678593i \(0.762590\pi\)
\(912\) −3.23607 −0.107157
\(913\) 3.23607 0.107098
\(914\) 2.20163 0.0728233
\(915\) 7.76393 0.256668
\(916\) 2.70820 0.0894816
\(917\) 1.70820 0.0564099
\(918\) −23.1803 −0.765065
\(919\) −36.6525 −1.20905 −0.604527 0.796585i \(-0.706638\pi\)
−0.604527 + 0.796585i \(0.706638\pi\)
\(920\) 5.20163 0.171493
\(921\) 13.1459 0.433172
\(922\) −19.1246 −0.629836
\(923\) 3.70820 0.122057
\(924\) 1.61803 0.0532294
\(925\) −14.5492 −0.478373
\(926\) −8.90983 −0.292795
\(927\) −1.58359 −0.0520120
\(928\) −5.85410 −0.192170
\(929\) 40.1803 1.31827 0.659137 0.752023i \(-0.270922\pi\)
0.659137 + 0.752023i \(0.270922\pi\)
\(930\) −3.29180 −0.107942
\(931\) −13.8885 −0.455179
\(932\) −0.236068 −0.00773266
\(933\) 20.6525 0.676132
\(934\) −33.4164 −1.09342
\(935\) −24.7984 −0.810994
\(936\) 0.381966 0.0124849
\(937\) −21.9098 −0.715763 −0.357881 0.933767i \(-0.616501\pi\)
−0.357881 + 0.933767i \(0.616501\pi\)
\(938\) 1.47214 0.0480669
\(939\) −32.1246 −1.04835
\(940\) 0.201626 0.00657632
\(941\) −0.0212862 −0.000693911 0 −0.000346956 1.00000i \(-0.500110\pi\)
−0.000346956 1.00000i \(0.500110\pi\)
\(942\) 14.0000 0.456145
\(943\) 21.4853 0.699657
\(944\) 13.5623 0.441415
\(945\) −1.78522 −0.0580732
\(946\) 9.47214 0.307966
\(947\) 1.21478 0.0394751 0.0197376 0.999805i \(-0.493717\pi\)
0.0197376 + 0.999805i \(0.493717\pi\)
\(948\) −17.4721 −0.567469
\(949\) −2.70820 −0.0879120
\(950\) −6.18034 −0.200517
\(951\) 41.6869 1.35179
\(952\) −1.00000 −0.0324102
\(953\) −37.3050 −1.20843 −0.604213 0.796823i \(-0.706512\pi\)
−0.604213 + 0.796823i \(0.706512\pi\)
\(954\) 0 0
\(955\) 31.7082 1.02605
\(956\) 14.2705 0.461541
\(957\) −40.1246 −1.29704
\(958\) 12.5967 0.406982
\(959\) −3.15905 −0.102011
\(960\) 2.23607 0.0721688
\(961\) −28.8328 −0.930091
\(962\) −4.70820 −0.151799
\(963\) 2.96556 0.0955638
\(964\) 8.70820 0.280472
\(965\) 5.20163 0.167446
\(966\) 1.43769 0.0462570
\(967\) −4.85410 −0.156097 −0.0780487 0.996950i \(-0.524869\pi\)
−0.0780487 + 0.996950i \(0.524869\pi\)
\(968\) 6.94427 0.223197
\(969\) 13.7082 0.440371
\(970\) 25.7771 0.827652
\(971\) 30.6525 0.983685 0.491842 0.870684i \(-0.336324\pi\)
0.491842 + 0.870684i \(0.336324\pi\)
\(972\) −3.94427 −0.126513
\(973\) −2.25735 −0.0723675
\(974\) 38.8328 1.24428
\(975\) −5.00000 −0.160128
\(976\) 3.47214 0.111140
\(977\) 37.8541 1.21106 0.605530 0.795822i \(-0.292961\pi\)
0.605530 + 0.795822i \(0.292961\pi\)
\(978\) −2.70820 −0.0865988
\(979\) −62.6869 −2.00348
\(980\) 9.59675 0.306557
\(981\) −3.32624 −0.106199
\(982\) 11.2361 0.358557
\(983\) −51.5623 −1.64458 −0.822291 0.569067i \(-0.807304\pi\)
−0.822291 + 0.569067i \(0.807304\pi\)
\(984\) 9.23607 0.294435
\(985\) 34.0213 1.08401
\(986\) 24.7984 0.789741
\(987\) 0.0557281 0.00177384
\(988\) −2.00000 −0.0636285
\(989\) 8.41641 0.267626
\(990\) −2.23607 −0.0710669
\(991\) −18.6180 −0.591421 −0.295711 0.955278i \(-0.595556\pi\)
−0.295711 + 0.955278i \(0.595556\pi\)
\(992\) −1.47214 −0.0467404
\(993\) −37.5066 −1.19024
\(994\) −0.875388 −0.0277656
\(995\) 10.5279 0.333756
\(996\) 1.23607 0.0391663
\(997\) −44.4164 −1.40668 −0.703341 0.710853i \(-0.748309\pi\)
−0.703341 + 0.710853i \(0.748309\pi\)
\(998\) −5.94427 −0.188163
\(999\) 25.7639 0.815135
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 538.2.a.a.1.1 2
3.2 odd 2 4842.2.a.f.1.1 2
4.3 odd 2 4304.2.a.d.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
538.2.a.a.1.1 2 1.1 even 1 trivial
4304.2.a.d.1.2 2 4.3 odd 2
4842.2.a.f.1.1 2 3.2 odd 2