Properties

Label 538.2.a.c.1.4
Level $538$
Weight $2$
Character 538.1
Self dual yes
Analytic conductor $4.296$
Analytic rank $0$
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] = [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: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.4913.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 6x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-2.04948\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +3.04948 q^{3} +1.00000 q^{4} +1.48793 q^{5} -3.04948 q^{6} +0.344151 q^{7} -1.00000 q^{8} +6.29934 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +3.04948 q^{3} +1.00000 q^{4} +1.48793 q^{5} -3.04948 q^{6} +0.344151 q^{7} -1.00000 q^{8} +6.29934 q^{9} -1.48793 q^{10} -0.905704 q^{11} +3.04948 q^{12} -2.24985 q^{13} -0.344151 q^{14} +4.53741 q^{15} +1.00000 q^{16} +2.58570 q^{17} -6.29934 q^{18} -0.688301 q^{19} +1.48793 q^{20} +1.04948 q^{21} +0.905704 q^{22} +1.46259 q^{23} -3.04948 q^{24} -2.78607 q^{25} +2.24985 q^{26} +10.0613 q^{27} +0.344151 q^{28} -4.24274 q^{29} -4.53741 q^{30} +1.92985 q^{31} -1.00000 q^{32} -2.76193 q^{33} -2.58570 q^{34} +0.512072 q^{35} +6.29934 q^{36} -1.84911 q^{37} +0.688301 q^{38} -6.86089 q^{39} -1.48793 q^{40} +1.21740 q^{41} -1.04948 q^{42} -6.59036 q^{43} -0.905704 q^{44} +9.37296 q^{45} -1.46259 q^{46} -6.42244 q^{47} +3.04948 q^{48} -6.88156 q^{49} +2.78607 q^{50} +7.88503 q^{51} -2.24985 q^{52} +2.87689 q^{53} -10.0613 q^{54} -1.34762 q^{55} -0.344151 q^{56} -2.09896 q^{57} +4.24274 q^{58} +11.0118 q^{59} +4.53741 q^{60} -1.75015 q^{61} -1.92985 q^{62} +2.16792 q^{63} +1.00000 q^{64} -3.34762 q^{65} +2.76193 q^{66} +4.34415 q^{67} +2.58570 q^{68} +4.46014 q^{69} -0.512072 q^{70} +5.29586 q^{71} -6.29934 q^{72} +5.90206 q^{73} +1.84911 q^{74} -8.49607 q^{75} -0.688301 q^{76} -0.311699 q^{77} +6.86089 q^{78} -15.1107 q^{79} +1.48793 q^{80} +11.7836 q^{81} -1.21740 q^{82} +15.7701 q^{83} +1.04948 q^{84} +3.84733 q^{85} +6.59036 q^{86} -12.9382 q^{87} +0.905704 q^{88} -1.61934 q^{89} -9.37296 q^{90} -0.774289 q^{91} +1.46259 q^{92} +5.88503 q^{93} +6.42244 q^{94} -1.02414 q^{95} -3.04948 q^{96} +14.3436 q^{97} +6.88156 q^{98} -5.70533 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 3 q^{3} + 4 q^{4} + 5 q^{5} - 3 q^{6} - q^{7} - 4 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + 3 q^{3} + 4 q^{4} + 5 q^{5} - 3 q^{6} - q^{7} - 4 q^{8} + 3 q^{9} - 5 q^{10} + 7 q^{11} + 3 q^{12} + 4 q^{13} + q^{14} + 8 q^{15} + 4 q^{16} + 4 q^{17} - 3 q^{18} + 2 q^{19} + 5 q^{20} - 5 q^{21} - 7 q^{22} + 16 q^{23} - 3 q^{24} - q^{25} - 4 q^{26} + 6 q^{27} - q^{28} - 8 q^{30} - q^{31} - 4 q^{32} + q^{33} - 4 q^{34} + 3 q^{35} + 3 q^{36} - 2 q^{37} - 2 q^{38} + 3 q^{39} - 5 q^{40} - q^{41} + 5 q^{42} + 9 q^{43} + 7 q^{44} + 8 q^{45} - 16 q^{46} + 13 q^{47} + 3 q^{48} - 15 q^{49} + q^{50} + 3 q^{51} + 4 q^{52} + 28 q^{53} - 6 q^{54} + 13 q^{55} + q^{56} + 10 q^{57} + 19 q^{59} + 8 q^{60} - 20 q^{61} + q^{62} + 12 q^{63} + 4 q^{64} + 5 q^{65} - q^{66} + 15 q^{67} + 4 q^{68} - 5 q^{69} - 3 q^{70} + 15 q^{71} - 3 q^{72} - 7 q^{73} + 2 q^{74} + 12 q^{75} + 2 q^{76} - 6 q^{77} - 3 q^{78} - 17 q^{79} + 5 q^{80} + 4 q^{81} + q^{82} + 6 q^{83} - 5 q^{84} - 29 q^{85} - 9 q^{86} - 34 q^{87} - 7 q^{88} + 20 q^{89} - 8 q^{90} - 18 q^{91} + 16 q^{92} - 5 q^{93} - 13 q^{94} - 6 q^{95} - 3 q^{96} + 3 q^{97} + 15 q^{98} - 16 q^{99}+O(q^{100}) \) 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
\(3\) 3.04948 1.76062 0.880309 0.474400i \(-0.157335\pi\)
0.880309 + 0.474400i \(0.157335\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.48793 0.665422 0.332711 0.943029i \(-0.392037\pi\)
0.332711 + 0.943029i \(0.392037\pi\)
\(6\) −3.04948 −1.24495
\(7\) 0.344151 0.130077 0.0650384 0.997883i \(-0.479283\pi\)
0.0650384 + 0.997883i \(0.479283\pi\)
\(8\) −1.00000 −0.353553
\(9\) 6.29934 2.09978
\(10\) −1.48793 −0.470524
\(11\) −0.905704 −0.273080 −0.136540 0.990635i \(-0.543598\pi\)
−0.136540 + 0.990635i \(0.543598\pi\)
\(12\) 3.04948 0.880309
\(13\) −2.24985 −0.623997 −0.311999 0.950083i \(-0.600998\pi\)
−0.311999 + 0.950083i \(0.600998\pi\)
\(14\) −0.344151 −0.0919782
\(15\) 4.53741 1.17155
\(16\) 1.00000 0.250000
\(17\) 2.58570 0.627123 0.313562 0.949568i \(-0.398478\pi\)
0.313562 + 0.949568i \(0.398478\pi\)
\(18\) −6.29934 −1.48477
\(19\) −0.688301 −0.157907 −0.0789536 0.996878i \(-0.525158\pi\)
−0.0789536 + 0.996878i \(0.525158\pi\)
\(20\) 1.48793 0.332711
\(21\) 1.04948 0.229016
\(22\) 0.905704 0.193097
\(23\) 1.46259 0.304971 0.152486 0.988306i \(-0.451272\pi\)
0.152486 + 0.988306i \(0.451272\pi\)
\(24\) −3.04948 −0.622473
\(25\) −2.78607 −0.557214
\(26\) 2.24985 0.441233
\(27\) 10.0613 1.93629
\(28\) 0.344151 0.0650384
\(29\) −4.24274 −0.787857 −0.393929 0.919141i \(-0.628884\pi\)
−0.393929 + 0.919141i \(0.628884\pi\)
\(30\) −4.53741 −0.828414
\(31\) 1.92985 0.346611 0.173305 0.984868i \(-0.444555\pi\)
0.173305 + 0.984868i \(0.444555\pi\)
\(32\) −1.00000 −0.176777
\(33\) −2.76193 −0.480790
\(34\) −2.58570 −0.443443
\(35\) 0.512072 0.0865559
\(36\) 6.29934 1.04989
\(37\) −1.84911 −0.303991 −0.151996 0.988381i \(-0.548570\pi\)
−0.151996 + 0.988381i \(0.548570\pi\)
\(38\) 0.688301 0.111657
\(39\) −6.86089 −1.09862
\(40\) −1.48793 −0.235262
\(41\) 1.21740 0.190126 0.0950631 0.995471i \(-0.469695\pi\)
0.0950631 + 0.995471i \(0.469695\pi\)
\(42\) −1.04948 −0.161938
\(43\) −6.59036 −1.00502 −0.502510 0.864571i \(-0.667590\pi\)
−0.502510 + 0.864571i \(0.667590\pi\)
\(44\) −0.905704 −0.136540
\(45\) 9.37296 1.39724
\(46\) −1.46259 −0.215647
\(47\) −6.42244 −0.936809 −0.468405 0.883514i \(-0.655171\pi\)
−0.468405 + 0.883514i \(0.655171\pi\)
\(48\) 3.04948 0.440155
\(49\) −6.88156 −0.983080
\(50\) 2.78607 0.394010
\(51\) 7.88503 1.10413
\(52\) −2.24985 −0.311999
\(53\) 2.87689 0.395172 0.197586 0.980286i \(-0.436690\pi\)
0.197586 + 0.980286i \(0.436690\pi\)
\(54\) −10.0613 −1.36916
\(55\) −1.34762 −0.181713
\(56\) −0.344151 −0.0459891
\(57\) −2.09896 −0.278014
\(58\) 4.24274 0.557099
\(59\) 11.0118 1.43361 0.716806 0.697273i \(-0.245603\pi\)
0.716806 + 0.697273i \(0.245603\pi\)
\(60\) 4.53741 0.585777
\(61\) −1.75015 −0.224083 −0.112042 0.993704i \(-0.535739\pi\)
−0.112042 + 0.993704i \(0.535739\pi\)
\(62\) −1.92985 −0.245091
\(63\) 2.16792 0.273132
\(64\) 1.00000 0.125000
\(65\) −3.34762 −0.415221
\(66\) 2.76193 0.339970
\(67\) 4.34415 0.530722 0.265361 0.964149i \(-0.414509\pi\)
0.265361 + 0.964149i \(0.414509\pi\)
\(68\) 2.58570 0.313562
\(69\) 4.46014 0.536938
\(70\) −0.512072 −0.0612043
\(71\) 5.29586 0.628503 0.314252 0.949340i \(-0.398246\pi\)
0.314252 + 0.949340i \(0.398246\pi\)
\(72\) −6.29934 −0.742384
\(73\) 5.90206 0.690784 0.345392 0.938459i \(-0.387746\pi\)
0.345392 + 0.938459i \(0.387746\pi\)
\(74\) 1.84911 0.214954
\(75\) −8.49607 −0.981041
\(76\) −0.688301 −0.0789536
\(77\) −0.311699 −0.0355213
\(78\) 6.86089 0.776843
\(79\) −15.1107 −1.70009 −0.850046 0.526709i \(-0.823426\pi\)
−0.850046 + 0.526709i \(0.823426\pi\)
\(80\) 1.48793 0.166355
\(81\) 11.7836 1.30929
\(82\) −1.21740 −0.134440
\(83\) 15.7701 1.73099 0.865495 0.500918i \(-0.167004\pi\)
0.865495 + 0.500918i \(0.167004\pi\)
\(84\) 1.04948 0.114508
\(85\) 3.84733 0.417302
\(86\) 6.59036 0.710657
\(87\) −12.9382 −1.38712
\(88\) 0.905704 0.0965483
\(89\) −1.61934 −0.171650 −0.0858250 0.996310i \(-0.527353\pi\)
−0.0858250 + 0.996310i \(0.527353\pi\)
\(90\) −9.37296 −0.987997
\(91\) −0.774289 −0.0811675
\(92\) 1.46259 0.152486
\(93\) 5.88503 0.610249
\(94\) 6.42244 0.662424
\(95\) −1.02414 −0.105075
\(96\) −3.04948 −0.311236
\(97\) 14.3436 1.45637 0.728184 0.685381i \(-0.240364\pi\)
0.728184 + 0.685381i \(0.240364\pi\)
\(98\) 6.88156 0.695143
\(99\) −5.70533 −0.573407
\(100\) −2.78607 −0.278607
\(101\) 2.90690 0.289247 0.144624 0.989487i \(-0.453803\pi\)
0.144624 + 0.989487i \(0.453803\pi\)
\(102\) −7.88503 −0.780734
\(103\) −12.0866 −1.19093 −0.595464 0.803382i \(-0.703032\pi\)
−0.595464 + 0.803382i \(0.703032\pi\)
\(104\) 2.24985 0.220616
\(105\) 1.56155 0.152392
\(106\) −2.87689 −0.279429
\(107\) −7.03712 −0.680304 −0.340152 0.940370i \(-0.610479\pi\)
−0.340152 + 0.940370i \(0.610479\pi\)
\(108\) 10.0613 0.968145
\(109\) −4.91926 −0.471180 −0.235590 0.971853i \(-0.575702\pi\)
−0.235590 + 0.971853i \(0.575702\pi\)
\(110\) 1.34762 0.128491
\(111\) −5.63882 −0.535213
\(112\) 0.344151 0.0325192
\(113\) 10.5108 0.988775 0.494387 0.869242i \(-0.335392\pi\)
0.494387 + 0.869242i \(0.335392\pi\)
\(114\) 2.09896 0.196586
\(115\) 2.17623 0.202934
\(116\) −4.24274 −0.393929
\(117\) −14.1726 −1.31026
\(118\) −11.0118 −1.01372
\(119\) 0.889869 0.0815742
\(120\) −4.53741 −0.414207
\(121\) −10.1797 −0.925427
\(122\) 1.75015 0.158451
\(123\) 3.71244 0.334740
\(124\) 1.92985 0.173305
\(125\) −11.5851 −1.03620
\(126\) −2.16792 −0.193134
\(127\) −20.0830 −1.78207 −0.891037 0.453930i \(-0.850021\pi\)
−0.891037 + 0.453930i \(0.850021\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −20.0972 −1.76946
\(130\) 3.34762 0.293606
\(131\) −8.54572 −0.746643 −0.373321 0.927702i \(-0.621781\pi\)
−0.373321 + 0.927702i \(0.621781\pi\)
\(132\) −2.76193 −0.240395
\(133\) −0.236879 −0.0205401
\(134\) −4.34415 −0.375277
\(135\) 14.9704 1.28845
\(136\) −2.58570 −0.221722
\(137\) −15.0625 −1.28687 −0.643436 0.765500i \(-0.722492\pi\)
−0.643436 + 0.765500i \(0.722492\pi\)
\(138\) −4.46014 −0.379673
\(139\) −2.85030 −0.241760 −0.120880 0.992667i \(-0.538572\pi\)
−0.120880 + 0.992667i \(0.538572\pi\)
\(140\) 0.512072 0.0432780
\(141\) −19.5851 −1.64936
\(142\) −5.29586 −0.444419
\(143\) 2.03770 0.170401
\(144\) 6.29934 0.524945
\(145\) −6.31289 −0.524257
\(146\) −5.90206 −0.488458
\(147\) −20.9852 −1.73083
\(148\) −1.84911 −0.151996
\(149\) −5.24502 −0.429689 −0.214844 0.976648i \(-0.568924\pi\)
−0.214844 + 0.976648i \(0.568924\pi\)
\(150\) 8.49607 0.693701
\(151\) 12.0613 0.981532 0.490766 0.871292i \(-0.336717\pi\)
0.490766 + 0.871292i \(0.336717\pi\)
\(152\) 0.688301 0.0558286
\(153\) 16.2882 1.31682
\(154\) 0.311699 0.0251174
\(155\) 2.87147 0.230642
\(156\) −6.86089 −0.549311
\(157\) −14.4190 −1.15076 −0.575380 0.817887i \(-0.695146\pi\)
−0.575380 + 0.817887i \(0.695146\pi\)
\(158\) 15.1107 1.20215
\(159\) 8.77304 0.695747
\(160\) −1.48793 −0.117631
\(161\) 0.503352 0.0396697
\(162\) −11.7836 −0.925809
\(163\) 20.9562 1.64142 0.820708 0.571347i \(-0.193579\pi\)
0.820708 + 0.571347i \(0.193579\pi\)
\(164\) 1.21740 0.0950631
\(165\) −4.10955 −0.319928
\(166\) −15.7701 −1.22399
\(167\) −2.59520 −0.200823 −0.100411 0.994946i \(-0.532016\pi\)
−0.100411 + 0.994946i \(0.532016\pi\)
\(168\) −1.04948 −0.0809692
\(169\) −7.93816 −0.610627
\(170\) −3.84733 −0.295077
\(171\) −4.33584 −0.331570
\(172\) −6.59036 −0.502510
\(173\) −15.6629 −1.19083 −0.595414 0.803419i \(-0.703012\pi\)
−0.595414 + 0.803419i \(0.703012\pi\)
\(174\) 12.9382 0.980839
\(175\) −0.958828 −0.0724806
\(176\) −0.905704 −0.0682700
\(177\) 33.5802 2.52404
\(178\) 1.61934 0.121375
\(179\) 25.1868 1.88255 0.941273 0.337646i \(-0.109631\pi\)
0.941273 + 0.337646i \(0.109631\pi\)
\(180\) 9.37296 0.698619
\(181\) 6.06957 0.451148 0.225574 0.974226i \(-0.427574\pi\)
0.225574 + 0.974226i \(0.427574\pi\)
\(182\) 0.774289 0.0573941
\(183\) −5.33704 −0.394525
\(184\) −1.46259 −0.107824
\(185\) −2.75134 −0.202283
\(186\) −5.88503 −0.431511
\(187\) −2.34187 −0.171255
\(188\) −6.42244 −0.468405
\(189\) 3.46259 0.251866
\(190\) 1.02414 0.0742992
\(191\) 8.11319 0.587050 0.293525 0.955951i \(-0.405172\pi\)
0.293525 + 0.955951i \(0.405172\pi\)
\(192\) 3.04948 0.220077
\(193\) −12.8968 −0.928333 −0.464166 0.885748i \(-0.653646\pi\)
−0.464166 + 0.885748i \(0.653646\pi\)
\(194\) −14.3436 −1.02981
\(195\) −10.2085 −0.731047
\(196\) −6.88156 −0.491540
\(197\) 14.4920 1.03251 0.516257 0.856434i \(-0.327325\pi\)
0.516257 + 0.856434i \(0.327325\pi\)
\(198\) 5.70533 0.405460
\(199\) 20.5344 1.45564 0.727822 0.685766i \(-0.240533\pi\)
0.727822 + 0.685766i \(0.240533\pi\)
\(200\) 2.78607 0.197005
\(201\) 13.2474 0.934400
\(202\) −2.90690 −0.204529
\(203\) −1.46014 −0.102482
\(204\) 7.88503 0.552063
\(205\) 1.81141 0.126514
\(206\) 12.0866 0.842113
\(207\) 9.21335 0.640372
\(208\) −2.24985 −0.155999
\(209\) 0.623397 0.0431213
\(210\) −1.56155 −0.107757
\(211\) −6.99356 −0.481456 −0.240728 0.970593i \(-0.577386\pi\)
−0.240728 + 0.970593i \(0.577386\pi\)
\(212\) 2.87689 0.197586
\(213\) 16.1496 1.10655
\(214\) 7.03712 0.481047
\(215\) −9.80599 −0.668763
\(216\) −10.0613 −0.684582
\(217\) 0.664158 0.0450860
\(218\) 4.91926 0.333174
\(219\) 17.9982 1.21621
\(220\) −1.34762 −0.0908567
\(221\) −5.81744 −0.391323
\(222\) 5.63882 0.378453
\(223\) −8.26630 −0.553552 −0.276776 0.960934i \(-0.589266\pi\)
−0.276776 + 0.960934i \(0.589266\pi\)
\(224\) −0.344151 −0.0229945
\(225\) −17.5504 −1.17003
\(226\) −10.5108 −0.699169
\(227\) −3.27586 −0.217427 −0.108713 0.994073i \(-0.534673\pi\)
−0.108713 + 0.994073i \(0.534673\pi\)
\(228\) −2.09896 −0.139007
\(229\) −3.04601 −0.201286 −0.100643 0.994923i \(-0.532090\pi\)
−0.100643 + 0.994923i \(0.532090\pi\)
\(230\) −2.17623 −0.143496
\(231\) −0.950519 −0.0625395
\(232\) 4.24274 0.278550
\(233\) 12.9540 0.848644 0.424322 0.905511i \(-0.360512\pi\)
0.424322 + 0.905511i \(0.360512\pi\)
\(234\) 14.1726 0.926491
\(235\) −9.55613 −0.623373
\(236\) 11.0118 0.716806
\(237\) −46.0799 −2.99321
\(238\) −0.889869 −0.0576817
\(239\) 8.68652 0.561885 0.280942 0.959725i \(-0.409353\pi\)
0.280942 + 0.959725i \(0.409353\pi\)
\(240\) 4.53741 0.292889
\(241\) −15.8491 −1.02093 −0.510465 0.859898i \(-0.670527\pi\)
−0.510465 + 0.859898i \(0.670527\pi\)
\(242\) 10.1797 0.654376
\(243\) 5.75015 0.368872
\(244\) −1.75015 −0.112042
\(245\) −10.2393 −0.654163
\(246\) −3.71244 −0.236697
\(247\) 1.54858 0.0985337
\(248\) −1.92985 −0.122545
\(249\) 48.0905 3.04761
\(250\) 11.5851 0.732707
\(251\) 4.61103 0.291046 0.145523 0.989355i \(-0.453514\pi\)
0.145523 + 0.989355i \(0.453514\pi\)
\(252\) 2.16792 0.136566
\(253\) −1.32467 −0.0832815
\(254\) 20.0830 1.26012
\(255\) 11.7324 0.734709
\(256\) 1.00000 0.0625000
\(257\) 20.2182 1.26118 0.630588 0.776118i \(-0.282814\pi\)
0.630588 + 0.776118i \(0.282814\pi\)
\(258\) 20.0972 1.25120
\(259\) −0.636372 −0.0395422
\(260\) −3.34762 −0.207611
\(261\) −26.7264 −1.65433
\(262\) 8.54572 0.527956
\(263\) 15.7471 0.971009 0.485504 0.874234i \(-0.338636\pi\)
0.485504 + 0.874234i \(0.338636\pi\)
\(264\) 2.76193 0.169985
\(265\) 4.28061 0.262956
\(266\) 0.236879 0.0145240
\(267\) −4.93816 −0.302210
\(268\) 4.34415 0.265361
\(269\) 1.00000 0.0609711
\(270\) −14.9704 −0.911072
\(271\) 17.3316 1.05282 0.526410 0.850231i \(-0.323538\pi\)
0.526410 + 0.850231i \(0.323538\pi\)
\(272\) 2.58570 0.156781
\(273\) −2.36118 −0.142905
\(274\) 15.0625 0.909956
\(275\) 2.52335 0.152164
\(276\) 4.46014 0.268469
\(277\) −23.1578 −1.39142 −0.695708 0.718325i \(-0.744909\pi\)
−0.695708 + 0.718325i \(0.744909\pi\)
\(278\) 2.85030 0.170950
\(279\) 12.1568 0.727806
\(280\) −0.512072 −0.0306021
\(281\) −14.0106 −0.835801 −0.417901 0.908493i \(-0.637234\pi\)
−0.417901 + 0.908493i \(0.637234\pi\)
\(282\) 19.5851 1.16628
\(283\) 26.1135 1.55229 0.776145 0.630555i \(-0.217173\pi\)
0.776145 + 0.630555i \(0.217173\pi\)
\(284\) 5.29586 0.314252
\(285\) −3.12311 −0.184997
\(286\) −2.03770 −0.120492
\(287\) 0.418970 0.0247310
\(288\) −6.29934 −0.371192
\(289\) −10.3142 −0.606716
\(290\) 6.31289 0.370706
\(291\) 43.7404 2.56411
\(292\) 5.90206 0.345392
\(293\) −8.14964 −0.476107 −0.238053 0.971252i \(-0.576509\pi\)
−0.238053 + 0.971252i \(0.576509\pi\)
\(294\) 20.9852 1.22388
\(295\) 16.3847 0.953956
\(296\) 1.84911 0.107477
\(297\) −9.11252 −0.528762
\(298\) 5.24502 0.303836
\(299\) −3.29062 −0.190301
\(300\) −8.49607 −0.490521
\(301\) −2.26808 −0.130730
\(302\) −12.0613 −0.694048
\(303\) 8.86453 0.509254
\(304\) −0.688301 −0.0394768
\(305\) −2.60409 −0.149110
\(306\) −16.2882 −0.931133
\(307\) 0.809130 0.0461795 0.0230898 0.999733i \(-0.492650\pi\)
0.0230898 + 0.999733i \(0.492650\pi\)
\(308\) −0.311699 −0.0177607
\(309\) −36.8579 −2.09677
\(310\) −2.87147 −0.163089
\(311\) −0.444367 −0.0251977 −0.0125989 0.999921i \(-0.504010\pi\)
−0.0125989 + 0.999921i \(0.504010\pi\)
\(312\) 6.86089 0.388421
\(313\) 1.05085 0.0593974 0.0296987 0.999559i \(-0.490545\pi\)
0.0296987 + 0.999559i \(0.490545\pi\)
\(314\) 14.4190 0.813710
\(315\) 3.22571 0.181748
\(316\) −15.1107 −0.850046
\(317\) 6.46784 0.363270 0.181635 0.983366i \(-0.441861\pi\)
0.181635 + 0.983366i \(0.441861\pi\)
\(318\) −8.77304 −0.491967
\(319\) 3.84266 0.215148
\(320\) 1.48793 0.0831777
\(321\) −21.4596 −1.19776
\(322\) −0.503352 −0.0280507
\(323\) −1.77974 −0.0990273
\(324\) 11.7836 0.654646
\(325\) 6.26825 0.347700
\(326\) −20.9562 −1.16066
\(327\) −15.0012 −0.829568
\(328\) −1.21740 −0.0672198
\(329\) −2.21029 −0.121857
\(330\) 4.10955 0.226223
\(331\) 22.0153 1.21007 0.605034 0.796200i \(-0.293160\pi\)
0.605034 + 0.796200i \(0.293160\pi\)
\(332\) 15.7701 0.865495
\(333\) −11.6482 −0.638315
\(334\) 2.59520 0.142003
\(335\) 6.46379 0.353154
\(336\) 1.04948 0.0572539
\(337\) 14.2647 0.777047 0.388524 0.921439i \(-0.372985\pi\)
0.388524 + 0.921439i \(0.372985\pi\)
\(338\) 7.93816 0.431779
\(339\) 32.0525 1.74086
\(340\) 3.84733 0.208651
\(341\) −1.74787 −0.0946524
\(342\) 4.33584 0.234455
\(343\) −4.77735 −0.257953
\(344\) 6.59036 0.355329
\(345\) 6.63637 0.357290
\(346\) 15.6629 0.842043
\(347\) 26.0825 1.40018 0.700092 0.714052i \(-0.253142\pi\)
0.700092 + 0.714052i \(0.253142\pi\)
\(348\) −12.9382 −0.693558
\(349\) 24.5684 1.31512 0.657559 0.753403i \(-0.271589\pi\)
0.657559 + 0.753403i \(0.271589\pi\)
\(350\) 0.958828 0.0512515
\(351\) −22.6364 −1.20824
\(352\) 0.905704 0.0482742
\(353\) 8.36823 0.445396 0.222698 0.974887i \(-0.428514\pi\)
0.222698 + 0.974887i \(0.428514\pi\)
\(354\) −33.5802 −1.78477
\(355\) 7.87987 0.418220
\(356\) −1.61934 −0.0858250
\(357\) 2.71364 0.143621
\(358\) −25.1868 −1.33116
\(359\) −2.12835 −0.112330 −0.0561651 0.998421i \(-0.517887\pi\)
−0.0561651 + 0.998421i \(0.517887\pi\)
\(360\) −9.37296 −0.493998
\(361\) −18.5262 −0.975065
\(362\) −6.06957 −0.319010
\(363\) −31.0428 −1.62932
\(364\) −0.774289 −0.0405838
\(365\) 8.78184 0.459663
\(366\) 5.33704 0.278971
\(367\) 9.53496 0.497721 0.248860 0.968539i \(-0.419944\pi\)
0.248860 + 0.968539i \(0.419944\pi\)
\(368\) 1.46259 0.0762428
\(369\) 7.66882 0.399223
\(370\) 2.75134 0.143035
\(371\) 0.990085 0.0514027
\(372\) 5.88503 0.305125
\(373\) 16.7808 0.868875 0.434437 0.900702i \(-0.356947\pi\)
0.434437 + 0.900702i \(0.356947\pi\)
\(374\) 2.34187 0.121095
\(375\) −35.3286 −1.82436
\(376\) 6.42244 0.331212
\(377\) 9.54555 0.491621
\(378\) −3.46259 −0.178096
\(379\) −29.0341 −1.49138 −0.745690 0.666293i \(-0.767880\pi\)
−0.745690 + 0.666293i \(0.767880\pi\)
\(380\) −1.02414 −0.0525374
\(381\) −61.2426 −3.13755
\(382\) −8.11319 −0.415107
\(383\) 35.5798 1.81804 0.909022 0.416749i \(-0.136831\pi\)
0.909022 + 0.416749i \(0.136831\pi\)
\(384\) −3.04948 −0.155618
\(385\) −0.463785 −0.0236367
\(386\) 12.8968 0.656430
\(387\) −41.5149 −2.11032
\(388\) 14.3436 0.728184
\(389\) −37.5839 −1.90558 −0.952789 0.303634i \(-0.901800\pi\)
−0.952789 + 0.303634i \(0.901800\pi\)
\(390\) 10.2085 0.516928
\(391\) 3.78181 0.191255
\(392\) 6.88156 0.347571
\(393\) −26.0600 −1.31455
\(394\) −14.4920 −0.730097
\(395\) −22.4837 −1.13128
\(396\) −5.70533 −0.286704
\(397\) −16.8913 −0.847750 −0.423875 0.905721i \(-0.639330\pi\)
−0.423875 + 0.905721i \(0.639330\pi\)
\(398\) −20.5344 −1.02930
\(399\) −0.722359 −0.0361632
\(400\) −2.78607 −0.139303
\(401\) 3.68585 0.184063 0.0920314 0.995756i \(-0.470664\pi\)
0.0920314 + 0.995756i \(0.470664\pi\)
\(402\) −13.2474 −0.660721
\(403\) −4.34187 −0.216284
\(404\) 2.90690 0.144624
\(405\) 17.5332 0.871231
\(406\) 1.46014 0.0724656
\(407\) 1.67474 0.0830140
\(408\) −7.88503 −0.390367
\(409\) 18.5784 0.918643 0.459322 0.888270i \(-0.348093\pi\)
0.459322 + 0.888270i \(0.348093\pi\)
\(410\) −1.81141 −0.0894590
\(411\) −45.9327 −2.26569
\(412\) −12.0866 −0.595464
\(413\) 3.78971 0.186480
\(414\) −9.21335 −0.452811
\(415\) 23.4647 1.15184
\(416\) 2.24985 0.110308
\(417\) −8.69194 −0.425646
\(418\) −0.623397 −0.0304913
\(419\) 3.30890 0.161650 0.0808251 0.996728i \(-0.474244\pi\)
0.0808251 + 0.996728i \(0.474244\pi\)
\(420\) 1.56155 0.0761960
\(421\) 7.23994 0.352853 0.176427 0.984314i \(-0.443546\pi\)
0.176427 + 0.984314i \(0.443546\pi\)
\(422\) 6.99356 0.340441
\(423\) −40.4571 −1.96709
\(424\) −2.87689 −0.139714
\(425\) −7.20393 −0.349442
\(426\) −16.1496 −0.782453
\(427\) −0.602314 −0.0291480
\(428\) −7.03712 −0.340152
\(429\) 6.21393 0.300011
\(430\) 9.80599 0.472887
\(431\) −11.1645 −0.537777 −0.268888 0.963171i \(-0.586656\pi\)
−0.268888 + 0.963171i \(0.586656\pi\)
\(432\) 10.0613 0.484073
\(433\) 28.8143 1.38473 0.692363 0.721549i \(-0.256570\pi\)
0.692363 + 0.721549i \(0.256570\pi\)
\(434\) −0.664158 −0.0318806
\(435\) −19.2510 −0.923017
\(436\) −4.91926 −0.235590
\(437\) −1.00670 −0.0481571
\(438\) −17.9982 −0.859988
\(439\) 20.4897 0.977922 0.488961 0.872306i \(-0.337376\pi\)
0.488961 + 0.872306i \(0.337376\pi\)
\(440\) 1.34762 0.0642454
\(441\) −43.3493 −2.06425
\(442\) 5.81744 0.276707
\(443\) 9.58503 0.455398 0.227699 0.973732i \(-0.426880\pi\)
0.227699 + 0.973732i \(0.426880\pi\)
\(444\) −5.63882 −0.267607
\(445\) −2.40947 −0.114220
\(446\) 8.26630 0.391421
\(447\) −15.9946 −0.756518
\(448\) 0.344151 0.0162596
\(449\) −21.4476 −1.01218 −0.506088 0.862482i \(-0.668909\pi\)
−0.506088 + 0.862482i \(0.668909\pi\)
\(450\) 17.5504 0.827333
\(451\) −1.10261 −0.0519197
\(452\) 10.5108 0.494387
\(453\) 36.7806 1.72810
\(454\) 3.27586 0.153744
\(455\) −1.15209 −0.0540107
\(456\) 2.09896 0.0982929
\(457\) 10.7176 0.501350 0.250675 0.968071i \(-0.419347\pi\)
0.250675 + 0.968071i \(0.419347\pi\)
\(458\) 3.04601 0.142331
\(459\) 26.0154 1.21429
\(460\) 2.17623 0.101467
\(461\) −19.6658 −0.915926 −0.457963 0.888971i \(-0.651421\pi\)
−0.457963 + 0.888971i \(0.651421\pi\)
\(462\) 0.950519 0.0442221
\(463\) 22.6226 1.05136 0.525682 0.850681i \(-0.323810\pi\)
0.525682 + 0.850681i \(0.323810\pi\)
\(464\) −4.24274 −0.196964
\(465\) 8.75651 0.406073
\(466\) −12.9540 −0.600082
\(467\) 30.0503 1.39056 0.695282 0.718737i \(-0.255280\pi\)
0.695282 + 0.718737i \(0.255280\pi\)
\(468\) −14.1726 −0.655128
\(469\) 1.49504 0.0690347
\(470\) 9.55613 0.440792
\(471\) −43.9704 −2.02605
\(472\) −11.0118 −0.506858
\(473\) 5.96891 0.274451
\(474\) 46.0799 2.11652
\(475\) 1.91766 0.0879881
\(476\) 0.889869 0.0407871
\(477\) 18.1225 0.829773
\(478\) −8.68652 −0.397312
\(479\) −32.6251 −1.49068 −0.745339 0.666685i \(-0.767712\pi\)
−0.745339 + 0.666685i \(0.767712\pi\)
\(480\) −4.53741 −0.207103
\(481\) 4.16022 0.189690
\(482\) 15.8491 0.721907
\(483\) 1.53496 0.0698432
\(484\) −10.1797 −0.462714
\(485\) 21.3422 0.969099
\(486\) −5.75015 −0.260832
\(487\) −4.12981 −0.187139 −0.0935697 0.995613i \(-0.529828\pi\)
−0.0935697 + 0.995613i \(0.529828\pi\)
\(488\) 1.75015 0.0792254
\(489\) 63.9056 2.88991
\(490\) 10.2393 0.462563
\(491\) −35.2225 −1.58957 −0.794785 0.606891i \(-0.792416\pi\)
−0.794785 + 0.606891i \(0.792416\pi\)
\(492\) 3.71244 0.167370
\(493\) −10.9704 −0.494084
\(494\) −1.54858 −0.0696738
\(495\) −8.48912 −0.381558
\(496\) 1.92985 0.0866527
\(497\) 1.82258 0.0817537
\(498\) −48.0905 −2.15499
\(499\) 16.3253 0.730819 0.365409 0.930847i \(-0.380929\pi\)
0.365409 + 0.930847i \(0.380929\pi\)
\(500\) −11.5851 −0.518102
\(501\) −7.91401 −0.353572
\(502\) −4.61103 −0.205801
\(503\) 39.0571 1.74147 0.870736 0.491751i \(-0.163643\pi\)
0.870736 + 0.491751i \(0.163643\pi\)
\(504\) −2.16792 −0.0965669
\(505\) 4.32526 0.192471
\(506\) 1.32467 0.0588889
\(507\) −24.2073 −1.07508
\(508\) −20.0830 −0.891037
\(509\) 8.19139 0.363077 0.181539 0.983384i \(-0.441892\pi\)
0.181539 + 0.983384i \(0.441892\pi\)
\(510\) −11.7324 −0.519518
\(511\) 2.03120 0.0898549
\(512\) −1.00000 −0.0441942
\(513\) −6.92518 −0.305754
\(514\) −20.2182 −0.891786
\(515\) −17.9840 −0.792469
\(516\) −20.0972 −0.884729
\(517\) 5.81683 0.255824
\(518\) 0.636372 0.0279606
\(519\) −47.7637 −2.09660
\(520\) 3.34762 0.146803
\(521\) −9.81960 −0.430205 −0.215102 0.976592i \(-0.569009\pi\)
−0.215102 + 0.976592i \(0.569009\pi\)
\(522\) 26.7264 1.16978
\(523\) 4.42973 0.193698 0.0968492 0.995299i \(-0.469124\pi\)
0.0968492 + 0.995299i \(0.469124\pi\)
\(524\) −8.54572 −0.373321
\(525\) −2.92393 −0.127611
\(526\) −15.7471 −0.686607
\(527\) 4.99000 0.217368
\(528\) −2.76193 −0.120197
\(529\) −20.8608 −0.906993
\(530\) −4.28061 −0.185938
\(531\) 69.3669 3.01027
\(532\) −0.236879 −0.0102700
\(533\) −2.73898 −0.118638
\(534\) 4.93816 0.213695
\(535\) −10.4707 −0.452689
\(536\) −4.34415 −0.187639
\(537\) 76.8065 3.31445
\(538\) −1.00000 −0.0431131
\(539\) 6.23265 0.268459
\(540\) 14.9704 0.644225
\(541\) 3.01458 0.129607 0.0648035 0.997898i \(-0.479358\pi\)
0.0648035 + 0.997898i \(0.479358\pi\)
\(542\) −17.3316 −0.744457
\(543\) 18.5090 0.794299
\(544\) −2.58570 −0.110861
\(545\) −7.31951 −0.313533
\(546\) 2.36118 0.101049
\(547\) 35.5411 1.51963 0.759813 0.650141i \(-0.225290\pi\)
0.759813 + 0.650141i \(0.225290\pi\)
\(548\) −15.0625 −0.643436
\(549\) −11.0248 −0.470525
\(550\) −2.52335 −0.107596
\(551\) 2.92028 0.124408
\(552\) −4.46014 −0.189836
\(553\) −5.20037 −0.221142
\(554\) 23.1578 0.983880
\(555\) −8.39016 −0.356142
\(556\) −2.85030 −0.120880
\(557\) 21.5510 0.913144 0.456572 0.889687i \(-0.349077\pi\)
0.456572 + 0.889687i \(0.349077\pi\)
\(558\) −12.1568 −0.514636
\(559\) 14.8274 0.627130
\(560\) 0.512072 0.0216390
\(561\) −7.14150 −0.301514
\(562\) 14.0106 0.591001
\(563\) 30.3310 1.27830 0.639150 0.769082i \(-0.279286\pi\)
0.639150 + 0.769082i \(0.279286\pi\)
\(564\) −19.5851 −0.824682
\(565\) 15.6393 0.657952
\(566\) −26.1135 −1.09763
\(567\) 4.05534 0.170308
\(568\) −5.29586 −0.222210
\(569\) 3.78038 0.158482 0.0792409 0.996855i \(-0.474750\pi\)
0.0792409 + 0.996855i \(0.474750\pi\)
\(570\) 3.12311 0.130812
\(571\) 18.4336 0.771424 0.385712 0.922619i \(-0.373956\pi\)
0.385712 + 0.922619i \(0.373956\pi\)
\(572\) 2.03770 0.0852006
\(573\) 24.7410 1.03357
\(574\) −0.418970 −0.0174875
\(575\) −4.07488 −0.169934
\(576\) 6.29934 0.262472
\(577\) 30.3922 1.26524 0.632622 0.774461i \(-0.281979\pi\)
0.632622 + 0.774461i \(0.281979\pi\)
\(578\) 10.3142 0.429013
\(579\) −39.3286 −1.63444
\(580\) −6.31289 −0.262129
\(581\) 5.42728 0.225161
\(582\) −43.7404 −1.81310
\(583\) −2.60561 −0.107913
\(584\) −5.90206 −0.244229
\(585\) −21.0878 −0.871873
\(586\) 8.14964 0.336658
\(587\) −38.4248 −1.58596 −0.792981 0.609246i \(-0.791472\pi\)
−0.792981 + 0.609246i \(0.791472\pi\)
\(588\) −20.9852 −0.865415
\(589\) −1.32832 −0.0547323
\(590\) −16.3847 −0.674549
\(591\) 44.1931 1.81786
\(592\) −1.84911 −0.0759979
\(593\) 15.0010 0.616018 0.308009 0.951383i \(-0.400337\pi\)
0.308009 + 0.951383i \(0.400337\pi\)
\(594\) 9.11252 0.373891
\(595\) 1.32406 0.0542812
\(596\) −5.24502 −0.214844
\(597\) 62.6192 2.56283
\(598\) 3.29062 0.134563
\(599\) −8.67407 −0.354413 −0.177207 0.984174i \(-0.556706\pi\)
−0.177207 + 0.984174i \(0.556706\pi\)
\(600\) 8.49607 0.346850
\(601\) −20.1244 −0.820891 −0.410445 0.911885i \(-0.634627\pi\)
−0.410445 + 0.911885i \(0.634627\pi\)
\(602\) 2.26808 0.0924400
\(603\) 27.3653 1.11440
\(604\) 12.0613 0.490766
\(605\) −15.1467 −0.615800
\(606\) −8.86453 −0.360097
\(607\) 5.35730 0.217446 0.108723 0.994072i \(-0.465324\pi\)
0.108723 + 0.994072i \(0.465324\pi\)
\(608\) 0.688301 0.0279143
\(609\) −4.45268 −0.180432
\(610\) 2.60409 0.105437
\(611\) 14.4496 0.584566
\(612\) 16.2882 0.658410
\(613\) 21.6009 0.872454 0.436227 0.899837i \(-0.356314\pi\)
0.436227 + 0.899837i \(0.356314\pi\)
\(614\) −0.809130 −0.0326538
\(615\) 5.52385 0.222743
\(616\) 0.311699 0.0125587
\(617\) −36.2231 −1.45829 −0.729144 0.684360i \(-0.760082\pi\)
−0.729144 + 0.684360i \(0.760082\pi\)
\(618\) 36.8579 1.48264
\(619\) 31.9927 1.28590 0.642948 0.765910i \(-0.277711\pi\)
0.642948 + 0.765910i \(0.277711\pi\)
\(620\) 2.87147 0.115321
\(621\) 14.7155 0.590513
\(622\) 0.444367 0.0178175
\(623\) −0.557298 −0.0223277
\(624\) −6.86089 −0.274655
\(625\) −3.30747 −0.132299
\(626\) −1.05085 −0.0420003
\(627\) 1.90104 0.0759201
\(628\) −14.4190 −0.575380
\(629\) −4.78123 −0.190640
\(630\) −3.22571 −0.128515
\(631\) −22.3539 −0.889894 −0.444947 0.895557i \(-0.646778\pi\)
−0.444947 + 0.895557i \(0.646778\pi\)
\(632\) 15.1107 0.601073
\(633\) −21.3267 −0.847661
\(634\) −6.46784 −0.256871
\(635\) −29.8820 −1.18583
\(636\) 8.77304 0.347873
\(637\) 15.4825 0.613439
\(638\) −3.84266 −0.152133
\(639\) 33.3604 1.31972
\(640\) −1.48793 −0.0588155
\(641\) −8.18573 −0.323317 −0.161659 0.986847i \(-0.551684\pi\)
−0.161659 + 0.986847i \(0.551684\pi\)
\(642\) 21.4596 0.846941
\(643\) 17.4703 0.688961 0.344480 0.938793i \(-0.388055\pi\)
0.344480 + 0.938793i \(0.388055\pi\)
\(644\) 0.503352 0.0198348
\(645\) −29.9032 −1.17744
\(646\) 1.77974 0.0700229
\(647\) −44.5260 −1.75050 −0.875249 0.483673i \(-0.839303\pi\)
−0.875249 + 0.483673i \(0.839303\pi\)
\(648\) −11.7836 −0.462904
\(649\) −9.97341 −0.391491
\(650\) −6.26825 −0.245861
\(651\) 2.02534 0.0793793
\(652\) 20.9562 0.820708
\(653\) 37.0875 1.45134 0.725672 0.688041i \(-0.241529\pi\)
0.725672 + 0.688041i \(0.241529\pi\)
\(654\) 15.0012 0.586593
\(655\) −12.7154 −0.496832
\(656\) 1.21740 0.0475316
\(657\) 37.1791 1.45049
\(658\) 2.21029 0.0861660
\(659\) 15.6928 0.611305 0.305652 0.952143i \(-0.401126\pi\)
0.305652 + 0.952143i \(0.401126\pi\)
\(660\) −4.10955 −0.159964
\(661\) 4.10083 0.159504 0.0797519 0.996815i \(-0.474587\pi\)
0.0797519 + 0.996815i \(0.474587\pi\)
\(662\) −22.0153 −0.855647
\(663\) −17.7402 −0.688971
\(664\) −15.7701 −0.611997
\(665\) −0.352460 −0.0136678
\(666\) 11.6482 0.451357
\(667\) −6.20539 −0.240274
\(668\) −2.59520 −0.100411
\(669\) −25.2079 −0.974595
\(670\) −6.46379 −0.249718
\(671\) 1.58511 0.0611926
\(672\) −1.04948 −0.0404846
\(673\) 8.56587 0.330190 0.165095 0.986278i \(-0.447207\pi\)
0.165095 + 0.986278i \(0.447207\pi\)
\(674\) −14.2647 −0.549456
\(675\) −28.0314 −1.07893
\(676\) −7.93816 −0.305314
\(677\) 15.2255 0.585165 0.292583 0.956240i \(-0.405485\pi\)
0.292583 + 0.956240i \(0.405485\pi\)
\(678\) −32.0525 −1.23097
\(679\) 4.93635 0.189440
\(680\) −3.84733 −0.147538
\(681\) −9.98968 −0.382805
\(682\) 1.74787 0.0669294
\(683\) −34.5265 −1.32112 −0.660560 0.750774i \(-0.729681\pi\)
−0.660560 + 0.750774i \(0.729681\pi\)
\(684\) −4.33584 −0.165785
\(685\) −22.4119 −0.856313
\(686\) 4.77735 0.182400
\(687\) −9.28875 −0.354388
\(688\) −6.59036 −0.251255
\(689\) −6.47259 −0.246586
\(690\) −6.63637 −0.252642
\(691\) 32.7649 1.24644 0.623218 0.782048i \(-0.285825\pi\)
0.623218 + 0.782048i \(0.285825\pi\)
\(692\) −15.6629 −0.595414
\(693\) −1.96349 −0.0745870
\(694\) −26.0825 −0.990080
\(695\) −4.24105 −0.160872
\(696\) 12.9382 0.490420
\(697\) 3.14783 0.119233
\(698\) −24.5684 −0.929929
\(699\) 39.5029 1.49414
\(700\) −0.958828 −0.0362403
\(701\) −4.80166 −0.181356 −0.0906782 0.995880i \(-0.528903\pi\)
−0.0906782 + 0.995880i \(0.528903\pi\)
\(702\) 22.6364 0.854355
\(703\) 1.27274 0.0480024
\(704\) −0.905704 −0.0341350
\(705\) −29.1412 −1.09752
\(706\) −8.36823 −0.314943
\(707\) 1.00041 0.0376243
\(708\) 33.5802 1.26202
\(709\) −34.4618 −1.29424 −0.647120 0.762389i \(-0.724027\pi\)
−0.647120 + 0.762389i \(0.724027\pi\)
\(710\) −7.87987 −0.295726
\(711\) −95.1876 −3.56982
\(712\) 1.61934 0.0606874
\(713\) 2.82258 0.105706
\(714\) −2.71364 −0.101555
\(715\) 3.03195 0.113389
\(716\) 25.1868 0.941273
\(717\) 26.4894 0.989264
\(718\) 2.12835 0.0794295
\(719\) −41.1653 −1.53521 −0.767604 0.640925i \(-0.778551\pi\)
−0.767604 + 0.640925i \(0.778551\pi\)
\(720\) 9.37296 0.349310
\(721\) −4.15961 −0.154912
\(722\) 18.5262 0.689475
\(723\) −48.3316 −1.79747
\(724\) 6.06957 0.225574
\(725\) 11.8206 0.439005
\(726\) 31.0428 1.15211
\(727\) −50.5549 −1.87498 −0.937489 0.348015i \(-0.886856\pi\)
−0.937489 + 0.348015i \(0.886856\pi\)
\(728\) 0.774289 0.0286971
\(729\) −17.8159 −0.659848
\(730\) −8.78184 −0.325031
\(731\) −17.0407 −0.630272
\(732\) −5.33704 −0.197263
\(733\) −29.3797 −1.08516 −0.542581 0.840003i \(-0.682553\pi\)
−0.542581 + 0.840003i \(0.682553\pi\)
\(734\) −9.53496 −0.351942
\(735\) −31.2245 −1.15173
\(736\) −1.46259 −0.0539118
\(737\) −3.93451 −0.144930
\(738\) −7.66882 −0.282293
\(739\) −22.4975 −0.827584 −0.413792 0.910371i \(-0.635796\pi\)
−0.413792 + 0.910371i \(0.635796\pi\)
\(740\) −2.75134 −0.101141
\(741\) 4.72236 0.173480
\(742\) −0.990085 −0.0363472
\(743\) −7.49123 −0.274826 −0.137413 0.990514i \(-0.543879\pi\)
−0.137413 + 0.990514i \(0.543879\pi\)
\(744\) −5.88503 −0.215756
\(745\) −7.80421 −0.285924
\(746\) −16.7808 −0.614387
\(747\) 99.3409 3.63469
\(748\) −2.34187 −0.0856274
\(749\) −2.42183 −0.0884917
\(750\) 35.3286 1.29002
\(751\) 31.3441 1.14376 0.571881 0.820337i \(-0.306214\pi\)
0.571881 + 0.820337i \(0.306214\pi\)
\(752\) −6.42244 −0.234202
\(753\) 14.0613 0.512421
\(754\) −9.54555 −0.347628
\(755\) 17.9463 0.653132
\(756\) 3.46259 0.125933
\(757\) −44.8626 −1.63056 −0.815280 0.579067i \(-0.803417\pi\)
−0.815280 + 0.579067i \(0.803417\pi\)
\(758\) 29.0341 1.05457
\(759\) −4.03957 −0.146627
\(760\) 1.02414 0.0371496
\(761\) −18.5983 −0.674189 −0.337094 0.941471i \(-0.609444\pi\)
−0.337094 + 0.941471i \(0.609444\pi\)
\(762\) 61.2426 2.21859
\(763\) −1.69297 −0.0612895
\(764\) 8.11319 0.293525
\(765\) 24.2356 0.876241
\(766\) −35.5798 −1.28555
\(767\) −24.7749 −0.894570
\(768\) 3.04948 0.110039
\(769\) 52.1929 1.88213 0.941063 0.338232i \(-0.109829\pi\)
0.941063 + 0.338232i \(0.109829\pi\)
\(770\) 0.463785 0.0167137
\(771\) 61.6550 2.22045
\(772\) −12.8968 −0.464166
\(773\) 41.7206 1.50059 0.750293 0.661105i \(-0.229912\pi\)
0.750293 + 0.661105i \(0.229912\pi\)
\(774\) 41.5149 1.49222
\(775\) −5.37669 −0.193136
\(776\) −14.3436 −0.514904
\(777\) −1.94060 −0.0696188
\(778\) 37.5839 1.34745
\(779\) −0.837940 −0.0300223
\(780\) −10.2085 −0.365523
\(781\) −4.79648 −0.171632
\(782\) −3.78181 −0.135237
\(783\) −42.6873 −1.52552
\(784\) −6.88156 −0.245770
\(785\) −21.4544 −0.765740
\(786\) 26.0600 0.929530
\(787\) 20.8879 0.744574 0.372287 0.928118i \(-0.378574\pi\)
0.372287 + 0.928118i \(0.378574\pi\)
\(788\) 14.4920 0.516257
\(789\) 48.0205 1.70958
\(790\) 22.4837 0.799934
\(791\) 3.61731 0.128617
\(792\) 5.70533 0.202730
\(793\) 3.93757 0.139827
\(794\) 16.8913 0.599450
\(795\) 13.0536 0.462965
\(796\) 20.5344 0.727822
\(797\) 20.6075 0.729955 0.364978 0.931016i \(-0.381077\pi\)
0.364978 + 0.931016i \(0.381077\pi\)
\(798\) 0.722359 0.0255712
\(799\) −16.6065 −0.587495
\(800\) 2.78607 0.0985024
\(801\) −10.2008 −0.360427
\(802\) −3.68585 −0.130152
\(803\) −5.34552 −0.188639
\(804\) 13.2474 0.467200
\(805\) 0.748951 0.0263971
\(806\) 4.34187 0.152936
\(807\) 3.04948 0.107347
\(808\) −2.90690 −0.102264
\(809\) 8.27012 0.290762 0.145381 0.989376i \(-0.453559\pi\)
0.145381 + 0.989376i \(0.453559\pi\)
\(810\) −17.5332 −0.616053
\(811\) 26.6200 0.934753 0.467377 0.884058i \(-0.345199\pi\)
0.467377 + 0.884058i \(0.345199\pi\)
\(812\) −1.46014 −0.0512409
\(813\) 52.8524 1.85362
\(814\) −1.67474 −0.0586997
\(815\) 31.1813 1.09223
\(816\) 7.88503 0.276031
\(817\) 4.53616 0.158700
\(818\) −18.5784 −0.649579
\(819\) −4.87751 −0.170434
\(820\) 1.81141 0.0632571
\(821\) 29.4026 1.02616 0.513078 0.858342i \(-0.328505\pi\)
0.513078 + 0.858342i \(0.328505\pi\)
\(822\) 45.9327 1.60209
\(823\) 15.4716 0.539305 0.269653 0.962958i \(-0.413091\pi\)
0.269653 + 0.962958i \(0.413091\pi\)
\(824\) 12.0866 0.421057
\(825\) 7.69492 0.267903
\(826\) −3.78971 −0.131861
\(827\) −24.5579 −0.853963 −0.426982 0.904260i \(-0.640423\pi\)
−0.426982 + 0.904260i \(0.640423\pi\)
\(828\) 9.21335 0.320186
\(829\) 20.1760 0.700741 0.350371 0.936611i \(-0.386056\pi\)
0.350371 + 0.936611i \(0.386056\pi\)
\(830\) −23.4647 −0.814472
\(831\) −70.6192 −2.44975
\(832\) −2.24985 −0.0779997
\(833\) −17.7936 −0.616513
\(834\) 8.69194 0.300977
\(835\) −3.86147 −0.133632
\(836\) 0.623397 0.0215606
\(837\) 19.4167 0.671139
\(838\) −3.30890 −0.114304
\(839\) −39.9874 −1.38052 −0.690259 0.723562i \(-0.742504\pi\)
−0.690259 + 0.723562i \(0.742504\pi\)
\(840\) −1.56155 −0.0538787
\(841\) −10.9992 −0.379281
\(842\) −7.23994 −0.249505
\(843\) −42.7250 −1.47153
\(844\) −6.99356 −0.240728
\(845\) −11.8114 −0.406325
\(846\) 40.4571 1.39094
\(847\) −3.50335 −0.120377
\(848\) 2.87689 0.0987930
\(849\) 79.6328 2.73299
\(850\) 7.20393 0.247093
\(851\) −2.70449 −0.0927086
\(852\) 16.1496 0.553277
\(853\) −38.2866 −1.31091 −0.655453 0.755236i \(-0.727522\pi\)
−0.655453 + 0.755236i \(0.727522\pi\)
\(854\) 0.602314 0.0206108
\(855\) −6.45142 −0.220634
\(856\) 7.03712 0.240524
\(857\) 10.4287 0.356239 0.178120 0.984009i \(-0.442999\pi\)
0.178120 + 0.984009i \(0.442999\pi\)
\(858\) −6.21393 −0.212140
\(859\) −31.3752 −1.07051 −0.535254 0.844691i \(-0.679784\pi\)
−0.535254 + 0.844691i \(0.679784\pi\)
\(860\) −9.80599 −0.334381
\(861\) 1.27764 0.0435419
\(862\) 11.1645 0.380266
\(863\) 22.9420 0.780954 0.390477 0.920613i \(-0.372310\pi\)
0.390477 + 0.920613i \(0.372310\pi\)
\(864\) −10.0613 −0.342291
\(865\) −23.3053 −0.792403
\(866\) −28.8143 −0.979150
\(867\) −31.4529 −1.06820
\(868\) 0.664158 0.0225430
\(869\) 13.6859 0.464261
\(870\) 19.2510 0.652672
\(871\) −9.77371 −0.331169
\(872\) 4.91926 0.166587
\(873\) 90.3549 3.05805
\(874\) 1.00670 0.0340522
\(875\) −3.98703 −0.134786
\(876\) 17.9982 0.608104
\(877\) 13.8521 0.467754 0.233877 0.972266i \(-0.424859\pi\)
0.233877 + 0.972266i \(0.424859\pi\)
\(878\) −20.4897 −0.691495
\(879\) −24.8522 −0.838243
\(880\) −1.34762 −0.0454283
\(881\) 20.0782 0.676451 0.338225 0.941065i \(-0.390173\pi\)
0.338225 + 0.941065i \(0.390173\pi\)
\(882\) 43.3493 1.45965
\(883\) 55.3481 1.86261 0.931306 0.364237i \(-0.118670\pi\)
0.931306 + 0.364237i \(0.118670\pi\)
\(884\) −5.81744 −0.195662
\(885\) 49.9650 1.67955
\(886\) −9.58503 −0.322015
\(887\) 1.28823 0.0432544 0.0216272 0.999766i \(-0.493115\pi\)
0.0216272 + 0.999766i \(0.493115\pi\)
\(888\) 5.63882 0.189226
\(889\) −6.91156 −0.231806
\(890\) 2.40947 0.0807655
\(891\) −10.6725 −0.357541
\(892\) −8.26630 −0.276776
\(893\) 4.42058 0.147929
\(894\) 15.9946 0.534939
\(895\) 37.4761 1.25269
\(896\) −0.344151 −0.0114973
\(897\) −10.0347 −0.335048
\(898\) 21.4476 0.715716
\(899\) −8.18784 −0.273080
\(900\) −17.5504 −0.585013
\(901\) 7.43877 0.247822
\(902\) 1.10261 0.0367127
\(903\) −6.91646 −0.230165
\(904\) −10.5108 −0.349585
\(905\) 9.03109 0.300203
\(906\) −36.7806 −1.22195
\(907\) 11.6760 0.387695 0.193848 0.981032i \(-0.437903\pi\)
0.193848 + 0.981032i \(0.437903\pi\)
\(908\) −3.27586 −0.108713
\(909\) 18.3115 0.607355
\(910\) 1.15209 0.0381913
\(911\) 31.6479 1.04854 0.524271 0.851551i \(-0.324338\pi\)
0.524271 + 0.851551i \(0.324338\pi\)
\(912\) −2.09896 −0.0695036
\(913\) −14.2830 −0.472698
\(914\) −10.7176 −0.354508
\(915\) −7.94113 −0.262526
\(916\) −3.04601 −0.100643
\(917\) −2.94102 −0.0971209
\(918\) −26.0154 −0.858635
\(919\) −49.3145 −1.62674 −0.813368 0.581749i \(-0.802368\pi\)
−0.813368 + 0.581749i \(0.802368\pi\)
\(920\) −2.17623 −0.0717482
\(921\) 2.46743 0.0813045
\(922\) 19.6658 0.647657
\(923\) −11.9149 −0.392184
\(924\) −0.950519 −0.0312698
\(925\) 5.15174 0.169388
\(926\) −22.6226 −0.743426
\(927\) −76.1375 −2.50069
\(928\) 4.24274 0.139275
\(929\) −25.7689 −0.845450 −0.422725 0.906258i \(-0.638926\pi\)
−0.422725 + 0.906258i \(0.638926\pi\)
\(930\) −8.75651 −0.287137
\(931\) 4.73659 0.155235
\(932\) 12.9540 0.424322
\(933\) −1.35509 −0.0443636
\(934\) −30.0503 −0.983277
\(935\) −3.48454 −0.113957
\(936\) 14.1726 0.463245
\(937\) −6.22816 −0.203465 −0.101733 0.994812i \(-0.532439\pi\)
−0.101733 + 0.994812i \(0.532439\pi\)
\(938\) −1.49504 −0.0488149
\(939\) 3.20454 0.104576
\(940\) −9.55613 −0.311687
\(941\) 15.3930 0.501799 0.250900 0.968013i \(-0.419274\pi\)
0.250900 + 0.968013i \(0.419274\pi\)
\(942\) 43.9704 1.43263
\(943\) 1.78056 0.0579830
\(944\) 11.0118 0.358403
\(945\) 5.15209 0.167597
\(946\) −5.96891 −0.194066
\(947\) −19.0683 −0.619635 −0.309818 0.950796i \(-0.600268\pi\)
−0.309818 + 0.950796i \(0.600268\pi\)
\(948\) −46.0799 −1.49661
\(949\) −13.2788 −0.431047
\(950\) −1.91766 −0.0622170
\(951\) 19.7236 0.639580
\(952\) −0.889869 −0.0288408
\(953\) −39.7141 −1.28647 −0.643234 0.765670i \(-0.722408\pi\)
−0.643234 + 0.765670i \(0.722408\pi\)
\(954\) −18.1225 −0.586738
\(955\) 12.0718 0.390636
\(956\) 8.68652 0.280942
\(957\) 11.7181 0.378793
\(958\) 32.6251 1.05407
\(959\) −5.18376 −0.167392
\(960\) 4.53741 0.146444
\(961\) −27.2757 −0.879861
\(962\) −4.16022 −0.134131
\(963\) −44.3292 −1.42849
\(964\) −15.8491 −0.510465
\(965\) −19.1895 −0.617733
\(966\) −1.53496 −0.0493866
\(967\) −54.5429 −1.75398 −0.876991 0.480507i \(-0.840453\pi\)
−0.876991 + 0.480507i \(0.840453\pi\)
\(968\) 10.1797 0.327188
\(969\) −5.42728 −0.174349
\(970\) −21.3422 −0.685257
\(971\) −20.6894 −0.663955 −0.331977 0.943287i \(-0.607716\pi\)
−0.331977 + 0.943287i \(0.607716\pi\)
\(972\) 5.75015 0.184436
\(973\) −0.980934 −0.0314473
\(974\) 4.12981 0.132328
\(975\) 19.1149 0.612167
\(976\) −1.75015 −0.0560208
\(977\) 3.49749 0.111895 0.0559473 0.998434i \(-0.482182\pi\)
0.0559473 + 0.998434i \(0.482182\pi\)
\(978\) −63.9056 −2.04347
\(979\) 1.46664 0.0468742
\(980\) −10.2393 −0.327081
\(981\) −30.9881 −0.989373
\(982\) 35.2225 1.12400
\(983\) 16.5884 0.529086 0.264543 0.964374i \(-0.414779\pi\)
0.264543 + 0.964374i \(0.414779\pi\)
\(984\) −3.71244 −0.118348
\(985\) 21.5631 0.687057
\(986\) 10.9704 0.349370
\(987\) −6.74023 −0.214544
\(988\) 1.54858 0.0492668
\(989\) −9.63900 −0.306502
\(990\) 8.48912 0.269802
\(991\) −27.1297 −0.861805 −0.430902 0.902399i \(-0.641805\pi\)
−0.430902 + 0.902399i \(0.641805\pi\)
\(992\) −1.92985 −0.0612727
\(993\) 67.1351 2.13047
\(994\) −1.82258 −0.0578086
\(995\) 30.5537 0.968617
\(996\) 48.0905 1.52381
\(997\) 56.5029 1.78946 0.894732 0.446603i \(-0.147366\pi\)
0.894732 + 0.446603i \(0.147366\pi\)
\(998\) −16.3253 −0.516767
\(999\) −18.6044 −0.588616
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.c.1.4 4
3.2 odd 2 4842.2.a.j.1.2 4
4.3 odd 2 4304.2.a.e.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
538.2.a.c.1.4 4 1.1 even 1 trivial
4304.2.a.e.1.1 4 4.3 odd 2
4842.2.a.j.1.2 4 3.2 odd 2