Properties

Label 2013.2.a.g.1.8
Level $2013$
Weight $2$
Character 2013.1
Self dual yes
Analytic conductor $16.074$
Analytic rank $0$
Dimension $13$
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] = [2013,2,Mod(1,2013)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2013, 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("2013.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2013 = 3 \cdot 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2013.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: \(16.0738859269\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 4 x^{12} - 11 x^{11} + 57 x^{10} + 28 x^{9} - 290 x^{8} + 51 x^{7} + 644 x^{6} - 259 x^{5} - 640 x^{4} + 274 x^{3} + 256 x^{2} - 74 x - 35 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 5 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(0.931518\) of defining polynomial
Character \(\chi\) \(=\) 2013.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.931518 q^{2} +1.00000 q^{3} -1.13227 q^{4} +1.13816 q^{5} +0.931518 q^{6} -2.87358 q^{7} -2.91777 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.931518 q^{2} +1.00000 q^{3} -1.13227 q^{4} +1.13816 q^{5} +0.931518 q^{6} -2.87358 q^{7} -2.91777 q^{8} +1.00000 q^{9} +1.06022 q^{10} +1.00000 q^{11} -1.13227 q^{12} +1.40430 q^{13} -2.67679 q^{14} +1.13816 q^{15} -0.453408 q^{16} -0.846049 q^{17} +0.931518 q^{18} +4.27664 q^{19} -1.28871 q^{20} -2.87358 q^{21} +0.931518 q^{22} +5.20869 q^{23} -2.91777 q^{24} -3.70459 q^{25} +1.30814 q^{26} +1.00000 q^{27} +3.25368 q^{28} +8.20744 q^{29} +1.06022 q^{30} +7.18706 q^{31} +5.41318 q^{32} +1.00000 q^{33} -0.788110 q^{34} -3.27059 q^{35} -1.13227 q^{36} +4.15083 q^{37} +3.98377 q^{38} +1.40430 q^{39} -3.32089 q^{40} +0.166336 q^{41} -2.67679 q^{42} +6.54819 q^{43} -1.13227 q^{44} +1.13816 q^{45} +4.85199 q^{46} +0.378293 q^{47} -0.453408 q^{48} +1.25745 q^{49} -3.45089 q^{50} -0.846049 q^{51} -1.59006 q^{52} -0.923656 q^{53} +0.931518 q^{54} +1.13816 q^{55} +8.38444 q^{56} +4.27664 q^{57} +7.64538 q^{58} -1.30403 q^{59} -1.28871 q^{60} +1.00000 q^{61} +6.69488 q^{62} -2.87358 q^{63} +5.94929 q^{64} +1.59832 q^{65} +0.931518 q^{66} -6.07605 q^{67} +0.957959 q^{68} +5.20869 q^{69} -3.04661 q^{70} -13.3483 q^{71} -2.91777 q^{72} +3.93967 q^{73} +3.86657 q^{74} -3.70459 q^{75} -4.84233 q^{76} -2.87358 q^{77} +1.30814 q^{78} +7.00809 q^{79} -0.516051 q^{80} +1.00000 q^{81} +0.154945 q^{82} +2.26262 q^{83} +3.25368 q^{84} -0.962939 q^{85} +6.09976 q^{86} +8.20744 q^{87} -2.91777 q^{88} +2.22445 q^{89} +1.06022 q^{90} -4.03538 q^{91} -5.89766 q^{92} +7.18706 q^{93} +0.352387 q^{94} +4.86751 q^{95} +5.41318 q^{96} +10.6596 q^{97} +1.17133 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q + 4 q^{2} + 13 q^{3} + 12 q^{4} + 7 q^{5} + 4 q^{6} + 7 q^{7} + 9 q^{8} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q + 4 q^{2} + 13 q^{3} + 12 q^{4} + 7 q^{5} + 4 q^{6} + 7 q^{7} + 9 q^{8} + 13 q^{9} + 2 q^{10} + 13 q^{11} + 12 q^{12} + 9 q^{13} + 7 q^{14} + 7 q^{15} + 2 q^{16} + 19 q^{17} + 4 q^{18} + 14 q^{19} + 19 q^{20} + 7 q^{21} + 4 q^{22} + 5 q^{23} + 9 q^{24} + 2 q^{25} - 4 q^{26} + 13 q^{27} + 7 q^{28} + 10 q^{29} + 2 q^{30} - q^{31} + 7 q^{32} + 13 q^{33} - 2 q^{34} + 16 q^{35} + 12 q^{36} - 8 q^{37} - 10 q^{38} + 9 q^{39} + 14 q^{40} + 21 q^{41} + 7 q^{42} + 11 q^{43} + 12 q^{44} + 7 q^{45} - 8 q^{46} + 22 q^{47} + 2 q^{48} + 19 q^{50} + 19 q^{51} - q^{52} + 16 q^{53} + 4 q^{54} + 7 q^{55} + 14 q^{57} - 13 q^{58} + 19 q^{59} + 19 q^{60} + 13 q^{61} + 3 q^{62} + 7 q^{63} - 13 q^{64} + 13 q^{65} + 4 q^{66} + 12 q^{67} + 36 q^{68} + 5 q^{69} - 20 q^{70} + 5 q^{71} + 9 q^{72} + 18 q^{73} + 6 q^{74} + 2 q^{75} - 5 q^{76} + 7 q^{77} - 4 q^{78} - q^{79} + 6 q^{80} + 13 q^{81} - 22 q^{82} + 48 q^{83} + 7 q^{84} - 2 q^{85} + 26 q^{86} + 10 q^{87} + 9 q^{88} + 15 q^{89} + 2 q^{90} - 11 q^{91} - 24 q^{92} - q^{93} - 23 q^{94} + 17 q^{95} + 7 q^{96} - 17 q^{97} - 15 q^{98} + 13 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.931518 0.658683 0.329341 0.944211i \(-0.393173\pi\)
0.329341 + 0.944211i \(0.393173\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.13227 −0.566137
\(5\) 1.13816 0.509001 0.254500 0.967073i \(-0.418089\pi\)
0.254500 + 0.967073i \(0.418089\pi\)
\(6\) 0.931518 0.380291
\(7\) −2.87358 −1.08611 −0.543055 0.839697i \(-0.682733\pi\)
−0.543055 + 0.839697i \(0.682733\pi\)
\(8\) −2.91777 −1.03159
\(9\) 1.00000 0.333333
\(10\) 1.06022 0.335270
\(11\) 1.00000 0.301511
\(12\) −1.13227 −0.326859
\(13\) 1.40430 0.389484 0.194742 0.980854i \(-0.437613\pi\)
0.194742 + 0.980854i \(0.437613\pi\)
\(14\) −2.67679 −0.715402
\(15\) 1.13816 0.293872
\(16\) −0.453408 −0.113352
\(17\) −0.846049 −0.205197 −0.102599 0.994723i \(-0.532716\pi\)
−0.102599 + 0.994723i \(0.532716\pi\)
\(18\) 0.931518 0.219561
\(19\) 4.27664 0.981130 0.490565 0.871405i \(-0.336790\pi\)
0.490565 + 0.871405i \(0.336790\pi\)
\(20\) −1.28871 −0.288164
\(21\) −2.87358 −0.627066
\(22\) 0.931518 0.198600
\(23\) 5.20869 1.08609 0.543043 0.839705i \(-0.317272\pi\)
0.543043 + 0.839705i \(0.317272\pi\)
\(24\) −2.91777 −0.595587
\(25\) −3.70459 −0.740918
\(26\) 1.30814 0.256546
\(27\) 1.00000 0.192450
\(28\) 3.25368 0.614887
\(29\) 8.20744 1.52408 0.762042 0.647528i \(-0.224197\pi\)
0.762042 + 0.647528i \(0.224197\pi\)
\(30\) 1.06022 0.193568
\(31\) 7.18706 1.29083 0.645417 0.763830i \(-0.276684\pi\)
0.645417 + 0.763830i \(0.276684\pi\)
\(32\) 5.41318 0.956924
\(33\) 1.00000 0.174078
\(34\) −0.788110 −0.135160
\(35\) −3.27059 −0.552831
\(36\) −1.13227 −0.188712
\(37\) 4.15083 0.682392 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(38\) 3.98377 0.646253
\(39\) 1.40430 0.224869
\(40\) −3.32089 −0.525079
\(41\) 0.166336 0.0259773 0.0129886 0.999916i \(-0.495865\pi\)
0.0129886 + 0.999916i \(0.495865\pi\)
\(42\) −2.67679 −0.413038
\(43\) 6.54819 0.998590 0.499295 0.866432i \(-0.333592\pi\)
0.499295 + 0.866432i \(0.333592\pi\)
\(44\) −1.13227 −0.170697
\(45\) 1.13816 0.169667
\(46\) 4.85199 0.715387
\(47\) 0.378293 0.0551797 0.0275899 0.999619i \(-0.491217\pi\)
0.0275899 + 0.999619i \(0.491217\pi\)
\(48\) −0.453408 −0.0654438
\(49\) 1.25745 0.179635
\(50\) −3.45089 −0.488030
\(51\) −0.846049 −0.118471
\(52\) −1.59006 −0.220501
\(53\) −0.923656 −0.126874 −0.0634370 0.997986i \(-0.520206\pi\)
−0.0634370 + 0.997986i \(0.520206\pi\)
\(54\) 0.931518 0.126764
\(55\) 1.13816 0.153469
\(56\) 8.38444 1.12042
\(57\) 4.27664 0.566455
\(58\) 7.64538 1.00389
\(59\) −1.30403 −0.169770 −0.0848852 0.996391i \(-0.527052\pi\)
−0.0848852 + 0.996391i \(0.527052\pi\)
\(60\) −1.28871 −0.166372
\(61\) 1.00000 0.128037
\(62\) 6.69488 0.850250
\(63\) −2.87358 −0.362037
\(64\) 5.94929 0.743662
\(65\) 1.59832 0.198248
\(66\) 0.931518 0.114662
\(67\) −6.07605 −0.742308 −0.371154 0.928571i \(-0.621038\pi\)
−0.371154 + 0.928571i \(0.621038\pi\)
\(68\) 0.957959 0.116170
\(69\) 5.20869 0.627052
\(70\) −3.04661 −0.364140
\(71\) −13.3483 −1.58415 −0.792075 0.610424i \(-0.790999\pi\)
−0.792075 + 0.610424i \(0.790999\pi\)
\(72\) −2.91777 −0.343862
\(73\) 3.93967 0.461104 0.230552 0.973060i \(-0.425947\pi\)
0.230552 + 0.973060i \(0.425947\pi\)
\(74\) 3.86657 0.449480
\(75\) −3.70459 −0.427769
\(76\) −4.84233 −0.555454
\(77\) −2.87358 −0.327475
\(78\) 1.30814 0.148117
\(79\) 7.00809 0.788472 0.394236 0.919009i \(-0.371009\pi\)
0.394236 + 0.919009i \(0.371009\pi\)
\(80\) −0.516051 −0.0576962
\(81\) 1.00000 0.111111
\(82\) 0.154945 0.0171108
\(83\) 2.26262 0.248355 0.124177 0.992260i \(-0.460371\pi\)
0.124177 + 0.992260i \(0.460371\pi\)
\(84\) 3.25368 0.355005
\(85\) −0.962939 −0.104445
\(86\) 6.09976 0.657754
\(87\) 8.20744 0.879930
\(88\) −2.91777 −0.311035
\(89\) 2.22445 0.235791 0.117896 0.993026i \(-0.462385\pi\)
0.117896 + 0.993026i \(0.462385\pi\)
\(90\) 1.06022 0.111757
\(91\) −4.03538 −0.423023
\(92\) −5.89766 −0.614874
\(93\) 7.18706 0.745263
\(94\) 0.352387 0.0363459
\(95\) 4.86751 0.499396
\(96\) 5.41318 0.552481
\(97\) 10.6596 1.08231 0.541157 0.840922i \(-0.317986\pi\)
0.541157 + 0.840922i \(0.317986\pi\)
\(98\) 1.17133 0.118323
\(99\) 1.00000 0.100504
\(100\) 4.19461 0.419461
\(101\) −3.11381 −0.309836 −0.154918 0.987927i \(-0.549511\pi\)
−0.154918 + 0.987927i \(0.549511\pi\)
\(102\) −0.788110 −0.0780345
\(103\) −0.547424 −0.0539393 −0.0269697 0.999636i \(-0.508586\pi\)
−0.0269697 + 0.999636i \(0.508586\pi\)
\(104\) −4.09744 −0.401787
\(105\) −3.27059 −0.319177
\(106\) −0.860402 −0.0835697
\(107\) −4.67778 −0.452219 −0.226109 0.974102i \(-0.572601\pi\)
−0.226109 + 0.974102i \(0.572601\pi\)
\(108\) −1.13227 −0.108953
\(109\) 15.4885 1.48353 0.741764 0.670661i \(-0.233989\pi\)
0.741764 + 0.670661i \(0.233989\pi\)
\(110\) 1.06022 0.101088
\(111\) 4.15083 0.393979
\(112\) 1.30290 0.123113
\(113\) −4.79624 −0.451192 −0.225596 0.974221i \(-0.572433\pi\)
−0.225596 + 0.974221i \(0.572433\pi\)
\(114\) 3.98377 0.373114
\(115\) 5.92832 0.552819
\(116\) −9.29307 −0.862840
\(117\) 1.40430 0.129828
\(118\) −1.21473 −0.111825
\(119\) 2.43119 0.222867
\(120\) −3.32089 −0.303154
\(121\) 1.00000 0.0909091
\(122\) 0.931518 0.0843357
\(123\) 0.166336 0.0149980
\(124\) −8.13772 −0.730789
\(125\) −9.90722 −0.886129
\(126\) −2.67679 −0.238467
\(127\) 2.84301 0.252276 0.126138 0.992013i \(-0.459742\pi\)
0.126138 + 0.992013i \(0.459742\pi\)
\(128\) −5.28449 −0.467087
\(129\) 6.54819 0.576536
\(130\) 1.48887 0.130582
\(131\) 5.03902 0.440262 0.220131 0.975470i \(-0.429352\pi\)
0.220131 + 0.975470i \(0.429352\pi\)
\(132\) −1.13227 −0.0985518
\(133\) −12.2893 −1.06561
\(134\) −5.65995 −0.488945
\(135\) 1.13816 0.0979572
\(136\) 2.46858 0.211679
\(137\) −14.5240 −1.24087 −0.620436 0.784257i \(-0.713044\pi\)
−0.620436 + 0.784257i \(0.713044\pi\)
\(138\) 4.85199 0.413029
\(139\) −20.5839 −1.74590 −0.872951 0.487807i \(-0.837797\pi\)
−0.872951 + 0.487807i \(0.837797\pi\)
\(140\) 3.70320 0.312978
\(141\) 0.378293 0.0318580
\(142\) −12.4342 −1.04345
\(143\) 1.40430 0.117434
\(144\) −0.453408 −0.0377840
\(145\) 9.34138 0.775759
\(146\) 3.66988 0.303721
\(147\) 1.25745 0.103712
\(148\) −4.69987 −0.386327
\(149\) 4.64026 0.380145 0.190073 0.981770i \(-0.439128\pi\)
0.190073 + 0.981770i \(0.439128\pi\)
\(150\) −3.45089 −0.281764
\(151\) −20.0189 −1.62912 −0.814558 0.580083i \(-0.803020\pi\)
−0.814558 + 0.580083i \(0.803020\pi\)
\(152\) −12.4783 −1.01212
\(153\) −0.846049 −0.0683990
\(154\) −2.67679 −0.215702
\(155\) 8.18002 0.657035
\(156\) −1.59006 −0.127306
\(157\) 18.2010 1.45260 0.726299 0.687379i \(-0.241239\pi\)
0.726299 + 0.687379i \(0.241239\pi\)
\(158\) 6.52817 0.519353
\(159\) −0.923656 −0.0732507
\(160\) 6.16107 0.487075
\(161\) −14.9676 −1.17961
\(162\) 0.931518 0.0731870
\(163\) −10.2340 −0.801589 −0.400794 0.916168i \(-0.631266\pi\)
−0.400794 + 0.916168i \(0.631266\pi\)
\(164\) −0.188338 −0.0147067
\(165\) 1.13816 0.0886056
\(166\) 2.10767 0.163587
\(167\) 22.1101 1.71093 0.855466 0.517859i \(-0.173271\pi\)
0.855466 + 0.517859i \(0.173271\pi\)
\(168\) 8.38444 0.646873
\(169\) −11.0279 −0.848302
\(170\) −0.896995 −0.0687964
\(171\) 4.27664 0.327043
\(172\) −7.41434 −0.565338
\(173\) −3.68124 −0.279880 −0.139940 0.990160i \(-0.544691\pi\)
−0.139940 + 0.990160i \(0.544691\pi\)
\(174\) 7.64538 0.579595
\(175\) 10.6454 0.804719
\(176\) −0.453408 −0.0341769
\(177\) −1.30403 −0.0980170
\(178\) 2.07212 0.155312
\(179\) 25.7165 1.92214 0.961070 0.276304i \(-0.0891098\pi\)
0.961070 + 0.276304i \(0.0891098\pi\)
\(180\) −1.28871 −0.0960547
\(181\) −22.6825 −1.68597 −0.842987 0.537934i \(-0.819205\pi\)
−0.842987 + 0.537934i \(0.819205\pi\)
\(182\) −3.75903 −0.278638
\(183\) 1.00000 0.0739221
\(184\) −15.1978 −1.12039
\(185\) 4.72430 0.347338
\(186\) 6.69488 0.490892
\(187\) −0.846049 −0.0618692
\(188\) −0.428332 −0.0312393
\(189\) −2.87358 −0.209022
\(190\) 4.53417 0.328943
\(191\) −7.49696 −0.542461 −0.271230 0.962514i \(-0.587431\pi\)
−0.271230 + 0.962514i \(0.587431\pi\)
\(192\) 5.94929 0.429353
\(193\) 0.393089 0.0282952 0.0141476 0.999900i \(-0.495497\pi\)
0.0141476 + 0.999900i \(0.495497\pi\)
\(194\) 9.92957 0.712902
\(195\) 1.59832 0.114458
\(196\) −1.42377 −0.101698
\(197\) −7.27055 −0.518005 −0.259003 0.965877i \(-0.583394\pi\)
−0.259003 + 0.965877i \(0.583394\pi\)
\(198\) 0.931518 0.0662001
\(199\) −9.86046 −0.698989 −0.349495 0.936938i \(-0.613647\pi\)
−0.349495 + 0.936938i \(0.613647\pi\)
\(200\) 10.8091 0.764322
\(201\) −6.07605 −0.428572
\(202\) −2.90058 −0.204084
\(203\) −23.5847 −1.65532
\(204\) 0.957959 0.0670706
\(205\) 0.189317 0.0132224
\(206\) −0.509936 −0.0355289
\(207\) 5.20869 0.362029
\(208\) −0.636723 −0.0441488
\(209\) 4.27664 0.295822
\(210\) −3.04661 −0.210236
\(211\) 22.7443 1.56578 0.782890 0.622160i \(-0.213744\pi\)
0.782890 + 0.622160i \(0.213744\pi\)
\(212\) 1.04583 0.0718280
\(213\) −13.3483 −0.914610
\(214\) −4.35744 −0.297869
\(215\) 7.45289 0.508283
\(216\) −2.91777 −0.198529
\(217\) −20.6526 −1.40199
\(218\) 14.4278 0.977174
\(219\) 3.93967 0.266218
\(220\) −1.28871 −0.0868847
\(221\) −1.18811 −0.0799210
\(222\) 3.86657 0.259507
\(223\) 6.37218 0.426712 0.213356 0.976974i \(-0.431561\pi\)
0.213356 + 0.976974i \(0.431561\pi\)
\(224\) −15.5552 −1.03933
\(225\) −3.70459 −0.246973
\(226\) −4.46778 −0.297192
\(227\) 17.5489 1.16476 0.582379 0.812917i \(-0.302122\pi\)
0.582379 + 0.812917i \(0.302122\pi\)
\(228\) −4.84233 −0.320691
\(229\) −8.33291 −0.550654 −0.275327 0.961351i \(-0.588786\pi\)
−0.275327 + 0.961351i \(0.588786\pi\)
\(230\) 5.52234 0.364132
\(231\) −2.87358 −0.189068
\(232\) −23.9474 −1.57223
\(233\) 22.9485 1.50341 0.751704 0.659501i \(-0.229232\pi\)
0.751704 + 0.659501i \(0.229232\pi\)
\(234\) 1.30814 0.0855155
\(235\) 0.430558 0.0280865
\(236\) 1.47652 0.0961133
\(237\) 7.00809 0.455225
\(238\) 2.26470 0.146798
\(239\) 5.29702 0.342636 0.171318 0.985216i \(-0.445197\pi\)
0.171318 + 0.985216i \(0.445197\pi\)
\(240\) −0.516051 −0.0333109
\(241\) 1.49502 0.0963030 0.0481515 0.998840i \(-0.484667\pi\)
0.0481515 + 0.998840i \(0.484667\pi\)
\(242\) 0.931518 0.0598803
\(243\) 1.00000 0.0641500
\(244\) −1.13227 −0.0724864
\(245\) 1.43117 0.0914344
\(246\) 0.154945 0.00987892
\(247\) 6.00571 0.382134
\(248\) −20.9702 −1.33161
\(249\) 2.26262 0.143388
\(250\) −9.22875 −0.583678
\(251\) −11.0035 −0.694534 −0.347267 0.937766i \(-0.612890\pi\)
−0.347267 + 0.937766i \(0.612890\pi\)
\(252\) 3.25368 0.204962
\(253\) 5.20869 0.327467
\(254\) 2.64831 0.166170
\(255\) −0.962939 −0.0603016
\(256\) −16.8212 −1.05132
\(257\) 20.5640 1.28275 0.641375 0.767228i \(-0.278364\pi\)
0.641375 + 0.767228i \(0.278364\pi\)
\(258\) 6.09976 0.379754
\(259\) −11.9277 −0.741152
\(260\) −1.80974 −0.112235
\(261\) 8.20744 0.508028
\(262\) 4.69394 0.289993
\(263\) 25.0818 1.54661 0.773306 0.634034i \(-0.218602\pi\)
0.773306 + 0.634034i \(0.218602\pi\)
\(264\) −2.91777 −0.179576
\(265\) −1.05127 −0.0645789
\(266\) −11.4477 −0.701902
\(267\) 2.22445 0.136134
\(268\) 6.87975 0.420248
\(269\) −19.5174 −1.19000 −0.595000 0.803726i \(-0.702848\pi\)
−0.595000 + 0.803726i \(0.702848\pi\)
\(270\) 1.06022 0.0645227
\(271\) −5.00015 −0.303737 −0.151869 0.988401i \(-0.548529\pi\)
−0.151869 + 0.988401i \(0.548529\pi\)
\(272\) 0.383605 0.0232595
\(273\) −4.03538 −0.244232
\(274\) −13.5294 −0.817341
\(275\) −3.70459 −0.223395
\(276\) −5.89766 −0.354998
\(277\) −15.9275 −0.956993 −0.478497 0.878089i \(-0.658818\pi\)
−0.478497 + 0.878089i \(0.658818\pi\)
\(278\) −19.1743 −1.15000
\(279\) 7.18706 0.430278
\(280\) 9.54283 0.570293
\(281\) 5.48295 0.327085 0.163543 0.986536i \(-0.447708\pi\)
0.163543 + 0.986536i \(0.447708\pi\)
\(282\) 0.352387 0.0209843
\(283\) −4.22686 −0.251261 −0.125630 0.992077i \(-0.540095\pi\)
−0.125630 + 0.992077i \(0.540095\pi\)
\(284\) 15.1139 0.896846
\(285\) 4.86751 0.288326
\(286\) 1.30814 0.0773517
\(287\) −0.477979 −0.0282142
\(288\) 5.41318 0.318975
\(289\) −16.2842 −0.957894
\(290\) 8.70166 0.510979
\(291\) 10.6596 0.624874
\(292\) −4.46079 −0.261048
\(293\) −15.7392 −0.919494 −0.459747 0.888050i \(-0.652060\pi\)
−0.459747 + 0.888050i \(0.652060\pi\)
\(294\) 1.17133 0.0683136
\(295\) −1.48420 −0.0864132
\(296\) −12.1112 −0.703947
\(297\) 1.00000 0.0580259
\(298\) 4.32249 0.250395
\(299\) 7.31459 0.423013
\(300\) 4.19461 0.242176
\(301\) −18.8167 −1.08458
\(302\) −18.6480 −1.07307
\(303\) −3.11381 −0.178884
\(304\) −1.93906 −0.111213
\(305\) 1.13816 0.0651709
\(306\) −0.788110 −0.0450533
\(307\) 22.6377 1.29200 0.646001 0.763337i \(-0.276440\pi\)
0.646001 + 0.763337i \(0.276440\pi\)
\(308\) 3.25368 0.185395
\(309\) −0.547424 −0.0311419
\(310\) 7.61984 0.432778
\(311\) 8.40924 0.476844 0.238422 0.971162i \(-0.423370\pi\)
0.238422 + 0.971162i \(0.423370\pi\)
\(312\) −4.09744 −0.231972
\(313\) −7.39027 −0.417723 −0.208861 0.977945i \(-0.566976\pi\)
−0.208861 + 0.977945i \(0.566976\pi\)
\(314\) 16.9546 0.956801
\(315\) −3.27059 −0.184277
\(316\) −7.93508 −0.446383
\(317\) 16.1084 0.904738 0.452369 0.891831i \(-0.350579\pi\)
0.452369 + 0.891831i \(0.350579\pi\)
\(318\) −0.860402 −0.0482490
\(319\) 8.20744 0.459528
\(320\) 6.77125 0.378524
\(321\) −4.67778 −0.261088
\(322\) −13.9426 −0.776989
\(323\) −3.61825 −0.201325
\(324\) −1.13227 −0.0629041
\(325\) −5.20238 −0.288576
\(326\) −9.53316 −0.527993
\(327\) 15.4885 0.856515
\(328\) −0.485329 −0.0267978
\(329\) −1.08705 −0.0599313
\(330\) 1.06022 0.0583630
\(331\) 23.9924 1.31874 0.659371 0.751818i \(-0.270823\pi\)
0.659371 + 0.751818i \(0.270823\pi\)
\(332\) −2.56191 −0.140603
\(333\) 4.15083 0.227464
\(334\) 20.5960 1.12696
\(335\) −6.91552 −0.377835
\(336\) 1.30290 0.0710792
\(337\) −2.36611 −0.128890 −0.0644452 0.997921i \(-0.520528\pi\)
−0.0644452 + 0.997921i \(0.520528\pi\)
\(338\) −10.2727 −0.558762
\(339\) −4.79624 −0.260496
\(340\) 1.09031 0.0591304
\(341\) 7.18706 0.389201
\(342\) 3.98377 0.215418
\(343\) 16.5017 0.891007
\(344\) −19.1061 −1.03013
\(345\) 5.92832 0.319170
\(346\) −3.42914 −0.184352
\(347\) −3.62961 −0.194848 −0.0974238 0.995243i \(-0.531060\pi\)
−0.0974238 + 0.995243i \(0.531060\pi\)
\(348\) −9.29307 −0.498161
\(349\) −9.39190 −0.502737 −0.251368 0.967892i \(-0.580881\pi\)
−0.251368 + 0.967892i \(0.580881\pi\)
\(350\) 9.91641 0.530055
\(351\) 1.40430 0.0749562
\(352\) 5.41318 0.288524
\(353\) −1.21621 −0.0647324 −0.0323662 0.999476i \(-0.510304\pi\)
−0.0323662 + 0.999476i \(0.510304\pi\)
\(354\) −1.21473 −0.0645621
\(355\) −15.1925 −0.806334
\(356\) −2.51869 −0.133490
\(357\) 2.43119 0.128672
\(358\) 23.9554 1.26608
\(359\) 10.9139 0.576016 0.288008 0.957628i \(-0.407007\pi\)
0.288008 + 0.957628i \(0.407007\pi\)
\(360\) −3.32089 −0.175026
\(361\) −0.710310 −0.0373847
\(362\) −21.1291 −1.11052
\(363\) 1.00000 0.0524864
\(364\) 4.56915 0.239489
\(365\) 4.48398 0.234702
\(366\) 0.931518 0.0486912
\(367\) −17.3256 −0.904390 −0.452195 0.891919i \(-0.649359\pi\)
−0.452195 + 0.891919i \(0.649359\pi\)
\(368\) −2.36166 −0.123110
\(369\) 0.166336 0.00865909
\(370\) 4.40078 0.228785
\(371\) 2.65420 0.137799
\(372\) −8.13772 −0.421921
\(373\) −26.9978 −1.39789 −0.698946 0.715174i \(-0.746347\pi\)
−0.698946 + 0.715174i \(0.746347\pi\)
\(374\) −0.788110 −0.0407522
\(375\) −9.90722 −0.511607
\(376\) −1.10377 −0.0569227
\(377\) 11.5257 0.593606
\(378\) −2.67679 −0.137679
\(379\) −13.0855 −0.672156 −0.336078 0.941834i \(-0.609101\pi\)
−0.336078 + 0.941834i \(0.609101\pi\)
\(380\) −5.51135 −0.282726
\(381\) 2.84301 0.145652
\(382\) −6.98355 −0.357310
\(383\) −7.46685 −0.381538 −0.190769 0.981635i \(-0.561098\pi\)
−0.190769 + 0.981635i \(0.561098\pi\)
\(384\) −5.28449 −0.269673
\(385\) −3.27059 −0.166685
\(386\) 0.366169 0.0186375
\(387\) 6.54819 0.332863
\(388\) −12.0695 −0.612738
\(389\) −0.429291 −0.0217659 −0.0108830 0.999941i \(-0.503464\pi\)
−0.0108830 + 0.999941i \(0.503464\pi\)
\(390\) 1.48887 0.0753917
\(391\) −4.40681 −0.222862
\(392\) −3.66894 −0.185309
\(393\) 5.03902 0.254185
\(394\) −6.77265 −0.341201
\(395\) 7.97633 0.401333
\(396\) −1.13227 −0.0568989
\(397\) −32.2072 −1.61643 −0.808217 0.588885i \(-0.799567\pi\)
−0.808217 + 0.588885i \(0.799567\pi\)
\(398\) −9.18519 −0.460412
\(399\) −12.2893 −0.615233
\(400\) 1.67969 0.0839846
\(401\) −13.0892 −0.653643 −0.326821 0.945086i \(-0.605978\pi\)
−0.326821 + 0.945086i \(0.605978\pi\)
\(402\) −5.65995 −0.282293
\(403\) 10.0928 0.502759
\(404\) 3.52569 0.175410
\(405\) 1.13816 0.0565556
\(406\) −21.9696 −1.09033
\(407\) 4.15083 0.205749
\(408\) 2.46858 0.122213
\(409\) −26.5459 −1.31261 −0.656304 0.754496i \(-0.727881\pi\)
−0.656304 + 0.754496i \(0.727881\pi\)
\(410\) 0.176352 0.00870940
\(411\) −14.5240 −0.716418
\(412\) 0.619834 0.0305370
\(413\) 3.74723 0.184389
\(414\) 4.85199 0.238462
\(415\) 2.57522 0.126413
\(416\) 7.60176 0.372707
\(417\) −20.5839 −1.00800
\(418\) 3.98377 0.194853
\(419\) −24.8530 −1.21415 −0.607073 0.794646i \(-0.707657\pi\)
−0.607073 + 0.794646i \(0.707657\pi\)
\(420\) 3.70320 0.180698
\(421\) −19.4476 −0.947817 −0.473908 0.880574i \(-0.657157\pi\)
−0.473908 + 0.880574i \(0.657157\pi\)
\(422\) 21.1867 1.03135
\(423\) 0.378293 0.0183932
\(424\) 2.69502 0.130882
\(425\) 3.13427 0.152034
\(426\) −12.4342 −0.602438
\(427\) −2.87358 −0.139062
\(428\) 5.29653 0.256018
\(429\) 1.40430 0.0678005
\(430\) 6.94250 0.334797
\(431\) −32.3889 −1.56012 −0.780058 0.625707i \(-0.784811\pi\)
−0.780058 + 0.625707i \(0.784811\pi\)
\(432\) −0.453408 −0.0218146
\(433\) −20.8652 −1.00272 −0.501359 0.865239i \(-0.667166\pi\)
−0.501359 + 0.865239i \(0.667166\pi\)
\(434\) −19.2382 −0.923465
\(435\) 9.34138 0.447885
\(436\) −17.5372 −0.839880
\(437\) 22.2757 1.06559
\(438\) 3.66988 0.175354
\(439\) 12.5132 0.597224 0.298612 0.954375i \(-0.403476\pi\)
0.298612 + 0.954375i \(0.403476\pi\)
\(440\) −3.32089 −0.158317
\(441\) 1.25745 0.0598784
\(442\) −1.10675 −0.0526426
\(443\) 28.5951 1.35859 0.679297 0.733863i \(-0.262285\pi\)
0.679297 + 0.733863i \(0.262285\pi\)
\(444\) −4.69987 −0.223046
\(445\) 2.53178 0.120018
\(446\) 5.93580 0.281068
\(447\) 4.64026 0.219477
\(448\) −17.0958 −0.807698
\(449\) 13.9163 0.656751 0.328376 0.944547i \(-0.393499\pi\)
0.328376 + 0.944547i \(0.393499\pi\)
\(450\) −3.45089 −0.162677
\(451\) 0.166336 0.00783244
\(452\) 5.43065 0.255436
\(453\) −20.0189 −0.940570
\(454\) 16.3471 0.767207
\(455\) −4.59291 −0.215319
\(456\) −12.4783 −0.584348
\(457\) 26.4741 1.23841 0.619204 0.785230i \(-0.287455\pi\)
0.619204 + 0.785230i \(0.287455\pi\)
\(458\) −7.76226 −0.362706
\(459\) −0.846049 −0.0394902
\(460\) −6.71248 −0.312971
\(461\) −34.1329 −1.58973 −0.794864 0.606787i \(-0.792458\pi\)
−0.794864 + 0.606787i \(0.792458\pi\)
\(462\) −2.67679 −0.124536
\(463\) 24.5877 1.14269 0.571344 0.820711i \(-0.306422\pi\)
0.571344 + 0.820711i \(0.306422\pi\)
\(464\) −3.72132 −0.172758
\(465\) 8.18002 0.379340
\(466\) 21.3770 0.990269
\(467\) −4.21020 −0.194825 −0.0974124 0.995244i \(-0.531057\pi\)
−0.0974124 + 0.995244i \(0.531057\pi\)
\(468\) −1.59006 −0.0735004
\(469\) 17.4600 0.806228
\(470\) 0.401073 0.0185001
\(471\) 18.2010 0.838658
\(472\) 3.80486 0.175133
\(473\) 6.54819 0.301086
\(474\) 6.52817 0.299849
\(475\) −15.8432 −0.726937
\(476\) −2.75277 −0.126173
\(477\) −0.923656 −0.0422913
\(478\) 4.93427 0.225688
\(479\) 25.1877 1.15086 0.575428 0.817853i \(-0.304836\pi\)
0.575428 + 0.817853i \(0.304836\pi\)
\(480\) 6.16107 0.281213
\(481\) 5.82902 0.265781
\(482\) 1.39264 0.0634331
\(483\) −14.9676 −0.681048
\(484\) −1.13227 −0.0514670
\(485\) 12.1323 0.550899
\(486\) 0.931518 0.0422545
\(487\) 21.0131 0.952194 0.476097 0.879393i \(-0.342051\pi\)
0.476097 + 0.879393i \(0.342051\pi\)
\(488\) −2.91777 −0.132081
\(489\) −10.2340 −0.462798
\(490\) 1.33317 0.0602263
\(491\) −4.29368 −0.193771 −0.0968856 0.995296i \(-0.530888\pi\)
−0.0968856 + 0.995296i \(0.530888\pi\)
\(492\) −0.188338 −0.00849091
\(493\) −6.94390 −0.312737
\(494\) 5.59443 0.251705
\(495\) 1.13816 0.0511565
\(496\) −3.25867 −0.146319
\(497\) 38.3573 1.72056
\(498\) 2.10767 0.0944470
\(499\) −16.7576 −0.750173 −0.375087 0.926990i \(-0.622387\pi\)
−0.375087 + 0.926990i \(0.622387\pi\)
\(500\) 11.2177 0.501670
\(501\) 22.1101 0.987807
\(502\) −10.2499 −0.457477
\(503\) −20.7558 −0.925456 −0.462728 0.886500i \(-0.653129\pi\)
−0.462728 + 0.886500i \(0.653129\pi\)
\(504\) 8.38444 0.373473
\(505\) −3.54402 −0.157707
\(506\) 4.85199 0.215697
\(507\) −11.0279 −0.489767
\(508\) −3.21906 −0.142823
\(509\) 14.7620 0.654315 0.327158 0.944970i \(-0.393909\pi\)
0.327158 + 0.944970i \(0.393909\pi\)
\(510\) −0.896995 −0.0397196
\(511\) −11.3210 −0.500810
\(512\) −5.10026 −0.225402
\(513\) 4.27664 0.188818
\(514\) 19.1558 0.844925
\(515\) −0.623056 −0.0274551
\(516\) −7.41434 −0.326398
\(517\) 0.378293 0.0166373
\(518\) −11.1109 −0.488184
\(519\) −3.68124 −0.161589
\(520\) −4.66354 −0.204510
\(521\) 3.40170 0.149031 0.0745155 0.997220i \(-0.476259\pi\)
0.0745155 + 0.997220i \(0.476259\pi\)
\(522\) 7.64538 0.334629
\(523\) 24.0147 1.05009 0.525045 0.851074i \(-0.324048\pi\)
0.525045 + 0.851074i \(0.324048\pi\)
\(524\) −5.70556 −0.249248
\(525\) 10.6454 0.464605
\(526\) 23.3642 1.01873
\(527\) −6.08061 −0.264875
\(528\) −0.453408 −0.0197320
\(529\) 4.13044 0.179584
\(530\) −0.979276 −0.0425370
\(531\) −1.30403 −0.0565901
\(532\) 13.9148 0.603284
\(533\) 0.233586 0.0101177
\(534\) 2.07212 0.0896692
\(535\) −5.32407 −0.230180
\(536\) 17.7285 0.765755
\(537\) 25.7165 1.10975
\(538\) −18.1809 −0.783832
\(539\) 1.25745 0.0541620
\(540\) −1.28871 −0.0554572
\(541\) −23.7949 −1.02302 −0.511512 0.859276i \(-0.670914\pi\)
−0.511512 + 0.859276i \(0.670914\pi\)
\(542\) −4.65773 −0.200067
\(543\) −22.6825 −0.973397
\(544\) −4.57982 −0.196358
\(545\) 17.6284 0.755117
\(546\) −3.75903 −0.160872
\(547\) −15.3604 −0.656763 −0.328382 0.944545i \(-0.606503\pi\)
−0.328382 + 0.944545i \(0.606503\pi\)
\(548\) 16.4452 0.702504
\(549\) 1.00000 0.0426790
\(550\) −3.45089 −0.147147
\(551\) 35.1003 1.49532
\(552\) −15.1978 −0.646859
\(553\) −20.1383 −0.856367
\(554\) −14.8368 −0.630355
\(555\) 4.72430 0.200536
\(556\) 23.3066 0.988420
\(557\) −19.2009 −0.813567 −0.406783 0.913525i \(-0.633350\pi\)
−0.406783 + 0.913525i \(0.633350\pi\)
\(558\) 6.69488 0.283417
\(559\) 9.19565 0.388935
\(560\) 1.48291 0.0626645
\(561\) −0.846049 −0.0357202
\(562\) 5.10747 0.215446
\(563\) −8.35263 −0.352021 −0.176011 0.984388i \(-0.556319\pi\)
−0.176011 + 0.984388i \(0.556319\pi\)
\(564\) −0.428332 −0.0180360
\(565\) −5.45888 −0.229657
\(566\) −3.93740 −0.165501
\(567\) −2.87358 −0.120679
\(568\) 38.9472 1.63419
\(569\) 39.5884 1.65963 0.829817 0.558036i \(-0.188445\pi\)
0.829817 + 0.558036i \(0.188445\pi\)
\(570\) 4.53417 0.189915
\(571\) −22.6971 −0.949843 −0.474921 0.880028i \(-0.657523\pi\)
−0.474921 + 0.880028i \(0.657523\pi\)
\(572\) −1.59006 −0.0664837
\(573\) −7.49696 −0.313190
\(574\) −0.445246 −0.0185842
\(575\) −19.2961 −0.804702
\(576\) 5.94929 0.247887
\(577\) 17.7404 0.738541 0.369271 0.929322i \(-0.379608\pi\)
0.369271 + 0.929322i \(0.379608\pi\)
\(578\) −15.1690 −0.630948
\(579\) 0.393089 0.0163362
\(580\) −10.5770 −0.439186
\(581\) −6.50181 −0.269741
\(582\) 9.92957 0.411594
\(583\) −0.923656 −0.0382539
\(584\) −11.4951 −0.475669
\(585\) 1.59832 0.0660825
\(586\) −14.6613 −0.605655
\(587\) 45.4916 1.87764 0.938820 0.344408i \(-0.111920\pi\)
0.938820 + 0.344408i \(0.111920\pi\)
\(588\) −1.42377 −0.0587154
\(589\) 30.7365 1.26648
\(590\) −1.38256 −0.0569189
\(591\) −7.27055 −0.299071
\(592\) −1.88202 −0.0773504
\(593\) 0.631502 0.0259327 0.0129663 0.999916i \(-0.495873\pi\)
0.0129663 + 0.999916i \(0.495873\pi\)
\(594\) 0.931518 0.0382207
\(595\) 2.76708 0.113439
\(596\) −5.25405 −0.215214
\(597\) −9.86046 −0.403562
\(598\) 6.81367 0.278632
\(599\) 7.97573 0.325880 0.162940 0.986636i \(-0.447902\pi\)
0.162940 + 0.986636i \(0.447902\pi\)
\(600\) 10.8091 0.441282
\(601\) −42.1385 −1.71886 −0.859432 0.511249i \(-0.829183\pi\)
−0.859432 + 0.511249i \(0.829183\pi\)
\(602\) −17.5281 −0.714393
\(603\) −6.07605 −0.247436
\(604\) 22.6669 0.922302
\(605\) 1.13816 0.0462728
\(606\) −2.90058 −0.117828
\(607\) 5.96984 0.242308 0.121154 0.992634i \(-0.461340\pi\)
0.121154 + 0.992634i \(0.461340\pi\)
\(608\) 23.1503 0.938867
\(609\) −23.5847 −0.955701
\(610\) 1.06022 0.0429269
\(611\) 0.531239 0.0214916
\(612\) 0.957959 0.0387232
\(613\) −19.5289 −0.788767 −0.394383 0.918946i \(-0.629042\pi\)
−0.394383 + 0.918946i \(0.629042\pi\)
\(614\) 21.0874 0.851019
\(615\) 0.189317 0.00763398
\(616\) 8.38444 0.337819
\(617\) −40.6362 −1.63595 −0.817975 0.575254i \(-0.804903\pi\)
−0.817975 + 0.575254i \(0.804903\pi\)
\(618\) −0.509936 −0.0205126
\(619\) 27.3664 1.09995 0.549974 0.835181i \(-0.314637\pi\)
0.549974 + 0.835181i \(0.314637\pi\)
\(620\) −9.26203 −0.371972
\(621\) 5.20869 0.209017
\(622\) 7.83336 0.314089
\(623\) −6.39213 −0.256095
\(624\) −0.636723 −0.0254893
\(625\) 7.24696 0.289878
\(626\) −6.88417 −0.275147
\(627\) 4.27664 0.170793
\(628\) −20.6085 −0.822370
\(629\) −3.51180 −0.140025
\(630\) −3.04661 −0.121380
\(631\) 1.65053 0.0657067 0.0328534 0.999460i \(-0.489541\pi\)
0.0328534 + 0.999460i \(0.489541\pi\)
\(632\) −20.4480 −0.813378
\(633\) 22.7443 0.904004
\(634\) 15.0053 0.595935
\(635\) 3.23580 0.128409
\(636\) 1.04583 0.0414699
\(637\) 1.76584 0.0699650
\(638\) 7.64538 0.302683
\(639\) −13.3483 −0.528050
\(640\) −6.01459 −0.237748
\(641\) −24.2765 −0.958866 −0.479433 0.877579i \(-0.659158\pi\)
−0.479433 + 0.877579i \(0.659158\pi\)
\(642\) −4.35744 −0.171974
\(643\) −27.9072 −1.10055 −0.550277 0.834982i \(-0.685478\pi\)
−0.550277 + 0.834982i \(0.685478\pi\)
\(644\) 16.9474 0.667821
\(645\) 7.45289 0.293457
\(646\) −3.37047 −0.132609
\(647\) 35.0312 1.37722 0.688609 0.725133i \(-0.258222\pi\)
0.688609 + 0.725133i \(0.258222\pi\)
\(648\) −2.91777 −0.114621
\(649\) −1.30403 −0.0511877
\(650\) −4.84611 −0.190080
\(651\) −20.6526 −0.809438
\(652\) 11.5877 0.453809
\(653\) −10.2826 −0.402390 −0.201195 0.979551i \(-0.564483\pi\)
−0.201195 + 0.979551i \(0.564483\pi\)
\(654\) 14.4278 0.564172
\(655\) 5.73522 0.224093
\(656\) −0.0754179 −0.00294458
\(657\) 3.93967 0.153701
\(658\) −1.01261 −0.0394757
\(659\) 16.9607 0.660697 0.330348 0.943859i \(-0.392834\pi\)
0.330348 + 0.943859i \(0.392834\pi\)
\(660\) −1.28871 −0.0501629
\(661\) −15.7229 −0.611551 −0.305776 0.952104i \(-0.598916\pi\)
−0.305776 + 0.952104i \(0.598916\pi\)
\(662\) 22.3494 0.868632
\(663\) −1.18811 −0.0461424
\(664\) −6.60180 −0.256200
\(665\) −13.9872 −0.542399
\(666\) 3.86657 0.149827
\(667\) 42.7500 1.65529
\(668\) −25.0347 −0.968622
\(669\) 6.37218 0.246363
\(670\) −6.44193 −0.248873
\(671\) 1.00000 0.0386046
\(672\) −15.5552 −0.600055
\(673\) 8.19652 0.315953 0.157976 0.987443i \(-0.449503\pi\)
0.157976 + 0.987443i \(0.449503\pi\)
\(674\) −2.20408 −0.0848979
\(675\) −3.70459 −0.142590
\(676\) 12.4866 0.480255
\(677\) 4.86904 0.187133 0.0935663 0.995613i \(-0.470173\pi\)
0.0935663 + 0.995613i \(0.470173\pi\)
\(678\) −4.46778 −0.171584
\(679\) −30.6311 −1.17551
\(680\) 2.80964 0.107745
\(681\) 17.5489 0.672474
\(682\) 6.69488 0.256360
\(683\) 19.5539 0.748208 0.374104 0.927387i \(-0.377950\pi\)
0.374104 + 0.927387i \(0.377950\pi\)
\(684\) −4.84233 −0.185151
\(685\) −16.5307 −0.631605
\(686\) 15.3716 0.586891
\(687\) −8.33291 −0.317920
\(688\) −2.96900 −0.113192
\(689\) −1.29709 −0.0494154
\(690\) 5.52234 0.210232
\(691\) −3.37313 −0.128320 −0.0641600 0.997940i \(-0.520437\pi\)
−0.0641600 + 0.997940i \(0.520437\pi\)
\(692\) 4.16817 0.158450
\(693\) −2.87358 −0.109158
\(694\) −3.38105 −0.128343
\(695\) −23.4278 −0.888665
\(696\) −23.9474 −0.907725
\(697\) −0.140728 −0.00533046
\(698\) −8.74872 −0.331144
\(699\) 22.9485 0.867993
\(700\) −12.0535 −0.455581
\(701\) −19.2156 −0.725765 −0.362882 0.931835i \(-0.618207\pi\)
−0.362882 + 0.931835i \(0.618207\pi\)
\(702\) 1.30814 0.0493724
\(703\) 17.7516 0.669515
\(704\) 5.94929 0.224222
\(705\) 0.430558 0.0162158
\(706\) −1.13292 −0.0426381
\(707\) 8.94779 0.336516
\(708\) 1.47652 0.0554910
\(709\) 16.3974 0.615817 0.307908 0.951416i \(-0.400371\pi\)
0.307908 + 0.951416i \(0.400371\pi\)
\(710\) −14.1521 −0.531118
\(711\) 7.00809 0.262824
\(712\) −6.49043 −0.243239
\(713\) 37.4352 1.40196
\(714\) 2.26470 0.0847541
\(715\) 1.59832 0.0597739
\(716\) −29.1181 −1.08819
\(717\) 5.29702 0.197821
\(718\) 10.1665 0.379412
\(719\) −44.2474 −1.65015 −0.825075 0.565024i \(-0.808867\pi\)
−0.825075 + 0.565024i \(0.808867\pi\)
\(720\) −0.516051 −0.0192321
\(721\) 1.57307 0.0585840
\(722\) −0.661667 −0.0246247
\(723\) 1.49502 0.0556006
\(724\) 25.6828 0.954492
\(725\) −30.4052 −1.12922
\(726\) 0.931518 0.0345719
\(727\) 8.20387 0.304265 0.152132 0.988360i \(-0.451386\pi\)
0.152132 + 0.988360i \(0.451386\pi\)
\(728\) 11.7743 0.436385
\(729\) 1.00000 0.0370370
\(730\) 4.17691 0.154594
\(731\) −5.54009 −0.204908
\(732\) −1.13227 −0.0418500
\(733\) 1.42155 0.0525062 0.0262531 0.999655i \(-0.491642\pi\)
0.0262531 + 0.999655i \(0.491642\pi\)
\(734\) −16.1391 −0.595706
\(735\) 1.43117 0.0527897
\(736\) 28.1956 1.03930
\(737\) −6.07605 −0.223814
\(738\) 0.154945 0.00570359
\(739\) −15.8808 −0.584184 −0.292092 0.956390i \(-0.594351\pi\)
−0.292092 + 0.956390i \(0.594351\pi\)
\(740\) −5.34921 −0.196641
\(741\) 6.00571 0.220625
\(742\) 2.47243 0.0907659
\(743\) −43.9673 −1.61300 −0.806502 0.591231i \(-0.798642\pi\)
−0.806502 + 0.591231i \(0.798642\pi\)
\(744\) −20.9702 −0.768804
\(745\) 5.28136 0.193494
\(746\) −25.1489 −0.920768
\(747\) 2.26262 0.0827849
\(748\) 0.957959 0.0350265
\(749\) 13.4420 0.491159
\(750\) −9.22875 −0.336986
\(751\) 7.36604 0.268791 0.134395 0.990928i \(-0.457091\pi\)
0.134395 + 0.990928i \(0.457091\pi\)
\(752\) −0.171521 −0.00625473
\(753\) −11.0035 −0.400989
\(754\) 10.7364 0.390998
\(755\) −22.7847 −0.829221
\(756\) 3.25368 0.118335
\(757\) 12.7893 0.464835 0.232418 0.972616i \(-0.425336\pi\)
0.232418 + 0.972616i \(0.425336\pi\)
\(758\) −12.1894 −0.442738
\(759\) 5.20869 0.189063
\(760\) −14.2023 −0.515170
\(761\) −24.5790 −0.890988 −0.445494 0.895285i \(-0.646972\pi\)
−0.445494 + 0.895285i \(0.646972\pi\)
\(762\) 2.64831 0.0959382
\(763\) −44.5074 −1.61127
\(764\) 8.48861 0.307107
\(765\) −0.962939 −0.0348151
\(766\) −6.95550 −0.251313
\(767\) −1.83126 −0.0661229
\(768\) −16.8212 −0.606982
\(769\) −21.8219 −0.786918 −0.393459 0.919342i \(-0.628722\pi\)
−0.393459 + 0.919342i \(0.628722\pi\)
\(770\) −3.04661 −0.109792
\(771\) 20.5640 0.740596
\(772\) −0.445084 −0.0160189
\(773\) 44.0296 1.58363 0.791817 0.610758i \(-0.209135\pi\)
0.791817 + 0.610758i \(0.209135\pi\)
\(774\) 6.09976 0.219251
\(775\) −26.6251 −0.956403
\(776\) −31.1021 −1.11650
\(777\) −11.9277 −0.427905
\(778\) −0.399892 −0.0143368
\(779\) 0.711359 0.0254871
\(780\) −1.80974 −0.0647991
\(781\) −13.3483 −0.477639
\(782\) −4.10502 −0.146795
\(783\) 8.20744 0.293310
\(784\) −0.570136 −0.0203620
\(785\) 20.7157 0.739373
\(786\) 4.69394 0.167427
\(787\) 0.872256 0.0310926 0.0155463 0.999879i \(-0.495051\pi\)
0.0155463 + 0.999879i \(0.495051\pi\)
\(788\) 8.23226 0.293262
\(789\) 25.0818 0.892936
\(790\) 7.43010 0.264351
\(791\) 13.7824 0.490044
\(792\) −2.91777 −0.103678
\(793\) 1.40430 0.0498683
\(794\) −30.0016 −1.06472
\(795\) −1.05127 −0.0372847
\(796\) 11.1647 0.395724
\(797\) 18.0615 0.639771 0.319886 0.947456i \(-0.396356\pi\)
0.319886 + 0.947456i \(0.396356\pi\)
\(798\) −11.4477 −0.405243
\(799\) −0.320055 −0.0113227
\(800\) −20.0536 −0.709003
\(801\) 2.22445 0.0785971
\(802\) −12.1928 −0.430543
\(803\) 3.93967 0.139028
\(804\) 6.87975 0.242630
\(805\) −17.0355 −0.600422
\(806\) 9.40165 0.331159
\(807\) −19.5174 −0.687046
\(808\) 9.08540 0.319623
\(809\) 47.4599 1.66860 0.834300 0.551310i \(-0.185872\pi\)
0.834300 + 0.551310i \(0.185872\pi\)
\(810\) 1.06022 0.0372522
\(811\) 17.9421 0.630034 0.315017 0.949086i \(-0.397990\pi\)
0.315017 + 0.949086i \(0.397990\pi\)
\(812\) 26.7043 0.937139
\(813\) −5.00015 −0.175363
\(814\) 3.86657 0.135523
\(815\) −11.6479 −0.408009
\(816\) 0.383605 0.0134289
\(817\) 28.0043 0.979746
\(818\) −24.7280 −0.864593
\(819\) −4.03538 −0.141008
\(820\) −0.214358 −0.00748572
\(821\) 18.7155 0.653176 0.326588 0.945167i \(-0.394101\pi\)
0.326588 + 0.945167i \(0.394101\pi\)
\(822\) −13.5294 −0.471892
\(823\) −15.1782 −0.529079 −0.264539 0.964375i \(-0.585220\pi\)
−0.264539 + 0.964375i \(0.585220\pi\)
\(824\) 1.59726 0.0556431
\(825\) −3.70459 −0.128977
\(826\) 3.49062 0.121454
\(827\) 7.54528 0.262375 0.131188 0.991358i \(-0.458121\pi\)
0.131188 + 0.991358i \(0.458121\pi\)
\(828\) −5.89766 −0.204958
\(829\) −38.8988 −1.35101 −0.675505 0.737355i \(-0.736074\pi\)
−0.675505 + 0.737355i \(0.736074\pi\)
\(830\) 2.39887 0.0832659
\(831\) −15.9275 −0.552520
\(832\) 8.35462 0.289644
\(833\) −1.06386 −0.0368606
\(834\) −19.1743 −0.663950
\(835\) 25.1648 0.870865
\(836\) −4.84233 −0.167476
\(837\) 7.18706 0.248421
\(838\) −23.1510 −0.799737
\(839\) −43.1046 −1.48814 −0.744068 0.668104i \(-0.767106\pi\)
−0.744068 + 0.668104i \(0.767106\pi\)
\(840\) 9.54283 0.329259
\(841\) 38.3620 1.32283
\(842\) −18.1158 −0.624311
\(843\) 5.48295 0.188843
\(844\) −25.7528 −0.886446
\(845\) −12.5515 −0.431786
\(846\) 0.352387 0.0121153
\(847\) −2.87358 −0.0987373
\(848\) 0.418793 0.0143814
\(849\) −4.22686 −0.145066
\(850\) 2.91963 0.100142
\(851\) 21.6204 0.741136
\(852\) 15.1139 0.517794
\(853\) −38.9132 −1.33236 −0.666181 0.745790i \(-0.732072\pi\)
−0.666181 + 0.745790i \(0.732072\pi\)
\(854\) −2.67679 −0.0915978
\(855\) 4.86751 0.166465
\(856\) 13.6487 0.466503
\(857\) −19.4681 −0.665018 −0.332509 0.943100i \(-0.607895\pi\)
−0.332509 + 0.943100i \(0.607895\pi\)
\(858\) 1.30814 0.0446590
\(859\) −46.4664 −1.58541 −0.792707 0.609603i \(-0.791329\pi\)
−0.792707 + 0.609603i \(0.791329\pi\)
\(860\) −8.43871 −0.287758
\(861\) −0.477979 −0.0162895
\(862\) −30.1708 −1.02762
\(863\) −6.69874 −0.228028 −0.114014 0.993479i \(-0.536371\pi\)
−0.114014 + 0.993479i \(0.536371\pi\)
\(864\) 5.41318 0.184160
\(865\) −4.18984 −0.142459
\(866\) −19.4363 −0.660473
\(867\) −16.2842 −0.553040
\(868\) 23.3844 0.793717
\(869\) 7.00809 0.237733
\(870\) 8.70166 0.295014
\(871\) −8.53263 −0.289117
\(872\) −45.1918 −1.53039
\(873\) 10.6596 0.360771
\(874\) 20.7502 0.701887
\(875\) 28.4692 0.962433
\(876\) −4.46079 −0.150716
\(877\) −2.94494 −0.0994436 −0.0497218 0.998763i \(-0.515833\pi\)
−0.0497218 + 0.998763i \(0.515833\pi\)
\(878\) 11.6563 0.393381
\(879\) −15.7392 −0.530870
\(880\) −0.516051 −0.0173961
\(881\) 52.7814 1.77825 0.889126 0.457663i \(-0.151313\pi\)
0.889126 + 0.457663i \(0.151313\pi\)
\(882\) 1.17133 0.0394409
\(883\) −13.1341 −0.441998 −0.220999 0.975274i \(-0.570932\pi\)
−0.220999 + 0.975274i \(0.570932\pi\)
\(884\) 1.34527 0.0452462
\(885\) −1.48420 −0.0498907
\(886\) 26.6369 0.894883
\(887\) −11.6115 −0.389875 −0.194938 0.980816i \(-0.562450\pi\)
−0.194938 + 0.980816i \(0.562450\pi\)
\(888\) −12.1112 −0.406424
\(889\) −8.16960 −0.274000
\(890\) 2.35840 0.0790537
\(891\) 1.00000 0.0335013
\(892\) −7.21505 −0.241578
\(893\) 1.61783 0.0541385
\(894\) 4.32249 0.144566
\(895\) 29.2695 0.978371
\(896\) 15.1854 0.507308
\(897\) 7.31459 0.244227
\(898\) 12.9633 0.432591
\(899\) 58.9874 1.96734
\(900\) 4.19461 0.139820
\(901\) 0.781459 0.0260342
\(902\) 0.154945 0.00515910
\(903\) −18.8167 −0.626182
\(904\) 13.9943 0.465444
\(905\) −25.8163 −0.858162
\(906\) −18.6480 −0.619537
\(907\) 17.2320 0.572179 0.286089 0.958203i \(-0.407645\pi\)
0.286089 + 0.958203i \(0.407645\pi\)
\(908\) −19.8701 −0.659413
\(909\) −3.11381 −0.103279
\(910\) −4.27838 −0.141827
\(911\) 7.53148 0.249529 0.124765 0.992186i \(-0.460182\pi\)
0.124765 + 0.992186i \(0.460182\pi\)
\(912\) −1.93906 −0.0642089
\(913\) 2.26262 0.0748818
\(914\) 24.6611 0.815718
\(915\) 1.13816 0.0376264
\(916\) 9.43514 0.311746
\(917\) −14.4800 −0.478173
\(918\) −0.788110 −0.0260115
\(919\) 34.0656 1.12372 0.561860 0.827232i \(-0.310086\pi\)
0.561860 + 0.827232i \(0.310086\pi\)
\(920\) −17.2975 −0.570281
\(921\) 22.6377 0.745937
\(922\) −31.7954 −1.04713
\(923\) −18.7451 −0.617001
\(924\) 3.25368 0.107038
\(925\) −15.3771 −0.505596
\(926\) 22.9039 0.752669
\(927\) −0.547424 −0.0179798
\(928\) 44.4284 1.45843
\(929\) 37.7346 1.23803 0.619015 0.785379i \(-0.287532\pi\)
0.619015 + 0.785379i \(0.287532\pi\)
\(930\) 7.61984 0.249864
\(931\) 5.37765 0.176245
\(932\) −25.9840 −0.851135
\(933\) 8.40924 0.275306
\(934\) −3.92188 −0.128328
\(935\) −0.962939 −0.0314915
\(936\) −4.09744 −0.133929
\(937\) −14.3169 −0.467713 −0.233856 0.972271i \(-0.575135\pi\)
−0.233856 + 0.972271i \(0.575135\pi\)
\(938\) 16.2643 0.531048
\(939\) −7.39027 −0.241172
\(940\) −0.487510 −0.0159008
\(941\) 16.2025 0.528187 0.264093 0.964497i \(-0.414927\pi\)
0.264093 + 0.964497i \(0.414927\pi\)
\(942\) 16.9546 0.552410
\(943\) 0.866391 0.0282136
\(944\) 0.591258 0.0192438
\(945\) −3.27059 −0.106392
\(946\) 6.09976 0.198320
\(947\) 22.4697 0.730166 0.365083 0.930975i \(-0.381041\pi\)
0.365083 + 0.930975i \(0.381041\pi\)
\(948\) −7.93508 −0.257719
\(949\) 5.53250 0.179593
\(950\) −14.7582 −0.478821
\(951\) 16.1084 0.522351
\(952\) −7.09365 −0.229906
\(953\) 23.0493 0.746639 0.373319 0.927703i \(-0.378220\pi\)
0.373319 + 0.927703i \(0.378220\pi\)
\(954\) −0.860402 −0.0278566
\(955\) −8.53274 −0.276113
\(956\) −5.99768 −0.193979
\(957\) 8.20744 0.265309
\(958\) 23.4628 0.758049
\(959\) 41.7359 1.34772
\(960\) 6.77125 0.218541
\(961\) 20.6538 0.666253
\(962\) 5.42984 0.175065
\(963\) −4.67778 −0.150740
\(964\) −1.69278 −0.0545207
\(965\) 0.447398 0.0144023
\(966\) −13.9426 −0.448595
\(967\) 53.4375 1.71843 0.859217 0.511611i \(-0.170951\pi\)
0.859217 + 0.511611i \(0.170951\pi\)
\(968\) −2.91777 −0.0937807
\(969\) −3.61825 −0.116235
\(970\) 11.3014 0.362867
\(971\) 19.8254 0.636227 0.318113 0.948053i \(-0.396951\pi\)
0.318113 + 0.948053i \(0.396951\pi\)
\(972\) −1.13227 −0.0363177
\(973\) 59.1494 1.89624
\(974\) 19.5741 0.627194
\(975\) −5.20238 −0.166609
\(976\) −0.453408 −0.0145132
\(977\) 46.6622 1.49286 0.746428 0.665466i \(-0.231767\pi\)
0.746428 + 0.665466i \(0.231767\pi\)
\(978\) −9.53316 −0.304837
\(979\) 2.22445 0.0710937
\(980\) −1.62048 −0.0517644
\(981\) 15.4885 0.494509
\(982\) −3.99964 −0.127634
\(983\) −23.3443 −0.744568 −0.372284 0.928119i \(-0.621425\pi\)
−0.372284 + 0.928119i \(0.621425\pi\)
\(984\) −0.485329 −0.0154717
\(985\) −8.27505 −0.263665
\(986\) −6.46837 −0.205995
\(987\) −1.08705 −0.0346013
\(988\) −6.80011 −0.216340
\(989\) 34.1075 1.08455
\(990\) 1.06022 0.0336959
\(991\) −32.4785 −1.03171 −0.515857 0.856675i \(-0.672526\pi\)
−0.515857 + 0.856675i \(0.672526\pi\)
\(992\) 38.9049 1.23523
\(993\) 23.9924 0.761376
\(994\) 35.7306 1.13330
\(995\) −11.2228 −0.355786
\(996\) −2.56191 −0.0811771
\(997\) −58.9975 −1.86847 −0.934234 0.356660i \(-0.883915\pi\)
−0.934234 + 0.356660i \(0.883915\pi\)
\(998\) −15.6100 −0.494126
\(999\) 4.15083 0.131326
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 2013.2.a.g.1.8 13
3.2 odd 2 6039.2.a.h.1.6 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2013.2.a.g.1.8 13 1.1 even 1 trivial
6039.2.a.h.1.6 13 3.2 odd 2