Properties

Label 4002.2.a.ba.1.2
Level $4002$
Weight $2$
Character 4002.1
Self dual yes
Analytic conductor $31.956$
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] = [4002,2,Mod(1,4002)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4002, 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("4002.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4002 = 2 \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4002.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: \(31.9561308889\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.16448.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 7x^{2} + 8x + 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.18398\) of defining polynomial
Character \(\chi\) \(=\) 4002.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +0.230234 q^{5} +1.00000 q^{6} +4.59819 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +0.230234 q^{5} +1.00000 q^{6} +4.59819 q^{7} -1.00000 q^{8} +1.00000 q^{9} -0.230234 q^{10} -4.50283 q^{11} -1.00000 q^{12} +1.15403 q^{13} -4.59819 q^{14} -0.230234 q^{15} +1.00000 q^{16} -7.42662 q^{17} -1.00000 q^{18} -5.94699 q^{19} +0.230234 q^{20} -4.59819 q^{21} +4.50283 q^{22} +1.00000 q^{23} +1.00000 q^{24} -4.94699 q^{25} -1.15403 q^{26} -1.00000 q^{27} +4.59819 q^{28} -1.00000 q^{29} +0.230234 q^{30} +6.50283 q^{31} -1.00000 q^{32} +4.50283 q^{33} +7.42662 q^{34} +1.05866 q^{35} +1.00000 q^{36} +7.29175 q^{37} +5.94699 q^{38} -1.15403 q^{39} -0.230234 q^{40} +0.0953661 q^{41} +4.59819 q^{42} +0.941339 q^{43} -4.50283 q^{44} +0.230234 q^{45} -1.00000 q^{46} -0.598193 q^{47} -1.00000 q^{48} +14.1434 q^{49} +4.94699 q^{50} +7.42662 q^{51} +1.15403 q^{52} +13.3736 q^{53} +1.00000 q^{54} -1.03670 q^{55} -4.59819 q^{56} +5.94699 q^{57} +1.00000 q^{58} -2.94134 q^{59} -0.230234 q^{60} +10.8883 q^{61} -6.50283 q^{62} +4.59819 q^{63} +1.00000 q^{64} +0.265697 q^{65} -4.50283 q^{66} +2.09251 q^{67} -7.42662 q^{68} -1.00000 q^{69} -1.05866 q^{70} +10.9633 q^{71} -1.00000 q^{72} -12.3928 q^{73} -7.29175 q^{74} +4.94699 q^{75} -5.94699 q^{76} -20.7049 q^{77} +1.15403 q^{78} +16.5276 q^{79} +0.230234 q^{80} +1.00000 q^{81} -0.0953661 q^{82} -1.71676 q^{83} -4.59819 q^{84} -1.70986 q^{85} -0.941339 q^{86} +1.00000 q^{87} +4.50283 q^{88} +12.5700 q^{89} -0.230234 q^{90} +5.30644 q^{91} +1.00000 q^{92} -6.50283 q^{93} +0.598193 q^{94} -1.36920 q^{95} +1.00000 q^{96} -1.82277 q^{97} -14.1434 q^{98} -4.50283 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 2 q^{5} + 4 q^{6} + 6 q^{7} - 4 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 2 q^{5} + 4 q^{6} + 6 q^{7} - 4 q^{8} + 4 q^{9} - 2 q^{10} - 4 q^{12} - 6 q^{14} - 2 q^{15} + 4 q^{16} - 6 q^{17} - 4 q^{18} + 2 q^{19} + 2 q^{20} - 6 q^{21} + 4 q^{23} + 4 q^{24} + 6 q^{25} - 4 q^{27} + 6 q^{28} - 4 q^{29} + 2 q^{30} + 8 q^{31} - 4 q^{32} + 6 q^{34} - 6 q^{35} + 4 q^{36} + 10 q^{37} - 2 q^{38} - 2 q^{40} + 6 q^{41} + 6 q^{42} + 14 q^{43} + 2 q^{45} - 4 q^{46} + 10 q^{47} - 4 q^{48} + 6 q^{49} - 6 q^{50} + 6 q^{51} + 4 q^{53} + 4 q^{54} - 20 q^{55} - 6 q^{56} - 2 q^{57} + 4 q^{58} - 22 q^{59} - 2 q^{60} + 28 q^{61} - 8 q^{62} + 6 q^{63} + 4 q^{64} + 12 q^{65} + 24 q^{67} - 6 q^{68} - 4 q^{69} + 6 q^{70} + 28 q^{71} - 4 q^{72} - 10 q^{74} - 6 q^{75} + 2 q^{76} - 4 q^{77} + 12 q^{79} + 2 q^{80} + 4 q^{81} - 6 q^{82} + 20 q^{83} - 6 q^{84} - 10 q^{85} - 14 q^{86} + 4 q^{87} - 24 q^{89} - 2 q^{90} + 28 q^{91} + 4 q^{92} - 8 q^{93} - 10 q^{94} + 2 q^{95} + 4 q^{96} - 32 q^{97} - 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 0.230234 0.102964 0.0514819 0.998674i \(-0.483606\pi\)
0.0514819 + 0.998674i \(0.483606\pi\)
\(6\) 1.00000 0.408248
\(7\) 4.59819 1.73795 0.868977 0.494853i \(-0.164778\pi\)
0.868977 + 0.494853i \(0.164778\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −0.230234 −0.0728065
\(11\) −4.50283 −1.35765 −0.678827 0.734299i \(-0.737511\pi\)
−0.678827 + 0.734299i \(0.737511\pi\)
\(12\) −1.00000 −0.288675
\(13\) 1.15403 0.320070 0.160035 0.987111i \(-0.448839\pi\)
0.160035 + 0.987111i \(0.448839\pi\)
\(14\) −4.59819 −1.22892
\(15\) −0.230234 −0.0594462
\(16\) 1.00000 0.250000
\(17\) −7.42662 −1.80122 −0.900610 0.434628i \(-0.856880\pi\)
−0.900610 + 0.434628i \(0.856880\pi\)
\(18\) −1.00000 −0.235702
\(19\) −5.94699 −1.36433 −0.682167 0.731197i \(-0.738962\pi\)
−0.682167 + 0.731197i \(0.738962\pi\)
\(20\) 0.230234 0.0514819
\(21\) −4.59819 −1.00341
\(22\) 4.50283 0.960006
\(23\) 1.00000 0.208514
\(24\) 1.00000 0.204124
\(25\) −4.94699 −0.989398
\(26\) −1.15403 −0.226323
\(27\) −1.00000 −0.192450
\(28\) 4.59819 0.868977
\(29\) −1.00000 −0.185695
\(30\) 0.230234 0.0420348
\(31\) 6.50283 1.16794 0.583971 0.811774i \(-0.301498\pi\)
0.583971 + 0.811774i \(0.301498\pi\)
\(32\) −1.00000 −0.176777
\(33\) 4.50283 0.783842
\(34\) 7.42662 1.27365
\(35\) 1.05866 0.178946
\(36\) 1.00000 0.166667
\(37\) 7.29175 1.19876 0.599378 0.800466i \(-0.295415\pi\)
0.599378 + 0.800466i \(0.295415\pi\)
\(38\) 5.94699 0.964730
\(39\) −1.15403 −0.184792
\(40\) −0.230234 −0.0364032
\(41\) 0.0953661 0.0148937 0.00744684 0.999972i \(-0.497630\pi\)
0.00744684 + 0.999972i \(0.497630\pi\)
\(42\) 4.59819 0.709517
\(43\) 0.941339 0.143553 0.0717764 0.997421i \(-0.477133\pi\)
0.0717764 + 0.997421i \(0.477133\pi\)
\(44\) −4.50283 −0.678827
\(45\) 0.230234 0.0343213
\(46\) −1.00000 −0.147442
\(47\) −0.598193 −0.0872554 −0.0436277 0.999048i \(-0.513892\pi\)
−0.0436277 + 0.999048i \(0.513892\pi\)
\(48\) −1.00000 −0.144338
\(49\) 14.1434 2.02048
\(50\) 4.94699 0.699610
\(51\) 7.42662 1.03993
\(52\) 1.15403 0.160035
\(53\) 13.3736 1.83701 0.918503 0.395413i \(-0.129399\pi\)
0.918503 + 0.395413i \(0.129399\pi\)
\(54\) 1.00000 0.136083
\(55\) −1.03670 −0.139789
\(56\) −4.59819 −0.614459
\(57\) 5.94699 0.787698
\(58\) 1.00000 0.131306
\(59\) −2.94134 −0.382930 −0.191465 0.981499i \(-0.561324\pi\)
−0.191465 + 0.981499i \(0.561324\pi\)
\(60\) −0.230234 −0.0297231
\(61\) 10.8883 1.39411 0.697054 0.717019i \(-0.254494\pi\)
0.697054 + 0.717019i \(0.254494\pi\)
\(62\) −6.50283 −0.825860
\(63\) 4.59819 0.579318
\(64\) 1.00000 0.125000
\(65\) 0.265697 0.0329556
\(66\) −4.50283 −0.554260
\(67\) 2.09251 0.255641 0.127820 0.991797i \(-0.459202\pi\)
0.127820 + 0.991797i \(0.459202\pi\)
\(68\) −7.42662 −0.900610
\(69\) −1.00000 −0.120386
\(70\) −1.05866 −0.126534
\(71\) 10.9633 1.30110 0.650552 0.759462i \(-0.274538\pi\)
0.650552 + 0.759462i \(0.274538\pi\)
\(72\) −1.00000 −0.117851
\(73\) −12.3928 −1.45046 −0.725232 0.688504i \(-0.758268\pi\)
−0.725232 + 0.688504i \(0.758268\pi\)
\(74\) −7.29175 −0.847649
\(75\) 4.94699 0.571229
\(76\) −5.94699 −0.682167
\(77\) −20.7049 −2.35954
\(78\) 1.15403 0.130668
\(79\) 16.5276 1.85950 0.929752 0.368185i \(-0.120021\pi\)
0.929752 + 0.368185i \(0.120021\pi\)
\(80\) 0.230234 0.0257410
\(81\) 1.00000 0.111111
\(82\) −0.0953661 −0.0105314
\(83\) −1.71676 −0.188439 −0.0942193 0.995551i \(-0.530035\pi\)
−0.0942193 + 0.995551i \(0.530035\pi\)
\(84\) −4.59819 −0.501704
\(85\) −1.70986 −0.185461
\(86\) −0.941339 −0.101507
\(87\) 1.00000 0.107211
\(88\) 4.50283 0.480003
\(89\) 12.5700 1.33242 0.666209 0.745765i \(-0.267916\pi\)
0.666209 + 0.745765i \(0.267916\pi\)
\(90\) −0.230234 −0.0242688
\(91\) 5.30644 0.556266
\(92\) 1.00000 0.104257
\(93\) −6.50283 −0.674312
\(94\) 0.598193 0.0616989
\(95\) −1.36920 −0.140477
\(96\) 1.00000 0.102062
\(97\) −1.82277 −0.185075 −0.0925373 0.995709i \(-0.529498\pi\)
−0.0925373 + 0.995709i \(0.529498\pi\)
\(98\) −14.1434 −1.42870
\(99\) −4.50283 −0.452551
\(100\) −4.94699 −0.494699
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) −7.42662 −0.735345
\(103\) −11.9119 −1.17371 −0.586857 0.809690i \(-0.699635\pi\)
−0.586857 + 0.809690i \(0.699635\pi\)
\(104\) −1.15403 −0.113162
\(105\) −1.05866 −0.103315
\(106\) −13.3736 −1.29896
\(107\) 12.0819 1.16800 0.583999 0.811755i \(-0.301487\pi\)
0.583999 + 0.811755i \(0.301487\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 7.71676 0.739131 0.369566 0.929205i \(-0.379506\pi\)
0.369566 + 0.929205i \(0.379506\pi\)
\(110\) 1.03670 0.0988459
\(111\) −7.29175 −0.692102
\(112\) 4.59819 0.434488
\(113\) 8.65810 0.814485 0.407243 0.913320i \(-0.366490\pi\)
0.407243 + 0.913320i \(0.366490\pi\)
\(114\) −5.94699 −0.556987
\(115\) 0.230234 0.0214695
\(116\) −1.00000 −0.0928477
\(117\) 1.15403 0.106690
\(118\) 2.94134 0.270772
\(119\) −34.1490 −3.13044
\(120\) 0.230234 0.0210174
\(121\) 9.27545 0.843223
\(122\) −10.8883 −0.985783
\(123\) −0.0953661 −0.00859887
\(124\) 6.50283 0.583971
\(125\) −2.29014 −0.204836
\(126\) −4.59819 −0.409640
\(127\) 5.11167 0.453587 0.226794 0.973943i \(-0.427176\pi\)
0.226794 + 0.973943i \(0.427176\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −0.941339 −0.0828802
\(130\) −0.265697 −0.0233031
\(131\) 10.3968 0.908374 0.454187 0.890906i \(-0.349930\pi\)
0.454187 + 0.890906i \(0.349930\pi\)
\(132\) 4.50283 0.391921
\(133\) −27.3454 −2.37115
\(134\) −2.09251 −0.180765
\(135\) −0.230234 −0.0198154
\(136\) 7.42662 0.636827
\(137\) 12.7958 1.09322 0.546610 0.837387i \(-0.315918\pi\)
0.546610 + 0.837387i \(0.315918\pi\)
\(138\) 1.00000 0.0851257
\(139\) −2.80361 −0.237799 −0.118900 0.992906i \(-0.537937\pi\)
−0.118900 + 0.992906i \(0.537937\pi\)
\(140\) 1.05866 0.0894732
\(141\) 0.598193 0.0503769
\(142\) −10.9633 −0.920019
\(143\) −5.19639 −0.434544
\(144\) 1.00000 0.0833333
\(145\) −0.230234 −0.0191199
\(146\) 12.3928 1.02563
\(147\) −14.1434 −1.16653
\(148\) 7.29175 0.599378
\(149\) −6.92783 −0.567550 −0.283775 0.958891i \(-0.591587\pi\)
−0.283775 + 0.958891i \(0.591587\pi\)
\(150\) −4.94699 −0.403920
\(151\) 17.5289 1.42648 0.713240 0.700920i \(-0.247227\pi\)
0.713240 + 0.700920i \(0.247227\pi\)
\(152\) 5.94699 0.482365
\(153\) −7.42662 −0.600407
\(154\) 20.7049 1.66845
\(155\) 1.49717 0.120256
\(156\) −1.15403 −0.0923961
\(157\) 6.09537 0.486463 0.243232 0.969968i \(-0.421792\pi\)
0.243232 + 0.969968i \(0.421792\pi\)
\(158\) −16.5276 −1.31487
\(159\) −13.3736 −1.06060
\(160\) −0.230234 −0.0182016
\(161\) 4.59819 0.362388
\(162\) −1.00000 −0.0785674
\(163\) −1.25343 −0.0981765 −0.0490882 0.998794i \(-0.515632\pi\)
−0.0490882 + 0.998794i \(0.515632\pi\)
\(164\) 0.0953661 0.00744684
\(165\) 1.03670 0.0807074
\(166\) 1.71676 0.133246
\(167\) −2.49879 −0.193362 −0.0966810 0.995315i \(-0.530823\pi\)
−0.0966810 + 0.995315i \(0.530823\pi\)
\(168\) 4.59819 0.354758
\(169\) −11.6682 −0.897555
\(170\) 1.70986 0.131140
\(171\) −5.94699 −0.454778
\(172\) 0.941339 0.0717764
\(173\) 6.21269 0.472342 0.236171 0.971712i \(-0.424107\pi\)
0.236171 + 0.971712i \(0.424107\pi\)
\(174\) −1.00000 −0.0758098
\(175\) −22.7472 −1.71953
\(176\) −4.50283 −0.339413
\(177\) 2.94134 0.221085
\(178\) −12.5700 −0.942161
\(179\) 9.55084 0.713863 0.356932 0.934131i \(-0.383823\pi\)
0.356932 + 0.934131i \(0.383823\pi\)
\(180\) 0.230234 0.0171606
\(181\) −14.2196 −1.05693 −0.528467 0.848954i \(-0.677233\pi\)
−0.528467 + 0.848954i \(0.677233\pi\)
\(182\) −5.30644 −0.393340
\(183\) −10.8883 −0.804889
\(184\) −1.00000 −0.0737210
\(185\) 1.67881 0.123429
\(186\) 6.50283 0.476810
\(187\) 33.4408 2.44543
\(188\) −0.598193 −0.0436277
\(189\) −4.59819 −0.334469
\(190\) 1.36920 0.0993323
\(191\) 2.00286 0.144922 0.0724608 0.997371i \(-0.476915\pi\)
0.0724608 + 0.997371i \(0.476915\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 21.6242 1.55655 0.778274 0.627925i \(-0.216096\pi\)
0.778274 + 0.627925i \(0.216096\pi\)
\(194\) 1.82277 0.130868
\(195\) −0.265697 −0.0190269
\(196\) 14.1434 1.01024
\(197\) 2.89898 0.206544 0.103272 0.994653i \(-0.467069\pi\)
0.103272 + 0.994653i \(0.467069\pi\)
\(198\) 4.50283 0.320002
\(199\) 12.2754 0.870184 0.435092 0.900386i \(-0.356716\pi\)
0.435092 + 0.900386i \(0.356716\pi\)
\(200\) 4.94699 0.349805
\(201\) −2.09251 −0.147594
\(202\) 10.0000 0.703598
\(203\) −4.59819 −0.322730
\(204\) 7.42662 0.519967
\(205\) 0.0219565 0.00153351
\(206\) 11.9119 0.829941
\(207\) 1.00000 0.0695048
\(208\) 1.15403 0.0800174
\(209\) 26.7783 1.85229
\(210\) 1.05866 0.0730546
\(211\) −15.8369 −1.09026 −0.545130 0.838352i \(-0.683520\pi\)
−0.545130 + 0.838352i \(0.683520\pi\)
\(212\) 13.3736 0.918503
\(213\) −10.9633 −0.751193
\(214\) −12.0819 −0.825899
\(215\) 0.216728 0.0147808
\(216\) 1.00000 0.0680414
\(217\) 29.9013 2.02983
\(218\) −7.71676 −0.522645
\(219\) 12.3928 0.837426
\(220\) −1.03670 −0.0698946
\(221\) −8.57052 −0.576516
\(222\) 7.29175 0.489390
\(223\) −13.6569 −0.914531 −0.457265 0.889330i \(-0.651171\pi\)
−0.457265 + 0.889330i \(0.651171\pi\)
\(224\) −4.59819 −0.307230
\(225\) −4.94699 −0.329799
\(226\) −8.65810 −0.575928
\(227\) 12.2726 0.814561 0.407280 0.913303i \(-0.366477\pi\)
0.407280 + 0.913303i \(0.366477\pi\)
\(228\) 5.94699 0.393849
\(229\) −7.37647 −0.487451 −0.243725 0.969844i \(-0.578370\pi\)
−0.243725 + 0.969844i \(0.578370\pi\)
\(230\) −0.230234 −0.0151812
\(231\) 20.7049 1.36228
\(232\) 1.00000 0.0656532
\(233\) 28.2020 1.84758 0.923789 0.382903i \(-0.125076\pi\)
0.923789 + 0.382903i \(0.125076\pi\)
\(234\) −1.15403 −0.0754411
\(235\) −0.137724 −0.00898415
\(236\) −2.94134 −0.191465
\(237\) −16.5276 −1.07359
\(238\) 34.1490 2.21355
\(239\) 1.14676 0.0741777 0.0370889 0.999312i \(-0.488192\pi\)
0.0370889 + 0.999312i \(0.488192\pi\)
\(240\) −0.230234 −0.0148616
\(241\) 12.6406 0.814249 0.407125 0.913373i \(-0.366531\pi\)
0.407125 + 0.913373i \(0.366531\pi\)
\(242\) −9.27545 −0.596248
\(243\) −1.00000 −0.0641500
\(244\) 10.8883 0.697054
\(245\) 3.25629 0.208037
\(246\) 0.0953661 0.00608032
\(247\) −6.86299 −0.436682
\(248\) −6.50283 −0.412930
\(249\) 1.71676 0.108795
\(250\) 2.29014 0.144841
\(251\) −5.62425 −0.354999 −0.177500 0.984121i \(-0.556801\pi\)
−0.177500 + 0.984121i \(0.556801\pi\)
\(252\) 4.59819 0.289659
\(253\) −4.50283 −0.283090
\(254\) −5.11167 −0.320735
\(255\) 1.70986 0.107076
\(256\) 1.00000 0.0625000
\(257\) −21.3177 −1.32976 −0.664882 0.746948i \(-0.731518\pi\)
−0.664882 + 0.746948i \(0.731518\pi\)
\(258\) 0.941339 0.0586052
\(259\) 33.5289 2.08338
\(260\) 0.265697 0.0164778
\(261\) −1.00000 −0.0618984
\(262\) −10.3968 −0.642317
\(263\) −1.96205 −0.120985 −0.0604927 0.998169i \(-0.519267\pi\)
−0.0604927 + 0.998169i \(0.519267\pi\)
\(264\) −4.50283 −0.277130
\(265\) 3.07906 0.189145
\(266\) 27.3454 1.67666
\(267\) −12.5700 −0.769271
\(268\) 2.09251 0.127820
\(269\) −3.85163 −0.234838 −0.117419 0.993082i \(-0.537462\pi\)
−0.117419 + 0.993082i \(0.537462\pi\)
\(270\) 0.230234 0.0140116
\(271\) −1.47183 −0.0894076 −0.0447038 0.999000i \(-0.514234\pi\)
−0.0447038 + 0.999000i \(0.514234\pi\)
\(272\) −7.42662 −0.450305
\(273\) −5.30644 −0.321160
\(274\) −12.7958 −0.773024
\(275\) 22.2754 1.34326
\(276\) −1.00000 −0.0601929
\(277\) −24.6665 −1.48207 −0.741035 0.671467i \(-0.765665\pi\)
−0.741035 + 0.671467i \(0.765665\pi\)
\(278\) 2.80361 0.168150
\(279\) 6.50283 0.389314
\(280\) −1.05866 −0.0632671
\(281\) −12.3081 −0.734237 −0.367119 0.930174i \(-0.619656\pi\)
−0.367119 + 0.930174i \(0.619656\pi\)
\(282\) −0.598193 −0.0356219
\(283\) −23.8087 −1.41528 −0.707641 0.706572i \(-0.750241\pi\)
−0.707641 + 0.706572i \(0.750241\pi\)
\(284\) 10.9633 0.650552
\(285\) 1.36920 0.0811045
\(286\) 5.19639 0.307269
\(287\) 0.438512 0.0258845
\(288\) −1.00000 −0.0589256
\(289\) 38.1547 2.24439
\(290\) 0.230234 0.0135198
\(291\) 1.82277 0.106853
\(292\) −12.3928 −0.725232
\(293\) 17.3162 1.01162 0.505811 0.862644i \(-0.331193\pi\)
0.505811 + 0.862644i \(0.331193\pi\)
\(294\) 14.1434 0.824859
\(295\) −0.677197 −0.0394279
\(296\) −7.29175 −0.423824
\(297\) 4.50283 0.261281
\(298\) 6.92783 0.401319
\(299\) 1.15403 0.0667391
\(300\) 4.94699 0.285615
\(301\) 4.32846 0.249488
\(302\) −17.5289 −1.00867
\(303\) 10.0000 0.574485
\(304\) −5.94699 −0.341083
\(305\) 2.50687 0.143543
\(306\) 7.42662 0.424552
\(307\) 0.920937 0.0525606 0.0262803 0.999655i \(-0.491634\pi\)
0.0262803 + 0.999655i \(0.491634\pi\)
\(308\) −20.7049 −1.17977
\(309\) 11.9119 0.677644
\(310\) −1.49717 −0.0850337
\(311\) 28.6829 1.62646 0.813229 0.581943i \(-0.197707\pi\)
0.813229 + 0.581943i \(0.197707\pi\)
\(312\) 1.15403 0.0653339
\(313\) −21.1506 −1.19551 −0.597753 0.801681i \(-0.703939\pi\)
−0.597753 + 0.801681i \(0.703939\pi\)
\(314\) −6.09537 −0.343981
\(315\) 1.05866 0.0596488
\(316\) 16.5276 0.929752
\(317\) −27.2061 −1.52805 −0.764024 0.645187i \(-0.776779\pi\)
−0.764024 + 0.645187i \(0.776779\pi\)
\(318\) 13.3736 0.749955
\(319\) 4.50283 0.252110
\(320\) 0.230234 0.0128705
\(321\) −12.0819 −0.674344
\(322\) −4.59819 −0.256247
\(323\) 44.1661 2.45746
\(324\) 1.00000 0.0555556
\(325\) −5.70896 −0.316676
\(326\) 1.25343 0.0694212
\(327\) −7.71676 −0.426738
\(328\) −0.0953661 −0.00526571
\(329\) −2.75061 −0.151646
\(330\) −1.03670 −0.0570687
\(331\) 21.1833 1.16434 0.582168 0.813068i \(-0.302204\pi\)
0.582168 + 0.813068i \(0.302204\pi\)
\(332\) −1.71676 −0.0942193
\(333\) 7.29175 0.399586
\(334\) 2.49879 0.136728
\(335\) 0.481767 0.0263218
\(336\) −4.59819 −0.250852
\(337\) −21.5317 −1.17291 −0.586454 0.809982i \(-0.699477\pi\)
−0.586454 + 0.809982i \(0.699477\pi\)
\(338\) 11.6682 0.634668
\(339\) −8.65810 −0.470243
\(340\) −1.70986 −0.0927303
\(341\) −29.2811 −1.58566
\(342\) 5.94699 0.321577
\(343\) 32.8466 1.77355
\(344\) −0.941339 −0.0507536
\(345\) −0.230234 −0.0123954
\(346\) −6.21269 −0.333996
\(347\) −13.2845 −0.713148 −0.356574 0.934267i \(-0.616055\pi\)
−0.356574 + 0.934267i \(0.616055\pi\)
\(348\) 1.00000 0.0536056
\(349\) 30.4392 1.62937 0.814686 0.579903i \(-0.196909\pi\)
0.814686 + 0.579903i \(0.196909\pi\)
\(350\) 22.7472 1.21589
\(351\) −1.15403 −0.0615974
\(352\) 4.50283 0.240001
\(353\) 26.0847 1.38835 0.694175 0.719807i \(-0.255770\pi\)
0.694175 + 0.719807i \(0.255770\pi\)
\(354\) −2.94134 −0.156330
\(355\) 2.52413 0.133967
\(356\) 12.5700 0.666209
\(357\) 34.1490 1.80736
\(358\) −9.55084 −0.504778
\(359\) 29.2456 1.54353 0.771763 0.635910i \(-0.219375\pi\)
0.771763 + 0.635910i \(0.219375\pi\)
\(360\) −0.230234 −0.0121344
\(361\) 16.3667 0.861406
\(362\) 14.2196 0.747365
\(363\) −9.27545 −0.486835
\(364\) 5.30644 0.278133
\(365\) −2.85324 −0.149345
\(366\) 10.8883 0.569142
\(367\) 25.0785 1.30909 0.654545 0.756023i \(-0.272860\pi\)
0.654545 + 0.756023i \(0.272860\pi\)
\(368\) 1.00000 0.0521286
\(369\) 0.0953661 0.00496456
\(370\) −1.67881 −0.0872772
\(371\) 61.4944 3.19263
\(372\) −6.50283 −0.337156
\(373\) 36.4257 1.88605 0.943025 0.332723i \(-0.107967\pi\)
0.943025 + 0.332723i \(0.107967\pi\)
\(374\) −33.4408 −1.72918
\(375\) 2.29014 0.118262
\(376\) 0.598193 0.0308494
\(377\) −1.15403 −0.0594354
\(378\) 4.59819 0.236506
\(379\) −22.0847 −1.13442 −0.567208 0.823575i \(-0.691976\pi\)
−0.567208 + 0.823575i \(0.691976\pi\)
\(380\) −1.36920 −0.0702385
\(381\) −5.11167 −0.261879
\(382\) −2.00286 −0.102475
\(383\) 30.2118 1.54375 0.771875 0.635774i \(-0.219319\pi\)
0.771875 + 0.635774i \(0.219319\pi\)
\(384\) 1.00000 0.0510310
\(385\) −4.76697 −0.242947
\(386\) −21.6242 −1.10065
\(387\) 0.941339 0.0478509
\(388\) −1.82277 −0.0925373
\(389\) −34.5622 −1.75237 −0.876187 0.481972i \(-0.839921\pi\)
−0.876187 + 0.481972i \(0.839921\pi\)
\(390\) 0.265697 0.0134541
\(391\) −7.42662 −0.375580
\(392\) −14.1434 −0.714348
\(393\) −10.3968 −0.524450
\(394\) −2.89898 −0.146049
\(395\) 3.80523 0.191462
\(396\) −4.50283 −0.226276
\(397\) −20.8981 −1.04884 −0.524422 0.851458i \(-0.675719\pi\)
−0.524422 + 0.851458i \(0.675719\pi\)
\(398\) −12.2754 −0.615313
\(399\) 27.3454 1.36898
\(400\) −4.94699 −0.247350
\(401\) −6.97056 −0.348093 −0.174047 0.984737i \(-0.555684\pi\)
−0.174047 + 0.984737i \(0.555684\pi\)
\(402\) 2.09251 0.104365
\(403\) 7.50444 0.373823
\(404\) −10.0000 −0.497519
\(405\) 0.230234 0.0114404
\(406\) 4.59819 0.228204
\(407\) −32.8335 −1.62750
\(408\) −7.42662 −0.367672
\(409\) 0.857280 0.0423897 0.0211949 0.999775i \(-0.493253\pi\)
0.0211949 + 0.999775i \(0.493253\pi\)
\(410\) −0.0219565 −0.00108436
\(411\) −12.7958 −0.631171
\(412\) −11.9119 −0.586857
\(413\) −13.5248 −0.665514
\(414\) −1.00000 −0.0491473
\(415\) −0.395256 −0.0194024
\(416\) −1.15403 −0.0565809
\(417\) 2.80361 0.137294
\(418\) −26.7783 −1.30977
\(419\) −3.35166 −0.163739 −0.0818695 0.996643i \(-0.526089\pi\)
−0.0818695 + 0.996643i \(0.526089\pi\)
\(420\) −1.05866 −0.0516574
\(421\) −28.7798 −1.40264 −0.701322 0.712845i \(-0.747406\pi\)
−0.701322 + 0.712845i \(0.747406\pi\)
\(422\) 15.8369 0.770930
\(423\) −0.598193 −0.0290851
\(424\) −13.3736 −0.649480
\(425\) 36.7394 1.78212
\(426\) 10.9633 0.531173
\(427\) 50.0666 2.42289
\(428\) 12.0819 0.583999
\(429\) 5.19639 0.250884
\(430\) −0.216728 −0.0104516
\(431\) 8.84597 0.426096 0.213048 0.977042i \(-0.431661\pi\)
0.213048 + 0.977042i \(0.431661\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 2.74775 0.132048 0.0660242 0.997818i \(-0.478969\pi\)
0.0660242 + 0.997818i \(0.478969\pi\)
\(434\) −29.9013 −1.43531
\(435\) 0.230234 0.0110389
\(436\) 7.71676 0.369566
\(437\) −5.94699 −0.284483
\(438\) −12.3928 −0.592150
\(439\) 16.4385 0.784567 0.392284 0.919844i \(-0.371685\pi\)
0.392284 + 0.919844i \(0.371685\pi\)
\(440\) 1.03670 0.0494230
\(441\) 14.1434 0.673494
\(442\) 8.57052 0.407658
\(443\) −29.1751 −1.38615 −0.693075 0.720865i \(-0.743745\pi\)
−0.693075 + 0.720865i \(0.743745\pi\)
\(444\) −7.29175 −0.346051
\(445\) 2.89404 0.137191
\(446\) 13.6569 0.646671
\(447\) 6.92783 0.327675
\(448\) 4.59819 0.217244
\(449\) −30.8777 −1.45721 −0.728604 0.684935i \(-0.759830\pi\)
−0.728604 + 0.684935i \(0.759830\pi\)
\(450\) 4.94699 0.233203
\(451\) −0.429417 −0.0202205
\(452\) 8.65810 0.407243
\(453\) −17.5289 −0.823579
\(454\) −12.2726 −0.575981
\(455\) 1.22172 0.0572753
\(456\) −5.94699 −0.278493
\(457\) −21.0797 −0.986068 −0.493034 0.870010i \(-0.664112\pi\)
−0.493034 + 0.870010i \(0.664112\pi\)
\(458\) 7.37647 0.344680
\(459\) 7.42662 0.346645
\(460\) 0.230234 0.0107347
\(461\) −14.3504 −0.668365 −0.334183 0.942508i \(-0.608460\pi\)
−0.334183 + 0.942508i \(0.608460\pi\)
\(462\) −20.7049 −0.963278
\(463\) 31.9436 1.48455 0.742273 0.670098i \(-0.233748\pi\)
0.742273 + 0.670098i \(0.233748\pi\)
\(464\) −1.00000 −0.0464238
\(465\) −1.49717 −0.0694297
\(466\) −28.2020 −1.30643
\(467\) −3.07180 −0.142146 −0.0710729 0.997471i \(-0.522642\pi\)
−0.0710729 + 0.997471i \(0.522642\pi\)
\(468\) 1.15403 0.0533449
\(469\) 9.62176 0.444292
\(470\) 0.137724 0.00635276
\(471\) −6.09537 −0.280860
\(472\) 2.94134 0.135386
\(473\) −4.23868 −0.194895
\(474\) 16.5276 0.759140
\(475\) 29.4197 1.34987
\(476\) −34.1490 −1.56522
\(477\) 13.3736 0.612336
\(478\) −1.14676 −0.0524516
\(479\) −0.637695 −0.0291370 −0.0145685 0.999894i \(-0.504637\pi\)
−0.0145685 + 0.999894i \(0.504637\pi\)
\(480\) 0.230234 0.0105087
\(481\) 8.41488 0.383686
\(482\) −12.6406 −0.575761
\(483\) −4.59819 −0.209225
\(484\) 9.27545 0.421611
\(485\) −0.419665 −0.0190560
\(486\) 1.00000 0.0453609
\(487\) 20.6430 0.935425 0.467713 0.883881i \(-0.345078\pi\)
0.467713 + 0.883881i \(0.345078\pi\)
\(488\) −10.8883 −0.492892
\(489\) 1.25343 0.0566822
\(490\) −3.25629 −0.147104
\(491\) 17.5819 0.793460 0.396730 0.917935i \(-0.370145\pi\)
0.396730 + 0.917935i \(0.370145\pi\)
\(492\) −0.0953661 −0.00429944
\(493\) 7.42662 0.334478
\(494\) 6.86299 0.308781
\(495\) −1.03670 −0.0465964
\(496\) 6.50283 0.291986
\(497\) 50.4113 2.26126
\(498\) −1.71676 −0.0769298
\(499\) −9.35607 −0.418835 −0.209418 0.977826i \(-0.567157\pi\)
−0.209418 + 0.977826i \(0.567157\pi\)
\(500\) −2.29014 −0.102418
\(501\) 2.49879 0.111638
\(502\) 5.62425 0.251022
\(503\) −43.1722 −1.92495 −0.962477 0.271362i \(-0.912526\pi\)
−0.962477 + 0.271362i \(0.912526\pi\)
\(504\) −4.59819 −0.204820
\(505\) −2.30234 −0.102453
\(506\) 4.50283 0.200175
\(507\) 11.6682 0.518204
\(508\) 5.11167 0.226794
\(509\) −31.7692 −1.40815 −0.704073 0.710127i \(-0.748637\pi\)
−0.704073 + 0.710127i \(0.748637\pi\)
\(510\) −1.70986 −0.0757140
\(511\) −56.9844 −2.52084
\(512\) −1.00000 −0.0441942
\(513\) 5.94699 0.262566
\(514\) 21.3177 0.940286
\(515\) −2.74253 −0.120850
\(516\) −0.941339 −0.0414401
\(517\) 2.69356 0.118463
\(518\) −33.5289 −1.47317
\(519\) −6.21269 −0.272707
\(520\) −0.265697 −0.0116516
\(521\) −35.1440 −1.53969 −0.769844 0.638232i \(-0.779666\pi\)
−0.769844 + 0.638232i \(0.779666\pi\)
\(522\) 1.00000 0.0437688
\(523\) 30.6026 1.33816 0.669079 0.743191i \(-0.266689\pi\)
0.669079 + 0.743191i \(0.266689\pi\)
\(524\) 10.3968 0.454187
\(525\) 22.7472 0.992770
\(526\) 1.96205 0.0855496
\(527\) −48.2940 −2.10372
\(528\) 4.50283 0.195960
\(529\) 1.00000 0.0434783
\(530\) −3.07906 −0.133746
\(531\) −2.94134 −0.127643
\(532\) −27.3454 −1.18557
\(533\) 0.110055 0.00476702
\(534\) 12.5700 0.543957
\(535\) 2.78166 0.120262
\(536\) −2.09251 −0.0903827
\(537\) −9.55084 −0.412149
\(538\) 3.85163 0.166055
\(539\) −63.6852 −2.74311
\(540\) −0.230234 −0.00990770
\(541\) −5.76197 −0.247727 −0.123863 0.992299i \(-0.539528\pi\)
−0.123863 + 0.992299i \(0.539528\pi\)
\(542\) 1.47183 0.0632207
\(543\) 14.2196 0.610221
\(544\) 7.42662 0.318414
\(545\) 1.77666 0.0761038
\(546\) 5.30644 0.227095
\(547\) −19.5044 −0.833950 −0.416975 0.908918i \(-0.636910\pi\)
−0.416975 + 0.908918i \(0.636910\pi\)
\(548\) 12.7958 0.546610
\(549\) 10.8883 0.464703
\(550\) −22.2754 −0.949828
\(551\) 5.94699 0.253350
\(552\) 1.00000 0.0425628
\(553\) 75.9973 3.23173
\(554\) 24.6665 1.04798
\(555\) −1.67881 −0.0712615
\(556\) −2.80361 −0.118900
\(557\) 32.8193 1.39060 0.695300 0.718720i \(-0.255272\pi\)
0.695300 + 0.718720i \(0.255272\pi\)
\(558\) −6.50283 −0.275287
\(559\) 1.08633 0.0459469
\(560\) 1.05866 0.0447366
\(561\) −33.4408 −1.41187
\(562\) 12.3081 0.519184
\(563\) 25.0440 1.05548 0.527739 0.849407i \(-0.323040\pi\)
0.527739 + 0.849407i \(0.323040\pi\)
\(564\) 0.598193 0.0251885
\(565\) 1.99339 0.0838626
\(566\) 23.8087 1.00076
\(567\) 4.59819 0.193106
\(568\) −10.9633 −0.460010
\(569\) 3.80808 0.159643 0.0798216 0.996809i \(-0.474565\pi\)
0.0798216 + 0.996809i \(0.474565\pi\)
\(570\) −1.36920 −0.0573495
\(571\) −2.36392 −0.0989269 −0.0494635 0.998776i \(-0.515751\pi\)
−0.0494635 + 0.998776i \(0.515751\pi\)
\(572\) −5.19639 −0.217272
\(573\) −2.00286 −0.0836705
\(574\) −0.438512 −0.0183031
\(575\) −4.94699 −0.206304
\(576\) 1.00000 0.0416667
\(577\) 22.8956 0.953156 0.476578 0.879132i \(-0.341877\pi\)
0.476578 + 0.879132i \(0.341877\pi\)
\(578\) −38.1547 −1.58703
\(579\) −21.6242 −0.898673
\(580\) −0.230234 −0.00955996
\(581\) −7.89398 −0.327498
\(582\) −1.82277 −0.0755564
\(583\) −60.2191 −2.49402
\(584\) 12.3928 0.512817
\(585\) 0.265697 0.0109852
\(586\) −17.3162 −0.715325
\(587\) 14.2550 0.588369 0.294184 0.955749i \(-0.404952\pi\)
0.294184 + 0.955749i \(0.404952\pi\)
\(588\) −14.1434 −0.583263
\(589\) −38.6723 −1.59346
\(590\) 0.677197 0.0278798
\(591\) −2.89898 −0.119248
\(592\) 7.29175 0.299689
\(593\) 13.5044 0.554561 0.277280 0.960789i \(-0.410567\pi\)
0.277280 + 0.960789i \(0.410567\pi\)
\(594\) −4.50283 −0.184753
\(595\) −7.86228 −0.322322
\(596\) −6.92783 −0.283775
\(597\) −12.2754 −0.502401
\(598\) −1.15403 −0.0471917
\(599\) −29.7855 −1.21700 −0.608501 0.793553i \(-0.708229\pi\)
−0.608501 + 0.793553i \(0.708229\pi\)
\(600\) −4.94699 −0.201960
\(601\) 35.9819 1.46773 0.733867 0.679293i \(-0.237714\pi\)
0.733867 + 0.679293i \(0.237714\pi\)
\(602\) −4.32846 −0.176415
\(603\) 2.09251 0.0852136
\(604\) 17.5289 0.713240
\(605\) 2.13553 0.0868215
\(606\) −10.0000 −0.406222
\(607\) 30.7871 1.24961 0.624805 0.780781i \(-0.285179\pi\)
0.624805 + 0.780781i \(0.285179\pi\)
\(608\) 5.94699 0.241182
\(609\) 4.59819 0.186328
\(610\) −2.50687 −0.101500
\(611\) −0.690331 −0.0279278
\(612\) −7.42662 −0.300203
\(613\) 18.2155 0.735719 0.367859 0.929881i \(-0.380091\pi\)
0.367859 + 0.929881i \(0.380091\pi\)
\(614\) −0.920937 −0.0371660
\(615\) −0.0219565 −0.000885373 0
\(616\) 20.7049 0.834223
\(617\) −17.4592 −0.702882 −0.351441 0.936210i \(-0.614308\pi\)
−0.351441 + 0.936210i \(0.614308\pi\)
\(618\) −11.9119 −0.479167
\(619\) −14.3781 −0.577904 −0.288952 0.957344i \(-0.593307\pi\)
−0.288952 + 0.957344i \(0.593307\pi\)
\(620\) 1.49717 0.0601279
\(621\) −1.00000 −0.0401286
\(622\) −28.6829 −1.15008
\(623\) 57.7993 2.31568
\(624\) −1.15403 −0.0461981
\(625\) 24.2077 0.968308
\(626\) 21.1506 0.845350
\(627\) −26.7783 −1.06942
\(628\) 6.09537 0.243232
\(629\) −54.1531 −2.15922
\(630\) −1.05866 −0.0421781
\(631\) −9.02357 −0.359223 −0.179611 0.983738i \(-0.557484\pi\)
−0.179611 + 0.983738i \(0.557484\pi\)
\(632\) −16.5276 −0.657434
\(633\) 15.8369 0.629462
\(634\) 27.2061 1.08049
\(635\) 1.17688 0.0467031
\(636\) −13.3736 −0.530298
\(637\) 16.3218 0.646695
\(638\) −4.50283 −0.178269
\(639\) 10.9633 0.433701
\(640\) −0.230234 −0.00910081
\(641\) −16.3826 −0.647076 −0.323538 0.946215i \(-0.604872\pi\)
−0.323538 + 0.946215i \(0.604872\pi\)
\(642\) 12.0819 0.476833
\(643\) −36.6733 −1.44625 −0.723127 0.690715i \(-0.757296\pi\)
−0.723127 + 0.690715i \(0.757296\pi\)
\(644\) 4.59819 0.181194
\(645\) −0.216728 −0.00853367
\(646\) −44.1661 −1.73769
\(647\) 30.5028 1.19919 0.599595 0.800304i \(-0.295328\pi\)
0.599595 + 0.800304i \(0.295328\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 13.2443 0.519886
\(650\) 5.70896 0.223924
\(651\) −29.9013 −1.17192
\(652\) −1.25343 −0.0490882
\(653\) −17.6601 −0.691092 −0.345546 0.938402i \(-0.612306\pi\)
−0.345546 + 0.938402i \(0.612306\pi\)
\(654\) 7.71676 0.301749
\(655\) 2.39370 0.0935297
\(656\) 0.0953661 0.00372342
\(657\) −12.3928 −0.483488
\(658\) 2.75061 0.107230
\(659\) 15.7473 0.613427 0.306714 0.951802i \(-0.400771\pi\)
0.306714 + 0.951802i \(0.400771\pi\)
\(660\) 1.03670 0.0403537
\(661\) −34.4640 −1.34049 −0.670247 0.742138i \(-0.733812\pi\)
−0.670247 + 0.742138i \(0.733812\pi\)
\(662\) −21.1833 −0.823310
\(663\) 8.57052 0.332852
\(664\) 1.71676 0.0666231
\(665\) −6.29585 −0.244143
\(666\) −7.29175 −0.282550
\(667\) −1.00000 −0.0387202
\(668\) −2.49879 −0.0966810
\(669\) 13.6569 0.528004
\(670\) −0.481767 −0.0186123
\(671\) −49.0283 −1.89272
\(672\) 4.59819 0.177379
\(673\) 47.2426 1.82107 0.910535 0.413432i \(-0.135670\pi\)
0.910535 + 0.413432i \(0.135670\pi\)
\(674\) 21.5317 0.829372
\(675\) 4.94699 0.190410
\(676\) −11.6682 −0.448778
\(677\) −5.38141 −0.206824 −0.103412 0.994639i \(-0.532976\pi\)
−0.103412 + 0.994639i \(0.532976\pi\)
\(678\) 8.65810 0.332512
\(679\) −8.38146 −0.321651
\(680\) 1.70986 0.0655702
\(681\) −12.2726 −0.470287
\(682\) 29.2811 1.12123
\(683\) −14.7945 −0.566097 −0.283048 0.959106i \(-0.591346\pi\)
−0.283048 + 0.959106i \(0.591346\pi\)
\(684\) −5.94699 −0.227389
\(685\) 2.94604 0.112562
\(686\) −32.8466 −1.25409
\(687\) 7.37647 0.281430
\(688\) 0.941339 0.0358882
\(689\) 15.4335 0.587970
\(690\) 0.230234 0.00876487
\(691\) −17.0310 −0.647889 −0.323945 0.946076i \(-0.605009\pi\)
−0.323945 + 0.946076i \(0.605009\pi\)
\(692\) 6.21269 0.236171
\(693\) −20.7049 −0.786513
\(694\) 13.2845 0.504272
\(695\) −0.645488 −0.0244848
\(696\) −1.00000 −0.0379049
\(697\) −0.708248 −0.0268268
\(698\) −30.4392 −1.15214
\(699\) −28.2020 −1.06670
\(700\) −22.7472 −0.859764
\(701\) 38.9065 1.46948 0.734740 0.678349i \(-0.237304\pi\)
0.734740 + 0.678349i \(0.237304\pi\)
\(702\) 1.15403 0.0435560
\(703\) −43.3640 −1.63550
\(704\) −4.50283 −0.169707
\(705\) 0.137724 0.00518700
\(706\) −26.0847 −0.981711
\(707\) −45.9819 −1.72933
\(708\) 2.94134 0.110542
\(709\) −13.2849 −0.498923 −0.249462 0.968385i \(-0.580254\pi\)
−0.249462 + 0.968385i \(0.580254\pi\)
\(710\) −2.52413 −0.0947287
\(711\) 16.5276 0.619835
\(712\) −12.5700 −0.471081
\(713\) 6.50283 0.243533
\(714\) −34.1490 −1.27800
\(715\) −1.19639 −0.0447423
\(716\) 9.55084 0.356932
\(717\) −1.14676 −0.0428265
\(718\) −29.2456 −1.09144
\(719\) −3.73430 −0.139266 −0.0696330 0.997573i \(-0.522183\pi\)
−0.0696330 + 0.997573i \(0.522183\pi\)
\(720\) 0.230234 0.00858032
\(721\) −54.7732 −2.03986
\(722\) −16.3667 −0.609106
\(723\) −12.6406 −0.470107
\(724\) −14.2196 −0.528467
\(725\) 4.94699 0.183727
\(726\) 9.27545 0.344244
\(727\) 26.6547 0.988569 0.494284 0.869300i \(-0.335430\pi\)
0.494284 + 0.869300i \(0.335430\pi\)
\(728\) −5.30644 −0.196670
\(729\) 1.00000 0.0370370
\(730\) 2.85324 0.105603
\(731\) −6.99096 −0.258570
\(732\) −10.8883 −0.402444
\(733\) 37.7912 1.39585 0.697925 0.716171i \(-0.254107\pi\)
0.697925 + 0.716171i \(0.254107\pi\)
\(734\) −25.0785 −0.925666
\(735\) −3.25629 −0.120110
\(736\) −1.00000 −0.0368605
\(737\) −9.42221 −0.347072
\(738\) −0.0953661 −0.00351048
\(739\) 14.5477 0.535145 0.267572 0.963538i \(-0.413779\pi\)
0.267572 + 0.963538i \(0.413779\pi\)
\(740\) 1.67881 0.0617143
\(741\) 6.86299 0.252118
\(742\) −61.4944 −2.25753
\(743\) −27.2725 −1.00053 −0.500266 0.865872i \(-0.666764\pi\)
−0.500266 + 0.865872i \(0.666764\pi\)
\(744\) 6.50283 0.238405
\(745\) −1.59502 −0.0584372
\(746\) −36.4257 −1.33364
\(747\) −1.71676 −0.0628129
\(748\) 33.4408 1.22272
\(749\) 55.5547 2.02993
\(750\) −2.29014 −0.0836240
\(751\) 46.6547 1.70245 0.851227 0.524797i \(-0.175859\pi\)
0.851227 + 0.524797i \(0.175859\pi\)
\(752\) −0.598193 −0.0218139
\(753\) 5.62425 0.204959
\(754\) 1.15403 0.0420272
\(755\) 4.03575 0.146876
\(756\) −4.59819 −0.167235
\(757\) −44.5955 −1.62085 −0.810425 0.585842i \(-0.800764\pi\)
−0.810425 + 0.585842i \(0.800764\pi\)
\(758\) 22.0847 0.802153
\(759\) 4.50283 0.163442
\(760\) 1.36920 0.0496661
\(761\) 12.0153 0.435556 0.217778 0.975998i \(-0.430119\pi\)
0.217778 + 0.975998i \(0.430119\pi\)
\(762\) 5.11167 0.185176
\(763\) 35.4831 1.28458
\(764\) 2.00286 0.0724608
\(765\) −1.70986 −0.0618202
\(766\) −30.2118 −1.09160
\(767\) −3.39439 −0.122564
\(768\) −1.00000 −0.0360844
\(769\) −19.5603 −0.705363 −0.352681 0.935743i \(-0.614730\pi\)
−0.352681 + 0.935743i \(0.614730\pi\)
\(770\) 4.76697 0.171790
\(771\) 21.3177 0.767740
\(772\) 21.6242 0.778274
\(773\) −14.7871 −0.531855 −0.265927 0.963993i \(-0.585678\pi\)
−0.265927 + 0.963993i \(0.585678\pi\)
\(774\) −0.941339 −0.0338357
\(775\) −32.1694 −1.15556
\(776\) 1.82277 0.0654338
\(777\) −33.5289 −1.20284
\(778\) 34.5622 1.23912
\(779\) −0.567142 −0.0203200
\(780\) −0.265697 −0.00951346
\(781\) −49.3658 −1.76645
\(782\) 7.42662 0.265575
\(783\) 1.00000 0.0357371
\(784\) 14.1434 0.505121
\(785\) 1.40336 0.0500881
\(786\) 10.3968 0.370842
\(787\) 4.83569 0.172374 0.0861869 0.996279i \(-0.472532\pi\)
0.0861869 + 0.996279i \(0.472532\pi\)
\(788\) 2.89898 0.103272
\(789\) 1.96205 0.0698509
\(790\) −3.80523 −0.135384
\(791\) 39.8116 1.41554
\(792\) 4.50283 0.160001
\(793\) 12.5654 0.446212
\(794\) 20.8981 0.741645
\(795\) −3.07906 −0.109203
\(796\) 12.2754 0.435092
\(797\) −5.42128 −0.192032 −0.0960158 0.995380i \(-0.530610\pi\)
−0.0960158 + 0.995380i \(0.530610\pi\)
\(798\) −27.3454 −0.968017
\(799\) 4.44255 0.157166
\(800\) 4.94699 0.174903
\(801\) 12.5700 0.444139
\(802\) 6.97056 0.246139
\(803\) 55.8025 1.96923
\(804\) −2.09251 −0.0737971
\(805\) 1.05866 0.0373129
\(806\) −7.50444 −0.264333
\(807\) 3.85163 0.135584
\(808\) 10.0000 0.351799
\(809\) 31.5215 1.10824 0.554118 0.832438i \(-0.313056\pi\)
0.554118 + 0.832438i \(0.313056\pi\)
\(810\) −0.230234 −0.00808961
\(811\) 34.4856 1.21095 0.605476 0.795864i \(-0.292983\pi\)
0.605476 + 0.795864i \(0.292983\pi\)
\(812\) −4.59819 −0.161365
\(813\) 1.47183 0.0516195
\(814\) 32.8335 1.15081
\(815\) −0.288583 −0.0101086
\(816\) 7.42662 0.259984
\(817\) −5.59813 −0.195854
\(818\) −0.857280 −0.0299741
\(819\) 5.30644 0.185422
\(820\) 0.0219565 0.000766756 0
\(821\) 18.8206 0.656845 0.328422 0.944531i \(-0.393483\pi\)
0.328422 + 0.944531i \(0.393483\pi\)
\(822\) 12.7958 0.446306
\(823\) 29.0593 1.01294 0.506472 0.862256i \(-0.330949\pi\)
0.506472 + 0.862256i \(0.330949\pi\)
\(824\) 11.9119 0.414971
\(825\) −22.2754 −0.775532
\(826\) 13.5248 0.470589
\(827\) 14.7447 0.512725 0.256362 0.966581i \(-0.417476\pi\)
0.256362 + 0.966581i \(0.417476\pi\)
\(828\) 1.00000 0.0347524
\(829\) −45.6422 −1.58522 −0.792609 0.609730i \(-0.791278\pi\)
−0.792609 + 0.609730i \(0.791278\pi\)
\(830\) 0.395256 0.0137196
\(831\) 24.6665 0.855673
\(832\) 1.15403 0.0400087
\(833\) −105.037 −3.63933
\(834\) −2.80361 −0.0970812
\(835\) −0.575306 −0.0199093
\(836\) 26.7783 0.926146
\(837\) −6.50283 −0.224771
\(838\) 3.35166 0.115781
\(839\) −5.37295 −0.185495 −0.0927475 0.995690i \(-0.529565\pi\)
−0.0927475 + 0.995690i \(0.529565\pi\)
\(840\) 1.05866 0.0365273
\(841\) 1.00000 0.0344828
\(842\) 28.7798 0.991818
\(843\) 12.3081 0.423912
\(844\) −15.8369 −0.545130
\(845\) −2.68642 −0.0924158
\(846\) 0.598193 0.0205663
\(847\) 42.6503 1.46548
\(848\) 13.3736 0.459252
\(849\) 23.8087 0.817114
\(850\) −36.7394 −1.26015
\(851\) 7.29175 0.249958
\(852\) −10.9633 −0.375596
\(853\) 42.6974 1.46193 0.730966 0.682414i \(-0.239070\pi\)
0.730966 + 0.682414i \(0.239070\pi\)
\(854\) −50.0666 −1.71325
\(855\) −1.36920 −0.0468257
\(856\) −12.0819 −0.412949
\(857\) 48.5254 1.65760 0.828799 0.559547i \(-0.189025\pi\)
0.828799 + 0.559547i \(0.189025\pi\)
\(858\) −5.19639 −0.177402
\(859\) 24.3112 0.829487 0.414744 0.909938i \(-0.363871\pi\)
0.414744 + 0.909938i \(0.363871\pi\)
\(860\) 0.216728 0.00739038
\(861\) −0.438512 −0.0149444
\(862\) −8.84597 −0.301295
\(863\) 41.2092 1.40278 0.701390 0.712778i \(-0.252563\pi\)
0.701390 + 0.712778i \(0.252563\pi\)
\(864\) 1.00000 0.0340207
\(865\) 1.43037 0.0486342
\(866\) −2.74775 −0.0933723
\(867\) −38.1547 −1.29580
\(868\) 29.9013 1.01491
\(869\) −74.4211 −2.52456
\(870\) −0.230234 −0.00780567
\(871\) 2.41481 0.0818228
\(872\) −7.71676 −0.261322
\(873\) −1.82277 −0.0616915
\(874\) 5.94699 0.201160
\(875\) −10.5305 −0.355996
\(876\) 12.3928 0.418713
\(877\) −24.3674 −0.822830 −0.411415 0.911448i \(-0.634965\pi\)
−0.411415 + 0.911448i \(0.634965\pi\)
\(878\) −16.4385 −0.554773
\(879\) −17.3162 −0.584061
\(880\) −1.03670 −0.0349473
\(881\) −11.9890 −0.403919 −0.201959 0.979394i \(-0.564731\pi\)
−0.201959 + 0.979394i \(0.564731\pi\)
\(882\) −14.1434 −0.476232
\(883\) −48.3787 −1.62807 −0.814037 0.580813i \(-0.802735\pi\)
−0.814037 + 0.580813i \(0.802735\pi\)
\(884\) −8.57052 −0.288258
\(885\) 0.677197 0.0227637
\(886\) 29.1751 0.980156
\(887\) 41.8351 1.40469 0.702343 0.711839i \(-0.252137\pi\)
0.702343 + 0.711839i \(0.252137\pi\)
\(888\) 7.29175 0.244695
\(889\) 23.5044 0.788314
\(890\) −2.89404 −0.0970086
\(891\) −4.50283 −0.150850
\(892\) −13.6569 −0.457265
\(893\) 3.55745 0.119045
\(894\) −6.92783 −0.231701
\(895\) 2.19893 0.0735021
\(896\) −4.59819 −0.153615
\(897\) −1.15403 −0.0385319
\(898\) 30.8777 1.03040
\(899\) −6.50283 −0.216881
\(900\) −4.94699 −0.164900
\(901\) −99.3207 −3.30885
\(902\) 0.429417 0.0142980
\(903\) −4.32846 −0.144042
\(904\) −8.65810 −0.287964
\(905\) −3.27383 −0.108826
\(906\) 17.5289 0.582358
\(907\) −0.443449 −0.0147245 −0.00736223 0.999973i \(-0.502343\pi\)
−0.00736223 + 0.999973i \(0.502343\pi\)
\(908\) 12.2726 0.407280
\(909\) −10.0000 −0.331679
\(910\) −1.22172 −0.0404998
\(911\) 44.0010 1.45782 0.728910 0.684610i \(-0.240027\pi\)
0.728910 + 0.684610i \(0.240027\pi\)
\(912\) 5.94699 0.196925
\(913\) 7.73026 0.255834
\(914\) 21.0797 0.697255
\(915\) −2.50687 −0.0828744
\(916\) −7.37647 −0.243725
\(917\) 47.8065 1.57871
\(918\) −7.42662 −0.245115
\(919\) −52.5556 −1.73365 −0.866825 0.498613i \(-0.833843\pi\)
−0.866825 + 0.498613i \(0.833843\pi\)
\(920\) −0.230234 −0.00759060
\(921\) −0.920937 −0.0303459
\(922\) 14.3504 0.472606
\(923\) 12.6519 0.416444
\(924\) 20.7049 0.681140
\(925\) −36.0722 −1.18605
\(926\) −31.9436 −1.04973
\(927\) −11.9119 −0.391238
\(928\) 1.00000 0.0328266
\(929\) −51.6298 −1.69392 −0.846960 0.531656i \(-0.821570\pi\)
−0.846960 + 0.531656i \(0.821570\pi\)
\(930\) 1.49717 0.0490942
\(931\) −84.1106 −2.75661
\(932\) 28.2020 0.923789
\(933\) −28.6829 −0.939036
\(934\) 3.07180 0.100512
\(935\) 7.69921 0.251791
\(936\) −1.15403 −0.0377206
\(937\) 12.2804 0.401183 0.200591 0.979675i \(-0.435714\pi\)
0.200591 + 0.979675i \(0.435714\pi\)
\(938\) −9.62176 −0.314162
\(939\) 21.1506 0.690225
\(940\) −0.137724 −0.00449208
\(941\) −36.4257 −1.18744 −0.593721 0.804671i \(-0.702342\pi\)
−0.593721 + 0.804671i \(0.702342\pi\)
\(942\) 6.09537 0.198598
\(943\) 0.0953661 0.00310555
\(944\) −2.94134 −0.0957324
\(945\) −1.05866 −0.0344383
\(946\) 4.23868 0.137812
\(947\) 2.22738 0.0723801 0.0361900 0.999345i \(-0.488478\pi\)
0.0361900 + 0.999345i \(0.488478\pi\)
\(948\) −16.5276 −0.536793
\(949\) −14.3016 −0.464250
\(950\) −29.4197 −0.954502
\(951\) 27.2061 0.882219
\(952\) 34.1490 1.10678
\(953\) 16.1524 0.523228 0.261614 0.965173i \(-0.415745\pi\)
0.261614 + 0.965173i \(0.415745\pi\)
\(954\) −13.3736 −0.432987
\(955\) 0.461126 0.0149217
\(956\) 1.14676 0.0370889
\(957\) −4.50283 −0.145556
\(958\) 0.637695 0.0206030
\(959\) 58.8377 1.89997
\(960\) −0.230234 −0.00743078
\(961\) 11.2868 0.364089
\(962\) −8.41488 −0.271307
\(963\) 12.0819 0.389333
\(964\) 12.6406 0.407125
\(965\) 4.97864 0.160268
\(966\) 4.59819 0.147944
\(967\) −1.46208 −0.0470174 −0.0235087 0.999724i \(-0.507484\pi\)
−0.0235087 + 0.999724i \(0.507484\pi\)
\(968\) −9.27545 −0.298124
\(969\) −44.1661 −1.41882
\(970\) 0.419665 0.0134746
\(971\) 6.24440 0.200392 0.100196 0.994968i \(-0.468053\pi\)
0.100196 + 0.994968i \(0.468053\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −12.8916 −0.413284
\(974\) −20.6430 −0.661446
\(975\) 5.70896 0.182833
\(976\) 10.8883 0.348527
\(977\) −13.8899 −0.444379 −0.222189 0.975004i \(-0.571320\pi\)
−0.222189 + 0.975004i \(0.571320\pi\)
\(978\) −1.25343 −0.0400804
\(979\) −56.6005 −1.80896
\(980\) 3.25629 0.104018
\(981\) 7.71676 0.246377
\(982\) −17.5819 −0.561061
\(983\) 27.5406 0.878407 0.439204 0.898387i \(-0.355261\pi\)
0.439204 + 0.898387i \(0.355261\pi\)
\(984\) 0.0953661 0.00304016
\(985\) 0.667444 0.0212665
\(986\) −7.42662 −0.236512
\(987\) 2.75061 0.0875528
\(988\) −6.86299 −0.218341
\(989\) 0.941339 0.0299328
\(990\) 1.03670 0.0329486
\(991\) −4.94289 −0.157016 −0.0785081 0.996913i \(-0.525016\pi\)
−0.0785081 + 0.996913i \(0.525016\pi\)
\(992\) −6.50283 −0.206465
\(993\) −21.1833 −0.672230
\(994\) −50.4113 −1.59895
\(995\) 2.82623 0.0895975
\(996\) 1.71676 0.0543976
\(997\) 10.2437 0.324423 0.162211 0.986756i \(-0.448137\pi\)
0.162211 + 0.986756i \(0.448137\pi\)
\(998\) 9.35607 0.296161
\(999\) −7.29175 −0.230701
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 4002.2.a.ba.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4002.2.a.ba.1.2 4 1.1 even 1 trivial