Properties

Label 8038.2.a.d.1.12
Level $8038$
Weight $2$
Character 8038.1
Self dual yes
Analytic conductor $64.184$
Analytic rank $0$
Dimension $92$
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] = [8038,2,Mod(1,8038)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8038, 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("8038.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8038 = 2 \cdot 4019 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8038.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: \(64.1837531447\)
Analytic rank: \(0\)
Dimension: \(92\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.12
Character \(\chi\) \(=\) 8038.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.41917 q^{3} +1.00000 q^{4} +2.79360 q^{5} -2.41917 q^{6} +0.842715 q^{7} +1.00000 q^{8} +2.85238 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -2.41917 q^{3} +1.00000 q^{4} +2.79360 q^{5} -2.41917 q^{6} +0.842715 q^{7} +1.00000 q^{8} +2.85238 q^{9} +2.79360 q^{10} -3.67980 q^{11} -2.41917 q^{12} -1.86090 q^{13} +0.842715 q^{14} -6.75818 q^{15} +1.00000 q^{16} +5.11357 q^{17} +2.85238 q^{18} +6.79587 q^{19} +2.79360 q^{20} -2.03867 q^{21} -3.67980 q^{22} +8.42119 q^{23} -2.41917 q^{24} +2.80418 q^{25} -1.86090 q^{26} +0.357110 q^{27} +0.842715 q^{28} +3.74696 q^{29} -6.75818 q^{30} -3.61114 q^{31} +1.00000 q^{32} +8.90207 q^{33} +5.11357 q^{34} +2.35420 q^{35} +2.85238 q^{36} -9.86990 q^{37} +6.79587 q^{38} +4.50183 q^{39} +2.79360 q^{40} +1.78244 q^{41} -2.03867 q^{42} -4.12943 q^{43} -3.67980 q^{44} +7.96841 q^{45} +8.42119 q^{46} +0.876797 q^{47} -2.41917 q^{48} -6.28983 q^{49} +2.80418 q^{50} -12.3706 q^{51} -1.86090 q^{52} -4.42347 q^{53} +0.357110 q^{54} -10.2799 q^{55} +0.842715 q^{56} -16.4404 q^{57} +3.74696 q^{58} +6.65122 q^{59} -6.75818 q^{60} -5.63927 q^{61} -3.61114 q^{62} +2.40375 q^{63} +1.00000 q^{64} -5.19860 q^{65} +8.90207 q^{66} +12.9969 q^{67} +5.11357 q^{68} -20.3723 q^{69} +2.35420 q^{70} +4.51649 q^{71} +2.85238 q^{72} +10.9440 q^{73} -9.86990 q^{74} -6.78378 q^{75} +6.79587 q^{76} -3.10103 q^{77} +4.50183 q^{78} +9.36761 q^{79} +2.79360 q^{80} -9.42106 q^{81} +1.78244 q^{82} -5.58958 q^{83} -2.03867 q^{84} +14.2852 q^{85} -4.12943 q^{86} -9.06452 q^{87} -3.67980 q^{88} -10.2932 q^{89} +7.96841 q^{90} -1.56821 q^{91} +8.42119 q^{92} +8.73595 q^{93} +0.876797 q^{94} +18.9849 q^{95} -2.41917 q^{96} +4.84073 q^{97} -6.28983 q^{98} -10.4962 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 92 q + 92 q^{2} + 31 q^{3} + 92 q^{4} + 28 q^{5} + 31 q^{6} + 29 q^{7} + 92 q^{8} + 113 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 92 q + 92 q^{2} + 31 q^{3} + 92 q^{4} + 28 q^{5} + 31 q^{6} + 29 q^{7} + 92 q^{8} + 113 q^{9} + 28 q^{10} + 37 q^{11} + 31 q^{12} + 20 q^{13} + 29 q^{14} + 30 q^{15} + 92 q^{16} + 52 q^{17} + 113 q^{18} + 61 q^{19} + 28 q^{20} + 5 q^{21} + 37 q^{22} + 71 q^{23} + 31 q^{24} + 118 q^{25} + 20 q^{26} + 112 q^{27} + 29 q^{28} + 30 q^{29} + 30 q^{30} + 89 q^{31} + 92 q^{32} + 52 q^{33} + 52 q^{34} + 58 q^{35} + 113 q^{36} + 15 q^{37} + 61 q^{38} + 43 q^{39} + 28 q^{40} + 75 q^{41} + 5 q^{42} + 46 q^{43} + 37 q^{44} + 63 q^{45} + 71 q^{46} + 92 q^{47} + 31 q^{48} + 131 q^{49} + 118 q^{50} + 45 q^{51} + 20 q^{52} + 72 q^{53} + 112 q^{54} + 86 q^{55} + 29 q^{56} + 44 q^{57} + 30 q^{58} + 95 q^{59} + 30 q^{60} - 4 q^{61} + 89 q^{62} + 67 q^{63} + 92 q^{64} + 55 q^{65} + 52 q^{66} + 40 q^{67} + 52 q^{68} + 25 q^{69} + 58 q^{70} + 84 q^{71} + 113 q^{72} + 87 q^{73} + 15 q^{74} + 132 q^{75} + 61 q^{76} + 96 q^{77} + 43 q^{78} + 68 q^{79} + 28 q^{80} + 156 q^{81} + 75 q^{82} + 120 q^{83} + 5 q^{84} - 14 q^{85} + 46 q^{86} + 73 q^{87} + 37 q^{88} + 86 q^{89} + 63 q^{90} + 93 q^{91} + 71 q^{92} + 29 q^{93} + 92 q^{94} + 67 q^{95} + 31 q^{96} + 65 q^{97} + 131 q^{98} + 94 q^{99}+O(q^{100}) \) 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
\(3\) −2.41917 −1.39671 −0.698354 0.715752i \(-0.746084\pi\)
−0.698354 + 0.715752i \(0.746084\pi\)
\(4\) 1.00000 0.500000
\(5\) 2.79360 1.24933 0.624667 0.780891i \(-0.285235\pi\)
0.624667 + 0.780891i \(0.285235\pi\)
\(6\) −2.41917 −0.987622
\(7\) 0.842715 0.318516 0.159258 0.987237i \(-0.449090\pi\)
0.159258 + 0.987237i \(0.449090\pi\)
\(8\) 1.00000 0.353553
\(9\) 2.85238 0.950794
\(10\) 2.79360 0.883412
\(11\) −3.67980 −1.10950 −0.554751 0.832016i \(-0.687187\pi\)
−0.554751 + 0.832016i \(0.687187\pi\)
\(12\) −2.41917 −0.698354
\(13\) −1.86090 −0.516121 −0.258060 0.966129i \(-0.583083\pi\)
−0.258060 + 0.966129i \(0.583083\pi\)
\(14\) 0.842715 0.225225
\(15\) −6.75818 −1.74496
\(16\) 1.00000 0.250000
\(17\) 5.11357 1.24022 0.620111 0.784514i \(-0.287087\pi\)
0.620111 + 0.784514i \(0.287087\pi\)
\(18\) 2.85238 0.672313
\(19\) 6.79587 1.55908 0.779540 0.626352i \(-0.215453\pi\)
0.779540 + 0.626352i \(0.215453\pi\)
\(20\) 2.79360 0.624667
\(21\) −2.03867 −0.444874
\(22\) −3.67980 −0.784537
\(23\) 8.42119 1.75594 0.877970 0.478716i \(-0.158898\pi\)
0.877970 + 0.478716i \(0.158898\pi\)
\(24\) −2.41917 −0.493811
\(25\) 2.80418 0.560835
\(26\) −1.86090 −0.364952
\(27\) 0.357110 0.0687258
\(28\) 0.842715 0.159258
\(29\) 3.74696 0.695792 0.347896 0.937533i \(-0.386896\pi\)
0.347896 + 0.937533i \(0.386896\pi\)
\(30\) −6.75818 −1.23387
\(31\) −3.61114 −0.648579 −0.324290 0.945958i \(-0.605125\pi\)
−0.324290 + 0.945958i \(0.605125\pi\)
\(32\) 1.00000 0.176777
\(33\) 8.90207 1.54965
\(34\) 5.11357 0.876970
\(35\) 2.35420 0.397933
\(36\) 2.85238 0.475397
\(37\) −9.86990 −1.62260 −0.811301 0.584629i \(-0.801240\pi\)
−0.811301 + 0.584629i \(0.801240\pi\)
\(38\) 6.79587 1.10244
\(39\) 4.50183 0.720870
\(40\) 2.79360 0.441706
\(41\) 1.78244 0.278370 0.139185 0.990266i \(-0.455552\pi\)
0.139185 + 0.990266i \(0.455552\pi\)
\(42\) −2.03867 −0.314574
\(43\) −4.12943 −0.629732 −0.314866 0.949136i \(-0.601960\pi\)
−0.314866 + 0.949136i \(0.601960\pi\)
\(44\) −3.67980 −0.554751
\(45\) 7.96841 1.18786
\(46\) 8.42119 1.24164
\(47\) 0.876797 0.127894 0.0639470 0.997953i \(-0.479631\pi\)
0.0639470 + 0.997953i \(0.479631\pi\)
\(48\) −2.41917 −0.349177
\(49\) −6.28983 −0.898547
\(50\) 2.80418 0.396570
\(51\) −12.3706 −1.73223
\(52\) −1.86090 −0.258060
\(53\) −4.42347 −0.607611 −0.303805 0.952734i \(-0.598257\pi\)
−0.303805 + 0.952734i \(0.598257\pi\)
\(54\) 0.357110 0.0485965
\(55\) −10.2799 −1.38614
\(56\) 0.842715 0.112613
\(57\) −16.4404 −2.17758
\(58\) 3.74696 0.491999
\(59\) 6.65122 0.865915 0.432957 0.901414i \(-0.357470\pi\)
0.432957 + 0.901414i \(0.357470\pi\)
\(60\) −6.75818 −0.872478
\(61\) −5.63927 −0.722034 −0.361017 0.932559i \(-0.617570\pi\)
−0.361017 + 0.932559i \(0.617570\pi\)
\(62\) −3.61114 −0.458615
\(63\) 2.40375 0.302844
\(64\) 1.00000 0.125000
\(65\) −5.19860 −0.644807
\(66\) 8.90207 1.09577
\(67\) 12.9969 1.58782 0.793912 0.608033i \(-0.208041\pi\)
0.793912 + 0.608033i \(0.208041\pi\)
\(68\) 5.11357 0.620111
\(69\) −20.3723 −2.45254
\(70\) 2.35420 0.281381
\(71\) 4.51649 0.536008 0.268004 0.963418i \(-0.413636\pi\)
0.268004 + 0.963418i \(0.413636\pi\)
\(72\) 2.85238 0.336157
\(73\) 10.9440 1.28090 0.640451 0.767999i \(-0.278748\pi\)
0.640451 + 0.767999i \(0.278748\pi\)
\(74\) −9.86990 −1.14735
\(75\) −6.78378 −0.783323
\(76\) 6.79587 0.779540
\(77\) −3.10103 −0.353395
\(78\) 4.50183 0.509732
\(79\) 9.36761 1.05394 0.526969 0.849884i \(-0.323328\pi\)
0.526969 + 0.849884i \(0.323328\pi\)
\(80\) 2.79360 0.312333
\(81\) −9.42106 −1.04678
\(82\) 1.78244 0.196837
\(83\) −5.58958 −0.613536 −0.306768 0.951784i \(-0.599248\pi\)
−0.306768 + 0.951784i \(0.599248\pi\)
\(84\) −2.03867 −0.222437
\(85\) 14.2852 1.54945
\(86\) −4.12943 −0.445288
\(87\) −9.06452 −0.971819
\(88\) −3.67980 −0.392268
\(89\) −10.2932 −1.09108 −0.545539 0.838085i \(-0.683675\pi\)
−0.545539 + 0.838085i \(0.683675\pi\)
\(90\) 7.96841 0.839944
\(91\) −1.56821 −0.164393
\(92\) 8.42119 0.877970
\(93\) 8.73595 0.905876
\(94\) 0.876797 0.0904347
\(95\) 18.9849 1.94781
\(96\) −2.41917 −0.246906
\(97\) 4.84073 0.491501 0.245751 0.969333i \(-0.420966\pi\)
0.245751 + 0.969333i \(0.420966\pi\)
\(98\) −6.28983 −0.635369
\(99\) −10.4962 −1.05491
\(100\) 2.80418 0.280418
\(101\) 6.86476 0.683069 0.341535 0.939869i \(-0.389053\pi\)
0.341535 + 0.939869i \(0.389053\pi\)
\(102\) −12.3706 −1.22487
\(103\) −6.85383 −0.675328 −0.337664 0.941267i \(-0.609637\pi\)
−0.337664 + 0.941267i \(0.609637\pi\)
\(104\) −1.86090 −0.182476
\(105\) −5.69522 −0.555797
\(106\) −4.42347 −0.429646
\(107\) 3.28550 0.317622 0.158811 0.987309i \(-0.449234\pi\)
0.158811 + 0.987309i \(0.449234\pi\)
\(108\) 0.357110 0.0343629
\(109\) 19.2307 1.84197 0.920985 0.389597i \(-0.127386\pi\)
0.920985 + 0.389597i \(0.127386\pi\)
\(110\) −10.2799 −0.980149
\(111\) 23.8770 2.26630
\(112\) 0.842715 0.0796291
\(113\) −9.87276 −0.928751 −0.464375 0.885638i \(-0.653721\pi\)
−0.464375 + 0.885638i \(0.653721\pi\)
\(114\) −16.4404 −1.53978
\(115\) 23.5254 2.19376
\(116\) 3.74696 0.347896
\(117\) −5.30800 −0.490725
\(118\) 6.65122 0.612294
\(119\) 4.30928 0.395031
\(120\) −6.75818 −0.616935
\(121\) 2.54096 0.230996
\(122\) −5.63927 −0.510555
\(123\) −4.31201 −0.388801
\(124\) −3.61114 −0.324290
\(125\) −6.13425 −0.548664
\(126\) 2.40375 0.214143
\(127\) 13.2507 1.17581 0.587903 0.808932i \(-0.299954\pi\)
0.587903 + 0.808932i \(0.299954\pi\)
\(128\) 1.00000 0.0883883
\(129\) 9.98978 0.879551
\(130\) −5.19860 −0.455947
\(131\) −19.2692 −1.68355 −0.841777 0.539826i \(-0.818490\pi\)
−0.841777 + 0.539826i \(0.818490\pi\)
\(132\) 8.90207 0.774826
\(133\) 5.72699 0.496593
\(134\) 12.9969 1.12276
\(135\) 0.997620 0.0858615
\(136\) 5.11357 0.438485
\(137\) 9.03398 0.771825 0.385912 0.922535i \(-0.373887\pi\)
0.385912 + 0.922535i \(0.373887\pi\)
\(138\) −20.3723 −1.73420
\(139\) 20.5078 1.73945 0.869724 0.493538i \(-0.164297\pi\)
0.869724 + 0.493538i \(0.164297\pi\)
\(140\) 2.35420 0.198967
\(141\) −2.12112 −0.178631
\(142\) 4.51649 0.379015
\(143\) 6.84774 0.572637
\(144\) 2.85238 0.237699
\(145\) 10.4675 0.869277
\(146\) 10.9440 0.905734
\(147\) 15.2162 1.25501
\(148\) −9.86990 −0.811301
\(149\) −8.54883 −0.700348 −0.350174 0.936685i \(-0.613878\pi\)
−0.350174 + 0.936685i \(0.613878\pi\)
\(150\) −6.78378 −0.553893
\(151\) 8.98631 0.731296 0.365648 0.930753i \(-0.380847\pi\)
0.365648 + 0.930753i \(0.380847\pi\)
\(152\) 6.79587 0.551218
\(153\) 14.5859 1.17920
\(154\) −3.10103 −0.249888
\(155\) −10.0881 −0.810292
\(156\) 4.50183 0.360435
\(157\) −22.7106 −1.81250 −0.906250 0.422742i \(-0.861068\pi\)
−0.906250 + 0.422742i \(0.861068\pi\)
\(158\) 9.36761 0.745247
\(159\) 10.7011 0.848655
\(160\) 2.79360 0.220853
\(161\) 7.09666 0.559295
\(162\) −9.42106 −0.740188
\(163\) 8.96046 0.701837 0.350919 0.936406i \(-0.385869\pi\)
0.350919 + 0.936406i \(0.385869\pi\)
\(164\) 1.78244 0.139185
\(165\) 24.8688 1.93603
\(166\) −5.58958 −0.433835
\(167\) −16.9775 −1.31376 −0.656879 0.753996i \(-0.728124\pi\)
−0.656879 + 0.753996i \(0.728124\pi\)
\(168\) −2.03867 −0.157287
\(169\) −9.53705 −0.733620
\(170\) 14.2852 1.09563
\(171\) 19.3844 1.48237
\(172\) −4.12943 −0.314866
\(173\) 5.08477 0.386588 0.193294 0.981141i \(-0.438083\pi\)
0.193294 + 0.981141i \(0.438083\pi\)
\(174\) −9.06452 −0.687180
\(175\) 2.36312 0.178635
\(176\) −3.67980 −0.277376
\(177\) −16.0904 −1.20943
\(178\) −10.2932 −0.771509
\(179\) 11.0419 0.825314 0.412657 0.910886i \(-0.364601\pi\)
0.412657 + 0.910886i \(0.364601\pi\)
\(180\) 7.96841 0.593930
\(181\) −1.93916 −0.144136 −0.0720682 0.997400i \(-0.522960\pi\)
−0.0720682 + 0.997400i \(0.522960\pi\)
\(182\) −1.56821 −0.116243
\(183\) 13.6423 1.00847
\(184\) 8.42119 0.620818
\(185\) −27.5725 −2.02717
\(186\) 8.73595 0.640551
\(187\) −18.8169 −1.37603
\(188\) 0.876797 0.0639470
\(189\) 0.300942 0.0218903
\(190\) 18.9849 1.37731
\(191\) −11.4895 −0.831352 −0.415676 0.909513i \(-0.636455\pi\)
−0.415676 + 0.909513i \(0.636455\pi\)
\(192\) −2.41917 −0.174589
\(193\) 10.9398 0.787464 0.393732 0.919225i \(-0.371184\pi\)
0.393732 + 0.919225i \(0.371184\pi\)
\(194\) 4.84073 0.347544
\(195\) 12.5763 0.900607
\(196\) −6.28983 −0.449274
\(197\) −8.25382 −0.588060 −0.294030 0.955796i \(-0.594997\pi\)
−0.294030 + 0.955796i \(0.594997\pi\)
\(198\) −10.4962 −0.745933
\(199\) 18.6100 1.31923 0.659614 0.751604i \(-0.270720\pi\)
0.659614 + 0.751604i \(0.270720\pi\)
\(200\) 2.80418 0.198285
\(201\) −31.4417 −2.21773
\(202\) 6.86476 0.483003
\(203\) 3.15762 0.221621
\(204\) −12.3706 −0.866115
\(205\) 4.97940 0.347777
\(206\) −6.85383 −0.477529
\(207\) 24.0205 1.66954
\(208\) −1.86090 −0.129030
\(209\) −25.0075 −1.72980
\(210\) −5.69522 −0.393008
\(211\) −9.86959 −0.679451 −0.339725 0.940525i \(-0.610334\pi\)
−0.339725 + 0.940525i \(0.610334\pi\)
\(212\) −4.42347 −0.303805
\(213\) −10.9261 −0.748647
\(214\) 3.28550 0.224592
\(215\) −11.5359 −0.786745
\(216\) 0.357110 0.0242982
\(217\) −3.04316 −0.206583
\(218\) 19.2307 1.30247
\(219\) −26.4755 −1.78905
\(220\) −10.2799 −0.693070
\(221\) −9.51583 −0.640104
\(222\) 23.8770 1.60252
\(223\) 6.05175 0.405255 0.202627 0.979256i \(-0.435052\pi\)
0.202627 + 0.979256i \(0.435052\pi\)
\(224\) 0.842715 0.0563063
\(225\) 7.99858 0.533239
\(226\) −9.87276 −0.656726
\(227\) −13.2276 −0.877950 −0.438975 0.898499i \(-0.644658\pi\)
−0.438975 + 0.898499i \(0.644658\pi\)
\(228\) −16.4404 −1.08879
\(229\) 16.1571 1.06769 0.533845 0.845582i \(-0.320747\pi\)
0.533845 + 0.845582i \(0.320747\pi\)
\(230\) 23.5254 1.55122
\(231\) 7.50191 0.493589
\(232\) 3.74696 0.246000
\(233\) 10.0916 0.661125 0.330563 0.943784i \(-0.392762\pi\)
0.330563 + 0.943784i \(0.392762\pi\)
\(234\) −5.30800 −0.346995
\(235\) 2.44942 0.159782
\(236\) 6.65122 0.432957
\(237\) −22.6618 −1.47204
\(238\) 4.30928 0.279329
\(239\) 22.6209 1.46323 0.731613 0.681720i \(-0.238768\pi\)
0.731613 + 0.681720i \(0.238768\pi\)
\(240\) −6.75818 −0.436239
\(241\) 17.0708 1.09962 0.549812 0.835288i \(-0.314699\pi\)
0.549812 + 0.835288i \(0.314699\pi\)
\(242\) 2.54096 0.163339
\(243\) 21.7198 1.39333
\(244\) −5.63927 −0.361017
\(245\) −17.5712 −1.12259
\(246\) −4.31201 −0.274924
\(247\) −12.6464 −0.804674
\(248\) −3.61114 −0.229307
\(249\) 13.5221 0.856931
\(250\) −6.13425 −0.387964
\(251\) −5.94256 −0.375091 −0.187546 0.982256i \(-0.560053\pi\)
−0.187546 + 0.982256i \(0.560053\pi\)
\(252\) 2.40375 0.151422
\(253\) −30.9883 −1.94822
\(254\) 13.2507 0.831420
\(255\) −34.5584 −2.16413
\(256\) 1.00000 0.0625000
\(257\) 0.582514 0.0363362 0.0181681 0.999835i \(-0.494217\pi\)
0.0181681 + 0.999835i \(0.494217\pi\)
\(258\) 9.98978 0.621937
\(259\) −8.31751 −0.516825
\(260\) −5.19860 −0.322403
\(261\) 10.6878 0.661555
\(262\) −19.2692 −1.19045
\(263\) 7.64410 0.471356 0.235678 0.971831i \(-0.424269\pi\)
0.235678 + 0.971831i \(0.424269\pi\)
\(264\) 8.90207 0.547885
\(265\) −12.3574 −0.759108
\(266\) 5.72699 0.351144
\(267\) 24.9010 1.52392
\(268\) 12.9969 0.793912
\(269\) 1.56036 0.0951368 0.0475684 0.998868i \(-0.484853\pi\)
0.0475684 + 0.998868i \(0.484853\pi\)
\(270\) 0.997620 0.0607132
\(271\) 6.91822 0.420252 0.210126 0.977674i \(-0.432613\pi\)
0.210126 + 0.977674i \(0.432613\pi\)
\(272\) 5.11357 0.310056
\(273\) 3.79376 0.229609
\(274\) 9.03398 0.545763
\(275\) −10.3188 −0.622248
\(276\) −20.3723 −1.22627
\(277\) 7.48585 0.449781 0.224891 0.974384i \(-0.427798\pi\)
0.224891 + 0.974384i \(0.427798\pi\)
\(278\) 20.5078 1.22998
\(279\) −10.3003 −0.616666
\(280\) 2.35420 0.140691
\(281\) 26.1553 1.56029 0.780146 0.625598i \(-0.215145\pi\)
0.780146 + 0.625598i \(0.215145\pi\)
\(282\) −2.12112 −0.126311
\(283\) 6.31827 0.375582 0.187791 0.982209i \(-0.439867\pi\)
0.187791 + 0.982209i \(0.439867\pi\)
\(284\) 4.51649 0.268004
\(285\) −45.9278 −2.72053
\(286\) 6.84774 0.404916
\(287\) 1.50208 0.0886653
\(288\) 2.85238 0.168078
\(289\) 9.14858 0.538152
\(290\) 10.4675 0.614671
\(291\) −11.7105 −0.686484
\(292\) 10.9440 0.640451
\(293\) 18.9286 1.10582 0.552910 0.833241i \(-0.313517\pi\)
0.552910 + 0.833241i \(0.313517\pi\)
\(294\) 15.2162 0.887425
\(295\) 18.5808 1.08182
\(296\) −9.86990 −0.573676
\(297\) −1.31409 −0.0762515
\(298\) −8.54883 −0.495221
\(299\) −15.6710 −0.906277
\(300\) −6.78378 −0.391662
\(301\) −3.47993 −0.200580
\(302\) 8.98631 0.517104
\(303\) −16.6070 −0.954048
\(304\) 6.79587 0.389770
\(305\) −15.7538 −0.902061
\(306\) 14.5859 0.833818
\(307\) 27.9385 1.59453 0.797267 0.603627i \(-0.206278\pi\)
0.797267 + 0.603627i \(0.206278\pi\)
\(308\) −3.10103 −0.176697
\(309\) 16.5806 0.943237
\(310\) −10.0881 −0.572963
\(311\) 7.63033 0.432676 0.216338 0.976319i \(-0.430589\pi\)
0.216338 + 0.976319i \(0.430589\pi\)
\(312\) 4.50183 0.254866
\(313\) 16.7916 0.949115 0.474558 0.880224i \(-0.342608\pi\)
0.474558 + 0.880224i \(0.342608\pi\)
\(314\) −22.7106 −1.28163
\(315\) 6.71509 0.378353
\(316\) 9.36761 0.526969
\(317\) −21.8726 −1.22849 −0.614244 0.789116i \(-0.710539\pi\)
−0.614244 + 0.789116i \(0.710539\pi\)
\(318\) 10.7011 0.600090
\(319\) −13.7881 −0.771983
\(320\) 2.79360 0.156167
\(321\) −7.94819 −0.443625
\(322\) 7.09666 0.395482
\(323\) 34.7512 1.93361
\(324\) −9.42106 −0.523392
\(325\) −5.21829 −0.289459
\(326\) 8.96046 0.496274
\(327\) −46.5224 −2.57270
\(328\) 1.78244 0.0984185
\(329\) 0.738890 0.0407363
\(330\) 24.8688 1.36898
\(331\) 27.1766 1.49376 0.746881 0.664958i \(-0.231550\pi\)
0.746881 + 0.664958i \(0.231550\pi\)
\(332\) −5.58958 −0.306768
\(333\) −28.1527 −1.54276
\(334\) −16.9775 −0.928967
\(335\) 36.3081 1.98372
\(336\) −2.03867 −0.111219
\(337\) −20.9394 −1.14064 −0.570320 0.821423i \(-0.693181\pi\)
−0.570320 + 0.821423i \(0.693181\pi\)
\(338\) −9.53705 −0.518747
\(339\) 23.8839 1.29719
\(340\) 14.2852 0.774726
\(341\) 13.2883 0.719601
\(342\) 19.3844 1.04819
\(343\) −11.1995 −0.604718
\(344\) −4.12943 −0.222644
\(345\) −56.9119 −3.06404
\(346\) 5.08477 0.273359
\(347\) −2.21501 −0.118908 −0.0594539 0.998231i \(-0.518936\pi\)
−0.0594539 + 0.998231i \(0.518936\pi\)
\(348\) −9.06452 −0.485909
\(349\) −26.1240 −1.39839 −0.699193 0.714933i \(-0.746457\pi\)
−0.699193 + 0.714933i \(0.746457\pi\)
\(350\) 2.36312 0.126314
\(351\) −0.664545 −0.0354708
\(352\) −3.67980 −0.196134
\(353\) 17.6451 0.939155 0.469578 0.882891i \(-0.344406\pi\)
0.469578 + 0.882891i \(0.344406\pi\)
\(354\) −16.0904 −0.855196
\(355\) 12.6172 0.669653
\(356\) −10.2932 −0.545539
\(357\) −10.4249 −0.551743
\(358\) 11.0419 0.583585
\(359\) 12.1707 0.642343 0.321172 0.947021i \(-0.395923\pi\)
0.321172 + 0.947021i \(0.395923\pi\)
\(360\) 7.96841 0.419972
\(361\) 27.1839 1.43073
\(362\) −1.93916 −0.101920
\(363\) −6.14701 −0.322635
\(364\) −1.56821 −0.0821964
\(365\) 30.5732 1.60027
\(366\) 13.6423 0.713097
\(367\) −16.8722 −0.880723 −0.440361 0.897821i \(-0.645150\pi\)
−0.440361 + 0.897821i \(0.645150\pi\)
\(368\) 8.42119 0.438985
\(369\) 5.08419 0.264672
\(370\) −27.5725 −1.43343
\(371\) −3.72773 −0.193534
\(372\) 8.73595 0.452938
\(373\) −15.6919 −0.812493 −0.406247 0.913763i \(-0.633163\pi\)
−0.406247 + 0.913763i \(0.633163\pi\)
\(374\) −18.8169 −0.973000
\(375\) 14.8398 0.766323
\(376\) 0.876797 0.0452173
\(377\) −6.97271 −0.359113
\(378\) 0.300942 0.0154788
\(379\) 10.6109 0.545044 0.272522 0.962149i \(-0.412142\pi\)
0.272522 + 0.962149i \(0.412142\pi\)
\(380\) 18.9849 0.973906
\(381\) −32.0556 −1.64226
\(382\) −11.4895 −0.587855
\(383\) 2.07089 0.105817 0.0529087 0.998599i \(-0.483151\pi\)
0.0529087 + 0.998599i \(0.483151\pi\)
\(384\) −2.41917 −0.123453
\(385\) −8.66301 −0.441508
\(386\) 10.9398 0.556821
\(387\) −11.7787 −0.598745
\(388\) 4.84073 0.245751
\(389\) −10.6659 −0.540782 −0.270391 0.962751i \(-0.587153\pi\)
−0.270391 + 0.962751i \(0.587153\pi\)
\(390\) 12.5763 0.636825
\(391\) 43.0623 2.17776
\(392\) −6.28983 −0.317684
\(393\) 46.6154 2.35143
\(394\) −8.25382 −0.415821
\(395\) 26.1693 1.31672
\(396\) −10.4962 −0.527455
\(397\) −28.4239 −1.42656 −0.713278 0.700881i \(-0.752790\pi\)
−0.713278 + 0.700881i \(0.752790\pi\)
\(398\) 18.6100 0.932835
\(399\) −13.8546 −0.693595
\(400\) 2.80418 0.140209
\(401\) 27.7211 1.38432 0.692162 0.721742i \(-0.256658\pi\)
0.692162 + 0.721742i \(0.256658\pi\)
\(402\) −31.4417 −1.56817
\(403\) 6.71996 0.334745
\(404\) 6.86476 0.341535
\(405\) −26.3186 −1.30778
\(406\) 3.15762 0.156710
\(407\) 36.3193 1.80028
\(408\) −12.3706 −0.612436
\(409\) −6.29387 −0.311212 −0.155606 0.987819i \(-0.549733\pi\)
−0.155606 + 0.987819i \(0.549733\pi\)
\(410\) 4.97940 0.245915
\(411\) −21.8547 −1.07801
\(412\) −6.85383 −0.337664
\(413\) 5.60508 0.275808
\(414\) 24.0205 1.18054
\(415\) −15.6150 −0.766511
\(416\) −1.86090 −0.0912381
\(417\) −49.6118 −2.42950
\(418\) −25.0075 −1.22316
\(419\) −4.13463 −0.201990 −0.100995 0.994887i \(-0.532203\pi\)
−0.100995 + 0.994887i \(0.532203\pi\)
\(420\) −5.69522 −0.277898
\(421\) −2.38583 −0.116278 −0.0581391 0.998308i \(-0.518517\pi\)
−0.0581391 + 0.998308i \(0.518517\pi\)
\(422\) −9.86959 −0.480444
\(423\) 2.50096 0.121601
\(424\) −4.42347 −0.214823
\(425\) 14.3393 0.695560
\(426\) −10.9261 −0.529373
\(427\) −4.75229 −0.229980
\(428\) 3.28550 0.158811
\(429\) −16.5659 −0.799807
\(430\) −11.5359 −0.556313
\(431\) −23.7381 −1.14343 −0.571713 0.820454i \(-0.693721\pi\)
−0.571713 + 0.820454i \(0.693721\pi\)
\(432\) 0.357110 0.0171815
\(433\) −14.5976 −0.701515 −0.350758 0.936466i \(-0.614076\pi\)
−0.350758 + 0.936466i \(0.614076\pi\)
\(434\) −3.04316 −0.146076
\(435\) −25.3226 −1.21413
\(436\) 19.2307 0.920985
\(437\) 57.2294 2.73765
\(438\) −26.4755 −1.26505
\(439\) −18.0585 −0.861886 −0.430943 0.902379i \(-0.641819\pi\)
−0.430943 + 0.902379i \(0.641819\pi\)
\(440\) −10.2799 −0.490074
\(441\) −17.9410 −0.854334
\(442\) −9.51583 −0.452622
\(443\) 22.3392 1.06137 0.530684 0.847570i \(-0.321935\pi\)
0.530684 + 0.847570i \(0.321935\pi\)
\(444\) 23.8770 1.13315
\(445\) −28.7551 −1.36312
\(446\) 6.05175 0.286558
\(447\) 20.6811 0.978182
\(448\) 0.842715 0.0398145
\(449\) −23.4665 −1.10745 −0.553727 0.832698i \(-0.686795\pi\)
−0.553727 + 0.832698i \(0.686795\pi\)
\(450\) 7.99858 0.377057
\(451\) −6.55901 −0.308852
\(452\) −9.87276 −0.464375
\(453\) −21.7394 −1.02141
\(454\) −13.2276 −0.620804
\(455\) −4.38094 −0.205382
\(456\) −16.4404 −0.769891
\(457\) 38.5252 1.80213 0.901066 0.433681i \(-0.142786\pi\)
0.901066 + 0.433681i \(0.142786\pi\)
\(458\) 16.1571 0.754971
\(459\) 1.82611 0.0852353
\(460\) 23.5254 1.09688
\(461\) 16.4972 0.768352 0.384176 0.923260i \(-0.374486\pi\)
0.384176 + 0.923260i \(0.374486\pi\)
\(462\) 7.50191 0.349020
\(463\) 7.94502 0.369236 0.184618 0.982810i \(-0.440895\pi\)
0.184618 + 0.982810i \(0.440895\pi\)
\(464\) 3.74696 0.173948
\(465\) 24.4047 1.13174
\(466\) 10.0916 0.467486
\(467\) −13.0541 −0.604070 −0.302035 0.953297i \(-0.597666\pi\)
−0.302035 + 0.953297i \(0.597666\pi\)
\(468\) −5.30800 −0.245362
\(469\) 10.9527 0.505748
\(470\) 2.44942 0.112983
\(471\) 54.9407 2.53153
\(472\) 6.65122 0.306147
\(473\) 15.1955 0.698689
\(474\) −22.6618 −1.04089
\(475\) 19.0568 0.874387
\(476\) 4.30928 0.197516
\(477\) −12.6174 −0.577713
\(478\) 22.6209 1.03466
\(479\) 33.5455 1.53273 0.766367 0.642403i \(-0.222063\pi\)
0.766367 + 0.642403i \(0.222063\pi\)
\(480\) −6.75818 −0.308467
\(481\) 18.3669 0.837458
\(482\) 17.0708 0.777552
\(483\) −17.1680 −0.781173
\(484\) 2.54096 0.115498
\(485\) 13.5230 0.614049
\(486\) 21.7198 0.985231
\(487\) −31.8262 −1.44218 −0.721091 0.692840i \(-0.756359\pi\)
−0.721091 + 0.692840i \(0.756359\pi\)
\(488\) −5.63927 −0.255278
\(489\) −21.6769 −0.980262
\(490\) −17.5712 −0.793788
\(491\) −7.28748 −0.328879 −0.164440 0.986387i \(-0.552582\pi\)
−0.164440 + 0.986387i \(0.552582\pi\)
\(492\) −4.31201 −0.194401
\(493\) 19.1603 0.862937
\(494\) −12.6464 −0.568990
\(495\) −29.3222 −1.31793
\(496\) −3.61114 −0.162145
\(497\) 3.80611 0.170727
\(498\) 13.5221 0.605941
\(499\) −21.8055 −0.976149 −0.488074 0.872802i \(-0.662300\pi\)
−0.488074 + 0.872802i \(0.662300\pi\)
\(500\) −6.13425 −0.274332
\(501\) 41.0714 1.83494
\(502\) −5.94256 −0.265229
\(503\) −24.7440 −1.10328 −0.551641 0.834081i \(-0.685998\pi\)
−0.551641 + 0.834081i \(0.685998\pi\)
\(504\) 2.40375 0.107071
\(505\) 19.1774 0.853381
\(506\) −30.9883 −1.37760
\(507\) 23.0718 1.02465
\(508\) 13.2507 0.587903
\(509\) 2.17944 0.0966020 0.0483010 0.998833i \(-0.484619\pi\)
0.0483010 + 0.998833i \(0.484619\pi\)
\(510\) −34.5584 −1.53027
\(511\) 9.22270 0.407988
\(512\) 1.00000 0.0441942
\(513\) 2.42687 0.107149
\(514\) 0.582514 0.0256936
\(515\) −19.1468 −0.843710
\(516\) 9.98978 0.439776
\(517\) −3.22644 −0.141899
\(518\) −8.31751 −0.365450
\(519\) −12.3009 −0.539950
\(520\) −5.19860 −0.227974
\(521\) −28.4321 −1.24563 −0.622816 0.782369i \(-0.714011\pi\)
−0.622816 + 0.782369i \(0.714011\pi\)
\(522\) 10.6878 0.467790
\(523\) 18.5870 0.812754 0.406377 0.913706i \(-0.366792\pi\)
0.406377 + 0.913706i \(0.366792\pi\)
\(524\) −19.2692 −0.841777
\(525\) −5.71679 −0.249501
\(526\) 7.64410 0.333299
\(527\) −18.4658 −0.804383
\(528\) 8.90207 0.387413
\(529\) 47.9165 2.08332
\(530\) −12.3574 −0.536771
\(531\) 18.9718 0.823307
\(532\) 5.72699 0.248296
\(533\) −3.31693 −0.143672
\(534\) 24.9010 1.07757
\(535\) 9.17836 0.396815
\(536\) 12.9969 0.561380
\(537\) −26.7123 −1.15272
\(538\) 1.56036 0.0672719
\(539\) 23.1453 0.996941
\(540\) 0.997620 0.0429307
\(541\) −13.6590 −0.587247 −0.293623 0.955921i \(-0.594861\pi\)
−0.293623 + 0.955921i \(0.594861\pi\)
\(542\) 6.91822 0.297163
\(543\) 4.69115 0.201316
\(544\) 5.11357 0.219242
\(545\) 53.7229 2.30124
\(546\) 3.79376 0.162358
\(547\) −10.5516 −0.451152 −0.225576 0.974226i \(-0.572426\pi\)
−0.225576 + 0.974226i \(0.572426\pi\)
\(548\) 9.03398 0.385912
\(549\) −16.0853 −0.686506
\(550\) −10.3188 −0.439996
\(551\) 25.4638 1.08480
\(552\) −20.3723 −0.867102
\(553\) 7.89423 0.335697
\(554\) 7.48585 0.318043
\(555\) 66.7026 2.83137
\(556\) 20.5078 0.869724
\(557\) 37.2798 1.57959 0.789797 0.613368i \(-0.210186\pi\)
0.789797 + 0.613368i \(0.210186\pi\)
\(558\) −10.3003 −0.436048
\(559\) 7.68445 0.325017
\(560\) 2.35420 0.0994833
\(561\) 45.5214 1.92191
\(562\) 26.1553 1.10329
\(563\) 8.91337 0.375654 0.187827 0.982202i \(-0.439856\pi\)
0.187827 + 0.982202i \(0.439856\pi\)
\(564\) −2.12112 −0.0893153
\(565\) −27.5805 −1.16032
\(566\) 6.31827 0.265577
\(567\) −7.93927 −0.333418
\(568\) 4.51649 0.189508
\(569\) 34.9216 1.46399 0.731995 0.681310i \(-0.238589\pi\)
0.731995 + 0.681310i \(0.238589\pi\)
\(570\) −45.9278 −1.92370
\(571\) −5.47098 −0.228954 −0.114477 0.993426i \(-0.536519\pi\)
−0.114477 + 0.993426i \(0.536519\pi\)
\(572\) 6.84774 0.286319
\(573\) 27.7951 1.16116
\(574\) 1.50208 0.0626958
\(575\) 23.6145 0.984793
\(576\) 2.85238 0.118849
\(577\) 14.5236 0.604627 0.302313 0.953209i \(-0.402241\pi\)
0.302313 + 0.953209i \(0.402241\pi\)
\(578\) 9.14858 0.380531
\(579\) −26.4652 −1.09986
\(580\) 10.4675 0.434638
\(581\) −4.71042 −0.195421
\(582\) −11.7105 −0.485418
\(583\) 16.2775 0.674146
\(584\) 10.9440 0.452867
\(585\) −14.8284 −0.613079
\(586\) 18.9286 0.781933
\(587\) 14.3448 0.592072 0.296036 0.955177i \(-0.404335\pi\)
0.296036 + 0.955177i \(0.404335\pi\)
\(588\) 15.2162 0.627504
\(589\) −24.5408 −1.01119
\(590\) 18.5808 0.764960
\(591\) 19.9674 0.821349
\(592\) −9.86990 −0.405650
\(593\) 13.9969 0.574782 0.287391 0.957813i \(-0.407212\pi\)
0.287391 + 0.957813i \(0.407212\pi\)
\(594\) −1.31409 −0.0539179
\(595\) 12.0384 0.493526
\(596\) −8.54883 −0.350174
\(597\) −45.0208 −1.84258
\(598\) −15.6710 −0.640834
\(599\) −13.4346 −0.548921 −0.274461 0.961598i \(-0.588499\pi\)
−0.274461 + 0.961598i \(0.588499\pi\)
\(600\) −6.78378 −0.276947
\(601\) 33.5650 1.36914 0.684572 0.728945i \(-0.259989\pi\)
0.684572 + 0.728945i \(0.259989\pi\)
\(602\) −3.47993 −0.141831
\(603\) 37.0721 1.50969
\(604\) 8.98631 0.365648
\(605\) 7.09841 0.288592
\(606\) −16.6070 −0.674614
\(607\) 23.1932 0.941382 0.470691 0.882298i \(-0.344005\pi\)
0.470691 + 0.882298i \(0.344005\pi\)
\(608\) 6.79587 0.275609
\(609\) −7.63881 −0.309540
\(610\) −15.7538 −0.637854
\(611\) −1.63163 −0.0660087
\(612\) 14.5859 0.589598
\(613\) 44.2697 1.78803 0.894017 0.448033i \(-0.147875\pi\)
0.894017 + 0.448033i \(0.147875\pi\)
\(614\) 27.9385 1.12751
\(615\) −12.0460 −0.485743
\(616\) −3.10103 −0.124944
\(617\) −12.2658 −0.493803 −0.246902 0.969041i \(-0.579412\pi\)
−0.246902 + 0.969041i \(0.579412\pi\)
\(618\) 16.5806 0.666969
\(619\) −4.52193 −0.181752 −0.0908758 0.995862i \(-0.528967\pi\)
−0.0908758 + 0.995862i \(0.528967\pi\)
\(620\) −10.0881 −0.405146
\(621\) 3.00729 0.120678
\(622\) 7.63033 0.305948
\(623\) −8.67424 −0.347526
\(624\) 4.50183 0.180217
\(625\) −31.1575 −1.24630
\(626\) 16.7916 0.671126
\(627\) 60.4974 2.41603
\(628\) −22.7106 −0.906250
\(629\) −50.4704 −2.01239
\(630\) 6.71509 0.267536
\(631\) 28.8137 1.14706 0.573528 0.819186i \(-0.305574\pi\)
0.573528 + 0.819186i \(0.305574\pi\)
\(632\) 9.36761 0.372624
\(633\) 23.8762 0.948995
\(634\) −21.8726 −0.868672
\(635\) 37.0170 1.46897
\(636\) 10.7011 0.424327
\(637\) 11.7047 0.463759
\(638\) −13.7881 −0.545875
\(639\) 12.8827 0.509634
\(640\) 2.79360 0.110427
\(641\) −44.3055 −1.74996 −0.874982 0.484156i \(-0.839127\pi\)
−0.874982 + 0.484156i \(0.839127\pi\)
\(642\) −7.94819 −0.313690
\(643\) 3.92968 0.154972 0.0774858 0.996993i \(-0.475311\pi\)
0.0774858 + 0.996993i \(0.475311\pi\)
\(644\) 7.09666 0.279648
\(645\) 27.9074 1.09885
\(646\) 34.7512 1.36727
\(647\) −3.40195 −0.133745 −0.0668723 0.997762i \(-0.521302\pi\)
−0.0668723 + 0.997762i \(0.521302\pi\)
\(648\) −9.42106 −0.370094
\(649\) −24.4752 −0.960735
\(650\) −5.21829 −0.204678
\(651\) 7.36192 0.288536
\(652\) 8.96046 0.350919
\(653\) 28.2526 1.10561 0.552805 0.833311i \(-0.313557\pi\)
0.552805 + 0.833311i \(0.313557\pi\)
\(654\) −46.5224 −1.81917
\(655\) −53.8302 −2.10332
\(656\) 1.78244 0.0695924
\(657\) 31.2166 1.21787
\(658\) 0.738890 0.0288049
\(659\) −14.2526 −0.555204 −0.277602 0.960696i \(-0.589540\pi\)
−0.277602 + 0.960696i \(0.589540\pi\)
\(660\) 24.8688 0.968016
\(661\) −23.9811 −0.932757 −0.466378 0.884585i \(-0.654441\pi\)
−0.466378 + 0.884585i \(0.654441\pi\)
\(662\) 27.1766 1.05625
\(663\) 23.0204 0.894039
\(664\) −5.58958 −0.216918
\(665\) 15.9989 0.620410
\(666\) −28.1527 −1.09090
\(667\) 31.5538 1.22177
\(668\) −16.9775 −0.656879
\(669\) −14.6402 −0.566023
\(670\) 36.3081 1.40270
\(671\) 20.7514 0.801099
\(672\) −2.03867 −0.0786434
\(673\) 23.6826 0.912896 0.456448 0.889750i \(-0.349121\pi\)
0.456448 + 0.889750i \(0.349121\pi\)
\(674\) −20.9394 −0.806554
\(675\) 1.00140 0.0385439
\(676\) −9.53705 −0.366810
\(677\) −14.9125 −0.573134 −0.286567 0.958060i \(-0.592514\pi\)
−0.286567 + 0.958060i \(0.592514\pi\)
\(678\) 23.8839 0.917255
\(679\) 4.07935 0.156551
\(680\) 14.2852 0.547814
\(681\) 31.9999 1.22624
\(682\) 13.2883 0.508834
\(683\) 1.63412 0.0625277 0.0312639 0.999511i \(-0.490047\pi\)
0.0312639 + 0.999511i \(0.490047\pi\)
\(684\) 19.3844 0.741183
\(685\) 25.2373 0.964267
\(686\) −11.1995 −0.427600
\(687\) −39.0867 −1.49125
\(688\) −4.12943 −0.157433
\(689\) 8.23163 0.313600
\(690\) −56.9119 −2.16660
\(691\) −29.3989 −1.11839 −0.559193 0.829038i \(-0.688889\pi\)
−0.559193 + 0.829038i \(0.688889\pi\)
\(692\) 5.08477 0.193294
\(693\) −8.84532 −0.336006
\(694\) −2.21501 −0.0840806
\(695\) 57.2905 2.17315
\(696\) −9.06452 −0.343590
\(697\) 9.11461 0.345240
\(698\) −26.1240 −0.988808
\(699\) −24.4134 −0.923399
\(700\) 2.36312 0.0893176
\(701\) −46.1423 −1.74277 −0.871384 0.490601i \(-0.836777\pi\)
−0.871384 + 0.490601i \(0.836777\pi\)
\(702\) −0.664545 −0.0250816
\(703\) −67.0746 −2.52977
\(704\) −3.67980 −0.138688
\(705\) −5.92555 −0.223169
\(706\) 17.6451 0.664083
\(707\) 5.78504 0.217569
\(708\) −16.0904 −0.604715
\(709\) −41.2886 −1.55063 −0.775313 0.631577i \(-0.782408\pi\)
−0.775313 + 0.631577i \(0.782408\pi\)
\(710\) 12.6172 0.473516
\(711\) 26.7200 1.00208
\(712\) −10.2932 −0.385754
\(713\) −30.4101 −1.13887
\(714\) −10.4249 −0.390141
\(715\) 19.1298 0.715415
\(716\) 11.0419 0.412657
\(717\) −54.7239 −2.04370
\(718\) 12.1707 0.454205
\(719\) 27.0402 1.00843 0.504214 0.863579i \(-0.331782\pi\)
0.504214 + 0.863579i \(0.331782\pi\)
\(720\) 7.96841 0.296965
\(721\) −5.77583 −0.215103
\(722\) 27.1839 1.01168
\(723\) −41.2971 −1.53586
\(724\) −1.93916 −0.0720682
\(725\) 10.5071 0.390225
\(726\) −6.14701 −0.228137
\(727\) −1.80457 −0.0669277 −0.0334639 0.999440i \(-0.510654\pi\)
−0.0334639 + 0.999440i \(0.510654\pi\)
\(728\) −1.56821 −0.0581216
\(729\) −24.2807 −0.899287
\(730\) 30.5732 1.13156
\(731\) −21.1161 −0.781007
\(732\) 13.6423 0.504235
\(733\) −16.6822 −0.616170 −0.308085 0.951359i \(-0.599688\pi\)
−0.308085 + 0.951359i \(0.599688\pi\)
\(734\) −16.8722 −0.622765
\(735\) 42.5078 1.56792
\(736\) 8.42119 0.310409
\(737\) −47.8260 −1.76169
\(738\) 5.08419 0.187152
\(739\) −31.1134 −1.14452 −0.572262 0.820071i \(-0.693934\pi\)
−0.572262 + 0.820071i \(0.693934\pi\)
\(740\) −27.5725 −1.01359
\(741\) 30.5939 1.12389
\(742\) −3.72773 −0.136849
\(743\) 24.5738 0.901526 0.450763 0.892644i \(-0.351152\pi\)
0.450763 + 0.892644i \(0.351152\pi\)
\(744\) 8.73595 0.320276
\(745\) −23.8820 −0.874968
\(746\) −15.6919 −0.574520
\(747\) −15.9436 −0.583346
\(748\) −18.8169 −0.688015
\(749\) 2.76874 0.101168
\(750\) 14.8398 0.541872
\(751\) −24.4679 −0.892847 −0.446423 0.894822i \(-0.647302\pi\)
−0.446423 + 0.894822i \(0.647302\pi\)
\(752\) 0.876797 0.0319735
\(753\) 14.3761 0.523893
\(754\) −6.97271 −0.253931
\(755\) 25.1041 0.913633
\(756\) 0.300942 0.0109451
\(757\) −2.70227 −0.0982158 −0.0491079 0.998793i \(-0.515638\pi\)
−0.0491079 + 0.998793i \(0.515638\pi\)
\(758\) 10.6109 0.385405
\(759\) 74.9661 2.72110
\(760\) 18.9849 0.688656
\(761\) 38.8424 1.40804 0.704018 0.710182i \(-0.251388\pi\)
0.704018 + 0.710182i \(0.251388\pi\)
\(762\) −32.0556 −1.16125
\(763\) 16.2060 0.586698
\(764\) −11.4895 −0.415676
\(765\) 40.7470 1.47321
\(766\) 2.07089 0.0748243
\(767\) −12.3772 −0.446916
\(768\) −2.41917 −0.0872943
\(769\) −49.1490 −1.77236 −0.886179 0.463343i \(-0.846650\pi\)
−0.886179 + 0.463343i \(0.846650\pi\)
\(770\) −8.66301 −0.312193
\(771\) −1.40920 −0.0507511
\(772\) 10.9398 0.393732
\(773\) −21.5096 −0.773645 −0.386823 0.922154i \(-0.626427\pi\)
−0.386823 + 0.922154i \(0.626427\pi\)
\(774\) −11.7787 −0.423377
\(775\) −10.1263 −0.363746
\(776\) 4.84073 0.173772
\(777\) 20.1215 0.721854
\(778\) −10.6659 −0.382390
\(779\) 12.1132 0.434001
\(780\) 12.5763 0.450304
\(781\) −16.6198 −0.594703
\(782\) 43.0623 1.53991
\(783\) 1.33807 0.0478189
\(784\) −6.28983 −0.224637
\(785\) −63.4441 −2.26442
\(786\) 46.6154 1.66271
\(787\) 34.2989 1.22263 0.611313 0.791389i \(-0.290642\pi\)
0.611313 + 0.791389i \(0.290642\pi\)
\(788\) −8.25382 −0.294030
\(789\) −18.4924 −0.658346
\(790\) 26.1693 0.931062
\(791\) −8.31992 −0.295822
\(792\) −10.4962 −0.372967
\(793\) 10.4941 0.372657
\(794\) −28.4239 −1.00873
\(795\) 29.8946 1.06025
\(796\) 18.6100 0.659614
\(797\) −35.8883 −1.27123 −0.635615 0.772006i \(-0.719253\pi\)
−0.635615 + 0.772006i \(0.719253\pi\)
\(798\) −13.8546 −0.490446
\(799\) 4.48356 0.158617
\(800\) 2.80418 0.0991426
\(801\) −29.3602 −1.03739
\(802\) 27.7211 0.978865
\(803\) −40.2719 −1.42116
\(804\) −31.4417 −1.10886
\(805\) 19.8252 0.698747
\(806\) 6.71996 0.236701
\(807\) −3.77477 −0.132878
\(808\) 6.86476 0.241501
\(809\) −12.5378 −0.440806 −0.220403 0.975409i \(-0.570737\pi\)
−0.220403 + 0.975409i \(0.570737\pi\)
\(810\) −26.3186 −0.924742
\(811\) 42.5345 1.49359 0.746795 0.665055i \(-0.231592\pi\)
0.746795 + 0.665055i \(0.231592\pi\)
\(812\) 3.15762 0.110811
\(813\) −16.7364 −0.586970
\(814\) 36.3193 1.27299
\(815\) 25.0319 0.876829
\(816\) −12.3706 −0.433057
\(817\) −28.0631 −0.981802
\(818\) −6.29387 −0.220060
\(819\) −4.47313 −0.156304
\(820\) 4.97940 0.173888
\(821\) −3.80109 −0.132659 −0.0663294 0.997798i \(-0.521129\pi\)
−0.0663294 + 0.997798i \(0.521129\pi\)
\(822\) −21.8547 −0.762271
\(823\) −49.7181 −1.73306 −0.866532 0.499122i \(-0.833656\pi\)
−0.866532 + 0.499122i \(0.833656\pi\)
\(824\) −6.85383 −0.238765
\(825\) 24.9630 0.869099
\(826\) 5.60508 0.195026
\(827\) 46.7135 1.62439 0.812193 0.583388i \(-0.198273\pi\)
0.812193 + 0.583388i \(0.198273\pi\)
\(828\) 24.0205 0.834769
\(829\) 47.3406 1.64421 0.822104 0.569337i \(-0.192800\pi\)
0.822104 + 0.569337i \(0.192800\pi\)
\(830\) −15.6150 −0.542005
\(831\) −18.1095 −0.628213
\(832\) −1.86090 −0.0645151
\(833\) −32.1635 −1.11440
\(834\) −49.6118 −1.71792
\(835\) −47.4283 −1.64132
\(836\) −25.0075 −0.864902
\(837\) −1.28957 −0.0445742
\(838\) −4.13463 −0.142828
\(839\) 28.4914 0.983632 0.491816 0.870699i \(-0.336333\pi\)
0.491816 + 0.870699i \(0.336333\pi\)
\(840\) −5.69522 −0.196504
\(841\) −14.9603 −0.515873
\(842\) −2.38583 −0.0822211
\(843\) −63.2740 −2.17927
\(844\) −9.86959 −0.339725
\(845\) −26.6427 −0.916536
\(846\) 2.50096 0.0859848
\(847\) 2.14130 0.0735761
\(848\) −4.42347 −0.151903
\(849\) −15.2850 −0.524579
\(850\) 14.3393 0.491835
\(851\) −83.1163 −2.84919
\(852\) −10.9261 −0.374324
\(853\) 38.0377 1.30239 0.651193 0.758912i \(-0.274269\pi\)
0.651193 + 0.758912i \(0.274269\pi\)
\(854\) −4.75229 −0.162620
\(855\) 54.1523 1.85197
\(856\) 3.28550 0.112296
\(857\) 26.8011 0.915508 0.457754 0.889079i \(-0.348654\pi\)
0.457754 + 0.889079i \(0.348654\pi\)
\(858\) −16.5659 −0.565549
\(859\) −26.8221 −0.915159 −0.457580 0.889169i \(-0.651284\pi\)
−0.457580 + 0.889169i \(0.651284\pi\)
\(860\) −11.5359 −0.393373
\(861\) −3.63380 −0.123840
\(862\) −23.7381 −0.808524
\(863\) 15.8736 0.540345 0.270172 0.962812i \(-0.412919\pi\)
0.270172 + 0.962812i \(0.412919\pi\)
\(864\) 0.357110 0.0121491
\(865\) 14.2048 0.482977
\(866\) −14.5976 −0.496046
\(867\) −22.1320 −0.751641
\(868\) −3.04316 −0.103292
\(869\) −34.4710 −1.16935
\(870\) −25.3226 −0.858517
\(871\) −24.1859 −0.819508
\(872\) 19.2307 0.651235
\(873\) 13.8076 0.467317
\(874\) 57.2294 1.93581
\(875\) −5.16942 −0.174758
\(876\) −26.4755 −0.894523
\(877\) 25.6418 0.865861 0.432931 0.901427i \(-0.357479\pi\)
0.432931 + 0.901427i \(0.357479\pi\)
\(878\) −18.0585 −0.609445
\(879\) −45.7915 −1.54451
\(880\) −10.2799 −0.346535
\(881\) 50.7600 1.71015 0.855073 0.518507i \(-0.173512\pi\)
0.855073 + 0.518507i \(0.173512\pi\)
\(882\) −17.9410 −0.604105
\(883\) 15.4942 0.521421 0.260710 0.965417i \(-0.416043\pi\)
0.260710 + 0.965417i \(0.416043\pi\)
\(884\) −9.51583 −0.320052
\(885\) −44.9501 −1.51098
\(886\) 22.3392 0.750500
\(887\) 37.9254 1.27341 0.636705 0.771108i \(-0.280297\pi\)
0.636705 + 0.771108i \(0.280297\pi\)
\(888\) 23.8770 0.801258
\(889\) 11.1665 0.374513
\(890\) −28.7551 −0.963872
\(891\) 34.6677 1.16141
\(892\) 6.05175 0.202627
\(893\) 5.95860 0.199397
\(894\) 20.6811 0.691679
\(895\) 30.8467 1.03109
\(896\) 0.842715 0.0281531
\(897\) 37.9108 1.26580
\(898\) −23.4665 −0.783088
\(899\) −13.5308 −0.451276
\(900\) 7.99858 0.266619
\(901\) −22.6197 −0.753572
\(902\) −6.55901 −0.218391
\(903\) 8.41854 0.280151
\(904\) −9.87276 −0.328363
\(905\) −5.41722 −0.180074
\(906\) −21.7394 −0.722244
\(907\) 15.1805 0.504062 0.252031 0.967719i \(-0.418902\pi\)
0.252031 + 0.967719i \(0.418902\pi\)
\(908\) −13.2276 −0.438975
\(909\) 19.5809 0.649458
\(910\) −4.38094 −0.145227
\(911\) −42.3483 −1.40306 −0.701532 0.712638i \(-0.747500\pi\)
−0.701532 + 0.712638i \(0.747500\pi\)
\(912\) −16.4404 −0.544395
\(913\) 20.5685 0.680720
\(914\) 38.5252 1.27430
\(915\) 38.1112 1.25992
\(916\) 16.1571 0.533845
\(917\) −16.2384 −0.536239
\(918\) 1.82611 0.0602705
\(919\) −32.0102 −1.05592 −0.527960 0.849270i \(-0.677043\pi\)
−0.527960 + 0.849270i \(0.677043\pi\)
\(920\) 23.5254 0.775610
\(921\) −67.5879 −2.22710
\(922\) 16.4972 0.543307
\(923\) −8.40472 −0.276645
\(924\) 7.50191 0.246795
\(925\) −27.6769 −0.910012
\(926\) 7.94502 0.261090
\(927\) −19.5498 −0.642098
\(928\) 3.74696 0.123000
\(929\) 56.3336 1.84824 0.924122 0.382097i \(-0.124798\pi\)
0.924122 + 0.382097i \(0.124798\pi\)
\(930\) 24.4047 0.800262
\(931\) −42.7449 −1.40091
\(932\) 10.0916 0.330563
\(933\) −18.4591 −0.604322
\(934\) −13.0541 −0.427142
\(935\) −52.5669 −1.71912
\(936\) −5.30800 −0.173497
\(937\) −44.5931 −1.45679 −0.728397 0.685156i \(-0.759734\pi\)
−0.728397 + 0.685156i \(0.759734\pi\)
\(938\) 10.9527 0.357618
\(939\) −40.6216 −1.32564
\(940\) 2.44942 0.0798911
\(941\) −25.2209 −0.822179 −0.411090 0.911595i \(-0.634852\pi\)
−0.411090 + 0.911595i \(0.634852\pi\)
\(942\) 54.9407 1.79006
\(943\) 15.0102 0.488800
\(944\) 6.65122 0.216479
\(945\) 0.840710 0.0273483
\(946\) 15.1955 0.494048
\(947\) 19.9357 0.647823 0.323912 0.946087i \(-0.395002\pi\)
0.323912 + 0.946087i \(0.395002\pi\)
\(948\) −22.6618 −0.736022
\(949\) −20.3657 −0.661100
\(950\) 19.0568 0.618285
\(951\) 52.9136 1.71584
\(952\) 4.30928 0.139665
\(953\) −57.0318 −1.84744 −0.923720 0.383068i \(-0.874868\pi\)
−0.923720 + 0.383068i \(0.874868\pi\)
\(954\) −12.6174 −0.408505
\(955\) −32.0971 −1.03864
\(956\) 22.6209 0.731613
\(957\) 33.3557 1.07824
\(958\) 33.5455 1.08381
\(959\) 7.61307 0.245839
\(960\) −6.75818 −0.218119
\(961\) −17.9597 −0.579345
\(962\) 18.3669 0.592172
\(963\) 9.37151 0.301993
\(964\) 17.0708 0.549812
\(965\) 30.5614 0.983806
\(966\) −17.1680 −0.552373
\(967\) 27.8890 0.896850 0.448425 0.893820i \(-0.351985\pi\)
0.448425 + 0.893820i \(0.351985\pi\)
\(968\) 2.54096 0.0816695
\(969\) −84.0690 −2.70068
\(970\) 13.5230 0.434199
\(971\) −43.9258 −1.40965 −0.704824 0.709382i \(-0.748974\pi\)
−0.704824 + 0.709382i \(0.748974\pi\)
\(972\) 21.7198 0.696663
\(973\) 17.2822 0.554043
\(974\) −31.8262 −1.01978
\(975\) 12.6239 0.404289
\(976\) −5.63927 −0.180508
\(977\) 9.57667 0.306385 0.153192 0.988196i \(-0.451045\pi\)
0.153192 + 0.988196i \(0.451045\pi\)
\(978\) −21.6769 −0.693150
\(979\) 37.8770 1.21055
\(980\) −17.5712 −0.561293
\(981\) 54.8534 1.75134
\(982\) −7.28748 −0.232553
\(983\) −11.0213 −0.351525 −0.175763 0.984433i \(-0.556239\pi\)
−0.175763 + 0.984433i \(0.556239\pi\)
\(984\) −4.31201 −0.137462
\(985\) −23.0578 −0.734684
\(986\) 19.1603 0.610189
\(987\) −1.78750 −0.0568968
\(988\) −12.6464 −0.402337
\(989\) −34.7747 −1.10577
\(990\) −29.3222 −0.931920
\(991\) 4.24190 0.134748 0.0673742 0.997728i \(-0.478538\pi\)
0.0673742 + 0.997728i \(0.478538\pi\)
\(992\) −3.61114 −0.114654
\(993\) −65.7449 −2.08635
\(994\) 3.80611 0.120722
\(995\) 51.9888 1.64816
\(996\) 13.5221 0.428465
\(997\) −50.6449 −1.60394 −0.801971 0.597363i \(-0.796215\pi\)
−0.801971 + 0.597363i \(0.796215\pi\)
\(998\) −21.8055 −0.690241
\(999\) −3.52464 −0.111515
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 8038.2.a.d.1.12 92
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8038.2.a.d.1.12 92 1.1 even 1 trivial