Properties

Label 6026.2.a.g.1.6
Level $6026$
Weight $2$
Character 6026.1
Self dual yes
Analytic conductor $48.118$
Analytic rank $1$
Dimension $21$
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] = [6026,2,Mod(1,6026)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6026, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("6026.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6026 = 2 \cdot 23 \cdot 131 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6026.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: \(48.1178522580\)
Analytic rank: \(1\)
Dimension: \(21\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Character \(\chi\) \(=\) 6026.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.31043 q^{3} +1.00000 q^{4} -3.45176 q^{5} -1.31043 q^{6} -4.90131 q^{7} +1.00000 q^{8} -1.28277 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.31043 q^{3} +1.00000 q^{4} -3.45176 q^{5} -1.31043 q^{6} -4.90131 q^{7} +1.00000 q^{8} -1.28277 q^{9} -3.45176 q^{10} +4.94848 q^{11} -1.31043 q^{12} +2.49647 q^{13} -4.90131 q^{14} +4.52329 q^{15} +1.00000 q^{16} +0.593016 q^{17} -1.28277 q^{18} -1.79083 q^{19} -3.45176 q^{20} +6.42283 q^{21} +4.94848 q^{22} -1.00000 q^{23} -1.31043 q^{24} +6.91463 q^{25} +2.49647 q^{26} +5.61228 q^{27} -4.90131 q^{28} +1.30090 q^{29} +4.52329 q^{30} +2.94314 q^{31} +1.00000 q^{32} -6.48465 q^{33} +0.593016 q^{34} +16.9181 q^{35} -1.28277 q^{36} +0.952399 q^{37} -1.79083 q^{38} -3.27145 q^{39} -3.45176 q^{40} +1.20314 q^{41} +6.42283 q^{42} -1.70995 q^{43} +4.94848 q^{44} +4.42781 q^{45} -1.00000 q^{46} +3.32071 q^{47} -1.31043 q^{48} +17.0229 q^{49} +6.91463 q^{50} -0.777106 q^{51} +2.49647 q^{52} +0.423826 q^{53} +5.61228 q^{54} -17.0810 q^{55} -4.90131 q^{56} +2.34676 q^{57} +1.30090 q^{58} +9.38527 q^{59} +4.52329 q^{60} -6.59544 q^{61} +2.94314 q^{62} +6.28726 q^{63} +1.00000 q^{64} -8.61719 q^{65} -6.48465 q^{66} -1.84298 q^{67} +0.593016 q^{68} +1.31043 q^{69} +16.9181 q^{70} -9.81033 q^{71} -1.28277 q^{72} -3.83908 q^{73} +0.952399 q^{74} -9.06115 q^{75} -1.79083 q^{76} -24.2541 q^{77} -3.27145 q^{78} -7.96524 q^{79} -3.45176 q^{80} -3.50619 q^{81} +1.20314 q^{82} -8.43352 q^{83} +6.42283 q^{84} -2.04695 q^{85} -1.70995 q^{86} -1.70474 q^{87} +4.94848 q^{88} -7.28886 q^{89} +4.42781 q^{90} -12.2360 q^{91} -1.00000 q^{92} -3.85678 q^{93} +3.32071 q^{94} +6.18152 q^{95} -1.31043 q^{96} +14.6055 q^{97} +17.0229 q^{98} -6.34777 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 21 q + 21 q^{2} + 21 q^{4} - 13 q^{5} - 18 q^{7} + 21 q^{8} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 21 q + 21 q^{2} + 21 q^{4} - 13 q^{5} - 18 q^{7} + 21 q^{8} + 7 q^{9} - 13 q^{10} - 4 q^{11} - 4 q^{13} - 18 q^{14} - 16 q^{15} + 21 q^{16} - 12 q^{17} + 7 q^{18} - 18 q^{19} - 13 q^{20} - 24 q^{21} - 4 q^{22} - 21 q^{23} + 2 q^{25} - 4 q^{26} - 9 q^{27} - 18 q^{28} - 16 q^{29} - 16 q^{30} - 7 q^{31} + 21 q^{32} - 15 q^{33} - 12 q^{34} + 7 q^{36} - 44 q^{37} - 18 q^{38} - 14 q^{39} - 13 q^{40} - 23 q^{41} - 24 q^{42} - 18 q^{43} - 4 q^{44} - 36 q^{45} - 21 q^{46} + 2 q^{47} - 13 q^{49} + 2 q^{50} - 26 q^{51} - 4 q^{52} - 39 q^{53} - 9 q^{54} - 32 q^{55} - 18 q^{56} - 22 q^{57} - 16 q^{58} - 27 q^{59} - 16 q^{60} - 34 q^{61} - 7 q^{62} - 28 q^{63} + 21 q^{64} - 25 q^{65} - 15 q^{66} - 19 q^{67} - 12 q^{68} - 24 q^{71} + 7 q^{72} - 8 q^{73} - 44 q^{74} + 50 q^{75} - 18 q^{76} - 16 q^{77} - 14 q^{78} - 27 q^{79} - 13 q^{80} + 33 q^{81} - 23 q^{82} + 7 q^{83} - 24 q^{84} - 22 q^{85} - 18 q^{86} - 15 q^{87} - 4 q^{88} - 12 q^{89} - 36 q^{90} - 20 q^{91} - 21 q^{92} - 43 q^{93} + 2 q^{94} - 14 q^{95} - 52 q^{97} - 13 q^{98} - 30 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.31043 −0.756578 −0.378289 0.925688i \(-0.623487\pi\)
−0.378289 + 0.925688i \(0.623487\pi\)
\(4\) 1.00000 0.500000
\(5\) −3.45176 −1.54367 −0.771837 0.635821i \(-0.780662\pi\)
−0.771837 + 0.635821i \(0.780662\pi\)
\(6\) −1.31043 −0.534981
\(7\) −4.90131 −1.85252 −0.926261 0.376883i \(-0.876996\pi\)
−0.926261 + 0.376883i \(0.876996\pi\)
\(8\) 1.00000 0.353553
\(9\) −1.28277 −0.427590
\(10\) −3.45176 −1.09154
\(11\) 4.94848 1.49202 0.746012 0.665933i \(-0.231966\pi\)
0.746012 + 0.665933i \(0.231966\pi\)
\(12\) −1.31043 −0.378289
\(13\) 2.49647 0.692395 0.346197 0.938162i \(-0.387473\pi\)
0.346197 + 0.938162i \(0.387473\pi\)
\(14\) −4.90131 −1.30993
\(15\) 4.52329 1.16791
\(16\) 1.00000 0.250000
\(17\) 0.593016 0.143827 0.0719137 0.997411i \(-0.477089\pi\)
0.0719137 + 0.997411i \(0.477089\pi\)
\(18\) −1.28277 −0.302352
\(19\) −1.79083 −0.410845 −0.205423 0.978673i \(-0.565857\pi\)
−0.205423 + 0.978673i \(0.565857\pi\)
\(20\) −3.45176 −0.771837
\(21\) 6.42283 1.40158
\(22\) 4.94848 1.05502
\(23\) −1.00000 −0.208514
\(24\) −1.31043 −0.267491
\(25\) 6.91463 1.38293
\(26\) 2.49647 0.489597
\(27\) 5.61228 1.08008
\(28\) −4.90131 −0.926261
\(29\) 1.30090 0.241570 0.120785 0.992679i \(-0.461459\pi\)
0.120785 + 0.992679i \(0.461459\pi\)
\(30\) 4.52329 0.825836
\(31\) 2.94314 0.528604 0.264302 0.964440i \(-0.414858\pi\)
0.264302 + 0.964440i \(0.414858\pi\)
\(32\) 1.00000 0.176777
\(33\) −6.48465 −1.12883
\(34\) 0.593016 0.101701
\(35\) 16.9181 2.85969
\(36\) −1.28277 −0.213795
\(37\) 0.952399 0.156573 0.0782867 0.996931i \(-0.475055\pi\)
0.0782867 + 0.996931i \(0.475055\pi\)
\(38\) −1.79083 −0.290512
\(39\) −3.27145 −0.523851
\(40\) −3.45176 −0.545771
\(41\) 1.20314 0.187898 0.0939492 0.995577i \(-0.470051\pi\)
0.0939492 + 0.995577i \(0.470051\pi\)
\(42\) 6.42283 0.991064
\(43\) −1.70995 −0.260766 −0.130383 0.991464i \(-0.541621\pi\)
−0.130383 + 0.991464i \(0.541621\pi\)
\(44\) 4.94848 0.746012
\(45\) 4.42781 0.660060
\(46\) −1.00000 −0.147442
\(47\) 3.32071 0.484376 0.242188 0.970229i \(-0.422135\pi\)
0.242188 + 0.970229i \(0.422135\pi\)
\(48\) −1.31043 −0.189144
\(49\) 17.0229 2.43184
\(50\) 6.91463 0.977877
\(51\) −0.777106 −0.108817
\(52\) 2.49647 0.346197
\(53\) 0.423826 0.0582170 0.0291085 0.999576i \(-0.490733\pi\)
0.0291085 + 0.999576i \(0.490733\pi\)
\(54\) 5.61228 0.763734
\(55\) −17.0810 −2.30320
\(56\) −4.90131 −0.654965
\(57\) 2.34676 0.310836
\(58\) 1.30090 0.170816
\(59\) 9.38527 1.22186 0.610929 0.791685i \(-0.290796\pi\)
0.610929 + 0.791685i \(0.290796\pi\)
\(60\) 4.52329 0.583954
\(61\) −6.59544 −0.844460 −0.422230 0.906489i \(-0.638753\pi\)
−0.422230 + 0.906489i \(0.638753\pi\)
\(62\) 2.94314 0.373779
\(63\) 6.28726 0.792120
\(64\) 1.00000 0.125000
\(65\) −8.61719 −1.06883
\(66\) −6.48465 −0.798205
\(67\) −1.84298 −0.225156 −0.112578 0.993643i \(-0.535911\pi\)
−0.112578 + 0.993643i \(0.535911\pi\)
\(68\) 0.593016 0.0719137
\(69\) 1.31043 0.157757
\(70\) 16.9181 2.02210
\(71\) −9.81033 −1.16427 −0.582136 0.813091i \(-0.697783\pi\)
−0.582136 + 0.813091i \(0.697783\pi\)
\(72\) −1.28277 −0.151176
\(73\) −3.83908 −0.449330 −0.224665 0.974436i \(-0.572129\pi\)
−0.224665 + 0.974436i \(0.572129\pi\)
\(74\) 0.952399 0.110714
\(75\) −9.06115 −1.04629
\(76\) −1.79083 −0.205423
\(77\) −24.2541 −2.76401
\(78\) −3.27145 −0.370418
\(79\) −7.96524 −0.896160 −0.448080 0.893994i \(-0.647892\pi\)
−0.448080 + 0.893994i \(0.647892\pi\)
\(80\) −3.45176 −0.385918
\(81\) −3.50619 −0.389576
\(82\) 1.20314 0.132864
\(83\) −8.43352 −0.925699 −0.462849 0.886437i \(-0.653173\pi\)
−0.462849 + 0.886437i \(0.653173\pi\)
\(84\) 6.42283 0.700788
\(85\) −2.04695 −0.222023
\(86\) −1.70995 −0.184389
\(87\) −1.70474 −0.182767
\(88\) 4.94848 0.527510
\(89\) −7.28886 −0.772618 −0.386309 0.922370i \(-0.626250\pi\)
−0.386309 + 0.922370i \(0.626250\pi\)
\(90\) 4.42781 0.466733
\(91\) −12.2360 −1.28268
\(92\) −1.00000 −0.104257
\(93\) −3.85678 −0.399930
\(94\) 3.32071 0.342505
\(95\) 6.18152 0.634211
\(96\) −1.31043 −0.133745
\(97\) 14.6055 1.48296 0.741481 0.670974i \(-0.234124\pi\)
0.741481 + 0.670974i \(0.234124\pi\)
\(98\) 17.0229 1.71957
\(99\) −6.34777 −0.637975
\(100\) 6.91463 0.691463
\(101\) −4.20650 −0.418562 −0.209281 0.977856i \(-0.567112\pi\)
−0.209281 + 0.977856i \(0.567112\pi\)
\(102\) −0.777106 −0.0769450
\(103\) 19.7304 1.94409 0.972045 0.234795i \(-0.0754420\pi\)
0.972045 + 0.234795i \(0.0754420\pi\)
\(104\) 2.49647 0.244799
\(105\) −22.1701 −2.16358
\(106\) 0.423826 0.0411657
\(107\) −0.287210 −0.0277656 −0.0138828 0.999904i \(-0.504419\pi\)
−0.0138828 + 0.999904i \(0.504419\pi\)
\(108\) 5.61228 0.540041
\(109\) −6.02172 −0.576776 −0.288388 0.957514i \(-0.593119\pi\)
−0.288388 + 0.957514i \(0.593119\pi\)
\(110\) −17.0810 −1.62861
\(111\) −1.24805 −0.118460
\(112\) −4.90131 −0.463130
\(113\) 3.26064 0.306736 0.153368 0.988169i \(-0.450988\pi\)
0.153368 + 0.988169i \(0.450988\pi\)
\(114\) 2.34676 0.219795
\(115\) 3.45176 0.321878
\(116\) 1.30090 0.120785
\(117\) −3.20239 −0.296061
\(118\) 9.38527 0.863984
\(119\) −2.90656 −0.266444
\(120\) 4.52329 0.412918
\(121\) 13.4875 1.22614
\(122\) −6.59544 −0.597123
\(123\) −1.57663 −0.142160
\(124\) 2.94314 0.264302
\(125\) −6.60885 −0.591113
\(126\) 6.28726 0.560114
\(127\) −5.97251 −0.529975 −0.264987 0.964252i \(-0.585368\pi\)
−0.264987 + 0.964252i \(0.585368\pi\)
\(128\) 1.00000 0.0883883
\(129\) 2.24078 0.197289
\(130\) −8.61719 −0.755778
\(131\) 1.00000 0.0873704
\(132\) −6.48465 −0.564416
\(133\) 8.77743 0.761100
\(134\) −1.84298 −0.159209
\(135\) −19.3722 −1.66729
\(136\) 0.593016 0.0508507
\(137\) −11.1566 −0.953171 −0.476585 0.879128i \(-0.658126\pi\)
−0.476585 + 0.879128i \(0.658126\pi\)
\(138\) 1.31043 0.111551
\(139\) 16.1131 1.36669 0.683347 0.730093i \(-0.260524\pi\)
0.683347 + 0.730093i \(0.260524\pi\)
\(140\) 16.9181 1.42984
\(141\) −4.35156 −0.366468
\(142\) −9.81033 −0.823265
\(143\) 12.3537 1.03307
\(144\) −1.28277 −0.106898
\(145\) −4.49038 −0.372906
\(146\) −3.83908 −0.317724
\(147\) −22.3073 −1.83987
\(148\) 0.952399 0.0782867
\(149\) −22.2413 −1.82208 −0.911039 0.412321i \(-0.864718\pi\)
−0.911039 + 0.412321i \(0.864718\pi\)
\(150\) −9.06115 −0.739840
\(151\) −12.2866 −0.999872 −0.499936 0.866062i \(-0.666643\pi\)
−0.499936 + 0.866062i \(0.666643\pi\)
\(152\) −1.79083 −0.145256
\(153\) −0.760704 −0.0614992
\(154\) −24.2541 −1.95445
\(155\) −10.1590 −0.815991
\(156\) −3.27145 −0.261925
\(157\) 11.1869 0.892815 0.446408 0.894830i \(-0.352703\pi\)
0.446408 + 0.894830i \(0.352703\pi\)
\(158\) −7.96524 −0.633681
\(159\) −0.555395 −0.0440457
\(160\) −3.45176 −0.272885
\(161\) 4.90131 0.386277
\(162\) −3.50619 −0.275472
\(163\) −13.8524 −1.08501 −0.542503 0.840054i \(-0.682523\pi\)
−0.542503 + 0.840054i \(0.682523\pi\)
\(164\) 1.20314 0.0939492
\(165\) 22.3834 1.74255
\(166\) −8.43352 −0.654568
\(167\) 20.4886 1.58546 0.792728 0.609575i \(-0.208660\pi\)
0.792728 + 0.609575i \(0.208660\pi\)
\(168\) 6.42283 0.495532
\(169\) −6.76766 −0.520589
\(170\) −2.04695 −0.156994
\(171\) 2.29723 0.175673
\(172\) −1.70995 −0.130383
\(173\) −23.0727 −1.75419 −0.877094 0.480318i \(-0.840521\pi\)
−0.877094 + 0.480318i \(0.840521\pi\)
\(174\) −1.70474 −0.129236
\(175\) −33.8908 −2.56190
\(176\) 4.94848 0.373006
\(177\) −12.2987 −0.924430
\(178\) −7.28886 −0.546323
\(179\) −16.8870 −1.26219 −0.631095 0.775706i \(-0.717394\pi\)
−0.631095 + 0.775706i \(0.717394\pi\)
\(180\) 4.42781 0.330030
\(181\) 9.20541 0.684232 0.342116 0.939658i \(-0.388856\pi\)
0.342116 + 0.939658i \(0.388856\pi\)
\(182\) −12.2360 −0.906989
\(183\) 8.64287 0.638900
\(184\) −1.00000 −0.0737210
\(185\) −3.28745 −0.241698
\(186\) −3.85678 −0.282793
\(187\) 2.93453 0.214594
\(188\) 3.32071 0.242188
\(189\) −27.5075 −2.00088
\(190\) 6.18152 0.448455
\(191\) −22.3547 −1.61753 −0.808765 0.588131i \(-0.799864\pi\)
−0.808765 + 0.588131i \(0.799864\pi\)
\(192\) −1.31043 −0.0945722
\(193\) 10.8674 0.782251 0.391125 0.920337i \(-0.372086\pi\)
0.391125 + 0.920337i \(0.372086\pi\)
\(194\) 14.6055 1.04861
\(195\) 11.2922 0.808654
\(196\) 17.0229 1.21592
\(197\) −8.37225 −0.596498 −0.298249 0.954488i \(-0.596403\pi\)
−0.298249 + 0.954488i \(0.596403\pi\)
\(198\) −6.34777 −0.451116
\(199\) −21.0862 −1.49476 −0.747379 0.664398i \(-0.768688\pi\)
−0.747379 + 0.664398i \(0.768688\pi\)
\(200\) 6.91463 0.488938
\(201\) 2.41510 0.170348
\(202\) −4.20650 −0.295968
\(203\) −6.37610 −0.447515
\(204\) −0.777106 −0.0544083
\(205\) −4.15294 −0.290054
\(206\) 19.7304 1.37468
\(207\) 1.28277 0.0891587
\(208\) 2.49647 0.173099
\(209\) −8.86191 −0.612991
\(210\) −22.1701 −1.52988
\(211\) 11.6134 0.799499 0.399749 0.916624i \(-0.369097\pi\)
0.399749 + 0.916624i \(0.369097\pi\)
\(212\) 0.423826 0.0291085
\(213\) 12.8558 0.880862
\(214\) −0.287210 −0.0196333
\(215\) 5.90235 0.402537
\(216\) 5.61228 0.381867
\(217\) −14.4253 −0.979250
\(218\) −6.02172 −0.407842
\(219\) 5.03085 0.339953
\(220\) −17.0810 −1.15160
\(221\) 1.48044 0.0995854
\(222\) −1.24805 −0.0837638
\(223\) −5.30447 −0.355213 −0.177607 0.984102i \(-0.556835\pi\)
−0.177607 + 0.984102i \(0.556835\pi\)
\(224\) −4.90131 −0.327483
\(225\) −8.86989 −0.591326
\(226\) 3.26064 0.216895
\(227\) 8.22379 0.545832 0.272916 0.962038i \(-0.412012\pi\)
0.272916 + 0.962038i \(0.412012\pi\)
\(228\) 2.34676 0.155418
\(229\) −27.6764 −1.82891 −0.914453 0.404691i \(-0.867379\pi\)
−0.914453 + 0.404691i \(0.867379\pi\)
\(230\) 3.45176 0.227602
\(231\) 31.7833 2.09119
\(232\) 1.30090 0.0854080
\(233\) 15.4503 1.01218 0.506092 0.862479i \(-0.331090\pi\)
0.506092 + 0.862479i \(0.331090\pi\)
\(234\) −3.20239 −0.209347
\(235\) −11.4623 −0.747717
\(236\) 9.38527 0.610929
\(237\) 10.4379 0.678014
\(238\) −2.90656 −0.188404
\(239\) 3.46983 0.224444 0.112222 0.993683i \(-0.464203\pi\)
0.112222 + 0.993683i \(0.464203\pi\)
\(240\) 4.52329 0.291977
\(241\) 8.42022 0.542394 0.271197 0.962524i \(-0.412580\pi\)
0.271197 + 0.962524i \(0.412580\pi\)
\(242\) 13.4875 0.867009
\(243\) −12.2422 −0.785338
\(244\) −6.59544 −0.422230
\(245\) −58.7588 −3.75396
\(246\) −1.57663 −0.100522
\(247\) −4.47075 −0.284467
\(248\) 2.94314 0.186890
\(249\) 11.0515 0.700363
\(250\) −6.60885 −0.417980
\(251\) −15.8996 −1.00358 −0.501788 0.864991i \(-0.667324\pi\)
−0.501788 + 0.864991i \(0.667324\pi\)
\(252\) 6.28726 0.396060
\(253\) −4.94848 −0.311108
\(254\) −5.97251 −0.374749
\(255\) 2.68238 0.167977
\(256\) 1.00000 0.0625000
\(257\) 7.19665 0.448915 0.224457 0.974484i \(-0.427939\pi\)
0.224457 + 0.974484i \(0.427939\pi\)
\(258\) 2.24078 0.139505
\(259\) −4.66800 −0.290056
\(260\) −8.61719 −0.534416
\(261\) −1.66875 −0.103293
\(262\) 1.00000 0.0617802
\(263\) 12.2294 0.754099 0.377050 0.926193i \(-0.376939\pi\)
0.377050 + 0.926193i \(0.376939\pi\)
\(264\) −6.48465 −0.399102
\(265\) −1.46295 −0.0898681
\(266\) 8.77743 0.538179
\(267\) 9.55155 0.584545
\(268\) −1.84298 −0.112578
\(269\) 4.07445 0.248424 0.124212 0.992256i \(-0.460360\pi\)
0.124212 + 0.992256i \(0.460360\pi\)
\(270\) −19.3722 −1.17896
\(271\) −10.8342 −0.658130 −0.329065 0.944307i \(-0.606734\pi\)
−0.329065 + 0.944307i \(0.606734\pi\)
\(272\) 0.593016 0.0359569
\(273\) 16.0344 0.970445
\(274\) −11.1566 −0.673993
\(275\) 34.2169 2.06336
\(276\) 1.31043 0.0788787
\(277\) 10.0518 0.603954 0.301977 0.953315i \(-0.402353\pi\)
0.301977 + 0.953315i \(0.402353\pi\)
\(278\) 16.1131 0.966399
\(279\) −3.77537 −0.226026
\(280\) 16.9181 1.01105
\(281\) −3.01978 −0.180145 −0.0900726 0.995935i \(-0.528710\pi\)
−0.0900726 + 0.995935i \(0.528710\pi\)
\(282\) −4.35156 −0.259132
\(283\) −5.32077 −0.316287 −0.158143 0.987416i \(-0.550551\pi\)
−0.158143 + 0.987416i \(0.550551\pi\)
\(284\) −9.81033 −0.582136
\(285\) −8.10046 −0.479830
\(286\) 12.3537 0.730491
\(287\) −5.89695 −0.348086
\(288\) −1.28277 −0.0755880
\(289\) −16.6483 −0.979314
\(290\) −4.49038 −0.263684
\(291\) −19.1395 −1.12198
\(292\) −3.83908 −0.224665
\(293\) −17.2528 −1.00792 −0.503959 0.863728i \(-0.668124\pi\)
−0.503959 + 0.863728i \(0.668124\pi\)
\(294\) −22.3073 −1.30099
\(295\) −32.3957 −1.88615
\(296\) 0.952399 0.0553571
\(297\) 27.7723 1.61151
\(298\) −22.2413 −1.28840
\(299\) −2.49647 −0.144374
\(300\) −9.06115 −0.523146
\(301\) 8.38102 0.483074
\(302\) −12.2866 −0.707016
\(303\) 5.51233 0.316675
\(304\) −1.79083 −0.102711
\(305\) 22.7659 1.30357
\(306\) −0.760704 −0.0434865
\(307\) −14.8867 −0.849632 −0.424816 0.905280i \(-0.639661\pi\)
−0.424816 + 0.905280i \(0.639661\pi\)
\(308\) −24.2541 −1.38200
\(309\) −25.8553 −1.47085
\(310\) −10.1590 −0.576993
\(311\) 11.9385 0.676971 0.338485 0.940972i \(-0.390085\pi\)
0.338485 + 0.940972i \(0.390085\pi\)
\(312\) −3.27145 −0.185209
\(313\) 11.2800 0.637583 0.318791 0.947825i \(-0.396723\pi\)
0.318791 + 0.947825i \(0.396723\pi\)
\(314\) 11.1869 0.631316
\(315\) −21.7021 −1.22277
\(316\) −7.96524 −0.448080
\(317\) 12.7630 0.716844 0.358422 0.933560i \(-0.383315\pi\)
0.358422 + 0.933560i \(0.383315\pi\)
\(318\) −0.555395 −0.0311450
\(319\) 6.43747 0.360429
\(320\) −3.45176 −0.192959
\(321\) 0.376369 0.0210069
\(322\) 4.90131 0.273139
\(323\) −1.06199 −0.0590909
\(324\) −3.50619 −0.194788
\(325\) 17.2621 0.957531
\(326\) −13.8524 −0.767215
\(327\) 7.89104 0.436376
\(328\) 1.20314 0.0664321
\(329\) −16.2758 −0.897316
\(330\) 22.3834 1.23217
\(331\) −3.76790 −0.207103 −0.103551 0.994624i \(-0.533021\pi\)
−0.103551 + 0.994624i \(0.533021\pi\)
\(332\) −8.43352 −0.462849
\(333\) −1.22171 −0.0669493
\(334\) 20.4886 1.12109
\(335\) 6.36152 0.347567
\(336\) 6.42283 0.350394
\(337\) 2.83747 0.154567 0.0772835 0.997009i \(-0.475375\pi\)
0.0772835 + 0.997009i \(0.475375\pi\)
\(338\) −6.76766 −0.368112
\(339\) −4.27285 −0.232069
\(340\) −2.04695 −0.111011
\(341\) 14.5641 0.788689
\(342\) 2.29723 0.124220
\(343\) −49.1252 −2.65251
\(344\) −1.70995 −0.0921946
\(345\) −4.52329 −0.243526
\(346\) −23.0727 −1.24040
\(347\) 21.9432 1.17797 0.588985 0.808144i \(-0.299528\pi\)
0.588985 + 0.808144i \(0.299528\pi\)
\(348\) −1.70474 −0.0913834
\(349\) 25.6240 1.37162 0.685811 0.727780i \(-0.259448\pi\)
0.685811 + 0.727780i \(0.259448\pi\)
\(350\) −33.8908 −1.81154
\(351\) 14.0109 0.747844
\(352\) 4.94848 0.263755
\(353\) 10.4939 0.558533 0.279266 0.960214i \(-0.409909\pi\)
0.279266 + 0.960214i \(0.409909\pi\)
\(354\) −12.2987 −0.653671
\(355\) 33.8629 1.79726
\(356\) −7.28886 −0.386309
\(357\) 3.80884 0.201585
\(358\) −16.8870 −0.892503
\(359\) −23.9118 −1.26201 −0.631007 0.775777i \(-0.717358\pi\)
−0.631007 + 0.775777i \(0.717358\pi\)
\(360\) 4.42781 0.233366
\(361\) −15.7929 −0.831206
\(362\) 9.20541 0.483825
\(363\) −17.6744 −0.927667
\(364\) −12.2360 −0.641338
\(365\) 13.2516 0.693619
\(366\) 8.64287 0.451770
\(367\) 21.6754 1.13145 0.565724 0.824595i \(-0.308597\pi\)
0.565724 + 0.824595i \(0.308597\pi\)
\(368\) −1.00000 −0.0521286
\(369\) −1.54335 −0.0803435
\(370\) −3.28745 −0.170906
\(371\) −2.07731 −0.107848
\(372\) −3.85678 −0.199965
\(373\) −30.3084 −1.56931 −0.784655 0.619933i \(-0.787160\pi\)
−0.784655 + 0.619933i \(0.787160\pi\)
\(374\) 2.93453 0.151741
\(375\) 8.66044 0.447223
\(376\) 3.32071 0.171253
\(377\) 3.24764 0.167262
\(378\) −27.5075 −1.41483
\(379\) 5.38087 0.276397 0.138198 0.990405i \(-0.455869\pi\)
0.138198 + 0.990405i \(0.455869\pi\)
\(380\) 6.18152 0.317105
\(381\) 7.82656 0.400967
\(382\) −22.3547 −1.14377
\(383\) 25.9080 1.32384 0.661919 0.749576i \(-0.269742\pi\)
0.661919 + 0.749576i \(0.269742\pi\)
\(384\) −1.31043 −0.0668726
\(385\) 83.7191 4.26672
\(386\) 10.8674 0.553135
\(387\) 2.19348 0.111501
\(388\) 14.6055 0.741481
\(389\) −16.5442 −0.838824 −0.419412 0.907796i \(-0.637764\pi\)
−0.419412 + 0.907796i \(0.637764\pi\)
\(390\) 11.2922 0.571805
\(391\) −0.593016 −0.0299901
\(392\) 17.0229 0.859784
\(393\) −1.31043 −0.0661025
\(394\) −8.37225 −0.421788
\(395\) 27.4941 1.38338
\(396\) −6.34777 −0.318987
\(397\) −1.48149 −0.0743539 −0.0371770 0.999309i \(-0.511837\pi\)
−0.0371770 + 0.999309i \(0.511837\pi\)
\(398\) −21.0862 −1.05695
\(399\) −11.5022 −0.575831
\(400\) 6.91463 0.345732
\(401\) 14.5304 0.725616 0.362808 0.931864i \(-0.381818\pi\)
0.362808 + 0.931864i \(0.381818\pi\)
\(402\) 2.41510 0.120454
\(403\) 7.34745 0.366003
\(404\) −4.20650 −0.209281
\(405\) 12.1025 0.601378
\(406\) −6.37610 −0.316441
\(407\) 4.71293 0.233611
\(408\) −0.777106 −0.0384725
\(409\) −35.4460 −1.75269 −0.876346 0.481682i \(-0.840026\pi\)
−0.876346 + 0.481682i \(0.840026\pi\)
\(410\) −4.15294 −0.205099
\(411\) 14.6199 0.721148
\(412\) 19.7304 0.972045
\(413\) −46.0001 −2.26352
\(414\) 1.28277 0.0630447
\(415\) 29.1105 1.42898
\(416\) 2.49647 0.122399
\(417\) −21.1151 −1.03401
\(418\) −8.86191 −0.433450
\(419\) 3.75478 0.183433 0.0917166 0.995785i \(-0.470765\pi\)
0.0917166 + 0.995785i \(0.470765\pi\)
\(420\) −22.1701 −1.08179
\(421\) −19.8396 −0.966925 −0.483462 0.875365i \(-0.660621\pi\)
−0.483462 + 0.875365i \(0.660621\pi\)
\(422\) 11.6134 0.565331
\(423\) −4.25971 −0.207114
\(424\) 0.423826 0.0205828
\(425\) 4.10049 0.198903
\(426\) 12.8558 0.622864
\(427\) 32.3263 1.56438
\(428\) −0.287210 −0.0138828
\(429\) −16.1887 −0.781598
\(430\) 5.90235 0.284637
\(431\) 17.6804 0.851636 0.425818 0.904809i \(-0.359986\pi\)
0.425818 + 0.904809i \(0.359986\pi\)
\(432\) 5.61228 0.270021
\(433\) 0.156994 0.00754466 0.00377233 0.999993i \(-0.498799\pi\)
0.00377233 + 0.999993i \(0.498799\pi\)
\(434\) −14.4253 −0.692434
\(435\) 5.88433 0.282132
\(436\) −6.02172 −0.288388
\(437\) 1.79083 0.0856672
\(438\) 5.03085 0.240383
\(439\) −9.59119 −0.457762 −0.228881 0.973454i \(-0.573507\pi\)
−0.228881 + 0.973454i \(0.573507\pi\)
\(440\) −17.0810 −0.814303
\(441\) −21.8364 −1.03983
\(442\) 1.48044 0.0704175
\(443\) 29.6953 1.41087 0.705433 0.708776i \(-0.250752\pi\)
0.705433 + 0.708776i \(0.250752\pi\)
\(444\) −1.24805 −0.0592300
\(445\) 25.1594 1.19267
\(446\) −5.30447 −0.251174
\(447\) 29.1457 1.37854
\(448\) −4.90131 −0.231565
\(449\) −15.1481 −0.714882 −0.357441 0.933936i \(-0.616351\pi\)
−0.357441 + 0.933936i \(0.616351\pi\)
\(450\) −8.86989 −0.418131
\(451\) 5.95370 0.280349
\(452\) 3.26064 0.153368
\(453\) 16.1008 0.756481
\(454\) 8.22379 0.385962
\(455\) 42.2356 1.98003
\(456\) 2.34676 0.109897
\(457\) −35.0201 −1.63817 −0.819086 0.573670i \(-0.805519\pi\)
−0.819086 + 0.573670i \(0.805519\pi\)
\(458\) −27.6764 −1.29323
\(459\) 3.32817 0.155346
\(460\) 3.45176 0.160939
\(461\) 22.6200 1.05352 0.526759 0.850015i \(-0.323407\pi\)
0.526759 + 0.850015i \(0.323407\pi\)
\(462\) 31.7833 1.47869
\(463\) 10.4348 0.484946 0.242473 0.970158i \(-0.422041\pi\)
0.242473 + 0.970158i \(0.422041\pi\)
\(464\) 1.30090 0.0603926
\(465\) 13.3127 0.617361
\(466\) 15.4503 0.715722
\(467\) 38.3231 1.77338 0.886690 0.462364i \(-0.152999\pi\)
0.886690 + 0.462364i \(0.152999\pi\)
\(468\) −3.20239 −0.148031
\(469\) 9.03302 0.417106
\(470\) −11.4623 −0.528716
\(471\) −14.6597 −0.675484
\(472\) 9.38527 0.431992
\(473\) −8.46168 −0.389069
\(474\) 10.4379 0.479429
\(475\) −12.3830 −0.568169
\(476\) −2.90656 −0.133222
\(477\) −0.543672 −0.0248930
\(478\) 3.46983 0.158706
\(479\) −4.13937 −0.189132 −0.0945662 0.995519i \(-0.530146\pi\)
−0.0945662 + 0.995519i \(0.530146\pi\)
\(480\) 4.52329 0.206459
\(481\) 2.37763 0.108411
\(482\) 8.42022 0.383531
\(483\) −6.42283 −0.292249
\(484\) 13.4875 0.613068
\(485\) −50.4146 −2.28921
\(486\) −12.2422 −0.555318
\(487\) 5.91980 0.268252 0.134126 0.990964i \(-0.457177\pi\)
0.134126 + 0.990964i \(0.457177\pi\)
\(488\) −6.59544 −0.298562
\(489\) 18.1527 0.820892
\(490\) −58.7588 −2.65445
\(491\) −40.9334 −1.84730 −0.923650 0.383237i \(-0.874809\pi\)
−0.923650 + 0.383237i \(0.874809\pi\)
\(492\) −1.57663 −0.0710799
\(493\) 0.771452 0.0347445
\(494\) −4.47075 −0.201149
\(495\) 21.9110 0.984825
\(496\) 2.94314 0.132151
\(497\) 48.0835 2.15684
\(498\) 11.0515 0.495232
\(499\) −4.81726 −0.215650 −0.107825 0.994170i \(-0.534389\pi\)
−0.107825 + 0.994170i \(0.534389\pi\)
\(500\) −6.60885 −0.295557
\(501\) −26.8489 −1.19952
\(502\) −15.8996 −0.709635
\(503\) 5.13742 0.229066 0.114533 0.993419i \(-0.463463\pi\)
0.114533 + 0.993419i \(0.463463\pi\)
\(504\) 6.28726 0.280057
\(505\) 14.5198 0.646123
\(506\) −4.94848 −0.219987
\(507\) 8.86855 0.393866
\(508\) −5.97251 −0.264987
\(509\) −32.1917 −1.42687 −0.713437 0.700719i \(-0.752863\pi\)
−0.713437 + 0.700719i \(0.752863\pi\)
\(510\) 2.68238 0.118778
\(511\) 18.8165 0.832394
\(512\) 1.00000 0.0441942
\(513\) −10.0506 −0.443747
\(514\) 7.19665 0.317431
\(515\) −68.1044 −3.00104
\(516\) 2.24078 0.0986447
\(517\) 16.4325 0.722700
\(518\) −4.66800 −0.205100
\(519\) 30.2352 1.32718
\(520\) −8.61719 −0.377889
\(521\) 17.0519 0.747057 0.373528 0.927619i \(-0.378148\pi\)
0.373528 + 0.927619i \(0.378148\pi\)
\(522\) −1.66875 −0.0730393
\(523\) 13.6682 0.597669 0.298835 0.954305i \(-0.403402\pi\)
0.298835 + 0.954305i \(0.403402\pi\)
\(524\) 1.00000 0.0436852
\(525\) 44.4115 1.93828
\(526\) 12.2294 0.533229
\(527\) 1.74533 0.0760277
\(528\) −6.48465 −0.282208
\(529\) 1.00000 0.0434783
\(530\) −1.46295 −0.0635463
\(531\) −12.0391 −0.522455
\(532\) 8.77743 0.380550
\(533\) 3.00359 0.130100
\(534\) 9.55155 0.413336
\(535\) 0.991379 0.0428611
\(536\) −1.84298 −0.0796047
\(537\) 22.1292 0.954945
\(538\) 4.07445 0.175662
\(539\) 84.2373 3.62836
\(540\) −19.3722 −0.833647
\(541\) 22.2262 0.955578 0.477789 0.878475i \(-0.341438\pi\)
0.477789 + 0.878475i \(0.341438\pi\)
\(542\) −10.8342 −0.465368
\(543\) −12.0630 −0.517675
\(544\) 0.593016 0.0254253
\(545\) 20.7855 0.890353
\(546\) 16.0344 0.686208
\(547\) 18.9703 0.811112 0.405556 0.914070i \(-0.367078\pi\)
0.405556 + 0.914070i \(0.367078\pi\)
\(548\) −11.1566 −0.476585
\(549\) 8.46044 0.361083
\(550\) 34.2169 1.45902
\(551\) −2.32969 −0.0992481
\(552\) 1.31043 0.0557756
\(553\) 39.0401 1.66016
\(554\) 10.0518 0.427060
\(555\) 4.30798 0.182863
\(556\) 16.1131 0.683347
\(557\) −1.82598 −0.0773691 −0.0386845 0.999251i \(-0.512317\pi\)
−0.0386845 + 0.999251i \(0.512317\pi\)
\(558\) −3.77537 −0.159824
\(559\) −4.26884 −0.180553
\(560\) 16.9181 0.714922
\(561\) −3.84550 −0.162357
\(562\) −3.01978 −0.127382
\(563\) −22.0226 −0.928141 −0.464070 0.885798i \(-0.653612\pi\)
−0.464070 + 0.885798i \(0.653612\pi\)
\(564\) −4.35156 −0.183234
\(565\) −11.2550 −0.473499
\(566\) −5.32077 −0.223648
\(567\) 17.1849 0.721699
\(568\) −9.81033 −0.411632
\(569\) 16.7467 0.702057 0.351029 0.936365i \(-0.385832\pi\)
0.351029 + 0.936365i \(0.385832\pi\)
\(570\) −8.10046 −0.339291
\(571\) 22.8993 0.958306 0.479153 0.877731i \(-0.340944\pi\)
0.479153 + 0.877731i \(0.340944\pi\)
\(572\) 12.3537 0.516535
\(573\) 29.2943 1.22379
\(574\) −5.89695 −0.246134
\(575\) −6.91463 −0.288360
\(576\) −1.28277 −0.0534488
\(577\) 21.5531 0.897267 0.448634 0.893716i \(-0.351911\pi\)
0.448634 + 0.893716i \(0.351911\pi\)
\(578\) −16.6483 −0.692479
\(579\) −14.2409 −0.591833
\(580\) −4.49038 −0.186453
\(581\) 41.3353 1.71488
\(582\) −19.1395 −0.793357
\(583\) 2.09730 0.0868612
\(584\) −3.83908 −0.158862
\(585\) 11.0539 0.457022
\(586\) −17.2528 −0.712705
\(587\) −1.95945 −0.0808750 −0.0404375 0.999182i \(-0.512875\pi\)
−0.0404375 + 0.999182i \(0.512875\pi\)
\(588\) −22.3073 −0.919937
\(589\) −5.27067 −0.217174
\(590\) −32.3957 −1.33371
\(591\) 10.9713 0.451297
\(592\) 0.952399 0.0391434
\(593\) −11.0941 −0.455579 −0.227790 0.973710i \(-0.573150\pi\)
−0.227790 + 0.973710i \(0.573150\pi\)
\(594\) 27.7723 1.13951
\(595\) 10.0327 0.411302
\(596\) −22.2413 −0.911039
\(597\) 27.6320 1.13090
\(598\) −2.49647 −0.102088
\(599\) −10.0001 −0.408594 −0.204297 0.978909i \(-0.565491\pi\)
−0.204297 + 0.978909i \(0.565491\pi\)
\(600\) −9.06115 −0.369920
\(601\) 8.40553 0.342869 0.171434 0.985196i \(-0.445160\pi\)
0.171434 + 0.985196i \(0.445160\pi\)
\(602\) 8.38102 0.341585
\(603\) 2.36412 0.0962745
\(604\) −12.2866 −0.499936
\(605\) −46.5555 −1.89275
\(606\) 5.51233 0.223923
\(607\) −35.2727 −1.43168 −0.715838 0.698267i \(-0.753955\pi\)
−0.715838 + 0.698267i \(0.753955\pi\)
\(608\) −1.79083 −0.0726279
\(609\) 8.35544 0.338579
\(610\) 22.7659 0.921763
\(611\) 8.29004 0.335379
\(612\) −0.760704 −0.0307496
\(613\) −31.1599 −1.25854 −0.629268 0.777188i \(-0.716645\pi\)
−0.629268 + 0.777188i \(0.716645\pi\)
\(614\) −14.8867 −0.600780
\(615\) 5.44214 0.219448
\(616\) −24.2541 −0.977224
\(617\) −18.1554 −0.730909 −0.365455 0.930829i \(-0.619086\pi\)
−0.365455 + 0.930829i \(0.619086\pi\)
\(618\) −25.8553 −1.04005
\(619\) −23.9300 −0.961828 −0.480914 0.876768i \(-0.659695\pi\)
−0.480914 + 0.876768i \(0.659695\pi\)
\(620\) −10.1590 −0.407996
\(621\) −5.61228 −0.225213
\(622\) 11.9385 0.478691
\(623\) 35.7250 1.43129
\(624\) −3.27145 −0.130963
\(625\) −11.7610 −0.470441
\(626\) 11.2800 0.450839
\(627\) 11.6129 0.463775
\(628\) 11.1869 0.446408
\(629\) 0.564788 0.0225196
\(630\) −21.7021 −0.864632
\(631\) 11.7862 0.469201 0.234601 0.972092i \(-0.424622\pi\)
0.234601 + 0.972092i \(0.424622\pi\)
\(632\) −7.96524 −0.316840
\(633\) −15.2185 −0.604883
\(634\) 12.7630 0.506885
\(635\) 20.6157 0.818107
\(636\) −0.555395 −0.0220229
\(637\) 42.4970 1.68379
\(638\) 6.43747 0.254862
\(639\) 12.5844 0.497831
\(640\) −3.45176 −0.136443
\(641\) −15.3250 −0.605300 −0.302650 0.953102i \(-0.597871\pi\)
−0.302650 + 0.953102i \(0.597871\pi\)
\(642\) 0.376369 0.0148541
\(643\) −42.2883 −1.66769 −0.833844 0.552000i \(-0.813865\pi\)
−0.833844 + 0.552000i \(0.813865\pi\)
\(644\) 4.90131 0.193139
\(645\) −7.73462 −0.304550
\(646\) −1.06199 −0.0417835
\(647\) −6.98442 −0.274586 −0.137293 0.990530i \(-0.543840\pi\)
−0.137293 + 0.990530i \(0.543840\pi\)
\(648\) −3.50619 −0.137736
\(649\) 46.4428 1.82304
\(650\) 17.2621 0.677077
\(651\) 18.9033 0.740879
\(652\) −13.8524 −0.542503
\(653\) −41.4687 −1.62280 −0.811399 0.584493i \(-0.801293\pi\)
−0.811399 + 0.584493i \(0.801293\pi\)
\(654\) 7.89104 0.308564
\(655\) −3.45176 −0.134871
\(656\) 1.20314 0.0469746
\(657\) 4.92466 0.192129
\(658\) −16.2758 −0.634498
\(659\) −27.7424 −1.08069 −0.540345 0.841443i \(-0.681706\pi\)
−0.540345 + 0.841443i \(0.681706\pi\)
\(660\) 22.3834 0.871274
\(661\) 7.04195 0.273900 0.136950 0.990578i \(-0.456270\pi\)
0.136950 + 0.990578i \(0.456270\pi\)
\(662\) −3.76790 −0.146444
\(663\) −1.94002 −0.0753441
\(664\) −8.43352 −0.327284
\(665\) −30.2976 −1.17489
\(666\) −1.22171 −0.0473403
\(667\) −1.30090 −0.0503709
\(668\) 20.4886 0.792728
\(669\) 6.95114 0.268746
\(670\) 6.36152 0.245767
\(671\) −32.6374 −1.25995
\(672\) 6.42283 0.247766
\(673\) −34.4211 −1.32684 −0.663418 0.748249i \(-0.730895\pi\)
−0.663418 + 0.748249i \(0.730895\pi\)
\(674\) 2.83747 0.109295
\(675\) 38.8068 1.49368
\(676\) −6.76766 −0.260295
\(677\) −48.3048 −1.85651 −0.928253 0.371950i \(-0.878689\pi\)
−0.928253 + 0.371950i \(0.878689\pi\)
\(678\) −4.27285 −0.164098
\(679\) −71.5860 −2.74722
\(680\) −2.04695 −0.0784969
\(681\) −10.7767 −0.412965
\(682\) 14.5641 0.557688
\(683\) 7.92791 0.303353 0.151677 0.988430i \(-0.451533\pi\)
0.151677 + 0.988430i \(0.451533\pi\)
\(684\) 2.29723 0.0878367
\(685\) 38.5098 1.47138
\(686\) −49.1252 −1.87561
\(687\) 36.2680 1.38371
\(688\) −1.70995 −0.0651914
\(689\) 1.05807 0.0403092
\(690\) −4.52329 −0.172199
\(691\) 4.26846 0.162380 0.0811899 0.996699i \(-0.474128\pi\)
0.0811899 + 0.996699i \(0.474128\pi\)
\(692\) −23.0727 −0.877094
\(693\) 31.1124 1.18186
\(694\) 21.9432 0.832951
\(695\) −55.6185 −2.10973
\(696\) −1.70474 −0.0646178
\(697\) 0.713479 0.0270250
\(698\) 25.6240 0.969883
\(699\) −20.2466 −0.765796
\(700\) −33.8908 −1.28095
\(701\) 17.4524 0.659166 0.329583 0.944127i \(-0.393092\pi\)
0.329583 + 0.944127i \(0.393092\pi\)
\(702\) 14.0109 0.528806
\(703\) −1.70559 −0.0643275
\(704\) 4.94848 0.186503
\(705\) 15.0205 0.565706
\(706\) 10.4939 0.394942
\(707\) 20.6174 0.775396
\(708\) −12.2987 −0.462215
\(709\) 28.6904 1.07749 0.538745 0.842469i \(-0.318899\pi\)
0.538745 + 0.842469i \(0.318899\pi\)
\(710\) 33.8629 1.27085
\(711\) 10.2176 0.383189
\(712\) −7.28886 −0.273162
\(713\) −2.94314 −0.110221
\(714\) 3.80884 0.142542
\(715\) −42.6420 −1.59472
\(716\) −16.8870 −0.631095
\(717\) −4.54697 −0.169810
\(718\) −23.9118 −0.892379
\(719\) 9.41139 0.350985 0.175493 0.984481i \(-0.443848\pi\)
0.175493 + 0.984481i \(0.443848\pi\)
\(720\) 4.42781 0.165015
\(721\) −96.7046 −3.60147
\(722\) −15.7929 −0.587751
\(723\) −11.0341 −0.410363
\(724\) 9.20541 0.342116
\(725\) 8.99522 0.334074
\(726\) −17.6744 −0.655959
\(727\) 35.3000 1.30921 0.654603 0.755973i \(-0.272836\pi\)
0.654603 + 0.755973i \(0.272836\pi\)
\(728\) −12.2360 −0.453495
\(729\) 26.5611 0.983746
\(730\) 13.2516 0.490463
\(731\) −1.01403 −0.0375053
\(732\) 8.64287 0.319450
\(733\) −22.4260 −0.828321 −0.414161 0.910204i \(-0.635925\pi\)
−0.414161 + 0.910204i \(0.635925\pi\)
\(734\) 21.6754 0.800054
\(735\) 76.9993 2.84016
\(736\) −1.00000 −0.0368605
\(737\) −9.11996 −0.335938
\(738\) −1.54335 −0.0568114
\(739\) −18.1709 −0.668428 −0.334214 0.942497i \(-0.608471\pi\)
−0.334214 + 0.942497i \(0.608471\pi\)
\(740\) −3.28745 −0.120849
\(741\) 5.85861 0.215222
\(742\) −2.07731 −0.0762603
\(743\) −13.7407 −0.504099 −0.252049 0.967714i \(-0.581105\pi\)
−0.252049 + 0.967714i \(0.581105\pi\)
\(744\) −3.85678 −0.141397
\(745\) 76.7715 2.81269
\(746\) −30.3084 −1.10967
\(747\) 10.8183 0.395820
\(748\) 2.93453 0.107297
\(749\) 1.40771 0.0514365
\(750\) 8.66044 0.316234
\(751\) 21.1294 0.771021 0.385511 0.922703i \(-0.374025\pi\)
0.385511 + 0.922703i \(0.374025\pi\)
\(752\) 3.32071 0.121094
\(753\) 20.8354 0.759283
\(754\) 3.24764 0.118272
\(755\) 42.4105 1.54347
\(756\) −27.5075 −1.00044
\(757\) −31.1843 −1.13341 −0.566706 0.823920i \(-0.691783\pi\)
−0.566706 + 0.823920i \(0.691783\pi\)
\(758\) 5.38087 0.195442
\(759\) 6.48465 0.235378
\(760\) 6.18152 0.224227
\(761\) −5.31043 −0.192503 −0.0962514 0.995357i \(-0.530685\pi\)
−0.0962514 + 0.995357i \(0.530685\pi\)
\(762\) 7.82656 0.283526
\(763\) 29.5143 1.06849
\(764\) −22.3547 −0.808765
\(765\) 2.62576 0.0949347
\(766\) 25.9080 0.936095
\(767\) 23.4300 0.846008
\(768\) −1.31043 −0.0472861
\(769\) 3.96990 0.143158 0.0715791 0.997435i \(-0.477196\pi\)
0.0715791 + 0.997435i \(0.477196\pi\)
\(770\) 83.7191 3.01703
\(771\) −9.43071 −0.339639
\(772\) 10.8674 0.391125
\(773\) 36.5290 1.31386 0.656929 0.753953i \(-0.271855\pi\)
0.656929 + 0.753953i \(0.271855\pi\)
\(774\) 2.19348 0.0788430
\(775\) 20.3507 0.731020
\(776\) 14.6055 0.524306
\(777\) 6.11710 0.219450
\(778\) −16.5442 −0.593138
\(779\) −2.15462 −0.0771972
\(780\) 11.2922 0.404327
\(781\) −48.5463 −1.73712
\(782\) −0.593016 −0.0212062
\(783\) 7.30099 0.260916
\(784\) 17.0229 0.607959
\(785\) −38.6146 −1.37821
\(786\) −1.31043 −0.0467415
\(787\) −27.1371 −0.967335 −0.483667 0.875252i \(-0.660696\pi\)
−0.483667 + 0.875252i \(0.660696\pi\)
\(788\) −8.37225 −0.298249
\(789\) −16.0258 −0.570535
\(790\) 27.4941 0.978196
\(791\) −15.9814 −0.568234
\(792\) −6.34777 −0.225558
\(793\) −16.4653 −0.584700
\(794\) −1.48149 −0.0525762
\(795\) 1.91709 0.0679922
\(796\) −21.0862 −0.747379
\(797\) 20.5804 0.728995 0.364497 0.931204i \(-0.381241\pi\)
0.364497 + 0.931204i \(0.381241\pi\)
\(798\) −11.5022 −0.407174
\(799\) 1.96923 0.0696665
\(800\) 6.91463 0.244469
\(801\) 9.34994 0.330364
\(802\) 14.5304 0.513088
\(803\) −18.9976 −0.670411
\(804\) 2.41510 0.0851740
\(805\) −16.9181 −0.596286
\(806\) 7.34745 0.258803
\(807\) −5.33929 −0.187952
\(808\) −4.20650 −0.147984
\(809\) −22.7174 −0.798700 −0.399350 0.916799i \(-0.630764\pi\)
−0.399350 + 0.916799i \(0.630764\pi\)
\(810\) 12.1025 0.425239
\(811\) −15.2314 −0.534845 −0.267423 0.963579i \(-0.586172\pi\)
−0.267423 + 0.963579i \(0.586172\pi\)
\(812\) −6.37610 −0.223757
\(813\) 14.1975 0.497927
\(814\) 4.71293 0.165188
\(815\) 47.8152 1.67490
\(816\) −0.777106 −0.0272042
\(817\) 3.06224 0.107134
\(818\) −35.4460 −1.23934
\(819\) 15.6959 0.548460
\(820\) −4.15294 −0.145027
\(821\) 39.7752 1.38816 0.694082 0.719896i \(-0.255810\pi\)
0.694082 + 0.719896i \(0.255810\pi\)
\(822\) 14.6199 0.509928
\(823\) 28.3875 0.989525 0.494762 0.869028i \(-0.335255\pi\)
0.494762 + 0.869028i \(0.335255\pi\)
\(824\) 19.7304 0.687339
\(825\) −44.8389 −1.56109
\(826\) −46.0001 −1.60055
\(827\) −18.5832 −0.646202 −0.323101 0.946365i \(-0.604725\pi\)
−0.323101 + 0.946365i \(0.604725\pi\)
\(828\) 1.28277 0.0445794
\(829\) 45.4197 1.57749 0.788746 0.614719i \(-0.210731\pi\)
0.788746 + 0.614719i \(0.210731\pi\)
\(830\) 29.1105 1.01044
\(831\) −13.1722 −0.456938
\(832\) 2.49647 0.0865494
\(833\) 10.0948 0.349765
\(834\) −21.1151 −0.731156
\(835\) −70.7218 −2.44743
\(836\) −8.86191 −0.306496
\(837\) 16.5177 0.570936
\(838\) 3.75478 0.129707
\(839\) −34.7039 −1.19811 −0.599057 0.800707i \(-0.704458\pi\)
−0.599057 + 0.800707i \(0.704458\pi\)
\(840\) −22.1701 −0.764940
\(841\) −27.3077 −0.941644
\(842\) −19.8396 −0.683719
\(843\) 3.95722 0.136294
\(844\) 11.6134 0.399749
\(845\) 23.3603 0.803620
\(846\) −4.25971 −0.146452
\(847\) −66.1064 −2.27144
\(848\) 0.423826 0.0145543
\(849\) 6.97250 0.239295
\(850\) 4.10049 0.140646
\(851\) −0.952399 −0.0326478
\(852\) 12.8558 0.440431
\(853\) −44.0052 −1.50671 −0.753354 0.657615i \(-0.771565\pi\)
−0.753354 + 0.657615i \(0.771565\pi\)
\(854\) 32.3263 1.10618
\(855\) −7.92948 −0.271182
\(856\) −0.287210 −0.00981664
\(857\) −1.21865 −0.0416282 −0.0208141 0.999783i \(-0.506626\pi\)
−0.0208141 + 0.999783i \(0.506626\pi\)
\(858\) −16.1887 −0.552673
\(859\) −49.1353 −1.67648 −0.838238 0.545305i \(-0.816414\pi\)
−0.838238 + 0.545305i \(0.816414\pi\)
\(860\) 5.90235 0.201268
\(861\) 7.72754 0.263354
\(862\) 17.6804 0.602198
\(863\) 20.0701 0.683194 0.341597 0.939847i \(-0.389032\pi\)
0.341597 + 0.939847i \(0.389032\pi\)
\(864\) 5.61228 0.190933
\(865\) 79.6415 2.70789
\(866\) 0.156994 0.00533488
\(867\) 21.8165 0.740927
\(868\) −14.4253 −0.489625
\(869\) −39.4159 −1.33709
\(870\) 5.88433 0.199498
\(871\) −4.60094 −0.155897
\(872\) −6.02172 −0.203921
\(873\) −18.7355 −0.634100
\(874\) 1.79083 0.0605758
\(875\) 32.3920 1.09505
\(876\) 5.03085 0.169977
\(877\) −55.5082 −1.87438 −0.937189 0.348822i \(-0.886582\pi\)
−0.937189 + 0.348822i \(0.886582\pi\)
\(878\) −9.59119 −0.323687
\(879\) 22.6086 0.762568
\(880\) −17.0810 −0.575799
\(881\) −42.6026 −1.43532 −0.717659 0.696395i \(-0.754786\pi\)
−0.717659 + 0.696395i \(0.754786\pi\)
\(882\) −21.8364 −0.735271
\(883\) 12.2124 0.410982 0.205491 0.978659i \(-0.434121\pi\)
0.205491 + 0.978659i \(0.434121\pi\)
\(884\) 1.48044 0.0497927
\(885\) 42.4523 1.42702
\(886\) 29.6953 0.997633
\(887\) −52.8154 −1.77337 −0.886684 0.462376i \(-0.846997\pi\)
−0.886684 + 0.462376i \(0.846997\pi\)
\(888\) −1.24805 −0.0418819
\(889\) 29.2731 0.981789
\(890\) 25.1594 0.843344
\(891\) −17.3503 −0.581257
\(892\) −5.30447 −0.177607
\(893\) −5.94684 −0.199003
\(894\) 29.1457 0.974777
\(895\) 58.2897 1.94841
\(896\) −4.90131 −0.163741
\(897\) 3.27145 0.109230
\(898\) −15.1481 −0.505498
\(899\) 3.82872 0.127695
\(900\) −8.86989 −0.295663
\(901\) 0.251336 0.00837321
\(902\) 5.95370 0.198237
\(903\) −10.9828 −0.365483
\(904\) 3.26064 0.108447
\(905\) −31.7748 −1.05623
\(906\) 16.1008 0.534913
\(907\) 57.3579 1.90454 0.952269 0.305262i \(-0.0987439\pi\)
0.952269 + 0.305262i \(0.0987439\pi\)
\(908\) 8.22379 0.272916
\(909\) 5.39597 0.178973
\(910\) 42.2356 1.40010
\(911\) 32.6540 1.08188 0.540938 0.841063i \(-0.318069\pi\)
0.540938 + 0.841063i \(0.318069\pi\)
\(912\) 2.34676 0.0777091
\(913\) −41.7331 −1.38117
\(914\) −35.0201 −1.15836
\(915\) −29.8331 −0.986252
\(916\) −27.6764 −0.914453
\(917\) −4.90131 −0.161856
\(918\) 3.32817 0.109846
\(919\) −53.7776 −1.77396 −0.886980 0.461808i \(-0.847201\pi\)
−0.886980 + 0.461808i \(0.847201\pi\)
\(920\) 3.45176 0.113801
\(921\) 19.5081 0.642812
\(922\) 22.6200 0.744949
\(923\) −24.4912 −0.806136
\(924\) 31.7833 1.04559
\(925\) 6.58549 0.216530
\(926\) 10.4348 0.342909
\(927\) −25.3095 −0.831274
\(928\) 1.30090 0.0427040
\(929\) 6.47143 0.212321 0.106160 0.994349i \(-0.466144\pi\)
0.106160 + 0.994349i \(0.466144\pi\)
\(930\) 13.3127 0.436540
\(931\) −30.4851 −0.999109
\(932\) 15.4503 0.506092
\(933\) −15.6446 −0.512181
\(934\) 38.3231 1.25397
\(935\) −10.1293 −0.331263
\(936\) −3.20239 −0.104673
\(937\) −50.8228 −1.66031 −0.830154 0.557534i \(-0.811747\pi\)
−0.830154 + 0.557534i \(0.811747\pi\)
\(938\) 9.03302 0.294939
\(939\) −14.7816 −0.482381
\(940\) −11.4623 −0.373859
\(941\) −0.933595 −0.0304343 −0.0152172 0.999884i \(-0.504844\pi\)
−0.0152172 + 0.999884i \(0.504844\pi\)
\(942\) −14.6597 −0.477639
\(943\) −1.20314 −0.0391795
\(944\) 9.38527 0.305464
\(945\) 94.9493 3.08870
\(946\) −8.46168 −0.275113
\(947\) 55.0764 1.78974 0.894871 0.446325i \(-0.147267\pi\)
0.894871 + 0.446325i \(0.147267\pi\)
\(948\) 10.4379 0.339007
\(949\) −9.58413 −0.311114
\(950\) −12.3830 −0.401756
\(951\) −16.7251 −0.542348
\(952\) −2.90656 −0.0942020
\(953\) 24.5155 0.794135 0.397068 0.917789i \(-0.370028\pi\)
0.397068 + 0.917789i \(0.370028\pi\)
\(954\) −0.543672 −0.0176020
\(955\) 77.1631 2.49694
\(956\) 3.46983 0.112222
\(957\) −8.43585 −0.272692
\(958\) −4.13937 −0.133737
\(959\) 54.6819 1.76577
\(960\) 4.52329 0.145989
\(961\) −22.3379 −0.720578
\(962\) 2.37763 0.0766579
\(963\) 0.368425 0.0118723
\(964\) 8.42022 0.271197
\(965\) −37.5115 −1.20754
\(966\) −6.42283 −0.206651
\(967\) 9.95960 0.320279 0.160140 0.987094i \(-0.448806\pi\)
0.160140 + 0.987094i \(0.448806\pi\)
\(968\) 13.4875 0.433504
\(969\) 1.39167 0.0447068
\(970\) −50.4146 −1.61872
\(971\) 5.63403 0.180805 0.0904024 0.995905i \(-0.471185\pi\)
0.0904024 + 0.995905i \(0.471185\pi\)
\(972\) −12.2422 −0.392669
\(973\) −78.9753 −2.53183
\(974\) 5.91980 0.189683
\(975\) −22.6208 −0.724447
\(976\) −6.59544 −0.211115
\(977\) −36.7141 −1.17459 −0.587295 0.809373i \(-0.699807\pi\)
−0.587295 + 0.809373i \(0.699807\pi\)
\(978\) 18.1527 0.580458
\(979\) −36.0688 −1.15276
\(980\) −58.7588 −1.87698
\(981\) 7.72448 0.246624
\(982\) −40.9334 −1.30624
\(983\) −32.6183 −1.04036 −0.520181 0.854056i \(-0.674136\pi\)
−0.520181 + 0.854056i \(0.674136\pi\)
\(984\) −1.57663 −0.0502610
\(985\) 28.8990 0.920798
\(986\) 0.771452 0.0245681
\(987\) 21.3284 0.678889
\(988\) −4.47075 −0.142234
\(989\) 1.70995 0.0543734
\(990\) 21.9110 0.696376
\(991\) −8.80535 −0.279711 −0.139856 0.990172i \(-0.544664\pi\)
−0.139856 + 0.990172i \(0.544664\pi\)
\(992\) 2.94314 0.0934448
\(993\) 4.93757 0.156689
\(994\) 48.0835 1.52512
\(995\) 72.7843 2.30742
\(996\) 11.0515 0.350182
\(997\) 25.3789 0.803758 0.401879 0.915693i \(-0.368357\pi\)
0.401879 + 0.915693i \(0.368357\pi\)
\(998\) −4.81726 −0.152488
\(999\) 5.34513 0.169112
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 6026.2.a.g.1.6 21
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6026.2.a.g.1.6 21 1.1 even 1 trivial