Properties

Label 4022.2.a.d.1.7
Level $4022$
Weight $2$
Character 4022.1
Self dual yes
Analytic conductor $32.116$
Analytic rank $1$
Dimension $37$
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] = [4022,2,Mod(1,4022)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4022, 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("4022.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4022 = 2 \cdot 2011 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4022.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: \(32.1158316930\)
Analytic rank: \(1\)
Dimension: \(37\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Character \(\chi\) \(=\) 4022.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.35142 q^{3} +1.00000 q^{4} -3.94455 q^{5} +2.35142 q^{6} -0.732372 q^{7} -1.00000 q^{8} +2.52919 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -2.35142 q^{3} +1.00000 q^{4} -3.94455 q^{5} +2.35142 q^{6} -0.732372 q^{7} -1.00000 q^{8} +2.52919 q^{9} +3.94455 q^{10} -5.60227 q^{11} -2.35142 q^{12} -6.57678 q^{13} +0.732372 q^{14} +9.27531 q^{15} +1.00000 q^{16} +7.88851 q^{17} -2.52919 q^{18} -3.58994 q^{19} -3.94455 q^{20} +1.72211 q^{21} +5.60227 q^{22} +0.597342 q^{23} +2.35142 q^{24} +10.5595 q^{25} +6.57678 q^{26} +1.10708 q^{27} -0.732372 q^{28} -5.16201 q^{29} -9.27531 q^{30} +2.38291 q^{31} -1.00000 q^{32} +13.1733 q^{33} -7.88851 q^{34} +2.88888 q^{35} +2.52919 q^{36} -8.53492 q^{37} +3.58994 q^{38} +15.4648 q^{39} +3.94455 q^{40} +5.16216 q^{41} -1.72211 q^{42} +4.23985 q^{43} -5.60227 q^{44} -9.97651 q^{45} -0.597342 q^{46} -2.11544 q^{47} -2.35142 q^{48} -6.46363 q^{49} -10.5595 q^{50} -18.5492 q^{51} -6.57678 q^{52} +13.3381 q^{53} -1.10708 q^{54} +22.0984 q^{55} +0.732372 q^{56} +8.44146 q^{57} +5.16201 q^{58} +14.6005 q^{59} +9.27531 q^{60} +3.15824 q^{61} -2.38291 q^{62} -1.85231 q^{63} +1.00000 q^{64} +25.9425 q^{65} -13.1733 q^{66} +15.0462 q^{67} +7.88851 q^{68} -1.40460 q^{69} -2.88888 q^{70} -0.278234 q^{71} -2.52919 q^{72} -15.4340 q^{73} +8.53492 q^{74} -24.8298 q^{75} -3.58994 q^{76} +4.10294 q^{77} -15.4648 q^{78} -3.94569 q^{79} -3.94455 q^{80} -10.1908 q^{81} -5.16216 q^{82} -13.3933 q^{83} +1.72211 q^{84} -31.1166 q^{85} -4.23985 q^{86} +12.1381 q^{87} +5.60227 q^{88} +1.59595 q^{89} +9.97651 q^{90} +4.81665 q^{91} +0.597342 q^{92} -5.60324 q^{93} +2.11544 q^{94} +14.1607 q^{95} +2.35142 q^{96} -11.1601 q^{97} +6.46363 q^{98} -14.1692 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 37 q - 37 q^{2} - 5 q^{3} + 37 q^{4} - 13 q^{5} + 5 q^{6} - 22 q^{7} - 37 q^{8} + 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 37 q - 37 q^{2} - 5 q^{3} + 37 q^{4} - 13 q^{5} + 5 q^{6} - 22 q^{7} - 37 q^{8} + 32 q^{9} + 13 q^{10} + 12 q^{11} - 5 q^{12} - 36 q^{13} + 22 q^{14} - 7 q^{15} + 37 q^{16} + 4 q^{17} - 32 q^{18} - 10 q^{19} - 13 q^{20} - 11 q^{21} - 12 q^{22} - q^{23} + 5 q^{24} + 8 q^{25} + 36 q^{26} - 20 q^{27} - 22 q^{28} - 6 q^{29} + 7 q^{30} - 23 q^{31} - 37 q^{32} - 27 q^{33} - 4 q^{34} + 24 q^{35} + 32 q^{36} - 46 q^{37} + 10 q^{38} + 5 q^{39} + 13 q^{40} + 11 q^{41} + 11 q^{42} - 22 q^{43} + 12 q^{44} - 57 q^{45} + q^{46} - 18 q^{47} - 5 q^{48} - q^{49} - 8 q^{50} + 18 q^{51} - 36 q^{52} - 25 q^{53} + 20 q^{54} - 25 q^{55} + 22 q^{56} - 25 q^{57} + 6 q^{58} + 24 q^{59} - 7 q^{60} - 40 q^{61} + 23 q^{62} - 38 q^{63} + 37 q^{64} + 14 q^{65} + 27 q^{66} - 49 q^{67} + 4 q^{68} - 19 q^{69} - 24 q^{70} + 27 q^{71} - 32 q^{72} - 87 q^{73} + 46 q^{74} - 5 q^{75} - 10 q^{76} - 50 q^{77} - 5 q^{78} + 11 q^{79} - 13 q^{80} + 5 q^{81} - 11 q^{82} - 9 q^{83} - 11 q^{84} - 68 q^{85} + 22 q^{86} - 12 q^{87} - 12 q^{88} - 3 q^{89} + 57 q^{90} - 19 q^{91} - q^{92} - 69 q^{93} + 18 q^{94} + 27 q^{95} + 5 q^{96} - 98 q^{97} + q^{98} + 33 q^{99}+O(q^{100}) \) 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
\(3\) −2.35142 −1.35759 −0.678797 0.734326i \(-0.737498\pi\)
−0.678797 + 0.734326i \(0.737498\pi\)
\(4\) 1.00000 0.500000
\(5\) −3.94455 −1.76406 −0.882029 0.471196i \(-0.843823\pi\)
−0.882029 + 0.471196i \(0.843823\pi\)
\(6\) 2.35142 0.959964
\(7\) −0.732372 −0.276810 −0.138405 0.990376i \(-0.544198\pi\)
−0.138405 + 0.990376i \(0.544198\pi\)
\(8\) −1.00000 −0.353553
\(9\) 2.52919 0.843063
\(10\) 3.94455 1.24738
\(11\) −5.60227 −1.68915 −0.844574 0.535440i \(-0.820146\pi\)
−0.844574 + 0.535440i \(0.820146\pi\)
\(12\) −2.35142 −0.678797
\(13\) −6.57678 −1.82407 −0.912035 0.410112i \(-0.865490\pi\)
−0.912035 + 0.410112i \(0.865490\pi\)
\(14\) 0.732372 0.195735
\(15\) 9.27531 2.39487
\(16\) 1.00000 0.250000
\(17\) 7.88851 1.91324 0.956622 0.291331i \(-0.0940980\pi\)
0.956622 + 0.291331i \(0.0940980\pi\)
\(18\) −2.52919 −0.596135
\(19\) −3.58994 −0.823588 −0.411794 0.911277i \(-0.635098\pi\)
−0.411794 + 0.911277i \(0.635098\pi\)
\(20\) −3.94455 −0.882029
\(21\) 1.72211 0.375796
\(22\) 5.60227 1.19441
\(23\) 0.597342 0.124554 0.0622772 0.998059i \(-0.480164\pi\)
0.0622772 + 0.998059i \(0.480164\pi\)
\(24\) 2.35142 0.479982
\(25\) 10.5595 2.11190
\(26\) 6.57678 1.28981
\(27\) 1.10708 0.213057
\(28\) −0.732372 −0.138405
\(29\) −5.16201 −0.958561 −0.479281 0.877662i \(-0.659102\pi\)
−0.479281 + 0.877662i \(0.659102\pi\)
\(30\) −9.27531 −1.69343
\(31\) 2.38291 0.427984 0.213992 0.976835i \(-0.431353\pi\)
0.213992 + 0.976835i \(0.431353\pi\)
\(32\) −1.00000 −0.176777
\(33\) 13.1733 2.29318
\(34\) −7.88851 −1.35287
\(35\) 2.88888 0.488309
\(36\) 2.52919 0.421531
\(37\) −8.53492 −1.40313 −0.701566 0.712604i \(-0.747516\pi\)
−0.701566 + 0.712604i \(0.747516\pi\)
\(38\) 3.58994 0.582365
\(39\) 15.4648 2.47635
\(40\) 3.94455 0.623688
\(41\) 5.16216 0.806194 0.403097 0.915157i \(-0.367934\pi\)
0.403097 + 0.915157i \(0.367934\pi\)
\(42\) −1.72211 −0.265728
\(43\) 4.23985 0.646571 0.323286 0.946301i \(-0.395213\pi\)
0.323286 + 0.946301i \(0.395213\pi\)
\(44\) −5.60227 −0.844574
\(45\) −9.97651 −1.48721
\(46\) −0.597342 −0.0880733
\(47\) −2.11544 −0.308569 −0.154284 0.988026i \(-0.549307\pi\)
−0.154284 + 0.988026i \(0.549307\pi\)
\(48\) −2.35142 −0.339399
\(49\) −6.46363 −0.923376
\(50\) −10.5595 −1.49334
\(51\) −18.5492 −2.59741
\(52\) −6.57678 −0.912035
\(53\) 13.3381 1.83213 0.916063 0.401034i \(-0.131349\pi\)
0.916063 + 0.401034i \(0.131349\pi\)
\(54\) −1.10708 −0.150654
\(55\) 22.0984 2.97975
\(56\) 0.732372 0.0978673
\(57\) 8.44146 1.11810
\(58\) 5.16201 0.677805
\(59\) 14.6005 1.90083 0.950413 0.310991i \(-0.100661\pi\)
0.950413 + 0.310991i \(0.100661\pi\)
\(60\) 9.27531 1.19744
\(61\) 3.15824 0.404372 0.202186 0.979347i \(-0.435195\pi\)
0.202186 + 0.979347i \(0.435195\pi\)
\(62\) −2.38291 −0.302630
\(63\) −1.85231 −0.233369
\(64\) 1.00000 0.125000
\(65\) 25.9425 3.21776
\(66\) −13.1733 −1.62152
\(67\) 15.0462 1.83818 0.919091 0.394045i \(-0.128925\pi\)
0.919091 + 0.394045i \(0.128925\pi\)
\(68\) 7.88851 0.956622
\(69\) −1.40460 −0.169094
\(70\) −2.88888 −0.345287
\(71\) −0.278234 −0.0330203 −0.0165102 0.999864i \(-0.505256\pi\)
−0.0165102 + 0.999864i \(0.505256\pi\)
\(72\) −2.52919 −0.298068
\(73\) −15.4340 −1.80641 −0.903205 0.429209i \(-0.858792\pi\)
−0.903205 + 0.429209i \(0.858792\pi\)
\(74\) 8.53492 0.992165
\(75\) −24.8298 −2.86710
\(76\) −3.58994 −0.411794
\(77\) 4.10294 0.467573
\(78\) −15.4648 −1.75104
\(79\) −3.94569 −0.443925 −0.221962 0.975055i \(-0.571246\pi\)
−0.221962 + 0.975055i \(0.571246\pi\)
\(80\) −3.94455 −0.441014
\(81\) −10.1908 −1.13231
\(82\) −5.16216 −0.570065
\(83\) −13.3933 −1.47010 −0.735052 0.678010i \(-0.762842\pi\)
−0.735052 + 0.678010i \(0.762842\pi\)
\(84\) 1.72211 0.187898
\(85\) −31.1166 −3.37507
\(86\) −4.23985 −0.457195
\(87\) 12.1381 1.30134
\(88\) 5.60227 0.597204
\(89\) 1.59595 0.169170 0.0845850 0.996416i \(-0.473044\pi\)
0.0845850 + 0.996416i \(0.473044\pi\)
\(90\) 9.97651 1.05162
\(91\) 4.81665 0.504922
\(92\) 0.597342 0.0622772
\(93\) −5.60324 −0.581029
\(94\) 2.11544 0.218191
\(95\) 14.1607 1.45286
\(96\) 2.35142 0.239991
\(97\) −11.1601 −1.13314 −0.566569 0.824014i \(-0.691729\pi\)
−0.566569 + 0.824014i \(0.691729\pi\)
\(98\) 6.46363 0.652925
\(99\) −14.1692 −1.42406
\(100\) 10.5595 1.05595
\(101\) 15.8952 1.58163 0.790815 0.612055i \(-0.209657\pi\)
0.790815 + 0.612055i \(0.209657\pi\)
\(102\) 18.5492 1.83665
\(103\) 4.52520 0.445881 0.222940 0.974832i \(-0.428434\pi\)
0.222940 + 0.974832i \(0.428434\pi\)
\(104\) 6.57678 0.644906
\(105\) −6.79297 −0.662926
\(106\) −13.3381 −1.29551
\(107\) −2.00963 −0.194278 −0.0971392 0.995271i \(-0.530969\pi\)
−0.0971392 + 0.995271i \(0.530969\pi\)
\(108\) 1.10708 0.106529
\(109\) 3.39488 0.325170 0.162585 0.986695i \(-0.448017\pi\)
0.162585 + 0.986695i \(0.448017\pi\)
\(110\) −22.0984 −2.10700
\(111\) 20.0692 1.90489
\(112\) −0.732372 −0.0692026
\(113\) −2.44327 −0.229843 −0.114922 0.993375i \(-0.536662\pi\)
−0.114922 + 0.993375i \(0.536662\pi\)
\(114\) −8.44146 −0.790615
\(115\) −2.35625 −0.219721
\(116\) −5.16201 −0.479281
\(117\) −16.6339 −1.53781
\(118\) −14.6005 −1.34409
\(119\) −5.77732 −0.529606
\(120\) −9.27531 −0.846716
\(121\) 20.3854 1.85322
\(122\) −3.15824 −0.285934
\(123\) −12.1384 −1.09448
\(124\) 2.38291 0.213992
\(125\) −21.9297 −1.96145
\(126\) 1.85231 0.165016
\(127\) 5.11200 0.453616 0.226808 0.973939i \(-0.427171\pi\)
0.226808 + 0.973939i \(0.427171\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −9.96968 −0.877782
\(130\) −25.9425 −2.27530
\(131\) 7.31053 0.638724 0.319362 0.947633i \(-0.396531\pi\)
0.319362 + 0.947633i \(0.396531\pi\)
\(132\) 13.1733 1.14659
\(133\) 2.62917 0.227978
\(134\) −15.0462 −1.29979
\(135\) −4.36693 −0.375845
\(136\) −7.88851 −0.676434
\(137\) 12.7625 1.09038 0.545189 0.838313i \(-0.316458\pi\)
0.545189 + 0.838313i \(0.316458\pi\)
\(138\) 1.40460 0.119568
\(139\) 5.16815 0.438357 0.219179 0.975685i \(-0.429662\pi\)
0.219179 + 0.975685i \(0.429662\pi\)
\(140\) 2.88888 0.244155
\(141\) 4.97429 0.418911
\(142\) 0.278234 0.0233489
\(143\) 36.8449 3.08112
\(144\) 2.52919 0.210766
\(145\) 20.3618 1.69096
\(146\) 15.4340 1.27732
\(147\) 15.1987 1.25357
\(148\) −8.53492 −0.701566
\(149\) 11.8339 0.969471 0.484735 0.874661i \(-0.338916\pi\)
0.484735 + 0.874661i \(0.338916\pi\)
\(150\) 24.8298 2.02735
\(151\) 2.37792 0.193512 0.0967561 0.995308i \(-0.469153\pi\)
0.0967561 + 0.995308i \(0.469153\pi\)
\(152\) 3.58994 0.291182
\(153\) 19.9515 1.61299
\(154\) −4.10294 −0.330624
\(155\) −9.39953 −0.754988
\(156\) 15.4648 1.23817
\(157\) −3.17462 −0.253362 −0.126681 0.991943i \(-0.540432\pi\)
−0.126681 + 0.991943i \(0.540432\pi\)
\(158\) 3.94569 0.313902
\(159\) −31.3635 −2.48728
\(160\) 3.94455 0.311844
\(161\) −0.437476 −0.0344780
\(162\) 10.1908 0.800663
\(163\) 0.771085 0.0603961 0.0301980 0.999544i \(-0.490386\pi\)
0.0301980 + 0.999544i \(0.490386\pi\)
\(164\) 5.16216 0.403097
\(165\) −51.9628 −4.04530
\(166\) 13.3933 1.03952
\(167\) 4.40690 0.341016 0.170508 0.985356i \(-0.445459\pi\)
0.170508 + 0.985356i \(0.445459\pi\)
\(168\) −1.72211 −0.132864
\(169\) 30.2540 2.32723
\(170\) 31.1166 2.38654
\(171\) −9.07963 −0.694337
\(172\) 4.23985 0.323286
\(173\) −9.05042 −0.688091 −0.344045 0.938953i \(-0.611797\pi\)
−0.344045 + 0.938953i \(0.611797\pi\)
\(174\) −12.1381 −0.920184
\(175\) −7.73347 −0.584595
\(176\) −5.60227 −0.422287
\(177\) −34.3320 −2.58055
\(178\) −1.59595 −0.119621
\(179\) −8.97070 −0.670502 −0.335251 0.942129i \(-0.608821\pi\)
−0.335251 + 0.942129i \(0.608821\pi\)
\(180\) −9.97651 −0.743605
\(181\) −3.91600 −0.291074 −0.145537 0.989353i \(-0.546491\pi\)
−0.145537 + 0.989353i \(0.546491\pi\)
\(182\) −4.81665 −0.357034
\(183\) −7.42637 −0.548973
\(184\) −0.597342 −0.0440367
\(185\) 33.6665 2.47521
\(186\) 5.60324 0.410849
\(187\) −44.1935 −3.23175
\(188\) −2.11544 −0.154284
\(189\) −0.810793 −0.0589765
\(190\) −14.1607 −1.02733
\(191\) 10.4873 0.758836 0.379418 0.925225i \(-0.376124\pi\)
0.379418 + 0.925225i \(0.376124\pi\)
\(192\) −2.35142 −0.169699
\(193\) −22.1435 −1.59393 −0.796963 0.604028i \(-0.793562\pi\)
−0.796963 + 0.604028i \(0.793562\pi\)
\(194\) 11.1601 0.801249
\(195\) −61.0017 −4.36842
\(196\) −6.46363 −0.461688
\(197\) 6.89885 0.491523 0.245761 0.969330i \(-0.420962\pi\)
0.245761 + 0.969330i \(0.420962\pi\)
\(198\) 14.1692 1.00696
\(199\) −7.26553 −0.515040 −0.257520 0.966273i \(-0.582905\pi\)
−0.257520 + 0.966273i \(0.582905\pi\)
\(200\) −10.5595 −0.746669
\(201\) −35.3799 −2.49551
\(202\) −15.8952 −1.11838
\(203\) 3.78051 0.265340
\(204\) −18.5492 −1.29871
\(205\) −20.3624 −1.42217
\(206\) −4.52520 −0.315285
\(207\) 1.51079 0.105007
\(208\) −6.57678 −0.456018
\(209\) 20.1118 1.39116
\(210\) 6.79297 0.468760
\(211\) 15.9296 1.09664 0.548320 0.836269i \(-0.315268\pi\)
0.548320 + 0.836269i \(0.315268\pi\)
\(212\) 13.3381 0.916063
\(213\) 0.654246 0.0448282
\(214\) 2.00963 0.137376
\(215\) −16.7243 −1.14059
\(216\) −1.10708 −0.0753271
\(217\) −1.74518 −0.118470
\(218\) −3.39488 −0.229930
\(219\) 36.2918 2.45237
\(220\) 22.0984 1.48988
\(221\) −51.8810 −3.48989
\(222\) −20.0692 −1.34696
\(223\) −27.6617 −1.85236 −0.926182 0.377078i \(-0.876929\pi\)
−0.926182 + 0.377078i \(0.876929\pi\)
\(224\) 0.732372 0.0489336
\(225\) 26.7069 1.78046
\(226\) 2.44327 0.162524
\(227\) −3.93286 −0.261033 −0.130517 0.991446i \(-0.541664\pi\)
−0.130517 + 0.991446i \(0.541664\pi\)
\(228\) 8.44146 0.559049
\(229\) −8.11720 −0.536400 −0.268200 0.963363i \(-0.586429\pi\)
−0.268200 + 0.963363i \(0.586429\pi\)
\(230\) 2.35625 0.155366
\(231\) −9.64775 −0.634775
\(232\) 5.16201 0.338903
\(233\) −5.35936 −0.351103 −0.175552 0.984470i \(-0.556171\pi\)
−0.175552 + 0.984470i \(0.556171\pi\)
\(234\) 16.6339 1.08739
\(235\) 8.34446 0.544333
\(236\) 14.6005 0.950413
\(237\) 9.27798 0.602670
\(238\) 5.77732 0.374488
\(239\) −19.4979 −1.26121 −0.630607 0.776102i \(-0.717194\pi\)
−0.630607 + 0.776102i \(0.717194\pi\)
\(240\) 9.27531 0.598719
\(241\) 0.671107 0.0432298 0.0216149 0.999766i \(-0.493119\pi\)
0.0216149 + 0.999766i \(0.493119\pi\)
\(242\) −20.3854 −1.31042
\(243\) 20.6416 1.32416
\(244\) 3.15824 0.202186
\(245\) 25.4961 1.62889
\(246\) 12.1384 0.773917
\(247\) 23.6102 1.50228
\(248\) −2.38291 −0.151315
\(249\) 31.4933 1.99581
\(250\) 21.9297 1.38696
\(251\) 24.4575 1.54374 0.771871 0.635779i \(-0.219321\pi\)
0.771871 + 0.635779i \(0.219321\pi\)
\(252\) −1.85231 −0.116684
\(253\) −3.34647 −0.210391
\(254\) −5.11200 −0.320755
\(255\) 73.1684 4.58198
\(256\) 1.00000 0.0625000
\(257\) 1.94806 0.121516 0.0607582 0.998153i \(-0.480648\pi\)
0.0607582 + 0.998153i \(0.480648\pi\)
\(258\) 9.96968 0.620685
\(259\) 6.25074 0.388402
\(260\) 25.9425 1.60888
\(261\) −13.0557 −0.808127
\(262\) −7.31053 −0.451646
\(263\) −9.59385 −0.591582 −0.295791 0.955253i \(-0.595583\pi\)
−0.295791 + 0.955253i \(0.595583\pi\)
\(264\) −13.1733 −0.810760
\(265\) −52.6128 −3.23198
\(266\) −2.62917 −0.161205
\(267\) −3.75274 −0.229664
\(268\) 15.0462 0.919091
\(269\) −24.0217 −1.46463 −0.732315 0.680966i \(-0.761560\pi\)
−0.732315 + 0.680966i \(0.761560\pi\)
\(270\) 4.36693 0.265763
\(271\) 25.6421 1.55765 0.778823 0.627243i \(-0.215817\pi\)
0.778823 + 0.627243i \(0.215817\pi\)
\(272\) 7.88851 0.478311
\(273\) −11.3260 −0.685479
\(274\) −12.7625 −0.771013
\(275\) −59.1571 −3.56731
\(276\) −1.40460 −0.0845472
\(277\) −15.9545 −0.958610 −0.479305 0.877648i \(-0.659111\pi\)
−0.479305 + 0.877648i \(0.659111\pi\)
\(278\) −5.16815 −0.309965
\(279\) 6.02684 0.360817
\(280\) −2.88888 −0.172643
\(281\) −9.92019 −0.591789 −0.295894 0.955221i \(-0.595618\pi\)
−0.295894 + 0.955221i \(0.595618\pi\)
\(282\) −4.97429 −0.296215
\(283\) −14.0807 −0.837011 −0.418505 0.908214i \(-0.637446\pi\)
−0.418505 + 0.908214i \(0.637446\pi\)
\(284\) −0.278234 −0.0165102
\(285\) −33.2978 −1.97239
\(286\) −36.8449 −2.17868
\(287\) −3.78062 −0.223163
\(288\) −2.52919 −0.149034
\(289\) 45.2286 2.66051
\(290\) −20.3618 −1.19569
\(291\) 26.2421 1.53834
\(292\) −15.4340 −0.903205
\(293\) −11.4487 −0.668840 −0.334420 0.942424i \(-0.608540\pi\)
−0.334420 + 0.942424i \(0.608540\pi\)
\(294\) −15.1987 −0.886408
\(295\) −57.5925 −3.35317
\(296\) 8.53492 0.496082
\(297\) −6.20215 −0.359885
\(298\) −11.8339 −0.685519
\(299\) −3.92859 −0.227196
\(300\) −24.8298 −1.43355
\(301\) −3.10515 −0.178978
\(302\) −2.37792 −0.136834
\(303\) −37.3763 −2.14721
\(304\) −3.58994 −0.205897
\(305\) −12.4579 −0.713335
\(306\) −19.9515 −1.14055
\(307\) −8.16040 −0.465739 −0.232869 0.972508i \(-0.574811\pi\)
−0.232869 + 0.972508i \(0.574811\pi\)
\(308\) 4.10294 0.233787
\(309\) −10.6407 −0.605325
\(310\) 9.39953 0.533857
\(311\) 7.06502 0.400621 0.200310 0.979733i \(-0.435805\pi\)
0.200310 + 0.979733i \(0.435805\pi\)
\(312\) −15.4648 −0.875521
\(313\) −11.0683 −0.625616 −0.312808 0.949816i \(-0.601270\pi\)
−0.312808 + 0.949816i \(0.601270\pi\)
\(314\) 3.17462 0.179154
\(315\) 7.30651 0.411675
\(316\) −3.94569 −0.221962
\(317\) 14.8709 0.835231 0.417616 0.908624i \(-0.362866\pi\)
0.417616 + 0.908624i \(0.362866\pi\)
\(318\) 31.3635 1.75878
\(319\) 28.9190 1.61915
\(320\) −3.94455 −0.220507
\(321\) 4.72549 0.263751
\(322\) 0.437476 0.0243796
\(323\) −28.3193 −1.57573
\(324\) −10.1908 −0.566154
\(325\) −69.4474 −3.85225
\(326\) −0.771085 −0.0427065
\(327\) −7.98279 −0.441449
\(328\) −5.16216 −0.285033
\(329\) 1.54929 0.0854150
\(330\) 51.9628 2.86046
\(331\) 16.8787 0.927735 0.463868 0.885905i \(-0.346461\pi\)
0.463868 + 0.885905i \(0.346461\pi\)
\(332\) −13.3933 −0.735052
\(333\) −21.5864 −1.18293
\(334\) −4.40690 −0.241135
\(335\) −59.3504 −3.24266
\(336\) 1.72211 0.0939491
\(337\) 26.6008 1.44904 0.724519 0.689255i \(-0.242062\pi\)
0.724519 + 0.689255i \(0.242062\pi\)
\(338\) −30.2540 −1.64560
\(339\) 5.74515 0.312034
\(340\) −31.1166 −1.68754
\(341\) −13.3497 −0.722928
\(342\) 9.07963 0.490970
\(343\) 9.86038 0.532411
\(344\) −4.23985 −0.228598
\(345\) 5.54053 0.298292
\(346\) 9.05042 0.486553
\(347\) 22.9759 1.23341 0.616705 0.787194i \(-0.288467\pi\)
0.616705 + 0.787194i \(0.288467\pi\)
\(348\) 12.1381 0.650669
\(349\) −18.5403 −0.992440 −0.496220 0.868197i \(-0.665279\pi\)
−0.496220 + 0.868197i \(0.665279\pi\)
\(350\) 7.73347 0.413371
\(351\) −7.28101 −0.388632
\(352\) 5.60227 0.298602
\(353\) 22.9134 1.21956 0.609778 0.792572i \(-0.291258\pi\)
0.609778 + 0.792572i \(0.291258\pi\)
\(354\) 34.3320 1.82472
\(355\) 1.09751 0.0582497
\(356\) 1.59595 0.0845850
\(357\) 13.5849 0.718990
\(358\) 8.97070 0.474116
\(359\) −25.6182 −1.35208 −0.676039 0.736866i \(-0.736305\pi\)
−0.676039 + 0.736866i \(0.736305\pi\)
\(360\) 9.97651 0.525808
\(361\) −6.11234 −0.321702
\(362\) 3.91600 0.205820
\(363\) −47.9347 −2.51592
\(364\) 4.81665 0.252461
\(365\) 60.8801 3.18661
\(366\) 7.42637 0.388182
\(367\) −23.9395 −1.24963 −0.624817 0.780771i \(-0.714826\pi\)
−0.624817 + 0.780771i \(0.714826\pi\)
\(368\) 0.597342 0.0311386
\(369\) 13.0561 0.679672
\(370\) −33.6665 −1.75024
\(371\) −9.76843 −0.507152
\(372\) −5.60324 −0.290514
\(373\) −11.4642 −0.593595 −0.296798 0.954940i \(-0.595919\pi\)
−0.296798 + 0.954940i \(0.595919\pi\)
\(374\) 44.1935 2.28519
\(375\) 51.5660 2.66286
\(376\) 2.11544 0.109095
\(377\) 33.9494 1.74848
\(378\) 0.810793 0.0417027
\(379\) −21.4613 −1.10239 −0.551195 0.834376i \(-0.685828\pi\)
−0.551195 + 0.834376i \(0.685828\pi\)
\(380\) 14.1607 0.726429
\(381\) −12.0205 −0.615827
\(382\) −10.4873 −0.536578
\(383\) −26.2934 −1.34353 −0.671764 0.740765i \(-0.734463\pi\)
−0.671764 + 0.740765i \(0.734463\pi\)
\(384\) 2.35142 0.119996
\(385\) −16.1843 −0.824826
\(386\) 22.1435 1.12708
\(387\) 10.7234 0.545100
\(388\) −11.1601 −0.566569
\(389\) −11.3139 −0.573638 −0.286819 0.957985i \(-0.592598\pi\)
−0.286819 + 0.957985i \(0.592598\pi\)
\(390\) 61.0017 3.08894
\(391\) 4.71214 0.238303
\(392\) 6.46363 0.326463
\(393\) −17.1901 −0.867128
\(394\) −6.89885 −0.347559
\(395\) 15.5640 0.783109
\(396\) −14.1692 −0.712028
\(397\) −7.68422 −0.385660 −0.192830 0.981232i \(-0.561767\pi\)
−0.192830 + 0.981232i \(0.561767\pi\)
\(398\) 7.26553 0.364188
\(399\) −6.18229 −0.309501
\(400\) 10.5595 0.527975
\(401\) −1.72129 −0.0859573 −0.0429786 0.999076i \(-0.513685\pi\)
−0.0429786 + 0.999076i \(0.513685\pi\)
\(402\) 35.3799 1.76459
\(403\) −15.6719 −0.780673
\(404\) 15.8952 0.790815
\(405\) 40.1980 1.99746
\(406\) −3.78051 −0.187623
\(407\) 47.8149 2.37010
\(408\) 18.5492 0.918323
\(409\) 11.0519 0.546482 0.273241 0.961946i \(-0.411904\pi\)
0.273241 + 0.961946i \(0.411904\pi\)
\(410\) 20.3624 1.00563
\(411\) −30.0101 −1.48029
\(412\) 4.52520 0.222940
\(413\) −10.6930 −0.526168
\(414\) −1.51079 −0.0742513
\(415\) 52.8305 2.59335
\(416\) 6.57678 0.322453
\(417\) −12.1525 −0.595111
\(418\) −20.1118 −0.983700
\(419\) −14.5494 −0.710783 −0.355391 0.934718i \(-0.615652\pi\)
−0.355391 + 0.934718i \(0.615652\pi\)
\(420\) −6.79297 −0.331463
\(421\) −3.33663 −0.162617 −0.0813087 0.996689i \(-0.525910\pi\)
−0.0813087 + 0.996689i \(0.525910\pi\)
\(422\) −15.9296 −0.775442
\(423\) −5.35035 −0.260143
\(424\) −13.3381 −0.647755
\(425\) 83.2987 4.04058
\(426\) −0.654246 −0.0316983
\(427\) −2.31301 −0.111934
\(428\) −2.00963 −0.0971392
\(429\) −86.6379 −4.18292
\(430\) 16.7243 0.806518
\(431\) 19.7302 0.950372 0.475186 0.879885i \(-0.342381\pi\)
0.475186 + 0.879885i \(0.342381\pi\)
\(432\) 1.10708 0.0532643
\(433\) −17.5306 −0.842465 −0.421232 0.906953i \(-0.638402\pi\)
−0.421232 + 0.906953i \(0.638402\pi\)
\(434\) 1.74518 0.0837713
\(435\) −47.8792 −2.29563
\(436\) 3.39488 0.162585
\(437\) −2.14442 −0.102582
\(438\) −36.2918 −1.73409
\(439\) 33.2764 1.58820 0.794099 0.607789i \(-0.207943\pi\)
0.794099 + 0.607789i \(0.207943\pi\)
\(440\) −22.0984 −1.05350
\(441\) −16.3477 −0.778464
\(442\) 51.8810 2.46773
\(443\) 11.2269 0.533408 0.266704 0.963778i \(-0.414065\pi\)
0.266704 + 0.963778i \(0.414065\pi\)
\(444\) 20.0692 0.952443
\(445\) −6.29529 −0.298426
\(446\) 27.6617 1.30982
\(447\) −27.8265 −1.31615
\(448\) −0.732372 −0.0346013
\(449\) 37.8756 1.78746 0.893730 0.448606i \(-0.148079\pi\)
0.893730 + 0.448606i \(0.148079\pi\)
\(450\) −26.7069 −1.25898
\(451\) −28.9198 −1.36178
\(452\) −2.44327 −0.114922
\(453\) −5.59149 −0.262711
\(454\) 3.93286 0.184578
\(455\) −18.9995 −0.890711
\(456\) −8.44146 −0.395308
\(457\) −8.93795 −0.418100 −0.209050 0.977905i \(-0.567037\pi\)
−0.209050 + 0.977905i \(0.567037\pi\)
\(458\) 8.11720 0.379292
\(459\) 8.73320 0.407631
\(460\) −2.35625 −0.109861
\(461\) −9.44331 −0.439819 −0.219909 0.975520i \(-0.570576\pi\)
−0.219909 + 0.975520i \(0.570576\pi\)
\(462\) 9.64775 0.448854
\(463\) −2.79255 −0.129781 −0.0648903 0.997892i \(-0.520670\pi\)
−0.0648903 + 0.997892i \(0.520670\pi\)
\(464\) −5.16201 −0.239640
\(465\) 22.1023 1.02497
\(466\) 5.35936 0.248268
\(467\) 20.4023 0.944104 0.472052 0.881571i \(-0.343513\pi\)
0.472052 + 0.881571i \(0.343513\pi\)
\(468\) −16.6339 −0.768903
\(469\) −11.0194 −0.508828
\(470\) −8.34446 −0.384901
\(471\) 7.46487 0.343963
\(472\) −14.6005 −0.672043
\(473\) −23.7528 −1.09215
\(474\) −9.27798 −0.426152
\(475\) −37.9079 −1.73933
\(476\) −5.77732 −0.264803
\(477\) 33.7345 1.54460
\(478\) 19.4979 0.891813
\(479\) 35.9555 1.64285 0.821425 0.570317i \(-0.193180\pi\)
0.821425 + 0.570317i \(0.193180\pi\)
\(480\) −9.27531 −0.423358
\(481\) 56.1323 2.55941
\(482\) −0.671107 −0.0305681
\(483\) 1.02869 0.0468071
\(484\) 20.3854 0.926609
\(485\) 44.0216 1.99892
\(486\) −20.6416 −0.936321
\(487\) −4.43948 −0.201172 −0.100586 0.994928i \(-0.532072\pi\)
−0.100586 + 0.994928i \(0.532072\pi\)
\(488\) −3.15824 −0.142967
\(489\) −1.81315 −0.0819934
\(490\) −25.4961 −1.15180
\(491\) 32.9313 1.48617 0.743085 0.669198i \(-0.233362\pi\)
0.743085 + 0.669198i \(0.233362\pi\)
\(492\) −12.1384 −0.547242
\(493\) −40.7206 −1.83396
\(494\) −23.6102 −1.06227
\(495\) 55.8911 2.51212
\(496\) 2.38291 0.106996
\(497\) 0.203771 0.00914037
\(498\) −31.4933 −1.41125
\(499\) 13.2766 0.594342 0.297171 0.954824i \(-0.403957\pi\)
0.297171 + 0.954824i \(0.403957\pi\)
\(500\) −21.9297 −0.980726
\(501\) −10.3625 −0.462962
\(502\) −24.4575 −1.09159
\(503\) −31.0346 −1.38377 −0.691883 0.722010i \(-0.743218\pi\)
−0.691883 + 0.722010i \(0.743218\pi\)
\(504\) 1.85231 0.0825082
\(505\) −62.6994 −2.79009
\(506\) 3.34647 0.148769
\(507\) −71.1400 −3.15944
\(508\) 5.11200 0.226808
\(509\) −12.9425 −0.573668 −0.286834 0.957980i \(-0.592603\pi\)
−0.286834 + 0.957980i \(0.592603\pi\)
\(510\) −73.1684 −3.23995
\(511\) 11.3034 0.500033
\(512\) −1.00000 −0.0441942
\(513\) −3.97434 −0.175472
\(514\) −1.94806 −0.0859251
\(515\) −17.8499 −0.786560
\(516\) −9.96968 −0.438891
\(517\) 11.8513 0.521218
\(518\) −6.25074 −0.274642
\(519\) 21.2814 0.934148
\(520\) −25.9425 −1.13765
\(521\) −2.99380 −0.131161 −0.0655803 0.997847i \(-0.520890\pi\)
−0.0655803 + 0.997847i \(0.520890\pi\)
\(522\) 13.0557 0.571432
\(523\) 33.6906 1.47319 0.736593 0.676336i \(-0.236433\pi\)
0.736593 + 0.676336i \(0.236433\pi\)
\(524\) 7.31053 0.319362
\(525\) 18.1847 0.793644
\(526\) 9.59385 0.418312
\(527\) 18.7976 0.818838
\(528\) 13.1733 0.573294
\(529\) −22.6432 −0.984486
\(530\) 52.6128 2.28535
\(531\) 36.9275 1.60252
\(532\) 2.62917 0.113989
\(533\) −33.9504 −1.47055
\(534\) 3.75274 0.162397
\(535\) 7.92710 0.342718
\(536\) −15.0462 −0.649896
\(537\) 21.0939 0.910269
\(538\) 24.0217 1.03565
\(539\) 36.2110 1.55972
\(540\) −4.36693 −0.187923
\(541\) 2.49802 0.107398 0.0536991 0.998557i \(-0.482899\pi\)
0.0536991 + 0.998557i \(0.482899\pi\)
\(542\) −25.6421 −1.10142
\(543\) 9.20817 0.395160
\(544\) −7.88851 −0.338217
\(545\) −13.3913 −0.573619
\(546\) 11.3260 0.484707
\(547\) 25.4749 1.08923 0.544615 0.838686i \(-0.316676\pi\)
0.544615 + 0.838686i \(0.316676\pi\)
\(548\) 12.7625 0.545189
\(549\) 7.98779 0.340911
\(550\) 59.1571 2.52247
\(551\) 18.5313 0.789460
\(552\) 1.40460 0.0597839
\(553\) 2.88971 0.122883
\(554\) 15.9545 0.677840
\(555\) −79.1641 −3.36033
\(556\) 5.16815 0.219179
\(557\) 29.6288 1.25541 0.627707 0.778450i \(-0.283994\pi\)
0.627707 + 0.778450i \(0.283994\pi\)
\(558\) −6.02684 −0.255136
\(559\) −27.8846 −1.17939
\(560\) 2.88888 0.122077
\(561\) 103.918 4.38741
\(562\) 9.92019 0.418458
\(563\) 3.05150 0.128606 0.0643028 0.997930i \(-0.479518\pi\)
0.0643028 + 0.997930i \(0.479518\pi\)
\(564\) 4.97429 0.209456
\(565\) 9.63760 0.405457
\(566\) 14.0807 0.591856
\(567\) 7.46343 0.313435
\(568\) 0.278234 0.0116744
\(569\) −34.2607 −1.43628 −0.718142 0.695897i \(-0.755007\pi\)
−0.718142 + 0.695897i \(0.755007\pi\)
\(570\) 33.2978 1.39469
\(571\) 1.81874 0.0761117 0.0380559 0.999276i \(-0.487884\pi\)
0.0380559 + 0.999276i \(0.487884\pi\)
\(572\) 36.8449 1.54056
\(573\) −24.6601 −1.03019
\(574\) 3.78062 0.157800
\(575\) 6.30763 0.263046
\(576\) 2.52919 0.105383
\(577\) −37.1690 −1.54736 −0.773682 0.633574i \(-0.781587\pi\)
−0.773682 + 0.633574i \(0.781587\pi\)
\(578\) −45.2286 −1.88126
\(579\) 52.0688 2.16391
\(580\) 20.3618 0.845478
\(581\) 9.80886 0.406940
\(582\) −26.2421 −1.08777
\(583\) −74.7235 −3.09473
\(584\) 15.4340 0.638662
\(585\) 65.6133 2.71278
\(586\) 11.4487 0.472942
\(587\) −6.79243 −0.280354 −0.140177 0.990126i \(-0.544767\pi\)
−0.140177 + 0.990126i \(0.544767\pi\)
\(588\) 15.1987 0.626785
\(589\) −8.55452 −0.352483
\(590\) 57.5925 2.37105
\(591\) −16.2221 −0.667289
\(592\) −8.53492 −0.350783
\(593\) −13.8884 −0.570329 −0.285164 0.958479i \(-0.592048\pi\)
−0.285164 + 0.958479i \(0.592048\pi\)
\(594\) 6.20215 0.254477
\(595\) 22.7889 0.934255
\(596\) 11.8339 0.484735
\(597\) 17.0843 0.699215
\(598\) 3.92859 0.160652
\(599\) 8.75043 0.357533 0.178767 0.983892i \(-0.442789\pi\)
0.178767 + 0.983892i \(0.442789\pi\)
\(600\) 24.8298 1.01367
\(601\) 9.32602 0.380416 0.190208 0.981744i \(-0.439084\pi\)
0.190208 + 0.981744i \(0.439084\pi\)
\(602\) 3.10515 0.126556
\(603\) 38.0546 1.54970
\(604\) 2.37792 0.0967561
\(605\) −80.4113 −3.26918
\(606\) 37.3763 1.51831
\(607\) −47.1463 −1.91361 −0.956804 0.290732i \(-0.906101\pi\)
−0.956804 + 0.290732i \(0.906101\pi\)
\(608\) 3.58994 0.145591
\(609\) −8.88957 −0.360224
\(610\) 12.4579 0.504404
\(611\) 13.9128 0.562851
\(612\) 19.9515 0.806493
\(613\) −24.0231 −0.970285 −0.485143 0.874435i \(-0.661232\pi\)
−0.485143 + 0.874435i \(0.661232\pi\)
\(614\) 8.16040 0.329327
\(615\) 47.8806 1.93073
\(616\) −4.10294 −0.165312
\(617\) 32.1413 1.29396 0.646981 0.762507i \(-0.276031\pi\)
0.646981 + 0.762507i \(0.276031\pi\)
\(618\) 10.6407 0.428030
\(619\) 4.93499 0.198354 0.0991770 0.995070i \(-0.468379\pi\)
0.0991770 + 0.995070i \(0.468379\pi\)
\(620\) −9.39953 −0.377494
\(621\) 0.661305 0.0265372
\(622\) −7.06502 −0.283282
\(623\) −1.16883 −0.0468280
\(624\) 15.4648 0.619087
\(625\) 33.7054 1.34822
\(626\) 11.0683 0.442377
\(627\) −47.2913 −1.88863
\(628\) −3.17462 −0.126681
\(629\) −67.3278 −2.68454
\(630\) −7.30651 −0.291098
\(631\) 35.8146 1.42576 0.712878 0.701288i \(-0.247391\pi\)
0.712878 + 0.701288i \(0.247391\pi\)
\(632\) 3.94569 0.156951
\(633\) −37.4573 −1.48879
\(634\) −14.8709 −0.590598
\(635\) −20.1645 −0.800205
\(636\) −31.3635 −1.24364
\(637\) 42.5099 1.68430
\(638\) −28.9190 −1.14491
\(639\) −0.703706 −0.0278382
\(640\) 3.94455 0.155922
\(641\) 19.2972 0.762192 0.381096 0.924535i \(-0.375547\pi\)
0.381096 + 0.924535i \(0.375547\pi\)
\(642\) −4.72549 −0.186500
\(643\) 49.8612 1.96633 0.983166 0.182713i \(-0.0584880\pi\)
0.983166 + 0.182713i \(0.0584880\pi\)
\(644\) −0.437476 −0.0172390
\(645\) 39.3259 1.54846
\(646\) 28.3193 1.11421
\(647\) −2.26143 −0.0889060 −0.0444530 0.999011i \(-0.514154\pi\)
−0.0444530 + 0.999011i \(0.514154\pi\)
\(648\) 10.1908 0.400331
\(649\) −81.7960 −3.21077
\(650\) 69.4474 2.72395
\(651\) 4.10365 0.160835
\(652\) 0.771085 0.0301980
\(653\) −47.5403 −1.86040 −0.930198 0.367059i \(-0.880365\pi\)
−0.930198 + 0.367059i \(0.880365\pi\)
\(654\) 7.98279 0.312152
\(655\) −28.8368 −1.12675
\(656\) 5.16216 0.201548
\(657\) −39.0354 −1.52292
\(658\) −1.54929 −0.0603975
\(659\) 27.6405 1.07672 0.538361 0.842714i \(-0.319044\pi\)
0.538361 + 0.842714i \(0.319044\pi\)
\(660\) −51.9628 −2.02265
\(661\) 43.9554 1.70967 0.854834 0.518902i \(-0.173659\pi\)
0.854834 + 0.518902i \(0.173659\pi\)
\(662\) −16.8787 −0.656008
\(663\) 121.994 4.73786
\(664\) 13.3933 0.519761
\(665\) −10.3709 −0.402166
\(666\) 21.5864 0.836457
\(667\) −3.08349 −0.119393
\(668\) 4.40690 0.170508
\(669\) 65.0443 2.51476
\(670\) 59.3504 2.29291
\(671\) −17.6933 −0.683043
\(672\) −1.72211 −0.0664320
\(673\) 20.3098 0.782885 0.391442 0.920203i \(-0.371976\pi\)
0.391442 + 0.920203i \(0.371976\pi\)
\(674\) −26.6008 −1.02462
\(675\) 11.6902 0.449955
\(676\) 30.2540 1.16362
\(677\) 3.67264 0.141151 0.0705755 0.997506i \(-0.477516\pi\)
0.0705755 + 0.997506i \(0.477516\pi\)
\(678\) −5.74515 −0.220641
\(679\) 8.17335 0.313664
\(680\) 31.1166 1.19327
\(681\) 9.24782 0.354377
\(682\) 13.3497 0.511187
\(683\) −6.18538 −0.236677 −0.118339 0.992973i \(-0.537757\pi\)
−0.118339 + 0.992973i \(0.537757\pi\)
\(684\) −9.07963 −0.347168
\(685\) −50.3425 −1.92349
\(686\) −9.86038 −0.376471
\(687\) 19.0870 0.728213
\(688\) 4.23985 0.161643
\(689\) −87.7216 −3.34193
\(690\) −5.54053 −0.210924
\(691\) −28.0612 −1.06750 −0.533749 0.845643i \(-0.679217\pi\)
−0.533749 + 0.845643i \(0.679217\pi\)
\(692\) −9.05042 −0.344045
\(693\) 10.3771 0.394194
\(694\) −22.9759 −0.872153
\(695\) −20.3860 −0.773287
\(696\) −12.1381 −0.460092
\(697\) 40.7218 1.54245
\(698\) 18.5403 0.701761
\(699\) 12.6021 0.476656
\(700\) −7.73347 −0.292298
\(701\) 44.9400 1.69736 0.848681 0.528906i \(-0.177397\pi\)
0.848681 + 0.528906i \(0.177397\pi\)
\(702\) 7.28101 0.274804
\(703\) 30.6398 1.15560
\(704\) −5.60227 −0.211143
\(705\) −19.6214 −0.738983
\(706\) −22.9134 −0.862357
\(707\) −11.6412 −0.437812
\(708\) −34.3320 −1.29028
\(709\) −37.3761 −1.40369 −0.701845 0.712330i \(-0.747640\pi\)
−0.701845 + 0.712330i \(0.747640\pi\)
\(710\) −1.09751 −0.0411888
\(711\) −9.97939 −0.374256
\(712\) −1.59595 −0.0598106
\(713\) 1.42342 0.0533073
\(714\) −13.5849 −0.508403
\(715\) −145.337 −5.43528
\(716\) −8.97070 −0.335251
\(717\) 45.8478 1.71222
\(718\) 25.6182 0.956064
\(719\) 38.1675 1.42341 0.711704 0.702479i \(-0.247924\pi\)
0.711704 + 0.702479i \(0.247924\pi\)
\(720\) −9.97651 −0.371803
\(721\) −3.31413 −0.123424
\(722\) 6.11234 0.227478
\(723\) −1.57806 −0.0586886
\(724\) −3.91600 −0.145537
\(725\) −54.5082 −2.02438
\(726\) 47.9347 1.77902
\(727\) 12.9750 0.481216 0.240608 0.970622i \(-0.422653\pi\)
0.240608 + 0.970622i \(0.422653\pi\)
\(728\) −4.81665 −0.178517
\(729\) −17.9647 −0.665361
\(730\) −60.8801 −2.25327
\(731\) 33.4461 1.23705
\(732\) −7.42637 −0.274486
\(733\) 39.9082 1.47404 0.737022 0.675869i \(-0.236232\pi\)
0.737022 + 0.675869i \(0.236232\pi\)
\(734\) 23.9395 0.883625
\(735\) −59.9522 −2.21137
\(736\) −0.597342 −0.0220183
\(737\) −84.2927 −3.10496
\(738\) −13.0561 −0.480601
\(739\) −4.55175 −0.167439 −0.0837194 0.996489i \(-0.526680\pi\)
−0.0837194 + 0.996489i \(0.526680\pi\)
\(740\) 33.6665 1.23760
\(741\) −55.5176 −2.03949
\(742\) 9.76843 0.358610
\(743\) 34.2614 1.25693 0.628464 0.777839i \(-0.283684\pi\)
0.628464 + 0.777839i \(0.283684\pi\)
\(744\) 5.60324 0.205425
\(745\) −46.6794 −1.71020
\(746\) 11.4642 0.419735
\(747\) −33.8741 −1.23939
\(748\) −44.1935 −1.61588
\(749\) 1.47180 0.0537783
\(750\) −51.5660 −1.88292
\(751\) 32.0113 1.16811 0.584054 0.811715i \(-0.301466\pi\)
0.584054 + 0.811715i \(0.301466\pi\)
\(752\) −2.11544 −0.0771422
\(753\) −57.5099 −2.09578
\(754\) −33.9494 −1.23636
\(755\) −9.37982 −0.341366
\(756\) −0.810793 −0.0294882
\(757\) −8.45348 −0.307247 −0.153624 0.988129i \(-0.549094\pi\)
−0.153624 + 0.988129i \(0.549094\pi\)
\(758\) 21.4613 0.779508
\(759\) 7.86897 0.285625
\(760\) −14.1607 −0.513663
\(761\) 36.8514 1.33586 0.667932 0.744223i \(-0.267180\pi\)
0.667932 + 0.744223i \(0.267180\pi\)
\(762\) 12.0205 0.435455
\(763\) −2.48631 −0.0900105
\(764\) 10.4873 0.379418
\(765\) −78.6998 −2.84540
\(766\) 26.2934 0.950018
\(767\) −96.0244 −3.46724
\(768\) −2.35142 −0.0848497
\(769\) 16.7762 0.604965 0.302483 0.953155i \(-0.402185\pi\)
0.302483 + 0.953155i \(0.402185\pi\)
\(770\) 16.1843 0.583240
\(771\) −4.58070 −0.164970
\(772\) −22.1435 −0.796963
\(773\) −7.81619 −0.281129 −0.140564 0.990072i \(-0.544892\pi\)
−0.140564 + 0.990072i \(0.544892\pi\)
\(774\) −10.7234 −0.385444
\(775\) 25.1624 0.903859
\(776\) 11.1601 0.400625
\(777\) −14.6981 −0.527292
\(778\) 11.3139 0.405623
\(779\) −18.5318 −0.663972
\(780\) −61.0017 −2.18421
\(781\) 1.55874 0.0557762
\(782\) −4.71214 −0.168506
\(783\) −5.71475 −0.204228
\(784\) −6.46363 −0.230844
\(785\) 12.5225 0.446946
\(786\) 17.1901 0.613152
\(787\) 20.7337 0.739077 0.369538 0.929216i \(-0.379516\pi\)
0.369538 + 0.929216i \(0.379516\pi\)
\(788\) 6.89885 0.245761
\(789\) 22.5592 0.803129
\(790\) −15.5640 −0.553741
\(791\) 1.78938 0.0636230
\(792\) 14.1692 0.503480
\(793\) −20.7711 −0.737603
\(794\) 7.68422 0.272703
\(795\) 123.715 4.38771
\(796\) −7.26553 −0.257520
\(797\) 15.1196 0.535563 0.267782 0.963480i \(-0.413709\pi\)
0.267782 + 0.963480i \(0.413709\pi\)
\(798\) 6.18229 0.218851
\(799\) −16.6877 −0.590367
\(800\) −10.5595 −0.373334
\(801\) 4.03645 0.142621
\(802\) 1.72129 0.0607810
\(803\) 86.4653 3.05129
\(804\) −35.3799 −1.24775
\(805\) 1.72565 0.0608211
\(806\) 15.6719 0.552019
\(807\) 56.4852 1.98837
\(808\) −15.8952 −0.559191
\(809\) −19.0803 −0.670828 −0.335414 0.942071i \(-0.608876\pi\)
−0.335414 + 0.942071i \(0.608876\pi\)
\(810\) −40.1980 −1.41241
\(811\) 1.02027 0.0358264 0.0179132 0.999840i \(-0.494298\pi\)
0.0179132 + 0.999840i \(0.494298\pi\)
\(812\) 3.78051 0.132670
\(813\) −60.2954 −2.11465
\(814\) −47.8149 −1.67591
\(815\) −3.04159 −0.106542
\(816\) −18.5492 −0.649353
\(817\) −15.2208 −0.532509
\(818\) −11.0519 −0.386421
\(819\) 12.1822 0.425681
\(820\) −20.3624 −0.711086
\(821\) 17.6393 0.615617 0.307809 0.951448i \(-0.400404\pi\)
0.307809 + 0.951448i \(0.400404\pi\)
\(822\) 30.0101 1.04672
\(823\) 22.0045 0.767028 0.383514 0.923535i \(-0.374714\pi\)
0.383514 + 0.923535i \(0.374714\pi\)
\(824\) −4.52520 −0.157643
\(825\) 139.103 4.84296
\(826\) 10.6930 0.372057
\(827\) 20.8310 0.724364 0.362182 0.932107i \(-0.382032\pi\)
0.362182 + 0.932107i \(0.382032\pi\)
\(828\) 1.51079 0.0525036
\(829\) 6.45825 0.224304 0.112152 0.993691i \(-0.464226\pi\)
0.112152 + 0.993691i \(0.464226\pi\)
\(830\) −52.8305 −1.83377
\(831\) 37.5157 1.30140
\(832\) −6.57678 −0.228009
\(833\) −50.9884 −1.76664
\(834\) 12.1525 0.420807
\(835\) −17.3832 −0.601572
\(836\) 20.1118 0.695581
\(837\) 2.63807 0.0911851
\(838\) 14.5494 0.502599
\(839\) 15.3261 0.529115 0.264558 0.964370i \(-0.414774\pi\)
0.264558 + 0.964370i \(0.414774\pi\)
\(840\) 6.79297 0.234380
\(841\) −2.35366 −0.0811606
\(842\) 3.33663 0.114988
\(843\) 23.3266 0.803409
\(844\) 15.9296 0.548320
\(845\) −119.339 −4.10537
\(846\) 5.35035 0.183949
\(847\) −14.9297 −0.512990
\(848\) 13.3381 0.458032
\(849\) 33.1097 1.13632
\(850\) −83.2987 −2.85712
\(851\) −5.09827 −0.174766
\(852\) 0.654246 0.0224141
\(853\) −3.33584 −0.114217 −0.0571086 0.998368i \(-0.518188\pi\)
−0.0571086 + 0.998368i \(0.518188\pi\)
\(854\) 2.31301 0.0791495
\(855\) 35.8151 1.22485
\(856\) 2.00963 0.0686878
\(857\) −4.55116 −0.155465 −0.0777324 0.996974i \(-0.524768\pi\)
−0.0777324 + 0.996974i \(0.524768\pi\)
\(858\) 86.6379 2.95777
\(859\) 11.6336 0.396933 0.198467 0.980108i \(-0.436404\pi\)
0.198467 + 0.980108i \(0.436404\pi\)
\(860\) −16.7243 −0.570295
\(861\) 8.88983 0.302965
\(862\) −19.7302 −0.672014
\(863\) 24.9565 0.849529 0.424765 0.905304i \(-0.360357\pi\)
0.424765 + 0.905304i \(0.360357\pi\)
\(864\) −1.10708 −0.0376636
\(865\) 35.6998 1.21383
\(866\) 17.5306 0.595713
\(867\) −106.352 −3.61189
\(868\) −1.74518 −0.0592352
\(869\) 22.1048 0.749854
\(870\) 47.8792 1.62326
\(871\) −98.9553 −3.35297
\(872\) −3.39488 −0.114965
\(873\) −28.2260 −0.955306
\(874\) 2.14442 0.0725361
\(875\) 16.0607 0.542950
\(876\) 36.2918 1.22619
\(877\) 53.4352 1.80438 0.902189 0.431342i \(-0.141960\pi\)
0.902189 + 0.431342i \(0.141960\pi\)
\(878\) −33.2764 −1.12303
\(879\) 26.9207 0.908014
\(880\) 22.0984 0.744938
\(881\) −32.1974 −1.08476 −0.542378 0.840134i \(-0.682476\pi\)
−0.542378 + 0.840134i \(0.682476\pi\)
\(882\) 16.3477 0.550457
\(883\) 8.65897 0.291398 0.145699 0.989329i \(-0.453457\pi\)
0.145699 + 0.989329i \(0.453457\pi\)
\(884\) −51.8810 −1.74495
\(885\) 135.424 4.55224
\(886\) −11.2269 −0.377177
\(887\) 39.5447 1.32778 0.663890 0.747830i \(-0.268904\pi\)
0.663890 + 0.747830i \(0.268904\pi\)
\(888\) −20.0692 −0.673479
\(889\) −3.74388 −0.125566
\(890\) 6.29529 0.211019
\(891\) 57.0914 1.91263
\(892\) −27.6617 −0.926182
\(893\) 7.59430 0.254134
\(894\) 27.8265 0.930657
\(895\) 35.3854 1.18280
\(896\) 0.732372 0.0244668
\(897\) 9.23777 0.308440
\(898\) −37.8756 −1.26392
\(899\) −12.3006 −0.410249
\(900\) 26.7069 0.890231
\(901\) 105.218 3.50531
\(902\) 28.9198 0.962924
\(903\) 7.30151 0.242979
\(904\) 2.44327 0.0812619
\(905\) 15.4469 0.513471
\(906\) 5.59149 0.185765
\(907\) −37.6257 −1.24934 −0.624671 0.780888i \(-0.714767\pi\)
−0.624671 + 0.780888i \(0.714767\pi\)
\(908\) −3.93286 −0.130517
\(909\) 40.2019 1.33341
\(910\) 18.9995 0.629828
\(911\) −25.5304 −0.845859 −0.422930 0.906162i \(-0.638998\pi\)
−0.422930 + 0.906162i \(0.638998\pi\)
\(912\) 8.44146 0.279525
\(913\) 75.0328 2.48322
\(914\) 8.93795 0.295641
\(915\) 29.2937 0.968420
\(916\) −8.11720 −0.268200
\(917\) −5.35403 −0.176805
\(918\) −8.73320 −0.288239
\(919\) 24.4882 0.807792 0.403896 0.914805i \(-0.367656\pi\)
0.403896 + 0.914805i \(0.367656\pi\)
\(920\) 2.35625 0.0776832
\(921\) 19.1885 0.632284
\(922\) 9.44331 0.310999
\(923\) 1.82988 0.0602314
\(924\) −9.64775 −0.317388
\(925\) −90.1245 −2.96327
\(926\) 2.79255 0.0917687
\(927\) 11.4451 0.375906
\(928\) 5.16201 0.169451
\(929\) −41.9376 −1.37593 −0.687965 0.725744i \(-0.741496\pi\)
−0.687965 + 0.725744i \(0.741496\pi\)
\(930\) −22.1023 −0.724762
\(931\) 23.2040 0.760482
\(932\) −5.35936 −0.175552
\(933\) −16.6129 −0.543880
\(934\) −20.4023 −0.667583
\(935\) 174.324 5.70100
\(936\) 16.6339 0.543696
\(937\) 17.2560 0.563730 0.281865 0.959454i \(-0.409047\pi\)
0.281865 + 0.959454i \(0.409047\pi\)
\(938\) 11.0194 0.359796
\(939\) 26.0262 0.849333
\(940\) 8.34446 0.272166
\(941\) −16.3442 −0.532805 −0.266402 0.963862i \(-0.585835\pi\)
−0.266402 + 0.963862i \(0.585835\pi\)
\(942\) −7.46487 −0.243219
\(943\) 3.08358 0.100415
\(944\) 14.6005 0.475206
\(945\) 3.19821 0.104038
\(946\) 23.7528 0.772270
\(947\) −12.5846 −0.408946 −0.204473 0.978872i \(-0.565548\pi\)
−0.204473 + 0.978872i \(0.565548\pi\)
\(948\) 9.27798 0.301335
\(949\) 101.506 3.29502
\(950\) 37.9079 1.22990
\(951\) −34.9677 −1.13391
\(952\) 5.77732 0.187244
\(953\) −32.6168 −1.05656 −0.528281 0.849069i \(-0.677163\pi\)
−0.528281 + 0.849069i \(0.677163\pi\)
\(954\) −33.7345 −1.09220
\(955\) −41.3678 −1.33863
\(956\) −19.4979 −0.630607
\(957\) −68.0007 −2.19815
\(958\) −35.9555 −1.16167
\(959\) −9.34692 −0.301828
\(960\) 9.27531 0.299359
\(961\) −25.3217 −0.816830
\(962\) −56.1323 −1.80978
\(963\) −5.08274 −0.163789
\(964\) 0.671107 0.0216149
\(965\) 87.3463 2.81178
\(966\) −1.02869 −0.0330976
\(967\) −17.1058 −0.550087 −0.275043 0.961432i \(-0.588692\pi\)
−0.275043 + 0.961432i \(0.588692\pi\)
\(968\) −20.3854 −0.655211
\(969\) 66.5906 2.13920
\(970\) −44.0216 −1.41345
\(971\) −46.2310 −1.48362 −0.741811 0.670609i \(-0.766033\pi\)
−0.741811 + 0.670609i \(0.766033\pi\)
\(972\) 20.6416 0.662079
\(973\) −3.78501 −0.121342
\(974\) 4.43948 0.142250
\(975\) 163.300 5.22979
\(976\) 3.15824 0.101093
\(977\) −49.2388 −1.57529 −0.787645 0.616129i \(-0.788700\pi\)
−0.787645 + 0.616129i \(0.788700\pi\)
\(978\) 1.81315 0.0579781
\(979\) −8.94092 −0.285753
\(980\) 25.4961 0.814444
\(981\) 8.58628 0.274139
\(982\) −32.9313 −1.05088
\(983\) 26.3514 0.840479 0.420240 0.907413i \(-0.361946\pi\)
0.420240 + 0.907413i \(0.361946\pi\)
\(984\) 12.1384 0.386959
\(985\) −27.2129 −0.867074
\(986\) 40.7206 1.29681
\(987\) −3.64303 −0.115959
\(988\) 23.6102 0.751142
\(989\) 2.53264 0.0805333
\(990\) −55.8911 −1.77634
\(991\) 52.6326 1.67193 0.835964 0.548784i \(-0.184909\pi\)
0.835964 + 0.548784i \(0.184909\pi\)
\(992\) −2.38291 −0.0756576
\(993\) −39.6889 −1.25949
\(994\) −0.203771 −0.00646321
\(995\) 28.6593 0.908560
\(996\) 31.4933 0.997903
\(997\) −24.0893 −0.762917 −0.381459 0.924386i \(-0.624578\pi\)
−0.381459 + 0.924386i \(0.624578\pi\)
\(998\) −13.2766 −0.420264
\(999\) −9.44883 −0.298948
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 4022.2.a.d.1.7 37
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4022.2.a.d.1.7 37 1.1 even 1 trivial