Properties

Label 8002.2.a.e.1.37
Level $8002$
Weight $2$
Character 8002.1
Self dual yes
Analytic conductor $63.896$
Analytic rank $0$
Dimension $77$
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] = [8002,2,Mod(1,8002)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8002, 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("8002.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8002 = 2 \cdot 4001 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8002.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: \(63.8962916974\)
Analytic rank: \(0\)
Dimension: \(77\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.37
Character \(\chi\) \(=\) 8002.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -0.169015 q^{3} +1.00000 q^{4} -0.814742 q^{5} +0.169015 q^{6} +3.07096 q^{7} -1.00000 q^{8} -2.97143 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -0.169015 q^{3} +1.00000 q^{4} -0.814742 q^{5} +0.169015 q^{6} +3.07096 q^{7} -1.00000 q^{8} -2.97143 q^{9} +0.814742 q^{10} -2.65913 q^{11} -0.169015 q^{12} -3.78538 q^{13} -3.07096 q^{14} +0.137704 q^{15} +1.00000 q^{16} -2.48508 q^{17} +2.97143 q^{18} -7.18899 q^{19} -0.814742 q^{20} -0.519040 q^{21} +2.65913 q^{22} +3.37191 q^{23} +0.169015 q^{24} -4.33620 q^{25} +3.78538 q^{26} +1.00926 q^{27} +3.07096 q^{28} -5.80122 q^{29} -0.137704 q^{30} +4.83287 q^{31} -1.00000 q^{32} +0.449434 q^{33} +2.48508 q^{34} -2.50204 q^{35} -2.97143 q^{36} -7.94827 q^{37} +7.18899 q^{38} +0.639787 q^{39} +0.814742 q^{40} -4.40611 q^{41} +0.519040 q^{42} -2.92068 q^{43} -2.65913 q^{44} +2.42095 q^{45} -3.37191 q^{46} +0.0383619 q^{47} -0.169015 q^{48} +2.43081 q^{49} +4.33620 q^{50} +0.420017 q^{51} -3.78538 q^{52} +8.11722 q^{53} -1.00926 q^{54} +2.16650 q^{55} -3.07096 q^{56} +1.21505 q^{57} +5.80122 q^{58} -6.63420 q^{59} +0.137704 q^{60} +13.5883 q^{61} -4.83287 q^{62} -9.12516 q^{63} +1.00000 q^{64} +3.08410 q^{65} -0.449434 q^{66} -10.0202 q^{67} -2.48508 q^{68} -0.569904 q^{69} +2.50204 q^{70} +9.18773 q^{71} +2.97143 q^{72} -4.01614 q^{73} +7.94827 q^{74} +0.732884 q^{75} -7.18899 q^{76} -8.16609 q^{77} -0.639787 q^{78} -2.45176 q^{79} -0.814742 q^{80} +8.74372 q^{81} +4.40611 q^{82} -1.24187 q^{83} -0.519040 q^{84} +2.02470 q^{85} +2.92068 q^{86} +0.980494 q^{87} +2.65913 q^{88} +0.972868 q^{89} -2.42095 q^{90} -11.6247 q^{91} +3.37191 q^{92} -0.816829 q^{93} -0.0383619 q^{94} +5.85717 q^{95} +0.169015 q^{96} +18.0126 q^{97} -2.43081 q^{98} +7.90143 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9} - 18 q^{10} + 30 q^{11} + 10 q^{12} - 2 q^{13} - 21 q^{14} + 21 q^{15} + 77 q^{16} + 60 q^{17} - 71 q^{18} - 3 q^{19} + 18 q^{20} + 10 q^{21} - 30 q^{22} + 53 q^{23} - 10 q^{24} + 59 q^{25} + 2 q^{26} + 43 q^{27} + 21 q^{28} + 30 q^{29} - 21 q^{30} + 22 q^{31} - 77 q^{32} + 31 q^{33} - 60 q^{34} + 41 q^{35} + 71 q^{36} - 3 q^{37} + 3 q^{38} + 44 q^{39} - 18 q^{40} + 48 q^{41} - 10 q^{42} + 21 q^{43} + 30 q^{44} + 33 q^{45} - 53 q^{46} + 107 q^{47} + 10 q^{48} + 24 q^{49} - 59 q^{50} + 18 q^{51} - 2 q^{52} + 42 q^{53} - 43 q^{54} + 49 q^{55} - 21 q^{56} + 32 q^{57} - 30 q^{58} + 42 q^{59} + 21 q^{60} - 31 q^{61} - 22 q^{62} + 109 q^{63} + 77 q^{64} + 39 q^{65} - 31 q^{66} - q^{67} + 60 q^{68} - 33 q^{69} - 41 q^{70} + 58 q^{71} - 71 q^{72} + 35 q^{73} + 3 q^{74} + 34 q^{75} - 3 q^{76} + 86 q^{77} - 44 q^{78} + 25 q^{79} + 18 q^{80} + 53 q^{81} - 48 q^{82} + 107 q^{83} + 10 q^{84} + 21 q^{85} - 21 q^{86} + 100 q^{87} - 30 q^{88} + 34 q^{89} - 33 q^{90} - 51 q^{91} + 53 q^{92} + 48 q^{93} - 107 q^{94} + 118 q^{95} - 10 q^{96} - 13 q^{97} - 24 q^{98} + 63 q^{99}+O(q^{100}) \) 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
\(3\) −0.169015 −0.0975810 −0.0487905 0.998809i \(-0.515537\pi\)
−0.0487905 + 0.998809i \(0.515537\pi\)
\(4\) 1.00000 0.500000
\(5\) −0.814742 −0.364364 −0.182182 0.983265i \(-0.558316\pi\)
−0.182182 + 0.983265i \(0.558316\pi\)
\(6\) 0.169015 0.0690002
\(7\) 3.07096 1.16071 0.580357 0.814362i \(-0.302913\pi\)
0.580357 + 0.814362i \(0.302913\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.97143 −0.990478
\(10\) 0.814742 0.257644
\(11\) −2.65913 −0.801758 −0.400879 0.916131i \(-0.631295\pi\)
−0.400879 + 0.916131i \(0.631295\pi\)
\(12\) −0.169015 −0.0487905
\(13\) −3.78538 −1.04987 −0.524937 0.851141i \(-0.675911\pi\)
−0.524937 + 0.851141i \(0.675911\pi\)
\(14\) −3.07096 −0.820749
\(15\) 0.137704 0.0355550
\(16\) 1.00000 0.250000
\(17\) −2.48508 −0.602721 −0.301360 0.953510i \(-0.597441\pi\)
−0.301360 + 0.953510i \(0.597441\pi\)
\(18\) 2.97143 0.700374
\(19\) −7.18899 −1.64927 −0.824634 0.565667i \(-0.808619\pi\)
−0.824634 + 0.565667i \(0.808619\pi\)
\(20\) −0.814742 −0.182182
\(21\) −0.519040 −0.113264
\(22\) 2.65913 0.566928
\(23\) 3.37191 0.703092 0.351546 0.936171i \(-0.385656\pi\)
0.351546 + 0.936171i \(0.385656\pi\)
\(24\) 0.169015 0.0345001
\(25\) −4.33620 −0.867239
\(26\) 3.78538 0.742373
\(27\) 1.00926 0.194233
\(28\) 3.07096 0.580357
\(29\) −5.80122 −1.07726 −0.538629 0.842543i \(-0.681058\pi\)
−0.538629 + 0.842543i \(0.681058\pi\)
\(30\) −0.137704 −0.0251412
\(31\) 4.83287 0.868009 0.434005 0.900911i \(-0.357100\pi\)
0.434005 + 0.900911i \(0.357100\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0.449434 0.0782364
\(34\) 2.48508 0.426188
\(35\) −2.50204 −0.422922
\(36\) −2.97143 −0.495239
\(37\) −7.94827 −1.30669 −0.653343 0.757062i \(-0.726634\pi\)
−0.653343 + 0.757062i \(0.726634\pi\)
\(38\) 7.18899 1.16621
\(39\) 0.639787 0.102448
\(40\) 0.814742 0.128822
\(41\) −4.40611 −0.688119 −0.344059 0.938948i \(-0.611802\pi\)
−0.344059 + 0.938948i \(0.611802\pi\)
\(42\) 0.519040 0.0800896
\(43\) −2.92068 −0.445399 −0.222700 0.974887i \(-0.571487\pi\)
−0.222700 + 0.974887i \(0.571487\pi\)
\(44\) −2.65913 −0.400879
\(45\) 2.42095 0.360894
\(46\) −3.37191 −0.497161
\(47\) 0.0383619 0.00559566 0.00279783 0.999996i \(-0.499109\pi\)
0.00279783 + 0.999996i \(0.499109\pi\)
\(48\) −0.169015 −0.0243953
\(49\) 2.43081 0.347259
\(50\) 4.33620 0.613231
\(51\) 0.420017 0.0588141
\(52\) −3.78538 −0.524937
\(53\) 8.11722 1.11499 0.557493 0.830182i \(-0.311763\pi\)
0.557493 + 0.830182i \(0.311763\pi\)
\(54\) −1.00926 −0.137343
\(55\) 2.16650 0.292131
\(56\) −3.07096 −0.410375
\(57\) 1.21505 0.160937
\(58\) 5.80122 0.761737
\(59\) −6.63420 −0.863699 −0.431849 0.901946i \(-0.642139\pi\)
−0.431849 + 0.901946i \(0.642139\pi\)
\(60\) 0.137704 0.0177775
\(61\) 13.5883 1.73980 0.869900 0.493228i \(-0.164183\pi\)
0.869900 + 0.493228i \(0.164183\pi\)
\(62\) −4.83287 −0.613775
\(63\) −9.12516 −1.14966
\(64\) 1.00000 0.125000
\(65\) 3.08410 0.382536
\(66\) −0.449434 −0.0553215
\(67\) −10.0202 −1.22417 −0.612084 0.790793i \(-0.709668\pi\)
−0.612084 + 0.790793i \(0.709668\pi\)
\(68\) −2.48508 −0.301360
\(69\) −0.569904 −0.0686084
\(70\) 2.50204 0.299051
\(71\) 9.18773 1.09038 0.545191 0.838312i \(-0.316457\pi\)
0.545191 + 0.838312i \(0.316457\pi\)
\(72\) 2.97143 0.350187
\(73\) −4.01614 −0.470053 −0.235027 0.971989i \(-0.575518\pi\)
−0.235027 + 0.971989i \(0.575518\pi\)
\(74\) 7.94827 0.923967
\(75\) 0.732884 0.0846261
\(76\) −7.18899 −0.824634
\(77\) −8.16609 −0.930612
\(78\) −0.639787 −0.0724416
\(79\) −2.45176 −0.275845 −0.137922 0.990443i \(-0.544042\pi\)
−0.137922 + 0.990443i \(0.544042\pi\)
\(80\) −0.814742 −0.0910909
\(81\) 8.74372 0.971524
\(82\) 4.40611 0.486573
\(83\) −1.24187 −0.136313 −0.0681565 0.997675i \(-0.521712\pi\)
−0.0681565 + 0.997675i \(0.521712\pi\)
\(84\) −0.519040 −0.0566319
\(85\) 2.02470 0.219609
\(86\) 2.92068 0.314945
\(87\) 0.980494 0.105120
\(88\) 2.65913 0.283464
\(89\) 0.972868 0.103124 0.0515619 0.998670i \(-0.483580\pi\)
0.0515619 + 0.998670i \(0.483580\pi\)
\(90\) −2.42095 −0.255191
\(91\) −11.6247 −1.21860
\(92\) 3.37191 0.351546
\(93\) −0.816829 −0.0847012
\(94\) −0.0383619 −0.00395673
\(95\) 5.85717 0.600933
\(96\) 0.169015 0.0172501
\(97\) 18.0126 1.82890 0.914450 0.404700i \(-0.132624\pi\)
0.914450 + 0.404700i \(0.132624\pi\)
\(98\) −2.43081 −0.245549
\(99\) 7.90143 0.794124
\(100\) −4.33620 −0.433620
\(101\) 10.1712 1.01207 0.506036 0.862512i \(-0.331110\pi\)
0.506036 + 0.862512i \(0.331110\pi\)
\(102\) −0.420017 −0.0415879
\(103\) −14.0294 −1.38236 −0.691179 0.722684i \(-0.742908\pi\)
−0.691179 + 0.722684i \(0.742908\pi\)
\(104\) 3.78538 0.371187
\(105\) 0.422883 0.0412692
\(106\) −8.11722 −0.788414
\(107\) 15.0755 1.45740 0.728700 0.684833i \(-0.240125\pi\)
0.728700 + 0.684833i \(0.240125\pi\)
\(108\) 1.00926 0.0971165
\(109\) 4.89644 0.468994 0.234497 0.972117i \(-0.424656\pi\)
0.234497 + 0.972117i \(0.424656\pi\)
\(110\) −2.16650 −0.206568
\(111\) 1.34338 0.127508
\(112\) 3.07096 0.290179
\(113\) 14.5861 1.37215 0.686075 0.727531i \(-0.259332\pi\)
0.686075 + 0.727531i \(0.259332\pi\)
\(114\) −1.21505 −0.113800
\(115\) −2.74723 −0.256181
\(116\) −5.80122 −0.538629
\(117\) 11.2480 1.03988
\(118\) 6.63420 0.610727
\(119\) −7.63159 −0.699587
\(120\) −0.137704 −0.0125706
\(121\) −3.92903 −0.357184
\(122\) −13.5883 −1.23022
\(123\) 0.744700 0.0671473
\(124\) 4.83287 0.434005
\(125\) 7.60659 0.680354
\(126\) 9.12516 0.812934
\(127\) −10.1576 −0.901337 −0.450669 0.892691i \(-0.648814\pi\)
−0.450669 + 0.892691i \(0.648814\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0.493639 0.0434625
\(130\) −3.08410 −0.270494
\(131\) 6.40092 0.559251 0.279625 0.960109i \(-0.409790\pi\)
0.279625 + 0.960109i \(0.409790\pi\)
\(132\) 0.449434 0.0391182
\(133\) −22.0771 −1.91433
\(134\) 10.0202 0.865617
\(135\) −0.822289 −0.0707714
\(136\) 2.48508 0.213094
\(137\) 12.3450 1.05470 0.527351 0.849648i \(-0.323185\pi\)
0.527351 + 0.849648i \(0.323185\pi\)
\(138\) 0.569904 0.0485135
\(139\) 8.03274 0.681328 0.340664 0.940185i \(-0.389348\pi\)
0.340664 + 0.940185i \(0.389348\pi\)
\(140\) −2.50204 −0.211461
\(141\) −0.00648375 −0.000546031 0
\(142\) −9.18773 −0.771017
\(143\) 10.0658 0.841745
\(144\) −2.97143 −0.247619
\(145\) 4.72649 0.392514
\(146\) 4.01614 0.332378
\(147\) −0.410844 −0.0338859
\(148\) −7.94827 −0.653343
\(149\) −11.1522 −0.913625 −0.456813 0.889563i \(-0.651009\pi\)
−0.456813 + 0.889563i \(0.651009\pi\)
\(150\) −0.732884 −0.0598397
\(151\) 7.09851 0.577668 0.288834 0.957379i \(-0.406732\pi\)
0.288834 + 0.957379i \(0.406732\pi\)
\(152\) 7.18899 0.583104
\(153\) 7.38426 0.596982
\(154\) 8.16609 0.658042
\(155\) −3.93754 −0.316271
\(156\) 0.639787 0.0512239
\(157\) 5.41650 0.432284 0.216142 0.976362i \(-0.430653\pi\)
0.216142 + 0.976362i \(0.430653\pi\)
\(158\) 2.45176 0.195052
\(159\) −1.37194 −0.108802
\(160\) 0.814742 0.0644110
\(161\) 10.3550 0.816089
\(162\) −8.74372 −0.686972
\(163\) 23.5997 1.84847 0.924234 0.381827i \(-0.124705\pi\)
0.924234 + 0.381827i \(0.124705\pi\)
\(164\) −4.40611 −0.344059
\(165\) −0.366172 −0.0285065
\(166\) 1.24187 0.0963878
\(167\) −16.7950 −1.29964 −0.649819 0.760089i \(-0.725156\pi\)
−0.649819 + 0.760089i \(0.725156\pi\)
\(168\) 0.519040 0.0400448
\(169\) 1.32908 0.102237
\(170\) −2.02470 −0.155287
\(171\) 21.3616 1.63356
\(172\) −2.92068 −0.222700
\(173\) 8.69062 0.660735 0.330368 0.943852i \(-0.392827\pi\)
0.330368 + 0.943852i \(0.392827\pi\)
\(174\) −0.980494 −0.0743311
\(175\) −13.3163 −1.00662
\(176\) −2.65913 −0.200439
\(177\) 1.12128 0.0842806
\(178\) −0.972868 −0.0729196
\(179\) 0.867658 0.0648518 0.0324259 0.999474i \(-0.489677\pi\)
0.0324259 + 0.999474i \(0.489677\pi\)
\(180\) 2.42095 0.180447
\(181\) −2.06989 −0.153854 −0.0769270 0.997037i \(-0.524511\pi\)
−0.0769270 + 0.997037i \(0.524511\pi\)
\(182\) 11.6247 0.861684
\(183\) −2.29663 −0.169771
\(184\) −3.37191 −0.248580
\(185\) 6.47578 0.476109
\(186\) 0.816829 0.0598928
\(187\) 6.60816 0.483236
\(188\) 0.0383619 0.00279783
\(189\) 3.09941 0.225449
\(190\) −5.85717 −0.424924
\(191\) 1.35941 0.0983632 0.0491816 0.998790i \(-0.484339\pi\)
0.0491816 + 0.998790i \(0.484339\pi\)
\(192\) −0.169015 −0.0121976
\(193\) −15.9083 −1.14510 −0.572551 0.819869i \(-0.694046\pi\)
−0.572551 + 0.819869i \(0.694046\pi\)
\(194\) −18.0126 −1.29323
\(195\) −0.521261 −0.0373283
\(196\) 2.43081 0.173629
\(197\) 14.9004 1.06161 0.530805 0.847494i \(-0.321890\pi\)
0.530805 + 0.847494i \(0.321890\pi\)
\(198\) −7.90143 −0.561530
\(199\) −11.8598 −0.840719 −0.420359 0.907358i \(-0.638096\pi\)
−0.420359 + 0.907358i \(0.638096\pi\)
\(200\) 4.33620 0.306615
\(201\) 1.69357 0.119455
\(202\) −10.1712 −0.715644
\(203\) −17.8153 −1.25039
\(204\) 0.420017 0.0294071
\(205\) 3.58984 0.250725
\(206\) 14.0294 0.977475
\(207\) −10.0194 −0.696397
\(208\) −3.78538 −0.262469
\(209\) 19.1165 1.32231
\(210\) −0.422883 −0.0291817
\(211\) −9.12273 −0.628034 −0.314017 0.949417i \(-0.601675\pi\)
−0.314017 + 0.949417i \(0.601675\pi\)
\(212\) 8.11722 0.557493
\(213\) −1.55287 −0.106401
\(214\) −15.0755 −1.03054
\(215\) 2.37960 0.162287
\(216\) −1.00926 −0.0686717
\(217\) 14.8416 1.00751
\(218\) −4.89644 −0.331629
\(219\) 0.678788 0.0458683
\(220\) 2.16650 0.146066
\(221\) 9.40697 0.632781
\(222\) −1.34338 −0.0901617
\(223\) 2.63638 0.176545 0.0882725 0.996096i \(-0.471865\pi\)
0.0882725 + 0.996096i \(0.471865\pi\)
\(224\) −3.07096 −0.205187
\(225\) 12.8847 0.858981
\(226\) −14.5861 −0.970256
\(227\) −8.03189 −0.533095 −0.266548 0.963822i \(-0.585883\pi\)
−0.266548 + 0.963822i \(0.585883\pi\)
\(228\) 1.21505 0.0804686
\(229\) 12.1310 0.801640 0.400820 0.916157i \(-0.368725\pi\)
0.400820 + 0.916157i \(0.368725\pi\)
\(230\) 2.74723 0.181147
\(231\) 1.38019 0.0908101
\(232\) 5.80122 0.380868
\(233\) 2.20968 0.144761 0.0723805 0.997377i \(-0.476940\pi\)
0.0723805 + 0.997377i \(0.476940\pi\)
\(234\) −11.2480 −0.735305
\(235\) −0.0312551 −0.00203886
\(236\) −6.63420 −0.431849
\(237\) 0.414385 0.0269172
\(238\) 7.63159 0.494683
\(239\) 7.70534 0.498417 0.249208 0.968450i \(-0.419830\pi\)
0.249208 + 0.968450i \(0.419830\pi\)
\(240\) 0.137704 0.00888874
\(241\) 23.9056 1.53990 0.769949 0.638106i \(-0.220282\pi\)
0.769949 + 0.638106i \(0.220282\pi\)
\(242\) 3.92903 0.252567
\(243\) −4.50561 −0.289035
\(244\) 13.5883 0.869900
\(245\) −1.98048 −0.126528
\(246\) −0.744700 −0.0474803
\(247\) 27.2130 1.73152
\(248\) −4.83287 −0.306888
\(249\) 0.209895 0.0133016
\(250\) −7.60659 −0.481083
\(251\) −23.1486 −1.46112 −0.730562 0.682846i \(-0.760742\pi\)
−0.730562 + 0.682846i \(0.760742\pi\)
\(252\) −9.12516 −0.574831
\(253\) −8.96634 −0.563709
\(254\) 10.1576 0.637342
\(255\) −0.342205 −0.0214297
\(256\) 1.00000 0.0625000
\(257\) 5.33938 0.333062 0.166531 0.986036i \(-0.446743\pi\)
0.166531 + 0.986036i \(0.446743\pi\)
\(258\) −0.493639 −0.0307326
\(259\) −24.4088 −1.51669
\(260\) 3.08410 0.191268
\(261\) 17.2379 1.06700
\(262\) −6.40092 −0.395450
\(263\) 3.15421 0.194497 0.0972483 0.995260i \(-0.468996\pi\)
0.0972483 + 0.995260i \(0.468996\pi\)
\(264\) −0.449434 −0.0276607
\(265\) −6.61344 −0.406260
\(266\) 22.0771 1.35363
\(267\) −0.164430 −0.0100629
\(268\) −10.0202 −0.612084
\(269\) −13.1864 −0.803988 −0.401994 0.915642i \(-0.631683\pi\)
−0.401994 + 0.915642i \(0.631683\pi\)
\(270\) 0.822289 0.0500429
\(271\) 10.5521 0.640995 0.320497 0.947249i \(-0.396150\pi\)
0.320497 + 0.947249i \(0.396150\pi\)
\(272\) −2.48508 −0.150680
\(273\) 1.96476 0.118913
\(274\) −12.3450 −0.745787
\(275\) 11.5305 0.695316
\(276\) −0.569904 −0.0343042
\(277\) −25.7575 −1.54762 −0.773808 0.633420i \(-0.781651\pi\)
−0.773808 + 0.633420i \(0.781651\pi\)
\(278\) −8.03274 −0.481772
\(279\) −14.3606 −0.859744
\(280\) 2.50204 0.149526
\(281\) 4.45765 0.265921 0.132961 0.991121i \(-0.457552\pi\)
0.132961 + 0.991121i \(0.457552\pi\)
\(282\) 0.00648375 0.000386102 0
\(283\) 1.07317 0.0637935 0.0318968 0.999491i \(-0.489845\pi\)
0.0318968 + 0.999491i \(0.489845\pi\)
\(284\) 9.18773 0.545191
\(285\) −0.989951 −0.0586397
\(286\) −10.0658 −0.595204
\(287\) −13.5310 −0.798709
\(288\) 2.97143 0.175093
\(289\) −10.8244 −0.636728
\(290\) −4.72649 −0.277549
\(291\) −3.04440 −0.178466
\(292\) −4.01614 −0.235027
\(293\) −21.7034 −1.26793 −0.633964 0.773363i \(-0.718573\pi\)
−0.633964 + 0.773363i \(0.718573\pi\)
\(294\) 0.410844 0.0239609
\(295\) 5.40516 0.314700
\(296\) 7.94827 0.461984
\(297\) −2.68376 −0.155728
\(298\) 11.1522 0.646031
\(299\) −12.7639 −0.738158
\(300\) 0.732884 0.0423131
\(301\) −8.96929 −0.516981
\(302\) −7.09851 −0.408473
\(303\) −1.71909 −0.0987591
\(304\) −7.18899 −0.412317
\(305\) −11.0709 −0.633920
\(306\) −7.38426 −0.422130
\(307\) −8.66351 −0.494453 −0.247226 0.968958i \(-0.579519\pi\)
−0.247226 + 0.968958i \(0.579519\pi\)
\(308\) −8.16609 −0.465306
\(309\) 2.37118 0.134892
\(310\) 3.93754 0.223637
\(311\) 15.2654 0.865623 0.432812 0.901484i \(-0.357522\pi\)
0.432812 + 0.901484i \(0.357522\pi\)
\(312\) −0.639787 −0.0362208
\(313\) −17.2240 −0.973557 −0.486778 0.873526i \(-0.661828\pi\)
−0.486778 + 0.873526i \(0.661828\pi\)
\(314\) −5.41650 −0.305671
\(315\) 7.43465 0.418895
\(316\) −2.45176 −0.137922
\(317\) −14.5111 −0.815023 −0.407512 0.913200i \(-0.633603\pi\)
−0.407512 + 0.913200i \(0.633603\pi\)
\(318\) 1.37194 0.0769343
\(319\) 15.4262 0.863701
\(320\) −0.814742 −0.0455454
\(321\) −2.54798 −0.142215
\(322\) −10.3550 −0.577062
\(323\) 17.8652 0.994048
\(324\) 8.74372 0.485762
\(325\) 16.4141 0.910492
\(326\) −23.5997 −1.30706
\(327\) −0.827573 −0.0457649
\(328\) 4.40611 0.243287
\(329\) 0.117808 0.00649497
\(330\) 0.366172 0.0201571
\(331\) −19.6185 −1.07833 −0.539165 0.842200i \(-0.681260\pi\)
−0.539165 + 0.842200i \(0.681260\pi\)
\(332\) −1.24187 −0.0681565
\(333\) 23.6177 1.29424
\(334\) 16.7950 0.918984
\(335\) 8.16391 0.446042
\(336\) −0.519040 −0.0283159
\(337\) 5.02933 0.273965 0.136983 0.990573i \(-0.456260\pi\)
0.136983 + 0.990573i \(0.456260\pi\)
\(338\) −1.32908 −0.0722923
\(339\) −2.46528 −0.133896
\(340\) 2.02470 0.109805
\(341\) −12.8512 −0.695933
\(342\) −21.3616 −1.15510
\(343\) −14.0318 −0.757647
\(344\) 2.92068 0.157472
\(345\) 0.464325 0.0249984
\(346\) −8.69062 −0.467210
\(347\) 2.17304 0.116655 0.0583275 0.998298i \(-0.481423\pi\)
0.0583275 + 0.998298i \(0.481423\pi\)
\(348\) 0.980494 0.0525600
\(349\) −21.0770 −1.12823 −0.564113 0.825698i \(-0.690782\pi\)
−0.564113 + 0.825698i \(0.690782\pi\)
\(350\) 13.3163 0.711786
\(351\) −3.82044 −0.203920
\(352\) 2.65913 0.141732
\(353\) 12.5787 0.669495 0.334748 0.942308i \(-0.391349\pi\)
0.334748 + 0.942308i \(0.391349\pi\)
\(354\) −1.12128 −0.0595954
\(355\) −7.48562 −0.397296
\(356\) 0.972868 0.0515619
\(357\) 1.28986 0.0682664
\(358\) −0.867658 −0.0458571
\(359\) −2.31276 −0.122063 −0.0610315 0.998136i \(-0.519439\pi\)
−0.0610315 + 0.998136i \(0.519439\pi\)
\(360\) −2.42095 −0.127595
\(361\) 32.6816 1.72008
\(362\) 2.06989 0.108791
\(363\) 0.664066 0.0348544
\(364\) −11.6247 −0.609302
\(365\) 3.27211 0.171270
\(366\) 2.29663 0.120047
\(367\) −0.557091 −0.0290799 −0.0145400 0.999894i \(-0.504628\pi\)
−0.0145400 + 0.999894i \(0.504628\pi\)
\(368\) 3.37191 0.175773
\(369\) 13.0925 0.681566
\(370\) −6.47578 −0.336660
\(371\) 24.9277 1.29418
\(372\) −0.816829 −0.0423506
\(373\) −10.9109 −0.564943 −0.282471 0.959276i \(-0.591154\pi\)
−0.282471 + 0.959276i \(0.591154\pi\)
\(374\) −6.60816 −0.341700
\(375\) −1.28563 −0.0663896
\(376\) −0.0383619 −0.00197837
\(377\) 21.9598 1.13099
\(378\) −3.09941 −0.159417
\(379\) 18.4773 0.949113 0.474557 0.880225i \(-0.342608\pi\)
0.474557 + 0.880225i \(0.342608\pi\)
\(380\) 5.85717 0.300466
\(381\) 1.71678 0.0879534
\(382\) −1.35941 −0.0695533
\(383\) −19.6251 −1.00279 −0.501397 0.865217i \(-0.667180\pi\)
−0.501397 + 0.865217i \(0.667180\pi\)
\(384\) 0.169015 0.00862503
\(385\) 6.65325 0.339081
\(386\) 15.9083 0.809709
\(387\) 8.67860 0.441158
\(388\) 18.0126 0.914450
\(389\) −25.5547 −1.29567 −0.647837 0.761779i \(-0.724326\pi\)
−0.647837 + 0.761779i \(0.724326\pi\)
\(390\) 0.521261 0.0263951
\(391\) −8.37947 −0.423768
\(392\) −2.43081 −0.122774
\(393\) −1.08185 −0.0545723
\(394\) −14.9004 −0.750672
\(395\) 1.99755 0.100508
\(396\) 7.90143 0.397062
\(397\) −24.6808 −1.23870 −0.619348 0.785117i \(-0.712603\pi\)
−0.619348 + 0.785117i \(0.712603\pi\)
\(398\) 11.8598 0.594478
\(399\) 3.73137 0.186802
\(400\) −4.33620 −0.216810
\(401\) 20.4996 1.02370 0.511850 0.859075i \(-0.328960\pi\)
0.511850 + 0.859075i \(0.328960\pi\)
\(402\) −1.69357 −0.0844678
\(403\) −18.2942 −0.911301
\(404\) 10.1712 0.506036
\(405\) −7.12387 −0.353988
\(406\) 17.8153 0.884159
\(407\) 21.1355 1.04765
\(408\) −0.420017 −0.0207939
\(409\) 13.6234 0.673632 0.336816 0.941570i \(-0.390650\pi\)
0.336816 + 0.941570i \(0.390650\pi\)
\(410\) −3.58984 −0.177290
\(411\) −2.08649 −0.102919
\(412\) −14.0294 −0.691179
\(413\) −20.3734 −1.00251
\(414\) 10.0194 0.492427
\(415\) 1.01180 0.0496675
\(416\) 3.78538 0.185593
\(417\) −1.35766 −0.0664847
\(418\) −19.1165 −0.935017
\(419\) −0.236224 −0.0115403 −0.00577015 0.999983i \(-0.501837\pi\)
−0.00577015 + 0.999983i \(0.501837\pi\)
\(420\) 0.422883 0.0206346
\(421\) −40.4372 −1.97079 −0.985394 0.170288i \(-0.945530\pi\)
−0.985394 + 0.170288i \(0.945530\pi\)
\(422\) 9.12273 0.444087
\(423\) −0.113990 −0.00554238
\(424\) −8.11722 −0.394207
\(425\) 10.7758 0.522703
\(426\) 1.55287 0.0752366
\(427\) 41.7291 2.01941
\(428\) 15.0755 0.728700
\(429\) −1.70128 −0.0821384
\(430\) −2.37960 −0.114754
\(431\) 38.0549 1.83304 0.916520 0.399988i \(-0.130986\pi\)
0.916520 + 0.399988i \(0.130986\pi\)
\(432\) 1.00926 0.0485582
\(433\) 15.0745 0.724437 0.362218 0.932093i \(-0.382019\pi\)
0.362218 + 0.932093i \(0.382019\pi\)
\(434\) −14.8416 −0.712418
\(435\) −0.798850 −0.0383019
\(436\) 4.89644 0.234497
\(437\) −24.2406 −1.15959
\(438\) −0.678788 −0.0324338
\(439\) 31.0098 1.48002 0.740008 0.672598i \(-0.234822\pi\)
0.740008 + 0.672598i \(0.234822\pi\)
\(440\) −2.16650 −0.103284
\(441\) −7.22299 −0.343952
\(442\) −9.40697 −0.447444
\(443\) −25.9257 −1.23177 −0.615883 0.787837i \(-0.711201\pi\)
−0.615883 + 0.787837i \(0.711201\pi\)
\(444\) 1.34338 0.0637539
\(445\) −0.792636 −0.0375746
\(446\) −2.63638 −0.124836
\(447\) 1.88490 0.0891525
\(448\) 3.07096 0.145089
\(449\) 18.0562 0.852124 0.426062 0.904694i \(-0.359901\pi\)
0.426062 + 0.904694i \(0.359901\pi\)
\(450\) −12.8847 −0.607391
\(451\) 11.7164 0.551705
\(452\) 14.5861 0.686075
\(453\) −1.19976 −0.0563695
\(454\) 8.03189 0.376955
\(455\) 9.47117 0.444015
\(456\) −1.21505 −0.0568999
\(457\) 9.37221 0.438413 0.219207 0.975678i \(-0.429653\pi\)
0.219207 + 0.975678i \(0.429653\pi\)
\(458\) −12.1310 −0.566845
\(459\) −2.50810 −0.117068
\(460\) −2.74723 −0.128090
\(461\) 28.3684 1.32125 0.660623 0.750718i \(-0.270292\pi\)
0.660623 + 0.750718i \(0.270292\pi\)
\(462\) −1.38019 −0.0642124
\(463\) 11.6167 0.539873 0.269936 0.962878i \(-0.412997\pi\)
0.269936 + 0.962878i \(0.412997\pi\)
\(464\) −5.80122 −0.269315
\(465\) 0.665505 0.0308620
\(466\) −2.20968 −0.102362
\(467\) 28.8799 1.33640 0.668200 0.743981i \(-0.267065\pi\)
0.668200 + 0.743981i \(0.267065\pi\)
\(468\) 11.2480 0.519939
\(469\) −30.7718 −1.42091
\(470\) 0.0312551 0.00144169
\(471\) −0.915471 −0.0421827
\(472\) 6.63420 0.305364
\(473\) 7.76646 0.357102
\(474\) −0.414385 −0.0190334
\(475\) 31.1729 1.43031
\(476\) −7.63159 −0.349793
\(477\) −24.1198 −1.10437
\(478\) −7.70534 −0.352434
\(479\) 14.5624 0.665372 0.332686 0.943038i \(-0.392045\pi\)
0.332686 + 0.943038i \(0.392045\pi\)
\(480\) −0.137704 −0.00628529
\(481\) 30.0872 1.37186
\(482\) −23.9056 −1.08887
\(483\) −1.75015 −0.0796348
\(484\) −3.92903 −0.178592
\(485\) −14.6756 −0.666384
\(486\) 4.50561 0.204379
\(487\) −29.8995 −1.35487 −0.677437 0.735581i \(-0.736909\pi\)
−0.677437 + 0.735581i \(0.736909\pi\)
\(488\) −13.5883 −0.615112
\(489\) −3.98870 −0.180375
\(490\) 1.98048 0.0894691
\(491\) 30.1072 1.35872 0.679359 0.733806i \(-0.262258\pi\)
0.679359 + 0.733806i \(0.262258\pi\)
\(492\) 0.744700 0.0335737
\(493\) 14.4165 0.649286
\(494\) −27.2130 −1.22437
\(495\) −6.43762 −0.289350
\(496\) 4.83287 0.217002
\(497\) 28.2152 1.26562
\(498\) −0.209895 −0.00940562
\(499\) 35.4216 1.58569 0.792845 0.609424i \(-0.208599\pi\)
0.792845 + 0.609424i \(0.208599\pi\)
\(500\) 7.60659 0.340177
\(501\) 2.83862 0.126820
\(502\) 23.1486 1.03317
\(503\) 7.53970 0.336179 0.168089 0.985772i \(-0.446240\pi\)
0.168089 + 0.985772i \(0.446240\pi\)
\(504\) 9.12516 0.406467
\(505\) −8.28691 −0.368763
\(506\) 8.96634 0.398603
\(507\) −0.224634 −0.00997637
\(508\) −10.1576 −0.450669
\(509\) 2.55867 0.113411 0.0567055 0.998391i \(-0.481940\pi\)
0.0567055 + 0.998391i \(0.481940\pi\)
\(510\) 0.342205 0.0151531
\(511\) −12.3334 −0.545597
\(512\) −1.00000 −0.0441942
\(513\) −7.25559 −0.320342
\(514\) −5.33938 −0.235510
\(515\) 11.4303 0.503681
\(516\) 0.493639 0.0217312
\(517\) −0.102009 −0.00448637
\(518\) 24.4088 1.07246
\(519\) −1.46885 −0.0644752
\(520\) −3.08410 −0.135247
\(521\) −7.45017 −0.326398 −0.163199 0.986593i \(-0.552181\pi\)
−0.163199 + 0.986593i \(0.552181\pi\)
\(522\) −17.2379 −0.754484
\(523\) 3.51052 0.153504 0.0767521 0.997050i \(-0.475545\pi\)
0.0767521 + 0.997050i \(0.475545\pi\)
\(524\) 6.40092 0.279625
\(525\) 2.25066 0.0982268
\(526\) −3.15421 −0.137530
\(527\) −12.0101 −0.523167
\(528\) 0.449434 0.0195591
\(529\) −11.6302 −0.505662
\(530\) 6.61344 0.287269
\(531\) 19.7131 0.855475
\(532\) −22.0771 −0.957164
\(533\) 16.6788 0.722438
\(534\) 0.164430 0.00711557
\(535\) −12.2826 −0.531023
\(536\) 10.0202 0.432808
\(537\) −0.146647 −0.00632831
\(538\) 13.1864 0.568505
\(539\) −6.46384 −0.278417
\(540\) −0.822289 −0.0353857
\(541\) 4.26827 0.183507 0.0917537 0.995782i \(-0.470753\pi\)
0.0917537 + 0.995782i \(0.470753\pi\)
\(542\) −10.5521 −0.453252
\(543\) 0.349844 0.0150132
\(544\) 2.48508 0.106547
\(545\) −3.98933 −0.170884
\(546\) −1.96476 −0.0840840
\(547\) −30.3139 −1.29613 −0.648065 0.761585i \(-0.724421\pi\)
−0.648065 + 0.761585i \(0.724421\pi\)
\(548\) 12.3450 0.527351
\(549\) −40.3766 −1.72323
\(550\) −11.5305 −0.491663
\(551\) 41.7049 1.77669
\(552\) 0.569904 0.0242567
\(553\) −7.52927 −0.320177
\(554\) 25.7575 1.09433
\(555\) −1.09451 −0.0464592
\(556\) 8.03274 0.340664
\(557\) 40.5246 1.71708 0.858540 0.512746i \(-0.171372\pi\)
0.858540 + 0.512746i \(0.171372\pi\)
\(558\) 14.3606 0.607931
\(559\) 11.0559 0.467613
\(560\) −2.50204 −0.105731
\(561\) −1.11688 −0.0471547
\(562\) −4.45765 −0.188035
\(563\) −9.44003 −0.397850 −0.198925 0.980015i \(-0.563745\pi\)
−0.198925 + 0.980015i \(0.563745\pi\)
\(564\) −0.00648375 −0.000273015 0
\(565\) −11.8839 −0.499961
\(566\) −1.07317 −0.0451088
\(567\) 26.8516 1.12766
\(568\) −9.18773 −0.385508
\(569\) 26.1437 1.09600 0.548000 0.836478i \(-0.315389\pi\)
0.548000 + 0.836478i \(0.315389\pi\)
\(570\) 0.989951 0.0414645
\(571\) 1.60146 0.0670192 0.0335096 0.999438i \(-0.489332\pi\)
0.0335096 + 0.999438i \(0.489332\pi\)
\(572\) 10.0658 0.420873
\(573\) −0.229761 −0.00959839
\(574\) 13.5310 0.564773
\(575\) −14.6213 −0.609749
\(576\) −2.97143 −0.123810
\(577\) −31.9091 −1.32839 −0.664197 0.747558i \(-0.731226\pi\)
−0.664197 + 0.747558i \(0.731226\pi\)
\(578\) 10.8244 0.450234
\(579\) 2.68874 0.111740
\(580\) 4.72649 0.196257
\(581\) −3.81374 −0.158220
\(582\) 3.04440 0.126194
\(583\) −21.5848 −0.893949
\(584\) 4.01614 0.166189
\(585\) −9.16421 −0.378894
\(586\) 21.7034 0.896560
\(587\) −11.1644 −0.460806 −0.230403 0.973095i \(-0.574004\pi\)
−0.230403 + 0.973095i \(0.574004\pi\)
\(588\) −0.410844 −0.0169429
\(589\) −34.7435 −1.43158
\(590\) −5.40516 −0.222527
\(591\) −2.51840 −0.103593
\(592\) −7.94827 −0.326672
\(593\) 38.4459 1.57879 0.789393 0.613888i \(-0.210395\pi\)
0.789393 + 0.613888i \(0.210395\pi\)
\(594\) 2.68376 0.110116
\(595\) 6.21778 0.254904
\(596\) −11.1522 −0.456813
\(597\) 2.00449 0.0820382
\(598\) 12.7639 0.521956
\(599\) −37.0267 −1.51287 −0.756434 0.654070i \(-0.773060\pi\)
−0.756434 + 0.654070i \(0.773060\pi\)
\(600\) −0.732884 −0.0299198
\(601\) −2.15567 −0.0879316 −0.0439658 0.999033i \(-0.513999\pi\)
−0.0439658 + 0.999033i \(0.513999\pi\)
\(602\) 8.96929 0.365561
\(603\) 29.7745 1.21251
\(604\) 7.09851 0.288834
\(605\) 3.20114 0.130145
\(606\) 1.71909 0.0698333
\(607\) −6.24324 −0.253405 −0.126703 0.991941i \(-0.540439\pi\)
−0.126703 + 0.991941i \(0.540439\pi\)
\(608\) 7.18899 0.291552
\(609\) 3.01106 0.122014
\(610\) 11.0709 0.448249
\(611\) −0.145214 −0.00587474
\(612\) 7.38426 0.298491
\(613\) 19.4918 0.787266 0.393633 0.919268i \(-0.371218\pi\)
0.393633 + 0.919268i \(0.371218\pi\)
\(614\) 8.66351 0.349631
\(615\) −0.606738 −0.0244660
\(616\) 8.16609 0.329021
\(617\) −36.1792 −1.45652 −0.728260 0.685301i \(-0.759671\pi\)
−0.728260 + 0.685301i \(0.759671\pi\)
\(618\) −2.37118 −0.0953830
\(619\) −29.1030 −1.16975 −0.584875 0.811124i \(-0.698856\pi\)
−0.584875 + 0.811124i \(0.698856\pi\)
\(620\) −3.93754 −0.158135
\(621\) 3.40315 0.136564
\(622\) −15.2654 −0.612088
\(623\) 2.98764 0.119697
\(624\) 0.639787 0.0256120
\(625\) 15.4836 0.619343
\(626\) 17.2240 0.688408
\(627\) −3.23097 −0.129033
\(628\) 5.41650 0.216142
\(629\) 19.7521 0.787567
\(630\) −7.43465 −0.296204
\(631\) 7.13286 0.283955 0.141977 0.989870i \(-0.454654\pi\)
0.141977 + 0.989870i \(0.454654\pi\)
\(632\) 2.45176 0.0975259
\(633\) 1.54188 0.0612843
\(634\) 14.5111 0.576309
\(635\) 8.27579 0.328415
\(636\) −1.37194 −0.0544008
\(637\) −9.20153 −0.364578
\(638\) −15.4262 −0.610729
\(639\) −27.3007 −1.08000
\(640\) 0.814742 0.0322055
\(641\) 32.3133 1.27630 0.638150 0.769912i \(-0.279700\pi\)
0.638150 + 0.769912i \(0.279700\pi\)
\(642\) 2.54798 0.100561
\(643\) −36.6551 −1.44554 −0.722769 0.691090i \(-0.757131\pi\)
−0.722769 + 0.691090i \(0.757131\pi\)
\(644\) 10.3550 0.408044
\(645\) −0.402188 −0.0158362
\(646\) −17.8652 −0.702898
\(647\) 9.96599 0.391804 0.195902 0.980624i \(-0.437237\pi\)
0.195902 + 0.980624i \(0.437237\pi\)
\(648\) −8.74372 −0.343486
\(649\) 17.6412 0.692477
\(650\) −16.4141 −0.643815
\(651\) −2.50845 −0.0983140
\(652\) 23.5997 0.924234
\(653\) 3.84625 0.150515 0.0752576 0.997164i \(-0.476022\pi\)
0.0752576 + 0.997164i \(0.476022\pi\)
\(654\) 0.827573 0.0323607
\(655\) −5.21509 −0.203771
\(656\) −4.40611 −0.172030
\(657\) 11.9337 0.465577
\(658\) −0.117808 −0.00459264
\(659\) −7.45776 −0.290513 −0.145256 0.989394i \(-0.546401\pi\)
−0.145256 + 0.989394i \(0.546401\pi\)
\(660\) −0.366172 −0.0142532
\(661\) 19.6195 0.763109 0.381554 0.924346i \(-0.375389\pi\)
0.381554 + 0.924346i \(0.375389\pi\)
\(662\) 19.6185 0.762495
\(663\) −1.58992 −0.0617475
\(664\) 1.24187 0.0481939
\(665\) 17.9871 0.697512
\(666\) −23.6177 −0.915169
\(667\) −19.5612 −0.757411
\(668\) −16.7950 −0.649819
\(669\) −0.445588 −0.0172274
\(670\) −8.16391 −0.315399
\(671\) −36.1330 −1.39490
\(672\) 0.519040 0.0200224
\(673\) −1.84451 −0.0711007 −0.0355504 0.999368i \(-0.511318\pi\)
−0.0355504 + 0.999368i \(0.511318\pi\)
\(674\) −5.02933 −0.193723
\(675\) −4.37637 −0.168446
\(676\) 1.32908 0.0511184
\(677\) −19.0178 −0.730914 −0.365457 0.930828i \(-0.619087\pi\)
−0.365457 + 0.930828i \(0.619087\pi\)
\(678\) 2.46528 0.0946786
\(679\) 55.3159 2.12283
\(680\) −2.02470 −0.0776437
\(681\) 1.35751 0.0520200
\(682\) 12.8512 0.492099
\(683\) 32.5115 1.24402 0.622010 0.783010i \(-0.286317\pi\)
0.622010 + 0.783010i \(0.286317\pi\)
\(684\) 21.3616 0.816781
\(685\) −10.0580 −0.384295
\(686\) 14.0318 0.535737
\(687\) −2.05033 −0.0782248
\(688\) −2.92068 −0.111350
\(689\) −30.7267 −1.17060
\(690\) −0.464325 −0.0176765
\(691\) 2.91815 0.111012 0.0555058 0.998458i \(-0.482323\pi\)
0.0555058 + 0.998458i \(0.482323\pi\)
\(692\) 8.69062 0.330368
\(693\) 24.2650 0.921751
\(694\) −2.17304 −0.0824875
\(695\) −6.54461 −0.248251
\(696\) −0.980494 −0.0371655
\(697\) 10.9495 0.414743
\(698\) 21.0770 0.797776
\(699\) −0.373470 −0.0141259
\(700\) −13.3163 −0.503309
\(701\) −5.30444 −0.200346 −0.100173 0.994970i \(-0.531940\pi\)
−0.100173 + 0.994970i \(0.531940\pi\)
\(702\) 3.82044 0.144193
\(703\) 57.1400 2.15508
\(704\) −2.65913 −0.100220
\(705\) 0.00528258 0.000198954 0
\(706\) −12.5787 −0.473404
\(707\) 31.2354 1.17473
\(708\) 1.12128 0.0421403
\(709\) 12.4448 0.467374 0.233687 0.972312i \(-0.424921\pi\)
0.233687 + 0.972312i \(0.424921\pi\)
\(710\) 7.48562 0.280930
\(711\) 7.28525 0.273218
\(712\) −0.972868 −0.0364598
\(713\) 16.2960 0.610290
\(714\) −1.28986 −0.0482716
\(715\) −8.20103 −0.306701
\(716\) 0.867658 0.0324259
\(717\) −1.30232 −0.0486360
\(718\) 2.31276 0.0863116
\(719\) −12.6156 −0.470483 −0.235241 0.971937i \(-0.575588\pi\)
−0.235241 + 0.971937i \(0.575588\pi\)
\(720\) 2.42095 0.0902235
\(721\) −43.0838 −1.60452
\(722\) −32.6816 −1.21628
\(723\) −4.04042 −0.150265
\(724\) −2.06989 −0.0769270
\(725\) 25.1552 0.934241
\(726\) −0.664066 −0.0246458
\(727\) −50.3308 −1.86667 −0.933333 0.359011i \(-0.883114\pi\)
−0.933333 + 0.359011i \(0.883114\pi\)
\(728\) 11.6247 0.430842
\(729\) −25.4696 −0.943320
\(730\) −3.27211 −0.121106
\(731\) 7.25812 0.268451
\(732\) −2.29663 −0.0848857
\(733\) 16.8222 0.621343 0.310671 0.950517i \(-0.399446\pi\)
0.310671 + 0.950517i \(0.399446\pi\)
\(734\) 0.557091 0.0205626
\(735\) 0.334732 0.0123468
\(736\) −3.37191 −0.124290
\(737\) 26.6451 0.981486
\(738\) −13.0925 −0.481940
\(739\) 9.38710 0.345310 0.172655 0.984982i \(-0.444765\pi\)
0.172655 + 0.984982i \(0.444765\pi\)
\(740\) 6.47578 0.238055
\(741\) −4.59942 −0.168964
\(742\) −24.9277 −0.915124
\(743\) 7.13519 0.261765 0.130882 0.991398i \(-0.458219\pi\)
0.130882 + 0.991398i \(0.458219\pi\)
\(744\) 0.816829 0.0299464
\(745\) 9.08618 0.332892
\(746\) 10.9109 0.399475
\(747\) 3.69013 0.135015
\(748\) 6.60816 0.241618
\(749\) 46.2962 1.69163
\(750\) 1.28563 0.0469446
\(751\) 3.62153 0.132151 0.0660757 0.997815i \(-0.478952\pi\)
0.0660757 + 0.997815i \(0.478952\pi\)
\(752\) 0.0383619 0.00139892
\(753\) 3.91246 0.142578
\(754\) −21.9598 −0.799728
\(755\) −5.78345 −0.210481
\(756\) 3.09941 0.112724
\(757\) −31.8496 −1.15759 −0.578797 0.815472i \(-0.696478\pi\)
−0.578797 + 0.815472i \(0.696478\pi\)
\(758\) −18.4773 −0.671124
\(759\) 1.51545 0.0550073
\(760\) −5.85717 −0.212462
\(761\) 44.7731 1.62302 0.811511 0.584337i \(-0.198645\pi\)
0.811511 + 0.584337i \(0.198645\pi\)
\(762\) −1.71678 −0.0621925
\(763\) 15.0368 0.544368
\(764\) 1.35941 0.0491816
\(765\) −6.01626 −0.217518
\(766\) 19.6251 0.709082
\(767\) 25.1129 0.906776
\(768\) −0.169015 −0.00609882
\(769\) 40.3762 1.45600 0.728002 0.685575i \(-0.240449\pi\)
0.728002 + 0.685575i \(0.240449\pi\)
\(770\) −6.65325 −0.239767
\(771\) −0.902438 −0.0325005
\(772\) −15.9083 −0.572551
\(773\) 3.98968 0.143499 0.0717495 0.997423i \(-0.477142\pi\)
0.0717495 + 0.997423i \(0.477142\pi\)
\(774\) −8.67860 −0.311946
\(775\) −20.9563 −0.752772
\(776\) −18.0126 −0.646614
\(777\) 4.12547 0.148000
\(778\) 25.5547 0.916180
\(779\) 31.6755 1.13489
\(780\) −0.521261 −0.0186641
\(781\) −24.4314 −0.874223
\(782\) 8.37947 0.299649
\(783\) −5.85496 −0.209239
\(784\) 2.43081 0.0868146
\(785\) −4.41305 −0.157508
\(786\) 1.08185 0.0385884
\(787\) −24.2218 −0.863416 −0.431708 0.902014i \(-0.642089\pi\)
−0.431708 + 0.902014i \(0.642089\pi\)
\(788\) 14.9004 0.530805
\(789\) −0.533109 −0.0189792
\(790\) −1.99755 −0.0710697
\(791\) 44.7935 1.59267
\(792\) −7.90143 −0.280765
\(793\) −51.4367 −1.82657
\(794\) 24.6808 0.875890
\(795\) 1.11777 0.0396433
\(796\) −11.8598 −0.420359
\(797\) 39.5164 1.39974 0.699871 0.714270i \(-0.253241\pi\)
0.699871 + 0.714270i \(0.253241\pi\)
\(798\) −3.73137 −0.132089
\(799\) −0.0953325 −0.00337262
\(800\) 4.33620 0.153308
\(801\) −2.89081 −0.102142
\(802\) −20.4996 −0.723865
\(803\) 10.6794 0.376869
\(804\) 1.69357 0.0597277
\(805\) −8.43665 −0.297353
\(806\) 18.2942 0.644387
\(807\) 2.22870 0.0784540
\(808\) −10.1712 −0.357822
\(809\) 26.8957 0.945604 0.472802 0.881169i \(-0.343243\pi\)
0.472802 + 0.881169i \(0.343243\pi\)
\(810\) 7.12387 0.250307
\(811\) 13.5426 0.475547 0.237773 0.971321i \(-0.423582\pi\)
0.237773 + 0.971321i \(0.423582\pi\)
\(812\) −17.8153 −0.625195
\(813\) −1.78347 −0.0625489
\(814\) −21.1355 −0.740798
\(815\) −19.2276 −0.673514
\(816\) 0.420017 0.0147035
\(817\) 20.9967 0.734582
\(818\) −13.6234 −0.476330
\(819\) 34.5422 1.20700
\(820\) 3.58984 0.125363
\(821\) −53.7470 −1.87578 −0.937892 0.346927i \(-0.887225\pi\)
−0.937892 + 0.346927i \(0.887225\pi\)
\(822\) 2.08649 0.0727747
\(823\) 41.2143 1.43664 0.718320 0.695713i \(-0.244911\pi\)
0.718320 + 0.695713i \(0.244911\pi\)
\(824\) 14.0294 0.488737
\(825\) −1.94883 −0.0678496
\(826\) 20.3734 0.708880
\(827\) −25.6253 −0.891078 −0.445539 0.895263i \(-0.646988\pi\)
−0.445539 + 0.895263i \(0.646988\pi\)
\(828\) −10.0194 −0.348198
\(829\) 7.20305 0.250172 0.125086 0.992146i \(-0.460079\pi\)
0.125086 + 0.992146i \(0.460079\pi\)
\(830\) −1.01180 −0.0351202
\(831\) 4.35341 0.151018
\(832\) −3.78538 −0.131234
\(833\) −6.04076 −0.209300
\(834\) 1.35766 0.0470118
\(835\) 13.6836 0.473541
\(836\) 19.1165 0.661157
\(837\) 4.87764 0.168596
\(838\) 0.236224 0.00816023
\(839\) 40.6571 1.40364 0.701820 0.712355i \(-0.252371\pi\)
0.701820 + 0.712355i \(0.252371\pi\)
\(840\) −0.422883 −0.0145909
\(841\) 4.65410 0.160486
\(842\) 40.4372 1.39356
\(843\) −0.753412 −0.0259489
\(844\) −9.12273 −0.314017
\(845\) −1.08285 −0.0372513
\(846\) 0.113990 0.00391905
\(847\) −12.0659 −0.414589
\(848\) 8.11722 0.278747
\(849\) −0.181383 −0.00622504
\(850\) −10.7758 −0.369607
\(851\) −26.8008 −0.918720
\(852\) −1.55287 −0.0532003
\(853\) −13.5529 −0.464044 −0.232022 0.972711i \(-0.574534\pi\)
−0.232022 + 0.972711i \(0.574534\pi\)
\(854\) −41.7291 −1.42794
\(855\) −17.4042 −0.595211
\(856\) −15.0755 −0.515269
\(857\) −27.2976 −0.932467 −0.466234 0.884662i \(-0.654389\pi\)
−0.466234 + 0.884662i \(0.654389\pi\)
\(858\) 1.70128 0.0580806
\(859\) 0.705554 0.0240732 0.0120366 0.999928i \(-0.496169\pi\)
0.0120366 + 0.999928i \(0.496169\pi\)
\(860\) 2.37960 0.0811436
\(861\) 2.28695 0.0779389
\(862\) −38.0549 −1.29616
\(863\) 24.6526 0.839183 0.419591 0.907713i \(-0.362173\pi\)
0.419591 + 0.907713i \(0.362173\pi\)
\(864\) −1.00926 −0.0343359
\(865\) −7.08061 −0.240748
\(866\) −15.0745 −0.512254
\(867\) 1.82948 0.0621325
\(868\) 14.8416 0.503756
\(869\) 6.51956 0.221161
\(870\) 0.798850 0.0270835
\(871\) 37.9304 1.28522
\(872\) −4.89644 −0.165814
\(873\) −53.5232 −1.81148
\(874\) 24.2406 0.819951
\(875\) 23.3595 0.789697
\(876\) 0.678788 0.0229341
\(877\) 49.5757 1.67405 0.837025 0.547164i \(-0.184293\pi\)
0.837025 + 0.547164i \(0.184293\pi\)
\(878\) −31.0098 −1.04653
\(879\) 3.66821 0.123726
\(880\) 2.16650 0.0730328
\(881\) −19.7620 −0.665798 −0.332899 0.942963i \(-0.608027\pi\)
−0.332899 + 0.942963i \(0.608027\pi\)
\(882\) 7.22299 0.243211
\(883\) 5.58022 0.187789 0.0938946 0.995582i \(-0.470068\pi\)
0.0938946 + 0.995582i \(0.470068\pi\)
\(884\) 9.40697 0.316391
\(885\) −0.913554 −0.0307088
\(886\) 25.9257 0.870991
\(887\) 27.8003 0.933441 0.466721 0.884405i \(-0.345435\pi\)
0.466721 + 0.884405i \(0.345435\pi\)
\(888\) −1.34338 −0.0450808
\(889\) −31.1935 −1.04620
\(890\) 0.792636 0.0265692
\(891\) −23.2507 −0.778927
\(892\) 2.63638 0.0882725
\(893\) −0.275783 −0.00922874
\(894\) −1.88490 −0.0630404
\(895\) −0.706917 −0.0236296
\(896\) −3.07096 −0.102594
\(897\) 2.15730 0.0720302
\(898\) −18.0562 −0.602543
\(899\) −28.0365 −0.935070
\(900\) 12.8847 0.429491
\(901\) −20.1720 −0.672025
\(902\) −11.7164 −0.390114
\(903\) 1.51595 0.0504476
\(904\) −14.5861 −0.485128
\(905\) 1.68643 0.0560588
\(906\) 1.19976 0.0398592
\(907\) 3.99944 0.132799 0.0663997 0.997793i \(-0.478849\pi\)
0.0663997 + 0.997793i \(0.478849\pi\)
\(908\) −8.03189 −0.266548
\(909\) −30.2231 −1.00244
\(910\) −9.47117 −0.313966
\(911\) −28.1964 −0.934190 −0.467095 0.884207i \(-0.654699\pi\)
−0.467095 + 0.884207i \(0.654699\pi\)
\(912\) 1.21505 0.0402343
\(913\) 3.30229 0.109290
\(914\) −9.37221 −0.310005
\(915\) 1.87116 0.0618585
\(916\) 12.1310 0.400820
\(917\) 19.6570 0.649130
\(918\) 2.50810 0.0827797
\(919\) 25.6955 0.847616 0.423808 0.905752i \(-0.360693\pi\)
0.423808 + 0.905752i \(0.360693\pi\)
\(920\) 2.74723 0.0905736
\(921\) 1.46427 0.0482492
\(922\) −28.3684 −0.934262
\(923\) −34.7790 −1.14476
\(924\) 1.38019 0.0454051
\(925\) 34.4652 1.13321
\(926\) −11.6167 −0.381748
\(927\) 41.6874 1.36919
\(928\) 5.80122 0.190434
\(929\) −47.9780 −1.57411 −0.787053 0.616885i \(-0.788394\pi\)
−0.787053 + 0.616885i \(0.788394\pi\)
\(930\) −0.665505 −0.0218228
\(931\) −17.4751 −0.572722
\(932\) 2.20968 0.0723805
\(933\) −2.58009 −0.0844684
\(934\) −28.8799 −0.944978
\(935\) −5.38394 −0.176074
\(936\) −11.2480 −0.367652
\(937\) 3.12031 0.101936 0.0509680 0.998700i \(-0.483769\pi\)
0.0509680 + 0.998700i \(0.483769\pi\)
\(938\) 30.7718 1.00473
\(939\) 2.91112 0.0950007
\(940\) −0.0312551 −0.00101943
\(941\) 48.9185 1.59470 0.797349 0.603519i \(-0.206235\pi\)
0.797349 + 0.603519i \(0.206235\pi\)
\(942\) 0.915471 0.0298277
\(943\) −14.8570 −0.483810
\(944\) −6.63420 −0.215925
\(945\) −2.52522 −0.0821454
\(946\) −7.76646 −0.252509
\(947\) 21.1680 0.687867 0.343933 0.938994i \(-0.388241\pi\)
0.343933 + 0.938994i \(0.388241\pi\)
\(948\) 0.414385 0.0134586
\(949\) 15.2026 0.493497
\(950\) −31.1729 −1.01138
\(951\) 2.45259 0.0795308
\(952\) 7.63159 0.247341
\(953\) −7.17049 −0.232275 −0.116137 0.993233i \(-0.537051\pi\)
−0.116137 + 0.993233i \(0.537051\pi\)
\(954\) 24.1198 0.780907
\(955\) −1.10757 −0.0358400
\(956\) 7.70534 0.249208
\(957\) −2.60726 −0.0842808
\(958\) −14.5624 −0.470489
\(959\) 37.9109 1.22421
\(960\) 0.137704 0.00444437
\(961\) −7.64336 −0.246560
\(962\) −30.0872 −0.970050
\(963\) −44.7957 −1.44352
\(964\) 23.9056 0.769949
\(965\) 12.9611 0.417233
\(966\) 1.75015 0.0563103
\(967\) 45.5372 1.46438 0.732188 0.681102i \(-0.238499\pi\)
0.732188 + 0.681102i \(0.238499\pi\)
\(968\) 3.92903 0.126284
\(969\) −3.01950 −0.0970002
\(970\) 14.6756 0.471205
\(971\) −15.9227 −0.510985 −0.255492 0.966811i \(-0.582238\pi\)
−0.255492 + 0.966811i \(0.582238\pi\)
\(972\) −4.50561 −0.144518
\(973\) 24.6682 0.790827
\(974\) 29.8995 0.958041
\(975\) −2.77424 −0.0888468
\(976\) 13.5883 0.434950
\(977\) −8.23938 −0.263601 −0.131801 0.991276i \(-0.542076\pi\)
−0.131801 + 0.991276i \(0.542076\pi\)
\(978\) 3.98870 0.127545
\(979\) −2.58698 −0.0826803
\(980\) −1.98048 −0.0632642
\(981\) −14.5494 −0.464528
\(982\) −30.1072 −0.960759
\(983\) 19.2525 0.614060 0.307030 0.951700i \(-0.400665\pi\)
0.307030 + 0.951700i \(0.400665\pi\)
\(984\) −0.744700 −0.0237402
\(985\) −12.1400 −0.386812
\(986\) −14.4165 −0.459115
\(987\) −0.0199114 −0.000633786 0
\(988\) 27.2130 0.865762
\(989\) −9.84825 −0.313156
\(990\) 6.43762 0.204601
\(991\) 15.1501 0.481260 0.240630 0.970617i \(-0.422646\pi\)
0.240630 + 0.970617i \(0.422646\pi\)
\(992\) −4.83287 −0.153444
\(993\) 3.31583 0.105225
\(994\) −28.2152 −0.894931
\(995\) 9.66267 0.306327
\(996\) 0.209895 0.00665078
\(997\) 39.9948 1.26665 0.633324 0.773887i \(-0.281690\pi\)
0.633324 + 0.773887i \(0.281690\pi\)
\(998\) −35.4216 −1.12125
\(999\) −8.02190 −0.253802
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 8002.2.a.e.1.37 77
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8002.2.a.e.1.37 77 1.1 even 1 trivial