Properties

Label 8043.2.a.p.1.11
Level $8043$
Weight $2$
Character 8043.1
Self dual yes
Analytic conductor $64.224$
Analytic rank $0$
Dimension $41$
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] = [8043,2,Mod(1,8043)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8043, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8043.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8043 = 3 \cdot 7 \cdot 383 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8043.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(64.2236783457\)
Analytic rank: \(0\)
Dimension: \(41\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.11
Character \(\chi\) \(=\) 8043.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.54858 q^{2} +1.00000 q^{3} +0.398108 q^{4} -1.62644 q^{5} -1.54858 q^{6} -1.00000 q^{7} +2.48066 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.54858 q^{2} +1.00000 q^{3} +0.398108 q^{4} -1.62644 q^{5} -1.54858 q^{6} -1.00000 q^{7} +2.48066 q^{8} +1.00000 q^{9} +2.51868 q^{10} +0.532524 q^{11} +0.398108 q^{12} +4.90309 q^{13} +1.54858 q^{14} -1.62644 q^{15} -4.63773 q^{16} +5.08151 q^{17} -1.54858 q^{18} +8.23700 q^{19} -0.647500 q^{20} -1.00000 q^{21} -0.824657 q^{22} +5.63311 q^{23} +2.48066 q^{24} -2.35468 q^{25} -7.59285 q^{26} +1.00000 q^{27} -0.398108 q^{28} -0.0589117 q^{29} +2.51868 q^{30} +0.876882 q^{31} +2.22058 q^{32} +0.532524 q^{33} -7.86914 q^{34} +1.62644 q^{35} +0.398108 q^{36} +6.54693 q^{37} -12.7557 q^{38} +4.90309 q^{39} -4.03465 q^{40} +6.91105 q^{41} +1.54858 q^{42} -7.06674 q^{43} +0.212002 q^{44} -1.62644 q^{45} -8.72334 q^{46} +6.26639 q^{47} -4.63773 q^{48} +1.00000 q^{49} +3.64642 q^{50} +5.08151 q^{51} +1.95196 q^{52} +8.18967 q^{53} -1.54858 q^{54} -0.866120 q^{55} -2.48066 q^{56} +8.23700 q^{57} +0.0912297 q^{58} -11.6469 q^{59} -0.647500 q^{60} +6.00758 q^{61} -1.35792 q^{62} -1.00000 q^{63} +5.83670 q^{64} -7.97460 q^{65} -0.824657 q^{66} +15.6926 q^{67} +2.02299 q^{68} +5.63311 q^{69} -2.51868 q^{70} +2.52323 q^{71} +2.48066 q^{72} +12.5230 q^{73} -10.1385 q^{74} -2.35468 q^{75} +3.27922 q^{76} -0.532524 q^{77} -7.59285 q^{78} -11.4267 q^{79} +7.54300 q^{80} +1.00000 q^{81} -10.7023 q^{82} -4.97930 q^{83} -0.398108 q^{84} -8.26479 q^{85} +10.9434 q^{86} -0.0589117 q^{87} +1.32101 q^{88} +15.5049 q^{89} +2.51868 q^{90} -4.90309 q^{91} +2.24259 q^{92} +0.876882 q^{93} -9.70402 q^{94} -13.3970 q^{95} +2.22058 q^{96} -15.1390 q^{97} -1.54858 q^{98} +0.532524 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 41 q + 7 q^{2} + 41 q^{3} + 45 q^{4} + 17 q^{5} + 7 q^{6} - 41 q^{7} + 12 q^{8} + 41 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 41 q + 7 q^{2} + 41 q^{3} + 45 q^{4} + 17 q^{5} + 7 q^{6} - 41 q^{7} + 12 q^{8} + 41 q^{9} + 18 q^{10} + 8 q^{11} + 45 q^{12} + 23 q^{13} - 7 q^{14} + 17 q^{15} + 37 q^{16} + 15 q^{17} + 7 q^{18} + 15 q^{19} + 53 q^{20} - 41 q^{21} + 13 q^{22} + 44 q^{23} + 12 q^{24} + 58 q^{25} + 9 q^{26} + 41 q^{27} - 45 q^{28} + 21 q^{29} + 18 q^{30} + 39 q^{31} + 61 q^{32} + 8 q^{33} + 9 q^{34} - 17 q^{35} + 45 q^{36} + 11 q^{37} + 44 q^{38} + 23 q^{39} + 24 q^{40} + 17 q^{41} - 7 q^{42} + 7 q^{43} + 30 q^{44} + 17 q^{45} - 12 q^{46} + 36 q^{47} + 37 q^{48} + 41 q^{49} + 28 q^{50} + 15 q^{51} + 58 q^{52} + 26 q^{53} + 7 q^{54} + 32 q^{55} - 12 q^{56} + 15 q^{57} - 4 q^{58} + 33 q^{59} + 53 q^{60} + 59 q^{61} - q^{62} - 41 q^{63} + 16 q^{64} + 72 q^{65} + 13 q^{66} + 12 q^{67} + 52 q^{68} + 44 q^{69} - 18 q^{70} + 33 q^{71} + 12 q^{72} + 18 q^{73} + 42 q^{74} + 58 q^{75} + 7 q^{76} - 8 q^{77} + 9 q^{78} + 22 q^{79} + 69 q^{80} + 41 q^{81} + 41 q^{82} + 32 q^{83} - 45 q^{84} - 44 q^{85} + 11 q^{86} + 21 q^{87} + 52 q^{88} + 63 q^{89} + 18 q^{90} - 23 q^{91} + 52 q^{92} + 39 q^{93} + 17 q^{94} + 37 q^{95} + 61 q^{96} + 8 q^{97} + 7 q^{98} + 8 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.54858 −1.09501 −0.547507 0.836801i \(-0.684423\pi\)
−0.547507 + 0.836801i \(0.684423\pi\)
\(3\) 1.00000 0.577350
\(4\) 0.398108 0.199054
\(5\) −1.62644 −0.727367 −0.363684 0.931523i \(-0.618481\pi\)
−0.363684 + 0.931523i \(0.618481\pi\)
\(6\) −1.54858 −0.632206
\(7\) −1.00000 −0.377964
\(8\) 2.48066 0.877046
\(9\) 1.00000 0.333333
\(10\) 2.51868 0.796477
\(11\) 0.532524 0.160562 0.0802810 0.996772i \(-0.474418\pi\)
0.0802810 + 0.996772i \(0.474418\pi\)
\(12\) 0.398108 0.114924
\(13\) 4.90309 1.35987 0.679937 0.733271i \(-0.262007\pi\)
0.679937 + 0.733271i \(0.262007\pi\)
\(14\) 1.54858 0.413876
\(15\) −1.62644 −0.419946
\(16\) −4.63773 −1.15943
\(17\) 5.08151 1.23245 0.616224 0.787571i \(-0.288662\pi\)
0.616224 + 0.787571i \(0.288662\pi\)
\(18\) −1.54858 −0.365004
\(19\) 8.23700 1.88970 0.944849 0.327507i \(-0.106209\pi\)
0.944849 + 0.327507i \(0.106209\pi\)
\(20\) −0.647500 −0.144785
\(21\) −1.00000 −0.218218
\(22\) −0.824657 −0.175818
\(23\) 5.63311 1.17459 0.587293 0.809375i \(-0.300194\pi\)
0.587293 + 0.809375i \(0.300194\pi\)
\(24\) 2.48066 0.506363
\(25\) −2.35468 −0.470937
\(26\) −7.59285 −1.48908
\(27\) 1.00000 0.192450
\(28\) −0.398108 −0.0752353
\(29\) −0.0589117 −0.0109396 −0.00546982 0.999985i \(-0.501741\pi\)
−0.00546982 + 0.999985i \(0.501741\pi\)
\(30\) 2.51868 0.459846
\(31\) 0.876882 0.157493 0.0787463 0.996895i \(-0.474908\pi\)
0.0787463 + 0.996895i \(0.474908\pi\)
\(32\) 2.22058 0.392546
\(33\) 0.532524 0.0927005
\(34\) −7.86914 −1.34955
\(35\) 1.62644 0.274919
\(36\) 0.398108 0.0663513
\(37\) 6.54693 1.07631 0.538154 0.842846i \(-0.319122\pi\)
0.538154 + 0.842846i \(0.319122\pi\)
\(38\) −12.7557 −2.06924
\(39\) 4.90309 0.785124
\(40\) −4.03465 −0.637935
\(41\) 6.91105 1.07932 0.539662 0.841882i \(-0.318552\pi\)
0.539662 + 0.841882i \(0.318552\pi\)
\(42\) 1.54858 0.238951
\(43\) −7.06674 −1.07767 −0.538834 0.842412i \(-0.681135\pi\)
−0.538834 + 0.842412i \(0.681135\pi\)
\(44\) 0.212002 0.0319605
\(45\) −1.62644 −0.242456
\(46\) −8.72334 −1.28619
\(47\) 6.26639 0.914046 0.457023 0.889455i \(-0.348916\pi\)
0.457023 + 0.889455i \(0.348916\pi\)
\(48\) −4.63773 −0.669398
\(49\) 1.00000 0.142857
\(50\) 3.64642 0.515682
\(51\) 5.08151 0.711554
\(52\) 1.95196 0.270688
\(53\) 8.18967 1.12494 0.562469 0.826819i \(-0.309852\pi\)
0.562469 + 0.826819i \(0.309852\pi\)
\(54\) −1.54858 −0.210735
\(55\) −0.866120 −0.116788
\(56\) −2.48066 −0.331492
\(57\) 8.23700 1.09102
\(58\) 0.0912297 0.0119790
\(59\) −11.6469 −1.51630 −0.758148 0.652082i \(-0.773896\pi\)
−0.758148 + 0.652082i \(0.773896\pi\)
\(60\) −0.647500 −0.0835919
\(61\) 6.00758 0.769192 0.384596 0.923085i \(-0.374341\pi\)
0.384596 + 0.923085i \(0.374341\pi\)
\(62\) −1.35792 −0.172457
\(63\) −1.00000 −0.125988
\(64\) 5.83670 0.729588
\(65\) −7.97460 −0.989128
\(66\) −0.824657 −0.101508
\(67\) 15.6926 1.91716 0.958579 0.284828i \(-0.0919363\pi\)
0.958579 + 0.284828i \(0.0919363\pi\)
\(68\) 2.02299 0.245324
\(69\) 5.63311 0.678147
\(70\) −2.51868 −0.301040
\(71\) 2.52323 0.299453 0.149726 0.988727i \(-0.452161\pi\)
0.149726 + 0.988727i \(0.452161\pi\)
\(72\) 2.48066 0.292349
\(73\) 12.5230 1.46571 0.732855 0.680385i \(-0.238187\pi\)
0.732855 + 0.680385i \(0.238187\pi\)
\(74\) −10.1385 −1.17857
\(75\) −2.35468 −0.271895
\(76\) 3.27922 0.376152
\(77\) −0.532524 −0.0606867
\(78\) −7.59285 −0.859721
\(79\) −11.4267 −1.28561 −0.642804 0.766031i \(-0.722229\pi\)
−0.642804 + 0.766031i \(0.722229\pi\)
\(80\) 7.54300 0.843333
\(81\) 1.00000 0.111111
\(82\) −10.7023 −1.18187
\(83\) −4.97930 −0.546549 −0.273274 0.961936i \(-0.588107\pi\)
−0.273274 + 0.961936i \(0.588107\pi\)
\(84\) −0.398108 −0.0434371
\(85\) −8.26479 −0.896442
\(86\) 10.9434 1.18006
\(87\) −0.0589117 −0.00631600
\(88\) 1.32101 0.140820
\(89\) 15.5049 1.64352 0.821760 0.569834i \(-0.192993\pi\)
0.821760 + 0.569834i \(0.192993\pi\)
\(90\) 2.51868 0.265492
\(91\) −4.90309 −0.513984
\(92\) 2.24259 0.233806
\(93\) 0.876882 0.0909284
\(94\) −9.70402 −1.00089
\(95\) −13.3970 −1.37450
\(96\) 2.22058 0.226637
\(97\) −15.1390 −1.53714 −0.768568 0.639769i \(-0.779030\pi\)
−0.768568 + 0.639769i \(0.779030\pi\)
\(98\) −1.54858 −0.156430
\(99\) 0.532524 0.0535207
\(100\) −0.937419 −0.0937419
\(101\) 1.97335 0.196355 0.0981776 0.995169i \(-0.468699\pi\)
0.0981776 + 0.995169i \(0.468699\pi\)
\(102\) −7.86914 −0.779161
\(103\) −19.1282 −1.88476 −0.942378 0.334550i \(-0.891416\pi\)
−0.942378 + 0.334550i \(0.891416\pi\)
\(104\) 12.1629 1.19267
\(105\) 1.62644 0.158725
\(106\) −12.6824 −1.23182
\(107\) −15.8203 −1.52941 −0.764703 0.644383i \(-0.777114\pi\)
−0.764703 + 0.644383i \(0.777114\pi\)
\(108\) 0.398108 0.0383080
\(109\) 7.82766 0.749754 0.374877 0.927075i \(-0.377685\pi\)
0.374877 + 0.927075i \(0.377685\pi\)
\(110\) 1.34126 0.127884
\(111\) 6.54693 0.621407
\(112\) 4.63773 0.438224
\(113\) 9.84192 0.925850 0.462925 0.886398i \(-0.346800\pi\)
0.462925 + 0.886398i \(0.346800\pi\)
\(114\) −12.7557 −1.19468
\(115\) −9.16194 −0.854355
\(116\) −0.0234532 −0.00217758
\(117\) 4.90309 0.453291
\(118\) 18.0362 1.66036
\(119\) −5.08151 −0.465821
\(120\) −4.03465 −0.368312
\(121\) −10.7164 −0.974220
\(122\) −9.30323 −0.842275
\(123\) 6.91105 0.623148
\(124\) 0.349094 0.0313495
\(125\) 11.9620 1.06991
\(126\) 1.54858 0.137959
\(127\) −7.22997 −0.641556 −0.320778 0.947154i \(-0.603944\pi\)
−0.320778 + 0.947154i \(0.603944\pi\)
\(128\) −13.4798 −1.19145
\(129\) −7.06674 −0.622192
\(130\) 12.3493 1.08311
\(131\) 1.58006 0.138050 0.0690252 0.997615i \(-0.478011\pi\)
0.0690252 + 0.997615i \(0.478011\pi\)
\(132\) 0.212002 0.0184524
\(133\) −8.23700 −0.714239
\(134\) −24.3013 −2.09931
\(135\) −1.62644 −0.139982
\(136\) 12.6055 1.08091
\(137\) −9.63658 −0.823309 −0.411654 0.911340i \(-0.635049\pi\)
−0.411654 + 0.911340i \(0.635049\pi\)
\(138\) −8.72334 −0.742580
\(139\) −17.9361 −1.52132 −0.760660 0.649151i \(-0.775124\pi\)
−0.760660 + 0.649151i \(0.775124\pi\)
\(140\) 0.647500 0.0547237
\(141\) 6.26639 0.527725
\(142\) −3.90744 −0.327905
\(143\) 2.61102 0.218344
\(144\) −4.63773 −0.386477
\(145\) 0.0958166 0.00795713
\(146\) −19.3929 −1.60497
\(147\) 1.00000 0.0824786
\(148\) 2.60639 0.214244
\(149\) 16.3820 1.34206 0.671031 0.741429i \(-0.265852\pi\)
0.671031 + 0.741429i \(0.265852\pi\)
\(150\) 3.64642 0.297729
\(151\) 17.2602 1.40462 0.702309 0.711872i \(-0.252152\pi\)
0.702309 + 0.711872i \(0.252152\pi\)
\(152\) 20.4332 1.65735
\(153\) 5.08151 0.410816
\(154\) 0.824657 0.0664528
\(155\) −1.42620 −0.114555
\(156\) 1.95196 0.156282
\(157\) −12.6930 −1.01302 −0.506508 0.862235i \(-0.669064\pi\)
−0.506508 + 0.862235i \(0.669064\pi\)
\(158\) 17.6952 1.40776
\(159\) 8.18967 0.649483
\(160\) −3.61164 −0.285525
\(161\) −5.63311 −0.443952
\(162\) −1.54858 −0.121668
\(163\) 8.70476 0.681810 0.340905 0.940098i \(-0.389267\pi\)
0.340905 + 0.940098i \(0.389267\pi\)
\(164\) 2.75134 0.214844
\(165\) −0.866120 −0.0674273
\(166\) 7.71085 0.598478
\(167\) −15.8115 −1.22353 −0.611765 0.791039i \(-0.709540\pi\)
−0.611765 + 0.791039i \(0.709540\pi\)
\(168\) −2.48066 −0.191387
\(169\) 11.0403 0.849257
\(170\) 12.7987 0.981616
\(171\) 8.23700 0.629899
\(172\) −2.81333 −0.214514
\(173\) −5.24891 −0.399067 −0.199533 0.979891i \(-0.563943\pi\)
−0.199533 + 0.979891i \(0.563943\pi\)
\(174\) 0.0912297 0.00691611
\(175\) 2.35468 0.177997
\(176\) −2.46970 −0.186161
\(177\) −11.6469 −0.875434
\(178\) −24.0107 −1.79968
\(179\) −5.54185 −0.414217 −0.207109 0.978318i \(-0.566405\pi\)
−0.207109 + 0.978318i \(0.566405\pi\)
\(180\) −0.647500 −0.0482618
\(181\) −12.3875 −0.920753 −0.460376 0.887724i \(-0.652286\pi\)
−0.460376 + 0.887724i \(0.652286\pi\)
\(182\) 7.59285 0.562819
\(183\) 6.00758 0.444093
\(184\) 13.9739 1.03017
\(185\) −10.6482 −0.782872
\(186\) −1.35792 −0.0995678
\(187\) 2.70603 0.197884
\(188\) 2.49470 0.181945
\(189\) −1.00000 −0.0727393
\(190\) 20.7464 1.50510
\(191\) 12.1231 0.877199 0.438600 0.898683i \(-0.355475\pi\)
0.438600 + 0.898683i \(0.355475\pi\)
\(192\) 5.83670 0.421228
\(193\) −19.9176 −1.43370 −0.716850 0.697228i \(-0.754417\pi\)
−0.716850 + 0.697228i \(0.754417\pi\)
\(194\) 23.4440 1.68318
\(195\) −7.97460 −0.571073
\(196\) 0.398108 0.0284363
\(197\) −8.45913 −0.602688 −0.301344 0.953515i \(-0.597435\pi\)
−0.301344 + 0.953515i \(0.597435\pi\)
\(198\) −0.824657 −0.0586059
\(199\) −11.9298 −0.845681 −0.422841 0.906204i \(-0.638967\pi\)
−0.422841 + 0.906204i \(0.638967\pi\)
\(200\) −5.84117 −0.413033
\(201\) 15.6926 1.10687
\(202\) −3.05589 −0.215012
\(203\) 0.0589117 0.00413479
\(204\) 2.02299 0.141638
\(205\) −11.2404 −0.785065
\(206\) 29.6216 2.06383
\(207\) 5.63311 0.391529
\(208\) −22.7392 −1.57668
\(209\) 4.38640 0.303414
\(210\) −2.51868 −0.173805
\(211\) 8.29673 0.571170 0.285585 0.958353i \(-0.407812\pi\)
0.285585 + 0.958353i \(0.407812\pi\)
\(212\) 3.26037 0.223923
\(213\) 2.52323 0.172889
\(214\) 24.4990 1.67472
\(215\) 11.4936 0.783860
\(216\) 2.48066 0.168788
\(217\) −0.876882 −0.0595266
\(218\) −12.1218 −0.820991
\(219\) 12.5230 0.846228
\(220\) −0.344809 −0.0232470
\(221\) 24.9151 1.67597
\(222\) −10.1385 −0.680449
\(223\) −16.8337 −1.12727 −0.563634 0.826025i \(-0.690597\pi\)
−0.563634 + 0.826025i \(0.690597\pi\)
\(224\) −2.22058 −0.148369
\(225\) −2.35468 −0.156979
\(226\) −15.2410 −1.01382
\(227\) 8.26899 0.548832 0.274416 0.961611i \(-0.411515\pi\)
0.274416 + 0.961611i \(0.411515\pi\)
\(228\) 3.27922 0.217171
\(229\) −21.2886 −1.40679 −0.703396 0.710798i \(-0.748334\pi\)
−0.703396 + 0.710798i \(0.748334\pi\)
\(230\) 14.1880 0.935530
\(231\) −0.532524 −0.0350375
\(232\) −0.146140 −0.00959457
\(233\) 6.91299 0.452885 0.226442 0.974025i \(-0.427290\pi\)
0.226442 + 0.974025i \(0.427290\pi\)
\(234\) −7.59285 −0.496360
\(235\) −10.1919 −0.664847
\(236\) −4.63672 −0.301825
\(237\) −11.4267 −0.742246
\(238\) 7.86914 0.510081
\(239\) 8.17338 0.528692 0.264346 0.964428i \(-0.414844\pi\)
0.264346 + 0.964428i \(0.414844\pi\)
\(240\) 7.54300 0.486898
\(241\) −8.16882 −0.526200 −0.263100 0.964769i \(-0.584745\pi\)
−0.263100 + 0.964769i \(0.584745\pi\)
\(242\) 16.5953 1.06678
\(243\) 1.00000 0.0641500
\(244\) 2.39167 0.153111
\(245\) −1.62644 −0.103910
\(246\) −10.7023 −0.682356
\(247\) 40.3868 2.56975
\(248\) 2.17525 0.138128
\(249\) −4.97930 −0.315550
\(250\) −18.5241 −1.17157
\(251\) −5.03779 −0.317982 −0.158991 0.987280i \(-0.550824\pi\)
−0.158991 + 0.987280i \(0.550824\pi\)
\(252\) −0.398108 −0.0250784
\(253\) 2.99977 0.188594
\(254\) 11.1962 0.702512
\(255\) −8.26479 −0.517561
\(256\) 9.20113 0.575071
\(257\) 13.8553 0.864271 0.432135 0.901809i \(-0.357760\pi\)
0.432135 + 0.901809i \(0.357760\pi\)
\(258\) 10.9434 0.681308
\(259\) −6.54693 −0.406807
\(260\) −3.17475 −0.196890
\(261\) −0.0589117 −0.00364655
\(262\) −2.44685 −0.151167
\(263\) −17.6071 −1.08570 −0.542849 0.839830i \(-0.682655\pi\)
−0.542849 + 0.839830i \(0.682655\pi\)
\(264\) 1.32101 0.0813027
\(265\) −13.3200 −0.818243
\(266\) 12.7557 0.782101
\(267\) 15.5049 0.948886
\(268\) 6.24735 0.381618
\(269\) −20.5693 −1.25414 −0.627068 0.778965i \(-0.715745\pi\)
−0.627068 + 0.778965i \(0.715745\pi\)
\(270\) 2.51868 0.153282
\(271\) 16.2853 0.989263 0.494632 0.869103i \(-0.335303\pi\)
0.494632 + 0.869103i \(0.335303\pi\)
\(272\) −23.5667 −1.42894
\(273\) −4.90309 −0.296749
\(274\) 14.9230 0.901534
\(275\) −1.25393 −0.0756146
\(276\) 2.24259 0.134988
\(277\) 0.403491 0.0242434 0.0121217 0.999927i \(-0.496141\pi\)
0.0121217 + 0.999927i \(0.496141\pi\)
\(278\) 27.7755 1.66586
\(279\) 0.876882 0.0524975
\(280\) 4.03465 0.241117
\(281\) 25.3008 1.50932 0.754658 0.656118i \(-0.227803\pi\)
0.754658 + 0.656118i \(0.227803\pi\)
\(282\) −9.70402 −0.577866
\(283\) −9.98933 −0.593804 −0.296902 0.954908i \(-0.595953\pi\)
−0.296902 + 0.954908i \(0.595953\pi\)
\(284\) 1.00452 0.0596073
\(285\) −13.3970 −0.793570
\(286\) −4.04337 −0.239090
\(287\) −6.91105 −0.407946
\(288\) 2.22058 0.130849
\(289\) 8.82176 0.518927
\(290\) −0.148380 −0.00871317
\(291\) −15.1390 −0.887465
\(292\) 4.98552 0.291755
\(293\) −3.63791 −0.212529 −0.106265 0.994338i \(-0.533889\pi\)
−0.106265 + 0.994338i \(0.533889\pi\)
\(294\) −1.54858 −0.0903152
\(295\) 18.9430 1.10290
\(296\) 16.2407 0.943973
\(297\) 0.532524 0.0309002
\(298\) −25.3688 −1.46958
\(299\) 27.6197 1.59729
\(300\) −0.937419 −0.0541219
\(301\) 7.06674 0.407320
\(302\) −26.7289 −1.53808
\(303\) 1.97335 0.113366
\(304\) −38.2010 −2.19097
\(305\) −9.77098 −0.559485
\(306\) −7.86914 −0.449849
\(307\) 10.6335 0.606885 0.303442 0.952850i \(-0.401864\pi\)
0.303442 + 0.952850i \(0.401864\pi\)
\(308\) −0.212002 −0.0120799
\(309\) −19.1282 −1.08816
\(310\) 2.20859 0.125439
\(311\) 11.5142 0.652909 0.326454 0.945213i \(-0.394146\pi\)
0.326454 + 0.945213i \(0.394146\pi\)
\(312\) 12.1629 0.688590
\(313\) 22.5354 1.27378 0.636889 0.770955i \(-0.280221\pi\)
0.636889 + 0.770955i \(0.280221\pi\)
\(314\) 19.6562 1.10927
\(315\) 1.62644 0.0916397
\(316\) −4.54908 −0.255905
\(317\) 7.99031 0.448780 0.224390 0.974499i \(-0.427961\pi\)
0.224390 + 0.974499i \(0.427961\pi\)
\(318\) −12.6824 −0.711192
\(319\) −0.0313719 −0.00175649
\(320\) −9.49307 −0.530678
\(321\) −15.8203 −0.883003
\(322\) 8.72334 0.486133
\(323\) 41.8564 2.32895
\(324\) 0.398108 0.0221171
\(325\) −11.5452 −0.640415
\(326\) −13.4800 −0.746591
\(327\) 7.82766 0.432871
\(328\) 17.1440 0.946618
\(329\) −6.26639 −0.345477
\(330\) 1.34126 0.0738338
\(331\) 23.4689 1.28997 0.644984 0.764196i \(-0.276864\pi\)
0.644984 + 0.764196i \(0.276864\pi\)
\(332\) −1.98230 −0.108793
\(333\) 6.54693 0.358770
\(334\) 24.4854 1.33978
\(335\) −25.5231 −1.39448
\(336\) 4.63773 0.253009
\(337\) −5.40122 −0.294223 −0.147112 0.989120i \(-0.546998\pi\)
−0.147112 + 0.989120i \(0.546998\pi\)
\(338\) −17.0969 −0.929948
\(339\) 9.84192 0.534540
\(340\) −3.29028 −0.178440
\(341\) 0.466961 0.0252873
\(342\) −12.7557 −0.689748
\(343\) −1.00000 −0.0539949
\(344\) −17.5302 −0.945165
\(345\) −9.16194 −0.493262
\(346\) 8.12836 0.436984
\(347\) 35.3855 1.89959 0.949796 0.312871i \(-0.101291\pi\)
0.949796 + 0.312871i \(0.101291\pi\)
\(348\) −0.0234532 −0.00125723
\(349\) 0.416052 0.0222708 0.0111354 0.999938i \(-0.496455\pi\)
0.0111354 + 0.999938i \(0.496455\pi\)
\(350\) −3.64642 −0.194909
\(351\) 4.90309 0.261708
\(352\) 1.18251 0.0630281
\(353\) 9.79867 0.521531 0.260765 0.965402i \(-0.416025\pi\)
0.260765 + 0.965402i \(0.416025\pi\)
\(354\) 18.0362 0.958612
\(355\) −4.10390 −0.217812
\(356\) 6.17264 0.327149
\(357\) −5.08151 −0.268942
\(358\) 8.58201 0.453573
\(359\) 22.0624 1.16441 0.582205 0.813042i \(-0.302190\pi\)
0.582205 + 0.813042i \(0.302190\pi\)
\(360\) −4.03465 −0.212645
\(361\) 48.8482 2.57096
\(362\) 19.1830 1.00824
\(363\) −10.7164 −0.562466
\(364\) −1.95196 −0.102311
\(365\) −20.3680 −1.06611
\(366\) −9.30323 −0.486288
\(367\) −25.5898 −1.33578 −0.667888 0.744262i \(-0.732802\pi\)
−0.667888 + 0.744262i \(0.732802\pi\)
\(368\) −26.1248 −1.36185
\(369\) 6.91105 0.359775
\(370\) 16.4896 0.857255
\(371\) −8.18967 −0.425186
\(372\) 0.349094 0.0180997
\(373\) 12.9959 0.672901 0.336451 0.941701i \(-0.390773\pi\)
0.336451 + 0.941701i \(0.390773\pi\)
\(374\) −4.19051 −0.216686
\(375\) 11.9620 0.617714
\(376\) 15.5448 0.801661
\(377\) −0.288850 −0.0148765
\(378\) 1.54858 0.0796505
\(379\) −15.5549 −0.799003 −0.399502 0.916733i \(-0.630817\pi\)
−0.399502 + 0.916733i \(0.630817\pi\)
\(380\) −5.33346 −0.273601
\(381\) −7.22997 −0.370403
\(382\) −18.7737 −0.960545
\(383\) −1.00000 −0.0510976
\(384\) −13.4798 −0.687887
\(385\) 0.866120 0.0441416
\(386\) 30.8440 1.56992
\(387\) −7.06674 −0.359223
\(388\) −6.02697 −0.305973
\(389\) −21.1533 −1.07251 −0.536257 0.844055i \(-0.680162\pi\)
−0.536257 + 0.844055i \(0.680162\pi\)
\(390\) 12.3493 0.625333
\(391\) 28.6247 1.44762
\(392\) 2.48066 0.125292
\(393\) 1.58006 0.0797034
\(394\) 13.0997 0.659951
\(395\) 18.5849 0.935109
\(396\) 0.212002 0.0106535
\(397\) 37.1404 1.86403 0.932013 0.362425i \(-0.118051\pi\)
0.932013 + 0.362425i \(0.118051\pi\)
\(398\) 18.4743 0.926032
\(399\) −8.23700 −0.412366
\(400\) 10.9204 0.546019
\(401\) −9.97826 −0.498290 −0.249145 0.968466i \(-0.580150\pi\)
−0.249145 + 0.968466i \(0.580150\pi\)
\(402\) −24.3013 −1.21204
\(403\) 4.29943 0.214170
\(404\) 0.785605 0.0390853
\(405\) −1.62644 −0.0808186
\(406\) −0.0912297 −0.00452765
\(407\) 3.48640 0.172814
\(408\) 12.6055 0.624066
\(409\) 5.83605 0.288574 0.144287 0.989536i \(-0.453911\pi\)
0.144287 + 0.989536i \(0.453911\pi\)
\(410\) 17.4067 0.859657
\(411\) −9.63658 −0.475338
\(412\) −7.61508 −0.375168
\(413\) 11.6469 0.573106
\(414\) −8.72334 −0.428729
\(415\) 8.09854 0.397542
\(416\) 10.8877 0.533814
\(417\) −17.9361 −0.878334
\(418\) −6.79270 −0.332242
\(419\) −34.4415 −1.68258 −0.841288 0.540586i \(-0.818202\pi\)
−0.841288 + 0.540586i \(0.818202\pi\)
\(420\) 0.647500 0.0315948
\(421\) 33.7371 1.64425 0.822123 0.569310i \(-0.192790\pi\)
0.822123 + 0.569310i \(0.192790\pi\)
\(422\) −12.8482 −0.625439
\(423\) 6.26639 0.304682
\(424\) 20.3158 0.986622
\(425\) −11.9654 −0.580405
\(426\) −3.90744 −0.189316
\(427\) −6.00758 −0.290727
\(428\) −6.29819 −0.304434
\(429\) 2.61102 0.126061
\(430\) −17.7989 −0.858337
\(431\) 26.7297 1.28752 0.643762 0.765226i \(-0.277373\pi\)
0.643762 + 0.765226i \(0.277373\pi\)
\(432\) −4.63773 −0.223133
\(433\) −14.6626 −0.704640 −0.352320 0.935880i \(-0.614607\pi\)
−0.352320 + 0.935880i \(0.614607\pi\)
\(434\) 1.35792 0.0651824
\(435\) 0.0958166 0.00459405
\(436\) 3.11626 0.149242
\(437\) 46.4000 2.21961
\(438\) −19.3929 −0.926631
\(439\) −20.1759 −0.962944 −0.481472 0.876462i \(-0.659898\pi\)
−0.481472 + 0.876462i \(0.659898\pi\)
\(440\) −2.14855 −0.102428
\(441\) 1.00000 0.0476190
\(442\) −38.5831 −1.83521
\(443\) −18.3348 −0.871113 −0.435556 0.900161i \(-0.643448\pi\)
−0.435556 + 0.900161i \(0.643448\pi\)
\(444\) 2.60639 0.123694
\(445\) −25.2179 −1.19544
\(446\) 26.0684 1.23437
\(447\) 16.3820 0.774840
\(448\) −5.83670 −0.275758
\(449\) −16.9010 −0.797608 −0.398804 0.917036i \(-0.630575\pi\)
−0.398804 + 0.917036i \(0.630575\pi\)
\(450\) 3.64642 0.171894
\(451\) 3.68030 0.173299
\(452\) 3.91815 0.184294
\(453\) 17.2602 0.810957
\(454\) −12.8052 −0.600978
\(455\) 7.97460 0.373855
\(456\) 20.4332 0.956873
\(457\) −10.5701 −0.494449 −0.247224 0.968958i \(-0.579518\pi\)
−0.247224 + 0.968958i \(0.579518\pi\)
\(458\) 32.9672 1.54046
\(459\) 5.08151 0.237185
\(460\) −3.64744 −0.170063
\(461\) −21.0826 −0.981913 −0.490957 0.871184i \(-0.663353\pi\)
−0.490957 + 0.871184i \(0.663353\pi\)
\(462\) 0.824657 0.0383665
\(463\) −28.0464 −1.30342 −0.651712 0.758466i \(-0.725949\pi\)
−0.651712 + 0.758466i \(0.725949\pi\)
\(464\) 0.273217 0.0126838
\(465\) −1.42620 −0.0661384
\(466\) −10.7053 −0.495915
\(467\) 32.5560 1.50651 0.753255 0.657728i \(-0.228483\pi\)
0.753255 + 0.657728i \(0.228483\pi\)
\(468\) 1.95196 0.0902294
\(469\) −15.6926 −0.724617
\(470\) 15.7830 0.728017
\(471\) −12.6930 −0.584865
\(472\) −28.8920 −1.32986
\(473\) −3.76321 −0.173033
\(474\) 17.6952 0.812770
\(475\) −19.3955 −0.889928
\(476\) −2.02299 −0.0927236
\(477\) 8.18967 0.374979
\(478\) −12.6571 −0.578924
\(479\) 20.3345 0.929106 0.464553 0.885545i \(-0.346215\pi\)
0.464553 + 0.885545i \(0.346215\pi\)
\(480\) −3.61164 −0.164848
\(481\) 32.1002 1.46364
\(482\) 12.6501 0.576196
\(483\) −5.63311 −0.256316
\(484\) −4.26629 −0.193922
\(485\) 24.6228 1.11806
\(486\) −1.54858 −0.0702451
\(487\) −9.03081 −0.409225 −0.204613 0.978843i \(-0.565593\pi\)
−0.204613 + 0.978843i \(0.565593\pi\)
\(488\) 14.9028 0.674617
\(489\) 8.70476 0.393643
\(490\) 2.51868 0.113782
\(491\) −2.83301 −0.127852 −0.0639260 0.997955i \(-0.520362\pi\)
−0.0639260 + 0.997955i \(0.520362\pi\)
\(492\) 2.75134 0.124040
\(493\) −0.299361 −0.0134825
\(494\) −62.5423 −2.81391
\(495\) −0.866120 −0.0389292
\(496\) −4.06674 −0.182602
\(497\) −2.52323 −0.113183
\(498\) 7.71085 0.345532
\(499\) −27.5225 −1.23208 −0.616039 0.787716i \(-0.711264\pi\)
−0.616039 + 0.787716i \(0.711264\pi\)
\(500\) 4.76216 0.212970
\(501\) −15.8115 −0.706406
\(502\) 7.80143 0.348195
\(503\) 33.3770 1.48821 0.744104 0.668064i \(-0.232877\pi\)
0.744104 + 0.668064i \(0.232877\pi\)
\(504\) −2.48066 −0.110497
\(505\) −3.20953 −0.142822
\(506\) −4.64539 −0.206513
\(507\) 11.0403 0.490319
\(508\) −2.87831 −0.127704
\(509\) −9.06947 −0.401997 −0.200999 0.979592i \(-0.564419\pi\)
−0.200999 + 0.979592i \(0.564419\pi\)
\(510\) 12.7987 0.566736
\(511\) −12.5230 −0.553986
\(512\) 12.7108 0.561745
\(513\) 8.23700 0.363672
\(514\) −21.4561 −0.946388
\(515\) 31.1109 1.37091
\(516\) −2.81333 −0.123850
\(517\) 3.33700 0.146761
\(518\) 10.1385 0.445459
\(519\) −5.24891 −0.230401
\(520\) −19.7823 −0.867511
\(521\) 14.4705 0.633965 0.316982 0.948431i \(-0.397330\pi\)
0.316982 + 0.948431i \(0.397330\pi\)
\(522\) 0.0912297 0.00399302
\(523\) 29.6323 1.29573 0.647865 0.761755i \(-0.275662\pi\)
0.647865 + 0.761755i \(0.275662\pi\)
\(524\) 0.629034 0.0274795
\(525\) 2.35468 0.102767
\(526\) 27.2660 1.18885
\(527\) 4.45588 0.194101
\(528\) −2.46970 −0.107480
\(529\) 8.73198 0.379651
\(530\) 20.6272 0.895986
\(531\) −11.6469 −0.505432
\(532\) −3.27922 −0.142172
\(533\) 33.8855 1.46774
\(534\) −24.0107 −1.03904
\(535\) 25.7308 1.11244
\(536\) 38.9281 1.68144
\(537\) −5.54185 −0.239148
\(538\) 31.8533 1.37329
\(539\) 0.532524 0.0229374
\(540\) −0.647500 −0.0278640
\(541\) 2.02251 0.0869545 0.0434773 0.999054i \(-0.486156\pi\)
0.0434773 + 0.999054i \(0.486156\pi\)
\(542\) −25.2192 −1.08326
\(543\) −12.3875 −0.531597
\(544\) 11.2839 0.483793
\(545\) −12.7312 −0.545347
\(546\) 7.59285 0.324944
\(547\) 10.0940 0.431589 0.215795 0.976439i \(-0.430766\pi\)
0.215795 + 0.976439i \(0.430766\pi\)
\(548\) −3.83640 −0.163883
\(549\) 6.00758 0.256397
\(550\) 1.94181 0.0827990
\(551\) −0.485256 −0.0206726
\(552\) 13.9739 0.594767
\(553\) 11.4267 0.485914
\(554\) −0.624839 −0.0265469
\(555\) −10.6482 −0.451991
\(556\) −7.14050 −0.302825
\(557\) −20.8793 −0.884682 −0.442341 0.896847i \(-0.645852\pi\)
−0.442341 + 0.896847i \(0.645852\pi\)
\(558\) −1.35792 −0.0574855
\(559\) −34.6489 −1.46549
\(560\) −7.54300 −0.318750
\(561\) 2.70603 0.114249
\(562\) −39.1803 −1.65272
\(563\) 28.7929 1.21348 0.606738 0.794902i \(-0.292478\pi\)
0.606738 + 0.794902i \(0.292478\pi\)
\(564\) 2.49470 0.105046
\(565\) −16.0073 −0.673433
\(566\) 15.4693 0.650223
\(567\) −1.00000 −0.0419961
\(568\) 6.25929 0.262634
\(569\) 29.9344 1.25491 0.627457 0.778651i \(-0.284096\pi\)
0.627457 + 0.778651i \(0.284096\pi\)
\(570\) 20.7464 0.868970
\(571\) −43.2703 −1.81080 −0.905402 0.424555i \(-0.860431\pi\)
−0.905402 + 0.424555i \(0.860431\pi\)
\(572\) 1.03947 0.0434623
\(573\) 12.1231 0.506451
\(574\) 10.7023 0.446707
\(575\) −13.2642 −0.553156
\(576\) 5.83670 0.243196
\(577\) −4.00163 −0.166590 −0.0832949 0.996525i \(-0.526544\pi\)
−0.0832949 + 0.996525i \(0.526544\pi\)
\(578\) −13.6612 −0.568232
\(579\) −19.9176 −0.827747
\(580\) 0.0381453 0.00158390
\(581\) 4.97930 0.206576
\(582\) 23.4440 0.971786
\(583\) 4.36119 0.180622
\(584\) 31.0654 1.28550
\(585\) −7.97460 −0.329709
\(586\) 5.63361 0.232722
\(587\) −30.1078 −1.24268 −0.621342 0.783540i \(-0.713412\pi\)
−0.621342 + 0.783540i \(0.713412\pi\)
\(588\) 0.398108 0.0164177
\(589\) 7.22288 0.297613
\(590\) −29.3348 −1.20770
\(591\) −8.45913 −0.347962
\(592\) −30.3629 −1.24791
\(593\) −21.0063 −0.862624 −0.431312 0.902203i \(-0.641949\pi\)
−0.431312 + 0.902203i \(0.641949\pi\)
\(594\) −0.824657 −0.0338361
\(595\) 8.26479 0.338823
\(596\) 6.52179 0.267143
\(597\) −11.9298 −0.488254
\(598\) −42.7714 −1.74905
\(599\) 38.6936 1.58098 0.790490 0.612475i \(-0.209826\pi\)
0.790490 + 0.612475i \(0.209826\pi\)
\(600\) −5.84117 −0.238465
\(601\) −6.29247 −0.256675 −0.128338 0.991731i \(-0.540964\pi\)
−0.128338 + 0.991731i \(0.540964\pi\)
\(602\) −10.9434 −0.446021
\(603\) 15.6926 0.639053
\(604\) 6.87144 0.279595
\(605\) 17.4296 0.708616
\(606\) −3.05589 −0.124137
\(607\) −32.0932 −1.30262 −0.651310 0.758811i \(-0.725780\pi\)
−0.651310 + 0.758811i \(0.725780\pi\)
\(608\) 18.2909 0.741794
\(609\) 0.0589117 0.00238722
\(610\) 15.1312 0.612643
\(611\) 30.7247 1.24299
\(612\) 2.02299 0.0817745
\(613\) −20.5436 −0.829748 −0.414874 0.909879i \(-0.636174\pi\)
−0.414874 + 0.909879i \(0.636174\pi\)
\(614\) −16.4668 −0.664547
\(615\) −11.2404 −0.453258
\(616\) −1.32101 −0.0532251
\(617\) −40.6182 −1.63523 −0.817613 0.575768i \(-0.804703\pi\)
−0.817613 + 0.575768i \(0.804703\pi\)
\(618\) 29.6216 1.19155
\(619\) 2.50189 0.100559 0.0502797 0.998735i \(-0.483989\pi\)
0.0502797 + 0.998735i \(0.483989\pi\)
\(620\) −0.567781 −0.0228026
\(621\) 5.63311 0.226049
\(622\) −17.8307 −0.714944
\(623\) −15.5049 −0.621192
\(624\) −22.7392 −0.910297
\(625\) −7.68204 −0.307282
\(626\) −34.8980 −1.39480
\(627\) 4.38640 0.175176
\(628\) −5.05320 −0.201645
\(629\) 33.2683 1.32649
\(630\) −2.51868 −0.100347
\(631\) −39.9728 −1.59129 −0.795646 0.605761i \(-0.792869\pi\)
−0.795646 + 0.605761i \(0.792869\pi\)
\(632\) −28.3459 −1.12754
\(633\) 8.29673 0.329765
\(634\) −12.3736 −0.491420
\(635\) 11.7591 0.466647
\(636\) 3.26037 0.129282
\(637\) 4.90309 0.194268
\(638\) 0.0485820 0.00192338
\(639\) 2.52323 0.0998176
\(640\) 21.9241 0.866625
\(641\) 19.5537 0.772323 0.386161 0.922431i \(-0.373801\pi\)
0.386161 + 0.922431i \(0.373801\pi\)
\(642\) 24.4990 0.966900
\(643\) −39.9311 −1.57473 −0.787364 0.616489i \(-0.788555\pi\)
−0.787364 + 0.616489i \(0.788555\pi\)
\(644\) −2.24259 −0.0883703
\(645\) 11.4936 0.452562
\(646\) −64.8181 −2.55023
\(647\) 33.3241 1.31011 0.655053 0.755583i \(-0.272646\pi\)
0.655053 + 0.755583i \(0.272646\pi\)
\(648\) 2.48066 0.0974496
\(649\) −6.20225 −0.243460
\(650\) 17.8788 0.701262
\(651\) −0.876882 −0.0343677
\(652\) 3.46544 0.135717
\(653\) 34.0377 1.33200 0.666000 0.745952i \(-0.268005\pi\)
0.666000 + 0.745952i \(0.268005\pi\)
\(654\) −12.1218 −0.473999
\(655\) −2.56988 −0.100413
\(656\) −32.0515 −1.25140
\(657\) 12.5230 0.488570
\(658\) 9.70402 0.378302
\(659\) 18.5205 0.721457 0.360729 0.932671i \(-0.382528\pi\)
0.360729 + 0.932671i \(0.382528\pi\)
\(660\) −0.344809 −0.0134217
\(661\) −22.0614 −0.858087 −0.429044 0.903284i \(-0.641149\pi\)
−0.429044 + 0.903284i \(0.641149\pi\)
\(662\) −36.3436 −1.41253
\(663\) 24.9151 0.967624
\(664\) −12.3520 −0.479349
\(665\) 13.3970 0.519514
\(666\) −10.1385 −0.392858
\(667\) −0.331857 −0.0128495
\(668\) −6.29469 −0.243549
\(669\) −16.8337 −0.650828
\(670\) 39.5247 1.52697
\(671\) 3.19918 0.123503
\(672\) −2.22058 −0.0856606
\(673\) 41.0041 1.58059 0.790297 0.612724i \(-0.209926\pi\)
0.790297 + 0.612724i \(0.209926\pi\)
\(674\) 8.36424 0.322179
\(675\) −2.35468 −0.0906318
\(676\) 4.39525 0.169048
\(677\) 31.3599 1.20526 0.602630 0.798021i \(-0.294120\pi\)
0.602630 + 0.798021i \(0.294120\pi\)
\(678\) −15.2410 −0.585328
\(679\) 15.1390 0.580982
\(680\) −20.5021 −0.786221
\(681\) 8.26899 0.316868
\(682\) −0.723127 −0.0276900
\(683\) −51.2962 −1.96279 −0.981397 0.191990i \(-0.938506\pi\)
−0.981397 + 0.191990i \(0.938506\pi\)
\(684\) 3.27922 0.125384
\(685\) 15.6733 0.598848
\(686\) 1.54858 0.0591252
\(687\) −21.2886 −0.812212
\(688\) 32.7736 1.24948
\(689\) 40.1547 1.52977
\(690\) 14.1880 0.540129
\(691\) 48.6136 1.84935 0.924675 0.380757i \(-0.124337\pi\)
0.924675 + 0.380757i \(0.124337\pi\)
\(692\) −2.08963 −0.0794359
\(693\) −0.532524 −0.0202289
\(694\) −54.7973 −2.08008
\(695\) 29.1720 1.10656
\(696\) −0.146140 −0.00553943
\(697\) 35.1186 1.33021
\(698\) −0.644291 −0.0243868
\(699\) 6.91299 0.261473
\(700\) 0.937419 0.0354311
\(701\) −42.0716 −1.58902 −0.794512 0.607249i \(-0.792273\pi\)
−0.794512 + 0.607249i \(0.792273\pi\)
\(702\) −7.59285 −0.286574
\(703\) 53.9271 2.03390
\(704\) 3.10819 0.117144
\(705\) −10.1919 −0.383850
\(706\) −15.1740 −0.571083
\(707\) −1.97335 −0.0742153
\(708\) −4.63672 −0.174259
\(709\) −17.4595 −0.655706 −0.327853 0.944729i \(-0.606325\pi\)
−0.327853 + 0.944729i \(0.606325\pi\)
\(710\) 6.35522 0.238507
\(711\) −11.4267 −0.428536
\(712\) 38.4625 1.44144
\(713\) 4.93958 0.184989
\(714\) 7.86914 0.294495
\(715\) −4.24667 −0.158816
\(716\) −2.20625 −0.0824516
\(717\) 8.17338 0.305240
\(718\) −34.1655 −1.27504
\(719\) 20.4883 0.764084 0.382042 0.924145i \(-0.375221\pi\)
0.382042 + 0.924145i \(0.375221\pi\)
\(720\) 7.54300 0.281111
\(721\) 19.1282 0.712371
\(722\) −75.6454 −2.81523
\(723\) −8.16882 −0.303802
\(724\) −4.93155 −0.183280
\(725\) 0.138719 0.00515188
\(726\) 16.5953 0.615908
\(727\) 9.06133 0.336066 0.168033 0.985781i \(-0.446258\pi\)
0.168033 + 0.985781i \(0.446258\pi\)
\(728\) −12.1629 −0.450788
\(729\) 1.00000 0.0370370
\(730\) 31.5415 1.16740
\(731\) −35.9097 −1.32817
\(732\) 2.39167 0.0883985
\(733\) −10.7353 −0.396518 −0.198259 0.980150i \(-0.563529\pi\)
−0.198259 + 0.980150i \(0.563529\pi\)
\(734\) 39.6279 1.46269
\(735\) −1.62644 −0.0599922
\(736\) 12.5088 0.461079
\(737\) 8.35669 0.307823
\(738\) −10.7023 −0.393958
\(739\) −4.89606 −0.180104 −0.0900522 0.995937i \(-0.528703\pi\)
−0.0900522 + 0.995937i \(0.528703\pi\)
\(740\) −4.23914 −0.155834
\(741\) 40.3868 1.48365
\(742\) 12.6824 0.465585
\(743\) 5.10457 0.187269 0.0936343 0.995607i \(-0.470152\pi\)
0.0936343 + 0.995607i \(0.470152\pi\)
\(744\) 2.17525 0.0797484
\(745\) −26.6443 −0.976172
\(746\) −20.1252 −0.736836
\(747\) −4.97930 −0.182183
\(748\) 1.07729 0.0393897
\(749\) 15.8203 0.578061
\(750\) −18.5241 −0.676405
\(751\) 2.87503 0.104911 0.0524557 0.998623i \(-0.483295\pi\)
0.0524557 + 0.998623i \(0.483295\pi\)
\(752\) −29.0618 −1.05977
\(753\) −5.03779 −0.183587
\(754\) 0.447308 0.0162900
\(755\) −28.0728 −1.02167
\(756\) −0.398108 −0.0144790
\(757\) −2.44746 −0.0889543 −0.0444772 0.999010i \(-0.514162\pi\)
−0.0444772 + 0.999010i \(0.514162\pi\)
\(758\) 24.0881 0.874919
\(759\) 2.99977 0.108885
\(760\) −33.2335 −1.20550
\(761\) 6.98471 0.253196 0.126598 0.991954i \(-0.459594\pi\)
0.126598 + 0.991954i \(0.459594\pi\)
\(762\) 11.1962 0.405596
\(763\) −7.82766 −0.283380
\(764\) 4.82632 0.174610
\(765\) −8.26479 −0.298814
\(766\) 1.54858 0.0559526
\(767\) −57.1058 −2.06197
\(768\) 9.20113 0.332017
\(769\) 44.7197 1.61263 0.806317 0.591484i \(-0.201458\pi\)
0.806317 + 0.591484i \(0.201458\pi\)
\(770\) −1.34126 −0.0483356
\(771\) 13.8553 0.498987
\(772\) −7.92935 −0.285384
\(773\) 13.2121 0.475208 0.237604 0.971362i \(-0.423638\pi\)
0.237604 + 0.971362i \(0.423638\pi\)
\(774\) 10.9434 0.393353
\(775\) −2.06478 −0.0741691
\(776\) −37.5548 −1.34814
\(777\) −6.54693 −0.234870
\(778\) 32.7576 1.17442
\(779\) 56.9263 2.03960
\(780\) −3.17475 −0.113674
\(781\) 1.34368 0.0480808
\(782\) −44.3278 −1.58516
\(783\) −0.0589117 −0.00210533
\(784\) −4.63773 −0.165633
\(785\) 20.6445 0.736834
\(786\) −2.44685 −0.0872763
\(787\) −22.7415 −0.810646 −0.405323 0.914174i \(-0.632841\pi\)
−0.405323 + 0.914174i \(0.632841\pi\)
\(788\) −3.36765 −0.119967
\(789\) −17.6071 −0.626828
\(790\) −28.7803 −1.02396
\(791\) −9.84192 −0.349938
\(792\) 1.32101 0.0469401
\(793\) 29.4557 1.04600
\(794\) −57.5151 −2.04113
\(795\) −13.3200 −0.472413
\(796\) −4.74935 −0.168336
\(797\) 35.0326 1.24092 0.620459 0.784239i \(-0.286946\pi\)
0.620459 + 0.784239i \(0.286946\pi\)
\(798\) 12.7557 0.451546
\(799\) 31.8427 1.12651
\(800\) −5.22876 −0.184865
\(801\) 15.5049 0.547840
\(802\) 15.4522 0.545634
\(803\) 6.66881 0.235337
\(804\) 6.24735 0.220327
\(805\) 9.16194 0.322916
\(806\) −6.65803 −0.234519
\(807\) −20.5693 −0.724075
\(808\) 4.89520 0.172213
\(809\) −13.2260 −0.465001 −0.232501 0.972596i \(-0.574691\pi\)
−0.232501 + 0.972596i \(0.574691\pi\)
\(810\) 2.51868 0.0884974
\(811\) −35.3500 −1.24131 −0.620653 0.784085i \(-0.713133\pi\)
−0.620653 + 0.784085i \(0.713133\pi\)
\(812\) 0.0234532 0.000823047 0
\(813\) 16.2853 0.571151
\(814\) −5.39898 −0.189234
\(815\) −14.1578 −0.495926
\(816\) −23.5667 −0.824998
\(817\) −58.2087 −2.03647
\(818\) −9.03761 −0.315993
\(819\) −4.90309 −0.171328
\(820\) −4.47490 −0.156270
\(821\) −20.0595 −0.700082 −0.350041 0.936734i \(-0.613832\pi\)
−0.350041 + 0.936734i \(0.613832\pi\)
\(822\) 14.9230 0.520501
\(823\) 6.63964 0.231443 0.115722 0.993282i \(-0.463082\pi\)
0.115722 + 0.993282i \(0.463082\pi\)
\(824\) −47.4506 −1.65302
\(825\) −1.25393 −0.0436561
\(826\) −18.0362 −0.627559
\(827\) 7.61754 0.264888 0.132444 0.991190i \(-0.457718\pi\)
0.132444 + 0.991190i \(0.457718\pi\)
\(828\) 2.24259 0.0779353
\(829\) −38.5832 −1.34005 −0.670026 0.742338i \(-0.733717\pi\)
−0.670026 + 0.742338i \(0.733717\pi\)
\(830\) −12.5413 −0.435314
\(831\) 0.403491 0.0139970
\(832\) 28.6179 0.992148
\(833\) 5.08151 0.176064
\(834\) 27.7755 0.961788
\(835\) 25.7165 0.889956
\(836\) 1.74626 0.0603957
\(837\) 0.876882 0.0303095
\(838\) 53.3355 1.84244
\(839\) −28.3458 −0.978604 −0.489302 0.872114i \(-0.662748\pi\)
−0.489302 + 0.872114i \(0.662748\pi\)
\(840\) 4.03465 0.139209
\(841\) −28.9965 −0.999880
\(842\) −52.2447 −1.80047
\(843\) 25.3008 0.871404
\(844\) 3.30299 0.113694
\(845\) −17.9565 −0.617722
\(846\) −9.70402 −0.333631
\(847\) 10.7164 0.368220
\(848\) −37.9814 −1.30429
\(849\) −9.98933 −0.342833
\(850\) 18.5293 0.635551
\(851\) 36.8796 1.26422
\(852\) 1.00452 0.0344143
\(853\) 29.9905 1.02685 0.513427 0.858133i \(-0.328376\pi\)
0.513427 + 0.858133i \(0.328376\pi\)
\(854\) 9.30323 0.318350
\(855\) −13.3970 −0.458168
\(856\) −39.2448 −1.34136
\(857\) −40.1009 −1.36982 −0.684910 0.728627i \(-0.740159\pi\)
−0.684910 + 0.728627i \(0.740159\pi\)
\(858\) −4.04337 −0.138038
\(859\) −30.5786 −1.04333 −0.521664 0.853151i \(-0.674689\pi\)
−0.521664 + 0.853151i \(0.674689\pi\)
\(860\) 4.57571 0.156031
\(861\) −6.91105 −0.235528
\(862\) −41.3931 −1.40986
\(863\) −18.5672 −0.632035 −0.316017 0.948753i \(-0.602346\pi\)
−0.316017 + 0.948753i \(0.602346\pi\)
\(864\) 2.22058 0.0755456
\(865\) 8.53704 0.290268
\(866\) 22.7063 0.771591
\(867\) 8.82176 0.299603
\(868\) −0.349094 −0.0118490
\(869\) −6.08501 −0.206420
\(870\) −0.148380 −0.00503055
\(871\) 76.9424 2.60709
\(872\) 19.4178 0.657569
\(873\) −15.1390 −0.512378
\(874\) −71.8542 −2.43050
\(875\) −11.9620 −0.404388
\(876\) 4.98552 0.168445
\(877\) −32.4123 −1.09448 −0.547242 0.836974i \(-0.684322\pi\)
−0.547242 + 0.836974i \(0.684322\pi\)
\(878\) 31.2441 1.05444
\(879\) −3.63791 −0.122704
\(880\) 4.01683 0.135407
\(881\) −11.6118 −0.391211 −0.195605 0.980683i \(-0.562667\pi\)
−0.195605 + 0.980683i \(0.562667\pi\)
\(882\) −1.54858 −0.0521435
\(883\) 9.68838 0.326040 0.163020 0.986623i \(-0.447877\pi\)
0.163020 + 0.986623i \(0.447877\pi\)
\(884\) 9.91891 0.333609
\(885\) 18.9430 0.636762
\(886\) 28.3930 0.953880
\(887\) 32.0555 1.07632 0.538160 0.842843i \(-0.319120\pi\)
0.538160 + 0.842843i \(0.319120\pi\)
\(888\) 16.2407 0.545003
\(889\) 7.22997 0.242485
\(890\) 39.0520 1.30902
\(891\) 0.532524 0.0178402
\(892\) −6.70163 −0.224387
\(893\) 51.6162 1.72727
\(894\) −25.3688 −0.848460
\(895\) 9.01350 0.301288
\(896\) 13.4798 0.450328
\(897\) 27.6197 0.922195
\(898\) 26.1726 0.873391
\(899\) −0.0516586 −0.00172291
\(900\) −0.937419 −0.0312473
\(901\) 41.6159 1.38643
\(902\) −5.69925 −0.189764
\(903\) 7.06674 0.235166
\(904\) 24.4145 0.812013
\(905\) 20.1475 0.669725
\(906\) −26.7289 −0.888009
\(907\) −16.4128 −0.544977 −0.272489 0.962159i \(-0.587847\pi\)
−0.272489 + 0.962159i \(0.587847\pi\)
\(908\) 3.29195 0.109247
\(909\) 1.97335 0.0654517
\(910\) −12.3493 −0.409376
\(911\) −46.0638 −1.52616 −0.763080 0.646304i \(-0.776314\pi\)
−0.763080 + 0.646304i \(0.776314\pi\)
\(912\) −38.2010 −1.26496
\(913\) −2.65160 −0.0877550
\(914\) 16.3687 0.541428
\(915\) −9.77098 −0.323019
\(916\) −8.47518 −0.280028
\(917\) −1.58006 −0.0521781
\(918\) −7.86914 −0.259720
\(919\) −25.1784 −0.830558 −0.415279 0.909694i \(-0.636316\pi\)
−0.415279 + 0.909694i \(0.636316\pi\)
\(920\) −22.7277 −0.749309
\(921\) 10.6335 0.350385
\(922\) 32.6481 1.07521
\(923\) 12.3717 0.407218
\(924\) −0.212002 −0.00697436
\(925\) −15.4160 −0.506873
\(926\) 43.4321 1.42727
\(927\) −19.1282 −0.628252
\(928\) −0.130818 −0.00429431
\(929\) 12.7155 0.417181 0.208590 0.978003i \(-0.433112\pi\)
0.208590 + 0.978003i \(0.433112\pi\)
\(930\) 2.20859 0.0724224
\(931\) 8.23700 0.269957
\(932\) 2.75212 0.0901485
\(933\) 11.5142 0.376957
\(934\) −50.4156 −1.64965
\(935\) −4.40120 −0.143935
\(936\) 12.1629 0.397558
\(937\) −37.7721 −1.23396 −0.616980 0.786978i \(-0.711644\pi\)
−0.616980 + 0.786978i \(0.711644\pi\)
\(938\) 24.3013 0.793466
\(939\) 22.5354 0.735416
\(940\) −4.05748 −0.132341
\(941\) 37.9554 1.23731 0.618655 0.785663i \(-0.287678\pi\)
0.618655 + 0.785663i \(0.287678\pi\)
\(942\) 19.6562 0.640435
\(943\) 38.9307 1.26776
\(944\) 54.0151 1.75804
\(945\) 1.62644 0.0529082
\(946\) 5.82764 0.189473
\(947\) −49.3020 −1.60210 −0.801050 0.598597i \(-0.795725\pi\)
−0.801050 + 0.598597i \(0.795725\pi\)
\(948\) −4.54908 −0.147747
\(949\) 61.4016 1.99318
\(950\) 30.0356 0.974483
\(951\) 7.99031 0.259103
\(952\) −12.6055 −0.408547
\(953\) 11.4742 0.371686 0.185843 0.982579i \(-0.440498\pi\)
0.185843 + 0.982579i \(0.440498\pi\)
\(954\) −12.6824 −0.410607
\(955\) −19.7176 −0.638046
\(956\) 3.25389 0.105238
\(957\) −0.0313719 −0.00101411
\(958\) −31.4896 −1.01738
\(959\) 9.63658 0.311181
\(960\) −9.49307 −0.306387
\(961\) −30.2311 −0.975196
\(962\) −49.7099 −1.60271
\(963\) −15.8203 −0.509802
\(964\) −3.25207 −0.104742
\(965\) 32.3948 1.04283
\(966\) 8.72334 0.280669
\(967\) −23.6001 −0.758927 −0.379464 0.925207i \(-0.623891\pi\)
−0.379464 + 0.925207i \(0.623891\pi\)
\(968\) −26.5838 −0.854436
\(969\) 41.8564 1.34462
\(970\) −38.1304 −1.22429
\(971\) 5.66897 0.181926 0.0909630 0.995854i \(-0.471006\pi\)
0.0909630 + 0.995854i \(0.471006\pi\)
\(972\) 0.398108 0.0127693
\(973\) 17.9361 0.575005
\(974\) 13.9850 0.448107
\(975\) −11.5452 −0.369744
\(976\) −27.8615 −0.891825
\(977\) −43.7258 −1.39891 −0.699456 0.714676i \(-0.746574\pi\)
−0.699456 + 0.714676i \(0.746574\pi\)
\(978\) −13.4800 −0.431044
\(979\) 8.25675 0.263887
\(980\) −0.647500 −0.0206836
\(981\) 7.82766 0.249918
\(982\) 4.38715 0.140000
\(983\) −2.67540 −0.0853319 −0.0426660 0.999089i \(-0.513585\pi\)
−0.0426660 + 0.999089i \(0.513585\pi\)
\(984\) 17.1440 0.546530
\(985\) 13.7583 0.438376
\(986\) 0.463585 0.0147635
\(987\) −6.26639 −0.199461
\(988\) 16.0783 0.511519
\(989\) −39.8078 −1.26581
\(990\) 1.34126 0.0426280
\(991\) −46.9775 −1.49229 −0.746145 0.665783i \(-0.768098\pi\)
−0.746145 + 0.665783i \(0.768098\pi\)
\(992\) 1.94718 0.0618232
\(993\) 23.4689 0.744764
\(994\) 3.90744 0.123936
\(995\) 19.4031 0.615121
\(996\) −1.98230 −0.0628115
\(997\) 3.13084 0.0991548 0.0495774 0.998770i \(-0.484213\pi\)
0.0495774 + 0.998770i \(0.484213\pi\)
\(998\) 42.6209 1.34914
\(999\) 6.54693 0.207136
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 8043.2.a.p.1.11 41
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8043.2.a.p.1.11 41 1.1 even 1 trivial