Properties

Label 619.2.a.a.1.9
Level $619$
Weight $2$
Character 619.1
Self dual yes
Analytic conductor $4.943$
Analytic rank $1$
Dimension $21$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [619,2,Mod(1,619)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(619, 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("619.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 619 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 619.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: \(4.94273988512\)
Analytic rank: \(1\)
Dimension: \(21\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Character \(\chi\) \(=\) 619.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.887456 q^{2} -2.66181 q^{3} -1.21242 q^{4} -0.744827 q^{5} +2.36224 q^{6} -0.702395 q^{7} +2.85088 q^{8} +4.08525 q^{9} +O(q^{10})\) \(q-0.887456 q^{2} -2.66181 q^{3} -1.21242 q^{4} -0.744827 q^{5} +2.36224 q^{6} -0.702395 q^{7} +2.85088 q^{8} +4.08525 q^{9} +0.661001 q^{10} +1.89013 q^{11} +3.22724 q^{12} +5.87931 q^{13} +0.623345 q^{14} +1.98259 q^{15} -0.105192 q^{16} -6.29540 q^{17} -3.62548 q^{18} -0.340965 q^{19} +0.903044 q^{20} +1.86964 q^{21} -1.67741 q^{22} +4.82124 q^{23} -7.58852 q^{24} -4.44523 q^{25} -5.21763 q^{26} -2.88873 q^{27} +0.851598 q^{28} -3.65200 q^{29} -1.75946 q^{30} -1.90470 q^{31} -5.60841 q^{32} -5.03118 q^{33} +5.58689 q^{34} +0.523162 q^{35} -4.95305 q^{36} -3.11217 q^{37} +0.302591 q^{38} -15.6496 q^{39} -2.12341 q^{40} -3.21126 q^{41} -1.65923 q^{42} +7.84913 q^{43} -2.29164 q^{44} -3.04280 q^{45} -4.27864 q^{46} +12.3713 q^{47} +0.280001 q^{48} -6.50664 q^{49} +3.94495 q^{50} +16.7572 q^{51} -7.12820 q^{52} -11.6789 q^{53} +2.56363 q^{54} -1.40782 q^{55} -2.00245 q^{56} +0.907585 q^{57} +3.24099 q^{58} +6.85518 q^{59} -2.40373 q^{60} -9.12686 q^{61} +1.69034 q^{62} -2.86946 q^{63} +5.18761 q^{64} -4.37907 q^{65} +4.46495 q^{66} -5.99837 q^{67} +7.63268 q^{68} -12.8332 q^{69} -0.464284 q^{70} -9.17613 q^{71} +11.6466 q^{72} +7.10331 q^{73} +2.76192 q^{74} +11.8324 q^{75} +0.413393 q^{76} -1.32762 q^{77} +13.8884 q^{78} -3.80975 q^{79} +0.0783497 q^{80} -4.56648 q^{81} +2.84985 q^{82} -16.5815 q^{83} -2.26680 q^{84} +4.68898 q^{85} -6.96576 q^{86} +9.72095 q^{87} +5.38854 q^{88} -1.94358 q^{89} +2.70036 q^{90} -4.12960 q^{91} -5.84537 q^{92} +5.06995 q^{93} -10.9790 q^{94} +0.253960 q^{95} +14.9286 q^{96} +9.71119 q^{97} +5.77436 q^{98} +7.72166 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 21 q - 9 q^{2} - 5 q^{3} + 15 q^{4} - 21 q^{5} - 6 q^{6} - 4 q^{7} - 21 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 21 q - 9 q^{2} - 5 q^{3} + 15 q^{4} - 21 q^{5} - 6 q^{6} - 4 q^{7} - 21 q^{8} + 6 q^{9} + q^{10} - 27 q^{11} - 8 q^{12} - 11 q^{13} - 19 q^{14} - 10 q^{15} + 11 q^{16} - 14 q^{17} - 14 q^{18} - 15 q^{19} - 25 q^{20} - 42 q^{21} + 12 q^{22} - 14 q^{23} - 8 q^{24} + 16 q^{25} - 11 q^{26} - 5 q^{27} + q^{28} - 78 q^{29} + q^{30} - 8 q^{31} - 41 q^{32} - 6 q^{33} + 7 q^{34} - 3 q^{35} - q^{36} - 23 q^{37} + 21 q^{38} - 4 q^{39} + 12 q^{40} - 59 q^{41} + 39 q^{42} + 2 q^{43} - 50 q^{44} - 36 q^{45} - 15 q^{46} - 12 q^{47} + 10 q^{48} + 17 q^{49} - 23 q^{50} - 8 q^{51} + 18 q^{52} - 36 q^{53} - 4 q^{54} + 23 q^{55} - 28 q^{56} - 24 q^{57} + 46 q^{58} - 17 q^{59} + 8 q^{60} - 22 q^{61} + 42 q^{62} - 6 q^{63} + 49 q^{64} - 53 q^{65} + 29 q^{66} + 15 q^{67} - 16 q^{68} - 30 q^{69} + 44 q^{70} - 56 q^{71} + 12 q^{72} - 2 q^{73} - 12 q^{74} + 2 q^{75} - 4 q^{76} - 47 q^{77} + 36 q^{78} + 5 q^{79} + 15 q^{80} - 19 q^{81} + 47 q^{82} - q^{83} - 20 q^{84} - 29 q^{85} - 23 q^{86} + 44 q^{87} + 61 q^{88} - 12 q^{89} + 91 q^{90} + 5 q^{91} + 35 q^{92} - 15 q^{93} + 34 q^{94} - 17 q^{95} + 14 q^{96} + 21 q^{97} + 24 q^{98} - 38 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.887456 −0.627526 −0.313763 0.949501i \(-0.601590\pi\)
−0.313763 + 0.949501i \(0.601590\pi\)
\(3\) −2.66181 −1.53680 −0.768399 0.639971i \(-0.778946\pi\)
−0.768399 + 0.639971i \(0.778946\pi\)
\(4\) −1.21242 −0.606211
\(5\) −0.744827 −0.333097 −0.166548 0.986033i \(-0.553262\pi\)
−0.166548 + 0.986033i \(0.553262\pi\)
\(6\) 2.36224 0.964382
\(7\) −0.702395 −0.265480 −0.132740 0.991151i \(-0.542378\pi\)
−0.132740 + 0.991151i \(0.542378\pi\)
\(8\) 2.85088 1.00794
\(9\) 4.08525 1.36175
\(10\) 0.661001 0.209027
\(11\) 1.89013 0.569896 0.284948 0.958543i \(-0.408024\pi\)
0.284948 + 0.958543i \(0.408024\pi\)
\(12\) 3.22724 0.931624
\(13\) 5.87931 1.63063 0.815314 0.579020i \(-0.196565\pi\)
0.815314 + 0.579020i \(0.196565\pi\)
\(14\) 0.623345 0.166596
\(15\) 1.98259 0.511903
\(16\) −0.105192 −0.0262980
\(17\) −6.29540 −1.52686 −0.763429 0.645892i \(-0.776486\pi\)
−0.763429 + 0.645892i \(0.776486\pi\)
\(18\) −3.62548 −0.854534
\(19\) −0.340965 −0.0782227 −0.0391114 0.999235i \(-0.512453\pi\)
−0.0391114 + 0.999235i \(0.512453\pi\)
\(20\) 0.903044 0.201927
\(21\) 1.86964 0.407990
\(22\) −1.67741 −0.357625
\(23\) 4.82124 1.00530 0.502649 0.864491i \(-0.332359\pi\)
0.502649 + 0.864491i \(0.332359\pi\)
\(24\) −7.58852 −1.54900
\(25\) −4.44523 −0.889047
\(26\) −5.21763 −1.02326
\(27\) −2.88873 −0.555937
\(28\) 0.851598 0.160937
\(29\) −3.65200 −0.678160 −0.339080 0.940758i \(-0.610116\pi\)
−0.339080 + 0.940758i \(0.610116\pi\)
\(30\) −1.75946 −0.321232
\(31\) −1.90470 −0.342094 −0.171047 0.985263i \(-0.554715\pi\)
−0.171047 + 0.985263i \(0.554715\pi\)
\(32\) −5.60841 −0.991437
\(33\) −5.03118 −0.875816
\(34\) 5.58689 0.958144
\(35\) 0.523162 0.0884306
\(36\) −4.95305 −0.825508
\(37\) −3.11217 −0.511638 −0.255819 0.966725i \(-0.582345\pi\)
−0.255819 + 0.966725i \(0.582345\pi\)
\(38\) 0.302591 0.0490868
\(39\) −15.6496 −2.50595
\(40\) −2.12341 −0.335741
\(41\) −3.21126 −0.501515 −0.250757 0.968050i \(-0.580680\pi\)
−0.250757 + 0.968050i \(0.580680\pi\)
\(42\) −1.65923 −0.256024
\(43\) 7.84913 1.19698 0.598491 0.801130i \(-0.295767\pi\)
0.598491 + 0.801130i \(0.295767\pi\)
\(44\) −2.29164 −0.345477
\(45\) −3.04280 −0.453595
\(46\) −4.27864 −0.630851
\(47\) 12.3713 1.80453 0.902267 0.431177i \(-0.141902\pi\)
0.902267 + 0.431177i \(0.141902\pi\)
\(48\) 0.280001 0.0404147
\(49\) −6.50664 −0.929520
\(50\) 3.94495 0.557900
\(51\) 16.7572 2.34647
\(52\) −7.12820 −0.988504
\(53\) −11.6789 −1.60423 −0.802113 0.597172i \(-0.796291\pi\)
−0.802113 + 0.597172i \(0.796291\pi\)
\(54\) 2.56363 0.348865
\(55\) −1.40782 −0.189830
\(56\) −2.00245 −0.267588
\(57\) 0.907585 0.120213
\(58\) 3.24099 0.425563
\(59\) 6.85518 0.892468 0.446234 0.894916i \(-0.352765\pi\)
0.446234 + 0.894916i \(0.352765\pi\)
\(60\) −2.40373 −0.310321
\(61\) −9.12686 −1.16858 −0.584288 0.811547i \(-0.698626\pi\)
−0.584288 + 0.811547i \(0.698626\pi\)
\(62\) 1.69034 0.214673
\(63\) −2.86946 −0.361518
\(64\) 5.18761 0.648451
\(65\) −4.37907 −0.543157
\(66\) 4.46495 0.549597
\(67\) −5.99837 −0.732817 −0.366409 0.930454i \(-0.619413\pi\)
−0.366409 + 0.930454i \(0.619413\pi\)
\(68\) 7.63268 0.925598
\(69\) −12.8332 −1.54494
\(70\) −0.464284 −0.0554925
\(71\) −9.17613 −1.08901 −0.544503 0.838759i \(-0.683282\pi\)
−0.544503 + 0.838759i \(0.683282\pi\)
\(72\) 11.6466 1.37256
\(73\) 7.10331 0.831380 0.415690 0.909506i \(-0.363540\pi\)
0.415690 + 0.909506i \(0.363540\pi\)
\(74\) 2.76192 0.321066
\(75\) 11.8324 1.36629
\(76\) 0.413393 0.0474194
\(77\) −1.32762 −0.151296
\(78\) 13.8884 1.57255
\(79\) −3.80975 −0.428631 −0.214315 0.976765i \(-0.568752\pi\)
−0.214315 + 0.976765i \(0.568752\pi\)
\(80\) 0.0783497 0.00875976
\(81\) −4.56648 −0.507387
\(82\) 2.84985 0.314714
\(83\) −16.5815 −1.82006 −0.910031 0.414541i \(-0.863942\pi\)
−0.910031 + 0.414541i \(0.863942\pi\)
\(84\) −2.26680 −0.247328
\(85\) 4.68898 0.508592
\(86\) −6.96576 −0.751137
\(87\) 9.72095 1.04219
\(88\) 5.38854 0.574421
\(89\) −1.94358 −0.206019 −0.103010 0.994680i \(-0.532847\pi\)
−0.103010 + 0.994680i \(0.532847\pi\)
\(90\) 2.70036 0.284643
\(91\) −4.12960 −0.432899
\(92\) −5.84537 −0.609422
\(93\) 5.06995 0.525729
\(94\) −10.9790 −1.13239
\(95\) 0.253960 0.0260557
\(96\) 14.9286 1.52364
\(97\) 9.71119 0.986022 0.493011 0.870023i \(-0.335896\pi\)
0.493011 + 0.870023i \(0.335896\pi\)
\(98\) 5.77436 0.583298
\(99\) 7.72166 0.776056
\(100\) 5.38950 0.538950
\(101\) −2.16605 −0.215530 −0.107765 0.994176i \(-0.534369\pi\)
−0.107765 + 0.994176i \(0.534369\pi\)
\(102\) −14.8713 −1.47247
\(103\) 16.4622 1.62207 0.811037 0.584995i \(-0.198904\pi\)
0.811037 + 0.584995i \(0.198904\pi\)
\(104\) 16.7612 1.64357
\(105\) −1.39256 −0.135900
\(106\) 10.3645 1.00669
\(107\) −15.2236 −1.47173 −0.735863 0.677131i \(-0.763223\pi\)
−0.735863 + 0.677131i \(0.763223\pi\)
\(108\) 3.50236 0.337015
\(109\) 13.4044 1.28391 0.641956 0.766741i \(-0.278123\pi\)
0.641956 + 0.766741i \(0.278123\pi\)
\(110\) 1.24938 0.119124
\(111\) 8.28402 0.786285
\(112\) 0.0738862 0.00698159
\(113\) −8.18271 −0.769764 −0.384882 0.922966i \(-0.625758\pi\)
−0.384882 + 0.922966i \(0.625758\pi\)
\(114\) −0.805442 −0.0754366
\(115\) −3.59099 −0.334861
\(116\) 4.42776 0.411108
\(117\) 24.0185 2.22051
\(118\) −6.08367 −0.560047
\(119\) 4.42185 0.405351
\(120\) 5.65213 0.515967
\(121\) −7.42740 −0.675218
\(122\) 8.09969 0.733312
\(123\) 8.54778 0.770727
\(124\) 2.30930 0.207381
\(125\) 7.03506 0.629235
\(126\) 2.54652 0.226862
\(127\) 5.96470 0.529281 0.264641 0.964347i \(-0.414747\pi\)
0.264641 + 0.964347i \(0.414747\pi\)
\(128\) 6.61305 0.584517
\(129\) −20.8929 −1.83952
\(130\) 3.88623 0.340845
\(131\) −19.4936 −1.70316 −0.851580 0.524224i \(-0.824355\pi\)
−0.851580 + 0.524224i \(0.824355\pi\)
\(132\) 6.09991 0.530929
\(133\) 0.239492 0.0207666
\(134\) 5.32329 0.459862
\(135\) 2.15161 0.185181
\(136\) −17.9474 −1.53898
\(137\) −11.6564 −0.995875 −0.497938 0.867213i \(-0.665909\pi\)
−0.497938 + 0.867213i \(0.665909\pi\)
\(138\) 11.3889 0.969491
\(139\) −9.44498 −0.801113 −0.400557 0.916272i \(-0.631183\pi\)
−0.400557 + 0.916272i \(0.631183\pi\)
\(140\) −0.634293 −0.0536076
\(141\) −32.9300 −2.77321
\(142\) 8.14341 0.683380
\(143\) 11.1127 0.929288
\(144\) −0.429735 −0.0358113
\(145\) 2.72011 0.225893
\(146\) −6.30388 −0.521713
\(147\) 17.3195 1.42849
\(148\) 3.77326 0.310160
\(149\) −12.7368 −1.04344 −0.521718 0.853118i \(-0.674709\pi\)
−0.521718 + 0.853118i \(0.674709\pi\)
\(150\) −10.5007 −0.857380
\(151\) −11.1386 −0.906448 −0.453224 0.891397i \(-0.649726\pi\)
−0.453224 + 0.891397i \(0.649726\pi\)
\(152\) −0.972051 −0.0788438
\(153\) −25.7183 −2.07920
\(154\) 1.17820 0.0949423
\(155\) 1.41867 0.113950
\(156\) 18.9739 1.51913
\(157\) 14.5199 1.15881 0.579405 0.815040i \(-0.303285\pi\)
0.579405 + 0.815040i \(0.303285\pi\)
\(158\) 3.38099 0.268977
\(159\) 31.0872 2.46537
\(160\) 4.17730 0.330244
\(161\) −3.38641 −0.266887
\(162\) 4.05255 0.318398
\(163\) −6.98817 −0.547356 −0.273678 0.961821i \(-0.588240\pi\)
−0.273678 + 0.961821i \(0.588240\pi\)
\(164\) 3.89340 0.304024
\(165\) 3.74736 0.291731
\(166\) 14.7154 1.14214
\(167\) −7.52881 −0.582596 −0.291298 0.956632i \(-0.594087\pi\)
−0.291298 + 0.956632i \(0.594087\pi\)
\(168\) 5.33014 0.411229
\(169\) 21.5663 1.65895
\(170\) −4.16127 −0.319155
\(171\) −1.39293 −0.106520
\(172\) −9.51645 −0.725623
\(173\) −6.03121 −0.458545 −0.229272 0.973362i \(-0.573635\pi\)
−0.229272 + 0.973362i \(0.573635\pi\)
\(174\) −8.62691 −0.654005
\(175\) 3.12231 0.236024
\(176\) −0.198826 −0.0149871
\(177\) −18.2472 −1.37154
\(178\) 1.72484 0.129282
\(179\) 12.2546 0.915952 0.457976 0.888965i \(-0.348575\pi\)
0.457976 + 0.888965i \(0.348575\pi\)
\(180\) 3.68916 0.274974
\(181\) 2.64714 0.196760 0.0983800 0.995149i \(-0.468634\pi\)
0.0983800 + 0.995149i \(0.468634\pi\)
\(182\) 3.66484 0.271656
\(183\) 24.2940 1.79587
\(184\) 13.7448 1.01328
\(185\) 2.31803 0.170425
\(186\) −4.49936 −0.329909
\(187\) −11.8991 −0.870151
\(188\) −14.9992 −1.09393
\(189\) 2.02903 0.147590
\(190\) −0.225378 −0.0163507
\(191\) −8.59383 −0.621828 −0.310914 0.950438i \(-0.600635\pi\)
−0.310914 + 0.950438i \(0.600635\pi\)
\(192\) −13.8084 −0.996538
\(193\) 2.19690 0.158137 0.0790683 0.996869i \(-0.474805\pi\)
0.0790683 + 0.996869i \(0.474805\pi\)
\(194\) −8.61826 −0.618755
\(195\) 11.6563 0.834722
\(196\) 7.88879 0.563485
\(197\) −5.74909 −0.409606 −0.204803 0.978803i \(-0.565655\pi\)
−0.204803 + 0.978803i \(0.565655\pi\)
\(198\) −6.85264 −0.486996
\(199\) 0.107419 0.00761471 0.00380736 0.999993i \(-0.498788\pi\)
0.00380736 + 0.999993i \(0.498788\pi\)
\(200\) −12.6728 −0.896105
\(201\) 15.9665 1.12619
\(202\) 1.92227 0.135251
\(203\) 2.56515 0.180038
\(204\) −20.3168 −1.42246
\(205\) 2.39183 0.167053
\(206\) −14.6095 −1.01789
\(207\) 19.6960 1.36896
\(208\) −0.618455 −0.0428822
\(209\) −0.644468 −0.0445788
\(210\) 1.23584 0.0852808
\(211\) −25.1728 −1.73297 −0.866483 0.499206i \(-0.833625\pi\)
−0.866483 + 0.499206i \(0.833625\pi\)
\(212\) 14.1598 0.972499
\(213\) 24.4251 1.67358
\(214\) 13.5103 0.923547
\(215\) −5.84624 −0.398711
\(216\) −8.23545 −0.560351
\(217\) 1.33785 0.0908191
\(218\) −11.8959 −0.805689
\(219\) −18.9077 −1.27766
\(220\) 1.70687 0.115077
\(221\) −37.0126 −2.48974
\(222\) −7.35171 −0.493414
\(223\) 19.7309 1.32128 0.660639 0.750704i \(-0.270286\pi\)
0.660639 + 0.750704i \(0.270286\pi\)
\(224\) 3.93932 0.263207
\(225\) −18.1599 −1.21066
\(226\) 7.26180 0.483047
\(227\) −10.9814 −0.728862 −0.364431 0.931230i \(-0.618737\pi\)
−0.364431 + 0.931230i \(0.618737\pi\)
\(228\) −1.10038 −0.0728741
\(229\) −16.6925 −1.10307 −0.551537 0.834151i \(-0.685958\pi\)
−0.551537 + 0.834151i \(0.685958\pi\)
\(230\) 3.18684 0.210134
\(231\) 3.53387 0.232512
\(232\) −10.4114 −0.683544
\(233\) 6.27483 0.411077 0.205539 0.978649i \(-0.434105\pi\)
0.205539 + 0.978649i \(0.434105\pi\)
\(234\) −21.3153 −1.39343
\(235\) −9.21445 −0.601085
\(236\) −8.31136 −0.541024
\(237\) 10.1409 0.658719
\(238\) −3.92420 −0.254368
\(239\) −14.7364 −0.953217 −0.476608 0.879116i \(-0.658134\pi\)
−0.476608 + 0.879116i \(0.658134\pi\)
\(240\) −0.208552 −0.0134620
\(241\) 14.0484 0.904940 0.452470 0.891780i \(-0.350543\pi\)
0.452470 + 0.891780i \(0.350543\pi\)
\(242\) 6.59150 0.423717
\(243\) 20.8213 1.33569
\(244\) 11.0656 0.708403
\(245\) 4.84632 0.309620
\(246\) −7.58578 −0.483652
\(247\) −2.00464 −0.127552
\(248\) −5.43007 −0.344810
\(249\) 44.1370 2.79707
\(250\) −6.24331 −0.394862
\(251\) 9.94157 0.627506 0.313753 0.949505i \(-0.398414\pi\)
0.313753 + 0.949505i \(0.398414\pi\)
\(252\) 3.47899 0.219156
\(253\) 9.11277 0.572915
\(254\) −5.29341 −0.332138
\(255\) −12.4812 −0.781603
\(256\) −16.2440 −1.01525
\(257\) −29.6057 −1.84675 −0.923376 0.383896i \(-0.874582\pi\)
−0.923376 + 0.383896i \(0.874582\pi\)
\(258\) 18.5416 1.15435
\(259\) 2.18597 0.135830
\(260\) 5.30928 0.329267
\(261\) −14.9193 −0.923484
\(262\) 17.2997 1.06878
\(263\) 1.35298 0.0834280 0.0417140 0.999130i \(-0.486718\pi\)
0.0417140 + 0.999130i \(0.486718\pi\)
\(264\) −14.3433 −0.882769
\(265\) 8.69879 0.534362
\(266\) −0.212539 −0.0130316
\(267\) 5.17345 0.316610
\(268\) 7.27255 0.444242
\(269\) −11.0204 −0.671926 −0.335963 0.941875i \(-0.609062\pi\)
−0.335963 + 0.941875i \(0.609062\pi\)
\(270\) −1.90946 −0.116206
\(271\) 18.9404 1.15055 0.575275 0.817960i \(-0.304895\pi\)
0.575275 + 0.817960i \(0.304895\pi\)
\(272\) 0.662225 0.0401533
\(273\) 10.9922 0.665279
\(274\) 10.3446 0.624938
\(275\) −8.40207 −0.506664
\(276\) 15.5593 0.936559
\(277\) −11.5489 −0.693908 −0.346954 0.937882i \(-0.612784\pi\)
−0.346954 + 0.937882i \(0.612784\pi\)
\(278\) 8.38201 0.502720
\(279\) −7.78117 −0.465846
\(280\) 1.49148 0.0891327
\(281\) 0.553133 0.0329971 0.0164986 0.999864i \(-0.494748\pi\)
0.0164986 + 0.999864i \(0.494748\pi\)
\(282\) 29.2239 1.74026
\(283\) −26.5908 −1.58066 −0.790330 0.612681i \(-0.790091\pi\)
−0.790330 + 0.612681i \(0.790091\pi\)
\(284\) 11.1253 0.660167
\(285\) −0.675994 −0.0400424
\(286\) −9.86201 −0.583153
\(287\) 2.25557 0.133142
\(288\) −22.9118 −1.35009
\(289\) 22.6320 1.33130
\(290\) −2.41398 −0.141754
\(291\) −25.8494 −1.51532
\(292\) −8.61220 −0.503991
\(293\) 10.8142 0.631773 0.315886 0.948797i \(-0.397698\pi\)
0.315886 + 0.948797i \(0.397698\pi\)
\(294\) −15.3703 −0.896412
\(295\) −5.10592 −0.297278
\(296\) −8.87244 −0.515700
\(297\) −5.46009 −0.316826
\(298\) 11.3033 0.654784
\(299\) 28.3456 1.63927
\(300\) −14.3458 −0.828257
\(301\) −5.51319 −0.317775
\(302\) 9.88504 0.568820
\(303\) 5.76561 0.331226
\(304\) 0.0358667 0.00205710
\(305\) 6.79793 0.389249
\(306\) 22.8239 1.30475
\(307\) 4.14390 0.236505 0.118252 0.992984i \(-0.462271\pi\)
0.118252 + 0.992984i \(0.462271\pi\)
\(308\) 1.60963 0.0917173
\(309\) −43.8194 −2.49280
\(310\) −1.25901 −0.0715068
\(311\) −17.2875 −0.980283 −0.490141 0.871643i \(-0.663055\pi\)
−0.490141 + 0.871643i \(0.663055\pi\)
\(312\) −44.6153 −2.52584
\(313\) −1.70690 −0.0964795 −0.0482397 0.998836i \(-0.515361\pi\)
−0.0482397 + 0.998836i \(0.515361\pi\)
\(314\) −12.8857 −0.727184
\(315\) 2.13725 0.120420
\(316\) 4.61903 0.259841
\(317\) 5.39482 0.303003 0.151502 0.988457i \(-0.451589\pi\)
0.151502 + 0.988457i \(0.451589\pi\)
\(318\) −27.5885 −1.54709
\(319\) −6.90276 −0.386480
\(320\) −3.86387 −0.215997
\(321\) 40.5225 2.26175
\(322\) 3.00529 0.167478
\(323\) 2.14651 0.119435
\(324\) 5.53650 0.307583
\(325\) −26.1349 −1.44970
\(326\) 6.20170 0.343480
\(327\) −35.6801 −1.97312
\(328\) −9.15493 −0.505497
\(329\) −8.68951 −0.479068
\(330\) −3.32561 −0.183069
\(331\) −25.3872 −1.39541 −0.697703 0.716387i \(-0.745795\pi\)
−0.697703 + 0.716387i \(0.745795\pi\)
\(332\) 20.1038 1.10334
\(333\) −12.7140 −0.696723
\(334\) 6.68149 0.365595
\(335\) 4.46775 0.244099
\(336\) −0.196671 −0.0107293
\(337\) 16.3438 0.890306 0.445153 0.895455i \(-0.353149\pi\)
0.445153 + 0.895455i \(0.353149\pi\)
\(338\) −19.1391 −1.04103
\(339\) 21.7808 1.18297
\(340\) −5.68502 −0.308314
\(341\) −3.60013 −0.194958
\(342\) 1.23616 0.0668440
\(343\) 9.48699 0.512249
\(344\) 22.3770 1.20648
\(345\) 9.55854 0.514615
\(346\) 5.35244 0.287749
\(347\) 21.5466 1.15668 0.578341 0.815795i \(-0.303700\pi\)
0.578341 + 0.815795i \(0.303700\pi\)
\(348\) −11.7859 −0.631790
\(349\) −3.97176 −0.212603 −0.106302 0.994334i \(-0.533901\pi\)
−0.106302 + 0.994334i \(0.533901\pi\)
\(350\) −2.77091 −0.148111
\(351\) −16.9838 −0.906526
\(352\) −10.6006 −0.565016
\(353\) 11.5765 0.616152 0.308076 0.951362i \(-0.400315\pi\)
0.308076 + 0.951362i \(0.400315\pi\)
\(354\) 16.1936 0.860680
\(355\) 6.83463 0.362744
\(356\) 2.35644 0.124891
\(357\) −11.7702 −0.622942
\(358\) −10.8754 −0.574784
\(359\) 20.0579 1.05862 0.529308 0.848430i \(-0.322452\pi\)
0.529308 + 0.848430i \(0.322452\pi\)
\(360\) −8.67468 −0.457196
\(361\) −18.8837 −0.993881
\(362\) −2.34922 −0.123472
\(363\) 19.7704 1.03767
\(364\) 5.00681 0.262428
\(365\) −5.29074 −0.276930
\(366\) −21.5599 −1.12695
\(367\) −4.02946 −0.210336 −0.105168 0.994454i \(-0.533538\pi\)
−0.105168 + 0.994454i \(0.533538\pi\)
\(368\) −0.507155 −0.0264373
\(369\) −13.1188 −0.682938
\(370\) −2.05715 −0.106946
\(371\) 8.20322 0.425890
\(372\) −6.14691 −0.318703
\(373\) −10.2886 −0.532725 −0.266362 0.963873i \(-0.585822\pi\)
−0.266362 + 0.963873i \(0.585822\pi\)
\(374\) 10.5600 0.546042
\(375\) −18.7260 −0.967008
\(376\) 35.2690 1.81886
\(377\) −21.4712 −1.10583
\(378\) −1.80068 −0.0926168
\(379\) 15.1331 0.777335 0.388667 0.921378i \(-0.372935\pi\)
0.388667 + 0.921378i \(0.372935\pi\)
\(380\) −0.307906 −0.0157953
\(381\) −15.8769 −0.813399
\(382\) 7.62665 0.390213
\(383\) 15.1141 0.772292 0.386146 0.922438i \(-0.373806\pi\)
0.386146 + 0.922438i \(0.373806\pi\)
\(384\) −17.6027 −0.898285
\(385\) 0.988846 0.0503962
\(386\) −1.94966 −0.0992349
\(387\) 32.0657 1.62999
\(388\) −11.7741 −0.597737
\(389\) −21.8318 −1.10692 −0.553459 0.832876i \(-0.686693\pi\)
−0.553459 + 0.832876i \(0.686693\pi\)
\(390\) −10.3444 −0.523810
\(391\) −30.3516 −1.53495
\(392\) −18.5497 −0.936900
\(393\) 51.8882 2.61741
\(394\) 5.10206 0.257038
\(395\) 2.83761 0.142776
\(396\) −9.36191 −0.470453
\(397\) 6.59713 0.331101 0.165550 0.986201i \(-0.447060\pi\)
0.165550 + 0.986201i \(0.447060\pi\)
\(398\) −0.0953294 −0.00477843
\(399\) −0.637483 −0.0319141
\(400\) 0.467602 0.0233801
\(401\) −3.00932 −0.150278 −0.0751391 0.997173i \(-0.523940\pi\)
−0.0751391 + 0.997173i \(0.523940\pi\)
\(402\) −14.1696 −0.706716
\(403\) −11.1983 −0.557827
\(404\) 2.62616 0.130656
\(405\) 3.40124 0.169009
\(406\) −2.27645 −0.112979
\(407\) −5.88242 −0.291580
\(408\) 47.7728 2.36510
\(409\) −28.4495 −1.40674 −0.703370 0.710824i \(-0.748322\pi\)
−0.703370 + 0.710824i \(0.748322\pi\)
\(410\) −2.12265 −0.104830
\(411\) 31.0272 1.53046
\(412\) −19.9592 −0.983318
\(413\) −4.81504 −0.236933
\(414\) −17.4793 −0.859061
\(415\) 12.3504 0.606256
\(416\) −32.9736 −1.61666
\(417\) 25.1408 1.23115
\(418\) 0.571938 0.0279744
\(419\) −17.2544 −0.842930 −0.421465 0.906845i \(-0.638484\pi\)
−0.421465 + 0.906845i \(0.638484\pi\)
\(420\) 1.68837 0.0823840
\(421\) −12.2245 −0.595786 −0.297893 0.954599i \(-0.596284\pi\)
−0.297893 + 0.954599i \(0.596284\pi\)
\(422\) 22.3398 1.08748
\(423\) 50.5397 2.45733
\(424\) −33.2953 −1.61696
\(425\) 27.9845 1.35745
\(426\) −21.6762 −1.05022
\(427\) 6.41066 0.310234
\(428\) 18.4575 0.892176
\(429\) −29.5798 −1.42813
\(430\) 5.18829 0.250201
\(431\) 37.6535 1.81370 0.906852 0.421448i \(-0.138478\pi\)
0.906852 + 0.421448i \(0.138478\pi\)
\(432\) 0.303871 0.0146200
\(433\) −16.3693 −0.786659 −0.393329 0.919398i \(-0.628677\pi\)
−0.393329 + 0.919398i \(0.628677\pi\)
\(434\) −1.18728 −0.0569914
\(435\) −7.24042 −0.347152
\(436\) −16.2518 −0.778322
\(437\) −1.64387 −0.0786371
\(438\) 16.7797 0.801767
\(439\) 40.1000 1.91387 0.956935 0.290302i \(-0.0937556\pi\)
0.956935 + 0.290302i \(0.0937556\pi\)
\(440\) −4.01353 −0.191338
\(441\) −26.5813 −1.26577
\(442\) 32.8471 1.56238
\(443\) 22.2005 1.05478 0.527389 0.849624i \(-0.323171\pi\)
0.527389 + 0.849624i \(0.323171\pi\)
\(444\) −10.0437 −0.476654
\(445\) 1.44763 0.0686243
\(446\) −17.5103 −0.829137
\(447\) 33.9029 1.60355
\(448\) −3.64375 −0.172151
\(449\) −31.0197 −1.46391 −0.731955 0.681353i \(-0.761392\pi\)
−0.731955 + 0.681353i \(0.761392\pi\)
\(450\) 16.1161 0.759721
\(451\) −6.06971 −0.285811
\(452\) 9.92089 0.466639
\(453\) 29.6489 1.39303
\(454\) 9.74553 0.457380
\(455\) 3.07583 0.144197
\(456\) 2.58742 0.121167
\(457\) 36.3469 1.70024 0.850118 0.526593i \(-0.176531\pi\)
0.850118 + 0.526593i \(0.176531\pi\)
\(458\) 14.8139 0.692208
\(459\) 18.1857 0.848838
\(460\) 4.35379 0.202997
\(461\) 6.97833 0.325013 0.162507 0.986707i \(-0.448042\pi\)
0.162507 + 0.986707i \(0.448042\pi\)
\(462\) −3.13616 −0.145907
\(463\) −6.18167 −0.287287 −0.143643 0.989630i \(-0.545882\pi\)
−0.143643 + 0.989630i \(0.545882\pi\)
\(464\) 0.384161 0.0178342
\(465\) −3.77623 −0.175119
\(466\) −5.56863 −0.257962
\(467\) −14.4609 −0.669171 −0.334585 0.942365i \(-0.608596\pi\)
−0.334585 + 0.942365i \(0.608596\pi\)
\(468\) −29.1205 −1.34610
\(469\) 4.21322 0.194549
\(470\) 8.17742 0.377196
\(471\) −38.6491 −1.78086
\(472\) 19.5433 0.899554
\(473\) 14.8359 0.682155
\(474\) −8.99957 −0.413364
\(475\) 1.51567 0.0695436
\(476\) −5.36115 −0.245728
\(477\) −47.7114 −2.18455
\(478\) 13.0779 0.598169
\(479\) −28.0812 −1.28306 −0.641531 0.767097i \(-0.721701\pi\)
−0.641531 + 0.767097i \(0.721701\pi\)
\(480\) −11.1192 −0.507519
\(481\) −18.2974 −0.834291
\(482\) −12.4674 −0.567874
\(483\) 9.01400 0.410151
\(484\) 9.00514 0.409325
\(485\) −7.23316 −0.328441
\(486\) −18.4780 −0.838180
\(487\) −1.25353 −0.0568031 −0.0284015 0.999597i \(-0.509042\pi\)
−0.0284015 + 0.999597i \(0.509042\pi\)
\(488\) −26.0196 −1.17785
\(489\) 18.6012 0.841176
\(490\) −4.30090 −0.194295
\(491\) 27.8138 1.25522 0.627610 0.778528i \(-0.284033\pi\)
0.627610 + 0.778528i \(0.284033\pi\)
\(492\) −10.3635 −0.467223
\(493\) 22.9908 1.03545
\(494\) 1.77903 0.0800423
\(495\) −5.75130 −0.258502
\(496\) 0.200359 0.00899637
\(497\) 6.44526 0.289110
\(498\) −39.1696 −1.75523
\(499\) 15.5317 0.695294 0.347647 0.937626i \(-0.386981\pi\)
0.347647 + 0.937626i \(0.386981\pi\)
\(500\) −8.52946 −0.381449
\(501\) 20.0403 0.895334
\(502\) −8.82271 −0.393777
\(503\) 5.79797 0.258519 0.129259 0.991611i \(-0.458740\pi\)
0.129259 + 0.991611i \(0.458740\pi\)
\(504\) −8.18049 −0.364388
\(505\) 1.61333 0.0717922
\(506\) −8.08719 −0.359519
\(507\) −57.4054 −2.54946
\(508\) −7.23173 −0.320856
\(509\) 27.8607 1.23491 0.617453 0.786608i \(-0.288165\pi\)
0.617453 + 0.786608i \(0.288165\pi\)
\(510\) 11.0765 0.490476
\(511\) −4.98933 −0.220715
\(512\) 1.18974 0.0525795
\(513\) 0.984957 0.0434869
\(514\) 26.2738 1.15889
\(515\) −12.2615 −0.540307
\(516\) 25.3310 1.11514
\(517\) 23.3833 1.02840
\(518\) −1.93996 −0.0852368
\(519\) 16.0540 0.704691
\(520\) −12.4842 −0.547469
\(521\) 43.8886 1.92279 0.961397 0.275165i \(-0.0887325\pi\)
0.961397 + 0.275165i \(0.0887325\pi\)
\(522\) 13.2403 0.579511
\(523\) 34.5381 1.51025 0.755123 0.655583i \(-0.227577\pi\)
0.755123 + 0.655583i \(0.227577\pi\)
\(524\) 23.6344 1.03247
\(525\) −8.31100 −0.362722
\(526\) −1.20071 −0.0523533
\(527\) 11.9908 0.522329
\(528\) 0.529239 0.0230322
\(529\) 0.244341 0.0106235
\(530\) −7.71979 −0.335326
\(531\) 28.0051 1.21532
\(532\) −0.290365 −0.0125889
\(533\) −18.8800 −0.817784
\(534\) −4.59121 −0.198681
\(535\) 11.3390 0.490227
\(536\) −17.1007 −0.738636
\(537\) −32.6195 −1.40763
\(538\) 9.78013 0.421651
\(539\) −12.2984 −0.529730
\(540\) −2.60865 −0.112259
\(541\) −18.2238 −0.783501 −0.391751 0.920071i \(-0.628130\pi\)
−0.391751 + 0.920071i \(0.628130\pi\)
\(542\) −16.8088 −0.722000
\(543\) −7.04618 −0.302381
\(544\) 35.3072 1.51378
\(545\) −9.98399 −0.427667
\(546\) −9.75511 −0.417480
\(547\) −45.4436 −1.94303 −0.971514 0.236982i \(-0.923842\pi\)
−0.971514 + 0.236982i \(0.923842\pi\)
\(548\) 14.1325 0.603710
\(549\) −37.2855 −1.59131
\(550\) 7.45647 0.317945
\(551\) 1.24520 0.0530475
\(552\) −36.5861 −1.55721
\(553\) 2.67595 0.113793
\(554\) 10.2492 0.435446
\(555\) −6.17016 −0.261909
\(556\) 11.4513 0.485643
\(557\) 30.7625 1.30345 0.651725 0.758455i \(-0.274046\pi\)
0.651725 + 0.758455i \(0.274046\pi\)
\(558\) 6.90544 0.292331
\(559\) 46.1475 1.95183
\(560\) −0.0550324 −0.00232554
\(561\) 31.6733 1.33725
\(562\) −0.490881 −0.0207066
\(563\) −38.5265 −1.62370 −0.811849 0.583867i \(-0.801539\pi\)
−0.811849 + 0.583867i \(0.801539\pi\)
\(564\) 39.9250 1.68115
\(565\) 6.09470 0.256406
\(566\) 23.5982 0.991906
\(567\) 3.20747 0.134701
\(568\) −26.1601 −1.09765
\(569\) −38.2664 −1.60421 −0.802105 0.597183i \(-0.796287\pi\)
−0.802105 + 0.597183i \(0.796287\pi\)
\(570\) 0.599915 0.0251277
\(571\) −4.57864 −0.191610 −0.0958051 0.995400i \(-0.530543\pi\)
−0.0958051 + 0.995400i \(0.530543\pi\)
\(572\) −13.4732 −0.563344
\(573\) 22.8752 0.955624
\(574\) −2.00172 −0.0835503
\(575\) −21.4315 −0.893757
\(576\) 21.1927 0.883028
\(577\) 40.3948 1.68166 0.840828 0.541302i \(-0.182068\pi\)
0.840828 + 0.541302i \(0.182068\pi\)
\(578\) −20.0849 −0.835424
\(579\) −5.84775 −0.243024
\(580\) −3.29792 −0.136939
\(581\) 11.6468 0.483190
\(582\) 22.9402 0.950902
\(583\) −22.0747 −0.914242
\(584\) 20.2507 0.837980
\(585\) −17.8896 −0.739644
\(586\) −9.59714 −0.396454
\(587\) 7.39675 0.305296 0.152648 0.988281i \(-0.451220\pi\)
0.152648 + 0.988281i \(0.451220\pi\)
\(588\) −20.9985 −0.865963
\(589\) 0.649435 0.0267595
\(590\) 4.53128 0.186550
\(591\) 15.3030 0.629481
\(592\) 0.327375 0.0134550
\(593\) −1.30863 −0.0537390 −0.0268695 0.999639i \(-0.508554\pi\)
−0.0268695 + 0.999639i \(0.508554\pi\)
\(594\) 4.84559 0.198817
\(595\) −3.29352 −0.135021
\(596\) 15.4423 0.632542
\(597\) −0.285929 −0.0117023
\(598\) −25.1554 −1.02868
\(599\) 27.8919 1.13963 0.569816 0.821773i \(-0.307015\pi\)
0.569816 + 0.821773i \(0.307015\pi\)
\(600\) 33.7327 1.37713
\(601\) −12.3512 −0.503817 −0.251908 0.967751i \(-0.581058\pi\)
−0.251908 + 0.967751i \(0.581058\pi\)
\(602\) 4.89271 0.199412
\(603\) −24.5048 −0.997914
\(604\) 13.5047 0.549498
\(605\) 5.53213 0.224913
\(606\) −5.11673 −0.207853
\(607\) 37.5057 1.52231 0.761154 0.648571i \(-0.224633\pi\)
0.761154 + 0.648571i \(0.224633\pi\)
\(608\) 1.91227 0.0775529
\(609\) −6.82794 −0.276682
\(610\) −6.03287 −0.244264
\(611\) 72.7345 2.94252
\(612\) 31.1814 1.26043
\(613\) 38.7247 1.56407 0.782037 0.623232i \(-0.214181\pi\)
0.782037 + 0.623232i \(0.214181\pi\)
\(614\) −3.67753 −0.148413
\(615\) −6.36661 −0.256727
\(616\) −3.78488 −0.152497
\(617\) −1.44876 −0.0583247 −0.0291623 0.999575i \(-0.509284\pi\)
−0.0291623 + 0.999575i \(0.509284\pi\)
\(618\) 38.8878 1.56430
\(619\) −1.00000 −0.0401934
\(620\) −1.72003 −0.0690779
\(621\) −13.9273 −0.558882
\(622\) 15.3419 0.615153
\(623\) 1.36516 0.0546940
\(624\) 1.64621 0.0659013
\(625\) 16.9863 0.679450
\(626\) 1.51480 0.0605434
\(627\) 1.71545 0.0685087
\(628\) −17.6042 −0.702483
\(629\) 19.5924 0.781199
\(630\) −1.89672 −0.0755670
\(631\) −3.22709 −0.128469 −0.0642343 0.997935i \(-0.520460\pi\)
−0.0642343 + 0.997935i \(0.520460\pi\)
\(632\) −10.8612 −0.432034
\(633\) 67.0053 2.66322
\(634\) −4.78767 −0.190143
\(635\) −4.44267 −0.176302
\(636\) −37.6907 −1.49453
\(637\) −38.2546 −1.51570
\(638\) 6.12590 0.242527
\(639\) −37.4868 −1.48295
\(640\) −4.92558 −0.194701
\(641\) 33.9282 1.34008 0.670042 0.742323i \(-0.266276\pi\)
0.670042 + 0.742323i \(0.266276\pi\)
\(642\) −35.9620 −1.41931
\(643\) −1.59209 −0.0627860 −0.0313930 0.999507i \(-0.509994\pi\)
−0.0313930 + 0.999507i \(0.509994\pi\)
\(644\) 4.10576 0.161790
\(645\) 15.5616 0.612738
\(646\) −1.90493 −0.0749486
\(647\) −0.934541 −0.0367406 −0.0183703 0.999831i \(-0.505848\pi\)
−0.0183703 + 0.999831i \(0.505848\pi\)
\(648\) −13.0185 −0.511415
\(649\) 12.9572 0.508614
\(650\) 23.1936 0.909727
\(651\) −3.56110 −0.139571
\(652\) 8.47261 0.331813
\(653\) −25.8591 −1.01195 −0.505973 0.862549i \(-0.668866\pi\)
−0.505973 + 0.862549i \(0.668866\pi\)
\(654\) 31.6646 1.23818
\(655\) 14.5193 0.567317
\(656\) 0.337798 0.0131888
\(657\) 29.0188 1.13213
\(658\) 7.71156 0.300628
\(659\) −3.49910 −0.136306 −0.0681528 0.997675i \(-0.521711\pi\)
−0.0681528 + 0.997675i \(0.521711\pi\)
\(660\) −4.54337 −0.176851
\(661\) 31.4489 1.22322 0.611611 0.791159i \(-0.290522\pi\)
0.611611 + 0.791159i \(0.290522\pi\)
\(662\) 22.5300 0.875654
\(663\) 98.5206 3.82622
\(664\) −47.2721 −1.83451
\(665\) −0.178380 −0.00691728
\(666\) 11.2831 0.437212
\(667\) −17.6072 −0.681752
\(668\) 9.12808 0.353176
\(669\) −52.5199 −2.03054
\(670\) −3.96493 −0.153179
\(671\) −17.2510 −0.665966
\(672\) −10.4857 −0.404496
\(673\) −7.73543 −0.298179 −0.149089 0.988824i \(-0.547634\pi\)
−0.149089 + 0.988824i \(0.547634\pi\)
\(674\) −14.5044 −0.558690
\(675\) 12.8411 0.494254
\(676\) −26.1474 −1.00567
\(677\) −27.4352 −1.05442 −0.527210 0.849735i \(-0.676762\pi\)
−0.527210 + 0.849735i \(0.676762\pi\)
\(678\) −19.3295 −0.742347
\(679\) −6.82109 −0.261769
\(680\) 13.3677 0.512629
\(681\) 29.2305 1.12011
\(682\) 3.19496 0.122341
\(683\) −13.9103 −0.532261 −0.266131 0.963937i \(-0.585745\pi\)
−0.266131 + 0.963937i \(0.585745\pi\)
\(684\) 1.68881 0.0645734
\(685\) 8.68202 0.331723
\(686\) −8.41929 −0.321450
\(687\) 44.4324 1.69520
\(688\) −0.825665 −0.0314782
\(689\) −68.6641 −2.61589
\(690\) −8.48279 −0.322934
\(691\) 3.45766 0.131536 0.0657678 0.997835i \(-0.479050\pi\)
0.0657678 + 0.997835i \(0.479050\pi\)
\(692\) 7.31237 0.277975
\(693\) −5.42365 −0.206028
\(694\) −19.1217 −0.725848
\(695\) 7.03488 0.266848
\(696\) 27.7133 1.05047
\(697\) 20.2162 0.765742
\(698\) 3.52476 0.133414
\(699\) −16.7024 −0.631743
\(700\) −3.78555 −0.143080
\(701\) −5.79713 −0.218955 −0.109477 0.993989i \(-0.534918\pi\)
−0.109477 + 0.993989i \(0.534918\pi\)
\(702\) 15.0724 0.568869
\(703\) 1.06114 0.0400217
\(704\) 9.80526 0.369550
\(705\) 24.5272 0.923746
\(706\) −10.2736 −0.386652
\(707\) 1.52142 0.0572189
\(708\) 22.1233 0.831445
\(709\) −30.0393 −1.12815 −0.564075 0.825724i \(-0.690767\pi\)
−0.564075 + 0.825724i \(0.690767\pi\)
\(710\) −6.06543 −0.227632
\(711\) −15.5638 −0.583688
\(712\) −5.54092 −0.207655
\(713\) −9.18300 −0.343906
\(714\) 10.4455 0.390913
\(715\) −8.27701 −0.309543
\(716\) −14.8577 −0.555260
\(717\) 39.2255 1.46490
\(718\) −17.8005 −0.664310
\(719\) 1.18923 0.0443507 0.0221754 0.999754i \(-0.492941\pi\)
0.0221754 + 0.999754i \(0.492941\pi\)
\(720\) 0.320078 0.0119286
\(721\) −11.5630 −0.430628
\(722\) 16.7585 0.623687
\(723\) −37.3944 −1.39071
\(724\) −3.20944 −0.119278
\(725\) 16.2340 0.602915
\(726\) −17.5453 −0.651168
\(727\) −10.3813 −0.385021 −0.192511 0.981295i \(-0.561663\pi\)
−0.192511 + 0.981295i \(0.561663\pi\)
\(728\) −11.7730 −0.436336
\(729\) −41.7230 −1.54530
\(730\) 4.69530 0.173781
\(731\) −49.4134 −1.82762
\(732\) −29.4546 −1.08867
\(733\) −8.96405 −0.331095 −0.165547 0.986202i \(-0.552939\pi\)
−0.165547 + 0.986202i \(0.552939\pi\)
\(734\) 3.57597 0.131991
\(735\) −12.9000 −0.475824
\(736\) −27.0395 −0.996689
\(737\) −11.3377 −0.417630
\(738\) 11.6424 0.428561
\(739\) −16.7451 −0.615979 −0.307989 0.951390i \(-0.599656\pi\)
−0.307989 + 0.951390i \(0.599656\pi\)
\(740\) −2.81043 −0.103313
\(741\) 5.33597 0.196022
\(742\) −7.28000 −0.267257
\(743\) −25.6229 −0.940011 −0.470006 0.882663i \(-0.655748\pi\)
−0.470006 + 0.882663i \(0.655748\pi\)
\(744\) 14.4538 0.529903
\(745\) 9.48669 0.347565
\(746\) 9.13071 0.334299
\(747\) −67.7398 −2.47847
\(748\) 14.4268 0.527495
\(749\) 10.6930 0.390714
\(750\) 16.6185 0.606823
\(751\) −36.6073 −1.33582 −0.667910 0.744242i \(-0.732811\pi\)
−0.667910 + 0.744242i \(0.732811\pi\)
\(752\) −1.30136 −0.0474556
\(753\) −26.4626 −0.964351
\(754\) 19.0548 0.693935
\(755\) 8.29634 0.301935
\(756\) −2.46004 −0.0894708
\(757\) 5.69461 0.206974 0.103487 0.994631i \(-0.467000\pi\)
0.103487 + 0.994631i \(0.467000\pi\)
\(758\) −13.4300 −0.487798
\(759\) −24.2565 −0.880455
\(760\) 0.724010 0.0262626
\(761\) 22.4126 0.812457 0.406228 0.913772i \(-0.366844\pi\)
0.406228 + 0.913772i \(0.366844\pi\)
\(762\) 14.0901 0.510429
\(763\) −9.41521 −0.340853
\(764\) 10.4193 0.376959
\(765\) 19.1557 0.692575
\(766\) −13.4131 −0.484634
\(767\) 40.3037 1.45528
\(768\) 43.2385 1.56024
\(769\) 15.8950 0.573187 0.286594 0.958052i \(-0.407477\pi\)
0.286594 + 0.958052i \(0.407477\pi\)
\(770\) −0.877557 −0.0316250
\(771\) 78.8048 2.83809
\(772\) −2.66357 −0.0958641
\(773\) −53.8368 −1.93637 −0.968187 0.250228i \(-0.919494\pi\)
−0.968187 + 0.250228i \(0.919494\pi\)
\(774\) −28.4569 −1.02286
\(775\) 8.46682 0.304137
\(776\) 27.6855 0.993851
\(777\) −5.81865 −0.208743
\(778\) 19.3748 0.694620
\(779\) 1.09493 0.0392298
\(780\) −14.1323 −0.506018
\(781\) −17.3441 −0.620620
\(782\) 26.9357 0.963220
\(783\) 10.5497 0.377014
\(784\) 0.684446 0.0244445
\(785\) −10.8148 −0.385996
\(786\) −46.0485 −1.64250
\(787\) −24.9789 −0.890401 −0.445200 0.895431i \(-0.646868\pi\)
−0.445200 + 0.895431i \(0.646868\pi\)
\(788\) 6.97032 0.248307
\(789\) −3.60137 −0.128212
\(790\) −2.51825 −0.0895954
\(791\) 5.74749 0.204357
\(792\) 22.0136 0.782218
\(793\) −53.6597 −1.90551
\(794\) −5.85467 −0.207774
\(795\) −23.1546 −0.821207
\(796\) −0.130237 −0.00461612
\(797\) −44.5255 −1.57717 −0.788587 0.614923i \(-0.789187\pi\)
−0.788587 + 0.614923i \(0.789187\pi\)
\(798\) 0.565738 0.0200269
\(799\) −77.8821 −2.75527
\(800\) 24.9307 0.881434
\(801\) −7.94001 −0.280547
\(802\) 2.67064 0.0943035
\(803\) 13.4262 0.473800
\(804\) −19.3582 −0.682710
\(805\) 2.52229 0.0888991
\(806\) 9.93801 0.350051
\(807\) 29.3343 1.03262
\(808\) −6.17515 −0.217241
\(809\) −14.4853 −0.509276 −0.254638 0.967037i \(-0.581956\pi\)
−0.254638 + 0.967037i \(0.581956\pi\)
\(810\) −3.01845 −0.106057
\(811\) 47.9375 1.68331 0.841657 0.540012i \(-0.181580\pi\)
0.841657 + 0.540012i \(0.181580\pi\)
\(812\) −3.11004 −0.109141
\(813\) −50.4159 −1.76816
\(814\) 5.22039 0.182974
\(815\) 5.20498 0.182322
\(816\) −1.76272 −0.0617075
\(817\) −2.67628 −0.0936311
\(818\) 25.2477 0.882766
\(819\) −16.8704 −0.589501
\(820\) −2.89991 −0.101269
\(821\) 25.2123 0.879916 0.439958 0.898018i \(-0.354993\pi\)
0.439958 + 0.898018i \(0.354993\pi\)
\(822\) −27.5353 −0.960404
\(823\) −18.1629 −0.633117 −0.316559 0.948573i \(-0.602527\pi\)
−0.316559 + 0.948573i \(0.602527\pi\)
\(824\) 46.9319 1.63495
\(825\) 22.3648 0.778641
\(826\) 4.27314 0.148681
\(827\) −24.1281 −0.839015 −0.419508 0.907752i \(-0.637797\pi\)
−0.419508 + 0.907752i \(0.637797\pi\)
\(828\) −23.8798 −0.829881
\(829\) −30.8815 −1.07256 −0.536280 0.844040i \(-0.680171\pi\)
−0.536280 + 0.844040i \(0.680171\pi\)
\(830\) −10.9604 −0.380442
\(831\) 30.7411 1.06640
\(832\) 30.4995 1.05738
\(833\) 40.9619 1.41925
\(834\) −22.3113 −0.772579
\(835\) 5.60766 0.194061
\(836\) 0.781367 0.0270242
\(837\) 5.50217 0.190183
\(838\) 15.3125 0.528961
\(839\) 6.41671 0.221529 0.110765 0.993847i \(-0.464670\pi\)
0.110765 + 0.993847i \(0.464670\pi\)
\(840\) −3.97003 −0.136979
\(841\) −15.6629 −0.540100
\(842\) 10.8487 0.373871
\(843\) −1.47234 −0.0507100
\(844\) 30.5200 1.05054
\(845\) −16.0631 −0.552589
\(846\) −44.8518 −1.54204
\(847\) 5.21697 0.179257
\(848\) 1.22853 0.0421879
\(849\) 70.7798 2.42916
\(850\) −24.8350 −0.851835
\(851\) −15.0045 −0.514349
\(852\) −29.6136 −1.01454
\(853\) −10.6345 −0.364119 −0.182060 0.983288i \(-0.558276\pi\)
−0.182060 + 0.983288i \(0.558276\pi\)
\(854\) −5.68918 −0.194680
\(855\) 1.03749 0.0354814
\(856\) −43.4008 −1.48341
\(857\) 4.45614 0.152219 0.0761095 0.997099i \(-0.475750\pi\)
0.0761095 + 0.997099i \(0.475750\pi\)
\(858\) 26.2508 0.896188
\(859\) 5.10952 0.174335 0.0871673 0.996194i \(-0.472219\pi\)
0.0871673 + 0.996194i \(0.472219\pi\)
\(860\) 7.08811 0.241703
\(861\) −6.00391 −0.204613
\(862\) −33.4158 −1.13815
\(863\) −17.1442 −0.583597 −0.291798 0.956480i \(-0.594254\pi\)
−0.291798 + 0.956480i \(0.594254\pi\)
\(864\) 16.2012 0.551177
\(865\) 4.49221 0.152740
\(866\) 14.5270 0.493649
\(867\) −60.2423 −2.04593
\(868\) −1.62204 −0.0550555
\(869\) −7.20094 −0.244275
\(870\) 6.42556 0.217847
\(871\) −35.2663 −1.19495
\(872\) 38.2145 1.29411
\(873\) 39.6727 1.34272
\(874\) 1.45887 0.0493469
\(875\) −4.94139 −0.167049
\(876\) 22.9241 0.774533
\(877\) −39.8771 −1.34655 −0.673277 0.739391i \(-0.735114\pi\)
−0.673277 + 0.739391i \(0.735114\pi\)
\(878\) −35.5870 −1.20100
\(879\) −28.7854 −0.970907
\(880\) 0.148091 0.00499216
\(881\) −6.42351 −0.216413 −0.108207 0.994128i \(-0.534511\pi\)
−0.108207 + 0.994128i \(0.534511\pi\)
\(882\) 23.5897 0.794307
\(883\) 40.3184 1.35682 0.678410 0.734683i \(-0.262669\pi\)
0.678410 + 0.734683i \(0.262669\pi\)
\(884\) 44.8749 1.50931
\(885\) 13.5910 0.456857
\(886\) −19.7020 −0.661900
\(887\) −19.3059 −0.648230 −0.324115 0.946018i \(-0.605066\pi\)
−0.324115 + 0.946018i \(0.605066\pi\)
\(888\) 23.6168 0.792527
\(889\) −4.18957 −0.140514
\(890\) −1.28471 −0.0430635
\(891\) −8.63124 −0.289158
\(892\) −23.9221 −0.800973
\(893\) −4.21817 −0.141156
\(894\) −30.0873 −1.00627
\(895\) −9.12756 −0.305101
\(896\) −4.64497 −0.155178
\(897\) −75.4506 −2.51922
\(898\) 27.5286 0.918642
\(899\) 6.95596 0.231994
\(900\) 22.0174 0.733915
\(901\) 73.5236 2.44943
\(902\) 5.38660 0.179354
\(903\) 14.6751 0.488356
\(904\) −23.3279 −0.775876
\(905\) −1.97166 −0.0655401
\(906\) −26.3121 −0.874162
\(907\) −15.5038 −0.514794 −0.257397 0.966306i \(-0.582865\pi\)
−0.257397 + 0.966306i \(0.582865\pi\)
\(908\) 13.3141 0.441844
\(909\) −8.84885 −0.293498
\(910\) −2.72967 −0.0904876
\(911\) 43.5095 1.44154 0.720768 0.693177i \(-0.243789\pi\)
0.720768 + 0.693177i \(0.243789\pi\)
\(912\) −0.0954705 −0.00316135
\(913\) −31.3413 −1.03725
\(914\) −32.2563 −1.06694
\(915\) −18.0948 −0.598197
\(916\) 20.2384 0.668695
\(917\) 13.6922 0.452155
\(918\) −16.1390 −0.532668
\(919\) −12.4316 −0.410081 −0.205040 0.978754i \(-0.565733\pi\)
−0.205040 + 0.978754i \(0.565733\pi\)
\(920\) −10.2375 −0.337520
\(921\) −11.0303 −0.363460
\(922\) −6.19296 −0.203954
\(923\) −53.9493 −1.77576
\(924\) −4.28454 −0.140951
\(925\) 13.8343 0.454870
\(926\) 5.48596 0.180280
\(927\) 67.2524 2.20886
\(928\) 20.4819 0.672352
\(929\) −8.03582 −0.263647 −0.131823 0.991273i \(-0.542083\pi\)
−0.131823 + 0.991273i \(0.542083\pi\)
\(930\) 3.35124 0.109892
\(931\) 2.21854 0.0727096
\(932\) −7.60773 −0.249200
\(933\) 46.0160 1.50650
\(934\) 12.8334 0.419922
\(935\) 8.86279 0.289844
\(936\) 68.4738 2.23814
\(937\) 48.8183 1.59483 0.797413 0.603434i \(-0.206201\pi\)
0.797413 + 0.603434i \(0.206201\pi\)
\(938\) −3.73905 −0.122084
\(939\) 4.54344 0.148270
\(940\) 11.1718 0.364384
\(941\) −39.8127 −1.29786 −0.648928 0.760850i \(-0.724782\pi\)
−0.648928 + 0.760850i \(0.724782\pi\)
\(942\) 34.2994 1.11754
\(943\) −15.4823 −0.504172
\(944\) −0.721109 −0.0234701
\(945\) −1.51128 −0.0491619
\(946\) −13.1662 −0.428070
\(947\) −58.4316 −1.89877 −0.949386 0.314111i \(-0.898294\pi\)
−0.949386 + 0.314111i \(0.898294\pi\)
\(948\) −12.2950 −0.399323
\(949\) 41.7626 1.35567
\(950\) −1.34509 −0.0436405
\(951\) −14.3600 −0.465655
\(952\) 12.6062 0.408569
\(953\) 15.4235 0.499616 0.249808 0.968295i \(-0.419633\pi\)
0.249808 + 0.968295i \(0.419633\pi\)
\(954\) 42.3418 1.37087
\(955\) 6.40092 0.207129
\(956\) 17.8667 0.577850
\(957\) 18.3739 0.593943
\(958\) 24.9208 0.805156
\(959\) 8.18741 0.264385
\(960\) 10.2849 0.331944
\(961\) −27.3721 −0.882972
\(962\) 16.2382 0.523540
\(963\) −62.1924 −2.00412
\(964\) −17.0326 −0.548584
\(965\) −1.63631 −0.0526748
\(966\) −7.99953 −0.257381
\(967\) −26.6967 −0.858508 −0.429254 0.903184i \(-0.641224\pi\)
−0.429254 + 0.903184i \(0.641224\pi\)
\(968\) −21.1747 −0.680579
\(969\) −5.71361 −0.183548
\(970\) 6.41911 0.206105
\(971\) −14.7951 −0.474798 −0.237399 0.971412i \(-0.576295\pi\)
−0.237399 + 0.971412i \(0.576295\pi\)
\(972\) −25.2442 −0.809708
\(973\) 6.63411 0.212680
\(974\) 1.11246 0.0356454
\(975\) 69.5662 2.22790
\(976\) 0.960072 0.0307311
\(977\) 0.662748 0.0212032 0.0106016 0.999944i \(-0.496625\pi\)
0.0106016 + 0.999944i \(0.496625\pi\)
\(978\) −16.5078 −0.527860
\(979\) −3.67362 −0.117409
\(980\) −5.87578 −0.187695
\(981\) 54.7605 1.74837
\(982\) −24.6836 −0.787684
\(983\) −11.3042 −0.360548 −0.180274 0.983616i \(-0.557698\pi\)
−0.180274 + 0.983616i \(0.557698\pi\)
\(984\) 24.3687 0.776846
\(985\) 4.28207 0.136438
\(986\) −20.4033 −0.649774
\(987\) 23.1299 0.736232
\(988\) 2.43047 0.0773234
\(989\) 37.8425 1.20332
\(990\) 5.10403 0.162217
\(991\) 1.44897 0.0460281 0.0230140 0.999735i \(-0.492674\pi\)
0.0230140 + 0.999735i \(0.492674\pi\)
\(992\) 10.6823 0.339164
\(993\) 67.5760 2.14446
\(994\) −5.71989 −0.181424
\(995\) −0.0800084 −0.00253644
\(996\) −53.5126 −1.69561
\(997\) −37.0709 −1.17405 −0.587024 0.809570i \(-0.699701\pi\)
−0.587024 + 0.809570i \(0.699701\pi\)
\(998\) −13.7837 −0.436315
\(999\) 8.99024 0.284439
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 619.2.a.a.1.9 21
3.2 odd 2 5571.2.a.e.1.13 21
4.3 odd 2 9904.2.a.j.1.19 21
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
619.2.a.a.1.9 21 1.1 even 1 trivial
5571.2.a.e.1.13 21 3.2 odd 2
9904.2.a.j.1.19 21 4.3 odd 2