Properties

Label 2671.2.a.b.1.91
Level $2671$
Weight $2$
Character 2671.1
Self dual yes
Analytic conductor $21.328$
Analytic rank $0$
Dimension $122$
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] = [2671,2,Mod(1,2671)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2671, 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("2671.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2671 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2671.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: \(21.3280423799\)
Analytic rank: \(0\)
Dimension: \(122\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.91
Character \(\chi\) \(=\) 2671.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.59269 q^{2} +0.133072 q^{3} +0.536661 q^{4} +3.85218 q^{5} +0.211942 q^{6} -3.78245 q^{7} -2.33064 q^{8} -2.98229 q^{9} +O(q^{10})\) \(q+1.59269 q^{2} +0.133072 q^{3} +0.536661 q^{4} +3.85218 q^{5} +0.211942 q^{6} -3.78245 q^{7} -2.33064 q^{8} -2.98229 q^{9} +6.13534 q^{10} +3.09744 q^{11} +0.0714145 q^{12} +5.35946 q^{13} -6.02427 q^{14} +0.512617 q^{15} -4.78532 q^{16} +4.63008 q^{17} -4.74987 q^{18} +7.98897 q^{19} +2.06732 q^{20} -0.503337 q^{21} +4.93326 q^{22} -7.87336 q^{23} -0.310143 q^{24} +9.83932 q^{25} +8.53596 q^{26} -0.796074 q^{27} -2.02990 q^{28} +4.15752 q^{29} +0.816440 q^{30} +1.46572 q^{31} -2.96024 q^{32} +0.412181 q^{33} +7.37429 q^{34} -14.5707 q^{35} -1.60048 q^{36} -4.72073 q^{37} +12.7240 q^{38} +0.713193 q^{39} -8.97807 q^{40} +7.47802 q^{41} -0.801660 q^{42} -9.48799 q^{43} +1.66227 q^{44} -11.4883 q^{45} -12.5398 q^{46} +6.78344 q^{47} -0.636791 q^{48} +7.30693 q^{49} +15.6710 q^{50} +0.616134 q^{51} +2.87621 q^{52} -1.40954 q^{53} -1.26790 q^{54} +11.9319 q^{55} +8.81555 q^{56} +1.06311 q^{57} +6.62164 q^{58} +6.62436 q^{59} +0.275102 q^{60} -0.305843 q^{61} +2.33443 q^{62} +11.2804 q^{63} +4.85589 q^{64} +20.6456 q^{65} +0.656477 q^{66} +13.7914 q^{67} +2.48479 q^{68} -1.04772 q^{69} -23.2066 q^{70} +9.58840 q^{71} +6.95066 q^{72} +5.49821 q^{73} -7.51866 q^{74} +1.30934 q^{75} +4.28737 q^{76} -11.7159 q^{77} +1.13589 q^{78} -8.56076 q^{79} -18.4339 q^{80} +8.84094 q^{81} +11.9102 q^{82} +7.47403 q^{83} -0.270122 q^{84} +17.8359 q^{85} -15.1114 q^{86} +0.553249 q^{87} -7.21902 q^{88} -5.28174 q^{89} -18.2974 q^{90} -20.2719 q^{91} -4.22533 q^{92} +0.195046 q^{93} +10.8039 q^{94} +30.7750 q^{95} -0.393924 q^{96} -8.88605 q^{97} +11.6377 q^{98} -9.23746 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 122 q + 14 q^{2} + 10 q^{3} + 128 q^{4} + 33 q^{5} + 22 q^{6} + 6 q^{7} + 36 q^{8} + 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 122 q + 14 q^{2} + 10 q^{3} + 128 q^{4} + 33 q^{5} + 22 q^{6} + 6 q^{7} + 36 q^{8} + 162 q^{9} + 16 q^{10} + 43 q^{11} + 23 q^{12} + 25 q^{13} + 45 q^{14} + 12 q^{15} + 132 q^{16} + 103 q^{17} + 30 q^{18} + 37 q^{19} + 63 q^{20} + 81 q^{21} + 15 q^{23} + 60 q^{24} + 151 q^{25} + 59 q^{26} + 22 q^{27} - 3 q^{28} + 80 q^{29} - 9 q^{30} + 15 q^{31} + 66 q^{32} + 93 q^{33} + 30 q^{34} + 23 q^{35} + 162 q^{36} + 18 q^{37} + 41 q^{38} + 10 q^{39} + 29 q^{40} + 249 q^{41} - 8 q^{42} + 14 q^{43} + 100 q^{44} + 59 q^{45} + 11 q^{46} + 57 q^{47} + 33 q^{48} + 180 q^{49} + 63 q^{50} + 26 q^{51} + 31 q^{52} + 65 q^{53} + 65 q^{54} - 8 q^{55} + 120 q^{56} + 57 q^{57} - 31 q^{58} + 108 q^{59} - q^{60} + 70 q^{61} + 25 q^{62} - 7 q^{63} + 100 q^{64} + 171 q^{65} + 12 q^{66} - 6 q^{67} + 184 q^{68} + 64 q^{69} - 24 q^{70} + 47 q^{71} + 53 q^{72} + 76 q^{73} + 66 q^{74} + 40 q^{75} + 32 q^{76} + 73 q^{77} - 19 q^{78} + 8 q^{79} + 115 q^{80} + 250 q^{81} - 13 q^{82} + 116 q^{83} + 159 q^{84} + 31 q^{85} + 91 q^{86} + 25 q^{87} - 43 q^{88} + 361 q^{89} + 32 q^{90} + 7 q^{91} + 5 q^{92} + 18 q^{93} + 23 q^{94} + 42 q^{95} + 77 q^{96} + 79 q^{97} + 56 q^{98} + 74 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.59269 1.12620 0.563101 0.826388i \(-0.309608\pi\)
0.563101 + 0.826388i \(0.309608\pi\)
\(3\) 0.133072 0.0768290 0.0384145 0.999262i \(-0.487769\pi\)
0.0384145 + 0.999262i \(0.487769\pi\)
\(4\) 0.536661 0.268331
\(5\) 3.85218 1.72275 0.861375 0.507970i \(-0.169604\pi\)
0.861375 + 0.507970i \(0.169604\pi\)
\(6\) 0.211942 0.0865250
\(7\) −3.78245 −1.42963 −0.714816 0.699313i \(-0.753489\pi\)
−0.714816 + 0.699313i \(0.753489\pi\)
\(8\) −2.33064 −0.824007
\(9\) −2.98229 −0.994097
\(10\) 6.13534 1.94016
\(11\) 3.09744 0.933912 0.466956 0.884280i \(-0.345351\pi\)
0.466956 + 0.884280i \(0.345351\pi\)
\(12\) 0.0714145 0.0206156
\(13\) 5.35946 1.48645 0.743223 0.669044i \(-0.233296\pi\)
0.743223 + 0.669044i \(0.233296\pi\)
\(14\) −6.02427 −1.61005
\(15\) 0.512617 0.132357
\(16\) −4.78532 −1.19633
\(17\) 4.63008 1.12296 0.561480 0.827490i \(-0.310232\pi\)
0.561480 + 0.827490i \(0.310232\pi\)
\(18\) −4.74987 −1.11955
\(19\) 7.98897 1.83280 0.916398 0.400269i \(-0.131083\pi\)
0.916398 + 0.400269i \(0.131083\pi\)
\(20\) 2.06732 0.462267
\(21\) −0.503337 −0.109837
\(22\) 4.93326 1.05177
\(23\) −7.87336 −1.64171 −0.820854 0.571138i \(-0.806502\pi\)
−0.820854 + 0.571138i \(0.806502\pi\)
\(24\) −0.310143 −0.0633077
\(25\) 9.83932 1.96786
\(26\) 8.53596 1.67404
\(27\) −0.796074 −0.153205
\(28\) −2.02990 −0.383614
\(29\) 4.15752 0.772032 0.386016 0.922492i \(-0.373851\pi\)
0.386016 + 0.922492i \(0.373851\pi\)
\(30\) 0.816440 0.149061
\(31\) 1.46572 0.263251 0.131625 0.991300i \(-0.457980\pi\)
0.131625 + 0.991300i \(0.457980\pi\)
\(32\) −2.96024 −0.523301
\(33\) 0.412181 0.0717516
\(34\) 7.37429 1.26468
\(35\) −14.5707 −2.46290
\(36\) −1.60048 −0.266747
\(37\) −4.72073 −0.776083 −0.388042 0.921642i \(-0.626848\pi\)
−0.388042 + 0.921642i \(0.626848\pi\)
\(38\) 12.7240 2.06410
\(39\) 0.713193 0.114202
\(40\) −8.97807 −1.41956
\(41\) 7.47802 1.16787 0.583935 0.811800i \(-0.301512\pi\)
0.583935 + 0.811800i \(0.301512\pi\)
\(42\) −0.801660 −0.123699
\(43\) −9.48799 −1.44690 −0.723452 0.690374i \(-0.757446\pi\)
−0.723452 + 0.690374i \(0.757446\pi\)
\(44\) 1.66227 0.250597
\(45\) −11.4883 −1.71258
\(46\) −12.5398 −1.84889
\(47\) 6.78344 0.989466 0.494733 0.869045i \(-0.335266\pi\)
0.494733 + 0.869045i \(0.335266\pi\)
\(48\) −0.636791 −0.0919128
\(49\) 7.30693 1.04385
\(50\) 15.6710 2.21621
\(51\) 0.616134 0.0862760
\(52\) 2.87621 0.398859
\(53\) −1.40954 −0.193615 −0.0968074 0.995303i \(-0.530863\pi\)
−0.0968074 + 0.995303i \(0.530863\pi\)
\(54\) −1.26790 −0.172539
\(55\) 11.9319 1.60890
\(56\) 8.81555 1.17803
\(57\) 1.06311 0.140812
\(58\) 6.62164 0.869464
\(59\) 6.62436 0.862418 0.431209 0.902252i \(-0.358087\pi\)
0.431209 + 0.902252i \(0.358087\pi\)
\(60\) 0.275102 0.0355155
\(61\) −0.305843 −0.0391592 −0.0195796 0.999808i \(-0.506233\pi\)
−0.0195796 + 0.999808i \(0.506233\pi\)
\(62\) 2.33443 0.296473
\(63\) 11.2804 1.42119
\(64\) 4.85589 0.606987
\(65\) 20.6456 2.56077
\(66\) 0.656477 0.0808068
\(67\) 13.7914 1.68489 0.842446 0.538780i \(-0.181115\pi\)
0.842446 + 0.538780i \(0.181115\pi\)
\(68\) 2.48479 0.301325
\(69\) −1.04772 −0.126131
\(70\) −23.2066 −2.77372
\(71\) 9.58840 1.13793 0.568967 0.822361i \(-0.307343\pi\)
0.568967 + 0.822361i \(0.307343\pi\)
\(72\) 6.95066 0.819143
\(73\) 5.49821 0.643517 0.321758 0.946822i \(-0.395726\pi\)
0.321758 + 0.946822i \(0.395726\pi\)
\(74\) −7.51866 −0.874026
\(75\) 1.30934 0.151189
\(76\) 4.28737 0.491795
\(77\) −11.7159 −1.33515
\(78\) 1.13589 0.128615
\(79\) −8.56076 −0.963160 −0.481580 0.876402i \(-0.659937\pi\)
−0.481580 + 0.876402i \(0.659937\pi\)
\(80\) −18.4339 −2.06098
\(81\) 8.84094 0.982327
\(82\) 11.9102 1.31526
\(83\) 7.47403 0.820381 0.410190 0.912000i \(-0.365462\pi\)
0.410190 + 0.912000i \(0.365462\pi\)
\(84\) −0.270122 −0.0294727
\(85\) 17.8359 1.93458
\(86\) −15.1114 −1.62951
\(87\) 0.553249 0.0593145
\(88\) −7.21902 −0.769551
\(89\) −5.28174 −0.559863 −0.279932 0.960020i \(-0.590312\pi\)
−0.279932 + 0.960020i \(0.590312\pi\)
\(90\) −18.2974 −1.92871
\(91\) −20.2719 −2.12507
\(92\) −4.22533 −0.440521
\(93\) 0.195046 0.0202253
\(94\) 10.8039 1.11434
\(95\) 30.7750 3.15745
\(96\) −0.393924 −0.0402047
\(97\) −8.88605 −0.902242 −0.451121 0.892463i \(-0.648976\pi\)
−0.451121 + 0.892463i \(0.648976\pi\)
\(98\) 11.6377 1.17558
\(99\) −9.23746 −0.928400
\(100\) 5.28039 0.528039
\(101\) −3.50694 −0.348954 −0.174477 0.984661i \(-0.555823\pi\)
−0.174477 + 0.984661i \(0.555823\pi\)
\(102\) 0.981310 0.0971641
\(103\) −15.4205 −1.51943 −0.759714 0.650257i \(-0.774661\pi\)
−0.759714 + 0.650257i \(0.774661\pi\)
\(104\) −12.4910 −1.22484
\(105\) −1.93895 −0.189222
\(106\) −2.24496 −0.218049
\(107\) −20.2044 −1.95323 −0.976614 0.214999i \(-0.931025\pi\)
−0.976614 + 0.214999i \(0.931025\pi\)
\(108\) −0.427222 −0.0411095
\(109\) 19.6582 1.88291 0.941456 0.337135i \(-0.109458\pi\)
0.941456 + 0.337135i \(0.109458\pi\)
\(110\) 19.0038 1.81194
\(111\) −0.628196 −0.0596257
\(112\) 18.1002 1.71031
\(113\) 7.21759 0.678974 0.339487 0.940611i \(-0.389747\pi\)
0.339487 + 0.940611i \(0.389747\pi\)
\(114\) 1.69320 0.158583
\(115\) −30.3296 −2.82825
\(116\) 2.23118 0.207160
\(117\) −15.9835 −1.47767
\(118\) 10.5506 0.971257
\(119\) −17.5131 −1.60542
\(120\) −1.19473 −0.109063
\(121\) −1.40589 −0.127808
\(122\) −0.487113 −0.0441011
\(123\) 0.995114 0.0897264
\(124\) 0.786594 0.0706382
\(125\) 18.6420 1.66739
\(126\) 17.9661 1.60055
\(127\) −4.79569 −0.425549 −0.212774 0.977101i \(-0.568250\pi\)
−0.212774 + 0.977101i \(0.568250\pi\)
\(128\) 13.6544 1.20689
\(129\) −1.26258 −0.111164
\(130\) 32.8821 2.88395
\(131\) −16.9873 −1.48419 −0.742095 0.670295i \(-0.766167\pi\)
−0.742095 + 0.670295i \(0.766167\pi\)
\(132\) 0.221202 0.0192532
\(133\) −30.2179 −2.62022
\(134\) 21.9655 1.89753
\(135\) −3.06662 −0.263933
\(136\) −10.7911 −0.925328
\(137\) 12.1572 1.03866 0.519331 0.854573i \(-0.326181\pi\)
0.519331 + 0.854573i \(0.326181\pi\)
\(138\) −1.66870 −0.142049
\(139\) 13.8845 1.17766 0.588832 0.808256i \(-0.299588\pi\)
0.588832 + 0.808256i \(0.299588\pi\)
\(140\) −7.81953 −0.660871
\(141\) 0.902684 0.0760197
\(142\) 15.2713 1.28154
\(143\) 16.6006 1.38821
\(144\) 14.2712 1.18927
\(145\) 16.0155 1.33002
\(146\) 8.75695 0.724730
\(147\) 0.972346 0.0801977
\(148\) −2.53343 −0.208247
\(149\) 9.26574 0.759079 0.379539 0.925176i \(-0.376082\pi\)
0.379539 + 0.925176i \(0.376082\pi\)
\(150\) 2.08537 0.170269
\(151\) −6.58802 −0.536126 −0.268063 0.963401i \(-0.586383\pi\)
−0.268063 + 0.963401i \(0.586383\pi\)
\(152\) −18.6195 −1.51024
\(153\) −13.8083 −1.11633
\(154\) −18.6598 −1.50365
\(155\) 5.64621 0.453515
\(156\) 0.382743 0.0306440
\(157\) −14.5013 −1.15733 −0.578665 0.815565i \(-0.696426\pi\)
−0.578665 + 0.815565i \(0.696426\pi\)
\(158\) −13.6346 −1.08471
\(159\) −0.187570 −0.0148752
\(160\) −11.4034 −0.901517
\(161\) 29.7806 2.34704
\(162\) 14.0809 1.10630
\(163\) −21.8843 −1.71411 −0.857054 0.515226i \(-0.827708\pi\)
−0.857054 + 0.515226i \(0.827708\pi\)
\(164\) 4.01317 0.313376
\(165\) 1.58780 0.123610
\(166\) 11.9038 0.923915
\(167\) −20.0378 −1.55057 −0.775287 0.631609i \(-0.782395\pi\)
−0.775287 + 0.631609i \(0.782395\pi\)
\(168\) 1.17310 0.0905067
\(169\) 15.7238 1.20952
\(170\) 28.4071 2.17873
\(171\) −23.8254 −1.82198
\(172\) −5.09184 −0.388249
\(173\) −21.0940 −1.60375 −0.801875 0.597492i \(-0.796164\pi\)
−0.801875 + 0.597492i \(0.796164\pi\)
\(174\) 0.881154 0.0668001
\(175\) −37.2167 −2.81332
\(176\) −14.8222 −1.11727
\(177\) 0.881515 0.0662588
\(178\) −8.41218 −0.630519
\(179\) 12.7991 0.956649 0.478324 0.878183i \(-0.341244\pi\)
0.478324 + 0.878183i \(0.341244\pi\)
\(180\) −6.16535 −0.459538
\(181\) −8.45270 −0.628284 −0.314142 0.949376i \(-0.601717\pi\)
−0.314142 + 0.949376i \(0.601717\pi\)
\(182\) −32.2868 −2.39326
\(183\) −0.0406991 −0.00300856
\(184\) 18.3500 1.35278
\(185\) −18.1851 −1.33700
\(186\) 0.310647 0.0227778
\(187\) 14.3414 1.04875
\(188\) 3.64041 0.265504
\(189\) 3.01111 0.219026
\(190\) 49.0150 3.55592
\(191\) −4.36910 −0.316137 −0.158069 0.987428i \(-0.550527\pi\)
−0.158069 + 0.987428i \(0.550527\pi\)
\(192\) 0.646182 0.0466342
\(193\) 18.3763 1.32275 0.661377 0.750054i \(-0.269973\pi\)
0.661377 + 0.750054i \(0.269973\pi\)
\(194\) −14.1527 −1.01611
\(195\) 2.74735 0.196742
\(196\) 3.92135 0.280096
\(197\) −15.6143 −1.11247 −0.556237 0.831024i \(-0.687755\pi\)
−0.556237 + 0.831024i \(0.687755\pi\)
\(198\) −14.7124 −1.04557
\(199\) 11.9493 0.847065 0.423533 0.905881i \(-0.360790\pi\)
0.423533 + 0.905881i \(0.360790\pi\)
\(200\) −22.9320 −1.62153
\(201\) 1.83525 0.129449
\(202\) −5.58547 −0.392992
\(203\) −15.7256 −1.10372
\(204\) 0.330655 0.0231505
\(205\) 28.8067 2.01195
\(206\) −24.5601 −1.71118
\(207\) 23.4806 1.63202
\(208\) −25.6467 −1.77828
\(209\) 24.7453 1.71167
\(210\) −3.08814 −0.213102
\(211\) −0.549275 −0.0378137 −0.0189068 0.999821i \(-0.506019\pi\)
−0.0189068 + 0.999821i \(0.506019\pi\)
\(212\) −0.756444 −0.0519528
\(213\) 1.27594 0.0874263
\(214\) −32.1793 −2.19973
\(215\) −36.5495 −2.49265
\(216\) 1.85537 0.126242
\(217\) −5.54400 −0.376351
\(218\) 31.3094 2.12054
\(219\) 0.731657 0.0494408
\(220\) 6.40339 0.431716
\(221\) 24.8147 1.66922
\(222\) −1.00052 −0.0671506
\(223\) 14.9690 1.00240 0.501199 0.865332i \(-0.332892\pi\)
0.501199 + 0.865332i \(0.332892\pi\)
\(224\) 11.1970 0.748128
\(225\) −29.3437 −1.95625
\(226\) 11.4954 0.764661
\(227\) 4.38784 0.291231 0.145615 0.989341i \(-0.453484\pi\)
0.145615 + 0.989341i \(0.453484\pi\)
\(228\) 0.570528 0.0377842
\(229\) −1.43584 −0.0948827 −0.0474413 0.998874i \(-0.515107\pi\)
−0.0474413 + 0.998874i \(0.515107\pi\)
\(230\) −48.3057 −3.18518
\(231\) −1.55906 −0.102578
\(232\) −9.68970 −0.636160
\(233\) −8.51135 −0.557597 −0.278798 0.960350i \(-0.589936\pi\)
−0.278798 + 0.960350i \(0.589936\pi\)
\(234\) −25.4567 −1.66416
\(235\) 26.1311 1.70460
\(236\) 3.55504 0.231413
\(237\) −1.13919 −0.0739987
\(238\) −27.8929 −1.80803
\(239\) −24.7635 −1.60182 −0.800908 0.598788i \(-0.795649\pi\)
−0.800908 + 0.598788i \(0.795649\pi\)
\(240\) −2.45304 −0.158343
\(241\) 5.53580 0.356592 0.178296 0.983977i \(-0.442941\pi\)
0.178296 + 0.983977i \(0.442941\pi\)
\(242\) −2.23914 −0.143938
\(243\) 3.56470 0.228676
\(244\) −0.164134 −0.0105076
\(245\) 28.1476 1.79829
\(246\) 1.58491 0.101050
\(247\) 42.8166 2.72435
\(248\) −3.41607 −0.216920
\(249\) 0.994582 0.0630291
\(250\) 29.6909 1.87782
\(251\) 13.4878 0.851340 0.425670 0.904878i \(-0.360038\pi\)
0.425670 + 0.904878i \(0.360038\pi\)
\(252\) 6.05374 0.381350
\(253\) −24.3872 −1.53321
\(254\) −7.63805 −0.479254
\(255\) 2.37346 0.148632
\(256\) 12.0355 0.752216
\(257\) −4.54466 −0.283488 −0.141744 0.989903i \(-0.545271\pi\)
−0.141744 + 0.989903i \(0.545271\pi\)
\(258\) −2.01090 −0.125193
\(259\) 17.8559 1.10951
\(260\) 11.0797 0.687134
\(261\) −12.3989 −0.767475
\(262\) −27.0555 −1.67150
\(263\) −25.9294 −1.59888 −0.799438 0.600749i \(-0.794869\pi\)
−0.799438 + 0.600749i \(0.794869\pi\)
\(264\) −0.960648 −0.0591238
\(265\) −5.42980 −0.333550
\(266\) −48.1277 −2.95090
\(267\) −0.702851 −0.0430138
\(268\) 7.40133 0.452108
\(269\) 12.7042 0.774590 0.387295 0.921956i \(-0.373409\pi\)
0.387295 + 0.921956i \(0.373409\pi\)
\(270\) −4.88418 −0.297242
\(271\) −18.3009 −1.11170 −0.555849 0.831283i \(-0.687607\pi\)
−0.555849 + 0.831283i \(0.687607\pi\)
\(272\) −22.1564 −1.34343
\(273\) −2.69762 −0.163267
\(274\) 19.3627 1.16974
\(275\) 30.4767 1.83781
\(276\) −0.562272 −0.0338448
\(277\) −19.6057 −1.17799 −0.588996 0.808136i \(-0.700477\pi\)
−0.588996 + 0.808136i \(0.700477\pi\)
\(278\) 22.1136 1.32629
\(279\) −4.37120 −0.261697
\(280\) 33.9591 2.02944
\(281\) −33.1876 −1.97981 −0.989904 0.141738i \(-0.954731\pi\)
−0.989904 + 0.141738i \(0.954731\pi\)
\(282\) 1.43770 0.0856136
\(283\) 5.71017 0.339434 0.169717 0.985493i \(-0.445715\pi\)
0.169717 + 0.985493i \(0.445715\pi\)
\(284\) 5.14572 0.305342
\(285\) 4.09528 0.242584
\(286\) 26.4396 1.56341
\(287\) −28.2852 −1.66963
\(288\) 8.82829 0.520212
\(289\) 4.43768 0.261040
\(290\) 25.5078 1.49787
\(291\) −1.18248 −0.0693184
\(292\) 2.95068 0.172675
\(293\) 2.84237 0.166053 0.0830265 0.996547i \(-0.473541\pi\)
0.0830265 + 0.996547i \(0.473541\pi\)
\(294\) 1.54865 0.0903188
\(295\) 25.5183 1.48573
\(296\) 11.0023 0.639498
\(297\) −2.46579 −0.143080
\(298\) 14.7574 0.854876
\(299\) −42.1969 −2.44031
\(300\) 0.702670 0.0405687
\(301\) 35.8878 2.06854
\(302\) −10.4927 −0.603786
\(303\) −0.466675 −0.0268098
\(304\) −38.2298 −2.19263
\(305\) −1.17816 −0.0674614
\(306\) −21.9923 −1.25722
\(307\) 7.93047 0.452616 0.226308 0.974056i \(-0.427334\pi\)
0.226308 + 0.974056i \(0.427334\pi\)
\(308\) −6.28747 −0.358262
\(309\) −2.05204 −0.116736
\(310\) 8.99266 0.510749
\(311\) −14.6132 −0.828637 −0.414319 0.910132i \(-0.635980\pi\)
−0.414319 + 0.910132i \(0.635980\pi\)
\(312\) −1.66220 −0.0941035
\(313\) 6.64186 0.375420 0.187710 0.982224i \(-0.439893\pi\)
0.187710 + 0.982224i \(0.439893\pi\)
\(314\) −23.0961 −1.30339
\(315\) 43.4541 2.44836
\(316\) −4.59423 −0.258445
\(317\) −25.4620 −1.43009 −0.715045 0.699079i \(-0.753594\pi\)
−0.715045 + 0.699079i \(0.753594\pi\)
\(318\) −0.298740 −0.0167525
\(319\) 12.8777 0.721010
\(320\) 18.7058 1.04569
\(321\) −2.68863 −0.150065
\(322\) 47.4312 2.64324
\(323\) 36.9896 2.05816
\(324\) 4.74459 0.263588
\(325\) 52.7334 2.92513
\(326\) −34.8549 −1.93043
\(327\) 2.61595 0.144662
\(328\) −17.4286 −0.962334
\(329\) −25.6580 −1.41457
\(330\) 2.52887 0.139210
\(331\) −18.5575 −1.02001 −0.510005 0.860171i \(-0.670357\pi\)
−0.510005 + 0.860171i \(0.670357\pi\)
\(332\) 4.01102 0.220133
\(333\) 14.0786 0.771502
\(334\) −31.9141 −1.74626
\(335\) 53.1272 2.90265
\(336\) 2.40863 0.131401
\(337\) 4.03353 0.219720 0.109860 0.993947i \(-0.464960\pi\)
0.109860 + 0.993947i \(0.464960\pi\)
\(338\) 25.0431 1.36217
\(339\) 0.960457 0.0521649
\(340\) 9.57186 0.519107
\(341\) 4.53996 0.245853
\(342\) −37.9465 −2.05191
\(343\) −1.16093 −0.0626844
\(344\) 22.1131 1.19226
\(345\) −4.03602 −0.217292
\(346\) −33.5963 −1.80615
\(347\) 6.38494 0.342762 0.171381 0.985205i \(-0.445177\pi\)
0.171381 + 0.985205i \(0.445177\pi\)
\(348\) 0.296907 0.0159159
\(349\) 6.50148 0.348016 0.174008 0.984744i \(-0.444328\pi\)
0.174008 + 0.984744i \(0.444328\pi\)
\(350\) −59.2747 −3.16837
\(351\) −4.26653 −0.227730
\(352\) −9.16915 −0.488717
\(353\) 10.0775 0.536370 0.268185 0.963367i \(-0.413576\pi\)
0.268185 + 0.963367i \(0.413576\pi\)
\(354\) 1.40398 0.0746207
\(355\) 36.9363 1.96037
\(356\) −2.83451 −0.150229
\(357\) −2.33049 −0.123343
\(358\) 20.3850 1.07738
\(359\) 6.31234 0.333153 0.166576 0.986029i \(-0.446729\pi\)
0.166576 + 0.986029i \(0.446729\pi\)
\(360\) 26.7752 1.41118
\(361\) 44.8237 2.35914
\(362\) −13.4625 −0.707575
\(363\) −0.187084 −0.00981936
\(364\) −10.8791 −0.570222
\(365\) 21.1801 1.10862
\(366\) −0.0648210 −0.00338825
\(367\) 5.17229 0.269991 0.134996 0.990846i \(-0.456898\pi\)
0.134996 + 0.990846i \(0.456898\pi\)
\(368\) 37.6765 1.96402
\(369\) −22.3016 −1.16098
\(370\) −28.9633 −1.50573
\(371\) 5.33150 0.276798
\(372\) 0.104673 0.00542706
\(373\) −6.62790 −0.343180 −0.171590 0.985168i \(-0.554890\pi\)
−0.171590 + 0.985168i \(0.554890\pi\)
\(374\) 22.8414 1.18110
\(375\) 2.48072 0.128104
\(376\) −15.8098 −0.815327
\(377\) 22.2821 1.14758
\(378\) 4.79577 0.246668
\(379\) −22.4008 −1.15065 −0.575326 0.817924i \(-0.695125\pi\)
−0.575326 + 0.817924i \(0.695125\pi\)
\(380\) 16.5158 0.847240
\(381\) −0.638171 −0.0326945
\(382\) −6.95863 −0.356034
\(383\) 0.242974 0.0124154 0.00620768 0.999981i \(-0.498024\pi\)
0.00620768 + 0.999981i \(0.498024\pi\)
\(384\) 1.81702 0.0927242
\(385\) −45.1318 −2.30013
\(386\) 29.2677 1.48969
\(387\) 28.2960 1.43836
\(388\) −4.76880 −0.242099
\(389\) 17.9266 0.908913 0.454456 0.890769i \(-0.349834\pi\)
0.454456 + 0.890769i \(0.349834\pi\)
\(390\) 4.37568 0.221571
\(391\) −36.4543 −1.84357
\(392\) −17.0298 −0.860137
\(393\) −2.26053 −0.114029
\(394\) −24.8688 −1.25287
\(395\) −32.9776 −1.65928
\(396\) −4.95739 −0.249118
\(397\) −6.28866 −0.315619 −0.157809 0.987470i \(-0.550443\pi\)
−0.157809 + 0.987470i \(0.550443\pi\)
\(398\) 19.0316 0.953967
\(399\) −4.02115 −0.201309
\(400\) −47.0843 −2.35421
\(401\) −2.80781 −0.140215 −0.0701077 0.997539i \(-0.522334\pi\)
−0.0701077 + 0.997539i \(0.522334\pi\)
\(402\) 2.92299 0.145785
\(403\) 7.85545 0.391308
\(404\) −1.88204 −0.0936350
\(405\) 34.0569 1.69230
\(406\) −25.0460 −1.24301
\(407\) −14.6222 −0.724794
\(408\) −1.43599 −0.0710920
\(409\) 1.27846 0.0632156 0.0316078 0.999500i \(-0.489937\pi\)
0.0316078 + 0.999500i \(0.489937\pi\)
\(410\) 45.8802 2.26586
\(411\) 1.61778 0.0797994
\(412\) −8.27560 −0.407709
\(413\) −25.0563 −1.23294
\(414\) 37.3974 1.83798
\(415\) 28.7913 1.41331
\(416\) −15.8653 −0.777859
\(417\) 1.84763 0.0904788
\(418\) 39.4116 1.92769
\(419\) −22.9895 −1.12311 −0.561555 0.827440i \(-0.689797\pi\)
−0.561555 + 0.827440i \(0.689797\pi\)
\(420\) −1.04056 −0.0507741
\(421\) 11.3435 0.552850 0.276425 0.961036i \(-0.410850\pi\)
0.276425 + 0.961036i \(0.410850\pi\)
\(422\) −0.874825 −0.0425858
\(423\) −20.2302 −0.983626
\(424\) 3.28513 0.159540
\(425\) 45.5569 2.20983
\(426\) 2.03218 0.0984597
\(427\) 1.15684 0.0559832
\(428\) −10.8429 −0.524111
\(429\) 2.20907 0.106655
\(430\) −58.2120 −2.80723
\(431\) −24.3947 −1.17505 −0.587526 0.809205i \(-0.699898\pi\)
−0.587526 + 0.809205i \(0.699898\pi\)
\(432\) 3.80947 0.183283
\(433\) 10.9727 0.527315 0.263658 0.964616i \(-0.415071\pi\)
0.263658 + 0.964616i \(0.415071\pi\)
\(434\) −8.82987 −0.423848
\(435\) 2.13122 0.102184
\(436\) 10.5498 0.505243
\(437\) −62.9000 −3.00892
\(438\) 1.16530 0.0556803
\(439\) 25.3593 1.21034 0.605168 0.796098i \(-0.293106\pi\)
0.605168 + 0.796098i \(0.293106\pi\)
\(440\) −27.8090 −1.32574
\(441\) −21.7914 −1.03769
\(442\) 39.5222 1.87988
\(443\) 33.3777 1.58582 0.792912 0.609336i \(-0.208564\pi\)
0.792912 + 0.609336i \(0.208564\pi\)
\(444\) −0.337129 −0.0159994
\(445\) −20.3462 −0.964504
\(446\) 23.8410 1.12890
\(447\) 1.23301 0.0583193
\(448\) −18.3672 −0.867767
\(449\) −2.75384 −0.129962 −0.0649808 0.997887i \(-0.520699\pi\)
−0.0649808 + 0.997887i \(0.520699\pi\)
\(450\) −46.7355 −2.20313
\(451\) 23.1627 1.09069
\(452\) 3.87340 0.182189
\(453\) −0.876680 −0.0411900
\(454\) 6.98846 0.327985
\(455\) −78.0910 −3.66096
\(456\) −2.47772 −0.116030
\(457\) 10.5646 0.494190 0.247095 0.968991i \(-0.420524\pi\)
0.247095 + 0.968991i \(0.420524\pi\)
\(458\) −2.28684 −0.106857
\(459\) −3.68589 −0.172043
\(460\) −16.2767 −0.758907
\(461\) 2.71693 0.126540 0.0632699 0.997996i \(-0.479847\pi\)
0.0632699 + 0.997996i \(0.479847\pi\)
\(462\) −2.48309 −0.115524
\(463\) −4.82893 −0.224419 −0.112210 0.993685i \(-0.535793\pi\)
−0.112210 + 0.993685i \(0.535793\pi\)
\(464\) −19.8951 −0.923605
\(465\) 0.751351 0.0348431
\(466\) −13.5559 −0.627967
\(467\) 12.3089 0.569588 0.284794 0.958589i \(-0.408075\pi\)
0.284794 + 0.958589i \(0.408075\pi\)
\(468\) −8.57771 −0.396505
\(469\) −52.1654 −2.40878
\(470\) 41.6187 1.91973
\(471\) −1.92971 −0.0889165
\(472\) −15.4390 −0.710639
\(473\) −29.3884 −1.35128
\(474\) −1.81438 −0.0833374
\(475\) 78.6061 3.60669
\(476\) −9.39858 −0.430783
\(477\) 4.20365 0.192472
\(478\) −39.4405 −1.80397
\(479\) 5.97832 0.273156 0.136578 0.990629i \(-0.456390\pi\)
0.136578 + 0.990629i \(0.456390\pi\)
\(480\) −1.51747 −0.0692626
\(481\) −25.3006 −1.15361
\(482\) 8.81682 0.401595
\(483\) 3.96295 0.180321
\(484\) −0.754486 −0.0342948
\(485\) −34.2307 −1.55434
\(486\) 5.67747 0.257535
\(487\) −32.6657 −1.48022 −0.740112 0.672484i \(-0.765228\pi\)
−0.740112 + 0.672484i \(0.765228\pi\)
\(488\) 0.712811 0.0322674
\(489\) −2.91218 −0.131693
\(490\) 44.8304 2.02523
\(491\) 7.55392 0.340904 0.170452 0.985366i \(-0.445477\pi\)
0.170452 + 0.985366i \(0.445477\pi\)
\(492\) 0.534039 0.0240763
\(493\) 19.2497 0.866962
\(494\) 68.1935 3.06817
\(495\) −35.5844 −1.59940
\(496\) −7.01392 −0.314934
\(497\) −36.2676 −1.62683
\(498\) 1.58406 0.0709835
\(499\) −19.9072 −0.891169 −0.445585 0.895240i \(-0.647004\pi\)
−0.445585 + 0.895240i \(0.647004\pi\)
\(500\) 10.0044 0.447411
\(501\) −2.66647 −0.119129
\(502\) 21.4818 0.958781
\(503\) −22.0686 −0.983991 −0.491996 0.870598i \(-0.663732\pi\)
−0.491996 + 0.870598i \(0.663732\pi\)
\(504\) −26.2905 −1.17107
\(505\) −13.5094 −0.601160
\(506\) −38.8413 −1.72671
\(507\) 2.09239 0.0929264
\(508\) −2.57366 −0.114188
\(509\) 18.8213 0.834239 0.417119 0.908852i \(-0.363040\pi\)
0.417119 + 0.908852i \(0.363040\pi\)
\(510\) 3.78019 0.167389
\(511\) −20.7967 −0.919992
\(512\) −8.14007 −0.359744
\(513\) −6.35981 −0.280793
\(514\) −7.23823 −0.319265
\(515\) −59.4027 −2.61759
\(516\) −0.677580 −0.0298288
\(517\) 21.0113 0.924075
\(518\) 28.4389 1.24954
\(519\) −2.80702 −0.123215
\(520\) −48.1176 −2.11010
\(521\) −29.7858 −1.30494 −0.652470 0.757814i \(-0.726267\pi\)
−0.652470 + 0.757814i \(0.726267\pi\)
\(522\) −19.7477 −0.864332
\(523\) 9.68696 0.423581 0.211790 0.977315i \(-0.432071\pi\)
0.211790 + 0.977315i \(0.432071\pi\)
\(524\) −9.11644 −0.398254
\(525\) −4.95250 −0.216145
\(526\) −41.2975 −1.80066
\(527\) 6.78639 0.295620
\(528\) −1.97242 −0.0858385
\(529\) 38.9897 1.69521
\(530\) −8.64798 −0.375644
\(531\) −19.7558 −0.857328
\(532\) −16.2168 −0.703086
\(533\) 40.0781 1.73598
\(534\) −1.11942 −0.0484422
\(535\) −77.8309 −3.36492
\(536\) −32.1429 −1.38836
\(537\) 1.70320 0.0734984
\(538\) 20.2339 0.872345
\(539\) 22.6327 0.974861
\(540\) −1.64574 −0.0708213
\(541\) 27.3382 1.17536 0.587680 0.809094i \(-0.300041\pi\)
0.587680 + 0.809094i \(0.300041\pi\)
\(542\) −29.1476 −1.25200
\(543\) −1.12482 −0.0482705
\(544\) −13.7062 −0.587646
\(545\) 75.7270 3.24379
\(546\) −4.29646 −0.183872
\(547\) 5.26107 0.224947 0.112473 0.993655i \(-0.464123\pi\)
0.112473 + 0.993655i \(0.464123\pi\)
\(548\) 6.52432 0.278705
\(549\) 0.912113 0.0389280
\(550\) 48.5399 2.06975
\(551\) 33.2143 1.41498
\(552\) 2.44187 0.103933
\(553\) 32.3806 1.37696
\(554\) −31.2258 −1.32666
\(555\) −2.41993 −0.102720
\(556\) 7.45125 0.316003
\(557\) −8.87242 −0.375936 −0.187968 0.982175i \(-0.560190\pi\)
−0.187968 + 0.982175i \(0.560190\pi\)
\(558\) −6.96196 −0.294723
\(559\) −50.8505 −2.15075
\(560\) 69.7254 2.94644
\(561\) 1.90843 0.0805742
\(562\) −52.8576 −2.22966
\(563\) 37.0268 1.56049 0.780246 0.625473i \(-0.215094\pi\)
0.780246 + 0.625473i \(0.215094\pi\)
\(564\) 0.484436 0.0203984
\(565\) 27.8035 1.16970
\(566\) 9.09453 0.382272
\(567\) −33.4404 −1.40437
\(568\) −22.3471 −0.937665
\(569\) −2.28965 −0.0959870 −0.0479935 0.998848i \(-0.515283\pi\)
−0.0479935 + 0.998848i \(0.515283\pi\)
\(570\) 6.52252 0.273198
\(571\) 16.1987 0.677896 0.338948 0.940805i \(-0.389929\pi\)
0.338948 + 0.940805i \(0.389929\pi\)
\(572\) 8.90889 0.372499
\(573\) −0.581404 −0.0242885
\(574\) −45.0496 −1.88033
\(575\) −77.4685 −3.23066
\(576\) −14.4817 −0.603404
\(577\) −22.1486 −0.922057 −0.461028 0.887385i \(-0.652519\pi\)
−0.461028 + 0.887385i \(0.652519\pi\)
\(578\) 7.06785 0.293984
\(579\) 2.44536 0.101626
\(580\) 8.59492 0.356885
\(581\) −28.2701 −1.17284
\(582\) −1.88333 −0.0780665
\(583\) −4.36595 −0.180819
\(584\) −12.8144 −0.530263
\(585\) −61.5713 −2.54566
\(586\) 4.52701 0.187009
\(587\) −38.2594 −1.57914 −0.789568 0.613663i \(-0.789695\pi\)
−0.789568 + 0.613663i \(0.789695\pi\)
\(588\) 0.521820 0.0215195
\(589\) 11.7096 0.482484
\(590\) 40.6427 1.67323
\(591\) −2.07783 −0.0854703
\(592\) 22.5902 0.928451
\(593\) −16.2128 −0.665781 −0.332890 0.942965i \(-0.608024\pi\)
−0.332890 + 0.942965i \(0.608024\pi\)
\(594\) −3.92724 −0.161137
\(595\) −67.4635 −2.76574
\(596\) 4.97257 0.203684
\(597\) 1.59012 0.0650792
\(598\) −67.2066 −2.74828
\(599\) 17.4063 0.711201 0.355600 0.934638i \(-0.384276\pi\)
0.355600 + 0.934638i \(0.384276\pi\)
\(600\) −3.05160 −0.124581
\(601\) −7.31569 −0.298413 −0.149207 0.988806i \(-0.547672\pi\)
−0.149207 + 0.988806i \(0.547672\pi\)
\(602\) 57.1582 2.32959
\(603\) −41.1301 −1.67495
\(604\) −3.53554 −0.143859
\(605\) −5.41574 −0.220181
\(606\) −0.743269 −0.0301932
\(607\) −7.70672 −0.312806 −0.156403 0.987693i \(-0.549990\pi\)
−0.156403 + 0.987693i \(0.549990\pi\)
\(608\) −23.6493 −0.959104
\(609\) −2.09264 −0.0847979
\(610\) −1.87645 −0.0759752
\(611\) 36.3556 1.47079
\(612\) −7.41036 −0.299546
\(613\) −6.42614 −0.259549 −0.129775 0.991544i \(-0.541425\pi\)
−0.129775 + 0.991544i \(0.541425\pi\)
\(614\) 12.6308 0.509737
\(615\) 3.83336 0.154576
\(616\) 27.3056 1.10017
\(617\) 44.7049 1.79975 0.899877 0.436144i \(-0.143656\pi\)
0.899877 + 0.436144i \(0.143656\pi\)
\(618\) −3.26826 −0.131469
\(619\) 14.6653 0.589450 0.294725 0.955582i \(-0.404772\pi\)
0.294725 + 0.955582i \(0.404772\pi\)
\(620\) 3.03010 0.121692
\(621\) 6.26778 0.251517
\(622\) −23.2743 −0.933213
\(623\) 19.9779 0.800398
\(624\) −3.41285 −0.136623
\(625\) 22.6157 0.904627
\(626\) 10.5784 0.422799
\(627\) 3.29291 0.131506
\(628\) −7.78229 −0.310547
\(629\) −21.8574 −0.871511
\(630\) 69.2088 2.75735
\(631\) 17.7486 0.706561 0.353280 0.935517i \(-0.385066\pi\)
0.353280 + 0.935517i \(0.385066\pi\)
\(632\) 19.9521 0.793651
\(633\) −0.0730931 −0.00290519
\(634\) −40.5531 −1.61057
\(635\) −18.4739 −0.733114
\(636\) −0.100661 −0.00399148
\(637\) 39.1612 1.55162
\(638\) 20.5101 0.812003
\(639\) −28.5954 −1.13122
\(640\) 52.5993 2.07917
\(641\) −4.16534 −0.164521 −0.0822605 0.996611i \(-0.526214\pi\)
−0.0822605 + 0.996611i \(0.526214\pi\)
\(642\) −4.28215 −0.169003
\(643\) −32.4839 −1.28104 −0.640520 0.767941i \(-0.721281\pi\)
−0.640520 + 0.767941i \(0.721281\pi\)
\(644\) 15.9821 0.629782
\(645\) −4.86371 −0.191508
\(646\) 58.9130 2.31790
\(647\) −20.5270 −0.806998 −0.403499 0.914980i \(-0.632206\pi\)
−0.403499 + 0.914980i \(0.632206\pi\)
\(648\) −20.6051 −0.809444
\(649\) 20.5185 0.805423
\(650\) 83.9880 3.29428
\(651\) −0.737750 −0.0289147
\(652\) −11.7444 −0.459948
\(653\) 9.24507 0.361788 0.180894 0.983503i \(-0.442101\pi\)
0.180894 + 0.983503i \(0.442101\pi\)
\(654\) 4.16640 0.162919
\(655\) −65.4383 −2.55689
\(656\) −35.7847 −1.39716
\(657\) −16.3973 −0.639718
\(658\) −40.8653 −1.59309
\(659\) −0.114905 −0.00447606 −0.00223803 0.999997i \(-0.500712\pi\)
−0.00223803 + 0.999997i \(0.500712\pi\)
\(660\) 0.852110 0.0331684
\(661\) −10.9411 −0.425560 −0.212780 0.977100i \(-0.568252\pi\)
−0.212780 + 0.977100i \(0.568252\pi\)
\(662\) −29.5563 −1.14874
\(663\) 3.30214 0.128245
\(664\) −17.4193 −0.676000
\(665\) −116.405 −4.51399
\(666\) 22.4228 0.868867
\(667\) −32.7336 −1.26745
\(668\) −10.7535 −0.416067
\(669\) 1.99195 0.0770133
\(670\) 84.6151 3.26897
\(671\) −0.947329 −0.0365712
\(672\) 1.49000 0.0574779
\(673\) 44.8362 1.72831 0.864154 0.503228i \(-0.167854\pi\)
0.864154 + 0.503228i \(0.167854\pi\)
\(674\) 6.42417 0.247450
\(675\) −7.83283 −0.301486
\(676\) 8.43835 0.324552
\(677\) 12.2871 0.472232 0.236116 0.971725i \(-0.424125\pi\)
0.236116 + 0.971725i \(0.424125\pi\)
\(678\) 1.52971 0.0587482
\(679\) 33.6110 1.28987
\(680\) −41.5692 −1.59411
\(681\) 0.583897 0.0223750
\(682\) 7.23076 0.276880
\(683\) 15.7722 0.603506 0.301753 0.953386i \(-0.402428\pi\)
0.301753 + 0.953386i \(0.402428\pi\)
\(684\) −12.7862 −0.488892
\(685\) 46.8319 1.78935
\(686\) −1.84900 −0.0705953
\(687\) −0.191069 −0.00728975
\(688\) 45.4030 1.73097
\(689\) −7.55436 −0.287798
\(690\) −6.42812 −0.244714
\(691\) −24.7877 −0.942968 −0.471484 0.881875i \(-0.656281\pi\)
−0.471484 + 0.881875i \(0.656281\pi\)
\(692\) −11.3204 −0.430335
\(693\) 34.9402 1.32727
\(694\) 10.1692 0.386019
\(695\) 53.4855 2.02882
\(696\) −1.28943 −0.0488756
\(697\) 34.6239 1.31147
\(698\) 10.3548 0.391936
\(699\) −1.13262 −0.0428396
\(700\) −19.9728 −0.754901
\(701\) −22.4515 −0.847980 −0.423990 0.905667i \(-0.639371\pi\)
−0.423990 + 0.905667i \(0.639371\pi\)
\(702\) −6.79525 −0.256470
\(703\) −37.7138 −1.42240
\(704\) 15.0408 0.566872
\(705\) 3.47731 0.130963
\(706\) 16.0503 0.604061
\(707\) 13.2648 0.498875
\(708\) 0.473075 0.0177793
\(709\) 6.09525 0.228912 0.114456 0.993428i \(-0.463487\pi\)
0.114456 + 0.993428i \(0.463487\pi\)
\(710\) 58.8280 2.20778
\(711\) 25.5307 0.957475
\(712\) 12.3099 0.461332
\(713\) −11.5401 −0.432180
\(714\) −3.71175 −0.138909
\(715\) 63.9485 2.39154
\(716\) 6.86878 0.256698
\(717\) −3.29532 −0.123066
\(718\) 10.0536 0.375197
\(719\) 50.3385 1.87731 0.938655 0.344857i \(-0.112073\pi\)
0.938655 + 0.344857i \(0.112073\pi\)
\(720\) 54.9753 2.04881
\(721\) 58.3273 2.17222
\(722\) 71.3902 2.65687
\(723\) 0.736659 0.0273966
\(724\) −4.53624 −0.168588
\(725\) 40.9072 1.51926
\(726\) −0.297967 −0.0110586
\(727\) 3.07810 0.114160 0.0570802 0.998370i \(-0.481821\pi\)
0.0570802 + 0.998370i \(0.481821\pi\)
\(728\) 47.2466 1.75107
\(729\) −26.0485 −0.964758
\(730\) 33.7334 1.24853
\(731\) −43.9302 −1.62482
\(732\) −0.0218416 −0.000807289 0
\(733\) 29.9429 1.10596 0.552982 0.833193i \(-0.313490\pi\)
0.552982 + 0.833193i \(0.313490\pi\)
\(734\) 8.23785 0.304065
\(735\) 3.74565 0.138161
\(736\) 23.3070 0.859108
\(737\) 42.7181 1.57354
\(738\) −35.5196 −1.30749
\(739\) 50.0635 1.84162 0.920808 0.390016i \(-0.127530\pi\)
0.920808 + 0.390016i \(0.127530\pi\)
\(740\) −9.75925 −0.358757
\(741\) 5.69768 0.209309
\(742\) 8.49143 0.311730
\(743\) 21.8551 0.801785 0.400892 0.916125i \(-0.368700\pi\)
0.400892 + 0.916125i \(0.368700\pi\)
\(744\) −0.454582 −0.0166658
\(745\) 35.6933 1.30770
\(746\) −10.5562 −0.386490
\(747\) −22.2897 −0.815538
\(748\) 7.69647 0.281411
\(749\) 76.4220 2.79240
\(750\) 3.95102 0.144271
\(751\) 15.5073 0.565869 0.282935 0.959139i \(-0.408692\pi\)
0.282935 + 0.959139i \(0.408692\pi\)
\(752\) −32.4609 −1.18373
\(753\) 1.79484 0.0654076
\(754\) 35.4884 1.29241
\(755\) −25.3783 −0.923610
\(756\) 1.61595 0.0587714
\(757\) 28.7870 1.04628 0.523140 0.852247i \(-0.324760\pi\)
0.523140 + 0.852247i \(0.324760\pi\)
\(758\) −35.6776 −1.29587
\(759\) −3.24525 −0.117795
\(760\) −71.7256 −2.60176
\(761\) 33.2039 1.20364 0.601819 0.798632i \(-0.294443\pi\)
0.601819 + 0.798632i \(0.294443\pi\)
\(762\) −1.01641 −0.0368206
\(763\) −74.3561 −2.69187
\(764\) −2.34473 −0.0848293
\(765\) −53.1920 −1.92316
\(766\) 0.386981 0.0139822
\(767\) 35.5030 1.28194
\(768\) 1.60158 0.0577920
\(769\) −42.5825 −1.53557 −0.767783 0.640710i \(-0.778640\pi\)
−0.767783 + 0.640710i \(0.778640\pi\)
\(770\) −71.8810 −2.59041
\(771\) −0.604766 −0.0217801
\(772\) 9.86184 0.354935
\(773\) 12.3322 0.443559 0.221779 0.975097i \(-0.428814\pi\)
0.221779 + 0.975097i \(0.428814\pi\)
\(774\) 45.0667 1.61989
\(775\) 14.4217 0.518041
\(776\) 20.7102 0.743454
\(777\) 2.37612 0.0852428
\(778\) 28.5515 1.02362
\(779\) 59.7417 2.14047
\(780\) 1.47440 0.0527919
\(781\) 29.6994 1.06273
\(782\) −58.0604 −2.07624
\(783\) −3.30970 −0.118279
\(784\) −34.9660 −1.24878
\(785\) −55.8617 −1.99379
\(786\) −3.60033 −0.128419
\(787\) −25.7076 −0.916377 −0.458189 0.888855i \(-0.651502\pi\)
−0.458189 + 0.888855i \(0.651502\pi\)
\(788\) −8.37961 −0.298511
\(789\) −3.45047 −0.122840
\(790\) −52.5231 −1.86869
\(791\) −27.3002 −0.970682
\(792\) 21.5292 0.765008
\(793\) −1.63915 −0.0582080
\(794\) −10.0159 −0.355450
\(795\) −0.722553 −0.0256263
\(796\) 6.41274 0.227294
\(797\) 21.7793 0.771461 0.385730 0.922612i \(-0.373949\pi\)
0.385730 + 0.922612i \(0.373949\pi\)
\(798\) −6.40444 −0.226715
\(799\) 31.4079 1.11113
\(800\) −29.1267 −1.02979
\(801\) 15.7517 0.556559
\(802\) −4.47197 −0.157911
\(803\) 17.0304 0.600988
\(804\) 0.984909 0.0347351
\(805\) 114.720 4.04336
\(806\) 12.5113 0.440692
\(807\) 1.69057 0.0595110
\(808\) 8.17343 0.287540
\(809\) −36.5836 −1.28621 −0.643105 0.765778i \(-0.722354\pi\)
−0.643105 + 0.765778i \(0.722354\pi\)
\(810\) 54.2421 1.90587
\(811\) −3.31224 −0.116308 −0.0581542 0.998308i \(-0.518521\pi\)
−0.0581542 + 0.998308i \(0.518521\pi\)
\(812\) −8.43933 −0.296162
\(813\) −2.43533 −0.0854107
\(814\) −23.2886 −0.816264
\(815\) −84.3023 −2.95298
\(816\) −2.94839 −0.103214
\(817\) −75.7993 −2.65188
\(818\) 2.03618 0.0711935
\(819\) 60.4567 2.11253
\(820\) 15.4595 0.539868
\(821\) −20.4794 −0.714736 −0.357368 0.933964i \(-0.616326\pi\)
−0.357368 + 0.933964i \(0.616326\pi\)
\(822\) 2.57663 0.0898702
\(823\) −25.5204 −0.889586 −0.444793 0.895633i \(-0.646723\pi\)
−0.444793 + 0.895633i \(0.646723\pi\)
\(824\) 35.9397 1.25202
\(825\) 4.05559 0.141197
\(826\) −39.9069 −1.38854
\(827\) −9.88107 −0.343598 −0.171799 0.985132i \(-0.554958\pi\)
−0.171799 + 0.985132i \(0.554958\pi\)
\(828\) 12.6012 0.437920
\(829\) −13.3413 −0.463363 −0.231682 0.972792i \(-0.574423\pi\)
−0.231682 + 0.972792i \(0.574423\pi\)
\(830\) 45.8557 1.59167
\(831\) −2.60897 −0.0905040
\(832\) 26.0250 0.902253
\(833\) 33.8317 1.17220
\(834\) 2.94270 0.101897
\(835\) −77.1894 −2.67125
\(836\) 13.2799 0.459294
\(837\) −1.16682 −0.0403312
\(838\) −36.6151 −1.26485
\(839\) −2.87114 −0.0991228 −0.0495614 0.998771i \(-0.515782\pi\)
−0.0495614 + 0.998771i \(0.515782\pi\)
\(840\) 4.51900 0.155920
\(841\) −11.7150 −0.403966
\(842\) 18.0667 0.622621
\(843\) −4.41634 −0.152107
\(844\) −0.294775 −0.0101466
\(845\) 60.5709 2.08370
\(846\) −32.2204 −1.10776
\(847\) 5.31770 0.182718
\(848\) 6.74508 0.231627
\(849\) 0.759862 0.0260784
\(850\) 72.5580 2.48872
\(851\) 37.1680 1.27410
\(852\) 0.684750 0.0234592
\(853\) −19.3900 −0.663902 −0.331951 0.943297i \(-0.607707\pi\)
−0.331951 + 0.943297i \(0.607707\pi\)
\(854\) 1.84248 0.0630484
\(855\) −91.7800 −3.13881
\(856\) 47.0892 1.60947
\(857\) −42.5804 −1.45452 −0.727259 0.686363i \(-0.759206\pi\)
−0.727259 + 0.686363i \(0.759206\pi\)
\(858\) 3.51836 0.120115
\(859\) 36.4148 1.24246 0.621228 0.783630i \(-0.286634\pi\)
0.621228 + 0.783630i \(0.286634\pi\)
\(860\) −19.6147 −0.668856
\(861\) −3.76397 −0.128276
\(862\) −38.8532 −1.32335
\(863\) −32.8207 −1.11723 −0.558615 0.829427i \(-0.688667\pi\)
−0.558615 + 0.829427i \(0.688667\pi\)
\(864\) 2.35657 0.0801721
\(865\) −81.2581 −2.76286
\(866\) 17.4761 0.593863
\(867\) 0.590530 0.0200554
\(868\) −2.97525 −0.100987
\(869\) −26.5164 −0.899507
\(870\) 3.39437 0.115080
\(871\) 73.9146 2.50450
\(872\) −45.8162 −1.55153
\(873\) 26.5008 0.896916
\(874\) −100.180 −3.38865
\(875\) −70.5123 −2.38375
\(876\) 0.392652 0.0132665
\(877\) −6.13368 −0.207120 −0.103560 0.994623i \(-0.533023\pi\)
−0.103560 + 0.994623i \(0.533023\pi\)
\(878\) 40.3896 1.36308
\(879\) 0.378239 0.0127577
\(880\) −57.0979 −1.92477
\(881\) −4.37391 −0.147361 −0.0736803 0.997282i \(-0.523474\pi\)
−0.0736803 + 0.997282i \(0.523474\pi\)
\(882\) −34.7069 −1.16864
\(883\) 28.8998 0.972555 0.486278 0.873804i \(-0.338354\pi\)
0.486278 + 0.873804i \(0.338354\pi\)
\(884\) 13.3171 0.447903
\(885\) 3.39576 0.114147
\(886\) 53.1604 1.78596
\(887\) −29.8072 −1.00083 −0.500414 0.865787i \(-0.666819\pi\)
−0.500414 + 0.865787i \(0.666819\pi\)
\(888\) 1.46410 0.0491320
\(889\) 18.1395 0.608378
\(890\) −32.4053 −1.08623
\(891\) 27.3843 0.917407
\(892\) 8.03329 0.268974
\(893\) 54.1927 1.81349
\(894\) 1.96380 0.0656793
\(895\) 49.3044 1.64807
\(896\) −51.6471 −1.72541
\(897\) −5.61522 −0.187487
\(898\) −4.38601 −0.146363
\(899\) 6.09375 0.203238
\(900\) −15.7477 −0.524922
\(901\) −6.52628 −0.217422
\(902\) 36.8910 1.22834
\(903\) 4.77566 0.158924
\(904\) −16.8216 −0.559479
\(905\) −32.5614 −1.08238
\(906\) −1.39628 −0.0463883
\(907\) 27.7853 0.922596 0.461298 0.887245i \(-0.347384\pi\)
0.461298 + 0.887245i \(0.347384\pi\)
\(908\) 2.35478 0.0781462
\(909\) 10.4587 0.346894
\(910\) −124.375 −4.12298
\(911\) −23.1095 −0.765651 −0.382825 0.923821i \(-0.625049\pi\)
−0.382825 + 0.923821i \(0.625049\pi\)
\(912\) −5.08730 −0.168457
\(913\) 23.1503 0.766164
\(914\) 16.8261 0.556558
\(915\) −0.156780 −0.00518300
\(916\) −0.770558 −0.0254599
\(917\) 64.2537 2.12184
\(918\) −5.87048 −0.193755
\(919\) −10.9408 −0.360905 −0.180452 0.983584i \(-0.557756\pi\)
−0.180452 + 0.983584i \(0.557756\pi\)
\(920\) 70.6876 2.33050
\(921\) 1.05532 0.0347740
\(922\) 4.32722 0.142509
\(923\) 51.3886 1.69148
\(924\) −0.836685 −0.0275249
\(925\) −46.4488 −1.52723
\(926\) −7.69098 −0.252741
\(927\) 45.9885 1.51046
\(928\) −12.3073 −0.404005
\(929\) −40.4546 −1.32727 −0.663637 0.748055i \(-0.730988\pi\)
−0.663637 + 0.748055i \(0.730988\pi\)
\(930\) 1.19667 0.0392404
\(931\) 58.3748 1.91316
\(932\) −4.56771 −0.149620
\(933\) −1.94460 −0.0636634
\(934\) 19.6043 0.641471
\(935\) 55.2457 1.80673
\(936\) 37.2518 1.21761
\(937\) 9.08113 0.296668 0.148334 0.988937i \(-0.452609\pi\)
0.148334 + 0.988937i \(0.452609\pi\)
\(938\) −83.0834 −2.71277
\(939\) 0.883845 0.0288432
\(940\) 14.0235 0.457397
\(941\) 12.8520 0.418964 0.209482 0.977812i \(-0.432822\pi\)
0.209482 + 0.977812i \(0.432822\pi\)
\(942\) −3.07344 −0.100138
\(943\) −58.8771 −1.91730
\(944\) −31.6997 −1.03174
\(945\) 11.5994 0.377327
\(946\) −46.8067 −1.52182
\(947\) 44.6938 1.45235 0.726177 0.687508i \(-0.241295\pi\)
0.726177 + 0.687508i \(0.241295\pi\)
\(948\) −0.611362 −0.0198561
\(949\) 29.4674 0.956553
\(950\) 125.195 4.06187
\(951\) −3.38828 −0.109872
\(952\) 40.8167 1.32288
\(953\) −9.11417 −0.295237 −0.147618 0.989044i \(-0.547161\pi\)
−0.147618 + 0.989044i \(0.547161\pi\)
\(954\) 6.69511 0.216762
\(955\) −16.8306 −0.544625
\(956\) −13.2896 −0.429816
\(957\) 1.71365 0.0553945
\(958\) 9.52161 0.307629
\(959\) −45.9841 −1.48490
\(960\) 2.48921 0.0803390
\(961\) −28.8517 −0.930699
\(962\) −40.2959 −1.29919
\(963\) 60.2553 1.94170
\(964\) 2.97085 0.0956847
\(965\) 70.7888 2.27877
\(966\) 6.31176 0.203077
\(967\) 39.3216 1.26450 0.632249 0.774765i \(-0.282132\pi\)
0.632249 + 0.774765i \(0.282132\pi\)
\(968\) 3.27662 0.105315
\(969\) 4.92227 0.158126
\(970\) −54.5189 −1.75050
\(971\) 32.9797 1.05837 0.529184 0.848507i \(-0.322498\pi\)
0.529184 + 0.848507i \(0.322498\pi\)
\(972\) 1.91304 0.0613607
\(973\) −52.5172 −1.68363
\(974\) −52.0263 −1.66703
\(975\) 7.01733 0.224735
\(976\) 1.46356 0.0468473
\(977\) 34.8469 1.11485 0.557425 0.830227i \(-0.311789\pi\)
0.557425 + 0.830227i \(0.311789\pi\)
\(978\) −4.63820 −0.148313
\(979\) −16.3599 −0.522863
\(980\) 15.1057 0.482535
\(981\) −58.6264 −1.87180
\(982\) 12.0310 0.383926
\(983\) −1.47944 −0.0471868 −0.0235934 0.999722i \(-0.507511\pi\)
−0.0235934 + 0.999722i \(0.507511\pi\)
\(984\) −2.31926 −0.0739352
\(985\) −60.1493 −1.91651
\(986\) 30.6588 0.976374
\(987\) −3.41436 −0.108680
\(988\) 22.9780 0.731027
\(989\) 74.7023 2.37540
\(990\) −56.6749 −1.80125
\(991\) −48.6649 −1.54589 −0.772946 0.634472i \(-0.781217\pi\)
−0.772946 + 0.634472i \(0.781217\pi\)
\(992\) −4.33887 −0.137759
\(993\) −2.46948 −0.0783665
\(994\) −57.7631 −1.83213
\(995\) 46.0310 1.45928
\(996\) 0.533754 0.0169126
\(997\) 17.8052 0.563897 0.281949 0.959430i \(-0.409019\pi\)
0.281949 + 0.959430i \(0.409019\pi\)
\(998\) −31.7060 −1.00364
\(999\) 3.75805 0.118899
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 2671.2.a.b.1.91 122
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2671.2.a.b.1.91 122 1.1 even 1 trivial