Properties

Label 6026.2.a.g.1.8
Level $6026$
Weight $2$
Character 6026.1
Self dual yes
Analytic conductor $48.118$
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] = [6026,2,Mod(1,6026)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6026, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("6026.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6026 = 2 \cdot 23 \cdot 131 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6026.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: \(48.1178522580\)
Analytic rank: \(1\)
Dimension: \(21\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Character \(\chi\) \(=\) 6026.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.08114 q^{3} +1.00000 q^{4} +0.379081 q^{5} -1.08114 q^{6} +0.614832 q^{7} +1.00000 q^{8} -1.83113 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.08114 q^{3} +1.00000 q^{4} +0.379081 q^{5} -1.08114 q^{6} +0.614832 q^{7} +1.00000 q^{8} -1.83113 q^{9} +0.379081 q^{10} +5.27578 q^{11} -1.08114 q^{12} -0.100390 q^{13} +0.614832 q^{14} -0.409841 q^{15} +1.00000 q^{16} -4.27441 q^{17} -1.83113 q^{18} -3.59007 q^{19} +0.379081 q^{20} -0.664722 q^{21} +5.27578 q^{22} -1.00000 q^{23} -1.08114 q^{24} -4.85630 q^{25} -0.100390 q^{26} +5.22315 q^{27} +0.614832 q^{28} -6.70272 q^{29} -0.409841 q^{30} -4.30760 q^{31} +1.00000 q^{32} -5.70388 q^{33} -4.27441 q^{34} +0.233071 q^{35} -1.83113 q^{36} +7.48078 q^{37} -3.59007 q^{38} +0.108536 q^{39} +0.379081 q^{40} +8.15498 q^{41} -0.664722 q^{42} -7.73581 q^{43} +5.27578 q^{44} -0.694145 q^{45} -1.00000 q^{46} -8.71811 q^{47} -1.08114 q^{48} -6.62198 q^{49} -4.85630 q^{50} +4.62126 q^{51} -0.100390 q^{52} +1.61946 q^{53} +5.22315 q^{54} +1.99995 q^{55} +0.614832 q^{56} +3.88139 q^{57} -6.70272 q^{58} -7.31903 q^{59} -0.409841 q^{60} +4.58610 q^{61} -4.30760 q^{62} -1.12584 q^{63} +1.00000 q^{64} -0.0380559 q^{65} -5.70388 q^{66} -12.7060 q^{67} -4.27441 q^{68} +1.08114 q^{69} +0.233071 q^{70} -3.66475 q^{71} -1.83113 q^{72} +2.16897 q^{73} +7.48078 q^{74} +5.25036 q^{75} -3.59007 q^{76} +3.24372 q^{77} +0.108536 q^{78} +1.41071 q^{79} +0.379081 q^{80} -0.153594 q^{81} +8.15498 q^{82} +12.3096 q^{83} -0.664722 q^{84} -1.62035 q^{85} -7.73581 q^{86} +7.24660 q^{87} +5.27578 q^{88} -0.0472708 q^{89} -0.694145 q^{90} -0.0617228 q^{91} -1.00000 q^{92} +4.65714 q^{93} -8.71811 q^{94} -1.36093 q^{95} -1.08114 q^{96} +0.462560 q^{97} -6.62198 q^{98} -9.66063 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 21 q + 21 q^{2} + 21 q^{4} - 13 q^{5} - 18 q^{7} + 21 q^{8} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 21 q + 21 q^{2} + 21 q^{4} - 13 q^{5} - 18 q^{7} + 21 q^{8} + 7 q^{9} - 13 q^{10} - 4 q^{11} - 4 q^{13} - 18 q^{14} - 16 q^{15} + 21 q^{16} - 12 q^{17} + 7 q^{18} - 18 q^{19} - 13 q^{20} - 24 q^{21} - 4 q^{22} - 21 q^{23} + 2 q^{25} - 4 q^{26} - 9 q^{27} - 18 q^{28} - 16 q^{29} - 16 q^{30} - 7 q^{31} + 21 q^{32} - 15 q^{33} - 12 q^{34} + 7 q^{36} - 44 q^{37} - 18 q^{38} - 14 q^{39} - 13 q^{40} - 23 q^{41} - 24 q^{42} - 18 q^{43} - 4 q^{44} - 36 q^{45} - 21 q^{46} + 2 q^{47} - 13 q^{49} + 2 q^{50} - 26 q^{51} - 4 q^{52} - 39 q^{53} - 9 q^{54} - 32 q^{55} - 18 q^{56} - 22 q^{57} - 16 q^{58} - 27 q^{59} - 16 q^{60} - 34 q^{61} - 7 q^{62} - 28 q^{63} + 21 q^{64} - 25 q^{65} - 15 q^{66} - 19 q^{67} - 12 q^{68} - 24 q^{71} + 7 q^{72} - 8 q^{73} - 44 q^{74} + 50 q^{75} - 18 q^{76} - 16 q^{77} - 14 q^{78} - 27 q^{79} - 13 q^{80} + 33 q^{81} - 23 q^{82} + 7 q^{83} - 24 q^{84} - 22 q^{85} - 18 q^{86} - 15 q^{87} - 4 q^{88} - 12 q^{89} - 36 q^{90} - 20 q^{91} - 21 q^{92} - 43 q^{93} + 2 q^{94} - 14 q^{95} - 52 q^{97} - 13 q^{98} - 30 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.08114 −0.624199 −0.312099 0.950049i \(-0.601032\pi\)
−0.312099 + 0.950049i \(0.601032\pi\)
\(4\) 1.00000 0.500000
\(5\) 0.379081 0.169530 0.0847651 0.996401i \(-0.472986\pi\)
0.0847651 + 0.996401i \(0.472986\pi\)
\(6\) −1.08114 −0.441375
\(7\) 0.614832 0.232385 0.116192 0.993227i \(-0.462931\pi\)
0.116192 + 0.993227i \(0.462931\pi\)
\(8\) 1.00000 0.353553
\(9\) −1.83113 −0.610376
\(10\) 0.379081 0.119876
\(11\) 5.27578 1.59071 0.795354 0.606145i \(-0.207285\pi\)
0.795354 + 0.606145i \(0.207285\pi\)
\(12\) −1.08114 −0.312099
\(13\) −0.100390 −0.0278431 −0.0139216 0.999903i \(-0.504432\pi\)
−0.0139216 + 0.999903i \(0.504432\pi\)
\(14\) 0.614832 0.164321
\(15\) −0.409841 −0.105821
\(16\) 1.00000 0.250000
\(17\) −4.27441 −1.03670 −0.518349 0.855169i \(-0.673453\pi\)
−0.518349 + 0.855169i \(0.673453\pi\)
\(18\) −1.83113 −0.431601
\(19\) −3.59007 −0.823619 −0.411810 0.911270i \(-0.635103\pi\)
−0.411810 + 0.911270i \(0.635103\pi\)
\(20\) 0.379081 0.0847651
\(21\) −0.664722 −0.145054
\(22\) 5.27578 1.12480
\(23\) −1.00000 −0.208514
\(24\) −1.08114 −0.220688
\(25\) −4.85630 −0.971260
\(26\) −0.100390 −0.0196881
\(27\) 5.22315 1.00519
\(28\) 0.614832 0.116192
\(29\) −6.70272 −1.24466 −0.622332 0.782754i \(-0.713814\pi\)
−0.622332 + 0.782754i \(0.713814\pi\)
\(30\) −0.409841 −0.0748265
\(31\) −4.30760 −0.773668 −0.386834 0.922149i \(-0.626431\pi\)
−0.386834 + 0.922149i \(0.626431\pi\)
\(32\) 1.00000 0.176777
\(33\) −5.70388 −0.992919
\(34\) −4.27441 −0.733056
\(35\) 0.233071 0.0393962
\(36\) −1.83113 −0.305188
\(37\) 7.48078 1.22983 0.614916 0.788593i \(-0.289190\pi\)
0.614916 + 0.788593i \(0.289190\pi\)
\(38\) −3.59007 −0.582387
\(39\) 0.108536 0.0173796
\(40\) 0.379081 0.0599380
\(41\) 8.15498 1.27359 0.636797 0.771031i \(-0.280259\pi\)
0.636797 + 0.771031i \(0.280259\pi\)
\(42\) −0.664722 −0.102569
\(43\) −7.73581 −1.17970 −0.589850 0.807513i \(-0.700813\pi\)
−0.589850 + 0.807513i \(0.700813\pi\)
\(44\) 5.27578 0.795354
\(45\) −0.694145 −0.103477
\(46\) −1.00000 −0.147442
\(47\) −8.71811 −1.27167 −0.635834 0.771826i \(-0.719344\pi\)
−0.635834 + 0.771826i \(0.719344\pi\)
\(48\) −1.08114 −0.156050
\(49\) −6.62198 −0.945997
\(50\) −4.85630 −0.686784
\(51\) 4.62126 0.647106
\(52\) −0.100390 −0.0139216
\(53\) 1.61946 0.222450 0.111225 0.993795i \(-0.464523\pi\)
0.111225 + 0.993795i \(0.464523\pi\)
\(54\) 5.22315 0.710780
\(55\) 1.99995 0.269673
\(56\) 0.614832 0.0821604
\(57\) 3.88139 0.514102
\(58\) −6.70272 −0.880110
\(59\) −7.31903 −0.952857 −0.476429 0.879213i \(-0.658069\pi\)
−0.476429 + 0.879213i \(0.658069\pi\)
\(60\) −0.409841 −0.0529103
\(61\) 4.58610 0.587190 0.293595 0.955930i \(-0.405148\pi\)
0.293595 + 0.955930i \(0.405148\pi\)
\(62\) −4.30760 −0.547066
\(63\) −1.12584 −0.141842
\(64\) 1.00000 0.125000
\(65\) −0.0380559 −0.00472025
\(66\) −5.70388 −0.702100
\(67\) −12.7060 −1.55229 −0.776145 0.630554i \(-0.782828\pi\)
−0.776145 + 0.630554i \(0.782828\pi\)
\(68\) −4.27441 −0.518349
\(69\) 1.08114 0.130154
\(70\) 0.233071 0.0278573
\(71\) −3.66475 −0.434926 −0.217463 0.976069i \(-0.569778\pi\)
−0.217463 + 0.976069i \(0.569778\pi\)
\(72\) −1.83113 −0.215800
\(73\) 2.16897 0.253859 0.126929 0.991912i \(-0.459488\pi\)
0.126929 + 0.991912i \(0.459488\pi\)
\(74\) 7.48078 0.869623
\(75\) 5.25036 0.606259
\(76\) −3.59007 −0.411810
\(77\) 3.24372 0.369656
\(78\) 0.108536 0.0122893
\(79\) 1.41071 0.158718 0.0793588 0.996846i \(-0.474713\pi\)
0.0793588 + 0.996846i \(0.474713\pi\)
\(80\) 0.379081 0.0423826
\(81\) −0.153594 −0.0170660
\(82\) 8.15498 0.900567
\(83\) 12.3096 1.35116 0.675580 0.737287i \(-0.263893\pi\)
0.675580 + 0.737287i \(0.263893\pi\)
\(84\) −0.664722 −0.0725271
\(85\) −1.62035 −0.175752
\(86\) −7.73581 −0.834174
\(87\) 7.24660 0.776918
\(88\) 5.27578 0.562400
\(89\) −0.0472708 −0.00501070 −0.00250535 0.999997i \(-0.500797\pi\)
−0.00250535 + 0.999997i \(0.500797\pi\)
\(90\) −0.694145 −0.0731694
\(91\) −0.0617228 −0.00647031
\(92\) −1.00000 −0.104257
\(93\) 4.65714 0.482923
\(94\) −8.71811 −0.899205
\(95\) −1.36093 −0.139628
\(96\) −1.08114 −0.110344
\(97\) 0.462560 0.0469658 0.0234829 0.999724i \(-0.492524\pi\)
0.0234829 + 0.999724i \(0.492524\pi\)
\(98\) −6.62198 −0.668921
\(99\) −9.66063 −0.970930
\(100\) −4.85630 −0.485630
\(101\) −9.05936 −0.901440 −0.450720 0.892665i \(-0.648833\pi\)
−0.450720 + 0.892665i \(0.648833\pi\)
\(102\) 4.62126 0.457573
\(103\) 12.9499 1.27599 0.637994 0.770041i \(-0.279764\pi\)
0.637994 + 0.770041i \(0.279764\pi\)
\(104\) −0.100390 −0.00984403
\(105\) −0.251984 −0.0245911
\(106\) 1.61946 0.157296
\(107\) −17.3147 −1.67387 −0.836936 0.547301i \(-0.815655\pi\)
−0.836936 + 0.547301i \(0.815655\pi\)
\(108\) 5.22315 0.502597
\(109\) 2.98980 0.286371 0.143186 0.989696i \(-0.454265\pi\)
0.143186 + 0.989696i \(0.454265\pi\)
\(110\) 1.99995 0.190688
\(111\) −8.08780 −0.767660
\(112\) 0.614832 0.0580961
\(113\) −12.3950 −1.16602 −0.583011 0.812464i \(-0.698125\pi\)
−0.583011 + 0.812464i \(0.698125\pi\)
\(114\) 3.88139 0.363525
\(115\) −0.379081 −0.0353495
\(116\) −6.70272 −0.622332
\(117\) 0.183826 0.0169948
\(118\) −7.31903 −0.673772
\(119\) −2.62805 −0.240913
\(120\) −0.409841 −0.0374132
\(121\) 16.8339 1.53035
\(122\) 4.58610 0.415206
\(123\) −8.81671 −0.794976
\(124\) −4.30760 −0.386834
\(125\) −3.73634 −0.334188
\(126\) −1.12584 −0.100297
\(127\) 15.5795 1.38246 0.691229 0.722636i \(-0.257070\pi\)
0.691229 + 0.722636i \(0.257070\pi\)
\(128\) 1.00000 0.0883883
\(129\) 8.36353 0.736368
\(130\) −0.0380559 −0.00333772
\(131\) 1.00000 0.0873704
\(132\) −5.70388 −0.496459
\(133\) −2.20729 −0.191396
\(134\) −12.7060 −1.09764
\(135\) 1.98000 0.170411
\(136\) −4.27441 −0.366528
\(137\) −4.52366 −0.386483 −0.193241 0.981151i \(-0.561900\pi\)
−0.193241 + 0.981151i \(0.561900\pi\)
\(138\) 1.08114 0.0920331
\(139\) −13.2164 −1.12100 −0.560501 0.828154i \(-0.689391\pi\)
−0.560501 + 0.828154i \(0.689391\pi\)
\(140\) 0.233071 0.0196981
\(141\) 9.42554 0.793774
\(142\) −3.66475 −0.307539
\(143\) −0.529635 −0.0442903
\(144\) −1.83113 −0.152594
\(145\) −2.54087 −0.211008
\(146\) 2.16897 0.179505
\(147\) 7.15932 0.590491
\(148\) 7.48078 0.614916
\(149\) 19.2562 1.57753 0.788764 0.614697i \(-0.210722\pi\)
0.788764 + 0.614697i \(0.210722\pi\)
\(150\) 5.25036 0.428690
\(151\) 8.36267 0.680545 0.340272 0.940327i \(-0.389481\pi\)
0.340272 + 0.940327i \(0.389481\pi\)
\(152\) −3.59007 −0.291193
\(153\) 7.82699 0.632775
\(154\) 3.24372 0.261386
\(155\) −1.63293 −0.131160
\(156\) 0.108536 0.00868982
\(157\) −6.72793 −0.536948 −0.268474 0.963287i \(-0.586519\pi\)
−0.268474 + 0.963287i \(0.586519\pi\)
\(158\) 1.41071 0.112230
\(159\) −1.75087 −0.138853
\(160\) 0.379081 0.0299690
\(161\) −0.614832 −0.0484555
\(162\) −0.153594 −0.0120675
\(163\) 0.451346 0.0353521 0.0176761 0.999844i \(-0.494373\pi\)
0.0176761 + 0.999844i \(0.494373\pi\)
\(164\) 8.15498 0.636797
\(165\) −2.16223 −0.168330
\(166\) 12.3096 0.955414
\(167\) −1.79938 −0.139240 −0.0696200 0.997574i \(-0.522179\pi\)
−0.0696200 + 0.997574i \(0.522179\pi\)
\(168\) −0.664722 −0.0512844
\(169\) −12.9899 −0.999225
\(170\) −1.62035 −0.124275
\(171\) 6.57388 0.502717
\(172\) −7.73581 −0.589850
\(173\) 4.48264 0.340809 0.170404 0.985374i \(-0.445493\pi\)
0.170404 + 0.985374i \(0.445493\pi\)
\(174\) 7.24660 0.549364
\(175\) −2.98581 −0.225706
\(176\) 5.27578 0.397677
\(177\) 7.91293 0.594772
\(178\) −0.0472708 −0.00354310
\(179\) −26.4904 −1.97999 −0.989993 0.141113i \(-0.954932\pi\)
−0.989993 + 0.141113i \(0.954932\pi\)
\(180\) −0.694145 −0.0517386
\(181\) 0.316097 0.0234953 0.0117477 0.999931i \(-0.496261\pi\)
0.0117477 + 0.999931i \(0.496261\pi\)
\(182\) −0.0617228 −0.00457520
\(183\) −4.95824 −0.366523
\(184\) −1.00000 −0.0737210
\(185\) 2.83582 0.208494
\(186\) 4.65714 0.341478
\(187\) −22.5509 −1.64908
\(188\) −8.71811 −0.635834
\(189\) 3.21136 0.233592
\(190\) −1.36093 −0.0987322
\(191\) 9.18166 0.664362 0.332181 0.943216i \(-0.392216\pi\)
0.332181 + 0.943216i \(0.392216\pi\)
\(192\) −1.08114 −0.0780249
\(193\) −9.13123 −0.657280 −0.328640 0.944455i \(-0.606590\pi\)
−0.328640 + 0.944455i \(0.606590\pi\)
\(194\) 0.462560 0.0332098
\(195\) 0.0411439 0.00294637
\(196\) −6.62198 −0.472999
\(197\) −25.9547 −1.84920 −0.924599 0.380941i \(-0.875600\pi\)
−0.924599 + 0.380941i \(0.875600\pi\)
\(198\) −9.66063 −0.686551
\(199\) −15.7517 −1.11661 −0.558306 0.829635i \(-0.688548\pi\)
−0.558306 + 0.829635i \(0.688548\pi\)
\(200\) −4.85630 −0.343392
\(201\) 13.7371 0.968938
\(202\) −9.05936 −0.637415
\(203\) −4.12104 −0.289241
\(204\) 4.62126 0.323553
\(205\) 3.09140 0.215913
\(206\) 12.9499 0.902260
\(207\) 1.83113 0.127272
\(208\) −0.100390 −0.00696078
\(209\) −18.9405 −1.31014
\(210\) −0.251984 −0.0173885
\(211\) 5.85101 0.402800 0.201400 0.979509i \(-0.435451\pi\)
0.201400 + 0.979509i \(0.435451\pi\)
\(212\) 1.61946 0.111225
\(213\) 3.96213 0.271480
\(214\) −17.3147 −1.18361
\(215\) −2.93250 −0.199995
\(216\) 5.22315 0.355390
\(217\) −2.64845 −0.179789
\(218\) 2.98980 0.202495
\(219\) −2.34497 −0.158458
\(220\) 1.99995 0.134837
\(221\) 0.429107 0.0288649
\(222\) −8.08780 −0.542818
\(223\) −8.04417 −0.538678 −0.269339 0.963045i \(-0.586805\pi\)
−0.269339 + 0.963045i \(0.586805\pi\)
\(224\) 0.614832 0.0410802
\(225\) 8.89250 0.592833
\(226\) −12.3950 −0.824502
\(227\) 7.78958 0.517013 0.258506 0.966010i \(-0.416770\pi\)
0.258506 + 0.966010i \(0.416770\pi\)
\(228\) 3.88139 0.257051
\(229\) −0.972106 −0.0642386 −0.0321193 0.999484i \(-0.510226\pi\)
−0.0321193 + 0.999484i \(0.510226\pi\)
\(230\) −0.379081 −0.0249959
\(231\) −3.50693 −0.230739
\(232\) −6.70272 −0.440055
\(233\) 19.7699 1.29517 0.647583 0.761995i \(-0.275780\pi\)
0.647583 + 0.761995i \(0.275780\pi\)
\(234\) 0.183826 0.0120171
\(235\) −3.30487 −0.215586
\(236\) −7.31903 −0.476429
\(237\) −1.52518 −0.0990714
\(238\) −2.62805 −0.170351
\(239\) −13.9374 −0.901534 −0.450767 0.892642i \(-0.648849\pi\)
−0.450767 + 0.892642i \(0.648849\pi\)
\(240\) −0.409841 −0.0264551
\(241\) 9.00899 0.580320 0.290160 0.956978i \(-0.406291\pi\)
0.290160 + 0.956978i \(0.406291\pi\)
\(242\) 16.8339 1.08212
\(243\) −15.5034 −0.994542
\(244\) 4.58610 0.293595
\(245\) −2.51027 −0.160375
\(246\) −8.81671 −0.562133
\(247\) 0.360407 0.0229321
\(248\) −4.30760 −0.273533
\(249\) −13.3085 −0.843392
\(250\) −3.73634 −0.236307
\(251\) 9.80227 0.618714 0.309357 0.950946i \(-0.399886\pi\)
0.309357 + 0.950946i \(0.399886\pi\)
\(252\) −1.12584 −0.0709209
\(253\) −5.27578 −0.331686
\(254\) 15.5795 0.977546
\(255\) 1.75183 0.109704
\(256\) 1.00000 0.0625000
\(257\) 5.10256 0.318289 0.159145 0.987255i \(-0.449126\pi\)
0.159145 + 0.987255i \(0.449126\pi\)
\(258\) 8.36353 0.520691
\(259\) 4.59942 0.285794
\(260\) −0.0380559 −0.00236012
\(261\) 12.2735 0.759712
\(262\) 1.00000 0.0617802
\(263\) 0.989009 0.0609849 0.0304925 0.999535i \(-0.490292\pi\)
0.0304925 + 0.999535i \(0.490292\pi\)
\(264\) −5.70388 −0.351050
\(265\) 0.613907 0.0377120
\(266\) −2.20729 −0.135338
\(267\) 0.0511066 0.00312767
\(268\) −12.7060 −0.776145
\(269\) −8.24587 −0.502759 −0.251380 0.967889i \(-0.580884\pi\)
−0.251380 + 0.967889i \(0.580884\pi\)
\(270\) 1.98000 0.120499
\(271\) −0.251100 −0.0152532 −0.00762661 0.999971i \(-0.502428\pi\)
−0.00762661 + 0.999971i \(0.502428\pi\)
\(272\) −4.27441 −0.259174
\(273\) 0.0667313 0.00403876
\(274\) −4.52366 −0.273285
\(275\) −25.6208 −1.54499
\(276\) 1.08114 0.0650772
\(277\) −30.6713 −1.84286 −0.921429 0.388547i \(-0.872977\pi\)
−0.921429 + 0.388547i \(0.872977\pi\)
\(278\) −13.2164 −0.792668
\(279\) 7.88777 0.472228
\(280\) 0.233071 0.0139287
\(281\) −8.71800 −0.520072 −0.260036 0.965599i \(-0.583734\pi\)
−0.260036 + 0.965599i \(0.583734\pi\)
\(282\) 9.42554 0.561283
\(283\) 25.7018 1.52782 0.763908 0.645325i \(-0.223278\pi\)
0.763908 + 0.645325i \(0.223278\pi\)
\(284\) −3.66475 −0.217463
\(285\) 1.47136 0.0871559
\(286\) −0.529635 −0.0313180
\(287\) 5.01394 0.295964
\(288\) −1.83113 −0.107900
\(289\) 1.27061 0.0747417
\(290\) −2.54087 −0.149205
\(291\) −0.500094 −0.0293160
\(292\) 2.16897 0.126929
\(293\) −17.2397 −1.00715 −0.503576 0.863951i \(-0.667983\pi\)
−0.503576 + 0.863951i \(0.667983\pi\)
\(294\) 7.15932 0.417540
\(295\) −2.77451 −0.161538
\(296\) 7.48078 0.434811
\(297\) 27.5562 1.59897
\(298\) 19.2562 1.11548
\(299\) 0.100390 0.00580569
\(300\) 5.25036 0.303130
\(301\) −4.75622 −0.274144
\(302\) 8.36267 0.481218
\(303\) 9.79448 0.562678
\(304\) −3.59007 −0.205905
\(305\) 1.73850 0.0995465
\(306\) 7.82699 0.447439
\(307\) −15.5417 −0.887012 −0.443506 0.896271i \(-0.646265\pi\)
−0.443506 + 0.896271i \(0.646265\pi\)
\(308\) 3.24372 0.184828
\(309\) −14.0007 −0.796470
\(310\) −1.63293 −0.0927442
\(311\) −15.0946 −0.855935 −0.427967 0.903794i \(-0.640770\pi\)
−0.427967 + 0.903794i \(0.640770\pi\)
\(312\) 0.108536 0.00614463
\(313\) −5.25969 −0.297295 −0.148648 0.988890i \(-0.547492\pi\)
−0.148648 + 0.988890i \(0.547492\pi\)
\(314\) −6.72793 −0.379679
\(315\) −0.426783 −0.0240465
\(316\) 1.41071 0.0793588
\(317\) 4.28608 0.240730 0.120365 0.992730i \(-0.461593\pi\)
0.120365 + 0.992730i \(0.461593\pi\)
\(318\) −1.75087 −0.0981840
\(319\) −35.3621 −1.97990
\(320\) 0.379081 0.0211913
\(321\) 18.7196 1.04483
\(322\) −0.614832 −0.0342632
\(323\) 15.3455 0.853844
\(324\) −0.153594 −0.00853300
\(325\) 0.487523 0.0270429
\(326\) 0.451346 0.0249977
\(327\) −3.23241 −0.178753
\(328\) 8.15498 0.450284
\(329\) −5.36017 −0.295516
\(330\) −2.16223 −0.119027
\(331\) −17.1562 −0.942991 −0.471495 0.881869i \(-0.656286\pi\)
−0.471495 + 0.881869i \(0.656286\pi\)
\(332\) 12.3096 0.675580
\(333\) −13.6983 −0.750660
\(334\) −1.79938 −0.0984576
\(335\) −4.81662 −0.263160
\(336\) −0.664722 −0.0362636
\(337\) −9.74530 −0.530860 −0.265430 0.964130i \(-0.585514\pi\)
−0.265430 + 0.964130i \(0.585514\pi\)
\(338\) −12.9899 −0.706559
\(339\) 13.4008 0.727830
\(340\) −1.62035 −0.0878758
\(341\) −22.7260 −1.23068
\(342\) 6.57388 0.355475
\(343\) −8.37523 −0.452220
\(344\) −7.73581 −0.417087
\(345\) 0.409841 0.0220651
\(346\) 4.48264 0.240988
\(347\) 20.3165 1.09065 0.545325 0.838225i \(-0.316406\pi\)
0.545325 + 0.838225i \(0.316406\pi\)
\(348\) 7.24660 0.388459
\(349\) 16.5982 0.888483 0.444242 0.895907i \(-0.353473\pi\)
0.444242 + 0.895907i \(0.353473\pi\)
\(350\) −2.98581 −0.159598
\(351\) −0.524350 −0.0279878
\(352\) 5.27578 0.281200
\(353\) −15.2653 −0.812491 −0.406246 0.913764i \(-0.633162\pi\)
−0.406246 + 0.913764i \(0.633162\pi\)
\(354\) 7.91293 0.420568
\(355\) −1.38924 −0.0737331
\(356\) −0.0472708 −0.00250535
\(357\) 2.84130 0.150377
\(358\) −26.4904 −1.40006
\(359\) −0.668603 −0.0352875 −0.0176438 0.999844i \(-0.505616\pi\)
−0.0176438 + 0.999844i \(0.505616\pi\)
\(360\) −0.694145 −0.0365847
\(361\) −6.11137 −0.321651
\(362\) 0.316097 0.0166137
\(363\) −18.1999 −0.955246
\(364\) −0.0617228 −0.00323516
\(365\) 0.822215 0.0430367
\(366\) −4.95824 −0.259171
\(367\) 13.8383 0.722353 0.361176 0.932498i \(-0.382375\pi\)
0.361176 + 0.932498i \(0.382375\pi\)
\(368\) −1.00000 −0.0521286
\(369\) −14.9328 −0.777371
\(370\) 2.83582 0.147427
\(371\) 0.995696 0.0516940
\(372\) 4.65714 0.241461
\(373\) −11.2191 −0.580906 −0.290453 0.956889i \(-0.593806\pi\)
−0.290453 + 0.956889i \(0.593806\pi\)
\(374\) −22.5509 −1.16608
\(375\) 4.03952 0.208600
\(376\) −8.71811 −0.449602
\(377\) 0.672884 0.0346553
\(378\) 3.21136 0.165174
\(379\) 9.03697 0.464198 0.232099 0.972692i \(-0.425441\pi\)
0.232099 + 0.972692i \(0.425441\pi\)
\(380\) −1.36093 −0.0698142
\(381\) −16.8437 −0.862929
\(382\) 9.18166 0.469775
\(383\) −13.2562 −0.677359 −0.338679 0.940902i \(-0.609980\pi\)
−0.338679 + 0.940902i \(0.609980\pi\)
\(384\) −1.08114 −0.0551719
\(385\) 1.22963 0.0626679
\(386\) −9.13123 −0.464767
\(387\) 14.1653 0.720060
\(388\) 0.462560 0.0234829
\(389\) −5.85650 −0.296937 −0.148468 0.988917i \(-0.547434\pi\)
−0.148468 + 0.988917i \(0.547434\pi\)
\(390\) 0.0411439 0.00208340
\(391\) 4.27441 0.216166
\(392\) −6.62198 −0.334461
\(393\) −1.08114 −0.0545365
\(394\) −25.9547 −1.30758
\(395\) 0.534775 0.0269074
\(396\) −9.66063 −0.485465
\(397\) −23.3470 −1.17175 −0.585876 0.810401i \(-0.699249\pi\)
−0.585876 + 0.810401i \(0.699249\pi\)
\(398\) −15.7517 −0.789564
\(399\) 2.38640 0.119469
\(400\) −4.85630 −0.242815
\(401\) −8.27895 −0.413431 −0.206716 0.978401i \(-0.566277\pi\)
−0.206716 + 0.978401i \(0.566277\pi\)
\(402\) 13.7371 0.685143
\(403\) 0.432439 0.0215413
\(404\) −9.05936 −0.450720
\(405\) −0.0582246 −0.00289320
\(406\) −4.12104 −0.204524
\(407\) 39.4670 1.95630
\(408\) 4.62126 0.228786
\(409\) 20.9817 1.03748 0.518740 0.854932i \(-0.326401\pi\)
0.518740 + 0.854932i \(0.326401\pi\)
\(410\) 3.09140 0.152673
\(411\) 4.89073 0.241242
\(412\) 12.9499 0.637994
\(413\) −4.49997 −0.221429
\(414\) 1.83113 0.0899950
\(415\) 4.66635 0.229062
\(416\) −0.100390 −0.00492201
\(417\) 14.2889 0.699728
\(418\) −18.9405 −0.926408
\(419\) −9.20606 −0.449745 −0.224873 0.974388i \(-0.572197\pi\)
−0.224873 + 0.974388i \(0.572197\pi\)
\(420\) −0.251984 −0.0122955
\(421\) −27.5113 −1.34082 −0.670409 0.741991i \(-0.733881\pi\)
−0.670409 + 0.741991i \(0.733881\pi\)
\(422\) 5.85101 0.284823
\(423\) 15.9640 0.776195
\(424\) 1.61946 0.0786480
\(425\) 20.7578 1.00690
\(426\) 3.96213 0.191966
\(427\) 2.81968 0.136454
\(428\) −17.3147 −0.836936
\(429\) 0.572612 0.0276459
\(430\) −2.93250 −0.141418
\(431\) 35.7431 1.72168 0.860842 0.508872i \(-0.169937\pi\)
0.860842 + 0.508872i \(0.169937\pi\)
\(432\) 5.22315 0.251299
\(433\) 38.9427 1.87147 0.935733 0.352708i \(-0.114739\pi\)
0.935733 + 0.352708i \(0.114739\pi\)
\(434\) −2.64845 −0.127130
\(435\) 2.74705 0.131711
\(436\) 2.98980 0.143186
\(437\) 3.59007 0.171737
\(438\) −2.34497 −0.112047
\(439\) 13.5949 0.648848 0.324424 0.945912i \(-0.394830\pi\)
0.324424 + 0.945912i \(0.394830\pi\)
\(440\) 1.99995 0.0953439
\(441\) 12.1257 0.577414
\(442\) 0.429107 0.0204106
\(443\) −15.2013 −0.722236 −0.361118 0.932520i \(-0.617605\pi\)
−0.361118 + 0.932520i \(0.617605\pi\)
\(444\) −8.08780 −0.383830
\(445\) −0.0179195 −0.000849465 0
\(446\) −8.04417 −0.380903
\(447\) −20.8187 −0.984691
\(448\) 0.614832 0.0290481
\(449\) −29.9533 −1.41358 −0.706791 0.707422i \(-0.749858\pi\)
−0.706791 + 0.707422i \(0.749858\pi\)
\(450\) 8.89250 0.419196
\(451\) 43.0239 2.02592
\(452\) −12.3950 −0.583011
\(453\) −9.04126 −0.424795
\(454\) 7.78958 0.365583
\(455\) −0.0233980 −0.00109691
\(456\) 3.88139 0.181763
\(457\) 31.0148 1.45081 0.725406 0.688321i \(-0.241652\pi\)
0.725406 + 0.688321i \(0.241652\pi\)
\(458\) −0.972106 −0.0454235
\(459\) −22.3259 −1.04208
\(460\) −0.379081 −0.0176747
\(461\) 4.55154 0.211986 0.105993 0.994367i \(-0.466198\pi\)
0.105993 + 0.994367i \(0.466198\pi\)
\(462\) −3.50693 −0.163157
\(463\) 5.14068 0.238908 0.119454 0.992840i \(-0.461886\pi\)
0.119454 + 0.992840i \(0.461886\pi\)
\(464\) −6.70272 −0.311166
\(465\) 1.76543 0.0818700
\(466\) 19.7699 0.915821
\(467\) −11.6226 −0.537830 −0.268915 0.963164i \(-0.586665\pi\)
−0.268915 + 0.963164i \(0.586665\pi\)
\(468\) 0.183826 0.00849738
\(469\) −7.81208 −0.360728
\(470\) −3.30487 −0.152442
\(471\) 7.27387 0.335162
\(472\) −7.31903 −0.336886
\(473\) −40.8125 −1.87656
\(474\) −1.52518 −0.0700541
\(475\) 17.4345 0.799948
\(476\) −2.62805 −0.120456
\(477\) −2.96544 −0.135778
\(478\) −13.9374 −0.637481
\(479\) 28.5323 1.30367 0.651837 0.758359i \(-0.273999\pi\)
0.651837 + 0.758359i \(0.273999\pi\)
\(480\) −0.409841 −0.0187066
\(481\) −0.750994 −0.0342424
\(482\) 9.00899 0.410348
\(483\) 0.664722 0.0302459
\(484\) 16.8339 0.765177
\(485\) 0.175348 0.00796212
\(486\) −15.5034 −0.703248
\(487\) 27.6020 1.25076 0.625382 0.780318i \(-0.284943\pi\)
0.625382 + 0.780318i \(0.284943\pi\)
\(488\) 4.58610 0.207603
\(489\) −0.487970 −0.0220668
\(490\) −2.51027 −0.113402
\(491\) 8.63792 0.389824 0.194912 0.980821i \(-0.437558\pi\)
0.194912 + 0.980821i \(0.437558\pi\)
\(492\) −8.81671 −0.397488
\(493\) 28.6502 1.29034
\(494\) 0.360407 0.0162155
\(495\) −3.66216 −0.164602
\(496\) −4.30760 −0.193417
\(497\) −2.25321 −0.101070
\(498\) −13.3085 −0.596368
\(499\) 0.577779 0.0258649 0.0129325 0.999916i \(-0.495883\pi\)
0.0129325 + 0.999916i \(0.495883\pi\)
\(500\) −3.73634 −0.167094
\(501\) 1.94539 0.0869135
\(502\) 9.80227 0.437497
\(503\) 37.5020 1.67213 0.836065 0.548630i \(-0.184851\pi\)
0.836065 + 0.548630i \(0.184851\pi\)
\(504\) −1.12584 −0.0501487
\(505\) −3.43423 −0.152821
\(506\) −5.27578 −0.234537
\(507\) 14.0440 0.623715
\(508\) 15.5795 0.691229
\(509\) 11.3872 0.504730 0.252365 0.967632i \(-0.418792\pi\)
0.252365 + 0.967632i \(0.418792\pi\)
\(510\) 1.75183 0.0775724
\(511\) 1.33355 0.0589928
\(512\) 1.00000 0.0441942
\(513\) −18.7515 −0.827898
\(514\) 5.10256 0.225064
\(515\) 4.90905 0.216318
\(516\) 8.36353 0.368184
\(517\) −45.9949 −2.02285
\(518\) 4.59942 0.202087
\(519\) −4.84638 −0.212732
\(520\) −0.0380559 −0.00166886
\(521\) −35.2190 −1.54297 −0.771486 0.636246i \(-0.780486\pi\)
−0.771486 + 0.636246i \(0.780486\pi\)
\(522\) 12.2735 0.537198
\(523\) −19.4966 −0.852526 −0.426263 0.904599i \(-0.640170\pi\)
−0.426263 + 0.904599i \(0.640170\pi\)
\(524\) 1.00000 0.0436852
\(525\) 3.22809 0.140885
\(526\) 0.989009 0.0431228
\(527\) 18.4125 0.802060
\(528\) −5.70388 −0.248230
\(529\) 1.00000 0.0434783
\(530\) 0.613907 0.0266664
\(531\) 13.4021 0.581601
\(532\) −2.20729 −0.0956982
\(533\) −0.818677 −0.0354608
\(534\) 0.0511066 0.00221160
\(535\) −6.56366 −0.283772
\(536\) −12.7060 −0.548818
\(537\) 28.6400 1.23591
\(538\) −8.24587 −0.355505
\(539\) −34.9361 −1.50481
\(540\) 1.98000 0.0852054
\(541\) 12.0331 0.517343 0.258671 0.965965i \(-0.416715\pi\)
0.258671 + 0.965965i \(0.416715\pi\)
\(542\) −0.251100 −0.0107857
\(543\) −0.341747 −0.0146658
\(544\) −4.27441 −0.183264
\(545\) 1.13338 0.0485486
\(546\) 0.0667313 0.00285584
\(547\) −26.0326 −1.11307 −0.556536 0.830824i \(-0.687870\pi\)
−0.556536 + 0.830824i \(0.687870\pi\)
\(548\) −4.52366 −0.193241
\(549\) −8.39773 −0.358407
\(550\) −25.6208 −1.09247
\(551\) 24.0632 1.02513
\(552\) 1.08114 0.0460166
\(553\) 0.867352 0.0368835
\(554\) −30.6713 −1.30310
\(555\) −3.06593 −0.130142
\(556\) −13.2164 −0.560501
\(557\) −3.15232 −0.133568 −0.0667841 0.997767i \(-0.521274\pi\)
−0.0667841 + 0.997767i \(0.521274\pi\)
\(558\) 7.88777 0.333916
\(559\) 0.776597 0.0328465
\(560\) 0.233071 0.00984905
\(561\) 24.3808 1.02936
\(562\) −8.71800 −0.367747
\(563\) 43.3718 1.82790 0.913952 0.405823i \(-0.133015\pi\)
0.913952 + 0.405823i \(0.133015\pi\)
\(564\) 9.42554 0.396887
\(565\) −4.69870 −0.197676
\(566\) 25.7018 1.08033
\(567\) −0.0944345 −0.00396588
\(568\) −3.66475 −0.153770
\(569\) −19.9713 −0.837240 −0.418620 0.908161i \(-0.637486\pi\)
−0.418620 + 0.908161i \(0.637486\pi\)
\(570\) 1.47136 0.0616285
\(571\) −13.0736 −0.547112 −0.273556 0.961856i \(-0.588200\pi\)
−0.273556 + 0.961856i \(0.588200\pi\)
\(572\) −0.529635 −0.0221451
\(573\) −9.92671 −0.414694
\(574\) 5.01394 0.209278
\(575\) 4.85630 0.202522
\(576\) −1.83113 −0.0762970
\(577\) −26.2810 −1.09409 −0.547045 0.837103i \(-0.684247\pi\)
−0.547045 + 0.837103i \(0.684247\pi\)
\(578\) 1.27061 0.0528504
\(579\) 9.87217 0.410274
\(580\) −2.54087 −0.105504
\(581\) 7.56836 0.313989
\(582\) −0.500094 −0.0207296
\(583\) 8.54393 0.353853
\(584\) 2.16897 0.0897526
\(585\) 0.0696851 0.00288112
\(586\) −17.2397 −0.712164
\(587\) 17.3173 0.714762 0.357381 0.933959i \(-0.383670\pi\)
0.357381 + 0.933959i \(0.383670\pi\)
\(588\) 7.15932 0.295245
\(589\) 15.4646 0.637208
\(590\) −2.77451 −0.114225
\(591\) 28.0608 1.15427
\(592\) 7.48078 0.307458
\(593\) 32.9152 1.35166 0.675832 0.737055i \(-0.263784\pi\)
0.675832 + 0.737055i \(0.263784\pi\)
\(594\) 27.5562 1.13064
\(595\) −0.996242 −0.0408420
\(596\) 19.2562 0.788764
\(597\) 17.0299 0.696988
\(598\) 0.100390 0.00410524
\(599\) 11.7553 0.480309 0.240154 0.970735i \(-0.422802\pi\)
0.240154 + 0.970735i \(0.422802\pi\)
\(600\) 5.25036 0.214345
\(601\) 5.69853 0.232448 0.116224 0.993223i \(-0.462921\pi\)
0.116224 + 0.993223i \(0.462921\pi\)
\(602\) −4.75622 −0.193849
\(603\) 23.2664 0.947480
\(604\) 8.36267 0.340272
\(605\) 6.38141 0.259441
\(606\) 9.79448 0.397874
\(607\) −17.3964 −0.706099 −0.353050 0.935605i \(-0.614855\pi\)
−0.353050 + 0.935605i \(0.614855\pi\)
\(608\) −3.59007 −0.145597
\(609\) 4.45544 0.180544
\(610\) 1.73850 0.0703900
\(611\) 0.875210 0.0354072
\(612\) 7.82699 0.316387
\(613\) 17.7128 0.715414 0.357707 0.933834i \(-0.383559\pi\)
0.357707 + 0.933834i \(0.383559\pi\)
\(614\) −15.5417 −0.627212
\(615\) −3.34225 −0.134772
\(616\) 3.24372 0.130693
\(617\) −37.4340 −1.50704 −0.753519 0.657426i \(-0.771645\pi\)
−0.753519 + 0.657426i \(0.771645\pi\)
\(618\) −14.0007 −0.563190
\(619\) 31.8540 1.28032 0.640160 0.768242i \(-0.278868\pi\)
0.640160 + 0.768242i \(0.278868\pi\)
\(620\) −1.63293 −0.0655801
\(621\) −5.22315 −0.209598
\(622\) −15.0946 −0.605237
\(623\) −0.0290636 −0.00116441
\(624\) 0.108536 0.00434491
\(625\) 22.8651 0.914605
\(626\) −5.25969 −0.210219
\(627\) 20.4774 0.817787
\(628\) −6.72793 −0.268474
\(629\) −31.9759 −1.27496
\(630\) −0.426783 −0.0170034
\(631\) −42.0682 −1.67471 −0.837355 0.546659i \(-0.815899\pi\)
−0.837355 + 0.546659i \(0.815899\pi\)
\(632\) 1.41071 0.0561152
\(633\) −6.32579 −0.251428
\(634\) 4.28608 0.170222
\(635\) 5.90590 0.234368
\(636\) −1.75087 −0.0694266
\(637\) 0.664779 0.0263395
\(638\) −35.3621 −1.40000
\(639\) 6.71063 0.265468
\(640\) 0.379081 0.0149845
\(641\) −3.99106 −0.157637 −0.0788187 0.996889i \(-0.525115\pi\)
−0.0788187 + 0.996889i \(0.525115\pi\)
\(642\) 18.7196 0.738805
\(643\) 41.3942 1.63243 0.816213 0.577750i \(-0.196069\pi\)
0.816213 + 0.577750i \(0.196069\pi\)
\(644\) −0.614832 −0.0242278
\(645\) 3.17046 0.124837
\(646\) 15.3455 0.603759
\(647\) 15.8086 0.621501 0.310750 0.950492i \(-0.399420\pi\)
0.310750 + 0.950492i \(0.399420\pi\)
\(648\) −0.153594 −0.00603374
\(649\) −38.6136 −1.51572
\(650\) 0.487523 0.0191222
\(651\) 2.86336 0.112224
\(652\) 0.451346 0.0176761
\(653\) 42.4282 1.66034 0.830171 0.557508i \(-0.188243\pi\)
0.830171 + 0.557508i \(0.188243\pi\)
\(654\) −3.23241 −0.126397
\(655\) 0.379081 0.0148119
\(656\) 8.15498 0.318399
\(657\) −3.97166 −0.154949
\(658\) −5.36017 −0.208961
\(659\) 13.1469 0.512130 0.256065 0.966660i \(-0.417574\pi\)
0.256065 + 0.966660i \(0.417574\pi\)
\(660\) −2.16223 −0.0841649
\(661\) 35.4668 1.37950 0.689749 0.724049i \(-0.257721\pi\)
0.689749 + 0.724049i \(0.257721\pi\)
\(662\) −17.1562 −0.666795
\(663\) −0.463927 −0.0180174
\(664\) 12.3096 0.477707
\(665\) −0.836742 −0.0324475
\(666\) −13.6983 −0.530796
\(667\) 6.70272 0.259530
\(668\) −1.79938 −0.0696200
\(669\) 8.69691 0.336242
\(670\) −4.81662 −0.186082
\(671\) 24.1953 0.934048
\(672\) −0.664722 −0.0256422
\(673\) −19.3611 −0.746317 −0.373159 0.927768i \(-0.621725\pi\)
−0.373159 + 0.927768i \(0.621725\pi\)
\(674\) −9.74530 −0.375375
\(675\) −25.3651 −0.976305
\(676\) −12.9899 −0.499612
\(677\) 13.7856 0.529824 0.264912 0.964273i \(-0.414657\pi\)
0.264912 + 0.964273i \(0.414657\pi\)
\(678\) 13.4008 0.514653
\(679\) 0.284396 0.0109141
\(680\) −1.62035 −0.0621376
\(681\) −8.42166 −0.322719
\(682\) −22.7260 −0.870223
\(683\) 17.3188 0.662687 0.331343 0.943510i \(-0.392498\pi\)
0.331343 + 0.943510i \(0.392498\pi\)
\(684\) 6.57388 0.251359
\(685\) −1.71484 −0.0655205
\(686\) −8.37523 −0.319768
\(687\) 1.05099 0.0400977
\(688\) −7.73581 −0.294925
\(689\) −0.162577 −0.00619370
\(690\) 0.409841 0.0156024
\(691\) −6.95936 −0.264746 −0.132373 0.991200i \(-0.542260\pi\)
−0.132373 + 0.991200i \(0.542260\pi\)
\(692\) 4.48264 0.170404
\(693\) −5.93966 −0.225629
\(694\) 20.3165 0.771205
\(695\) −5.01009 −0.190044
\(696\) 7.24660 0.274682
\(697\) −34.8578 −1.32033
\(698\) 16.5982 0.628252
\(699\) −21.3741 −0.808442
\(700\) −2.98581 −0.112853
\(701\) −8.33027 −0.314630 −0.157315 0.987548i \(-0.550284\pi\)
−0.157315 + 0.987548i \(0.550284\pi\)
\(702\) −0.524350 −0.0197903
\(703\) −26.8565 −1.01291
\(704\) 5.27578 0.198839
\(705\) 3.57304 0.134569
\(706\) −15.2653 −0.574518
\(707\) −5.56998 −0.209481
\(708\) 7.91293 0.297386
\(709\) 15.9516 0.599076 0.299538 0.954084i \(-0.403167\pi\)
0.299538 + 0.954084i \(0.403167\pi\)
\(710\) −1.38924 −0.0521372
\(711\) −2.58320 −0.0968774
\(712\) −0.0472708 −0.00177155
\(713\) 4.30760 0.161321
\(714\) 2.84130 0.106333
\(715\) −0.200774 −0.00750854
\(716\) −26.4904 −0.989993
\(717\) 15.0683 0.562737
\(718\) −0.668603 −0.0249520
\(719\) 8.10670 0.302329 0.151164 0.988509i \(-0.451698\pi\)
0.151164 + 0.988509i \(0.451698\pi\)
\(720\) −0.694145 −0.0258693
\(721\) 7.96199 0.296520
\(722\) −6.11137 −0.227442
\(723\) −9.74002 −0.362235
\(724\) 0.316097 0.0117477
\(725\) 32.5504 1.20889
\(726\) −18.1999 −0.675461
\(727\) 38.7631 1.43764 0.718821 0.695195i \(-0.244682\pi\)
0.718821 + 0.695195i \(0.244682\pi\)
\(728\) −0.0617228 −0.00228760
\(729\) 17.2222 0.637858
\(730\) 0.822215 0.0304315
\(731\) 33.0661 1.22299
\(732\) −4.95824 −0.183262
\(733\) −36.6878 −1.35509 −0.677547 0.735480i \(-0.736957\pi\)
−0.677547 + 0.735480i \(0.736957\pi\)
\(734\) 13.8383 0.510780
\(735\) 2.71396 0.100106
\(736\) −1.00000 −0.0368605
\(737\) −67.0344 −2.46924
\(738\) −14.9328 −0.549684
\(739\) 42.8080 1.57472 0.787359 0.616495i \(-0.211448\pi\)
0.787359 + 0.616495i \(0.211448\pi\)
\(740\) 2.83582 0.104247
\(741\) −0.389652 −0.0143142
\(742\) 0.995696 0.0365532
\(743\) −37.6738 −1.38212 −0.691058 0.722799i \(-0.742855\pi\)
−0.691058 + 0.722799i \(0.742855\pi\)
\(744\) 4.65714 0.170739
\(745\) 7.29965 0.267438
\(746\) −11.2191 −0.410762
\(747\) −22.5405 −0.824715
\(748\) −22.5509 −0.824542
\(749\) −10.6456 −0.388982
\(750\) 4.03952 0.147502
\(751\) −29.6729 −1.08278 −0.541389 0.840772i \(-0.682101\pi\)
−0.541389 + 0.840772i \(0.682101\pi\)
\(752\) −8.71811 −0.317917
\(753\) −10.5977 −0.386201
\(754\) 0.672884 0.0245050
\(755\) 3.17013 0.115373
\(756\) 3.21136 0.116796
\(757\) −40.2256 −1.46202 −0.731012 0.682365i \(-0.760951\pi\)
−0.731012 + 0.682365i \(0.760951\pi\)
\(758\) 9.03697 0.328238
\(759\) 5.70388 0.207038
\(760\) −1.36093 −0.0493661
\(761\) 22.2108 0.805142 0.402571 0.915389i \(-0.368117\pi\)
0.402571 + 0.915389i \(0.368117\pi\)
\(762\) −16.8437 −0.610183
\(763\) 1.83823 0.0665483
\(764\) 9.18166 0.332181
\(765\) 2.96706 0.107274
\(766\) −13.2562 −0.478965
\(767\) 0.734756 0.0265305
\(768\) −1.08114 −0.0390124
\(769\) 1.36151 0.0490974 0.0245487 0.999699i \(-0.492185\pi\)
0.0245487 + 0.999699i \(0.492185\pi\)
\(770\) 1.22963 0.0443129
\(771\) −5.51661 −0.198676
\(772\) −9.13123 −0.328640
\(773\) −19.4923 −0.701090 −0.350545 0.936546i \(-0.614004\pi\)
−0.350545 + 0.936546i \(0.614004\pi\)
\(774\) 14.1653 0.509160
\(775\) 20.9190 0.751433
\(776\) 0.462560 0.0166049
\(777\) −4.97264 −0.178392
\(778\) −5.85650 −0.209966
\(779\) −29.2770 −1.04896
\(780\) 0.0411439 0.00147319
\(781\) −19.3344 −0.691841
\(782\) 4.27441 0.152853
\(783\) −35.0093 −1.25113
\(784\) −6.62198 −0.236499
\(785\) −2.55043 −0.0910288
\(786\) −1.08114 −0.0385631
\(787\) −29.4836 −1.05098 −0.525488 0.850801i \(-0.676117\pi\)
−0.525488 + 0.850801i \(0.676117\pi\)
\(788\) −25.9547 −0.924599
\(789\) −1.06926 −0.0380667
\(790\) 0.534775 0.0190264
\(791\) −7.62083 −0.270965
\(792\) −9.66063 −0.343276
\(793\) −0.460398 −0.0163492
\(794\) −23.3470 −0.828554
\(795\) −0.663722 −0.0235398
\(796\) −15.7517 −0.558306
\(797\) 22.5052 0.797174 0.398587 0.917130i \(-0.369501\pi\)
0.398587 + 0.917130i \(0.369501\pi\)
\(798\) 2.38640 0.0844777
\(799\) 37.2648 1.31833
\(800\) −4.85630 −0.171696
\(801\) 0.0865589 0.00305841
\(802\) −8.27895 −0.292340
\(803\) 11.4430 0.403815
\(804\) 13.7371 0.484469
\(805\) −0.233071 −0.00821468
\(806\) 0.432439 0.0152320
\(807\) 8.91497 0.313822
\(808\) −9.05936 −0.318707
\(809\) −18.2134 −0.640349 −0.320175 0.947359i \(-0.603742\pi\)
−0.320175 + 0.947359i \(0.603742\pi\)
\(810\) −0.0582246 −0.00204580
\(811\) −9.21285 −0.323507 −0.161753 0.986831i \(-0.551715\pi\)
−0.161753 + 0.986831i \(0.551715\pi\)
\(812\) −4.12104 −0.144620
\(813\) 0.271475 0.00952105
\(814\) 39.4670 1.38332
\(815\) 0.171097 0.00599325
\(816\) 4.62126 0.161776
\(817\) 27.7721 0.971624
\(818\) 20.9817 0.733609
\(819\) 0.113022 0.00394932
\(820\) 3.09140 0.107956
\(821\) 15.1297 0.528031 0.264015 0.964518i \(-0.414953\pi\)
0.264015 + 0.964518i \(0.414953\pi\)
\(822\) 4.89073 0.170584
\(823\) −38.9952 −1.35929 −0.679643 0.733543i \(-0.737865\pi\)
−0.679643 + 0.733543i \(0.737865\pi\)
\(824\) 12.9499 0.451130
\(825\) 27.6998 0.964382
\(826\) −4.49997 −0.156574
\(827\) −22.2480 −0.773639 −0.386820 0.922155i \(-0.626426\pi\)
−0.386820 + 0.922155i \(0.626426\pi\)
\(828\) 1.83113 0.0636361
\(829\) −3.16991 −0.110096 −0.0550478 0.998484i \(-0.517531\pi\)
−0.0550478 + 0.998484i \(0.517531\pi\)
\(830\) 4.66635 0.161972
\(831\) 33.1601 1.15031
\(832\) −0.100390 −0.00348039
\(833\) 28.3051 0.980713
\(834\) 14.2889 0.494782
\(835\) −0.682110 −0.0236054
\(836\) −18.9405 −0.655069
\(837\) −22.4992 −0.777687
\(838\) −9.20606 −0.318018
\(839\) −24.3391 −0.840279 −0.420139 0.907460i \(-0.638019\pi\)
−0.420139 + 0.907460i \(0.638019\pi\)
\(840\) −0.251984 −0.00869426
\(841\) 15.9264 0.549187
\(842\) −27.5113 −0.948102
\(843\) 9.42541 0.324629
\(844\) 5.85101 0.201400
\(845\) −4.92423 −0.169399
\(846\) 15.9640 0.548853
\(847\) 10.3500 0.355631
\(848\) 1.61946 0.0556125
\(849\) −27.7874 −0.953661
\(850\) 20.7578 0.711987
\(851\) −7.48078 −0.256438
\(852\) 3.96213 0.135740
\(853\) −12.6255 −0.432290 −0.216145 0.976361i \(-0.569348\pi\)
−0.216145 + 0.976361i \(0.569348\pi\)
\(854\) 2.81968 0.0964875
\(855\) 2.49203 0.0852258
\(856\) −17.3147 −0.591803
\(857\) −13.1920 −0.450630 −0.225315 0.974286i \(-0.572341\pi\)
−0.225315 + 0.974286i \(0.572341\pi\)
\(858\) 0.572612 0.0195486
\(859\) −24.6661 −0.841595 −0.420797 0.907155i \(-0.638250\pi\)
−0.420797 + 0.907155i \(0.638250\pi\)
\(860\) −2.93250 −0.0999974
\(861\) −5.42080 −0.184740
\(862\) 35.7431 1.21741
\(863\) 47.5123 1.61734 0.808668 0.588265i \(-0.200189\pi\)
0.808668 + 0.588265i \(0.200189\pi\)
\(864\) 5.22315 0.177695
\(865\) 1.69928 0.0577773
\(866\) 38.9427 1.32333
\(867\) −1.37371 −0.0466537
\(868\) −2.64845 −0.0898943
\(869\) 7.44262 0.252474
\(870\) 2.74705 0.0931337
\(871\) 1.27556 0.0432206
\(872\) 2.98980 0.101248
\(873\) −0.847005 −0.0286668
\(874\) 3.59007 0.121436
\(875\) −2.29722 −0.0776601
\(876\) −2.34497 −0.0792291
\(877\) −32.4295 −1.09506 −0.547532 0.836785i \(-0.684433\pi\)
−0.547532 + 0.836785i \(0.684433\pi\)
\(878\) 13.5949 0.458804
\(879\) 18.6386 0.628663
\(880\) 1.99995 0.0674183
\(881\) −23.3163 −0.785548 −0.392774 0.919635i \(-0.628484\pi\)
−0.392774 + 0.919635i \(0.628484\pi\)
\(882\) 12.1257 0.408293
\(883\) −11.7070 −0.393972 −0.196986 0.980406i \(-0.563115\pi\)
−0.196986 + 0.980406i \(0.563115\pi\)
\(884\) 0.429107 0.0144324
\(885\) 2.99964 0.100832
\(886\) −15.2013 −0.510698
\(887\) 50.3212 1.68962 0.844809 0.535067i \(-0.179714\pi\)
0.844809 + 0.535067i \(0.179714\pi\)
\(888\) −8.08780 −0.271409
\(889\) 9.57878 0.321262
\(890\) −0.0179195 −0.000600662 0
\(891\) −0.810329 −0.0271470
\(892\) −8.04417 −0.269339
\(893\) 31.2987 1.04737
\(894\) −20.8187 −0.696282
\(895\) −10.0420 −0.335668
\(896\) 0.614832 0.0205401
\(897\) −0.108536 −0.00362391
\(898\) −29.9533 −0.999554
\(899\) 28.8726 0.962956
\(900\) 8.89250 0.296417
\(901\) −6.92225 −0.230613
\(902\) 43.0239 1.43254
\(903\) 5.14217 0.171121
\(904\) −12.3950 −0.412251
\(905\) 0.119826 0.00398317
\(906\) −9.04126 −0.300376
\(907\) 14.1432 0.469617 0.234808 0.972042i \(-0.424554\pi\)
0.234808 + 0.972042i \(0.424554\pi\)
\(908\) 7.78958 0.258506
\(909\) 16.5888 0.550217
\(910\) −0.0233980 −0.000775635 0
\(911\) −30.7231 −1.01790 −0.508951 0.860796i \(-0.669966\pi\)
−0.508951 + 0.860796i \(0.669966\pi\)
\(912\) 3.88139 0.128526
\(913\) 64.9430 2.14930
\(914\) 31.0148 1.02588
\(915\) −1.87957 −0.0621368
\(916\) −0.972106 −0.0321193
\(917\) 0.614832 0.0203035
\(918\) −22.3259 −0.736864
\(919\) 25.6938 0.847560 0.423780 0.905765i \(-0.360703\pi\)
0.423780 + 0.905765i \(0.360703\pi\)
\(920\) −0.379081 −0.0124979
\(921\) 16.8028 0.553672
\(922\) 4.55154 0.149897
\(923\) 0.367904 0.0121097
\(924\) −3.50693 −0.115370
\(925\) −36.3289 −1.19449
\(926\) 5.14068 0.168933
\(927\) −23.7128 −0.778832
\(928\) −6.70272 −0.220027
\(929\) −8.91222 −0.292401 −0.146200 0.989255i \(-0.546704\pi\)
−0.146200 + 0.989255i \(0.546704\pi\)
\(930\) 1.76543 0.0578909
\(931\) 23.7734 0.779142
\(932\) 19.7699 0.647583
\(933\) 16.3194 0.534274
\(934\) −11.6226 −0.380303
\(935\) −8.54861 −0.279570
\(936\) 0.183826 0.00600855
\(937\) 50.7293 1.65726 0.828628 0.559800i \(-0.189122\pi\)
0.828628 + 0.559800i \(0.189122\pi\)
\(938\) −7.81208 −0.255074
\(939\) 5.68648 0.185571
\(940\) −3.30487 −0.107793
\(941\) 31.5893 1.02978 0.514891 0.857256i \(-0.327832\pi\)
0.514891 + 0.857256i \(0.327832\pi\)
\(942\) 7.27387 0.236995
\(943\) −8.15498 −0.265563
\(944\) −7.31903 −0.238214
\(945\) 1.21736 0.0396009
\(946\) −40.8125 −1.32693
\(947\) −12.7920 −0.415685 −0.207842 0.978162i \(-0.566644\pi\)
−0.207842 + 0.978162i \(0.566644\pi\)
\(948\) −1.52518 −0.0495357
\(949\) −0.217742 −0.00706821
\(950\) 17.4345 0.565649
\(951\) −4.63388 −0.150264
\(952\) −2.62805 −0.0851754
\(953\) 23.4487 0.759578 0.379789 0.925073i \(-0.375997\pi\)
0.379789 + 0.925073i \(0.375997\pi\)
\(954\) −2.96544 −0.0960096
\(955\) 3.48060 0.112629
\(956\) −13.9374 −0.450767
\(957\) 38.2315 1.23585
\(958\) 28.5323 0.921837
\(959\) −2.78129 −0.0898126
\(960\) −0.409841 −0.0132276
\(961\) −12.4446 −0.401437
\(962\) −0.750994 −0.0242130
\(963\) 31.7053 1.02169
\(964\) 9.00899 0.290160
\(965\) −3.46147 −0.111429
\(966\) 0.664722 0.0213871
\(967\) 48.7326 1.56714 0.783568 0.621306i \(-0.213398\pi\)
0.783568 + 0.621306i \(0.213398\pi\)
\(968\) 16.8339 0.541062
\(969\) −16.5907 −0.532969
\(970\) 0.175348 0.00563007
\(971\) 37.2671 1.19596 0.597979 0.801512i \(-0.295971\pi\)
0.597979 + 0.801512i \(0.295971\pi\)
\(972\) −15.5034 −0.497271
\(973\) −8.12587 −0.260504
\(974\) 27.6020 0.884424
\(975\) −0.527082 −0.0168801
\(976\) 4.58610 0.146798
\(977\) 25.3556 0.811197 0.405599 0.914051i \(-0.367063\pi\)
0.405599 + 0.914051i \(0.367063\pi\)
\(978\) −0.487970 −0.0156036
\(979\) −0.249391 −0.00797056
\(980\) −2.51027 −0.0801876
\(981\) −5.47471 −0.174794
\(982\) 8.63792 0.275647
\(983\) 41.5682 1.32582 0.662909 0.748700i \(-0.269321\pi\)
0.662909 + 0.748700i \(0.269321\pi\)
\(984\) −8.81671 −0.281067
\(985\) −9.83895 −0.313495
\(986\) 28.6502 0.912408
\(987\) 5.79512 0.184461
\(988\) 0.360407 0.0114661
\(989\) 7.73581 0.245985
\(990\) −3.66216 −0.116391
\(991\) −22.0387 −0.700082 −0.350041 0.936734i \(-0.613832\pi\)
−0.350041 + 0.936734i \(0.613832\pi\)
\(992\) −4.30760 −0.136767
\(993\) 18.5483 0.588614
\(994\) −2.25321 −0.0714674
\(995\) −5.97119 −0.189299
\(996\) −13.3085 −0.421696
\(997\) 49.9169 1.58088 0.790441 0.612538i \(-0.209851\pi\)
0.790441 + 0.612538i \(0.209851\pi\)
\(998\) 0.577779 0.0182893
\(999\) 39.0732 1.23622
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 6026.2.a.g.1.8 21
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6026.2.a.g.1.8 21 1.1 even 1 trivial