Properties

Label 5586.2.a.ce.1.3
Level $5586$
Weight $2$
Character 5586.1
Self dual yes
Analytic conductor $44.604$
Analytic rank $0$
Dimension $8$
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] = [5586,2,Mod(1,5586)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5586, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5586.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5586 = 2 \cdot 3 \cdot 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5586.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: \(44.6044345691\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 22x^{6} + 60x^{5} + 87x^{4} - 176x^{3} - 40x^{2} + 64x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-3.71976\) of defining polynomial
Character \(\chi\) \(=\) 5586.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.63704 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.63704 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -2.63704 q^{10} -3.26053 q^{11} -1.00000 q^{12} -4.44345 q^{13} +2.63704 q^{15} +1.00000 q^{16} +2.36962 q^{17} +1.00000 q^{18} -1.00000 q^{19} -2.63704 q^{20} -3.26053 q^{22} -1.98320 q^{23} -1.00000 q^{24} +1.95397 q^{25} -4.44345 q^{26} -1.00000 q^{27} +0.0962338 q^{29} +2.63704 q^{30} +2.43285 q^{31} +1.00000 q^{32} +3.26053 q^{33} +2.36962 q^{34} +1.00000 q^{36} -0.452007 q^{37} -1.00000 q^{38} +4.44345 q^{39} -2.63704 q^{40} -9.53392 q^{41} +12.1761 q^{43} -3.26053 q^{44} -2.63704 q^{45} -1.98320 q^{46} -13.3321 q^{47} -1.00000 q^{48} +1.95397 q^{50} -2.36962 q^{51} -4.44345 q^{52} +7.24699 q^{53} -1.00000 q^{54} +8.59814 q^{55} +1.00000 q^{57} +0.0962338 q^{58} +1.31981 q^{59} +2.63704 q^{60} +5.52115 q^{61} +2.43285 q^{62} +1.00000 q^{64} +11.7175 q^{65} +3.26053 q^{66} +5.74748 q^{67} +2.36962 q^{68} +1.98320 q^{69} +11.1849 q^{71} +1.00000 q^{72} -4.16583 q^{73} -0.452007 q^{74} -1.95397 q^{75} -1.00000 q^{76} +4.44345 q^{78} -12.9020 q^{79} -2.63704 q^{80} +1.00000 q^{81} -9.53392 q^{82} -2.69233 q^{83} -6.24879 q^{85} +12.1761 q^{86} -0.0962338 q^{87} -3.26053 q^{88} -0.282443 q^{89} -2.63704 q^{90} -1.98320 q^{92} -2.43285 q^{93} -13.3321 q^{94} +2.63704 q^{95} -1.00000 q^{96} +3.67729 q^{97} -3.26053 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} - 4 q^{5} - 8 q^{6} + 8 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} - 4 q^{5} - 8 q^{6} + 8 q^{8} + 8 q^{9} - 4 q^{10} + 8 q^{11} - 8 q^{12} + 4 q^{13} + 4 q^{15} + 8 q^{16} - 8 q^{17} + 8 q^{18} - 8 q^{19} - 4 q^{20} + 8 q^{22} + 4 q^{23} - 8 q^{24} + 24 q^{25} + 4 q^{26} - 8 q^{27} + 16 q^{29} + 4 q^{30} - 4 q^{31} + 8 q^{32} - 8 q^{33} - 8 q^{34} + 8 q^{36} + 12 q^{37} - 8 q^{38} - 4 q^{39} - 4 q^{40} + 16 q^{43} + 8 q^{44} - 4 q^{45} + 4 q^{46} - 12 q^{47} - 8 q^{48} + 24 q^{50} + 8 q^{51} + 4 q^{52} + 24 q^{53} - 8 q^{54} + 8 q^{55} + 8 q^{57} + 16 q^{58} + 8 q^{59} + 4 q^{60} - 4 q^{62} + 8 q^{64} + 8 q^{65} - 8 q^{66} + 8 q^{67} - 8 q^{68} - 4 q^{69} + 24 q^{71} + 8 q^{72} + 16 q^{73} + 12 q^{74} - 24 q^{75} - 8 q^{76} - 4 q^{78} + 20 q^{79} - 4 q^{80} + 8 q^{81} - 16 q^{83} + 48 q^{85} + 16 q^{86} - 16 q^{87} + 8 q^{88} - 4 q^{90} + 4 q^{92} + 4 q^{93} - 12 q^{94} + 4 q^{95} - 8 q^{96} + 8 q^{97} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −2.63704 −1.17932 −0.589659 0.807652i \(-0.700738\pi\)
−0.589659 + 0.807652i \(0.700738\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −2.63704 −0.833904
\(11\) −3.26053 −0.983087 −0.491543 0.870853i \(-0.663567\pi\)
−0.491543 + 0.870853i \(0.663567\pi\)
\(12\) −1.00000 −0.288675
\(13\) −4.44345 −1.23239 −0.616196 0.787593i \(-0.711327\pi\)
−0.616196 + 0.787593i \(0.711327\pi\)
\(14\) 0 0
\(15\) 2.63704 0.680880
\(16\) 1.00000 0.250000
\(17\) 2.36962 0.574718 0.287359 0.957823i \(-0.407223\pi\)
0.287359 + 0.957823i \(0.407223\pi\)
\(18\) 1.00000 0.235702
\(19\) −1.00000 −0.229416
\(20\) −2.63704 −0.589659
\(21\) 0 0
\(22\) −3.26053 −0.695147
\(23\) −1.98320 −0.413526 −0.206763 0.978391i \(-0.566293\pi\)
−0.206763 + 0.978391i \(0.566293\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.95397 0.390793
\(26\) −4.44345 −0.871432
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 0.0962338 0.0178702 0.00893508 0.999960i \(-0.497156\pi\)
0.00893508 + 0.999960i \(0.497156\pi\)
\(30\) 2.63704 0.481455
\(31\) 2.43285 0.436953 0.218477 0.975842i \(-0.429891\pi\)
0.218477 + 0.975842i \(0.429891\pi\)
\(32\) 1.00000 0.176777
\(33\) 3.26053 0.567586
\(34\) 2.36962 0.406387
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −0.452007 −0.0743094 −0.0371547 0.999310i \(-0.511829\pi\)
−0.0371547 + 0.999310i \(0.511829\pi\)
\(38\) −1.00000 −0.162221
\(39\) 4.44345 0.711522
\(40\) −2.63704 −0.416952
\(41\) −9.53392 −1.48895 −0.744474 0.667651i \(-0.767300\pi\)
−0.744474 + 0.667651i \(0.767300\pi\)
\(42\) 0 0
\(43\) 12.1761 1.85684 0.928422 0.371528i \(-0.121166\pi\)
0.928422 + 0.371528i \(0.121166\pi\)
\(44\) −3.26053 −0.491543
\(45\) −2.63704 −0.393106
\(46\) −1.98320 −0.292407
\(47\) −13.3321 −1.94469 −0.972343 0.233558i \(-0.924963\pi\)
−0.972343 + 0.233558i \(0.924963\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) 1.95397 0.276333
\(51\) −2.36962 −0.331814
\(52\) −4.44345 −0.616196
\(53\) 7.24699 0.995450 0.497725 0.867335i \(-0.334169\pi\)
0.497725 + 0.867335i \(0.334169\pi\)
\(54\) −1.00000 −0.136083
\(55\) 8.59814 1.15937
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) 0.0962338 0.0126361
\(59\) 1.31981 0.171824 0.0859121 0.996303i \(-0.472620\pi\)
0.0859121 + 0.996303i \(0.472620\pi\)
\(60\) 2.63704 0.340440
\(61\) 5.52115 0.706910 0.353455 0.935451i \(-0.385007\pi\)
0.353455 + 0.935451i \(0.385007\pi\)
\(62\) 2.43285 0.308973
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 11.7175 1.45338
\(66\) 3.26053 0.401344
\(67\) 5.74748 0.702166 0.351083 0.936344i \(-0.385813\pi\)
0.351083 + 0.936344i \(0.385813\pi\)
\(68\) 2.36962 0.287359
\(69\) 1.98320 0.238750
\(70\) 0 0
\(71\) 11.1849 1.32740 0.663701 0.747998i \(-0.268985\pi\)
0.663701 + 0.747998i \(0.268985\pi\)
\(72\) 1.00000 0.117851
\(73\) −4.16583 −0.487573 −0.243787 0.969829i \(-0.578390\pi\)
−0.243787 + 0.969829i \(0.578390\pi\)
\(74\) −0.452007 −0.0525447
\(75\) −1.95397 −0.225625
\(76\) −1.00000 −0.114708
\(77\) 0 0
\(78\) 4.44345 0.503122
\(79\) −12.9020 −1.45159 −0.725796 0.687910i \(-0.758528\pi\)
−0.725796 + 0.687910i \(0.758528\pi\)
\(80\) −2.63704 −0.294830
\(81\) 1.00000 0.111111
\(82\) −9.53392 −1.05285
\(83\) −2.69233 −0.295522 −0.147761 0.989023i \(-0.547207\pi\)
−0.147761 + 0.989023i \(0.547207\pi\)
\(84\) 0 0
\(85\) −6.24879 −0.677776
\(86\) 12.1761 1.31299
\(87\) −0.0962338 −0.0103173
\(88\) −3.26053 −0.347574
\(89\) −0.282443 −0.0299388 −0.0149694 0.999888i \(-0.504765\pi\)
−0.0149694 + 0.999888i \(0.504765\pi\)
\(90\) −2.63704 −0.277968
\(91\) 0 0
\(92\) −1.98320 −0.206763
\(93\) −2.43285 −0.252275
\(94\) −13.3321 −1.37510
\(95\) 2.63704 0.270554
\(96\) −1.00000 −0.102062
\(97\) 3.67729 0.373372 0.186686 0.982420i \(-0.440225\pi\)
0.186686 + 0.982420i \(0.440225\pi\)
\(98\) 0 0
\(99\) −3.26053 −0.327696
\(100\) 1.95397 0.195397
\(101\) 13.0331 1.29685 0.648423 0.761280i \(-0.275429\pi\)
0.648423 + 0.761280i \(0.275429\pi\)
\(102\) −2.36962 −0.234628
\(103\) 11.7777 1.16050 0.580248 0.814440i \(-0.302956\pi\)
0.580248 + 0.814440i \(0.302956\pi\)
\(104\) −4.44345 −0.435716
\(105\) 0 0
\(106\) 7.24699 0.703890
\(107\) −2.34941 −0.227126 −0.113563 0.993531i \(-0.536226\pi\)
−0.113563 + 0.993531i \(0.536226\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 8.67251 0.830676 0.415338 0.909667i \(-0.363663\pi\)
0.415338 + 0.909667i \(0.363663\pi\)
\(110\) 8.59814 0.819801
\(111\) 0.452007 0.0429026
\(112\) 0 0
\(113\) 11.1344 1.04744 0.523718 0.851892i \(-0.324545\pi\)
0.523718 + 0.851892i \(0.324545\pi\)
\(114\) 1.00000 0.0936586
\(115\) 5.22978 0.487680
\(116\) 0.0962338 0.00893508
\(117\) −4.44345 −0.410797
\(118\) 1.31981 0.121498
\(119\) 0 0
\(120\) 2.63704 0.240727
\(121\) −0.368940 −0.0335400
\(122\) 5.52115 0.499861
\(123\) 9.53392 0.859645
\(124\) 2.43285 0.218477
\(125\) 8.03251 0.718449
\(126\) 0 0
\(127\) −11.6003 −1.02936 −0.514678 0.857383i \(-0.672089\pi\)
−0.514678 + 0.857383i \(0.672089\pi\)
\(128\) 1.00000 0.0883883
\(129\) −12.1761 −1.07205
\(130\) 11.7175 1.02770
\(131\) 10.3929 0.908032 0.454016 0.890994i \(-0.349991\pi\)
0.454016 + 0.890994i \(0.349991\pi\)
\(132\) 3.26053 0.283793
\(133\) 0 0
\(134\) 5.74748 0.496506
\(135\) 2.63704 0.226960
\(136\) 2.36962 0.203194
\(137\) −7.75165 −0.662268 −0.331134 0.943584i \(-0.607431\pi\)
−0.331134 + 0.943584i \(0.607431\pi\)
\(138\) 1.98320 0.168821
\(139\) 11.7319 0.995083 0.497542 0.867440i \(-0.334236\pi\)
0.497542 + 0.867440i \(0.334236\pi\)
\(140\) 0 0
\(141\) 13.3321 1.12276
\(142\) 11.1849 0.938616
\(143\) 14.4880 1.21155
\(144\) 1.00000 0.0833333
\(145\) −0.253772 −0.0210746
\(146\) −4.16583 −0.344766
\(147\) 0 0
\(148\) −0.452007 −0.0371547
\(149\) 0.858768 0.0703530 0.0351765 0.999381i \(-0.488801\pi\)
0.0351765 + 0.999381i \(0.488801\pi\)
\(150\) −1.95397 −0.159541
\(151\) 8.05804 0.655754 0.327877 0.944720i \(-0.393667\pi\)
0.327877 + 0.944720i \(0.393667\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 2.36962 0.191573
\(154\) 0 0
\(155\) −6.41552 −0.515307
\(156\) 4.44345 0.355761
\(157\) 9.13601 0.729133 0.364566 0.931177i \(-0.381217\pi\)
0.364566 + 0.931177i \(0.381217\pi\)
\(158\) −12.9020 −1.02643
\(159\) −7.24699 −0.574723
\(160\) −2.63704 −0.208476
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 8.30168 0.650238 0.325119 0.945673i \(-0.394596\pi\)
0.325119 + 0.945673i \(0.394596\pi\)
\(164\) −9.53392 −0.744474
\(165\) −8.59814 −0.669364
\(166\) −2.69233 −0.208965
\(167\) 1.27909 0.0989787 0.0494893 0.998775i \(-0.484241\pi\)
0.0494893 + 0.998775i \(0.484241\pi\)
\(168\) 0 0
\(169\) 6.74426 0.518789
\(170\) −6.24879 −0.479260
\(171\) −1.00000 −0.0764719
\(172\) 12.1761 0.928422
\(173\) −22.0420 −1.67583 −0.837913 0.545804i \(-0.816224\pi\)
−0.837913 + 0.545804i \(0.816224\pi\)
\(174\) −0.0962338 −0.00729547
\(175\) 0 0
\(176\) −3.26053 −0.245772
\(177\) −1.31981 −0.0992027
\(178\) −0.282443 −0.0211700
\(179\) −0.655972 −0.0490297 −0.0245148 0.999699i \(-0.507804\pi\)
−0.0245148 + 0.999699i \(0.507804\pi\)
\(180\) −2.63704 −0.196553
\(181\) 0.349942 0.0260110 0.0130055 0.999915i \(-0.495860\pi\)
0.0130055 + 0.999915i \(0.495860\pi\)
\(182\) 0 0
\(183\) −5.52115 −0.408135
\(184\) −1.98320 −0.146204
\(185\) 1.19196 0.0876345
\(186\) −2.43285 −0.178385
\(187\) −7.72623 −0.564998
\(188\) −13.3321 −0.972343
\(189\) 0 0
\(190\) 2.63704 0.191311
\(191\) −5.30321 −0.383727 −0.191863 0.981422i \(-0.561453\pi\)
−0.191863 + 0.981422i \(0.561453\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −0.418560 −0.0301286 −0.0150643 0.999887i \(-0.504795\pi\)
−0.0150643 + 0.999887i \(0.504795\pi\)
\(194\) 3.67729 0.264014
\(195\) −11.7175 −0.839111
\(196\) 0 0
\(197\) 25.6356 1.82646 0.913230 0.407445i \(-0.133580\pi\)
0.913230 + 0.407445i \(0.133580\pi\)
\(198\) −3.26053 −0.231716
\(199\) 27.6516 1.96017 0.980084 0.198583i \(-0.0636341\pi\)
0.980084 + 0.198583i \(0.0636341\pi\)
\(200\) 1.95397 0.138166
\(201\) −5.74748 −0.405396
\(202\) 13.0331 0.917009
\(203\) 0 0
\(204\) −2.36962 −0.165907
\(205\) 25.1413 1.75594
\(206\) 11.7777 0.820594
\(207\) −1.98320 −0.137842
\(208\) −4.44345 −0.308098
\(209\) 3.26053 0.225536
\(210\) 0 0
\(211\) −9.66873 −0.665623 −0.332811 0.942993i \(-0.607997\pi\)
−0.332811 + 0.942993i \(0.607997\pi\)
\(212\) 7.24699 0.497725
\(213\) −11.1849 −0.766376
\(214\) −2.34941 −0.160603
\(215\) −32.1089 −2.18981
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 8.67251 0.587376
\(219\) 4.16583 0.281501
\(220\) 8.59814 0.579687
\(221\) −10.5293 −0.708278
\(222\) 0.452007 0.0303367
\(223\) −0.367428 −0.0246048 −0.0123024 0.999924i \(-0.503916\pi\)
−0.0123024 + 0.999924i \(0.503916\pi\)
\(224\) 0 0
\(225\) 1.95397 0.130264
\(226\) 11.1344 0.740649
\(227\) 15.9079 1.05584 0.527921 0.849294i \(-0.322972\pi\)
0.527921 + 0.849294i \(0.322972\pi\)
\(228\) 1.00000 0.0662266
\(229\) −11.3895 −0.752640 −0.376320 0.926490i \(-0.622811\pi\)
−0.376320 + 0.926490i \(0.622811\pi\)
\(230\) 5.22978 0.344842
\(231\) 0 0
\(232\) 0.0962338 0.00631806
\(233\) −3.01687 −0.197642 −0.0988210 0.995105i \(-0.531507\pi\)
−0.0988210 + 0.995105i \(0.531507\pi\)
\(234\) −4.44345 −0.290477
\(235\) 35.1572 2.29340
\(236\) 1.31981 0.0859121
\(237\) 12.9020 0.838077
\(238\) 0 0
\(239\) −28.2992 −1.83052 −0.915261 0.402861i \(-0.868016\pi\)
−0.915261 + 0.402861i \(0.868016\pi\)
\(240\) 2.63704 0.170220
\(241\) 21.4909 1.38435 0.692174 0.721731i \(-0.256653\pi\)
0.692174 + 0.721731i \(0.256653\pi\)
\(242\) −0.368940 −0.0237163
\(243\) −1.00000 −0.0641500
\(244\) 5.52115 0.353455
\(245\) 0 0
\(246\) 9.53392 0.607861
\(247\) 4.44345 0.282730
\(248\) 2.43285 0.154486
\(249\) 2.69233 0.170620
\(250\) 8.03251 0.508020
\(251\) 17.1974 1.08549 0.542744 0.839898i \(-0.317385\pi\)
0.542744 + 0.839898i \(0.317385\pi\)
\(252\) 0 0
\(253\) 6.46630 0.406533
\(254\) −11.6003 −0.727865
\(255\) 6.24879 0.391314
\(256\) 1.00000 0.0625000
\(257\) 12.5578 0.783335 0.391667 0.920107i \(-0.371898\pi\)
0.391667 + 0.920107i \(0.371898\pi\)
\(258\) −12.1761 −0.758053
\(259\) 0 0
\(260\) 11.7175 0.726691
\(261\) 0.0962338 0.00595672
\(262\) 10.3929 0.642076
\(263\) −10.6077 −0.654098 −0.327049 0.945007i \(-0.606054\pi\)
−0.327049 + 0.945007i \(0.606054\pi\)
\(264\) 3.26053 0.200672
\(265\) −19.1106 −1.17395
\(266\) 0 0
\(267\) 0.282443 0.0172852
\(268\) 5.74748 0.351083
\(269\) 26.0968 1.59115 0.795576 0.605854i \(-0.207168\pi\)
0.795576 + 0.605854i \(0.207168\pi\)
\(270\) 2.63704 0.160485
\(271\) −12.2869 −0.746377 −0.373189 0.927755i \(-0.621736\pi\)
−0.373189 + 0.927755i \(0.621736\pi\)
\(272\) 2.36962 0.143680
\(273\) 0 0
\(274\) −7.75165 −0.468294
\(275\) −6.37097 −0.384184
\(276\) 1.98320 0.119375
\(277\) 19.7258 1.18521 0.592603 0.805495i \(-0.298100\pi\)
0.592603 + 0.805495i \(0.298100\pi\)
\(278\) 11.7319 0.703630
\(279\) 2.43285 0.145651
\(280\) 0 0
\(281\) 1.56281 0.0932292 0.0466146 0.998913i \(-0.485157\pi\)
0.0466146 + 0.998913i \(0.485157\pi\)
\(282\) 13.3321 0.793915
\(283\) 26.3439 1.56598 0.782990 0.622034i \(-0.213694\pi\)
0.782990 + 0.622034i \(0.213694\pi\)
\(284\) 11.1849 0.663701
\(285\) −2.63704 −0.156205
\(286\) 14.4880 0.856694
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −11.3849 −0.669699
\(290\) −0.253772 −0.0149020
\(291\) −3.67729 −0.215567
\(292\) −4.16583 −0.243787
\(293\) −10.6076 −0.619700 −0.309850 0.950785i \(-0.600279\pi\)
−0.309850 + 0.950785i \(0.600279\pi\)
\(294\) 0 0
\(295\) −3.48038 −0.202636
\(296\) −0.452007 −0.0262723
\(297\) 3.26053 0.189195
\(298\) 0.858768 0.0497471
\(299\) 8.81227 0.509627
\(300\) −1.95397 −0.112812
\(301\) 0 0
\(302\) 8.05804 0.463688
\(303\) −13.0331 −0.748734
\(304\) −1.00000 −0.0573539
\(305\) −14.5595 −0.833673
\(306\) 2.36962 0.135462
\(307\) 16.2375 0.926723 0.463362 0.886169i \(-0.346643\pi\)
0.463362 + 0.886169i \(0.346643\pi\)
\(308\) 0 0
\(309\) −11.7777 −0.670012
\(310\) −6.41552 −0.364377
\(311\) −23.0812 −1.30881 −0.654406 0.756143i \(-0.727081\pi\)
−0.654406 + 0.756143i \(0.727081\pi\)
\(312\) 4.44345 0.251561
\(313\) −13.2774 −0.750485 −0.375243 0.926927i \(-0.622441\pi\)
−0.375243 + 0.926927i \(0.622441\pi\)
\(314\) 9.13601 0.515575
\(315\) 0 0
\(316\) −12.9020 −0.725796
\(317\) −13.3284 −0.748597 −0.374298 0.927308i \(-0.622116\pi\)
−0.374298 + 0.927308i \(0.622116\pi\)
\(318\) −7.24699 −0.406391
\(319\) −0.313773 −0.0175679
\(320\) −2.63704 −0.147415
\(321\) 2.34941 0.131131
\(322\) 0 0
\(323\) −2.36962 −0.131849
\(324\) 1.00000 0.0555556
\(325\) −8.68235 −0.481610
\(326\) 8.30168 0.459787
\(327\) −8.67251 −0.479591
\(328\) −9.53392 −0.526423
\(329\) 0 0
\(330\) −8.59814 −0.473312
\(331\) −28.1155 −1.54537 −0.772685 0.634790i \(-0.781087\pi\)
−0.772685 + 0.634790i \(0.781087\pi\)
\(332\) −2.69233 −0.147761
\(333\) −0.452007 −0.0247698
\(334\) 1.27909 0.0699885
\(335\) −15.1563 −0.828078
\(336\) 0 0
\(337\) 30.5947 1.66660 0.833301 0.552820i \(-0.186448\pi\)
0.833301 + 0.552820i \(0.186448\pi\)
\(338\) 6.74426 0.366839
\(339\) −11.1344 −0.604737
\(340\) −6.24879 −0.338888
\(341\) −7.93239 −0.429563
\(342\) −1.00000 −0.0540738
\(343\) 0 0
\(344\) 12.1761 0.656493
\(345\) −5.22978 −0.281562
\(346\) −22.0420 −1.18499
\(347\) −31.0492 −1.66681 −0.833403 0.552666i \(-0.813611\pi\)
−0.833403 + 0.552666i \(0.813611\pi\)
\(348\) −0.0962338 −0.00515867
\(349\) 14.0808 0.753727 0.376863 0.926269i \(-0.377003\pi\)
0.376863 + 0.926269i \(0.377003\pi\)
\(350\) 0 0
\(351\) 4.44345 0.237174
\(352\) −3.26053 −0.173787
\(353\) −1.99562 −0.106216 −0.0531082 0.998589i \(-0.516913\pi\)
−0.0531082 + 0.998589i \(0.516913\pi\)
\(354\) −1.31981 −0.0701469
\(355\) −29.4950 −1.56543
\(356\) −0.282443 −0.0149694
\(357\) 0 0
\(358\) −0.655972 −0.0346692
\(359\) 18.6935 0.986603 0.493301 0.869858i \(-0.335790\pi\)
0.493301 + 0.869858i \(0.335790\pi\)
\(360\) −2.63704 −0.138984
\(361\) 1.00000 0.0526316
\(362\) 0.349942 0.0183926
\(363\) 0.368940 0.0193643
\(364\) 0 0
\(365\) 10.9854 0.575004
\(366\) −5.52115 −0.288595
\(367\) 10.8045 0.563989 0.281994 0.959416i \(-0.409004\pi\)
0.281994 + 0.959416i \(0.409004\pi\)
\(368\) −1.98320 −0.103382
\(369\) −9.53392 −0.496316
\(370\) 1.19196 0.0619670
\(371\) 0 0
\(372\) −2.43285 −0.126138
\(373\) 6.33223 0.327870 0.163935 0.986471i \(-0.447581\pi\)
0.163935 + 0.986471i \(0.447581\pi\)
\(374\) −7.72623 −0.399514
\(375\) −8.03251 −0.414797
\(376\) −13.3321 −0.687550
\(377\) −0.427610 −0.0220230
\(378\) 0 0
\(379\) 15.1117 0.776237 0.388119 0.921609i \(-0.373125\pi\)
0.388119 + 0.921609i \(0.373125\pi\)
\(380\) 2.63704 0.135277
\(381\) 11.6003 0.594299
\(382\) −5.30321 −0.271336
\(383\) −7.15839 −0.365777 −0.182888 0.983134i \(-0.558545\pi\)
−0.182888 + 0.983134i \(0.558545\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −0.418560 −0.0213041
\(387\) 12.1761 0.618948
\(388\) 3.67729 0.186686
\(389\) −19.6296 −0.995261 −0.497631 0.867389i \(-0.665796\pi\)
−0.497631 + 0.867389i \(0.665796\pi\)
\(390\) −11.7175 −0.593341
\(391\) −4.69945 −0.237661
\(392\) 0 0
\(393\) −10.3929 −0.524252
\(394\) 25.6356 1.29150
\(395\) 34.0231 1.71189
\(396\) −3.26053 −0.163848
\(397\) 3.05488 0.153320 0.0766599 0.997057i \(-0.475574\pi\)
0.0766599 + 0.997057i \(0.475574\pi\)
\(398\) 27.6516 1.38605
\(399\) 0 0
\(400\) 1.95397 0.0976983
\(401\) 27.6978 1.38316 0.691581 0.722299i \(-0.256915\pi\)
0.691581 + 0.722299i \(0.256915\pi\)
\(402\) −5.74748 −0.286658
\(403\) −10.8103 −0.538497
\(404\) 13.0331 0.648423
\(405\) −2.63704 −0.131035
\(406\) 0 0
\(407\) 1.47378 0.0730526
\(408\) −2.36962 −0.117314
\(409\) 19.5395 0.966166 0.483083 0.875575i \(-0.339517\pi\)
0.483083 + 0.875575i \(0.339517\pi\)
\(410\) 25.1413 1.24164
\(411\) 7.75165 0.382361
\(412\) 11.7777 0.580248
\(413\) 0 0
\(414\) −1.98320 −0.0974691
\(415\) 7.09978 0.348514
\(416\) −4.44345 −0.217858
\(417\) −11.7319 −0.574512
\(418\) 3.26053 0.159478
\(419\) 19.3022 0.942972 0.471486 0.881874i \(-0.343718\pi\)
0.471486 + 0.881874i \(0.343718\pi\)
\(420\) 0 0
\(421\) 5.33795 0.260156 0.130078 0.991504i \(-0.458477\pi\)
0.130078 + 0.991504i \(0.458477\pi\)
\(422\) −9.66873 −0.470667
\(423\) −13.3321 −0.648229
\(424\) 7.24699 0.351945
\(425\) 4.63016 0.224596
\(426\) −11.1849 −0.541910
\(427\) 0 0
\(428\) −2.34941 −0.113563
\(429\) −14.4880 −0.699488
\(430\) −32.1089 −1.54843
\(431\) 8.06973 0.388705 0.194353 0.980932i \(-0.437739\pi\)
0.194353 + 0.980932i \(0.437739\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −0.207726 −0.00998266 −0.00499133 0.999988i \(-0.501589\pi\)
−0.00499133 + 0.999988i \(0.501589\pi\)
\(434\) 0 0
\(435\) 0.253772 0.0121674
\(436\) 8.67251 0.415338
\(437\) 1.98320 0.0948695
\(438\) 4.16583 0.199051
\(439\) 28.9616 1.38226 0.691131 0.722729i \(-0.257113\pi\)
0.691131 + 0.722729i \(0.257113\pi\)
\(440\) 8.59814 0.409900
\(441\) 0 0
\(442\) −10.5293 −0.500828
\(443\) −5.50387 −0.261497 −0.130748 0.991416i \(-0.541738\pi\)
−0.130748 + 0.991416i \(0.541738\pi\)
\(444\) 0.452007 0.0214513
\(445\) 0.744811 0.0353074
\(446\) −0.367428 −0.0173982
\(447\) −0.858768 −0.0406183
\(448\) 0 0
\(449\) 0.546522 0.0257920 0.0128960 0.999917i \(-0.495895\pi\)
0.0128960 + 0.999917i \(0.495895\pi\)
\(450\) 1.95397 0.0921108
\(451\) 31.0856 1.46377
\(452\) 11.1344 0.523718
\(453\) −8.05804 −0.378600
\(454\) 15.9079 0.746593
\(455\) 0 0
\(456\) 1.00000 0.0468293
\(457\) 38.6840 1.80956 0.904781 0.425878i \(-0.140035\pi\)
0.904781 + 0.425878i \(0.140035\pi\)
\(458\) −11.3895 −0.532197
\(459\) −2.36962 −0.110605
\(460\) 5.22978 0.243840
\(461\) −29.2112 −1.36050 −0.680250 0.732980i \(-0.738129\pi\)
−0.680250 + 0.732980i \(0.738129\pi\)
\(462\) 0 0
\(463\) −21.1418 −0.982544 −0.491272 0.871006i \(-0.663468\pi\)
−0.491272 + 0.871006i \(0.663468\pi\)
\(464\) 0.0962338 0.00446754
\(465\) 6.41552 0.297513
\(466\) −3.01687 −0.139754
\(467\) 20.4481 0.946224 0.473112 0.881002i \(-0.343131\pi\)
0.473112 + 0.881002i \(0.343131\pi\)
\(468\) −4.44345 −0.205399
\(469\) 0 0
\(470\) 35.1572 1.62168
\(471\) −9.13601 −0.420965
\(472\) 1.31981 0.0607490
\(473\) −39.7007 −1.82544
\(474\) 12.9020 0.592610
\(475\) −1.95397 −0.0896541
\(476\) 0 0
\(477\) 7.24699 0.331817
\(478\) −28.2992 −1.29437
\(479\) −23.9563 −1.09459 −0.547296 0.836939i \(-0.684343\pi\)
−0.547296 + 0.836939i \(0.684343\pi\)
\(480\) 2.63704 0.120364
\(481\) 2.00847 0.0915783
\(482\) 21.4909 0.978882
\(483\) 0 0
\(484\) −0.368940 −0.0167700
\(485\) −9.69716 −0.440325
\(486\) −1.00000 −0.0453609
\(487\) 14.0906 0.638507 0.319253 0.947669i \(-0.396568\pi\)
0.319253 + 0.947669i \(0.396568\pi\)
\(488\) 5.52115 0.249931
\(489\) −8.30168 −0.375415
\(490\) 0 0
\(491\) 12.3018 0.555174 0.277587 0.960700i \(-0.410465\pi\)
0.277587 + 0.960700i \(0.410465\pi\)
\(492\) 9.53392 0.429822
\(493\) 0.228038 0.0102703
\(494\) 4.44345 0.199920
\(495\) 8.59814 0.386458
\(496\) 2.43285 0.109238
\(497\) 0 0
\(498\) 2.69233 0.120646
\(499\) 5.60115 0.250742 0.125371 0.992110i \(-0.459988\pi\)
0.125371 + 0.992110i \(0.459988\pi\)
\(500\) 8.03251 0.359225
\(501\) −1.27909 −0.0571454
\(502\) 17.1974 0.767556
\(503\) 2.52742 0.112692 0.0563461 0.998411i \(-0.482055\pi\)
0.0563461 + 0.998411i \(0.482055\pi\)
\(504\) 0 0
\(505\) −34.3689 −1.52939
\(506\) 6.46630 0.287462
\(507\) −6.74426 −0.299523
\(508\) −11.6003 −0.514678
\(509\) −18.1208 −0.803190 −0.401595 0.915817i \(-0.631544\pi\)
−0.401595 + 0.915817i \(0.631544\pi\)
\(510\) 6.24879 0.276701
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 1.00000 0.0441511
\(514\) 12.5578 0.553901
\(515\) −31.0583 −1.36859
\(516\) −12.1761 −0.536025
\(517\) 43.4697 1.91180
\(518\) 0 0
\(519\) 22.0420 0.967538
\(520\) 11.7175 0.513848
\(521\) −23.0907 −1.01162 −0.505812 0.862644i \(-0.668807\pi\)
−0.505812 + 0.862644i \(0.668807\pi\)
\(522\) 0.0962338 0.00421204
\(523\) 21.0628 0.921011 0.460505 0.887657i \(-0.347668\pi\)
0.460505 + 0.887657i \(0.347668\pi\)
\(524\) 10.3929 0.454016
\(525\) 0 0
\(526\) −10.6077 −0.462517
\(527\) 5.76495 0.251125
\(528\) 3.26053 0.141896
\(529\) −19.0669 −0.828996
\(530\) −19.1106 −0.830110
\(531\) 1.31981 0.0572747
\(532\) 0 0
\(533\) 42.3635 1.83497
\(534\) 0.282443 0.0122225
\(535\) 6.19549 0.267854
\(536\) 5.74748 0.248253
\(537\) 0.655972 0.0283073
\(538\) 26.0968 1.12511
\(539\) 0 0
\(540\) 2.63704 0.113480
\(541\) 34.2111 1.47085 0.735425 0.677606i \(-0.236983\pi\)
0.735425 + 0.677606i \(0.236983\pi\)
\(542\) −12.2869 −0.527769
\(543\) −0.349942 −0.0150175
\(544\) 2.36962 0.101597
\(545\) −22.8697 −0.979632
\(546\) 0 0
\(547\) −9.99258 −0.427252 −0.213626 0.976916i \(-0.568527\pi\)
−0.213626 + 0.976916i \(0.568527\pi\)
\(548\) −7.75165 −0.331134
\(549\) 5.52115 0.235637
\(550\) −6.37097 −0.271659
\(551\) −0.0962338 −0.00409970
\(552\) 1.98320 0.0844107
\(553\) 0 0
\(554\) 19.7258 0.838067
\(555\) −1.19196 −0.0505958
\(556\) 11.7319 0.497542
\(557\) −36.7882 −1.55877 −0.779384 0.626546i \(-0.784468\pi\)
−0.779384 + 0.626546i \(0.784468\pi\)
\(558\) 2.43285 0.102991
\(559\) −54.1041 −2.28836
\(560\) 0 0
\(561\) 7.72623 0.326202
\(562\) 1.56281 0.0659230
\(563\) −26.7311 −1.12658 −0.563291 0.826259i \(-0.690465\pi\)
−0.563291 + 0.826259i \(0.690465\pi\)
\(564\) 13.3321 0.561382
\(565\) −29.3618 −1.23526
\(566\) 26.3439 1.10732
\(567\) 0 0
\(568\) 11.1849 0.469308
\(569\) −5.65805 −0.237198 −0.118599 0.992942i \(-0.537840\pi\)
−0.118599 + 0.992942i \(0.537840\pi\)
\(570\) −2.63704 −0.110453
\(571\) −13.8124 −0.578030 −0.289015 0.957325i \(-0.593328\pi\)
−0.289015 + 0.957325i \(0.593328\pi\)
\(572\) 14.4880 0.605774
\(573\) 5.30321 0.221545
\(574\) 0 0
\(575\) −3.87511 −0.161603
\(576\) 1.00000 0.0416667
\(577\) −21.2759 −0.885727 −0.442864 0.896589i \(-0.646037\pi\)
−0.442864 + 0.896589i \(0.646037\pi\)
\(578\) −11.3849 −0.473549
\(579\) 0.418560 0.0173947
\(580\) −0.253772 −0.0105373
\(581\) 0 0
\(582\) −3.67729 −0.152429
\(583\) −23.6290 −0.978614
\(584\) −4.16583 −0.172383
\(585\) 11.7175 0.484461
\(586\) −10.6076 −0.438194
\(587\) −2.14686 −0.0886103 −0.0443052 0.999018i \(-0.514107\pi\)
−0.0443052 + 0.999018i \(0.514107\pi\)
\(588\) 0 0
\(589\) −2.43285 −0.100244
\(590\) −3.48038 −0.143285
\(591\) −25.6356 −1.05451
\(592\) −0.452007 −0.0185774
\(593\) −29.3692 −1.20605 −0.603025 0.797723i \(-0.706038\pi\)
−0.603025 + 0.797723i \(0.706038\pi\)
\(594\) 3.26053 0.133781
\(595\) 0 0
\(596\) 0.858768 0.0351765
\(597\) −27.6516 −1.13170
\(598\) 8.81227 0.360360
\(599\) 10.9089 0.445725 0.222863 0.974850i \(-0.428460\pi\)
0.222863 + 0.974850i \(0.428460\pi\)
\(600\) −1.95397 −0.0797703
\(601\) 6.34523 0.258827 0.129414 0.991591i \(-0.458690\pi\)
0.129414 + 0.991591i \(0.458690\pi\)
\(602\) 0 0
\(603\) 5.74748 0.234055
\(604\) 8.05804 0.327877
\(605\) 0.972907 0.0395543
\(606\) −13.0331 −0.529435
\(607\) −43.4796 −1.76478 −0.882392 0.470516i \(-0.844068\pi\)
−0.882392 + 0.470516i \(0.844068\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 0 0
\(610\) −14.5595 −0.589496
\(611\) 59.2405 2.39661
\(612\) 2.36962 0.0957864
\(613\) 5.58771 0.225685 0.112843 0.993613i \(-0.464004\pi\)
0.112843 + 0.993613i \(0.464004\pi\)
\(614\) 16.2375 0.655292
\(615\) −25.1413 −1.01380
\(616\) 0 0
\(617\) 17.4172 0.701190 0.350595 0.936527i \(-0.385979\pi\)
0.350595 + 0.936527i \(0.385979\pi\)
\(618\) −11.7777 −0.473770
\(619\) 16.0433 0.644834 0.322417 0.946598i \(-0.395505\pi\)
0.322417 + 0.946598i \(0.395505\pi\)
\(620\) −6.41552 −0.257654
\(621\) 1.98320 0.0795832
\(622\) −23.0812 −0.925470
\(623\) 0 0
\(624\) 4.44345 0.177880
\(625\) −30.9518 −1.23807
\(626\) −13.2774 −0.530673
\(627\) −3.26053 −0.130213
\(628\) 9.13601 0.364566
\(629\) −1.07109 −0.0427070
\(630\) 0 0
\(631\) −12.0407 −0.479332 −0.239666 0.970855i \(-0.577038\pi\)
−0.239666 + 0.970855i \(0.577038\pi\)
\(632\) −12.9020 −0.513215
\(633\) 9.66873 0.384298
\(634\) −13.3284 −0.529338
\(635\) 30.5903 1.21394
\(636\) −7.24699 −0.287362
\(637\) 0 0
\(638\) −0.313773 −0.0124224
\(639\) 11.1849 0.442468
\(640\) −2.63704 −0.104238
\(641\) 18.7444 0.740358 0.370179 0.928961i \(-0.379296\pi\)
0.370179 + 0.928961i \(0.379296\pi\)
\(642\) 2.34941 0.0927240
\(643\) −1.01673 −0.0400960 −0.0200480 0.999799i \(-0.506382\pi\)
−0.0200480 + 0.999799i \(0.506382\pi\)
\(644\) 0 0
\(645\) 32.1089 1.26429
\(646\) −2.36962 −0.0932316
\(647\) −47.7699 −1.87803 −0.939014 0.343880i \(-0.888259\pi\)
−0.939014 + 0.343880i \(0.888259\pi\)
\(648\) 1.00000 0.0392837
\(649\) −4.30327 −0.168918
\(650\) −8.68235 −0.340550
\(651\) 0 0
\(652\) 8.30168 0.325119
\(653\) 35.2589 1.37979 0.689893 0.723911i \(-0.257658\pi\)
0.689893 + 0.723911i \(0.257658\pi\)
\(654\) −8.67251 −0.339122
\(655\) −27.4065 −1.07086
\(656\) −9.53392 −0.372237
\(657\) −4.16583 −0.162524
\(658\) 0 0
\(659\) −45.3855 −1.76797 −0.883985 0.467516i \(-0.845149\pi\)
−0.883985 + 0.467516i \(0.845149\pi\)
\(660\) −8.59814 −0.334682
\(661\) −10.1740 −0.395723 −0.197862 0.980230i \(-0.563400\pi\)
−0.197862 + 0.980230i \(0.563400\pi\)
\(662\) −28.1155 −1.09274
\(663\) 10.5293 0.408924
\(664\) −2.69233 −0.104483
\(665\) 0 0
\(666\) −0.452007 −0.0175149
\(667\) −0.190851 −0.00738979
\(668\) 1.27909 0.0494893
\(669\) 0.367428 0.0142056
\(670\) −15.1563 −0.585539
\(671\) −18.0019 −0.694955
\(672\) 0 0
\(673\) −29.3657 −1.13196 −0.565982 0.824417i \(-0.691503\pi\)
−0.565982 + 0.824417i \(0.691503\pi\)
\(674\) 30.5947 1.17846
\(675\) −1.95397 −0.0752082
\(676\) 6.74426 0.259395
\(677\) 5.24950 0.201755 0.100877 0.994899i \(-0.467835\pi\)
0.100877 + 0.994899i \(0.467835\pi\)
\(678\) −11.1344 −0.427614
\(679\) 0 0
\(680\) −6.24879 −0.239630
\(681\) −15.9079 −0.609590
\(682\) −7.93239 −0.303747
\(683\) 10.8055 0.413462 0.206731 0.978398i \(-0.433718\pi\)
0.206731 + 0.978398i \(0.433718\pi\)
\(684\) −1.00000 −0.0382360
\(685\) 20.4414 0.781025
\(686\) 0 0
\(687\) 11.3895 0.434537
\(688\) 12.1761 0.464211
\(689\) −32.2016 −1.22678
\(690\) −5.22978 −0.199094
\(691\) −34.2368 −1.30243 −0.651215 0.758893i \(-0.725740\pi\)
−0.651215 + 0.758893i \(0.725740\pi\)
\(692\) −22.0420 −0.837913
\(693\) 0 0
\(694\) −31.0492 −1.17861
\(695\) −30.9374 −1.17352
\(696\) −0.0962338 −0.00364773
\(697\) −22.5918 −0.855726
\(698\) 14.0808 0.532965
\(699\) 3.01687 0.114109
\(700\) 0 0
\(701\) 36.0826 1.36282 0.681411 0.731901i \(-0.261367\pi\)
0.681411 + 0.731901i \(0.261367\pi\)
\(702\) 4.44345 0.167707
\(703\) 0.452007 0.0170478
\(704\) −3.26053 −0.122886
\(705\) −35.1572 −1.32410
\(706\) −1.99562 −0.0751063
\(707\) 0 0
\(708\) −1.31981 −0.0496014
\(709\) −28.6794 −1.07708 −0.538539 0.842601i \(-0.681024\pi\)
−0.538539 + 0.842601i \(0.681024\pi\)
\(710\) −29.4950 −1.10693
\(711\) −12.9020 −0.483864
\(712\) −0.282443 −0.0105850
\(713\) −4.82484 −0.180692
\(714\) 0 0
\(715\) −38.2054 −1.42880
\(716\) −0.655972 −0.0245148
\(717\) 28.2992 1.05685
\(718\) 18.6935 0.697634
\(719\) 3.74972 0.139841 0.0699205 0.997553i \(-0.477725\pi\)
0.0699205 + 0.997553i \(0.477725\pi\)
\(720\) −2.63704 −0.0982766
\(721\) 0 0
\(722\) 1.00000 0.0372161
\(723\) −21.4909 −0.799254
\(724\) 0.349942 0.0130055
\(725\) 0.188038 0.00698354
\(726\) 0.368940 0.0136926
\(727\) 52.4402 1.94490 0.972450 0.233111i \(-0.0748904\pi\)
0.972450 + 0.233111i \(0.0748904\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 10.9854 0.406589
\(731\) 28.8529 1.06716
\(732\) −5.52115 −0.204067
\(733\) 33.4542 1.23566 0.617830 0.786312i \(-0.288012\pi\)
0.617830 + 0.786312i \(0.288012\pi\)
\(734\) 10.8045 0.398800
\(735\) 0 0
\(736\) −1.98320 −0.0731018
\(737\) −18.7398 −0.690290
\(738\) −9.53392 −0.350948
\(739\) 45.5330 1.67496 0.837479 0.546469i \(-0.184028\pi\)
0.837479 + 0.546469i \(0.184028\pi\)
\(740\) 1.19196 0.0438173
\(741\) −4.44345 −0.163234
\(742\) 0 0
\(743\) −27.9381 −1.02495 −0.512474 0.858703i \(-0.671271\pi\)
−0.512474 + 0.858703i \(0.671271\pi\)
\(744\) −2.43285 −0.0891927
\(745\) −2.26460 −0.0829686
\(746\) 6.33223 0.231839
\(747\) −2.69233 −0.0985073
\(748\) −7.72623 −0.282499
\(749\) 0 0
\(750\) −8.03251 −0.293306
\(751\) −34.3955 −1.25511 −0.627554 0.778573i \(-0.715944\pi\)
−0.627554 + 0.778573i \(0.715944\pi\)
\(752\) −13.3321 −0.486171
\(753\) −17.1974 −0.626707
\(754\) −0.427610 −0.0155726
\(755\) −21.2493 −0.773343
\(756\) 0 0
\(757\) −21.4913 −0.781113 −0.390557 0.920579i \(-0.627717\pi\)
−0.390557 + 0.920579i \(0.627717\pi\)
\(758\) 15.1117 0.548882
\(759\) −6.46630 −0.234712
\(760\) 2.63704 0.0956554
\(761\) 17.5599 0.636545 0.318272 0.947999i \(-0.396897\pi\)
0.318272 + 0.947999i \(0.396897\pi\)
\(762\) 11.6003 0.420233
\(763\) 0 0
\(764\) −5.30321 −0.191863
\(765\) −6.24879 −0.225925
\(766\) −7.15839 −0.258643
\(767\) −5.86450 −0.211755
\(768\) −1.00000 −0.0360844
\(769\) 44.6752 1.61103 0.805515 0.592575i \(-0.201889\pi\)
0.805515 + 0.592575i \(0.201889\pi\)
\(770\) 0 0
\(771\) −12.5578 −0.452258
\(772\) −0.418560 −0.0150643
\(773\) −30.7531 −1.10611 −0.553057 0.833143i \(-0.686539\pi\)
−0.553057 + 0.833143i \(0.686539\pi\)
\(774\) 12.1761 0.437662
\(775\) 4.75371 0.170758
\(776\) 3.67729 0.132007
\(777\) 0 0
\(778\) −19.6296 −0.703756
\(779\) 9.53392 0.341588
\(780\) −11.7175 −0.419555
\(781\) −36.4687 −1.30495
\(782\) −4.69945 −0.168052
\(783\) −0.0962338 −0.00343912
\(784\) 0 0
\(785\) −24.0920 −0.859880
\(786\) −10.3929 −0.370702
\(787\) −24.3598 −0.868333 −0.434167 0.900833i \(-0.642957\pi\)
−0.434167 + 0.900833i \(0.642957\pi\)
\(788\) 25.6356 0.913230
\(789\) 10.6077 0.377643
\(790\) 34.0231 1.21049
\(791\) 0 0
\(792\) −3.26053 −0.115858
\(793\) −24.5329 −0.871191
\(794\) 3.05488 0.108414
\(795\) 19.1106 0.677782
\(796\) 27.6516 0.980084
\(797\) −14.4957 −0.513465 −0.256732 0.966483i \(-0.582646\pi\)
−0.256732 + 0.966483i \(0.582646\pi\)
\(798\) 0 0
\(799\) −31.5920 −1.11765
\(800\) 1.95397 0.0690831
\(801\) −0.282443 −0.00997962
\(802\) 27.6978 0.978043
\(803\) 13.5828 0.479327
\(804\) −5.74748 −0.202698
\(805\) 0 0
\(806\) −10.8103 −0.380775
\(807\) −26.0968 −0.918652
\(808\) 13.0331 0.458504
\(809\) 7.30039 0.256668 0.128334 0.991731i \(-0.459037\pi\)
0.128334 + 0.991731i \(0.459037\pi\)
\(810\) −2.63704 −0.0926560
\(811\) −48.0207 −1.68624 −0.843118 0.537729i \(-0.819283\pi\)
−0.843118 + 0.537729i \(0.819283\pi\)
\(812\) 0 0
\(813\) 12.2869 0.430921
\(814\) 1.47378 0.0516560
\(815\) −21.8918 −0.766838
\(816\) −2.36962 −0.0829534
\(817\) −12.1761 −0.425989
\(818\) 19.5395 0.683183
\(819\) 0 0
\(820\) 25.1413 0.877972
\(821\) −55.0416 −1.92096 −0.960482 0.278342i \(-0.910215\pi\)
−0.960482 + 0.278342i \(0.910215\pi\)
\(822\) 7.75165 0.270370
\(823\) −4.50808 −0.157142 −0.0785709 0.996909i \(-0.525036\pi\)
−0.0785709 + 0.996909i \(0.525036\pi\)
\(824\) 11.7777 0.410297
\(825\) 6.37097 0.221809
\(826\) 0 0
\(827\) −10.7492 −0.373786 −0.186893 0.982380i \(-0.559842\pi\)
−0.186893 + 0.982380i \(0.559842\pi\)
\(828\) −1.98320 −0.0689211
\(829\) 23.9667 0.832399 0.416199 0.909273i \(-0.363362\pi\)
0.416199 + 0.909273i \(0.363362\pi\)
\(830\) 7.09978 0.246437
\(831\) −19.7258 −0.684279
\(832\) −4.44345 −0.154049
\(833\) 0 0
\(834\) −11.7319 −0.406241
\(835\) −3.37300 −0.116727
\(836\) 3.26053 0.112768
\(837\) −2.43285 −0.0840917
\(838\) 19.3022 0.666782
\(839\) 13.5218 0.466826 0.233413 0.972378i \(-0.425011\pi\)
0.233413 + 0.972378i \(0.425011\pi\)
\(840\) 0 0
\(841\) −28.9907 −0.999681
\(842\) 5.33795 0.183958
\(843\) −1.56281 −0.0538259
\(844\) −9.66873 −0.332811
\(845\) −17.7849 −0.611818
\(846\) −13.3321 −0.458367
\(847\) 0 0
\(848\) 7.24699 0.248863
\(849\) −26.3439 −0.904119
\(850\) 4.63016 0.158813
\(851\) 0.896421 0.0307289
\(852\) −11.1849 −0.383188
\(853\) −44.1133 −1.51041 −0.755206 0.655487i \(-0.772463\pi\)
−0.755206 + 0.655487i \(0.772463\pi\)
\(854\) 0 0
\(855\) 2.63704 0.0901848
\(856\) −2.34941 −0.0803013
\(857\) 24.3784 0.832750 0.416375 0.909193i \(-0.363300\pi\)
0.416375 + 0.909193i \(0.363300\pi\)
\(858\) −14.4880 −0.494612
\(859\) 42.2602 1.44190 0.720950 0.692987i \(-0.243706\pi\)
0.720950 + 0.692987i \(0.243706\pi\)
\(860\) −32.1089 −1.09491
\(861\) 0 0
\(862\) 8.06973 0.274856
\(863\) 11.1263 0.378743 0.189372 0.981905i \(-0.439355\pi\)
0.189372 + 0.981905i \(0.439355\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 58.1257 1.97633
\(866\) −0.207726 −0.00705881
\(867\) 11.3849 0.386651
\(868\) 0 0
\(869\) 42.0675 1.42704
\(870\) 0.253772 0.00860368
\(871\) −25.5386 −0.865344
\(872\) 8.67251 0.293688
\(873\) 3.67729 0.124457
\(874\) 1.98320 0.0670829
\(875\) 0 0
\(876\) 4.16583 0.140750
\(877\) 22.5809 0.762502 0.381251 0.924471i \(-0.375493\pi\)
0.381251 + 0.924471i \(0.375493\pi\)
\(878\) 28.9616 0.977407
\(879\) 10.6076 0.357784
\(880\) 8.59814 0.289843
\(881\) 22.5436 0.759513 0.379756 0.925086i \(-0.376008\pi\)
0.379756 + 0.925086i \(0.376008\pi\)
\(882\) 0 0
\(883\) −9.41767 −0.316930 −0.158465 0.987365i \(-0.550655\pi\)
−0.158465 + 0.987365i \(0.550655\pi\)
\(884\) −10.5293 −0.354139
\(885\) 3.48038 0.116992
\(886\) −5.50387 −0.184906
\(887\) −3.88840 −0.130560 −0.0652798 0.997867i \(-0.520794\pi\)
−0.0652798 + 0.997867i \(0.520794\pi\)
\(888\) 0.452007 0.0151683
\(889\) 0 0
\(890\) 0.744811 0.0249661
\(891\) −3.26053 −0.109232
\(892\) −0.367428 −0.0123024
\(893\) 13.3321 0.446141
\(894\) −0.858768 −0.0287215
\(895\) 1.72982 0.0578216
\(896\) 0 0
\(897\) −8.81227 −0.294233
\(898\) 0.546522 0.0182377
\(899\) 0.234123 0.00780843
\(900\) 1.95397 0.0651322
\(901\) 17.1726 0.572103
\(902\) 31.0856 1.03504
\(903\) 0 0
\(904\) 11.1344 0.370324
\(905\) −0.922811 −0.0306753
\(906\) −8.05804 −0.267710
\(907\) −8.38922 −0.278560 −0.139280 0.990253i \(-0.544479\pi\)
−0.139280 + 0.990253i \(0.544479\pi\)
\(908\) 15.9079 0.527921
\(909\) 13.0331 0.432282
\(910\) 0 0
\(911\) 10.5150 0.348377 0.174188 0.984712i \(-0.444270\pi\)
0.174188 + 0.984712i \(0.444270\pi\)
\(912\) 1.00000 0.0331133
\(913\) 8.77843 0.290524
\(914\) 38.6840 1.27955
\(915\) 14.5595 0.481321
\(916\) −11.3895 −0.376320
\(917\) 0 0
\(918\) −2.36962 −0.0782092
\(919\) −27.1316 −0.894990 −0.447495 0.894287i \(-0.647684\pi\)
−0.447495 + 0.894287i \(0.647684\pi\)
\(920\) 5.22978 0.172421
\(921\) −16.2375 −0.535044
\(922\) −29.2112 −0.962019
\(923\) −49.6995 −1.63588
\(924\) 0 0
\(925\) −0.883205 −0.0290396
\(926\) −21.1418 −0.694764
\(927\) 11.7777 0.386832
\(928\) 0.0962338 0.00315903
\(929\) 34.9902 1.14799 0.573995 0.818859i \(-0.305393\pi\)
0.573995 + 0.818859i \(0.305393\pi\)
\(930\) 6.41552 0.210373
\(931\) 0 0
\(932\) −3.01687 −0.0988210
\(933\) 23.0812 0.755643
\(934\) 20.4481 0.669081
\(935\) 20.3744 0.666313
\(936\) −4.44345 −0.145239
\(937\) 0.164326 0.00536828 0.00268414 0.999996i \(-0.499146\pi\)
0.00268414 + 0.999996i \(0.499146\pi\)
\(938\) 0 0
\(939\) 13.2774 0.433293
\(940\) 35.1572 1.14670
\(941\) 21.8637 0.712738 0.356369 0.934345i \(-0.384015\pi\)
0.356369 + 0.934345i \(0.384015\pi\)
\(942\) −9.13601 −0.297667
\(943\) 18.9077 0.615720
\(944\) 1.31981 0.0429560
\(945\) 0 0
\(946\) −39.7007 −1.29078
\(947\) 29.1135 0.946063 0.473031 0.881046i \(-0.343160\pi\)
0.473031 + 0.881046i \(0.343160\pi\)
\(948\) 12.9020 0.419039
\(949\) 18.5107 0.600881
\(950\) −1.95397 −0.0633950
\(951\) 13.3284 0.432202
\(952\) 0 0
\(953\) −3.81327 −0.123524 −0.0617619 0.998091i \(-0.519672\pi\)
−0.0617619 + 0.998091i \(0.519672\pi\)
\(954\) 7.24699 0.234630
\(955\) 13.9848 0.452536
\(956\) −28.2992 −0.915261
\(957\) 0.313773 0.0101428
\(958\) −23.9563 −0.773993
\(959\) 0 0
\(960\) 2.63704 0.0851100
\(961\) −25.0812 −0.809072
\(962\) 2.00847 0.0647556
\(963\) −2.34941 −0.0757088
\(964\) 21.4909 0.692174
\(965\) 1.10376 0.0355312
\(966\) 0 0
\(967\) −43.4232 −1.39639 −0.698197 0.715905i \(-0.746014\pi\)
−0.698197 + 0.715905i \(0.746014\pi\)
\(968\) −0.368940 −0.0118582
\(969\) 2.36962 0.0761233
\(970\) −9.69716 −0.311357
\(971\) −10.4462 −0.335234 −0.167617 0.985852i \(-0.553607\pi\)
−0.167617 + 0.985852i \(0.553607\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 14.0906 0.451492
\(975\) 8.68235 0.278058
\(976\) 5.52115 0.176728
\(977\) −3.59658 −0.115065 −0.0575324 0.998344i \(-0.518323\pi\)
−0.0575324 + 0.998344i \(0.518323\pi\)
\(978\) −8.30168 −0.265458
\(979\) 0.920912 0.0294325
\(980\) 0 0
\(981\) 8.67251 0.276892
\(982\) 12.3018 0.392567
\(983\) −47.5539 −1.51673 −0.758367 0.651828i \(-0.774002\pi\)
−0.758367 + 0.651828i \(0.774002\pi\)
\(984\) 9.53392 0.303930
\(985\) −67.6020 −2.15398
\(986\) 0.228038 0.00726221
\(987\) 0 0
\(988\) 4.44345 0.141365
\(989\) −24.1478 −0.767854
\(990\) 8.59814 0.273267
\(991\) 9.75973 0.310028 0.155014 0.987912i \(-0.450458\pi\)
0.155014 + 0.987912i \(0.450458\pi\)
\(992\) 2.43285 0.0772431
\(993\) 28.1155 0.892220
\(994\) 0 0
\(995\) −72.9182 −2.31166
\(996\) 2.69233 0.0853098
\(997\) 8.52971 0.270139 0.135069 0.990836i \(-0.456874\pi\)
0.135069 + 0.990836i \(0.456874\pi\)
\(998\) 5.60115 0.177301
\(999\) 0.452007 0.0143009
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 5586.2.a.ce.1.3 8
7.6 odd 2 5586.2.a.cf.1.6 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5586.2.a.ce.1.3 8 1.1 even 1 trivial
5586.2.a.cf.1.6 yes 8 7.6 odd 2