Properties

Label 3822.2.a.bz.1.2
Level $3822$
Weight $2$
Character 3822.1
Self dual yes
Analytic conductor $30.519$
Analytic rank $0$
Dimension $4$
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] = [3822,2,Mod(1,3822)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3822, 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("3822.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3822.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: \(30.5188236525\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.31808.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 11x^{2} + 12x + 28 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.34975\) of defining polynomial
Character \(\chi\) \(=\) 3822.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.34975 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.34975 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -1.34975 q^{10} +5.25858 q^{11} -1.00000 q^{12} -1.00000 q^{13} +1.34975 q^{15} +1.00000 q^{16} +3.25858 q^{17} +1.00000 q^{18} -1.34975 q^{19} -1.34975 q^{20} +5.25858 q^{22} +0.650251 q^{23} -1.00000 q^{24} -3.17818 q^{25} -1.00000 q^{26} -1.00000 q^{27} +0.155630 q^{29} +1.34975 q^{30} -1.90883 q^{31} +1.00000 q^{32} -5.25858 q^{33} +3.25858 q^{34} +1.00000 q^{36} +7.59239 q^{37} -1.34975 q^{38} +1.00000 q^{39} -1.34975 q^{40} -3.32305 q^{41} -3.16742 q^{43} +5.25858 q^{44} -1.34975 q^{45} +0.650251 q^{46} +8.15147 q^{47} -1.00000 q^{48} -3.17818 q^{50} -3.25858 q^{51} -1.00000 q^{52} +4.49462 q^{53} -1.00000 q^{54} -7.09777 q^{55} +1.34975 q^{57} +0.155630 q^{58} -1.90883 q^{59} +1.34975 q^{60} +2.26934 q^{61} -1.90883 q^{62} +1.00000 q^{64} +1.34975 q^{65} -5.25858 q^{66} +1.88629 q^{67} +3.25858 q^{68} -0.650251 q^{69} -5.30783 q^{71} +1.00000 q^{72} +5.34975 q^{73} +7.59239 q^{74} +3.17818 q^{75} -1.34975 q^{76} +1.00000 q^{78} +5.74802 q^{79} -1.34975 q^{80} +1.00000 q^{81} -3.32305 q^{82} -2.95736 q^{83} -4.39827 q^{85} -3.16742 q^{86} -0.155630 q^{87} +5.25858 q^{88} +10.7598 q^{89} -1.34975 q^{90} +0.650251 q^{92} +1.90883 q^{93} +8.15147 q^{94} +1.82182 q^{95} -1.00000 q^{96} -11.3456 q^{97} +5.25858 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} + 2 q^{5} - 4 q^{6} + 4 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} + 2 q^{5} - 4 q^{6} + 4 q^{8} + 4 q^{9} + 2 q^{10} + 6 q^{11} - 4 q^{12} - 4 q^{13} - 2 q^{15} + 4 q^{16} - 2 q^{17} + 4 q^{18} + 2 q^{19} + 2 q^{20} + 6 q^{22} + 10 q^{23} - 4 q^{24} + 6 q^{25} - 4 q^{26} - 4 q^{27} + 10 q^{29} - 2 q^{30} + 4 q^{32} - 6 q^{33} - 2 q^{34} + 4 q^{36} + 6 q^{37} + 2 q^{38} + 4 q^{39} + 2 q^{40} + 10 q^{43} + 6 q^{44} + 2 q^{45} + 10 q^{46} + 8 q^{47} - 4 q^{48} + 6 q^{50} + 2 q^{51} - 4 q^{52} + 16 q^{53} - 4 q^{54} - 6 q^{55} - 2 q^{57} + 10 q^{58} - 2 q^{60} - 2 q^{61} + 4 q^{64} - 2 q^{65} - 6 q^{66} + 28 q^{67} - 2 q^{68} - 10 q^{69} + 16 q^{71} + 4 q^{72} + 14 q^{73} + 6 q^{74} - 6 q^{75} + 2 q^{76} + 4 q^{78} + 8 q^{79} + 2 q^{80} + 4 q^{81} - 4 q^{83} - 10 q^{85} + 10 q^{86} - 10 q^{87} + 6 q^{88} - 4 q^{89} + 2 q^{90} + 10 q^{92} + 8 q^{94} + 26 q^{95} - 4 q^{96} - 4 q^{97} + 6 q^{99}+O(q^{100}) \) 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
\(5\) −1.34975 −0.603626 −0.301813 0.953367i \(-0.597592\pi\)
−0.301813 + 0.953367i \(0.597592\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.34975 −0.426828
\(11\) 5.25858 1.58552 0.792761 0.609532i \(-0.208643\pi\)
0.792761 + 0.609532i \(0.208643\pi\)
\(12\) −1.00000 −0.288675
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 1.34975 0.348504
\(16\) 1.00000 0.250000
\(17\) 3.25858 0.790323 0.395161 0.918612i \(-0.370689\pi\)
0.395161 + 0.918612i \(0.370689\pi\)
\(18\) 1.00000 0.235702
\(19\) −1.34975 −0.309654 −0.154827 0.987942i \(-0.549482\pi\)
−0.154827 + 0.987942i \(0.549482\pi\)
\(20\) −1.34975 −0.301813
\(21\) 0 0
\(22\) 5.25858 1.12113
\(23\) 0.650251 0.135587 0.0677933 0.997699i \(-0.478404\pi\)
0.0677933 + 0.997699i \(0.478404\pi\)
\(24\) −1.00000 −0.204124
\(25\) −3.17818 −0.635635
\(26\) −1.00000 −0.196116
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 0.155630 0.0288998 0.0144499 0.999896i \(-0.495400\pi\)
0.0144499 + 0.999896i \(0.495400\pi\)
\(30\) 1.34975 0.246429
\(31\) −1.90883 −0.342837 −0.171418 0.985198i \(-0.554835\pi\)
−0.171418 + 0.985198i \(0.554835\pi\)
\(32\) 1.00000 0.176777
\(33\) −5.25858 −0.915402
\(34\) 3.25858 0.558842
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 7.59239 1.24818 0.624091 0.781352i \(-0.285470\pi\)
0.624091 + 0.781352i \(0.285470\pi\)
\(38\) −1.34975 −0.218958
\(39\) 1.00000 0.160128
\(40\) −1.34975 −0.213414
\(41\) −3.32305 −0.518973 −0.259486 0.965747i \(-0.583553\pi\)
−0.259486 + 0.965747i \(0.583553\pi\)
\(42\) 0 0
\(43\) −3.16742 −0.483027 −0.241513 0.970398i \(-0.577644\pi\)
−0.241513 + 0.970398i \(0.577644\pi\)
\(44\) 5.25858 0.792761
\(45\) −1.34975 −0.201209
\(46\) 0.650251 0.0958742
\(47\) 8.15147 1.18901 0.594507 0.804090i \(-0.297347\pi\)
0.594507 + 0.804090i \(0.297347\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) −3.17818 −0.449462
\(51\) −3.25858 −0.456293
\(52\) −1.00000 −0.138675
\(53\) 4.49462 0.617384 0.308692 0.951162i \(-0.400109\pi\)
0.308692 + 0.951162i \(0.400109\pi\)
\(54\) −1.00000 −0.136083
\(55\) −7.09777 −0.957063
\(56\) 0 0
\(57\) 1.34975 0.178779
\(58\) 0.155630 0.0204353
\(59\) −1.90883 −0.248509 −0.124255 0.992250i \(-0.539654\pi\)
−0.124255 + 0.992250i \(0.539654\pi\)
\(60\) 1.34975 0.174252
\(61\) 2.26934 0.290560 0.145280 0.989391i \(-0.453592\pi\)
0.145280 + 0.989391i \(0.453592\pi\)
\(62\) −1.90883 −0.242422
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 1.34975 0.167416
\(66\) −5.25858 −0.647287
\(67\) 1.88629 0.230447 0.115223 0.993340i \(-0.463242\pi\)
0.115223 + 0.993340i \(0.463242\pi\)
\(68\) 3.25858 0.395161
\(69\) −0.650251 −0.0782810
\(70\) 0 0
\(71\) −5.30783 −0.629924 −0.314962 0.949104i \(-0.601992\pi\)
−0.314962 + 0.949104i \(0.601992\pi\)
\(72\) 1.00000 0.117851
\(73\) 5.34975 0.626141 0.313070 0.949730i \(-0.398642\pi\)
0.313070 + 0.949730i \(0.398642\pi\)
\(74\) 7.59239 0.882597
\(75\) 3.17818 0.366984
\(76\) −1.34975 −0.154827
\(77\) 0 0
\(78\) 1.00000 0.113228
\(79\) 5.74802 0.646703 0.323351 0.946279i \(-0.395190\pi\)
0.323351 + 0.946279i \(0.395190\pi\)
\(80\) −1.34975 −0.150907
\(81\) 1.00000 0.111111
\(82\) −3.32305 −0.366969
\(83\) −2.95736 −0.324612 −0.162306 0.986740i \(-0.551893\pi\)
−0.162306 + 0.986740i \(0.551893\pi\)
\(84\) 0 0
\(85\) −4.39827 −0.477059
\(86\) −3.16742 −0.341551
\(87\) −0.155630 −0.0166853
\(88\) 5.25858 0.560567
\(89\) 10.7598 1.14054 0.570269 0.821458i \(-0.306839\pi\)
0.570269 + 0.821458i \(0.306839\pi\)
\(90\) −1.34975 −0.142276
\(91\) 0 0
\(92\) 0.650251 0.0677933
\(93\) 1.90883 0.197937
\(94\) 8.15147 0.840760
\(95\) 1.82182 0.186915
\(96\) −1.00000 −0.102062
\(97\) −11.3456 −1.15197 −0.575985 0.817460i \(-0.695381\pi\)
−0.575985 + 0.817460i \(0.695381\pi\)
\(98\) 0 0
\(99\) 5.25858 0.528508
\(100\) −3.17818 −0.317818
\(101\) 1.77991 0.177107 0.0885536 0.996071i \(-0.471776\pi\)
0.0885536 + 0.996071i \(0.471776\pi\)
\(102\) −3.25858 −0.322648
\(103\) 17.9487 1.76854 0.884271 0.466974i \(-0.154656\pi\)
0.884271 + 0.466974i \(0.154656\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) 4.49462 0.436556
\(107\) −11.4593 −1.10781 −0.553906 0.832579i \(-0.686863\pi\)
−0.553906 + 0.832579i \(0.686863\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −1.02182 −0.0978726 −0.0489363 0.998802i \(-0.515583\pi\)
−0.0489363 + 0.998802i \(0.515583\pi\)
\(110\) −7.09777 −0.676746
\(111\) −7.59239 −0.720638
\(112\) 0 0
\(113\) 15.9539 1.50082 0.750410 0.660973i \(-0.229856\pi\)
0.750410 + 0.660973i \(0.229856\pi\)
\(114\) 1.34975 0.126416
\(115\) −0.877675 −0.0818436
\(116\) 0.155630 0.0144499
\(117\) −1.00000 −0.0924500
\(118\) −1.90883 −0.175722
\(119\) 0 0
\(120\) 1.34975 0.123215
\(121\) 16.6527 1.51388
\(122\) 2.26934 0.205457
\(123\) 3.32305 0.299629
\(124\) −1.90883 −0.171418
\(125\) 11.0385 0.987312
\(126\) 0 0
\(127\) −2.03776 −0.180822 −0.0904111 0.995905i \(-0.528818\pi\)
−0.0904111 + 0.995905i \(0.528818\pi\)
\(128\) 1.00000 0.0883883
\(129\) 3.16742 0.278876
\(130\) 1.34975 0.118381
\(131\) 15.5615 1.35962 0.679809 0.733389i \(-0.262063\pi\)
0.679809 + 0.733389i \(0.262063\pi\)
\(132\) −5.25858 −0.457701
\(133\) 0 0
\(134\) 1.88629 0.162950
\(135\) 1.34975 0.116168
\(136\) 3.25858 0.279421
\(137\) 2.58060 0.220476 0.110238 0.993905i \(-0.464839\pi\)
0.110238 + 0.993905i \(0.464839\pi\)
\(138\) −0.650251 −0.0553530
\(139\) 18.2059 1.54420 0.772102 0.635498i \(-0.219205\pi\)
0.772102 + 0.635498i \(0.219205\pi\)
\(140\) 0 0
\(141\) −8.15147 −0.686478
\(142\) −5.30783 −0.445423
\(143\) −5.25858 −0.439745
\(144\) 1.00000 0.0833333
\(145\) −0.210062 −0.0174447
\(146\) 5.34975 0.442748
\(147\) 0 0
\(148\) 7.59239 0.624091
\(149\) 9.50641 0.778795 0.389398 0.921070i \(-0.372683\pi\)
0.389398 + 0.921070i \(0.372683\pi\)
\(150\) 3.17818 0.259497
\(151\) 13.1297 1.06848 0.534238 0.845334i \(-0.320599\pi\)
0.534238 + 0.845334i \(0.320599\pi\)
\(152\) −1.34975 −0.109479
\(153\) 3.25858 0.263441
\(154\) 0 0
\(155\) 2.57645 0.206945
\(156\) 1.00000 0.0800641
\(157\) 2.32275 0.185375 0.0926877 0.995695i \(-0.470454\pi\)
0.0926877 + 0.995695i \(0.470454\pi\)
\(158\) 5.74802 0.457288
\(159\) −4.49462 −0.356447
\(160\) −1.34975 −0.106707
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 21.5956 1.69150 0.845748 0.533583i \(-0.179155\pi\)
0.845748 + 0.533583i \(0.179155\pi\)
\(164\) −3.32305 −0.259486
\(165\) 7.09777 0.552561
\(166\) −2.95736 −0.229535
\(167\) −25.3321 −1.96026 −0.980128 0.198364i \(-0.936437\pi\)
−0.980128 + 0.198364i \(0.936437\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) −4.39827 −0.337332
\(171\) −1.34975 −0.103218
\(172\) −3.16742 −0.241513
\(173\) 1.80142 0.136960 0.0684799 0.997652i \(-0.478185\pi\)
0.0684799 + 0.997652i \(0.478185\pi\)
\(174\) −0.155630 −0.0117983
\(175\) 0 0
\(176\) 5.25858 0.396381
\(177\) 1.90883 0.143477
\(178\) 10.7598 0.806482
\(179\) −0.274526 −0.0205190 −0.0102595 0.999947i \(-0.503266\pi\)
−0.0102595 + 0.999947i \(0.503266\pi\)
\(180\) −1.34975 −0.100604
\(181\) −12.7750 −0.949560 −0.474780 0.880105i \(-0.657472\pi\)
−0.474780 + 0.880105i \(0.657472\pi\)
\(182\) 0 0
\(183\) −2.26934 −0.167755
\(184\) 0.650251 0.0479371
\(185\) −10.2478 −0.753435
\(186\) 1.90883 0.139962
\(187\) 17.1355 1.25307
\(188\) 8.15147 0.594507
\(189\) 0 0
\(190\) 1.82182 0.132169
\(191\) −19.1487 −1.38555 −0.692777 0.721152i \(-0.743613\pi\)
−0.692777 + 0.721152i \(0.743613\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 9.49016 0.683117 0.341558 0.939861i \(-0.389045\pi\)
0.341558 + 0.939861i \(0.389045\pi\)
\(194\) −11.3456 −0.814566
\(195\) −1.34975 −0.0966576
\(196\) 0 0
\(197\) 1.33826 0.0953473 0.0476737 0.998863i \(-0.484819\pi\)
0.0476737 + 0.998863i \(0.484819\pi\)
\(198\) 5.25858 0.373711
\(199\) 21.0988 1.49565 0.747827 0.663894i \(-0.231097\pi\)
0.747827 + 0.663894i \(0.231097\pi\)
\(200\) −3.17818 −0.224731
\(201\) −1.88629 −0.133048
\(202\) 1.77991 0.125234
\(203\) 0 0
\(204\) −3.25858 −0.228146
\(205\) 4.48528 0.313266
\(206\) 17.9487 1.25055
\(207\) 0.650251 0.0451955
\(208\) −1.00000 −0.0693375
\(209\) −7.09777 −0.490963
\(210\) 0 0
\(211\) 7.16742 0.493425 0.246713 0.969089i \(-0.420650\pi\)
0.246713 + 0.969089i \(0.420650\pi\)
\(212\) 4.49462 0.308692
\(213\) 5.30783 0.363687
\(214\) −11.4593 −0.783342
\(215\) 4.27522 0.291568
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) −1.02182 −0.0692064
\(219\) −5.34975 −0.361503
\(220\) −7.09777 −0.478532
\(221\) −3.25858 −0.219196
\(222\) −7.59239 −0.509568
\(223\) −12.6593 −0.847730 −0.423865 0.905725i \(-0.639327\pi\)
−0.423865 + 0.905725i \(0.639327\pi\)
\(224\) 0 0
\(225\) −3.17818 −0.211878
\(226\) 15.9539 1.06124
\(227\) −9.77991 −0.649115 −0.324558 0.945866i \(-0.605215\pi\)
−0.324558 + 0.945866i \(0.605215\pi\)
\(228\) 1.34975 0.0893893
\(229\) 27.0721 1.78897 0.894487 0.447094i \(-0.147541\pi\)
0.894487 + 0.447094i \(0.147541\pi\)
\(230\) −0.877675 −0.0578722
\(231\) 0 0
\(232\) 0.155630 0.0102176
\(233\) −22.5838 −1.47951 −0.739756 0.672875i \(-0.765059\pi\)
−0.739756 + 0.672875i \(0.765059\pi\)
\(234\) −1.00000 −0.0653720
\(235\) −11.0024 −0.717721
\(236\) −1.90883 −0.124255
\(237\) −5.74802 −0.373374
\(238\) 0 0
\(239\) 13.0617 0.844893 0.422447 0.906388i \(-0.361171\pi\)
0.422447 + 0.906388i \(0.361171\pi\)
\(240\) 1.34975 0.0871259
\(241\) −6.04509 −0.389399 −0.194699 0.980863i \(-0.562373\pi\)
−0.194699 + 0.980863i \(0.562373\pi\)
\(242\) 16.6527 1.07048
\(243\) −1.00000 −0.0641500
\(244\) 2.26934 0.145280
\(245\) 0 0
\(246\) 3.32305 0.211870
\(247\) 1.34975 0.0858825
\(248\) −1.90883 −0.121211
\(249\) 2.95736 0.187415
\(250\) 11.0385 0.698135
\(251\) −10.7331 −0.677468 −0.338734 0.940882i \(-0.609999\pi\)
−0.338734 + 0.940882i \(0.609999\pi\)
\(252\) 0 0
\(253\) 3.41940 0.214976
\(254\) −2.03776 −0.127861
\(255\) 4.39827 0.275430
\(256\) 1.00000 0.0625000
\(257\) 1.13969 0.0710918 0.0355459 0.999368i \(-0.488683\pi\)
0.0355459 + 0.999368i \(0.488683\pi\)
\(258\) 3.16742 0.197195
\(259\) 0 0
\(260\) 1.34975 0.0837079
\(261\) 0.155630 0.00963327
\(262\) 15.5615 0.961395
\(263\) 25.2922 1.55958 0.779792 0.626039i \(-0.215325\pi\)
0.779792 + 0.626039i \(0.215325\pi\)
\(264\) −5.25858 −0.323643
\(265\) −6.06661 −0.372669
\(266\) 0 0
\(267\) −10.7598 −0.658490
\(268\) 1.88629 0.115223
\(269\) −4.66174 −0.284231 −0.142116 0.989850i \(-0.545390\pi\)
−0.142116 + 0.989850i \(0.545390\pi\)
\(270\) 1.34975 0.0821431
\(271\) 6.70538 0.407323 0.203661 0.979041i \(-0.434716\pi\)
0.203661 + 0.979041i \(0.434716\pi\)
\(272\) 3.25858 0.197581
\(273\) 0 0
\(274\) 2.58060 0.155900
\(275\) −16.7127 −1.00781
\(276\) −0.650251 −0.0391405
\(277\) −29.1068 −1.74886 −0.874430 0.485152i \(-0.838764\pi\)
−0.874430 + 0.485152i \(0.838764\pi\)
\(278\) 18.2059 1.09192
\(279\) −1.90883 −0.114279
\(280\) 0 0
\(281\) 0.356353 0.0212582 0.0106291 0.999944i \(-0.496617\pi\)
0.0106291 + 0.999944i \(0.496617\pi\)
\(282\) −8.15147 −0.485413
\(283\) 21.8392 1.29820 0.649102 0.760701i \(-0.275145\pi\)
0.649102 + 0.760701i \(0.275145\pi\)
\(284\) −5.30783 −0.314962
\(285\) −1.82182 −0.107916
\(286\) −5.25858 −0.310947
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −6.38163 −0.375390
\(290\) −0.210062 −0.0123353
\(291\) 11.3456 0.665090
\(292\) 5.34975 0.313070
\(293\) −9.18478 −0.536581 −0.268290 0.963338i \(-0.586459\pi\)
−0.268290 + 0.963338i \(0.586459\pi\)
\(294\) 0 0
\(295\) 2.57645 0.150007
\(296\) 7.59239 0.441299
\(297\) −5.25858 −0.305134
\(298\) 9.50641 0.550691
\(299\) −0.650251 −0.0376050
\(300\) 3.17818 0.183492
\(301\) 0 0
\(302\) 13.1297 0.755527
\(303\) −1.77991 −0.102253
\(304\) −1.34975 −0.0774134
\(305\) −3.06304 −0.175389
\(306\) 3.25858 0.186281
\(307\) −6.01321 −0.343192 −0.171596 0.985167i \(-0.554892\pi\)
−0.171596 + 0.985167i \(0.554892\pi\)
\(308\) 0 0
\(309\) −17.9487 −1.02107
\(310\) 2.57645 0.146332
\(311\) −13.8926 −0.787777 −0.393888 0.919158i \(-0.628870\pi\)
−0.393888 + 0.919158i \(0.628870\pi\)
\(312\) 1.00000 0.0566139
\(313\) 12.9206 0.730317 0.365158 0.930945i \(-0.381015\pi\)
0.365158 + 0.930945i \(0.381015\pi\)
\(314\) 2.32275 0.131080
\(315\) 0 0
\(316\) 5.74802 0.323351
\(317\) −28.4696 −1.59901 −0.799507 0.600657i \(-0.794906\pi\)
−0.799507 + 0.600657i \(0.794906\pi\)
\(318\) −4.49462 −0.252046
\(319\) 0.818395 0.0458213
\(320\) −1.34975 −0.0754533
\(321\) 11.4593 0.639596
\(322\) 0 0
\(323\) −4.39827 −0.244726
\(324\) 1.00000 0.0555556
\(325\) 3.17818 0.176294
\(326\) 21.5956 1.19607
\(327\) 1.02182 0.0565068
\(328\) −3.32305 −0.183485
\(329\) 0 0
\(330\) 7.09777 0.390719
\(331\) −18.7383 −1.02995 −0.514975 0.857205i \(-0.672199\pi\)
−0.514975 + 0.857205i \(0.672199\pi\)
\(332\) −2.95736 −0.162306
\(333\) 7.59239 0.416060
\(334\) −25.3321 −1.38611
\(335\) −2.54602 −0.139104
\(336\) 0 0
\(337\) 5.30955 0.289230 0.144615 0.989488i \(-0.453806\pi\)
0.144615 + 0.989488i \(0.453806\pi\)
\(338\) 1.00000 0.0543928
\(339\) −15.9539 −0.866498
\(340\) −4.39827 −0.238530
\(341\) −10.0378 −0.543575
\(342\) −1.34975 −0.0729861
\(343\) 0 0
\(344\) −3.16742 −0.170776
\(345\) 0.877675 0.0472525
\(346\) 1.80142 0.0968452
\(347\) 29.0628 1.56017 0.780085 0.625673i \(-0.215176\pi\)
0.780085 + 0.625673i \(0.215176\pi\)
\(348\) −0.155630 −0.00834266
\(349\) 8.83576 0.472967 0.236484 0.971635i \(-0.424005\pi\)
0.236484 + 0.971635i \(0.424005\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 5.25858 0.280283
\(353\) 25.7030 1.36803 0.684016 0.729467i \(-0.260232\pi\)
0.684016 + 0.729467i \(0.260232\pi\)
\(354\) 1.90883 0.101453
\(355\) 7.16424 0.380239
\(356\) 10.7598 0.570269
\(357\) 0 0
\(358\) −0.274526 −0.0145091
\(359\) −23.4211 −1.23612 −0.618060 0.786131i \(-0.712081\pi\)
−0.618060 + 0.786131i \(0.712081\pi\)
\(360\) −1.34975 −0.0711380
\(361\) −17.1782 −0.904115
\(362\) −12.7750 −0.671440
\(363\) −16.6527 −0.874040
\(364\) 0 0
\(365\) −7.22082 −0.377955
\(366\) −2.26934 −0.118620
\(367\) 18.7442 0.978437 0.489219 0.872161i \(-0.337282\pi\)
0.489219 + 0.872161i \(0.337282\pi\)
\(368\) 0.650251 0.0338967
\(369\) −3.32305 −0.172991
\(370\) −10.2478 −0.532759
\(371\) 0 0
\(372\) 1.90883 0.0989684
\(373\) −19.9005 −1.03041 −0.515205 0.857067i \(-0.672284\pi\)
−0.515205 + 0.857067i \(0.672284\pi\)
\(374\) 17.1355 0.886057
\(375\) −11.0385 −0.570025
\(376\) 8.15147 0.420380
\(377\) −0.155630 −0.00801536
\(378\) 0 0
\(379\) −24.6760 −1.26752 −0.633760 0.773530i \(-0.718489\pi\)
−0.633760 + 0.773530i \(0.718489\pi\)
\(380\) 1.82182 0.0934576
\(381\) 2.03776 0.104398
\(382\) −19.1487 −0.979735
\(383\) −21.2115 −1.08386 −0.541928 0.840425i \(-0.682305\pi\)
−0.541928 + 0.840425i \(0.682305\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 9.49016 0.483037
\(387\) −3.16742 −0.161009
\(388\) −11.3456 −0.575985
\(389\) −33.2073 −1.68368 −0.841839 0.539729i \(-0.818527\pi\)
−0.841839 + 0.539729i \(0.818527\pi\)
\(390\) −1.34975 −0.0683472
\(391\) 2.11890 0.107157
\(392\) 0 0
\(393\) −15.5615 −0.784975
\(394\) 1.33826 0.0674207
\(395\) −7.75839 −0.390367
\(396\) 5.25858 0.264254
\(397\) 21.3907 1.07357 0.536784 0.843720i \(-0.319639\pi\)
0.536784 + 0.843720i \(0.319639\pi\)
\(398\) 21.0988 1.05759
\(399\) 0 0
\(400\) −3.17818 −0.158909
\(401\) −21.4745 −1.07239 −0.536193 0.844095i \(-0.680138\pi\)
−0.536193 + 0.844095i \(0.680138\pi\)
\(402\) −1.88629 −0.0940795
\(403\) 1.90883 0.0950858
\(404\) 1.77991 0.0885536
\(405\) −1.34975 −0.0670696
\(406\) 0 0
\(407\) 39.9252 1.97902
\(408\) −3.25858 −0.161324
\(409\) −4.18854 −0.207110 −0.103555 0.994624i \(-0.533022\pi\)
−0.103555 + 0.994624i \(0.533022\pi\)
\(410\) 4.48528 0.221512
\(411\) −2.58060 −0.127292
\(412\) 17.9487 0.884271
\(413\) 0 0
\(414\) 0.650251 0.0319581
\(415\) 3.99169 0.195944
\(416\) −1.00000 −0.0490290
\(417\) −18.2059 −0.891547
\(418\) −7.09777 −0.347163
\(419\) 8.88501 0.434061 0.217030 0.976165i \(-0.430363\pi\)
0.217030 + 0.976165i \(0.430363\pi\)
\(420\) 0 0
\(421\) 9.59411 0.467588 0.233794 0.972286i \(-0.424886\pi\)
0.233794 + 0.972286i \(0.424886\pi\)
\(422\) 7.16742 0.348904
\(423\) 8.15147 0.396338
\(424\) 4.49462 0.218278
\(425\) −10.3564 −0.502357
\(426\) 5.30783 0.257165
\(427\) 0 0
\(428\) −11.4593 −0.553906
\(429\) 5.25858 0.253887
\(430\) 4.27522 0.206169
\(431\) 16.6897 0.803916 0.401958 0.915658i \(-0.368330\pi\)
0.401958 + 0.915658i \(0.368330\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −12.8613 −0.618077 −0.309038 0.951050i \(-0.600007\pi\)
−0.309038 + 0.951050i \(0.600007\pi\)
\(434\) 0 0
\(435\) 0.210062 0.0100717
\(436\) −1.02182 −0.0489363
\(437\) −0.877675 −0.0419849
\(438\) −5.34975 −0.255621
\(439\) 2.23359 0.106603 0.0533017 0.998578i \(-0.483025\pi\)
0.0533017 + 0.998578i \(0.483025\pi\)
\(440\) −7.09777 −0.338373
\(441\) 0 0
\(442\) −3.25858 −0.154995
\(443\) 26.5093 1.25949 0.629747 0.776800i \(-0.283159\pi\)
0.629747 + 0.776800i \(0.283159\pi\)
\(444\) −7.59239 −0.360319
\(445\) −14.5230 −0.688458
\(446\) −12.6593 −0.599435
\(447\) −9.50641 −0.449638
\(448\) 0 0
\(449\) −28.1306 −1.32757 −0.663784 0.747925i \(-0.731050\pi\)
−0.663784 + 0.747925i \(0.731050\pi\)
\(450\) −3.17818 −0.149821
\(451\) −17.4745 −0.822843
\(452\) 15.9539 0.750410
\(453\) −13.1297 −0.616885
\(454\) −9.77991 −0.458994
\(455\) 0 0
\(456\) 1.34975 0.0632078
\(457\) −15.0715 −0.705015 −0.352507 0.935809i \(-0.614671\pi\)
−0.352507 + 0.935809i \(0.614671\pi\)
\(458\) 27.0721 1.26500
\(459\) −3.25858 −0.152098
\(460\) −0.877675 −0.0409218
\(461\) 10.4492 0.486670 0.243335 0.969942i \(-0.421759\pi\)
0.243335 + 0.969942i \(0.421759\pi\)
\(462\) 0 0
\(463\) 12.0115 0.558221 0.279111 0.960259i \(-0.409960\pi\)
0.279111 + 0.960259i \(0.409960\pi\)
\(464\) 0.155630 0.00722495
\(465\) −2.57645 −0.119480
\(466\) −22.5838 −1.04617
\(467\) −21.6468 −1.00170 −0.500848 0.865535i \(-0.666978\pi\)
−0.500848 + 0.865535i \(0.666978\pi\)
\(468\) −1.00000 −0.0462250
\(469\) 0 0
\(470\) −11.0024 −0.507505
\(471\) −2.32275 −0.107027
\(472\) −1.90883 −0.0878612
\(473\) −16.6561 −0.765850
\(474\) −5.74802 −0.264015
\(475\) 4.28974 0.196827
\(476\) 0 0
\(477\) 4.49462 0.205795
\(478\) 13.0617 0.597430
\(479\) −3.28211 −0.149963 −0.0749817 0.997185i \(-0.523890\pi\)
−0.0749817 + 0.997185i \(0.523890\pi\)
\(480\) 1.34975 0.0616073
\(481\) −7.59239 −0.346183
\(482\) −6.04509 −0.275346
\(483\) 0 0
\(484\) 16.6527 0.756941
\(485\) 15.3137 0.695360
\(486\) −1.00000 −0.0453609
\(487\) 25.3907 1.15056 0.575281 0.817956i \(-0.304893\pi\)
0.575281 + 0.817956i \(0.304893\pi\)
\(488\) 2.26934 0.102728
\(489\) −21.5956 −0.976585
\(490\) 0 0
\(491\) −16.6368 −0.750806 −0.375403 0.926862i \(-0.622496\pi\)
−0.375403 + 0.926862i \(0.622496\pi\)
\(492\) 3.32305 0.149815
\(493\) 0.507134 0.0228402
\(494\) 1.34975 0.0607281
\(495\) −7.09777 −0.319021
\(496\) −1.90883 −0.0857092
\(497\) 0 0
\(498\) 2.95736 0.132522
\(499\) −15.0495 −0.673710 −0.336855 0.941556i \(-0.609363\pi\)
−0.336855 + 0.941556i \(0.609363\pi\)
\(500\) 11.0385 0.493656
\(501\) 25.3321 1.13175
\(502\) −10.7331 −0.479042
\(503\) −23.3529 −1.04126 −0.520628 0.853784i \(-0.674302\pi\)
−0.520628 + 0.853784i \(0.674302\pi\)
\(504\) 0 0
\(505\) −2.40243 −0.106907
\(506\) 3.41940 0.152011
\(507\) −1.00000 −0.0444116
\(508\) −2.03776 −0.0904111
\(509\) −6.67237 −0.295748 −0.147874 0.989006i \(-0.547243\pi\)
−0.147874 + 0.989006i \(0.547243\pi\)
\(510\) 4.39827 0.194759
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 1.34975 0.0595929
\(514\) 1.13969 0.0502695
\(515\) −24.2263 −1.06754
\(516\) 3.16742 0.139438
\(517\) 42.8652 1.88521
\(518\) 0 0
\(519\) −1.80142 −0.0790737
\(520\) 1.34975 0.0591904
\(521\) −13.4846 −0.590769 −0.295385 0.955378i \(-0.595448\pi\)
−0.295385 + 0.955378i \(0.595448\pi\)
\(522\) 0.155630 0.00681175
\(523\) −36.5162 −1.59674 −0.798371 0.602166i \(-0.794304\pi\)
−0.798371 + 0.602166i \(0.794304\pi\)
\(524\) 15.5615 0.679809
\(525\) 0 0
\(526\) 25.2922 1.10279
\(527\) −6.22009 −0.270952
\(528\) −5.25858 −0.228850
\(529\) −22.5772 −0.981616
\(530\) −6.06661 −0.263517
\(531\) −1.90883 −0.0828363
\(532\) 0 0
\(533\) 3.32305 0.143937
\(534\) −10.7598 −0.465622
\(535\) 15.4672 0.668705
\(536\) 1.88629 0.0814752
\(537\) 0.274526 0.0118467
\(538\) −4.66174 −0.200982
\(539\) 0 0
\(540\) 1.34975 0.0580840
\(541\) 31.7130 1.36345 0.681724 0.731609i \(-0.261230\pi\)
0.681724 + 0.731609i \(0.261230\pi\)
\(542\) 6.70538 0.288021
\(543\) 12.7750 0.548229
\(544\) 3.25858 0.139711
\(545\) 1.37920 0.0590785
\(546\) 0 0
\(547\) 11.1848 0.478227 0.239113 0.970992i \(-0.423143\pi\)
0.239113 + 0.970992i \(0.423143\pi\)
\(548\) 2.58060 0.110238
\(549\) 2.26934 0.0968532
\(550\) −16.7127 −0.712632
\(551\) −0.210062 −0.00894893
\(552\) −0.650251 −0.0276765
\(553\) 0 0
\(554\) −29.1068 −1.23663
\(555\) 10.2478 0.434996
\(556\) 18.2059 0.772102
\(557\) 44.9103 1.90291 0.951455 0.307787i \(-0.0995884\pi\)
0.951455 + 0.307787i \(0.0995884\pi\)
\(558\) −1.90883 −0.0808074
\(559\) 3.16742 0.133967
\(560\) 0 0
\(561\) −17.1355 −0.723463
\(562\) 0.356353 0.0150318
\(563\) 18.9910 0.800374 0.400187 0.916434i \(-0.368945\pi\)
0.400187 + 0.916434i \(0.368945\pi\)
\(564\) −8.15147 −0.343239
\(565\) −21.5338 −0.905934
\(566\) 21.8392 0.917969
\(567\) 0 0
\(568\) −5.30783 −0.222712
\(569\) 22.1657 0.929235 0.464617 0.885512i \(-0.346192\pi\)
0.464617 + 0.885512i \(0.346192\pi\)
\(570\) −1.82182 −0.0763078
\(571\) −37.3269 −1.56208 −0.781041 0.624479i \(-0.785311\pi\)
−0.781041 + 0.624479i \(0.785311\pi\)
\(572\) −5.25858 −0.219872
\(573\) 19.1487 0.799950
\(574\) 0 0
\(575\) −2.06661 −0.0861836
\(576\) 1.00000 0.0416667
\(577\) −27.2084 −1.13270 −0.566349 0.824165i \(-0.691645\pi\)
−0.566349 + 0.824165i \(0.691645\pi\)
\(578\) −6.38163 −0.265441
\(579\) −9.49016 −0.394398
\(580\) −0.210062 −0.00872234
\(581\) 0 0
\(582\) 11.3456 0.470290
\(583\) 23.6353 0.978876
\(584\) 5.34975 0.221374
\(585\) 1.34975 0.0558053
\(586\) −9.18478 −0.379420
\(587\) 30.5623 1.26144 0.630720 0.776011i \(-0.282760\pi\)
0.630720 + 0.776011i \(0.282760\pi\)
\(588\) 0 0
\(589\) 2.57645 0.106161
\(590\) 2.57645 0.106071
\(591\) −1.33826 −0.0550488
\(592\) 7.59239 0.312045
\(593\) −16.4167 −0.674151 −0.337076 0.941478i \(-0.609438\pi\)
−0.337076 + 0.941478i \(0.609438\pi\)
\(594\) −5.25858 −0.215762
\(595\) 0 0
\(596\) 9.50641 0.389398
\(597\) −21.0988 −0.863516
\(598\) −0.650251 −0.0265907
\(599\) 2.96296 0.121063 0.0605317 0.998166i \(-0.480720\pi\)
0.0605317 + 0.998166i \(0.480720\pi\)
\(600\) 3.17818 0.129749
\(601\) −19.7966 −0.807519 −0.403760 0.914865i \(-0.632297\pi\)
−0.403760 + 0.914865i \(0.632297\pi\)
\(602\) 0 0
\(603\) 1.88629 0.0768156
\(604\) 13.1297 0.534238
\(605\) −22.4770 −0.913819
\(606\) −1.77991 −0.0723037
\(607\) 29.0150 1.17768 0.588840 0.808249i \(-0.299585\pi\)
0.588840 + 0.808249i \(0.299585\pi\)
\(608\) −1.34975 −0.0547396
\(609\) 0 0
\(610\) −3.06304 −0.124019
\(611\) −8.15147 −0.329773
\(612\) 3.25858 0.131720
\(613\) −21.2826 −0.859595 −0.429798 0.902925i \(-0.641415\pi\)
−0.429798 + 0.902925i \(0.641415\pi\)
\(614\) −6.01321 −0.242673
\(615\) −4.48528 −0.180864
\(616\) 0 0
\(617\) −46.3601 −1.86639 −0.933194 0.359372i \(-0.882991\pi\)
−0.933194 + 0.359372i \(0.882991\pi\)
\(618\) −17.9487 −0.722004
\(619\) 33.2340 1.33579 0.667894 0.744256i \(-0.267196\pi\)
0.667894 + 0.744256i \(0.267196\pi\)
\(620\) 2.57645 0.103473
\(621\) −0.650251 −0.0260937
\(622\) −13.8926 −0.557042
\(623\) 0 0
\(624\) 1.00000 0.0400320
\(625\) 0.991689 0.0396675
\(626\) 12.9206 0.516412
\(627\) 7.09777 0.283458
\(628\) 2.32275 0.0926877
\(629\) 24.7404 0.986466
\(630\) 0 0
\(631\) −31.4007 −1.25004 −0.625021 0.780608i \(-0.714910\pi\)
−0.625021 + 0.780608i \(0.714910\pi\)
\(632\) 5.74802 0.228644
\(633\) −7.16742 −0.284879
\(634\) −28.4696 −1.13067
\(635\) 2.75047 0.109149
\(636\) −4.49462 −0.178223
\(637\) 0 0
\(638\) 0.818395 0.0324006
\(639\) −5.30783 −0.209975
\(640\) −1.34975 −0.0533535
\(641\) −39.3432 −1.55396 −0.776981 0.629524i \(-0.783250\pi\)
−0.776981 + 0.629524i \(0.783250\pi\)
\(642\) 11.4593 0.452263
\(643\) 25.9120 1.02187 0.510935 0.859619i \(-0.329299\pi\)
0.510935 + 0.859619i \(0.329299\pi\)
\(644\) 0 0
\(645\) −4.27522 −0.168337
\(646\) −4.39827 −0.173048
\(647\) 1.30881 0.0514547 0.0257274 0.999669i \(-0.491810\pi\)
0.0257274 + 0.999669i \(0.491810\pi\)
\(648\) 1.00000 0.0392837
\(649\) −10.0378 −0.394017
\(650\) 3.17818 0.124658
\(651\) 0 0
\(652\) 21.5956 0.845748
\(653\) 43.0639 1.68522 0.842610 0.538525i \(-0.181018\pi\)
0.842610 + 0.538525i \(0.181018\pi\)
\(654\) 1.02182 0.0399563
\(655\) −21.0042 −0.820701
\(656\) −3.32305 −0.129743
\(657\) 5.34975 0.208714
\(658\) 0 0
\(659\) 22.6583 0.882641 0.441321 0.897350i \(-0.354510\pi\)
0.441321 + 0.897350i \(0.354510\pi\)
\(660\) 7.09777 0.276280
\(661\) −35.9411 −1.39795 −0.698974 0.715147i \(-0.746360\pi\)
−0.698974 + 0.715147i \(0.746360\pi\)
\(662\) −18.7383 −0.728284
\(663\) 3.25858 0.126553
\(664\) −2.95736 −0.114768
\(665\) 0 0
\(666\) 7.59239 0.294199
\(667\) 0.101199 0.00391843
\(668\) −25.3321 −0.980128
\(669\) 12.6593 0.489437
\(670\) −2.54602 −0.0983612
\(671\) 11.9335 0.460689
\(672\) 0 0
\(673\) −31.4296 −1.21152 −0.605760 0.795647i \(-0.707131\pi\)
−0.605760 + 0.795647i \(0.707131\pi\)
\(674\) 5.30955 0.204516
\(675\) 3.17818 0.122328
\(676\) 1.00000 0.0384615
\(677\) 11.8524 0.455524 0.227762 0.973717i \(-0.426859\pi\)
0.227762 + 0.973717i \(0.426859\pi\)
\(678\) −15.9539 −0.612707
\(679\) 0 0
\(680\) −4.39827 −0.168666
\(681\) 9.77991 0.374767
\(682\) −10.0378 −0.384366
\(683\) 25.0576 0.958801 0.479401 0.877596i \(-0.340854\pi\)
0.479401 + 0.877596i \(0.340854\pi\)
\(684\) −1.34975 −0.0516090
\(685\) −3.48317 −0.133085
\(686\) 0 0
\(687\) −27.0721 −1.03286
\(688\) −3.16742 −0.120757
\(689\) −4.49462 −0.171231
\(690\) 0.877675 0.0334125
\(691\) 36.9935 1.40730 0.703650 0.710547i \(-0.251552\pi\)
0.703650 + 0.710547i \(0.251552\pi\)
\(692\) 1.80142 0.0684799
\(693\) 0 0
\(694\) 29.0628 1.10321
\(695\) −24.5734 −0.932123
\(696\) −0.155630 −0.00589915
\(697\) −10.8284 −0.410156
\(698\) 8.83576 0.334438
\(699\) 22.5838 0.854197
\(700\) 0 0
\(701\) −12.9986 −0.490950 −0.245475 0.969403i \(-0.578944\pi\)
−0.245475 + 0.969403i \(0.578944\pi\)
\(702\) 1.00000 0.0377426
\(703\) −10.2478 −0.386504
\(704\) 5.25858 0.198190
\(705\) 11.0024 0.414376
\(706\) 25.7030 0.967344
\(707\) 0 0
\(708\) 1.90883 0.0717384
\(709\) −37.1873 −1.39660 −0.698298 0.715807i \(-0.746059\pi\)
−0.698298 + 0.715807i \(0.746059\pi\)
\(710\) 7.16424 0.268869
\(711\) 5.74802 0.215568
\(712\) 10.7598 0.403241
\(713\) −1.24122 −0.0464841
\(714\) 0 0
\(715\) 7.09777 0.265442
\(716\) −0.274526 −0.0102595
\(717\) −13.0617 −0.487799
\(718\) −23.4211 −0.874068
\(719\) −6.51268 −0.242882 −0.121441 0.992599i \(-0.538752\pi\)
−0.121441 + 0.992599i \(0.538752\pi\)
\(720\) −1.34975 −0.0503022
\(721\) 0 0
\(722\) −17.1782 −0.639306
\(723\) 6.04509 0.224819
\(724\) −12.7750 −0.474780
\(725\) −0.494620 −0.0183697
\(726\) −16.6527 −0.618040
\(727\) −25.7968 −0.956752 −0.478376 0.878155i \(-0.658774\pi\)
−0.478376 + 0.878155i \(0.658774\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −7.22082 −0.267255
\(731\) −10.3213 −0.381747
\(732\) −2.26934 −0.0838773
\(733\) 46.2888 1.70971 0.854857 0.518863i \(-0.173645\pi\)
0.854857 + 0.518863i \(0.173645\pi\)
\(734\) 18.7442 0.691860
\(735\) 0 0
\(736\) 0.650251 0.0239686
\(737\) 9.91920 0.365378
\(738\) −3.32305 −0.122323
\(739\) −1.87210 −0.0688663 −0.0344331 0.999407i \(-0.510963\pi\)
−0.0344331 + 0.999407i \(0.510963\pi\)
\(740\) −10.2478 −0.376718
\(741\) −1.34975 −0.0495843
\(742\) 0 0
\(743\) 11.0211 0.404326 0.202163 0.979352i \(-0.435203\pi\)
0.202163 + 0.979352i \(0.435203\pi\)
\(744\) 1.90883 0.0699812
\(745\) −12.8313 −0.470101
\(746\) −19.9005 −0.728610
\(747\) −2.95736 −0.108204
\(748\) 17.1355 0.626537
\(749\) 0 0
\(750\) −11.0385 −0.403069
\(751\) 27.2892 0.995795 0.497898 0.867236i \(-0.334106\pi\)
0.497898 + 0.867236i \(0.334106\pi\)
\(752\) 8.15147 0.297254
\(753\) 10.7331 0.391136
\(754\) −0.155630 −0.00566772
\(755\) −17.7217 −0.644960
\(756\) 0 0
\(757\) −13.9754 −0.507946 −0.253973 0.967211i \(-0.581737\pi\)
−0.253973 + 0.967211i \(0.581737\pi\)
\(758\) −24.6760 −0.896272
\(759\) −3.41940 −0.124116
\(760\) 1.82182 0.0660845
\(761\) 6.01424 0.218016 0.109008 0.994041i \(-0.465233\pi\)
0.109008 + 0.994041i \(0.465233\pi\)
\(762\) 2.03776 0.0738203
\(763\) 0 0
\(764\) −19.1487 −0.692777
\(765\) −4.39827 −0.159020
\(766\) −21.2115 −0.766402
\(767\) 1.90883 0.0689240
\(768\) −1.00000 −0.0360844
\(769\) 9.25125 0.333609 0.166804 0.985990i \(-0.446655\pi\)
0.166804 + 0.985990i \(0.446655\pi\)
\(770\) 0 0
\(771\) −1.13969 −0.0410448
\(772\) 9.49016 0.341558
\(773\) −22.2218 −0.799263 −0.399632 0.916676i \(-0.630862\pi\)
−0.399632 + 0.916676i \(0.630862\pi\)
\(774\) −3.16742 −0.113850
\(775\) 6.06661 0.217919
\(776\) −11.3456 −0.407283
\(777\) 0 0
\(778\) −33.2073 −1.19054
\(779\) 4.48528 0.160702
\(780\) −1.34975 −0.0483288
\(781\) −27.9117 −0.998758
\(782\) 2.11890 0.0757716
\(783\) −0.155630 −0.00556177
\(784\) 0 0
\(785\) −3.13513 −0.111897
\(786\) −15.5615 −0.555061
\(787\) 29.5786 1.05436 0.527182 0.849752i \(-0.323249\pi\)
0.527182 + 0.849752i \(0.323249\pi\)
\(788\) 1.33826 0.0476737
\(789\) −25.2922 −0.900426
\(790\) −7.75839 −0.276031
\(791\) 0 0
\(792\) 5.25858 0.186856
\(793\) −2.26934 −0.0805867
\(794\) 21.3907 0.759127
\(795\) 6.06661 0.215161
\(796\) 21.0988 0.747827
\(797\) −5.40203 −0.191350 −0.0956749 0.995413i \(-0.530501\pi\)
−0.0956749 + 0.995413i \(0.530501\pi\)
\(798\) 0 0
\(799\) 26.5623 0.939705
\(800\) −3.17818 −0.112366
\(801\) 10.7598 0.380179
\(802\) −21.4745 −0.758292
\(803\) 28.1321 0.992760
\(804\) −1.88629 −0.0665242
\(805\) 0 0
\(806\) 1.90883 0.0672358
\(807\) 4.66174 0.164101
\(808\) 1.77991 0.0626169
\(809\) −44.1176 −1.55109 −0.775546 0.631291i \(-0.782525\pi\)
−0.775546 + 0.631291i \(0.782525\pi\)
\(810\) −1.34975 −0.0474254
\(811\) −2.21006 −0.0776058 −0.0388029 0.999247i \(-0.512354\pi\)
−0.0388029 + 0.999247i \(0.512354\pi\)
\(812\) 0 0
\(813\) −6.70538 −0.235168
\(814\) 39.9252 1.39938
\(815\) −29.1486 −1.02103
\(816\) −3.25858 −0.114073
\(817\) 4.27522 0.149571
\(818\) −4.18854 −0.146449
\(819\) 0 0
\(820\) 4.48528 0.156633
\(821\) −27.5113 −0.960151 −0.480076 0.877227i \(-0.659391\pi\)
−0.480076 + 0.877227i \(0.659391\pi\)
\(822\) −2.58060 −0.0900089
\(823\) 16.0711 0.560203 0.280102 0.959970i \(-0.409632\pi\)
0.280102 + 0.959970i \(0.409632\pi\)
\(824\) 17.9487 0.625274
\(825\) 16.7127 0.581862
\(826\) 0 0
\(827\) −53.9526 −1.87612 −0.938058 0.346478i \(-0.887377\pi\)
−0.938058 + 0.346478i \(0.887377\pi\)
\(828\) 0.650251 0.0225978
\(829\) 24.6125 0.854827 0.427414 0.904056i \(-0.359425\pi\)
0.427414 + 0.904056i \(0.359425\pi\)
\(830\) 3.99169 0.138554
\(831\) 29.1068 1.00970
\(832\) −1.00000 −0.0346688
\(833\) 0 0
\(834\) −18.2059 −0.630419
\(835\) 34.1920 1.18326
\(836\) −7.09777 −0.245481
\(837\) 1.90883 0.0659790
\(838\) 8.88501 0.306927
\(839\) −24.6672 −0.851606 −0.425803 0.904816i \(-0.640008\pi\)
−0.425803 + 0.904816i \(0.640008\pi\)
\(840\) 0 0
\(841\) −28.9758 −0.999165
\(842\) 9.59411 0.330635
\(843\) −0.356353 −0.0122735
\(844\) 7.16742 0.246713
\(845\) −1.34975 −0.0464328
\(846\) 8.15147 0.280253
\(847\) 0 0
\(848\) 4.49462 0.154346
\(849\) −21.8392 −0.749519
\(850\) −10.3564 −0.355220
\(851\) 4.93696 0.169237
\(852\) 5.30783 0.181843
\(853\) 22.0347 0.754455 0.377227 0.926121i \(-0.376878\pi\)
0.377227 + 0.926121i \(0.376878\pi\)
\(854\) 0 0
\(855\) 1.82182 0.0623050
\(856\) −11.4593 −0.391671
\(857\) −57.3284 −1.95830 −0.979150 0.203139i \(-0.934886\pi\)
−0.979150 + 0.203139i \(0.934886\pi\)
\(858\) 5.25858 0.179525
\(859\) −8.34804 −0.284832 −0.142416 0.989807i \(-0.545487\pi\)
−0.142416 + 0.989807i \(0.545487\pi\)
\(860\) 4.27522 0.145784
\(861\) 0 0
\(862\) 16.6897 0.568455
\(863\) −8.06661 −0.274591 −0.137295 0.990530i \(-0.543841\pi\)
−0.137295 + 0.990530i \(0.543841\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −2.43147 −0.0826725
\(866\) −12.8613 −0.437046
\(867\) 6.38163 0.216732
\(868\) 0 0
\(869\) 30.2264 1.02536
\(870\) 0.210062 0.00712176
\(871\) −1.88629 −0.0639144
\(872\) −1.02182 −0.0346032
\(873\) −11.3456 −0.383990
\(874\) −0.877675 −0.0296878
\(875\) 0 0
\(876\) −5.34975 −0.180751
\(877\) 26.4857 0.894359 0.447179 0.894444i \(-0.352429\pi\)
0.447179 + 0.894444i \(0.352429\pi\)
\(878\) 2.23359 0.0753800
\(879\) 9.18478 0.309795
\(880\) −7.09777 −0.239266
\(881\) 9.38606 0.316224 0.158112 0.987421i \(-0.449459\pi\)
0.158112 + 0.987421i \(0.449459\pi\)
\(882\) 0 0
\(883\) 0.824271 0.0277389 0.0138695 0.999904i \(-0.495585\pi\)
0.0138695 + 0.999904i \(0.495585\pi\)
\(884\) −3.25858 −0.109598
\(885\) −2.57645 −0.0866063
\(886\) 26.5093 0.890597
\(887\) −19.4079 −0.651654 −0.325827 0.945429i \(-0.605643\pi\)
−0.325827 + 0.945429i \(0.605643\pi\)
\(888\) −7.59239 −0.254784
\(889\) 0 0
\(890\) −14.5230 −0.486814
\(891\) 5.25858 0.176169
\(892\) −12.6593 −0.423865
\(893\) −11.0024 −0.368183
\(894\) −9.50641 −0.317942
\(895\) 0.370541 0.0123858
\(896\) 0 0
\(897\) 0.650251 0.0217112
\(898\) −28.1306 −0.938732
\(899\) −0.297072 −0.00990791
\(900\) −3.17818 −0.105939
\(901\) 14.6461 0.487932
\(902\) −17.4745 −0.581838
\(903\) 0 0
\(904\) 15.9539 0.530620
\(905\) 17.2431 0.573179
\(906\) −13.1297 −0.436204
\(907\) −13.9377 −0.462793 −0.231397 0.972860i \(-0.574329\pi\)
−0.231397 + 0.972860i \(0.574329\pi\)
\(908\) −9.77991 −0.324558
\(909\) 1.77991 0.0590357
\(910\) 0 0
\(911\) 51.9821 1.72224 0.861121 0.508400i \(-0.169763\pi\)
0.861121 + 0.508400i \(0.169763\pi\)
\(912\) 1.34975 0.0446947
\(913\) −15.5515 −0.514680
\(914\) −15.0715 −0.498521
\(915\) 3.06304 0.101261
\(916\) 27.0721 0.894487
\(917\) 0 0
\(918\) −3.25858 −0.107549
\(919\) −22.0568 −0.727588 −0.363794 0.931479i \(-0.618519\pi\)
−0.363794 + 0.931479i \(0.618519\pi\)
\(920\) −0.877675 −0.0289361
\(921\) 6.01321 0.198142
\(922\) 10.4492 0.344127
\(923\) 5.30783 0.174709
\(924\) 0 0
\(925\) −24.1300 −0.793388
\(926\) 12.0115 0.394722
\(927\) 17.9487 0.589514
\(928\) 0.155630 0.00510881
\(929\) −44.7485 −1.46815 −0.734075 0.679069i \(-0.762384\pi\)
−0.734075 + 0.679069i \(0.762384\pi\)
\(930\) −2.57645 −0.0844850
\(931\) 0 0
\(932\) −22.5838 −0.739756
\(933\) 13.8926 0.454823
\(934\) −21.6468 −0.708306
\(935\) −23.1287 −0.756389
\(936\) −1.00000 −0.0326860
\(937\) 9.50440 0.310495 0.155248 0.987876i \(-0.450382\pi\)
0.155248 + 0.987876i \(0.450382\pi\)
\(938\) 0 0
\(939\) −12.9206 −0.421649
\(940\) −11.0024 −0.358860
\(941\) −26.2392 −0.855373 −0.427686 0.903927i \(-0.640671\pi\)
−0.427686 + 0.903927i \(0.640671\pi\)
\(942\) −2.32275 −0.0756792
\(943\) −2.16081 −0.0703658
\(944\) −1.90883 −0.0621273
\(945\) 0 0
\(946\) −16.6561 −0.541537
\(947\) −22.1757 −0.720615 −0.360307 0.932834i \(-0.617328\pi\)
−0.360307 + 0.932834i \(0.617328\pi\)
\(948\) −5.74802 −0.186687
\(949\) −5.34975 −0.173660
\(950\) 4.28974 0.139178
\(951\) 28.4696 0.923191
\(952\) 0 0
\(953\) 30.2045 0.978418 0.489209 0.872167i \(-0.337285\pi\)
0.489209 + 0.872167i \(0.337285\pi\)
\(954\) 4.49462 0.145519
\(955\) 25.8460 0.836357
\(956\) 13.0617 0.422447
\(957\) −0.818395 −0.0264549
\(958\) −3.28211 −0.106040
\(959\) 0 0
\(960\) 1.34975 0.0435630
\(961\) −27.3564 −0.882463
\(962\) −7.59239 −0.244788
\(963\) −11.4593 −0.369271
\(964\) −6.04509 −0.194699
\(965\) −12.8093 −0.412347
\(966\) 0 0
\(967\) 60.7193 1.95260 0.976301 0.216418i \(-0.0694372\pi\)
0.976301 + 0.216418i \(0.0694372\pi\)
\(968\) 16.6527 0.535238
\(969\) 4.39827 0.141293
\(970\) 15.3137 0.491694
\(971\) −35.5279 −1.14015 −0.570073 0.821594i \(-0.693085\pi\)
−0.570073 + 0.821594i \(0.693085\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 25.3907 0.813570
\(975\) −3.17818 −0.101783
\(976\) 2.26934 0.0726399
\(977\) −6.06344 −0.193987 −0.0969933 0.995285i \(-0.530923\pi\)
−0.0969933 + 0.995285i \(0.530923\pi\)
\(978\) −21.5956 −0.690550
\(979\) 56.5813 1.80835
\(980\) 0 0
\(981\) −1.02182 −0.0326242
\(982\) −16.6368 −0.530900
\(983\) −6.78021 −0.216255 −0.108128 0.994137i \(-0.534485\pi\)
−0.108128 + 0.994137i \(0.534485\pi\)
\(984\) 3.32305 0.105935
\(985\) −1.80632 −0.0575541
\(986\) 0.507134 0.0161504
\(987\) 0 0
\(988\) 1.34975 0.0429412
\(989\) −2.05961 −0.0654919
\(990\) −7.09777 −0.225582
\(991\) −23.1225 −0.734509 −0.367255 0.930120i \(-0.619702\pi\)
−0.367255 + 0.930120i \(0.619702\pi\)
\(992\) −1.90883 −0.0606055
\(993\) 18.7383 0.594642
\(994\) 0 0
\(995\) −28.4781 −0.902816
\(996\) 2.95736 0.0937074
\(997\) −22.3460 −0.707704 −0.353852 0.935301i \(-0.615128\pi\)
−0.353852 + 0.935301i \(0.615128\pi\)
\(998\) −15.0495 −0.476385
\(999\) −7.59239 −0.240213
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 3822.2.a.bz.1.2 4
7.6 odd 2 3822.2.a.ca.1.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3822.2.a.bz.1.2 4 1.1 even 1 trivial
3822.2.a.ca.1.3 yes 4 7.6 odd 2