Properties

Label 666.2.a.h.1.1
Level $666$
Weight $2$
Character 666.1
Self dual yes
Analytic conductor $5.318$
Analytic rank $1$
Dimension $2$
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] = [666,2,Mod(1,666)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(666, 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("666.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 666 = 2 \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 666.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: \(5.31803677462\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 666.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} -1.56155 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} -1.56155 q^{7} -1.00000 q^{8} +2.00000 q^{10} +1.56155 q^{11} +6.68466 q^{13} +1.56155 q^{14} +1.00000 q^{16} -3.56155 q^{17} -3.56155 q^{19} -2.00000 q^{20} -1.56155 q^{22} -8.68466 q^{23} -1.00000 q^{25} -6.68466 q^{26} -1.56155 q^{28} +1.12311 q^{29} -9.12311 q^{31} -1.00000 q^{32} +3.56155 q^{34} +3.12311 q^{35} -1.00000 q^{37} +3.56155 q^{38} +2.00000 q^{40} -11.1231 q^{41} +1.12311 q^{43} +1.56155 q^{44} +8.68466 q^{46} +10.2462 q^{47} -4.56155 q^{49} +1.00000 q^{50} +6.68466 q^{52} +1.56155 q^{53} -3.12311 q^{55} +1.56155 q^{56} -1.12311 q^{58} -0.876894 q^{59} -12.2462 q^{61} +9.12311 q^{62} +1.00000 q^{64} -13.3693 q^{65} +2.24621 q^{67} -3.56155 q^{68} -3.12311 q^{70} -2.24621 q^{71} +3.56155 q^{73} +1.00000 q^{74} -3.56155 q^{76} -2.43845 q^{77} -6.00000 q^{79} -2.00000 q^{80} +11.1231 q^{82} -14.9309 q^{83} +7.12311 q^{85} -1.12311 q^{86} -1.56155 q^{88} +12.9309 q^{89} -10.4384 q^{91} -8.68466 q^{92} -10.2462 q^{94} +7.12311 q^{95} +2.87689 q^{97} +4.56155 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} + q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} + q^{7} - 2 q^{8} + 4 q^{10} - q^{11} + q^{13} - q^{14} + 2 q^{16} - 3 q^{17} - 3 q^{19} - 4 q^{20} + q^{22} - 5 q^{23} - 2 q^{25} - q^{26} + q^{28} - 6 q^{29} - 10 q^{31} - 2 q^{32} + 3 q^{34} - 2 q^{35} - 2 q^{37} + 3 q^{38} + 4 q^{40} - 14 q^{41} - 6 q^{43} - q^{44} + 5 q^{46} + 4 q^{47} - 5 q^{49} + 2 q^{50} + q^{52} - q^{53} + 2 q^{55} - q^{56} + 6 q^{58} - 10 q^{59} - 8 q^{61} + 10 q^{62} + 2 q^{64} - 2 q^{65} - 12 q^{67} - 3 q^{68} + 2 q^{70} + 12 q^{71} + 3 q^{73} + 2 q^{74} - 3 q^{76} - 9 q^{77} - 12 q^{79} - 4 q^{80} + 14 q^{82} - q^{83} + 6 q^{85} + 6 q^{86} + q^{88} - 3 q^{89} - 25 q^{91} - 5 q^{92} - 4 q^{94} + 6 q^{95} + 14 q^{97} + 5 q^{98}+O(q^{100}) \) 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
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −2.00000 −0.894427 −0.447214 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 0 0
\(7\) −1.56155 −0.590211 −0.295106 0.955465i \(-0.595355\pi\)
−0.295106 + 0.955465i \(0.595355\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 2.00000 0.632456
\(11\) 1.56155 0.470826 0.235413 0.971895i \(-0.424356\pi\)
0.235413 + 0.971895i \(0.424356\pi\)
\(12\) 0 0
\(13\) 6.68466 1.85399 0.926995 0.375073i \(-0.122382\pi\)
0.926995 + 0.375073i \(0.122382\pi\)
\(14\) 1.56155 0.417343
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.56155 −0.863803 −0.431902 0.901921i \(-0.642157\pi\)
−0.431902 + 0.901921i \(0.642157\pi\)
\(18\) 0 0
\(19\) −3.56155 −0.817076 −0.408538 0.912741i \(-0.633961\pi\)
−0.408538 + 0.912741i \(0.633961\pi\)
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) −1.56155 −0.332924
\(23\) −8.68466 −1.81088 −0.905438 0.424478i \(-0.860458\pi\)
−0.905438 + 0.424478i \(0.860458\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) −6.68466 −1.31097
\(27\) 0 0
\(28\) −1.56155 −0.295106
\(29\) 1.12311 0.208555 0.104278 0.994548i \(-0.466747\pi\)
0.104278 + 0.994548i \(0.466747\pi\)
\(30\) 0 0
\(31\) −9.12311 −1.63856 −0.819279 0.573395i \(-0.805626\pi\)
−0.819279 + 0.573395i \(0.805626\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 3.56155 0.610801
\(35\) 3.12311 0.527901
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) 3.56155 0.577760
\(39\) 0 0
\(40\) 2.00000 0.316228
\(41\) −11.1231 −1.73714 −0.868569 0.495569i \(-0.834960\pi\)
−0.868569 + 0.495569i \(0.834960\pi\)
\(42\) 0 0
\(43\) 1.12311 0.171272 0.0856360 0.996326i \(-0.472708\pi\)
0.0856360 + 0.996326i \(0.472708\pi\)
\(44\) 1.56155 0.235413
\(45\) 0 0
\(46\) 8.68466 1.28048
\(47\) 10.2462 1.49456 0.747282 0.664507i \(-0.231359\pi\)
0.747282 + 0.664507i \(0.231359\pi\)
\(48\) 0 0
\(49\) −4.56155 −0.651650
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 6.68466 0.926995
\(53\) 1.56155 0.214496 0.107248 0.994232i \(-0.465796\pi\)
0.107248 + 0.994232i \(0.465796\pi\)
\(54\) 0 0
\(55\) −3.12311 −0.421119
\(56\) 1.56155 0.208671
\(57\) 0 0
\(58\) −1.12311 −0.147471
\(59\) −0.876894 −0.114162 −0.0570810 0.998370i \(-0.518179\pi\)
−0.0570810 + 0.998370i \(0.518179\pi\)
\(60\) 0 0
\(61\) −12.2462 −1.56797 −0.783983 0.620782i \(-0.786815\pi\)
−0.783983 + 0.620782i \(0.786815\pi\)
\(62\) 9.12311 1.15864
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −13.3693 −1.65826
\(66\) 0 0
\(67\) 2.24621 0.274418 0.137209 0.990542i \(-0.456187\pi\)
0.137209 + 0.990542i \(0.456187\pi\)
\(68\) −3.56155 −0.431902
\(69\) 0 0
\(70\) −3.12311 −0.373283
\(71\) −2.24621 −0.266576 −0.133288 0.991077i \(-0.542554\pi\)
−0.133288 + 0.991077i \(0.542554\pi\)
\(72\) 0 0
\(73\) 3.56155 0.416848 0.208424 0.978039i \(-0.433167\pi\)
0.208424 + 0.978039i \(0.433167\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) −3.56155 −0.408538
\(77\) −2.43845 −0.277887
\(78\) 0 0
\(79\) −6.00000 −0.675053 −0.337526 0.941316i \(-0.609590\pi\)
−0.337526 + 0.941316i \(0.609590\pi\)
\(80\) −2.00000 −0.223607
\(81\) 0 0
\(82\) 11.1231 1.22834
\(83\) −14.9309 −1.63888 −0.819438 0.573168i \(-0.805714\pi\)
−0.819438 + 0.573168i \(0.805714\pi\)
\(84\) 0 0
\(85\) 7.12311 0.772609
\(86\) −1.12311 −0.121108
\(87\) 0 0
\(88\) −1.56155 −0.166462
\(89\) 12.9309 1.37067 0.685335 0.728228i \(-0.259656\pi\)
0.685335 + 0.728228i \(0.259656\pi\)
\(90\) 0 0
\(91\) −10.4384 −1.09425
\(92\) −8.68466 −0.905438
\(93\) 0 0
\(94\) −10.2462 −1.05682
\(95\) 7.12311 0.730815
\(96\) 0 0
\(97\) 2.87689 0.292104 0.146052 0.989277i \(-0.453343\pi\)
0.146052 + 0.989277i \(0.453343\pi\)
\(98\) 4.56155 0.460786
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) 11.1231 1.10679 0.553395 0.832919i \(-0.313332\pi\)
0.553395 + 0.832919i \(0.313332\pi\)
\(102\) 0 0
\(103\) −10.0000 −0.985329 −0.492665 0.870219i \(-0.663977\pi\)
−0.492665 + 0.870219i \(0.663977\pi\)
\(104\) −6.68466 −0.655485
\(105\) 0 0
\(106\) −1.56155 −0.151671
\(107\) 6.43845 0.622428 0.311214 0.950340i \(-0.399264\pi\)
0.311214 + 0.950340i \(0.399264\pi\)
\(108\) 0 0
\(109\) 3.56155 0.341135 0.170567 0.985346i \(-0.445440\pi\)
0.170567 + 0.985346i \(0.445440\pi\)
\(110\) 3.12311 0.297776
\(111\) 0 0
\(112\) −1.56155 −0.147553
\(113\) 12.2462 1.15203 0.576013 0.817440i \(-0.304608\pi\)
0.576013 + 0.817440i \(0.304608\pi\)
\(114\) 0 0
\(115\) 17.3693 1.61970
\(116\) 1.12311 0.104278
\(117\) 0 0
\(118\) 0.876894 0.0807247
\(119\) 5.56155 0.509827
\(120\) 0 0
\(121\) −8.56155 −0.778323
\(122\) 12.2462 1.10872
\(123\) 0 0
\(124\) −9.12311 −0.819279
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) 5.56155 0.493508 0.246754 0.969078i \(-0.420636\pi\)
0.246754 + 0.969078i \(0.420636\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 13.3693 1.17257
\(131\) −15.1231 −1.32131 −0.660656 0.750689i \(-0.729722\pi\)
−0.660656 + 0.750689i \(0.729722\pi\)
\(132\) 0 0
\(133\) 5.56155 0.482248
\(134\) −2.24621 −0.194043
\(135\) 0 0
\(136\) 3.56155 0.305401
\(137\) 5.75379 0.491579 0.245790 0.969323i \(-0.420953\pi\)
0.245790 + 0.969323i \(0.420953\pi\)
\(138\) 0 0
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 3.12311 0.263951
\(141\) 0 0
\(142\) 2.24621 0.188498
\(143\) 10.4384 0.872907
\(144\) 0 0
\(145\) −2.24621 −0.186538
\(146\) −3.56155 −0.294756
\(147\) 0 0
\(148\) −1.00000 −0.0821995
\(149\) −3.12311 −0.255855 −0.127927 0.991784i \(-0.540832\pi\)
−0.127927 + 0.991784i \(0.540832\pi\)
\(150\) 0 0
\(151\) 3.31534 0.269799 0.134899 0.990859i \(-0.456929\pi\)
0.134899 + 0.990859i \(0.456929\pi\)
\(152\) 3.56155 0.288880
\(153\) 0 0
\(154\) 2.43845 0.196496
\(155\) 18.2462 1.46557
\(156\) 0 0
\(157\) −10.4924 −0.837386 −0.418693 0.908128i \(-0.637512\pi\)
−0.418693 + 0.908128i \(0.637512\pi\)
\(158\) 6.00000 0.477334
\(159\) 0 0
\(160\) 2.00000 0.158114
\(161\) 13.5616 1.06880
\(162\) 0 0
\(163\) −20.9309 −1.63943 −0.819716 0.572770i \(-0.805869\pi\)
−0.819716 + 0.572770i \(0.805869\pi\)
\(164\) −11.1231 −0.868569
\(165\) 0 0
\(166\) 14.9309 1.15886
\(167\) 16.6847 1.29110 0.645549 0.763719i \(-0.276629\pi\)
0.645549 + 0.763719i \(0.276629\pi\)
\(168\) 0 0
\(169\) 31.6847 2.43728
\(170\) −7.12311 −0.546317
\(171\) 0 0
\(172\) 1.12311 0.0856360
\(173\) −1.56155 −0.118723 −0.0593613 0.998237i \(-0.518906\pi\)
−0.0593613 + 0.998237i \(0.518906\pi\)
\(174\) 0 0
\(175\) 1.56155 0.118042
\(176\) 1.56155 0.117706
\(177\) 0 0
\(178\) −12.9309 −0.969210
\(179\) −7.12311 −0.532406 −0.266203 0.963917i \(-0.585769\pi\)
−0.266203 + 0.963917i \(0.585769\pi\)
\(180\) 0 0
\(181\) 14.4924 1.07721 0.538607 0.842557i \(-0.318951\pi\)
0.538607 + 0.842557i \(0.318951\pi\)
\(182\) 10.4384 0.773749
\(183\) 0 0
\(184\) 8.68466 0.640242
\(185\) 2.00000 0.147043
\(186\) 0 0
\(187\) −5.56155 −0.406701
\(188\) 10.2462 0.747282
\(189\) 0 0
\(190\) −7.12311 −0.516764
\(191\) −2.43845 −0.176440 −0.0882199 0.996101i \(-0.528118\pi\)
−0.0882199 + 0.996101i \(0.528118\pi\)
\(192\) 0 0
\(193\) 0.630683 0.0453976 0.0226988 0.999742i \(-0.492774\pi\)
0.0226988 + 0.999742i \(0.492774\pi\)
\(194\) −2.87689 −0.206549
\(195\) 0 0
\(196\) −4.56155 −0.325825
\(197\) −11.8078 −0.841268 −0.420634 0.907230i \(-0.638192\pi\)
−0.420634 + 0.907230i \(0.638192\pi\)
\(198\) 0 0
\(199\) 13.1231 0.930272 0.465136 0.885239i \(-0.346005\pi\)
0.465136 + 0.885239i \(0.346005\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) −11.1231 −0.782619
\(203\) −1.75379 −0.123092
\(204\) 0 0
\(205\) 22.2462 1.55374
\(206\) 10.0000 0.696733
\(207\) 0 0
\(208\) 6.68466 0.463498
\(209\) −5.56155 −0.384701
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 1.56155 0.107248
\(213\) 0 0
\(214\) −6.43845 −0.440123
\(215\) −2.24621 −0.153190
\(216\) 0 0
\(217\) 14.2462 0.967096
\(218\) −3.56155 −0.241219
\(219\) 0 0
\(220\) −3.12311 −0.210560
\(221\) −23.8078 −1.60148
\(222\) 0 0
\(223\) 28.4924 1.90799 0.953997 0.299817i \(-0.0969255\pi\)
0.953997 + 0.299817i \(0.0969255\pi\)
\(224\) 1.56155 0.104336
\(225\) 0 0
\(226\) −12.2462 −0.814606
\(227\) 5.36932 0.356374 0.178187 0.983997i \(-0.442977\pi\)
0.178187 + 0.983997i \(0.442977\pi\)
\(228\) 0 0
\(229\) 27.3693 1.80862 0.904308 0.426881i \(-0.140388\pi\)
0.904308 + 0.426881i \(0.140388\pi\)
\(230\) −17.3693 −1.14530
\(231\) 0 0
\(232\) −1.12311 −0.0737355
\(233\) 16.0000 1.04819 0.524097 0.851658i \(-0.324403\pi\)
0.524097 + 0.851658i \(0.324403\pi\)
\(234\) 0 0
\(235\) −20.4924 −1.33678
\(236\) −0.876894 −0.0570810
\(237\) 0 0
\(238\) −5.56155 −0.360502
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 0 0
\(241\) −8.24621 −0.531185 −0.265593 0.964085i \(-0.585568\pi\)
−0.265593 + 0.964085i \(0.585568\pi\)
\(242\) 8.56155 0.550357
\(243\) 0 0
\(244\) −12.2462 −0.783983
\(245\) 9.12311 0.582854
\(246\) 0 0
\(247\) −23.8078 −1.51485
\(248\) 9.12311 0.579318
\(249\) 0 0
\(250\) −12.0000 −0.758947
\(251\) −7.12311 −0.449606 −0.224803 0.974404i \(-0.572174\pi\)
−0.224803 + 0.974404i \(0.572174\pi\)
\(252\) 0 0
\(253\) −13.5616 −0.852608
\(254\) −5.56155 −0.348963
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −23.5616 −1.46973 −0.734865 0.678214i \(-0.762754\pi\)
−0.734865 + 0.678214i \(0.762754\pi\)
\(258\) 0 0
\(259\) 1.56155 0.0970302
\(260\) −13.3693 −0.829130
\(261\) 0 0
\(262\) 15.1231 0.934309
\(263\) −13.7538 −0.848095 −0.424047 0.905640i \(-0.639391\pi\)
−0.424047 + 0.905640i \(0.639391\pi\)
\(264\) 0 0
\(265\) −3.12311 −0.191851
\(266\) −5.56155 −0.341001
\(267\) 0 0
\(268\) 2.24621 0.137209
\(269\) −18.0540 −1.10077 −0.550385 0.834911i \(-0.685519\pi\)
−0.550385 + 0.834911i \(0.685519\pi\)
\(270\) 0 0
\(271\) −5.75379 −0.349518 −0.174759 0.984611i \(-0.555915\pi\)
−0.174759 + 0.984611i \(0.555915\pi\)
\(272\) −3.56155 −0.215951
\(273\) 0 0
\(274\) −5.75379 −0.347599
\(275\) −1.56155 −0.0941652
\(276\) 0 0
\(277\) 27.5616 1.65601 0.828007 0.560718i \(-0.189475\pi\)
0.828007 + 0.560718i \(0.189475\pi\)
\(278\) 12.0000 0.719712
\(279\) 0 0
\(280\) −3.12311 −0.186641
\(281\) 12.9309 0.771391 0.385696 0.922626i \(-0.373962\pi\)
0.385696 + 0.922626i \(0.373962\pi\)
\(282\) 0 0
\(283\) −6.19224 −0.368090 −0.184045 0.982918i \(-0.558919\pi\)
−0.184045 + 0.982918i \(0.558919\pi\)
\(284\) −2.24621 −0.133288
\(285\) 0 0
\(286\) −10.4384 −0.617238
\(287\) 17.3693 1.02528
\(288\) 0 0
\(289\) −4.31534 −0.253844
\(290\) 2.24621 0.131902
\(291\) 0 0
\(292\) 3.56155 0.208424
\(293\) 28.3002 1.65331 0.826657 0.562706i \(-0.190240\pi\)
0.826657 + 0.562706i \(0.190240\pi\)
\(294\) 0 0
\(295\) 1.75379 0.102110
\(296\) 1.00000 0.0581238
\(297\) 0 0
\(298\) 3.12311 0.180917
\(299\) −58.0540 −3.35735
\(300\) 0 0
\(301\) −1.75379 −0.101087
\(302\) −3.31534 −0.190776
\(303\) 0 0
\(304\) −3.56155 −0.204269
\(305\) 24.4924 1.40243
\(306\) 0 0
\(307\) −17.3693 −0.991319 −0.495660 0.868517i \(-0.665074\pi\)
−0.495660 + 0.868517i \(0.665074\pi\)
\(308\) −2.43845 −0.138943
\(309\) 0 0
\(310\) −18.2462 −1.03632
\(311\) −30.2462 −1.71511 −0.857553 0.514396i \(-0.828016\pi\)
−0.857553 + 0.514396i \(0.828016\pi\)
\(312\) 0 0
\(313\) −2.87689 −0.162612 −0.0813058 0.996689i \(-0.525909\pi\)
−0.0813058 + 0.996689i \(0.525909\pi\)
\(314\) 10.4924 0.592122
\(315\) 0 0
\(316\) −6.00000 −0.337526
\(317\) 7.12311 0.400073 0.200037 0.979788i \(-0.435894\pi\)
0.200037 + 0.979788i \(0.435894\pi\)
\(318\) 0 0
\(319\) 1.75379 0.0981933
\(320\) −2.00000 −0.111803
\(321\) 0 0
\(322\) −13.5616 −0.755756
\(323\) 12.6847 0.705793
\(324\) 0 0
\(325\) −6.68466 −0.370798
\(326\) 20.9309 1.15925
\(327\) 0 0
\(328\) 11.1231 0.614171
\(329\) −16.0000 −0.882109
\(330\) 0 0
\(331\) −23.3693 −1.28449 −0.642247 0.766498i \(-0.721998\pi\)
−0.642247 + 0.766498i \(0.721998\pi\)
\(332\) −14.9309 −0.819438
\(333\) 0 0
\(334\) −16.6847 −0.912944
\(335\) −4.49242 −0.245447
\(336\) 0 0
\(337\) −4.93087 −0.268602 −0.134301 0.990941i \(-0.542879\pi\)
−0.134301 + 0.990941i \(0.542879\pi\)
\(338\) −31.6847 −1.72342
\(339\) 0 0
\(340\) 7.12311 0.386305
\(341\) −14.2462 −0.771476
\(342\) 0 0
\(343\) 18.0540 0.974823
\(344\) −1.12311 −0.0605538
\(345\) 0 0
\(346\) 1.56155 0.0839496
\(347\) −11.6155 −0.623554 −0.311777 0.950155i \(-0.600924\pi\)
−0.311777 + 0.950155i \(0.600924\pi\)
\(348\) 0 0
\(349\) −6.49242 −0.347531 −0.173766 0.984787i \(-0.555594\pi\)
−0.173766 + 0.984787i \(0.555594\pi\)
\(350\) −1.56155 −0.0834685
\(351\) 0 0
\(352\) −1.56155 −0.0832310
\(353\) −26.0000 −1.38384 −0.691920 0.721974i \(-0.743235\pi\)
−0.691920 + 0.721974i \(0.743235\pi\)
\(354\) 0 0
\(355\) 4.49242 0.238433
\(356\) 12.9309 0.685335
\(357\) 0 0
\(358\) 7.12311 0.376468
\(359\) 28.4924 1.50377 0.751886 0.659293i \(-0.229144\pi\)
0.751886 + 0.659293i \(0.229144\pi\)
\(360\) 0 0
\(361\) −6.31534 −0.332386
\(362\) −14.4924 −0.761705
\(363\) 0 0
\(364\) −10.4384 −0.547123
\(365\) −7.12311 −0.372840
\(366\) 0 0
\(367\) 24.6847 1.28853 0.644264 0.764803i \(-0.277164\pi\)
0.644264 + 0.764803i \(0.277164\pi\)
\(368\) −8.68466 −0.452719
\(369\) 0 0
\(370\) −2.00000 −0.103975
\(371\) −2.43845 −0.126598
\(372\) 0 0
\(373\) 27.3693 1.41713 0.708565 0.705646i \(-0.249343\pi\)
0.708565 + 0.705646i \(0.249343\pi\)
\(374\) 5.56155 0.287581
\(375\) 0 0
\(376\) −10.2462 −0.528408
\(377\) 7.50758 0.386660
\(378\) 0 0
\(379\) 14.2462 0.731779 0.365889 0.930658i \(-0.380765\pi\)
0.365889 + 0.930658i \(0.380765\pi\)
\(380\) 7.12311 0.365408
\(381\) 0 0
\(382\) 2.43845 0.124762
\(383\) −11.8078 −0.603349 −0.301674 0.953411i \(-0.597546\pi\)
−0.301674 + 0.953411i \(0.597546\pi\)
\(384\) 0 0
\(385\) 4.87689 0.248550
\(386\) −0.630683 −0.0321009
\(387\) 0 0
\(388\) 2.87689 0.146052
\(389\) −16.7386 −0.848682 −0.424341 0.905502i \(-0.639494\pi\)
−0.424341 + 0.905502i \(0.639494\pi\)
\(390\) 0 0
\(391\) 30.9309 1.56424
\(392\) 4.56155 0.230393
\(393\) 0 0
\(394\) 11.8078 0.594867
\(395\) 12.0000 0.603786
\(396\) 0 0
\(397\) −14.4924 −0.727354 −0.363677 0.931525i \(-0.618479\pi\)
−0.363677 + 0.931525i \(0.618479\pi\)
\(398\) −13.1231 −0.657802
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) −11.5616 −0.577356 −0.288678 0.957426i \(-0.593216\pi\)
−0.288678 + 0.957426i \(0.593216\pi\)
\(402\) 0 0
\(403\) −60.9848 −3.03787
\(404\) 11.1231 0.553395
\(405\) 0 0
\(406\) 1.75379 0.0870391
\(407\) −1.56155 −0.0774033
\(408\) 0 0
\(409\) 20.2462 1.00111 0.500555 0.865705i \(-0.333129\pi\)
0.500555 + 0.865705i \(0.333129\pi\)
\(410\) −22.2462 −1.09866
\(411\) 0 0
\(412\) −10.0000 −0.492665
\(413\) 1.36932 0.0673797
\(414\) 0 0
\(415\) 29.8617 1.46586
\(416\) −6.68466 −0.327742
\(417\) 0 0
\(418\) 5.56155 0.272024
\(419\) −23.3153 −1.13903 −0.569514 0.821981i \(-0.692869\pi\)
−0.569514 + 0.821981i \(0.692869\pi\)
\(420\) 0 0
\(421\) 20.2462 0.986740 0.493370 0.869820i \(-0.335765\pi\)
0.493370 + 0.869820i \(0.335765\pi\)
\(422\) 4.00000 0.194717
\(423\) 0 0
\(424\) −1.56155 −0.0758357
\(425\) 3.56155 0.172761
\(426\) 0 0
\(427\) 19.1231 0.925432
\(428\) 6.43845 0.311214
\(429\) 0 0
\(430\) 2.24621 0.108322
\(431\) 37.1771 1.79076 0.895378 0.445306i \(-0.146905\pi\)
0.895378 + 0.445306i \(0.146905\pi\)
\(432\) 0 0
\(433\) 20.4384 0.982209 0.491105 0.871101i \(-0.336593\pi\)
0.491105 + 0.871101i \(0.336593\pi\)
\(434\) −14.2462 −0.683840
\(435\) 0 0
\(436\) 3.56155 0.170567
\(437\) 30.9309 1.47962
\(438\) 0 0
\(439\) −10.4924 −0.500776 −0.250388 0.968146i \(-0.580558\pi\)
−0.250388 + 0.968146i \(0.580558\pi\)
\(440\) 3.12311 0.148888
\(441\) 0 0
\(442\) 23.8078 1.13242
\(443\) 18.7386 0.890299 0.445150 0.895456i \(-0.353150\pi\)
0.445150 + 0.895456i \(0.353150\pi\)
\(444\) 0 0
\(445\) −25.8617 −1.22596
\(446\) −28.4924 −1.34916
\(447\) 0 0
\(448\) −1.56155 −0.0737764
\(449\) −32.2462 −1.52179 −0.760896 0.648873i \(-0.775240\pi\)
−0.760896 + 0.648873i \(0.775240\pi\)
\(450\) 0 0
\(451\) −17.3693 −0.817889
\(452\) 12.2462 0.576013
\(453\) 0 0
\(454\) −5.36932 −0.251995
\(455\) 20.8769 0.978724
\(456\) 0 0
\(457\) 5.61553 0.262683 0.131342 0.991337i \(-0.458072\pi\)
0.131342 + 0.991337i \(0.458072\pi\)
\(458\) −27.3693 −1.27888
\(459\) 0 0
\(460\) 17.3693 0.809849
\(461\) 26.9848 1.25681 0.628405 0.777887i \(-0.283708\pi\)
0.628405 + 0.777887i \(0.283708\pi\)
\(462\) 0 0
\(463\) −11.3693 −0.528377 −0.264188 0.964471i \(-0.585104\pi\)
−0.264188 + 0.964471i \(0.585104\pi\)
\(464\) 1.12311 0.0521389
\(465\) 0 0
\(466\) −16.0000 −0.741186
\(467\) 19.6155 0.907698 0.453849 0.891079i \(-0.350050\pi\)
0.453849 + 0.891079i \(0.350050\pi\)
\(468\) 0 0
\(469\) −3.50758 −0.161965
\(470\) 20.4924 0.945245
\(471\) 0 0
\(472\) 0.876894 0.0403623
\(473\) 1.75379 0.0806393
\(474\) 0 0
\(475\) 3.56155 0.163415
\(476\) 5.56155 0.254913
\(477\) 0 0
\(478\) 8.00000 0.365911
\(479\) −42.0540 −1.92150 −0.960748 0.277424i \(-0.910519\pi\)
−0.960748 + 0.277424i \(0.910519\pi\)
\(480\) 0 0
\(481\) −6.68466 −0.304794
\(482\) 8.24621 0.375605
\(483\) 0 0
\(484\) −8.56155 −0.389161
\(485\) −5.75379 −0.261266
\(486\) 0 0
\(487\) 13.1231 0.594665 0.297332 0.954774i \(-0.403903\pi\)
0.297332 + 0.954774i \(0.403903\pi\)
\(488\) 12.2462 0.554360
\(489\) 0 0
\(490\) −9.12311 −0.412140
\(491\) 1.06913 0.0482492 0.0241246 0.999709i \(-0.492320\pi\)
0.0241246 + 0.999709i \(0.492320\pi\)
\(492\) 0 0
\(493\) −4.00000 −0.180151
\(494\) 23.8078 1.07116
\(495\) 0 0
\(496\) −9.12311 −0.409640
\(497\) 3.50758 0.157336
\(498\) 0 0
\(499\) −33.4233 −1.49623 −0.748116 0.663568i \(-0.769041\pi\)
−0.748116 + 0.663568i \(0.769041\pi\)
\(500\) 12.0000 0.536656
\(501\) 0 0
\(502\) 7.12311 0.317920
\(503\) −32.9848 −1.47072 −0.735361 0.677676i \(-0.762987\pi\)
−0.735361 + 0.677676i \(0.762987\pi\)
\(504\) 0 0
\(505\) −22.2462 −0.989943
\(506\) 13.5616 0.602885
\(507\) 0 0
\(508\) 5.56155 0.246754
\(509\) −1.94602 −0.0862560 −0.0431280 0.999070i \(-0.513732\pi\)
−0.0431280 + 0.999070i \(0.513732\pi\)
\(510\) 0 0
\(511\) −5.56155 −0.246029
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 23.5616 1.03926
\(515\) 20.0000 0.881305
\(516\) 0 0
\(517\) 16.0000 0.703679
\(518\) −1.56155 −0.0686107
\(519\) 0 0
\(520\) 13.3693 0.586283
\(521\) −24.0000 −1.05146 −0.525730 0.850652i \(-0.676208\pi\)
−0.525730 + 0.850652i \(0.676208\pi\)
\(522\) 0 0
\(523\) −29.1231 −1.27346 −0.636732 0.771085i \(-0.719714\pi\)
−0.636732 + 0.771085i \(0.719714\pi\)
\(524\) −15.1231 −0.660656
\(525\) 0 0
\(526\) 13.7538 0.599694
\(527\) 32.4924 1.41539
\(528\) 0 0
\(529\) 52.4233 2.27927
\(530\) 3.12311 0.135659
\(531\) 0 0
\(532\) 5.56155 0.241124
\(533\) −74.3542 −3.22064
\(534\) 0 0
\(535\) −12.8769 −0.556717
\(536\) −2.24621 −0.0970215
\(537\) 0 0
\(538\) 18.0540 0.778362
\(539\) −7.12311 −0.306814
\(540\) 0 0
\(541\) −3.06913 −0.131952 −0.0659761 0.997821i \(-0.521016\pi\)
−0.0659761 + 0.997821i \(0.521016\pi\)
\(542\) 5.75379 0.247146
\(543\) 0 0
\(544\) 3.56155 0.152700
\(545\) −7.12311 −0.305120
\(546\) 0 0
\(547\) −2.19224 −0.0937332 −0.0468666 0.998901i \(-0.514924\pi\)
−0.0468666 + 0.998901i \(0.514924\pi\)
\(548\) 5.75379 0.245790
\(549\) 0 0
\(550\) 1.56155 0.0665848
\(551\) −4.00000 −0.170406
\(552\) 0 0
\(553\) 9.36932 0.398424
\(554\) −27.5616 −1.17098
\(555\) 0 0
\(556\) −12.0000 −0.508913
\(557\) 34.0000 1.44063 0.720313 0.693649i \(-0.243998\pi\)
0.720313 + 0.693649i \(0.243998\pi\)
\(558\) 0 0
\(559\) 7.50758 0.317537
\(560\) 3.12311 0.131975
\(561\) 0 0
\(562\) −12.9309 −0.545456
\(563\) 8.87689 0.374116 0.187058 0.982349i \(-0.440105\pi\)
0.187058 + 0.982349i \(0.440105\pi\)
\(564\) 0 0
\(565\) −24.4924 −1.03040
\(566\) 6.19224 0.260279
\(567\) 0 0
\(568\) 2.24621 0.0942489
\(569\) 38.3002 1.60563 0.802814 0.596230i \(-0.203335\pi\)
0.802814 + 0.596230i \(0.203335\pi\)
\(570\) 0 0
\(571\) 10.7386 0.449398 0.224699 0.974428i \(-0.427860\pi\)
0.224699 + 0.974428i \(0.427860\pi\)
\(572\) 10.4384 0.436453
\(573\) 0 0
\(574\) −17.3693 −0.724981
\(575\) 8.68466 0.362175
\(576\) 0 0
\(577\) −9.12311 −0.379800 −0.189900 0.981803i \(-0.560816\pi\)
−0.189900 + 0.981803i \(0.560816\pi\)
\(578\) 4.31534 0.179495
\(579\) 0 0
\(580\) −2.24621 −0.0932688
\(581\) 23.3153 0.967283
\(582\) 0 0
\(583\) 2.43845 0.100990
\(584\) −3.56155 −0.147378
\(585\) 0 0
\(586\) −28.3002 −1.16907
\(587\) 13.7538 0.567680 0.283840 0.958872i \(-0.408392\pi\)
0.283840 + 0.958872i \(0.408392\pi\)
\(588\) 0 0
\(589\) 32.4924 1.33883
\(590\) −1.75379 −0.0722023
\(591\) 0 0
\(592\) −1.00000 −0.0410997
\(593\) −33.3693 −1.37031 −0.685157 0.728396i \(-0.740266\pi\)
−0.685157 + 0.728396i \(0.740266\pi\)
\(594\) 0 0
\(595\) −11.1231 −0.456003
\(596\) −3.12311 −0.127927
\(597\) 0 0
\(598\) 58.0540 2.37400
\(599\) 34.2462 1.39926 0.699631 0.714504i \(-0.253348\pi\)
0.699631 + 0.714504i \(0.253348\pi\)
\(600\) 0 0
\(601\) −9.31534 −0.379981 −0.189990 0.981786i \(-0.560846\pi\)
−0.189990 + 0.981786i \(0.560846\pi\)
\(602\) 1.75379 0.0714791
\(603\) 0 0
\(604\) 3.31534 0.134899
\(605\) 17.1231 0.696153
\(606\) 0 0
\(607\) 35.8617 1.45558 0.727792 0.685798i \(-0.240547\pi\)
0.727792 + 0.685798i \(0.240547\pi\)
\(608\) 3.56155 0.144440
\(609\) 0 0
\(610\) −24.4924 −0.991669
\(611\) 68.4924 2.77091
\(612\) 0 0
\(613\) −30.9848 −1.25147 −0.625733 0.780037i \(-0.715200\pi\)
−0.625733 + 0.780037i \(0.715200\pi\)
\(614\) 17.3693 0.700969
\(615\) 0 0
\(616\) 2.43845 0.0982478
\(617\) 23.1231 0.930901 0.465451 0.885074i \(-0.345892\pi\)
0.465451 + 0.885074i \(0.345892\pi\)
\(618\) 0 0
\(619\) 18.7386 0.753169 0.376585 0.926382i \(-0.377098\pi\)
0.376585 + 0.926382i \(0.377098\pi\)
\(620\) 18.2462 0.732785
\(621\) 0 0
\(622\) 30.2462 1.21276
\(623\) −20.1922 −0.808985
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 2.87689 0.114984
\(627\) 0 0
\(628\) −10.4924 −0.418693
\(629\) 3.56155 0.142008
\(630\) 0 0
\(631\) −12.7386 −0.507117 −0.253559 0.967320i \(-0.581601\pi\)
−0.253559 + 0.967320i \(0.581601\pi\)
\(632\) 6.00000 0.238667
\(633\) 0 0
\(634\) −7.12311 −0.282895
\(635\) −11.1231 −0.441407
\(636\) 0 0
\(637\) −30.4924 −1.20815
\(638\) −1.75379 −0.0694332
\(639\) 0 0
\(640\) 2.00000 0.0790569
\(641\) 21.8617 0.863487 0.431743 0.901996i \(-0.357899\pi\)
0.431743 + 0.901996i \(0.357899\pi\)
\(642\) 0 0
\(643\) 2.19224 0.0864533 0.0432267 0.999065i \(-0.486236\pi\)
0.0432267 + 0.999065i \(0.486236\pi\)
\(644\) 13.5616 0.534400
\(645\) 0 0
\(646\) −12.6847 −0.499071
\(647\) −14.9309 −0.586993 −0.293497 0.955960i \(-0.594819\pi\)
−0.293497 + 0.955960i \(0.594819\pi\)
\(648\) 0 0
\(649\) −1.36932 −0.0537504
\(650\) 6.68466 0.262194
\(651\) 0 0
\(652\) −20.9309 −0.819716
\(653\) −16.7386 −0.655033 −0.327517 0.944845i \(-0.606212\pi\)
−0.327517 + 0.944845i \(0.606212\pi\)
\(654\) 0 0
\(655\) 30.2462 1.18182
\(656\) −11.1231 −0.434284
\(657\) 0 0
\(658\) 16.0000 0.623745
\(659\) 18.2462 0.710771 0.355386 0.934720i \(-0.384350\pi\)
0.355386 + 0.934720i \(0.384350\pi\)
\(660\) 0 0
\(661\) −19.5616 −0.760856 −0.380428 0.924810i \(-0.624223\pi\)
−0.380428 + 0.924810i \(0.624223\pi\)
\(662\) 23.3693 0.908274
\(663\) 0 0
\(664\) 14.9309 0.579430
\(665\) −11.1231 −0.431336
\(666\) 0 0
\(667\) −9.75379 −0.377668
\(668\) 16.6847 0.645549
\(669\) 0 0
\(670\) 4.49242 0.173557
\(671\) −19.1231 −0.738239
\(672\) 0 0
\(673\) 0.0539753 0.00208060 0.00104030 0.999999i \(-0.499669\pi\)
0.00104030 + 0.999999i \(0.499669\pi\)
\(674\) 4.93087 0.189930
\(675\) 0 0
\(676\) 31.6847 1.21864
\(677\) 22.4384 0.862380 0.431190 0.902261i \(-0.358094\pi\)
0.431190 + 0.902261i \(0.358094\pi\)
\(678\) 0 0
\(679\) −4.49242 −0.172403
\(680\) −7.12311 −0.273159
\(681\) 0 0
\(682\) 14.2462 0.545516
\(683\) −24.1080 −0.922465 −0.461233 0.887279i \(-0.652593\pi\)
−0.461233 + 0.887279i \(0.652593\pi\)
\(684\) 0 0
\(685\) −11.5076 −0.439682
\(686\) −18.0540 −0.689304
\(687\) 0 0
\(688\) 1.12311 0.0428180
\(689\) 10.4384 0.397673
\(690\) 0 0
\(691\) −31.1231 −1.18398 −0.591989 0.805946i \(-0.701657\pi\)
−0.591989 + 0.805946i \(0.701657\pi\)
\(692\) −1.56155 −0.0593613
\(693\) 0 0
\(694\) 11.6155 0.440919
\(695\) 24.0000 0.910372
\(696\) 0 0
\(697\) 39.6155 1.50055
\(698\) 6.49242 0.245742
\(699\) 0 0
\(700\) 1.56155 0.0590211
\(701\) −20.6307 −0.779210 −0.389605 0.920982i \(-0.627388\pi\)
−0.389605 + 0.920982i \(0.627388\pi\)
\(702\) 0 0
\(703\) 3.56155 0.134327
\(704\) 1.56155 0.0588532
\(705\) 0 0
\(706\) 26.0000 0.978523
\(707\) −17.3693 −0.653240
\(708\) 0 0
\(709\) −42.3002 −1.58862 −0.794308 0.607515i \(-0.792167\pi\)
−0.794308 + 0.607515i \(0.792167\pi\)
\(710\) −4.49242 −0.168598
\(711\) 0 0
\(712\) −12.9309 −0.484605
\(713\) 79.2311 2.96723
\(714\) 0 0
\(715\) −20.8769 −0.780752
\(716\) −7.12311 −0.266203
\(717\) 0 0
\(718\) −28.4924 −1.06333
\(719\) −22.6307 −0.843982 −0.421991 0.906600i \(-0.638669\pi\)
−0.421991 + 0.906600i \(0.638669\pi\)
\(720\) 0 0
\(721\) 15.6155 0.581553
\(722\) 6.31534 0.235033
\(723\) 0 0
\(724\) 14.4924 0.538607
\(725\) −1.12311 −0.0417111
\(726\) 0 0
\(727\) −31.3693 −1.16342 −0.581712 0.813395i \(-0.697617\pi\)
−0.581712 + 0.813395i \(0.697617\pi\)
\(728\) 10.4384 0.386875
\(729\) 0 0
\(730\) 7.12311 0.263638
\(731\) −4.00000 −0.147945
\(732\) 0 0
\(733\) −47.8617 −1.76781 −0.883907 0.467663i \(-0.845096\pi\)
−0.883907 + 0.467663i \(0.845096\pi\)
\(734\) −24.6847 −0.911127
\(735\) 0 0
\(736\) 8.68466 0.320121
\(737\) 3.50758 0.129203
\(738\) 0 0
\(739\) 47.6155 1.75157 0.875783 0.482705i \(-0.160346\pi\)
0.875783 + 0.482705i \(0.160346\pi\)
\(740\) 2.00000 0.0735215
\(741\) 0 0
\(742\) 2.43845 0.0895182
\(743\) 12.9848 0.476368 0.238184 0.971220i \(-0.423448\pi\)
0.238184 + 0.971220i \(0.423448\pi\)
\(744\) 0 0
\(745\) 6.24621 0.228843
\(746\) −27.3693 −1.00206
\(747\) 0 0
\(748\) −5.56155 −0.203351
\(749\) −10.0540 −0.367364
\(750\) 0 0
\(751\) −39.2311 −1.43156 −0.715781 0.698325i \(-0.753929\pi\)
−0.715781 + 0.698325i \(0.753929\pi\)
\(752\) 10.2462 0.373641
\(753\) 0 0
\(754\) −7.50758 −0.273410
\(755\) −6.63068 −0.241315
\(756\) 0 0
\(757\) −10.6847 −0.388341 −0.194170 0.980968i \(-0.562201\pi\)
−0.194170 + 0.980968i \(0.562201\pi\)
\(758\) −14.2462 −0.517446
\(759\) 0 0
\(760\) −7.12311 −0.258382
\(761\) −40.1080 −1.45391 −0.726956 0.686684i \(-0.759066\pi\)
−0.726956 + 0.686684i \(0.759066\pi\)
\(762\) 0 0
\(763\) −5.56155 −0.201342
\(764\) −2.43845 −0.0882199
\(765\) 0 0
\(766\) 11.8078 0.426632
\(767\) −5.86174 −0.211655
\(768\) 0 0
\(769\) −5.61553 −0.202501 −0.101251 0.994861i \(-0.532284\pi\)
−0.101251 + 0.994861i \(0.532284\pi\)
\(770\) −4.87689 −0.175751
\(771\) 0 0
\(772\) 0.630683 0.0226988
\(773\) 16.6847 0.600105 0.300053 0.953923i \(-0.402996\pi\)
0.300053 + 0.953923i \(0.402996\pi\)
\(774\) 0 0
\(775\) 9.12311 0.327712
\(776\) −2.87689 −0.103274
\(777\) 0 0
\(778\) 16.7386 0.600109
\(779\) 39.6155 1.41937
\(780\) 0 0
\(781\) −3.50758 −0.125511
\(782\) −30.9309 −1.10609
\(783\) 0 0
\(784\) −4.56155 −0.162913
\(785\) 20.9848 0.748981
\(786\) 0 0
\(787\) 35.2311 1.25585 0.627926 0.778273i \(-0.283904\pi\)
0.627926 + 0.778273i \(0.283904\pi\)
\(788\) −11.8078 −0.420634
\(789\) 0 0
\(790\) −12.0000 −0.426941
\(791\) −19.1231 −0.679939
\(792\) 0 0
\(793\) −81.8617 −2.90700
\(794\) 14.4924 0.514317
\(795\) 0 0
\(796\) 13.1231 0.465136
\(797\) −34.0000 −1.20434 −0.602171 0.798367i \(-0.705697\pi\)
−0.602171 + 0.798367i \(0.705697\pi\)
\(798\) 0 0
\(799\) −36.4924 −1.29101
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) 11.5616 0.408253
\(803\) 5.56155 0.196263
\(804\) 0 0
\(805\) −27.1231 −0.955964
\(806\) 60.9848 2.14810
\(807\) 0 0
\(808\) −11.1231 −0.391309
\(809\) 37.4233 1.31573 0.657866 0.753135i \(-0.271459\pi\)
0.657866 + 0.753135i \(0.271459\pi\)
\(810\) 0 0
\(811\) 17.3693 0.609919 0.304960 0.952365i \(-0.401357\pi\)
0.304960 + 0.952365i \(0.401357\pi\)
\(812\) −1.75379 −0.0615459
\(813\) 0 0
\(814\) 1.56155 0.0547324
\(815\) 41.8617 1.46635
\(816\) 0 0
\(817\) −4.00000 −0.139942
\(818\) −20.2462 −0.707892
\(819\) 0 0
\(820\) 22.2462 0.776871
\(821\) 35.3153 1.23251 0.616257 0.787545i \(-0.288648\pi\)
0.616257 + 0.787545i \(0.288648\pi\)
\(822\) 0 0
\(823\) 19.8078 0.690455 0.345228 0.938519i \(-0.387802\pi\)
0.345228 + 0.938519i \(0.387802\pi\)
\(824\) 10.0000 0.348367
\(825\) 0 0
\(826\) −1.36932 −0.0476446
\(827\) 50.2462 1.74723 0.873616 0.486616i \(-0.161769\pi\)
0.873616 + 0.486616i \(0.161769\pi\)
\(828\) 0 0
\(829\) 3.56155 0.123698 0.0618489 0.998086i \(-0.480300\pi\)
0.0618489 + 0.998086i \(0.480300\pi\)
\(830\) −29.8617 −1.03652
\(831\) 0 0
\(832\) 6.68466 0.231749
\(833\) 16.2462 0.562898
\(834\) 0 0
\(835\) −33.3693 −1.15479
\(836\) −5.56155 −0.192350
\(837\) 0 0
\(838\) 23.3153 0.805415
\(839\) −31.1231 −1.07449 −0.537244 0.843427i \(-0.680535\pi\)
−0.537244 + 0.843427i \(0.680535\pi\)
\(840\) 0 0
\(841\) −27.7386 −0.956505
\(842\) −20.2462 −0.697731
\(843\) 0 0
\(844\) −4.00000 −0.137686
\(845\) −63.3693 −2.17997
\(846\) 0 0
\(847\) 13.3693 0.459375
\(848\) 1.56155 0.0536239
\(849\) 0 0
\(850\) −3.56155 −0.122160
\(851\) 8.68466 0.297706
\(852\) 0 0
\(853\) 36.4384 1.24763 0.623814 0.781573i \(-0.285582\pi\)
0.623814 + 0.781573i \(0.285582\pi\)
\(854\) −19.1231 −0.654379
\(855\) 0 0
\(856\) −6.43845 −0.220062
\(857\) −43.6695 −1.49172 −0.745861 0.666102i \(-0.767962\pi\)
−0.745861 + 0.666102i \(0.767962\pi\)
\(858\) 0 0
\(859\) 0.0539753 0.00184161 0.000920807 1.00000i \(-0.499707\pi\)
0.000920807 1.00000i \(0.499707\pi\)
\(860\) −2.24621 −0.0765952
\(861\) 0 0
\(862\) −37.1771 −1.26626
\(863\) −57.3693 −1.95287 −0.976437 0.215802i \(-0.930763\pi\)
−0.976437 + 0.215802i \(0.930763\pi\)
\(864\) 0 0
\(865\) 3.12311 0.106189
\(866\) −20.4384 −0.694527
\(867\) 0 0
\(868\) 14.2462 0.483548
\(869\) −9.36932 −0.317832
\(870\) 0 0
\(871\) 15.0152 0.508769
\(872\) −3.56155 −0.120609
\(873\) 0 0
\(874\) −30.9309 −1.04625
\(875\) −18.7386 −0.633481
\(876\) 0 0
\(877\) 34.0000 1.14810 0.574049 0.818821i \(-0.305372\pi\)
0.574049 + 0.818821i \(0.305372\pi\)
\(878\) 10.4924 0.354102
\(879\) 0 0
\(880\) −3.12311 −0.105280
\(881\) −16.4924 −0.555644 −0.277822 0.960633i \(-0.589613\pi\)
−0.277822 + 0.960633i \(0.589613\pi\)
\(882\) 0 0
\(883\) −28.4384 −0.957030 −0.478515 0.878079i \(-0.658825\pi\)
−0.478515 + 0.878079i \(0.658825\pi\)
\(884\) −23.8078 −0.800742
\(885\) 0 0
\(886\) −18.7386 −0.629537
\(887\) −23.1231 −0.776398 −0.388199 0.921576i \(-0.626903\pi\)
−0.388199 + 0.921576i \(0.626903\pi\)
\(888\) 0 0
\(889\) −8.68466 −0.291274
\(890\) 25.8617 0.866887
\(891\) 0 0
\(892\) 28.4924 0.953997
\(893\) −36.4924 −1.22117
\(894\) 0 0
\(895\) 14.2462 0.476198
\(896\) 1.56155 0.0521678
\(897\) 0 0
\(898\) 32.2462 1.07607
\(899\) −10.2462 −0.341730
\(900\) 0 0
\(901\) −5.56155 −0.185282
\(902\) 17.3693 0.578335
\(903\) 0 0
\(904\) −12.2462 −0.407303
\(905\) −28.9848 −0.963489
\(906\) 0 0
\(907\) 38.3002 1.27174 0.635868 0.771797i \(-0.280642\pi\)
0.635868 + 0.771797i \(0.280642\pi\)
\(908\) 5.36932 0.178187
\(909\) 0 0
\(910\) −20.8769 −0.692062
\(911\) −36.4924 −1.20905 −0.604524 0.796587i \(-0.706637\pi\)
−0.604524 + 0.796587i \(0.706637\pi\)
\(912\) 0 0
\(913\) −23.3153 −0.771625
\(914\) −5.61553 −0.185745
\(915\) 0 0
\(916\) 27.3693 0.904308
\(917\) 23.6155 0.779853
\(918\) 0 0
\(919\) 44.2462 1.45955 0.729774 0.683689i \(-0.239625\pi\)
0.729774 + 0.683689i \(0.239625\pi\)
\(920\) −17.3693 −0.572649
\(921\) 0 0
\(922\) −26.9848 −0.888699
\(923\) −15.0152 −0.494230
\(924\) 0 0
\(925\) 1.00000 0.0328798
\(926\) 11.3693 0.373619
\(927\) 0 0
\(928\) −1.12311 −0.0368677
\(929\) −46.2462 −1.51729 −0.758644 0.651505i \(-0.774138\pi\)
−0.758644 + 0.651505i \(0.774138\pi\)
\(930\) 0 0
\(931\) 16.2462 0.532448
\(932\) 16.0000 0.524097
\(933\) 0 0
\(934\) −19.6155 −0.641840
\(935\) 11.1231 0.363764
\(936\) 0 0
\(937\) −32.2462 −1.05344 −0.526719 0.850040i \(-0.676578\pi\)
−0.526719 + 0.850040i \(0.676578\pi\)
\(938\) 3.50758 0.114526
\(939\) 0 0
\(940\) −20.4924 −0.668389
\(941\) 49.8617 1.62545 0.812723 0.582650i \(-0.197984\pi\)
0.812723 + 0.582650i \(0.197984\pi\)
\(942\) 0 0
\(943\) 96.6004 3.14574
\(944\) −0.876894 −0.0285405
\(945\) 0 0
\(946\) −1.75379 −0.0570206
\(947\) 5.36932 0.174479 0.0872397 0.996187i \(-0.472195\pi\)
0.0872397 + 0.996187i \(0.472195\pi\)
\(948\) 0 0
\(949\) 23.8078 0.772833
\(950\) −3.56155 −0.115552
\(951\) 0 0
\(952\) −5.56155 −0.180251
\(953\) −49.8617 −1.61518 −0.807590 0.589744i \(-0.799229\pi\)
−0.807590 + 0.589744i \(0.799229\pi\)
\(954\) 0 0
\(955\) 4.87689 0.157813
\(956\) −8.00000 −0.258738
\(957\) 0 0
\(958\) 42.0540 1.35870
\(959\) −8.98485 −0.290136
\(960\) 0 0
\(961\) 52.2311 1.68487
\(962\) 6.68466 0.215522
\(963\) 0 0
\(964\) −8.24621 −0.265593
\(965\) −1.26137 −0.0406048
\(966\) 0 0
\(967\) −3.36932 −0.108350 −0.0541750 0.998531i \(-0.517253\pi\)
−0.0541750 + 0.998531i \(0.517253\pi\)
\(968\) 8.56155 0.275179
\(969\) 0 0
\(970\) 5.75379 0.184743
\(971\) −9.75379 −0.313014 −0.156507 0.987677i \(-0.550023\pi\)
−0.156507 + 0.987677i \(0.550023\pi\)
\(972\) 0 0
\(973\) 18.7386 0.600733
\(974\) −13.1231 −0.420491
\(975\) 0 0
\(976\) −12.2462 −0.391992
\(977\) 33.9157 1.08506 0.542530 0.840036i \(-0.317467\pi\)
0.542530 + 0.840036i \(0.317467\pi\)
\(978\) 0 0
\(979\) 20.1922 0.645347
\(980\) 9.12311 0.291427
\(981\) 0 0
\(982\) −1.06913 −0.0341173
\(983\) 19.6155 0.625638 0.312819 0.949813i \(-0.398727\pi\)
0.312819 + 0.949813i \(0.398727\pi\)
\(984\) 0 0
\(985\) 23.6155 0.752453
\(986\) 4.00000 0.127386
\(987\) 0 0
\(988\) −23.8078 −0.757426
\(989\) −9.75379 −0.310152
\(990\) 0 0
\(991\) 60.7386 1.92943 0.964713 0.263303i \(-0.0848119\pi\)
0.964713 + 0.263303i \(0.0848119\pi\)
\(992\) 9.12311 0.289659
\(993\) 0 0
\(994\) −3.50758 −0.111254
\(995\) −26.2462 −0.832061
\(996\) 0 0
\(997\) −36.9309 −1.16961 −0.584806 0.811173i \(-0.698829\pi\)
−0.584806 + 0.811173i \(0.698829\pi\)
\(998\) 33.4233 1.05800
\(999\) 0 0
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 666.2.a.h.1.1 2
3.2 odd 2 666.2.a.k.1.1 yes 2
4.3 odd 2 5328.2.a.y.1.2 2
12.11 even 2 5328.2.a.bh.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
666.2.a.h.1.1 2 1.1 even 1 trivial
666.2.a.k.1.1 yes 2 3.2 odd 2
5328.2.a.y.1.2 2 4.3 odd 2
5328.2.a.bh.1.2 2 12.11 even 2