Properties

Label 6036.2.a.h.1.4
Level $6036$
Weight $2$
Character 6036.1
Self dual yes
Analytic conductor $48.198$
Analytic rank $0$
Dimension $24$
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] = [6036,2,Mod(1,6036)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6036, 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("6036.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6036 = 2^{2} \cdot 3 \cdot 503 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6036.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(48.1977026600\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Character \(\chi\) \(=\) 6036.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -2.71451 q^{5} -1.53717 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -2.71451 q^{5} -1.53717 q^{7} +1.00000 q^{9} +3.37579 q^{11} +3.94514 q^{13} -2.71451 q^{15} +2.09305 q^{17} -1.92796 q^{19} -1.53717 q^{21} +0.550188 q^{23} +2.36854 q^{25} +1.00000 q^{27} +5.84336 q^{29} -8.35732 q^{31} +3.37579 q^{33} +4.17267 q^{35} +9.61307 q^{37} +3.94514 q^{39} -11.4801 q^{41} -2.25876 q^{43} -2.71451 q^{45} -7.28095 q^{47} -4.63710 q^{49} +2.09305 q^{51} +6.23615 q^{53} -9.16359 q^{55} -1.92796 q^{57} +0.619676 q^{59} +14.4658 q^{61} -1.53717 q^{63} -10.7091 q^{65} -7.35747 q^{67} +0.550188 q^{69} +2.63521 q^{71} +13.8616 q^{73} +2.36854 q^{75} -5.18917 q^{77} +2.77019 q^{79} +1.00000 q^{81} +4.13535 q^{83} -5.68160 q^{85} +5.84336 q^{87} -12.3614 q^{89} -6.06436 q^{91} -8.35732 q^{93} +5.23346 q^{95} +10.5240 q^{97} +3.37579 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 24 q^{3} + 18 q^{5} + 9 q^{7} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q + 24 q^{3} + 18 q^{5} + 9 q^{7} + 24 q^{9} + 9 q^{11} + 7 q^{13} + 18 q^{15} + 12 q^{17} - q^{19} + 9 q^{21} + 22 q^{23} + 48 q^{25} + 24 q^{27} + 44 q^{29} + 15 q^{31} + 9 q^{33} + 3 q^{35} + 18 q^{37} + 7 q^{39} + 37 q^{41} - 8 q^{43} + 18 q^{45} + 24 q^{47} + 63 q^{49} + 12 q^{51} + 59 q^{53} + 20 q^{55} - q^{57} + 44 q^{59} + 33 q^{61} + 9 q^{63} + 15 q^{65} + 13 q^{67} + 22 q^{69} + 47 q^{71} + 28 q^{73} + 48 q^{75} + 19 q^{77} + 12 q^{79} + 24 q^{81} + 17 q^{83} + 31 q^{85} + 44 q^{87} + 41 q^{89} - q^{91} + 15 q^{93} + 58 q^{95} + 51 q^{97} + 9 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −2.71451 −1.21396 −0.606982 0.794716i \(-0.707620\pi\)
−0.606982 + 0.794716i \(0.707620\pi\)
\(6\) 0 0
\(7\) −1.53717 −0.580997 −0.290498 0.956875i \(-0.593821\pi\)
−0.290498 + 0.956875i \(0.593821\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.37579 1.01784 0.508919 0.860815i \(-0.330045\pi\)
0.508919 + 0.860815i \(0.330045\pi\)
\(12\) 0 0
\(13\) 3.94514 1.09418 0.547092 0.837072i \(-0.315735\pi\)
0.547092 + 0.837072i \(0.315735\pi\)
\(14\) 0 0
\(15\) −2.71451 −0.700882
\(16\) 0 0
\(17\) 2.09305 0.507640 0.253820 0.967251i \(-0.418313\pi\)
0.253820 + 0.967251i \(0.418313\pi\)
\(18\) 0 0
\(19\) −1.92796 −0.442305 −0.221152 0.975239i \(-0.570982\pi\)
−0.221152 + 0.975239i \(0.570982\pi\)
\(20\) 0 0
\(21\) −1.53717 −0.335439
\(22\) 0 0
\(23\) 0.550188 0.114722 0.0573611 0.998353i \(-0.481731\pi\)
0.0573611 + 0.998353i \(0.481731\pi\)
\(24\) 0 0
\(25\) 2.36854 0.473709
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 5.84336 1.08508 0.542542 0.840029i \(-0.317462\pi\)
0.542542 + 0.840029i \(0.317462\pi\)
\(30\) 0 0
\(31\) −8.35732 −1.50102 −0.750510 0.660859i \(-0.770192\pi\)
−0.750510 + 0.660859i \(0.770192\pi\)
\(32\) 0 0
\(33\) 3.37579 0.587649
\(34\) 0 0
\(35\) 4.17267 0.705309
\(36\) 0 0
\(37\) 9.61307 1.58038 0.790189 0.612863i \(-0.209982\pi\)
0.790189 + 0.612863i \(0.209982\pi\)
\(38\) 0 0
\(39\) 3.94514 0.631728
\(40\) 0 0
\(41\) −11.4801 −1.79289 −0.896444 0.443158i \(-0.853858\pi\)
−0.896444 + 0.443158i \(0.853858\pi\)
\(42\) 0 0
\(43\) −2.25876 −0.344457 −0.172229 0.985057i \(-0.555097\pi\)
−0.172229 + 0.985057i \(0.555097\pi\)
\(44\) 0 0
\(45\) −2.71451 −0.404655
\(46\) 0 0
\(47\) −7.28095 −1.06204 −0.531018 0.847361i \(-0.678190\pi\)
−0.531018 + 0.847361i \(0.678190\pi\)
\(48\) 0 0
\(49\) −4.63710 −0.662443
\(50\) 0 0
\(51\) 2.09305 0.293086
\(52\) 0 0
\(53\) 6.23615 0.856601 0.428300 0.903636i \(-0.359113\pi\)
0.428300 + 0.903636i \(0.359113\pi\)
\(54\) 0 0
\(55\) −9.16359 −1.23562
\(56\) 0 0
\(57\) −1.92796 −0.255365
\(58\) 0 0
\(59\) 0.619676 0.0806750 0.0403375 0.999186i \(-0.487157\pi\)
0.0403375 + 0.999186i \(0.487157\pi\)
\(60\) 0 0
\(61\) 14.4658 1.85216 0.926078 0.377332i \(-0.123158\pi\)
0.926078 + 0.377332i \(0.123158\pi\)
\(62\) 0 0
\(63\) −1.53717 −0.193666
\(64\) 0 0
\(65\) −10.7091 −1.32830
\(66\) 0 0
\(67\) −7.35747 −0.898858 −0.449429 0.893316i \(-0.648372\pi\)
−0.449429 + 0.893316i \(0.648372\pi\)
\(68\) 0 0
\(69\) 0.550188 0.0662349
\(70\) 0 0
\(71\) 2.63521 0.312742 0.156371 0.987698i \(-0.450020\pi\)
0.156371 + 0.987698i \(0.450020\pi\)
\(72\) 0 0
\(73\) 13.8616 1.62238 0.811188 0.584785i \(-0.198821\pi\)
0.811188 + 0.584785i \(0.198821\pi\)
\(74\) 0 0
\(75\) 2.36854 0.273496
\(76\) 0 0
\(77\) −5.18917 −0.591360
\(78\) 0 0
\(79\) 2.77019 0.311671 0.155835 0.987783i \(-0.450193\pi\)
0.155835 + 0.987783i \(0.450193\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 4.13535 0.453914 0.226957 0.973905i \(-0.427122\pi\)
0.226957 + 0.973905i \(0.427122\pi\)
\(84\) 0 0
\(85\) −5.68160 −0.616257
\(86\) 0 0
\(87\) 5.84336 0.626474
\(88\) 0 0
\(89\) −12.3614 −1.31030 −0.655152 0.755497i \(-0.727396\pi\)
−0.655152 + 0.755497i \(0.727396\pi\)
\(90\) 0 0
\(91\) −6.06436 −0.635718
\(92\) 0 0
\(93\) −8.35732 −0.866614
\(94\) 0 0
\(95\) 5.23346 0.536942
\(96\) 0 0
\(97\) 10.5240 1.06855 0.534275 0.845311i \(-0.320585\pi\)
0.534275 + 0.845311i \(0.320585\pi\)
\(98\) 0 0
\(99\) 3.37579 0.339279
\(100\) 0 0
\(101\) 15.1634 1.50882 0.754409 0.656405i \(-0.227924\pi\)
0.754409 + 0.656405i \(0.227924\pi\)
\(102\) 0 0
\(103\) 4.09047 0.403046 0.201523 0.979484i \(-0.435411\pi\)
0.201523 + 0.979484i \(0.435411\pi\)
\(104\) 0 0
\(105\) 4.17267 0.407210
\(106\) 0 0
\(107\) 11.2461 1.08720 0.543600 0.839344i \(-0.317061\pi\)
0.543600 + 0.839344i \(0.317061\pi\)
\(108\) 0 0
\(109\) −15.9901 −1.53158 −0.765788 0.643094i \(-0.777651\pi\)
−0.765788 + 0.643094i \(0.777651\pi\)
\(110\) 0 0
\(111\) 9.61307 0.912432
\(112\) 0 0
\(113\) −6.33778 −0.596209 −0.298104 0.954533i \(-0.596354\pi\)
−0.298104 + 0.954533i \(0.596354\pi\)
\(114\) 0 0
\(115\) −1.49349 −0.139269
\(116\) 0 0
\(117\) 3.94514 0.364728
\(118\) 0 0
\(119\) −3.21738 −0.294937
\(120\) 0 0
\(121\) 0.395928 0.0359935
\(122\) 0 0
\(123\) −11.4801 −1.03512
\(124\) 0 0
\(125\) 7.14311 0.638899
\(126\) 0 0
\(127\) −1.52367 −0.135204 −0.0676020 0.997712i \(-0.521535\pi\)
−0.0676020 + 0.997712i \(0.521535\pi\)
\(128\) 0 0
\(129\) −2.25876 −0.198872
\(130\) 0 0
\(131\) 20.7658 1.81431 0.907157 0.420793i \(-0.138248\pi\)
0.907157 + 0.420793i \(0.138248\pi\)
\(132\) 0 0
\(133\) 2.96361 0.256978
\(134\) 0 0
\(135\) −2.71451 −0.233627
\(136\) 0 0
\(137\) 11.4083 0.974681 0.487340 0.873212i \(-0.337967\pi\)
0.487340 + 0.873212i \(0.337967\pi\)
\(138\) 0 0
\(139\) 2.71570 0.230342 0.115171 0.993346i \(-0.463258\pi\)
0.115171 + 0.993346i \(0.463258\pi\)
\(140\) 0 0
\(141\) −7.28095 −0.613166
\(142\) 0 0
\(143\) 13.3179 1.11370
\(144\) 0 0
\(145\) −15.8618 −1.31725
\(146\) 0 0
\(147\) −4.63710 −0.382461
\(148\) 0 0
\(149\) 10.6937 0.876064 0.438032 0.898959i \(-0.355676\pi\)
0.438032 + 0.898959i \(0.355676\pi\)
\(150\) 0 0
\(151\) −10.5063 −0.854987 −0.427493 0.904018i \(-0.640603\pi\)
−0.427493 + 0.904018i \(0.640603\pi\)
\(152\) 0 0
\(153\) 2.09305 0.169213
\(154\) 0 0
\(155\) 22.6860 1.82218
\(156\) 0 0
\(157\) −10.3552 −0.826436 −0.413218 0.910632i \(-0.635595\pi\)
−0.413218 + 0.910632i \(0.635595\pi\)
\(158\) 0 0
\(159\) 6.23615 0.494559
\(160\) 0 0
\(161\) −0.845735 −0.0666532
\(162\) 0 0
\(163\) −6.33893 −0.496504 −0.248252 0.968696i \(-0.579856\pi\)
−0.248252 + 0.968696i \(0.579856\pi\)
\(164\) 0 0
\(165\) −9.16359 −0.713385
\(166\) 0 0
\(167\) 0.395120 0.0305753 0.0152877 0.999883i \(-0.495134\pi\)
0.0152877 + 0.999883i \(0.495134\pi\)
\(168\) 0 0
\(169\) 2.56411 0.197239
\(170\) 0 0
\(171\) −1.92796 −0.147435
\(172\) 0 0
\(173\) 17.3781 1.32123 0.660614 0.750725i \(-0.270296\pi\)
0.660614 + 0.750725i \(0.270296\pi\)
\(174\) 0 0
\(175\) −3.64086 −0.275223
\(176\) 0 0
\(177\) 0.619676 0.0465777
\(178\) 0 0
\(179\) −3.15569 −0.235867 −0.117934 0.993021i \(-0.537627\pi\)
−0.117934 + 0.993021i \(0.537627\pi\)
\(180\) 0 0
\(181\) 13.9671 1.03817 0.519085 0.854723i \(-0.326273\pi\)
0.519085 + 0.854723i \(0.326273\pi\)
\(182\) 0 0
\(183\) 14.4658 1.06934
\(184\) 0 0
\(185\) −26.0947 −1.91852
\(186\) 0 0
\(187\) 7.06570 0.516695
\(188\) 0 0
\(189\) −1.53717 −0.111813
\(190\) 0 0
\(191\) −17.7722 −1.28595 −0.642974 0.765888i \(-0.722300\pi\)
−0.642974 + 0.765888i \(0.722300\pi\)
\(192\) 0 0
\(193\) 8.00421 0.576155 0.288078 0.957607i \(-0.406984\pi\)
0.288078 + 0.957607i \(0.406984\pi\)
\(194\) 0 0
\(195\) −10.7091 −0.766895
\(196\) 0 0
\(197\) 11.1777 0.796376 0.398188 0.917304i \(-0.369639\pi\)
0.398188 + 0.917304i \(0.369639\pi\)
\(198\) 0 0
\(199\) 7.15795 0.507414 0.253707 0.967281i \(-0.418350\pi\)
0.253707 + 0.967281i \(0.418350\pi\)
\(200\) 0 0
\(201\) −7.35747 −0.518956
\(202\) 0 0
\(203\) −8.98225 −0.630430
\(204\) 0 0
\(205\) 31.1627 2.17650
\(206\) 0 0
\(207\) 0.550188 0.0382407
\(208\) 0 0
\(209\) −6.50838 −0.450194
\(210\) 0 0
\(211\) 16.2867 1.12122 0.560611 0.828079i \(-0.310566\pi\)
0.560611 + 0.828079i \(0.310566\pi\)
\(212\) 0 0
\(213\) 2.63521 0.180562
\(214\) 0 0
\(215\) 6.13141 0.418158
\(216\) 0 0
\(217\) 12.8467 0.872088
\(218\) 0 0
\(219\) 13.8616 0.936679
\(220\) 0 0
\(221\) 8.25738 0.555452
\(222\) 0 0
\(223\) −2.74204 −0.183621 −0.0918104 0.995777i \(-0.529265\pi\)
−0.0918104 + 0.995777i \(0.529265\pi\)
\(224\) 0 0
\(225\) 2.36854 0.157903
\(226\) 0 0
\(227\) 8.70484 0.577760 0.288880 0.957365i \(-0.406717\pi\)
0.288880 + 0.957365i \(0.406717\pi\)
\(228\) 0 0
\(229\) −23.0163 −1.52096 −0.760481 0.649360i \(-0.775037\pi\)
−0.760481 + 0.649360i \(0.775037\pi\)
\(230\) 0 0
\(231\) −5.18917 −0.341422
\(232\) 0 0
\(233\) −12.5846 −0.824447 −0.412224 0.911083i \(-0.635248\pi\)
−0.412224 + 0.911083i \(0.635248\pi\)
\(234\) 0 0
\(235\) 19.7642 1.28927
\(236\) 0 0
\(237\) 2.77019 0.179943
\(238\) 0 0
\(239\) 20.2452 1.30955 0.654777 0.755822i \(-0.272763\pi\)
0.654777 + 0.755822i \(0.272763\pi\)
\(240\) 0 0
\(241\) −7.77341 −0.500730 −0.250365 0.968152i \(-0.580551\pi\)
−0.250365 + 0.968152i \(0.580551\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 12.5874 0.804182
\(246\) 0 0
\(247\) −7.60607 −0.483963
\(248\) 0 0
\(249\) 4.13535 0.262067
\(250\) 0 0
\(251\) 30.3141 1.91341 0.956706 0.291058i \(-0.0940071\pi\)
0.956706 + 0.291058i \(0.0940071\pi\)
\(252\) 0 0
\(253\) 1.85732 0.116769
\(254\) 0 0
\(255\) −5.68160 −0.355796
\(256\) 0 0
\(257\) 28.4899 1.77715 0.888576 0.458730i \(-0.151696\pi\)
0.888576 + 0.458730i \(0.151696\pi\)
\(258\) 0 0
\(259\) −14.7769 −0.918195
\(260\) 0 0
\(261\) 5.84336 0.361695
\(262\) 0 0
\(263\) 20.3946 1.25759 0.628793 0.777573i \(-0.283549\pi\)
0.628793 + 0.777573i \(0.283549\pi\)
\(264\) 0 0
\(265\) −16.9281 −1.03988
\(266\) 0 0
\(267\) −12.3614 −0.756505
\(268\) 0 0
\(269\) 18.4449 1.12461 0.562304 0.826931i \(-0.309915\pi\)
0.562304 + 0.826931i \(0.309915\pi\)
\(270\) 0 0
\(271\) 14.8716 0.903386 0.451693 0.892173i \(-0.350820\pi\)
0.451693 + 0.892173i \(0.350820\pi\)
\(272\) 0 0
\(273\) −6.06436 −0.367032
\(274\) 0 0
\(275\) 7.99569 0.482158
\(276\) 0 0
\(277\) 28.8774 1.73508 0.867538 0.497371i \(-0.165701\pi\)
0.867538 + 0.497371i \(0.165701\pi\)
\(278\) 0 0
\(279\) −8.35732 −0.500340
\(280\) 0 0
\(281\) 30.1244 1.79707 0.898537 0.438898i \(-0.144631\pi\)
0.898537 + 0.438898i \(0.144631\pi\)
\(282\) 0 0
\(283\) −1.93521 −0.115037 −0.0575183 0.998344i \(-0.518319\pi\)
−0.0575183 + 0.998344i \(0.518319\pi\)
\(284\) 0 0
\(285\) 5.23346 0.310003
\(286\) 0 0
\(287\) 17.6469 1.04166
\(288\) 0 0
\(289\) −12.6191 −0.742302
\(290\) 0 0
\(291\) 10.5240 0.616927
\(292\) 0 0
\(293\) 11.9502 0.698140 0.349070 0.937097i \(-0.386497\pi\)
0.349070 + 0.937097i \(0.386497\pi\)
\(294\) 0 0
\(295\) −1.68211 −0.0979365
\(296\) 0 0
\(297\) 3.37579 0.195883
\(298\) 0 0
\(299\) 2.17057 0.125527
\(300\) 0 0
\(301\) 3.47210 0.200128
\(302\) 0 0
\(303\) 15.1634 0.871117
\(304\) 0 0
\(305\) −39.2675 −2.24845
\(306\) 0 0
\(307\) 2.04353 0.116631 0.0583153 0.998298i \(-0.481427\pi\)
0.0583153 + 0.998298i \(0.481427\pi\)
\(308\) 0 0
\(309\) 4.09047 0.232699
\(310\) 0 0
\(311\) 16.5423 0.938026 0.469013 0.883191i \(-0.344610\pi\)
0.469013 + 0.883191i \(0.344610\pi\)
\(312\) 0 0
\(313\) −0.304714 −0.0172235 −0.00861174 0.999963i \(-0.502741\pi\)
−0.00861174 + 0.999963i \(0.502741\pi\)
\(314\) 0 0
\(315\) 4.17267 0.235103
\(316\) 0 0
\(317\) −12.5420 −0.704428 −0.352214 0.935919i \(-0.614571\pi\)
−0.352214 + 0.935919i \(0.614571\pi\)
\(318\) 0 0
\(319\) 19.7259 1.10444
\(320\) 0 0
\(321\) 11.2461 0.627695
\(322\) 0 0
\(323\) −4.03532 −0.224531
\(324\) 0 0
\(325\) 9.34423 0.518325
\(326\) 0 0
\(327\) −15.9901 −0.884255
\(328\) 0 0
\(329\) 11.1921 0.617039
\(330\) 0 0
\(331\) −2.32409 −0.127744 −0.0638718 0.997958i \(-0.520345\pi\)
−0.0638718 + 0.997958i \(0.520345\pi\)
\(332\) 0 0
\(333\) 9.61307 0.526793
\(334\) 0 0
\(335\) 19.9719 1.09118
\(336\) 0 0
\(337\) −6.42488 −0.349986 −0.174993 0.984570i \(-0.555990\pi\)
−0.174993 + 0.984570i \(0.555990\pi\)
\(338\) 0 0
\(339\) −6.33778 −0.344221
\(340\) 0 0
\(341\) −28.2125 −1.52779
\(342\) 0 0
\(343\) 17.8882 0.965874
\(344\) 0 0
\(345\) −1.49349 −0.0804068
\(346\) 0 0
\(347\) −22.7362 −1.22054 −0.610271 0.792193i \(-0.708940\pi\)
−0.610271 + 0.792193i \(0.708940\pi\)
\(348\) 0 0
\(349\) 13.9408 0.746236 0.373118 0.927784i \(-0.378289\pi\)
0.373118 + 0.927784i \(0.378289\pi\)
\(350\) 0 0
\(351\) 3.94514 0.210576
\(352\) 0 0
\(353\) 8.21088 0.437021 0.218511 0.975835i \(-0.429880\pi\)
0.218511 + 0.975835i \(0.429880\pi\)
\(354\) 0 0
\(355\) −7.15330 −0.379658
\(356\) 0 0
\(357\) −3.21738 −0.170282
\(358\) 0 0
\(359\) −33.3585 −1.76059 −0.880297 0.474423i \(-0.842657\pi\)
−0.880297 + 0.474423i \(0.842657\pi\)
\(360\) 0 0
\(361\) −15.2830 −0.804367
\(362\) 0 0
\(363\) 0.395928 0.0207808
\(364\) 0 0
\(365\) −37.6274 −1.96951
\(366\) 0 0
\(367\) 34.9325 1.82346 0.911730 0.410789i \(-0.134747\pi\)
0.911730 + 0.410789i \(0.134747\pi\)
\(368\) 0 0
\(369\) −11.4801 −0.597629
\(370\) 0 0
\(371\) −9.58604 −0.497682
\(372\) 0 0
\(373\) −21.8544 −1.13158 −0.565789 0.824550i \(-0.691428\pi\)
−0.565789 + 0.824550i \(0.691428\pi\)
\(374\) 0 0
\(375\) 7.14311 0.368868
\(376\) 0 0
\(377\) 23.0528 1.18728
\(378\) 0 0
\(379\) −23.7463 −1.21976 −0.609881 0.792493i \(-0.708783\pi\)
−0.609881 + 0.792493i \(0.708783\pi\)
\(380\) 0 0
\(381\) −1.52367 −0.0780601
\(382\) 0 0
\(383\) −5.03462 −0.257257 −0.128628 0.991693i \(-0.541057\pi\)
−0.128628 + 0.991693i \(0.541057\pi\)
\(384\) 0 0
\(385\) 14.0860 0.717890
\(386\) 0 0
\(387\) −2.25876 −0.114819
\(388\) 0 0
\(389\) 2.05244 0.104063 0.0520313 0.998645i \(-0.483430\pi\)
0.0520313 + 0.998645i \(0.483430\pi\)
\(390\) 0 0
\(391\) 1.15157 0.0582376
\(392\) 0 0
\(393\) 20.7658 1.04749
\(394\) 0 0
\(395\) −7.51969 −0.378357
\(396\) 0 0
\(397\) −4.59868 −0.230801 −0.115400 0.993319i \(-0.536815\pi\)
−0.115400 + 0.993319i \(0.536815\pi\)
\(398\) 0 0
\(399\) 2.96361 0.148366
\(400\) 0 0
\(401\) 7.33823 0.366454 0.183227 0.983071i \(-0.441346\pi\)
0.183227 + 0.983071i \(0.441346\pi\)
\(402\) 0 0
\(403\) −32.9708 −1.64239
\(404\) 0 0
\(405\) −2.71451 −0.134885
\(406\) 0 0
\(407\) 32.4516 1.60857
\(408\) 0 0
\(409\) −34.6193 −1.71181 −0.855907 0.517130i \(-0.827001\pi\)
−0.855907 + 0.517130i \(0.827001\pi\)
\(410\) 0 0
\(411\) 11.4083 0.562732
\(412\) 0 0
\(413\) −0.952549 −0.0468719
\(414\) 0 0
\(415\) −11.2254 −0.551035
\(416\) 0 0
\(417\) 2.71570 0.132988
\(418\) 0 0
\(419\) 22.0494 1.07719 0.538593 0.842566i \(-0.318956\pi\)
0.538593 + 0.842566i \(0.318956\pi\)
\(420\) 0 0
\(421\) −11.7226 −0.571323 −0.285661 0.958331i \(-0.592213\pi\)
−0.285661 + 0.958331i \(0.592213\pi\)
\(422\) 0 0
\(423\) −7.28095 −0.354012
\(424\) 0 0
\(425\) 4.95749 0.240473
\(426\) 0 0
\(427\) −22.2364 −1.07610
\(428\) 0 0
\(429\) 13.3179 0.642996
\(430\) 0 0
\(431\) 28.2016 1.35842 0.679212 0.733942i \(-0.262321\pi\)
0.679212 + 0.733942i \(0.262321\pi\)
\(432\) 0 0
\(433\) −1.97163 −0.0947503 −0.0473751 0.998877i \(-0.515086\pi\)
−0.0473751 + 0.998877i \(0.515086\pi\)
\(434\) 0 0
\(435\) −15.8618 −0.760516
\(436\) 0 0
\(437\) −1.06074 −0.0507422
\(438\) 0 0
\(439\) −6.84816 −0.326845 −0.163422 0.986556i \(-0.552253\pi\)
−0.163422 + 0.986556i \(0.552253\pi\)
\(440\) 0 0
\(441\) −4.63710 −0.220814
\(442\) 0 0
\(443\) −29.0831 −1.38178 −0.690890 0.722960i \(-0.742781\pi\)
−0.690890 + 0.722960i \(0.742781\pi\)
\(444\) 0 0
\(445\) 33.5551 1.59066
\(446\) 0 0
\(447\) 10.6937 0.505796
\(448\) 0 0
\(449\) −29.5580 −1.39493 −0.697463 0.716621i \(-0.745688\pi\)
−0.697463 + 0.716621i \(0.745688\pi\)
\(450\) 0 0
\(451\) −38.7543 −1.82487
\(452\) 0 0
\(453\) −10.5063 −0.493627
\(454\) 0 0
\(455\) 16.4617 0.771738
\(456\) 0 0
\(457\) −7.44120 −0.348085 −0.174042 0.984738i \(-0.555683\pi\)
−0.174042 + 0.984738i \(0.555683\pi\)
\(458\) 0 0
\(459\) 2.09305 0.0976953
\(460\) 0 0
\(461\) −17.0065 −0.792071 −0.396036 0.918235i \(-0.629614\pi\)
−0.396036 + 0.918235i \(0.629614\pi\)
\(462\) 0 0
\(463\) −21.9204 −1.01873 −0.509363 0.860552i \(-0.670119\pi\)
−0.509363 + 0.860552i \(0.670119\pi\)
\(464\) 0 0
\(465\) 22.6860 1.05204
\(466\) 0 0
\(467\) −13.0467 −0.603729 −0.301864 0.953351i \(-0.597609\pi\)
−0.301864 + 0.953351i \(0.597609\pi\)
\(468\) 0 0
\(469\) 11.3097 0.522233
\(470\) 0 0
\(471\) −10.3552 −0.477143
\(472\) 0 0
\(473\) −7.62508 −0.350601
\(474\) 0 0
\(475\) −4.56646 −0.209523
\(476\) 0 0
\(477\) 6.23615 0.285534
\(478\) 0 0
\(479\) 20.6455 0.943319 0.471660 0.881781i \(-0.343655\pi\)
0.471660 + 0.881781i \(0.343655\pi\)
\(480\) 0 0
\(481\) 37.9249 1.72923
\(482\) 0 0
\(483\) −0.845735 −0.0384823
\(484\) 0 0
\(485\) −28.5674 −1.29718
\(486\) 0 0
\(487\) 28.6425 1.29792 0.648958 0.760824i \(-0.275205\pi\)
0.648958 + 0.760824i \(0.275205\pi\)
\(488\) 0 0
\(489\) −6.33893 −0.286657
\(490\) 0 0
\(491\) 15.3406 0.692313 0.346157 0.938177i \(-0.387487\pi\)
0.346157 + 0.938177i \(0.387487\pi\)
\(492\) 0 0
\(493\) 12.2305 0.550832
\(494\) 0 0
\(495\) −9.16359 −0.411873
\(496\) 0 0
\(497\) −4.05078 −0.181702
\(498\) 0 0
\(499\) −17.5833 −0.787137 −0.393568 0.919295i \(-0.628760\pi\)
−0.393568 + 0.919295i \(0.628760\pi\)
\(500\) 0 0
\(501\) 0.395120 0.0176527
\(502\) 0 0
\(503\) 1.00000 0.0445878
\(504\) 0 0
\(505\) −41.1612 −1.83165
\(506\) 0 0
\(507\) 2.56411 0.113876
\(508\) 0 0
\(509\) 23.3822 1.03640 0.518200 0.855260i \(-0.326602\pi\)
0.518200 + 0.855260i \(0.326602\pi\)
\(510\) 0 0
\(511\) −21.3077 −0.942595
\(512\) 0 0
\(513\) −1.92796 −0.0851216
\(514\) 0 0
\(515\) −11.1036 −0.489284
\(516\) 0 0
\(517\) −24.5789 −1.08098
\(518\) 0 0
\(519\) 17.3781 0.762812
\(520\) 0 0
\(521\) −4.77654 −0.209264 −0.104632 0.994511i \(-0.533366\pi\)
−0.104632 + 0.994511i \(0.533366\pi\)
\(522\) 0 0
\(523\) 40.6939 1.77942 0.889710 0.456526i \(-0.150906\pi\)
0.889710 + 0.456526i \(0.150906\pi\)
\(524\) 0 0
\(525\) −3.64086 −0.158900
\(526\) 0 0
\(527\) −17.4923 −0.761977
\(528\) 0 0
\(529\) −22.6973 −0.986839
\(530\) 0 0
\(531\) 0.619676 0.0268917
\(532\) 0 0
\(533\) −45.2905 −1.96175
\(534\) 0 0
\(535\) −30.5276 −1.31982
\(536\) 0 0
\(537\) −3.15569 −0.136178
\(538\) 0 0
\(539\) −15.6539 −0.674259
\(540\) 0 0
\(541\) 16.8769 0.725596 0.362798 0.931868i \(-0.381821\pi\)
0.362798 + 0.931868i \(0.381821\pi\)
\(542\) 0 0
\(543\) 13.9671 0.599387
\(544\) 0 0
\(545\) 43.4053 1.85928
\(546\) 0 0
\(547\) −35.0868 −1.50020 −0.750102 0.661323i \(-0.769995\pi\)
−0.750102 + 0.661323i \(0.769995\pi\)
\(548\) 0 0
\(549\) 14.4658 0.617385
\(550\) 0 0
\(551\) −11.2658 −0.479938
\(552\) 0 0
\(553\) −4.25826 −0.181080
\(554\) 0 0
\(555\) −26.0947 −1.10766
\(556\) 0 0
\(557\) 13.1312 0.556389 0.278194 0.960525i \(-0.410264\pi\)
0.278194 + 0.960525i \(0.410264\pi\)
\(558\) 0 0
\(559\) −8.91110 −0.376900
\(560\) 0 0
\(561\) 7.06570 0.298314
\(562\) 0 0
\(563\) 13.5270 0.570095 0.285047 0.958513i \(-0.407991\pi\)
0.285047 + 0.958513i \(0.407991\pi\)
\(564\) 0 0
\(565\) 17.2040 0.723776
\(566\) 0 0
\(567\) −1.53717 −0.0645552
\(568\) 0 0
\(569\) −10.5449 −0.442064 −0.221032 0.975267i \(-0.570943\pi\)
−0.221032 + 0.975267i \(0.570943\pi\)
\(570\) 0 0
\(571\) −34.8265 −1.45744 −0.728721 0.684811i \(-0.759885\pi\)
−0.728721 + 0.684811i \(0.759885\pi\)
\(572\) 0 0
\(573\) −17.7722 −0.742442
\(574\) 0 0
\(575\) 1.30314 0.0543449
\(576\) 0 0
\(577\) 19.7659 0.822865 0.411432 0.911440i \(-0.365029\pi\)
0.411432 + 0.911440i \(0.365029\pi\)
\(578\) 0 0
\(579\) 8.00421 0.332643
\(580\) 0 0
\(581\) −6.35675 −0.263722
\(582\) 0 0
\(583\) 21.0519 0.871880
\(584\) 0 0
\(585\) −10.7091 −0.442767
\(586\) 0 0
\(587\) −37.0343 −1.52857 −0.764285 0.644878i \(-0.776908\pi\)
−0.764285 + 0.644878i \(0.776908\pi\)
\(588\) 0 0
\(589\) 16.1126 0.663908
\(590\) 0 0
\(591\) 11.1777 0.459788
\(592\) 0 0
\(593\) 8.72990 0.358494 0.179247 0.983804i \(-0.442634\pi\)
0.179247 + 0.983804i \(0.442634\pi\)
\(594\) 0 0
\(595\) 8.73361 0.358043
\(596\) 0 0
\(597\) 7.15795 0.292955
\(598\) 0 0
\(599\) −27.3967 −1.11940 −0.559699 0.828696i \(-0.689083\pi\)
−0.559699 + 0.828696i \(0.689083\pi\)
\(600\) 0 0
\(601\) 32.6974 1.33376 0.666878 0.745167i \(-0.267630\pi\)
0.666878 + 0.745167i \(0.267630\pi\)
\(602\) 0 0
\(603\) −7.35747 −0.299619
\(604\) 0 0
\(605\) −1.07475 −0.0436948
\(606\) 0 0
\(607\) −27.1537 −1.10213 −0.551067 0.834461i \(-0.685779\pi\)
−0.551067 + 0.834461i \(0.685779\pi\)
\(608\) 0 0
\(609\) −8.98225 −0.363979
\(610\) 0 0
\(611\) −28.7243 −1.16206
\(612\) 0 0
\(613\) 40.7922 1.64758 0.823791 0.566894i \(-0.191855\pi\)
0.823791 + 0.566894i \(0.191855\pi\)
\(614\) 0 0
\(615\) 31.1627 1.25660
\(616\) 0 0
\(617\) 23.0594 0.928336 0.464168 0.885747i \(-0.346353\pi\)
0.464168 + 0.885747i \(0.346353\pi\)
\(618\) 0 0
\(619\) 18.6326 0.748908 0.374454 0.927246i \(-0.377830\pi\)
0.374454 + 0.927246i \(0.377830\pi\)
\(620\) 0 0
\(621\) 0.550188 0.0220783
\(622\) 0 0
\(623\) 19.0016 0.761283
\(624\) 0 0
\(625\) −31.2327 −1.24931
\(626\) 0 0
\(627\) −6.50838 −0.259920
\(628\) 0 0
\(629\) 20.1207 0.802263
\(630\) 0 0
\(631\) −19.2473 −0.766222 −0.383111 0.923702i \(-0.625147\pi\)
−0.383111 + 0.923702i \(0.625147\pi\)
\(632\) 0 0
\(633\) 16.2867 0.647338
\(634\) 0 0
\(635\) 4.13602 0.164133
\(636\) 0 0
\(637\) −18.2940 −0.724834
\(638\) 0 0
\(639\) 2.63521 0.104247
\(640\) 0 0
\(641\) −43.0701 −1.70117 −0.850584 0.525840i \(-0.823751\pi\)
−0.850584 + 0.525840i \(0.823751\pi\)
\(642\) 0 0
\(643\) −41.1497 −1.62279 −0.811393 0.584502i \(-0.801290\pi\)
−0.811393 + 0.584502i \(0.801290\pi\)
\(644\) 0 0
\(645\) 6.13141 0.241424
\(646\) 0 0
\(647\) 10.4482 0.410761 0.205380 0.978682i \(-0.434157\pi\)
0.205380 + 0.978682i \(0.434157\pi\)
\(648\) 0 0
\(649\) 2.09189 0.0821140
\(650\) 0 0
\(651\) 12.8467 0.503500
\(652\) 0 0
\(653\) 39.1008 1.53013 0.765066 0.643952i \(-0.222706\pi\)
0.765066 + 0.643952i \(0.222706\pi\)
\(654\) 0 0
\(655\) −56.3688 −2.20251
\(656\) 0 0
\(657\) 13.8616 0.540792
\(658\) 0 0
\(659\) 4.40413 0.171561 0.0857803 0.996314i \(-0.472662\pi\)
0.0857803 + 0.996314i \(0.472662\pi\)
\(660\) 0 0
\(661\) −17.4676 −0.679411 −0.339706 0.940532i \(-0.610327\pi\)
−0.339706 + 0.940532i \(0.610327\pi\)
\(662\) 0 0
\(663\) 8.25738 0.320690
\(664\) 0 0
\(665\) −8.04474 −0.311961
\(666\) 0 0
\(667\) 3.21495 0.124483
\(668\) 0 0
\(669\) −2.74204 −0.106013
\(670\) 0 0
\(671\) 48.8335 1.88519
\(672\) 0 0
\(673\) −11.0200 −0.424790 −0.212395 0.977184i \(-0.568126\pi\)
−0.212395 + 0.977184i \(0.568126\pi\)
\(674\) 0 0
\(675\) 2.36854 0.0911653
\(676\) 0 0
\(677\) −30.5546 −1.17431 −0.587154 0.809476i \(-0.699752\pi\)
−0.587154 + 0.809476i \(0.699752\pi\)
\(678\) 0 0
\(679\) −16.1772 −0.620824
\(680\) 0 0
\(681\) 8.70484 0.333570
\(682\) 0 0
\(683\) −1.71406 −0.0655867 −0.0327934 0.999462i \(-0.510440\pi\)
−0.0327934 + 0.999462i \(0.510440\pi\)
\(684\) 0 0
\(685\) −30.9680 −1.18323
\(686\) 0 0
\(687\) −23.0163 −0.878128
\(688\) 0 0
\(689\) 24.6025 0.937279
\(690\) 0 0
\(691\) −33.6363 −1.27959 −0.639793 0.768547i \(-0.720980\pi\)
−0.639793 + 0.768547i \(0.720980\pi\)
\(692\) 0 0
\(693\) −5.18917 −0.197120
\(694\) 0 0
\(695\) −7.37177 −0.279627
\(696\) 0 0
\(697\) −24.0284 −0.910141
\(698\) 0 0
\(699\) −12.5846 −0.475995
\(700\) 0 0
\(701\) 6.76444 0.255489 0.127745 0.991807i \(-0.459226\pi\)
0.127745 + 0.991807i \(0.459226\pi\)
\(702\) 0 0
\(703\) −18.5336 −0.699008
\(704\) 0 0
\(705\) 19.7642 0.744362
\(706\) 0 0
\(707\) −23.3088 −0.876618
\(708\) 0 0
\(709\) −19.0188 −0.714265 −0.357132 0.934054i \(-0.616245\pi\)
−0.357132 + 0.934054i \(0.616245\pi\)
\(710\) 0 0
\(711\) 2.77019 0.103890
\(712\) 0 0
\(713\) −4.59810 −0.172200
\(714\) 0 0
\(715\) −36.1516 −1.35199
\(716\) 0 0
\(717\) 20.2452 0.756071
\(718\) 0 0
\(719\) 40.0401 1.49324 0.746622 0.665249i \(-0.231674\pi\)
0.746622 + 0.665249i \(0.231674\pi\)
\(720\) 0 0
\(721\) −6.28776 −0.234169
\(722\) 0 0
\(723\) −7.77341 −0.289096
\(724\) 0 0
\(725\) 13.8402 0.514014
\(726\) 0 0
\(727\) 8.68094 0.321958 0.160979 0.986958i \(-0.448535\pi\)
0.160979 + 0.986958i \(0.448535\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −4.72770 −0.174860
\(732\) 0 0
\(733\) 33.8498 1.25027 0.625136 0.780516i \(-0.285044\pi\)
0.625136 + 0.780516i \(0.285044\pi\)
\(734\) 0 0
\(735\) 12.5874 0.464294
\(736\) 0 0
\(737\) −24.8372 −0.914891
\(738\) 0 0
\(739\) 26.5103 0.975197 0.487599 0.873068i \(-0.337873\pi\)
0.487599 + 0.873068i \(0.337873\pi\)
\(740\) 0 0
\(741\) −7.60607 −0.279416
\(742\) 0 0
\(743\) −42.7990 −1.57014 −0.785071 0.619405i \(-0.787374\pi\)
−0.785071 + 0.619405i \(0.787374\pi\)
\(744\) 0 0
\(745\) −29.0282 −1.06351
\(746\) 0 0
\(747\) 4.13535 0.151305
\(748\) 0 0
\(749\) −17.2872 −0.631660
\(750\) 0 0
\(751\) 35.7938 1.30613 0.653067 0.757300i \(-0.273482\pi\)
0.653067 + 0.757300i \(0.273482\pi\)
\(752\) 0 0
\(753\) 30.3141 1.10471
\(754\) 0 0
\(755\) 28.5193 1.03792
\(756\) 0 0
\(757\) −15.0664 −0.547597 −0.273799 0.961787i \(-0.588280\pi\)
−0.273799 + 0.961787i \(0.588280\pi\)
\(758\) 0 0
\(759\) 1.85732 0.0674164
\(760\) 0 0
\(761\) −2.20573 −0.0799578 −0.0399789 0.999201i \(-0.512729\pi\)
−0.0399789 + 0.999201i \(0.512729\pi\)
\(762\) 0 0
\(763\) 24.5796 0.889840
\(764\) 0 0
\(765\) −5.68160 −0.205419
\(766\) 0 0
\(767\) 2.44471 0.0882733
\(768\) 0 0
\(769\) 10.1805 0.367119 0.183559 0.983009i \(-0.441238\pi\)
0.183559 + 0.983009i \(0.441238\pi\)
\(770\) 0 0
\(771\) 28.4899 1.02604
\(772\) 0 0
\(773\) 4.56876 0.164327 0.0821635 0.996619i \(-0.473817\pi\)
0.0821635 + 0.996619i \(0.473817\pi\)
\(774\) 0 0
\(775\) −19.7947 −0.711046
\(776\) 0 0
\(777\) −14.7769 −0.530120
\(778\) 0 0
\(779\) 22.1331 0.793002
\(780\) 0 0
\(781\) 8.89591 0.318321
\(782\) 0 0
\(783\) 5.84336 0.208825
\(784\) 0 0
\(785\) 28.1093 1.00326
\(786\) 0 0
\(787\) 10.5764 0.377008 0.188504 0.982072i \(-0.439636\pi\)
0.188504 + 0.982072i \(0.439636\pi\)
\(788\) 0 0
\(789\) 20.3946 0.726067
\(790\) 0 0
\(791\) 9.74227 0.346395
\(792\) 0 0
\(793\) 57.0696 2.02660
\(794\) 0 0
\(795\) −16.9281 −0.600376
\(796\) 0 0
\(797\) −13.8051 −0.489003 −0.244501 0.969649i \(-0.578624\pi\)
−0.244501 + 0.969649i \(0.578624\pi\)
\(798\) 0 0
\(799\) −15.2394 −0.539132
\(800\) 0 0
\(801\) −12.3614 −0.436768
\(802\) 0 0
\(803\) 46.7937 1.65132
\(804\) 0 0
\(805\) 2.29575 0.0809146
\(806\) 0 0
\(807\) 18.4449 0.649293
\(808\) 0 0
\(809\) −0.857865 −0.0301609 −0.0150805 0.999886i \(-0.504800\pi\)
−0.0150805 + 0.999886i \(0.504800\pi\)
\(810\) 0 0
\(811\) 26.8986 0.944536 0.472268 0.881455i \(-0.343435\pi\)
0.472268 + 0.881455i \(0.343435\pi\)
\(812\) 0 0
\(813\) 14.8716 0.521570
\(814\) 0 0
\(815\) 17.2071 0.602738
\(816\) 0 0
\(817\) 4.35479 0.152355
\(818\) 0 0
\(819\) −6.06436 −0.211906
\(820\) 0 0
\(821\) −32.4200 −1.13147 −0.565733 0.824588i \(-0.691407\pi\)
−0.565733 + 0.824588i \(0.691407\pi\)
\(822\) 0 0
\(823\) 30.6923 1.06986 0.534932 0.844895i \(-0.320337\pi\)
0.534932 + 0.844895i \(0.320337\pi\)
\(824\) 0 0
\(825\) 7.99569 0.278374
\(826\) 0 0
\(827\) −12.7231 −0.442427 −0.221214 0.975225i \(-0.571002\pi\)
−0.221214 + 0.975225i \(0.571002\pi\)
\(828\) 0 0
\(829\) 48.2117 1.67446 0.837231 0.546849i \(-0.184173\pi\)
0.837231 + 0.546849i \(0.184173\pi\)
\(830\) 0 0
\(831\) 28.8774 1.00175
\(832\) 0 0
\(833\) −9.70569 −0.336282
\(834\) 0 0
\(835\) −1.07256 −0.0371173
\(836\) 0 0
\(837\) −8.35732 −0.288871
\(838\) 0 0
\(839\) 48.9341 1.68939 0.844696 0.535247i \(-0.179781\pi\)
0.844696 + 0.535247i \(0.179781\pi\)
\(840\) 0 0
\(841\) 5.14481 0.177407
\(842\) 0 0
\(843\) 30.1244 1.03754
\(844\) 0 0
\(845\) −6.96030 −0.239441
\(846\) 0 0
\(847\) −0.608610 −0.0209121
\(848\) 0 0
\(849\) −1.93521 −0.0664164
\(850\) 0 0
\(851\) 5.28900 0.181304
\(852\) 0 0
\(853\) 3.43338 0.117557 0.0587783 0.998271i \(-0.481279\pi\)
0.0587783 + 0.998271i \(0.481279\pi\)
\(854\) 0 0
\(855\) 5.23346 0.178981
\(856\) 0 0
\(857\) −14.2818 −0.487857 −0.243929 0.969793i \(-0.578436\pi\)
−0.243929 + 0.969793i \(0.578436\pi\)
\(858\) 0 0
\(859\) 0.155049 0.00529020 0.00264510 0.999997i \(-0.499158\pi\)
0.00264510 + 0.999997i \(0.499158\pi\)
\(860\) 0 0
\(861\) 17.6469 0.601404
\(862\) 0 0
\(863\) −40.3250 −1.37268 −0.686339 0.727282i \(-0.740783\pi\)
−0.686339 + 0.727282i \(0.740783\pi\)
\(864\) 0 0
\(865\) −47.1728 −1.60392
\(866\) 0 0
\(867\) −12.6191 −0.428568
\(868\) 0 0
\(869\) 9.35156 0.317230
\(870\) 0 0
\(871\) −29.0262 −0.983516
\(872\) 0 0
\(873\) 10.5240 0.356183
\(874\) 0 0
\(875\) −10.9802 −0.371198
\(876\) 0 0
\(877\) −23.0702 −0.779025 −0.389513 0.921021i \(-0.627357\pi\)
−0.389513 + 0.921021i \(0.627357\pi\)
\(878\) 0 0
\(879\) 11.9502 0.403071
\(880\) 0 0
\(881\) 42.0445 1.41652 0.708258 0.705954i \(-0.249481\pi\)
0.708258 + 0.705954i \(0.249481\pi\)
\(882\) 0 0
\(883\) −0.215055 −0.00723719 −0.00361860 0.999993i \(-0.501152\pi\)
−0.00361860 + 0.999993i \(0.501152\pi\)
\(884\) 0 0
\(885\) −1.68211 −0.0565437
\(886\) 0 0
\(887\) −30.4124 −1.02115 −0.510575 0.859833i \(-0.670567\pi\)
−0.510575 + 0.859833i \(0.670567\pi\)
\(888\) 0 0
\(889\) 2.34215 0.0785531
\(890\) 0 0
\(891\) 3.37579 0.113093
\(892\) 0 0
\(893\) 14.0374 0.469743
\(894\) 0 0
\(895\) 8.56614 0.286334
\(896\) 0 0
\(897\) 2.17057 0.0724732
\(898\) 0 0
\(899\) −48.8348 −1.62873
\(900\) 0 0
\(901\) 13.0526 0.434845
\(902\) 0 0
\(903\) 3.47210 0.115544
\(904\) 0 0
\(905\) −37.9139 −1.26030
\(906\) 0 0
\(907\) −54.7078 −1.81654 −0.908271 0.418381i \(-0.862598\pi\)
−0.908271 + 0.418381i \(0.862598\pi\)
\(908\) 0 0
\(909\) 15.1634 0.502939
\(910\) 0 0
\(911\) −54.8367 −1.81682 −0.908410 0.418080i \(-0.862703\pi\)
−0.908410 + 0.418080i \(0.862703\pi\)
\(912\) 0 0
\(913\) 13.9601 0.462011
\(914\) 0 0
\(915\) −39.2675 −1.29814
\(916\) 0 0
\(917\) −31.9206 −1.05411
\(918\) 0 0
\(919\) 10.2671 0.338680 0.169340 0.985558i \(-0.445836\pi\)
0.169340 + 0.985558i \(0.445836\pi\)
\(920\) 0 0
\(921\) 2.04353 0.0673367
\(922\) 0 0
\(923\) 10.3963 0.342197
\(924\) 0 0
\(925\) 22.7690 0.748639
\(926\) 0 0
\(927\) 4.09047 0.134349
\(928\) 0 0
\(929\) 35.5010 1.16475 0.582375 0.812920i \(-0.302124\pi\)
0.582375 + 0.812920i \(0.302124\pi\)
\(930\) 0 0
\(931\) 8.94015 0.293001
\(932\) 0 0
\(933\) 16.5423 0.541570
\(934\) 0 0
\(935\) −19.1799 −0.627249
\(936\) 0 0
\(937\) −39.1731 −1.27973 −0.639865 0.768488i \(-0.721010\pi\)
−0.639865 + 0.768488i \(0.721010\pi\)
\(938\) 0 0
\(939\) −0.304714 −0.00994398
\(940\) 0 0
\(941\) 30.8375 1.00527 0.502637 0.864498i \(-0.332363\pi\)
0.502637 + 0.864498i \(0.332363\pi\)
\(942\) 0 0
\(943\) −6.31621 −0.205684
\(944\) 0 0
\(945\) 4.17267 0.135737
\(946\) 0 0
\(947\) −43.3454 −1.40854 −0.704269 0.709934i \(-0.748725\pi\)
−0.704269 + 0.709934i \(0.748725\pi\)
\(948\) 0 0
\(949\) 54.6859 1.77518
\(950\) 0 0
\(951\) −12.5420 −0.406702
\(952\) 0 0
\(953\) −34.2923 −1.11084 −0.555418 0.831571i \(-0.687442\pi\)
−0.555418 + 0.831571i \(0.687442\pi\)
\(954\) 0 0
\(955\) 48.2426 1.56109
\(956\) 0 0
\(957\) 19.7259 0.637648
\(958\) 0 0
\(959\) −17.5366 −0.566287
\(960\) 0 0
\(961\) 38.8449 1.25306
\(962\) 0 0
\(963\) 11.2461 0.362400
\(964\) 0 0
\(965\) −21.7275 −0.699432
\(966\) 0 0
\(967\) −15.9359 −0.512464 −0.256232 0.966615i \(-0.582481\pi\)
−0.256232 + 0.966615i \(0.582481\pi\)
\(968\) 0 0
\(969\) −4.03532 −0.129633
\(970\) 0 0
\(971\) −5.60353 −0.179826 −0.0899130 0.995950i \(-0.528659\pi\)
−0.0899130 + 0.995950i \(0.528659\pi\)
\(972\) 0 0
\(973\) −4.17449 −0.133828
\(974\) 0 0
\(975\) 9.34423 0.299255
\(976\) 0 0
\(977\) −4.31006 −0.137891 −0.0689454 0.997620i \(-0.521963\pi\)
−0.0689454 + 0.997620i \(0.521963\pi\)
\(978\) 0 0
\(979\) −41.7294 −1.33368
\(980\) 0 0
\(981\) −15.9901 −0.510525
\(982\) 0 0
\(983\) −43.5685 −1.38962 −0.694809 0.719194i \(-0.744511\pi\)
−0.694809 + 0.719194i \(0.744511\pi\)
\(984\) 0 0
\(985\) −30.3419 −0.966772
\(986\) 0 0
\(987\) 11.1921 0.356248
\(988\) 0 0
\(989\) −1.24274 −0.0395169
\(990\) 0 0
\(991\) −39.3251 −1.24920 −0.624602 0.780944i \(-0.714739\pi\)
−0.624602 + 0.780944i \(0.714739\pi\)
\(992\) 0 0
\(993\) −2.32409 −0.0737527
\(994\) 0 0
\(995\) −19.4303 −0.615982
\(996\) 0 0
\(997\) −46.5764 −1.47509 −0.737545 0.675298i \(-0.764015\pi\)
−0.737545 + 0.675298i \(0.764015\pi\)
\(998\) 0 0
\(999\) 9.61307 0.304144
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 6036.2.a.h.1.4 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6036.2.a.h.1.4 24 1.1 even 1 trivial