Properties

Label 4840.2.a.n.1.1
Level $4840$
Weight $2$
Character 4840.1
Self dual yes
Analytic conductor $38.648$
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] = [4840,2,Mod(1,4840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4840, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4840.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4840 = 2^{3} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4840.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: \(38.6475945783\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 440)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 4840.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.56155 q^{3} -1.00000 q^{5} -3.56155 q^{7} -0.561553 q^{9} +O(q^{10})\) \(q-1.56155 q^{3} -1.00000 q^{5} -3.56155 q^{7} -0.561553 q^{9} +3.12311 q^{13} +1.56155 q^{15} -5.56155 q^{17} -2.43845 q^{19} +5.56155 q^{21} +7.12311 q^{23} +1.00000 q^{25} +5.56155 q^{27} +0.438447 q^{29} +8.68466 q^{31} +3.56155 q^{35} +9.80776 q^{37} -4.87689 q^{39} +10.0000 q^{41} -5.12311 q^{43} +0.561553 q^{45} -7.12311 q^{47} +5.68466 q^{49} +8.68466 q^{51} -4.43845 q^{53} +3.80776 q^{57} -13.3693 q^{59} +3.56155 q^{61} +2.00000 q^{63} -3.12311 q^{65} -11.1231 q^{69} -2.43845 q^{71} -4.87689 q^{73} -1.56155 q^{75} -0.876894 q^{79} -7.00000 q^{81} -10.0000 q^{83} +5.56155 q^{85} -0.684658 q^{87} +9.80776 q^{89} -11.1231 q^{91} -13.5616 q^{93} +2.43845 q^{95} +17.1231 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} - 2 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} - 2 q^{5} - 3 q^{7} + 3 q^{9} - 2 q^{13} - q^{15} - 7 q^{17} - 9 q^{19} + 7 q^{21} + 6 q^{23} + 2 q^{25} + 7 q^{27} + 5 q^{29} + 5 q^{31} + 3 q^{35} - q^{37} - 18 q^{39} + 20 q^{41} - 2 q^{43} - 3 q^{45} - 6 q^{47} - q^{49} + 5 q^{51} - 13 q^{53} - 13 q^{57} - 2 q^{59} + 3 q^{61} + 4 q^{63} + 2 q^{65} - 14 q^{69} - 9 q^{71} - 18 q^{73} + q^{75} - 10 q^{79} - 14 q^{81} - 20 q^{83} + 7 q^{85} + 11 q^{87} - q^{89} - 14 q^{91} - 23 q^{93} + 9 q^{95} + 26 q^{97}+O(q^{100}) \) 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
\(3\) −1.56155 −0.901563 −0.450781 0.892634i \(-0.648855\pi\)
−0.450781 + 0.892634i \(0.648855\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −3.56155 −1.34614 −0.673070 0.739579i \(-0.735025\pi\)
−0.673070 + 0.739579i \(0.735025\pi\)
\(8\) 0 0
\(9\) −0.561553 −0.187184
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) 3.12311 0.866194 0.433097 0.901347i \(-0.357421\pi\)
0.433097 + 0.901347i \(0.357421\pi\)
\(14\) 0 0
\(15\) 1.56155 0.403191
\(16\) 0 0
\(17\) −5.56155 −1.34887 −0.674437 0.738332i \(-0.735614\pi\)
−0.674437 + 0.738332i \(0.735614\pi\)
\(18\) 0 0
\(19\) −2.43845 −0.559418 −0.279709 0.960085i \(-0.590238\pi\)
−0.279709 + 0.960085i \(0.590238\pi\)
\(20\) 0 0
\(21\) 5.56155 1.21363
\(22\) 0 0
\(23\) 7.12311 1.48527 0.742635 0.669696i \(-0.233576\pi\)
0.742635 + 0.669696i \(0.233576\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 5.56155 1.07032
\(28\) 0 0
\(29\) 0.438447 0.0814176 0.0407088 0.999171i \(-0.487038\pi\)
0.0407088 + 0.999171i \(0.487038\pi\)
\(30\) 0 0
\(31\) 8.68466 1.55981 0.779905 0.625897i \(-0.215267\pi\)
0.779905 + 0.625897i \(0.215267\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.56155 0.602012
\(36\) 0 0
\(37\) 9.80776 1.61239 0.806193 0.591652i \(-0.201524\pi\)
0.806193 + 0.591652i \(0.201524\pi\)
\(38\) 0 0
\(39\) −4.87689 −0.780928
\(40\) 0 0
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) −5.12311 −0.781266 −0.390633 0.920546i \(-0.627744\pi\)
−0.390633 + 0.920546i \(0.627744\pi\)
\(44\) 0 0
\(45\) 0.561553 0.0837114
\(46\) 0 0
\(47\) −7.12311 −1.03901 −0.519506 0.854467i \(-0.673884\pi\)
−0.519506 + 0.854467i \(0.673884\pi\)
\(48\) 0 0
\(49\) 5.68466 0.812094
\(50\) 0 0
\(51\) 8.68466 1.21610
\(52\) 0 0
\(53\) −4.43845 −0.609668 −0.304834 0.952406i \(-0.598601\pi\)
−0.304834 + 0.952406i \(0.598601\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 3.80776 0.504351
\(58\) 0 0
\(59\) −13.3693 −1.74054 −0.870268 0.492578i \(-0.836055\pi\)
−0.870268 + 0.492578i \(0.836055\pi\)
\(60\) 0 0
\(61\) 3.56155 0.456010 0.228005 0.973660i \(-0.426780\pi\)
0.228005 + 0.973660i \(0.426780\pi\)
\(62\) 0 0
\(63\) 2.00000 0.251976
\(64\) 0 0
\(65\) −3.12311 −0.387374
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) −11.1231 −1.33906
\(70\) 0 0
\(71\) −2.43845 −0.289390 −0.144695 0.989476i \(-0.546220\pi\)
−0.144695 + 0.989476i \(0.546220\pi\)
\(72\) 0 0
\(73\) −4.87689 −0.570797 −0.285399 0.958409i \(-0.592126\pi\)
−0.285399 + 0.958409i \(0.592126\pi\)
\(74\) 0 0
\(75\) −1.56155 −0.180313
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −0.876894 −0.0986583 −0.0493292 0.998783i \(-0.515708\pi\)
−0.0493292 + 0.998783i \(0.515708\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) −10.0000 −1.09764 −0.548821 0.835940i \(-0.684923\pi\)
−0.548821 + 0.835940i \(0.684923\pi\)
\(84\) 0 0
\(85\) 5.56155 0.603235
\(86\) 0 0
\(87\) −0.684658 −0.0734031
\(88\) 0 0
\(89\) 9.80776 1.03962 0.519810 0.854282i \(-0.326003\pi\)
0.519810 + 0.854282i \(0.326003\pi\)
\(90\) 0 0
\(91\) −11.1231 −1.16602
\(92\) 0 0
\(93\) −13.5616 −1.40627
\(94\) 0 0
\(95\) 2.43845 0.250179
\(96\) 0 0
\(97\) 17.1231 1.73859 0.869294 0.494295i \(-0.164574\pi\)
0.869294 + 0.494295i \(0.164574\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 16.2462 1.61656 0.808279 0.588799i \(-0.200399\pi\)
0.808279 + 0.588799i \(0.200399\pi\)
\(102\) 0 0
\(103\) −14.2462 −1.40372 −0.701860 0.712314i \(-0.747647\pi\)
−0.701860 + 0.712314i \(0.747647\pi\)
\(104\) 0 0
\(105\) −5.56155 −0.542752
\(106\) 0 0
\(107\) −5.12311 −0.495269 −0.247635 0.968853i \(-0.579653\pi\)
−0.247635 + 0.968853i \(0.579653\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) −15.3153 −1.45367
\(112\) 0 0
\(113\) 1.12311 0.105653 0.0528264 0.998604i \(-0.483177\pi\)
0.0528264 + 0.998604i \(0.483177\pi\)
\(114\) 0 0
\(115\) −7.12311 −0.664233
\(116\) 0 0
\(117\) −1.75379 −0.162138
\(118\) 0 0
\(119\) 19.8078 1.81577
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) −15.6155 −1.40800
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 7.36932 0.653921 0.326961 0.945038i \(-0.393976\pi\)
0.326961 + 0.945038i \(0.393976\pi\)
\(128\) 0 0
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) −0.684658 −0.0598189 −0.0299094 0.999553i \(-0.509522\pi\)
−0.0299094 + 0.999553i \(0.509522\pi\)
\(132\) 0 0
\(133\) 8.68466 0.753055
\(134\) 0 0
\(135\) −5.56155 −0.478662
\(136\) 0 0
\(137\) −19.3693 −1.65483 −0.827416 0.561589i \(-0.810190\pi\)
−0.827416 + 0.561589i \(0.810190\pi\)
\(138\) 0 0
\(139\) −16.4924 −1.39887 −0.699435 0.714697i \(-0.746565\pi\)
−0.699435 + 0.714697i \(0.746565\pi\)
\(140\) 0 0
\(141\) 11.1231 0.936734
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −0.438447 −0.0364111
\(146\) 0 0
\(147\) −8.87689 −0.732154
\(148\) 0 0
\(149\) 1.31534 0.107757 0.0538785 0.998547i \(-0.482842\pi\)
0.0538785 + 0.998547i \(0.482842\pi\)
\(150\) 0 0
\(151\) 0.876894 0.0713607 0.0356803 0.999363i \(-0.488640\pi\)
0.0356803 + 0.999363i \(0.488640\pi\)
\(152\) 0 0
\(153\) 3.12311 0.252488
\(154\) 0 0
\(155\) −8.68466 −0.697569
\(156\) 0 0
\(157\) −12.9309 −1.03200 −0.515998 0.856590i \(-0.672579\pi\)
−0.515998 + 0.856590i \(0.672579\pi\)
\(158\) 0 0
\(159\) 6.93087 0.549654
\(160\) 0 0
\(161\) −25.3693 −1.99938
\(162\) 0 0
\(163\) 0.192236 0.0150571 0.00752854 0.999972i \(-0.497604\pi\)
0.00752854 + 0.999972i \(0.497604\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −1.31534 −0.101784 −0.0508921 0.998704i \(-0.516206\pi\)
−0.0508921 + 0.998704i \(0.516206\pi\)
\(168\) 0 0
\(169\) −3.24621 −0.249709
\(170\) 0 0
\(171\) 1.36932 0.104714
\(172\) 0 0
\(173\) 21.3693 1.62468 0.812340 0.583185i \(-0.198194\pi\)
0.812340 + 0.583185i \(0.198194\pi\)
\(174\) 0 0
\(175\) −3.56155 −0.269228
\(176\) 0 0
\(177\) 20.8769 1.56920
\(178\) 0 0
\(179\) 8.49242 0.634753 0.317377 0.948300i \(-0.397198\pi\)
0.317377 + 0.948300i \(0.397198\pi\)
\(180\) 0 0
\(181\) 14.4924 1.07721 0.538607 0.842557i \(-0.318951\pi\)
0.538607 + 0.842557i \(0.318951\pi\)
\(182\) 0 0
\(183\) −5.56155 −0.411122
\(184\) 0 0
\(185\) −9.80776 −0.721081
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −19.8078 −1.44080
\(190\) 0 0
\(191\) 1.75379 0.126900 0.0634499 0.997985i \(-0.479790\pi\)
0.0634499 + 0.997985i \(0.479790\pi\)
\(192\) 0 0
\(193\) −19.8078 −1.42579 −0.712897 0.701269i \(-0.752617\pi\)
−0.712897 + 0.701269i \(0.752617\pi\)
\(194\) 0 0
\(195\) 4.87689 0.349242
\(196\) 0 0
\(197\) −18.2462 −1.29999 −0.649994 0.759939i \(-0.725229\pi\)
−0.649994 + 0.759939i \(0.725229\pi\)
\(198\) 0 0
\(199\) −6.93087 −0.491316 −0.245658 0.969357i \(-0.579004\pi\)
−0.245658 + 0.969357i \(0.579004\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −1.56155 −0.109600
\(204\) 0 0
\(205\) −10.0000 −0.698430
\(206\) 0 0
\(207\) −4.00000 −0.278019
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −0.684658 −0.0471338 −0.0235669 0.999722i \(-0.507502\pi\)
−0.0235669 + 0.999722i \(0.507502\pi\)
\(212\) 0 0
\(213\) 3.80776 0.260904
\(214\) 0 0
\(215\) 5.12311 0.349393
\(216\) 0 0
\(217\) −30.9309 −2.09972
\(218\) 0 0
\(219\) 7.61553 0.514610
\(220\) 0 0
\(221\) −17.3693 −1.16839
\(222\) 0 0
\(223\) −23.1231 −1.54844 −0.774219 0.632918i \(-0.781857\pi\)
−0.774219 + 0.632918i \(0.781857\pi\)
\(224\) 0 0
\(225\) −0.561553 −0.0374369
\(226\) 0 0
\(227\) 18.0000 1.19470 0.597351 0.801980i \(-0.296220\pi\)
0.597351 + 0.801980i \(0.296220\pi\)
\(228\) 0 0
\(229\) −6.49242 −0.429031 −0.214516 0.976721i \(-0.568817\pi\)
−0.214516 + 0.976721i \(0.568817\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −11.3153 −0.741293 −0.370646 0.928774i \(-0.620864\pi\)
−0.370646 + 0.928774i \(0.620864\pi\)
\(234\) 0 0
\(235\) 7.12311 0.464660
\(236\) 0 0
\(237\) 1.36932 0.0889467
\(238\) 0 0
\(239\) −0.876894 −0.0567216 −0.0283608 0.999598i \(-0.509029\pi\)
−0.0283608 + 0.999598i \(0.509029\pi\)
\(240\) 0 0
\(241\) 28.7386 1.85122 0.925609 0.378481i \(-0.123553\pi\)
0.925609 + 0.378481i \(0.123553\pi\)
\(242\) 0 0
\(243\) −5.75379 −0.369106
\(244\) 0 0
\(245\) −5.68466 −0.363180
\(246\) 0 0
\(247\) −7.61553 −0.484564
\(248\) 0 0
\(249\) 15.6155 0.989594
\(250\) 0 0
\(251\) 23.1231 1.45952 0.729759 0.683705i \(-0.239632\pi\)
0.729759 + 0.683705i \(0.239632\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −8.68466 −0.543854
\(256\) 0 0
\(257\) −7.75379 −0.483668 −0.241834 0.970318i \(-0.577749\pi\)
−0.241834 + 0.970318i \(0.577749\pi\)
\(258\) 0 0
\(259\) −34.9309 −2.17050
\(260\) 0 0
\(261\) −0.246211 −0.0152401
\(262\) 0 0
\(263\) −24.4384 −1.50694 −0.753470 0.657483i \(-0.771621\pi\)
−0.753470 + 0.657483i \(0.771621\pi\)
\(264\) 0 0
\(265\) 4.43845 0.272652
\(266\) 0 0
\(267\) −15.3153 −0.937284
\(268\) 0 0
\(269\) −22.8769 −1.39483 −0.697414 0.716668i \(-0.745666\pi\)
−0.697414 + 0.716668i \(0.745666\pi\)
\(270\) 0 0
\(271\) −12.0000 −0.728948 −0.364474 0.931214i \(-0.618751\pi\)
−0.364474 + 0.931214i \(0.618751\pi\)
\(272\) 0 0
\(273\) 17.3693 1.05124
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −18.7386 −1.12590 −0.562948 0.826493i \(-0.690333\pi\)
−0.562948 + 0.826493i \(0.690333\pi\)
\(278\) 0 0
\(279\) −4.87689 −0.291972
\(280\) 0 0
\(281\) −18.4924 −1.10317 −0.551583 0.834120i \(-0.685976\pi\)
−0.551583 + 0.834120i \(0.685976\pi\)
\(282\) 0 0
\(283\) 18.4924 1.09926 0.549630 0.835408i \(-0.314769\pi\)
0.549630 + 0.835408i \(0.314769\pi\)
\(284\) 0 0
\(285\) −3.80776 −0.225552
\(286\) 0 0
\(287\) −35.6155 −2.10232
\(288\) 0 0
\(289\) 13.9309 0.819463
\(290\) 0 0
\(291\) −26.7386 −1.56745
\(292\) 0 0
\(293\) 11.1231 0.649819 0.324909 0.945745i \(-0.394666\pi\)
0.324909 + 0.945745i \(0.394666\pi\)
\(294\) 0 0
\(295\) 13.3693 0.778392
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 22.2462 1.28653
\(300\) 0 0
\(301\) 18.2462 1.05169
\(302\) 0 0
\(303\) −25.3693 −1.45743
\(304\) 0 0
\(305\) −3.56155 −0.203934
\(306\) 0 0
\(307\) −20.7386 −1.18362 −0.591808 0.806079i \(-0.701586\pi\)
−0.591808 + 0.806079i \(0.701586\pi\)
\(308\) 0 0
\(309\) 22.2462 1.26554
\(310\) 0 0
\(311\) 21.1771 1.20084 0.600421 0.799684i \(-0.295000\pi\)
0.600421 + 0.799684i \(0.295000\pi\)
\(312\) 0 0
\(313\) 16.2462 0.918290 0.459145 0.888361i \(-0.348156\pi\)
0.459145 + 0.888361i \(0.348156\pi\)
\(314\) 0 0
\(315\) −2.00000 −0.112687
\(316\) 0 0
\(317\) −3.56155 −0.200037 −0.100018 0.994986i \(-0.531890\pi\)
−0.100018 + 0.994986i \(0.531890\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 8.00000 0.446516
\(322\) 0 0
\(323\) 13.5616 0.754585
\(324\) 0 0
\(325\) 3.12311 0.173239
\(326\) 0 0
\(327\) −3.12311 −0.172708
\(328\) 0 0
\(329\) 25.3693 1.39866
\(330\) 0 0
\(331\) −2.24621 −0.123463 −0.0617315 0.998093i \(-0.519662\pi\)
−0.0617315 + 0.998093i \(0.519662\pi\)
\(332\) 0 0
\(333\) −5.50758 −0.301813
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −6.93087 −0.377549 −0.188774 0.982021i \(-0.560451\pi\)
−0.188774 + 0.982021i \(0.560451\pi\)
\(338\) 0 0
\(339\) −1.75379 −0.0952527
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 4.68466 0.252948
\(344\) 0 0
\(345\) 11.1231 0.598848
\(346\) 0 0
\(347\) −24.2462 −1.30160 −0.650802 0.759247i \(-0.725567\pi\)
−0.650802 + 0.759247i \(0.725567\pi\)
\(348\) 0 0
\(349\) 24.7386 1.32423 0.662114 0.749403i \(-0.269659\pi\)
0.662114 + 0.749403i \(0.269659\pi\)
\(350\) 0 0
\(351\) 17.3693 0.927106
\(352\) 0 0
\(353\) 3.75379 0.199794 0.0998970 0.994998i \(-0.468149\pi\)
0.0998970 + 0.994998i \(0.468149\pi\)
\(354\) 0 0
\(355\) 2.43845 0.129419
\(356\) 0 0
\(357\) −30.9309 −1.63704
\(358\) 0 0
\(359\) 8.49242 0.448213 0.224106 0.974565i \(-0.428054\pi\)
0.224106 + 0.974565i \(0.428054\pi\)
\(360\) 0 0
\(361\) −13.0540 −0.687051
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 4.87689 0.255268
\(366\) 0 0
\(367\) −28.0000 −1.46159 −0.730794 0.682598i \(-0.760850\pi\)
−0.730794 + 0.682598i \(0.760850\pi\)
\(368\) 0 0
\(369\) −5.61553 −0.292333
\(370\) 0 0
\(371\) 15.8078 0.820698
\(372\) 0 0
\(373\) 29.3693 1.52069 0.760343 0.649522i \(-0.225031\pi\)
0.760343 + 0.649522i \(0.225031\pi\)
\(374\) 0 0
\(375\) 1.56155 0.0806382
\(376\) 0 0
\(377\) 1.36932 0.0705234
\(378\) 0 0
\(379\) 27.6155 1.41851 0.709257 0.704950i \(-0.249030\pi\)
0.709257 + 0.704950i \(0.249030\pi\)
\(380\) 0 0
\(381\) −11.5076 −0.589551
\(382\) 0 0
\(383\) −29.8617 −1.52586 −0.762932 0.646479i \(-0.776241\pi\)
−0.762932 + 0.646479i \(0.776241\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 2.87689 0.146241
\(388\) 0 0
\(389\) −32.2462 −1.63495 −0.817474 0.575966i \(-0.804626\pi\)
−0.817474 + 0.575966i \(0.804626\pi\)
\(390\) 0 0
\(391\) −39.6155 −2.00344
\(392\) 0 0
\(393\) 1.06913 0.0539305
\(394\) 0 0
\(395\) 0.876894 0.0441213
\(396\) 0 0
\(397\) 5.50758 0.276417 0.138209 0.990403i \(-0.455866\pi\)
0.138209 + 0.990403i \(0.455866\pi\)
\(398\) 0 0
\(399\) −13.5616 −0.678927
\(400\) 0 0
\(401\) −21.8078 −1.08903 −0.544514 0.838752i \(-0.683286\pi\)
−0.544514 + 0.838752i \(0.683286\pi\)
\(402\) 0 0
\(403\) 27.1231 1.35110
\(404\) 0 0
\(405\) 7.00000 0.347833
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 17.1231 0.846683 0.423342 0.905970i \(-0.360857\pi\)
0.423342 + 0.905970i \(0.360857\pi\)
\(410\) 0 0
\(411\) 30.2462 1.49194
\(412\) 0 0
\(413\) 47.6155 2.34301
\(414\) 0 0
\(415\) 10.0000 0.490881
\(416\) 0 0
\(417\) 25.7538 1.26117
\(418\) 0 0
\(419\) −5.75379 −0.281091 −0.140545 0.990074i \(-0.544886\pi\)
−0.140545 + 0.990074i \(0.544886\pi\)
\(420\) 0 0
\(421\) 21.1231 1.02948 0.514739 0.857347i \(-0.327889\pi\)
0.514739 + 0.857347i \(0.327889\pi\)
\(422\) 0 0
\(423\) 4.00000 0.194487
\(424\) 0 0
\(425\) −5.56155 −0.269775
\(426\) 0 0
\(427\) −12.6847 −0.613854
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −20.4924 −0.987085 −0.493543 0.869722i \(-0.664298\pi\)
−0.493543 + 0.869722i \(0.664298\pi\)
\(432\) 0 0
\(433\) −23.8617 −1.14672 −0.573361 0.819303i \(-0.694361\pi\)
−0.573361 + 0.819303i \(0.694361\pi\)
\(434\) 0 0
\(435\) 0.684658 0.0328269
\(436\) 0 0
\(437\) −17.3693 −0.830887
\(438\) 0 0
\(439\) 24.9848 1.19246 0.596231 0.802813i \(-0.296664\pi\)
0.596231 + 0.802813i \(0.296664\pi\)
\(440\) 0 0
\(441\) −3.19224 −0.152011
\(442\) 0 0
\(443\) 20.0000 0.950229 0.475114 0.879924i \(-0.342407\pi\)
0.475114 + 0.879924i \(0.342407\pi\)
\(444\) 0 0
\(445\) −9.80776 −0.464933
\(446\) 0 0
\(447\) −2.05398 −0.0971497
\(448\) 0 0
\(449\) −28.7386 −1.35626 −0.678130 0.734942i \(-0.737209\pi\)
−0.678130 + 0.734942i \(0.737209\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −1.36932 −0.0643361
\(454\) 0 0
\(455\) 11.1231 0.521459
\(456\) 0 0
\(457\) 38.0540 1.78009 0.890045 0.455873i \(-0.150673\pi\)
0.890045 + 0.455873i \(0.150673\pi\)
\(458\) 0 0
\(459\) −30.9309 −1.44373
\(460\) 0 0
\(461\) −28.9309 −1.34744 −0.673722 0.738984i \(-0.735306\pi\)
−0.673722 + 0.738984i \(0.735306\pi\)
\(462\) 0 0
\(463\) −0.876894 −0.0407527 −0.0203764 0.999792i \(-0.506486\pi\)
−0.0203764 + 0.999792i \(0.506486\pi\)
\(464\) 0 0
\(465\) 13.5616 0.628902
\(466\) 0 0
\(467\) 8.68466 0.401878 0.200939 0.979604i \(-0.435601\pi\)
0.200939 + 0.979604i \(0.435601\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 20.1922 0.930409
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −2.43845 −0.111884
\(476\) 0 0
\(477\) 2.49242 0.114120
\(478\) 0 0
\(479\) 12.0000 0.548294 0.274147 0.961688i \(-0.411605\pi\)
0.274147 + 0.961688i \(0.411605\pi\)
\(480\) 0 0
\(481\) 30.6307 1.39664
\(482\) 0 0
\(483\) 39.6155 1.80257
\(484\) 0 0
\(485\) −17.1231 −0.777520
\(486\) 0 0
\(487\) −34.2462 −1.55184 −0.775922 0.630829i \(-0.782715\pi\)
−0.775922 + 0.630829i \(0.782715\pi\)
\(488\) 0 0
\(489\) −0.300187 −0.0135749
\(490\) 0 0
\(491\) 15.8078 0.713394 0.356697 0.934220i \(-0.383903\pi\)
0.356697 + 0.934220i \(0.383903\pi\)
\(492\) 0 0
\(493\) −2.43845 −0.109822
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 8.68466 0.389560
\(498\) 0 0
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 0 0
\(501\) 2.05398 0.0917648
\(502\) 0 0
\(503\) −29.1231 −1.29854 −0.649268 0.760560i \(-0.724924\pi\)
−0.649268 + 0.760560i \(0.724924\pi\)
\(504\) 0 0
\(505\) −16.2462 −0.722947
\(506\) 0 0
\(507\) 5.06913 0.225128
\(508\) 0 0
\(509\) −35.3693 −1.56772 −0.783859 0.620939i \(-0.786751\pi\)
−0.783859 + 0.620939i \(0.786751\pi\)
\(510\) 0 0
\(511\) 17.3693 0.768373
\(512\) 0 0
\(513\) −13.5616 −0.598757
\(514\) 0 0
\(515\) 14.2462 0.627763
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −33.3693 −1.46475
\(520\) 0 0
\(521\) 9.50758 0.416535 0.208267 0.978072i \(-0.433218\pi\)
0.208267 + 0.978072i \(0.433218\pi\)
\(522\) 0 0
\(523\) −23.3693 −1.02187 −0.510934 0.859620i \(-0.670701\pi\)
−0.510934 + 0.859620i \(0.670701\pi\)
\(524\) 0 0
\(525\) 5.56155 0.242726
\(526\) 0 0
\(527\) −48.3002 −2.10399
\(528\) 0 0
\(529\) 27.7386 1.20603
\(530\) 0 0
\(531\) 7.50758 0.325801
\(532\) 0 0
\(533\) 31.2311 1.35277
\(534\) 0 0
\(535\) 5.12311 0.221491
\(536\) 0 0
\(537\) −13.2614 −0.572270
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −11.5616 −0.497070 −0.248535 0.968623i \(-0.579949\pi\)
−0.248535 + 0.968623i \(0.579949\pi\)
\(542\) 0 0
\(543\) −22.6307 −0.971176
\(544\) 0 0
\(545\) −2.00000 −0.0856706
\(546\) 0 0
\(547\) 13.1231 0.561103 0.280552 0.959839i \(-0.409483\pi\)
0.280552 + 0.959839i \(0.409483\pi\)
\(548\) 0 0
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) −1.06913 −0.0455465
\(552\) 0 0
\(553\) 3.12311 0.132808
\(554\) 0 0
\(555\) 15.3153 0.650100
\(556\) 0 0
\(557\) −7.12311 −0.301816 −0.150908 0.988548i \(-0.548220\pi\)
−0.150908 + 0.988548i \(0.548220\pi\)
\(558\) 0 0
\(559\) −16.0000 −0.676728
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 8.24621 0.347536 0.173768 0.984787i \(-0.444406\pi\)
0.173768 + 0.984787i \(0.444406\pi\)
\(564\) 0 0
\(565\) −1.12311 −0.0472494
\(566\) 0 0
\(567\) 24.9309 1.04700
\(568\) 0 0
\(569\) 2.87689 0.120606 0.0603028 0.998180i \(-0.480793\pi\)
0.0603028 + 0.998180i \(0.480793\pi\)
\(570\) 0 0
\(571\) −31.8078 −1.33111 −0.665557 0.746347i \(-0.731806\pi\)
−0.665557 + 0.746347i \(0.731806\pi\)
\(572\) 0 0
\(573\) −2.73863 −0.114408
\(574\) 0 0
\(575\) 7.12311 0.297054
\(576\) 0 0
\(577\) −30.4924 −1.26942 −0.634708 0.772752i \(-0.718880\pi\)
−0.634708 + 0.772752i \(0.718880\pi\)
\(578\) 0 0
\(579\) 30.9309 1.28544
\(580\) 0 0
\(581\) 35.6155 1.47758
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 1.75379 0.0725102
\(586\) 0 0
\(587\) 21.5616 0.889941 0.444970 0.895545i \(-0.353214\pi\)
0.444970 + 0.895545i \(0.353214\pi\)
\(588\) 0 0
\(589\) −21.1771 −0.872586
\(590\) 0 0
\(591\) 28.4924 1.17202
\(592\) 0 0
\(593\) 7.61553 0.312732 0.156366 0.987699i \(-0.450022\pi\)
0.156366 + 0.987699i \(0.450022\pi\)
\(594\) 0 0
\(595\) −19.8078 −0.812039
\(596\) 0 0
\(597\) 10.8229 0.442953
\(598\) 0 0
\(599\) −15.3153 −0.625768 −0.312884 0.949791i \(-0.601295\pi\)
−0.312884 + 0.949791i \(0.601295\pi\)
\(600\) 0 0
\(601\) −37.2311 −1.51869 −0.759343 0.650690i \(-0.774480\pi\)
−0.759343 + 0.650690i \(0.774480\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −34.6847 −1.40781 −0.703903 0.710296i \(-0.748561\pi\)
−0.703903 + 0.710296i \(0.748561\pi\)
\(608\) 0 0
\(609\) 2.43845 0.0988109
\(610\) 0 0
\(611\) −22.2462 −0.899985
\(612\) 0 0
\(613\) −27.6155 −1.11538 −0.557690 0.830049i \(-0.688312\pi\)
−0.557690 + 0.830049i \(0.688312\pi\)
\(614\) 0 0
\(615\) 15.6155 0.629679
\(616\) 0 0
\(617\) 2.87689 0.115819 0.0579097 0.998322i \(-0.481556\pi\)
0.0579097 + 0.998322i \(0.481556\pi\)
\(618\) 0 0
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) 39.6155 1.58972
\(622\) 0 0
\(623\) −34.9309 −1.39948
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −54.5464 −2.17491
\(630\) 0 0
\(631\) −2.43845 −0.0970730 −0.0485365 0.998821i \(-0.515456\pi\)
−0.0485365 + 0.998821i \(0.515456\pi\)
\(632\) 0 0
\(633\) 1.06913 0.0424941
\(634\) 0 0
\(635\) −7.36932 −0.292442
\(636\) 0 0
\(637\) 17.7538 0.703431
\(638\) 0 0
\(639\) 1.36932 0.0541693
\(640\) 0 0
\(641\) 3.94602 0.155859 0.0779293 0.996959i \(-0.475169\pi\)
0.0779293 + 0.996959i \(0.475169\pi\)
\(642\) 0 0
\(643\) −19.3153 −0.761723 −0.380861 0.924632i \(-0.624372\pi\)
−0.380861 + 0.924632i \(0.624372\pi\)
\(644\) 0 0
\(645\) −8.00000 −0.315000
\(646\) 0 0
\(647\) −9.36932 −0.368346 −0.184173 0.982894i \(-0.558961\pi\)
−0.184173 + 0.982894i \(0.558961\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 48.3002 1.89303
\(652\) 0 0
\(653\) 10.1922 0.398853 0.199427 0.979913i \(-0.436092\pi\)
0.199427 + 0.979913i \(0.436092\pi\)
\(654\) 0 0
\(655\) 0.684658 0.0267518
\(656\) 0 0
\(657\) 2.73863 0.106844
\(658\) 0 0
\(659\) −5.06913 −0.197465 −0.0987326 0.995114i \(-0.531479\pi\)
−0.0987326 + 0.995114i \(0.531479\pi\)
\(660\) 0 0
\(661\) −14.4924 −0.563690 −0.281845 0.959460i \(-0.590946\pi\)
−0.281845 + 0.959460i \(0.590946\pi\)
\(662\) 0 0
\(663\) 27.1231 1.05337
\(664\) 0 0
\(665\) −8.68466 −0.336777
\(666\) 0 0
\(667\) 3.12311 0.120927
\(668\) 0 0
\(669\) 36.1080 1.39601
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 21.1771 0.816316 0.408158 0.912911i \(-0.366171\pi\)
0.408158 + 0.912911i \(0.366171\pi\)
\(674\) 0 0
\(675\) 5.56155 0.214064
\(676\) 0 0
\(677\) 6.24621 0.240061 0.120031 0.992770i \(-0.461701\pi\)
0.120031 + 0.992770i \(0.461701\pi\)
\(678\) 0 0
\(679\) −60.9848 −2.34038
\(680\) 0 0
\(681\) −28.1080 −1.07710
\(682\) 0 0
\(683\) 3.80776 0.145700 0.0728500 0.997343i \(-0.476791\pi\)
0.0728500 + 0.997343i \(0.476791\pi\)
\(684\) 0 0
\(685\) 19.3693 0.740064
\(686\) 0 0
\(687\) 10.1383 0.386799
\(688\) 0 0
\(689\) −13.8617 −0.528090
\(690\) 0 0
\(691\) −35.6155 −1.35488 −0.677439 0.735579i \(-0.736910\pi\)
−0.677439 + 0.735579i \(0.736910\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 16.4924 0.625593
\(696\) 0 0
\(697\) −55.6155 −2.10659
\(698\) 0 0
\(699\) 17.6695 0.668322
\(700\) 0 0
\(701\) 36.5464 1.38034 0.690169 0.723648i \(-0.257536\pi\)
0.690169 + 0.723648i \(0.257536\pi\)
\(702\) 0 0
\(703\) −23.9157 −0.901998
\(704\) 0 0
\(705\) −11.1231 −0.418920
\(706\) 0 0
\(707\) −57.8617 −2.17611
\(708\) 0 0
\(709\) 48.2462 1.81192 0.905962 0.423358i \(-0.139149\pi\)
0.905962 + 0.423358i \(0.139149\pi\)
\(710\) 0 0
\(711\) 0.492423 0.0184673
\(712\) 0 0
\(713\) 61.8617 2.31674
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 1.36932 0.0511381
\(718\) 0 0
\(719\) −42.0540 −1.56835 −0.784174 0.620541i \(-0.786913\pi\)
−0.784174 + 0.620541i \(0.786913\pi\)
\(720\) 0 0
\(721\) 50.7386 1.88961
\(722\) 0 0
\(723\) −44.8769 −1.66899
\(724\) 0 0
\(725\) 0.438447 0.0162835
\(726\) 0 0
\(727\) 23.6155 0.875851 0.437926 0.899011i \(-0.355713\pi\)
0.437926 + 0.899011i \(0.355713\pi\)
\(728\) 0 0
\(729\) 29.9848 1.11055
\(730\) 0 0
\(731\) 28.4924 1.05383
\(732\) 0 0
\(733\) 4.00000 0.147743 0.0738717 0.997268i \(-0.476464\pi\)
0.0738717 + 0.997268i \(0.476464\pi\)
\(734\) 0 0
\(735\) 8.87689 0.327429
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 46.7386 1.71931 0.859654 0.510876i \(-0.170679\pi\)
0.859654 + 0.510876i \(0.170679\pi\)
\(740\) 0 0
\(741\) 11.8920 0.436865
\(742\) 0 0
\(743\) 28.4384 1.04331 0.521653 0.853158i \(-0.325316\pi\)
0.521653 + 0.853158i \(0.325316\pi\)
\(744\) 0 0
\(745\) −1.31534 −0.0481904
\(746\) 0 0
\(747\) 5.61553 0.205461
\(748\) 0 0
\(749\) 18.2462 0.666702
\(750\) 0 0
\(751\) 18.4384 0.672828 0.336414 0.941714i \(-0.390786\pi\)
0.336414 + 0.941714i \(0.390786\pi\)
\(752\) 0 0
\(753\) −36.1080 −1.31585
\(754\) 0 0
\(755\) −0.876894 −0.0319135
\(756\) 0 0
\(757\) −26.4924 −0.962883 −0.481442 0.876478i \(-0.659887\pi\)
−0.481442 + 0.876478i \(0.659887\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −2.00000 −0.0724999 −0.0362500 0.999343i \(-0.511541\pi\)
−0.0362500 + 0.999343i \(0.511541\pi\)
\(762\) 0 0
\(763\) −7.12311 −0.257874
\(764\) 0 0
\(765\) −3.12311 −0.112916
\(766\) 0 0
\(767\) −41.7538 −1.50764
\(768\) 0 0
\(769\) −30.9848 −1.11734 −0.558671 0.829389i \(-0.688689\pi\)
−0.558671 + 0.829389i \(0.688689\pi\)
\(770\) 0 0
\(771\) 12.1080 0.436057
\(772\) 0 0
\(773\) 0.822919 0.0295983 0.0147992 0.999890i \(-0.495289\pi\)
0.0147992 + 0.999890i \(0.495289\pi\)
\(774\) 0 0
\(775\) 8.68466 0.311962
\(776\) 0 0
\(777\) 54.5464 1.95684
\(778\) 0 0
\(779\) −24.3845 −0.873664
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 2.43845 0.0871430
\(784\) 0 0
\(785\) 12.9309 0.461523
\(786\) 0 0
\(787\) 26.8769 0.958058 0.479029 0.877799i \(-0.340989\pi\)
0.479029 + 0.877799i \(0.340989\pi\)
\(788\) 0 0
\(789\) 38.1619 1.35860
\(790\) 0 0
\(791\) −4.00000 −0.142224
\(792\) 0 0
\(793\) 11.1231 0.394993
\(794\) 0 0
\(795\) −6.93087 −0.245813
\(796\) 0 0
\(797\) 12.7386 0.451226 0.225613 0.974217i \(-0.427562\pi\)
0.225613 + 0.974217i \(0.427562\pi\)
\(798\) 0 0
\(799\) 39.6155 1.40150
\(800\) 0 0
\(801\) −5.50758 −0.194601
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 25.3693 0.894151
\(806\) 0 0
\(807\) 35.7235 1.25753
\(808\) 0 0
\(809\) 52.7386 1.85419 0.927096 0.374824i \(-0.122297\pi\)
0.927096 + 0.374824i \(0.122297\pi\)
\(810\) 0 0
\(811\) 46.4384 1.63067 0.815337 0.578986i \(-0.196552\pi\)
0.815337 + 0.578986i \(0.196552\pi\)
\(812\) 0 0
\(813\) 18.7386 0.657193
\(814\) 0 0
\(815\) −0.192236 −0.00673373
\(816\) 0 0
\(817\) 12.4924 0.437055
\(818\) 0 0
\(819\) 6.24621 0.218260
\(820\) 0 0
\(821\) −19.7538 −0.689412 −0.344706 0.938711i \(-0.612021\pi\)
−0.344706 + 0.938711i \(0.612021\pi\)
\(822\) 0 0
\(823\) 18.7386 0.653188 0.326594 0.945165i \(-0.394099\pi\)
0.326594 + 0.945165i \(0.394099\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −10.3845 −0.361103 −0.180552 0.983565i \(-0.557788\pi\)
−0.180552 + 0.983565i \(0.557788\pi\)
\(828\) 0 0
\(829\) 47.8617 1.66231 0.831153 0.556043i \(-0.187681\pi\)
0.831153 + 0.556043i \(0.187681\pi\)
\(830\) 0 0
\(831\) 29.2614 1.01507
\(832\) 0 0
\(833\) −31.6155 −1.09541
\(834\) 0 0
\(835\) 1.31534 0.0455193
\(836\) 0 0
\(837\) 48.3002 1.66950
\(838\) 0 0
\(839\) 8.00000 0.276191 0.138095 0.990419i \(-0.455902\pi\)
0.138095 + 0.990419i \(0.455902\pi\)
\(840\) 0 0
\(841\) −28.8078 −0.993371
\(842\) 0 0
\(843\) 28.8769 0.994573
\(844\) 0 0
\(845\) 3.24621 0.111673
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −28.8769 −0.991052
\(850\) 0 0
\(851\) 69.8617 2.39483
\(852\) 0 0
\(853\) 40.1080 1.37327 0.686635 0.727002i \(-0.259087\pi\)
0.686635 + 0.727002i \(0.259087\pi\)
\(854\) 0 0
\(855\) −1.36932 −0.0468296
\(856\) 0 0
\(857\) −50.0540 −1.70981 −0.854906 0.518784i \(-0.826385\pi\)
−0.854906 + 0.518784i \(0.826385\pi\)
\(858\) 0 0
\(859\) 21.3693 0.729112 0.364556 0.931182i \(-0.381221\pi\)
0.364556 + 0.931182i \(0.381221\pi\)
\(860\) 0 0
\(861\) 55.6155 1.89537
\(862\) 0 0
\(863\) 9.36932 0.318935 0.159468 0.987203i \(-0.449022\pi\)
0.159468 + 0.987203i \(0.449022\pi\)
\(864\) 0 0
\(865\) −21.3693 −0.726579
\(866\) 0 0
\(867\) −21.7538 −0.738797
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −9.61553 −0.325436
\(874\) 0 0
\(875\) 3.56155 0.120402
\(876\) 0 0
\(877\) −39.1231 −1.32109 −0.660547 0.750785i \(-0.729675\pi\)
−0.660547 + 0.750785i \(0.729675\pi\)
\(878\) 0 0
\(879\) −17.3693 −0.585853
\(880\) 0 0
\(881\) −34.4924 −1.16208 −0.581040 0.813875i \(-0.697354\pi\)
−0.581040 + 0.813875i \(0.697354\pi\)
\(882\) 0 0
\(883\) −17.9460 −0.603932 −0.301966 0.953319i \(-0.597643\pi\)
−0.301966 + 0.953319i \(0.597643\pi\)
\(884\) 0 0
\(885\) −20.8769 −0.701769
\(886\) 0 0
\(887\) 55.3693 1.85912 0.929560 0.368671i \(-0.120187\pi\)
0.929560 + 0.368671i \(0.120187\pi\)
\(888\) 0 0
\(889\) −26.2462 −0.880270
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 17.3693 0.581242
\(894\) 0 0
\(895\) −8.49242 −0.283870
\(896\) 0 0
\(897\) −34.7386 −1.15989
\(898\) 0 0
\(899\) 3.80776 0.126996
\(900\) 0 0
\(901\) 24.6847 0.822365
\(902\) 0 0
\(903\) −28.4924 −0.948168
\(904\) 0 0
\(905\) −14.4924 −0.481744
\(906\) 0 0
\(907\) 45.1771 1.50008 0.750040 0.661392i \(-0.230034\pi\)
0.750040 + 0.661392i \(0.230034\pi\)
\(908\) 0 0
\(909\) −9.12311 −0.302594
\(910\) 0 0
\(911\) −41.6695 −1.38057 −0.690286 0.723537i \(-0.742515\pi\)
−0.690286 + 0.723537i \(0.742515\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 5.56155 0.183859
\(916\) 0 0
\(917\) 2.43845 0.0805246
\(918\) 0 0
\(919\) 9.75379 0.321748 0.160874 0.986975i \(-0.448569\pi\)
0.160874 + 0.986975i \(0.448569\pi\)
\(920\) 0 0
\(921\) 32.3845 1.06710
\(922\) 0 0
\(923\) −7.61553 −0.250668
\(924\) 0 0
\(925\) 9.80776 0.322477
\(926\) 0 0
\(927\) 8.00000 0.262754
\(928\) 0 0
\(929\) 0.930870 0.0305408 0.0152704 0.999883i \(-0.495139\pi\)
0.0152704 + 0.999883i \(0.495139\pi\)
\(930\) 0 0
\(931\) −13.8617 −0.454300
\(932\) 0 0
\(933\) −33.0691 −1.08263
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 29.3693 0.959454 0.479727 0.877418i \(-0.340736\pi\)
0.479727 + 0.877418i \(0.340736\pi\)
\(938\) 0 0
\(939\) −25.3693 −0.827896
\(940\) 0 0
\(941\) 28.4384 0.927067 0.463533 0.886079i \(-0.346581\pi\)
0.463533 + 0.886079i \(0.346581\pi\)
\(942\) 0 0
\(943\) 71.2311 2.31960
\(944\) 0 0
\(945\) 19.8078 0.644347
\(946\) 0 0
\(947\) 50.0540 1.62654 0.813268 0.581890i \(-0.197686\pi\)
0.813268 + 0.581890i \(0.197686\pi\)
\(948\) 0 0
\(949\) −15.2311 −0.494421
\(950\) 0 0
\(951\) 5.56155 0.180346
\(952\) 0 0
\(953\) −37.6695 −1.22023 −0.610117 0.792311i \(-0.708878\pi\)
−0.610117 + 0.792311i \(0.708878\pi\)
\(954\) 0 0
\(955\) −1.75379 −0.0567513
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 68.9848 2.22764
\(960\) 0 0
\(961\) 44.4233 1.43301
\(962\) 0 0
\(963\) 2.87689 0.0927066
\(964\) 0 0
\(965\) 19.8078 0.637634
\(966\) 0 0
\(967\) 8.05398 0.258998 0.129499 0.991580i \(-0.458663\pi\)
0.129499 + 0.991580i \(0.458663\pi\)
\(968\) 0 0
\(969\) −21.1771 −0.680306
\(970\) 0 0
\(971\) 2.24621 0.0720843 0.0360422 0.999350i \(-0.488525\pi\)
0.0360422 + 0.999350i \(0.488525\pi\)
\(972\) 0 0
\(973\) 58.7386 1.88307
\(974\) 0 0
\(975\) −4.87689 −0.156186
\(976\) 0 0
\(977\) −51.3693 −1.64345 −0.821725 0.569884i \(-0.806988\pi\)
−0.821725 + 0.569884i \(0.806988\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −1.12311 −0.0358580
\(982\) 0 0
\(983\) 20.8769 0.665870 0.332935 0.942950i \(-0.391961\pi\)
0.332935 + 0.942950i \(0.391961\pi\)
\(984\) 0 0
\(985\) 18.2462 0.581373
\(986\) 0 0
\(987\) −39.6155 −1.26098
\(988\) 0 0
\(989\) −36.4924 −1.16039
\(990\) 0 0
\(991\) −52.4924 −1.66748 −0.833738 0.552160i \(-0.813804\pi\)
−0.833738 + 0.552160i \(0.813804\pi\)
\(992\) 0 0
\(993\) 3.50758 0.111310
\(994\) 0 0
\(995\) 6.93087 0.219723
\(996\) 0 0
\(997\) −12.9848 −0.411234 −0.205617 0.978633i \(-0.565920\pi\)
−0.205617 + 0.978633i \(0.565920\pi\)
\(998\) 0 0
\(999\) 54.5464 1.72577
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 4840.2.a.n.1.1 2
4.3 odd 2 9680.2.a.bl.1.2 2
11.10 odd 2 440.2.a.f.1.1 2
33.32 even 2 3960.2.a.be.1.2 2
44.43 even 2 880.2.a.l.1.2 2
55.32 even 4 2200.2.b.h.1849.3 4
55.43 even 4 2200.2.b.h.1849.2 4
55.54 odd 2 2200.2.a.m.1.2 2
88.21 odd 2 3520.2.a.bl.1.2 2
88.43 even 2 3520.2.a.bs.1.1 2
132.131 odd 2 7920.2.a.ca.1.1 2
220.43 odd 4 4400.2.b.u.4049.3 4
220.87 odd 4 4400.2.b.u.4049.2 4
220.219 even 2 4400.2.a.br.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.a.f.1.1 2 11.10 odd 2
880.2.a.l.1.2 2 44.43 even 2
2200.2.a.m.1.2 2 55.54 odd 2
2200.2.b.h.1849.2 4 55.43 even 4
2200.2.b.h.1849.3 4 55.32 even 4
3520.2.a.bl.1.2 2 88.21 odd 2
3520.2.a.bs.1.1 2 88.43 even 2
3960.2.a.be.1.2 2 33.32 even 2
4400.2.a.br.1.1 2 220.219 even 2
4400.2.b.u.4049.2 4 220.87 odd 4
4400.2.b.u.4049.3 4 220.43 odd 4
4840.2.a.n.1.1 2 1.1 even 1 trivial
7920.2.a.ca.1.1 2 132.131 odd 2
9680.2.a.bl.1.2 2 4.3 odd 2