Properties

Label 4840.2.a.n.1.2
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 1.2
Root \(2.56155\) 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)
 
\(f(q)\) \(=\) \(q+2.56155 q^{3} -1.00000 q^{5} +0.561553 q^{7} +3.56155 q^{9} +O(q^{10})\) \(q+2.56155 q^{3} -1.00000 q^{5} +0.561553 q^{7} +3.56155 q^{9} -5.12311 q^{13} -2.56155 q^{15} -1.43845 q^{17} -6.56155 q^{19} +1.43845 q^{21} -1.12311 q^{23} +1.00000 q^{25} +1.43845 q^{27} +4.56155 q^{29} -3.68466 q^{31} -0.561553 q^{35} -10.8078 q^{37} -13.1231 q^{39} +10.0000 q^{41} +3.12311 q^{43} -3.56155 q^{45} +1.12311 q^{47} -6.68466 q^{49} -3.68466 q^{51} -8.56155 q^{53} -16.8078 q^{57} +11.3693 q^{59} -0.561553 q^{61} +2.00000 q^{63} +5.12311 q^{65} -2.87689 q^{69} -6.56155 q^{71} -13.1231 q^{73} +2.56155 q^{75} -9.12311 q^{79} -7.00000 q^{81} -10.0000 q^{83} +1.43845 q^{85} +11.6847 q^{87} -10.8078 q^{89} -2.87689 q^{91} -9.43845 q^{93} +6.56155 q^{95} +8.87689 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} - 2 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) 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
\(3\) 2.56155 1.47891 0.739457 0.673204i \(-0.235083\pi\)
0.739457 + 0.673204i \(0.235083\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 0.561553 0.212247 0.106124 0.994353i \(-0.466156\pi\)
0.106124 + 0.994353i \(0.466156\pi\)
\(8\) 0 0
\(9\) 3.56155 1.18718
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) −5.12311 −1.42089 −0.710447 0.703751i \(-0.751507\pi\)
−0.710447 + 0.703751i \(0.751507\pi\)
\(14\) 0 0
\(15\) −2.56155 −0.661390
\(16\) 0 0
\(17\) −1.43845 −0.348875 −0.174437 0.984668i \(-0.555811\pi\)
−0.174437 + 0.984668i \(0.555811\pi\)
\(18\) 0 0
\(19\) −6.56155 −1.50532 −0.752662 0.658407i \(-0.771230\pi\)
−0.752662 + 0.658407i \(0.771230\pi\)
\(20\) 0 0
\(21\) 1.43845 0.313895
\(22\) 0 0
\(23\) −1.12311 −0.234184 −0.117092 0.993121i \(-0.537357\pi\)
−0.117092 + 0.993121i \(0.537357\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.43845 0.276829
\(28\) 0 0
\(29\) 4.56155 0.847059 0.423530 0.905882i \(-0.360791\pi\)
0.423530 + 0.905882i \(0.360791\pi\)
\(30\) 0 0
\(31\) −3.68466 −0.661784 −0.330892 0.943669i \(-0.607350\pi\)
−0.330892 + 0.943669i \(0.607350\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.561553 −0.0949197
\(36\) 0 0
\(37\) −10.8078 −1.77679 −0.888393 0.459084i \(-0.848178\pi\)
−0.888393 + 0.459084i \(0.848178\pi\)
\(38\) 0 0
\(39\) −13.1231 −2.10138
\(40\) 0 0
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) 3.12311 0.476269 0.238135 0.971232i \(-0.423464\pi\)
0.238135 + 0.971232i \(0.423464\pi\)
\(44\) 0 0
\(45\) −3.56155 −0.530925
\(46\) 0 0
\(47\) 1.12311 0.163822 0.0819109 0.996640i \(-0.473898\pi\)
0.0819109 + 0.996640i \(0.473898\pi\)
\(48\) 0 0
\(49\) −6.68466 −0.954951
\(50\) 0 0
\(51\) −3.68466 −0.515955
\(52\) 0 0
\(53\) −8.56155 −1.17602 −0.588010 0.808854i \(-0.700088\pi\)
−0.588010 + 0.808854i \(0.700088\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −16.8078 −2.22624
\(58\) 0 0
\(59\) 11.3693 1.48016 0.740079 0.672519i \(-0.234788\pi\)
0.740079 + 0.672519i \(0.234788\pi\)
\(60\) 0 0
\(61\) −0.561553 −0.0718995 −0.0359497 0.999354i \(-0.511446\pi\)
−0.0359497 + 0.999354i \(0.511446\pi\)
\(62\) 0 0
\(63\) 2.00000 0.251976
\(64\) 0 0
\(65\) 5.12311 0.635443
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) −2.87689 −0.346337
\(70\) 0 0
\(71\) −6.56155 −0.778713 −0.389357 0.921087i \(-0.627303\pi\)
−0.389357 + 0.921087i \(0.627303\pi\)
\(72\) 0 0
\(73\) −13.1231 −1.53594 −0.767972 0.640484i \(-0.778734\pi\)
−0.767972 + 0.640484i \(0.778734\pi\)
\(74\) 0 0
\(75\) 2.56155 0.295783
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −9.12311 −1.02643 −0.513215 0.858260i \(-0.671546\pi\)
−0.513215 + 0.858260i \(0.671546\pi\)
\(80\) 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
\(85\) 1.43845 0.156022
\(86\) 0 0
\(87\) 11.6847 1.25273
\(88\) 0 0
\(89\) −10.8078 −1.14562 −0.572810 0.819688i \(-0.694147\pi\)
−0.572810 + 0.819688i \(0.694147\pi\)
\(90\) 0 0
\(91\) −2.87689 −0.301580
\(92\) 0 0
\(93\) −9.43845 −0.978721
\(94\) 0 0
\(95\) 6.56155 0.673201
\(96\) 0 0
\(97\) 8.87689 0.901312 0.450656 0.892698i \(-0.351190\pi\)
0.450656 + 0.892698i \(0.351190\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −0.246211 −0.0244989 −0.0122495 0.999925i \(-0.503899\pi\)
−0.0122495 + 0.999925i \(0.503899\pi\)
\(102\) 0 0
\(103\) 2.24621 0.221326 0.110663 0.993858i \(-0.464703\pi\)
0.110663 + 0.993858i \(0.464703\pi\)
\(104\) 0 0
\(105\) −1.43845 −0.140378
\(106\) 0 0
\(107\) 3.12311 0.301922 0.150961 0.988540i \(-0.451763\pi\)
0.150961 + 0.988540i \(0.451763\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) −27.6847 −2.62771
\(112\) 0 0
\(113\) −7.12311 −0.670085 −0.335043 0.942203i \(-0.608751\pi\)
−0.335043 + 0.942203i \(0.608751\pi\)
\(114\) 0 0
\(115\) 1.12311 0.104730
\(116\) 0 0
\(117\) −18.2462 −1.68686
\(118\) 0 0
\(119\) −0.807764 −0.0740476
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 25.6155 2.30967
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −17.3693 −1.54128 −0.770639 0.637272i \(-0.780063\pi\)
−0.770639 + 0.637272i \(0.780063\pi\)
\(128\) 0 0
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) 11.6847 1.02089 0.510447 0.859909i \(-0.329480\pi\)
0.510447 + 0.859909i \(0.329480\pi\)
\(132\) 0 0
\(133\) −3.68466 −0.319500
\(134\) 0 0
\(135\) −1.43845 −0.123802
\(136\) 0 0
\(137\) 5.36932 0.458732 0.229366 0.973340i \(-0.426335\pi\)
0.229366 + 0.973340i \(0.426335\pi\)
\(138\) 0 0
\(139\) 16.4924 1.39887 0.699435 0.714697i \(-0.253435\pi\)
0.699435 + 0.714697i \(0.253435\pi\)
\(140\) 0 0
\(141\) 2.87689 0.242278
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −4.56155 −0.378816
\(146\) 0 0
\(147\) −17.1231 −1.41229
\(148\) 0 0
\(149\) 13.6847 1.12109 0.560545 0.828124i \(-0.310592\pi\)
0.560545 + 0.828124i \(0.310592\pi\)
\(150\) 0 0
\(151\) 9.12311 0.742428 0.371214 0.928547i \(-0.378942\pi\)
0.371214 + 0.928547i \(0.378942\pi\)
\(152\) 0 0
\(153\) −5.12311 −0.414179
\(154\) 0 0
\(155\) 3.68466 0.295959
\(156\) 0 0
\(157\) 15.9309 1.27142 0.635711 0.771927i \(-0.280707\pi\)
0.635711 + 0.771927i \(0.280707\pi\)
\(158\) 0 0
\(159\) −21.9309 −1.73923
\(160\) 0 0
\(161\) −0.630683 −0.0497048
\(162\) 0 0
\(163\) 20.8078 1.62979 0.814895 0.579609i \(-0.196795\pi\)
0.814895 + 0.579609i \(0.196795\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −13.6847 −1.05895 −0.529475 0.848325i \(-0.677611\pi\)
−0.529475 + 0.848325i \(0.677611\pi\)
\(168\) 0 0
\(169\) 13.2462 1.01894
\(170\) 0 0
\(171\) −23.3693 −1.78710
\(172\) 0 0
\(173\) −3.36932 −0.256164 −0.128082 0.991764i \(-0.540882\pi\)
−0.128082 + 0.991764i \(0.540882\pi\)
\(174\) 0 0
\(175\) 0.561553 0.0424494
\(176\) 0 0
\(177\) 29.1231 2.18903
\(178\) 0 0
\(179\) −24.4924 −1.83065 −0.915325 0.402716i \(-0.868066\pi\)
−0.915325 + 0.402716i \(0.868066\pi\)
\(180\) 0 0
\(181\) −18.4924 −1.37453 −0.687265 0.726406i \(-0.741189\pi\)
−0.687265 + 0.726406i \(0.741189\pi\)
\(182\) 0 0
\(183\) −1.43845 −0.106333
\(184\) 0 0
\(185\) 10.8078 0.794603
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0.807764 0.0587562
\(190\) 0 0
\(191\) 18.2462 1.32025 0.660125 0.751156i \(-0.270503\pi\)
0.660125 + 0.751156i \(0.270503\pi\)
\(192\) 0 0
\(193\) 0.807764 0.0581441 0.0290721 0.999577i \(-0.490745\pi\)
0.0290721 + 0.999577i \(0.490745\pi\)
\(194\) 0 0
\(195\) 13.1231 0.939765
\(196\) 0 0
\(197\) −1.75379 −0.124952 −0.0624761 0.998046i \(-0.519900\pi\)
−0.0624761 + 0.998046i \(0.519900\pi\)
\(198\) 0 0
\(199\) 21.9309 1.55464 0.777319 0.629107i \(-0.216579\pi\)
0.777319 + 0.629107i \(0.216579\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 2.56155 0.179786
\(204\) 0 0
\(205\) −10.0000 −0.698430
\(206\) 0 0
\(207\) −4.00000 −0.278019
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 11.6847 0.804405 0.402203 0.915551i \(-0.368245\pi\)
0.402203 + 0.915551i \(0.368245\pi\)
\(212\) 0 0
\(213\) −16.8078 −1.15165
\(214\) 0 0
\(215\) −3.12311 −0.212994
\(216\) 0 0
\(217\) −2.06913 −0.140462
\(218\) 0 0
\(219\) −33.6155 −2.27153
\(220\) 0 0
\(221\) 7.36932 0.495714
\(222\) 0 0
\(223\) −14.8769 −0.996231 −0.498115 0.867111i \(-0.665974\pi\)
−0.498115 + 0.867111i \(0.665974\pi\)
\(224\) 0 0
\(225\) 3.56155 0.237437
\(226\) 0 0
\(227\) 18.0000 1.19470 0.597351 0.801980i \(-0.296220\pi\)
0.597351 + 0.801980i \(0.296220\pi\)
\(228\) 0 0
\(229\) 26.4924 1.75067 0.875334 0.483518i \(-0.160641\pi\)
0.875334 + 0.483518i \(0.160641\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −23.6847 −1.55163 −0.775817 0.630958i \(-0.782662\pi\)
−0.775817 + 0.630958i \(0.782662\pi\)
\(234\) 0 0
\(235\) −1.12311 −0.0732633
\(236\) 0 0
\(237\) −23.3693 −1.51800
\(238\) 0 0
\(239\) −9.12311 −0.590125 −0.295062 0.955478i \(-0.595340\pi\)
−0.295062 + 0.955478i \(0.595340\pi\)
\(240\) 0 0
\(241\) −20.7386 −1.33589 −0.667946 0.744209i \(-0.732826\pi\)
−0.667946 + 0.744209i \(0.732826\pi\)
\(242\) 0 0
\(243\) −22.2462 −1.42710
\(244\) 0 0
\(245\) 6.68466 0.427067
\(246\) 0 0
\(247\) 33.6155 2.13890
\(248\) 0 0
\(249\) −25.6155 −1.62332
\(250\) 0 0
\(251\) 14.8769 0.939021 0.469511 0.882927i \(-0.344430\pi\)
0.469511 + 0.882927i \(0.344430\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 3.68466 0.230742
\(256\) 0 0
\(257\) −24.2462 −1.51244 −0.756219 0.654319i \(-0.772955\pi\)
−0.756219 + 0.654319i \(0.772955\pi\)
\(258\) 0 0
\(259\) −6.06913 −0.377117
\(260\) 0 0
\(261\) 16.2462 1.00562
\(262\) 0 0
\(263\) −28.5616 −1.76118 −0.880590 0.473878i \(-0.842854\pi\)
−0.880590 + 0.473878i \(0.842854\pi\)
\(264\) 0 0
\(265\) 8.56155 0.525932
\(266\) 0 0
\(267\) −27.6847 −1.69427
\(268\) 0 0
\(269\) −31.1231 −1.89761 −0.948805 0.315864i \(-0.897706\pi\)
−0.948805 + 0.315864i \(0.897706\pi\)
\(270\) 0 0
\(271\) −12.0000 −0.728948 −0.364474 0.931214i \(-0.618751\pi\)
−0.364474 + 0.931214i \(0.618751\pi\)
\(272\) 0 0
\(273\) −7.36932 −0.446011
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 30.7386 1.84691 0.923453 0.383712i \(-0.125355\pi\)
0.923453 + 0.383712i \(0.125355\pi\)
\(278\) 0 0
\(279\) −13.1231 −0.785660
\(280\) 0 0
\(281\) 14.4924 0.864545 0.432273 0.901743i \(-0.357712\pi\)
0.432273 + 0.901743i \(0.357712\pi\)
\(282\) 0 0
\(283\) −14.4924 −0.861485 −0.430743 0.902475i \(-0.641748\pi\)
−0.430743 + 0.902475i \(0.641748\pi\)
\(284\) 0 0
\(285\) 16.8078 0.995606
\(286\) 0 0
\(287\) 5.61553 0.331474
\(288\) 0 0
\(289\) −14.9309 −0.878286
\(290\) 0 0
\(291\) 22.7386 1.33296
\(292\) 0 0
\(293\) 2.87689 0.168070 0.0840350 0.996463i \(-0.473219\pi\)
0.0840350 + 0.996463i \(0.473219\pi\)
\(294\) 0 0
\(295\) −11.3693 −0.661947
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 5.75379 0.332750
\(300\) 0 0
\(301\) 1.75379 0.101087
\(302\) 0 0
\(303\) −0.630683 −0.0362318
\(304\) 0 0
\(305\) 0.561553 0.0321544
\(306\) 0 0
\(307\) 28.7386 1.64020 0.820100 0.572220i \(-0.193918\pi\)
0.820100 + 0.572220i \(0.193918\pi\)
\(308\) 0 0
\(309\) 5.75379 0.327322
\(310\) 0 0
\(311\) −24.1771 −1.37096 −0.685478 0.728093i \(-0.740407\pi\)
−0.685478 + 0.728093i \(0.740407\pi\)
\(312\) 0 0
\(313\) −0.246211 −0.0139167 −0.00695834 0.999976i \(-0.502215\pi\)
−0.00695834 + 0.999976i \(0.502215\pi\)
\(314\) 0 0
\(315\) −2.00000 −0.112687
\(316\) 0 0
\(317\) 0.561553 0.0315399 0.0157700 0.999876i \(-0.494980\pi\)
0.0157700 + 0.999876i \(0.494980\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 8.00000 0.446516
\(322\) 0 0
\(323\) 9.43845 0.525169
\(324\) 0 0
\(325\) −5.12311 −0.284179
\(326\) 0 0
\(327\) 5.12311 0.283308
\(328\) 0 0
\(329\) 0.630683 0.0347707
\(330\) 0 0
\(331\) 14.2462 0.783043 0.391521 0.920169i \(-0.371949\pi\)
0.391521 + 0.920169i \(0.371949\pi\)
\(332\) 0 0
\(333\) −38.4924 −2.10937
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 21.9309 1.19465 0.597325 0.801999i \(-0.296230\pi\)
0.597325 + 0.801999i \(0.296230\pi\)
\(338\) 0 0
\(339\) −18.2462 −0.990998
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −7.68466 −0.414933
\(344\) 0 0
\(345\) 2.87689 0.154887
\(346\) 0 0
\(347\) −7.75379 −0.416245 −0.208123 0.978103i \(-0.566735\pi\)
−0.208123 + 0.978103i \(0.566735\pi\)
\(348\) 0 0
\(349\) −24.7386 −1.32423 −0.662114 0.749403i \(-0.730341\pi\)
−0.662114 + 0.749403i \(0.730341\pi\)
\(350\) 0 0
\(351\) −7.36932 −0.393345
\(352\) 0 0
\(353\) 20.2462 1.07760 0.538799 0.842435i \(-0.318878\pi\)
0.538799 + 0.842435i \(0.318878\pi\)
\(354\) 0 0
\(355\) 6.56155 0.348251
\(356\) 0 0
\(357\) −2.06913 −0.109510
\(358\) 0 0
\(359\) −24.4924 −1.29266 −0.646330 0.763058i \(-0.723697\pi\)
−0.646330 + 0.763058i \(0.723697\pi\)
\(360\) 0 0
\(361\) 24.0540 1.26600
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 13.1231 0.686895
\(366\) 0 0
\(367\) −28.0000 −1.46159 −0.730794 0.682598i \(-0.760850\pi\)
−0.730794 + 0.682598i \(0.760850\pi\)
\(368\) 0 0
\(369\) 35.6155 1.85407
\(370\) 0 0
\(371\) −4.80776 −0.249607
\(372\) 0 0
\(373\) 4.63068 0.239768 0.119884 0.992788i \(-0.461748\pi\)
0.119884 + 0.992788i \(0.461748\pi\)
\(374\) 0 0
\(375\) −2.56155 −0.132278
\(376\) 0 0
\(377\) −23.3693 −1.20358
\(378\) 0 0
\(379\) −13.6155 −0.699383 −0.349691 0.936865i \(-0.613714\pi\)
−0.349691 + 0.936865i \(0.613714\pi\)
\(380\) 0 0
\(381\) −44.4924 −2.27942
\(382\) 0 0
\(383\) 27.8617 1.42367 0.711834 0.702348i \(-0.247865\pi\)
0.711834 + 0.702348i \(0.247865\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 11.1231 0.565419
\(388\) 0 0
\(389\) −15.7538 −0.798749 −0.399374 0.916788i \(-0.630773\pi\)
−0.399374 + 0.916788i \(0.630773\pi\)
\(390\) 0 0
\(391\) 1.61553 0.0817008
\(392\) 0 0
\(393\) 29.9309 1.50981
\(394\) 0 0
\(395\) 9.12311 0.459033
\(396\) 0 0
\(397\) 38.4924 1.93188 0.965940 0.258767i \(-0.0833163\pi\)
0.965940 + 0.258767i \(0.0833163\pi\)
\(398\) 0 0
\(399\) −9.43845 −0.472513
\(400\) 0 0
\(401\) −1.19224 −0.0595374 −0.0297687 0.999557i \(-0.509477\pi\)
−0.0297687 + 0.999557i \(0.509477\pi\)
\(402\) 0 0
\(403\) 18.8769 0.940325
\(404\) 0 0
\(405\) 7.00000 0.347833
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 8.87689 0.438934 0.219467 0.975620i \(-0.429568\pi\)
0.219467 + 0.975620i \(0.429568\pi\)
\(410\) 0 0
\(411\) 13.7538 0.678424
\(412\) 0 0
\(413\) 6.38447 0.314159
\(414\) 0 0
\(415\) 10.0000 0.490881
\(416\) 0 0
\(417\) 42.2462 2.06881
\(418\) 0 0
\(419\) −22.2462 −1.08680 −0.543399 0.839474i \(-0.682863\pi\)
−0.543399 + 0.839474i \(0.682863\pi\)
\(420\) 0 0
\(421\) 12.8769 0.627581 0.313791 0.949492i \(-0.398401\pi\)
0.313791 + 0.949492i \(0.398401\pi\)
\(422\) 0 0
\(423\) 4.00000 0.194487
\(424\) 0 0
\(425\) −1.43845 −0.0697749
\(426\) 0 0
\(427\) −0.315342 −0.0152604
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 12.4924 0.601739 0.300869 0.953665i \(-0.402723\pi\)
0.300869 + 0.953665i \(0.402723\pi\)
\(432\) 0 0
\(433\) 33.8617 1.62729 0.813646 0.581361i \(-0.197480\pi\)
0.813646 + 0.581361i \(0.197480\pi\)
\(434\) 0 0
\(435\) −11.6847 −0.560236
\(436\) 0 0
\(437\) 7.36932 0.352522
\(438\) 0 0
\(439\) −40.9848 −1.95610 −0.978050 0.208371i \(-0.933184\pi\)
−0.978050 + 0.208371i \(0.933184\pi\)
\(440\) 0 0
\(441\) −23.8078 −1.13370
\(442\) 0 0
\(443\) 20.0000 0.950229 0.475114 0.879924i \(-0.342407\pi\)
0.475114 + 0.879924i \(0.342407\pi\)
\(444\) 0 0
\(445\) 10.8078 0.512337
\(446\) 0 0
\(447\) 35.0540 1.65800
\(448\) 0 0
\(449\) 20.7386 0.978717 0.489358 0.872083i \(-0.337231\pi\)
0.489358 + 0.872083i \(0.337231\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 23.3693 1.09799
\(454\) 0 0
\(455\) 2.87689 0.134871
\(456\) 0 0
\(457\) 0.946025 0.0442532 0.0221266 0.999755i \(-0.492956\pi\)
0.0221266 + 0.999755i \(0.492956\pi\)
\(458\) 0 0
\(459\) −2.06913 −0.0965787
\(460\) 0 0
\(461\) −0.0691303 −0.00321972 −0.00160986 0.999999i \(-0.500512\pi\)
−0.00160986 + 0.999999i \(0.500512\pi\)
\(462\) 0 0
\(463\) −9.12311 −0.423987 −0.211993 0.977271i \(-0.567996\pi\)
−0.211993 + 0.977271i \(0.567996\pi\)
\(464\) 0 0
\(465\) 9.43845 0.437698
\(466\) 0 0
\(467\) −3.68466 −0.170506 −0.0852528 0.996359i \(-0.527170\pi\)
−0.0852528 + 0.996359i \(0.527170\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 40.8078 1.88032
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −6.56155 −0.301065
\(476\) 0 0
\(477\) −30.4924 −1.39615
\(478\) 0 0
\(479\) 12.0000 0.548294 0.274147 0.961688i \(-0.411605\pi\)
0.274147 + 0.961688i \(0.411605\pi\)
\(480\) 0 0
\(481\) 55.3693 2.52462
\(482\) 0 0
\(483\) −1.61553 −0.0735091
\(484\) 0 0
\(485\) −8.87689 −0.403079
\(486\) 0 0
\(487\) −17.7538 −0.804501 −0.402250 0.915530i \(-0.631772\pi\)
−0.402250 + 0.915530i \(0.631772\pi\)
\(488\) 0 0
\(489\) 53.3002 2.41032
\(490\) 0 0
\(491\) −4.80776 −0.216971 −0.108486 0.994098i \(-0.534600\pi\)
−0.108486 + 0.994098i \(0.534600\pi\)
\(492\) 0 0
\(493\) −6.56155 −0.295517
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −3.68466 −0.165280
\(498\) 0 0
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 0 0
\(501\) −35.0540 −1.56610
\(502\) 0 0
\(503\) −20.8769 −0.930855 −0.465427 0.885086i \(-0.654099\pi\)
−0.465427 + 0.885086i \(0.654099\pi\)
\(504\) 0 0
\(505\) 0.246211 0.0109563
\(506\) 0 0
\(507\) 33.9309 1.50692
\(508\) 0 0
\(509\) −10.6307 −0.471197 −0.235598 0.971851i \(-0.575705\pi\)
−0.235598 + 0.971851i \(0.575705\pi\)
\(510\) 0 0
\(511\) −7.36932 −0.325999
\(512\) 0 0
\(513\) −9.43845 −0.416718
\(514\) 0 0
\(515\) −2.24621 −0.0989799
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −8.63068 −0.378845
\(520\) 0 0
\(521\) 42.4924 1.86163 0.930813 0.365495i \(-0.119100\pi\)
0.930813 + 0.365495i \(0.119100\pi\)
\(522\) 0 0
\(523\) 1.36932 0.0598760 0.0299380 0.999552i \(-0.490469\pi\)
0.0299380 + 0.999552i \(0.490469\pi\)
\(524\) 0 0
\(525\) 1.43845 0.0627790
\(526\) 0 0
\(527\) 5.30019 0.230880
\(528\) 0 0
\(529\) −21.7386 −0.945158
\(530\) 0 0
\(531\) 40.4924 1.75722
\(532\) 0 0
\(533\) −51.2311 −2.21906
\(534\) 0 0
\(535\) −3.12311 −0.135024
\(536\) 0 0
\(537\) −62.7386 −2.70737
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −7.43845 −0.319804 −0.159902 0.987133i \(-0.551118\pi\)
−0.159902 + 0.987133i \(0.551118\pi\)
\(542\) 0 0
\(543\) −47.3693 −2.03281
\(544\) 0 0
\(545\) −2.00000 −0.0856706
\(546\) 0 0
\(547\) 4.87689 0.208521 0.104260 0.994550i \(-0.466752\pi\)
0.104260 + 0.994550i \(0.466752\pi\)
\(548\) 0 0
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) −29.9309 −1.27510
\(552\) 0 0
\(553\) −5.12311 −0.217857
\(554\) 0 0
\(555\) 27.6847 1.17515
\(556\) 0 0
\(557\) 1.12311 0.0475875 0.0237938 0.999717i \(-0.492425\pi\)
0.0237938 + 0.999717i \(0.492425\pi\)
\(558\) 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 \(-0.555594\pi\)
−0.173768 + 0.984787i \(0.555594\pi\)
\(564\) 0 0
\(565\) 7.12311 0.299671
\(566\) 0 0
\(567\) −3.93087 −0.165081
\(568\) 0 0
\(569\) 11.1231 0.466305 0.233152 0.972440i \(-0.425096\pi\)
0.233152 + 0.972440i \(0.425096\pi\)
\(570\) 0 0
\(571\) −11.1922 −0.468380 −0.234190 0.972191i \(-0.575244\pi\)
−0.234190 + 0.972191i \(0.575244\pi\)
\(572\) 0 0
\(573\) 46.7386 1.95253
\(574\) 0 0
\(575\) −1.12311 −0.0468367
\(576\) 0 0
\(577\) 2.49242 0.103761 0.0518805 0.998653i \(-0.483479\pi\)
0.0518805 + 0.998653i \(0.483479\pi\)
\(578\) 0 0
\(579\) 2.06913 0.0859901
\(580\) 0 0
\(581\) −5.61553 −0.232971
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 18.2462 0.754388
\(586\) 0 0
\(587\) 17.4384 0.719762 0.359881 0.932998i \(-0.382817\pi\)
0.359881 + 0.932998i \(0.382817\pi\)
\(588\) 0 0
\(589\) 24.1771 0.996199
\(590\) 0 0
\(591\) −4.49242 −0.184794
\(592\) 0 0
\(593\) −33.6155 −1.38042 −0.690212 0.723607i \(-0.742483\pi\)
−0.690212 + 0.723607i \(0.742483\pi\)
\(594\) 0 0
\(595\) 0.807764 0.0331151
\(596\) 0 0
\(597\) 56.1771 2.29917
\(598\) 0 0
\(599\) −27.6847 −1.13116 −0.565582 0.824692i \(-0.691349\pi\)
−0.565582 + 0.824692i \(0.691349\pi\)
\(600\) 0 0
\(601\) 45.2311 1.84501 0.922507 0.385981i \(-0.126137\pi\)
0.922507 + 0.385981i \(0.126137\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −22.3153 −0.905752 −0.452876 0.891574i \(-0.649602\pi\)
−0.452876 + 0.891574i \(0.649602\pi\)
\(608\) 0 0
\(609\) 6.56155 0.265888
\(610\) 0 0
\(611\) −5.75379 −0.232773
\(612\) 0 0
\(613\) 13.6155 0.549926 0.274963 0.961455i \(-0.411334\pi\)
0.274963 + 0.961455i \(0.411334\pi\)
\(614\) 0 0
\(615\) −25.6155 −1.03292
\(616\) 0 0
\(617\) 11.1231 0.447799 0.223900 0.974612i \(-0.428121\pi\)
0.223900 + 0.974612i \(0.428121\pi\)
\(618\) 0 0
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) −1.61553 −0.0648289
\(622\) 0 0
\(623\) −6.06913 −0.243155
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 15.5464 0.619875
\(630\) 0 0
\(631\) −6.56155 −0.261211 −0.130606 0.991434i \(-0.541692\pi\)
−0.130606 + 0.991434i \(0.541692\pi\)
\(632\) 0 0
\(633\) 29.9309 1.18965
\(634\) 0 0
\(635\) 17.3693 0.689280
\(636\) 0 0
\(637\) 34.2462 1.35688
\(638\) 0 0
\(639\) −23.3693 −0.924476
\(640\) 0 0
\(641\) 41.0540 1.62153 0.810767 0.585369i \(-0.199050\pi\)
0.810767 + 0.585369i \(0.199050\pi\)
\(642\) 0 0
\(643\) −31.6847 −1.24952 −0.624760 0.780816i \(-0.714803\pi\)
−0.624760 + 0.780816i \(0.714803\pi\)
\(644\) 0 0
\(645\) −8.00000 −0.315000
\(646\) 0 0
\(647\) 15.3693 0.604230 0.302115 0.953271i \(-0.402307\pi\)
0.302115 + 0.953271i \(0.402307\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −5.30019 −0.207731
\(652\) 0 0
\(653\) 30.8078 1.20560 0.602800 0.797892i \(-0.294051\pi\)
0.602800 + 0.797892i \(0.294051\pi\)
\(654\) 0 0
\(655\) −11.6847 −0.456557
\(656\) 0 0
\(657\) −46.7386 −1.82345
\(658\) 0 0
\(659\) −33.9309 −1.32176 −0.660880 0.750492i \(-0.729817\pi\)
−0.660880 + 0.750492i \(0.729817\pi\)
\(660\) 0 0
\(661\) 18.4924 0.719272 0.359636 0.933093i \(-0.382901\pi\)
0.359636 + 0.933093i \(0.382901\pi\)
\(662\) 0 0
\(663\) 18.8769 0.733118
\(664\) 0 0
\(665\) 3.68466 0.142885
\(666\) 0 0
\(667\) −5.12311 −0.198367
\(668\) 0 0
\(669\) −38.1080 −1.47334
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −24.1771 −0.931958 −0.465979 0.884796i \(-0.654298\pi\)
−0.465979 + 0.884796i \(0.654298\pi\)
\(674\) 0 0
\(675\) 1.43845 0.0553659
\(676\) 0 0
\(677\) −10.2462 −0.393794 −0.196897 0.980424i \(-0.563086\pi\)
−0.196897 + 0.980424i \(0.563086\pi\)
\(678\) 0 0
\(679\) 4.98485 0.191301
\(680\) 0 0
\(681\) 46.1080 1.76686
\(682\) 0 0
\(683\) −16.8078 −0.643131 −0.321566 0.946887i \(-0.604209\pi\)
−0.321566 + 0.946887i \(0.604209\pi\)
\(684\) 0 0
\(685\) −5.36932 −0.205151
\(686\) 0 0
\(687\) 67.8617 2.58909
\(688\) 0 0
\(689\) 43.8617 1.67100
\(690\) 0 0
\(691\) 5.61553 0.213625 0.106812 0.994279i \(-0.465936\pi\)
0.106812 + 0.994279i \(0.465936\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −16.4924 −0.625593
\(696\) 0 0
\(697\) −14.3845 −0.544851
\(698\) 0 0
\(699\) −60.6695 −2.29473
\(700\) 0 0
\(701\) −33.5464 −1.26703 −0.633515 0.773730i \(-0.718389\pi\)
−0.633515 + 0.773730i \(0.718389\pi\)
\(702\) 0 0
\(703\) 70.9157 2.67464
\(704\) 0 0
\(705\) −2.87689 −0.108350
\(706\) 0 0
\(707\) −0.138261 −0.00519983
\(708\) 0 0
\(709\) 31.7538 1.19254 0.596269 0.802784i \(-0.296649\pi\)
0.596269 + 0.802784i \(0.296649\pi\)
\(710\) 0 0
\(711\) −32.4924 −1.21856
\(712\) 0 0
\(713\) 4.13826 0.154979
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −23.3693 −0.872743
\(718\) 0 0
\(719\) −4.94602 −0.184456 −0.0922278 0.995738i \(-0.529399\pi\)
−0.0922278 + 0.995738i \(0.529399\pi\)
\(720\) 0 0
\(721\) 1.26137 0.0469757
\(722\) 0 0
\(723\) −53.1231 −1.97567
\(724\) 0 0
\(725\) 4.56155 0.169412
\(726\) 0 0
\(727\) −17.6155 −0.653324 −0.326662 0.945141i \(-0.605924\pi\)
−0.326662 + 0.945141i \(0.605924\pi\)
\(728\) 0 0
\(729\) −35.9848 −1.33277
\(730\) 0 0
\(731\) −4.49242 −0.166158
\(732\) 0 0
\(733\) 4.00000 0.147743 0.0738717 0.997268i \(-0.476464\pi\)
0.0738717 + 0.997268i \(0.476464\pi\)
\(734\) 0 0
\(735\) 17.1231 0.631595
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −2.73863 −0.100742 −0.0503711 0.998731i \(-0.516040\pi\)
−0.0503711 + 0.998731i \(0.516040\pi\)
\(740\) 0 0
\(741\) 86.1080 3.16325
\(742\) 0 0
\(743\) 32.5616 1.19457 0.597284 0.802030i \(-0.296247\pi\)
0.597284 + 0.802030i \(0.296247\pi\)
\(744\) 0 0
\(745\) −13.6847 −0.501367
\(746\) 0 0
\(747\) −35.6155 −1.30310
\(748\) 0 0
\(749\) 1.75379 0.0640820
\(750\) 0 0
\(751\) 22.5616 0.823283 0.411641 0.911346i \(-0.364956\pi\)
0.411641 + 0.911346i \(0.364956\pi\)
\(752\) 0 0
\(753\) 38.1080 1.38873
\(754\) 0 0
\(755\) −9.12311 −0.332024
\(756\) 0 0
\(757\) 6.49242 0.235971 0.117986 0.993015i \(-0.462356\pi\)
0.117986 + 0.993015i \(0.462356\pi\)
\(758\) 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
\(763\) 1.12311 0.0406592
\(764\) 0 0
\(765\) 5.12311 0.185226
\(766\) 0 0
\(767\) −58.2462 −2.10315
\(768\) 0 0
\(769\) 34.9848 1.26159 0.630793 0.775951i \(-0.282730\pi\)
0.630793 + 0.775951i \(0.282730\pi\)
\(770\) 0 0
\(771\) −62.1080 −2.23676
\(772\) 0 0
\(773\) 46.1771 1.66087 0.830437 0.557112i \(-0.188091\pi\)
0.830437 + 0.557112i \(0.188091\pi\)
\(774\) 0 0
\(775\) −3.68466 −0.132357
\(776\) 0 0
\(777\) −15.5464 −0.557724
\(778\) 0 0
\(779\) −65.6155 −2.35092
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 6.56155 0.234491
\(784\) 0 0
\(785\) −15.9309 −0.568597
\(786\) 0 0
\(787\) 35.1231 1.25200 0.626002 0.779822i \(-0.284690\pi\)
0.626002 + 0.779822i \(0.284690\pi\)
\(788\) 0 0
\(789\) −73.1619 −2.60463
\(790\) 0 0
\(791\) −4.00000 −0.142224
\(792\) 0 0
\(793\) 2.87689 0.102162
\(794\) 0 0
\(795\) 21.9309 0.777808
\(796\) 0 0
\(797\) −36.7386 −1.30135 −0.650675 0.759357i \(-0.725514\pi\)
−0.650675 + 0.759357i \(0.725514\pi\)
\(798\) 0 0
\(799\) −1.61553 −0.0571533
\(800\) 0 0
\(801\) −38.4924 −1.36006
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0.630683 0.0222287
\(806\) 0 0
\(807\) −79.7235 −2.80640
\(808\) 0 0
\(809\) 3.26137 0.114664 0.0573318 0.998355i \(-0.481741\pi\)
0.0573318 + 0.998355i \(0.481741\pi\)
\(810\) 0 0
\(811\) 50.5616 1.77546 0.887728 0.460368i \(-0.152283\pi\)
0.887728 + 0.460368i \(0.152283\pi\)
\(812\) 0 0
\(813\) −30.7386 −1.07805
\(814\) 0 0
\(815\) −20.8078 −0.728864
\(816\) 0 0
\(817\) −20.4924 −0.716939
\(818\) 0 0
\(819\) −10.2462 −0.358032
\(820\) 0 0
\(821\) −36.2462 −1.26500 −0.632501 0.774560i \(-0.717971\pi\)
−0.632501 + 0.774560i \(0.717971\pi\)
\(822\) 0 0
\(823\) −30.7386 −1.07148 −0.535741 0.844383i \(-0.679968\pi\)
−0.535741 + 0.844383i \(0.679968\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −51.6155 −1.79485 −0.897424 0.441169i \(-0.854564\pi\)
−0.897424 + 0.441169i \(0.854564\pi\)
\(828\) 0 0
\(829\) −9.86174 −0.342512 −0.171256 0.985227i \(-0.554783\pi\)
−0.171256 + 0.985227i \(0.554783\pi\)
\(830\) 0 0
\(831\) 78.7386 2.73141
\(832\) 0 0
\(833\) 9.61553 0.333158
\(834\) 0 0
\(835\) 13.6847 0.473577
\(836\) 0 0
\(837\) −5.30019 −0.183201
\(838\) 0 0
\(839\) 8.00000 0.276191 0.138095 0.990419i \(-0.455902\pi\)
0.138095 + 0.990419i \(0.455902\pi\)
\(840\) 0 0
\(841\) −8.19224 −0.282491
\(842\) 0 0
\(843\) 37.1231 1.27859
\(844\) 0 0
\(845\) −13.2462 −0.455684
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −37.1231 −1.27406
\(850\) 0 0
\(851\) 12.1383 0.416094
\(852\) 0 0
\(853\) −34.1080 −1.16783 −0.583917 0.811813i \(-0.698481\pi\)
−0.583917 + 0.811813i \(0.698481\pi\)
\(854\) 0 0
\(855\) 23.3693 0.799214
\(856\) 0 0
\(857\) −12.9460 −0.442228 −0.221114 0.975248i \(-0.570969\pi\)
−0.221114 + 0.975248i \(0.570969\pi\)
\(858\) 0 0
\(859\) −3.36932 −0.114960 −0.0574798 0.998347i \(-0.518306\pi\)
−0.0574798 + 0.998347i \(0.518306\pi\)
\(860\) 0 0
\(861\) 14.3845 0.490221
\(862\) 0 0
\(863\) −15.3693 −0.523178 −0.261589 0.965179i \(-0.584246\pi\)
−0.261589 + 0.965179i \(0.584246\pi\)
\(864\) 0 0
\(865\) 3.36932 0.114560
\(866\) 0 0
\(867\) −38.2462 −1.29891
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 31.6155 1.07002
\(874\) 0 0
\(875\) −0.561553 −0.0189839
\(876\) 0 0
\(877\) −30.8769 −1.04264 −0.521319 0.853362i \(-0.674560\pi\)
−0.521319 + 0.853362i \(0.674560\pi\)
\(878\) 0 0
\(879\) 7.36932 0.248561
\(880\) 0 0
\(881\) −1.50758 −0.0507916 −0.0253958 0.999677i \(-0.508085\pi\)
−0.0253958 + 0.999677i \(0.508085\pi\)
\(882\) 0 0
\(883\) −55.0540 −1.85271 −0.926357 0.376647i \(-0.877077\pi\)
−0.926357 + 0.376647i \(0.877077\pi\)
\(884\) 0 0
\(885\) −29.1231 −0.978962
\(886\) 0 0
\(887\) 30.6307 1.02848 0.514239 0.857647i \(-0.328074\pi\)
0.514239 + 0.857647i \(0.328074\pi\)
\(888\) 0 0
\(889\) −9.75379 −0.327132
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −7.36932 −0.246605
\(894\) 0 0
\(895\) 24.4924 0.818691
\(896\) 0 0
\(897\) 14.7386 0.492109
\(898\) 0 0
\(899\) −16.8078 −0.560570
\(900\) 0 0
\(901\) 12.3153 0.410284
\(902\) 0 0
\(903\) 4.49242 0.149498
\(904\) 0 0
\(905\) 18.4924 0.614709
\(906\) 0 0
\(907\) −0.177081 −0.00587988 −0.00293994 0.999996i \(-0.500936\pi\)
−0.00293994 + 0.999996i \(0.500936\pi\)
\(908\) 0 0
\(909\) −0.876894 −0.0290848
\(910\) 0 0
\(911\) 36.6695 1.21491 0.607457 0.794352i \(-0.292190\pi\)
0.607457 + 0.794352i \(0.292190\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 1.43845 0.0475536
\(916\) 0 0
\(917\) 6.56155 0.216682
\(918\) 0 0
\(919\) 26.2462 0.865783 0.432891 0.901446i \(-0.357493\pi\)
0.432891 + 0.901446i \(0.357493\pi\)
\(920\) 0 0
\(921\) 73.6155 2.42571
\(922\) 0 0
\(923\) 33.6155 1.10647
\(924\) 0 0
\(925\) −10.8078 −0.355357
\(926\) 0 0
\(927\) 8.00000 0.262754
\(928\) 0 0
\(929\) −27.9309 −0.916382 −0.458191 0.888854i \(-0.651502\pi\)
−0.458191 + 0.888854i \(0.651502\pi\)
\(930\) 0 0
\(931\) 43.8617 1.43751
\(932\) 0 0
\(933\) −61.9309 −2.02753
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 4.63068 0.151278 0.0756389 0.997135i \(-0.475900\pi\)
0.0756389 + 0.997135i \(0.475900\pi\)
\(938\) 0 0
\(939\) −0.630683 −0.0205816
\(940\) 0 0
\(941\) 32.5616 1.06148 0.530738 0.847536i \(-0.321915\pi\)
0.530738 + 0.847536i \(0.321915\pi\)
\(942\) 0 0
\(943\) −11.2311 −0.365734
\(944\) 0 0
\(945\) −0.807764 −0.0262766
\(946\) 0 0
\(947\) 12.9460 0.420689 0.210345 0.977627i \(-0.432541\pi\)
0.210345 + 0.977627i \(0.432541\pi\)
\(948\) 0 0
\(949\) 67.2311 2.18241
\(950\) 0 0
\(951\) 1.43845 0.0466448
\(952\) 0 0
\(953\) 40.6695 1.31741 0.658707 0.752399i \(-0.271104\pi\)
0.658707 + 0.752399i \(0.271104\pi\)
\(954\) 0 0
\(955\) −18.2462 −0.590434
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 3.01515 0.0973644
\(960\) 0 0
\(961\) −17.4233 −0.562042
\(962\) 0 0
\(963\) 11.1231 0.358437
\(964\) 0 0
\(965\) −0.807764 −0.0260028
\(966\) 0 0
\(967\) −29.0540 −0.934313 −0.467156 0.884175i \(-0.654722\pi\)
−0.467156 + 0.884175i \(0.654722\pi\)
\(968\) 0 0
\(969\) 24.1771 0.776680
\(970\) 0 0
\(971\) −14.2462 −0.457183 −0.228591 0.973522i \(-0.573412\pi\)
−0.228591 + 0.973522i \(0.573412\pi\)
\(972\) 0 0
\(973\) 9.26137 0.296906
\(974\) 0 0
\(975\) −13.1231 −0.420276
\(976\) 0 0
\(977\) −26.6307 −0.851991 −0.425996 0.904725i \(-0.640076\pi\)
−0.425996 + 0.904725i \(0.640076\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 7.12311 0.227423
\(982\) 0 0
\(983\) 29.1231 0.928883 0.464441 0.885604i \(-0.346255\pi\)
0.464441 + 0.885604i \(0.346255\pi\)
\(984\) 0 0
\(985\) 1.75379 0.0558804
\(986\) 0 0
\(987\) 1.61553 0.0514228
\(988\) 0 0
\(989\) −3.50758 −0.111534
\(990\) 0 0
\(991\) −19.5076 −0.619679 −0.309839 0.950789i \(-0.600275\pi\)
−0.309839 + 0.950789i \(0.600275\pi\)
\(992\) 0 0
\(993\) 36.4924 1.15805
\(994\) 0 0
\(995\) −21.9309 −0.695255
\(996\) 0 0
\(997\) 52.9848 1.67805 0.839023 0.544095i \(-0.183127\pi\)
0.839023 + 0.544095i \(0.183127\pi\)
\(998\) 0 0
\(999\) −15.5464 −0.491866
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.1.2 2
4.3 odd 2 9680.2.a.bl.1.1 2
11.10 odd 2 440.2.a.f.1.2 2
33.32 even 2 3960.2.a.be.1.1 2
44.43 even 2 880.2.a.l.1.1 2
55.32 even 4 2200.2.b.h.1849.1 4
55.43 even 4 2200.2.b.h.1849.4 4
55.54 odd 2 2200.2.a.m.1.1 2
88.21 odd 2 3520.2.a.bl.1.1 2
88.43 even 2 3520.2.a.bs.1.2 2
132.131 odd 2 7920.2.a.ca.1.2 2
220.43 odd 4 4400.2.b.u.4049.1 4
220.87 odd 4 4400.2.b.u.4049.4 4
220.219 even 2 4400.2.a.br.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.a.f.1.2 2 11.10 odd 2
880.2.a.l.1.1 2 44.43 even 2
2200.2.a.m.1.1 2 55.54 odd 2
2200.2.b.h.1849.1 4 55.32 even 4
2200.2.b.h.1849.4 4 55.43 even 4
3520.2.a.bl.1.1 2 88.21 odd 2
3520.2.a.bs.1.2 2 88.43 even 2
3960.2.a.be.1.1 2 33.32 even 2
4400.2.a.br.1.2 2 220.219 even 2
4400.2.b.u.4049.1 4 220.43 odd 4
4400.2.b.u.4049.4 4 220.87 odd 4
4840.2.a.n.1.2 2 1.1 even 1 trivial
7920.2.a.ca.1.2 2 132.131 odd 2
9680.2.a.bl.1.1 2 4.3 odd 2