Properties

Label 2268.2.a.g.1.2
Level $2268$
Weight $2$
Character 2268.1
Self dual yes
Analytic conductor $18.110$
Analytic rank $1$
Dimension $3$
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] = [2268,2,Mod(1,2268)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2268, 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("2268.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2268.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: \(18.1100711784\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.321.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 252)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.69963\) of defining polynomial
Character \(\chi\) \(=\) 2268.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.11126 q^{5} +1.00000 q^{7} +O(q^{10})\) \(q-1.11126 q^{5} +1.00000 q^{7} -1.88874 q^{11} -1.00000 q^{13} -5.87636 q^{17} +7.09888 q^{19} +3.98762 q^{23} -3.76509 q^{25} +0.987620 q^{29} -0.666208 q^{31} -1.11126 q^{35} -1.33379 q^{37} -1.88874 q^{41} -10.8640 q^{43} -11.0989 q^{47} +1.00000 q^{49} -12.2101 q^{53} +2.09888 q^{55} +4.76509 q^{59} +3.76509 q^{61} +1.11126 q^{65} +4.09888 q^{67} -10.7651 q^{71} -3.09888 q^{73} -1.88874 q^{77} +6.43130 q^{79} -11.8764 q^{83} +6.53018 q^{85} +14.3090 q^{89} -1.00000 q^{91} -7.88874 q^{95} +0.765092 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{5} + 3 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{5} + 3 q^{7} - 6 q^{11} - 3 q^{13} + 3 q^{19} - 6 q^{23} + 6 q^{25} - 15 q^{29} - 3 q^{31} - 3 q^{35} - 3 q^{37} - 6 q^{41} + 3 q^{43} - 15 q^{47} + 3 q^{49} - 18 q^{53} - 12 q^{55} - 3 q^{59} - 6 q^{61} + 3 q^{65} - 6 q^{67} - 15 q^{71} + 9 q^{73} - 6 q^{77} + 3 q^{79} - 18 q^{83} - 15 q^{85} + 6 q^{89} - 3 q^{91} - 24 q^{95} - 15 q^{97}+O(q^{100}) \) 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
\(3\) 0 0
\(4\) 0 0
\(5\) −1.11126 −0.496972 −0.248486 0.968635i \(-0.579933\pi\)
−0.248486 + 0.968635i \(0.579933\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.88874 −0.569475 −0.284738 0.958605i \(-0.591906\pi\)
−0.284738 + 0.958605i \(0.591906\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350 −0.138675 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.87636 −1.42523 −0.712613 0.701557i \(-0.752488\pi\)
−0.712613 + 0.701557i \(0.752488\pi\)
\(18\) 0 0
\(19\) 7.09888 1.62860 0.814298 0.580447i \(-0.197122\pi\)
0.814298 + 0.580447i \(0.197122\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.98762 0.831476 0.415738 0.909484i \(-0.363523\pi\)
0.415738 + 0.909484i \(0.363523\pi\)
\(24\) 0 0
\(25\) −3.76509 −0.753018
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0.987620 0.183396 0.0916982 0.995787i \(-0.470770\pi\)
0.0916982 + 0.995787i \(0.470770\pi\)
\(30\) 0 0
\(31\) −0.666208 −0.119654 −0.0598272 0.998209i \(-0.519055\pi\)
−0.0598272 + 0.998209i \(0.519055\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.11126 −0.187838
\(36\) 0 0
\(37\) −1.33379 −0.219274 −0.109637 0.993972i \(-0.534969\pi\)
−0.109637 + 0.993972i \(0.534969\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −1.88874 −0.294971 −0.147485 0.989064i \(-0.547118\pi\)
−0.147485 + 0.989064i \(0.547118\pi\)
\(42\) 0 0
\(43\) −10.8640 −1.65674 −0.828370 0.560181i \(-0.810732\pi\)
−0.828370 + 0.560181i \(0.810732\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −11.0989 −1.61894 −0.809469 0.587162i \(-0.800245\pi\)
−0.809469 + 0.587162i \(0.800245\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −12.2101 −1.67719 −0.838596 0.544753i \(-0.816623\pi\)
−0.838596 + 0.544753i \(0.816623\pi\)
\(54\) 0 0
\(55\) 2.09888 0.283014
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.76509 0.620362 0.310181 0.950677i \(-0.399610\pi\)
0.310181 + 0.950677i \(0.399610\pi\)
\(60\) 0 0
\(61\) 3.76509 0.482071 0.241035 0.970516i \(-0.422513\pi\)
0.241035 + 0.970516i \(0.422513\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.11126 0.137835
\(66\) 0 0
\(67\) 4.09888 0.500758 0.250379 0.968148i \(-0.419445\pi\)
0.250379 + 0.968148i \(0.419445\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −10.7651 −1.27758 −0.638791 0.769381i \(-0.720565\pi\)
−0.638791 + 0.769381i \(0.720565\pi\)
\(72\) 0 0
\(73\) −3.09888 −0.362697 −0.181348 0.983419i \(-0.558046\pi\)
−0.181348 + 0.983419i \(0.558046\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.88874 −0.215241
\(78\) 0 0
\(79\) 6.43130 0.723578 0.361789 0.932260i \(-0.382166\pi\)
0.361789 + 0.932260i \(0.382166\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −11.8764 −1.30360 −0.651800 0.758391i \(-0.725986\pi\)
−0.651800 + 0.758391i \(0.725986\pi\)
\(84\) 0 0
\(85\) 6.53018 0.708298
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 14.3090 1.51675 0.758377 0.651816i \(-0.225993\pi\)
0.758377 + 0.651816i \(0.225993\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −7.88874 −0.809367
\(96\) 0 0
\(97\) 0.765092 0.0776833 0.0388417 0.999245i \(-0.487633\pi\)
0.0388417 + 0.999245i \(0.487633\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 8.87636 0.883230 0.441615 0.897205i \(-0.354406\pi\)
0.441615 + 0.897205i \(0.354406\pi\)
\(102\) 0 0
\(103\) −9.96286 −0.981670 −0.490835 0.871253i \(-0.663308\pi\)
−0.490835 + 0.871253i \(0.663308\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −5.22253 −0.504881 −0.252440 0.967612i \(-0.581233\pi\)
−0.252440 + 0.967612i \(0.581233\pi\)
\(108\) 0 0
\(109\) −9.09888 −0.871515 −0.435758 0.900064i \(-0.643519\pi\)
−0.435758 + 0.900064i \(0.643519\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −12.4203 −1.16840 −0.584202 0.811609i \(-0.698592\pi\)
−0.584202 + 0.811609i \(0.698592\pi\)
\(114\) 0 0
\(115\) −4.43130 −0.413221
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −5.87636 −0.538685
\(120\) 0 0
\(121\) −7.43268 −0.675698
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 9.74033 0.871202
\(126\) 0 0
\(127\) 7.66621 0.680266 0.340133 0.940377i \(-0.389528\pi\)
0.340133 + 0.940377i \(0.389528\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −14.5426 −1.27059 −0.635295 0.772270i \(-0.719121\pi\)
−0.635295 + 0.772270i \(0.719121\pi\)
\(132\) 0 0
\(133\) 7.09888 0.615551
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 4.64145 0.396546 0.198273 0.980147i \(-0.436467\pi\)
0.198273 + 0.980147i \(0.436467\pi\)
\(138\) 0 0
\(139\) −19.2967 −1.63672 −0.818360 0.574705i \(-0.805117\pi\)
−0.818360 + 0.574705i \(0.805117\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1.88874 0.157944
\(144\) 0 0
\(145\) −1.09751 −0.0911430
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −13.6414 −1.11755 −0.558775 0.829319i \(-0.688729\pi\)
−0.558775 + 0.829319i \(0.688729\pi\)
\(150\) 0 0
\(151\) 16.9629 1.38042 0.690209 0.723610i \(-0.257519\pi\)
0.690209 + 0.723610i \(0.257519\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0.740333 0.0594649
\(156\) 0 0
\(157\) 22.6291 1.80600 0.902998 0.429644i \(-0.141361\pi\)
0.902998 + 0.429644i \(0.141361\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 3.98762 0.314269
\(162\) 0 0
\(163\) −18.0989 −1.41761 −0.708807 0.705402i \(-0.750766\pi\)
−0.708807 + 0.705402i \(0.750766\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −17.2225 −1.33272 −0.666360 0.745631i \(-0.732148\pi\)
−0.666360 + 0.745631i \(0.732148\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −0.863976 −0.0656869 −0.0328435 0.999461i \(-0.510456\pi\)
−0.0328435 + 0.999461i \(0.510456\pi\)
\(174\) 0 0
\(175\) −3.76509 −0.284614
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −4.67859 −0.349694 −0.174847 0.984596i \(-0.555943\pi\)
−0.174847 + 0.984596i \(0.555943\pi\)
\(180\) 0 0
\(181\) −16.8640 −1.25349 −0.626745 0.779225i \(-0.715613\pi\)
−0.626745 + 0.779225i \(0.715613\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 1.48220 0.108973
\(186\) 0 0
\(187\) 11.0989 0.811631
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −17.0617 −1.23454 −0.617272 0.786750i \(-0.711762\pi\)
−0.617272 + 0.786750i \(0.711762\pi\)
\(192\) 0 0
\(193\) 7.43268 0.535016 0.267508 0.963556i \(-0.413800\pi\)
0.267508 + 0.963556i \(0.413800\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −26.2953 −1.87346 −0.936730 0.350052i \(-0.886164\pi\)
−0.936730 + 0.350052i \(0.886164\pi\)
\(198\) 0 0
\(199\) 10.4327 0.739553 0.369776 0.929121i \(-0.379434\pi\)
0.369776 + 0.929121i \(0.379434\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0.987620 0.0693174
\(204\) 0 0
\(205\) 2.09888 0.146592
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −13.4079 −0.927445
\(210\) 0 0
\(211\) 25.3955 1.74830 0.874150 0.485655i \(-0.161419\pi\)
0.874150 + 0.485655i \(0.161419\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 12.0727 0.823355
\(216\) 0 0
\(217\) −0.666208 −0.0452251
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 5.87636 0.395286
\(222\) 0 0
\(223\) 14.5302 0.973013 0.486507 0.873677i \(-0.338271\pi\)
0.486507 + 0.873677i \(0.338271\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 19.4079 1.28815 0.644074 0.764963i \(-0.277243\pi\)
0.644074 + 0.764963i \(0.277243\pi\)
\(228\) 0 0
\(229\) −10.5302 −0.695854 −0.347927 0.937522i \(-0.613114\pi\)
−0.347927 + 0.937522i \(0.613114\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 22.6414 1.48329 0.741645 0.670792i \(-0.234046\pi\)
0.741645 + 0.670792i \(0.234046\pi\)
\(234\) 0 0
\(235\) 12.3338 0.804568
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 4.97524 0.321822 0.160911 0.986969i \(-0.448557\pi\)
0.160911 + 0.986969i \(0.448557\pi\)
\(240\) 0 0
\(241\) 0.234908 0.0151318 0.00756588 0.999971i \(-0.497592\pi\)
0.00756588 + 0.999971i \(0.497592\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −1.11126 −0.0709961
\(246\) 0 0
\(247\) −7.09888 −0.451691
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −8.09888 −0.511197 −0.255599 0.966783i \(-0.582273\pi\)
−0.255599 + 0.966783i \(0.582273\pi\)
\(252\) 0 0
\(253\) −7.53156 −0.473505
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 23.7527 1.48165 0.740827 0.671696i \(-0.234434\pi\)
0.740827 + 0.671696i \(0.234434\pi\)
\(258\) 0 0
\(259\) −1.33379 −0.0828778
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −6.65383 −0.410293 −0.205146 0.978731i \(-0.565767\pi\)
−0.205146 + 0.978731i \(0.565767\pi\)
\(264\) 0 0
\(265\) 13.5687 0.833519
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 18.9876 1.15770 0.578848 0.815436i \(-0.303503\pi\)
0.578848 + 0.815436i \(0.303503\pi\)
\(270\) 0 0
\(271\) −22.3338 −1.35668 −0.678341 0.734748i \(-0.737301\pi\)
−0.678341 + 0.734748i \(0.737301\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 7.11126 0.428825
\(276\) 0 0
\(277\) −31.8268 −1.91229 −0.956145 0.292895i \(-0.905381\pi\)
−0.956145 + 0.292895i \(0.905381\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 19.8516 1.18425 0.592123 0.805847i \(-0.298290\pi\)
0.592123 + 0.805847i \(0.298290\pi\)
\(282\) 0 0
\(283\) −3.13602 −0.186417 −0.0932086 0.995647i \(-0.529712\pi\)
−0.0932086 + 0.995647i \(0.529712\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −1.88874 −0.111489
\(288\) 0 0
\(289\) 17.5316 1.03127
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 9.87773 0.577063 0.288532 0.957470i \(-0.406833\pi\)
0.288532 + 0.957470i \(0.406833\pi\)
\(294\) 0 0
\(295\) −5.29528 −0.308303
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −3.98762 −0.230610
\(300\) 0 0
\(301\) −10.8640 −0.626189
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −4.18401 −0.239576
\(306\) 0 0
\(307\) 15.4313 0.880711 0.440355 0.897824i \(-0.354852\pi\)
0.440355 + 0.897824i \(0.354852\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 17.6662 1.00176 0.500879 0.865517i \(-0.333010\pi\)
0.500879 + 0.865517i \(0.333010\pi\)
\(312\) 0 0
\(313\) 13.4327 0.759260 0.379630 0.925138i \(-0.376051\pi\)
0.379630 + 0.925138i \(0.376051\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 15.8640 0.891010 0.445505 0.895280i \(-0.353024\pi\)
0.445505 + 0.895280i \(0.353024\pi\)
\(318\) 0 0
\(319\) −1.86535 −0.104440
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −41.7156 −2.32112
\(324\) 0 0
\(325\) 3.76509 0.208850
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −11.0989 −0.611901
\(330\) 0 0
\(331\) −18.0989 −0.994805 −0.497402 0.867520i \(-0.665713\pi\)
−0.497402 + 0.867520i \(0.665713\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −4.55494 −0.248863
\(336\) 0 0
\(337\) −1.03714 −0.0564966 −0.0282483 0.999601i \(-0.508993\pi\)
−0.0282483 + 0.999601i \(0.508993\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 1.25829 0.0681402
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −21.9642 −1.17910 −0.589551 0.807731i \(-0.700695\pi\)
−0.589551 + 0.807731i \(0.700695\pi\)
\(348\) 0 0
\(349\) 23.5302 1.25954 0.629771 0.776781i \(-0.283149\pi\)
0.629771 + 0.776781i \(0.283149\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 29.5426 1.57239 0.786196 0.617977i \(-0.212048\pi\)
0.786196 + 0.617977i \(0.212048\pi\)
\(354\) 0 0
\(355\) 11.9629 0.634923
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 34.0741 1.79836 0.899182 0.437575i \(-0.144163\pi\)
0.899182 + 0.437575i \(0.144163\pi\)
\(360\) 0 0
\(361\) 31.3942 1.65232
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 3.44368 0.180250
\(366\) 0 0
\(367\) −27.0617 −1.41261 −0.706306 0.707907i \(-0.749640\pi\)
−0.706306 + 0.707907i \(0.749640\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −12.2101 −0.633919
\(372\) 0 0
\(373\) −10.8640 −0.562515 −0.281258 0.959632i \(-0.590752\pi\)
−0.281258 + 0.959632i \(0.590752\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −0.987620 −0.0508650
\(378\) 0 0
\(379\) 0.765092 0.0393001 0.0196501 0.999807i \(-0.493745\pi\)
0.0196501 + 0.999807i \(0.493745\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 9.75409 0.498411 0.249205 0.968451i \(-0.419831\pi\)
0.249205 + 0.968451i \(0.419831\pi\)
\(384\) 0 0
\(385\) 2.09888 0.106969
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −24.6538 −1.25000 −0.624999 0.780625i \(-0.714901\pi\)
−0.624999 + 0.780625i \(0.714901\pi\)
\(390\) 0 0
\(391\) −23.4327 −1.18504
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −7.14687 −0.359598
\(396\) 0 0
\(397\) 14.1964 0.712496 0.356248 0.934391i \(-0.384056\pi\)
0.356248 + 0.934391i \(0.384056\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 38.9491 1.94503 0.972513 0.232850i \(-0.0748051\pi\)
0.972513 + 0.232850i \(0.0748051\pi\)
\(402\) 0 0
\(403\) 0.666208 0.0331862
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 2.51918 0.124871
\(408\) 0 0
\(409\) −8.19777 −0.405354 −0.202677 0.979246i \(-0.564964\pi\)
−0.202677 + 0.979246i \(0.564964\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 4.76509 0.234475
\(414\) 0 0
\(415\) 13.1978 0.647853
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −0.457436 −0.0223472 −0.0111736 0.999938i \(-0.503557\pi\)
−0.0111736 + 0.999938i \(0.503557\pi\)
\(420\) 0 0
\(421\) −18.6291 −0.907925 −0.453963 0.891021i \(-0.649990\pi\)
−0.453963 + 0.891021i \(0.649990\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 22.1250 1.07322
\(426\) 0 0
\(427\) 3.76509 0.182206
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 10.5178 0.506625 0.253312 0.967385i \(-0.418480\pi\)
0.253312 + 0.967385i \(0.418480\pi\)
\(432\) 0 0
\(433\) −2.80223 −0.134667 −0.0673333 0.997731i \(-0.521449\pi\)
−0.0673333 + 0.997731i \(0.521449\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 28.3077 1.35414
\(438\) 0 0
\(439\) 4.29528 0.205002 0.102501 0.994733i \(-0.467315\pi\)
0.102501 + 0.994733i \(0.467315\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 22.1978 1.05465 0.527324 0.849664i \(-0.323195\pi\)
0.527324 + 0.849664i \(0.323195\pi\)
\(444\) 0 0
\(445\) −15.9011 −0.753785
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 13.4313 0.633862 0.316931 0.948449i \(-0.397348\pi\)
0.316931 + 0.948449i \(0.397348\pi\)
\(450\) 0 0
\(451\) 3.56732 0.167979
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 1.11126 0.0520969
\(456\) 0 0
\(457\) 14.9011 0.697045 0.348522 0.937300i \(-0.386684\pi\)
0.348522 + 0.937300i \(0.386684\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 6.79123 0.316299 0.158150 0.987415i \(-0.449447\pi\)
0.158150 + 0.987415i \(0.449447\pi\)
\(462\) 0 0
\(463\) 4.43268 0.206004 0.103002 0.994681i \(-0.467155\pi\)
0.103002 + 0.994681i \(0.467155\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −26.4189 −1.22252 −0.611261 0.791429i \(-0.709337\pi\)
−0.611261 + 0.791429i \(0.709337\pi\)
\(468\) 0 0
\(469\) 4.09888 0.189269
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 20.5192 0.943473
\(474\) 0 0
\(475\) −26.7280 −1.22636
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −5.51918 −0.252178 −0.126089 0.992019i \(-0.540242\pi\)
−0.126089 + 0.992019i \(0.540242\pi\)
\(480\) 0 0
\(481\) 1.33379 0.0608157
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −0.850219 −0.0386065
\(486\) 0 0
\(487\) −40.4930 −1.83492 −0.917458 0.397834i \(-0.869762\pi\)
−0.917458 + 0.397834i \(0.869762\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 1.11126 0.0501506 0.0250753 0.999686i \(-0.492017\pi\)
0.0250753 + 0.999686i \(0.492017\pi\)
\(492\) 0 0
\(493\) −5.80361 −0.261381
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −10.7651 −0.482880
\(498\) 0 0
\(499\) 2.66758 0.119418 0.0597088 0.998216i \(-0.480983\pi\)
0.0597088 + 0.998216i \(0.480983\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 9.86398 0.439813 0.219906 0.975521i \(-0.429425\pi\)
0.219906 + 0.975521i \(0.429425\pi\)
\(504\) 0 0
\(505\) −9.86398 −0.438941
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 9.98762 0.442693 0.221347 0.975195i \(-0.428955\pi\)
0.221347 + 0.975195i \(0.428955\pi\)
\(510\) 0 0
\(511\) −3.09888 −0.137087
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 11.0714 0.487863
\(516\) 0 0
\(517\) 20.9629 0.921946
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 38.7527 1.69779 0.848894 0.528564i \(-0.177269\pi\)
0.848894 + 0.528564i \(0.177269\pi\)
\(522\) 0 0
\(523\) −19.6304 −0.858379 −0.429190 0.903214i \(-0.641201\pi\)
−0.429190 + 0.903214i \(0.641201\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 3.91487 0.170535
\(528\) 0 0
\(529\) −7.09888 −0.308647
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 1.88874 0.0818102
\(534\) 0 0
\(535\) 5.80361 0.250912
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −1.88874 −0.0813536
\(540\) 0 0
\(541\) −0.332415 −0.0142916 −0.00714582 0.999974i \(-0.502275\pi\)
−0.00714582 + 0.999974i \(0.502275\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 10.1113 0.433119
\(546\) 0 0
\(547\) −27.2953 −1.16706 −0.583531 0.812091i \(-0.698329\pi\)
−0.583531 + 0.812091i \(0.698329\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 7.01100 0.298679
\(552\) 0 0
\(553\) 6.43130 0.273487
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −31.4079 −1.33080 −0.665398 0.746489i \(-0.731738\pi\)
−0.665398 + 0.746489i \(0.731738\pi\)
\(558\) 0 0
\(559\) 10.8640 0.459497
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 23.6662 0.997412 0.498706 0.866771i \(-0.333809\pi\)
0.498706 + 0.866771i \(0.333809\pi\)
\(564\) 0 0
\(565\) 13.8022 0.580664
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −4.72795 −0.198206 −0.0991030 0.995077i \(-0.531597\pi\)
−0.0991030 + 0.995077i \(0.531597\pi\)
\(570\) 0 0
\(571\) 12.7651 0.534202 0.267101 0.963668i \(-0.413934\pi\)
0.267101 + 0.963668i \(0.413934\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −15.0138 −0.626117
\(576\) 0 0
\(577\) −10.6304 −0.442551 −0.221276 0.975211i \(-0.571022\pi\)
−0.221276 + 0.975211i \(0.571022\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −11.8764 −0.492714
\(582\) 0 0
\(583\) 23.0617 0.955120
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 3.61669 0.149277 0.0746384 0.997211i \(-0.476220\pi\)
0.0746384 + 0.997211i \(0.476220\pi\)
\(588\) 0 0
\(589\) −4.72933 −0.194869
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 22.6786 0.931298 0.465649 0.884970i \(-0.345821\pi\)
0.465649 + 0.884970i \(0.345821\pi\)
\(594\) 0 0
\(595\) 6.53018 0.267711
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −40.5906 −1.65848 −0.829242 0.558889i \(-0.811228\pi\)
−0.829242 + 0.558889i \(0.811228\pi\)
\(600\) 0 0
\(601\) 13.1991 0.538404 0.269202 0.963084i \(-0.413240\pi\)
0.269202 + 0.963084i \(0.413240\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 8.25967 0.335803
\(606\) 0 0
\(607\) −1.66758 −0.0676852 −0.0338426 0.999427i \(-0.510774\pi\)
−0.0338426 + 0.999427i \(0.510774\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 11.0989 0.449013
\(612\) 0 0
\(613\) 22.9257 0.925961 0.462981 0.886368i \(-0.346780\pi\)
0.462981 + 0.886368i \(0.346780\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −10.7513 −0.432832 −0.216416 0.976301i \(-0.569437\pi\)
−0.216416 + 0.976301i \(0.569437\pi\)
\(618\) 0 0
\(619\) 25.9629 1.04354 0.521768 0.853088i \(-0.325273\pi\)
0.521768 + 0.853088i \(0.325273\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 14.3090 0.573279
\(624\) 0 0
\(625\) 8.00138 0.320055
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 7.83784 0.312515
\(630\) 0 0
\(631\) 3.00138 0.119483 0.0597415 0.998214i \(-0.480972\pi\)
0.0597415 + 0.998214i \(0.480972\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −8.51918 −0.338073
\(636\) 0 0
\(637\) −1.00000 −0.0396214
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 13.5453 0.535008 0.267504 0.963557i \(-0.413801\pi\)
0.267504 + 0.963557i \(0.413801\pi\)
\(642\) 0 0
\(643\) 34.9629 1.37880 0.689400 0.724381i \(-0.257874\pi\)
0.689400 + 0.724381i \(0.257874\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 25.9890 1.02173 0.510866 0.859660i \(-0.329325\pi\)
0.510866 + 0.859660i \(0.329325\pi\)
\(648\) 0 0
\(649\) −9.00000 −0.353281
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −5.87636 −0.229960 −0.114980 0.993368i \(-0.536680\pi\)
−0.114980 + 0.993368i \(0.536680\pi\)
\(654\) 0 0
\(655\) 16.1606 0.631448
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −41.2719 −1.60772 −0.803862 0.594815i \(-0.797225\pi\)
−0.803862 + 0.594815i \(0.797225\pi\)
\(660\) 0 0
\(661\) −36.9629 −1.43769 −0.718844 0.695171i \(-0.755329\pi\)
−0.718844 + 0.695171i \(0.755329\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −7.88874 −0.305912
\(666\) 0 0
\(667\) 3.93825 0.152490
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −7.11126 −0.274527
\(672\) 0 0
\(673\) 45.3942 1.74982 0.874908 0.484289i \(-0.160922\pi\)
0.874908 + 0.484289i \(0.160922\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 2.29528 0.0882146 0.0441073 0.999027i \(-0.485956\pi\)
0.0441073 + 0.999027i \(0.485956\pi\)
\(678\) 0 0
\(679\) 0.765092 0.0293615
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 32.1483 1.23012 0.615059 0.788481i \(-0.289132\pi\)
0.615059 + 0.788481i \(0.289132\pi\)
\(684\) 0 0
\(685\) −5.15787 −0.197072
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 12.2101 0.465170
\(690\) 0 0
\(691\) −6.80361 −0.258821 −0.129411 0.991591i \(-0.541309\pi\)
−0.129411 + 0.991591i \(0.541309\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 21.4437 0.813405
\(696\) 0 0
\(697\) 11.0989 0.420400
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 42.5933 1.60873 0.804363 0.594137i \(-0.202507\pi\)
0.804363 + 0.594137i \(0.202507\pi\)
\(702\) 0 0
\(703\) −9.46844 −0.357109
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 8.87636 0.333830
\(708\) 0 0
\(709\) −5.13465 −0.192836 −0.0964178 0.995341i \(-0.530739\pi\)
−0.0964178 + 0.995341i \(0.530739\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −2.65658 −0.0994898
\(714\) 0 0
\(715\) −2.09888 −0.0784938
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 23.6057 0.880344 0.440172 0.897914i \(-0.354918\pi\)
0.440172 + 0.897914i \(0.354918\pi\)
\(720\) 0 0
\(721\) −9.96286 −0.371036
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −3.71848 −0.138101
\(726\) 0 0
\(727\) 0.00137742 5.10856e−5 0 2.55428e−5 1.00000i \(-0.499992\pi\)
2.55428e−5 1.00000i \(0.499992\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 63.8406 2.36123
\(732\) 0 0
\(733\) −21.3955 −0.790262 −0.395131 0.918625i \(-0.629301\pi\)
−0.395131 + 0.918625i \(0.629301\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −7.74171 −0.285170
\(738\) 0 0
\(739\) −32.0232 −1.17799 −0.588997 0.808135i \(-0.700477\pi\)
−0.588997 + 0.808135i \(0.700477\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −50.5192 −1.85337 −0.926685 0.375840i \(-0.877354\pi\)
−0.926685 + 0.375840i \(0.877354\pi\)
\(744\) 0 0
\(745\) 15.1593 0.555392
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −5.22253 −0.190827
\(750\) 0 0
\(751\) 40.6291 1.48258 0.741288 0.671187i \(-0.234215\pi\)
0.741288 + 0.671187i \(0.234215\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −18.8502 −0.686030
\(756\) 0 0
\(757\) −7.90112 −0.287171 −0.143585 0.989638i \(-0.545863\pi\)
−0.143585 + 0.989638i \(0.545863\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 22.0248 0.798397 0.399198 0.916865i \(-0.369288\pi\)
0.399198 + 0.916865i \(0.369288\pi\)
\(762\) 0 0
\(763\) −9.09888 −0.329402
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −4.76509 −0.172057
\(768\) 0 0
\(769\) −48.8255 −1.76069 −0.880346 0.474333i \(-0.842689\pi\)
−0.880346 + 0.474333i \(0.842689\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 34.4807 1.24018 0.620092 0.784529i \(-0.287095\pi\)
0.620092 + 0.784529i \(0.287095\pi\)
\(774\) 0 0
\(775\) 2.50833 0.0901020
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −13.4079 −0.480388
\(780\) 0 0
\(781\) 20.3324 0.727551
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −25.1469 −0.897530
\(786\) 0 0
\(787\) 11.7266 0.418007 0.209004 0.977915i \(-0.432978\pi\)
0.209004 + 0.977915i \(0.432978\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −12.4203 −0.441615
\(792\) 0 0
\(793\) −3.76509 −0.133702
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −52.2471 −1.85069 −0.925344 0.379128i \(-0.876224\pi\)
−0.925344 + 0.379128i \(0.876224\pi\)
\(798\) 0 0
\(799\) 65.2210 2.30735
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 5.85297 0.206547
\(804\) 0 0
\(805\) −4.43130 −0.156183
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −30.5439 −1.07387 −0.536934 0.843624i \(-0.680418\pi\)
−0.536934 + 0.843624i \(0.680418\pi\)
\(810\) 0 0
\(811\) −27.9629 −0.981909 −0.490954 0.871185i \(-0.663352\pi\)
−0.490954 + 0.871185i \(0.663352\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 20.1126 0.704515
\(816\) 0 0
\(817\) −77.1221 −2.69816
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −7.56870 −0.264149 −0.132075 0.991240i \(-0.542164\pi\)
−0.132075 + 0.991240i \(0.542164\pi\)
\(822\) 0 0
\(823\) −36.3955 −1.26867 −0.634334 0.773059i \(-0.718726\pi\)
−0.634334 + 0.773059i \(0.718726\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −2.36955 −0.0823975 −0.0411987 0.999151i \(-0.513118\pi\)
−0.0411987 + 0.999151i \(0.513118\pi\)
\(828\) 0 0
\(829\) −22.2335 −0.772202 −0.386101 0.922456i \(-0.626178\pi\)
−0.386101 + 0.922456i \(0.626178\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −5.87636 −0.203604
\(834\) 0 0
\(835\) 19.1388 0.662325
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −12.1099 −0.418080 −0.209040 0.977907i \(-0.567034\pi\)
−0.209040 + 0.977907i \(0.567034\pi\)
\(840\) 0 0
\(841\) −28.0246 −0.966366
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 13.3352 0.458744
\(846\) 0 0
\(847\) −7.43268 −0.255390
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −5.31866 −0.182321
\(852\) 0 0
\(853\) −33.0989 −1.13328 −0.566642 0.823964i \(-0.691758\pi\)
−0.566642 + 0.823964i \(0.691758\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 17.5426 0.599243 0.299621 0.954058i \(-0.403140\pi\)
0.299621 + 0.954058i \(0.403140\pi\)
\(858\) 0 0
\(859\) 13.8626 0.472986 0.236493 0.971633i \(-0.424002\pi\)
0.236493 + 0.971633i \(0.424002\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −10.9890 −0.374070 −0.187035 0.982353i \(-0.559888\pi\)
−0.187035 + 0.982353i \(0.559888\pi\)
\(864\) 0 0
\(865\) 0.960106 0.0326446
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −12.1470 −0.412060
\(870\) 0 0
\(871\) −4.09888 −0.138885
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 9.74033 0.329283
\(876\) 0 0
\(877\) 24.4944 0.827118 0.413559 0.910477i \(-0.364286\pi\)
0.413559 + 0.910477i \(0.364286\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −19.9243 −0.671268 −0.335634 0.941992i \(-0.608951\pi\)
−0.335634 + 0.941992i \(0.608951\pi\)
\(882\) 0 0
\(883\) 39.5316 1.33034 0.665171 0.746691i \(-0.268358\pi\)
0.665171 + 0.746691i \(0.268358\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −41.0755 −1.37918 −0.689590 0.724200i \(-0.742209\pi\)
−0.689590 + 0.724200i \(0.742209\pi\)
\(888\) 0 0
\(889\) 7.66621 0.257116
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −78.7897 −2.63660
\(894\) 0 0
\(895\) 5.19915 0.173788
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −0.657960 −0.0219442
\(900\) 0 0
\(901\) 71.7512 2.39038
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 18.7403 0.622950
\(906\) 0 0
\(907\) −11.6648 −0.387324 −0.193662 0.981068i \(-0.562037\pi\)
−0.193662 + 0.981068i \(0.562037\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −41.6057 −1.37846 −0.689229 0.724544i \(-0.742051\pi\)
−0.689229 + 0.724544i \(0.742051\pi\)
\(912\) 0 0
\(913\) 22.4313 0.742368
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −14.5426 −0.480238
\(918\) 0 0
\(919\) −28.3338 −0.934646 −0.467323 0.884087i \(-0.654781\pi\)
−0.467323 + 0.884087i \(0.654781\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 10.7651 0.354337
\(924\) 0 0
\(925\) 5.02185 0.165117
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 29.8255 0.978542 0.489271 0.872132i \(-0.337263\pi\)
0.489271 + 0.872132i \(0.337263\pi\)
\(930\) 0 0
\(931\) 7.09888 0.232657
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −12.3338 −0.403358
\(936\) 0 0
\(937\) −2.56870 −0.0839158 −0.0419579 0.999119i \(-0.513360\pi\)
−0.0419579 + 0.999119i \(0.513360\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 4.66896 0.152204 0.0761019 0.997100i \(-0.475753\pi\)
0.0761019 + 0.997100i \(0.475753\pi\)
\(942\) 0 0
\(943\) −7.53156 −0.245261
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 31.6908 1.02981 0.514907 0.857246i \(-0.327827\pi\)
0.514907 + 0.857246i \(0.327827\pi\)
\(948\) 0 0
\(949\) 3.09888 0.100594
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 13.0604 0.423067 0.211533 0.977371i \(-0.432154\pi\)
0.211533 + 0.977371i \(0.432154\pi\)
\(954\) 0 0
\(955\) 18.9601 0.613535
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 4.64145 0.149880
\(960\) 0 0
\(961\) −30.5562 −0.985683
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −8.25967 −0.265888
\(966\) 0 0
\(967\) −8.96148 −0.288182 −0.144091 0.989564i \(-0.546026\pi\)
−0.144091 + 0.989564i \(0.546026\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0.543941 0.0174559 0.00872795 0.999962i \(-0.497222\pi\)
0.00872795 + 0.999962i \(0.497222\pi\)
\(972\) 0 0
\(973\) −19.2967 −0.618622
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 36.4559 1.16633 0.583164 0.812355i \(-0.301815\pi\)
0.583164 + 0.812355i \(0.301815\pi\)
\(978\) 0 0
\(979\) −27.0260 −0.863754
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −36.3832 −1.16044 −0.580221 0.814459i \(-0.697034\pi\)
−0.580221 + 0.814459i \(0.697034\pi\)
\(984\) 0 0
\(985\) 29.2210 0.931058
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −43.3214 −1.37754
\(990\) 0 0
\(991\) −31.9642 −1.01538 −0.507689 0.861541i \(-0.669500\pi\)
−0.507689 + 0.861541i \(0.669500\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −11.5935 −0.367537
\(996\) 0 0
\(997\) 26.0000 0.823428 0.411714 0.911313i \(-0.364930\pi\)
0.411714 + 0.911313i \(0.364930\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2268.2.a.g.1.2 3
3.2 odd 2 2268.2.a.j.1.2 3
4.3 odd 2 9072.2.a.bt.1.2 3
9.2 odd 6 756.2.j.a.253.2 6
9.4 even 3 252.2.j.b.169.3 yes 6
9.5 odd 6 756.2.j.a.505.2 6
9.7 even 3 252.2.j.b.85.3 6
12.11 even 2 9072.2.a.bz.1.2 3
36.7 odd 6 1008.2.r.g.337.1 6
36.11 even 6 3024.2.r.i.1009.2 6
36.23 even 6 3024.2.r.i.2017.2 6
36.31 odd 6 1008.2.r.g.673.1 6
63.2 odd 6 5292.2.l.g.361.2 6
63.4 even 3 1764.2.l.d.961.3 6
63.5 even 6 5292.2.i.g.2125.2 6
63.11 odd 6 5292.2.i.d.1549.2 6
63.13 odd 6 1764.2.j.d.1177.1 6
63.16 even 3 1764.2.l.d.949.3 6
63.20 even 6 5292.2.j.e.1765.2 6
63.23 odd 6 5292.2.i.d.2125.2 6
63.25 even 3 1764.2.i.f.373.1 6
63.31 odd 6 1764.2.l.g.961.1 6
63.32 odd 6 5292.2.l.g.3313.2 6
63.34 odd 6 1764.2.j.d.589.1 6
63.38 even 6 5292.2.i.g.1549.2 6
63.40 odd 6 1764.2.i.e.1537.3 6
63.41 even 6 5292.2.j.e.3529.2 6
63.47 even 6 5292.2.l.d.361.2 6
63.52 odd 6 1764.2.i.e.373.3 6
63.58 even 3 1764.2.i.f.1537.1 6
63.59 even 6 5292.2.l.d.3313.2 6
63.61 odd 6 1764.2.l.g.949.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.2.j.b.85.3 6 9.7 even 3
252.2.j.b.169.3 yes 6 9.4 even 3
756.2.j.a.253.2 6 9.2 odd 6
756.2.j.a.505.2 6 9.5 odd 6
1008.2.r.g.337.1 6 36.7 odd 6
1008.2.r.g.673.1 6 36.31 odd 6
1764.2.i.e.373.3 6 63.52 odd 6
1764.2.i.e.1537.3 6 63.40 odd 6
1764.2.i.f.373.1 6 63.25 even 3
1764.2.i.f.1537.1 6 63.58 even 3
1764.2.j.d.589.1 6 63.34 odd 6
1764.2.j.d.1177.1 6 63.13 odd 6
1764.2.l.d.949.3 6 63.16 even 3
1764.2.l.d.961.3 6 63.4 even 3
1764.2.l.g.949.1 6 63.61 odd 6
1764.2.l.g.961.1 6 63.31 odd 6
2268.2.a.g.1.2 3 1.1 even 1 trivial
2268.2.a.j.1.2 3 3.2 odd 2
3024.2.r.i.1009.2 6 36.11 even 6
3024.2.r.i.2017.2 6 36.23 even 6
5292.2.i.d.1549.2 6 63.11 odd 6
5292.2.i.d.2125.2 6 63.23 odd 6
5292.2.i.g.1549.2 6 63.38 even 6
5292.2.i.g.2125.2 6 63.5 even 6
5292.2.j.e.1765.2 6 63.20 even 6
5292.2.j.e.3529.2 6 63.41 even 6
5292.2.l.d.361.2 6 63.47 even 6
5292.2.l.d.3313.2 6 63.59 even 6
5292.2.l.g.361.2 6 63.2 odd 6
5292.2.l.g.3313.2 6 63.32 odd 6
9072.2.a.bt.1.2 3 4.3 odd 2
9072.2.a.bz.1.2 3 12.11 even 2