Properties

Label 6480.2.a.bm.1.2
Level $6480$
Weight $2$
Character 6480.1
Self dual yes
Analytic conductor $51.743$
Analytic rank $0$
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] = [6480,2,Mod(1,6480)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6480, 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("6480.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6480 = 2^{4} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6480.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: \(51.7430605098\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3240)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-3.27492\) of defining polynomial
Character \(\chi\) \(=\) 6480.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{5} +3.27492 q^{7} +O(q^{10})\) \(q+1.00000 q^{5} +3.27492 q^{7} -6.27492 q^{11} -1.27492 q^{13} +2.00000 q^{17} -1.00000 q^{19} +7.27492 q^{23} +1.00000 q^{25} +6.27492 q^{29} -6.27492 q^{31} +3.27492 q^{35} -10.5498 q^{37} +7.54983 q^{41} -4.00000 q^{43} +1.27492 q^{47} +3.72508 q^{49} +0.725083 q^{53} -6.27492 q^{55} +13.0000 q^{59} +8.54983 q^{61} -1.27492 q^{65} -0.549834 q^{67} +8.27492 q^{71} +15.0997 q^{73} -20.5498 q^{77} -10.5498 q^{79} +2.54983 q^{83} +2.00000 q^{85} +12.8248 q^{89} -4.17525 q^{91} -1.00000 q^{95} +16.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} - q^{7} - 5 q^{11} + 5 q^{13} + 4 q^{17} - 2 q^{19} + 7 q^{23} + 2 q^{25} + 5 q^{29} - 5 q^{31} - q^{35} - 6 q^{37} - 8 q^{43} - 5 q^{47} + 15 q^{49} + 9 q^{53} - 5 q^{55} + 26 q^{59} + 2 q^{61} + 5 q^{65} + 14 q^{67} + 9 q^{71} - 26 q^{77} - 6 q^{79} - 10 q^{83} + 4 q^{85} + 3 q^{89} - 31 q^{91} - 2 q^{95} + 32 q^{97}+O(q^{100}) \) 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
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 3.27492 1.23780 0.618901 0.785469i \(-0.287578\pi\)
0.618901 + 0.785469i \(0.287578\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −6.27492 −1.89196 −0.945979 0.324227i \(-0.894896\pi\)
−0.945979 + 0.324227i \(0.894896\pi\)
\(12\) 0 0
\(13\) −1.27492 −0.353598 −0.176799 0.984247i \(-0.556574\pi\)
−0.176799 + 0.984247i \(0.556574\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 7.27492 1.51693 0.758463 0.651717i \(-0.225951\pi\)
0.758463 + 0.651717i \(0.225951\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.27492 1.16522 0.582611 0.812751i \(-0.302031\pi\)
0.582611 + 0.812751i \(0.302031\pi\)
\(30\) 0 0
\(31\) −6.27492 −1.12701 −0.563504 0.826113i \(-0.690547\pi\)
−0.563504 + 0.826113i \(0.690547\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.27492 0.553562
\(36\) 0 0
\(37\) −10.5498 −1.73438 −0.867191 0.497976i \(-0.834077\pi\)
−0.867191 + 0.497976i \(0.834077\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 7.54983 1.17909 0.589543 0.807737i \(-0.299308\pi\)
0.589543 + 0.807737i \(0.299308\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.27492 0.185966 0.0929829 0.995668i \(-0.470360\pi\)
0.0929829 + 0.995668i \(0.470360\pi\)
\(48\) 0 0
\(49\) 3.72508 0.532155
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0.725083 0.0995978 0.0497989 0.998759i \(-0.484142\pi\)
0.0497989 + 0.998759i \(0.484142\pi\)
\(54\) 0 0
\(55\) −6.27492 −0.846110
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 13.0000 1.69246 0.846228 0.532821i \(-0.178868\pi\)
0.846228 + 0.532821i \(0.178868\pi\)
\(60\) 0 0
\(61\) 8.54983 1.09469 0.547347 0.836906i \(-0.315638\pi\)
0.547347 + 0.836906i \(0.315638\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.27492 −0.158134
\(66\) 0 0
\(67\) −0.549834 −0.0671730 −0.0335865 0.999436i \(-0.510693\pi\)
−0.0335865 + 0.999436i \(0.510693\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 8.27492 0.982052 0.491026 0.871145i \(-0.336622\pi\)
0.491026 + 0.871145i \(0.336622\pi\)
\(72\) 0 0
\(73\) 15.0997 1.76728 0.883641 0.468165i \(-0.155085\pi\)
0.883641 + 0.468165i \(0.155085\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −20.5498 −2.34187
\(78\) 0 0
\(79\) −10.5498 −1.18695 −0.593475 0.804853i \(-0.702244\pi\)
−0.593475 + 0.804853i \(0.702244\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 2.54983 0.279881 0.139940 0.990160i \(-0.455309\pi\)
0.139940 + 0.990160i \(0.455309\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 12.8248 1.35942 0.679710 0.733481i \(-0.262105\pi\)
0.679710 + 0.733481i \(0.262105\pi\)
\(90\) 0 0
\(91\) −4.17525 −0.437685
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) 16.0000 1.62455 0.812277 0.583272i \(-0.198228\pi\)
0.812277 + 0.583272i \(0.198228\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 9.72508 0.967682 0.483841 0.875156i \(-0.339241\pi\)
0.483841 + 0.875156i \(0.339241\pi\)
\(102\) 0 0
\(103\) 5.82475 0.573930 0.286965 0.957941i \(-0.407354\pi\)
0.286965 + 0.957941i \(0.407354\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −15.0997 −1.45974 −0.729870 0.683586i \(-0.760419\pi\)
−0.729870 + 0.683586i \(0.760419\pi\)
\(108\) 0 0
\(109\) −6.27492 −0.601028 −0.300514 0.953777i \(-0.597158\pi\)
−0.300514 + 0.953777i \(0.597158\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 7.27492 0.678390
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 6.54983 0.600422
\(120\) 0 0
\(121\) 28.3746 2.57951
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −11.2749 −1.00049 −0.500244 0.865885i \(-0.666756\pi\)
−0.500244 + 0.865885i \(0.666756\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −1.54983 −0.135410 −0.0677048 0.997705i \(-0.521568\pi\)
−0.0677048 + 0.997705i \(0.521568\pi\)
\(132\) 0 0
\(133\) −3.27492 −0.283971
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 14.5498 1.24308 0.621538 0.783384i \(-0.286508\pi\)
0.621538 + 0.783384i \(0.286508\pi\)
\(138\) 0 0
\(139\) 9.00000 0.763370 0.381685 0.924292i \(-0.375344\pi\)
0.381685 + 0.924292i \(0.375344\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 8.00000 0.668994
\(144\) 0 0
\(145\) 6.27492 0.521104
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 2.00000 0.163846 0.0819232 0.996639i \(-0.473894\pi\)
0.0819232 + 0.996639i \(0.473894\pi\)
\(150\) 0 0
\(151\) 12.2749 0.998919 0.499459 0.866337i \(-0.333532\pi\)
0.499459 + 0.866337i \(0.333532\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −6.27492 −0.504013
\(156\) 0 0
\(157\) 10.7251 0.855955 0.427977 0.903789i \(-0.359226\pi\)
0.427977 + 0.903789i \(0.359226\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 23.8248 1.87765
\(162\) 0 0
\(163\) −20.5498 −1.60959 −0.804794 0.593555i \(-0.797724\pi\)
−0.804794 + 0.593555i \(0.797724\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 7.45017 0.576511 0.288256 0.957554i \(-0.406925\pi\)
0.288256 + 0.957554i \(0.406925\pi\)
\(168\) 0 0
\(169\) −11.3746 −0.874968
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 8.72508 0.663356 0.331678 0.943393i \(-0.392385\pi\)
0.331678 + 0.943393i \(0.392385\pi\)
\(174\) 0 0
\(175\) 3.27492 0.247560
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 11.5498 0.863275 0.431638 0.902047i \(-0.357936\pi\)
0.431638 + 0.902047i \(0.357936\pi\)
\(180\) 0 0
\(181\) 16.2749 1.20971 0.604853 0.796337i \(-0.293232\pi\)
0.604853 + 0.796337i \(0.293232\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −10.5498 −0.775639
\(186\) 0 0
\(187\) −12.5498 −0.917735
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −19.3746 −1.40190 −0.700948 0.713212i \(-0.747239\pi\)
−0.700948 + 0.713212i \(0.747239\pi\)
\(192\) 0 0
\(193\) 4.00000 0.287926 0.143963 0.989583i \(-0.454015\pi\)
0.143963 + 0.989583i \(0.454015\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −22.3746 −1.59412 −0.797062 0.603898i \(-0.793613\pi\)
−0.797062 + 0.603898i \(0.793613\pi\)
\(198\) 0 0
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 20.5498 1.44232
\(204\) 0 0
\(205\) 7.54983 0.527303
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 6.27492 0.434045
\(210\) 0 0
\(211\) 16.0997 1.10835 0.554173 0.832401i \(-0.313034\pi\)
0.554173 + 0.832401i \(0.313034\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −4.00000 −0.272798
\(216\) 0 0
\(217\) −20.5498 −1.39501
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −2.54983 −0.171520
\(222\) 0 0
\(223\) −8.00000 −0.535720 −0.267860 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 7.45017 0.494485 0.247242 0.968954i \(-0.420476\pi\)
0.247242 + 0.968954i \(0.420476\pi\)
\(228\) 0 0
\(229\) 8.54983 0.564989 0.282494 0.959269i \(-0.408838\pi\)
0.282494 + 0.959269i \(0.408838\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −9.09967 −0.596139 −0.298070 0.954544i \(-0.596343\pi\)
−0.298070 + 0.954544i \(0.596343\pi\)
\(234\) 0 0
\(235\) 1.27492 0.0831664
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 13.4502 0.870019 0.435009 0.900426i \(-0.356745\pi\)
0.435009 + 0.900426i \(0.356745\pi\)
\(240\) 0 0
\(241\) −24.2749 −1.56368 −0.781842 0.623476i \(-0.785720\pi\)
−0.781842 + 0.623476i \(0.785720\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 3.72508 0.237987
\(246\) 0 0
\(247\) 1.27492 0.0811210
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −18.7251 −1.18192 −0.590958 0.806702i \(-0.701250\pi\)
−0.590958 + 0.806702i \(0.701250\pi\)
\(252\) 0 0
\(253\) −45.6495 −2.86996
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 3.09967 0.193352 0.0966760 0.995316i \(-0.469179\pi\)
0.0966760 + 0.995316i \(0.469179\pi\)
\(258\) 0 0
\(259\) −34.5498 −2.14682
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0.725083 0.0447105 0.0223553 0.999750i \(-0.492884\pi\)
0.0223553 + 0.999750i \(0.492884\pi\)
\(264\) 0 0
\(265\) 0.725083 0.0445415
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −25.9244 −1.58064 −0.790320 0.612694i \(-0.790086\pi\)
−0.790320 + 0.612694i \(0.790086\pi\)
\(270\) 0 0
\(271\) −13.4502 −0.817039 −0.408520 0.912750i \(-0.633955\pi\)
−0.408520 + 0.912750i \(0.633955\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −6.27492 −0.378392
\(276\) 0 0
\(277\) 25.2749 1.51862 0.759311 0.650728i \(-0.225536\pi\)
0.759311 + 0.650728i \(0.225536\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 6.17525 0.368384 0.184192 0.982890i \(-0.441033\pi\)
0.184192 + 0.982890i \(0.441033\pi\)
\(282\) 0 0
\(283\) 11.4502 0.680642 0.340321 0.940309i \(-0.389464\pi\)
0.340321 + 0.940309i \(0.389464\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 24.7251 1.45948
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 10.7251 0.626566 0.313283 0.949660i \(-0.398571\pi\)
0.313283 + 0.949660i \(0.398571\pi\)
\(294\) 0 0
\(295\) 13.0000 0.756889
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −9.27492 −0.536382
\(300\) 0 0
\(301\) −13.0997 −0.755052
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 8.54983 0.489562
\(306\) 0 0
\(307\) −1.09967 −0.0627614 −0.0313807 0.999508i \(-0.509990\pi\)
−0.0313807 + 0.999508i \(0.509990\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 25.9244 1.47004 0.735020 0.678046i \(-0.237173\pi\)
0.735020 + 0.678046i \(0.237173\pi\)
\(312\) 0 0
\(313\) 8.00000 0.452187 0.226093 0.974106i \(-0.427405\pi\)
0.226093 + 0.974106i \(0.427405\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 5.27492 0.296269 0.148134 0.988967i \(-0.452673\pi\)
0.148134 + 0.988967i \(0.452673\pi\)
\(318\) 0 0
\(319\) −39.3746 −2.20455
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −2.00000 −0.111283
\(324\) 0 0
\(325\) −1.27492 −0.0707197
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 4.17525 0.230189
\(330\) 0 0
\(331\) 18.8248 1.03470 0.517351 0.855773i \(-0.326918\pi\)
0.517351 + 0.855773i \(0.326918\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −0.549834 −0.0300407
\(336\) 0 0
\(337\) 5.45017 0.296889 0.148445 0.988921i \(-0.452573\pi\)
0.148445 + 0.988921i \(0.452573\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 39.3746 2.13225
\(342\) 0 0
\(343\) −10.7251 −0.579100
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −14.0000 −0.751559 −0.375780 0.926709i \(-0.622625\pi\)
−0.375780 + 0.926709i \(0.622625\pi\)
\(348\) 0 0
\(349\) 12.8248 0.686493 0.343247 0.939245i \(-0.388473\pi\)
0.343247 + 0.939245i \(0.388473\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 6.54983 0.348613 0.174306 0.984691i \(-0.444232\pi\)
0.174306 + 0.984691i \(0.444232\pi\)
\(354\) 0 0
\(355\) 8.27492 0.439187
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −0.824752 −0.0435287 −0.0217644 0.999763i \(-0.506928\pi\)
−0.0217644 + 0.999763i \(0.506928\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 15.0997 0.790353
\(366\) 0 0
\(367\) −12.5498 −0.655096 −0.327548 0.944835i \(-0.606222\pi\)
−0.327548 + 0.944835i \(0.606222\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 2.37459 0.123282
\(372\) 0 0
\(373\) 31.0997 1.61028 0.805140 0.593085i \(-0.202090\pi\)
0.805140 + 0.593085i \(0.202090\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −8.00000 −0.412021
\(378\) 0 0
\(379\) −9.27492 −0.476420 −0.238210 0.971214i \(-0.576561\pi\)
−0.238210 + 0.971214i \(0.576561\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −2.37459 −0.121336 −0.0606678 0.998158i \(-0.519323\pi\)
−0.0606678 + 0.998158i \(0.519323\pi\)
\(384\) 0 0
\(385\) −20.5498 −1.04732
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −16.5498 −0.839110 −0.419555 0.907730i \(-0.637814\pi\)
−0.419555 + 0.907730i \(0.637814\pi\)
\(390\) 0 0
\(391\) 14.5498 0.735817
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −10.5498 −0.530820
\(396\) 0 0
\(397\) 7.09967 0.356322 0.178161 0.984001i \(-0.442985\pi\)
0.178161 + 0.984001i \(0.442985\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 23.2749 1.16229 0.581147 0.813799i \(-0.302604\pi\)
0.581147 + 0.813799i \(0.302604\pi\)
\(402\) 0 0
\(403\) 8.00000 0.398508
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 66.1993 3.28138
\(408\) 0 0
\(409\) −29.8248 −1.47474 −0.737370 0.675490i \(-0.763932\pi\)
−0.737370 + 0.675490i \(0.763932\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 42.5739 2.09493
\(414\) 0 0
\(415\) 2.54983 0.125166
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 9.09967 0.444548 0.222274 0.974984i \(-0.428652\pi\)
0.222274 + 0.974984i \(0.428652\pi\)
\(420\) 0 0
\(421\) −32.2749 −1.57298 −0.786492 0.617601i \(-0.788105\pi\)
−0.786492 + 0.617601i \(0.788105\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 2.00000 0.0970143
\(426\) 0 0
\(427\) 28.0000 1.35501
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 6.27492 0.302252 0.151126 0.988514i \(-0.451710\pi\)
0.151126 + 0.988514i \(0.451710\pi\)
\(432\) 0 0
\(433\) 12.5498 0.603107 0.301553 0.953449i \(-0.402495\pi\)
0.301553 + 0.953449i \(0.402495\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −7.27492 −0.348006
\(438\) 0 0
\(439\) 25.3746 1.21106 0.605531 0.795821i \(-0.292961\pi\)
0.605531 + 0.795821i \(0.292961\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −10.5498 −0.501238 −0.250619 0.968086i \(-0.580634\pi\)
−0.250619 + 0.968086i \(0.580634\pi\)
\(444\) 0 0
\(445\) 12.8248 0.607952
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 1.54983 0.0731412 0.0365706 0.999331i \(-0.488357\pi\)
0.0365706 + 0.999331i \(0.488357\pi\)
\(450\) 0 0
\(451\) −47.3746 −2.23078
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −4.17525 −0.195739
\(456\) 0 0
\(457\) 16.0000 0.748448 0.374224 0.927338i \(-0.377909\pi\)
0.374224 + 0.927338i \(0.377909\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −23.3746 −1.08866 −0.544332 0.838870i \(-0.683217\pi\)
−0.544332 + 0.838870i \(0.683217\pi\)
\(462\) 0 0
\(463\) −41.4743 −1.92747 −0.963736 0.266857i \(-0.914015\pi\)
−0.963736 + 0.266857i \(0.914015\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 40.1993 1.86020 0.930102 0.367302i \(-0.119718\pi\)
0.930102 + 0.367302i \(0.119718\pi\)
\(468\) 0 0
\(469\) −1.80066 −0.0831469
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 25.0997 1.15408
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 4.27492 0.195326 0.0976630 0.995220i \(-0.468863\pi\)
0.0976630 + 0.995220i \(0.468863\pi\)
\(480\) 0 0
\(481\) 13.4502 0.613275
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 16.0000 0.726523
\(486\) 0 0
\(487\) 37.2749 1.68909 0.844544 0.535486i \(-0.179872\pi\)
0.844544 + 0.535486i \(0.179872\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −15.0000 −0.676941 −0.338470 0.940977i \(-0.609909\pi\)
−0.338470 + 0.940977i \(0.609909\pi\)
\(492\) 0 0
\(493\) 12.5498 0.565216
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 27.0997 1.21559
\(498\) 0 0
\(499\) 29.5498 1.32283 0.661416 0.750019i \(-0.269956\pi\)
0.661416 + 0.750019i \(0.269956\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 9.72508 0.432761
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 28.1993 1.24991 0.624957 0.780659i \(-0.285117\pi\)
0.624957 + 0.780659i \(0.285117\pi\)
\(510\) 0 0
\(511\) 49.4502 2.18755
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 5.82475 0.256669
\(516\) 0 0
\(517\) −8.00000 −0.351840
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −26.9244 −1.17958 −0.589790 0.807557i \(-0.700790\pi\)
−0.589790 + 0.807557i \(0.700790\pi\)
\(522\) 0 0
\(523\) 39.6495 1.73375 0.866876 0.498524i \(-0.166124\pi\)
0.866876 + 0.498524i \(0.166124\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −12.5498 −0.546679
\(528\) 0 0
\(529\) 29.9244 1.30106
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −9.62541 −0.416923
\(534\) 0 0
\(535\) −15.0997 −0.652816
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −23.3746 −1.00681
\(540\) 0 0
\(541\) 43.9244 1.88846 0.944229 0.329289i \(-0.106809\pi\)
0.944229 + 0.329289i \(0.106809\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −6.27492 −0.268788
\(546\) 0 0
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −6.27492 −0.267320
\(552\) 0 0
\(553\) −34.5498 −1.46921
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 32.9244 1.39505 0.697526 0.716559i \(-0.254284\pi\)
0.697526 + 0.716559i \(0.254284\pi\)
\(558\) 0 0
\(559\) 5.09967 0.215693
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −37.0997 −1.56356 −0.781782 0.623551i \(-0.785689\pi\)
−0.781782 + 0.623551i \(0.785689\pi\)
\(564\) 0 0
\(565\) −6.00000 −0.252422
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −26.4502 −1.10885 −0.554424 0.832234i \(-0.687062\pi\)
−0.554424 + 0.832234i \(0.687062\pi\)
\(570\) 0 0
\(571\) 1.17525 0.0491826 0.0245913 0.999698i \(-0.492172\pi\)
0.0245913 + 0.999698i \(0.492172\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 7.27492 0.303385
\(576\) 0 0
\(577\) −11.4502 −0.476677 −0.238338 0.971182i \(-0.576603\pi\)
−0.238338 + 0.971182i \(0.576603\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 8.35050 0.346437
\(582\) 0 0
\(583\) −4.54983 −0.188435
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 34.5498 1.42602 0.713012 0.701152i \(-0.247330\pi\)
0.713012 + 0.701152i \(0.247330\pi\)
\(588\) 0 0
\(589\) 6.27492 0.258553
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −48.1993 −1.97931 −0.989655 0.143469i \(-0.954174\pi\)
−0.989655 + 0.143469i \(0.954174\pi\)
\(594\) 0 0
\(595\) 6.54983 0.268517
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 15.7251 0.642509 0.321255 0.946993i \(-0.395895\pi\)
0.321255 + 0.946993i \(0.395895\pi\)
\(600\) 0 0
\(601\) −1.90033 −0.0775161 −0.0387581 0.999249i \(-0.512340\pi\)
−0.0387581 + 0.999249i \(0.512340\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 28.3746 1.15359
\(606\) 0 0
\(607\) 24.0000 0.974130 0.487065 0.873366i \(-0.338067\pi\)
0.487065 + 0.873366i \(0.338067\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −1.62541 −0.0657572
\(612\) 0 0
\(613\) −46.0241 −1.85890 −0.929448 0.368954i \(-0.879716\pi\)
−0.929448 + 0.368954i \(0.879716\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −43.6495 −1.75726 −0.878631 0.477501i \(-0.841543\pi\)
−0.878631 + 0.477501i \(0.841543\pi\)
\(618\) 0 0
\(619\) 18.3746 0.738537 0.369268 0.929323i \(-0.379608\pi\)
0.369268 + 0.929323i \(0.379608\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 42.0000 1.68269
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −21.0997 −0.841299
\(630\) 0 0
\(631\) −24.4743 −0.974305 −0.487152 0.873317i \(-0.661964\pi\)
−0.487152 + 0.873317i \(0.661964\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −11.2749 −0.447431
\(636\) 0 0
\(637\) −4.74917 −0.188169
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 33.9244 1.33993 0.669967 0.742391i \(-0.266308\pi\)
0.669967 + 0.742391i \(0.266308\pi\)
\(642\) 0 0
\(643\) 33.0997 1.30532 0.652662 0.757649i \(-0.273652\pi\)
0.652662 + 0.757649i \(0.273652\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 28.5498 1.12241 0.561205 0.827677i \(-0.310338\pi\)
0.561205 + 0.827677i \(0.310338\pi\)
\(648\) 0 0
\(649\) −81.5739 −3.20206
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 9.45017 0.369814 0.184907 0.982756i \(-0.440802\pi\)
0.184907 + 0.982756i \(0.440802\pi\)
\(654\) 0 0
\(655\) −1.54983 −0.0605570
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 17.2749 0.672935 0.336468 0.941695i \(-0.390768\pi\)
0.336468 + 0.941695i \(0.390768\pi\)
\(660\) 0 0
\(661\) −39.9244 −1.55288 −0.776440 0.630191i \(-0.782977\pi\)
−0.776440 + 0.630191i \(0.782977\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −3.27492 −0.126996
\(666\) 0 0
\(667\) 45.6495 1.76756
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −53.6495 −2.07112
\(672\) 0 0
\(673\) −8.90033 −0.343083 −0.171541 0.985177i \(-0.554875\pi\)
−0.171541 + 0.985177i \(0.554875\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −26.9244 −1.03479 −0.517395 0.855747i \(-0.673098\pi\)
−0.517395 + 0.855747i \(0.673098\pi\)
\(678\) 0 0
\(679\) 52.3987 2.01088
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1.45017 0.0554890 0.0277445 0.999615i \(-0.491168\pi\)
0.0277445 + 0.999615i \(0.491168\pi\)
\(684\) 0 0
\(685\) 14.5498 0.555921
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −0.924421 −0.0352176
\(690\) 0 0
\(691\) 7.82475 0.297668 0.148834 0.988862i \(-0.452448\pi\)
0.148834 + 0.988862i \(0.452448\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 9.00000 0.341389
\(696\) 0 0
\(697\) 15.0997 0.571941
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −33.3746 −1.26054 −0.630270 0.776376i \(-0.717056\pi\)
−0.630270 + 0.776376i \(0.717056\pi\)
\(702\) 0 0
\(703\) 10.5498 0.397895
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 31.8488 1.19780
\(708\) 0 0
\(709\) −0.900331 −0.0338126 −0.0169063 0.999857i \(-0.505382\pi\)
−0.0169063 + 0.999857i \(0.505382\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −45.6495 −1.70959
\(714\) 0 0
\(715\) 8.00000 0.299183
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 44.8248 1.67168 0.835841 0.548972i \(-0.184981\pi\)
0.835841 + 0.548972i \(0.184981\pi\)
\(720\) 0 0
\(721\) 19.0756 0.710412
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 6.27492 0.233045
\(726\) 0 0
\(727\) 30.3746 1.12653 0.563266 0.826276i \(-0.309545\pi\)
0.563266 + 0.826276i \(0.309545\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −8.00000 −0.295891
\(732\) 0 0
\(733\) −6.00000 −0.221615 −0.110808 0.993842i \(-0.535344\pi\)
−0.110808 + 0.993842i \(0.535344\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 3.45017 0.127088
\(738\) 0 0
\(739\) −15.9244 −0.585789 −0.292895 0.956145i \(-0.594619\pi\)
−0.292895 + 0.956145i \(0.594619\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 28.5498 1.04739 0.523696 0.851905i \(-0.324553\pi\)
0.523696 + 0.851905i \(0.324553\pi\)
\(744\) 0 0
\(745\) 2.00000 0.0732743
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −49.4502 −1.80687
\(750\) 0 0
\(751\) −34.5498 −1.26074 −0.630371 0.776294i \(-0.717097\pi\)
−0.630371 + 0.776294i \(0.717097\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 12.2749 0.446730
\(756\) 0 0
\(757\) 34.3746 1.24937 0.624683 0.780879i \(-0.285228\pi\)
0.624683 + 0.780879i \(0.285228\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 49.5498 1.79618 0.898090 0.439812i \(-0.144955\pi\)
0.898090 + 0.439812i \(0.144955\pi\)
\(762\) 0 0
\(763\) −20.5498 −0.743954
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −16.5739 −0.598450
\(768\) 0 0
\(769\) −16.2749 −0.586889 −0.293444 0.955976i \(-0.594802\pi\)
−0.293444 + 0.955976i \(0.594802\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 31.6495 1.13835 0.569177 0.822215i \(-0.307262\pi\)
0.569177 + 0.822215i \(0.307262\pi\)
\(774\) 0 0
\(775\) −6.27492 −0.225402
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −7.54983 −0.270501
\(780\) 0 0
\(781\) −51.9244 −1.85800
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 10.7251 0.382795
\(786\) 0 0
\(787\) 34.0000 1.21197 0.605985 0.795476i \(-0.292779\pi\)
0.605985 + 0.795476i \(0.292779\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −19.6495 −0.698656
\(792\) 0 0
\(793\) −10.9003 −0.387082
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −25.4502 −0.901491 −0.450746 0.892652i \(-0.648842\pi\)
−0.450746 + 0.892652i \(0.648842\pi\)
\(798\) 0 0
\(799\) 2.54983 0.0902067
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −94.7492 −3.34363
\(804\) 0 0
\(805\) 23.8248 0.839712
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 16.6495 0.585365 0.292683 0.956210i \(-0.405452\pi\)
0.292683 + 0.956210i \(0.405452\pi\)
\(810\) 0 0
\(811\) −13.1752 −0.462646 −0.231323 0.972877i \(-0.574305\pi\)
−0.231323 + 0.972877i \(0.574305\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −20.5498 −0.719829
\(816\) 0 0
\(817\) 4.00000 0.139942
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 4.82475 0.168385 0.0841925 0.996450i \(-0.473169\pi\)
0.0841925 + 0.996450i \(0.473169\pi\)
\(822\) 0 0
\(823\) −23.4502 −0.817421 −0.408711 0.912664i \(-0.634021\pi\)
−0.408711 + 0.912664i \(0.634021\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −21.0997 −0.733707 −0.366854 0.930279i \(-0.619565\pi\)
−0.366854 + 0.930279i \(0.619565\pi\)
\(828\) 0 0
\(829\) −13.1752 −0.457595 −0.228798 0.973474i \(-0.573479\pi\)
−0.228798 + 0.973474i \(0.573479\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 7.45017 0.258133
\(834\) 0 0
\(835\) 7.45017 0.257824
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −52.4743 −1.81161 −0.905806 0.423692i \(-0.860734\pi\)
−0.905806 + 0.423692i \(0.860734\pi\)
\(840\) 0 0
\(841\) 10.3746 0.357744
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −11.3746 −0.391298
\(846\) 0 0
\(847\) 92.9244 3.19292
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −76.7492 −2.63093
\(852\) 0 0
\(853\) −49.2990 −1.68797 −0.843983 0.536370i \(-0.819795\pi\)
−0.843983 + 0.536370i \(0.819795\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −52.7492 −1.80188 −0.900939 0.433946i \(-0.857121\pi\)
−0.900939 + 0.433946i \(0.857121\pi\)
\(858\) 0 0
\(859\) −52.4743 −1.79040 −0.895199 0.445666i \(-0.852967\pi\)
−0.895199 + 0.445666i \(0.852967\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −23.2749 −0.792287 −0.396144 0.918189i \(-0.629652\pi\)
−0.396144 + 0.918189i \(0.629652\pi\)
\(864\) 0 0
\(865\) 8.72508 0.296662
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 66.1993 2.24566
\(870\) 0 0
\(871\) 0.700993 0.0237523
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 3.27492 0.110712
\(876\) 0 0
\(877\) −13.2749 −0.448262 −0.224131 0.974559i \(-0.571954\pi\)
−0.224131 + 0.974559i \(0.571954\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 18.6254 0.627506 0.313753 0.949505i \(-0.398414\pi\)
0.313753 + 0.949505i \(0.398414\pi\)
\(882\) 0 0
\(883\) −32.1993 −1.08359 −0.541797 0.840509i \(-0.682256\pi\)
−0.541797 + 0.840509i \(0.682256\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −0.175248 −0.00588426 −0.00294213 0.999996i \(-0.500937\pi\)
−0.00294213 + 0.999996i \(0.500937\pi\)
\(888\) 0 0
\(889\) −36.9244 −1.23841
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −1.27492 −0.0426635
\(894\) 0 0
\(895\) 11.5498 0.386068
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −39.3746 −1.31322
\(900\) 0 0
\(901\) 1.45017 0.0483120
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 16.2749 0.540997
\(906\) 0 0
\(907\) −51.0997 −1.69674 −0.848368 0.529406i \(-0.822415\pi\)
−0.848368 + 0.529406i \(0.822415\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 27.0241 0.895348 0.447674 0.894197i \(-0.352253\pi\)
0.447674 + 0.894197i \(0.352253\pi\)
\(912\) 0 0
\(913\) −16.0000 −0.529523
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −5.07558 −0.167610
\(918\) 0 0
\(919\) −11.9244 −0.393350 −0.196675 0.980469i \(-0.563014\pi\)
−0.196675 + 0.980469i \(0.563014\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −10.5498 −0.347252
\(924\) 0 0
\(925\) −10.5498 −0.346876
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −60.1238 −1.97260 −0.986298 0.164972i \(-0.947247\pi\)
−0.986298 + 0.164972i \(0.947247\pi\)
\(930\) 0 0
\(931\) −3.72508 −0.122085
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −12.5498 −0.410423
\(936\) 0 0
\(937\) −0.549834 −0.0179623 −0.00898115 0.999960i \(-0.502859\pi\)
−0.00898115 + 0.999960i \(0.502859\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 7.09967 0.231443 0.115721 0.993282i \(-0.463082\pi\)
0.115721 + 0.993282i \(0.463082\pi\)
\(942\) 0 0
\(943\) 54.9244 1.78859
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 27.4502 0.892011 0.446005 0.895030i \(-0.352846\pi\)
0.446005 + 0.895030i \(0.352846\pi\)
\(948\) 0 0
\(949\) −19.2508 −0.624908
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −33.0997 −1.07220 −0.536102 0.844153i \(-0.680104\pi\)
−0.536102 + 0.844153i \(0.680104\pi\)
\(954\) 0 0
\(955\) −19.3746 −0.626947
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 47.6495 1.53868
\(960\) 0 0
\(961\) 8.37459 0.270148
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 4.00000 0.128765
\(966\) 0 0
\(967\) −10.3505 −0.332850 −0.166425 0.986054i \(-0.553222\pi\)
−0.166425 + 0.986054i \(0.553222\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −36.6495 −1.17614 −0.588069 0.808811i \(-0.700112\pi\)
−0.588069 + 0.808811i \(0.700112\pi\)
\(972\) 0 0
\(973\) 29.4743 0.944901
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 3.64950 0.116758 0.0583790 0.998294i \(-0.481407\pi\)
0.0583790 + 0.998294i \(0.481407\pi\)
\(978\) 0 0
\(979\) −80.4743 −2.57197
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 46.7492 1.49107 0.745534 0.666468i \(-0.232195\pi\)
0.745534 + 0.666468i \(0.232195\pi\)
\(984\) 0 0
\(985\) −22.3746 −0.712914
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −29.0997 −0.925316
\(990\) 0 0
\(991\) 1.72508 0.0547991 0.0273995 0.999625i \(-0.491277\pi\)
0.0273995 + 0.999625i \(0.491277\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −8.00000 −0.253617
\(996\) 0 0
\(997\) 8.37459 0.265226 0.132613 0.991168i \(-0.457663\pi\)
0.132613 + 0.991168i \(0.457663\pi\)
\(998\) 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 6480.2.a.bm.1.2 2
3.2 odd 2 6480.2.a.bf.1.2 2
4.3 odd 2 3240.2.a.n.1.1 yes 2
12.11 even 2 3240.2.a.h.1.1 2
36.7 odd 6 3240.2.q.y.1081.2 4
36.11 even 6 3240.2.q.be.1081.2 4
36.23 even 6 3240.2.q.be.2161.2 4
36.31 odd 6 3240.2.q.y.2161.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3240.2.a.h.1.1 2 12.11 even 2
3240.2.a.n.1.1 yes 2 4.3 odd 2
3240.2.q.y.1081.2 4 36.7 odd 6
3240.2.q.y.2161.2 4 36.31 odd 6
3240.2.q.be.1081.2 4 36.11 even 6
3240.2.q.be.2161.2 4 36.23 even 6
6480.2.a.bf.1.2 2 3.2 odd 2
6480.2.a.bm.1.2 2 1.1 even 1 trivial