Properties

Label 648.2.a.f.1.1
Level $648$
Weight $2$
Character 648.1
Self dual yes
Analytic conductor $5.174$
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] = [648,2,Mod(1,648)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(648, 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("648.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 648 = 2^{3} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 648.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: \(5.17430605098\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 72)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.37228\) of defining polynomial
Character \(\chi\) \(=\) 648.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.37228 q^{5} -1.37228 q^{7} +O(q^{10})\) \(q-3.37228 q^{5} -1.37228 q^{7} +1.00000 q^{11} +5.37228 q^{13} +0.372281 q^{17} +6.37228 q^{19} +5.37228 q^{23} +6.37228 q^{25} +1.37228 q^{29} +0.627719 q^{31} +4.62772 q^{35} -2.74456 q^{37} -0.255437 q^{41} +9.74456 q^{43} -1.37228 q^{47} -5.11684 q^{49} -10.7446 q^{53} -3.37228 q^{55} +7.00000 q^{59} -3.37228 q^{61} -18.1168 q^{65} +7.74456 q^{67} -4.00000 q^{71} +5.11684 q^{73} -1.37228 q^{77} -0.627719 q^{79} +15.3723 q^{83} -1.25544 q^{85} +6.00000 q^{89} -7.37228 q^{91} -21.4891 q^{95} -9.74456 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{5} + 3 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{5} + 3 q^{7} + 2 q^{11} + 5 q^{13} - 5 q^{17} + 7 q^{19} + 5 q^{23} + 7 q^{25} - 3 q^{29} + 7 q^{31} + 15 q^{35} + 6 q^{37} - 12 q^{41} + 8 q^{43} + 3 q^{47} + 7 q^{49} - 10 q^{53} - q^{55} + 14 q^{59} - q^{61} - 19 q^{65} + 4 q^{67} - 8 q^{71} - 7 q^{73} + 3 q^{77} - 7 q^{79} + 25 q^{83} - 14 q^{85} + 12 q^{89} - 9 q^{91} - 20 q^{95} - 8 q^{97}+O(q^{100}) \) 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
\(5\) −3.37228 −1.50813 −0.754065 0.656800i \(-0.771910\pi\)
−0.754065 + 0.656800i \(0.771910\pi\)
\(6\) 0 0
\(7\) −1.37228 −0.518674 −0.259337 0.965787i \(-0.583504\pi\)
−0.259337 + 0.965787i \(0.583504\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.00000 0.301511 0.150756 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) 0 0
\(13\) 5.37228 1.49000 0.745001 0.667063i \(-0.232449\pi\)
0.745001 + 0.667063i \(0.232449\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.372281 0.0902915 0.0451457 0.998980i \(-0.485625\pi\)
0.0451457 + 0.998980i \(0.485625\pi\)
\(18\) 0 0
\(19\) 6.37228 1.46190 0.730951 0.682430i \(-0.239077\pi\)
0.730951 + 0.682430i \(0.239077\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.37228 1.12020 0.560099 0.828426i \(-0.310763\pi\)
0.560099 + 0.828426i \(0.310763\pi\)
\(24\) 0 0
\(25\) 6.37228 1.27446
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.37228 0.254826 0.127413 0.991850i \(-0.459333\pi\)
0.127413 + 0.991850i \(0.459333\pi\)
\(30\) 0 0
\(31\) 0.627719 0.112742 0.0563708 0.998410i \(-0.482047\pi\)
0.0563708 + 0.998410i \(0.482047\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 4.62772 0.782227
\(36\) 0 0
\(37\) −2.74456 −0.451203 −0.225602 0.974220i \(-0.572435\pi\)
−0.225602 + 0.974220i \(0.572435\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −0.255437 −0.0398926 −0.0199463 0.999801i \(-0.506350\pi\)
−0.0199463 + 0.999801i \(0.506350\pi\)
\(42\) 0 0
\(43\) 9.74456 1.48603 0.743016 0.669274i \(-0.233395\pi\)
0.743016 + 0.669274i \(0.233395\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.37228 −0.200168 −0.100084 0.994979i \(-0.531911\pi\)
−0.100084 + 0.994979i \(0.531911\pi\)
\(48\) 0 0
\(49\) −5.11684 −0.730978
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −10.7446 −1.47588 −0.737940 0.674867i \(-0.764201\pi\)
−0.737940 + 0.674867i \(0.764201\pi\)
\(54\) 0 0
\(55\) −3.37228 −0.454718
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 7.00000 0.911322 0.455661 0.890153i \(-0.349403\pi\)
0.455661 + 0.890153i \(0.349403\pi\)
\(60\) 0 0
\(61\) −3.37228 −0.431776 −0.215888 0.976418i \(-0.569265\pi\)
−0.215888 + 0.976418i \(0.569265\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −18.1168 −2.24712
\(66\) 0 0
\(67\) 7.74456 0.946149 0.473074 0.881022i \(-0.343144\pi\)
0.473074 + 0.881022i \(0.343144\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −4.00000 −0.474713 −0.237356 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) 0 0
\(73\) 5.11684 0.598881 0.299441 0.954115i \(-0.403200\pi\)
0.299441 + 0.954115i \(0.403200\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.37228 −0.156386
\(78\) 0 0
\(79\) −0.627719 −0.0706239 −0.0353119 0.999376i \(-0.511242\pi\)
−0.0353119 + 0.999376i \(0.511242\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 15.3723 1.68733 0.843664 0.536872i \(-0.180394\pi\)
0.843664 + 0.536872i \(0.180394\pi\)
\(84\) 0 0
\(85\) −1.25544 −0.136171
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −7.37228 −0.772825
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −21.4891 −2.20474
\(96\) 0 0
\(97\) −9.74456 −0.989410 −0.494705 0.869061i \(-0.664724\pi\)
−0.494705 + 0.869061i \(0.664724\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 10.1168 1.00666 0.503332 0.864093i \(-0.332107\pi\)
0.503332 + 0.864093i \(0.332107\pi\)
\(102\) 0 0
\(103\) −6.62772 −0.653049 −0.326524 0.945189i \(-0.605877\pi\)
−0.326524 + 0.945189i \(0.605877\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 15.8614 1.53338 0.766690 0.642017i \(-0.221902\pi\)
0.766690 + 0.642017i \(0.221902\pi\)
\(108\) 0 0
\(109\) 6.74456 0.646012 0.323006 0.946397i \(-0.395307\pi\)
0.323006 + 0.946397i \(0.395307\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1.37228 0.129093 0.0645467 0.997915i \(-0.479440\pi\)
0.0645467 + 0.997915i \(0.479440\pi\)
\(114\) 0 0
\(115\) −18.1168 −1.68940
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −0.510875 −0.0468318
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −4.62772 −0.413916
\(126\) 0 0
\(127\) −4.74456 −0.421012 −0.210506 0.977593i \(-0.567511\pi\)
−0.210506 + 0.977593i \(0.567511\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 12.6277 1.10329 0.551644 0.834079i \(-0.314001\pi\)
0.551644 + 0.834079i \(0.314001\pi\)
\(132\) 0 0
\(133\) −8.74456 −0.758250
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 6.25544 0.534438 0.267219 0.963636i \(-0.413895\pi\)
0.267219 + 0.963636i \(0.413895\pi\)
\(138\) 0 0
\(139\) −5.74456 −0.487247 −0.243624 0.969870i \(-0.578336\pi\)
−0.243624 + 0.969870i \(0.578336\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 5.37228 0.449253
\(144\) 0 0
\(145\) −4.62772 −0.384311
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −2.62772 −0.215271 −0.107636 0.994190i \(-0.534328\pi\)
−0.107636 + 0.994190i \(0.534328\pi\)
\(150\) 0 0
\(151\) −5.37228 −0.437190 −0.218595 0.975816i \(-0.570147\pi\)
−0.218595 + 0.975816i \(0.570147\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −2.11684 −0.170029
\(156\) 0 0
\(157\) 14.1168 1.12665 0.563323 0.826236i \(-0.309523\pi\)
0.563323 + 0.826236i \(0.309523\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −7.37228 −0.581017
\(162\) 0 0
\(163\) −12.0000 −0.939913 −0.469956 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −1.88316 −0.145723 −0.0728615 0.997342i \(-0.523213\pi\)
−0.0728615 + 0.997342i \(0.523213\pi\)
\(168\) 0 0
\(169\) 15.8614 1.22011
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −10.6277 −0.808010 −0.404005 0.914757i \(-0.632382\pi\)
−0.404005 + 0.914757i \(0.632382\pi\)
\(174\) 0 0
\(175\) −8.74456 −0.661027
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −22.9783 −1.71748 −0.858738 0.512416i \(-0.828751\pi\)
−0.858738 + 0.512416i \(0.828751\pi\)
\(180\) 0 0
\(181\) −23.4891 −1.74593 −0.872966 0.487780i \(-0.837807\pi\)
−0.872966 + 0.487780i \(0.837807\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 9.25544 0.680473
\(186\) 0 0
\(187\) 0.372281 0.0272239
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −18.8614 −1.36476 −0.682382 0.730996i \(-0.739056\pi\)
−0.682382 + 0.730996i \(0.739056\pi\)
\(192\) 0 0
\(193\) −9.74456 −0.701429 −0.350714 0.936482i \(-0.614061\pi\)
−0.350714 + 0.936482i \(0.614061\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 26.7446 1.90547 0.952736 0.303801i \(-0.0982557\pi\)
0.952736 + 0.303801i \(0.0982557\pi\)
\(198\) 0 0
\(199\) 18.2337 1.29255 0.646276 0.763104i \(-0.276326\pi\)
0.646276 + 0.763104i \(0.276326\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −1.88316 −0.132172
\(204\) 0 0
\(205\) 0.861407 0.0601632
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 6.37228 0.440780
\(210\) 0 0
\(211\) −15.3723 −1.05827 −0.529136 0.848537i \(-0.677484\pi\)
−0.529136 + 0.848537i \(0.677484\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −32.8614 −2.24113
\(216\) 0 0
\(217\) −0.861407 −0.0584761
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 2.00000 0.134535
\(222\) 0 0
\(223\) 25.6060 1.71470 0.857351 0.514732i \(-0.172108\pi\)
0.857351 + 0.514732i \(0.172108\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −15.0000 −0.995585 −0.497792 0.867296i \(-0.665856\pi\)
−0.497792 + 0.867296i \(0.665856\pi\)
\(228\) 0 0
\(229\) −17.6060 −1.16344 −0.581718 0.813391i \(-0.697619\pi\)
−0.581718 + 0.813391i \(0.697619\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −0.372281 −0.0243890 −0.0121945 0.999926i \(-0.503882\pi\)
−0.0121945 + 0.999926i \(0.503882\pi\)
\(234\) 0 0
\(235\) 4.62772 0.301879
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 14.8614 0.961304 0.480652 0.876911i \(-0.340400\pi\)
0.480652 + 0.876911i \(0.340400\pi\)
\(240\) 0 0
\(241\) −5.74456 −0.370040 −0.185020 0.982735i \(-0.559235\pi\)
−0.185020 + 0.982735i \(0.559235\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 17.2554 1.10241
\(246\) 0 0
\(247\) 34.2337 2.17824
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 27.1168 1.71160 0.855800 0.517307i \(-0.173065\pi\)
0.855800 + 0.517307i \(0.173065\pi\)
\(252\) 0 0
\(253\) 5.37228 0.337752
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −18.4891 −1.15332 −0.576660 0.816984i \(-0.695644\pi\)
−0.576660 + 0.816984i \(0.695644\pi\)
\(258\) 0 0
\(259\) 3.76631 0.234027
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −3.88316 −0.239446 −0.119723 0.992807i \(-0.538201\pi\)
−0.119723 + 0.992807i \(0.538201\pi\)
\(264\) 0 0
\(265\) 36.2337 2.22582
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −1.25544 −0.0765454 −0.0382727 0.999267i \(-0.512186\pi\)
−0.0382727 + 0.999267i \(0.512186\pi\)
\(270\) 0 0
\(271\) 1.48913 0.0904579 0.0452290 0.998977i \(-0.485598\pi\)
0.0452290 + 0.998977i \(0.485598\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 6.37228 0.384263
\(276\) 0 0
\(277\) 11.8832 0.713990 0.356995 0.934106i \(-0.383801\pi\)
0.356995 + 0.934106i \(0.383801\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 14.8614 0.886557 0.443279 0.896384i \(-0.353815\pi\)
0.443279 + 0.896384i \(0.353815\pi\)
\(282\) 0 0
\(283\) −4.86141 −0.288981 −0.144490 0.989506i \(-0.546154\pi\)
−0.144490 + 0.989506i \(0.546154\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0.350532 0.0206912
\(288\) 0 0
\(289\) −16.8614 −0.991847
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 25.3723 1.48226 0.741132 0.671359i \(-0.234289\pi\)
0.741132 + 0.671359i \(0.234289\pi\)
\(294\) 0 0
\(295\) −23.6060 −1.37439
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 28.8614 1.66910
\(300\) 0 0
\(301\) −13.3723 −0.770765
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 11.3723 0.651175
\(306\) 0 0
\(307\) 25.6277 1.46265 0.731326 0.682029i \(-0.238902\pi\)
0.731326 + 0.682029i \(0.238902\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −28.1168 −1.59436 −0.797180 0.603742i \(-0.793676\pi\)
−0.797180 + 0.603742i \(0.793676\pi\)
\(312\) 0 0
\(313\) −23.2337 −1.31325 −0.656623 0.754219i \(-0.728016\pi\)
−0.656623 + 0.754219i \(0.728016\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −9.60597 −0.539525 −0.269762 0.962927i \(-0.586945\pi\)
−0.269762 + 0.962927i \(0.586945\pi\)
\(318\) 0 0
\(319\) 1.37228 0.0768330
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 2.37228 0.131997
\(324\) 0 0
\(325\) 34.2337 1.89894
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 1.88316 0.103822
\(330\) 0 0
\(331\) −3.13859 −0.172513 −0.0862563 0.996273i \(-0.527490\pi\)
−0.0862563 + 0.996273i \(0.527490\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −26.1168 −1.42692
\(336\) 0 0
\(337\) 25.9783 1.41513 0.707563 0.706651i \(-0.249795\pi\)
0.707563 + 0.706651i \(0.249795\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0.627719 0.0339929
\(342\) 0 0
\(343\) 16.6277 0.897812
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −28.7228 −1.54192 −0.770961 0.636883i \(-0.780224\pi\)
−0.770961 + 0.636883i \(0.780224\pi\)
\(348\) 0 0
\(349\) 34.1168 1.82623 0.913116 0.407699i \(-0.133669\pi\)
0.913116 + 0.407699i \(0.133669\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 21.9783 1.16978 0.584892 0.811111i \(-0.301137\pi\)
0.584892 + 0.811111i \(0.301137\pi\)
\(354\) 0 0
\(355\) 13.4891 0.715928
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −14.2337 −0.751225 −0.375613 0.926777i \(-0.622568\pi\)
−0.375613 + 0.926777i \(0.622568\pi\)
\(360\) 0 0
\(361\) 21.6060 1.13716
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −17.2554 −0.903191
\(366\) 0 0
\(367\) 23.3723 1.22002 0.610012 0.792393i \(-0.291165\pi\)
0.610012 + 0.792393i \(0.291165\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 14.7446 0.765500
\(372\) 0 0
\(373\) 3.88316 0.201062 0.100531 0.994934i \(-0.467946\pi\)
0.100531 + 0.994934i \(0.467946\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 7.37228 0.379692
\(378\) 0 0
\(379\) −23.1168 −1.18743 −0.593716 0.804674i \(-0.702340\pi\)
−0.593716 + 0.804674i \(0.702340\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −30.3505 −1.55084 −0.775420 0.631446i \(-0.782462\pi\)
−0.775420 + 0.631446i \(0.782462\pi\)
\(384\) 0 0
\(385\) 4.62772 0.235850
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −29.6060 −1.50108 −0.750541 0.660824i \(-0.770207\pi\)
−0.750541 + 0.660824i \(0.770207\pi\)
\(390\) 0 0
\(391\) 2.00000 0.101144
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 2.11684 0.106510
\(396\) 0 0
\(397\) 16.2337 0.814745 0.407373 0.913262i \(-0.366445\pi\)
0.407373 + 0.913262i \(0.366445\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 17.2337 0.860609 0.430305 0.902684i \(-0.358406\pi\)
0.430305 + 0.902684i \(0.358406\pi\)
\(402\) 0 0
\(403\) 3.37228 0.167985
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −2.74456 −0.136043
\(408\) 0 0
\(409\) −5.74456 −0.284050 −0.142025 0.989863i \(-0.545361\pi\)
−0.142025 + 0.989863i \(0.545361\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −9.60597 −0.472679
\(414\) 0 0
\(415\) −51.8397 −2.54471
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 27.6060 1.34864 0.674320 0.738439i \(-0.264437\pi\)
0.674320 + 0.738439i \(0.264437\pi\)
\(420\) 0 0
\(421\) −17.8832 −0.871572 −0.435786 0.900050i \(-0.643529\pi\)
−0.435786 + 0.900050i \(0.643529\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 2.37228 0.115073
\(426\) 0 0
\(427\) 4.62772 0.223951
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −12.7446 −0.613884 −0.306942 0.951728i \(-0.599306\pi\)
−0.306942 + 0.951728i \(0.599306\pi\)
\(432\) 0 0
\(433\) 14.8832 0.715239 0.357619 0.933867i \(-0.383589\pi\)
0.357619 + 0.933867i \(0.383589\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 34.2337 1.63762
\(438\) 0 0
\(439\) −10.1168 −0.482851 −0.241425 0.970419i \(-0.577615\pi\)
−0.241425 + 0.970419i \(0.577615\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −26.7228 −1.26964 −0.634820 0.772660i \(-0.718926\pi\)
−0.634820 + 0.772660i \(0.718926\pi\)
\(444\) 0 0
\(445\) −20.2337 −0.959169
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −33.1168 −1.56288 −0.781440 0.623980i \(-0.785515\pi\)
−0.781440 + 0.623980i \(0.785515\pi\)
\(450\) 0 0
\(451\) −0.255437 −0.0120281
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 24.8614 1.16552
\(456\) 0 0
\(457\) 7.74456 0.362275 0.181138 0.983458i \(-0.442022\pi\)
0.181138 + 0.983458i \(0.442022\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −24.1168 −1.12323 −0.561617 0.827398i \(-0.689820\pi\)
−0.561617 + 0.827398i \(0.689820\pi\)
\(462\) 0 0
\(463\) −10.3505 −0.481030 −0.240515 0.970645i \(-0.577316\pi\)
−0.240515 + 0.970645i \(0.577316\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 1.62772 0.0753218 0.0376609 0.999291i \(-0.488009\pi\)
0.0376609 + 0.999291i \(0.488009\pi\)
\(468\) 0 0
\(469\) −10.6277 −0.490742
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 9.74456 0.448055
\(474\) 0 0
\(475\) 40.6060 1.86313
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 29.6060 1.35273 0.676366 0.736566i \(-0.263554\pi\)
0.676366 + 0.736566i \(0.263554\pi\)
\(480\) 0 0
\(481\) −14.7446 −0.672294
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 32.8614 1.49216
\(486\) 0 0
\(487\) −4.74456 −0.214997 −0.107498 0.994205i \(-0.534284\pi\)
−0.107498 + 0.994205i \(0.534284\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 11.7446 0.530025 0.265012 0.964245i \(-0.414624\pi\)
0.265012 + 0.964245i \(0.414624\pi\)
\(492\) 0 0
\(493\) 0.510875 0.0230086
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 5.48913 0.246221
\(498\) 0 0
\(499\) 25.9783 1.16295 0.581473 0.813566i \(-0.302477\pi\)
0.581473 + 0.813566i \(0.302477\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 29.4891 1.31486 0.657428 0.753518i \(-0.271645\pi\)
0.657428 + 0.753518i \(0.271645\pi\)
\(504\) 0 0
\(505\) −34.1168 −1.51818
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −13.8832 −0.615360 −0.307680 0.951490i \(-0.599553\pi\)
−0.307680 + 0.951490i \(0.599553\pi\)
\(510\) 0 0
\(511\) −7.02175 −0.310624
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 22.3505 0.984882
\(516\) 0 0
\(517\) −1.37228 −0.0603529
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −5.11684 −0.224173 −0.112087 0.993698i \(-0.535753\pi\)
−0.112087 + 0.993698i \(0.535753\pi\)
\(522\) 0 0
\(523\) 9.48913 0.414930 0.207465 0.978242i \(-0.433479\pi\)
0.207465 + 0.978242i \(0.433479\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0.233688 0.0101796
\(528\) 0 0
\(529\) 5.86141 0.254844
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −1.37228 −0.0594401
\(534\) 0 0
\(535\) −53.4891 −2.31254
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −5.11684 −0.220398
\(540\) 0 0
\(541\) −32.2337 −1.38583 −0.692917 0.721017i \(-0.743675\pi\)
−0.692917 + 0.721017i \(0.743675\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −22.7446 −0.974270
\(546\) 0 0
\(547\) −27.7446 −1.18627 −0.593136 0.805102i \(-0.702110\pi\)
−0.593136 + 0.805102i \(0.702110\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 8.74456 0.372531
\(552\) 0 0
\(553\) 0.861407 0.0366307
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −14.0000 −0.593199 −0.296600 0.955002i \(-0.595853\pi\)
−0.296600 + 0.955002i \(0.595853\pi\)
\(558\) 0 0
\(559\) 52.3505 2.21419
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −1.74456 −0.0735245 −0.0367623 0.999324i \(-0.511704\pi\)
−0.0367623 + 0.999324i \(0.511704\pi\)
\(564\) 0 0
\(565\) −4.62772 −0.194690
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −7.23369 −0.303252 −0.151626 0.988438i \(-0.548451\pi\)
−0.151626 + 0.988438i \(0.548451\pi\)
\(570\) 0 0
\(571\) −3.51087 −0.146926 −0.0734628 0.997298i \(-0.523405\pi\)
−0.0734628 + 0.997298i \(0.523405\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 34.2337 1.42764
\(576\) 0 0
\(577\) −13.8614 −0.577058 −0.288529 0.957471i \(-0.593166\pi\)
−0.288529 + 0.957471i \(0.593166\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −21.0951 −0.875172
\(582\) 0 0
\(583\) −10.7446 −0.444994
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −25.2337 −1.04151 −0.520753 0.853707i \(-0.674349\pi\)
−0.520753 + 0.853707i \(0.674349\pi\)
\(588\) 0 0
\(589\) 4.00000 0.164817
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −26.0000 −1.06769 −0.533846 0.845582i \(-0.679254\pi\)
−0.533846 + 0.845582i \(0.679254\pi\)
\(594\) 0 0
\(595\) 1.72281 0.0706285
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 11.1386 0.455111 0.227555 0.973765i \(-0.426927\pi\)
0.227555 + 0.973765i \(0.426927\pi\)
\(600\) 0 0
\(601\) −39.9783 −1.63075 −0.815373 0.578935i \(-0.803468\pi\)
−0.815373 + 0.578935i \(0.803468\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 33.7228 1.37103
\(606\) 0 0
\(607\) 6.11684 0.248275 0.124138 0.992265i \(-0.460384\pi\)
0.124138 + 0.992265i \(0.460384\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −7.37228 −0.298251
\(612\) 0 0
\(613\) −1.25544 −0.0507066 −0.0253533 0.999679i \(-0.508071\pi\)
−0.0253533 + 0.999679i \(0.508071\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −7.97825 −0.321192 −0.160596 0.987020i \(-0.551342\pi\)
−0.160596 + 0.987020i \(0.551342\pi\)
\(618\) 0 0
\(619\) −13.2337 −0.531907 −0.265953 0.963986i \(-0.585687\pi\)
−0.265953 + 0.963986i \(0.585687\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −8.23369 −0.329876
\(624\) 0 0
\(625\) −16.2554 −0.650217
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −1.02175 −0.0407398
\(630\) 0 0
\(631\) −32.0000 −1.27390 −0.636950 0.770905i \(-0.719804\pi\)
−0.636950 + 0.770905i \(0.719804\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 16.0000 0.634941
\(636\) 0 0
\(637\) −27.4891 −1.08916
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −31.2337 −1.23366 −0.616828 0.787098i \(-0.711583\pi\)
−0.616828 + 0.787098i \(0.711583\pi\)
\(642\) 0 0
\(643\) 3.00000 0.118308 0.0591542 0.998249i \(-0.481160\pi\)
0.0591542 + 0.998249i \(0.481160\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −23.7228 −0.932640 −0.466320 0.884616i \(-0.654420\pi\)
−0.466320 + 0.884616i \(0.654420\pi\)
\(648\) 0 0
\(649\) 7.00000 0.274774
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −40.1168 −1.56989 −0.784947 0.619563i \(-0.787310\pi\)
−0.784947 + 0.619563i \(0.787310\pi\)
\(654\) 0 0
\(655\) −42.5842 −1.66390
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 1.88316 0.0733573 0.0366787 0.999327i \(-0.488322\pi\)
0.0366787 + 0.999327i \(0.488322\pi\)
\(660\) 0 0
\(661\) 17.0951 0.664922 0.332461 0.943117i \(-0.392121\pi\)
0.332461 + 0.943117i \(0.392121\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 29.4891 1.14354
\(666\) 0 0
\(667\) 7.37228 0.285456
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −3.37228 −0.130185
\(672\) 0 0
\(673\) 5.37228 0.207086 0.103543 0.994625i \(-0.466982\pi\)
0.103543 + 0.994625i \(0.466982\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 19.6060 0.753519 0.376759 0.926311i \(-0.377038\pi\)
0.376759 + 0.926311i \(0.377038\pi\)
\(678\) 0 0
\(679\) 13.3723 0.513181
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −9.62772 −0.368394 −0.184197 0.982889i \(-0.558969\pi\)
−0.184197 + 0.982889i \(0.558969\pi\)
\(684\) 0 0
\(685\) −21.0951 −0.806002
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −57.7228 −2.19906
\(690\) 0 0
\(691\) −2.11684 −0.0805285 −0.0402643 0.999189i \(-0.512820\pi\)
−0.0402643 + 0.999189i \(0.512820\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 19.3723 0.734833
\(696\) 0 0
\(697\) −0.0950946 −0.00360196
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −12.5109 −0.472529 −0.236265 0.971689i \(-0.575923\pi\)
−0.236265 + 0.971689i \(0.575923\pi\)
\(702\) 0 0
\(703\) −17.4891 −0.659615
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −13.8832 −0.522130
\(708\) 0 0
\(709\) 11.6060 0.435871 0.217936 0.975963i \(-0.430068\pi\)
0.217936 + 0.975963i \(0.430068\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 3.37228 0.126293
\(714\) 0 0
\(715\) −18.1168 −0.677532
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 22.5109 0.839514 0.419757 0.907637i \(-0.362115\pi\)
0.419757 + 0.907637i \(0.362115\pi\)
\(720\) 0 0
\(721\) 9.09509 0.338719
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 8.74456 0.324765
\(726\) 0 0
\(727\) 28.1168 1.04280 0.521398 0.853314i \(-0.325411\pi\)
0.521398 + 0.853314i \(0.325411\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 3.62772 0.134176
\(732\) 0 0
\(733\) −9.13859 −0.337542 −0.168771 0.985655i \(-0.553980\pi\)
−0.168771 + 0.985655i \(0.553980\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 7.74456 0.285275
\(738\) 0 0
\(739\) 24.8832 0.915342 0.457671 0.889122i \(-0.348684\pi\)
0.457671 + 0.889122i \(0.348684\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 41.3723 1.51780 0.758901 0.651206i \(-0.225737\pi\)
0.758901 + 0.651206i \(0.225737\pi\)
\(744\) 0 0
\(745\) 8.86141 0.324657
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −21.7663 −0.795324
\(750\) 0 0
\(751\) −43.3723 −1.58268 −0.791339 0.611378i \(-0.790615\pi\)
−0.791339 + 0.611378i \(0.790615\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 18.1168 0.659339
\(756\) 0 0
\(757\) 31.4891 1.14449 0.572246 0.820082i \(-0.306072\pi\)
0.572246 + 0.820082i \(0.306072\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 32.3505 1.17271 0.586353 0.810056i \(-0.300563\pi\)
0.586353 + 0.810056i \(0.300563\pi\)
\(762\) 0 0
\(763\) −9.25544 −0.335069
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 37.6060 1.35787
\(768\) 0 0
\(769\) 15.8832 0.572761 0.286381 0.958116i \(-0.407548\pi\)
0.286381 + 0.958116i \(0.407548\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −16.9783 −0.610665 −0.305333 0.952246i \(-0.598768\pi\)
−0.305333 + 0.952246i \(0.598768\pi\)
\(774\) 0 0
\(775\) 4.00000 0.143684
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −1.62772 −0.0583191
\(780\) 0 0
\(781\) −4.00000 −0.143131
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −47.6060 −1.69913
\(786\) 0 0
\(787\) −31.8397 −1.13496 −0.567481 0.823387i \(-0.692082\pi\)
−0.567481 + 0.823387i \(0.692082\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −1.88316 −0.0669573
\(792\) 0 0
\(793\) −18.1168 −0.643348
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 43.6060 1.54460 0.772301 0.635256i \(-0.219106\pi\)
0.772301 + 0.635256i \(0.219106\pi\)
\(798\) 0 0
\(799\) −0.510875 −0.0180734
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 5.11684 0.180570
\(804\) 0 0
\(805\) 24.8614 0.876249
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 41.8614 1.47177 0.735884 0.677107i \(-0.236767\pi\)
0.735884 + 0.677107i \(0.236767\pi\)
\(810\) 0 0
\(811\) −41.3505 −1.45201 −0.726007 0.687688i \(-0.758626\pi\)
−0.726007 + 0.687688i \(0.758626\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 40.4674 1.41751
\(816\) 0 0
\(817\) 62.0951 2.17243
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 25.3723 0.885499 0.442749 0.896645i \(-0.354003\pi\)
0.442749 + 0.896645i \(0.354003\pi\)
\(822\) 0 0
\(823\) 42.1168 1.46810 0.734050 0.679095i \(-0.237628\pi\)
0.734050 + 0.679095i \(0.237628\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −14.9783 −0.520845 −0.260422 0.965495i \(-0.583862\pi\)
−0.260422 + 0.965495i \(0.583862\pi\)
\(828\) 0 0
\(829\) −7.48913 −0.260108 −0.130054 0.991507i \(-0.541515\pi\)
−0.130054 + 0.991507i \(0.541515\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −1.90491 −0.0660011
\(834\) 0 0
\(835\) 6.35053 0.219769
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −33.0951 −1.14257 −0.571285 0.820752i \(-0.693555\pi\)
−0.571285 + 0.820752i \(0.693555\pi\)
\(840\) 0 0
\(841\) −27.1168 −0.935064
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −53.4891 −1.84008
\(846\) 0 0
\(847\) 13.7228 0.471521
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −14.7446 −0.505437
\(852\) 0 0
\(853\) 15.8832 0.543829 0.271914 0.962321i \(-0.412343\pi\)
0.271914 + 0.962321i \(0.412343\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 19.8832 0.679196 0.339598 0.940571i \(-0.389709\pi\)
0.339598 + 0.940571i \(0.389709\pi\)
\(858\) 0 0
\(859\) 20.4891 0.699080 0.349540 0.936921i \(-0.386338\pi\)
0.349540 + 0.936921i \(0.386338\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 26.9783 0.918350 0.459175 0.888346i \(-0.348145\pi\)
0.459175 + 0.888346i \(0.348145\pi\)
\(864\) 0 0
\(865\) 35.8397 1.21858
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −0.627719 −0.0212939
\(870\) 0 0
\(871\) 41.6060 1.40976
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 6.35053 0.214687
\(876\) 0 0
\(877\) −0.861407 −0.0290876 −0.0145438 0.999894i \(-0.504630\pi\)
−0.0145438 + 0.999894i \(0.504630\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −20.9783 −0.706775 −0.353388 0.935477i \(-0.614970\pi\)
−0.353388 + 0.935477i \(0.614970\pi\)
\(882\) 0 0
\(883\) −42.8397 −1.44167 −0.720835 0.693107i \(-0.756241\pi\)
−0.720835 + 0.693107i \(0.756241\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −16.8614 −0.566151 −0.283075 0.959098i \(-0.591355\pi\)
−0.283075 + 0.959098i \(0.591355\pi\)
\(888\) 0 0
\(889\) 6.51087 0.218368
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −8.74456 −0.292626
\(894\) 0 0
\(895\) 77.4891 2.59018
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0.861407 0.0287295
\(900\) 0 0
\(901\) −4.00000 −0.133259
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 79.2119 2.63309
\(906\) 0 0
\(907\) 37.0000 1.22856 0.614282 0.789086i \(-0.289446\pi\)
0.614282 + 0.789086i \(0.289446\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −18.1168 −0.600238 −0.300119 0.953902i \(-0.597026\pi\)
−0.300119 + 0.953902i \(0.597026\pi\)
\(912\) 0 0
\(913\) 15.3723 0.508748
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −17.3288 −0.572247
\(918\) 0 0
\(919\) 1.76631 0.0582653 0.0291326 0.999576i \(-0.490725\pi\)
0.0291326 + 0.999576i \(0.490725\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −21.4891 −0.707323
\(924\) 0 0
\(925\) −17.4891 −0.575039
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −48.5842 −1.59400 −0.796998 0.603982i \(-0.793580\pi\)
−0.796998 + 0.603982i \(0.793580\pi\)
\(930\) 0 0
\(931\) −32.6060 −1.06862
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −1.25544 −0.0410572
\(936\) 0 0
\(937\) 30.0000 0.980057 0.490029 0.871706i \(-0.336986\pi\)
0.490029 + 0.871706i \(0.336986\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −47.8397 −1.55953 −0.779764 0.626073i \(-0.784661\pi\)
−0.779764 + 0.626073i \(0.784661\pi\)
\(942\) 0 0
\(943\) −1.37228 −0.0446876
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −42.2554 −1.37312 −0.686559 0.727074i \(-0.740879\pi\)
−0.686559 + 0.727074i \(0.740879\pi\)
\(948\) 0 0
\(949\) 27.4891 0.892335
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −25.1168 −0.813614 −0.406807 0.913514i \(-0.633358\pi\)
−0.406807 + 0.913514i \(0.633358\pi\)
\(954\) 0 0
\(955\) 63.6060 2.05824
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −8.58422 −0.277199
\(960\) 0 0
\(961\) −30.6060 −0.987289
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 32.8614 1.05785
\(966\) 0 0
\(967\) −2.62772 −0.0845017 −0.0422509 0.999107i \(-0.513453\pi\)
−0.0422509 + 0.999107i \(0.513453\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 17.4891 0.561253 0.280626 0.959817i \(-0.409458\pi\)
0.280626 + 0.959817i \(0.409458\pi\)
\(972\) 0 0
\(973\) 7.88316 0.252722
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 11.7446 0.375742 0.187871 0.982194i \(-0.439841\pi\)
0.187871 + 0.982194i \(0.439841\pi\)
\(978\) 0 0
\(979\) 6.00000 0.191761
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 37.8832 1.20829 0.604143 0.796876i \(-0.293516\pi\)
0.604143 + 0.796876i \(0.293516\pi\)
\(984\) 0 0
\(985\) −90.1902 −2.87370
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 52.3505 1.66465
\(990\) 0 0
\(991\) 5.02175 0.159521 0.0797606 0.996814i \(-0.474584\pi\)
0.0797606 + 0.996814i \(0.474584\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −61.4891 −1.94934
\(996\) 0 0
\(997\) −2.35053 −0.0744421 −0.0372210 0.999307i \(-0.511851\pi\)
−0.0372210 + 0.999307i \(0.511851\pi\)
\(998\) 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 648.2.a.f.1.1 2
3.2 odd 2 648.2.a.g.1.2 2
4.3 odd 2 1296.2.a.n.1.1 2
8.3 odd 2 5184.2.a.bs.1.2 2
8.5 even 2 5184.2.a.bt.1.2 2
9.2 odd 6 216.2.i.b.145.1 4
9.4 even 3 72.2.i.b.25.1 4
9.5 odd 6 216.2.i.b.73.1 4
9.7 even 3 72.2.i.b.49.1 yes 4
12.11 even 2 1296.2.a.p.1.2 2
24.5 odd 2 5184.2.a.bp.1.1 2
24.11 even 2 5184.2.a.bo.1.1 2
36.7 odd 6 144.2.i.d.49.2 4
36.11 even 6 432.2.i.d.145.1 4
36.23 even 6 432.2.i.d.289.1 4
36.31 odd 6 144.2.i.d.97.2 4
72.5 odd 6 1728.2.i.i.1153.2 4
72.11 even 6 1728.2.i.j.577.2 4
72.13 even 6 576.2.i.j.385.2 4
72.29 odd 6 1728.2.i.i.577.2 4
72.43 odd 6 576.2.i.l.193.1 4
72.59 even 6 1728.2.i.j.1153.2 4
72.61 even 6 576.2.i.j.193.2 4
72.67 odd 6 576.2.i.l.385.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.2.i.b.25.1 4 9.4 even 3
72.2.i.b.49.1 yes 4 9.7 even 3
144.2.i.d.49.2 4 36.7 odd 6
144.2.i.d.97.2 4 36.31 odd 6
216.2.i.b.73.1 4 9.5 odd 6
216.2.i.b.145.1 4 9.2 odd 6
432.2.i.d.145.1 4 36.11 even 6
432.2.i.d.289.1 4 36.23 even 6
576.2.i.j.193.2 4 72.61 even 6
576.2.i.j.385.2 4 72.13 even 6
576.2.i.l.193.1 4 72.43 odd 6
576.2.i.l.385.1 4 72.67 odd 6
648.2.a.f.1.1 2 1.1 even 1 trivial
648.2.a.g.1.2 2 3.2 odd 2
1296.2.a.n.1.1 2 4.3 odd 2
1296.2.a.p.1.2 2 12.11 even 2
1728.2.i.i.577.2 4 72.29 odd 6
1728.2.i.i.1153.2 4 72.5 odd 6
1728.2.i.j.577.2 4 72.11 even 6
1728.2.i.j.1153.2 4 72.59 even 6
5184.2.a.bo.1.1 2 24.11 even 2
5184.2.a.bp.1.1 2 24.5 odd 2
5184.2.a.bs.1.2 2 8.3 odd 2
5184.2.a.bt.1.2 2 8.5 even 2