Properties

Label 6048.2.j.d.5615.27
Level $6048$
Weight $2$
Character 6048.5615
Analytic conductor $48.294$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6048,2,Mod(5615,6048)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6048, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6048.5615");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6048.j (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(48.2935231425\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: no (minimal twist has level 1512)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 5615.27
Character \(\chi\) \(=\) 6048.5615
Dual form 6048.2.j.d.5615.28

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.530075 q^{5} +1.00000i q^{7} +O(q^{10})\) \(q+0.530075 q^{5} +1.00000i q^{7} +0.905144i q^{11} -4.38398i q^{13} +1.25430i q^{17} -0.00364927 q^{19} +0.778929 q^{23} -4.71902 q^{25} -5.43228 q^{29} +3.49953i q^{31} +0.530075i q^{35} -5.34322i q^{37} -1.74538i q^{41} +0.259493 q^{43} -3.75927 q^{47} -1.00000 q^{49} -3.08087 q^{53} +0.479794i q^{55} -5.73302i q^{59} +0.555629i q^{61} -2.32384i q^{65} -11.5465 q^{67} +16.0043 q^{71} -13.1552 q^{73} -0.905144 q^{77} -1.96889i q^{79} +5.82561i q^{83} +0.664872i q^{85} +1.94491i q^{89} +4.38398 q^{91} -0.00193439 q^{95} -13.8042 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q+O(q^{10}) \) Copy content Toggle raw display \( 48 q - 16 q^{19} + 48 q^{25} - 64 q^{43} - 48 q^{49} - 32 q^{67}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6048\mathbb{Z}\right)^\times\).

\(n\) \(2593\) \(3781\) \(3809\) \(4159\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\)

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
\(5\) 0.530075 0.237057 0.118528 0.992951i \(-0.462182\pi\)
0.118528 + 0.992951i \(0.462182\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.905144i 0.272911i 0.990646 + 0.136456i \(0.0435711\pi\)
−0.990646 + 0.136456i \(0.956429\pi\)
\(12\) 0 0
\(13\) − 4.38398i − 1.21590i −0.793976 0.607949i \(-0.791992\pi\)
0.793976 0.607949i \(-0.208008\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.25430i 0.304212i 0.988364 + 0.152106i \(0.0486055\pi\)
−0.988364 + 0.152106i \(0.951395\pi\)
\(18\) 0 0
\(19\) −0.00364927 −0.000837200 0 −0.000418600 1.00000i \(-0.500133\pi\)
−0.000418600 1.00000i \(0.500133\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.778929 0.162418 0.0812090 0.996697i \(-0.474122\pi\)
0.0812090 + 0.996697i \(0.474122\pi\)
\(24\) 0 0
\(25\) −4.71902 −0.943804
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −5.43228 −1.00875 −0.504375 0.863485i \(-0.668277\pi\)
−0.504375 + 0.863485i \(0.668277\pi\)
\(30\) 0 0
\(31\) 3.49953i 0.628533i 0.949335 + 0.314267i \(0.101759\pi\)
−0.949335 + 0.314267i \(0.898241\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.530075i 0.0895990i
\(36\) 0 0
\(37\) − 5.34322i − 0.878421i −0.898384 0.439210i \(-0.855258\pi\)
0.898384 0.439210i \(-0.144742\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 1.74538i − 0.272582i −0.990669 0.136291i \(-0.956482\pi\)
0.990669 0.136291i \(-0.0435183\pi\)
\(42\) 0 0
\(43\) 0.259493 0.0395723 0.0197862 0.999804i \(-0.493701\pi\)
0.0197862 + 0.999804i \(0.493701\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.75927 −0.548346 −0.274173 0.961680i \(-0.588404\pi\)
−0.274173 + 0.961680i \(0.588404\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −3.08087 −0.423190 −0.211595 0.977357i \(-0.567866\pi\)
−0.211595 + 0.977357i \(0.567866\pi\)
\(54\) 0 0
\(55\) 0.479794i 0.0646955i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 5.73302i − 0.746376i −0.927756 0.373188i \(-0.878265\pi\)
0.927756 0.373188i \(-0.121735\pi\)
\(60\) 0 0
\(61\) 0.555629i 0.0711410i 0.999367 + 0.0355705i \(0.0113248\pi\)
−0.999367 + 0.0355705i \(0.988675\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 2.32384i − 0.288237i
\(66\) 0 0
\(67\) −11.5465 −1.41063 −0.705313 0.708896i \(-0.749194\pi\)
−0.705313 + 0.708896i \(0.749194\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 16.0043 1.89936 0.949681 0.313218i \(-0.101407\pi\)
0.949681 + 0.313218i \(0.101407\pi\)
\(72\) 0 0
\(73\) −13.1552 −1.53970 −0.769851 0.638224i \(-0.779669\pi\)
−0.769851 + 0.638224i \(0.779669\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −0.905144 −0.103151
\(78\) 0 0
\(79\) − 1.96889i − 0.221517i −0.993847 0.110759i \(-0.964672\pi\)
0.993847 0.110759i \(-0.0353280\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 5.82561i 0.639444i 0.947511 + 0.319722i \(0.103589\pi\)
−0.947511 + 0.319722i \(0.896411\pi\)
\(84\) 0 0
\(85\) 0.664872i 0.0721155i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.94491i 0.206160i 0.994673 + 0.103080i \(0.0328697\pi\)
−0.994673 + 0.103080i \(0.967130\pi\)
\(90\) 0 0
\(91\) 4.38398 0.459566
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −0.00193439 −0.000198464 0
\(96\) 0 0
\(97\) −13.8042 −1.40161 −0.700803 0.713355i \(-0.747175\pi\)
−0.700803 + 0.713355i \(0.747175\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −5.41639 −0.538951 −0.269475 0.963007i \(-0.586850\pi\)
−0.269475 + 0.963007i \(0.586850\pi\)
\(102\) 0 0
\(103\) 2.14390i 0.211245i 0.994406 + 0.105623i \(0.0336835\pi\)
−0.994406 + 0.105623i \(0.966316\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 9.10325i 0.880045i 0.897987 + 0.440022i \(0.145029\pi\)
−0.897987 + 0.440022i \(0.854971\pi\)
\(108\) 0 0
\(109\) − 1.29567i − 0.124103i −0.998073 0.0620515i \(-0.980236\pi\)
0.998073 0.0620515i \(-0.0197643\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 14.0728i − 1.32386i −0.749567 0.661928i \(-0.769738\pi\)
0.749567 0.661928i \(-0.230262\pi\)
\(114\) 0 0
\(115\) 0.412891 0.0385023
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −1.25430 −0.114981
\(120\) 0 0
\(121\) 10.1807 0.925519
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −5.15181 −0.460792
\(126\) 0 0
\(127\) 11.5015i 1.02060i 0.859997 + 0.510298i \(0.170465\pi\)
−0.859997 + 0.510298i \(0.829535\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 10.2828i 0.898411i 0.893429 + 0.449205i \(0.148293\pi\)
−0.893429 + 0.449205i \(0.851707\pi\)
\(132\) 0 0
\(133\) − 0.00364927i 0 0.000316432i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 19.3027i − 1.64914i −0.565761 0.824570i \(-0.691417\pi\)
0.565761 0.824570i \(-0.308583\pi\)
\(138\) 0 0
\(139\) −15.3098 −1.29856 −0.649280 0.760550i \(-0.724930\pi\)
−0.649280 + 0.760550i \(0.724930\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 3.96814 0.331832
\(144\) 0 0
\(145\) −2.87952 −0.239131
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1.53793 −0.125992 −0.0629962 0.998014i \(-0.520066\pi\)
−0.0629962 + 0.998014i \(0.520066\pi\)
\(150\) 0 0
\(151\) − 10.1304i − 0.824398i −0.911094 0.412199i \(-0.864761\pi\)
0.911094 0.412199i \(-0.135239\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1.85501i 0.148998i
\(156\) 0 0
\(157\) − 2.61643i − 0.208814i −0.994535 0.104407i \(-0.966706\pi\)
0.994535 0.104407i \(-0.0332945\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0.778929i 0.0613882i
\(162\) 0 0
\(163\) −14.4604 −1.13262 −0.566312 0.824191i \(-0.691630\pi\)
−0.566312 + 0.824191i \(0.691630\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 7.65107 0.592058 0.296029 0.955179i \(-0.404338\pi\)
0.296029 + 0.955179i \(0.404338\pi\)
\(168\) 0 0
\(169\) −6.21931 −0.478409
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 9.98644 0.759255 0.379628 0.925139i \(-0.376052\pi\)
0.379628 + 0.925139i \(0.376052\pi\)
\(174\) 0 0
\(175\) − 4.71902i − 0.356724i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 16.1143i 1.20444i 0.798331 + 0.602219i \(0.205717\pi\)
−0.798331 + 0.602219i \(0.794283\pi\)
\(180\) 0 0
\(181\) − 20.4077i − 1.51689i −0.651736 0.758446i \(-0.725959\pi\)
0.651736 0.758446i \(-0.274041\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 2.83231i − 0.208236i
\(186\) 0 0
\(187\) −1.13532 −0.0830228
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −4.05526 −0.293428 −0.146714 0.989179i \(-0.546870\pi\)
−0.146714 + 0.989179i \(0.546870\pi\)
\(192\) 0 0
\(193\) −5.82186 −0.419066 −0.209533 0.977802i \(-0.567194\pi\)
−0.209533 + 0.977802i \(0.567194\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −15.4557 −1.10118 −0.550588 0.834777i \(-0.685596\pi\)
−0.550588 + 0.834777i \(0.685596\pi\)
\(198\) 0 0
\(199\) − 13.6803i − 0.969771i −0.874578 0.484885i \(-0.838861\pi\)
0.874578 0.484885i \(-0.161139\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 5.43228i − 0.381271i
\(204\) 0 0
\(205\) − 0.925182i − 0.0646175i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 0.00330311i 0 0.000228481i
\(210\) 0 0
\(211\) −10.9961 −0.757000 −0.378500 0.925601i \(-0.623560\pi\)
−0.378500 + 0.925601i \(0.623560\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0.137551 0.00938089
\(216\) 0 0
\(217\) −3.49953 −0.237563
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 5.49882 0.369891
\(222\) 0 0
\(223\) − 18.7956i − 1.25864i −0.777144 0.629322i \(-0.783332\pi\)
0.777144 0.629322i \(-0.216668\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 18.8278i 1.24965i 0.780767 + 0.624823i \(0.214829\pi\)
−0.780767 + 0.624823i \(0.785171\pi\)
\(228\) 0 0
\(229\) − 15.7721i − 1.04225i −0.853480 0.521126i \(-0.825512\pi\)
0.853480 0.521126i \(-0.174488\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 4.23089i 0.277175i 0.990350 + 0.138587i \(0.0442562\pi\)
−0.990350 + 0.138587i \(0.955744\pi\)
\(234\) 0 0
\(235\) −1.99269 −0.129989
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 7.80160 0.504644 0.252322 0.967643i \(-0.418806\pi\)
0.252322 + 0.967643i \(0.418806\pi\)
\(240\) 0 0
\(241\) −20.2165 −1.30226 −0.651129 0.758967i \(-0.725704\pi\)
−0.651129 + 0.758967i \(0.725704\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −0.530075 −0.0338653
\(246\) 0 0
\(247\) 0.0159983i 0.00101795i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 12.7749i − 0.806345i −0.915124 0.403172i \(-0.867907\pi\)
0.915124 0.403172i \(-0.132093\pi\)
\(252\) 0 0
\(253\) 0.705043i 0.0443257i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 20.3263i − 1.26792i −0.773365 0.633961i \(-0.781428\pi\)
0.773365 0.633961i \(-0.218572\pi\)
\(258\) 0 0
\(259\) 5.34322 0.332012
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −4.85204 −0.299190 −0.149595 0.988747i \(-0.547797\pi\)
−0.149595 + 0.988747i \(0.547797\pi\)
\(264\) 0 0
\(265\) −1.63309 −0.100320
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −31.3976 −1.91435 −0.957174 0.289514i \(-0.906506\pi\)
−0.957174 + 0.289514i \(0.906506\pi\)
\(270\) 0 0
\(271\) − 24.4088i − 1.48273i −0.671103 0.741364i \(-0.734179\pi\)
0.671103 0.741364i \(-0.265821\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 4.27139i − 0.257575i
\(276\) 0 0
\(277\) 12.8387i 0.771402i 0.922624 + 0.385701i \(0.126040\pi\)
−0.922624 + 0.385701i \(0.873960\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 27.8988i 1.66431i 0.554547 + 0.832153i \(0.312892\pi\)
−0.554547 + 0.832153i \(0.687108\pi\)
\(282\) 0 0
\(283\) −6.65773 −0.395761 −0.197880 0.980226i \(-0.563406\pi\)
−0.197880 + 0.980226i \(0.563406\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.74538 0.103026
\(288\) 0 0
\(289\) 15.4267 0.907455
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 24.0566 1.40540 0.702699 0.711487i \(-0.251978\pi\)
0.702699 + 0.711487i \(0.251978\pi\)
\(294\) 0 0
\(295\) − 3.03893i − 0.176933i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 3.41481i − 0.197484i
\(300\) 0 0
\(301\) 0.259493i 0.0149569i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0.294525i 0.0168645i
\(306\) 0 0
\(307\) −23.6311 −1.34870 −0.674350 0.738411i \(-0.735576\pi\)
−0.674350 + 0.738411i \(0.735576\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 25.7037 1.45752 0.728761 0.684768i \(-0.240096\pi\)
0.728761 + 0.684768i \(0.240096\pi\)
\(312\) 0 0
\(313\) −9.99902 −0.565178 −0.282589 0.959241i \(-0.591193\pi\)
−0.282589 + 0.959241i \(0.591193\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 23.7177 1.33212 0.666059 0.745899i \(-0.267980\pi\)
0.666059 + 0.745899i \(0.267980\pi\)
\(318\) 0 0
\(319\) − 4.91700i − 0.275299i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 0.00457727i 0 0.000254686i
\(324\) 0 0
\(325\) 20.6881i 1.14757i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 3.75927i − 0.207255i
\(330\) 0 0
\(331\) −12.8674 −0.707254 −0.353627 0.935386i \(-0.615052\pi\)
−0.353627 + 0.935386i \(0.615052\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −6.12050 −0.334398
\(336\) 0 0
\(337\) −10.2878 −0.560412 −0.280206 0.959940i \(-0.590403\pi\)
−0.280206 + 0.959940i \(0.590403\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −3.16758 −0.171534
\(342\) 0 0
\(343\) − 1.00000i − 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 2.21263i − 0.118780i −0.998235 0.0593901i \(-0.981084\pi\)
0.998235 0.0593901i \(-0.0189156\pi\)
\(348\) 0 0
\(349\) 8.10593i 0.433900i 0.976183 + 0.216950i \(0.0696109\pi\)
−0.976183 + 0.216950i \(0.930389\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 16.8821i 0.898546i 0.893395 + 0.449273i \(0.148317\pi\)
−0.893395 + 0.449273i \(0.851683\pi\)
\(354\) 0 0
\(355\) 8.48349 0.450257
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −6.06059 −0.319866 −0.159933 0.987128i \(-0.551128\pi\)
−0.159933 + 0.987128i \(0.551128\pi\)
\(360\) 0 0
\(361\) −19.0000 −0.999999
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −6.97325 −0.364997
\(366\) 0 0
\(367\) 4.36701i 0.227956i 0.993483 + 0.113978i \(0.0363594\pi\)
−0.993483 + 0.113978i \(0.963641\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 3.08087i − 0.159951i
\(372\) 0 0
\(373\) − 10.3873i − 0.537835i −0.963163 0.268918i \(-0.913334\pi\)
0.963163 0.268918i \(-0.0866660\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 23.8150i 1.22654i
\(378\) 0 0
\(379\) −28.3525 −1.45637 −0.728186 0.685380i \(-0.759636\pi\)
−0.728186 + 0.685380i \(0.759636\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −26.3164 −1.34470 −0.672352 0.740232i \(-0.734716\pi\)
−0.672352 + 0.740232i \(0.734716\pi\)
\(384\) 0 0
\(385\) −0.479794 −0.0244526
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 17.5966 0.892182 0.446091 0.894988i \(-0.352816\pi\)
0.446091 + 0.894988i \(0.352816\pi\)
\(390\) 0 0
\(391\) 0.977009i 0.0494095i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 1.04366i − 0.0525121i
\(396\) 0 0
\(397\) 5.06951i 0.254431i 0.991875 + 0.127216i \(0.0406040\pi\)
−0.991875 + 0.127216i \(0.959396\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 16.6708i − 0.832498i −0.909251 0.416249i \(-0.863345\pi\)
0.909251 0.416249i \(-0.136655\pi\)
\(402\) 0 0
\(403\) 15.3419 0.764233
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 4.83639 0.239731
\(408\) 0 0
\(409\) −5.09197 −0.251782 −0.125891 0.992044i \(-0.540179\pi\)
−0.125891 + 0.992044i \(0.540179\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 5.73302 0.282104
\(414\) 0 0
\(415\) 3.08801i 0.151584i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 37.4571i − 1.82990i −0.403570 0.914949i \(-0.632231\pi\)
0.403570 0.914949i \(-0.367769\pi\)
\(420\) 0 0
\(421\) 35.9612i 1.75264i 0.481728 + 0.876321i \(0.340009\pi\)
−0.481728 + 0.876321i \(0.659991\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 5.91906i − 0.287116i
\(426\) 0 0
\(427\) −0.555629 −0.0268888
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −20.6573 −0.995025 −0.497513 0.867457i \(-0.665753\pi\)
−0.497513 + 0.867457i \(0.665753\pi\)
\(432\) 0 0
\(433\) 23.8116 1.14431 0.572157 0.820144i \(-0.306107\pi\)
0.572157 + 0.820144i \(0.306107\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −0.00284252 −0.000135976 0
\(438\) 0 0
\(439\) − 26.5607i − 1.26767i −0.773467 0.633836i \(-0.781479\pi\)
0.773467 0.633836i \(-0.218521\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 0.346564i − 0.0164658i −0.999966 0.00823288i \(-0.997379\pi\)
0.999966 0.00823288i \(-0.00262064\pi\)
\(444\) 0 0
\(445\) 1.03095i 0.0488716i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 10.4601i 0.493641i 0.969061 + 0.246821i \(0.0793859\pi\)
−0.969061 + 0.246821i \(0.920614\pi\)
\(450\) 0 0
\(451\) 1.57982 0.0743908
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 2.32384 0.108943
\(456\) 0 0
\(457\) −11.0889 −0.518717 −0.259359 0.965781i \(-0.583511\pi\)
−0.259359 + 0.965781i \(0.583511\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −37.8447 −1.76261 −0.881303 0.472553i \(-0.843333\pi\)
−0.881303 + 0.472553i \(0.843333\pi\)
\(462\) 0 0
\(463\) 18.1959i 0.845634i 0.906215 + 0.422817i \(0.138959\pi\)
−0.906215 + 0.422817i \(0.861041\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 6.19184i 0.286524i 0.989685 + 0.143262i \(0.0457592\pi\)
−0.989685 + 0.143262i \(0.954241\pi\)
\(468\) 0 0
\(469\) − 11.5465i − 0.533167i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0.234879i 0.0107997i
\(474\) 0 0
\(475\) 0.0172210 0.000790152 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 4.17517 0.190769 0.0953843 0.995441i \(-0.469592\pi\)
0.0953843 + 0.995441i \(0.469592\pi\)
\(480\) 0 0
\(481\) −23.4246 −1.06807
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −7.31727 −0.332260
\(486\) 0 0
\(487\) 16.4966i 0.747534i 0.927523 + 0.373767i \(0.121934\pi\)
−0.927523 + 0.373767i \(0.878066\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 25.1039i − 1.13292i −0.824089 0.566461i \(-0.808312\pi\)
0.824089 0.566461i \(-0.191688\pi\)
\(492\) 0 0
\(493\) − 6.81370i − 0.306873i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 16.0043i 0.717892i
\(498\) 0 0
\(499\) −37.1869 −1.66472 −0.832358 0.554238i \(-0.813010\pi\)
−0.832358 + 0.554238i \(0.813010\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 8.24722 0.367725 0.183863 0.982952i \(-0.441140\pi\)
0.183863 + 0.982952i \(0.441140\pi\)
\(504\) 0 0
\(505\) −2.87109 −0.127762
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 29.7128 1.31700 0.658498 0.752582i \(-0.271192\pi\)
0.658498 + 0.752582i \(0.271192\pi\)
\(510\) 0 0
\(511\) − 13.1552i − 0.581953i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 1.13643i 0.0500771i
\(516\) 0 0
\(517\) − 3.40268i − 0.149650i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 9.22757i 0.404267i 0.979358 + 0.202134i \(0.0647875\pi\)
−0.979358 + 0.202134i \(0.935212\pi\)
\(522\) 0 0
\(523\) −1.55380 −0.0679431 −0.0339716 0.999423i \(-0.510816\pi\)
−0.0339716 + 0.999423i \(0.510816\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −4.38945 −0.191207
\(528\) 0 0
\(529\) −22.3933 −0.973620
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −7.65171 −0.331433
\(534\) 0 0
\(535\) 4.82541i 0.208621i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 0.905144i − 0.0389873i
\(540\) 0 0
\(541\) − 7.33522i − 0.315366i −0.987490 0.157683i \(-0.949598\pi\)
0.987490 0.157683i \(-0.0504024\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 0.686805i − 0.0294195i
\(546\) 0 0
\(547\) −21.4488 −0.917086 −0.458543 0.888672i \(-0.651628\pi\)
−0.458543 + 0.888672i \(0.651628\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0.0198238 0.000844524 0
\(552\) 0 0
\(553\) 1.96889 0.0837256
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 29.9101 1.26733 0.633666 0.773607i \(-0.281549\pi\)
0.633666 + 0.773607i \(0.281549\pi\)
\(558\) 0 0
\(559\) − 1.13761i − 0.0481159i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 42.7436i − 1.80143i −0.434414 0.900713i \(-0.643044\pi\)
0.434414 0.900713i \(-0.356956\pi\)
\(564\) 0 0
\(565\) − 7.45963i − 0.313829i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 20.9180i 0.876927i 0.898749 + 0.438464i \(0.144477\pi\)
−0.898749 + 0.438464i \(0.855523\pi\)
\(570\) 0 0
\(571\) 5.23949 0.219266 0.109633 0.993972i \(-0.465032\pi\)
0.109633 + 0.993972i \(0.465032\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −3.67578 −0.153291
\(576\) 0 0
\(577\) −5.71213 −0.237799 −0.118900 0.992906i \(-0.537937\pi\)
−0.118900 + 0.992906i \(0.537937\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −5.82561 −0.241687
\(582\) 0 0
\(583\) − 2.78863i − 0.115493i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 28.8731i 1.19172i 0.803088 + 0.595860i \(0.203189\pi\)
−0.803088 + 0.595860i \(0.796811\pi\)
\(588\) 0 0
\(589\) − 0.0127707i 0 0.000526208i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 32.7752i 1.34592i 0.739680 + 0.672958i \(0.234977\pi\)
−0.739680 + 0.672958i \(0.765023\pi\)
\(594\) 0 0
\(595\) −0.664872 −0.0272571
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −18.9096 −0.772625 −0.386313 0.922368i \(-0.626251\pi\)
−0.386313 + 0.922368i \(0.626251\pi\)
\(600\) 0 0
\(601\) 36.4186 1.48555 0.742773 0.669543i \(-0.233510\pi\)
0.742773 + 0.669543i \(0.233510\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 5.39654 0.219401
\(606\) 0 0
\(607\) 8.21476i 0.333427i 0.986005 + 0.166713i \(0.0533155\pi\)
−0.986005 + 0.166713i \(0.946685\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 16.4806i 0.666732i
\(612\) 0 0
\(613\) − 27.6348i − 1.11616i −0.829787 0.558080i \(-0.811538\pi\)
0.829787 0.558080i \(-0.188462\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 23.3314i − 0.939287i −0.882856 0.469643i \(-0.844383\pi\)
0.882856 0.469643i \(-0.155617\pi\)
\(618\) 0 0
\(619\) 22.9819 0.923719 0.461859 0.886953i \(-0.347182\pi\)
0.461859 + 0.886953i \(0.347182\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −1.94491 −0.0779211
\(624\) 0 0
\(625\) 20.8643 0.834570
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 6.70199 0.267226
\(630\) 0 0
\(631\) 44.4382i 1.76906i 0.466486 + 0.884528i \(0.345520\pi\)
−0.466486 + 0.884528i \(0.654480\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 6.09668i 0.241939i
\(636\) 0 0
\(637\) 4.38398i 0.173700i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 37.6567i − 1.48735i −0.668541 0.743675i \(-0.733081\pi\)
0.668541 0.743675i \(-0.266919\pi\)
\(642\) 0 0
\(643\) 22.1342 0.872886 0.436443 0.899732i \(-0.356238\pi\)
0.436443 + 0.899732i \(0.356238\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −20.8560 −0.819936 −0.409968 0.912100i \(-0.634460\pi\)
−0.409968 + 0.912100i \(0.634460\pi\)
\(648\) 0 0
\(649\) 5.18921 0.203694
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 12.5052 0.489366 0.244683 0.969603i \(-0.421316\pi\)
0.244683 + 0.969603i \(0.421316\pi\)
\(654\) 0 0
\(655\) 5.45065i 0.212974i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 27.7704i − 1.08178i −0.841092 0.540891i \(-0.818087\pi\)
0.841092 0.540891i \(-0.181913\pi\)
\(660\) 0 0
\(661\) 17.8134i 0.692862i 0.938075 + 0.346431i \(0.112607\pi\)
−0.938075 + 0.346431i \(0.887393\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 0.00193439i 0 7.50123e-5i
\(666\) 0 0
\(667\) −4.23136 −0.163839
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −0.502924 −0.0194152
\(672\) 0 0
\(673\) −16.2911 −0.627975 −0.313987 0.949427i \(-0.601665\pi\)
−0.313987 + 0.949427i \(0.601665\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 4.61803 0.177485 0.0887426 0.996055i \(-0.471715\pi\)
0.0887426 + 0.996055i \(0.471715\pi\)
\(678\) 0 0
\(679\) − 13.8042i − 0.529757i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 30.9791i 1.18538i 0.805429 + 0.592692i \(0.201935\pi\)
−0.805429 + 0.592692i \(0.798065\pi\)
\(684\) 0 0
\(685\) − 10.2319i − 0.390940i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 13.5065i 0.514556i
\(690\) 0 0
\(691\) 29.0382 1.10467 0.552333 0.833624i \(-0.313738\pi\)
0.552333 + 0.833624i \(0.313738\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −8.11534 −0.307832
\(696\) 0 0
\(697\) 2.18923 0.0829228
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 27.6010 1.04248 0.521238 0.853411i \(-0.325470\pi\)
0.521238 + 0.853411i \(0.325470\pi\)
\(702\) 0 0
\(703\) 0.0194989i 0 0.000735413i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 5.41639i − 0.203704i
\(708\) 0 0
\(709\) − 37.8212i − 1.42040i −0.703998 0.710202i \(-0.748604\pi\)
0.703998 0.710202i \(-0.251396\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 2.72588i 0.102085i
\(714\) 0 0
\(715\) 2.10341 0.0786631
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −41.2974 −1.54013 −0.770067 0.637963i \(-0.779777\pi\)
−0.770067 + 0.637963i \(0.779777\pi\)
\(720\) 0 0
\(721\) −2.14390 −0.0798432
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 25.6350 0.952061
\(726\) 0 0
\(727\) − 11.9168i − 0.441971i −0.975277 0.220985i \(-0.929073\pi\)
0.975277 0.220985i \(-0.0709273\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0.325482i 0.0120384i
\(732\) 0 0
\(733\) − 47.2142i − 1.74390i −0.489597 0.871949i \(-0.662856\pi\)
0.489597 0.871949i \(-0.337144\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 10.4512i − 0.384976i
\(738\) 0 0
\(739\) 13.4529 0.494874 0.247437 0.968904i \(-0.420412\pi\)
0.247437 + 0.968904i \(0.420412\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 10.7698 0.395106 0.197553 0.980292i \(-0.436701\pi\)
0.197553 + 0.980292i \(0.436701\pi\)
\(744\) 0 0
\(745\) −0.815220 −0.0298673
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −9.10325 −0.332626
\(750\) 0 0
\(751\) − 14.1004i − 0.514530i −0.966341 0.257265i \(-0.917179\pi\)
0.966341 0.257265i \(-0.0828213\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 5.36986i − 0.195429i
\(756\) 0 0
\(757\) 13.8374i 0.502929i 0.967866 + 0.251465i \(0.0809122\pi\)
−0.967866 + 0.251465i \(0.919088\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 33.1367i − 1.20121i −0.799548 0.600603i \(-0.794927\pi\)
0.799548 0.600603i \(-0.205073\pi\)
\(762\) 0 0
\(763\) 1.29567 0.0469066
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −25.1335 −0.907517
\(768\) 0 0
\(769\) 30.1190 1.08612 0.543060 0.839694i \(-0.317266\pi\)
0.543060 + 0.839694i \(0.317266\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 16.0579 0.577561 0.288780 0.957395i \(-0.406750\pi\)
0.288780 + 0.957395i \(0.406750\pi\)
\(774\) 0 0
\(775\) − 16.5143i − 0.593212i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0.00636936i 0 0.000228206i
\(780\) 0 0
\(781\) 14.4862i 0.518357i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 1.38691i − 0.0495008i
\(786\) 0 0
\(787\) −11.2848 −0.402258 −0.201129 0.979565i \(-0.564461\pi\)
−0.201129 + 0.979565i \(0.564461\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 14.0728 0.500371
\(792\) 0 0
\(793\) 2.43587 0.0865002
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −39.3545 −1.39401 −0.697004 0.717068i \(-0.745484\pi\)
−0.697004 + 0.717068i \(0.745484\pi\)
\(798\) 0 0
\(799\) − 4.71524i − 0.166813i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 11.9074i − 0.420202i
\(804\) 0 0
\(805\) 0.412891i 0.0145525i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 0.183323i − 0.00644529i −0.999995 0.00322265i \(-0.998974\pi\)
0.999995 0.00322265i \(-0.00102580\pi\)
\(810\) 0 0
\(811\) 46.7136 1.64033 0.820167 0.572124i \(-0.193880\pi\)
0.820167 + 0.572124i \(0.193880\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −7.66508 −0.268496
\(816\) 0 0
\(817\) −0.000946960 0 −3.31299e−5 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −19.7408 −0.688959 −0.344480 0.938794i \(-0.611945\pi\)
−0.344480 + 0.938794i \(0.611945\pi\)
\(822\) 0 0
\(823\) 11.6881i 0.407421i 0.979031 + 0.203711i \(0.0653002\pi\)
−0.979031 + 0.203711i \(0.934700\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 30.8036i 1.07115i 0.844489 + 0.535573i \(0.179904\pi\)
−0.844489 + 0.535573i \(0.820096\pi\)
\(828\) 0 0
\(829\) 33.2395i 1.15446i 0.816583 + 0.577228i \(0.195866\pi\)
−0.816583 + 0.577228i \(0.804134\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 1.25430i − 0.0434588i
\(834\) 0 0
\(835\) 4.05564 0.140351
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 42.9568 1.48303 0.741517 0.670934i \(-0.234107\pi\)
0.741517 + 0.670934i \(0.234107\pi\)
\(840\) 0 0
\(841\) 0.509663 0.0175746
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −3.29670 −0.113410
\(846\) 0 0
\(847\) 10.1807i 0.349813i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 4.16199i − 0.142671i
\(852\) 0 0
\(853\) − 31.5917i − 1.08168i −0.841126 0.540840i \(-0.818107\pi\)
0.841126 0.540840i \(-0.181893\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 37.4806i 1.28031i 0.768245 + 0.640156i \(0.221130\pi\)
−0.768245 + 0.640156i \(0.778870\pi\)
\(858\) 0 0
\(859\) 41.4525 1.41434 0.707171 0.707042i \(-0.249971\pi\)
0.707171 + 0.707042i \(0.249971\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −43.8203 −1.49166 −0.745830 0.666136i \(-0.767947\pi\)
−0.745830 + 0.666136i \(0.767947\pi\)
\(864\) 0 0
\(865\) 5.29356 0.179987
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 1.78213 0.0604545
\(870\) 0 0
\(871\) 50.6195i 1.71518i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 5.15181i − 0.174163i
\(876\) 0 0
\(877\) − 33.3035i − 1.12458i −0.826940 0.562289i \(-0.809921\pi\)
0.826940 0.562289i \(-0.190079\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 35.6627i 1.20151i 0.799434 + 0.600754i \(0.205133\pi\)
−0.799434 + 0.600754i \(0.794867\pi\)
\(882\) 0 0
\(883\) −8.32201 −0.280058 −0.140029 0.990147i \(-0.544720\pi\)
−0.140029 + 0.990147i \(0.544720\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 43.3551 1.45572 0.727861 0.685725i \(-0.240515\pi\)
0.727861 + 0.685725i \(0.240515\pi\)
\(888\) 0 0
\(889\) −11.5015 −0.385749
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0.0137186 0.000459075 0
\(894\) 0 0
\(895\) 8.54178i 0.285520i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 19.0104i − 0.634032i
\(900\) 0 0
\(901\) − 3.86433i − 0.128739i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 10.8176i − 0.359590i
\(906\) 0 0
\(907\) 7.89354 0.262101 0.131050 0.991376i \(-0.458165\pi\)
0.131050 + 0.991376i \(0.458165\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −47.9212 −1.58770 −0.793850 0.608113i \(-0.791927\pi\)
−0.793850 + 0.608113i \(0.791927\pi\)
\(912\) 0 0
\(913\) −5.27302 −0.174511
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −10.2828 −0.339567
\(918\) 0 0
\(919\) − 22.2192i − 0.732943i −0.930429 0.366471i \(-0.880566\pi\)
0.930429 0.366471i \(-0.119434\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 70.1627i − 2.30943i
\(924\) 0 0
\(925\) 25.2148i 0.829057i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 34.2658i 1.12422i 0.827061 + 0.562112i \(0.190011\pi\)
−0.827061 + 0.562112i \(0.809989\pi\)
\(930\) 0 0
\(931\) 0.00364927 0.000119600 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −0.601805 −0.0196811
\(936\) 0 0
\(937\) 5.43172 0.177447 0.0887233 0.996056i \(-0.471721\pi\)
0.0887233 + 0.996056i \(0.471721\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 18.1273 0.590932 0.295466 0.955353i \(-0.404525\pi\)
0.295466 + 0.955353i \(0.404525\pi\)
\(942\) 0 0
\(943\) − 1.35953i − 0.0442723i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 11.9799i − 0.389295i −0.980873 0.194647i \(-0.937644\pi\)
0.980873 0.194647i \(-0.0623562\pi\)
\(948\) 0 0
\(949\) 57.6723i 1.87212i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 22.7578i 0.737197i 0.929589 + 0.368599i \(0.120162\pi\)
−0.929589 + 0.368599i \(0.879838\pi\)
\(954\) 0 0
\(955\) −2.14959 −0.0695592
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 19.3027 0.623316
\(960\) 0 0
\(961\) 18.7533 0.604946
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −3.08602 −0.0993425
\(966\) 0 0
\(967\) 41.9989i 1.35060i 0.737545 + 0.675298i \(0.235985\pi\)
−0.737545 + 0.675298i \(0.764015\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 36.5508i 1.17297i 0.809959 + 0.586486i \(0.199489\pi\)
−0.809959 + 0.586486i \(0.800511\pi\)
\(972\) 0 0
\(973\) − 15.3098i − 0.490809i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 30.6094i − 0.979281i −0.871925 0.489640i \(-0.837128\pi\)
0.871925 0.489640i \(-0.162872\pi\)
\(978\) 0 0
\(979\) −1.76042 −0.0562633
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 36.1930 1.15438 0.577189 0.816610i \(-0.304150\pi\)
0.577189 + 0.816610i \(0.304150\pi\)
\(984\) 0 0
\(985\) −8.19270 −0.261041
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0.202127 0.00642726
\(990\) 0 0
\(991\) 52.1154i 1.65550i 0.561097 + 0.827750i \(0.310379\pi\)
−0.561097 + 0.827750i \(0.689621\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 7.25159i − 0.229891i
\(996\) 0 0
\(997\) 20.2906i 0.642609i 0.946976 + 0.321305i \(0.104121\pi\)
−0.946976 + 0.321305i \(0.895879\pi\)
\(998\) 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 6048.2.j.d.5615.27 48
3.2 odd 2 inner 6048.2.j.d.5615.22 48
4.3 odd 2 1512.2.j.d.323.6 yes 48
8.3 odd 2 inner 6048.2.j.d.5615.21 48
8.5 even 2 1512.2.j.d.323.44 yes 48
12.11 even 2 1512.2.j.d.323.43 yes 48
24.5 odd 2 1512.2.j.d.323.5 48
24.11 even 2 inner 6048.2.j.d.5615.28 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.j.d.323.5 48 24.5 odd 2
1512.2.j.d.323.6 yes 48 4.3 odd 2
1512.2.j.d.323.43 yes 48 12.11 even 2
1512.2.j.d.323.44 yes 48 8.5 even 2
6048.2.j.d.5615.21 48 8.3 odd 2 inner
6048.2.j.d.5615.22 48 3.2 odd 2 inner
6048.2.j.d.5615.27 48 1.1 even 1 trivial
6048.2.j.d.5615.28 48 24.11 even 2 inner