Properties

Label 4840.2.a.bd.1.4
Level $4840$
Weight $2$
Character 4840.1
Self dual yes
Analytic conductor $38.648$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4840,2,Mod(1,4840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4840, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4840.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4840 = 2^{3} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4840.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(38.6475945783\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.22733568.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 8x^{4} - 2x^{3} + 16x^{2} + 8x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(1.90131\) of defining polynomial
Character \(\chi\) \(=\) 4840.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.614975 q^{3} +1.00000 q^{5} +2.24598 q^{7} -2.62181 q^{9} +O(q^{10})\) \(q+0.614975 q^{3} +1.00000 q^{5} +2.24598 q^{7} -2.62181 q^{9} -5.19447 q^{13} +0.614975 q^{15} +3.22995 q^{17} +7.19447 q^{19} +1.38122 q^{21} +5.11377 q^{23} +1.00000 q^{25} -3.45727 q^{27} -9.61890 q^{29} +0.491468 q^{31} +2.24598 q^{35} +0.350828 q^{37} -3.19447 q^{39} +4.42090 q^{41} +10.9537 q^{43} -2.62181 q^{45} -1.03420 q^{47} -1.95557 q^{49} +1.98634 q^{51} +3.75348 q^{53} +4.42442 q^{57} +12.7082 q^{59} +6.38983 q^{61} -5.88852 q^{63} -5.19447 q^{65} -8.44020 q^{67} +3.14484 q^{69} -12.4475 q^{71} +12.0273 q^{73} +0.614975 q^{75} -2.13034 q^{79} +5.73928 q^{81} +10.7270 q^{83} +3.22995 q^{85} -5.91538 q^{87} +9.90118 q^{89} -11.6667 q^{91} +0.302241 q^{93} +7.19447 q^{95} -15.9253 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{3} + 6 q^{5} + 4 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{3} + 6 q^{5} + 4 q^{7} + 4 q^{9} - 2 q^{15} + 8 q^{17} + 12 q^{19} - 8 q^{21} - 8 q^{23} + 6 q^{25} - 14 q^{27} + 16 q^{29} - 4 q^{31} + 4 q^{35} + 8 q^{37} + 12 q^{39} + 32 q^{41} - 4 q^{43} + 4 q^{45} - 6 q^{47} + 16 q^{49} + 40 q^{51} + 8 q^{53} - 16 q^{57} + 4 q^{59} + 16 q^{61} + 28 q^{63} - 2 q^{67} + 8 q^{69} - 28 q^{71} + 16 q^{73} - 2 q^{75} - 10 q^{81} + 12 q^{83} + 8 q^{85} - 24 q^{87} + 18 q^{89} - 24 q^{91} + 20 q^{93} + 12 q^{95}+O(q^{100}) \) 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
\(3\) 0.614975 0.355056 0.177528 0.984116i \(-0.443190\pi\)
0.177528 + 0.984116i \(0.443190\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 2.24598 0.848901 0.424450 0.905451i \(-0.360467\pi\)
0.424450 + 0.905451i \(0.360467\pi\)
\(8\) 0 0
\(9\) −2.62181 −0.873935
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) −5.19447 −1.44069 −0.720344 0.693617i \(-0.756016\pi\)
−0.720344 + 0.693617i \(0.756016\pi\)
\(14\) 0 0
\(15\) 0.614975 0.158786
\(16\) 0 0
\(17\) 3.22995 0.783378 0.391689 0.920098i \(-0.371891\pi\)
0.391689 + 0.920098i \(0.371891\pi\)
\(18\) 0 0
\(19\) 7.19447 1.65053 0.825263 0.564749i \(-0.191027\pi\)
0.825263 + 0.564749i \(0.191027\pi\)
\(20\) 0 0
\(21\) 1.38122 0.301408
\(22\) 0 0
\(23\) 5.11377 1.06629 0.533147 0.846023i \(-0.321009\pi\)
0.533147 + 0.846023i \(0.321009\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −3.45727 −0.665352
\(28\) 0 0
\(29\) −9.61890 −1.78618 −0.893092 0.449874i \(-0.851469\pi\)
−0.893092 + 0.449874i \(0.851469\pi\)
\(30\) 0 0
\(31\) 0.491468 0.0882703 0.0441352 0.999026i \(-0.485947\pi\)
0.0441352 + 0.999026i \(0.485947\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.24598 0.379640
\(36\) 0 0
\(37\) 0.350828 0.0576758 0.0288379 0.999584i \(-0.490819\pi\)
0.0288379 + 0.999584i \(0.490819\pi\)
\(38\) 0 0
\(39\) −3.19447 −0.511525
\(40\) 0 0
\(41\) 4.42090 0.690429 0.345214 0.938524i \(-0.387806\pi\)
0.345214 + 0.938524i \(0.387806\pi\)
\(42\) 0 0
\(43\) 10.9537 1.67042 0.835211 0.549929i \(-0.185345\pi\)
0.835211 + 0.549929i \(0.185345\pi\)
\(44\) 0 0
\(45\) −2.62181 −0.390836
\(46\) 0 0
\(47\) −1.03420 −0.150853 −0.0754265 0.997151i \(-0.524032\pi\)
−0.0754265 + 0.997151i \(0.524032\pi\)
\(48\) 0 0
\(49\) −1.95557 −0.279367
\(50\) 0 0
\(51\) 1.98634 0.278143
\(52\) 0 0
\(53\) 3.75348 0.515580 0.257790 0.966201i \(-0.417006\pi\)
0.257790 + 0.966201i \(0.417006\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4.42442 0.586029
\(58\) 0 0
\(59\) 12.7082 1.65447 0.827234 0.561858i \(-0.189913\pi\)
0.827234 + 0.561858i \(0.189913\pi\)
\(60\) 0 0
\(61\) 6.38983 0.818134 0.409067 0.912504i \(-0.365854\pi\)
0.409067 + 0.912504i \(0.365854\pi\)
\(62\) 0 0
\(63\) −5.88852 −0.741884
\(64\) 0 0
\(65\) −5.19447 −0.644295
\(66\) 0 0
\(67\) −8.44020 −1.03114 −0.515568 0.856849i \(-0.672419\pi\)
−0.515568 + 0.856849i \(0.672419\pi\)
\(68\) 0 0
\(69\) 3.14484 0.378594
\(70\) 0 0
\(71\) −12.4475 −1.47725 −0.738625 0.674116i \(-0.764525\pi\)
−0.738625 + 0.674116i \(0.764525\pi\)
\(72\) 0 0
\(73\) 12.0273 1.40769 0.703846 0.710353i \(-0.251465\pi\)
0.703846 + 0.710353i \(0.251465\pi\)
\(74\) 0 0
\(75\) 0.614975 0.0710112
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −2.13034 −0.239682 −0.119841 0.992793i \(-0.538238\pi\)
−0.119841 + 0.992793i \(0.538238\pi\)
\(80\) 0 0
\(81\) 5.73928 0.637698
\(82\) 0 0
\(83\) 10.7270 1.17744 0.588719 0.808338i \(-0.299633\pi\)
0.588719 + 0.808338i \(0.299633\pi\)
\(84\) 0 0
\(85\) 3.22995 0.350337
\(86\) 0 0
\(87\) −5.91538 −0.634196
\(88\) 0 0
\(89\) 9.90118 1.04952 0.524762 0.851249i \(-0.324154\pi\)
0.524762 + 0.851249i \(0.324154\pi\)
\(90\) 0 0
\(91\) −11.6667 −1.22300
\(92\) 0 0
\(93\) 0.302241 0.0313409
\(94\) 0 0
\(95\) 7.19447 0.738137
\(96\) 0 0
\(97\) −15.9253 −1.61697 −0.808484 0.588518i \(-0.799712\pi\)
−0.808484 + 0.588518i \(0.799712\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 16.8726 1.67889 0.839445 0.543444i \(-0.182880\pi\)
0.839445 + 0.543444i \(0.182880\pi\)
\(102\) 0 0
\(103\) 7.05805 0.695450 0.347725 0.937597i \(-0.386954\pi\)
0.347725 + 0.937597i \(0.386954\pi\)
\(104\) 0 0
\(105\) 1.38122 0.134794
\(106\) 0 0
\(107\) 7.64175 0.738756 0.369378 0.929279i \(-0.379571\pi\)
0.369378 + 0.929279i \(0.379571\pi\)
\(108\) 0 0
\(109\) −11.2590 −1.07841 −0.539206 0.842174i \(-0.681276\pi\)
−0.539206 + 0.842174i \(0.681276\pi\)
\(110\) 0 0
\(111\) 0.215751 0.0204781
\(112\) 0 0
\(113\) −0.783709 −0.0737252 −0.0368626 0.999320i \(-0.511736\pi\)
−0.0368626 + 0.999320i \(0.511736\pi\)
\(114\) 0 0
\(115\) 5.11377 0.476861
\(116\) 0 0
\(117\) 13.6189 1.25907
\(118\) 0 0
\(119\) 7.25441 0.665010
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 2.71875 0.245141
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 8.20546 0.728117 0.364059 0.931376i \(-0.381391\pi\)
0.364059 + 0.931376i \(0.381391\pi\)
\(128\) 0 0
\(129\) 6.73625 0.593094
\(130\) 0 0
\(131\) 6.35603 0.555329 0.277664 0.960678i \(-0.410440\pi\)
0.277664 + 0.960678i \(0.410440\pi\)
\(132\) 0 0
\(133\) 16.1586 1.40113
\(134\) 0 0
\(135\) −3.45727 −0.297555
\(136\) 0 0
\(137\) −8.27612 −0.707077 −0.353538 0.935420i \(-0.615022\pi\)
−0.353538 + 0.935420i \(0.615022\pi\)
\(138\) 0 0
\(139\) 11.0778 0.939606 0.469803 0.882771i \(-0.344325\pi\)
0.469803 + 0.882771i \(0.344325\pi\)
\(140\) 0 0
\(141\) −0.636005 −0.0535613
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −9.61890 −0.798806
\(146\) 0 0
\(147\) −1.20263 −0.0991910
\(148\) 0 0
\(149\) 8.46662 0.693613 0.346806 0.937937i \(-0.387266\pi\)
0.346806 + 0.937937i \(0.387266\pi\)
\(150\) 0 0
\(151\) 6.39661 0.520548 0.260274 0.965535i \(-0.416187\pi\)
0.260274 + 0.965535i \(0.416187\pi\)
\(152\) 0 0
\(153\) −8.46830 −0.684622
\(154\) 0 0
\(155\) 0.491468 0.0394757
\(156\) 0 0
\(157\) 16.0273 1.27912 0.639560 0.768741i \(-0.279117\pi\)
0.639560 + 0.768741i \(0.279117\pi\)
\(158\) 0 0
\(159\) 2.30830 0.183060
\(160\) 0 0
\(161\) 11.4854 0.905178
\(162\) 0 0
\(163\) −10.1970 −0.798694 −0.399347 0.916800i \(-0.630763\pi\)
−0.399347 + 0.916800i \(0.630763\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −13.3145 −1.03031 −0.515154 0.857097i \(-0.672265\pi\)
−0.515154 + 0.857097i \(0.672265\pi\)
\(168\) 0 0
\(169\) 13.9825 1.07558
\(170\) 0 0
\(171\) −18.8625 −1.44245
\(172\) 0 0
\(173\) 18.7177 1.42308 0.711538 0.702647i \(-0.247999\pi\)
0.711538 + 0.702647i \(0.247999\pi\)
\(174\) 0 0
\(175\) 2.24598 0.169780
\(176\) 0 0
\(177\) 7.81523 0.587429
\(178\) 0 0
\(179\) −19.5287 −1.45965 −0.729823 0.683636i \(-0.760398\pi\)
−0.729823 + 0.683636i \(0.760398\pi\)
\(180\) 0 0
\(181\) 6.39136 0.475066 0.237533 0.971379i \(-0.423661\pi\)
0.237533 + 0.971379i \(0.423661\pi\)
\(182\) 0 0
\(183\) 3.92959 0.290483
\(184\) 0 0
\(185\) 0.350828 0.0257934
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −7.76497 −0.564818
\(190\) 0 0
\(191\) −6.58718 −0.476632 −0.238316 0.971188i \(-0.576595\pi\)
−0.238316 + 0.971188i \(0.576595\pi\)
\(192\) 0 0
\(193\) −7.20967 −0.518964 −0.259482 0.965748i \(-0.583552\pi\)
−0.259482 + 0.965748i \(0.583552\pi\)
\(194\) 0 0
\(195\) −3.19447 −0.228761
\(196\) 0 0
\(197\) −1.87253 −0.133412 −0.0667060 0.997773i \(-0.521249\pi\)
−0.0667060 + 0.997773i \(0.521249\pi\)
\(198\) 0 0
\(199\) 0.607258 0.0430474 0.0215237 0.999768i \(-0.493148\pi\)
0.0215237 + 0.999768i \(0.493148\pi\)
\(200\) 0 0
\(201\) −5.19052 −0.366111
\(202\) 0 0
\(203\) −21.6039 −1.51629
\(204\) 0 0
\(205\) 4.42090 0.308769
\(206\) 0 0
\(207\) −13.4073 −0.931872
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 12.1496 0.836412 0.418206 0.908352i \(-0.362659\pi\)
0.418206 + 0.908352i \(0.362659\pi\)
\(212\) 0 0
\(213\) −7.65492 −0.524507
\(214\) 0 0
\(215\) 10.9537 0.747036
\(216\) 0 0
\(217\) 1.10383 0.0749328
\(218\) 0 0
\(219\) 7.39651 0.499809
\(220\) 0 0
\(221\) −16.7779 −1.12860
\(222\) 0 0
\(223\) 0.918013 0.0614747 0.0307374 0.999527i \(-0.490214\pi\)
0.0307374 + 0.999527i \(0.490214\pi\)
\(224\) 0 0
\(225\) −2.62181 −0.174787
\(226\) 0 0
\(227\) −21.9513 −1.45696 −0.728479 0.685068i \(-0.759772\pi\)
−0.728479 + 0.685068i \(0.759772\pi\)
\(228\) 0 0
\(229\) −21.9249 −1.44884 −0.724421 0.689358i \(-0.757893\pi\)
−0.724421 + 0.689358i \(0.757893\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 11.2543 0.737297 0.368648 0.929569i \(-0.379821\pi\)
0.368648 + 0.929569i \(0.379821\pi\)
\(234\) 0 0
\(235\) −1.03420 −0.0674636
\(236\) 0 0
\(237\) −1.31010 −0.0851004
\(238\) 0 0
\(239\) −15.5108 −1.00331 −0.501654 0.865068i \(-0.667275\pi\)
−0.501654 + 0.865068i \(0.667275\pi\)
\(240\) 0 0
\(241\) −18.7929 −1.21056 −0.605280 0.796013i \(-0.706939\pi\)
−0.605280 + 0.796013i \(0.706939\pi\)
\(242\) 0 0
\(243\) 13.9013 0.891771
\(244\) 0 0
\(245\) −1.95557 −0.124937
\(246\) 0 0
\(247\) −37.3715 −2.37789
\(248\) 0 0
\(249\) 6.59682 0.418057
\(250\) 0 0
\(251\) −4.84499 −0.305813 −0.152907 0.988241i \(-0.548863\pi\)
−0.152907 + 0.988241i \(0.548863\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 1.98634 0.124389
\(256\) 0 0
\(257\) 21.6015 1.34746 0.673732 0.738976i \(-0.264690\pi\)
0.673732 + 0.738976i \(0.264690\pi\)
\(258\) 0 0
\(259\) 0.787953 0.0489610
\(260\) 0 0
\(261\) 25.2189 1.56101
\(262\) 0 0
\(263\) −16.2554 −1.00235 −0.501176 0.865346i \(-0.667099\pi\)
−0.501176 + 0.865346i \(0.667099\pi\)
\(264\) 0 0
\(265\) 3.75348 0.230574
\(266\) 0 0
\(267\) 6.08898 0.372640
\(268\) 0 0
\(269\) −20.1959 −1.23137 −0.615684 0.787993i \(-0.711120\pi\)
−0.615684 + 0.787993i \(0.711120\pi\)
\(270\) 0 0
\(271\) 23.5368 1.42976 0.714881 0.699246i \(-0.246481\pi\)
0.714881 + 0.699246i \(0.246481\pi\)
\(272\) 0 0
\(273\) −7.17472 −0.434234
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 2.75817 0.165723 0.0828613 0.996561i \(-0.473594\pi\)
0.0828613 + 0.996561i \(0.473594\pi\)
\(278\) 0 0
\(279\) −1.28853 −0.0771425
\(280\) 0 0
\(281\) −9.55050 −0.569735 −0.284868 0.958567i \(-0.591950\pi\)
−0.284868 + 0.958567i \(0.591950\pi\)
\(282\) 0 0
\(283\) 2.48859 0.147931 0.0739657 0.997261i \(-0.476434\pi\)
0.0739657 + 0.997261i \(0.476434\pi\)
\(284\) 0 0
\(285\) 4.42442 0.262080
\(286\) 0 0
\(287\) 9.92926 0.586106
\(288\) 0 0
\(289\) −6.56742 −0.386319
\(290\) 0 0
\(291\) −9.79366 −0.574115
\(292\) 0 0
\(293\) 7.01604 0.409882 0.204941 0.978774i \(-0.434300\pi\)
0.204941 + 0.978774i \(0.434300\pi\)
\(294\) 0 0
\(295\) 12.7082 0.739900
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −26.5633 −1.53620
\(300\) 0 0
\(301\) 24.6018 1.41802
\(302\) 0 0
\(303\) 10.3763 0.596101
\(304\) 0 0
\(305\) 6.38983 0.365880
\(306\) 0 0
\(307\) 15.7618 0.899576 0.449788 0.893135i \(-0.351500\pi\)
0.449788 + 0.893135i \(0.351500\pi\)
\(308\) 0 0
\(309\) 4.34052 0.246924
\(310\) 0 0
\(311\) 2.37143 0.134472 0.0672358 0.997737i \(-0.478582\pi\)
0.0672358 + 0.997737i \(0.478582\pi\)
\(312\) 0 0
\(313\) 22.8728 1.29285 0.646424 0.762979i \(-0.276264\pi\)
0.646424 + 0.762979i \(0.276264\pi\)
\(314\) 0 0
\(315\) −5.88852 −0.331781
\(316\) 0 0
\(317\) 16.1104 0.904848 0.452424 0.891803i \(-0.350559\pi\)
0.452424 + 0.891803i \(0.350559\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 4.69949 0.262300
\(322\) 0 0
\(323\) 23.2378 1.29299
\(324\) 0 0
\(325\) −5.19447 −0.288137
\(326\) 0 0
\(327\) −6.92398 −0.382897
\(328\) 0 0
\(329\) −2.32279 −0.128059
\(330\) 0 0
\(331\) 26.8231 1.47433 0.737166 0.675712i \(-0.236164\pi\)
0.737166 + 0.675712i \(0.236164\pi\)
\(332\) 0 0
\(333\) −0.919803 −0.0504049
\(334\) 0 0
\(335\) −8.44020 −0.461138
\(336\) 0 0
\(337\) −26.6484 −1.45163 −0.725816 0.687889i \(-0.758537\pi\)
−0.725816 + 0.687889i \(0.758537\pi\)
\(338\) 0 0
\(339\) −0.481962 −0.0261766
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −20.1140 −1.08606
\(344\) 0 0
\(345\) 3.14484 0.169313
\(346\) 0 0
\(347\) 17.2459 0.925810 0.462905 0.886408i \(-0.346807\pi\)
0.462905 + 0.886408i \(0.346807\pi\)
\(348\) 0 0
\(349\) −7.51222 −0.402120 −0.201060 0.979579i \(-0.564439\pi\)
−0.201060 + 0.979579i \(0.564439\pi\)
\(350\) 0 0
\(351\) 17.9587 0.958565
\(352\) 0 0
\(353\) 12.1232 0.645254 0.322627 0.946526i \(-0.395434\pi\)
0.322627 + 0.946526i \(0.395434\pi\)
\(354\) 0 0
\(355\) −12.4475 −0.660646
\(356\) 0 0
\(357\) 4.46128 0.236116
\(358\) 0 0
\(359\) 15.8372 0.835853 0.417926 0.908481i \(-0.362757\pi\)
0.417926 + 0.908481i \(0.362757\pi\)
\(360\) 0 0
\(361\) 32.7604 1.72423
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 12.0273 0.629539
\(366\) 0 0
\(367\) 23.4664 1.22494 0.612468 0.790495i \(-0.290177\pi\)
0.612468 + 0.790495i \(0.290177\pi\)
\(368\) 0 0
\(369\) −11.5907 −0.603390
\(370\) 0 0
\(371\) 8.43024 0.437676
\(372\) 0 0
\(373\) −17.3749 −0.899639 −0.449819 0.893120i \(-0.648512\pi\)
−0.449819 + 0.893120i \(0.648512\pi\)
\(374\) 0 0
\(375\) 0.614975 0.0317572
\(376\) 0 0
\(377\) 49.9651 2.57333
\(378\) 0 0
\(379\) −35.7198 −1.83480 −0.917400 0.397966i \(-0.869716\pi\)
−0.917400 + 0.397966i \(0.869716\pi\)
\(380\) 0 0
\(381\) 5.04616 0.258522
\(382\) 0 0
\(383\) 14.8706 0.759852 0.379926 0.925017i \(-0.375949\pi\)
0.379926 + 0.925017i \(0.375949\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −28.7185 −1.45984
\(388\) 0 0
\(389\) −15.2397 −0.772684 −0.386342 0.922356i \(-0.626262\pi\)
−0.386342 + 0.922356i \(0.626262\pi\)
\(390\) 0 0
\(391\) 16.5172 0.835311
\(392\) 0 0
\(393\) 3.90880 0.197173
\(394\) 0 0
\(395\) −2.13034 −0.107189
\(396\) 0 0
\(397\) 17.8780 0.897271 0.448636 0.893715i \(-0.351910\pi\)
0.448636 + 0.893715i \(0.351910\pi\)
\(398\) 0 0
\(399\) 9.93717 0.497481
\(400\) 0 0
\(401\) 24.6977 1.23334 0.616672 0.787220i \(-0.288481\pi\)
0.616672 + 0.787220i \(0.288481\pi\)
\(402\) 0 0
\(403\) −2.55292 −0.127170
\(404\) 0 0
\(405\) 5.73928 0.285187
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 37.4699 1.85277 0.926383 0.376583i \(-0.122901\pi\)
0.926383 + 0.376583i \(0.122901\pi\)
\(410\) 0 0
\(411\) −5.08961 −0.251052
\(412\) 0 0
\(413\) 28.5424 1.40448
\(414\) 0 0
\(415\) 10.7270 0.526566
\(416\) 0 0
\(417\) 6.81257 0.333613
\(418\) 0 0
\(419\) −6.28017 −0.306807 −0.153403 0.988164i \(-0.549023\pi\)
−0.153403 + 0.988164i \(0.549023\pi\)
\(420\) 0 0
\(421\) −24.2587 −1.18229 −0.591147 0.806563i \(-0.701325\pi\)
−0.591147 + 0.806563i \(0.701325\pi\)
\(422\) 0 0
\(423\) 2.71146 0.131836
\(424\) 0 0
\(425\) 3.22995 0.156676
\(426\) 0 0
\(427\) 14.3514 0.694514
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 14.0321 0.675901 0.337950 0.941164i \(-0.390266\pi\)
0.337950 + 0.941164i \(0.390266\pi\)
\(432\) 0 0
\(433\) −24.3312 −1.16928 −0.584640 0.811293i \(-0.698764\pi\)
−0.584640 + 0.811293i \(0.698764\pi\)
\(434\) 0 0
\(435\) −5.91538 −0.283621
\(436\) 0 0
\(437\) 36.7909 1.75995
\(438\) 0 0
\(439\) −6.52928 −0.311626 −0.155813 0.987787i \(-0.549800\pi\)
−0.155813 + 0.987787i \(0.549800\pi\)
\(440\) 0 0
\(441\) 5.12712 0.244149
\(442\) 0 0
\(443\) 0.922170 0.0438136 0.0219068 0.999760i \(-0.493026\pi\)
0.0219068 + 0.999760i \(0.493026\pi\)
\(444\) 0 0
\(445\) 9.90118 0.469361
\(446\) 0 0
\(447\) 5.20676 0.246271
\(448\) 0 0
\(449\) −8.48757 −0.400553 −0.200277 0.979739i \(-0.564184\pi\)
−0.200277 + 0.979739i \(0.564184\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 3.93376 0.184824
\(454\) 0 0
\(455\) −11.6667 −0.546943
\(456\) 0 0
\(457\) −15.5158 −0.725799 −0.362900 0.931828i \(-0.618213\pi\)
−0.362900 + 0.931828i \(0.618213\pi\)
\(458\) 0 0
\(459\) −11.1668 −0.521222
\(460\) 0 0
\(461\) −26.9388 −1.25466 −0.627332 0.778752i \(-0.715853\pi\)
−0.627332 + 0.778752i \(0.715853\pi\)
\(462\) 0 0
\(463\) −33.9970 −1.57997 −0.789986 0.613125i \(-0.789912\pi\)
−0.789986 + 0.613125i \(0.789912\pi\)
\(464\) 0 0
\(465\) 0.302241 0.0140161
\(466\) 0 0
\(467\) 4.26684 0.197446 0.0987230 0.995115i \(-0.468524\pi\)
0.0987230 + 0.995115i \(0.468524\pi\)
\(468\) 0 0
\(469\) −18.9565 −0.875332
\(470\) 0 0
\(471\) 9.85641 0.454159
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 7.19447 0.330105
\(476\) 0 0
\(477\) −9.84089 −0.450584
\(478\) 0 0
\(479\) 2.84891 0.130170 0.0650850 0.997880i \(-0.479268\pi\)
0.0650850 + 0.997880i \(0.479268\pi\)
\(480\) 0 0
\(481\) −1.82237 −0.0830927
\(482\) 0 0
\(483\) 7.06325 0.321389
\(484\) 0 0
\(485\) −15.9253 −0.723130
\(486\) 0 0
\(487\) −39.5054 −1.79016 −0.895080 0.445905i \(-0.852882\pi\)
−0.895080 + 0.445905i \(0.852882\pi\)
\(488\) 0 0
\(489\) −6.27093 −0.283581
\(490\) 0 0
\(491\) 38.7458 1.74857 0.874287 0.485409i \(-0.161329\pi\)
0.874287 + 0.485409i \(0.161329\pi\)
\(492\) 0 0
\(493\) −31.0686 −1.39926
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −27.9569 −1.25404
\(498\) 0 0
\(499\) 12.0103 0.537654 0.268827 0.963189i \(-0.413364\pi\)
0.268827 + 0.963189i \(0.413364\pi\)
\(500\) 0 0
\(501\) −8.18810 −0.365817
\(502\) 0 0
\(503\) −26.7888 −1.19445 −0.597226 0.802073i \(-0.703731\pi\)
−0.597226 + 0.802073i \(0.703731\pi\)
\(504\) 0 0
\(505\) 16.8726 0.750823
\(506\) 0 0
\(507\) 8.59892 0.381891
\(508\) 0 0
\(509\) 42.4779 1.88280 0.941400 0.337294i \(-0.109511\pi\)
0.941400 + 0.337294i \(0.109511\pi\)
\(510\) 0 0
\(511\) 27.0131 1.19499
\(512\) 0 0
\(513\) −24.8732 −1.09818
\(514\) 0 0
\(515\) 7.05805 0.311015
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 11.5109 0.505272
\(520\) 0 0
\(521\) −17.0296 −0.746080 −0.373040 0.927815i \(-0.621685\pi\)
−0.373040 + 0.927815i \(0.621685\pi\)
\(522\) 0 0
\(523\) −44.4082 −1.94184 −0.970918 0.239413i \(-0.923045\pi\)
−0.970918 + 0.239413i \(0.923045\pi\)
\(524\) 0 0
\(525\) 1.38122 0.0602815
\(526\) 0 0
\(527\) 1.58742 0.0691490
\(528\) 0 0
\(529\) 3.15061 0.136983
\(530\) 0 0
\(531\) −33.3184 −1.44590
\(532\) 0 0
\(533\) −22.9643 −0.994692
\(534\) 0 0
\(535\) 7.64175 0.330382
\(536\) 0 0
\(537\) −12.0097 −0.518257
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 4.25636 0.182995 0.0914975 0.995805i \(-0.470835\pi\)
0.0914975 + 0.995805i \(0.470835\pi\)
\(542\) 0 0
\(543\) 3.93053 0.168675
\(544\) 0 0
\(545\) −11.2590 −0.482280
\(546\) 0 0
\(547\) 15.7213 0.672194 0.336097 0.941827i \(-0.390893\pi\)
0.336097 + 0.941827i \(0.390893\pi\)
\(548\) 0 0
\(549\) −16.7529 −0.714996
\(550\) 0 0
\(551\) −69.2029 −2.94814
\(552\) 0 0
\(553\) −4.78470 −0.203466
\(554\) 0 0
\(555\) 0.215751 0.00915810
\(556\) 0 0
\(557\) 4.21547 0.178615 0.0893076 0.996004i \(-0.471535\pi\)
0.0893076 + 0.996004i \(0.471535\pi\)
\(558\) 0 0
\(559\) −56.8987 −2.40656
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −12.7102 −0.535672 −0.267836 0.963464i \(-0.586309\pi\)
−0.267836 + 0.963464i \(0.586309\pi\)
\(564\) 0 0
\(565\) −0.783709 −0.0329709
\(566\) 0 0
\(567\) 12.8903 0.541342
\(568\) 0 0
\(569\) 23.4920 0.984837 0.492418 0.870359i \(-0.336113\pi\)
0.492418 + 0.870359i \(0.336113\pi\)
\(570\) 0 0
\(571\) 21.4826 0.899019 0.449510 0.893276i \(-0.351599\pi\)
0.449510 + 0.893276i \(0.351599\pi\)
\(572\) 0 0
\(573\) −4.05095 −0.169231
\(574\) 0 0
\(575\) 5.11377 0.213259
\(576\) 0 0
\(577\) −5.57447 −0.232068 −0.116034 0.993245i \(-0.537018\pi\)
−0.116034 + 0.993245i \(0.537018\pi\)
\(578\) 0 0
\(579\) −4.43377 −0.184261
\(580\) 0 0
\(581\) 24.0926 0.999528
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 13.6189 0.563072
\(586\) 0 0
\(587\) −39.6780 −1.63769 −0.818843 0.574017i \(-0.805384\pi\)
−0.818843 + 0.574017i \(0.805384\pi\)
\(588\) 0 0
\(589\) 3.53586 0.145692
\(590\) 0 0
\(591\) −1.15156 −0.0473687
\(592\) 0 0
\(593\) 6.42867 0.263994 0.131997 0.991250i \(-0.457861\pi\)
0.131997 + 0.991250i \(0.457861\pi\)
\(594\) 0 0
\(595\) 7.25441 0.297402
\(596\) 0 0
\(597\) 0.373449 0.0152842
\(598\) 0 0
\(599\) 25.7321 1.05139 0.525693 0.850674i \(-0.323806\pi\)
0.525693 + 0.850674i \(0.323806\pi\)
\(600\) 0 0
\(601\) 21.9690 0.896136 0.448068 0.894000i \(-0.352112\pi\)
0.448068 + 0.894000i \(0.352112\pi\)
\(602\) 0 0
\(603\) 22.1286 0.901145
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −1.29521 −0.0525709 −0.0262855 0.999654i \(-0.508368\pi\)
−0.0262855 + 0.999654i \(0.508368\pi\)
\(608\) 0 0
\(609\) −13.2858 −0.538369
\(610\) 0 0
\(611\) 5.37211 0.217332
\(612\) 0 0
\(613\) 1.08300 0.0437418 0.0218709 0.999761i \(-0.493038\pi\)
0.0218709 + 0.999761i \(0.493038\pi\)
\(614\) 0 0
\(615\) 2.71875 0.109630
\(616\) 0 0
\(617\) −12.3102 −0.495592 −0.247796 0.968812i \(-0.579706\pi\)
−0.247796 + 0.968812i \(0.579706\pi\)
\(618\) 0 0
\(619\) 31.9610 1.28462 0.642311 0.766444i \(-0.277976\pi\)
0.642311 + 0.766444i \(0.277976\pi\)
\(620\) 0 0
\(621\) −17.6797 −0.709461
\(622\) 0 0
\(623\) 22.2379 0.890941
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 1.13316 0.0451819
\(630\) 0 0
\(631\) −1.80177 −0.0717272 −0.0358636 0.999357i \(-0.511418\pi\)
−0.0358636 + 0.999357i \(0.511418\pi\)
\(632\) 0 0
\(633\) 7.47170 0.296973
\(634\) 0 0
\(635\) 8.20546 0.325624
\(636\) 0 0
\(637\) 10.1582 0.402481
\(638\) 0 0
\(639\) 32.6350 1.29102
\(640\) 0 0
\(641\) −25.5940 −1.01090 −0.505452 0.862855i \(-0.668674\pi\)
−0.505452 + 0.862855i \(0.668674\pi\)
\(642\) 0 0
\(643\) −16.8113 −0.662974 −0.331487 0.943460i \(-0.607550\pi\)
−0.331487 + 0.943460i \(0.607550\pi\)
\(644\) 0 0
\(645\) 6.73625 0.265240
\(646\) 0 0
\(647\) −33.9970 −1.33656 −0.668279 0.743911i \(-0.732969\pi\)
−0.668279 + 0.743911i \(0.732969\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0.678827 0.0266053
\(652\) 0 0
\(653\) 42.9973 1.68262 0.841308 0.540557i \(-0.181786\pi\)
0.841308 + 0.540557i \(0.181786\pi\)
\(654\) 0 0
\(655\) 6.35603 0.248351
\(656\) 0 0
\(657\) −31.5333 −1.23023
\(658\) 0 0
\(659\) 29.3748 1.14428 0.572141 0.820155i \(-0.306113\pi\)
0.572141 + 0.820155i \(0.306113\pi\)
\(660\) 0 0
\(661\) 23.7553 0.923975 0.461988 0.886886i \(-0.347136\pi\)
0.461988 + 0.886886i \(0.347136\pi\)
\(662\) 0 0
\(663\) −10.3180 −0.400717
\(664\) 0 0
\(665\) 16.1586 0.626605
\(666\) 0 0
\(667\) −49.1888 −1.90460
\(668\) 0 0
\(669\) 0.564555 0.0218270
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −16.5964 −0.639745 −0.319873 0.947461i \(-0.603640\pi\)
−0.319873 + 0.947461i \(0.603640\pi\)
\(674\) 0 0
\(675\) −3.45727 −0.133070
\(676\) 0 0
\(677\) 14.6950 0.564773 0.282387 0.959301i \(-0.408874\pi\)
0.282387 + 0.959301i \(0.408874\pi\)
\(678\) 0 0
\(679\) −35.7679 −1.37265
\(680\) 0 0
\(681\) −13.4995 −0.517302
\(682\) 0 0
\(683\) −42.2066 −1.61499 −0.807495 0.589874i \(-0.799177\pi\)
−0.807495 + 0.589874i \(0.799177\pi\)
\(684\) 0 0
\(685\) −8.27612 −0.316214
\(686\) 0 0
\(687\) −13.4833 −0.514420
\(688\) 0 0
\(689\) −19.4973 −0.742790
\(690\) 0 0
\(691\) 48.6795 1.85186 0.925928 0.377700i \(-0.123285\pi\)
0.925928 + 0.377700i \(0.123285\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 11.0778 0.420205
\(696\) 0 0
\(697\) 14.2793 0.540867
\(698\) 0 0
\(699\) 6.92115 0.261782
\(700\) 0 0
\(701\) 0.580332 0.0219188 0.0109594 0.999940i \(-0.496511\pi\)
0.0109594 + 0.999940i \(0.496511\pi\)
\(702\) 0 0
\(703\) 2.52402 0.0951953
\(704\) 0 0
\(705\) −0.636005 −0.0239534
\(706\) 0 0
\(707\) 37.8956 1.42521
\(708\) 0 0
\(709\) 40.9372 1.53743 0.768714 0.639593i \(-0.220897\pi\)
0.768714 + 0.639593i \(0.220897\pi\)
\(710\) 0 0
\(711\) 5.58533 0.209466
\(712\) 0 0
\(713\) 2.51325 0.0941221
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −9.53875 −0.356231
\(718\) 0 0
\(719\) −10.0377 −0.374344 −0.187172 0.982327i \(-0.559932\pi\)
−0.187172 + 0.982327i \(0.559932\pi\)
\(720\) 0 0
\(721\) 15.8522 0.590368
\(722\) 0 0
\(723\) −11.5572 −0.429816
\(724\) 0 0
\(725\) −9.61890 −0.357237
\(726\) 0 0
\(727\) −22.1551 −0.821687 −0.410844 0.911706i \(-0.634766\pi\)
−0.410844 + 0.911706i \(0.634766\pi\)
\(728\) 0 0
\(729\) −8.66886 −0.321069
\(730\) 0 0
\(731\) 35.3799 1.30857
\(732\) 0 0
\(733\) −43.5669 −1.60918 −0.804589 0.593832i \(-0.797615\pi\)
−0.804589 + 0.593832i \(0.797615\pi\)
\(734\) 0 0
\(735\) −1.20263 −0.0443596
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −46.7110 −1.71829 −0.859146 0.511731i \(-0.829005\pi\)
−0.859146 + 0.511731i \(0.829005\pi\)
\(740\) 0 0
\(741\) −22.9825 −0.844285
\(742\) 0 0
\(743\) −33.3576 −1.22377 −0.611887 0.790945i \(-0.709589\pi\)
−0.611887 + 0.790945i \(0.709589\pi\)
\(744\) 0 0
\(745\) 8.46662 0.310193
\(746\) 0 0
\(747\) −28.1240 −1.02900
\(748\) 0 0
\(749\) 17.1632 0.627130
\(750\) 0 0
\(751\) −24.0038 −0.875910 −0.437955 0.898997i \(-0.644297\pi\)
−0.437955 + 0.898997i \(0.644297\pi\)
\(752\) 0 0
\(753\) −2.97955 −0.108581
\(754\) 0 0
\(755\) 6.39661 0.232796
\(756\) 0 0
\(757\) −23.0230 −0.836786 −0.418393 0.908266i \(-0.637407\pi\)
−0.418393 + 0.908266i \(0.637407\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 6.87356 0.249166 0.124583 0.992209i \(-0.460241\pi\)
0.124583 + 0.992209i \(0.460241\pi\)
\(762\) 0 0
\(763\) −25.2874 −0.915465
\(764\) 0 0
\(765\) −8.46830 −0.306172
\(766\) 0 0
\(767\) −66.0124 −2.38357
\(768\) 0 0
\(769\) −24.6838 −0.890120 −0.445060 0.895501i \(-0.646818\pi\)
−0.445060 + 0.895501i \(0.646818\pi\)
\(770\) 0 0
\(771\) 13.2844 0.478425
\(772\) 0 0
\(773\) 15.2591 0.548830 0.274415 0.961611i \(-0.411516\pi\)
0.274415 + 0.961611i \(0.411516\pi\)
\(774\) 0 0
\(775\) 0.491468 0.0176541
\(776\) 0 0
\(777\) 0.484572 0.0173839
\(778\) 0 0
\(779\) 31.8061 1.13957
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 33.2551 1.18844
\(784\) 0 0
\(785\) 16.0273 0.572040
\(786\) 0 0
\(787\) 25.9701 0.925735 0.462868 0.886427i \(-0.346820\pi\)
0.462868 + 0.886427i \(0.346820\pi\)
\(788\) 0 0
\(789\) −9.99668 −0.355891
\(790\) 0 0
\(791\) −1.76020 −0.0625854
\(792\) 0 0
\(793\) −33.1918 −1.17867
\(794\) 0 0
\(795\) 2.30830 0.0818669
\(796\) 0 0
\(797\) 19.1403 0.677984 0.338992 0.940789i \(-0.389914\pi\)
0.338992 + 0.940789i \(0.389914\pi\)
\(798\) 0 0
\(799\) −3.34040 −0.118175
\(800\) 0 0
\(801\) −25.9590 −0.917215
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 11.4854 0.404808
\(806\) 0 0
\(807\) −12.4200 −0.437205
\(808\) 0 0
\(809\) 19.9636 0.701883 0.350941 0.936397i \(-0.385862\pi\)
0.350941 + 0.936397i \(0.385862\pi\)
\(810\) 0 0
\(811\) −33.4667 −1.17518 −0.587588 0.809160i \(-0.699922\pi\)
−0.587588 + 0.809160i \(0.699922\pi\)
\(812\) 0 0
\(813\) 14.4746 0.507646
\(814\) 0 0
\(815\) −10.1970 −0.357187
\(816\) 0 0
\(817\) 78.8060 2.75707
\(818\) 0 0
\(819\) 30.5878 1.06882
\(820\) 0 0
\(821\) −13.6778 −0.477358 −0.238679 0.971099i \(-0.576714\pi\)
−0.238679 + 0.971099i \(0.576714\pi\)
\(822\) 0 0
\(823\) −47.5149 −1.65627 −0.828133 0.560532i \(-0.810597\pi\)
−0.828133 + 0.560532i \(0.810597\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −3.58722 −0.124740 −0.0623699 0.998053i \(-0.519866\pi\)
−0.0623699 + 0.998053i \(0.519866\pi\)
\(828\) 0 0
\(829\) −37.2896 −1.29512 −0.647561 0.762014i \(-0.724211\pi\)
−0.647561 + 0.762014i \(0.724211\pi\)
\(830\) 0 0
\(831\) 1.69621 0.0588408
\(832\) 0 0
\(833\) −6.31639 −0.218850
\(834\) 0 0
\(835\) −13.3145 −0.460768
\(836\) 0 0
\(837\) −1.69914 −0.0587308
\(838\) 0 0
\(839\) 37.2210 1.28501 0.642507 0.766280i \(-0.277895\pi\)
0.642507 + 0.766280i \(0.277895\pi\)
\(840\) 0 0
\(841\) 63.5231 2.19045
\(842\) 0 0
\(843\) −5.87332 −0.202288
\(844\) 0 0
\(845\) 13.9825 0.481014
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 1.53042 0.0525239
\(850\) 0 0
\(851\) 1.79405 0.0614993
\(852\) 0 0
\(853\) −19.3642 −0.663017 −0.331508 0.943452i \(-0.607558\pi\)
−0.331508 + 0.943452i \(0.607558\pi\)
\(854\) 0 0
\(855\) −18.8625 −0.645084
\(856\) 0 0
\(857\) 0.844366 0.0288430 0.0144215 0.999896i \(-0.495409\pi\)
0.0144215 + 0.999896i \(0.495409\pi\)
\(858\) 0 0
\(859\) 21.8675 0.746110 0.373055 0.927809i \(-0.378310\pi\)
0.373055 + 0.927809i \(0.378310\pi\)
\(860\) 0 0
\(861\) 6.10625 0.208100
\(862\) 0 0
\(863\) −4.99125 −0.169904 −0.0849520 0.996385i \(-0.527074\pi\)
−0.0849520 + 0.996385i \(0.527074\pi\)
\(864\) 0 0
\(865\) 18.7177 0.636419
\(866\) 0 0
\(867\) −4.03880 −0.137165
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 43.8424 1.48554
\(872\) 0 0
\(873\) 41.7530 1.41313
\(874\) 0 0
\(875\) 2.24598 0.0759280
\(876\) 0 0
\(877\) −54.7015 −1.84714 −0.923570 0.383430i \(-0.874743\pi\)
−0.923570 + 0.383430i \(0.874743\pi\)
\(878\) 0 0
\(879\) 4.31469 0.145531
\(880\) 0 0
\(881\) −13.7463 −0.463125 −0.231562 0.972820i \(-0.574384\pi\)
−0.231562 + 0.972820i \(0.574384\pi\)
\(882\) 0 0
\(883\) −23.6026 −0.794290 −0.397145 0.917756i \(-0.629999\pi\)
−0.397145 + 0.917756i \(0.629999\pi\)
\(884\) 0 0
\(885\) 7.81523 0.262706
\(886\) 0 0
\(887\) −35.4265 −1.18950 −0.594752 0.803909i \(-0.702750\pi\)
−0.594752 + 0.803909i \(0.702750\pi\)
\(888\) 0 0
\(889\) 18.4293 0.618099
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −7.44050 −0.248987
\(894\) 0 0
\(895\) −19.5287 −0.652774
\(896\) 0 0
\(897\) −16.3358 −0.545436
\(898\) 0 0
\(899\) −4.72738 −0.157667
\(900\) 0 0
\(901\) 12.1236 0.403894
\(902\) 0 0
\(903\) 15.1295 0.503478
\(904\) 0 0
\(905\) 6.39136 0.212456
\(906\) 0 0
\(907\) −13.6049 −0.451742 −0.225871 0.974157i \(-0.572523\pi\)
−0.225871 + 0.974157i \(0.572523\pi\)
\(908\) 0 0
\(909\) −44.2368 −1.46724
\(910\) 0 0
\(911\) 35.4052 1.17303 0.586514 0.809939i \(-0.300500\pi\)
0.586514 + 0.809939i \(0.300500\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 3.92959 0.129908
\(916\) 0 0
\(917\) 14.2755 0.471419
\(918\) 0 0
\(919\) −33.9654 −1.12042 −0.560208 0.828352i \(-0.689279\pi\)
−0.560208 + 0.828352i \(0.689279\pi\)
\(920\) 0 0
\(921\) 9.69315 0.319400
\(922\) 0 0
\(923\) 64.6584 2.12826
\(924\) 0 0
\(925\) 0.350828 0.0115352
\(926\) 0 0
\(927\) −18.5048 −0.607778
\(928\) 0 0
\(929\) −32.5351 −1.06744 −0.533722 0.845660i \(-0.679207\pi\)
−0.533722 + 0.845660i \(0.679207\pi\)
\(930\) 0 0
\(931\) −14.0693 −0.461102
\(932\) 0 0
\(933\) 1.45837 0.0477449
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −46.0027 −1.50284 −0.751421 0.659823i \(-0.770631\pi\)
−0.751421 + 0.659823i \(0.770631\pi\)
\(938\) 0 0
\(939\) 14.0662 0.459034
\(940\) 0 0
\(941\) −20.3541 −0.663524 −0.331762 0.943363i \(-0.607643\pi\)
−0.331762 + 0.943363i \(0.607643\pi\)
\(942\) 0 0
\(943\) 22.6075 0.736200
\(944\) 0 0
\(945\) −7.76497 −0.252594
\(946\) 0 0
\(947\) 15.6423 0.508308 0.254154 0.967164i \(-0.418203\pi\)
0.254154 + 0.967164i \(0.418203\pi\)
\(948\) 0 0
\(949\) −62.4756 −2.02804
\(950\) 0 0
\(951\) 9.90747 0.321272
\(952\) 0 0
\(953\) −34.4420 −1.11569 −0.557843 0.829946i \(-0.688371\pi\)
−0.557843 + 0.829946i \(0.688371\pi\)
\(954\) 0 0
\(955\) −6.58718 −0.213156
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −18.5880 −0.600238
\(960\) 0 0
\(961\) −30.7585 −0.992208
\(962\) 0 0
\(963\) −20.0352 −0.645624
\(964\) 0 0
\(965\) −7.20967 −0.232088
\(966\) 0 0
\(967\) −20.3210 −0.653479 −0.326739 0.945114i \(-0.605950\pi\)
−0.326739 + 0.945114i \(0.605950\pi\)
\(968\) 0 0
\(969\) 14.2907 0.459082
\(970\) 0 0
\(971\) −60.6592 −1.94665 −0.973323 0.229440i \(-0.926311\pi\)
−0.973323 + 0.229440i \(0.926311\pi\)
\(972\) 0 0
\(973\) 24.8805 0.797633
\(974\) 0 0
\(975\) −3.19447 −0.102305
\(976\) 0 0
\(977\) −27.9407 −0.893903 −0.446952 0.894558i \(-0.647490\pi\)
−0.446952 + 0.894558i \(0.647490\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 29.5188 0.942462
\(982\) 0 0
\(983\) −14.8724 −0.474356 −0.237178 0.971466i \(-0.576222\pi\)
−0.237178 + 0.971466i \(0.576222\pi\)
\(984\) 0 0
\(985\) −1.87253 −0.0596636
\(986\) 0 0
\(987\) −1.42846 −0.0454683
\(988\) 0 0
\(989\) 56.0146 1.78116
\(990\) 0 0
\(991\) −54.1090 −1.71883 −0.859414 0.511280i \(-0.829172\pi\)
−0.859414 + 0.511280i \(0.829172\pi\)
\(992\) 0 0
\(993\) 16.4956 0.523471
\(994\) 0 0
\(995\) 0.607258 0.0192514
\(996\) 0 0
\(997\) −25.8227 −0.817814 −0.408907 0.912576i \(-0.634090\pi\)
−0.408907 + 0.912576i \(0.634090\pi\)
\(998\) 0 0
\(999\) −1.21291 −0.0383747
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 4840.2.a.bd.1.4 yes 6
4.3 odd 2 9680.2.a.da.1.3 6
11.10 odd 2 4840.2.a.bc.1.4 6
44.43 even 2 9680.2.a.db.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4840.2.a.bc.1.4 6 11.10 odd 2
4840.2.a.bd.1.4 yes 6 1.1 even 1 trivial
9680.2.a.da.1.3 6 4.3 odd 2
9680.2.a.db.1.3 6 44.43 even 2