Properties

Label 5292.2.x.b.881.3
Level $5292$
Weight $2$
Character 5292.881
Analytic conductor $42.257$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5292,2,Mod(881,5292)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5292, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5292.881");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5292 = 2^{2} \cdot 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5292.x (of order \(6\), degree \(2\), 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: \(42.2568327497\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} + 5 x^{14} - 17 x^{13} + 22 x^{12} - 31 x^{11} + 62 x^{10} - 52 x^{9} + 52 x^{8} + \cdots + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 3^{6} \)
Twist minimal: no (minimal twist has level 252)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 881.3
Root \(-0.268067 - 1.71118i\) of defining polynomial
Character \(\chi\) \(=\) 5292.881
Dual form 5292.2.x.b.4409.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.842869 + 1.45989i) q^{5} +(3.38216 - 1.95269i) q^{11} +(5.24391 + 3.02757i) q^{13} +0.402488 q^{17} -0.168144i q^{19} +(7.69373 + 4.44198i) q^{23} +(1.07914 + 1.86913i) q^{25} +(6.15380 - 3.55290i) q^{29} +(-5.44527 - 3.14383i) q^{31} -6.26515 q^{37} +(1.64707 - 2.85281i) q^{41} +(1.80474 + 3.12590i) q^{43} +(-4.38482 - 7.59474i) q^{47} -5.71148i q^{53} +6.58345i q^{55} +(-2.25163 + 3.89994i) q^{59} +(-4.43678 + 2.56157i) q^{61} +(-8.83986 + 5.10369i) q^{65} +(2.95521 - 5.11857i) q^{67} -11.4308i q^{71} +6.99239i q^{73} +(-0.603968 - 1.04610i) q^{79} +(0.181350 + 0.314108i) q^{83} +(-0.339244 + 0.587588i) q^{85} +2.77052 q^{89} +(0.245471 + 0.141723i) q^{95} +(0.508914 - 0.293821i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 6 q^{11} + 3 q^{13} - 18 q^{17} + 21 q^{23} - 8 q^{25} - 6 q^{29} - 6 q^{31} - 2 q^{37} - 6 q^{41} - 2 q^{43} + 18 q^{47} + 15 q^{59} - 3 q^{61} - 39 q^{65} - 7 q^{67} - q^{79} + 6 q^{85} - 42 q^{89}+ \cdots + 3 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5292\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\) \(2647\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(-1\) \(1\)

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
\(5\) −0.842869 + 1.45989i −0.376942 + 0.652883i −0.990616 0.136677i \(-0.956358\pi\)
0.613673 + 0.789560i \(0.289691\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.38216 1.95269i 1.01976 0.588758i 0.105725 0.994395i \(-0.466284\pi\)
0.914034 + 0.405637i \(0.132950\pi\)
\(12\) 0 0
\(13\) 5.24391 + 3.02757i 1.45440 + 0.839698i 0.998727 0.0504496i \(-0.0160654\pi\)
0.455673 + 0.890147i \(0.349399\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.402488 0.0976176 0.0488088 0.998808i \(-0.484458\pi\)
0.0488088 + 0.998808i \(0.484458\pi\)
\(18\) 0 0
\(19\) 0.168144i 0.0385748i −0.999814 0.0192874i \(-0.993860\pi\)
0.999814 0.0192874i \(-0.00613975\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 7.69373 + 4.44198i 1.60425 + 0.926216i 0.990623 + 0.136623i \(0.0436248\pi\)
0.613630 + 0.789593i \(0.289709\pi\)
\(24\) 0 0
\(25\) 1.07914 + 1.86913i 0.215829 + 0.373827i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.15380 3.55290i 1.14273 0.659757i 0.195627 0.980678i \(-0.437326\pi\)
0.947106 + 0.320921i \(0.103993\pi\)
\(30\) 0 0
\(31\) −5.44527 3.14383i −0.978000 0.564649i −0.0763342 0.997082i \(-0.524322\pi\)
−0.901666 + 0.432434i \(0.857655\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −6.26515 −1.02998 −0.514992 0.857195i \(-0.672205\pi\)
−0.514992 + 0.857195i \(0.672205\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.64707 2.85281i 0.257229 0.445534i −0.708269 0.705942i \(-0.750524\pi\)
0.965499 + 0.260408i \(0.0838571\pi\)
\(42\) 0 0
\(43\) 1.80474 + 3.12590i 0.275220 + 0.476695i 0.970191 0.242343i \(-0.0779161\pi\)
−0.694971 + 0.719038i \(0.744583\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.38482 7.59474i −0.639592 1.10781i −0.985522 0.169546i \(-0.945770\pi\)
0.345930 0.938260i \(-0.387563\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 5.71148i 0.784532i −0.919852 0.392266i \(-0.871691\pi\)
0.919852 0.392266i \(-0.128309\pi\)
\(54\) 0 0
\(55\) 6.58345i 0.887712i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −2.25163 + 3.89994i −0.293138 + 0.507729i −0.974550 0.224171i \(-0.928033\pi\)
0.681412 + 0.731900i \(0.261366\pi\)
\(60\) 0 0
\(61\) −4.43678 + 2.56157i −0.568071 + 0.327976i −0.756379 0.654134i \(-0.773033\pi\)
0.188308 + 0.982110i \(0.439700\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −8.83986 + 5.10369i −1.09645 + 0.633035i
\(66\) 0 0
\(67\) 2.95521 5.11857i 0.361036 0.625332i −0.627096 0.778942i \(-0.715757\pi\)
0.988132 + 0.153610i \(0.0490899\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 11.4308i 1.35658i −0.734792 0.678292i \(-0.762720\pi\)
0.734792 0.678292i \(-0.237280\pi\)
\(72\) 0 0
\(73\) 6.99239i 0.818397i 0.912445 + 0.409199i \(0.134192\pi\)
−0.912445 + 0.409199i \(0.865808\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −0.603968 1.04610i −0.0679517 0.117696i 0.830048 0.557692i \(-0.188313\pi\)
−0.898000 + 0.439996i \(0.854980\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0.181350 + 0.314108i 0.0199058 + 0.0344779i 0.875807 0.482662i \(-0.160330\pi\)
−0.855901 + 0.517140i \(0.826997\pi\)
\(84\) 0 0
\(85\) −0.339244 + 0.587588i −0.0367962 + 0.0637329i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2.77052 0.293674 0.146837 0.989161i \(-0.453091\pi\)
0.146837 + 0.989161i \(0.453091\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0.245471 + 0.141723i 0.0251848 + 0.0145405i
\(96\) 0 0
\(97\) 0.508914 0.293821i 0.0516723 0.0298330i −0.473941 0.880556i \(-0.657169\pi\)
0.525614 + 0.850723i \(0.323836\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 6.92329 + 11.9915i 0.688893 + 1.19320i 0.972196 + 0.234167i \(0.0752364\pi\)
−0.283303 + 0.959030i \(0.591430\pi\)
\(102\) 0 0
\(103\) 10.4610 + 6.03967i 1.03075 + 0.595106i 0.917201 0.398425i \(-0.130443\pi\)
0.113554 + 0.993532i \(0.463777\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 18.3942i 1.77824i −0.457678 0.889118i \(-0.651319\pi\)
0.457678 0.889118i \(-0.348681\pi\)
\(108\) 0 0
\(109\) 11.0207 1.05559 0.527796 0.849371i \(-0.323018\pi\)
0.527796 + 0.849371i \(0.323018\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 7.36811 + 4.25398i 0.693133 + 0.400181i 0.804785 0.593567i \(-0.202281\pi\)
−0.111652 + 0.993747i \(0.535614\pi\)
\(114\) 0 0
\(115\) −12.9696 + 7.48801i −1.20942 + 0.698260i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 2.12600 3.68234i 0.193273 0.334758i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −12.0670 −1.07930
\(126\) 0 0
\(127\) −10.6312 −0.943365 −0.471682 0.881769i \(-0.656353\pi\)
−0.471682 + 0.881769i \(0.656353\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 3.16740 5.48610i 0.276737 0.479322i −0.693835 0.720134i \(-0.744080\pi\)
0.970572 + 0.240812i \(0.0774136\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 14.4158 8.32296i 1.23162 0.711078i 0.264255 0.964453i \(-0.414874\pi\)
0.967368 + 0.253375i \(0.0815406\pi\)
\(138\) 0 0
\(139\) −4.24007 2.44800i −0.359638 0.207637i 0.309284 0.950970i \(-0.399911\pi\)
−0.668922 + 0.743333i \(0.733244\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 23.6477 1.97752
\(144\) 0 0
\(145\) 11.9785i 0.994761i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −4.57864 2.64348i −0.375097 0.216562i 0.300586 0.953755i \(-0.402818\pi\)
−0.675683 + 0.737192i \(0.736151\pi\)
\(150\) 0 0
\(151\) 7.29163 + 12.6295i 0.593385 + 1.02777i 0.993773 + 0.111427i \(0.0355421\pi\)
−0.400388 + 0.916346i \(0.631125\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 9.17930 5.29967i 0.737299 0.425680i
\(156\) 0 0
\(157\) 15.4160 + 8.90044i 1.23033 + 0.710332i 0.967099 0.254400i \(-0.0818781\pi\)
0.263232 + 0.964732i \(0.415211\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −0.0964456 −0.00755420 −0.00377710 0.999993i \(-0.501202\pi\)
−0.00377710 + 0.999993i \(0.501202\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2.47872 4.29327i 0.191809 0.332224i −0.754041 0.656828i \(-0.771898\pi\)
0.945850 + 0.324604i \(0.105231\pi\)
\(168\) 0 0
\(169\) 11.8324 + 20.4943i 0.910185 + 1.57649i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 7.40033 + 12.8177i 0.562637 + 0.974515i 0.997265 + 0.0739055i \(0.0235463\pi\)
−0.434629 + 0.900610i \(0.643120\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0.684450i 0.0511582i 0.999673 + 0.0255791i \(0.00814297\pi\)
−0.999673 + 0.0255791i \(0.991857\pi\)
\(180\) 0 0
\(181\) 7.84745i 0.583297i −0.956526 0.291648i \(-0.905796\pi\)
0.956526 0.291648i \(-0.0942037\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 5.28070 9.14644i 0.388245 0.672459i
\(186\) 0 0
\(187\) 1.36128 0.785934i 0.0995464 0.0574732i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −16.9694 + 9.79729i −1.22786 + 0.708907i −0.966582 0.256356i \(-0.917478\pi\)
−0.261281 + 0.965263i \(0.584145\pi\)
\(192\) 0 0
\(193\) −9.18116 + 15.9022i −0.660875 + 1.14467i 0.319512 + 0.947582i \(0.396481\pi\)
−0.980386 + 0.197086i \(0.936852\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 5.92313i 0.422006i −0.977485 0.211003i \(-0.932327\pi\)
0.977485 0.211003i \(-0.0676730\pi\)
\(198\) 0 0
\(199\) 15.7348i 1.11541i 0.830039 + 0.557706i \(0.188318\pi\)
−0.830039 + 0.557706i \(0.811682\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 2.77653 + 4.80909i 0.193921 + 0.335881i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −0.328332 0.568688i −0.0227112 0.0393370i
\(210\) 0 0
\(211\) 5.06619 8.77489i 0.348771 0.604088i −0.637261 0.770648i \(-0.719933\pi\)
0.986031 + 0.166560i \(0.0532659\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −6.08462 −0.414968
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 2.11061 + 1.21856i 0.141975 + 0.0819693i
\(222\) 0 0
\(223\) −13.3944 + 7.73325i −0.896955 + 0.517857i −0.876211 0.481928i \(-0.839937\pi\)
−0.0207437 + 0.999785i \(0.506603\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 14.0360 + 24.3110i 0.931600 + 1.61358i 0.780588 + 0.625046i \(0.214920\pi\)
0.151011 + 0.988532i \(0.451747\pi\)
\(228\) 0 0
\(229\) −14.7453 8.51319i −0.974396 0.562568i −0.0738222 0.997271i \(-0.523520\pi\)
−0.900573 + 0.434704i \(0.856853\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 18.4769i 1.21047i 0.796049 + 0.605233i \(0.206920\pi\)
−0.796049 + 0.605233i \(0.793080\pi\)
\(234\) 0 0
\(235\) 14.7833 0.964358
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 6.06656 + 3.50253i 0.392413 + 0.226560i 0.683205 0.730226i \(-0.260585\pi\)
−0.290792 + 0.956786i \(0.593919\pi\)
\(240\) 0 0
\(241\) 5.38459 3.10879i 0.346852 0.200255i −0.316446 0.948611i \(-0.602490\pi\)
0.663298 + 0.748355i \(0.269156\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0.509067 0.881730i 0.0323912 0.0561031i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 9.81844 0.619734 0.309867 0.950780i \(-0.399715\pi\)
0.309867 + 0.950780i \(0.399715\pi\)
\(252\) 0 0
\(253\) 34.6952 2.18127
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −0.667904 + 1.15684i −0.0416627 + 0.0721619i −0.886105 0.463485i \(-0.846599\pi\)
0.844442 + 0.535647i \(0.179932\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 17.6238 10.1751i 1.08673 0.627424i 0.154026 0.988067i \(-0.450776\pi\)
0.932704 + 0.360643i \(0.117443\pi\)
\(264\) 0 0
\(265\) 8.33814 + 4.81402i 0.512208 + 0.295723i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −26.7228 −1.62932 −0.814659 0.579940i \(-0.803076\pi\)
−0.814659 + 0.579940i \(0.803076\pi\)
\(270\) 0 0
\(271\) 4.34764i 0.264100i 0.991243 + 0.132050i \(0.0421560\pi\)
−0.991243 + 0.132050i \(0.957844\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 7.29968 + 4.21447i 0.440187 + 0.254142i
\(276\) 0 0
\(277\) 2.19901 + 3.80880i 0.132126 + 0.228849i 0.924496 0.381192i \(-0.124486\pi\)
−0.792370 + 0.610041i \(0.791153\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −4.62273 + 2.66893i −0.275769 + 0.159215i −0.631506 0.775371i \(-0.717563\pi\)
0.355738 + 0.934586i \(0.384230\pi\)
\(282\) 0 0
\(283\) 15.5431 + 8.97381i 0.923941 + 0.533437i 0.884890 0.465800i \(-0.154233\pi\)
0.0390505 + 0.999237i \(0.487567\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −16.8380 −0.990471
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −13.1126 + 22.7117i −0.766048 + 1.32683i 0.173642 + 0.984809i \(0.444446\pi\)
−0.939691 + 0.342026i \(0.888887\pi\)
\(294\) 0 0
\(295\) −3.79566 6.57428i −0.220992 0.382769i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 26.8968 + 46.5867i 1.55548 + 2.69418i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 8.63628i 0.494512i
\(306\) 0 0
\(307\) 7.19520i 0.410652i −0.978694 0.205326i \(-0.934175\pi\)
0.978694 0.205326i \(-0.0658254\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −1.08721 + 1.88311i −0.0616503 + 0.106781i −0.895203 0.445658i \(-0.852970\pi\)
0.833553 + 0.552440i \(0.186303\pi\)
\(312\) 0 0
\(313\) 10.2870 5.93922i 0.581457 0.335704i −0.180255 0.983620i \(-0.557692\pi\)
0.761712 + 0.647916i \(0.224359\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 7.09969 4.09901i 0.398758 0.230223i −0.287190 0.957874i \(-0.592721\pi\)
0.685948 + 0.727651i \(0.259388\pi\)
\(318\) 0 0
\(319\) 13.8754 24.0329i 0.776875 1.34559i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0.0676757i 0.00376558i
\(324\) 0 0
\(325\) 13.0688i 0.724924i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −8.58540 14.8704i −0.471897 0.817349i 0.527586 0.849501i \(-0.323097\pi\)
−0.999483 + 0.0321526i \(0.989764\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 4.98170 + 8.62856i 0.272179 + 0.471428i
\(336\) 0 0
\(337\) 3.95399 6.84850i 0.215387 0.373062i −0.738005 0.674795i \(-0.764232\pi\)
0.953392 + 0.301733i \(0.0975653\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −24.5557 −1.32977
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −0.443850 0.256257i −0.0238271 0.0137566i 0.488039 0.872822i \(-0.337712\pi\)
−0.511866 + 0.859065i \(0.671046\pi\)
\(348\) 0 0
\(349\) −5.74612 + 3.31752i −0.307583 + 0.177583i −0.645844 0.763469i \(-0.723494\pi\)
0.338262 + 0.941052i \(0.390161\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 9.03437 + 15.6480i 0.480851 + 0.832858i 0.999759 0.0219721i \(-0.00699449\pi\)
−0.518908 + 0.854830i \(0.673661\pi\)
\(354\) 0 0
\(355\) 16.6877 + 9.63465i 0.885691 + 0.511354i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 1.76296i 0.0930453i 0.998917 + 0.0465227i \(0.0148140\pi\)
−0.998917 + 0.0465227i \(0.985186\pi\)
\(360\) 0 0
\(361\) 18.9717 0.998512
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −10.2081 5.89367i −0.534318 0.308489i
\(366\) 0 0
\(367\) −28.9614 + 16.7209i −1.51177 + 0.872822i −0.511867 + 0.859065i \(0.671046\pi\)
−0.999905 + 0.0137576i \(0.995621\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −12.7844 + 22.1433i −0.661952 + 1.14653i 0.318150 + 0.948040i \(0.396938\pi\)
−0.980102 + 0.198494i \(0.936395\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 43.0267 2.21599
\(378\) 0 0
\(379\) 25.7920 1.32485 0.662423 0.749130i \(-0.269528\pi\)
0.662423 + 0.749130i \(0.269528\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 16.4158 28.4330i 0.838808 1.45286i −0.0520838 0.998643i \(-0.516586\pi\)
0.890892 0.454215i \(-0.150080\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −17.4542 + 10.0772i −0.884965 + 0.510935i −0.872292 0.488985i \(-0.837367\pi\)
−0.0126730 + 0.999920i \(0.504034\pi\)
\(390\) 0 0
\(391\) 3.09663 + 1.78784i 0.156603 + 0.0904150i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 2.03626 0.102456
\(396\) 0 0
\(397\) 34.8864i 1.75090i −0.483311 0.875449i \(-0.660566\pi\)
0.483311 0.875449i \(-0.339434\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 8.36793 + 4.83122i 0.417874 + 0.241260i 0.694167 0.719814i \(-0.255773\pi\)
−0.276293 + 0.961073i \(0.589106\pi\)
\(402\) 0 0
\(403\) −19.0364 32.9719i −0.948268 1.64245i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −21.1897 + 12.2339i −1.05034 + 0.606412i
\(408\) 0 0
\(409\) −32.1202 18.5446i −1.58824 0.916973i −0.993597 0.112986i \(-0.963958\pi\)
−0.594647 0.803987i \(-0.702708\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −0.611419 −0.0300134
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 1.84193 3.19031i 0.0899841 0.155857i −0.817520 0.575900i \(-0.804652\pi\)
0.907504 + 0.420043i \(0.137985\pi\)
\(420\) 0 0
\(421\) −8.55139 14.8114i −0.416769 0.721866i 0.578843 0.815439i \(-0.303504\pi\)
−0.995612 + 0.0935732i \(0.970171\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0.434342 + 0.752303i 0.0210687 + 0.0364921i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 31.5512i 1.51977i −0.650058 0.759885i \(-0.725255\pi\)
0.650058 0.759885i \(-0.274745\pi\)
\(432\) 0 0
\(433\) 10.0692i 0.483893i −0.970290 0.241947i \(-0.922214\pi\)
0.970290 0.241947i \(-0.0777859\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0.746890 1.29365i 0.0357286 0.0618837i
\(438\) 0 0
\(439\) −24.1966 + 13.9699i −1.15484 + 0.666748i −0.950062 0.312060i \(-0.898981\pi\)
−0.204779 + 0.978808i \(0.565648\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −30.1930 + 17.4319i −1.43451 + 0.828215i −0.997460 0.0712223i \(-0.977310\pi\)
−0.437050 + 0.899437i \(0.643977\pi\)
\(444\) 0 0
\(445\) −2.33518 + 4.04466i −0.110698 + 0.191735i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 23.2411i 1.09682i −0.836211 0.548408i \(-0.815234\pi\)
0.836211 0.548408i \(-0.184766\pi\)
\(450\) 0 0
\(451\) 12.8649i 0.605783i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 3.10938 + 5.38560i 0.145451 + 0.251928i 0.929541 0.368719i \(-0.120203\pi\)
−0.784090 + 0.620647i \(0.786870\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −2.17165 3.76140i −0.101144 0.175186i 0.811012 0.585029i \(-0.198917\pi\)
−0.912156 + 0.409843i \(0.865584\pi\)
\(462\) 0 0
\(463\) 3.57451 6.19124i 0.166122 0.287731i −0.770931 0.636918i \(-0.780209\pi\)
0.937053 + 0.349187i \(0.113542\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −1.88890 −0.0874080 −0.0437040 0.999045i \(-0.513916\pi\)
−0.0437040 + 0.999045i \(0.513916\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 12.2078 + 7.04818i 0.561316 + 0.324076i
\(474\) 0 0
\(475\) 0.314283 0.181451i 0.0144203 0.00832555i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −5.22491 9.04981i −0.238732 0.413497i 0.721618 0.692291i \(-0.243399\pi\)
−0.960351 + 0.278794i \(0.910065\pi\)
\(480\) 0 0
\(481\) −32.8539 18.9682i −1.49801 0.864875i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0.990611i 0.0449813i
\(486\) 0 0
\(487\) 23.6596 1.07212 0.536060 0.844180i \(-0.319912\pi\)
0.536060 + 0.844180i \(0.319912\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 11.6767 + 6.74152i 0.526960 + 0.304241i 0.739778 0.672851i \(-0.234931\pi\)
−0.212817 + 0.977092i \(0.568264\pi\)
\(492\) 0 0
\(493\) 2.47683 1.43000i 0.111551 0.0644039i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 6.04035 10.4622i 0.270403 0.468352i −0.698562 0.715550i \(-0.746176\pi\)
0.968965 + 0.247197i \(0.0795096\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 20.5283 0.915310 0.457655 0.889130i \(-0.348690\pi\)
0.457655 + 0.889130i \(0.348690\pi\)
\(504\) 0 0
\(505\) −23.3417 −1.03869
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 4.09043 7.08483i 0.181305 0.314029i −0.761020 0.648728i \(-0.775301\pi\)
0.942325 + 0.334699i \(0.108635\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −17.6345 + 10.1813i −0.777070 + 0.448642i
\(516\) 0 0
\(517\) −29.6603 17.1244i −1.30446 0.753131i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −27.7491 −1.21571 −0.607856 0.794048i \(-0.707970\pi\)
−0.607856 + 0.794048i \(0.707970\pi\)
\(522\) 0 0
\(523\) 22.9604i 1.00399i 0.864872 + 0.501993i \(0.167400\pi\)
−0.864872 + 0.501993i \(0.832600\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −2.19166 1.26535i −0.0954700 0.0551196i
\(528\) 0 0
\(529\) 27.9623 + 48.4322i 1.21575 + 2.10575i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 17.2742 9.97325i 0.748228 0.431990i
\(534\) 0 0
\(535\) 26.8536 + 15.5039i 1.16098 + 0.670292i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 5.20810 0.223914 0.111957 0.993713i \(-0.464288\pi\)
0.111957 + 0.993713i \(0.464288\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −9.28902 + 16.0890i −0.397898 + 0.689179i
\(546\) 0 0
\(547\) 10.6224 + 18.3985i 0.454181 + 0.786664i 0.998641 0.0521229i \(-0.0165988\pi\)
−0.544460 + 0.838787i \(0.683265\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −0.597397 1.03472i −0.0254500 0.0440807i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 12.8109i 0.542813i −0.962465 0.271407i \(-0.912511\pi\)
0.962465 0.271407i \(-0.0874888\pi\)
\(558\) 0 0
\(559\) 21.8559i 0.924406i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 18.7396 32.4580i 0.789781 1.36794i −0.136319 0.990665i \(-0.543527\pi\)
0.926101 0.377277i \(-0.123139\pi\)
\(564\) 0 0
\(565\) −12.4207 + 7.17109i −0.522542 + 0.301690i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 5.94906 3.43469i 0.249397 0.143990i −0.370091 0.928996i \(-0.620673\pi\)
0.619488 + 0.785006i \(0.287340\pi\)
\(570\) 0 0
\(571\) −0.0847909 + 0.146862i −0.00354839 + 0.00614599i −0.867794 0.496924i \(-0.834463\pi\)
0.864246 + 0.503070i \(0.167796\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 19.1741i 0.799617i
\(576\) 0 0
\(577\) 6.24916i 0.260156i 0.991504 + 0.130078i \(0.0415228\pi\)
−0.991504 + 0.130078i \(0.958477\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −11.1527 19.3171i −0.461900 0.800033i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −10.7881 18.6855i −0.445273 0.771235i 0.552799 0.833315i \(-0.313560\pi\)
−0.998071 + 0.0620801i \(0.980227\pi\)
\(588\) 0 0
\(589\) −0.528615 + 0.915588i −0.0217812 + 0.0377261i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 8.26071 0.339227 0.169613 0.985511i \(-0.445748\pi\)
0.169613 + 0.985511i \(0.445748\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 30.7618 + 17.7603i 1.25689 + 0.725667i 0.972469 0.233033i \(-0.0748649\pi\)
0.284422 + 0.958699i \(0.408198\pi\)
\(600\) 0 0
\(601\) 35.8981 20.7258i 1.46432 0.845423i 0.465109 0.885254i \(-0.346015\pi\)
0.999206 + 0.0398308i \(0.0126819\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 3.58388 + 6.20746i 0.145705 + 0.252369i
\(606\) 0 0
\(607\) −2.09569 1.20995i −0.0850616 0.0491103i 0.456866 0.889536i \(-0.348972\pi\)
−0.541927 + 0.840425i \(0.682305\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 53.1015i 2.14826i
\(612\) 0 0
\(613\) −42.6456 −1.72244 −0.861219 0.508234i \(-0.830298\pi\)
−0.861219 + 0.508234i \(0.830298\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −13.2535 7.65193i −0.533567 0.308055i 0.208901 0.977937i \(-0.433011\pi\)
−0.742468 + 0.669882i \(0.766345\pi\)
\(618\) 0 0
\(619\) 23.9177 13.8089i 0.961334 0.555026i 0.0647505 0.997901i \(-0.479375\pi\)
0.896583 + 0.442875i \(0.146042\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 4.77517 8.27084i 0.191007 0.330834i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −2.52164 −0.100545
\(630\) 0 0
\(631\) 8.28775 0.329930 0.164965 0.986299i \(-0.447249\pi\)
0.164965 + 0.986299i \(0.447249\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 8.96069 15.5204i 0.355594 0.615907i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −8.58307 + 4.95544i −0.339011 + 0.195728i −0.659835 0.751411i \(-0.729374\pi\)
0.320824 + 0.947139i \(0.396040\pi\)
\(642\) 0 0
\(643\) 6.83668 + 3.94716i 0.269612 + 0.155661i 0.628711 0.777639i \(-0.283583\pi\)
−0.359099 + 0.933299i \(0.616916\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 4.31931 0.169810 0.0849049 0.996389i \(-0.472941\pi\)
0.0849049 + 0.996389i \(0.472941\pi\)
\(648\) 0 0
\(649\) 17.5870i 0.690349i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 37.5853 + 21.6999i 1.47082 + 0.849181i 0.999463 0.0327591i \(-0.0104294\pi\)
0.471361 + 0.881940i \(0.343763\pi\)
\(654\) 0 0
\(655\) 5.33940 + 9.24812i 0.208628 + 0.361354i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 9.34894 5.39761i 0.364183 0.210261i −0.306731 0.951796i \(-0.599235\pi\)
0.670914 + 0.741535i \(0.265902\pi\)
\(660\) 0 0
\(661\) −3.39495 1.96008i −0.132048 0.0762381i 0.432521 0.901624i \(-0.357624\pi\)
−0.564569 + 0.825386i \(0.690958\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 63.1276 2.44431
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −10.0039 + 17.3273i −0.386197 + 0.668913i
\(672\) 0 0
\(673\) −12.3404 21.3742i −0.475687 0.823915i 0.523925 0.851765i \(-0.324467\pi\)
−0.999612 + 0.0278497i \(0.991134\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −7.36327 12.7536i −0.282994 0.490159i 0.689127 0.724641i \(-0.257994\pi\)
−0.972121 + 0.234481i \(0.924661\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1.84900i 0.0707499i −0.999374 0.0353750i \(-0.988737\pi\)
0.999374 0.0353750i \(-0.0112625\pi\)
\(684\) 0 0
\(685\) 28.0606i 1.07214i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 17.2919 29.9505i 0.658769 1.14102i
\(690\) 0 0
\(691\) 33.7613 19.4921i 1.28434 0.741514i 0.306701 0.951806i \(-0.400775\pi\)
0.977639 + 0.210292i \(0.0674415\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 7.14764 4.12669i 0.271126 0.156534i
\(696\) 0 0
\(697\) 0.662926 1.14822i 0.0251101 0.0434920i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 25.4389i 0.960813i 0.877046 + 0.480406i \(0.159511\pi\)
−0.877046 + 0.480406i \(0.840489\pi\)
\(702\) 0 0
\(703\) 1.05344i 0.0397314i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 7.14517 + 12.3758i 0.268342 + 0.464783i 0.968434 0.249270i \(-0.0801908\pi\)
−0.700092 + 0.714053i \(0.746857\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −27.9296 48.3756i −1.04597 1.81168i
\(714\) 0 0
\(715\) −19.9319 + 34.5230i −0.745410 + 1.29109i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −33.4688 −1.24818 −0.624088 0.781354i \(-0.714529\pi\)
−0.624088 + 0.781354i \(0.714529\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 13.2817 + 7.66819i 0.493269 + 0.284789i
\(726\) 0 0
\(727\) 12.1354 7.00636i 0.450076 0.259851i −0.257786 0.966202i \(-0.582993\pi\)
0.707862 + 0.706350i \(0.249660\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0.726384 + 1.25813i 0.0268663 + 0.0465338i
\(732\) 0 0
\(733\) 23.6491 + 13.6538i 0.873501 + 0.504316i 0.868510 0.495672i \(-0.165078\pi\)
0.00499085 + 0.999988i \(0.498411\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 23.0824i 0.850251i
\(738\) 0 0
\(739\) −52.6314 −1.93608 −0.968039 0.250801i \(-0.919306\pi\)
−0.968039 + 0.250801i \(0.919306\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 30.9523 + 17.8703i 1.13553 + 0.655599i 0.945320 0.326144i \(-0.105750\pi\)
0.190211 + 0.981743i \(0.439083\pi\)
\(744\) 0 0
\(745\) 7.71839 4.45621i 0.282780 0.163263i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 16.5641 28.6899i 0.604433 1.04691i −0.387708 0.921782i \(-0.626733\pi\)
0.992141 0.125126i \(-0.0399336\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −24.5836 −0.894687
\(756\) 0 0
\(757\) −13.6903 −0.497584 −0.248792 0.968557i \(-0.580034\pi\)
−0.248792 + 0.968557i \(0.580034\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −6.51737 + 11.2884i −0.236255 + 0.409205i −0.959637 0.281243i \(-0.909253\pi\)
0.723382 + 0.690448i \(0.242587\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −23.6147 + 13.6340i −0.852678 + 0.492294i
\(768\) 0 0
\(769\) −18.4866 10.6732i −0.666642 0.384886i 0.128161 0.991753i \(-0.459093\pi\)
−0.794803 + 0.606867i \(0.792426\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −11.4788 −0.412864 −0.206432 0.978461i \(-0.566185\pi\)
−0.206432 + 0.978461i \(0.566185\pi\)
\(774\) 0 0
\(775\) 13.5706i 0.487470i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −0.479682 0.276944i −0.0171864 0.00992256i
\(780\) 0 0
\(781\) −22.3208 38.6607i −0.798701 1.38339i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −25.9873 + 15.0038i −0.927528 + 0.535509i
\(786\) 0 0
\(787\) −35.6808 20.6003i −1.27188 0.734322i −0.296541 0.955020i \(-0.595833\pi\)
−0.975342 + 0.220698i \(0.929166\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −31.0214 −1.10160
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −25.0066 + 43.3127i −0.885779 + 1.53421i −0.0409600 + 0.999161i \(0.513042\pi\)
−0.844819 + 0.535053i \(0.820292\pi\)
\(798\) 0 0
\(799\) −1.76484 3.05679i −0.0624355 0.108141i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 13.6540 + 23.6494i 0.481838 + 0.834568i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 50.6908i 1.78219i 0.453812 + 0.891097i \(0.350064\pi\)
−0.453812 + 0.891097i \(0.649936\pi\)
\(810\) 0 0
\(811\) 8.96566i 0.314827i −0.987533 0.157413i \(-0.949684\pi\)
0.987533 0.157413i \(-0.0503155\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0.0812910 0.140800i 0.00284750 0.00493201i
\(816\) 0 0
\(817\) 0.525599 0.303455i 0.0183884 0.0106165i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −28.2190 + 16.2922i −0.984849 + 0.568603i −0.903731 0.428102i \(-0.859183\pi\)
−0.0811184 + 0.996704i \(0.525849\pi\)
\(822\) 0 0
\(823\) 10.0877 17.4724i 0.351636 0.609051i −0.634901 0.772594i \(-0.718959\pi\)
0.986536 + 0.163543i \(0.0522923\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0.253288i 0.00880770i 0.999990 + 0.00440385i \(0.00140179\pi\)
−0.999990 + 0.00440385i \(0.998598\pi\)
\(828\) 0 0
\(829\) 7.05416i 0.245001i 0.992468 + 0.122501i \(0.0390913\pi\)
−0.992468 + 0.122501i \(0.960909\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 4.17848 + 7.23733i 0.144602 + 0.250458i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 17.0936 + 29.6069i 0.590136 + 1.02215i 0.994214 + 0.107420i \(0.0342591\pi\)
−0.404078 + 0.914725i \(0.632408\pi\)
\(840\) 0 0
\(841\) 10.7462 18.6130i 0.370558 0.641826i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −39.8926 −1.37235
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −48.2024 27.8296i −1.65236 0.953988i
\(852\) 0 0
\(853\) −21.7586 + 12.5623i −0.745000 + 0.430126i −0.823884 0.566758i \(-0.808198\pi\)
0.0788844 + 0.996884i \(0.474864\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 21.0954 + 36.5383i 0.720604 + 1.24812i 0.960758 + 0.277388i \(0.0894688\pi\)
−0.240154 + 0.970735i \(0.577198\pi\)
\(858\) 0 0
\(859\) 4.08139 + 2.35639i 0.139255 + 0.0803990i 0.568009 0.823022i \(-0.307714\pi\)
−0.428754 + 0.903421i \(0.641047\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 35.6120i 1.21225i −0.795371 0.606123i \(-0.792724\pi\)
0.795371 0.606123i \(-0.207276\pi\)
\(864\) 0 0
\(865\) −24.9500 −0.848326
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −4.08543 2.35873i −0.138589 0.0800143i
\(870\) 0 0
\(871\) 30.9937 17.8942i 1.05018 0.606322i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −20.4532 + 35.4260i −0.690655 + 1.19625i 0.280969 + 0.959717i \(0.409344\pi\)
−0.971624 + 0.236532i \(0.923989\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −37.4443 −1.26153 −0.630765 0.775974i \(-0.717259\pi\)
−0.630765 + 0.775974i \(0.717259\pi\)
\(882\) 0 0
\(883\) −49.8357 −1.67711 −0.838553 0.544821i \(-0.816598\pi\)
−0.838553 + 0.544821i \(0.816598\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 14.4482 25.0251i 0.485124 0.840260i −0.514730 0.857352i \(-0.672108\pi\)
0.999854 + 0.0170929i \(0.00544110\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −1.27701 + 0.737280i −0.0427334 + 0.0246721i
\(894\) 0 0
\(895\) −0.999223 0.576902i −0.0334003 0.0192837i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −44.6789 −1.49012
\(900\) 0 0
\(901\) 2.29880i 0.0765841i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 11.4564 + 6.61437i 0.380825 + 0.219869i
\(906\) 0 0
\(907\) 7.43498 + 12.8778i 0.246874 + 0.427599i 0.962657 0.270724i \(-0.0872632\pi\)
−0.715783 + 0.698323i \(0.753930\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −7.81616 + 4.51266i −0.258961 + 0.149511i −0.623861 0.781536i \(-0.714437\pi\)
0.364899 + 0.931047i \(0.381103\pi\)
\(912\) 0 0
\(913\) 1.22671 + 0.708243i 0.0405982 + 0.0234394i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −26.4166 −0.871403 −0.435702 0.900091i \(-0.643500\pi\)
−0.435702 + 0.900091i \(0.643500\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 34.6075 59.9420i 1.13912 1.97302i
\(924\) 0 0
\(925\) −6.76100 11.7104i −0.222300 0.385036i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −11.1259 19.2706i −0.365029 0.632249i 0.623752 0.781623i \(-0.285608\pi\)
−0.988781 + 0.149373i \(0.952274\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 2.64976i 0.0866563i
\(936\) 0 0
\(937\) 14.6822i 0.479647i 0.970817 + 0.239823i \(0.0770896\pi\)
−0.970817 + 0.239823i \(0.922910\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −23.0396 + 39.9058i −0.751070 + 1.30089i 0.196235 + 0.980557i \(0.437129\pi\)
−0.947305 + 0.320334i \(0.896205\pi\)
\(942\) 0 0
\(943\) 25.3442 14.6325i 0.825322 0.476500i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 6.96116 4.01903i 0.226207 0.130601i −0.382614 0.923908i \(-0.624976\pi\)
0.608821 + 0.793308i \(0.291643\pi\)
\(948\) 0 0
\(949\) −21.1700 + 36.6675i −0.687206 + 1.19028i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 54.9348i 1.77951i −0.456437 0.889756i \(-0.650875\pi\)
0.456437 0.889756i \(-0.349125\pi\)
\(954\) 0 0
\(955\) 33.0313i 1.06887i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 4.26733 + 7.39124i 0.137656 + 0.238427i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −15.4770 26.8070i −0.498223 0.862948i
\(966\) 0 0
\(967\) −26.5917 + 46.0582i −0.855132 + 1.48113i 0.0213900 + 0.999771i \(0.493191\pi\)
−0.876522 + 0.481361i \(0.840143\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 15.2281 0.488692 0.244346 0.969688i \(-0.421427\pi\)
0.244346 + 0.969688i \(0.421427\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1.49418 + 0.862667i 0.0478031 + 0.0275992i 0.523711 0.851896i \(-0.324547\pi\)
−0.475908 + 0.879495i \(0.657880\pi\)
\(978\) 0 0
\(979\) 9.37033 5.40997i 0.299477 0.172903i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −30.1191 52.1679i −0.960651 1.66390i −0.720871 0.693070i \(-0.756258\pi\)
−0.239780 0.970827i \(-0.577075\pi\)
\(984\) 0 0
\(985\) 8.64713 + 4.99242i 0.275521 + 0.159072i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 32.0664i 1.01965i
\(990\) 0 0
\(991\) 5.74624 0.182535 0.0912676 0.995826i \(-0.470908\pi\)
0.0912676 + 0.995826i \(0.470908\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −22.9711 13.2624i −0.728234 0.420446i
\(996\) 0 0
\(997\) 0.0224508 0.0129620i 0.000711024 0.000410510i −0.499644 0.866231i \(-0.666536\pi\)
0.500355 + 0.865820i \(0.333203\pi\)
\(998\) 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 5292.2.x.b.881.3 16
3.2 odd 2 1764.2.x.b.293.1 16
7.2 even 3 756.2.bm.a.17.6 16
7.3 odd 6 756.2.w.a.341.6 16
7.4 even 3 5292.2.w.b.1097.3 16
7.5 odd 6 5292.2.bm.a.2285.3 16
7.6 odd 2 5292.2.x.a.881.6 16
9.2 odd 6 5292.2.x.a.4409.6 16
9.7 even 3 1764.2.x.a.1469.8 16
21.2 odd 6 252.2.bm.a.185.7 yes 16
21.5 even 6 1764.2.bm.a.1697.2 16
21.11 odd 6 1764.2.w.b.509.5 16
21.17 even 6 252.2.w.a.5.4 16
21.20 even 2 1764.2.x.a.293.8 16
28.3 even 6 3024.2.ca.d.2609.6 16
28.23 odd 6 3024.2.df.d.17.6 16
63.2 odd 6 756.2.w.a.521.6 16
63.11 odd 6 5292.2.bm.a.4625.3 16
63.16 even 3 252.2.w.a.101.4 yes 16
63.20 even 6 inner 5292.2.x.b.4409.3 16
63.23 odd 6 2268.2.t.a.1781.6 16
63.25 even 3 1764.2.bm.a.1685.2 16
63.31 odd 6 2268.2.t.a.2105.6 16
63.34 odd 6 1764.2.x.b.1469.1 16
63.38 even 6 756.2.bm.a.89.6 16
63.47 even 6 5292.2.w.b.521.3 16
63.52 odd 6 252.2.bm.a.173.7 yes 16
63.58 even 3 2268.2.t.b.1781.3 16
63.59 even 6 2268.2.t.b.2105.3 16
63.61 odd 6 1764.2.w.b.1109.5 16
84.23 even 6 1008.2.df.d.689.2 16
84.59 odd 6 1008.2.ca.d.257.5 16
252.79 odd 6 1008.2.ca.d.353.5 16
252.115 even 6 1008.2.df.d.929.2 16
252.191 even 6 3024.2.ca.d.2033.6 16
252.227 odd 6 3024.2.df.d.1601.6 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.2.w.a.5.4 16 21.17 even 6
252.2.w.a.101.4 yes 16 63.16 even 3
252.2.bm.a.173.7 yes 16 63.52 odd 6
252.2.bm.a.185.7 yes 16 21.2 odd 6
756.2.w.a.341.6 16 7.3 odd 6
756.2.w.a.521.6 16 63.2 odd 6
756.2.bm.a.17.6 16 7.2 even 3
756.2.bm.a.89.6 16 63.38 even 6
1008.2.ca.d.257.5 16 84.59 odd 6
1008.2.ca.d.353.5 16 252.79 odd 6
1008.2.df.d.689.2 16 84.23 even 6
1008.2.df.d.929.2 16 252.115 even 6
1764.2.w.b.509.5 16 21.11 odd 6
1764.2.w.b.1109.5 16 63.61 odd 6
1764.2.x.a.293.8 16 21.20 even 2
1764.2.x.a.1469.8 16 9.7 even 3
1764.2.x.b.293.1 16 3.2 odd 2
1764.2.x.b.1469.1 16 63.34 odd 6
1764.2.bm.a.1685.2 16 63.25 even 3
1764.2.bm.a.1697.2 16 21.5 even 6
2268.2.t.a.1781.6 16 63.23 odd 6
2268.2.t.a.2105.6 16 63.31 odd 6
2268.2.t.b.1781.3 16 63.58 even 3
2268.2.t.b.2105.3 16 63.59 even 6
3024.2.ca.d.2033.6 16 252.191 even 6
3024.2.ca.d.2609.6 16 28.3 even 6
3024.2.df.d.17.6 16 28.23 odd 6
3024.2.df.d.1601.6 16 252.227 odd 6
5292.2.w.b.521.3 16 63.47 even 6
5292.2.w.b.1097.3 16 7.4 even 3
5292.2.x.a.881.6 16 7.6 odd 2
5292.2.x.a.4409.6 16 9.2 odd 6
5292.2.x.b.881.3 16 1.1 even 1 trivial
5292.2.x.b.4409.3 16 63.20 even 6 inner
5292.2.bm.a.2285.3 16 7.5 odd 6
5292.2.bm.a.4625.3 16 63.11 odd 6