Centers of components with n-periodic attractor are roots of equation:
P(n)=0
which degree is 2n-1

Number of centers = number of components of M-set for given period
Number of centers < = number of roots = 2n-1
number of roots:
(%i3) makelist(length(allroots(%i*P(n))),n,1,10);
(%o3) [1,2,4,8,16,32,64,128,256,512]


All code is in Maxima  with fpprec : 16;

Definition of complex quadratic polynomial for z0=0 :
P(n):=if n=0 then 0 else P(n-1)^2+c;

Equation:
P(n)=0

So centers for period n are roots of equations:
period equation
1         c=0
2          c2 + c = 0
3          (c2 + c)2+ c = 0
4          ((c2 + c)2+ c)2+ c = 0
...

PeterPein: "according to the Maxima-help results may become more accurate, when solving
for %i*P(7), because the polynom-solver uses a different algorithm for polynomials with complex coefficients."
so :

Real centers are in F(1/2) family or in cardioids on real axis

allroots(%i*P(1));
there is only 1 period 1 component with center :
c=0.0 ( center of the main cardioid )

divisors(1);
{1}

allroots(%i*P(2));
c=0.0
c=-1.0

divisors(2);
{1,2} so one should remove center for period 1 component c=0 from list

there is only 1 period 2 component with center ( and it is not cardioid) :
c=-1.0

allroots(%i*P(3));
c=0.0
c=0.74486176661974*%i-0.12256116687665
c=-0.74486176661974*%i-0.12256116687665
c=-1.754877666246693

divisors(3);
{1,3} so we remove c=0 from list and

there are 3 period 3 components :
2 circular regions attached to the main cardioid with centers :
c=0.74486176661974*%i-0.12256116687665
c=-0.74486176661974*%i-0.12256116687665
and one miniature copy of M on main antenna with center:
c=-1.754877666246693

allroots(%i*P(4));
c=0.0
c=0.53006061757853*%i+0.28227139076691
c=0.28227139076691-0.53006061757852*%i
c=-1.032247108922828*%i-0.15652016683375
c=-1.8607834444415842*10^-14*%i-0.99999999999996
c=1.032247108922827*%i-0.15652016683376
c=1.1449662215926455*10^-14*%i-1.310702641336853
c=3.1870184510020416*10^-16*%i-1.940799806529488

divisors(4);
{1,2,4}
we remove centers for period 1:
c=0.0
and period=2 :
c=-1.8607834444415842*10^-14*%i-0.99999999999996 = -1

so there are 6 period 4 components :
c=1.1449662215926455*10^-14*%i-1.310702641336853 = -1.310702641 which belongs to the F(1/2) family
2 components attached to the main cardioid with centers :
c=0.28227139076691+0.53006061757853*%i
c=0.28227139076691-0.53006061757852*%i
3 cardioids in miniature copy of M with centers :
c=-1.032247108922828*%i-0.15652016683375 = -0.1565201668 - 1.0322471089 I
c=1.032247108922827*%i-0.15652016683376 = -0.1565201668 + 1.0322471089 I
c=3.1870184510020416*10^-16*%i-1.940799806529488 = -1.940799806529488 which lays on main antenna


allroots(%i*P(5));

divisors(5);
{1,5}
so one should remove only center for period =1 which is c=0.0
and there are 15 period 5 components
4 disk components with centers :
c=0.33493230559749*%i+0.37951358801592
c=0.56276576145299*%i-0.50434017544624
c=0.37951358801593-0.3349323055975*%i
c=-0.56276576145299*%i-0.50434017544625
and 11 cardioids
8 cardioids not lying on real axis with centers :
c=-1.25636793006831-0.38032096347279*%i
c=-1.25636793006853+0.38032096347303*%i
c=-0.19804209936426+1.100269537292705*%i
c=-0.19804209936427-1.100269537292689*%i
c=-0.044212357704065-0.98658097628091*%i
c=-0.044212357704058+0.98658097628089*%i
c=0.35925922475801-0.64251373713854*%i
c=0.35925922475801+0.64251373713854*%i
3 cardioids in miniature copy of M lying on real axis with centers :
c=-3.7125482369056422*10^-13*%i-1.62541372512043 = -1.62541372512043
c=3.2353666472991833*10^-13*%i-1.860782522209003 = -1.860782522209003
c=-1.8065471720474881*10^-13*%i-1.985424253052442 = -1.98542425305


allroots(%i*P(6));
c=0.0
c=0.21585065087082*%i+0.38900684056977
c=0.38900684056977-0.21585065087082*%i
c=0.44332563339962-0.37296241666285*%i
c=0.37296241666285*%i+0.44332563339962
c=0.60418181048899*%i+0.39653457003241
c=-9.6008352803251748*10^-10*%i-0.9999999970598
c=0.39653457003103-0.60418181048979*%i
c=0.7448617666176*%i-0.1225611668439
c=-0.74486176675623*%i-0.12256116676608
c=0.35989273901232-0.68476202020949*%i
c=0.68476202021174*%i+0.35989273901251
c=0.86056947251121*%i-0.11341865597063
c=-0.86056947239579*%i-0.11341865611029
c=0.66298074461135*%i-0.59689164465885
c=-0.66298074452848*%i-0.59689164468062
c=-1.020497366488349*%i-0.015570385995175
c=1.097780642914231*%i-0.16359826153093
c=1.020497366492764*%i-0.015570386016637
c=-1.097780642917053*%i-0.16359826149894
c=1.114454265857003*%i-0.21752674705594
c=-0.24033240708634*%i-1.138000668993346
c=-1.11445426587117*%i-0.21752674706338
c=0.2403324027092*%i-1.13800066848188
c=3.2048611067021922*10^-8*%i-1.476014600660105
c=0.42726889601584*%i-1.284084926217826
c=-0.42726889602331*%i-1.284084927217583
c=-2.0028034035711978*10^-6*%i-1.754878686853023
c=-1.0838749405127502*10^-6*%i-1.907279868506552
c=1.2443431020326344*10^-6*%i-1.966773825052916
c=2.3751736937535738*10^-6*%i-1.772891887893593
c=-5.5959089846013193*10^-7*%i-1.996375784899055

divisors(6);
{1,2,3,6}
so one should remove :
c=0.0
center of period 2 component :
c=-9.6008352803251748*10^-10*%i-0.9999999970598 = -1
centers of period 3 components :
c=-2.0028034035711978*10^-6*%i-1.754878686853023 = -1.754878686853023
c=0.7448617666176*%i-0.1225611668439
c=-0.74486176675623*%i-0.12256116676608

so for period=6 thera are 27 components
(and 20 of them are cardioids )
with centers :
c=0.21585065087082*%i+0.38900684056977
c=0.38900684056977-0.21585065087082*%i
c=-0.11341865597063+0.86056947251121*%i
c=-0.11341865611029-0.86056947239579*%i
c=-1.138000668993346-0.24033240708634*%i
c=-1.13800066848188+0.2403324027092*%i
c=0.44332563339962-0.37296241666285*%i cardioid
c=0.44332563339962+0.37296241666285*%i cardioid
c=0.39653457003241+0.60418181048899*%i cardioid
c=0.39653457003103-0.60418181048979*%i cardioid
c=0.35989273901232-0.68476202020949*%i cardioid
c=0.35989273901251+0.68476202021174*%i cardioid
c=-0.015570385995175-1.020497366488349*%i cardioid
c=-0.015570386016637+1.020497366492764*%i cardioid
c=-0.16359826153093+1.097780642914231*%i cardioid
c=-0.16359826149894-1.097780642917053*%i cardioid
c=-0.21752674705594+1.114454265857003*%i cardioid
c=-0.21752674706338-1.11445426587117*%i cardioid
c=-0.59689164465885+0.66298074461135*%i cardioid
c=-0.59689164468062-0.66298074452848*%i cardioid
c=-1.284084926217826+0.42726889601584*%i cardioid
c=-1.284084927217583-0.42726889602331*%i cardioid
and 5 components lying on real axis ( main antenna) with centers :
c=-1.0838749405127502*10^-6*%i-1.907279868506552 = -1.9072798 cardioid
c=1.2443431020326344*10^-6*%i-1.966773825052916 = -1.966773  cardioid
c=2.3751736937535738*10^-6*%i-1.772891887893593 = -1.772891887893593
c=-5.5959089846013193*10^-7*%i-1.996375784899055 = -1.99637578 cardioid
c=3.2048611067021922*10^-8*%i-1.476014600660105 = -1.476014600660105  cardioid




Problem : Mandelbrot set contains points c : Abs(c)< =2, but among roots for period >6 computed with Maxima are c: Abs(c)> 2 .


period 7 ( probably bad roots !!!!!!!!!!! ) see above
allroots(%i*P(7));
c=0.0
c=0.14474937132163*%i+0.37600868184677
c=0.37600868184677-0.14474937132163*%i
c=0.43237619264199-0.22675990443535*%i
c=0.22675990443535*%i+0.43237619264199
c=0.34775870088348*%i+0.45682328582332
c=0.45682328582332-0.34775870088348*%i
c=0.39617012803317*%i+0.45277449872492
c=0.45277449872493-0.39617012803315*%i
c=0.12119278609178-0.61061169220516*%i
c=0.56932471130781*%i+0.38653917659553
c=0.6106116922083*%i+0.12119278620677
c=0.38653917659283-0.56932471130283*%i
c=0.61480676013913*%i+0.4129160247172
c=0.14152619689199*%i-1.458700483246053
c=0.013288383473933*%i-0.28677977987114
c=-0.40898083002731*%i-0.20720330190092
c=0.43606694732151*%i-0.24190600036444
c=0.69833073363318*%i+0.35246655247607
c=0.044418620975236-0.88587974483496*%i
c=0.41293521759259-0.61481949126974*%i
c=-0.75122072517616*%i-0.01682322376233
c=0.37688687479681-0.67873395288664*%i
c=0.35233896607439-0.69816908477277*%i
c=0.67857675822296*%i+0.37689796941967
c=0.786238987229*%i-0.059608726502816
c=0.8699260606637*%i+0.024457767375424
c=0.97829762261065*%i-0.34822961681632
c=1.000594099632147*%i+0.020182578309324
c=0.032936276551309-1.008474570735644*%i
c=-1.06710069580134*%i-0.006396995002131
c=0.84164019955104*%i-0.57075579957122
c=1.154977335904774*%i-0.21564291510383
c=0.26135701321722*%i-1.766667328367312
c=-0.84725686009173*%i-0.58646848078042
c=0.7576374195106*%i-0.76850294927528
c=-0.44511079741985*%i-0.92918186205081
c=1.054125748451466*%i-0.0072853668949859
c=-0.98737990625706*%i-0.3788003161858
c=0.66732713142163*%i-0.98134174182844
c=1.146953640150215*%i-0.15279631898407
c=-0.79763758510621*%i-0.76739438761301
c=1.101220621709238*%i-0.090613406166806
c=0.15523299843152*%i-1.121052347600134
c=-1.11792924521294*%i-0.087789606543876
c=1.117658267192613*%i-0.2696280948415
c=-1.158098904936238*%i-0.15548423014013
c=-1.162061916464406*%i-0.22335118329413
c=-1.117097712371963*%i-0.28305341040574
c=0.68760425246826*%i-1.161445265392643
c=-0.77376961958581*%i-1.32199843508228
c=-0.78629952902958*%i-1.060621739871458
c=0.69195734973778*%i-1.368243717420777
c=-0.25599802859445*%i-2.169681218903451
c=-0.59221697720935*%i-1.747015369504228
c=-0.71144697213007*%i-1.55874819525607
c=0.6521665183425*%i-1.586262455459096
c=-0.4055527211104*%i-2.13214450533245
c=0.58539437525699*%i-1.833824054179376
c=-0.15147880196692*%i-2.318595950326579
c=-0.54613579596639*%i-1.920477347060587
c=0.47990575796663*%i-2.076877025434899
c=0.3109673589573*%i-2.264618525502887
c=0.08507193086182*%i-2.355854414008519


allroots(%i*P(8));
c=0.0
c=0.1009348768643*%i+0.35903106283661
c=0.35903106283661-0.1009348768643*%i
c=0.40489966517512-0.14582036376659*%i
c=0.14582036376659*%i+0.40489966517512
c=0.21038120059848*%i+0.43728392949718
c=0.43728392949717-0.21038120059847*%i
c=0.46695920563629-0.35154672458297*%i
c=0.23942604028655*%i+0.44207453001541
c=0.44207453001556-0.23942604028663*%i
c=0.28227271573749-0.53006363898614*%i
c=0.40652098405419*%i+0.45114581418109
c=0.33193138934452*%i+0.45052885729312
c=0.45052885956398-0.33193138762847*%i
c=0.70675892157877*%i+0.046623865479682
c=0.04447045851916*%i-0.28094042034854
c=-0.32433701728071*%i-0.23948105434755
c=0.39315674138852*%i-0.16960396581043
c=0.53127348719266*%i+0.28294132970234
c=0.35154671833136*%i+0.46695919966527
c=0.45114581196813-0.4065209673506*%i
c=-0.703941305283*%i-0.11379471286602
c=0.91188764408357*%i+0.19157852996819
c=0.12119958185057-0.087772165029298*%i
c=0.15225300705247*%i+0.1285394612771
c=0.20027873770765-0.29597652768793*%i
c=0.35399895949102*%i+0.22377896575234
c=0.31633117745537-0.43291451095819*%i
c=0.46174046888455-0.39013794154169*%i
c=0.47124106880489*%i+0.34459097910324
c=0.39012075747003*%i+0.46167164332181
c=0.39974242055532-0.51076928265698*%i
c=0.53347974640832*%i+0.41355099916702
c=0.43586613994694-0.56048813818755*%i
c=0.57860352006051*%i+0.43849605981703
c=0.64844328327724*%i-2.473107081400534
c=0.55667121607892*%i-2.543700413951774
c=0.92682652296755*%i+1.330138826107012
c=1.35475980137841*%i+1.128718045330038
c=0.61164424670165*%i+1.416959194683648
c=0.44703035866671*%i+1.446563209916518
c=0.27901087007102*%i+1.467401103397619
c=0.1088169480961*%i+1.479763389547626
c=1.483789569966605-0.062165363963818*%
i c=1.4663842698732-0.4008670190612*%i
c=1.479414876428265-0.23250824587604*%i
c=1.444326005151156-0.566075235322*%i
c=1.412853860563851-0.72718419930465*%i
c=0.77184035268127*%i+1.378246853022241
c=1.371661376772095-0.88344658942624*%i
c=1.075981829676153*%i+1.272480709211295
c=1.320582880383484-1.034263934133797*%i
c=1.218786725602607*%i+1.205284037095295
c=1.25961461676074-1.179123944062059*%i
c=1.188903932818713-1.31754783183073*%i
c=0.52544228656707*%i-1.751895650344574
c=-0.7204483675075*%i-2.341506907960749
c=-0.51057760867517*%i-0.48849146624951
c=0.40903529933035*%i-0.58379907296353
c=3.031486123991353*%i-1.881444804373956
c=1.010939171077592-0.15722631380221*%i
c=0.36099325292845*%i+1.016655897542853
c=0.82051714824404*%i+1.106056835931806
c=1.101414028949689-0.64024770886122*%i
c=1.22656858293115-1.028003345966226*%i
c=1.179436106313368*%i+1.211475765075658
c=1.681494638193593*%i+1.254551590479107
c=1.318786192769876-1.31999634465237*%i
c=0.79303509534803-2.564892623573454*%i
c=1.44986601886061*%i+1.268884605757609
c=1.337983628197473-1.558909565003132*%i
c=1.297631156936559-1.783502563641077*%i
c=1.900331665624172*%i+1.189153067464539
c=2.997763959016507*%i-0.043803130101234
c=1.216244326668931-1.998452874038776*%i
c=-0.52549650311784*%i-2.272671481299655
c=2.107239143110463*%i+1.086735010158356
c=1.102071916716153-2.201770201844492*%i
c=2.300304404744414*%i+0.95397911253791
c=0.95985821931041-2.39125198983471*%i
c=2.637317250628422*%i+0.61357030048855
c=1.412675026056877*%i-2.255105411670275
c=-3.262184167008823*%i-1.743989476517722
c=2.477564817288966*%i+0.79515181874106
c=-2.127459530985297*%i-1.586165933077359
c=0.60445022540825-2.720971142715763*%i
c=2.778244982113434*%i+0.41194810861639
c=-0.79742474221886*%i-3.326217971847459
c=-3.133673914161008*%i-0.31403050412428
c=-0.40369666151466*%i-4.511529544550431
c=0.050423747305793*%i-1.887952086307056
c=1.432214685906702*%i-1.533836893259122
c=-1.60004025915665*%i-1.755026984748295
c=2.462951469624905*%i-0.99573644661906
c=0.39631734621006-2.857968483832601*%i
c=-2.858411133680237*%i-0.84525564352989
c=0.17161273281901-2.973200826282943*%i
c=2.899111469774935*%i+0.19219948258487
c=3.07238483786018*%i-0.30399298492317
c=-3.068901352548297*%i-0.066416784334888
c=3.050707700692243*%i-0.49303296065255
c=-3.224051077573923*%i-0.54758321647355
c=3.426984215300141*%i-1.447641577847429
c=3.241763418060661*%i-0.68747168121434
c=-3.350772101333931*%i-0.87597652369955
c=-1.72220139169066*%i-3.857832092150365
c=3.370517965450475*%i-1.039243780505915
c=-3.431948794616503*%i-1.273370905218336
c=-2.891316574182917*%i-3.019965235650469
c=1.11751808137272*%i-4.745190147887588
c=-3.403556414562095*%i-1.784560125485866
c=3.415137860607642*%i-1.918501562502018
c=-3.448348601279481*%i-2.293058889517882
c=3.378078767989817*%i-2.412619269736374
c=2.507516852734714*%i-4.12770784743365
c=2.145578991377567*%i-4.471862609743579
c=-3.39999644860823*%i-2.839212533394794
c=3.0828565308939*%i-3.336923684101179
c=3.274328126505209*%i-2.87775605479715
c=-3.213677302471269*%i-3.561304193622279
c=1.673718265650381*%i-4.835752321876791
c=2.816957455616037*%i-3.741736998792531
c=-1.46399841282173*%i-5.307871787646453
c=-2.255353850399216*%i-4.873799838056543
c=-2.854526536185064*%i-4.25927647661863
c=0.216136450166*%i-5.386811569921569
c=-0.58031296266934*%i-5.4598585059451
c=0.9126960882921*%i-5.17799095681513


allroots(%i*P(9))

(%o2) [
c=0.0,
c=0.072842377887596*%i+0.34320449315556,
c=0.34320449315556-0.072842377887596*%i,
c= 0.37860812413293-0.098558011118382*%i,
c=0.098558011118382*%i+0.37860812413293,
c=0.23447582880105*%i+ 0.44689100319239,
c=0.15284628188966*%i+0.41296293975205,
c=0.41296293969717-0.15284628187934*%i,
c= 0.13490209083556*%i+0.40644723702239,
c=0.40644723701436-0.13490209086347*%i,
c=0.32848944541709- 0.41041834636023*%i,
c=-0.058384369889365*%i-0.25355480037597,
c=0.374849335032*%i-0.16035173479806,
c= 0.17686660605415-0.64944836762081*%i,
c=0.33888697198379-0.02522445148147*%i,
c=0.07561931619356*%i+ 0.34739080944913,
c=0.15198210202821*%i+0.39269667633431,
c=0.37305620276026-0.11434645611142*%i,
c= 0.18462557132479*%i+0.4381683176865,
c=0.42331952009279-0.16398119807959*%i,
c=-0.68479709353011*%i- 0.52659502018666,
c=-0.68482077765819*%i-0.52659660234373,
c=-0.75153459042639*%i-0.60313332451467,
c=- 0.86989857002913*%i-0.43856667731784,
c=-0.99445013348067*%i-0.92431230011862,
c=-0.90033744638496*%i- 0.99236302035698,
c=-0.94612897846786*%i-0.95495978705887,
c=-1.014352255544788*%i-0.89604008175544,
c=- 0.85537799407042*%i-1.02914153216818,
c=-0.81027744668996*%i-1.064843284644687,
c=-1.306176080211782*%i- 0.28762142473598,
c=-0.6703103602242*%i-1.164412942231494,
c=1.152507852635348*%i-2.292445316246894,
c= 0.46996073263628*%i-2.796082483276573,
c=0.02842476562294*%i-2.800914437529971,
c=-0.26094804061594*%i- 2.789291556671118,
c=0.31781438735017*%i-2.80486323904311,
c=1.284983385017911*%i-2.500352477615652,
c= 1.790407032376613*%i+1.787995919841667,
c=3.036237307880918*%i-0.12478737260014,
c=3.439520848842796*%i- 0.81080190015173,
c=3.39215108792*%i-0.77257000657123,
c=3.738840511617319*%i-1.329898250804217,
c= 3.574325304821171*%i-1.311487073074154,
c=3.67201579318444*%i-0.57233157959631,
c=3.634014829903079*%i- 0.93294881543557,
c=2.111791050092925-2.736914920077573*%i,
c=3.098259286028883-1.168031728353044*%i,
c= 3.258223573960632-0.41293382822175*%i,
c=2.267069020781644-2.581114945776313*%i,
c=3.63015113036831*%i+ 0.31992389674437,
c=3.244864454249196-0.52260660833111*%i,
c=3.609158509069998*%i+0.4277905194906,
c= 3.20858426952822-0.74053473644801*%i,
c=3.413553415675678*%i+1.051694927440135,
c=3.130508441148387- 1.062440093877199*%i,
c=3.279046073196338-0.082247529907196*%i,
c=-2.085082052973973*%i-3.12145378043056,
c=- 3.466204554464297*%i-2.605231163402215,
c=-3.621894717507038*%i-2.722852821473585,
c=-3.741451106850608*%i- 2.574096702801256, c=-3.946523778755823*%i-2.290351023395608,
c=-4.035610505034121*%i-2.14402415969511,
c= 0.42040179489491*%i+0.079337444803998,
c=-3.849186437724348*%i-2.433001471203761,
c=-4.445989993846932*%i- 2.987496678911279,
c=-4.381865259518644*%i-2.201525895152078,
c=-4.470918347606096*%i-2.241832741520415,
c=- 4.486418250254408*%i-2.000484773959188,
c=-4.629608262383633*%i-1.691399033116698,
c=-4.564370692531868*%i- 1.84129808060658,
c=-4.682373765923783*%i-1.542220805032206,
c=-4.763846736563135*%i-1.236396002745319,
c= 0.77141267435501*%i+4.28411585762266,
c=3.511343344140504-2.685876575482919*%i,
c=2.44999871630944- 3.872163051636269*%i,
c=4.085239754197526*%i+2.147997718294608,
c=0.29176766628673*%i+4.311725177176814,
c= 4.290719306094834-0.19782394977447*%i,
c=3.8100116023665*%i-3.265307370076227,
c=4.197038276141803- 0.8596187589312*%i,
c=3.877489125819797-1.972934849931576*%i,
c=-4.859172917533236*%i-0.14377381238801,
c= 4.0848649075669-1.349788030267135*%i,
c=1.160327941251452*%i-4.787630913743948,
c=4.793702067303816*%i- 1.630821471374324,
c=2.770102556512495*%i-1.169503209540362,
c=-2.191672910105681*%i-5.293386333824479,
c= 4.192080459938538-1.332943732819951*%i,
c=4.29097211401001-0.92329175992532*%i,
c=-4.043490852015741*%i- 1.403863412018048,
c=4.257592917486884-1.132457683663214*%i,
c=4.35487848370887-0.70601141053965*%i,
c=- 3.962259579352785*%i-2.131188047408566,
c=4.395610534267416-0.51304705453194*%i,
c=-4.119448192587421*%i- 2.593915282013699,
c=4.41300842992754-0.31195749803815*%i,
c=4.101019232497553-1.740916011187071*%i,
c= 0.53745982854316*%i+4.492504002306767,
c=5.790674687315771*%i-3.356372296104312,
c=6.067148347497238*%i- 3.462393294348415,
c=6.227689278200474*%i-3.201249661600443,
c=5.610014193118431*%i-4.389451098672639,
c= 6.336813339027403*%i-2.973519621792839,
c=6.498981493346799*%i-2.46183916024713,
c=6.571239100343453*%i- 2.202790251327131,
c=6.418761777010987*%i-2.721715078365243,
c=6.631647882341805*%i-1.94359634295295,
c= 6.717585973250126*%i-1.418336329439499,
c=6.745375768141111*%i-1.153193535399106,
c=6.762889194391087*%i- 0.88777615943615,
c=5.145422075764434-1.090399081591649*%i,
c=5.083699145132592-1.407376636396206*%i,
c= 4.907028065633257-2.283252753080535*%i,
c=5.027883704978974-1.713783207022705*%i,
c=5.216505748218217- 0.77319078349964*%i,
c=0.11458435150339*%i+5.420908554019312,
c=4.970756679975467-2.005610797947009*%i,
c= 0.39316629457807*%i+5.468260367392709,
c=4.548054457725299-3.282619645855831*%i,
c=5.291565461251193- 0.46582171511211*%i,
c=5.361525545904623-0.17082457070255*%i,
c=4.833776642348872-2.548299828330541*%i,
c= 0.66604147720223*%i+5.503370266497021,
c=1.196350605341276*%i+5.53512664149721,
c=-2.488818238250194*%i- 2.08422029032695,
c=-1.475182970673571*%i-6.988034240951217,
c=0.069422639674808*%i+0.59134963817681,
c= 3.248282648200813*%i-1.77031200203834,
c=1.58229376768217-2.1642669014812*%i,
c=2.70563450429769- 3.412947102925957*%i, c=4.094675839429497*%i+4.730930061053529,
c=4.29487223733775*%i-3.551065704672253,
c= 3.854898898263711*%i+4.863504936294233,
c=3.757266603639596*%i+3.480832425003853,
c=3.820520112635582- 4.064080943093618*%i,
c=4.749656865081732-2.802419414064591*%i,
c=-6.091177677757463*%i-3.879444535986754,
c=- 1.403688266585051*%i-0.7510883524753,
c=1.474233371717173-4.137790747633834*%i,
c=3.602545553222968*%i- 2.663851250240023,
c=4.689669307308418*%i+1.621526862552723,
c=4.654338443747795-3.046925630700683*%i,
c= 3.993237733627378-4.193016089353537*%i,
c=-3.547169681800456*%i-6.417080529260469,
c=3.581118057184563- 4.851047438123347*%i,
c=4.431386200588756-3.510046069452621*%i,
c=2.150990218287649*%i-5.170681723922374,
c= 4.16323268280467-3.951772059069115*%i,
c=0.93361166388015*%i+5.525821462170559,
c=2.91535033809141*%i+ 5.241667449212194,
c=1.454810285104196*%i+5.531087808091963,
c=1.709529987890067*%i+5.513812864235753,
c= 1.961020556170166*%i+5.483243117937631,
c=2.20885603968155*%i+5.438694303412855,
c=2.450458596000323*%i+ 5.380648547486985,
c=3.613482854529293*%i+4.9770242343694,
c=3.146341330201975*%i+5.161308414311717,
c=- 1.36330706786191*%i-1.522729749998572,
c=1.754194739286586*%i-1.324023951360116,
c=-4.020887021767583*%i- 0.35321380286578,
c=4.423169255702019*%i+0.56725645895501,
c=7.547307441627732*%i-0.3529074374535,
c=- 0.56922655644581*%i-0.30086350857637,
c=2.528909580475892*%i+0.55715351361886,
c=0.90542520328461- 3.167191431095619*%i,
c=3.941549265056169*%i+2.490447376552838,
c=2.363199611659919-4.28645295300972*%i,
c= 5.444510905163702*%i+4.048843498720551,
c=4.376329911911939*%i+3.756741552070025,
c=2.684337388473447*%i+ 5.313802068098448,
c=3.435399966600405-4.891217349496063*%i,
c=3.838809017463235-4.449352550293011*%i,
c= 4.304691532589437-3.731094795693537*%i,
c=3.669356702624342-4.635782200614946*%i,
c=3.377162498543522*%i+ 5.073590505756731,
c=-3.989275445667734*%i-7.678105773458505,
c=-1.196449700341274*%i-1.660514216027552,
c= 1.868647355389478*%i-1.515326289248046,
c=-3.94532321384216*%i-0.76005243045798,
c=4.877521165754379*%i- 0.23437870202349,
c=0.90845277063954-5.906439507249057*%i,
c=5.116787243309529*%i+4.22990373868049,
c= 4.524301962114626*%i+4.441076707018393,
c=3.344532562243454-5.590713359390787*%i,
c=4.327565742811285*%i+ 4.579079502210148,
c=3.149850673992412*%i-8.63660682474001,
c=0.54267041084941*%i-4.083429231592574,
c= 4.837005133846772*%i-2.019160866326359,
c=-4.851176885456276*%i-2.659078602926866,
c=3.488511305468701- 5.245967481439283*%i,
c=3.14322319983347-5.919672865991981*%i,
c=4.790840592631549*%i+4.36247411070773,
c= 5.219273835987922*%i-7.599004012504567,
c=6.231373516619107*%i+1.814586091202588,
c=9.238460206462154*%i- 9.473592523984021,
c=0.8017054912922*%i+2.993151457492147,
c=2.994815009221894-1.41481830068442*%i,
c= 3.395748757154831-3.327349799655693*%i, c=2.824026476901597*%i+3.499263504744925,
c=4.306613298185843*%i+ 4.144583603431915,
c=3.828909877882513-4.783820900553586*%i,
c=4.052233835207945-5.897363821634007*%i,
c= 7.682789614543275*%i+3.998599497096879,
c=5.341570892724384*%i+4.545479285393436,
c=4.007124855831226- 6.831642179642602*%i,
c=6.955904222030479*%i+4.389680322766176,
c=6.179346032865692*%i+4.597466793779183,
c= 3.741907147969519-7.677752217437768*%i,
c=-10.70259774375449*%i-5.135083026168701,
c=8.94401899954795*%i+ 2.797483964674912,
c=3.308031012946459-8.45563836523308*%i,
c=8.349319222651809*%i+3.460780184891878,
c= 2.737655950249461-9.163066543095795*%i,
c=10.68447713611862*%i-1.961632823683914,
c=2.052586228654581- 9.793163169132065*%i,
c=9.457358809206889*%i+2.022342948164543,
c=10.49513388304639*%i-0.86598877948274,
c=- 3.201431872574153*%i-10.68641105987934,
c=-6.952569489348675*%i-10.7843547383378,
c=10.78252435794592*%i- 4.190428001674925,
c=2.561797941351202*%i-9.183626127463763,
c=10.22814547786931*%i+0.17661827190645,
c= 6.359015724202811*%i-9.750150460450646,
c=9.884522305766314*%i+1.145057521804813,
c=1.270522750850396- 10.33798491675218*%i,
c=-11.61190857552919*%i-2.574183809262431,
c=0.40957521982753-10.79099194023796*%i,
c=- 11.15578891622423*%i-0.51283363601867,
c=10.78241017558887*%i-3.082705792925521,
c=10.71774975054352*%i- 5.274061313775681,
c=-11.43944565506188*%i-1.503395356757265,
c=-11.23180004632175*%i-7.590794510035775,
c=- 11.62461042892892*%i-3.700893622195842,
c=8.164434934563111*%i-9.082712430799294,
c=10.57404495601466*%i- 6.399234125340157,
c=-4.160677867960226*%i-11.84673407740144,
c=-11.52300282881227*%i-4.90207130487737,
c= 8.011881614252385*%i-12.08915449899951,
c=-11.47966391975784*%i-6.403387052823983,
c=10.23030114688655*%i- 7.582492045971357,
c=9.587704799458223*%i-8.559784959014504,
c=-0.12173266930205*%i-13.85597207806127,
c= 8.831661748900196*%i-10.873231231718,
c=-9.827444485861481*%i-10.32804171751151,
c=-11.02119216574917*%i- 9.152138978000439,
c=1.498884551967825*%i-17.44443668825174,
c=4.064757212040989*%i-13.98213038335175,
c= 7.330625833393392*%i-13.2318242228388,
c=-6.399227039737753*%i-15.4159357343508,
c=-4.274087075180561*%i- 16.83889713437826,
c=6.103557963680922*%i-14.75559798321698,
c=-8.22099227298585*%i-13.87140889247179,
c=- 9.686951739293999*%i-11.75910895576927,
c=4.074463778576021*%i-16.33297352365103,
c=-1.471547812505635*%i- 17.64428835253731
]
On my computer Maxima stops at allroots(%i*P(10));
so I go with real roots only

realroots(P(10));
[c=-1.999985903501511,
c=-1.999872893095017,
c=-1.999646931886673,
c=-1.999307483434677,
c=-1.998855739831924,
c=-1.998288899660111,
c=-1.997608095407486,
c=-1.996804565191269,
c=-1.995924144983292,
c=-1.994889169931412,
c=-1.993747860193253,
c=-1.992479354143143,
c=-1.991120904684067,
c=-1.989600569009781,
c=-1.987940579652786,
c=-1.985482305288315,
c=-1.985424250364304,
c=-1.982719391584396,
c=-1.980577200651169,
c=-1.97829332947731,
c=-1.976042181253433,
c=-1.973497420549393,
c=-1.970857888460159,
c=-1.967742770910263,
c=-1.965821772813797,
c=-1.962378710508347,
c=-1.959098070859909,
c=-1.955423325300217,
c=-1.951899975538254
,c=-1.9468734562397,
c=-1.935390740633011,
c=-1.929320305585861,
c=-1.925034254789352,
c=-1.919635504484177,
c=-1.91447988152504,
c=-1.899832338094711,
c=-1.894002050161362,
c=-1.887172192335129,
c=-1.882407933473587,
c=-1.874314695596695,
c=-1.86155840754509,
c=-1.860782533884049
c=-1.846627205610275,
c=-1.835158854722977,
c=-1.829509645700455,
c=-1.816294878721237,
c=-1.802436143159866,
c=-1.72191509604454,
c=-1.701700896024704,
c=-1.629432529211044,
c=-1.625413745641708,
c=-1.536243289709091,
c=-1.501716822385788,
c=-1.447008818387985
,c=-1.0,
c=0.0
]

realroots(P(11));
Maxima encountered a Lisp error: Error in PROGN [or a callee]:
Bind stack overflow.Automatically continuing.To reenable the Lisp debugger set *debugger-hook* to nil.

How to draw centers in Maxima :
# definition
P(n):=if n=0 then 0 else P(n-1)^2+c;
# compute centers and put in table
c:allroots(P(4));

# make list with real part of centers
rec:makelist(realpart(rhs(c[i])),i,1,length(c));

# make a list with imaginary part of centers:
imc:makelist(imagpart(rhs(c[i])),i,1,length(c));

# draw on the screen
plot2d([discrete,rec,imc],[gnuplot_curve_styles, ["with points"]])$
# or to the eps file C:\Program Files\wxMaxima\m1.eps

plot2d([discrete,rec,imc],[gnuplot_curve_styles, ["with points"]],[gnuplot_term,ps],[gnuplot_out_file,"m1.eps"]);
--------------------------
# all in one procedure (draw on the screen) :
draw_centers(p):=block( P(n):=if n=0 then 0 else P(n-1)^2+c, r:allroots(P(p)), rec:makelist(realpart(rhs(r[i])),i,1,length(r)), imc:makelist(imagpart(rhs(r[i])),i,1,length(r)), plot2d([discrete,rec,imc],[gnuplot_curve_styles, ["with points"]]) );

# all in one procedure (draw on the screen) with the same range of x and y :
draw_centers(p):=block( P(n):=if n=0 then 0 else P(n-1)^2+c, r:allroots(P(p)), rec:makelist(realpart(rhs(r[i])),i,1,length(r)), imc:makelist(imagpart(rhs(r[i])),i,1,length(r)), plot2d([discrete,rec,imc],[style,points],[x, -2.5,1.5],[y,-2,2]))$
# all in one procedure to the eps file C:\Program Files\wxMaxima\m1.eps
draw_centers_to_file(p,filename):=block( P(n):=if n=0 then 0 else P(n-1)^2+c, r:allroots(P(p)), rec:makelist(realpart(rhs(r[i])),i,1,length(r)), imc:makelist(imagpart(rhs(r[i])),i,1,length(r)), plot2d([discrete,rec,imc],[style,points],[x, -2.5,1.5],[y,-2,2],[gnuplot_term,ps],[gnuplot_out_file,filename]))$;
# example of use :
draw_centers_to_file(6,"m6.eps");


How to get list of coefficients:
definition of polynomial ( by recurence relation)
P(n):=if n=0 then 0 else P(n-1)^2+c;
degree of polynomial:
degreeOfP(n):=hipow(expand(P(n)),c);
procedure:
give_coefficients(n):=block( P(n):=if n=0 then 0 else P(n-1)^2+c, Pn:expand(P(n)), degree:hipow(Pn,c), a:makelist(coeff(Pn,c,degree-i),i,0,degree) );
example of use:
(%i7) give_coefficients(3);
(%o7) [1,2,1,1,0]
check:
(%i8) display(z:ratsimp(P(3)));
z=c^4+2*c^3+c^2+c