Given the IFS defined by A1,A2,...,A(k0),... and a proposed dimension q
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For each Aj, let n range from 0 to Nc.

For each n, let s range from 0 to Nc and calculate the series for
(a11z+a12)^n/(a21z+a22)^n+q
storing the sth coefficient as M[j,n,s]

Create the 4 dimensional array fancyM as follows, letting all variables range from 0 to Nc.
fancyM[m,n,r,s] = sum j=0 to k0-1 of M[j,m,r]*M[j,n,s]
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Noticing that in our case Ak = R+kG, we can create a separate function to deal with terms Ak0 and above
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For each Aj, let n range from 0 to Nc.

For each n, let s range from 0 to Nc and let l range from 0 to Lc. Calculate the series for
((tr11+g11)z+(tr12+g12))^n/((tr21+g21)z+(tr22+g22))^n+q
storing the [s,l]th coefficient as F[j,n,s,l]

Create the 4 dimensional array fancyF as follows, letting all variables range from 0 to Nc.
fancyF[m,n,r,s] = sum l=0 to Lc of Zeta[l+2q,k0] sum lp=0 to l F[j,m,r,l-lp]*F[j,n,s,l’]
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fancyL =2Re[ fancyM+fancyF]

R=(1/12)(({{-6+8I, 11I}, {4I, -6-8I}}));
G=(1/12)(({{-2-2I, -1-5I}, {4-4I, -2-10I}}));
(Subscript[A, k_]):=R+k G

ϕL[Φ_,Λ_]:=With[{dim=Length[Φ]-1},N[(∑_(m=0)^dim)((∑_(n=0)^dim)(Φ[[m+1,n+1]]Table[Λ[[m+1,n+1,r+1,s+1]],{r,0,dim},{s,0,dim}]))]]

λfunc[q_,Nc_,Lc_,k0_]:=Block[{M,fancyM,F,fancyF,fancyL,Φ0,Φ1,count},
M=Table[With[{series =N[ Series[(((((Subscript[A, k])[[1,1]]z+(Subscript[A, k])[[1,2]]))^n)/((((Subscript[A, k])[[2,1]]z+(Subscript[A, k])[[2,2]]))^(n+q))),{z,0,Nc}]]},Table[SeriesCoefficient[series,s],{s,0,Nc}]],{k,1,k0-1},{n,0,Nc}];
fancyM=Table[0,{m,0,Nc},{n,0,Nc},{r,0,Nc},{s,0,Nc}];
For[m=0,m≤Nc,m++,
For[n=0,n≤Nc,n++,
For[r=0,r≤Nc,r++,
For[s=0,s≤Nc,s++,
If[m≤n,
fancyM[[m+1,n+1,r+1,s+1]]=(∑_(k=1)^(k0-1))(M[[k,m+1,r+1]]\[Conjugate]M[[k,n+1,s+1]]);,
fancyM[[m+1,n+1,r+1,s+1]]=fancyM[[n+1,m+1,s+1,r+1]]\[Conjugate];
]]]]];
F=Table[With[{series =N[ Series[(((((t R[[1,1]]+G[[1,1]])z+(t R[[1,2]]+G[[1,2]])))^n)/((((t R[[2,1]]+G[[2,1]])z+(t R[[2,2]]+G[[2,2]])))^(n+q))),{z,0,Nc},{t,0,Lc}]]},Table[SeriesCoefficient[series,{s,l}],{s,0,Nc},{l,0,Lc}]],{n,0,Nc}];
fancyF=Table[0,{m,0,Nc},{n,0,Nc},{r,0,Nc},{s,0,Nc}];
For[m=0,m≤Nc,m++,
For[n=0,n≤Nc,n++,
For[r=0,r≤Nc,r++,
For[s=0,s≤Nc,s++,
If[m≤n,
fancyF[[m+1,n+1,r+1,s+1]]=(∑_(l=0)^Lc)(Zeta[l+2q,k0](∑_(lp=0)^l)(F[[m+1,r+1,l-lp+1]]\[Conjugate]F[[n+1,s+1,lp+1]]));,
fancyF[[m+1,n+1,r+1,s+1]]=fancyF[[n+1,m+1,s+1,r+1]]\[Conjugate];
]]]]];
fancyL=2Re[fancyM+fancyF];

Φ0=Table[0,{m,0,Nc},{n,0,Nc},{r,0,Nc},{s,0,Nc}];
Φ0[[1,1,1,1]]=1;

For[count=1,count≤52,count++,Φ0=ϕL[Φ0,fancyL];];
Φ1=ϕL[Φ0,fancyL];
((Φ1[[1,1]])/(Φ0[[1,1]]))]

secant[x0_,y0_,x1_,y1_,z_]:=x0-(y0-z)(((x1-x0)/(y1-y0)))

secantmethod[f_,z_,x1_,x2_,ϵ_,iterations_]:=Block[{k1,k2,k3,y1,y2,count},
k1=x1;
k2=x2;
y1=f[k1];
y2=f[k2];
count=1;
Print[{count,k1,y1}];

While[Abs[y1-z]>ϵ&&count<iterations,
k3=secant[k1,y1,k2,y2,z];
k1=k2;
y1=y2;

k2=k3;
y2=f[k2];

count=count+1;
Print[{count,k1,y1}];
];
]

secantmethod[λfunc[#,10,5,100]&,1,1.3,1.31,(10^(-8)),10]