Getting the eigenvalue of a large matrix

It’s required to see the first posted version of two-dimensional
square lattice case to understand this question, “Poor rendering of
fractals” (Go for Hofstadter butterfly)
Poor rendering of fractals, Now,
I m trying to plot the Hofstadter butterfly spectra but for another system and I have a large matrix that depends on $p$,
$q$. I tried to do the same things but still, something messing, this the part of the code

F(p_, q_) := F1(p, q) + F2(p, q) + F3(p, q);

matrix(p_, q_) := Module({sigma}, sigma = p/q;(*which numerical function is required here in this case???*)  );

 attachsigma(sigma_, lst_) := {sigma, #} & /@ lst

frac = Table(p/q, {q, 2, 10}, {p, 2, q}) // Flatten // DeleteDuplicates

pq = {Numerator@#, Denominator@#} & /@ frac

Ens = Eigenvalues(#) & /@ (matrix(#((1)), #((2))) & /@ pq)

pts = Flatten(#, 1) &@MapThread(attachsigma, {frac, Ens})

plot = ListPlot(pts, 
  PlotMarkers -> Graphics({PointSize(Tiny), Point({0, 0.1})}))```