Dear Mr. Michael,

Thank you for your reply. All your comments were very valuable and indeed I have been able to reach higher orders in a faster time.
I would like to ask about the following FindRoot routine (the last to number in INT are the orders at the horizon and infinity)

\[Omega]guess1 = 311/100 - (274/100) I;
\[Omega]guess2 = 516/100 - (476/100) I;
f4[y_?NumericQ] :=
INT[y, 10, 1/10, 1/10, 1, 0, 0, 0, 1, 1/10000, 10^2, 15, 15][[1]];
y0n = y /.
FindRoot[f4[y] == 0, {y, \[Omega]guess1, \[Omega]guess2}][[1]];
check = Abs[f4[y0n]];
{y0n, check}

by using the secant method I have obtained a 10^-6 error for a certain frequency. Do you know how can I reduce the error and if there is a systematic way to find the first frequencies consecutively? I have been trying to hunt the frequencies by putting the quasi normal modes of the table as my initial guesses, but this is not quite efficient.
Have a nice week.

Best,
Julián.