Consider a power series
$$f(q)=\sum_{n=0}^\infty a_nq^n.$$
Suppose that $f$ has a positive radius of convergence. For an automorphism $\sigma\in \operatorname{Aut}(\mathbb C)$ define
$$f^\sigma(q)=\sum_{n=0}^\infty a_n^\sigma q^n.$$
Does $f^\sigma$ have a positive radius of convergence?
What if all coefficients $a_n$ lie in a number field $K$ (that is, a finite extension of $\mathbb Q$)?