In this note, the main focus is on a question about transcendental entire functions mapping $\mathbf{Q}$ into $\mathbf{Q}$ (which is related to a Mahler's problem). In particular, we prove that, for any $t>0$, there is no a transcendental entire function $f\in\mathbf{Q}[[z]]$ such that $f(\mathbf{Q})\subseteq\mathbf{Q}$ and whose denominator of $f(p/q)$ is $O(q^{t})$, for all rational numbers $p/q$, with $q$ sufficiently large.

We construct modular forms on $\Gamma_0(2)$ in which all of their zeros in the fundamental domain for $\Gamma_0(2)$ are on a part of a specific geodesic inside the domain not on the boundary of the domain.