Let $f$ be a modular form of weight 2 for the group $\Gamma_0(N)$, with Fourier expansion $$ \sum_{n\geq 0} a(n)\ q^n. $$
Let $d$ be an integer dividing $N$, and consider the Fourier series $\sum b(n)q^n$, where $$ b(n) = a \left( n / gcd(n,d) \right). $$
I want to believe that this latter series is the Fourier expansion of another modular form, perhaps it's some combination of Hecke operators acting on $f$? In my example, $N$ and $d$ are squarefree, if that helps.