Let $f(z)$ be an automorphic/modular form on $\Gamma_0(p)/\mathbb{H}$, where $p$ is some prime.
I know that the Hecke operators $T_n$ act on this space of functions whenever $(n,p)=1$.
Assume that $f(z)$ is a Hecke eigenforms for all $T_n$ with $(n,p)=1$, is it plausible that there exists a $\sigma\in SL_2(\mathbb{Z})$ such that $f(\sigma z)$ would also be a Hecke eigenform for all $T_n$ with $(n,p)=1$, and, of course $\sigma \not\in \Gamma_0(p)$.
(I understand that $f(Sz)$ is not automorphic for $\Gamma_0(p)/\mathbb{H}$.)