Consider the space of modular forms $M_k(N)$. Any modular form $f \in M_k(N)$ is determined by a finite number of Fourier coefficients (e.g., Sturm's bound), thus there is a finite set of Hecke operators that lets us distinguish eigenforms from each other. In fact this is true for $T_p$'s rather than $T_n$'s
Question: Does there always exist a Hecke operator $T_p$ that distinguishes eigenforms? I.e., is there always some $T_p$ acting on $M_k(N)$ with distinct eigenvalues?
This is not true for all $p$ certainly (e.g., this question), but I want to know if you can have strange situations like $f_1, f_2, f_3$ are distinct eigenforms with $a_p(f_1) = a_p(f_2)$ for $p \equiv 1, 2$ mod $4$ and $a_p(f_1) = a_p(f_3)$ for $p \equiv 3$ mod $4$, say.
I would also be interested in partial results, e.g., cuspidal newforms in weight 2.
Edit: As pointed out in a comment and an answer, it's easy to come up with counterexamples using quadratic twists. I would still like to know what happens if one restricts to "minimal" modular forms, say newforms of prime level.