This is certainly known.

For any prime, any reasonable sense in which $T_p$ acts on cuspforms for $\Gamma_0(p^\ell N)$ would project to $\Gamma_0(N)$ (for $p$ prime to $N$), compatibly with everything else going on. One part of the characterization of newforms is being in the kernel of such a projection.

The inevitability of this is better seen thinking of automorphic forms on adele groups, because then the corresponding $T_p$ is an integral operator that produces right $K_p=GL_2(\mathbb Z_p)$-invariant functions. The representation generated by a newform has no $K_p$-invariant vectors in it, so this projection is $0$.

This specific statement may not be explicit, but the situation is described in a number of places: Gelbart's old Princeton book, Bump's book, for example.

Edit: and there is some potential for misunderstanding about what is meant by $T_2$... Especially given this vanishing, and given that there is the $U_p$ operator ("Atkin-Lehner"), as mentioned in the other answer. Perhaps confirmation about the intention and context of the question would be good.

Edit-edit: Yes, the above discussion applies equally to waveforms (whether or not it succeeds in addressing the intended question). This is already clear classically, and is even clearer looking at automorphic forms on adele groups, because the Hecke operators at finite places have no interaction with the archimedean phenomena (holomorphic discrete series, principal series, whatever type the repn is at archimedean places).