Let $f\in S\_k(\Gamma\_0(N))$, $(p,N)=1$ and $T\_r:S\_k(\Gamma\_0(Np))\longrightarrow S\_k(\Gamma\_0(N))$ the trace map. Does then $T\_r(f\mid V\_p) = f\mid T\_p$ hold?



Relation between Hecke Operator and Trace map
You don't define your notations, as Olivier points out, but I'm going to assume that $V\_p$ is the map $S\_k(\Gamma\_0(N)) \to S\_k(\Gamma\_0(Np))$ given by $f(z) \mapsto f(pz)$.


Then the answer is "yes" (at least up to a constant factor, something like $p^{k-1}$, that depends on your choice of conventions); in fact, this is essentially the **definition** of the $T\_p$ operator -- pullback along one degeneracy map followed by pushforward along the other.


