Let $\text{S}_k^{new}(\Gamma_0(N),\chi)$ be the space of newforms. We call $f\in\text{S}_k^{new}(\Gamma_0(N))$ a newform if $f$ is a Hecke eigenform i.e $\text{T}_nf=\lambda_nf$ ($\text{T}_n$ hecke operator) for all $n$ and $f$ is normalized by setting the first coefficient $a(1)=1$ in the Fourier expansion of $f$. It is well known, that two different newforms in $\text{S}_k^{new}(\Gamma_0(N),\chi)$ are orthogonal relative to the Petersson inner product.
Let $f\in\text{S}_k^{new}(\Gamma_0(N),\chi)$, $g\in \text{S}_k^{new}(\Gamma_0(M),\chi)$ be newforms and $n$, $m$ positive integers, is then $<V_nf,V_mg>=0$? Where $V_d$ is the map $V_d:\text{S}_k^{new}(\Gamma_0(N),\chi)\to \text{S}_k^{new}(\Gamma_0(dN),\chi)$ given by $f(z)\mapsto f(dz)$.