Suppose $f$ is a weight $k$ cuspidal Hecke eigenform on $\Gamma_0(N)$. Then $f(2z)$ is a weight $k$ cuspform on $\Gamma_0(2N)$.

Is it possible that $f(z)$ and $f(2z)$ can be orthogonal (regarded as forms on $\Gamma_0(2N)$)? That is, can the Petersson inner product $\langle f(z), f(2z) \rangle = 0$, where the product is taken over $\Gamma_0(2N) \backslash\mathcal{H}$?

More generally, can $\langle f(z), f(nz) \rangle = 0$ (where the product is regarded over the appropriate quotient of the upper half plane)?