Products of fourier coefficients of distinct eigenforms.

Question

Let $f(z)=\sum\limits_{n=1}^{\infty}a_f(n)q^n$ and $g(z)=\sum\limits_{m=1}^{\infty}b_g(m)q^m$ be two distinct eigenforms in $S(k,N)$, the space of holomorphic cusp forms of weight $k$ and level $N$. Can something be said about the product $$\sum\limits_{m=1}^{M}\sum\limits_{n=1}^{M}a_f(n)b_g(m)$$ when $M$ is finite or $M= \infty$? If not a general result, is there a result for specific cases, e.g. $\sum\limits_{n=1}^{M}a_f(n)b_g(n)$ or for the case where N=1?