Independence between coefficients of two modular forms

Let $k$ be an even integer and $S_k$ be the space of cusp forms of weight $k$ on ${\text{SL}}_2(\text{ZZ})$. Let $S={\oplus }_{k\in 2\text{ZZ}}{S}_k$. For $f,g\in S$, we let R(f, g) = \{ (a_f(p), a_g(p)) \in \mathbb{P}^1(\CC)\ |\ \text{p$ is a prime} }$bethesetofratiosoftheFouriercoefficientsof$f$and$g$,where$a_f(n)$(resp\mathrm{.}$a_g(n)$)isthe$n$thFouriercoefficientof$f$(resp\mathrm{.}$g$). In this paper, we prove that if $f$ and $g$ are nonzero and $R(f,g)$ is finite, then $f=cg$ for some constant $c$. This result is extended to the space of weakly holomorphic modular forms on ${\text{SL}}_2(\text{ZZ})$. We apply it to studying the number of representations of a positive integer by a quadratic form.