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Let $k$ be an even integer and $S_k$ be the space of cusp forms of weight $k$ on $SL_2(ZZ)$. Let $S = oplus_{kin 2ZZ} S_k$. For $f, gin S$, we let $R(f, g) = { (a_f(p), a_g(p)) in mathbb{P}^1(CC) | text{$p$ is a prime} }$ be the set of ratios of the Fourier coefficients of $f$ and $g$, where $a_f(n)$ (resp. $a_g(n)$) is the $n$th Fourier coefficient of $f$ (resp. $g$). 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 $SL_2(ZZ)$. We apply it to studying the number of representations of a positive integer by a quadratic form.
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In this note, we consider discriminant forms that are given by the norm form of real quadratic fields and their induced Weil representations. We prove that there exists an isomorphism between the space of vector-valued modular forms for the Weil representations that are invariant under the action of the automorphism group and the space of scalar-valued modular forms that satisfy some epsilon-condition, with which we translate Borcherdss theorem of obstructions to scalar-valued modular forms. In the end, we consider an example in the case of level 12.
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