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 SL₂(Z). Let S=⊕_k∈2ZS_k. For f,g∈S, we let R(f,g)be the set of ratios of the Fourier coefficients of f and g defined by R(f,g):={x∈P¹(C)|x=[a_f(p):a_g(p)]for some primep}, where a_f(n)(resp. a_g(n))denotes the nth 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 SL₂(Z). We apply it to study the number of representations of a positive integer by a quadratic form.