Parity of Fourier coefficients of modular forms (Q1358724)

If \(p(n)\) denotes the number of partitions of a nonnegative integer \(n\), then it has been conjectured by Subbarao that in an arithmetic progression \(r\pmod t\) there are infinitely many integers \(N\equiv r\pmod t\) for which \(p(N)\) is even, and there are infinitely many integers \(M\equiv r\pmod t\) for which \(p(M)\) is odd. In a previous article, the first author proved that there are infinitely many such \(N\), and that there are infinitely many such \(M\) provided there is at least one such \(M\). In this note, we generalize this result to include many modular forms \(f(z)= \sum_{n\geq N_0}a(n)q^n\) with integer Fourier coefficients. Specifically if \(f(z)\) is an integral or half-integral weight meromorphic modular form with integer coefficients, all of whose poles are at cusps, then in an arithmetic progression \(r\pmod t\) there are infinitely many \(N\equiv r\pmod t\) for which \(a(N)\) is even, and there are infinitely many \(M\equiv r\pmod t\) for which \(a(M)\) is odd provided that there is at least one such non-zero \(M\). The proof depends on the theory of \(\ell\)-adic Galois representations attached to modular forms as developed by Deligne and Serre.