The number of representations of arithmetic progressions by integral quadratic forms

Let f be a positive definite integral quadratic forms and let r (n, f) be the number of representations of an integer n by f. In this article, we prove that if f (z) is a modular form of weight k 2 and level N, then f(m,r)(z) is a modular form of weight k/2 and level Nm(2) (see Definition 2.3 for the definition of f(m,r)(z)). As applications, we prove that if n = 3 (mod 8), then r (n, x(2) + 7y(2) + 7z(2)) = r (n, 2x(2) + 4y(2) + 2xy + 7z(2)), and if n = 1 (mod 3), then r (n, x(2) + y(2) + 2z(2) + 3t(2) + 3w(2)) = r (n, x(2) + y(2) + 2z(2) + 2t(2) + 2zt + 6w(2)).