Schneider-Siegel theorem for a family of values of a harmonic weak Maass form at Hecke orbits

Let j(z) be the modular j-invariant function. Let τ be an algebraic number in the complex upper half plane ℍ. It was proved by Schneider and Siegel that if τ is not a CM point, i.e., [ℚ(τ): ℚ] ≠ 2, then j(τ) is transcendental. Let f be a harmonic weak Maass form of weight 0 on Γ₀(N). In this paper, we consider an extension of the results of Schneider and Siegel to a family of values of f on Hecke orbits of τ. For a positive integer m, let T_m denote the m-th Hecke operator. Suppose that the coefficients of the principal part of f at the cusp i∞ are algebraic, and that f has its poles only at cusps equivalent to i∞. We prove, under a mild assumption on f, that, for any fixed τ, if N is a prime such that N ≥ 23 and N ∉ {23, 29, 31, 41, 47, 59, 71}, then f(T_m.τ) are transcendental for infinitely many positive integers m prime to N.