  Let $j(z)$ be the modular $j$-invariant function. Let $\tau$ be an algebraic
number in the complex upper half plane $\mathbb{H}$. It was proved by Schneider
and Siegel that if $\tau$ is not a CM point, i.e.,
$[\mathbb{Q}(\tau):\mathbb{Q}]\neq2$, then $j(\tau)$ is transcendental. Let $f$
be a harmonic weak Maass form of weight $0$ on $\Gamma_0(N)$. 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 $\tau$.
  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 $i
\infty$ are algebraic, and that $f$ has its poles only at cusps equivalent to
$i \infty$. We prove, under a mild assumption on $f$, that for any fixed
$\tau$, if $N$ is a prime such that $ N\geq 23 \text{ and } N \not \in \{23,
29, 31, 41, 47, 59, 71\},$ then $f(T_m.\tau)$ are transcendental for infinitely
many positive integers $m$ prime to $N$.
