Let $S^{\infty}_{k}(\Gamma_{0}(N))$ be the space of weakly holomorphic modular functions for $\Gamma_{0}(N)$ whose only possible poles lie at the cusp $\infty$ and vanish at all other cusps. There is a "reduced row echelon basis" for this space (see e.g. this paper) of the form $$g_{n} = q^{-n} + \sum_{m} b(n, m)q^{m},$$ where $m$ runs through some indexing set.
I am wondering if the $g_{n}$ necessarily have algebraic integer coefficients. A comment in this overflow post references Shimura's well-known text to point out that weak forms should have bases consisting of forms with algebraic coefficients. Does this imply the above reduced echelon basis must also share this property?