# Differential Equations Satisfied by Modular Forms

In Verrill's paper preprint here, she has the following theorem which is from a paper of Stiller. It states that

Let $\Gamma$ be a discrete subgroup of $SL_{2}(\mathbb{R})$ commensurable with $SL_{2}(\mathbb{Z})$. For $f \in M_{k}(\Gamma)$ (the space of weight $k$ modular forms) and $t \in M_{0}(\Gamma)$ (the space of meromorphic weight 0 modular forms), if $f = \sum_{n \geq 0}b_{n}t^{n}$ near $t = 0$, then there is a linear order $k + 1$ differential equation satisfied by $g(x) = \sum_{n \geq 0} b_{n}x^{n}$, of the form $$P_{k + 1}(x)\frac{d^{k + 1}g}{dx^{k + 1}} + P_{k}(x)\frac{d^{k}g}{dx^{k}} + \cdots + P_{0}(x)g = 0$$ where $P_{i}(x)$ are algebraic functions in $x$.

If we take $t$ to be a Hauptmodul for $\Gamma$, then $P_{i}(x)$ are rational functions. Hence by multiplying by a suitable polynomial, we can in fact assume that the $P_{i}(x)$'s are polynomials. My question is that how does one get explicit bounds on the degrees of these $P_{i}(x)$'s (specifically in the case when $k = 1$)?

-