Let $\mathfrak{P}$ be a place of $\bar{\mathbb{Q}}$ above a prime number $p$, $k \geq 1$, $N \geq 1$ prime to $p$ and $\varepsilon$ a Dirichlet character mod $N$.
It is well known that a modular form in $S_{k}(N,\varepsilon)$ with algebraic integer Fourier coefficients is characterized mod $\mathfrak{P}$ by the coefficients $a_{n}$ with $n \leq \frac{Nk}{12} \prod\limits_{q \mid N} \left(1+\frac{1}{q}\right)$ (see for example https://wstein.org/books/modform/modform/newforms.html#Sturm'sT).
I’m looking for the same result on spaces of Katz modular forms mod $\mathfrak{P}$ (if it’s even true). I know reductions of characteristic zero modular forms are the same as Katz modular forms if $k \geq 2$ and either $N \neq 1$ or $p \geq 5$ (see Edixhoven, Serre’s Conjecture) and the Sturm bound for classical modular forms is therefore also true, but what happen is the other cases (namely $k = 1$ and $k \geq 2$, $N = 1$, $p \leq 3$)?
