On the trace and norm maps from $Γ_0(\mathfrak{p})$ to $\operatorname{GL}_2(A)$

Let $f$ be a Drinfeld modular form for $Γ_0(\mathfrak{p})$. From such a form, one can obtain two forms for the full modular group $\operatorname{GL}_2(A)$: by taking the trace or the norm from $Γ_0(\mathfrak{p})$ to $\operatorname{GL}_2(A)$. In this paper we show some connections between the arithmetic modulo $\mathfrak{p}$ of the coefficients of the $u$-series expansion of $f$ and those of a form closely related to its trace, and of the coefficients of $f$ and those of its norm.

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