Low-lying zeros of Hilbert modular $L$-functions weighted by powers of central $L$-values

Let $\mathcal{F}(\mathbf{\text{k}},\mathfrak{q})$ be the set of primitive Hilbert modular forms of weight $\mathbf{\text{k}}$ and prime level $\mathfrak{q}$, with trivial central character. We study the one-level density of low-lying zeros of $L(s,\pi )$ weighted by powers of central $L$-values $L(1\mathrm{/}2,\pi {)}^r$, where $\pi$ runs through $\mathcal{F}(\mathbf{\text{k}},\mathfrak{q})$. For $r=1,2,3$, we show that the resulting distributions $W_r$ match with predictions from Random Matrix Theory. For general $r\ge 1$, we also formulate a conjectural formula for $W_r$ based on the ``recipe’’ method.