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For prime $p\equiv-1\bmod d$ and $q$ a power of $p$, we obtain the slopes ofthe $q$-adic Newton polygons of $L$-functions of $x^d+ax^{d-1}\in\mathbb{F}_q[x]$ with respect to finite characters $\chi$ when $p$ is largerthan an explicit bound depending only on $d$ and $\log_p q$. The main tools areDwork's trace formula and Zhu's rigid transform theorem.

Newton polygons of $L$-functions of polynomials $x^d+ax^{d-1}$ with $p\equiv-1\bmod d$