The following peculiar p-adic integrals have arisen in my work, and I would be interested if anyone can see how to tackle them.
Let $F$ be a $p$-adic field, $\mathfrak{o}$ its ring of integers, $\mathfrak{o}^{\times}$ the units in $\mathfrak{o}$, and $\varpi$ a fixed uniformizer in $F$. Let $\psi$ be an additive character of $F$, trivial on $\mathfrak{p}$, but nontrivial on $\mathfrak{o}$. Let $\gamma_{\psi}$ be the Weil index associated to $\psi$.
Integral 1) Suppose that $p = 2$. Let $n$ be a non-negative integer. Compute $$\int_{\mathfrak{o}^{\times}} \gamma_{\psi}(\varpi^n z) d^{\times} z$$
Integral 2) Suppose that $p \neq 2$ and let $n$ be a non-negative integer. Compute $$\int_{\mathfrak{o}^{\times}} \gamma_{\psi}(\varpi^n z) \tau(z) d^{\times} z,$$ where $\tau$ is a (possibly trivial) character of $\mathfrak{o}^{\times}$ that is trivial on $1 + \mathfrak{p}$.
