Being in need of some understanding about p-adic Mellin transforms, I got myself a copy of Automorphic Representations and L-Functions by Goldfeld & Hundley. I have two questions regarding the content there.
(1) Let $f$ be of the form: $$f\left(\mathfrak{z}\right)=\sum_{m=0}^{p^{M}-1}a_{m}\left[\mathfrak{z}\overset{p^{M}}{\equiv}m\right]$$ where $M$ is a non-negative integer, where $\mathfrak{z}\in\mathbb{Q}_{p}$, where $a_{0},\ldots,a_{p^{M}-1}$ are complex constants with $a_{0}=0$, and where $\left[\mathfrak{z}\overset{p^{M}}{\equiv}m\right]$ is equal to $1$ whenever $\mathfrak{z}$ is congruent to $m$ modulo $p^{M}$, and is equal to $0$ otherwise.
In order to proceed, I need to find the conductor of $f$. I follow the book's instructions like so.
First, it says that for all but finitely many integers $m$, $f$ will be identically zero on $p^{m}\mathbb{Z}_{p}^{\times}$. This is true: when $m<0$, every element of $p^{m}\mathbb{Z}_{p}^{\times}$ has a $p$-adic magnitude $>1$, and is thus outside of $\mathbb{Z}_{p}$—the support of $f$. On the other hand, since $m\geq M$ and $\mathfrak{z}\in p^{m}\mathbb{Z}_{p}^{\times}$ implies $\mathfrak{z}\overset{p^{M}}{\equiv}0$, we have that $f\left(p^{m}\mathbb{Z}_{p}^{\times}\right)=f\left(0\right)=0$ for all $m\geq M$. Thus, $f$ is non-identically zero on $p^{m}\mathbb{Z}_{p}^{\times}$ for $m\in\left\{ 0,\ldots,M-1\right\}$.
For each $m$ for which $f$ is not identically zero on $p^{m}\mathbb{Z}_{p}^{\times}$, the book then says to find an integer $N_{m}$ so that, for each $\mathfrak{a}\in\mathbb{Z}_{p}^{\times}$, $f\left(p^{m}\mathfrak{z}\right)$ is constant for all $\mathfrak{z}\in\mathfrak{a}\left(1+p^{N_{m}}\mathbb{Z}_{p}\right)$. I find that $N_{m}=M-m$. The book says that the conductor $N$ is then the largest of these$ N_{m}$s, which is $M$. Hence: $f$ has conductor M$.
IS THIS CORRECT? Please answer “Yes, it is correct,” or “No, it is not correct”, and—if the latter—please explain what is wrong.
(2) In order to compute the $p$-adic Mellin transform and inverse transform for an $f$ of conductor $N$, I need to use “normalized unitary multiplicative characters” (NUMC) on $\mathbb{Q}_{p}^{\times}$ which are “characters modulo $p^{N}$”. Let $u_{p}:\mathbb{Q}_{p}\rightarrow\mathbb{Z}_{p}^{\times}$ be the function $u_{p}\left(\mathfrak{z}\right)\overset{\textrm{def}}{=}\mathfrak{z}\left|\mathfrak{z}\right|_{p}$, where $\mathfrak{z}\in\mathbb{Q}_{p}$ and $\left|\cdot\right|_{p}$ is the $p$-adic absolute value; let $v_{p}\left(\mathfrak{z}\right)$ denote the $p$-adic valuation of $\mathfrak{z}$. I found some notes which said that every NUMC $\psi:\mathbb{Q}_{p}^{\times}\rightarrow\mathbb{C}^{\times}$ is of the form: $$\psi\left(\mathfrak{z}\right)=\omega\left(u_{p}\left(\mathfrak{z}\right)\right)e^{-2\pi itv_{p}\left(\mathfrak{z}\right)}$$ for some $t\in\left[0,1\right)$ and some continuous $\omega:\mathbb{Z}_{p}^{\times}\rightarrow\mathbb{T}$ satisfying: $$\omega\left(\mathfrak{a}\mathfrak{b}\right)=\omega\left(\mathfrak{a}\right)\omega\left(\mathfrak{b}\right),\textrm{ }\forall\mathfrak{a},\mathfrak{b}\in\mathbb{Z}_{p}^{\times}$$ Thus, I ask: what are the range of values of $t\in\left[0,1\right)$ and formulas for the $\omega$s (expressed in the form $\omega\left(\mathfrak{z}\right)=e^{2\pi iX}$, where $X$ is some expression in terms of $u_{p}\left(\mathfrak{z}\right)$ and other parameters) so that I get an explicit formula:$$\psi\left(\mathfrak{z}\right)=\omega\left(u_{p}\left(\mathfrak{z}\right)\right)e^{-2\pi itv_{p}\left(\mathfrak{z}\right)}$$ for every NUMC $\psi$ mod $p^{N}$? I cannot compute integrals and evaluate sums without formulas for the quantities invovled, so assistance in obtaining these formulas would be much appreciated. A list of all NUMCs mod $p^{N}$ would also suffice.