Existence and uniqueness of the ordinary p-root

For every prime natural number $p$, every positive natural number $N$, every natural number $k\ge 2$, and every pair of unital ring homomorphisms $\iota:\overline{\mathbb Q}\to\mathbb C$ and $\iota_p:\overline{\mathbb Q}\to\mathbb C_p$, where $\overline{\mathbb Q}$ is the algebraic closure of $\mathbb Q$ and $\mathbb C_p$ is the completed algebraic closure of $\mathbb Q_p$, let $f$ consist of a complex cusp form $F$ of weight $k$ for $\Gamma_1(N)$ (viewed in $\mathrm{GL}_2(\mathbb R)$), a Dirichlet character $\varepsilon$ modulo $N$ with values in $\overline{\mathbb Q}$, and coefficients $a_n\in\overline{\mathbb Q}$ for every natural number $n$, such that the coefficient of degree $n$ of the width-one $q$-expansion of $F$ is $\iota(a_n)$, $a_1=1$, and, for every $\gamma=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\Gamma_0(N)$ and every $z$ in the complex upper half-plane, $F(\gamma z)=\iota(\varepsilon(d\bmod N))(cz+d)^kF(z)$; further require, for every prime natural number $\ell$ and every such $z$, the identity $\ell^{-1}\sum_{b=0}^{\ell-1}F((z+b)/\ell)+\iota(\varepsilon(\ell\bmod N))\ell^{k-1}F(\ell z)=\iota(a_\ell)F(z)$. If $\|\iota_p(a_p)\|=1$, then there exists exactly one $\alpha\in\mathbb C_p$ satisfying both $\|\alpha\|=1$ and $\alpha^2-\iota_p(a_p)\alpha+\iota_p(\varepsilon(p\bmod N))p^{k-1}=0$, where the norms are the $p$-adic norms on $\mathbb C_p$ and $p$ in the polynomial is its natural image there. Uniqueness is among roots of norm one, rather than among all polynomial roots. The quantifiers include $N=1$, $k=2$, $p=2$, and primes dividing $N$; the character is evaluated with its extension by zero on nonunits, so when $p\mid N$ the constant term is zero. No coprimality of $p$ and $N$, primitivity of the character, or compatibility between the two homomorphisms beyond their common domain is assumed. The hypotheses exclude $N=0$ and $k<2$, so the natural-number subtraction $k-1$ is the usual positive integer subtraction here.