Fix an odd prime $p$ and an integer $r\geq 1$. Fix a Dirichlet character $\chi$ of primitive conductor $p^r$. Let $f\in S_2(p^r,\chi)$ be a normalized newform of weight $2$, level $p^r$ and nebentype $\chi$. Assume $f$ is ordinary at $p$.

Let $K_{f,p}$ be the finite extension of $\mathbb{Q}_p$ generated by the fourier coefficients of $f$. Associated to $f$ there is a Galois representation $$ \varrho_f: G_{\mathbb{Q}} \longrightarrow \mathrm{GL}_2(K_{f,p}) $$ which is unramified at all primes different from $p$. By a theorem of Mazur and Wiles, the restriction of $\varrho_f$ to a decomposition subgroup $D_p$ of $G_{\mathbb Q}$ is of the form $$ \varrho_{f|D_p}: \begin{pmatrix} \alpha_p & \star \\ 0 & \beta_p \end{pmatrix} $$ on a suitable basis. Here $\alpha_p$ and $\beta_p$ are characters of $D_p$.

Pretty much is known about the behavior of these two characters, but a precise description of them depends on the way one has constructed $\varrho_f$. To be sure what are we talking about, let me sketch (and fix) one of the various possible such constructions (all them are the same up to twist):

Let $X_r:=X_1(p^r)/\mathbb{Q}$ denote the modular curve associated to the moduli problem of classifying pairs $(E,i)$ where $E$ is an elliptic curve and $i$ is an isomorphism between the group scheme $\mu_{p^r}$ and a closed finite flat subgroup scheme of $E$. This model of $X_1(p^r)$ over $\mathbb{Q}$ satisfies that the cusp at infinity is a rational point. Don't confuse this model with another natural model $X'_1(p^r)$ this curve has over $\mathbb{Q}$, which arises by replacing the above level structure $i$ by the level structure consisting of the choice of a point of exact order $p^r$ on $E$. Curves $X_1(p^r)$ and $X'_1(p^r)$ are not isomorphic over $\mathbb{Q}$, but one is the twist of the other over $\mathbb{Q}(\mu_{p^r})$.

Let $A_f/\mathbb{Q}$ denote the abelian variety associated by Eichler-Shimura to $f$. This is a simple abelian variety over $\mathbb{Q}$ equipped with a surjection $\mathrm{Jac}(X_r) \rightarrow A_f$. The endomorphism algebra $\mathrm{End}(A_f)\otimes \mathbb{Q}$ is isomorphic to a number field $K$ such that $d=[K:\mathbb{Q}]=\mathrm{dim}(A_f)$. Let $V_p(A_f):=H^1_{\mathrm{et}}(\bar{A}_f,\mathbb{Q}_p)(1)$ denote the $p$-adic Tate module of $A_f$. The Tate twist "$(1)$" by the cyclotomic character is taken here so that $V_p(A_f)$ is Kummer self-dual, that is to say, there is an isomorphism of Galois modules $V_p(A_f) \simeq \mathrm{Hom}(V_p(A_f),\mathbb{Q}_p(1))$. The Galois group $G_{\mathbb{Q}}$ acts on $V_p(A_f)$ in a natural way, giving rise to a representation $r: G_{\mathbb{Q}} \longrightarrow \mathrm{GL}(V_p(A_f))\simeq \mathrm{GL}_{2d}(\mathbb{Q}_p)$. Since the action of $G_{\mathbb{Q}}$ commutes with the endomorphisms of $A_f$, it follows that $r$ factors through $\prod_{\wp} \mathrm{GL}_{2}(K_{\wp})$, where $\wp$ runs over the prime ideals of $K$ over $p$ and $K_{\wp}$ denotes the completion of $K$ with respect to this prime. Projecting to one of these factors yields the sough-after representation, denoted above $$ \varrho_f: G_{\mathbb{Q}} \longrightarrow \mathrm{GL}_2(K_{f,p}). $$

The question I would like to ask is: with this so-defined $\varrho_f$, what is $\alpha_p$ and what $\beta_p$? One of the two is surely the unramified character $\eta_f$ sending $\mathrm{Frob}_p$ to $a_p(f)$ (or its inverse, $\eta_f^{-1}$), and the other encodes also the cyclotomic character and the finite character $\chi$.

There are many papers where one can find explicit formulae describing $\alpha_p$ and $\beta_p$, but most often they state something like "associated to an ordinary newform $f$ there is a Galois representation whose restriction to $D_p$ is ... with $\alpha_p = ...$ and $\beta_p= ...$". My question asks for something more precise: if $\varrho_f$ is the *one* constructed above, who is $\alpha_p$ and who is $\beta_p$?