Let ${\Bbb Q}_p(\zeta_{p^n})/{\Bbb Q}_p$ be a cyclotomic abelian extension of degree $p^n - p^{n-1}$ and define $G \colon= {\mathrm{Gal}}({\Bbb Q}_p(\zeta_{p^n})/{\Bbb Q}_p)$, for which we set $G = \langle \sigma \rangle$.
For the representation $\rho \colon G \to {\Bbb C}$, ${\frak f}_{\rho}$ is defined as follows$\colon$ \begin{equation*} {\frak f}_{\rho} \colon= \underset{i = 0,\cdots}{\Sigma} \phantom{i} \frac{g_i}{g_0}(\chi_{\rho}(1) - \chi_{\rho}(G_i)) \quad(\chi_{\rho} ~{\mathrm{is~the ~character~ of}}~\rho), \end{equation*} where $G_i \colon= \{ \sigma \in G ~|~ \sigma(\pi) \equiv \pi \phantom{I} {\mathrm{mod}}( \pi^{i+1}) \}$ and $g_i = |G_i|$.
Let us $\sharp(k)$ be the number of irreducible representations $\rho's$ of $G$ having their Artin conductor ${\frak f}_{\chi}$ exactly $p^k$,