Serre's conjecture says that given any odd, irreducible, continuous representation $\rho:G_{\mathbb{Q}}\rightarrow GL_2(\overline{\mathbb{F}_p})$ there is some eigenform $f$ of weight $k(\rho)$, level $N(\rho)$, and nebentype $\epsilon(\rho)$, such that $\rho$ is isomorphic to the mod $p$ representation $\bar \rho_f$ associated to $f$. Up until now, I have intuitively thought of the Serre weight $k(\rho)$ as being the *minimal* weight among all weights of eigenforms whose representations are isomorphic to $\rho$. I am no longer sure whether this is correct: I am currently reading Edixhoven's survey of Serre's conjecture, and it seems like one might be able to cook up an example where the Serre weight $k(\bar\rho_f)$ of some weight $k$ eigenform $f$ is actually *greater* than the weight of $f$, which seems strange to me... Can this happen? I.e., are there examples where the Serre weight of a representation coming from a weight $k$ eigenform is actually greater that $k$?

To give this a bit more context, fix $p\nmid N$ and let $f \in S_{k}(\Gamma_0(N))$ be an eigenform with mod $p$ representation $\bar\rho_f$. Fix an inertia subgroup $I_p\subseteq G_{\mathbb{Q}}$ at $p$ and write $I_{p,w}\subseteq I_p$ for its wild subgroup. If $\bar\rho_f\mid_{I_{p,w}}$ is nontrivial, we can uniquely write $$ \bar\rho_f\mid_{I_{p}}=\begin{pmatrix}\chi^\beta &* \\ 0& \chi^{\alpha} \end{pmatrix} $$ where $\chi$ is the mod $p$ cyclotomic character and $\alpha$, $ \beta$ are integers such that $0\leq \alpha\leq p-2$, $1\leq\beta\leq p-1$. As stated in the Edixhoven article, setting $a=\min(\alpha,\beta)$ and $b=\max(\alpha,\beta)$, we define $k(\bar\rho_f)$ to be $1+pa+b+p-1$ if $\chi^{\beta-\alpha}=\chi$ and $\bar \rho_f\mid_{G_{\mathbb{Q}_p}} \otimes \chi^{-\alpha}$ is not finite at $p$, otherwise we define it to be $1+pa+b$.

Now consider the case where $p=3$, and suppose that we have some weight 6 eigenform $f$ on $\Gamma_0(N)$ such that
$$
\bar\rho_f\mid_{I_{3}}=\begin{pmatrix}1 &* \\ 0& \chi \end{pmatrix}
$$
and $\rho_f\mid_{G_{\mathbb{Q}_3}}
\otimes \chi^{-1}$ is not finite at $3$. Then the formula above would say that $f$ has Serre weight $k(\bar \rho_f)=8$, which is *larger* than the weight of $f$. I am of course assuming that we can find such a form satisfying these conditions, but even so, I see no obvious reason why this shouldn't occur...