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Let $E/\mathbb{Q}$ be an elliptic curve of conductor $N$ and let $f$ be its newform. Suppose $p \geq 5$ is a prime such that $p^2 \mid N$. We assume $f$ is $p$-minimal, which is equivalent to that the reduction type of $E$ at $p$ is potentially good, or that the local representation $\pi_{f,p}$ is supercuspidal. Let $\alpha = \left[\begin{matrix} 1 &0 \\ N/p &1 \end{matrix}\right]$ and write $f|[\alpha](q) = \sum_{n \geq 1} a_n q^n$. Note that $a_1 \in \mathbb{Q}(\zeta_p) = K$, so we can put $T = Norm_{K/\mathbb{Q}}(a_1)$. It turns out that $a_1$, and hence the rational number $T$, are determined by the representation $\pi_{f,p}$, which, in turn, is determined by a character of $\mathbb{F}_{p^2}^{\times}$ whose order is in the set $\{3,4,6\} \cap \{a \in \mathbb{Z}: a \mbox{ divides } p+1\}$ (For proof of these claims, see this paper by @François Brunault).

I computed $T$ in various cases and found that its prime factors always consist of $p$ and primes that are $\pm1$ mod $p$. The table of these prime divisors for $5 \leq p \leq 300$ can be found here.

My question is: is this always the case? i.e., is it true that for every $p \geq 5$ and every possible order of the character, we have the facotrization $T = p^k \prod q_i^{k_i}$, where $q_i \equiv \pm 1 \pmod{p}$? Any idea/suggestion on whether this claim holds, or on how to go about proving it, will be appreciated!

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