An upper bound on the least prime for certain modular congruences

Let $l\geq 5$ be a prime and $m$ be a positive integer. Recently Ahlgren, Allen and Tang proved several congruences for newforms, such as $f\vert_{T(Q)} \equiv f\ ({\rm mod}\ l^m)$ for all $f\in S_{l-3}^{{\rm new}}(6)$, for a set of primes $Q \equiv 1\ ({\rm mod}\ l^m)$ of positive density. We obtain a bound on the first such prime. As an application, we obtain an upper bound on primes $Q$ which satisfy certain congruences for the partition function.