A note on congruences for weakly holomorphic modular forms

Let O_L be the ring of integers of a number field L. Write q = e^{2 \pi i z}, and suppose that \begin{equation*} f(z) = \sum _{n \gg - \infty } a_f(n) q^n \in M_{k}^{!}(\operatorname {SL}_2(\mathbb {Z})) \cap O_L[[q]] \end{equation*} is a weakly holomorphic modular form of even weight k \leq 2. We answer a question of Ono by showing that if p \geq 5 is prime and 2-k = r(p-1) + 2 p^t for some r \geq 0 and t > 0, then a_f(p^t) \equiv 0 \pmod p. For p = 2,3, we show the same result, under the condition that 2 - k - 2 p^t is even and at least 4. This represents the “missing case” of Theorem 2.5 from [Proc. Amer. Math. Soc. 144 (2016), pp. 4591–4597].