The effective Chebotarev density theorem and modular forms modulo $\mathfrak{m}$

Suppose that $f$ (resp. $g$) is a modular form of integral (resp. half-integral) weight with coefficients in the ring of integers $\mathcal{O}_K$ of a number field $K$. For any ideal $\mathfrak{m}\subset \mathcal{O}_K$, we bound the first prime $p$ for which $f\mid T_p$ (resp. $g\mid T_{p^2}$) is zero ( $\mod\mathfrak{m}$). Applications include the solution to a question of Ono (2001) concerning partitions.