Let \(d \in {\mathbb N}\) be such that \(-d\) is a discriminant and denote by \({\mathcal Q}_d\) the set of equivalence classes of integral positive definite binary quadratic forms \(Q(x,y) = ax^2+bxy+cy^2\) of discriminant \(-d = b^2-4ac\). Writing \(\Gamma = \text{PSL}_2({\mathbb Z})\) the \(m\)-th Hecke trace of the singular modulus of discriminant \(-d\) is defined as \[ t_m(d) = \sum_{Q\in {\mathcal Q}_{d/\Gamma}} {{(j(z)-744)| T(m) (\alpha_Q)}\over{\omega_Q}}, \] where \(\alpha_Q\in {\mathbb H}\) is the CM point of \({\mathbb H}\) associated to \({\mathbb Q}\) and \(q\omega_Q\) the number of integral automorphs of \(Q\). Congruence properties of the \(t_m(d)\) have recently received some attention. In this paper the author proves \[ t_m(4^nd) \equiv 0 \bmod 2 \cdot 16^n \] for odd \(m\) and \(d \equiv 7 \bmod 8\).