Brent's nonvanishing conjecture for interpolated Hecke-group cusp-form coefficients

Let $\Delta_m$ be one of $\Delta_m^{\star}$, $\Delta_m^{\dagger}$, or $\Delta_m^{\diamond}$, with Fourier expansion

$$\Delta_m(z)=\sum_{n=1}^{\infty}\tau_m(n)q_m(z)^n.$$

Brent's cusp-coefficient nonvanishing conjecture. There are corresponding polynomials $T_n\in\mathbb{Q}[x]$ satisfying $\tau_m(n)=T_n(m)$ for $m=3,4,\ldots$, together with the factorization and coefficient descriptions stated for $T_n^{\star}$, $T_n^{\dagger}$, and $T_n^{\diamond}$; none of these polynomials takes an integer greater than two to zero. Consequently, none of the coefficients $\tau_m(n)$ vanishes for $m=3,4,\ldots$. The claim concerns nonvanishing of Fourier coefficients across the three normalized cusp-form families and is supported only by the conjectured interpolating factorizations and numerical observations.