Brent's polynomial interpolation conjecture for the normalized Hecke-group form $H_{6,m}$

Let $H_{6,m}$ have Fourier expansion

$$H_{6,m}(z)=\sum_{n=0}^{\infty}\gamma_n(m)q_m(z)^n,$$

where $q_m(z)=\exp(2\pi i z/\lambda_m)$ and $m\geq3$. Brent's $H_{6,m}$ interpolation conjecture. For every $n$, there is a polynomial $C_n(x)$ such that $\gamma_n(m)=C_n(m)$ for $m=3,4,\ldots$; $C_0(x)\equiv1$, the stated formulas hold for $C_1$ and $C_2$, and for $n>2$ one has

$$C_n(x)=(x-2)(3x-2)x^{n+1}\overline d_n d_n(x),$$

where $d_n(x)$ is irreducible over $\mathbb{Q}$ of degree $2n-3$ with integer coefficients, $\overline d_n\in\mathbb{Q}$ has sign $(-1)^n$, and the asserted leading-coefficient, divisibility, and denominator formulas hold for primes. The conjecture is part of the paper's numerical program on polynomial interpolation of Fourier coefficients of Hecke-group modular forms; it is not proved.