Brent's polynomial interpolation conjecture for the normalized Hecke-group form $H_{4,m}$

Let $H_{4,m}$ have Fourier expansion

$$H_{4,m}(z)=\sum_{n=0}^{\infty}\beta_n(m)q_m(z)^n,$$

where $q_m(z)=\exp(2\pi i z/\lambda_m)$ and $m\geq3$. Brent's $H_{4,m}$ interpolation conjecture. For every $n$, there is a polynomial $B_n(x)\in\mathbb{Q}[x]$ such that $\beta_n(m)=B_n(m)$ for $m=3,4,\ldots$; $B_0(x)\equiv1$, and for positive $n$ the degree of $B_n$ is $3n-1$. The conjecture further gives the displayed formulas for $B_1$ and $B_2$, the factorization and sign conditions according as $n\in\mathscr Q$ or not, and the leading-coefficient and denominator divisibility assertions for primes. Here $\mathscr Q$ is the set of positive integers not represented by $x^2+xy+y^2$. Numerical computations support the claim in the tested range, with vanishing coefficients predicted precisely at $m=6$ for indices in $\mathscr Q$.