Modular Forms and Numerical Explorations of Rational Approximations to $\zeta (3)$

We revisit Beukers’ modular-form proof of the irrationality of $\zeta (3)$ from the point of view of the auxiliary weight two modular form. For the Fricke group ${\mathrm{\Gamma }}_0(6{)}^{\star }$, we show that Beukers’ choice is not isolated: it belongs to a one-parameter affine family. These approximations have the same exponential decay as the classical Apéry approximations and satisfy the same denominator-growth estimate needed in Beukers’ irrationality argument. We then apply the same construction to several other genus-zero Fricke groups.