  We study the accessory parameter problem for four-punctured spheres from the
point of view of modular forms. The value of the accessory parameter giving the
uniformization is characterized as the unique zero of a system of equations.
This gives an effective method to compute the uniformizing differential
equation. As an application, we compute numerically and study the local
expansion of the real-analytic function associating to a four-punctured sphere
the value of its uniformizing parameter, and make some observations on its
coefficients.
