Explicit formulas for pre-modular forms Z_{r,s}^{(n,0,0)}(τ) in the Lamé case for n ≥ 5

Derive explicit expressions for the pre-modular forms Z_{r,s}^{(n,0,0)}(τ) associated with the classical Lamé equation y''(z) = [n(n + 1)℘(z; τ) + B] y(z) for integer n ≥ 5.

Background

The paper constructs pre-modular forms Z_{r,s}^{(n)}(τ) that characterize monodromy data for generalized Lamé equations and provides explicit formulas for several low-degree cases, including the Lamé equation for small n (see (1.19)).

However, for larger Lamé parameters, deriving explicit closed-form expressions becomes increasingly difficult due to high weights and complexity. The authors note that beyond certain small values of n, explicit formulas have not been obtained.