Explicit hypergeometric modularity of certain weight two and four Hecke eigenforms

Recently, Allen et al. developed the Explicit Hypergeometric Modularity Method (EHMM) that establishes the modularity of a large class of hypergeometric Galois representations in dimensions two and three. Motivated by this framework, we construct two explicit families of eta-quotients, which we call the $\mathbb{K}_4$ and $\mathbb{K}_5$ functions, from the hypergeometric background. These $\mathbb{K}_4$ and $\mathbb{K}_5$ functions are constructed using the theory of weight $1/2$ Jacobi theta functions and their cubic analogues, respectively. Using these constructions, we then express the Fourier coefficients of certain Hecke eigenforms of weight two and four in terms of finite field period functions. As an application, we obtain new identities relating the Fourier coefficients of modular forms to special values of the finite field Appell series $F_1^p$ and $F_2^p$.