  The theories of hypergeometric functions and modular forms are highly
intertwined. For example, particular values of truncated hypergeometric
functions and hypergeometric character sums are often congruent or equal to
Fourier coefficients of modular forms. In this series of papers, we develop and
explore an explicit "Hypergeometric-Modularity" method for associating a
modular form to a given hypergeometric datum. In particular, for certain length
three and four hypergeometric data we give an explicit method for finding a
modular form $f$ such that the corresponding hypergeometric Galois
representation has a subrepresentation isomorphic to the Deligne representation
of $f$. Our method utilizes Ramanujan's theory of elliptic functions to
alternative bases, commutative formal group laws, and supercongruences. As a
byproduct, we give a collection of eta quotients with multiplicative
coefficients constructed from hypergeometric functions. In the second paper, we
discuss a number of applications, including explicit connections between
hypergeometric values and periods of these explicit eta quotients as well as
evaluation formulae for certain special $L$-values.
