  A well-known formula of Whipple relates certain hypergeometric values
$_7F_6(1)$ and $_4F_3(1)$. In this paper we revisit this relation from the
viewpoint of the underlying hypergeometric data $HD$, to which there are also
associated hypergeometric character sums and Galois representations. We explain
a special structure behind Whipple's formula when the hypergeometric data $HD$
are primitive and defined over $\mathbb Q$. In this case, by the work of Katz,
Beukers, Cohen, and Mellit, there are compatible families of $\ell$-adic
representations of the absolute Galois group of $\mathbb Q$ attached to $HD$.
For specialized choices of $HD$, these Galois representations are shown to be
decomposable and automorphic. As a consequence, the values of the corresponding
hypergeometric character sums can be explicitly expressed in terms of Fourier
coefficients of certain modular forms. We further relate the hypergeometric
values $_7F_6(1)$ in Whipple's formula to the periods of modular forms
occurred.
