  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 self-dual. If the data are also defined over $\mathbb Q$, 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 these
modular forms.
