  Conjectured links between the distribution of values taken by the
characteristic polynomials of random orthogonal matrices and that for certain
families of L-functions at the centre of the critical strip are used to
motivate a series of conjectures concerning the value-distribution of the
Fourier coefficients of half-integral weight modular forms related to these
L-functions. Our conjectures may be viewed as being analogous to the Sato-Tate
conjecture for integral weight modular forms. Numerical evidence is presented
in support of them.
