  Shimura's conjectire (1963) concerns the rationality of the generating series
for Hecke operators for the symplectic group of genus g. This conjecture wes
proved by Andrianov for arbitrary genus g. For genus g=4, we explicify the
rational fraction in this conjecture. Using formulas for images of double
cosets, we first compute the sum of the generating series under the Satake
spherical map, which is a rational fraction with polynomial coefficients. Then
we recover the coefficients of this fraction as elements of the Hecke algebra
using polynomial representation of basic Hecke operators under spherical map.
Numerical examples of these fractions for special choice of Satake parameters
are given.
