Let
R
R
be a polynomial ring over a field and
M
=
⨁
n
M
n
M= \bigoplus _n M_n
be a finitely generated graded
R
R
-module, minimally generated by homogeneous elements of degree zero with a graded
R
R
-minimal free resolution
F
\mathbf {F}
. A Cohen-Macaulay module
M
M
is Gorenstein when the graded resolution is symmetric. We give an upper bound for the first Hilbert coefficient,
e
1
e_1
in terms of the shifts in the graded resolution of
M
M
. When
M
=
R
/
I
M = R/I
, a Gorenstein algebra, this bound agrees with the bound obtained in [ES09] in Gorenstein algebras with quasi-pure resolution. We conjecture a similar bound for the higher coefficients.