<p>Let <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R">
<mml:semantics>
<mml:mi>R</mml:mi>
<mml:annotation encoding="application/x-tex">R</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> be a polynomial ring over a field and <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M equals circled-plus Underscript n Endscripts upper M Subscript n">
<mml:semantics>
<mml:mrow>
<mml:mi>M</mml:mi>
<mml:mo>=</mml:mo>
<mml:munder>
<mml:mo>⨁<!-- ⨁ --></mml:mo>
<mml:mi>n</mml:mi>
</mml:munder>
<mml:msub>
<mml:mi>M</mml:mi>
<mml:mi>n</mml:mi>
</mml:msub>
</mml:mrow>
<mml:annotation encoding="application/x-tex">M= \bigoplus _n M_n</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> be a finitely generated graded <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R">
<mml:semantics>
<mml:mi>R</mml:mi>
<mml:annotation encoding="application/x-tex">R</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>-module, minimally generated by homogeneous elements of degree zero with a graded <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R">
<mml:semantics>
<mml:mi>R</mml:mi>
<mml:annotation encoding="application/x-tex">R</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>-minimal free resolution <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper F">
<mml:semantics>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi mathvariant="bold">F</mml:mi>
</mml:mrow>
<mml:annotation encoding="application/x-tex">\mathbf {F}</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>. A Cohen-Macaulay module <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M">
<mml:semantics>
<mml:mi>M</mml:mi>
<mml:annotation encoding="application/x-tex">M</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> is Gorenstein when the graded resolution is symmetric. We give an upper bound for the first Hilbert coefficient, <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="e 1">
<mml:semantics>
<mml:msub>
<mml:mi>e</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
<mml:annotation encoding="application/x-tex">e_1</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> in terms of the shifts in the graded resolution of <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M">
<mml:semantics>
<mml:mi>M</mml:mi>
<mml:annotation encoding="application/x-tex">M</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>. When <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M equals upper R slash upper I">
<mml:semantics>
<mml:mrow>
<mml:mi>M</mml:mi>
<mml:mo>=</mml:mo>
<mml:mi>R</mml:mi>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mo>/</mml:mo>
</mml:mrow>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:annotation encoding="application/x-tex">M = R/I</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>, a Gorenstein algebra, this bound agrees with the bound obtained in <bold>[ES09]</bold> in Gorenstein algebras with quasi-pure resolution. We conjecture a similar bound for the higher coefficients.</p>
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