where the integers e„, e l , ed depend on q and are known as normalized Hilbert coefficients. ei is sometimes written a s ei(q) to emphasize its dependence on q. It is easily seen that e , is positive . In case Q is a Cohen-Macaulay ring, Northcott [2] showed that e l is non-negative and Narita [1] showed that in this case e2 is non-negative a s w e ll. N arita gave an example of a Cohen-Macaulay ring and an m-primary ideal q such that e3 (q) is negative. The purpose of this note is to show that for any integer d 3, it is possible to construct an example of a Cohen-Macaulay ring (Q, m) of dimension d and an mprimary ideal q such that e ( q ) is negative. W e use the a rguem ents given i n [ 1 ] to o b ta in e x p lic it values of the normalized Hilbert coefficients and use them subsequently to test our examples for the claim m ade above . T o g ive a general treatment we m ust introduce some auxilliary notations which are explained at the appropriate p la c e . Throughout this note (Q, m) denotes a Cohen-Macaulay ring with infinite residue field.