For the case of cusp forms, each Hecke eigenform (newform) corresponds to a cuspidal representation.
Let G be a reductive algebraic group over a number field K and let A denote the adeles of K. The group G(K) embeds diagonally in the group G(A) by sending g in G(K) to the tuple (gp)p in G(A) with g = gp for all (finite and infinite) primes p. Let Z denote the center of G and let ω be a continuous unitary character from Z(K) \ Z(A)× to C×.
A cuspidal function generates a unitary representation of the group G(A) on the complex Hilbert space
is given by The space of cusp forms with central character ω decomposes into a direct sum of Hilbert spaces where the sum is over irreducible subrepresentations of L20(G(K) \ G(A), ω) and the mπ are positive integers (i.e. each irreducible subrepresentation occurs with finite multiplicity).
A cuspidal representation of G(A) is such a subrepresentation (π, Vπ) for some ω.