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New congruences are found for Andrews' smallest parts partition functionspt(n). The generating function for spt(n) is related to the holomorphic partalpha(24z) of a certain weak Maass form M(z) of weight 3/2. We show that anormalized form of the generating function for spt(n) is an eigenform modulo 72for the Hecke operators T(p^2) for primes p > 3, and an eigenform modulo t fort = 5, 7 or 13 provided that (t, 6p) = 1. The result for the modulus 3 wasobserved earlier by the author and considered by Ono and Folsom. Similarcongruences for higher powers of t (namely 5^6, 7^4 and 13^2) occur for thecoefficients of the function alpha(z). Analogous results for the partitionfunction were found by Atkin in 1966. Our results depend on the recent resultof Ono that M[p](z/24) is a weakly holomorphic modular form of weight 3/2 forthe full modular group where  M[p](z) = M(z)|T(p^2) - chi(p)(1 + p)M(z).

Congruences for Andrews' spt-function modulo 32760 and extension of Atkin's Hecke-type partition congruences