On a conjecture of Atkin.

Let j be the modular invariant. For the primes p ≦ 23 the q-expansion coefficients of Um( j – 744) are multiplicative as it was a Hecke eigenform modulo a power of p which increases with m. This was conjectured by Atkin on the basis of extensive numerical experiments, and is proved in this paper. The cases p = 5, 7 and 11 are under special consideration in this paper. [ABSTRACT FROM AUTHOR]