Congruences for the coefficients of modular forms and for the coefficients of 𝑗(𝜏)

Abstract

x = exp 2ri-T, im r > 0, have been given by D. H. Lehmer [1], J. Lehner [2; 3], and A. van Wijngaarden [4]. The moduli for which congruence properties have been determined are products of powers of 2, 3, 5, 7, 11. Thus Lehner has shown that if n>I is divisible by 2a3b5c7dl le, where a, b, c, d> 1 and e= 1, 2, 3 then c(n) is divisible by 23a+832b+35c+1 7dl 1e. In this note we give several congruence properties modulo 13, derived from some general congruences for the coefficients of certain modular forms and an explicit formula for the coefficients c(n). These general congruences are of interest in themselves and will be proved here as well. If n is a non-negative integer, define p,(n) as the coefficient of xn in fl(i -xn)r; otherwise define Pr(n) as zero.2 (Here and in what follows all products are extended from 1 to oo and all sums from 0 to 0o, unless otherwise stated.) Special cases of identities proved by the author in [5] and [6] follow: Let p be a prime >3. Set 8=(p-1)/12, A=(p2-1)/12. Then