An Interpretation of some congruences concerning Ramanujan's -function

Some properties of τ If we put ∆(z) = D(e), Im (z) > 0, (3) then it is known that the function ∆ is, up to a constant factor, the unique cusp form of weight 12 for the group SL (2,Z). In particular, the function ∆ is, for each prime number p, an eigenfunction of the Hecke operator Tp, with corresponding eigenvalue τ(p) (cf. e.g. Hecke [6], p. 644–671). This implies the following properties, which have been conjectured by Ramanujan [16] and proved by Mordell [14]: τ(mn) = τ(m)τ(n), if (m,n) = 1 (4) τ(p) = τ(p)τ(p)− p11τ(pn−1), if p is prime. (5) These formulas allow us to compute τ(n) from the values of τ(p) for primes p.