In Merel's recent proof [7] of the uniform boundedness conjecture for the torsion of elliptic curves over number fields, a key step is to show that for sufficiently large primes N, the Hecke operators T1, T2, …, TD are linearly independent in their actions on the cycle e from 0 to i∞ in H1(X0(N) (C), Q). In particular, he shows independence when max(D8, 400D4) < N/(log N)4. In this paper we use analytic techniques to show that one can choose D considerably larger than this, provided that N is large.