 
Summary: Appendix by A. Agashe and W. Stein.
In this appendix, we apply a result of J. Sturm* to obtain a bound on the number of
Hecke operators needed to generate the Hecke algebra as an abelian group. This bound
was suggested to the authors of this appendix by Loc Merel and Ken Ribet.
Theorem. The ring T of Hecke operators acting on the space of cusp forms of weight k
and level N is generated as an abelian group by the Hecke operators T n with
n #
kN
12 #
pN
# 1 + 1
p
# .
Proof. For any ring R, let S k (N ; R) = S k (N ; Z) #R, where S k (N ; Z) is the subgroup of
cusp forms with integer Fourier expansion at the cusp #, and let TR = T#Z R. There is
a perfect pairing S k (N ; R) #R TR # R given by #f, T # ## a 1 (T (f )).
Let M be the submodule of T generated by T 1 , T 2 , . . . , T r , where r is the largest
integer # kN
12 # pN # 1 + 1
p # . Consider the exact sequence of additive abelian groups
