 
Summary: Generating the Hecke algebra
Amod Agashe, William A. Stein
November 11, 2004
Abstract
Let T be the Hecke algebra associate to weight k modular forms for
X0 (N ). We give a bound for the number of Hecke operators Tn needed
to generate T as a Zmodule.
Introduction
In this note we apply a theorem of Sturm [S] to prove a bound on the number
of Hecke operators needed to generate the Hecke algebra as a Zmodule. This
bound was observed by to Ken Ribet, but has not been written down. In section
2 we record our notation and some standard theorems. In section 3 we state
Sturm's theorem and use it to deduce a bound on the number of generators of
the Hecke algebra.
1 Modular forms and Hecke operators
Let N and k be positive integers and let M k (N) = M k (# 0 (N)) be the Cvector
space of weight k modular forms on X 0 (N ). This space can be viewed as the
set of functions f(z), holomorphic on the upper halfplane, such that
f(z) = f [#] k (z) := (cz + d) k f # az + b
cz + d
