  We wish to give a new proof of one of the main results of Atkin and Lehner.
Their theory depends, among other things, on a theorem characterizing forms in
S_k(Gamma_0(N)) whose Fourier coefficients satisfy a certain vanishing
condition. Our proof involves rephrasing this vanishing condition in terms of
representation theory; this, together with an elementary linear algebra
argument, allows us to rewrite the problem as a collection of local problems.
Furthermore, the classical phrasing of the theorem makes the resulting local
problems trivial; this is in contrast to the method of Casselman, whose local
problem relies upon knowledge of the structure of irreducible representations
of GL_2(Q_p). Our proof is therefore much more accessible to mathematicians who
aren't specialists in the representation theory of p-adic groups; the method is
also applicable to other Atkin-Lehner-style problems.
