Hi.


For a Dirichlet character $\chi$ mod $N$, the (weak) multiplicity one theorem states the following:


Let $f, g \in S\_k^{\text{new}}(\Gamma\_0(N), \chi)$ be such that for every $m \in \mathbb{N}$ s.t. $(m,N)=1$, we have
$$T(m)f = \lambda\_m f,~~ T(m)g = \lambda\_m g$$
i.e. $f,g$ are simultaneous eigenforms with the same eigenvalues for all the $m$-th Hecke operators, then actually, provided that $f \neq 0, g \neq 0$, $f$ and $g$ coincide up to a constant, i.e. $\mathbb{C}f = \mathbb{C}g$.


What i am wondering about is: How can one construct counterexamples when one drops the keyword 'new'? I am in particular interested in $S\_k(\Gamma\_0(p))$ (trivial character) for odd primes $p$. Is there a general 'method' for doing that?


Thanks in advance,


Fabian Werner



Counterexamples for (weak) multiplicity one
Very concretely: if $f\in S\_k^{\text{new}}(\Gamma\_0(N), \chi)$, and $d>1$ is an integer, then $f(z)$ and $f(dz)$ are elements of $S\_k(\Gamma\_0(dN), \chi)$ with the same Hecke eigenvalues at $m$ coprime with $dN$, yet they are not multiples of each other. Of course this is the same what Marc was saying, but in more classical terms.


In particular, if $dN=p$ (i.e. $d=p$, $N=1$, $\chi$ is trivial), then you get a counterexample for $S\_k(\Gamma\_0(p))$.


I suggest that you read [this famous paper](http://www.digizeitschriften.de/dms/img/?PPN=PPN235181684_0185&DMDID=dmdlog35) by Atkin and Lehner.


