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

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