Is every eigenvector sequence for the Hecke operators a eigenform?

Question

Let $f(n) = a_n$ be a sequence taking values in $\mathbb C$ for $n=1,2,...$. Let $T_m$ be the Hecke operators (of a fixed weight $k$) defined as usual in terms of the $a_n$. That is: $$T_m(f)(n) = \sum_{r>0, r|(m,n)}r^{k-1}a_{mn/r^2}.$$

Suppose $f$ is such that it is an eigenvalue for all the $T_m$, then is it true that $f$ is actually a cusp form for $SL_2(\mathbb Z)$? That is, is $\sum_{n\geq 1}a_nq^n$ the q-expansion of some modular form?

Also, is it true that such sequences always have an analytic (meromorphic?) continuation to the entire plane? This is certainly true if the answer to the first question is yes.

One can also generalize a little and consider $f$ that is an eigenvalue for all $T_m$ for $(m,N) = 1$ for some fixed number $N$. In this case, do they necessarily come from some congruence subgroup of level $n$?

Answer 1

The answer is no. Your form $f$ would have to be of weight $k$, and of level $SL_2(\mathbb Z)$, so should be in a finite dimensional vector space of dimension $d$. That would mean that the Hecke operators $T_n$ acting on the space of all sequences would have at most $d$ different systems of eigenvalues.

But this is not true. It is certainly possible to see this by hand. But to avoid any computation: If you take $N$ large enough, the number of different systems of eigenvalues, for the Hecke operators $T_\ell$, $\ell \nmid N$ of forms of weight $k$ and level $N$ goes to infinity. Choose $N$ such that this number is $>d$. Let $(a_\ell)_{\ell \nmid Np}$ be such a system. Since the $T_\ell$'s commute, there exists eigenforms (in your big space of sequences) for all the $T_\ell$ ($\ell$ prime, with no condition) with eigenvalues $a_\ell$ when $\ell \nmid N$. Hence there are strictly more that $d$ systems of eigenvalues appearing in the space of sequence, a contradiction.

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You could ask a slightly more difficult question by asking if $f$ is always a modular form of some level $N$ instead of $1$. But the answer would still be no: there are sequences which are eigenvectors for all the $T_n$ with system of eigenvalues different from any classical modular forms of weight $k$ and some level $N$. You can get some for instance using $p$-adic modular form of weight $k$.