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 $n$?