Let $f=\sum a(n)q^n$ a modular cuspidal form of integer weight on some congruece subgroup. Suppose that $a(1)=1$ and that $f$ is proper to Hecke operators. Its is known that the coefficients $a(n)$ give a multiplitive function.
My question look naive :
If $\alpha(n)$ is a multiplicative function, can we find a modular form like $f$ in $\mathbb{Z}[[q]]$ with coefficients the $\alpha(n)$ ?
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2$\begingroup$ Hecke eigenvalues form very special multiplicative functions, this is why we love them! Another reason, besides David Loeffler's excellent answer, is that the Hecke multiplicativity relations say much more than just usual multiplicativity, and also Hecke eigenvalues grow at most polynomially by easy estimates (and much more is true, see the Ramanujan conjecture and its proof by Deligne). $\endgroup$– GH from MOCommented Mar 21, 2017 at 13:23
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Trivially not, for cardinality reasons. Any function from the set of prime powers to $\mathbf{Z}$ extends uniquely to a multiplicative function $\mathbf{Z} \to \mathbf{Z}$, and there are obviously uncountably many of these (there's a $\{\pm 1\}$-valued one for every subset of the primes). Since the set of Hecke eigenforms is obviously countable, it can't hit every multiplicative function.
answered Mar 21, 2017 at 10:25