Periodicity over the prime indices of a multiplicative sequence implies periodicity?

Question

I have a real sequence $(a_p)$ indexed by the prime numbers which takes values -1, 0, or 1, having the property that $a_p=a_q$ whenever $p\equiv q \pmod m$, where $m$ is a fixed integer $>2$. I'm wondering if by extending multiplicatively the sequence over the natural numbers, that is, by imposing $a_{rs}=a_r a_s$ for every natural numbers $r$ and $s$, I obtain a periodic sequence, i.e., $(a_n)$ such that $a_i=a_j$ whenever $i\equiv j \pmod m$.

What kind of sequence should I start with for the extended sequence be periodic modulo $m$?

Do that conditions suffice?

Thanks in advance for any ideas or suggestions.

Answer 1

Look for sequences with values in $\{-1,0,1\}$.

Assume that $|a_p|>1$ for some integer. Then the subsequence $a_{p^n}$ tends to $\pm\infty$, which is not possible for a periodic sequence.

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