Let $A$ and $B$ be Hecke eigenforms of some weight $k$ and level $N$. We know that there are irreducible representations $\rho_a$, $\rho_b$ of the absolute Galois group of $\mathbb{Q}$ whose trace of Frobenius of $p$ is given by $a_p$ or $b_p$, the Fourier coefficients of $A$ or $B$.

$AB$ is a modular form of weight $2k$. It isn't clearly a Hecke eigenform, but its Fourier coefficients are still algebraic numbers, and it can be expressed as a sum of Hecke eigenforms and possibly an Eisenstein series of weight $2k$.

We also have $\rho_a \otimes \rho_b$ a representation. It isn't irreducible, but can be expressed as a sum of irreducibles. The obvious hope is that this would correspond to the product, but that cannot be true because the characters don't match up. But convolution if appropriately defined would produce a form related to $\rho_a \otimes \rho_b$.

So my two questions: what is multiplication of modular forms representation-theoretically, and what is tensoring of representations on the modular forms side?