 Euler factor of a coefficientwise product of Hecke eigenforms

For normalized level-one Hecke eigenforms, let ${\alpha }_p+{\beta }_p=a_p$, ${\alpha }_p{\beta }_p=p^{k-1}$, and similarly ${\gamma }_p+{\delta }_p=b_p$, ${\gamma }_p{\delta }_p=p^{k-1}$. The Hecke multiplication relations give $a_{p^r}=({\alpha }_p^{r+1}-{\beta }_p^{r+1})/({\alpha }_p-{\beta }_p)$, with the repeated-root case obtained by continuity. Multiplying the two expressions and summing four geometric series proves the displayed local factor. Multiplicativity gives an Euler product; multiplying by $\zeta (2s-2k+2)$ cancels its local numerator and produces the degree-four convolution factors.