= Euler factor of a coefficientwise product of Hecke eigenforms
{c}
{title2=$\sum_{r\geq0}a_{p^r}\overline b_{p^r}X^r=\frac{1-p^{2k-2}X^2}{\prod_{u\in\{\alpha_p,\beta_p\},v\in\{\overline\gamma_p,\overline\delta_p\}}(1-uvX)}$}
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.
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