Satake parameters (reviewed)

Let $f$ be newform of level $N$, weight $k$ and character $\chi$. Let $p$ be a good prime, i.e., $p \nmid N$.

The Satake parameters are the reciprocal roots of $L_p\left(p^{-(k-1)/2} t \right)$, where $$ L_p\left( t \right) = 1 -a_p t + \chi(p) p^{k-1} t^2 = \det(1 - t \, T_p),$$ $T_p$ is Hecke operator, and $a_p$ its trace.