Automorphic Form and Hecke Eigenvalues

Let $F$ be a number field and $\mathbb{A}_F$ its adele ring; write $G(\mathbb{A}_F)$ for adelic points of $G$.

An automorphic form (informally, as in the letter) is a function $\phi$ on $G(F)\backslash G(\mathbb{A}_F)$ on which the spherical Hecke algebras at almost all finite places act by convolution.

Being a Hecke eigenfunction at $p$ means there is a homomorphism $\chi_p:\mathcal H(G(F_p),K_p)\to\mathbb{C}$ with $T\phi=\chi_p(T)\phi$ for all $T$.

Via Satake , $\chi_p$ determines the conjugacy class $\alpha_p$ used in the Euler product.

Example: Classical modular forms correspond to automorphic forms on $\mathrm{GL}_2$.