What makes the theory of automorphic forms more complicated is the lack of names of individual elements. For modular forms, one can label all the various congruence subgroups as $\Gamma(N)$, $\Gamma_1(N)$, $\Gamma_0(N)$; one can label the Hecke operators as $T_p$-operators, diamond operators $$, Atkin-Lehner operators $w_d$ etc. With the explicit names come all the slew of explicit/messy computations. Now we try to label/parametrise things by equally complicated but somehow slightly more manageable index sets. All those correspondences are an attempt at labelling things.