Modular Forms and Automorphic Forms

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.