Kiddie Talk: the Five Elementary Operations According to Eichler

Then Γ0(N) is a discrete subgroup of G. Note that Γ0(1) = SL2(Z). Now we make three choices that determine our space of modular forms. We pick an integer k, called the weight, a positive integer N , called the level, and a Dirichlet character χ : Z/NZ → C, called the nebentypus. If you’ve never seen modular forms before, you should immediately specialize to the easiest case N = 1, in which case χ is also necessarily trivial. Given these three choices, a modular form f : H → C is a function satisfying the following three criteria: • The function f is holomorphic on H, • The function f is holomorphic “at the cusps” (to be explained below), and • The function f transforms nicely in the following way: for every ( a b c d ) ∈ Γ0(N)