8 1. Modular Forms
Finding coset representatives for Γ0(N), Γ1(N) and Γ(N) in SL2(Z) is
straightforward and will be discussed at length later in this book. To make
the problem more explicit, note that you can quotient out by Γ(N) first.
Then the question amounts to finding coset representatives for a subgroup
of SL2(Z/NZ) (and lifting), which is reasonably straightforward.
Given coset representatives for a finite index subgroup G of SL2(Z), we
can compute generators for G as follows. Let R be a set of coset represen-
tatives for G. Let σ, τ ∈ SL2(Z) be the matrices denoted by S and T in
(1.1.2). Define maps s, t : R → G as follows. If r ∈ R, then there exists a
unique αr ∈ R such that Grσ = Gαr. Let s(r) = rσαr
Likewise, there is
a unique βr such that Grτ = Gβr and we let t(r) = rτβr
Note that s(r)
and t(r) are in G for all r. Then G is generated by s(R) ∪ t(R).
Proposition 1.17. The above procedure computes generators for G.
Proof. Without loss of generality, assume that I = ( 1 0
) represents the
coset of G. Let g be an element of G. Since σ and τ generate SL2(Z), it is
possible to write g as a product of powers of σ and τ. There is a procedure,
which we explain below with an example in order to avoid cumbersome
notation, which writes g as a product of elements of s(R) ∪ t(R) times a
right coset representative r ∈ R. For example, if
g = στ
then g = Iστ
for some y ∈ R. Continuing,
= s(I)(yτ)τστ = s(I)(t(y)z)τστ
for some z ∈ R. Again,
s(I)(t(y)z)τστ = s(I)t(y)(zτ)στ = · · · .
The procedure illustrated above (with an example) makes sense for arbitrary
g and, after carrying it out, writes g as a product of elements of s(R) ∪ t(R)
times a right coset representative r ∈ R. But g ∈ G and I is the right coset
representative for G, so this right coset representative must be I.
Remark 1.18. We could also apply the proof of Proposition 1.17 to write
any element of G in terms of the given generators. Moreover, we could use
it to write any element γ ∈ SL2(Z) in the form gr, where g ∈ G and r ∈ R,
so we can decide whether or not γ ∈ G.
1.4.1. Computing Widths of Cusps. Let Γ be a congruence subgroup
of level N. Suppose α ∈ C(Γ) is a cusp, and choose γ ∈ SL2(Z) such that
γ(∞) = α. Recall that the minimal h such that
the width of the cusp α for the group Γ. In this section we discuss how to