nt.number theory – When does the double coset representative for a congruence subgroup contain a conjugation class \$ text {SL} _2 ( mathbb {Z})?

In the newspaper L p-adic functions and p-adic periods of modular forms, Greenberg / Stevens say that if $$sigma_l: = begin {pmatrix} l & 0 \ 0 & 1 end {pmatrix}$$ is the usual Hecke operator at $$l$$ double coset representative, and $$Gamma$$ is the congruence group $$Gamma_1 (N)$$then $$g sigma_lg ^ {- 1}$$ continues to be in the same double coset class if $$l equiv 1 pmod {N}$$, for all $$g in text {SL} _2 ( mathbb {Z})$$. I thought it would be simple to see some kind of argument of the Euclidean / Restricted Lines operations, but I have a hard time doing that. What is the argument here, and what is wrong when $$N$$ do not divide $$l-1$$?