## nt.number theory – Double Cosets and Weber's function

Leave $$n$$ be an odd positive integer leave $$mathcal M_n$$ is the set of all $$2$$-by-$$2$$ Primitive matrices with integral entries and with determinant. $$n$$.

Leave $$Gamma$$ be the subgroup of $$operatorname {SL} _2 ( mathbb Z)$$ generated by the matrices $$T ^ 2 = begin {pmatrix} 1 & 2 \ 0 & 1 end {pmatrix}$$ Y $$S = begin {pmatrix} 0 & -1 \ 1 & 0 end {pmatrix}$$.

So
$$Gamma = bigg lbrace begin {pmatrix} a & b \ c & d end {pmatrix}: begin {pmatrix} a & b \ c & d end {pmatrix} equiv begin {pmatrix} 1 & 0 \ 0 & 1 end {pmatrix} text {or} begin {pmatrix} a & b \ c & d end {pmatrix} equiv begin pmatrix} 0 & 1 1 & 0 end {pmatrix} text {mod} 2 bigg rbrace.$$

How many cosets are there in $$Gamma backslash mathcal M_n / Gamma$$ ?

Leave $$r, s, t$$ be positive integers suppose that $$rt = n$$, $$s <2t$$, cast $$s$$ even. There are matrices $$A, B in Gamma$$ such that $$A begin {pmatrix} n & 0 \ 0 & 1 end {pmatrix} B = begin {pmatrix} r & s \ 0 & t end {pmatrix}$$?

Motivation.

The Hauptmodul for the group. $$Gamma$$ is the function
$$mathfrak f ( tau) ^ {24} = q ^ {- 1/2} prod_ {k = 1} ^ { infty} (1 + q ^ {n-1/2}).$$
Leave $$Phi_n (X)$$ be the minimum polynomial of $$mathfrak f (n tau) 24$$ finished $$mathbb C ( mathfrak f ^ {24})$$. Is $$mathfrak f left ( frac {r tau + s} {t} right)$$ a root of $$Phi_n (X)$$?