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) $?