Let $f$ be a square-free integer and denote by $\Gamma_0(f)^+$ the normalizer of $\Gamma_0(f)$ in $\SL(2,\R)$. We find the analogues of the cusp form $\Delta$ for the groups $\Gamma_0(f)^+$.

We begin by reviewing Monstrous Moonshine. The impact of Moonshine on algebra has been profound, but so far it has had little to teach number theory. We introduce (using `postcards') a much larger context in which Monstrous Moonshine naturally sits. This context suggests Moonshine should indeed have consequences for number theory. We provide some humble examples of this: new generalisations of Gauss sums and quadratic reciprocity.

Let $G$ be a discrete subgroup of $\SL(2,\R)$ which contains $\Gamma(N)$ for some $N$. If the genus of $X(G)$ is zero, then there is a unique normalised generator of the field of $G$-automorphic functions which is known as a normalised Hauptmodul. This paper gives a characterisation of normalised Hauptmoduls as formal $q$ series using modular polynomials.