Construction of modular function bases for Γ ₀(121) related to p(11n+6)

Motivated by arithmetic properties of partition numbers $p(n)$, our goal is to find algorithmically a Ramanujan type identity of the form $sum_{n=0}^{infty}p(11n+6)q^n=R$, where $R$ is a polynomial in products of the form $e_alpha:=prod_{n=1}^{infty}(1-q^{11^alpha n})$ with $alpha=0,1,2$. To this end we multiply the left side by an appropriate factor such the result is a modular function for $Gamma_0(121)$ having only poles at infinity. It turns out that polynomials in the $e_alpha$ do not generate the full space of such functions, so we were led to modify our goal. More concretely, we give three different ways to construct the space of modular functions for $Gamma_0(121)$ having only poles at infinity. This in turn leads to three different representations of $R$ not solely in terms of the $e_alpha$ but, for example, by using as generators also other functions like the modular invariant $j$.