The ring of modular forms for $\Gamma_0(11)$

Question

Let $\mathcal M(11) = \oplus \mathcal M_k(11)$ be a graded algebra of modular forms for congruence group $\Gamma_0(11)$. I want to find generators and relations between them. I proved that $\dim \mathcal M_k = 2k$. There is two algebraically independent Poincare series $\phi_0, \phi_1 \in \mathcal M_1$, and also I proved that there is some form $C \in \mathcal M_2$ such that $C$ is not rational function of $\phi_0, \phi_1$, but $C^2=P(\phi_0,\phi_1),$ where $P \in \mathbb C[T_0,T_1], \deg P=2$, moreover there are no other relations on $C, \phi_0, \phi_1$ and these modular forms generate $\mathcal M$. My question is how can I find $C$ and this polynomial $P$.