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 are 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 $P$.
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1$\begingroup$ How did you prove the existence of such a $C$? That might give a hint about how to show that $C^{2} = P(\phi_{0},\phi_{1})$. By the way, I think you want $\deg P = 4$. It's also more common to use $\mathcal{M}_{k}$ to denote the space of modular forms of weight $k$ (rather than of weight $2k$). $\endgroup$– Jeremy RouseCommented May 30, 2015 at 18:45
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Use the sage command
ModularFormsRing(Gamma0(11)).generators()
(for example at https://sagecell.sagemath.org)
which gives you three generators $\phi_0,\phi_1,\tilde{C}$ of degrees 2,2,4 (corresponding to 1,1,2 in your notation). Moreover it is then a simple matter of computation to find the relation $$\tilde{C}^2+a(\phi_0,\phi_1)\tilde{C}+b(\phi_0,\phi_1)=0,$$ where $a,b\in\mathbb C[T_0,T_1]$, with $a$ homogeneous of degree 2, $b$ homogeneous of degree 4. Then $$C=\tilde{C}+\frac a2,\quad P =\frac {a^2}4-b.$$
