I want to find a basis of space of modular forms in $M_5\left(\Gamma_0(11),\left(\frac{\bullet}{11}\right)\right)$, which is of dimension $5$. Additionally, I want the basis to have explicit forms.
Here is my approach:
Using $\eta$-quotient, I found two basis elements as follows
$$\frac{\eta^{11}(z)}{\eta(11z)}\quad\text{and}\quad\frac{\eta^{11}(11z)}{\eta(z)}.$$
Using the $U$ operator, I constructed the remaining basis elements
$$\frac{\eta^{11}(z)}{\eta(11z)}\ |\ U(11),\quad\frac{\eta^{11}(11z)}{\eta(z)}\ |\ U(11),\quad\frac{\eta^{11}(11z)}{\eta(z)}\ |\ U(121),$$
where $\sum a(n)q^n\ |\ u(m):=\sum a(mn)q^n$.
However, the basis I want looks like
$$\frac{\eta^{11}(z)}{\eta(11z)}\quad\text{and}\quad\frac{\eta^{11}(11z)}{\eta(z)},$$
which have explicit forms
$$\frac{\eta^{11}(z)}{\eta(11z)}=\prod_{n=1}^\infty\frac{(1-q^n)^{11}}{1-q^{11n}}\quad\text{and}\quad\frac{\eta^{11}(11z)}{\eta(z)}=q^5\prod_{n=1}^\infty\frac{(1-q^{11n})^{11}}{1-q^n}.$$
Maybe we can use Eisenstein series or theta functions to construct such modular forms, but I am not familiar with those theories.
Any comments are appreciated!