I'm trying to realize the spaces of half-integral weight modular forms for $\Gamma_{0}(8)$ as the spaces of polynomials in some modular forms of level 8. For every integer $k$, it is known that every modular form $f$ in $M_{k}(SL_{2}(\mathbb{Z}))$ can be written in the form $$f=\sum_{4a+6b=k} c_{a,b}E_{4}^{a}E_{6}^{b},$$ where the $c_{a,b}$ are complex numbers determined by $f$ and $E_{4}$ and $E_{6}$ are normalized Eisenstein series of weight 4 and 6, respectively. Now let $k\in\mathbb{Z}+1/2$. Then it is also known that every modular form $g\in M_{k}(\Gamma_{0}(4))$ is written as $$g=\sum_{\frac{1}{2}a+2b=k} d_{a,b}\theta^{a}F^{b}.$$ Here, $\theta(z)=\sum_{n=-\infty}^{\infty} q^{n^{2}}$ and $F(z)=-\frac{1}{24}(E_{2}(z)-3E_{2}(2z)+2E_{2}(4z))$. My goal is to describe the spaces $M_{k}(\Gamma_{0}(8))~(k\in\mathbb{Z}+1/2)$ in a similar way.
First I used $\theta$ and a linear combination of $E_{2}(z), E_{2}(2z), E_{2}(4z)$ and $E_{2}(8z)$. But I realized that this is not a proper choice because I found a weight 3/2 modular form with Fourier expansion $q+2q^{2}+4q^{5}+\cdots$ by MAGMA.
I guess there should be a theory on construction of bases for spaces of half-integral weight modular forms on which MAGMA is based. So my question is: what is the theory on which MAGMA algorithm for computing those bases is based?
The standard algorithm for computing half-integral weight forms is Basmaji's thesis. Basmaji explains only the case of $N$ divisible by 16, and also the thesis is in German, both of which might be obstacles for some.
There's an exposition of the algorithm in English, and an explanation how to remove the restriction on the level, in section 2.4 of Soma Purkait's thesis. Apparently her description is based on reading the code of the relevant parts of MAGMA.