(Explicit) Basis for Kohnen's plus-space of modular forms of half integral weight

Question

Sorry if this is trivial, but I could not find any reference.

Let $k,a,b$ be integers. The space of modular forms of integer weight $M_k(\text{SL}_2(\mathbb{Z}))$ admits a basis of the form $\{ E_4^aE_6^b : 4a+6b = k \}$ where $E_4$ and $E_6$ are normalized Eisenstein series of weight 4 and 6 respectively. Similarly, the space of modular forms of half integer weight $M_{k+\frac{1}{2}}(4)$ for $\Gamma_0(4)$ has a basis of the form $ \{ \theta^aF^b : \frac{a}{2}+2b = k+ \frac{1}{2} \},$ where $\theta = 1+\sum_{n\geq1}q^{n^2}$ and $F=\sum_{n\geq1, n \text{ odd}}\sigma_1(n)q^n$ (see page 254 of Kohnen's paper).

Question: Does there exists similar 'explicit' basis for the Kohnen's plus subspace : $$M_{k+\frac{1}{2}}^+(4) = \{ f \in M_{k+\frac{1}{2}}(4) : a_f(n) = 0 \text{ if } (-1)^kn\equiv 2,3\pmod4 \,\} ?$$

Extras: What about the generalized cases of forms of level $N$ in respective spaces? For $M_k(\Gamma_0(N))$ this is answered here. What about $M_{k+\frac{1}{2}}(4N)$ and $M_{k+\frac{1}{2}}^+(4N)$ ?

Answer 1

There does exist an explicit basis when $k$ is EVEN: denote by $E_{k,4}$ the Eisenstein series $E_k(4\tau)$. Then the Rankin-Cohen brackets $[\theta,E_{k-2j,4}]_j$ for $0\le j\le\lfloor k/6\rfloor$ (with a small modification for $k=2$) form a basis of $M^+_{k+1/2}(4)$, where of course $\theta$ is the usual generator of $M_{1/2}(4)$.

The case $k$ odd is more difficult. The best I could come up with is to use brackets of $\theta$ with the two Eisenstein series of weight $k$ and quadratic character modulo 4, but the Kohnen space is only a subset of this.