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](https://link.springer.com/article/10.1007/BF01420529)).


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](https://math.stackexchange.com/questions/2467734/basis-for-modular-forms-over-congruence-subgroup). What about $M\_{k+\frac{1}{2}}(4N)$ and $M\_{k+\frac{1}{2}}^+(4N)$ ?



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


