I'll begin with some background that is unnecessary for the actual question, but that might be interesting to the reader:

*Topological modular forms ($TMF$) is a generalized cohomology theory whose coefficient ring $TMF^*(pt)$ is closely related to the ring $MF_*:=\mathbb Z[c_4,c_6,\Delta]/c_4^3-c_6^2-1728\Delta$ of classical modular forms.
The main goal of the Stolz-Teichner program is to construct $TMF$ via methods of functorial quantum field theory.
More precisely, Stolz-Teichner want to realize cocycles for $TMF^*(X)$ as extended supersymmetric field theories over $X$, that is, functors from the 2-category of 0-, 1-, and 2-dimensional supermanifolds over $X$ (i.e., equipped with a map to $X$) to some algebraic target 2-category (e.g. algebras and bimodules).
In particular, an element in $TMF^*(pt)$ should be represented by a functor $Z$ from the 2-category of 0-, 1-, and 2-dimensional supermanifolds (no map to $X$) to the 2-category of algebras.*

There is a natural map from $TMF^*(pt)$ to $MF_*$ (a $\mathbb Q$-isomorphism), and, correspondingly, there is a natural way of extracting a modular form from a supersymmetric field theory $Z$. The construction goes roughly as follows. Given a supersymmetric field theory, consider the value $V:=Z(S^1_{per})$ of $Z$ on the manifold $S^1$, equipped with the periodic supermanifold structure (Ramond sector). The vector space $V$ comes equipped with an action of the semigroup of annuli, which, at the infinitesimal level, means that there are two operators $L_0$ and $\bar L_0$ acting on $V$. Moreover, the theory being supersymmetric, there is also an odd square root of $\bar L_0$, called $\bar G_0$ (the theories considered here are only half-supersymmetric: no square root of $L_0$).

*The modular form associated to $Z$ is given by evaluating the field theory on elliptic curves (with their Ramond-Ramond supermanifold structure). Letting $E_q:= \mathbb C^\times/\mathbb Z^q$, the value of the modular form at the point $q$ is the supertrace of the operator $q^{L_0}+\bar q^{\bar L_0}$ on $V$. A priori, this doesn't look holomorphic... this is where supersymmetry comes to help: the existence of an odd square root of $\bar L_0$ implies that the coefficient of $\bar q^n$ is zero whenever $n\not =0$, and so $Z(E_q)=str(q^{L_0}+\bar q^{\bar L_0})$ is indeed holomorphic as a function of $q$.*

My question is about the existence of situations (ignoring $q$) where $str(\bar q^{\bar L_0})$ is a *non-zero* constant.

(

*Notational warning:*$q$, $L_0$, $M$ below correspond to $\bar q$, $\bar L_0$, and $V$ above)

**The question:**

Let $V$ be an $N=1$ super vertex algebra that is holomorphic, in the sense that $V$ has a unique irreducible module (namely $V$ itself). Let $M$ be its unique irreducible parity-twisted module (the Ramond sector of $V$). The Ramond algebra (spanned by $L_n$ and $G_n$, $n\in\mathbb Z$) acts on $M$. In particular, we get an even operator $L_0$ and an odd operator $G_0$ acting on $M$, subject to the relation $G_0^2 = L_0 - c/24$.
That relation implies that the supertrace $str_M(q^{L_0-c/24})$ is a constant (as opposed to a power series in $q$).

Is there an example of a vertex algebra $V$ as above such that $str_M(q^{L_0-c/24})\not =0$?