I have a weak jacobi form: $\phi(\tau,z)$ of weight $0$ and index $M=3$.
We know it has a "q-expansion":
$$\phi(\tau,z)=\sum_{n\geq 0,~l}c(n,l)q^ny^l$$
where $q=e^{2\pi i \tau}$ and $y=e^{2\pi i z}$.
I furthermore know its q-expansion for some special values of the variables:
\begin{align}\label{eq1}\phi(\tau,0)=C,~C\in\mathbb{R} \text{ (q-independent)}\tag{1}\end{align}
and:
\begin{align}\label{eq2}e^{2\pi i \tau}\phi(2\tau,\tau+\frac{1}{2})=q^2\phi(q^2,-q)=1-20q-62q^2-216q^3-641q^4-1636q^5-...\tag{2}\end{align}
Here the coefficients on the R.H.S. are given by the absolute value of those appearing in the Mackay-Thompson series for conjugacy class $4C$ of the Monster group. It also turns out (via moonshine) that these are the coefficients of the q-expansion for the Hauptmodul of $\Gamma_0(4)$.
My question is the following: Can I say something about the q-expansion of
$$q^M\phi(\tau,\tau+\frac{1}{2})=\phi(\tau,\frac{1}{2})$$
(The equality comes from the "elliptic" transformation law for Jacobi forms.)
In fact I know that the above (viewed as a function of $\tau$) is a modular form of weight $0$ for the congruence subgroup $\Gamma_0(4)$.
Attempts:
Numerics seems to indicate that the coefficients I am looking for are given by differences of those for Mackay-Thompson series $4A$ and $4C$ or by those of the q-expansion of $\frac{256}{T_{4C}(q)}$ where $T_{4C}$ the Mackay- Thompson series of type $4C$.
I know I have been quite vague on the specific definition of the series. I am only asking about some general direction to look at, my thoughts were:
Specific properties of Jacobi forms allowing to relate different expansions.
Relations between different Mackay-Thompson series ("replication identities" are mentioned in the sources but I don't understand this well-enough to decide if it might be relevant).
- Exploiting the fact that the function whose q-expansion whe want to constrain is a weight $0$ modular form for $\Gamma_0(4)$.
The last one seems the most promising, since we then know it can be expressed as a rational function of the Hauptmodule (related to $T_{4C}$). I am just looking for some hints about how to nail down which precise rational function. Maybe eq. \ref{eq1} and eq. \ref{eq2} can help.
Sources:
T.Gannon: "Moonshine beyond the Monster", CUP.
J. H. Conway, S. P. Norton: "Monstrous Moonshine".
R.Borcherds: "Monstrous Moonshine and monstrous Lie Superalgebras".
Thanks in advance.