calculus and analysis – How to expand Lie characters?

The following involves characters of affine Lie algebras, and I will be using as reference the book on CFT by Francesco et at (here are some screenshots if useful). But hopefully the post will be self-contained.

Let
$$
Theta^k_lambda=sum_{ninmathbb Z}expbig(-2pi i(knz+frac12lambda z-kn^2tau-nlambdatau-lambda^2tau/4k)big)tag{14.176}
$$

and
$$
chi^k_lambda=frac{Theta^{k+2}_{lambda+1}-Theta^{k+2}_{-lambda-1}}{Theta^2_1-Theta^2_{-1}}tag{14.174}
$$

The idea is to expand $chi$ for $lambda=k=1$, $z=0$, in powers of $q=e^{2pi i tau}$. The expected result is
$$
chi^1_1=q^{5/24}(2+2q+6q^2+8q^3+cdots)tag{14.179}
$$

How can I use Mathematica to recover eq.$14.179$ from the other two equations? The naive approach does not quite work, because $chi$ yields an indeterminate form if we take $lambda=1,z=0$ directly. And the sum does not converge for $|mathrm{re}(q)|<1$ (and it cannot be analytically continued), so the expansion around $q=0$ is only asymptotic (not a proper power series in the strict sense).