Suppose $\Lambda$ is a root lattice (the integral lattice generated by a crystallographic root system). Consider its theta series
$$\theta_{\Lambda}(q) = \sum_{a\in \Lambda} q^{(a,a)/2},$$
where $(\cdot,\cdot)$ denotes the Euclidean inner product.
My question is:
For which $\Lambda$ do we have
$$\theta_{\Lambda}(q) = 1+m\sum_{n>0}\frac{f(n)\: q^n}{1-q^n}$$
where $m$ is a nonzero integer and $f$ is a totally multiplicative arithmetic function?
Equivalently, $\theta_{\Lambda}(q) = 1+m\sum_{n>0} \sigma_f(n)\: q^n$, where the coefficients $\sigma_f(n) = \sum_{d|n} f(d)$ factorize as $\sigma_f(n) = \prod_{p|n} \frac{f(p)^{1+\nu_p(n)}-1}{f(p)-1}$ for prime $p$.
I know that the lattices $A_1, A_1^2, A_2, A_2^2, D_4$ and $E_8$ have that property, with
$$f(n) = \lambda(n),\: \left(\frac{-4}{n}\right),\: \left(\frac{-3}{n}\right),\: n \chi_3(n),\: n \chi_2(n), \: n^3$$
and
$$m = 2,\: 4,\: 6,\: 12,\: 24,\: 240$$
respectively (where $\lambda$ is the Liouville function, $\left(\frac{a}{n}\right)$ is a Kronecker symbol and $\chi_a(n)$ is the principal Dirichlet character modulo $a$). These lattices also appear in this paper as the six maximal systems of "hypercomplex integers". For the other four nonmaximal systems ($A_1^4, A_1^8, A_2^4, D_4^2$), $f$ is multiplicative but not totally so.
I also found the lattice $E_8\times E_8$, with $f(n)=n^7$ and $m=480$.
I have looked at some other theta series in the OEIS, including those for indecomposable root lattices in low dimensions, but so far I haven't found any other such lattice satisfying all the requirements. In every case I checked, either some coefficient fails to be divisible by $m$, or $\sigma_f(4)$ is not equal to $(\sigma_f(2)-1)^2+\sigma_f(2)$ as complete multiplicativity demands. It's possible that the seven I found are the only ones, but I don't know how to prove it.
I self-answered a generalization of this question on MathOverflow, which implies that the seven root lattices above are the only ones with that property.
Here is my answer for anyone interested.