This is a slight generalization of a question I asked in Math StackExchange and which remains unanswered after a month. So I decided to post it here. I am sorry in advance if this is inappropriate for this site.
assume $ Lambda $ is an even trellis. Consider his theta series
$$ theta _ { Lambda} (q) = sum_ {a in Lambda} q ^ {(a, a) / 2}, $$
or $ ( cdot, cdot) $ means the Euclidean domestic product.
My question is:
For who $ Lambda $ have we
$$ theta _ { Lambda} (q) = 1 + m sum_ {n> 0} frac {f (n) : q ^ n} {1q ^ n} $$
or $ m $ is not zero and $ f $ is a fully multiplicative arithmetic function?
Examples
I only know two types of networks with this property:

Maximum orders in rational division algebras with class number 1, scaled by $ sqrt {2} $:

Dimension 1: The integers, with $ m = $ 2 and $ f (n) = lambda (n) $ is the function of Liouville.

Dimension 2: The rings of imaginary quadratic fields of discriminants $ D = 3, 4, 7, 8, 11, 19, 43, 67, 163 $. Right here $ m = frac {2} {L (0, f)} $ and $ f (n) = left ( frac {D} {n} right) $ is a Kronecker symbol, and $ L (0, f) $ is given by $ sum_ {n = 0} ^ { D } frac {n} {D} left ( frac {D} {n} right) $.

Dimension 4: The maximal orders of fully defined quaternion discriminant algebras $ D = 4, 9, 25, 49, 169 $. Right here $ m = frac {24} { sqrt {D} 1} $ and $ f (n) = n left ( frac {D} {n} right) $.

Dimension 8: The order of Coxeter in rational octonions, with $ m = $ 240 and $ f (n) = n ^ 3 $.


The two 16dimensional networks of Heterotic String Theory, $ E_8 times E_8 $ and $ D_ {16} ^ + $. Both networks have the same theta series, with $ m = $ 480 and $ f (n) = n ^ 7 $.
These include in particular all the root networks I've mentioned in the original Math.SE article.
Attempt
(Do not hesitate to skip this part)
I do not know much about modular forms, so it can contain errors. Theorem 4 in these notes implies that in an even dimension there is a level $ N $ and a character $ chi $ take values in $ { 1,0,1 } $ For who $ theta _ { Lambda} $ is a modular form of weight $ k = ( mathrm {dim} : Lambda) / $ 2. The requested property in turn implies that the Epstein zeta function of the network has an Euler product.
$$ zeta _ { Lambda} (s) propo prod_p frac {1} {1 (1 + f (p)) p ^ { s} + f (p) p ^ { 2s}} = zeta (s) prod_p frac {1} {1f (p) p ^ { s}}, $$
which in the very dimension means that $ theta _ { Lambda} $ is a proper form of Hecke (non cumulate, given the main coefficient 1); so we see that it has to act from a series of weights from Eisenstein $ klevel $ N $ and the character $ chi $, by the decomposition of the space of modular forms into subspaces Eisenstein + cuspidal.
This series Eisenstein has Fourier expansion $ E_ {k, chi} (q) = 1 (2k / B_ {k, chi}) sum ( cdots) $ or $ B_ {k, chi} $ is a generalized number of Bernoulli and the $ ( cdots) $ the part has integral coefficients. So, one possible solution would be to find those generalized Bernoulli numbers for which $ 2k / B_ {k, chi} = m $ is a negative integer (since in $ Lambda $ there must be an even number of vectors of the standard 2) and check on a casebycase basis if the associated Eisenstein series is the theta series of a network.
If this approach is correct, we can then use Tables 1 to 3 of this article, which show that the only cases of this type with $ mathrm {dim} : Lambda ge $ 4 are those given in the Examples section, as well as a certain series of Eisenstein weights 2 and 42, which do not appear to correspond to a network.
Moreover, I do not understand what happens in the case of odd dimensions, where the modular forms involved have a semiintegral weight. It seems that the concept of Hecke eigenform is defined a little differently, so the above approach may not work here. I found this answer which says that zeta functions associated with modular forms of an entire halfweight are usually devoid of Euler products. Here are also some potentially relevant questions (1, 2) about particular cases.