Let $P$ and $Q$ be two even, unimodular, positive definite quadratic forms of rank $n$. Let $r\_{k}(P)$ be the number of vectors of norm $k$, in symbols:


$$
r\_k(P)=\textrm{cardinality of }\{v\in \mathbb{Z}^n \; | \; P(v,v)=k\}
$$


It is well known that there exists a constant $c$, which is explicitly known and depends just on the rank $n$, such that if
$$ r\_k(P)=r\_k(Q) \quad \forall \, k \leq c $$
then
$$ r\_k(P)=r\_k(Q) \quad \forall \, k $$


The proof I know relies on the theory of modular forms. In particular, one looks at the associated theta series, and uses the fact that the dimension of the vector space of modular forms of given weight is known, actually one uses even an explicit basis of this vector space.


I would like to know if there exists an alternative proof which does not use modular forms.



Number of vectors of fixed norm
It is known at least since Hermite that, given integers $N$ and $D$ there are only finitely many equivalence classes of positive definite quadratic modules rank $n<N$ and discriminant $d<D$. 
(His proof (as well as the modern proof using Minkowski's convex body theorem) doesn't even allude to theta series.)


