Loop-space cohomology quotients

Let $X=\Omega S^2$ be the based loop space of the $2$-sphere, and let $R=H^*(X;\mathbb Z)$ with its usual cup product and cohomological grading. For a $12$-element subset $A$ of $\{2,3,\ldots,70\}$, define $V_A=\bigoplus_{k\in A} R^{2k}$. For each $k\in A$, let $i_k:R^{2k}\to V_A$ be the canonical inclusion into the $R^{2k}$ summand.

Define $D_A$ to be the subgroup of $V_A$ generated by all elements $i_k(u\smile v)$, where $k\in A$, and $u$ and $v$ are homogeneous elements of $R$ of positive cohomological degree with $\deg(u)+\deg(v)=2k$. In other words, $D_A$ is the direct sum, over $k\in A$, of the subgroup of $R^{2k}$ generated by all decomposable cup products in degree $2k$.

For an abelian group $G$ and a prime $p$, let $G[p^\infty]$ denote the $p$-primary torsion subgroup of $G$, meaning the subgroup of elements whose order is a power of $p$. The trivial group is considered cyclic. Call $A$ admissible if, for every prime $p$, the group $(V_A/D_A)[p^\infty]$ is cyclic.

Let $m=1000003$, which is prime. Determine the number of admissible $12$-element subsets $A$ of $\{2,3,\ldots,70\}$ such that $\sum_{k\in A} k \equiv 0 \pmod{37}$, and report the answer modulo $m$.