Determine which divisor occurs in the general Hecke-level relation

Determine the precise divisor relating the weight-4 newform level to the discriminant data of each lattice in the census, beyond the established divisibility by gcd(exp D(L), |det L|/2).

Background

For the chain (1,2,1,3), the newform level equals |det L|/2. Across the full census, the paper proves computationally that the level divides gcd(exp D(L), |det L|/2), while also noting that equality holds only in some cases.

The authors state that the exact divisor selected by the general vector-valued Shimura-lift construction remains unresolved.