Additional multiplicities for the \(\Gamma_0(6)\) congruence family

Determine whether the internal congruence family represented by the congruences for \(\Phi_9=\varphi(-q^9)/\varphi(-q)\) and \(\Psi_9=\psi(q^9)/\psi(q)\) manifests in any additional modular-congruence family beyond these two equivalent realizations over \(\Gamma_0(6)\).

Background

The paper studies multiplicities in internal congruence families: the same coefficient-divisibility pattern can occur in generating functions associated with different modular forms. For the level-6 setting, the Atkin–Lehner involution $V$ exchanges the Hauptmoduln $\xi$ and $\zeta$, corresponding respectively to the $\varphi(-q)$- and $\psi(q)$-based generating functions. The invariant function $t=\xi\zeta$ yields a quadratic extension $\mathbb{Q}(t)(\xi)$, whose two roots $\xi$ and $\zeta$ account for the two congruence realizations established in the paper.

The authors ask whether this quadratic-function-field framework permits the same congruence family to occur in any further modular form or congruence family. They note that the extension has only the identity and the involution exchanging $\xi$ and $\zeta$ as automorphisms fixing $\mathbb{Q}(t)$, suggesting—but not formally presenting as a general theorem—that no additional equivalent family exists over $\Gamma_0(6)$.