Walsh eta-units at squarefree level-integral structure and Fricke values

For squarefree $M>1$, we determine the quotient of the principal cuspidal lattice of $X_0(M)$ by the sum of its primitive Atkin-Lehner character sublattices. Its invariant factors are explicit at every level. The quotient is a 2-group, and with $k\ge3$ prime divisors its exponent is exactly $2^{k-2}