Liouville convolution recipe: H(d,d) term puts characters on the cofactor; fit with mixed Eisenstein bases

In the two-variable Liouville recipe (sum over ax+by=n of G(a,b) = sum_{d|n} ((n/d - 1) H(d,d) - Q(d,d))), do not insist on H(d,d)=0. The term sum (n/d)H(d,d) becomes sum d*H(n/d,n/d) under d -> n/d, so a character on the cofactor in degree 1 is reachable. Example: x^2+y^2+z^2+2w^2, with weight chi_{-4}(a)chi_{-8}(b) and mod-8 tables, gives count 8 sum chi_8(n/d) d - 2 sum chi_8(d) d, with H(d,d) = chi_8(d) - [d odd]. Leftover parity terms die by the reindexing d -> 2d. Only degree-2 cofactor characters (like r6) need the 4-variable setup. When guessing the target formula by least squares, include mixed-character Eisenstein terms sum chi1(d)chi2(n/d) d in the basis. Without them x^2+y^2+(z^2+zw+w^2) looked like it had a cusp form, but it is 12chi12(n/d)-chi12(d)+3chi_{-4}(d)chi_{-3}(n/d)-4chi_{-3}(d)chi_{-4}(n/d), all times d.