Solve the aequatio for eta(q)^8 (c86) (fancy_display:false, Solve( 16*ETA(Q)^8*ETA(Q^4)^16+ETA(Q)^16*ETA(Q^4)^8=ETA(Q^2)^24,eta(q)^8)) 24 4 24 2 12 4 8 sqrt(64 eta (q ) + eta (q )) + 8 eta (q ) (d86) [eta (q) = - --------------------------------------------, 4 4 eta (q ) 24 4 24 2 12 4 8 sqrt(64 eta (q ) + eta (q )) - 8 eta (q ) eta (q) = --------------------------------------------] 4 4 eta (q ) (Don't you wish Mathematica would let you.) By the infinite product formula, eta(positive)>0. For real q, the first rhs is strictly negative (hence the wrong one for positive q) and both roots are real. Also by the product formula, eta(negative) = negative^(1/24)*positive product. Thus the phase of eta(negative)^8 = 8 pi/24, not 0. The aequatio generally fails off the real unit interval. The magnitude is usually wrong, too. --rwg