Let H={-r=x+iy:y>0}. Let A>0, k>O, y=I1. Let M(Q, k, y) denote the set of functions f for which f(r)= .D=o ane2'i"rli and f(-1/T)=y(&/i)kf(T), for all T r H. Let MO(A, k, y) denote the set of feM(A, k. y) for which f((T)=O(yc) uniformly for all x as y-+, for some real c. We give a new proof that if A=2 cos(7r/q) for an integer q_3, then M(A, k, y)= MO(A, k, y). Petersson [5, p. 176] and Ogg [4] filled a gap in Hecke's work [2, p. 21] by establishing analytically the theorem below. We present here a short, elementary proof which uses no non-Euclidean geometry. THEOREM. Let A=2 cos(r/q) for an integer q>3. Then M(2, k, y)= MO (2 k, y). PROOF. Let feM(2,k,y). Let H1={reH:Ixlj?2/2,y 1} and let Cl(B(A)) denote the closure of B(A). For large y, If(r)I