Weyl-type bounds for twisted GL(2) short character sums

Let f be a Hecke-Maass or holomorphic primitive cusp form of full level for SL (2, Z) with normalized Fourier coefficients lambda(f)(n). Let chi be a primitive Dirichlet character of modulus p, a prime. In this article, we shorten the range of cancellation for N in the twisted GL (2) short character sum. Here, we consider the problem of cancellation in short character sum of the form S-f,S-chi(N) := Sigma(n is an element of Z)lambda(f)(n)chi(n)W(n/N). We show that, for 0< theta < 1/10, S-f,S-chi(N) <<(f,is an element of) N(3/4+theta/2)p(1/6)(pN)(is an element of) + N1-theta(pN)(is an element of), which is non-trivial if N >= p(2/3+alpha+is an element of) where alpha == 4 theta/1-6 theta. Previously, such a bound was known for N >= p(3/4+is an element of).