Additive bases with coefficients of newforms

Abstract

Let f (z) = ∞ n=1 a(n)e 2πinz be a normalized Hecke eigenform in S new 2k (Γ 0 (N )) with integer Fourier coefficients. We prove that there exists a constant C(f ) > 0 such that any integer is a sum of at most C(f ) coefficients a(n). It holds C(f ) ≪ ε,k N 6k−3 16 +ε .

Exponential sums in prime fields for modular forms

“…Later Garcia and Nicolae (see [11]) extended such result for coefficients a(n) of normalized Hecke eigenforms of weight k in S new k (Γ 0 (N )). For any λ ∈ Z, the equation The proof of the last two results are strongly attached to the Waring-Goldbach problem.…”

“…Tr F/K (az + w) = a Tr F/K (z) + Tr F/K (w), for all a ∈ K, z, w ∈ F. (10) Tr F/K (a) = ra, for any a ∈ K. (11) Tr F/K (z p ) = Tr F/K (z), for any z ∈ F. (12) Throughout this section, F = F q , K = F p with q = p r and we will simply write Tr (z) instead Tr F/K (z).…”

“…We proceed by induction over ν. Before that, following properties (10) and (11) of trace function we get…”

Exponential sums in prime fields for modular forms

“…Later García and Nicolae [12] extended this result for coefficients a(n) of normalized Hecke eigenforms of weight k in S new k ( 0 (N )). More precisely, they proved that for any λ ∈ Z, the equation The proof of the above two results are connected to the identity a(p 2 ) = a 2 (p) − p k−1 and the solubility of the equation…”

“…We are studying the finite field version of this additivity problem by obtaining nontrivial exponential sums associated with coefficients of modular forms, in the sense of [24]. We are working with the class of forms that García and Nicolae [12] considered, but with Fourier coefficients evaluated only at prime powers. Our results are recorded in Corollaries 15 and 16.…”