Additive bases with coefficients of newforms

Abstract

Let $f(z)={\sum }_{n=1}^{\infty }a(n)e^{2\pi inz}$ be a normalized Hecke eigenform in $S_{2k}^{\text{new}}({\Gamma }_0(N))$ 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 $C(f){\ll }_{\epsilon ,k}{N}^{\frac{6k-3}{16}+\epsilon }$.