# Additive bases with coefficients of newforms

## TL;DR

This paper proves that for certain modular forms with integer coefficients, every integer can be expressed as a sum of a bounded number of these coefficients, with the bound depending on the form’s level and weight.

## Contribution

It establishes a uniform bound on the number of Fourier coefficients needed to represent any integer as their sum, depending explicitly on the form's level and weight.

## Key findings

- Every integer is a sum of at most C(f) Fourier coefficients.
- The bound C(f) depends polynomially on the level N.
- Explicit bound C(f) ollows from the form's parameters.

## Abstract

Let $f(z)=\sum_{n=1}^{\infty}a(n) e^{2\pi i nz}$ 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 $C(f)\ll_{\varepsilon,k}N^{\frac{6k-3}{16}+\varepsilon}$.