# On Eisenstein series in $M_{2k}(\Gamma_0(N))$ and their applications

## TL;DR

This paper derives formulas for Fourier coefficients of Eisenstein series in modular forms, enabling applications to divisor sums, quadratic forms, and extending to more general modular forms.

## Contribution

It introduces an orthogonal relation for Eisenstein series in $M_{2k}(\Gamma_0(N))$ and uses it to explicitly compute Fourier coefficients, extending classical divisor sum and quadratic form results.

## Key findings

- Formulas for Fourier coefficients of Eisenstein series in terms of divisor sums.
- Extensions of Ramanujan's convolution sum results.
- Expressions for the number of representations of integers by quadratic forms.

## Abstract

Let $k,N \in \mathbb{N}$ with $N$ square-free and $k>1$. We prove an orthogonal relation and use this to compute the Fourier coefficients of the Eisenstein part of any $f(z) \in M_{2k}(\Gamma_0(N))$ in terms of sum of divisors function. In particular, if $f(z) \in E_{2k}(\Gamma_0(N))$, then the computation will to yield to an expression for the Fourier coefficients of $f(z)$. Then we apply our main theorem to give formulas for convolution sums of the divisor function to extend the result by Ramanujan, and to eta quotients which yields to formulas for number of representations of integers by certain families of quadratic forms. At last we give essential results to derive similar results for modular forms in a more general setting.