On Eisenstein series in $M_{2k}(Γ_0(N))$ and their applications

Zafer Selcuk Aygin

Published 2017-05-17Version 1

Let $k,N \in \mathbb{N}$ with $N$ square-free and $k>1$. Let $f(z) \in M_{2k}(\Gamma_0(N))$ be a modular form. We prove an orthogonal relation, and use this to compute the coefficients of Eisenstein part of $f(z)$ 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 Fourier coefficients of $f(z)$. We give three applications of the results. First, we give formulas for convolution sums of the divisor function to extend the result by Ramanujan. Second, we give formulas for number of representations of integers by certain infinite families of quadratic forms. And at last, we determine a formula for Fourier coefficients of $f(z)\in E_{2k}(\Gamma_0(N))$, where $f(z)$ is an eta quotient, and then we show that the set $\{ f(z) \in E_{2k}(\Gamma_0(N)), k \geq 1 \}$ is finite for all $N \in \mathbb{N}$ square-free.