# On the number of representations of certain quadratic forms and a formula for the Ramanujan Tau function

## TL;DR

This paper computes the number of representations of specific quadratic forms using modular forms theory and derives a new formula for the Ramanujan Tau function by comparing these results with existing formulas.

## Contribution

It introduces formulas for counting representations of certain quadratic forms and provides a new expression for the Ramanujan Tau function through modular forms analysis.

## Key findings

- Formulas for representations of quadratic forms with 14 variables.
- Fourier coefficients of newforms of level 3 and weights 7, 9, 11.
- A novel formula for the Ramanujan Tau function.

## Abstract

In this paper, we find the number of representations of the quadratic form $x_1^2+ x_1x_2 + x_2^2 + \ldots + x_{2k-1}^2 + x_{2k-1}x_{2k} + x_{2k}^2,$ for $k=7,9,11,12,14$ using the theory of modular forms. By comparing our formulas with the formulas obtained by G. A. Lomadze, we obtain the Fourier coefficients of certain newforms of level $3$ and weights $7,9,11$ in terms of certain finite sums involving the solutions of similar quadratic forms of lower variables. In the case of $24$ variables, comparison of these formulas gives rise to a new formula for the Ramanujan Tau function.