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

Abstract

In this paper, we find the number of representations of the quadratic form $x_1^2+x_1{x}_2+x_2^2+\dots +x_{2k-1}^2+x_{2k-1}{x}_{2k}+x_{2k}^2,$ 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.

arXiv comment11 pages, minor revision, certain terms are combined to simplify formulas in Theorem 2.3