Representations by sexternary quadratic forms with coefficients 1,2,5 and 10

TL;DR

This paper develops bases for certain modular form spaces and derives formulas for counting representations of integers by specific sextenary quadratic forms with coefficients 1, 2, 5, and 10.

Contribution

It provides explicit bases for modular forms spaces and new formulas for counting representations by particular sextenary quadratic forms.

Findings

Explicit bases for modular forms spaces $M_{3}(\Gamma _{0}(40),(rac{d}{ullet}))$ for specified $d$.

Formulas for the number of representations of positive integers by quadratic forms with coefficients 1, 2, 5, and 10.

Abstract

In this article, we find bases for the spaces of modular forms $M_{3}(\Gamma _{0}(40),\left( \frac{d}{\cdot }\right) )$ for $d=-4,-8,-20\text{ and }-40.$ We then derive formulas for the number of representations of a positive integer by all the diagonal sextenary quadratic forms with coefficients $% 1,2,5$ and $10$.