# Representations by sextenary quadratic forms with coefficients $1,2,3$ and $6$ and on newforms in $S_{3} (\Gamma_0 (24), \chi )$

## TL;DR

This paper employs modular form theory to derive formulas for representations by sextenary quadratic forms with coefficients 1, 2, 3, and 6, and expresses newforms in a specific modular space as eta quotients.

## Contribution

It provides explicit formulas for counting representations by certain sextenary quadratic forms and expresses newforms in terms of eta quotients.

## Key findings

- Formulas for $N(1^{l_1},2^{l_2},3^{l_3},6^{l_6};n)$ for all $l_i$ with sum 6.
- Representation counts are explicitly computed using modular forms.
- Newforms in $S_{3} ( ext{Gamma}_0(24), ext{chi})$ are expressed as eta quotients.

## Abstract

We use theory of modular forms to give formulas for $N(1^{l_1},2^{l_2},3^{l_3},6^{l_6};n)$ for all $l_1,l_2,l_3,l_6 \in \mathbb{N}_0$, with $l_1+l_2+l_3+l_6=6$. We also apply our results to write newforms in $S_{3} (\Gamma_0 (24), \chi )$ in terms of eta quotients.