Representations by sextenary quadratic forms with coefficients $1,2,3$ and $6$ and on newforms in $S_3({\Gamma }_0(24),\chi )$

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.