Certain quaternary quadratic forms of level 48 and their representation numbers

TL;DR

This paper develops formulas for counting representations of integers by specific quaternary quadratic forms using modular forms of weight 2 on (48), advancing understanding of quadratic form representations.

Contribution

It constructs a basis for modular forms on (48) and derives explicit formulas for representation numbers of certain quadratic forms.

Findings

Formulas for representation numbers of specific quadratic forms derived.

Basis for modular forms on (48) established.

Enhanced methods for counting quadratic form representations.

Abstract

In this paper, we find a basis for the space of modular forms of weight $2$ on $\Gamma_1(48)$. We use this basis to find formulas for the number of representations of a positive integer $n$ by certain quaternary quadratic forms of the form $\sum_{i=1}^4 a_i x_i^2$, $\sum_{i=1}^2 b_i(x_{2i-1}^2 + x_{2i-1}x_{2i}+x_{2i}^2)$ and $a_1x_1^2 + a_2 x_2^2 + b_1(x_3^2+x_3x_4+x_4^2)$, where $a_i$'s belong to $\{1,2,3,4,6,12\}$ and $b_i$'s belong to $\{1,2,4,8,16\}$.