Representations by $x_1^2+2x_2^2+x_3^2+x_4^2+x_1x_3+x_1x_4+x_2x_4$

TL;DR

This paper proves a key identity for the representation numbers of a specific quaternary quadratic form and derives explicit formulas for these numbers, advancing understanding in quadratic form theory and modular forms.

Contribution

It establishes a new identity for representation numbers of a quadratic form and provides explicit formulas, confirming a conjecture and extending previous results.

Findings

Proved the identity $r_Q(p^2n)=r_Q(p^2)r_Q(n)/r_Q(1)$ for primes $p eq 13$.

Derived explicit formulas for $r_Q(n^2)$ for all integers $n$.

Confirmed a conjecture from prior research on quadratic forms.

Abstract

Let $r_Q(n)$ be the representation number of a nonnegative integer $n$ by the quaternary quadratic form $Q=x_1^2+2x_2^2+x_3^2+x_4^2+x_1x_3+x_1x_4+x_2x_4$. We first prove the identity $r_Q(p^2n)=r_Q(p^2)r_Q(n)/r_Q(1)$ for any prime $p$ different from 13 and any positive integer $n$ prime to $p$, which was conjectured in [Eum et al, A modularity criterion for Klein forms, with an application to modular forms of level 13, J. Math. Anal. Appl. 375 (2011), 28--41]. And, we explicitly determine a concise formula for the number $r_Q(n^2)$ as well for any integer $n$.