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Let $r_Q(n)$ be the representation number of a nonnegative integer $n$ by thequaternary quadratic form $Q=x_1^2+2x_2^2+x_3^2+x_4^2+x_1x_3+x_1x_4+x_2x_4$. Wefirst 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 wasconjectured in [Eum et al, A modularity criterion for Klein forms, with anapplication 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 $n$.

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