Certain quaternary quadratic forms of level 48 and their representation numbers

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 $a_1{x}_1^2+a_2{x}_2^2+b_1(x_3^2+x_3{x}_4+x_4^2)$, where $a_i$'s belong to $\{1,2,3,4,6,12\}$ and $b_i$'s belong to $\{1,2,4,8,16\}$.

16 pages, 12 tables