Certain quaternary quadratic forms of level 48 and their representation numbers

In this paper, we find a basis for the space of modular forms of weight $2$ on ${\mathrm{\Gamma }}_1(48)$. 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 $\{1,2,4,8,16\}$.