  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
$\{1,2,4,8,16\}$.
