**N. Kachakhidze**

## On the Representation of Numbers by the Direct Sums of Some Quaternary Quadratic Forms

**abstract:**

The systems of bases are constructed for the spaces of cusp forms
$S_{2m}(\Gm_0(5),\chi^m)$ and $S_{2m}(\Gm_0(13),\chi^m)$
for an arbitrary integer $m\geq 2$.
Formulas are obtained for the number of representations of a positive integers
by the direct sums of some quaternary quadratic forms