= Solution
The drawn $x,y$ axes pass through opposite sites, rather than along the square edges. In the <D4h molecular point group> class order $(E,2C_4,C_2,2C_2',2C_2'',i,2S_4,\sigma_h,2\sigma_v,2\sigma_d)$, an <s orbital> contributes $+1$ whenever its centre is fixed and zero otherwise. Thus
$$
\chi_\Gamma=(4,0,0,2,0,0,0,4,2,0).
$$
For example, a transverse twofold rotation about the printed $x$ axis fixes sites 1 and 3, while its diagonal counterpart exchanges all four sites. Reflection in the square plane fixes every site. Taking <character inner products>, with class sizes $(1,2,1,2,2,1,2,1,2,2)$, gives
$$
\boxed{\Gamma=A_{1g}\oplus B_{1g}\oplus E_u.}
$$
These have dimensions $1+1+2=4$. The $B_{1g}$ label is tied to the printed diagonal-axis convention; rotating the transverse coordinate labels by $45^\circ$ would interchange $B_{1g}$ and $B_{2g}$.
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