n<0}} b_m(n,1) e(n\bar\tau).$$ This implies that $$\overline{\tfrac{\partial}{\partial\bar\tau} F_{m}(\tau,1)}=-2\pi i \sum_{\substack{ n<0}} \overline{b_m(n,1)} n
e(-n\tau)$$ is a holomorphic modular form of weight $2$ for $\Gamma'$ and therefore has to vanish identically. We obtain the assertion.

We have $b_m(0,1)=24\sigma_m(1)$.

Hilbert modular surfaces and Hirzebruch-Zagier divisors {#sect:3}
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Let $p\equiv 1 \pmod{4}$ be a prime and consider the real quadratic field $F={\mathbb{Q}}(\sqrt{p})$. We write ${\mathcal{O}}_F$ for the ring of integers in $F$ and $\partial_F=(\sqrt{p})$ for the different ideal. Moreover, we write $\epsilon_p(x)=(\frac{x}{p})$ for the quadratic character of $F$. The conjugation in $F$ is denoted by $x\mapsto x'$ and the norm of $x\in F$ by ${\operatorname{N}}(x)=xx'$.

The Hilbert modular group $\Gamma={\operatorname{SL}}_2({\mathcal{O}}_F)$ acts on ${\mathbb{H}}^2$ in the usual way. We denote by $X=\Gamma{\backslash}{\mathbb{H}}^2$ the Hilbert modular surface associated to $F$. We will use $z=(z_1,z_2)$ as a standard variable on ${\mathbb{H}}^2$ and write $z_1=x_1+iy_1$, $z_2=x_2+iy_2$ for the decomposition in real and imaginary parts. Recall that a Hilbert modular form $H(z_1,z_2)$ is called symmetric if $H(z_1,z_2)=H(z_2,z_1)$. It is called skew-symmetric if $H(z_1,z_2)=-H(z_2,z_1)$.

It is well known that the Hilbert modular group can also be viewed as a discrete subgroup of the orthogonal group of the rational quadratic space $$V=\{M\in \operatorname{Mat}_2(F);\; {}^tM=M'\}
= \{{\left(\begin{smallmatrix} a & \lambda \\ 