are determined by Poincaré duality.)

[**Lemma 9 **]{}: In cases \#1, \#2, \#5, and \#6 in Table 2, we have $n_+=14$ and $n_-=6$ (so $n_+-n_-=2+6=8$). Furthermore, we have $$\begin{aligned}
(n_+^{\prime},n_+^{\prime\prime})=\begin{cases}
(12,2)\quad &\mbox{for \#1} \\
(14,0)\quad &\mbox{for \#2} \\
(13,1)\quad &\mbox{for \#5,\, \#6}\end{cases}\end{aligned}$$ and $$\begin{aligned}
(n_-^{\prime},n_-^{\prime\prime})=\begin{cases}
(3,3)\quad &\mbox{for \#1} \\
(6,0)\quad &\mbox{for \#2} \\
(5,1)\quad &\mbox{for \#5,\,\#6}\end{cases}\end{aligned}$$ In case \#3, $n_+= 11$ and $n_-=9$. (For the last case, confer the article of Hulek and Verrill \[HV\] for more detailed discussion.)

For the computations of $n_+$ and $n_-$, confer Proposition 2.4 of Hulek and Verrill \[HV\]. The Proof of Lemma 9 will be given at the end of Section 4.

Explicit Formulas
=================

We shall now give explicit formulas for the weight $3$ forms $g_Y = g_{3,\Gamma}$ for the examples in Table 2. We will denote the weight $3$ form in the $i$th case by $h_i$. By Remark 3 it suffices to compute $h_i$ for $i=8,7,3,4$, and then $h_8(\tau) = h_5(\tau/2) = h_1(\tau/4)
$, $h_7(\tau) = h_2(\tau/2)$, and $h_6(\tau) =
h_9(\tau/2) = h_4(\tau/2 -1/2)$. Two kinds of formulas suggest themselves for the $h_i$’s: as a product of $\eta$-functions or as inverse Mellin transforms of the Dirichlet series attached to Hecke characters. The 