In Kubert-Lang's "Units in the modular function field II", for a fixed integer $N$, the Klein form is defined as, for $r, s \in \mathbb{Z}$ (at least one of them is not $\equiv 0 \mod N$) and for $\omega_i \in \mathbb{C}$ with $\Im \omega_1/\omega_2 \gt 0$,
$$ k_{r,s}(\omega_1, \omega_2) = \exp( - \frac{r\eta(\omega_1) + s\eta(\omega_2)}{N} \frac{r\omega_1 + s\omega_2}{N}) \sigma(\frac{r\omega_1 + s\omega_2}{N}),$$
where $\eta$ is the Weierstrass eta function, $\sigma$ is the Weierstrass sigma function, with respect to the latice $ \Lambda = \omega_1 \mathbb{Z} + \omega_2 \mathbb{Z}$.
(i.e., $\eta(\omega) = \zeta(z + \omega) - \zeta(z)$ for any $z \in \mathbb{C}$, where $\zeta$ is the Weierstrass zeta function with respect to the latice, and $$ \sigma(z) = z \Pi_{(a,b) \in \mathbb{Z}^2 - \{ 0 \}}(1 - \frac{z}{a\omega_1 + b\omega_2}) \exp(\frac{z}{a\omega_1 + b\omega_2} + \frac{1}{2} (\frac{z}{a\omega_1 + b\omega_2})^2).$$)
The authors says that, for integers $a, b \in \mathbb{Z}$, the Klein form satisfies
$$k_{r + aN, s + bN} (\omega_1, \omega_2) = (-1)^{ab + a +b } \exp(-2 \pi i \frac {(as - br)}{2N}) k_{r,s}(\omega_1, \omega_2).$$
Using the relation
$$ \sigma(z + \omega)/\sigma(z) = (-1)^{ab + a + b}\exp(\eta(\omega)(z + \frac{1}{2} \omega))$$
for $\omega = a\omega_1 + b\omega_2 \in \Lambda$, and using the Legendre relation,
in order to show the relation, we must show that
$$\exp( - \frac{(r+aN)\eta(\omega_1) + (s + bN)\eta(\omega_2)}{N} \frac{(r+aN)\omega_1 + (s + bN)\omega_2}{N} + \frac{r\eta(\omega_1) + s\eta(\omega_2)}{N} \frac{r\omega_1 + s\omega_2}{N} + \eta(a\omega_1 + b\omega_2)( \frac{r\omega_1 + s\omega_2}{N} + \frac{1}{2} (a\omega_1 + b\omega_2))) \\ = \exp(-\frac{2 \pi i (as - br)}{2N}). $$
I tried again and again, but I can't get the desired result. If the term $\eta(a\omega_1 + b\omega_2)( \frac{r\omega_1 + s\omega_2}{N} + \frac{1}{2} (a\omega_1 + b\omega_2))$ were replaced by $2\eta(a\omega_1 + b\omega_2)( \frac{r\omega_1 + s\omega_2}{N} + \frac{1}{2} (a\omega_1 + b\omega_2))$, then I could show the relation.
And the authors says that for $\gamma \in \Gamma(N)$, $k_{0,a}(\gamma \tau)/k_{0,a}(\tau)$ is a $2N$ root of $1$. But p108 of the Mazur's paper "Modular curves and the Eisenstein ideal", the authors says that $k_{0,a}(\gamma \tau)/k_{0,a}(\tau)$ is not constant.
And, in that Mazur's paper, the author also says that the $q$ expansion of $k_{0,a}(\tau)$ is ($q = \exp( 2 \pi i \tau)$) $$ - \frac{1}{2 \pi i}\exp(-\frac{2 \pi i}{2N} a)(1 - \zeta_N^a) \Pi_{m \ge 1} (1 - \zeta_N^a q^m)( 1 - \zeta_N^{-a}q^m)(1 - q^m)^{-2}.$$ (where $\zeta_N = \exp(\frac{2 \pi i}{N}).$)
But computing it, I've got the wrong result: $$ - \exp(- \frac{1}{2}\eta(1) (\frac{a}{N})^2) \frac{1}{2 \pi i}\exp(-\frac{2 \pi i}{2N} a)(1 - \zeta_N^a) \Pi_{m \ge 1} (1 - \zeta_N^a q^m)( 1 - \zeta_N^{-a}q^m)(1 - q^m)^{-2}.$$
Now the $q$ expansion of $\sigma(a/N, \tau \mathbb{Z} + \mathbb{Z})$ is $$ - \frac{1}{2 \pi i} \exp(\frac{1}{2}\eta(1) (\frac{a}{N})^2)\exp(-\frac{2 \pi i}{2N} a)(1 - \zeta_N^a) \Pi_{m \ge 1} (1 - \zeta_N^a q^m)( 1 - \zeta_N^{-a}q^m)(1 - q^m)^{-2}.$$ So if the term $\exp(\frac{1}{2}\eta(1) (\frac{a}{N})^2)$ were replaced by $\exp(\eta(1) (\frac{a}{N})^2)$, then I could show it.
What is wrong in my opinion? Or are there another definitions of the functions? (It seems that if the definition of the Klein form is $$ k_{r,s}(\omega_1, \omega_2) = \exp( - 2 \frac{r\eta(\omega_1) + s\eta(\omega_2)}{N} \frac{r\omega_1 + s\omega_2}{N}) \sigma(\frac{r\omega_1 + s\omega_2}{N}),$$ then I can show everything, except the "transformation law" described in p108 of Mazur's paper.)