I am considering a family of lattices $\Lambda_n$ in $\mathbb{C}$ generated by $\langle 1,\tau_n\rangle$ for $n\geq 2$ with $$\tau_n = -\left(\frac{n^2}{4}+1\right)^{-1} \left(\frac{n^2}{2}+1-\frac{n}{2}i\right)$$ which comes from the complex numbers $\omega_{1;n}=1+\frac{n}{2}i$ and $\omega_{2;n}=-1+n i$ and $\tau_n=\frac{\omega_{2;n}}{\omega_{1;n}}$.
Then, $\tau_n\to -2$ in the topology of the complex numbers as $n\to\infty$. The real part is monotonously decreasing to $-2$ and the imaginary part monotonously decreasing to $0$. By the logic of the fundamental domain that for any $\tau$ the translation $\tau\pm 1$ spans the same lattice, I imagine that the $\tau_n$ move to the left in the fundamental domain and then "come back" from the right if $\mathrm{real}(\tau_n)\leq -\frac{1}{2}$ by a map $\tau_n\mapsto \tau_n+1$.
But for the imaginary part, I am stuck. Already, after translation into the fundamental domain, $\mathrm{real}(\tau_2)=-\frac{1}{2}$ and $\mathrm{imag}(\tau_2) =\frac{1}{2}$, so we start on the boundary of the fundamental domain for the real part and within the radius $1$ for the imaginary part and then decrease to $0$. So this is already not in the fundamental domain.
I followed the online course on modular forms by Borcherds and I understood that by choosing $\omega_{1;n}$ as the smaller basis vector of the lattice, I can always find a $\tau_n$ in the fundamental domain such that $\langle 1,\tau_n\rangle\stackrel{\sim}{=}\langle \omega_{1;n},\omega_{2;n}\rangle$. But this seems to fail here. Where am I going wrong with the calculation of $\tau_n$? And how can I find the structure of the implied lattice $\Lambda_{\infty}$?