Non-freeness of parabolic two-generator groups

We want to check whether a subgroup of $\text{SL}(2, \mathbb{C})$ $$ G_\lambda = \langle \begin{pmatrix}1 & 1 \\ 0 & 1 \end{pmatrix}, \begin{pmatrix}1 & 0 \\ \lambda & 1 \end{pmatrix}\rangle $$ is a free group of rank $2$ or not for a given $\lambda \in \mathbb{C}$. $\lambda$ is called a relation number if $G_\lambda$ is not a free group of rank $2$.
In the paper, we show $\lambda = -u^2$ is a relation number if and only if $u$ is a root of special kinds of univariate polynomial, called generalized Chebyshev polynomials with a specific initial condition, defined by recursion $$ s^{\bf n}_{k+1}(t) = n_kt s_k^{\bf n}(t) - s_{k-1}^{\bf n}(t) $$ for an integer sequence ${\bf n} = (n_1, n_2, \cdots)$ with $s^{\bf n}_0(t)=0$ and $s^{\bf n}_1(t)=1$. For example, we have the following table for small $k$.

For a given $u$, to find a integer sequence ${\bf n} = (n_1, n_2, \cdots, n_{k})$ such that $s^{\bf n}_{k+1}(u) = 0$ is a difficult question. It is well-known that $\lambda$ cannot be a relation number if $| \lambda | \geq 4$ and, on the other hand, it is conjectured that all the rational numbers whose absolute value is smaller than $4$ are relation number.

We developed two algorithms, one for finding the shortest sequence ${\bf n}$ such that $s^{\bf n}_{k+1}(u)=0$ (the exhaustive algorithm) and the other for finding any such a sequence (the greedy algorithm). Using these algorithms, we list $\lambda=p/q \in \mathbb{Q}$ for $0 < q < 23$ and $0 < p < 4q$ and its corresponding sequence ${\bf n}$ except $6$ cases $\lambda = 51/13, 67/17, 74/19, 75/19, 83/21, 87/22$.

You can download a Mathematica notebook containing the algorithms HERE. You can also calculate for a complex number $\lambda$ with the same algorithm.

The following is our result $(q \leq 22)$. Click HERE for the result in .csv format. Here $l(\lambda)$ and $u(\lambda)$ is a lower bound and a upper bound, respectively, for the minimum length of ${\bf n}$ such that $s^{{\bf n}}(u) = 0$.