Let $\zeta$ be a primitive $N$-th root of unity and $\Gamma \subset (\mathbb{Z}/N)^\times$ a subgroup. Let $|\Gamma|$ be the cardinality of $\Gamma$ and consider the linear map $M_\Gamma\colon \mathbb{Q}^{N-1}\rightarrow \mathbb{Q}(\zeta)^{(N-1)\cdot |\Gamma|}$ given by the matrix \begin{align*} M_\Gamma = (\zeta^{nir}-\zeta^{ni})_{i=1,...,N-1 \atop (n,r)\in \{1,...,N-1\}\times \Gamma}. \end{align*} My question(s): What is the kernel of $M_\Gamma$? Is there any known result about this that I am missing?

Remark: There is THM 2.2 in the paper of Lam and Leung “On Vanishing Sums of Roots of Unity”, which gives a description of all $\mathbb{Z}$-linear relations among $N$-th roots of unity. However, I'm interested in those $\mathbb{Z}$-linear combinations given by the set of equations encoded in $M_\Gamma$ for a subgroup $\Gamma\subset (\mathbb{Z}/N\mathbb{Z})^\times$.

Let $\zeta$ be a primitive $N$-th root of unity and $\Gamma \subset (\mathbb{Z}/N)^\times$ a subgroup. Let $|\Gamma|$ be the cardinality of $\Gamma$ and consider the linear map $M_\Gamma\colon \mathbb{Q}^{N-1}\rightarrow \mathbb{Q}(\zeta)^{(N-1)\cdot |\Gamma|}$ given by the matrix \begin{align*} M_\Gamma = (\zeta^{nir}-\zeta^{ni})_{i=1,...,N-1 \atop (n,r)\in \{1,...,N-1\}\times \Gamma}. \end{align*} My question(s): What is the kernel of $M_\Gamma$? Is there any known result about this that I am missing?

Remark: There is Theorem 2.2 in the paper of Lam and Leung “On vanishing sums of roots of unity”, which gives a description of all $\mathbb{Z}$-linear relations among $N$-th roots of unity. However, I'm interested in those $\mathbb{Z}$-linear combinations given by the set of equations encoded in $M_\Gamma$ for a subgroup $\Gamma\subset (\mathbb{Z}/N\mathbb{Z})^\times$.