Suppose I have a motive $M$ over $\mathbb{Q}$, and can compute the Euler factor of the associated $L$-function for any good prime $p$. How can I compute the Zariski closure of the image of the Galois group? In particular I am interested in the case where $M$ is a hypergeometric motive as implemented in Magma.

Suppose I have a motive $M$ over $\mathbb{Q}$, and can compute the Euler factor of the associated $L$-function for any good prime $p$. How can I compute the Zariski closure of the image of the Galois group? In particular I am interested in the case where $M$ is a hypergeometric motive as implemented in Magma. I only have access to the characteristic polynomial of the Frobenius operator for any prime $p$, so ideally any characterization would need only this information. Sato-Tate data indicates it is $\mathrm{USp}(4)$.

Suppose I have a motive $M$, and can compute the Euler factor of the associated $L$-function for any good prime $p$. How can I compute the Zariski closure of the image of the Galois group?

Suppose I have a motive $M$ over $\mathbb{Q}$, and can compute the Euler factor of the associated $L$-function for any good prime $p$. How can I compute the Zariski closure of the image of the Galois group? In particular I am interested in the case where $M$ is a hypergeometric motive as implemented in Magma.