We have a reductive group $G/\mathbb{Q}$ and a representation space $V$ of this group.
Let $K$ be an open subgroup of $G(\mathbb{A}_{\mathbb{Q}})$ (where $\mathbb{A}_{\mathbb{Q}}$ are adeles of $\mathbb{Q}$) with some nice properties that wont concern us.
Define the space of "algebraic modular forms":
$\{f: G(\mathbb{A}_{\mathbb{Q}})\rightarrow V\,|\,f(gk) = f(g) \text{ for all } k\in K, g\in G(\mathbb{A}_{\mathbb{Q}}) \text{ and } f(\gamma g) = \gamma f(g) \text{ for all } \gamma\in G(\mathbb{Q})\}$
Now assume that $G(\mathbb{Q}) \backslash G(\mathbb{A}_{\mathbb{Q}})/K$ is finite, with reps $z_1, z_2, ..., z_h\in G(\mathbb{A}_{\mathbb{Q}})$.
The paper I am reading claims that you determine an algebraic modular form $f$ as soon as you specify the values $f(z_1), f(z_2), ..., f(z_m)\in V$.
I can't see why though.
Suppose $g\in G(\mathbb{A}_{\mathbb{Q}})$. Then $g = \gamma z_i k$ tells us that $f(g) = \gamma f(z_i)$.
Surely we have a dependence on $\gamma$ too? I hope I haven't missed anything simple.