Definition 63.31.1. An *unramified cusp form on $\text{GL}_2(\mathbf{A})$ with values in $\Lambda $*^{1} is a function

such that

$f(x\gamma ) = f(x)$ for all $x\in \text{GL}_2(\mathbf{A})$ and all $\gamma \in \text{GL}_2(K)$

$f(ux) = f(x)$ for all $x\in \text{GL}_2(\mathbf{A})$ and all $u\in \text{GL}_2(O)$

for all $x\in \text{GL}_2(\mathbf{A})$,

\[ \int _{\mathbf{A} \mod K} f \left(x \left( \begin{matrix} 1 & z \\ 0 & 1 \end{matrix} \right) \right) dz = 0 \]see [Section 4.1, dJ-conjecture] for an explanation of how to make sense out of this for a general ring $\Lambda $ in which $p$ is invertible.

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