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

$f : \text{GL}_2(\mathbf{A}) \to \Lambda$

such that

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

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

3. 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.

[1] This is likely nonstandard notation.

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