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1. ('''automorphy condition''') For any <math>\gamma \in \Gamma</math> there is the equality<ref group="note">Some authors use different conventions, allowing an additional constant depending only on <math>\gamma</math>, see e.g. https://dlmf.nist.gov/23.15#E5</ref> <math>f(\gamma(z)) = (cz + d)^k f(z)</math>
2. ('''growth condition''') For any <math>\gamma \in \text{SL}_2(\mathbb{Z})</math> the function <math>(cz + d)^{-k} f(\gamma(z))</math> is bounded for <math>\text{im}(z) \to \infty</math></blockquote> where <math display=inline> \gamma(z) = \frac{az+b}{cz+d} </math> and the function <math display=inline> \gamma </math> is identified with the matrix <math display=inline>\gamma = \begin{pmatrix}
a & b \\
c & d
\end{pmatrix} \in \text{SL}_2(\mathbb{Z}).\,</math> (The identification of such functions with such matrices causes composition of such functions to correspond to matrix multiplication.) In addition, it is called a '''cusp form''' if it satisfies the following growth condition:<blockquote>3. ('''cuspidal condition''') For any <math>\gamma \in \text{SL}_2(\mathbb{Z})</math> the function <math>(cz + d)^{-k}f(\gamma(z)) \to 0</math> as <math>\text{im}(z) \to \infty</math></blockquote>
=== As sections of a line bundle ===
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