Definition of a modular form
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We define the modular group $\text{SL}_2(ZZ)$ to be the group of matrices
of the
form
$\{[[a,b],[c,d]] \in \text{SL}_2(RR): a,b,c,d \in ZZ \}$,

and we define *modular forms of weight $k$ and level $1$* to be
holomorphic
functions on the upper half plane which satisfy the following criteria:
- $f(\frac{az+b}{cz+d})=(cz+d)^{-k} f(z)$ if $[[a,b],[c,d]] \in
\text{SL}_2(ZZ)$.
- The Fourier expansion of $f$ has the form
$f(z)=\sum_{n=0}^\infty a_n e^{2\pi i z}$.

If $a_0$ is zero, then we say that $f$ is a *cusp form*.
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