Level-one modular forms and cusp forms

Fix an integer $k$. A modular form of weight $k$ for $PS{L}_2(\mathbb{Z})$ is a holomorphic function $f:\mathfrak{H}\to \mathbb{C}$ (Complex differentiability at a point, the complex derivative, holomorphic functions, and entire functions) such that

$f(\gamma \cdot \tau )=(c\tau +d{)}^{k}f(\tau )$

for every $\gamma =\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in S{L}_2(\mathbb{Z})$ and every $\tau \in \mathfrak{H}$ (The modular group and its action on the upper half-plane), and such that the associated periodic function $F$ of The q-expansion principle at the cusp is holomorphic at $q=0$. Since $T=\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$ acts by $\tau \mapsto \tau +1$ with factor $1$, the law gives $f(\tau +1)=f(\tau )$, so the q-expansion principle applies and $F$ is the unique holomorphic function on $0<\mathrm{\mid }q\mathrm{\mid }<1$ with $f(\tau )=F(e^{2\pi i\tau })$. The form $f$ is a cusp form if additionally $F(0)=0$.

Write $M_k$ and $S_k$ for the sets of modular and cusp forms of weight $k$. They are $\mathbb{C}$-subspaces of the space of holomorphic functions on $\mathfrak{H}$: sums and scalar multiples of functions satisfying the transformation law satisfy it again, and the q-expansion condition is preserved because $F_{af+bg}=a{F}_f+b{F}_g$ by uniqueness (Vector space over a field, Meromorphic functions on a plane domain). The transformation law is well posed: the factor $(c\tau +d{)}^k$ depends only on $\gamma$ and, by the cocycle identity for the Möbius action, the conditions for all $\gamma$ are consistent (Möbius transformations form a group and identify with the projective linear quotient of GL_2(C)). Since $-I\in S{L}_2(\mathbb{Z})$ acts as the identity with factor $(-1{)}^k$, one has $f=(-1{)}^{k}f$, so $M_k=\{0\}$ for odd $k$. The weight condition is compatible with multiplication: $M_k{M}_{\mathrm{\ell }}\subseteq M_{k+\mathrm{\ell }}$ and $S_k{M}_{\mathrm{\ell }}\subseteq S_{k+\mathrm{\ell }}$, because the products of the factors are $(c\tau +d{)}^{k+\mathrm{\ell }}$ and the q-expansion of a product has constant term $F_f(0)F_g(0)$.