Hecke operator (reviewed)

Let $f$ be a modular form of weight $k$, level $N$, and character $\chi$.

For each positive integer $n$ the Hecke operator $T_n$ is a linear operator on the vector space $M_k(N,\chi)$ whose action on $f\in M_k(N,\chi)$ can be defined as follows. If $f(z)=\sum a_n (f)q^n$ is the $q$-expansion of $f\in M_k(N,\chi)$, where $q=e^{2\pi i z}$, then the $q$-expansion of $T_nf\in M_k(N,\chi)$ has coefficients \[ a_m(T_nf) := \sum_{d|\gcd(m,n)}\chi(d)d^{k-1}a_{mn/d^2}(f). \] The Hecke operators pairwise commute, and when restricted to the subspace $S_k(N,\chi)$ of cusp forms, they commute with their adjoints with respect to the Petersson scalar product. This implies that $S_k(N,\chi)$ has a canonical basis whose elements are eigenforms for all the Hecke operators. If we normalize such an eigenform $f(z)=\sum a_n q^n$ so that $a_1=1$, then for all $n\ge 1$ we have \[ T_n f = a_nf. \] The newspace $S_k^{\rm new}(N,\chi)\subseteq S_k(N,\chi)$ is invariant under the action of the Hecke operators, so the canonical basis of normalized eigenforms for $S_k(N,\chi)$ includes a basis of newforms for $S_k^{\rm new}(N,\chi)$.