Classical newforms

Let $M_k(N,\chi)$ be the space of modular forms of weight $k$, level $N$, and character $\chi$.

If $M$ is a proper divisor of $N$ and $\psi$ is a Dirichlet character of modulus $M$ that induces $\chi$, then for every divisor $D \mid (N/M)$, there is a map from $M_k(M,\psi)$ to $S_k(N,\chi)$ via $f(z) \mapsto f(Dz)$. Such modular forms are said to be old, and together they span a subspace $M_k^{\rm old}(N,\chi) \subseteq M_k(N,\chi)$.

The orthogonal complement of the $M_k^{\rm old}(N,\chi)$ in $M_k(N,\chi)$ with respect to the Petersson scalar product is denoted $M_k^{\rm new}(N,\chi)$, and we have the decomposition $$M_k(N,\chi)=M_k^{\rm old}(N,\chi)\oplus M_k^{\rm new}(N,\chi).$$

Restricting to cusp forms gives a corresponding decomposition into old forms and new forms $$S_k(N,\chi)=S_k^{\rm old}(N,\chi)\oplus S_k^{\rm new}(N,\chi).$$

A newform is a cusp form $f\in S_k^{\rm new}(N,\chi)$ that is also an eigenform of all Hecke operators, normalized so that the $q$-expansion $f(z)=\sum a_n q^n$, where $q=e^{2\pi i z}$, begins with the coefficient $a_1=1$. The newforms are a standard basis for the vector space $S_k^{\rm new}(N,\chi)$.