newforms

Newforms are a distinguished class of cusp forms in the theory of modular forms. For a fixed even weight k ≥ 2 and a positive integer N, a newform is a cusp form in S_k(Γ0(N)) that is new at level N, i.e., it is not obtained from a cusp form of lower level by the standard level-raising maps. The space S_k(Γ0(N)) decomposes into oldforms and newforms: S_k(Γ0(N)) = S_k^old(Γ0(N)) ⊕ S_k^new(Γ0(N)). Oldforms arise from cusp forms of levels M | N with M < N via degeneracy maps; newforms form the orthogonal complement and are finite in number.

Newforms have good arithmetic properties: they are eigenforms for all Hecke operators T_p with p ∤ N and for the Atkin–Lehner operators W_p with p | N; together these give a common eigenbasis of S_k^new(Γ0(N)). For a normalized newform f = ∑ a_n q^n, a_1 = 1, and the Fourier coefficients a_n satisfy multiplicativity relations that determine the Hecke action. The L-function L(f, s) = ∑ a_n n^{-s} has an Euler product, analytic continuation, and a functional equation with conductor N.

Newforms correspond to cuspidal automorphic representations of GL2 over Q that are unramified outside N and are new at N. This links to the Langlands program and the modularity of arithmetic objects: the eigenvalues encode arithmetic data such as the number of points on reductions of associated elliptic curves or other motives.