eigenform

An eigenform is a modular form that serves as an eigenvector for the Hecke operators acting on a given space of modular forms. In the common holomorphic setting, fix a weight k, a level N, and a nebentypus χ. Let f be a nonzero cusp form in S_k(Γ0(N), χ) with Fourier expansion f(z) = ∑_{n≥1} a_n q^n. If there exist scalars λ_n such that T_n f = λ_n f for all n (with the appropriate Hecke operators included for primes dividing N), then f is called a Hecke eigenform. When f is normalized so that a_1 = 1, the eigenvalues λ_n coincide with the Fourier coefficients a_n.

Eigenforms exhibit multiplicative Fourier coefficients: for coprime m and n, a_{mn} = a_m a_n. For prime powers, the coefficients satisfy recurrence relations determined by the weight, level, and character; in particular, for a prime p not dividing N one has a_p as the eigenvalue of T_p, and the p-power coefficients are determined by a_p and the relation a_{p^r+1} = a_p a_{p^r} − χ(p) p^{k−1} a_{p^{r−1}}. Associated to an eigenform f are complex numbers α_p and β_p (the Satake parameters) with a_p = α_p + β_p and α_p β_p = χ(p) p^{k−1} for p ∤ N. The L-function L(f, s) has an Euler product ∏_p (1 − α_p p^{−s})^{−1} (1 − β_p p^{−s})^{−1} for p ∤ N, generalizing appropriately at primes dividing N.

Newforms are primitive eigenforms: Hecke eigenforms that are new at their level. The space S_k(Γ0(N), χ) decomposes into oldforms and newforms, with newforms forming a canonical basis of eigenforms. Eigenforms play a central role in number theory, connecting modular forms to Galois representations and arithmetic geometry, and they satisfy deep properties such as the Ramanujan–Petersson bounds proved by Deligne.