Hecke operators and eigenforms

For a prime p, define Tₚf as p to the power k−1 times f(pτ), plus one over p times the sum of f((τ+b)/p) over b from zero to p−1. The first term dilates exponents, while the average selects those original exponents divisible by p. In the average, a sum of pth roots of unity vanishes unless the exponent is divisible by p. This elementary filter explains the coefficient formula, but preservation of modularity follows from the double-coset correspondence and is an additional structural assertion.

For a cusp expansion with coefficients aₙ, the coefficient of qⁿ in Tₚf is aₚₙ+p^(k−1)aₙ/ₚ, where the second coefficient is zero unless p divides n. For general positive r the coefficient involves a divisor sum, and these operators preserve both the holomorphic and cuspidal spaces. Their definition is linear; an arbitrary cusp form need not be an eigenvector. A single q-series manipulated by this formula must first be known to lie in the modular space before one can infer a modular correspondence.

At full level, the Hecke operators commute and are self-adjoint on the finite-dimensional cusp space for the Petersson pairing. They therefore admit a basis of simultaneous eigenvectors. A nonzero simultaneous eigenform has nonzero first Fourier coefficient: the first coefficient of Tᵣf is aᵣ(f), so if a₁ vanished the eigenvalue equation would force every positive coefficient to vanish. Dividing by a₁ gives the normalized eigenform with a₁=1. Its rth coefficient is then exactly its Tᵣ eigenvalue.

The normalization is coefficient one, not Petersson norm one; these choices usually give different vectors. For a normalized eigenform, coprime indices satisfy aₘₙ=aₘaₙ, while powers of one prime satisfy aₚʳ⁺¹=aₚaₚʳ−p^(k−1)aₚʳ⁻¹. The subtraction term prevents complete multiplicativity. It reflects the overlap in composing two prime correspondences. These conclusions use the Hecke algebra relations and the eigenform hypothesis, rather than holding for every linear combination of modular forms.

For a congruence subgroup of level N, primes not dividing N have analogous good Hecke operators; with character χ the recurrence contains χ(p)p^(k−1). At primes dividing N, U-type operators and the distinction between old and new subspaces enter. Self-adjointness or the full-level prime formula should not be asserted indiscriminately for all such operators. The later modularity theorem will use a normalized newform, which records an exact arithmetic level rather than merely a form viewed at a larger compatible level.

Low-dimensional spaces make the eigenform condition easy to establish. Since S₁₂ is one dimensional and Hecke-stable, its normalized generator Δ is automatically a simultaneous eigenform. Thus its Ramanujan coefficients obey the recurrence without diagonalizing a large matrix. In larger dimensions, one must compute or prove simultaneous eigenspace membership before imposing those arithmetic identities. Comparing enough output coefficients can certify an operator calculation, but one must allow for the larger input precision required by the aₚₙ term.