Linear Dependence Among Hecke Eigenvalues

We prove an explicit upper bound on the absolute value of the coefficients of a non-trivial integral linear relation among Hecke eigenvalues of a given cuspidal eigenform. Our motivation lies in its algorithmic application. For any fixed positive integer n, the bound established here yields an algorithm that computes cuspidal Hecke eigenforms with a given weight k whose Hecke eigenvalues generate a number field of degree n. The resulting algorithm reduces to Cremona’s when n = 1 and k = 2.