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Distinguishing Hecke eigenvalues of primitive cusp forms

Here λf (n) (resp. λg(n)) are the Hecke eigenvalues of f (resp. g) normalised so that the standard L function L(s, f) (resp. L(s, g)) of f (resp. g) which is given by the analytic continuation of the Dirichlet series ∑∞ n=1 λf (n)/n s (resp. ∑∞ n=1 λg(n)/n ) has a functional equation under s 7→ 1− s. The strong multiplicity one theorem for GL(2) says that there are infinitely many primes p such that λf (p) 6= λg(p). A refinement of the strong multiplicity one theorem due to Ramakrishnan (see [15]) says that the set of primes p such that λf (p) 6= λg(p) has density ≥ 1/8. In this article we will be interested in “finding” the smallest prime p such that λf (p) 6= λg(p). We will be specifically interested in the case when k1 6= k2. We will establish an upper bound C(k1, k2) depending on k1, k2 such that for some prime p, p ≤ C(k1, k2), we must have λf (p) 6= λg(p). Our motivation for this problem comes from the strong multiplicity one theorem and our result should be viewed as an effective version of the theorem in a special case. The exact statement appears below as Theorem 1. In the third section of this paper we consider a stronger version of the problem above which is as follows. The strong multiplicity one theorem for SL(2) says that there are infinitely many primes p such that |λf (p)| 6= |λg(p)|. In analogy with our earlier result we will give an upper bound C̃(k1, k2) depending on k1, k2 such that for some prime p, p ≤ C̃(k1, k2), we must have |λf (p)| 6= |λg(p)|.