PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 142, Number 3, March 2014, Pages 761–764 S 0002-9939(2013)11797-1 Article electronically published on November 19, 2013 AN OMEGA-RESULT FOR SAITO-KUROKAWA LIFTS SOUMYA DAS AND JYOTI SENGUPTA (Communicated by Kathrin Bringmann) Abstract. We prove an Omega-result for the Hecke eigenvalues of Hecke operators acting on the space of Siegel modular forms of degree 2 which are Saito–Kurokawa lifts. 1. Introduction A central problem in the theory of modular forms is the estimation of the Hecke eigenvalues λ(n) of the Hecke operators T (n) defined on the space of appropriate modular forms. Especially in the case of cuspidal Siegel modular forms of weight k and degree g, which we denote by S g k , the “generalised Ramanujan–Petersson conjecture”, meaning that all the Satake p-parameters of F have absolute value 1, implies (see [5]) that λ F (p)  g,ε p gk/2−g(g+1)/4+ε for all > 0. (1.1) It is well known by the result of Deligne that when g = 1 we have |λ f (n)|≤ τ (n)n (k−1)/2 , where τ (n) denotes the number of divisors of n. For two positive arithmetical functions a(n) and b(n), we say that a(n) = Ω(b(n)) when a(n)= o(b(n)) does not hold; i.e., there is a subsequence (n k ) and a positive constant c such that a(n k ) >c · b(n k ) for all k ≥ 1. We note here that an Omega-result for Ellptic cusp forms was obtained by Rankin [6]. With the Sato-Tate conjecture being settled recently [3], one can now prove the desired conjecture about the Omega-result for eigenvalues for the elliptic cusp forms (see [4]): |λ f (n)| =Ω  n (k−1)/2 exp{c log n/ log log n}  for some c> 0. However when g = 2, (1.1) is known to be false (and also in some cases of higher genus), and the elements of the Maaß space in S 2 k precisely fail to satisfy it. Nevertheless Weissauer [7] has proved the above conjecture for g = 2 for F not a Saito–Kurokawa lift, i.e., not belonging to the Maaß space of degree 2. In the present paper, we concentrate on the space of Maaß cusp forms denoted by S ∗ k (i.e., Saito–Kurokawa lifts of elliptic modular forms) in S 2 k . The aim of this paper is to establish an Omega-result for S ∗ k . By a result of Breulman [1], we know Received by the editors January 5, 2012 and, in revised form, April 4, 2012. 2010 Mathematics Subject Classification. Primary 11F46; Secondary 11F30. Key words and phrases. Eigenvalues, Saito–Kurokawa lifts, Omega-result. c 2013 American Mathematical Society Reverts to public domain 28 years from publication 761