A note on signs of Fourier coefficients of two cusp forms

Kohnen and Sengupta (Proc. Am. Math. Soc. 137(11) (2009) 3563–3567) showed that if two Hecke eigencusp forms of weight \(k_1\) and \(k_2\) respectively, with \(1<k_1<k_2\) over \(\Gamma _0({N})\), have totally real algebraic Fourier coefficients \(\lbrace a(n) \rbrace \) and \(\lbrace b(n) \rbrace \) respectively for \(n \ge 1\) with \(a(1)=1=b(1)\), then there exists an element \(\sigma \) of the absolute Galois group \(\mathrm{Gal}({\bar{\mathbb {Q}}}/{\mathbb {Q}})\) such that \(a(n)^{\sigma } b(n)^{\sigma } < 0\) for infinitely many n. Later Gun et al. (Arch. Math. (Basel) 105(5) (2015) 413–424) extended their result by showing that if two Hecke eigen cusp forms, with \(1<k_1<k_2\), have real Fourier-coefficients \(\lbrace a(n)\rbrace \) and \(\lbrace b(n)\rbrace \) for \(n \ge 1\) and \(a(1)b(1) \ne 0\), then there exists infinitely many n such that \(a(n)b(n) > 0\) and infinitely many n such that \(a(n)b(n) < 0\). When \(k_1=k_2\), the simultaneous sign changes of Fourier coefficients of two normalized Hecke eigen cusp forms follow from an earlier work of Ram Murty (Math. Ann. 262 (1983) 431–446). In this note, we compare the signs of the Fourier coefficients of two cusp forms simultaneously for the congruence subgroup \(\Gamma _0({N})\) where the coefficients lie in an arithmetic progression. Next, we consider an analogous question for the particular sparse sequences of Fourier coefficients of normalized Hecke eigencusp forms for the full modular group.