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In this article, we give some results on simultaneous non-vanishing andsimultaneous sign-changes for the Fourier coefficients of two modular forms.More precisely, given two modular forms $f$ and $g$ with Fourier coefficients$a_n$ and $b_n$ respectively, we consider the following questions: existence ofinfinitely many primes $p$ such that $a_p b_p\neq 0$; simultaneousnon-vanishing in the short intervals and in arithmetic progressions;simultaneous sign changes in short intervals.

Simultaneous non-vanishing and sign changes of Fourier coefficients of modular forms