  In this article, we give some results on simultaneous non-vanishing and
simultaneous 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 of
infinitely many primes $p$ such that $a_p b_p\neq 0$; simultaneous
non-vanishing in the short intervals and in arithmetic progressions;
simultaneous sign changes in short intervals.
