This thesis consists of three parts. In the first part, we study the gaps between
non-zero Fourier coefficients of cuspdial CM eigenforms in the short intervals. In
the second part, we study the sign changes for the Fourier coefficients of Hilbert
modular forms of half-integral weight. In the third part, we study the simultaneous
behaviour of Fourier coefficients of two different Hilbert modular cusp forms
of integral weight.
In Chapter 1, we present the definitions and some preliminaries on classical
modular forms. We shall also recall some relevant results from the literature, which
are useful in the subsequent chapters.
In Chapter 2, we show that for an elliptic curve E over Q of conductor N with
complex multiplication (CM) by Q(i), the n-th Fourier coefficient of fE is non-zero
in the short interval (X;X + cX
1
4 ) for all X � 0 and for some c > 0, where fE is
the corresponding cuspidal Hecke eigenform in S2
