Modular forms and modular symbols for noncongruence groups
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Modular forms for congruence groups are a major area of research in number theory and have been studied extensively. Modular forms for noncongruence groups are less understood. In this thesis, we look at noncongruence groups from two points of view. The first is computational: We look at a method of computation with finite index subgroups of the modular group called Farey symbols. This method allows us to work with noncongruence groups as easily as with congruence groups. We present an algorithm that uses Farey symbols to calculate modular symbols, from which we can calculate cusp forms. The second point of view involves the modular forms of noncongruence groups and the unbounded denominator property. We show that large families of noncongruence groups can be constructed which satisfy the unbounded denominator property.