School of Mathematics and Statistics - Theses
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ItemAnalytic and computational approaches to comparing the Fourier coefficients of Hecke eigenforms( 2012)Consider the Fourier expansions of two elements of a given space of modular forms. How many leading coefficients must agree in order to guarantee that the two expansions are the same? Sturm (1987) gave an upper bound for modular forms of a given weight and level. This was generalised by Ram Murty (1997) and Ghitza (2011) to the case of two modular forms of the same level but having potentially different weights. We consider their expansions modulo a prime, presenting a new bound of Ghitza, which is the maximum of two expressions. We describe the asymptotics of these two expressions. We then calculate the smallest level beyond which the second term is larger. In order to achieve this, we first establish a bound and then use the software Sage to check up to that bound. The bound is gotten by generalising a result of Bach and Sorenson (1996), who provide a practical upper bound for the smallest prime in an arithmetic progression. Our second problem is a study (using Sage) of how many leading Fourier coefficients are needed to distinguish newforms of level N and even integer weight k (say up to degree n_0). We provide extensive empirical evidence that the Sturm bound can be drastically improved in this setting. We prove that if k is sufficiently large then n_0 is greater than or equal to the smallest prime that does not divide N. We conjecture that if N is squarefree and k is sufficiently large then n_0 is in fact equal to the smallest prime that does not divide N. This is strongly supported by our data. We provide some intuition for this apparent stability phenomenon, and discuss what can happen if N is not squarefree. We present some original results of greater generality. We show that increasing concave functions on the positive reals preserve asymptotic equivalence, and generalise this. We also extend Bach and Sorenson's work to provide explicit bounds on the distribution of primes in arithmetic progression.