| Summary: | In computational number theory I developed an algorithm to compute simultaneous Hecke eigenforms. This problem transforms to a computational problem in the theory of lattices in Euclidean spaces, which, as I discovered, is connected to classical group such as the special orthogonal group over a finite field. By extending the Kneser neighboring process, I introduced a family of commuting, self-adjoint linear operators that act on the vector spaces of the isometry classes of lattices. To get their representation explicitly, instead of using a trivial, slow technique of enumerating all neighbor lattices of a given lattice, which consists in brute-force search in a finite set of candidates for a solution of the problem and in rather a simple test for any such candidate whether it is a solution, I designed an efficient algorithm that is also easy to implement. In this work I focused towards advancing theory that is also verified computationally by implementation of the algorithms in Magma. Thereby I computed the action of algebra of the Hecke operators on the Hilbert spaces of modular forms spanned by the Siegel theta series and produced simultaneous eigenfunctions explicitly as linear combinations of Siegel theta series attached to positive definite integral quadratic forms of nontrivial level. Not only could I then numerically confirm the results that were theoretically derived by Walling for eigenvalues of the average Siegel theta series, but also explain them via the aforementioned connection between the problem to construct neighbor lattices and the theory of buildings of groups. Moreover, in order to identify nonzero cusp forms among eigenfunctions, I extended the Poor-Yuen method for determining support sets for the Fourier coefficients. Last but not least, my numerical observations suggest a strategy for constructing new cases of Hecke eigenforms that are cusp forms and describing their eigenvalues by a formula.
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