| The primary arithmetic information attached to a Maass cusp form is its Laplace eigenvalue. However, in the case of cuspidal Maass forms, the range that these eigenvalues can take is not well-understood. In particular it is unknown if, given a real number r, one can prove that there exists a primitive Maass cusp form with Laplace eigenvalue 1/4 + r2. Conversely, given the Fourier coefficients of a primitive Maass cusp form f on a congruence subgroup of SL(2,Z) of level D, it is not clear whether or not one can determine its Laplace eigenvalue. In this paper we show that given only a finite number of Fourier coefficients one can first determine the level D, and then compute the Laplace eigenvalue to arbitrarily high precision.;The key to our results will be understanding the resonance and rapid decay properties of Maass cusp forms. Let f be a primitive Maass cusp form with Fourier coefficients lambdaf(n ). The resonance sum for f is a smoothly weighted exponential sum of the Fourier coefficients of f..;Sums of this form have been studied for many different classes of functions f, including holomorphic modular forms for SL(2, Z), and Maass cusp forms for SL(n,Z). In this paper we take f to be a primitive Maass cusp form for a congruence subgroup of SL(2,Z) of level D. Thus our result extends the family of automorphic forms for which their resonance properties are understood. Similar analysis and algorithms can be easily implemented for holomorphic cusp forms for congruence subgroups. Our techniques include Voronoi summation, weighted exponential sums, and asymptotics expansions of Bessel functions.;We then use these estimates in a new application of resonance sums. In particular we show that given only limited information about a Maass cusp form f (in particular a finite list of high Fourier coefficients), one can determine its level and estimate its spectral parameter, and thus its Laplace eigenvalue. This is done using a large parallel computing cluster running MATLAB and Mathematica. |
