  Let $f$ be a primitive Maass cusp form for a congruence subgroup $\Gamma_0(D)
\subset $ SL($2,\mathbb{Z}$) and $\lambda_f(n)$ its $n$-th Fourier coefficient.
In this paper it is shown that with knowledge of only finitely many
$\lambda_f(n)$ one can often solve for the level $D$, and in some cases,
estimate the Laplace eigenvalue to arbitrarily high precision. This is done by
analyzing the resonance and rapid decay of smoothly weighted sums of
$\lambda_f(n)e(\alpha n^{\beta})$ for $X \leq n \leq 2X$ and any choice of
$\alpha \in \mathbb{R}$, and $\beta>0$. The methods include the Voronoi
summation formula, asymptotic expansions of Bessel functions, weighted
stationary phase, and computational software. These algorithms manifest the
belief that the resonance and rapid decay nature uniquely characterizes the
underlying cusp form. They also demonstrate that the Fourier coefficients of a
cusp form contain all arithmetic information of the form.
