Abstract. We describe numerical calculations which examine the PhillipsSarnak conjecture concerning the disappearance of cusp forms on a noncompact finite volume Riemann surface S under deformation of the surface. Our calculations indicate that if the Teichmüller space of S is not trivial, then each cusp form has a set of deformations under which either the cusp form remains a cusp form or else it dissolves into a resonance whose constant term is uniformly a factor of 10 8 smaller than a typical Fourier coefficient of the form. We give explicit examples of those deformations in several cases.

We describe a practical method for finding an L−function without first finding the associated underlying object. The procedure involves using the Euler product and the approximate functional equation in a new way. No use is made of the functional equation of twists of the L−function. The method is used to find a large number of Maass forms on SL(3, Z) and to give the first examples of Maass forms of higher level on GL(3), and on GL(4) and Sp(4).

Stephen D. Miller showed that, assuming the generalized Riemann Hypothesis, every entire L-function of real archimedian type has a zero in the interval 1 2 + it with −t 0 < t < t 0 , where t 0 ≈ 14.13 corresponds to the first zero of the Riemann zeta function. We give a numerical example of a self-dual degree-4 L-function whose first positive imaginary zero is at t 1 ≈ 14.496. In particular, Miller's result does not hold for general L-functions. We show that all L-functions satisfying some additional (conjecturally true) conditions have a zero in the interval (−t 2 , t 2 ) with t 2 ≈ 22.661.

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